id sid tid token lemma pos cana-5460 1 1 communications communication NOUN cana-5460 1 2 on on ADP cana-5460 1 3 applied apply VERB cana-5460 1 4 nonlinear nonlinear ADJ cana-5460 1 5 analysis analysis NOUN cana-5460 1 6 issn issn NOUN cana-5460 1 7 : : PUNCT cana-5460 1 8 1074 1074 NUM cana-5460 1 9 - - PUNCT cana-5460 1 10 133x 133x NUM cana-5460 1 11 vol vol NOUN cana-5460 1 12 32 32 NUM cana-5460 1 13 no.3 no.3 NOUN cana-5460 1 14 ( ( PUNCT cana-5460 1 15 2025 2025 NUM cana-5460 1 16 ) ) PUNCT cana-5460 1 17 939 939 NUM cana-5460 1 18 https://internationalpubls.com https://internationalpubls.com X cana-5460 1 19 method method NOUN cana-5460 1 20 of of ADP cana-5460 1 21 a a DET cana-5460 1 22 priori priori ADJ cana-5460 1 23 estimate estimate NOUN cana-5460 1 24 in in ADP cana-5460 1 25 the the DET cana-5460 1 26 case case NOUN cana-5460 1 27 of of ADP cana-5460 1 28 fractional fractional ADJ cana-5460 1 29 order order NOUN cana-5460 1 30 differential differential ADJ cana-5460 1 31 equations equation NOUN cana-5460 1 32 with with ADP cana-5460 1 33 integral integral ADJ cana-5460 1 34 conditions condition NOUN cana-5460 1 35 1djebaili 1djebaili PROPN cana-5460 1 36 manel manel VERB cana-5460 1 37 1university 1university NUM cana-5460 1 38 abbes abbe NOUN cana-5460 1 39 - - PUNCT cana-5460 1 40 laghrour laghrour ADJ cana-5460 1 41 , , PUNCT cana-5460 1 42 khenchela khenchela PROPN cana-5460 1 43 , , PUNCT cana-5460 1 44 knowledge knowledge NOUN cana-5460 1 45 engineering engineering NOUN cana-5460 1 46 and and CCONJ cana-5460 1 47 information information NOUN cana-5460 1 48 security security NOUN cana-5460 1 49 ( ( PUNCT cana-5460 1 50 icosi icosi NOUN cana-5460 1 51 ) ) PUNCT cana-5460 1 52 , , PUNCT cana-5460 1 53 faculty faculty NOUN cana-5460 1 54 of of ADP cana-5460 1 55 sciences science NOUN cana-5460 1 56 and and CCONJ cana-5460 1 57 technology technology NOUN cana-5460 1 58 ( ( PUNCT cana-5460 1 59 algeria algeria PROPN cana-5460 1 60 ) ) PUNCT cana-5460 1 61 . . PUNCT cana-5460 2 1 manel.djebaili@univ-khenchela.dz manel.djebaili@univ-khenchela.dz NOUN cana-5460 2 2 article article NOUN cana-5460 2 3 history history NOUN cana-5460 2 4 : : PUNCT cana-5460 2 5 received:12 received:12 PROPN cana-5460 2 6 - - PUNCT cana-5460 2 7 01 01 NUM cana-5460 2 8 - - PUNCT cana-5460 2 9 2025 2025 NUM cana-5460 2 10 revised:15 revised:15 ADJ cana-5460 2 11 - - PUNCT cana-5460 2 12 02 02 NUM cana-5460 2 13 - - PUNCT cana-5460 2 14 2025 2025 NUM cana-5460 2 15 accepted:01 accepted:01 VERB cana-5460 2 16 - - PUNCT cana-5460 2 17 03 03 NUM cana-5460 2 18 - - PUNCT cana-5460 2 19 2025 2025 NUM cana-5460 2 20 abstract abstract NOUN cana-5460 2 21 : : PUNCT cana-5460 2 22 this this DET cana-5460 2 23 paper paper NOUN cana-5460 2 24 concentrate concentrate NOUN cana-5460 2 25 on on ADP cana-5460 2 26 exploring explore VERB cana-5460 2 27 the the DET cana-5460 2 28 existence existence NOUN cana-5460 2 29 and and CCONJ cana-5460 2 30 uniqueness uniqueness NOUN cana-5460 2 31 of of ADP cana-5460 2 32 a a DET cana-5460 2 33 solution solution NOUN cana-5460 2 34 for for ADP cana-5460 2 35 a a DET cana-5460 2 36 non non ADJ cana-5460 2 37 - - ADJ cana-5460 2 38 linear linear ADJ cana-5460 2 39 boundary boundary ADJ cana-5460 2 40 value value NOUN cana-5460 2 41 problem problem NOUN cana-5460 2 42 that that PRON cana-5460 2 43 has have VERB cana-5460 2 44 integral integral ADJ cana-5460 2 45 conditions condition NOUN cana-5460 2 46 in in ADP cana-5460 2 47 the the DET cana-5460 2 48 case case NOUN cana-5460 2 49 of of ADP cana-5460 2 50 fractional fractional ADJ cana-5460 2 51 partial partial ADJ cana-5460 2 52 differential differential NOUN cana-5460 2 53 equations equation NOUN cana-5460 2 54 . . PUNCT cana-5460 3 1 for for ADP cana-5460 3 2 this this PRON cana-5460 3 3 we we PRON cana-5460 3 4 split split VERB cana-5460 3 5 the the DET cana-5460 3 6 proof proof NOUN cana-5460 3 7 into into ADP cana-5460 3 8 two two NUM cana-5460 3 9 sections section NOUN cana-5460 3 10 : : PUNCT cana-5460 3 11 linear linear ADJ cana-5460 3 12 and and CCONJ cana-5460 3 13 non non ADJ cana-5460 3 14 - - ADJ cana-5460 3 15 linear linear ADJ cana-5460 3 16 problem problem NOUN cana-5460 3 17 ; ; PUNCT cana-5460 3 18 for for ADP cana-5460 3 19 the the DET cana-5460 3 20 associated associated ADJ cana-5460 3 21 linear linear PROPN cana-5460 3 22 problem problem NOUN cana-5460 3 23 , , PUNCT cana-5460 3 24 we we PRON cana-5460 3 25 derive derive VERB cana-5460 3 26 the the DET cana-5460 3 27 a a PRON cana-5460 3 28 priori priori ADV cana-5460 3 29 bound bind VERB cana-5460 3 30 and and CCONJ cana-5460 3 31 demonstrate demonstrate VERB cana-5460 3 32 the the DET cana-5460 3 33 density density NOUN cana-5460 3 34 of of ADP cana-5460 3 35 the the DET cana-5460 3 36 operator operator NOUN cana-5460 3 37 generated generate VERB cana-5460 3 38 by by ADP cana-5460 3 39 the the DET cana-5460 3 40 problem problem NOUN cana-5460 3 41 posed pose VERB cana-5460 3 42 ; ; PUNCT cana-5460 3 43 we we PRON cana-5460 3 44 solve solve VERB cana-5460 3 45 the the DET cana-5460 3 46 non non ADJ cana-5460 3 47 - - ADJ cana-5460 3 48 linear linear ADJ cana-5460 3 49 problem problem NOUN cana-5460 3 50 by by ADP cana-5460 3 51 introducing introduce VERB cana-5460 3 52 a a DET cana-5460 3 53 iterative iterative NOUN cana-5460 3 54 process process NOUN cana-5460 3 55 . . PUNCT cana-5460 4 1 the the DET cana-5460 4 2 results result NOUN cana-5460 4 3 show show VERB cana-5460 4 4 the the DET cana-5460 4 5 efficiency efficiency NOUN cana-5460 4 6 of of ADP cana-5460 4 7 energy energy NOUN cana-5460 4 8 inequality inequality NOUN cana-5460 4 9 method method NOUN cana-5460 4 10 in in ADP cana-5460 4 11 the the DET cana-5460 4 12 case case NOUN cana-5460 4 13 of of ADP cana-5460 4 14 time time NOUN cana-5460 4 15 fractional fractional ADJ cana-5460 4 16 order order NOUN cana-5460 4 17 differential differential ADJ cana-5460 4 18 equations equation NOUN cana-5460 4 19 with with ADP cana-5460 4 20 integral integral ADJ cana-5460 4 21 conditions condition NOUN cana-5460 4 22 our our PRON cana-5460 4 23 results result NOUN cana-5460 4 24 illustrate illustrate VERB cana-5460 4 25 the the DET cana-5460 4 26 existence existence NOUN cana-5460 4 27 and and CCONJ cana-5460 4 28 uniqueness uniqueness NOUN cana-5460 4 29 of of ADP cana-5460 4 30 the the DET cana-5460 4 31 continuous continuous ADJ cana-5460 4 32 dependence dependence NOUN cana-5460 4 33 of of ADP cana-5460 4 34 solution solution NOUN cana-5460 4 35 on on ADP cana-5460 4 36 fractional fractional ADJ cana-5460 4 37 order order NOUN cana-5460 4 38 . . PUNCT cana-5460 5 1 keywords keyword NOUN cana-5460 5 2 : : PUNCT cana-5460 5 3 fractional fractional ADJ cana-5460 5 4 partial partial ADJ cana-5460 5 5 differential differential NOUN cana-5460 5 6 equations equation NOUN cana-5460 5 7 ; ; PUNCT cana-5460 5 8 integral integral ADJ cana-5460 5 9 conditions condition NOUN cana-5460 5 10 ; ; PUNCT cana-5460 5 11 priori priori X cana-5460 5 12 estimate estimate NOUN cana-5460 5 13 ; ; PUNCT cana-5460 5 14 density density NOUN cana-5460 5 15 of of ADP cana-5460 5 16 the the DET cana-5460 5 17 operator operator NOUN cana-5460 5 18 ; ; PUNCT cana-5460 5 19 non non ADJ cana-5460 5 20 - - ADJ cana-5460 5 21 linear linear ADJ cana-5460 5 22 problem problem NOUN cana-5460 5 23 . . PUNCT cana-5460 6 1 1.introduction 1.introduction NUM cana-5460 6 2 in in ADP cana-5460 6 3 mathematics mathematic NOUN cana-5460 6 4 , , PUNCT cana-5460 6 5 fractional fractional ADJ cana-5460 6 6 calculus calculus NOUN cana-5460 6 7 is be AUX cana-5460 6 8 a a DET cana-5460 6 9 field field NOUN cana-5460 6 10 of of ADP cana-5460 6 11 analysis analysis NOUN cana-5460 6 12 that that PRON cana-5460 6 13 investigates investigate VERB cana-5460 6 14 the the DET cana-5460 6 15 extension extension NOUN cana-5460 6 16 of of ADP cana-5460 6 17 differentiation differentiation NOUN cana-5460 6 18 and and CCONJ cana-5460 6 19 integration integration NOUN cana-5460 6 20 from from ADP cana-5460 6 21 integers integer NOUN cana-5460 6 22 to to ADP cana-5460 6 23 non non NOUN cana-5460 6 24 - - NOUN cana-5460 6 25 integers integer NOUN cana-5460 6 26 , , PUNCT cana-5460 6 27 commonly commonly ADV cana-5460 6 28 referred refer VERB cana-5460 6 29 to to ADP cana-5460 6 30 as as ADP cana-5460 6 31 fractional fractional ADJ cana-5460 6 32 orders order NOUN cana-5460 6 33 . . PUNCT cana-5460 7 1 fractional fractional ADJ cana-5460 7 2 differentiation differentiation NOUN cana-5460 7 3 , , PUNCT cana-5460 7 4 in in ADP cana-5460 7 5 particular particular ADJ cana-5460 7 6 , , PUNCT cana-5460 7 7 has have AUX cana-5460 7 8 been be AUX cana-5460 7 9 a a DET cana-5460 7 10 subject subject NOUN cana-5460 7 11 of of ADP cana-5460 7 12 interest interest NOUN cana-5460 7 13 for for ADP cana-5460 7 14 almost almost ADV cana-5460 7 15 as as ADV cana-5460 7 16 long long ADV cana-5460 7 17 as as ADP cana-5460 7 18 the the DET cana-5460 7 19 classical classical ADJ cana-5460 7 20 calculus calculus NOUN cana-5460 7 21 that that PRON cana-5460 7 22 we we PRON cana-5460 7 23 know know VERB cana-5460 7 24 today today NOUN cana-5460 7 25 . . PUNCT cana-5460 8 1 in in ADP cana-5460 8 2 addition addition NOUN cana-5460 8 3 , , PUNCT cana-5460 8 4 many many ADJ cana-5460 8 5 problems problem NOUN cana-5460 8 6 in in ADP cana-5460 8 7 physics physics NOUN cana-5460 8 8 and and CCONJ cana-5460 8 9 modern modern ADJ cana-5460 8 10 technology technology NOUN cana-5460 8 11 are be AUX cana-5460 8 12 formulated formulate VERB cana-5460 8 13 using use VERB cana-5460 8 14 non non ADJ cana-5460 8 15 - - ADJ cana-5460 8 16 local local ADJ cana-5460 8 17 conditions condition NOUN cana-5460 8 18 for for ADP cana-5460 8 19 partial partial ADJ cana-5460 8 20 differential differential ADJ cana-5460 8 21 equations equation NOUN cana-5460 8 22 , , PUNCT cana-5460 8 23 which which PRON cana-5460 8 24 are be AUX cana-5460 8 25 described describe VERB cana-5460 8 26 by by ADP cana-5460 8 27 integral integral ADJ cana-5460 8 28 conditions condition NOUN cana-5460 8 29 . . PUNCT cana-5460 9 1 these these DET cana-5460 9 2 conditions condition NOUN cana-5460 9 3 have have AUX cana-5460 9 4 gained gain VERB cana-5460 9 5 significant significant ADJ cana-5460 9 6 attention attention NOUN cana-5460 9 7 due due ADP cana-5460 9 8 to to ADP cana-5460 9 9 their their PRON cana-5460 9 10 applications application NOUN cana-5460 9 11 in in ADP cana-5460 9 12 a a DET cana-5460 9 13 variety variety NOUN cana-5460 9 14 of of ADP cana-5460 9 15 fields field NOUN cana-5460 9 16 , , PUNCT cana-5460 9 17 such such ADJ cana-5460 9 18 as as ADP cana-5460 9 19 population population NOUN cana-5460 9 20 dynamics dynamic NOUN cana-5460 9 21 , , PUNCT cana-5460 9 22 blood blood NOUN cana-5460 9 23 flow flow NOUN cana-5460 9 24 models model NOUN cana-5460 9 25 , , PUNCT cana-5460 9 26 chemical chemical NOUN cana-5460 9 27 engineering engineering NOUN cana-5460 9 28 , , PUNCT cana-5460 9 29 and and CCONJ cana-5460 9 30 cellular cellular ADJ cana-5460 9 31 systems system NOUN cana-5460 9 32 ( ( PUNCT cana-5460 9 33 a. a. NOUN cana-5460 9 34 bouziani bouziani PROPN cana-5460 9 35 , , PUNCT cana-5460 9 36 2002 2002 NUM cana-5460 9 37 ; ; PUNCT cana-5460 9 38 a.bouziani,2003 a.bouziani,2003 NOUN cana-5460 9 39 ; ; PUNCT cana-5460 9 40 a.bouziani a.bouziani NOUN cana-5460 9 41 , , PUNCT cana-5460 9 42 n.merazga n.merazga PROPN cana-5460 9 43 , , PUNCT cana-5460 9 44 a.bouziani,2003 a.bouziani,2003 NOUN cana-5460 9 45 ; ; PUNCT cana-5460 9 46 n.merazga n.merazga PROPN cana-5460 9 47 , , PUNCT cana-5460 9 48 a. a. PROPN cana-5460 9 49 bouziani,2005 bouziani,2005 PROPN cana-5460 9 50 ) ) PUNCT cana-5460 9 51 . . PUNCT cana-5460 10 1 numerous numerous ADJ cana-5460 10 2 authors author NOUN cana-5460 10 3 have have AUX cana-5460 10 4 studied study VERB cana-5460 10 5 the the DET cana-5460 10 6 existence existence NOUN cana-5460 10 7 and and CCONJ cana-5460 10 8 uniqueness uniqueness NOUN cana-5460 10 9 of of ADP cana-5460 10 10 solutions solution NOUN cana-5460 10 11 to to ADP cana-5460 10 12 problems problem NOUN cana-5460 10 13 involving involve VERB cana-5460 10 14 fractional fractional ADJ cana-5460 10 15 differential differential ADJ cana-5460 10 16 equations equation NOUN cana-5460 10 17 , , PUNCT cana-5460 10 18 including include VERB cana-5460 10 19 initial initial ADJ cana-5460 10 20 and and CCONJ cana-5460 10 21 boundary boundary ADJ cana-5460 10 22 value value NOUN cana-5460 10 23 problems problem NOUN cana-5460 10 24 ( ( PUNCT cana-5460 10 25 a. a. NOUN cana-5460 10 26 anguraj anguraj PROPN cana-5460 10 27 , , PUNCT cana-5460 10 28 p.karthikeyan p.karthikeyan PROPN cana-5460 10 29 , , PUNCT cana-5460 10 30 2010 2010 NUM cana-5460 10 31 ; ; PUNCT cana-5460 10 32 b. b. PROPN cana-5460 10 33 ahmad ahmad PROPN cana-5460 10 34 , , PUNCT cana-5460 10 35 j. j. PROPN cana-5460 10 36 nieto,2009 nieto,2009 PROPN cana-5460 10 37 ; ; PUNCT cana-5460 10 38 m. m. NOUN cana-5460 10 39 benchohra benchohra NOUN cana-5460 10 40 , , PUNCT cana-5460 10 41 j. j. PROPN cana-5460 10 42 r. r. PROPN cana-5460 10 43 graef graef PROPN cana-5460 10 44 , , PUNCT cana-5460 10 45 s. s. PROPN cana-5460 10 46 hamani,2008 hamani,2008 PROPN cana-5460 10 47 ; ; PUNCT cana-5460 10 48 m. m. NOUN cana-5460 10 49 belmekki belmekki PROPN cana-5460 10 50 , , PUNCT cana-5460 10 51 m. m. NOUN cana-5460 10 52 benchohra,2008 benchohra,2008 NOUN cana-5460 10 53 ; ; PUNCT cana-5460 10 54 r. r. PROPN cana-5460 10 55 p. p. PROPN cana-5460 10 56 agarwal agarwal PROPN cana-5460 10 57 , , PUNCT cana-5460 10 58 m. m. NOUN cana-5460 10 59 benchohra benchohra NOUN cana-5460 10 60 , , PUNCT cana-5460 10 61 s. s. PROPN cana-5460 10 62 hamani,2005 hamani,2005 PROPN cana-5460 10 63 ; ; PUNCT cana-5460 10 64 r. r. PROPN cana-5460 10 65 w. w. PROPN cana-5460 10 66 ibrahim ibrahim PROPN cana-5460 10 67 , , PUNCT cana-5460 10 68 s. s. PROPN cana-5460 10 69 momani,2007 momani,2007 PROPN cana-5460 10 70 ; ; PUNCT cana-5460 10 71 x. x. PROPN cana-5460 10 72 j. j. PROPN cana-5460 10 73 li li PROPN cana-5460 10 74 , , PUNCT cana-5460 10 75 c. c. PROPN cana-5460 10 76 j. j. PROPN cana-5460 10 77 xu,2010 xu,2010 PROPN cana-5460 10 78 ) ) PUNCT cana-5460 10 79 . . PUNCT cana-5460 11 1 mailto:manel.djebaili@univ-khenchela.dz mailto:manel.djebaili@univ-khenchela.dz NOUN cana-5460 11 2 communications communication NOUN cana-5460 11 3 on on ADP cana-5460 11 4 applied apply VERB cana-5460 11 5 nonlinear nonlinear ADJ cana-5460 11 6 analysis analysis NOUN cana-5460 11 7 issn issn NOUN cana-5460 11 8 : : PUNCT cana-5460 11 9 1074 1074 NUM cana-5460 11 10 - - PUNCT cana-5460 11 11 133x 133x NUM cana-5460 11 12 vol vol NOUN cana-5460 11 13 32 32 NUM cana-5460 11 14 no.3 no.3 NOUN cana-5460 11 15 ( ( PUNCT cana-5460 11 16 2025 2025 NUM cana-5460 11 17 ) ) PUNCT cana-5460 11 18 940 940 NUM cana-5460 11 19 https://internationalpubls.com https://internationalpubls.com X cana-5460 11 20 for for ADP cana-5460 11 21 this this DET cana-5460 11 22 purpose purpose NOUN cana-5460 11 23 , , PUNCT cana-5460 11 24 we we PRON cana-5460 11 25 employed employ VERB cana-5460 11 26 the the DET cana-5460 11 27 energy energy NOUN cana-5460 11 28 inequality inequality NOUN cana-5460 11 29 method method NOUN cana-5460 11 30 , , PUNCT cana-5460 11 31 which which PRON cana-5460 11 32 is be AUX cana-5460 11 33 a a DET cana-5460 11 34 useful useful ADJ cana-5460 11 35 tool tool NOUN cana-5460 11 36 for for ADP cana-5460 11 37 studying study VERB cana-5460 11 38 fractional fractional ADJ cana-5460 11 39 and and CCONJ cana-5460 11 40 non non ADJ cana-5460 11 41 - - ADJ cana-5460 11 42 local local ADJ cana-5460 11 43 classical classical ADJ cana-5460 11 44 problems problem NOUN cana-5460 11 45 . . PUNCT cana-5460 12 1 compared compare VERB cana-5460 12 2 to to ADP cana-5460 12 3 other other ADJ cana-5460 12 4 techniques technique NOUN cana-5460 12 5 , , PUNCT cana-5460 12 6 this this DET cana-5460 12 7 method method NOUN cana-5460 12 8 plays play VERB cana-5460 12 9 an an DET cana-5460 12 10 essential essential ADJ cana-5460 12 11 role role NOUN cana-5460 12 12 in in ADP cana-5460 12 13 proving prove VERB cana-5460 12 14 the the DET cana-5460 12 15 existence existence NOUN cana-5460 12 16 and and CCONJ cana-5460 12 17 uniqueness uniqueness NOUN cana-5460 12 18 of of ADP cana-5460 12 19 the the DET cana-5460 12 20 solution solution NOUN cana-5460 12 21 . . PUNCT cana-5460 13 1 it it PRON cana-5460 13 2 depends depend VERB cana-5460 13 3 on on ADP cana-5460 13 4 density density NOUN cana-5460 13 5 arguments argument NOUN cana-5460 13 6 and and CCONJ cana-5460 13 7 certain certain ADJ cana-5460 13 8 a a DET cana-5460 13 9 priori priori ADJ cana-5460 13 10 bounds bound NOUN cana-5460 13 11 . . PUNCT cana-5460 14 1 2 2 X cana-5460 14 2 . . X cana-5460 14 3 preliminary preliminary ADJ cana-5460 14 4 definition definition NOUN cana-5460 14 5 2.1 2.1 NUM cana-5460 14 6 ( ( PUNCT cana-5460 14 7 i.podlubny i.podlubny ADP cana-5460 14 8 , , PUNCT cana-5460 14 9 1999)(gamma 1999)(gamma NOUN cana-5460 14 10 function function NOUN cana-5460 14 11 ) ) PUNCT cana-5460 14 12 for for ADP cana-5460 14 13 any any DET cana-5460 14 14 complex complex ADJ cana-5460 14 15 number number NOUN cana-5460 14 16 𝐳 𝐳 PRON cana-5460 14 17 such such ADJ cana-5460 14 18 that that DET cana-5460 14 19 re(𝐳 re(𝐳 NOUN cana-5460 14 20 ) ) PUNCT cana-5460 14 21 > > X cana-5460 14 22 0 0 NUM cana-5460 14 23 , , PUNCT cana-5460 14 24 we we PRON cana-5460 14 25 define define VERB cana-5460 14 26 the the DET cana-5460 14 27 following follow VERB cana-5460 14 28 function function NOUN cana-5460 14 29 called call VERB cana-5460 14 30 gamma gamma NOUN cana-5460 14 31 and and CCONJ cana-5460 14 32 denoted denote VERB cana-5460 14 33 by by ADP cana-5460 14 34 the the DET cana-5460 14 35 greek greek ADJ cana-5460 14 36 letter letter NOUN cana-5460 14 37 " " PUNCT cana-5460 14 38 πšͺ πšͺ PROPN cana-5460 14 39 " " PUNCT cana-5460 14 40 . . PUNCT cana-5460 15 1 πšͺ πšͺ NOUN cana-5460 15 2 ∢ ∢ NOUN cana-5460 15 3 π‘βˆ—+ π‘βˆ—+ PROPN cana-5460 15 4 β†’ β†’ SYM cana-5460 15 5 𝐑 𝐑 NOUN cana-5460 15 6 𝐳 𝐳 X cana-5460 15 7 β†’ β†’ SYM cana-5460 15 8 πšͺ(𝐳 πšͺ(𝐳 NUM cana-5460 15 9 ) ) PUNCT cana-5460 15 10 = = SYM cana-5460 15 11 ∫ ∫ PROPN cana-5460 15 12 π­π³βˆ’πŸ π­π³βˆ’πŸ PROPN cana-5460 15 13 + + PROPN cana-5460 15 14 ∞ ∞ PROPN cana-5460 15 15 𝟎 𝟎 NUM cana-5460 15 16 πžπ±π©βˆ’π­ππ­. πžπ±π©βˆ’π­ππ­. VERB cana-5460 15 17 ( ( PUNCT cana-5460 15 18 1 1 NUM cana-5460 15 19 ) ) PUNCT cana-5460 15 20 definition definition NOUN cana-5460 15 21 2.2 2.2 NUM cana-5460 15 22 ( ( PUNCT cana-5460 15 23 i.podlubny i.podlubny ADP cana-5460 15 24 , , PUNCT cana-5460 15 25 1999 1999 NUM cana-5460 15 26 ) ) PUNCT cana-5460 15 27 ( ( PUNCT cana-5460 15 28 the the DET cana-5460 15 29 riemann riemann PROPN cana-5460 15 30 - - PUNCT cana-5460 15 31 liouville liouville NOUN cana-5460 15 32 integral integral ADJ cana-5460 15 33 ) ) PUNCT cana-5460 15 34 the the DET cana-5460 15 35 riemann riemann PROPN cana-5460 15 36 - - PUNCT cana-5460 15 37 liouville liouville NOUN cana-5460 15 38 integral integral ADJ cana-5460 15 39 of of ADP cana-5460 15 40 order order NOUN cana-5460 15 41 Ξ± Ξ± X cana-5460 15 42 > > X cana-5460 15 43 0 0 NUM cana-5460 15 44 , , PUNCT cana-5460 15 45 for for ADP cana-5460 15 46 an an DET cana-5460 15 47 integral integral ADJ cana-5460 15 48 function function NOUN cana-5460 15 49 s s PART cana-5460 15 50 , , PUNCT cana-5460 15 51 is be AUX cana-5460 15 52 defined define VERB cana-5460 15 53 by by ADP cana-5460 15 54 dt dt X cana-5460 15 55 βˆ’Ξ±s(t βˆ’Ξ±s(t PROPN cana-5460 15 56 ) ) PUNCT cana-5460 15 57 = = SYM cana-5460 15 58 1 1 NUM cana-5460 15 59 Ξ³(Ξ± Ξ³(Ξ± NOUN cana-5460 15 60 ) ) PUNCT cana-5460 15 61 ∫ ∫ PROPN cana-5460 16 1 s(Ο„ s(Ο„ PROPN cana-5460 16 2 ) ) PUNCT cana-5460 16 3 ( ( PUNCT cana-5460 16 4 tβˆ’Ο„)1βˆ’Ξ± tβˆ’Ο„)1βˆ’Ξ± PROPN cana-5460 16 5 t t PROPN cana-5460 16 6 a a DET cana-5460 16 7 dΟ„ dΟ„ PROPN cana-5460 16 8 . . PUNCT cana-5460 16 9 ( ( PUNCT cana-5460 16 10 2 2 X cana-5460 16 11 ) ) PUNCT cana-5460 16 12 definition definition NOUN cana-5460 16 13 2.3 2.3 NUM cana-5460 16 14 ( ( PUNCT cana-5460 16 15 haim haim PROPN cana-5460 16 16 brezis brezis NOUN cana-5460 16 17 , , PUNCT cana-5460 16 18 1983 1983 NUM cana-5460 16 19 ) ) PUNCT cana-5460 16 20 let let VERB cana-5460 16 21 r r PRON cana-5460 16 22 be be AUX cana-5460 16 23 a a DET cana-5460 16 24 subspace subspace NOUN cana-5460 16 25 vector vector NOUN cana-5460 16 26 of of ADP cana-5460 16 27 the the DET cana-5460 16 28 hilbert hilbert PROPN cana-5460 16 29 space space PROPN cana-5460 16 30 h h PROPN cana-5460 16 31 , , PUNCT cana-5460 16 32 then then ADV cana-5460 16 33 𝐑 𝐑 PROPN cana-5460 16 34 ⏊ ⏊ PROPN cana-5460 16 35 the the DET cana-5460 16 36 orthogonal orthogonal ADJ cana-5460 16 37 complement complement NOUN cana-5460 16 38 of of ADP cana-5460 16 39 r r NOUN cana-5460 16 40 is be AUX cana-5460 16 41 defined define VERB cana-5460 16 42 as as ADP cana-5460 16 43 : : PUNCT cana-5460 16 44 𝐑 𝐑 PROPN cana-5460 16 45 ⏊ ⏊ PROPN cana-5460 16 46 = = PUNCT cana-5460 16 47 { { PUNCT cana-5460 16 48 𝐟 𝐟 PROPN cana-5460 16 49 ∈ ∈ PROPN cana-5460 16 50 𝐇 𝐇 PROPN cana-5460 16 51 , , PUNCT cana-5460 16 52 ( ( PUNCT cana-5460 16 53 𝐟 𝐟 NOUN cana-5460 16 54 , , PUNCT cana-5460 16 55 𝐠)𝐇 𝐠)𝐇 VERB cana-5460 16 56 = = SYM cana-5460 16 57 𝟎 𝟎 NUM cana-5460 16 58 , , PUNCT cana-5460 16 59 βˆ€π  βˆ€π  X cana-5460 16 60 ∈ ∈ PROPN cana-5460 16 61 𝐑 𝐑 PROPN cana-5460 16 62 } } PUNCT cana-5460 16 63 . . PUNCT cana-5460 17 1 proposition proposition NOUN cana-5460 17 2 2.1 2.1 NUM cana-5460 17 3 let let VERB cana-5460 17 4 r r PRON cana-5460 17 5 be be AUX cana-5460 17 6 a a DET cana-5460 17 7 subspace subspace NOUN cana-5460 17 8 vector vector NOUN cana-5460 17 9 of of ADP cana-5460 17 10 the the DET cana-5460 17 11 hilbert hilbert PROPN cana-5460 17 12 space space NOUN cana-5460 17 13 h. h. PROPN cana-5460 17 14 r r NOUN cana-5460 17 15 is be AUX cana-5460 17 16 dense dense ADJ cana-5460 17 17 in in ADP cana-5460 17 18 h h NOUN cana-5460 17 19 if if SCONJ cana-5460 18 1 and and CCONJ cana-5460 18 2 only only ADV cana-5460 18 3 if if SCONJ cana-5460 18 4 : : PUNCT cana-5460 18 5 𝐑 𝐑 NOUN cana-5460 18 6 ⏊ ⏊ PROPN cana-5460 18 7 = = PUNCT cana-5460 18 8 { { PUNCT cana-5460 18 9 𝟎 𝟎 PROPN cana-5460 18 10 } } PUNCT cana-5460 18 11 . . PUNCT cana-5460 19 1 definition definition NOUN cana-5460 19 2 2.4 2.4 NUM cana-5460 19 3 ( ( PUNCT cana-5460 19 4 bertram bertram PROPN cana-5460 19 5 ross ross PROPN cana-5460 19 6 , , PUNCT cana-5460 19 7 2006 2006 NUM cana-5460 19 8 ) ) PUNCT cana-5460 19 9 ( ( PUNCT cana-5460 19 10 left leave VERB cana-5460 19 11 caputo caputo PROPN cana-5460 19 12 derivative derivative PROPN cana-5460 19 13 ) ) PUNCT cana-5460 19 14 βˆ‚0 βˆ‚0 NOUN cana-5460 19 15 c c PROPN cana-5460 19 16 t t PROPN cana-5460 19 17 Ξ±u(x Ξ±u(x PROPN cana-5460 19 18 , , PUNCT cana-5460 19 19 t t PROPN cana-5460 19 20 ) ) PUNCT cana-5460 19 21 = = SYM cana-5460 19 22 1 1 NUM cana-5460 19 23 Ξ³(1βˆ’Ξ± Ξ³(1βˆ’Ξ± NOUN cana-5460 19 24 ) ) PUNCT cana-5460 19 25 ∫ ∫ PROPN cana-5460 20 1 βˆ‚u(x βˆ‚u(x PROPN cana-5460 20 2 , , PUNCT cana-5460 20 3 Ο„ Ο„ X cana-5460 20 4 ) ) PUNCT cana-5460 20 5 βˆ‚Ο„ βˆ‚Ο„ PROPN cana-5460 20 6 t t NOUN cana-5460 20 7 0 0 NUM cana-5460 20 8 1 1 NUM cana-5460 20 9 ( ( PUNCT cana-5460 20 10 tβˆ’Ο„)Ξ± tβˆ’Ο„)Ξ± PROPN cana-5460 20 11 dΟ„ dΟ„ PROPN cana-5460 20 12 . . PROPN cana-5460 20 13 ( ( PUNCT cana-5460 20 14 3 3 X cana-5460 20 15 ) ) PUNCT cana-5460 20 16 definition definition NOUN cana-5460 20 17 2.5 2.5 NUM cana-5460 20 18 ( ( PUNCT cana-5460 20 19 stefan stefan PROPN cana-5460 20 20 g g PROPN cana-5460 20 21 samko samko PROPN cana-5460 20 22 , , PUNCT cana-5460 20 23 anatoly anatoly PROPN cana-5460 20 24 a a DET cana-5460 20 25 kilbas kilbas PROPN cana-5460 20 26 , , PUNCT cana-5460 20 27 oleg oleg PROPN cana-5460 20 28 i i PRON cana-5460 20 29 marichev marichev PROPN cana-5460 20 30 , , PUNCT cana-5460 20 31 1993 1993 NUM cana-5460 20 32 ) ) PUNCT cana-5460 20 33 ( ( PUNCT cana-5460 20 34 mittagleffler mittagleffler NOUN cana-5460 20 35 function function NOUN cana-5460 20 36 ) ) PUNCT cana-5460 20 37 for for ADP cana-5460 20 38 𝐳 𝐳 DET cana-5460 20 39 ∈ ∈ PROPN cana-5460 20 40 𝕔 𝕔 NOUN cana-5460 20 41 , , PUNCT cana-5460 20 42 mittag mittag ADJ cana-5460 20 43 - - PUNCT cana-5460 20 44 leffler leffler NOUN cana-5460 20 45 function function NOUN cana-5460 20 46 𝐄𝛂(𝐳 𝐄𝛂(𝐳 PROPN cana-5460 20 47 ) ) PUNCT cana-5460 20 48 is be AUX cana-5460 20 49 defined define VERB cana-5460 20 50 as as SCONJ cana-5460 20 51 follows follow VERB cana-5460 20 52 : : PUNCT cana-5460 20 53 communications communication NOUN cana-5460 20 54 on on ADP cana-5460 20 55 applied apply VERB cana-5460 20 56 nonlinear nonlinear ADJ cana-5460 20 57 analysis analysis NOUN cana-5460 20 58 issn issn NOUN cana-5460 20 59 : : PUNCT cana-5460 20 60 1074 1074 NUM cana-5460 20 61 - - PUNCT cana-5460 20 62 133x 133x NUM cana-5460 20 63 vol vol NOUN cana-5460 20 64 32 32 NUM cana-5460 20 65 no.3 no.3 NOUN cana-5460 20 66 ( ( PUNCT cana-5460 20 67 2025 2025 NUM cana-5460 20 68 ) ) PUNCT cana-5460 20 69 941 941 NUM cana-5460 21 1 https://internationalpubls.com https://internationalpubls.com X cana-5460 21 2 eΞ±(z)= eΞ±(z)= PROPN cana-5460 21 3 βˆ‘ βˆ‘ PROPN cana-5460 21 4 zk zk X cana-5460 21 5 Ξ³(Ξ±k+1 Ξ³(Ξ±k+1 NOUN cana-5460 21 6 ) ) PUNCT cana-5460 22 1 ∞ ∞ NUM cana-5460 22 2 k=0 k=0 PROPN cana-5460 22 3 , , PUNCT cana-5460 22 4 Ξ± Ξ± PROPN cana-5460 22 5 > > X cana-5460 22 6 0 0 NUM cana-5460 22 7 . . PUNCT cana-5460 23 1 ( ( PUNCT cana-5460 23 2 4 4 X cana-5460 23 3 ) ) PUNCT cana-5460 23 4 the the DET cana-5460 23 5 two two NUM cana-5460 23 6 - - PUNCT cana-5460 23 7 term term NOUN cana-5460 23 8 mittag mittag ADJ cana-5460 23 9 - - PUNCT cana-5460 23 10 leffler leffler NOUN cana-5460 23 11 function function NOUN cana-5460 23 12 is be AUX cana-5460 23 13 vital vital ADJ cana-5460 23 14 in in ADP cana-5460 23 15 the the DET cana-5460 23 16 fractional fractional ADJ cana-5460 23 17 calculus calculus NOUN cana-5460 23 18 theory theory NOUN cana-5460 23 19 and and CCONJ cana-5460 23 20 defined define VERB cana-5460 23 21 as as ADP cana-5460 23 22 : : PUNCT cana-5460 23 23 eΞ± eΞ± NOUN cana-5460 23 24 , , PUNCT cana-5460 23 25 Ξ²(z)= Ξ²(z)= ADP cana-5460 23 26 βˆ‘ βˆ‘ PROPN cana-5460 23 27 zk zk PROPN cana-5460 23 28 Ξ³(Ξ±k+Ξ² Ξ³(Ξ±k+Ξ² PROPN cana-5460 23 29 ) ) PUNCT cana-5460 23 30 ∞ ∞ NUM cana-5460 23 31 k=0 k=0 PROPN cana-5460 23 32 , , PUNCT cana-5460 23 33 ( ( PUNCT cana-5460 23 34 Ξ± Ξ± X cana-5460 23 35 > > X cana-5460 23 36 0 0 PROPN cana-5460 23 37 , , PUNCT cana-5460 23 38 𝛽 𝛽 NOUN cana-5460 23 39 > > X cana-5460 23 40 0 0 NUM cana-5460 23 41 ) ) PUNCT cana-5460 23 42 . . PUNCT cana-5460 24 1 ( ( PUNCT cana-5460 24 2 5 5 X cana-5460 24 3 ) ) PUNCT cana-5460 24 4 cauchy cauchy NOUN cana-5460 24 5 - - PUNCT cana-5460 24 6 schwarz schwarz PROPN cana-5460 24 7 inequality inequality NOUN cana-5460 24 8 : : PUNCT cana-5460 24 9 βˆ€(f βˆ€(f NUM cana-5460 24 10 , , PUNCT cana-5460 24 11 g g NOUN cana-5460 24 12 ) ) PUNCT cana-5460 24 13 ∈ ∈ PROPN cana-5460 24 14 𝕃2(Ο‰ 𝕃2(Ο‰ PROPN cana-5460 24 15 ) ) PUNCT cana-5460 24 16 Γ— Γ— PROPN cana-5460 24 17 𝕃2(Ο‰ 𝕃2(Ο‰ PROPN cana-5460 24 18 ) ) PUNCT cana-5460 24 19 , , PUNCT cana-5460 24 20 we we PRON cana-5460 24 21 have have VERB cana-5460 24 22 : : PUNCT cana-5460 24 23 ∫ ∫ PROPN cana-5460 24 24 |f(t)g(t)|Ο‰ |f(t)g(t)|Ο‰ PROPN cana-5460 24 25 dt dt PROPN cana-5460 24 26 ≀ ≀ PROPN cana-5460 24 27 ( ( PUNCT cana-5460 24 28 ∫ ∫ PROPN cana-5460 24 29 ( ( PUNCT cana-5460 24 30 f(t))2Ο‰ f(t))2Ο‰ PROPN cana-5460 24 31 dt dt PROPN cana-5460 24 32 ) ) PUNCT cana-5460 24 33 1 1 NUM cana-5460 24 34 2(∫ 2(∫ NUM cana-5460 24 35 ( ( PUNCT cana-5460 24 36 g(t))2Ο‰ g(t))2Ο‰ PROPN cana-5460 24 37 dt dt NOUN cana-5460 24 38 ) ) PUNCT cana-5460 24 39 1 1 NUM cana-5460 24 40 2 2 NUM cana-5460 24 41 , , PUNCT cana-5460 24 42 ( ( PUNCT cana-5460 24 43 6 6 X cana-5460 24 44 ) ) PUNCT cana-5460 24 45 cauchy cauchy NOUN cana-5460 24 46 inequality inequality NOUN cana-5460 24 47 : : PUNCT cana-5460 24 48 βˆ€(a βˆ€(a PROPN cana-5460 24 49 , , PUNCT cana-5460 24 50 b b X cana-5460 24 51 ) ) PUNCT cana-5460 24 52 ∈ ∈ PROPN cana-5460 24 53 ℝ2 ℝ2 NOUN cana-5460 24 54 : : PUNCT cana-5460 24 55 |ab| |ab| VERB cana-5460 24 56 ≀ ≀ NUM cana-5460 24 57 1 1 NUM cana-5460 24 58 2 2 NUM cana-5460 24 59 a2 a2 NOUN cana-5460 24 60 + + CCONJ cana-5460 24 61 1 1 NUM cana-5460 24 62 2 2 NUM cana-5460 24 63 b2 b2 NOUN cana-5460 24 64 . . PUNCT cana-5460 25 1 ( ( PUNCT cana-5460 25 2 7 7 X cana-5460 25 3 ) ) PUNCT cana-5460 25 4 cauchy cauchy NOUN cana-5460 25 5 inequality inequality NOUN cana-5460 25 6 with with ADP cana-5460 25 7 π›œ π›œ NOUN cana-5460 25 8 : : PUNCT cana-5460 25 9 let let VERB cana-5460 25 10 Ο΅ Ο΅ PART cana-5460 25 11 be be AUX cana-5460 25 12 a a DET cana-5460 25 13 strictly strictly ADV cana-5460 25 14 positive positive ADJ cana-5460 25 15 number number NOUN cana-5460 25 16 , , PUNCT cana-5460 25 17 thenβˆ€(a thenβˆ€(a X cana-5460 25 18 , , PUNCT cana-5460 25 19 b b X cana-5460 25 20 ) ) PUNCT cana-5460 25 21 ∈ ∈ PROPN cana-5460 25 22 ℝ2 ℝ2 PRON cana-5460 25 23 : : PUNCT cana-5460 25 24 |π‘Žπ‘| |π‘Žπ‘| ADJ cana-5460 25 25 ≀ ≀ PUNCT cana-5460 25 26 πœ€a2 πœ€a2 NOUN cana-5460 25 27 2 2 NUM cana-5460 25 28 + + NUM cana-5460 25 29 b2 b2 NOUN cana-5460 25 30 2πœ€ 2πœ€ NOUN cana-5460 25 31 . . PUNCT cana-5460 26 1 ( ( PUNCT cana-5460 26 2 8) 8) NUM cana-5460 26 3 poincarΓ© poincarΓ© PROPN cana-5460 26 4 inequality inequality PROPN cana-5460 26 5 lemma lemma PROPN cana-5460 26 6 2.1(a.a 2.1(a.a PROPN cana-5460 26 7 . . PUNCT cana-5460 27 1 alikhanov alikhanov PROPN cana-5460 27 2 , , PUNCT cana-5460 27 3 2010 2010 NUM cana-5460 27 4 ) ) PUNCT cana-5460 27 5 for for ADP cana-5460 27 6 any any DET cana-5460 27 7 function function NOUN cana-5460 27 8 s(t s(t PROPN cana-5460 27 9 ) ) PUNCT cana-5460 27 10 that that PRON cana-5460 27 11 is be AUX cana-5460 27 12 absolutely absolutely ADV cana-5460 27 13 continuous continuous ADJ cana-5460 27 14 on on ADP cana-5460 27 15 the the DET cana-5460 27 16 interval interval NOUN cana-5460 28 1 [ [ X cana-5460 28 2 0,t 0,t X cana-5460 28 3 ] ] X cana-5460 28 4 , , PUNCT cana-5460 28 5 the the DET cana-5460 28 6 following follow VERB cana-5460 28 7 inequality inequality NOUN cana-5460 28 8 holds hold VERB cana-5460 28 9 : : PUNCT cana-5460 28 10 s(t s(t PROPN cana-5460 28 11 ) ) PUNCT cana-5460 28 12 βˆ‚t βˆ‚t PROPN cana-5460 28 13 Ξ² Ξ² X cana-5460 28 14 s(t s(t PROPN cana-5460 28 15 ) ) PUNCT cana-5460 28 16 β‰₯ β‰₯ NOUN cana-5460 28 17 1 1 NUM cana-5460 28 18 2 2 NUM cana-5460 28 19 βˆ‚t βˆ‚t PROPN cana-5460 28 20 Ξ² Ξ² X cana-5460 28 21 s2(t s2(t PROPN cana-5460 28 22 ) ) PUNCT cana-5460 28 23 , , PUNCT cana-5460 28 24 0 0 PUNCT cana-5460 28 25 < < X cana-5460 28 26 𝛼 𝛼 X cana-5460 28 27 < < X cana-5460 28 28 1 1 NUM cana-5460 28 29 . . PUNCT cana-5460 29 1 ( ( PUNCT cana-5460 29 2 9 9 X cana-5460 29 3 ) ) PUNCT cana-5460 29 4 lemma lemma PROPN cana-5460 29 5 2.2(a.a 2.2(a.a PROPN cana-5460 29 6 . . PUNCT cana-5460 29 7 alikhanov alikhanov PROPN cana-5460 29 8 , , PUNCT cana-5460 29 9 2010 2010 NUM cana-5460 29 10 ) ) PUNCT cana-5460 29 11 ( ( PUNCT cana-5460 29 12 gnonwall gnonwall PROPN cana-5460 29 13 lemma lemma PROPN cana-5460 29 14 ) ) PUNCT cana-5460 29 15 let let VERB cana-5460 29 16 a a DET cana-5460 29 17 non non ADJ cana-5460 29 18 - - ADJ cana-5460 29 19 negative negative ADJ cana-5460 29 20 , , PUNCT cana-5460 29 21 absolutely absolutely ADV cana-5460 29 22 continuous continuous ADJ cana-5460 29 23 function function NOUN cana-5460 29 24 y(t y(t NUM cana-5460 29 25 ) ) PUNCT cana-5460 29 26 satisfy satisfy VERB cana-5460 29 27 the the DET cana-5460 29 28 inequality inequality NOUN cana-5460 29 29 : : PUNCT cana-5460 29 30 βˆ‚t βˆ‚t PROPN cana-5460 29 31 Ξ±y(t Ξ±y(t NOUN cana-5460 29 32 ) ) PUNCT cana-5460 29 33 ≀ ≀ NUM cana-5460 29 34 k1u(t k1u(t PROPN cana-5460 29 35 ) ) PUNCT cana-5460 30 1 + + NUM cana-5460 31 1 k2(t k2(t PROPN cana-5460 31 2 ) ) PUNCT cana-5460 31 3 , , PUNCT cana-5460 32 1 0 0 PUNCT cana-5460 32 2 < < X cana-5460 32 3 𝛼 𝛼 X cana-5460 32 4 < < X cana-5460 32 5 1 1 NUM cana-5460 32 6 . . PUNCT cana-5460 32 7 ( ( PUNCT cana-5460 32 8 10 10 NUM cana-5460 32 9 ) ) PUNCT cana-5460 32 10 for for ADP cana-5460 32 11 all all DET cana-5460 32 12 t t NOUN cana-5460 32 13 ∈ ∈ PROPN cana-5460 33 1 [ [ X cana-5460 33 2 0 0 NUM cana-5460 33 3 , , PUNCT cana-5460 33 4 t t PROPN cana-5460 33 5 ] ] X cana-5460 33 6 , , PUNCT cana-5460 33 7 where where SCONJ cana-5460 33 8 k1 k1 PROPN cana-5460 33 9 is be AUX cana-5460 33 10 a a DET cana-5460 33 11 positive positive ADJ cana-5460 33 12 constant constant NOUN cana-5460 33 13 and and CCONJ cana-5460 33 14 k2(t k2(t PROPN cana-5460 33 15 ) ) PUNCT cana-5460 33 16 a a DET cana-5460 33 17 non non ADJ cana-5460 33 18 - - ADJ cana-5460 33 19 negative negative ADJ cana-5460 33 20 integrale integrale NOUN cana-5460 33 21 function function NOUN cana-5460 33 22 over over ADP cana-5460 33 23 [ [ X cana-5460 33 24 0,t 0,t X cana-5460 33 25 ] ] X cana-5460 33 26 . . PUNCT cana-5460 34 1 then then ADV cana-5460 34 2 , , PUNCT cana-5460 34 3 y(t y(t PROPN cana-5460 34 4 ) ) PUNCT cana-5460 34 5 ≀ ≀ NUM cana-5460 34 6 y(0)eΞ±(k1 y(0)eΞ±(k1 PROPN cana-5460 34 7 t t PROPN cana-5460 34 8 Ξ± Ξ± NOUN cana-5460 34 9 ) ) PUNCT cana-5460 34 10 + + CCONJ cana-5460 34 11 Ξ³(Ξ±)eΞ± Ξ³(Ξ±)eΞ± X cana-5460 34 12 , , PUNCT cana-5460 34 13 Ξ±(k1 Ξ±(k1 NOUN cana-5460 34 14 t t PROPN cana-5460 34 15 Ξ±)dt Ξ±)dt PROPN cana-5460 34 16 βˆ’Ξ±k2(t βˆ’Ξ±k2(t NOUN cana-5460 34 17 ) ) PUNCT cana-5460 34 18 , , PUNCT cana-5460 34 19 ( ( PUNCT cana-5460 34 20 11 11 X cana-5460 34 21 ) ) PUNCT cana-5460 34 22 communications communication NOUN cana-5460 34 23 on on ADP cana-5460 34 24 applied apply VERB cana-5460 34 25 nonlinear nonlinear ADJ cana-5460 34 26 analysis analysis NOUN cana-5460 34 27 issn issn NOUN cana-5460 34 28 : : PUNCT cana-5460 34 29 1074 1074 NUM cana-5460 34 30 - - PUNCT cana-5460 34 31 133x 133x NUM cana-5460 34 32 vol vol NOUN cana-5460 34 33 32 32 NUM cana-5460 34 34 no.3 no.3 NOUN cana-5460 34 35 ( ( PUNCT cana-5460 34 36 2025 2025 NUM cana-5460 34 37 ) ) PUNCT cana-5460 34 38 942 942 NUM cana-5460 34 39 https://internationalpubls.com https://internationalpubls.com X cana-5460 35 1 where∢ where∢ PROPN cana-5460 35 2 eΞ± eΞ± VERB cana-5460 35 3 et et NOUN cana-5460 35 4 eΞ± eΞ± NOUN cana-5460 35 5 , , PUNCT cana-5460 35 6 Ξ±are Ξ±are VERB cana-5460 35 7 mittag mittag ADJ cana-5460 35 8 - - PUNCT cana-5460 35 9 leffler leffler NOUN cana-5460 35 10 functions function NOUN cana-5460 35 11 . . PUNCT cana-5460 36 1 lemma lemma PROPN cana-5460 36 2 2.3 2.3 NUM cana-5460 36 3 ( ( PUNCT cana-5460 36 4 ladyzhenskaya ladyzhenskaya PROPN cana-5460 36 5 , , PUNCT cana-5460 36 6 1985 1985 NUM cana-5460 36 7 ) ) PUNCT cana-5460 36 8 let let VERB cana-5460 36 9 u(t u(t NOUN cana-5460 36 10 ) ) PUNCT cana-5460 36 11 be be AUX cana-5460 36 12 a a DET cana-5460 36 13 non non ADJ cana-5460 36 14 - - ADJ cana-5460 36 15 negative negative ADJ cana-5460 36 16 , , PUNCT cana-5460 36 17 absolutely absolutely ADV cana-5460 36 18 continuous continuous ADJ cana-5460 36 19 function function NOUN cana-5460 36 20 on on ADP cana-5460 36 21 [ [ X cana-5460 36 22 0,t 0,t X cana-5460 36 23 ] ] X cana-5460 36 24 , , PUNCT cana-5460 36 25 and and CCONJ cana-5460 36 26 for for ADP cana-5460 36 27 all all DET cana-5460 36 28 t t NOUN cana-5460 36 29 ∈ ∈ PROPN cana-5460 37 1 [ [ X cana-5460 37 2 0 0 NUM cana-5460 37 3 , , PUNCT cana-5460 37 4 t t PROPN cana-5460 37 5 ] ] PUNCT cana-5460 37 6 , , PUNCT cana-5460 37 7 satisfies satisfy VERB cana-5460 37 8 the the DET cana-5460 37 9 inequality inequality NOUN cana-5460 37 10 : : PUNCT cana-5460 37 11 dΟ† dΟ† ADP cana-5460 37 12 dt dt X cana-5460 37 13 ≀ ≀ NOUN cana-5460 37 14 c(t)Ο†(t c(t)Ο†(t NUM cana-5460 37 15 ) ) PUNCT cana-5460 37 16 + + NUM cana-5460 37 17 b(t b(t NOUN cana-5460 37 18 ) ) PUNCT cana-5460 37 19 . . PUNCT cana-5460 38 1 ( ( PUNCT cana-5460 38 2 12 12 NUM cana-5460 38 3 ) ) PUNCT cana-5460 38 4 such such ADJ cana-5460 38 5 that that SCONJ cana-5460 38 6 the the DET cana-5460 38 7 functions function NOUN cana-5460 38 8 c(t c(t NOUN cana-5460 38 9 ) ) PUNCT cana-5460 38 10 and and CCONJ cana-5460 38 11 b(t b(t NOUN cana-5460 38 12 ) ) PUNCT cana-5460 38 13 are be AUX cana-5460 38 14 summable summable ADJ cana-5460 38 15 and and CCONJ cana-5460 38 16 non non ADJ cana-5460 38 17 - - ADJ cana-5460 38 18 negative negative ADJ cana-5460 38 19 on on ADP cana-5460 38 20 [ [ X cana-5460 38 21 0,t 0,t X cana-5460 38 22 ] ] X cana-5460 38 23 . . PUNCT cana-5460 39 1 here here ADV cana-5460 39 2 : : PUNCT cana-5460 39 3 πœ‘(𝑑 πœ‘(𝑑 X cana-5460 39 4 ) ) PUNCT cana-5460 39 5 ≀ ≀ NUM cana-5460 39 6 π‘’βˆ« π‘’βˆ« NUM cana-5460 39 7 𝐢(𝜏)π‘‘πœ 𝐢(𝜏)π‘‘πœ NOUN cana-5460 39 8 𝑑 𝑑 X cana-5460 39 9 0 0 NUM cana-5460 39 10 πœ‘(0 πœ‘(0 NOUN cana-5460 39 11 ) ) PUNCT cana-5460 40 1 + + NUM cana-5460 40 2 ∫ ∫ PROPN cana-5460 40 3 𝐡(πœ‰)π‘’βˆ« 𝐡(πœ‰)π‘’βˆ« PROPN cana-5460 40 4 𝐢(𝜏)π‘‘πœ 𝐢(𝜏)π‘‘πœ NOUN cana-5460 40 5 πœ‰ πœ‰ NOUN cana-5460 40 6 0 0 NUM cana-5460 40 7 𝑑 𝑑 NOUN cana-5460 40 8 0 0 NUM cana-5460 40 9 π‘‘πœ‰ π‘‘πœ‰ ADP cana-5460 40 10 ( ( PUNCT cana-5460 40 11 13 13 NUM cana-5460 40 12 ) ) PUNCT cana-5460 40 13 lemma lemma PROPN cana-5460 40 14 2.4 2.4 NUM cana-5460 40 15 ( ( PUNCT cana-5460 40 16 mesloub mesloub PROPN cana-5460 40 17 s s NOUN cana-5460 40 18 , , PUNCT cana-5460 40 19 mezhoudi mezhoudi ADJ cana-5460 40 20 r r NOUN cana-5460 40 21 , , PUNCT cana-5460 40 22 medjeden medjeden X cana-5460 40 23 m m PROPN cana-5460 40 24 , , PUNCT cana-5460 40 25 2002 2002 NUM cana-5460 40 26 ) ) PUNCT cana-5460 40 27 for for ADP cana-5460 40 28 any any DET cana-5460 40 29 n n PRON cana-5460 40 30 ∈ ∈ PROPN cana-5460 40 31 β„• β„• PROPN cana-5460 40 32 , , PUNCT cana-5460 40 33 we we PRON cana-5460 40 34 have have VERB cana-5460 40 35 β€–β„‘x β€–β„‘x NOUN cana-5460 41 1 2nuβ€– 2nuβ€– NUM cana-5460 41 2 2 2 NUM cana-5460 41 3 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 41 4 ) ) PUNCT cana-5460 41 5 ≀ ≀ NOUN cana-5460 41 6 ( ( PUNCT cana-5460 41 7 1 1 NUM cana-5460 41 8 2 2 NUM cana-5460 41 9 ) ) PUNCT cana-5460 41 10 2n 2n NUM cana-5460 41 11 β€–uβ€–2𝕃2(0,1 β€–uβ€–2𝕃2(0,1 NOUN cana-5460 41 12 ) ) PUNCT cana-5460 41 13 . . PUNCT cana-5460 42 1 ( ( PUNCT cana-5460 42 2 14 14 NUM cana-5460 42 3 ) ) PUNCT cana-5460 42 4 where where SCONJ cana-5460 42 5 : : PUNCT cana-5460 43 1 β„‘x β„‘x PROPN cana-5460 43 2 2nu 2nu NOUN cana-5460 43 3 = = SYM cana-5460 44 1 ∫ ∫ PROPN cana-5460 44 2 ∫ ∫ PROPN cana-5460 44 3 … … PUNCT cana-5460 45 1 ΞΎ1 ΞΎ1 NOUN cana-5460 45 2 0 0 NUM cana-5460 45 3 ∫ ∫ PROPN cana-5460 45 4 u(Ξ· u(Ξ· PROPN cana-5460 45 5 , , PUNCT cana-5460 45 6 t t PROPN cana-5460 45 7 ) ) PUNCT cana-5460 45 8 ΞΎ2nβˆ’1 ΞΎ2nβˆ’1 PROPN cana-5460 45 9 0 0 NUM cana-5460 45 10 x x SYM cana-5460 45 11 0 0 NUM cana-5460 45 12 dΞ·dΞΎ2nβˆ’1 dΞ·dΞΎ2nβˆ’1 PROPN cana-5460 45 13 … … SYM cana-5460 45 14 dΞΎ1 dΞΎ1 NOUN cana-5460 45 15 = = SYM cana-5460 45 16 ∫ ∫ PROPN cana-5460 45 17 ( ( PUNCT cana-5460 45 18 x x PART cana-5460 45 19 βˆ’ βˆ’ NOUN cana-5460 45 20 ΞΎ)2nβˆ’1 ΞΎ)2nβˆ’1 ADP cana-5460 45 21 ( ( PUNCT cana-5460 45 22 2n 2n NUM cana-5460 45 23 βˆ’ βˆ’ NOUN cana-5460 45 24 1 1 NUM cana-5460 45 25 ) ) PUNCT cana-5460 45 26 ! ! PUNCT cana-5460 46 1 x x X cana-5460 46 2 0 0 PUNCT cana-5460 47 1 u(ΞΎ u(ΞΎ NOUN cana-5460 47 2 , , PUNCT cana-5460 47 3 t)dΞΎ t)dΞΎ PROPN cana-5460 47 4 3 3 NUM cana-5460 47 5 . . NOUN cana-5460 47 6 problem problem NOUN cana-5460 47 7 statement statement NOUN cana-5460 47 8 in in ADP cana-5460 47 9 a a DET cana-5460 47 10 rectangular rectangular ADJ cana-5460 47 11 domain domain NOUN cana-5460 47 12 : : PUNCT cana-5460 47 13 Ο‰ Ο‰ NUM cana-5460 47 14 = = SYM cana-5460 47 15 ( ( PUNCT cana-5460 47 16 0,1 0,1 NUM cana-5460 47 17 ) ) PUNCT cana-5460 47 18 Γ— Γ— NOUN cana-5460 47 19 ( ( PUNCT cana-5460 47 20 0 0 NUM cana-5460 47 21 , , PUNCT cana-5460 47 22 t t PROPN cana-5460 47 23 ) ) PUNCT cana-5460 47 24 , , PUNCT cana-5460 47 25 0 0 NUM cana-5460 48 1 ≀ ≀ NUM cana-5460 48 2 t t X cana-5460 48 3 ≀ ≀ NUM cana-5460 48 4 ∞ ∞ PROPN cana-5460 48 5 , , PUNCT cana-5460 48 6 consider consider VERB cana-5460 48 7 the the DET cana-5460 48 8 following follow VERB cana-5460 48 9 fractional fractional ADJ cana-5460 48 10 partial partial ADJ cana-5460 48 11 differential differential NOUN cana-5460 48 12 equation equation NOUN cana-5460 48 13 : : PUNCT cana-5460 49 1 β„’v β„’v PROPN cana-5460 49 2 = = PRON cana-5460 49 3 βˆ‚0 βˆ‚0 NOUN cana-5460 49 4 c c PROPN cana-5460 49 5 t t PROPN cana-5460 49 6 Ξ΄v(x Ξ΄v(x NUM cana-5460 49 7 , , PUNCT cana-5460 49 8 t t PROPN cana-5460 49 9 ) ) PUNCT cana-5460 49 10 βˆ’ βˆ’ PROPN cana-5460 49 11 Ξ± Ξ± PROPN cana-5460 49 12 βˆ‚2v βˆ‚2v PROPN cana-5460 49 13 βˆ‚x2 βˆ‚x2 PROPN cana-5460 49 14 βˆ’ βˆ’ NOUN cana-5460 49 15 Ξ² Ξ² NOUN cana-5460 49 16 βˆ‚3v βˆ‚3v NOUN cana-5460 49 17 βˆ‚t βˆ‚t PROPN cana-5460 49 18 βˆ‚x2 βˆ‚x2 PROPN cana-5460 49 19 + + NOUN cana-5460 49 20 Ξ³v Ξ³v NOUN cana-5460 49 21 βˆ’ βˆ’ NOUN cana-5460 49 22 ∫ ∫ NOUN cana-5460 49 23 a(t a(t NOUN cana-5460 49 24 βˆ’ βˆ’ NOUN cana-5460 49 25 s)v(x s)v(x NOUN cana-5460 49 26 , , PUNCT cana-5460 49 27 s)ds s)ds PROPN cana-5460 49 28 = = SYM cana-5460 49 29 g(x g(x PROPN cana-5460 49 30 , , PUNCT cana-5460 49 31 t t PROPN cana-5460 49 32 , , PUNCT cana-5460 49 33 v v NOUN cana-5460 49 34 , , PUNCT cana-5460 49 35 r r NOUN cana-5460 49 36 ) ) PUNCT cana-5460 49 37 , , PUNCT cana-5460 49 38 t t PROPN cana-5460 49 39 0 0 NUM cana-5460 49 40 0 0 PUNCT cana-5460 49 41 < < X cana-5460 49 42 𝑑 𝑑 X cana-5460 49 43 < < X cana-5460 49 44 𝑇 𝑇 PROPN cana-5460 49 45 , , PUNCT cana-5460 49 46 0 0 PUNCT cana-5460 49 47 < < X cana-5460 49 48 𝛿 𝛿 X cana-5460 49 49 < < X cana-5460 49 50 1 1 NUM cana-5460 49 51 ( ( PUNCT cana-5460 49 52 15 15 NUM cana-5460 49 53 ) ) PUNCT cana-5460 49 54 where where SCONJ cana-5460 49 55 : : PUNCT cana-5460 49 56 r(x r(x PROPN cana-5460 49 57 , , PUNCT cana-5460 49 58 t t PROPN cana-5460 49 59 ) ) PUNCT cana-5460 49 60 = = SYM cana-5460 50 1 ∫ ∫ PROPN cana-5460 50 2 g(x g(x PROPN cana-5460 50 3 , , PUNCT cana-5460 50 4 s s PROPN cana-5460 50 5 , , PUNCT cana-5460 50 6 v(x v(x PROPN cana-5460 50 7 , , PUNCT cana-5460 50 8 s s NOUN cana-5460 50 9 ) ) PUNCT cana-5460 50 10 , , PUNCT cana-5460 50 11 r(x r(x PROPN cana-5460 50 12 , , PUNCT cana-5460 50 13 s))ds s))d VERB cana-5460 50 14 t t PROPN cana-5460 50 15 0 0 NUM cana-5460 50 16 . . PUNCT cana-5460 51 1 where where SCONJ cana-5460 51 2 a(t a(t NOUN cana-5460 51 3 ) ) PUNCT cana-5460 51 4 is be AUX cana-5460 51 5 a a DET cana-5460 51 6 function function NOUN cana-5460 51 7 of of ADP cana-5460 51 8 t t NOUN cana-5460 51 9 and and CCONJ cana-5460 51 10 satisfies satisfy VERB cana-5460 51 11 the the DET cana-5460 51 12 condition condition NOUN cana-5460 51 13 0 0 PUNCT cana-5460 51 14 < < X cana-5460 51 15 a0 a0 PROPN cana-5460 51 16 < < X cana-5460 51 17 π‘Ž(𝑑 π‘Ž(𝑑 PROPN cana-5460 51 18 ) ) PUNCT cana-5460 51 19 < < X cana-5460 51 20 a1 a1 NOUN cana-5460 51 21 and and CCONJ cana-5460 51 22 Ξ± Ξ± NOUN cana-5460 51 23 , , PUNCT cana-5460 51 24 Ξ² Ξ² X cana-5460 51 25 and and CCONJ cana-5460 51 26 Ξ³ Ξ³ NOUN cana-5460 51 27 are be AUX cana-5460 51 28 strictly strictly ADV cana-5460 51 29 positive positive ADJ cana-5460 51 30 constants constant NOUN cana-5460 51 31 . . PUNCT cana-5460 52 1 with with ADP cana-5460 52 2 initial initial ADJ cana-5460 52 3 conditions condition NOUN cana-5460 52 4 , , PUNCT cana-5460 52 5 β„“v β„“v PROPN cana-5460 52 6 = = SYM cana-5460 52 7 v(x v(x PROPN cana-5460 52 8 , , PUNCT cana-5460 52 9 0 0 NUM cana-5460 52 10 ) ) PUNCT cana-5460 52 11 = = SYM cana-5460 52 12 ΙΈ(x ΙΈ(x PROPN cana-5460 52 13 ) ) PUNCT cana-5460 52 14 , , PUNCT cana-5460 52 15 qv qv X cana-5460 52 16 = = SYM cana-5460 52 17 βˆ‚v(x,0 βˆ‚v(x,0 ADJ cana-5460 52 18 ) ) PUNCT cana-5460 52 19 βˆ‚t βˆ‚t PROPN cana-5460 52 20 = = PUNCT cana-5460 52 21 ψ(x ψ(x PROPN cana-5460 52 22 ) ) PUNCT cana-5460 52 23 , , PUNCT cana-5460 52 24 0 0 PUNCT cana-5460 52 25 < < X cana-5460 52 26 π‘₯ π‘₯ X cana-5460 52 27 < < X cana-5460 52 28 1 1 NUM cana-5460 52 29 , , PUNCT cana-5460 52 30 ( ( PUNCT cana-5460 52 31 16 16 NUM cana-5460 52 32 ) ) PUNCT cana-5460 52 33 and and CCONJ cana-5460 52 34 integral integral ADJ cana-5460 52 35 conditions condition NOUN cana-5460 52 36 , , PUNCT cana-5460 52 37 ∫ ∫ PROPN cana-5460 52 38 v(x v(x PROPN cana-5460 52 39 , , PUNCT cana-5460 52 40 t)dx t)dx PROPN cana-5460 52 41 = = SYM cana-5460 52 42 m(t m(t NOUN cana-5460 52 43 ) ) PUNCT cana-5460 52 44 , , PUNCT cana-5460 52 45 1 1 NUM cana-5460 52 46 0 0 NUM cana-5460 52 47 ∫ ∫ PROPN cana-5460 52 48 xv(x xv(x PROPN cana-5460 52 49 , , PUNCT cana-5460 52 50 t)dx t)dx PROPN cana-5460 52 51 = = SYM cana-5460 52 52 n(t n(t PROPN cana-5460 52 53 ) ) PUNCT cana-5460 52 54 , , PUNCT cana-5460 52 55 1 1 NUM cana-5460 52 56 0 0 NUM cana-5460 52 57 0 0 NUM cana-5460 52 58 < < X cana-5460 52 59 𝑑 𝑑 PROPN cana-5460 52 60 ≀ ≀ PROPN cana-5460 52 61 𝑇 𝑇 PROPN cana-5460 52 62 , , PUNCT cana-5460 52 63 ( ( PUNCT cana-5460 52 64 17 17 NUM cana-5460 52 65 ) ) PUNCT cana-5460 52 66 communications communication NOUN cana-5460 52 67 on on ADP cana-5460 52 68 applied apply VERB cana-5460 52 69 nonlinear nonlinear ADJ cana-5460 52 70 analysis analysis NOUN cana-5460 52 71 issn issn NOUN cana-5460 52 72 : : PUNCT cana-5460 52 73 1074 1074 NUM cana-5460 52 74 - - PUNCT cana-5460 52 75 133x 133x NUM cana-5460 52 76 vol vol NOUN cana-5460 52 77 32 32 NUM cana-5460 52 78 no.3 no.3 NOUN cana-5460 52 79 ( ( PUNCT cana-5460 52 80 2025 2025 NUM cana-5460 52 81 ) ) PUNCT cana-5460 52 82 943 943 NUM cana-5460 52 83 https://internationalpubls.com https://internationalpubls.com X cana-5460 52 84 where where SCONJ cana-5460 52 85 Ο† Ο† PROPN cana-5460 52 86 , , PUNCT cana-5460 52 87 ψ ψ PROPN cana-5460 52 88 , , PUNCT cana-5460 52 89 m m PROPN cana-5460 52 90 , , PUNCT cana-5460 52 91 n n PROPN cana-5460 52 92 and and CCONJ cana-5460 52 93 g g PROPN cana-5460 52 94 are be AUX cana-5460 52 95 known know VERB cana-5460 52 96 functions function NOUN cana-5460 52 97 . . PUNCT cana-5460 53 1 since since SCONJ cana-5460 53 2 the the DET cana-5460 53 3 boundary boundary ADJ cana-5460 53 4 conditions condition NOUN cana-5460 53 5 are be AUX cana-5460 53 6 non non ADJ cana-5460 53 7 - - ADJ cana-5460 53 8 homogeneous homogeneous ADJ cana-5460 53 9 , , PUNCT cana-5460 53 10 we we PRON cana-5460 53 11 construct construct VERB cana-5460 53 12 the the DET cana-5460 53 13 function function NOUN cana-5460 53 14 u(x u(x NOUN cana-5460 53 15 , , PUNCT cana-5460 53 16 t t NOUN cana-5460 53 17 ) ) PUNCT cana-5460 53 18 = = SYM cana-5460 53 19 6(2n(t 6(2n(t NUM cana-5460 53 20 ) ) PUNCT cana-5460 54 1 βˆ’ βˆ’ PROPN cana-5460 54 2 m(t))x m(t))x PROPN cana-5460 54 3 βˆ’ βˆ’ PROPN cana-5460 54 4 2(3n(t 2(3n(t NUM cana-5460 54 5 ) ) PUNCT cana-5460 54 6 βˆ’ βˆ’ ADP cana-5460 54 7 2m(t 2m(t NUM cana-5460 54 8 ) ) PUNCT cana-5460 54 9 ) ) PUNCT cana-5460 54 10 . . PUNCT cana-5460 55 1 and and CCONJ cana-5460 55 2 we we PRON cana-5460 55 3 introduce introduce VERB cana-5460 55 4 a a DET cana-5460 55 5 new new ADJ cana-5460 55 6 function function NOUN cana-5460 55 7 : : PUNCT cana-5460 55 8 u(x u(x PROPN cana-5460 55 9 , , PUNCT cana-5460 55 10 t t NOUN cana-5460 55 11 ) ) PUNCT cana-5460 55 12 = = SYM cana-5460 56 1 v(x v(x PROPN cana-5460 56 2 , , PUNCT cana-5460 56 3 t t PROPN cana-5460 56 4 ) ) PUNCT cana-5460 56 5 βˆ’ βˆ’ PROPN cana-5460 57 1 u(x u(x PROPN cana-5460 57 2 , , PUNCT cana-5460 57 3 t t PROPN cana-5460 57 4 ) ) PUNCT cana-5460 57 5 . . PUNCT cana-5460 58 1 then then ADV cana-5460 58 2 , , PUNCT cana-5460 58 3 the the DET cana-5460 58 4 problem problem NOUN cana-5460 58 5 ( ( PUNCT cana-5460 58 6 15)-(17 15)-(17 NUM cana-5460 58 7 ) ) PUNCT cana-5460 58 8 can can AUX cana-5460 58 9 be be AUX cana-5460 58 10 reformulated reformulate VERB cana-5460 58 11 as as SCONJ cana-5460 58 12 follows follow VERB cana-5460 58 13 : : PUNCT cana-5460 58 14 β„’u β„’u NOUN cana-5460 58 15 = = PUNCT cana-5460 58 16 βˆ‚0 βˆ‚0 NOUN cana-5460 58 17 c c PROPN cana-5460 58 18 t t PROPN cana-5460 58 19 Ξ΄u(x Ξ΄u(x PROPN cana-5460 58 20 , , PUNCT cana-5460 58 21 t t PROPN cana-5460 58 22 ) ) PUNCT cana-5460 58 23 βˆ’ βˆ’ PROPN cana-5460 58 24 Ξ± Ξ± PROPN cana-5460 58 25 βˆ‚2u βˆ‚2u PROPN cana-5460 58 26 βˆ‚x2 βˆ‚x2 PROPN cana-5460 58 27 βˆ’ βˆ’ PROPN cana-5460 58 28 Ξ² Ξ² X cana-5460 58 29 βˆ‚3u βˆ‚3u ADJ cana-5460 58 30 βˆ‚tβˆ‚x2 βˆ‚tβˆ‚x2 NOUN cana-5460 58 31 + + NUM cana-5460 58 32 Ξ³u Ξ³u INTJ cana-5460 58 33 βˆ’ βˆ’ PROPN cana-5460 58 34 ∫ ∫ NOUN cana-5460 58 35 a(t a(t NOUN cana-5460 58 36 βˆ’ βˆ’ NOUN cana-5460 58 37 s)u(x s)u(x NOUN cana-5460 58 38 , , PUNCT cana-5460 58 39 s)ds s)ds PROPN cana-5460 58 40 = = SYM cana-5460 58 41 f(x f(x PROPN cana-5460 58 42 , , PUNCT cana-5460 58 43 t t PROPN cana-5460 58 44 , , PUNCT cana-5460 58 45 u u NOUN cana-5460 58 46 , , PUNCT cana-5460 58 47 r r NOUN cana-5460 58 48 ) ) PUNCT cana-5460 58 49 t t NOUN cana-5460 58 50 0 0 NUM cana-5460 58 51 , , PUNCT cana-5460 58 52 ( ( PUNCT cana-5460 58 53 18 18 NUM cana-5460 58 54 ) ) PUNCT cana-5460 58 55 where where SCONJ cana-5460 58 56 : : PUNCT cana-5460 58 57 f(x f(x PROPN cana-5460 58 58 , , PUNCT cana-5460 58 59 t t PROPN cana-5460 58 60 , , PUNCT cana-5460 58 61 u(x u(x PROPN cana-5460 58 62 , , PUNCT cana-5460 58 63 t t PROPN cana-5460 58 64 ) ) PUNCT cana-5460 58 65 , , PUNCT cana-5460 58 66 r(x r(x PROPN cana-5460 58 67 , , PUNCT cana-5460 58 68 t t PROPN cana-5460 58 69 ) ) PUNCT cana-5460 58 70 ) ) PUNCT cana-5460 59 1 = = SYM cana-5460 59 2 g(x g(x PROPN cana-5460 59 3 , , PUNCT cana-5460 59 4 t t PROPN cana-5460 59 5 , , PUNCT cana-5460 59 6 v v NOUN cana-5460 59 7 , , PUNCT cana-5460 59 8 r r NOUN cana-5460 59 9 ) ) PUNCT cana-5460 59 10 βˆ’ βˆ’ ADP cana-5460 60 1 β„’v β„’v ADJ cana-5460 60 2 + + NUM cana-5460 60 3 ∫ ∫ NOUN cana-5460 60 4 a(t a(t NOUN cana-5460 60 5 βˆ’ βˆ’ NOUN cana-5460 60 6 s)u(x s)u(x NOUN cana-5460 60 7 , , PUNCT cana-5460 60 8 s)ds s)ds PROPN cana-5460 60 9 t t PROPN cana-5460 60 10 0 0 NUM cana-5460 60 11 . . PUNCT cana-5460 61 1 the the DET cana-5460 61 2 initial initial ADJ cana-5460 61 3 conditions condition NOUN cana-5460 61 4 β„“u β„“u X cana-5460 61 5 = = SYM cana-5460 61 6 u(x u(x NOUN cana-5460 61 7 , , PUNCT cana-5460 61 8 0 0 NUM cana-5460 61 9 ) ) PUNCT cana-5460 61 10 = = SYM cana-5460 61 11 ΙΈ(x ΙΈ(x ADJ cana-5460 61 12 ) ) PUNCT cana-5460 61 13 βˆ’β„“u βˆ’β„“u PROPN cana-5460 61 14 = = SYM cana-5460 61 15 Ο†(x Ο†(x PROPN cana-5460 61 16 ) ) PUNCT cana-5460 61 17 , , PUNCT cana-5460 61 18 qu qu PROPN cana-5460 61 19 = = PUNCT cana-5460 61 20 βˆ‚u(x,0 βˆ‚u(x,0 PROPN cana-5460 61 21 ) ) PUNCT cana-5460 61 22 βˆ‚t βˆ‚t PROPN cana-5460 61 23 = = PUNCT cana-5460 61 24 ψ(x ψ(x PROPN cana-5460 61 25 ) ) PUNCT cana-5460 61 26 βˆ’ βˆ’ PROPN cana-5460 61 27 qu qu PROPN cana-5460 61 28 = = PROPN cana-5460 61 29 ψ(x ψ(x PROPN cana-5460 61 30 ) ) PUNCT cana-5460 61 31 , , PUNCT cana-5460 61 32 0 0 PUNCT cana-5460 61 33 < < X cana-5460 61 34 π‘₯ π‘₯ X cana-5460 61 35 < < X cana-5460 61 36 1 1 NUM cana-5460 61 37 . . PUNCT cana-5460 61 38 ( ( PUNCT cana-5460 61 39 19 19 NUM cana-5460 61 40 ) ) PUNCT cana-5460 61 41 the the DET cana-5460 61 42 integral integral ADJ cana-5460 61 43 conditions condition NOUN cana-5460 61 44 : : PUNCT cana-5460 61 45 ∫ ∫ PROPN cana-5460 61 46 u(x u(x PROPN cana-5460 61 47 , , PUNCT cana-5460 61 48 t)dx t)dx PROPN cana-5460 61 49 = = SYM cana-5460 61 50 0 0 NUM cana-5460 61 51 , , PUNCT cana-5460 61 52 t t PROPN cana-5460 61 53 0 0 NUM cana-5460 61 54 ∫ ∫ PROPN cana-5460 61 55 xu(x xu(x NOUN cana-5460 61 56 , , PUNCT cana-5460 61 57 t)dx t)dx PROPN cana-5460 61 58 = = SYM cana-5460 61 59 0 0 NUM cana-5460 61 60 , , PUNCT cana-5460 61 61 t t PROPN cana-5460 61 62 0 0 NUM cana-5460 61 63 0 0 NUM cana-5460 61 64 < < X cana-5460 61 65 𝑑 𝑑 PROPN cana-5460 61 66 ≀ ≀ NOUN cana-5460 61 67 𝑇 𝑇 PROPN cana-5460 61 68 . . PUNCT cana-5460 62 1 ( ( PUNCT cana-5460 62 2 20 20 NUM cana-5460 62 3 ) ) PUNCT cana-5460 62 4 4.technical 4.technical NUM cana-5460 62 5 tools tool NOUN cana-5460 62 6 and and CCONJ cana-5460 62 7 associated associate VERB cana-5460 62 8 linear linear PROPN cana-5460 62 9 problem problem NOUN cana-5460 62 10 we we PRON cana-5460 62 11 define define VERB cana-5460 62 12 some some DET cana-5460 62 13 function function NOUN cana-5460 62 14 spaces space NOUN cana-5460 62 15 and and CCONJ cana-5460 62 16 tools tool NOUN cana-5460 62 17 required require VERB cana-5460 62 18 to to PART cana-5460 62 19 investigate investigate VERB cana-5460 62 20 the the DET cana-5460 62 21 following following ADJ cana-5460 62 22 linear linear ADJ cana-5460 62 23 problem problem NOUN cana-5460 62 24 associated associate VERB cana-5460 62 25 with with ADP cana-5460 62 26 problems problem NOUN cana-5460 62 27 ( ( PUNCT cana-5460 62 28 18)-(20 18)-(20 X cana-5460 62 29 ) ) PUNCT cana-5460 62 30 β„’u β„’u NOUN cana-5460 62 31 = = PUNCT cana-5460 62 32 βˆ‚0 βˆ‚0 NOUN cana-5460 62 33 c c PROPN cana-5460 62 34 t t PROPN cana-5460 62 35 Ξ΄u(x Ξ΄u(x PROPN cana-5460 62 36 , , PUNCT cana-5460 62 37 t t PROPN cana-5460 62 38 ) ) PUNCT cana-5460 63 1 βˆ’ βˆ’ PROPN cana-5460 63 2 Ξ± Ξ± PROPN cana-5460 63 3 βˆ‚2u βˆ‚2u PROPN cana-5460 63 4 βˆ‚x2 βˆ‚x2 PROPN cana-5460 64 1 βˆ’ βˆ’ PROPN cana-5460 64 2 Ξ² Ξ² X cana-5460 64 3 βˆ‚3u βˆ‚3u ADJ cana-5460 64 4 βˆ‚tβˆ‚x2 βˆ‚tβˆ‚x2 NOUN cana-5460 64 5 + + NUM cana-5460 64 6 Ξ³u Ξ³u INTJ cana-5460 64 7 βˆ’ βˆ’ PROPN cana-5460 64 8 ∫ ∫ NOUN cana-5460 64 9 a(t a(t NOUN cana-5460 64 10 βˆ’ βˆ’ NOUN cana-5460 64 11 s)u(x s)u(x NOUN cana-5460 64 12 , , PUNCT cana-5460 64 13 s)ds s)ds PROPN cana-5460 64 14 = = SYM cana-5460 64 15 f(x f(x PROPN cana-5460 64 16 , , PUNCT cana-5460 64 17 t t PROPN cana-5460 64 18 ) ) PUNCT cana-5460 64 19 t t NOUN cana-5460 64 20 0 0 NUM cana-5460 64 21 , , PUNCT cana-5460 64 22 ( ( PUNCT cana-5460 64 23 21 21 NUM cana-5460 64 24 ) ) PUNCT cana-5460 64 25 the the DET cana-5460 64 26 initial initial ADJ cana-5460 64 27 conditions condition NOUN cana-5460 64 28 β„“u β„“u X cana-5460 64 29 = = SYM cana-5460 64 30 u(x u(x NOUN cana-5460 64 31 , , PUNCT cana-5460 64 32 0 0 NUM cana-5460 64 33 ) ) PUNCT cana-5460 64 34 = = SYM cana-5460 64 35 ΙΈ(x ΙΈ(x ADJ cana-5460 64 36 ) ) PUNCT cana-5460 64 37 βˆ’β„“u βˆ’β„“u PROPN cana-5460 64 38 = = SYM cana-5460 64 39 Ο†(x Ο†(x PROPN cana-5460 64 40 ) ) PUNCT cana-5460 64 41 , , PUNCT cana-5460 64 42 qu qu PROPN cana-5460 64 43 = = PUNCT cana-5460 64 44 βˆ‚u(x,0 βˆ‚u(x,0 PROPN cana-5460 64 45 ) ) PUNCT cana-5460 64 46 βˆ‚t βˆ‚t PROPN cana-5460 64 47 = = PUNCT cana-5460 64 48 ψ(x ψ(x PROPN cana-5460 64 49 ) ) PUNCT cana-5460 64 50 βˆ’ βˆ’ PROPN cana-5460 64 51 qu qu PROPN cana-5460 64 52 = = PROPN cana-5460 64 53 ψ(x ψ(x PROPN cana-5460 64 54 ) ) PUNCT cana-5460 64 55 , , PUNCT cana-5460 65 1 0 0 PUNCT cana-5460 65 2 < < X cana-5460 65 3 π‘₯ π‘₯ X cana-5460 65 4 < < X cana-5460 65 5 1 1 NUM cana-5460 65 6 . . PUNCT cana-5460 66 1 ( ( PUNCT cana-5460 66 2 22 22 NUM cana-5460 66 3 ) ) PUNCT cana-5460 66 4 the the DET cana-5460 66 5 integrals integral NOUN cana-5460 66 6 conditions condition NOUN cana-5460 66 7 : : PUNCT cana-5460 66 8 ∫ ∫ PROPN cana-5460 66 9 u(x u(x PROPN cana-5460 66 10 , , PUNCT cana-5460 66 11 t)dx t)dx PROPN cana-5460 66 12 = = SYM cana-5460 66 13 0 0 NUM cana-5460 66 14 , , PUNCT cana-5460 66 15 t t PROPN cana-5460 66 16 0 0 NUM cana-5460 66 17 ∫ ∫ PROPN cana-5460 66 18 xu(x xu(x NOUN cana-5460 66 19 , , PUNCT cana-5460 66 20 t)dx t)dx PROPN cana-5460 66 21 = = SYM cana-5460 66 22 0 0 NUM cana-5460 66 23 , , PUNCT cana-5460 66 24 t t PROPN cana-5460 66 25 0 0 NUM cana-5460 66 26 0 0 NUM cana-5460 66 27 < < X cana-5460 66 28 𝑑 𝑑 PROPN cana-5460 66 29 ≀ ≀ NOUN cana-5460 66 30 𝑇 𝑇 PROPN cana-5460 66 31 . . PUNCT cana-5460 67 1 ( ( PUNCT cana-5460 67 2 23 23 NUM cana-5460 67 3 ) ) PUNCT cana-5460 67 4 we we PRON cana-5460 67 5 will will AUX cana-5460 67 6 show show VERB cana-5460 67 7 the the DET cana-5460 67 8 existence existence NOUN cana-5460 67 9 and and CCONJ cana-5460 67 10 uniqueness uniqueness NOUN cana-5460 67 11 of of ADP cana-5460 67 12 the the DET cana-5460 67 13 solution solution NOUN cana-5460 67 14 of of ADP cana-5460 67 15 problem problem NOUN cana-5460 67 16 ( ( PUNCT cana-5460 67 17 21 21 NUM cana-5460 67 18 ) ) PUNCT cana-5460 67 19 – – PUNCT cana-5460 67 20 ( ( PUNCT cana-5460 67 21 23 23 NUM cana-5460 67 22 ) ) PUNCT cana-5460 67 23 , , PUNCT cana-5460 67 24 the the DET cana-5460 67 25 proof proof NOUN cana-5460 67 26 will will AUX cana-5460 67 27 be be AUX cana-5460 67 28 based base VERB cana-5460 67 29 on on ADP cana-5460 67 30 a a DET cana-5460 67 31 priori priori ADJ cana-5460 67 32 estimates estimate NOUN cana-5460 67 33 and and CCONJ cana-5460 67 34 on on ADP cana-5460 67 35 the the DET cana-5460 67 36 density density NOUN cana-5460 67 37 of of ADP cana-5460 67 38 the the DET cana-5460 67 39 set set NOUN cana-5460 67 40 of of ADP cana-5460 67 41 values value NOUN cana-5460 67 42 of of ADP cana-5460 67 43 the the DET cana-5460 67 44 operator operator NOUN cana-5460 67 45 generated generate VERB cana-5460 67 46 by by ADP cana-5460 67 47 problem problem NOUN cana-5460 67 48 ( ( PUNCT cana-5460 67 49 21 21 NUM cana-5460 67 50 ) ) PUNCT cana-5460 67 51 – – PUNCT cana-5460 67 52 ( ( PUNCT cana-5460 67 53 23 23 NUM cana-5460 67 54 ) ) PUNCT cana-5460 67 55 . . PUNCT cana-5460 68 1 for for ADP cana-5460 68 2 this this PRON cana-5460 68 3 , , PUNCT cana-5460 68 4 we we PRON cana-5460 68 5 must must AUX cana-5460 68 6 first first ADV cana-5460 68 7 convert convert VERB cana-5460 68 8 problem problem NOUN cana-5460 68 9 ( ( PUNCT cana-5460 68 10 21 21 NUM cana-5460 68 11 ) ) PUNCT cana-5460 68 12 – – PUNCT cana-5460 68 13 ( ( PUNCT cana-5460 68 14 23 23 NUM cana-5460 68 15 ) ) PUNCT cana-5460 68 16 into into ADP cana-5460 68 17 an an DET cana-5460 68 18 equivalent equivalent ADJ cana-5460 68 19 operational operational ADJ cana-5460 68 20 form form NOUN cana-5460 68 21 : : PUNCT cana-5460 68 22 lu lu NOUN cana-5460 68 23 = = SYM cana-5460 68 24 β„± β„± PROPN cana-5460 68 25 = = SYM cana-5460 68 26 ( ( PUNCT cana-5460 68 27 f f X cana-5460 68 28 , , PUNCT cana-5460 68 29 Ο† Ο† PROPN cana-5460 68 30 , , PUNCT cana-5460 68 31 ψ ψ NOUN cana-5460 68 32 ) ) PUNCT cana-5460 68 33 . . PUNCT cana-5460 69 1 ( ( PUNCT cana-5460 69 2 24 24 NUM cana-5460 69 3 ) ) PUNCT cana-5460 69 4 communications communication NOUN cana-5460 69 5 on on ADP cana-5460 69 6 applied apply VERB cana-5460 69 7 nonlinear nonlinear ADJ cana-5460 69 8 analysis analysis NOUN cana-5460 69 9 issn issn NOUN cana-5460 69 10 : : PUNCT cana-5460 69 11 1074 1074 NUM cana-5460 69 12 - - PUNCT cana-5460 69 13 133x 133x NUM cana-5460 69 14 vol vol NOUN cana-5460 69 15 32 32 NUM cana-5460 69 16 no.3 no.3 NOUN cana-5460 69 17 ( ( PUNCT cana-5460 69 18 2025 2025 NUM cana-5460 69 19 ) ) PUNCT cana-5460 69 20 944 944 NUM cana-5460 69 21 https://internationalpubls.com https://internationalpubls.com X cana-5460 70 1 where where SCONJ cana-5460 70 2 the the DET cana-5460 70 3 operator operator NOUN cana-5460 70 4 l l NOUN cana-5460 70 5 = = SYM cana-5460 70 6 ( ( PUNCT cana-5460 70 7 β„’ β„’ PROPN cana-5460 70 8 , , PUNCT cana-5460 70 9 β„“ β„“ PROPN cana-5460 70 10 , , PUNCT cana-5460 70 11 q q NOUN cana-5460 70 12 ) ) PUNCT cana-5460 70 13 with with ADP cana-5460 70 14 l l NOUN cana-5460 70 15 : : PUNCT cana-5460 70 16 e e X cana-5460 70 17 β†’ β†’ SYM cana-5460 70 18 f f PROPN cana-5460 70 19 is be AUX cana-5460 70 20 defined define VERB cana-5460 70 21 in in ADP cana-5460 70 22 d(l d(l ADJ cana-5460 70 23 ) ) PUNCT cana-5460 70 24 suchaway suchaway NOUN cana-5460 71 1 that that SCONJ cana-5460 71 2 : : PUNCT cana-5460 71 3 d(l d(l ADJ cana-5460 71 4 ) ) PUNCT cana-5460 71 5 = = SYM cana-5460 71 6 { { PUNCT cana-5460 71 7 u u NOUN cana-5460 71 8 ∈ ∈ PROPN cana-5460 71 9 𝕃2(d 𝕃2(d PROPN cana-5460 71 10 ) ) PUNCT cana-5460 71 11 , , PUNCT cana-5460 71 12 βˆ‚0 βˆ‚0 NOUN cana-5460 71 13 c c PROPN cana-5460 71 14 t t PROPN cana-5460 71 15 Ξ΄u Ξ΄u NOUN cana-5460 71 16 , , PUNCT cana-5460 71 17 βˆ‚u βˆ‚u PROPN cana-5460 71 18 βˆ‚t βˆ‚t PROPN cana-5460 71 19 , , PUNCT cana-5460 71 20 βˆ‚u βˆ‚u PROPN cana-5460 71 21 βˆ‚x βˆ‚x PROPN cana-5460 71 22 , , PUNCT cana-5460 71 23 βˆ‚2u βˆ‚2u PROPN cana-5460 71 24 βˆ‚x2 βˆ‚x2 PROPN cana-5460 71 25 , , PUNCT cana-5460 71 26 βˆ‚3u βˆ‚3u ADJ cana-5460 71 27 βˆ‚t βˆ‚t PROPN cana-5460 71 28 βˆ‚x2 βˆ‚x2 PROPN cana-5460 71 29 ∈ ∈ PROPN cana-5460 71 30 𝕃2(d 𝕃2(d PROPN cana-5460 71 31 ) ) PUNCT cana-5460 71 32 ∫ ∫ PROPN cana-5460 72 1 u(x u(x PROPN cana-5460 72 2 , , PUNCT cana-5460 72 3 t)dx t)dx PROPN cana-5460 72 4 = = SYM cana-5460 72 5 0 0 NUM cana-5460 72 6 , , PUNCT cana-5460 72 7 t t PROPN cana-5460 72 8 0 0 NUM cana-5460 72 9 ∫ ∫ PROPN cana-5460 72 10 xu(x xu(x NOUN cana-5460 72 11 , , PUNCT cana-5460 72 12 t)dx t)dx PROPN cana-5460 72 13 = = SYM cana-5460 72 14 0 0 NUM cana-5460 72 15 , , PUNCT cana-5460 72 16 t t PROPN cana-5460 72 17 0 0 NUM cana-5460 72 18 0 0 NUM cana-5460 72 19 < < X cana-5460 72 20 𝑑 𝑑 PROPN cana-5460 72 21 ≀ ≀ NOUN cana-5460 72 22 𝑇 𝑇 PROPN cana-5460 72 23 } } PUNCT cana-5460 72 24 ( ( PUNCT cana-5460 72 25 25 25 NUM cana-5460 72 26 ) ) PUNCT cana-5460 72 27 and and CCONJ cana-5460 72 28 u u NOUN cana-5460 72 29 satisfies satisfy VERB cana-5460 72 30 the the DET cana-5460 72 31 initial initial ADJ cana-5460 72 32 condition condition NOUN cana-5460 72 33 ( ( PUNCT cana-5460 72 34 4.2 4.2 NUM cana-5460 72 35 ) ) PUNCT cana-5460 72 36 . . PUNCT cana-5460 73 1 e e NOUN cana-5460 73 2 is be AUX cana-5460 73 3 the the DET cana-5460 73 4 banach banach NOUN cana-5460 73 5 space space NOUN cana-5460 73 6 equipped equip VERB cana-5460 73 7 with with ADP cana-5460 73 8 the the DET cana-5460 73 9 following follow VERB cana-5460 73 10 norm norm NOUN cana-5460 73 11 : : PUNCT cana-5460 73 12 β€–uβ€–2e β€–uβ€–2e PROPN cana-5460 73 13 = = SYM cana-5460 73 14 sup sup NOUN cana-5460 73 15 ( ( PUNCT cana-5460 73 16 dt dt NOUN cana-5460 73 17 Ξ΄βˆ’1β€–β„‘x Ξ΄βˆ’1β€–β„‘x PROPN cana-5460 73 18 βˆ‚u βˆ‚u PROPN cana-5460 73 19 βˆ‚Ο„ βˆ‚Ο„ PROPN cana-5460 73 20 β€– β€– PROPN cana-5460 73 21 2 2 NUM cana-5460 73 22 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 73 23 ) ) PUNCT cana-5460 74 1 + + CCONJ cana-5460 74 2 ∫ ∫ PROPN cana-5460 74 3 β€– β€– PROPN cana-5460 74 4 βˆ‚u βˆ‚u PROPN cana-5460 74 5 βˆ‚Ο„ βˆ‚Ο„ PROPN cana-5460 74 6 β€– β€– PROPN cana-5460 74 7 2 2 NUM cana-5460 74 8 𝕃2(Ο‰ 𝕃2(Ο‰ NOUN cana-5460 74 9 ) ) PUNCT cana-5460 74 10 dΟ„ dΟ„ NOUN cana-5460 75 1 + + CCONJ cana-5460 75 2 ∫ ∫ PROPN cana-5460 75 3 u2 u2 PROPN cana-5460 75 4 t t PROPN cana-5460 75 5 0 0 NUM cana-5460 75 6 t t PROPN cana-5460 75 7 0 0 NUM cana-5460 75 8 dx dx PROPN cana-5460 75 9 ) ) PUNCT cana-5460 75 10 , , PUNCT cana-5460 75 11 ( ( PUNCT cana-5460 75 12 26 26 NUM cana-5460 75 13 ) ) PUNCT cana-5460 75 14 and and CCONJ cana-5460 75 15 f f PROPN cana-5460 75 16 is be AUX cana-5460 75 17 the the DET cana-5460 75 18 hilbert hilbert PROPN cana-5460 75 19 space space NOUN cana-5460 75 20 composed compose VERB cana-5460 75 21 of of ADP cana-5460 75 22 functions function NOUN cana-5460 75 23 with with ADP cana-5460 75 24 the the DET cana-5460 75 25 norm norm NOUN cana-5460 75 26 : : PUNCT cana-5460 75 27 β€–luβ€–2f β€–luβ€–2f X cana-5460 75 28 = = SYM cana-5460 75 29 β€–Ο†β€– β€–Ο†β€– ADJ cana-5460 75 30 2 2 NUM cana-5460 75 31 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 75 32 ) ) PUNCT cana-5460 76 1 + + CCONJ cana-5460 76 2 β€–β„‘xΟˆβ€– β€–β„‘xΟˆβ€– PROPN cana-5460 76 3 2 2 NUM cana-5460 76 4 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 76 5 ) ) PUNCT cana-5460 77 1 + + CCONJ cana-5460 77 2 β€–β„‘xfβ€– β€–β„‘xfβ€– PROPN cana-5460 77 3 2 2 NUM cana-5460 77 4 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 77 5 ) ) PUNCT cana-5460 77 6 . . PUNCT cana-5460 78 1 ( ( PUNCT cana-5460 78 2 27 27 NUM cana-5460 78 3 ) ) PUNCT cana-5460 78 4 5 5 NUM cana-5460 78 5 . . PUNCT cana-5460 78 6 prior prior ADJ cana-5460 78 7 estimation estimation NOUN cana-5460 78 8 and and CCONJ cana-5460 78 9 uniqueness uniqueness NOUN cana-5460 78 10 of of ADP cana-5460 78 11 the the DET cana-5460 78 12 solution solution NOUN cana-5460 78 13 : : PUNCT cana-5460 78 14 the the DET cana-5460 78 15 priori priori ADJ cana-5460 78 16 estimation estimation NOUN cana-5460 78 17 method method NOUN cana-5460 78 18 , , PUNCT cana-5460 78 19 also also ADV cana-5460 78 20 known know VERB cana-5460 78 21 as as ADP cana-5460 78 22 the the DET cana-5460 78 23 energy energy NOUN cana-5460 78 24 integral integral ADJ cana-5460 78 25 method method NOUN cana-5460 78 26 , , PUNCT cana-5460 78 27 is be AUX cana-5460 78 28 one one NUM cana-5460 78 29 of of ADP cana-5460 78 30 the the DET cana-5460 78 31 most most ADV cana-5460 78 32 effective effective ADJ cana-5460 78 33 functional functional ADJ cana-5460 78 34 analysis analysis NOUN cana-5460 78 35 methods method NOUN cana-5460 78 36 for for ADP cana-5460 78 37 solving solve VERB cana-5460 78 38 partial partial ADJ cana-5460 78 39 differential differential ADJ cana-5460 78 40 equations equation NOUN cana-5460 78 41 with with ADP cana-5460 78 42 integral integral ADJ cana-5460 78 43 conditions condition NOUN cana-5460 78 44 , , PUNCT cana-5460 78 45 and and CCONJ cana-5460 78 46 is be AUX cana-5460 78 47 an an DET cana-5460 78 48 important important ADJ cana-5460 78 49 technique technique NOUN cana-5460 78 50 for for ADP cana-5460 78 51 proving prove VERB cana-5460 78 52 the the DET cana-5460 78 53 existence existence NOUN cana-5460 78 54 , , PUNCT cana-5460 78 55 uniqueness uniqueness NOUN cana-5460 78 56 , , PUNCT cana-5460 78 57 and and CCONJ cana-5460 78 58 continuous continuous ADJ cana-5460 78 59 dependence dependence NOUN cana-5460 78 60 of of ADP cana-5460 78 61 solutions solution NOUN cana-5460 78 62 to to PART cana-5460 78 63 pde pde VERB cana-5460 78 64 . . PUNCT cana-5460 79 1 theorem theorem VERB cana-5460 79 2 5.1 5.1 NUM cana-5460 79 3 for for ADP cana-5460 79 4 any any DET cana-5460 79 5 function function NOUN cana-5460 79 6 u u NOUN cana-5460 79 7 ∈ ∈ PROPN cana-5460 79 8 d(l d(l ADJ cana-5460 79 9 ) ) PUNCT cana-5460 79 10 , , PUNCT cana-5460 79 11 we we PRON cana-5460 79 12 have have VERB cana-5460 79 13 the the DET cana-5460 79 14 a a DET cana-5460 79 15 priori priori ADJ cana-5460 79 16 estimation estimation NOUN cana-5460 79 17 β€–uβ€–e β€–uβ€–e VERB cana-5460 79 18 ≀ ≀ NOUN cana-5460 79 19 c c X cana-5460 79 20 β€–luβ€–f β€–luβ€–f PROPN cana-5460 79 21 , , PUNCT cana-5460 79 22 ( ( PUNCT cana-5460 79 23 28 28 NUM cana-5460 79 24 ) ) PUNCT cana-5460 79 25 where where SCONJ cana-5460 79 26 c c NOUN cana-5460 79 27 is be AUX cana-5460 79 28 a a DET cana-5460 79 29 constant constant ADJ cana-5460 79 30 that that PRON cana-5460 79 31 is be AUX cana-5460 79 32 independent independent ADJ cana-5460 79 33 of of ADP cana-5460 79 34 u. u. PROPN cana-5460 79 35 proof proof NOUN cana-5460 79 36 : : PUNCT cana-5460 79 37 we we PRON cana-5460 79 38 multiply multiply VERB cana-5460 79 39 ( ( PUNCT cana-5460 79 40 21 21 NUM cana-5460 79 41 ) ) PUNCT cana-5460 79 42 by by ADP cana-5460 79 43 mu mu NOUN cana-5460 79 44 = = SYM cana-5460 79 45 βˆ’β„‘x βˆ’β„‘x PROPN cana-5460 79 46 2 2 NUM cana-5460 79 47 βˆ‚u βˆ‚u PROPN cana-5460 79 48 βˆ‚t βˆ‚t PROPN cana-5460 79 49 = = PUNCT cana-5460 79 50 βˆ’βˆ« βˆ’βˆ« NOUN cana-5460 79 51 ∫ ∫ PROPN cana-5460 79 52 βˆ‚u βˆ‚u PROPN cana-5460 79 53 βˆ‚t βˆ‚t PROPN cana-5460 79 54 ( ( PUNCT cana-5460 79 55 ΞΎ ΞΎ PROPN cana-5460 79 56 , , PUNCT cana-5460 79 57 t)dΞΎdΙ³ t)dΞΎdΙ³ PRON cana-5460 79 58 Ι³ Ι³ ADP cana-5460 79 59 0 0 NUM cana-5460 79 60 t t NOUN cana-5460 79 61 0 0 NUM cana-5460 79 62 , , PUNCT cana-5460 79 63 and and CCONJ cana-5460 79 64 integrate integrate VERB cana-5460 79 65 over over ADP cana-5460 79 66 the the DET cana-5460 79 67 subdomain subdomain NOUN cana-5460 79 68 to to PART cana-5460 79 69 obtain obtain VERB cana-5460 79 70 Ο‰ Ο‰ NUM cana-5460 79 71 = = SYM cana-5460 79 72 ( ( PUNCT cana-5460 79 73 0,1 0,1 NUM cana-5460 79 74 ) ) PUNCT cana-5460 79 75 Γ— Γ— NOUN cana-5460 79 76 ( ( PUNCT cana-5460 79 77 0 0 NUM cana-5460 79 78 , , PUNCT cana-5460 79 79 Ο„ Ο„ PROPN cana-5460 79 80 ) ) PUNCT cana-5460 79 81 , , PUNCT cana-5460 79 82 we we PRON cana-5460 79 83 obtain obtain VERB cana-5460 79 84 : : PUNCT cana-5460 79 85 ( ( PUNCT cana-5460 79 86 β„’u β„’u NOUN cana-5460 79 87 , , PUNCT cana-5460 79 88 mu)𝕃2(Ο‰ mu)𝕃2(Ο‰ NUM cana-5460 79 89 ) ) PUNCT cana-5460 80 1 = = SYM cana-5460 80 2 βˆ’ βˆ’ PROPN cana-5460 80 3 ( ( PUNCT cana-5460 80 4 βˆ‚0 βˆ‚0 NOUN cana-5460 80 5 c c PROPN cana-5460 80 6 t t NOUN cana-5460 80 7 Ξ΄u Ξ΄u NOUN cana-5460 80 8 , , PUNCT cana-5460 80 9 β„‘x β„‘x PROPN cana-5460 80 10 2 2 NUM cana-5460 80 11 βˆ‚u βˆ‚u PROPN cana-5460 80 12 βˆ‚t βˆ‚t PROPN cana-5460 80 13 ) ) PUNCT cana-5460 80 14 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 80 15 ) ) PUNCT cana-5460 81 1 + + CCONJ cana-5460 81 2 Ξ± Ξ± X cana-5460 81 3 ( ( PUNCT cana-5460 81 4 βˆ‚2u βˆ‚2u ADJ cana-5460 81 5 βˆ‚x2 βˆ‚x2 NOUN cana-5460 81 6 , , PUNCT cana-5460 81 7 β„‘x β„‘x PROPN cana-5460 81 8 2 2 NUM cana-5460 81 9 βˆ‚u βˆ‚u PROPN cana-5460 81 10 βˆ‚t βˆ‚t PROPN cana-5460 81 11 ) ) PUNCT cana-5460 81 12 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 81 13 ) ) PUNCT cana-5460 82 1 + + X cana-5460 82 2 Ξ² Ξ² X cana-5460 82 3 ( ( PUNCT cana-5460 82 4 βˆ‚3u βˆ‚3u ADJ cana-5460 82 5 βˆ‚t βˆ‚t PROPN cana-5460 82 6 βˆ‚x2 βˆ‚x2 NOUN cana-5460 82 7 , , PUNCT cana-5460 82 8 β„‘x β„‘x PROPN cana-5460 82 9 2 2 NUM cana-5460 82 10 βˆ‚u βˆ‚u PROPN cana-5460 82 11 βˆ‚t βˆ‚t PROPN cana-5460 82 12 ) ) PUNCT cana-5460 82 13 𝕃2(Ο‰ 𝕃2(Ο‰ PROPN cana-5460 82 14 ) ) PUNCT cana-5460 82 15 βˆ’ βˆ’ PROPN cana-5460 82 16 Ξ³(u Ξ³(u PROPN cana-5460 82 17 , , PUNCT cana-5460 82 18 β„‘x β„‘x PROPN cana-5460 82 19 2 2 NUM cana-5460 82 20 βˆ‚u βˆ‚u PROPN cana-5460 82 21 βˆ‚t βˆ‚t PROPN cana-5460 82 22 ) ) PUNCT cana-5460 82 23 𝕃2(Ο‰ 𝕃2(Ο‰ PROPN cana-5460 82 24 ) ) PUNCT cana-5460 82 25 = = SYM cana-5460 83 1 βˆ’ βˆ’ PROPN cana-5460 83 2 ( ( PUNCT cana-5460 83 3 ∫ ∫ PROPN cana-5460 83 4 a(t a(t NOUN cana-5460 83 5 βˆ’ βˆ’ NOUN cana-5460 83 6 s)u(x s)u(x NOUN cana-5460 83 7 , , PUNCT cana-5460 83 8 s)ds s)ds PROPN cana-5460 83 9 t t PROPN cana-5460 83 10 0 0 NUM cana-5460 83 11 , , PUNCT cana-5460 83 12 β„‘x β„‘x PROPN cana-5460 83 13 2 2 NUM cana-5460 83 14 βˆ‚u βˆ‚u PROPN cana-5460 83 15 βˆ‚t βˆ‚t PROPN cana-5460 83 16 ) ) PUNCT cana-5460 83 17 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 83 18 ) ) PUNCT cana-5460 83 19 βˆ’ βˆ’ PROPN cana-5460 83 20 ( ( PUNCT cana-5460 83 21 f(x f(x PROPN cana-5460 83 22 , , PUNCT cana-5460 83 23 t t PROPN cana-5460 83 24 ) ) PUNCT cana-5460 83 25 , , PUNCT cana-5460 83 26 β„‘x β„‘x PROPN cana-5460 83 27 2 2 NUM cana-5460 83 28 βˆ‚u βˆ‚u PROPN cana-5460 83 29 βˆ‚t βˆ‚t PROPN cana-5460 83 30 ) ) PUNCT cana-5460 83 31 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 83 32 ) ) PUNCT cana-5460 83 33 ( ( PUNCT cana-5460 83 34 29 29 NUM cana-5460 83 35 ) ) PUNCT cana-5460 83 36 by by ADP cana-5460 83 37 integrating integrate VERB cana-5460 83 38 by by ADP cana-5460 83 39 parts part NOUN cana-5460 83 40 for for ADP cana-5460 83 41 each each DET cana-5460 83 42 term term NOUN cana-5460 83 43 on on ADP cana-5460 83 44 the the DET cana-5460 83 45 left left ADJ cana-5460 83 46 - - PUNCT cana-5460 83 47 hand hand NOUN cana-5460 83 48 side side NOUN cana-5460 83 49 of of ADP cana-5460 83 50 ( ( PUNCT cana-5460 83 51 22 22 NUM cana-5460 83 52 ) ) PUNCT cana-5460 83 53 , , PUNCT cana-5460 83 54 and and CCONJ cana-5460 83 55 using use VERB cana-5460 83 56 the the DET cana-5460 83 57 conditions condition NOUN cana-5460 83 58 ( ( PUNCT cana-5460 83 59 20 20 NUM cana-5460 83 60 ) ) PUNCT cana-5460 83 61 , , PUNCT cana-5460 83 62 we we PRON cana-5460 83 63 obtain obtain VERB cana-5460 83 64 communications communication NOUN cana-5460 83 65 on on ADP cana-5460 83 66 applied apply VERB cana-5460 83 67 nonlinear nonlinear ADJ cana-5460 83 68 analysis analysis NOUN cana-5460 83 69 issn issn NOUN cana-5460 83 70 : : PUNCT cana-5460 83 71 1074 1074 NUM cana-5460 83 72 - - PUNCT cana-5460 83 73 133x 133x NUM cana-5460 83 74 vol vol NOUN cana-5460 83 75 32 32 NUM cana-5460 83 76 no.3 no.3 NOUN cana-5460 83 77 ( ( PUNCT cana-5460 83 78 2025 2025 NUM cana-5460 83 79 ) ) PUNCT cana-5460 83 80 945 945 NUM cana-5460 83 81 https://internationalpubls.com https://internationalpubls.com X cana-5460 83 82 βˆ’ βˆ’ PROPN cana-5460 83 83 ( ( PUNCT cana-5460 83 84 βˆ‚0 βˆ‚0 NOUN cana-5460 83 85 c c PROPN cana-5460 83 86 t t NOUN cana-5460 83 87 Ξ΄u Ξ΄u NOUN cana-5460 83 88 , , PUNCT cana-5460 83 89 β„‘x β„‘x PROPN cana-5460 83 90 2 2 NUM cana-5460 83 91 βˆ‚u βˆ‚u PROPN cana-5460 83 92 βˆ‚t βˆ‚t PROPN cana-5460 83 93 ) ) PUNCT cana-5460 83 94 𝕃2(Ο‰ 𝕃2(Ο‰ PROPN cana-5460 83 95 ) ) PUNCT cana-5460 83 96 = = PUNCT cana-5460 84 1 βˆ’βˆ« βˆ’βˆ« NOUN cana-5460 84 2 ∫ ∫ PROPN cana-5460 84 3 ( ( PUNCT cana-5460 84 4 βˆ‚0 βˆ‚0 NOUN cana-5460 84 5 c c PROPN cana-5460 84 6 t t NOUN cana-5460 84 7 Ξ΄u Ξ΄u ADP cana-5460 84 8 β„‘x β„‘x PROPN cana-5460 84 9 2 2 NUM cana-5460 84 10 βˆ‚u βˆ‚u PROPN cana-5460 84 11 βˆ‚t βˆ‚t PROPN cana-5460 84 12 ) ) PUNCT cana-5460 84 13 1 1 NUM cana-5460 84 14 0 0 NUM cana-5460 84 15 Ο„ Ο„ X cana-5460 84 16 0 0 NUM cana-5460 84 17 dxdt dxdt NOUN cana-5460 84 18 = = SYM cana-5460 84 19 ∫ ∫ PROPN cana-5460 84 20 ∫ ∫ PROPN cana-5460 84 21 ( ( PUNCT cana-5460 84 22 βˆ‚0 βˆ‚0 NOUN cana-5460 84 23 c c PROPN cana-5460 84 24 t t PROPN cana-5460 84 25 Ξ΄β„‘x Ξ΄β„‘x NOUN cana-5460 84 26 βˆ‚u βˆ‚u PROPN cana-5460 84 27 βˆ‚t βˆ‚t PROPN cana-5460 84 28 ) ) PUNCT cana-5460 85 1 ( ( PUNCT cana-5460 85 2 β„‘x β„‘x PROPN cana-5460 85 3 βˆ‚u βˆ‚u PROPN cana-5460 85 4 βˆ‚t βˆ‚t PROPN cana-5460 85 5 ) ) PUNCT cana-5460 85 6 1 1 NUM cana-5460 85 7 0 0 NUM cana-5460 85 8 Ο„ Ο„ PROPN cana-5460 85 9 0 0 NUM cana-5460 85 10 dxdt dxdt NOUN cana-5460 85 11 . . PUNCT cana-5460 86 1 ( ( PUNCT cana-5460 86 2 30 30 X cana-5460 86 3 ) ) PUNCT cana-5460 86 4 Ξ± Ξ± PROPN cana-5460 86 5 ( ( PUNCT cana-5460 86 6 βˆ‚2u βˆ‚2u PROPN cana-5460 86 7 βˆ‚x2 βˆ‚x2 NOUN cana-5460 86 8 , , PUNCT cana-5460 86 9 β„‘x β„‘x PROPN cana-5460 86 10 2 2 NUM cana-5460 86 11 βˆ‚u βˆ‚u PROPN cana-5460 86 12 βˆ‚t βˆ‚t PROPN cana-5460 86 13 ) ) PUNCT cana-5460 86 14 𝕃2(Ο‰ 𝕃2(Ο‰ PROPN cana-5460 86 15 ) ) PUNCT cana-5460 86 16 = = SYM cana-5460 87 1 α∫ α∫ NUM cana-5460 87 2 ∫ ∫ PROPN cana-5460 87 3 ( ( PUNCT cana-5460 87 4 βˆ‚2u βˆ‚2u ADJ cana-5460 87 5 βˆ‚x2 βˆ‚x2 NOUN cana-5460 87 6 β„‘x β„‘x PROPN cana-5460 87 7 2 2 NUM cana-5460 87 8 βˆ‚u βˆ‚u PROPN cana-5460 87 9 βˆ‚t βˆ‚t PROPN cana-5460 87 10 ) ) PUNCT cana-5460 88 1 1 1 NUM cana-5460 88 2 0 0 NUM cana-5460 88 3 Ο„ Ο„ X cana-5460 88 4 0 0 NUM cana-5460 88 5 dxdt dxdt NOUN cana-5460 88 6 = = SYM cana-5460 88 7 α∫ α∫ NUM cana-5460 88 8 ∫ ∫ PROPN cana-5460 88 9 u u NOUN cana-5460 88 10 1 1 NUM cana-5460 88 11 0 0 NUM cana-5460 88 12 Ο„ Ο„ X cana-5460 88 13 0 0 NUM cana-5460 88 14 βˆ‚u βˆ‚u PROPN cana-5460 88 15 βˆ‚t βˆ‚t PROPN cana-5460 88 16 dxdt dxdt NOUN cana-5460 88 17 . . PUNCT cana-5460 89 1 ( ( PUNCT cana-5460 89 2 31 31 NUM cana-5460 89 3 ) ) PUNCT cana-5460 89 4 Ξ² Ξ² NOUN cana-5460 89 5 ( ( PUNCT cana-5460 89 6 βˆ‚3u βˆ‚3u ADJ cana-5460 89 7 βˆ‚t βˆ‚t PROPN cana-5460 89 8 βˆ‚x2 βˆ‚x2 NOUN cana-5460 89 9 , , PUNCT cana-5460 89 10 β„‘x β„‘x PROPN cana-5460 89 11 2 2 NUM cana-5460 89 12 βˆ‚u βˆ‚u PROPN cana-5460 89 13 βˆ‚t βˆ‚t PROPN cana-5460 89 14 ) ) PUNCT cana-5460 89 15 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 89 16 ) ) PUNCT cana-5460 90 1 = = SYM cana-5460 91 1 β∫ β∫ PROPN cana-5460 91 2 ∫ ∫ NOUN cana-5460 91 3 ( ( PUNCT cana-5460 91 4 βˆ‚3u βˆ‚3u ADJ cana-5460 91 5 βˆ‚t βˆ‚t PROPN cana-5460 91 6 βˆ‚x2 βˆ‚x2 NOUN cana-5460 91 7 β„‘x β„‘x PROPN cana-5460 91 8 2 2 NUM cana-5460 91 9 βˆ‚u βˆ‚u PROPN cana-5460 91 10 βˆ‚t βˆ‚t PROPN cana-5460 91 11 ) ) PUNCT cana-5460 91 12 1 1 NUM cana-5460 91 13 0 0 NUM cana-5460 91 14 Ο„ Ο„ X cana-5460 91 15 0 0 NUM cana-5460 91 16 dxdt dxdt NOUN cana-5460 91 17 = = SYM cana-5460 91 18 β∫ β∫ PROPN cana-5460 91 19 ∫ ∫ PROPN cana-5460 91 20 ( ( PUNCT cana-5460 91 21 βˆ‚u βˆ‚u PROPN cana-5460 91 22 βˆ‚t βˆ‚t PROPN cana-5460 91 23 ) ) PUNCT cana-5460 91 24 21 21 NUM cana-5460 91 25 0 0 NUM cana-5460 91 26 Ο„ Ο„ PROPN cana-5460 91 27 0 0 NUM cana-5460 91 28 dxdt dxdt NOUN cana-5460 91 29 = = SYM cana-5460 91 30 β∫ β∫ PROPN cana-5460 91 31 β€– β€– PROPN cana-5460 91 32 βˆ‚u βˆ‚u PROPN cana-5460 91 33 βˆ‚t βˆ‚t PROPN cana-5460 91 34 β€– β€– PROPN cana-5460 91 35 2 2 NUM cana-5460 91 36 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 91 37 ) ) PUNCT cana-5460 91 38 dt dt PUNCT cana-5460 91 39 Ο„ Ο„ PROPN cana-5460 91 40 0 0 NUM cana-5460 91 41 . . PUNCT cana-5460 92 1 ( ( PUNCT cana-5460 92 2 32 32 NUM cana-5460 92 3 ) ) PUNCT cana-5460 92 4 βˆ’Ξ³(u βˆ’Ξ³(u NOUN cana-5460 92 5 , , PUNCT cana-5460 92 6 β„‘x β„‘x PROPN cana-5460 92 7 2 2 NUM cana-5460 92 8 βˆ‚u βˆ‚u PROPN cana-5460 92 9 βˆ‚t βˆ‚t PROPN cana-5460 92 10 ) ) PUNCT cana-5460 92 11 𝕃2(Ο‰ 𝕃2(Ο‰ PROPN cana-5460 92 12 ) ) PUNCT cana-5460 92 13 = = SYM cana-5460 93 1 βˆ’Ξ³βˆ« βˆ’Ξ³βˆ« PROPN cana-5460 93 2 ∫ ∫ NOUN cana-5460 93 3 ( ( PUNCT cana-5460 93 4 u u NOUN cana-5460 93 5 β„‘x β„‘x PROPN cana-5460 93 6 2 2 NUM cana-5460 93 7 βˆ‚u βˆ‚u PROPN cana-5460 93 8 βˆ‚t βˆ‚t PROPN cana-5460 93 9 ) ) PUNCT cana-5460 93 10 1 1 NUM cana-5460 93 11 0 0 NUM cana-5460 93 12 Ο„ Ο„ X cana-5460 93 13 0 0 NUM cana-5460 93 14 dxdt dxdt NOUN cana-5460 93 15 = = NOUN cana-5460 93 16 βˆ’ βˆ’ NOUN cana-5460 93 17 Ξ³ Ξ³ X cana-5460 93 18 2 2 NUM cana-5460 93 19 ∫ ∫ NOUN cana-5460 93 20 ( ( PUNCT cana-5460 93 21 β„‘x β„‘x PROPN cana-5460 93 22 2u(x 2u(x NOUN cana-5460 93 23 , , PUNCT cana-5460 93 24 Ο„ Ο„ PROPN cana-5460 93 25 ) ) PUNCT cana-5460 93 26 ) ) PUNCT cana-5460 93 27 21 21 NUM cana-5460 93 28 0 0 NUM cana-5460 93 29 dx dx PROPN cana-5460 93 30 + + CCONJ cana-5460 93 31 Ξ³ Ξ³ PROPN cana-5460 93 32 2 2 NUM cana-5460 93 33 ∫ ∫ NOUN cana-5460 93 34 ( ( PUNCT cana-5460 93 35 Ο†(x))2 Ο†(x))2 PROPN cana-5460 93 36 1 1 NUM cana-5460 93 37 0 0 NUM cana-5460 93 38 dx dx PROPN cana-5460 93 39 . . PUNCT cana-5460 94 1 ( ( PUNCT cana-5460 94 2 33 33 NUM cana-5460 94 3 ) ) PUNCT cana-5460 94 4 applying apply VERB cana-5460 94 5 the the DET cana-5460 94 6 cauchy cauchy ADJ cana-5460 94 7 inequalities inequality NOUN cana-5460 94 8 ( ( PUNCT cana-5460 94 9 7 7 NUM cana-5460 94 10 ) ) PUNCT cana-5460 94 11 and and CCONJ cana-5460 94 12 ( ( PUNCT cana-5460 94 13 8) 8) NUM cana-5460 94 14 , , PUNCT cana-5460 94 15 and and CCONJ cana-5460 94 16 integrating integrate VERB cana-5460 94 17 by by ADP cana-5460 94 18 parts part NOUN cana-5460 94 19 for for ADP cana-5460 94 20 the the DET cana-5460 94 21 two two NUM cana-5460 94 22 terms term NOUN cana-5460 94 23 on on ADP cana-5460 94 24 the the DET cana-5460 94 25 right right ADJ cana-5460 94 26 - - PUNCT cana-5460 94 27 hand hand NOUN cana-5460 94 28 side side NOUN cana-5460 94 29 of of ADP cana-5460 94 30 ( ( PUNCT cana-5460 94 31 29 29 NUM cana-5460 94 32 ) ) PUNCT cana-5460 94 33 , , PUNCT cana-5460 94 34 we we PRON cana-5460 94 35 obtain obtain VERB cana-5460 94 36 : : PUNCT cana-5460 94 37 βˆ’(f βˆ’(f PROPN cana-5460 94 38 , , PUNCT cana-5460 94 39 β„‘x β„‘x PROPN cana-5460 94 40 2 2 NUM cana-5460 94 41 βˆ‚u βˆ‚u PROPN cana-5460 94 42 βˆ‚t βˆ‚t PROPN cana-5460 94 43 ) ) PUNCT cana-5460 94 44 𝕃2(Ο‰ 𝕃2(Ο‰ NOUN cana-5460 94 45 ) ) PUNCT cana-5460 94 46 ≀ ≀ NUM cana-5460 94 47 Ξ΅ Ξ΅ PROPN cana-5460 94 48 2 2 NUM cana-5460 94 49 ∫ ∫ PROPN cana-5460 94 50 ∫ ∫ PROPN cana-5460 94 51 ( ( PUNCT cana-5460 94 52 β„‘xf β„‘xf PROPN cana-5460 94 53 ) ) PUNCT cana-5460 94 54 21 21 NUM cana-5460 94 55 0 0 NUM cana-5460 94 56 Ο„ Ο„ PROPN cana-5460 94 57 0 0 NUM cana-5460 94 58 dxdt dxdt NOUN cana-5460 94 59 + + CCONJ cana-5460 94 60 1 1 NUM cana-5460 94 61 2Ξ΅ 2Ξ΅ NUM cana-5460 94 62 ∫ ∫ PROPN cana-5460 94 63 ∫ ∫ PROPN cana-5460 95 1 ( ( PUNCT cana-5460 95 2 β„‘x β„‘x PROPN cana-5460 95 3 βˆ‚u βˆ‚u PROPN cana-5460 95 4 βˆ‚t βˆ‚t PROPN cana-5460 95 5 ) ) PUNCT cana-5460 95 6 21 21 NUM cana-5460 95 7 0 0 NUM cana-5460 95 8 Ο„ Ο„ PROPN cana-5460 95 9 0 0 NUM cana-5460 95 10 dxdt dxdt NOUN cana-5460 95 11 ( ( PUNCT cana-5460 95 12 34 34 NUM cana-5460 95 13 ) ) PUNCT cana-5460 95 14 ( ( PUNCT cana-5460 95 15 ∫ ∫ PROPN cana-5460 95 16 a(t a(t NOUN cana-5460 95 17 βˆ’ βˆ’ NOUN cana-5460 95 18 s)u(x s)u(x NOUN cana-5460 95 19 , , PUNCT cana-5460 95 20 s)ds s)ds PROPN cana-5460 95 21 t t PROPN cana-5460 95 22 0 0 NUM cana-5460 95 23 , , PUNCT cana-5460 95 24 β„‘x β„‘x PROPN cana-5460 95 25 2 2 NUM cana-5460 95 26 βˆ‚u βˆ‚u PROPN cana-5460 95 27 βˆ‚t βˆ‚t PROPN cana-5460 95 28 ) ) PUNCT cana-5460 95 29 𝕃2(Ο‰ 𝕃2(Ο‰ PROPN cana-5460 95 30 ) ) PUNCT cana-5460 95 31 = = SYM cana-5460 96 1 ∫ ∫ PROPN cana-5460 96 2 ∫ ∫ PROPN cana-5460 96 3 ( ( PUNCT cana-5460 96 4 ∫ ∫ PROPN cana-5460 96 5 a(t a(t NOUN cana-5460 96 6 βˆ’ βˆ’ NOUN cana-5460 96 7 s)u(x s)u(x NOUN cana-5460 96 8 , , PUNCT cana-5460 96 9 s)ds s)ds PROPN cana-5460 96 10 t t PROPN cana-5460 96 11 0 0 NUM cana-5460 96 12 ) ) PUNCT cana-5460 96 13 1 1 NUM cana-5460 96 14 0 0 NUM cana-5460 96 15 Ο„ Ο„ X cana-5460 96 16 0 0 X cana-5460 97 1 β„‘x β„‘x PROPN cana-5460 97 2 2 2 NUM cana-5460 97 3 βˆ‚u βˆ‚u PROPN cana-5460 97 4 βˆ‚t βˆ‚t PROPN cana-5460 97 5 dxdt dxdt NOUN cana-5460 97 6 ≀ ≀ NUM cana-5460 97 7 a1 a1 NOUN cana-5460 97 8 t t NOUN cana-5460 97 9 2β€–uβ€–2𝕃2(0,1 2β€–uβ€–2𝕃2(0,1 NUM cana-5460 97 10 ) ) PUNCT cana-5460 98 1 + + NUM cana-5460 98 2 ∫ ∫ PROPN cana-5460 98 3 β€–β„‘x β€–β„‘x NOUN cana-5460 99 1 βˆ‚u βˆ‚u PROPN cana-5460 100 1 βˆ‚t βˆ‚t PROPN cana-5460 100 2 β€– β€– PROPN cana-5460 100 3 2 2 NUM cana-5460 100 4 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 100 5 ) ) PUNCT cana-5460 100 6 dt dt PROPN cana-5460 100 7 , , PUNCT cana-5460 100 8 Ο„ Ο„ PROPN cana-5460 100 9 0 0 NUM cana-5460 100 10 ( ( PUNCT cana-5460 100 11 35 35 NUM cana-5460 100 12 ) ) PUNCT cana-5460 100 13 by by ADP cana-5460 100 14 substituting substitute VERB cana-5460 100 15 ( ( PUNCT cana-5460 100 16 30)-(35 30)-(35 NUM cana-5460 100 17 ) ) PUNCT cana-5460 100 18 into into ADP cana-5460 100 19 ( ( PUNCT cana-5460 100 20 29 29 NUM cana-5460 100 21 ) ) PUNCT cana-5460 100 22 , , PUNCT cana-5460 100 23 and and CCONJ cana-5460 100 24 applying apply VERB cana-5460 100 25 lemma lemma PROPN cana-5460 100 26 ( ( PUNCT cana-5460 100 27 2.1 2.1 NUM cana-5460 100 28 ) ) PUNCT cana-5460 100 29 , , PUNCT cana-5460 100 30 we we PRON cana-5460 100 31 obtain obtain VERB cana-5460 100 32 : : PUNCT cana-5460 100 33 communications communication NOUN cana-5460 100 34 on on ADP cana-5460 100 35 applied apply VERB cana-5460 100 36 nonlinear nonlinear ADJ cana-5460 100 37 analysis analysis NOUN cana-5460 100 38 issn issn NOUN cana-5460 100 39 : : PUNCT cana-5460 100 40 1074 1074 NUM cana-5460 100 41 - - PUNCT cana-5460 100 42 133x 133x NUM cana-5460 100 43 vol vol NOUN cana-5460 100 44 32 32 NUM cana-5460 100 45 no.3 no.3 NOUN cana-5460 100 46 ( ( PUNCT cana-5460 100 47 2025 2025 NUM cana-5460 100 48 ) ) PUNCT cana-5460 100 49 946 946 NUM cana-5460 100 50 https://internationalpubls.com https://internationalpubls.com X cana-5460 100 51 1 1 NUM cana-5460 100 52 2 2 NUM cana-5460 100 53 ∫ ∫ NOUN cana-5460 100 54 ∫ ∫ PROPN cana-5460 100 55 ( ( PUNCT cana-5460 100 56 βˆ‚0 βˆ‚0 NOUN cana-5460 100 57 c c PROPN cana-5460 100 58 t t PROPN cana-5460 100 59 Ξ΄β„‘x Ξ΄β„‘x NOUN cana-5460 100 60 βˆ‚u βˆ‚u PROPN cana-5460 100 61 βˆ‚t βˆ‚t PROPN cana-5460 100 62 ) ) PUNCT cana-5460 101 1 21 21 NUM cana-5460 101 2 0 0 NUM cana-5460 101 3 Ο„ Ο„ PROPN cana-5460 101 4 0 0 NUM cana-5460 101 5 dxdt dxdt NOUN cana-5460 101 6 + + CCONJ cana-5460 102 1 α∫ α∫ NUM cana-5460 102 2 ∫ ∫ NOUN cana-5460 102 3 u u NOUN cana-5460 102 4 1 1 NUM cana-5460 102 5 0 0 NUM cana-5460 102 6 Ο„ Ο„ X cana-5460 102 7 0 0 NUM cana-5460 102 8 βˆ‚u βˆ‚u PROPN cana-5460 102 9 βˆ‚t βˆ‚t PROPN cana-5460 102 10 dxdt dxdt NOUN cana-5460 102 11 + + CCONJ cana-5460 102 12 β∫ β∫ PROPN cana-5460 102 13 β€– β€– PROPN cana-5460 102 14 βˆ‚u βˆ‚u PROPN cana-5460 102 15 βˆ‚t βˆ‚t PROPN cana-5460 102 16 β€– β€– PROPN cana-5460 102 17 2 2 NUM cana-5460 102 18 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 102 19 ) ) PUNCT cana-5460 102 20 dt dt PUNCT cana-5460 103 1 Ο„ Ο„ PROPN cana-5460 103 2 0 0 NUM cana-5460 103 3 βˆ’ βˆ’ PROPN cana-5460 103 4 Ξ³ Ξ³ PROPN cana-5460 103 5 2 2 NUM cana-5460 103 6 ∫ ∫ NOUN cana-5460 103 7 ( ( PUNCT cana-5460 103 8 β„‘x β„‘x PROPN cana-5460 103 9 2u(x 2u(x NOUN cana-5460 103 10 , , PUNCT cana-5460 103 11 Ο„ Ο„ PROPN cana-5460 103 12 ) ) PUNCT cana-5460 103 13 ) ) PUNCT cana-5460 103 14 2 2 NUM cana-5460 103 15 1 1 NUM cana-5460 103 16 0 0 NUM cana-5460 103 17 dx dx PROPN cana-5460 103 18 + + CCONJ cana-5460 103 19 Ξ³ Ξ³ PROPN cana-5460 103 20 2 2 NUM cana-5460 103 21 ∫ ∫ NOUN cana-5460 103 22 ( ( PUNCT cana-5460 103 23 Ο†(x))2 Ο†(x))2 PROPN cana-5460 103 24 1 1 NUM cana-5460 103 25 0 0 NUM cana-5460 103 26 dx dx PROPN cana-5460 103 27 ≀ ≀ NUM cana-5460 103 28 1 1 NUM cana-5460 103 29 2Ξ΅ 2Ξ΅ NUM cana-5460 103 30 ∫ ∫ PROPN cana-5460 103 31 ∫ ∫ PROPN cana-5460 103 32 ( ( PUNCT cana-5460 103 33 β„‘xf β„‘xf PROPN cana-5460 103 34 ) ) PUNCT cana-5460 103 35 2 2 NUM cana-5460 103 36 1 1 NUM cana-5460 103 37 0 0 NUM cana-5460 103 38 Ο„ Ο„ PROPN cana-5460 103 39 0 0 NUM cana-5460 103 40 dxdt dxdt NOUN cana-5460 103 41 + + CCONJ cana-5460 103 42 Ξ΅ Ξ΅ PROPN cana-5460 103 43 2 2 NUM cana-5460 103 44 ∫ ∫ NOUN cana-5460 103 45 ∫ ∫ PROPN cana-5460 104 1 ( ( PUNCT cana-5460 104 2 β„‘x β„‘x PROPN cana-5460 104 3 βˆ‚u βˆ‚u PROPN cana-5460 104 4 βˆ‚t βˆ‚t PROPN cana-5460 104 5 ) ) PUNCT cana-5460 104 6 21 21 NUM cana-5460 104 7 0 0 NUM cana-5460 104 8 Ο„ Ο„ PROPN cana-5460 104 9 0 0 NUM cana-5460 104 10 dxdt dxdt NOUN cana-5460 104 11 + + NOUN cana-5460 104 12 t2β€–uβ€–2𝕃2(0,1 t2β€–uβ€–2𝕃2(0,1 NOUN cana-5460 104 13 ) ) PUNCT cana-5460 105 1 + + CCONJ cana-5460 105 2 ∫ ∫ PROPN cana-5460 105 3 β€–β„‘x β€–β„‘x NOUN cana-5460 106 1 βˆ‚u βˆ‚u PROPN cana-5460 107 1 βˆ‚t βˆ‚t PROPN cana-5460 107 2 β€– β€– PROPN cana-5460 107 3 2 2 NUM cana-5460 107 4 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 107 5 ) ) PUNCT cana-5460 107 6 dt dt PROPN cana-5460 107 7 , , PUNCT cana-5460 107 8 Ο„ Ο„ PROPN cana-5460 107 9 0 0 NUM cana-5460 107 10 ( ( PUNCT cana-5460 107 11 36 36 NUM cana-5460 107 12 ) ) PUNCT cana-5460 107 13 evaluating evaluate VERB cana-5460 107 14 the the DET cana-5460 107 15 first first ADJ cana-5460 107 16 and and CCONJ cana-5460 107 17 third third ADJ cana-5460 107 18 terms term NOUN cana-5460 107 19 on on ADP cana-5460 107 20 the the DET cana-5460 107 21 left left ADJ cana-5460 107 22 - - PUNCT cana-5460 107 23 hand hand NOUN cana-5460 107 24 side side NOUN cana-5460 107 25 , , PUNCT cana-5460 107 26 we we PRON cana-5460 107 27 have have AUX cana-5460 107 28 : : PUNCT cana-5460 107 29 α∫ α∫ NUM cana-5460 107 30 ∫ ∫ PROPN cana-5460 107 31 u u NOUN cana-5460 107 32 1 1 NUM cana-5460 107 33 0 0 NUM cana-5460 107 34 Ο„ Ο„ X cana-5460 107 35 0 0 NUM cana-5460 107 36 βˆ‚u βˆ‚u PROPN cana-5460 107 37 βˆ‚t βˆ‚t PROPN cana-5460 107 38 dxdt dxdt NOUN cana-5460 107 39 = = SYM cana-5460 107 40 α∫ α∫ NUM cana-5460 107 41 ∫ ∫ PROPN cana-5460 107 42 u u NOUN cana-5460 107 43 Ο„ Ο„ PROPN cana-5460 107 44 0 0 NUM cana-5460 107 45 1 1 NUM cana-5460 107 46 0 0 NUM cana-5460 107 47 βˆ‚u βˆ‚u PROPN cana-5460 108 1 βˆ‚t βˆ‚t PROPN cana-5460 108 2 dtdx dtdx NOUN cana-5460 109 1 = = NOUN cana-5460 109 2 Ξ± Ξ± PROPN cana-5460 109 3 2 2 NUM cana-5460 109 4 ∫ ∫ NOUN cana-5460 109 5 u2(x u2(x PROPN cana-5460 109 6 , , PUNCT cana-5460 109 7 Ο„)dx Ο„)dx PROPN cana-5460 109 8 βˆ’ βˆ’ PROPN cana-5460 109 9 1 1 NUM cana-5460 109 10 0 0 NUM cana-5460 109 11 Ξ± Ξ± PRON cana-5460 109 12 2 2 NUM cana-5460 109 13 ∫ ∫ NOUN cana-5460 109 14 Ο†2(x Ο†2(x NUM cana-5460 109 15 ) ) PUNCT cana-5460 109 16 1 1 NUM cana-5460 109 17 0 0 NUM cana-5460 109 18 dx dx PROPN cana-5460 110 1 βˆ’ βˆ’ PROPN cana-5460 110 2 Ξ± Ξ± NOUN cana-5460 110 3 2 2 NUM cana-5460 110 4 ∫ ∫ NOUN cana-5460 110 5 ∫ ∫ PROPN cana-5460 110 6 u2 u2 PROPN cana-5460 110 7 Ο„ Ο„ PROPN cana-5460 110 8 o o NOUN cana-5460 110 9 1 1 NUM cana-5460 110 10 0 0 NUM cana-5460 110 11 ( ( PUNCT cana-5460 110 12 x x NOUN cana-5460 110 13 , , PUNCT cana-5460 110 14 t)dtdx t)dtdx PROPN cana-5460 110 15 ( ( PUNCT cana-5460 110 16 37 37 NUM cana-5460 110 17 ) ) SYM cana-5460 110 18 1 1 NUM cana-5460 110 19 2 2 NUM cana-5460 110 20 ∫ ∫ NOUN cana-5460 110 21 ∫ ∫ PROPN cana-5460 110 22 ( ( PUNCT cana-5460 110 23 βˆ‚0 βˆ‚0 NOUN cana-5460 110 24 c c PROPN cana-5460 110 25 t t PROPN cana-5460 110 26 Ξ΄β„‘x Ξ΄β„‘x NOUN cana-5460 110 27 βˆ‚u βˆ‚u PROPN cana-5460 110 28 βˆ‚t βˆ‚t PROPN cana-5460 110 29 ) ) PUNCT cana-5460 110 30 21 21 NUM cana-5460 110 31 0 0 NUM cana-5460 110 32 Ο„ Ο„ PROPN cana-5460 110 33 0 0 NUM cana-5460 110 34 dxdt dxdt NOUN cana-5460 110 35 = = NOUN cana-5460 110 36 dt dt NOUN cana-5460 111 1 Ξ΄βˆ’1 Ξ΄βˆ’1 PROPN cana-5460 111 2 β€–β„‘x β€–β„‘x NOUN cana-5460 112 1 βˆ‚u βˆ‚u PROPN cana-5460 112 2 βˆ‚t βˆ‚t PROPN cana-5460 112 3 β€– β€– PROPN cana-5460 112 4 2 2 NUM cana-5460 112 5 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 112 6 ) ) PUNCT cana-5460 112 7 βˆ’ βˆ’ PROPN cana-5460 112 8 t1βˆ’Ξ΄ t1βˆ’Ξ΄ PROPN cana-5460 112 9 Ξ³(1βˆ’Ξ΄ Ξ³(1βˆ’Ξ΄ NOUN cana-5460 112 10 ) ) PUNCT cana-5460 112 11 β€–β„‘xΟˆβ€– β€–β„‘xΟˆβ€– PROPN cana-5460 112 12 2 2 NUM cana-5460 112 13 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 112 14 ) ) PUNCT cana-5460 112 15 . . PUNCT cana-5460 113 1 ( ( PUNCT cana-5460 113 2 38 38 NUM cana-5460 113 3 ) ) PUNCT cana-5460 113 4 substituting substituting NOUN cana-5460 113 5 ( ( PUNCT cana-5460 113 6 37 37 NUM cana-5460 113 7 ) ) PUNCT cana-5460 113 8 and and CCONJ cana-5460 113 9 ( ( PUNCT cana-5460 113 10 38 38 NUM cana-5460 113 11 ) ) PUNCT cana-5460 113 12 as as ADV cana-5460 113 13 well well ADV cana-5460 113 14 as as ADP cana-5460 113 15 the the DET cana-5460 113 16 conditions condition NOUN cana-5460 113 17 ( ( PUNCT cana-5460 113 18 22 22 NUM cana-5460 113 19 ) ) PUNCT cana-5460 113 20 into into ADP cana-5460 113 21 inequality inequality NOUN cana-5460 113 22 ( ( PUNCT cana-5460 113 23 36 36 NUM cana-5460 113 24 ) ) PUNCT cana-5460 113 25 , , PUNCT cana-5460 113 26 we we PRON cana-5460 113 27 obtain obtain VERB cana-5460 113 28 : : PUNCT cana-5460 113 29 dt dt PROPN cana-5460 113 30 Ξ΄βˆ’1 Ξ΄βˆ’1 PROPN cana-5460 113 31 β€–β„‘x β€–β„‘x NOUN cana-5460 114 1 βˆ‚u βˆ‚u PROPN cana-5460 114 2 βˆ‚t βˆ‚t PROPN cana-5460 114 3 β€– β€– PROPN cana-5460 114 4 2 2 NUM cana-5460 114 5 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 114 6 ) ) PUNCT cana-5460 115 1 + + NOUN cana-5460 115 2 ∫ ∫ PROPN cana-5460 115 3 β€– β€– PROPN cana-5460 115 4 βˆ‚u βˆ‚u PROPN cana-5460 115 5 βˆ‚t βˆ‚t PROPN cana-5460 115 6 β€– β€– PROPN cana-5460 115 7 2 2 NUM cana-5460 115 8 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 115 9 ) ) PUNCT cana-5460 115 10 dt dt PUNCT cana-5460 116 1 Ο„ Ο„ PROPN cana-5460 116 2 0 0 NUM cana-5460 117 1 + + NUM cana-5460 117 2 ∫ ∫ PROPN cana-5460 117 3 u2dx u2dx X cana-5460 117 4 1 1 NUM cana-5460 117 5 0 0 NUM cana-5460 117 6 ≀ ≀ NUM cana-5460 117 7 Ι³ Ι³ ADP cana-5460 117 8 1 1 NUM cana-5460 117 9 ( ( PUNCT cana-5460 117 10 ∫ ∫ PROPN cana-5460 117 11 Ο†2(x Ο†2(x NUM cana-5460 117 12 ) ) PUNCT cana-5460 117 13 1 1 NUM cana-5460 117 14 0 0 NUM cana-5460 117 15 dx dx PROPN cana-5460 117 16 + + CCONJ cana-5460 117 17 ∫ ∫ PROPN cana-5460 117 18 ∫ ∫ PROPN cana-5460 117 19 u2 u2 PROPN cana-5460 117 20 Ο„ Ο„ PROPN cana-5460 117 21 o o NOUN cana-5460 117 22 1 1 NUM cana-5460 117 23 0 0 NUM cana-5460 117 24 dtdx dtdx NOUN cana-5460 118 1 + + CCONJ cana-5460 118 2 ∫ ∫ PROPN cana-5460 118 3 ∫ ∫ PROPN cana-5460 118 4 ( ( PUNCT cana-5460 118 5 β„‘xf β„‘xf PROPN cana-5460 118 6 ) ) PUNCT cana-5460 118 7 2 2 NUM cana-5460 118 8 1 1 NUM cana-5460 118 9 0 0 NUM cana-5460 118 10 Ο„ Ο„ PROPN cana-5460 118 11 0 0 NUM cana-5460 118 12 dxdt dxdt PROPN cana-5460 118 13 + + CCONJ cana-5460 118 14 β€–β„‘xΟˆβ€– β€–β„‘xΟˆβ€– PROPN cana-5460 118 15 2 2 NUM cana-5460 118 16 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 118 17 ) ) PUNCT cana-5460 119 1 + + NUM cana-5460 119 2 ∫ ∫ PROPN cana-5460 119 3 ∫ ∫ PROPN cana-5460 119 4 ( ( PUNCT cana-5460 119 5 β„‘x β„‘x PROPN cana-5460 119 6 βˆ‚u βˆ‚u PROPN cana-5460 119 7 βˆ‚t βˆ‚t PROPN cana-5460 119 8 ) ) PUNCT cana-5460 119 9 21 21 NUM cana-5460 119 10 0 0 NUM cana-5460 119 11 Ο„ Ο„ PROPN cana-5460 119 12 0 0 NUM cana-5460 119 13 dxdt dxdt NOUN cana-5460 119 14 ) ) PUNCT cana-5460 119 15 , , PUNCT cana-5460 119 16 ( ( PUNCT cana-5460 119 17 39 39 NUM cana-5460 119 18 ) ) PUNCT cana-5460 119 19 where where SCONJ cana-5460 119 20 : : PUNCT cana-5460 119 21 Ι³1 Ι³1 PROPN cana-5460 119 22 = = SYM cana-5460 119 23 max max PROPN cana-5460 119 24 ( ( PUNCT cana-5460 119 25 Ξ±+Ξ³ Ξ±+Ξ³ NUM cana-5460 119 26 2 2 NUM cana-5460 119 27 , , PUNCT cana-5460 119 28 a1 a1 NOUN cana-5460 119 29 t t NOUN cana-5460 119 30 2 2 NUM cana-5460 119 31 + + NUM cana-5460 119 32 Ξ± Ξ± NOUN cana-5460 119 33 2 2 NUM cana-5460 119 34 , , PUNCT cana-5460 119 35 1 1 NUM cana-5460 119 36 2Ξ΅ 2Ξ΅ NOUN cana-5460 119 37 , , PUNCT cana-5460 119 38 Ξ΅ Ξ΅ PROPN cana-5460 119 39 2 2 NUM cana-5460 119 40 , , PUNCT cana-5460 119 41 t1βˆ’Ξ΄ t1βˆ’Ξ΄ PROPN cana-5460 119 42 Ξ³(1βˆ’Ξ΄ Ξ³(1βˆ’Ξ΄ NOUN cana-5460 119 43 ) ) PUNCT cana-5460 119 44 ) ) PUNCT cana-5460 119 45 min min NOUN cana-5460 119 46 ( ( PUNCT cana-5460 119 47 1 1 NUM cana-5460 119 48 2 2 NUM cana-5460 119 49 , , PUNCT cana-5460 119 50 Ξ² Ξ² X cana-5460 119 51 , , PUNCT cana-5460 119 52 Ξ±+Ξ³ Ξ±+Ξ³ NUM cana-5460 119 53 2 2 NUM cana-5460 119 54 ) ) PUNCT cana-5460 119 55 . . PUNCT cana-5460 120 1 ( ( PUNCT cana-5460 120 2 40 40 NUM cana-5460 120 3 ) ) PUNCT cana-5460 120 4 for for ADP cana-5460 120 5 the the DET cana-5460 120 6 second second ADJ cana-5460 120 7 term term NOUN cana-5460 120 8 on on ADP cana-5460 120 9 the the DET cana-5460 120 10 right right ADJ cana-5460 120 11 - - PUNCT cana-5460 120 12 hand hand NOUN cana-5460 120 13 side side NOUN cana-5460 120 14 of of ADP cana-5460 120 15 ( ( PUNCT cana-5460 120 16 39 39 NUM cana-5460 120 17 ) ) PUNCT cana-5460 120 18 , , PUNCT cana-5460 120 19 we we PRON cana-5460 120 20 apply apply VERB cana-5460 120 21 the the DET cana-5460 120 22 lemma lemma PROPN cana-5460 120 23 ( ( PUNCT cana-5460 120 24 2.3 2.3 NUM cana-5460 120 25 ) ) PUNCT cana-5460 120 26 by by ADP cana-5460 120 27 letting let VERB cana-5460 120 28 Ο†(t Ο†(t PROPN cana-5460 120 29 ) ) PUNCT cana-5460 121 1 = = PUNCT cana-5460 122 1 ∫ ∫ PROPN cana-5460 122 2 ∫ ∫ PROPN cana-5460 122 3 u2dxdt u2dxdt NOUN cana-5460 122 4 ; ; PUNCT cana-5460 122 5 1 1 NUM cana-5460 122 6 0 0 NUM cana-5460 122 7 t t NOUN cana-5460 122 8 0 0 NUM cana-5460 122 9 βˆ‚Ο† βˆ‚Ο† PROPN cana-5460 123 1 βˆ‚t βˆ‚t PROPN cana-5460 123 2 = = SYM cana-5460 123 3 ∫ ∫ PROPN cana-5460 123 4 u2 u2 PROPN cana-5460 123 5 1 1 NUM cana-5460 123 6 0 0 NUM cana-5460 123 7 dx dx NOUN cana-5460 123 8 ; ; PUNCT cana-5460 123 9 Ο†(0 Ο†(0 ADJ cana-5460 123 10 ) ) PUNCT cana-5460 123 11 = = SYM cana-5460 123 12 0 0 NUM cana-5460 123 13 , , PUNCT cana-5460 123 14 ( ( PUNCT cana-5460 123 15 41 41 NUM cana-5460 123 16 ) ) PUNCT cana-5460 123 17 communications communication NOUN cana-5460 123 18 on on ADP cana-5460 123 19 applied apply VERB cana-5460 123 20 nonlinear nonlinear ADJ cana-5460 123 21 analysis analysis NOUN cana-5460 123 22 issn issn NOUN cana-5460 123 23 : : PUNCT cana-5460 123 24 1074 1074 NUM cana-5460 123 25 - - PUNCT cana-5460 123 26 133x 133x NUM cana-5460 123 27 vol vol NOUN cana-5460 123 28 32 32 NUM cana-5460 123 29 no.3 no.3 NOUN cana-5460 123 30 ( ( PUNCT cana-5460 123 31 2025 2025 NUM cana-5460 123 32 ) ) PUNCT cana-5460 123 33 947 947 NUM cana-5460 123 34 https://internationalpubls.com https://internationalpubls.com X cana-5460 123 35 this this PRON cana-5460 123 36 leads lead VERB cana-5460 123 37 to to ADP cana-5460 123 38 ∫ ∫ PROPN cana-5460 124 1 ∫ ∫ PROPN cana-5460 124 2 u2 u2 PROPN cana-5460 124 3 Ο„ Ο„ PROPN cana-5460 124 4 o o NOUN cana-5460 124 5 1 1 NUM cana-5460 124 6 0 0 NUM cana-5460 124 7 dtdx dtdx NOUN cana-5460 124 8 ≀ ≀ PROPN cana-5460 124 9 tΙ³1e tΙ³1e PROPN cana-5460 124 10 Ι³1 Ι³1 PROPN cana-5460 124 11 t t PROPN cana-5460 124 12 ( ( PUNCT cana-5460 124 13 ∫ ∫ PROPN cana-5460 124 14 Ο†2(x Ο†2(x NUM cana-5460 124 15 ) ) PUNCT cana-5460 124 16 1 1 NUM cana-5460 124 17 0 0 NUM cana-5460 124 18 dx dx PROPN cana-5460 125 1 + + CCONJ cana-5460 125 2 ∫ ∫ PROPN cana-5460 125 3 ∫ ∫ PROPN cana-5460 125 4 ( ( PUNCT cana-5460 125 5 β„‘xf β„‘xf PROPN cana-5460 125 6 ) ) PUNCT cana-5460 125 7 21 21 NUM cana-5460 125 8 0 0 NUM cana-5460 125 9 Ο„ Ο„ PROPN cana-5460 125 10 0 0 NUM cana-5460 125 11 dxdt dxdt PROPN cana-5460 125 12 + + CCONJ cana-5460 125 13 β€–β„‘xΟˆβ€– β€–β„‘xΟˆβ€– PROPN cana-5460 125 14 2 2 NUM cana-5460 125 15 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 125 16 ) ) PUNCT cana-5460 126 1 + + CCONJ cana-5460 126 2 ∫ ∫ PROPN cana-5460 126 3 ∫ ∫ PROPN cana-5460 126 4 ( ( PUNCT cana-5460 126 5 β„‘x β„‘x PROPN cana-5460 126 6 βˆ‚u βˆ‚u PROPN cana-5460 126 7 βˆ‚t βˆ‚t PROPN cana-5460 126 8 ) ) PUNCT cana-5460 126 9 21 21 NUM cana-5460 126 10 0 0 NUM cana-5460 126 11 Ο„ Ο„ PROPN cana-5460 126 12 0 0 NUM cana-5460 126 13 dxdt dxdt NOUN cana-5460 126 14 ) ) PUNCT cana-5460 126 15 . . PUNCT cana-5460 127 1 ( ( PUNCT cana-5460 127 2 42 42 NUM cana-5460 127 3 ) ) PUNCT cana-5460 127 4 thus thus ADV cana-5460 127 5 , , PUNCT cana-5460 127 6 the the DET cana-5460 127 7 inequality inequality NOUN cana-5460 127 8 ( ( PUNCT cana-5460 127 9 39 39 NUM cana-5460 127 10 ) ) PUNCT cana-5460 127 11 becomes become VERB cana-5460 127 12 dt dt ADP cana-5460 127 13 Ξ΄βˆ’1 Ξ΄βˆ’1 PROPN cana-5460 127 14 β€–β„‘x β€–β„‘x NOUN cana-5460 128 1 βˆ‚u βˆ‚u PROPN cana-5460 128 2 βˆ‚t βˆ‚t PROPN cana-5460 128 3 β€– β€– PROPN cana-5460 128 4 2 2 NUM cana-5460 128 5 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 128 6 ) ) PUNCT cana-5460 129 1 + + NOUN cana-5460 129 2 ∫ ∫ PROPN cana-5460 129 3 β€– β€– PROPN cana-5460 129 4 βˆ‚u βˆ‚u PROPN cana-5460 129 5 βˆ‚t βˆ‚t PROPN cana-5460 129 6 β€– β€– PROPN cana-5460 129 7 2 2 NUM cana-5460 129 8 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 129 9 ) ) PUNCT cana-5460 129 10 dt dt PUNCT cana-5460 130 1 Ο„ Ο„ PROPN cana-5460 130 2 0 0 NUM cana-5460 131 1 + + NUM cana-5460 131 2 ∫ ∫ PROPN cana-5460 131 3 u2dx u2dx X cana-5460 131 4 1 1 NUM cana-5460 131 5 0 0 NUM cana-5460 131 6 ≀ ≀ NOUN cana-5460 131 7 Ι³2 Ι³2 NOUN cana-5460 131 8 ( ( PUNCT cana-5460 131 9 ∫ ∫ PROPN cana-5460 131 10 Ο†2(x Ο†2(x NUM cana-5460 131 11 ) ) PUNCT cana-5460 131 12 1 1 NUM cana-5460 131 13 0 0 NUM cana-5460 131 14 dx dx PROPN cana-5460 132 1 + + CCONJ cana-5460 132 2 ∫ ∫ PROPN cana-5460 132 3 ∫ ∫ PROPN cana-5460 132 4 ( ( PUNCT cana-5460 132 5 β„‘xf β„‘xf PROPN cana-5460 132 6 ) ) PUNCT cana-5460 132 7 2 2 NUM cana-5460 132 8 1 1 NUM cana-5460 132 9 0 0 NUM cana-5460 132 10 Ο„ Ο„ PROPN cana-5460 132 11 0 0 NUM cana-5460 132 12 dxdt dxdt PROPN cana-5460 132 13 + + CCONJ cana-5460 132 14 β€–β„‘xΟˆβ€– β€–β„‘xΟˆβ€– PROPN cana-5460 132 15 2 2 NUM cana-5460 132 16 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 132 17 ) ) PUNCT cana-5460 133 1 + + NUM cana-5460 133 2 ∫ ∫ PROPN cana-5460 133 3 ∫ ∫ PROPN cana-5460 133 4 ( ( PUNCT cana-5460 133 5 β„‘x β„‘x PROPN cana-5460 133 6 βˆ‚u βˆ‚u PROPN cana-5460 133 7 βˆ‚t βˆ‚t PROPN cana-5460 133 8 ) ) PUNCT cana-5460 133 9 21 21 NUM cana-5460 133 10 0 0 NUM cana-5460 133 11 Ο„ Ο„ PROPN cana-5460 133 12 0 0 NUM cana-5460 133 13 dxdt dxdt NOUN cana-5460 133 14 ) ) PUNCT cana-5460 133 15 , , PUNCT cana-5460 133 16 ( ( PUNCT cana-5460 133 17 43 43 NUM cana-5460 133 18 ) ) PUNCT cana-5460 133 19 such such ADJ cana-5460 133 20 that that SCONJ cana-5460 133 21 : : PUNCT cana-5460 133 22 Ι³2 Ι³2 NOUN cana-5460 133 23 = = PUNCT cana-5460 133 24 max(Ι³1 max(Ι³1 PROPN cana-5460 133 25 , , PUNCT cana-5460 133 26 Ι³1 Ι³1 PROPN cana-5460 133 27 2 2 NUM cana-5460 133 28 t t NOUN cana-5460 133 29 eΙ³1 eΙ³1 NOUN cana-5460 133 30 t t PROPN cana-5460 133 31 ) ) PUNCT cana-5460 133 32 . . PUNCT cana-5460 134 1 ( ( PUNCT cana-5460 134 2 44 44 NUM cana-5460 134 3 ) ) PUNCT cana-5460 134 4 finally finally ADV cana-5460 134 5 , , PUNCT cana-5460 134 6 we we PRON cana-5460 134 7 apply apply VERB cana-5460 134 8 lemma lemma PROPN cana-5460 134 9 ( ( PUNCT cana-5460 134 10 2.2 2.2 NUM cana-5460 134 11 ) ) PUNCT cana-5460 134 12 to to ADP cana-5460 134 13 the the DET cana-5460 134 14 last last ADJ cana-5460 134 15 term term NOUN cana-5460 134 16 on on ADP cana-5460 134 17 the the DET cana-5460 134 18 right right ADJ cana-5460 134 19 - - PUNCT cana-5460 134 20 hand hand NOUN cana-5460 134 21 side side NOUN cana-5460 134 22 of of ADP cana-5460 134 23 ( ( PUNCT cana-5460 134 24 43 43 NUM cana-5460 134 25 ) ) PUNCT cana-5460 134 26 by by ADP cana-5460 134 27 letting let VERB cana-5460 134 28 : : PUNCT cana-5460 134 29 y(t y(t X cana-5460 134 30 ) ) PUNCT cana-5460 135 1 = = SYM cana-5460 136 1 ∫ ∫ PROPN cana-5460 136 2 ∫ ∫ PROPN cana-5460 136 3 ( ( PUNCT cana-5460 136 4 β„‘x β„‘x PROPN cana-5460 136 5 βˆ‚u βˆ‚u PROPN cana-5460 136 6 βˆ‚t βˆ‚t PROPN cana-5460 136 7 ) ) PUNCT cana-5460 136 8 21 21 NUM cana-5460 136 9 0 0 NUM cana-5460 136 10 Ο„ Ο„ PROPN cana-5460 136 11 0 0 NUM cana-5460 136 12 dxdt dxdt NOUN cana-5460 136 13 ; ; PUNCT cana-5460 136 14 βˆ‚t βˆ‚t PROPN cana-5460 136 15 Ξ΄y(t Ξ΄y(t PUNCT cana-5460 136 16 ) ) PUNCT cana-5460 137 1 = = PUNCT cana-5460 138 1 dΞ΄βˆ’1 dΞ΄βˆ’1 NOUN cana-5460 138 2 β€–β„‘x β€–β„‘x NOUN cana-5460 139 1 βˆ‚u βˆ‚u PROPN cana-5460 139 2 βˆ‚t βˆ‚t PROPN cana-5460 139 3 β€– β€– PROPN cana-5460 139 4 2 2 NUM cana-5460 139 5 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 139 6 ) ) PUNCT cana-5460 139 7 . . PUNCT cana-5460 140 1 ( ( PUNCT cana-5460 140 2 45 45 NUM cana-5460 140 3 ) ) PUNCT cana-5460 140 4 thus thus ADV cana-5460 140 5 , , PUNCT cana-5460 140 6 we we PRON cana-5460 140 7 obtain obtain VERB cana-5460 140 8 : : PUNCT cana-5460 140 9 ∫ ∫ PROPN cana-5460 140 10 ∫ ∫ PROPN cana-5460 141 1 ( ( PUNCT cana-5460 141 2 β„‘x β„‘x PROPN cana-5460 141 3 2 2 NUM cana-5460 141 4 βˆ‚u βˆ‚u PROPN cana-5460 141 5 βˆ‚t βˆ‚t PROPN cana-5460 141 6 ) ) PUNCT cana-5460 141 7 21 21 NUM cana-5460 141 8 0 0 NUM cana-5460 141 9 Ο„ Ο„ X cana-5460 141 10 0 0 NUM cana-5460 141 11 dxdt dxdt NOUN cana-5460 141 12 ≀ ≀ NUM cana-5460 141 13 Ι³2Ξ³(Ξ΄)eΞ΄ Ι³2Ξ³(Ξ΄)eΞ΄ NOUN cana-5460 141 14 , , PUNCT cana-5460 141 15 Ξ΄(Ι³2 Ξ΄(Ι³2 PROPN cana-5460 141 16 t t PROPN cana-5460 141 17 Ξ΄ Ξ΄ PROPN cana-5460 141 18 ) ) PUNCT cana-5460 141 19 ( ( PUNCT cana-5460 141 20 t t PROPN cana-5460 141 21 Ξ΄ Ξ΄ PROPN cana-5460 141 22 Ξ³(Ξ΄ Ξ³(Ξ΄ PROPN cana-5460 141 23 ) ) PUNCT cana-5460 141 24 β€–Ο†β€–2𝕃2(0,1 β€–Ο†β€–2𝕃2(0,1 NOUN cana-5460 141 25 ) ) PUNCT cana-5460 142 1 + + NUM cana-5460 142 2 t t PROPN cana-5460 142 3 Ξ΄ Ξ΄ PROPN cana-5460 142 4 Ξ³(Ξ΄ Ξ³(Ξ΄ PROPN cana-5460 142 5 ) ) PUNCT cana-5460 142 6 β€–β„‘xΟˆβ€– β€–β„‘xΟˆβ€– PROPN cana-5460 142 7 2 2 NUM cana-5460 142 8 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 142 9 ) ) PUNCT cana-5460 143 1 + + NOUN cana-5460 143 2 dt dt NOUN cana-5460 143 3 βˆ’Ξ΄βˆ’1β€–β„‘xfβ€– βˆ’Ξ΄βˆ’1β€–β„‘xfβ€– ADP cana-5460 143 4 2 2 NUM cana-5460 143 5 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 143 6 ) ) PUNCT cana-5460 143 7 ) ) PUNCT cana-5460 144 1 ≀ ≀ NUM cana-5460 144 2 Ι³2Ξ³(Ξ΄)eΞ΄ Ι³2Ξ³(Ξ΄)eΞ΄ NOUN cana-5460 144 3 , , PUNCT cana-5460 144 4 Ξ΄(Ι³2 Ξ΄(Ι³2 PROPN cana-5460 144 5 t t PROPN cana-5460 144 6 Ξ΄)max Ξ΄)max PROPN cana-5460 144 7 ( ( PUNCT cana-5460 144 8 1 1 NUM cana-5460 144 9 , , PUNCT cana-5460 144 10 t t PROPN cana-5460 144 11 Ξ΄ Ξ΄ PROPN cana-5460 144 12 Ξ³(Ξ΄ Ξ³(Ξ΄ PROPN cana-5460 144 13 ) ) PUNCT cana-5460 144 14 ) ) PUNCT cana-5460 144 15 ( ( PUNCT cana-5460 144 16 dt dt X cana-5460 144 17 βˆ’Ξ΄βˆ’1β€–β„‘xfβ€– βˆ’Ξ΄βˆ’1β€–β„‘xfβ€– X cana-5460 144 18 2 2 NUM cana-5460 144 19 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 144 20 ) ) PUNCT cana-5460 145 1 + + NOUN cana-5460 145 2 β€–Ο†β€–2𝕃2(0,1 β€–Ο†β€–2𝕃2(0,1 NUM cana-5460 145 3 ) ) PUNCT cana-5460 146 1 + + CCONJ cana-5460 146 2 β€–β„‘xΟˆβ€– β€–β„‘xΟˆβ€– PROPN cana-5460 146 3 2 2 NUM cana-5460 146 4 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 146 5 ) ) PUNCT cana-5460 146 6 ) ) PUNCT cana-5460 146 7 . . PUNCT cana-5460 147 1 ( ( PUNCT cana-5460 147 2 46 46 X cana-5460 147 3 ) ) PUNCT cana-5460 147 4 substituting substitute VERB cana-5460 147 5 equation equation NOUN cana-5460 147 6 ( ( PUNCT cana-5460 147 7 46 46 NUM cana-5460 147 8 ) ) PUNCT cana-5460 147 9 into into ADP cana-5460 147 10 equation equation NOUN cana-5460 147 11 ( ( PUNCT cana-5460 147 12 43 43 NUM cana-5460 147 13 ) ) PUNCT cana-5460 147 14 , , PUNCT cana-5460 147 15 we we PRON cana-5460 147 16 obtain obtain VERB cana-5460 147 17 : : PUNCT cana-5460 147 18 dt dt PROPN cana-5460 148 1 Ξ΄βˆ’1 Ξ΄βˆ’1 PROPN cana-5460 148 2 β€–β„‘x β€–β„‘x NOUN cana-5460 149 1 βˆ‚u βˆ‚u PROPN cana-5460 149 2 βˆ‚t βˆ‚t PROPN cana-5460 149 3 β€– β€– PROPN cana-5460 149 4 2 2 NUM cana-5460 149 5 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 149 6 ) ) PUNCT cana-5460 150 1 + + NOUN cana-5460 150 2 ∫ ∫ PROPN cana-5460 150 3 β€– β€– PROPN cana-5460 150 4 βˆ‚u βˆ‚u PROPN cana-5460 150 5 βˆ‚t βˆ‚t PROPN cana-5460 150 6 β€– β€– PROPN cana-5460 150 7 2 2 NUM cana-5460 150 8 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 150 9 ) ) PUNCT cana-5460 150 10 dt dt PUNCT cana-5460 151 1 Ο„ Ο„ PROPN cana-5460 151 2 0 0 NUM cana-5460 152 1 + + NUM cana-5460 152 2 ∫ ∫ PROPN cana-5460 152 3 u2dx u2dx ADJ cana-5460 152 4 1 1 NUM cana-5460 152 5 0 0 NUM cana-5460 152 6 ≀ ≀ NOUN cana-5460 152 7 Ι³3 Ι³3 NOUN cana-5460 152 8 ( ( PUNCT cana-5460 152 9 ∫ ∫ PROPN cana-5460 152 10 Ο†2(x Ο†2(x NUM cana-5460 152 11 ) ) PUNCT cana-5460 152 12 1 1 NUM cana-5460 152 13 0 0 NUM cana-5460 152 14 dx dx PROPN cana-5460 153 1 + + CCONJ cana-5460 153 2 ∫ ∫ PROPN cana-5460 153 3 ∫ ∫ PROPN cana-5460 153 4 ( ( PUNCT cana-5460 153 5 β„‘xf β„‘xf PROPN cana-5460 153 6 ) ) PUNCT cana-5460 153 7 21 21 NUM cana-5460 153 8 0 0 NUM cana-5460 153 9 Ο„ Ο„ PROPN cana-5460 153 10 0 0 NUM cana-5460 153 11 dxdt dxdt PROPN cana-5460 153 12 + + CCONJ cana-5460 153 13 β€–β„‘xΟˆβ€– β€–β„‘xΟˆβ€– PROPN cana-5460 153 14 2 2 NUM cana-5460 153 15 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 153 16 ) ) PUNCT cana-5460 154 1 + + CCONJ cana-5460 154 2 dt dt X cana-5460 154 3 βˆ’Ξ΄βˆ’1β€–β„‘xfβ€– βˆ’Ξ΄βˆ’1β€–β„‘xfβ€– ADP cana-5460 154 4 2 2 NUM cana-5460 154 5 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 154 6 ) ) PUNCT cana-5460 154 7 ) ) PUNCT cana-5460 154 8 , , PUNCT cana-5460 154 9 ( ( PUNCT cana-5460 154 10 47 47 NUM cana-5460 154 11 ) ) PUNCT cana-5460 154 12 communications communication NOUN cana-5460 154 13 on on ADP cana-5460 154 14 applied apply VERB cana-5460 154 15 nonlinear nonlinear ADJ cana-5460 154 16 analysis analysis NOUN cana-5460 154 17 issn issn NOUN cana-5460 154 18 : : PUNCT cana-5460 154 19 1074 1074 NUM cana-5460 154 20 - - PUNCT cana-5460 154 21 133x 133x NUM cana-5460 154 22 vol vol NOUN cana-5460 154 23 32 32 NUM cana-5460 154 24 no.3 no.3 NOUN cana-5460 154 25 ( ( PUNCT cana-5460 154 26 2025 2025 NUM cana-5460 154 27 ) ) PUNCT cana-5460 154 28 948 948 NUM cana-5460 154 29 https://internationalpubls.com https://internationalpubls.com X cana-5460 154 30 where where SCONJ cana-5460 154 31 : : PUNCT cana-5460 154 32 Ι³3 Ι³3 NOUN cana-5460 154 33 = = SYM cana-5460 154 34 Ι³2max(Ι³2 Ι³2max(Ι³2 PROPN cana-5460 154 35 , , PUNCT cana-5460 154 36 Ι³2Ξ³(Ξ΄ Ι³2Ξ³(Ξ΄ PROPN cana-5460 154 37 ) ) PUNCT cana-5460 154 38 , , PUNCT cana-5460 154 39 eΞ΄ eΞ΄ PROPN cana-5460 154 40 , , PUNCT cana-5460 154 41 Ξ΄(Ι³2 Ξ΄(Ι³2 PROPN cana-5460 154 42 t t PROPN cana-5460 154 43 Ξ΄)max Ξ΄)max PROPN cana-5460 154 44 ( ( PUNCT cana-5460 154 45 1 1 NUM cana-5460 154 46 , , PUNCT cana-5460 154 47 t t PROPN cana-5460 154 48 Ξ΄ Ξ΄ PROPN cana-5460 154 49 Ξ³(Ξ΄ Ξ³(Ξ΄ PROPN cana-5460 154 50 ) ) PUNCT cana-5460 154 51 ) ) PUNCT cana-5460 154 52 ) ) PUNCT cana-5460 154 53 . . PUNCT cana-5460 155 1 ( ( PUNCT cana-5460 155 2 48 48 NUM cana-5460 155 3 ) ) PUNCT cana-5460 155 4 on on ADP cana-5460 155 5 the the DET cana-5460 155 6 other other ADJ cana-5460 155 7 hand hand NOUN cana-5460 155 8 : : PUNCT cana-5460 155 9 dt dt X cana-5460 155 10 βˆ’Ξ΄βˆ’1β€–β„‘xfβ€– βˆ’Ξ΄βˆ’1β€–β„‘xfβ€– X cana-5460 155 11 2 2 NUM cana-5460 155 12 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 155 13 ) ) PUNCT cana-5460 155 14 ≀ ≀ NOUN cana-5460 155 15 tΞ΄ tΞ΄ ADP cana-5460 155 16 Ξ³(1+Ξ΄ Ξ³(1+Ξ΄ NOUN cana-5460 155 17 ) ) PUNCT cana-5460 156 1 ∫ ∫ PROPN cana-5460 156 2 β€–β„‘xfβ€– β€–β„‘xfβ€– PROPN cana-5460 156 3 2 2 NUM cana-5460 156 4 𝕃2(0,1)dt 𝕃2(0,1)dt NUM cana-5460 156 5 t t PROPN cana-5460 156 6 0 0 NUM cana-5460 156 7 . . PUNCT cana-5460 157 1 ( ( PUNCT cana-5460 157 2 49 49 NUM cana-5460 157 3 ) ) PUNCT cana-5460 157 4 hence hence ADV cana-5460 157 5 , , PUNCT cana-5460 157 6 inequality inequality NOUN cana-5460 157 7 ( ( PUNCT cana-5460 157 8 47 47 NUM cana-5460 157 9 ) ) PUNCT cana-5460 157 10 becomes become VERB cana-5460 157 11 dt dt ADP cana-5460 157 12 Ξ΄βˆ’1 Ξ΄βˆ’1 PROPN cana-5460 157 13 β€–β„‘x β€–β„‘x NOUN cana-5460 158 1 βˆ‚u βˆ‚u PROPN cana-5460 158 2 βˆ‚t βˆ‚t PROPN cana-5460 158 3 β€– β€– PROPN cana-5460 158 4 2 2 NUM cana-5460 158 5 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 158 6 ) ) PUNCT cana-5460 159 1 + + NOUN cana-5460 159 2 ∫ ∫ PROPN cana-5460 159 3 β€– β€– PROPN cana-5460 159 4 βˆ‚u βˆ‚u PROPN cana-5460 159 5 βˆ‚t βˆ‚t PROPN cana-5460 159 6 β€– β€– PROPN cana-5460 159 7 2 2 NUM cana-5460 159 8 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 159 9 ) ) PUNCT cana-5460 159 10 dt dt PUNCT cana-5460 160 1 Ο„ Ο„ PROPN cana-5460 160 2 0 0 NUM cana-5460 161 1 + + NUM cana-5460 161 2 ∫ ∫ PROPN cana-5460 161 3 u2dx u2dx X cana-5460 161 4 1 1 NUM cana-5460 161 5 0 0 NUM cana-5460 161 6 ≀ ≀ NUM cana-5460 161 7 c(∫ c(∫ NOUN cana-5460 161 8 Ο†2(x Ο†2(x NOUN cana-5460 161 9 ) ) PUNCT cana-5460 161 10 1 1 NUM cana-5460 161 11 0 0 NUM cana-5460 161 12 dx dx PROPN cana-5460 162 1 + + CCONJ cana-5460 162 2 ∫ ∫ PROPN cana-5460 162 3 ∫ ∫ PROPN cana-5460 162 4 ( ( PUNCT cana-5460 162 5 β„‘xf β„‘xf PROPN cana-5460 162 6 ) ) PUNCT cana-5460 162 7 21 21 NUM cana-5460 162 8 0 0 NUM cana-5460 162 9 Ο„ Ο„ PROPN cana-5460 162 10 0 0 NUM cana-5460 162 11 dxdt dxdt PROPN cana-5460 162 12 + + CCONJ cana-5460 162 13 β€–β„‘xΟˆβ€– β€–β„‘xΟˆβ€– PROPN cana-5460 162 14 2 2 NUM cana-5460 162 15 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 162 16 ) ) PUNCT cana-5460 162 17 ) ) PUNCT cana-5460 162 18 , , PUNCT cana-5460 162 19 ( ( PUNCT cana-5460 162 20 50 50 NUM cana-5460 162 21 ) ) PUNCT cana-5460 162 22 such such ADJ cana-5460 162 23 that that PRON cana-5460 162 24 : : PUNCT cana-5460 162 25 c c X cana-5460 162 26 = = SYM cana-5460 162 27 Ι³3 Ι³3 NOUN cana-5460 162 28 ( ( PUNCT cana-5460 162 29 1 1 NUM cana-5460 162 30 + + CCONJ cana-5460 162 31 tΞ΄ tΞ΄ ADP cana-5460 162 32 Ξ³(1+Ξ΄ Ξ³(1+Ξ΄ NOUN cana-5460 162 33 ) ) PUNCT cana-5460 162 34 ) ) PUNCT cana-5460 162 35 . . PUNCT cana-5460 163 1 ( ( PUNCT cana-5460 163 2 51 51 NUM cana-5460 163 3 ) ) PUNCT cana-5460 163 4 we we PRON cana-5460 163 5 observe observe VERB cana-5460 163 6 that that SCONJ cana-5460 163 7 the the DET cana-5460 163 8 right right ADJ cana-5460 163 9 - - PUNCT cana-5460 163 10 hand hand NOUN cana-5460 163 11 side side NOUN cana-5460 163 12 of of ADP cana-5460 163 13 inequality inequality NOUN cana-5460 163 14 ( ( PUNCT cana-5460 163 15 50 50 NUM cana-5460 163 16 ) ) PUNCT cana-5460 163 17 is be AUX cana-5460 163 18 independent independent ADJ cana-5460 163 19 of of ADP cana-5460 163 20 Ο„ Ο„ PROPN cana-5460 163 21 , , PUNCT cana-5460 163 22 by by ADP cana-5460 163 23 taking take VERB cana-5460 163 24 the the DET cana-5460 163 25 supremum supremum NOUN cana-5460 163 26 of of ADP cana-5460 163 27 the the DET cana-5460 163 28 left left ADJ cana-5460 163 29 - - PUNCT cana-5460 163 30 hand hand NOUN cana-5460 163 31 side side NOUN cana-5460 163 32 with with ADP cana-5460 163 33 respect respect NOUN cana-5460 163 34 to to ADP cana-5460 163 35 Ο„ Ο„ PROPN cana-5460 163 36 ∈ ∈ PROPN cana-5460 164 1 [ [ X cana-5460 164 2 0 0 NUM cana-5460 164 3 , , PUNCT cana-5460 164 4 t t PROPN cana-5460 164 5 ] ] PUNCT cana-5460 164 6 . . PUNCT cana-5460 165 1 we we PRON cana-5460 165 2 obtain obtain VERB cana-5460 165 3 the the DET cana-5460 165 4 desired desire VERB cana-5460 165 5 inequality inequality NOUN cana-5460 165 6 , , PUNCT cana-5460 165 7 which which PRON cana-5460 165 8 concludes conclude VERB cana-5460 165 9 the the DET cana-5460 165 10 proof proof NOUN cana-5460 165 11 . . PUNCT cana-5460 166 1 proposition proposition NOUN cana-5460 166 2 5.1 5.1 NUM cana-5460 166 3 : : PUNCT cana-5460 166 4 the the DET cana-5460 166 5 operator operator NOUN cana-5460 166 6 l l NOUN cana-5460 166 7 which which PRON cana-5460 166 8 is be AUX cana-5460 166 9 defined define VERB cana-5460 166 10 from from ADP cana-5460 166 11 e e NOUN cana-5460 166 12 to to ADP cana-5460 166 13 f f PROPN cana-5460 166 14 has have VERB cana-5460 166 15 a a DET cana-5460 166 16 closure closure NOUN cana-5460 166 17 . . PUNCT cana-5460 167 1 theorem theorem VERB cana-5460 167 2 5.1 5.1 NUM cana-5460 167 3 holds hold NOUN cana-5460 167 4 for for ADP cana-5460 167 5 strong strong ADJ cana-5460 167 6 solutions solution NOUN cana-5460 167 7 , , PUNCT cana-5460 167 8 and and CCONJ cana-5460 167 9 we we PRON cana-5460 167 10 have have VERB cana-5460 167 11 the the DET cana-5460 167 12 inequality inequality NOUN cana-5460 167 13 : : PUNCT cana-5460 167 14 β€–uβ€–e β€–uβ€–e VERB cana-5460 167 15 ≀ ≀ NUM cana-5460 167 16 ć‖lΜ…uβ€–f ć‖lΜ…uβ€–f NOUN cana-5460 167 17 , , PUNCT cana-5460 167 18 ( ( PUNCT cana-5460 167 19 52 52 NUM cana-5460 167 20 ) ) PUNCT cana-5460 167 21 thus thus ADV cana-5460 167 22 , , PUNCT cana-5460 167 23 we we PRON cana-5460 167 24 obtain obtain VERB cana-5460 167 25 : : PUNCT cana-5460 167 26 corollary corollary ADJ cana-5460 167 27 5.1 5.1 NUM cana-5460 167 28 : : PUNCT cana-5460 167 29 the the DET cana-5460 167 30 strong strong ADJ cana-5460 167 31 solution solution NOUN cana-5460 167 32 of of ADP cana-5460 167 33 ( ( PUNCT cana-5460 167 34 21)-(23 21)-(23 NOUN cana-5460 167 35 ) ) PUNCT cana-5460 167 36 is be AUX cana-5460 167 37 unique unique ADJ cana-5460 167 38 if if SCONJ cana-5460 167 39 it it PRON cana-5460 167 40 exists exist VERB cana-5460 167 41 , , PUNCT cana-5460 167 42 and and CCONJ cana-5460 167 43 depends depend VERB cana-5460 167 44 continuously continuously ADV cana-5460 167 45 on on ADP cana-5460 167 46 β„± β„± PROPN cana-5460 167 47 ∈ ∈ PROPN cana-5460 167 48 f. f. PROPN cana-5460 167 49 corollary corollary PROPN cana-5460 167 50 5.2 5.2 NUM cana-5460 167 51 : : PUNCT cana-5460 167 52 the the DET cana-5460 167 53 set set NOUN cana-5460 167 54 of of ADP cana-5460 167 55 values value NOUN cana-5460 167 56 r(lΜ… r(lΜ… VERB cana-5460 167 57 ) ) PUNCT cana-5460 167 58 of of ADP cana-5460 167 59 the the DET cana-5460 167 60 operator operator NOUN cana-5460 167 61 lΜ…is lΜ…is AUX cana-5460 167 62 closed close VERB cana-5460 167 63 in in ADP cana-5460 167 64 f. f. PROPN cana-5460 167 65 ∎ ∎ PROPN cana-5460 167 66 6 6 NUM cana-5460 167 67 . . PUNCT cana-5460 167 68 existence existence NOUN cana-5460 167 69 of of ADP cana-5460 167 70 the the DET cana-5460 167 71 solution solution NOUN cana-5460 167 72 communications communication NOUN cana-5460 167 73 on on ADP cana-5460 167 74 applied apply VERB cana-5460 167 75 nonlinear nonlinear ADJ cana-5460 167 76 analysis analysis NOUN cana-5460 167 77 issn issn NOUN cana-5460 167 78 : : PUNCT cana-5460 167 79 1074 1074 NUM cana-5460 167 80 - - PUNCT cana-5460 167 81 133x 133x NUM cana-5460 167 82 vol vol NOUN cana-5460 167 83 32 32 NUM cana-5460 167 84 no.3 no.3 NOUN cana-5460 167 85 ( ( PUNCT cana-5460 167 86 2025 2025 NUM cana-5460 167 87 ) ) PUNCT cana-5460 167 88 949 949 NUM cana-5460 167 89 https://internationalpubls.com https://internationalpubls.com X cana-5460 167 90 to to PART cana-5460 167 91 prove prove VERB cana-5460 167 92 the the DET cana-5460 167 93 existence existence NOUN cana-5460 167 94 of of ADP cana-5460 167 95 the the DET cana-5460 167 96 solution solution NOUN cana-5460 167 97 , , PUNCT cana-5460 167 98 we we PRON cana-5460 167 99 must must AUX cana-5460 167 100 demonstrate demonstrate VERB cana-5460 167 101 that that SCONJ cana-5460 167 102 : : PUNCT cana-5460 167 103 r(l r(l NOUN cana-5460 167 104 ) ) PUNCT cana-5460 167 105 is be AUX cana-5460 167 106 dense dense ADJ cana-5460 167 107 in in ADP cana-5460 167 108 f f PROPN cana-5460 167 109 , , PUNCT cana-5460 167 110 for for ADP cana-5460 167 111 all all PRON cana-5460 167 112 : : PUNCT cana-5460 167 113 u u PROPN cana-5460 167 114 ∈ ∈ PROPN cana-5460 167 115 e e NOUN cana-5460 167 116 , , PUNCT cana-5460 167 117 and and CCONJ cana-5460 167 118 β„± β„± PROPN cana-5460 167 119 = = SYM cana-5460 167 120 ( ( PUNCT cana-5460 167 121 f f PROPN cana-5460 167 122 , , PUNCT cana-5460 167 123 Ο† Ο† PROPN cana-5460 167 124 , , PUNCT cana-5460 167 125 ψ ψ NOUN cana-5460 167 126 ) ) PUNCT cana-5460 167 127 ∈ ∈ PROPN cana-5460 167 128 f. f. PROPN cana-5460 167 129 theorem theorem VERB cana-5460 167 130 5.1 5.1 NUM cana-5460 167 131 for for ADP cana-5460 167 132 z z PROPN cana-5460 167 133 ∈ ∈ PROPN cana-5460 167 134 l2(Ο‰ l2(Ο‰ PROPN cana-5460 167 135 ) ) PUNCT cana-5460 167 136 and and CCONJ cana-5460 167 137 all all DET cana-5460 167 138 u u NOUN cana-5460 167 139 ∈ ∈ PROPN cana-5460 167 140 e e NOUN cana-5460 167 141 , , PUNCT cana-5460 167 142 we we PRON cana-5460 167 143 have have VERB cana-5460 167 144 : : PUNCT cana-5460 167 145 ∫ ∫ PROPN cana-5460 167 146 lu lu PROPN cana-5460 167 147 . . PUNCT cana-5460 168 1 z z PROPN cana-5460 168 2 dxdt dxdt NOUN cana-5460 168 3 = = NOUN cana-5460 168 4 0Ο‰ 0Ο‰ NOUN cana-5460 168 5 , , PUNCT cana-5460 168 6 ( ( PUNCT cana-5460 168 7 53 53 NUM cana-5460 168 8 ) ) PUNCT cana-5460 168 9 then then ADV cana-5460 168 10 : : PUNCT cana-5460 168 11 z z NOUN cana-5460 168 12 disappears disappear VERB cana-5460 168 13 almost almost ADV cana-5460 168 14 everywhere everywhere ADV cana-5460 168 15 in in ADP cana-5460 168 16 Ο‰ Ο‰ PROPN cana-5460 168 17 , , PUNCT cana-5460 168 18 this this PRON cana-5460 168 19 implies imply VERB cana-5460 168 20 that that SCONJ cana-5460 168 21 the the DET cana-5460 168 22 problem problem NOUN cana-5460 168 23 ( ( PUNCT cana-5460 168 24 21)-(23 21)-(23 NOUN cana-5460 168 25 ) ) PUNCT cana-5460 168 26 has have VERB cana-5460 168 27 a a DET cana-5460 168 28 unique unique ADJ cana-5460 168 29 solution solution NOUN cana-5460 168 30 . . PUNCT cana-5460 169 1 proof proof NOUN cana-5460 169 2 : : PUNCT cana-5460 169 3 the the DET cana-5460 169 4 proof proof NOUN cana-5460 169 5 of of ADP cana-5460 169 6 this this DET cana-5460 169 7 theorem theorem NOUN cana-5460 169 8 consists consist NOUN cana-5460 169 9 of of ADP cana-5460 169 10 choosing choose VERB cana-5460 169 11 z z PROPN cana-5460 169 12 ∈ ∈ PROPN cana-5460 169 13 r(l r(l NOUN cana-5460 169 14 ) ) PUNCT cana-5460 169 15 ⏊ ⏊ PROPN cana-5460 169 16 , , PUNCT cana-5460 169 17 we we PRON cana-5460 169 18 demonstrate demonstrate VERB cana-5460 169 19 that that SCONJ cana-5460 169 20 : : PUNCT cana-5460 169 21 r(l r(l NOUN cana-5460 169 22 ) ) PUNCT cana-5460 169 23 ⏊ ⏊ PROPN cana-5460 169 24 = = SYM cana-5460 169 25 { { PUNCT cana-5460 169 26 0 0 NUM cana-5460 169 27 } } PUNCT cana-5460 169 28 ⟺ ⟺ PROPN cana-5460 169 29 r(l)Μ…Μ… r(l)Μ…Μ… PROPN cana-5460 169 30 Μ…Μ… Μ…Μ… PROPN cana-5460 169 31 Μ…Μ… Μ…Μ… PROPN cana-5460 169 32 = = PROPN cana-5460 170 1 f. f. PROPN cana-5460 170 2 the the DET cana-5460 170 3 scalar scalar ADJ cana-5460 170 4 product product NOUN cana-5460 170 5 in in ADP cana-5460 170 6 f f PROPN cana-5460 170 7 is be AUX cana-5460 170 8 defined define VERB cana-5460 170 9 by by ADP cana-5460 170 10 : : PUNCT cana-5460 170 11 ( ( PUNCT cana-5460 170 12 lu lu INTJ cana-5460 170 13 , , PUNCT cana-5460 170 14 z)f z)f X cana-5460 170 15 = = SYM cana-5460 170 16 ∫ ∫ PROPN cana-5460 171 1 lu lu PROPN cana-5460 171 2 . . PUNCT cana-5460 172 1 z z PROPN cana-5460 172 2 dxdtΟ‰ dxdtΟ‰ PROPN cana-5460 172 3 . . PUNCT cana-5460 173 1 ( ( PUNCT cana-5460 173 2 54 54 NUM cana-5460 173 3 ) ) PUNCT cana-5460 173 4 then then ADV cana-5460 173 5 ( ( PUNCT cana-5460 173 6 53 53 NUM cana-5460 173 7 ) ) PUNCT cana-5460 173 8 can can AUX cana-5460 173 9 be be AUX cana-5460 173 10 written write VERB cana-5460 173 11 as as ADP cana-5460 173 12 : : PUNCT cana-5460 173 13 ∫ ∫ PROPN cana-5460 174 1 ( ( PUNCT cana-5460 174 2 βˆ‚0 βˆ‚0 NOUN cana-5460 174 3 c c PROPN cana-5460 174 4 t t PROPN cana-5460 174 5 Ξ΄u(x Ξ΄u(x PROPN cana-5460 174 6 , , PUNCT cana-5460 174 7 t t PROPN cana-5460 174 8 ) ) PUNCT cana-5460 174 9 βˆ’ βˆ’ PROPN cana-5460 174 10 Ξ± Ξ± PROPN cana-5460 174 11 βˆ‚2u βˆ‚2u PROPN cana-5460 174 12 βˆ‚x2 βˆ‚x2 PROPN cana-5460 174 13 βˆ’ βˆ’ PROPN cana-5460 174 14 Ξ² Ξ² X cana-5460 174 15 βˆ‚3u βˆ‚3u NOUN cana-5460 174 16 βˆ‚t βˆ‚t PROPN cana-5460 174 17 βˆ‚x2 βˆ‚x2 PROPN cana-5460 174 18 + + SYM cana-5460 174 19 Ξ³u Ξ³u PROPN cana-5460 174 20 , , PUNCT cana-5460 174 21 z z NOUN cana-5460 174 22 ) ) PUNCT cana-5460 174 23 dxdt dxdt NOUN cana-5460 174 24 = = SYM cana-5460 174 25 0 0 PROPN cana-5460 174 26 . . PUNCT cana-5460 175 1 Ο‰ Ο‰ NOUN cana-5460 175 2 ( ( PUNCT cana-5460 175 3 55 55 NUM cana-5460 175 4 ) ) PUNCT cana-5460 175 5 if if SCONJ cana-5460 175 6 letting let VERB cana-5460 175 7 : : PUNCT cana-5460 175 8 u(x u(x PROPN cana-5460 175 9 , , PUNCT cana-5460 175 10 t t NOUN cana-5460 175 11 ) ) PUNCT cana-5460 175 12 = = SYM cana-5460 176 1 β„‘t β„‘t NUM cana-5460 176 2 2ΞΌ 2ΞΌ NOUN cana-5460 176 3 = = SYM cana-5460 176 4 ∫ ∫ PROPN cana-5460 176 5 ∫ ∫ PROPN cana-5460 176 6 ΞΌ(x ΞΌ(x PROPN cana-5460 176 7 , , PUNCT cana-5460 176 8 ΞΎ)dΞΎds ΞΎ)dΞΎds PROPN cana-5460 176 9 s s PART cana-5460 176 10 0 0 NUM cana-5460 176 11 t t NOUN cana-5460 176 12 0 0 NUM cana-5460 176 13 , , PUNCT cana-5460 176 14 ( ( PUNCT cana-5460 176 15 56 56 NUM cana-5460 176 16 ) ) PUNCT cana-5460 176 17 where where SCONJ cana-5460 176 18 βˆ‚0 βˆ‚0 NOUN cana-5460 176 19 c c PROPN cana-5460 176 20 t t PROPN cana-5460 176 21 δμ δμ PROPN cana-5460 176 22 , , PUNCT cana-5460 176 23 βˆ‚2ΞΌ βˆ‚2ΞΌ ADJ cana-5460 176 24 βˆ‚x2 βˆ‚x2 PROPN cana-5460 176 25 , , PUNCT cana-5460 176 26 βˆ‚3ΞΌ βˆ‚3ΞΌ PROPN cana-5460 176 27 βˆ‚tβˆ‚x2 βˆ‚tβˆ‚x2 PROPN cana-5460 176 28 , , PUNCT cana-5460 176 29 ΞΌ ΞΌ PROPN cana-5460 176 30 ∈ ∈ PROPN cana-5460 176 31 l2(Ο‰ l2(Ο‰ NOUN cana-5460 176 32 ) ) PUNCT cana-5460 176 33 and and CCONJ cana-5460 176 34 it it PRON cana-5460 176 35 also also ADV cana-5460 176 36 satisfies satisfy VERB cana-5460 176 37 the the DET cana-5460 176 38 initial initial ADJ cana-5460 176 39 boundary boundary ADJ cana-5460 176 40 conditions condition NOUN cana-5460 176 41 ( ( PUNCT cana-5460 176 42 19 19 NUM cana-5460 176 43 ) ) PUNCT cana-5460 176 44 and and CCONJ cana-5460 176 45 ( ( PUNCT cana-5460 176 46 20 20 NUM cana-5460 176 47 ) ) PUNCT cana-5460 176 48 , , PUNCT cana-5460 176 49 we we PRON cana-5460 176 50 can can AUX cana-5460 176 51 write write VERB cana-5460 176 52 equation equation NOUN cana-5460 176 53 ( ( PUNCT cana-5460 176 54 55 55 NUM cana-5460 176 55 ) ) PUNCT cana-5460 176 56 as as ADP cana-5460 176 57 : : PUNCT cana-5460 176 58 ∫ ∫ PROPN cana-5460 176 59 ( ( PUNCT cana-5460 176 60 βˆ‚0 βˆ‚0 NOUN cana-5460 176 61 c c PROPN cana-5460 176 62 t t PROPN cana-5460 176 63 Ξ΄β„‘t Ξ΄β„‘t NOUN cana-5460 176 64 2ΞΌ 2ΞΌ NUM cana-5460 176 65 βˆ’ βˆ’ PROPN cana-5460 176 66 Ξ± Ξ± PROPN cana-5460 176 67 βˆ‚2β„‘t βˆ‚2β„‘t PROPN cana-5460 176 68 2ΞΌ 2ΞΌ NUM cana-5460 176 69 βˆ‚x2 βˆ‚x2 NOUN cana-5460 176 70 βˆ’ βˆ’ NOUN cana-5460 177 1 Ξ² Ξ² X cana-5460 177 2 βˆ‚3β„‘t βˆ‚3β„‘t NUM cana-5460 177 3 2ΞΌ 2ΞΌ NUM cana-5460 177 4 βˆ‚t βˆ‚t PROPN cana-5460 177 5 βˆ‚x2 βˆ‚x2 NOUN cana-5460 177 6 + + SYM cana-5460 177 7 Ξ³β„‘t Ξ³β„‘t NOUN cana-5460 177 8 2ΞΌ 2ΞΌ NOUN cana-5460 177 9 , , PUNCT cana-5460 177 10 z z NOUN cana-5460 177 11 ) ) PUNCT cana-5460 177 12 dxdt dxdt NOUN cana-5460 177 13 = = SYM cana-5460 177 14 0 0 PROPN cana-5460 177 15 . . PUNCT cana-5460 178 1 Ο‰ Ο‰ NOUN cana-5460 178 2 ( ( PUNCT cana-5460 178 3 57 57 NUM cana-5460 178 4 ) ) PUNCT cana-5460 178 5 we we PRON cana-5460 178 6 can can AUX cana-5460 178 7 express express VERB cana-5460 178 8 z z NOUN cana-5460 178 9 as as ADP cana-5460 178 10 a a DET cana-5460 178 11 function function NOUN cana-5460 178 12 of of ADP cana-5460 178 13 ΞΌ ΞΌ PROPN cana-5460 178 14 as as SCONJ cana-5460 178 15 follows follow VERB cana-5460 178 16 : : PUNCT cana-5460 178 17 z(x z(x NUM cana-5460 178 18 , , PUNCT cana-5460 178 19 t t PROPN cana-5460 178 20 ) ) PUNCT cana-5460 178 21 = = PUNCT cana-5460 179 1 β„‘tΞΌ β„‘tΞΌ NOUN cana-5460 179 2 βˆ’ βˆ’ NOUN cana-5460 179 3 β„‘x β„‘x PROPN cana-5460 179 4 2β„‘tΞΌ 2β„‘tΞΌ NUM cana-5460 179 5 . . PUNCT cana-5460 180 1 ( ( PUNCT cana-5460 180 2 58 58 X cana-5460 180 3 ) ) PUNCT cana-5460 180 4 we we PRON cana-5460 180 5 can can AUX cana-5460 180 6 then then ADV cana-5460 180 7 substitute substitute VERB cana-5460 180 8 ( ( PUNCT cana-5460 180 9 58 58 NUM cana-5460 180 10 ) ) PUNCT cana-5460 180 11 into into ADP cana-5460 180 12 ( ( PUNCT cana-5460 180 13 57 57 NUM cana-5460 180 14 ) ) PUNCT cana-5460 180 15 and and CCONJ cana-5460 180 16 do do VERB cana-5460 180 17 integrations integration NOUN cana-5460 180 18 by by ADP cana-5460 180 19 parts part NOUN cana-5460 180 20 on on ADP cana-5460 180 21 each each DET cana-5460 180 22 term term NOUN cana-5460 180 23 communications communication NOUN cana-5460 180 24 on on ADP cana-5460 180 25 applied apply VERB cana-5460 180 26 nonlinear nonlinear ADJ cana-5460 180 27 analysis analysis NOUN cana-5460 180 28 issn issn NOUN cana-5460 180 29 : : PUNCT cana-5460 180 30 1074 1074 NUM cana-5460 180 31 - - PUNCT cana-5460 180 32 133x 133x NUM cana-5460 180 33 vol vol NOUN cana-5460 180 34 32 32 NUM cana-5460 180 35 no.3 no.3 NOUN cana-5460 180 36 ( ( PUNCT cana-5460 180 37 2025 2025 NUM cana-5460 180 38 ) ) PUNCT cana-5460 180 39 950 950 NUM cana-5460 180 40 https://internationalpubls.com https://internationalpubls.com X cana-5460 180 41 ∫ ∫ PROPN cana-5460 180 42 ( ( PUNCT cana-5460 180 43 βˆ‚0 βˆ‚0 NOUN cana-5460 180 44 c c PROPN cana-5460 180 45 t t NOUN cana-5460 180 46 Ξ΄β„‘t Ξ΄β„‘t NOUN cana-5460 180 47 2ΞΌ 2ΞΌ NOUN cana-5460 180 48 . . PUNCT cana-5460 181 1 β„‘tΞΌ β„‘tΞΌ NOUN cana-5460 181 2 ) ) PUNCT cana-5460 181 3 Ο‰ Ο‰ NUM cana-5460 181 4 dxdt dxdt NOUN cana-5460 181 5 = = SYM cana-5460 181 6 ∫ ∫ PROPN cana-5460 181 7 ( ( PUNCT cana-5460 181 8 βˆ‚0 βˆ‚0 NOUN cana-5460 181 9 c c NOUN cana-5460 181 10 t t PROPN cana-5460 182 1 Ξ΄β„‘tΞΌ Ξ΄β„‘tΞΌ PROPN cana-5460 182 2 . . PUNCT cana-5460 183 1 β„‘tΞΌ β„‘tΞΌ NOUN cana-5460 183 2 ) ) PUNCT cana-5460 183 3 Ο‰ Ο‰ PROPN cana-5460 183 4 dxdt dxdt NOUN cana-5460 183 5 β‰₯ β‰₯ NOUN cana-5460 183 6 1 1 NUM cana-5460 183 7 2 2 NUM cana-5460 183 8 ∫ ∫ NOUN cana-5460 183 9 βˆ‚0 βˆ‚0 NOUN cana-5460 183 10 c c PROPN cana-5460 183 11 t t PROPN cana-5460 183 12 Ξ΄β€–β„‘tμ‖𝕃2(0,1 Ξ΄β€–β„‘tμ‖𝕃2(0,1 NOUN cana-5460 183 13 ) ) PUNCT cana-5460 183 14 Ο„ Ο„ PROPN cana-5460 183 15 0 0 NUM cana-5460 183 16 dt dt NOUN cana-5460 183 17 , , PUNCT cana-5460 183 18 ( ( PUNCT cana-5460 183 19 59 59 NUM cana-5460 183 20 ) ) PUNCT cana-5460 183 21 βˆ’βˆ« βˆ’βˆ« NOUN cana-5460 183 22 ( ( PUNCT cana-5460 183 23 βˆ‚0 βˆ‚0 NOUN cana-5460 183 24 c c PROPN cana-5460 183 25 t t NOUN cana-5460 183 26 Ξ΄β„‘t Ξ΄β„‘t NOUN cana-5460 183 27 2ΞΌ 2ΞΌ NOUN cana-5460 183 28 . . PUNCT cana-5460 184 1 β„‘x β„‘x PROPN cana-5460 184 2 2β„‘tΞΌ 2β„‘tΞΌ NUM cana-5460 184 3 ) ) PUNCT cana-5460 184 4 Ο‰ Ο‰ NUM cana-5460 184 5 dxdt dxdt NOUN cana-5460 184 6 = = SYM cana-5460 184 7 ∫ ∫ PROPN cana-5460 184 8 ∫ ∫ PROPN cana-5460 184 9 βˆ‚0 βˆ‚0 PROPN cana-5460 184 10 c c PROPN cana-5460 184 11 t t PROPN cana-5460 184 12 Ξ΄β„‘x(β„‘tΞΌ Ξ΄β„‘x(β„‘tΞΌ PROPN cana-5460 184 13 ) ) PUNCT cana-5460 184 14 . . PUNCT cana-5460 185 1 β„‘x(β„‘tΞΌ β„‘x(β„‘tΞΌ X cana-5460 185 2 ) ) PUNCT cana-5460 186 1 1 1 NUM cana-5460 186 2 0 0 NUM cana-5460 186 3 Ο„ Ο„ X cana-5460 186 4 0 0 NUM cana-5460 186 5 dxdt dxdt NOUN cana-5460 186 6 β‰₯ β‰₯ NOUN cana-5460 186 7 1 1 NUM cana-5460 186 8 2 2 NUM cana-5460 186 9 ∫ ∫ NOUN cana-5460 186 10 βˆ‚0 βˆ‚0 NOUN cana-5460 186 11 c c PROPN cana-5460 186 12 t t PROPN cana-5460 186 13 Ξ΄β€–β„‘x(β„‘tΞΌ)‖𝕃2(0,1 Ξ΄β€–β„‘x(β„‘tΞΌ)‖𝕃2(0,1 NOUN cana-5460 186 14 ) ) PUNCT cana-5460 186 15 Ο„ Ο„ X cana-5460 186 16 0 0 NUM cana-5460 186 17 , , PUNCT cana-5460 186 18 ( ( PUNCT cana-5460 186 19 60 60 NUM cana-5460 186 20 ) ) PUNCT cana-5460 186 21 βˆ’ βˆ’ NOUN cana-5460 187 1 ∫ ∫ PROPN cana-5460 187 2 ( ( PUNCT cana-5460 187 3 Ξ± Ξ± PROPN cana-5460 187 4 βˆ‚2(β„‘t βˆ‚2(β„‘t PROPN cana-5460 187 5 2ΞΌ 2ΞΌ NOUN cana-5460 187 6 ) ) PUNCT cana-5460 187 7 βˆ‚x2 βˆ‚x2 NOUN cana-5460 187 8 . . PUNCT cana-5460 187 9 β„‘tΞΌ)dxdtΟ‰ β„‘tΞΌ)dxdtΟ‰ PUNCT cana-5460 188 1 = = PUNCT cana-5460 188 2 βˆ’Ξ±βˆ« βˆ’Ξ±βˆ« PROPN cana-5460 188 3 ∫ ∫ PROPN cana-5460 188 4 ( ( PUNCT cana-5460 188 5 βˆ‚ βˆ‚ X cana-5460 188 6 βˆ‚x βˆ‚x PROPN cana-5460 188 7 ( ( PUNCT cana-5460 188 8 β„‘t β„‘t PROPN cana-5460 188 9 2ΞΌ 2ΞΌ NOUN cana-5460 188 10 ) ) PUNCT cana-5460 188 11 ) ) PUNCT cana-5460 188 12 2 2 NUM cana-5460 188 13 dx dx PROPN cana-5460 188 14 1 1 NUM cana-5460 188 15 0 0 NUM cana-5460 188 16 dt dt NOUN cana-5460 188 17 Ο„ Ο„ PROPN cana-5460 188 18 0 0 NUM cana-5460 188 19 , , PUNCT cana-5460 188 20 ( ( PUNCT cana-5460 188 21 61 61 NUM cana-5460 188 22 ) ) PUNCT cana-5460 188 23 ∫ ∫ PROPN cana-5460 188 24 ( ( PUNCT cana-5460 188 25 Ξ± Ξ± PROPN cana-5460 188 26 βˆ‚2(β„‘t βˆ‚2(β„‘t PROPN cana-5460 188 27 2ΞΌ 2ΞΌ NOUN cana-5460 188 28 ) ) PUNCT cana-5460 188 29 βˆ‚x2 βˆ‚x2 NOUN cana-5460 188 30 . . PUNCT cana-5460 189 1 β„‘x β„‘x PROPN cana-5460 189 2 2β„‘tΞΌ)dxdt 2β„‘tΞΌ)dxdt NUM cana-5460 189 3 = = SYM cana-5460 189 4 Ξ± Ξ± PROPN cana-5460 189 5 ∫ ∫ PROPN cana-5460 189 6 ∫ ∫ PROPN cana-5460 189 7 ( ( PUNCT cana-5460 189 8 β„‘t β„‘t PROPN cana-5460 189 9 2ΞΌ 2ΞΌ NOUN cana-5460 189 10 ) ) PUNCT cana-5460 189 11 21 21 NUM cana-5460 189 12 0 0 NUM cana-5460 189 13 Ο„ Ο„ PROPN cana-5460 189 14 0Ο‰ 0Ο‰ NOUN cana-5460 189 15 dxdt dxdt NOUN cana-5460 189 16 , , PUNCT cana-5460 189 17 ( ( PUNCT cana-5460 189 18 62 62 NUM cana-5460 189 19 ) ) PUNCT cana-5460 189 20 βˆ’βˆ« βˆ’βˆ« NOUN cana-5460 189 21 ( ( PUNCT cana-5460 189 22 Ξ² Ξ² X cana-5460 189 23 βˆ‚3β„‘t βˆ‚3β„‘t NUM cana-5460 189 24 2ΞΌ 2ΞΌ NUM cana-5460 189 25 βˆ‚t βˆ‚t PROPN cana-5460 189 26 βˆ‚x2 βˆ‚x2 NOUN cana-5460 189 27 . . PUNCT cana-5460 189 28 β„‘tΞΌ)dxdt β„‘tΞΌ)dxdt X cana-5460 190 1 = = NOUN cana-5460 190 2 Ο‰ Ο‰ NUM cana-5460 190 3 β∫ β∫ PROPN cana-5460 190 4 ∫ ∫ PROPN cana-5460 190 5 ( ( PUNCT cana-5460 190 6 βˆ‚ βˆ‚ X cana-5460 190 7 βˆ‚x βˆ‚x PROPN cana-5460 190 8 ( ( PUNCT cana-5460 190 9 β„‘tΞΌ β„‘tΞΌ PROPN cana-5460 190 10 ) ) PUNCT cana-5460 190 11 ) ) PUNCT cana-5460 190 12 2 2 NUM cana-5460 190 13 1 1 NUM cana-5460 190 14 0 0 NUM cana-5460 190 15 Ο„ Ο„ PROPN cana-5460 190 16 0 0 NUM cana-5460 190 17 dxdt dxdt NOUN cana-5460 190 18 , , PUNCT cana-5460 190 19 ( ( PUNCT cana-5460 190 20 63 63 NUM cana-5460 190 21 ) ) PUNCT cana-5460 190 22 βˆ’βˆ« βˆ’βˆ« NOUN cana-5460 190 23 ( ( PUNCT cana-5460 190 24 βˆ’Ξ² βˆ’Ξ² PROPN cana-5460 190 25 βˆ‚3β„‘t βˆ‚3β„‘t PROPN cana-5460 190 26 2ΞΌ 2ΞΌ NUM cana-5460 190 27 βˆ‚t βˆ‚t PROPN cana-5460 190 28 βˆ‚x2 βˆ‚x2 NOUN cana-5460 190 29 . . PUNCT cana-5460 191 1 β„‘x β„‘x PROPN cana-5460 191 2 2β„‘tΞΌ)dxdt 2β„‘tΞΌ)dxdt NUM cana-5460 191 3 = = SYM cana-5460 191 4 Ο‰ Ο‰ X cana-5460 191 5 β∫ β∫ PROPN cana-5460 191 6 ∫ ∫ PROPN cana-5460 191 7 ( ( PUNCT cana-5460 191 8 β„‘tΞΌ β„‘tΞΌ PROPN cana-5460 191 9 ) ) PUNCT cana-5460 191 10 21 21 NUM cana-5460 191 11 0 0 NUM cana-5460 191 12 Ο„ Ο„ PROPN cana-5460 191 13 0 0 NUM cana-5460 191 14 dxdt dxdt NOUN cana-5460 191 15 , , PUNCT cana-5460 191 16 ( ( PUNCT cana-5460 191 17 64 64 NUM cana-5460 191 18 ) ) PUNCT cana-5460 191 19 ∫ ∫ PROPN cana-5460 191 20 ( ( PUNCT cana-5460 191 21 Ξ³β„‘t Ξ³β„‘t NOUN cana-5460 191 22 2ΞΌ 2ΞΌ NOUN cana-5460 191 23 . . PUNCT cana-5460 192 1 β„‘tΞΌ)Ο‰ β„‘tΞΌ)Ο‰ PROPN cana-5460 192 2 dxdt dxdt NOUN cana-5460 192 3 = = SYM cana-5460 192 4 Ξ³ Ξ³ X cana-5460 192 5 2 2 NUM cana-5460 192 6 ∫ ∫ NOUN cana-5460 192 7 ∫ ∫ PROPN cana-5460 192 8 ( ( PUNCT cana-5460 192 9 β„‘t β„‘t PROPN cana-5460 192 10 2ΞΌ(x 2ΞΌ(x PROPN cana-5460 192 11 , , PUNCT cana-5460 192 12 t t PROPN cana-5460 192 13 ) ) PUNCT cana-5460 192 14 ) ) PUNCT cana-5460 193 1 21 21 NUM cana-5460 193 2 0 0 NUM cana-5460 193 3 Ο„ Ο„ PROPN cana-5460 193 4 0 0 NUM cana-5460 193 5 dxdt dxdt NOUN cana-5460 193 6 βˆ’ βˆ’ PROPN cana-5460 193 7 Ξ³ Ξ³ NOUN cana-5460 193 8 2 2 NUM cana-5460 193 9 ∫ ∫ NOUN cana-5460 193 10 ∫ ∫ PROPN cana-5460 193 11 Ο†2 Ο†2 PROPN cana-5460 193 12 1 1 NUM cana-5460 193 13 0 0 NUM cana-5460 193 14 Ο„ Ο„ PROPN cana-5460 193 15 0 0 NUM cana-5460 193 16 dxdt dxdt NOUN cana-5460 193 17 , , PUNCT cana-5460 193 18 ( ( PUNCT cana-5460 193 19 65 65 NUM cana-5460 193 20 ) ) PUNCT cana-5460 193 21 βˆ’βˆ« βˆ’βˆ« NOUN cana-5460 193 22 ( ( PUNCT cana-5460 193 23 Ξ³β„‘t Ξ³β„‘t NOUN cana-5460 193 24 2ΞΌ 2ΞΌ NOUN cana-5460 193 25 . . PUNCT cana-5460 194 1 β„‘x β„‘x PROPN cana-5460 194 2 2β„‘tΞΌ)Ο‰ 2β„‘tΞΌ)Ο‰ NUM cana-5460 194 3 dxdt dxdt NOUN cana-5460 194 4 = = PUNCT cana-5460 194 5 βˆ’ βˆ’ NOUN cana-5460 194 6 Ξ³ Ξ³ SYM cana-5460 194 7 2 2 NUM cana-5460 194 8 ∫ ∫ NOUN cana-5460 194 9 ∫ ∫ PROPN cana-5460 194 10 ( ( PUNCT cana-5460 194 11 β„‘x β„‘x PROPN cana-5460 194 12 2(β„‘t 2(β„‘t PROPN cana-5460 194 13 2ΞΌ(x 2ΞΌ(x PROPN cana-5460 194 14 , , PUNCT cana-5460 194 15 0 0 NUM cana-5460 194 16 ) ) PUNCT cana-5460 194 17 ) ) PUNCT cana-5460 194 18 ) ) PUNCT cana-5460 195 1 21 21 NUM cana-5460 195 2 0 0 NUM cana-5460 195 3 Ο„ Ο„ PROPN cana-5460 195 4 0 0 NUM cana-5460 195 5 dxdt dxdt NOUN cana-5460 195 6 + + CCONJ cana-5460 195 7 Ξ³ Ξ³ PROPN cana-5460 195 8 2 2 NUM cana-5460 195 9 ∫ ∫ NOUN cana-5460 195 10 ∫ ∫ PROPN cana-5460 195 11 ( ( PUNCT cana-5460 195 12 β„‘x β„‘x PROPN cana-5460 195 13 2(β„‘t 2(β„‘t PROPN cana-5460 195 14 2ΞΌ(x 2ΞΌ(x PROPN cana-5460 195 15 , , PUNCT cana-5460 195 16 0 0 NUM cana-5460 195 17 ) ) PUNCT cana-5460 195 18 ) ) PUNCT cana-5460 195 19 ) ) PUNCT cana-5460 196 1 2 2 NUM cana-5460 196 2 1 1 NUM cana-5460 196 3 0 0 NUM cana-5460 196 4 Ο„ Ο„ PROPN cana-5460 196 5 0 0 NUM cana-5460 196 6 dxdt dxdt NOUN cana-5460 196 7 = = SYM cana-5460 196 8 βˆ’ βˆ’ NOUN cana-5460 196 9 Ξ³ Ξ³ SYM cana-5460 196 10 2 2 NUM cana-5460 196 11 ∫ ∫ NOUN cana-5460 196 12 ∫ ∫ PROPN cana-5460 196 13 ( ( PUNCT cana-5460 196 14 β„‘t β„‘t PROPN cana-5460 196 15 2ΞΌ(x 2ΞΌ(x PROPN cana-5460 196 16 , , PUNCT cana-5460 196 17 t t PROPN cana-5460 196 18 ) ) PUNCT cana-5460 196 19 ) ) PUNCT cana-5460 196 20 21 21 NUM cana-5460 196 21 0 0 NUM cana-5460 196 22 Ο„ Ο„ PROPN cana-5460 196 23 0 0 NUM cana-5460 196 24 dxdt dxdt NOUN cana-5460 196 25 + + CCONJ cana-5460 196 26 Ξ³ Ξ³ PROPN cana-5460 196 27 2 2 NUM cana-5460 196 28 ∫ ∫ PROPN cana-5460 196 29 ∫ ∫ PROPN cana-5460 196 30 Ο†2 Ο†2 PROPN cana-5460 196 31 1 1 NUM cana-5460 196 32 0 0 NUM cana-5460 196 33 Ο„ Ο„ PROPN cana-5460 196 34 0 0 NUM cana-5460 196 35 dxdt dxdt NOUN cana-5460 196 36 . . PUNCT cana-5460 197 1 ( ( PUNCT cana-5460 197 2 66 66 NUM cana-5460 197 3 ) ) PUNCT cana-5460 197 4 by by ADP cana-5460 197 5 substituting substitute VERB cana-5460 197 6 ( ( PUNCT cana-5460 197 7 59)-(66 59)-(66 NUM cana-5460 197 8 ) ) PUNCT cana-5460 197 9 into into ADP cana-5460 197 10 ( ( PUNCT cana-5460 197 11 57 57 NUM cana-5460 197 12 ) ) PUNCT cana-5460 197 13 , , PUNCT cana-5460 197 14 we we PRON cana-5460 197 15 obtain obtain VERB cana-5460 197 16 : : PUNCT cana-5460 197 17 communications communication NOUN cana-5460 197 18 on on ADP cana-5460 197 19 applied apply VERB cana-5460 197 20 nonlinear nonlinear ADJ cana-5460 197 21 analysis analysis NOUN cana-5460 197 22 issn issn NOUN cana-5460 197 23 : : PUNCT cana-5460 197 24 1074 1074 NUM cana-5460 197 25 - - PUNCT cana-5460 197 26 133x 133x NUM cana-5460 197 27 vol vol NOUN cana-5460 197 28 32 32 NUM cana-5460 197 29 no.3 no.3 NOUN cana-5460 197 30 ( ( PUNCT cana-5460 197 31 2025 2025 NUM cana-5460 197 32 ) ) PUNCT cana-5460 198 1 951 951 NUM cana-5460 198 2 https://internationalpubls.com https://internationalpubls.com X cana-5460 198 3 dt dt X cana-5460 198 4 Ξ΄βˆ’1β€–β„‘tΞΌβ€– Ξ΄βˆ’1β€–β„‘tΞΌβ€– VERB cana-5460 198 5 2 2 NUM cana-5460 198 6 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 198 7 ) ) PUNCT cana-5460 199 1 + + NUM cana-5460 199 2 dt dt PUNCT cana-5460 199 3 Ξ΄βˆ’1β€–β„‘xβ„‘tΞΌβ€– Ξ΄βˆ’1β€–β„‘xβ„‘tΞΌβ€– PRON cana-5460 199 4 2 2 NUM cana-5460 199 5 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 199 6 ) ) PUNCT cana-5460 200 1 + + CCONJ cana-5460 201 1 ∫ ∫ PROPN cana-5460 201 2 ∫ ∫ PROPN cana-5460 201 3 ( ( PUNCT cana-5460 201 4 βˆ‚ βˆ‚ NUM cana-5460 201 5 βˆ‚x βˆ‚x PROPN cana-5460 201 6 ( ( PUNCT cana-5460 201 7 β„‘tΞΌ β„‘tΞΌ PROPN cana-5460 201 8 ) ) PUNCT cana-5460 201 9 ) ) PUNCT cana-5460 201 10 2 2 NUM cana-5460 201 11 1 1 NUM cana-5460 201 12 0 0 NUM cana-5460 201 13 Ο„ Ο„ PROPN cana-5460 201 14 0 0 NUM cana-5460 201 15 dxdt dxdt NOUN cana-5460 201 16 + + CCONJ cana-5460 201 17 ∫ ∫ PROPN cana-5460 201 18 ∫ ∫ PROPN cana-5460 201 19 ( ( PUNCT cana-5460 201 20 β„‘tΞΌ β„‘tΞΌ PROPN cana-5460 201 21 ) ) PUNCT cana-5460 201 22 2 2 NUM cana-5460 201 23 1 1 NUM cana-5460 201 24 0 0 NUM cana-5460 201 25 Ο„ Ο„ X cana-5460 201 26 0 0 NUM cana-5460 201 27 dxdt dxdt NOUN cana-5460 201 28 ≀ ≀ PROPN cana-5460 201 29 ρ1 ρ1 NOUN cana-5460 201 30 ( ( PUNCT cana-5460 201 31 ∫ ∫ PROPN cana-5460 201 32 ∫ ∫ PROPN cana-5460 201 33 ( ( PUNCT cana-5460 201 34 βˆ‚ βˆ‚ NUM cana-5460 201 35 βˆ‚x βˆ‚x PROPN cana-5460 201 36 ( ( PUNCT cana-5460 201 37 β„‘t β„‘t PROPN cana-5460 201 38 2ΞΌ 2ΞΌ NOUN cana-5460 201 39 ) ) PUNCT cana-5460 201 40 ) ) PUNCT cana-5460 201 41 2 2 NUM cana-5460 201 42 dx dx PROPN cana-5460 201 43 1 1 NUM cana-5460 201 44 0 0 NUM cana-5460 201 45 dt dt NOUN cana-5460 201 46 Ο„ Ο„ PROPN cana-5460 201 47 0 0 NUM cana-5460 202 1 + + NUM cana-5460 202 2 ∫ ∫ PROPN cana-5460 202 3 ∫ ∫ PROPN cana-5460 202 4 ( ( PUNCT cana-5460 202 5 β„‘t β„‘t PROPN cana-5460 202 6 2ΞΌ 2ΞΌ NOUN cana-5460 202 7 ) ) PUNCT cana-5460 202 8 21 21 NUM cana-5460 202 9 0 0 NUM cana-5460 202 10 Ο„ Ο„ PROPN cana-5460 202 11 0 0 NUM cana-5460 202 12 dxdt dxdt NOUN cana-5460 202 13 ) ) PUNCT cana-5460 202 14 , , PUNCT cana-5460 202 15 ( ( PUNCT cana-5460 202 16 67 67 NUM cana-5460 202 17 ) ) PUNCT cana-5460 202 18 where where SCONJ cana-5460 202 19 : : PUNCT cana-5460 202 20 ρ1 ρ1 NOUN cana-5460 202 21 = = SYM cana-5460 202 22 1 1 NUM cana-5460 202 23 ( ( PUNCT cana-5460 202 24 1,2Ξ² 1,2Ξ² NUM cana-5460 202 25 , , PUNCT cana-5460 202 26 Ξ± Ξ± NOUN cana-5460 202 27 ) ) PUNCT cana-5460 202 28 . . PUNCT cana-5460 203 1 ( ( PUNCT cana-5460 203 2 68 68 NUM cana-5460 203 3 ) ) PUNCT cana-5460 203 4 we we PRON cana-5460 203 5 apply apply VERB cana-5460 203 6 lemma lemma PROPN cana-5460 203 7 ( ( PUNCT cana-5460 203 8 2.3 2.3 NUM cana-5460 203 9 ) ) PUNCT cana-5460 203 10 and and CCONJ cana-5460 203 11 we we PRON cana-5460 203 12 obtain obtain VERB cana-5460 203 13 : : PUNCT cana-5460 203 14 Ο†1(t Ο†1(t X cana-5460 203 15 ) ) PUNCT cana-5460 203 16 = = SYM cana-5460 203 17 ∫ ∫ PROPN cana-5460 203 18 ∫ ∫ PROPN cana-5460 203 19 ( ( PUNCT cana-5460 203 20 βˆ‚ βˆ‚ NUM cana-5460 203 21 βˆ‚x βˆ‚x PROPN cana-5460 203 22 ( ( PUNCT cana-5460 203 23 β„‘t β„‘t PROPN cana-5460 203 24 2ΞΌ 2ΞΌ NOUN cana-5460 203 25 ) ) PUNCT cana-5460 203 26 ) ) PUNCT cana-5460 203 27 2 2 NUM cana-5460 203 28 dx dx PROPN cana-5460 203 29 1 1 NUM cana-5460 203 30 0 0 NUM cana-5460 203 31 dt dt NOUN cana-5460 203 32 Ο„ Ο„ PROPN cana-5460 203 33 0 0 NUM cana-5460 203 34 ; ; PUNCT cana-5460 203 35 βˆ‚Ο†1(t βˆ‚Ο†1(t PROPN cana-5460 203 36 ) ) PUNCT cana-5460 203 37 βˆ‚t βˆ‚t PROPN cana-5460 203 38 = = SYM cana-5460 203 39 ∫ ∫ PROPN cana-5460 203 40 ( ( PUNCT cana-5460 203 41 βˆ‚ βˆ‚ NOUN cana-5460 203 42 βˆ‚x βˆ‚x PROPN cana-5460 203 43 ( ( PUNCT cana-5460 203 44 β„‘t β„‘t PROPN cana-5460 203 45 2ΞΌ 2ΞΌ NOUN cana-5460 203 46 ) ) PUNCT cana-5460 203 47 ) ) PUNCT cana-5460 203 48 2 2 NUM cana-5460 203 49 dx dx PROPN cana-5460 203 50 1 1 NUM cana-5460 203 51 0 0 NUM cana-5460 203 52 ; ; PUNCT cana-5460 203 53 Ο†1(0 Ο†1(0 PROPN cana-5460 203 54 ) ) PUNCT cana-5460 203 55 = = SYM cana-5460 203 56 0 0 NUM cana-5460 203 57 , , PUNCT cana-5460 203 58 ( ( PUNCT cana-5460 203 59 69 69 NUM cana-5460 203 60 ) ) PUNCT cana-5460 203 61 so so ADV cana-5460 203 62 Ο†1(t Ο†1(t X cana-5460 203 63 ) ) PUNCT cana-5460 203 64 ≀ ≀ NOUN cana-5460 203 65 ρ1te ρ1te PUNCT cana-5460 203 66 tρ1 tρ1 PROPN cana-5460 203 67 ∫ ∫ PROPN cana-5460 203 68 ∫ ∫ PROPN cana-5460 203 69 ( ( PUNCT cana-5460 203 70 ( ( PUNCT cana-5460 203 71 β„‘t β„‘t PROPN cana-5460 203 72 2ΞΌ 2ΞΌ NOUN cana-5460 203 73 ) ) PUNCT cana-5460 203 74 ) ) PUNCT cana-5460 203 75 2 2 NUM cana-5460 203 76 dx dx PROPN cana-5460 203 77 1 1 NUM cana-5460 203 78 0 0 NUM cana-5460 203 79 dt dt NOUN cana-5460 203 80 Ο„ Ο„ PROPN cana-5460 203 81 0 0 NUM cana-5460 203 82 , , PUNCT cana-5460 203 83 ( ( PUNCT cana-5460 203 84 70 70 NUM cana-5460 203 85 ) ) PUNCT cana-5460 203 86 then then ADV cana-5460 203 87 , , PUNCT cana-5460 203 88 equation equation NOUN cana-5460 203 89 ( ( PUNCT cana-5460 203 90 67 67 NUM cana-5460 203 91 ) ) PUNCT cana-5460 203 92 can can AUX cana-5460 203 93 be be AUX cana-5460 203 94 transformed transform VERB cana-5460 203 95 as as SCONJ cana-5460 203 96 follows follow VERB cana-5460 203 97 : : PUNCT cana-5460 203 98 dt dt PUNCT cana-5460 203 99 Ξ΄βˆ’1β€–β„‘tΞΌβ€– Ξ΄βˆ’1β€–β„‘tΞΌβ€– VERB cana-5460 203 100 2 2 NUM cana-5460 203 101 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 203 102 ) ) PUNCT cana-5460 204 1 + + NUM cana-5460 204 2 dt dt PUNCT cana-5460 204 3 Ξ΄βˆ’1β€–β„‘xβ„‘tΞΌβ€– Ξ΄βˆ’1β€–β„‘xβ„‘tΞΌβ€– PRON cana-5460 204 4 2 2 NUM cana-5460 204 5 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 204 6 ) ) PUNCT cana-5460 205 1 + + CCONJ cana-5460 206 1 ∫ ∫ PROPN cana-5460 206 2 ∫ ∫ PROPN cana-5460 206 3 ( ( PUNCT cana-5460 206 4 βˆ‚ βˆ‚ NUM cana-5460 206 5 βˆ‚x βˆ‚x PROPN cana-5460 206 6 ( ( PUNCT cana-5460 206 7 β„‘tΞΌ β„‘tΞΌ PROPN cana-5460 206 8 ) ) PUNCT cana-5460 206 9 ) ) PUNCT cana-5460 206 10 2 2 NUM cana-5460 206 11 1 1 NUM cana-5460 206 12 0 0 NUM cana-5460 206 13 Ο„ Ο„ PROPN cana-5460 206 14 0 0 NUM cana-5460 206 15 dxdt dxdt NOUN cana-5460 206 16 + + CCONJ cana-5460 206 17 ∫ ∫ PROPN cana-5460 206 18 ∫ ∫ PROPN cana-5460 206 19 ( ( PUNCT cana-5460 206 20 β„‘tΞΌ β„‘tΞΌ PROPN cana-5460 206 21 ) ) PUNCT cana-5460 206 22 2 2 NUM cana-5460 206 23 1 1 NUM cana-5460 206 24 0 0 NUM cana-5460 206 25 Ο„ Ο„ X cana-5460 206 26 0 0 NUM cana-5460 206 27 dxdt dxdt NOUN cana-5460 206 28 ≀ ≀ ADJ cana-5460 206 29 ρ2 ρ2 NOUN cana-5460 206 30 ( ( PUNCT cana-5460 206 31 ∫ ∫ PROPN cana-5460 206 32 ∫ ∫ PROPN cana-5460 206 33 ( ( PUNCT cana-5460 206 34 ( ( PUNCT cana-5460 206 35 β„‘t β„‘t PROPN cana-5460 206 36 2ΞΌ 2ΞΌ NOUN cana-5460 206 37 ) ) PUNCT cana-5460 206 38 ) ) PUNCT cana-5460 206 39 2 2 NUM cana-5460 206 40 dx dx PROPN cana-5460 206 41 1 1 NUM cana-5460 206 42 0 0 NUM cana-5460 206 43 dt dt NOUN cana-5460 206 44 Ο„ Ο„ PROPN cana-5460 206 45 0 0 NUM cana-5460 206 46 ) ) PUNCT cana-5460 206 47 , , PUNCT cana-5460 206 48 ( ( PUNCT cana-5460 206 49 71 71 NUM cana-5460 206 50 ) ) PUNCT cana-5460 206 51 such such ADJ cana-5460 206 52 that that SCONJ cana-5460 206 53 : : PUNCT cana-5460 206 54 ρ2 ρ2 NOUN cana-5460 206 55 = = SYM cana-5460 206 56 max(ρ1 max(ρ1 PROPN cana-5460 206 57 2tetρ1 2tetρ1 NUM cana-5460 206 58 , , PUNCT cana-5460 206 59 ρ1 ρ1 NOUN cana-5460 206 60 ) ) PUNCT cana-5460 206 61 . . PUNCT cana-5460 207 1 ( ( PUNCT cana-5460 207 2 72 72 X cana-5460 207 3 ) ) PUNCT cana-5460 207 4 applying apply VERB cana-5460 207 5 lemma lemma PROPN cana-5460 207 6 ( ( PUNCT cana-5460 207 7 2.3 2.3 NUM cana-5460 207 8 ) ) PUNCT cana-5460 207 9 , , PUNCT cana-5460 207 10 we we PRON cana-5460 207 11 arrive arrive VERB cana-5460 207 12 at at ADP cana-5460 207 13 equation equation NOUN cana-5460 207 14 Ο†2(t Ο†2(t NUM cana-5460 207 15 ) ) PUNCT cana-5460 207 16 = = SYM cana-5460 208 1 ∫ ∫ PROPN cana-5460 208 2 ∫ ∫ PROPN cana-5460 208 3 ( ( PUNCT cana-5460 208 4 β„‘t β„‘t PROPN cana-5460 208 5 2ΞΌ 2ΞΌ NOUN cana-5460 208 6 ) ) PUNCT cana-5460 208 7 2 2 NUM cana-5460 208 8 dx dx PROPN cana-5460 208 9 1 1 NUM cana-5460 208 10 0 0 NUM cana-5460 208 11 dt dt NOUN cana-5460 208 12 Ο„ Ο„ PROPN cana-5460 208 13 0 0 NUM cana-5460 208 14 ; ; PUNCT cana-5460 208 15 βˆ‚Ο†2(t βˆ‚Ο†2(t PROPN cana-5460 208 16 ) ) PUNCT cana-5460 208 17 βˆ‚t βˆ‚t PROPN cana-5460 208 18 = = SYM cana-5460 208 19 ∫ ∫ PROPN cana-5460 208 20 ( ( PUNCT cana-5460 208 21 β„‘t β„‘t PROPN cana-5460 208 22 2ΞΌ 2ΞΌ NOUN cana-5460 208 23 ) ) PUNCT cana-5460 208 24 2 2 NUM cana-5460 208 25 dx dx PROPN cana-5460 208 26 1 1 NUM cana-5460 208 27 0 0 NUM cana-5460 208 28 ; ; PUNCT cana-5460 208 29 Ο†2(0 Ο†2(0 ADV cana-5460 208 30 ) ) PUNCT cana-5460 208 31 = = SYM cana-5460 208 32 0 0 NUM cana-5460 208 33 , , PUNCT cana-5460 208 34 ( ( PUNCT cana-5460 208 35 73 73 NUM cana-5460 208 36 ) ) PUNCT cana-5460 208 37 so so ADV cana-5460 208 38 : : PUNCT cana-5460 208 39 Ο†2(t Ο†2(t NUM cana-5460 208 40 ) ) PUNCT cana-5460 208 41 = = SYM cana-5460 209 1 ∫ ∫ PROPN cana-5460 209 2 ∫ ∫ PROPN cana-5460 209 3 ( ( PUNCT cana-5460 209 4 β„‘t β„‘t PROPN cana-5460 209 5 2ΞΌ 2ΞΌ NOUN cana-5460 209 6 ) ) PUNCT cana-5460 209 7 2 2 NUM cana-5460 209 8 dx dx PROPN cana-5460 209 9 1 1 NUM cana-5460 209 10 0 0 NUM cana-5460 209 11 dt dt NOUN cana-5460 209 12 Ο„ Ο„ PROPN cana-5460 209 13 0 0 NUM cana-5460 209 14 ≀ ≀ NUM cana-5460 209 15 0 0 NUM cana-5460 209 16 . . PUNCT cana-5460 210 1 ( ( PUNCT cana-5460 210 2 74 74 NUM cana-5460 210 3 ) ) PUNCT cana-5460 210 4 hence hence ADV cana-5460 210 5 , , PUNCT cana-5460 210 6 ( ( PUNCT cana-5460 210 7 71 71 NUM cana-5460 210 8 ) ) PUNCT cana-5460 210 9 is be AUX cana-5460 210 10 transformed transform VERB cana-5460 210 11 into into ADP cana-5460 210 12 dt dt PUNCT cana-5460 210 13 Ξ΄βˆ’1β€–β„‘tΞΌβ€– Ξ΄βˆ’1β€–β„‘tΞΌβ€– PROPN cana-5460 210 14 2 2 NUM cana-5460 210 15 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 210 16 ) ) PUNCT cana-5460 211 1 + + NUM cana-5460 211 2 dt dt PUNCT cana-5460 211 3 Ξ΄βˆ’1β€–β„‘xβ„‘tΞΌβ€– Ξ΄βˆ’1β€–β„‘xβ„‘tΞΌβ€– PRON cana-5460 211 4 2 2 NUM cana-5460 211 5 𝕃2(0,1 𝕃2(0,1 NOUN cana-5460 211 6 ) ) PUNCT cana-5460 212 1 + + CCONJ cana-5460 212 2 communications communication NOUN cana-5460 212 3 on on ADP cana-5460 212 4 applied apply VERB cana-5460 212 5 nonlinear nonlinear ADJ cana-5460 212 6 analysis analysis NOUN cana-5460 212 7 issn issn NOUN cana-5460 212 8 : : PUNCT cana-5460 212 9 1074 1074 NUM cana-5460 212 10 - - PUNCT cana-5460 212 11 133x 133x NUM cana-5460 212 12 vol vol NOUN cana-5460 212 13 32 32 NUM cana-5460 212 14 no.3 no.3 NOUN cana-5460 212 15 ( ( PUNCT cana-5460 212 16 2025 2025 NUM cana-5460 212 17 ) ) PUNCT cana-5460 212 18 952 952 NUM cana-5460 213 1 https://internationalpubls.com https://internationalpubls.com X cana-5460 213 2 ∫ ∫ PROPN cana-5460 213 3 ∫ ∫ PROPN cana-5460 213 4 ( ( PUNCT cana-5460 213 5 βˆ‚ βˆ‚ NUM cana-5460 213 6 βˆ‚x βˆ‚x PROPN cana-5460 213 7 ( ( PUNCT cana-5460 213 8 β„‘tΞΌ β„‘tΞΌ PROPN cana-5460 213 9 ) ) PUNCT cana-5460 213 10 ) ) PUNCT cana-5460 213 11 2 2 NUM cana-5460 213 12 1 1 NUM cana-5460 213 13 0 0 NUM cana-5460 213 14 Ο„ Ο„ PROPN cana-5460 213 15 0 0 NUM cana-5460 213 16 dxdt dxdt NOUN cana-5460 213 17 + + CCONJ cana-5460 213 18 ∫ ∫ PROPN cana-5460 213 19 ∫ ∫ PROPN cana-5460 213 20 ( ( PUNCT cana-5460 213 21 β„‘tΞΌ β„‘tΞΌ PROPN cana-5460 213 22 ) ) PUNCT cana-5460 213 23 21 21 NUM cana-5460 213 24 0 0 NUM cana-5460 213 25 Ο„ Ο„ X cana-5460 213 26 0 0 NUM cana-5460 213 27 dxdt dxdt NOUN cana-5460 213 28 ≀ ≀ NOUN cana-5460 213 29 0 0 NUM cana-5460 213 30 . . PUNCT cana-5460 214 1 ( ( PUNCT cana-5460 214 2 75 75 NUM cana-5460 214 3 ) ) PUNCT cana-5460 214 4 setting set VERB cana-5460 214 5 : : PUNCT cana-5460 214 6 ΞΌ=0 ΞΌ=0 PUNCT cana-5460 214 7 in in ADP cana-5460 214 8 ( ( PUNCT cana-5460 214 9 58 58 NUM cana-5460 214 10 ) ) PUNCT cana-5460 214 11 we we PRON cana-5460 214 12 conclude conclude VERB cana-5460 214 13 that that PRON cana-5460 214 14 : : PUNCT cana-5460 214 15 z z X cana-5460 214 16 = = SYM cana-5460 214 17 0 0 NUM cana-5460 214 18 inl2(Ο‰ inl2(Ο‰ NUM cana-5460 214 19 ) ) PUNCT cana-5460 214 20 . . PUNCT cana-5460 215 1 since since SCONJ cana-5460 215 2 z z PROPN cana-5460 215 3 ∈ ∈ PROPN cana-5460 215 4 r(l r(l NOUN cana-5460 215 5 ) ) PUNCT cana-5460 215 6 ⏊ ⏊ PROPN cana-5460 215 7 , , PUNCT cana-5460 215 8 so so ADV cana-5460 215 9 r(l r(l NOUN cana-5460 215 10 ) ) PUNCT cana-5460 215 11 ⏊ ⏊ PROPN cana-5460 215 12 = = PUNCT cana-5460 215 13 { { PUNCT cana-5460 215 14 0 0 NUM cana-5460 215 15 } } PUNCT cana-5460 215 16 . . PUNCT cana-5460 216 1 this this PRON cana-5460 216 2 proves prove VERB cana-5460 216 3 that that SCONJ cana-5460 216 4 r(l)Μ…Μ… r(l)Μ…Μ… PROPN cana-5460 216 5 Μ…Μ… Μ…Μ… PROPN cana-5460 216 6 Μ…Μ… Μ…Μ… PROPN cana-5460 217 1 = = PROPN cana-5460 217 2 f. f. PROPN cana-5460 217 3 this this PRON cana-5460 217 4 completes complete VERB cana-5460 217 5 the the DET cana-5460 217 6 proof proof NOUN cana-5460 217 7 . . PUNCT cana-5460 218 1 ∎ ∎ PROPN cana-5460 218 2 7 7 NUM cana-5460 218 3 . . PUNCT cana-5460 219 1 the the DET cana-5460 219 2 study study NOUN cana-5460 219 3 of of ADP cana-5460 219 4 the the DET cana-5460 219 5 nonlinear nonlinear ADJ cana-5460 219 6 problem problem NOUN cana-5460 219 7 this this DET cana-5460 219 8 section section NOUN cana-5460 219 9 is be AUX cana-5460 219 10 devoted devote VERB cana-5460 219 11 to to ADP cana-5460 219 12 solving solve VERB cana-5460 219 13 the the DET cana-5460 219 14 main main ADJ cana-5460 219 15 problems problem NOUN cana-5460 219 16 ( ( PUNCT cana-5460 219 17 15)–(17 15)–(17 NUM cana-5460 219 18 ) ) PUNCT cana-5460 219 19 . . PUNCT cana-5460 220 1 consider consider VERB cana-5460 220 2 now now ADV cana-5460 220 3 the the DET cana-5460 220 4 auxiliary auxiliary ADJ cana-5460 220 5 problem problem NOUN cana-5460 220 6 with with ADP cana-5460 220 7 the the DET cana-5460 220 8 homogenous homogenous ADJ cana-5460 220 9 equation equation NOUN cana-5460 220 10 : : PUNCT cana-5460 221 1 β„’u β„’u NOUN cana-5460 221 2 = = PUNCT cana-5460 221 3 βˆ‚0 βˆ‚0 NOUN cana-5460 221 4 c c PROPN cana-5460 221 5 t t PROPN cana-5460 221 6 Ξ΄u(x Ξ΄u(x PROPN cana-5460 221 7 , , PUNCT cana-5460 221 8 t t PROPN cana-5460 221 9 ) ) PUNCT cana-5460 221 10 βˆ’ βˆ’ PROPN cana-5460 221 11 Ξ± Ξ± PROPN cana-5460 221 12 βˆ‚2u βˆ‚2u PROPN cana-5460 221 13 βˆ‚x2 βˆ‚x2 PROPN cana-5460 221 14 βˆ’ βˆ’ PROPN cana-5460 221 15 Ξ² Ξ² X cana-5460 221 16 βˆ‚3u βˆ‚3u ADJ cana-5460 221 17 βˆ‚tβˆ‚x2 βˆ‚tβˆ‚x2 NOUN cana-5460 221 18 + + NUM cana-5460 221 19 Ξ³u Ξ³u INTJ cana-5460 221 20 βˆ’ βˆ’ PROPN cana-5460 221 21 ∫ ∫ NOUN cana-5460 221 22 a(t a(t NOUN cana-5460 221 23 βˆ’ βˆ’ NOUN cana-5460 221 24 s)u(x s)u(x NOUN cana-5460 221 25 , , PUNCT cana-5460 221 26 s)ds s)ds PROPN cana-5460 221 27 = = SYM cana-5460 221 28 0 0 NUM cana-5460 221 29 t t NOUN cana-5460 221 30 0 0 NUM cana-5460 221 31 , , PUNCT cana-5460 221 32 ( ( PUNCT cana-5460 221 33 76 76 NUM cana-5460 221 34 ) ) PUNCT cana-5460 221 35 β„“u β„“u NOUN cana-5460 221 36 = = SYM cana-5460 221 37 u(x u(x PROPN cana-5460 221 38 , , PUNCT cana-5460 221 39 0 0 NUM cana-5460 221 40 ) ) PUNCT cana-5460 221 41 = = SYM cana-5460 221 42 Ο†(x Ο†(x NOUN cana-5460 221 43 ) ) PUNCT cana-5460 221 44 , , PUNCT cana-5460 221 45 qu qu PROPN cana-5460 221 46 = = PUNCT cana-5460 221 47 βˆ‚u(x,0 βˆ‚u(x,0 PROPN cana-5460 221 48 ) ) PUNCT cana-5460 221 49 βˆ‚t βˆ‚t PROPN cana-5460 221 50 = = PUNCT cana-5460 221 51 ψ(x ψ(x PROPN cana-5460 221 52 ) ) PUNCT cana-5460 221 53 , , PUNCT cana-5460 221 54 0 0 PUNCT cana-5460 221 55 < < X cana-5460 221 56 π‘₯ π‘₯ X cana-5460 221 57 < < X cana-5460 221 58 1 1 NUM cana-5460 221 59 . . PUNCT cana-5460 222 1 ( ( PUNCT cana-5460 222 2 77 77 NUM cana-5460 222 3 ) ) PUNCT cana-5460 222 4 ∫ ∫ PROPN cana-5460 222 5 u(x u(x PROPN cana-5460 222 6 , , PUNCT cana-5460 222 7 t)dx t)dx PROPN cana-5460 222 8 = = SYM cana-5460 222 9 0 0 NUM cana-5460 222 10 , , PUNCT cana-5460 222 11 t t PROPN cana-5460 222 12 0 0 NUM cana-5460 222 13 ∫ ∫ PROPN cana-5460 222 14 xu(x xu(x NOUN cana-5460 222 15 , , PUNCT cana-5460 222 16 t)dx t)dx PROPN cana-5460 222 17 = = SYM cana-5460 222 18 0 0 NUM cana-5460 222 19 , , PUNCT cana-5460 222 20 t t PROPN cana-5460 222 21 0 0 NUM cana-5460 222 22 0 0 NUM cana-5460 222 23 < < X cana-5460 222 24 𝑑 𝑑 PROPN cana-5460 222 25 ≀ ≀ NOUN cana-5460 222 26 𝑇 𝑇 PROPN cana-5460 222 27 . . PUNCT cana-5460 223 1 ( ( PUNCT cana-5460 223 2 78 78 NUM cana-5460 223 3 ) ) PUNCT cana-5460 223 4 if if SCONJ cana-5460 223 5 v v NOUN cana-5460 223 6 and and CCONJ cana-5460 223 7 u u NOUN cana-5460 223 8 are be AUX cana-5460 223 9 solutions solution NOUN cana-5460 223 10 of of ADP cana-5460 223 11 problems problem NOUN cana-5460 223 12 ( ( PUNCT cana-5460 223 13 18)-(20),(21)-(23 18)-(20),(21)-(23 NOUN cana-5460 223 14 ) ) PUNCT cana-5460 223 15 , , PUNCT cana-5460 223 16 respectively respectively ADV cana-5460 223 17 , , PUNCT cana-5460 223 18 then then ADV cana-5460 223 19 h h NOUN cana-5460 223 20 = = PUNCT cana-5460 223 21 u u PROPN cana-5460 223 22 βˆ’ βˆ’ PROPN cana-5460 223 23 v v NUM cana-5460 223 24 satisfies satisfie NOUN cana-5460 223 25 β„’w β„’w PROPN cana-5460 223 26 = = PUNCT cana-5460 223 27 βˆ‚0 βˆ‚0 NOUN cana-5460 223 28 c c PROPN cana-5460 223 29 t t PROPN cana-5460 223 30 Ξ΄w(x Ξ΄w(x PROPN cana-5460 223 31 , , PUNCT cana-5460 223 32 t t PROPN cana-5460 223 33 ) ) PUNCT cana-5460 223 34 βˆ’ βˆ’ NOUN cana-5460 224 1 Ξ± Ξ± PRON cana-5460 224 2 βˆ‚2w βˆ‚2w VERB cana-5460 224 3 βˆ‚x2 βˆ‚x2 NOUN cana-5460 224 4 βˆ’ βˆ’ NOUN cana-5460 224 5 Ξ² Ξ² X cana-5460 224 6 βˆ‚3w βˆ‚3w NOUN cana-5460 224 7 βˆ‚tβˆ‚x2 βˆ‚tβˆ‚x2 PROPN cana-5460 225 1 + + CCONJ cana-5460 225 2 Ξ³w Ξ³w NUM cana-5460 225 3 βˆ’ βˆ’ NOUN cana-5460 225 4 ∫ ∫ PROPN cana-5460 225 5 a(t a(t NOUN cana-5460 225 6 βˆ’ βˆ’ PROPN cana-5460 225 7 s)w(x s)w(x ADJ cana-5460 225 8 , , PUNCT cana-5460 225 9 s)ds s)ds PROPN cana-5460 225 10 = = SYM cana-5460 225 11 Ο‡(x Ο‡(x PROPN cana-5460 225 12 , , PUNCT cana-5460 225 13 t t PROPN cana-5460 225 14 , , PUNCT cana-5460 225 15 w w PROPN cana-5460 225 16 , , PUNCT cana-5460 225 17 βˆ‚w βˆ‚w PROPN cana-5460 225 18 βˆ‚x βˆ‚x PROPN cana-5460 225 19 t t PROPN cana-5460 225 20 0 0 NUM cana-5460 225 21 ) ) PUNCT cana-5460 225 22 , , PUNCT cana-5460 225 23 ( ( PUNCT cana-5460 225 24 79 79 NUM cana-5460 225 25 ) ) PUNCT cana-5460 225 26 β„“w β„“w NOUN cana-5460 226 1 = = SYM cana-5460 226 2 w(x w(x PROPN cana-5460 226 3 , , PUNCT cana-5460 226 4 0 0 NUM cana-5460 226 5 ) ) PUNCT cana-5460 226 6 = = SYM cana-5460 226 7 Ο†(x Ο†(x NOUN cana-5460 226 8 ) ) PUNCT cana-5460 226 9 , , PUNCT cana-5460 226 10 qw qw X cana-5460 226 11 = = SYM cana-5460 226 12 βˆ‚w(x,0 βˆ‚w(x,0 PROPN cana-5460 226 13 ) ) PUNCT cana-5460 227 1 βˆ‚t βˆ‚t PROPN cana-5460 227 2 = = PUNCT cana-5460 227 3 ψ(x ψ(x PROPN cana-5460 227 4 ) ) PUNCT cana-5460 227 5 , , PUNCT cana-5460 227 6 0 0 PUNCT cana-5460 227 7 < < X cana-5460 227 8 π‘₯ π‘₯ X cana-5460 227 9 < < X cana-5460 227 10 1 1 NUM cana-5460 227 11 . . PUNCT cana-5460 227 12 ( ( PUNCT cana-5460 227 13 80 80 NUM cana-5460 227 14 ) ) PUNCT cana-5460 227 15 ∫ ∫ PROPN cana-5460 228 1 w(x w(x NOUN cana-5460 228 2 , , PUNCT cana-5460 228 3 t)dx t)dx PROPN cana-5460 228 4 = = SYM cana-5460 228 5 0 0 NUM cana-5460 228 6 , , PUNCT cana-5460 228 7 t t PROPN cana-5460 228 8 0 0 NUM cana-5460 228 9 ∫ ∫ PROPN cana-5460 228 10 xw(x xw(x PROPN cana-5460 228 11 , , PUNCT cana-5460 228 12 t)dx t)dx PROPN cana-5460 228 13 = = SYM cana-5460 228 14 0 0 NUM cana-5460 228 15 , , PUNCT cana-5460 228 16 t t PROPN cana-5460 228 17 0 0 NUM cana-5460 228 18 0 0 NUM cana-5460 228 19 < < X cana-5460 228 20 𝑑 𝑑 PROPN cana-5460 228 21 ≀ ≀ NOUN cana-5460 228 22 𝑇 𝑇 PROPN cana-5460 228 23 . . PUNCT cana-5460 229 1 ( ( PUNCT cana-5460 229 2 81 81 NUM cana-5460 229 3 ) ) PUNCT cana-5460 229 4 such such ADJ cana-5460 229 5 that that SCONJ cana-5460 229 6 the the DET cana-5460 229 7 function function NOUN cana-5460 229 8 Ο‡ Ο‡ X cana-5460 229 9 ( ( PUNCT cana-5460 229 10 x x PROPN cana-5460 229 11 , , PUNCT cana-5460 229 12 t t PROPN cana-5460 229 13 , , PUNCT cana-5460 229 14 w w PROPN cana-5460 229 15 , , PUNCT cana-5460 229 16 βˆ‚w βˆ‚w PROPN cana-5460 229 17 βˆ‚x βˆ‚x PROPN cana-5460 229 18 ) ) PUNCT cana-5460 230 1 = = SYM cana-5460 230 2 Ο‡ Ο‡ X cana-5460 230 3 ( ( PUNCT cana-5460 230 4 x x PROPN cana-5460 230 5 , , PUNCT cana-5460 230 6 t t PROPN cana-5460 230 7 , , PUNCT cana-5460 230 8 w w PROPN cana-5460 230 9 + + NUM cana-5460 230 10 u u NOUN cana-5460 230 11 , , PUNCT cana-5460 230 12 βˆ‚w βˆ‚w PROPN cana-5460 230 13 βˆ‚x βˆ‚x PROPN cana-5460 230 14 + + CCONJ cana-5460 230 15 βˆ‚u βˆ‚u PROPN cana-5460 230 16 βˆ‚x βˆ‚x PROPN cana-5460 230 17 ) ) PUNCT cana-5460 230 18 , , PUNCT cana-5460 230 19 verifies verify VERB cana-5460 230 20 the the DET cana-5460 230 21 following follow VERB cana-5460 230 22 condition condition NOUN cana-5460 230 23 : : PUNCT cana-5460 230 24 |Ο‡(x |Ο‡(x PROPN cana-5460 230 25 , , PUNCT cana-5460 230 26 t t PROPN cana-5460 230 27 , , PUNCT cana-5460 230 28 w1 w1 NOUN cana-5460 230 29 , , PUNCT cana-5460 230 30 y1 y1 PROPN cana-5460 230 31 ) ) PUNCT cana-5460 230 32 βˆ’ βˆ’ PROPN cana-5460 231 1 Ο‡(x Ο‡(x PROPN cana-5460 231 2 , , PUNCT cana-5460 231 3 t t PROPN cana-5460 231 4 , , PUNCT cana-5460 231 5 w2 w2 NOUN cana-5460 231 6 , , PUNCT cana-5460 231 7 y2)| y2)| PROPN cana-5460 231 8 ≀ ≀ PROPN cana-5460 231 9 m(|w1 m(|w1 PROPN cana-5460 231 10 βˆ’ βˆ’ PUNCT cana-5460 231 11 w2| w2| X cana-5460 232 1 + + CCONJ cana-5460 232 2 |y1 |y1 AUX cana-5460 232 3 βˆ’ βˆ’ PROPN cana-5460 232 4 y2| y2| NOUN cana-5460 232 5 ) ) PUNCT cana-5460 232 6 , , PUNCT cana-5460 232 7 βˆ€ βˆ€ X cana-5460 232 8 ( ( PUNCT cana-5460 232 9 x x NOUN cana-5460 232 10 , , PUNCT cana-5460 232 11 t t PROPN cana-5460 232 12 ) ) PUNCT cana-5460 232 13 ∈ ∈ PROPN cana-5460 232 14 β„š β„š PROPN cana-5460 232 15 ( ( PUNCT cana-5460 232 16 82 82 NUM cana-5460 232 17 ) ) PUNCT cana-5460 232 18 now now ADV cana-5460 232 19 we we PRON cana-5460 232 20 will will AUX cana-5460 232 21 show show VERB cana-5460 232 22 that that SCONJ cana-5460 232 23 the the DET cana-5460 232 24 solution solution NOUN cana-5460 232 25 of of ADP cana-5460 232 26 problems problem NOUN cana-5460 232 27 ( ( PUNCT cana-5460 232 28 79)-(81 79)-(81 ADV cana-5460 232 29 ) ) PUNCT cana-5460 232 30 is be AUX cana-5460 232 31 unique unique ADJ cana-5460 232 32 . . PUNCT cana-5460 233 1 we we PRON cana-5460 233 2 will will AUX cana-5460 233 3 establish establish VERB cana-5460 233 4 a a DET cana-5460 233 5 similar similar ADJ cana-5460 233 6 proof proof NOUN cana-5460 233 7 for for ADP cana-5460 233 8 problems problem NOUN cana-5460 233 9 ( ( PUNCT cana-5460 233 10 21)-(23 21)-(23 NOUN cana-5460 233 11 ) ) PUNCT cana-5460 233 12 . . PUNCT cana-5460 234 1 first first ADV cana-5460 234 2 we we PRON cana-5460 234 3 introduce introduce VERB cana-5460 234 4 the the DET cana-5460 234 5 following following ADJ cana-5460 234 6 space space NOUN cana-5460 234 7 : : PUNCT cana-5460 234 8 β€²c1(β„š)= β€²c1(β„š)= NOUN cana-5460 234 9 { { PUNCT cana-5460 234 10 w w PROPN cana-5460 234 11 ∈ ∈ PROPN cana-5460 234 12 c1(β„š c1(β„š NOUN cana-5460 234 13 ) ) PUNCT cana-5460 234 14 , , PUNCT cana-5460 234 15 βˆ‚w2 βˆ‚w2 PROPN cana-5460 234 16 βˆ‚t βˆ‚t PROPN cana-5460 234 17 βˆ‚x2 βˆ‚x2 PROPN cana-5460 234 18 ∈ ∈ PROPN cana-5460 234 19 c(β„š c(β„š PROPN cana-5460 234 20 ) ) PUNCT cana-5460 234 21 } } PUNCT cana-5460 234 22 ( ( PUNCT cana-5460 234 23 83 83 NUM cana-5460 234 24 ) ) PUNCT cana-5460 234 25 we we PRON cana-5460 234 26 suppose suppose VERB cana-5460 234 27 that that SCONJ cana-5460 234 28 : : PUNCT cana-5460 234 29 w w X cana-5460 234 30 , , PUNCT cana-5460 234 31 u u PROPN cana-5460 234 32 ∈ ∈ NOUN cana-5460 234 33 β€²c1(β„š β€²c1(β„š NOUN cana-5460 234 34 ) ) PUNCT cana-5460 234 35 verify verify VERB cana-5460 234 36 homogenous homogenous ADJ cana-5460 234 37 initiale initiale NOUN cana-5460 234 38 and and CCONJ cana-5460 234 39 boundary boundary ADJ cana-5460 234 40 conditions condition NOUN cana-5460 234 41 , , PUNCT cana-5460 234 42 we we PRON cana-5460 234 43 have have VERB cana-5460 234 44 : : PUNCT cana-5460 234 45 communications communication NOUN cana-5460 234 46 on on ADP cana-5460 234 47 applied apply VERB cana-5460 234 48 nonlinear nonlinear ADJ cana-5460 234 49 analysis analysis NOUN cana-5460 234 50 issn issn NOUN cana-5460 234 51 : : PUNCT cana-5460 234 52 1074 1074 NUM cana-5460 234 53 - - PUNCT cana-5460 234 54 133x 133x NUM cana-5460 234 55 vol vol NOUN cana-5460 234 56 32 32 NUM cana-5460 234 57 no.3 no.3 NOUN cana-5460 234 58 ( ( PUNCT cana-5460 234 59 2025 2025 NUM cana-5460 234 60 ) ) PUNCT cana-5460 234 61 953 953 NUM cana-5460 234 62 https://internationalpubls.com https://internationalpubls.com X cana-5460 234 63 ( ( PUNCT cana-5460 234 64 β„’w,β„‘xu β„’w,β„‘xu PROPN cana-5460 234 65 ) ) PUNCT cana-5460 234 66 = = SYM cana-5460 234 67 ( ( PUNCT cana-5460 234 68 βˆ‚0 βˆ‚0 NOUN cana-5460 234 69 c c PROPN cana-5460 234 70 t t PROPN cana-5460 234 71 Ξ΄w,β„‘xu)𝕃2(Ο‰ Ξ΄w,β„‘xu)𝕃2(Ο‰ PROPN cana-5460 234 72 ) ) PUNCT cana-5460 235 1 βˆ’ βˆ’ PROPN cana-5460 235 2 Ξ± Ξ± INTJ cana-5460 235 3 ( ( PUNCT cana-5460 235 4 βˆ‚2w βˆ‚2w ADP cana-5460 235 5 βˆ‚x2 βˆ‚x2 NOUN cana-5460 235 6 , , PUNCT cana-5460 235 7 β„‘xu β„‘xu ADJ cana-5460 235 8 ) ) PUNCT cana-5460 235 9 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 235 10 ) ) PUNCT cana-5460 235 11 βˆ’ βˆ’ NOUN cana-5460 236 1 Ξ² Ξ² X cana-5460 236 2 ( ( PUNCT cana-5460 236 3 βˆ‚3w βˆ‚3w NOUN cana-5460 236 4 βˆ‚tβˆ‚x2 βˆ‚tβˆ‚x2 PROPN cana-5460 236 5 , , PUNCT cana-5460 236 6 β„‘xu β„‘xu ADJ cana-5460 236 7 ) ) PUNCT cana-5460 236 8 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 236 9 ) ) PUNCT cana-5460 236 10 + + CCONJ cana-5460 236 11 Ξ³(w,β„‘xu)𝕃2(Ο‰ Ξ³(w,β„‘xu)𝕃2(Ο‰ NUM cana-5460 236 12 ) ) PUNCT cana-5460 236 13 βˆ’ βˆ’ PROPN cana-5460 237 1 ( ( PUNCT cana-5460 237 2 ∫ ∫ PROPN cana-5460 237 3 a(t a(t NOUN cana-5460 237 4 βˆ’ βˆ’ PROPN cana-5460 237 5 t t PROPN cana-5460 237 6 0 0 NUM cana-5460 237 7 s)w(x s)w(x ADJ cana-5460 237 8 , , PUNCT cana-5460 237 9 s)ds s)ds PROPN cana-5460 237 10 , , PUNCT cana-5460 237 11 β„‘xu β„‘xu ADJ cana-5460 237 12 ) ) PUNCT cana-5460 237 13 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 237 14 ) ) PUNCT cana-5460 237 15 . . PUNCT cana-5460 238 1 ( ( PUNCT cana-5460 238 2 84 84 NUM cana-5460 238 3 ) ) PUNCT cana-5460 239 1 where where SCONJ cana-5460 239 2 : : PUNCT cana-5460 239 3 ( ( PUNCT cana-5460 239 4 βˆ‚0 βˆ‚0 NOUN cana-5460 239 5 c c PROPN cana-5460 239 6 t t NOUN cana-5460 239 7 Ξ΄w,β„‘xu)𝕃2(Ο‰ Ξ΄w,β„‘xu)𝕃2(Ο‰ PROPN cana-5460 239 8 ) ) PUNCT cana-5460 239 9 = = SYM cana-5460 240 1 βˆ’ βˆ’ PROPN cana-5460 240 2 ( ( PUNCT cana-5460 240 3 βˆ‚0 βˆ‚0 NOUN cana-5460 240 4 c c PROPN cana-5460 240 5 t t PROPN cana-5460 240 6 Ξ΄β„‘xw Ξ΄β„‘xw PROPN cana-5460 240 7 , , PUNCT cana-5460 240 8 u)𝕃2(Ο‰ u)𝕃2(Ο‰ NUM cana-5460 240 9 ) ) PUNCT cana-5460 240 10 ( ( PUNCT cana-5460 240 11 85 85 NUM cana-5460 240 12 ) ) PUNCT cana-5460 240 13 βˆ’Ξ± βˆ’Ξ± NOUN cana-5460 240 14 ( ( PUNCT cana-5460 240 15 βˆ‚2w βˆ‚2w NOUN cana-5460 240 16 βˆ‚x2 βˆ‚x2 NOUN cana-5460 240 17 , , PUNCT cana-5460 240 18 β„‘xu β„‘xu ADJ cana-5460 240 19 ) ) PUNCT cana-5460 240 20 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 240 21 ) ) PUNCT cana-5460 240 22 = = SYM cana-5460 240 23 βˆ’Ξ± βˆ’Ξ± NOUN cana-5460 240 24 ( ( PUNCT cana-5460 240 25 βˆ‚ βˆ‚ NOUN cana-5460 240 26 βˆ‚x βˆ‚x PROPN cana-5460 240 27 ( ( PUNCT cana-5460 240 28 βˆ‚w βˆ‚w PROPN cana-5460 240 29 βˆ‚x βˆ‚x PROPN cana-5460 240 30 ) ) PUNCT cana-5460 240 31 , , PUNCT cana-5460 240 32 β„‘xu β„‘xu ADJ cana-5460 240 33 ) ) PUNCT cana-5460 240 34 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 240 35 ) ) PUNCT cana-5460 240 36 = = SYM cana-5460 241 1 Ξ± Ξ± PROPN cana-5460 241 2 ( ( PUNCT cana-5460 241 3 βˆ‚w βˆ‚w PROPN cana-5460 241 4 βˆ‚x βˆ‚x PROPN cana-5460 241 5 , , PUNCT cana-5460 241 6 u u NOUN cana-5460 241 7 ) ) PUNCT cana-5460 241 8 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 241 9 ) ) PUNCT cana-5460 241 10 ( ( PUNCT cana-5460 241 11 86 86 NUM cana-5460 241 12 ) ) PUNCT cana-5460 241 13 βˆ’Ξ² βˆ’Ξ² PROPN cana-5460 241 14 ( ( PUNCT cana-5460 241 15 βˆ‚3w βˆ‚3w NOUN cana-5460 241 16 βˆ‚tβˆ‚x2 βˆ‚tβˆ‚x2 PROPN cana-5460 241 17 , , PUNCT cana-5460 241 18 β„‘xu β„‘xu ADJ cana-5460 241 19 ) ) PUNCT cana-5460 241 20 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 241 21 ) ) PUNCT cana-5460 242 1 = = SYM cana-5460 242 2 βˆ’Ξ² βˆ’Ξ² NOUN cana-5460 242 3 ( ( PUNCT cana-5460 242 4 βˆ‚ βˆ‚ X cana-5460 242 5 βˆ‚t βˆ‚t PROPN cana-5460 242 6 ( ( PUNCT cana-5460 242 7 βˆ‚w βˆ‚w PROPN cana-5460 242 8 βˆ‚x βˆ‚x PROPN cana-5460 242 9 ) ) PUNCT cana-5460 242 10 , , PUNCT cana-5460 242 11 u u NOUN cana-5460 242 12 ) ) PUNCT cana-5460 242 13 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 242 14 ) ) PUNCT cana-5460 242 15 ( ( PUNCT cana-5460 242 16 87 87 NUM cana-5460 242 17 ) ) PUNCT cana-5460 242 18 βˆ’(∫ βˆ’(∫ NOUN cana-5460 242 19 a(t a(t PROPN cana-5460 242 20 βˆ’ βˆ’ PROPN cana-5460 242 21 s)w(x s)w(x ADJ cana-5460 242 22 , , PUNCT cana-5460 242 23 s)ds s)ds PROPN cana-5460 242 24 t t PROPN cana-5460 242 25 0 0 NUM cana-5460 242 26 , , PUNCT cana-5460 242 27 β„‘xu β„‘xu ADJ cana-5460 242 28 ) ) PUNCT cana-5460 242 29 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 242 30 ) ) PUNCT cana-5460 242 31 = = PUNCT cana-5460 243 1 ( ( PUNCT cana-5460 243 2 ∫ ∫ PROPN cana-5460 243 3 a(t a(t NOUN cana-5460 243 4 βˆ’ βˆ’ NOUN cana-5460 243 5 s)β„‘xw(x s)β„‘xw(x NOUN cana-5460 243 6 , , PUNCT cana-5460 243 7 s)ds s)ds PROPN cana-5460 243 8 t t PROPN cana-5460 243 9 0 0 NUM cana-5460 243 10 , , PUNCT cana-5460 243 11 u u NOUN cana-5460 243 12 ) ) PUNCT cana-5460 243 13 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 243 14 ) ) PUNCT cana-5460 243 15 . . PUNCT cana-5460 244 1 ( ( PUNCT cana-5460 244 2 88 88 NUM cana-5460 244 3 ) ) PUNCT cana-5460 244 4 we we PRON cana-5460 244 5 obtain obtain VERB cana-5460 244 6 : : PUNCT cana-5460 244 7 βˆ’ βˆ’ PROPN cana-5460 244 8 ( ( PUNCT cana-5460 244 9 βˆ‚0 βˆ‚0 NOUN cana-5460 244 10 c c PROPN cana-5460 244 11 t t PROPN cana-5460 244 12 Ξ΄β„‘xw Ξ΄β„‘xw PROPN cana-5460 244 13 , , PUNCT cana-5460 244 14 u)𝕃2 u)𝕃2 NOUN cana-5460 244 15 + + X cana-5460 244 16 Ξ± Ξ± PROPN cana-5460 244 17 ( ( PUNCT cana-5460 244 18 βˆ‚w βˆ‚w PROPN cana-5460 244 19 βˆ‚x βˆ‚x PROPN cana-5460 244 20 , , PUNCT cana-5460 244 21 u u NOUN cana-5460 244 22 ) ) PUNCT cana-5460 244 23 𝕃2 𝕃2 NOUN cana-5460 244 24 + + CCONJ cana-5460 245 1 Ξ² Ξ² X cana-5460 245 2 ( ( PUNCT cana-5460 245 3 βˆ‚ βˆ‚ X cana-5460 245 4 βˆ‚t βˆ‚t PROPN cana-5460 245 5 ( ( PUNCT cana-5460 245 6 βˆ‚w βˆ‚w PROPN cana-5460 245 7 βˆ‚x βˆ‚x PROPN cana-5460 245 8 ) ) PUNCT cana-5460 245 9 , , PUNCT cana-5460 245 10 u u NOUN cana-5460 245 11 ) ) PUNCT cana-5460 245 12 𝕃2 𝕃2 NOUN cana-5460 246 1 + + CCONJ cana-5460 246 2 ( ( PUNCT cana-5460 246 3 ∫ ∫ PROPN cana-5460 246 4 a(t a(t NOUN cana-5460 246 5 βˆ’ βˆ’ NOUN cana-5460 246 6 s)β„‘xw(x s)β„‘xw(x NOUN cana-5460 246 7 , , PUNCT cana-5460 246 8 s)ds s)ds PROPN cana-5460 246 9 t t PROPN cana-5460 246 10 0 0 NUM cana-5460 246 11 , , PUNCT cana-5460 246 12 u u NOUN cana-5460 246 13 ) ) PUNCT cana-5460 246 14 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 246 15 ) ) PUNCT cana-5460 246 16 = = PRON cana-5460 246 17 ( ( PUNCT cana-5460 246 18 u u NOUN cana-5460 246 19 , , PUNCT cana-5460 246 20 β„‘xΟ‡)𝕃2(Ο‰)(89 β„‘xΟ‡)𝕃2(Ο‰)(89 NUM cana-5460 246 21 ) ) PUNCT cana-5460 246 22 such such ADJ cana-5460 246 23 that that SCONJ cana-5460 246 24 : : PUNCT cana-5460 246 25 ΞΊ(w ΞΊ(w NOUN cana-5460 246 26 , , PUNCT cana-5460 246 27 u u NOUN cana-5460 246 28 ) ) PUNCT cana-5460 246 29 = = SYM cana-5460 246 30 ( ( PUNCT cana-5460 246 31 u u NOUN cana-5460 246 32 , , PUNCT cana-5460 246 33 β„‘xΟ‡)𝕃2(Ο‰ β„‘xΟ‡)𝕃2(Ο‰ ADV cana-5460 246 34 ) ) PUNCT cana-5460 246 35 ( ( PUNCT cana-5460 246 36 90 90 NUM cana-5460 246 37 ) ) PUNCT cana-5460 246 38 definition definition NOUN cana-5460 246 39 : : PUNCT cana-5460 246 40 a a DET cana-5460 246 41 function function NOUN cana-5460 246 42 w w NOUN cana-5460 246 43 ∈ ∈ NOUN cana-5460 246 44 𝕃2(0 𝕃2(0 NOUN cana-5460 246 45 , , PUNCT cana-5460 246 46 t t PROPN cana-5460 246 47 , , PUNCT cana-5460 246 48 h1(Ο‰ h1(Ο‰ PROPN cana-5460 246 49 ) ) PUNCT cana-5460 246 50 ) ) PUNCT cana-5460 246 51 is be AUX cana-5460 246 52 considered consider VERB cana-5460 246 53 as as ADP cana-5460 246 54 the the DET cana-5460 246 55 weak weak ADJ cana-5460 246 56 solution solution NOUN cana-5460 246 57 of of ADP cana-5460 246 58 the the DET cana-5460 246 59 problem problem NOUN cana-5460 246 60 ( ( PUNCT cana-5460 246 61 79)-(81 79)-(81 X cana-5460 246 62 ) ) PUNCT cana-5460 246 63 if if SCONJ cana-5460 246 64 it it PRON cana-5460 246 65 satisfies satisfy VERB cana-5460 246 66 ( ( PUNCT cana-5460 246 67 89 89 NUM cana-5460 246 68 ) ) PUNCT cana-5460 246 69 and and CCONJ cana-5460 246 70 ( ( PUNCT cana-5460 246 71 90 90 NUM cana-5460 246 72 ) ) PUNCT cana-5460 246 73 holds hold VERB cana-5460 246 74 . . PUNCT cana-5460 247 1 we we PRON cana-5460 247 2 will will AUX cana-5460 247 3 construct construct VERB cana-5460 247 4 an an DET cana-5460 247 5 iteration iteration NOUN cana-5460 247 6 sequence sequence NOUN cana-5460 247 7 as as SCONJ cana-5460 247 8 follows follow VERB cana-5460 247 9 , , PUNCT cana-5460 247 10 let let VERB cana-5460 247 11 : : PUNCT cana-5460 247 12 w(0 w(0 PROPN cana-5460 247 13 ) ) PUNCT cana-5460 247 14 = = SYM cana-5460 247 15 0 0 NUM cana-5460 247 16 , , PUNCT cana-5460 247 17 and and CCONJ cana-5460 247 18 ( ( PUNCT cana-5460 247 19 w(n w(n NOUN cana-5460 247 20 ) ) PUNCT cana-5460 247 21 ) ) PUNCT cana-5460 248 1 n n CCONJ cana-5460 248 2 ∈ ∈ PROPN cana-5460 248 3 β„• β„• PROPN cana-5460 248 4 , , PUNCT cana-5460 248 5 if if SCONJ cana-5460 248 6 w(nβˆ’1 w(nβˆ’1 X cana-5460 248 7 ) ) PUNCT cana-5460 248 8 is be AUX cana-5460 248 9 given give VERB cana-5460 248 10 , , PUNCT cana-5460 248 11 then then ADV cana-5460 248 12 for for ADP cana-5460 248 13 n n PRON cana-5460 248 14 ∈ ∈ PROPN cana-5460 248 15 β„• β„• PROPN cana-5460 248 16 solve solve VERB cana-5460 248 17 the the DET cana-5460 248 18 following following ADJ cana-5460 248 19 problem problem NOUN cana-5460 248 20 : : PUNCT cana-5460 248 21 communications communication NOUN cana-5460 248 22 on on ADP cana-5460 248 23 applied apply VERB cana-5460 248 24 nonlinear nonlinear ADJ cana-5460 248 25 analysis analysis NOUN cana-5460 248 26 issn issn NOUN cana-5460 248 27 : : PUNCT cana-5460 248 28 1074 1074 NUM cana-5460 248 29 - - PUNCT cana-5460 248 30 133x 133x NUM cana-5460 248 31 vol vol NOUN cana-5460 248 32 32 32 NUM cana-5460 248 33 no.3 no.3 NOUN cana-5460 248 34 ( ( PUNCT cana-5460 248 35 2025 2025 NUM cana-5460 248 36 ) ) PUNCT cana-5460 248 37 954 954 NUM cana-5460 248 38 https://internationalpubls.com https://internationalpubls.com X cana-5460 248 39 β„’w(n β„’w(n ADJ cana-5460 248 40 ) ) PUNCT cana-5460 248 41 = = SYM cana-5460 248 42 βˆ‚ βˆ‚ NUM cana-5460 248 43 0 0 NUM cana-5460 248 44 c c PROPN cana-5460 248 45 t t PROPN cana-5460 248 46 Ξ΄w(n Ξ΄w(n PUNCT cana-5460 248 47 ) ) PUNCT cana-5460 248 48 βˆ’ βˆ’ PROPN cana-5460 248 49 Ξ± Ξ± PRON cana-5460 248 50 βˆ‚2w(n βˆ‚2w(n NOUN cana-5460 248 51 ) ) PUNCT cana-5460 248 52 βˆ‚x2 βˆ‚x2 NOUN cana-5460 248 53 βˆ’ βˆ’ NOUN cana-5460 248 54 Ξ² Ξ² X cana-5460 248 55 βˆ‚3w(n βˆ‚3w(n X cana-5460 248 56 ) ) PUNCT cana-5460 248 57 βˆ‚tβˆ‚x2 βˆ‚tβˆ‚x2 NOUN cana-5460 248 58 + + NUM cana-5460 248 59 Ξ³w(n Ξ³w(n PUNCT cana-5460 248 60 ) ) PUNCT cana-5460 248 61 βˆ’ βˆ’ NUM cana-5460 249 1 ∫ ∫ PROPN cana-5460 249 2 a(t a(t NOUN cana-5460 249 3 βˆ’ βˆ’ NUM cana-5460 249 4 s)w(n)(x s)w(n)(x PROPN cana-5460 249 5 , , PUNCT cana-5460 249 6 s)ds s)ds PROPN cana-5460 249 7 = = SYM cana-5460 249 8 Ο‡(x Ο‡(x PROPN cana-5460 249 9 , , PUNCT cana-5460 249 10 t t PROPN cana-5460 249 11 , , PUNCT cana-5460 249 12 w(nβˆ’1 w(nβˆ’1 PROPN cana-5460 249 13 ) ) PUNCT cana-5460 249 14 , , PUNCT cana-5460 249 15 βˆ‚w(nβˆ’1 βˆ‚w(nβˆ’1 PROPN cana-5460 249 16 ) ) PUNCT cana-5460 249 17 βˆ‚x βˆ‚x PROPN cana-5460 249 18 t t NOUN cana-5460 249 19 0 0 NUM cana-5460 249 20 ) ) PUNCT cana-5460 249 21 , , PUNCT cana-5460 249 22 ( ( PUNCT cana-5460 249 23 91 91 NUM cana-5460 249 24 ) ) PUNCT cana-5460 249 25 β„“w(n β„“w(n NOUN cana-5460 249 26 ) ) PUNCT cana-5460 249 27 = = SYM cana-5460 249 28 w(n)(x w(n)(x PROPN cana-5460 249 29 , , PUNCT cana-5460 249 30 0 0 NUM cana-5460 249 31 ) ) PUNCT cana-5460 249 32 = = SYM cana-5460 249 33 0 0 NUM cana-5460 249 34 , , PUNCT cana-5460 249 35 qw(n qw(n PUNCT cana-5460 249 36 ) ) PUNCT cana-5460 249 37 = = SYM cana-5460 250 1 βˆ‚w(n)(x,0 βˆ‚w(n)(x,0 PROPN cana-5460 250 2 ) ) PUNCT cana-5460 251 1 βˆ‚t βˆ‚t PROPN cana-5460 251 2 = = SYM cana-5460 251 3 0 0 PROPN cana-5460 251 4 , , PUNCT cana-5460 251 5 0 0 NUM cana-5460 251 6 < < X cana-5460 251 7 π‘₯ π‘₯ X cana-5460 251 8 < < X cana-5460 251 9 1 1 NUM cana-5460 251 10 . . PUNCT cana-5460 252 1 ( ( PUNCT cana-5460 252 2 92 92 NUM cana-5460 252 3 ) ) PUNCT cana-5460 252 4 ∫ ∫ PROPN cana-5460 252 5 w(n)(x w(n)(x PROPN cana-5460 252 6 , , PUNCT cana-5460 252 7 t)dx t)dx PROPN cana-5460 252 8 = = SYM cana-5460 252 9 0 0 NUM cana-5460 252 10 , , PUNCT cana-5460 252 11 t t PROPN cana-5460 252 12 0 0 NUM cana-5460 252 13 ∫ ∫ PROPN cana-5460 252 14 xw(n)(x xw(n)(x PROPN cana-5460 252 15 , , PUNCT cana-5460 252 16 t)dx t)dx PROPN cana-5460 252 17 = = SYM cana-5460 252 18 0 0 NUM cana-5460 252 19 , , PUNCT cana-5460 252 20 t t PROPN cana-5460 252 21 0 0 NUM cana-5460 252 22 0 0 NUM cana-5460 252 23 < < X cana-5460 252 24 𝑑 𝑑 PROPN cana-5460 252 25 ≀ ≀ NOUN cana-5460 252 26 𝑇 𝑇 PROPN cana-5460 252 27 . . PUNCT cana-5460 253 1 ( ( PUNCT cana-5460 253 2 93 93 NUM cana-5460 253 3 ) ) PUNCT cana-5460 253 4 theorem theorem VERB cana-5460 253 5 : : PUNCT cana-5460 253 6 for for ADP cana-5460 253 7 each each DET cana-5460 253 8 fixed fix VERB cana-5460 253 9 n n CCONJ cana-5460 253 10 assure assure VERB cana-5460 253 11 that that SCONJ cana-5460 253 12 the the DET cana-5460 253 13 solution solution NOUN cana-5460 253 14 of of ADP cana-5460 253 15 problem problem NOUN cana-5460 253 16 ( ( PUNCT cana-5460 253 17 91)-(93 91)-(93 NOUN cana-5460 253 18 ) ) PUNCT cana-5460 253 19 , , PUNCT cana-5460 253 20 w(n)(x w(n)(x PROPN cana-5460 253 21 , , PUNCT cana-5460 253 22 t t PROPN cana-5460 253 23 ) ) PUNCT cana-5460 253 24 is be AUX cana-5460 253 25 unique unique ADJ cana-5460 253 26 . . PUNCT cana-5460 254 1 we we PRON cana-5460 254 2 put put VERB cana-5460 254 3 : : PUNCT cana-5460 254 4 w(n)(x w(n)(x PROPN cana-5460 254 5 , , PUNCT cana-5460 254 6 t t PROPN cana-5460 254 7 ) ) PUNCT cana-5460 254 8 = = SYM cana-5460 254 9 w(n+1)(x w(n+1)(x PROPN cana-5460 254 10 , , PUNCT cana-5460 254 11 t t PROPN cana-5460 254 12 ) ) PUNCT cana-5460 254 13 βˆ’ βˆ’ PROPN cana-5460 255 1 w(n)(x w(n)(x PROPN cana-5460 255 2 , , PUNCT cana-5460 255 3 t t PROPN cana-5460 255 4 ) ) PUNCT cana-5460 255 5 , , PUNCT cana-5460 255 6 we we PRON cana-5460 255 7 obtain obtain VERB cana-5460 255 8 : : PUNCT cana-5460 255 9 β„’w(n β„’w(n ADJ cana-5460 255 10 ) ) PUNCT cana-5460 255 11 = = SYM cana-5460 255 12 βˆ‚ βˆ‚ NUM cana-5460 255 13 0 0 NUM cana-5460 255 14 c c PROPN cana-5460 255 15 t t PROPN cana-5460 255 16 Ξ΄w(n Ξ΄w(n PUNCT cana-5460 255 17 ) ) PUNCT cana-5460 255 18 βˆ’ βˆ’ PROPN cana-5460 255 19 Ξ± Ξ± PRON cana-5460 255 20 βˆ‚2w(n βˆ‚2w(n NOUN cana-5460 255 21 ) ) PUNCT cana-5460 255 22 βˆ‚x2 βˆ‚x2 NOUN cana-5460 255 23 βˆ’ βˆ’ NOUN cana-5460 255 24 Ξ² Ξ² X cana-5460 255 25 βˆ‚3w(n βˆ‚3w(n X cana-5460 255 26 ) ) PUNCT cana-5460 255 27 βˆ‚tβˆ‚x2 βˆ‚tβˆ‚x2 NOUN cana-5460 255 28 + + NUM cana-5460 255 29 Ξ³w(n Ξ³w(n PUNCT cana-5460 255 30 ) ) PUNCT cana-5460 255 31 βˆ’ βˆ’ NUM cana-5460 256 1 ∫ ∫ PROPN cana-5460 256 2 a(t a(t NOUN cana-5460 256 3 βˆ’ βˆ’ NUM cana-5460 256 4 s)w(n)(x s)w(n)(x PROPN cana-5460 256 5 , , PUNCT cana-5460 256 6 s)ds s)ds PROPN cana-5460 256 7 = = SYM cana-5460 256 8 t t PROPN cana-5460 256 9 0 0 NUM cana-5460 256 10 n(nβˆ’1)(x n(nβˆ’1)(x PROPN cana-5460 256 11 , , PUNCT cana-5460 256 12 t t PROPN cana-5460 256 13 ) ) PUNCT cana-5460 256 14 ( ( PUNCT cana-5460 256 15 94 94 NUM cana-5460 256 16 ) ) PUNCT cana-5460 256 17 β„“w(n β„“w(n NOUN cana-5460 256 18 ) ) PUNCT cana-5460 256 19 = = SYM cana-5460 256 20 w(n)(x w(n)(x PROPN cana-5460 256 21 , , PUNCT cana-5460 256 22 0 0 NUM cana-5460 256 23 ) ) PUNCT cana-5460 256 24 = = SYM cana-5460 256 25 0 0 NUM cana-5460 256 26 , , PUNCT cana-5460 256 27 qw(n qw(n PUNCT cana-5460 256 28 ) ) PUNCT cana-5460 256 29 = = SYM cana-5460 257 1 βˆ‚w(n)(x,0 βˆ‚w(n)(x,0 PROPN cana-5460 257 2 ) ) PUNCT cana-5460 258 1 βˆ‚t βˆ‚t PROPN cana-5460 258 2 = = SYM cana-5460 258 3 0 0 PROPN cana-5460 258 4 , , PUNCT cana-5460 258 5 0 0 NUM cana-5460 258 6 < < X cana-5460 258 7 π‘₯ π‘₯ X cana-5460 258 8 < < X cana-5460 258 9 1 1 NUM cana-5460 258 10 . . PUNCT cana-5460 259 1 ( ( PUNCT cana-5460 259 2 95 95 NUM cana-5460 259 3 ) ) PUNCT cana-5460 259 4 ∫ ∫ PROPN cana-5460 259 5 w(n)(x w(n)(x PROPN cana-5460 259 6 , , PUNCT cana-5460 259 7 t)dx t)dx PROPN cana-5460 259 8 = = SYM cana-5460 259 9 0 0 NUM cana-5460 259 10 , , PUNCT cana-5460 259 11 t t PROPN cana-5460 259 12 0 0 NUM cana-5460 259 13 ∫ ∫ PROPN cana-5460 259 14 xw(n)(x xw(n)(x PROPN cana-5460 259 15 , , PUNCT cana-5460 259 16 t)dx t)dx PROPN cana-5460 259 17 = = SYM cana-5460 259 18 0 0 NUM cana-5460 259 19 , , PUNCT cana-5460 259 20 t t PROPN cana-5460 259 21 0 0 NUM cana-5460 259 22 0 0 NUM cana-5460 259 23 < < X cana-5460 259 24 𝑑 𝑑 PROPN cana-5460 259 25 ≀ ≀ NOUN cana-5460 259 26 𝑇 𝑇 PROPN cana-5460 259 27 . . PUNCT cana-5460 260 1 ( ( PUNCT cana-5460 260 2 96 96 NUM cana-5460 260 3 ) ) PUNCT cana-5460 260 4 such such ADJ cana-5460 260 5 that that SCONJ cana-5460 260 6 : : PUNCT cana-5460 260 7 n(nβˆ’1)(x n(nβˆ’1)(x PROPN cana-5460 260 8 , , PUNCT cana-5460 260 9 t t PROPN cana-5460 260 10 ) ) PUNCT cana-5460 260 11 = = SYM cana-5460 261 1 Ο‡ Ο‡ X cana-5460 261 2 ( ( PUNCT cana-5460 261 3 x x PROPN cana-5460 261 4 , , PUNCT cana-5460 261 5 t t PROPN cana-5460 261 6 , , PUNCT cana-5460 261 7 w(n w(n PROPN cana-5460 261 8 ) ) PUNCT cana-5460 261 9 , , PUNCT cana-5460 261 10 βˆ‚w(n βˆ‚w(n X cana-5460 261 11 ) ) PUNCT cana-5460 261 12 βˆ‚x βˆ‚x PROPN cana-5460 261 13 ) ) PUNCT cana-5460 262 1 βˆ’ βˆ’ PROPN cana-5460 263 1 Ο‡ Ο‡ X cana-5460 263 2 ( ( PUNCT cana-5460 263 3 x x PROPN cana-5460 263 4 , , PUNCT cana-5460 263 5 t t PROPN cana-5460 263 6 , , PUNCT cana-5460 263 7 w(nβˆ’1 w(nβˆ’1 PROPN cana-5460 263 8 ) ) PUNCT cana-5460 263 9 , , PUNCT cana-5460 263 10 βˆ‚w(nβˆ’1 βˆ‚w(nβˆ’1 PROPN cana-5460 263 11 ) ) PUNCT cana-5460 263 12 βˆ‚x βˆ‚x PROPN cana-5460 263 13 ) ) PUNCT cana-5460 263 14 . . PUNCT cana-5460 264 1 ( ( PUNCT cana-5460 264 2 97 97 X cana-5460 264 3 ) ) PUNCT cana-5460 264 4 lemma lemma PROPN cana-5460 264 5 7.1 7.1 NUM cana-5460 264 6 supposing suppose VERB cana-5460 264 7 that that SCONJ cana-5460 264 8 the the DET cana-5460 264 9 condition condition NOUN cana-5460 264 10 ( ( PUNCT cana-5460 264 11 82 82 NUM cana-5460 264 12 ) ) PUNCT cana-5460 264 13 holds hold NOUN cana-5460 264 14 , , PUNCT cana-5460 264 15 then then ADV cana-5460 264 16 for for ADP cana-5460 264 17 the the DET cana-5460 264 18 linearized linearize VERB cana-5460 264 19 problem problem NOUN cana-5460 264 20 ( ( PUNCT cana-5460 264 21 94)-(96 94)-(96 NOUN cana-5460 264 22 ) ) PUNCT cana-5460 264 23 , , PUNCT cana-5460 264 24 we we PRON cana-5460 264 25 have have VERB cana-5460 264 26 : : PUNCT cana-5460 264 27 β€–w(n)β€– β€–w(n)β€– NOUN cana-5460 264 28 𝕃2(0,t 𝕃2(0,t NOUN cana-5460 264 29 , , PUNCT cana-5460 264 30 h1(Ο‰ h1(Ο‰ PROPN cana-5460 264 31 ) ) PUNCT cana-5460 264 32 ≀ ≀ NOUN cana-5460 264 33 cβ€–w(nβˆ’1)β€– cβ€–w(nβˆ’1)β€– NOUN cana-5460 264 34 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 264 35 ) ) PUNCT cana-5460 264 36 . . PUNCT cana-5460 265 1 ( ( PUNCT cana-5460 265 2 98 98 NUM cana-5460 265 3 ) ) PUNCT cana-5460 266 1 where where SCONJ cana-5460 266 2 : : PUNCT cana-5460 266 3 c c X cana-5460 266 4 > > X cana-5460 266 5 0 0 X cana-5460 266 6 . . PUNCT cana-5460 267 1 proof proof NOUN cana-5460 267 2 : : PUNCT cana-5460 267 3 we we PRON cana-5460 267 4 put put VERB cana-5460 267 5 : : PUNCT cana-5460 267 6 mw(n mw(n X cana-5460 267 7 ) ) PUNCT cana-5460 267 8 = = SYM cana-5460 268 1 βˆ’β„‘x βˆ’β„‘x PROPN cana-5460 268 2 2 2 NUM cana-5460 268 3 βˆ‚w βˆ‚w PROPN cana-5460 268 4 ( ( PUNCT cana-5460 268 5 n n CCONJ cana-5460 268 6 ) ) PUNCT cana-5460 268 7 βˆ‚t βˆ‚t PROPN cana-5460 268 8 , , PUNCT cana-5460 268 9 we we PRON cana-5460 268 10 get get VERB cana-5460 268 11 : : PUNCT cana-5460 268 12 communications communication NOUN cana-5460 268 13 on on ADP cana-5460 268 14 applied apply VERB cana-5460 268 15 nonlinear nonlinear ADJ cana-5460 268 16 analysis analysis NOUN cana-5460 268 17 issn issn NOUN cana-5460 268 18 : : PUNCT cana-5460 268 19 1074 1074 NUM cana-5460 268 20 - - PUNCT cana-5460 268 21 133x 133x NUM cana-5460 268 22 vol vol NOUN cana-5460 268 23 32 32 NUM cana-5460 268 24 no.3 no.3 NOUN cana-5460 268 25 ( ( PUNCT cana-5460 268 26 2025 2025 NUM cana-5460 268 27 ) ) PUNCT cana-5460 268 28 955 955 NUM cana-5460 268 29 https://internationalpubls.com https://internationalpubls.com X cana-5460 268 30 ( ( PUNCT cana-5460 268 31 βˆ‚ βˆ‚ NUM cana-5460 268 32 0 0 NUM cana-5460 268 33 c c PROPN cana-5460 268 34 t t PROPN cana-5460 268 35 Ξ΄w(n Ξ΄w(n PUNCT cana-5460 268 36 ) ) PUNCT cana-5460 268 37 , , PUNCT cana-5460 268 38 βˆ’β„‘x βˆ’β„‘x PROPN cana-5460 268 39 2 2 NUM cana-5460 268 40 βˆ‚w βˆ‚w PROPN cana-5460 268 41 ( ( PUNCT cana-5460 268 42 n n CCONJ cana-5460 268 43 ) ) PUNCT cana-5460 268 44 βˆ‚t βˆ‚t PROPN cana-5460 268 45 ) ) PUNCT cana-5460 268 46 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 268 47 ) ) PUNCT cana-5460 269 1 + + CCONJ cana-5460 269 2 Ξ± Ξ± PROPN cana-5460 269 3 ( ( PUNCT cana-5460 269 4 βˆ‚2w(n βˆ‚2w(n NOUN cana-5460 269 5 ) ) PUNCT cana-5460 269 6 βˆ‚x2 βˆ‚x2 NOUN cana-5460 269 7 , , PUNCT cana-5460 270 1 β„‘x β„‘x PROPN cana-5460 270 2 2 2 NUM cana-5460 270 3 βˆ‚w βˆ‚w PROPN cana-5460 270 4 ( ( PUNCT cana-5460 270 5 n n CCONJ cana-5460 270 6 ) ) PUNCT cana-5460 270 7 βˆ‚t βˆ‚t PROPN cana-5460 270 8 ) ) PUNCT cana-5460 270 9 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 270 10 ) ) PUNCT cana-5460 271 1 + + CCONJ cana-5460 271 2 Ξ² Ξ² X cana-5460 271 3 ( ( PUNCT cana-5460 271 4 βˆ‚3w(n βˆ‚3w(n ADJ cana-5460 271 5 ) ) PUNCT cana-5460 271 6 βˆ‚tβˆ‚x2 βˆ‚tβˆ‚x2 NOUN cana-5460 271 7 , , PUNCT cana-5460 271 8 β„‘x β„‘x PROPN cana-5460 271 9 2 2 NUM cana-5460 271 10 βˆ‚w βˆ‚w PROPN cana-5460 271 11 ( ( PUNCT cana-5460 271 12 n n CCONJ cana-5460 271 13 ) ) PUNCT cana-5460 271 14 βˆ‚t βˆ‚t PROPN cana-5460 271 15 ) ) PUNCT cana-5460 271 16 𝕃2(Ο‰ 𝕃2(Ο‰ PROPN cana-5460 271 17 ) ) PUNCT cana-5460 271 18 βˆ’ βˆ’ PROPN cana-5460 271 19 Ξ³ Ξ³ X cana-5460 271 20 ( ( PUNCT cana-5460 271 21 w(n w(n PROPN cana-5460 271 22 ) ) PUNCT cana-5460 271 23 , , PUNCT cana-5460 271 24 β„‘x β„‘x PROPN cana-5460 271 25 2 2 NUM cana-5460 271 26 βˆ‚w βˆ‚w PROPN cana-5460 271 27 ( ( PUNCT cana-5460 271 28 n n CCONJ cana-5460 271 29 ) ) PUNCT cana-5460 271 30 βˆ‚t βˆ‚t PROPN cana-5460 271 31 ) ) PUNCT cana-5460 271 32 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 271 33 ) ) PUNCT cana-5460 272 1 + + CCONJ cana-5460 272 2 ( ( PUNCT cana-5460 272 3 ∫ ∫ PROPN cana-5460 272 4 a(t a(t NOUN cana-5460 272 5 βˆ’ βˆ’ PRON cana-5460 272 6 s)w(n)(x s)w(n)(x PROPN cana-5460 272 7 , , PUNCT cana-5460 272 8 s)ds s)ds PROPN cana-5460 272 9 , , PUNCT cana-5460 272 10 t t NOUN cana-5460 272 11 0 0 NUM cana-5460 272 12 β„‘x β„‘x PROPN cana-5460 272 13 2 2 NUM cana-5460 272 14 βˆ‚w βˆ‚w PROPN cana-5460 272 15 ( ( PUNCT cana-5460 272 16 n n CCONJ cana-5460 272 17 ) ) PUNCT cana-5460 272 18 βˆ‚t βˆ‚t PROPN cana-5460 272 19 ) ) PUNCT cana-5460 272 20 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 272 21 ) ) PUNCT cana-5460 273 1 = = PRON cana-5460 273 2 ( ( PUNCT cana-5460 273 3 n(nβˆ’1)(x n(nβˆ’1)(x PROPN cana-5460 273 4 , , PUNCT cana-5460 273 5 t t PROPN cana-5460 273 6 ) ) PUNCT cana-5460 273 7 , , PUNCT cana-5460 273 8 βˆ’β„‘x βˆ’β„‘x PROPN cana-5460 273 9 2 2 NUM cana-5460 273 10 βˆ‚w βˆ‚w PROPN cana-5460 273 11 ( ( PUNCT cana-5460 273 12 n n CCONJ cana-5460 273 13 ) ) PUNCT cana-5460 273 14 βˆ‚t βˆ‚t PROPN cana-5460 273 15 ) ) PUNCT cana-5460 273 16 𝕃2(Ο‰ 𝕃2(Ο‰ NOUN cana-5460 273 17 ) ) PUNCT cana-5460 273 18 ( ( PUNCT cana-5460 273 19 99 99 NUM cana-5460 273 20 ) ) PUNCT cana-5460 273 21 after after ADP cana-5460 273 22 integrating integrate VERB cana-5460 273 23 by by ADP cana-5460 273 24 parts part NOUN cana-5460 273 25 all all DET cana-5460 273 26 terms term NOUN cana-5460 273 27 of of ADP cana-5460 273 28 ( ( PUNCT cana-5460 273 29 99 99 NUM cana-5460 273 30 ) ) PUNCT cana-5460 273 31 and and CCONJ cana-5460 273 32 using use VERB cana-5460 273 33 conditions condition NOUN cana-5460 273 34 ( ( PUNCT cana-5460 273 35 95 95 NUM cana-5460 273 36 ) ) PUNCT cana-5460 273 37 and and CCONJ cana-5460 273 38 ( ( PUNCT cana-5460 273 39 96 96 NUM cana-5460 273 40 ) ) PUNCT cana-5460 273 41 , , PUNCT cana-5460 273 42 proceding procede VERB cana-5460 273 43 as as ADP cana-5460 273 44 in in ADP cana-5460 273 45 the the DET cana-5460 273 46 establishment establishment NOUN cana-5460 273 47 of of ADP cana-5460 273 48 theorem theorem ADJ cana-5460 273 49 5.1 5.1 NUM cana-5460 273 50 dt dt NOUN cana-5460 273 51 Ξ΄βˆ’1 Ξ΄βˆ’1 PROPN cana-5460 273 52 β€–β„‘x β€–β„‘x NOUN cana-5460 273 53 βˆ‚w(n βˆ‚w(n PROPN cana-5460 273 54 ) ) PUNCT cana-5460 273 55 βˆ‚t βˆ‚t PROPN cana-5460 273 56 β€– β€– PROPN cana-5460 273 57 2 2 NUM cana-5460 273 58 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 273 59 ) ) PUNCT cana-5460 274 1 + + PROPN cana-5460 274 2 ( ( PUNCT cana-5460 274 3 a0 a0 NOUN cana-5460 274 4 + + CCONJ cana-5460 274 5 Ξ± Ξ± PROPN cana-5460 274 6 2 2 NUM cana-5460 274 7 ) ) PUNCT cana-5460 274 8 β€–w(n β€–w(n NUM cana-5460 274 9 ) ) PUNCT cana-5460 274 10 ( ( PUNCT cana-5460 274 11 . . PUNCT cana-5460 274 12 , , PUNCT cana-5460 274 13 Ο„)β€– Ο„)β€– DET cana-5460 274 14 2 2 NUM cana-5460 274 15 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 274 16 ) ) PUNCT cana-5460 274 17 ≀ ≀ NUM cana-5460 275 1 ∫ ∫ PROPN cana-5460 275 2 β€–β„‘xn β€–β„‘xn PROPN cana-5460 275 3 ( ( PUNCT cana-5460 275 4 nβˆ’1)(x nβˆ’1)(x PROPN cana-5460 275 5 , , PUNCT cana-5460 275 6 t)β€– t)β€– NOUN cana-5460 275 7 2 2 NUM cana-5460 275 8 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 275 9 ) ) PUNCT cana-5460 275 10 dt dt PUNCT cana-5460 276 1 Ο„ Ο„ PROPN cana-5460 276 2 0 0 NUM cana-5460 277 1 + + CCONJ cana-5460 277 2 ( ( PUNCT cana-5460 277 3 Ξ² Ξ² X cana-5460 277 4 + + X cana-5460 277 5 Ξ΅ Ξ΅ PROPN cana-5460 277 6 2 2 NUM cana-5460 277 7 ) ) PUNCT cana-5460 277 8 ∫ ∫ PROPN cana-5460 277 9 β€–β„‘x β€–β„‘x PROPN cana-5460 277 10 βˆ‚w(n)(.,t βˆ‚w(n)(.,t NOUN cana-5460 277 11 ) ) PUNCT cana-5460 278 1 βˆ‚t βˆ‚t PROPN cana-5460 278 2 β€– β€– PROPN cana-5460 278 3 2 2 NUM cana-5460 278 4 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 278 5 ) ) PUNCT cana-5460 278 6 dt dt PUNCT cana-5460 279 1 Ο„ Ο„ PROPN cana-5460 279 2 0 0 NUM cana-5460 280 1 + + CCONJ cana-5460 280 2 ( ( PUNCT cana-5460 280 3 βˆ’ βˆ’ PROPN cana-5460 280 4 Ξ± Ξ± NOUN cana-5460 280 5 2 2 NUM cana-5460 280 6 + + NUM cana-5460 280 7 a1 a1 PROPN cana-5460 280 8 t t NOUN cana-5460 280 9 2 2 NUM cana-5460 280 10 ) ) PUNCT cana-5460 280 11 ∫ ∫ NOUN cana-5460 280 12 β€–w(n β€–w(n PUNCT cana-5460 280 13 ) ) PUNCT cana-5460 280 14 ( ( PUNCT cana-5460 280 15 . . PUNCT cana-5460 280 16 , , PUNCT cana-5460 280 17 Ο„)β€– Ο„)β€– DET cana-5460 280 18 2 2 NUM cana-5460 280 19 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 280 20 ) ) PUNCT cana-5460 280 21 dt dt PUNCT cana-5460 281 1 Ο„ Ο„ PROPN cana-5460 281 2 0 0 NUM cana-5460 281 3 ( ( PUNCT cana-5460 281 4 100 100 NUM cana-5460 281 5 ) ) PUNCT cana-5460 281 6 we we PRON cana-5460 281 7 apply apply VERB cana-5460 281 8 β„‘x β„‘x PROPN cana-5460 281 9 to to ADP cana-5460 281 10 the the DET cana-5460 281 11 equation equation NOUN cana-5460 281 12 ( ( PUNCT cana-5460 281 13 94 94 NUM cana-5460 281 14 ) ) PUNCT cana-5460 281 15 , , PUNCT cana-5460 281 16 and and CCONJ cana-5460 281 17 we we PRON cana-5460 281 18 multiplying multiply VERB cana-5460 281 19 the the DET cana-5460 281 20 resulting result VERB cana-5460 281 21 equation equation NOUN cana-5460 281 22 by by ADP cana-5460 281 23 βˆ‚w(n βˆ‚w(n PROPN cana-5460 281 24 ) ) PUNCT cana-5460 281 25 βˆ‚x βˆ‚x PROPN cana-5460 281 26 , , PUNCT cana-5460 281 27 and and CCONJ cana-5460 281 28 we we PRON cana-5460 281 29 integrate integrate VERB cana-5460 281 30 by by ADP cana-5460 281 31 parts part NOUN cana-5460 281 32 over over ADP cana-5460 281 33 Ο‰ Ο‰ PROPN cana-5460 281 34 : : PUNCT cana-5460 281 35 ∫ ∫ PROPN cana-5460 281 36 βˆ‚ βˆ‚ NUM cana-5460 281 37 0 0 NUM cana-5460 282 1 c c NOUN cana-5460 282 2 t t PROPN cana-5460 282 3 Ξ΄β„‘xw Ξ΄β„‘xw PROPN cana-5460 282 4 ( ( PUNCT cana-5460 282 5 n n CCONJ cana-5460 282 6 ) ) PUNCT cana-5460 282 7 Ο‰ Ο‰ NOUN cana-5460 282 8 . . PUNCT cana-5460 283 1 βˆ‚w(n βˆ‚w(n X cana-5460 283 2 ) ) PUNCT cana-5460 283 3 βˆ‚x βˆ‚x PROPN cana-5460 283 4 dxdt dxdt NOUN cana-5460 283 5 βˆ’ βˆ’ PROPN cana-5460 283 6 α∫ α∫ NUM cana-5460 283 7 ( ( PUNCT cana-5460 283 8 βˆ‚w(n βˆ‚w(n PROPN cana-5460 283 9 ) ) PUNCT cana-5460 283 10 βˆ‚x βˆ‚x PROPN cana-5460 283 11 ) ) PUNCT cana-5460 283 12 2 2 NUM cana-5460 283 13 dxdt dxdt NOUN cana-5460 283 14 Ο‰ Ο‰ NUM cana-5460 283 15 βˆ’ βˆ’ NOUN cana-5460 283 16 β∫ β∫ PROPN cana-5460 283 17 βˆ‚ βˆ‚ NOUN cana-5460 283 18 βˆ‚x βˆ‚x PROPN cana-5460 283 19 ( ( PUNCT cana-5460 283 20 βˆ‚2w(n βˆ‚2w(n NOUN cana-5460 283 21 ) ) PUNCT cana-5460 283 22 βˆ‚t βˆ‚t PROPN cana-5460 283 23 βˆ‚x βˆ‚x PROPN cana-5460 283 24 ) ) PUNCT cana-5460 283 25 dxdt dxdt NOUN cana-5460 283 26 Ο‰ Ο‰ PROPN cana-5460 283 27 + + CCONJ cana-5460 283 28 Ξ³βˆ«β„‘x Ξ³βˆ«β„‘x NOUN cana-5460 283 29 Ο‰ Ο‰ NUM cana-5460 283 30 w(n w(n PROPN cana-5460 283 31 ) ) PUNCT cana-5460 283 32 . . PUNCT cana-5460 284 1 βˆ‚w(n βˆ‚w(n X cana-5460 284 2 ) ) PUNCT cana-5460 284 3 βˆ‚x βˆ‚x PROPN cana-5460 284 4 dxdt dxdt NOUN cana-5460 284 5 βˆ’ βˆ’ PROPN cana-5460 285 1 ∫ ∫ PROPN cana-5460 285 2 ∫ ∫ PROPN cana-5460 285 3 a(t a(t NOUN cana-5460 285 4 βˆ’ βˆ’ PROPN cana-5460 285 5 s)β„‘xw s)β„‘xw NOUN cana-5460 285 6 ( ( PUNCT cana-5460 285 7 n)(x n)(x PROPN cana-5460 285 8 , , PUNCT cana-5460 285 9 s s NOUN cana-5460 285 10 ) ) PUNCT cana-5460 285 11 . . PUNCT cana-5460 286 1 βˆ‚w(n βˆ‚w(n X cana-5460 286 2 ) ) PUNCT cana-5460 286 3 βˆ‚x βˆ‚x PROPN cana-5460 286 4 dsdxdt dsdxdt NOUN cana-5460 286 5 = = PUNCT cana-5460 286 6 βˆ«β„‘x βˆ«β„‘x PUNCT cana-5460 286 7 Ο‰ Ο‰ NUM cana-5460 286 8 t t PROPN cana-5460 286 9 0Ο‰ 0Ο‰ PROPN cana-5460 286 10 n(nβˆ’1)(x n(nβˆ’1)(x PROPN cana-5460 286 11 , , PUNCT cana-5460 286 12 t t PROPN cana-5460 286 13 ) ) PUNCT cana-5460 286 14 . . PUNCT cana-5460 287 1 βˆ‚w(n βˆ‚w(n X cana-5460 287 2 ) ) PUNCT cana-5460 287 3 βˆ‚x βˆ‚x PROPN cana-5460 287 4 dxdt dxdt NOUN cana-5460 287 5 . . PUNCT cana-5460 288 1 ( ( PUNCT cana-5460 288 2 101 101 NUM cana-5460 288 3 ) ) PUNCT cana-5460 289 1 where where SCONJ cana-5460 289 2 : : PUNCT cana-5460 289 3 ∫ ∫ PROPN cana-5460 289 4 βˆ‚ βˆ‚ NUM cana-5460 289 5 0 0 NUM cana-5460 289 6 c c NOUN cana-5460 289 7 t t PROPN cana-5460 289 8 Ξ΄β„‘xw Ξ΄β„‘xw PROPN cana-5460 289 9 ( ( PUNCT cana-5460 289 10 n n CCONJ cana-5460 289 11 ) ) PUNCT cana-5460 289 12 Ο‰ Ο‰ NOUN cana-5460 289 13 . . PUNCT cana-5460 290 1 βˆ‚w(n βˆ‚w(n X cana-5460 290 2 ) ) PUNCT cana-5460 290 3 βˆ‚x βˆ‚x PROPN cana-5460 290 4 dxdt dxdt NOUN cana-5460 290 5 = = SYM cana-5460 290 6 βˆ’βˆ« βˆ’βˆ« NOUN cana-5460 290 7 βˆ‚ βˆ‚ NOUN cana-5460 290 8 0 0 NUM cana-5460 290 9 c c PROPN cana-5460 290 10 t t PROPN cana-5460 290 11 Ξ΄w(n Ξ΄w(n PUNCT cana-5460 290 12 ) ) PUNCT cana-5460 290 13 . . PUNCT cana-5460 291 1 Ο‰ Ο‰ NUM cana-5460 291 2 w(n)dxdt w(n)dxdt X cana-5460 291 3 ( ( PUNCT cana-5460 291 4 102 102 NUM cana-5460 291 5 ) ) PUNCT cana-5460 291 6 α∫ α∫ NUM cana-5460 291 7 ( ( PUNCT cana-5460 291 8 βˆ‚w(n βˆ‚w(n PROPN cana-5460 291 9 ) ) PUNCT cana-5460 291 10 βˆ‚x βˆ‚x PROPN cana-5460 291 11 ) ) PUNCT cana-5460 291 12 2 2 NUM cana-5460 291 13 dxdt dxdt NOUN cana-5460 291 14 = = SYM cana-5460 291 15 α∫ α∫ PROPN cana-5460 291 16 β€– β€– PROPN cana-5460 291 17 βˆ‚w(n)(.,t βˆ‚w(n)(.,t NUM cana-5460 291 18 ) ) PUNCT cana-5460 291 19 βˆ‚x βˆ‚x PROPN cana-5460 291 20 β€– β€– PROPN cana-5460 291 21 2 2 NUM cana-5460 291 22 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 291 23 ) ) PUNCT cana-5460 292 1 Ο„ Ο„ PROPN cana-5460 292 2 0Ο‰ 0Ο‰ NOUN cana-5460 292 3 dt dt X cana-5460 292 4 ( ( PUNCT cana-5460 292 5 103 103 NUM cana-5460 292 6 ) ) PUNCT cana-5460 292 7 β∫ β∫ PROPN cana-5460 292 8 βˆ‚ βˆ‚ NOUN cana-5460 292 9 βˆ‚x βˆ‚x PROPN cana-5460 292 10 ( ( PUNCT cana-5460 292 11 βˆ‚2w(n βˆ‚2w(n NOUN cana-5460 292 12 ) ) PUNCT cana-5460 292 13 βˆ‚tβˆ‚x βˆ‚tβˆ‚x ADJ cana-5460 292 14 ) ) PUNCT cana-5460 292 15 dxdt dxdt NOUN cana-5460 292 16 Ο‰ Ο‰ PROPN cana-5460 292 17 = = PUNCT cana-5460 292 18 Ξ² Ξ² X cana-5460 292 19 β€– β€– PROPN cana-5460 292 20 βˆ‚w(n βˆ‚w(n PROPN cana-5460 292 21 ) ) PUNCT cana-5460 292 22 βˆ‚x βˆ‚x PROPN cana-5460 292 23 β€– β€– PROPN cana-5460 292 24 2 2 NUM cana-5460 292 25 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 292 26 ) ) PUNCT cana-5460 292 27 ( ( PUNCT cana-5460 292 28 104 104 X cana-5460 292 29 ) ) PUNCT cana-5460 292 30 γ∫ γ∫ VERB cana-5460 292 31 β„‘xΟ‰ β„‘xΟ‰ PROPN cana-5460 292 32 w(n w(n NOUN cana-5460 292 33 ) ) PUNCT cana-5460 292 34 . . PUNCT cana-5460 293 1 βˆ‚w(n βˆ‚w(n X cana-5460 293 2 ) ) PUNCT cana-5460 293 3 βˆ‚x βˆ‚x PROPN cana-5460 293 4 dxdt dxdt NOUN cana-5460 293 5 = = PUNCT cana-5460 293 6 βˆ’Ξ³βˆ« βˆ’Ξ³βˆ« PROPN cana-5460 293 7 β€–w(n)β€– β€–w(n)β€– X cana-5460 293 8 2 2 NUM cana-5460 293 9 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 293 10 ) ) PUNCT cana-5460 294 1 Ο„ Ο„ PROPN cana-5460 294 2 0 0 NUM cana-5460 294 3 dt dt X cana-5460 294 4 ( ( PUNCT cana-5460 294 5 105 105 NUM cana-5460 294 6 ) ) PUNCT cana-5460 294 7 ∫ ∫ PROPN cana-5460 295 1 ∫ ∫ PROPN cana-5460 295 2 a(t a(t NOUN cana-5460 295 3 βˆ’ βˆ’ PROPN cana-5460 295 4 s)β„‘xw s)β„‘xw NOUN cana-5460 295 5 ( ( PUNCT cana-5460 295 6 n)(x n)(x PROPN cana-5460 295 7 , , PUNCT cana-5460 295 8 s s NOUN cana-5460 295 9 ) ) PUNCT cana-5460 295 10 . . PUNCT cana-5460 296 1 βˆ‚w(n βˆ‚w(n X cana-5460 296 2 ) ) PUNCT cana-5460 296 3 βˆ‚x βˆ‚x PROPN cana-5460 296 4 dsdxdt dsdxdt NOUN cana-5460 296 5 ≀ ≀ NUM cana-5460 296 6 a1 a1 NOUN cana-5460 296 7 t t NOUN cana-5460 296 8 0Ο‰ 0Ο‰ NOUN cana-5460 296 9 t2 t2 PROPN cana-5460 296 10 ∫ ∫ PROPN cana-5460 296 11 β€–w(n β€–w(n PUNCT cana-5460 296 12 ) ) PUNCT cana-5460 296 13 ( ( PUNCT cana-5460 296 14 . . PUNCT cana-5460 296 15 , , PUNCT cana-5460 296 16 t)β€– t)β€– NOUN cana-5460 296 17 2 2 NUM cana-5460 296 18 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 296 19 ) ) PUNCT cana-5460 296 20 dt dt PUNCT cana-5460 297 1 Ο„ Ο„ PROPN cana-5460 297 2 0 0 NUM cana-5460 297 3 ( ( PUNCT cana-5460 297 4 106 106 NUM cana-5460 297 5 ) ) PUNCT cana-5460 297 6 communications communication NOUN cana-5460 297 7 on on ADP cana-5460 297 8 applied apply VERB cana-5460 297 9 nonlinear nonlinear ADJ cana-5460 297 10 analysis analysis NOUN cana-5460 297 11 issn issn NOUN cana-5460 297 12 : : PUNCT cana-5460 297 13 1074 1074 NUM cana-5460 297 14 - - PUNCT cana-5460 297 15 133x 133x NUM cana-5460 297 16 vol vol NOUN cana-5460 297 17 32 32 NUM cana-5460 297 18 no.3 no.3 NOUN cana-5460 297 19 ( ( PUNCT cana-5460 297 20 2025 2025 NUM cana-5460 297 21 ) ) PUNCT cana-5460 298 1 956 956 NUM cana-5460 299 1 https://internationalpubls.com https://internationalpubls.com X cana-5460 299 2 ∫ ∫ PROPN cana-5460 300 1 β„‘xn β„‘xn PROPN cana-5460 300 2 ( ( PUNCT cana-5460 300 3 nβˆ’1)(x nβˆ’1)(x PROPN cana-5460 300 4 , , PUNCT cana-5460 300 5 t t PROPN cana-5460 300 6 ) ) PUNCT cana-5460 300 7 . . PUNCT cana-5460 301 1 βˆ‚w(n βˆ‚w(n X cana-5460 301 2 ) ) PUNCT cana-5460 301 3 βˆ‚xΟ‰ βˆ‚xΟ‰ PROPN cana-5460 301 4 dxdt dxdt NOUN cana-5460 301 5 ≀ ≀ NUM cana-5460 301 6 1 1 NUM cana-5460 301 7 2 2 NUM cana-5460 301 8 ∫ ∫ NOUN cana-5460 301 9 β€–n(nβˆ’1)(x β€–n(nβˆ’1)(x NOUN cana-5460 301 10 , , PUNCT cana-5460 301 11 t)β€– t)β€– NOUN cana-5460 301 12 2 2 NUM cana-5460 301 13 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 301 14 ) ) PUNCT cana-5460 301 15 dt dt PUNCT cana-5460 302 1 Ο„ Ο„ PROPN cana-5460 302 2 0 0 NUM cana-5460 303 1 + + CCONJ cana-5460 303 2 1 1 NUM cana-5460 303 3 2 2 NUM cana-5460 303 4 ∫ ∫ NOUN cana-5460 303 5 β€–w(n β€–w(n NUM cana-5460 303 6 ) ) PUNCT cana-5460 303 7 ( ( PUNCT cana-5460 303 8 . . PUNCT cana-5460 303 9 , , PUNCT cana-5460 303 10 t)β€– t)β€– NOUN cana-5460 303 11 2 2 NUM cana-5460 303 12 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 303 13 ) ) PUNCT cana-5460 303 14 dt dt PUNCT cana-5460 304 1 Ο„ Ο„ PROPN cana-5460 304 2 0 0 NUM cana-5460 304 3 ( ( PUNCT cana-5460 304 4 107 107 NUM cana-5460 304 5 ) ) PUNCT cana-5460 304 6 after after ADP cana-5460 304 7 integration integration NOUN cana-5460 304 8 by by ADP cana-5460 304 9 parts part NOUN cana-5460 304 10 of of ADP cana-5460 304 11 all all DET cana-5460 304 12 the the DET cana-5460 304 13 terms term NOUN cana-5460 304 14 of of ADP cana-5460 304 15 ( ( PUNCT cana-5460 304 16 101 101 NUM cana-5460 304 17 ) ) PUNCT cana-5460 304 18 and and CCONJ cana-5460 304 19 taking take VERB cana-5460 304 20 into into ADP cana-5460 304 21 consideration consideration NOUN cana-5460 304 22 conditions condition NOUN cana-5460 304 23 ( ( PUNCT cana-5460 304 24 95 95 NUM cana-5460 304 25 ) ) PUNCT cana-5460 304 26 , , PUNCT cana-5460 304 27 ( ( PUNCT cana-5460 304 28 96 96 NUM cana-5460 304 29 ) ) PUNCT cana-5460 304 30 and and CCONJ cana-5460 304 31 using use VERB cana-5460 304 32 inequality inequality NOUN cana-5460 304 33 ( ( PUNCT cana-5460 304 34 8) 8) NUM cana-5460 304 35 , , PUNCT cana-5460 304 36 we we PRON cana-5460 304 37 have have VERB cana-5460 304 38 : : PUNCT cana-5460 304 39 ∫ ∫ PROPN cana-5460 304 40 βˆ‚ βˆ‚ NUM cana-5460 304 41 0 0 PUNCT cana-5460 304 42 c c PROPN cana-5460 304 43 t t PROPN cana-5460 304 44 Ξ΄w(n Ξ΄w(n PUNCT cana-5460 304 45 ) ) PUNCT cana-5460 304 46 . . PUNCT cana-5460 305 1 Ο‰ Ο‰ NUM cana-5460 305 2 w(n)dxdt w(n)dxdt NOUN cana-5460 306 1 + + X cana-5460 306 2 Ξ± Ξ± PROPN cana-5460 306 3 ∫ ∫ PROPN cana-5460 306 4 β€– β€– PROPN cana-5460 306 5 βˆ‚w(n)(.,t βˆ‚w(n)(.,t PROPN cana-5460 306 6 ) ) PUNCT cana-5460 306 7 βˆ‚x βˆ‚x PROPN cana-5460 306 8 β€– β€– PROPN cana-5460 306 9 2 2 NUM cana-5460 306 10 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 306 11 ) ) PUNCT cana-5460 306 12 Ο„ Ο„ X cana-5460 306 13 0 0 NUM cana-5460 306 14 dt dt NOUN cana-5460 306 15 + + X cana-5460 306 16 Ξ² Ξ² PROPN cana-5460 306 17 β€– β€– PROPN cana-5460 306 18 βˆ‚w(n βˆ‚w(n PROPN cana-5460 306 19 ) ) PUNCT cana-5460 306 20 βˆ‚x βˆ‚x PROPN cana-5460 306 21 β€– β€– PROPN cana-5460 306 22 2 2 NUM cana-5460 306 23 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 306 24 ) ) PUNCT cana-5460 306 25 ≀ ≀ NUM cana-5460 306 26 1 1 NUM cana-5460 306 27 2 2 NUM cana-5460 306 28 ∫ ∫ NOUN cana-5460 306 29 β€–n(nβˆ’1)(x β€–n(nβˆ’1)(x NOUN cana-5460 306 30 , , PUNCT cana-5460 306 31 t)β€– t)β€– NOUN cana-5460 306 32 2 2 NUM cana-5460 306 33 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 306 34 ) ) PUNCT cana-5460 306 35 dt dt PUNCT cana-5460 307 1 Ο„ Ο„ PROPN cana-5460 307 2 0 0 NUM cana-5460 308 1 + + CCONJ cana-5460 308 2 ( ( PUNCT cana-5460 308 3 Ξ³ Ξ³ X cana-5460 308 4 + + CCONJ cana-5460 308 5 1 1 NUM cana-5460 308 6 2 2 NUM cana-5460 308 7 + + NUM cana-5460 308 8 a1 a1 PROPN cana-5460 308 9 t t NOUN cana-5460 308 10 2)∫ 2)∫ NUM cana-5460 308 11 β€–w(n β€–w(n NOUN cana-5460 308 12 ) ) PUNCT cana-5460 308 13 ( ( PUNCT cana-5460 308 14 . . PUNCT cana-5460 308 15 , , PUNCT cana-5460 308 16 t)β€– t)β€– NOUN cana-5460 308 17 2 2 NUM cana-5460 308 18 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 308 19 ) ) PUNCT cana-5460 308 20 dt dt PUNCT cana-5460 309 1 Ο„ Ο„ PROPN cana-5460 309 2 0 0 NUM cana-5460 309 3 ( ( PUNCT cana-5460 309 4 108 108 NUM cana-5460 309 5 ) ) PUNCT cana-5460 309 6 after after ADP cana-5460 309 7 combination combination NOUN cana-5460 309 8 of of ADP cana-5460 309 9 inequalities inequality NOUN cana-5460 309 10 ( ( PUNCT cana-5460 309 11 100 100 NUM cana-5460 309 12 ) ) PUNCT cana-5460 309 13 and and CCONJ cana-5460 309 14 ( ( PUNCT cana-5460 309 15 108 108 NUM cana-5460 309 16 ) ) PUNCT cana-5460 309 17 , , PUNCT cana-5460 309 18 we we PRON cana-5460 309 19 obtain obtain VERB cana-5460 309 20 : : PUNCT cana-5460 309 21 dt dt PROPN cana-5460 310 1 Ξ΄βˆ’1 Ξ΄βˆ’1 PROPN cana-5460 310 2 β€–β„‘x β€–β„‘x NOUN cana-5460 310 3 βˆ‚w(n βˆ‚w(n PROPN cana-5460 310 4 ) ) PUNCT cana-5460 310 5 βˆ‚t βˆ‚t PROPN cana-5460 310 6 β€– β€– PROPN cana-5460 310 7 2 2 NUM cana-5460 310 8 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 310 9 ) ) PUNCT cana-5460 311 1 + + NOUN cana-5460 311 2 ∫ ∫ PROPN cana-5460 311 3 βˆ‚ βˆ‚ NOUN cana-5460 311 4 0 0 PUNCT cana-5460 311 5 c c PROPN cana-5460 311 6 t t PROPN cana-5460 311 7 Ξ΄w(n Ξ΄w(n PUNCT cana-5460 311 8 ) ) PUNCT cana-5460 311 9 . . PUNCT cana-5460 312 1 ωτ ωτ NUM cana-5460 312 2 w(n)dxdt w(n)dxdt VERB cana-5460 312 3 + + CCONJ cana-5460 312 4 α∫ α∫ PROPN cana-5460 312 5 β€– β€– PROPN cana-5460 312 6 βˆ‚w(n βˆ‚w(n PROPN cana-5460 312 7 ) ) PUNCT cana-5460 312 8 ( ( PUNCT cana-5460 312 9 . . PUNCT cana-5460 312 10 , , PUNCT cana-5460 312 11 t t X cana-5460 312 12 ) ) PUNCT cana-5460 312 13 βˆ‚x βˆ‚x PROPN cana-5460 312 14 β€– β€– PROPN cana-5460 312 15 2 2 NUM cana-5460 312 16 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 312 17 ) ) PUNCT cana-5460 312 18 Ο„ Ο„ X cana-5460 312 19 0 0 NUM cana-5460 313 1 + + CCONJ cana-5460 313 2 Ξ² Ξ² PROPN cana-5460 313 3 β€– β€– PROPN cana-5460 313 4 βˆ‚w(n βˆ‚w(n PROPN cana-5460 313 5 ) ) PUNCT cana-5460 313 6 βˆ‚x βˆ‚x PROPN cana-5460 313 7 β€– β€– PROPN cana-5460 313 8 2 2 NUM cana-5460 313 9 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 313 10 ) ) PUNCT cana-5460 314 1 + + CCONJ cana-5460 314 2 ( ( PUNCT cana-5460 314 3 a0 a0 NOUN cana-5460 314 4 + + CCONJ cana-5460 314 5 Ξ± Ξ± PROPN cana-5460 314 6 2 2 NUM cana-5460 314 7 ) ) PUNCT cana-5460 314 8 β€–w(n β€–w(n NUM cana-5460 314 9 ) ) PUNCT cana-5460 314 10 ( ( PUNCT cana-5460 314 11 . . PUNCT cana-5460 314 12 , , PUNCT cana-5460 314 13 Ο„)β€– Ο„)β€– PRON cana-5460 314 14 2 2 NUM cana-5460 314 15 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 314 16 ) ) PUNCT cana-5460 314 17 ≀ ≀ NUM cana-5460 314 18 ∫ ∫ PROPN cana-5460 314 19 β€–n(nβˆ’1)(x β€–n(nβˆ’1)(x PROPN cana-5460 314 20 , , PUNCT cana-5460 314 21 t)β€– t)β€– NOUN cana-5460 314 22 2 2 NUM cana-5460 314 23 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 314 24 ) ) PUNCT cana-5460 314 25 dt dt PUNCT cana-5460 315 1 Ο„ Ο„ PROPN cana-5460 315 2 0 0 NUM cana-5460 316 1 + + CCONJ cana-5460 316 2 ( ( PUNCT cana-5460 316 3 Ξ² Ξ² X cana-5460 316 4 + + X cana-5460 316 5 Ξ΅ Ξ΅ PROPN cana-5460 316 6 2 2 NUM cana-5460 316 7 ) ) PUNCT cana-5460 316 8 ∫ ∫ PROPN cana-5460 316 9 β€–β„‘x β€–β„‘x PROPN cana-5460 316 10 βˆ‚w(n)(.,t βˆ‚w(n)(.,t PRON cana-5460 316 11 ) ) PUNCT cana-5460 317 1 βˆ‚t βˆ‚t PROPN cana-5460 317 2 β€– β€– PROPN cana-5460 317 3 2 2 NUM cana-5460 317 4 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 317 5 ) ) PUNCT cana-5460 317 6 dt dt PUNCT cana-5460 318 1 Ο„ Ο„ PROPN cana-5460 318 2 0 0 NUM cana-5460 319 1 + + CCONJ cana-5460 319 2 ( ( PUNCT cana-5460 319 3 a1 a1 PROPN cana-5460 319 4 t t NOUN cana-5460 319 5 2 2 NUM cana-5460 319 6 βˆ’ βˆ’ NOUN cana-5460 319 7 Ξ± Ξ± NOUN cana-5460 319 8 2 2 NUM cana-5460 320 1 + + CCONJ cana-5460 320 2 Ξ³ Ξ³ X cana-5460 320 3 + + NOUN cana-5460 320 4 1 1 NUM cana-5460 320 5 2 2 NUM cana-5460 320 6 ) ) PUNCT cana-5460 320 7 ∫ ∫ PROPN cana-5460 320 8 β€–w(n β€–w(n PUNCT cana-5460 320 9 ) ) PUNCT cana-5460 320 10 ( ( PUNCT cana-5460 320 11 . . PUNCT cana-5460 320 12 , , PUNCT cana-5460 320 13 Ο„)β€– Ο„)β€– DET cana-5460 320 14 2 2 NUM cana-5460 320 15 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 320 16 ) ) PUNCT cana-5460 321 1 dt dt PUNCT cana-5460 322 1 Ο„ Ο„ PROPN cana-5460 322 2 0 0 NUM cana-5460 322 3 ( ( PUNCT cana-5460 322 4 109 109 NUM cana-5460 322 5 ) ) PUNCT cana-5460 322 6 now now ADV cana-5460 322 7 we we PRON cana-5460 322 8 will will AUX cana-5460 322 9 eliminate eliminate VERB cana-5460 322 10 the the DET cana-5460 322 11 last last ADJ cana-5460 322 12 term term NOUN cana-5460 322 13 in in ADP cana-5460 322 14 ( ( PUNCT cana-5460 322 15 109 109 NUM cana-5460 322 16 ) ) PUNCT cana-5460 322 17 by by ADP cana-5460 322 18 applying apply VERB cana-5460 322 19 the the DET cana-5460 322 20 gronwall gronwall ADJ cana-5460 322 21 lemma lemma PROPN cana-5460 322 22 : : PUNCT cana-5460 322 23 dt dt PROPN cana-5460 323 1 Ξ΄βˆ’1 Ξ΄βˆ’1 PROPN cana-5460 323 2 β€–β„‘x β€–β„‘x NOUN cana-5460 323 3 βˆ‚w(n βˆ‚w(n PROPN cana-5460 323 4 ) ) PUNCT cana-5460 323 5 βˆ‚t βˆ‚t PROPN cana-5460 323 6 β€– β€– PROPN cana-5460 323 7 2 2 NUM cana-5460 323 8 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 323 9 ) ) PUNCT cana-5460 324 1 + + CCONJ cana-5460 324 2 ∫ ∫ PROPN cana-5460 324 3 βˆ‚ βˆ‚ NOUN cana-5460 324 4 0 0 NUM cana-5460 324 5 c c PROPN cana-5460 324 6 t t PROPN cana-5460 324 7 Ξ΄w(n Ξ΄w(n PUNCT cana-5460 324 8 ) ) PUNCT cana-5460 324 9 . . PUNCT cana-5460 325 1 ωτ ωτ NUM cana-5460 325 2 w(n)dxdt w(n)dxdt NOUN cana-5460 326 1 + + X cana-5460 326 2 Ξ± Ξ± PROPN cana-5460 326 3 ∫ ∫ PROPN cana-5460 326 4 β€– β€– PROPN cana-5460 326 5 βˆ‚w(n)(.,t βˆ‚w(n)(.,t PROPN cana-5460 326 6 ) ) PUNCT cana-5460 326 7 βˆ‚x βˆ‚x PROPN cana-5460 326 8 β€– β€– PROPN cana-5460 326 9 2 2 NUM cana-5460 326 10 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 326 11 ) ) PUNCT cana-5460 326 12 Ο„ Ο„ X cana-5460 326 13 0 0 NUM cana-5460 327 1 + + CCONJ cana-5460 327 2 Ξ² Ξ² PROPN cana-5460 327 3 β€– β€– PROPN cana-5460 327 4 βˆ‚w(n βˆ‚w(n PROPN cana-5460 327 5 ) ) PUNCT cana-5460 327 6 βˆ‚x βˆ‚x PROPN cana-5460 327 7 β€– β€– PROPN cana-5460 327 8 2 2 NUM cana-5460 327 9 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 327 10 ) ) PUNCT cana-5460 328 1 + + CCONJ cana-5460 328 2 ( ( PUNCT cana-5460 328 3 a0 a0 NOUN cana-5460 328 4 + + CCONJ cana-5460 328 5 Ξ± Ξ± PROPN cana-5460 328 6 2 2 NUM cana-5460 328 7 ) ) PUNCT cana-5460 328 8 β€–w(n β€–w(n NUM cana-5460 328 9 ) ) PUNCT cana-5460 328 10 ( ( PUNCT cana-5460 328 11 . . PUNCT cana-5460 328 12 , , PUNCT cana-5460 328 13 Ο„)β€– Ο„)β€– DET cana-5460 328 14 2 2 NUM cana-5460 328 15 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 328 16 ) ) PUNCT cana-5460 328 17 ≀ ≀ NUM cana-5460 328 18 exp exp NOUN cana-5460 328 19 c0 c0 PROPN cana-5460 328 20 { { PUNCT cana-5460 328 21 ∫ ∫ PROPN cana-5460 328 22 β€–n β€–n PROPN cana-5460 328 23 ( ( PUNCT cana-5460 328 24 nβˆ’1)(x nβˆ’1)(x PROPN cana-5460 328 25 , , PUNCT cana-5460 328 26 t)β€– t)β€– NOUN cana-5460 328 27 2 2 NUM cana-5460 328 28 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 328 29 ) ) PUNCT cana-5460 328 30 dt dt PUNCT cana-5460 329 1 Ο„ Ο„ PROPN cana-5460 329 2 0 0 NUM cana-5460 330 1 + + CCONJ cana-5460 330 2 c1 c1 PROPN cana-5460 330 3 ∫ ∫ PROPN cana-5460 330 4 β€–β„‘x β€–β„‘x PROPN cana-5460 330 5 βˆ‚w(n)(.,t βˆ‚w(n)(.,t PRON cana-5460 330 6 ) ) PUNCT cana-5460 331 1 βˆ‚t βˆ‚t PROPN cana-5460 331 2 β€– β€– PROPN cana-5460 331 3 2 2 NUM cana-5460 331 4 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 331 5 ) ) PUNCT cana-5460 331 6 dt dt PUNCT cana-5460 331 7 Ο„ Ο„ PROPN cana-5460 331 8 0 0 NUM cana-5460 331 9 } } PUNCT cana-5460 331 10 . . PUNCT cana-5460 332 1 ( ( PUNCT cana-5460 332 2 110 110 NUM cana-5460 332 3 ) ) PUNCT cana-5460 332 4 where where SCONJ cana-5460 332 5 : : PUNCT cana-5460 332 6 { { PUNCT cana-5460 332 7 c0 c0 NOUN cana-5460 332 8 = = PUNCT cana-5460 332 9 Ξ² Ξ² X cana-5460 332 10 + + CCONJ cana-5460 332 11 Ξ΅ Ξ΅ PROPN cana-5460 332 12 2 2 NUM cana-5460 332 13 , , PUNCT cana-5460 332 14 c1 c1 NOUN cana-5460 332 15 = = PUNCT cana-5460 332 16 a1 a1 PROPN cana-5460 332 17 t t PROPN cana-5460 332 18 2 2 NUM cana-5460 332 19 βˆ’ βˆ’ NOUN cana-5460 333 1 Ξ± Ξ± NOUN cana-5460 334 1 2 2 NUM cana-5460 335 1 + + CCONJ cana-5460 335 2 Ξ³ Ξ³ X cana-5460 335 3 + + NOUN cana-5460 335 4 1 1 NUM cana-5460 335 5 2 2 NUM cana-5460 335 6 . . PUNCT cana-5460 336 1 communications communication NOUN cana-5460 336 2 on on ADP cana-5460 336 3 applied apply VERB cana-5460 336 4 nonlinear nonlinear ADJ cana-5460 336 5 analysis analysis NOUN cana-5460 336 6 issn issn NOUN cana-5460 336 7 : : PUNCT cana-5460 336 8 1074 1074 NUM cana-5460 336 9 - - PUNCT cana-5460 336 10 133x 133x NUM cana-5460 336 11 vol vol NOUN cana-5460 336 12 32 32 NUM cana-5460 336 13 no.3 no.3 NOUN cana-5460 336 14 ( ( PUNCT cana-5460 336 15 2025 2025 NUM cana-5460 336 16 ) ) PUNCT cana-5460 336 17 957 957 NUM cana-5460 336 18 https://internationalpubls.com https://internationalpubls.com X cana-5460 337 1 we we PRON cana-5460 337 2 apply apply VERB cana-5460 337 3 gronwall gronwall DET cana-5460 337 4 lemma lemma PROPN cana-5460 337 5 to to ADP cana-5460 337 6 the the DET cana-5460 337 7 last last ADJ cana-5460 337 8 term term NOUN cana-5460 337 9 of of ADP cana-5460 337 10 ( ( PUNCT cana-5460 337 11 110 110 NUM cana-5460 337 12 ) ) PUNCT cana-5460 337 13 ∫ ∫ PROPN cana-5460 337 14 β€–β„‘x β€–β„‘x PROPN cana-5460 337 15 βˆ‚w(n βˆ‚w(n PROPN cana-5460 337 16 ) ) PUNCT cana-5460 337 17 ( ( PUNCT cana-5460 337 18 . . PUNCT cana-5460 337 19 , , PUNCT cana-5460 337 20 t t X cana-5460 337 21 ) ) PUNCT cana-5460 337 22 βˆ‚t βˆ‚t PROPN cana-5460 337 23 β€– β€– PROPN cana-5460 337 24 2 2 NUM cana-5460 337 25 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 337 26 ) ) PUNCT cana-5460 337 27 dt dt PUNCT cana-5460 338 1 Ο„ Ο„ PROPN cana-5460 338 2 0 0 NUM cana-5460 338 3 ≀ ≀ NOUN cana-5460 338 4 Ξ³(Ξ΄)eδδ(c1 Ξ³(Ξ΄)eδδ(c1 ADP cana-5460 338 5 exp(c0 exp(c0 PROPN cana-5460 338 6 t t PROPN cana-5460 338 7 ) ) PUNCT cana-5460 338 8 t t PROPN cana-5460 338 9 Ξ΄ Ξ΄ PROPN cana-5460 338 10 ) ) PUNCT cana-5460 338 11 exp(c0 exp(c0 PROPN cana-5460 338 12 t t PROPN cana-5460 338 13 ) ) PUNCT cana-5460 338 14 . . PUNCT cana-5460 339 1 dt dt PUNCT cana-5460 340 1 βˆ’Ξ΄β€–n(nβˆ’1)(x βˆ’Ξ΄β€–n(nβˆ’1)(x PROPN cana-5460 340 2 , , PUNCT cana-5460 340 3 t)β€– t)β€– NOUN cana-5460 340 4 2 2 NUM cana-5460 340 5 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 340 6 ) ) PUNCT cana-5460 340 7 ( ( PUNCT cana-5460 340 8 111 111 NUM cana-5460 340 9 ) ) PUNCT cana-5460 340 10 on on ADP cana-5460 340 11 the the DET cana-5460 340 12 other other ADJ cana-5460 340 13 side side NOUN cana-5460 340 14 , , PUNCT cana-5460 340 15 we we PRON cana-5460 340 16 applying apply VERB cana-5460 340 17 the the DET cana-5460 340 18 condition condition NOUN cana-5460 340 19 ( ( PUNCT cana-5460 340 20 82 82 NUM cana-5460 340 21 ) ) PUNCT cana-5460 340 22 , , PUNCT cana-5460 340 23 we we PRON cana-5460 340 24 get get VERB cana-5460 340 25 : : PUNCT cana-5460 340 26 ∫ ∫ PROPN cana-5460 340 27 β€–n(nβˆ’1)(x β€–n(nβˆ’1)(x PROPN cana-5460 340 28 , , PUNCT cana-5460 340 29 t)β€– t)β€– NOUN cana-5460 340 30 2 2 NUM cana-5460 340 31 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 340 32 ) ) PUNCT cana-5460 340 33 dt dt PUNCT cana-5460 341 1 Ο„ Ο„ PROPN cana-5460 341 2 0 0 NUM cana-5460 341 3 ≀ ≀ NUM cana-5460 341 4 2m2∫ 2m2∫ NOUN cana-5460 341 5 ( ( PUNCT cana-5460 341 6 β€–w(nβˆ’1 β€–w(nβˆ’1 NOUN cana-5460 341 7 ) ) PUNCT cana-5460 341 8 ( ( PUNCT cana-5460 341 9 . . PUNCT cana-5460 341 10 , , PUNCT cana-5460 341 11 t)β€– t)β€– NOUN cana-5460 341 12 2 2 NUM cana-5460 341 13 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 341 14 ) ) PUNCT cana-5460 342 1 + + CCONJ cana-5460 343 1 β€– β€– ADJ cana-5460 343 2 βˆ‚w(nβˆ’1 βˆ‚w(nβˆ’1 NOUN cana-5460 343 3 ) ) PUNCT cana-5460 343 4 βˆ‚x βˆ‚x PROPN cana-5460 343 5 β€– β€– PROPN cana-5460 343 6 2 2 NUM cana-5460 343 7 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 343 8 ) ) PUNCT cana-5460 343 9 ) ) PUNCT cana-5460 343 10 dt dt PUNCT cana-5460 344 1 Ο„ Ο„ PROPN cana-5460 344 2 0 0 NUM cana-5460 344 3 ( ( PUNCT cana-5460 344 4 112 112 NUM cana-5460 344 5 ) ) PUNCT cana-5460 344 6 combining combine VERB cana-5460 344 7 ( ( PUNCT cana-5460 344 8 111)-(112 111)-(112 NUM cana-5460 344 9 ) ) PUNCT cana-5460 344 10 and and CCONJ cana-5460 344 11 using use VERB cana-5460 344 12 ( ( PUNCT cana-5460 344 13 49 49 NUM cana-5460 344 14 ) ) PUNCT cana-5460 344 15 , , PUNCT cana-5460 344 16 we we PRON cana-5460 344 17 get get VERB cana-5460 344 18 : : PUNCT cana-5460 344 19 dt dt PROPN cana-5460 345 1 Ξ΄βˆ’1 Ξ΄βˆ’1 PROPN cana-5460 345 2 β€–β„‘x β€–β„‘x NOUN cana-5460 345 3 βˆ‚w(n βˆ‚w(n PROPN cana-5460 345 4 ) ) PUNCT cana-5460 345 5 βˆ‚t βˆ‚t PROPN cana-5460 345 6 β€– β€– PROPN cana-5460 345 7 2 2 NUM cana-5460 345 8 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 345 9 ) ) PUNCT cana-5460 346 1 + + NOUN cana-5460 346 2 ∫ ∫ PROPN cana-5460 346 3 βˆ‚ βˆ‚ NOUN cana-5460 346 4 0 0 PUNCT cana-5460 346 5 c c PROPN cana-5460 346 6 t t PROPN cana-5460 346 7 Ξ΄w(n Ξ΄w(n PUNCT cana-5460 346 8 ) ) PUNCT cana-5460 346 9 . . PUNCT cana-5460 347 1 ωτ ωτ NUM cana-5460 347 2 w(n)dxdt w(n)dxdt X cana-5460 347 3 + + CCONJ cana-5460 347 4 ∫ ∫ PROPN cana-5460 347 5 β€– β€– PROPN cana-5460 347 6 βˆ‚w(n βˆ‚w(n PROPN cana-5460 347 7 ) ) PUNCT cana-5460 347 8 ( ( PUNCT cana-5460 347 9 . . PUNCT cana-5460 347 10 , , PUNCT cana-5460 347 11 t t X cana-5460 347 12 ) ) PUNCT cana-5460 347 13 βˆ‚x βˆ‚x PROPN cana-5460 347 14 β€– β€– PROPN cana-5460 347 15 2 2 NUM cana-5460 347 16 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 347 17 ) ) PUNCT cana-5460 347 18 Ο„ Ο„ X cana-5460 347 19 0 0 NUM cana-5460 348 1 + + CCONJ cana-5460 348 2 β€– β€– PROPN cana-5460 348 3 βˆ‚w(n βˆ‚w(n PROPN cana-5460 348 4 ) ) PUNCT cana-5460 348 5 βˆ‚x βˆ‚x PROPN cana-5460 348 6 β€– β€– PROPN cana-5460 348 7 2 2 NUM cana-5460 348 8 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 348 9 ) ) PUNCT cana-5460 349 1 + + NOUN cana-5460 349 2 β€–w(n β€–w(n NUM cana-5460 349 3 ) ) PUNCT cana-5460 349 4 ( ( PUNCT cana-5460 349 5 . . PUNCT cana-5460 350 1 , , PUNCT cana-5460 350 2 Ο„)β€– Ο„)β€– PRON cana-5460 350 3 2 2 NUM cana-5460 350 4 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 350 5 ) ) PUNCT cana-5460 350 6 ≀ ≀ NUM cana-5460 350 7 cβˆ—m2∫ cβˆ—m2∫ VERB cana-5460 350 8 ( ( PUNCT cana-5460 350 9 β€–w(nβˆ’1 β€–w(nβˆ’1 NOUN cana-5460 350 10 ) ) PUNCT cana-5460 350 11 ( ( PUNCT cana-5460 350 12 . . PUNCT cana-5460 350 13 , , PUNCT cana-5460 350 14 t)β€– t)β€– NOUN cana-5460 350 15 2 2 NUM cana-5460 350 16 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 350 17 ) ) PUNCT cana-5460 350 18 + + CCONJ cana-5460 350 19 β€– β€– ADJ cana-5460 350 20 βˆ‚w(nβˆ’1 βˆ‚w(nβˆ’1 NOUN cana-5460 350 21 ) ) PUNCT cana-5460 350 22 βˆ‚x βˆ‚x PROPN cana-5460 350 23 β€– β€– PROPN cana-5460 350 24 2 2 NUM cana-5460 350 25 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 350 26 ) ) PUNCT cana-5460 350 27 ) ) PUNCT cana-5460 350 28 dt dt PUNCT cana-5460 351 1 Ο„ Ο„ PROPN cana-5460 351 2 0 0 NUM cana-5460 351 3 ( ( PUNCT cana-5460 351 4 113 113 NUM cana-5460 351 5 ) ) PUNCT cana-5460 351 6 such such ADJ cana-5460 351 7 that that SCONJ cana-5460 351 8 : : PUNCT cana-5460 351 9 cβˆ— cβˆ— PROPN cana-5460 351 10 = = PROPN cana-5460 351 11 exp(c0 exp(c0 PROPN cana-5460 351 12 t t PROPN cana-5460 351 13 ) ) PUNCT cana-5460 351 14 ( ( PUNCT cana-5460 351 15 1 1 NUM cana-5460 351 16 + + NUM cana-5460 351 17 Ξ³(Ξ΄)eδδ(c1 Ξ³(Ξ΄)eδδ(c1 PROPN cana-5460 351 18 exp(c0 exp(c0 PROPN cana-5460 351 19 t t PROPN cana-5460 351 20 ) ) PUNCT cana-5460 351 21 t t PROPN cana-5460 351 22 Ξ΄ Ξ΄ PROPN cana-5460 351 23 ) ) PUNCT cana-5460 351 24 ) ) PUNCT cana-5460 351 25 . . PUNCT cana-5460 352 1 tΞ΄ tΞ΄ ADP cana-5460 352 2 Ξ³(1+Ξ΄ Ξ³(1+Ξ΄ NOUN cana-5460 352 3 ) ) PUNCT cana-5460 352 4 ( ( PUNCT cana-5460 352 5 114 114 NUM cana-5460 352 6 ) ) PUNCT cana-5460 352 7 after after ADP cana-5460 352 8 discarding discard VERB cana-5460 352 9 the the DET cana-5460 352 10 first first ADJ cana-5460 352 11 two two NUM cana-5460 352 12 terms term NOUN cana-5460 352 13 on on ADP cana-5460 352 14 ( ( PUNCT cana-5460 352 15 113 113 NUM cana-5460 352 16 ) ) PUNCT cana-5460 352 17 , , PUNCT cana-5460 352 18 we we PRON cana-5460 352 19 obtain obtain VERB cana-5460 352 20 : : PUNCT cana-5460 352 21 ∫ ∫ PROPN cana-5460 352 22 β€– β€– PROPN cana-5460 352 23 βˆ‚w(n βˆ‚w(n PROPN cana-5460 352 24 ) ) PUNCT cana-5460 352 25 ( ( PUNCT cana-5460 352 26 . . PUNCT cana-5460 352 27 , , PUNCT cana-5460 352 28 t t X cana-5460 352 29 ) ) PUNCT cana-5460 352 30 βˆ‚x βˆ‚x PROPN cana-5460 352 31 β€– β€– PROPN cana-5460 352 32 2 2 NUM cana-5460 352 33 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 352 34 ) ) PUNCT cana-5460 352 35 Ο„ Ο„ X cana-5460 352 36 0 0 NUM cana-5460 353 1 + + CCONJ cana-5460 353 2 β€– β€– PROPN cana-5460 353 3 βˆ‚w(n βˆ‚w(n PROPN cana-5460 353 4 ) ) PUNCT cana-5460 353 5 βˆ‚x βˆ‚x PROPN cana-5460 353 6 β€– β€– PROPN cana-5460 353 7 2 2 NUM cana-5460 353 8 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 353 9 ) ) PUNCT cana-5460 354 1 + + CCONJ cana-5460 354 2 β€–w(n β€–w(n NOUN cana-5460 354 3 ) ) PUNCT cana-5460 354 4 ( ( PUNCT cana-5460 354 5 . . PUNCT cana-5460 355 1 , , PUNCT cana-5460 355 2 Ο„)β€– Ο„)β€– PRON cana-5460 355 3 2 2 NUM cana-5460 355 4 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 355 5 ) ) PUNCT cana-5460 355 6 ≀ ≀ NUM cana-5460 355 7 cβˆ—m2∫ cβˆ—m2∫ VERB cana-5460 355 8 ( ( PUNCT cana-5460 355 9 β€–w(nβˆ’1 β€–w(nβˆ’1 NOUN cana-5460 355 10 ) ) PUNCT cana-5460 355 11 ( ( PUNCT cana-5460 355 12 . . PUNCT cana-5460 355 13 , , PUNCT cana-5460 355 14 t)β€– t)β€– NOUN cana-5460 355 15 2 2 NUM cana-5460 355 16 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 355 17 ) ) PUNCT cana-5460 355 18 + + CCONJ cana-5460 355 19 β€– β€– ADJ cana-5460 355 20 βˆ‚w(nβˆ’1 βˆ‚w(nβˆ’1 NOUN cana-5460 355 21 ) ) PUNCT cana-5460 355 22 βˆ‚x βˆ‚x PROPN cana-5460 355 23 β€– β€– PROPN cana-5460 355 24 2 2 NUM cana-5460 355 25 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 355 26 ) ) PUNCT cana-5460 355 27 ) ) PUNCT cana-5460 355 28 dt dt PUNCT cana-5460 356 1 Ο„ Ο„ PROPN cana-5460 356 2 0 0 NUM cana-5460 356 3 ( ( PUNCT cana-5460 356 4 115 115 NUM cana-5460 356 5 ) ) PUNCT cana-5460 356 6 communications communication NOUN cana-5460 356 7 on on ADP cana-5460 356 8 applied apply VERB cana-5460 356 9 nonlinear nonlinear ADJ cana-5460 356 10 analysis analysis NOUN cana-5460 356 11 issn issn NOUN cana-5460 356 12 : : PUNCT cana-5460 356 13 1074 1074 NUM cana-5460 356 14 - - PUNCT cana-5460 356 15 133x 133x NUM cana-5460 356 16 vol vol NOUN cana-5460 356 17 32 32 NUM cana-5460 356 18 no.3 no.3 NOUN cana-5460 356 19 ( ( PUNCT cana-5460 356 20 2025 2025 NUM cana-5460 356 21 ) ) PUNCT cana-5460 356 22 958 958 NUM cana-5460 356 23 https://internationalpubls.com https://internationalpubls.com X cana-5460 356 24 here here ADV cana-5460 356 25 , , PUNCT cana-5460 356 26 the the DET cana-5460 356 27 rhs rhs PROPN cana-5460 356 28 does do AUX cana-5460 356 29 n’t not PART cana-5460 356 30 depend depend VERB cana-5460 356 31 on on ADP cana-5460 356 32 Ο„ Ο„ PROPN cana-5460 356 33 so so ADV cana-5460 356 34 , , PUNCT cana-5460 356 35 we we PRON cana-5460 356 36 can can AUX cana-5460 356 37 replace replace VERB cana-5460 356 38 the the DET cana-5460 356 39 lhs lhs NOUN cana-5460 356 40 by by ADP cana-5460 356 41 upper upper ADJ cana-5460 356 42 bounds bound NOUN cana-5460 356 43 with with ADP cana-5460 356 44 respect respect NOUN cana-5460 356 45 to to ADP cana-5460 356 46 Ο„ Ο„ PROPN cana-5460 356 47 , , PUNCT cana-5460 356 48 we we PRON cana-5460 356 49 obtain obtain VERB cana-5460 356 50 : : PUNCT cana-5460 356 51 ∫ ∫ PROPN cana-5460 356 52 β€– β€– PROPN cana-5460 356 53 βˆ‚w(n βˆ‚w(n PROPN cana-5460 356 54 ) ) PUNCT cana-5460 356 55 ( ( PUNCT cana-5460 356 56 . . PUNCT cana-5460 356 57 , , PUNCT cana-5460 356 58 t t X cana-5460 356 59 ) ) PUNCT cana-5460 356 60 βˆ‚x βˆ‚x PROPN cana-5460 356 61 β€– β€– PROPN cana-5460 356 62 2 2 NUM cana-5460 356 63 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 356 64 ) ) PUNCT cana-5460 356 65 t t NOUN cana-5460 356 66 0 0 NUM cana-5460 357 1 + + CCONJ cana-5460 357 2 β€– β€– PROPN cana-5460 357 3 βˆ‚w(n βˆ‚w(n PROPN cana-5460 357 4 ) ) PUNCT cana-5460 357 5 βˆ‚x βˆ‚x PROPN cana-5460 357 6 β€– β€– PROPN cana-5460 357 7 2 2 NUM cana-5460 357 8 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 357 9 ) ) PUNCT cana-5460 358 1 + + CCONJ cana-5460 358 2 β€–w(n β€–w(n NOUN cana-5460 358 3 ) ) PUNCT cana-5460 358 4 ( ( PUNCT cana-5460 358 5 . . PUNCT cana-5460 359 1 , , PUNCT cana-5460 359 2 Ο„)β€– Ο„)β€– PRON cana-5460 359 3 2 2 NUM cana-5460 359 4 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 359 5 ) ) PUNCT cana-5460 359 6 ≀ ≀ NUM cana-5460 359 7 cβˆ—m2∫ cβˆ—m2∫ VERB cana-5460 359 8 ( ( PUNCT cana-5460 359 9 β€–w(nβˆ’1 β€–w(nβˆ’1 NOUN cana-5460 359 10 ) ) PUNCT cana-5460 359 11 ( ( PUNCT cana-5460 359 12 . . PUNCT cana-5460 359 13 , , PUNCT cana-5460 359 14 t)β€– t)β€– NOUN cana-5460 359 15 2 2 NUM cana-5460 359 16 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 359 17 ) ) PUNCT cana-5460 359 18 + + CCONJ cana-5460 359 19 β€– β€– ADJ cana-5460 359 20 βˆ‚w(nβˆ’1 βˆ‚w(nβˆ’1 NOUN cana-5460 359 21 ) ) PUNCT cana-5460 359 22 ( ( PUNCT cana-5460 359 23 . . PUNCT cana-5460 359 24 , , PUNCT cana-5460 359 25 t t X cana-5460 359 26 ) ) PUNCT cana-5460 359 27 βˆ‚x βˆ‚x PROPN cana-5460 359 28 β€– β€– PROPN cana-5460 359 29 2 2 NUM cana-5460 359 30 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 359 31 ) ) PUNCT cana-5460 359 32 ) ) PUNCT cana-5460 360 1 dt dt PROPN cana-5460 361 1 t t NOUN cana-5460 361 2 0 0 NUM cana-5460 361 3 ( ( PUNCT cana-5460 361 4 116 116 NUM cana-5460 361 5 ) ) PUNCT cana-5460 361 6 we we PRON cana-5460 361 7 integrate integrate VERB cana-5460 361 8 over over ADP cana-5460 361 9 ( ( PUNCT cana-5460 361 10 0,t 0,t PROPN cana-5460 361 11 ) ) PUNCT cana-5460 361 12 , , PUNCT cana-5460 361 13 we we PRON cana-5460 361 14 get get VERB cana-5460 361 15 : : PUNCT cana-5460 361 16 ∫ ∫ PROPN cana-5460 361 17 β€– β€– PROPN cana-5460 361 18 βˆ‚w(n βˆ‚w(n PROPN cana-5460 361 19 ) ) PUNCT cana-5460 361 20 ( ( PUNCT cana-5460 361 21 . . PUNCT cana-5460 361 22 , , PUNCT cana-5460 361 23 t t X cana-5460 361 24 ) ) PUNCT cana-5460 361 25 βˆ‚x βˆ‚x PROPN cana-5460 361 26 β€– β€– PROPN cana-5460 361 27 2 2 NUM cana-5460 361 28 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 361 29 ) ) PUNCT cana-5460 362 1 t t NOUN cana-5460 362 2 0 0 NUM cana-5460 363 1 + + NOUN cana-5460 363 2 ∫ ∫ PROPN cana-5460 363 3 β€–w(n β€–w(n NOUN cana-5460 363 4 ) ) PUNCT cana-5460 363 5 ( ( PUNCT cana-5460 363 6 . . PUNCT cana-5460 363 7 , , PUNCT cana-5460 363 8 Ο„)β€– Ο„)β€– DET cana-5460 363 9 2 2 NUM cana-5460 363 10 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 363 11 ) ) PUNCT cana-5460 363 12 dt dt PUNCT cana-5460 364 1 t t NOUN cana-5460 364 2 0 0 NUM cana-5460 364 3 ≀ ≀ PROPN cana-5460 364 4 Ξ»m2∫ Ξ»m2∫ PROPN cana-5460 364 5 ( ( PUNCT cana-5460 364 6 β€–w(nβˆ’1 β€–w(nβˆ’1 NOUN cana-5460 364 7 ) ) PUNCT cana-5460 364 8 ( ( PUNCT cana-5460 364 9 . . PUNCT cana-5460 364 10 , , PUNCT cana-5460 364 11 t)β€– t)β€– NOUN cana-5460 364 12 2 2 NUM cana-5460 364 13 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 364 14 ) ) PUNCT cana-5460 364 15 + + CCONJ cana-5460 364 16 β€– β€– ADJ cana-5460 364 17 βˆ‚w(nβˆ’1 βˆ‚w(nβˆ’1 NOUN cana-5460 364 18 ) ) PUNCT cana-5460 364 19 ( ( PUNCT cana-5460 364 20 . . PUNCT cana-5460 364 21 , , PUNCT cana-5460 364 22 t t X cana-5460 364 23 ) ) PUNCT cana-5460 364 24 βˆ‚x βˆ‚x PROPN cana-5460 364 25 β€– β€– PROPN cana-5460 364 26 2 2 NUM cana-5460 364 27 𝕃2(Ο‰ 𝕃2(Ο‰ NUM cana-5460 364 28 ) ) PUNCT cana-5460 364 29 ) ) PUNCT cana-5460 364 30 dt dt PROPN cana-5460 365 1 t t NOUN cana-5460 365 2 0 0 NUM cana-5460 365 3 ( ( PUNCT cana-5460 365 4 117 117 NUM cana-5460 365 5 ) ) PUNCT cana-5460 365 6 such such ADJ cana-5460 365 7 that that SCONJ cana-5460 365 8 : : PUNCT cana-5460 365 9 Ξ» Ξ» X cana-5460 365 10 = = SYM cana-5460 365 11 cβˆ—m2 cβˆ—m2 PROPN cana-5460 365 12 t t NOUN cana-5460 365 13 min min NOUN cana-5460 365 14 ( ( PUNCT cana-5460 365 15 1,t 1,t NUM cana-5460 365 16 ) ) PUNCT cana-5460 365 17 . . PUNCT cana-5460 366 1 we we PRON cana-5460 366 2 obtain obtain VERB cana-5460 366 3 the the DET cana-5460 366 4 inequality inequality NOUN cana-5460 366 5 : : PUNCT cana-5460 366 6 β€–w(n)β€– β€–w(n)β€– NUM cana-5460 366 7 2 2 NUM cana-5460 366 8 𝕃2(0,t 𝕃2(0,t NOUN cana-5460 366 9 , , PUNCT cana-5460 366 10 h1(Ο‰ h1(Ο‰ PROPN cana-5460 366 11 ) ) PUNCT cana-5460 366 12 ) ) PUNCT cana-5460 366 13 ≀ ≀ NUM cana-5460 366 14 Ξ»β€–w(nβˆ’1)β€– Ξ»β€–w(nβˆ’1)β€– NOUN cana-5460 366 15 2 2 NUM cana-5460 366 16 𝕃2(0,t 𝕃2(0,t NOUN cana-5460 366 17 , , PUNCT cana-5460 366 18 h1(Ο‰ h1(Ο‰ PROPN cana-5460 366 19 ) ) PUNCT cana-5460 366 20 ) ) PUNCT cana-5460 366 21 . . PUNCT cana-5460 367 1 ( ( PUNCT cana-5460 367 2 118 118 NUM cana-5460 367 3 ) ) PUNCT cana-5460 367 4 using use VERB cana-5460 367 5 the the DET cana-5460 367 6 convergence convergence NOUN cana-5460 367 7 of of ADP cana-5460 367 8 series series NOUN cana-5460 367 9 criteria criterion NOUN cana-5460 367 10 we we PRON cana-5460 367 11 conclude conclude VERB cana-5460 367 12 that that SCONJ cana-5460 367 13 βˆ‘ βˆ‘ PROPN cana-5460 367 14 w(n)∞ w(n)∞ PROPN cana-5460 367 15 nβˆ’1 nβˆ’1 PROPN cana-5460 367 16 converges converge VERB cana-5460 367 17 if if SCONJ cana-5460 367 18 Ξ» Ξ» X cana-5460 367 19 < < X cana-5460 367 20 1 1 NUM cana-5460 367 21 , , PUNCT cana-5460 367 22 in in ADP cana-5460 367 23 other other ADJ cana-5460 367 24 words word NOUN cana-5460 367 25 if if SCONJ cana-5460 367 26 m m VERB cana-5460 367 27 < < X cana-5460 367 28 √ √ ADJ cana-5460 367 29 min min NOUN cana-5460 367 30 ( ( PUNCT cana-5460 367 31 1,t 1,t NOUN cana-5460 367 32 ) ) PUNCT cana-5460 367 33 cβˆ—t cβˆ—t NOUN cana-5460 367 34 . . PUNCT cana-5460 368 1 since since SCONJ cana-5460 368 2 : : PUNCT cana-5460 368 3 w(n w(n X cana-5460 368 4 ) ) PUNCT cana-5460 368 5 = = SYM cana-5460 368 6 w(n+1)(x w(n+1)(x PROPN cana-5460 368 7 , , PUNCT cana-5460 368 8 t t PROPN cana-5460 368 9 ) ) PUNCT cana-5460 368 10 βˆ’ βˆ’ PROPN cana-5460 368 11 w(n)(x w(n)(x PROPN cana-5460 368 12 , , PUNCT cana-5460 368 13 t t PROPN cana-5460 368 14 ) ) PUNCT cana-5460 368 15 , , PUNCT cana-5460 368 16 then then ADV cana-5460 368 17 ( ( PUNCT cana-5460 368 18 w(n w(n NOUN cana-5460 368 19 ) ) PUNCT cana-5460 368 20 ) ) PUNCT cana-5460 368 21 nβˆˆβ„• nβˆˆβ„• NOUN cana-5460 368 22 converge converge VERB cana-5460 368 23 to to ADP cana-5460 368 24 a a DET cana-5460 368 25 function function NOUN cana-5460 368 26 w w NOUN cana-5460 368 27 ∈ ∈ NOUN cana-5460 368 28 𝕃2(0 𝕃2(0 NOUN cana-5460 368 29 , , PUNCT cana-5460 368 30 t t PROPN cana-5460 368 31 , , PUNCT cana-5460 368 32 h1(Ο‰ h1(Ο‰ PROPN cana-5460 368 33 ) ) PUNCT cana-5460 368 34 ) ) PUNCT cana-5460 368 35 . . PUNCT cana-5460 369 1 so so ADV cana-5460 369 2 to to PART cana-5460 369 3 prove prove VERB cana-5460 369 4 that that SCONJ cana-5460 369 5 w w NOUN cana-5460 369 6 is be AUX cana-5460 369 7 the the DET cana-5460 369 8 solution solution NOUN cana-5460 369 9 of of ADP cana-5460 369 10 problem problem NOUN cana-5460 369 11 ( ( PUNCT cana-5460 369 12 94)-(96 94)-(96 NOUN cana-5460 369 13 ) ) PUNCT cana-5460 369 14 , , PUNCT cana-5460 369 15 we we PRON cana-5460 369 16 have have VERB cana-5460 369 17 only only ADV cana-5460 369 18 to to PART cana-5460 369 19 prove prove VERB cana-5460 369 20 that that SCONJ cana-5460 369 21 w w ADJ cana-5460 369 22 verifies verifie NOUN cana-5460 369 23 ( ( PUNCT cana-5460 369 24 81 81 NUM cana-5460 369 25 ) ) PUNCT cana-5460 369 26 and and CCONJ cana-5460 369 27 ( ( PUNCT cana-5460 369 28 90 90 NUM cana-5460 369 29 ) ) PUNCT cana-5460 369 30 . . PUNCT cana-5460 370 1 we we PRON cana-5460 370 2 have have VERB cana-5460 370 3 from from ADP cana-5460 370 4 problem problem NOUN cana-5460 370 5 ( ( PUNCT cana-5460 370 6 91)-(93 91)-(93 NOUN cana-5460 370 7 ) ) PUNCT cana-5460 370 8 , , PUNCT cana-5460 370 9 that that SCONJ cana-5460 370 10 : : PUNCT cana-5460 370 11 ΞΊ(w(n ΞΊ(w(n PROPN cana-5460 370 12 ) ) PUNCT cana-5460 370 13 , , PUNCT cana-5460 370 14 u u NOUN cana-5460 370 15 ) ) PUNCT cana-5460 370 16 = = SYM cana-5460 370 17 ( ( PUNCT cana-5460 370 18 u u NOUN cana-5460 370 19 , , PUNCT cana-5460 370 20 β„‘xΟ‡ β„‘xΟ‡ PROPN cana-5460 370 21 ( ( PUNCT cana-5460 370 22 x x PROPN cana-5460 370 23 , , PUNCT cana-5460 370 24 t t PROPN cana-5460 370 25 , , PUNCT cana-5460 370 26 w w PROPN cana-5460 370 27 ( ( PUNCT cana-5460 370 28 nβˆ’1 nβˆ’1 PROPN cana-5460 370 29 ) ) PUNCT cana-5460 370 30 , , PUNCT cana-5460 370 31 βˆ‚w(nβˆ’1 βˆ‚w(nβˆ’1 PROPN cana-5460 370 32 ) ) PUNCT cana-5460 370 33 βˆ‚x βˆ‚x PROPN cana-5460 370 34 ) ) PUNCT cana-5460 370 35 ) ) PUNCT cana-5460 371 1 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 371 2 ) ) PUNCT cana-5460 371 3 ( ( PUNCT cana-5460 371 4 119 119 NUM cana-5460 371 5 ) ) PUNCT cana-5460 371 6 communications communication NOUN cana-5460 371 7 on on ADP cana-5460 371 8 applied apply VERB cana-5460 371 9 nonlinear nonlinear ADJ cana-5460 371 10 analysis analysis NOUN cana-5460 371 11 issn issn NOUN cana-5460 371 12 : : PUNCT cana-5460 371 13 1074 1074 NUM cana-5460 371 14 - - PUNCT cana-5460 371 15 133x 133x NUM cana-5460 371 16 vol vol NOUN cana-5460 371 17 32 32 NUM cana-5460 371 18 no.3 no.3 NOUN cana-5460 371 19 ( ( PUNCT cana-5460 371 20 2025 2025 NUM cana-5460 371 21 ) ) PUNCT cana-5460 371 22 959 959 NUM cana-5460 371 23 https://internationalpubls.com https://internationalpubls.com X cana-5460 371 24 more more ADV cana-5460 371 25 precisely precisely ADV cana-5460 371 26 ΞΊ(w(n ΞΊ(w(n NOUN cana-5460 371 27 ) ) PUNCT cana-5460 371 28 βˆ’w βˆ’w NOUN cana-5460 371 29 , , PUNCT cana-5460 371 30 u u NOUN cana-5460 371 31 ) ) PUNCT cana-5460 371 32 + + NUM cana-5460 371 33 ΞΊ(w ΞΊ(w NOUN cana-5460 371 34 , , PUNCT cana-5460 371 35 u u NOUN cana-5460 371 36 ) ) PUNCT cana-5460 371 37 = = SYM cana-5460 371 38 ( ( PUNCT cana-5460 371 39 u u NOUN cana-5460 371 40 , , PUNCT cana-5460 371 41 β„‘xΟ‡ β„‘xΟ‡ PROPN cana-5460 371 42 ( ( PUNCT cana-5460 371 43 x x PROPN cana-5460 371 44 , , PUNCT cana-5460 371 45 t t PROPN cana-5460 371 46 , , PUNCT cana-5460 371 47 w w PROPN cana-5460 371 48 ( ( PUNCT cana-5460 371 49 nβˆ’1 nβˆ’1 PROPN cana-5460 371 50 ) ) PUNCT cana-5460 371 51 , , PUNCT cana-5460 371 52 βˆ‚w(nβˆ’1 βˆ‚w(nβˆ’1 PROPN cana-5460 371 53 ) ) PUNCT cana-5460 371 54 βˆ‚x βˆ‚x PROPN cana-5460 371 55 ) ) PUNCT cana-5460 372 1 βˆ’ βˆ’ PROPN cana-5460 373 1 β„‘xΟ‡ β„‘xΟ‡ PROPN cana-5460 373 2 ( ( PUNCT cana-5460 373 3 x x PROPN cana-5460 373 4 , , PUNCT cana-5460 373 5 t t PROPN cana-5460 373 6 , , PUNCT cana-5460 373 7 w w PROPN cana-5460 373 8 , , PUNCT cana-5460 373 9 βˆ‚w βˆ‚w PROPN cana-5460 373 10 βˆ‚x βˆ‚x PROPN cana-5460 373 11 ) ) PUNCT cana-5460 373 12 ) ) PUNCT cana-5460 373 13 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 373 14 ) ) PUNCT cana-5460 374 1 + + CCONJ cana-5460 374 2 ( ( PUNCT cana-5460 374 3 u u NOUN cana-5460 374 4 , , PUNCT cana-5460 374 5 β„‘xΟ‡ β„‘xΟ‡ PROPN cana-5460 374 6 ( ( PUNCT cana-5460 374 7 x x PROPN cana-5460 374 8 , , PUNCT cana-5460 374 9 t t PROPN cana-5460 374 10 , , PUNCT cana-5460 374 11 w w PROPN cana-5460 374 12 , , PUNCT cana-5460 374 13 βˆ‚w βˆ‚w PROPN cana-5460 374 14 βˆ‚x βˆ‚x PROPN cana-5460 374 15 ) ) PUNCT cana-5460 374 16 ) ) PUNCT cana-5460 374 17 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 374 18 ) ) PUNCT cana-5460 374 19 , , PUNCT cana-5460 374 20 ( ( PUNCT cana-5460 374 21 120 120 NUM cana-5460 374 22 ) ) PUNCT cana-5460 374 23 after after ADP cana-5460 374 24 using use VERB cana-5460 374 25 ( ( PUNCT cana-5460 374 26 91 91 NUM cana-5460 374 27 ) ) PUNCT cana-5460 374 28 , , PUNCT cana-5460 374 29 then then ADV cana-5460 374 30 ( ( PUNCT cana-5460 374 31 120 120 NUM cana-5460 374 32 ) ) PUNCT cana-5460 374 33 becomes become VERB cana-5460 374 34 : : PUNCT cana-5460 374 35 ΞΊ(w(n ΞΊ(w(n PROPN cana-5460 374 36 ) ) PUNCT cana-5460 374 37 βˆ’w βˆ’w NOUN cana-5460 374 38 , , PUNCT cana-5460 374 39 u u NOUN cana-5460 374 40 ) ) PUNCT cana-5460 374 41 = = SYM cana-5460 375 1 βˆ’ βˆ’ PROPN cana-5460 375 2 ( ( PUNCT cana-5460 375 3 βˆ‚ βˆ‚ NUM cana-5460 375 4 0 0 NUM cana-5460 375 5 c c PROPN cana-5460 375 6 t t PROPN cana-5460 375 7 Ξ΄β„‘x(w Ξ΄β„‘x(w PROPN cana-5460 375 8 ( ( PUNCT cana-5460 375 9 n n CCONJ cana-5460 375 10 ) ) PUNCT cana-5460 375 11 βˆ’w βˆ’w ADV cana-5460 375 12 ) ) PUNCT cana-5460 375 13 , , PUNCT cana-5460 375 14 u u NOUN cana-5460 375 15 ) ) PUNCT cana-5460 375 16 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 375 17 ) ) PUNCT cana-5460 376 1 + + CCONJ cana-5460 376 2 Ξ± Ξ± X cana-5460 376 3 ( ( PUNCT cana-5460 376 4 βˆ‚(w(n βˆ‚(w(n ADJ cana-5460 376 5 ) ) PUNCT cana-5460 376 6 βˆ’w βˆ’w ADV cana-5460 376 7 ) ) PUNCT cana-5460 376 8 βˆ‚x βˆ‚x PROPN cana-5460 376 9 , , PUNCT cana-5460 376 10 u u NOUN cana-5460 376 11 ) ) PUNCT cana-5460 376 12 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 376 13 ) ) PUNCT cana-5460 377 1 + + CCONJ cana-5460 377 2 Ξ² Ξ² X cana-5460 377 3 ( ( PUNCT cana-5460 377 4 βˆ‚ βˆ‚ X cana-5460 377 5 βˆ‚t βˆ‚t PROPN cana-5460 377 6 ( ( PUNCT cana-5460 377 7 βˆ‚w(n βˆ‚w(n PROPN cana-5460 377 8 ) ) PUNCT cana-5460 377 9 βˆ‚x βˆ‚x PROPN cana-5460 377 10 ) ) PUNCT cana-5460 377 11 , , PUNCT cana-5460 377 12 u u NOUN cana-5460 377 13 ) ) PUNCT cana-5460 377 14 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 377 15 ) ) PUNCT cana-5460 378 1 + + CCONJ cana-5460 378 2 Ξ³(β„‘x(w Ξ³(β„‘x(w NUM cana-5460 378 3 ( ( PUNCT cana-5460 378 4 n n CCONJ cana-5460 378 5 ) ) PUNCT cana-5460 378 6 βˆ’w βˆ’w ADV cana-5460 378 7 ) ) PUNCT cana-5460 378 8 , , PUNCT cana-5460 378 9 u u NOUN cana-5460 378 10 ) ) PUNCT cana-5460 378 11 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 378 12 ) ) PUNCT cana-5460 379 1 + + CCONJ cana-5460 379 2 ( ( PUNCT cana-5460 379 3 ∫ ∫ PROPN cana-5460 379 4 a(t a(t PROPN cana-5460 379 5 βˆ’ βˆ’ PROPN cana-5460 379 6 s)β„‘x(w s)β„‘x(w NOUN cana-5460 379 7 ( ( PUNCT cana-5460 379 8 n n CCONJ cana-5460 379 9 ) ) PUNCT cana-5460 379 10 βˆ’w)(x βˆ’w)(x NOUN cana-5460 379 11 , , PUNCT cana-5460 379 12 s)ds s)ds PROPN cana-5460 379 13 t t PROPN cana-5460 379 14 0 0 NUM cana-5460 379 15 , , PUNCT cana-5460 379 16 u u NOUN cana-5460 379 17 ) ) PUNCT cana-5460 379 18 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 379 19 ) ) PUNCT cana-5460 379 20 ( ( PUNCT cana-5460 379 21 121 121 NUM cana-5460 379 22 ) ) PUNCT cana-5460 379 23 after after ADP cana-5460 379 24 applying apply VERB cana-5460 379 25 the the DET cana-5460 379 26 integration integration NOUN cana-5460 379 27 by by ADP cana-5460 379 28 parts part NOUN cana-5460 379 29 , , PUNCT cana-5460 379 30 and and CCONJ cana-5460 379 31 taking take VERB cana-5460 379 32 in in ADP cana-5460 379 33 consideration consideration NOUN cana-5460 379 34 conditions condition NOUN cana-5460 379 35 on on ADP cana-5460 379 36 : : PUNCT cana-5460 379 37 u u NOUN cana-5460 379 38 and and CCONJ cana-5460 379 39 w w PROPN cana-5460 379 40 , , PUNCT cana-5460 379 41 ( ( PUNCT cana-5460 379 42 121 121 NUM cana-5460 379 43 ) ) PUNCT cana-5460 379 44 will will AUX cana-5460 379 45 be be AUX cana-5460 379 46 transformed transform VERB cana-5460 379 47 as as ADP cana-5460 379 48 : : PUNCT cana-5460 379 49 ΞΊ(w(n ΞΊ(w(n PROPN cana-5460 379 50 ) ) PUNCT cana-5460 379 51 βˆ’w βˆ’w NOUN cana-5460 379 52 , , PUNCT cana-5460 379 53 u u NOUN cana-5460 379 54 ) ) PUNCT cana-5460 379 55 = = SYM cana-5460 380 1 βˆ’ βˆ’ PROPN cana-5460 380 2 ( ( PUNCT cana-5460 380 3 βˆ‚ βˆ‚ NUM cana-5460 380 4 0 0 NUM cana-5460 380 5 c c NOUN cana-5460 380 6 t t NOUN cana-5460 380 7 Ξ΄(w(n Ξ΄(w(n NOUN cana-5460 380 8 ) ) PUNCT cana-5460 380 9 βˆ’w βˆ’w ADV cana-5460 380 10 ) ) PUNCT cana-5460 380 11 , , PUNCT cana-5460 380 12 β„‘xu)𝕃2(Ο‰ β„‘xu)𝕃2(Ο‰ NOUN cana-5460 380 13 ) ) PUNCT cana-5460 381 1 + + CCONJ cana-5460 381 2 Ξ± Ξ± X cana-5460 381 3 ( ( PUNCT cana-5460 381 4 βˆ‚(w(n βˆ‚(w(n ADJ cana-5460 381 5 ) ) PUNCT cana-5460 381 6 βˆ’w βˆ’w ADV cana-5460 381 7 ) ) PUNCT cana-5460 381 8 βˆ‚x βˆ‚x PROPN cana-5460 381 9 , , PUNCT cana-5460 381 10 u u NOUN cana-5460 381 11 ) ) PUNCT cana-5460 381 12 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 381 13 ) ) PUNCT cana-5460 382 1 + + CCONJ cana-5460 382 2 Ξ² Ξ² X cana-5460 382 3 ( ( PUNCT cana-5460 382 4 ( ( PUNCT cana-5460 382 5 βˆ‚w(n βˆ‚w(n PROPN cana-5460 382 6 ) ) PUNCT cana-5460 382 7 βˆ‚x βˆ‚x PROPN cana-5460 382 8 ) ) PUNCT cana-5460 382 9 , , PUNCT cana-5460 382 10 βˆ‚u βˆ‚u PROPN cana-5460 382 11 βˆ‚t βˆ‚t PROPN cana-5460 382 12 ) ) PUNCT cana-5460 382 13 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 382 14 ) ) PUNCT cana-5460 383 1 + + CCONJ cana-5460 383 2 Ξ³(β„‘x(w Ξ³(β„‘x(w NUM cana-5460 383 3 ( ( PUNCT cana-5460 383 4 n n CCONJ cana-5460 383 5 ) ) PUNCT cana-5460 383 6 βˆ’ βˆ’ PROPN cana-5460 383 7 w w NOUN cana-5460 383 8 ) ) PUNCT cana-5460 383 9 , , PUNCT cana-5460 383 10 u u NOUN cana-5460 383 11 ) ) PUNCT cana-5460 383 12 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 383 13 ) ) PUNCT cana-5460 384 1 + + CCONJ cana-5460 384 2 ( ( PUNCT cana-5460 384 3 ∫ ∫ PROPN cana-5460 384 4 a(t a(t PROPN cana-5460 384 5 βˆ’ βˆ’ PROPN cana-5460 384 6 s)β„‘x(w s)β„‘x(w NOUN cana-5460 384 7 ( ( PUNCT cana-5460 384 8 n n CCONJ cana-5460 384 9 ) ) PUNCT cana-5460 384 10 βˆ’w)(x βˆ’w)(x NOUN cana-5460 384 11 , , PUNCT cana-5460 384 12 s)ds s)ds PROPN cana-5460 384 13 t t PROPN cana-5460 384 14 0 0 NUM cana-5460 384 15 , , PUNCT cana-5460 384 16 u u NOUN cana-5460 384 17 ) ) PUNCT cana-5460 384 18 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 384 19 ) ) PUNCT cana-5460 384 20 ( ( PUNCT cana-5460 384 21 122 122 X cana-5460 384 22 ) ) PUNCT cana-5460 384 23 we we PRON cana-5460 384 24 will will AUX cana-5460 384 25 apply apply VERB cana-5460 384 26 the the DET cana-5460 384 27 inequality inequality NOUN cana-5460 384 28 of of ADP cana-5460 384 29 cauchy cauchy PROPN cana-5460 384 30 - - PUNCT cana-5460 384 31 schwartz schwartz PROPN cana-5460 384 32 and and CCONJ cana-5460 384 33 lemma lemma PROPN cana-5460 384 34 ( ( PUNCT cana-5460 384 35 2.4 2.4 NUM cana-5460 384 36 ) ) PUNCT cana-5460 384 37 , , PUNCT cana-5460 384 38 we we PRON cana-5460 384 39 have have VERB cana-5460 384 40 : : PUNCT cana-5460 384 41 ΞΊ(w(n ΞΊ(w(n PROPN cana-5460 384 42 ) ) PUNCT cana-5460 384 43 βˆ’ βˆ’ PROPN cana-5460 384 44 w w PROPN cana-5460 384 45 , , PUNCT cana-5460 384 46 u u NOUN cana-5460 384 47 ) ) PUNCT cana-5460 384 48 ≀ ≀ NUM cana-5460 384 49 ΞΆβ€–w(n ΞΆβ€–w(n NOUN cana-5460 384 50 ) ) PUNCT cana-5460 384 51 βˆ’ βˆ’ PROPN cana-5460 384 52 wβ€– wβ€– PROPN cana-5460 384 53 𝕃2(0,t 𝕃2(0,t NOUN cana-5460 384 54 , , PUNCT cana-5460 384 55 h1(Ο‰ h1(Ο‰ PROPN cana-5460 384 56 ) ) PUNCT cana-5460 384 57 ) ) PUNCT cana-5460 384 58 . . PUNCT cana-5460 385 1 [ [ X cana-5460 385 2 β€–u‖𝕃2(Ο‰ β€–u‖𝕃2(Ο‰ ADV cana-5460 385 3 ) ) PUNCT cana-5460 385 4 + + CCONJ cana-5460 385 5 β€– β€– PROPN cana-5460 385 6 βˆ‚u βˆ‚u PROPN cana-5460 385 7 βˆ‚t βˆ‚t PROPN cana-5460 385 8 β€– β€– PROPN cana-5460 385 9 𝕃2(Ο‰ 𝕃2(Ο‰ PROPN cana-5460 385 10 ) ) PUNCT cana-5460 385 11 ] ] PUNCT cana-5460 385 12 ( ( PUNCT cana-5460 385 13 123 123 NUM cana-5460 385 14 ) ) PUNCT cana-5460 385 15 such such ADJ cana-5460 385 16 that that SCONJ cana-5460 385 17 : : PUNCT cana-5460 385 18 ΞΆ ΞΆ NOUN cana-5460 385 19 = = SYM cana-5460 385 20 max max PROPN cana-5460 385 21 ( ( PUNCT cana-5460 385 22 Ξ± Ξ± PROPN cana-5460 385 23 + + X cana-5460 385 24 t t PROPN cana-5460 385 25 2 2 NUM cana-5460 385 26 + + CCONJ cana-5460 385 27 Ξ³ Ξ³ X cana-5460 385 28 2 2 NUM cana-5460 385 29 , , PUNCT cana-5460 385 30 Ξ² Ξ² NOUN cana-5460 385 31 ) ) PUNCT cana-5460 385 32 , , PUNCT cana-5460 385 33 communications communication NOUN cana-5460 385 34 on on ADP cana-5460 385 35 applied apply VERB cana-5460 385 36 nonlinear nonlinear ADJ cana-5460 385 37 analysis analysis NOUN cana-5460 385 38 issn issn NOUN cana-5460 385 39 : : PUNCT cana-5460 385 40 1074 1074 NUM cana-5460 385 41 - - PUNCT cana-5460 385 42 133x 133x NUM cana-5460 385 43 vol vol NOUN cana-5460 385 44 32 32 NUM cana-5460 385 45 no.3 no.3 NOUN cana-5460 385 46 ( ( PUNCT cana-5460 385 47 2025 2025 NUM cana-5460 385 48 ) ) PUNCT cana-5460 385 49 960 960 NUM cana-5460 385 50 https://internationalpubls.com https://internationalpubls.com X cana-5460 385 51 and and CCONJ cana-5460 385 52 from from ADP cana-5460 385 53 ( ( PUNCT cana-5460 385 54 120 120 NUM cana-5460 385 55 ) ) PUNCT cana-5460 385 56 , , PUNCT cana-5460 385 57 we we PRON cana-5460 385 58 have have VERB cana-5460 385 59 the the DET cana-5460 385 60 following follow VERB cana-5460 385 61 estimation estimation NOUN cana-5460 385 62 : : PUNCT cana-5460 385 63 ( ( PUNCT cana-5460 385 64 u u NOUN cana-5460 385 65 , , PUNCT cana-5460 385 66 β„‘xΟ‡ β„‘xΟ‡ PROPN cana-5460 385 67 ( ( PUNCT cana-5460 385 68 x x PROPN cana-5460 385 69 , , PUNCT cana-5460 385 70 t t PROPN cana-5460 385 71 , , PUNCT cana-5460 385 72 w w PROPN cana-5460 385 73 ( ( PUNCT cana-5460 385 74 nβˆ’1 nβˆ’1 PROPN cana-5460 385 75 ) ) PUNCT cana-5460 385 76 , , PUNCT cana-5460 385 77 βˆ‚w(nβˆ’1 βˆ‚w(nβˆ’1 PROPN cana-5460 385 78 ) ) PUNCT cana-5460 385 79 βˆ‚x βˆ‚x PROPN cana-5460 385 80 ) ) PUNCT cana-5460 386 1 βˆ’ βˆ’ PROPN cana-5460 387 1 β„‘xΟ‡ β„‘xΟ‡ PROPN cana-5460 387 2 ( ( PUNCT cana-5460 387 3 x x PROPN cana-5460 387 4 , , PUNCT cana-5460 387 5 t t PROPN cana-5460 387 6 , , PUNCT cana-5460 387 7 w w PROPN cana-5460 387 8 , , PUNCT cana-5460 387 9 βˆ‚w βˆ‚w PROPN cana-5460 387 10 βˆ‚x βˆ‚x PROPN cana-5460 387 11 ) ) PUNCT cana-5460 387 12 ) ) PUNCT cana-5460 388 1 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 388 2 ) ) PUNCT cana-5460 388 3 ≀ ≀ NUM cana-5460 388 4 m m VERB cana-5460 388 5 √2 √2 ADP cana-5460 388 6 β€–w(n β€–w(n NOUN cana-5460 388 7 ) ) PUNCT cana-5460 388 8 βˆ’wβ€– βˆ’wβ€– ADJ cana-5460 388 9 𝕃2(0,t 𝕃2(0,t NOUN cana-5460 388 10 , , PUNCT cana-5460 388 11 h1(Ο‰ h1(Ο‰ PROPN cana-5460 388 12 ) ) PUNCT cana-5460 388 13 ) ) PUNCT cana-5460 388 14 . . PUNCT cana-5460 389 1 β€–u‖𝕃2(Ο‰ β€–u‖𝕃2(Ο‰ ADV cana-5460 389 2 ) ) PUNCT cana-5460 389 3 ( ( PUNCT cana-5460 389 4 124 124 NUM cana-5460 389 5 ) ) PUNCT cana-5460 389 6 when when SCONJ cana-5460 389 7 the the DET cana-5460 389 8 limit limit NOUN cana-5460 389 9 n n X cana-5460 389 10 β†’ β†’ SYM cana-5460 389 11 ∞ ∞ NUM cana-5460 389 12 in in ADP cana-5460 389 13 ( ( PUNCT cana-5460 389 14 122 122 NUM cana-5460 389 15 ) ) PUNCT cana-5460 389 16 , , PUNCT cana-5460 389 17 and and CCONJ cana-5460 389 18 we we PRON cana-5460 389 19 take take VERB cana-5460 389 20 in in ADP cana-5460 389 21 consideration consideration NOUN cana-5460 389 22 ( ( PUNCT cana-5460 389 23 123 123 NUM cana-5460 389 24 ) ) PUNCT cana-5460 389 25 and and CCONJ cana-5460 389 26 ( ( PUNCT cana-5460 389 27 124 124 NUM cana-5460 389 28 ) ) PUNCT cana-5460 389 29 , , PUNCT cana-5460 389 30 we we PRON cana-5460 389 31 obtain obtain VERB cana-5460 389 32 ΞΊ(w ΞΊ(w NOUN cana-5460 389 33 , , PUNCT cana-5460 389 34 u u NOUN cana-5460 389 35 ) ) PUNCT cana-5460 389 36 = = SYM cana-5460 389 37 ( ( PUNCT cana-5460 389 38 u u NOUN cana-5460 389 39 , , PUNCT cana-5460 389 40 β„‘xΟ‡ β„‘xΟ‡ PROPN cana-5460 389 41 ( ( PUNCT cana-5460 389 42 x x PROPN cana-5460 389 43 , , PUNCT cana-5460 389 44 t t PROPN cana-5460 389 45 , , PUNCT cana-5460 389 46 w w PROPN cana-5460 389 47 , , PUNCT cana-5460 389 48 βˆ‚w βˆ‚w PROPN cana-5460 389 49 βˆ‚x βˆ‚x PROPN cana-5460 389 50 ) ) PUNCT cana-5460 389 51 ) ) PUNCT cana-5460 389 52 𝕃2(Ο‰ 𝕃2(Ο‰ ADV cana-5460 389 53 ) ) PUNCT cana-5460 389 54 . . PUNCT cana-5460 390 1 ( ( PUNCT cana-5460 390 2 125 125 NUM cana-5460 390 3 ) ) PUNCT cana-5460 390 4 so so SCONJ cana-5460 390 5 the the DET cana-5460 390 6 problem problem NOUN cana-5460 390 7 ( ( PUNCT cana-5460 390 8 94)-(96 94)-(96 NOUN cana-5460 390 9 ) ) PUNCT cana-5460 390 10 admit admit VERB cana-5460 390 11 a a DET cana-5460 390 12 weak weak ADJ cana-5460 390 13 solution solution NOUN cana-5460 390 14 . . PUNCT cana-5460 391 1 now now ADV cana-5460 391 2 , , PUNCT cana-5460 391 3 we we PRON cana-5460 391 4 will will AUX cana-5460 391 5 prove prove VERB cana-5460 391 6 the the DET cana-5460 391 7 uniqueness uniqueness NOUN cana-5460 391 8 of of ADP cana-5460 391 9 problem problem NOUN cana-5460 391 10 ( ( PUNCT cana-5460 391 11 79)-(81 79)-(81 NUM cana-5460 391 12 ) ) PUNCT cana-5460 391 13 . . PUNCT cana-5460 392 1 theorem theorem VERB cana-5460 392 2 : : PUNCT cana-5460 392 3 under under ADP cana-5460 392 4 condition condition NOUN cana-5460 392 5 of of ADP cana-5460 392 6 lemma lemma PROPN cana-5460 392 7 ( ( PUNCT cana-5460 392 8 76 76 NUM cana-5460 392 9 ) ) PUNCT cana-5460 392 10 , , PUNCT cana-5460 392 11 the the DET cana-5460 392 12 problem problem NOUN cana-5460 392 13 ( ( PUNCT cana-5460 392 14 79)-(81 79)-(81 ADV cana-5460 392 15 ) ) PUNCT cana-5460 392 16 admits admit VERB cana-5460 392 17 a a DET cana-5460 392 18 unique unique ADJ cana-5460 392 19 solution solution NOUN cana-5460 392 20 . . PUNCT cana-5460 393 1 proof proof NOUN cana-5460 393 2 : : PUNCT cana-5460 393 3 we we PRON cana-5460 393 4 suppose suppose VERB cana-5460 393 5 that that SCONJ cana-5460 393 6 the the DET cana-5460 393 7 problem problem NOUN cana-5460 393 8 ( ( PUNCT cana-5460 393 9 79)-(81 79)-(81 ADV cana-5460 393 10 ) ) PUNCT cana-5460 393 11 admit admit PROPN cana-5460 393 12 u1 u1 NOUN cana-5460 393 13 , , PUNCT cana-5460 393 14 u2 u2 PROPN cana-5460 393 15 solutions solution NOUN cana-5460 393 16 in𝕃2(0 in𝕃2(0 PROPN cana-5460 393 17 , , PUNCT cana-5460 393 18 t t PROPN cana-5460 393 19 , , PUNCT cana-5460 393 20 h1(Ο‰ h1(Ο‰ PROPN cana-5460 393 21 ) ) PUNCT cana-5460 393 22 ) ) PUNCT cana-5460 393 23 , , PUNCT cana-5460 393 24 and and CCONJ cana-5460 393 25 w w NOUN cana-5460 393 26 = = NOUN cana-5460 393 27 u1 u1 NOUN cana-5460 393 28 βˆ’ βˆ’ PROPN cana-5460 393 29 u2 u2 PROPN cana-5460 393 30 , , PUNCT cana-5460 393 31 and and CCONJ cana-5460 393 32 verifies verifie NOUN cana-5460 393 33 : : PUNCT cana-5460 394 1 β„’w β„’w VERB cana-5460 394 2 = = SYM cana-5460 394 3 βˆ‚ βˆ‚ NUM cana-5460 394 4 0 0 NUM cana-5460 395 1 c c NOUN cana-5460 395 2 t t PROPN cana-5460 395 3 Ξ΄wβˆ’ Ξ΄wβˆ’ NUM cana-5460 395 4 Ξ± Ξ± PRON cana-5460 395 5 βˆ‚2w βˆ‚2w VERB cana-5460 395 6 βˆ‚x2 βˆ‚x2 NOUN cana-5460 395 7 βˆ’ βˆ’ NOUN cana-5460 395 8 Ξ² Ξ² X cana-5460 395 9 βˆ‚3w βˆ‚3w NOUN cana-5460 395 10 βˆ‚tβˆ‚x2 βˆ‚tβˆ‚x2 PROPN cana-5460 396 1 + + CCONJ cana-5460 396 2 Ξ³w Ξ³w NUM cana-5460 396 3 βˆ’ βˆ’ NOUN cana-5460 396 4 ∫ ∫ PROPN cana-5460 396 5 a(t a(t NOUN cana-5460 396 6 βˆ’ βˆ’ PROPN cana-5460 396 7 s)w(x s)w(x ADJ cana-5460 396 8 , , PUNCT cana-5460 396 9 s)ds s)ds PROPN cana-5460 396 10 = = SYM cana-5460 396 11 t t PROPN cana-5460 396 12 0 0 PUNCT cana-5460 397 1 n(x n(x PROPN cana-5460 397 2 , , PUNCT cana-5460 397 3 t t PROPN cana-5460 397 4 ) ) PUNCT cana-5460 397 5 ( ( PUNCT cana-5460 397 6 126 126 NUM cana-5460 397 7 ) ) PUNCT cana-5460 397 8 β„“w β„“w NOUN cana-5460 397 9 = = SYM cana-5460 397 10 w(x w(x PROPN cana-5460 397 11 , , PUNCT cana-5460 397 12 0 0 NUM cana-5460 397 13 ) ) PUNCT cana-5460 397 14 = = SYM cana-5460 397 15 0 0 NUM cana-5460 397 16 , , PUNCT cana-5460 397 17 qw qw X cana-5460 397 18 = = SYM cana-5460 397 19 βˆ‚w(x,0 βˆ‚w(x,0 PROPN cana-5460 397 20 ) ) PUNCT cana-5460 398 1 βˆ‚t βˆ‚t PROPN cana-5460 398 2 = = SYM cana-5460 398 3 0 0 PROPN cana-5460 398 4 , , PUNCT cana-5460 398 5 0 0 NUM cana-5460 398 6 < < X cana-5460 398 7 π‘₯ π‘₯ X cana-5460 398 8 < < X cana-5460 398 9 1 1 NUM cana-5460 398 10 . . PUNCT cana-5460 399 1 ( ( PUNCT cana-5460 399 2 127 127 NUM cana-5460 399 3 ) ) PUNCT cana-5460 399 4 ∫ ∫ PROPN cana-5460 400 1 w(x w(x NOUN cana-5460 400 2 , , PUNCT cana-5460 400 3 t)dx t)dx PROPN cana-5460 400 4 = = SYM cana-5460 400 5 0 0 NUM cana-5460 400 6 , , PUNCT cana-5460 400 7 t t PROPN cana-5460 400 8 0 0 NUM cana-5460 400 9 ∫ ∫ PROPN cana-5460 400 10 xw(x xw(x PROPN cana-5460 400 11 , , PUNCT cana-5460 400 12 t)dx t)dx PROPN cana-5460 400 13 = = SYM cana-5460 400 14 0 0 NUM cana-5460 400 15 , , PUNCT cana-5460 400 16 t t PROPN cana-5460 400 17 0 0 NUM cana-5460 400 18 0 0 NUM cana-5460 400 19 < < X cana-5460 400 20 𝑑 𝑑 PROPN cana-5460 400 21 ≀ ≀ NOUN cana-5460 400 22 𝑇 𝑇 PROPN cana-5460 400 23 . . PUNCT cana-5460 401 1 ( ( PUNCT cana-5460 401 2 128 128 NUM cana-5460 401 3 ) ) PUNCT cana-5460 402 1 where where SCONJ cana-5460 402 2 : : PUNCT cana-5460 402 3 n(x n(x PROPN cana-5460 402 4 , , PUNCT cana-5460 402 5 t t PROPN cana-5460 402 6 ) ) PUNCT cana-5460 402 7 = = PUNCT cana-5460 403 1 Ο‡(x Ο‡(x PROPN cana-5460 403 2 , , PUNCT cana-5460 403 3 t t PROPN cana-5460 403 4 , , PUNCT cana-5460 403 5 u1 u1 NOUN cana-5460 403 6 , , PUNCT cana-5460 403 7 r1 r1 PROPN cana-5460 403 8 ) ) PUNCT cana-5460 403 9 βˆ’ βˆ’ PROPN cana-5460 404 1 Ο‡(x Ο‡(x PROPN cana-5460 404 2 , , PUNCT cana-5460 404 3 t t PROPN cana-5460 404 4 , , PUNCT cana-5460 404 5 u2 u2 NOUN cana-5460 404 6 , , PUNCT cana-5460 404 7 r2 r2 PROPN cana-5460 404 8 ) ) PUNCT cana-5460 404 9 this this PRON cana-5460 404 10 will will AUX cana-5460 404 11 be be AUX cana-5460 404 12 done do VERB cana-5460 404 13 by by ADP cana-5460 404 14 establishing establish VERB cana-5460 404 15 the the DET cana-5460 404 16 same same ADJ cana-5460 404 17 proof proof NOUN cana-5460 404 18 of of ADP cana-5460 404 19 lemma lemma PROPN cana-5460 404 20 ( ( PUNCT cana-5460 404 21 76 76 NUM cana-5460 404 22 ) ) PUNCT cana-5460 404 23 , , PUNCT cana-5460 404 24 we we PRON cana-5460 404 25 obtain obtain VERB cana-5460 404 26 : : PUNCT cana-5460 404 27 β€–wβ€– β€–wβ€– PROPN cana-5460 404 28 𝕃2(0,t 𝕃2(0,t NOUN cana-5460 404 29 , , PUNCT cana-5460 404 30 h1(Ο‰ h1(Ο‰ PROPN cana-5460 404 31 ) ) PUNCT cana-5460 404 32 ) ) PUNCT cana-5460 404 33 ≀ ≀ NUM cana-5460 405 1 cβ€–wβ€– cβ€–wβ€– PROPN cana-5460 405 2 𝕃2(0,t 𝕃2(0,t NOUN cana-5460 405 3 , , PUNCT cana-5460 405 4 h1(Ο‰ h1(Ο‰ PROPN cana-5460 405 5 ) ) PUNCT cana-5460 405 6 ) ) PUNCT cana-5460 405 7 . . PUNCT cana-5460 406 1 ( ( PUNCT cana-5460 406 2 129 129 NUM cana-5460 406 3 ) ) PUNCT cana-5460 406 4 since since SCONJ cana-5460 406 5 c c X cana-5460 406 6 < < X cana-5460 406 7 1 1 NUM cana-5460 406 8 , , PUNCT cana-5460 406 9 then then ADV cana-5460 406 10 : : PUNCT cana-5460 406 11 ( ( PUNCT cana-5460 406 12 1 1 NUM cana-5460 406 13 βˆ’ βˆ’ NOUN cana-5460 406 14 c)β€–wβ€– c)β€–wβ€– NOUN cana-5460 406 15 𝕃2(0,t 𝕃2(0,t NOUN cana-5460 406 16 , , PUNCT cana-5460 406 17 h1(Ο‰ h1(Ο‰ PROPN cana-5460 406 18 ) ) PUNCT cana-5460 406 19 ) ) PUNCT cana-5460 406 20 ≀ ≀ NOUN cana-5460 406 21 0 0 NUM cana-5460 406 22 , , PUNCT cana-5460 406 23 we we PRON cana-5460 406 24 deduce deduce VERB cana-5460 406 25 finally finally ADV cana-5460 406 26 that that PRON cana-5460 406 27 : : PUNCT cana-5460 406 28 u1 u1 VERB cana-5460 406 29 βˆ’ βˆ’ PROPN cana-5460 406 30 u2 u2 PROPN cana-5460 406 31 = = PROPN cana-5460 406 32 0 0 NUM cana-5460 406 33 , , PUNCT cana-5460 406 34 so so ADV cana-5460 406 35 u1 u1 NOUN cana-5460 406 36 = = PROPN cana-5460 406 37 u2 u2 NOUN cana-5460 406 38 in in ADP cana-5460 406 39 𝕃2(0 𝕃2(0 NOUN cana-5460 406 40 , , PUNCT cana-5460 406 41 t t PROPN cana-5460 406 42 , , PUNCT cana-5460 406 43 h1(Ο‰)).∎ h1(Ο‰)).∎ NOUN cana-5460 406 44 communications communication NOUN cana-5460 406 45 on on ADP cana-5460 406 46 applied apply VERB cana-5460 406 47 nonlinear nonlinear ADJ cana-5460 406 48 analysis analysis NOUN cana-5460 406 49 issn issn NOUN cana-5460 406 50 : : PUNCT cana-5460 406 51 1074 1074 NUM cana-5460 406 52 - - PUNCT cana-5460 406 53 133x 133x NUM cana-5460 406 54 vol vol NOUN cana-5460 406 55 32 32 NUM cana-5460 406 56 no.3 no.3 NOUN cana-5460 406 57 ( ( PUNCT cana-5460 406 58 2025 2025 NUM cana-5460 406 59 ) ) PUNCT cana-5460 406 60 961 961 NUM cana-5460 406 61 https://internationalpubls.com https://internationalpubls.com X cana-5460 406 62 8 8 NUM cana-5460 406 63 . . PUNCT cana-5460 407 1 conclusion conclusion NOUN cana-5460 407 2 based base VERB cana-5460 407 3 on on ADP cana-5460 407 4 the the DET cana-5460 407 5 a a DET cana-5460 407 6 priori priori ADJ cana-5460 407 7 estimate estimate NOUN cana-5460 407 8 method method NOUN cana-5460 407 9 and and CCONJ cana-5460 407 10 the the DET cana-5460 407 11 density density NOUN cana-5460 407 12 of of ADP cana-5460 407 13 the the DET cana-5460 407 14 operator operator NOUN cana-5460 407 15 generated generate VERB cana-5460 407 16 by by ADP cana-5460 407 17 the the DET cana-5460 407 18 problem problem NOUN cana-5460 407 19 , , PUNCT cana-5460 407 20 and and CCONJ cana-5460 407 21 the the DET cana-5460 407 22 iterative iterative NOUN cana-5460 407 23 process process NOUN cana-5460 407 24 , , PUNCT cana-5460 407 25 we we PRON cana-5460 407 26 have have AUX cana-5460 407 27 successfully successfully ADV cana-5460 407 28 proven prove VERB cana-5460 407 29 the the DET cana-5460 407 30 problem problem NOUN cana-5460 407 31 's 's PART cana-5460 407 32 existence existence NOUN cana-5460 407 33 and and CCONJ cana-5460 407 34 uniqueness uniqueness NOUN cana-5460 407 35 of of ADP cana-5460 407 36 its its PRON cana-5460 407 37 solution solution NOUN cana-5460 407 38 . . PUNCT cana-5460 408 1 after after ADP cana-5460 408 2 carrying carry VERB cana-5460 408 3 out out ADP cana-5460 408 4 our our PRON cana-5460 408 5 experimental experimental ADJ cana-5460 408 6 research research NOUN cana-5460 408 7 , , PUNCT cana-5460 408 8 we we PRON cana-5460 408 9 were be AUX cana-5460 408 10 able able ADJ cana-5460 408 11 to to PART cana-5460 408 12 confirm confirm VERB cana-5460 408 13 our our PRON cana-5460 408 14 initial initial ADJ cana-5460 408 15 hypotheses hypothesis NOUN cana-5460 408 16 and and CCONJ cana-5460 408 17 respond respond VERB cana-5460 408 18 to to ADP cana-5460 408 19 our our PRON cana-5460 408 20 problem problem NOUN cana-5460 408 21 . . PUNCT cana-5460 409 1 this this DET cana-5460 409 2 work work NOUN cana-5460 409 3 could could AUX cana-5460 409 4 be be AUX cana-5460 409 5 considered consider VERB cana-5460 409 6 as as ADP cana-5460 409 7 a a DET cana-5460 409 8 contribution contribution NOUN cana-5460 409 9 to to ADP cana-5460 409 10 the the DET cana-5460 409 11 development development NOUN cana-5460 409 12 of of ADP cana-5460 409 13 the the DET cana-5460 409 14 functional functional ADJ cana-5460 409 15 analysis analysis NOUN cana-5460 409 16 method method NOUN cana-5460 409 17 . . PUNCT cana-5460 410 1 and and CCONJ cana-5460 410 2 to to PART cana-5460 410 3 conclude conclude VERB cana-5460 410 4 we we PRON cana-5460 410 5 argue argue VERB cana-5460 410 6 that that SCONJ cana-5460 410 7 the the DET cana-5460 410 8 research research NOUN cana-5460 410 9 carried carry VERB cana-5460 410 10 out out ADP cana-5460 410 11 in in ADP cana-5460 410 12 this this DET cana-5460 410 13 article article NOUN cana-5460 410 14 could could AUX cana-5460 410 15 serve serve VERB cana-5460 410 16 as as ADP cana-5460 410 17 a a DET cana-5460 410 18 contribution contribution NOUN cana-5460 410 19 for for ADP cana-5460 410 20 possible possible ADJ cana-5460 410 21 studies study NOUN cana-5460 410 22 in in ADP cana-5460 410 23 the the DET cana-5460 410 24 field field NOUN cana-5460 410 25 of of ADP cana-5460 410 26 applied apply VERB cana-5460 410 27 mathematics mathematic NOUN cana-5460 410 28 . . PUNCT cana-5460 411 1 our our PRON cana-5460 411 2 study study NOUN cana-5460 411 3 being be AUX cana-5460 411 4 based base VERB cana-5460 411 5 on on ADP cana-5460 411 6 the the DET cana-5460 411 7 a a DET cana-5460 411 8 priori priori ADJ cana-5460 411 9 estimation estimation NOUN cana-5460 411 10 method method NOUN cana-5460 411 11 is be AUX cana-5460 411 12 not not PART cana-5460 411 13 exhaustive exhaustive ADJ cana-5460 411 14 and and CCONJ cana-5460 411 15 opens open VERB cana-5460 411 16 several several ADJ cana-5460 411 17 research research NOUN cana-5460 411 18 perspectives perspective NOUN cana-5460 411 19 such such ADJ cana-5460 411 20 as as ADP cana-5460 411 21 : : PUNCT cana-5460 411 22 the the DET cana-5460 411 23 application application NOUN cana-5460 411 24 of of ADP cana-5460 411 25 the the DET cana-5460 411 26 latter latter ADJ cana-5460 411 27 to to ADP cana-5460 411 28 ordinary ordinary ADJ cana-5460 411 29 physical physical ADJ cana-5460 411 30 problems problem NOUN cana-5460 411 31 and and CCONJ cana-5460 411 32 other other ADJ cana-5460 411 33 purely purely ADV cana-5460 411 34 fractional fractional ADJ cana-5460 411 35 ones one NOUN cana-5460 411 36 . . PUNCT cana-5460 412 1 references reference NOUN cana-5460 412 2 [ [ X cana-5460 412 3 1 1 NUM cana-5460 412 4 ] ] PUNCT cana-5460 412 5 a. a. NOUN cana-5460 412 6 bouziani bouziani PROPN cana-5460 412 7 ; ; PUNCT cana-5460 412 8 on on ADP cana-5460 412 9 the the DET cana-5460 412 10 solvability solvability NOUN cana-5460 412 11 of of ADP cana-5460 412 12 a a DET cana-5460 412 13 class class NOUN cana-5460 412 14 of of ADP cana-5460 412 15 singular singular ADJ cana-5460 412 16 parabolic parabolic NOUN cana-5460 412 17 equations equation NOUN cana-5460 412 18 with with ADP cana-5460 412 19 nonlocal nonlocal ADJ cana-5460 412 20 boundary boundary ADJ cana-5460 412 21 conditions condition NOUN cana-5460 412 22 in in ADP cana-5460 412 23 non non ADJ cana-5460 412 24 classical classical ADJ cana-5460 412 25 function function NOUN cana-5460 412 26 spaces space NOUN cana-5460 412 27 , , PUNCT cana-5460 412 28 ijmms ijmms NOUN cana-5460 412 29 , , PUNCT cana-5460 412 30 v v NOUN cana-5460 412 31 30 30 NUM cana-5460 412 32 ( ( PUNCT cana-5460 412 33 2002 2002 NUM cana-5460 412 34 ) ) PUNCT cana-5460 412 35 , , PUNCT cana-5460 412 36 issue issue NOUN cana-5460 412 37 7 7 NUM cana-5460 412 38 , , PUNCT cana-5460 412 39 pages page NOUN cana-5460 412 40 435 435 NUM cana-5460 412 41 - - SYM cana-5460 412 42 447 447 NUM cana-5460 412 43 [ [ X cana-5460 412 44 2 2 NUM cana-5460 412 45 ] ] PUNCT cana-5460 412 46 a. a. NOUN cana-5460 412 47 bouziani bouziani PROPN cana-5460 412 48 ; ; PUNCT cana-5460 412 49 on on ADP cana-5460 412 50 the the DET cana-5460 412 51 solvability solvability NOUN cana-5460 412 52 of of ADP cana-5460 412 53 parabolic parabolic ADJ cana-5460 412 54 and and CCONJ cana-5460 412 55 hyperbolic hyperbolic ADJ cana-5460 412 56 problems problem NOUN cana-5460 412 57 with with ADP cana-5460 412 58 a a DET cana-5460 412 59 boundary boundary ADJ cana-5460 412 60 integral integral ADJ cana-5460 412 61 condition condition NOUN cana-5460 412 62 , , PUNCT cana-5460 412 63 ijmms ijmms NOUN cana-5460 412 64 journal journal NOUN cana-5460 412 65 , , PUNCT cana-5460 412 66 volume volume NOUN cana-5460 412 67 31 31 NUM cana-5460 412 68 ( ( PUNCT cana-5460 412 69 2002 2002 NUM cana-5460 412 70 ) ) PUNCT cana-5460 412 71 , , PUNCT cana-5460 412 72 issue issue NOUN cana-5460 412 73 4 4 NUM cana-5460 412 74 , , PUNCT cana-5460 412 75 pages page NOUN cana-5460 412 76 201 201 NUM cana-5460 412 77 - - SYM cana-5460 412 78 213 213 NUM cana-5460 412 79 [ [ SYM cana-5460 412 80 3 3 NUM cana-5460 412 81 ] ] PUNCT cana-5460 412 82 a. a. NOUN cana-5460 412 83 bouziani bouziani PROPN cana-5460 412 84 ; ; PUNCT cana-5460 412 85 on on ADP cana-5460 412 86 a a DET cana-5460 412 87 class class NOUN cana-5460 412 88 of of ADP cana-5460 412 89 nonlinear nonlinear ADJ cana-5460 412 90 reaction reaction NOUN cana-5460 412 91 - - PUNCT cana-5460 412 92 diffusion diffusion NOUN cana-5460 412 93 systems system NOUN cana-5460 412 94 with with ADP cana-5460 412 95 nonlocal nonlocal ADJ cana-5460 412 96 boundary boundary ADJ cana-5460 412 97 conditions condition NOUN cana-5460 412 98 , , PUNCT cana-5460 412 99 a. a. NOUN cana-5460 413 1 [ [ X cana-5460 413 2 4 4 NUM cana-5460 413 3 ] ] PUNCT cana-5460 413 4 a. a. NOUN cana-5460 413 5 bouziani bouziani PROPN cana-5460 413 6 ; ; PUNCT cana-5460 413 7 on on ADP cana-5460 413 8 the the DET cana-5460 413 9 weak weak ADJ cana-5460 413 10 solution solution NOUN cana-5460 413 11 of of ADP cana-5460 413 12 a a DET cana-5460 413 13 three three NUM cana-5460 413 14 - - PUNCT cana-5460 413 15 point point NOUN cana-5460 413 16 boundary boundary ADJ cana-5460 413 17 value value NOUN cana-5460 413 18 problem problem NOUN cana-5460 413 19 for for ADP cana-5460 413 20 a a DET cana-5460 413 21 class class NOUN cana-5460 413 22 of of ADP cana-5460 413 23 parabolic parabolic ADJ cana-5460 413 24 equations equation NOUN cana-5460 413 25 with with ADP cana-5460 413 26 energy energy NOUN cana-5460 413 27 specification specification NOUN cana-5460 413 28 , , PUNCT cana-5460 413 29 abstract abstract ADJ cana-5460 413 30 and and CCONJ cana-5460 413 31 applied apply VERB cana-5460 413 32 analysis analysis NOUN cana-5460 413 33 , , PUNCT cana-5460 413 34 v v ADP cana-5460 413 35 2003 2003 NUM cana-5460 413 36 ( ( PUNCT cana-5460 413 37 2003 2003 NUM cana-5460 413 38 ) ) PUNCT cana-5460 413 39 , , PUNCT cana-5460 413 40 issue issue NOUN cana-5460 413 41 10 10 NUM cana-5460 413 42 , , PUNCT cana-5460 413 43 pages page NOUN cana-5460 413 44 573 573 NUM cana-5460 413 45 - - SYM cana-5460 413 46 589 589 NUM cana-5460 413 47 . . PUNCT cana-5460 414 1 [ [ X cana-5460 414 2 5 5 NUM cana-5460 414 3 ] ] PUNCT cana-5460 414 4 a. a. NOUN cana-5460 414 5 bouziani bouziani PROPN cana-5460 414 6 ; ; PUNCT cana-5460 414 7 solution solution NOUN cana-5460 414 8 of of ADP cana-5460 414 9 a a DET cana-5460 414 10 transmission transmission NOUN cana-5460 414 11 problem problem NOUN cana-5460 414 12 for for ADP cana-5460 414 13 semilinear semilinear PROPN cana-5460 414 14 parabolic parabolic ADJ cana-5460 414 15 - - PUNCT cana-5460 414 16 hyperbolic hyperbolic ADJ cana-5460 414 17 equations equation NOUN cana-5460 414 18 by by ADP cana-5460 414 19 the the DET cana-5460 414 20 time time NOUN cana-5460 414 21 - - PUNCT cana-5460 414 22 discretization discretization NOUN cana-5460 414 23 method method NOUN cana-5460 414 24 , , PUNCT cana-5460 414 25 journal journal NOUN cana-5460 414 26 of of ADP cana-5460 414 27 applied apply VERB cana-5460 414 28 mathematics mathematic NOUN cana-5460 414 29 and and CCONJ cana-5460 414 30 stochastic stochastic ADJ cana-5460 414 31 analysis analysis NOUN cana-5460 414 32 , , PUNCT cana-5460 414 33 v v ADP cana-5460 414 34 2006 2006 NUM cana-5460 414 35 ( ( PUNCT cana-5460 414 36 2006 2006 NUM cana-5460 414 37 ) ) PUNCT cana-5460 414 38 , , PUNCT cana-5460 414 39 article article NOUN cana-5460 414 40 i i PROPN cana-5460 414 41 d d PROPN cana-5460 414 42 61439 61439 NUM cana-5460 414 43 , , PUNCT cana-5460 414 44 23 23 NUM cana-5460 414 45 pages page NOUN cana-5460 414 46 [ [ X cana-5460 414 47 6 6 NUM cana-5460 414 48 ] ] X cana-5460 414 49 n. n. PROPN cana-5460 414 50 merazga merazga PROPN cana-5460 414 51 , , PUNCT cana-5460 414 52 a. a. NOUN cana-5460 414 53 bouziani bouziani PROPN cana-5460 414 54 ; ; PUNCT cana-5460 414 55 rothe rothe NOUN cana-5460 414 56 method method NOUN cana-5460 414 57 for for ADP cana-5460 414 58 a a DET cana-5460 414 59 mixed mixed ADJ cana-5460 414 60 problem problem NOUN cana-5460 414 61 with with ADP cana-5460 414 62 an an DET cana-5460 414 63 integral integral ADJ cana-5460 414 64 condition condition NOUN cana-5460 414 65 for for ADP cana-5460 414 66 the the DET cana-5460 414 67 two two NUM cana-5460 414 68 - - PUNCT cana-5460 414 69 dimensional dimensional ADJ cana-5460 414 70 diffusion diffusion NOUN cana-5460 414 71 equation equation NOUN cana-5460 414 72 , , PUNCT cana-5460 414 73 abstract abstract ADJ cana-5460 414 74 and and CCONJ cana-5460 414 75 applied apply VERB cana-5460 414 76 analysis analysis NOUN cana-5460 414 77 v v ADP cana-5460 414 78 2003 2003 NUM cana-5460 414 79 ( ( PUNCT cana-5460 414 80 2003 2003 NUM cana-5460 414 81 ) ) PUNCT cana-5460 414 82 , , PUNCT cana-5460 414 83 issue issue NOUN cana-5460 414 84 16 16 NUM cana-5460 414 85 , , PUNCT cana-5460 414 86 pages page NOUN cana-5460 414 87 899 899 NUM cana-5460 414 88 - - SYM cana-5460 414 89 922 922 NUM cana-5460 414 90 . . PUNCT cana-5460 415 1 communications communication NOUN cana-5460 415 2 on on ADP cana-5460 415 3 applied apply VERB cana-5460 415 4 nonlinear nonlinear ADJ cana-5460 415 5 analysis analysis NOUN cana-5460 415 6 issn issn NOUN cana-5460 415 7 : : PUNCT cana-5460 415 8 1074 1074 NUM cana-5460 415 9 - - PUNCT cana-5460 415 10 133x 133x NUM cana-5460 415 11 vol vol NOUN cana-5460 415 12 32 32 NUM cana-5460 415 13 no.3 no.3 NOUN cana-5460 415 14 ( ( PUNCT cana-5460 415 15 2025 2025 NUM cana-5460 415 16 ) ) PUNCT cana-5460 415 17 962 962 NUM cana-5460 415 18 https://internationalpubls.com https://internationalpubls.com X cana-5460 416 1 [ [ X cana-5460 416 2 7 7 X cana-5460 416 3 ] ] X cana-5460 416 4 n. n. PROPN cana-5460 416 5 merazga merazga PROPN cana-5460 416 6 , , PUNCT cana-5460 416 7 a a DET cana-5460 416 8 bouziani bouziani NOUN cana-5460 416 9 ; ; PUNCT cana-5460 416 10 rothe rothe PROPN cana-5460 416 11 time time NOUN cana-5460 416 12 - - PUNCT cana-5460 416 13 discretization discretization NOUN cana-5460 416 14 method method NOUN cana-5460 416 15 for for ADP cana-5460 416 16 a a DET cana-5460 416 17 nonlocal nonlocal ADJ cana-5460 416 18 problem problem NOUN cana-5460 416 19 arising arise VERB cana-5460 416 20 in in ADP cana-5460 416 21 thermo thermo NOUN cana-5460 416 22 elasticity elasticity NOUN cana-5460 416 23 , , PUNCT cana-5460 416 24 journal journal NOUN cana-5460 416 25 of of ADP cana-5460 416 26 applied apply VERB cana-5460 416 27 mathematics mathematic NOUN cana-5460 416 28 and and CCONJ cana-5460 416 29 stochastic stochastic ADJ cana-5460 416 30 analysis analysis NOUN cana-5460 416 31 , , PUNCT cana-5460 416 32 v v ADP cana-5460 416 33 2005 2005 NUM cana-5460 417 1 [ [ X cana-5460 417 2 8 8 NUM cana-5460 417 3 ] ] PUNCT cana-5460 417 4 a. a. NOUN cana-5460 417 5 anguraj anguraj PROPN cana-5460 417 6 , , PUNCT cana-5460 417 7 p. p. PROPN cana-5460 417 8 karthikeyan karthikeyan PROPN cana-5460 417 9 ; ; PUNCT cana-5460 417 10 existence existence NOUN cana-5460 417 11 of of ADP cana-5460 417 12 solutions solution NOUN cana-5460 417 13 for for ADP cana-5460 417 14 fractional fractional ADJ cana-5460 417 15 semilinear semilinear PROPN cana-5460 417 16 evolution evolution PROPN cana-5460 417 17 boundary boundary ADJ cana-5460 417 18 value value NOUN cana-5460 417 19 problem problem NOUN cana-5460 417 20 , , PUNCT cana-5460 417 21 commun commun PROPN cana-5460 417 22 . . PUNCT cana-5460 418 1 appl appl PROPN cana-5460 418 2 . . PUNCT cana-5460 419 1 anal anal PROPN cana-5460 419 2 . . PUNCT cana-5460 420 1 14 14 NUM cana-5460 420 2 ( ( PUNCT cana-5460 420 3 2010 2010 NUM cana-5460 420 4 ) ) PUNCT cana-5460 421 1 505–514 505–514 NUM cana-5460 421 2 [ [ X cana-5460 421 3 9 9 NUM cana-5460 421 4 ] ] SYM cana-5460 421 5 a.a a.a PROPN cana-5460 421 6 . . PROPN cana-5460 421 7 alikhanov alikhanov PROPN cana-5460 421 8 . . PUNCT cana-5460 422 1 a a DET cana-5460 422 2 priori priori ADJ cana-5460 422 3 estimates estimate NOUN cana-5460 422 4 for for ADP cana-5460 422 5 solutions solution NOUN cana-5460 422 6 of of ADP cana-5460 422 7 boundary boundary ADJ cana-5460 422 8 value value NOUN cana-5460 422 9 problems problem NOUN cana-5460 422 10 for for ADP cana-5460 422 11 fractional fractional ADJ cana-5460 422 12 order order NOUN cana-5460 422 13 equations equation NOUN cana-5460 422 14 . . PUNCT cana-5460 423 1 differential differential ADJ cana-5460 423 2 equations equation NOUN cana-5460 423 3 , , PUNCT cana-5460 423 4 ( ( PUNCT cana-5460 423 5 46 46 NUM cana-5460 423 6 ) ) PUNCT cana-5460 423 7 , , PUNCT cana-5460 423 8 ( ( PUNCT cana-5460 423 9 5):660 5):660 X cana-5460 423 10 { { PUNCT cana-5460 423 11 666 666 NUM cana-5460 423 12 , , PUNCT cana-5460 423 13 2010 2010 NUM cana-5460 423 14 . . PUNCT cana-5460 424 1 [ [ X cana-5460 424 2 10 10 NUM cana-5460 424 3 ] ] PUNCT cana-5460 424 4 aghili aghili VERB cana-5460 424 5 a a DET cana-5460 424 6 , , PUNCT cana-5460 424 7 fractional fractional ADJ cana-5460 424 8 black black ADJ cana-5460 424 9 – – PUNCT cana-5460 424 10 scholes schole NOUN cana-5460 424 11 equation equation NOUN cana-5460 424 12 . . PUNCT cana-5460 425 1 int int PROPN cana-5460 426 1 j j PROPN cana-5460 426 2 financ financ PROPN cana-5460 426 3 eng eng PROPN cana-5460 426 4 . . PROPN cana-5460 426 5 2017 2017 NUM cana-5460 426 6 ; ; PUNCT cana-5460 426 7 4(1 4(1 NOUN cana-5460 426 8 ): ): PUNCT cana-5460 426 9 1750004 1750004 NUM cana-5460 426 10 . . PUNCT cana-5460 427 1 [ [ X cana-5460 427 2 11 11 NUM cana-5460 427 3 ] ] X cana-5460 427 4 b. b. PROPN cana-5460 427 5 ahmad ahmad PROPN cana-5460 427 6 , , PUNCT cana-5460 427 7 j. j. PROPN cana-5460 427 8 nieto nieto PROPN cana-5460 427 9 ; ; PUNCT cana-5460 427 10 existence existence NOUN cana-5460 427 11 results result VERB cana-5460 427 12 for for ADP cana-5460 427 13 nonlinear nonlinear ADJ cana-5460 427 14 boundary boundary ADJ cana-5460 427 15 value value NOUN cana-5460 427 16 problems problem NOUN cana-5460 427 17 of of ADP cana-5460 427 18 fractional fractional ADJ cana-5460 427 19 integro integro ADJ cana-5460 427 20 differential differential ADJ cana-5460 427 21 equations equation NOUN cana-5460 427 22 with with ADP cana-5460 427 23 integral integral ADJ cana-5460 427 24 boundary boundary ADJ cana-5460 427 25 conditions condition NOUN cana-5460 427 26 , , PUNCT cana-5460 427 27 boundary boundary ADJ cana-5460 427 28 value value NOUN cana-5460 427 29 problems problem NOUN cana-5460 427 30 vol vol NOUN cana-5460 427 31 . . PUNCT cana-5460 428 1 2009 2009 NUM cana-5460 428 2 ( ( PUNCT cana-5460 428 3 2009 2009 NUM cana-5460 428 4 ) ) PUNCT cana-5460 428 5 , , PUNCT cana-5460 428 6 article article NOUN cana-5460 428 7 i i PROPN cana-5460 428 8 d d PROPN cana-5460 428 9 708576 708576 NUM cana-5460 428 10 , , PUNCT cana-5460 428 11 11 11 NUM cana-5460 428 12 pages page NOUN cana-5460 428 13 . . PUNCT cana-5460 429 1 [ [ X cana-5460 429 2 12 12 NUM cana-5460 429 3 ] ] X cana-5460 429 4 bertram bertram PROPN cana-5460 429 5 ross ross PROPN cana-5460 429 6 . . PROPN cana-5460 429 7 fractional fractional ADJ cana-5460 429 8 calculus calculus NOUN cana-5460 429 9 and and CCONJ cana-5460 429 10 its its PRON cana-5460 429 11 applications application NOUN cana-5460 429 12 : : PUNCT cana-5460 429 13 proceeding proceed VERB cana-5460 429 14 of of ADP cana-5460 429 15 the the DET cana-5460 429 16 international international ADJ cana-5460 429 17 conference conference NOUN cana-5460 429 18 held hold VERB cana-5460 429 19 at at ADP cana-5460 429 20 the the DET cana-5460 429 21 university university NOUN cana-5460 429 22 of of ADP cana-5460 429 23 new new PROPN cana-5460 429 24 haven haven NOUN cana-5460 429 25 , , PUNCT cana-5460 429 26 june june PROPN cana-5460 429 27 1974 1974 NUM cana-5460 429 28 , , PUNCT cana-5460 429 29 volume volume NOUN cana-5460 429 30 457.springer 457.springer NUM cana-5460 429 31 , , PUNCT cana-5460 429 32 2006 2006 NUM cana-5460 429 33 . . PUNCT cana-5460 430 1 [ [ X cana-5460 430 2 13 13 NUM cana-5460 430 3 ] ] X cana-5460 430 4 cassani cassani PROPN cana-5460 430 5 d d PROPN cana-5460 430 6 , , PUNCT cana-5460 430 7 vilasi vilasi PROPN cana-5460 430 8 l l PROPN cana-5460 430 9 , , PUNCT cana-5460 430 10 wang wang PROPN cana-5460 430 11 y y PROPN cana-5460 430 12 , , PUNCT cana-5460 430 13 local local ADJ cana-5460 430 14 versus versus ADP cana-5460 430 15 nonlocal nonlocal ADJ cana-5460 430 16 elliptic elliptic ADJ cana-5460 430 17 equations equation NOUN cana-5460 430 18 : : PUNCT cana-5460 430 19 short short ADJ cana-5460 430 20 - - PUNCT cana-5460 430 21 long long ADJ cana-5460 430 22 range range NOUN cana-5460 430 23 field field NOUN cana-5460 430 24 interactions interaction NOUN cana-5460 430 25 . . PUNCT cana-5460 431 1 adv adv PROPN cana-5460 431 2 nonlinear nonlinear ADJ cana-5460 431 3 anal anal NOUN cana-5460 431 4 . . PUNCT cana-5460 431 5 2021 2021 NUM cana-5460 431 6 ; ; PUNCT cana-5460 431 7 10(1 10(1 NUM cana-5460 431 8 ): ): PUNCT cana-5460 431 9 895 895 NUM cana-5460 431 10 - - SYM cana-5460 431 11 921 921 NUM cana-5460 431 12 . . PUNCT cana-5460 432 1 [ [ X cana-5460 432 2 14 14 NUM cana-5460 432 3 ] ] PUNCT cana-5460 432 4 haim haim PROPN cana-5460 432 5 brezis brezis PROPN cana-5460 432 6 . . PUNCT cana-5460 433 1 analyse analyse NOUN cana-5460 433 2 fonctionnelle fonctionnelle PROPN cana-5460 433 3 . . PUNCT cana-5460 434 1 theorie theorie PROPN cana-5460 434 2 et et PROPN cana-5460 435 1 application.collection application.collection PROPN cana-5460 435 2 mathematiques mathematique VERB cana-5460 435 3 appliquΓ©s appliquΓ©s PROPN cana-5460 435 4 pour pour X cana-5460 435 5 la la X cana-5460 435 6 maitrise maitrise PROPN cana-5460 435 7 , , PUNCT cana-5460 435 8 1983 1983 NUM cana-5460 435 9 . . PUNCT cana-5460 436 1 [ [ X cana-5460 436 2 15 15 NUM cana-5460 436 3 ] ] PUNCT cana-5460 436 4 i.podlubny i.podlubny ADP cana-5460 436 5 , , PUNCT cana-5460 436 6 fractional fractional ADJ cana-5460 436 7 differential differential ADJ cana-5460 436 8 equation equation NOUN cana-5460 436 9 , , PUNCT cana-5460 436 10 san san PROPN cana-5460 436 11 diego diego PROPN cana-5460 436 12 : : PUNCT cana-5460 436 13 academic academic ADJ cana-5460 436 14 press press NOUN cana-5460 436 15 , , PUNCT cana-5460 436 16 1999 1999 NUM cana-5460 436 17 . . PUNCT cana-5460 437 1 [ [ X cana-5460 437 2 16 16 NUM cana-5460 437 3 ] ] X cana-5460 437 4 ladyzhenskaya ladyzhenskaya PROPN cana-5460 437 5 , , PUNCT cana-5460 437 6 o.a o.a PROPN cana-5460 437 7 . . PROPN cana-5460 438 1 the the DET cana-5460 438 2 boundary boundary ADJ cana-5460 438 3 value value NOUN cana-5460 438 4 problems problem NOUN cana-5460 438 5 of of ADP cana-5460 438 6 mathematical mathematical ADJ cana-5460 438 7 physics physics NOUN cana-5460 438 8 ; ; PUNCT cana-5460 438 9 springer springer NOUN cana-5460 438 10 : : PUNCT cana-5460 438 11 new new PROPN cana-5460 438 12 york york PROPN cana-5460 438 13 , , PUNCT cana-5460 438 14 ny ny PROPN cana-5460 438 15 , , PUNCT cana-5460 438 16 usa usa PROPN cana-5460 438 17 , , PUNCT cana-5460 438 18 1985 1985 NUM cana-5460 438 19 . . PUNCT cana-5460 439 1 [ [ X cana-5460 439 2 17 17 NUM cana-5460 439 3 ] ] PUNCT cana-5460 439 4 m. m. NOUN cana-5460 439 5 benchohra benchohra NOUN cana-5460 439 6 , , PUNCT cana-5460 439 7 j. j. PROPN cana-5460 439 8 r. r. PROPN cana-5460 439 9 graef graef PROPN cana-5460 439 10 , , PUNCT cana-5460 439 11 s. s. PROPN cana-5460 439 12 hamani hamani PROPN cana-5460 439 13 ; ; PUNCT cana-5460 439 14 existence existence NOUN cana-5460 439 15 results result VERB cana-5460 439 16 for for ADP cana-5460 439 17 boundary boundary ADJ cana-5460 439 18 value value NOUN cana-5460 439 19 problems problem NOUN cana-5460 439 20 with with ADP cana-5460 439 21 nonlinear nonlinear ADJ cana-5460 439 22 fractional fractional ADJ cana-5460 439 23 differential differential NOUN cana-5460 439 24 equations equation NOUN cana-5460 439 25 , , PUNCT cana-5460 439 26 appl appl PROPN cana-5460 439 27 . . PROPN cana-5460 440 1 anal anal PROPN cana-5460 440 2 . . PUNCT cana-5460 441 1 87 87 NUM cana-5460 441 2 ( ( PUNCT cana-5460 441 3 2008 2008 NUM cana-5460 441 4 ) ) PUNCT cana-5460 441 5 851–863 851–863 NUM cana-5460 441 6 . . PUNCT cana-5460 442 1 [ [ X cana-5460 442 2 18 18 NUM cana-5460 442 3 ] ] PUNCT cana-5460 442 4 m. m. NOUN cana-5460 442 5 belmekki belmekki PROPN cana-5460 442 6 , , PUNCT cana-5460 442 7 m. m. NOUN cana-5460 442 8 benchohra benchohra NOUN cana-5460 442 9 ; ; PUNCT cana-5460 442 10 existence existence NOUN cana-5460 442 11 results result VERB cana-5460 442 12 for for ADP cana-5460 442 13 fractional fractional ADJ cana-5460 442 14 order order NOUN cana-5460 442 15 semilinear semilinear PROPN cana-5460 442 16 functional functional ADJ cana-5460 442 17 differential differential NOUN cana-5460 442 18 equations equation NOUN cana-5460 442 19 , , PUNCT cana-5460 442 20 proc proc NOUN cana-5460 442 21 . . PUNCT cana-5460 443 1 a. a. PROPN cana-5460 443 2 razmadze razmadze PROPN cana-5460 443 3 math math PROPN cana-5460 443 4 . . PUNCT cana-5460 444 1 inst inst PROPN cana-5460 444 2 . . PUNCT cana-5460 445 1 146 146 NUM cana-5460 445 2 ( ( PUNCT cana-5460 445 3 2008 2008 NUM cana-5460 445 4 ) ) PUNCT cana-5460 445 5 9–20 9–20 NOUN cana-5460 445 6 . . PUNCT cana-5460 446 1 [ [ X cana-5460 446 2 19 19 NUM cana-5460 446 3 ] ] X cana-5460 446 4 mesloub mesloub PROPN cana-5460 446 5 s s NOUN cana-5460 446 6 , , PUNCT cana-5460 446 7 mezhoudi mezhoudi ADJ cana-5460 446 8 r r NOUN cana-5460 446 9 , , PUNCT cana-5460 446 10 medjeden medjeden X cana-5460 446 11 m m PROPN cana-5460 446 12 , , PUNCT cana-5460 446 13 8 8 NUM cana-5460 446 14 . . PUNCT cana-5460 447 1 a a DET cana-5460 447 2 mixed mixed ADJ cana-5460 447 3 problem problem NOUN cana-5460 447 4 for for ADP cana-5460 447 5 a a DET cana-5460 447 6 parabolic parabolic ADJ cana-5460 447 7 equation equation NOUN cana-5460 447 8 of of ADP cana-5460 447 9 higher high ADJ cana-5460 447 10 order order NOUN cana-5460 447 11 with with ADP cana-5460 447 12 integral integral ADJ cana-5460 447 13 conditions condition NOUN cana-5460 447 14 . . PUNCT cana-5460 448 1 bull bull NOUN cana-5460 448 2 polish polish PROPN cana-5460 448 3 acadsci acadsci PROPN cana-5460 448 4 math math NOUN cana-5460 448 5 . . PUNCT cana-5460 449 1 2002 2002 NUM cana-5460 449 2 ; ; PUNCT cana-5460 449 3 50(3 50(3 NUM cana-5460 449 4 ): ): PUNCT cana-5460 449 5 313–22 313–22 NUM cana-5460 449 6 . . PUNCT cana-5460 450 1 communications communication NOUN cana-5460 450 2 on on ADP cana-5460 450 3 applied apply VERB cana-5460 450 4 nonlinear nonlinear ADJ cana-5460 450 5 analysis analysis NOUN cana-5460 450 6 issn issn NOUN cana-5460 450 7 : : PUNCT cana-5460 450 8 1074 1074 NUM cana-5460 450 9 - - PUNCT cana-5460 450 10 133x 133x NUM cana-5460 450 11 vol vol NOUN cana-5460 450 12 32 32 NUM cana-5460 450 13 no.3 no.3 NOUN cana-5460 450 14 ( ( PUNCT cana-5460 450 15 2025 2025 NUM cana-5460 450 16 ) ) PUNCT cana-5460 451 1 963 963 NUM cana-5460 451 2 https://internationalpubls.com https://internationalpubls.com X cana-5460 451 3 [ [ X cana-5460 451 4 20 20 NUM cana-5460 451 5 ] ] PUNCT cana-5460 451 6 merad merad NOUN cana-5460 451 7 a a PRON cana-5460 451 8 , , PUNCT cana-5460 451 9 martΔ±n martΔ±n NOUN cana-5460 451 10 - - PUNCT cana-5460 451 11 vaquero vaquero NOUN cana-5460 451 12 j j PROPN cana-5460 451 13 , , PUNCT cana-5460 451 14 a a DET cana-5460 451 15 galerkin galerkin ADJ cana-5460 451 16 method method NOUN cana-5460 451 17 for for ADP cana-5460 451 18 two two NUM cana-5460 451 19 - - PUNCT cana-5460 451 20 dimensional dimensional ADJ cana-5460 451 21 hyperbolic hyperbolic ADJ cana-5460 451 22 integro integro ADJ cana-5460 451 23 differential differential ADJ cana-5460 451 24 equation equation NOUN cana-5460 451 25 with with ADP cana-5460 451 26 purely purely ADV cana-5460 451 27 integral integral ADJ cana-5460 451 28 conditions condition NOUN cana-5460 451 29 . . PUNCT cana-5460 452 1 appl appl PROPN cana-5460 452 2 math math PROPN cana-5460 452 3 comput comput NOUN cana-5460 452 4 . . PUNCT cana-5460 453 1 2016 2016 NUM cana-5460 453 2 ; ; PUNCT cana-5460 453 3 291 291 NUM cana-5460 453 4 : : PUNCT cana-5460 453 5 386 386 NUM cana-5460 453 6 - - SYM cana-5460 453 7 94 94 NUM cana-5460 453 8 [ [ X cana-5460 453 9 21 21 NUM cana-5460 453 10 ] ] X cana-5460 453 11 mesloub mesloub PROPN cana-5460 453 12 s s NOUN cana-5460 453 13 , , PUNCT cana-5460 453 14 aldosari aldosari ADJ cana-5460 453 15 f f X cana-5460 453 16 , , PUNCT cana-5460 453 17 even even ADV cana-5460 453 18 higher high ADJ cana-5460 453 19 order order NOUN cana-5460 453 20 fractional fractional ADJ cana-5460 453 21 initial initial ADJ cana-5460 453 22 boundary boundary ADJ cana-5460 453 23 value value NOUN cana-5460 453 24 problem problem NOUN cana-5460 453 25 with with ADP cana-5460 453 26 nonlocal nonlocal ADJ cana-5460 453 27 constraints constraint NOUN cana-5460 453 28 of of ADP cana-5460 453 29 purely purely ADV cana-5460 453 30 integral integral ADJ cana-5460 453 31 type type NOUN cana-5460 453 32 . . PUNCT cana-5460 453 33 symmetry symmetry NOUN cana-5460 453 34 . . PUNCT cana-5460 454 1 2019 2019 NUM cana-5460 454 2 ; ; PUNCT cana-5460 455 1 11(3 11(3 NUM cana-5460 455 2 ): ): PUNCT cana-5460 455 3 305 305 NUM cana-5460 455 4 . . PUNCT cana-5460 456 1 [ [ X cana-5460 456 2 22 22 NUM cana-5460 456 3 ] ] SYM cana-5460 456 4 mingqi mingqi NOUN cana-5460 456 5 x x NOUN cana-5460 456 6 , , PUNCT cana-5460 456 7 radulescu radulescu NOUN cana-5460 456 8 vd vd NOUN cana-5460 456 9 , , PUNCT cana-5460 456 10 zhang zhang PROPN cana-5460 456 11 b b PROPN cana-5460 456 12 , , PUNCT cana-5460 456 13 nonlocal nonlocal ADJ cana-5460 456 14 kirchhoff kirchhoff NOUN cana-5460 456 15 diffusion diffusion NOUN cana-5460 456 16 problems problem NOUN cana-5460 456 17 : : PUNCT cana-5460 456 18 local local ADJ cana-5460 456 19 existence existence NOUN cana-5460 456 20 and and CCONJ cana-5460 456 21 blow blow NOUN cana-5460 456 22 - - PUNCT cana-5460 456 23 up up NOUN cana-5460 456 24 of of ADP cana-5460 456 25 solutions solution NOUN cana-5460 456 26 . . PUNCT cana-5460 457 1 nonlinearity nonlinearity NOUN cana-5460 457 2 . . PUNCT cana-5460 458 1 2018 2018 NUM cana-5460 458 2 ; ; PUNCT cana-5460 458 3 31(7 31(7 NUM cana-5460 458 4 ): ): PUNCT cana-5460 458 5 3228 3228 NUM cana-5460 458 6 . . PUNCT cana-5460 459 1 [ [ X cana-5460 459 2 23 23 NUM cana-5460 459 3 ] ] SYM cana-5460 459 4 pandir pandir NOUN cana-5460 459 5 y y PROPN cana-5460 459 6 , , PUNCT cana-5460 459 7 duzgun duzgun PROPN cana-5460 459 8 hh hh PROPN cana-5460 459 9 , , PUNCT cana-5460 459 10 new new ADJ cana-5460 459 11 exact exact ADJ cana-5460 459 12 solutions solution NOUN cana-5460 459 13 of of ADP cana-5460 459 14 time time NOUN cana-5460 459 15 fractional fractional PROPN cana-5460 459 16 gardner gardner NOUN cana-5460 459 17 equation equation NOUN cana-5460 459 18 by by ADP cana-5460 459 19 using use VERB cana-5460 459 20 new new ADJ cana-5460 459 21 version version NOUN cana-5460 459 22 of of ADP cana-5460 459 23 f f NOUN cana-5460 459 24 - - PUNCT cana-5460 459 25 expansion expansion NOUN cana-5460 459 26 method method NOUN cana-5460 459 27 . . PUNCT cana-5460 460 1 commun commun PROPN cana-5460 460 2 theor theor PROPN cana-5460 460 3 phys phys PROPN cana-5460 460 4 . . PUNCT cana-5460 461 1 2017 2017 NUM cana-5460 461 2 ; ; PUNCT cana-5460 461 3 67(1 67(1 NUM cana-5460 461 4 ): ): PUNCT cana-5460 461 5 9 9 NUM cana-5460 461 6 . . PUNCT cana-5460 462 1 [ [ X cana-5460 462 2 24 24 NUM cana-5460 462 3 ] ] X cana-5460 462 4 r. r. PROPN cana-5460 462 5 p. p. PROPN cana-5460 462 6 agarwal agarwal PROPN cana-5460 462 7 , , PUNCT cana-5460 462 8 m. m. NOUN cana-5460 462 9 benchohra benchohra NOUN cana-5460 462 10 , , PUNCT cana-5460 462 11 s. s. PROPN cana-5460 462 12 hamani hamani PROPN cana-5460 462 13 ; ; PUNCT cana-5460 462 14 a a DET cana-5460 462 15 survey survey NOUN cana-5460 462 16 on on ADP cana-5460 462 17 existence existence NOUN cana-5460 462 18 results result NOUN cana-5460 462 19 for for ADP cana-5460 462 20 boundary boundary ADJ cana-5460 462 21 value value NOUN cana-5460 462 22 problems problem NOUN cana-5460 462 23 of of ADP cana-5460 462 24 nonlinear nonlinear ADJ cana-5460 462 25 fractional fractional ADJ cana-5460 462 26 differential differential ADJ cana-5460 462 27 equations equation NOUN cana-5460 462 28 and and CCONJ cana-5460 462 29 inclusions inclusion NOUN cana-5460 462 30 , , PUNCT cana-5460 462 31 acta acta PROPN cana-5460 462 32 appl appl PROPN cana-5460 462 33 . . PUNCT cana-5460 463 1 math.doi math.doi X cana-5460 463 2 10.1007 10.1007 NUM cana-5460 463 3 / / SYM cana-5460 463 4 s10440 s10440 NOUN cana-5460 463 5 - - PUNCT cana-5460 463 6 008 008 NUM cana-5460 463 7 - - PUNCT cana-5460 463 8 9356-(2005 9356-(2005 NUM cana-5460 463 9 ) ) PUNCT cana-5460 463 10 , , PUNCT cana-5460 463 11 issue issue NOUN cana-5460 463 12 1 1 NUM cana-5460 463 13 , , PUNCT cana-5460 463 14 pages page NOUN cana-5460 463 15 13 13 NUM cana-5460 463 16 - - SYM cana-5460 463 17 28 28 NUM cana-5460 463 18 . . PUNCT cana-5460 464 1 [ [ X cana-5460 464 2 25 25 NUM cana-5460 464 3 ] ] PUNCT cana-5460 464 4 r. r. PROPN cana-5460 464 5 w. w. PROPN cana-5460 464 6 ibrahim ibrahim PROPN cana-5460 464 7 , , PUNCT cana-5460 464 8 s. s. PROPN cana-5460 464 9 momani momani PROPN cana-5460 464 10 ; ; PUNCT cana-5460 464 11 on on ADP cana-5460 464 12 existence existence NOUN cana-5460 464 13 and and CCONJ cana-5460 464 14 uniqueness uniqueness NOUN cana-5460 464 15 of of ADP cana-5460 464 16 solutions solution NOUN cana-5460 464 17 of of ADP cana-5460 464 18 a a DET cana-5460 464 19 class class NOUN cana-5460 464 20 of of ADP cana-5460 464 21 fractional fractional ADJ cana-5460 464 22 differential differential ADJ cana-5460 464 23 equations equation NOUN cana-5460 464 24 , , PUNCT cana-5460 464 25 journal journal NOUN cana-5460 464 26 of of ADP cana-5460 464 27 mathematical mathematical ADJ cana-5460 464 28 analysis analysis NOUN cana-5460 464 29 and and CCONJ cana-5460 464 30 applications application NOUN cana-5460 464 31 , , PUNCT cana-5460 464 32 3334 3334 NUM cana-5460 464 33 ( ( PUNCT cana-5460 464 34 2007 2007 NUM cana-5460 464 35 ) ) PUNCT cana-5460 464 36 , , PUNCT cana-5460 464 37 1 1 NUM cana-5460 464 38 – – PUNCT cana-5460 464 39 [ [ X cana-5460 464 40 26 26 NUM cana-5460 464 41 ] ] X cana-5460 464 42 stefan stefan PROPN cana-5460 464 43 g g PROPN cana-5460 464 44 samko samko PROPN cana-5460 464 45 , , PUNCT cana-5460 464 46 anatoly anatoly PROPN cana-5460 464 47 a a DET cana-5460 464 48 kilbas kilbas PROPN cana-5460 464 49 , , PUNCT cana-5460 464 50 oleg oleg PROPN cana-5460 464 51 i i PRON cana-5460 464 52 marichev marichev PROPN cana-5460 464 53 , , PUNCT cana-5460 464 54 et et PROPN cana-5460 464 55 al al PROPN cana-5460 464 56 . . PUNCT cana-5460 464 57 fractional fractional ADJ cana-5460 464 58 integrals integral NOUN cana-5460 464 59 and and CCONJ cana-5460 464 60 derivatives derivative NOUN cana-5460 464 61 , , PUNCT cana-5460 464 62 volume volume NOUN cana-5460 464 63 i. i. PROPN cana-5460 464 64 gordon gordon PROPN cana-5460 464 65 and and CCONJ cana-5460 464 66 breach breach VERB cana-5460 464 67 science science NOUN cana-5460 464 68 publishers publisher NOUN cana-5460 464 69 , , PUNCT cana-5460 464 70 yverdon yverdon PROPN cana-5460 464 71 y y PROPN cana-5460 464 72 verdon verdon PROPN cana-5460 464 73 - - PUNCT cana-5460 464 74 lesbains lesbains PROPN cana-5460 464 75 , , PUNCT cana-5460 464 76 switzer switzer NOUN cana-5460 464 77 land land NOUN cana-5460 464 78 , , PUNCT cana-5460 464 79 1993 1993 NUM cana-5460 464 80 . . PUNCT cana-5460 465 1 [ [ X cana-5460 465 2 27 27 NUM cana-5460 465 3 ] ] SYM cana-5460 465 4 tang tang X cana-5460 465 5 xh xh PROPN cana-5460 465 6 , , PUNCT cana-5460 465 7 cheng cheng PROPN cana-5460 465 8 b b PROPN cana-5460 465 9 , , PUNCT cana-5460 465 10 ground ground NOUN cana-5460 465 11 state state NOUN cana-5460 465 12 sign sign NOUN cana-5460 465 13 - - PUNCT cana-5460 465 14 changing change VERB cana-5460 465 15 solutions solution NOUN cana-5460 465 16 for for ADP cana-5460 465 17 kirchhoff kirchhoff NOUN cana-5460 465 18 type type NOUN cana-5460 465 19 problems problem NOUN cana-5460 465 20 in in ADP cana-5460 465 21 bounded bounded ADJ cana-5460 465 22 domains domain NOUN cana-5460 465 23 , , PUNCT cana-5460 465 24 j j PROPN cana-5460 465 25 diff diff PROPN cana-5460 465 26 eq eq PROPN cana-5460 465 27 . . PROPN cana-5460 465 28 2016 2016 NUM cana-5460 465 29 ; ; PUNCT cana-5460 465 30 261(4 261(4 NUM cana-5460 465 31 ): ): PUNCT cana-5460 465 32 2384 2384 NUM cana-5460 465 33 - - SYM cana-5460 465 34 402 402 NUM cana-5460 465 35 . . PUNCT cana-5460 466 1 [ [ X cana-5460 466 2 28 28 NUM cana-5460 466 3 ] ] PUNCT cana-5460 466 4 x. x. PROPN cana-5460 466 5 j. j. PROPN cana-5460 466 6 li li PROPN cana-5460 466 7 , , PUNCT cana-5460 466 8 c. c. PROPN cana-5460 466 9 j. j. PROPN cana-5460 466 10 xu xu PROPN cana-5460 466 11 ; ; PUNCT cana-5460 466 12 existence existence NOUN cana-5460 466 13 and and CCONJ cana-5460 466 14 uniqueness uniqueness NOUN cana-5460 466 15 of of ADP cana-5460 466 16 the the DET cana-5460 466 17 weak weak ADJ cana-5460 466 18 solution solution NOUN cana-5460 466 19 of of ADP cana-5460 466 20 the the DET cana-5460 466 21 space space NOUN cana-5460 466 22 - - PUNCT cana-5460 466 23 time time NOUN cana-5460 466 24 fractional fractional ADJ cana-5460 466 25 diffusion diffusion NOUN cana-5460 466 26 equation equation NOUN cana-5460 466 27 and and CCONJ cana-5460 466 28 a a DET cana-5460 466 29 spectral spectral ADJ cana-5460 466 30 method method NOUN cana-5460 466 31 approximation approximation NOUN cana-5460 466 32 , , PUNCT cana-5460 466 33 communications communication NOUN cana-5460 466 34 in in ADP cana-5460 466 35 computational computational ADJ cana-5460 466 36 physics physics NOUN cana-5460 466 37 , , PUNCT cana-5460 466 38 vol vol NOUN cana-5460 466 39 . . PROPN cana-5460 466 40 8 8 NUM cana-5460 466 41 , , PUNCT cana-5460 466 42 no no INTJ cana-5460 466 43 . . NOUN cana-5460 466 44 5 5 NUM cana-5460 466 45 , , PUNCT cana-5460 466 46 pp pp ADJ cana-5460 466 47 . . PUNCT cana-5460 467 1 1016–1051 1016–1051 NUM cana-5460 467 2 , , PUNCT cana-5460 467 3 2010 2010 NUM cana-5460 467 4 . . PUNCT