id sid tid token lemma pos cana-4832 1 1 communications communication NOUN cana-4832 1 2 on on ADP cana-4832 1 3 applied apply VERB cana-4832 1 4 nonlinear nonlinear ADJ cana-4832 1 5 analysis analysis NOUN cana-4832 1 6 issn issn NOUN cana-4832 1 7 : : PUNCT cana-4832 1 8 1074 1074 NUM cana-4832 1 9 - - PUNCT cana-4832 1 10 133x 133x NUM cana-4832 1 11 vol vol VERB cana-4832 1 12 32 32 NUM cana-4832 1 13 no no NOUN cana-4832 1 14 . . PUNCT cana-4832 2 1 10s 10 NOUN cana-4832 2 2 ( ( PUNCT cana-4832 2 3 2025 2025 NUM cana-4832 2 4 ) ) PUNCT cana-4832 2 5 389 389 NUM cana-4832 2 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 2 7 some some DET cana-4832 2 8 fixed fix VERB cana-4832 2 9 point point NOUN cana-4832 2 10 results result NOUN cana-4832 2 11 for for ADP cana-4832 2 12 expansive expansive ADJ cana-4832 2 13 mappings mapping NOUN cana-4832 2 14 in in ADP cana-4832 2 15 dislocated dislocated ADJ cana-4832 2 16 quasi quasi ADJ cana-4832 2 17 - - ADJ cana-4832 2 18 b b ADJ cana-4832 2 19 - - PUNCT cana-4832 2 20 metric metric ADJ cana-4832 2 21 spaces space NOUN cana-4832 2 22 with with ADP cana-4832 2 23 an an DET cana-4832 2 24 application application NOUN cana-4832 2 25 to to ADP cana-4832 2 26 integral integral ADJ cana-4832 2 27 equations equation NOUN cana-4832 2 28 1dasari 1dasari NUM cana-4832 2 29 ratna ratna PROPN cana-4832 2 30 babu babu PROPN cana-4832 2 31 , , PUNCT cana-4832 2 32 2p 2p NOUN cana-4832 2 33 . . PUNCT cana-4832 3 1 sudheer sudheer NOUN cana-4832 3 2 kumar,3s kumar,3s PROPN cana-4832 3 3 . . PUNCT cana-4832 4 1 lakshmana lakshmana PROPN cana-4832 4 2 rao rao PROPN cana-4832 4 3 1*department 1*department NUM cana-4832 4 4 of of ADP cana-4832 4 5 mathematics mathematic NOUN cana-4832 4 6 , , PUNCT cana-4832 4 7 pscmrcet pscmrcet NOUN cana-4832 4 8 , , PUNCT cana-4832 4 9 vijayawada-520001 vijayawada-520001 ADJ cana-4832 4 10 , , PUNCT cana-4832 4 11 india india PROPN cana-4832 4 12 email email NOUN cana-4832 4 13 : : PUNCT cana-4832 4 14 ratnababud@gmail.com ratnababud@gmail.com X cana-4832 5 1 2department 2department NUM cana-4832 5 2 of of ADP cana-4832 5 3 information information NOUN cana-4832 5 4 technology technology NOUN cana-4832 5 5 , , PUNCT cana-4832 5 6 aditya aditya PROPN cana-4832 5 7 institute institute PROPN cana-4832 5 8 of of ADP cana-4832 5 9 technology technology NOUN cana-4832 5 10 and and CCONJ cana-4832 5 11 management management NOUN cana-4832 5 12 , , PUNCT cana-4832 5 13 tekklai tekklai NOUN cana-4832 5 14 -532201 -532201 PROPN cana-4832 5 15 , , PUNCT cana-4832 5 16 india india PROPN cana-4832 5 17 . . PUNCT cana-4832 6 1 e e X cana-4832 6 2 - - NOUN cana-4832 6 3 mail mail NOUN cana-4832 6 4 : : PUNCT cana-4832 6 5 sudheerkumar9732@gmail.com sudheerkumar9732@gmail.com X cana-4832 7 1 3lecturer 3lecturer NUM cana-4832 7 2 in in ADP cana-4832 7 3 mathematics mathematic NOUN cana-4832 7 4 , , PUNCT cana-4832 7 5 govt govt NOUN cana-4832 7 6 . . PUNCT cana-4832 8 1 degree degree NOUN cana-4832 8 2 college college PROPN cana-4832 8 3 , , PUNCT cana-4832 8 4 tekkali tekkali PROPN cana-4832 8 5 , , PUNCT cana-4832 8 6 dr.b.r.amedkar dr.b.r.amedkar PROPN cana-4832 8 7 university university PROPN cana-4832 8 8 , , PUNCT cana-4832 8 9 etcherla etcherla PROPN cana-4832 8 10 , , PUNCT cana-4832 8 11 srikakulam srikakulam NOUN cana-4832 8 12 , , PUNCT cana-4832 8 13 india,532201 india,532201 ADJ cana-4832 8 14 e e NOUN cana-4832 8 15 - - NOUN cana-4832 8 16 mail mail NOUN cana-4832 8 17 : : PUNCT cana-4832 9 1 laxmana.mat@gmail.com laxmana.mat@gmail.com PROPN cana-4832 9 2 article article PROPN cana-4832 9 3 history history NOUN cana-4832 9 4 : : PUNCT cana-4832 9 5 received receive VERB cana-4832 9 6 : : PUNCT cana-4832 9 7 12 12 NUM cana-4832 9 8 - - SYM cana-4832 9 9 01 01 NUM cana-4832 9 10 - - PUNCT cana-4832 9 11 2025 2025 NUM cana-4832 9 12 revised revise VERB cana-4832 9 13 : : PUNCT cana-4832 9 14 15 15 NUM cana-4832 9 15 - - NUM cana-4832 9 16 02 02 NUM cana-4832 9 17 - - PUNCT cana-4832 9 18 2025 2025 NUM cana-4832 9 19 accepted accept VERB cana-4832 9 20 : : PUNCT cana-4832 9 21 01 01 NUM cana-4832 9 22 - - SYM cana-4832 9 23 03 03 NUM cana-4832 9 24 - - PUNCT cana-4832 9 25 2025 2025 NUM cana-4832 9 26 abstract abstract NOUN cana-4832 9 27 : : PUNCT cana-4832 9 28 in in ADP cana-4832 9 29 this this DET cana-4832 9 30 paper paper NOUN cana-4832 9 31 , , PUNCT cana-4832 9 32 we we PRON cana-4832 9 33 prove prove VERB cana-4832 9 34 some some DET cana-4832 9 35 new new ADJ cana-4832 9 36 fixed fix VERB cana-4832 9 37 point point NOUN cana-4832 9 38 results result NOUN cana-4832 9 39 for for ADP cana-4832 9 40 expansive expansive ADJ cana-4832 9 41 type type NOUN cana-4832 9 42 mappings mapping NOUN cana-4832 9 43 in in ADP cana-4832 9 44 complete complete ADJ cana-4832 9 45 dislocated dislocate VERB cana-4832 9 46 quasi quasi ADJ cana-4832 9 47 𝑏-metric 𝑏-metric PROPN cana-4832 9 48 space space NOUN cana-4832 9 49 . . PUNCT cana-4832 10 1 a a DET cana-4832 10 2 common common ADJ cana-4832 10 3 fixed fix VERB cana-4832 10 4 point point NOUN cana-4832 10 5 result result NOUN cana-4832 10 6 is be AUX cana-4832 10 7 also also ADV cana-4832 10 8 established establish VERB cana-4832 10 9 considering consider VERB cana-4832 10 10 such such ADJ cana-4832 10 11 mappings mapping NOUN cana-4832 10 12 . . PUNCT cana-4832 11 1 our our PRON cana-4832 11 2 results result NOUN cana-4832 11 3 extend extend VERB cana-4832 11 4 and and CCONJ cana-4832 11 5 generalize generalize VERB cana-4832 11 6 the the DET cana-4832 11 7 results result NOUN cana-4832 11 8 of of ADP cana-4832 11 9 das das PROPN cana-4832 11 10 et et PROPN cana-4832 11 11 al al PROPN cana-4832 11 12 . . PUNCT cana-4832 12 1 [ [ X cana-4832 12 2 9 9 NUM cana-4832 12 3 ] ] PUNCT cana-4832 12 4 from from ADP cana-4832 12 5 the the DET cana-4832 12 6 dislocated dislocate VERB cana-4832 12 7 quasi quasi ADJ cana-4832 12 8 metric metric ADJ cana-4832 12 9 space space NOUN cana-4832 12 10 setting set VERB cana-4832 12 11 to to ADP cana-4832 12 12 dislocated dislocate VERB cana-4832 12 13 quasi-𝑏-metric quasi-𝑏-metric ADJ cana-4832 12 14 spaces space NOUN cana-4832 12 15 . . PUNCT cana-4832 13 1 suitable suitable ADJ cana-4832 13 2 examples example NOUN cana-4832 13 3 are be AUX cana-4832 13 4 provided provide VERB cana-4832 13 5 to to PART cana-4832 13 6 demonstrate demonstrate VERB cana-4832 13 7 our our PRON cana-4832 13 8 results result NOUN cana-4832 13 9 . . PUNCT cana-4832 14 1 the the DET cana-4832 14 2 solution solution NOUN cana-4832 14 3 to to ADP cana-4832 14 4 a a DET cana-4832 14 5 system system NOUN cana-4832 14 6 of of ADP cana-4832 14 7 fredholm fredholm ADJ cana-4832 14 8 integral integral ADJ cana-4832 14 9 equations equation NOUN cana-4832 14 10 is be AUX cana-4832 14 11 also also ADV cana-4832 14 12 established establish VERB cana-4832 14 13 to to PART cana-4832 14 14 show show VERB cana-4832 14 15 the the DET cana-4832 14 16 applicability applicability NOUN cana-4832 14 17 of of ADP cana-4832 14 18 our our PRON cana-4832 14 19 results result NOUN cana-4832 14 20 . . PUNCT cana-4832 15 1 keywords keyword NOUN cana-4832 15 2 : : PUNCT cana-4832 15 3 fixed fix VERB cana-4832 15 4 points point NOUN cana-4832 15 5 ; ; PUNCT cana-4832 15 6 dislocated dislocated ADJ cana-4832 15 7 qusi-𝑏-metricspace qusi-𝑏-metricspace NOUN cana-4832 15 8 ; ; PUNCT cana-4832 15 9 expansive expansive ADJ cana-4832 15 10 map map NOUN cana-4832 15 11 ; ; PUNCT cana-4832 15 12 integral integral ADJ cana-4832 15 13 equation equation NOUN cana-4832 15 14 . . PUNCT cana-4832 16 1 ams am NOUN cana-4832 16 2 subject subject ADJ cana-4832 16 3 classification classification NOUN cana-4832 16 4 ( ( PUNCT cana-4832 16 5 2020 2020 NUM cana-4832 16 6 ): ): PUNCT cana-4832 16 7 47h10,54h25 47h10,54h25 NOUN cana-4832 16 8 .. .. PROPN cana-4832 16 9 1 1 X cana-4832 16 10 . . X cana-4832 16 11 introduction introduction NOUN cana-4832 16 12 the the DET cana-4832 16 13 development development NOUN cana-4832 16 14 of of ADP cana-4832 16 15 fixed fix VERB cana-4832 16 16 point point NOUN cana-4832 16 17 theory theory NOUN cana-4832 16 18 is be AUX cana-4832 16 19 based base VERB cana-4832 16 20 on on ADP cana-4832 16 21 the the DET cana-4832 16 22 generalization generalization NOUN cana-4832 16 23 of of ADP cana-4832 16 24 contraction contraction NOUN cana-4832 16 25 conditions condition NOUN cana-4832 16 26 in in ADP cana-4832 16 27 one one NUM cana-4832 16 28 direction direction NOUN cana-4832 16 29 or or CCONJ cana-4832 16 30 / / SYM cana-4832 16 31 and and CCONJ cana-4832 16 32 generalization generalization NOUN cana-4832 16 33 of of ADP cana-4832 16 34 ambient ambient ADJ cana-4832 16 35 spaces space NOUN cana-4832 16 36 of of ADP cana-4832 16 37 the the DET cana-4832 16 38 operator operator NOUN cana-4832 16 39 under under ADP cana-4832 16 40 consideration consideration NOUN cana-4832 16 41 on on ADP cana-4832 16 42 the the DET cana-4832 16 43 other other ADJ cana-4832 16 44 . . PUNCT cana-4832 17 1 using use VERB cana-4832 17 2 the the DET cana-4832 17 3 picard picard NOUN cana-4832 17 4 iteration iteration NOUN cana-4832 17 5 approach approach NOUN cana-4832 17 6 , , PUNCT cana-4832 17 7 polish polish ADJ cana-4832 17 8 mathematician mathematician NOUN cana-4832 17 9 banach banach ADV cana-4832 17 10 developed develop VERB cana-4832 17 11 the the DET cana-4832 17 12 banach banach NOUN cana-4832 17 13 contraction contraction NOUN cana-4832 17 14 mapping mapping NOUN cana-4832 17 15 concept concept NOUN cana-4832 17 16 in in ADP cana-4832 17 17 1922 1922 NUM cana-4832 17 18 . . PUNCT cana-4832 18 1 the the DET cana-4832 18 2 existence existence NOUN cana-4832 18 3 of of ADP cana-4832 18 4 a a DET cana-4832 18 5 solution solution NOUN cana-4832 18 6 for for ADP cana-4832 18 7 a a DET cana-4832 18 8 differential differential ADJ cana-4832 18 9 equation equation NOUN cana-4832 18 10 with with ADP cana-4832 18 11 initial initial ADJ cana-4832 18 12 value value NOUN cana-4832 18 13 condition condition NOUN cana-4832 18 14 , , PUNCT cana-4832 18 15 the the DET cana-4832 18 16 implicit implicit ADJ cana-4832 18 17 function function NOUN cana-4832 18 18 existence existence NOUN cana-4832 18 19 theorem theorem VERB cana-4832 18 20 , , PUNCT cana-4832 18 21 and and CCONJ cana-4832 18 22 fixed fix VERB cana-4832 18 23 point point NOUN cana-4832 18 24 theory theory NOUN cana-4832 18 25 ’s ’s PART cana-4832 18 26 elegant elegant ADJ cana-4832 18 27 assertion assertion NOUN cana-4832 18 28 and and CCONJ cana-4832 18 29 effective effective ADJ cana-4832 18 30 method method NOUN cana-4832 18 31 of of ADP cana-4832 18 32 solving solve VERB cana-4832 18 33 it it PRON cana-4832 18 34 have have AUX cana-4832 18 35 drawn draw VERB cana-4832 18 36 the the DET cana-4832 18 37 attention attention NOUN cana-4832 18 38 of of ADP cana-4832 18 39 academics academic NOUN cana-4832 18 40 and and CCONJ cana-4832 18 41 inspired inspire VERB cana-4832 18 42 people people NOUN cana-4832 18 43 to to PART cana-4832 18 44 conduct conduct VERB cana-4832 18 45 in in ADP cana-4832 18 46 - - PUNCT cana-4832 18 47 depth depth NOUN cana-4832 18 48 , , PUNCT cana-4832 18 49 comprehensive comprehensive ADJ cana-4832 18 50 research research NOUN cana-4832 18 51 . . PUNCT cana-4832 19 1 with with ADP cana-4832 19 2 the the DET cana-4832 19 3 advent advent NOUN cana-4832 19 4 of of ADP cana-4832 19 5 the the DET cana-4832 19 6 computer computer NOUN cana-4832 19 7 , , PUNCT cana-4832 19 8 particularly particularly ADV cana-4832 19 9 in in ADP cana-4832 19 10 the the DET cana-4832 19 11 last last ADJ cana-4832 19 12 few few ADJ cana-4832 19 13 decades decade NOUN cana-4832 19 14 , , PUNCT cana-4832 19 15 many many ADJ cana-4832 19 16 individuals individual NOUN cana-4832 19 17 have have AUX cana-4832 19 18 dealt deal VERB cana-4832 19 19 with with ADP cana-4832 19 20 a a DET cana-4832 19 21 large large ADJ cana-4832 19 22 number number NOUN cana-4832 19 23 of of ADP cana-4832 19 24 applications application NOUN cana-4832 19 25 by by ADP cana-4832 19 26 using use VERB cana-4832 19 27 a a DET cana-4832 19 28 range range NOUN cana-4832 19 29 of of ADP cana-4832 19 30 iteration iteration NOUN cana-4832 19 31 techniques technique NOUN cana-4832 19 32 to to PART cana-4832 19 33 approach approach VERB cana-4832 19 34 the the DET cana-4832 19 35 fixed fix VERB cana-4832 19 36 point point NOUN cana-4832 19 37 . . PUNCT cana-4832 20 1 as as ADP cana-4832 20 2 a a DET cana-4832 20 3 result result NOUN cana-4832 20 4 , , PUNCT cana-4832 20 5 they they PRON cana-4832 20 6 made make VERB cana-4832 20 7 a a DET cana-4832 20 8 breakthrough breakthrough NOUN cana-4832 20 9 and and CCONJ cana-4832 20 10 gradually gradually ADV cana-4832 20 11 improved improve VERB cana-4832 20 12 this this DET cana-4832 20 13 subject subject NOUN cana-4832 20 14 . . PUNCT cana-4832 21 1 these these DET cana-4832 21 2 days day NOUN cana-4832 21 3 , , PUNCT cana-4832 21 4 nonlinear nonlinear ADJ cana-4832 21 5 functional functional ADJ cana-4832 21 6 analysis analysis NOUN cana-4832 21 7 relies rely VERB cana-4832 21 8 heavily heavily ADV cana-4832 21 9 on on ADP cana-4832 21 10 fixed fix VERB cana-4832 21 11 point point NOUN cana-4832 21 12 theory theory NOUN cana-4832 21 13 . . PUNCT cana-4832 22 1 the the DET cana-4832 22 2 notion notion NOUN cana-4832 22 3 of of ADP cana-4832 22 4 dislocated dislocate VERB cana-4832 22 5 metric metric ADJ cana-4832 22 6 space space NOUN cana-4832 22 7 initially initially ADV cana-4832 22 8 surfaced surface VERB cana-4832 22 9 in in ADP cana-4832 22 10 domain domain NOUN cana-4832 22 11 theory theory NOUN cana-4832 22 12 , , PUNCT cana-4832 22 13 which which PRON cana-4832 22 14 was be AUX cana-4832 22 15 proposed propose VERB cana-4832 22 16 by by ADP cana-4832 22 17 matthews matthews PROPN cana-4832 23 1 [ [ X cana-4832 23 2 15 15 NUM cana-4832 23 3 ] ] PUNCT cana-4832 23 4 in in ADP cana-4832 23 5 1986 1986 NUM cana-4832 23 6 along along ADP cana-4832 23 7 with with ADP cana-4832 23 8 various various ADJ cana-4832 23 9 concepts concept NOUN cana-4832 23 10 of of ADP cana-4832 23 11 metric metric ADJ cana-4832 23 12 domains domain NOUN cana-4832 23 13 . . PUNCT cana-4832 24 1 hitzler hitzler ADV cana-4832 24 2 et et NOUN cana-4832 24 3 al al PROPN cana-4832 24 4 . . PUNCT cana-4832 25 1 [ [ X cana-4832 25 2 13 13 NUM cana-4832 25 3 ] ] PUNCT cana-4832 25 4 presented present VERB cana-4832 25 5 the the DET cana-4832 25 6 idea idea NOUN cana-4832 25 7 of of ADP cana-4832 25 8 dislocated dislocate VERB cana-4832 25 9 metric metric ADJ cana-4832 25 10 space space NOUN cana-4832 25 11 later later ADV cana-4832 25 12 in in ADP cana-4832 25 13 2000 2000 NUM cana-4832 25 14 , , PUNCT cana-4832 25 15 where where SCONJ cana-4832 25 16 a a DET cana-4832 25 17 point point NOUN cana-4832 25 18 ’s ’s PART cana-4832 25 19 self self NOUN cana-4832 25 20 - - PUNCT cana-4832 25 21 distance distance NOUN cana-4832 25 22 is be AUX cana-4832 25 23 not not PART cana-4832 25 24 always always ADV cana-4832 25 25 zero zero NUM cana-4832 25 26 . . PUNCT cana-4832 26 1 in in ADP cana-4832 26 2 this this DET cana-4832 26 3 area area NOUN cana-4832 26 4 , , PUNCT cana-4832 26 5 they they PRON cana-4832 26 6 also also ADV cana-4832 26 7 extended extend VERB cana-4832 26 8 the the DET cana-4832 26 9 banach banach NOUN cana-4832 26 10 contraction contraction NOUN cana-4832 26 11 concept concept NOUN cana-4832 26 12 . . PUNCT cana-4832 27 1 topology topology NOUN cana-4832 27 2 , , PUNCT cana-4832 27 3 logical logical ADJ cana-4832 27 4 programming programming NOUN cana-4832 27 5 , , PUNCT cana-4832 27 6 computer computer NOUN cana-4832 27 7 science science NOUN cana-4832 27 8 , , PUNCT cana-4832 27 9 electronic electronic ADJ cana-4832 27 10 engineering engineering NOUN cana-4832 27 11 , , PUNCT cana-4832 27 12 and and CCONJ cana-4832 27 13 other other ADJ cana-4832 27 14 fields field NOUN cana-4832 27 15 all all PRON cana-4832 27 16 heavily heavily ADV cana-4832 27 17 rely rely VERB cana-4832 27 18 on on ADP cana-4832 27 19 dislocated dislocate VERB cana-4832 27 20 metric metric ADJ cana-4832 27 21 space space NOUN cana-4832 27 22 . . PUNCT cana-4832 28 1 zeyadaet zeyadaet PROPN cana-4832 28 2 al al PROPN cana-4832 28 3 . . PUNCT cana-4832 29 1 [ [ X cana-4832 29 2 26 26 NUM cana-4832 29 3 ] ] PUNCT cana-4832 29 4 expanded expand VERB cana-4832 29 5 hitzler hitzler ADJ cana-4832 29 6 ’s ’s PART cana-4832 29 7 [ [ X cana-4832 29 8 13 13 NUM cana-4832 29 9 ] ] PUNCT cana-4832 29 10 result result NOUN cana-4832 29 11 in in ADP cana-4832 29 12 dislocated dislocated ADJ cana-4832 29 13 quasi quasi ADJ cana-4832 29 14 - - ADJ cana-4832 29 15 metric metric ADJ cana-4832 29 16 space space NOUN cana-4832 29 17 and and CCONJ cana-4832 29 18 introduced introduce VERB cana-4832 29 19 the the DET cana-4832 29 20 mailto:ratnababud@gmail.com mailto:ratnababud@gmail.com PROPN cana-4832 29 21 mailto:laxmana.mat@gmail.com mailto:laxmana.mat@gmail.com X cana-4832 29 22 communications communication NOUN cana-4832 29 23 on on ADP cana-4832 29 24 applied apply VERB cana-4832 29 25 nonlinear nonlinear ADJ cana-4832 29 26 analysis analysis NOUN cana-4832 29 27 issn issn NOUN cana-4832 29 28 : : PUNCT cana-4832 29 29 1074 1074 NUM cana-4832 29 30 - - PUNCT cana-4832 29 31 133x 133x NUM cana-4832 29 32 vol vol VERB cana-4832 29 33 32 32 NUM cana-4832 29 34 no no NOUN cana-4832 29 35 . . PUNCT cana-4832 30 1 10s 10 NOUN cana-4832 30 2 ( ( PUNCT cana-4832 30 3 2025 2025 NUM cana-4832 30 4 ) ) PUNCT cana-4832 30 5 390 390 NUM cana-4832 30 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 30 7 entire entire ADJ cana-4832 30 8 dislocated dislocated ADJ cana-4832 30 9 quasi quasi ADJ cana-4832 30 10 - - ADJ cana-4832 30 11 metric metric ADJ cana-4832 30 12 space space NOUN cana-4832 30 13 . . PUNCT cana-4832 31 1 for for SCONJ cana-4832 31 2 more more ADJ cana-4832 31 3 details detail NOUN cana-4832 31 4 see see VERB cana-4832 31 5 [ [ X cana-4832 31 6 1 1 NUM cana-4832 31 7 , , PUNCT cana-4832 31 8 11 11 NUM cana-4832 31 9 , , PUNCT cana-4832 31 10 14 14 NUM cana-4832 31 11 , , PUNCT cana-4832 31 12 16 16 NUM cana-4832 31 13 , , PUNCT cana-4832 31 14 17 17 NUM cana-4832 31 15 , , PUNCT cana-4832 31 16 21 21 NUM cana-4832 31 17 , , PUNCT cana-4832 31 18 23 23 NUM cana-4832 31 19 ] ] PUNCT cana-4832 31 20 the the DET cana-4832 31 21 innovative innovative ADJ cana-4832 31 22 idea idea NOUN cana-4832 31 23 of of ADP cana-4832 31 24 expansive expansive ADJ cana-4832 31 25 mapping mapping NOUN cana-4832 31 26 was be AUX cana-4832 31 27 first first ADV cana-4832 31 28 presented present VERB cana-4832 31 29 by by ADP cana-4832 31 30 wang wang PROPN cana-4832 31 31 et et PROPN cana-4832 31 32 al al PROPN cana-4832 31 33 . . PUNCT cana-4832 32 1 [ [ X cana-4832 32 2 23 23 NUM cana-4832 32 3 ] ] PUNCT cana-4832 32 4 in in ADP cana-4832 32 5 1984 1984 NUM cana-4832 32 6 . . PUNCT cana-4832 33 1 they they PRON cana-4832 33 2 carried carry VERB cana-4832 33 3 out out ADP cana-4832 33 4 a a DET cana-4832 33 5 comprehensive comprehensive ADJ cana-4832 33 6 investigation investigation NOUN cana-4832 33 7 and and CCONJ cana-4832 33 8 revealed reveal VERB cana-4832 33 9 complex complex ADJ cana-4832 33 10 fixed fix VERB cana-4832 33 11 point point NOUN cana-4832 33 12 out out ADP cana-4832 33 13 comes come VERB cana-4832 33 14 in in ADP cana-4832 33 15 the the DET cana-4832 33 16 domain domain NOUN cana-4832 33 17 of of ADP cana-4832 33 18 entire entire ADJ cana-4832 33 19 metric metric ADJ cana-4832 33 20 space space NOUN cana-4832 33 21 . . PUNCT cana-4832 34 1 applications application NOUN cana-4832 34 2 for for ADP cana-4832 34 3 expansive expansive ADJ cana-4832 34 4 mappings mapping NOUN cana-4832 34 5 can can AUX cana-4832 34 6 be be AUX cana-4832 34 7 found find VERB cana-4832 34 8 in in ADP cana-4832 34 9 nonlinear nonlinear ADJ cana-4832 34 10 analysis analysis NOUN cana-4832 34 11 , , PUNCT cana-4832 34 12 dynamical dynamical ADJ cana-4832 34 13 system system NOUN cana-4832 34 14 theory theory NOUN cana-4832 34 15 , , PUNCT cana-4832 34 16 and and CCONJ cana-4832 34 17 chaos chaos NOUN cana-4832 34 18 theory theory NOUN cana-4832 34 19 . . PUNCT cana-4832 35 1 since since SCONJ cana-4832 35 2 then then ADV cana-4832 35 3 , , PUNCT cana-4832 35 4 other other ADJ cana-4832 35 5 academics academic NOUN cana-4832 35 6 have have AUX cana-4832 35 7 conducted conduct VERB cana-4832 35 8 thorough thorough ADJ cana-4832 35 9 studies study NOUN cana-4832 35 10 , , PUNCT cana-4832 35 11 methodically methodically ADV cana-4832 35 12 developing develop VERB cana-4832 35 13 and and CCONJ cana-4832 35 14 extending extend VERB cana-4832 35 15 fixed fix VERB cana-4832 35 16 point point NOUN cana-4832 35 17 theoretical theoretical ADJ cana-4832 35 18 results result NOUN cana-4832 35 19 in in ADP cana-4832 35 20 this this DET cana-4832 35 21 specific specific ADJ cana-4832 35 22 field field NOUN cana-4832 36 1 [ [ X cana-4832 36 2 3,10,12,18 3,10,12,18 NUM cana-4832 36 3 – – PUNCT cana-4832 36 4 20 20 NUM cana-4832 36 5 , , PUNCT cana-4832 36 6 22 22 NUM cana-4832 36 7 , , PUNCT cana-4832 36 8 25 25 NUM cana-4832 36 9 ] ] PUNCT cana-4832 36 10 . . PUNCT cana-4832 37 1 definition definition NOUN cana-4832 37 2 1.1 1.1 NUM cana-4832 37 3 . . PUNCT cana-4832 38 1 [ [ X cana-4832 38 2 8 8 NUM cana-4832 38 3 ] ] PUNCT cana-4832 38 4 let let VERB cana-4832 38 5 x x PRON cana-4832 38 6 be be AUX cana-4832 38 7 a a DET cana-4832 38 8 non non ADJ cana-4832 38 9 - - ADJ cana-4832 38 10 empty empty ADJ cana-4832 38 11 set set NOUN cana-4832 38 12 and and CCONJ cana-4832 38 13 𝑠 𝑠 DET cana-4832 38 14 β‰₯ β‰₯ NUM cana-4832 38 15 1 1 NUM cana-4832 38 16 be be AUX cana-4832 38 17 a a DET cana-4832 38 18 given give VERB cana-4832 38 19 real real ADJ cana-4832 38 20 number number NOUN cana-4832 38 21 . . PUNCT cana-4832 39 1 let let VERB cana-4832 39 2 𝑑 𝑑 PRON cana-4832 39 3 : : PUNCT cana-4832 39 4 𝑋 𝑋 NOUN cana-4832 39 5 Γ— Γ— NOUN cana-4832 39 6 𝑋 𝑋 PROPN cana-4832 39 7 β†’ β†’ SYM cana-4832 39 8 [ [ X cana-4832 39 9 0,∞ 0,∞ X cana-4832 39 10 ) ) PUNCT cana-4832 39 11 be be VERB cana-4832 39 12 a a DET cana-4832 39 13 mapping mapping NOUN cana-4832 39 14 and and CCONJ cana-4832 39 15 for for ADP cana-4832 39 16 any any DET cana-4832 39 17 π‘Ž π‘Ž NOUN cana-4832 39 18 , , PUNCT cana-4832 39 19 𝑏 𝑏 PROPN cana-4832 39 20 , , PUNCT cana-4832 39 21 𝑐 𝑐 PROPN cana-4832 39 22 πœ– πœ– PROPN cana-4832 39 23 𝑋 𝑋 PROPN cana-4832 39 24 : : PUNCT cana-4832 39 25 ( ( PUNCT cana-4832 39 26 i i NOUN cana-4832 39 27 ) ) PUNCT cana-4832 39 28 . . PUNCT cana-4832 40 1 0 0 NUM cana-4832 41 1 ≀ ≀ NUM cana-4832 41 2 𝑑(π‘Ž 𝑑(π‘Ž NOUN cana-4832 41 3 , , PUNCT cana-4832 41 4 𝑏 𝑏 NOUN cana-4832 41 5 ) ) PUNCT cana-4832 41 6 and and CCONJ cana-4832 41 7 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 41 8 , , PUNCT cana-4832 41 9 𝑏 𝑏 NOUN cana-4832 41 10 ) ) PUNCT cana-4832 41 11 = = SYM cana-4832 41 12 0 0 PUNCT cana-4832 42 1 if if SCONJ cana-4832 42 2 and and CCONJ cana-4832 42 3 only only ADV cana-4832 42 4 if if SCONJ cana-4832 42 5 π‘Ž π‘Ž PROPN cana-4832 42 6 = = SYM cana-4832 42 7 𝑏 𝑏 NOUN cana-4832 42 8 ; ; PUNCT cana-4832 42 9 ( ( PUNCT cana-4832 42 10 ii ii NOUN cana-4832 42 11 ) ) PUNCT cana-4832 42 12 . . PUNCT cana-4832 43 1 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 43 2 , , PUNCT cana-4832 43 3 𝑏 𝑏 NOUN cana-4832 43 4 ) ) PUNCT cana-4832 43 5 = = SYM cana-4832 43 6 0 0 NUM cana-4832 43 7 implies imply VERB cana-4832 43 8 π‘Ž π‘Ž PROPN cana-4832 43 9 = = SYM cana-4832 43 10 𝑏 𝑏 NOUN cana-4832 43 11 ; ; PUNCT cana-4832 43 12 ( ( PUNCT cana-4832 43 13 iii iii NOUN cana-4832 43 14 ) ) PUNCT cana-4832 43 15 . . PUNCT cana-4832 44 1 𝑑(π‘Ž 𝑑(π‘Ž NOUN cana-4832 44 2 , , PUNCT cana-4832 44 3 𝑏 𝑏 NOUN cana-4832 44 4 ) ) PUNCT cana-4832 44 5 = = SYM cana-4832 44 6 0 0 PUNCT cana-4832 45 1 = = SYM cana-4832 45 2 𝑑(𝑏 𝑑(𝑏 PROPN cana-4832 45 3 , , PUNCT cana-4832 45 4 π‘Ž π‘Ž NOUN cana-4832 45 5 ) ) PUNCT cana-4832 45 6 implies imply VERB cana-4832 45 7 π‘Ž π‘Ž PROPN cana-4832 45 8 = = SYM cana-4832 45 9 𝑏 𝑏 NOUN cana-4832 45 10 ; ; PUNCT cana-4832 45 11 ( ( PUNCT cana-4832 45 12 iv iv X cana-4832 45 13 ) ) PUNCT cana-4832 45 14 . . PUNCT cana-4832 46 1 𝑑(π‘Ž 𝑑(π‘Ž NOUN cana-4832 46 2 , , PUNCT cana-4832 46 3 𝑏 𝑏 NOUN cana-4832 46 4 ) ) PUNCT cana-4832 46 5 = = SYM cana-4832 46 6 𝑑(𝑏 𝑑(𝑏 PROPN cana-4832 46 7 , , PUNCT cana-4832 46 8 π‘Ž π‘Ž NOUN cana-4832 46 9 ) ) PUNCT cana-4832 46 10 ; ; PUNCT cana-4832 46 11 ( ( PUNCT cana-4832 46 12 v v NOUN cana-4832 46 13 ) ) PUNCT cana-4832 46 14 . . PUNCT cana-4832 47 1 𝑑(π‘Ž 𝑑(π‘Ž NOUN cana-4832 47 2 , , PUNCT cana-4832 47 3 𝑐 𝑐 NOUN cana-4832 47 4 ) ) PUNCT cana-4832 47 5 ≀ ≀ NOUN cana-4832 47 6 𝑑(π‘Ž 𝑑(π‘Ž NOUN cana-4832 47 7 , , PUNCT cana-4832 47 8 𝑏 𝑏 NOUN cana-4832 47 9 ) ) PUNCT cana-4832 47 10 + + CCONJ cana-4832 47 11 𝑑(𝑏 𝑑(𝑏 PROPN cana-4832 47 12 , , PUNCT cana-4832 47 13 𝑐 𝑐 NOUN cana-4832 47 14 ) ) PUNCT cana-4832 47 15 ; ; PUNCT cana-4832 47 16 ( ( PUNCT cana-4832 47 17 vi vi NOUN cana-4832 47 18 ) ) PUNCT cana-4832 47 19 . . PUNCT cana-4832 48 1 𝑑(π‘Ž 𝑑(π‘Ž NOUN cana-4832 48 2 , , PUNCT cana-4832 48 3 𝑐 𝑐 NOUN cana-4832 48 4 ) ) PUNCT cana-4832 48 5 ≀ ≀ NOUN cana-4832 48 6 𝑠[𝑑(π‘Ž 𝑠[𝑑(π‘Ž NOUN cana-4832 48 7 , , PUNCT cana-4832 48 8 𝑏 𝑏 NOUN cana-4832 48 9 ) ) PUNCT cana-4832 48 10 + + ADJ cana-4832 48 11 𝑑(𝑏 𝑑(𝑏 PROPN cana-4832 48 12 , , PUNCT cana-4832 48 13 𝑐 𝑐 NOUN cana-4832 48 14 ) ) PUNCT cana-4832 48 15 ] ] PUNCT cana-4832 48 16 . . PUNCT cana-4832 49 1 then then ADV cana-4832 49 2 ( ( PUNCT cana-4832 49 3 1 1 X cana-4832 49 4 ) ) PUNCT cana-4832 49 5 ( ( PUNCT cana-4832 49 6 𝑋 𝑋 PROPN cana-4832 49 7 , , PUNCT cana-4832 49 8 𝑑 𝑑 NOUN cana-4832 49 9 ) ) PUNCT cana-4832 49 10 is be AUX cana-4832 49 11 called call VERB cana-4832 49 12 a a DET cana-4832 49 13 metric metric ADJ cana-4832 49 14 space space NOUN cana-4832 49 15 if if SCONJ cana-4832 49 16 ( ( PUNCT cana-4832 49 17 i i NOUN cana-4832 49 18 ) ) PUNCT cana-4832 49 19 , , PUNCT cana-4832 49 20 ( ( PUNCT cana-4832 49 21 iv iv X cana-4832 49 22 ) ) PUNCT cana-4832 49 23 , , PUNCT cana-4832 49 24 and and CCONJ cana-4832 49 25 ( ( PUNCT cana-4832 49 26 v v NOUN cana-4832 49 27 ) ) PUNCT cana-4832 49 28 hold hold VERB cana-4832 49 29 ; ; PUNCT cana-4832 49 30 ( ( PUNCT cana-4832 49 31 2 2 X cana-4832 49 32 ) ) PUNCT cana-4832 49 33 ( ( PUNCT cana-4832 49 34 𝑋 𝑋 PROPN cana-4832 49 35 , , PUNCT cana-4832 49 36 𝑑 𝑑 NOUN cana-4832 49 37 ) ) PUNCT cana-4832 49 38 is be AUX cana-4832 49 39 called call VERB cana-4832 49 40 a a DET cana-4832 49 41 b b NOUN cana-4832 49 42 - - PUNCT cana-4832 49 43 metric metric ADJ cana-4832 49 44 space space NOUN cana-4832 49 45 if if SCONJ cana-4832 49 46 ( ( PUNCT cana-4832 49 47 i i NOUN cana-4832 49 48 ) ) PUNCT cana-4832 49 49 , , PUNCT cana-4832 49 50 ( ( PUNCT cana-4832 49 51 iv iv X cana-4832 49 52 ) ) PUNCT cana-4832 49 53 , , PUNCT cana-4832 49 54 and and CCONJ cana-4832 49 55 ( ( PUNCT cana-4832 49 56 vi vi NOUN cana-4832 49 57 ) ) PUNCT cana-4832 49 58 hold hold VERB cana-4832 49 59 ; ; PUNCT cana-4832 49 60 ( ( PUNCT cana-4832 49 61 3 3 X cana-4832 49 62 ) ) PUNCT cana-4832 49 63 ( ( PUNCT cana-4832 49 64 𝑋 𝑋 PROPN cana-4832 49 65 , , PUNCT cana-4832 49 66 𝑑 𝑑 NOUN cana-4832 49 67 ) ) PUNCT cana-4832 49 68 is be AUX cana-4832 49 69 called call VERB cana-4832 49 70 a a DET cana-4832 49 71 quasi quasi ADJ cana-4832 49 72 - - ADJ cana-4832 49 73 metric metric ADJ cana-4832 49 74 space space NOUN cana-4832 49 75 if if SCONJ cana-4832 49 76 ( ( PUNCT cana-4832 49 77 i i NOUN cana-4832 49 78 ) ) PUNCT cana-4832 49 79 , , PUNCT cana-4832 49 80 and and CCONJ cana-4832 49 81 ( ( PUNCT cana-4832 49 82 v v NOUN cana-4832 49 83 ) ) PUNCT cana-4832 49 84 hold hold VERB cana-4832 49 85 ; ; PUNCT cana-4832 49 86 ( ( PUNCT cana-4832 49 87 4 4 X cana-4832 49 88 ) ) PUNCT cana-4832 49 89 ( ( PUNCT cana-4832 49 90 𝑋 𝑋 PROPN cana-4832 49 91 , , PUNCT cana-4832 49 92 𝑑 𝑑 NOUN cana-4832 49 93 ) ) PUNCT cana-4832 49 94 is be AUX cana-4832 49 95 called call VERB cana-4832 49 96 a a DET cana-4832 49 97 quasi quasi ADJ cana-4832 49 98 - - ADJ cana-4832 49 99 b b ADJ cana-4832 49 100 - - PUNCT cana-4832 49 101 metric metric ADJ cana-4832 49 102 space space NOUN cana-4832 49 103 if if SCONJ cana-4832 49 104 ( ( PUNCT cana-4832 49 105 i i NOUN cana-4832 49 106 ) ) PUNCT cana-4832 49 107 , , PUNCT cana-4832 49 108 and and CCONJ cana-4832 49 109 ( ( PUNCT cana-4832 49 110 vi vi NOUN cana-4832 49 111 ) ) PUNCT cana-4832 49 112 hold hold VERB cana-4832 49 113 ; ; PUNCT cana-4832 49 114 ( ( PUNCT cana-4832 49 115 5 5 X cana-4832 49 116 ) ) PUNCT cana-4832 49 117 ( ( PUNCT cana-4832 49 118 𝑋 𝑋 PROPN cana-4832 49 119 , , PUNCT cana-4832 49 120 𝑑 𝑑 NOUN cana-4832 49 121 ) ) PUNCT cana-4832 49 122 is be AUX cana-4832 49 123 called call VERB cana-4832 49 124 a a DET cana-4832 49 125 dislocated dislocate VERB cana-4832 49 126 metric metric ADJ cana-4832 49 127 space space NOUN cana-4832 49 128 ( ( PUNCT cana-4832 49 129 𝑑-metric 𝑑-metric PROPN cana-4832 49 130 space space NOUN cana-4832 49 131 ) ) PUNCT cana-4832 50 1 if if SCONJ cana-4832 50 2 ( ( PUNCT cana-4832 50 3 ii ii NOUN cana-4832 50 4 ) ) PUNCT cana-4832 50 5 , , PUNCT cana-4832 50 6 ( ( PUNCT cana-4832 50 7 iv iv X cana-4832 50 8 ) ) PUNCT cana-4832 50 9 , , PUNCT cana-4832 50 10 and and CCONJ cana-4832 50 11 ( ( PUNCT cana-4832 50 12 v v NOUN cana-4832 50 13 ) ) PUNCT cana-4832 50 14 hold hold NOUN cana-4832 50 15 ; ; PUNCT cana-4832 50 16 ( ( PUNCT cana-4832 50 17 6 6 NUM cana-4832 50 18 ) ) PUNCT cana-4832 50 19 ( ( PUNCT cana-4832 50 20 𝑋 𝑋 PROPN cana-4832 50 21 , , PUNCT cana-4832 50 22 𝑑 𝑑 NOUN cana-4832 50 23 ) ) PUNCT cana-4832 50 24 is be AUX cana-4832 50 25 called call VERB cana-4832 50 26 a a DET cana-4832 50 27 dislocated dislocate VERB cana-4832 50 28 𝑏-metric 𝑏-metric NOUN cana-4832 50 29 space space NOUN cana-4832 50 30 ( ( PUNCT cana-4832 50 31 𝑑 𝑑 PROPN cana-4832 50 32 𝑏-metric 𝑏-metric PROPN cana-4832 50 33 space space NOUN cana-4832 50 34 ) ) PUNCT cana-4832 51 1 if if SCONJ cana-4832 51 2 ( ( PUNCT cana-4832 51 3 ii ii NOUN cana-4832 51 4 ) ) PUNCT cana-4832 51 5 , , PUNCT cana-4832 51 6 ( ( PUNCT cana-4832 51 7 iv iv X cana-4832 51 8 ) ) PUNCT cana-4832 51 9 , , PUNCT cana-4832 51 10 and and CCONJ cana-4832 51 11 ( ( PUNCT cana-4832 51 12 vi vi NOUN cana-4832 51 13 ) ) PUNCT cana-4832 51 14 hold hold VERB cana-4832 51 15 ; ; PUNCT cana-4832 51 16 ( ( PUNCT cana-4832 51 17 7 7 X cana-4832 51 18 ) ) PUNCT cana-4832 51 19 ( ( PUNCT cana-4832 51 20 𝑋 𝑋 PROPN cana-4832 51 21 , , PUNCT cana-4832 51 22 𝑑 𝑑 NOUN cana-4832 51 23 ) ) PUNCT cana-4832 51 24 is be AUX cana-4832 51 25 called call VERB cana-4832 51 26 a a DET cana-4832 51 27 dislocated dislocate VERB cana-4832 51 28 quasi quasi ADJ cana-4832 51 29 - - ADJ cana-4832 51 30 metric metric ADJ cana-4832 51 31 space space NOUN cana-4832 51 32 ( ( PUNCT cana-4832 51 33 π‘‘π‘ž-metric π‘‘π‘ž-metric ADJ cana-4832 51 34 space space NOUN cana-4832 51 35 ) ) PUNCT cana-4832 51 36 if if SCONJ cana-4832 51 37 ( ( PUNCT cana-4832 51 38 iii iii NOUN cana-4832 51 39 ) ) PUNCT cana-4832 51 40 and and CCONJ cana-4832 51 41 ( ( PUNCT cana-4832 51 42 v v NOUN cana-4832 51 43 ) ) PUNCT cana-4832 51 44 hold hold NOUN cana-4832 51 45 ; ; PUNCT cana-4832 51 46 ( ( PUNCT cana-4832 51 47 8) 8) NUM cana-4832 51 48 ( ( PUNCT cana-4832 51 49 𝑋 𝑋 PROPN cana-4832 51 50 , , PUNCT cana-4832 51 51 𝑑 𝑑 NOUN cana-4832 51 52 ) ) PUNCT cana-4832 51 53 is be AUX cana-4832 51 54 called call VERB cana-4832 51 55 a a DET cana-4832 51 56 dislocated dislocate VERB cana-4832 51 57 quasi quasi ADJ cana-4832 51 58 - - ADJ cana-4832 51 59 b b ADJ cana-4832 51 60 - - PUNCT cana-4832 51 61 metric metric ADJ cana-4832 51 62 space space NOUN cana-4832 51 63 ( ( PUNCT cana-4832 51 64 π‘‘π‘ž π‘‘π‘ž PROPN cana-4832 51 65 𝑏-metric 𝑏-metric PROPN cana-4832 51 66 space space NOUN cana-4832 51 67 ) ) PUNCT cana-4832 51 68 if if SCONJ cana-4832 51 69 ( ( PUNCT cana-4832 51 70 iii iii NOUN cana-4832 51 71 ) ) PUNCT cana-4832 51 72 and and CCONJ cana-4832 51 73 ( ( PUNCT cana-4832 51 74 vi vi NOUN cana-4832 51 75 ) ) PUNCT cana-4832 51 76 hold hold NOUN cana-4832 51 77 . . PUNCT cana-4832 52 1 even even ADV cana-4832 52 2 though though SCONJ cana-4832 52 3 the the DET cana-4832 52 4 examples example NOUN cana-4832 52 5 provided provide VERB cana-4832 52 6 were be AUX cana-4832 52 7 well well ADV cana-4832 52 8 - - PUNCT cana-4832 52 9 known know VERB cana-4832 52 10 , , PUNCT cana-4832 52 11 we we PRON cana-4832 52 12 felt feel VERB cana-4832 52 13 that that SCONJ cana-4832 52 14 providing provide VERB cana-4832 52 15 a a DET cana-4832 52 16 thorough thorough ADJ cana-4832 52 17 review review NOUN cana-4832 52 18 would would AUX cana-4832 52 19 be be AUX cana-4832 52 20 helpful helpful ADJ cana-4832 52 21 for for ADP cana-4832 52 22 convenient convenient ADJ cana-4832 52 23 reference reference NOUN cana-4832 52 24 . . PUNCT cana-4832 53 1 example example NOUN cana-4832 53 2 1.2 1.2 NUM cana-4832 53 3 . . PUNCT cana-4832 54 1 ( ( PUNCT cana-4832 54 2 π‘Ž π‘Ž X cana-4832 54 3 ) ) PUNCT cana-4832 54 4 let let VERB cana-4832 54 5 𝑋=ℝ 𝑋=ℝ NOUN cana-4832 54 6 and and CCONJ cana-4832 55 1 𝑑:𝑋 𝑑:𝑋 ADJ cana-4832 55 2 Γ— Γ— NOUN cana-4832 55 3 𝑋→ℝ+ 𝑋→ℝ+ VERB cana-4832 55 4 defined define VERB cana-4832 55 5 as as ADP cana-4832 55 6 𝑑(π‘Ž 𝑑(π‘Ž NOUN cana-4832 55 7 , , PUNCT cana-4832 55 8 𝑏 𝑏 NOUN cana-4832 55 9 ) ) PUNCT cana-4832 55 10 = = NOUN cana-4832 55 11 { { PUNCT cana-4832 55 12 π‘Ž π‘Ž X cana-4832 55 13 βˆ’ βˆ’ PROPN cana-4832 55 14 𝑏 𝑏 NOUN cana-4832 55 15 , , PUNCT cana-4832 55 16 π‘Ž π‘Ž DET cana-4832 55 17 β‰₯ β‰₯ NOUN cana-4832 55 18 𝑏 𝑏 SYM cana-4832 55 19 1 1 NUM cana-4832 55 20 , , PUNCT cana-4832 55 21 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘–π‘ π‘’. π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘–π‘ π‘’. NOUN cana-4832 55 22 then then ADV cana-4832 55 23 ( ( PUNCT cana-4832 55 24 𝑋 𝑋 PROPN cana-4832 55 25 , , PUNCT cana-4832 55 26 𝑑 𝑑 NOUN cana-4832 55 27 ) ) PUNCT cana-4832 55 28 is be AUX cana-4832 55 29 a a DET cana-4832 55 30 quasi quasi ADJ cana-4832 55 31 - - ADJ cana-4832 55 32 metric metric ADJ cana-4832 55 33 space space NOUN cana-4832 55 34 , , PUNCT cana-4832 55 35 but but CCONJ cana-4832 55 36 it it PRON cana-4832 55 37 is be AUX cana-4832 55 38 not not PART cana-4832 55 39 a a DET cana-4832 55 40 metric metric ADJ cana-4832 55 41 space space NOUN cana-4832 55 42 . . PUNCT cana-4832 56 1 ( ( PUNCT cana-4832 56 2 𝑏 𝑏 NOUN cana-4832 56 3 ) ) PUNCT cana-4832 56 4 let let VERB cana-4832 56 5 𝑋=ℝ+ 𝑋=ℝ+ PRON cana-4832 56 6 and and CCONJ cana-4832 56 7 𝑑 𝑑 NOUN cana-4832 56 8 : : PUNCT cana-4832 56 9 𝑋 𝑋 PROPN cana-4832 56 10 Γ— Γ— PROPN cana-4832 56 11 𝑋→ 𝑋→ PROPN cana-4832 56 12 ℝ+ ℝ+ PUNCT cana-4832 56 13 defined define VERB cana-4832 56 14 as as ADP cana-4832 56 15 𝑑(π‘Ž 𝑑(π‘Ž NOUN cana-4832 56 16 , , PUNCT cana-4832 56 17 𝑏 𝑏 NOUN cana-4832 56 18 ) ) PUNCT cana-4832 56 19 = = NOUN cana-4832 56 20 { { PUNCT cana-4832 56 21 0 0 NUM cana-4832 56 22 , , PUNCT cana-4832 56 23 π‘Ž π‘Ž PROPN cana-4832 56 24 = = SYM cana-4832 56 25 𝑏 𝑏 NOUN cana-4832 56 26 ( ( PUNCT cana-4832 56 27 π‘Ž π‘Ž X cana-4832 56 28 + + X cana-4832 56 29 𝑏)2 𝑏)2 PROPN cana-4832 56 30 , , PUNCT cana-4832 56 31 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘–π‘ π‘’. π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘–π‘ π‘’. NOUN cana-4832 56 32 then then ADV cana-4832 56 33 ( ( PUNCT cana-4832 56 34 𝑋 𝑋 PROPN cana-4832 56 35 , , PUNCT cana-4832 56 36 𝑑 𝑑 NOUN cana-4832 56 37 ) ) PUNCT cana-4832 56 38 is be AUX cana-4832 56 39 a a DET cana-4832 56 40 𝑏-metric 𝑏-metric ADJ cana-4832 56 41 space space NOUN cana-4832 56 42 , , PUNCT cana-4832 56 43 but but CCONJ cana-4832 56 44 it it PRON cana-4832 56 45 is be AUX cana-4832 56 46 not not PART cana-4832 56 47 a a DET cana-4832 56 48 metric metric ADJ cana-4832 56 49 space space NOUN cana-4832 56 50 . . PUNCT cana-4832 57 1 ( ( PUNCT cana-4832 57 2 𝑐 𝑐 X cana-4832 57 3 ) ) PUNCT cana-4832 57 4 let let VERB cana-4832 57 5 𝑋 𝑋 PROPN cana-4832 57 6 = = PUNCT cana-4832 57 7 𝐢([0,1],ℝ 𝐢([0,1],ℝ PROPN cana-4832 57 8 ) ) PUNCT cana-4832 57 9 with with ADP cana-4832 57 10 the the DET cana-4832 57 11 usual usual ADJ cana-4832 57 12 partial partial ADJ cana-4832 57 13 ordering ordering NOUN cana-4832 57 14 , , PUNCT cana-4832 57 15 and and CCONJ cana-4832 57 16 let let VERB cana-4832 57 17 𝑑:𝑋 𝑑:𝑋 PROPN cana-4832 57 18 Γ— Γ— NOUN cana-4832 57 19 𝑋→ℝ+ 𝑋→ℝ+ AUX cana-4832 57 20 be be AUX cana-4832 57 21 defined define VERB cana-4832 57 22 as as ADP cana-4832 57 23 communications communication NOUN cana-4832 57 24 on on ADP cana-4832 57 25 applied apply VERB cana-4832 57 26 nonlinear nonlinear ADJ cana-4832 57 27 analysis analysis NOUN cana-4832 57 28 issn issn NOUN cana-4832 57 29 : : PUNCT cana-4832 57 30 1074 1074 NUM cana-4832 57 31 - - PUNCT cana-4832 57 32 133x 133x NUM cana-4832 57 33 vol vol VERB cana-4832 57 34 32 32 NUM cana-4832 57 35 no no NOUN cana-4832 57 36 . . PUNCT cana-4832 58 1 10s 10 NOUN cana-4832 58 2 ( ( PUNCT cana-4832 58 3 2025 2025 NUM cana-4832 58 4 ) ) PUNCT cana-4832 58 5 391 391 NUM cana-4832 58 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 58 7 𝑑(𝑓 𝑑(𝑓 PROPN cana-4832 58 8 , , PUNCT cana-4832 58 9 𝑔 𝑔 NOUN cana-4832 58 10 ) ) PUNCT cana-4832 58 11 = = SYM cana-4832 58 12 { { PUNCT cana-4832 58 13 ∫(𝑔(𝑑 ∫(𝑔(𝑑 NOUN cana-4832 58 14 ) ) PUNCT cana-4832 58 15 βˆ’ βˆ’ NOUN cana-4832 59 1 𝑓(𝑑))3 𝑓(𝑑))3 NOUN cana-4832 59 2 𝑑𝑑 𝑑𝑑 ADP cana-4832 59 3 , , PUNCT cana-4832 59 4 𝑓 𝑓 PRON cana-4832 59 5 ≀ ≀ PROPN cana-4832 59 6 𝑔 𝑔 NOUN cana-4832 59 7 , , PUNCT cana-4832 59 8 1 1 NUM cana-4832 59 9 0 0 NUM cana-4832 59 10 ∫(𝑓(𝑑 ∫(𝑓(𝑑 NUM cana-4832 59 11 ) ) PUNCT cana-4832 59 12 βˆ’ βˆ’ NOUN cana-4832 60 1 𝑔(𝑑))3 𝑔(𝑑))3 VERB cana-4832 60 2 1 1 NUM cana-4832 60 3 0 0 NUM cana-4832 60 4 𝑑𝑑 𝑑𝑑 PROPN cana-4832 60 5 , , PUNCT cana-4832 60 6 𝑓 𝑓 DET cana-4832 60 7 β‰₯ β‰₯ NOUN cana-4832 60 8 𝑔. 𝑔. NOUN cana-4832 61 1 then then ADV cana-4832 61 2 ( ( PUNCT cana-4832 61 3 𝑋 𝑋 PROPN cana-4832 61 4 , , PUNCT cana-4832 61 5 𝑑 𝑑 NOUN cana-4832 61 6 ) ) PUNCT cana-4832 61 7 is be AUX cana-4832 61 8 a a DET cana-4832 61 9 quasi quasi ADJ cana-4832 61 10 - - ADJ cana-4832 61 11 b b ADJ cana-4832 61 12 - - PUNCT cana-4832 61 13 metric metric ADJ cana-4832 61 14 space space NOUN cana-4832 61 15 , , PUNCT cana-4832 61 16 but but CCONJ cana-4832 61 17 it it PRON cana-4832 61 18 is be AUX cana-4832 61 19 not not PART cana-4832 61 20 a a DET cana-4832 61 21 quasi quasi ADJ cana-4832 61 22 - - ADJ cana-4832 61 23 metric metric ADJ cana-4832 61 24 space space NOUN cana-4832 61 25 and and CCONJ cana-4832 61 26 𝑏-metric 𝑏-metric PROPN cana-4832 61 27 space space NOUN cana-4832 61 28 . . PUNCT cana-4832 62 1 ( ( PUNCT cana-4832 62 2 𝑑 𝑑 AUX cana-4832 62 3 ) ) PUNCT cana-4832 62 4 let let VERB cana-4832 62 5 𝑋=ℝ+ 𝑋=ℝ+ PRON cana-4832 62 6 and and CCONJ cana-4832 62 7 𝑑 𝑑 NOUN cana-4832 62 8 : : PUNCT cana-4832 62 9 𝑋 𝑋 NOUN cana-4832 62 10 Γ— Γ— NOUN cana-4832 62 11 𝑋 𝑋 PROPN cana-4832 62 12 β†’ β†’ SYM cana-4832 62 13 ℝ+ ℝ+ PUNCT cana-4832 62 14 defined define VERB cana-4832 62 15 as as ADP cana-4832 62 16 𝑑(π‘Ž 𝑑(π‘Ž NOUN cana-4832 62 17 , , PUNCT cana-4832 62 18 𝑏 𝑏 NOUN cana-4832 62 19 ) ) PUNCT cana-4832 62 20 = = PUNCT cana-4832 62 21 max{π‘Ž max{π‘Ž ADV cana-4832 62 22 , , PUNCT cana-4832 62 23 𝑏}.then 𝑏}.then X cana-4832 62 24 ( ( PUNCT cana-4832 62 25 𝑋 𝑋 PROPN cana-4832 62 26 , , PUNCT cana-4832 62 27 𝑑 𝑑 NOUN cana-4832 62 28 ) ) PUNCT cana-4832 62 29 is be AUX cana-4832 62 30 a a DET cana-4832 62 31 dislocated dislocate VERB cana-4832 62 32 metric metric ADJ cana-4832 62 33 space space NOUN cana-4832 62 34 , , PUNCT cana-4832 62 35 but but CCONJ cana-4832 62 36 it it PRON cana-4832 62 37 is be AUX cana-4832 62 38 not not PART cana-4832 62 39 a a DET cana-4832 62 40 metric metric ADJ cana-4832 62 41 space space NOUN cana-4832 62 42 . . PUNCT cana-4832 63 1 ( ( PUNCT cana-4832 63 2 𝑒 𝑒 X cana-4832 63 3 ) ) PUNCT cana-4832 63 4 let let VERB cana-4832 63 5 𝑋 𝑋 NOUN cana-4832 63 6 = = PUNCT cana-4832 64 1 [ [ X cana-4832 64 2 0,1 0,1 NUM cana-4832 64 3 ] ] PUNCT cana-4832 64 4 and and CCONJ cana-4832 64 5 𝑑:𝑋 𝑑:𝑋 ADJ cana-4832 64 6 Γ— Γ— NOUN cana-4832 64 7 𝑋 𝑋 PROPN cana-4832 64 8 β†’ β†’ SYM cana-4832 64 9 ℝ+ ℝ+ PUNCT cana-4832 64 10 be be AUX cana-4832 64 11 defined define VERB cana-4832 64 12 as as ADP cana-4832 64 13 𝑑(π‘Ž 𝑑(π‘Ž NOUN cana-4832 64 14 , , PUNCT cana-4832 64 15 𝑏 𝑏 NOUN cana-4832 64 16 ) ) PUNCT cana-4832 64 17 = = SYM cana-4832 64 18 |π‘Ž |π‘Ž X cana-4832 65 1 βˆ’ βˆ’ PROPN cana-4832 65 2 𝑏| 𝑏| PROPN cana-4832 65 3 + + CCONJ cana-4832 65 4 π‘Ž. π‘Ž. NOUN cana-4832 65 5 then then ADV cana-4832 65 6 ( ( PUNCT cana-4832 65 7 𝑋 𝑋 PROPN cana-4832 65 8 , , PUNCT cana-4832 65 9 𝑑 𝑑 NOUN cana-4832 65 10 ) ) PUNCT cana-4832 65 11 is be AUX cana-4832 65 12 a a DET cana-4832 65 13 dislocated dislocate VERB cana-4832 65 14 quasi quasi ADJ cana-4832 65 15 - - ADJ cana-4832 65 16 metric metric ADJ cana-4832 65 17 space space NOUN cana-4832 65 18 , , PUNCT cana-4832 65 19 but but CCONJ cana-4832 65 20 it it PRON cana-4832 65 21 is be AUX cana-4832 65 22 not not PART cana-4832 65 23 a a DET cana-4832 65 24 dislocated dislocate VERB cana-4832 65 25 metric metric ADJ cana-4832 65 26 space space NOUN cana-4832 65 27 , , PUNCT cana-4832 65 28 and and CCONJ cana-4832 65 29 it it PRON cana-4832 65 30 is be AUX cana-4832 65 31 not not PART cana-4832 65 32 a a DET cana-4832 65 33 quasimetric quasimetric ADJ cana-4832 65 34 space space NOUN cana-4832 65 35 . . PUNCT cana-4832 66 1 ( ( PUNCT cana-4832 66 2 𝑓 𝑓 X cana-4832 66 3 ) ) PUNCT cana-4832 66 4 let let VERB cana-4832 66 5 𝑋 𝑋 PROPN cana-4832 66 6 = = PUNCT cana-4832 67 1 [ [ X cana-4832 67 2 0,∞ 0,∞ NUM cana-4832 67 3 ) ) PUNCT cana-4832 67 4 and and CCONJ cana-4832 67 5 𝑑 𝑑 NOUN cana-4832 67 6 : : PUNCT cana-4832 67 7 𝑋 𝑋 NOUN cana-4832 67 8 Γ— Γ— NOUN cana-4832 67 9 𝑋 𝑋 PROPN cana-4832 67 10 β†’[0,∞ β†’[0,∞ NOUN cana-4832 67 11 ) ) PUNCT cana-4832 67 12 be be AUX cana-4832 67 13 defined define VERB cana-4832 67 14 as as ADP cana-4832 67 15 𝑑(π‘Ž 𝑑(π‘Ž NOUN cana-4832 67 16 , , PUNCT cana-4832 67 17 𝑏 𝑏 NOUN cana-4832 67 18 ) ) PUNCT cana-4832 67 19 = = NOUN cana-4832 67 20 ( ( PUNCT cana-4832 68 1 π‘Ž π‘Ž X cana-4832 68 2 + + X cana-4832 68 3 𝑏)2 𝑏)2 PROPN cana-4832 68 4 . . PUNCT cana-4832 69 1 then then ADV cana-4832 69 2 ( ( PUNCT cana-4832 69 3 𝑋 𝑋 PROPN cana-4832 69 4 , , PUNCT cana-4832 69 5 𝑑 𝑑 NOUN cana-4832 69 6 ) ) PUNCT cana-4832 69 7 is be AUX cana-4832 69 8 a a DET cana-4832 69 9 dislocated dislocate VERB cana-4832 69 10 𝑏-metric 𝑏-metric ADJ cana-4832 69 11 space space NOUN cana-4832 69 12 , , PUNCT cana-4832 69 13 but but CCONJ cana-4832 69 14 it it PRON cana-4832 69 15 is be AUX cana-4832 69 16 not not PART cana-4832 69 17 a a DET cana-4832 69 18 𝑏-metric 𝑏-metric ADJ cana-4832 69 19 space space NOUN cana-4832 69 20 . . PUNCT cana-4832 70 1 ( ( PUNCT cana-4832 70 2 𝑔 𝑔 X cana-4832 70 3 ) ) PUNCT cana-4832 70 4 let let VERB cana-4832 70 5 𝑋=ℝ 𝑋=ℝ NOUN cana-4832 70 6 and and CCONJ cana-4832 70 7 𝑑 𝑑 NOUN cana-4832 70 8 : : PUNCT cana-4832 70 9 𝑋 𝑋 NOUN cana-4832 70 10 Γ— Γ— NOUN cana-4832 70 11 𝑋 𝑋 PROPN cana-4832 70 12 β†’ β†’ SYM cana-4832 70 13 ℝ+ ℝ+ PUNCT cana-4832 70 14 be be AUX cana-4832 70 15 defined define VERB cana-4832 70 16 as as ADP cana-4832 70 17 𝑑(π‘Ž 𝑑(π‘Ž NOUN cana-4832 70 18 , , PUNCT cana-4832 70 19 𝑏 𝑏 NOUN cana-4832 70 20 ) ) PUNCT cana-4832 70 21 = = SYM cana-4832 70 22 |π‘Ž |π‘Ž X cana-4832 71 1 βˆ’ βˆ’ PROPN cana-4832 71 2 𝑏|2 𝑏|2 PROPN cana-4832 71 3 + + CCONJ cana-4832 71 4 |π‘Ž| |π‘Ž| PROPN cana-4832 71 5 𝑛 𝑛 DET cana-4832 71 6 + + NUM cana-4832 71 7 |𝑏| |𝑏| PROPN cana-4832 71 8 π‘š π‘š NOUN cana-4832 71 9 , , PUNCT cana-4832 71 10 where where SCONJ cana-4832 71 11 𝑛,π‘š 𝑛,π‘š NOUN cana-4832 71 12 ∈ ∈ PROPN cana-4832 71 13 β„•\{1 β„•\{1 PROPN cana-4832 71 14 } } PUNCT cana-4832 71 15 , , PUNCT cana-4832 71 16 𝑛 𝑛 DET cana-4832 71 17 β‰  β‰  PROPN cana-4832 71 18 π‘š.then π‘š.then ADV cana-4832 71 19 ( ( PUNCT cana-4832 71 20 𝑋 𝑋 PROPN cana-4832 71 21 , , PUNCT cana-4832 71 22 𝑑 𝑑 NOUN cana-4832 71 23 ) ) PUNCT cana-4832 71 24 is be AUX cana-4832 71 25 a a DET cana-4832 71 26 dislocated dislocate VERB cana-4832 71 27 quasi quasi ADJ cana-4832 71 28 - - ADJ cana-4832 71 29 metric metric ADJ cana-4832 71 30 space space NOUN cana-4832 71 31 , , PUNCT cana-4832 71 32 but but CCONJ cana-4832 71 33 it it PRON cana-4832 71 34 is be AUX cana-4832 71 35 not not PART cana-4832 71 36 a a DET cana-4832 71 37 quasi quasi ADJ cana-4832 71 38 𝑏metric 𝑏metric ADJ cana-4832 71 39 space space NOUN cana-4832 71 40 , , PUNCT cana-4832 71 41 dislocated dislocate VERB cana-4832 71 42 𝑏-metric 𝑏-metric ADJ cana-4832 71 43 space space NOUN cana-4832 71 44 and and CCONJ cana-4832 71 45 dislocated dislocated ADJ cana-4832 71 46 quasi quasi ADJ cana-4832 71 47 - - ADJ cana-4832 71 48 metric metric ADJ cana-4832 71 49 space space NOUN cana-4832 71 50 . . PUNCT cana-4832 72 1 thus thus ADV cana-4832 72 2 , , PUNCT cana-4832 72 3 we we PRON cana-4832 72 4 get get VERB cana-4832 72 5 the the DET cana-4832 72 6 process process NOUN cana-4832 72 7 diagram diagram NOUN cana-4832 72 8 ( ( PUNCT cana-4832 72 9 refer refer VERB cana-4832 72 10 to to PART cana-4832 72 11 figure figure VERB cana-4832 72 12 1 1 NUM cana-4832 72 13 ) ) PUNCT cana-4832 72 14 , , PUNCT cana-4832 72 15 in in ADP cana-4832 72 16 which which PRON cana-4832 72 17 generalization generalization NOUN cana-4832 72 18 relationships relationship NOUN cana-4832 72 19 are be AUX cana-4832 72 20 represented represent VERB cana-4832 72 21 by by ADP cana-4832 72 22 arrows arrow NOUN cana-4832 72 23 . . PUNCT cana-4832 73 1 figure figure NOUN cana-4832 73 2 1 1 NUM cana-4832 73 3 : : PUNCT cana-4832 73 4 process process NOUN cana-4832 73 5 diagram diagram NOUN cana-4832 73 6 . . PUNCT cana-4832 74 1 the the DET cana-4832 74 2 following follow VERB cana-4832 74 3 lemmas lemma NOUN cana-4832 74 4 are be AUX cana-4832 74 5 useful useful ADJ cana-4832 74 6 in in ADP cana-4832 74 7 proving prove VERB cana-4832 74 8 our our PRON cana-4832 74 9 main main ADJ cana-4832 74 10 results result NOUN cana-4832 74 11 . . PUNCT cana-4832 75 1 lemma lemma PROPN cana-4832 75 2 1.3 1.3 NUM cana-4832 75 3 . . PUNCT cana-4832 76 1 [ [ X cana-4832 76 2 2 2 NUM cana-4832 76 3 ] ] X cana-4832 76 4 let let VERB cana-4832 76 5 ( ( PUNCT cana-4832 76 6 𝑋 𝑋 PROPN cana-4832 76 7 , , PUNCT cana-4832 76 8 𝑑 𝑑 NOUN cana-4832 76 9 ) ) PUNCT cana-4832 76 10 be be VERB cana-4832 76 11 a a DET cana-4832 76 12 b b NOUN cana-4832 76 13 - - PUNCT cana-4832 76 14 metric metric ADJ cana-4832 76 15 space space NOUN cana-4832 76 16 with with ADP cana-4832 76 17 coefficients coefficient NOUN cana-4832 76 18 𝑠 𝑠 PROPN cana-4832 76 19 β‰₯1 β‰₯1 PROPN cana-4832 76 20 . . PUNCT cana-4832 76 21 suppose suppose VERB cana-4832 76 22 that that SCONJ cana-4832 76 23 { { PUNCT cana-4832 76 24 π‘Žπ‘› π‘Žπ‘› NOUN cana-4832 76 25 } } PUNCT cana-4832 76 26 and and CCONJ cana-4832 76 27 { { PUNCT cana-4832 76 28 𝑏𝑛 𝑏𝑛 NOUN cana-4832 76 29 } } PUNCT cana-4832 76 30 are be AUX cana-4832 76 31 𝑏-convergent 𝑏-convergent VERB cana-4832 76 32 to to ADP cana-4832 76 33 π‘₯ π‘₯ PROPN cana-4832 76 34 and and CCONJ cana-4832 76 35 𝑦 𝑦 NOUN cana-4832 76 36 respectively respectively ADV cana-4832 76 37 . . PUNCT cana-4832 77 1 then then ADV cana-4832 77 2 we we PRON cana-4832 77 3 have have VERB cana-4832 77 4 1 1 NUM cana-4832 77 5 𝑠2 𝑠2 NOUN cana-4832 77 6 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-4832 77 7 , , PUNCT cana-4832 77 8 𝑦 𝑦 NOUN cana-4832 77 9 ) ) PUNCT cana-4832 77 10 ≀ ≀ NOUN cana-4832 77 11 lim lim PROPN cana-4832 77 12 inf inf PROPN cana-4832 77 13 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4832 77 14 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 77 15 , , PUNCT cana-4832 77 16 𝑏𝑛 𝑏𝑛 NOUN cana-4832 77 17 ) ) PUNCT cana-4832 77 18 ≀ ≀ NOUN cana-4832 77 19 lim lim PROPN cana-4832 77 20 sup sup NOUN cana-4832 77 21 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4832 77 22 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NOUN cana-4832 77 23 , , PUNCT cana-4832 77 24 𝑏𝑛 𝑏𝑛 NOUN cana-4832 77 25 ) ) PUNCT cana-4832 77 26 ≀ ≀ PROPN cana-4832 77 27 𝑠2𝑑(π‘₯ 𝑠2𝑑(π‘₯ PROPN cana-4832 77 28 , , PUNCT cana-4832 77 29 𝑦 𝑦 NOUN cana-4832 77 30 ) ) PUNCT cana-4832 77 31 . . PUNCT cana-4832 78 1 in in ADP cana-4832 78 2 particular particular ADJ cana-4832 78 3 , , PUNCT cana-4832 78 4 if if SCONJ cana-4832 78 5 π‘₯ π‘₯ ADP cana-4832 78 6 = = SYM cana-4832 78 7 𝑦 𝑦 NOUN cana-4832 78 8 , , PUNCT cana-4832 78 9 then then ADV cana-4832 78 10 we we PRON cana-4832 78 11 have have VERB cana-4832 78 12 lim lim PROPN cana-4832 78 13 nβ†’βˆž nβ†’βˆž X cana-4832 78 14 d(an d(an PROPN cana-4832 78 15 , , PUNCT cana-4832 78 16 bn bn ADJ cana-4832 78 17 ) ) PUNCT cana-4832 78 18 = = SYM cana-4832 78 19 0 0 NUM cana-4832 78 20 .moreover .moreover NOUN cana-4832 78 21 for for ADP cana-4832 78 22 each each DET cana-4832 78 23 𝑧 𝑧 DET cana-4832 78 24 ∈ ∈ NOUN cana-4832 78 25 𝑋 𝑋 NOUN cana-4832 78 26 we we PRON cana-4832 78 27 have have VERB cana-4832 78 28 communications communication NOUN cana-4832 78 29 on on ADP cana-4832 78 30 applied apply VERB cana-4832 78 31 nonlinear nonlinear ADJ cana-4832 78 32 analysis analysis NOUN cana-4832 78 33 issn issn NOUN cana-4832 78 34 : : PUNCT cana-4832 78 35 1074 1074 NUM cana-4832 78 36 - - PUNCT cana-4832 78 37 133x 133x NUM cana-4832 78 38 vol vol VERB cana-4832 78 39 32 32 NUM cana-4832 78 40 no no NOUN cana-4832 78 41 . . PUNCT cana-4832 79 1 10s 10 NOUN cana-4832 79 2 ( ( PUNCT cana-4832 79 3 2025 2025 NUM cana-4832 79 4 ) ) PUNCT cana-4832 79 5 392 392 NUM cana-4832 79 6 https://internationalpubls.com https://internationalpubls.com SYM cana-4832 79 7 1 1 NUM cana-4832 79 8 𝑆 𝑆 PROPN cana-4832 79 9 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-4832 79 10 , , PUNCT cana-4832 79 11 𝑧 𝑧 NOUN cana-4832 79 12 ) ) PUNCT cana-4832 79 13 ≀ ≀ NOUN cana-4832 79 14 lim lim PROPN cana-4832 79 15 inf inf PROPN cana-4832 79 16 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4832 79 17 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 79 18 , , PUNCT cana-4832 79 19 𝑧 𝑧 NOUN cana-4832 79 20 ) ) PUNCT cana-4832 79 21 ≀ ≀ NOUN cana-4832 79 22 lim lim PROPN cana-4832 79 23 sup sup NOUN cana-4832 79 24 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4832 79 25 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NOUN cana-4832 79 26 , , PUNCT cana-4832 79 27 𝑧)≀ 𝑧)≀ PROPN cana-4832 79 28 𝑠 𝑠 PROPN cana-4832 79 29 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-4832 79 30 , , PUNCT cana-4832 79 31 𝑧 𝑧 NOUN cana-4832 79 32 ) ) PUNCT cana-4832 79 33 . . PUNCT cana-4832 80 1 lemma lemma PROPN cana-4832 80 2 1.4.let 1.4.let NUM cana-4832 80 3 π‘Ž π‘Ž NOUN cana-4832 80 4 be be AUX cana-4832 80 5 a a DET cana-4832 80 6 limit limit NOUN cana-4832 80 7 of of ADP cana-4832 80 8 some some DET cana-4832 80 9 sequence sequence NOUN cana-4832 80 10 { { PUNCT cana-4832 80 11 π‘Žπ‘› π‘Žπ‘› NOUN cana-4832 80 12 } } PUNCT cana-4832 80 13 in in ADP cana-4832 80 14 a a DET cana-4832 80 15 π‘‘π‘ž π‘‘π‘ž PROPN cana-4832 80 16 𝑏-metric 𝑏-metric PROPN cana-4832 80 17 space(𝑋 space(𝑋 PROPN cana-4832 80 18 , , PUNCT cana-4832 80 19 𝑑 𝑑 NOUN cana-4832 80 20 ) ) PUNCT cana-4832 80 21 , , PUNCT cana-4832 80 22 then then ADV cana-4832 80 23 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 80 24 , , PUNCT cana-4832 80 25 π‘Ž π‘Ž NOUN cana-4832 80 26 ) ) PUNCT cana-4832 80 27 = = SYM cana-4832 80 28 0 0 X cana-4832 80 29 . . PUNCT cana-4832 81 1 proof proof NOUN cana-4832 81 2 . . PUNCT cana-4832 82 1 let let VERB cana-4832 82 2 π‘Ž π‘Ž PROPN cana-4832 82 3 ∈ ∈ PROPN cana-4832 82 4 𝑋 𝑋 NOUN cana-4832 82 5 , , PUNCT cana-4832 82 6 { { PUNCT cana-4832 82 7 an an PRON cana-4832 82 8 } } PUNCT cana-4832 82 9 βŠ† βŠ† NUM cana-4832 82 10 𝑋 𝑋 NOUN cana-4832 82 11 and and CCONJ cana-4832 82 12 a a DET cana-4832 82 13 sequence sequence NOUN cana-4832 82 14 which which PRON cana-4832 82 15 converges converge VERB cana-4832 82 16 to to PART cana-4832 82 17 π‘Ž. π‘Ž. VERB cana-4832 82 18 then then ADV cana-4832 82 19 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 82 20 , , PUNCT cana-4832 82 21 𝑏 𝑏 NOUN cana-4832 82 22 ) ) PUNCT cana-4832 82 23 ≀ ≀ NOUN cana-4832 82 24 𝑠[𝑑(π‘Ž 𝑠[𝑑(π‘Ž NOUN cana-4832 82 25 , , PUNCT cana-4832 82 26 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 82 27 ) ) PUNCT cana-4832 82 28 + + NUM cana-4832 83 1 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 83 2 , , PUNCT cana-4832 83 3 π‘Ž π‘Ž NOUN cana-4832 83 4 ) ) PUNCT cana-4832 83 5 ] ] PUNCT cana-4832 83 6 , , PUNCT cana-4832 83 7 βˆ€ βˆ€ VERB cana-4832 83 8 𝑛 𝑛 DET cana-4832 83 9 ∈ ∈ NOUN cana-4832 83 10 𝑁. 𝑁. PROPN cana-4832 83 11 by by ADP cana-4832 83 12 taking take VERB cana-4832 83 13 limit limit NOUN cana-4832 83 14 superior superior ADJ cana-4832 83 15 as as ADP cana-4832 83 16 𝑛 𝑛 PROPN cana-4832 83 17 β†’ β†’ SYM cana-4832 83 18 ∞ ∞ NUM cana-4832 83 19 and and CCONJ cana-4832 83 20 using use VERB cana-4832 83 21 lemma lemma PROPN cana-4832 83 22 1.3 1.3 NUM cana-4832 83 23 , , PUNCT cana-4832 83 24 we we PRON cana-4832 83 25 get get VERB cana-4832 83 26 𝑑(π‘Ž 𝑑(π‘Ž NOUN cana-4832 83 27 , , PUNCT cana-4832 83 28 π‘Ž π‘Ž NOUN cana-4832 83 29 ) ) PUNCT cana-4832 83 30 = = SYM cana-4832 83 31 0 0 X cana-4832 83 32 . . PUNCT cana-4832 84 1 recently recently ADV cana-4832 84 2 , , PUNCT cana-4832 84 3 das das PROPN cana-4832 84 4 et et PROPN cana-4832 84 5 al al PROPN cana-4832 84 6 . . PUNCT cana-4832 85 1 [ [ X cana-4832 85 2 9 9 NUM cana-4832 85 3 ] ] PUNCT cana-4832 85 4 established establish VERB cana-4832 85 5 the the DET cana-4832 85 6 following follow VERB cana-4832 85 7 theorems theorem NOUN cana-4832 85 8 in in ADP cana-4832 85 9 π‘‘π‘ž-metric π‘‘π‘ž-metric ADJ cana-4832 85 10 spaces space NOUN cana-4832 85 11 . . PUNCT cana-4832 86 1 theorem theorem VERB cana-4832 86 2 1.5 1.5 NUM cana-4832 86 3 . . PUNCT cana-4832 87 1 [ [ X cana-4832 87 2 9 9 NUM cana-4832 87 3 ] ] X cana-4832 87 4 let let NOUN cana-4832 87 5 ( ( PUNCT cana-4832 87 6 𝑋 𝑋 PROPN cana-4832 87 7 , , PUNCT cana-4832 87 8 𝑑 𝑑 NOUN cana-4832 87 9 ) ) PUNCT cana-4832 87 10 be be VERB cana-4832 87 11 a a DET cana-4832 87 12 complete complete ADJ cana-4832 87 13 π‘‘π‘ž-metric π‘‘π‘ž-metric ADJ cana-4832 87 14 space space NOUN cana-4832 87 15 and and CCONJ cana-4832 87 16 𝑇 𝑇 PROPN cana-4832 87 17 be be AUX cana-4832 87 18 an an PRON cana-4832 87 19 onto onto ADP cana-4832 87 20 self self NOUN cana-4832 87 21 - - PUNCT cana-4832 87 22 mapping mapping NOUN cana-4832 87 23 on on ADP cana-4832 87 24 𝑋 𝑋 PROPN cana-4832 87 25 such such ADJ cana-4832 87 26 that that PRON cana-4832 87 27 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž NOUN cana-4832 87 28 , , PUNCT cana-4832 87 29 𝑇𝑏 𝑇𝑏 PROPN cana-4832 87 30 ) ) PUNCT cana-4832 87 31 β‰₯ β‰₯ NOUN cana-4832 87 32 π‘˜ π‘˜ DET cana-4832 87 33 min min PROPN cana-4832 87 34 { { PUNCT cana-4832 87 35 𝛼 𝛼 NOUN cana-4832 87 36 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 87 37 , , PUNCT cana-4832 87 38 𝑏 𝑏 NOUN cana-4832 87 39 ) ) PUNCT cana-4832 87 40 ; ; PUNCT cana-4832 87 41 𝛽1 𝛽1 NOUN cana-4832 87 42 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž NOUN cana-4832 87 43 , , PUNCT cana-4832 87 44 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 87 45 , , PUNCT cana-4832 87 46 𝑏 𝑏 NOUN cana-4832 87 47 ) ) PUNCT cana-4832 87 48 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 87 49 , , PUNCT cana-4832 87 50 𝑏 𝑏 NOUN cana-4832 87 51 ) ) PUNCT cana-4832 88 1 + + CCONJ cana-4832 88 2 𝛽2𝑑(π‘Ž 𝛽2𝑑(π‘Ž PROPN cana-4832 88 3 , , PUNCT cana-4832 88 4 𝑏 𝑏 NOUN cana-4832 88 5 ) ) PUNCT cana-4832 88 6 ; ; PUNCT cana-4832 88 7 𝛾1𝑑(π‘‡π‘Ž 𝛾1𝑑(π‘‡π‘Ž PROPN cana-4832 88 8 , , PUNCT cana-4832 88 9 π‘Ž π‘Ž NOUN cana-4832 88 10 ) ) PUNCT cana-4832 88 11 + + NUM cana-4832 88 12 𝛾2 𝛾2 NOUN cana-4832 88 13 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 88 14 , , PUNCT cana-4832 88 15 𝑏 𝑏 NOUN cana-4832 88 16 ) ) PUNCT cana-4832 88 17 + + CCONJ cana-4832 88 18 𝛾3 𝛾3 ADJ cana-4832 88 19 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 88 20 , , PUNCT cana-4832 88 21 𝑏 𝑏 NOUN cana-4832 88 22 ) ) PUNCT cana-4832 88 23 ; ; PUNCT cana-4832 88 24 𝛿1𝑑(π‘‡π‘Ž 𝛿1𝑑(π‘‡π‘Ž NOUN cana-4832 88 25 , , PUNCT cana-4832 88 26 𝑏 𝑏 NOUN cana-4832 88 27 ) ) PUNCT cana-4832 88 28 + + CCONJ cana-4832 88 29 𝛿2𝑑(𝑇𝑏 𝛿2𝑑(𝑇𝑏 PROPN cana-4832 88 30 , , PUNCT cana-4832 88 31 π‘Ž π‘Ž NOUN cana-4832 88 32 ) ) PUNCT cana-4832 88 33 + + CCONJ cana-4832 88 34 𝛿3𝑑(π‘Ž 𝛿3𝑑(π‘Ž PROPN cana-4832 88 35 , , PUNCT cana-4832 88 36 𝑏 𝑏 NOUN cana-4832 88 37 ) ) PUNCT cana-4832 88 38 } } PUNCT cana-4832 88 39 for for ADP cana-4832 88 40 all all DET cana-4832 88 41 π‘Ž π‘Ž NOUN cana-4832 88 42 , , PUNCT cana-4832 88 43 𝑏 𝑏 PROPN cana-4832 88 44 ∈ ∈ PROPN cana-4832 88 45 𝑋 𝑋 NOUN cana-4832 88 46 with with ADP cana-4832 88 47 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 88 48 , , PUNCT cana-4832 88 49 𝑏 𝑏 NOUN cana-4832 88 50 ) ) PUNCT cana-4832 88 51 β‰  β‰  PROPN cana-4832 88 52 0 0 NUM cana-4832 88 53 , , PUNCT cana-4832 88 54 π‘˜ π‘˜ X cana-4832 88 55 > > X cana-4832 88 56 1 1 NUM cana-4832 88 57 , , PUNCT cana-4832 88 58 nonnegative nonnegative ADJ cana-4832 88 59 real real ADJ cana-4832 88 60 numbers number NOUN cana-4832 88 61 𝛼 𝛼 ADJ cana-4832 88 62 , , PUNCT cana-4832 88 63 𝛽𝑖 𝛽𝑖 VERB cana-4832 88 64 , , PUNCT cana-4832 88 65 𝛾𝑗 𝛾𝑗 INTJ cana-4832 88 66 , , PUNCT cana-4832 88 67 𝛿𝑗 𝛿𝑗 PROPN cana-4832 88 68 for for ADP cana-4832 88 69 𝑖 𝑖 X cana-4832 88 70 = = SYM cana-4832 88 71 1 1 NUM cana-4832 88 72 , , PUNCT cana-4832 88 73 2 2 NUM cana-4832 88 74 ; ; PUNCT cana-4832 88 75 𝑗 𝑗 NOUN cana-4832 88 76 = = SYM cana-4832 88 77 1 1 NUM cana-4832 88 78 , , PUNCT cana-4832 88 79 2 2 NUM cana-4832 88 80 , , PUNCT cana-4832 88 81 3 3 NUM cana-4832 88 82 and and CCONJ cana-4832 88 83 1 1 NUM cana-4832 88 84 π‘˜ π‘˜ NOUN cana-4832 88 85 = = SYM cana-4832 88 86 min{𝛼 min{𝛼 NOUN cana-4832 88 87 , , PUNCT cana-4832 88 88 𝛽2 𝛽2 NOUN cana-4832 88 89 , , PUNCT cana-4832 88 90 𝛾2 𝛾2 VERB cana-4832 88 91 + + CCONJ cana-4832 88 92 𝛾3 𝛾3 NOUN cana-4832 88 93 , , PUNCT cana-4832 88 94 π‘˜ π‘˜ PRON cana-4832 88 95 2𝛿2(𝛿1 2𝛿2(𝛿1 NUM cana-4832 88 96 + + CCONJ cana-4832 88 97 𝛿2 𝛿2 NOUN cana-4832 88 98 ) ) PUNCT cana-4832 88 99 + + NUM cana-4832 88 100 π‘˜π›Ώ3 π‘˜π›Ώ3 NOUN cana-4832 88 101 } } PUNCT cana-4832 88 102 . . PUNCT cana-4832 89 1 then then ADV cana-4832 89 2 t t PROPN cana-4832 89 3 has have VERB cana-4832 89 4 a a DET cana-4832 89 5 unique unique ADJ cana-4832 89 6 fixed fix VERB cana-4832 89 7 point point NOUN cana-4832 89 8 . . PUNCT cana-4832 90 1 theorem theorem VERB cana-4832 90 2 1.6 1.6 NUM cana-4832 90 3 . . PUNCT cana-4832 91 1 [ [ X cana-4832 91 2 9 9 NUM cana-4832 91 3 ] ] X cana-4832 91 4 let let NOUN cana-4832 91 5 ( ( PUNCT cana-4832 91 6 𝑋 𝑋 PROPN cana-4832 91 7 , , PUNCT cana-4832 91 8 𝑑 𝑑 NOUN cana-4832 91 9 ) ) PUNCT cana-4832 91 10 be be VERB cana-4832 91 11 a a DET cana-4832 91 12 complete complete ADJ cana-4832 91 13 π‘‘π‘ž π‘‘π‘ž NOUN cana-4832 91 14 -metric -metric ADJ cana-4832 91 15 space space NOUN cana-4832 91 16 and and CCONJ cana-4832 91 17 s s PROPN cana-4832 91 18 , , PUNCT cana-4832 91 19 t t PROPN cana-4832 91 20 be be AUX cana-4832 91 21 two two NUM cana-4832 91 22 onto onto ADP cana-4832 91 23 self self NOUN cana-4832 91 24 - - PUNCT cana-4832 91 25 mapping mapping NOUN cana-4832 91 26 on on ADP cana-4832 91 27 𝑋 𝑋 PROPN cana-4832 91 28 such such ADJ cana-4832 91 29 that that SCONJ cana-4832 91 30 𝑑(π‘†π‘Ž 𝑑(π‘†π‘Ž NOUN cana-4832 91 31 , , PUNCT cana-4832 91 32 𝑇𝑏 𝑇𝑏 NOUN cana-4832 91 33 ) ) PUNCT cana-4832 91 34 β‰₯ β‰₯ NOUN cana-4832 91 35 π‘˜ π‘˜ DET cana-4832 91 36 min min PROPN cana-4832 91 37 { { PUNCT cana-4832 91 38 𝛼 𝛼 NOUN cana-4832 91 39 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 91 40 , , PUNCT cana-4832 91 41 𝑏 𝑏 NOUN cana-4832 91 42 ) ) PUNCT cana-4832 91 43 ; ; PUNCT cana-4832 91 44 𝛽1 𝛽1 NOUN cana-4832 91 45 𝑑(π‘†π‘Ž 𝑑(π‘†π‘Ž NOUN cana-4832 91 46 , , PUNCT cana-4832 91 47 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 91 48 , , PUNCT cana-4832 91 49 𝑏 𝑏 NOUN cana-4832 91 50 ) ) PUNCT cana-4832 91 51 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 91 52 , , PUNCT cana-4832 91 53 𝑏 𝑏 NOUN cana-4832 91 54 ) ) PUNCT cana-4832 92 1 + + CCONJ cana-4832 92 2 𝛽2𝑑(π‘Ž 𝛽2𝑑(π‘Ž PROPN cana-4832 92 3 , , PUNCT cana-4832 92 4 𝑏 𝑏 NOUN cana-4832 92 5 ) ) PUNCT cana-4832 92 6 ; ; PUNCT cana-4832 92 7 𝛾1𝑑(π‘†π‘Ž 𝛾1𝑑(π‘†π‘Ž VERB cana-4832 92 8 , , PUNCT cana-4832 92 9 π‘Ž π‘Ž NOUN cana-4832 92 10 ) ) PUNCT cana-4832 92 11 + + NUM cana-4832 92 12 𝛾2 𝛾2 NOUN cana-4832 92 13 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 92 14 , , PUNCT cana-4832 92 15 𝑏 𝑏 NOUN cana-4832 92 16 ) ) PUNCT cana-4832 92 17 + + CCONJ cana-4832 92 18 𝛾3 𝛾3 ADJ cana-4832 92 19 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 92 20 , , PUNCT cana-4832 92 21 𝑏 𝑏 NOUN cana-4832 92 22 ) ) PUNCT cana-4832 92 23 ; ; PUNCT cana-4832 92 24 𝛿1𝑑(π‘†π‘Ž 𝛿1𝑑(π‘†π‘Ž X cana-4832 92 25 , , PUNCT cana-4832 92 26 𝑏 𝑏 NOUN cana-4832 92 27 ) ) PUNCT cana-4832 92 28 + + CCONJ cana-4832 92 29 𝛿2𝑑(𝑇𝑏 𝛿2𝑑(𝑇𝑏 PROPN cana-4832 92 30 , , PUNCT cana-4832 92 31 π‘Ž π‘Ž NOUN cana-4832 92 32 ) ) PUNCT cana-4832 92 33 + + CCONJ cana-4832 92 34 𝛿3𝑑(π‘Ž 𝛿3𝑑(π‘Ž PROPN cana-4832 92 35 , , PUNCT cana-4832 92 36 𝑏 𝑏 NOUN cana-4832 92 37 ) ) PUNCT cana-4832 92 38 } } PUNCT cana-4832 92 39 for for ADP cana-4832 92 40 all all DET cana-4832 92 41 π‘Ž π‘Ž NOUN cana-4832 92 42 , , PUNCT cana-4832 92 43 𝑏 𝑏 PROPN cana-4832 92 44 ∈ ∈ PROPN cana-4832 92 45 𝑋 𝑋 NOUN cana-4832 92 46 with with ADP cana-4832 92 47 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 92 48 , , PUNCT cana-4832 92 49 𝑏 𝑏 NOUN cana-4832 92 50 ) ) PUNCT cana-4832 92 51 β‰  β‰  PROPN cana-4832 92 52 0 0 NUM cana-4832 92 53 , , PUNCT cana-4832 92 54 π‘˜ π‘˜ X cana-4832 92 55 > > X cana-4832 92 56 1 1 NUM cana-4832 92 57 , , PUNCT cana-4832 92 58 nonnegative nonnegative ADJ cana-4832 92 59 real real ADJ cana-4832 92 60 numbers number NOUN cana-4832 92 61 𝛼 𝛼 ADJ cana-4832 92 62 , , PUNCT cana-4832 92 63 𝛽𝑖 𝛽𝑖 VERB cana-4832 92 64 , , PUNCT cana-4832 92 65 𝛾𝑗 𝛾𝑗 INTJ cana-4832 92 66 , , PUNCT cana-4832 92 67 𝛿𝑗 𝛿𝑗 PROPN cana-4832 92 68 for for ADP cana-4832 92 69 𝑖 𝑖 X cana-4832 92 70 = = SYM cana-4832 92 71 1 1 NUM cana-4832 92 72 , , PUNCT cana-4832 92 73 2 2 NUM cana-4832 92 74 ; ; PUNCT cana-4832 92 75 𝑗 𝑗 NOUN cana-4832 92 76 = = SYM cana-4832 92 77 1 1 NUM cana-4832 92 78 , , PUNCT cana-4832 92 79 2 2 NUM cana-4832 92 80 , , PUNCT cana-4832 92 81 3 3 NUM cana-4832 92 82 and and CCONJ cana-4832 92 83 1 1 NUM cana-4832 92 84 π‘˜ π‘˜ NOUN cana-4832 92 85 = = SYM cana-4832 92 86 min{𝛼 min{𝛼 NOUN cana-4832 92 87 , , PUNCT cana-4832 92 88 𝛽2 𝛽2 NOUN cana-4832 92 89 , , PUNCT cana-4832 92 90 𝛾2 𝛾2 VERB cana-4832 92 91 + + CCONJ cana-4832 92 92 𝛾3 𝛾3 ADJ cana-4832 92 93 , , PUNCT cana-4832 92 94 𝛿3 𝛿3 NOUN cana-4832 92 95 } } PUNCT cana-4832 92 96 . . PUNCT cana-4832 93 1 then then ADV cana-4832 93 2 𝑆 𝑆 PROPN cana-4832 93 3 and and CCONJ cana-4832 93 4 𝑇 𝑇 PROPN cana-4832 93 5 have have VERB cana-4832 93 6 a a DET cana-4832 93 7 unique unique ADJ cana-4832 93 8 common common ADJ cana-4832 93 9 fixed fix VERB cana-4832 93 10 point point NOUN cana-4832 93 11 . . PUNCT cana-4832 94 1 2 2 X cana-4832 94 2 . . X cana-4832 94 3 main main ADJ cana-4832 94 4 results result NOUN cana-4832 94 5 in in ADP cana-4832 94 6 this this DET cana-4832 94 7 section section NOUN cana-4832 94 8 , , PUNCT cana-4832 94 9 we we PRON cana-4832 94 10 formulate formulate VERB cana-4832 94 11 some some DET cana-4832 94 12 fixed fix VERB cana-4832 94 13 point point NOUN cana-4832 94 14 results result NOUN cana-4832 94 15 for for ADP cana-4832 94 16 onto onto ADP cana-4832 94 17 expansive expansive ADJ cana-4832 94 18 type type NOUN cana-4832 94 19 mapping mapping NOUN cana-4832 94 20 in in ADP cana-4832 94 21 a a DET cana-4832 94 22 complete complete ADJ cana-4832 94 23 π‘‘π‘ž π‘‘π‘ž PROPN cana-4832 94 24 𝑏-metric 𝑏-metric PROPN cana-4832 94 25 space space NOUN cana-4832 94 26 . . PUNCT cana-4832 95 1 theorem theorem VERB cana-4832 95 2 2.1 2.1 NUM cana-4832 95 3 . . PUNCT cana-4832 96 1 let let VERB cana-4832 96 2 ( ( PUNCT cana-4832 96 3 x x X cana-4832 96 4 , , PUNCT cana-4832 96 5 d d NOUN cana-4832 96 6 ) ) PUNCT cana-4832 96 7 be be AUX cana-4832 96 8 a a DET cana-4832 96 9 complete complete ADJ cana-4832 96 10 dq dq NOUN cana-4832 96 11 b b X cana-4832 96 12 - - PUNCT cana-4832 96 13 metric metric ADJ cana-4832 96 14 space space NOUN cana-4832 96 15 and and CCONJ cana-4832 96 16 t t PROPN cana-4832 96 17 be be AUX cana-4832 96 18 an an PRON cana-4832 96 19 onto onto ADP cana-4832 96 20 self self NOUN cana-4832 96 21 - - PUNCT cana-4832 96 22 mapping mapping NOUN cana-4832 96 23 on on ADP cana-4832 96 24 x x SYM cana-4832 96 25 such such ADJ cana-4832 96 26 that that SCONJ cana-4832 96 27 communications communication NOUN cana-4832 96 28 on on ADP cana-4832 96 29 applied apply VERB cana-4832 96 30 nonlinear nonlinear ADJ cana-4832 96 31 analysis analysis NOUN cana-4832 96 32 issn issn NOUN cana-4832 96 33 : : PUNCT cana-4832 96 34 1074 1074 NUM cana-4832 96 35 - - PUNCT cana-4832 96 36 133x 133x NUM cana-4832 96 37 vol vol VERB cana-4832 96 38 32 32 NUM cana-4832 96 39 no no NOUN cana-4832 96 40 . . PUNCT cana-4832 97 1 10s 10 NOUN cana-4832 97 2 ( ( PUNCT cana-4832 97 3 2025 2025 NUM cana-4832 97 4 ) ) PUNCT cana-4832 97 5 393 393 NUM cana-4832 97 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 97 7 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž NOUN cana-4832 97 8 , , PUNCT cana-4832 97 9 𝑇𝑏 𝑇𝑏 PROPN cana-4832 97 10 ) ) PUNCT cana-4832 97 11 β‰₯ β‰₯ NOUN cana-4832 97 12 π‘˜ π‘˜ DET cana-4832 97 13 min min PROPN cana-4832 97 14 { { PUNCT cana-4832 97 15 𝛼 𝛼 NOUN cana-4832 97 16 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 97 17 , , PUNCT cana-4832 97 18 𝑏 𝑏 NOUN cana-4832 97 19 ) ) PUNCT cana-4832 97 20 ; ; PUNCT cana-4832 97 21 𝛽1 𝛽1 NOUN cana-4832 97 22 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž NOUN cana-4832 97 23 , , PUNCT cana-4832 97 24 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 97 25 , , PUNCT cana-4832 97 26 𝑏 𝑏 NOUN cana-4832 97 27 ) ) PUNCT cana-4832 97 28 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 97 29 , , PUNCT cana-4832 97 30 𝑏 𝑏 NOUN cana-4832 97 31 ) ) PUNCT cana-4832 97 32 + + CCONJ cana-4832 97 33 𝛽2𝑑(π‘Ž 𝛽2𝑑(π‘Ž PROPN cana-4832 97 34 , , PUNCT cana-4832 97 35 𝑏 𝑏 NOUN cana-4832 97 36 ) ) PUNCT cana-4832 97 37 ; ; PUNCT cana-4832 97 38 𝛾1𝑑(π‘‡π‘Ž 𝛾1𝑑(π‘‡π‘Ž PROPN cana-4832 97 39 , , PUNCT cana-4832 97 40 π‘Ž π‘Ž NOUN cana-4832 97 41 ) ) PUNCT cana-4832 97 42 + + NUM cana-4832 97 43 𝛾2 𝛾2 NOUN cana-4832 97 44 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 97 45 , , PUNCT cana-4832 97 46 𝑏 𝑏 NOUN cana-4832 97 47 ) ) PUNCT cana-4832 97 48 + + CCONJ cana-4832 97 49 𝛾3 𝛾3 ADJ cana-4832 97 50 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 97 51 , , PUNCT cana-4832 97 52 𝑏 𝑏 NOUN cana-4832 97 53 ) ) PUNCT cana-4832 97 54 ; ; PUNCT cana-4832 97 55 𝛿1𝑑(π‘‡π‘Ž 𝛿1𝑑(π‘‡π‘Ž NOUN cana-4832 97 56 , , PUNCT cana-4832 97 57 𝑏 𝑏 NOUN cana-4832 97 58 ) ) PUNCT cana-4832 97 59 + + CCONJ cana-4832 97 60 𝛿2𝑑(𝑇𝑏 𝛿2𝑑(𝑇𝑏 PROPN cana-4832 97 61 , , PUNCT cana-4832 97 62 π‘Ž π‘Ž NOUN cana-4832 97 63 ) ) PUNCT cana-4832 97 64 + + CCONJ cana-4832 97 65 𝛿3𝑑(π‘Ž 𝛿3𝑑(π‘Ž PROPN cana-4832 97 66 , , PUNCT cana-4832 97 67 𝑏 𝑏 NOUN cana-4832 97 68 ) ) PUNCT cana-4832 97 69 ; ; PUNCT cana-4832 97 70 πœ†1 πœ†1 VERB cana-4832 97 71 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž PROPN cana-4832 97 72 , , PUNCT cana-4832 97 73 𝑏)𝑑(𝑇𝑏 𝑏)𝑑(𝑇𝑏 PROPN cana-4832 97 74 , , PUNCT cana-4832 97 75 𝑏 𝑏 NOUN cana-4832 97 76 ) ) PUNCT cana-4832 97 77 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 97 78 , , PUNCT cana-4832 97 79 𝑏 𝑏 NOUN cana-4832 97 80 ) ) PUNCT cana-4832 97 81 + + NUM cana-4832 97 82 πœ†2 πœ†2 PROPN cana-4832 97 83 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 97 84 , , PUNCT cana-4832 97 85 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 97 86 , , PUNCT cana-4832 97 87 𝑏 𝑏 NOUN cana-4832 97 88 ) ) PUNCT cana-4832 97 89 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 97 90 , , PUNCT cana-4832 97 91 𝑏 𝑏 NOUN cana-4832 97 92 ) ) PUNCT cana-4832 97 93 + + CCONJ cana-4832 97 94 πœ†3𝑑(π‘Ž πœ†3𝑑(π‘Ž PROPN cana-4832 97 95 , , PUNCT cana-4832 97 96 𝑏 𝑏 NOUN cana-4832 97 97 ) ) PUNCT cana-4832 97 98 } } PUNCT cana-4832 97 99 ( ( PUNCT cana-4832 97 100 2.1 2.1 NUM cana-4832 97 101 ) ) PUNCT cana-4832 97 102 for for ADP cana-4832 97 103 all all DET cana-4832 97 104 π‘Ž π‘Ž NOUN cana-4832 97 105 , , PUNCT cana-4832 97 106 𝑏 𝑏 PROPN cana-4832 97 107 ∈ ∈ PROPN cana-4832 97 108 𝑋 𝑋 NOUN cana-4832 97 109 with with ADP cana-4832 97 110 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 97 111 , , PUNCT cana-4832 97 112 𝑏 𝑏 NOUN cana-4832 97 113 ) ) PUNCT cana-4832 97 114 β‰  β‰  PROPN cana-4832 97 115 0 0 NUM cana-4832 97 116 , , PUNCT cana-4832 97 117 π‘˜ π‘˜ X cana-4832 97 118 > > X cana-4832 97 119 1 1 NUM cana-4832 97 120 , , PUNCT cana-4832 97 121 nonnegative nonnegative ADJ cana-4832 97 122 real real ADJ cana-4832 97 123 numbers number NOUN cana-4832 97 124 𝛼 𝛼 ADJ cana-4832 97 125 , , PUNCT cana-4832 97 126 𝛽𝑖 𝛽𝑖 VERB cana-4832 97 127 , , PUNCT cana-4832 97 128 𝛾𝑗 𝛾𝑗 INTJ cana-4832 97 129 , , PUNCT cana-4832 97 130 𝛿𝑗 𝛿𝑗 PROPN cana-4832 97 131 , , PUNCT cana-4832 97 132 πœ†π‘— πœ†π‘— X cana-4832 97 133 for for ADP cana-4832 97 134 𝑖 𝑖 PRON cana-4832 97 135 = = SYM cana-4832 97 136 1 1 NUM cana-4832 97 137 , , PUNCT cana-4832 97 138 2 2 NUM cana-4832 97 139 ; ; PUNCT cana-4832 97 140 𝑗 𝑗 NOUN cana-4832 97 141 = = SYM cana-4832 97 142 1 1 NUM cana-4832 97 143 , , PUNCT cana-4832 97 144 2 2 NUM cana-4832 97 145 , , PUNCT cana-4832 97 146 3 3 NUM cana-4832 97 147 and and CCONJ cana-4832 97 148 1 1 NUM cana-4832 97 149 π‘˜ π‘˜ X cana-4832 97 150 = = SYM cana-4832 97 151 π‘šπ‘–π‘›{𝛼 π‘šπ‘–π‘›{𝛼 PROPN cana-4832 97 152 , , PUNCT cana-4832 97 153 𝛽2 𝛽2 NOUN cana-4832 97 154 , , PUNCT cana-4832 97 155 𝛾2 𝛾2 VERB cana-4832 97 156 + + CCONJ cana-4832 97 157 𝛾3 𝛾3 NOUN cana-4832 97 158 , , PUNCT cana-4832 97 159 π‘˜ π‘˜ PRON cana-4832 97 160 2𝛿2(𝛿1 2𝛿2(𝛿1 NUM cana-4832 97 161 + + CCONJ cana-4832 97 162 𝛿2 𝛿2 NOUN cana-4832 97 163 ) ) PUNCT cana-4832 97 164 + + NUM cana-4832 97 165 π‘˜π›Ώ3 π‘˜π›Ώ3 NOUN cana-4832 97 166 , , PUNCT cana-4832 97 167 πœ†3 πœ†3 NOUN cana-4832 97 168 } } PUNCT cana-4832 97 169 . . PUNCT cana-4832 98 1 then then ADV cana-4832 98 2 t t PROPN cana-4832 98 3 has have VERB cana-4832 98 4 a a DET cana-4832 98 5 unique unique ADJ cana-4832 98 6 fixed fix VERB cana-4832 98 7 point point NOUN cana-4832 98 8 . . PUNCT cana-4832 99 1 proof proof NOUN cana-4832 99 2 . . PUNCT cana-4832 100 1 let let VERB cana-4832 100 2 us we PRON cana-4832 100 3 take take VERB cana-4832 100 4 πœƒ(π‘Ž πœƒ(π‘Ž NOUN cana-4832 100 5 , , PUNCT cana-4832 100 6 𝑏 𝑏 NOUN cana-4832 100 7 ) ) PUNCT cana-4832 100 8 = = SYM cana-4832 100 9 min min NOUN cana-4832 100 10 { { PUNCT cana-4832 100 11 𝛼 𝛼 NOUN cana-4832 100 12 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 100 13 , , PUNCT cana-4832 100 14 𝑏 𝑏 NOUN cana-4832 100 15 ) ) PUNCT cana-4832 100 16 ; ; PUNCT cana-4832 100 17 𝛽1 𝛽1 NOUN cana-4832 100 18 𝑑(π‘‡π‘Ž,π‘Ž)𝑑(𝑇𝑏,𝑏 𝑑(π‘‡π‘Ž,π‘Ž)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 100 19 ) ) PUNCT cana-4832 100 20 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 NOUN cana-4832 100 21 ) ) PUNCT cana-4832 101 1 + + CCONJ cana-4832 101 2 𝛽2𝑑(π‘Ž 𝛽2𝑑(π‘Ž PROPN cana-4832 101 3 , , PUNCT cana-4832 101 4 𝑏 𝑏 NOUN cana-4832 101 5 ) ) PUNCT cana-4832 101 6 ; ; PUNCT cana-4832 101 7 𝛾1𝑑(π‘‡π‘Ž 𝛾1𝑑(π‘‡π‘Ž PROPN cana-4832 101 8 , , PUNCT cana-4832 101 9 π‘Ž π‘Ž NOUN cana-4832 101 10 ) ) PUNCT cana-4832 101 11 + + NUM cana-4832 101 12 𝛾2 𝛾2 NOUN cana-4832 101 13 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 101 14 , , PUNCT cana-4832 101 15 𝑏 𝑏 NOUN cana-4832 101 16 ) ) PUNCT cana-4832 101 17 + + CCONJ cana-4832 101 18 𝛾3 𝛾3 ADJ cana-4832 101 19 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 101 20 , , PUNCT cana-4832 101 21 𝑏 𝑏 NOUN cana-4832 101 22 ) ) PUNCT cana-4832 101 23 ; ; PUNCT cana-4832 101 24 𝛿1𝑑(π‘‡π‘Ž 𝛿1𝑑(π‘‡π‘Ž NOUN cana-4832 101 25 , , PUNCT cana-4832 101 26 𝑏 𝑏 NOUN cana-4832 101 27 ) ) PUNCT cana-4832 101 28 + + CCONJ cana-4832 101 29 𝛿2𝑑(𝑇𝑏 𝛿2𝑑(𝑇𝑏 PROPN cana-4832 101 30 , , PUNCT cana-4832 101 31 π‘Ž π‘Ž NOUN cana-4832 101 32 ) ) PUNCT cana-4832 101 33 + + CCONJ cana-4832 101 34 𝛿3𝑑(π‘Ž 𝛿3𝑑(π‘Ž PROPN cana-4832 101 35 , , PUNCT cana-4832 101 36 𝑏 𝑏 NOUN cana-4832 101 37 ) ) PUNCT cana-4832 101 38 ; ; PUNCT cana-4832 101 39 πœ†1 πœ†1 VERB cana-4832 101 40 𝑑(π‘‡π‘Ž,𝑏)𝑑(𝑇𝑏,𝑏 𝑑(π‘‡π‘Ž,𝑏)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 101 41 ) ) PUNCT cana-4832 101 42 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 NOUN cana-4832 101 43 ) ) PUNCT cana-4832 102 1 + + NUM cana-4832 102 2 πœ†2 πœ†2 NOUN cana-4832 102 3 𝑑(𝑇𝑏,π‘Ž)𝑑(𝑇𝑏,𝑏 𝑑(𝑇𝑏,π‘Ž)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 102 4 ) ) PUNCT cana-4832 102 5 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 VERB cana-4832 102 6 ) ) PUNCT cana-4832 103 1 + + CCONJ cana-4832 103 2 πœ†3𝑑(π‘Ž πœ†3𝑑(π‘Ž PROPN cana-4832 103 3 , , PUNCT cana-4832 103 4 𝑏 𝑏 NOUN cana-4832 103 5 ) ) PUNCT cana-4832 103 6 } } PUNCT cana-4832 103 7 . . PUNCT cana-4832 104 1 for for ADP cana-4832 104 2 π‘Ž0 π‘Ž0 PROPN cana-4832 104 3 ∈ ∈ PROPN cana-4832 104 4 𝑋 𝑋 PROPN cana-4832 104 5 , , PUNCT cana-4832 104 6 since since SCONJ cana-4832 104 7 𝑇 𝑇 PROPN cana-4832 104 8 are be AUX cana-4832 104 9 onto onto ADP cana-4832 104 10 , , PUNCT cana-4832 104 11 there there PRON cana-4832 104 12 exist exist VERB cana-4832 104 13 π‘Ž0 π‘Ž0 PROPN cana-4832 104 14 ∈ ∈ NOUN cana-4832 104 15 𝑋 𝑋 NOUN cana-4832 104 16 such such ADJ cana-4832 104 17 that that DET cana-4832 104 18 π‘Ž0 π‘Ž0 ADV cana-4832 104 19 = = SYM cana-4832 104 20 π‘‡π‘Ž1 π‘‡π‘Ž1 NOUN cana-4832 104 21 . . PUNCT cana-4832 105 1 continuing continue VERB cana-4832 105 2 this this DET cana-4832 105 3 process process NOUN cana-4832 105 4 , , PUNCT cana-4832 105 5 we we PRON cana-4832 105 6 define define VERB cana-4832 105 7 a a DET cana-4832 105 8 sequence sequence NOUN cana-4832 105 9 { { PUNCT cana-4832 105 10 π‘Žπ‘› π‘Žπ‘› NOUN cana-4832 105 11 } } PUNCT cana-4832 105 12 in in ADP cana-4832 105 13 𝑋 𝑋 PROPN cana-4832 105 14 with with ADP cana-4832 105 15 π‘Žπ‘›βˆ’1 π‘Žπ‘›βˆ’1 PROPN cana-4832 105 16 = = SYM cana-4832 105 17 π‘‡π‘Žπ‘› π‘‡π‘Žπ‘› PROPN cana-4832 105 18 , , PUNCT cana-4832 105 19 for for ADP cana-4832 105 20 all all DET cana-4832 105 21 𝑛 𝑛 DET cana-4832 105 22 ∈ ∈ PROPN cana-4832 105 23 β„•. β„•. PROPN cana-4832 105 24 the the DET cana-4832 105 25 cases case NOUN cana-4832 105 26 listed list VERB cana-4832 105 27 below below ADV cana-4832 105 28 will will AUX cana-4832 105 29 occur occur VERB cana-4832 105 30 . . PUNCT cana-4832 106 1 case case NOUN cana-4832 106 2 ( ( PUNCT cana-4832 106 3 i i NOUN cana-4832 106 4 ) ) PUNCT cana-4832 106 5 . . PUNCT cana-4832 107 1 if if SCONJ cana-4832 107 2 πœƒ(π‘Ž πœƒ(π‘Ž NOUN cana-4832 107 3 , , PUNCT cana-4832 107 4 𝑏 𝑏 NOUN cana-4832 107 5 ) ) PUNCT cana-4832 107 6 = = SYM cana-4832 107 7 𝛼 𝛼 PRON cana-4832 107 8 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 107 9 , , PUNCT cana-4832 107 10 𝑏 𝑏 NOUN cana-4832 107 11 ) ) PUNCT cana-4832 107 12 , , PUNCT cana-4832 107 13 then then ADV cana-4832 107 14 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž PROPN cana-4832 107 15 , , PUNCT cana-4832 107 16 𝑇𝑏 𝑇𝑏 PROPN cana-4832 107 17 ) ) PUNCT cana-4832 107 18 β‰₯ β‰₯ NOUN cana-4832 107 19 π‘˜ π‘˜ PRON cana-4832 107 20 𝛼 𝛼 X cana-4832 107 21 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 107 22 , , PUNCT cana-4832 107 23 𝑏 𝑏 NOUN cana-4832 107 24 ) ) PUNCT cana-4832 107 25 , , PUNCT cana-4832 107 26 for for ADP cana-4832 107 27 all all DET cana-4832 107 28 π‘Ž π‘Ž NOUN cana-4832 107 29 , , PUNCT cana-4832 107 30 𝑏 𝑏 PROPN cana-4832 107 31 ∈ ∈ PROPN cana-4832 107 32 𝑋 𝑋 NOUN cana-4832 107 33 . . PUNCT cana-4832 108 1 ( ( PUNCT cana-4832 108 2 2.2 2.2 NUM cana-4832 108 3 ) ) PUNCT cana-4832 108 4 now now ADV cana-4832 108 5 , , PUNCT cana-4832 108 6 using use VERB cana-4832 108 7 ( ( PUNCT cana-4832 108 8 2.2 2.2 NUM cana-4832 108 9 ) ) PUNCT cana-4832 108 10 , , PUNCT cana-4832 108 11 we we PRON cana-4832 108 12 get get VERB cana-4832 108 13 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 NOUN cana-4832 108 14 , , PUNCT cana-4832 108 15 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 108 16 ) ) PUNCT cana-4832 108 17 = = PUNCT cana-4832 108 18 𝑑(π‘‡π‘Žπ‘› 𝑑(π‘‡π‘Žπ‘› PROPN cana-4832 108 19 , , PUNCT cana-4832 108 20 π‘‡π‘Žπ‘›+1 π‘‡π‘Žπ‘›+1 NOUN cana-4832 108 21 ) ) PUNCT cana-4832 108 22 β‰₯ β‰₯ NOUN cana-4832 108 23 π‘˜ π‘˜ X cana-4832 108 24 𝛼 𝛼 NOUN cana-4832 108 25 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NOUN cana-4832 108 26 , , PUNCT cana-4832 108 27 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 108 28 ) ) PUNCT cana-4832 108 29 i. i. PROPN cana-4832 108 30 e. e. PROPN cana-4832 108 31 , , PUNCT cana-4832 108 32 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› PROPN cana-4832 108 33 , , PUNCT cana-4832 108 34 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 108 35 ) ) PUNCT cana-4832 108 36 ≀ ≀ NUM cana-4832 108 37 1 1 NUM cana-4832 108 38 π‘˜ π‘˜ PRON cana-4832 108 39 𝛼 𝛼 NOUN cana-4832 108 40 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 NOUN cana-4832 108 41 , , PUNCT cana-4832 108 42 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 108 43 ) ) PUNCT cana-4832 108 44 . . PUNCT cana-4832 109 1 let let VERB cana-4832 109 2 𝜏 𝜏 NOUN cana-4832 109 3 = = SYM cana-4832 109 4 1 1 NUM cana-4832 109 5 π‘˜ π‘˜ PRON cana-4832 109 6 𝛼 𝛼 X cana-4832 109 7 < < X cana-4832 109 8 1 1 NUM cana-4832 109 9 . . PUNCT cana-4832 110 1 then then ADV cana-4832 110 2 from from ADP cana-4832 110 3 the the DET cana-4832 110 4 above above ADJ cana-4832 110 5 inequality inequality NOUN cana-4832 110 6 , , PUNCT cana-4832 110 7 we we PRON cana-4832 110 8 have have VERB cana-4832 110 9 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 110 10 , , PUNCT cana-4832 110 11 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 110 12 ) ) PUNCT cana-4832 110 13 ≀ ≀ NOUN cana-4832 110 14 𝜏 𝜏 PRON cana-4832 110 15 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 PROPN cana-4832 110 16 , , PUNCT cana-4832 110 17 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 110 18 ) ) PUNCT cana-4832 110 19 . . PUNCT cana-4832 111 1 also also ADV cana-4832 111 2 , , PUNCT cana-4832 111 3 𝑑(π‘Žπ‘›+1 𝑑(π‘Žπ‘›+1 PROPN cana-4832 111 4 , , PUNCT cana-4832 111 5 π‘Žπ‘›+2 π‘Žπ‘›+2 NOUN cana-4832 111 6 ) ) PUNCT cana-4832 111 7 ≀ ≀ NOUN cana-4832 111 8 𝜏 𝜏 PRON cana-4832 111 9 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NOUN cana-4832 111 10 , , PUNCT cana-4832 111 11 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 111 12 ) ) PUNCT cana-4832 111 13 + + CCONJ cana-4832 111 14 𝜏 𝜏 PROPN cana-4832 111 15 2 2 NUM cana-4832 111 16 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 NUM cana-4832 111 17 , , PUNCT cana-4832 111 18 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 111 19 ) ) PUNCT cana-4832 111 20 . . PUNCT cana-4832 112 1 from from ADP cana-4832 112 2 this this PRON cana-4832 112 3 we we PRON cana-4832 112 4 get get VERB cana-4832 112 5 , , PUNCT cana-4832 112 6 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NOUN cana-4832 112 7 , , PUNCT cana-4832 112 8 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 112 9 ) ) PUNCT cana-4832 112 10 ≀ ≀ NOUN cana-4832 112 11 πœπ‘› πœπ‘› ADP cana-4832 112 12 𝑑(π‘Ž0 𝑑(π‘Ž0 ADJ cana-4832 112 13 , , PUNCT cana-4832 112 14 π‘Ž1 π‘Ž1 PROPN cana-4832 112 15 ) ) PUNCT cana-4832 112 16 . . PUNCT cana-4832 113 1 for for ADP cana-4832 113 2 𝑗 𝑗 PROPN cana-4832 113 3 > > X cana-4832 113 4 𝑖 𝑖 PROPN cana-4832 113 5 , , PUNCT cana-4832 113 6 𝑑(π‘Žπ‘– 𝑑(π‘Žπ‘– PROPN cana-4832 113 7 , , PUNCT cana-4832 113 8 π‘Žπ‘— π‘Žπ‘— NOUN cana-4832 113 9 ) ) PUNCT cana-4832 113 10 ≀ ≀ NOUN cana-4832 113 11 𝑠 𝑠 X cana-4832 113 12 𝑑(π‘Žπ‘– 𝑑(π‘Žπ‘– PROPN cana-4832 113 13 , , PUNCT cana-4832 113 14 π‘Žπ‘–+1 π‘Žπ‘–+1 NUM cana-4832 113 15 ) ) PUNCT cana-4832 113 16 + + CCONJ cana-4832 113 17 𝑠 𝑠 PROPN cana-4832 113 18 2 2 NUM cana-4832 113 19 𝑑(π‘Žπ‘–+1 𝑑(π‘Žπ‘–+1 NOUN cana-4832 113 20 , , PUNCT cana-4832 113 21 π‘Žπ‘–+2 π‘Žπ‘–+2 NUM cana-4832 113 22 ) ) PUNCT cana-4832 113 23 + + NUM cana-4832 113 24 β‹―+ β‹―+ NOUN cana-4832 113 25 π‘ π‘—βˆ’π‘– π‘ π‘—βˆ’π‘– ADJ cana-4832 113 26 𝑑(π‘Žπ‘—βˆ’1 𝑑(π‘Žπ‘—βˆ’1 NOUN cana-4832 113 27 , , PUNCT cana-4832 113 28 π‘Žπ‘— π‘Žπ‘— NOUN cana-4832 113 29 ) ) PUNCT cana-4832 113 30 ≀ ≀ NOUN cana-4832 113 31 𝑠 𝑠 ADP cana-4832 113 32 πœπ‘–π‘‘(π‘Ž0 πœπ‘–π‘‘(π‘Ž0 PROPN cana-4832 113 33 , , PUNCT cana-4832 113 34 π‘Ž1 π‘Ž1 NOUN cana-4832 113 35 ) ) PUNCT cana-4832 114 1 + + CCONJ cana-4832 114 2 𝑠 𝑠 NUM cana-4832 114 3 2πœπ‘–+1𝑑(π‘Ž0 2πœπ‘–+1𝑑(π‘Ž0 NUM cana-4832 114 4 , , PUNCT cana-4832 114 5 π‘Ž1 π‘Ž1 NOUN cana-4832 114 6 ) ) PUNCT cana-4832 115 1 + + NUM cana-4832 115 2 β‹―+ β‹―+ NOUN cana-4832 115 3 𝑠 𝑠 NOUN cana-4832 115 4 π‘—βˆ’π‘–πœπ‘—βˆ’1𝑑(π‘Ž0 π‘—βˆ’π‘–πœπ‘—βˆ’1𝑑(π‘Ž0 ADJ cana-4832 115 5 , , PUNCT cana-4832 115 6 π‘Ž1 π‘Ž1 NOUN cana-4832 115 7 ) ) PUNCT cana-4832 115 8 ≀ ≀ NOUN cana-4832 116 1 [ [ X cana-4832 116 2 𝑠 𝑠 X cana-4832 116 3 πœπ‘– πœπ‘– NOUN cana-4832 116 4 + + NOUN cana-4832 116 5 𝑠2πœπ‘–+1 𝑠2πœπ‘–+1 NOUN cana-4832 117 1 + + ADJ cana-4832 117 2 β‹―]𝑑(π‘Ž0 β‹―]𝑑(π‘Ž0 ADJ cana-4832 117 3 , , PUNCT cana-4832 117 4 π‘Ž1 π‘Ž1 NOUN cana-4832 117 5 ) ) PUNCT cana-4832 117 6 = = PUNCT cana-4832 118 1 𝑠 𝑠 PRON cana-4832 118 2 πœπ‘–[1 πœπ‘–[1 NOUN cana-4832 119 1 + + CCONJ cana-4832 119 2 π‘ πœ π‘ πœ PROPN cana-4832 119 3 + + CCONJ cana-4832 119 4 ( ( PUNCT cana-4832 119 5 π‘ πœ)2 π‘ πœ)2 PROPN cana-4832 119 6 + + PROPN cana-4832 119 7 β‹― β‹― NOUN cana-4832 119 8 ] ] SYM cana-4832 119 9 𝑑(π‘Ž0 𝑑(π‘Ž0 ADJ cana-4832 119 10 , , PUNCT cana-4832 119 11 π‘Ž1 π‘Ž1 ADJ cana-4832 119 12 ) ) PUNCT cana-4832 119 13 communications communication NOUN cana-4832 119 14 on on ADP cana-4832 119 15 applied apply VERB cana-4832 119 16 nonlinear nonlinear ADJ cana-4832 119 17 analysis analysis NOUN cana-4832 119 18 issn issn NOUN cana-4832 119 19 : : PUNCT cana-4832 119 20 1074 1074 NUM cana-4832 119 21 - - PUNCT cana-4832 119 22 133x 133x NUM cana-4832 119 23 vol vol VERB cana-4832 119 24 32 32 NUM cana-4832 119 25 no no NOUN cana-4832 119 26 . . PUNCT cana-4832 120 1 10s 10 NOUN cana-4832 120 2 ( ( PUNCT cana-4832 120 3 2025 2025 NUM cana-4832 120 4 ) ) PUNCT cana-4832 120 5 394 394 NUM cana-4832 120 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 120 7 = = PUNCT cana-4832 120 8 𝑠 𝑠 X cana-4832 120 9 πœπ‘– πœπ‘– INTJ cana-4832 120 10 1 1 NUM cana-4832 120 11 βˆ’ βˆ’ NOUN cana-4832 120 12 π‘ πœ π‘ πœ INTJ cana-4832 120 13 𝑑(π‘Ž0 𝑑(π‘Ž0 ADJ cana-4832 120 14 , , PUNCT cana-4832 120 15 π‘Ž1 π‘Ž1 NOUN cana-4832 120 16 ) ) PUNCT cana-4832 120 17 β†’ β†’ SYM cana-4832 120 18 0 0 NUM cana-4832 120 19 as as ADP cana-4832 120 20 𝑖 𝑖 X cana-4832 120 21 , , PUNCT cana-4832 120 22 𝑗 𝑗 PROPN cana-4832 120 23 β†’ β†’ SYM cana-4832 120 24 ∞ ∞ NUM cana-4832 120 25 therefore therefore ADV cana-4832 120 26 { { PUNCT cana-4832 120 27 π‘Žπ‘› π‘Žπ‘› NOUN cana-4832 120 28 } } PUNCT cana-4832 120 29 is be AUX cana-4832 120 30 a a DET cana-4832 120 31 𝑏-cauchy 𝑏-cauchy NOUN cana-4832 120 32 sequence sequence NOUN cana-4832 120 33 in in ADP cana-4832 120 34 𝑋. 𝑋. PROPN cana-4832 120 35 since since SCONJ cana-4832 120 36 𝑋 𝑋 PROPN cana-4832 120 37 is be AUX cana-4832 120 38 complete complete ADJ cana-4832 120 39 , , PUNCT cana-4832 120 40 there there PRON cana-4832 120 41 exists exist VERB cana-4832 120 42 𝑒 𝑒 PRON cana-4832 120 43 ∈ ∈ PROPN cana-4832 120 44 𝑋 𝑋 NOUN cana-4832 120 45 such such ADJ cana-4832 120 46 that that SCONJ cana-4832 120 47 lim lim PROPN cana-4832 120 48 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4832 120 49 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 120 50 = = PUNCT cana-4832 120 51 𝑒. 𝑒. NOUN cana-4832 120 52 since since ADV cana-4832 120 53 , , PUNCT cana-4832 120 54 𝑇 𝑇 PROPN cana-4832 120 55 is be AUX cana-4832 120 56 onto onto ADP cana-4832 120 57 , , PUNCT cana-4832 120 58 we we PRON cana-4832 120 59 can can AUX cana-4832 120 60 find find VERB cana-4832 120 61 𝑝 𝑝 ADP cana-4832 120 62 ∈ ∈ NOUN cana-4832 120 63 𝑋 𝑋 NOUN cana-4832 120 64 such such ADJ cana-4832 120 65 that that SCONJ cana-4832 120 66 𝑇𝑝 𝑇𝑝 PROPN cana-4832 120 67 = = SYM cana-4832 120 68 𝑒. 𝑒. NOUN cana-4832 120 69 now now ADV cana-4832 120 70 , , PUNCT cana-4832 120 71 for for ADP cana-4832 120 72 all all DET cana-4832 120 73 𝑛 𝑛 DET cana-4832 120 74 ∈ ∈ PROPN cana-4832 120 75 β„•. β„•. PROPN cana-4832 120 76 𝑑(𝑒 𝑑(𝑒 PROPN cana-4832 120 77 , , PUNCT cana-4832 120 78 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 120 79 ) ) PUNCT cana-4832 120 80 = = PUNCT cana-4832 120 81 𝑑(𝑇𝑝 𝑑(𝑇𝑝 X cana-4832 120 82 , , PUNCT cana-4832 120 83 π‘‡π‘Žπ‘›+1 π‘‡π‘Žπ‘›+1 NOUN cana-4832 120 84 ) ) PUNCT cana-4832 120 85 β‰₯ β‰₯ NOUN cana-4832 120 86 π‘˜ π‘˜ X cana-4832 120 87 𝛼 𝛼 X cana-4832 120 88 𝑑(𝑝 𝑑(𝑝 PROPN cana-4832 120 89 , , PUNCT cana-4832 120 90 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 120 91 ) ) PUNCT cana-4832 120 92 . . PUNCT cana-4832 121 1 taking take VERB cana-4832 121 2 limit limit NOUN cana-4832 121 3 superior superior ADJ cana-4832 121 4 as as ADP cana-4832 121 5 𝑛 𝑛 PROPN cana-4832 121 6 β†’ β†’ SYM cana-4832 121 7 ∞ ∞ PROPN cana-4832 121 8 , , PUNCT cana-4832 121 9 and and CCONJ cana-4832 121 10 using use VERB cana-4832 121 11 lemma lemma PROPN cana-4832 121 12 1.3 1.3 NUM cana-4832 121 13 , , PUNCT cana-4832 121 14 we we PRON cana-4832 121 15 get get VERB cana-4832 121 16 1 1 NUM cana-4832 121 17 𝑠 𝑠 PRON cana-4832 121 18 𝑑(𝑝 𝑑(𝑝 PROPN cana-4832 121 19 , , PUNCT cana-4832 121 20 𝑒 𝑒 NOUN cana-4832 121 21 ) ) PUNCT cana-4832 121 22 ≀ ≀ NOUN cana-4832 122 1 π‘˜ π‘˜ PRON cana-4832 122 2 𝛼 𝛼 NOUN cana-4832 122 3 𝑠 𝑠 PROPN cana-4832 122 4 lim lim PROPN cana-4832 122 5 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4832 122 6 sup𝑑(𝑝 sup𝑑(𝑝 PROPN cana-4832 122 7 , , PUNCT cana-4832 122 8 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 122 9 ) ) PUNCT cana-4832 122 10 ≀ ≀ NOUN cana-4832 122 11 lim lim PROPN cana-4832 122 12 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4832 122 13 sup𝑑(𝑒 sup𝑑(𝑒 PROPN cana-4832 122 14 , , PUNCT cana-4832 122 15 π‘Žπ‘› π‘Žπ‘› NOUN cana-4832 122 16 ) ) PUNCT cana-4832 122 17 ≀ ≀ NOUN cana-4832 122 18 𝑠 𝑠 ADP cana-4832 122 19 𝑑(𝑒 𝑑(𝑒 PROPN cana-4832 122 20 , , PUNCT cana-4832 122 21 𝑒 𝑒 NOUN cana-4832 122 22 ) ) PUNCT cana-4832 122 23 . . PUNCT cana-4832 123 1 from from ADP cana-4832 123 2 lemma lemma PROPN cana-4832 123 3 . . PROPN cana-4832 124 1 1.4 1.4 NUM cana-4832 124 2 , , PUNCT cana-4832 124 3 we we PRON cana-4832 124 4 get get VERB cana-4832 124 5 𝑑(𝑝 𝑑(𝑝 NOUN cana-4832 124 6 , , PUNCT cana-4832 124 7 𝑒 𝑒 NOUN cana-4832 124 8 ) ) PUNCT cana-4832 124 9 = = SYM cana-4832 124 10 0 0 NUM cana-4832 124 11 and and CCONJ cana-4832 124 12 similarly similarly ADV cana-4832 124 13 𝑑(𝑒 𝑑(𝑒 ADJ cana-4832 124 14 , , PUNCT cana-4832 124 15 𝑝 𝑝 NOUN cana-4832 124 16 ) ) PUNCT cana-4832 124 17 = = SYM cana-4832 125 1 0 0 X cana-4832 125 2 . . PUNCT cana-4832 126 1 thus thus ADV cana-4832 126 2 , , PUNCT cana-4832 126 3 𝑑(𝑝 𝑑(𝑝 PROPN cana-4832 126 4 , , PUNCT cana-4832 126 5 𝑒 𝑒 NOUN cana-4832 126 6 ) ) PUNCT cana-4832 126 7 = = PUNCT cana-4832 126 8 𝑑(𝑒 𝑑(𝑒 ADJ cana-4832 126 9 , , PUNCT cana-4832 126 10 𝑝 𝑝 NOUN cana-4832 126 11 ) ) PUNCT cana-4832 126 12 = = SYM cana-4832 127 1 0 0 X cana-4832 127 2 . . PUNCT cana-4832 128 1 so so ADV cana-4832 128 2 , , PUNCT cana-4832 128 3 𝑝 𝑝 PROPN cana-4832 128 4 = = SYM cana-4832 128 5 𝑒. 𝑒. NOUN cana-4832 128 6 uniqueness uniqueness NOUN cana-4832 128 7 . . PUNCT cana-4832 129 1 let let VERB cana-4832 129 2 𝑣(β‰  𝑣(β‰  PROPN cana-4832 129 3 𝑒 𝑒 NOUN cana-4832 129 4 ) ) PUNCT cana-4832 129 5 be be VERB cana-4832 129 6 another another DET cana-4832 129 7 fixed fix VERB cana-4832 129 8 point point NOUN cana-4832 129 9 of of ADP cana-4832 129 10 𝑇. 𝑇. PROPN cana-4832 129 11 then then ADV cana-4832 129 12 𝑑(𝑒 𝑑(𝑒 NUM cana-4832 129 13 , , PUNCT cana-4832 129 14 𝑣 𝑣 NOUN cana-4832 129 15 ) ) PUNCT cana-4832 129 16 = = SYM cana-4832 129 17 𝑑(𝑇𝑒 𝑑(𝑇𝑒 NOUN cana-4832 129 18 , , PUNCT cana-4832 129 19 𝑇𝑣 𝑇𝑣 PROPN cana-4832 129 20 ) ) PUNCT cana-4832 129 21 β‰₯ β‰₯ NOUN cana-4832 129 22 π‘˜ π‘˜ PRON cana-4832 129 23 𝛼 𝛼 X cana-4832 129 24 𝑑(𝑒 𝑑(𝑒 ADJ cana-4832 129 25 , , PUNCT cana-4832 129 26 𝑣 𝑣 NOUN cana-4832 129 27 ) ) PUNCT cana-4832 129 28 , , PUNCT cana-4832 129 29 i. i. PROPN cana-4832 129 30 e. e. PROPN cana-4832 129 31 , , PUNCT cana-4832 129 32 ( ( PUNCT cana-4832 129 33 1 1 NUM cana-4832 129 34 βˆ’ βˆ’ NOUN cana-4832 129 35 π‘˜ π‘˜ PROPN cana-4832 129 36 𝛼)𝑑(𝑒 𝛼)𝑑(𝑒 PROPN cana-4832 129 37 , , PUNCT cana-4832 129 38 𝑣 𝑣 NOUN cana-4832 129 39 ) ) PUNCT cana-4832 129 40 ≀ ≀ NOUN cana-4832 129 41 0 0 NUM cana-4832 129 42 which which PRON cana-4832 129 43 gives give VERB cana-4832 129 44 us we PRON cana-4832 129 45 𝑑(𝑒 𝑑(𝑒 NOUN cana-4832 129 46 , , PUNCT cana-4832 129 47 𝑣 𝑣 NOUN cana-4832 129 48 ) ) PUNCT cana-4832 129 49 = = SYM cana-4832 129 50 0 0 X cana-4832 129 51 . . PUNCT cana-4832 130 1 similarly similarly ADV cana-4832 130 2 , , PUNCT cana-4832 130 3 we we PRON cana-4832 130 4 can can AUX cana-4832 130 5 prove prove VERB cana-4832 130 6 that that SCONJ cana-4832 130 7 𝑑(𝑣 𝑑(𝑣 NOUN cana-4832 130 8 , , PUNCT cana-4832 130 9 𝑒 𝑒 NOUN cana-4832 130 10 ) ) PUNCT cana-4832 130 11 = = SYM cana-4832 130 12 0 0 NUM cana-4832 130 13 , , PUNCT cana-4832 130 14 and and CCONJ cana-4832 130 15 that that SCONJ cana-4832 130 16 𝑒 𝑒 X cana-4832 130 17 = = SYM cana-4832 130 18 𝑣. 𝑣. NOUN cana-4832 130 19 case case NOUN cana-4832 130 20 ( ( PUNCT cana-4832 130 21 ii ii NOUN cana-4832 130 22 ) ) PUNCT cana-4832 130 23 . . PUNCT cana-4832 131 1 πœƒ(π‘Ž πœƒ(π‘Ž NOUN cana-4832 131 2 , , PUNCT cana-4832 131 3 𝑏 𝑏 NOUN cana-4832 131 4 ) ) PUNCT cana-4832 131 5 = = SYM cana-4832 131 6 𝛽1 𝛽1 NOUN cana-4832 131 7 𝑑(π‘‡π‘Ž,π‘Ž)𝑑(𝑇𝑏,𝑏 𝑑(π‘‡π‘Ž,π‘Ž)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 131 8 ) ) PUNCT cana-4832 131 9 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 NOUN cana-4832 131 10 ) ) PUNCT cana-4832 132 1 + + CCONJ cana-4832 132 2 𝛽2𝑑(π‘Ž 𝛽2𝑑(π‘Ž PROPN cana-4832 132 3 , , PUNCT cana-4832 132 4 𝑏 𝑏 NOUN cana-4832 132 5 ) ) PUNCT cana-4832 132 6 , , PUNCT cana-4832 132 7 then then ADV cana-4832 132 8 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž PROPN cana-4832 132 9 , , PUNCT cana-4832 132 10 𝑇𝑏 𝑇𝑏 PROPN cana-4832 132 11 ) ) PUNCT cana-4832 132 12 β‰₯ β‰₯ NOUN cana-4832 132 13 π‘˜ π‘˜ X cana-4832 132 14 [ [ X cana-4832 132 15 𝛽1 𝛽1 ADJ cana-4832 132 16 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž NOUN cana-4832 132 17 , , PUNCT cana-4832 132 18 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 132 19 , , PUNCT cana-4832 132 20 𝑏 𝑏 NOUN cana-4832 132 21 ) ) PUNCT cana-4832 132 22 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 132 23 , , PUNCT cana-4832 132 24 𝑏 𝑏 NOUN cana-4832 132 25 ) ) PUNCT cana-4832 133 1 + + CCONJ cana-4832 133 2 𝛽2𝑑(π‘Ž 𝛽2𝑑(π‘Ž PROPN cana-4832 133 3 , , PUNCT cana-4832 133 4 𝑏 𝑏 NOUN cana-4832 133 5 ) ) PUNCT cana-4832 133 6 ] ] X cana-4832 133 7 ( ( PUNCT cana-4832 133 8 2.3 2.3 NUM cana-4832 133 9 ) ) PUNCT cana-4832 133 10 now now ADV cana-4832 133 11 , , PUNCT cana-4832 133 12 using use VERB cana-4832 133 13 ( ( PUNCT cana-4832 133 14 2.3 2.3 NUM cana-4832 133 15 ) ) PUNCT cana-4832 133 16 , , PUNCT cana-4832 133 17 we we PRON cana-4832 133 18 get get VERB cana-4832 133 19 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 NOUN cana-4832 133 20 , , PUNCT cana-4832 133 21 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 133 22 ) ) PUNCT cana-4832 133 23 = = PUNCT cana-4832 133 24 𝑑(π‘‡π‘Žπ‘› 𝑑(π‘‡π‘Žπ‘› PROPN cana-4832 133 25 , , PUNCT cana-4832 133 26 π‘‡π‘Žπ‘›+1 π‘‡π‘Žπ‘›+1 NOUN cana-4832 133 27 ) ) PUNCT cana-4832 133 28 β‰₯ β‰₯ NOUN cana-4832 133 29 π‘˜π›½1 π‘˜π›½1 PROPN cana-4832 133 30 𝑑(π‘‡π‘Žπ‘› 𝑑(π‘‡π‘Žπ‘› PROPN cana-4832 133 31 , , PUNCT cana-4832 133 32 π‘Žπ‘›)𝑑(π‘‡π‘Žπ‘›+1 π‘Žπ‘›)𝑑(π‘‡π‘Žπ‘›+1 NOUN cana-4832 133 33 , , PUNCT cana-4832 133 34 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 133 35 ) ) PUNCT cana-4832 133 36 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 133 37 , , PUNCT cana-4832 133 38 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 133 39 ) ) PUNCT cana-4832 133 40 + + CCONJ cana-4832 133 41 π‘˜π›½2𝑑(π‘Žπ‘› π‘˜π›½2𝑑(π‘Žπ‘› ADV cana-4832 133 42 , , PUNCT cana-4832 133 43 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 133 44 ) ) PUNCT cana-4832 133 45 = = PUNCT cana-4832 134 1 π‘˜π›½1 π‘˜π›½1 PROPN cana-4832 134 2 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 NOUN cana-4832 134 3 , , PUNCT cana-4832 134 4 π‘Žπ‘›)𝑑(π‘Žπ‘› π‘Žπ‘›)𝑑(π‘Žπ‘› PROPN cana-4832 134 5 , , PUNCT cana-4832 134 6 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 134 7 ) ) PUNCT cana-4832 134 8 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 134 9 , , PUNCT cana-4832 134 10 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 134 11 ) ) PUNCT cana-4832 134 12 + + CCONJ cana-4832 134 13 π‘˜π›½2𝑑(π‘Žπ‘› π‘˜π›½2𝑑(π‘Žπ‘› ADJ cana-4832 134 14 , , PUNCT cana-4832 134 15 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 134 16 ) ) PUNCT cana-4832 134 17 β‰₯ β‰₯ NOUN cana-4832 134 18 π‘˜π›½2𝑑(π‘Žπ‘› π‘˜π›½2𝑑(π‘Žπ‘› NOUN cana-4832 134 19 , , PUNCT cana-4832 134 20 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 134 21 ) ) PUNCT cana-4832 134 22 . . PUNCT cana-4832 135 1 i. i. PROPN cana-4832 135 2 e. e. PROPN cana-4832 135 3 , , PUNCT cana-4832 135 4 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› PROPN cana-4832 135 5 , , PUNCT cana-4832 135 6 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 135 7 ) ) PUNCT cana-4832 135 8 ≀ ≀ NOUN cana-4832 135 9 1 1 NUM cana-4832 135 10 π‘˜π›½2 π‘˜π›½2 PROPN cana-4832 135 11 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 PROPN cana-4832 135 12 , , PUNCT cana-4832 135 13 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 135 14 ) ) PUNCT cana-4832 135 15 which which PRON cana-4832 135 16 implies imply VERB cana-4832 135 17 that that SCONJ cana-4832 135 18 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NOUN cana-4832 135 19 , , PUNCT cana-4832 135 20 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 135 21 ) ) PUNCT cana-4832 135 22 ≀ ≀ NOUN cana-4832 135 23 πœ— πœ— ADP cana-4832 135 24 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 NUM cana-4832 135 25 , , PUNCT cana-4832 135 26 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 135 27 ) ) PUNCT cana-4832 135 28 , , PUNCT cana-4832 135 29 where where SCONJ cana-4832 135 30 πœ— πœ— NOUN cana-4832 135 31 = = SYM cana-4832 135 32 1 1 NUM cana-4832 135 33 π‘˜π›½2 π‘˜π›½2 NOUN cana-4832 135 34 < < X cana-4832 135 35 1 1 X cana-4832 135 36 . . X cana-4832 135 37 proceeding proceed VERB cana-4832 135 38 similar similar ADJ cana-4832 135 39 to to ADP cana-4832 135 40 case case NOUN cana-4832 135 41 ( ( PUNCT cana-4832 135 42 i i NOUN cana-4832 135 43 ) ) PUNCT cana-4832 135 44 , , PUNCT cana-4832 135 45 we we PRON cana-4832 135 46 get get AUX cana-4832 135 47 { { PUNCT cana-4832 135 48 π‘Žπ‘› π‘Žπ‘› PRON cana-4832 135 49 } } PUNCT cana-4832 135 50 is be AUX cana-4832 135 51 a a DET cana-4832 135 52 𝑏-cauchy 𝑏-cauchy NOUN cana-4832 135 53 sequence sequence NOUN cana-4832 135 54 in in ADP cana-4832 135 55 𝑋 𝑋 PROPN cana-4832 135 56 , , PUNCT cana-4832 135 57 which which PRON cana-4832 135 58 converges converge VERB cana-4832 135 59 to to ADP cana-4832 135 60 some some DET cana-4832 135 61 𝑒 𝑒 PRON cana-4832 135 62 ∈ ∈ PROPN cana-4832 135 63 𝑋 𝑋 PROPN cana-4832 135 64 , , PUNCT cana-4832 135 65 which which PRON cana-4832 135 66 can can AUX cana-4832 135 67 be be AUX cana-4832 135 68 shown show VERB cana-4832 135 69 to to PART cana-4832 135 70 be be AUX cana-4832 135 71 unique unique ADJ cana-4832 135 72 fixed fix VERB cana-4832 135 73 point point NOUN cana-4832 135 74 of of ADP cana-4832 135 75 𝑇. 𝑇. PROPN cana-4832 135 76 case case NOUN cana-4832 135 77 ( ( PUNCT cana-4832 135 78 iii iii NOUN cana-4832 135 79 ) ) PUNCT cana-4832 135 80 . . PUNCT cana-4832 136 1 πœƒ(π‘Ž πœƒ(π‘Ž NOUN cana-4832 136 2 , , PUNCT cana-4832 136 3 𝑏 𝑏 NOUN cana-4832 136 4 ) ) PUNCT cana-4832 136 5 = = SYM cana-4832 136 6 𝛾1𝑑(π‘‡π‘Ž 𝛾1𝑑(π‘‡π‘Ž ADJ cana-4832 136 7 , , PUNCT cana-4832 136 8 π‘Ž π‘Ž NOUN cana-4832 136 9 ) ) PUNCT cana-4832 136 10 + + NUM cana-4832 136 11 𝛾2 𝛾2 NOUN cana-4832 136 12 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 136 13 , , PUNCT cana-4832 136 14 𝑏 𝑏 NOUN cana-4832 136 15 ) ) PUNCT cana-4832 136 16 + + CCONJ cana-4832 136 17 𝛾3 𝛾3 ADJ cana-4832 136 18 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 136 19 , , PUNCT cana-4832 136 20 𝑏 𝑏 NOUN cana-4832 136 21 ) ) PUNCT cana-4832 136 22 , , PUNCT cana-4832 136 23 then then ADV cana-4832 136 24 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž PROPN cana-4832 136 25 , , PUNCT cana-4832 136 26 𝑇𝑏 𝑇𝑏 PROPN cana-4832 136 27 ) ) PUNCT cana-4832 136 28 β‰₯ β‰₯ NOUN cana-4832 136 29 π‘˜[𝛾1𝑑(π‘†π‘Ž π‘˜[𝛾1𝑑(π‘†π‘Ž PROPN cana-4832 136 30 , , PUNCT cana-4832 136 31 π‘Ž π‘Ž NOUN cana-4832 136 32 ) ) PUNCT cana-4832 136 33 + + NUM cana-4832 136 34 𝛾2 𝛾2 NOUN cana-4832 136 35 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 136 36 , , PUNCT cana-4832 136 37 𝑏 𝑏 NOUN cana-4832 136 38 ) ) PUNCT cana-4832 136 39 + + CCONJ cana-4832 136 40 𝛾3 𝛾3 ADJ cana-4832 136 41 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 136 42 , , PUNCT cana-4832 136 43 𝑏 𝑏 NOUN cana-4832 136 44 ) ) PUNCT cana-4832 136 45 ] ] PUNCT cana-4832 136 46 ( ( PUNCT cana-4832 136 47 2.4 2.4 NUM cana-4832 136 48 ) ) PUNCT cana-4832 136 49 now now ADV cana-4832 136 50 , , PUNCT cana-4832 136 51 using use VERB cana-4832 136 52 ( ( PUNCT cana-4832 136 53 2.4 2.4 NUM cana-4832 136 54 ) ) PUNCT cana-4832 136 55 , , PUNCT cana-4832 136 56 we we PRON cana-4832 136 57 get get VERB cana-4832 136 58 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 NOUN cana-4832 136 59 , , PUNCT cana-4832 136 60 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 136 61 ) ) PUNCT cana-4832 136 62 = = PUNCT cana-4832 136 63 𝑑(π‘‡π‘Žπ‘› 𝑑(π‘‡π‘Žπ‘› PROPN cana-4832 136 64 , , PUNCT cana-4832 136 65 π‘‡π‘Žπ‘›+1 π‘‡π‘Žπ‘›+1 NOUN cana-4832 136 66 ) ) PUNCT cana-4832 136 67 β‰₯ β‰₯ NOUN cana-4832 136 68 π‘˜π›Ύ1𝑑(π‘‡π‘Žπ‘› π‘˜π›Ύ1𝑑(π‘‡π‘Žπ‘› PROPN cana-4832 136 69 , , PUNCT cana-4832 136 70 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 136 71 ) ) PUNCT cana-4832 137 1 + + CCONJ cana-4832 137 2 π‘˜π›Ύ2 π‘˜π›Ύ2 PROPN cana-4832 137 3 𝑑(π‘‡π‘Žπ‘›+1 𝑑(π‘‡π‘Žπ‘›+1 PROPN cana-4832 137 4 , , PUNCT cana-4832 137 5 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 137 6 ) ) PUNCT cana-4832 137 7 + + NUM cana-4832 137 8 π‘˜π›Ύ3 π‘˜π›Ύ3 NOUN cana-4832 137 9 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 137 10 , , PUNCT cana-4832 137 11 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 137 12 ) ) PUNCT cana-4832 137 13 = = SYM cana-4832 138 1 π‘˜π›Ύ1𝑑(π‘Žπ‘›βˆ’1 π‘˜π›Ύ1𝑑(π‘Žπ‘›βˆ’1 PROPN cana-4832 138 2 , , PUNCT cana-4832 138 3 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 138 4 ) ) PUNCT cana-4832 139 1 + + CCONJ cana-4832 139 2 π‘˜π›Ύ2 π‘˜π›Ύ2 NOUN cana-4832 139 3 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NOUN cana-4832 139 4 , , PUNCT cana-4832 139 5 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 139 6 ) ) PUNCT cana-4832 140 1 + + NUM cana-4832 140 2 π‘˜π›Ύ3 π‘˜π›Ύ3 NOUN cana-4832 140 3 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 140 4 , , PUNCT cana-4832 140 5 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 140 6 ) ) PUNCT cana-4832 140 7 β‰₯ β‰₯ NOUN cana-4832 140 8 π‘˜[𝛾2 π‘˜[𝛾2 NOUN cana-4832 140 9 + + CCONJ cana-4832 140 10 𝛾3]𝑑(π‘Žπ‘› 𝛾3]𝑑(π‘Žπ‘› NOUN cana-4832 140 11 , , PUNCT cana-4832 140 12 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 140 13 ) ) PUNCT cana-4832 140 14 communications communication NOUN cana-4832 140 15 on on ADP cana-4832 140 16 applied apply VERB cana-4832 140 17 nonlinear nonlinear ADJ cana-4832 140 18 analysis analysis NOUN cana-4832 140 19 issn issn NOUN cana-4832 140 20 : : PUNCT cana-4832 140 21 1074 1074 NUM cana-4832 140 22 - - PUNCT cana-4832 140 23 133x 133x NUM cana-4832 140 24 vol vol VERB cana-4832 140 25 32 32 NUM cana-4832 140 26 no no NOUN cana-4832 140 27 . . PUNCT cana-4832 141 1 10s 10 NOUN cana-4832 141 2 ( ( PUNCT cana-4832 141 3 2025 2025 NUM cana-4832 141 4 ) ) PUNCT cana-4832 141 5 395 395 NUM cana-4832 141 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 141 7 i.e. i.e. X cana-4832 141 8 , , PUNCT cana-4832 141 9 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 141 10 , , PUNCT cana-4832 141 11 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 141 12 ) ) PUNCT cana-4832 141 13 ≀ ≀ NOUN cana-4832 141 14 1 1 NUM cana-4832 141 15 π‘˜[𝛾2+𝛾3 π‘˜[𝛾2+𝛾3 PROPN cana-4832 141 16 ] ] SYM cana-4832 141 17 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 PROPN cana-4832 141 18 , , PUNCT cana-4832 141 19 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 141 20 ) ) PUNCT cana-4832 141 21 implies imply VERB cana-4832 141 22 that that SCONJ cana-4832 141 23 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NOUN cana-4832 141 24 , , PUNCT cana-4832 141 25 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 141 26 ) ) PUNCT cana-4832 141 27 ≀ ≀ NOUN cana-4832 141 28 𝜌 𝜌 ADP cana-4832 141 29 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 PROPN cana-4832 141 30 , , PUNCT cana-4832 141 31 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 141 32 ) ) PUNCT cana-4832 141 33 , , PUNCT cana-4832 141 34 where where SCONJ cana-4832 141 35 𝜌 𝜌 ADP cana-4832 141 36 = = SYM cana-4832 141 37 1 1 NUM cana-4832 141 38 π‘˜[𝛾2+𝛾3 π‘˜[𝛾2+𝛾3 PROPN cana-4832 141 39 ] ] X cana-4832 141 40 < < X cana-4832 141 41 1 1 X cana-4832 141 42 . . X cana-4832 141 43 proceeding proceed VERB cana-4832 141 44 similar similar ADJ cana-4832 141 45 to to ADP cana-4832 141 46 case case NOUN cana-4832 141 47 ( ( PUNCT cana-4832 141 48 i i NOUN cana-4832 141 49 ) ) PUNCT cana-4832 141 50 , , PUNCT cana-4832 141 51 we we PRON cana-4832 141 52 get get AUX cana-4832 141 53 { { PUNCT cana-4832 141 54 π‘Žπ‘› π‘Žπ‘› PRON cana-4832 141 55 } } PUNCT cana-4832 141 56 is be AUX cana-4832 141 57 a a DET cana-4832 141 58 𝑏-cauchy 𝑏-cauchy NOUN cana-4832 141 59 sequence sequence NOUN cana-4832 141 60 in in ADP cana-4832 141 61 𝑋 𝑋 PROPN cana-4832 141 62 , , PUNCT cana-4832 141 63 and and CCONJ cana-4832 141 64 that that SCONJ cana-4832 141 65 converges converge VERB cana-4832 141 66 to to ADP cana-4832 141 67 some some DET cana-4832 141 68 𝑒 𝑒 PRON cana-4832 141 69 ∈ ∈ PROPN cana-4832 141 70 𝑋 𝑋 PROPN cana-4832 141 71 , , PUNCT cana-4832 141 72 which which PRON cana-4832 141 73 is be AUX cana-4832 141 74 a a DET cana-4832 141 75 unique unique ADJ cana-4832 141 76 fixed fix VERB cana-4832 141 77 point point NOUN cana-4832 141 78 of of ADP cana-4832 141 79 𝑇. 𝑇. PROPN cana-4832 141 80 case case NOUN cana-4832 141 81 ( ( PUNCT cana-4832 141 82 iv iv NUM cana-4832 141 83 ) ) PUNCT cana-4832 141 84 . . PUNCT cana-4832 142 1 πœƒ(π‘Ž πœƒ(π‘Ž NOUN cana-4832 142 2 , , PUNCT cana-4832 142 3 𝑏 𝑏 NOUN cana-4832 142 4 ) ) PUNCT cana-4832 142 5 = = SYM cana-4832 142 6 𝛿1𝑑(π‘‡π‘Ž 𝛿1𝑑(π‘‡π‘Ž NOUN cana-4832 142 7 , , PUNCT cana-4832 142 8 𝑏 𝑏 NOUN cana-4832 142 9 ) ) PUNCT cana-4832 142 10 + + CCONJ cana-4832 142 11 𝛿2𝑑(𝑇𝑏 𝛿2𝑑(𝑇𝑏 PROPN cana-4832 142 12 , , PUNCT cana-4832 142 13 π‘Ž π‘Ž NOUN cana-4832 142 14 ) ) PUNCT cana-4832 142 15 + + CCONJ cana-4832 142 16 𝛿3𝑑(π‘Ž 𝛿3𝑑(π‘Ž PROPN cana-4832 142 17 , , PUNCT cana-4832 142 18 𝑏 𝑏 NOUN cana-4832 142 19 ) ) PUNCT cana-4832 142 20 , , PUNCT cana-4832 142 21 then then ADV cana-4832 142 22 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž PROPN cana-4832 142 23 , , PUNCT cana-4832 142 24 𝑆𝑏 𝑆𝑏 PROPN cana-4832 142 25 ) ) PUNCT cana-4832 142 26 β‰₯ β‰₯ NOUN cana-4832 142 27 π‘˜[𝛿1𝑑(π‘‡π‘Ž π‘˜[𝛿1𝑑(π‘‡π‘Ž NOUN cana-4832 142 28 , , PUNCT cana-4832 142 29 𝑏 𝑏 NOUN cana-4832 142 30 ) ) PUNCT cana-4832 142 31 + + CCONJ cana-4832 142 32 𝛿2𝑑(𝑇𝑏 𝛿2𝑑(𝑇𝑏 PROPN cana-4832 142 33 , , PUNCT cana-4832 142 34 π‘Ž π‘Ž NOUN cana-4832 142 35 ) ) PUNCT cana-4832 142 36 + + CCONJ cana-4832 142 37 𝛿3𝑑(π‘Ž 𝛿3𝑑(π‘Ž PROPN cana-4832 142 38 , , PUNCT cana-4832 142 39 𝑏 𝑏 NOUN cana-4832 142 40 ) ) PUNCT cana-4832 142 41 ] ] X cana-4832 142 42 ( ( PUNCT cana-4832 142 43 2.5 2.5 NUM cana-4832 142 44 ) ) PUNCT cana-4832 142 45 from from ADP cana-4832 142 46 ( ( PUNCT cana-4832 142 47 2.5 2.5 NUM cana-4832 142 48 ) ) PUNCT cana-4832 142 49 , , PUNCT cana-4832 142 50 we we PRON cana-4832 142 51 get get VERB cana-4832 142 52 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 NOUN cana-4832 142 53 , , PUNCT cana-4832 142 54 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 142 55 ) ) PUNCT cana-4832 142 56 = = PUNCT cana-4832 143 1 𝑑(π‘‡π‘Žπ‘› 𝑑(π‘‡π‘Žπ‘› PROPN cana-4832 143 2 , , PUNCT cana-4832 143 3 π‘‡π‘Žπ‘›+1 π‘‡π‘Žπ‘›+1 NOUN cana-4832 143 4 ) ) PUNCT cana-4832 143 5 β‰₯ β‰₯ NOUN cana-4832 143 6 π‘˜π›Ώ1𝑑(π‘‡π‘Žπ‘› π‘˜π›Ώ1𝑑(π‘‡π‘Žπ‘› NOUN cana-4832 143 7 , , PUNCT cana-4832 143 8 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 143 9 ) ) PUNCT cana-4832 143 10 + + NUM cana-4832 143 11 π‘˜π›Ώ2 π‘˜π›Ώ2 NOUN cana-4832 143 12 𝑑(π‘‡π‘Žπ‘›+1 𝑑(π‘‡π‘Žπ‘›+1 PROPN cana-4832 143 13 , , PUNCT cana-4832 143 14 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 143 15 ) ) PUNCT cana-4832 143 16 + + CCONJ cana-4832 143 17 π‘˜π›Ώ3 π‘˜π›Ώ3 NOUN cana-4832 143 18 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 143 19 , , PUNCT cana-4832 143 20 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 143 21 ) ) PUNCT cana-4832 143 22 = = SYM cana-4832 143 23 π‘˜π›Ώ1𝑑(π‘Žπ‘›βˆ’1 π‘˜π›Ώ1𝑑(π‘Žπ‘›βˆ’1 PROPN cana-4832 143 24 , , PUNCT cana-4832 143 25 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 143 26 ) ) PUNCT cana-4832 143 27 + + NUM cana-4832 143 28 π‘˜π›Ώ2 π‘˜π›Ώ2 NOUN cana-4832 143 29 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 143 30 , , PUNCT cana-4832 143 31 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 143 32 ) ) PUNCT cana-4832 143 33 + + CCONJ cana-4832 143 34 π‘˜π›Ώ3 π‘˜π›Ώ3 NOUN cana-4832 143 35 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 143 36 , , PUNCT cana-4832 143 37 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 143 38 ) ) PUNCT cana-4832 143 39 β‰₯ β‰₯ NOUN cana-4832 143 40 π‘˜π›Ώ2 π‘˜π›Ώ2 NOUN cana-4832 143 41 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 143 42 , , PUNCT cana-4832 143 43 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 143 44 ) ) PUNCT cana-4832 143 45 + + CCONJ cana-4832 143 46 π‘˜π›Ώ3𝑑(π‘Žπ‘› π‘˜π›Ώ3𝑑(π‘Žπ‘› NOUN cana-4832 143 47 , , PUNCT cana-4832 143 48 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 143 49 ) ) PUNCT cana-4832 143 50 ( ( PUNCT cana-4832 143 51 2.6 2.6 NUM cana-4832 143 52 ) ) PUNCT cana-4832 143 53 where where SCONJ cana-4832 143 54 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› ADJ cana-4832 143 55 , , PUNCT cana-4832 143 56 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 143 57 ) ) PUNCT cana-4832 143 58 = = SYM cana-4832 143 59 𝑑(π‘‡π‘Žπ‘›+1 𝑑(π‘‡π‘Žπ‘›+1 PROPN cana-4832 143 60 , , PUNCT cana-4832 143 61 π‘‡π‘Žπ‘›+1 π‘‡π‘Žπ‘›+1 PROPN cana-4832 143 62 ) ) PUNCT cana-4832 143 63 β‰₯ β‰₯ NOUN cana-4832 143 64 π‘˜π›Ώ1𝑑(π‘‡π‘Žπ‘›+1 π‘˜π›Ώ1𝑑(π‘‡π‘Žπ‘›+1 PROPN cana-4832 143 65 , , PUNCT cana-4832 143 66 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 143 67 ) ) PUNCT cana-4832 143 68 + + NUM cana-4832 143 69 π‘˜π›Ώ2 π‘˜π›Ώ2 NOUN cana-4832 143 70 𝑑(π‘‡π‘Žπ‘›+1 𝑑(π‘‡π‘Žπ‘›+1 PROPN cana-4832 143 71 , , PUNCT cana-4832 143 72 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 143 73 ) ) PUNCT cana-4832 143 74 + + CCONJ cana-4832 143 75 π‘˜π›Ώ3 π‘˜π›Ώ3 PROPN cana-4832 143 76 𝑑(π‘Žπ‘›+1 𝑑(π‘Žπ‘›+1 PROPN cana-4832 143 77 , , PUNCT cana-4832 143 78 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 143 79 ) ) PUNCT cana-4832 143 80 β‰₯ β‰₯ NOUN cana-4832 143 81 π‘˜π›Ώ1𝑑(π‘Žπ‘› π‘˜π›Ώ1𝑑(π‘Žπ‘› NOUN cana-4832 143 82 , , PUNCT cana-4832 143 83 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 143 84 ) ) PUNCT cana-4832 143 85 + + NUM cana-4832 143 86 π‘˜π›Ώ2 π‘˜π›Ώ2 NOUN cana-4832 143 87 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 143 88 , , PUNCT cana-4832 143 89 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 143 90 ) ) PUNCT cana-4832 143 91 = = PRON cana-4832 143 92 π‘˜[𝛿1 π‘˜[𝛿1 NOUN cana-4832 143 93 + + CCONJ cana-4832 143 94 𝛿2]𝑑(π‘Žπ‘› 𝛿2]𝑑(π‘Žπ‘› ADJ cana-4832 143 95 , , PUNCT cana-4832 143 96 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 143 97 ) ) PUNCT cana-4832 143 98 . . PUNCT cana-4832 144 1 from from ADP cana-4832 144 2 the the DET cana-4832 144 3 inequality inequality NOUN cana-4832 144 4 ( ( PUNCT cana-4832 144 5 2.6 2.6 NUM cana-4832 144 6 ) ) PUNCT cana-4832 144 7 , , PUNCT cana-4832 144 8 we we PRON cana-4832 144 9 get get VERB cana-4832 144 10 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 NOUN cana-4832 144 11 , , PUNCT cana-4832 144 12 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 144 13 ) ) PUNCT cana-4832 144 14 β‰₯ β‰₯ NOUN cana-4832 145 1 [ [ X cana-4832 145 2 π‘˜2𝛿2(𝛿1 π‘˜2𝛿2(𝛿1 X cana-4832 145 3 + + X cana-4832 145 4 𝛿2 𝛿2 NOUN cana-4832 145 5 ) ) PUNCT cana-4832 146 1 + + CCONJ cana-4832 146 2 π‘˜ π‘˜ DET cana-4832 146 3 𝛿3]𝑑(π‘Žπ‘› 𝛿3]𝑑(π‘Žπ‘› NOUN cana-4832 146 4 , , PUNCT cana-4832 146 5 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 146 6 ) ) PUNCT cana-4832 146 7 which which PRON cana-4832 146 8 implies imply VERB cana-4832 146 9 that that SCONJ cana-4832 146 10 i.e. i.e. ADV cana-4832 146 11 , , PUNCT cana-4832 146 12 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 146 13 , , PUNCT cana-4832 146 14 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 146 15 ) ) PUNCT cana-4832 146 16 ≀ ≀ NUM cana-4832 146 17 1 1 NUM cana-4832 146 18 π‘˜2𝛿2(𝛿1+𝛿2)+π‘˜ π‘˜2𝛿2(𝛿1+𝛿2)+π‘˜ PROPN cana-4832 146 19 𝛿3 𝛿3 PROPN cana-4832 146 20 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 PROPN cana-4832 146 21 , , PUNCT cana-4832 146 22 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 146 23 ) ) PUNCT cana-4832 146 24 implies imply VERB cana-4832 146 25 that that SCONJ cana-4832 146 26 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NOUN cana-4832 146 27 , , PUNCT cana-4832 146 28 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 146 29 ) ) PUNCT cana-4832 146 30 ≀ ≀ NUM cana-4832 146 31 πœ” πœ” DET cana-4832 146 32 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 PROPN cana-4832 146 33 , , PUNCT cana-4832 146 34 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 146 35 ) ) PUNCT cana-4832 146 36 , , PUNCT cana-4832 146 37 where where SCONJ cana-4832 146 38 πœ” πœ” PART cana-4832 146 39 = = SYM cana-4832 146 40 1 1 NUM cana-4832 146 41 π‘˜2𝛿2(𝛿1+𝛿2)+π‘˜ π‘˜2𝛿2(𝛿1+𝛿2)+π‘˜ NOUN cana-4832 146 42 𝛿3 𝛿3 X cana-4832 146 43 < < X cana-4832 146 44 1 1 X cana-4832 146 45 . . X cana-4832 146 46 proceeding proceed VERB cana-4832 146 47 similar similar ADJ cana-4832 146 48 to to ADP cana-4832 146 49 case case NOUN cana-4832 146 50 ( ( PUNCT cana-4832 146 51 i i NOUN cana-4832 146 52 ) ) PUNCT cana-4832 146 53 , , PUNCT cana-4832 146 54 we we PRON cana-4832 146 55 get get AUX cana-4832 146 56 { { PUNCT cana-4832 146 57 π‘Žπ‘› π‘Žπ‘› PRON cana-4832 146 58 } } PUNCT cana-4832 146 59 is be AUX cana-4832 146 60 a a DET cana-4832 146 61 𝑏-cauchy 𝑏-cauchy NOUN cana-4832 146 62 sequence sequence NOUN cana-4832 146 63 in in ADP cana-4832 146 64 𝑋 𝑋 PROPN cana-4832 146 65 , , PUNCT cana-4832 146 66 and and CCONJ cana-4832 146 67 that that SCONJ cana-4832 146 68 converges converge VERB cana-4832 146 69 to to ADP cana-4832 146 70 some some DET cana-4832 146 71 𝑒 𝑒 PRON cana-4832 146 72 ∈ ∈ PROPN cana-4832 146 73 𝑋 𝑋 PROPN cana-4832 146 74 , , PUNCT cana-4832 146 75 which which PRON cana-4832 146 76 is be AUX cana-4832 146 77 a a DET cana-4832 146 78 unique unique ADJ cana-4832 146 79 fixed fix VERB cana-4832 146 80 point point NOUN cana-4832 146 81 of of ADP cana-4832 146 82 𝑇. 𝑇. PROPN cana-4832 146 83 case case NOUN cana-4832 146 84 ( ( PUNCT cana-4832 146 85 v v NOUN cana-4832 146 86 ) ) PUNCT cana-4832 146 87 . . PUNCT cana-4832 147 1 πœƒ(π‘Ž πœƒ(π‘Ž NOUN cana-4832 147 2 , , PUNCT cana-4832 147 3 𝑏 𝑏 NOUN cana-4832 147 4 ) ) PUNCT cana-4832 147 5 = = VERB cana-4832 147 6 πœ†1 πœ†1 VERB cana-4832 147 7 𝑑(π‘‡π‘Ž,𝑏)𝑑(𝑇𝑏,𝑏 𝑑(π‘‡π‘Ž,𝑏)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 147 8 ) ) PUNCT cana-4832 147 9 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 NOUN cana-4832 147 10 ) ) PUNCT cana-4832 148 1 + + NUM cana-4832 148 2 πœ†2 πœ†2 NOUN cana-4832 148 3 𝑑(𝑇𝑏,π‘Ž)𝑑(𝑇𝑏,𝑏 𝑑(𝑇𝑏,π‘Ž)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 148 4 ) ) PUNCT cana-4832 148 5 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 VERB cana-4832 148 6 ) ) PUNCT cana-4832 149 1 + + CCONJ cana-4832 149 2 πœ†3𝑑(π‘Ž πœ†3𝑑(π‘Ž PROPN cana-4832 149 3 , , PUNCT cana-4832 149 4 𝑏 𝑏 NOUN cana-4832 149 5 ) ) PUNCT cana-4832 149 6 , , PUNCT cana-4832 149 7 then then ADV cana-4832 149 8 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž PROPN cana-4832 149 9 , , PUNCT cana-4832 149 10 𝑇𝑏 𝑇𝑏 PROPN cana-4832 149 11 ) ) PUNCT cana-4832 149 12 β‰₯ β‰₯ NOUN cana-4832 149 13 π‘˜ π‘˜ X cana-4832 150 1 [ [ X cana-4832 150 2 πœ†1 πœ†1 NOUN cana-4832 150 3 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž NOUN cana-4832 150 4 , , PUNCT cana-4832 150 5 𝑏)𝑑(𝑇𝑏 𝑏)𝑑(𝑇𝑏 PROPN cana-4832 150 6 , , PUNCT cana-4832 150 7 𝑏 𝑏 NOUN cana-4832 150 8 ) ) PUNCT cana-4832 150 9 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 150 10 , , PUNCT cana-4832 150 11 𝑏 𝑏 NOUN cana-4832 150 12 ) ) PUNCT cana-4832 150 13 + + NUM cana-4832 150 14 πœ†2 πœ†2 PROPN cana-4832 150 15 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 150 16 , , PUNCT cana-4832 150 17 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 150 18 , , PUNCT cana-4832 150 19 𝑏 𝑏 NOUN cana-4832 150 20 ) ) PUNCT cana-4832 150 21 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 150 22 , , PUNCT cana-4832 150 23 𝑏 𝑏 NOUN cana-4832 150 24 ) ) PUNCT cana-4832 150 25 + + CCONJ cana-4832 150 26 πœ†3𝑑(π‘Ž πœ†3𝑑(π‘Ž PROPN cana-4832 150 27 , , PUNCT cana-4832 150 28 𝑏 𝑏 NOUN cana-4832 150 29 ) ) PUNCT cana-4832 150 30 ] ] PUNCT cana-4832 150 31 ( ( PUNCT cana-4832 150 32 2.7 2.7 NUM cana-4832 150 33 ) ) PUNCT cana-4832 150 34 now now ADV cana-4832 150 35 , , PUNCT cana-4832 150 36 using use VERB cana-4832 150 37 ( ( PUNCT cana-4832 150 38 2.6 2.6 NUM cana-4832 150 39 ) ) PUNCT cana-4832 150 40 , , PUNCT cana-4832 150 41 we we PRON cana-4832 150 42 get get VERB cana-4832 150 43 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 NOUN cana-4832 150 44 , , PUNCT cana-4832 150 45 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 150 46 ) ) PUNCT cana-4832 150 47 = = PUNCT cana-4832 150 48 𝑑(π‘‡π‘Žπ‘› 𝑑(π‘‡π‘Žπ‘› PROPN cana-4832 150 49 , , PUNCT cana-4832 150 50 π‘‡π‘Žπ‘›+1 π‘‡π‘Žπ‘›+1 NOUN cana-4832 150 51 ) ) PUNCT cana-4832 150 52 β‰₯ β‰₯ NOUN cana-4832 150 53 π‘˜ π‘˜ DET cana-4832 150 54 πœ†1 πœ†1 X cana-4832 150 55 𝑑(π‘‡π‘Žπ‘› 𝑑(π‘‡π‘Žπ‘› PROPN cana-4832 150 56 , , PUNCT cana-4832 150 57 π‘Žπ‘›+1)𝑑(π‘‡π‘Žπ‘›+1 π‘Žπ‘›+1)𝑑(π‘‡π‘Žπ‘›+1 ADJ cana-4832 150 58 , , PUNCT cana-4832 150 59 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 150 60 ) ) PUNCT cana-4832 150 61 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 150 62 , , PUNCT cana-4832 150 63 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 150 64 ) ) PUNCT cana-4832 150 65 + + CCONJ cana-4832 150 66 π‘˜ π‘˜ PRON cana-4832 150 67 πœ†2 πœ†2 PROPN cana-4832 150 68 𝑑(π‘‡π‘Žπ‘›+1 𝑑(π‘‡π‘Žπ‘›+1 PROPN cana-4832 150 69 , , PUNCT cana-4832 150 70 π‘Žπ‘›)𝑑(π‘‡π‘Žπ‘›+1 π‘Žπ‘›)𝑑(π‘‡π‘Žπ‘›+1 PROPN cana-4832 150 71 , , PUNCT cana-4832 150 72 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 150 73 ) ) PUNCT cana-4832 150 74 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 150 75 , , PUNCT cana-4832 150 76 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 150 77 ) ) PUNCT cana-4832 151 1 + + CCONJ cana-4832 151 2 π‘˜ π‘˜ DET cana-4832 151 3 πœ†3 πœ†3 NOUN cana-4832 151 4 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NOUN cana-4832 151 5 , , PUNCT cana-4832 151 6 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 151 7 ) ) PUNCT cana-4832 151 8 = = PUNCT cana-4832 152 1 π‘˜ π‘˜ DET cana-4832 152 2 πœ†1 πœ†1 PROPN cana-4832 152 3 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 ADP cana-4832 152 4 , , PUNCT cana-4832 152 5 π‘Žπ‘›+1)𝑑(π‘Žπ‘› π‘Žπ‘›+1)𝑑(π‘Žπ‘› NOUN cana-4832 152 6 , , PUNCT cana-4832 152 7 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 152 8 ) ) PUNCT cana-4832 152 9 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NOUN cana-4832 152 10 , , PUNCT cana-4832 152 11 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 152 12 ) ) PUNCT cana-4832 152 13 + + CCONJ cana-4832 152 14 π‘˜ π‘˜ PRON cana-4832 152 15 πœ†2 πœ†2 NOUN cana-4832 152 16 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 152 17 , , PUNCT cana-4832 152 18 π‘Žπ‘›)𝑑(π‘Žπ‘› π‘Žπ‘›)𝑑(π‘Žπ‘› NUM cana-4832 152 19 , , PUNCT cana-4832 152 20 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 152 21 ) ) PUNCT cana-4832 152 22 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 152 23 , , PUNCT cana-4832 152 24 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 152 25 ) ) PUNCT cana-4832 152 26 + + CCONJ cana-4832 152 27 π‘˜ π‘˜ DET cana-4832 152 28 πœ†3 πœ†3 NOUN cana-4832 152 29 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 152 30 , , PUNCT cana-4832 152 31 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 152 32 ) ) PUNCT cana-4832 152 33 β‰₯ β‰₯ NOUN cana-4832 152 34 π‘˜πœ†3𝑑(π‘Žπ‘› π‘˜πœ†3𝑑(π‘Žπ‘› NOUN cana-4832 152 35 , , PUNCT cana-4832 152 36 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 152 37 ) ) PUNCT cana-4832 152 38 . . PUNCT cana-4832 153 1 communications communication NOUN cana-4832 153 2 on on ADP cana-4832 153 3 applied apply VERB cana-4832 153 4 nonlinear nonlinear ADJ cana-4832 153 5 analysis analysis NOUN cana-4832 153 6 issn issn NOUN cana-4832 153 7 : : PUNCT cana-4832 153 8 1074 1074 NUM cana-4832 153 9 - - PUNCT cana-4832 153 10 133x 133x NUM cana-4832 153 11 vol vol VERB cana-4832 153 12 32 32 NUM cana-4832 153 13 no no NOUN cana-4832 153 14 . . PUNCT cana-4832 154 1 10s 10 NOUN cana-4832 154 2 ( ( PUNCT cana-4832 154 3 2025 2025 NUM cana-4832 154 4 ) ) PUNCT cana-4832 154 5 396 396 NUM cana-4832 154 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 154 7 i.e. i.e. X cana-4832 154 8 , , PUNCT cana-4832 154 9 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 154 10 , , PUNCT cana-4832 154 11 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 154 12 ) ) PUNCT cana-4832 154 13 ≀ ≀ NOUN cana-4832 154 14 1 1 NUM cana-4832 154 15 π‘˜πœ†3 π‘˜πœ†3 NOUN cana-4832 154 16 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 PROPN cana-4832 154 17 , , PUNCT cana-4832 154 18 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 154 19 ) ) PUNCT cana-4832 154 20 which which PRON cana-4832 154 21 implies imply VERB cana-4832 154 22 that that SCONJ cana-4832 154 23 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NOUN cana-4832 154 24 , , PUNCT cana-4832 154 25 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 154 26 ) ) PUNCT cana-4832 154 27 ≀ ≀ NUM cana-4832 154 28 πœ‘ πœ‘ ADP cana-4832 154 29 𝑑(π‘Žπ‘›βˆ’1 𝑑(π‘Žπ‘›βˆ’1 PROPN cana-4832 154 30 , , PUNCT cana-4832 154 31 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 154 32 ) ) PUNCT cana-4832 154 33 , , PUNCT cana-4832 154 34 where where SCONJ cana-4832 154 35 πœ‘ πœ‘ NOUN cana-4832 154 36 = = SYM cana-4832 154 37 1 1 NUM cana-4832 154 38 π‘˜πœ†3 π‘˜πœ†3 NOUN cana-4832 154 39 < < X cana-4832 154 40 1 1 X cana-4832 154 41 . . X cana-4832 154 42 proceeding proceed VERB cana-4832 154 43 similar similar ADJ cana-4832 154 44 to to ADP cana-4832 154 45 case case NOUN cana-4832 154 46 ( ( PUNCT cana-4832 154 47 i i NOUN cana-4832 154 48 ) ) PUNCT cana-4832 154 49 , , PUNCT cana-4832 154 50 we we PRON cana-4832 154 51 get get AUX cana-4832 154 52 { { PUNCT cana-4832 154 53 π‘Žπ‘› π‘Žπ‘› PRON cana-4832 154 54 } } PUNCT cana-4832 154 55 is be AUX cana-4832 154 56 a a DET cana-4832 154 57 𝑏-cauchy 𝑏-cauchy NOUN cana-4832 154 58 sequence sequence NOUN cana-4832 154 59 in in ADP cana-4832 154 60 𝑋 𝑋 PROPN cana-4832 154 61 , , PUNCT cana-4832 154 62 and and CCONJ cana-4832 154 63 that that SCONJ cana-4832 154 64 converges converge VERB cana-4832 154 65 to to ADP cana-4832 154 66 some some DET cana-4832 154 67 𝑒 𝑒 PRON cana-4832 154 68 ∈ ∈ PROPN cana-4832 154 69 𝑋 𝑋 PROPN cana-4832 154 70 , , PUNCT cana-4832 154 71 which which PRON cana-4832 154 72 is be AUX cana-4832 154 73 a a DET cana-4832 154 74 unique unique ADJ cana-4832 154 75 fixed fix VERB cana-4832 154 76 point point NOUN cana-4832 154 77 of of ADP cana-4832 154 78 𝑇. 𝑇. PROPN cana-4832 154 79 the the DET cana-4832 154 80 following follow VERB cana-4832 154 81 is be AUX cana-4832 154 82 an an DET cana-4832 154 83 example example NOUN cana-4832 154 84 in in ADP cana-4832 154 85 support support NOUN cana-4832 154 86 of of ADP cana-4832 154 87 theorem theorem ADJ cana-4832 154 88 2.1 2.1 NUM cana-4832 154 89 . . PUNCT cana-4832 154 90 example example NOUN cana-4832 154 91 2.2 2.2 NUM cana-4832 154 92 . . PUNCT cana-4832 155 1 let let VERB cana-4832 155 2 𝑋 𝑋 NOUN cana-4832 155 3 = = PUNCT cana-4832 155 4 ℝ+ ℝ+ PROPN cana-4832 155 5 . . PUNCT cana-4832 156 1 we we PRON cana-4832 156 2 define define VERB cana-4832 156 3 𝑑 𝑑 NOUN cana-4832 156 4 : : PUNCT cana-4832 156 5 𝑋 𝑋 NOUN cana-4832 156 6 Γ— Γ— NOUN cana-4832 156 7 𝑋 𝑋 PROPN cana-4832 156 8 β†’ β†’ SYM cana-4832 156 9 ℝ+ ℝ+ PUNCT cana-4832 156 10 by by ADP cana-4832 156 11 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 156 12 , , PUNCT cana-4832 156 13 𝑏 𝑏 NOUN cana-4832 156 14 ) ) PUNCT cana-4832 156 15 = = SYM cana-4832 156 16 |π‘Ž |π‘Ž X cana-4832 156 17 βˆ’ βˆ’ PROPN cana-4832 156 18 𝑏|2 𝑏|2 PROPN cana-4832 156 19 + + CCONJ cana-4832 156 20 |π‘Ž|2 |π‘Ž|2 PROPN cana-4832 156 21 . . PUNCT cana-4832 157 1 then then ADV cana-4832 157 2 clearly clearly ADV cana-4832 157 3 , , PUNCT cana-4832 157 4 ( ( PUNCT cana-4832 157 5 𝑋 𝑋 NOUN cana-4832 157 6 , , PUNCT cana-4832 157 7 𝑑 𝑑 NOUN cana-4832 157 8 ) ) PUNCT cana-4832 157 9 is be AUX cana-4832 157 10 a a DET cana-4832 157 11 complete complete ADJ cana-4832 157 12 𝑑𝑝 𝑑𝑝 ADJ cana-4832 157 13 𝑏-metric 𝑏-metric ADJ cana-4832 157 14 space space NOUN cana-4832 157 15 with with ADP cana-4832 157 16 𝑠 𝑠 PROPN cana-4832 157 17 = = SYM cana-4832 157 18 2 2 X cana-4832 157 19 . . X cana-4832 158 1 we we PRON cana-4832 158 2 define define VERB cana-4832 158 3 self self NOUN cana-4832 158 4 - - PUNCT cana-4832 158 5 mappings mapping NOUN cana-4832 158 6 𝑇 𝑇 NOUN cana-4832 158 7 : : PUNCT cana-4832 158 8 𝑋 𝑋 PROPN cana-4832 158 9 β†’ β†’ SYM cana-4832 158 10 𝑋 𝑋 PROPN cana-4832 158 11 by by ADP cana-4832 158 12 𝑇(π‘Ž 𝑇(π‘Ž PROPN cana-4832 158 13 ) ) PUNCT cana-4832 158 14 = = PUNCT cana-4832 158 15 π‘Ž(π‘Ž π‘Ž(π‘Ž PROPN cana-4832 158 16 + + CCONJ cana-4832 158 17 2 2 NUM cana-4832 158 18 ) ) PUNCT cana-4832 158 19 , , PUNCT cana-4832 158 20 for for ADP cana-4832 158 21 all all PRON cana-4832 158 22 π‘Ž π‘Ž PROPN cana-4832 158 23 ∈ ∈ NOUN cana-4832 158 24 𝑋. 𝑋. NOUN cana-4832 158 25 we we PRON cana-4832 158 26 take take VERB cana-4832 158 27 π‘˜ π‘˜ PRON cana-4832 158 28 = = NOUN cana-4832 158 29 3 3 NUM cana-4832 158 30 2 2 NUM cana-4832 158 31 , , PUNCT cana-4832 158 32 𝛼 𝛼 NOUN cana-4832 158 33 = = SYM cana-4832 158 34 𝛽2 𝛽2 PROPN cana-4832 158 35 = = SYM cana-4832 158 36 𝛾3 𝛾3 NOUN cana-4832 158 37 = = SYM cana-4832 158 38 𝛿3 𝛿3 NOUN cana-4832 158 39 = = SYM cana-4832 158 40 πœ†3 πœ†3 NOUN cana-4832 158 41 = = SYM cana-4832 158 42 1 1 NUM cana-4832 158 43 , , PUNCT cana-4832 158 44 𝛽1 𝛽1 NOUN cana-4832 158 45 = = SYM cana-4832 158 46 𝛾1 𝛾1 PROPN cana-4832 158 47 = = NOUN cana-4832 158 48 𝛾2 𝛾2 PROPN cana-4832 158 49 = = SYM cana-4832 158 50 𝛿1 𝛿1 NOUN cana-4832 158 51 = = SYM cana-4832 158 52 𝛿2 𝛿2 NOUN cana-4832 158 53 = = NOUN cana-4832 158 54 πœ†1 πœ†1 NOUN cana-4832 158 55 = = PUNCT cana-4832 158 56 πœ†2 πœ†2 NOUN cana-4832 158 57 = = SYM cana-4832 158 58 0 0 NUM cana-4832 158 59 . . PUNCT cana-4832 158 60 without without ADP cana-4832 158 61 loss loss NOUN cana-4832 158 62 of of ADP cana-4832 158 63 generality generality NOUN cana-4832 158 64 we we PRON cana-4832 158 65 assume assume VERB cana-4832 158 66 that that SCONJ cana-4832 158 67 π‘Ž π‘Ž DET cana-4832 158 68 β‰₯ β‰₯ NOUN cana-4832 158 69 𝑏. 𝑏. NOUN cana-4832 158 70 we we PRON cana-4832 158 71 consider consider VERB cana-4832 158 72 𝑑(π‘†π‘Ž 𝑑(π‘†π‘Ž NOUN cana-4832 158 73 , , PUNCT cana-4832 159 1 𝑇𝑏 𝑇𝑏 NOUN cana-4832 159 2 ) ) PUNCT cana-4832 159 3 = = SYM cana-4832 159 4 |π‘‡π‘Ž |π‘‡π‘Ž PROPN cana-4832 159 5 βˆ’ βˆ’ PROPN cana-4832 159 6 𝑇𝑏|2 𝑇𝑏|2 NOUN cana-4832 159 7 + + X cana-4832 159 8 |π‘†π‘Ž|2 |π‘†π‘Ž|2 PROPN cana-4832 159 9 = = SYM cana-4832 159 10 ( ( PUNCT cana-4832 159 11 π‘Ž2 π‘Ž2 NOUN cana-4832 159 12 + + CCONJ cana-4832 159 13 2π‘Ž 2π‘Ž NUM cana-4832 159 14 βˆ’ βˆ’ PROPN cana-4832 159 15 𝑏2 𝑏2 PROPN cana-4832 159 16 βˆ’ βˆ’ PROPN cana-4832 159 17 2𝑏)2 2𝑏)2 NUM cana-4832 159 18 + + CCONJ cana-4832 159 19 ( ( PUNCT cana-4832 159 20 π‘Ž2 π‘Ž2 NOUN cana-4832 159 21 + + CCONJ cana-4832 159 22 2π‘Ž)2 2π‘Ž)2 NUM cana-4832 159 23 = = SYM cana-4832 159 24 ( ( PUNCT cana-4832 159 25 π‘Ž π‘Ž X cana-4832 159 26 βˆ’ βˆ’ PROPN cana-4832 159 27 𝑏)2(π‘Ž 𝑏)2(π‘Ž PROPN cana-4832 159 28 + + CCONJ cana-4832 159 29 𝑏 𝑏 PROPN cana-4832 159 30 + + CCONJ cana-4832 159 31 2)2 2)2 NUM cana-4832 159 32 + + CCONJ cana-4832 159 33 π‘Ž2(a π‘Ž2(a PROPN cana-4832 160 1 + + CCONJ cana-4832 160 2 2)2 2)2 NUM cana-4832 160 3 β‰₯ β‰₯ NUM cana-4832 160 4 3 3 NUM cana-4832 160 5 2 2 NUM cana-4832 160 6 [ [ X cana-4832 160 7 ( ( PUNCT cana-4832 160 8 π‘Ž π‘Ž DET cana-4832 160 9 βˆ’ βˆ’ PROPN cana-4832 160 10 𝑏)2 𝑏)2 PROPN cana-4832 160 11 + + CCONJ cana-4832 160 12 π‘Ž2 π‘Ž2 VERB cana-4832 160 13 ] ] PUNCT cana-4832 160 14 = = SYM cana-4832 160 15 π‘˜ π‘˜ PRON cana-4832 160 16 min min NOUN cana-4832 160 17 { { PUNCT cana-4832 160 18 𝛼 𝛼 NOUN cana-4832 160 19 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 160 20 , , PUNCT cana-4832 160 21 𝑏 𝑏 NOUN cana-4832 160 22 ) ) PUNCT cana-4832 160 23 ; ; PUNCT cana-4832 160 24 𝛽1 𝛽1 NOUN cana-4832 160 25 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž NOUN cana-4832 160 26 , , PUNCT cana-4832 160 27 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 160 28 , , PUNCT cana-4832 160 29 𝑏 𝑏 NOUN cana-4832 160 30 ) ) PUNCT cana-4832 160 31 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 160 32 , , PUNCT cana-4832 160 33 𝑏 𝑏 NOUN cana-4832 160 34 ) ) PUNCT cana-4832 161 1 + + CCONJ cana-4832 161 2 𝛽2𝑑(π‘Ž 𝛽2𝑑(π‘Ž PROPN cana-4832 161 3 , , PUNCT cana-4832 161 4 𝑏 𝑏 NOUN cana-4832 161 5 ) ) PUNCT cana-4832 161 6 ; ; PUNCT cana-4832 161 7 𝛾1𝑑(π‘‡π‘Ž 𝛾1𝑑(π‘‡π‘Ž PROPN cana-4832 161 8 , , PUNCT cana-4832 161 9 π‘Ž π‘Ž NOUN cana-4832 161 10 ) ) PUNCT cana-4832 161 11 + + NUM cana-4832 161 12 𝛾2 𝛾2 NOUN cana-4832 161 13 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 161 14 , , PUNCT cana-4832 161 15 𝑏 𝑏 NOUN cana-4832 161 16 ) ) PUNCT cana-4832 161 17 + + CCONJ cana-4832 161 18 𝛾3 𝛾3 ADJ cana-4832 161 19 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 161 20 , , PUNCT cana-4832 161 21 𝑏 𝑏 NOUN cana-4832 161 22 ) ) PUNCT cana-4832 161 23 ; ; PUNCT cana-4832 161 24 𝛿1𝑑(π‘‡π‘Ž 𝛿1𝑑(π‘‡π‘Ž NOUN cana-4832 161 25 , , PUNCT cana-4832 161 26 𝑏 𝑏 NOUN cana-4832 161 27 ) ) PUNCT cana-4832 161 28 + + CCONJ cana-4832 161 29 𝛿2𝑑(𝑇𝑏 𝛿2𝑑(𝑇𝑏 PROPN cana-4832 161 30 , , PUNCT cana-4832 161 31 π‘Ž π‘Ž NOUN cana-4832 161 32 ) ) PUNCT cana-4832 161 33 + + CCONJ cana-4832 161 34 𝛿3𝑑(π‘Ž 𝛿3𝑑(π‘Ž PROPN cana-4832 161 35 , , PUNCT cana-4832 161 36 𝑏 𝑏 NOUN cana-4832 161 37 ) ) PUNCT cana-4832 161 38 ; ; PUNCT cana-4832 162 1 πœ†1 πœ†1 VERB cana-4832 162 2 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž PROPN cana-4832 162 3 , , PUNCT cana-4832 162 4 𝑏)𝑑(𝑇𝑏 𝑏)𝑑(𝑇𝑏 PROPN cana-4832 162 5 , , PUNCT cana-4832 162 6 𝑏 𝑏 NOUN cana-4832 162 7 ) ) PUNCT cana-4832 162 8 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 162 9 , , PUNCT cana-4832 162 10 𝑏 𝑏 NOUN cana-4832 162 11 ) ) PUNCT cana-4832 162 12 + + NUM cana-4832 162 13 πœ†2 πœ†2 PROPN cana-4832 162 14 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 162 15 , , PUNCT cana-4832 162 16 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 162 17 , , PUNCT cana-4832 162 18 𝑏 𝑏 NOUN cana-4832 162 19 ) ) PUNCT cana-4832 162 20 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 162 21 , , PUNCT cana-4832 162 22 𝑏 𝑏 NOUN cana-4832 162 23 ) ) PUNCT cana-4832 162 24 + + CCONJ cana-4832 162 25 πœ†3𝑑(π‘Ž πœ†3𝑑(π‘Ž PROPN cana-4832 162 26 , , PUNCT cana-4832 162 27 𝑏 𝑏 NOUN cana-4832 162 28 ) ) PUNCT cana-4832 162 29 } } PUNCT cana-4832 162 30 communications communication NOUN cana-4832 162 31 on on ADP cana-4832 162 32 applied apply VERB cana-4832 162 33 nonlinear nonlinear ADJ cana-4832 162 34 analysis analysis NOUN cana-4832 162 35 issn issn NOUN cana-4832 162 36 : : PUNCT cana-4832 162 37 1074 1074 NUM cana-4832 162 38 - - PUNCT cana-4832 162 39 133x 133x NUM cana-4832 162 40 vol vol VERB cana-4832 162 41 32 32 NUM cana-4832 162 42 no no NOUN cana-4832 162 43 . . PUNCT cana-4832 163 1 10s 10 NOUN cana-4832 163 2 ( ( PUNCT cana-4832 163 3 2025 2025 NUM cana-4832 163 4 ) ) PUNCT cana-4832 163 5 397 397 NUM cana-4832 163 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 163 7 table table NOUN cana-4832 163 8 1 1 NUM cana-4832 163 9 and and CCONJ cana-4832 163 10 figure figure NOUN cana-4832 163 11 2 2 NUM cana-4832 163 12 , , PUNCT cana-4832 163 13 illustrates illustrate VERB cana-4832 163 14 the the DET cana-4832 163 15 condition condition NOUN cana-4832 163 16 ( ( PUNCT cana-4832 163 17 2.1 2.1 NUM cana-4832 163 18 ) ) PUNCT cana-4832 163 19 of of ADP cana-4832 163 20 theorem theorem ADJ cana-4832 163 21 2.1 2.1 NUM cana-4832 163 22 , , PUNCT cana-4832 163 23 with with ADP cana-4832 163 24 blue blue ADJ cana-4832 163 25 line line NOUN cana-4832 163 26 representing represent VERB cana-4832 163 27 the the DET cana-4832 163 28 left left ADJ cana-4832 163 29 part part NOUN cana-4832 163 30 of of ADP cana-4832 163 31 the the DET cana-4832 163 32 condition condition NOUN cana-4832 163 33 and and CCONJ cana-4832 163 34 red red ADJ cana-4832 163 35 line line NOUN cana-4832 163 36 representing represent VERB cana-4832 163 37 the the DET cana-4832 163 38 right right ADJ cana-4832 163 39 part part NOUN cana-4832 163 40 of of ADP cana-4832 163 41 the the DET cana-4832 163 42 condition condition NOUN cana-4832 163 43 . . PUNCT cana-4832 164 1 thus thus ADV cana-4832 164 2 , , PUNCT cana-4832 164 3 all all DET cana-4832 164 4 the the DET cana-4832 164 5 conditions condition NOUN cana-4832 164 6 of of ADP cana-4832 164 7 theorem theorem ADJ cana-4832 164 8 2.8 2.8 NUM cana-4832 164 9 are be AUX cana-4832 164 10 satisfied satisfied ADJ cana-4832 164 11 . . PUNCT cana-4832 165 1 so so ADV cana-4832 165 2 𝑇 𝑇 PROPN cana-4832 165 3 has have VERB cana-4832 165 4 a a DET cana-4832 165 5 unique unique ADJ cana-4832 165 6 common common ADJ cana-4832 165 7 fixed fix VERB cana-4832 165 8 point point NOUN cana-4832 165 9 , , PUNCT cana-4832 165 10 which which PRON cana-4832 165 11 is be AUX cana-4832 165 12 clearly clearly ADV cana-4832 165 13 0 0 NUM cana-4832 165 14 here here ADV cana-4832 165 15 . . PUNCT cana-4832 166 1 corollary corollary ADJ cana-4832 166 2 2.3 2.3 NUM cana-4832 166 3 . . PUNCT cana-4832 167 1 let let VERB cana-4832 167 2 ( ( PUNCT cana-4832 167 3 𝑋 𝑋 NOUN cana-4832 167 4 , , PUNCT cana-4832 167 5 𝑑 𝑑 NOUN cana-4832 167 6 ) ) PUNCT cana-4832 167 7 be be VERB cana-4832 167 8 a a DET cana-4832 167 9 complete complete ADJ cana-4832 167 10 π‘‘π‘ž π‘‘π‘ž PROPN cana-4832 167 11 𝑏-metric 𝑏-metric PROPN cana-4832 167 12 space space NOUN cana-4832 167 13 and and CCONJ cana-4832 167 14 𝑇 𝑇 PROPN cana-4832 167 15 be be AUX cana-4832 167 16 an an DET cana-4832 167 17 onto onto ADP cana-4832 167 18 self self NOUN cana-4832 167 19 - - PUNCT cana-4832 167 20 mapping mapping NOUN cana-4832 167 21 on on ADP cana-4832 167 22 𝑋 𝑋 PROPN cana-4832 167 23 such such ADJ cana-4832 167 24 that that DET cana-4832 167 25 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž NOUN cana-4832 167 26 , , PUNCT cana-4832 167 27 𝑇𝑏 𝑇𝑏 PROPN cana-4832 167 28 ) ) PUNCT cana-4832 167 29 β‰₯ β‰₯ NOUN cana-4832 167 30 π‘˜π‘‘(π‘Ž π‘˜π‘‘(π‘Ž NOUN cana-4832 167 31 , , PUNCT cana-4832 167 32 𝑏 𝑏 NOUN cana-4832 167 33 ) ) PUNCT cana-4832 167 34 for for ADP cana-4832 167 35 all all DET cana-4832 167 36 π‘Ž π‘Ž NOUN cana-4832 167 37 , , PUNCT cana-4832 167 38 𝑏 𝑏 PROPN cana-4832 167 39 ∈ ∈ PROPN cana-4832 167 40 𝑋 𝑋 NOUN cana-4832 167 41 with with ADP cana-4832 167 42 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 167 43 , , PUNCT cana-4832 167 44 𝑏 𝑏 NOUN cana-4832 167 45 ) ) PUNCT cana-4832 167 46 β‰  β‰  PROPN cana-4832 167 47 0 0 NUM cana-4832 167 48 , , PUNCT cana-4832 167 49 π‘˜ π‘˜ X cana-4832 167 50 > > X cana-4832 167 51 1 1 X cana-4832 167 52 . . PUNCT cana-4832 168 1 then then ADV cana-4832 168 2 𝑇 𝑇 PROPN cana-4832 168 3 has have VERB cana-4832 168 4 a a DET cana-4832 168 5 unique unique ADJ cana-4832 168 6 fixed fix VERB cana-4832 168 7 point point NOUN cana-4832 168 8 . . PUNCT cana-4832 169 1 proof proof NOUN cana-4832 169 2 . . PUNCT cana-4832 170 1 by by ADP cana-4832 170 2 taking take VERB cana-4832 170 3 𝛼 𝛼 PRON cana-4832 170 4 = = SYM cana-4832 170 5 𝛽2 𝛽2 PROPN cana-4832 170 6 = = SYM cana-4832 170 7 𝛾3 𝛾3 NOUN cana-4832 170 8 = = SYM cana-4832 170 9 𝛿3 𝛿3 NOUN cana-4832 170 10 = = PUNCT cana-4832 170 11 πœ†3 πœ†3 NOUN cana-4832 170 12 = = SYM cana-4832 170 13 1 1 NUM cana-4832 170 14 and and CCONJ cana-4832 170 15 𝛽1 𝛽1 NOUN cana-4832 170 16 = = SYM cana-4832 170 17 𝛾1 𝛾1 NOUN cana-4832 170 18 = = NOUN cana-4832 170 19 𝛾2 𝛾2 PROPN cana-4832 170 20 = = SYM cana-4832 170 21 𝛿1 𝛿1 NOUN cana-4832 170 22 = = SYM cana-4832 170 23 𝛿2 𝛿2 NOUN cana-4832 170 24 = = NOUN cana-4832 170 25 πœ†1 πœ†1 NOUN cana-4832 170 26 = = PUNCT cana-4832 170 27 πœ†2 πœ†2 NOUN cana-4832 170 28 = = SYM cana-4832 170 29 0 0 NUM cana-4832 170 30 in in ADP cana-4832 170 31 theorem theorem NOUN cana-4832 170 32 2.1 2.1 NUM cana-4832 170 33 , , PUNCT cana-4832 170 34 the the DET cana-4832 170 35 result result NOUN cana-4832 170 36 follows follow VERB cana-4832 170 37 easily easily ADV cana-4832 170 38 . . PUNCT cana-4832 171 1 corollary corollary ADJ cana-4832 171 2 2.4 2.4 NUM cana-4832 171 3 . . PUNCT cana-4832 172 1 let let VERB cana-4832 172 2 ( ( PUNCT cana-4832 172 3 𝑋 𝑋 NOUN cana-4832 172 4 , , PUNCT cana-4832 172 5 𝑑 𝑑 NOUN cana-4832 172 6 ) ) PUNCT cana-4832 172 7 be be VERB cana-4832 172 8 a a DET cana-4832 172 9 complete complete ADJ cana-4832 172 10 π‘‘π‘ž π‘‘π‘ž PROPN cana-4832 172 11 𝑏-metric 𝑏-metric PROPN cana-4832 172 12 space space NOUN cana-4832 172 13 and and CCONJ cana-4832 172 14 𝑇 𝑇 PROPN cana-4832 172 15 be be AUX cana-4832 172 16 an an PRON cana-4832 172 17 onto onto ADP cana-4832 172 18 self self NOUN cana-4832 172 19 - - PUNCT cana-4832 172 20 mapping mapping NOUN cana-4832 172 21 on on ADP cana-4832 172 22 𝑋 𝑋 PROPN cana-4832 172 23 such such ADJ cana-4832 172 24 that that PRON cana-4832 172 25 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž NOUN cana-4832 172 26 , , PUNCT cana-4832 172 27 𝑇𝑏 𝑇𝑏 PROPN cana-4832 172 28 ) ) PUNCT cana-4832 172 29 β‰₯ β‰₯ NOUN cana-4832 172 30 π‘˜ π‘˜ DET cana-4832 172 31 min min NOUN cana-4832 172 32 { { PUNCT cana-4832 172 33 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 172 34 , , PUNCT cana-4832 172 35 𝑏 𝑏 NOUN cana-4832 172 36 ) ) PUNCT cana-4832 172 37 ; ; PUNCT cana-4832 172 38 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž X cana-4832 172 39 , , PUNCT cana-4832 172 40 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 172 41 , , PUNCT cana-4832 172 42 𝑏 𝑏 NOUN cana-4832 172 43 ) ) PUNCT cana-4832 172 44 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 172 45 , , PUNCT cana-4832 172 46 𝑏 𝑏 NOUN cana-4832 172 47 ) ) PUNCT cana-4832 173 1 + + CCONJ cana-4832 173 2 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 173 3 , , PUNCT cana-4832 173 4 𝑏 𝑏 NOUN cana-4832 173 5 ) ) PUNCT cana-4832 173 6 ; ; PUNCT cana-4832 173 7 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž X cana-4832 173 8 , , PUNCT cana-4832 173 9 π‘Ž π‘Ž NOUN cana-4832 173 10 ) ) PUNCT cana-4832 173 11 + + CCONJ cana-4832 173 12 𝑑(𝑇𝑏 𝑑(𝑇𝑏 PROPN cana-4832 173 13 , , PUNCT cana-4832 173 14 𝑏 𝑏 NOUN cana-4832 173 15 ) ) PUNCT cana-4832 173 16 + + CCONJ cana-4832 174 1 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 174 2 , , PUNCT cana-4832 174 3 𝑏 𝑏 NOUN cana-4832 174 4 ) ) PUNCT cana-4832 174 5 ; ; PUNCT cana-4832 174 6 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž X cana-4832 174 7 , , PUNCT cana-4832 174 8 𝑏 𝑏 NOUN cana-4832 174 9 ) ) PUNCT cana-4832 174 10 + + CCONJ cana-4832 174 11 𝑑(𝑇𝑏 𝑑(𝑇𝑏 PROPN cana-4832 174 12 , , PUNCT cana-4832 174 13 π‘Ž π‘Ž NOUN cana-4832 174 14 ) ) PUNCT cana-4832 174 15 + + CCONJ cana-4832 174 16 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 174 17 , , PUNCT cana-4832 174 18 𝑏 𝑏 NOUN cana-4832 174 19 ) ) PUNCT cana-4832 174 20 ; ; PUNCT cana-4832 174 21 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž PROPN cana-4832 174 22 , , PUNCT cana-4832 174 23 𝑏)𝑑(𝑇𝑏 𝑏)𝑑(𝑇𝑏 PROPN cana-4832 174 24 , , PUNCT cana-4832 174 25 𝑏 𝑏 NOUN cana-4832 174 26 ) ) PUNCT cana-4832 174 27 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 174 28 , , PUNCT cana-4832 174 29 𝑏 𝑏 NOUN cana-4832 174 30 ) ) PUNCT cana-4832 174 31 + + CCONJ cana-4832 174 32 𝑑(𝑇𝑏 𝑑(𝑇𝑏 PROPN cana-4832 174 33 , , PUNCT cana-4832 174 34 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 174 35 , , PUNCT cana-4832 174 36 𝑏 𝑏 NOUN cana-4832 174 37 ) ) PUNCT cana-4832 174 38 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 174 39 , , PUNCT cana-4832 174 40 𝑏 𝑏 NOUN cana-4832 174 41 ) ) PUNCT cana-4832 174 42 + + CCONJ cana-4832 174 43 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 174 44 , , PUNCT cana-4832 174 45 𝑏 𝑏 NOUN cana-4832 174 46 ) ) PUNCT cana-4832 174 47 } } PUNCT cana-4832 174 48 for for ADP cana-4832 174 49 all all DET cana-4832 174 50 π‘Ž π‘Ž NOUN cana-4832 174 51 , , PUNCT cana-4832 174 52 𝑏 𝑏 PROPN cana-4832 174 53 ∈ ∈ PROPN cana-4832 174 54 𝑋 𝑋 NOUN cana-4832 174 55 with with ADP cana-4832 174 56 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 174 57 , , PUNCT cana-4832 174 58 𝑏 𝑏 NOUN cana-4832 174 59 ) ) PUNCT cana-4832 174 60 β‰  β‰  PROPN cana-4832 174 61 0 0 NUM cana-4832 174 62 , , PUNCT cana-4832 174 63 π‘˜ π‘˜ X cana-4832 174 64 > > X cana-4832 174 65 1 1 X cana-4832 174 66 . . PUNCT cana-4832 175 1 then then ADV cana-4832 175 2 𝑇 𝑇 PROPN cana-4832 175 3 has have VERB cana-4832 175 4 a a DET cana-4832 175 5 unique unique ADJ cana-4832 175 6 fixed fix VERB cana-4832 175 7 point point NOUN cana-4832 175 8 . . PUNCT cana-4832 176 1 proof proof NOUN cana-4832 176 2 . . PUNCT cana-4832 177 1 putting put VERB cana-4832 177 2 𝛼 𝛼 NOUN cana-4832 177 3 = = NOUN cana-4832 177 4 𝛽1 𝛽1 NOUN cana-4832 177 5 = = SYM cana-4832 177 6 𝛽2 𝛽2 PROPN cana-4832 177 7 = = SYM cana-4832 177 8 𝛾1 𝛾1 PROPN cana-4832 177 9 = = NOUN cana-4832 177 10 𝛾2 𝛾2 PROPN cana-4832 177 11 = = SYM cana-4832 177 12 𝛾3 𝛾3 NOUN cana-4832 177 13 = = SYM cana-4832 177 14 𝛿1 𝛿1 NOUN cana-4832 177 15 = = SYM cana-4832 177 16 𝛿2 𝛿2 NOUN cana-4832 177 17 = = SYM cana-4832 177 18 𝛿3 𝛿3 NOUN cana-4832 177 19 = = SYM cana-4832 177 20 πœ†1 πœ†1 NOUN cana-4832 177 21 = = PUNCT cana-4832 177 22 πœ†2 πœ†2 NOUN cana-4832 177 23 = = SYM cana-4832 177 24 πœ†3 πœ†3 NOUN cana-4832 177 25 = = NOUN cana-4832 177 26 1 1 NUM cana-4832 177 27 in in ADP cana-4832 177 28 theorem theorem NOUN cana-4832 177 29 2.1 2.1 NUM cana-4832 177 30 , , PUNCT cana-4832 177 31 the the DET cana-4832 177 32 result result NOUN cana-4832 177 33 follows follow VERB cana-4832 177 34 easily easily ADV cana-4832 177 35 . . PUNCT cana-4832 178 1 theorem theorem VERB cana-4832 178 2 2.5 2.5 NUM cana-4832 178 3 . . PUNCT cana-4832 179 1 let let VERB cana-4832 179 2 ( ( PUNCT cana-4832 179 3 𝑋 𝑋 NOUN cana-4832 179 4 , , PUNCT cana-4832 179 5 𝑑 𝑑 NOUN cana-4832 179 6 ) ) PUNCT cana-4832 179 7 be be VERB cana-4832 179 8 a a DET cana-4832 179 9 complete complete ADJ cana-4832 179 10 π‘‘π‘ž π‘‘π‘ž PROPN cana-4832 179 11 𝑏-metric 𝑏-metric PROPN cana-4832 179 12 space space NOUN cana-4832 179 13 and and CCONJ cana-4832 179 14 𝑇 𝑇 PROPN cana-4832 179 15 be be AUX cana-4832 179 16 an an PRON cana-4832 179 17 onto onto ADP cana-4832 179 18 self self NOUN cana-4832 179 19 - - PUNCT cana-4832 179 20 mapping mapping NOUN cana-4832 179 21 on on ADP cana-4832 179 22 𝑋 𝑋 PROPN cana-4832 179 23 such such ADJ cana-4832 179 24 that that SCONJ cana-4832 179 25 communications communication NOUN cana-4832 179 26 on on ADP cana-4832 179 27 applied apply VERB cana-4832 179 28 nonlinear nonlinear ADJ cana-4832 179 29 analysis analysis NOUN cana-4832 179 30 issn issn NOUN cana-4832 179 31 : : PUNCT cana-4832 179 32 1074 1074 NUM cana-4832 179 33 - - PUNCT cana-4832 179 34 133x 133x NUM cana-4832 179 35 vol vol VERB cana-4832 179 36 32 32 NUM cana-4832 179 37 no no NOUN cana-4832 179 38 . . PUNCT cana-4832 180 1 10s 10 NOUN cana-4832 180 2 ( ( PUNCT cana-4832 180 3 2025 2025 NUM cana-4832 180 4 ) ) PUNCT cana-4832 180 5 398 398 NUM cana-4832 180 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 180 7 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž NOUN cana-4832 180 8 , , PUNCT cana-4832 180 9 𝑇𝑏 𝑇𝑏 PROPN cana-4832 180 10 ) ) PUNCT cana-4832 180 11 β‰₯ β‰₯ NOUN cana-4832 180 12 𝛼1𝑑(π‘Ž 𝛼1𝑑(π‘Ž PROPN cana-4832 180 13 , , PUNCT cana-4832 180 14 𝑏 𝑏 NOUN cana-4832 180 15 ) ) PUNCT cana-4832 180 16 + + CCONJ cana-4832 180 17 𝛼2 𝛼2 PROPN cana-4832 180 18 [ [ PUNCT cana-4832 180 19 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž NOUN cana-4832 180 20 , , PUNCT cana-4832 180 21 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 180 22 , , PUNCT cana-4832 180 23 𝑏 𝑏 NOUN cana-4832 180 24 ) ) PUNCT cana-4832 180 25 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 180 26 , , PUNCT cana-4832 180 27 𝑏 𝑏 NOUN cana-4832 180 28 ) ) PUNCT cana-4832 180 29 + + CCONJ cana-4832 180 30 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 180 31 , , PUNCT cana-4832 180 32 𝑏 𝑏 NOUN cana-4832 180 33 ) ) PUNCT cana-4832 180 34 ] ] PUNCT cana-4832 181 1 + + CCONJ cana-4832 181 2 𝛼3[𝑑(π‘‡π‘Ž 𝛼3[𝑑(π‘‡π‘Ž NOUN cana-4832 181 3 , , PUNCT cana-4832 181 4 π‘Ž π‘Ž NOUN cana-4832 181 5 ) ) PUNCT cana-4832 181 6 + + CCONJ cana-4832 181 7 𝑑(𝑇𝑏 𝑑(𝑇𝑏 PROPN cana-4832 181 8 , , PUNCT cana-4832 181 9 𝑏 𝑏 NOUN cana-4832 181 10 ) ) PUNCT cana-4832 181 11 + + CCONJ cana-4832 181 12 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 181 13 , , PUNCT cana-4832 181 14 𝑏 𝑏 NOUN cana-4832 181 15 ) ) PUNCT cana-4832 181 16 ] ] PUNCT cana-4832 182 1 + + CCONJ cana-4832 182 2 𝛼4[𝑑(π‘‡π‘Ž 𝛼4[𝑑(π‘‡π‘Ž NOUN cana-4832 182 3 , , PUNCT cana-4832 182 4 𝑏 𝑏 NOUN cana-4832 182 5 ) ) PUNCT cana-4832 182 6 + + CCONJ cana-4832 182 7 𝑑(𝑇𝑏 𝑑(𝑇𝑏 PROPN cana-4832 182 8 , , PUNCT cana-4832 182 9 π‘Ž π‘Ž NOUN cana-4832 182 10 ) ) PUNCT cana-4832 182 11 + + CCONJ cana-4832 182 12 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 182 13 , , PUNCT cana-4832 182 14 𝑏 𝑏 NOUN cana-4832 182 15 ) ) PUNCT cana-4832 182 16 ] ] PUNCT cana-4832 182 17 + + CCONJ cana-4832 182 18 𝛼5 𝛼5 NOUN cana-4832 182 19 [ [ PUNCT cana-4832 182 20 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž NOUN cana-4832 182 21 , , PUNCT cana-4832 182 22 𝑏)𝑑(𝑇𝑏 𝑏)𝑑(𝑇𝑏 PROPN cana-4832 182 23 , , PUNCT cana-4832 182 24 𝑏 𝑏 NOUN cana-4832 182 25 ) ) PUNCT cana-4832 182 26 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 182 27 , , PUNCT cana-4832 182 28 𝑏 𝑏 NOUN cana-4832 182 29 ) ) PUNCT cana-4832 182 30 + + CCONJ cana-4832 182 31 𝑑(𝑇𝑏 𝑑(𝑇𝑏 PROPN cana-4832 182 32 , , PUNCT cana-4832 182 33 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 182 34 , , PUNCT cana-4832 182 35 𝑏 𝑏 NOUN cana-4832 182 36 ) ) PUNCT cana-4832 182 37 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 182 38 , , PUNCT cana-4832 182 39 𝑏 𝑏 NOUN cana-4832 182 40 ) ) PUNCT cana-4832 182 41 + + CCONJ cana-4832 182 42 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 182 43 , , PUNCT cana-4832 182 44 𝑏 𝑏 NOUN cana-4832 182 45 ) ) PUNCT cana-4832 182 46 ] ] PUNCT cana-4832 182 47 ( ( PUNCT cana-4832 182 48 2.8 2.8 NUM cana-4832 182 49 ) ) PUNCT cana-4832 182 50 for for ADP cana-4832 182 51 all all DET cana-4832 182 52 π‘Ž π‘Ž NOUN cana-4832 182 53 , , PUNCT cana-4832 182 54 𝑏 𝑏 PROPN cana-4832 182 55 ∈ ∈ PROPN cana-4832 182 56 𝑋 𝑋 NOUN cana-4832 182 57 with with ADP cana-4832 182 58 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 182 59 , , PUNCT cana-4832 182 60 𝑏 𝑏 NOUN cana-4832 182 61 ) ) PUNCT cana-4832 182 62 β‰  β‰  PROPN cana-4832 182 63 0 0 NUM cana-4832 182 64 , , PUNCT cana-4832 182 65 𝛼1 𝛼1 NOUN cana-4832 182 66 + + CCONJ cana-4832 182 67 𝛼2 𝛼2 ADJ cana-4832 182 68 + + CCONJ cana-4832 182 69 𝛼3 𝛼3 NOUN cana-4832 182 70 + + CCONJ cana-4832 182 71 𝛼4 𝛼4 NOUN cana-4832 182 72 + + CCONJ cana-4832 182 73 𝛼5 𝛼5 NOUN cana-4832 182 74 > > X cana-4832 182 75 1 1 NUM cana-4832 182 76 . . PUNCT cana-4832 183 1 then then ADV cana-4832 183 2 𝑇 𝑇 PROPN cana-4832 183 3 has have VERB cana-4832 183 4 a a DET cana-4832 183 5 unique unique ADJ cana-4832 183 6 fixed fix VERB cana-4832 183 7 point point NOUN cana-4832 183 8 . . PUNCT cana-4832 184 1 proof proof NOUN cana-4832 184 2 . . PUNCT cana-4832 185 1 let let VERB cana-4832 185 2 πœƒ(π‘Ž πœƒ(π‘Ž VERB cana-4832 185 3 , , PUNCT cana-4832 185 4 𝑏 𝑏 NOUN cana-4832 185 5 ) ) PUNCT cana-4832 185 6 = = SYM cana-4832 185 7 min min NOUN cana-4832 185 8 { { PUNCT cana-4832 185 9 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 185 10 , , PUNCT cana-4832 185 11 𝑏 𝑏 NOUN cana-4832 185 12 ) ) PUNCT cana-4832 185 13 ; ; PUNCT cana-4832 185 14 𝑑(π‘‡π‘Ž,π‘Ž)𝑑(𝑇𝑏,𝑏 𝑑(π‘‡π‘Ž,π‘Ž)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 185 15 ) ) PUNCT cana-4832 185 16 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 NOUN cana-4832 185 17 ) ) PUNCT cana-4832 186 1 + + CCONJ cana-4832 186 2 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 186 3 , , PUNCT cana-4832 186 4 𝑏 𝑏 NOUN cana-4832 186 5 ) ) PUNCT cana-4832 186 6 ; ; PUNCT cana-4832 186 7 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž X cana-4832 186 8 , , PUNCT cana-4832 186 9 π‘Ž π‘Ž NOUN cana-4832 186 10 ) ) PUNCT cana-4832 186 11 + + CCONJ cana-4832 186 12 𝑑(𝑇𝑏 𝑑(𝑇𝑏 PROPN cana-4832 186 13 , , PUNCT cana-4832 186 14 𝑏 𝑏 NOUN cana-4832 186 15 ) ) PUNCT cana-4832 186 16 + + CCONJ cana-4832 187 1 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 187 2 , , PUNCT cana-4832 187 3 𝑏 𝑏 NOUN cana-4832 187 4 ) ) PUNCT cana-4832 187 5 ; ; PUNCT cana-4832 187 6 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž X cana-4832 187 7 , , PUNCT cana-4832 187 8 𝑏 𝑏 NOUN cana-4832 187 9 ) ) PUNCT cana-4832 187 10 + + CCONJ cana-4832 187 11 𝑑(𝑇𝑏 𝑑(𝑇𝑏 PROPN cana-4832 187 12 , , PUNCT cana-4832 187 13 π‘Ž π‘Ž NOUN cana-4832 187 14 ) ) PUNCT cana-4832 187 15 + + CCONJ cana-4832 187 16 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 187 17 , , PUNCT cana-4832 187 18 𝑏 𝑏 NOUN cana-4832 187 19 ) ) PUNCT cana-4832 187 20 ; ; PUNCT cana-4832 187 21 𝑑(π‘‡π‘Ž,𝑏)𝑑(𝑇𝑏,𝑏 𝑑(π‘‡π‘Ž,𝑏)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 187 22 ) ) PUNCT cana-4832 187 23 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 NOUN cana-4832 187 24 ) ) PUNCT cana-4832 188 1 + + NUM cana-4832 188 2 𝑑(𝑇𝑏,π‘Ž)𝑑(𝑇𝑏,𝑏 𝑑(𝑇𝑏,π‘Ž)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 188 3 ) ) PUNCT cana-4832 188 4 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 NOUN cana-4832 188 5 ) ) PUNCT cana-4832 189 1 + + CCONJ cana-4832 189 2 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 189 3 , , PUNCT cana-4832 189 4 𝑏 𝑏 NOUN cana-4832 189 5 ) ) PUNCT cana-4832 189 6 } } PUNCT cana-4832 189 7 . . PUNCT cana-4832 190 1 ( ( PUNCT cana-4832 190 2 2.9 2.9 NUM cana-4832 190 3 ) ) PUNCT cana-4832 190 4 using use VERB cana-4832 190 5 ( ( PUNCT cana-4832 190 6 2.8 2.8 NUM cana-4832 190 7 ) ) PUNCT cana-4832 190 8 and and CCONJ cana-4832 190 9 ( ( PUNCT cana-4832 190 10 2.9 2.9 NUM cana-4832 190 11 ) ) PUNCT cana-4832 190 12 , , PUNCT cana-4832 190 13 we we PRON cana-4832 190 14 get get VERB cana-4832 190 15 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž NOUN cana-4832 190 16 , , PUNCT cana-4832 190 17 𝑇𝑏 𝑇𝑏 NOUN cana-4832 190 18 ) ) PUNCT cana-4832 190 19 β‰₯ β‰₯ NOUN cana-4832 190 20 𝛼1πœƒ(π‘Ž 𝛼1πœƒ(π‘Ž PROPN cana-4832 190 21 , , PUNCT cana-4832 190 22 𝑏 𝑏 NOUN cana-4832 190 23 ) ) PUNCT cana-4832 191 1 + + CCONJ cana-4832 191 2 𝛼2πœƒ(π‘Ž 𝛼2πœƒ(π‘Ž NOUN cana-4832 191 3 , , PUNCT cana-4832 191 4 𝑏 𝑏 NOUN cana-4832 191 5 ) ) PUNCT cana-4832 192 1 + + CCONJ cana-4832 192 2 𝛼3πœƒ(π‘Ž 𝛼3πœƒ(π‘Ž PROPN cana-4832 192 3 , , PUNCT cana-4832 192 4 𝑏 𝑏 NOUN cana-4832 192 5 ) ) PUNCT cana-4832 192 6 + + CCONJ cana-4832 192 7 𝛼4πœƒ(π‘Ž 𝛼4πœƒ(π‘Ž NOUN cana-4832 192 8 , , PUNCT cana-4832 192 9 𝑏 𝑏 NOUN cana-4832 192 10 ) ) PUNCT cana-4832 193 1 + + CCONJ cana-4832 193 2 𝛼5πœƒ(π‘Ž 𝛼5πœƒ(π‘Ž PROPN cana-4832 193 3 , , PUNCT cana-4832 193 4 𝑏 𝑏 NOUN cana-4832 193 5 ) ) PUNCT cana-4832 193 6 = = SYM cana-4832 193 7 ( ( PUNCT cana-4832 193 8 𝛼1 𝛼1 NOUN cana-4832 193 9 + + CCONJ cana-4832 193 10 𝛼2 𝛼2 ADJ cana-4832 193 11 + + CCONJ cana-4832 193 12 𝛼3 𝛼3 NOUN cana-4832 193 13 + + CCONJ cana-4832 193 14 𝛼4 𝛼4 NOUN cana-4832 193 15 + + CCONJ cana-4832 193 16 𝛼5)πœƒ(π‘Ž 𝛼5)πœƒ(π‘Ž NUM cana-4832 193 17 , , PUNCT cana-4832 193 18 𝑏 𝑏 NOUN cana-4832 193 19 ) ) PUNCT cana-4832 193 20 . . PUNCT cana-4832 194 1 then then ADV cana-4832 194 2 the the DET cana-4832 194 3 inequality inequality NOUN cana-4832 194 4 becomes become VERB cana-4832 194 5 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž NOUN cana-4832 194 6 , , PUNCT cana-4832 194 7 𝑇𝑏 𝑇𝑏 PROPN cana-4832 194 8 ) ) PUNCT cana-4832 194 9 β‰₯ β‰₯ NOUN cana-4832 194 10 π‘˜ π‘˜ DET cana-4832 194 11 πœƒ(π‘Ž πœƒ(π‘Ž NOUN cana-4832 194 12 , , PUNCT cana-4832 194 13 𝑏 𝑏 NOUN cana-4832 194 14 ) ) PUNCT cana-4832 194 15 , , PUNCT cana-4832 194 16 where where SCONJ cana-4832 194 17 π‘˜ π‘˜ X cana-4832 194 18 > > X cana-4832 194 19 1 1 NUM cana-4832 194 20 . . PUNCT cana-4832 194 21 therefore therefore ADV cana-4832 194 22 by by ADP cana-4832 194 23 corollary corollary ADJ cana-4832 194 24 2.4 2.4 NUM cana-4832 194 25 , , PUNCT cana-4832 194 26 we we PRON cana-4832 194 27 conclude conclude VERB cana-4832 194 28 the the DET cana-4832 194 29 proof proof NOUN cana-4832 194 30 . . PUNCT cana-4832 195 1 remark remark VERB cana-4832 195 2 2.6 2.6 NUM cana-4832 195 3 . . PUNCT cana-4832 196 1 theorem theorem VERB cana-4832 196 2 2.1 2.1 NUM cana-4832 196 3 and and CCONJ cana-4832 196 4 example example NOUN cana-4832 196 5 2.2 2.2 NUM cana-4832 196 6 extend extend NOUN cana-4832 196 7 and and CCONJ cana-4832 196 8 generalize generalize VERB cana-4832 196 9 theorem theorem VERB cana-4832 196 10 1.5 1.5 NUM cana-4832 196 11 to to ADP cana-4832 196 12 π‘‘π‘ž π‘‘π‘ž PROPN cana-4832 196 13 𝑏$-metric 𝑏$-metric ADJ cana-4832 196 14 spaces space NOUN cana-4832 196 15 by by ADP cana-4832 196 16 taking take VERB cana-4832 196 17 πœ†π‘– πœ†π‘– PROPN cana-4832 196 18 = = SYM cana-4832 196 19 0 0 NUM cana-4832 196 20 , , PUNCT cana-4832 196 21 𝑖 𝑖 SYM cana-4832 197 1 = = NOUN cana-4832 197 2 1,2,3 1,2,3 NUM cana-4832 197 3 in in ADP cana-4832 197 4 theorem theorem ADJ cana-4832 197 5 2.1 2.1 NUM cana-4832 197 6 . . PUNCT cana-4832 198 1 remark remark PROPN cana-4832 198 2 2.7 2.7 NUM cana-4832 198 3 . . PUNCT cana-4832 199 1 corollary corollary ADJ cana-4832 199 2 2.3 2.3 NUM cana-4832 199 3 extend extend NOUN cana-4832 199 4 and and CCONJ cana-4832 199 5 generalize generalize VERB cana-4832 199 6 the the DET cana-4832 199 7 result result NOUN cana-4832 199 8 of of ADP cana-4832 199 9 [ [ X cana-4832 199 10 24 24 NUM cana-4832 199 11 ] ] PUNCT cana-4832 199 12 in in ADP cana-4832 199 13 the the DET cana-4832 199 14 framework framework NOUN cana-4832 199 15 of of ADP cana-4832 199 16 π‘‘π‘ž π‘‘π‘ž PROPN cana-4832 199 17 𝑏-metric 𝑏-metric PROPN cana-4832 199 18 space space NOUN cana-4832 199 19 . . PUNCT cana-4832 200 1 in in ADP cana-4832 200 2 the the DET cana-4832 200 3 following following NOUN cana-4832 200 4 , , PUNCT cana-4832 200 5 we we PRON cana-4832 200 6 deduce deduce VERB cana-4832 200 7 a a DET cana-4832 200 8 common common ADJ cana-4832 200 9 fixed fix VERB cana-4832 200 10 point point NOUN cana-4832 200 11 theorem theorem VERB cana-4832 200 12 to to ADP cana-4832 200 13 a a DET cana-4832 200 14 pair pair NOUN cana-4832 200 15 of of ADP cana-4832 200 16 onto onto ADP cana-4832 200 17 expansive expansive ADJ cana-4832 200 18 type type NOUN cana-4832 200 19 self self NOUN cana-4832 200 20 - - PUNCT cana-4832 200 21 mappings mapping NOUN cana-4832 200 22 . . PUNCT cana-4832 201 1 theorem theorem NOUN cana-4832 201 2 2.8 2.8 NUM cana-4832 201 3 . . PUNCT cana-4832 202 1 let let VERB cana-4832 202 2 ( ( PUNCT cana-4832 202 3 𝑋 𝑋 NOUN cana-4832 202 4 , , PUNCT cana-4832 202 5 𝑑 𝑑 NOUN cana-4832 202 6 ) ) PUNCT cana-4832 202 7 be be VERB cana-4832 202 8 a a DET cana-4832 202 9 complete complete ADJ cana-4832 202 10 π‘‘π‘ž π‘‘π‘ž PROPN cana-4832 202 11 𝑏-metric 𝑏-metric PROPN cana-4832 202 12 space space NOUN cana-4832 202 13 and and CCONJ cana-4832 202 14 𝑆 𝑆 PROPN cana-4832 202 15 , , PUNCT cana-4832 202 16 𝑇 𝑇 PROPN cana-4832 202 17 be be VERB cana-4832 202 18 two two NUM cana-4832 202 19 onto onto ADP cana-4832 202 20 self self NOUN cana-4832 202 21 - - PUNCT cana-4832 202 22 mapping mapping NOUN cana-4832 202 23 on on ADP cana-4832 202 24 𝑋 𝑋 PROPN cana-4832 202 25 such such ADJ cana-4832 202 26 that that SCONJ cana-4832 202 27 𝑑(π‘†π‘Ž 𝑑(π‘†π‘Ž NOUN cana-4832 202 28 , , PUNCT cana-4832 202 29 𝑇𝑏 𝑇𝑏 NOUN cana-4832 202 30 ) ) PUNCT cana-4832 202 31 β‰₯ β‰₯ NOUN cana-4832 202 32 π‘˜ π‘˜ DET cana-4832 202 33 min min PROPN cana-4832 202 34 { { PUNCT cana-4832 202 35 𝛼 𝛼 NOUN cana-4832 202 36 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 202 37 , , PUNCT cana-4832 202 38 𝑏 𝑏 NOUN cana-4832 202 39 ) ) PUNCT cana-4832 202 40 ; ; PUNCT cana-4832 202 41 𝛽1 𝛽1 NOUN cana-4832 202 42 𝑑(π‘†π‘Ž 𝑑(π‘†π‘Ž NOUN cana-4832 202 43 , , PUNCT cana-4832 202 44 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 202 45 , , PUNCT cana-4832 202 46 𝑏 𝑏 NOUN cana-4832 202 47 ) ) PUNCT cana-4832 202 48 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 202 49 , , PUNCT cana-4832 202 50 𝑏 𝑏 NOUN cana-4832 202 51 ) ) PUNCT cana-4832 203 1 + + CCONJ cana-4832 203 2 𝛽2𝑑(π‘Ž 𝛽2𝑑(π‘Ž PROPN cana-4832 203 3 , , PUNCT cana-4832 203 4 𝑏 𝑏 NOUN cana-4832 203 5 ) ) PUNCT cana-4832 203 6 ; ; PUNCT cana-4832 203 7 𝛾1𝑑(π‘†π‘Ž 𝛾1𝑑(π‘†π‘Ž VERB cana-4832 203 8 , , PUNCT cana-4832 203 9 π‘Ž π‘Ž NOUN cana-4832 203 10 ) ) PUNCT cana-4832 203 11 + + NUM cana-4832 203 12 𝛾2 𝛾2 NOUN cana-4832 203 13 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 203 14 , , PUNCT cana-4832 203 15 𝑏 𝑏 NOUN cana-4832 203 16 ) ) PUNCT cana-4832 203 17 + + CCONJ cana-4832 203 18 𝛾3 𝛾3 ADJ cana-4832 203 19 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 203 20 , , PUNCT cana-4832 203 21 𝑏 𝑏 NOUN cana-4832 203 22 ) ) PUNCT cana-4832 203 23 ; ; PUNCT cana-4832 203 24 𝛿1𝑑(π‘†π‘Ž 𝛿1𝑑(π‘†π‘Ž X cana-4832 203 25 , , PUNCT cana-4832 203 26 𝑏 𝑏 NOUN cana-4832 203 27 ) ) PUNCT cana-4832 203 28 + + CCONJ cana-4832 203 29 𝛿2𝑑(𝑇𝑏 𝛿2𝑑(𝑇𝑏 PROPN cana-4832 203 30 , , PUNCT cana-4832 203 31 π‘Ž π‘Ž NOUN cana-4832 203 32 ) ) PUNCT cana-4832 203 33 + + CCONJ cana-4832 203 34 𝛿3𝑑(π‘Ž 𝛿3𝑑(π‘Ž PROPN cana-4832 203 35 , , PUNCT cana-4832 203 36 𝑏 𝑏 NOUN cana-4832 203 37 ) ) PUNCT cana-4832 203 38 ; ; PUNCT cana-4832 203 39 πœ†1 πœ†1 VERB cana-4832 203 40 𝑑(π‘†π‘Ž 𝑑(π‘†π‘Ž ADV cana-4832 203 41 , , PUNCT cana-4832 203 42 𝑏)𝑑(𝑇𝑏 𝑏)𝑑(𝑇𝑏 PROPN cana-4832 203 43 , , PUNCT cana-4832 203 44 𝑏 𝑏 NOUN cana-4832 203 45 ) ) PUNCT cana-4832 203 46 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 203 47 , , PUNCT cana-4832 203 48 𝑏 𝑏 NOUN cana-4832 203 49 ) ) PUNCT cana-4832 203 50 + + NUM cana-4832 203 51 πœ†2 πœ†2 PROPN cana-4832 203 52 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 203 53 , , PUNCT cana-4832 203 54 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 203 55 , , PUNCT cana-4832 203 56 𝑏 𝑏 NOUN cana-4832 203 57 ) ) PUNCT cana-4832 203 58 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 203 59 , , PUNCT cana-4832 203 60 𝑏 𝑏 NOUN cana-4832 203 61 ) ) PUNCT cana-4832 203 62 + + CCONJ cana-4832 203 63 πœ†3𝑑(π‘Ž πœ†3𝑑(π‘Ž PROPN cana-4832 203 64 , , PUNCT cana-4832 203 65 𝑏 𝑏 NOUN cana-4832 203 66 ) ) PUNCT cana-4832 203 67 } } PUNCT cana-4832 203 68 ( ( PUNCT cana-4832 203 69 2.10 2.10 NUM cana-4832 203 70 ) ) PUNCT cana-4832 203 71 and and CCONJ cana-4832 203 72 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž PROPN cana-4832 203 73 , , PUNCT cana-4832 203 74 𝑆𝑏 𝑆𝑏 PROPN cana-4832 203 75 ) ) PUNCT cana-4832 203 76 β‰₯ β‰₯ NOUN cana-4832 203 77 π‘˜ π‘˜ DET cana-4832 203 78 min min PROPN cana-4832 203 79 { { PUNCT cana-4832 203 80 𝛼 𝛼 NOUN cana-4832 203 81 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 203 82 , , PUNCT cana-4832 203 83 𝑏 𝑏 NOUN cana-4832 203 84 ) ) PUNCT cana-4832 203 85 ; ; PUNCT cana-4832 203 86 𝛽1 𝛽1 NOUN cana-4832 203 87 𝑑(π‘†π‘Ž 𝑑(π‘†π‘Ž NOUN cana-4832 203 88 , , PUNCT cana-4832 203 89 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 203 90 , , PUNCT cana-4832 203 91 𝑏 𝑏 NOUN cana-4832 203 92 ) ) PUNCT cana-4832 203 93 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 203 94 , , PUNCT cana-4832 203 95 𝑏 𝑏 NOUN cana-4832 203 96 ) ) PUNCT cana-4832 203 97 + + CCONJ cana-4832 204 1 𝛽2𝑑(π‘Ž 𝛽2𝑑(π‘Ž PROPN cana-4832 204 2 , , PUNCT cana-4832 204 3 𝑏 𝑏 NOUN cana-4832 204 4 ) ) PUNCT cana-4832 204 5 ; ; PUNCT cana-4832 204 6 𝛾1𝑑(π‘†π‘Ž 𝛾1𝑑(π‘†π‘Ž VERB cana-4832 204 7 , , PUNCT cana-4832 204 8 π‘Ž π‘Ž NOUN cana-4832 204 9 ) ) PUNCT cana-4832 204 10 + + NUM cana-4832 204 11 𝛾2 𝛾2 NOUN cana-4832 204 12 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 204 13 , , PUNCT cana-4832 204 14 𝑏 𝑏 NOUN cana-4832 204 15 ) ) PUNCT cana-4832 204 16 + + CCONJ cana-4832 204 17 𝛾3 𝛾3 ADJ cana-4832 204 18 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 204 19 , , PUNCT cana-4832 204 20 𝑏 𝑏 NOUN cana-4832 204 21 ) ) PUNCT cana-4832 204 22 ; ; PUNCT cana-4832 204 23 𝛿1𝑑(π‘†π‘Ž 𝛿1𝑑(π‘†π‘Ž X cana-4832 204 24 , , PUNCT cana-4832 204 25 𝑏 𝑏 NOUN cana-4832 204 26 ) ) PUNCT cana-4832 204 27 + + CCONJ cana-4832 204 28 𝛿2𝑑(𝑇𝑏 𝛿2𝑑(𝑇𝑏 PROPN cana-4832 204 29 , , PUNCT cana-4832 204 30 π‘Ž π‘Ž NOUN cana-4832 204 31 ) ) PUNCT cana-4832 204 32 + + CCONJ cana-4832 204 33 𝛿3𝑑(π‘Ž 𝛿3𝑑(π‘Ž PROPN cana-4832 204 34 , , PUNCT cana-4832 204 35 𝑏 𝑏 NOUN cana-4832 204 36 ) ) PUNCT cana-4832 204 37 ; ; PUNCT cana-4832 204 38 πœ†1 πœ†1 VERB cana-4832 204 39 𝑑(π‘†π‘Ž 𝑑(π‘†π‘Ž ADV cana-4832 204 40 , , PUNCT cana-4832 204 41 𝑏)𝑑(𝑇𝑏 𝑏)𝑑(𝑇𝑏 PROPN cana-4832 204 42 , , PUNCT cana-4832 204 43 𝑏 𝑏 NOUN cana-4832 204 44 ) ) PUNCT cana-4832 204 45 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 204 46 , , PUNCT cana-4832 204 47 𝑏 𝑏 NOUN cana-4832 204 48 ) ) PUNCT cana-4832 204 49 + + NUM cana-4832 204 50 πœ†2 πœ†2 PROPN cana-4832 204 51 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 204 52 , , PUNCT cana-4832 204 53 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 204 54 , , PUNCT cana-4832 204 55 𝑏 𝑏 NOUN cana-4832 204 56 ) ) PUNCT cana-4832 204 57 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 204 58 , , PUNCT cana-4832 204 59 𝑏 𝑏 NOUN cana-4832 204 60 ) ) PUNCT cana-4832 204 61 + + CCONJ cana-4832 204 62 πœ†3𝑑(π‘Ž πœ†3𝑑(π‘Ž PROPN cana-4832 204 63 , , PUNCT cana-4832 204 64 𝑏 𝑏 NOUN cana-4832 204 65 ) ) PUNCT cana-4832 204 66 } } PUNCT cana-4832 204 67 ( ( PUNCT cana-4832 204 68 2.11 2.11 NUM cana-4832 204 69 ) ) PUNCT cana-4832 204 70 communications communication NOUN cana-4832 204 71 on on ADP cana-4832 204 72 applied apply VERB cana-4832 204 73 nonlinear nonlinear ADJ cana-4832 204 74 analysis analysis NOUN cana-4832 204 75 issn issn NOUN cana-4832 204 76 : : PUNCT cana-4832 204 77 1074 1074 NUM cana-4832 204 78 - - PUNCT cana-4832 204 79 133x 133x NUM cana-4832 204 80 vol vol VERB cana-4832 204 81 32 32 NUM cana-4832 204 82 no no NOUN cana-4832 204 83 . . PUNCT cana-4832 205 1 10s 10 NOUN cana-4832 205 2 ( ( PUNCT cana-4832 205 3 2025 2025 NUM cana-4832 205 4 ) ) PUNCT cana-4832 205 5 399 399 NUM cana-4832 205 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 205 7 for for ADP cana-4832 205 8 all all DET cana-4832 205 9 π‘Ž π‘Ž NOUN cana-4832 205 10 , , PUNCT cana-4832 205 11 𝑏 𝑏 PROPN cana-4832 205 12 ∈ ∈ PROPN cana-4832 205 13 𝑋 𝑋 NOUN cana-4832 205 14 with with ADP cana-4832 205 15 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 205 16 , , PUNCT cana-4832 205 17 𝑏 𝑏 NOUN cana-4832 205 18 ) ) PUNCT cana-4832 205 19 β‰  β‰  PROPN cana-4832 205 20 0 0 NUM cana-4832 205 21 , , PUNCT cana-4832 205 22 π‘˜ π‘˜ X cana-4832 205 23 > > X cana-4832 205 24 1 1 NUM cana-4832 205 25 , , PUNCT cana-4832 205 26 nonnegative nonnegative ADJ cana-4832 205 27 real real ADJ cana-4832 205 28 numbers number NOUN cana-4832 205 29 𝛼 𝛼 ADJ cana-4832 205 30 , , PUNCT cana-4832 205 31 𝛽𝑖 𝛽𝑖 VERB cana-4832 205 32 , , PUNCT cana-4832 205 33 𝛾𝑗 𝛾𝑗 INTJ cana-4832 205 34 , , PUNCT cana-4832 205 35 𝛿𝑗 𝛿𝑗 PROPN cana-4832 205 36 , , PUNCT cana-4832 205 37 πœ†π‘— πœ†π‘— X cana-4832 205 38 for for ADP cana-4832 205 39 𝑖 𝑖 PRON cana-4832 205 40 = = SYM cana-4832 205 41 1 1 NUM cana-4832 205 42 , , PUNCT cana-4832 205 43 2 2 NUM cana-4832 205 44 ; ; PUNCT cana-4832 205 45 𝑗 𝑗 NOUN cana-4832 205 46 = = SYM cana-4832 205 47 1 1 NUM cana-4832 205 48 , , PUNCT cana-4832 205 49 2 2 NUM cana-4832 205 50 , , PUNCT cana-4832 205 51 3 3 NUM cana-4832 205 52 and and CCONJ cana-4832 205 53 1 1 NUM cana-4832 205 54 π‘˜ π‘˜ X cana-4832 205 55 = = SYM cana-4832 205 56 π‘šπ‘–π‘›{𝛼 π‘šπ‘–π‘›{𝛼 PROPN cana-4832 205 57 , , PUNCT cana-4832 205 58 𝛽2 𝛽2 NOUN cana-4832 205 59 , , PUNCT cana-4832 205 60 𝛾2 𝛾2 VERB cana-4832 205 61 + + CCONJ cana-4832 205 62 𝛾3 𝛾3 NOUN cana-4832 205 63 , , PUNCT cana-4832 205 64 𝛿3 𝛿3 NOUN cana-4832 205 65 , , PUNCT cana-4832 205 66 πœ†3 πœ†3 NOUN cana-4832 205 67 } } PUNCT cana-4832 205 68 . . PUNCT cana-4832 206 1 then then ADV cana-4832 206 2 𝑆 𝑆 PROPN cana-4832 206 3 and and CCONJ cana-4832 206 4 𝑇 𝑇 PROPN cana-4832 206 5 have have VERB cana-4832 206 6 a a DET cana-4832 206 7 unique unique ADJ cana-4832 206 8 common common ADJ cana-4832 206 9 fixed fix VERB cana-4832 206 10 point point NOUN cana-4832 206 11 . . PUNCT cana-4832 207 1 proof proof NOUN cana-4832 207 2 . . PUNCT cana-4832 208 1 let let VERB cana-4832 208 2 us we PRON cana-4832 208 3 take take VERB cana-4832 208 4 πœƒ(π‘Ž πœƒ(π‘Ž NOUN cana-4832 208 5 , , PUNCT cana-4832 208 6 𝑏 𝑏 NOUN cana-4832 208 7 ) ) PUNCT cana-4832 208 8 = = SYM cana-4832 208 9 min min NOUN cana-4832 208 10 { { PUNCT cana-4832 208 11 𝛼 𝛼 NOUN cana-4832 208 12 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 208 13 , , PUNCT cana-4832 208 14 𝑏 𝑏 NOUN cana-4832 208 15 ) ) PUNCT cana-4832 208 16 ; ; PUNCT cana-4832 208 17 𝛽1 𝛽1 NOUN cana-4832 208 18 𝑑(π‘†π‘Ž,π‘Ž)𝑑(𝑇𝑏,𝑏 𝑑(π‘†π‘Ž,π‘Ž)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 208 19 ) ) PUNCT cana-4832 208 20 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 NOUN cana-4832 208 21 ) ) PUNCT cana-4832 209 1 + + CCONJ cana-4832 209 2 𝛽2𝑑(π‘Ž 𝛽2𝑑(π‘Ž PROPN cana-4832 209 3 , , PUNCT cana-4832 209 4 𝑏 𝑏 NOUN cana-4832 209 5 ) ) PUNCT cana-4832 209 6 ; ; PUNCT cana-4832 209 7 𝛾1𝑑(π‘†π‘Ž 𝛾1𝑑(π‘†π‘Ž VERB cana-4832 209 8 , , PUNCT cana-4832 209 9 π‘Ž π‘Ž NOUN cana-4832 209 10 ) ) PUNCT cana-4832 209 11 + + NUM cana-4832 209 12 𝛾2 𝛾2 NOUN cana-4832 209 13 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 209 14 , , PUNCT cana-4832 209 15 𝑏 𝑏 NOUN cana-4832 209 16 ) ) PUNCT cana-4832 209 17 + + CCONJ cana-4832 209 18 𝛾3 𝛾3 ADJ cana-4832 209 19 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 209 20 , , PUNCT cana-4832 209 21 𝑏 𝑏 NOUN cana-4832 209 22 ) ) PUNCT cana-4832 209 23 ; ; PUNCT cana-4832 209 24 𝛿1𝑑(π‘†π‘Ž 𝛿1𝑑(π‘†π‘Ž X cana-4832 209 25 , , PUNCT cana-4832 209 26 𝑏 𝑏 NOUN cana-4832 209 27 ) ) PUNCT cana-4832 209 28 + + CCONJ cana-4832 209 29 𝛿2𝑑(𝑇𝑏 𝛿2𝑑(𝑇𝑏 PROPN cana-4832 209 30 , , PUNCT cana-4832 209 31 π‘Ž π‘Ž NOUN cana-4832 209 32 ) ) PUNCT cana-4832 209 33 + + CCONJ cana-4832 209 34 𝛿3𝑑(π‘Ž 𝛿3𝑑(π‘Ž PROPN cana-4832 209 35 , , PUNCT cana-4832 209 36 𝑏 𝑏 NOUN cana-4832 209 37 ) ) PUNCT cana-4832 209 38 ; ; PUNCT cana-4832 209 39 πœ†1 πœ†1 PROPN cana-4832 209 40 𝑑(π‘†π‘Ž,𝑏)𝑑(𝑇𝑏,𝑏 𝑑(π‘†π‘Ž,𝑏)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 209 41 ) ) PUNCT cana-4832 209 42 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 NOUN cana-4832 209 43 ) ) PUNCT cana-4832 210 1 + + NUM cana-4832 210 2 πœ†2 πœ†2 NOUN cana-4832 210 3 𝑑(𝑇𝑏,π‘Ž)𝑑(𝑇𝑏,𝑏 𝑑(𝑇𝑏,π‘Ž)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 210 4 ) ) PUNCT cana-4832 210 5 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 VERB cana-4832 210 6 ) ) PUNCT cana-4832 211 1 + + CCONJ cana-4832 211 2 πœ†3𝑑(π‘Ž πœ†3𝑑(π‘Ž PROPN cana-4832 211 3 , , PUNCT cana-4832 211 4 𝑏 𝑏 NOUN cana-4832 211 5 ) ) PUNCT cana-4832 211 6 } } PUNCT cana-4832 211 7 . . PUNCT cana-4832 212 1 for for ADP cana-4832 212 2 π‘Ž0 π‘Ž0 PROPN cana-4832 212 3 ∈ ∈ PROPN cana-4832 212 4 𝑋 𝑋 PROPN cana-4832 212 5 , , PUNCT cana-4832 212 6 since since SCONJ cana-4832 212 7 𝑆 𝑆 PROPN cana-4832 212 8 , , PUNCT cana-4832 212 9 𝑇 𝑇 PROPN cana-4832 212 10 are be AUX cana-4832 212 11 onto onto ADP cana-4832 212 12 , , PUNCT cana-4832 212 13 there there PRON cana-4832 212 14 exist exist VERB cana-4832 212 15 π‘Ž0 π‘Ž0 PROPN cana-4832 212 16 , , PUNCT cana-4832 212 17 π‘Ž1 π‘Ž1 PROPN cana-4832 212 18 ∈ ∈ PROPN cana-4832 212 19 𝑋 𝑋 NOUN cana-4832 212 20 such such ADJ cana-4832 212 21 that that DET cana-4832 212 22 π‘Ž0 π‘Ž0 PROPN cana-4832 212 23 = = SYM cana-4832 212 24 π‘†π‘Ž1 π‘†π‘Ž1 NOUN cana-4832 212 25 , , PUNCT cana-4832 212 26 , , PUNCT cana-4832 212 27 π‘Ž1 π‘Ž1 NOUN cana-4832 212 28 = = PUNCT cana-4832 212 29 π‘‡π‘Ž2 π‘‡π‘Ž2 NOUN cana-4832 212 30 . . PUNCT cana-4832 213 1 continuing continue VERB cana-4832 213 2 this this DET cana-4832 213 3 process process NOUN cana-4832 213 4 , , PUNCT cana-4832 213 5 we we PRON cana-4832 213 6 define define VERB cana-4832 213 7 a a DET cana-4832 213 8 sequence sequence NOUN cana-4832 213 9 { { PUNCT cana-4832 213 10 π‘Žπ‘› π‘Žπ‘› NOUN cana-4832 213 11 } } PUNCT cana-4832 213 12 by by ADP cana-4832 213 13 π‘†π‘Ž2π‘›βˆ’1 π‘†π‘Ž2π‘›βˆ’1 X cana-4832 213 14 = = PUNCT cana-4832 213 15 π‘Ž2π‘›βˆ’2 π‘Ž2π‘›βˆ’2 NOUN cana-4832 213 16 π‘‡π‘Ž2𝑛 π‘‡π‘Ž2𝑛 ADP cana-4832 214 1 = = SYM cana-4832 214 2 π‘Ž2π‘›βˆ’1 π‘Ž2π‘›βˆ’1 PROPN cana-4832 214 3 , , PUNCT cana-4832 214 4 for for ADP cana-4832 214 5 all all DET cana-4832 214 6 𝑛 𝑛 DET cana-4832 214 7 ∈ ∈ PROPN cana-4832 214 8 β„•. β„•. PROPN cana-4832 214 9 ( ( PUNCT cana-4832 214 10 2.12 2.12 NUM cana-4832 214 11 ) ) PUNCT cana-4832 214 12 the the DET cana-4832 214 13 following follow VERB cana-4832 214 14 cases case NOUN cana-4832 214 15 will will AUX cana-4832 214 16 arise arise VERB cana-4832 214 17 . . PUNCT cana-4832 215 1 case case NOUN cana-4832 215 2 ( ( PUNCT cana-4832 215 3 i i NOUN cana-4832 215 4 ) ) PUNCT cana-4832 215 5 . . PUNCT cana-4832 216 1 if if SCONJ cana-4832 216 2 πœƒ(π‘Ž πœƒ(π‘Ž NOUN cana-4832 216 3 , , PUNCT cana-4832 216 4 𝑏 𝑏 NOUN cana-4832 216 5 ) ) PUNCT cana-4832 216 6 = = SYM cana-4832 216 7 𝛼 𝛼 PRON cana-4832 216 8 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 216 9 , , PUNCT cana-4832 216 10 𝑏 𝑏 NOUN cana-4832 216 11 ) ) PUNCT cana-4832 216 12 , , PUNCT cana-4832 216 13 then then ADV cana-4832 216 14 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž PROPN cana-4832 216 15 , , PUNCT cana-4832 216 16 𝑆𝑏 𝑆𝑏 PROPN cana-4832 216 17 ) ) PUNCT cana-4832 216 18 β‰₯ β‰₯ NOUN cana-4832 216 19 π‘˜ π‘˜ PRON cana-4832 216 20 𝛼 𝛼 X cana-4832 216 21 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 216 22 , , PUNCT cana-4832 216 23 𝑏 𝑏 NOUN cana-4832 216 24 ) ) PUNCT cana-4832 216 25 , , PUNCT cana-4832 216 26 for for ADP cana-4832 216 27 all all DET cana-4832 216 28 π‘Ž π‘Ž NOUN cana-4832 216 29 , , PUNCT cana-4832 216 30 𝑏 𝑏 PROPN cana-4832 216 31 ∈ ∈ PROPN cana-4832 216 32 𝑋 𝑋 NOUN cana-4832 216 33 . . PUNCT cana-4832 217 1 ( ( PUNCT cana-4832 217 2 2.13 2.13 NUM cana-4832 217 3 ) ) PUNCT cana-4832 217 4 now now ADV cana-4832 217 5 , , PUNCT cana-4832 217 6 using use VERB cana-4832 217 7 ( ( PUNCT cana-4832 217 8 2.12 2.12 NUM cana-4832 217 9 ) ) PUNCT cana-4832 217 10 and and CCONJ cana-4832 217 11 ( ( PUNCT cana-4832 217 12 2.13 2.13 NUM cana-4832 217 13 ) ) PUNCT cana-4832 217 14 , , PUNCT cana-4832 217 15 we we PRON cana-4832 217 16 get get AUX cana-4832 217 17 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 217 18 , , PUNCT cana-4832 217 19 π‘Ž2𝑛+1 π‘Ž2𝑛+1 ADJ cana-4832 217 20 ) ) PUNCT cana-4832 217 21 = = SYM cana-4832 217 22 𝑑(π‘†π‘Ž2𝑛+1 𝑑(π‘†π‘Ž2𝑛+1 NOUN cana-4832 217 23 , , PUNCT cana-4832 217 24 π‘‡π‘Ž2𝑛+2 π‘‡π‘Ž2𝑛+2 NUM cana-4832 217 25 ) ) PUNCT cana-4832 217 26 β‰₯ β‰₯ NOUN cana-4832 217 27 π‘˜ π‘˜ DET cana-4832 217 28 𝛼 𝛼 NOUN cana-4832 217 29 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 217 30 , , PUNCT cana-4832 217 31 π‘Ž2𝑛+2 π‘Ž2𝑛+2 PROPN cana-4832 217 32 ) ) PUNCT cana-4832 217 33 i. i. PROPN cana-4832 217 34 e. e. PROPN cana-4832 217 35 , , PUNCT cana-4832 217 36 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 217 37 , , PUNCT cana-4832 217 38 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 217 39 ) ) PUNCT cana-4832 217 40 ≀ ≀ NOUN cana-4832 218 1 1 1 NUM cana-4832 218 2 π‘˜ π‘˜ PRON cana-4832 218 3 𝛼 𝛼 PRON cana-4832 218 4 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 218 5 , , PUNCT cana-4832 218 6 π‘Ž2𝑛+1 π‘Ž2𝑛+1 PROPN cana-4832 218 7 ) ) PUNCT cana-4832 218 8 . . PUNCT cana-4832 219 1 let let VERB cana-4832 219 2 𝜏 𝜏 NOUN cana-4832 219 3 = = SYM cana-4832 219 4 1 1 NUM cana-4832 219 5 π‘˜ π‘˜ PRON cana-4832 219 6 𝛼 𝛼 X cana-4832 219 7 < < X cana-4832 219 8 1 1 NUM cana-4832 219 9 . . PUNCT cana-4832 220 1 then then ADV cana-4832 220 2 from from ADP cana-4832 220 3 the the DET cana-4832 220 4 above above ADJ cana-4832 220 5 inequality inequality NOUN cana-4832 220 6 , , PUNCT cana-4832 220 7 we we PRON cana-4832 220 8 have have VERB cana-4832 220 9 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 220 10 , , PUNCT cana-4832 220 11 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 220 12 ) ) PUNCT cana-4832 220 13 ≀ ≀ NOUN cana-4832 220 14 𝜏 𝜏 PRON cana-4832 220 15 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 220 16 , , PUNCT cana-4832 220 17 π‘Ž2𝑛+1 π‘Ž2𝑛+1 PROPN cana-4832 220 18 ) ) PUNCT cana-4832 220 19 . . PUNCT cana-4832 221 1 also also ADV cana-4832 221 2 , , PUNCT cana-4832 221 3 from from ADP cana-4832 221 4 ( ( PUNCT cana-4832 221 5 2.11 2.11 NUM cana-4832 221 6 ) ) PUNCT cana-4832 221 7 , , PUNCT cana-4832 221 8 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 221 9 , , PUNCT cana-4832 221 10 π‘Ž2𝑛+1 π‘Ž2𝑛+1 ADJ cana-4832 221 11 ) ) PUNCT cana-4832 221 12 ≀ ≀ NOUN cana-4832 221 13 𝜏 𝜏 DET cana-4832 221 14 𝑑(π‘Ž2π‘›βˆ’1 𝑑(π‘Ž2π‘›βˆ’1 NOUN cana-4832 221 15 , , PUNCT cana-4832 221 16 π‘Ž2𝑛 π‘Ž2𝑛 PROPN cana-4832 221 17 ) ) PUNCT cana-4832 221 18 . . PUNCT cana-4832 222 1 so so ADV cana-4832 222 2 , , PUNCT cana-4832 222 3 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 222 4 , , PUNCT cana-4832 222 5 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 222 6 ) ) PUNCT cana-4832 222 7 ≀ ≀ NOUN cana-4832 222 8 𝜏 𝜏 DET cana-4832 222 9 2 2 NUM cana-4832 222 10 𝑑(π‘Ž2π‘›βˆ’1 𝑑(π‘Ž2π‘›βˆ’1 NOUN cana-4832 222 11 , , PUNCT cana-4832 222 12 π‘Ž2𝑛 π‘Ž2𝑛 PROPN cana-4832 222 13 ) ) PUNCT cana-4832 222 14 . . PUNCT cana-4832 223 1 from from ADP cana-4832 223 2 this this PRON cana-4832 223 3 , , PUNCT cana-4832 223 4 we we PRON cana-4832 223 5 get get VERB cana-4832 223 6 𝑑(π‘Žπ‘› 𝑑(π‘Žπ‘› NUM cana-4832 223 7 , , PUNCT cana-4832 223 8 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-4832 223 9 ) ) PUNCT cana-4832 223 10 ≀ ≀ NOUN cana-4832 223 11 πœπ‘› πœπ‘› ADP cana-4832 223 12 𝑑(π‘Ž0 𝑑(π‘Ž0 ADJ cana-4832 223 13 , , PUNCT cana-4832 223 14 π‘Ž1 π‘Ž1 PROPN cana-4832 223 15 ) ) PUNCT cana-4832 223 16 . . PUNCT cana-4832 224 1 for for ADP cana-4832 224 2 𝑗 𝑗 PROPN cana-4832 224 3 > > X cana-4832 224 4 𝑖 𝑖 PROPN cana-4832 224 5 , , PUNCT cana-4832 224 6 𝑑(π‘Žπ‘– 𝑑(π‘Žπ‘– PROPN cana-4832 224 7 , , PUNCT cana-4832 224 8 π‘Žπ‘— π‘Žπ‘— NOUN cana-4832 224 9 ) ) PUNCT cana-4832 224 10 ≀ ≀ NOUN cana-4832 224 11 𝑠 𝑠 X cana-4832 224 12 𝑑(π‘Žπ‘– 𝑑(π‘Žπ‘– PROPN cana-4832 224 13 , , PUNCT cana-4832 224 14 π‘Žπ‘–+1 π‘Žπ‘–+1 NUM cana-4832 224 15 ) ) PUNCT cana-4832 224 16 + + CCONJ cana-4832 224 17 𝑠 𝑠 PROPN cana-4832 224 18 2 2 NUM cana-4832 224 19 𝑑(π‘Žπ‘–+1 𝑑(π‘Žπ‘–+1 NOUN cana-4832 224 20 , , PUNCT cana-4832 224 21 π‘Žπ‘–+2 π‘Žπ‘–+2 NUM cana-4832 224 22 ) ) PUNCT cana-4832 224 23 + + NUM cana-4832 224 24 β‹―+ β‹―+ NOUN cana-4832 224 25 π‘ π‘—βˆ’π‘– π‘ π‘—βˆ’π‘– ADJ cana-4832 224 26 𝑑(π‘Žπ‘—βˆ’1 𝑑(π‘Žπ‘—βˆ’1 NOUN cana-4832 224 27 , , PUNCT cana-4832 224 28 π‘Žπ‘— π‘Žπ‘— NOUN cana-4832 224 29 ) ) PUNCT cana-4832 224 30 ≀ ≀ NOUN cana-4832 224 31 𝑠 𝑠 ADP cana-4832 224 32 πœπ‘–π‘‘(π‘Ž0 πœπ‘–π‘‘(π‘Ž0 PROPN cana-4832 224 33 , , PUNCT cana-4832 224 34 π‘Ž1 π‘Ž1 NOUN cana-4832 224 35 ) ) PUNCT cana-4832 225 1 + + CCONJ cana-4832 225 2 𝑠 𝑠 NUM cana-4832 225 3 2πœπ‘–+1𝑑(π‘Ž0 2πœπ‘–+1𝑑(π‘Ž0 NUM cana-4832 225 4 , , PUNCT cana-4832 225 5 π‘Ž1 π‘Ž1 NOUN cana-4832 225 6 ) ) PUNCT cana-4832 226 1 + + NUM cana-4832 226 2 β‹―+ β‹―+ NOUN cana-4832 226 3 𝑠 𝑠 NOUN cana-4832 226 4 π‘—βˆ’π‘–πœπ‘—βˆ’1𝑑(π‘Ž0 π‘—βˆ’π‘–πœπ‘—βˆ’1𝑑(π‘Ž0 ADJ cana-4832 226 5 , , PUNCT cana-4832 226 6 π‘Ž1 π‘Ž1 NOUN cana-4832 226 7 ) ) PUNCT cana-4832 226 8 ≀ ≀ NOUN cana-4832 227 1 [ [ X cana-4832 227 2 𝑠 𝑠 X cana-4832 227 3 πœπ‘– πœπ‘– NOUN cana-4832 227 4 + + NOUN cana-4832 227 5 𝑠2πœπ‘–+1 𝑠2πœπ‘–+1 NOUN cana-4832 228 1 + + ADJ cana-4832 228 2 β‹―]𝑑(π‘Ž0 β‹―]𝑑(π‘Ž0 ADJ cana-4832 228 3 , , PUNCT cana-4832 228 4 π‘Ž1 π‘Ž1 NOUN cana-4832 228 5 ) ) PUNCT cana-4832 228 6 = = PUNCT cana-4832 229 1 𝑠 𝑠 PRON cana-4832 229 2 πœπ‘–[1 πœπ‘–[1 NOUN cana-4832 230 1 + + CCONJ cana-4832 230 2 π‘ πœ π‘ πœ PROPN cana-4832 230 3 + + CCONJ cana-4832 230 4 ( ( PUNCT cana-4832 230 5 π‘ πœ)2 π‘ πœ)2 PROPN cana-4832 230 6 + + PROPN cana-4832 230 7 β‹― β‹― NOUN cana-4832 230 8 ] ] SYM cana-4832 230 9 𝑑(π‘Ž0 𝑑(π‘Ž0 ADJ cana-4832 230 10 , , PUNCT cana-4832 230 11 π‘Ž1 π‘Ž1 NOUN cana-4832 230 12 ) ) PUNCT cana-4832 230 13 = = SYM cana-4832 231 1 𝑠 𝑠 X cana-4832 231 2 πœπ‘– πœπ‘– INTJ cana-4832 231 3 1 1 NUM cana-4832 231 4 βˆ’ βˆ’ NOUN cana-4832 231 5 π‘ πœ π‘ πœ INTJ cana-4832 231 6 𝑑(π‘Ž0 𝑑(π‘Ž0 ADJ cana-4832 231 7 , , PUNCT cana-4832 231 8 π‘Ž1 π‘Ž1 NOUN cana-4832 231 9 ) ) PUNCT cana-4832 231 10 β†’ β†’ SYM cana-4832 231 11 0 0 NUM cana-4832 231 12 as as ADP cana-4832 231 13 𝑖 𝑖 X cana-4832 231 14 , , PUNCT cana-4832 231 15 𝑗 𝑗 PROPN cana-4832 231 16 β†’ β†’ SYM cana-4832 231 17 ∞ ∞ NUM cana-4832 231 18 communications communication NOUN cana-4832 231 19 on on ADP cana-4832 231 20 applied apply VERB cana-4832 231 21 nonlinear nonlinear ADJ cana-4832 231 22 analysis analysis NOUN cana-4832 231 23 issn issn NOUN cana-4832 231 24 : : PUNCT cana-4832 231 25 1074 1074 NUM cana-4832 231 26 - - PUNCT cana-4832 231 27 133x 133x NUM cana-4832 231 28 vol vol VERB cana-4832 231 29 32 32 NUM cana-4832 231 30 no no NOUN cana-4832 231 31 . . PUNCT cana-4832 232 1 10s 10 NOUN cana-4832 232 2 ( ( PUNCT cana-4832 232 3 2025 2025 NUM cana-4832 232 4 ) ) PUNCT cana-4832 232 5 400 400 NUM cana-4832 232 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 232 7 therefore therefore ADV cana-4832 232 8 { { PUNCT cana-4832 232 9 π‘Žπ‘› π‘Žπ‘› NOUN cana-4832 232 10 } } PUNCT cana-4832 232 11 is be AUX cana-4832 232 12 a a DET cana-4832 232 13 𝑏-cauchy 𝑏-cauchy NOUN cana-4832 232 14 sequence sequence NOUN cana-4832 232 15 in in ADP cana-4832 232 16 𝑋. 𝑋. PROPN cana-4832 232 17 since since SCONJ cana-4832 232 18 𝑋 𝑋 PROPN cana-4832 232 19 is be AUX cana-4832 232 20 complete complete ADJ cana-4832 232 21 , , PUNCT cana-4832 232 22 there there PRON cana-4832 232 23 exists exist VERB cana-4832 232 24 𝑒 𝑒 PRON cana-4832 232 25 ∈ ∈ PROPN cana-4832 232 26 𝑋 𝑋 NOUN cana-4832 232 27 such such ADJ cana-4832 232 28 that that SCONJ cana-4832 232 29 lim lim PROPN cana-4832 232 30 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4832 232 31 π‘Žπ‘› π‘Žπ‘› PROPN cana-4832 232 32 = = SYM cana-4832 232 33 𝑒. 𝑒. NOUN cana-4832 232 34 since since ADV cana-4832 232 35 , , PUNCT cana-4832 232 36 𝑆 𝑆 PROPN cana-4832 232 37 and and CCONJ cana-4832 232 38 𝑇 𝑇 PROPN cana-4832 232 39 are be AUX cana-4832 232 40 onto onto ADP cana-4832 232 41 , , PUNCT cana-4832 232 42 we we PRON cana-4832 232 43 can can AUX cana-4832 232 44 find find VERB cana-4832 232 45 𝑝 𝑝 NOUN cana-4832 232 46 , , PUNCT cana-4832 232 47 π‘ž π‘ž PROPN cana-4832 232 48 ∈ ∈ PROPN cana-4832 232 49 𝑋 𝑋 PROPN cana-4832 232 50 such such ADJ cana-4832 232 51 that that SCONJ cana-4832 232 52 𝑆𝑝 𝑆𝑝 PROPN cana-4832 232 53 = = SYM cana-4832 232 54 π‘‡π‘ž π‘‡π‘ž PROPN cana-4832 232 55 = = PUNCT cana-4832 232 56 𝑒. 𝑒. NOUN cana-4832 232 57 now now ADV cana-4832 232 58 , , PUNCT cana-4832 232 59 for for ADP cana-4832 232 60 all all DET cana-4832 232 61 𝑛 𝑛 DET cana-4832 232 62 ∈ ∈ PROPN cana-4832 232 63 β„•. β„•. PROPN cana-4832 232 64 𝑑(𝑒 𝑑(𝑒 NOUN cana-4832 232 65 , , PUNCT cana-4832 232 66 π‘Ž2𝑛+1 π‘Ž2𝑛+1 ADJ cana-4832 232 67 ) ) PUNCT cana-4832 232 68 = = SYM cana-4832 232 69 𝑑(𝑆𝑝 𝑑(𝑆𝑝 NOUN cana-4832 232 70 , , PUNCT cana-4832 232 71 π‘‡π‘Ž2𝑛+2 π‘‡π‘Ž2𝑛+2 NUM cana-4832 232 72 ) ) PUNCT cana-4832 232 73 β‰₯ β‰₯ NOUN cana-4832 232 74 π‘˜ π‘˜ X cana-4832 232 75 𝛼 𝛼 X cana-4832 232 76 𝑑(𝑝 𝑑(𝑝 PROPN cana-4832 232 77 , , PUNCT cana-4832 232 78 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 232 79 ) ) PUNCT cana-4832 232 80 . . PUNCT cana-4832 233 1 taking take VERB cana-4832 233 2 limit limit NOUN cana-4832 233 3 superior superior ADJ cana-4832 233 4 as as ADP cana-4832 233 5 𝑛 𝑛 PROPN cana-4832 233 6 β†’ β†’ SYM cana-4832 233 7 ∞ ∞ PROPN cana-4832 233 8 , , PUNCT cana-4832 233 9 and and CCONJ cana-4832 233 10 using use VERB cana-4832 233 11 lemma lemma PROPN cana-4832 233 12 1.3 1.3 NUM cana-4832 233 13 , , PUNCT cana-4832 233 14 we we PRON cana-4832 233 15 get get VERB cana-4832 233 16 1 1 NUM cana-4832 233 17 𝑠 𝑠 PRON cana-4832 233 18 𝑑(𝑝 𝑑(𝑝 PROPN cana-4832 233 19 , , PUNCT cana-4832 233 20 𝑒 𝑒 NOUN cana-4832 233 21 ) ) PUNCT cana-4832 233 22 ≀ ≀ NOUN cana-4832 234 1 π‘˜ π‘˜ DET cana-4832 234 2 𝛼 𝛼 NOUN cana-4832 234 3 𝑠 𝑠 PROPN cana-4832 234 4 lim lim PROPN cana-4832 234 5 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4832 234 6 sup𝑑(𝑝 sup𝑑(𝑝 PROPN cana-4832 234 7 , , PUNCT cana-4832 234 8 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 234 9 ) ) PUNCT cana-4832 234 10 ≀ ≀ NOUN cana-4832 234 11 lim lim PROPN cana-4832 234 12 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4832 234 13 sup𝑑(𝑒 sup𝑑(𝑒 PROPN cana-4832 234 14 , , PUNCT cana-4832 234 15 π‘Ž2𝑛+1 π‘Ž2𝑛+1 ADJ cana-4832 234 16 ) ) PUNCT cana-4832 234 17 ≀ ≀ NOUN cana-4832 234 18 𝑠 𝑠 ADP cana-4832 234 19 𝑑(𝑒 𝑑(𝑒 PROPN cana-4832 234 20 , , PUNCT cana-4832 234 21 𝑒 𝑒 NOUN cana-4832 234 22 ) ) PUNCT cana-4832 234 23 . . PUNCT cana-4832 235 1 from from ADP cana-4832 235 2 lemma lemma PROPN cana-4832 235 3 . . PROPN cana-4832 236 1 1.4 1.4 NUM cana-4832 236 2 , , PUNCT cana-4832 236 3 we we PRON cana-4832 236 4 get get VERB cana-4832 236 5 𝑑(𝑝 𝑑(𝑝 NOUN cana-4832 236 6 , , PUNCT cana-4832 236 7 𝑒 𝑒 NOUN cana-4832 236 8 ) ) PUNCT cana-4832 236 9 = = SYM cana-4832 236 10 0 0 NUM cana-4832 236 11 and and CCONJ cana-4832 236 12 similarly similarly ADV cana-4832 236 13 𝑑(𝑒 𝑑(𝑒 ADJ cana-4832 236 14 , , PUNCT cana-4832 236 15 𝑝 𝑝 NOUN cana-4832 236 16 ) ) PUNCT cana-4832 236 17 = = SYM cana-4832 237 1 0 0 X cana-4832 237 2 . . PUNCT cana-4832 238 1 thus thus ADV cana-4832 238 2 , , PUNCT cana-4832 238 3 𝑑(𝑝 𝑑(𝑝 PROPN cana-4832 238 4 , , PUNCT cana-4832 238 5 𝑒 𝑒 NOUN cana-4832 238 6 ) ) PUNCT cana-4832 238 7 = = PUNCT cana-4832 238 8 𝑑(𝑒 𝑑(𝑒 ADJ cana-4832 238 9 , , PUNCT cana-4832 238 10 𝑝 𝑝 NOUN cana-4832 238 11 ) ) PUNCT cana-4832 238 12 = = SYM cana-4832 239 1 0 0 X cana-4832 239 2 . . PUNCT cana-4832 240 1 so so ADV cana-4832 240 2 , , PUNCT cana-4832 240 3 𝑝 𝑝 PROPN cana-4832 240 4 = = SYM cana-4832 240 5 𝑒. 𝑒. NOUN cana-4832 240 6 uniqueness uniqueness NOUN cana-4832 240 7 . . PUNCT cana-4832 241 1 let let VERB cana-4832 241 2 𝑣(β‰  𝑣(β‰  PROPN cana-4832 241 3 𝑒 𝑒 NOUN cana-4832 241 4 ) ) PUNCT cana-4832 241 5 be be VERB cana-4832 241 6 another another DET cana-4832 241 7 common common ADJ cana-4832 241 8 fixed fix VERB cana-4832 241 9 point point NOUN cana-4832 241 10 of of ADP cana-4832 241 11 𝑆 𝑆 PROPN cana-4832 241 12 and and CCONJ cana-4832 241 13 𝑇. 𝑇. PROPN cana-4832 241 14 then then ADV cana-4832 241 15 𝑑(𝑒 𝑑(𝑒 NUM cana-4832 241 16 , , PUNCT cana-4832 241 17 𝑣 𝑣 NOUN cana-4832 241 18 ) ) PUNCT cana-4832 242 1 = = SYM cana-4832 242 2 𝑑(𝑆𝑒 𝑑(𝑆𝑒 NOUN cana-4832 242 3 , , PUNCT cana-4832 242 4 𝑇𝑣 𝑇𝑣 PROPN cana-4832 242 5 ) ) PUNCT cana-4832 242 6 β‰₯ β‰₯ NOUN cana-4832 242 7 π‘˜ π‘˜ PRON cana-4832 242 8 𝛼 𝛼 X cana-4832 242 9 𝑑(𝑒 𝑑(𝑒 ADJ cana-4832 242 10 , , PUNCT cana-4832 242 11 𝑣 𝑣 NOUN cana-4832 242 12 ) ) PUNCT cana-4832 242 13 , , PUNCT cana-4832 242 14 i. i. PROPN cana-4832 242 15 e. e. PROPN cana-4832 242 16 , , PUNCT cana-4832 242 17 ( ( PUNCT cana-4832 242 18 1 1 NUM cana-4832 242 19 βˆ’ βˆ’ NOUN cana-4832 242 20 π‘˜ π‘˜ PROPN cana-4832 242 21 𝛼)𝑑(𝑒 𝛼)𝑑(𝑒 PROPN cana-4832 242 22 , , PUNCT cana-4832 242 23 𝑣 𝑣 NOUN cana-4832 242 24 ) ) PUNCT cana-4832 242 25 ≀ ≀ NOUN cana-4832 242 26 0 0 NUM cana-4832 242 27 which which PRON cana-4832 242 28 gives give VERB cana-4832 242 29 us we PRON cana-4832 242 30 𝑑(𝑒 𝑑(𝑒 NOUN cana-4832 242 31 , , PUNCT cana-4832 242 32 𝑣 𝑣 NOUN cana-4832 242 33 ) ) PUNCT cana-4832 242 34 = = SYM cana-4832 242 35 0 0 X cana-4832 242 36 . . PUNCT cana-4832 243 1 similarly similarly ADV cana-4832 243 2 , , PUNCT cana-4832 243 3 we we PRON cana-4832 243 4 can can AUX cana-4832 243 5 prove prove VERB cana-4832 243 6 that that SCONJ cana-4832 243 7 𝑑(𝑣 𝑑(𝑣 NOUN cana-4832 243 8 , , PUNCT cana-4832 243 9 𝑒 𝑒 NOUN cana-4832 243 10 ) ) PUNCT cana-4832 243 11 = = SYM cana-4832 243 12 0 0 NUM cana-4832 243 13 , , PUNCT cana-4832 243 14 and and CCONJ cana-4832 243 15 that that SCONJ cana-4832 243 16 𝑒 𝑒 X cana-4832 243 17 = = SYM cana-4832 243 18 𝑣. 𝑣. NOUN cana-4832 243 19 case case NOUN cana-4832 243 20 ( ( PUNCT cana-4832 243 21 ii ii NOUN cana-4832 243 22 ) ) PUNCT cana-4832 243 23 . . PUNCT cana-4832 244 1 πœƒ(π‘Ž πœƒ(π‘Ž NOUN cana-4832 244 2 , , PUNCT cana-4832 244 3 𝑏 𝑏 NOUN cana-4832 244 4 ) ) PUNCT cana-4832 244 5 = = SYM cana-4832 244 6 𝛽1 𝛽1 NOUN cana-4832 244 7 𝑑(π‘†π‘Ž,π‘Ž)𝑑(𝑇𝑏,𝑏 𝑑(π‘†π‘Ž,π‘Ž)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 244 8 ) ) PUNCT cana-4832 244 9 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 NOUN cana-4832 244 10 ) ) PUNCT cana-4832 245 1 + + CCONJ cana-4832 245 2 𝛽2𝑑(π‘Ž 𝛽2𝑑(π‘Ž PROPN cana-4832 245 3 , , PUNCT cana-4832 245 4 𝑏 𝑏 NOUN cana-4832 245 5 ) ) PUNCT cana-4832 245 6 , , PUNCT cana-4832 245 7 then then ADV cana-4832 245 8 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž PROPN cana-4832 245 9 , , PUNCT cana-4832 245 10 𝑆𝑏 𝑆𝑏 PROPN cana-4832 245 11 ) ) PUNCT cana-4832 245 12 β‰₯ β‰₯ NOUN cana-4832 245 13 π‘˜ π‘˜ X cana-4832 246 1 [ [ X cana-4832 246 2 𝛽1 𝛽1 NOUN cana-4832 246 3 𝑑(π‘†π‘Ž 𝑑(π‘†π‘Ž ADV cana-4832 246 4 , , PUNCT cana-4832 246 5 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 246 6 , , PUNCT cana-4832 246 7 𝑏 𝑏 NOUN cana-4832 246 8 ) ) PUNCT cana-4832 246 9 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 246 10 , , PUNCT cana-4832 246 11 𝑏 𝑏 NOUN cana-4832 246 12 ) ) PUNCT cana-4832 246 13 + + CCONJ cana-4832 246 14 𝛽2𝑑(π‘Ž 𝛽2𝑑(π‘Ž PROPN cana-4832 246 15 , , PUNCT cana-4832 246 16 𝑏 𝑏 NOUN cana-4832 246 17 ) ) PUNCT cana-4832 246 18 ] ] PUNCT cana-4832 246 19 ( ( PUNCT cana-4832 246 20 2.14 2.14 NUM cana-4832 246 21 ) ) PUNCT cana-4832 246 22 now now ADV cana-4832 246 23 , , PUNCT cana-4832 246 24 using use VERB cana-4832 246 25 ( ( PUNCT cana-4832 246 26 2.12 2.12 NUM cana-4832 246 27 ) ) PUNCT cana-4832 246 28 and and CCONJ cana-4832 246 29 ( ( PUNCT cana-4832 246 30 2.14 2.14 NUM cana-4832 246 31 ) ) PUNCT cana-4832 246 32 , , PUNCT cana-4832 246 33 we we PRON cana-4832 246 34 get get AUX cana-4832 246 35 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 246 36 , , PUNCT cana-4832 246 37 π‘Ž2𝑛+1 π‘Ž2𝑛+1 ADJ cana-4832 246 38 ) ) PUNCT cana-4832 246 39 = = SYM cana-4832 246 40 𝑑(π‘†π‘Ž2𝑛+1 𝑑(π‘†π‘Ž2𝑛+1 NOUN cana-4832 246 41 , , PUNCT cana-4832 246 42 π‘‡π‘Ž2𝑛+2 π‘‡π‘Ž2𝑛+2 NUM cana-4832 246 43 ) ) PUNCT cana-4832 246 44 β‰₯ β‰₯ PROPN cana-4832 246 45 π‘˜π›½1 π‘˜π›½1 PROPN cana-4832 246 46 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 PROPN cana-4832 246 47 , , PUNCT cana-4832 246 48 π‘Ž2𝑛+1)𝑑(π‘Ž2𝑛+1 π‘Ž2𝑛+1)𝑑(π‘Ž2𝑛+1 PROPN cana-4832 246 49 , , PUNCT cana-4832 246 50 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 246 51 ) ) PUNCT cana-4832 246 52 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 246 53 , , PUNCT cana-4832 246 54 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 246 55 ) ) PUNCT cana-4832 247 1 + + CCONJ cana-4832 247 2 π‘˜π›½2𝑑(π‘Ž2𝑛+1 π‘˜π›½2𝑑(π‘Ž2𝑛+1 NOUN cana-4832 247 3 , , PUNCT cana-4832 247 4 π‘Ž2𝑛+2 π‘Ž2𝑛+2 PROPN cana-4832 247 5 ) ) PUNCT cana-4832 247 6 β‰₯ β‰₯ NOUN cana-4832 247 7 π‘˜π›½2𝑑(π‘Ž2𝑛+1 π‘˜π›½2𝑑(π‘Ž2𝑛+1 NOUN cana-4832 247 8 , , PUNCT cana-4832 247 9 π‘Ž2𝑛+2 π‘Ž2𝑛+2 PROPN cana-4832 247 10 ) ) PUNCT cana-4832 247 11 i. i. PROPN cana-4832 247 12 e. e. PROPN cana-4832 247 13 , , PUNCT cana-4832 247 14 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 247 15 , , PUNCT cana-4832 247 16 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 247 17 ) ) PUNCT cana-4832 247 18 ≀ ≀ NUM cana-4832 247 19 1 1 NUM cana-4832 247 20 π‘˜π›½2 π‘˜π›½2 PROPN cana-4832 247 21 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 247 22 , , PUNCT cana-4832 247 23 π‘Ž2𝑛+1 π‘Ž2𝑛+1 PROPN cana-4832 247 24 ) ) PUNCT cana-4832 247 25 which which PRON cana-4832 247 26 implies imply VERB cana-4832 247 27 that that SCONJ cana-4832 247 28 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 247 29 , , PUNCT cana-4832 247 30 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 247 31 ) ) PUNCT cana-4832 247 32 ≀ ≀ NOUN cana-4832 247 33 πœ— πœ— X cana-4832 247 34 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 247 35 , , PUNCT cana-4832 247 36 π‘Ž2𝑛+1 π‘Ž2𝑛+1 PROPN cana-4832 247 37 ) ) PUNCT cana-4832 247 38 , , PUNCT cana-4832 247 39 where where SCONJ cana-4832 247 40 πœ— πœ— X cana-4832 247 41 < < X cana-4832 247 42 1 1 NUM cana-4832 247 43 . . X cana-4832 247 44 proceeding proceed VERB cana-4832 247 45 similar similar ADJ cana-4832 247 46 to to ADP cana-4832 247 47 case case NOUN cana-4832 247 48 ( ( PUNCT cana-4832 247 49 i i NOUN cana-4832 247 50 ) ) PUNCT cana-4832 247 51 , , PUNCT cana-4832 247 52 we we PRON cana-4832 247 53 get get AUX cana-4832 247 54 { { PUNCT cana-4832 247 55 π‘Žπ‘› π‘Žπ‘› PRON cana-4832 247 56 } } PUNCT cana-4832 247 57 is be AUX cana-4832 247 58 a a DET cana-4832 247 59 𝑏-cauchy 𝑏-cauchy NOUN cana-4832 247 60 sequence sequence NOUN cana-4832 247 61 in in ADP cana-4832 247 62 𝑋 𝑋 PROPN cana-4832 247 63 , , PUNCT cana-4832 247 64 which which PRON cana-4832 247 65 converges converge VERB cana-4832 247 66 to to ADP cana-4832 247 67 some some DET cana-4832 247 68 𝑒 𝑒 PRON cana-4832 247 69 ∈ ∈ PROPN cana-4832 247 70 𝑋 𝑋 PROPN cana-4832 247 71 , , PUNCT cana-4832 247 72 which which PRON cana-4832 247 73 can can AUX cana-4832 247 74 be be AUX cana-4832 247 75 shown show VERB cana-4832 247 76 to to PART cana-4832 247 77 be be AUX cana-4832 247 78 unique unique ADJ cana-4832 247 79 common common ADJ cana-4832 247 80 fixed fix VERB cana-4832 247 81 point point NOUN cana-4832 247 82 of of ADP cana-4832 247 83 𝑆 𝑆 PROPN cana-4832 247 84 and and CCONJ cana-4832 247 85 𝑇. 𝑇. PROPN cana-4832 247 86 case case NOUN cana-4832 247 87 ( ( PUNCT cana-4832 247 88 iii iii NOUN cana-4832 247 89 ) ) PUNCT cana-4832 247 90 . . PUNCT cana-4832 248 1 πœƒ(π‘Ž πœƒ(π‘Ž NOUN cana-4832 248 2 , , PUNCT cana-4832 248 3 𝑏 𝑏 NOUN cana-4832 248 4 ) ) PUNCT cana-4832 248 5 = = PUNCT cana-4832 248 6 𝛾1𝑑(π‘†π‘Ž 𝛾1𝑑(π‘†π‘Ž ADJ cana-4832 248 7 , , PUNCT cana-4832 248 8 π‘Ž π‘Ž NOUN cana-4832 248 9 ) ) PUNCT cana-4832 248 10 + + NUM cana-4832 248 11 𝛾2 𝛾2 NOUN cana-4832 248 12 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 248 13 , , PUNCT cana-4832 248 14 𝑏 𝑏 NOUN cana-4832 248 15 ) ) PUNCT cana-4832 248 16 + + CCONJ cana-4832 248 17 𝛾3 𝛾3 ADJ cana-4832 248 18 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 248 19 , , PUNCT cana-4832 248 20 𝑏 𝑏 NOUN cana-4832 248 21 ) ) PUNCT cana-4832 248 22 , , PUNCT cana-4832 248 23 then then ADV cana-4832 248 24 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž PROPN cana-4832 248 25 , , PUNCT cana-4832 248 26 𝑆𝑏 𝑆𝑏 PROPN cana-4832 248 27 ) ) PUNCT cana-4832 248 28 β‰₯ β‰₯ NOUN cana-4832 248 29 π‘˜[𝛾1𝑑(π‘†π‘Ž π‘˜[𝛾1𝑑(π‘†π‘Ž PROPN cana-4832 248 30 , , PUNCT cana-4832 248 31 π‘Ž π‘Ž NOUN cana-4832 248 32 ) ) PUNCT cana-4832 248 33 + + NUM cana-4832 248 34 𝛾2 𝛾2 NOUN cana-4832 248 35 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 248 36 , , PUNCT cana-4832 248 37 𝑏 𝑏 NOUN cana-4832 248 38 ) ) PUNCT cana-4832 248 39 + + CCONJ cana-4832 248 40 𝛾3 𝛾3 ADJ cana-4832 248 41 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 248 42 , , PUNCT cana-4832 248 43 𝑏 𝑏 NOUN cana-4832 248 44 ) ) PUNCT cana-4832 248 45 ] ] PUNCT cana-4832 248 46 ( ( PUNCT cana-4832 248 47 2.15 2.15 NUM cana-4832 248 48 ) ) PUNCT cana-4832 248 49 now now ADV cana-4832 248 50 , , PUNCT cana-4832 248 51 using use VERB cana-4832 248 52 ( ( PUNCT cana-4832 248 53 2.12 2.12 NUM cana-4832 248 54 ) ) PUNCT cana-4832 248 55 and and CCONJ cana-4832 248 56 ( ( PUNCT cana-4832 248 57 2.15 2.15 NUM cana-4832 248 58 ) ) PUNCT cana-4832 248 59 , , PUNCT cana-4832 248 60 we we PRON cana-4832 248 61 get get AUX cana-4832 248 62 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 248 63 , , PUNCT cana-4832 248 64 π‘Ž2𝑛+1 π‘Ž2𝑛+1 ADJ cana-4832 248 65 ) ) PUNCT cana-4832 248 66 = = SYM cana-4832 248 67 𝑑(π‘†π‘Ž2𝑛+1 𝑑(π‘†π‘Ž2𝑛+1 NOUN cana-4832 248 68 , , PUNCT cana-4832 248 69 π‘‡π‘Ž2𝑛+2 π‘‡π‘Ž2𝑛+2 NUM cana-4832 248 70 ) ) PUNCT cana-4832 248 71 β‰₯ β‰₯ NOUN cana-4832 248 72 π‘˜π›Ύ1𝑑(π‘Ž2𝑛 π‘˜π›Ύ1𝑑(π‘Ž2𝑛 PROPN cana-4832 248 73 , , PUNCT cana-4832 248 74 π‘Ž2𝑛+1 π‘Ž2𝑛+1 PROPN cana-4832 248 75 ) ) PUNCT cana-4832 249 1 + + NUM cana-4832 249 2 π‘˜π›Ύ2 π‘˜π›Ύ2 PROPN cana-4832 249 3 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 249 4 , , PUNCT cana-4832 249 5 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 249 6 ) ) PUNCT cana-4832 250 1 + + NUM cana-4832 250 2 π‘˜π›Ύ3 π‘˜π›Ύ3 PROPN cana-4832 250 3 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 250 4 , , PUNCT cana-4832 250 5 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 250 6 ) ) PUNCT cana-4832 250 7 β‰₯ β‰₯ NOUN cana-4832 250 8 π‘˜[𝛾2 π‘˜[𝛾2 NOUN cana-4832 250 9 + + CCONJ cana-4832 250 10 𝛾3]𝑑(π‘Ž2𝑛+1 𝛾3]𝑑(π‘Ž2𝑛+1 PROPN cana-4832 250 11 , , PUNCT cana-4832 250 12 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 250 13 ) ) PUNCT cana-4832 250 14 i.e. i.e. X cana-4832 250 15 , , PUNCT cana-4832 250 16 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 250 17 , , PUNCT cana-4832 250 18 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 250 19 ) ) PUNCT cana-4832 250 20 ≀ ≀ NOUN cana-4832 250 21 1 1 NUM cana-4832 250 22 π‘˜[𝛾2+𝛾3 π‘˜[𝛾2+𝛾3 NOUN cana-4832 250 23 ] ] PUNCT cana-4832 250 24 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 250 25 , , PUNCT cana-4832 250 26 π‘Ž2𝑛+1 π‘Ž2𝑛+1 ADJ cana-4832 250 27 ) ) PUNCT cana-4832 250 28 implies imply VERB cana-4832 250 29 that that SCONJ cana-4832 250 30 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 250 31 , , PUNCT cana-4832 250 32 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 250 33 ) ) PUNCT cana-4832 250 34 ≀ ≀ NOUN cana-4832 250 35 𝜌 𝜌 ADP cana-4832 250 36 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 250 37 , , PUNCT cana-4832 250 38 π‘Ž2𝑛+1 π‘Ž2𝑛+1 PROPN cana-4832 250 39 ) ) PUNCT cana-4832 250 40 , , PUNCT cana-4832 250 41 where where SCONJ cana-4832 250 42 𝜌 𝜌 ADP cana-4832 250 43 < < X cana-4832 250 44 1 1 NUM cana-4832 250 45 . . X cana-4832 250 46 proceeding proceed VERB cana-4832 250 47 similar similar ADJ cana-4832 250 48 to to ADP cana-4832 250 49 case case NOUN cana-4832 250 50 ( ( PUNCT cana-4832 250 51 i i NOUN cana-4832 250 52 ) ) PUNCT cana-4832 250 53 , , PUNCT cana-4832 250 54 we we PRON cana-4832 250 55 get get VERB cana-4832 250 56 { { PUNCT cana-4832 250 57 π‘Žπ‘› π‘Žπ‘› PRON cana-4832 250 58 } } PUNCT cana-4832 250 59 is be AUX cana-4832 250 60 a a DET cana-4832 250 61 𝑏-cauchy 𝑏-cauchy NOUN cana-4832 250 62 sequence sequence NOUN cana-4832 250 63 in in ADP cana-4832 250 64 𝑋 𝑋 PROPN cana-4832 250 65 , , PUNCT cana-4832 250 66 and and CCONJ cana-4832 250 67 that that SCONJ cana-4832 250 68 converges converge VERB cana-4832 250 69 to to ADP cana-4832 250 70 some some DET cana-4832 250 71 𝑒 𝑒 PRON cana-4832 250 72 ∈ ∈ PROPN cana-4832 250 73 𝑋 𝑋 PROPN cana-4832 250 74 , , PUNCT cana-4832 250 75 which which PRON cana-4832 250 76 is be AUX cana-4832 250 77 a a DET cana-4832 250 78 unique unique ADJ cana-4832 250 79 common common ADJ cana-4832 250 80 fixed fix VERB cana-4832 250 81 point point NOUN cana-4832 250 82 of of ADP cana-4832 250 83 𝑆 𝑆 PROPN cana-4832 250 84 and and CCONJ cana-4832 250 85 𝑇. 𝑇. PROPN cana-4832 250 86 communications communication NOUN cana-4832 250 87 on on ADP cana-4832 250 88 applied apply VERB cana-4832 250 89 nonlinear nonlinear ADJ cana-4832 250 90 analysis analysis NOUN cana-4832 250 91 issn issn NOUN cana-4832 250 92 : : PUNCT cana-4832 250 93 1074 1074 NUM cana-4832 250 94 - - PUNCT cana-4832 250 95 133x 133x NUM cana-4832 250 96 vol vol VERB cana-4832 250 97 32 32 NUM cana-4832 250 98 no no NOUN cana-4832 250 99 . . PUNCT cana-4832 251 1 10s 10 NOUN cana-4832 251 2 ( ( PUNCT cana-4832 251 3 2025 2025 NUM cana-4832 251 4 ) ) PUNCT cana-4832 251 5 401 401 NUM cana-4832 251 6 https://internationalpubls.com https://internationalpubls.com NOUN cana-4832 251 7 case case NOUN cana-4832 251 8 ( ( PUNCT cana-4832 251 9 iv iv NUM cana-4832 251 10 ) ) PUNCT cana-4832 251 11 . . PUNCT cana-4832 252 1 πœƒ(π‘Ž πœƒ(π‘Ž NOUN cana-4832 252 2 , , PUNCT cana-4832 252 3 𝑏 𝑏 NOUN cana-4832 252 4 ) ) PUNCT cana-4832 252 5 = = SYM cana-4832 253 1 𝛿1𝑑(π‘†π‘Ž 𝛿1𝑑(π‘†π‘Ž ADJ cana-4832 253 2 , , PUNCT cana-4832 253 3 𝑏 𝑏 NOUN cana-4832 253 4 ) ) PUNCT cana-4832 253 5 + + CCONJ cana-4832 253 6 𝛿2𝑑(𝑇𝑏 𝛿2𝑑(𝑇𝑏 PROPN cana-4832 253 7 , , PUNCT cana-4832 253 8 π‘Ž π‘Ž NOUN cana-4832 253 9 ) ) PUNCT cana-4832 253 10 + + CCONJ cana-4832 253 11 𝛿3𝑑(π‘Ž 𝛿3𝑑(π‘Ž PROPN cana-4832 253 12 , , PUNCT cana-4832 253 13 𝑏 𝑏 NOUN cana-4832 253 14 ) ) PUNCT cana-4832 253 15 , , PUNCT cana-4832 253 16 then then ADV cana-4832 253 17 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž PROPN cana-4832 253 18 , , PUNCT cana-4832 253 19 𝑆𝑏 𝑆𝑏 PROPN cana-4832 253 20 ) ) PUNCT cana-4832 253 21 β‰₯ β‰₯ NOUN cana-4832 253 22 π‘˜[𝛿1𝑑(π‘†π‘Ž π‘˜[𝛿1𝑑(π‘†π‘Ž NOUN cana-4832 253 23 , , PUNCT cana-4832 253 24 𝑏 𝑏 NOUN cana-4832 253 25 ) ) PUNCT cana-4832 253 26 + + CCONJ cana-4832 253 27 𝛿2𝑑(𝑇𝑏 𝛿2𝑑(𝑇𝑏 PROPN cana-4832 253 28 , , PUNCT cana-4832 253 29 π‘Ž π‘Ž NOUN cana-4832 253 30 ) ) PUNCT cana-4832 253 31 + + CCONJ cana-4832 253 32 𝛿3𝑑(π‘Ž 𝛿3𝑑(π‘Ž PROPN cana-4832 253 33 , , PUNCT cana-4832 253 34 𝑏 𝑏 NOUN cana-4832 253 35 ) ) PUNCT cana-4832 253 36 ] ] X cana-4832 253 37 ( ( PUNCT cana-4832 253 38 2.16 2.16 NUM cana-4832 253 39 ) ) PUNCT cana-4832 253 40 now now ADV cana-4832 253 41 , , PUNCT cana-4832 253 42 using use VERB cana-4832 253 43 ( ( PUNCT cana-4832 253 44 2.12 2.12 NUM cana-4832 253 45 ) ) PUNCT cana-4832 253 46 and and CCONJ cana-4832 253 47 ( ( PUNCT cana-4832 253 48 2.16 2.16 NUM cana-4832 253 49 ) ) PUNCT cana-4832 253 50 , , PUNCT cana-4832 253 51 we we PRON cana-4832 253 52 get get AUX cana-4832 253 53 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 253 54 , , PUNCT cana-4832 253 55 π‘Ž2𝑛+1 π‘Ž2𝑛+1 ADJ cana-4832 253 56 ) ) PUNCT cana-4832 253 57 = = SYM cana-4832 253 58 𝑑(π‘†π‘Ž2𝑛+1 𝑑(π‘†π‘Ž2𝑛+1 NOUN cana-4832 253 59 , , PUNCT cana-4832 253 60 π‘‡π‘Ž2𝑛+2 π‘‡π‘Ž2𝑛+2 NUM cana-4832 253 61 ) ) PUNCT cana-4832 253 62 β‰₯ β‰₯ NOUN cana-4832 253 63 π‘˜π›Ώ1𝑑(π‘Ž2𝑛 π‘˜π›Ώ1𝑑(π‘Ž2𝑛 NUM cana-4832 253 64 , , PUNCT cana-4832 253 65 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 253 66 ) ) PUNCT cana-4832 253 67 + + NUM cana-4832 253 68 π‘˜π›Ώ2 π‘˜π›Ώ2 NOUN cana-4832 253 69 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 NOUN cana-4832 253 70 , , PUNCT cana-4832 253 71 π‘Ž2𝑛+1 π‘Ž2𝑛+1 PROPN cana-4832 253 72 ) ) PUNCT cana-4832 254 1 + + CCONJ cana-4832 254 2 π‘˜π›Ώ3 π‘˜π›Ώ3 PROPN cana-4832 254 3 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 254 4 , , PUNCT cana-4832 254 5 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 254 6 ) ) PUNCT cana-4832 254 7 β‰₯ β‰₯ NOUN cana-4832 254 8 π‘˜π›Ώ3𝑑(π‘Ž2𝑛+1 π‘˜π›Ώ3𝑑(π‘Ž2𝑛+1 ADJ cana-4832 254 9 , , PUNCT cana-4832 254 10 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 254 11 ) ) PUNCT cana-4832 254 12 i.e. i.e. X cana-4832 254 13 , , PUNCT cana-4832 254 14 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 254 15 , , PUNCT cana-4832 254 16 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 254 17 ) ) PUNCT cana-4832 254 18 ≀ ≀ NUM cana-4832 254 19 1 1 NUM cana-4832 254 20 π‘˜π›Ώ3 π‘˜π›Ώ3 NOUN cana-4832 254 21 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 254 22 , , PUNCT cana-4832 254 23 π‘Ž2𝑛+1 π‘Ž2𝑛+1 ADJ cana-4832 254 24 ) ) PUNCT cana-4832 254 25 implies imply VERB cana-4832 254 26 that that SCONJ cana-4832 254 27 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 254 28 , , PUNCT cana-4832 254 29 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 254 30 ) ) PUNCT cana-4832 254 31 ≀ ≀ NUM cana-4832 254 32 πœ” πœ” X cana-4832 254 33 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 254 34 , , PUNCT cana-4832 254 35 π‘Ž2𝑛+1 π‘Ž2𝑛+1 PROPN cana-4832 254 36 ) ) PUNCT cana-4832 254 37 , , PUNCT cana-4832 254 38 where where SCONJ cana-4832 254 39 πœ” πœ” X cana-4832 254 40 < < X cana-4832 254 41 1 1 NUM cana-4832 254 42 . . X cana-4832 254 43 proceeding proceed VERB cana-4832 254 44 similar similar ADJ cana-4832 254 45 to to ADP cana-4832 254 46 case case NOUN cana-4832 254 47 ( ( PUNCT cana-4832 254 48 i i NOUN cana-4832 254 49 ) ) PUNCT cana-4832 254 50 , , PUNCT cana-4832 254 51 we we PRON cana-4832 254 52 get get AUX cana-4832 254 53 { { PUNCT cana-4832 254 54 π‘Žπ‘› π‘Žπ‘› PRON cana-4832 254 55 } } PUNCT cana-4832 254 56 is be AUX cana-4832 254 57 a a DET cana-4832 254 58 𝑏-cauchy 𝑏-cauchy NOUN cana-4832 254 59 sequence sequence NOUN cana-4832 254 60 in in ADP cana-4832 254 61 𝑋 𝑋 PROPN cana-4832 254 62 , , PUNCT cana-4832 254 63 and and CCONJ cana-4832 254 64 that that SCONJ cana-4832 254 65 converges converge VERB cana-4832 254 66 to to ADP cana-4832 254 67 some some DET cana-4832 254 68 𝑒 𝑒 PRON cana-4832 254 69 ∈ ∈ PROPN cana-4832 254 70 𝑋 𝑋 PROPN cana-4832 254 71 , , PUNCT cana-4832 254 72 which which PRON cana-4832 254 73 is be AUX cana-4832 254 74 a a DET cana-4832 254 75 unique unique ADJ cana-4832 254 76 common common ADJ cana-4832 254 77 fixed fix VERB cana-4832 254 78 point point NOUN cana-4832 254 79 of of ADP cana-4832 254 80 𝑆 𝑆 PROPN cana-4832 254 81 and and CCONJ cana-4832 254 82 𝑇. 𝑇. PROPN cana-4832 254 83 case case NOUN cana-4832 254 84 ( ( PUNCT cana-4832 254 85 v v NOUN cana-4832 254 86 ) ) PUNCT cana-4832 254 87 . . PUNCT cana-4832 255 1 πœƒ(π‘Ž πœƒ(π‘Ž NOUN cana-4832 255 2 , , PUNCT cana-4832 255 3 𝑏 𝑏 NOUN cana-4832 255 4 ) ) PUNCT cana-4832 255 5 = = PUNCT cana-4832 255 6 πœ†1 πœ†1 NUM cana-4832 255 7 𝑑(π‘†π‘Ž,𝑏)𝑑(𝑇𝑏,𝑏 𝑑(π‘†π‘Ž,𝑏)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 255 8 ) ) PUNCT cana-4832 255 9 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 NOUN cana-4832 255 10 ) ) PUNCT cana-4832 256 1 + + NUM cana-4832 256 2 πœ†2 πœ†2 NOUN cana-4832 256 3 𝑑(𝑇𝑏,π‘Ž)𝑑(𝑇𝑏,𝑏 𝑑(𝑇𝑏,π‘Ž)𝑑(𝑇𝑏,𝑏 NOUN cana-4832 256 4 ) ) PUNCT cana-4832 256 5 𝑑(π‘Ž,𝑏 𝑑(π‘Ž,𝑏 VERB cana-4832 256 6 ) ) PUNCT cana-4832 257 1 + + CCONJ cana-4832 257 2 πœ†3𝑑(π‘Ž πœ†3𝑑(π‘Ž PROPN cana-4832 257 3 , , PUNCT cana-4832 257 4 𝑏 𝑏 NOUN cana-4832 257 5 ) ) PUNCT cana-4832 257 6 , , PUNCT cana-4832 257 7 then then ADV cana-4832 257 8 𝑑(π‘‡π‘Ž 𝑑(π‘‡π‘Ž PROPN cana-4832 257 9 , , PUNCT cana-4832 257 10 𝑆𝑏 𝑆𝑏 PROPN cana-4832 257 11 ) ) PUNCT cana-4832 257 12 β‰₯ β‰₯ NOUN cana-4832 257 13 π‘˜ π‘˜ X cana-4832 258 1 [ [ X cana-4832 258 2 πœ†1 πœ†1 VERB cana-4832 258 3 𝑑(π‘†π‘Ž 𝑑(π‘†π‘Ž ADP cana-4832 258 4 , , PUNCT cana-4832 258 5 𝑏)𝑑(𝑇𝑏 𝑏)𝑑(𝑇𝑏 PROPN cana-4832 258 6 , , PUNCT cana-4832 258 7 𝑏 𝑏 NOUN cana-4832 258 8 ) ) PUNCT cana-4832 258 9 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 258 10 , , PUNCT cana-4832 258 11 𝑏 𝑏 NOUN cana-4832 258 12 ) ) PUNCT cana-4832 258 13 + + NUM cana-4832 258 14 πœ†2 πœ†2 PROPN cana-4832 258 15 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 258 16 , , PUNCT cana-4832 258 17 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 258 18 , , PUNCT cana-4832 258 19 𝑏 𝑏 NOUN cana-4832 258 20 ) ) PUNCT cana-4832 258 21 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 258 22 , , PUNCT cana-4832 258 23 𝑏 𝑏 NOUN cana-4832 258 24 ) ) PUNCT cana-4832 258 25 + + CCONJ cana-4832 259 1 πœ†3𝑑(π‘Ž πœ†3𝑑(π‘Ž PROPN cana-4832 259 2 , , PUNCT cana-4832 259 3 𝑏 𝑏 NOUN cana-4832 259 4 ) ) PUNCT cana-4832 259 5 ] ] PUNCT cana-4832 259 6 ( ( PUNCT cana-4832 259 7 2.17 2.17 NUM cana-4832 259 8 ) ) PUNCT cana-4832 259 9 now now ADV cana-4832 259 10 , , PUNCT cana-4832 259 11 using use VERB cana-4832 259 12 ( ( PUNCT cana-4832 259 13 2.12 2.12 NUM cana-4832 259 14 ) ) PUNCT cana-4832 259 15 and and CCONJ cana-4832 259 16 ( ( PUNCT cana-4832 259 17 2.17 2.17 NUM cana-4832 259 18 ) ) PUNCT cana-4832 259 19 , , PUNCT cana-4832 259 20 we we PRON cana-4832 259 21 get get AUX cana-4832 259 22 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 259 23 , , PUNCT cana-4832 259 24 π‘Ž2𝑛+1 π‘Ž2𝑛+1 ADJ cana-4832 259 25 ) ) PUNCT cana-4832 259 26 = = SYM cana-4832 259 27 𝑑(π‘†π‘Ž2𝑛+1 𝑑(π‘†π‘Ž2𝑛+1 NOUN cana-4832 259 28 , , PUNCT cana-4832 259 29 π‘‡π‘Ž2𝑛+2 π‘‡π‘Ž2𝑛+2 NUM cana-4832 259 30 ) ) PUNCT cana-4832 259 31 β‰₯ β‰₯ NOUN cana-4832 259 32 π‘˜πœ†1 π‘˜πœ†1 PROPN cana-4832 259 33 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 PROPN cana-4832 259 34 , , PUNCT cana-4832 259 35 π‘Ž2𝑛+2)𝑑(π‘Ž2𝑛+1 π‘Ž2𝑛+2)𝑑(π‘Ž2𝑛+1 NUM cana-4832 259 36 , , PUNCT cana-4832 259 37 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 259 38 ) ) PUNCT cana-4832 259 39 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 259 40 , , PUNCT cana-4832 259 41 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 259 42 ) ) PUNCT cana-4832 259 43 + + CCONJ cana-4832 259 44 π‘˜πœ†3 π‘˜πœ†3 PROPN cana-4832 259 45 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 259 46 , , PUNCT cana-4832 259 47 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 259 48 ) ) PUNCT cana-4832 259 49 β‰₯ β‰₯ NOUN cana-4832 259 50 π‘˜πœ†3𝑑(π‘Ž2𝑛+1 π‘˜πœ†3𝑑(π‘Ž2𝑛+1 NOUN cana-4832 259 51 , , PUNCT cana-4832 259 52 π‘Ž2𝑛+2 π‘Ž2𝑛+2 PROPN cana-4832 259 53 ) ) PUNCT cana-4832 259 54 i.e. i.e. X cana-4832 259 55 , , PUNCT cana-4832 259 56 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 259 57 , , PUNCT cana-4832 259 58 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 259 59 ) ) PUNCT cana-4832 259 60 ≀ ≀ NUM cana-4832 259 61 1 1 NUM cana-4832 259 62 π‘˜πœ†3 π‘˜πœ†3 NOUN cana-4832 259 63 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 259 64 , , PUNCT cana-4832 259 65 π‘Ž2𝑛+1 π‘Ž2𝑛+1 PROPN cana-4832 259 66 ) ) PUNCT cana-4832 259 67 which which PRON cana-4832 259 68 implies imply VERB cana-4832 259 69 that that SCONJ cana-4832 259 70 𝑑(π‘Ž2𝑛+1 𝑑(π‘Ž2𝑛+1 PROPN cana-4832 259 71 , , PUNCT cana-4832 259 72 π‘Ž2𝑛+2 π‘Ž2𝑛+2 ADV cana-4832 259 73 ) ) PUNCT cana-4832 259 74 ≀ ≀ NOUN cana-4832 259 75 πœ‘ πœ‘ PRON cana-4832 259 76 𝑑(π‘Ž2𝑛 𝑑(π‘Ž2𝑛 NOUN cana-4832 259 77 , , PUNCT cana-4832 259 78 π‘Ž2𝑛+1 π‘Ž2𝑛+1 PROPN cana-4832 259 79 ) ) PUNCT cana-4832 259 80 , , PUNCT cana-4832 259 81 where where SCONJ cana-4832 259 82 πœ‘ πœ‘ X cana-4832 259 83 < < X cana-4832 259 84 1 1 X cana-4832 259 85 . . X cana-4832 259 86 proceeding proceed VERB cana-4832 259 87 similar similar ADJ cana-4832 259 88 to to ADP cana-4832 259 89 case case NOUN cana-4832 259 90 ( ( PUNCT cana-4832 259 91 i i NOUN cana-4832 259 92 ) ) PUNCT cana-4832 259 93 , , PUNCT cana-4832 259 94 we we PRON cana-4832 259 95 get get VERB cana-4832 259 96 { { PUNCT cana-4832 259 97 π‘Žπ‘› π‘Žπ‘› PRON cana-4832 259 98 } } PUNCT cana-4832 259 99 is be AUX cana-4832 259 100 a a DET cana-4832 259 101 𝑏-cauchy 𝑏-cauchy NOUN cana-4832 259 102 sequence sequence NOUN cana-4832 259 103 in in ADP cana-4832 259 104 𝑋 𝑋 PROPN cana-4832 259 105 , , PUNCT cana-4832 259 106 and and CCONJ cana-4832 259 107 that that SCONJ cana-4832 259 108 converges converge VERB cana-4832 259 109 to to ADP cana-4832 259 110 some some DET cana-4832 259 111 𝑒 𝑒 PRON cana-4832 259 112 ∈ ∈ PROPN cana-4832 259 113 𝑋 𝑋 PROPN cana-4832 259 114 , , PUNCT cana-4832 259 115 which which PRON cana-4832 259 116 is be AUX cana-4832 259 117 a a DET cana-4832 259 118 unique unique ADJ cana-4832 259 119 common common ADJ cana-4832 259 120 fixed fix VERB cana-4832 259 121 point point NOUN cana-4832 259 122 of of ADP cana-4832 259 123 𝑆 𝑆 PROPN cana-4832 259 124 and and CCONJ cana-4832 259 125 𝑇. 𝑇. PROPN cana-4832 259 126 example example NOUN cana-4832 259 127 2.9 2.9 NUM cana-4832 259 128 . . PUNCT cana-4832 260 1 let let VERB cana-4832 260 2 𝑋 𝑋 NOUN cana-4832 260 3 = = PUNCT cana-4832 260 4 ℝ+ ℝ+ PROPN cana-4832 260 5 . . PUNCT cana-4832 261 1 we we PRON cana-4832 261 2 define define VERB cana-4832 261 3 𝑑 𝑑 NOUN cana-4832 261 4 : : PUNCT cana-4832 261 5 𝑋 𝑋 NOUN cana-4832 261 6 Γ— Γ— NOUN cana-4832 261 7 𝑋 𝑋 PROPN cana-4832 261 8 β†’ β†’ SYM cana-4832 261 9 ℝ+ ℝ+ PUNCT cana-4832 261 10 by by ADP cana-4832 261 11 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 261 12 , , PUNCT cana-4832 261 13 𝑏 𝑏 NOUN cana-4832 261 14 ) ) PUNCT cana-4832 261 15 = = SYM cana-4832 261 16 |π‘Ž |π‘Ž X cana-4832 261 17 βˆ’ βˆ’ PROPN cana-4832 261 18 𝑏|2 𝑏|2 PROPN cana-4832 261 19 + + CCONJ cana-4832 261 20 |π‘Ž|2 |π‘Ž|2 PROPN cana-4832 261 21 . . PUNCT cana-4832 262 1 then then ADV cana-4832 262 2 clearly clearly ADV cana-4832 262 3 , , PUNCT cana-4832 262 4 ( ( PUNCT cana-4832 262 5 𝑋 𝑋 NOUN cana-4832 262 6 , , PUNCT cana-4832 262 7 𝑑 𝑑 NOUN cana-4832 262 8 ) ) PUNCT cana-4832 262 9 is be AUX cana-4832 262 10 a a DET cana-4832 262 11 complete complete ADJ cana-4832 262 12 𝑑𝑝 𝑑𝑝 ADJ cana-4832 262 13 𝑏-metric 𝑏-metric ADJ cana-4832 262 14 space space NOUN cana-4832 262 15 with with ADP cana-4832 262 16 𝑠 𝑠 PROPN cana-4832 262 17 = = SYM cana-4832 262 18 2 2 X cana-4832 262 19 . . X cana-4832 263 1 we we PRON cana-4832 263 2 define define VERB cana-4832 263 3 self self NOUN cana-4832 263 4 - - PUNCT cana-4832 263 5 mappings mapping NOUN cana-4832 263 6 𝑆 𝑆 PROPN cana-4832 263 7 , , PUNCT cana-4832 263 8 𝑇 𝑇 PROPN cana-4832 263 9 : : PUNCT cana-4832 263 10 𝑋 𝑋 PROPN cana-4832 263 11 β†’ β†’ SYM cana-4832 263 12 𝑋 𝑋 NOUN cana-4832 263 13 by by ADP cana-4832 263 14 𝑆(π‘Ž 𝑆(π‘Ž NUM cana-4832 263 15 ) ) PUNCT cana-4832 263 16 = = PUNCT cana-4832 263 17 π‘Ž(π‘Ž π‘Ž(π‘Ž PROPN cana-4832 263 18 + + CCONJ cana-4832 263 19 2)and 2)and NUM cana-4832 263 20 𝑇(π‘Ž 𝑇(π‘Ž NUM cana-4832 263 21 ) ) PUNCT cana-4832 263 22 = = NUM cana-4832 263 23 2π‘Ž 2π‘Ž NOUN cana-4832 263 24 , , PUNCT cana-4832 263 25 for for ADP cana-4832 263 26 all all PRON cana-4832 263 27 π‘Ž π‘Ž PROPN cana-4832 263 28 ∈ ∈ NOUN cana-4832 263 29 𝑋. 𝑋. NOUN cana-4832 263 30 we we PRON cana-4832 263 31 take take VERB cana-4832 263 32 π‘˜ π‘˜ PRON cana-4832 263 33 = = NOUN cana-4832 263 34 3 3 NUM cana-4832 263 35 2 2 NUM cana-4832 263 36 , , PUNCT cana-4832 263 37 𝛼 𝛼 NOUN cana-4832 263 38 = = SYM cana-4832 263 39 𝛽2 𝛽2 PROPN cana-4832 263 40 = = SYM cana-4832 263 41 𝛾3 𝛾3 NOUN cana-4832 263 42 = = SYM cana-4832 263 43 𝛿3 𝛿3 NOUN cana-4832 263 44 = = SYM cana-4832 263 45 πœ†3 πœ†3 NOUN cana-4832 263 46 = = SYM cana-4832 263 47 1 1 NUM cana-4832 263 48 , , PUNCT cana-4832 263 49 𝛽1 𝛽1 NOUN cana-4832 263 50 = = SYM cana-4832 263 51 𝛾1 𝛾1 PROPN cana-4832 263 52 = = NOUN cana-4832 263 53 𝛾2 𝛾2 PROPN cana-4832 263 54 = = SYM cana-4832 263 55 𝛿1 𝛿1 NOUN cana-4832 263 56 = = SYM cana-4832 263 57 𝛿2 𝛿2 NOUN cana-4832 263 58 = = NOUN cana-4832 263 59 πœ†1 πœ†1 NOUN cana-4832 263 60 = = PUNCT cana-4832 263 61 πœ†2 πœ†2 NOUN cana-4832 263 62 = = SYM cana-4832 263 63 0 0 NUM cana-4832 263 64 . . PUNCT cana-4832 264 1 without without ADP cana-4832 264 2 loss loss NOUN cana-4832 264 3 of of ADP cana-4832 264 4 generality generality NOUN cana-4832 264 5 we we PRON cana-4832 264 6 assume assume VERB cana-4832 264 7 that that SCONJ cana-4832 264 8 π‘Ž π‘Ž DET cana-4832 264 9 β‰₯ β‰₯ NOUN cana-4832 264 10 𝑏. 𝑏. NOUN cana-4832 264 11 we we PRON cana-4832 264 12 consider consider VERB cana-4832 264 13 𝑑(π‘†π‘Ž 𝑑(π‘†π‘Ž NOUN cana-4832 264 14 , , PUNCT cana-4832 264 15 𝑇𝑏 𝑇𝑏 NOUN cana-4832 264 16 ) ) PUNCT cana-4832 264 17 = = VERB cana-4832 264 18 |π‘†π‘Ž |π‘†π‘Ž VERB cana-4832 264 19 βˆ’ βˆ’ PROPN cana-4832 264 20 𝑇𝑏|2 𝑇𝑏|2 NOUN cana-4832 265 1 + + X cana-4832 265 2 |π‘†π‘Ž|2 |π‘†π‘Ž|2 PROPN cana-4832 265 3 = = SYM cana-4832 265 4 ( ( PUNCT cana-4832 265 5 π‘Ž2 π‘Ž2 NOUN cana-4832 265 6 + + CCONJ cana-4832 265 7 2π‘Ž 2π‘Ž NUM cana-4832 265 8 βˆ’ βˆ’ NOUN cana-4832 265 9 2𝑏2)2 2𝑏2)2 NUM cana-4832 265 10 + + CCONJ cana-4832 265 11 ( ( PUNCT cana-4832 265 12 π‘Ž2 π‘Ž2 NOUN cana-4832 265 13 + + CCONJ cana-4832 265 14 2π‘Ž)2 2π‘Ž)2 NUM cana-4832 265 15 β‰₯ β‰₯ NOUN cana-4832 265 16 ( ( PUNCT cana-4832 265 17 2π‘Ž 2π‘Ž NUM cana-4832 265 18 βˆ’ βˆ’ NOUN cana-4832 265 19 2𝑏)2 2𝑏)2 NUM cana-4832 265 20 + + CCONJ cana-4832 265 21 2π‘Ž2 2π‘Ž2 NUM cana-4832 265 22 β‰₯ β‰₯ NOUN cana-4832 265 23 3 3 NUM cana-4832 265 24 2 2 NUM cana-4832 265 25 [ [ X cana-4832 265 26 ( ( PUNCT cana-4832 265 27 π‘Ž π‘Ž PRON cana-4832 265 28 βˆ’ βˆ’ PROPN cana-4832 265 29 𝑏)2 𝑏)2 PROPN cana-4832 265 30 + + CCONJ cana-4832 265 31 π‘Ž2 π‘Ž2 PROPN cana-4832 265 32 ] ] PUNCT cana-4832 265 33 communications communication NOUN cana-4832 265 34 on on ADP cana-4832 265 35 applied apply VERB cana-4832 265 36 nonlinear nonlinear ADJ cana-4832 265 37 analysis analysis NOUN cana-4832 265 38 issn issn NOUN cana-4832 265 39 : : PUNCT cana-4832 265 40 1074 1074 NUM cana-4832 265 41 - - PUNCT cana-4832 265 42 133x 133x NUM cana-4832 265 43 vol vol VERB cana-4832 265 44 32 32 NUM cana-4832 265 45 no no NOUN cana-4832 265 46 . . PUNCT cana-4832 266 1 10s 10 NOUN cana-4832 266 2 ( ( PUNCT cana-4832 266 3 2025 2025 NUM cana-4832 266 4 ) ) PUNCT cana-4832 266 5 402 402 NUM cana-4832 266 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 266 7 = = PUNCT cana-4832 267 1 π‘˜ π‘˜ DET cana-4832 267 2 min min NOUN cana-4832 267 3 { { PUNCT cana-4832 267 4 𝛼 𝛼 NOUN cana-4832 267 5 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 267 6 , , PUNCT cana-4832 267 7 𝑏 𝑏 NOUN cana-4832 267 8 ) ) PUNCT cana-4832 267 9 ; ; PUNCT cana-4832 267 10 𝛽1 𝛽1 NOUN cana-4832 267 11 𝑑(π‘†π‘Ž 𝑑(π‘†π‘Ž NOUN cana-4832 267 12 , , PUNCT cana-4832 267 13 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 267 14 , , PUNCT cana-4832 267 15 𝑏 𝑏 NOUN cana-4832 267 16 ) ) PUNCT cana-4832 267 17 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 267 18 , , PUNCT cana-4832 267 19 𝑏 𝑏 NOUN cana-4832 267 20 ) ) PUNCT cana-4832 267 21 + + CCONJ cana-4832 267 22 𝛽2𝑑(π‘Ž 𝛽2𝑑(π‘Ž PROPN cana-4832 267 23 , , PUNCT cana-4832 267 24 𝑏 𝑏 NOUN cana-4832 267 25 ) ) PUNCT cana-4832 267 26 ; ; PUNCT cana-4832 267 27 𝛾1𝑑(π‘†π‘Ž 𝛾1𝑑(π‘†π‘Ž VERB cana-4832 267 28 , , PUNCT cana-4832 267 29 π‘Ž π‘Ž NOUN cana-4832 267 30 ) ) PUNCT cana-4832 267 31 + + NUM cana-4832 267 32 𝛾2 𝛾2 NOUN cana-4832 267 33 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 267 34 , , PUNCT cana-4832 267 35 𝑏 𝑏 NOUN cana-4832 267 36 ) ) PUNCT cana-4832 267 37 + + CCONJ cana-4832 267 38 𝛾3 𝛾3 ADJ cana-4832 267 39 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 267 40 , , PUNCT cana-4832 267 41 𝑏 𝑏 NOUN cana-4832 267 42 ) ) PUNCT cana-4832 267 43 ; ; PUNCT cana-4832 267 44 𝛿1𝑑(π‘†π‘Ž 𝛿1𝑑(π‘†π‘Ž X cana-4832 267 45 , , PUNCT cana-4832 267 46 𝑏 𝑏 NOUN cana-4832 267 47 ) ) PUNCT cana-4832 267 48 + + CCONJ cana-4832 268 1 𝛿2𝑑(𝑇𝑏 𝛿2𝑑(𝑇𝑏 PROPN cana-4832 268 2 , , PUNCT cana-4832 268 3 π‘Ž π‘Ž NOUN cana-4832 268 4 ) ) PUNCT cana-4832 268 5 + + CCONJ cana-4832 268 6 𝛿3𝑑(π‘Ž 𝛿3𝑑(π‘Ž PROPN cana-4832 268 7 , , PUNCT cana-4832 268 8 𝑏 𝑏 NOUN cana-4832 268 9 ) ) PUNCT cana-4832 268 10 ; ; PUNCT cana-4832 268 11 πœ†1 πœ†1 VERB cana-4832 268 12 𝑑(π‘†π‘Ž 𝑑(π‘†π‘Ž ADV cana-4832 268 13 , , PUNCT cana-4832 268 14 𝑏)𝑑(𝑇𝑏 𝑏)𝑑(𝑇𝑏 PROPN cana-4832 268 15 , , PUNCT cana-4832 268 16 𝑏 𝑏 NOUN cana-4832 268 17 ) ) PUNCT cana-4832 268 18 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 268 19 , , PUNCT cana-4832 268 20 𝑏 𝑏 NOUN cana-4832 268 21 ) ) PUNCT cana-4832 268 22 + + NUM cana-4832 268 23 πœ†2 πœ†2 PROPN cana-4832 268 24 𝑑(𝑇𝑏 𝑑(𝑇𝑏 NOUN cana-4832 268 25 , , PUNCT cana-4832 268 26 π‘Ž)𝑑(𝑇𝑏 π‘Ž)𝑑(𝑇𝑏 PROPN cana-4832 268 27 , , PUNCT cana-4832 268 28 𝑏 𝑏 NOUN cana-4832 268 29 ) ) PUNCT cana-4832 268 30 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 268 31 , , PUNCT cana-4832 268 32 𝑏 𝑏 NOUN cana-4832 268 33 ) ) PUNCT cana-4832 268 34 + + CCONJ cana-4832 268 35 πœ†3𝑑(π‘Ž πœ†3𝑑(π‘Ž PROPN cana-4832 268 36 , , PUNCT cana-4832 268 37 𝑏 𝑏 NOUN cana-4832 268 38 ) ) PUNCT cana-4832 268 39 } } PUNCT cana-4832 268 40 table table NOUN cana-4832 268 41 2 2 NUM cana-4832 268 42 and and CCONJ cana-4832 268 43 figure figure NOUN cana-4832 268 44 3 3 NUM cana-4832 268 45 , , PUNCT cana-4832 268 46 illustrates illustrate VERB cana-4832 268 47 the the DET cana-4832 268 48 condition condition NOUN cana-4832 268 49 ( ( PUNCT cana-4832 268 50 2.10 2.10 NUM cana-4832 268 51 ) ) PUNCT cana-4832 268 52 and and CCONJ cana-4832 268 53 ( ( PUNCT cana-4832 268 54 2.11 2.11 NUM cana-4832 268 55 ) ) PUNCT cana-4832 268 56 of of ADP cana-4832 268 57 theorem theorem ADJ cana-4832 268 58 2.8 2.8 NUM cana-4832 268 59 , , PUNCT cana-4832 268 60 with with ADP cana-4832 268 61 blue blue ADJ cana-4832 268 62 line line NOUN cana-4832 268 63 representing represent VERB cana-4832 268 64 the the DET cana-4832 268 65 left left ADJ cana-4832 268 66 part part NOUN cana-4832 268 67 of of ADP cana-4832 268 68 the the DET cana-4832 268 69 condition condition NOUN cana-4832 268 70 and and CCONJ cana-4832 268 71 red red ADJ cana-4832 268 72 line line NOUN cana-4832 268 73 representing represent VERB cana-4832 268 74 the the DET cana-4832 268 75 right right ADJ cana-4832 268 76 part part NOUN cana-4832 268 77 of of ADP cana-4832 268 78 the the DET cana-4832 268 79 condition condition NOUN cana-4832 268 80 . . PUNCT cana-4832 269 1 thus thus ADV cana-4832 269 2 , , PUNCT cana-4832 269 3 all all DET cana-4832 269 4 the the DET cana-4832 269 5 conditions condition NOUN cana-4832 269 6 of of ADP cana-4832 269 7 theorem theorem ADJ cana-4832 269 8 2.8 2.8 NUM cana-4832 269 9 are be AUX cana-4832 269 10 satisfied satisfied ADJ cana-4832 269 11 . . PUNCT cana-4832 270 1 so so ADV cana-4832 270 2 , , PUNCT cana-4832 270 3 𝑆 𝑆 PROPN cana-4832 270 4 and and CCONJ cana-4832 270 5 𝑇 𝑇 PROPN cana-4832 270 6 have have VERB cana-4832 270 7 a a DET cana-4832 270 8 unique unique ADJ cana-4832 270 9 common common ADJ cana-4832 270 10 fixed fix VERB cana-4832 270 11 point point NOUN cana-4832 270 12 in in ADP cana-4832 270 13 𝑋 𝑋 PROPN cana-4832 270 14 which which PRON cana-4832 270 15 is be AUX cana-4832 270 16 clearly clearly ADV cana-4832 270 17 0 0 NUM cana-4832 270 18 here here ADV cana-4832 270 19 . . PUNCT cana-4832 271 1 communications communication NOUN cana-4832 271 2 on on ADP cana-4832 271 3 applied apply VERB cana-4832 271 4 nonlinear nonlinear ADJ cana-4832 271 5 analysis analysis NOUN cana-4832 271 6 issn issn NOUN cana-4832 271 7 : : PUNCT cana-4832 271 8 1074 1074 NUM cana-4832 271 9 - - PUNCT cana-4832 271 10 133x 133x NUM cana-4832 271 11 vol vol VERB cana-4832 271 12 32 32 NUM cana-4832 271 13 no no NOUN cana-4832 271 14 . . PUNCT cana-4832 272 1 10s 10 NOUN cana-4832 272 2 ( ( PUNCT cana-4832 272 3 2025 2025 NUM cana-4832 272 4 ) ) PUNCT cana-4832 272 5 403 403 NUM cana-4832 272 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 272 7 remark remark VERB cana-4832 272 8 2.10 2.10 NUM cana-4832 272 9 . . PUNCT cana-4832 273 1 theorem theorem VERB cana-4832 273 2 2.8 2.8 NUM cana-4832 273 3 and and CCONJ cana-4832 273 4 example example NOUN cana-4832 273 5 2.9 2.9 NUM cana-4832 273 6 extend extend VERB cana-4832 273 7 and and CCONJ cana-4832 273 8 generalize generalize VERB cana-4832 273 9 theorem theorem VERB cana-4832 273 10 1.6 1.6 NUM cana-4832 273 11 to to ADP cana-4832 273 12 π‘‘π‘ž π‘‘π‘ž PROPN cana-4832 273 13 𝑏-metric 𝑏-metric PROPN cana-4832 273 14 spaces space NOUN cana-4832 273 15 . . PUNCT cana-4832 274 1 corollary corollary ADJ cana-4832 274 2 2.11 2.11 NUM cana-4832 274 3 . . PUNCT cana-4832 275 1 let let VERB cana-4832 275 2 ( ( PUNCT cana-4832 275 3 𝑋 𝑋 NOUN cana-4832 275 4 , , PUNCT cana-4832 275 5 𝑑 𝑑 NOUN cana-4832 275 6 ) ) PUNCT cana-4832 275 7 be be VERB cana-4832 275 8 a a DET cana-4832 275 9 complete complete ADJ cana-4832 275 10 π‘‘π‘ž π‘‘π‘ž PROPN cana-4832 275 11 𝑏-metric 𝑏-metric PROPN cana-4832 275 12 space space NOUN cana-4832 275 13 and and CCONJ cana-4832 275 14 𝑆 𝑆 PROPN cana-4832 275 15 , , PUNCT cana-4832 275 16 𝑇 𝑇 PROPN cana-4832 275 17 be be VERB cana-4832 275 18 two two NUM cana-4832 275 19 onto onto ADP cana-4832 275 20 self self NOUN cana-4832 275 21 - - PUNCT cana-4832 275 22 mappings mapping NOUN cana-4832 275 23 on on ADP cana-4832 275 24 𝑋 𝑋 PROPN cana-4832 275 25 such such ADJ cana-4832 275 26 that that SCONJ cana-4832 275 27 𝑑(π‘†π‘Ž 𝑑(π‘†π‘Ž NOUN cana-4832 275 28 , , PUNCT cana-4832 275 29 𝑇 𝑇 PROPN cana-4832 275 30 𝑏 𝑏 NOUN cana-4832 275 31 ) ) PUNCT cana-4832 275 32 β‰₯ β‰₯ NOUN cana-4832 275 33 π‘˜ π‘˜ X cana-4832 275 34 𝑑(π‘Ž 𝑑(π‘Ž PROPN cana-4832 275 35 , , PUNCT cana-4832 275 36 𝑏 𝑏 NOUN cana-4832 275 37 ) ) PUNCT cana-4832 275 38 , , PUNCT cana-4832 275 39 for for ADP cana-4832 275 40 all all DET cana-4832 275 41 π‘Ž π‘Ž NOUN cana-4832 275 42 , , PUNCT cana-4832 275 43 𝑏 𝑏 PROPN cana-4832 275 44 ∈ ∈ PROPN cana-4832 275 45 𝑋 𝑋 NOUN cana-4832 275 46 with with ADP cana-4832 275 47 π‘˜ π‘˜ PROPN cana-4832 275 48 > > X cana-4832 275 49 1 1 NUM cana-4832 275 50 . . PUNCT cana-4832 276 1 then then ADV cana-4832 276 2 𝑆 𝑆 PROPN cana-4832 276 3 and and CCONJ cana-4832 276 4 𝑇 𝑇 PROPN cana-4832 276 5 have have VERB cana-4832 276 6 a a DET cana-4832 276 7 unique unique ADJ cana-4832 276 8 common common ADJ cana-4832 276 9 fixed fix VERB cana-4832 276 10 point point NOUN cana-4832 276 11 in in ADP cana-4832 276 12 𝑋. 𝑋. PROPN cana-4832 276 13 3 3 NUM cana-4832 276 14 . . PUNCT cana-4832 276 15 application application NOUN cana-4832 276 16 to to ADP cana-4832 276 17 nonlinear nonlinear ADJ cana-4832 276 18 integral integral ADJ cana-4832 276 19 equations equation NOUN cana-4832 276 20 let let VERB cana-4832 276 21 Ο‰= Ο‰= ADJ cana-4832 276 22 𝐢[π‘Ž 𝐢[π‘Ž NOUN cana-4832 276 23 , , PUNCT cana-4832 277 1 𝑏 𝑏 NOUN cana-4832 277 2 ] ] PUNCT cana-4832 277 3 be be AUX cana-4832 277 4 a a DET cana-4832 277 5 set set NOUN cana-4832 277 6 of of ADP cana-4832 277 7 real real ADV cana-4832 277 8 valued value VERB cana-4832 277 9 continuous continuous ADJ cana-4832 277 10 functions function NOUN cana-4832 277 11 on on ADP cana-4832 277 12 [ [ X cana-4832 277 13 π‘Ž π‘Ž X cana-4832 277 14 , , PUNCT cana-4832 277 15 𝑏],where 𝑏],where ADV cana-4832 277 16 [ [ X cana-4832 277 17 π‘Ž π‘Ž X cana-4832 277 18 , , PUNCT cana-4832 277 19 𝑏 𝑏 NOUN cana-4832 277 20 ] ] PUNCT cana-4832 277 21 is be AUX cana-4832 277 22 closed close VERB cana-4832 277 23 and and CCONJ cana-4832 277 24 bounded bound VERB cana-4832 277 25 integral integral ADJ cana-4832 277 26 in in ADP cana-4832 277 27 ℝ. ℝ. PROPN cana-4832 277 28 we we PRON cana-4832 277 29 define define VERB cana-4832 277 30 𝑑 𝑑 NOUN cana-4832 277 31 : : PUNCT cana-4832 277 32 Ο‰ Ο‰ PROPN cana-4832 277 33 Γ— Γ— PROPN cana-4832 277 34 Ο‰ Ο‰ PROPN cana-4832 277 35 β†’ β†’ PUNCT cana-4832 277 36 ℝ+ ℝ+ PUNCT cana-4832 277 37 by by ADP cana-4832 277 38 𝑑(πœ‰ 𝑑(πœ‰ PROPN cana-4832 277 39 , , PUNCT cana-4832 277 40 πœ‚ πœ‚ NOUN cana-4832 277 41 ) ) PUNCT cana-4832 277 42 = = SYM cana-4832 277 43 max max PROPN cana-4832 277 44 π‘Žβ‰€π‘‘β‰€π‘ π‘Žβ‰€π‘‘β‰€π‘ X cana-4832 277 45 { { PUNCT cana-4832 277 46 |πœ‰(𝑑 |πœ‰(𝑑 PROPN cana-4832 277 47 ) ) PUNCT cana-4832 277 48 βˆ’ βˆ’ PROPN cana-4832 277 49 πœ‚(𝑑)|𝑝 πœ‚(𝑑)|𝑝 PROPN cana-4832 277 50 + + CCONJ cana-4832 277 51 |πœ‰(𝑑)|𝑝 |πœ‰(𝑑)|𝑝 PROPN cana-4832 277 52 } } PUNCT cana-4832 277 53 , , PUNCT cana-4832 277 54 where where SCONJ cana-4832 277 55 𝑝 𝑝 X cana-4832 277 56 > > SYM cana-4832 277 57 1 1 NUM cana-4832 277 58 a a DET cana-4832 277 59 real real ADJ cana-4832 277 60 number number NOUN cana-4832 277 61 , , PUNCT cana-4832 277 62 for for ADP cana-4832 277 63 all all DET cana-4832 277 64 πœ‰ πœ‰ NOUN cana-4832 277 65 , , PUNCT cana-4832 277 66 πœ‚ πœ‚ PROPN cana-4832 277 67 πœ– πœ– PROPN cana-4832 277 68 Ο‰ Ο‰ PROPN cana-4832 277 69 . . PUNCT cana-4832 278 1 therefore therefore ADV cana-4832 278 2 ( ( PUNCT cana-4832 278 3 Ο‰ Ο‰ NOUN cana-4832 278 4 , , PUNCT cana-4832 278 5 𝑑 𝑑 NOUN cana-4832 278 6 ) ) PUNCT cana-4832 278 7 is be AUX cana-4832 278 8 a a DET cana-4832 278 9 complete complete ADJ cana-4832 278 10 𝑏-metric 𝑏-metric ADJ cana-4832 278 11 space space NOUN cana-4832 278 12 with with ADP cana-4832 278 13 𝑠 𝑠 PROPN cana-4832 278 14 = = SYM cana-4832 278 15 2𝑝. 2𝑝. NUM cana-4832 278 16 many many ADJ cana-4832 278 17 author author NOUN cana-4832 278 18 's 's PART cana-4832 278 19 studied study VERB cana-4832 278 20 unique unique ADJ cana-4832 278 21 solution solution NOUN cana-4832 278 22 of of ADP cana-4832 278 23 a a DET cana-4832 278 24 nonlinear nonlinear ADJ cana-4832 278 25 integral integral ADJ cana-4832 278 26 equations equation NOUN cana-4832 278 27 [ [ X cana-4832 278 28 4 4 NUM cana-4832 278 29 - - SYM cana-4832 278 30 7 7 NUM cana-4832 278 31 ] ] PUNCT cana-4832 278 32 . . PUNCT cana-4832 279 1 in in ADP cana-4832 279 2 this this DET cana-4832 279 3 section section NOUN cana-4832 279 4 , , PUNCT cana-4832 279 5 we we PRON cana-4832 279 6 obtain obtain VERB cana-4832 279 7 the the DET cana-4832 279 8 existence existence NOUN cana-4832 279 9 of of ADP cana-4832 279 10 unique unique ADJ cana-4832 279 11 solution solution NOUN cana-4832 279 12 of of ADP cana-4832 279 13 a a DET cana-4832 279 14 nonlinear nonlinear ADJ cana-4832 279 15 integral integral ADJ cana-4832 279 16 equation equation NOUN cana-4832 279 17 of of ADP cana-4832 279 18 fredholm fredholm NOUN cana-4832 279 19 type type NOUN cana-4832 279 20 defined define VERB cana-4832 279 21 by by ADP cana-4832 279 22 πœ‰(𝑑 πœ‰(𝑑 NOUN cana-4832 279 23 ) ) PUNCT cana-4832 279 24 = = PUNCT cana-4832 280 1 𝑓(𝑑 𝑓(𝑑 VERB cana-4832 280 2 ) ) PUNCT cana-4832 281 1 + + CCONJ cana-4832 281 2 πœ‡ πœ‡ ADP cana-4832 281 3 ∫ ∫ PROPN cana-4832 281 4 β„³(𝑑 β„³(𝑑 X cana-4832 281 5 , , PUNCT cana-4832 281 6 π‘Ÿ π‘Ÿ NOUN cana-4832 281 7 , , PUNCT cana-4832 281 8 πœ‰(π‘Ÿ))π‘‘π‘Ÿ πœ‰(π‘Ÿ))π‘‘π‘Ÿ PROPN cana-4832 281 9 𝑏 𝑏 PROPN cana-4832 281 10 π‘Ž π‘Ž X cana-4832 281 11 ( ( PUNCT cana-4832 281 12 3.1 3.1 NUM cana-4832 281 13 ) ) PUNCT cana-4832 281 14 where where SCONJ cana-4832 281 15 πœ‰ πœ‰ PROPN cana-4832 281 16 πœ– πœ– PROPN cana-4832 281 17 Ο‰ Ο‰ PROPN cana-4832 281 18 is be AUX cana-4832 281 19 the the DET cana-4832 281 20 unknown unknown ADJ cana-4832 281 21 function function NOUN cana-4832 281 22 , , PUNCT cana-4832 281 23 πœ‡ πœ‡ X cana-4832 281 24 in in ADP cana-4832 281 25 ℝ ℝ PROPN cana-4832 281 26 , , PUNCT cana-4832 281 27 𝑑 𝑑 PROPN cana-4832 281 28 , , PUNCT cana-4832 281 29 π‘Ÿ π‘Ÿ X cana-4832 281 30 πœ– πœ– X cana-4832 281 31 [ [ X cana-4832 281 32 π‘Ž π‘Ž X cana-4832 281 33 , , PUNCT cana-4832 281 34 𝑏 𝑏 NOUN cana-4832 281 35 ] ] X cana-4832 281 36 , , PUNCT cana-4832 281 37 β„³:[π‘Ž β„³:[π‘Ž PROPN cana-4832 281 38 , , PUNCT cana-4832 281 39 𝑏 𝑏 NOUN cana-4832 281 40 ] ] X cana-4832 281 41 Γ— Γ— NOUN cana-4832 282 1 [ [ X cana-4832 282 2 π‘Ž π‘Ž X cana-4832 282 3 , , PUNCT cana-4832 282 4 𝑏 𝑏 NOUN cana-4832 282 5 ] ] X cana-4832 282 6 Γ— Γ— NOUN cana-4832 282 7 ℝ ℝ PROPN cana-4832 282 8 β†’ β†’ SYM cana-4832 282 9 ℝ ℝ PROPN cana-4832 282 10 and and CCONJ cana-4832 282 11 𝑓 𝑓 NOUN cana-4832 282 12 : : PUNCT cana-4832 283 1 [ [ X cana-4832 283 2 π‘Ž π‘Ž X cana-4832 283 3 , , PUNCT cana-4832 283 4 𝑏 𝑏 NOUN cana-4832 283 5 ] ] X cana-4832 283 6 β†’ β†’ PUNCT cana-4832 283 7 ℝ ℝ PROPN cana-4832 283 8 are be AUX cana-4832 283 9 continuous continuous ADJ cana-4832 283 10 functions function NOUN cana-4832 283 11 . . PUNCT cana-4832 284 1 let let VERB cana-4832 284 2 β„±:Ο‰ β„±:Ο‰ NOUN cana-4832 284 3 β†’ β†’ SYM cana-4832 284 4 Ο‰ Ο‰ PROPN cana-4832 284 5 be be AUX cana-4832 284 6 a a DET cana-4832 284 7 mapping mapping NOUN cana-4832 284 8 defined define VERB cana-4832 284 9 by by ADP cana-4832 284 10 β„±(πœ‰(𝑑 β„±(πœ‰(𝑑 NOUN cana-4832 284 11 ) ) PUNCT cana-4832 284 12 ) ) PUNCT cana-4832 285 1 = = SYM cana-4832 285 2 𝑓(𝑑 𝑓(𝑑 VERB cana-4832 285 3 ) ) PUNCT cana-4832 286 1 + + CCONJ cana-4832 286 2 πœ‡ πœ‡ ADP cana-4832 286 3 ∫ ∫ PROPN cana-4832 286 4 β„³(𝑑 β„³(𝑑 X cana-4832 286 5 , , PUNCT cana-4832 286 6 π‘Ÿ π‘Ÿ NOUN cana-4832 286 7 , , PUNCT cana-4832 286 8 πœ‰(π‘Ÿ))π‘‘π‘Ÿ πœ‰(π‘Ÿ))π‘‘π‘Ÿ PROPN cana-4832 287 1 𝑏 𝑏 PROPN cana-4832 287 2 π‘Ž π‘Ž PROPN cana-4832 287 3 ( ( PUNCT cana-4832 287 4 3.2 3.2 NUM cana-4832 287 5 ) ) PUNCT cana-4832 287 6 theorem theorem VERB cana-4832 287 7 3.1 3.1 NUM cana-4832 287 8 . . PUNCT cana-4832 288 1 let let VERB cana-4832 288 2 β„±:Ο‰ β„±:Ο‰ NOUN cana-4832 288 3 β†’ β†’ SYM cana-4832 288 4 Ο‰ Ο‰ PROPN cana-4832 288 5 be be AUX cana-4832 288 6 a a DET cana-4832 288 7 mapping mapping NOUN cana-4832 288 8 defined define VERB cana-4832 288 9 by by ADP cana-4832 288 10 ( ( PUNCT cana-4832 288 11 3.2 3.2 NUM cana-4832 288 12 ) ) PUNCT cana-4832 288 13 and and CCONJ cana-4832 288 14 there there PRON cana-4832 288 15 exists exist VERB cana-4832 288 16 a a DET cana-4832 288 17 constant constant ADJ cana-4832 288 18 𝐾 𝐾 NOUN cana-4832 288 19 > > X cana-4832 288 20 1 1 NUM cana-4832 288 21 such such ADJ cana-4832 288 22 that that PRON cana-4832 288 23 for for ADP cana-4832 288 24 all all DET cana-4832 288 25 𝑑 𝑑 NOUN cana-4832 288 26 , , PUNCT cana-4832 288 27 π‘Ÿ π‘Ÿ X cana-4832 288 28 πœ– πœ– X cana-4832 288 29 [ [ X cana-4832 288 30 π‘Ž π‘Ž X cana-4832 288 31 , , PUNCT cana-4832 288 32 𝑏 𝑏 NOUN cana-4832 288 33 ] ] PUNCT cana-4832 288 34 and and CCONJ cana-4832 288 35 πœ‰1 πœ‰1 PROPN cana-4832 288 36 , , PUNCT cana-4832 288 37 πœ‰2 πœ‰2 PROPN cana-4832 288 38 πœ– πœ– PROPN cana-4832 288 39 Ο‰ Ο‰ PROPN cana-4832 288 40 with with ADP cana-4832 288 41 |πœ‡| |πœ‡| PROPN cana-4832 288 42 β‰₯ β‰₯ PROPN cana-4832 288 43 1 1 NUM cana-4832 288 44 , , PUNCT cana-4832 288 45 the the DET cana-4832 288 46 following follow VERB cana-4832 288 47 condition condition NOUN cana-4832 288 48 is be AUX cana-4832 288 49 satisfied satisfied ADJ cana-4832 288 50 : : PUNCT cana-4832 288 51 |∫ |∫ X cana-4832 288 52 [ [ X cana-4832 288 53 β„³(𝑑 β„³(𝑑 X cana-4832 288 54 , , PUNCT cana-4832 288 55 π‘Ÿ π‘Ÿ X cana-4832 288 56 , , PUNCT cana-4832 288 57 πœ‰1(π‘Ÿ πœ‰1(π‘Ÿ PROPN cana-4832 288 58 ) ) PUNCT cana-4832 288 59 ) ) PUNCT cana-4832 289 1 βˆ’β„³(𝑑 βˆ’β„³(𝑑 PROPN cana-4832 289 2 , , PUNCT cana-4832 289 3 π‘Ÿ π‘Ÿ NOUN cana-4832 289 4 , , PUNCT cana-4832 289 5 πœ‰2(π‘Ÿ))]π‘‘π‘Ÿ πœ‰2(π‘Ÿ))]π‘‘π‘Ÿ NOUN cana-4832 290 1 𝑏 𝑏 PROPN cana-4832 290 2 π‘Ž π‘Ž PROPN cana-4832 290 3 | | NOUN cana-4832 290 4 𝑝 𝑝 ADJ cana-4832 290 5 β‰₯ β‰₯ NOUN cana-4832 290 6 𝐾|πœ‰1(𝑑 𝐾|πœ‰1(𝑑 NOUN cana-4832 290 7 ) ) PUNCT cana-4832 291 1 βˆ’ βˆ’ PROPN cana-4832 291 2 πœ‰2(𝑑)| πœ‰2(𝑑)| PROPN cana-4832 291 3 𝑝 𝑝 PROPN cana-4832 291 4 + + CCONJ cana-4832 291 5 𝐾|πœ‰1(𝑑)| 𝐾|πœ‰1(𝑑)| NOUN cana-4832 291 6 𝑝. 𝑝. NOUN cana-4832 291 7 then then ADV cana-4832 291 8 the the DET cana-4832 291 9 system system NOUN cana-4832 291 10 of of ADP cana-4832 291 11 nonlinear nonlinear ADJ cana-4832 291 12 integral integral ADJ cana-4832 291 13 equations equation NOUN cana-4832 291 14 ( ( PUNCT cana-4832 291 15 3.1 3.1 NUM cana-4832 291 16 ) ) PUNCT cana-4832 291 17 has have VERB cana-4832 291 18 a a DET cana-4832 291 19 unique unique ADJ cana-4832 291 20 solution solution NOUN cana-4832 291 21 in in ADP cana-4832 291 22 Ο‰ Ο‰ PROPN cana-4832 291 23 . . PUNCT cana-4832 292 1 proof proof NOUN cana-4832 292 2 . . PUNCT cana-4832 293 1 let let VERB cana-4832 293 2 πœ‰1 πœ‰1 PROPN cana-4832 293 3 , , PUNCT cana-4832 293 4 πœ‰2 πœ‰2 PROPN cana-4832 293 5 πœ– πœ– PROPN cana-4832 293 6 Ο‰ Ο‰ PROPN cana-4832 293 7 and and CCONJ cana-4832 293 8 for for ADP cana-4832 293 9 all all DET cana-4832 293 10 𝑑 𝑑 X cana-4832 293 11 πœ– πœ– X cana-4832 293 12 [ [ X cana-4832 293 13 π‘Ž π‘Ž X cana-4832 293 14 , , PUNCT cana-4832 293 15 𝑏 𝑏 NOUN cana-4832 293 16 ] ] PUNCT cana-4832 293 17 , , PUNCT cana-4832 293 18 we we PRON cana-4832 293 19 have have VERB cana-4832 293 20 𝑑(β„±πœ‰1 𝑑(β„±πœ‰1 VERB cana-4832 293 21 , , PUNCT cana-4832 293 22 β„±πœ‰2 β„±πœ‰2 ADJ cana-4832 293 23 ) ) PUNCT cana-4832 294 1 = = SYM cana-4832 294 2 max max PROPN cana-4832 294 3 π‘Žβ‰€π‘‘β‰€π‘ π‘Žβ‰€π‘‘β‰€π‘ X cana-4832 294 4 { { PUNCT cana-4832 294 5 |β„±πœ‰1(𝑑 |β„±πœ‰1(𝑑 NOUN cana-4832 294 6 ) ) PUNCT cana-4832 294 7 βˆ’ βˆ’ PROPN cana-4832 294 8 β„±πœ‰2(𝑑)| β„±πœ‰2(𝑑)| X cana-4832 294 9 𝑝 𝑝 PROPN cana-4832 294 10 + + CCONJ cana-4832 294 11 |β„±πœ‰1(𝑑)| |β„±πœ‰1(𝑑)| PROPN cana-4832 294 12 𝑝 𝑝 PROPN cana-4832 294 13 } } PUNCT cana-4832 294 14 β‰₯ β‰₯ NOUN cana-4832 294 15 |πœ‡|𝑝max |πœ‡|𝑝max X cana-4832 294 16 π‘Žβ‰€π‘‘β‰€π‘ π‘Žβ‰€π‘‘β‰€π‘ X cana-4832 294 17 { { PUNCT cana-4832 294 18 |∫ |∫ X cana-4832 294 19 β„³(𝑑 β„³(𝑑 X cana-4832 294 20 , , PUNCT cana-4832 294 21 π‘Ÿ π‘Ÿ NOUN cana-4832 294 22 , , PUNCT cana-4832 294 23 πœ‰1(π‘Ÿ))π‘‘π‘Ÿ πœ‰1(π‘Ÿ))π‘‘π‘Ÿ ADV cana-4832 294 24 𝑏 𝑏 ADP cana-4832 294 25 π‘Ž π‘Ž PRON cana-4832 294 26 βˆ’βˆ« βˆ’βˆ« NOUN cana-4832 294 27 β„³(𝑑 β„³(𝑑 NOUN cana-4832 294 28 , , PUNCT cana-4832 294 29 π‘Ÿ π‘Ÿ NOUN cana-4832 294 30 , , PUNCT cana-4832 294 31 πœ‰2(π‘Ÿ))π‘‘π‘Ÿ πœ‰2(π‘Ÿ))π‘‘π‘Ÿ PUNCT cana-4832 295 1 𝑏 𝑏 PROPN cana-4832 295 2 π‘Ž π‘Ž PROPN cana-4832 295 3 | | NOUN cana-4832 295 4 𝑝 𝑝 NOUN cana-4832 295 5 + + CCONJ cana-4832 295 6 |∫ |∫ X cana-4832 295 7 β„³(𝑑 β„³(𝑑 X cana-4832 295 8 , , PUNCT cana-4832 295 9 π‘Ÿ π‘Ÿ NOUN cana-4832 295 10 , , PUNCT cana-4832 295 11 πœ‰1(π‘Ÿ))π‘‘π‘Ÿ πœ‰1(π‘Ÿ))π‘‘π‘Ÿ ADV cana-4832 295 12 𝑏 𝑏 ADP cana-4832 295 13 π‘Ž π‘Ž ADP cana-4832 295 14 | | NOUN cana-4832 295 15 𝑝 𝑝 NOUN cana-4832 295 16 } } PUNCT cana-4832 295 17 β‰₯ β‰₯ NOUN cana-4832 295 18 |πœ‡|𝑝max |πœ‡|𝑝max X cana-4832 295 19 π‘Žβ‰€π‘‘β‰€π‘ π‘Žβ‰€π‘‘β‰€π‘ X cana-4832 295 20 { { PUNCT cana-4832 295 21 |∫ |∫ X cana-4832 295 22 [ [ X cana-4832 295 23 β„³(𝑑 β„³(𝑑 X cana-4832 295 24 , , PUNCT cana-4832 295 25 π‘Ÿ π‘Ÿ X cana-4832 295 26 , , PUNCT cana-4832 295 27 πœ‰1(π‘Ÿ πœ‰1(π‘Ÿ PROPN cana-4832 295 28 ) ) PUNCT cana-4832 295 29 ) ) PUNCT cana-4832 296 1 βˆ’β„³(𝑑 βˆ’β„³(𝑑 PROPN cana-4832 296 2 , , PUNCT cana-4832 296 3 π‘Ÿ π‘Ÿ NOUN cana-4832 296 4 , , PUNCT cana-4832 296 5 πœ‰2(π‘Ÿ))]π‘‘π‘Ÿ πœ‰2(π‘Ÿ))]π‘‘π‘Ÿ NOUN cana-4832 297 1 𝑏 𝑏 PROPN cana-4832 297 2 π‘Ž π‘Ž PROPN cana-4832 297 3 | | NOUN cana-4832 297 4 𝑝 𝑝 ADJ cana-4832 297 5 } } PUNCT cana-4832 297 6 β‰₯ β‰₯ NOUN cana-4832 297 7 𝐾max 𝐾max PROPN cana-4832 297 8 π‘Žβ‰€π‘‘β‰€π‘ π‘Žβ‰€π‘‘β‰€π‘ PROPN cana-4832 297 9 { { PUNCT cana-4832 297 10 |πœ‰1(𝑑 |πœ‰1(𝑑 NOUN cana-4832 297 11 ) ) PUNCT cana-4832 297 12 βˆ’ βˆ’ PROPN cana-4832 297 13 πœ‰2(𝑑)| πœ‰2(𝑑)| PROPN cana-4832 297 14 𝑝 𝑝 PROPN cana-4832 297 15 + + CCONJ cana-4832 297 16 𝐾|πœ‰1(𝑑)| 𝐾|πœ‰1(𝑑)| NOUN cana-4832 297 17 𝑝 𝑝 NOUN cana-4832 297 18 } } PUNCT cana-4832 297 19 = = SYM cana-4832 297 20 𝐾𝑑(πœ‰1 𝐾𝑑(πœ‰1 NOUN cana-4832 297 21 , , PUNCT cana-4832 297 22 πœ‰2 πœ‰2 PROPN cana-4832 297 23 ) ) PUNCT cana-4832 297 24 which which PRON cana-4832 297 25 implies imply VERB cana-4832 297 26 that that SCONJ cana-4832 297 27 𝑑(β„±πœ‰1 𝑑(β„±πœ‰1 PROPN cana-4832 297 28 , , PUNCT cana-4832 297 29 β„±πœ‰2 β„±πœ‰2 X cana-4832 297 30 ) ) PUNCT cana-4832 297 31 β‰₯ β‰₯ PROPN cana-4832 297 32 𝐾𝑑(πœ‰1 𝐾𝑑(πœ‰1 NOUN cana-4832 297 33 , , PUNCT cana-4832 297 34 πœ‰2 πœ‰2 PROPN cana-4832 297 35 ) ) PUNCT cana-4832 297 36 . . PUNCT cana-4832 298 1 communications communication NOUN cana-4832 298 2 on on ADP cana-4832 298 3 applied apply VERB cana-4832 298 4 nonlinear nonlinear ADJ cana-4832 298 5 analysis analysis NOUN cana-4832 298 6 issn issn NOUN cana-4832 298 7 : : PUNCT cana-4832 298 8 1074 1074 NUM cana-4832 298 9 - - PUNCT cana-4832 298 10 133x 133x NUM cana-4832 298 11 vol vol VERB cana-4832 298 12 32 32 NUM cana-4832 298 13 no no NOUN cana-4832 298 14 . . PUNCT cana-4832 299 1 10s 10 NOUN cana-4832 299 2 ( ( PUNCT cana-4832 299 3 2025 2025 NUM cana-4832 299 4 ) ) PUNCT cana-4832 299 5 404 404 NUM cana-4832 299 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 299 7 therefore therefore ADV cana-4832 299 8 , , PUNCT cana-4832 299 9 all all DET cana-4832 299 10 the the DET cana-4832 299 11 conditions condition NOUN cana-4832 299 12 of of ADP cana-4832 299 13 corollary corollary ADJ cana-4832 299 14 2.3 2.3 NUM cana-4832 299 15 are be AUX cana-4832 299 16 satisfied satisfied ADJ cana-4832 299 17 , , PUNCT cana-4832 299 18 and and CCONJ cana-4832 299 19 hence hence ADV cana-4832 299 20 β„± β„± PROPN cana-4832 299 21 has have VERB cana-4832 299 22 a a DET cana-4832 299 23 unique unique ADJ cana-4832 299 24 solution solution NOUN cana-4832 299 25 for for ADP cana-4832 299 26 nonlinear nonlinear ADJ cana-4832 299 27 integral integral ADJ cana-4832 299 28 equations equation NOUN cana-4832 299 29 defined define VERB cana-4832 299 30 in in ADP cana-4832 299 31 ( ( PUNCT cana-4832 299 32 3.1 3.1 NUM cana-4832 299 33 ) ) PUNCT cana-4832 299 34 . . PUNCT cana-4832 300 1 4 4 X cana-4832 300 2 . . X cana-4832 300 3 conclusion conclusion NOUN cana-4832 300 4 and and CCONJ cana-4832 300 5 future future ADJ cana-4832 300 6 work work NOUN cana-4832 300 7 in in ADP cana-4832 300 8 this this DET cana-4832 300 9 paper paper NOUN cana-4832 300 10 , , PUNCT cana-4832 300 11 we we PRON cana-4832 300 12 studied study VERB cana-4832 300 13 fixed fix VERB cana-4832 300 14 point point NOUN cana-4832 300 15 results result NOUN cana-4832 300 16 for for ADP cana-4832 300 17 expansive expansive ADJ cana-4832 300 18 mappings mapping NOUN cana-4832 300 19 in in ADP cana-4832 300 20 dislocated dislocated ADJ cana-4832 300 21 quasi-𝑏-metric quasi-𝑏-metric ADJ cana-4832 300 22 spaces space NOUN cana-4832 300 23 . . PUNCT cana-4832 301 1 using use VERB cana-4832 301 2 similar similar ADJ cana-4832 301 3 approaches approach NOUN cana-4832 301 4 , , PUNCT cana-4832 301 5 it it PRON cana-4832 301 6 can can AUX cana-4832 301 7 be be AUX cana-4832 301 8 studied study VERB cana-4832 301 9 new new ADJ cana-4832 301 10 fixed fix VERB cana-4832 301 11 point point NOUN cana-4832 301 12 results result NOUN cana-4832 301 13 on on ADP cana-4832 301 14 metric metric ADJ cana-4832 301 15 and and CCONJ cana-4832 301 16 some some DET cana-4832 301 17 generalized generalized ADJ cana-4832 301 18 metric metric ADJ cana-4832 301 19 spaces space NOUN cana-4832 301 20 . . PUNCT cana-4832 302 1 the the DET cana-4832 302 2 investigation investigation NOUN cana-4832 302 3 of of ADP cana-4832 302 4 certain certain ADJ cana-4832 302 5 circumstances circumstance NOUN cana-4832 302 6 to to PART cana-4832 302 7 exclude exclude VERB cana-4832 302 8 the the DET cana-4832 302 9 identity identity NOUN cana-4832 302 10 map map NOUN cana-4832 302 11 of of ADP cana-4832 302 12 𝑋 𝑋 PROPN cana-4832 302 13 from from ADP cana-4832 302 14 theorem theorem ADJ cana-4832 302 15 2.1 2.1 NUM cana-4832 302 16 and and CCONJ cana-4832 302 17 theorem theorem VERB cana-4832 302 18 2.8 2.8 NUM cana-4832 302 19 and and CCONJ cana-4832 302 20 related related ADJ cana-4832 302 21 results result NOUN cana-4832 302 22 is be AUX cana-4832 302 23 a a DET cana-4832 302 24 worthwhile worthwhile ADJ cana-4832 302 25 problem problem NOUN cana-4832 302 26 for for ADP cana-4832 302 27 future future ADJ cana-4832 302 28 effort effort NOUN cana-4832 302 29 . . PUNCT cana-4832 303 1 refrences refrence VERB cana-4832 303 2 [ [ X cana-4832 303 3 1 1 X cana-4832 303 4 ] ] PUNCT cana-4832 303 5 c. c. NOUN cana-4832 303 6 aage aage PROPN cana-4832 303 7 and and CCONJ cana-4832 303 8 j. j. PROPN cana-4832 303 9 salunke salunke PROPN cana-4832 303 10 , , PUNCT cana-4832 303 11 the the DET cana-4832 303 12 results result NOUN cana-4832 303 13 on on ADP cana-4832 303 14 fixed fix VERB cana-4832 303 15 points point NOUN cana-4832 303 16 in in ADP cana-4832 303 17 dislocated dislocated ADJ cana-4832 303 18 and and CCONJ cana-4832 303 19 dislocated dislocated ADJ cana-4832 303 20 quasi quasi ADJ cana-4832 303 21 - - ADJ cana-4832 303 22 metric metric ADJ cana-4832 303 23 space space NOUN cana-4832 303 24 , , PUNCT cana-4832 303 25 appl appl PROPN cana-4832 303 26 . . PROPN cana-4832 303 27 math math PROPN cana-4832 303 28 . . PUNCT cana-4832 304 1 sci sci PROPN cana-4832 304 2 . . PROPN cana-4832 304 3 2 2 NUM cana-4832 304 4 ( ( PUNCT cana-4832 304 5 59 59 NUM cana-4832 304 6 ) ) PUNCT cana-4832 304 7 ( ( PUNCT cana-4832 304 8 2008 2008 NUM cana-4832 304 9 ) ) PUNCT cana-4832 304 10 , , PUNCT cana-4832 304 11 2941 2941 NUM cana-4832 304 12 - - SYM cana-4832 304 13 2948 2948 NUM cana-4832 304 14 . . PUNCT cana-4832 305 1 [ [ X cana-4832 305 2 2 2 NUM cana-4832 305 3 ] ] PUNCT cana-4832 305 4 a. a. NOUN cana-4832 305 5 aghajani aghajani PROPN cana-4832 305 6 , , PUNCT cana-4832 305 7 m. m. NOUN cana-4832 305 8 abbas abbas PROPN cana-4832 305 9 and and CCONJ cana-4832 305 10 j. j. PROPN cana-4832 305 11 r. r. PROPN cana-4832 305 12 roshan roshan PROPN cana-4832 305 13 , , PUNCT cana-4832 305 14 common common ADJ cana-4832 305 15 fixed fix VERB cana-4832 305 16 point point NOUN cana-4832 305 17 of of ADP cana-4832 305 18 generalized generalized ADJ cana-4832 305 19 weak weak ADJ cana-4832 305 20 contractive contractive ADJ cana-4832 305 21 mappings mapping NOUN cana-4832 305 22 in in ADP cana-4832 305 23 partially partially ADV cana-4832 305 24 ordered order VERB cana-4832 305 25 𝑏-metric 𝑏-metric PROPN cana-4832 305 26 spaces space NOUN cana-4832 305 27 , , PUNCT cana-4832 305 28 mathematica mathematica PROPN cana-4832 305 29 slovaca slovaca PROPN cana-4832 305 30 , , PUNCT cana-4832 305 31 64(4)(2014 64(4)(2014 NUM cana-4832 305 32 ) ) PUNCT cana-4832 305 33 , , PUNCT cana-4832 305 34 941 941 NUM cana-4832 305 35 - - SYM cana-4832 305 36 960 960 NUM cana-4832 305 37 . . PUNCT cana-4832 306 1 [ [ X cana-4832 306 2 3 3 X cana-4832 306 3 ] ] X cana-4832 306 4 m. m. NOUN cana-4832 306 5 ahmed ahmed PROPN cana-4832 306 6 , , PUNCT cana-4832 306 7 a a DET cana-4832 306 8 common common ADJ cana-4832 306 9 fixed fix VERB cana-4832 306 10 point point NOUN cana-4832 306 11 theorem theorem VERB cana-4832 306 12 for for ADP cana-4832 306 13 expansive expansive ADJ cana-4832 306 14 mappings mapping NOUN cana-4832 306 15 in in ADP cana-4832 306 16 2 2 NUM cana-4832 306 17 - - PUNCT cana-4832 306 18 metric metric ADJ cana-4832 306 19 spaces space NOUN cana-4832 306 20 and and CCONJ cana-4832 306 21 its its PRON cana-4832 306 22 application application NOUN cana-4832 306 23 , , PUNCT cana-4832 306 24 chaos chaos NOUN cana-4832 306 25 , , PUNCT cana-4832 306 26 solitons soliton NOUN cana-4832 306 27 fractals fractal VERB cana-4832 306 28 42 42 NUM cana-4832 306 29 ( ( PUNCT cana-4832 306 30 5 5 NUM cana-4832 306 31 ) ) PUNCT cana-4832 306 32 ( ( PUNCT cana-4832 306 33 2009 2009 NUM cana-4832 306 34 ) ) PUNCT cana-4832 306 35 , , PUNCT cana-4832 306 36 2914 2914 NUM cana-4832 306 37 - - SYM cana-4832 306 38 2920 2920 NUM cana-4832 306 39 . . PUNCT cana-4832 307 1 [ [ X cana-4832 307 2 4 4 X cana-4832 307 3 ] ] X cana-4832 307 4 d. d. PROPN cana-4832 307 5 r. r. PROPN cana-4832 307 6 babu babu PROPN cana-4832 307 7 , , PUNCT cana-4832 307 8 some some DET cana-4832 307 9 best good ADJ cana-4832 307 10 proximity proximity NOUN cana-4832 307 11 theorems theorem NOUN cana-4832 307 12 for for ADP cana-4832 307 13 generalized generalized ADJ cana-4832 307 14 proximal proximal PROPN cana-4832 307 15 𝒡-contraction 𝒡-contraction PROPN cana-4832 307 16 maps map NOUN cana-4832 307 17 in in ADP cana-4832 307 18 𝑏metric 𝑏metric ADJ cana-4832 307 19 spaces space NOUN cana-4832 307 20 with with ADP cana-4832 307 21 applications application NOUN cana-4832 307 22 , , PUNCT cana-4832 307 23 sahand sahand NOUN cana-4832 307 24 commun commun PROPN cana-4832 307 25 . . PUNCT cana-4832 307 26 math math PROPN cana-4832 307 27 . . PUNCT cana-4832 308 1 anal anal PROPN cana-4832 308 2 . . PUNCT cana-4832 308 3 , , PUNCT cana-4832 308 4 ( ( PUNCT cana-4832 308 5 in in ADP cana-4832 308 6 press press NOUN cana-4832 308 7 ) ) PUNCT cana-4832 308 8 , , PUNCT cana-4832 308 9 https://doi.org/10.22130/scma.2024.2042087.1910 https://doi.org/10.22130/scma.2024.2042087.1910 X cana-4832 308 10 [ [ X cana-4832 308 11 5 5 X cana-4832 308 12 ] ] PUNCT cana-4832 308 13 d. d. PROPN cana-4832 308 14 r. r. PROPN cana-4832 308 15 babu babu PROPN cana-4832 308 16 , , PUNCT cana-4832 308 17 k. k. PROPN cana-4832 308 18 b. b. PROPN cana-4832 308 19 chander chander PROPN cana-4832 308 20 , , PUNCT cana-4832 308 21 t. t. PROPN cana-4832 308 22 v. v. PROPN cana-4832 308 23 p. p. PROPN cana-4832 308 24 kumar kumar PROPN cana-4832 308 25 , , PUNCT cana-4832 308 26 n.siva n.siva NOUN cana-4832 308 27 prasad prasad PROPN cana-4832 308 28 and and CCONJ cana-4832 308 29 k. k. PROPN cana-4832 308 30 narayana narayana PROPN cana-4832 308 31 , , PUNCT cana-4832 308 32 fixed fix VERB cana-4832 308 33 points point NOUN cana-4832 308 34 of of ADP cana-4832 308 35 cyclic cyclic ADJ cana-4832 308 36 ( ( PUNCT cana-4832 308 37 οΏ½ οΏ½ PROPN cana-4832 308 38 ̈ ̈ SYM cana-4832 308 39 οΏ½ οΏ½ PROPN cana-4832 308 40 , , PUNCT cana-4832 308 41 οΏ½ οΏ½ PROPN cana-4832 308 42 ̈ ̈ SYM cana-4832 308 43 οΏ½ οΏ½ SYM cana-4832 308 44 )-admissible )-admissible ADJ cana-4832 308 45 generalized generalized ADJ cana-4832 308 46 contraction contraction NOUN cana-4832 308 47 type type NOUN cana-4832 308 48 maps map NOUN cana-4832 308 49 in in ADP cana-4832 308 50 𝑏-metric 𝑏-metric PROPN cana-4832 308 51 spaces space NOUN cana-4832 308 52 with with ADP cana-4832 308 53 applications application NOUN cana-4832 308 54 , , PUNCT cana-4832 308 55 appl appl PROPN cana-4832 308 56 . . PROPN cana-4832 308 57 math math NOUN cana-4832 308 58 . . PUNCT cana-4832 309 1 e e X cana-4832 309 2 - - NOUN cana-4832 309 3 notes note NOUN cana-4832 309 4 , , PUNCT cana-4832 309 5 24(2024 24(2024 NUM cana-4832 309 6 ) ) PUNCT cana-4832 309 7 , , PUNCT cana-4832 309 8 379 379 NUM cana-4832 309 9 - - SYM cana-4832 309 10 398 398 NUM cana-4832 309 11 . . PUNCT cana-4832 310 1 [ [ X cana-4832 310 2 6 6 NUM cana-4832 310 3 ] ] PUNCT cana-4832 310 4 d. d. PROPN cana-4832 310 5 r. r. PROPN cana-4832 310 6 babu babu PROPN cana-4832 310 7 , , PUNCT cana-4832 310 8 k. k. PROPN cana-4832 310 9 b. b. PROPN cana-4832 310 10 chander chander PROPN cana-4832 310 11 , , PUNCT cana-4832 310 12 n. n. PROPN cana-4832 310 13 siva siva PROPN cana-4832 310 14 prasad prasad PROPN cana-4832 310 15 , , PUNCT cana-4832 310 16 shaik shaik PROPN cana-4832 310 17 asha asha PROPN cana-4832 310 18 , , PUNCT cana-4832 310 19 e. e. PROPN cana-4832 310 20 sundesh sundesh PROPN cana-4832 310 21 babu babu PROPN cana-4832 310 22 and and CCONJ cana-4832 310 23 t. t. PROPN cana-4832 310 24 v. v. PROPN cana-4832 310 25 p. p. PROPN cana-4832 310 26 kumar kumar PROPN cana-4832 310 27 , , PUNCT cana-4832 310 28 some some DET cana-4832 310 29 coupled couple VERB cana-4832 310 30 fixed fix VERB cana-4832 310 31 point point NOUN cana-4832 310 32 theorems theorem NOUN cana-4832 310 33 on on ADP cana-4832 310 34 orthogonal orthogonal ADJ cana-4832 310 35 𝑏-metric 𝑏-metric PROPN cana-4832 310 36 spaces space NOUN cana-4832 310 37 with with ADP cana-4832 310 38 applications application NOUN cana-4832 310 39 , , PUNCT cana-4832 310 40 bull bull NOUN cana-4832 310 41 . . PUNCT cana-4832 310 42 math math NOUN cana-4832 310 43 . . PUNCT cana-4832 311 1 anal anal PROPN cana-4832 311 2 . . PUNCT cana-4832 311 3 appl appl PROPN cana-4832 311 4 . . PROPN cana-4832 311 5 , , PUNCT cana-4832 311 6 16(3)(2024 16(3)(2024 NUM cana-4832 311 7 ) ) PUNCT cana-4832 311 8 , , PUNCT cana-4832 311 9 45 45 NUM cana-4832 311 10 - - SYM cana-4832 311 11 61 61 NUM cana-4832 311 12 . . PUNCT cana-4832 312 1 [ [ X cana-4832 312 2 7 7 X cana-4832 312 3 ] ] X cana-4832 312 4 d. d. PROPN cana-4832 312 5 r. r. PROPN cana-4832 312 6 babu babu PROPN cana-4832 312 7 , , PUNCT cana-4832 312 8 n. n. PROPN cana-4832 312 9 siva siva PROPN cana-4832 312 10 prasad prasad PROPN cana-4832 312 11 , , PUNCT cana-4832 312 12 v. v. ADP cana-4832 312 13 a. a. NOUN cana-4832 312 14 babu babu PROPN cana-4832 312 15 and and CCONJ cana-4832 312 16 k. k. PROPN cana-4832 312 17 b. b. PROPN cana-4832 312 18 chander chander PROPN cana-4832 312 19 , , PUNCT cana-4832 312 20 some some DET cana-4832 312 21 common common ADJ cana-4832 312 22 fixed fix VERB cana-4832 312 23 point point NOUN cana-4832 312 24 theorems theorem NOUN cana-4832 312 25 in in ADP cana-4832 312 26 𝑏-metric 𝑏-metric PROPN cana-4832 312 27 spaces space NOUN cana-4832 312 28 via via ADP cana-4832 312 29 β„±-class β„±-class PROPN cana-4832 312 30 function function NOUN cana-4832 312 31 with with ADP cana-4832 312 32 applications application NOUN cana-4832 312 33 , , PUNCT cana-4832 312 34 adv adv PROPN cana-4832 312 35 . . PUNCT cana-4832 312 36 fixed fix VERB cana-4832 312 37 point point NOUN cana-4832 312 38 theory theory NOUN cana-4832 312 39 , , PUNCT cana-4832 312 40 14 14 NUM cana-4832 312 41 ( ( PUNCT cana-4832 312 42 24 24 NUM cana-4832 312 43 ) ) PUNCT cana-4832 312 44 ( ( PUNCT cana-4832 312 45 2024 2024 NUM cana-4832 312 46 ) ) PUNCT cana-4832 312 47 , , PUNCT cana-4832 312 48 38 38 NUM cana-4832 312 49 pages page NOUN cana-4832 312 50 , , PUNCT cana-4832 312 51 https://doi.org/10.28919/afpt/8515 https://doi.org/10.28919/afpt/8515 NOUN cana-4832 312 52 [ [ X cana-4832 312 53 8 8 NUM cana-4832 312 54 ] ] PUNCT cana-4832 312 55 s. s. PROPN cana-4832 312 56 czerwik czerwik PROPN cana-4832 312 57 , , PUNCT cana-4832 312 58 contraction contraction NOUN cana-4832 312 59 mappings mapping NOUN cana-4832 312 60 in in ADP cana-4832 312 61 𝑏-metric 𝑏-metric PROPN cana-4832 312 62 spaces space NOUN cana-4832 312 63 , , PUNCT cana-4832 312 64 acta acta PROPN cana-4832 312 65 math math PROPN cana-4832 312 66 . . PUNCT cana-4832 313 1 inform inform NOUN cana-4832 313 2 . . PUNCT cana-4832 314 1 univ univ PROPN cana-4832 314 2 . . PUNCT cana-4832 314 3 ostraviensis ostraviensis NOUN cana-4832 314 4 , , PUNCT cana-4832 314 5 1(1993 1(1993 NUM cana-4832 314 6 ) ) PUNCT cana-4832 314 7 , , PUNCT cana-4832 314 8 5 5 NUM cana-4832 314 9 - - SYM cana-4832 314 10 11 11 NUM cana-4832 314 11 . . PUNCT cana-4832 315 1 [ [ X cana-4832 315 2 9 9 NUM cana-4832 315 3 ] ] X cana-4832 315 4 h. h. PROPN cana-4832 315 5 das das PROPN cana-4832 315 6 and and CCONJ cana-4832 315 7 n. n. PROPN cana-4832 315 8 goswami goswami PROPN cana-4832 315 9 , , PUNCT cana-4832 315 10 expansive expansive ADJ cana-4832 315 11 type type NOUN cana-4832 315 12 mappings mapping NOUN cana-4832 315 13 in in ADP cana-4832 315 14 dislocated dislocated ADJ cana-4832 315 15 quasi quasi ADJ cana-4832 315 16 - - ADJ cana-4832 315 17 metric metric ADJ cana-4832 315 18 space space NOUN cana-4832 315 19 with with ADP cana-4832 315 20 some some DET cana-4832 315 21 fixed fix VERB cana-4832 315 22 point point NOUN cana-4832 315 23 results result NOUN cana-4832 315 24 and and CCONJ cana-4832 315 25 application application NOUN cana-4832 315 26 , , PUNCT cana-4832 315 27 korean korean PROPN cana-4832 315 28 j. j. PROPN cana-4832 315 29 math math PROPN cana-4832 315 30 . . PUNCT cana-4832 316 1 32(2)(2024 32(2)(2024 NUM cana-4832 316 2 ) ) PUNCT cana-4832 316 3 , , PUNCT cana-4832 316 4 245 245 NUM cana-4832 316 5 - - SYM cana-4832 316 6 257 257 NUM cana-4832 316 7 . . PUNCT cana-4832 317 1 [ [ X cana-4832 317 2 10 10 NUM cana-4832 317 3 ] ] X cana-4832 317 4 r. r. PROPN cana-4832 317 5 daheriya daheriya PROPN cana-4832 317 6 , , PUNCT cana-4832 317 7 r. r. PROPN cana-4832 317 8 jain jain PROPN cana-4832 317 9 and and CCONJ cana-4832 317 10 m. m. PROPN cana-4832 317 11 ughade ughade PROPN cana-4832 317 12 , , PUNCT cana-4832 317 13 some some DET cana-4832 317 14 fixed fix VERB cana-4832 317 15 point point NOUN cana-4832 317 16 theorem theorem VERB cana-4832 317 17 for for ADP cana-4832 317 18 expansive expansive ADJ cana-4832 317 19 type type NOUN cana-4832 317 20 mapping mapping NOUN cana-4832 317 21 in in ADP cana-4832 317 22 dislocated dislocate VERB cana-4832 317 23 metric metric ADJ cana-4832 317 24 space space NOUN cana-4832 317 25 , , PUNCT cana-4832 317 26 isrn isrn PROPN cana-4832 317 27 math math NOUN cana-4832 317 28 . . PUNCT cana-4832 318 1 anal anal ADJ cana-4832 318 2 . . PUNCT cana-4832 319 1 2012 2012 NUM cana-4832 319 2 ( ( PUNCT cana-4832 319 3 2012 2012 NUM cana-4832 319 4 ) ) PUNCT cana-4832 319 5 , , PUNCT cana-4832 319 6 article article NOUN cana-4832 319 7 i i PROPN cana-4832 319 8 d d PROPN cana-4832 319 9 376832 376832 NUM cana-4832 319 10 . . PUNCT cana-4832 320 1 [ [ X cana-4832 320 2 11 11 NUM cana-4832 320 3 ] ] X cana-4832 320 4 diana diana PROPN cana-4832 320 5 dolicanin dolicanin PROPN cana-4832 320 6 - - PUNCT cana-4832 320 7 dekic dekic PROPN cana-4832 320 8 , , PUNCT cana-4832 320 9 tatjana tatjana PROPN cana-4832 320 10 doΕ‘enovic doΕ‘enovic PROPN cana-4832 320 11 , , PUNCT cana-4832 320 12 huaping huape VERB cana-4832 320 13 huang huang PROPN cana-4832 320 14 and and CCONJ cana-4832 320 15 stojan stojan ADP cana-4832 320 16 radenovic radenovic PROPN cana-4832 320 17 , , PUNCT cana-4832 320 18 a a DET cana-4832 320 19 note note NOUN cana-4832 320 20 on on ADP cana-4832 320 21 recent recent ADJ cana-4832 320 22 cyclic cyclic ADJ cana-4832 320 23 fixed fix VERB cana-4832 320 24 point point NOUN cana-4832 320 25 results result NOUN cana-4832 320 26 in in ADP cana-4832 320 27 dislocated dislocated ADJ cana-4832 320 28 quasi-𝑏-metric quasi-𝑏-metric ADJ cana-4832 320 29 spaces space NOUN cana-4832 320 30 , , PUNCT cana-4832 320 31 fixed fix VERB cana-4832 320 32 point point NOUN cana-4832 320 33 theory theory NOUN cana-4832 320 34 and and CCONJ cana-4832 320 35 appl appl NOUN cana-4832 320 36 . . PUNCT cana-4832 321 1 ( ( PUNCT cana-4832 321 2 2016)(74 2016)(74 NUM cana-4832 321 3 ) ) PUNCT cana-4832 321 4 2016 2016 NUM cana-4832 321 5 , , PUNCT cana-4832 321 6 10 10 NUM cana-4832 321 7 pages page NOUN cana-4832 321 8 . . PUNCT cana-4832 322 1 [ [ X cana-4832 322 2 12 12 NUM cana-4832 322 3 ] ] X cana-4832 322 4 y. y. PROPN cana-4832 322 5 han han PROPN cana-4832 322 6 and and CCONJ cana-4832 322 7 s. s. PROPN cana-4832 322 8 xu xu PROPN cana-4832 322 9 , , PUNCT cana-4832 322 10 some some DET cana-4832 322 11 new new ADJ cana-4832 322 12 theorems theorem NOUN cana-4832 322 13 of of ADP cana-4832 322 14 expanding expand VERB cana-4832 322 15 mappings mapping NOUN cana-4832 322 16 without without ADP cana-4832 322 17 continuity continuity NOUN cana-4832 322 18 in in ADP cana-4832 322 19 cone cone NOUN cana-4832 322 20 metric metric ADJ cana-4832 322 21 spaces space NOUN cana-4832 322 22 , , PUNCT cana-4832 322 23 fixed fix VERB cana-4832 322 24 point point NOUN cana-4832 322 25 theory theory NOUN cana-4832 322 26 appl appl NOUN cana-4832 322 27 . . PUNCT cana-4832 323 1 2013(3 2013(3 NUM cana-4832 323 2 ) ) PUNCT cana-4832 323 3 ( ( PUNCT cana-4832 323 4 2013 2013 NUM cana-4832 323 5 ) ) PUNCT cana-4832 323 6 , , PUNCT cana-4832 323 7 9 9 NUM cana-4832 323 8 pages page NOUN cana-4832 323 9 . . PUNCT cana-4832 324 1 [ [ X cana-4832 324 2 13 13 NUM cana-4832 324 3 ] ] PUNCT cana-4832 324 4 p. p. NOUN cana-4832 324 5 hitzler hitzler NOUN cana-4832 324 6 and and CCONJ cana-4832 324 7 a. a. PROPN cana-4832 324 8 k. k. PROPN cana-4832 324 9 seda seda PROPN cana-4832 324 10 , , PUNCT cana-4832 324 11 dislocated dislocated ADJ cana-4832 324 12 topologies topology NOUN cana-4832 324 13 , , PUNCT cana-4832 324 14 j. j. PROPN cana-4832 324 15 electr electr PROPN cana-4832 324 16 . . PUNCT cana-4832 325 1 eng eng PROPN cana-4832 325 2 . . PROPN cana-4832 326 1 51 51 NUM cana-4832 326 2 ( ( PUNCT cana-4832 326 3 12 12 NUM cana-4832 326 4 ) ) PUNCT cana-4832 326 5 ( ( PUNCT cana-4832 326 6 2000 2000 NUM cana-4832 326 7 ) ) PUNCT cana-4832 326 8 , , PUNCT cana-4832 326 9 3 3 NUM cana-4832 326 10 - - SYM cana-4832 326 11 7 7 NUM cana-4832 326 12 . . PUNCT cana-4832 327 1 [ [ X cana-4832 327 2 14 14 NUM cana-4832 327 3 ] ] X cana-4832 327 4 c. c. PROPN cana-4832 327 5 klin klin PROPN cana-4832 327 6 - - PROPN cana-4832 327 7 eam eam PROPN cana-4832 327 8 and and CCONJ cana-4832 327 9 c. c. PROPN cana-4832 327 10 suanoom suanoom NOUN cana-4832 327 11 , , PUNCT cana-4832 327 12 dislocated dislocate VERB cana-4832 327 13 quasi-𝑏-metric quasi-𝑏-metric ADJ cana-4832 327 14 spaces space NOUN cana-4832 327 15 and and CCONJ cana-4832 327 16 fixed fix VERB cana-4832 327 17 point point NOUN cana-4832 327 18 theorems theorem NOUN cana-4832 327 19 for for ADP cana-4832 327 20 cyclic cyclic ADJ cana-4832 327 21 contractions contraction NOUN cana-4832 327 22 , , PUNCT cana-4832 327 23 fixed fix VERB cana-4832 327 24 point point NOUN cana-4832 327 25 theory theory NOUN cana-4832 327 26 and and CCONJ cana-4832 327 27 applications application NOUN cana-4832 327 28 ( ( PUNCT cana-4832 327 29 2015)(74 2015)(74 NUM cana-4832 327 30 ) ) PUNCT cana-4832 327 31 2015 2015 NUM cana-4832 327 32 , , PUNCT cana-4832 327 33 12 12 NUM cana-4832 327 34 pages page NOUN cana-4832 327 35 . . PUNCT cana-4832 328 1 https://doi.org/10.22130/scma.2024.2042087.1910 https://doi.org/10.22130/scma.2024.2042087.1910 PROPN cana-4832 328 2 https://doi.org/10.28919/afpt/8515 https://doi.org/10.28919/afpt/8515 PROPN cana-4832 328 3 communications communication NOUN cana-4832 328 4 on on ADP cana-4832 328 5 applied apply VERB cana-4832 328 6 nonlinear nonlinear ADJ cana-4832 328 7 analysis analysis NOUN cana-4832 328 8 issn issn NOUN cana-4832 328 9 : : PUNCT cana-4832 328 10 1074 1074 NUM cana-4832 328 11 - - PUNCT cana-4832 328 12 133x 133x NUM cana-4832 328 13 vol vol VERB cana-4832 328 14 32 32 NUM cana-4832 328 15 no no NOUN cana-4832 328 16 . . PUNCT cana-4832 329 1 10s 10 NOUN cana-4832 329 2 ( ( PUNCT cana-4832 329 3 2025 2025 NUM cana-4832 329 4 ) ) PUNCT cana-4832 329 5 405 405 NUM cana-4832 329 6 https://internationalpubls.com https://internationalpubls.com X cana-4832 330 1 [ [ X cana-4832 330 2 15 15 NUM cana-4832 330 3 ] ] X cana-4832 330 4 s. s. PROPN cana-4832 330 5 matthews matthews PROPN cana-4832 330 6 , , PUNCT cana-4832 330 7 metric metric ADJ cana-4832 330 8 domains domain NOUN cana-4832 330 9 for for ADP cana-4832 330 10 completeness completeness NOUN cana-4832 330 11 , , PUNCT cana-4832 330 12 phd phd NOUN cana-4832 330 13 thesis thesis NOUN cana-4832 330 14 , , PUNCT cana-4832 330 15 department department NOUN cana-4832 330 16 of of ADP cana-4832 330 17 computer computer NOUN cana-4832 330 18 science science NOUN cana-4832 330 19 , , PUNCT cana-4832 330 20 university university PROPN cana-4832 330 21 of of ADP cana-4832 330 22 warwick warwick PROPN cana-4832 330 23 coventry coventry PROPN cana-4832 330 24 , , PUNCT cana-4832 330 25 uk uk PROPN cana-4832 330 26 , , PUNCT cana-4832 330 27 ( ( PUNCT cana-4832 330 28 1986 1986 NUM cana-4832 330 29 ) ) PUNCT cana-4832 330 30 1 1 NUM cana-4832 330 31 - - SYM cana-4832 330 32 127 127 NUM cana-4832 330 33 . . PUNCT cana-4832 331 1 [ [ X cana-4832 331 2 16 16 NUM cana-4832 331 3 ] ] X cana-4832 331 4 s. s. PROPN cana-4832 331 5 mhannaa mhannaa PROPN cana-4832 331 6 , , PUNCT cana-4832 331 7 o. o. PROPN cana-4832 331 8 baizb baizb PROPN cana-4832 331 9 , , PUNCT cana-4832 331 10 h. h. PROPN cana-4832 331 11 benaissac benaissac NOUN cana-4832 331 12 , , PUNCT cana-4832 331 13 and and CCONJ cana-4832 331 14 d. d. PROPN cana-4832 331 15 el el PROPN cana-4832 331 16 moutawakil moutawakil PROPN cana-4832 331 17 , , PUNCT cana-4832 331 18 some some DET cana-4832 331 19 new new ADJ cana-4832 331 20 results result NOUN cana-4832 331 21 of of ADP cana-4832 331 22 fixed fix VERB cana-4832 331 23 point point NOUN cana-4832 331 24 in in ADP cana-4832 331 25 dislocated dislocated ADJ cana-4832 331 26 quasi quasi ADJ cana-4832 331 27 - - ADJ cana-4832 331 28 metric metric ADJ cana-4832 331 29 spaces space NOUN cana-4832 331 30 , , PUNCT cana-4832 331 31 j. j. PROPN cana-4832 331 32 math math PROPN cana-4832 331 33 . . PUNCT cana-4832 332 1 computer computer NOUN cana-4832 332 2 sci sci PROPN cana-4832 332 3 . . PROPN cana-4832 332 4 , , PUNCT cana-4832 332 5 24 24 NUM cana-4832 332 6 ( ( PUNCT cana-4832 332 7 2022 2022 NUM cana-4832 332 8 ) ) PUNCT cana-4832 332 9 , , PUNCT cana-4832 332 10 22 22 NUM cana-4832 332 11 - - SYM cana-4832 332 12 32 32 NUM cana-4832 332 13 . . PUNCT cana-4832 333 1 [ [ X cana-4832 333 2 17 17 NUM cana-4832 333 3 ] ] PUNCT cana-4832 333 4 m. m. NOUN cana-4832 333 5 raji raji NOUN cana-4832 333 6 and and CCONJ cana-4832 333 7 m. m. PROPN cana-4832 333 8 a. a. PROPN cana-4832 333 9 ibrahim ibrahim PROPN cana-4832 333 10 , , PUNCT cana-4832 333 11 fixed fix VERB cana-4832 333 12 point point NOUN cana-4832 333 13 theorems theorem NOUN cana-4832 333 14 for for ADP cana-4832 333 15 fuzzy fuzzy ADJ cana-4832 333 16 contractions contraction NOUN cana-4832 333 17 mappings mapping NOUN cana-4832 333 18 in in ADP cana-4832 333 19 a a DET cana-4832 333 20 dislocated dislocate VERB cana-4832 333 21 𝑏-metric 𝑏-metric NOUN cana-4832 333 22 spaces space NOUN cana-4832 333 23 with with ADP cana-4832 333 24 applications application NOUN cana-4832 333 25 , , PUNCT cana-4832 333 26 annals annal VERB cana-4832 333 27 math math NOUN cana-4832 333 28 . . PUNCT cana-4832 334 1 computer computer PROPN cana-4832 334 2 sci sci PROPN cana-4832 334 3 . . PROPN cana-4832 334 4 , , PUNCT cana-4832 334 5 21 21 NUM cana-4832 334 6 ( ( PUNCT cana-4832 334 7 2024 2024 NUM cana-4832 334 8 ) ) PUNCT cana-4832 334 9 1 1 NUM cana-4832 334 10 - - SYM cana-4832 334 11 13 13 NUM cana-4832 334 12 . . PUNCT cana-4832 335 1 [ [ X cana-4832 335 2 18 18 NUM cana-4832 335 3 ] ] PUNCT cana-4832 335 4 m. m. NOUN cana-4832 335 5 rahman rahman PROPN cana-4832 335 6 and and CCONJ cana-4832 335 7 m. m. NOUN cana-4832 335 8 sarwar sarwar PROPN cana-4832 335 9 , , PUNCT cana-4832 335 10 fixed fix VERB cana-4832 335 11 point point NOUN cana-4832 335 12 theorems theorem NOUN cana-4832 335 13 for for ADP cana-4832 335 14 expanding expand VERB cana-4832 335 15 mappings mapping NOUN cana-4832 335 16 in in ADP cana-4832 335 17 dislocated dislocated ADJ cana-4832 335 18 metric metric ADJ cana-4832 335 19 space space NOUN cana-4832 335 20 , , PUNCT cana-4832 335 21 math math NOUN cana-4832 335 22 . . PUNCT cana-4832 336 1 sci sci PROPN cana-4832 336 2 . . PROPN cana-4832 336 3 lett lett PROPN cana-4832 336 4 . . PROPN cana-4832 337 1 4 4 NUM cana-4832 337 2 ( ( PUNCT cana-4832 337 3 1 1 NUM cana-4832 337 4 ) ) PUNCT cana-4832 337 5 ( ( PUNCT cana-4832 337 6 2015 2015 NUM cana-4832 337 7 ) ) PUNCT cana-4832 337 8 , , PUNCT cana-4832 337 9 69 69 NUM cana-4832 337 10 - - SYM cana-4832 337 11 73 73 NUM cana-4832 337 12 . . PUNCT cana-4832 338 1 [ [ X cana-4832 338 2 19 19 NUM cana-4832 338 3 ] ] PUNCT cana-4832 338 4 a. a. NOUN cana-4832 338 5 rani rani PROPN cana-4832 338 6 , , PUNCT cana-4832 338 7 a. a. NOUN cana-4832 338 8 rani rani PROPN cana-4832 338 9 and and CCONJ cana-4832 338 10 k. k. PROPN cana-4832 338 11 jyoti jyoti PROPN cana-4832 338 12 , , PUNCT cana-4832 338 13 𝑑 𝑑 PROPN cana-4832 338 14 βˆ’ βˆ’ PROPN cana-4832 338 15 𝛼 𝛼 NOUN cana-4832 338 16 βˆ’ βˆ’ PROPN cana-4832 338 17 πœ“ πœ“ PROPN cana-4832 338 18 expansive expansive ADJ cana-4832 338 19 mapping mapping NOUN cana-4832 338 20 in in ADP cana-4832 338 21 dislocated dislocate VERB cana-4832 338 22 metric metric ADJ cana-4832 338 23 space space NOUN cana-4832 338 24 , , PUNCT cana-4832 338 25 asian asian PROPN cana-4832 338 26 j. j. PROPN cana-4832 338 27 math math PROPN cana-4832 338 28 . . PUNCT cana-4832 339 1 comput comput NOUN cana-4832 339 2 . . PUNCT cana-4832 340 1 research research NOUN cana-4832 340 2 15 15 NUM cana-4832 340 3 ( ( PUNCT cana-4832 340 4 2 2 NUM cana-4832 340 5 ) ) PUNCT cana-4832 340 6 ( ( PUNCT cana-4832 340 7 2017 2017 NUM cana-4832 340 8 ) ) PUNCT cana-4832 340 9 , , PUNCT cana-4832 340 10 103 103 NUM cana-4832 340 11 - - SYM cana-4832 340 12 112 112 NUM cana-4832 340 13 . . PUNCT cana-4832 341 1 [ [ X cana-4832 341 2 20 20 NUM cana-4832 341 3 ] ] PUNCT cana-4832 341 4 p. p. NOUN cana-4832 341 5 shahi shahi PROPN cana-4832 341 6 , , PUNCT cana-4832 341 7 j. j. PROPN cana-4832 341 8 kaur kaur PROPN cana-4832 341 9 and and CCONJ cana-4832 341 10 s. s. PROPN cana-4832 341 11 bhatia bhatia PROPN cana-4832 341 12 , , PUNCT cana-4832 341 13 fixed fix VERB cana-4832 341 14 point point NOUN cana-4832 341 15 theorems theorem NOUN cana-4832 341 16 for for ADP cana-4832 341 17 ( ( PUNCT cana-4832 341 18 πœ‰ πœ‰ X cana-4832 341 19 , , PUNCT cana-4832 341 20 𝛼)-expansive 𝛼)-expansive ADJ cana-4832 341 21 mappings mapping NOUN cana-4832 341 22 in in ADP cana-4832 341 23 complete complete ADJ cana-4832 341 24 metric metric ADJ cana-4832 341 25 spaces space NOUN cana-4832 341 26 , , PUNCT cana-4832 341 27 fixed fix VERB cana-4832 341 28 point point NOUN cana-4832 341 29 theory theory NOUN cana-4832 341 30 appl appl NOUN cana-4832 341 31 . . PUNCT cana-4832 342 1 2012(157)(2012 2012(157)(2012 NUM cana-4832 342 2 ) ) PUNCT cana-4832 342 3 , , PUNCT cana-4832 342 4 12 12 NUM cana-4832 342 5 pages page NOUN cana-4832 342 6 . . PUNCT cana-4832 343 1 [ [ X cana-4832 343 2 21 21 NUM cana-4832 343 3 ] ] X cana-4832 343 4 c. c. PROPN cana-4832 343 5 suanoom suanoom PROPN cana-4832 343 6 , , PUNCT cana-4832 343 7 c. c. PROPN cana-4832 343 8 klin klin PROPN cana-4832 343 9 - - PROPN cana-4832 343 10 eam eam PROPN cana-4832 343 11 , , PUNCT cana-4832 343 12 and and CCONJ cana-4832 343 13 s. s. PROPN cana-4832 343 14 suantai suantai PROPN cana-4832 343 15 , , PUNCT cana-4832 343 16 dislocated dislocate VERB cana-4832 343 17 quasi-𝑏-metric quasi-𝑏-metric ADJ cana-4832 343 18 spaces space NOUN cana-4832 343 19 and and CCONJ cana-4832 343 20 fixed fix VERB cana-4832 343 21 point point NOUN cana-4832 343 22 theorems theorem NOUN cana-4832 343 23 for for ADP cana-4832 343 24 cyclic cyclic ADJ cana-4832 343 25 weakly weakly ADJ cana-4832 343 26 contractions contraction NOUN cana-4832 343 27 , , PUNCT cana-4832 343 28 j. j. PROPN cana-4832 343 29 nonlinear nonlinear PROPN cana-4832 343 30 sci sci PROPN cana-4832 343 31 . . PUNCT cana-4832 343 32 appl appl PROPN cana-4832 343 33 . . PROPN cana-4832 343 34 9 9 NUM cana-4832 343 35 ( ( PUNCT cana-4832 343 36 2016 2016 NUM cana-4832 343 37 ) ) PUNCT cana-4832 343 38 , , PUNCT cana-4832 343 39 2779–2788 2779–2788 NOUN cana-4832 343 40 . . PUNCT cana-4832 344 1 [ [ X cana-4832 344 2 22 22 NUM cana-4832 344 3 ] ] PUNCT cana-4832 344 4 t. t. PROPN cana-4832 344 5 taniguchi taniguchi PROPN cana-4832 344 6 , , PUNCT cana-4832 344 7 common common ADJ cana-4832 344 8 fixed fix VERB cana-4832 344 9 point point NOUN cana-4832 344 10 theorems theorem NOUN cana-4832 344 11 on on ADP cana-4832 344 12 expansion expansion NOUN cana-4832 344 13 type type NOUN cana-4832 344 14 mappings mapping NOUN cana-4832 344 15 on on ADP cana-4832 344 16 complete complete ADJ cana-4832 344 17 metric metric ADJ cana-4832 344 18 spaces space NOUN cana-4832 344 19 , , PUNCT cana-4832 344 20 math math NOUN cana-4832 344 21 . . PUNCT cana-4832 345 1 japon japon PROPN cana-4832 345 2 . . PROPN cana-4832 345 3 , , PUNCT cana-4832 345 4 34 34 NUM cana-4832 345 5 ( ( PUNCT cana-4832 345 6 1989 1989 NUM cana-4832 345 7 ) ) PUNCT cana-4832 345 8 , , PUNCT cana-4832 345 9 139–142 139–142 NUM cana-4832 345 10 . . PUNCT cana-4832 346 1 [ [ X cana-4832 346 2 23 23 NUM cana-4832 346 3 ] ] PUNCT cana-4832 346 4 m. m. NOUN cana-4832 346 5 ur ur INTJ cana-4832 346 6 - - PROPN cana-4832 346 7 rahman rahman PROPN cana-4832 346 8 , , PUNCT cana-4832 346 9 a. a. PROPN cana-4832 346 10 ali ali PROPN cana-4832 346 11 , , PUNCT cana-4832 346 12 o. o. PROPN cana-4832 346 13 aziz aziz PROPN cana-4832 346 14 , , PUNCT cana-4832 346 15 m. m. NOUN cana-4832 346 16 ur ur PROPN cana-4832 346 17 - - PROPN cana-4832 346 18 rahman rahman PROPN cana-4832 346 19 , , PUNCT cana-4832 346 20 fixed fix VERB cana-4832 346 21 point point NOUN cana-4832 346 22 theorems theorem NOUN cana-4832 346 23 in in ADP cana-4832 346 24 dislocated dislocate VERB cana-4832 346 25 quasi quasi ADJ cana-4832 346 26 metric metric ADJ cana-4832 346 27 and and CCONJ cana-4832 346 28 dislocated dislocated ADJ cana-4832 346 29 quasi quasi NOUN cana-4832 346 30 $ $ SYM cana-4832 346 31 b$-metric b$-metric ADJ cana-4832 346 32 spaces space NOUN cana-4832 346 33 , , PUNCT cana-4832 346 34 commun commun PROPN cana-4832 346 35 . . PUNCT cana-4832 347 1 nonlinear nonlinear PROPN cana-4832 347 2 anal anal NOUN cana-4832 347 3 . . PUNCT cana-4832 348 1 1 1 NUM cana-4832 348 2 ( ( PUNCT cana-4832 348 3 2024 2024 NUM cana-4832 348 4 ) ) PUNCT cana-4832 348 5 , , PUNCT cana-4832 348 6 1 1 NUM cana-4832 348 7 - - SYM cana-4832 348 8 10 10 NUM cana-4832 348 9 . . PUNCT cana-4832 349 1 [ [ X cana-4832 349 2 24 24 NUM cana-4832 349 3 ] ] PUNCT cana-4832 349 4 s. s. PROPN cana-4832 349 5 z. z. PROPN cana-4832 349 6 wang wang PROPN cana-4832 349 7 , , PUNCT cana-4832 349 8 b. b. PROPN cana-4832 349 9 y. y. PROPN cana-4832 349 10 li li PROPN cana-4832 349 11 , , PUNCT cana-4832 349 12 z. z. PROPN cana-4832 349 13 m. m. PROPN cana-4832 349 14 gao gao PROPN cana-4832 349 15 and and CCONJ cana-4832 349 16 k. k. PROPN cana-4832 349 17 iseki iseki PROPN cana-4832 349 18 , , PUNCT cana-4832 349 19 some some DET cana-4832 349 20 fixed fix VERB cana-4832 349 21 point point NOUN cana-4832 349 22 theorems theorem NOUN cana-4832 349 23 on on ADP cana-4832 349 24 expansion expansion NOUN cana-4832 349 25 mappings mapping NOUN cana-4832 349 26 , , PUNCT cana-4832 349 27 math math NOUN cana-4832 349 28 . . PUNCT cana-4832 350 1 japon japon PROPN cana-4832 350 2 . . PROPN cana-4832 350 3 , , PUNCT cana-4832 350 4 29 29 NUM cana-4832 350 5 ( ( PUNCT cana-4832 350 6 1984 1984 NUM cana-4832 350 7 ) ) PUNCT cana-4832 350 8 , , PUNCT cana-4832 350 9 631 631 NUM cana-4832 350 10 - - SYM cana-4832 350 11 636 636 NUM cana-4832 350 12 . . PUNCT cana-4832 351 1 [ [ X cana-4832 351 2 25 25 NUM cana-4832 351 3 ] ] PUNCT cana-4832 351 4 s. s. PROPN cana-4832 351 5 s. s. PROPN cana-4832 351 6 yesilkaya yesilkaya PROPN cana-4832 351 7 and and CCONJ cana-4832 351 8 c. c. PROPN cana-4832 351 9 aydΔ±n aydΔ±n PROPN cana-4832 351 10 , , PUNCT cana-4832 351 11 fixed fix VERB cana-4832 351 12 point point NOUN cana-4832 351 13 results result NOUN cana-4832 351 14 of of ADP cana-4832 351 15 expansive expansive ADJ cana-4832 351 16 mappings mapping NOUN cana-4832 351 17 in in ADP cana-4832 351 18 metric metric ADJ cana-4832 351 19 spaces space NOUN cana-4832 351 20 , , PUNCT cana-4832 351 21 math math NOUN cana-4832 351 22 . . PUNCT cana-4832 351 23 , , PUNCT cana-4832 351 24 8 8 NUM cana-4832 351 25 ( ( PUNCT cana-4832 351 26 10)(1800 10)(1800 NUM cana-4832 351 27 ) ) PUNCT cana-4832 351 28 ( ( PUNCT cana-4832 351 29 2020 2020 NUM cana-4832 351 30 ) ) PUNCT cana-4832 351 31 , , PUNCT cana-4832 351 32 10 10 NUM cana-4832 351 33 pages page NOUN cana-4832 351 34 . . PUNCT cana-4832 352 1 [ [ X cana-4832 352 2 26 26 NUM cana-4832 352 3 ] ] X cana-4832 352 4 f. f. PROPN cana-4832 352 5 zeyada zeyada PROPN cana-4832 352 6 , , PUNCT cana-4832 352 7 g. g. PROPN cana-4832 352 8 hassan hassan PROPN cana-4832 352 9 and and CCONJ cana-4832 352 10 m. m. PROPN cana-4832 352 11 ahmed ahmed PROPN cana-4832 352 12 , , PUNCT cana-4832 352 13 a a DET cana-4832 352 14 generalization generalization NOUN cana-4832 352 15 of of ADP cana-4832 352 16 a a DET cana-4832 352 17 fixed fix VERB cana-4832 352 18 point point NOUN cana-4832 352 19 theorem theorem ADJ cana-4832 352 20 due due ADP cana-4832 352 21 to to ADP cana-4832 352 22 hitzler hitzler NOUN cana-4832 352 23 and and CCONJ cana-4832 352 24 seda seda NOUN cana-4832 352 25 in in ADP cana-4832 352 26 dislocated dislocated ADJ cana-4832 352 27 quasi quasi ADJ cana-4832 352 28 - - ADJ cana-4832 352 29 metric metric ADJ cana-4832 352 30 spaces space NOUN cana-4832 352 31 , , PUNCT cana-4832 352 32 arab arab PROPN cana-4832 352 33 . . PUNCT cana-4832 353 1 j. j. PROPN cana-4832 353 2 sci sci PROPN cana-4832 353 3 . . PUNCT cana-4832 354 1 eng eng PROPN cana-4832 354 2 . . PROPN cana-4832 354 3 , , PUNCT cana-4832 354 4 31 31 NUM cana-4832 354 5 ( ( PUNCT cana-4832 354 6 1a 1a NUM cana-4832 354 7 ) ) PUNCT cana-4832 354 8 ( ( PUNCT cana-4832 354 9 2005 2005 NUM cana-4832 354 10 ) ) PUNCT cana-4832 354 11 , , PUNCT cana-4832 354 12 111 111 NUM cana-4832 354 13 - - SYM cana-4832 354 14 114 114 NUM cana-4832 354 15 . . PUNCT