id sid tid token lemma pos cana-3309 1 1 communications communication NOUN cana-3309 1 2 on on ADP cana-3309 1 3 applied apply VERB cana-3309 1 4 nonlinear nonlinear ADJ cana-3309 1 5 analysis analysis NOUN cana-3309 1 6 issn issn NOUN cana-3309 1 7 : : PUNCT cana-3309 1 8 1074 1074 NUM cana-3309 1 9 - - PUNCT cana-3309 1 10 133x 133x NUM cana-3309 1 11 vol vol NOUN cana-3309 1 12 32 32 NUM cana-3309 1 13 no no NOUN cana-3309 1 14 . . PUNCT cana-3309 2 1 6s 6s NUM cana-3309 2 2 ( ( PUNCT cana-3309 2 3 2025 2025 NUM cana-3309 2 4 ) ) PUNCT cana-3309 2 5 454 454 NUM cana-3309 2 6 https://internationalpubls.com https://internationalpubls.com X cana-3309 2 7 some some DET cana-3309 2 8 fixed fix VERB cana-3309 2 9 point point NOUN cana-3309 2 10 theorems theorem NOUN cana-3309 2 11 using use VERB cana-3309 2 12 𝚽𝐩 𝚽𝐩 PROPN cana-3309 2 13 operator operator NOUN cana-3309 2 14 k k PROPN cana-3309 2 15 dinesh dinesh PROPN cana-3309 2 16 1 1 NUM cana-3309 2 17 * * PUNCT cana-3309 2 18 , , PUNCT cana-3309 2 19 kastriot kastriot PROPN cana-3309 2 20 zoto zoto PROPN cana-3309 2 21 2 2 NUM cana-3309 2 22 , , PUNCT cana-3309 2 23 adriana adriana PROPN cana-3309 2 24 topi topi VERB cana-3309 2 25 3 3 NUM cana-3309 2 26 , , PUNCT cana-3309 2 27 b b PROPN cana-3309 2 28 shoba shoba VERB cana-3309 2 29 4 4 NUM cana-3309 2 30 , , PUNCT cana-3309 2 31 doaa doaa VERB cana-3309 2 32 rizk rizk VERB cana-3309 2 33 5 5 NUM cana-3309 2 34 1 1 NUM cana-3309 2 35 department department NOUN cana-3309 2 36 of of ADP cana-3309 2 37 mathematics mathematics PROPN cana-3309 2 38 , , PUNCT cana-3309 2 39 k.ramakrishnan k.ramakrishnan ADP cana-3309 2 40 college college NOUN cana-3309 2 41 of of ADP cana-3309 2 42 engineering engineering PROPN cana-3309 2 43 , , PUNCT cana-3309 2 44 tiruchirappalli tiruchirappalli PROPN cana-3309 2 45 , , PUNCT cana-3309 2 46 tamilnadu tamilnadu PROPN cana-3309 2 47 , , PUNCT cana-3309 2 48 india india PROPN cana-3309 2 49 . . PUNCT cana-3309 3 1 email email NOUN cana-3309 4 1 i i PROPN cana-3309 4 2 d d PROPN cana-3309 4 3 : : PUNCT cana-3309 4 4 dinesh.skksv93@gmail.com dinesh.skksv93@gmail.com X cana-3309 4 5 2 2 NUM cana-3309 4 6 department department NOUN cana-3309 4 7 of of ADP cana-3309 4 8 mathematics mathematic NOUN cana-3309 4 9 , , PUNCT cana-3309 4 10 informatics informatic NOUN cana-3309 4 11 and and CCONJ cana-3309 4 12 physics physics PROPN cana-3309 4 13 , , PUNCT cana-3309 4 14 faculty faculty NOUN cana-3309 4 15 of of ADP cana-3309 4 16 natural natural ADJ cana-3309 4 17 sciences science NOUN cana-3309 4 18 , , PUNCT cana-3309 4 19 university university NOUN cana-3309 4 20 of of ADP cana-3309 4 21 gjirokastra gjirokastra PROPN cana-3309 4 22 , , PUNCT cana-3309 4 23 albania albania PROPN cana-3309 4 24 . . PUNCT cana-3309 5 1 email email NOUN cana-3309 5 2 i i PROPN cana-3309 5 3 d d PROPN cana-3309 5 4 : : PUNCT cana-3309 5 5 kzoto@uogj.edu.al kzoto@uogj.edu.al PROPN cana-3309 5 6 3 3 NUM cana-3309 5 7 department department NOUN cana-3309 5 8 of of ADP cana-3309 5 9 informatics informatic NOUN cana-3309 5 10 and and CCONJ cana-3309 5 11 technology technology NOUN cana-3309 5 12 , , PUNCT cana-3309 5 13 faculty faculty NOUN cana-3309 5 14 of of ADP cana-3309 5 15 engineering engineering NOUN cana-3309 5 16 , , PUNCT cana-3309 5 17 informatics informatic NOUN cana-3309 5 18 , , PUNCT cana-3309 5 19 and and CCONJ cana-3309 5 20 architecture architecture NOUN cana-3309 5 21 , , PUNCT cana-3309 5 22 european european PROPN cana-3309 5 23 university university PROPN cana-3309 5 24 of of ADP cana-3309 5 25 tirana tirana PROPN cana-3309 5 26 , , PUNCT cana-3309 5 27 tirana tirana ADJ cana-3309 5 28 , , PUNCT cana-3309 5 29 1000 1000 NUM cana-3309 5 30 , , PUNCT cana-3309 5 31 albania albania PROPN cana-3309 5 32 email email NOUN cana-3309 5 33 i i PROPN cana-3309 5 34 d d PROPN cana-3309 5 35 : : PUNCT cana-3309 5 36 ardiana.topi@uet.edu.al ardiana.topi@uet.edu.al PROPN cana-3309 5 37 4 4 NUM cana-3309 5 38 department department NOUN cana-3309 5 39 of of ADP cana-3309 5 40 mathematics mathematics PROPN cana-3309 5 41 , , PUNCT cana-3309 5 42 st st PROPN cana-3309 5 43 joseph joseph PROPN cana-3309 5 44 's 's PART cana-3309 5 45 college college PROPN cana-3309 5 46 of of ADP cana-3309 5 47 engineering engineering PROPN cana-3309 5 48 , , PUNCT cana-3309 5 49 omr omr PROPN cana-3309 5 50 chennai chennai NOUN cana-3309 5 51 600 600 NUM cana-3309 5 52 019 019 NUM cana-3309 5 53 , , PUNCT cana-3309 5 54 india india PROPN cana-3309 5 55 email email NOUN cana-3309 6 1 i i PROPN cana-3309 6 2 d d PROPN cana-3309 6 3 shobabalasubramaniyam@gmail.com shobabalasubramaniyam@gmail.com PROPN cana-3309 7 1 5 5 NUM cana-3309 8 1 department department NOUN cana-3309 8 2 of of ADP cana-3309 8 3 mathematics mathematic NOUN cana-3309 8 4 , , PUNCT cana-3309 8 5 college college NOUN cana-3309 8 6 of of ADP cana-3309 8 7 science science NOUN cana-3309 8 8 , , PUNCT cana-3309 8 9 qassim qassim PROPN cana-3309 8 10 university university PROPN cana-3309 8 11 , , PUNCT cana-3309 8 12 buraydah buraydah PROPN cana-3309 8 13 , , PUNCT cana-3309 8 14 51452 51452 NUM cana-3309 8 15 , , PUNCT cana-3309 8 16 saudi saudi PROPN cana-3309 8 17 arabia arabia PROPN cana-3309 8 18 . . PUNCT cana-3309 9 1 email email NOUN cana-3309 10 1 i i PROPN cana-3309 10 2 d d PROPN cana-3309 10 3 : : PUNCT cana-3309 10 4 d.hussien@qu.edu.sa d.hussien@qu.edu.sa PROPN cana-3309 10 5 article article NOUN cana-3309 10 6 history history NOUN cana-3309 10 7 : : PUNCT cana-3309 10 8 received receive VERB cana-3309 10 9 : : PUNCT cana-3309 10 10 22 22 NUM cana-3309 10 11 - - SYM cana-3309 10 12 10 10 NUM cana-3309 10 13 - - PUNCT cana-3309 10 14 2024 2024 NUM cana-3309 10 15 revised revise VERB cana-3309 10 16 : : PUNCT cana-3309 10 17 06 06 NUM cana-3309 10 18 - - SYM cana-3309 10 19 12 12 NUM cana-3309 10 20 - - PUNCT cana-3309 10 21 2024 2024 NUM cana-3309 10 22 accepted accept VERB cana-3309 10 23 : : PUNCT cana-3309 10 24 13 13 NUM cana-3309 10 25 - - SYM cana-3309 10 26 12 12 NUM cana-3309 10 27 - - PUNCT cana-3309 10 28 2024 2024 NUM cana-3309 10 29 abstract abstract NOUN cana-3309 10 30 : : PUNCT cana-3309 10 31 this this DET cana-3309 10 32 article article NOUN cana-3309 10 33 's 's PART cana-3309 10 34 goal goal NOUN cana-3309 10 35 is be AUX cana-3309 10 36 to to PART cana-3309 10 37 use use VERB cana-3309 10 38 the the DET cana-3309 10 39 Ο†p Ο†p NOUN cana-3309 10 40 operator operator NOUN cana-3309 10 41 to to PART cana-3309 10 42 prove prove VERB cana-3309 10 43 a a DET cana-3309 10 44 few few ADJ cana-3309 10 45 fixed fix VERB cana-3309 10 46 point point NOUN cana-3309 10 47 theorems theorem NOUN cana-3309 10 48 in in ADP cana-3309 10 49 complete complete ADJ cana-3309 10 50 metric metric ADJ cana-3309 10 51 space space NOUN cana-3309 10 52 . . PUNCT cana-3309 11 1 furthermore furthermore ADV cana-3309 11 2 , , PUNCT cana-3309 11 3 we we PRON cana-3309 11 4 investigate investigate VERB cana-3309 11 5 whether whether SCONJ cana-3309 11 6 fixed fix VERB cana-3309 11 7 points point NOUN cana-3309 11 8 for for ADP cana-3309 11 9 self self NOUN cana-3309 11 10 mappings mapping NOUN cana-3309 11 11 that that PRON cana-3309 11 12 meet meet VERB cana-3309 11 13 the the DET cana-3309 11 14 requirements requirement NOUN cana-3309 11 15 of of ADP cana-3309 11 16 rational rational ADJ cana-3309 11 17 expression expression NOUN cana-3309 11 18 exist exist VERB cana-3309 11 19 and and CCONJ cana-3309 11 20 are be AUX cana-3309 11 21 unique unique ADJ cana-3309 11 22 in in ADP cana-3309 11 23 a a DET cana-3309 11 24 complete complete ADJ cana-3309 11 25 metric metric ADJ cana-3309 11 26 space space NOUN cana-3309 11 27 . . PUNCT cana-3309 12 1 our our PRON cana-3309 12 2 findings finding NOUN cana-3309 12 3 expand expand VERB cana-3309 12 4 upon upon SCONJ cana-3309 12 5 and and CCONJ cana-3309 12 6 generalize generalize VERB cana-3309 12 7 a a DET cana-3309 12 8 great great ADJ cana-3309 12 9 deal deal NOUN cana-3309 12 10 of of ADP cana-3309 12 11 previously previously ADV cana-3309 12 12 published publish VERB cana-3309 12 13 research research NOUN cana-3309 12 14 . . PUNCT cana-3309 13 1 keywords keyword NOUN cana-3309 13 2 : : PUNCT cana-3309 13 3 rational rational ADJ cana-3309 13 4 expression expression NOUN cana-3309 13 5 , , PUNCT cana-3309 13 6 complete complete ADJ cana-3309 13 7 metric metric ADJ cana-3309 13 8 spaces space NOUN cana-3309 13 9 , , PUNCT cana-3309 13 10 fixed fix VERB cana-3309 13 11 point point NOUN cana-3309 13 12 , , PUNCT cana-3309 13 13 Ο†p Ο†p ADP cana-3309 13 14 operator operator NOUN cana-3309 13 15 , , PUNCT cana-3309 13 16 self self NOUN cana-3309 13 17 mapping mapping NOUN cana-3309 13 18 1 1 NUM cana-3309 13 19 . . PUNCT cana-3309 13 20 introduction introduction NOUN cana-3309 13 21 in in ADP cana-3309 13 22 1889 1889 NUM cana-3309 13 23 , , PUNCT cana-3309 13 24 h. h. PROPN cana-3309 13 25 poincare poincare PROPN cana-3309 13 26 the the DET cana-3309 13 27 french french ADJ cana-3309 13 28 mathematician mathematician NOUN cana-3309 13 29 , , PUNCT cana-3309 13 30 introduced introduce VERB cana-3309 13 31 the the DET cana-3309 13 32 fixed fix VERB cana-3309 13 33 points point NOUN cana-3309 13 34 in in ADP cana-3309 13 35 different different ADJ cana-3309 13 36 version version NOUN cana-3309 13 37 the the DET cana-3309 13 38 origin origin NOUN cana-3309 13 39 of of ADP cana-3309 13 40 fixed fix VERB cana-3309 13 41 point point NOUN cana-3309 13 42 theory theory NOUN cana-3309 13 43 , , PUNCT cana-3309 13 44 in in ADP cana-3309 13 45 the the DET cana-3309 13 46 19th 19th ADJ cana-3309 13 47 century century NOUN cana-3309 13 48 was be AUX cana-3309 13 49 notorious notorious ADJ cana-3309 13 50 by by ADP cana-3309 13 51 mathematicians mathematician NOUN cana-3309 13 52 like like ADP cana-3309 13 53 cauchy cauchy NOUN cana-3309 13 54 , , PUNCT cana-3309 13 55 fredholm fredholm NOUN cana-3309 13 56 , , PUNCT cana-3309 13 57 caristi caristi NOUN cana-3309 13 58 , , PUNCT cana-3309 13 59 liouville liouville ADJ cana-3309 13 60 , , PUNCT cana-3309 13 61 lipschitz lipschitz NOUN cana-3309 13 62 , , PUNCT cana-3309 13 63 peano peano NOUN cana-3309 13 64 and and CCONJ cana-3309 13 65 picard picard PROPN cana-3309 13 66 . . PUNCT cana-3309 14 1 banach banach NOUN cana-3309 14 2 ’s ’s PART cana-3309 14 3 contribution contribution NOUN cana-3309 14 4 to to ADP cana-3309 14 5 metric metric ADJ cana-3309 14 6 fixed fix VERB cana-3309 14 7 point point NOUN cana-3309 14 8 theory theory NOUN cana-3309 14 9 was be AUX cana-3309 14 10 not not PART cana-3309 14 11 recognized recognize VERB cana-3309 14 12 until until ADP cana-3309 14 13 f. f. PROPN cana-3309 14 14 brouwer brouwer PROPN cana-3309 14 15 ’s ’s PART cana-3309 14 16 work work NOUN cana-3309 14 17 and and CCONJ cana-3309 14 18 contribution contribution NOUN cana-3309 14 19 to to ADP cana-3309 14 20 the the DET cana-3309 14 21 development development NOUN cana-3309 14 22 of of ADP cana-3309 14 23 the the DET cana-3309 14 24 nonlinear nonlinear ADJ cana-3309 14 25 functional functional ADJ cana-3309 14 26 analysis analysis NOUN cana-3309 14 27 as as ADP cana-3309 14 28 an an DET cana-3309 14 29 active active ADJ cana-3309 14 30 and and CCONJ cana-3309 14 31 vital vital ADJ cana-3309 14 32 branch branch NOUN cana-3309 14 33 of of ADP cana-3309 14 34 mathematics mathematic NOUN cana-3309 14 35 . . PUNCT cana-3309 15 1 in in ADP cana-3309 15 2 this this DET cana-3309 15 3 paper paper NOUN cana-3309 15 4 , , PUNCT cana-3309 15 5 we we PRON cana-3309 15 6 investigate investigate VERB cana-3309 15 7 some some DET cana-3309 15 8 fixed fix VERB cana-3309 15 9 point point NOUN cana-3309 15 10 theorems theorem NOUN cana-3309 15 11 of of ADP cana-3309 15 12 complete complete ADJ cana-3309 15 13 metric metric ADJ cana-3309 15 14 sapces sapce NOUN cana-3309 15 15 using use VERB cana-3309 15 16 Ο†p Ο†p ADP cana-3309 15 17 operator operator NOUN cana-3309 15 18 . . PUNCT cana-3309 16 1 definition definition NOUN cana-3309 16 2 1.1 1.1 NUM cana-3309 16 3 : : PUNCT cana-3309 16 4 let let VERB cana-3309 16 5 𝑋 𝑋 NOUN cana-3309 16 6 be be AUX cana-3309 16 7 a a DET cana-3309 16 8 none none NOUN cana-3309 16 9 - - PUNCT cana-3309 16 10 empty empty ADJ cana-3309 16 11 set set NOUN cana-3309 16 12 , , PUNCT cana-3309 16 13 a a DET cana-3309 16 14 function function NOUN cana-3309 16 15 𝑑 𝑑 NOUN cana-3309 16 16 : : PUNCT cana-3309 16 17 𝑋 𝑋 NOUN cana-3309 16 18 Γ— Γ— NOUN cana-3309 16 19 𝑋 𝑋 PROPN cana-3309 16 20 β†’ β†’ SYM cana-3309 16 21 𝑅 𝑅 PROPN cana-3309 16 22 is be AUX cana-3309 16 23 called call VERB cana-3309 16 24 a a DET cana-3309 16 25 metric metric NOUN cana-3309 16 26 on on ADP cana-3309 16 27 𝑋 𝑋 PROPN cana-3309 16 28 , , PUNCT cana-3309 16 29 if if SCONJ cana-3309 16 30 it it PRON cana-3309 16 31 satisfies satisfy VERB cana-3309 16 32 the the DET cana-3309 16 33 following follow VERB cana-3309 16 34 conditions condition NOUN cana-3309 16 35 , , PUNCT cana-3309 16 36 ( ( PUNCT cana-3309 16 37 i i NOUN cana-3309 16 38 ) ) PUNCT cana-3309 16 39 𝑑(πœ› 𝑑(πœ› PROPN cana-3309 16 40 , , PUNCT cana-3309 16 41 휁 휁 NOUN cana-3309 16 42 ) ) PUNCT cana-3309 16 43 β‰₯ β‰₯ NOUN cana-3309 16 44 0 0 NUM cana-3309 16 45 and and CCONJ cana-3309 16 46 𝑑(πœ› 𝑑(πœ› PROPN cana-3309 16 47 , , PUNCT cana-3309 16 48 휁 휁 NOUN cana-3309 16 49 ) ) PUNCT cana-3309 16 50 = = SYM cana-3309 16 51 0 0 PUNCT cana-3309 17 1 if if SCONJ cana-3309 17 2 and and CCONJ cana-3309 17 3 only only ADV cana-3309 17 4 if if SCONJ cana-3309 17 5 πœ› πœ› PROPN cana-3309 17 6 = = SYM cana-3309 17 7 휁 휁 PROPN cana-3309 17 8 , , PUNCT cana-3309 17 9 βˆ€πœ› βˆ€πœ› NUM cana-3309 17 10 , , PUNCT cana-3309 17 11 휁 휁 PROPN cana-3309 17 12 ∈ ∈ PROPN cana-3309 17 13 𝑋 𝑋 PROPN cana-3309 17 14 ( ( PUNCT cana-3309 17 15 ii ii NOUN cana-3309 17 16 ) ) PUNCT cana-3309 17 17 𝑑(πœ› 𝑑(πœ› PROPN cana-3309 17 18 , , PUNCT cana-3309 17 19 휁 휁 NOUN cana-3309 17 20 ) ) PUNCT cana-3309 17 21 = = SYM cana-3309 17 22 𝑑(휁 𝑑(휁 PROPN cana-3309 17 23 , , PUNCT cana-3309 17 24 πœ›)βˆ€ πœ›)βˆ€ PROPN cana-3309 17 25 πœ› πœ› PROPN cana-3309 17 26 , , PUNCT cana-3309 17 27 휁 휁 PROPN cana-3309 17 28 ∈ ∈ PROPN cana-3309 17 29 𝑋 𝑋 PROPN cana-3309 17 30 ( ( PUNCT cana-3309 17 31 iii iii NOUN cana-3309 17 32 ) ) PUNCT cana-3309 17 33 𝑑(πœ› 𝑑(πœ› PROPN cana-3309 17 34 , , PUNCT cana-3309 17 35 휁 휁 NOUN cana-3309 17 36 ) ) PUNCT cana-3309 17 37 ≀ ≀ NOUN cana-3309 17 38 𝑑(πœ› 𝑑(πœ› PROPN cana-3309 17 39 , , PUNCT cana-3309 17 40 𝑧 𝑧 NOUN cana-3309 17 41 ) ) PUNCT cana-3309 17 42 + + X cana-3309 17 43 𝑑(𝑧 𝑑(𝑧 ADJ cana-3309 17 44 , , PUNCT cana-3309 17 45 휁 휁 NOUN cana-3309 17 46 ) ) PUNCT cana-3309 17 47 βˆ€ βˆ€ PUNCT cana-3309 17 48 πœ› πœ› ADP cana-3309 17 49 , , PUNCT cana-3309 17 50 휁 휁 X cana-3309 17 51 , , PUNCT cana-3309 17 52 𝑧 𝑧 DET cana-3309 17 53 ∈ ∈ NOUN cana-3309 17 54 𝑋 𝑋 NOUN cana-3309 17 55 then then ADV cana-3309 17 56 ( ( PUNCT cana-3309 17 57 𝑋 𝑋 PROPN cana-3309 17 58 , , PUNCT cana-3309 17 59 𝑑 𝑑 NOUN cana-3309 17 60 ) ) PUNCT cana-3309 17 61 is be AUX cana-3309 17 62 called call VERB cana-3309 17 63 metric metric ADJ cana-3309 17 64 space space NOUN cana-3309 17 65 . . PUNCT cana-3309 18 1 definition definition NOUN cana-3309 18 2 1.2 1.2 NUM cana-3309 18 3 : : PUNCT cana-3309 18 4 a a DET cana-3309 18 5 sequence sequence NOUN cana-3309 18 6 { { PUNCT cana-3309 18 7 πœ›π‘› πœ›π‘› NOUN cana-3309 18 8 } } PUNCT cana-3309 18 9 is be AUX cana-3309 18 10 said say VERB cana-3309 18 11 to to PART cana-3309 18 12 be be AUX cana-3309 18 13 a a DET cana-3309 18 14 cauchy cauchy ADJ cana-3309 18 15 sequence sequence NOUN cana-3309 18 16 if if SCONJ cana-3309 18 17 give give VERB cana-3309 18 18 νœ€ νœ€ PRON cana-3309 18 19 > > X cana-3309 18 20 0 0 NUM cana-3309 18 21 , , PUNCT cana-3309 18 22 there there PRON cana-3309 18 23 exists exist VERB cana-3309 18 24 a a DET cana-3309 18 25 positive positive ADJ cana-3309 18 26 integer integer NOUN cana-3309 18 27 π‘š π‘š ADP cana-3309 19 1 such such ADJ cana-3309 19 2 that that PRON cana-3309 20 1 |πœ›π‘› |πœ›π‘› NOUN cana-3309 21 1 βˆ’ βˆ’ PROPN cana-3309 22 1 πœ›π‘š| πœ›π‘š| INTJ cana-3309 22 2 < < X cana-3309 22 3 νœ€ νœ€ X cana-3309 22 4 whenever whenever SCONJ cana-3309 22 5 𝑛 𝑛 DET cana-3309 22 6 β‰₯ β‰₯ NOUN cana-3309 22 7 π‘š. π‘š. ADV cana-3309 22 8 mailto:dinesh.skksv93@gmail.com mailto:dinesh.skksv93@gmail.com X cana-3309 22 9 mailto:kzoto@uogj.edu.al mailto:kzoto@uogj.edu.al PROPN cana-3309 22 10 mailto:ardiana.topi@uet.edu.al mailto:ardiana.topi@uet.edu.al NOUN cana-3309 22 11 mailto:shobabalasubramaniyam@gmail.com mailto:shobabalasubramaniyam@gmail.com PROPN cana-3309 22 12 communications communication NOUN cana-3309 22 13 on on ADP cana-3309 22 14 applied apply VERB cana-3309 22 15 nonlinear nonlinear ADJ cana-3309 22 16 analysis analysis NOUN cana-3309 22 17 issn issn NOUN cana-3309 22 18 : : PUNCT cana-3309 22 19 1074 1074 NUM cana-3309 22 20 - - PUNCT cana-3309 22 21 133x 133x NUM cana-3309 22 22 vol vol NOUN cana-3309 22 23 32 32 NUM cana-3309 22 24 no no NOUN cana-3309 22 25 . . PUNCT cana-3309 23 1 6s 6s NUM cana-3309 23 2 ( ( PUNCT cana-3309 23 3 2025 2025 NUM cana-3309 23 4 ) ) PUNCT cana-3309 23 5 455 455 NUM cana-3309 24 1 https://internationalpubls.com https://internationalpubls.com X cana-3309 24 2 definition definition NOUN cana-3309 24 3 1.3 1.3 NUM cana-3309 24 4 : : PUNCT cana-3309 24 5 a a DET cana-3309 24 6 metric metric ADJ cana-3309 24 7 space space NOUN cana-3309 24 8 ( ( PUNCT cana-3309 24 9 𝑋 𝑋 PROPN cana-3309 24 10 , , PUNCT cana-3309 24 11 𝑑 𝑑 NOUN cana-3309 24 12 ) ) PUNCT cana-3309 24 13 is be AUX cana-3309 24 14 said say VERB cana-3309 24 15 to to PART cana-3309 24 16 be be AUX cana-3309 24 17 complete complete ADJ cana-3309 24 18 if if SCONJ cana-3309 24 19 every every DET cana-3309 24 20 cauchy cauchy ADJ cana-3309 24 21 sequence sequence NOUN cana-3309 24 22 in in ADP cana-3309 24 23 𝑋 𝑋 PROPN cana-3309 24 24 is be AUX cana-3309 24 25 convergent convergent NOUN cana-3309 24 26 in in ADP cana-3309 24 27 𝑋 𝑋 ADJ cana-3309 24 28 definition definition NOUN cana-3309 24 29 1.4 1.4 NUM cana-3309 24 30 : : PUNCT cana-3309 24 31 let let VERB cana-3309 24 32 ( ( PUNCT cana-3309 24 33 𝑋 𝑋 NOUN cana-3309 24 34 , , PUNCT cana-3309 24 35 𝑑 𝑑 NOUN cana-3309 24 36 ) ) PUNCT cana-3309 24 37 be be VERB cana-3309 24 38 a a DET cana-3309 24 39 complete complete ADJ cana-3309 24 40 metric metric ADJ cana-3309 24 41 space space NOUN cana-3309 24 42 Ο†p Ο†p ADP cana-3309 24 43 : : PUNCT cana-3309 24 44 𝑋 𝑋 PROPN cana-3309 24 45 β†’ β†’ SYM cana-3309 24 46 𝑋 𝑋 PROPN cana-3309 24 47 is be AUX cana-3309 24 48 an an DET cana-3309 24 49 increasing increase VERB cana-3309 24 50 and and CCONJ cana-3309 24 51 positive positive ADJ cana-3309 24 52 mapping mapping NOUN cana-3309 24 53 . . PUNCT cana-3309 25 1 if if SCONJ cana-3309 25 2 𝑋 𝑋 PROPN cana-3309 25 3 = = SYM cana-3309 25 4 𝑅 𝑅 PROPN cana-3309 25 5 , , PUNCT cana-3309 25 6 then then ADV cana-3309 25 7 Ο†p Ο†p ADP cana-3309 25 8 : : PUNCT cana-3309 25 9 𝑅 𝑅 PROPN cana-3309 25 10 β†’ β†’ SYM cana-3309 25 11 𝑅 𝑅 PROPN cana-3309 25 12 is be AUX cana-3309 25 13 a a DET cana-3309 25 14 p p ADJ cana-3309 25 15 - - PUNCT cana-3309 25 16 lapalcian lapalcian ADJ cana-3309 25 17 operator operator NOUN cana-3309 25 18 , , PUNCT cana-3309 25 19 Ο†p(πœ› Ο†p(πœ› ADV cana-3309 25 20 ) ) PUNCT cana-3309 26 1 = = SYM cana-3309 26 2 |πœ›|π‘βˆ’2πœ› |πœ›|π‘βˆ’2πœ› PROPN cana-3309 26 3 for for ADP cana-3309 26 4 some some DET cana-3309 26 5 𝑝 𝑝 PROPN cana-3309 26 6 > > SYM cana-3309 26 7 1 1 NUM cana-3309 26 8 lemma lemma PROPN cana-3309 26 9 1.5 1.5 NUM cana-3309 26 10 : : PUNCT cana-3309 26 11 show show VERB cana-3309 26 12 that that SCONJ cana-3309 26 13 the the DET cana-3309 26 14 operator operator NOUN cana-3309 26 15 Ο†p Ο†p ADP cana-3309 26 16 : : PUNCT cana-3309 26 17 𝑋 𝑋 PROPN cana-3309 26 18 β†’ β†’ SYM cana-3309 26 19 𝑋 𝑋 PROPN cana-3309 26 20 holds hold VERB cana-3309 26 21 the the DET cana-3309 26 22 following follow VERB cana-3309 26 23 properties property NOUN cana-3309 26 24 ( ( PUNCT cana-3309 26 25 i i NOUN cana-3309 26 26 ) ) PUNCT cana-3309 26 27 if if SCONJ cana-3309 26 28 πœ› πœ› PROPN cana-3309 26 29 ≀ ≀ X cana-3309 26 30 휁 휁 NOUN cana-3309 26 31 then then ADV cana-3309 26 32 , , PUNCT cana-3309 26 33 Ο†p(πœ› Ο†p(πœ› X cana-3309 26 34 ) ) PUNCT cana-3309 26 35 ≀ ≀ NOUN cana-3309 26 36 Ο†p(휁 Ο†p(휁 NUM cana-3309 26 37 ) ) PUNCT cana-3309 26 38 βˆ€πœ› βˆ€πœ› NUM cana-3309 26 39 , , PUNCT cana-3309 26 40 휁 휁 PROPN cana-3309 26 41 ∈ ∈ PROPN cana-3309 26 42 𝑋 𝑋 PROPN cana-3309 26 43 ( ( PUNCT cana-3309 26 44 ii ii NOUN cana-3309 26 45 ) ) PUNCT cana-3309 26 46 Ο†p Ο†p ADP cana-3309 26 47 is be AUX cana-3309 26 48 continuous continuous ADJ cana-3309 26 49 bijection bijection NOUN cana-3309 26 50 and and CCONJ cana-3309 26 51 its its PRON cana-3309 26 52 inverse inverse NOUN cana-3309 26 53 mapping mapping NOUN cana-3309 26 54 is be AUX cana-3309 26 55 also also ADV cana-3309 26 56 continuous continuous ADJ cana-3309 26 57 . . PUNCT cana-3309 27 1 that that PRON cana-3309 27 2 is be AUX cana-3309 27 3 Ο†p Ο†p ADP cana-3309 27 4 is be AUX cana-3309 27 5 homeomorphishm homeomorphishm NOUN cana-3309 27 6 ( ( PUNCT cana-3309 27 7 iii iii NOUN cana-3309 27 8 ) ) PUNCT cana-3309 27 9 Ο†p(πœ›νœ Ο†p(πœ›νœ NOUN cana-3309 27 10 ) ) PUNCT cana-3309 27 11 = = SYM cana-3309 28 1 Ο†p(πœ›)Ο†p(휁 Ο†p(πœ›)Ο†p(휁 ADJ cana-3309 28 2 ) ) PUNCT cana-3309 28 3 βˆ€ βˆ€ PUNCT cana-3309 29 1 πœ› πœ› NOUN cana-3309 29 2 , , PUNCT cana-3309 29 3 휁 휁 PROPN cana-3309 29 4 ∈ ∈ PROPN cana-3309 29 5 𝑋 𝑋 PROPN cana-3309 29 6 ( ( PUNCT cana-3309 29 7 iv iv NOUN cana-3309 29 8 ) ) PUNCT cana-3309 29 9 Ο†p Ο†p ADP cana-3309 29 10 ( ( PUNCT cana-3309 29 11 πœ› πœ› PROPN cana-3309 29 12 + + PROPN cana-3309 29 13 휁 휁 NOUN cana-3309 29 14 ) ) PUNCT cana-3309 29 15 ≀ ≀ NOUN cana-3309 29 16 Ο†p(πœ› Ο†p(πœ› PUNCT cana-3309 29 17 ) ) PUNCT cana-3309 29 18 + + CCONJ cana-3309 29 19 Ο†p(휁 Ο†p(휁 X cana-3309 29 20 ) ) PUNCT cana-3309 29 21 βˆ€ βˆ€ PUNCT cana-3309 29 22 πœ› πœ› ADP cana-3309 29 23 , , PUNCT cana-3309 29 24 휁 휁 PROPN cana-3309 29 25 ∈ ∈ PROPN cana-3309 29 26 𝑋 𝑋 PROPN cana-3309 29 27 2 2 NUM cana-3309 29 28 . . PUNCT cana-3309 29 29 main main ADJ cana-3309 29 30 result result NOUN cana-3309 29 31 theorem theorem VERB cana-3309 29 32 2.1 2.1 NUM cana-3309 29 33 : : PUNCT cana-3309 29 34 let let VERB cana-3309 29 35 𝑇 𝑇 PROPN cana-3309 29 36 be be AUX cana-3309 29 37 continuous continuous ADJ cana-3309 29 38 self self NOUN cana-3309 29 39 map map NOUN cana-3309 29 40 , , PUNCT cana-3309 29 41 defined define VERB cana-3309 29 42 on on ADP cana-3309 29 43 a a DET cana-3309 29 44 complete complete ADJ cana-3309 29 45 metric metric ADJ cana-3309 29 46 space space NOUN cana-3309 29 47 𝑋 𝑋 NOUN cana-3309 29 48 and and CCONJ cana-3309 29 49 Ο†p Ο†p ADP cana-3309 29 50 : : PUNCT cana-3309 29 51 𝑋 𝑋 PROPN cana-3309 29 52 β†’ β†’ SYM cana-3309 29 53 𝑋 𝑋 PROPN cana-3309 29 54 further further ADJ cana-3309 29 55 𝑇 𝑇 PROPN cana-3309 29 56 satisfies satisfy VERB cana-3309 29 57 the the DET cana-3309 29 58 following follow VERB cana-3309 29 59 conditions condition NOUN cana-3309 29 60 Ο†p(𝑑(π‘‡πœ› Ο†p(𝑑(π‘‡πœ› PROPN cana-3309 29 61 , , PUNCT cana-3309 29 62 π‘‡νœ π‘‡νœ PROPN cana-3309 29 63 ) ) PUNCT cana-3309 29 64 ) ) PUNCT cana-3309 29 65 ≀ ≀ NUM cana-3309 29 66 πœ†1Ο†p(𝑑(πœ› πœ†1Ο†p(𝑑(πœ› PROPN cana-3309 29 67 , , PUNCT cana-3309 29 68 휁 휁 NOUN cana-3309 29 69 ) ) PUNCT cana-3309 29 70 ) ) PUNCT cana-3309 30 1 + + CCONJ cana-3309 30 2 πœ†2Ο†p[𝑑(π‘‡πœ› πœ†2Ο†p[𝑑(π‘‡πœ› NUM cana-3309 30 3 , , PUNCT cana-3309 30 4 πœ› πœ› PROPN cana-3309 30 5 ) ) PUNCT cana-3309 30 6 + + X cana-3309 30 7 𝑑(π‘‡νœ 𝑑(π‘‡νœ X cana-3309 30 8 , , PUNCT cana-3309 30 9 휁 휁 NOUN cana-3309 30 10 ) ) PUNCT cana-3309 30 11 ] ] PUNCT cana-3309 31 1 + + X cana-3309 31 2 πœ†3Ο†p[𝑑(π‘‡νœ πœ†3Ο†p[𝑑(π‘‡νœ X cana-3309 31 3 , , PUNCT cana-3309 31 4 πœ› πœ› X cana-3309 31 5 ) ) PUNCT cana-3309 31 6 + + CCONJ cana-3309 31 7 𝑑(π‘‡πœ› 𝑑(π‘‡πœ› ADJ cana-3309 31 8 , , PUNCT cana-3309 31 9 휁 휁 NOUN cana-3309 31 10 ) ) PUNCT cana-3309 31 11 ] ] PUNCT cana-3309 32 1 + + ADJ cana-3309 32 2 πœ†4Ο†p πœ†4Ο†p PUNCT cana-3309 32 3 [ [ PUNCT cana-3309 32 4 𝑑(πœ›,π‘‡πœ›)𝑑(𝜁,π‘‡πœ 𝑑(πœ›,π‘‡πœ›)𝑑(𝜁,π‘‡πœ NOUN cana-3309 32 5 ) ) PUNCT cana-3309 32 6 𝑑(πœ›,𝜁 𝑑(πœ›,𝜁 NOUN cana-3309 32 7 ) ) PUNCT cana-3309 32 8 ] ] PUNCT cana-3309 33 1 + + X cana-3309 33 2 πœ†5Ο†p πœ†5Ο†p PUNCT cana-3309 33 3 [ [ PUNCT cana-3309 33 4 𝑑(πœ›,π‘‡πœ)𝑑(𝜁,π‘‡πœ› 𝑑(πœ›,π‘‡πœ)𝑑(𝜁,π‘‡πœ› NUM cana-3309 33 5 ) ) PUNCT cana-3309 33 6 𝑑(πœ›,𝜁 𝑑(πœ›,𝜁 NOUN cana-3309 33 7 ) ) PUNCT cana-3309 33 8 ] ] PUNCT cana-3309 34 1 + + X cana-3309 34 2 πœ†6Ο†p πœ†6Ο†p X cana-3309 34 3 [ [ PUNCT cana-3309 34 4 𝑑(πœ›,π‘‡πœ›)𝑑(𝜁,π‘‡πœ)+𝑑(πœ›,π‘‡πœ)𝑑(𝜁,π‘‡πœ› 𝑑(πœ›,π‘‡πœ›)𝑑(𝜁,π‘‡πœ)+𝑑(πœ›,π‘‡πœ)𝑑(𝜁,π‘‡πœ› NOUN cana-3309 34 5 ) ) PUNCT cana-3309 34 6 𝑑(πœ›,𝜁 𝑑(πœ›,𝜁 NOUN cana-3309 34 7 ) ) PUNCT cana-3309 34 8 ] ] PUNCT cana-3309 35 1 + + CCONJ cana-3309 35 2 πœ†7Ο†p πœ†7Ο†p PUNCT cana-3309 35 3 [ [ PUNCT cana-3309 35 4 𝑑(πœ›,π‘‡πœ›)𝑑(𝜁,π‘‡πœ)+𝑑(πœ›,π‘‡πœ)𝑑(𝜁,π‘‡πœ› 𝑑(πœ›,π‘‡πœ›)𝑑(𝜁,π‘‡πœ)+𝑑(πœ›,π‘‡πœ)𝑑(𝜁,π‘‡πœ› NOUN cana-3309 35 5 ) ) PUNCT cana-3309 35 6 𝑑(πœ›,𝜁 𝑑(πœ›,𝜁 NOUN cana-3309 35 7 ) ) PUNCT cana-3309 35 8 ] ] PUNCT cana-3309 36 1 + + X cana-3309 36 2 πœ†8Ο†p πœ†8Ο†p X cana-3309 36 3 [ [ PUNCT cana-3309 36 4 𝑑(πœ›,π‘‡πœ)[𝑑(πœ›,π‘‡πœ›)+𝑑(𝜁,π‘‡πœ)+𝑑(πœ›,π‘‡πœ)+𝑑(𝜁,π‘‡πœ› 𝑑(πœ›,π‘‡πœ)[𝑑(πœ›,π‘‡πœ›)+𝑑(𝜁,π‘‡πœ)+𝑑(πœ›,π‘‡πœ)+𝑑(𝜁,π‘‡πœ› NOUN cana-3309 36 5 ) ) PUNCT cana-3309 36 6 ] ] PUNCT cana-3309 36 7 𝑑(πœ›,π‘‡πœ›)+𝑑(𝜁,π‘‡πœ)+𝑑(πœ›,π‘‡πœ)+𝑑(𝜁,π‘‡πœ› 𝑑(πœ›,π‘‡πœ›)+𝑑(𝜁,π‘‡πœ)+𝑑(πœ›,π‘‡πœ)+𝑑(𝜁,π‘‡πœ› NOUN cana-3309 36 8 ) ) PUNCT cana-3309 36 9 ] ] PUNCT cana-3309 36 10 for for ADP cana-3309 36 11 all all DET cana-3309 36 12 πœ› πœ› PROPN cana-3309 36 13 , , PUNCT cana-3309 36 14 휁 휁 PROPN cana-3309 36 15 ∈ ∈ PROPN cana-3309 36 16 𝑋 𝑋 PROPN cana-3309 36 17 , , PUNCT cana-3309 36 18 πœ› πœ› PROPN cana-3309 36 19 β‰  β‰  PROPN cana-3309 36 20 휁 휁 NOUN cana-3309 36 21 and and CCONJ cana-3309 36 22 Ο†p(πœ†1 Ο†p(πœ†1 ADJ cana-3309 36 23 ) ) PUNCT cana-3309 36 24 + + CCONJ cana-3309 36 25 2Ο†p(πœ†2 2Ο†p(πœ†2 NUM cana-3309 36 26 ) ) PUNCT cana-3309 36 27 + + NUM cana-3309 36 28 2Ο†p(πœ†3 2Ο†p(πœ†3 NUM cana-3309 36 29 ) ) PUNCT cana-3309 36 30 + + NUM cana-3309 36 31 Ο†p(πœ†4 Ο†p(πœ†4 NOUN cana-3309 36 32 ) ) PUNCT cana-3309 36 33 + + CCONJ cana-3309 36 34 Ο†p(πœ†6 Ο†p(πœ†6 NOUN cana-3309 36 35 ) ) PUNCT cana-3309 36 36 + + X cana-3309 36 37 Ο†p(πœ†7 Ο†p(πœ†7 NOUN cana-3309 36 38 ) ) PUNCT cana-3309 36 39 < < X cana-3309 36 40 1 1 NUM cana-3309 36 41 then then ADV cana-3309 36 42 𝑇 𝑇 PROPN cana-3309 36 43 has have AUX cana-3309 36 44 unique unique ADJ cana-3309 36 45 fixed fix VERB cana-3309 36 46 point point NOUN cana-3309 36 47 in in ADP cana-3309 36 48 𝑇. 𝑇. PROPN cana-3309 36 49 proof proof NOUN cana-3309 36 50 : : PUNCT cana-3309 36 51 let let VERB cana-3309 36 52 πœ›0 πœ›0 NOUN cana-3309 36 53 be be AUX cana-3309 36 54 an an DET cana-3309 36 55 arbitrary arbitrary ADJ cana-3309 36 56 point point NOUN cana-3309 36 57 in in ADP cana-3309 36 58 𝑋 𝑋 PROPN cana-3309 36 59 , , PUNCT cana-3309 36 60 and and CCONJ cana-3309 36 61 we we PRON cana-3309 36 62 define define VERB cana-3309 36 63 a a DET cana-3309 36 64 sequence sequence NOUN cana-3309 36 65 { { PUNCT cana-3309 36 66 πœ›π‘› πœ›π‘› NOUN cana-3309 36 67 } } PUNCT cana-3309 36 68 by by ADP cana-3309 36 69 means mean NOUN cana-3309 36 70 of of ADP cana-3309 36 71 iterates iterate NOUN cana-3309 36 72 of of ADP cana-3309 36 73 𝑇. 𝑇. PROPN cana-3309 36 74 by by ADP cana-3309 36 75 setting set VERB cana-3309 36 76 π‘‡π‘›πœ›0 π‘‡π‘›πœ›0 NOUN cana-3309 36 77 = = SYM cana-3309 36 78 πœ›π‘› πœ›π‘› NOUN cana-3309 36 79 , , PUNCT cana-3309 36 80 where where SCONJ cana-3309 36 81 𝑛 𝑛 PROPN cana-3309 36 82 is be AUX cana-3309 36 83 a a DET cana-3309 36 84 positive positive ADJ cana-3309 36 85 integers integer NOUN cana-3309 36 86 . . PUNCT cana-3309 37 1 if if SCONJ cana-3309 37 2 πœ›π‘› πœ›π‘› NOUN cana-3309 37 3 = = SYM cana-3309 37 4 πœ›π‘›+1 πœ›π‘›+1 PROPN cana-3309 37 5 , , PUNCT cana-3309 37 6 for for ADP cana-3309 37 7 some some DET cana-3309 37 8 𝑛 𝑛 NOUN cana-3309 37 9 , , PUNCT cana-3309 37 10 then then ADV cana-3309 37 11 we we PRON cana-3309 37 12 have have VERB cana-3309 37 13 π‘‡πœ›π‘› π‘‡πœ›π‘› PROPN cana-3309 37 14 = = NOUN cana-3309 37 15 πœ›π‘› πœ›π‘› NOUN cana-3309 37 16 , , PUNCT cana-3309 37 17 then then ADV cana-3309 37 18 πœ›π‘› πœ›π‘› NOUN cana-3309 37 19 is be AUX cana-3309 37 20 a a DET cana-3309 37 21 fixed fix VERB cana-3309 37 22 point point NOUN cana-3309 37 23 of of ADP cana-3309 37 24 𝑇 𝑇 NOUN cana-3309 37 25 taking take VERB cana-3309 37 26 πœ›π‘› πœ›π‘› NOUN cana-3309 37 27 β‰  β‰  PROPN cana-3309 37 28 πœ›π‘›+1 πœ›π‘›+1 NOUN cana-3309 37 29 for for ADP cana-3309 37 30 all all DET cana-3309 37 31 𝑛 𝑛 DET cana-3309 37 32 Ο†p(𝑑(πœ›π‘›+1 Ο†p(𝑑(πœ›π‘›+1 ADP cana-3309 37 33 , , PUNCT cana-3309 37 34 πœ›π‘› πœ›π‘› NOUN cana-3309 37 35 ) ) PUNCT cana-3309 37 36 ) ) PUNCT cana-3309 38 1 = = SYM cana-3309 38 2 Ο†p(𝑑(π‘‡πœ›π‘› Ο†p(𝑑(π‘‡πœ›π‘› VERB cana-3309 38 3 , , PUNCT cana-3309 38 4 π‘‡πœ›π‘›βˆ’1 π‘‡πœ›π‘›βˆ’1 NOUN cana-3309 38 5 ) ) PUNCT cana-3309 38 6 ) ) PUNCT cana-3309 38 7 Ο†p(𝑑(π‘‡πœ›π‘› Ο†p(𝑑(π‘‡πœ›π‘› VERB cana-3309 38 8 , , PUNCT cana-3309 38 9 π‘‡πœ›π‘›βˆ’1 π‘‡πœ›π‘›βˆ’1 NOUN cana-3309 38 10 ) ) PUNCT cana-3309 38 11 ) ) PUNCT cana-3309 39 1 ≀ ≀ PROPN cana-3309 39 2 πœ†1Ο†p(𝑑(πœ›π‘› πœ†1Ο†p(𝑑(πœ›π‘› PROPN cana-3309 39 3 , , PUNCT cana-3309 39 4 πœ›π‘›βˆ’1 πœ›π‘›βˆ’1 PROPN cana-3309 39 5 ) ) PUNCT cana-3309 39 6 ) ) PUNCT cana-3309 40 1 + + CCONJ cana-3309 40 2 πœ†2Ο†p([𝑑(π‘‡πœ›π‘› πœ†2Ο†p([𝑑(π‘‡πœ›π‘› ADJ cana-3309 40 3 , , PUNCT cana-3309 40 4 πœ›π‘› πœ›π‘› NOUN cana-3309 40 5 ) ) PUNCT cana-3309 41 1 + + CCONJ cana-3309 41 2 𝑑(π‘‡πœ›π‘›βˆ’1 𝑑(π‘‡πœ›π‘›βˆ’1 NOUN cana-3309 41 3 , , PUNCT cana-3309 41 4 πœ›π‘›βˆ’1 πœ›π‘›βˆ’1 PROPN cana-3309 41 5 ) ) PUNCT cana-3309 41 6 ] ] PUNCT cana-3309 41 7 ) ) PUNCT cana-3309 41 8 communications communication NOUN cana-3309 41 9 on on ADP cana-3309 41 10 applied apply VERB cana-3309 41 11 nonlinear nonlinear ADJ cana-3309 41 12 analysis analysis NOUN cana-3309 41 13 issn issn NOUN cana-3309 41 14 : : PUNCT cana-3309 41 15 1074 1074 NUM cana-3309 41 16 - - PUNCT cana-3309 41 17 133x 133x NUM cana-3309 41 18 vol vol NOUN cana-3309 41 19 32 32 NUM cana-3309 41 20 no no NOUN cana-3309 41 21 . . PUNCT cana-3309 42 1 6s 6s NUM cana-3309 42 2 ( ( PUNCT cana-3309 42 3 2025 2025 NUM cana-3309 42 4 ) ) PUNCT cana-3309 42 5 456 456 NUM cana-3309 42 6 https://internationalpubls.com https://internationalpubls.com X cana-3309 43 1 + + NOUN cana-3309 43 2 πœ†3Ο†p([𝑑(π‘‡πœ›π‘›βˆ’1 πœ†3Ο†p([𝑑(π‘‡πœ›π‘›βˆ’1 NOUN cana-3309 43 3 , , PUNCT cana-3309 43 4 πœ›π‘› πœ›π‘› NOUN cana-3309 43 5 ) ) PUNCT cana-3309 44 1 + + CCONJ cana-3309 44 2 𝑑(π‘‡πœ›π‘› 𝑑(π‘‡πœ›π‘› PROPN cana-3309 44 3 , , PUNCT cana-3309 44 4 πœ›π‘›βˆ’1 πœ›π‘›βˆ’1 PROPN cana-3309 44 5 ) ) PUNCT cana-3309 44 6 ] ] PUNCT cana-3309 44 7 ) ) PUNCT cana-3309 45 1 + + X cana-3309 45 2 πœ†4Ο†p πœ†4Ο†p PUNCT cana-3309 45 3 ( ( PUNCT cana-3309 45 4 [ [ PUNCT cana-3309 45 5 𝑑(πœ›π‘›,π‘‡πœ›π‘›)𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘›βˆ’1 𝑑(πœ›π‘›,π‘‡πœ›π‘›)𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘›βˆ’1 NOUN cana-3309 45 6 ) ) PUNCT cana-3309 45 7 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 NUM cana-3309 45 8 ) ) PUNCT cana-3309 45 9 ] ] PUNCT cana-3309 45 10 ) ) PUNCT cana-3309 46 1 + + X cana-3309 46 2 πœ†5Ο†p πœ†5Ο†p PUNCT cana-3309 46 3 ( ( PUNCT cana-3309 46 4 [ [ PUNCT cana-3309 46 5 𝑑(πœ›π‘›,π‘‡πœ›π‘›βˆ’1)𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘› 𝑑(πœ›π‘›,π‘‡πœ›π‘›βˆ’1)𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘› PROPN cana-3309 46 6 ) ) PUNCT cana-3309 46 7 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 PROPN cana-3309 46 8 ) ) PUNCT cana-3309 46 9 ] ] PUNCT cana-3309 46 10 ) ) PUNCT cana-3309 47 1 + + X cana-3309 47 2 πœ†6Ο†p πœ†6Ο†p X cana-3309 47 3 ( ( PUNCT cana-3309 47 4 [ [ PUNCT cana-3309 47 5 𝑑(πœ›π‘›,π‘‡πœ›π‘›)𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘›βˆ’1)+𝑑(πœ›π‘›,π‘‡πœ›π‘›βˆ’1)𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘› 𝑑(πœ›π‘›,π‘‡πœ›π‘›)𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘›βˆ’1)+𝑑(πœ›π‘›,π‘‡πœ›π‘›βˆ’1)𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘› ADJ cana-3309 47 6 ) ) PUNCT cana-3309 47 7 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 NOUN cana-3309 47 8 ) ) PUNCT cana-3309 47 9 ] ] PUNCT cana-3309 47 10 ) ) PUNCT cana-3309 48 1 + + X cana-3309 48 2 πœ†7Ο†p πœ†7Ο†p X cana-3309 48 3 ( ( PUNCT cana-3309 48 4 [ [ PUNCT cana-3309 48 5 𝑑(πœ›π‘›,π‘‡πœ›π‘›)𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘›βˆ’1)+𝑑(πœ›π‘›,π‘‡πœ›π‘›βˆ’1)𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘› 𝑑(πœ›π‘›,π‘‡πœ›π‘›)𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘›βˆ’1)+𝑑(πœ›π‘›,π‘‡πœ›π‘›βˆ’1)𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘› ADJ cana-3309 48 6 ) ) PUNCT cana-3309 48 7 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 NOUN cana-3309 48 8 ) ) PUNCT cana-3309 48 9 ] ] PUNCT cana-3309 48 10 ) ) PUNCT cana-3309 49 1 + + X cana-3309 49 2 πœ†8Ο†p πœ†8Ο†p PUNCT cana-3309 49 3 ( ( PUNCT cana-3309 49 4 [ [ PUNCT cana-3309 49 5 𝑑(πœ›π‘›,π‘‡πœ›π‘›βˆ’1)[𝑑(πœ›π‘›,π‘‡πœ›π‘›)+𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘›βˆ’1)+𝑑(πœ›π‘›,π‘‡πœ›π‘›βˆ’1)+𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘› 𝑑(πœ›π‘›,π‘‡πœ›π‘›βˆ’1)[𝑑(πœ›π‘›,π‘‡πœ›π‘›)+𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘›βˆ’1)+𝑑(πœ›π‘›,π‘‡πœ›π‘›βˆ’1)+𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘› PROPN cana-3309 49 6 ) ) PUNCT cana-3309 49 7 ] ] PUNCT cana-3309 49 8 𝑑(πœ›π‘›,π‘‡πœ›π‘›)+𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘›βˆ’1)+𝑑(πœ›π‘›,π‘‡πœ›π‘›βˆ’1)+𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘› 𝑑(πœ›π‘›,π‘‡πœ›π‘›)+𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘›βˆ’1)+𝑑(πœ›π‘›,π‘‡πœ›π‘›βˆ’1)+𝑑(πœ›π‘›βˆ’1,π‘‡πœ›π‘› X cana-3309 49 9 ) ) PUNCT cana-3309 49 10 ] ] PUNCT cana-3309 49 11 ) ) PUNCT cana-3309 49 12 ≀ ≀ PROPN cana-3309 49 13 πœ†1Ο†p(𝑑(πœ›π‘› πœ†1Ο†p(𝑑(πœ›π‘› PROPN cana-3309 49 14 , , PUNCT cana-3309 49 15 πœ›π‘›βˆ’1 πœ›π‘›βˆ’1 PROPN cana-3309 49 16 ) ) PUNCT cana-3309 50 1 + + X cana-3309 50 2 πœ†2[𝑑(πœ›π‘›+1 πœ†2[𝑑(πœ›π‘›+1 ADJ cana-3309 50 3 , , PUNCT cana-3309 50 4 πœ›π‘› πœ›π‘› NOUN cana-3309 50 5 ) ) PUNCT cana-3309 51 1 + + CCONJ cana-3309 51 2 𝑑(πœ›π‘› 𝑑(πœ›π‘› PROPN cana-3309 51 3 , , PUNCT cana-3309 51 4 πœ›π‘›βˆ’1 πœ›π‘›βˆ’1 PROPN cana-3309 51 5 ) ) PUNCT cana-3309 51 6 ] ] PUNCT cana-3309 52 1 + + PUNCT cana-3309 52 2 πœ†3Ο†p[𝑑(πœ›π‘› πœ†3Ο†p[𝑑(πœ›π‘› ADJ cana-3309 52 3 , , PUNCT cana-3309 52 4 πœ›π‘› πœ›π‘› NOUN cana-3309 52 5 ) ) PUNCT cana-3309 52 6 + + CCONJ cana-3309 52 7 𝑑(πœ›π‘›+1 𝑑(πœ›π‘›+1 PROPN cana-3309 52 8 , , PUNCT cana-3309 52 9 πœ›π‘›βˆ’1 πœ›π‘›βˆ’1 PROPN cana-3309 52 10 ) ) PUNCT cana-3309 52 11 ] ] PUNCT cana-3309 52 12 + + NUM cana-3309 52 13 πœ†4Ο†p πœ†4Ο†p PUNCT cana-3309 52 14 [ [ PUNCT cana-3309 52 15 𝑑(πœ›π‘›,πœ›π‘›+1)𝑑(πœ›π‘›βˆ’1,πœ›π‘› 𝑑(πœ›π‘›,πœ›π‘›+1)𝑑(πœ›π‘›βˆ’1,πœ›π‘› NOUN cana-3309 52 16 ) ) PUNCT cana-3309 52 17 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 PROPN cana-3309 52 18 ) ) PUNCT cana-3309 52 19 ] ] PUNCT cana-3309 53 1 + + X cana-3309 53 2 πœ†5Ο†p πœ†5Ο†p PUNCT cana-3309 53 3 [ [ PUNCT cana-3309 53 4 𝑑(πœ›π‘›,πœ›π‘›)𝑑(πœ›π‘›βˆ’1,πœ›π‘›+1 𝑑(πœ›π‘›,πœ›π‘›)𝑑(πœ›π‘›βˆ’1,πœ›π‘›+1 PROPN cana-3309 53 5 ) ) PUNCT cana-3309 53 6 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 PROPN cana-3309 53 7 ) ) PUNCT cana-3309 53 8 ] ] PUNCT cana-3309 54 1 + + X cana-3309 54 2 πœ†6Ο†p πœ†6Ο†p X cana-3309 54 3 [ [ PUNCT cana-3309 54 4 𝑑(πœ›π‘›,πœ›π‘›+1)𝑑(πœ›π‘›βˆ’1,πœ›π‘›)+𝑑(πœ›π‘›,πœ›π‘›)𝑑(πœ›π‘›βˆ’1,πœ›π‘›+1 𝑑(πœ›π‘›,πœ›π‘›+1)𝑑(πœ›π‘›βˆ’1,πœ›π‘›)+𝑑(πœ›π‘›,πœ›π‘›)𝑑(πœ›π‘›βˆ’1,πœ›π‘›+1 NOUN cana-3309 54 5 ) ) PUNCT cana-3309 54 6 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 NUM cana-3309 54 7 ) ) PUNCT cana-3309 54 8 ] ] PUNCT cana-3309 55 1 + + X cana-3309 55 2 πœ†7Ο†p πœ†7Ο†p X cana-3309 55 3 [ [ PUNCT cana-3309 55 4 𝑑(πœ›π‘›,πœ›π‘›+1)𝑑(πœ›π‘›βˆ’1,πœ›π‘›)+𝑑(πœ›π‘›,πœ›π‘›)𝑑(πœ›π‘›βˆ’1,πœ›π‘›+1 𝑑(πœ›π‘›,πœ›π‘›+1)𝑑(πœ›π‘›βˆ’1,πœ›π‘›)+𝑑(πœ›π‘›,πœ›π‘›)𝑑(πœ›π‘›βˆ’1,πœ›π‘›+1 NOUN cana-3309 55 5 ) ) PUNCT cana-3309 55 6 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 𝑑(πœ›π‘›,πœ›π‘›βˆ’1 NUM cana-3309 55 7 ) ) PUNCT cana-3309 55 8 ] ] PUNCT cana-3309 56 1 + + X cana-3309 56 2 πœ†8Ο†p πœ†8Ο†p X cana-3309 56 3 [ [ PUNCT cana-3309 56 4 𝑑(πœ›π‘›,πœ›π‘›)[𝑑(πœ›π‘›,πœ›π‘›+1)+𝑑(πœ›π‘›βˆ’1,πœ›π‘›)+𝑑(πœ›π‘›,πœ›π‘›)+𝑑(πœ›π‘›βˆ’1,πœ›π‘›+1 𝑑(πœ›π‘›,πœ›π‘›)[𝑑(πœ›π‘›,πœ›π‘›+1)+𝑑(πœ›π‘›βˆ’1,πœ›π‘›)+𝑑(πœ›π‘›,πœ›π‘›)+𝑑(πœ›π‘›βˆ’1,πœ›π‘›+1 NOUN cana-3309 56 5 ) ) PUNCT cana-3309 56 6 ] ] PUNCT cana-3309 56 7 𝑑(πœ›π‘›,πœ›π‘›+1)+𝑑(πœ›π‘›βˆ’1,πœ›π‘›)+𝑑(πœ›π‘›,πœ›π‘›)+𝑑(πœ›π‘›βˆ’1,πœ›π‘›+1 𝑑(πœ›π‘›,πœ›π‘›+1)+𝑑(πœ›π‘›βˆ’1,πœ›π‘›)+𝑑(πœ›π‘›,πœ›π‘›)+𝑑(πœ›π‘›βˆ’1,πœ›π‘›+1 X cana-3309 56 8 ) ) PUNCT cana-3309 56 9 ] ] PUNCT cana-3309 56 10 from from ADP cana-3309 56 11 the the DET cana-3309 56 12 property property NOUN cana-3309 56 13 of of ADP cana-3309 56 14 Ο†p Ο†p ADP cana-3309 56 15 operator operator NOUN cana-3309 56 16 , , PUNCT cana-3309 56 17 Ο†p(𝑑(πœ›π‘›+1 Ο†p(𝑑(πœ›π‘›+1 ADP cana-3309 56 18 , , PUNCT cana-3309 56 19 πœ›π‘› πœ›π‘› NOUN cana-3309 56 20 ) ) PUNCT cana-3309 56 21 ) ) PUNCT cana-3309 56 22 ≀ ≀ PROPN cana-3309 56 23 πœ†1Ο†p(𝑑(πœ›π‘› πœ†1Ο†p(𝑑(πœ›π‘› PROPN cana-3309 56 24 , , PUNCT cana-3309 56 25 πœ›π‘›βˆ’1 πœ›π‘›βˆ’1 PROPN cana-3309 56 26 ) ) PUNCT cana-3309 56 27 ) ) PUNCT cana-3309 57 1 + + CCONJ cana-3309 57 2 πœ†2Ο†p(𝑑(πœ›π‘›+1 πœ†2Ο†p(𝑑(πœ›π‘›+1 ADJ cana-3309 57 3 , , PUNCT cana-3309 57 4 πœ›π‘› πœ›π‘› NOUN cana-3309 57 5 ) ) PUNCT cana-3309 57 6 ) ) PUNCT cana-3309 58 1 + + CCONJ cana-3309 58 2 πœ†2Ο†p(𝑑(πœ›π‘› πœ†2Ο†p(𝑑(πœ›π‘› ADJ cana-3309 58 3 , , PUNCT cana-3309 58 4 πœ›π‘›βˆ’1 πœ›π‘›βˆ’1 PROPN cana-3309 58 5 ) ) PUNCT cana-3309 58 6 ) ) PUNCT cana-3309 59 1 + + ADP cana-3309 59 2 πœ†3Ο†p[𝑑(πœ›π‘›+1 πœ†3Ο†p[𝑑(πœ›π‘›+1 ADJ cana-3309 59 3 , , PUNCT cana-3309 59 4 πœ›π‘› πœ›π‘› NOUN cana-3309 59 5 ) ) PUNCT cana-3309 59 6 + + CCONJ cana-3309 59 7 𝑑(πœ›π‘› 𝑑(πœ›π‘› PROPN cana-3309 59 8 , , PUNCT cana-3309 59 9 πœ›π‘›βˆ’1 πœ›π‘›βˆ’1 PROPN cana-3309 59 10 ) ) PUNCT cana-3309 59 11 ] ] PUNCT cana-3309 60 1 + + CCONJ cana-3309 60 2 πœ†4Ο†p(𝑑(πœ›π‘› πœ†4Ο†p(𝑑(πœ›π‘› PROPN cana-3309 60 3 , , PUNCT cana-3309 60 4 πœ›π‘›+1 πœ›π‘›+1 NUM cana-3309 60 5 ) ) PUNCT cana-3309 60 6 ) ) PUNCT cana-3309 61 1 + + X cana-3309 61 2 πœ†6Ο†p(𝑑(πœ›π‘› πœ†6Ο†p(𝑑(πœ›π‘› ADJ cana-3309 61 3 , , PUNCT cana-3309 61 4 πœ›π‘›+1 πœ›π‘›+1 NUM cana-3309 61 5 ) ) PUNCT cana-3309 61 6 ) ) PUNCT cana-3309 62 1 + + CCONJ cana-3309 62 2 πœ†7Ο†p(𝑑(πœ›π‘› πœ†7Ο†p(𝑑(πœ›π‘› ADJ cana-3309 62 3 , , PUNCT cana-3309 62 4 πœ›π‘›+1 πœ›π‘›+1 NUM cana-3309 62 5 ) ) PUNCT cana-3309 62 6 ) ) PUNCT cana-3309 62 7 Ο†p(𝑑(πœ›π‘›+1 Ο†p(𝑑(πœ›π‘›+1 ADP cana-3309 62 8 , , PUNCT cana-3309 62 9 πœ›π‘› πœ›π‘› NOUN cana-3309 62 10 ) ) PUNCT cana-3309 62 11 ) ) PUNCT cana-3309 63 1 βˆ’ βˆ’ PROPN cana-3309 63 2 πœ†2Ο†p(𝑑(πœ›π‘›+1 πœ†2Ο†p(𝑑(πœ›π‘›+1 PROPN cana-3309 63 3 , , PUNCT cana-3309 63 4 πœ›π‘› πœ›π‘› NOUN cana-3309 63 5 ) ) PUNCT cana-3309 63 6 ) ) PUNCT cana-3309 64 1 βˆ’ βˆ’ PROPN cana-3309 64 2 πœ†3Ο†p(𝑑(πœ›π‘›+1 πœ†3Ο†p(𝑑(πœ›π‘›+1 NOUN cana-3309 64 3 , , PUNCT cana-3309 64 4 πœ›π‘› πœ›π‘› NOUN cana-3309 64 5 ) ) PUNCT cana-3309 64 6 ) ) PUNCT cana-3309 65 1 βˆ’ βˆ’ PROPN cana-3309 65 2 πœ†4Ο†p(𝑑(πœ›π‘› πœ†4Ο†p(𝑑(πœ›π‘› PROPN cana-3309 65 3 , , PUNCT cana-3309 65 4 πœ›π‘›+1 πœ›π‘›+1 NUM cana-3309 65 5 ) ) PUNCT cana-3309 65 6 ) ) PUNCT cana-3309 66 1 βˆ’πœ†6Ο†p(𝑑(πœ›π‘› βˆ’πœ†6Ο†p(𝑑(πœ›π‘› PROPN cana-3309 66 2 , , PUNCT cana-3309 66 3 πœ›π‘›+1 πœ›π‘›+1 NUM cana-3309 66 4 ) ) PUNCT cana-3309 66 5 ) ) PUNCT cana-3309 67 1 βˆ’ βˆ’ PROPN cana-3309 67 2 πœ†7Ο†p(𝑑(πœ›π‘› πœ†7Ο†p(𝑑(πœ›π‘› PROPN cana-3309 67 3 , , PUNCT cana-3309 67 4 πœ›π‘›+1 πœ›π‘›+1 NUM cana-3309 67 5 ) ) PUNCT cana-3309 67 6 ) ) PUNCT cana-3309 67 7 ≀ ≀ PROPN cana-3309 67 8 πœ†1Ο†p(𝑑(πœ›π‘› πœ†1Ο†p(𝑑(πœ›π‘› PROPN cana-3309 67 9 , , PUNCT cana-3309 67 10 πœ›π‘›βˆ’1 πœ›π‘›βˆ’1 PROPN cana-3309 67 11 ) ) PUNCT cana-3309 67 12 ) ) PUNCT cana-3309 68 1 + + CCONJ cana-3309 68 2 πœ†2Ο†p(𝑑(πœ›π‘› πœ†2Ο†p(𝑑(πœ›π‘› ADJ cana-3309 68 3 , , PUNCT cana-3309 68 4 πœ›π‘›βˆ’1 πœ›π‘›βˆ’1 PROPN cana-3309 68 5 ) ) PUNCT cana-3309 68 6 ) ) PUNCT cana-3309 69 1 + + CCONJ cana-3309 69 2 πœ†3Ο†p(𝑑(πœ›π‘› πœ†3Ο†p(𝑑(πœ›π‘› PROPN cana-3309 69 3 , , PUNCT cana-3309 69 4 πœ›π‘›βˆ’1 πœ›π‘›βˆ’1 PROPN cana-3309 69 5 ) ) PUNCT cana-3309 69 6 ) ) PUNCT cana-3309 69 7 𝑑(πœ›π‘›+1 𝑑(πœ›π‘›+1 PROPN cana-3309 69 8 , , PUNCT cana-3309 69 9 πœ›π‘› πœ›π‘› NOUN cana-3309 69 10 ) ) PUNCT cana-3309 69 11 ≀ ≀ NOUN cana-3309 69 12 ( ( PUNCT cana-3309 69 13 Ο†p(πœ†1 Ο†p(πœ†1 NOUN cana-3309 69 14 ) ) PUNCT cana-3309 69 15 + + CCONJ cana-3309 69 16 Ο†p(πœ†2 Ο†p(πœ†2 NOUN cana-3309 69 17 ) ) PUNCT cana-3309 69 18 + + NUM cana-3309 69 19 Ο†p(πœ†3 Ο†p(πœ†3 X cana-3309 69 20 ) ) PUNCT cana-3309 69 21 1 1 NUM cana-3309 69 22 βˆ’ βˆ’ NOUN cana-3309 69 23 Ο†p(πœ†2 Ο†p(πœ†2 NOUN cana-3309 69 24 ) ) PUNCT cana-3309 69 25 βˆ’ βˆ’ ADP cana-3309 69 26 Ο†p(πœ†3 Ο†p(πœ†3 NOUN cana-3309 69 27 ) ) PUNCT cana-3309 69 28 βˆ’ βˆ’ NOUN cana-3309 69 29 Ο†p(πœ†4 Ο†p(πœ†4 NOUN cana-3309 69 30 ) ) PUNCT cana-3309 69 31 βˆ’ βˆ’ NOUN cana-3309 69 32 Ο†p(πœ†6 Ο†p(πœ†6 NOUN cana-3309 69 33 ) ) PUNCT cana-3309 69 34 βˆ’ βˆ’ PROPN cana-3309 69 35 Ο†p(πœ†7 Ο†p(πœ†7 NOUN cana-3309 69 36 ) ) PUNCT cana-3309 69 37 ) ) PUNCT cana-3309 70 1 𝑑(πœ›π‘› 𝑑(πœ›π‘› PROPN cana-3309 70 2 , , PUNCT cana-3309 70 3 πœ›π‘›βˆ’1 πœ›π‘›βˆ’1 PROPN cana-3309 70 4 ) ) PUNCT cana-3309 70 5 on on ADP cana-3309 70 6 applying apply VERB cana-3309 70 7 the the DET cana-3309 70 8 same same ADJ cana-3309 70 9 process process NOUN cana-3309 70 10 , , PUNCT cana-3309 70 11 we we PRON cana-3309 70 12 get get VERB cana-3309 70 13 𝑑(πœ›π‘›+1 𝑑(πœ›π‘›+1 PROPN cana-3309 70 14 , , PUNCT cana-3309 70 15 πœ›π‘› πœ›π‘› NOUN cana-3309 70 16 ) ) PUNCT cana-3309 70 17 ≀ ≀ NOUN cana-3309 70 18 ( ( PUNCT cana-3309 70 19 Ο†p(πœ†1 Ο†p(πœ†1 NOUN cana-3309 70 20 ) ) PUNCT cana-3309 70 21 + + CCONJ cana-3309 70 22 Ο†p(πœ†2 Ο†p(πœ†2 NOUN cana-3309 70 23 ) ) PUNCT cana-3309 70 24 + + NUM cana-3309 70 25 Ο†p(πœ†3 Ο†p(πœ†3 X cana-3309 70 26 ) ) PUNCT cana-3309 70 27 1 1 NUM cana-3309 70 28 βˆ’ βˆ’ NOUN cana-3309 70 29 Ο†p(πœ†2 Ο†p(πœ†2 NOUN cana-3309 70 30 ) ) PUNCT cana-3309 70 31 βˆ’ βˆ’ ADP cana-3309 70 32 Ο†p(πœ†3 Ο†p(πœ†3 NOUN cana-3309 70 33 ) ) PUNCT cana-3309 70 34 βˆ’ βˆ’ NOUN cana-3309 70 35 Ο†p(πœ†4 Ο†p(πœ†4 NOUN cana-3309 70 36 ) ) PUNCT cana-3309 70 37 βˆ’ βˆ’ NOUN cana-3309 70 38 Ο†p(πœ†6 Ο†p(πœ†6 NOUN cana-3309 70 39 ) ) PUNCT cana-3309 70 40 βˆ’ βˆ’ PROPN cana-3309 70 41 Ο†p(πœ†7 Ο†p(πœ†7 NOUN cana-3309 70 42 ) ) PUNCT cana-3309 70 43 ) ) PUNCT cana-3309 70 44 𝑛 𝑛 PRON cana-3309 70 45 𝑑(πœ›1 𝑑(πœ›1 NOUN cana-3309 70 46 , , PUNCT cana-3309 70 47 πœ›0 πœ›0 NOUN cana-3309 70 48 ) ) PUNCT cana-3309 70 49 𝑑(πœ›π‘›+1 𝑑(πœ›π‘›+1 PROPN cana-3309 70 50 , , PUNCT cana-3309 70 51 πœ›π‘› πœ›π‘› NOUN cana-3309 70 52 ) ) PUNCT cana-3309 70 53 ≀ ≀ NUM cana-3309 70 54 𝛿𝑛 𝛿𝑛 ADP cana-3309 70 55 𝑑(πœ›1 𝑑(πœ›1 PROPN cana-3309 70 56 , , PUNCT cana-3309 70 57 πœ›0 πœ›0 NOUN cana-3309 70 58 ) ) PUNCT cana-3309 70 59 where where SCONJ cana-3309 70 60 𝛿 𝛿 ADJ cana-3309 70 61 = = SYM cana-3309 70 62 ( ( PUNCT cana-3309 70 63 Ο†p(πœ†1)+Ο†p(πœ†2)+Ο†p(πœ†3 Ο†p(πœ†1)+Ο†p(πœ†2)+Ο†p(πœ†3 PROPN cana-3309 70 64 ) ) PUNCT cana-3309 70 65 1βˆ’Ο†p(πœ†2)βˆ’Ο†p(πœ†3)βˆ’Ο†p(πœ†4)βˆ’Ο†p(πœ†6)βˆ’Ο†p(πœ†7 1βˆ’Ο†p(πœ†2)βˆ’Ο†p(πœ†3)βˆ’Ο†p(πœ†4)βˆ’Ο†p(πœ†6)βˆ’Ο†p(πœ†7 NUM cana-3309 70 66 ) ) PUNCT cana-3309 70 67 ) ) PUNCT cana-3309 71 1 < < X cana-3309 71 2 1 1 NUM cana-3309 71 3 communications communication NOUN cana-3309 71 4 on on ADP cana-3309 71 5 applied apply VERB cana-3309 71 6 nonlinear nonlinear ADJ cana-3309 71 7 analysis analysis NOUN cana-3309 71 8 issn issn NOUN cana-3309 71 9 : : PUNCT cana-3309 71 10 1074 1074 NUM cana-3309 71 11 - - PUNCT cana-3309 71 12 133x 133x NUM cana-3309 71 13 vol vol NOUN cana-3309 71 14 32 32 NUM cana-3309 71 15 no no NOUN cana-3309 71 16 . . PUNCT cana-3309 72 1 6s 6s NUM cana-3309 72 2 ( ( PUNCT cana-3309 72 3 2025 2025 NUM cana-3309 72 4 ) ) PUNCT cana-3309 72 5 457 457 NUM cana-3309 72 6 https://internationalpubls.com https://internationalpubls.com X cana-3309 72 7 by by ADP cana-3309 72 8 triangular triangular NOUN cana-3309 72 9 inequality inequality NOUN cana-3309 72 10 , , PUNCT cana-3309 72 11 we we PRON cana-3309 72 12 have have VERB cana-3309 72 13 for for ADP cana-3309 72 14 π‘š π‘š X cana-3309 72 15 > > X cana-3309 72 16 𝑛 𝑛 PRON cana-3309 72 17 𝑑(πœ›π‘› 𝑑(πœ›π‘› PROPN cana-3309 72 18 , , PUNCT cana-3309 72 19 πœ›π‘š πœ›π‘š VERB cana-3309 72 20 ) ) PUNCT cana-3309 72 21 ≀ ≀ NOUN cana-3309 72 22 𝑑(πœ›π‘› 𝑑(πœ›π‘› NUM cana-3309 72 23 , , PUNCT cana-3309 72 24 πœ›π‘›+1 πœ›π‘›+1 NUM cana-3309 72 25 ) ) PUNCT cana-3309 73 1 + + CCONJ cana-3309 73 2 𝑑(πœ›π‘›+1 𝑑(πœ›π‘›+1 PROPN cana-3309 73 3 , , PUNCT cana-3309 73 4 πœ›π‘›+2 πœ›π‘›+2 NUM cana-3309 73 5 ) ) PUNCT cana-3309 73 6 + + CCONJ cana-3309 73 7 𝑑(πœ›π‘›+2 𝑑(πœ›π‘›+2 NOUN cana-3309 73 8 , , PUNCT cana-3309 73 9 πœ›π‘›+3 πœ›π‘›+3 NOUN cana-3309 73 10 ) ) PUNCT cana-3309 74 1 + + CCONJ cana-3309 74 2 β‹― β‹― VERB cana-3309 74 3 + + CCONJ cana-3309 74 4 𝑑(πœ›π‘šβˆ’1 𝑑(πœ›π‘šβˆ’1 NOUN cana-3309 74 5 , , PUNCT cana-3309 74 6 πœ›π‘› πœ›π‘› NOUN cana-3309 74 7 ) ) PUNCT cana-3309 74 8 ≀ ≀ NOUN cana-3309 74 9 ( ( PUNCT cana-3309 74 10 𝛿𝑛 𝛿𝑛 ADP cana-3309 74 11 + + CCONJ cana-3309 74 12 𝛿𝑛+1 𝛿𝑛+1 X cana-3309 74 13 + + CCONJ cana-3309 74 14 β‹― β‹― PROPN cana-3309 74 15 + + CCONJ cana-3309 74 16 π›Ώπ‘šβˆ’1)𝑑(πœ›1 π›Ώπ‘šβˆ’1)𝑑(πœ›1 PROPN cana-3309 74 17 , , PUNCT cana-3309 74 18 πœ›0 πœ›0 NOUN cana-3309 74 19 ) ) PUNCT cana-3309 74 20 therefore therefore ADV cana-3309 74 21 , , PUNCT cana-3309 74 22 𝑑(πœ›π‘› 𝑑(πœ›π‘› PROPN cana-3309 74 23 , , PUNCT cana-3309 74 24 πœ›π‘š πœ›π‘š VERB cana-3309 74 25 ) ) PUNCT cana-3309 74 26 ≀ ≀ NOUN cana-3309 74 27 𝛿𝑛 𝛿𝑛 ADP cana-3309 74 28 1βˆ’π›Ώ 1βˆ’π›Ώ NUM cana-3309 74 29 𝑑(πœ›0 𝑑(πœ›0 NOUN cana-3309 74 30 , , PUNCT cana-3309 74 31 π‘‡πœ›0 π‘‡πœ›0 ADV cana-3309 74 32 ) ) PUNCT cana-3309 75 1 β†’ β†’ SYM cana-3309 75 2 0 0 NUM cana-3309 75 3 as as ADP cana-3309 75 4 π‘š π‘š PROPN cana-3309 75 5 , , PUNCT cana-3309 75 6 𝑛 𝑛 PROPN cana-3309 75 7 β†’ β†’ SYM cana-3309 75 8 ∞ ∞ PROPN cana-3309 75 9 so so ADV cana-3309 75 10 , , PUNCT cana-3309 75 11 { { PUNCT cana-3309 75 12 πœ›π‘› πœ›π‘› NOUN cana-3309 75 13 } } PUNCT cana-3309 75 14 is be AUX cana-3309 75 15 cauchy cauchy ADJ cana-3309 75 16 sequence sequence NOUN cana-3309 75 17 in in ADP cana-3309 75 18 πœ› πœ› PROPN cana-3309 75 19 , , PUNCT cana-3309 75 20 so so ADV cana-3309 75 21 by by ADP cana-3309 75 22 completeness completeness NOUN cana-3309 75 23 of of ADP cana-3309 75 24 𝑋 𝑋 PROPN cana-3309 75 25 , , PUNCT cana-3309 75 26 there there PRON cana-3309 75 27 is be VERB cana-3309 75 28 a a DET cana-3309 75 29 point point NOUN cana-3309 75 30 𝑒 𝑒 ADP cana-3309 75 31 ∈ ∈ PROPN cana-3309 75 32 𝑋 𝑋 PROPN cana-3309 75 33 , , PUNCT cana-3309 75 34 such such ADJ cana-3309 75 35 that that SCONJ cana-3309 75 36 πœ›π‘› πœ›π‘› NOUN cana-3309 75 37 β†’ β†’ SYM cana-3309 75 38 𝑒 𝑒 NOUN cana-3309 75 39 as as ADP cana-3309 75 40 𝑛 𝑛 PROPN cana-3309 75 41 β†’ β†’ SYM cana-3309 75 42 ∞. ∞. PROPN cana-3309 75 43 further far ADV cana-3309 75 44 , , PUNCT cana-3309 75 45 the the DET cana-3309 75 46 continuity continuity NOUN cana-3309 75 47 of of ADP cana-3309 75 48 𝑇 𝑇 PROPN cana-3309 75 49 in in ADP cana-3309 75 50 𝑋 𝑋 PROPN cana-3309 75 51 implies imply VERB cana-3309 75 52 𝑇(𝑒 𝑇(𝑒 NOUN cana-3309 75 53 ) ) PUNCT cana-3309 75 54 = = SYM cana-3309 75 55 𝑇 𝑇 PROPN cana-3309 75 56 ( ( PUNCT cana-3309 75 57 lim lim PROPN cana-3309 75 58 π‘›β†’βˆž π‘›β†’βˆž PUNCT cana-3309 75 59 πœ›π‘› πœ›π‘› NOUN cana-3309 75 60 ) ) PUNCT cana-3309 75 61 = = SYM cana-3309 75 62 lim lim NOUN cana-3309 75 63 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3309 75 64 π‘‡πœ›π‘› π‘‡πœ›π‘› PROPN cana-3309 75 65 = = PROPN cana-3309 75 66 lim lim NOUN cana-3309 75 67 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3309 75 68 πœ›π‘›+1 πœ›π‘›+1 NUM cana-3309 76 1 = = SYM cana-3309 76 2 𝑒 𝑒 PROPN cana-3309 76 3 therefore therefore ADV cana-3309 76 4 , , PUNCT cana-3309 76 5 𝑒 𝑒 PRON cana-3309 76 6 is be AUX cana-3309 76 7 a a DET cana-3309 76 8 fixed fix VERB cana-3309 76 9 point point NOUN cana-3309 76 10 of of ADP cana-3309 76 11 𝑇 𝑇 PROPN cana-3309 76 12 in in ADP cana-3309 76 13 πœ›. πœ›. NOUN cana-3309 76 14 suppose suppose VERB cana-3309 77 1 if if SCONJ cana-3309 77 2 there there PRON cana-3309 77 3 is be VERB cana-3309 77 4 any any DET cana-3309 77 5 other other ADJ cana-3309 77 6 πœ›1 πœ›1 NOUN cana-3309 77 7 β‰  β‰  PROPN cana-3309 77 8 πœ›2 πœ›2 NOUN cana-3309 77 9 in in ADP cana-3309 77 10 𝑋 𝑋 PROPN cana-3309 77 11 such such ADJ cana-3309 77 12 that that DET cana-3309 77 13 𝑇(πœ›2 𝑇(πœ›2 NOUN cana-3309 77 14 ) ) PUNCT cana-3309 78 1 = = SYM cana-3309 78 2 πœ›2 πœ›2 NOUN cana-3309 78 3 , , PUNCT cana-3309 78 4 then then ADV cana-3309 78 5 𝑑(πœ›1 𝑑(πœ›1 NOUN cana-3309 78 6 , , PUNCT cana-3309 78 7 πœ›2 πœ›2 NOUN cana-3309 78 8 ) ) PUNCT cana-3309 78 9 = = SYM cana-3309 79 1 𝑑(π‘‡πœ›1 𝑑(π‘‡πœ›1 NOUN cana-3309 79 2 , , PUNCT cana-3309 79 3 π‘‡πœ›2 π‘‡πœ›2 ADJ cana-3309 79 4 ) ) PUNCT cana-3309 79 5 Ο†p(𝑑(πœ›1 Ο†p(𝑑(πœ›1 PROPN cana-3309 79 6 , , PUNCT cana-3309 79 7 πœ›2 πœ›2 NOUN cana-3309 79 8 ) ) PUNCT cana-3309 79 9 ) ) PUNCT cana-3309 80 1 ≀ ≀ NUM cana-3309 80 2 πœ†1Ο†p(𝑑(πœ›1 πœ†1Ο†p(𝑑(πœ›1 NOUN cana-3309 80 3 , , PUNCT cana-3309 80 4 πœ›2 πœ›2 NOUN cana-3309 80 5 ) ) PUNCT cana-3309 80 6 ) ) PUNCT cana-3309 81 1 + + CCONJ cana-3309 81 2 πœ†2Ο†p([𝑑(π‘‡πœ›1 πœ†2Ο†p([𝑑(π‘‡πœ›1 NOUN cana-3309 81 3 , , PUNCT cana-3309 81 4 πœ›2 πœ›2 NOUN cana-3309 81 5 ) ) PUNCT cana-3309 82 1 + + CCONJ cana-3309 82 2 𝑑(π‘‡πœ›2 𝑑(π‘‡πœ›2 PROPN cana-3309 82 3 , , PUNCT cana-3309 82 4 πœ›2 πœ›2 NOUN cana-3309 82 5 ) ) PUNCT cana-3309 82 6 ] ] PUNCT cana-3309 82 7 ) ) PUNCT cana-3309 83 1 + + ADJ cana-3309 83 2 πœ†3Ο†p([𝑑(π‘‡πœ›2 πœ†3Ο†p([𝑑(π‘‡πœ›2 PROPN cana-3309 83 3 , , PUNCT cana-3309 83 4 πœ›1 πœ›1 NOUN cana-3309 83 5 ) ) PUNCT cana-3309 83 6 + + CCONJ cana-3309 83 7 𝑑(π‘‡πœ›1 𝑑(π‘‡πœ›1 PROPN cana-3309 83 8 , , PUNCT cana-3309 83 9 πœ›2 πœ›2 NOUN cana-3309 83 10 ) ) PUNCT cana-3309 83 11 ] ] PUNCT cana-3309 83 12 ) ) PUNCT cana-3309 83 13 + + CCONJ cana-3309 83 14 πœ†4Ο†p πœ†4Ο†p PUNCT cana-3309 83 15 ( ( PUNCT cana-3309 83 16 [ [ PUNCT cana-3309 83 17 𝑑(πœ›1,π‘‡πœ›2)𝑑(πœ›2,π‘‡πœ›2 𝑑(πœ›1,π‘‡πœ›2)𝑑(πœ›2,π‘‡πœ›2 PROPN cana-3309 83 18 ) ) PUNCT cana-3309 83 19 𝑑(πœ›1,πœ›2 𝑑(πœ›1,πœ›2 NOUN cana-3309 83 20 ) ) PUNCT cana-3309 83 21 ] ] PUNCT cana-3309 83 22 ) ) PUNCT cana-3309 84 1 + + X cana-3309 84 2 πœ†5Ο†p πœ†5Ο†p PUNCT cana-3309 84 3 ( ( PUNCT cana-3309 84 4 [ [ PUNCT cana-3309 84 5 𝑑(πœ›1,π‘‡πœ›2)𝑑(πœ›2,π‘‡πœ›1 𝑑(πœ›1,π‘‡πœ›2)𝑑(πœ›2,π‘‡πœ›1 PROPN cana-3309 84 6 ) ) PUNCT cana-3309 84 7 𝑑(πœ›1,πœ›2 𝑑(πœ›1,πœ›2 NOUN cana-3309 84 8 ) ) PUNCT cana-3309 84 9 ] ] PUNCT cana-3309 84 10 ) ) PUNCT cana-3309 85 1 + + X cana-3309 85 2 πœ†6Ο†p πœ†6Ο†p X cana-3309 85 3 ( ( PUNCT cana-3309 85 4 [ [ PUNCT cana-3309 85 5 𝑑(πœ›1,π‘‡πœ›1)𝑑(πœ›2,π‘‡πœ›2)+𝑑(πœ›1,π‘‡πœ›2)𝑑(πœ›2,π‘‡πœ›1 𝑑(πœ›1,π‘‡πœ›1)𝑑(πœ›2,π‘‡πœ›2)+𝑑(πœ›1,π‘‡πœ›2)𝑑(πœ›2,π‘‡πœ›1 ADP cana-3309 85 6 ) ) PUNCT cana-3309 85 7 𝑑(πœ›1,πœ›2 𝑑(πœ›1,πœ›2 NOUN cana-3309 85 8 ) ) PUNCT cana-3309 85 9 ] ] PUNCT cana-3309 85 10 ) ) PUNCT cana-3309 86 1 + + X cana-3309 86 2 πœ†7Ο†p πœ†7Ο†p X cana-3309 86 3 ( ( PUNCT cana-3309 86 4 [ [ PUNCT cana-3309 86 5 𝑑(πœ›1,π‘‡πœ›1)𝑑(πœ›2,π‘‡πœ›2)+𝑑(πœ›1,π‘‡πœ›2)𝑑(πœ›2,π‘‡πœ›1 𝑑(πœ›1,π‘‡πœ›1)𝑑(πœ›2,π‘‡πœ›2)+𝑑(πœ›1,π‘‡πœ›2)𝑑(πœ›2,π‘‡πœ›1 NOUN cana-3309 86 6 ) ) PUNCT cana-3309 86 7 𝑑(πœ›1,𝑣 𝑑(πœ›1,𝑣 NOUN cana-3309 86 8 ) ) PUNCT cana-3309 86 9 ] ] PUNCT cana-3309 86 10 ) ) PUNCT cana-3309 87 1 + + X cana-3309 87 2 πœ†8Ο†p πœ†8Ο†p PUNCT cana-3309 87 3 ( ( PUNCT cana-3309 87 4 [ [ PUNCT cana-3309 87 5 𝑑(πœ›1,π‘‡πœ›2)[𝑑(πœ›1,π‘‡πœ›1)+𝑑(πœ›2,π‘‡πœ›2)+𝑑(πœ›1,π‘‡πœ›2)+𝑑(πœ›2,π‘‡πœ›1 𝑑(πœ›1,π‘‡πœ›2)[𝑑(πœ›1,π‘‡πœ›1)+𝑑(πœ›2,π‘‡πœ›2)+𝑑(πœ›1,π‘‡πœ›2)+𝑑(πœ›2,π‘‡πœ›1 NOUN cana-3309 87 6 ) ) PUNCT cana-3309 87 7 ] ] PUNCT cana-3309 87 8 𝑑(πœ›1,π‘‡πœ›2)+𝑑(πœ›2,π‘‡πœ›2)+𝑑(πœ›1,π‘‡πœ›2)+𝑑(πœ›2,π‘‡πœ›1 𝑑(πœ›1,π‘‡πœ›2)+𝑑(πœ›2,π‘‡πœ›2)+𝑑(πœ›1,π‘‡πœ›2)+𝑑(πœ›2,π‘‡πœ›1 PROPN cana-3309 87 9 ) ) PUNCT cana-3309 87 10 ] ] PUNCT cana-3309 87 11 ) ) PUNCT cana-3309 87 12 Ο†p(𝑑(πœ›1 Ο†p(𝑑(πœ›1 PROPN cana-3309 87 13 , , PUNCT cana-3309 87 14 πœ›2 πœ›2 NOUN cana-3309 87 15 ) ) PUNCT cana-3309 87 16 ) ) PUNCT cana-3309 87 17 ≀ ≀ NUM cana-3309 87 18 πœ†1Ο†p(𝑑(πœ›1 πœ†1Ο†p(𝑑(πœ›1 NOUN cana-3309 87 19 , , PUNCT cana-3309 87 20 πœ›2 πœ›2 NOUN cana-3309 87 21 ) ) PUNCT cana-3309 87 22 ) ) PUNCT cana-3309 88 1 + + CCONJ cana-3309 88 2 πœ†2Ο†p([𝑑(πœ›1 πœ†2Ο†p([𝑑(πœ›1 NOUN cana-3309 88 3 , , PUNCT cana-3309 88 4 πœ›1 πœ›1 NOUN cana-3309 88 5 ) ) PUNCT cana-3309 89 1 + + CCONJ cana-3309 89 2 𝑑(πœ›2 𝑑(πœ›2 ADJ cana-3309 89 3 , , PUNCT cana-3309 89 4 πœ›2 πœ›2 NOUN cana-3309 89 5 ) ) PUNCT cana-3309 89 6 ] ] PUNCT cana-3309 89 7 ) ) PUNCT cana-3309 90 1 + + NOUN cana-3309 90 2 πœ†3Ο†p([𝑑(πœ›2 πœ†3Ο†p([𝑑(πœ›2 NOUN cana-3309 90 3 , , PUNCT cana-3309 90 4 πœ›1 πœ›1 NOUN cana-3309 90 5 ) ) PUNCT cana-3309 90 6 + + NUM cana-3309 90 7 𝑑(πœ›1 𝑑(πœ›1 ADJ cana-3309 90 8 , , PUNCT cana-3309 90 9 πœ›2 πœ›2 NOUN cana-3309 90 10 ) ) PUNCT cana-3309 90 11 ] ] PUNCT cana-3309 90 12 ) ) PUNCT cana-3309 91 1 + + CCONJ cana-3309 91 2 πœ†4Ο†p πœ†4Ο†p PUNCT cana-3309 91 3 ( ( PUNCT cana-3309 91 4 [ [ PUNCT cana-3309 91 5 𝑑(πœ›1,πœ›1)𝑑(πœ›2,πœ›2 𝑑(πœ›1,πœ›1)𝑑(πœ›2,πœ›2 NOUN cana-3309 91 6 ) ) PUNCT cana-3309 91 7 𝑑(πœ›1,πœ›2 𝑑(πœ›1,πœ›2 NOUN cana-3309 91 8 ) ) PUNCT cana-3309 91 9 ] ] PUNCT cana-3309 91 10 ) ) PUNCT cana-3309 92 1 + + X cana-3309 92 2 πœ†5Ο†p πœ†5Ο†p PUNCT cana-3309 92 3 ( ( PUNCT cana-3309 92 4 [ [ PUNCT cana-3309 92 5 𝑑(πœ›1,πœ›2)𝑑(πœ›2,πœ›1 𝑑(πœ›1,πœ›2)𝑑(πœ›2,πœ›1 NOUN cana-3309 92 6 ) ) PUNCT cana-3309 92 7 𝑑(πœ›1,πœ›2 𝑑(πœ›1,πœ›2 NOUN cana-3309 92 8 ) ) PUNCT cana-3309 92 9 ] ] PUNCT cana-3309 92 10 ) ) PUNCT cana-3309 92 11 + + CCONJ cana-3309 92 12 πœ†6Ο†p πœ†6Ο†p PUNCT cana-3309 92 13 ( ( PUNCT cana-3309 92 14 [ [ PUNCT cana-3309 92 15 𝑑(πœ›1,πœ›1)𝑑(πœ›2,πœ›2)+𝑑(πœ›1,πœ›2)𝑑(πœ›2,πœ›1 𝑑(πœ›1,πœ›1)𝑑(πœ›2,πœ›2)+𝑑(πœ›1,πœ›2)𝑑(πœ›2,πœ›1 PROPN cana-3309 92 16 ) ) PUNCT cana-3309 92 17 𝑑(πœ›1,πœ›2 𝑑(πœ›1,πœ›2 NOUN cana-3309 92 18 ) ) PUNCT cana-3309 92 19 ] ] PUNCT cana-3309 92 20 ) ) PUNCT cana-3309 92 21 + + X cana-3309 92 22 πœ†7Ο†p πœ†7Ο†p X cana-3309 92 23 ( ( PUNCT cana-3309 92 24 [ [ PUNCT cana-3309 92 25 𝑑(πœ›1,πœ›1)𝑑(πœ›2,πœ›2)+𝑑(πœ›1,πœ›2)𝑑(πœ›2,πœ›1 𝑑(πœ›1,πœ›1)𝑑(πœ›2,πœ›2)+𝑑(πœ›1,πœ›2)𝑑(πœ›2,πœ›1 PROPN cana-3309 92 26 ) ) PUNCT cana-3309 92 27 𝑑(πœ›1,πœ›2 𝑑(πœ›1,πœ›2 NOUN cana-3309 92 28 ) ) PUNCT cana-3309 92 29 ] ] PUNCT cana-3309 92 30 ) ) PUNCT cana-3309 93 1 + + X cana-3309 93 2 πœ†8Ο†p πœ†8Ο†p PUNCT cana-3309 93 3 ( ( PUNCT cana-3309 93 4 [ [ PUNCT cana-3309 93 5 𝑑(πœ›1,πœ›2)[𝑑(πœ›1,πœ›1)+𝑑(πœ›2,πœ›2)+𝑑(πœ›1,πœ›2)+𝑑(πœ›2,πœ›1 𝑑(πœ›1,πœ›2)[𝑑(πœ›1,πœ›1)+𝑑(πœ›2,πœ›2)+𝑑(πœ›1,πœ›2)+𝑑(πœ›2,πœ›1 X cana-3309 93 6 ) ) PUNCT cana-3309 93 7 ] ] PUNCT cana-3309 93 8 𝑑(πœ›1,πœ›2)+𝑑(πœ›2,πœ›2)+𝑑(πœ›1,πœ›2)+𝑑(πœ›2,πœ›1 𝑑(πœ›1,πœ›2)+𝑑(πœ›2,πœ›2)+𝑑(πœ›1,πœ›2)+𝑑(πœ›2,πœ›1 PROPN cana-3309 93 9 ) ) PUNCT cana-3309 93 10 ] ] PUNCT cana-3309 93 11 ) ) PUNCT cana-3309 93 12 Ο†p(𝑑(πœ›1 Ο†p(𝑑(πœ›1 PROPN cana-3309 93 13 , , PUNCT cana-3309 93 14 πœ›2 πœ›2 NOUN cana-3309 93 15 ) ) PUNCT cana-3309 93 16 ) ) PUNCT cana-3309 93 17 ≀ ≀ NUM cana-3309 93 18 πœ†1Ο†p(𝑑(πœ›1 πœ†1Ο†p(𝑑(πœ›1 NOUN cana-3309 93 19 , , PUNCT cana-3309 93 20 πœ›2 πœ›2 NOUN cana-3309 93 21 ) ) PUNCT cana-3309 93 22 ) ) PUNCT cana-3309 94 1 + + CCONJ cana-3309 94 2 2πœ†3Ο†p(𝑑(πœ›1 2πœ†3Ο†p(𝑑(πœ›1 NUM cana-3309 94 3 , , PUNCT cana-3309 94 4 πœ›2 πœ›2 NOUN cana-3309 94 5 ) ) PUNCT cana-3309 94 6 ) ) PUNCT cana-3309 95 1 + + CCONJ cana-3309 95 2 πœ†5Ο†p(𝑑(πœ›1 πœ†5Ο†p(𝑑(πœ›1 NOUN cana-3309 95 3 , , PUNCT cana-3309 95 4 πœ›2 πœ›2 NOUN cana-3309 95 5 ) ) PUNCT cana-3309 95 6 ) ) PUNCT cana-3309 96 1 + + X cana-3309 96 2 πœ†6(𝑑(πœ›1 πœ†6(𝑑(πœ›1 NUM cana-3309 96 3 , , PUNCT cana-3309 96 4 πœ›2 πœ›2 NOUN cana-3309 96 5 ) ) PUNCT cana-3309 96 6 ) ) PUNCT cana-3309 97 1 + + CCONJ cana-3309 97 2 πœ†7Ο†p(𝑑(πœ›1 πœ†7Ο†p(𝑑(πœ›1 PROPN cana-3309 97 3 , , PUNCT cana-3309 97 4 πœ›2 πœ›2 PROPN cana-3309 97 5 ) ) PUNCT cana-3309 97 6 ) ) PUNCT cana-3309 98 1 + + CCONJ cana-3309 98 2 πœ†8Ο†p(𝑑(πœ›1 πœ†8Ο†p(𝑑(πœ›1 SYM cana-3309 98 3 , , PUNCT cana-3309 98 4 πœ›2 πœ›2 NOUN cana-3309 98 5 ) ) PUNCT cana-3309 98 6 ) ) PUNCT cana-3309 98 7 𝑑(πœ›1 𝑑(πœ›1 NOUN cana-3309 98 8 , , PUNCT cana-3309 98 9 πœ›2 πœ›2 NOUN cana-3309 98 10 ) ) PUNCT cana-3309 98 11 ≀ ≀ NOUN cana-3309 98 12 ( ( PUNCT cana-3309 98 13 Ο†p(πœ†1 Ο†p(πœ†1 NOUN cana-3309 98 14 ) ) PUNCT cana-3309 98 15 + + CCONJ cana-3309 98 16 2Ο†p(πœ†3 2Ο†p(πœ†3 NUM cana-3309 98 17 ) ) PUNCT cana-3309 99 1 + + CCONJ cana-3309 99 2 Ο†p(πœ†5 Ο†p(πœ†5 NOUN cana-3309 99 3 ) ) PUNCT cana-3309 99 4 + + CCONJ cana-3309 99 5 Ο†p(πœ†6 Ο†p(πœ†6 NOUN cana-3309 99 6 ) ) PUNCT cana-3309 99 7 + + X cana-3309 99 8 Ο†p(πœ†7 Ο†p(πœ†7 NOUN cana-3309 99 9 ) ) PUNCT cana-3309 99 10 + + CCONJ cana-3309 99 11 Ο†p(πœ†8))𝑑(πœ›1 Ο†p(πœ†8))𝑑(πœ›1 PROPN cana-3309 99 12 , , PUNCT cana-3309 99 13 πœ›2 πœ›2 PROPN cana-3309 99 14 ) ) PUNCT cana-3309 99 15 which which PRON cana-3309 99 16 is be AUX cana-3309 99 17 contradiction contradiction NOUN cana-3309 99 18 . . PUNCT cana-3309 100 1 hence hence ADV cana-3309 100 2 πœ›1 πœ›1 PROPN cana-3309 100 3 is be AUX cana-3309 100 4 a a DET cana-3309 100 5 fixed fix VERB cana-3309 100 6 point point NOUN cana-3309 100 7 communications communication NOUN cana-3309 100 8 on on ADP cana-3309 100 9 applied apply VERB cana-3309 100 10 nonlinear nonlinear ADJ cana-3309 100 11 analysis analysis NOUN cana-3309 100 12 issn issn NOUN cana-3309 100 13 : : PUNCT cana-3309 100 14 1074 1074 NUM cana-3309 100 15 - - PUNCT cana-3309 100 16 133x 133x NUM cana-3309 100 17 vol vol NOUN cana-3309 100 18 32 32 NUM cana-3309 100 19 no no NOUN cana-3309 100 20 . . PUNCT cana-3309 101 1 6s 6s NUM cana-3309 101 2 ( ( PUNCT cana-3309 101 3 2025 2025 NUM cana-3309 101 4 ) ) PUNCT cana-3309 101 5 458 458 NUM cana-3309 101 6 https://internationalpubls.com https://internationalpubls.com X cana-3309 101 7 corollary corollary NOUN cana-3309 101 8 2.2 2.2 NUM cana-3309 101 9 : : PUNCT cana-3309 101 10 let let VERB cana-3309 101 11 𝑇 𝑇 PROPN cana-3309 101 12 be be AUX cana-3309 101 13 continues continue VERB cana-3309 101 14 self self NOUN cana-3309 101 15 map map NOUN cana-3309 101 16 , , PUNCT cana-3309 101 17 defined define VERB cana-3309 101 18 on on ADP cana-3309 101 19 a a DET cana-3309 101 20 complete complete ADJ cana-3309 101 21 metric metric ADJ cana-3309 101 22 space space NOUN cana-3309 101 23 𝑋. 𝑋. PROPN cana-3309 101 24 further far ADV cana-3309 101 25 𝑇 𝑇 PROPN cana-3309 101 26 satisfies satisfy VERB cana-3309 101 27 the the DET cana-3309 101 28 following follow VERB cana-3309 101 29 conditions condition NOUN cana-3309 101 30 Ο†p(𝑑(π‘‡πœ› Ο†p(𝑑(π‘‡πœ› PROPN cana-3309 101 31 , , PUNCT cana-3309 101 32 π‘‡νœ π‘‡νœ PROPN cana-3309 101 33 ) ) PUNCT cana-3309 101 34 ) ) PUNCT cana-3309 101 35 ≀ ≀ NUM cana-3309 101 36 πœ†1Ο†p(𝑑(πœ› πœ†1Ο†p(𝑑(πœ› PROPN cana-3309 101 37 , , PUNCT cana-3309 101 38 휁 휁 NOUN cana-3309 101 39 ) ) PUNCT cana-3309 101 40 ) ) PUNCT cana-3309 102 1 + + CCONJ cana-3309 103 1 πœ†2([𝑑(π‘‡πœ› πœ†2([𝑑(π‘‡πœ› ADJ cana-3309 103 2 , , PUNCT cana-3309 103 3 πœ› πœ› PROPN cana-3309 103 4 ) ) PUNCT cana-3309 103 5 + + X cana-3309 103 6 𝑑(π‘‡νœ 𝑑(π‘‡νœ X cana-3309 103 7 , , PUNCT cana-3309 103 8 휁 휁 NOUN cana-3309 103 9 ) ) PUNCT cana-3309 103 10 ] ] PUNCT cana-3309 103 11 ) ) PUNCT cana-3309 104 1 + + ADJ cana-3309 104 2 πœ†3Ο†p([𝑑(π‘‡νœ πœ†3Ο†p([𝑑(π‘‡νœ PROPN cana-3309 104 3 , , PUNCT cana-3309 104 4 πœ› πœ› PROPN cana-3309 104 5 ) ) PUNCT cana-3309 104 6 + + CCONJ cana-3309 104 7 𝑑(π‘‡πœ› 𝑑(π‘‡πœ› ADJ cana-3309 104 8 , , PUNCT cana-3309 104 9 휁 휁 NOUN cana-3309 104 10 ) ) PUNCT cana-3309 104 11 ] ] PUNCT cana-3309 104 12 ) ) PUNCT cana-3309 105 1 + + NOUN cana-3309 105 2 πœ†4Ο†p πœ†4Ο†p PUNCT cana-3309 105 3 ( ( PUNCT cana-3309 105 4 [ [ PUNCT cana-3309 105 5 𝑑(πœ›,π‘‡πœ›)𝑑(𝑦,π‘‡πœ 𝑑(πœ›,π‘‡πœ›)𝑑(𝑦,π‘‡πœ NOUN cana-3309 105 6 ) ) PUNCT cana-3309 105 7 𝑑(πœ›,𝜁 𝑑(πœ›,𝜁 NOUN cana-3309 105 8 ) ) PUNCT cana-3309 105 9 ] ] PUNCT cana-3309 105 10 ) ) PUNCT cana-3309 106 1 + + CCONJ cana-3309 106 2 πœ†5Ο†p πœ†5Ο†p PUNCT cana-3309 106 3 ( ( PUNCT cana-3309 106 4 [ [ PUNCT cana-3309 106 5 𝑑(πœ›,π‘‡πœ)𝑑(𝜁,π‘‡πœ› 𝑑(πœ›,π‘‡πœ)𝑑(𝜁,π‘‡πœ› NUM cana-3309 106 6 ) ) PUNCT cana-3309 106 7 𝑑(πœ›,𝜁 𝑑(πœ›,𝜁 NOUN cana-3309 106 8 ) ) PUNCT cana-3309 106 9 ] ] PUNCT cana-3309 106 10 ) ) PUNCT cana-3309 106 11 for for ADP cana-3309 106 12 all all DET cana-3309 106 13 πœ› πœ› PROPN cana-3309 106 14 , , PUNCT cana-3309 106 15 휁 휁 PROPN cana-3309 106 16 ∈ ∈ PROPN cana-3309 106 17 𝑋 𝑋 PROPN cana-3309 106 18 , , PUNCT cana-3309 106 19 πœ› πœ› PROPN cana-3309 106 20 β‰  β‰  PROPN cana-3309 106 21 휁 휁 NOUN cana-3309 106 22 and and CCONJ cana-3309 106 23 Ο†p(πœ†1 Ο†p(πœ†1 ADJ cana-3309 106 24 ) ) PUNCT cana-3309 106 25 + + CCONJ cana-3309 106 26 2Ο†p(πœ†2 2Ο†p(πœ†2 NUM cana-3309 106 27 ) ) PUNCT cana-3309 107 1 + + NUM cana-3309 107 2 2Ο†p(πœ†3 2Ο†p(πœ†3 NUM cana-3309 107 3 ) ) PUNCT cana-3309 107 4 + + NUM cana-3309 107 5 Ο†p(πœ†4 Ο†p(πœ†4 NOUN cana-3309 107 6 ) ) PUNCT cana-3309 107 7 < < X cana-3309 108 1 1 1 NUM cana-3309 108 2 then then ADV cana-3309 108 3 𝑇 𝑇 PROPN cana-3309 108 4 has have AUX cana-3309 108 5 unique unique ADJ cana-3309 108 6 fixed fix VERB cana-3309 108 7 point point NOUN cana-3309 108 8 in in ADP cana-3309 108 9 𝑇. 𝑇. PROPN cana-3309 108 10 proof proof NOUN cana-3309 108 11 : : PUNCT cana-3309 108 12 the the DET cana-3309 108 13 proof proof NOUN cana-3309 108 14 is be AUX cana-3309 108 15 comes come VERB cana-3309 108 16 instead instead ADV cana-3309 108 17 of of ADP cana-3309 109 1 πœ†6 πœ†6 PROPN cana-3309 109 2 = = PROPN cana-3309 109 3 πœ†7 πœ†7 PROPN cana-3309 109 4 = = PUNCT cana-3309 109 5 πœ†8 πœ†8 PROPN cana-3309 109 6 = = SYM cana-3309 109 7 0 0 NUM cana-3309 109 8 instead instead ADV cana-3309 109 9 of of ADP cana-3309 109 10 above above ADP cana-3309 109 11 theorem theorem ADJ cana-3309 109 12 . . PUNCT cana-3309 110 1 references reference NOUN cana-3309 110 2 [ [ X cana-3309 110 3 1 1 NUM cana-3309 110 4 ] ] X cana-3309 110 5 banach banach NOUN cana-3309 110 6 , , PUNCT cana-3309 110 7 s(1922 s(1922 PROPN cana-3309 110 8 ) ) PUNCT cana-3309 110 9 : : PUNCT cana-3309 110 10 surles surle VERB cana-3309 110 11 operation operation NOUN cana-3309 110 12 dans dan NOUN cana-3309 110 13 les les PROPN cana-3309 110 14 ensembles ensemble NOUN cana-3309 110 15 abstracts abstract NOUN cana-3309 110 16 etleur etleur NOUN cana-3309 110 17 application application PROPN cana-3309 110 18 aux aux PROPN cana-3309 110 19 equations equations PROPN cana-3309 110 20 integrals.fun.math integrals.fun.math PROPN cana-3309 110 21 , , PUNCT cana-3309 110 22 vol.3,pp vol.3,pp PROPN cana-3309 110 23 133 133 NUM cana-3309 110 24 – – PUNCT cana-3309 110 25 181 181 NUM cana-3309 110 26 . . PUNCT cana-3309 111 1 [ [ X cana-3309 111 2 2 2 NUM cana-3309 111 3 ] ] PUNCT cana-3309 111 4 k. k. PROPN cana-3309 111 5 dinesh dinesh PROPN cana-3309 111 6 , , PUNCT cana-3309 111 7 r.krishnakumar r.krishnakumar PROPN cana-3309 111 8 , , PUNCT cana-3309 111 9 nagaral nagaral ADJ cana-3309 111 10 pandit pandit PROPN cana-3309 111 11 sanatammappa sanatammappa NOUN cana-3309 111 12 , , PUNCT cana-3309 111 13 some some DET cana-3309 111 14 results result NOUN cana-3309 111 15 in in ADP cana-3309 111 16 b b NOUN cana-3309 111 17 - - PUNCT cana-3309 111 18 metric metric ADJ cana-3309 111 19 space space NOUN cana-3309 111 20 , , PUNCT cana-3309 111 21 malaya malaya PROPN cana-3309 111 22 journal journal PROPN cana-3309 111 23 of of ADP cana-3309 111 24 matematik matematik PROPN cana-3309 111 25 , , PUNCT cana-3309 111 26 vol.9(1 vol.9(1 NOUN cana-3309 111 27 ) ) PUNCT cana-3309 111 28 , , PUNCT cana-3309 111 29 pp:539 pp:539 NOUN cana-3309 111 30 - - SYM cana-3309 111 31 541 541 NUM cana-3309 111 32 , , PUNCT cana-3309 111 33 2021 2021 NUM cana-3309 111 34 [ [ X cana-3309 111 35 3 3 X cana-3309 111 36 ] ] X cana-3309 111 37 n. n. PROPN cana-3309 111 38 hussain hussain PROPN cana-3309 111 39 , , PUNCT cana-3309 111 40 k. k. PROPN cana-3309 111 41 zoto zoto PROPN cana-3309 111 42 , , PUNCT cana-3309 111 43 s. s. PROPN cana-3309 111 44 radenovic radenovic PROPN cana-3309 111 45 , , PUNCT cana-3309 111 46 β€œ " PUNCT cana-3309 111 47 common common ADJ cana-3309 111 48 fixed fix VERB cana-3309 111 49 point point NOUN cana-3309 111 50 results result NOUN cana-3309 111 51 of of ADP cana-3309 111 52 ( ( PUNCT cana-3309 111 53 Ξ± Ξ± NOUN cana-3309 111 54 βˆ’ βˆ’ NOUN cana-3309 111 55 ψ ψ PROPN cana-3309 111 56 , , PUNCT cana-3309 111 57 Ο†)contractions Ο†)contraction NOUN cana-3309 111 58 for for ADP cana-3309 111 59 a a DET cana-3309 111 60 pair pair NOUN cana-3309 111 61 of of ADP cana-3309 111 62 mappings mapping NOUN cana-3309 111 63 and and CCONJ cana-3309 111 64 applications application NOUN cana-3309 111 65 ” " PUNCT cana-3309 111 66 mathematics mathematic NOUN cana-3309 111 67 , , PUNCT cana-3309 111 68 6 6 NUM cana-3309 111 69 , , PUNCT cana-3309 111 70 182 182 NUM cana-3309 111 71 , , PUNCT cana-3309 111 72 ( ( PUNCT cana-3309 111 73 2018 2018 NUM cana-3309 111 74 ) ) PUNCT cana-3309 111 75 . . PUNCT cana-3309 112 1 [ [ X cana-3309 112 2 4 4 X cana-3309 112 3 ] ] PUNCT cana-3309 112 4 r. r. PROPN cana-3309 112 5 krishnakumar krishnakumar PROPN cana-3309 112 6 , , PUNCT cana-3309 112 7 d. d. PROPN cana-3309 112 8 dhamodharan dhamodharan VERB cana-3309 112 9 , , PUNCT cana-3309 112 10 β€œ " PUNCT cana-3309 112 11 some some DET cana-3309 112 12 fixed fix VERB cana-3309 112 13 point point NOUN cana-3309 112 14 theorems theorem NOUN cana-3309 112 15 in in ADP cana-3309 112 16 cone cone NOUN cana-3309 112 17 banach banach NOUN cana-3309 112 18 spaces space VERB cana-3309 112 19 using use VERB cana-3309 112 20 Ο†p Ο†p ADP cana-3309 112 21 operator operator NOUN cana-3309 112 22 ” " PUNCT cana-3309 112 23 , , PUNCT cana-3309 112 24 international international ADJ cana-3309 112 25 journal journal NOUN cana-3309 112 26 of of ADP cana-3309 112 27 mathematics mathematic NOUN cana-3309 112 28 and and CCONJ cana-3309 112 29 its its PRON cana-3309 112 30 applications application NOUN cana-3309 112 31 , , PUNCT cana-3309 112 32 volume volume NOUN cana-3309 112 33 4 4 NUM cana-3309 112 34 , , PUNCT cana-3309 112 35 issue issue NOUN cana-3309 112 36 2 2 NUM cana-3309 112 37 - - PUNCT cana-3309 112 38 b(2016 b(2016 PROPN cana-3309 112 39 ) ) PUNCT cana-3309 112 40 , , PUNCT cana-3309 112 41 105 105 NUM cana-3309 112 42 - - SYM cana-3309 112 43 112 112 NUM cana-3309 112 44 [ [ X cana-3309 112 45 5 5 NUM cana-3309 112 46 ] ] PUNCT cana-3309 112 47 chatterjee chatterjee NOUN cana-3309 112 48 , , PUNCT cana-3309 112 49 s.k s.k PROPN cana-3309 112 50 ( ( PUNCT cana-3309 112 51 1972):fixed 1972):fixed ADJ cana-3309 112 52 point point NOUN cana-3309 112 53 theorems theorem NOUN cana-3309 112 54 . . PUNCT cana-3309 113 1 comptes.rend comptes.rend PROPN cana-3309 113 2 . . PUNCT cana-3309 113 3 acad acad PROPN cana-3309 113 4 . . PUNCT cana-3309 114 1 bulgaria bulgaria PROPN cana-3309 115 1 sci.vol.25,pp sci.vol.25,pp ADV cana-3309 115 2 727 727 NUM cana-3309 115 3 - - SYM cana-3309 115 4 730 730 NUM cana-3309 115 5 . . PUNCT cana-3309 116 1 [ [ X cana-3309 116 2 6 6 NUM cana-3309 116 3 ] ] PUNCT cana-3309 116 4 k. k. PROPN cana-3309 116 5 zoto zoto PROPN cana-3309 116 6 , , PUNCT cana-3309 116 7 b.e b.e PROPN cana-3309 116 8 . . PROPN cana-3309 116 9 rhoades rhoades PROPN cana-3309 116 10 , , PUNCT cana-3309 116 11 s. s. PROPN cana-3309 116 12 radenovic radenovic PROPN cana-3309 116 13 , , PUNCT cana-3309 116 14 β€œ " PUNCT cana-3309 116 15 some some DET cana-3309 116 16 generalizations generalization NOUN cana-3309 116 17 for for ADP cana-3309 116 18 ( ( PUNCT cana-3309 116 19 ) ) PUNCT cana-3309 116 20 , , PUNCT cana-3309 116 21   X cana-3309 116 22   PROPN cana-3309 116 23 οͺβˆ’ οͺβˆ’ PROPN cana-3309 116 24 contractions contraction NOUN cana-3309 116 25 in in ADP cana-3309 116 26 bβˆ’metric bβˆ’metric ADJ cana-3309 116 27 - - PUNCT cana-3309 116 28 like like ADJ cana-3309 116 29 spaces space NOUN cana-3309 116 30 and and CCONJ cana-3309 116 31 application application NOUN cana-3309 116 32 ” " PUNCT cana-3309 116 33 fixed fix VERB cana-3309 116 34 point point NOUN cana-3309 116 35 theory theory NOUN cana-3309 116 36 appl appl NOUN cana-3309 116 37 , , PUNCT cana-3309 116 38 26 26 NUM cana-3309 116 39 , , PUNCT cana-3309 116 40 ( ( PUNCT cana-3309 116 41 2017 2017 NUM cana-3309 116 42 ) ) PUNCT cana-3309 116 43 . . PUNCT cana-3309 117 1 [ [ X cana-3309 117 2 7 7 X cana-3309 117 3 ] ] X cana-3309 117 4 swapna swapna PROPN cana-3309 117 5 siddamsetti siddamsetti PROPN cana-3309 117 6 , , PUNCT cana-3309 117 7 hussein hussein PROPN cana-3309 117 8 z. z. PROPN cana-3309 117 9 almngoshi almngoshi PROPN cana-3309 117 10 , , PUNCT cana-3309 117 11 k. k. PROPN cana-3309 117 12 dinesh dinesh PROPN cana-3309 117 13 , , PUNCT cana-3309 117 14 g. g. PROPN cana-3309 117 15 c. c. PROPN cana-3309 117 16 prashant prashant PROPN cana-3309 117 17 , , PUNCT cana-3309 117 18 m. m. NOUN cana-3309 117 19 anto anto PROPN cana-3309 117 20 bennet bennet PROPN cana-3309 117 21 , , PUNCT cana-3309 117 22 pankaj pankaj PROPN cana-3309 117 23 dadheech dadheech NOUN cana-3309 117 24 , , PUNCT cana-3309 117 25 β€œ " PUNCT cana-3309 117 26 modular modular ADJ cana-3309 117 27 metric metric ADJ cana-3309 117 28 spaces space NOUN cana-3309 117 29 : : PUNCT cana-3309 117 30 some some DET cana-3309 117 31 fixed fix VERB cana-3309 117 32 - - PUNCT cana-3309 117 33 point point NOUN cana-3309 117 34 theorems theorem NOUN cana-3309 117 35 and and CCONJ cana-3309 117 36 application application NOUN cana-3309 117 37 of of ADP cana-3309 117 38 secure secure ADJ cana-3309 117 39 dynamic dynamic ADJ cana-3309 117 40 routing routing NOUN cana-3309 117 41 for for ADP cana-3309 117 42 wsn wsn PROPN cana-3309 117 43 ” " PUNCT cana-3309 117 44 , , PUNCT cana-3309 117 45 journal journal NOUN cana-3309 117 46 of of ADP cana-3309 117 47 interdisciplinary interdisciplinary ADJ cana-3309 117 48 mathematics mathematic NOUN cana-3309 117 49 , , PUNCT cana-3309 117 50 pages page NOUN cana-3309 117 51 : : PUNCT cana-3309 117 52 393–401 393–401 NUM cana-3309 117 53 , , PUNCT cana-3309 117 54 volume volume NOUN cana-3309 117 55 27(2 27(2 NUM cana-3309 117 56 ) ) PUNCT cana-3309 117 57 , , PUNCT cana-3309 117 58 2024 2024 NUM cana-3309 117 59 [ [ X cana-3309 117 60 8 8 NUM cana-3309 117 61 ] ] X cana-3309 117 62 fisher fisher NOUN cana-3309 117 63 , , PUNCT cana-3309 117 64 b(1976):a b(1976):a NUM cana-3309 117 65 fixed fix VERB cana-3309 117 66 point point NOUN cana-3309 117 67 theorem theorem VERB cana-3309 117 68 for for ADP cana-3309 117 69 compact compact ADJ cana-3309 117 70 metric metric ADJ cana-3309 117 71 space space NOUN cana-3309 117 72 publ publ NOUN cana-3309 117 73 . . PUNCT cana-3309 118 1 inst inst PROPN cana-3309 118 2 math math NOUN cana-3309 118 3 , , PUNCT cana-3309 118 4 vol.25 vol.25 ADJ cana-3309 118 5 , , PUNCT cana-3309 118 6 193 193 NUM cana-3309 118 7 - - SYM cana-3309 118 8 194 194 NUM cana-3309 118 9 . . PUNCT cana-3309 119 1 [ [ X cana-3309 119 2 9 9 NUM cana-3309 119 3 ] ] X cana-3309 119 4 kannan kannan PROPN cana-3309 119 5 , , PUNCT cana-3309 119 6 r(1969 r(1969 PROPN cana-3309 119 7 ) ) PUNCT cana-3309 119 8 : : PUNCT cana-3309 119 9 some some DET cana-3309 119 10 results result NOUN cana-3309 119 11 on on ADP cana-3309 119 12 fixed fix VERB cana-3309 119 13 points point NOUN cana-3309 119 14 -ii -ii INTJ cana-3309 119 15 .bulletin .bulletin NOUN cana-3309 119 16 of of ADP cana-3309 119 17 calcutta calcutta NOUN cana-3309 119 18 math.soc.vol.60,pp71 math.soc.vol.60,pp71 NOUN cana-3309 119 19 - - PUNCT cana-3309 119 20 76 76 NUM cana-3309 119 21 . . PUNCT cana-3309 120 1 [ [ X cana-3309 120 2 10 10 NUM cana-3309 120 3 ] ] PUNCT cana-3309 120 4 k. k. PROPN cana-3309 120 5 zoto zoto PROPN cana-3309 120 6 , , PUNCT cana-3309 120 7 n.mlaiki n.mlaiki NOUN cana-3309 120 8 , , PUNCT cana-3309 120 9 h.aydi h.aydi ADJ cana-3309 120 10 , , PUNCT cana-3309 120 11 β€œ " PUNCT cana-3309 120 12 related relate VERB cana-3309 120 13 fixed fix VERB cana-3309 120 14 point point NOUN cana-3309 120 15 theorems theorem NOUN cana-3309 120 16 via via ADP cana-3309 120 17 general general ADJ cana-3309 120 18 approach approach NOUN cana-3309 120 19 of of ADP cana-3309 120 20 simulations simulation NOUN cana-3309 120 21 functions function NOUN cana-3309 120 22 ” " PUNCT cana-3309 120 23 , , PUNCT cana-3309 120 24 journal journal NOUN cana-3309 120 25 of of ADP cana-3309 120 26 mathematics mathematic NOUN cana-3309 120 27 , , PUNCT cana-3309 120 28 vol vol NOUN cana-3309 120 29 . . PUNCT cana-3309 120 30 2020 2020 NUM cana-3309 120 31 , , PUNCT cana-3309 120 32 article article NOUN cana-3309 120 33 i i PROPN cana-3309 120 34 d d PROPN cana-3309 120 35 4820191 4820191 NUM cana-3309 121 1 [ [ X cana-3309 121 2 11 11 NUM cana-3309 121 3 ] ] X cana-3309 121 4 h. h. PROPN cana-3309 121 5 huang huang PROPN cana-3309 121 6 , , PUNCT cana-3309 121 7 k.zoto k.zoto NOUN cana-3309 121 8 , , PUNCT cana-3309 121 9 zd.mitroviΔ‡ zd.mitroviΔ‡ NUM cana-3309 121 10 , , PUNCT cana-3309 121 11 s.radenoviΔ‡ s.radenoviΔ‡ ADV cana-3309 121 12 , , PUNCT cana-3309 121 13 β€œ " PUNCT cana-3309 121 14 fixed fix VERB cana-3309 121 15 point point NOUN cana-3309 121 16 results result NOUN cana-3309 121 17 for for ADP cana-3309 121 18 generalized generalized ADJ cana-3309 121 19 f f NOUN cana-3309 121 20 - - PUNCT cana-3309 121 21 contractions contraction NOUN cana-3309 121 22 in in ADP cana-3309 121 23 b b NOUN cana-3309 121 24 - - PUNCT cana-3309 121 25 metric metric ADJ cana-3309 121 26 - - PUNCT cana-3309 121 27 like like ADJ cana-3309 121 28 spaces space NOUN cana-3309 121 29 ” " PUNCT cana-3309 121 30 fractal fractal ADJ cana-3309 121 31 and and CCONJ cana-3309 121 32 fractional fractional ADJ cana-3309 121 33 . . PUNCT cana-3309 122 1 6(5),272 6(5),272 NUM cana-3309 122 2 , , PUNCT cana-3309 122 3 ( ( PUNCT cana-3309 122 4 2022 2022 NUM cana-3309 122 5 ) ) PUNCT cana-3309 122 6 . . PUNCT cana-3309 123 1 https://doi.org/10.3390/fractalfract6050272 https://doi.org/10.3390/fractalfract6050272 NOUN cana-3309 123 2 . . PUNCT cana-3309 124 1 https://doi.org/10.3390/fractalfract6050272 https://doi.org/10.3390/fractalfract6050272 NOUN