id sid tid token lemma pos cana-761 1 1 communications communication NOUN cana-761 1 2 on on ADP cana-761 1 3 applied apply VERB cana-761 1 4 nonlinear nonlinear ADJ cana-761 1 5 analysis analysis NOUN cana-761 1 6 issn issn NOUN cana-761 1 7 : : PUNCT cana-761 1 8 1074 1074 NUM cana-761 1 9 - - PUNCT cana-761 1 10 133x 133x NUM cana-761 1 11 vol vol NOUN cana-761 1 12 31 31 NUM cana-761 1 13 no no NOUN cana-761 1 14 . . PUNCT cana-761 2 1 3s 3s NUM cana-761 2 2 ( ( PUNCT cana-761 2 3 2024 2024 NUM cana-761 2 4 ) ) PUNCT cana-761 2 5 227 227 NUM cana-761 2 6 https://internationalpubls.com https://internationalpubls.com NUM cana-761 2 7 fixed fix VERB cana-761 2 8 point point NOUN cana-761 2 9 theorems theorem NOUN cana-761 2 10 in in ADP cana-761 2 11 revised revise VERB cana-761 2 12 fuzzy fuzzy ADJ cana-761 2 13 metric metric ADJ cana-761 2 14 space space NOUN cana-761 2 15 via via ADP cana-761 2 16 𝑹𝑭 𝑹𝑭 PROPN cana-761 2 17 −contraction −contraction NUM cana-761 2 18 parakath parakath PROPN cana-761 2 19 nisha nisha PROPN cana-761 2 20 bagam bagam PROPN cana-761 2 21 p1 p1 PROPN cana-761 2 22 , , PUNCT cana-761 2 23 sandhya sandhya PROPN cana-761 2 24 p2 p2 PROPN cana-761 2 25 , , PUNCT cana-761 2 26 thangathamizh thangathamizh PROPN cana-761 2 27 r3 r3 PROPN cana-761 2 28 , , PUNCT cana-761 2 29 shanmugavel shanmugavel NOUN cana-761 2 30 p4 p4 ADJ cana-761 2 31 , , PUNCT cana-761 2 32 sarathbabu sarathbabu PROPN cana-761 2 33 k5 k5 PROPN cana-761 2 34 , , PUNCT cana-761 2 35 anusuya anusuya PROPN cana-761 2 36 r6 r6 NOUN cana-761 2 37 1 1 NUM cana-761 2 38 pg pg PROPN cana-761 2 39 & & CCONJ cana-761 2 40 research research PROPN cana-761 2 41 department department PROPN cana-761 2 42 of of ADP cana-761 2 43 mathematics mathematics PROPN cana-761 2 44 , , PUNCT cana-761 2 45 urumu urumu PROPN cana-761 2 46 dhanalakshmi dhanalakshmi PROPN cana-761 2 47 college college NOUN cana-761 2 48 , , PUNCT cana-761 2 49 bharathidasan bharathidasan ADJ cana-761 2 50 university university NOUN cana-761 2 51 , , PUNCT cana-761 2 52 trichy trichy NOUN cana-761 2 53 620019 620019 NUM cana-761 2 54 , , PUNCT cana-761 2 55 india india PROPN cana-761 2 56 ; ; PUNCT cana-761 2 57 parakath.nisha979@gmail.com parakath.nisha979@gmail.com PROPN cana-761 2 58 2 2 NUM cana-761 2 59 department department NOUN cana-761 2 60 of of ADP cana-761 2 61 mathematics mathematic NOUN cana-761 2 62 , , PUNCT cana-761 2 63 srm srm PROPN cana-761 2 64 trichy trichy PROPN cana-761 2 65 arts arts PROPN cana-761 2 66 & & CCONJ cana-761 2 67 science science PROPN cana-761 2 68 college college PROPN cana-761 2 69 , , PUNCT cana-761 2 70 trichy trichy PROPN cana-761 2 71 , , PUNCT cana-761 2 72 india india PROPN cana-761 2 73 ; ; PUNCT cana-761 2 74 sandhyaprasad2684@gmail.com sandhyaprasad2684@gmail.com X cana-761 2 75 3 3 NUM cana-761 2 76 department department NOUN cana-761 2 77 of of ADP cana-761 2 78 mathematics mathematics PROPN cana-761 2 79 , , PUNCT cana-761 2 80 k. k. PROPN cana-761 2 81 ramakrishnan ramakrishnan PROPN cana-761 2 82 college college PROPN cana-761 2 83 of of ADP cana-761 2 84 engineering engineering PROPN cana-761 2 85 , , PUNCT cana-761 2 86 samayapuram samayapuram PROPN cana-761 2 87 , , PUNCT cana-761 2 88 trichy trichy NOUN cana-761 2 89 621112 621112 NUM cana-761 2 90 , , PUNCT cana-761 2 91 india india PROPN cana-761 2 92 ; ; PUNCT cana-761 2 93 thamizh1418@gmail.com thamizh1418@gmail.com X cana-761 2 94 4 4 NUM cana-761 2 95 department department NOUN cana-761 2 96 of of ADP cana-761 2 97 mathematics mathematic NOUN cana-761 2 98 , , PUNCT cana-761 2 99 selvamm selvamm PROPN cana-761 2 100 arts arts PROPN cana-761 2 101 & & CCONJ cana-761 2 102 science science PROPN cana-761 2 103 college college PROPN cana-761 2 104 , , PUNCT cana-761 2 105 periyar periyar PROPN cana-761 2 106 university university NOUN cana-761 2 107 , , PUNCT cana-761 2 108 namakkal namakkal NOUN cana-761 2 109 , , PUNCT cana-761 2 110 india india PROPN cana-761 2 111 ; ; PUNCT cana-761 2 112 p.sham1988@gmail.com p.sham1988@gmail.com X cana-761 2 113 5 5 NUM cana-761 2 114 department department NOUN cana-761 2 115 of of ADP cana-761 2 116 mathematics mathematic NOUN cana-761 2 117 , , PUNCT cana-761 2 118 selvamm selvamm PROPN cana-761 2 119 arts arts PROPN cana-761 2 120 & & CCONJ cana-761 2 121 science science PROPN cana-761 2 122 college college PROPN cana-761 2 123 , , PUNCT cana-761 2 124 periyar periyar PROPN cana-761 2 125 university university NOUN cana-761 2 126 , , PUNCT cana-761 2 127 namakkal namakkal NOUN cana-761 2 128 , , PUNCT cana-761 2 129 india india PROPN cana-761 2 130 ; ; PUNCT cana-761 2 131 babusarath234@gmail.com babusarath234@gmail.com NUM cana-761 2 132 6 6 NUM cana-761 2 133 pg pg PROPN cana-761 2 134 & & CCONJ cana-761 2 135 research research PROPN cana-761 2 136 department department PROPN cana-761 2 137 of of ADP cana-761 2 138 mathematics mathematics PROPN cana-761 2 139 , , PUNCT cana-761 2 140 urumu urumu PROPN cana-761 2 141 dhanalakshmi dhanalakshmi PROPN cana-761 2 142 college college NOUN cana-761 2 143 , , PUNCT cana-761 2 144 bharathidasan bharathidasan ADJ cana-761 2 145 university university NOUN cana-761 2 146 , , PUNCT cana-761 2 147 trichy trichy NOUN cana-761 2 148 620019 620019 NUM cana-761 2 149 , , PUNCT cana-761 2 150 india india PROPN cana-761 2 151 ; ; PUNCT cana-761 2 152 anusuyar23@gmail.com anusuyar23@gmail.com X cana-761 2 153 article article NOUN cana-761 2 154 history history NOUN cana-761 2 155 : : PUNCT cana-761 2 156 received receive VERB cana-761 2 157 : : PUNCT cana-761 2 158 14 14 NUM cana-761 2 159 - - PUNCT cana-761 2 160 04 04 NUM cana-761 2 161 - - PUNCT cana-761 2 162 2024 2024 NUM cana-761 2 163 revised revise VERB cana-761 2 164 : : PUNCT cana-761 2 165 29 29 NUM cana-761 2 166 - - SYM cana-761 2 167 05 05 NUM cana-761 2 168 - - PUNCT cana-761 2 169 2024 2024 NUM cana-761 2 170 accepted accept VERB cana-761 2 171 : : PUNCT cana-761 2 172 15 15 NUM cana-761 2 173 - - SYM cana-761 2 174 06 06 NUM cana-761 2 175 - - PUNCT cana-761 2 176 2024 2024 NUM cana-761 2 177 abstract abstract NOUN cana-761 2 178 : : PUNCT cana-761 2 179 the the DET cana-761 2 180 aim aim NOUN cana-761 2 181 of of ADP cana-761 2 182 this this DET cana-761 2 183 paper paper NOUN cana-761 2 184 is be AUX cana-761 2 185 to to PART cana-761 2 186 introduce introduce VERB cana-761 2 187 a a DET cana-761 2 188 new new ADJ cana-761 2 189 type type NOUN cana-761 2 190 of of ADP cana-761 2 191 contraction contraction NOUN cana-761 2 192 called call VERB cana-761 2 193 𝑅𝐹 𝑅𝐹 PROPN cana-761 2 194 −contraction −contraction PROPN cana-761 2 195 . . PUNCT cana-761 3 1 as as SCONJ cana-761 3 2 compared compare VERB cana-761 3 3 to to ADP cana-761 3 4 the the DET cana-761 3 5 f f NOUN cana-761 3 6 - - PUNCT cana-761 3 7 contraction contraction NOUN cana-761 3 8 in in ADP cana-761 3 9 the the DET cana-761 3 10 existing exist VERB cana-761 3 11 literature literature NOUN cana-761 3 12 , , PUNCT cana-761 3 13 our our PRON cana-761 3 14 𝑅𝐹 𝑅𝐹 PROPN cana-761 3 15 −contraction −contraction NOUN cana-761 3 16 is be AUX cana-761 3 17 much much ADV cana-761 3 18 simpler simple ADJ cana-761 3 19 and and CCONJ cana-761 3 20 more more ADV cana-761 3 21 straightforward straightforward ADJ cana-761 3 22 , , PUNCT cana-761 3 23 since since SCONJ cana-761 3 24 it it PRON cana-761 3 25 contains contain VERB cana-761 3 26 only only ADV cana-761 3 27 one one NUM cana-761 3 28 condition condition NOUN cana-761 3 29 that that PRON cana-761 3 30 is be AUX cana-761 3 31 , , PUNCT cana-761 3 32 the the DET cana-761 3 33 function function NOUN cana-761 3 34 revised revise VERB cana-761 3 35 fuzzy fuzzy ADJ cana-761 3 36 is be AUX cana-761 3 37 strictly strictly ADV cana-761 3 38 non non ADJ cana-761 3 39 - - ADJ cana-761 3 40 increasing increase VERB cana-761 3 41 . . PUNCT cana-761 4 1 moreover moreover ADV cana-761 4 2 , , PUNCT cana-761 4 3 some some DET cana-761 4 4 fixed fix VERB cana-761 4 5 - - PUNCT cana-761 4 6 point point NOUN cana-761 4 7 theorems theorem NOUN cana-761 4 8 for for ADP cana-761 4 9 𝑅𝐹 𝑅𝐹 PROPN cana-761 4 10 −contraction −contraction NOUN cana-761 4 11 are be AUX cana-761 4 12 presented present VERB cana-761 4 13 . . PUNCT cana-761 5 1 further far ADV cana-761 5 2 , , PUNCT cana-761 5 3 some some DET cana-761 5 4 examples example NOUN cana-761 5 5 are be AUX cana-761 5 6 given give VERB cana-761 5 7 to to PART cana-761 5 8 illustrate illustrate VERB cana-761 5 9 its its PRON cana-761 5 10 validity validity NOUN cana-761 5 11 and and CCONJ cana-761 5 12 superiority superiority NOUN cana-761 5 13 . . PUNCT cana-761 6 1 in in ADP cana-761 6 2 addition addition NOUN cana-761 6 3 , , PUNCT cana-761 6 4 by by ADP cana-761 6 5 applying apply VERB cana-761 6 6 a a DET cana-761 6 7 very very ADV cana-761 6 8 significant significant ADJ cana-761 6 9 lemma lemma NOUN cana-761 6 10 , , PUNCT cana-761 6 11 we we PRON cana-761 6 12 show show VERB cana-761 6 13 that that SCONJ cana-761 6 14 our our PRON cana-761 6 15 proofs proof NOUN cana-761 6 16 of of ADP cana-761 6 17 most most ADJ cana-761 6 18 fixed fix VERB cana-761 6 19 - - PUNCT cana-761 6 20 point point NOUN cana-761 6 21 theorems theorem NOUN cana-761 6 22 are be AUX cana-761 6 23 shorter short ADJ cana-761 6 24 and and CCONJ cana-761 6 25 more more ADV cana-761 6 26 elegant elegant ADJ cana-761 6 27 than than ADP cana-761 6 28 ones one NOUN cana-761 6 29 in in ADP cana-761 6 30 the the DET cana-761 6 31 literature literature NOUN cana-761 6 32 . . PUNCT cana-761 7 1 keywords keyword NOUN cana-761 7 2 and and CCONJ cana-761 7 3 phrases phrase NOUN cana-761 7 4 : : PUNCT cana-761 7 5 t t NOUN cana-761 7 6 - - PUNCT cana-761 7 7 conorm conorm NOUN cana-761 7 8 , , PUNCT cana-761 7 9 revised revise VERB cana-761 7 10 fuzzy fuzzy ADJ cana-761 7 11 metric metric ADJ cana-761 7 12 , , PUNCT cana-761 7 13 𝑅𝐹 𝑅𝐹 PROPN cana-761 7 14 −contraction −contraction NOUN cana-761 7 15 , , PUNCT cana-761 7 16 fixed fix VERB cana-761 7 17 point point NOUN cana-761 7 18 . . PUNCT cana-761 8 1 2020 2020 NUM cana-761 8 2 mathematical mathematical ADJ cana-761 8 3 classification classification NOUN cana-761 8 4 : : PUNCT cana-761 8 5 46n20 46n20 NUM cana-761 8 6 , , PUNCT cana-761 8 7 46s40 46s40 NUM cana-761 8 8 , , PUNCT cana-761 8 9 47h10 47h10 NUM cana-761 8 10 , , PUNCT cana-761 8 11 37c25 37c25 NUM cana-761 8 12 . . NOUN cana-761 9 1 1 1 X cana-761 9 2 . . X cana-761 9 3 introduction introduction NOUN cana-761 9 4 george george PROPN cana-761 9 5 and and CCONJ cana-761 9 6 veeramani veeramani NOUN cana-761 10 1 [ [ X cana-761 10 2 3 3 X cana-761 10 3 ] ] PUNCT cana-761 10 4 proposed propose VERB cana-761 10 5 axioms axiom NOUN cana-761 10 6 to to ADP cana-761 10 7 fuzzy fuzzy ADJ cana-761 10 8 metric metric ADJ cana-761 10 9 spaces space NOUN cana-761 10 10 [ [ X cana-761 10 11 for for ADP cana-761 10 12 short short ADJ cana-761 10 13 , , PUNCT cana-761 10 14 fms fms PROPN cana-761 10 15 ] ] PUNCT cana-761 10 16 based base VERB cana-761 10 17 on on ADP cana-761 10 18 zadeh zadeh PROPN cana-761 10 19 's 's PART cana-761 10 20 theory theory NOUN cana-761 10 21 of of ADP cana-761 10 22 fuzzy fuzzy ADJ cana-761 10 23 sets set NOUN cana-761 10 24 [ [ X cana-761 10 25 25 25 NUM cana-761 10 26 ] ] PUNCT cana-761 10 27 . . PUNCT cana-761 11 1 the the DET cana-761 11 2 triangular triangular NOUN cana-761 11 3 norm norm NOUN cana-761 11 4 ( ( PUNCT cana-761 11 5 for for ADP cana-761 11 6 short short ADJ cana-761 11 7 , , PUNCT cana-761 11 8 t t NOUN cana-761 11 9 - - PUNCT cana-761 11 10 norm norm NOUN cana-761 11 11 ) ) PUNCT cana-761 11 12 , , PUNCT cana-761 11 13 initially initially ADV cana-761 11 14 proposed propose VERB cana-761 11 15 by by ADP cana-761 11 16 schweizer schweizer PROPN cana-761 11 17 and and CCONJ cana-761 11 18 sklar sklar PROPN cana-761 11 19 [ [ X cana-761 11 20 21 21 NUM cana-761 11 21 ] ] PUNCT cana-761 11 22 , , PUNCT cana-761 11 23 is be AUX cana-761 11 24 one one NUM cana-761 11 25 of of ADP cana-761 11 26 the the DET cana-761 11 27 most most ADV cana-761 11 28 significant significant ADJ cana-761 11 29 axioms axiom NOUN cana-761 11 30 in in ADP cana-761 11 31 binary binary ADJ cana-761 11 32 functions function NOUN cana-761 11 33 . . PUNCT cana-761 12 1 in in ADP cana-761 12 2 various various ADJ cana-761 12 3 domains domain NOUN cana-761 12 4 , , PUNCT cana-761 12 5 such such ADJ cana-761 12 6 as as ADP cana-761 12 7 fuzzy fuzzy ADJ cana-761 12 8 sets set NOUN cana-761 12 9 , , PUNCT cana-761 12 10 fuzzy fuzzy ADJ cana-761 12 11 logic logic NOUN cana-761 12 12 , , PUNCT cana-761 12 13 and and CCONJ cana-761 12 14 its its PRON cana-761 12 15 applications application NOUN cana-761 12 16 , , PUNCT cana-761 12 17 this this PRON cana-761 12 18 is be AUX cana-761 12 19 a a DET cana-761 12 20 critical critical ADJ cana-761 12 21 process process NOUN cana-761 12 22 . . PUNCT cana-761 13 1 the the DET cana-761 13 2 fixed fix VERB cana-761 13 3 - - PUNCT cana-761 13 4 point point NOUN cana-761 13 5 [ [ X cana-761 13 6 for for ADP cana-761 13 7 short short ADJ cana-761 13 8 , , PUNCT cana-761 13 9 fp fp X cana-761 13 10 ] ] X cana-761 13 11 theory theory NOUN cana-761 13 12 constructed construct VERB cana-761 13 13 in in ADP cana-761 13 14 fmss fmss PROPN cana-761 13 15 , , PUNCT cana-761 13 16 which which PRON cana-761 13 17 was be AUX cana-761 13 18 pioneered pioneer VERB cana-761 13 19 by by ADP cana-761 13 20 grabiec grabiec PROPN cana-761 14 1 [ [ X cana-761 14 2 4 4 NUM cana-761 14 3 ] ] PUNCT cana-761 14 4 , , PUNCT cana-761 14 5 where where SCONJ cana-761 14 6 a a DET cana-761 14 7 fm fm NOUN cana-761 14 8 version version NOUN cana-761 14 9 of of ADP cana-761 14 10 the the DET cana-761 14 11 banach banach NOUN cana-761 14 12 contraction contraction NOUN cana-761 14 13 principle principle NOUN cana-761 14 14 was be AUX cana-761 14 15 introduced introduce VERB cana-761 14 16 , , PUNCT cana-761 14 17 is be AUX cana-761 14 18 one one NUM cana-761 14 19 of of ADP cana-761 14 20 the the DET cana-761 14 21 most most ADV cana-761 14 22 fascinating fascinating ADJ cana-761 14 23 motives motive NOUN cana-761 14 24 . . PUNCT cana-761 15 1 following follow VERB cana-761 15 2 that that PRON cana-761 15 3 , , PUNCT cana-761 15 4 gregori gregori PROPN cana-761 15 5 and and CCONJ cana-761 15 6 his his PRON cana-761 15 7 coauthors coauthor NOUN cana-761 15 8 presented present VERB cana-761 15 9 a a DET cana-761 15 10 number number NOUN cana-761 15 11 of of ADP cana-761 15 12 fuzzy fuzzy ADJ cana-761 15 13 contractive contractive ADJ cana-761 15 14 mappings mapping NOUN cana-761 15 15 [ [ X cana-761 15 16 for for ADP cana-761 15 17 short short ADJ cana-761 15 18 , , PUNCT cana-761 15 19 fcm fcm X cana-761 15 20 ] ] PUNCT cana-761 15 21 in in ADP cana-761 15 22 fms fms PROPN cana-761 15 23 ( ( PUNCT cana-761 15 24 see see VERB cana-761 15 25 [ [ X cana-761 15 26 5 5 NUM cana-761 15 27 ] ] NUM cana-761 15 28 ) ) PUNCT cana-761 15 29 . . PUNCT cana-761 16 1 mihet mihet PROPN cana-761 17 1 [ [ X cana-761 17 2 11 11 NUM cana-761 17 3 ] ] PUNCT cana-761 17 4 , , PUNCT cana-761 17 5 on on ADP cana-761 17 6 the the DET cana-761 17 7 other other ADJ cana-761 17 8 hand hand NOUN cana-761 17 9 , , PUNCT cana-761 17 10 established establish VERB cana-761 17 11 a a DET cana-761 17 12 fixed fix VERB cana-761 17 13 - - PUNCT cana-761 17 14 point point NOUN cana-761 17 15 theorem theorem NOUN cana-761 17 16 for for ADP cana-761 17 17 weak weak ADJ cana-761 17 18 banach banach NOUN cana-761 17 19 contraction contraction NOUN cana-761 17 20 in in ADP cana-761 17 21 w w NOUN cana-761 17 22 - - PUNCT cana-761 17 23 complete complete ADJ cana-761 17 24 fms fms PROPN cana-761 17 25 , , PUNCT cana-761 17 26 and and CCONJ cana-761 17 27 expanded expand VERB cana-761 17 28 prior prior ADJ cana-761 17 29 results result NOUN cana-761 17 30 including include VERB cana-761 17 31 additional additional ADJ cana-761 17 32 types type NOUN cana-761 17 33 of of ADP cana-761 17 34 contractions contraction NOUN cana-761 17 35 such such ADJ cana-761 17 36 as as ADP cana-761 17 37 edelstein edelstein PROPN cana-761 17 38 fuzzy fuzzy ADJ cana-761 17 39 contractive contractive ADJ cana-761 17 40 mappings mapping NOUN cana-761 17 41 , , PUNCT cana-761 17 42 fuzzy fuzzy ADJ cana-761 17 43 y y PROPN cana-761 17 44 - - PUNCT cana-761 17 45 contractive contractive ADJ cana-761 17 46 mappings mapping NOUN cana-761 17 47 , , PUNCT cana-761 17 48 and and CCONJ cana-761 17 49 so so ADV cana-761 17 50 on on ADV cana-761 17 51 ( ( PUNCT cana-761 17 52 for for ADP cana-761 17 53 details detail NOUN cana-761 17 54 , , PUNCT cana-761 17 55 see see VERB cana-761 17 56 [ [ X cana-761 17 57 11 11 NUM cana-761 17 58 ] ] NUM cana-761 17 59 ) ) PUNCT cana-761 17 60 . . PUNCT cana-761 18 1 wardowski wardowski VERB cana-761 19 1 [ [ X cana-761 19 2 23 23 NUM cana-761 19 3 ] ] PUNCT cana-761 19 4 recently recently ADV cana-761 19 5 developed develop VERB cana-761 19 6 a a DET cana-761 19 7 novel novel ADJ cana-761 19 8 idea idea NOUN cana-761 19 9 of of ADP cana-761 19 10 fuzzy fuzzy ADJ cana-761 19 11 h h ADJ cana-761 19 12 - - PUNCT cana-761 19 13 contractive contractive ADJ cana-761 19 14 mapping mapping NOUN cana-761 19 15 and and CCONJ cana-761 19 16 deduced deduce VERB cana-761 19 17 some some DET cana-761 19 18 interesting interesting ADJ cana-761 19 19 fp fp NOUN cana-761 19 20 theorems theorem NOUN cana-761 19 21 . . PUNCT cana-761 20 1 wardowski wardowski PROPN cana-761 21 1 [ [ X cana-761 21 2 24 24 NUM cana-761 21 3 ] ] PUNCT cana-761 21 4 also also ADV cana-761 21 5 developed develop VERB cana-761 21 6 a a DET cana-761 21 7 fp fp X cana-761 21 8 theorem theorem NOUN cana-761 21 9 in in ADP cana-761 21 10 metric metric ADJ cana-761 21 11 spaces space NOUN cana-761 21 12 https://cran.r-project.org/web/classifications/msc-2010.html#code:37c25 https://cran.r-project.org/web/classifications/msc-2010.html#code:37c25 NOUN cana-761 21 13 communications communication NOUN cana-761 21 14 on on ADP cana-761 21 15 applied apply VERB cana-761 21 16 nonlinear nonlinear ADJ cana-761 21 17 analysis analysis NOUN cana-761 21 18 issn issn NOUN cana-761 21 19 : : PUNCT cana-761 21 20 1074 1074 NUM cana-761 21 21 - - PUNCT cana-761 21 22 133x 133x NUM cana-761 21 23 vol vol NOUN cana-761 21 24 31 31 NUM cana-761 21 25 no no NOUN cana-761 21 26 . . PUNCT cana-761 22 1 3s 3s NUM cana-761 22 2 ( ( PUNCT cana-761 22 3 2024 2024 NUM cana-761 22 4 ) ) PUNCT cana-761 22 5 228 228 NUM cana-761 22 6 https://internationalpubls.com https://internationalpubls.com X cana-761 22 7 and and CCONJ cana-761 22 8 introduced introduce VERB cana-761 22 9 the the DET cana-761 22 10 f f PROPN cana-761 22 11 - - PUNCT cana-761 22 12 contraction contraction NOUN cana-761 22 13 contraction contraction NOUN cana-761 22 14 . . PUNCT cana-761 23 1 [ [ X cana-761 23 2 7,11,22 7,11,22 NUM cana-761 23 3 ] ] PUNCT cana-761 23 4 recently recently ADV cana-761 23 5 proposed propose VERB cana-761 23 6 further further ADJ cana-761 23 7 contractions contraction NOUN cana-761 23 8 in in ADP cana-761 23 9 fms fms PROPN cana-761 23 10 . . PUNCT cana-761 24 1 alexander alexander PROPN cana-761 24 2 sostak sostak PROPN cana-761 25 1 [ [ X cana-761 25 2 1 1 NUM cana-761 25 3 ] ] PUNCT cana-761 25 4 introduced introduce VERB cana-761 25 5 the the DET cana-761 25 6 notion notion NOUN cana-761 25 7 of of ADP cana-761 25 8 revised revise VERB cana-761 25 9 fuzzy fuzzy ADJ cana-761 25 10 metric metric ADJ cana-761 25 11 [ [ X cana-761 25 12 for for ADP cana-761 25 13 short short ADJ cana-761 25 14 , , PUNCT cana-761 25 15 rfm rfm PROPN cana-761 25 16 ] ] PUNCT cana-761 25 17 in in ADP cana-761 25 18 the the DET cana-761 25 19 year year NOUN cana-761 25 20 2018 2018 NUM cana-761 25 21 , , PUNCT cana-761 25 22 which which PRON cana-761 25 23 allows allow VERB cana-761 25 24 for for ADP cana-761 25 25 the the DET cana-761 25 26 incremental incremental ADJ cana-761 25 27 assessment assessment NOUN cana-761 25 28 of of ADP cana-761 25 29 the the DET cana-761 25 30 membership membership NOUN cana-761 25 31 of of ADP cana-761 25 32 components component NOUN cana-761 25 33 in in ADP cana-761 25 34 a a DET cana-761 25 35 set set NOUN cana-761 25 36 . . PUNCT cana-761 26 1 muraliraj muraliraj PROPN cana-761 26 2 and and CCONJ cana-761 26 3 thangathamizh thangathamizh PROPN cana-761 26 4 developed develop VERB cana-761 26 5 revised revise VERB cana-761 26 6 fuzzy fuzzy ADJ cana-761 26 7 contraction contraction NOUN cana-761 26 8 mappings mapping NOUN cana-761 27 1 [ [ X cana-761 27 2 for for ADP cana-761 27 3 short short ADJ cana-761 27 4 , , PUNCT cana-761 27 5 rfcm rfcm NOUN cana-761 27 6 ] ] PUNCT cana-761 27 7 and and CCONJ cana-761 27 8 established establish VERB cana-761 27 9 fp fp NOUN cana-761 27 10 findings finding NOUN cana-761 27 11 for for ADP cana-761 27 12 them they PRON cana-761 28 1 [ [ X cana-761 28 2 11 11 NUM cana-761 28 3 - - SYM cana-761 28 4 14 14 NUM cana-761 28 5 , , PUNCT cana-761 28 6 24 24 NUM cana-761 28 7 - - SYM cana-761 28 8 26 26 NUM cana-761 28 9 ] ] PUNCT cana-761 28 10 . . PUNCT cana-761 29 1 many many ADJ cana-761 29 2 generic generic ADJ cana-761 29 3 topological topological ADJ cana-761 29 4 ideas idea NOUN cana-761 29 5 and and CCONJ cana-761 29 6 conclusions conclusion NOUN cana-761 29 7 were be AUX cana-761 29 8 then then ADV cana-761 29 9 applied apply VERB cana-761 29 10 to to ADP cana-761 29 11 the the DET cana-761 29 12 revised revise VERB cana-761 29 13 fuzzy fuzzy ADJ cana-761 29 14 topological topological ADJ cana-761 29 15 space space NOUN cana-761 29 16 . . PUNCT cana-761 30 1 we we PRON cana-761 30 2 intraduce intraduce VERB cana-761 30 3 a a DET cana-761 30 4 new new ADJ cana-761 30 5 contraction contraction NOUN cana-761 30 6 called call VERB cana-761 30 7 rf rf NOUN cana-761 30 8 - - NOUN cana-761 30 9 contraction contraction NOUN cana-761 30 10 throughout throughout ADP cana-761 30 11 this this DET cana-761 30 12 research research NOUN cana-761 30 13 , , PUNCT cana-761 30 14 which which PRON cana-761 30 15 differs differ VERB cana-761 30 16 from from ADP cana-761 30 17 [ [ X cana-761 30 18 7,17,23 7,17,23 NUM cana-761 30 19 ] ] PUNCT cana-761 30 20 in in SCONJ cana-761 30 21 that that SCONJ cana-761 30 22 it it PRON cana-761 30 23 incorporates incorporate VERB cana-761 30 24 a a DET cana-761 30 25 simpler simple ADJ cana-761 30 26 criterion criterion NOUN cana-761 30 27 , , PUNCT cana-761 30 28 namely namely ADV cana-761 30 29 that that SCONJ cana-761 30 30 the the DET cana-761 30 31 mapping mapping NOUN cana-761 30 32 is be AUX cana-761 30 33 strictly strictly ADV cana-761 30 34 nonincreasing nonincrease VERB cana-761 30 35 . . PUNCT cana-761 31 1 furthermore furthermore ADV cana-761 31 2 , , PUNCT cana-761 31 3 in in ADP cana-761 31 4 the the DET cana-761 31 5 context context NOUN cana-761 31 6 of of ADP cana-761 31 7 rfms rfms PROPN cana-761 31 8 , , PUNCT cana-761 31 9 we we PRON cana-761 31 10 deal deal VERB cana-761 31 11 with with ADP cana-761 31 12 fp fp X cana-761 31 13 theorems theorem NOUN cana-761 31 14 for for ADP cana-761 31 15 𝑅𝐹 𝑅𝐹 PROPN cana-761 31 16 −contraction −contraction NOUN cana-761 31 17 . . PUNCT cana-761 32 1 in in ADP cana-761 32 2 particular particular ADJ cana-761 32 3 , , PUNCT cana-761 32 4 we we PRON cana-761 32 5 prove prove VERB cana-761 32 6 a a DET cana-761 32 7 lemma lemma PROPN cana-761 32 8 in in ADP cana-761 32 9 rfms rfms NOUN cana-761 32 10 with with ADP cana-761 32 11 regard regard NOUN cana-761 32 12 to to ADP cana-761 32 13 the the DET cana-761 32 14 cauchy cauchy ADJ cana-761 32 15 sequence sequence NOUN cana-761 32 16 . . PUNCT cana-761 33 1 second second ADJ cana-761 33 2 , , PUNCT cana-761 33 3 we we PRON cana-761 33 4 introduce introduce VERB cana-761 33 5 the the DET cana-761 33 6 idea idea NOUN cana-761 33 7 of of ADP cana-761 33 8 𝑅𝐹 𝑅𝐹 PROPN cana-761 33 9 −contraction −contraction PROPN cana-761 33 10 , , PUNCT cana-761 33 11 which which PRON cana-761 33 12 requires require VERB cana-761 33 13 just just ADV cana-761 33 14 that that SCONJ cana-761 33 15 the the DET cana-761 33 16 function function NOUN cana-761 33 17 be be AUX cana-761 33 18 strictly strictly ADV cana-761 33 19 non non ADJ cana-761 33 20 - - ADJ cana-761 33 21 increasing increase VERB cana-761 33 22 . . PUNCT cana-761 34 1 third third ADV cana-761 34 2 , , PUNCT cana-761 34 3 we we PRON cana-761 34 4 get get VERB cana-761 34 5 certain certain ADJ cana-761 34 6 fp fp NOUN cana-761 34 7 theorems theorem NOUN cana-761 34 8 for for ADP cana-761 34 9 𝑅𝐹 𝑅𝐹 PROPN cana-761 34 10 −contraction −contraction NOUN cana-761 34 11 with with ADP cana-761 34 12 shorter short ADJ cana-761 34 13 requirements requirement NOUN cana-761 34 14 and and CCONJ cana-761 34 15 simple simple ADJ cana-761 34 16 proofs proof NOUN cana-761 34 17 using use VERB cana-761 34 18 the the DET cana-761 34 19 preceding precede VERB cana-761 34 20 lemma lemma PROPN cana-761 34 21 . . PUNCT cana-761 35 1 fourth fourth ADJ cana-761 35 2 , , PUNCT cana-761 35 3 we we PRON cana-761 35 4 provide provide VERB cana-761 35 5 some some DET cana-761 35 6 instances instance NOUN cana-761 35 7 to to PART cana-761 35 8 back back VERB cana-761 35 9 up up ADP cana-761 35 10 our our PRON cana-761 35 11 findings finding NOUN cana-761 35 12 . . PUNCT cana-761 36 1 our our PRON cana-761 36 2 examples example NOUN cana-761 36 3 demonstrate demonstrate VERB cana-761 36 4 that that SCONJ cana-761 36 5 our our PRON cana-761 36 6 findings finding NOUN cana-761 36 7 are be AUX cana-761 36 8 true true ADJ cana-761 36 9 generalizations generalization NOUN cana-761 36 10 in in ADP cana-761 36 11 the the DET cana-761 36 12 literature literature NOUN cana-761 36 13 . . PUNCT cana-761 37 1 2 2 X cana-761 37 2 . . X cana-761 37 3 preliminaries preliminary NOUN cana-761 37 4 we we PRON cana-761 37 5 'll will AUX cana-761 37 6 go go VERB cana-761 37 7 over over ADP cana-761 37 8 a a DET cana-761 37 9 few few ADJ cana-761 37 10 fundamental fundamental ADJ cana-761 37 11 definitions definition NOUN cana-761 37 12 and and CCONJ cana-761 37 13 ideas idea NOUN cana-761 37 14 in in ADP cana-761 37 15 the the DET cana-761 37 16 sections section NOUN cana-761 37 17 that that PRON cana-761 37 18 follow follow VERB cana-761 37 19 . . PUNCT cana-761 38 1 definition definition NOUN cana-761 38 2 2.1 2.1 NUM cana-761 38 3 ( ( PUNCT cana-761 38 4 [ [ X cana-761 38 5 21 21 NUM cana-761 38 6 ] ] PUNCT cana-761 38 7 ) ) PUNCT cana-761 38 8 . . PUNCT cana-761 39 1 a a DET cana-761 39 2 binary binary PROPN cana-761 39 3 operation operation NOUN cana-761 39 4   PROPN cana-761 39 5   PROPN cana-761 39 6   PROPN cana-761 39 7   ADP cana-761 39 8 2 2 NUM cana-761 39 9 : : SYM cana-761 39 10 0,1 0,1 NUM cana-761 39 11     SPACE cana-761 39 12 0,1 0,1 NUM cana-761 39 13     SPACE cana-761 39 14 t t NOUN cana-761 39 15 → → PUNCT cana-761 39 16 is be AUX cana-761 39 17 called call VERB cana-761 39 18 a a DET cana-761 39 19 triangular triangular NOUN cana-761 39 20 norm norm NOUN cana-761 39 21 ( ( PUNCT cana-761 39 22 for for ADP cana-761 39 23 short short ADJ cana-761 39 24 , , PUNCT cana-761 39 25 t t NOUN cana-761 39 26 - - PUNCT cana-761 39 27 norm norm NOUN cana-761 39 28 ) ) PUNCT cana-761 39 29 if if SCONJ cana-761 39 30 the the DET cana-761 39 31 following follow VERB cana-761 39 32 conditions condition NOUN cana-761 39 33 hold hold VERB cana-761 39 34 : : PUNCT cana-761 39 35 ( ( PUNCT cana-761 39 36 i i NOUN cana-761 39 37 ) ) PUNCT cana-761 39 38 ( ( PUNCT cana-761 39 39 ) ) PUNCT cana-761 39 40 , , PUNCT cana-761 39 41     SPACE cana-761 39 42 0 0 NUM cana-761 39 43 t t NOUN cana-761 39 44 g g NOUN cana-761 39 45 g= g= NOUN cana-761 39 46 , , PUNCT cana-761 39 47 for for ADP cana-761 39 48 each each DET cana-761 39 49   ADJ cana-761 39 50 0,1 0,1 NOUN cana-761 39 51     SPACE cana-761 39 52 g g NOUN cana-761 39 53   NOUN cana-761 39 54 ; ; PUNCT cana-761 39 55 ( ( PUNCT cana-761 39 56 ii ii NOUN cana-761 39 57 ) ) PUNCT cana-761 39 58 ( ( PUNCT cana-761 39 59 ) ) PUNCT cana-761 39 60 ( ( PUNCT cana-761 39 61 ) ) PUNCT cana-761 39 62 , , PUNCT cana-761 39 63     SPACE cana-761 39 64 , , PUNCT cana-761 39 65     SPACE cana-761 39 66 t t PROPN cana-761 39 67 g g PROPN cana-761 39 68 d d PROPN cana-761 39 69 t t PROPN cana-761 39 70 l l NOUN cana-761 39 71 m m NOUN cana-761 39 72 , , PUNCT cana-761 39 73 for for ADP cana-761 39 74 any any PRON cana-761 39 75         SPACE cana-761 39 76 , , PUNCT cana-761 39 77             SPACE cana-761 39 78 g g NOUN cana-761 39 79 l l PROPN cana-761 39 80 d d NOUN cana-761 39 81 m m VERB cana-761 39 82   NOUN cana-761 39 83 and and CCONJ cana-761 39 84   ADJ cana-761 39 85   PROPN cana-761 39 86 , , PUNCT cana-761 39 87     SPACE cana-761 39 88 , , PUNCT cana-761 39 89     SPACE cana-761 39 90 , , PUNCT cana-761 39 91     SPACE cana-761 39 92 0,1 0,1 NUM cana-761 39 93     SPACE cana-761 39 94 g g NOUN cana-761 39 95 d d PROPN cana-761 39 96 l l NOUN cana-761 39 97 m m PROPN cana-761 39 98 ; ; PUNCT cana-761 39 99 ( ( PUNCT cana-761 39 100 iii iii X cana-761 39 101 ) ) PUNCT cana-761 39 102 t t PROPN cana-761 39 103 is be AUX cana-761 39 104 associative associative ADJ cana-761 39 105 and and CCONJ cana-761 39 106 commutative commutative ADJ cana-761 39 107 . . PUNCT cana-761 40 1 three three NUM cana-761 40 2 basic basic ADJ cana-761 40 3 examples example NOUN cana-761 40 4 of of ADP cana-761 40 5 continuous continuous ADJ cana-761 40 6 t t PROPN cana-761 40 7 - - PUNCT cana-761 40 8 conorms conorm NOUN cana-761 40 9 are be AUX cana-761 40 10 as as SCONJ cana-761 40 11 follows follow VERB cana-761 40 12 : : PUNCT cana-761 40 13 ( ( PUNCT cana-761 40 14 ) ) PUNCT cana-761 41 1   PROPN cana-761 41 2   PROPN cana-761 41 3 , , PUNCT cana-761 41 4             SPACE cana-761 41 5 , , PUNCT cana-761 41 6     SPACE cana-761 41 7 maxt maxt NOUN cana-761 41 8 g g PROPN cana-761 42 1 d d PROPN cana-761 42 2 max max PROPN cana-761 42 3 g g PROPN cana-761 42 4 d= d= NOUN cana-761 42 5 , , PUNCT cana-761 42 6 ( ( PUNCT cana-761 42 7 ) ) PUNCT cana-761 42 8 , , PUNCT cana-761 42 9             SPACE cana-761 42 10 pt pt X cana-761 42 11 g g PROPN cana-761 42 12 d d PROPN cana-761 42 13 g g PROPN cana-761 42 14 d d PROPN cana-761 42 15 gd= gd= X cana-761 42 16 + + CCONJ cana-761 42 17 − − PROPN cana-761 42 18 and and CCONJ cana-761 42 19 ( ( PUNCT cana-761 42 20 ) ) PUNCT cana-761 42 21   PROPN cana-761 42 22   PROPN cana-761 42 23 , , PUNCT cana-761 42 24                 SPACE cana-761 42 25 , , PUNCT cana-761 42 26 1lt 1lt ADJ cana-761 42 27 g g PROPN cana-761 42 28 d d X cana-761 42 29 ming ming PROPN cana-761 43 1 g g PROPN cana-761 43 2 d= d= NOUN cana-761 43 3 + + CCONJ cana-761 43 4 ( ( PUNCT cana-761 43 5 maximum maximum ADJ cana-761 43 6 , , PUNCT cana-761 43 7 product product NOUN cana-761 43 8 , , PUNCT cana-761 43 9 and and CCONJ cana-761 43 10 lukasiewicz lukasiewicz VERB cana-761 43 11 t t PROPN cana-761 43 12 - - PUNCT cana-761 43 13 conorm conorm NOUN cana-761 43 14 , , PUNCT cana-761 43 15 respectively respectively ADV cana-761 43 16 ) ) PUNCT cana-761 43 17 . . PUNCT cana-761 44 1 definition definition NOUN cana-761 44 2 2.2 2.2 NUM cana-761 44 3 ( ( PUNCT cana-761 44 4 [ [ X cana-761 44 5 1 1 NUM cana-761 44 6 ] ] NUM cana-761 44 7 ) ) PUNCT cana-761 44 8 . . PUNCT cana-761 45 1 a a DET cana-761 45 2 triple triple ADJ cana-761 45 3 ( ( PUNCT cana-761 45 4 ) ) PUNCT cana-761 45 5 , , PUNCT cana-761 45 6 , , PUNCT cana-761 45 7 x x X cana-761 45 8 w w PROPN cana-761 45 9 t t PROPN cana-761 45 10 is be AUX cana-761 45 11 called call VERB cana-761 45 12 a a DET cana-761 45 13 rfms rfms NOUN cana-761 45 14 if if SCONJ cana-761 45 15 x x PRON cana-761 45 16 is be AUX cana-761 45 17 a a DET cana-761 45 18 nonempty nonempty ADJ cana-761 45 19 set set NOUN cana-761 45 20 , , PUNCT cana-761 45 21 t t PROPN cana-761 45 22 is be AUX cana-761 45 23 a a DET cana-761 45 24 continuous continuous ADJ cana-761 45 25 t t NOUN cana-761 45 26 - - PUNCT cana-761 45 27 conorm conorm NOUN cana-761 45 28 , , PUNCT cana-761 45 29 and and CCONJ cana-761 45 30 ( ( PUNCT cana-761 45 31 ) ) PUNCT cana-761 45 32   PROPN cana-761 45 33 2 2 NOUN cana-761 45 34     SPACE cana-761 45 35 : : PUNCT cana-761 45 36     SPACE cana-761 45 37 0 0 NUM cana-761 45 38 , , PUNCT cana-761 45 39     SPACE cana-761 45 40 0,1 0,1 NUM cana-761 45 41     SPACE cana-761 45 42 w w NOUN cana-761 45 43 x x PUNCT cana-761 45 44   NOUN cana-761 46 1 + + NOUN cana-761 46 2   X cana-761 46 3 → → PUNCT cana-761 46 4 be be AUX cana-761 46 5 a a DET cana-761 46 6 rf rf NOUN cana-761 46 7 satisfying satisfy VERB cana-761 46 8 the the DET cana-761 46 9 following follow VERB cana-761 46 10 conditions condition NOUN cana-761 46 11 : : PUNCT cana-761 46 12 ( ( PUNCT cana-761 46 13 rf rf NOUN cana-761 46 14 1 1 NUM cana-761 46 15 ) ) PUNCT cana-761 46 16 ( ( PUNCT cana-761 46 17 ) ) PUNCT cana-761 46 18 , , PUNCT cana-761 46 19     SPACE cana-761 46 20 , , PUNCT cana-761 46 21         SPACE cana-761 46 22 1w 1w VERB cana-761 46 23 q q PROPN cana-761 46 24 r r NOUN cana-761 46 25 t t X cana-761 46 26   PROPN cana-761 46 27 for for ADP cana-761 46 28 all all PRON cana-761 46 29 , , PUNCT cana-761 46 30     SPACE cana-761 46 31 q q NOUN cana-761 46 32 r r NOUN cana-761 46 33 x x PROPN cana-761 46 34 and and CCONJ cana-761 46 35         SPACE cana-761 46 36 0t 0t NOUN cana-761 46 37 ; ; PUNCT cana-761 46 38 ( ( PUNCT cana-761 46 39 rf rf NOUN cana-761 46 40 2 2 NUM cana-761 46 41 ) ) PUNCT cana-761 46 42 ( ( PUNCT cana-761 46 43 ) ) PUNCT cana-761 46 44 , , PUNCT cana-761 46 45     SPACE cana-761 46 46 , , PUNCT cana-761 46 47             SPACE cana-761 46 48 0 0 PROPN cana-761 46 49 ( ( PUNCT cana-761 46 50         SPACE cana-761 46 51 0)w 0)w PROPN cana-761 46 52 q q NOUN cana-761 47 1 r r NOUN cana-761 47 2 t t X cana-761 47 3 t= t= NOUN cana-761 47 4   INTJ cana-761 47 5 if if SCONJ cana-761 48 1 and and CCONJ cana-761 48 2 only only ADV cana-761 48 3 if if SCONJ cana-761 48 4         SPACE cana-761 48 5 q q X cana-761 48 6 r= r= ADJ cana-761 48 7 ; ; PUNCT cana-761 48 8 ( ( PUNCT cana-761 48 9 rf rf NOUN cana-761 48 10 3 3 NUM cana-761 48 11 ) ) PUNCT cana-761 48 12 ( ( PUNCT cana-761 48 13 ) ) PUNCT cana-761 48 14 ( ( PUNCT cana-761 48 15 ) ) PUNCT cana-761 48 16 , , PUNCT cana-761 48 17     SPACE cana-761 48 18 , , PUNCT cana-761 48 19             SPACE cana-761 48 20 , , PUNCT cana-761 48 21     SPACE cana-761 48 22 , , PUNCT cana-761 48 23     SPACE cana-761 48 24 w w NOUN cana-761 48 25 q q NOUN cana-761 48 26 r r NOUN cana-761 48 27 t t NOUN cana-761 48 28 w w NOUN cana-761 48 29 r r NOUN cana-761 48 30 q q NOUN cana-761 48 31 t= t= NOUN cana-761 48 32 for for ADP cana-761 48 33 all all PRON cana-761 48 34 , , PUNCT cana-761 48 35     SPACE cana-761 48 36 q q NOUN cana-761 48 37 r r NOUN cana-761 48 38 x x PROPN cana-761 48 39 and and CCONJ cana-761 48 40         SPACE cana-761 48 41 0t 0t NOUN cana-761 48 42 ; ; PUNCT cana-761 48 43 communications communication NOUN cana-761 48 44 on on ADP cana-761 48 45 applied apply VERB cana-761 48 46 nonlinear nonlinear ADJ cana-761 48 47 analysis analysis NOUN cana-761 48 48 issn issn NOUN cana-761 48 49 : : PUNCT cana-761 48 50 1074 1074 NUM cana-761 48 51 - - PUNCT cana-761 48 52 133x 133x NUM cana-761 48 53 vol vol NOUN cana-761 48 54 31 31 NUM cana-761 48 55 no no NOUN cana-761 48 56 . . PUNCT cana-761 49 1 3s 3s NUM cana-761 49 2 ( ( PUNCT cana-761 49 3 2024 2024 NUM cana-761 49 4 ) ) PUNCT cana-761 49 5 229 229 NUM cana-761 49 6 https://internationalpubls.com https://internationalpubls.com X cana-761 49 7 ( ( PUNCT cana-761 49 8 rf rf NOUN cana-761 49 9 4 4 NUM cana-761 49 10 ) ) PUNCT cana-761 49 11 ( ( PUNCT cana-761 49 12 ) ) PUNCT cana-761 49 13 ( ( PUNCT cana-761 49 14 ) ) PUNCT cana-761 49 15 ( ( PUNCT cana-761 49 16 ) ) PUNCT cana-761 49 17 ( ( PUNCT cana-761 49 18 ) ) PUNCT cana-761 49 19 , , PUNCT cana-761 49 20     SPACE cana-761 49 21 , , PUNCT cana-761 49 22                 SPACE cana-761 49 23 , , PUNCT cana-761 49 24     SPACE cana-761 49 25 , , PUNCT cana-761 49 26     SPACE cana-761 49 27 , , PUNCT cana-761 49 28 , , PUNCT cana-761 49 29     SPACE cana-761 49 30 , , PUNCT cana-761 49 31     SPACE cana-761 49 32 w w NOUN cana-761 49 33 q q VERB cana-761 49 34 l l NOUN cana-761 49 35 t t NOUN cana-761 49 36 s s PROPN cana-761 49 37 t t PROPN cana-761 49 38 w w NOUN cana-761 49 39 q q NOUN cana-761 49 40 r r NOUN cana-761 49 41 t t NOUN cana-761 49 42 w w NOUN cana-761 49 43 r r NOUN cana-761 49 44 l l NOUN cana-761 49 45 s+ s+ PUNCT cana-761 49 46   PROPN cana-761 49 47 for for ADP cana-761 49 48 all all PRON cana-761 49 49 , , PUNCT cana-761 49 50     SPACE cana-761 49 51 , , PUNCT cana-761 49 52 q q NOUN cana-761 49 53 r r NOUN cana-761 49 54 l l NOUN cana-761 49 55 x x PROPN cana-761 49 56 and and CCONJ cana-761 49 57 , , PUNCT cana-761 49 58         SPACE cana-761 49 59 0 0 NUM cana-761 49 60 t t NOUN cana-761 49 61 s s NOUN cana-761 49 62 ; ; PUNCT cana-761 49 63 ( ( PUNCT cana-761 49 64 rf rf NUM cana-761 49 65 5 5 NUM cana-761 49 66 ) ) PUNCT cana-761 49 67 ( ( PUNCT cana-761 49 68 ) ) PUNCT cana-761 49 69 ( ( PUNCT cana-761 49 70 ) ) PUNCT cana-761 49 71   PROPN cana-761 49 72   PROPN cana-761 49 73 , , PUNCT cana-761 49 74     SPACE cana-761 49 75 , , PUNCT cana-761 49 76         SPACE cana-761 49 77 : : PUNCT cana-761 49 78     SPACE cana-761 49 79 0 0 NUM cana-761 49 80 , , PUNCT cana-761 49 81     SPACE cana-761 49 82 0,1 0,1 NUM cana-761 49 83     SPACE cana-761 49 84 w w NOUN cana-761 49 85 q q NOUN cana-761 49 86 r r NOUN cana-761 50 1 − − NOUN cana-761 50 2 + + NOUN cana-761 50 3   NOUN cana-761 50 4 → → PUNCT cana-761 50 5 is be AUX cana-761 50 6 continuous continuous ADJ cana-761 50 7 for for ADP cana-761 50 8 all all PRON cana-761 50 9 , , PUNCT cana-761 50 10     SPACE cana-761 50 11 q q NOUN cana-761 50 12 r r NOUN cana-761 50 13 x x PROPN cana-761 50 14 . . PUNCT cana-761 51 1 if if SCONJ cana-761 51 2 ( ( PUNCT cana-761 51 3 rf rf NOUN cana-761 51 4 4 4 NUM cana-761 51 5 ) ) PUNCT cana-761 51 6 is be AUX cana-761 51 7 replaced replace VERB cana-761 51 8 by by ADP cana-761 51 9 the the DET cana-761 51 10 following follow VERB cana-761 51 11 condition condition NOUN cana-761 51 12 : : PUNCT cana-761 51 13 ( ( PUNCT cana-761 51 14 rf rf NOUN cana-761 51 15 4 4 NUM cana-761 51 16 ) ) PUNCT cana-761 51 17 ’ ' PUNCT cana-761 51 18 ( ( PUNCT cana-761 51 19 ) ) PUNCT cana-761 51 20 ( ( PUNCT cana-761 51 21 ) ) PUNCT cana-761 51 22 ( ( PUNCT cana-761 51 23 ) ) PUNCT cana-761 51 24 ( ( PUNCT cana-761 51 25 ) ) PUNCT cana-761 51 26 , , PUNCT cana-761 51 27     SPACE cana-761 51 28 , , PUNCT cana-761 51 29         SPACE cana-761 51 30 , , PUNCT cana-761 51 31     SPACE cana-761 51 32 , , PUNCT cana-761 51 33     SPACE cana-761 51 34 , , PUNCT cana-761 51 35 , , PUNCT cana-761 51 36     SPACE cana-761 51 37 , , PUNCT cana-761 51 38     SPACE cana-761 51 39 w w NOUN cana-761 51 40 q q VERB cana-761 51 41 l l NOUN cana-761 51 42 t t NOUN cana-761 52 1 t t PROPN cana-761 52 2 w w NOUN cana-761 52 3 q q NOUN cana-761 52 4 r r NOUN cana-761 52 5 t t NOUN cana-761 52 6 w w NOUN cana-761 52 7 r r NOUN cana-761 52 8 l l NOUN cana-761 52 9 t t NOUN cana-761 52 10 for for ADP cana-761 52 11 all all PRON cana-761 52 12 , , PUNCT cana-761 52 13     SPACE cana-761 52 14 , , PUNCT cana-761 52 15 q q NOUN cana-761 52 16 r r NOUN cana-761 52 17 l l NOUN cana-761 52 18 x x NOUN cana-761 52 19 and and CCONJ cana-761 52 20 0 0 NUM cana-761 52 21 t t NOUN cana-761 52 22   X cana-761 52 23 ; ; PUNCT cana-761 52 24 then then ADV cana-761 52 25 ( ( PUNCT cana-761 52 26 ) ) PUNCT cana-761 52 27 , , PUNCT cana-761 52 28 , , PUNCT cana-761 52 29     SPACE cana-761 52 30 x x PRON cana-761 52 31 w w PROPN cana-761 52 32 t t PROPN cana-761 52 33 is be AUX cana-761 52 34 called call VERB cana-761 52 35 a a DET cana-761 52 36 strong strong ADJ cana-761 52 37 rfms rfms NOUN cana-761 52 38 . . PUNCT cana-761 53 1 moreover moreover ADV cana-761 53 2 , , PUNCT cana-761 53 3 if if SCONJ cana-761 53 4 ( ( PUNCT cana-761 53 5 ) ) PUNCT cana-761 53 6 , , PUNCT cana-761 53 7 , , PUNCT cana-761 53 8     SPACE cana-761 53 9 x x VERB cana-761 53 10 w w PROPN cana-761 53 11 t t PROPN cana-761 53 12 is be AUX cana-761 53 13 a a DET cana-761 53 14 rfms rfms NOUN cana-761 53 15 , , PUNCT cana-761 53 16 then then ADV cana-761 53 17 w w PROPN cana-761 53 18 is be AUX cana-761 53 19 a a DET cana-761 53 20 continuous continuous ADJ cana-761 53 21 function function NOUN cana-761 53 22 on on ADP cana-761 53 23 ( ( PUNCT cana-761 53 24 ) ) PUNCT cana-761 53 25 2 2 NUM cana-761 53 26 0,x 0,x NOUN cana-761 53 27   VERB cana-761 54 1 + + NOUN cana-761 54 2   NOUN cana-761 54 3 and and CCONJ cana-761 54 4 ( ( PUNCT cana-761 54 5 ) ) PUNCT cana-761 54 6 , , PUNCT cana-761 54 7     SPACE cana-761 54 8 , , PUNCT cana-761 54 9     SPACE cana-761 54 10 w w NOUN cana-761 54 11 q q NOUN cana-761 54 12 r r NOUN cana-761 54 13 − − PROPN cana-761 54 14 is be AUX cana-761 54 15 non non ADJ cana-761 54 16 - - ADJ cana-761 54 17 increasing increase VERB cana-761 54 18 for for ADP cana-761 54 19 all all PRON cana-761 54 20 , , PUNCT cana-761 54 21     SPACE cana-761 54 22 q q NOUN cana-761 55 1 r r NOUN cana-761 55 2 x x PROPN cana-761 55 3 . . PUNCT cana-761 56 1 in in ADP cana-761 56 2 the the DET cana-761 56 3 sequel sequel NOUN cana-761 56 4 , , PUNCT cana-761 56 5 unless unless SCONJ cana-761 56 6 there there PRON cana-761 56 7 is be VERB cana-761 56 8 a a DET cana-761 56 9 special special ADJ cana-761 56 10 explanation explanation NOUN cana-761 56 11 , , PUNCT cana-761 56 12 we we PRON cana-761 56 13 always always ADV cana-761 56 14 denote denote VERB cana-761 56 15 by by ADP cana-761 56 16 n n PRON cana-761 56 17 , , PUNCT cana-761 56 18 the the DET cana-761 56 19 set set NOUN cana-761 56 20 of of ADP cana-761 56 21 all all DET cana-761 56 22 positive positive ADJ cana-761 56 23 integers integer NOUN cana-761 56 24 ; ; PUNCT cana-761 56 25 0n 0n NUM cana-761 56 26 , , PUNCT cana-761 56 27 the the DET cana-761 56 28 set set NOUN cana-761 56 29 of of ADP cana-761 56 30 all all DET cana-761 56 31 nonnegative nonnegative ADJ cana-761 56 32 integers integer NOUN cana-761 56 33 ; ; PUNCT cana-761 57 1 r r NOUN cana-761 57 2 , , PUNCT cana-761 57 3 the the DET cana-761 57 4 set set NOUN cana-761 57 5 of of ADP cana-761 57 6 all all DET cana-761 57 7 real real ADJ cana-761 57 8 numbers number NOUN cana-761 57 9 and and CCONJ cana-761 57 10 + + NOUN cana-761 57 11 r r NOUN cana-761 57 12 , , PUNCT cana-761 57 13 the the DET cana-761 57 14 set set NOUN cana-761 57 15 of of ADP cana-761 57 16 all all DET cana-761 57 17 positive positive ADJ cana-761 57 18 real real ADJ cana-761 57 19 numbers number NOUN cana-761 57 20 . . PUNCT cana-761 58 1 definition definition NOUN cana-761 58 2 2.3 2.3 NUM cana-761 58 3 ( ( PUNCT cana-761 58 4 [ [ X cana-761 58 5 12 12 NUM cana-761 58 6 ] ] PUNCT cana-761 58 7 ) ) PUNCT cana-761 58 8 . . PUNCT cana-761 59 1 let let VERB cana-761 59 2 ( ( PUNCT cana-761 59 3 ) ) PUNCT cana-761 59 4 , , PUNCT cana-761 59 5 , , PUNCT cana-761 59 6 x x X cana-761 59 7 w w PROPN cana-761 59 8 t t PROPN cana-761 59 9 be be AUX cana-761 59 10 a a DET cana-761 59 11 rfms rfms NOUN cana-761 59 12 and and CCONJ cana-761 59 13   PROPN cana-761 59 14 n n PROPN cana-761 59 15 n n CCONJ cana-761 59 16 q q PROPN cana-761 59 17 n n NOUN cana-761 59 18 be be AUX cana-761 59 19 a a DET cana-761 59 20 sequence sequence NOUN cana-761 59 21 in in ADP cana-761 59 22 x x X cana-761 59 23 . . PUNCT cana-761 60 1 then then ADV cana-761 60 2 , , PUNCT cana-761 60 3 we we PRON cana-761 60 4 say say VERB cana-761 60 5 the the DET cana-761 60 6 following follow VERB cana-761 60 7 : : PUNCT cana-761 60 8 ( ( PUNCT cana-761 60 9 i i NOUN cana-761 60 10 ) ) PUNCT cana-761 60 11   PROPN cana-761 60 12 n n VERB cana-761 60 13 n n CCONJ cana-761 60 14 q q PROPN cana-761 60 15 n n PROPN cana-761 60 16 converges converge VERB cana-761 60 17 to to ADP cana-761 60 18 q q PROPN cana-761 60 19 x x PROPN cana-761 60 20 ( ( PUNCT cana-761 60 21 say say VERB cana-761 60 22 lim lim PROPN cana-761 60 23 n n CCONJ cana-761 60 24 n n PROPN cana-761 60 25 q q NOUN cana-761 60 26 q q NOUN cana-761 60 27 → → PUNCT cana-761 60 28 = = PUNCT cana-761 60 29 ) ) PUNCT cana-761 60 30 , , PUNCT cana-761 60 31 if if SCONJ cana-761 60 32 ( ( PUNCT cana-761 60 33 ) ) PUNCT cana-761 60 34 lim lim PROPN cana-761 60 35     SPACE cana-761 60 36 , , PUNCT cana-761 60 37 , , PUNCT cana-761 60 38 0n 0n NOUN cana-761 60 39 n n PROPN cana-761 60 40 w w PROPN cana-761 60 41 q q X cana-761 60 42 q q PROPN cana-761 60 43 t t NOUN cana-761 60 44 → → PUNCT cana-761 60 45 = = PUNCT cana-761 60 46 for for ADP cana-761 60 47 any any DET cana-761 60 48 0 0 NUM cana-761 60 49 t t NOUN cana-761 60 50   NOUN cana-761 60 51 ; ; PUNCT cana-761 60 52 ( ( PUNCT cana-761 60 53 ii ii X cana-761 60 54 ) ) PUNCT cana-761 60 55   PROPN cana-761 60 56 n n VERB cana-761 60 57 n n CCONJ cana-761 60 58 q q PROPN cana-761 60 59 n n NOUN cana-761 60 60 is be AUX cana-761 60 61 a a DET cana-761 60 62 cauchy cauchy ADJ cana-761 60 63 sequence sequence NOUN cana-761 60 64 if if SCONJ cana-761 60 65 , , PUNCT cana-761 60 66 for for ADP cana-761 60 67 any any DET cana-761 60 68 ( ( PUNCT cana-761 60 69 ) ) SYM cana-761 60 70 0,1 0,1 NUM cana-761 60 71     SPACE cana-761 60 72   PROPN cana-761 60 73   NOUN cana-761 60 74 and and CCONJ cana-761 60 75 0 0 NUM cana-761 60 76 t t NOUN cana-761 60 77   INTJ cana-761 60 78 , , PUNCT cana-761 60 79 there there PRON cana-761 60 80 exists exist VERB cana-761 60 81 0n 0n NOUN cana-761 60 82 n n NOUN cana-761 60 83 such such ADJ cana-761 60 84 that that PRON cana-761 60 85 ( ( PUNCT cana-761 60 86 ) ) PUNCT cana-761 60 87 , , PUNCT cana-761 60 88     SPACE cana-761 60 89 , , PUNCT cana-761 60 90     SPACE cana-761 60 91 m m VERB cana-761 60 92 nw nw INTJ cana-761 60 93 q q PROPN cana-761 60 94 q q PROPN cana-761 60 95 t t PROPN cana-761 60 96   X cana-761 60 97 for for ADP cana-761 60 98 any any DET cana-761 60 99 0 0 NUM cana-761 60 100 , , PUNCT cana-761 60 101     SPACE cana-761 60 102 m m VERB cana-761 60 103 n n PRON cana-761 60 104 n n PROPN cana-761 60 105 ; ; PUNCT cana-761 60 106 ( ( PUNCT cana-761 60 107 iii iii X cana-761 60 108 ) ) PUNCT cana-761 60 109 ( ( PUNCT cana-761 60 110 ) ) PUNCT cana-761 60 111 , , PUNCT cana-761 60 112 , , PUNCT cana-761 60 113     SPACE cana-761 60 114 x x VERB cana-761 60 115 w w PROPN cana-761 60 116 t t PROPN cana-761 60 117 is be AUX cana-761 60 118 complete complete ADJ cana-761 60 119 if if SCONJ cana-761 60 120 every every DET cana-761 60 121 cauchy cauchy ADJ cana-761 60 122 sequence sequence NOUN cana-761 60 123 is be AUX cana-761 60 124 convergent convergent ADJ cana-761 60 125 . . PUNCT cana-761 61 1 definition definition NOUN cana-761 61 2 2.4 2.4 NUM cana-761 61 3 let let VERB cana-761 61 4 ( ( PUNCT cana-761 61 5 ) ) PUNCT cana-761 61 6 , , PUNCT cana-761 61 7 , , PUNCT cana-761 61 8     SPACE cana-761 61 9 x x PUNCT cana-761 61 10 w w PROPN cana-761 61 11 t t PROPN cana-761 61 12 be be AUX cana-761 61 13 a a DET cana-761 61 14 rfms rfms NOUN cana-761 61 15 and and CCONJ cana-761 61 16     SPACE cana-761 61 17 : : PUNCT cana-761 61 18         SPACE cana-761 61 19 f f X cana-761 62 1 x x PROPN cana-761 62 2 x→ x→ PROPN cana-761 62 3 a a DET cana-761 62 4 mapping mapping NOUN cana-761 62 5 . . PUNCT cana-761 63 1 then then ADV cana-761 63 2 f f PROPN cana-761 63 3 is be AUX cana-761 63 4 called call VERB cana-761 63 5 a a DET cana-761 63 6 revised revise VERB cana-761 63 7 fuzzy fuzzy ADJ cana-761 63 8 contraction contraction NOUN cana-761 63 9 if if SCONJ cana-761 63 10 there there PRON cana-761 63 11 exists exist VERB cana-761 63 12 ( ( PUNCT cana-761 63 13 ) ) PUNCT cana-761 63 14 0,1 0,1 NUM cana-761 63 15     SPACE cana-761 63 16 k k NOUN cana-761 63 17   NOUN cana-761 63 18 such such ADJ cana-761 63 19 that that SCONJ cana-761 63 20 ( ( PUNCT cana-761 63 21 ) ) PUNCT cana-761 63 22 ( ( PUNCT cana-761 63 23 ) ) PUNCT cana-761 63 24 ( ( PUNCT cana-761 63 25 ) ) PUNCT cana-761 63 26 ( ( PUNCT cana-761 63 27 ) ) PUNCT cana-761 63 28 ( ( PUNCT cana-761 63 29 ) ) PUNCT cana-761 63 30 , , PUNCT cana-761 63 31     SPACE cana-761 63 32 , , PUNCT cana-761 63 33     SPACE cana-761 63 34 , , PUNCT cana-761 63 35     SPACE cana-761 63 36 , , PUNCT cana-761 63 37     SPACE cana-761 63 38 w w PROPN cana-761 63 39 f f PROPN cana-761 64 1 q q X cana-761 65 1 f f PROPN cana-761 66 1 r r NOUN cana-761 67 1 t t PROPN cana-761 68 1 k k NOUN cana-761 68 2 w w PROPN cana-761 68 3 q q NOUN cana-761 68 4 r r PROPN cana-761 68 5 t t NOUN cana-761 68 6 ( ( PUNCT cana-761 68 7 1 1 NUM cana-761 68 8 ) ) PUNCT cana-761 68 9 for for ADP cana-761 68 10 all all PRON cana-761 68 11 , , PUNCT cana-761 68 12     SPACE cana-761 68 13 q q NOUN cana-761 68 14 r r NOUN cana-761 68 15 x x PROPN cana-761 68 16 and and CCONJ cana-761 68 17 0 0 NUM cana-761 68 18 t t NOUN cana-761 68 19   X cana-761 68 20 . . PUNCT cana-761 69 1 in in ADP cana-761 69 2 this this DET cana-761 69 3 case case NOUN cana-761 69 4 , , PUNCT cana-761 69 5 k k PROPN cana-761 69 6 is be AUX cana-761 69 7 called call VERB cana-761 69 8 the the DET cana-761 69 9 contractive contractive ADJ cana-761 69 10 constant constant NOUN cana-761 69 11 of of ADP cana-761 69 12 f f PROPN cana-761 69 13 . . PUNCT cana-761 70 1 we we PRON cana-761 70 2 say say VERB cana-761 70 3 that that SCONJ cana-761 70 4 the the DET cana-761 70 5 mapping mapping NOUN cana-761 70 6 : : PUNCT cana-761 70 7         SPACE cana-761 70 8 t t NOUN cana-761 70 9 x x X cana-761 70 10 x→ x→ PROPN cana-761 70 11 is be AUX cana-761 70 12 called call VERB cana-761 70 13 a a DET cana-761 70 14 tirado tirado NOUN cana-761 70 15 contraction contraction NOUN cana-761 70 16 if if SCONJ cana-761 70 17 there there PRON cana-761 70 18 exists exist VERB cana-761 70 19 ( ( PUNCT cana-761 70 20 ) ) PUNCT cana-761 70 21 0,1 0,1 NUM cana-761 70 22     SPACE cana-761 70 23 k k NOUN cana-761 70 24   NOUN cana-761 70 25 such such ADJ cana-761 70 26 that that SCONJ cana-761 70 27 ( ( PUNCT cana-761 70 28 ) ) PUNCT cana-761 70 29 ( ( PUNCT cana-761 70 30 ) ) PUNCT cana-761 70 31 ( ( PUNCT cana-761 70 32 ) ) PUNCT cana-761 70 33 , , PUNCT cana-761 70 34     SPACE cana-761 70 35 , , PUNCT cana-761 70 36     SPACE cana-761 70 37 , , PUNCT cana-761 70 38     SPACE cana-761 70 39 , , PUNCT cana-761 70 40     SPACE cana-761 70 41 w w NOUN cana-761 70 42 tq tq ADP cana-761 70 43 tr tr PROPN cana-761 70 44 t t PROPN cana-761 70 45 k k PROPN cana-761 70 46 w w PROPN cana-761 70 47 q q NOUN cana-761 70 48 r r NOUN cana-761 70 49 t t NOUN cana-761 70 50 for for ADP cana-761 70 51 all all PRON cana-761 70 52 , , PUNCT cana-761 70 53     SPACE cana-761 70 54 q q NOUN cana-761 70 55 r r NOUN cana-761 70 56 x x PROPN cana-761 70 57 and and CCONJ cana-761 70 58 0 0 NUM cana-761 70 59 t t NOUN cana-761 70 60   X cana-761 70 61 . . PUNCT cana-761 71 1 ( ( PUNCT cana-761 71 2 2 2 NUM cana-761 71 3 ) ) PUNCT cana-761 71 4 3 3 NUM cana-761 71 5 . . PUNCT cana-761 72 1 methods method NOUN cana-761 72 2 lemma lemma PROPN cana-761 72 3 3.1 3.1 NUM cana-761 72 4 . . PUNCT cana-761 73 1 let let VERB cana-761 73 2 ( ( PUNCT cana-761 73 3 ) ) PUNCT cana-761 73 4 , , PUNCT cana-761 73 5 , , PUNCT cana-761 73 6 x x X cana-761 73 7 w w PROPN cana-761 73 8 t t PROPN cana-761 73 9 be be AUX cana-761 73 10 a a DET cana-761 73 11 rfms rfms NOUN cana-761 73 12 and and CCONJ cana-761 73 13   PROPN cana-761 73 14 nq nq PROPN cana-761 73 15 be be AUX cana-761 73 16 a a DET cana-761 73 17 sequence sequence NOUN cana-761 73 18 in in ADP cana-761 73 19 x x INTJ cana-761 73 20 such such ADJ cana-761 73 21 that that PRON cana-761 73 22 for for ADP cana-761 73 23 each each DET cana-761 73 24 n n PROPN cana-761 73 25 n n PROPN cana-761 73 26 , , PUNCT cana-761 73 27 communications communication NOUN cana-761 73 28 on on ADP cana-761 73 29 applied apply VERB cana-761 73 30 nonlinear nonlinear ADJ cana-761 73 31 analysis analysis NOUN cana-761 73 32 issn issn NOUN cana-761 73 33 : : PUNCT cana-761 73 34 1074 1074 NUM cana-761 73 35 - - PUNCT cana-761 73 36 133x 133x NUM cana-761 73 37 vol vol NOUN cana-761 73 38 31 31 NUM cana-761 73 39 no no NOUN cana-761 73 40 . . PUNCT cana-761 74 1 3s 3s NUM cana-761 74 2 ( ( PUNCT cana-761 74 3 2024 2024 NUM cana-761 74 4 ) ) PUNCT cana-761 74 5 230 230 NUM cana-761 74 6 https://internationalpubls.com https://internationalpubls.com X cana-761 74 7 ( ( PUNCT cana-761 74 8 ) ) PUNCT cana-761 74 9 1 1 NUM cana-761 74 10 0 0 NUM cana-761 74 11 lim lim PROPN cana-761 74 12 , , PUNCT cana-761 74 13     SPACE cana-761 74 14 , , PUNCT cana-761 74 15     SPACE cana-761 74 16 1n 1n PROPN cana-761 74 17 n n ADP cana-761 74 18 t t NOUN cana-761 74 19 w w NOUN cana-761 74 20 q q X cana-761 74 21 q q PROPN cana-761 74 22 t t NOUN cana-761 74 23 + + CCONJ cana-761 75 1 + + CCONJ cana-761 75 2 → → SYM cana-761 75 3   PROPN cana-761 75 4 , , PUNCT cana-761 75 5 and and CCONJ cana-761 75 6 for for ADP cana-761 75 7 any any DET cana-761 75 8         SPACE cana-761 75 9 0t 0t NOUN cana-761 75 10 , , PUNCT cana-761 75 11 ( ( PUNCT cana-761 75 12 3 3 X cana-761 75 13 ) ) PUNCT cana-761 75 14 ( ( PUNCT cana-761 75 15 ) ) PUNCT cana-761 75 16 1lim 1lim NUM cana-761 75 17 , , PUNCT cana-761 75 18     SPACE cana-761 75 19 , , PUNCT cana-761 75 20     SPACE cana-761 75 21 0n 0n NOUN cana-761 75 22 n n PROPN cana-761 75 23 t t PROPN cana-761 75 24 w w NOUN cana-761 75 25 q q X cana-761 75 26 q q X cana-761 75 27 t+ t+ NOUN cana-761 75 28 → → PUNCT cana-761 75 29 = = PRON cana-761 75 30 . . PUNCT cana-761 76 1 ( ( PUNCT cana-761 76 2 4 4 X cana-761 76 3 ) ) PUNCT cana-761 76 4 if if SCONJ cana-761 76 5   PROPN cana-761 76 6 nq nq PROPN cana-761 76 7 is be AUX cana-761 76 8 not not PART cana-761 76 9 a a DET cana-761 76 10 cauchy cauchy ADJ cana-761 76 11 sequence sequence NOUN cana-761 76 12 in in ADP cana-761 76 13 x x SYM cana-761 76 14 , , PUNCT cana-761 76 15 then then ADV cana-761 76 16 there there PRON cana-761 76 17 exist exist VERB cana-761 76 18 ( ( PUNCT cana-761 76 19 ) ) PUNCT cana-761 76 20 0 0 NUM cana-761 76 21     SPACE cana-761 76 22 0,1 0,1 NUM cana-761 76 23     SPACE cana-761 76 24 , , PUNCT cana-761 76 25     SPACE cana-761 76 26 0t 0t NOUN cana-761 76 27   NOUN cana-761 76 28   VERB cana-761 76 29 , , PUNCT cana-761 76 30 and and CCONJ cana-761 76 31 two two NUM cana-761 76 32 sequences sequence NOUN cana-761 76 33 of of ADP cana-761 76 34 positive positive ADJ cana-761 76 35 integers integer NOUN cana-761 76 36   NOUN cana-761 76 37 kn kn NOUN cana-761 76 38 , , PUNCT cana-761 76 39   PROPN cana-761 76 40 km km ADJ cana-761 76 41 , , PUNCT cana-761 76 42     SPACE cana-761 76 43 , , PUNCT cana-761 76 44     SPACE cana-761 76 45 k k PROPN cana-761 77 1 kn kn PROPN cana-761 77 2 m m PROPN cana-761 77 3 k k PROPN cana-761 77 4 k k PROPN cana-761 77 5 n n PROPN cana-761 77 6   VERB cana-761 77 7   PROPN cana-761 77 8 , , PUNCT cana-761 77 9 such such ADJ cana-761 77 10 that that SCONJ cana-761 77 11 the the DET cana-761 77 12 following follow VERB cana-761 77 13 sequences sequence NOUN cana-761 77 14 ( ( PUNCT cana-761 77 15 ) ) PUNCT cana-761 77 16   PROPN cana-761 77 17   PROPN cana-761 77 18 ( ( PUNCT cana-761 77 19 ) ) PUNCT cana-761 77 20   PROPN cana-761 77 21   PROPN cana-761 77 22 ( ( PUNCT cana-761 77 23 ) ) PUNCT cana-761 77 24   NOUN cana-761 77 25 0 0 VERB cana-761 78 1 1 1 NUM cana-761 78 2 0 0 NUM cana-761 78 3 1 1 NUM cana-761 78 4 0 0 NUM cana-761 78 5 , , PUNCT cana-761 78 6 , , PUNCT cana-761 78 7 , , PUNCT cana-761 78 8 , , PUNCT cana-761 78 9 , , PUNCT cana-761 78 10 , , PUNCT cana-761 78 11 , , PUNCT cana-761 78 12 , , PUNCT cana-761 78 13 , , PUNCT cana-761 78 14 mk mk PROPN cana-761 78 15 nk nk PROPN cana-761 78 16 mk mk PROPN cana-761 78 17 nk nk PROPN cana-761 78 18 mk mk PROPN cana-761 78 19 nkw nkw PROPN cana-761 78 20 q q PROPN cana-761 78 21 q q PROPN cana-761 79 1 t t PROPN cana-761 80 1 w w PROPN cana-761 81 1 q q X cana-761 82 1 q q PROPN cana-761 83 1 t t PROPN cana-761 83 2 w w NOUN cana-761 83 3 q q X cana-761 83 4 q q X cana-761 83 5 t+ t+ NOUN cana-761 83 6 − − PROPN cana-761 83 7 ( ( PUNCT cana-761 83 8 ) ) PUNCT cana-761 83 9   NOUN cana-761 83 10 1 1 NOUN cana-761 83 11 1 1 NUM cana-761 83 12 0 0 NUM cana-761 83 13 , , PUNCT cana-761 83 14 , , PUNCT cana-761 83 15 mk mk PROPN cana-761 83 16 nkw nkw PROPN cana-761 83 17 q q PROPN cana-761 83 18 q q X cana-761 83 19 t− t− PROPN cana-761 84 1 + + PROPN cana-761 84 2 , , PUNCT cana-761 84 3 ( ( PUNCT cana-761 84 4 ) ) PUNCT cana-761 84 5   NOUN cana-761 84 6 1 1 NOUN cana-761 84 7 1 1 NUM cana-761 84 8 0 0 NUM cana-761 84 9 , , PUNCT cana-761 84 10 , , PUNCT cana-761 84 11 mk mk PROPN cana-761 84 12 nkw nkw PROPN cana-761 84 13 q q PROPN cana-761 84 14 q q X cana-761 84 15 t+ t+ X cana-761 84 16 + + CCONJ cana-761 84 17 tend tend VERB cana-761 84 18 to to PART cana-761 84 19   VERB cana-761 84 20 as as ADP cana-761 84 21 k k PROPN cana-761 84 22 → → PUNCT cana-761 84 23 . . PUNCT cana-761 85 1 proof proof NOUN cana-761 85 2 . . PUNCT cana-761 86 1 let let VERB cana-761 86 2   PRON cana-761 86 3 nq nq PROPN cana-761 86 4 be be AUX cana-761 86 5 a a DET cana-761 86 6 sequence sequence NOUN cana-761 86 7 in in ADP cana-761 86 8 x x SYM cana-761 86 9 , , PUNCT cana-761 86 10 which which PRON cana-761 86 11 is be AUX cana-761 86 12 not not PART cana-761 86 13 a a DET cana-761 86 14 cauchy cauchy ADJ cana-761 86 15 sequence sequence NOUN cana-761 86 16 . . PUNCT cana-761 87 1 then then ADV cana-761 87 2 , , PUNCT cana-761 87 3 by by ADP cana-761 87 4 definition definition NOUN cana-761 87 5 2.3 2.3 NUM cana-761 87 6 , , PUNCT cana-761 87 7 there there PRON cana-761 87 8 exists exist VERB cana-761 87 9 ( ( PUNCT cana-761 87 10 ) ) PUNCT cana-761 87 11 00,1 00,1 NOUN cana-761 87 12     SPACE cana-761 87 13 , , PUNCT cana-761 87 14 0 0 NUM cana-761 87 15     SPACE cana-761 87 16 t t PRON cana-761 87 17   NOUN cana-761 87 18   NOUN cana-761 87 19 and and CCONJ cana-761 87 20 sequence sequence NOUN cana-761 87 21   PROPN cana-761 87 22 kn kn NOUN cana-761 87 23 , , PUNCT cana-761 87 24   PROPN cana-761 87 25 km km ADJ cana-761 87 26 , , PUNCT cana-761 87 27     SPACE cana-761 87 28 , , PUNCT cana-761 87 29     SPACE cana-761 87 30 k k PROPN cana-761 88 1 kn kn PROPN cana-761 88 2 m m PROPN cana-761 88 3 k k PROPN cana-761 88 4 k k PROPN cana-761 88 5 n n PROPN cana-761 88 6   VERB cana-761 88 7   PROPN cana-761 88 8 , , PUNCT cana-761 88 9 such such ADJ cana-761 88 10 that that PRON cana-761 88 11 for for ADP cana-761 88 12 any any DET cana-761 88 13 k k PROPN cana-761 88 14 n n PROPN cana-761 88 15 , , PUNCT cana-761 88 16 we we PRON cana-761 88 17 have have VERB cana-761 88 18 ( ( PUNCT cana-761 88 19 ) ) PUNCT cana-761 88 20 0 0 NUM cana-761 88 21 , , PUNCT cana-761 88 22 , , PUNCT cana-761 88 23 mk mk PROPN cana-761 88 24 nkw nkw PROPN cana-761 88 25 q q PROPN cana-761 88 26 q q PROPN cana-761 88 27 t t NOUN cana-761 88 28   NOUN cana-761 88 29 . . PUNCT cana-761 89 1 ( ( PUNCT cana-761 89 2 5 5 NUM cana-761 89 3 ) ) PUNCT cana-761 89 4 and and CCONJ cana-761 89 5 ( ( PUNCT cana-761 89 6 ) ) PUNCT cana-761 89 7 0 0 NUM cana-761 89 8 , , PUNCT cana-761 89 9 , , PUNCT cana-761 89 10 mk mk PROPN cana-761 89 11 nkw nkw PROPN cana-761 89 12 q q PROPN cana-761 89 13 q q PROPN cana-761 89 14 t t PROPN cana-761 89 15   PROPN cana-761 89 16 . . PUNCT cana-761 90 1 ( ( PUNCT cana-761 90 2 6 6 NUM cana-761 90 3 ) ) PUNCT cana-761 90 4 clearly clearly ADV cana-761 90 5 , , PUNCT cana-761 90 6 by by ADP cana-761 90 7 ( ( PUNCT cana-761 90 8 5 5 NUM cana-761 90 9 ) ) PUNCT cana-761 90 10 , , PUNCT cana-761 90 11 one one PRON cana-761 90 12 has have VERB cana-761 90 13 ( ( PUNCT cana-761 90 14 ) ) PUNCT cana-761 90 15 0lim 0lim PROPN cana-761 90 16 inf inf NOUN cana-761 90 17 , , PUNCT cana-761 90 18 , , PUNCT cana-761 90 19 mk mk PROPN cana-761 91 1 nk nk PROPN cana-761 91 2 k k PROPN cana-761 92 1 w w PROPN cana-761 92 2 q q X cana-761 92 3 q q PROPN cana-761 92 4 t t PROPN cana-761 92 5   PROPN cana-761 92 6 → → PUNCT cana-761 92 7   INTJ cana-761 92 8 . . PUNCT cana-761 93 1 ( ( PUNCT cana-761 93 2 7 7 X cana-761 93 3 ) ) PUNCT cana-761 93 4 using use VERB cana-761 93 5 condition condition NOUN cana-761 93 6 ( ( PUNCT cana-761 93 7 rf rf NOUN cana-761 93 8 4 4 NUM cana-761 93 9 ) ) PUNCT cana-761 93 10 , , PUNCT cana-761 93 11 for for ADP cana-761 93 12 any any DET cana-761 93 13 k k PROPN cana-761 93 14 n n PROPN cana-761 93 15 and and CCONJ cana-761 93 16 ( ( PUNCT cana-761 93 17 ) ) PUNCT cana-761 93 18 00 00 NUM cana-761 93 19 , , PUNCT cana-761 93 20     SPACE cana-761 93 21 p p PROPN cana-761 93 22 t t PROPN cana-761 93 23 , , PUNCT cana-761 93 24 it it PRON cana-761 93 25 is be AUX cana-761 93 26 not not PART cana-761 93 27 hard hard ADJ cana-761 93 28 to to PART cana-761 93 29 verify verify VERB cana-761 93 30 that that SCONJ cana-761 93 31 ( ( PUNCT cana-761 93 32 ) ) PUNCT cana-761 93 33 ( ( PUNCT cana-761 93 34 ) ) PUNCT cana-761 93 35 ( ( PUNCT cana-761 93 36 ) ) PUNCT cana-761 93 37 ( ( PUNCT cana-761 93 38 ) ) PUNCT cana-761 93 39 0 0 NUM cana-761 93 40 1 1 NUM cana-761 93 41 1 1 NUM cana-761 93 42 0 0 NUM cana-761 93 43 , , PUNCT cana-761 93 44 , , PUNCT cana-761 93 45 , , PUNCT cana-761 93 46 , , PUNCT cana-761 93 47 , , PUNCT cana-761 93 48 , , PUNCT cana-761 93 49 , , PUNCT cana-761 93 50 mk mk PROPN cana-761 93 51 nk nk PROPN cana-761 93 52 mk mk PROPN cana-761 93 53 mk mk PROPN cana-761 93 54 mk mk PROPN cana-761 93 55 nkw nkw PROPN cana-761 93 56 q q PROPN cana-761 93 57 q q PROPN cana-761 94 1 t t PROPN cana-761 94 2 t t PROPN cana-761 95 1 w w NOUN cana-761 95 2 q q PROPN cana-761 95 3 q q X cana-761 95 4 p p X cana-761 95 5 w w PROPN cana-761 95 6 q q X cana-761 95 7 q q PROPN cana-761 95 8 t t NOUN cana-761 95 9 p− p− NOUN cana-761 95 10 − − NOUN cana-761 95 11 − − PROPN cana-761 95 12 . . PUNCT cana-761 96 1 ( ( PUNCT cana-761 96 2 8) 8) NUM cana-761 96 3 note note VERB cana-761 96 4 that that SCONJ cana-761 96 5 , , PUNCT cana-761 96 6 by by ADP cana-761 96 7 ( ( PUNCT cana-761 96 8 3 3 NUM cana-761 96 9 ) ) PUNCT cana-761 96 10 and and CCONJ cana-761 96 11 ( ( PUNCT cana-761 96 12 4 4 NUM cana-761 96 13 ) ) PUNCT cana-761 96 14 , , PUNCT cana-761 96 15 it it PRON cana-761 96 16 follows follow VERB cana-761 96 17 that that SCONJ cana-761 96 18 ( ( PUNCT cana-761 96 19 ) ) SYM cana-761 96 20 1 1 X cana-761 96 21 0 0 NUM cana-761 96 22 lim lim PROPN cana-761 96 23 lim lim PROPN cana-761 96 24 , , PUNCT cana-761 96 25     SPACE cana-761 96 26 , , PUNCT cana-761 96 27     SPACE cana-761 96 28 0n 0n NOUN cana-761 96 29 n n CCONJ cana-761 96 30 n n NOUN cana-761 96 31 p p NOUN cana-761 96 32 w w NOUN cana-761 96 33 q q X cana-761 96 34 q q X cana-761 96 35 p p X cana-761 96 36 + + PROPN cana-761 96 37 + + NUM cana-761 96 38 → → NOUN cana-761 96 39 → → NOUN cana-761 96 40   NOUN cana-761 96 41   NOUN cana-761 96 42 = = SYM cana-761 96 43   PROPN cana-761 96 44   PROPN cana-761 96 45   PROPN cana-761 96 46   PROPN cana-761 96 47 .. .. PUNCT cana-761 96 48 ( ( PUNCT cana-761 96 49 9 9 X cana-761 96 50 ) ) PUNCT cana-761 96 51 if if SCONJ cana-761 96 52 we we PRON cana-761 96 53 take take VERB cana-761 96 54 0p 0p NOUN cana-761 97 1 + + PROPN cana-761 97 2 → → SYM cana-761 97 3 in in ADP cana-761 97 4 ( ( PUNCT cana-761 97 5 8) 8) NUM cana-761 97 6 , , PUNCT cana-761 97 7 then then ADV cana-761 97 8 by by ADP cana-761 97 9 ( ( PUNCT cana-761 97 10 9 9 NUM cana-761 97 11 ) ) PUNCT cana-761 97 12 , , PUNCT cana-761 97 13 ( ( PUNCT cana-761 97 14 6 6 NUM cana-761 97 15 ) ) PUNCT cana-761 97 16 and and CCONJ cana-761 97 17 the the DET cana-761 97 18 continuity continuity NOUN cana-761 97 19 of of ADP cana-761 97 20 t t PROPN cana-761 97 21 , , PUNCT cana-761 97 22 we we PRON cana-761 97 23 obtain obtain VERB cana-761 97 24 ( ( PUNCT cana-761 97 25 ) ) PUNCT cana-761 97 26 ( ( PUNCT cana-761 97 27 ) ) PUNCT cana-761 97 28 ( ( PUNCT cana-761 97 29 ) ) PUNCT cana-761 97 30 ( ( PUNCT cana-761 97 31 ) ) PUNCT cana-761 97 32 0 0 NUM cana-761 97 33 1 1 NUM cana-761 97 34 1 1 NUM cana-761 97 35 0 0 NUM cana-761 97 36 0 0 NUM cana-761 98 1 lim lim PROPN cana-761 98 2 , , PUNCT cana-761 98 3 , , PUNCT cana-761 98 4 lim lim PROPN cana-761 98 5 lim lim PROPN cana-761 98 6 , , PUNCT cana-761 98 7 , , PUNCT cana-761 98 8 , , PUNCT cana-761 98 9 , , PUNCT cana-761 98 10 , , PUNCT cana-761 98 11 mk mk PROPN cana-761 98 12 nk nk PROPN cana-761 98 13 mk mk PROPN cana-761 98 14 mk mk PROPN cana-761 98 15 mk mk PROPN cana-761 99 1 nk nk PROPN cana-761 99 2 k k PROPN cana-761 100 1 k k PROPN cana-761 101 1 p p X cana-761 101 2 w w PROPN cana-761 101 3 q q X cana-761 101 4 q q PROPN cana-761 101 5 t t NOUN cana-761 101 6 t t PROPN cana-761 101 7 w w NOUN cana-761 101 8 q q PROPN cana-761 101 9 q q X cana-761 101 10 p p X cana-761 101 11 w w PROPN cana-761 101 12 q q X cana-761 101 13 q q PROPN cana-761 101 14 t t NOUN cana-761 101 15 p p NOUN cana-761 102 1 + + CCONJ cana-761 102 2 − − PROPN cana-761 102 3 − − PROPN cana-761 102 4 → → PUNCT cana-761 102 5 → → PUNCT cana-761 102 6 → → SYM cana-761 102 7   NOUN cana-761 102 8   X cana-761 102 9 − − PROPN cana-761 102 10   PROPN cana-761 102 11   PROPN cana-761 102 12   NOUN cana-761 102 13 . . PUNCT cana-761 103 1 ( ( PUNCT cana-761 103 2 ) ) PUNCT cana-761 103 3 ( ( PUNCT cana-761 103 4 ) ) SYM cana-761 103 5 1 1 NUM cana-761 103 6 1 1 NUM cana-761 103 7 0 0 NUM cana-761 103 8 0 0 NUM cana-761 103 9 0 0 NUM cana-761 103 10 lim lim PROPN cana-761 103 11 lim lim PROPN cana-761 103 12 , , PUNCT cana-761 103 13 , , PUNCT cana-761 103 14 , , PUNCT cana-761 103 15 lim lim PROPN cana-761 103 16 lim lim PROPN cana-761 103 17 , , PUNCT cana-761 103 18 , , PUNCT cana-761 103 19 mk mk PROPN cana-761 103 20 mk mk PROPN cana-761 103 21 mk mk PROPN cana-761 104 1 nk nk PROPN cana-761 104 2 k k PROPN cana-761 104 3 kp kp PROPN cana-761 105 1 p p PROPN cana-761 105 2 t t PROPN cana-761 106 1 w w NOUN cana-761 106 2 q q PROPN cana-761 106 3 q q X cana-761 107 1 p p X cana-761 107 2 w w PROPN cana-761 107 3 q q X cana-761 107 4 q q PROPN cana-761 107 5 t t NOUN cana-761 107 6 p p NOUN cana-761 108 1 + + CCONJ cana-761 109 1 + + ADJ cana-761 109 2 − − NOUN cana-761 109 3 − − PROPN cana-761 109 4 → → PUNCT cana-761 109 5 →→ →→ NOUN cana-761 109 6 → → SYM cana-761 109 7   NOUN cana-761 109 8   PROPN cana-761 109 9   PROPN cana-761 109 10   VERB cana-761 109 11 = = NUM cana-761 109 12 − − PROPN cana-761 110 1   PROPN cana-761 110 2   PROPN cana-761 110 3   NOUN cana-761 111 1   PROPN cana-761 111 2   PROPN cana-761 111 3   PROPN cana-761 111 4   ADJ cana-761 111 5   NOUN cana-761 111 6   NOUN cana-761 111 7 ( ( PUNCT cana-761 111 8 ) ) PUNCT cana-761 111 9 ( ( PUNCT cana-761 111 10 ) ) PUNCT cana-761 111 11 1 1 NUM cana-761 111 12 00 00 NUM cana-761 111 13 , , PUNCT cana-761 111 14 lim lim PROPN cana-761 111 15 , , PUNCT cana-761 111 16 , , PUNCT cana-761 111 17 mk mk PROPN cana-761 112 1 nk nk PROPN cana-761 112 2 k k PROPN cana-761 113 1 t t PROPN cana-761 113 2 w w PROPN cana-761 113 3 q q X cana-761 113 4 q q X cana-761 113 5 t− t− PROPN cana-761 113 6 → → PUNCT cana-761 113 7 = = NUM cana-761 113 8 ( ( PUNCT cana-761 113 9 ) ) PUNCT cana-761 113 10 1 1 NUM cana-761 113 11 0lim 0lim NUM cana-761 113 12 , , PUNCT cana-761 113 13 , , PUNCT cana-761 113 14 mk mk PROPN cana-761 113 15 nk nk PROPN cana-761 113 16 k k PROPN cana-761 114 1 w w PROPN cana-761 114 2 q q X cana-761 114 3 q q X cana-761 114 4 t− t− PROPN cana-761 114 5 → → PUNCT cana-761 114 6 = = NOUN cana-761 114 7   X cana-761 114 8 . . PUNCT cana-761 115 1 this this DET cana-761 115 2 inequality inequality NOUN cana-761 115 3 and and CCONJ cana-761 115 4 ( ( PUNCT cana-761 115 5 7 7 X cana-761 115 6 ) ) PUNCT cana-761 115 7 imply imply VERB cana-761 115 8 communications communication NOUN cana-761 115 9 on on ADP cana-761 115 10 applied apply VERB cana-761 115 11 nonlinear nonlinear ADJ cana-761 115 12 analysis analysis NOUN cana-761 115 13 issn issn NOUN cana-761 115 14 : : PUNCT cana-761 115 15 1074 1074 NUM cana-761 115 16 - - PUNCT cana-761 115 17 133x 133x NUM cana-761 115 18 vol vol NOUN cana-761 115 19 31 31 NUM cana-761 115 20 no no NOUN cana-761 115 21 . . PUNCT cana-761 116 1 3s 3s NUM cana-761 116 2 ( ( PUNCT cana-761 116 3 2024 2024 NUM cana-761 116 4 ) ) PUNCT cana-761 116 5 231 231 NUM cana-761 116 6 https://internationalpubls.com https://internationalpubls.com X cana-761 116 7 ( ( PUNCT cana-761 116 8 ) ) PUNCT cana-761 116 9 0lim 0lim NUM cana-761 116 10 , , PUNCT cana-761 116 11 , , PUNCT cana-761 116 12 mk mk PROPN cana-761 116 13 nk nk PROPN cana-761 116 14 k k PROPN cana-761 117 1 w w PROPN cana-761 117 2 q q X cana-761 117 3 q q PROPN cana-761 117 4 t t PROPN cana-761 117 5   PROPN cana-761 117 6 → → PUNCT cana-761 117 7 = = PUNCT cana-761 117 8 . . PUNCT cana-761 118 1 ( ( PUNCT cana-761 118 2 10 10 NUM cana-761 118 3 ) ) PUNCT cana-761 118 4 let let VERB cana-761 118 5 us we PRON cana-761 118 6 prove prove VERB cana-761 118 7 that that SCONJ cana-761 118 8 ( ( PUNCT cana-761 118 9 ) ) PUNCT cana-761 118 10 1 1 NUM cana-761 118 11 0lim 0lim NUM cana-761 118 12 , , PUNCT cana-761 118 13 , , PUNCT cana-761 118 14 mk mk PROPN cana-761 118 15 nk nk PROPN cana-761 118 16 k k PROPN cana-761 119 1 w w PROPN cana-761 119 2 q q X cana-761 119 3 q q PROPN cana-761 119 4 t t NOUN cana-761 119 5 + + PROPN cana-761 119 6 → → PUNCT cana-761 119 7 = = PUNCT cana-761 119 8 . . PUNCT cana-761 120 1 ( ( PUNCT cana-761 120 2 11 11 NUM cana-761 120 3 ) ) PUNCT cana-761 120 4 ( ( PUNCT cana-761 120 5 ) ) PUNCT cana-761 120 6 ( ( PUNCT cana-761 120 7 ) ) PUNCT cana-761 120 8 ( ( PUNCT cana-761 120 9 ) ) PUNCT cana-761 120 10 ( ( PUNCT cana-761 120 11 ) ) PUNCT cana-761 120 12 1 1 NUM cana-761 120 13 0 0 NUM cana-761 120 14 0 0 NUM cana-761 120 15 1 1 NUM cana-761 120 16 0 0 NUM cana-761 120 17 lim lim NOUN cana-761 120 18 , , PUNCT cana-761 120 19 , , PUNCT cana-761 120 20 lim lim PROPN cana-761 120 21 lim lim PROPN cana-761 120 22 , , PUNCT cana-761 120 23 , , PUNCT cana-761 120 24 , , PUNCT cana-761 120 25 , , PUNCT cana-761 120 26 , , PUNCT cana-761 120 27 mk mk PROPN cana-761 120 28 nk nk PROPN cana-761 120 29 mk mk PROPN cana-761 121 1 nk nk PROPN cana-761 121 2 nk nk PROPN cana-761 121 3 nk nk PROPN cana-761 122 1 k k PROPN cana-761 123 1 k k PROPN cana-761 124 1 p p X cana-761 124 2 w w PROPN cana-761 124 3 q q X cana-761 124 4 q q PROPN cana-761 124 5 t t NOUN cana-761 124 6 t t PROPN cana-761 124 7 w w PROPN cana-761 124 8 q q X cana-761 124 9 q q PROPN cana-761 124 10 t t PROPN cana-761 124 11 p p NOUN cana-761 124 12 w w PROPN cana-761 124 13 q q NOUN cana-761 124 14 q q X cana-761 124 15 p p X cana-761 124 16 + + PROPN cana-761 124 17 + + ADJ cana-761 124 18 + + NUM cana-761 124 19 → → NUM cana-761 124 20 → → PUNCT cana-761 124 21 → → SYM cana-761 124 22   NOUN cana-761 124 23   X cana-761 124 24 − − PROPN cana-761 124 25   PROPN cana-761 125 1   PROPN cana-761 125 2   NOUN cana-761 125 3 . . PUNCT cana-761 126 1 ( ( PUNCT cana-761 126 2 ) ) PUNCT cana-761 126 3 ( ( PUNCT cana-761 126 4 ) ) PUNCT cana-761 126 5 0lim 0lim NUM cana-761 126 6 , , PUNCT cana-761 126 7 , , PUNCT cana-761 126 8 , , PUNCT cana-761 127 1 0mk 0mk ADJ cana-761 127 2 nk nk PROPN cana-761 127 3 k k PROPN cana-761 128 1 t t PROPN cana-761 128 2 w w PROPN cana-761 128 3 q q X cana-761 128 4 q q PROPN cana-761 128 5 t t NOUN cana-761 128 6 → → PUNCT cana-761 128 7 = = SYM cana-761 128 8 ( ( PUNCT cana-761 128 9 ) ) PUNCT cana-761 128 10 0lim 0lim NUM cana-761 128 11 , , PUNCT cana-761 128 12 , , PUNCT cana-761 128 13 mk mk PROPN cana-761 129 1 nk nk PROPN cana-761 129 2 k k PROPN cana-761 130 1 w w PROPN cana-761 130 2 q q X cana-761 130 3 q q PROPN cana-761 130 4 t t NOUN cana-761 130 5 → → PUNCT cana-761 130 6 = = VERB cana-761 130 7 = = NOUN cana-761 130 8 . . PUNCT cana-761 131 1 ( ( PUNCT cana-761 131 2 12 12 NUM cana-761 131 3 ) ) PUNCT cana-761 131 4 on on ADP cana-761 131 5 the the DET cana-761 131 6 other other ADJ cana-761 131 7 hand hand NOUN cana-761 131 8 , , PUNCT cana-761 131 9 by by ADP cana-761 131 10 ( ( PUNCT cana-761 131 11 10 10 NUM cana-761 131 12 ) ) PUNCT cana-761 131 13 and and CCONJ cana-761 131 14 ( ( PUNCT cana-761 131 15 4 4 NUM cana-761 131 16 ) ) PUNCT cana-761 131 17 , , PUNCT cana-761 131 18 we we PRON cana-761 131 19 have have VERB cana-761 131 20 ( ( PUNCT cana-761 131 21 ) ) PUNCT cana-761 131 22 0lim 0lim NUM cana-761 131 23 , , PUNCT cana-761 131 24 , , PUNCT cana-761 132 1 mk mk PROPN cana-761 132 2 nk nk PROPN cana-761 132 3 k k PROPN cana-761 133 1 w w PROPN cana-761 133 2 q q X cana-761 133 3 q q PROPN cana-761 133 4 t t PROPN cana-761 133 5 → → PUNCT cana-761 133 6 = = SYM cana-761 133 7 ( ( PUNCT cana-761 133 8 ) ) PUNCT cana-761 133 9 ( ( PUNCT cana-761 133 10 ) ) PUNCT cana-761 133 11 ( ( PUNCT cana-761 133 12 ) ) PUNCT cana-761 133 13 1 1 NUM cana-761 133 14 0 0 NUM cana-761 133 15 1 1 NUM cana-761 133 16 0 0 NUM cana-761 133 17 lim lim PROPN cana-761 133 18 lim lim PROPN cana-761 133 19 , , PUNCT cana-761 133 20 , , PUNCT cana-761 133 21 , , PUNCT cana-761 133 22 , , PUNCT cana-761 133 23 , , PUNCT cana-761 133 24 mk mk PROPN cana-761 134 1 nk nk PROPN cana-761 134 2 nk nk PROPN cana-761 134 3 nk nk PROPN cana-761 135 1 k k PROPN cana-761 136 1 p p PROPN cana-761 136 2 t t PROPN cana-761 136 3 w w NOUN cana-761 137 1 q q X cana-761 137 2 q q PROPN cana-761 137 3 t t PROPN cana-761 137 4 p p NOUN cana-761 137 5 w w PROPN cana-761 137 6 q q NOUN cana-761 137 7 q q X cana-761 138 1 p p X cana-761 138 2 + + PROPN cana-761 138 3 + + CCONJ cana-761 138 4 + + NUM cana-761 138 5 → → NOUN cana-761 138 6 → → SYM cana-761 138 7   NOUN cana-761 138 8   X cana-761 139 1 − − PROPN cana-761 139 2   PROPN cana-761 139 3   PROPN cana-761 139 4   PROPN cana-761 139 5 ( ( PUNCT cana-761 139 6 ) ) PUNCT cana-761 139 7 ( ( PUNCT cana-761 139 8 ) ) PUNCT cana-761 139 9 ( ( PUNCT cana-761 139 10 ) ) PUNCT cana-761 139 11 1 1 NUM cana-761 139 12 0 0 NUM cana-761 139 13 1 1 NUM cana-761 139 14 0lim 0lim NUM cana-761 139 15 , , PUNCT cana-761 139 16 , , PUNCT cana-761 139 17 , , PUNCT cana-761 139 18 0 0 NUM cana-761 139 19 lim lim PROPN cana-761 139 20 , , PUNCT cana-761 139 21 , , PUNCT cana-761 139 22 mk mk PROPN cana-761 139 23 nk nk PROPN cana-761 139 24 mk mk PROPN cana-761 140 1 nk nk PROPN cana-761 140 2 k k PROPN cana-761 141 1 k k PROPN cana-761 141 2 t t PROPN cana-761 141 3 w w PROPN cana-761 141 4 q q X cana-761 141 5 q q PROPN cana-761 141 6 t t PROPN cana-761 142 1 w w NOUN cana-761 142 2 q q X cana-761 142 3 q q X cana-761 142 4 t+ t+ X cana-761 142 5 + + NOUN cana-761 142 6 → → PUNCT cana-761 142 7 → → NUM cana-761 142 8   NUM cana-761 142 9 = = PUNCT cana-761 142 10 . . PUNCT cana-761 143 1 ( ( PUNCT cana-761 143 2 13 13 NUM cana-761 143 3 ) ) PUNCT cana-761 143 4 then then ADV cana-761 143 5 , , PUNCT cana-761 143 6 by by ADP cana-761 143 7 ( ( PUNCT cana-761 143 8 12 12 NUM cana-761 143 9 ) ) PUNCT cana-761 143 10 and and CCONJ cana-761 143 11 ( ( PUNCT cana-761 143 12 13 13 NUM cana-761 143 13 ) ) PUNCT cana-761 143 14 , , PUNCT cana-761 143 15 we we PRON cana-761 143 16 obtain obtain VERB cana-761 143 17 ( ( PUNCT cana-761 143 18 11 11 NUM cana-761 143 19 ) ) PUNCT cana-761 143 20 . . PUNCT cana-761 144 1 the the DET cana-761 144 2 left left ADJ cana-761 144 3 proofs proof NOUN cana-761 144 4 are be AUX cana-761 144 5 similar similar ADJ cana-761 144 6 to to ADP cana-761 144 7 the the DET cana-761 144 8 above above ADJ cana-761 144 9 argument argument NOUN cana-761 144 10 , , PUNCT cana-761 144 11 and and CCONJ cana-761 144 12 therefore therefore ADV cana-761 144 13 we we PRON cana-761 144 14 omit omit VERB cana-761 144 15 them they PRON cana-761 144 16 . . PUNCT cana-761 145 1 remark remark PROPN cana-761 145 2 3.2 3.2 NUM cana-761 145 3 . . PUNCT cana-761 146 1 condition condition NOUN cana-761 146 2 ( ( PUNCT cana-761 146 3 3 3 NUM cana-761 146 4 ) ) PUNCT cana-761 146 5 in in ADP cana-761 146 6 lemma lemma PROPN cana-761 146 7 3.1 3.1 NUM cana-761 146 8 can can AUX cana-761 146 9 be be AUX cana-761 146 10 omitted omit VERB cana-761 146 11 if if SCONJ cana-761 146 12 ( ( PUNCT cana-761 146 13 ) ) PUNCT cana-761 146 14 , , PUNCT cana-761 146 15 , , PUNCT cana-761 146 16     SPACE cana-761 146 17 x x VERB cana-761 146 18 w w PROPN cana-761 146 19 t t PROPN cana-761 146 20 is be AUX cana-761 146 21 a a DET cana-761 146 22 strong strong ADJ cana-761 146 23 rfms rfms NOUN cana-761 146 24 . . PUNCT cana-761 147 1 in in ADP cana-761 147 2 this this DET cana-761 147 3 case case NOUN cana-761 147 4 , , PUNCT cana-761 147 5 instead instead ADV cana-761 147 6 of of ADP cana-761 147 7 ( ( PUNCT cana-761 147 8 8) 8) NUM cana-761 147 9 , , PUNCT cana-761 147 10 we we PRON cana-761 147 11 have have VERB cana-761 147 12 ( ( PUNCT cana-761 147 13 ) ) PUNCT cana-761 147 14 ( ( PUNCT cana-761 147 15 ) ) PUNCT cana-761 147 16 ( ( PUNCT cana-761 147 17 ) ) PUNCT cana-761 147 18 ( ( PUNCT cana-761 147 19 ) ) PUNCT cana-761 147 20 0 0 NUM cana-761 147 21 1 1 NUM cana-761 147 22 1 1 NUM cana-761 147 23 0 0 NUM cana-761 147 24 , , PUNCT cana-761 147 25 , , PUNCT cana-761 147 26 , , PUNCT cana-761 147 27 , , PUNCT cana-761 147 28 , , PUNCT cana-761 147 29 , , PUNCT cana-761 147 30 , , PUNCT cana-761 147 31 mk mk PROPN cana-761 147 32 nk nk PROPN cana-761 147 33 mk mk PROPN cana-761 147 34 mk mk PROPN cana-761 147 35 mk mk PROPN cana-761 147 36 nkw nkw PROPN cana-761 147 37 q q PROPN cana-761 148 1 q q PROPN cana-761 148 2 t t PROPN cana-761 148 3 t t PROPN cana-761 149 1 w w NOUN cana-761 149 2 q q PROPN cana-761 149 3 q q X cana-761 149 4 p p X cana-761 149 5 w w PROPN cana-761 149 6 q q X cana-761 149 7 q q PROPN cana-761 149 8 t t NOUN cana-761 149 9 p− p− NOUN cana-761 149 10 − − NOUN cana-761 149 11 − − PROPN cana-761 149 12 . . PUNCT cana-761 150 1 in in ADP cana-761 150 2 the the DET cana-761 150 3 following following NOUN cana-761 150 4 , , PUNCT cana-761 150 5 denote denote VERB cana-761 150 6 by by ADP cana-761 150 7 f f PROPN cana-761 150 8 the the DET cana-761 150 9 class class NOUN cana-761 150 10 of of ADP cana-761 150 11 all all DET cana-761 150 12 mappings mapping NOUN cana-761 150 13   ADJ cana-761 150 14   PROPN cana-761 150 15 ( ( PUNCT cana-761 150 16 ) ) PUNCT cana-761 150 17     SPACE cana-761 150 18 : : PUNCT cana-761 150 19     SPACE cana-761 150 20 0,1 0,1 NUM cana-761 150 21         SPACE cana-761 150 22 0,f 0,f NOUN cana-761 150 23 → → PUNCT cana-761 151 1 + + ADP cana-761 151 2   VERB cana-761 151 3 satisfying satisfy VERB cana-761 151 4 the the DET cana-761 151 5 following follow VERB cana-761 151 6 condition condition NOUN cana-761 151 7 : : PUNCT cana-761 151 8 for for ADP cana-761 151 9 all all PRON cana-761 151 10 , , PUNCT cana-761 151 11     SPACE cana-761 151 12 q q NOUN cana-761 151 13 r r NOUN cana-761 151 14 x x PROPN cana-761 151 15 , , PUNCT cana-761 151 16 q q X cana-761 151 17 r r PROPN cana-761 151 18 implies imply VERB cana-761 151 19 ( ( PUNCT cana-761 151 20 ) ) PUNCT cana-761 151 21 ( ( PUNCT cana-761 151 22 ) ) PUNCT cana-761 151 23         SPACE cana-761 152 1 f f PROPN cana-761 153 1 q q X cana-761 153 2 f f X cana-761 153 3 r r PROPN cana-761 153 4 . . PUNCT cana-761 154 1 that that PRON cana-761 154 2 is be AUX cana-761 154 3 to to PART cana-761 154 4 say say VERB cana-761 154 5 , , PUNCT cana-761 154 6 f f PROPN cana-761 154 7 is be AUX cana-761 154 8 strictly strictly ADV cana-761 154 9 non non ADJ cana-761 154 10 - - ADJ cana-761 154 11 increasing increase VERB cana-761 154 12 on on ADP cana-761 154 13   ADJ cana-761 154 14 0,1 0,1 NOUN cana-761 154 15     SPACE cana-761 154 16 . . PUNCT cana-761 155 1 definition definition NOUN cana-761 155 2 3.3 3.3 NUM cana-761 155 3 . . PUNCT cana-761 156 1 let let VERB cana-761 156 2 ( ( PUNCT cana-761 156 3 ) ) PUNCT cana-761 156 4 , , PUNCT cana-761 156 5     SPACE cana-761 156 6 x x PUNCT cana-761 157 1 d d NOUN cana-761 157 2 be be AUX cana-761 157 3 a a DET cana-761 157 4 metric metric ADJ cana-761 157 5 space space NOUN cana-761 157 6 and and CCONJ cana-761 157 7     SPACE cana-761 157 8 : : PUNCT cana-761 157 9     SPACE cana-761 157 10 f f NOUN cana-761 157 11 r r NOUN cana-761 157 12 r+ r+ NOUN cana-761 157 13 → → PUNCT cana-761 157 14 be be AUX cana-761 157 15 a a DET cana-761 157 16 mapping mapping NOUN cana-761 157 17 , , PUNCT cana-761 157 18 satisfying satisfy VERB cana-761 157 19 the the DET cana-761 157 20 following following NOUN cana-761 157 21 : : PUNCT cana-761 157 22 ( ( PUNCT cana-761 157 23 rf rf NOUN cana-761 157 24 1 1 NUM cana-761 157 25 ) ) PUNCT cana-761 157 26 f f PROPN cana-761 157 27 is be AUX cana-761 157 28 strictly strictly ADV cana-761 157 29 non non ADJ cana-761 157 30 increasing increase VERB cana-761 157 31 on on ADP cana-761 157 32 r+ r+ NOUN cana-761 157 33 ; ; PUNCT cana-761 157 34 ( ( PUNCT cana-761 157 35 rf rf NOUN cana-761 157 36 2 2 NUM cana-761 157 37 ) ) PUNCT cana-761 157 38 for for ADP cana-761 157 39 each each DET cana-761 157 40 sequence sequence NOUN cana-761 157 41   PROPN cana-761 157 42 n n VERB cana-761 157 43 n n CCONJ cana-761 157 44   X cana-761 157 45 n n NOUN cana-761 157 46 of of ADP cana-761 157 47 positive positive ADJ cana-761 157 48 numbers number NOUN cana-761 157 49 , , PUNCT cana-761 157 50 ( ( PUNCT cana-761 157 51 ) ) PUNCT cana-761 157 52 lim lim PROPN cana-761 157 53 n n PROPN cana-761 157 54 k k PROPN cana-761 157 55 f f X cana-761 157 56   X cana-761 157 57 → → PUNCT cana-761 157 58 = = SYM cana-761 157 59 − − NOUN cana-761 157 60 if if SCONJ cana-761 157 61 , , PUNCT cana-761 157 62 and and CCONJ cana-761 157 63 only only ADV cana-761 157 64 if if SCONJ cana-761 157 65 lim lim PROPN cana-761 157 66 0n 0n NOUN cana-761 157 67 k k PROPN cana-761 157 68   PUNCT cana-761 157 69 → → PUNCT cana-761 157 70 = = NUM cana-761 157 71 ; ; PUNCT cana-761 157 72 ( ( PUNCT cana-761 157 73 rf rf NOUN cana-761 157 74 3 3 X cana-761 157 75 ) ) PUNCT cana-761 157 76 there there PRON cana-761 157 77 exists exist VERB cana-761 157 78 ( ( PUNCT cana-761 157 79 ) ) PUNCT cana-761 157 80 0,1 0,1 NUM cana-761 157 81     SPACE cana-761 157 82 k k NOUN cana-761 157 83   NOUN cana-761 157 84 such such ADJ cana-761 157 85 that that SCONJ cana-761 157 86 ( ( PUNCT cana-761 157 87 ) ) PUNCT cana-761 157 88 0 0 NUM cana-761 157 89 lim lim PROPN cana-761 157 90         SPACE cana-761 157 91 0k 0k NOUN cana-761 157 92 k k NOUN cana-761 157 93 f f PROPN cana-761 158 1 a a ADV cana-761 158 2 + + PROPN cana-761 158 3 → → SYM cana-761 158 4 = = SYM cana-761 158 5 . . PUNCT cana-761 159 1 the the DET cana-761 159 2 mapping mapping NOUN cana-761 159 3         SPACE cana-761 159 4 : : PUNCT cana-761 159 5         SPACE cana-761 159 6 j j X cana-761 159 7 x x X cana-761 159 8 x→ x→ PROPN cana-761 159 9 is be AUX cana-761 159 10 said say VERB cana-761 159 11 to to PART cana-761 159 12 be be AUX cana-761 159 13 an an DET cana-761 159 14 f−contraction f−contraction NOUN cana-761 159 15 if if SCONJ cana-761 159 16 there there PRON cana-761 159 17 exists exist VERB cana-761 159 18         SPACE cana-761 159 19 0t 0t NOUN cana-761 159 20 such such ADJ cana-761 159 21 that that SCONJ cana-761 159 22 communications communication NOUN cana-761 159 23 on on ADP cana-761 159 24 applied apply VERB cana-761 159 25 nonlinear nonlinear ADJ cana-761 159 26 analysis analysis NOUN cana-761 159 27 issn issn NOUN cana-761 159 28 : : PUNCT cana-761 159 29 1074 1074 NUM cana-761 159 30 - - PUNCT cana-761 159 31 133x 133x NUM cana-761 159 32 vol vol NOUN cana-761 159 33 31 31 NUM cana-761 159 34 no no NOUN cana-761 159 35 . . PUNCT cana-761 160 1 3s 3s NUM cana-761 160 2 ( ( PUNCT cana-761 160 3 2024 2024 NUM cana-761 160 4 ) ) PUNCT cana-761 160 5 232 232 NUM cana-761 160 6 https://internationalpubls.com https://internationalpubls.com X cana-761 160 7 ( ( PUNCT cana-761 160 8 ) ) PUNCT cana-761 160 9 ( ( PUNCT cana-761 160 10 ) ) PUNCT cana-761 160 11 ( ( PUNCT cana-761 160 12 ) ) PUNCT cana-761 160 13 ( ( PUNCT cana-761 160 14 ) ) PUNCT cana-761 160 15 ( ( PUNCT cana-761 160 16 ) ) PUNCT cana-761 160 17 ( ( PUNCT cana-761 160 18 ) ) PUNCT cana-761 160 19     SPACE cana-761 160 20 , , PUNCT cana-761 160 21     SPACE cana-761 160 22 , , PUNCT cana-761 160 23     SPACE cana-761 160 24 f f PROPN cana-761 161 1 d d X cana-761 161 2 j j PROPN cana-761 161 3 q q X cana-761 161 4 j j PROPN cana-761 161 5 r r NOUN cana-761 161 6 f f PROPN cana-761 161 7 d d X cana-761 161 8 q q PROPN cana-761 161 9 r r PROPN cana-761 161 10 + + CCONJ cana-761 161 11   NOUN cana-761 161 12 for for ADP cana-761 161 13 all all PRON cana-761 161 14 , , PUNCT cana-761 161 15 q q NOUN cana-761 161 16 r r NOUN cana-761 161 17 x x NOUN cana-761 161 18 with with ADP cana-761 161 19 ( ( PUNCT cana-761 161 20 ) ) PUNCT cana-761 161 21 ( ( PUNCT cana-761 161 22 ) ) PUNCT cana-761 161 23 ( ( PUNCT cana-761 161 24 ) ) PUNCT cana-761 161 25 , , PUNCT cana-761 161 26     SPACE cana-761 161 27 0d 0d X cana-761 162 1 j j PROPN cana-761 162 2 q q X cana-761 162 3 j j PROPN cana-761 162 4 r r NOUN cana-761 162 5   PROPN cana-761 162 6 . . PUNCT cana-761 163 1 definition definition NOUN cana-761 163 2 3.4 3.4 NUM cana-761 163 3 . . PUNCT cana-761 164 1 let let VERB cana-761 164 2 ( ( PUNCT cana-761 164 3 ) ) PUNCT cana-761 164 4 , , PUNCT cana-761 164 5 , , PUNCT cana-761 164 6 x x X cana-761 164 7 w w PROPN cana-761 164 8 t t PROPN cana-761 164 9 be be AUX cana-761 164 10 a a DET cana-761 164 11 rfms rfms NOUN cana-761 164 12 and and CCONJ cana-761 164 13 ff ff PROPN cana-761 164 14 . . PUNCT cana-761 165 1 the the DET cana-761 165 2 mapping mapping NOUN cana-761 165 3     SPACE cana-761 165 4 : : PUNCT cana-761 165 5         SPACE cana-761 165 6 j j X cana-761 165 7 x x X cana-761 165 8 x→ x→ PROPN cana-761 165 9 is be AUX cana-761 165 10 said say VERB cana-761 165 11 to to PART cana-761 165 12 be be AUX cana-761 165 13 a a DET cana-761 165 14 𝑅𝐹 𝑅𝐹 NOUN cana-761 165 15 −contraction −contraction NOUN cana-761 165 16 if if SCONJ cana-761 165 17 there there PRON cana-761 165 18 exists exist VERB cana-761 165 19 ( ( PUNCT cana-761 165 20 ) ) PUNCT cana-761 165 21 0,1 0,1 NUM cana-761 165 22     SPACE cana-761 165 23   NOUN cana-761 165 24   NOUN cana-761 165 25 such such ADJ cana-761 165 26 that that SCONJ cana-761 165 27 ( ( PUNCT cana-761 165 28 ) ) PUNCT cana-761 165 29 ( ( PUNCT cana-761 165 30 ) ) PUNCT cana-761 165 31 ( ( PUNCT cana-761 165 32 ) ) PUNCT cana-761 165 33 ( ( PUNCT cana-761 165 34 ) ) PUNCT cana-761 165 35 ( ( PUNCT cana-761 165 36 ) ) PUNCT cana-761 165 37 ( ( PUNCT cana-761 165 38 ) ) PUNCT cana-761 165 39 ( ( PUNCT cana-761 165 40 ) ) PUNCT cana-761 165 41 1 1 NUM cana-761 165 42 , , PUNCT cana-761 165 43     SPACE cana-761 165 44 , , PUNCT cana-761 165 45     SPACE cana-761 165 46 f f PROPN cana-761 166 1 d d X cana-761 166 2 f f X cana-761 166 3 q q X cana-761 166 4 f f PROPN cana-761 166 5 r r NOUN cana-761 166 6 f f PROPN cana-761 166 7 d d X cana-761 166 8 q q NOUN cana-761 166 9 r r NOUN cana-761 166 10   NOUN cana-761 166 11   NOUN cana-761 166 12 for for ADP cana-761 166 13 all all PRON cana-761 166 14 , , PUNCT cana-761 166 15         SPACE cana-761 166 16 , , PUNCT cana-761 166 17     SPACE cana-761 166 18 q q NOUN cana-761 167 1 r r NOUN cana-761 167 2 x x X cana-761 167 3 q q X cana-761 167 4 r r PROPN cana-761 167 5   PROPN cana-761 167 6 , , PUNCT cana-761 167 7 and and CCONJ cana-761 167 8 0 0 NUM cana-761 167 9 t t NOUN cana-761 167 10   X cana-761 167 11 . . PUNCT cana-761 168 1 ( ( PUNCT cana-761 168 2 14 14 NUM cana-761 168 3 ) ) PUNCT cana-761 168 4 theorem theorem VERB cana-761 168 5 3.5 3.5 NUM cana-761 168 6 . . PUNCT cana-761 169 1 let let VERB cana-761 169 2 ( ( PUNCT cana-761 169 3 ) ) PUNCT cana-761 169 4 , , PUNCT cana-761 169 5 , , PUNCT cana-761 169 6 x x X cana-761 169 7 w w PROPN cana-761 169 8 t t PROPN cana-761 169 9 be be AUX cana-761 169 10 a a DET cana-761 169 11 complete complete ADJ cana-761 169 12 rfms rfms NOUN cana-761 170 1 such such ADJ cana-761 170 2 that that SCONJ cana-761 170 3 ( ( PUNCT cana-761 170 4 ) ) PUNCT cana-761 170 5 0 0 NUM cana-761 170 6 lim lim PROPN cana-761 170 7 , , PUNCT cana-761 170 8     SPACE cana-761 170 9 , , PUNCT cana-761 170 10     SPACE cana-761 170 11 1 1 NUM cana-761 170 12 t t NOUN cana-761 170 13 w w NOUN cana-761 170 14 q q NOUN cana-761 170 15 r r NOUN cana-761 170 16 t t NOUN cana-761 170 17 + + PROPN cana-761 170 18 → → PROPN cana-761 170 19   PROPN cana-761 170 20 , , PUNCT cana-761 170 21 for for ADP cana-761 170 22 all all PRON cana-761 170 23 , , PUNCT cana-761 170 24         SPACE cana-761 170 25 q q PUNCT cana-761 170 26 r r NOUN cana-761 170 27 x x PROPN cana-761 170 28 . . PUNCT cana-761 171 1 if if SCONJ cana-761 171 2     SPACE cana-761 171 3 : : PUNCT cana-761 171 4         SPACE cana-761 171 5 j j X cana-761 171 6 x x X cana-761 171 7 x→ x→ PROPN cana-761 171 8 is be AUX cana-761 171 9 a a DET cana-761 171 10 continuous continuous ADJ cana-761 171 11 rf rf ADJ cana-761 171 12 f f NOUN cana-761 171 13 - - PUNCT cana-761 171 14 contraction contraction NOUN cana-761 171 15 , , PUNCT cana-761 171 16 then then ADV cana-761 171 17 j j PROPN cana-761 171 18 has have VERB cana-761 171 19 a a DET cana-761 171 20 unique unique ADJ cana-761 171 21 fixed fix VERB cana-761 171 22 - - PUNCT cana-761 171 23 point point NOUN cana-761 171 24 in in ADP cana-761 171 25 x x X cana-761 171 26 . . PUNCT cana-761 172 1 proof proof NOUN cana-761 172 2 . . PUNCT cana-761 173 1 choose choose VERB cana-761 173 2 0q 0q PROPN cana-761 173 3 x x PROPN cana-761 173 4 and and CCONJ cana-761 173 5 ( ( PUNCT cana-761 173 6 ) ) PUNCT cana-761 173 7 1n 1n PROPN cana-761 173 8 nq nq PROPN cana-761 173 9 j j PROPN cana-761 173 10 q+ q+ ADP cana-761 173 11 = = PUNCT cana-761 173 12 for for SCONJ cana-761 173 13 all all DET cana-761 173 14 0n 0n NOUN cana-761 173 15 n n PROPN cana-761 173 16 . . PUNCT cana-761 174 1 suppose suppose VERB cana-761 174 2 that that SCONJ cana-761 174 3 : : PUNCT cana-761 174 4         SPACE cana-761 174 5 j j X cana-761 174 6 x x X cana-761 174 7 x→ x→ PROPN cana-761 174 8 is be AUX cana-761 174 9 a a DET cana-761 174 10 𝑅𝐹 𝑅𝐹 PROPN cana-761 174 11 −contractive −contractive NOUN cana-761 174 12 mapping mapping NOUN cana-761 174 13 . . PUNCT cana-761 175 1 if if SCONJ cana-761 175 2 ( ( PUNCT cana-761 175 3 ) ) PUNCT cana-761 175 4 1n 1n NUM cana-761 175 5 n n CCONJ cana-761 175 6 nq nq PROPN cana-761 175 7 q q PROPN cana-761 175 8 j j PROPN cana-761 175 9 q+= q+= CCONJ cana-761 175 10 = = PRON cana-761 175 11 holds hold VERB cana-761 175 12 for for ADP cana-761 175 13 some some DET cana-761 175 14 0n 0n NOUN cana-761 175 15 n n PROPN cana-761 175 16 , , PUNCT cana-761 175 17 then then ADV cana-761 175 18 nq nq PROPN cana-761 175 19 is be AUX cana-761 175 20 a a DET cana-761 175 21 fp fp PROPN cana-761 175 22 . . PROPN cana-761 175 23 assume assume VERB cana-761 176 1 that that SCONJ cana-761 176 2 1n 1n NUM cana-761 176 3 nq nq PROPN cana-761 176 4 q q X cana-761 177 1 + + PROPN cana-761 177 2   PROPN cana-761 177 3 for for ADP cana-761 177 4 any any DET cana-761 177 5 0n 0n NOUN cana-761 177 6 n n PROPN cana-761 177 7 . . PUNCT cana-761 178 1 by by ADP cana-761 178 2 ( ( PUNCT cana-761 178 3 14 14 NUM cana-761 178 4 ) ) PUNCT cana-761 178 5 , , PUNCT cana-761 178 6 for for ADP cana-761 178 7 every every DET cana-761 178 8 n n NOUN cana-761 178 9 n n PROPN cana-761 178 10 and and CCONJ cana-761 178 11 0 0 NUM cana-761 178 12 t t NOUN cana-761 178 13   NOUN cana-761 178 14 , , PUNCT cana-761 178 15 one one PRON cana-761 178 16 has have VERB cana-761 178 17 ( ( PUNCT cana-761 178 18 ) ) PUNCT cana-761 178 19 ( ( PUNCT cana-761 178 20 ) ) PUNCT cana-761 178 21 ( ( PUNCT cana-761 178 22 ) ) PUNCT cana-761 178 23 ( ( PUNCT cana-761 178 24 ) ) PUNCT cana-761 178 25 ( ( PUNCT cana-761 178 26 ) ) PUNCT cana-761 178 27 ( ( PUNCT cana-761 178 28 ) ) PUNCT cana-761 178 29 ( ( PUNCT cana-761 178 30 ) ) PUNCT cana-761 178 31 1 1 NUM cana-761 178 32 1 1 NUM cana-761 178 33 1 1 NUM cana-761 178 34 1 1 NUM cana-761 178 35 , , PUNCT cana-761 178 36     SPACE cana-761 178 37 , , PUNCT cana-761 178 38         SPACE cana-761 178 39 , , PUNCT cana-761 178 40     SPACE cana-761 178 41 , , PUNCT cana-761 178 42     SPACE cana-761 178 43 , , PUNCT cana-761 178 44     SPACE cana-761 178 45 , , PUNCT cana-761 178 46     SPACE cana-761 178 47 n n CCONJ cana-761 178 48 n n CCONJ cana-761 178 49 n n CCONJ cana-761 178 50 n n CCONJ cana-761 178 51 n n ADV cana-761 178 52 nf nf NOUN cana-761 179 1 w w PROPN cana-761 179 2 q q X cana-761 179 3 q q PROPN cana-761 179 4 t t X cana-761 179 5 f f PROPN cana-761 179 6 w w PROPN cana-761 179 7 q q X cana-761 179 8 q q PROPN cana-761 179 9 t t X cana-761 179 10 f f PROPN cana-761 179 11 w w PROPN cana-761 179 12 q q X cana-761 179 13 q q PROPN cana-761 179 14 t t NOUN cana-761 179 15   NOUN cana-761 179 16 + + CCONJ cana-761 179 17 + + CCONJ cana-761 179 18 − − NOUN cana-761 179 19   NOUN cana-761 179 20 . . PUNCT cana-761 180 1 then then ADV cana-761 180 2 , , PUNCT cana-761 180 3 we we PRON cana-761 180 4 get get VERB cana-761 180 5 ( ( PUNCT cana-761 180 6 ) ) PUNCT cana-761 180 7 ( ( PUNCT cana-761 180 8 ) ) PUNCT cana-761 180 9 1 1 NUM cana-761 180 10 1 1 NUM cana-761 180 11 , , PUNCT cana-761 180 12     SPACE cana-761 180 13 , , PUNCT cana-761 180 14     SPACE cana-761 180 15 , , PUNCT cana-761 180 16     SPACE cana-761 180 17 , , PUNCT cana-761 180 18     SPACE cana-761 180 19 n n ADP cana-761 180 20 n n CCONJ cana-761 180 21 n n ADV cana-761 180 22 nw nw NOUN cana-761 180 23 q q PROPN cana-761 181 1 q q PROPN cana-761 182 1 t t PROPN cana-761 183 1 w w NOUN cana-761 184 1 q q X cana-761 185 1 q q X cana-761 186 1 t− t− PROPN cana-761 187 1 + + NOUN cana-761 187 2   ADJ cana-761 187 3 . . PUNCT cana-761 188 1 thus thus ADV cana-761 188 2 , , PUNCT cana-761 188 3 ( ( PUNCT cana-761 188 4 ) ) PUNCT cana-761 188 5   NOUN cana-761 188 6 1 1 PROPN cana-761 188 7 , , PUNCT cana-761 188 8     SPACE cana-761 188 9 , , PUNCT cana-761 188 10     SPACE cana-761 188 11 n n PRON cana-761 188 12 nw nw NOUN cana-761 188 13 q q NOUN cana-761 189 1 q q X cana-761 190 1 t+ t+ INTJ cana-761 191 1 ( ( PUNCT cana-761 191 2 0)t 0)t INTJ cana-761 191 3   PROPN cana-761 191 4 is be AUX cana-761 191 5 a a DET cana-761 191 6 strictly strictly ADV cana-761 191 7 non non ADJ cana-761 191 8 - - ADJ cana-761 191 9 increasing increase VERB cana-761 191 10 sequence sequence NOUN cana-761 191 11 bounded bound VERB cana-761 191 12 from from ADP cana-761 191 13 above above ADV cana-761 191 14 , , PUNCT cana-761 191 15 so so CCONJ cana-761 191 16 ( ( PUNCT cana-761 191 17 ) ) PUNCT cana-761 191 18   NOUN cana-761 191 19 1 1 PROPN cana-761 191 20 , , PUNCT cana-761 191 21     SPACE cana-761 191 22 , , PUNCT cana-761 191 23     SPACE cana-761 191 24 n n PRON cana-761 191 25 nw nw NOUN cana-761 191 26 q q NOUN cana-761 191 27 q q X cana-761 191 28 t+ t+ INTJ cana-761 191 29 ( ( PUNCT cana-761 191 30 0)t 0)t INTJ cana-761 191 31   PROPN cana-761 191 32 is be AUX cana-761 191 33 convergent convergent ADJ cana-761 191 34 . . PUNCT cana-761 192 1 in in ADP cana-761 192 2 other other ADJ cana-761 192 3 words word NOUN cana-761 192 4 , , PUNCT cana-761 192 5 there there PRON cana-761 192 6 exists exist VERB cana-761 192 7 ( ( PUNCT cana-761 192 8 ) ) PUNCT cana-761 192 9   ADJ cana-761 192 10 0,1 0,1 NOUN cana-761 192 11     SPACE cana-761 192 12 a a DET cana-761 192 13 t t NOUN cana-761 192 14   NOUN cana-761 192 15 such such ADJ cana-761 192 16 that that PRON cana-761 192 17 for for ADP cana-761 192 18 any any DET cana-761 192 19         SPACE cana-761 192 20 0t 0t NOUN cana-761 192 21 , , PUNCT cana-761 192 22 one one PRON cana-761 192 23 has have VERB cana-761 192 24 ( ( PUNCT cana-761 192 25 ) ) PUNCT cana-761 192 26 ( ( PUNCT cana-761 192 27 ) ) PUNCT cana-761 192 28 1lim 1lim NUM cana-761 192 29 , , PUNCT cana-761 192 30     SPACE cana-761 192 31 , , PUNCT cana-761 192 32             SPACE cana-761 192 33 n n CCONJ cana-761 192 34 n n CCONJ cana-761 192 35 n n PROPN cana-761 192 36 w w PROPN cana-761 192 37 q q X cana-761 192 38 q q PROPN cana-761 192 39 t t PROPN cana-761 192 40 a a DET cana-761 192 41 t+ t+ NOUN cana-761 192 42 → → PUNCT cana-761 192 43 = = PRON cana-761 192 44 . . PUNCT cana-761 193 1 ( ( PUNCT cana-761 193 2 15 15 NUM cana-761 193 3 ) ) PUNCT cana-761 193 4 clearly clearly ADV cana-761 193 5 , , PUNCT cana-761 193 6 for for ADP cana-761 193 7 any any DET cana-761 193 8         SPACE cana-761 193 9 0t 0t NOUN cana-761 193 10 and and CCONJ cana-761 193 11 n n PROPN cana-761 193 12 n n PROPN cana-761 193 13 , , PUNCT cana-761 193 14 it it PRON cana-761 193 15 follows follow VERB cana-761 193 16 that that PRON cana-761 193 17 ( ( PUNCT cana-761 193 18 ) ) PUNCT cana-761 193 19 ( ( PUNCT cana-761 193 20 ) ) PUNCT cana-761 193 21 1 1 NUM cana-761 193 22 , , PUNCT cana-761 193 23     SPACE cana-761 193 24 , , PUNCT cana-761 193 25         SPACE cana-761 193 26 n n PROPN cana-761 193 27 nw nw NOUN cana-761 193 28 q q PROPN cana-761 193 29 q q PROPN cana-761 193 30 t t PROPN cana-761 193 31 a a DET cana-761 193 32 t+ t+ NOUN cana-761 193 33   PROPN cana-761 193 34 . . PUNCT cana-761 194 1 ( ( PUNCT cana-761 194 2 16 16 X cana-761 194 3 ) ) PUNCT cana-761 194 4 note note VERB cana-761 194 5 that that SCONJ cana-761 194 6 , , PUNCT cana-761 194 7 by by ADP cana-761 194 8 ( ( PUNCT cana-761 194 9 15 15 NUM cana-761 194 10 ) ) PUNCT cana-761 194 11 and and CCONJ cana-761 194 12 ( ( PUNCT cana-761 194 13 16 16 NUM cana-761 194 14 ) ) PUNCT cana-761 194 15 , , PUNCT cana-761 194 16 for for ADP cana-761 194 17 any any DET cana-761 194 18         SPACE cana-761 194 19 0t 0t NOUN cana-761 194 20 , , PUNCT cana-761 194 21 we we PRON cana-761 194 22 have have VERB cana-761 194 23 ( ( PUNCT cana-761 194 24 ) ) PUNCT cana-761 194 25 ( ( PUNCT cana-761 194 26 ) ) PUNCT cana-761 194 27 ( ( PUNCT cana-761 194 28 ) ) PUNCT cana-761 194 29 ( ( PUNCT cana-761 194 30 ) ) PUNCT cana-761 194 31 1lim 1lim NUM cana-761 194 32     SPACE cana-761 194 33 , , PUNCT cana-761 194 34     SPACE cana-761 194 35 , , PUNCT cana-761 194 36         SPACE cana-761 194 37 0n 0n NOUN cana-761 194 38 n n CCONJ cana-761 194 39 n n NOUN cana-761 194 40 f f PROPN cana-761 194 41 w w PROPN cana-761 194 42 q q X cana-761 195 1 q q PROPN cana-761 195 2 t t X cana-761 195 3 f f PROPN cana-761 195 4 a a DET cana-761 195 5 t+ t+ NOUN cana-761 195 6 → → PUNCT cana-761 195 7 = = NOUN cana-761 195 8 − − PROPN cana-761 195 9 . . PUNCT cana-761 196 1 ( ( PUNCT cana-761 196 2 17 17 NUM cana-761 196 3 ) ) PUNCT cana-761 196 4 assume assume VERB cana-761 196 5 that that SCONJ cana-761 196 6 ( ( PUNCT cana-761 196 7 ) ) PUNCT cana-761 196 8     SPACE cana-761 196 9 1 1 NUM cana-761 196 10     SPACE cana-761 196 11 a a DET cana-761 196 12 t t NOUN cana-761 196 13   PROPN cana-761 196 14 for for ADP cana-761 196 15 some some DET cana-761 196 16 0 0 NUM cana-761 196 17 t t NOUN cana-761 196 18   NOUN cana-761 196 19 . . PUNCT cana-761 197 1 by by ADP cana-761 197 2 ( ( PUNCT cana-761 197 3 14 14 NUM cana-761 197 4 ) ) PUNCT cana-761 197 5 , , PUNCT cana-761 197 6 it it PRON cana-761 197 7 implies imply VERB cana-761 197 8 that that SCONJ cana-761 197 9 communications communication NOUN cana-761 197 10 on on ADP cana-761 197 11 applied apply VERB cana-761 197 12 nonlinear nonlinear ADJ cana-761 197 13 analysis analysis NOUN cana-761 197 14 issn issn NOUN cana-761 197 15 : : PUNCT cana-761 197 16 1074 1074 NUM cana-761 197 17 - - PUNCT cana-761 197 18 133x 133x NUM cana-761 197 19 vol vol NOUN cana-761 197 20 31 31 NUM cana-761 197 21 no no NOUN cana-761 197 22 . . PUNCT cana-761 198 1 3s 3s NUM cana-761 198 2 ( ( PUNCT cana-761 198 3 2024 2024 NUM cana-761 198 4 ) ) PUNCT cana-761 198 5 233 233 NUM cana-761 198 6 https://internationalpubls.com https://internationalpubls.com X cana-761 198 7 ( ( PUNCT cana-761 198 8 ) ) PUNCT cana-761 198 9 ( ( PUNCT cana-761 198 10 ) ) PUNCT cana-761 198 11 ( ( PUNCT cana-761 198 12 ) ) PUNCT cana-761 198 13 ( ( PUNCT cana-761 198 14 ) ) PUNCT cana-761 198 15 ( ( PUNCT cana-761 198 16 ) ) PUNCT cana-761 198 17 ( ( PUNCT cana-761 198 18 ) ) PUNCT cana-761 198 19 ( ( PUNCT cana-761 198 20 ) ) PUNCT cana-761 198 21 1 1 NUM cana-761 198 22 1 1 NUM cana-761 198 23 1 1 NUM cana-761 198 24 1 1 NUM cana-761 198 25 , , PUNCT cana-761 198 26     SPACE cana-761 198 27 , , PUNCT cana-761 198 28         SPACE cana-761 198 29 , , PUNCT cana-761 198 30     SPACE cana-761 198 31 , , PUNCT cana-761 198 32     SPACE cana-761 198 33 , , PUNCT cana-761 198 34     SPACE cana-761 198 35 , , PUNCT cana-761 198 36     SPACE cana-761 198 37 n n CCONJ cana-761 198 38 n n CCONJ cana-761 198 39 n n CCONJ cana-761 198 40 n n CCONJ cana-761 198 41 n n ADV cana-761 198 42 nf nf NOUN cana-761 198 43 w w PROPN cana-761 198 44 q q X cana-761 198 45 q q PROPN cana-761 199 1 t t X cana-761 200 1 f f PROPN cana-761 201 1 w w PROPN cana-761 201 2 q q X cana-761 201 3 q q PROPN cana-761 201 4 t t X cana-761 201 5 f f PROPN cana-761 201 6 w w PROPN cana-761 201 7 q q X cana-761 201 8 q q PROPN cana-761 201 9 t t NOUN cana-761 201 10   NOUN cana-761 201 11 + + CCONJ cana-761 201 12 + + CCONJ cana-761 201 13 − − NOUN cana-761 201 14   NOUN cana-761 201 15 . . PUNCT cana-761 202 1 ( ( PUNCT cana-761 202 2 18 18 NUM cana-761 202 3 ) ) PUNCT cana-761 202 4 taking take VERB cana-761 202 5 the the DET cana-761 202 6 limit limit NOUN cana-761 202 7 from from ADP cana-761 202 8 both both DET cana-761 202 9 sides side NOUN cana-761 202 10 of of ADP cana-761 202 11 ( ( PUNCT cana-761 202 12 18 18 NUM cana-761 202 13 ) ) PUNCT cana-761 202 14 together together ADV cana-761 202 15 with with ADP cana-761 202 16 ( ( PUNCT cana-761 202 17 17 17 NUM cana-761 202 18 ) ) PUNCT cana-761 202 19 , , PUNCT cana-761 202 20 we we PRON cana-761 202 21 get get VERB cana-761 202 22 ( ( PUNCT cana-761 202 23 ) ) PUNCT cana-761 202 24 ( ( PUNCT cana-761 202 25 ) ) PUNCT cana-761 202 26 ( ( PUNCT cana-761 202 27 ) ) PUNCT cana-761 202 28 ( ( PUNCT cana-761 202 29 ) ) PUNCT cana-761 202 30 ( ( PUNCT cana-761 202 31 ) ) PUNCT cana-761 202 32 ( ( PUNCT cana-761 202 33 ) ) PUNCT cana-761 202 34 ( ( PUNCT cana-761 202 35 ) ) PUNCT cana-761 202 36 1 1 NUM cana-761 202 37     SPACE cana-761 202 38 0 0 NUM cana-761 202 39         SPACE cana-761 202 40 0 0 NUM cana-761 202 41 0 0 NUM cana-761 202 42 , , PUNCT cana-761 202 43 f f PROPN cana-761 202 44 a a PRON cana-761 202 45 t t X cana-761 202 46 f f PROPN cana-761 202 47 a a DET cana-761 202 48 t t X cana-761 202 49 f f PROPN cana-761 202 50 a a DET cana-761 202 51 t t NOUN cana-761 202 52   NOUN cana-761 202 53 − − PROPN cana-761 202 54   NOUN cana-761 202 55 − − PROPN cana-761 202 56   NOUN cana-761 202 57 − − NOUN cana-761 202 58 which which PRON cana-761 202 59 means mean VERB cana-761 202 60 that that SCONJ cana-761 202 61 ( ( PUNCT cana-761 202 62 ) ) PUNCT cana-761 202 63 ( ( PUNCT cana-761 202 64 ) ) PUNCT cana-761 202 65 0 0 NUM cana-761 202 66     SPACE cana-761 202 67 0f 0f NOUN cana-761 202 68 a a DET cana-761 202 69 t t NOUN cana-761 202 70 − − PROPN cana-761 203 1 = = PUNCT cana-761 203 2 . . PUNCT cana-761 204 1 this this PRON cana-761 204 2 is be AUX cana-761 204 3 a a DET cana-761 204 4 contradiction contradiction NOUN cana-761 204 5 with with ADP cana-761 204 6 ( ( PUNCT cana-761 204 7 ) ) PUNCT cana-761 204 8 ( ( PUNCT cana-761 204 9 ) ) PUNCT cana-761 204 10 0 0 NUM cana-761 204 11 0f 0f PROPN cana-761 204 12 a a DET cana-761 204 13 t t NOUN cana-761 204 14 − − NOUN cana-761 204 15   INTJ cana-761 204 16 . . PUNCT cana-761 205 1 therefore therefore ADV cana-761 205 2 , , PUNCT cana-761 205 3 we we PRON cana-761 205 4 have have VERB cana-761 205 5 ( ( PUNCT cana-761 205 6 ) ) PUNCT cana-761 205 7 1lim 1lim NUM cana-761 205 8     SPACE cana-761 205 9 , , PUNCT cana-761 205 10     SPACE cana-761 205 11 , , PUNCT cana-761 205 12     SPACE cana-761 205 13 0n 0n NOUN cana-761 205 14 n n CCONJ cana-761 205 15 n n PROPN cana-761 205 16 w w PROPN cana-761 205 17 q q X cana-761 205 18 q q X cana-761 205 19 t+ t+ NOUN cana-761 205 20 → → PUNCT cana-761 205 21 = = PRON cana-761 205 22 . . PUNCT cana-761 206 1 ( ( PUNCT cana-761 206 2 19 19 NUM cana-761 206 3 ) ) PUNCT cana-761 206 4 further far ADV cana-761 206 5 , , PUNCT cana-761 206 6 we we PRON cana-761 206 7 need need VERB cana-761 206 8 to to PART cana-761 206 9 prove prove VERB cana-761 206 10 that that SCONJ cana-761 206 11   PROPN cana-761 206 12 nq nq PROPN cana-761 206 13 is be AUX cana-761 206 14 a a DET cana-761 206 15 cauchy cauchy ADJ cana-761 206 16 sequence sequence NOUN cana-761 206 17 . . PUNCT cana-761 207 1 suppose suppose VERB cana-761 207 2 that that SCONJ cana-761 207 3 this this DET cana-761 207 4 claim claim NOUN cana-761 207 5 is be AUX cana-761 207 6 not not PART cana-761 207 7 true true ADJ cana-761 207 8 . . PUNCT cana-761 208 1 using use VERB cana-761 208 2 lemma lemma PROPN cana-761 208 3 3.1 3.1 NUM cana-761 208 4 and and CCONJ cana-761 208 5 noting note VERB cana-761 208 6 that that SCONJ cana-761 208 7 ( ( PUNCT cana-761 208 8 19 19 NUM cana-761 208 9 ) ) PUNCT cana-761 208 10 is be AUX cana-761 208 11 in in ADP cana-761 208 12 fact fact NOUN cana-761 208 13 condition condition NOUN cana-761 208 14 ( ( PUNCT cana-761 208 15 4 4 NUM cana-761 208 16 ) ) PUNCT cana-761 208 17 , , PUNCT cana-761 208 18 then then ADV cana-761 208 19 there there PRON cana-761 208 20 exist exist VERB cana-761 208 21 ( ( PUNCT cana-761 208 22 ) ) PUNCT cana-761 208 23 00,1 00,1 NOUN cana-761 208 24     SPACE cana-761 208 25 , , PUNCT cana-761 208 26     SPACE cana-761 208 27 0t 0t NOUN cana-761 209 1   NOUN cana-761 209 2   VERB cana-761 209 3 and and CCONJ cana-761 209 4 sequences sequence NOUN cana-761 209 5   PROPN cana-761 209 6   PROPN cana-761 209 7 kmq kmq PROPN cana-761 209 8 and and CCONJ cana-761 209 9   PROPN cana-761 209 10   PROPN cana-761 209 11 knq knq NOUN cana-761 209 12 such such ADJ cana-761 209 13 that that PRON cana-761 209 14 ( ( PUNCT cana-761 209 15 ) ) PUNCT cana-761 209 16 0lim 0lim NOUN cana-761 209 17     SPACE cana-761 209 18 , , PUNCT cana-761 209 19     SPACE cana-761 209 20 , , PUNCT cana-761 209 21     SPACE cana-761 209 22 k k PROPN cana-761 210 1 km km PROPN cana-761 211 1 n n CCONJ cana-761 212 1 k k NOUN cana-761 212 2 w w PROPN cana-761 212 3 q q X cana-761 212 4 q q PROPN cana-761 212 5 t t PROPN cana-761 212 6   PROPN cana-761 212 7 → → PUNCT cana-761 212 8 = = X cana-761 212 9 . . PUNCT cana-761 213 1 by by ADP cana-761 213 2 ( ( PUNCT cana-761 213 3 14 14 NUM cana-761 213 4 ) ) PUNCT cana-761 213 5 , , PUNCT cana-761 213 6 we we PRON cana-761 213 7 have have VERB cana-761 213 8 ( ( PUNCT cana-761 213 9 ) ) PUNCT cana-761 213 10 ( ( PUNCT cana-761 213 11 ) ) PUNCT cana-761 213 12 ( ( PUNCT cana-761 213 13 ) ) PUNCT cana-761 213 14 ( ( PUNCT cana-761 213 15 ) ) PUNCT cana-761 213 16 ( ( PUNCT cana-761 213 17 ) ) PUNCT cana-761 213 18 ( ( PUNCT cana-761 213 19 ) ) PUNCT cana-761 213 20 ( ( PUNCT cana-761 213 21 ) ) PUNCT cana-761 213 22 ( ( PUNCT cana-761 213 23 ) ) PUNCT cana-761 213 24 ( ( PUNCT cana-761 213 25 ) ) PUNCT cana-761 213 26 ( ( PUNCT cana-761 213 27 ) ) PUNCT cana-761 213 28 ( ( PUNCT cana-761 213 29 ) ) PUNCT cana-761 213 30 0 0 NUM cana-761 213 31 0 0 NUM cana-761 213 32 0 0 NUM cana-761 213 33 1 1 NUM cana-761 213 34     SPACE cana-761 213 35 , , PUNCT cana-761 213 36         SPACE cana-761 213 37 , , PUNCT cana-761 213 38         SPACE cana-761 213 39 , , PUNCT cana-761 213 40         SPACE cana-761 213 41 , , PUNCT cana-761 213 42         SPACE cana-761 213 43 , , PUNCT cana-761 213 44     SPACE cana-761 213 45 , , PUNCT cana-761 213 46     SPACE cana-761 213 47 . . PUNCT cana-761 214 1 k k PROPN cana-761 215 1 k k PROPN cana-761 215 2 k k PROPN cana-761 215 3 k k PROPN cana-761 215 4 k k PROPN cana-761 215 5 km km PROPN cana-761 215 6 n n VERB cana-761 215 7 m m VERB cana-761 215 8 n n ADV cana-761 215 9 m m PROPN cana-761 215 10 nf nf PROPN cana-761 215 11 w w PROPN cana-761 215 12 j j PROPN cana-761 215 13 q q PROPN cana-761 215 14 j j PROPN cana-761 215 15 q q PROPN cana-761 215 16 t t PROPN cana-761 215 17 f f PROPN cana-761 215 18 w w PROPN cana-761 215 19 j j PROPN cana-761 215 20 q q PROPN cana-761 215 21 j j PROPN cana-761 215 22 q q PROPN cana-761 216 1 t t PROPN cana-761 216 2 f f PROPN cana-761 216 3 w w PROPN cana-761 216 4 q q X cana-761 216 5 q q PROPN cana-761 216 6 t t PROPN cana-761 216 7   NOUN cana-761 216 8   PROPN cana-761 216 9   NOUN cana-761 216 10 letting let VERB cana-761 216 11 k k PROPN cana-761 216 12 → → PUNCT cana-761 216 13 from from ADP cana-761 216 14 both both DET cana-761 216 15 sides side NOUN cana-761 216 16 of of ADP cana-761 216 17 the the DET cana-761 216 18 above above ADJ cana-761 216 19 inequality inequality NOUN cana-761 216 20 , , PUNCT cana-761 216 21 we we PRON cana-761 216 22 have have VERB cana-761 216 23 ( ( PUNCT cana-761 216 24 ) ) PUNCT cana-761 216 25 ( ( PUNCT cana-761 216 26 ) ) PUNCT cana-761 216 27 ( ( PUNCT cana-761 216 28 ) ) PUNCT cana-761 216 29 ( ( PUNCT cana-761 216 30 ) ) PUNCT cana-761 216 31 ( ( PUNCT cana-761 216 32 ) ) PUNCT cana-761 216 33 ( ( PUNCT cana-761 216 34 ) ) PUNCT cana-761 216 35 ( ( PUNCT cana-761 216 36 ) ) PUNCT cana-761 216 37 1 1 NUM cana-761 216 38 0 0 NUM cana-761 216 39         SPACE cana-761 216 40 0 0 NUM cana-761 216 41             SPACE cana-761 216 42 0f 0f NOUN cana-761 216 43 f f PROPN cana-761 216 44 f f PROPN cana-761 216 45   PROPN cana-761 216 46   PROPN cana-761 216 47   VERB cana-761 216 48 − − PROPN cana-761 216 49   NOUN cana-761 216 50 − − NOUN cana-761 216 51   NOUN cana-761 216 52 − − PROPN cana-761 216 53 , , PUNCT cana-761 216 54 which which PRON cana-761 216 55 establishes establish VERB cana-761 216 56 that that PRON cana-761 216 57 ( ( PUNCT cana-761 216 58 ) ) PUNCT cana-761 216 59 ( ( PUNCT cana-761 216 60 ) ) PUNCT cana-761 216 61     SPACE cana-761 216 62 0 0 NUM cana-761 216 63 0f 0f NOUN cana-761 216 64   PROPN cana-761 216 65 − − PROPN cana-761 217 1 = = PUNCT cana-761 217 2 . . PUNCT cana-761 218 1 this this PRON cana-761 218 2 is be AUX cana-761 218 3 in in ADP cana-761 218 4 contradiction contradiction NOUN cana-761 218 5 with with ADP cana-761 218 6 ( ( PUNCT cana-761 218 7 ) ) PUNCT cana-761 218 8 ( ( PUNCT cana-761 218 9 ) ) PUNCT cana-761 218 10     SPACE cana-761 218 11 0 0 NUM cana-761 218 12 0f 0f NOUN cana-761 218 13   PROPN cana-761 218 14 − − PROPN cana-761 219 1   INTJ cana-761 219 2 . . PUNCT cana-761 220 1 hence hence ADV cana-761 220 2 , , PUNCT cana-761 220 3   PROPN cana-761 220 4 nq nq PROPN cana-761 220 5 is be AUX cana-761 220 6 a a DET cana-761 220 7 cauchy cauchy ADJ cana-761 220 8 sequence sequence NOUN cana-761 220 9 . . PUNCT cana-761 221 1 since since SCONJ cana-761 221 2 ( ( PUNCT cana-761 221 3 ) ) PUNCT cana-761 221 4 , , PUNCT cana-761 221 5 , , PUNCT cana-761 221 6 x x X cana-761 221 7 w w PROPN cana-761 221 8 t t PROPN cana-761 221 9 is be AUX cana-761 221 10 complete complete ADJ cana-761 221 11 , , PUNCT cana-761 221 12 then then ADV cana-761 221 13 there there PRON cana-761 221 14 exists exist VERB cana-761 221 15 q q PUNCT cana-761 221 16 x x PROPN cana-761 221 17 such such ADJ cana-761 221 18 that that SCONJ cana-761 221 19 lim lim PROPN cana-761 221 20     SPACE cana-761 221 21 n n CCONJ cana-761 221 22 n n CCONJ cana-761 221 23 q q NOUN cana-761 221 24 q q NOUN cana-761 221 25 → → PUNCT cana-761 221 26 = = NOUN cana-761 221 27 . . PUNCT cana-761 222 1 ( ( PUNCT cana-761 222 2 20 20 NUM cana-761 222 3 ) ) PUNCT cana-761 222 4 let let VERB cana-761 222 5 us we PRON cana-761 222 6 prove prove VERB cana-761 222 7 that that SCONJ cana-761 222 8 q q NOUN cana-761 222 9 is be AUX cana-761 222 10 a a DET cana-761 222 11 fp fp X cana-761 222 12 of of ADP cana-761 222 13 j j PROPN cana-761 222 14 . . PUNCT cana-761 223 1 as as ADP cana-761 223 2 a a DET cana-761 223 3 matter matter NOUN cana-761 223 4 of of ADP cana-761 223 5 fact fact NOUN cana-761 223 6 , , PUNCT cana-761 223 7 it it PRON cana-761 223 8 follows follow VERB cana-761 223 9 immediately immediately ADV cana-761 223 10 from from ADP cana-761 223 11 ( ( PUNCT cana-761 223 12 20 20 NUM cana-761 223 13 ) ) PUNCT cana-761 223 14 and and CCONJ cana-761 223 15 the the DET cana-761 223 16 continuity continuity NOUN cana-761 223 17 of of ADP cana-761 223 18 j j PROPN cana-761 223 19 that that SCONJ cana-761 223 20 ( ( PUNCT cana-761 223 21 ) ) PUNCT cana-761 223 22 ( ( PUNCT cana-761 223 23 ) ) PUNCT cana-761 223 24 1lim 1lim NUM cana-761 223 25         SPACE cana-761 223 26 lim lim PROPN cana-761 223 27         SPACE cana-761 223 28 n n CCONJ cana-761 223 29 n n CCONJ cana-761 223 30 n n CCONJ cana-761 223 31 n n NOUN cana-761 223 32 q q NOUN cana-761 223 33 q q X cana-761 223 34 j j PROPN cana-761 223 35 q q X cana-761 223 36 f f PROPN cana-761 223 37 q+ q+ PROPN cana-761 223 38 → → PUNCT cana-761 223 39 → → PUNCT cana-761 223 40 = = NOUN cana-761 224 1 = = PUNCT cana-761 224 2 = = NOUN cana-761 224 3 . . PUNCT cana-761 225 1 finally finally ADV cana-761 225 2 , , PUNCT cana-761 225 3 we we PRON cana-761 225 4 prove prove VERB cana-761 225 5 the the DET cana-761 225 6 uniqueness uniqueness NOUN cana-761 225 7 of of ADP cana-761 225 8 the the DET cana-761 225 9 fp fp AUX cana-761 225 10 . . PROPN cana-761 225 11 suppose suppose VERB cana-761 225 12 that that SCONJ cana-761 225 13 q q PROPN cana-761 225 14 and and CCONJ cana-761 225 15 r r NOUN cana-761 225 16 are be AUX cana-761 225 17 distinct distinct ADJ cana-761 225 18 fps fps NOUN cana-761 225 19 of of ADP cana-761 225 20 j j PROPN cana-761 225 21 . . PUNCT cana-761 226 1 again again ADV cana-761 226 2 , , PUNCT cana-761 226 3 by by ADP cana-761 226 4 using use VERB cana-761 226 5 ( ( PUNCT cana-761 226 6 14 14 NUM cana-761 226 7 ) ) PUNCT cana-761 226 8 , , PUNCT cana-761 226 9 we we PRON cana-761 226 10 easily easily ADV cana-761 226 11 obtain obtain VERB cana-761 226 12 that that PRON cana-761 226 13 ( ( PUNCT cana-761 226 14 ) ) PUNCT cana-761 226 15 ( ( PUNCT cana-761 226 16 ) ) PUNCT cana-761 226 17 ( ( PUNCT cana-761 226 18 ) ) PUNCT cana-761 226 19 ( ( PUNCT cana-761 226 20 ) ) PUNCT cana-761 226 21 ( ( PUNCT cana-761 226 22 ) ) PUNCT cana-761 226 23 ( ( PUNCT cana-761 226 24 ) ) PUNCT cana-761 226 25 ( ( PUNCT cana-761 226 26 ) ) PUNCT cana-761 226 27 ( ( PUNCT cana-761 226 28 ) ) PUNCT cana-761 226 29 ( ( PUNCT cana-761 226 30 ) ) PUNCT cana-761 226 31 ( ( PUNCT cana-761 226 32 ) ) PUNCT cana-761 226 33 ( ( PUNCT cana-761 226 34 ) ) PUNCT cana-761 226 35 1 1 NUM cana-761 226 36 ( ( PUNCT cana-761 226 37 , , PUNCT cana-761 226 38     SPACE cana-761 226 39 , , PUNCT cana-761 226 40     SPACE cana-761 226 41 ( ( PUNCT cana-761 226 42 , , PUNCT cana-761 226 43     SPACE cana-761 226 44 , , PUNCT cana-761 226 45     SPACE cana-761 226 46 , , PUNCT cana-761 226 47     SPACE cana-761 226 48 , , PUNCT cana-761 226 49     SPACE cana-761 226 50 f f PROPN cana-761 227 1 w w PROPN cana-761 227 2 j j PROPN cana-761 227 3 q q X cana-761 227 4 j j PROPN cana-761 228 1 r r NOUN cana-761 228 2 t t PROPN cana-761 228 3 f f PROPN cana-761 228 4 w w PROPN cana-761 228 5 j j PROPN cana-761 228 6 q q X cana-761 228 7 j j PROPN cana-761 228 8 r r NOUN cana-761 228 9 t t PROPN cana-761 228 10 f f PROPN cana-761 228 11 w w PROPN cana-761 228 12 q q NOUN cana-761 228 13 r r NOUN cana-761 228 14 t t NOUN cana-761 228 15   NOUN cana-761 228 16   PROPN cana-761 228 17   NOUN cana-761 228 18 . . PUNCT cana-761 229 1 as as ADP cana-761 229 2 a a DET cana-761 229 3 consequence consequence NOUN cana-761 229 4 , , PUNCT cana-761 229 5 we we PRON cana-761 229 6 have have VERB cana-761 229 7 communications communication NOUN cana-761 229 8 on on ADP cana-761 229 9 applied apply VERB cana-761 229 10 nonlinear nonlinear ADJ cana-761 229 11 analysis analysis NOUN cana-761 229 12 issn issn NOUN cana-761 229 13 : : PUNCT cana-761 229 14 1074 1074 NUM cana-761 229 15 - - PUNCT cana-761 229 16 133x 133x NUM cana-761 229 17 vol vol NOUN cana-761 229 18 31 31 NUM cana-761 229 19 no no NOUN cana-761 229 20 . . PUNCT cana-761 230 1 3s 3s NUM cana-761 230 2 ( ( PUNCT cana-761 230 3 2024 2024 NUM cana-761 230 4 ) ) PUNCT cana-761 230 5 234 234 NUM cana-761 230 6 https://internationalpubls.com https://internationalpubls.com X cana-761 230 7 ( ( PUNCT cana-761 230 8 ) ) PUNCT cana-761 230 9 ( ( PUNCT cana-761 230 10 ) ) PUNCT cana-761 230 11 ( ( PUNCT cana-761 230 12 ) ) PUNCT cana-761 230 13 ( ( PUNCT cana-761 230 14 ) ) PUNCT cana-761 230 15 ( ( PUNCT cana-761 230 16 ) ) PUNCT cana-761 230 17 ( ( PUNCT cana-761 230 18 ) ) PUNCT cana-761 230 19 ( ( PUNCT cana-761 230 20 ) ) PUNCT cana-761 230 21 ( ( PUNCT cana-761 230 22 , , PUNCT cana-761 230 23     SPACE cana-761 230 24 , , PUNCT cana-761 230 25     SPACE cana-761 230 26 , , PUNCT cana-761 230 27     SPACE cana-761 230 28 , , PUNCT cana-761 230 29         SPACE cana-761 230 30 ( ( PUNCT cana-761 230 31 , , PUNCT cana-761 230 32     SPACE cana-761 230 33 , , PUNCT cana-761 230 34     SPACE cana-761 230 35 w w PROPN cana-761 230 36 j j PROPN cana-761 230 37 q q X cana-761 230 38 j j PROPN cana-761 230 39 r r NOUN cana-761 230 40 t t PROPN cana-761 230 41 w w NOUN cana-761 230 42 q q NOUN cana-761 230 43 r r NOUN cana-761 230 44 t t PROPN cana-761 230 45 w w PROPN cana-761 230 46 j j PROPN cana-761 230 47 q q X cana-761 230 48 j j PROPN cana-761 230 49 r r NOUN cana-761 230 50 t t PROPN cana-761 230 51 = = PUNCT cana-761 230 52 . . PUNCT cana-761 231 1 this this PRON cana-761 231 2 is be AUX cana-761 231 3 a a DET cana-761 231 4 contradiction contradiction NOUN cana-761 231 5 . . PUNCT cana-761 232 1 remark remark NOUN cana-761 232 2 3.6 3.6 NUM cana-761 232 3 . . PUNCT cana-761 233 1 let let VERB cana-761 233 2 ( ( PUNCT cana-761 233 3 ) ) PUNCT cana-761 233 4 , , PUNCT cana-761 233 5 , , PUNCT cana-761 233 6 x x X cana-761 233 7 w w PROPN cana-761 233 8 t t PROPN cana-761 233 9 be be AUX cana-761 233 10 a a DET cana-761 233 11 rfms rfms NOUN cana-761 233 12 . . PUNCT cana-761 234 1 ( ( PUNCT cana-761 234 2 i i NOUN cana-761 234 3 ) ) PUNCT cana-761 234 4 define define VERB cana-761 234 5 a a DET cana-761 234 6 strictly strictly ADV cana-761 234 7 nonincreasing nonincrease VERB cana-761 234 8 function function NOUN cana-761 234 9 ( ( PUNCT cana-761 234 10 ) ) PUNCT cana-761 234 11 1 1 NUM cana-761 234 12     SPACE cana-761 234 13 1 1 NUM cana-761 234 14 f f NOUN cana-761 234 15 t t NOUN cana-761 234 16 t t PROPN cana-761 234 17 = = PUNCT cana-761 235 1 + + CCONJ cana-761 235 2 for for ADP cana-761 235 3 any any DET cana-761 235 4 ( ( PUNCT cana-761 235 5 ) ) SYM cana-761 235 6 0,1 0,1 NUM cana-761 235 7     SPACE cana-761 235 8 t t NOUN cana-761 235 9 and and CCONJ cana-761 235 10 let let VERB cana-761 235 11 j j PROPN cana-761 235 12 be be AUX cana-761 235 13 a a DET cana-761 235 14 rf rf NOUN cana-761 235 15 - - NOUN cana-761 235 16 contraction contraction NOUN cana-761 235 17 . . PUNCT cana-761 236 1 then then ADV cana-761 236 2 , , PUNCT cana-761 236 3 the the DET cana-761 236 4 rf rf NOUN cana-761 236 5 contraction contraction NOUN cana-761 236 6 ( ( PUNCT cana-761 236 7 1 1 X cana-761 236 8 ) ) PUNCT cana-761 236 9 is be AUX cana-761 236 10 obtained obtain VERB cana-761 236 11 . . PUNCT cana-761 237 1 indeed indeed ADV cana-761 237 2 , , PUNCT cana-761 237 3 since since SCONJ cana-761 237 4 j j PROPN cana-761 237 5 is be AUX cana-761 237 6 rf rf VERB cana-761 237 7 - - ADJ cana-761 237 8 contractive contractive ADJ cana-761 237 9 , , PUNCT cana-761 237 10 then then ADV cana-761 237 11 there there PRON cana-761 237 12 exists exist VERB cana-761 237 13 ( ( PUNCT cana-761 237 14 ) ) PUNCT cana-761 237 15 0,1 0,1 NUM cana-761 237 16     SPACE cana-761 237 17 t t NOUN cana-761 237 18 such such ADJ cana-761 237 19 that that PRON cana-761 237 20 ( ( PUNCT cana-761 237 21 ) ) PUNCT cana-761 237 22 ( ( PUNCT cana-761 237 23 ) ) PUNCT cana-761 237 24 ( ( PUNCT cana-761 237 25 ) ) PUNCT cana-761 237 26 ( ( PUNCT cana-761 237 27 ) ) PUNCT cana-761 237 28 1 1 NUM cana-761 237 29 1 1 NUM cana-761 237 30 1 1 NUM cana-761 237 31 1 1 NUM cana-761 237 32 , , PUNCT cana-761 237 33     SPACE cana-761 237 34 , , PUNCT cana-761 237 35     SPACE cana-761 237 36 1 1 NUM cana-761 237 37 , , PUNCT cana-761 237 38     SPACE cana-761 237 39 , , PUNCT cana-761 237 40     SPACE cana-761 237 41 w w NOUN cana-761 237 42 q q NOUN cana-761 237 43 r r NOUN cana-761 237 44 tw tw NUM cana-761 238 1 j j PROPN cana-761 238 2 q q X cana-761 238 3 j j PROPN cana-761 238 4 r r NOUN cana-761 238 5 t t NOUN cana-761 238 6   NOUN cana-761 238 7   NOUN cana-761 238 8   ADJ cana-761 238 9   NUM cana-761 238 10   PROPN cana-761 238 11   NOUN cana-761 239 1 ++ ++ PROPN cana-761 239 2   PROPN cana-761 239 3   NOUN cana-761 239 4 that that PRON cana-761 239 5 is be AUX cana-761 239 6 , , PUNCT cana-761 239 7 ( ( PUNCT cana-761 239 8 ) ) PUNCT cana-761 239 9 ( ( PUNCT cana-761 239 10 ) ) PUNCT cana-761 239 11 ( ( PUNCT cana-761 239 12 ) ) PUNCT cana-761 239 13 ( ( PUNCT cana-761 239 14 ) ) PUNCT cana-761 239 15 ( ( PUNCT cana-761 239 16 ) ) PUNCT cana-761 239 17 1 1 NUM cana-761 239 18 1 1 NUM cana-761 239 19 , , PUNCT cana-761 239 20     SPACE cana-761 239 21 , , PUNCT cana-761 239 22     SPACE cana-761 239 23 1 1 NUM cana-761 239 24 , , PUNCT cana-761 239 25     SPACE cana-761 239 26 , , PUNCT cana-761 239 27     SPACE cana-761 239 28 w w PROPN cana-761 239 29 j j PROPN cana-761 239 30 q q X cana-761 240 1 j j PROPN cana-761 240 2 r r NOUN cana-761 240 3 t t PROPN cana-761 240 4 w w NOUN cana-761 240 5 q q NOUN cana-761 240 6 r r NOUN cana-761 240 7 t t NOUN cana-761 240 8   NOUN cana-761 240 9 + + CCONJ cana-761 240 10   NUM cana-761 240 11 + + CCONJ cana-761 240 12 therefore therefore ADV cana-761 240 13 , , PUNCT cana-761 240 14 ( ( PUNCT cana-761 240 15 ) ) PUNCT cana-761 240 16 ( ( PUNCT cana-761 240 17 ) ) PUNCT cana-761 240 18 ( ( PUNCT cana-761 240 19 ) ) PUNCT cana-761 240 20 ( ( PUNCT cana-761 240 21 ) ) PUNCT cana-761 240 22 ( ( PUNCT cana-761 240 23 ) ) PUNCT cana-761 240 24 , , PUNCT cana-761 240 25     SPACE cana-761 240 26 , , PUNCT cana-761 240 27     SPACE cana-761 240 28 , , PUNCT cana-761 240 29     SPACE cana-761 240 30 , , PUNCT cana-761 240 31     SPACE cana-761 240 32 w w PROPN cana-761 240 33 j j PROPN cana-761 240 34 q q X cana-761 240 35 j j PROPN cana-761 240 36 r r NOUN cana-761 240 37 t t PROPN cana-761 240 38 w w NOUN cana-761 240 39 q q NOUN cana-761 240 40 r r NOUN cana-761 240 41 t t NOUN cana-761 240 42 holds hold VERB cana-761 240 43 for for ADP cana-761 240 44 all all PRON cana-761 240 45 , , PUNCT cana-761 240 46 , , PUNCT cana-761 240 47 q q NOUN cana-761 240 48 r r NOUN cana-761 240 49 x x NOUN cana-761 240 50 and and CCONJ cana-761 240 51 0 0 NUM cana-761 240 52 t t NOUN cana-761 240 53   X cana-761 240 54 . . PUNCT cana-761 241 1 ( ( PUNCT cana-761 241 2 ii ii NOUN cana-761 241 3 ) ) PUNCT cana-761 241 4 let let VERB cana-761 241 5 ( ( PUNCT cana-761 241 6 ) ) PUNCT cana-761 241 7 1 1 NUM cana-761 241 8     SPACE cana-761 241 9 1 1 NUM cana-761 241 10 t t NOUN cana-761 241 11 t t NOUN cana-761 241 12 = = PUNCT cana-761 242 1 + + CCONJ cana-761 242 2 , , PUNCT cana-761 242 3 where where SCONJ cana-761 242 4 ( ( PUNCT cana-761 242 5 ) ) PUNCT cana-761 242 6 0,1 0,1 NUM cana-761 242 7     SPACE cana-761 242 8 t t NOUN cana-761 242 9 , , PUNCT cana-761 242 10 and and CCONJ cana-761 242 11 suppose suppose VERB cana-761 242 12 that that SCONJ cana-761 242 13 j j PROPN cana-761 242 14 is be AUX cana-761 242 15 a a DET cana-761 242 16 rf rf NOUN cana-761 242 17 - - NOUN cana-761 242 18 contraction contraction NOUN cana-761 242 19 . . PUNCT cana-761 243 1 then then ADV cana-761 243 2 we we PRON cana-761 243 3 easily easily ADV cana-761 243 4 obtain obtain VERB cana-761 243 5 the the DET cana-761 243 6 tirado tirado NOUN cana-761 243 7 contraction contraction NOUN cana-761 243 8 . . PUNCT cana-761 244 1 corollary corollary ADJ cana-761 244 2 3.7 3.7 NUM cana-761 244 3 . . PUNCT cana-761 245 1 let let VERB cana-761 245 2 ( ( PUNCT cana-761 245 3 ) ) PUNCT cana-761 245 4 , , PUNCT cana-761 245 5     SPACE cana-761 245 6 x x PUNCT cana-761 246 1 d d NOUN cana-761 246 2 be be AUX cana-761 246 3 a a DET cana-761 246 4 complete complete ADJ cana-761 246 5 rfms rfms NOUN cana-761 246 6 , , PUNCT cana-761 246 7 and and CCONJ cana-761 246 8 : : PUNCT cana-761 246 9     SPACE cana-761 246 10 j j PROPN cana-761 246 11 x x X cana-761 246 12 x→ x→ PUNCT cana-761 246 13 be be AUX cana-761 246 14 a a DET cana-761 246 15 function function NOUN cana-761 246 16 such such ADJ cana-761 246 17 that that SCONJ cana-761 246 18 there there PRON cana-761 246 19 exists exist VERB cana-761 246 20 ( ( PUNCT cana-761 246 21 ) ) PUNCT cana-761 246 22 ( ( PUNCT cana-761 246 23 ) ) PUNCT cana-761 246 24 0,1 0,1 NUM cana-761 246 25             SPACE cana-761 246 26 1 1 NUM cana-761 246 27     SPACE cana-761 246 28 , , PUNCT cana-761 246 29     SPACE cana-761 246 30 2 2 NUM cana-761 246 31 , , PUNCT cana-761 246 32     SPACE cana-761 246 33 3 3 NUM cana-761 246 34 , , PUNCT cana-761 246 35     SPACE cana-761 246 36 4ik 4ik ADJ cana-761 246 37 i i PROPN cana-761 246 38 = = PUNCT cana-761 246 39 and and CCONJ cana-761 246 40 for for ADP cana-761 246 41 all all PRON cana-761 246 42 , , PUNCT cana-761 246 43     SPACE cana-761 246 44 , , PUNCT cana-761 246 45         SPACE cana-761 246 46 q q NOUN cana-761 246 47 r r NOUN cana-761 246 48 x x X cana-761 246 49 q q X cana-761 246 50 r r PROPN cana-761 246 51   PROPN cana-761 246 52 , , PUNCT cana-761 246 53 and and CCONJ cana-761 246 54 one one NUM cana-761 246 55 of of ADP cana-761 246 56 the the DET cana-761 246 57 followings following NOUN cana-761 246 58 : : PUNCT cana-761 246 59 ( ( PUNCT cana-761 246 60 1 1 X cana-761 246 61 ) ) PUNCT cana-761 246 62 ( ( PUNCT cana-761 246 63 ) ) PUNCT cana-761 246 64 ( ( PUNCT cana-761 246 65 ) ) PUNCT cana-761 246 66 ( ( PUNCT cana-761 246 67 ) ) PUNCT cana-761 246 68 ( ( PUNCT cana-761 246 69 ) ) PUNCT cana-761 246 70 ( ( PUNCT cana-761 246 71 ) ) PUNCT cana-761 246 72 1 1 NUM cana-761 246 73 1 1 NUM cana-761 246 74 , , PUNCT cana-761 246 75     SPACE cana-761 246 76 , , PUNCT cana-761 246 77             SPACE cana-761 246 78 , , PUNCT cana-761 246 79     SPACE cana-761 246 80 , , PUNCT cana-761 246 81     SPACE cana-761 246 82 w w PROPN cana-761 246 83 j j PROPN cana-761 246 84 q q X cana-761 246 85 j j PROPN cana-761 246 86 r r NOUN cana-761 246 87 t t PROPN cana-761 246 88 w w NOUN cana-761 246 89 q q NOUN cana-761 246 90 r r NOUN cana-761 246 91 t t X cana-761 246 92 k k X cana-761 246 93   NOUN cana-761 246 94 ; ; PUNCT cana-761 246 95 ( ( PUNCT cana-761 246 96 2 2 X cana-761 246 97 ) ) PUNCT cana-761 246 98 ( ( PUNCT cana-761 246 99 ) ) PUNCT cana-761 246 100 ( ( PUNCT cana-761 246 101 ) ) PUNCT cana-761 246 102 ( ( PUNCT cana-761 246 103 ) ) PUNCT cana-761 246 104   PROPN cana-761 246 105   PROPN cana-761 246 106 ( ( PUNCT cana-761 246 107 ) ) PUNCT cana-761 246 108 ( ( PUNCT cana-761 246 109 ) ) PUNCT cana-761 246 110 ( ( PUNCT cana-761 246 111 ) ) PUNCT cana-761 246 112   PROPN cana-761 246 113   PROPN cana-761 246 114 ( ( PUNCT cana-761 246 115 ) ) PUNCT cana-761 246 116   PROPN cana-761 246 117   PROPN cana-761 246 118 ( ( PUNCT cana-761 246 119 ) ) PUNCT cana-761 246 120   PROPN cana-761 246 121   PROPN cana-761 246 122 , , PUNCT cana-761 246 123 , , PUNCT cana-761 246 124 , , PUNCT cana-761 246 125 , , PUNCT cana-761 246 126 , , PUNCT cana-761 246 127 , , PUNCT cana-761 246 128 , , PUNCT cana-761 246 129 , , PUNCT cana-761 246 130 2 2 NUM cana-761 246 131 1 1 NUM cana-761 246 132             SPACE cana-761 246 133 11 11 NUM cana-761 246 134 w w PROPN cana-761 246 135 j j PROPN cana-761 246 136 q q X cana-761 246 137 j j PROPN cana-761 246 138 r r NOUN cana-761 246 139 t t PROPN cana-761 246 140 w w NOUN cana-761 246 141 q q NOUN cana-761 246 142 r r NOUN cana-761 246 143 t t PROPN cana-761 246 144 w w NOUN cana-761 246 145 q q NOUN cana-761 246 146 r r NOUN cana-761 246 147 tw tw X cana-761 246 148 j j PROPN cana-761 246 149 q q X cana-761 246 150 j j PROPN cana-761 246 151 r r NOUN cana-761 246 152 t t X cana-761 246 153 ex ex X cana-761 246 154 e e X cana-761 246 155 e e X cana-761 246 156 k k PROPN cana-761 246 157 e e PROPN cana-761 246 158 p p NOUN cana-761 246 159 − − PROPN cana-761 246 160 − − PROPN cana-761 246 161 −− −− PROPN cana-761 246 162   NOUN cana-761 246 163   NOUN cana-761 246 164   NUM cana-761 246 165   PROPN cana-761 247 1   PROPN cana-761 247 2   PROPN cana-761 247 3 ++ ++ PROPN cana-761 247 4   PROPN cana-761 247 5   NOUN cana-761 247 6 ; ; PUNCT cana-761 247 7 ( ( PUNCT cana-761 247 8 3 3 X cana-761 247 9 ) ) PUNCT cana-761 247 10 ( ( PUNCT cana-761 247 11 ) ) PUNCT cana-761 247 12 ( ( PUNCT cana-761 247 13 ) ) PUNCT cana-761 247 14 ( ( PUNCT cana-761 247 15 ) ) PUNCT cana-761 247 16   PROPN cana-761 247 17   PROPN cana-761 247 18 ( ( PUNCT cana-761 247 19 ) ) PUNCT cana-761 247 20 ( ( PUNCT cana-761 247 21 ) ) PUNCT cana-761 247 22 ( ( PUNCT cana-761 247 23 ) ) PUNCT cana-761 247 24   PROPN cana-761 247 25   PROPN cana-761 247 26 ( ( PUNCT cana-761 247 27 ) ) PUNCT cana-761 247 28 ( ( PUNCT cana-761 247 29 ) ) PUNCT cana-761 247 30 ( ( PUNCT cana-761 247 31 ) ) PUNCT cana-761 247 32   PROPN cana-761 247 33   PROPN cana-761 247 34 ( ( PUNCT cana-761 247 35 ) ) PUNCT cana-761 247 36 ( ( PUNCT cana-761 247 37 ) ) PUNCT cana-761 247 38 ( ( PUNCT cana-761 247 39 ) ) PUNCT cana-761 247 40   PROPN cana-761 247 41   PROPN cana-761 247 42 , , PUNCT cana-761 247 43 , , PUNCT cana-761 247 44 , , PUNCT cana-761 247 45 , , PUNCT cana-761 247 46 , , PUNCT cana-761 247 47 , , PUNCT cana-761 247 48 , , PUNCT cana-761 247 49 , , PUNCT cana-761 247 50 3 3 NUM cana-761 247 51 w w PROPN cana-761 247 52 j j PROPN cana-761 247 53 q q X cana-761 248 1 j j PROPN cana-761 248 2 r r NOUN cana-761 248 3 t t PROPN cana-761 248 4 w w PROPN cana-761 248 5 j j PROPN cana-761 248 6 q q X cana-761 248 7 j j PROPN cana-761 248 8 r r NOUN cana-761 248 9 t t PROPN cana-761 248 10 w w PROPN cana-761 248 11 j j PROPN cana-761 248 12 q q X cana-761 248 13 j j PROPN cana-761 248 14 r r NOUN cana-761 248 15 t t PROPN cana-761 248 16 w w PROPN cana-761 248 17 j j PROPN cana-761 248 18 q q X cana-761 248 19 j j PROPN cana-761 248 20 r r NOUN cana-761 248 21 t t NOUN cana-761 248 22 e e X cana-761 248 23 e e X cana-761 248 24 e e X cana-761 248 25 e e X cana-761 248 26 − − PROPN cana-761 248 27 − − PROPN cana-761 248 28 − − PROPN cana-761 249 1 + + CCONJ cana-761 249 2 − − PROPN cana-761 249 3   NOUN cana-761 249 4 ( ( PUNCT cana-761 249 5 ) ) PUNCT cana-761 249 6   PROPN cana-761 249 7   PROPN cana-761 249 8 ( ( PUNCT cana-761 249 9 ) ) PUNCT cana-761 249 10   PROPN cana-761 249 11   PROPN cana-761 249 12 ( ( PUNCT cana-761 249 13 ) ) PUNCT cana-761 249 14   PROPN cana-761 249 15   PROPN cana-761 249 16 ( ( PUNCT cana-761 249 17 ) ) PUNCT cana-761 249 18   PROPN cana-761 249 19   PROPN cana-761 249 20 , , PUNCT cana-761 249 21 , , PUNCT cana-761 249 22 , , PUNCT cana-761 249 23 , , PUNCT cana-761 249 24 , , PUNCT cana-761 249 25 , , PUNCT cana-761 249 26 , , PUNCT cana-761 249 27 , , PUNCT cana-761 249 28 1 1 NUM cana-761 249 29 w w NOUN cana-761 249 30 q q NOUN cana-761 249 31 r r NOUN cana-761 249 32 t t PROPN cana-761 249 33 w w NOUN cana-761 249 34 q q NOUN cana-761 249 35 r r NOUN cana-761 249 36 t t PROPN cana-761 249 37 w w NOUN cana-761 249 38 q q NOUN cana-761 249 39 r r NOUN cana-761 249 40 t t PROPN cana-761 249 41 w w NOUN cana-761 249 42 q q NOUN cana-761 249 43 r r NOUN cana-761 249 44 t t NOUN cana-761 249 45 e e X cana-761 249 46 e e X cana-761 249 47 e e X cana-761 249 48 e e X cana-761 249 49 − − PROPN cana-761 249 50 − − PROPN cana-761 249 51 − − PROPN cana-761 250 1 + + CCONJ cana-761 250 2 − − PROPN cana-761 250 3 ; ; PUNCT cana-761 250 4 ( ( PUNCT cana-761 250 5 4 4 X cana-761 250 6 ) ) PUNCT cana-761 250 7 ( ( PUNCT cana-761 250 8 ) ) PUNCT cana-761 250 9 ( ( PUNCT cana-761 250 10 ) ) PUNCT cana-761 250 11 ( ( PUNCT cana-761 250 12 ) ) PUNCT cana-761 250 13   PROPN cana-761 250 14   PROPN cana-761 250 15 , , PUNCT cana-761 250 16 , , PUNCT cana-761 250 17 exp exp PROPN cana-761 250 18 w w PROPN cana-761 250 19 j j PROPN cana-761 250 20 q q PROPN cana-761 250 21 j j PROPN cana-761 250 22 r r NOUN cana-761 250 23 t t PROPN cana-761 250 24 . . PUNCT cana-761 251 1 then then ADV cana-761 251 2 , , PUNCT cana-761 251 3 j j PROPN cana-761 251 4 has have VERB cana-761 251 5 a a DET cana-761 251 6 unique unique ADJ cana-761 251 7 fp fp NOUN cana-761 251 8 in in ADP cana-761 251 9 x x X cana-761 251 10 . . PUNCT cana-761 252 1 proof proof NOUN cana-761 252 2 . . PUNCT cana-761 253 1 for for ADP cana-761 253 2 cases case NOUN cana-761 253 3 ( ( PUNCT cana-761 253 4 1)–(4 1)–(4 NUM cana-761 253 5 ) ) PUNCT cana-761 253 6 , , PUNCT cana-761 253 7 put put VERB cana-761 253 8 communications communication NOUN cana-761 253 9 on on ADP cana-761 253 10 applied apply VERB cana-761 253 11 nonlinear nonlinear ADJ cana-761 253 12 analysis analysis NOUN cana-761 253 13 issn issn NOUN cana-761 253 14 : : PUNCT cana-761 253 15 1074 1074 NUM cana-761 253 16 - - PUNCT cana-761 253 17 133x 133x NUM cana-761 253 18 vol vol NOUN cana-761 253 19 31 31 NUM cana-761 253 20 no no NOUN cana-761 253 21 . . PUNCT cana-761 254 1 3s 3s NUM cana-761 254 2 ( ( PUNCT cana-761 254 3 2024 2024 NUM cana-761 254 4 ) ) PUNCT cana-761 254 5 235 235 NUM cana-761 254 6 https://internationalpubls.com https://internationalpubls.com X cana-761 254 7 ( ( PUNCT cana-761 254 8 ) ) PUNCT cana-761 254 9 ( ( PUNCT cana-761 254 10 ) ) PUNCT cana-761 254 11   PROPN cana-761 254 12   PROPN cana-761 254 13   PROPN cana-761 254 14   PROPN cana-761 254 15 ( ( PUNCT cana-761 254 16 ) ) PUNCT cana-761 254 17   PROPN cana-761 254 18   PROPN cana-761 254 19   PROPN cana-761 254 20   PROPN cana-761 254 21   PROPN cana-761 254 22   PROPN cana-761 254 23   PROPN cana-761 254 24   PROPN cana-761 254 25 ( ( PUNCT cana-761 254 26 ) ) PUNCT cana-761 254 27   PROPN cana-761 254 28   PROPN cana-761 254 29 1 1 NUM cana-761 254 30 2 2 NUM cana-761 254 31 3 3 NUM cana-761 254 32 4 4 NUM cana-761 254 33 , , PUNCT cana-761 254 34     SPACE cana-761 254 35 , , PUNCT cana-761 254 36     SPACE cana-761 254 37 , , PUNCT cana-761 254 38     SPACE cana-761 254 39 , , PUNCT cana-761 254 40 1 1 NUM cana-761 254 41 3 3 NUM cana-761 254 42 t t NOUN cana-761 254 43 t t PROPN cana-761 254 44 t t PROPN cana-761 254 45 t t PROPN cana-761 254 46 t t PROPN cana-761 254 47 t t PROPN cana-761 254 48 t t PROPN cana-761 254 49 exp exp NOUN cana-761 254 50 exp exp NOUN cana-761 254 51 exp exp NOUN cana-761 254 52 f f PROPN cana-761 254 53 t t PROPN cana-761 254 54 t t PROPN cana-761 255 1 f f PROPN cana-761 255 2 t t PROPN cana-761 256 1 f f PROPN cana-761 256 2 t t PROPN cana-761 256 3 f f PROPN cana-761 256 4 t t PROPN cana-761 256 5 exp exp NOUN cana-761 256 6 exp exp NOUN cana-761 256 7 exp exp NOUN cana-761 256 8 exp exp NOUN cana-761 256 9 − − PROPN cana-761 256 10 − − PROPN cana-761 256 11 − − PROPN cana-761 256 12 − − NOUN cana-761 256 13 − − PROPN cana-761 257 1 = = PUNCT cana-761 257 2 = = PUNCT cana-761 257 3 = = PUNCT cana-761 257 4 = = PUNCT cana-761 258 1 + + PUNCT cana-761 258 2 + + CCONJ cana-761 258 3 − − NOUN cana-761 258 4 respectively respectively ADV cana-761 258 5 . . PUNCT cana-761 259 1 using use VERB cana-761 259 2 theorem theorem NOUN cana-761 259 3 3.5 3.5 NUM cana-761 259 4 , , PUNCT cana-761 259 5 we we PRON cana-761 259 6 claim claim VERB cana-761 259 7 that that SCONJ cana-761 259 8 j j PROPN cana-761 259 9 has have VERB cana-761 259 10 a a DET cana-761 259 11 unique unique ADJ cana-761 259 12 fp fp NOUN cana-761 259 13 . . PROPN cana-761 259 14 example example NOUN cana-761 259 15 3.8 3.8 NUM cana-761 259 16 . . PUNCT cana-761 260 1 let let VERB cana-761 260 2         SPACE cana-761 260 3 x x SYM cana-761 260 4 r= r= ADJ cana-761 260 5 and and CCONJ cana-761 260 6 define define VERB cana-761 260 7 the the DET cana-761 260 8 usual usual ADJ cana-761 260 9 metric metric NOUN cana-761 260 10 ( ( PUNCT cana-761 260 11 ) ) PUNCT cana-761 260 12 , , PUNCT cana-761 260 13     SPACE cana-761 260 14 d d NOUN cana-761 260 15 q q X cana-761 260 16 r r NOUN cana-761 260 17 q q X cana-761 260 18 r= r= ADJ cana-761 260 19 − − PROPN cana-761 260 20 for for ADP cana-761 260 21 all all PRON cana-761 260 22 , , PUNCT cana-761 260 23     SPACE cana-761 260 24 q q NOUN cana-761 261 1 r r NOUN cana-761 261 2 x x PROPN cana-761 261 3 . . PUNCT cana-761 262 1 let let VERB cana-761 262 2 t t NOUN cana-761 262 3 be be AUX cana-761 262 4 a a DET cana-761 262 5 product product NOUN cana-761 262 6 t t NOUN cana-761 262 7 - - PUNCT cana-761 262 8 conorm conorm NOUN cana-761 262 9 . . PUNCT cana-761 263 1 define define VERB cana-761 263 2 a a DET cana-761 263 3 rfm rfm PROPN cana-761 263 4 as as SCONJ cana-761 263 5 follows follow VERB cana-761 263 6 : : PUNCT cana-761 263 7 ( ( PUNCT cana-761 263 8 ) ) PUNCT cana-761 263 9 ( ( PUNCT cana-761 263 10 ) ) PUNCT cana-761 263 11 ( ( PUNCT cana-761 263 12 ) ) PUNCT cana-761 263 13 , , PUNCT cana-761 263 14     SPACE cana-761 263 15 , , PUNCT cana-761 263 16             SPACE cana-761 263 17 1 1 NUM cana-761 263 18         SPACE cana-761 263 19 1 1 NUM cana-761 263 20 , , PUNCT cana-761 263 21     SPACE cana-761 263 22 , , PUNCT cana-761 263 23         SPACE cana-761 263 24 1 1 NUM cana-761 263 25 d d NOUN cana-761 263 26 q q NOUN cana-761 263 27 r r NOUN cana-761 263 28 d d X cana-761 263 29 q q NOUN cana-761 263 30 r r NOUN cana-761 263 31 t t NOUN cana-761 263 32 t t NOUN cana-761 263 33 w w NOUN cana-761 263 34 q q NOUN cana-761 263 35 r r NOUN cana-761 263 36 t t NOUN cana-761 263 37 exp exp NOUN cana-761 263 38 exp exp NOUN cana-761 263 39   ADV cana-761 263 40   NOUN cana-761 263 41   NOUN cana-761 264 1   ADJ cana-761 264 2   NUM cana-761 264 3   NUM cana-761 264 4   NUM cana-761 264 5 − − NOUN cana-761 264 6   INTJ cana-761 265 1   NUM cana-761 265 2   PUNCT cana-761 266 1 + + PUNCT cana-761 267 1 + + ADJ cana-761 267 2   NUM cana-761 267 3   NUM cana-761 267 4   NUM cana-761 267 5   PROPN cana-761 267 6   PROPN cana-761 268 1   PROPN cana-761 268 2   PROPN cana-761 268 3   NOUN cana-761 268 4   PROPN cana-761 268 5   PROPN cana-761 268 6 = = PROPN cana-761 268 7 − − PROPN cana-761 269 1   PROPN cana-761 269 2   PROPN cana-761 270 1   PROPN cana-761 270 2   NOUN cana-761 270 3 , , PUNCT cana-761 270 4 where where SCONJ cana-761 270 5 , , PUNCT cana-761 270 6     SPACE cana-761 270 7 q q NOUN cana-761 270 8 r r NOUN cana-761 270 9 x x NOUN cana-761 270 10 , , PUNCT cana-761 270 11 and and CCONJ cana-761 270 12         SPACE cana-761 270 13 0t 0t NOUN cana-761 270 14 . . PUNCT cana-761 271 1 clearly clearly ADV cana-761 271 2 , , PUNCT cana-761 271 3 ( ( PUNCT cana-761 271 4 ) ) PUNCT cana-761 271 5 , , PUNCT cana-761 271 6     SPACE cana-761 271 7 , , PUNCT cana-761 271 8     SPACE cana-761 271 9 w w NOUN cana-761 271 10 q q NOUN cana-761 271 11 r r NOUN cana-761 271 12 t t NOUN cana-761 271 13 satisfies satisfy VERB cana-761 271 14 the the DET cana-761 271 15 conditions condition NOUN cana-761 271 16 of of ADP cana-761 271 17 ( ( PUNCT cana-761 271 18 rf rf NUM cana-761 271 19 1)–(rf 1)–(rf NUM cana-761 271 20 3 3 NUM cana-761 271 21 ) ) PUNCT cana-761 271 22 and and CCONJ cana-761 271 23 ( ( PUNCT cana-761 271 24 rf rf NUM cana-761 271 25 5 5 NUM cana-761 271 26 ) ) PUNCT cana-761 271 27 . . PUNCT cana-761 272 1 moreover moreover ADV cana-761 272 2 , , PUNCT cana-761 272 3 for for ADP cana-761 272 4 all all PRON cana-761 272 5 , , PUNCT cana-761 272 6     SPACE cana-761 272 7 , , PUNCT cana-761 272 8 q q NOUN cana-761 272 9 r r NOUN cana-761 272 10 l l NOUN cana-761 272 11 x x NOUN cana-761 272 12 and and CCONJ cana-761 272 13 , , PUNCT cana-761 272 14             SPACE cana-761 272 15 0 0 NUM cana-761 272 16 t t NOUN cana-761 272 17 s s NOUN cana-761 272 18 , , PUNCT cana-761 272 19 it it PRON cana-761 272 20 is be AUX cana-761 272 21 clear clear ADJ cana-761 272 22 that that SCONJ cana-761 272 23 ( ( PUNCT cana-761 272 24 ) ) PUNCT cana-761 272 25 ( ( PUNCT cana-761 272 26 ) ) PUNCT cana-761 272 27 ( ( PUNCT cana-761 272 28 ) ) PUNCT cana-761 272 29 , , PUNCT cana-761 272 30     SPACE cana-761 272 31 , , PUNCT cana-761 272 32         SPACE cana-761 272 33 1 1 NUM cana-761 272 34     SPACE cana-761 272 35 1 1 NUM cana-761 272 36 , , PUNCT cana-761 272 37     SPACE cana-761 272 38 , , PUNCT cana-761 272 39         SPACE cana-761 273 1 1 1 NUM cana-761 273 2 d d NOUN cana-761 273 3 q q PROPN cana-761 273 4 l l NOUN cana-761 273 5 d d X cana-761 273 6 q q X cana-761 273 7 l l NOUN cana-761 273 8 t t NOUN cana-761 273 9 s s PROPN cana-761 273 10 t t PROPN cana-761 273 11 s s PROPN cana-761 273 12 w w PROPN cana-761 273 13 q q PROPN cana-761 273 14 l l NOUN cana-761 273 15 t t NOUN cana-761 273 16 s s X cana-761 273 17 e e X cana-761 273 18 e e X cana-761 273 19   NOUN cana-761 274 1   NOUN cana-761 274 2   NOUN cana-761 275 1   ADJ cana-761 275 2   NUM cana-761 275 3   NUM cana-761 275 4   NUM cana-761 275 5 − − NOUN cana-761 275 6   INTJ cana-761 276 1   NUM cana-761 276 2   PUNCT cana-761 277 1 + + PUNCT cana-761 278 1 + + PUNCT cana-761 279 1 + + PUNCT cana-761 280 1 + + ADJ cana-761 280 2   NUM cana-761 280 3   NUM cana-761 280 4   NUM cana-761 280 5   PROPN cana-761 280 6   PROPN cana-761 281 1   PROPN cana-761 281 2   PROPN cana-761 281 3   NOUN cana-761 281 4   NOUN cana-761 281 5   PROPN cana-761 281 6 + + PROPN cana-761 281 7 = = SYM cana-761 281 8 − − PROPN cana-761 281 9   PROPN cana-761 281 10   PROPN cana-761 282 1   PROPN cana-761 282 2   PROPN cana-761 282 3 ( ( PUNCT cana-761 282 4 ) ) PUNCT cana-761 282 5 ( ( PUNCT cana-761 282 6 ) ) PUNCT cana-761 282 7     SPACE cana-761 282 8 , , PUNCT cana-761 282 9     SPACE cana-761 282 10 , , PUNCT cana-761 282 11         SPACE cana-761 282 12 . . PUNCT cana-761 282 13     SPACE cana-761 282 14 , , PUNCT cana-761 282 15     SPACE cana-761 282 16 , , PUNCT cana-761 282 17     SPACE cana-761 282 18 , , PUNCT cana-761 282 19 w w PROPN cana-761 282 20 q q X cana-761 282 21 r r NOUN cana-761 282 22 t t NOUN cana-761 282 23 w w NOUN cana-761 282 24 r r NOUN cana-761 282 25 l l NOUN cana-761 282 26 s= s= NOUN cana-761 282 27 that that PRON cana-761 282 28 is be AUX cana-761 282 29 , , PUNCT cana-761 282 30 condition condition NOUN cana-761 282 31 ( ( PUNCT cana-761 282 32 rf rf NOUN cana-761 282 33 4 4 NUM cana-761 282 34 ) ) PUNCT cana-761 282 35 holds hold VERB cana-761 282 36 . . PUNCT cana-761 283 1 let let VERB cana-761 283 2 ( ( PUNCT cana-761 283 3 ) ) PUNCT cana-761 283 4 ( ( PUNCT cana-761 283 5 ) ) PUNCT cana-761 283 6 ( ( PUNCT cana-761 283 7 ) ) PUNCT cana-761 283 8 1 1 NUM cana-761 283 9 1 1 NUM cana-761 283 10 1 1 NUM cana-761 283 11     SPACE cana-761 283 12 , , PUNCT cana-761 283 13         SPACE cana-761 283 14 ( ( PUNCT cana-761 283 15 0 0 NUM cana-761 283 16             SPACE cana-761 283 17 1 1 NUM cana-761 283 18     SPACE cana-761 283 19 ) ) PUNCT cana-761 283 20         SPACE cana-761 283 21 4 4 NUM cana-761 283 22     SPACE cana-761 283 23 2 2 NUM cana-761 283 24 j j NOUN cana-761 283 25 x x PUNCT cana-761 283 26 x x PUNCT cana-761 283 27 x x PUNCT cana-761 283 28 x x SYM cana-761 283 29 f f X cana-761 283 30 y y PROPN cana-761 283 31 y y PROPN cana-761 283 32 and and CCONJ cana-761 283 33 ln ln PROPN cana-761 283 34 y y PROPN cana-761 283 35 = = NOUN cana-761 283 36   PROPN cana-761 283 37 = = PUNCT cana-761 283 38 − − PROPN cana-761 283 39   PROPN cana-761 283 40   PROPN cana-761 283 41 = = PUNCT cana-761 283 42 . . PUNCT cana-761 284 1 since since SCONJ cana-761 284 2 ( ( PUNCT cana-761 284 3 ) ) PUNCT cana-761 284 4 ( ( PUNCT cana-761 284 5 ) ) PUNCT cana-761 284 6 ( ( PUNCT cana-761 284 7 ) ) PUNCT cana-761 284 8 ( ( PUNCT cana-761 284 9 ) ) PUNCT cana-761 284 10 ( ( PUNCT cana-761 284 11 ) ) PUNCT cana-761 284 12 ( ( PUNCT cana-761 284 13 ) ) PUNCT cana-761 284 14 , , PUNCT cana-761 284 15     SPACE cana-761 284 16 , , PUNCT cana-761 284 17     SPACE cana-761 284 18 , , PUNCT cana-761 284 19     SPACE cana-761 284 20 , , PUNCT cana-761 284 21         SPACE cana-761 284 22 f f PROPN cana-761 284 23 w w PROPN cana-761 284 24 f f PROPN cana-761 284 25 q q X cana-761 284 26 f f PROPN cana-761 284 27 r r NOUN cana-761 284 28 t t PROPN cana-761 284 29 f f PROPN cana-761 284 30 w w NOUN cana-761 284 31 q q NOUN cana-761 284 32 r r NOUN cana-761 284 33 t= t= NOUN cana-761 284 34 holds hold VERB cana-761 284 35 for for ADP cana-761 284 36 all all PRON cana-761 284 37 , , PUNCT cana-761 284 38     SPACE cana-761 284 39 q q NOUN cana-761 284 40 r r NOUN cana-761 284 41 x x PROPN cana-761 284 42 , , PUNCT cana-761 284 43     SPACE cana-761 284 44 q q NOUN cana-761 284 45 r r NOUN cana-761 284 46 and and CCONJ cana-761 284 47         SPACE cana-761 284 48 0t 0t NOUN cana-761 284 49 , , PUNCT cana-761 284 50 then then ADV cana-761 284 51 condition condition NOUN cana-761 284 52 ( ( PUNCT cana-761 284 53 14 14 NUM cana-761 284 54 ) ) PUNCT cana-761 284 55 is be AUX cana-761 284 56 fulfilled fulfil VERB cana-761 284 57 . . PUNCT cana-761 285 1 hence hence ADV cana-761 285 2 , , PUNCT cana-761 285 3 by by ADP cana-761 285 4 theorem theorem NOUN cana-761 285 5 3.5 3.5 NUM cana-761 285 6 , , PUNCT cana-761 285 7 it it PRON cana-761 285 8 follows follow VERB cana-761 285 9 that that SCONJ cana-761 285 10 j j PROPN cana-761 285 11 has have VERB cana-761 285 12 a a DET cana-761 285 13 unique unique ADJ cana-761 285 14 fp fp NOUN cana-761 285 15 . . PUNCT cana-761 286 1 it it PRON cana-761 286 2 is be AUX cana-761 286 3 worth worth ADJ cana-761 286 4 mentioning mention VERB cana-761 286 5 that that SCONJ cana-761 286 6 this this DET cana-761 286 7 example example NOUN cana-761 286 8 is be AUX cana-761 286 9 true true ADJ cana-761 286 10 for for ADP cana-761 286 11 arbitrary arbitrary ADJ cana-761 286 12 function function NOUN cana-761 286 13 ( ( PUNCT cana-761 286 14 ) ) PUNCT cana-761 286 15         SPACE cana-761 286 16 j j PROPN cana-761 286 17 q q PROPN cana-761 286 18 kq= kq= PROPN cana-761 286 19 , , PUNCT cana-761 286 20 where where SCONJ cana-761 286 21 0 0 NUM cana-761 286 22             SPACE cana-761 286 23 1 1 NUM cana-761 286 24     SPACE cana-761 286 25 k k NOUN cana-761 286 26   PROPN cana-761 286 27 is be AUX cana-761 286 28 a a DET cana-761 286 29 constant constant ADJ cana-761 286 30 with with ADP cana-761 286 31         SPACE cana-761 286 32 t t PROPN cana-761 286 33 k k PROPN cana-761 286 34 . . PUNCT cana-761 287 1 example example NOUN cana-761 287 2 3.9 3.9 NUM cana-761 287 3 . . PUNCT cana-761 288 1 let let VERB cana-761 288 2 ( ( PUNCT cana-761 288 3 ) ) PUNCT cana-761 288 4 , , PUNCT cana-761 288 5     SPACE cana-761 288 6 x x PUNCT cana-761 289 1 d d NOUN cana-761 289 2 be be AUX cana-761 289 3 a a DET cana-761 289 4 metric metric ADJ cana-761 289 5 space space NOUN cana-761 289 6 and and CCONJ cana-761 289 7 t t PROPN cana-761 289 8 a a DET cana-761 289 9 t t NOUN cana-761 289 10 - - PUNCT cana-761 289 11 conorm conorm NOUN cana-761 289 12 . . PUNCT cana-761 290 1 then then ADV cana-761 290 2 for for SCONJ cana-761 290 3 all all PRON cana-761 290 4 , , PUNCT cana-761 290 5     SPACE cana-761 290 6 q q NOUN cana-761 290 7 r r NOUN cana-761 290 8 x x PROPN cana-761 290 9 and and CCONJ cana-761 290 10         SPACE cana-761 290 11 0t 0t NOUN cana-761 290 12 , , PUNCT cana-761 290 13 ( ( PUNCT cana-761 290 14 ) ) PUNCT cana-761 290 15 ( ( PUNCT cana-761 290 16 ) ) PUNCT cana-761 290 17 ( ( PUNCT cana-761 290 18 ) ) PUNCT cana-761 290 19 , , PUNCT cana-761 290 20     SPACE cana-761 290 21 , , PUNCT cana-761 290 22     SPACE cana-761 290 23 , , PUNCT cana-761 290 24             SPACE cana-761 290 25 1 1 NUM cana-761 290 26             SPACE cana-761 290 27 , , PUNCT cana-761 290 28     SPACE cana-761 290 29 d d NOUN cana-761 290 30 q q NOUN cana-761 290 31 r r NOUN cana-761 290 32 w w NOUN cana-761 290 33 q q NOUN cana-761 290 34 r r NOUN cana-761 290 35 t t NOUN cana-761 290 36 t t NOUN cana-761 291 1 d d X cana-761 291 2 q q NOUN cana-761 291 3 r r NOUN cana-761 291 4 = = PUNCT cana-761 291 5 + + CCONJ cana-761 291 6 + + NUM cana-761 291 7 defines define VERB cana-761 291 8 a a DET cana-761 291 9 rfm rfm PROPN cana-761 291 10 . . PUNCT cana-761 292 1 define define VERB cana-761 292 2 a a DET cana-761 292 3 function function NOUN cana-761 292 4 ( ( PUNCT cana-761 292 5 ) ) PUNCT cana-761 292 6         SPACE cana-761 292 7 f f PROPN cana-761 293 1 x x X cana-761 293 2 x= x= PUNCT cana-761 293 3 on on ADP cana-761 293 4   ADJ cana-761 293 5 0,1 0,1 NOUN cana-761 293 6     SPACE cana-761 293 7 and and CCONJ cana-761 293 8 let let VERB cana-761 293 9 ( ( PUNCT cana-761 293 10 ) ) PUNCT cana-761 293 11 0,1 0,1 NUM cana-761 293 12     SPACE cana-761 293 13   NOUN cana-761 293 14   NOUN cana-761 293 15 be be AUX cana-761 293 16 a a DET cana-761 293 17 constant constant ADJ cana-761 293 18 . . PUNCT cana-761 294 1 if if SCONJ cana-761 294 2 condition condition NOUN cana-761 294 3 ( ( PUNCT cana-761 294 4 14 14 NUM cana-761 294 5 ) ) PUNCT cana-761 294 6 is be AUX cana-761 294 7 fulfilled fulfil VERB cana-761 294 8 , , PUNCT cana-761 294 9 then then ADV cana-761 294 10 ( ( PUNCT cana-761 294 11 ) ) PUNCT cana-761 294 12 ( ( PUNCT cana-761 294 13 ) ) PUNCT cana-761 294 14 ( ( PUNCT cana-761 294 15 ) ) PUNCT cana-761 294 16 ( ( PUNCT cana-761 294 17 ) ) PUNCT cana-761 294 18 ( ( PUNCT cana-761 294 19 ) ) PUNCT cana-761 294 20 ( ( PUNCT cana-761 294 21 ) ) PUNCT cana-761 294 22 ( ( PUNCT cana-761 294 23 ) ) PUNCT cana-761 294 24 ( ( PUNCT cana-761 294 25 ) ) PUNCT cana-761 294 26 ( ( PUNCT cana-761 294 27 ) ) PUNCT cana-761 294 28 ( ( PUNCT cana-761 294 29 ) ) PUNCT cana-761 294 30 ( ( PUNCT cana-761 294 31 ) ) PUNCT cana-761 294 32 ( ( PUNCT cana-761 294 33 ) ) PUNCT cana-761 294 34 , , PUNCT cana-761 294 35     SPACE cana-761 294 36 , , PUNCT cana-761 294 37     SPACE cana-761 294 38 , , PUNCT cana-761 294 39             SPACE cana-761 294 40 , , PUNCT cana-761 294 41     SPACE cana-761 294 42 , , PUNCT cana-761 294 43     SPACE cana-761 294 44 , , PUNCT cana-761 294 45     SPACE cana-761 294 46 , , PUNCT cana-761 294 47     SPACE cana-761 294 48 , , PUNCT cana-761 294 49     SPACE cana-761 294 50 , , PUNCT cana-761 294 51         SPACE cana-761 294 52 1 1 NUM cana-761 294 53             SPACE cana-761 294 54 , , PUNCT cana-761 294 55     SPACE cana-761 294 56 d d NOUN cana-761 294 57 q q NOUN cana-761 294 58 r r NOUN cana-761 294 59 t t PROPN cana-761 294 60 w w NOUN cana-761 294 61 f f PROPN cana-761 294 62 q q X cana-761 295 1 f f PROPN cana-761 295 2 r r NOUN cana-761 295 3 t t PROPN cana-761 296 1 f f PROPN cana-761 296 2 w w PROPN cana-761 296 3 f f PROPN cana-761 296 4 q q X cana-761 297 1 f f PROPN cana-761 297 2 r r NOUN cana-761 297 3 t t PROPN cana-761 298 1 f f PROPN cana-761 298 2 w w PROPN cana-761 298 3 q q NOUN cana-761 298 4 r r NOUN cana-761 298 5 t t PROPN cana-761 298 6 w w NOUN cana-761 298 7 q q NOUN cana-761 298 8 r r NOUN cana-761 298 9 t t NOUN cana-761 298 10 d d X cana-761 298 11 q q NOUN cana-761 298 12 r r NOUN cana-761 298 13   NOUN cana-761 298 14   NOUN cana-761 298 15   NOUN cana-761 298 16   NOUN cana-761 298 17   PROPN cana-761 298 18   NOUN cana-761 298 19   NOUN cana-761 298 20 = = PUNCT cana-761 298 21   NUM cana-761 298 22 = = PUNCT cana-761 299 1 = = X cana-761 299 2   PROPN cana-761 299 3   PROPN cana-761 299 4   PROPN cana-761 299 5   VERB cana-761 299 6 + + PROPN cana-761 299 7 + + CCONJ cana-761 299 8   ADJ cana-761 299 9   NOUN cana-761 299 10   NOUN cana-761 299 11 communications communication NOUN cana-761 299 12 on on ADP cana-761 299 13 applied apply VERB cana-761 299 14 nonlinear nonlinear ADJ cana-761 299 15 analysis analysis NOUN cana-761 299 16 issn issn NOUN cana-761 299 17 : : PUNCT cana-761 299 18 1074 1074 NUM cana-761 299 19 - - PUNCT cana-761 299 20 133x 133x NUM cana-761 299 21 vol vol NOUN cana-761 299 22 31 31 NUM cana-761 299 23 no no NOUN cana-761 299 24 . . PUNCT cana-761 300 1 3s 3s NUM cana-761 300 2 ( ( PUNCT cana-761 300 3 2024 2024 NUM cana-761 300 4 ) ) PUNCT cana-761 300 5 236 236 NUM cana-761 300 6 https://internationalpubls.com https://internationalpubls.com X cana-761 300 7 holds hold VERB cana-761 300 8 for for ADP cana-761 300 9 all all PRON cana-761 300 10 , , PUNCT cana-761 300 11     SPACE cana-761 300 12 q q NOUN cana-761 300 13 r r NOUN cana-761 300 14 x x PROPN cana-761 300 15 and and CCONJ cana-761 300 16 0 0 NUM cana-761 300 17 t t NOUN cana-761 300 18   X cana-761 300 19 . . PUNCT cana-761 301 1 that that PRON cana-761 301 2 is be AUX cana-761 301 3 to to PART cana-761 301 4 say say VERB cana-761 301 5 , , PUNCT cana-761 301 6 we we PRON cana-761 301 7 obtain obtain VERB cana-761 301 8 the the DET cana-761 301 9 contractive contractive ADJ cana-761 301 10 condition condition NOUN cana-761 301 11 ( ( PUNCT cana-761 301 12 1 1 NUM cana-761 301 13 ) ) PUNCT cana-761 301 14 from from ADP cana-761 301 15 [ [ X cana-761 301 16 22 22 NUM cana-761 301 17 ] ] PUNCT cana-761 301 18 . . PUNCT cana-761 302 1 theorem theorem VERB cana-761 302 2 3.10 3.10 NUM cana-761 302 3 . . PUNCT cana-761 303 1 let let AUX cana-761 303 2 ( ( PUNCT cana-761 303 3 ) ) PUNCT cana-761 303 4 , , PUNCT cana-761 303 5 , , PUNCT cana-761 303 6 x x X cana-761 303 7 w w PROPN cana-761 303 8 t t PROPN cana-761 303 9 be be AUX cana-761 303 10 a a DET cana-761 303 11 complete complete ADJ cana-761 303 12 rfms rfms NOUN cana-761 303 13 and and CCONJ cana-761 303 14 ff ff NOUN cana-761 303 15 be be AUX cana-761 303 16 a a DET cana-761 303 17 continuous continuous ADJ cana-761 303 18 mapping mapping NOUN cana-761 303 19 . . PUNCT cana-761 304 1 if if SCONJ cana-761 304 2     SPACE cana-761 304 3 : : PUNCT cana-761 304 4         SPACE cana-761 304 5 j j X cana-761 304 6 x x X cana-761 304 7 x→ x→ PROPN cana-761 304 8 is be AUX cana-761 304 9 a a DET cana-761 304 10 rfm rfm PROPN cana-761 304 11 , , PUNCT cana-761 304 12 𝐹 𝐹 PROPN cana-761 304 13 −contraction −contraction NOUN cana-761 304 14 , , PUNCT cana-761 304 15 then then ADV cana-761 304 16 j j PROPN cana-761 304 17 has have VERB cana-761 304 18 a a DET cana-761 304 19 unique unique ADJ cana-761 304 20 fixed fix VERB cana-761 304 21 - - PUNCT cana-761 304 22 point point NOUN cana-761 304 23 in in ADP cana-761 304 24 x. x. NOUN cana-761 304 25 proof proof NOUN cana-761 304 26 . . PUNCT cana-761 305 1 choose choose VERB cana-761 305 2 0q 0q PROPN cana-761 305 3 x x PROPN cana-761 305 4 and and CCONJ cana-761 305 5 define define VERB cana-761 305 6 a a DET cana-761 305 7 sequenced sequence VERB cana-761 305 8   PROPN cana-761 305 9 nq nq PROPN cana-761 305 10 by by ADP cana-761 305 11 ( ( PUNCT cana-761 305 12 ) ) PUNCT cana-761 305 13 1n 1n PROPN cana-761 305 14 nq nq PROPN cana-761 305 15 j j PROPN cana-761 305 16 q+ q+ ADP cana-761 305 17 = = PUNCT cana-761 305 18 for for SCONJ cana-761 305 19 all all DET cana-761 305 20 0n 0n NOUN cana-761 305 21 n n PROPN cana-761 305 22 . . PUNCT cana-761 306 1 if if SCONJ cana-761 306 2 1n 1n NUM cana-761 306 3 nq nq X cana-761 306 4 q q NOUN cana-761 307 1 + + NOUN cana-761 307 2 = = NOUN cana-761 307 3 for for ADP cana-761 307 4 some some DET cana-761 307 5 0n 0n NOUN cana-761 307 6 n n PROPN cana-761 307 7 , , PUNCT cana-761 307 8 then then ADV cana-761 307 9 the the DET cana-761 307 10 proof proof NOUN cana-761 307 11 is be AUX cana-761 307 12 finished finish VERB cana-761 307 13 . . PUNCT cana-761 308 1 assume assume VERB cana-761 308 2 that that SCONJ cana-761 308 3 1n 1n NUM cana-761 308 4 nq nq PROPN cana-761 309 1 q q X cana-761 309 2 + + PROPN cana-761 309 3   PROPN cana-761 309 4 for for ADP cana-761 309 5 any any DET cana-761 309 6 0n 0n NOUN cana-761 309 7 n n PROPN cana-761 309 8 . . PUNCT cana-761 310 1 from from ADP cana-761 310 2 the the DET cana-761 310 3 definition definition NOUN cana-761 310 4 of of ADP cana-761 310 5 the the DET cana-761 310 6 f f PROPN cana-761 310 7 - - PUNCT cana-761 310 8 contractive contractive ADJ cana-761 310 9 , , PUNCT cana-761 310 10 we we PRON cana-761 310 11 have have AUX cana-761 310 12 ( ( PUNCT cana-761 310 13 ) ) PUNCT cana-761 310 14 ( ( PUNCT cana-761 310 15 ) ) PUNCT cana-761 310 16 ( ( PUNCT cana-761 310 17 ) ) PUNCT cana-761 310 18 ( ( PUNCT cana-761 310 19 ) ) PUNCT cana-761 310 20 ( ( PUNCT cana-761 310 21 ) ) PUNCT cana-761 310 22 ( ( PUNCT cana-761 310 23 ) ) PUNCT cana-761 310 24 ( ( PUNCT cana-761 310 25 ) ) PUNCT cana-761 310 26 1 1 NUM cana-761 310 27 1 1 NUM cana-761 310 28 1 1 NUM cana-761 310 29 1 1 NUM cana-761 310 30 1 1 NUM cana-761 310 31 , , PUNCT cana-761 310 32     SPACE cana-761 310 33 , , PUNCT cana-761 310 34         SPACE cana-761 310 35 , , PUNCT cana-761 310 36     SPACE cana-761 310 37 , , PUNCT cana-761 310 38     SPACE cana-761 310 39 , , PUNCT cana-761 310 40     SPACE cana-761 310 41 , , PUNCT cana-761 310 42     SPACE cana-761 310 43 n n PRON cana-761 310 44 m m VERB cana-761 310 45 n n VERB cana-761 310 46 m m VERB cana-761 310 47 n n ADV cana-761 310 48 mf mf X cana-761 311 1 w w PROPN cana-761 311 2 q q X cana-761 311 3 q q PROPN cana-761 311 4 t t X cana-761 311 5 f f PROPN cana-761 311 6 w w PROPN cana-761 311 7 q q X cana-761 311 8 q q PROPN cana-761 311 9 t t X cana-761 312 1 f f PROPN cana-761 312 2 w w PROPN cana-761 312 3 q q X cana-761 312 4 q q PROPN cana-761 312 5 t t NOUN cana-761 312 6   NOUN cana-761 312 7 + + X cana-761 313 1 + + PUNCT cana-761 313 2 + + CCONJ cana-761 313 3 + + ADJ cana-761 313 4   PROPN cana-761 313 5   NOUN cana-761 313 6 . . PUNCT cana-761 314 1 then then ADV cana-761 314 2 , , PUNCT cana-761 314 3 ( ( PUNCT cana-761 314 4 ) ) PUNCT cana-761 314 5 ( ( PUNCT cana-761 314 6 ) ) PUNCT cana-761 314 7 1 1 NUM cana-761 314 8 1 1 NUM cana-761 314 9 , , PUNCT cana-761 314 10     SPACE cana-761 314 11 , , PUNCT cana-761 314 12     SPACE cana-761 314 13 , , PUNCT cana-761 314 14     SPACE cana-761 314 15 , , PUNCT cana-761 314 16     SPACE cana-761 314 17 n n PRON cana-761 314 18 m m VERB cana-761 314 19 n n VERB cana-761 314 20 mw mw NOUN cana-761 314 21 q q PROPN cana-761 314 22 q q PROPN cana-761 314 23 t t PROPN cana-761 314 24 w w PROPN cana-761 314 25 q q X cana-761 314 26 q q X cana-761 314 27 t+ t+ NOUN cana-761 314 28 + + CCONJ cana-761 314 29   X cana-761 314 30 . . PUNCT cana-761 315 1 for for ADP cana-761 315 2 any any DET cana-761 315 3 n n ADJ cana-761 315 4 m m NOUN cana-761 315 5 . . PUNCT cana-761 316 1 let let VERB cana-761 316 2 ( ( PUNCT cana-761 316 3 ) ) PUNCT cana-761 316 4 ( ( PUNCT cana-761 316 5 ) ) PUNCT cana-761 316 6 inf inf PROPN cana-761 316 7 , , PUNCT cana-761 316 8     SPACE cana-761 316 9 , , PUNCT cana-761 316 10     SPACE cana-761 316 11 m m VERB cana-761 316 12 n n VERB cana-761 316 13 m m VERB cana-761 316 14 n n ADV cana-761 316 15 m m PROPN cana-761 316 16 a a DET cana-761 316 17 t t NOUN cana-761 316 18 w w NOUN cana-761 316 19 q q X cana-761 316 20 q q PROPN cana-761 316 21 t t NOUN cana-761 316 22   X cana-761 316 23 = = PUNCT cana-761 316 24 . . PUNCT cana-761 317 1 notice notice VERB cana-761 317 2 that that SCONJ cana-761 317 3 ( ( PUNCT cana-761 317 4 ) ) PUNCT cana-761 317 5 ( ( PUNCT cana-761 317 6 ) ) PUNCT cana-761 317 7 1 1 NUM cana-761 317 8 1inf 1inf NUM cana-761 317 9 , , PUNCT cana-761 317 10     SPACE cana-761 317 11 , , PUNCT cana-761 317 12     SPACE cana-761 317 13 inf inf NOUN cana-761 317 14 , , PUNCT cana-761 317 15     SPACE cana-761 317 16 , , PUNCT cana-761 317 17     SPACE cana-761 317 18 n n PRON cana-761 317 19 m m VERB cana-761 317 20 n n VERB cana-761 317 21 m m VERB cana-761 317 22 n n PRON cana-761 317 23 m m NOUN cana-761 317 24 n n PRON cana-761 317 25 m m NOUN cana-761 317 26 w w NOUN cana-761 317 27 q q X cana-761 317 28 q q PROPN cana-761 317 29 t t PROPN cana-761 317 30 w w NOUN cana-761 317 31 q q X cana-761 317 32 q q X cana-761 317 33 t+ t+ NOUN cana-761 317 34 + + CCONJ cana-761 317 35   PROPN cana-761 317 36   PROPN cana-761 317 37   NOUN cana-761 317 38 then then ADV cana-761 317 39 ( ( PUNCT cana-761 317 40 ) ) PUNCT cana-761 317 41 ( ( PUNCT cana-761 317 42 ) ) PUNCT cana-761 317 43 1 1 NUM cana-761 317 44 m m NOUN cana-761 317 45 ma ma PROPN cana-761 317 46 t t PROPN cana-761 317 47 a a DET cana-761 317 48 t+ t+ NOUN cana-761 317 49   NUM cana-761 317 50 , , PUNCT cana-761 317 51 for for ADP cana-761 317 52 any any DET cana-761 317 53 m m NOUN cana-761 317 54 n n PROPN cana-761 317 55 . . PUNCT cana-761 318 1 since since SCONJ cana-761 318 2 ( ( PUNCT cana-761 318 3 ) ) PUNCT cana-761 318 4   PROPN cana-761 318 5 ma ma PROPN cana-761 318 6 t t PROPN cana-761 318 7 is be AUX cana-761 318 8 bounded bound VERB cana-761 318 9 , , PUNCT cana-761 318 10 then then ADV cana-761 318 11 there there PRON cana-761 318 12 exists exist VERB cana-761 318 13 ( ( PUNCT cana-761 318 14 ) ) PUNCT cana-761 318 15   ADJ cana-761 318 16 0,1a 0,1a PROPN cana-761 318 17 t t PROPN cana-761 318 18   NOUN cana-761 318 19 such such ADJ cana-761 318 20 that that SCONJ cana-761 318 21 ( ( PUNCT cana-761 318 22 ) ) PUNCT cana-761 318 23 ( ( PUNCT cana-761 318 24 ) ) PUNCT cana-761 318 25 lim lim PROPN cana-761 318 26 m m PROPN cana-761 318 27 n n ADV cana-761 318 28 a a DET cana-761 318 29 t t NOUN cana-761 318 30 a a DET cana-761 318 31 t t NOUN cana-761 318 32 → → PUNCT cana-761 318 33 = = PUNCT cana-761 318 34 for for ADP cana-761 318 35 all all DET cana-761 318 36 0 0 NUM cana-761 318 37 t t NOUN cana-761 318 38   NOUN cana-761 318 39 . . PUNCT cana-761 319 1 let let VERB cana-761 319 2 us we PRON cana-761 319 3 prove prove VERB cana-761 319 4 that that SCONJ cana-761 319 5 ( ( PUNCT cana-761 319 6 ) ) PUNCT cana-761 319 7 0a 0a PROPN cana-761 319 8 t t NOUN cana-761 319 9 = = PUNCT cana-761 319 10 for for ADP cana-761 319 11 all all DET cana-761 319 12         SPACE cana-761 319 13 0t 0t NOUN cana-761 319 14 . . PUNCT cana-761 320 1 suppose suppose VERB cana-761 320 2 the the DET cana-761 320 3 contrary contrary NOUN cana-761 320 4 , , PUNCT cana-761 320 5 and and CCONJ cana-761 320 6 there there PRON cana-761 320 7 exists exist VERB cana-761 320 8 0s 0s NOUN cana-761 320 9   PRON cana-761 320 10 such such ADJ cana-761 320 11 that that PRON cana-761 320 12 ( ( PUNCT cana-761 320 13 ) ) PUNCT cana-761 320 14 0 0 NUM cana-761 320 15 1a 1a PROPN cana-761 320 16 s s PROPN cana-761 320 17   PROPN cana-761 320 18 . . PUNCT cana-761 321 1 then then ADV cana-761 321 2 by by ADP cana-761 321 3 ( ( PUNCT cana-761 321 4 14 14 NUM cana-761 321 5 ) ) PUNCT cana-761 321 6   ADJ cana-761 321 7 0,1 0,1 NOUN cana-761 321 8     SPACE cana-761 321 9   NOUN cana-761 321 10   NOUN cana-761 321 11 such such ADJ cana-761 321 12 that that PRON cana-761 321 13 for for ADP cana-761 321 14 any any DET cana-761 321 15         SPACE cana-761 321 16 0t 0t NOUN cana-761 321 17 , , PUNCT cana-761 321 18 one one PRON cana-761 321 19 has have VERB cana-761 321 20 ( ( PUNCT cana-761 321 21 ) ) PUNCT cana-761 321 22 ( ( PUNCT cana-761 321 23 ) ) PUNCT cana-761 321 24 ( ( PUNCT cana-761 321 25 ) ) PUNCT cana-761 321 26 ( ( PUNCT cana-761 321 27 ) ) PUNCT cana-761 321 28 1 1 NUM cana-761 321 29 1 1 NUM cana-761 321 30 1 1 NUM cana-761 321 31 1 1 NUM cana-761 321 32 1 1 NUM cana-761 321 33 lim lim PROPN cana-761 321 34 inf inf PROPN cana-761 321 35 , , PUNCT cana-761 321 36     SPACE cana-761 321 37 , , PUNCT cana-761 321 38           SPACE cana-761 321 39 lim lim PROPN cana-761 321 40 inf inf PROPN cana-761 321 41 , , PUNCT cana-761 321 42     SPACE cana-761 321 43 , , PUNCT cana-761 321 44     SPACE cana-761 321 45 n n PRON cana-761 321 46 m m VERB cana-761 321 47 n n PRON cana-761 321 48 m m NOUN cana-761 321 49 m m VERB cana-761 321 50 n n ADV cana-761 321 51 m m NOUN cana-761 321 52 m m VERB cana-761 321 53 n n AUX cana-761 321 54 m m NOUN cana-761 321 55 f f NOUN cana-761 321 56 w w PROPN cana-761 321 57 q q X cana-761 321 58 q q PROPN cana-761 321 59 t t X cana-761 322 1 f f PROPN cana-761 322 2 w w PROPN cana-761 322 3 q q X cana-761 322 4 q q PROPN cana-761 322 5 t t NOUN cana-761 322 6   NOUN cana-761 322 7 + + X cana-761 323 1 + + PUNCT cana-761 323 2 + + CCONJ cana-761 323 3 + + NUM cana-761 323 4 → → NOUN cana-761 323 5   NOUN cana-761 323 6 → → PUNCT cana-761 323 7   PROPN cana-761 323 8   NOUN cana-761 323 9 ( ( PUNCT cana-761 323 10 ) ) PUNCT cana-761 323 11 ( ( PUNCT cana-761 323 12 ) ) PUNCT cana-761 323 13       SPACE cana-761 323 14 lim lim PROPN cana-761 323 15 inf inf PROPN cana-761 323 16 , , PUNCT cana-761 323 17     SPACE cana-761 323 18 , , PUNCT cana-761 323 19     SPACE cana-761 323 20 n n PRON cana-761 323 21 m m VERB cana-761 323 22 m m VERB cana-761 323 23 n n ADV cana-761 323 24 m m NOUN cana-761 323 25 f f NOUN cana-761 323 26 w w NOUN cana-761 323 27 q q X cana-761 323 28 q q X cana-761 323 29 s s PART cana-761 323 30 → → PUNCT cana-761 323 31   ADJ cana-761 323 32   NOUN cana-761 323 33 using use VERB cana-761 323 34 the the DET cana-761 323 35 assumption assumption NOUN cana-761 323 36 that that SCONJ cana-761 323 37 f f PROPN cana-761 323 38 is be AUX cana-761 323 39 continuous continuous ADJ cana-761 323 40 , , PUNCT cana-761 323 41 we we PRON cana-761 323 42 have have VERB cana-761 323 43 ( ( PUNCT cana-761 323 44 ) ) PUNCT cana-761 323 45 ( ( PUNCT cana-761 323 46 ) ) PUNCT cana-761 323 47 ( ( PUNCT cana-761 323 48 ) ) PUNCT cana-761 323 49 ( ( PUNCT cana-761 323 50 ) ) PUNCT cana-761 323 51 ( ( PUNCT cana-761 323 52 ) ) PUNCT cana-761 323 53 ( ( PUNCT cana-761 323 54 ) ) PUNCT cana-761 323 55 1 1 NUM cana-761 323 56 f f PROPN cana-761 323 57 a a PRON cana-761 323 58 s s X cana-761 323 59 f f X cana-761 323 60 a a PRON cana-761 323 61 s s X cana-761 323 62 f f X cana-761 323 63 a a DET cana-761 323 64 s s NOUN cana-761 323 65   NOUN cana-761 323 66   NUM cana-761 323 67   NOUN cana-761 323 68 , , PUNCT cana-761 323 69 which which PRON cana-761 323 70 means mean VERB cana-761 323 71 that that SCONJ cana-761 323 72 ( ( PUNCT cana-761 323 73 ) ) PUNCT cana-761 323 74 ( ( PUNCT cana-761 323 75 ) ) PUNCT cana-761 323 76     SPACE cana-761 323 77 1f 1f NOUN cana-761 323 78 a a DET cana-761 323 79 s s NOUN cana-761 323 80 = = NOUN cana-761 323 81 . . PUNCT cana-761 324 1 this this PRON cana-761 324 2 is be AUX cana-761 324 3 in in ADP cana-761 324 4 contraction contraction NOUN cana-761 324 5 with with ADP cana-761 324 6 ( ( PUNCT cana-761 324 7 ) ) PUNCT cana-761 324 8 ( ( PUNCT cana-761 324 9 ) ) PUNCT cana-761 324 10 1f 1f PROPN cana-761 324 11 a a PRON cana-761 324 12 s s PROPN cana-761 324 13   PROPN cana-761 324 14 . . PUNCT cana-761 325 1 thus thus ADV cana-761 325 2 ( ( PUNCT cana-761 325 3 ) ) PUNCT cana-761 325 4 ( ( PUNCT cana-761 325 5 ) ) PUNCT cana-761 325 6 lim lim PROPN cana-761 325 7 inf inf PROPN cana-761 325 8 , , PUNCT cana-761 325 9     SPACE cana-761 325 10 , , PUNCT cana-761 325 11     SPACE cana-761 325 12 0n 0n NOUN cana-761 325 13 m m VERB cana-761 325 14 m m VERB cana-761 325 15 n n ADV cana-761 325 16 m m PROPN cana-761 325 17 w w NOUN cana-761 325 18 q q PROPN cana-761 325 19 q q X cana-761 325 20 s s VERB cana-761 325 21 → → PUNCT cana-761 325 22   ADJ cana-761 325 23 = = X cana-761 325 24 . . PUNCT cana-761 326 1 for for ADP cana-761 326 2 any any DET cana-761 326 3 0 0 NUM cana-761 326 4 t t NOUN cana-761 326 5   NOUN cana-761 326 6 . . PUNCT cana-761 327 1 consequently consequently ADV cana-761 327 2 , , PUNCT cana-761 327 3 ( ( PUNCT cana-761 327 4 ) ) PUNCT cana-761 327 5 , , PUNCT cana-761 327 6 lim lim PROPN cana-761 327 7       SPACE cana-761 327 8 , , PUNCT cana-761 327 9     SPACE cana-761 327 10 , , PUNCT cana-761 327 11     SPACE cana-761 327 12 0n 0n NOUN cana-761 327 13 m m VERB cana-761 327 14 m m VERB cana-761 327 15 n n PRON cana-761 327 16 w w NOUN cana-761 327 17 q q X cana-761 327 18 q q X cana-761 327 19 s s NOUN cana-761 327 20 → → PUNCT cana-761 327 21 = = PUNCT cana-761 327 22 , , PUNCT cana-761 327 23 for for ADP cana-761 327 24 any any DET cana-761 327 25 0 0 NUM cana-761 327 26 t t NOUN cana-761 327 27   NOUN cana-761 327 28 . . PUNCT cana-761 328 1 thus thus ADV cana-761 328 2   PROPN cana-761 328 3 nq nq PROPN cana-761 328 4 is be AUX cana-761 328 5 a a DET cana-761 328 6 cauchy cauchy ADJ cana-761 328 7 sequence sequence NOUN cana-761 328 8 . . PUNCT cana-761 329 1 since since SCONJ cana-761 329 2 ( ( PUNCT cana-761 329 3 ) ) PUNCT cana-761 329 4 , , PUNCT cana-761 329 5 , , PUNCT cana-761 329 6 w w PROPN cana-761 329 7 x x SYM cana-761 329 8 t t PROPN cana-761 329 9 is be AUX cana-761 329 10 complete complete ADJ cana-761 329 11 , , PUNCT cana-761 329 12 then then ADV cana-761 329 13 there there PRON cana-761 329 14 is be VERB cana-761 329 15 q q PROPN cana-761 329 16 x x PROPN cana-761 329 17 such such ADJ cana-761 329 18 that that SCONJ cana-761 329 19 lim lim PROPN cana-761 329 20 n n CCONJ cana-761 329 21 n n CCONJ cana-761 329 22 n n ADV cana-761 329 23 q q NOUN cana-761 329 24 q q NOUN cana-761 329 25 → → PUNCT cana-761 329 26 = = NOUN cana-761 329 27 . . PUNCT cana-761 330 1 communications communication NOUN cana-761 330 2 on on ADP cana-761 330 3 applied apply VERB cana-761 330 4 nonlinear nonlinear ADJ cana-761 330 5 analysis analysis NOUN cana-761 330 6 issn issn NOUN cana-761 330 7 : : PUNCT cana-761 330 8 1074 1074 NUM cana-761 330 9 - - PUNCT cana-761 330 10 133x 133x NUM cana-761 330 11 vol vol NOUN cana-761 330 12 31 31 NUM cana-761 330 13 no no NOUN cana-761 330 14 . . PUNCT cana-761 331 1 3s 3s NUM cana-761 331 2 ( ( PUNCT cana-761 331 3 2024 2024 NUM cana-761 331 4 ) ) PUNCT cana-761 331 5 237 237 NUM cana-761 331 6 https://internationalpubls.com https://internationalpubls.com X cana-761 331 7 taking take VERB cana-761 331 8 advantage advantage NOUN cana-761 331 9 of of ADP cana-761 331 10 ( ( PUNCT cana-761 331 11 14 14 NUM cana-761 331 12 ) ) PUNCT cana-761 331 13 , , PUNCT cana-761 331 14 we we PRON cana-761 331 15 have have VERB cana-761 331 16 ( ( PUNCT cana-761 331 17 ) ) PUNCT cana-761 331 18 ( ( PUNCT cana-761 331 19 ) ) PUNCT cana-761 331 20 ( ( PUNCT cana-761 331 21 ) ) PUNCT cana-761 331 22 ( ( PUNCT cana-761 331 23 ) ) PUNCT cana-761 331 24 ( ( PUNCT cana-761 331 25 ) ) PUNCT cana-761 331 26 ( ( PUNCT cana-761 331 27 ) ) PUNCT cana-761 331 28 ( ( PUNCT cana-761 331 29 ) ) SYM cana-761 331 30 1 1 NUM cana-761 331 31 1 1 NUM cana-761 331 32 1 1 NUM cana-761 331 33 , , PUNCT cana-761 331 34     SPACE cana-761 331 35 , , PUNCT cana-761 331 36         SPACE cana-761 331 37 , , PUNCT cana-761 331 38     SPACE cana-761 331 39 , , PUNCT cana-761 331 40     SPACE cana-761 331 41 , , PUNCT cana-761 331 42     SPACE cana-761 331 43 , , PUNCT cana-761 331 44 n n PROPN cana-761 331 45 n n CCONJ cana-761 332 1 nf nf VERB cana-761 332 2 w w PROPN cana-761 332 3 q q PROPN cana-761 332 4 j j PROPN cana-761 332 5 q q PROPN cana-761 332 6 t t PROPN cana-761 333 1 f f PROPN cana-761 333 2 w w PROPN cana-761 333 3 q q PROPN cana-761 333 4 j j PROPN cana-761 333 5 q q PROPN cana-761 333 6 t t PROPN cana-761 333 7 w w PROPN cana-761 333 8 q q X cana-761 333 9 q q PROPN cana-761 333 10 t t NOUN cana-761 333 11   NOUN cana-761 333 12 + + CCONJ cana-761 333 13 + + PROPN cana-761 333 14   PROPN cana-761 333 15   NOUN cana-761 333 16 . . PUNCT cana-761 334 1 for for ADP cana-761 334 2 any any DET cana-761 334 3 0 0 NUM cana-761 334 4 t t NOUN cana-761 334 5   NOUN cana-761 334 6 . . PUNCT cana-761 335 1 consequently consequently ADV cana-761 335 2 , , PUNCT cana-761 335 3 ( ( PUNCT cana-761 335 4 ) ) PUNCT cana-761 335 5 , , PUNCT cana-761 335 6 lim lim PROPN cana-761 335 7       SPACE cana-761 335 8 , , PUNCT cana-761 335 9     SPACE cana-761 335 10 , , PUNCT cana-761 335 11     SPACE cana-761 335 12 0n 0n NOUN cana-761 335 13 m m VERB cana-761 335 14 m m VERB cana-761 335 15 n n PRON cana-761 335 16 w w NOUN cana-761 335 17 q q X cana-761 335 18 q q X cana-761 335 19 s s NOUN cana-761 335 20 → → PUNCT cana-761 335 21 = = PUNCT cana-761 335 22 for for ADP cana-761 335 23 any any DET cana-761 335 24 0 0 NUM cana-761 335 25 t t NOUN cana-761 335 26   NOUN cana-761 335 27 . . PUNCT cana-761 336 1 thus thus ADV cana-761 336 2 , , PUNCT cana-761 336 3   PROPN cana-761 336 4 nq nq PROPN cana-761 336 5 ( ( PUNCT cana-761 336 6 ) ) PUNCT cana-761 336 7 ( ( PUNCT cana-761 336 8 ) ) PUNCT cana-761 336 9 ( ( PUNCT cana-761 336 10 ) ) PUNCT cana-761 336 11 ( ( PUNCT cana-761 336 12 ) ) PUNCT cana-761 336 13 ( ( PUNCT cana-761 336 14 ) ) PUNCT cana-761 336 15 ( ( PUNCT cana-761 336 16 ) ) PUNCT cana-761 336 17 ( ( PUNCT cana-761 336 18 ) ) PUNCT cana-761 336 19 1 1 NUM cana-761 336 20 1 1 NUM cana-761 336 21 1 1 NUM cana-761 336 22 1 1 NUM cana-761 336 23 , , PUNCT cana-761 336 24     SPACE cana-761 336 25 , , PUNCT cana-761 336 26         SPACE cana-761 336 27 , , PUNCT cana-761 336 28     SPACE cana-761 336 29 , , PUNCT cana-761 336 30     SPACE cana-761 336 31 , , PUNCT cana-761 336 32     SPACE cana-761 336 33 , , PUNCT cana-761 336 34     SPACE cana-761 336 35 n n CCONJ cana-761 336 36 n n CCONJ cana-761 336 37 n n CCONJ cana-761 336 38 n n CCONJ cana-761 336 39 n n ADV cana-761 336 40 nf nf NOUN cana-761 336 41 w w PROPN cana-761 336 42 q q X cana-761 336 43 q q PROPN cana-761 336 44 t t X cana-761 337 1 f f PROPN cana-761 337 2 w w PROPN cana-761 337 3 q q X cana-761 337 4 q q PROPN cana-761 337 5 t t X cana-761 337 6 f f PROPN cana-761 337 7 w w PROPN cana-761 337 8 q q X cana-761 337 9 q q PROPN cana-761 337 10 t t NOUN cana-761 337 11   NOUN cana-761 337 12 + + CCONJ cana-761 337 13 + + CCONJ cana-761 337 14 − − NOUN cana-761 337 15   NOUN cana-761 337 16 for for ADP cana-761 337 17 all all DET cana-761 337 18 n n PROPN cana-761 337 19 n n PROPN cana-761 337 20 . . PUNCT cana-761 338 1 letting let VERB cana-761 338 2 n→ n→ NOUN cana-761 338 3 and and CCONJ cana-761 338 4 using use VERB cana-761 338 5 the the DET cana-761 338 6 assumption assumption NOUN cana-761 338 7 that that SCONJ cana-761 338 8 f f PROPN cana-761 338 9 is be AUX cana-761 338 10 continuous continuous ADJ cana-761 338 11 , , PUNCT cana-761 338 12 we we PRON cana-761 338 13 have have VERB cana-761 338 14 ( ( PUNCT cana-761 338 15 ) ) PUNCT cana-761 338 16 ( ( PUNCT cana-761 338 17 ) ) PUNCT cana-761 338 18 ( ( PUNCT cana-761 338 19 ) ) PUNCT cana-761 338 20 ( ( PUNCT cana-761 338 21 ) ) PUNCT cana-761 338 22 ( ( PUNCT cana-761 338 23 ) ) PUNCT cana-761 338 24 ( ( PUNCT cana-761 338 25 ) ) PUNCT cana-761 338 26 ( ( PUNCT cana-761 338 27 ) ) PUNCT cana-761 338 28 1 1 NUM cana-761 338 29 , , PUNCT cana-761 338 30     SPACE cana-761 338 31 , , PUNCT cana-761 338 32           SPACE cana-761 338 33 , , PUNCT cana-761 338 34     SPACE cana-761 338 35 , , PUNCT cana-761 338 36         SPACE cana-761 338 37 0f 0f NOUN cana-761 339 1 w w PROPN cana-761 339 2 q q PROPN cana-761 339 3 j j PROPN cana-761 339 4 q q PROPN cana-761 339 5 t t PROPN cana-761 339 6 f f PROPN cana-761 339 7 w w PROPN cana-761 339 8 q q PROPN cana-761 339 9 j j PROPN cana-761 339 10 q q PROPN cana-761 339 11 t t PROPN cana-761 339 12 f f PROPN cana-761 339 13   PROPN cana-761 339 14   PROPN cana-761 339 15   NOUN cana-761 339 16 . . PUNCT cana-761 340 1 thus thus ADV cana-761 340 2 , , PUNCT cana-761 340 3 it it PRON cana-761 340 4 leads lead VERB cana-761 340 5 to to ADP cana-761 340 6 ( ( PUNCT cana-761 340 7 ) ) PUNCT cana-761 340 8 ( ( PUNCT cana-761 340 9 ) ) PUNCT cana-761 340 10 , , PUNCT cana-761 340 11     SPACE cana-761 340 12 , , PUNCT cana-761 340 13         SPACE cana-761 340 14 0w 0w X cana-761 341 1 q q X cana-761 341 2 f f X cana-761 341 3 q q PROPN cana-761 341 4 t t PROPN cana-761 341 5   PROPN cana-761 341 6 . . PUNCT cana-761 342 1 therefore therefore ADV cana-761 342 2 , , PUNCT cana-761 342 3 ( ( PUNCT cana-761 342 4 ) ) PUNCT cana-761 342 5     SPACE cana-761 342 6 q q PROPN cana-761 342 7 j j PROPN cana-761 342 8 q= q= ADV cana-761 342 9 . . PUNCT cana-761 343 1 suppose suppose VERB cana-761 343 2 now now ADV cana-761 343 3 that that SCONJ cana-761 343 4 j j PROPN cana-761 343 5 has have VERB cana-761 343 6 distinct distinct ADJ cana-761 343 7 fps fps NOUN cana-761 343 8 , , PUNCT cana-761 343 9     SPACE cana-761 343 10 q q NOUN cana-761 343 11 r r NOUN cana-761 343 12 x x NOUN cana-761 343 13 , , PUNCT cana-761 343 14 then then ADV cana-761 343 15 by by ADP cana-761 343 16 ( ( PUNCT cana-761 343 17 14 14 NUM cana-761 343 18 ) ) PUNCT cana-761 343 19 , , PUNCT cana-761 343 20 we we PRON cana-761 343 21 obtain obtain VERB cana-761 343 22 ( ( PUNCT cana-761 343 23 ) ) PUNCT cana-761 343 24 ( ( PUNCT cana-761 343 25 ) ) PUNCT cana-761 343 26 ( ( PUNCT cana-761 343 27 ) ) PUNCT cana-761 343 28 ( ( PUNCT cana-761 343 29 ) ) PUNCT cana-761 343 30 ( ( PUNCT cana-761 343 31 ) ) PUNCT cana-761 343 32 ( ( PUNCT cana-761 343 33 ) ) PUNCT cana-761 343 34 ( ( PUNCT cana-761 343 35 ) ) PUNCT cana-761 343 36 ( ( PUNCT cana-761 343 37 ) ) PUNCT cana-761 343 38 ( ( PUNCT cana-761 343 39 ) ) PUNCT cana-761 343 40 ( ( PUNCT cana-761 343 41 ) ) PUNCT cana-761 343 42 ( ( PUNCT cana-761 343 43 ) ) PUNCT cana-761 343 44 ( ( PUNCT cana-761 343 45 ) ) PUNCT cana-761 343 46 1 1 NUM cana-761 343 47 , , PUNCT cana-761 343 48     SPACE cana-761 343 49 , , PUNCT cana-761 343 50         SPACE cana-761 343 51 , , PUNCT cana-761 343 52     SPACE cana-761 343 53 , , PUNCT cana-761 343 54         SPACE cana-761 343 55 , , PUNCT cana-761 343 56     SPACE cana-761 343 57 , , PUNCT cana-761 343 58             SPACE cana-761 343 59 , , PUNCT cana-761 343 60     SPACE cana-761 343 61 , , PUNCT cana-761 343 62     SPACE cana-761 343 63 f f PROPN cana-761 344 1 w w PROPN cana-761 344 2 q q NOUN cana-761 344 3 r r NOUN cana-761 344 4 t t X cana-761 344 5 f f PROPN cana-761 344 6 w w PROPN cana-761 344 7 j j PROPN cana-761 344 8 q q X cana-761 344 9 j j PROPN cana-761 344 10 r r NOUN cana-761 344 11 t t PROPN cana-761 344 12 f f PROPN cana-761 344 13 w w PROPN cana-761 344 14 j j PROPN cana-761 344 15 q q X cana-761 344 16 j j PROPN cana-761 344 17 r r NOUN cana-761 344 18 t t PROPN cana-761 345 1 f f PROPN cana-761 345 2 w w PROPN cana-761 345 3 q q NOUN cana-761 345 4 r r NOUN cana-761 345 5 t t NOUN cana-761 345 6   NOUN cana-761 345 7 = = SYM cana-761 345 8   PROPN cana-761 345 9   NOUN cana-761 345 10 . . PUNCT cana-761 346 1 this this PRON cana-761 346 2 is be AUX cana-761 346 3 a a DET cana-761 346 4 contradiction contradiction NOUN cana-761 346 5 . . PUNCT cana-761 347 1 hence hence ADV cana-761 347 2 , , PUNCT cana-761 347 3         SPACE cana-761 347 4 q q PROPN cana-761 347 5 r= r= ADJ cana-761 347 6 . . PUNCT cana-761 348 1 theorem theorem VERB cana-761 348 2 3.10 3.10 NUM cana-761 348 3 . . PUNCT cana-761 349 1 let let VERB cana-761 349 2 ( ( PUNCT cana-761 349 3 ) ) PUNCT cana-761 349 4 , , PUNCT cana-761 349 5 , , PUNCT cana-761 349 6 x x X cana-761 349 7 w w PROPN cana-761 349 8 t t PROPN cana-761 349 9 be be AUX cana-761 349 10 a a DET cana-761 349 11 complete complete ADJ cana-761 349 12 rfms rfms NOUN cana-761 350 1 such such ADJ cana-761 350 2 that that SCONJ cana-761 350 3 ( ( PUNCT cana-761 350 4 ) ) PUNCT cana-761 350 5 0 0 NUM cana-761 350 6 lim lim PROPN cana-761 350 7 , , PUNCT cana-761 350 8     SPACE cana-761 350 9 , , PUNCT cana-761 350 10     SPACE cana-761 350 11 1 1 NUM cana-761 350 12 t t NOUN cana-761 350 13 w w NOUN cana-761 350 14 q q NOUN cana-761 350 15 r r NOUN cana-761 350 16 t t NOUN cana-761 350 17 + + PROPN cana-761 350 18 → → PROPN cana-761 350 19   PROPN cana-761 350 20 for for ADP cana-761 350 21 all all PRON cana-761 350 22 , , PUNCT cana-761 350 23     SPACE cana-761 350 24 q q NOUN cana-761 350 25 r r NOUN cana-761 350 26 x x PROPN cana-761 350 27 . . PUNCT cana-761 351 1 let let VERB cana-761 351 2     SPACE cana-761 351 3 : : PUNCT cana-761 351 4     SPACE cana-761 351 5 f f X cana-761 351 6 x x PUNCT cana-761 351 7 x→ x→ AUX cana-761 351 8 be be AUX cana-761 351 9 a a DET cana-761 351 10 mapping mapping NOUN cana-761 351 11 and and CCONJ cana-761 351 12 ff ff PROPN cana-761 351 13 . . PUNCT cana-761 352 1 suppose suppose VERB cana-761 352 2 that that SCONJ cana-761 352 3 for for ADP cana-761 352 4 all all PRON cana-761 352 5 , , PUNCT cana-761 352 6     SPACE cana-761 352 7 q q NOUN cana-761 352 8 r r NOUN cana-761 352 9 x x PROPN cana-761 352 10 , , PUNCT cana-761 352 11     SPACE cana-761 352 12 q q NOUN cana-761 352 13 r r NOUN cana-761 352 14 and and CCONJ cana-761 352 15         SPACE cana-761 352 16 0t 0t NOUN cana-761 352 17 , , PUNCT cana-761 352 18 there there PRON cana-761 352 19 exists exist VERB cana-761 352 20 ( ( PUNCT cana-761 352 21 ) ) PUNCT cana-761 352 22 0,1 0,1 NUM cana-761 352 23     SPACE cana-761 352 24   NOUN cana-761 352 25   NOUN cana-761 352 26 such such ADJ cana-761 352 27 that that SCONJ cana-761 352 28 ( ( PUNCT cana-761 352 29 ) ) PUNCT cana-761 352 30 ( ( PUNCT cana-761 352 31 ) ) PUNCT cana-761 352 32 ( ( PUNCT cana-761 352 33 ) ) PUNCT cana-761 352 34 ( ( PUNCT cana-761 352 35 ) ) PUNCT cana-761 352 36 ( ( PUNCT cana-761 352 37 ) ) PUNCT cana-761 352 38 ( ( PUNCT cana-761 352 39 ) ) PUNCT cana-761 352 40 ( ( PUNCT cana-761 352 41 ) ) PUNCT cana-761 352 42 ( ( PUNCT cana-761 352 43 ) ) PUNCT cana-761 352 44 ( ( PUNCT cana-761 352 45 ) ) PUNCT cana-761 352 46   PROPN cana-761 352 47   PROPN cana-761 352 48 ( ( PUNCT cana-761 352 49 ) ) PUNCT cana-761 352 50 1 1 NUM cana-761 352 51 , , PUNCT cana-761 352 52     SPACE cana-761 352 53 , , PUNCT cana-761 352 54         SPACE cana-761 352 55 , , PUNCT cana-761 352 56     SPACE cana-761 352 57 , , PUNCT cana-761 352 58     SPACE cana-761 352 59 , , PUNCT cana-761 352 60 , , PUNCT cana-761 352 61     SPACE cana-761 352 62 , , PUNCT cana-761 352 63     SPACE cana-761 352 64 , , PUNCT cana-761 352 65 , , PUNCT cana-761 352 66     SPACE cana-761 352 67 , , PUNCT cana-761 352 68     SPACE cana-761 352 69 f f PROPN cana-761 352 70 w w PROPN cana-761 352 71 f f PROPN cana-761 352 72 q q X cana-761 352 73 f f PROPN cana-761 352 74 r r NOUN cana-761 352 75 t t PROPN cana-761 352 76 f f PROPN cana-761 352 77 max max PROPN cana-761 352 78 w w PROPN cana-761 352 79 q q X cana-761 352 80 r r NOUN cana-761 352 81 t t PROPN cana-761 352 82 w w NOUN cana-761 352 83 q q PROPN cana-761 352 84 j j PROPN cana-761 352 85 q q PROPN cana-761 352 86 t t PROPN cana-761 352 87 w w PROPN cana-761 352 88 r r PROPN cana-761 352 89 j j PROPN cana-761 352 90 r r NOUN cana-761 352 91 t t NOUN cana-761 352 92   NOUN cana-761 352 93   NOUN cana-761 352 94 . . PUNCT cana-761 353 1 ( ( PUNCT cana-761 353 2 21 21 NUM cana-761 353 3 ) ) PUNCT cana-761 353 4 then then ADV cana-761 353 5 , , PUNCT cana-761 353 6 j j PROPN cana-761 353 7 has have VERB cana-761 353 8 a a DET cana-761 353 9 unique unique ADJ cana-761 353 10 fp fp NOUN cana-761 353 11 , , PUNCT cana-761 353 12 provided provide VERB cana-761 353 13 that that SCONJ cana-761 353 14 j j PROPN cana-761 353 15 or or CCONJ cana-761 353 16 f f PROPN cana-761 353 17 is be AUX cana-761 353 18 continuous continuous ADJ cana-761 353 19 . . PUNCT cana-761 354 1 proof proof NOUN cana-761 354 2 . . PUNCT cana-761 355 1 choose choose VERB cana-761 355 2 0q 0q PROPN cana-761 355 3 x x PROPN cana-761 355 4 and and CCONJ cana-761 355 5 define define VERB cana-761 355 6 a a DET cana-761 355 7 sequence sequence NOUN cana-761 355 8   PROPN cana-761 355 9 nq nq PROPN cana-761 355 10 as as SCONJ cana-761 355 11 follows follow VERB cana-761 355 12 : : PUNCT cana-761 355 13 ( ( PUNCT cana-761 355 14 ) ) PUNCT cana-761 355 15 ( ( PUNCT cana-761 355 16 ) ) PUNCT cana-761 355 17 1 1 NUM cana-761 355 18 0 0 NUM cana-761 355 19     SPACE cana-761 355 20 n n PRON cana-761 355 21 nq nq PROPN cana-761 355 22 j j PROPN cana-761 355 23 q q PROPN cana-761 355 24 n n PROPN cana-761 355 25 n+ n+ NUM cana-761 355 26 = = SYM cana-761 355 27   NOUN cana-761 355 28 . . PUNCT cana-761 356 1 by by ADP cana-761 356 2 ( ( PUNCT cana-761 356 3 21 21 NUM cana-761 356 4 ) ) PUNCT cana-761 356 5 , , PUNCT cana-761 356 6 we we PRON cana-761 356 7 have have VERB cana-761 356 8 ( ( PUNCT cana-761 356 9 ) ) PUNCT cana-761 356 10 ( ( PUNCT cana-761 356 11 ) ) PUNCT cana-761 356 12 ( ( PUNCT cana-761 356 13 ) ) PUNCT cana-761 356 14 ( ( PUNCT cana-761 356 15 ) ) PUNCT cana-761 356 16 ( ( PUNCT cana-761 356 17 ) ) PUNCT cana-761 356 18 ( ( PUNCT cana-761 356 19 ) ) PUNCT cana-761 356 20 1 1 NUM cana-761 356 21 1 1 NUM cana-761 356 22 , , PUNCT cana-761 356 23     SPACE cana-761 356 24 , , PUNCT cana-761 356 25             SPACE cana-761 356 26 , , PUNCT cana-761 356 27     SPACE cana-761 356 28 , , PUNCT cana-761 356 29     SPACE cana-761 356 30 n n CCONJ cana-761 356 31 n n ADV cana-761 356 32 nf nf VERB cana-761 357 1 w w PROPN cana-761 357 2 q q X cana-761 357 3 q q PROPN cana-761 357 4 t t X cana-761 357 5 f f PROPN cana-761 357 6 w w PROPN cana-761 357 7 j j PROPN cana-761 357 8 q q PROPN cana-761 357 9 j j PROPN cana-761 357 10 q q X cana-761 357 11 t+ t+ PUNCT cana-761 357 12 −= −= NOUN cana-761 357 13 ( ( PUNCT cana-761 357 14 ) ) PUNCT cana-761 357 15 ( ( PUNCT cana-761 357 16 ) ) PUNCT cana-761 357 17 ( ( PUNCT cana-761 357 18 ) ) PUNCT cana-761 357 19 ( ( PUNCT cana-761 357 20 ) ) PUNCT cana-761 357 21 1 1 NUM cana-761 357 22 1 1 NUM cana-761 357 23     SPACE cana-761 357 24 , , PUNCT cana-761 357 25     SPACE cana-761 357 26 , , PUNCT cana-761 357 27     SPACE cana-761 357 28 n n PROPN cana-761 357 29 nf nf VERB cana-761 357 30 w w PROPN cana-761 357 31 j j PROPN cana-761 357 32 q q PROPN cana-761 357 33 j j PROPN cana-761 357 34 q q PROPN cana-761 357 35 t t PROPN cana-761 357 36   NOUN cana-761 357 37 − − NOUN cana-761 357 38 ( ( PUNCT cana-761 357 39 ) ) PUNCT cana-761 357 40 ( ( PUNCT cana-761 357 41 ) ) PUNCT cana-761 357 42 ( ( PUNCT cana-761 357 43 ) ) PUNCT cana-761 357 44   PROPN cana-761 357 45   PROPN cana-761 357 46 ( ( PUNCT cana-761 357 47 ) ) SYM cana-761 357 48 1 1 NUM cana-761 357 49 1 1 NUM cana-761 357 50 1 1 NUM cana-761 357 51     SPACE cana-761 357 52 , , PUNCT cana-761 357 53     SPACE cana-761 357 54 , , PUNCT cana-761 357 55     SPACE cana-761 357 56 , , PUNCT cana-761 357 57 , , PUNCT cana-761 357 58     SPACE cana-761 357 59 , , PUNCT cana-761 357 60     SPACE cana-761 357 61 , , PUNCT cana-761 357 62 , , PUNCT cana-761 357 63     SPACE cana-761 357 64 , , PUNCT cana-761 357 65     SPACE cana-761 357 66 n n CCONJ cana-761 357 67 n n CCONJ cana-761 357 68 n n CCONJ cana-761 357 69 n n CCONJ cana-761 357 70 n n CCONJ cana-761 357 71 nf nf NOUN cana-761 357 72 max max PROPN cana-761 357 73 w w PROPN cana-761 357 74 q q PROPN cana-761 357 75 q q PROPN cana-761 357 76 t t PROPN cana-761 357 77 w w PROPN cana-761 357 78 q q X cana-761 357 79 q q PROPN cana-761 357 80 t t PROPN cana-761 357 81 w w PROPN cana-761 357 82 q q X cana-761 357 83 q q NOUN cana-761 357 84 t− t− PROPN cana-761 357 85 − − PROPN cana-761 357 86 + + NOUN cana-761 357 87   NOUN cana-761 357 88 ( ( PUNCT cana-761 357 89 ) ) PUNCT cana-761 357 90 ( ( PUNCT cana-761 357 91 ) ) PUNCT cana-761 357 92   PROPN cana-761 357 93   PROPN cana-761 357 94 ( ( PUNCT cana-761 357 95 ) ) PUNCT cana-761 357 96 1 1 NUM cana-761 357 97 1 1 NUM cana-761 357 98     SPACE cana-761 357 99 , , PUNCT cana-761 357 100     SPACE cana-761 357 101 , , PUNCT cana-761 357 102     SPACE cana-761 357 103 , , PUNCT cana-761 357 104 , , PUNCT cana-761 357 105     SPACE cana-761 357 106 , , PUNCT cana-761 357 107     SPACE cana-761 357 108 n n ADP cana-761 357 109 n n CCONJ cana-761 357 110 n n CCONJ cana-761 357 111 nf nf NOUN cana-761 357 112 max max PROPN cana-761 357 113 w w PROPN cana-761 357 114 q q PROPN cana-761 357 115 q q PROPN cana-761 357 116 t t PROPN cana-761 357 117 w w NOUN cana-761 357 118 q q X cana-761 357 119 q q X cana-761 357 120 t− t− PROPN cana-761 357 121 + + NOUN cana-761 357 122 = = NOUN cana-761 357 123 , , PUNCT cana-761 357 124 for for ADP cana-761 357 125 all all DET cana-761 357 126 n n PROPN cana-761 357 127 n n PROPN cana-761 357 128 and and CCONJ cana-761 357 129 0 0 NUM cana-761 357 130 t t NOUN cana-761 357 131   X cana-761 357 132 . . PUNCT cana-761 358 1 ( ( PUNCT cana-761 358 2 22 22 NUM cana-761 358 3 ) ) PUNCT cana-761 358 4 communications communication NOUN cana-761 358 5 on on ADP cana-761 358 6 applied apply VERB cana-761 358 7 nonlinear nonlinear ADJ cana-761 358 8 analysis analysis NOUN cana-761 358 9 issn issn NOUN cana-761 358 10 : : PUNCT cana-761 358 11 1074 1074 NUM cana-761 358 12 - - PUNCT cana-761 358 13 133x 133x NUM cana-761 358 14 vol vol NOUN cana-761 358 15 31 31 NUM cana-761 358 16 no no NOUN cana-761 358 17 . . PUNCT cana-761 359 1 3s 3s NUM cana-761 359 2 ( ( PUNCT cana-761 359 3 2024 2024 NUM cana-761 359 4 ) ) PUNCT cana-761 359 5 238 238 NUM cana-761 359 6 https://internationalpubls.com https://internationalpubls.com X cana-761 359 7 if if SCONJ cana-761 359 8 ( ( PUNCT cana-761 359 9 ) ) PUNCT cana-761 359 10 ( ( PUNCT cana-761 359 11 ) ) PUNCT cana-761 359 12   PROPN cana-761 359 13   PROPN cana-761 359 14 ( ( PUNCT cana-761 359 15 ) ) SYM cana-761 359 16 1 1 NUM cana-761 359 17 1 1 NUM cana-761 359 18 1 1 NUM cana-761 359 19 , , PUNCT cana-761 359 20     SPACE cana-761 359 21 , , PUNCT cana-761 359 22     SPACE cana-761 359 23 , , PUNCT cana-761 359 24 , , PUNCT cana-761 359 25     SPACE cana-761 359 26 , , PUNCT cana-761 359 27     SPACE cana-761 359 28 , , PUNCT cana-761 359 29     SPACE cana-761 359 30 , , PUNCT cana-761 359 31     SPACE cana-761 359 32 n n CCONJ cana-761 359 33 n n CCONJ cana-761 359 34 n n CCONJ cana-761 359 35 n n CCONJ cana-761 359 36 n n CCONJ cana-761 359 37 nmax nmax PROPN cana-761 360 1 w w PROPN cana-761 360 2 q q X cana-761 360 3 q q PROPN cana-761 360 4 t t PROPN cana-761 360 5 w w PROPN cana-761 360 6 q q X cana-761 360 7 q q PROPN cana-761 360 8 t t PROPN cana-761 360 9 w w NOUN cana-761 360 10 q q X cana-761 360 11 q q X cana-761 360 12 t− t− PROPN cana-761 361 1 + + PROPN cana-761 361 2 + + PROPN cana-761 361 3 = = X cana-761 361 4 , , PUNCT cana-761 361 5 then then ADV cana-761 361 6 by by ADP cana-761 361 7 ( ( PUNCT cana-761 361 8 22 22 NUM cana-761 361 9 ) ) PUNCT cana-761 361 10 , , PUNCT cana-761 361 11 we we PRON cana-761 361 12 get get VERB cana-761 361 13 ( ( PUNCT cana-761 361 14 ) ) PUNCT cana-761 361 15 ( ( PUNCT cana-761 361 16 ) ) PUNCT cana-761 361 17 1 1 NUM cana-761 361 18 1 1 NUM cana-761 361 19 , , PUNCT cana-761 361 20     SPACE cana-761 361 21 , , PUNCT cana-761 361 22     SPACE cana-761 361 23 , , PUNCT cana-761 361 24     SPACE cana-761 361 25 , , PUNCT cana-761 361 26     SPACE cana-761 361 27 n n ADP cana-761 361 28 n n CCONJ cana-761 361 29 n n ADV cana-761 361 30 nw nw NOUN cana-761 361 31 q q PROPN cana-761 362 1 q q PROPN cana-761 363 1 t t PROPN cana-761 363 2 w w PROPN cana-761 363 3 q q X cana-761 363 4 q q X cana-761 363 5 t+ t+ PUNCT cana-761 363 6 + + NOUN cana-761 363 7   PROPN cana-761 363 8 , , PUNCT cana-761 363 9 which which PRON cana-761 363 10 is be AUX cana-761 363 11 a a DET cana-761 363 12 contradiction contradiction NOUN cana-761 363 13 . . PUNCT cana-761 364 1 if if SCONJ cana-761 364 2 ( ( PUNCT cana-761 364 3 ) ) PUNCT cana-761 364 4 ( ( PUNCT cana-761 364 5 ) ) PUNCT cana-761 364 6   PROPN cana-761 364 7   PROPN cana-761 364 8 ( ( PUNCT cana-761 364 9 ) ) SYM cana-761 364 10 1 1 NUM cana-761 364 11 1 1 NUM cana-761 364 12 1 1 NUM cana-761 364 13 , , PUNCT cana-761 364 14     SPACE cana-761 364 15 , , PUNCT cana-761 364 16     SPACE cana-761 364 17 , , PUNCT cana-761 364 18 , , PUNCT cana-761 364 19     SPACE cana-761 364 20 , , PUNCT cana-761 364 21     SPACE cana-761 364 22 , , PUNCT cana-761 364 23     SPACE cana-761 364 24 , , PUNCT cana-761 364 25     SPACE cana-761 364 26 n n CCONJ cana-761 364 27 n n CCONJ cana-761 364 28 n n CCONJ cana-761 364 29 n n CCONJ cana-761 364 30 n n CCONJ cana-761 364 31 nmax nmax PROPN cana-761 364 32 w w PROPN cana-761 364 33 q q X cana-761 364 34 q q PROPN cana-761 364 35 t t PROPN cana-761 364 36 w w PROPN cana-761 364 37 q q X cana-761 364 38 q q PROPN cana-761 364 39 t t PROPN cana-761 364 40 w w NOUN cana-761 364 41 q q X cana-761 364 42 q q X cana-761 364 43 t− t− PROPN cana-761 364 44 + + CCONJ cana-761 364 45 −= −= NOUN cana-761 364 46 , , PUNCT cana-761 364 47 then then ADV cana-761 364 48 by by ADP cana-761 364 49 ( ( PUNCT cana-761 364 50 22 22 NUM cana-761 364 51 ) ) PUNCT cana-761 364 52 , , PUNCT cana-761 364 53 we we PRON cana-761 364 54 have have VERB cana-761 364 55 ( ( PUNCT cana-761 364 56 ) ) PUNCT cana-761 364 57 ( ( PUNCT cana-761 364 58 ) ) PUNCT cana-761 364 59 1 1 NUM cana-761 364 60 1 1 NUM cana-761 364 61 , , PUNCT cana-761 364 62     SPACE cana-761 364 63 , , PUNCT cana-761 364 64     SPACE cana-761 364 65 , , PUNCT cana-761 364 66     SPACE cana-761 364 67 , , PUNCT cana-761 364 68     SPACE cana-761 364 69 n n ADP cana-761 364 70 n n CCONJ cana-761 364 71 n n ADV cana-761 364 72 nw nw NOUN cana-761 364 73 q q PROPN cana-761 364 74 q q PROPN cana-761 365 1 t t PROPN cana-761 365 2 w w NOUN cana-761 365 3 q q X cana-761 365 4 q q X cana-761 365 5 t+ t+ NOUN cana-761 365 6 − − NOUN cana-761 365 7 . . PUNCT cana-761 366 1 following follow VERB cana-761 366 2 the the DET cana-761 366 3 proof proof NOUN cana-761 366 4 of of ADP cana-761 366 5 theorem theorem NOUN cana-761 366 6 3.5 3.5 NUM cana-761 366 7 , , PUNCT cana-761 366 8 we we PRON cana-761 366 9 find find VERB cana-761 366 10 q q PUNCT cana-761 366 11 x x PROPN cana-761 366 12 such such ADJ cana-761 366 13 that that SCONJ cana-761 366 14 lim lim PROPN cana-761 366 15     SPACE cana-761 366 16 n n CCONJ cana-761 366 17 n n CCONJ cana-761 366 18 q q NOUN cana-761 366 19 q q NOUN cana-761 366 20 → → PUNCT cana-761 366 21 = = PUNCT cana-761 366 22 . . PUNCT cana-761 367 1 suppose suppose VERB cana-761 367 2 first first ADV cana-761 367 3 that that SCONJ cana-761 367 4 j j PROPN cana-761 367 5 is be AUX cana-761 367 6 continuous continuous ADJ cana-761 367 7 . . PUNCT cana-761 368 1 then then ADV cana-761 368 2 , , PUNCT cana-761 368 3 by by ADP cana-761 368 4 the the DET cana-761 368 5 construction construction NOUN cana-761 368 6 of of ADP cana-761 368 7 sequence sequence NOUN cana-761 368 8   PROPN cana-761 368 9 nq nq PROPN cana-761 369 1 it it PRON cana-761 369 2 follows follow VERB cana-761 369 3 that that SCONJ cana-761 369 4 j j PROPN cana-761 369 5 has have VERB cana-761 369 6 a a DET cana-761 369 7 fp fp ADJ cana-761 369 8 q q PROPN cana-761 369 9 . . PUNCT cana-761 370 1 suppose suppose VERB cana-761 370 2 that that SCONJ cana-761 370 3 f f PROPN cana-761 370 4 is be AUX cana-761 370 5 continuous continuous ADJ cana-761 370 6 . . PUNCT cana-761 371 1 then then ADV cana-761 371 2 , , PUNCT cana-761 371 3 by by ADP cana-761 371 4 ( ( PUNCT cana-761 371 5 21 21 NUM cana-761 371 6 ) ) PUNCT cana-761 371 7 , , PUNCT cana-761 371 8 we we PRON cana-761 371 9 have have VERB cana-761 371 10 ( ( PUNCT cana-761 371 11 ) ) PUNCT cana-761 371 12 ( ( PUNCT cana-761 371 13 ) ) PUNCT cana-761 371 14 ( ( PUNCT cana-761 371 15 ) ) PUNCT cana-761 371 16 ( ( PUNCT cana-761 371 17 ) ) PUNCT cana-761 371 18 ( ( PUNCT cana-761 371 19 ) ) PUNCT cana-761 371 20 ( ( PUNCT cana-761 371 21 ) ) PUNCT cana-761 371 22 1 1 NUM cana-761 371 23 1 1 NUM cana-761 371 24 1 1 NUM cana-761 371 25 , , PUNCT cana-761 371 26     SPACE cana-761 371 27 , , PUNCT cana-761 371 28             SPACE cana-761 371 29 , , PUNCT cana-761 371 30     SPACE cana-761 371 31 , , PUNCT cana-761 371 32     SPACE cana-761 371 33 n n PROPN cana-761 372 1 nf nf VERB cana-761 372 2 w w PROPN cana-761 372 3 q q PROPN cana-761 372 4 j j PROPN cana-761 372 5 q q PROPN cana-761 372 6 t t PROPN cana-761 372 7 f f PROPN cana-761 372 8 w w PROPN cana-761 372 9 q q PROPN cana-761 372 10 j j PROPN cana-761 372 11 q q PROPN cana-761 372 12 t t PROPN cana-761 372 13   NOUN cana-761 372 14 + + CCONJ cana-761 373 1 + + ADJ cana-761 373 2   PROPN cana-761 373 3 ( ( PUNCT cana-761 373 4 ) ) PUNCT cana-761 373 5 ( ( PUNCT cana-761 373 6 ) ) PUNCT cana-761 373 7 ( ( PUNCT cana-761 373 8 ) ) PUNCT cana-761 373 9 ( ( PUNCT cana-761 373 10 ) ) PUNCT cana-761 373 11   PROPN cana-761 373 12   PROPN cana-761 373 13 ( ( PUNCT cana-761 373 14 ) ) PUNCT cana-761 373 15 1 1 NUM cana-761 373 16     SPACE cana-761 373 17 , , PUNCT cana-761 373 18     SPACE cana-761 373 19 , , PUNCT cana-761 373 20     SPACE cana-761 373 21 , , PUNCT cana-761 373 22 , , PUNCT cana-761 373 23     SPACE cana-761 373 24 , , PUNCT cana-761 373 25     SPACE cana-761 373 26 , , PUNCT cana-761 373 27 , , PUNCT cana-761 373 28     SPACE cana-761 373 29 , , PUNCT cana-761 373 30     SPACE cana-761 373 31 n n CCONJ cana-761 373 32 n n CCONJ cana-761 373 33 nf nf NOUN cana-761 374 1 max max PROPN cana-761 374 2 w w PROPN cana-761 374 3 q q PROPN cana-761 374 4 q q PROPN cana-761 374 5 t t PROPN cana-761 374 6 w w PROPN cana-761 374 7 q q X cana-761 374 8 q q PROPN cana-761 374 9 t t PROPN cana-761 374 10 w w PROPN cana-761 374 11 q q PROPN cana-761 374 12 j j PROPN cana-761 374 13 q q PROPN cana-761 374 14 t+ t+ PROPN cana-761 374 15 , , PUNCT cana-761 374 16 ( ( PUNCT cana-761 374 17 23 23 NUM cana-761 374 18 ) ) PUNCT cana-761 374 19 for for ADP cana-761 374 20 all all DET cana-761 374 21 n n PROPN cana-761 374 22 n n PROPN cana-761 374 23 and and CCONJ cana-761 374 24 0 0 NUM cana-761 374 25 t t NOUN cana-761 374 26   X cana-761 374 27 . . PUNCT cana-761 375 1 if if SCONJ cana-761 375 2 ( ( PUNCT cana-761 375 3 ) ) PUNCT cana-761 375 4 j j PROPN cana-761 375 5 q q NOUN cana-761 376 1 q= q= ADV cana-761 376 2 , , PUNCT cana-761 376 3 then then ADV cana-761 376 4 taking take VERB cana-761 376 5 n→ n→ NOUN cana-761 376 6 from from ADP cana-761 376 7 both both DET cana-761 376 8 sides side NOUN cana-761 376 9 of of ADP cana-761 376 10 ( ( PUNCT cana-761 376 11 23 23 NUM cana-761 376 12 ) ) PUNCT cana-761 376 13 , , PUNCT cana-761 376 14 we we PRON cana-761 376 15 have have VERB cana-761 376 16 ( ( PUNCT cana-761 376 17 ) ) PUNCT cana-761 376 18 ( ( PUNCT cana-761 376 19 ) ) PUNCT cana-761 376 20 ( ( PUNCT cana-761 376 21 ) ) PUNCT cana-761 376 22 ( ( PUNCT cana-761 376 23 ) ) PUNCT cana-761 376 24 ( ( PUNCT cana-761 376 25 ) ) PUNCT cana-761 376 26 ( ( PUNCT cana-761 376 27 ) ) PUNCT cana-761 376 28 ( ( PUNCT cana-761 376 29 ) ) PUNCT cana-761 376 30 ( ( PUNCT cana-761 376 31 ) ) PUNCT cana-761 376 32   PROPN cana-761 376 33   PROPN cana-761 376 34 ( ( PUNCT cana-761 376 35 ) ) PUNCT cana-761 376 36 ( ( PUNCT cana-761 376 37 ) ) PUNCT cana-761 376 38 1 1 NUM cana-761 376 39 , , PUNCT cana-761 376 40     SPACE cana-761 376 41 , , PUNCT cana-761 376 42             SPACE cana-761 376 43 , , PUNCT cana-761 376 44     SPACE cana-761 376 45 , , PUNCT cana-761 376 46         SPACE cana-761 376 47 ( ( PUNCT cana-761 376 48 0,0 0,0 NOUN cana-761 376 49 , , PUNCT cana-761 376 50 , , PUNCT cana-761 376 51     SPACE cana-761 376 52 , , PUNCT cana-761 376 53         SPACE cana-761 376 54 ( ( PUNCT cana-761 376 55 , , PUNCT cana-761 376 56     SPACE cana-761 376 57 , , PUNCT cana-761 376 58 f f PROPN cana-761 377 1 w w PROPN cana-761 377 2 q q PROPN cana-761 377 3 j j PROPN cana-761 377 4 q q PROPN cana-761 377 5 t t PROPN cana-761 378 1 f f PROPN cana-761 378 2 w w PROPN cana-761 378 3 q q PROPN cana-761 378 4 j j PROPN cana-761 378 5 q q PROPN cana-761 378 6 t t PROPN cana-761 379 1 f f PROPN cana-761 379 2 max max PROPN cana-761 379 3 w w PROPN cana-761 379 4 q q PROPN cana-761 379 5 j j PROPN cana-761 379 6 q q PROPN cana-761 379 7 t t PROPN cana-761 379 8 f f PROPN cana-761 379 9 w w PROPN cana-761 379 10 q q PROPN cana-761 379 11 j j PROPN cana-761 379 12 q q PROPN cana-761 379 13 t t PROPN cana-761 379 14   NOUN cana-761 379 15   NOUN cana-761 379 16   NOUN cana-761 379 17 = = SYM cana-761 379 18 , , PUNCT cana-761 379 19 which which PRON cana-761 379 20 means mean VERB cana-761 379 21 that that SCONJ cana-761 379 22 ( ( PUNCT cana-761 379 23 ) ) PUNCT cana-761 379 24 ( ( PUNCT cana-761 379 25 ) ) PUNCT cana-761 379 26 ( ( PUNCT cana-761 379 27 , , PUNCT cana-761 379 28     SPACE cana-761 379 29 , , PUNCT cana-761 379 30 1f 1f PROPN cana-761 379 31 w w PROPN cana-761 379 32 q q X cana-761 379 33 j j PROPN cana-761 379 34 q q PROPN cana-761 379 35 t t PROPN cana-761 379 36 = = PUNCT cana-761 379 37 . . PUNCT cana-761 380 1 this this PRON cana-761 380 2 is be AUX cana-761 380 3 in in ADP cana-761 380 4 contradiction contradiction NOUN cana-761 380 5 with with ADP cana-761 380 6 ( ( PUNCT cana-761 380 7 ) ) PUNCT cana-761 380 8 ( ( PUNCT cana-761 380 9 ) ) PUNCT cana-761 380 10 ( ( PUNCT cana-761 380 11 , , PUNCT cana-761 380 12     SPACE cana-761 380 13 , , PUNCT cana-761 380 14 1f 1f PROPN cana-761 380 15 w w PROPN cana-761 380 16 q q PROPN cana-761 380 17 j j PROPN cana-761 380 18 q q PROPN cana-761 380 19 t t PROPN cana-761 380 20   PROPN cana-761 380 21 . . PUNCT cana-761 381 1 finally finally ADV cana-761 381 2 , , PUNCT cana-761 381 3 we we PRON cana-761 381 4 prove prove VERB cana-761 381 5 the the DET cana-761 381 6 uniqueness uniqueness NOUN cana-761 381 7 of of ADP cana-761 381 8 the the DET cana-761 381 9 fp fp PROPN cana-761 381 10 . . PROPN cana-761 381 11 assume assume VERB cana-761 381 12 that that SCONJ cana-761 381 13 j j PROPN cana-761 381 14 has have VERB cana-761 381 15 two two NUM cana-761 381 16 distinct distinct ADJ cana-761 381 17 fps fps PROPN cana-761 381 18 , , PUNCT cana-761 381 19 , , PUNCT cana-761 381 20 p p X cana-761 381 21 q q X cana-761 381 22 . . PUNCT cana-761 382 1 then then ADV cana-761 382 2 , , PUNCT cana-761 382 3 by by ADP cana-761 382 4 ( ( PUNCT cana-761 382 5 21 21 NUM cana-761 382 6 ) ) PUNCT cana-761 382 7 , , PUNCT cana-761 382 8 we we PRON cana-761 382 9 have have VERB cana-761 382 10 ( ( PUNCT cana-761 382 11 ) ) PUNCT cana-761 382 12 ( ( PUNCT cana-761 382 13 ) ) PUNCT cana-761 382 14 ( ( PUNCT cana-761 382 15 ) ) PUNCT cana-761 382 16 ( ( PUNCT cana-761 382 17 ) ) PUNCT cana-761 382 18 ( ( PUNCT cana-761 382 19 ) ) PUNCT cana-761 382 20 ( ( PUNCT cana-761 382 21 ) ) PUNCT cana-761 382 22 ( ( PUNCT cana-761 382 23 ) ) PUNCT cana-761 382 24 ( ( PUNCT cana-761 382 25 ) ) PUNCT cana-761 382 26 ( ( PUNCT cana-761 382 27 ) ) PUNCT cana-761 382 28 ( ( PUNCT cana-761 382 29 ) ) PUNCT cana-761 382 30 1 1 NUM cana-761 382 31 , , PUNCT cana-761 382 32     SPACE cana-761 382 33 , , PUNCT cana-761 382 34     SPACE cana-761 382 35 ( ( PUNCT cana-761 382 36 , , PUNCT cana-761 382 37     SPACE cana-761 382 38 , , PUNCT cana-761 382 39         SPACE cana-761 382 40 ( ( PUNCT cana-761 382 41 , , PUNCT cana-761 382 42     SPACE cana-761 382 43 , , PUNCT cana-761 382 44     SPACE cana-761 382 45 f f PROPN cana-761 383 1 w w PROPN cana-761 383 2 p p X cana-761 383 3 q q PROPN cana-761 383 4 t t PROPN cana-761 383 5 f f PROPN cana-761 383 6 w w PROPN cana-761 383 7 j j PROPN cana-761 383 8 p p PROPN cana-761 383 9 j j PROPN cana-761 383 10 q q PROPN cana-761 383 11 t t PROPN cana-761 383 12 f f PROPN cana-761 383 13 w w PROPN cana-761 383 14 j j PROPN cana-761 384 1 p p PROPN cana-761 384 2 j j PROPN cana-761 384 3 q q PROPN cana-761 384 4 t t PROPN cana-761 384 5   NOUN cana-761 384 6 = = PROPN cana-761 384 7   PROPN cana-761 384 8 . . PUNCT cana-761 385 1 ( ( PUNCT cana-761 385 2 ) ) PUNCT cana-761 385 3 ( ( PUNCT cana-761 385 4 ) ) PUNCT cana-761 385 5 ( ( PUNCT cana-761 385 6 ) ) PUNCT cana-761 385 7 ( ( PUNCT cana-761 385 8 ) ) PUNCT cana-761 385 9 ( ( PUNCT cana-761 385 10 ) ) PUNCT cana-761 385 11   PROPN cana-761 385 12   PROPN cana-761 385 13 ( ( PUNCT cana-761 385 14 ) ) PUNCT cana-761 385 15     SPACE cana-761 385 16 , , PUNCT cana-761 385 17     SPACE cana-761 385 18 , , PUNCT cana-761 385 19     SPACE cana-761 385 20 , , PUNCT cana-761 385 21 , , PUNCT cana-761 385 22     SPACE cana-761 385 23 , , PUNCT cana-761 385 24     SPACE cana-761 385 25 , , PUNCT cana-761 385 26 , , PUNCT cana-761 385 27     SPACE cana-761 385 28 , , PUNCT cana-761 385 29     SPACE cana-761 385 30 f f PROPN cana-761 386 1 max max PROPN cana-761 386 2 w w PROPN cana-761 386 3 p p PROPN cana-761 386 4 q q PROPN cana-761 386 5 t t PROPN cana-761 387 1 w w PROPN cana-761 387 2 p p PROPN cana-761 387 3 j j PROPN cana-761 387 4 p p PROPN cana-761 387 5 t t PROPN cana-761 387 6 w w PROPN cana-761 387 7 q q PROPN cana-761 387 8 j j PROPN cana-761 387 9 q q PROPN cana-761 387 10 t t NOUN cana-761 387 11 , , PUNCT cana-761 387 12 ( ( PUNCT cana-761 387 13 ) ) PUNCT cana-761 387 14 ( ( PUNCT cana-761 387 15 ) ) PUNCT cana-761 387 16 ( ( PUNCT cana-761 387 17 ) ) PUNCT cana-761 387 18   PROPN cana-761 387 19   PROPN cana-761 387 20 ( ( PUNCT cana-761 387 21 ) ) PUNCT cana-761 387 22 , , PUNCT cana-761 387 23     SPACE cana-761 387 24 , , PUNCT cana-761 387 25     SPACE cana-761 387 26 , , PUNCT cana-761 387 27 , , PUNCT cana-761 387 28     SPACE cana-761 387 29 , , PUNCT cana-761 387 30     SPACE cana-761 387 31 , , PUNCT cana-761 387 32 , , PUNCT cana-761 387 33     SPACE cana-761 387 34 , , PUNCT cana-761 387 35     SPACE cana-761 387 36 f f PROPN cana-761 388 1 max max PROPN cana-761 388 2 w w PROPN cana-761 388 3 p p PROPN cana-761 388 4 q q PROPN cana-761 388 5 t t PROPN cana-761 389 1 w w PROPN cana-761 389 2 p p X cana-761 389 3 p p PROPN cana-761 389 4 t t PROPN cana-761 389 5 w w NOUN cana-761 389 6 q q X cana-761 389 7 q q NOUN cana-761 389 8 t= t= ADJ cana-761 389 9 ( ( PUNCT cana-761 389 10 ) ) PUNCT cana-761 389 11   PROPN cana-761 389 12   PROPN cana-761 389 13 ( ( PUNCT cana-761 389 14 ) ) PUNCT cana-761 389 15 , , PUNCT cana-761 389 16     SPACE cana-761 389 17 , , PUNCT cana-761 389 18     SPACE cana-761 389 19 , , PUNCT cana-761 389 20 0,0f 0,0f NOUN cana-761 390 1 max max PROPN cana-761 390 2 w w PROPN cana-761 390 3 p p PROPN cana-761 390 4 q q PROPN cana-761 390 5 t= t= ADJ cana-761 390 6 ( ( PUNCT cana-761 390 7 ) ) PUNCT cana-761 390 8 ( ( PUNCT cana-761 390 9 ) ) PUNCT cana-761 390 10 , , PUNCT cana-761 390 11     SPACE cana-761 390 12 , , PUNCT cana-761 390 13     SPACE cana-761 390 14 f f PROPN cana-761 391 1 w w PROPN cana-761 391 2 p p X cana-761 391 3 q q X cana-761 391 4 t= t= NOUN cana-761 391 5 , , PUNCT cana-761 391 6 this this PRON cana-761 391 7 is be AUX cana-761 391 8 a a DET cana-761 391 9 contradiction contradiction NOUN cana-761 391 10 . . PUNCT cana-761 392 1 therefore therefore ADV cana-761 392 2 , , PUNCT cana-761 392 3 p p NOUN cana-761 392 4 q= q= ADV cana-761 392 5 . . PUNCT cana-761 393 1 refrences refrence VERB cana-761 394 1 [ [ X cana-761 394 2 1 1 X cana-761 394 3 ] ] X cana-761 394 4 alexander alexander PROPN cana-761 394 5 sostak sostak PROPN cana-761 394 6 “ " PUNCT cana-761 394 7 george george NOUN cana-761 394 8 - - PUNCT cana-761 394 9 veeramani veeramani NOUN cana-761 394 10 fuzzy fuzzy ADJ cana-761 394 11 metrics metric NOUN cana-761 394 12 revised revise VERB cana-761 394 13 ” " PUNCT cana-761 394 14 axioms axiom NOUN cana-761 394 15 2018,7,60 2018,7,60 NUM cana-761 394 16 . . PUNCT cana-761 395 1 [ [ X cana-761 395 2 2 2 NUM cana-761 395 3 ] ] PUNCT cana-761 395 4 chandok chandok NOUN cana-761 395 5 , , PUNCT cana-761 395 6 s. s. PROPN cana-761 395 7 ; ; PUNCT cana-761 395 8 huang huang PROPN cana-761 395 9 , , PUNCT cana-761 395 10 h. h. PROPN cana-761 395 11 ; ; PUNCT cana-761 395 12 radenovi´c radenovi´c PROPN cana-761 395 13 , , PUNCT cana-761 395 14 s. s. PROPN cana-761 395 15 “ " PUNCT cana-761 395 16 some some DET cana-761 395 17 fixed fix VERB cana-761 395 18 - - PUNCT cana-761 395 19 point point NOUN cana-761 395 20 results result NOUN cana-761 395 21 for for ADP cana-761 395 22 the the DET cana-761 395 23 generalized generalized ADJ cana-761 395 24 f f PROPN cana-761 395 25 - - PUNCT cana-761 395 26 suzuki suzuki NOUN cana-761 395 27 type type NOUN cana-761 395 28 contractions contraction NOUN cana-761 395 29 in in ADP cana-761 395 30 b b NOUN cana-761 395 31 - - PUNCT cana-761 395 32 metric metric ADJ cana-761 395 33 spaces”.sahand spaces”.sahand CCONJ cana-761 395 34 commun commun PROPN cana-761 395 35 . . PUNCT cana-761 395 36 math math PROPN cana-761 395 37 . . PUNCT cana-761 396 1 anal anal ADJ cana-761 396 2 . . PUNCT cana-761 397 1 2018 2018 NUM cana-761 397 2 , , PUNCT cana-761 397 3 11 11 NUM cana-761 397 4 , , PUNCT cana-761 397 5 81–89 81–89 NUM cana-761 397 6 . . PUNCT cana-761 398 1 [ [ X cana-761 398 2 3 3 NUM cana-761 398 3 ] ] X cana-761 398 4 george.a george.a PROPN cana-761 398 5 , , PUNCT cana-761 398 6 veeramani veeramani PROPN cana-761 398 7 . . PUNCT cana-761 399 1 p. p. NOUN cana-761 399 2 “ " PUNCT cana-761 399 3 on on ADP cana-761 399 4 some some DET cana-761 399 5 results result NOUN cana-761 399 6 in in ADP cana-761 399 7 fuzzy fuzzy ADJ cana-761 399 8 metric metric ADJ cana-761 399 9 spaces space NOUN cana-761 399 10 ” " PUNCT cana-761 399 11 , , PUNCT cana-761 399 12 fuzzy fuzzy ADJ cana-761 399 13 sets set NOUN cana-761 399 14 and and CCONJ cana-761 399 15 syst syst NOUN cana-761 399 16 . . PUNCT cana-761 400 1 64(1994 64(1994 NUM cana-761 400 2 ) ) PUNCT cana-761 400 3 , , PUNCT cana-761 400 4 395 395 NUM cana-761 400 5 - - SYM cana-761 400 6 399 399 NUM cana-761 400 7 . . PUNCT cana-761 401 1 [ [ X cana-761 401 2 4 4 NUM cana-761 401 3 ] ] X cana-761 401 4 grabiec grabiec PROPN cana-761 401 5 . . PUNCT cana-761 402 1 m m PROPN cana-761 402 2 , , PUNCT cana-761 402 3 fixed fix VERB cana-761 402 4 points point NOUN cana-761 402 5 in in ADP cana-761 402 6 fuzzy fuzzy ADJ cana-761 402 7 metric metric ADJ cana-761 402 8 spaces space NOUN cana-761 402 9 . . PUNCT cana-761 403 1 fuzzy fuzzy ADJ cana-761 403 2 sets set NOUN cana-761 403 3 and and CCONJ cana-761 403 4 syst syst NOUN cana-761 403 5 . . PUNCT cana-761 403 6 27(1988 27(1988 NUM cana-761 403 7 ) ) PUNCT cana-761 403 8 , , PUNCT cana-761 403 9 385 385 NUM cana-761 403 10 - - SYM cana-761 403 11 389 389 NUM cana-761 403 12 . . PUNCT cana-761 404 1 communications communication NOUN cana-761 404 2 on on ADP cana-761 404 3 applied apply VERB cana-761 404 4 nonlinear nonlinear ADJ cana-761 404 5 analysis analysis NOUN cana-761 404 6 issn issn NOUN cana-761 404 7 : : PUNCT cana-761 404 8 1074 1074 NUM cana-761 404 9 - - PUNCT cana-761 404 10 133x 133x NUM cana-761 404 11 vol vol NOUN cana-761 404 12 31 31 NUM cana-761 404 13 no no NOUN cana-761 404 14 . . PUNCT cana-761 405 1 3s 3s NUM cana-761 405 2 ( ( PUNCT cana-761 405 3 2024 2024 NUM cana-761 405 4 ) ) PUNCT cana-761 405 5 239 239 NUM cana-761 405 6 https://internationalpubls.com https://internationalpubls.com X cana-761 406 1 [ [ X cana-761 406 2 5 5 NUM cana-761 406 3 ] ] X cana-761 406 4 gregori gregori X cana-761 406 5 , , PUNCT cana-761 406 6 v. v. PROPN cana-761 406 7 , , PUNCT cana-761 406 8 miñana miñana PROPN cana-761 406 9 , , PUNCT cana-761 406 10 j.-j j.-j PROPN cana-761 406 11 . . PROPN cana-761 406 12 , , PUNCT cana-761 406 13 miravet miravet PROPN cana-761 406 14 , , PUNCT cana-761 406 15 d. d. PROPN cana-761 406 16 contractive contractive ADJ cana-761 406 17 sequences sequence NOUN cana-761 406 18 in in ADP cana-761 406 19 fuzzy fuzzy ADJ cana-761 406 20 metric metric ADJ cana-761 406 21 spaces space NOUN cana-761 406 22 . . PUNCT cana-761 407 1 fuzzy fuzzy ADJ cana-761 407 2 sets set NOUN cana-761 407 3 syst syst PROPN cana-761 407 4 . . PUNCT cana-761 408 1 2020 2020 NUM cana-761 408 2 , , PUNCT cana-761 408 3 379 379 NUM cana-761 408 4 , , PUNCT cana-761 408 5 125–133 125–133 NUM cana-761 408 6 . . PUNCT cana-761 409 1 [ [ X cana-761 409 2 6 6 NUM cana-761 409 3 ] ] X cana-761 409 4 gupta gupta PROPN cana-761 409 5 . . PUNCT cana-761 410 1 v v X cana-761 410 2 , , PUNCT cana-761 410 3 mani mani PROPN cana-761 410 4 . . PUNCT cana-761 411 1 n n X cana-761 411 2 , , PUNCT cana-761 411 3 a. a. PROPN cana-761 411 4 saini saini PROPN cana-761 411 5 , , PUNCT cana-761 411 6 fixed fix VERB cana-761 411 7 points point NOUN cana-761 411 8 theorems theorem NOUN cana-761 411 9 and and CCONJ cana-761 411 10 its its PRON cana-761 411 11 applications application NOUN cana-761 411 12 in in ADP cana-761 411 13 fuzzy fuzzy ADJ cana-761 411 14 metric metric ADJ cana-761 411 15 spaces space NOUN cana-761 411 16 , , PUNCT cana-761 411 17 conference conference NOUN cana-761 411 18 paper paper NOUN cana-761 411 19 , , PUNCT cana-761 411 20 2013 2013 NUM cana-761 411 21 ( ( PUNCT cana-761 411 22 2013 2013 NUM cana-761 411 23 ) ) PUNCT cana-761 411 24 , , PUNCT cana-761 411 25 11 11 NUM cana-761 411 26 pages page NOUN cana-761 411 27 . . PUNCT cana-761 412 1 1 1 NUM cana-761 412 2 , , PUNCT cana-761 412 3 1.6 1.6 NUM cana-761 412 4 . . PUNCT cana-761 413 1 [ [ X cana-761 413 2 7 7 X cana-761 413 3 ] ] X cana-761 413 4 huaping huape VERB cana-761 413 5 huang huang PROPN cana-761 413 6 , , PUNCT cana-761 413 7 biljana biljana ADJ cana-761 413 8 cari´c cari´c PROPN cana-761 413 9 , , PUNCT cana-761 413 10 tatjana tatjana PROPN cana-761 413 11 došenovi´c došenovi´c PROPN cana-761 413 12 , , PUNCT cana-761 413 13 dušan dušan PROPN cana-761 413 14 raki´c raki´c PROPN cana-761 413 15 and and CCONJ cana-761 413 16 mirjana mirjana PROPN cana-761 413 17 brdar brdar PROPN cana-761 413 18 , , PUNCT cana-761 413 19 “ " PUNCT cana-761 413 20 fixed fix VERB cana-761 413 21 - - PUNCT cana-761 413 22 point point NOUN cana-761 413 23 theorems theorem NOUN cana-761 413 24 in in ADP cana-761 413 25 fuzzy fuzzy ADJ cana-761 413 26 metric metric ADJ cana-761 413 27 spaces space NOUN cana-761 413 28 via via ADP cana-761 413 29 fuzzy fuzzy ADJ cana-761 413 30 f f NOUN cana-761 413 31 - - PUNCT cana-761 413 32 contraction contraction NOUN cana-761 413 33 ” " PUNCT cana-761 413 34 , , PUNCT cana-761 413 35 mathematics mathematic NOUN cana-761 413 36 2021 2021 NUM cana-761 413 37 , , PUNCT cana-761 413 38 9 9 NUM cana-761 413 39 , , PUNCT cana-761 413 40 641 641 NUM cana-761 413 41 . . PUNCT cana-761 414 1 [ [ X cana-761 414 2 8 8 NUM cana-761 414 3 ] ] X cana-761 414 4 jehad jehad PROPN cana-761 414 5 r r NOUN cana-761 414 6 , , PUNCT cana-761 414 7 madhan madhan NOUN cana-761 414 8 kider kider NOUN cana-761 414 9 “ " PUNCT cana-761 414 10 some some DET cana-761 414 11 properties property NOUN cana-761 414 12 of of ADP cana-761 414 13 algebra algebra NOUN cana-761 414 14 fuzzy fuzzy ADJ cana-761 414 15 metric metric ADJ cana-761 414 16 space space NOUN cana-761 414 17 ” " PUNCT cana-761 414 18 , , PUNCT cana-761 414 19 journal journal NOUN cana-761 414 20 of of ADP cana-761 414 21 al al PROPN cana-761 414 22 - - PUNCT cana-761 414 23 qadisiyah qadisiyah NOUN cana-761 414 24 for for ADP cana-761 414 25 computer computer NOUN cana-761 414 26 science science NOUN cana-761 414 27 and and CCONJ cana-761 414 28 mathematics mathematic NOUN cana-761 414 29 vol.12(2 vol.12(2 ADV cana-761 414 30 ) ) PUNCT cana-761 414 31 2020 2020 NUM cana-761 414 32 , , PUNCT cana-761 414 33 pp pp ADP cana-761 414 34 math math NOUN cana-761 414 35 43–56 43–56 PROPN cana-761 414 36 43 43 NUM cana-761 414 37 . . PUNCT cana-761 415 1 [ [ X cana-761 415 2 9 9 NUM cana-761 415 3 ] ] X cana-761 415 4 jehad jehad PROPN cana-761 415 5 r r NOUN cana-761 415 6 , , PUNCT cana-761 415 7 madhan madhan NOUN cana-761 415 8 kider kider NOUN cana-761 415 9 “ " PUNCT cana-761 415 10 application application NOUN cana-761 415 11 of of ADP cana-761 415 12 fixed fix VERB cana-761 415 13 points point NOUN cana-761 415 14 in in ADP cana-761 415 15 algebra algebra PROPN cana-761 415 16 fuzzy fuzzy ADJ cana-761 415 17 normed norme VERB cana-761 415 18 spaces space NOUN cana-761 415 19 ” " PUNCT cana-761 415 20 , , PUNCT cana-761 415 21 journal journal NOUN cana-761 415 22 of of ADP cana-761 415 23 physics physics PROPN cana-761 415 24 : : PUNCT cana-761 415 25 conference conference NOUN cana-761 415 26 series series NOUN cana-761 415 27 1879 1879 NUM cana-761 415 28 ( ( PUNCT cana-761 415 29 2021 2021 NUM cana-761 415 30 ) ) PUNCT cana-761 415 31 022099 022099 NUM cana-761 415 32 , , PUNCT cana-761 415 33 iop iop NOUN cana-761 415 34 publishing publishing NOUN cana-761 415 35 . . PUNCT cana-761 416 1 [ [ X cana-761 416 2 10 10 NUM cana-761 416 3 ] ] X cana-761 416 4 j. j. PROPN cana-761 416 5 kramosil kramosil PROPN cana-761 416 6 , , PUNCT cana-761 416 7 j. j. PROPN cana-761 416 8 michalek michalek PROPN cana-761 416 9 : : PUNCT cana-761 416 10 fuzzy fuzzy ADJ cana-761 416 11 metric metric ADJ cana-761 416 12 and and CCONJ cana-761 416 13 statistical statistical ADJ cana-761 416 14 metric metric ADJ cana-761 416 15 spaces space NOUN cana-761 416 16 . . PUNCT cana-761 417 1 kibernetika.11(1975 kibernetika.11(1975 PROPN cana-761 417 2 ) ) PUNCT cana-761 417 3 , , PUNCT cana-761 418 1 336–344 336–344 NUM cana-761 418 2 . . PUNCT cana-761 419 1 [ [ X cana-761 419 2 11 11 NUM cana-761 419 3 ] ] X cana-761 419 4 mihet mihet PROPN cana-761 419 5 d. d. PROPN cana-761 419 6 erratum erratum PROPN cana-761 419 7 to to ADP cana-761 419 8 “ " PUNCT cana-761 419 9 fuzzy fuzzy ADJ cana-761 419 10 y y PROPN cana-761 419 11 - - PUNCT cana-761 419 12 contractive contractive ADJ cana-761 419 13 mappings mapping NOUN cana-761 419 14 in in ADP cana-761 419 15 non non ADJ cana-761 419 16 - - ADJ cana-761 419 17 archimedean archimedean ADJ cana-761 419 18 fuzzy fuzzy ADJ cana-761 419 19 metric metric ADJ cana-761 419 20 spaces space NOUN cana-761 419 21 , , PUNCT cana-761 419 22 fuzzy fuzzy ADJ cana-761 419 23 sets set NOUN cana-761 419 24 and and CCONJ cana-761 419 25 systems system NOUN cana-761 419 26 . . PUNCT cana-761 420 1 159(2008 159(2008 NUM cana-761 420 2 ) ) PUNCT cana-761 420 3 , , PUNCT cana-761 420 4 739 739 NUM cana-761 420 5 - - SYM cana-761 420 6 744 744 NUM cana-761 420 7 ] ] PUNCT cana-761 420 8 ” " PUNCT cana-761 420 9 . . PUNCT cana-761 421 1 fuzzy fuzzy ADJ cana-761 421 2 sets set VERB cana-761 421 3 syst syst PROPN cana-761 421 4 . . PUNCT cana-761 422 1 2010 2010 NUM cana-761 422 2 , , PUNCT cana-761 422 3 161 161 NUM cana-761 422 4 , , PUNCT cana-761 422 5 1150–1151 1150–1151 NUM cana-761 422 6 [ [ X cana-761 422 7 12 12 NUM cana-761 422 8 ] ] X cana-761 422 9 muraliraj muraliraj NOUN cana-761 422 10 a a DET cana-761 422 11 , , PUNCT cana-761 422 12 thangathamizh thangathamizh ADJ cana-761 422 13 r r NOUN cana-761 422 14 , , PUNCT cana-761 422 15 popovic popovic NOUN cana-761 422 16 n n CCONJ cana-761 422 17 , , PUNCT cana-761 422 18 savic savic PROPN cana-761 422 19 a a X cana-761 422 20 , , PUNCT cana-761 422 21 radenovic radenovic PROPN cana-761 422 22 s. s. PROPN cana-761 422 23 the the DET cana-761 422 24 first first ADJ cana-761 422 25 rational rational ADJ cana-761 422 26 type type NOUN cana-761 422 27 revisd revisd NOUN cana-761 422 28 fuzzycontractions fuzzycontraction NOUN cana-761 422 29 in in ADP cana-761 422 30 revisd revisd NOUN cana-761 422 31 fuzzy fuzzy ADJ cana-761 422 32 metric metric ADJ cana-761 422 33 spaces space NOUN cana-761 422 34 with with ADP cana-761 422 35 an an DET cana-761 422 36 applications application NOUN cana-761 422 37 . . PUNCT cana-761 423 1 mathematics mathematic NOUN cana-761 423 2 . . PUNCT cana-761 424 1 2023;11(10):2244 2023;11(10):2244 X cana-761 424 2 . . PUNCT cana-761 425 1 [ [ X cana-761 425 2 13 13 NUM cana-761 425 3 ] ] SYM cana-761 425 4 muraliraj muraliraj NOUN cana-761 425 5 . . PUNCT cana-761 426 1 a a PRON cana-761 426 2 and and CCONJ cana-761 426 3 thangathamizh thangathamizh ADJ cana-761 426 4 . . PUNCT cana-761 427 1 r r X cana-761 427 2 , , PUNCT cana-761 427 3 “ " PUNCT cana-761 427 4 some some DET cana-761 427 5 topological topological ADJ cana-761 427 6 properties property NOUN cana-761 427 7 of of ADP cana-761 427 8 revised revise VERB cana-761 427 9 fuzzy fuzzy ADJ cana-761 427 10 cone cone NOUN cana-761 427 11 metric metric ADJ cana-761 427 12 space space NOUN cana-761 427 13 ” " PUNCT cana-761 427 14 , , PUNCT cana-761 427 15 ratio ratio PROPN cana-761 427 16 mathematica mathematica PROPN cana-761 427 17 , , PUNCT cana-761 427 18 volume volume NOUN cana-761 427 19 26 26 NUM cana-761 427 20 , , PUNCT cana-761 427 21 number number NOUN cana-761 427 22 2 2 NUM cana-761 427 23 , , PUNCT cana-761 427 24 ( ( PUNCT cana-761 427 25 2023 2023 NUM cana-761 427 26 ) ) PUNCT cana-761 428 1 [ [ X cana-761 428 2 14 14 NUM cana-761 428 3 ] ] X cana-761 428 4 muraliraj muraliraj VERB cana-761 428 5 a a PRON cana-761 428 6 and and CCONJ cana-761 428 7 thangathamizh thangathamizh ADJ cana-761 428 8 , , PUNCT cana-761 428 9 “ " PUNCT cana-761 428 10 new new ADJ cana-761 428 11 relation relation NOUN cana-761 428 12 - - PUNCT cana-761 428 13 theoretic theoretic NOUN cana-761 428 14 fixed fix VERB cana-761 428 15 point point NOUN cana-761 428 16 theorems theorem NOUN cana-761 428 17 in in ADP cana-761 428 18 revised revise VERB cana-761 428 19 fuzzy fuzzy ADJ cana-761 428 20 metric metric ADJ cana-761 428 21 spaces space NOUN cana-761 428 22 with with ADP cana-761 428 23 an an DET cana-761 428 24 application application NOUN cana-761 428 25 to to ADP cana-761 428 26 fractional fractional ADJ cana-761 428 27 differential differential ADJ cana-761 428 28 equations equation NOUN cana-761 428 29 ” " PUNCT cana-761 428 30 , , PUNCT cana-761 428 31 communications communication NOUN cana-761 428 32 in in ADP cana-761 428 33 mathematics mathematic NOUN cana-761 428 34 and and CCONJ cana-761 428 35 applications application NOUN cana-761 428 36 , , PUNCT cana-761 428 37 vol.12 vol.12 NOUN cana-761 428 38 , , PUNCT cana-761 428 39 no no DET cana-761 428 40 22 22 NUM cana-761 428 41 . . PUNCT cana-761 429 1 ( ( PUNCT cana-761 429 2 2023 2023 NUM cana-761 429 3 ) ) PUNCT cana-761 430 1 [ [ X cana-761 430 2 15 15 NUM cana-761 430 3 ] ] X cana-761 430 4 muraliraj muraliraj NOUN cana-761 430 5 . . PUNCT cana-761 431 1 a a PRON cana-761 431 2 and and CCONJ cana-761 431 3 thangathamizh thangathamizh ADJ cana-761 431 4 . . PUNCT cana-761 432 1 r r X cana-761 432 2 , , PUNCT cana-761 432 3 “ " PUNCT cana-761 432 4 fixed fix VERB cana-761 432 5 point point NOUN cana-761 432 6 theorems theorem NOUN cana-761 432 7 in in ADP cana-761 432 8 revised revise VERB cana-761 432 9 fuzzy fuzzy ADJ cana-761 432 10 metric metric ADJ cana-761 432 11 space space NOUN cana-761 432 12 ” " PUNCT cana-761 432 13 , , PUNCT cana-761 432 14 advances advance NOUN cana-761 432 15 in in ADP cana-761 432 16 fuzzy fuzzy ADJ cana-761 432 17 sets set NOUN cana-761 432 18 and and CCONJ cana-761 432 19 systems system NOUN cana-761 432 20 , , PUNCT cana-761 432 21 volume volume NOUN cana-761 432 22 26 26 NUM cana-761 432 23 , , PUNCT cana-761 432 24 number number NOUN cana-761 432 25 2 2 NUM cana-761 432 26 , , PUNCT cana-761 432 27 2021 2021 NUM cana-761 432 28 . . PUNCT cana-761 433 1 [ [ X cana-761 433 2 16 16 NUM cana-761 433 3 ] ] PUNCT cana-761 433 4 a. a. NOUN cana-761 433 5 muraliraj muraliraj PROPN cana-761 433 6 . . PUNCT cana-761 434 1 and and CCONJ cana-761 434 2 r. r. PROPN cana-761 434 3 thangathamizh thangathamizh PROPN cana-761 434 4 , , PUNCT cana-761 434 5 “ " PUNCT cana-761 434 6 introduction introduction NOUN cana-761 434 7 on on ADP cana-761 434 8 revised revise VERB cana-761 434 9 fuzzy fuzzy ADJ cana-761 434 10 modular modular ADJ cana-761 434 11 spaces space NOUN cana-761 434 12 ” " PUNCT cana-761 434 13 , , PUNCT cana-761 434 14 issn issn PROPN cana-761 434 15 0973 0973 NUM cana-761 434 16 - - SYM cana-761 434 17 1768 1768 NUM cana-761 434 18 volume volume NOUN cana-761 434 19 17 17 NUM cana-761 434 20 , , PUNCT cana-761 434 21 number number NOUN cana-761 434 22 2 2 NUM cana-761 434 23 ( ( PUNCT cana-761 434 24 2021 2021 NUM cana-761 434 25 ) ) PUNCT cana-761 434 26 , , PUNCT cana-761 434 27 pp pp ADP cana-761 434 28 . . PUNCT cana-761 435 1 303 303 NUM cana-761 435 2 - - SYM cana-761 435 3 317 317 NUM cana-761 435 4 . . PUNCT cana-761 436 1 [ [ X cana-761 436 2 17 17 NUM cana-761 436 3 ] ] PUNCT cana-761 436 4 thangathamizh thangathamizh PROPN cana-761 436 5 r r NOUN cana-761 436 6 , , PUNCT cana-761 436 7 muraliraj muraliraj VERB cana-761 436 8 a a PRON cana-761 436 9 and and CCONJ cana-761 436 10 shanmugavel shanmugavel NOUN cana-761 436 11 p p X cana-761 436 12 “ " PUNCT cana-761 436 13 new new ADJ cana-761 436 14 approach approach NOUN cana-761 436 15 of of ADP cana-761 436 16 lebesgue lebesgue PROPN cana-761 436 17 integral integral ADJ cana-761 436 18 in in ADP cana-761 436 19 revised revise VERB cana-761 436 20 fuzzy fuzzy ADJ cana-761 436 21 cone cone NOUN cana-761 436 22 metric metric ADJ cana-761 436 23 spaces space NOUN cana-761 436 24 vie vie X cana-761 436 25 unique unique ADJ cana-761 436 26 coupled couple VERB cana-761 436 27 fixed fix VERB cana-761 436 28 point point NOUN cana-761 436 29 theorems theorem NOUN cana-761 436 30 ” " PUNCT cana-761 436 31 , , PUNCT cana-761 436 32 military military ADJ cana-761 436 33 technical technical ADJ cana-761 436 34 courier courier NOUN cana-761 436 35 , , PUNCT cana-761 436 36 3 3 NUM cana-761 436 37 , , PUNCT cana-761 436 38 2024 2024 NUM cana-761 436 39 . . PUNCT cana-761 437 1 doi doi NOUN cana-761 437 2 : : PUNCT cana-761 437 3 10.5937 10.5937 NUM cana-761 437 4 / / SYM cana-761 437 5 vojtechg72 vojtechg72 NOUN cana-761 437 6 - - PUNCT cana-761 437 7 48816 48816 NUM cana-761 437 8 . . PUNCT cana-761 438 1 [ [ X cana-761 438 2 18 18 NUM cana-761 438 3 ] ] PUNCT cana-761 438 4 rhoades rhoade NOUN cana-761 438 5 . . PUNCT cana-761 439 1 e.b e.b PROPN cana-761 439 2 , , PUNCT cana-761 439 3 a a DET cana-761 439 4 comparison comparison NOUN cana-761 439 5 of of ADP cana-761 439 6 various various ADJ cana-761 439 7 definitions definition NOUN cana-761 439 8 of of ADP cana-761 439 9 contractive contractive ADJ cana-761 439 10 mappings mapping NOUN cana-761 439 11 , , PUNCT cana-761 439 12 trans trans PROPN cana-761 439 13 . . PROPN cana-761 440 1 amer amer PROPN cana-761 440 2 . . PUNCT cana-761 440 3 math math PROPN cana-761 440 4 . . PUNCT cana-761 441 1 soc soc PROPN cana-761 441 2 . . PUNCT cana-761 442 1 226 226 NUM cana-761 442 2 ( ( PUNCT cana-761 442 3 1977 1977 NUM cana-761 442 4 ) ) PUNCT cana-761 442 5 , , PUNCT cana-761 442 6 257 257 NUM cana-761 442 7 - - SYM cana-761 442 8 290 290 NUM cana-761 442 9 . . PUNCT cana-761 443 1 [ [ X cana-761 443 2 19 19 NUM cana-761 443 3 ] ] X cana-761 443 4 secelean secelean PROPN cana-761 443 5 , , PUNCT cana-761 443 6 n.a n.a PROPN cana-761 443 7 . . PROPN cana-761 443 8 iterated iterate VERB cana-761 443 9 function function NOUN cana-761 443 10 system system NOUN cana-761 443 11 consisting consist VERB cana-761 443 12 of of ADP cana-761 443 13 f f NOUN cana-761 443 14 - - PUNCT cana-761 443 15 contractions contraction NOUN cana-761 443 16 . . PUNCT cana-761 444 1 fixed fix VERB cana-761 444 2 point point NOUN cana-761 444 3 theory theory NOUN cana-761 444 4 appl appl NOUN cana-761 444 5 . . PROPN cana-761 445 1 2013 2013 NUM cana-761 445 2 , , PUNCT cana-761 445 3 2013 2013 NUM cana-761 445 4 , , PUNCT cana-761 445 5 277 277 NUM cana-761 445 6 . . PUNCT cana-761 446 1 [ [ X cana-761 446 2 20 20 NUM cana-761 446 3 ] ] PUNCT cana-761 446 4 b. b. PROPN cana-761 446 5 schweizer schweizer PROPN cana-761 446 6 , , PUNCT cana-761 446 7 a. a. NOUN cana-761 446 8 sklar sklar NOUN cana-761 446 9 : : PUNCT cana-761 446 10 statistical statistical ADJ cana-761 446 11 metric metric ADJ cana-761 446 12 spaces space NOUN cana-761 446 13 . . PUNCT cana-761 447 1 pacific pacific PROPN cana-761 447 2 j. j. PROPN cana-761 447 3 math math PROPN cana-761 447 4 . . PUNCT cana-761 448 1 10(1960 10(1960 NUM cana-761 448 2 ) ) PUNCT cana-761 448 3 , , PUNCT cana-761 449 1 314–334 314–334 NUM cana-761 449 2 . . PUNCT cana-761 450 1 [ [ X cana-761 450 2 21 21 NUM cana-761 450 3 ] ] X cana-761 450 4 shukla shukla NOUN cana-761 450 5 , , PUNCT cana-761 450 6 s. s. PROPN cana-761 450 7 ; ; PUNCT cana-761 450 8 gopal gopal PROPN cana-761 450 9 , , PUNCT cana-761 450 10 d. d. PROPN cana-761 450 11 ; ; PUNCT cana-761 450 12 sintunavarat sintunavarat PROPN cana-761 450 13 , , PUNCT cana-761 450 14 w. w. PROPN cana-761 450 15 a a DET cana-761 450 16 new new ADJ cana-761 450 17 class class NOUN cana-761 450 18 of of ADP cana-761 450 19 fuzzy fuzzy ADJ cana-761 450 20 contractive contractive ADJ cana-761 450 21 mappings mapping NOUN cana-761 450 22 and and CCONJ cana-761 450 23 fixed fix VERB cana-761 450 24 - - PUNCT cana-761 450 25 point point NOUN cana-761 450 26 theorems theorem NOUN cana-761 450 27 . . PUNCT cana-761 451 1 fuzzy fuzzy ADJ cana-761 451 2 sets set VERB cana-761 451 3 syst.2018 syst.2018 PROPN cana-761 451 4 , , PUNCT cana-761 451 5 350 350 NUM cana-761 451 6 , , PUNCT cana-761 451 7 85–94 85–94 NUM cana-761 451 8 . . PUNCT cana-761 452 1 [ [ X cana-761 452 2 22 22 NUM cana-761 452 3 ] ] PUNCT cana-761 452 4 wardowski wardowski NOUN cana-761 452 5 , , PUNCT cana-761 452 6 d. d. PROPN cana-761 452 7 fuzzy fuzzy PROPN cana-761 452 8 contractive contractive ADJ cana-761 452 9 mappings mapping NOUN cana-761 452 10 and and CCONJ cana-761 452 11 fixed fix VERB cana-761 452 12 - - PUNCT cana-761 452 13 points point NOUN cana-761 452 14 in in ADP cana-761 452 15 fuzzy fuzzy ADJ cana-761 452 16 metric metric ADJ cana-761 452 17 space space NOUN cana-761 452 18 . . PUNCT cana-761 453 1 fuzzy fuzzy ADJ cana-761 453 2 sets set NOUN cana-761 453 3 syst syst PROPN cana-761 453 4 . . PUNCT cana-761 454 1 2013 2013 NUM cana-761 454 2 , , PUNCT cana-761 454 3 222 222 NUM cana-761 454 4 , , PUNCT cana-761 454 5 108 108 NUM cana-761 454 6 – – SYM cana-761 454 7 114 114 NUM cana-761 454 8 . . PUNCT cana-761 455 1 [ [ X cana-761 455 2 23 23 NUM cana-761 455 3 ] ] PUNCT cana-761 455 4 wardowski wardowski PROPN cana-761 455 5 , , PUNCT cana-761 455 6 d. d. PROPN cana-761 455 7 fixed fix VERB cana-761 455 8 points point NOUN cana-761 455 9 of of ADP cana-761 455 10 a a DET cana-761 455 11 new new ADJ cana-761 455 12 type type NOUN cana-761 455 13 of of ADP cana-761 455 14 contractive contractive ADJ cana-761 455 15 mappings mapping NOUN cana-761 455 16 in in ADP cana-761 455 17 complete complete ADJ cana-761 455 18 metric metric ADJ cana-761 455 19 spaces space NOUN cana-761 455 20 . . PUNCT cana-761 456 1 fixed fix VERB cana-761 456 2 point point NOUN cana-761 456 3 theory theory NOUN cana-761 456 4 appl appl PROPN cana-761 456 5 . . PROPN cana-761 456 6 2012 2012 NUM cana-761 456 7 , , PUNCT cana-761 456 8 2012,94 2012,94 NUM cana-761 456 9 . . PUNCT cana-761 457 1 [ [ X cana-761 457 2 24 24 NUM cana-761 457 3 ] ] PUNCT cana-761 457 4 l. l. PROPN cana-761 457 5 a. a. PROPN cana-761 457 6 zadeh zadeh PROPN cana-761 457 7 : : PUNCT cana-761 457 8 fuzzy fuzzy ADJ cana-761 457 9 sets set NOUN cana-761 457 10 . . PUNCT cana-761 458 1 inf inf PROPN cana-761 458 2 . . PUNCT cana-761 458 3 control control PROPN cana-761 458 4 . . PUNCT cana-761 458 5 , , PUNCT cana-761 458 6 8(1965 8(1965 NUM cana-761 458 7 ) ) PUNCT cana-761 458 8 , , PUNCT cana-761 458 9 338–353 338–353 NUM cana-761 458 10 . . PUNCT cana-761 459 1 [ [ X cana-761 459 2 25 25 NUM cana-761 459 3 ] ] X cana-761 459 4 muraliraj muraliraj VERB cana-761 459 5 a a PRON cana-761 459 6 and and CCONJ cana-761 459 7 shanmugavel shanmugavel NOUN cana-761 459 8 p p X cana-761 459 9 , , PUNCT cana-761 459 10 thangathamizh thangathamizh ADJ cana-761 459 11 r r NOUN cana-761 459 12 “ " PUNCT cana-761 459 13 existence existence NOUN cana-761 459 14 of of ADP cana-761 459 15 fixed fix VERB cana-761 459 16 point point NOUN cana-761 459 17 theorems theorem NOUN cana-761 459 18 in in ADP cana-761 459 19 revised revise VERB cana-761 459 20 fuzzy fuzzy ADJ cana-761 459 21 modular modular ADJ cana-761 459 22 spaces space NOUN cana-761 459 23 ” " PUNCT cana-761 459 24 , , PUNCT cana-761 459 25 advances advance NOUN cana-761 459 26 in in ADP cana-761 459 27 nonlinear nonlinear ADJ cana-761 459 28 variational variational ADJ cana-761 459 29 inequalities inequality NOUN cana-761 459 30 , , PUNCT cana-761 459 31 2024 2024 NUM cana-761 459 32 . . PUNCT cana-761 460 1 ( ( PUNCT cana-761 460 2 accepted accept VERB cana-761 460 3 for for ADP cana-761 460 4 publication publication NOUN cana-761 460 5 ) ) PUNCT cana-761 461 1 [ [ X cana-761 461 2 26 26 NUM cana-761 461 3 ] ] PUNCT cana-761 461 4 thangathamizh thangathamizh PROPN cana-761 461 5 r r NOUN cana-761 461 6 , , PUNCT cana-761 461 7 balamurugan balamurugan VERB cana-761 461 8 k k PROPN cana-761 461 9 , , PUNCT cana-761 461 10 karnan karnan PROPN cana-761 461 11 c c NOUN cana-761 461 12 , , PUNCT cana-761 461 13 shanmugavel shanmugavel NOUN cana-761 461 14 p p NOUN cana-761 461 15 , , PUNCT cana-761 461 16 and and CCONJ cana-761 461 17 balraj balraj X cana-761 461 18 d d NOUN cana-761 461 19 , , PUNCT cana-761 461 20 “ " PUNCT cana-761 461 21 revised revise VERB cana-761 461 22 fuzzy fuzzy ADJ cana-761 461 23 differential differential ADJ cana-761 461 24 equations equation NOUN cana-761 461 25 using use VERB cana-761 461 26 weakly weakly ADJ cana-761 461 27 compatible compatible ADJ cana-761 461 28 self self NOUN cana-761 461 29 - - PUNCT cana-761 461 30 mappings mapping NOUN cana-761 461 31 in in ADP cana-761 461 32 revised revise VERB cana-761 461 33 fuzzy fuzzy ADJ cana-761 461 34 metric metric ADJ cana-761 461 35 spaces space NOUN cana-761 461 36 ” " PUNCT cana-761 461 37 , , PUNCT cana-761 461 38 advances advance NOUN cana-761 461 39 in in ADP cana-761 461 40 nonlinear nonlinear ADJ cana-761 461 41 variational variational ADJ cana-761 461 42 inequalities inequality NOUN cana-761 461 43 , , PUNCT cana-761 461 44 2024 2024 NUM cana-761 461 45 . . PUNCT cana-761 462 1 ( ( PUNCT cana-761 462 2 accepted accept VERB cana-761 462 3 for for ADP cana-761 462 4 publication publication NOUN cana-761 462 5 ) ) PUNCT cana-761 463 1 [ [ X cana-761 463 2 27 27 NUM cana-761 463 3 ] ] PUNCT cana-761 463 4 thangathamizh thangathamizh PROPN cana-761 463 5 r r NOUN cana-761 463 6 , , PUNCT cana-761 463 7 abdelhamid abdelhamid ADP cana-761 463 8 moussaoui moussaoui NOUN cana-761 463 9 , , PUNCT cana-761 463 10 tatjana tatjana PROPN cana-761 463 11 dosenovic dosenovic PROPN cana-761 463 12 , , PUNCT cana-761 463 13 stojan stojan ADP cana-761 463 14 radenovic radenovic PROPN cana-761 463 15 “ " PUNCT cana-761 463 16 fixed fix VERB cana-761 463 17 point point NOUN cana-761 463 18 results result NOUN cana-761 463 19 in in ADP cana-761 463 20 controlled control VERB cana-761 463 21 revised revise VERB cana-761 463 22 fuzzy fuzzy ADJ cana-761 463 23 metric metric ADJ cana-761 463 24 spaces space NOUN cana-761 463 25 with with ADP cana-761 463 26 an an DET cana-761 463 27 application application NOUN cana-761 463 28 to to ADP cana-761 463 29 the the DET cana-761 463 30 transformation transformation NOUN cana-761 463 31 of of ADP cana-761 463 32 solar solar ADJ cana-761 463 33 energy energy NOUN cana-761 463 34 to to ADP cana-761 463 35 electric electric ADJ cana-761 463 36 power power NOUN cana-761 463 37 ” " PUNCT cana-761 463 38 , , PUNCT cana-761 463 39 military military ADJ cana-761 463 40 technical technical ADJ cana-761 463 41 courier courier NOUN cana-761 463 42 , , PUNCT cana-761 463 43 4,2024 4,2024 NUM cana-761 463 44 . . PUNCT cana-761 464 1 doi.10.5937 doi.10.5937 NOUN cana-761 464 2 / / SYM cana-761 464 3 vojtehg72 vojtehg72 NOUN cana-761 464 4 - - PUNCT cana-761 464 5 49064 49064 NUM cana-761 464 6 . . PUNCT cana-761 465 1 [ [ X cana-761 465 2 28 28 NUM cana-761 465 3 ] ] X cana-761 465 4 muraliraj muraliraj VERB cana-761 465 5 a a PRON cana-761 465 6 and and CCONJ cana-761 465 7 shanmugavel shanmugavel NOUN cana-761 465 8 p p X cana-761 465 9 , , PUNCT cana-761 465 10 thangathamizh thangathamizh ADJ cana-761 465 11 r r NOUN cana-761 465 12 “ " PUNCT cana-761 465 13 fixed fix VERB cana-761 465 14 point point NOUN cana-761 465 15 theorems theorem NOUN cana-761 465 16 in in ADP cana-761 465 17 molular molular PROPN cana-761 465 18 revised revise VERB cana-761 465 19 fuzzy fuzzy ADJ cana-761 465 20 modular modular ADJ cana-761 465 21 spaces space NOUN cana-761 465 22 ” " PUNCT cana-761 465 23 , , PUNCT cana-761 465 24 advances advance NOUN cana-761 465 25 in in ADP cana-761 465 26 nonlinear nonlinear ADJ cana-761 465 27 variational variational ADJ cana-761 465 28 inequalities inequality NOUN cana-761 465 29 , , PUNCT cana-761 465 30 2024 2024 NUM cana-761 465 31 . . PUNCT cana-761 466 1 ( ( PUNCT cana-761 466 2 accepted accept VERB cana-761 466 3 for for ADP cana-761 466 4 publication publication NOUN cana-761 466 5 ) ) PUNCT