id sid tid token lemma pos cana-697 1 1 communications communication NOUN cana-697 1 2 on on ADP cana-697 1 3 applied apply VERB cana-697 1 4 nonlinear nonlinear ADJ cana-697 1 5 analysis analysis NOUN cana-697 1 6 issn issn NOUN cana-697 1 7 : : PUNCT cana-697 1 8 1074 1074 NUM cana-697 1 9 - - PUNCT cana-697 1 10 133x 133x NUM cana-697 1 11 vol vol NOUN cana-697 1 12 31 31 NUM cana-697 1 13 no no NOUN cana-697 1 14 . . PUNCT cana-697 2 1 2s 2s NUM cana-697 2 2 ( ( PUNCT cana-697 2 3 2024 2024 NUM cana-697 2 4 ) ) PUNCT cana-697 2 5 676 676 NUM cana-697 2 6 https://internationalpubls.com https://internationalpubls.com X cana-697 2 7 on on ADP cana-697 2 8 fixed fix VERB cana-697 2 9 point point NOUN cana-697 2 10 for for ADP cana-697 2 11 nonexpansive nonexpansive ADJ cana-697 2 12 mappings mapping NOUN cana-697 2 13 in in ADP cana-697 2 14 partial partial ADJ cana-697 2 15 metric metric ADJ cana-697 2 16 spaces space NOUN cana-697 2 17 arta arta PROPN cana-697 2 18 ekayanti1 ekayanti1 PROPN cana-697 2 19 , , PUNCT cana-697 2 20 jumadi2 jumadi2 PROPN cana-697 2 21 , , PUNCT cana-697 2 22 erika erika PROPN cana-697 2 23 eka eka PROPN cana-697 2 24 santi3 santi3 PROPN cana-697 2 25 department department PROPN cana-697 2 26 of of ADP cana-697 2 27 mathematics mathematics PROPN cana-697 2 28 education education NOUN cana-697 2 29 , , PUNCT cana-697 2 30 universitas universita NOUN cana-697 2 31 muhammadiyah muhammadiyah ADP cana-697 2 32 ponorogo ponorogo PROPN cana-697 2 33 , , PUNCT cana-697 2 34 east east PROPN cana-697 2 35 java java PROPN cana-697 2 36 , , PUNCT cana-697 2 37 indonesia indonesia PROPN cana-697 2 38 arta_ekayanti@umpo.ac.id1 arta_ekayanti@umpo.ac.id1 PROPN cana-697 2 39 , , PUNCT cana-697 2 40 jumadi@umpo.ac.id2 jumadi@umpo.ac.id2 PROPN cana-697 2 41 , , PUNCT cana-697 2 42 erika_ekasanti@umpo.ac.id3 erika_ekasanti@umpo.ac.id3 ADJ cana-697 2 43 article article NOUN cana-697 2 44 history history NOUN cana-697 2 45 : : PUNCT cana-697 2 46 received receive VERB cana-697 2 47 : : PUNCT cana-697 2 48 12 12 NUM cana-697 2 49 - - PUNCT cana-697 2 50 04 04 NUM cana-697 2 51 - - PUNCT cana-697 2 52 2024 2024 NUM cana-697 2 53 revised revise VERB cana-697 2 54 : : PUNCT cana-697 2 55 19 19 NUM cana-697 2 56 - - PUNCT cana-697 2 57 05 05 NUM cana-697 2 58 - - PUNCT cana-697 2 59 2024 2024 NUM cana-697 2 60 accepted accept VERB cana-697 2 61 : : PUNCT cana-697 2 62 02 02 NUM cana-697 2 63 - - PUNCT cana-697 2 64 06 06 NUM cana-697 2 65 - - PUNCT cana-697 2 66 2024 2024 NUM cana-697 2 67 abstract abstract NOUN cana-697 2 68 : : PUNCT cana-697 2 69 in in ADP cana-697 2 70 this this DET cana-697 2 71 paper paper NOUN cana-697 2 72 , , PUNCT cana-697 2 73 we we PRON cana-697 2 74 establish establish VERB cana-697 2 75 some some DET cana-697 2 76 fixed fix VERB cana-697 2 77 points point NOUN cana-697 2 78 theorem theorem VERB cana-697 2 79 for for ADP cana-697 2 80 nonexpansive nonexpansive ADJ cana-697 2 81 mappings mapping NOUN cana-697 2 82 in in ADP cana-697 2 83 partial partial ADJ cana-697 2 84 metric metric ADJ cana-697 2 85 spaces space NOUN cana-697 2 86 . . PUNCT cana-697 3 1 our our PRON cana-697 3 2 result result NOUN cana-697 3 3 generalizes generalize VERB cana-697 3 4 vetro vetro PROPN cana-697 3 5 ’s ’s PART cana-697 3 6 results result NOUN cana-697 3 7 ( ( PUNCT cana-697 3 8 2015 2015 NUM cana-697 3 9 ) ) PUNCT cana-697 3 10 in in ADP cana-697 3 11 the the DET cana-697 3 12 setting setting NOUN cana-697 3 13 of of ADP cana-697 3 14 partial partial ADJ cana-697 3 15 metric metric ADJ cana-697 3 16 spaces space NOUN cana-697 3 17 . . PUNCT cana-697 4 1 this this DET cana-697 4 2 work work NOUN cana-697 4 3 proves prove VERB cana-697 4 4 and and CCONJ cana-697 4 5 generalizes generalize VERB cana-697 4 6 some some DET cana-697 4 7 results result NOUN cana-697 4 8 of of ADP cana-697 4 9 aydi aydi VERB cana-697 4 10 ( ( PUNCT cana-697 4 11 2017 2017 NUM cana-697 4 12 ) ) PUNCT cana-697 4 13 . . PUNCT cana-697 5 1 suitable suitable ADJ cana-697 5 2 example example NOUN cana-697 5 3 is be AUX cana-697 5 4 provided provide VERB cana-697 5 5 to to PART cana-697 5 6 illustrate illustrate VERB cana-697 5 7 the the DET cana-697 5 8 usability usability NOUN cana-697 5 9 of of ADP cana-697 5 10 our our PRON cana-697 5 11 results result NOUN cana-697 5 12 . . PUNCT cana-697 6 1 keywords keyword NOUN cana-697 6 2 : : PUNCT cana-697 6 3 fixed fix VERB cana-697 6 4 points point NOUN cana-697 6 5 , , PUNCT cana-697 6 6 nonexpansive nonexpansive ADJ cana-697 6 7 mapping mapping NOUN cana-697 6 8 , , PUNCT cana-697 6 9 partial partial ADJ cana-697 6 10 metric metric ADJ cana-697 6 11 spaces space NOUN cana-697 6 12 . . PUNCT cana-697 7 1 2020 2020 NUM cana-697 7 2 mathematics mathematic NOUN cana-697 7 3 subject subject ADJ cana-697 7 4 classification classification NOUN cana-697 7 5 : : PUNCT cana-697 7 6 47h10 47h10 NUM cana-697 7 7 , , PUNCT cana-697 7 8 47h09 47h09 NUM cana-697 7 9 1 1 NUM cana-697 7 10 . . PUNCT cana-697 8 1 introduction introduction NOUN cana-697 8 2 the the DET cana-697 8 3 banach banach NOUN cana-697 8 4 contraction contraction NOUN cana-697 8 5 principle principle NOUN cana-697 8 6 is be AUX cana-697 8 7 a a DET cana-697 8 8 fundamental fundamental ADJ cana-697 8 9 topic topic NOUN cana-697 8 10 in in ADP cana-697 8 11 mathematics mathematic NOUN cana-697 8 12 , , PUNCT cana-697 8 13 especially especially ADV cana-697 8 14 in in ADP cana-697 8 15 the the DET cana-697 8 16 focus focus NOUN cana-697 8 17 of of ADP cana-697 8 18 fixed fix VERB cana-697 8 19 point point NOUN cana-697 8 20 theory theory NOUN cana-697 8 21 . . PUNCT cana-697 9 1 s. s. PROPN cana-697 9 2 banach banach PROPN cana-697 9 3 introduced introduce VERB cana-697 9 4 the the DET cana-697 9 5 banach banach NOUN cana-697 9 6 contraction contraction NOUN cana-697 9 7 principle principle NOUN cana-697 9 8 in in ADP cana-697 9 9 1922 1922 NUM cana-697 9 10 [ [ X cana-697 9 11 2 2 NUM cana-697 9 12 ] ] PUNCT cana-697 9 13 . . PUNCT cana-697 10 1 the the DET cana-697 10 2 banach banach NOUN cana-697 10 3 contraction contraction NOUN cana-697 10 4 principle principle NOUN cana-697 10 5 guarantees guarantee VERB cana-697 10 6 the the DET cana-697 10 7 existence existence NOUN cana-697 10 8 of of ADP cana-697 10 9 fixed fix VERB cana-697 10 10 points point NOUN cana-697 10 11 from from ADP cana-697 10 12 a a DET cana-697 10 13 contraction contraction NOUN cana-697 10 14 mapping mapping NOUN cana-697 10 15 . . PUNCT cana-697 11 1 in in ADP cana-697 11 2 its its PRON cana-697 11 3 development development NOUN cana-697 11 4 , , PUNCT cana-697 11 5 studies study NOUN cana-697 11 6 regarding regard VERB cana-697 11 7 the the DET cana-697 11 8 existence existence NOUN cana-697 11 9 of of ADP cana-697 11 10 fixed fix VERB cana-697 11 11 points point NOUN cana-697 11 12 from from ADP cana-697 11 13 contraction contraction NOUN cana-697 11 14 mapping mapping NOUN cana-697 11 15 have have AUX cana-697 11 16 attracted attract VERB cana-697 11 17 much much ADJ cana-697 11 18 interest interest NOUN cana-697 11 19 from from ADP cana-697 11 20 researchers researcher NOUN cana-697 11 21 . . PUNCT cana-697 12 1 various various ADJ cana-697 12 2 studies study NOUN cana-697 12 3 were be AUX cana-697 12 4 carried carry VERB cana-697 12 5 out out ADP cana-697 12 6 in in ADP cana-697 12 7 an an DET cana-697 12 8 effort effort NOUN cana-697 12 9 to to PART cana-697 12 10 develop develop VERB cana-697 12 11 fixed fix VERB cana-697 12 12 point point NOUN cana-697 12 13 theory theory NOUN cana-697 12 14 , , PUNCT cana-697 12 15 including include VERB cana-697 12 16 by by ADP cana-697 12 17 providing provide VERB cana-697 12 18 a a DET cana-697 12 19 new new ADJ cana-697 12 20 definition definition NOUN cana-697 12 21 of of ADP cana-697 12 22 contraction contraction NOUN cana-697 12 23 mapping mapping NOUN cana-697 12 24 in in ADP cana-697 12 25 various various ADJ cana-697 12 26 applications application NOUN cana-697 12 27 [ [ X cana-697 12 28 11 11 NUM cana-697 12 29 , , PUNCT cana-697 12 30 12,14 12,14 ADV cana-697 12 31 ] ] PUNCT cana-697 12 32 as as ADP cana-697 12 33 a a DET cana-697 12 34 generalization generalization NOUN cana-697 12 35 of of ADP cana-697 12 36 the the DET cana-697 12 37 banach banach NOUN cana-697 12 38 contraction contraction NOUN cana-697 12 39 principle principle NOUN cana-697 12 40 [ [ X cana-697 12 41 3 3 NUM cana-697 12 42 ] ] PUNCT cana-697 12 43 . . PUNCT cana-697 13 1 contraction contraction NOUN cana-697 13 2 mapping mapping NOUN cana-697 13 3 is be AUX cana-697 13 4 a a DET cana-697 13 5 special special ADJ cana-697 13 6 case case NOUN cana-697 13 7 of of ADP cana-697 13 8 lipschitz-πœ† lipschitz-πœ† NOUN cana-697 13 9 mapping mapping NOUN cana-697 13 10 [ [ X cana-697 13 11 16 16 NUM cana-697 13 12 ] ] PUNCT cana-697 13 13 . . PUNCT cana-697 14 1 note note VERB cana-697 14 2 that that SCONJ cana-697 14 3 if if SCONJ cana-697 14 4 we we PRON cana-697 14 5 are be AUX cana-697 14 6 given give VERB cana-697 14 7 a a DET cana-697 14 8 metric metric ADJ cana-697 14 9 space space NOUN cana-697 14 10 ( ( PUNCT cana-697 14 11 𝑋 𝑋 PROPN cana-697 14 12 , , PUNCT cana-697 14 13 𝑑 𝑑 NOUN cana-697 14 14 ) ) PUNCT cana-697 14 15 and and CCONJ cana-697 14 16 given give VERB cana-697 14 17 a a DET cana-697 14 18 mapping mapping NOUN cana-697 14 19 𝑓 𝑓 X cana-697 14 20 : : PUNCT cana-697 14 21 𝑋 𝑋 PROPN cana-697 14 22 β†’ β†’ SYM cana-697 14 23 𝑋 𝑋 PROPN cana-697 14 24 , , PUNCT cana-697 14 25 then then ADV cana-697 14 26 the the DET cana-697 14 27 mapping mapping NOUN cana-697 14 28 𝑓 𝑓 NOUN cana-697 14 29 is be AUX cana-697 14 30 said say VERB cana-697 14 31 to to PART cana-697 14 32 be be AUX cana-697 14 33 a a DET cana-697 14 34 lipschitz-πœ† lipschitz-πœ† ADJ cana-697 14 35 mapping mapping NOUN cana-697 14 36 if if SCONJ cana-697 14 37 𝑑(𝑓(π‘₯ 𝑑(𝑓(π‘₯ NOUN cana-697 14 38 ) ) PUNCT cana-697 14 39 , , PUNCT cana-697 14 40 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 14 41 ) ) PUNCT cana-697 14 42 ) ) PUNCT cana-697 14 43 ≀ ≀ NUM cana-697 14 44 πœ†π‘‘(π‘₯ πœ†π‘‘(π‘₯ NOUN cana-697 14 45 , , PUNCT cana-697 14 46 𝑦 𝑦 NOUN cana-697 14 47 ) ) PUNCT cana-697 14 48 for for ADP cana-697 14 49 every every DET cana-697 14 50 π‘₯ π‘₯ PROPN cana-697 14 51 , , PUNCT cana-697 14 52 𝑦 𝑦 NOUN cana-697 14 53 ∈ ∈ NOUN cana-697 14 54 𝑋 𝑋 NOUN cana-697 14 55 with with ADP cana-697 14 56 πœ† πœ† DET cana-697 14 57 β‰₯ β‰₯ NOUN cana-697 14 58 0 0 NUM cana-697 14 59 . . PUNCT cana-697 15 1 in in ADP cana-697 15 2 this this DET cana-697 15 3 case case NOUN cana-697 15 4 , , PUNCT cana-697 15 5 if if SCONJ cana-697 15 6 πœ† πœ† PRON cana-697 15 7 ∈ ∈ PROPN cana-697 15 8 [ [ X cana-697 15 9 0,1 0,1 NUM cana-697 15 10 ) ) PUNCT cana-697 15 11 , , PUNCT cana-697 15 12 then then ADV cana-697 15 13 the the DET cana-697 15 14 mapping mapping NOUN cana-697 15 15 𝑓 𝑓 NOUN cana-697 15 16 is be AUX cana-697 15 17 said say VERB cana-697 15 18 to to PART cana-697 15 19 be be AUX cana-697 15 20 a a DET cana-697 15 21 contraction contraction NOUN cana-697 15 22 mapping mapping NOUN cana-697 15 23 , , PUNCT cana-697 15 24 whereas whereas SCONJ cana-697 15 25 if if SCONJ cana-697 15 26 πœ† πœ† PRON cana-697 15 27 = = SYM cana-697 15 28 1 1 NUM cana-697 15 29 , , PUNCT cana-697 15 30 the the DET cana-697 15 31 mapping mapping NOUN cana-697 15 32 𝑓 𝑓 NOUN cana-697 15 33 is be AUX cana-697 15 34 said say VERB cana-697 15 35 to to PART cana-697 15 36 be be AUX cana-697 15 37 a a DET cana-697 15 38 non non ADJ cana-697 15 39 - - ADJ cana-697 15 40 expansive expansive ADJ cana-697 15 41 mapping mapping NOUN cana-697 15 42 . . PUNCT cana-697 16 1 like like ADP cana-697 16 2 contraction contraction NOUN cana-697 16 3 mapping mapping NOUN cana-697 16 4 widely widely ADV cana-697 16 5 studied study VERB cana-697 16 6 in in ADP cana-697 16 7 relation relation NOUN cana-697 16 8 to to ADP cana-697 16 9 fixed fix VERB cana-697 16 10 points point NOUN cana-697 16 11 , , PUNCT cana-697 16 12 non non ADJ cana-697 16 13 - - ADJ cana-697 16 14 expansive expansive ADJ cana-697 16 15 mapping mapping NOUN cana-697 16 16 also also ADV cana-697 16 17 has have VERB cana-697 16 18 an an DET cana-697 16 19 equally equally ADV cana-697 16 20 important important ADJ cana-697 16 21 role role NOUN cana-697 16 22 in in ADP cana-697 16 23 fixed fix VERB cana-697 16 24 point point NOUN cana-697 16 25 studies study NOUN cana-697 16 26 [ [ X cana-697 16 27 4,8,13,15 4,8,13,15 NUM cana-697 16 28 ] ] PUNCT cana-697 16 29 . . PUNCT cana-697 17 1 one one NUM cana-697 17 2 of of ADP cana-697 17 3 them they PRON cana-697 17 4 was be AUX cana-697 17 5 carried carry VERB cana-697 17 6 out out ADP cana-697 17 7 by by ADP cana-697 17 8 vetro vetro NOUN cana-697 17 9 [ [ X cana-697 17 10 16 16 NUM cana-697 17 11 ] ] PUNCT cana-697 17 12 , , PUNCT cana-697 17 13 and and CCONJ cana-697 17 14 the the DET cana-697 17 15 existence existence NOUN cana-697 17 16 of of ADP cana-697 17 17 fixed fix VERB cana-697 17 18 points point NOUN cana-697 17 19 from from ADP cana-697 17 20 non non ADJ cana-697 17 21 - - ADJ cana-697 17 22 expansive expansive ADJ cana-697 17 23 mapping mapping NOUN cana-697 17 24 in in ADP cana-697 17 25 metric metric ADJ cana-697 17 26 space space NOUN cana-697 17 27 was be AUX cana-697 17 28 successfully successfully ADV cana-697 17 29 demonstrated demonstrate VERB cana-697 17 30 . . PUNCT cana-697 18 1 furthermore furthermore ADV cana-697 18 2 , , PUNCT cana-697 18 3 aydi aydi VERB cana-697 18 4 [ [ X cana-697 18 5 1 1 NUM cana-697 18 6 ] ] PUNCT cana-697 18 7 extend extend VERB cana-697 18 8 a a DET cana-697 18 9 fixed fix VERB cana-697 18 10 point point NOUN cana-697 18 11 theorem theorem NOUN cana-697 18 12 for for ADP cana-697 18 13 𝛼-nonexpansive 𝛼-nonexpansive ADJ cana-697 18 14 mappings mapping NOUN cana-697 18 15 on on ADP cana-697 18 16 partial partial ADJ cana-697 18 17 metric metric ADJ cana-697 18 18 spaces space NOUN cana-697 18 19 . . PUNCT cana-697 19 1 motivated motivate VERB cana-697 19 2 by by ADP cana-697 19 3 vetro vetro NOUN cana-697 19 4 [ [ X cana-697 19 5 16 16 NUM cana-697 19 6 ] ] PUNCT cana-697 19 7 and and CCONJ cana-697 19 8 aydi aydi VERB cana-697 19 9 [ [ X cana-697 19 10 1 1 NUM cana-697 19 11 ] ] PUNCT cana-697 19 12 , , PUNCT cana-697 19 13 we we PRON cana-697 19 14 will will AUX cana-697 19 15 prove prove VERB cana-697 19 16 the the DET cana-697 19 17 fixed fix VERB cana-697 19 18 point point NOUN cana-697 19 19 theorem theorem VERB cana-697 19 20 for for ADP cana-697 19 21 nonexpansive nonexpansive ADJ cana-697 19 22 mapping mapping NOUN cana-697 19 23 in in ADP cana-697 19 24 partial partial ADJ cana-697 19 25 metric metric ADJ cana-697 19 26 spaces space NOUN cana-697 19 27 . . PUNCT cana-697 20 1 and and CCONJ cana-697 20 2 we we PRON cana-697 20 3 also also ADV cana-697 20 4 prove prove VERB cana-697 20 5 a a DET cana-697 20 6 more more ADV cana-697 20 7 general general ADJ cana-697 20 8 theorem theorem NOUN cana-697 20 9 . . PROPN cana-697 21 1 2 2 NUM cana-697 21 2 . . NUM cana-697 21 3 preliminaries preliminary NOUN cana-697 21 4 in in ADP cana-697 21 5 1992 1992 NUM cana-697 21 6 , , PUNCT cana-697 21 7 matthews matthews PROPN cana-697 21 8 introduced introduce VERB cana-697 21 9 a a DET cana-697 21 10 new new ADJ cana-697 21 11 concept concept NOUN cana-697 21 12 as as ADP cana-697 21 13 a a DET cana-697 21 14 generalization generalization NOUN cana-697 21 15 of of ADP cana-697 21 16 standard standard ADJ cana-697 21 17 metrics metric NOUN cana-697 21 18 , , PUNCT cana-697 21 19 namely namely ADV cana-697 21 20 partial partial ADJ cana-697 21 21 metrics metric NOUN cana-697 21 22 . . PUNCT cana-697 22 1 let let VERB cana-697 22 2 we we PRON cana-697 22 3 consider consider VERB cana-697 22 4 the the DET cana-697 22 5 following follow VERB cana-697 22 6 definitions definition NOUN cana-697 22 7 . . PUNCT cana-697 23 1 communications communication NOUN cana-697 23 2 on on ADP cana-697 23 3 applied apply VERB cana-697 23 4 nonlinear nonlinear ADJ cana-697 23 5 analysis analysis NOUN cana-697 23 6 issn issn NOUN cana-697 23 7 : : PUNCT cana-697 23 8 1074 1074 NUM cana-697 23 9 - - PUNCT cana-697 23 10 133x 133x NUM cana-697 23 11 vol vol NOUN cana-697 23 12 31 31 NUM cana-697 23 13 no no NOUN cana-697 23 14 . . PUNCT cana-697 24 1 2s 2s NUM cana-697 24 2 ( ( PUNCT cana-697 24 3 2024 2024 NUM cana-697 24 4 ) ) PUNCT cana-697 24 5 677 677 NUM cana-697 24 6 https://internationalpubls.com https://internationalpubls.com X cana-697 24 7 definition definition NOUN cana-697 24 8 2.1 2.1 NUM cana-697 24 9 [ [ X cana-697 24 10 9,10 9,10 NUM cana-697 24 11 ] ] X cana-697 24 12 let let VERB cana-697 24 13 x x PRON cana-697 24 14 be be AUX cana-697 24 15 any any PRON cana-697 24 16 nonempty nonempty ADV cana-697 24 17 set set VERB cana-697 24 18 . . PUNCT cana-697 25 1 a a DET cana-697 25 2 partial partial ADJ cana-697 25 3 metric metric NOUN cana-697 25 4 on on ADP cana-697 25 5 𝑋 𝑋 PROPN cana-697 25 6 is be AUX cana-697 25 7 a a DET cana-697 25 8 mapping mapping NOUN cana-697 25 9 𝑝 𝑝 NOUN cana-697 25 10 : : PUNCT cana-697 25 11 𝑋 𝑋 PROPN cana-697 25 12 Γ— Γ— NOUN cana-697 25 13 𝑋 𝑋 PROPN cana-697 25 14 β†’ β†’ SYM cana-697 25 15 [ [ X cana-697 25 16 0 0 NUM cana-697 25 17 , , PUNCT cana-697 25 18 ∞ ∞ PROPN cana-697 25 19 ) ) PUNCT cana-697 25 20 which which PRON cana-697 25 21 satisfies satisfy VERB cana-697 25 22 following follow VERB cana-697 25 23 conditions condition NOUN cana-697 25 24 : : PUNCT cana-697 25 25 ( ( PUNCT cana-697 25 26 p1 p1 NOUN cana-697 25 27 ) ) PUNCT cana-697 25 28 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 25 29 , , PUNCT cana-697 25 30 𝑦 𝑦 NOUN cana-697 25 31 ) ) PUNCT cana-697 25 32 = = SYM cana-697 25 33 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 25 34 , , PUNCT cana-697 25 35 π‘₯ π‘₯ NOUN cana-697 25 36 ) ) PUNCT cana-697 25 37 , , PUNCT cana-697 25 38 ( ( PUNCT cana-697 25 39 p2 p2 X cana-697 25 40 ) ) PUNCT cana-697 25 41 if if SCONJ cana-697 25 42 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 25 43 , , PUNCT cana-697 25 44 π‘₯ π‘₯ NOUN cana-697 25 45 ) ) PUNCT cana-697 25 46 = = SYM cana-697 25 47 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 25 48 , , PUNCT cana-697 25 49 𝑦 𝑦 NOUN cana-697 25 50 ) ) PUNCT cana-697 25 51 = = SYM cana-697 25 52 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 25 53 , , PUNCT cana-697 25 54 𝑦 𝑦 NOUN cana-697 25 55 ) ) PUNCT cana-697 25 56 then then ADV cana-697 25 57 π‘₯ π‘₯ X cana-697 25 58 = = SYM cana-697 25 59 𝑦 𝑦 PROPN cana-697 25 60 , , PUNCT cana-697 25 61 ( ( PUNCT cana-697 25 62 p3 p3 NOUN cana-697 25 63 ) ) PUNCT cana-697 25 64 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 25 65 , , PUNCT cana-697 25 66 π‘₯ π‘₯ NOUN cana-697 25 67 ) ) PUNCT cana-697 25 68 ≀ ≀ NOUN cana-697 25 69 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 25 70 , , PUNCT cana-697 25 71 𝑦 𝑦 NOUN cana-697 25 72 ) ) PUNCT cana-697 25 73 , , PUNCT cana-697 25 74 ( ( PUNCT cana-697 25 75 p4 p4 ADJ cana-697 25 76 ) ) PUNCT cana-697 25 77 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 25 78 , , PUNCT cana-697 25 79 𝑧 𝑧 NOUN cana-697 25 80 ) ) PUNCT cana-697 25 81 + + ADP cana-697 25 82 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 25 83 , , PUNCT cana-697 25 84 𝑦 𝑦 NOUN cana-697 25 85 ) ) PUNCT cana-697 25 86 ≀ ≀ NOUN cana-697 25 87 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 25 88 , , PUNCT cana-697 25 89 𝑦 𝑦 NOUN cana-697 25 90 ) ) PUNCT cana-697 25 91 + + CCONJ cana-697 25 92 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 25 93 , , PUNCT cana-697 25 94 𝑧 𝑧 NOUN cana-697 25 95 ) ) PUNCT cana-697 25 96 , , PUNCT cana-697 25 97 for for ADP cana-697 25 98 all all DET cana-697 25 99 π‘₯ π‘₯ PROPN cana-697 25 100 , , PUNCT cana-697 25 101 𝑦 𝑦 NOUN cana-697 25 102 , , PUNCT cana-697 25 103 𝑧 𝑧 DET cana-697 25 104 ∈ ∈ PROPN cana-697 25 105 𝑋. 𝑋. PROPN cana-697 25 106 then then ADV cana-697 25 107 pair pair NOUN cana-697 25 108 ( ( PUNCT cana-697 25 109 𝑋 𝑋 PROPN cana-697 25 110 , , PUNCT cana-697 25 111 𝑝 𝑝 NOUN cana-697 25 112 ) ) PUNCT cana-697 25 113 is be AUX cana-697 25 114 called call VERB cana-697 25 115 a a DET cana-697 25 116 partial partial ADJ cana-697 25 117 metric metric ADJ cana-697 25 118 space space NOUN cana-697 25 119 . . PUNCT cana-697 26 1 partial partial ADJ cana-697 26 2 metric metric PROPN cana-697 26 3 𝑝 𝑝 PROPN cana-697 26 4 will will AUX cana-697 26 5 be be AUX cana-697 26 6 a a DET cana-697 26 7 metric metric ADJ cana-697 26 8 if if SCONJ cana-697 26 9 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 26 10 , , PUNCT cana-697 26 11 π‘₯ π‘₯ NOUN cana-697 26 12 ) ) PUNCT cana-697 26 13 = = SYM cana-697 27 1 0 0 X cana-697 27 2 . . PUNCT cana-697 28 1 several several ADJ cana-697 28 2 properties property NOUN cana-697 28 3 of of ADP cana-697 28 4 partial partial ADJ cana-697 28 5 metric metric ADJ cana-697 28 6 spaces space NOUN cana-697 28 7 are be AUX cana-697 28 8 given give VERB cana-697 28 9 as as ADP cana-697 28 10 follow follow VERB cana-697 28 11 [ [ X cana-697 28 12 5,6,7,9,10 5,6,7,9,10 PROPN cana-697 28 13 ] ] PUNCT cana-697 28 14 . . PUNCT cana-697 29 1 definition definition NOUN cana-697 29 2 2.2 2.2 NUM cana-697 29 3 . . PUNCT cana-697 30 1 let let AUX cana-697 30 2 ( ( PUNCT cana-697 30 3 𝑋 𝑋 PROPN cana-697 30 4 , , PUNCT cana-697 30 5 𝑝 𝑝 NOUN cana-697 30 6 ) ) PUNCT cana-697 30 7 be be AUX cana-697 30 8 a a DET cana-697 30 9 partial partial ADJ cana-697 30 10 metric metric ADJ cana-697 30 11 space space NOUN cana-697 30 12 . . PUNCT cana-697 31 1 a a DET cana-697 31 2 sequence sequence NOUN cana-697 31 3 ( ( PUNCT cana-697 31 4 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 31 5 ) ) PUNCT cana-697 31 6 is be AUX cana-697 31 7 said say VERB cana-697 31 8 to to PART cana-697 31 9 converges converge NOUN cana-697 31 10 to to ADP cana-697 31 11 a a DET cana-697 31 12 point point NOUN cana-697 31 13 π‘₯ π‘₯ DET cana-697 31 14 ∈ ∈ NOUN cana-697 31 15 𝑋 𝑋 NOUN cana-697 31 16 if if SCONJ cana-697 31 17 and and CCONJ cana-697 31 18 only only ADV cana-697 31 19 if if SCONJ cana-697 31 20 π‘™π‘–π‘šπ‘›β†’βˆž π‘™π‘–π‘šπ‘›β†’βˆž PROPN cana-697 31 21 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 PROPN cana-697 31 22 , , PUNCT cana-697 31 23 π‘₯ π‘₯ NOUN cana-697 31 24 ) ) PUNCT cana-697 31 25 = = SYM cana-697 31 26 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 31 27 , , PUNCT cana-697 31 28 π‘₯ π‘₯ NOUN cana-697 31 29 ) ) PUNCT cana-697 31 30 . . PUNCT cana-697 32 1 definition definition NOUN cana-697 32 2 2.3 2.3 NUM cana-697 32 3 . . PUNCT cana-697 33 1 let let AUX cana-697 33 2 ( ( PUNCT cana-697 33 3 𝑋 𝑋 PROPN cana-697 33 4 , , PUNCT cana-697 33 5 𝑝 𝑝 NOUN cana-697 33 6 ) ) PUNCT cana-697 33 7 be be AUX cana-697 33 8 a a DET cana-697 33 9 partial partial ADJ cana-697 33 10 metric metric ADJ cana-697 33 11 space space NOUN cana-697 33 12 . . PUNCT cana-697 34 1 a a DET cana-697 34 2 sequence sequence NOUN cana-697 34 3 ( ( PUNCT cana-697 34 4 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 34 5 ) ) PUNCT cana-697 34 6 is be AUX cana-697 34 7 called call VERB cana-697 34 8 cauchy cauchy ADJ cana-697 34 9 sequence sequence NOUN cana-697 34 10 if if SCONJ cana-697 34 11 and and CCONJ cana-697 34 12 only only ADV cana-697 34 13 if if SCONJ cana-697 34 14 π‘™π‘–π‘šπ‘›β†’βˆž π‘™π‘–π‘šπ‘›β†’βˆž PROPN cana-697 34 15 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 34 16 , , PUNCT cana-697 34 17 π‘₯π‘š π‘₯π‘š NOUN cana-697 34 18 ) ) PUNCT cana-697 34 19 is be AUX cana-697 34 20 finite finite ADJ cana-697 34 21 . . PUNCT cana-697 35 1 definition definition NOUN cana-697 35 2 2.4 2.4 NUM cana-697 35 3 . . PUNCT cana-697 36 1 let let AUX cana-697 36 2 ( ( PUNCT cana-697 36 3 𝑋 𝑋 PROPN cana-697 36 4 , , PUNCT cana-697 36 5 𝑝 𝑝 NOUN cana-697 36 6 ) ) PUNCT cana-697 36 7 be be AUX cana-697 36 8 a a DET cana-697 36 9 partial partial ADJ cana-697 36 10 metric metric ADJ cana-697 36 11 space space NOUN cana-697 36 12 . . PUNCT cana-697 37 1 if if SCONJ cana-697 37 2 every every DET cana-697 37 3 cauchy cauchy ADJ cana-697 37 4 sequence sequence NOUN cana-697 37 5 ( ( PUNCT cana-697 37 6 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 37 7 ) ) PUNCT cana-697 37 8 converges converge VERB cana-697 37 9 to to ADP cana-697 37 10 a a DET cana-697 37 11 point point NOUN cana-697 37 12 π‘₯ π‘₯ DET cana-697 37 13 ∈ ∈ NOUN cana-697 37 14 𝑋 𝑋 NOUN cana-697 37 15 such such ADJ cana-697 37 16 that that DET cana-697 37 17 π‘™π‘–π‘šπ‘›β†’βˆž π‘™π‘–π‘šπ‘›β†’βˆž PROPN cana-697 37 18 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 37 19 , , PUNCT cana-697 37 20 π‘₯π‘š π‘₯π‘š ADJ cana-697 37 21 ) ) PUNCT cana-697 37 22 = = SYM cana-697 37 23 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 37 24 , , PUNCT cana-697 37 25 π‘₯ π‘₯ NOUN cana-697 37 26 ) ) PUNCT cana-697 37 27 , , PUNCT cana-697 37 28 then then ADV cana-697 37 29 ( ( PUNCT cana-697 37 30 𝑋 𝑋 PROPN cana-697 37 31 , , PUNCT cana-697 37 32 𝑝 𝑝 NOUN cana-697 37 33 ) ) PUNCT cana-697 37 34 is be AUX cana-697 37 35 known know VERB cana-697 37 36 as as ADP cana-697 37 37 complete complete ADJ cana-697 37 38 partial partial ADJ cana-697 37 39 metric metric ADJ cana-697 37 40 space space NOUN cana-697 37 41 . . PUNCT cana-697 38 1 for for ADP cana-697 38 2 𝑝 𝑝 ADP cana-697 38 3 metric metric ADJ cana-697 38 4 spaces space NOUN cana-697 38 5 on on ADP cana-697 38 6 𝑋 𝑋 PROPN cana-697 38 7 , , PUNCT cana-697 38 8 the the DET cana-697 38 9 mapping mapping NOUN cana-697 38 10 𝑝𝑠 𝑝𝑠 NUM cana-697 38 11 : : PUNCT cana-697 38 12 𝑋 𝑋 PROPN cana-697 38 13 Γ— Γ— NOUN cana-697 38 14 𝑋 𝑋 PROPN cana-697 38 15 β†’ β†’ SYM cana-697 38 16 [ [ X cana-697 38 17 0 0 NUM cana-697 38 18 , , PUNCT cana-697 38 19 ∞ ∞ PROPN cana-697 38 20 ) ) PUNCT cana-697 38 21 defined define VERB cana-697 38 22 by by ADP cana-697 38 23 𝑝𝑠(π‘₯ 𝑝𝑠(π‘₯ ADJ cana-697 38 24 , , PUNCT cana-697 38 25 𝑦 𝑦 NOUN cana-697 38 26 ) ) PUNCT cana-697 38 27 = = SYM cana-697 38 28 2𝑝(π‘₯ 2𝑝(π‘₯ NUM cana-697 38 29 , , PUNCT cana-697 38 30 𝑦 𝑦 NOUN cana-697 38 31 ) ) PUNCT cana-697 38 32 βˆ’ βˆ’ PROPN cana-697 38 33 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 38 34 , , PUNCT cana-697 38 35 π‘₯ π‘₯ NOUN cana-697 38 36 ) ) PUNCT cana-697 38 37 βˆ’ βˆ’ ADP cana-697 38 38 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 38 39 , , PUNCT cana-697 38 40 𝑦 𝑦 NOUN cana-697 38 41 ) ) PUNCT cana-697 38 42 for for ADP cana-697 38 43 each each DET cana-697 38 44 π‘₯ π‘₯ PROPN cana-697 38 45 , , PUNCT cana-697 38 46 𝑦 𝑦 NOUN cana-697 38 47 ∈ ∈ NOUN cana-697 38 48 𝑋 𝑋 NOUN cana-697 38 49 is be AUX cana-697 38 50 a a DET cana-697 38 51 metric metric NOUN cana-697 38 52 on on ADP cana-697 38 53 𝑋. 𝑋. PROPN cana-697 38 54 a a DET cana-697 38 55 sequence sequence NOUN cana-697 38 56 ( ( PUNCT cana-697 38 57 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 38 58 ) ) PUNCT cana-697 38 59 is be AUX cana-697 38 60 cauchy cauchy ADJ cana-697 38 61 in in ADP cana-697 38 62 partial partial ADJ cana-697 38 63 metric metric ADJ cana-697 38 64 spaces space NOUN cana-697 38 65 ( ( PUNCT cana-697 38 66 𝑋 𝑋 PROPN cana-697 38 67 , , PUNCT cana-697 38 68 𝑝 𝑝 NOUN cana-697 38 69 ) ) PUNCT cana-697 38 70 if if SCONJ cana-697 39 1 and and CCONJ cana-697 39 2 only only ADV cana-697 39 3 if if SCONJ cana-697 39 4 ( ( PUNCT cana-697 39 5 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 39 6 ) ) PUNCT cana-697 39 7 is be AUX cana-697 39 8 cauchy cauchy ADJ cana-697 39 9 sequence sequence NOUN cana-697 39 10 in in ADP cana-697 39 11 metric metric ADJ cana-697 39 12 space space NOUN cana-697 39 13 ( ( PUNCT cana-697 39 14 𝑋 𝑋 NOUN cana-697 39 15 , , PUNCT cana-697 39 16 𝑝𝑠 𝑝𝑠 CCONJ cana-697 39 17 ) ) PUNCT cana-697 39 18 . . PUNCT cana-697 40 1 it it PRON cana-697 40 2 implies imply VERB cana-697 40 3 , , PUNCT cana-697 40 4 a a DET cana-697 40 5 partial partial ADJ cana-697 40 6 metric metric ADJ cana-697 40 7 space space NOUN cana-697 40 8 ( ( PUNCT cana-697 40 9 𝑋 𝑋 PROPN cana-697 40 10 , , PUNCT cana-697 40 11 𝑝 𝑝 NOUN cana-697 40 12 ) ) PUNCT cana-697 40 13 is be AUX cana-697 40 14 complete complete ADJ cana-697 40 15 if if SCONJ cana-697 40 16 and and CCONJ cana-697 40 17 only only ADV cana-697 40 18 if if SCONJ cana-697 40 19 metric metric ADJ cana-697 40 20 spaces space NOUN cana-697 40 21 ( ( PUNCT cana-697 40 22 𝑋 𝑋 NOUN cana-697 40 23 , , PUNCT cana-697 40 24 𝑝𝑠 𝑝𝑠 CCONJ cana-697 40 25 ) ) PUNCT cana-697 40 26 is be AUX cana-697 40 27 complete complete ADJ cana-697 40 28 . . PUNCT cana-697 41 1 therefore therefore ADV cana-697 41 2 , , PUNCT cana-697 41 3 for for ADP cana-697 41 4 ( ( PUNCT cana-697 41 5 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 41 6 ) ) PUNCT cana-697 41 7 is be AUX cana-697 41 8 sequence sequence NOUN cana-697 41 9 in in ADP cana-697 41 10 partial partial ADJ cana-697 41 11 metric metric ADJ cana-697 41 12 spaces space NOUN cana-697 41 13 ( ( PUNCT cana-697 41 14 𝑋 𝑋 PROPN cana-697 41 15 , , PUNCT cana-697 41 16 𝑝 𝑝 NOUN cana-697 41 17 ) ) PUNCT cana-697 41 18 and and CCONJ cana-697 41 19 π‘₯ π‘₯ PRON cana-697 41 20 ∈ ∈ PROPN cana-697 41 21 𝑋 𝑋 PROPN cana-697 41 22 , , PUNCT cana-697 41 23 we we PRON cana-697 41 24 have have VERB cana-697 41 25 lim lim PROPN cana-697 41 26 π‘›β†’βˆž π‘›β†’βˆž NUM cana-697 41 27 𝑝𝑠(π‘₯𝑛 𝑝𝑠(π‘₯𝑛 PROPN cana-697 41 28 , , PUNCT cana-697 41 29 π‘₯ π‘₯ NOUN cana-697 41 30 ) ) PUNCT cana-697 41 31 = = SYM cana-697 41 32 0 0 PUNCT cana-697 42 1 if if SCONJ cana-697 42 2 and and CCONJ cana-697 42 3 only only ADV cana-697 42 4 if if SCONJ cana-697 42 5 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 42 6 , , PUNCT cana-697 42 7 π‘₯ π‘₯ NOUN cana-697 42 8 ) ) PUNCT cana-697 42 9 = = SYM cana-697 42 10 lim lim NOUN cana-697 42 11 π‘›β†’βˆž π‘›β†’βˆž NUM cana-697 42 12 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 PROPN cana-697 42 13 , , PUNCT cana-697 42 14 π‘₯ π‘₯ NOUN cana-697 42 15 ) ) PUNCT cana-697 42 16 = = VERB cana-697 42 17 lim lim PROPN cana-697 42 18 𝑛,π‘šβ†’βˆž 𝑛,π‘šβ†’βˆž PROPN cana-697 42 19 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 42 20 , , PUNCT cana-697 42 21 π‘₯π‘š π‘₯π‘š NOUN cana-697 42 22 ) ) PUNCT cana-697 42 23 definition definition NOUN cana-697 42 24 2.5 2.5 NUM cana-697 42 25 . . PUNCT cana-697 43 1 let let AUX cana-697 43 2 ( ( PUNCT cana-697 43 3 𝑋 𝑋 PROPN cana-697 43 4 , , PUNCT cana-697 43 5 𝑝 𝑝 NOUN cana-697 43 6 ) ) PUNCT cana-697 43 7 be be AUX cana-697 43 8 a a DET cana-697 43 9 partial partial ADJ cana-697 43 10 metric metric ADJ cana-697 43 11 space space NOUN cana-697 43 12 . . PUNCT cana-697 44 1 mapping map VERB cana-697 45 1 𝑓 𝑓 X cana-697 45 2 : : PUNCT cana-697 45 3 𝑋 𝑋 PROPN cana-697 45 4 β†’ β†’ SYM cana-697 45 5 𝑋 𝑋 PROPN cana-697 45 6 is be AUX cana-697 45 7 ( ( PUNCT cana-697 45 8 sequentially sequentially ADV cana-697 45 9 ) ) PUNCT cana-697 45 10 continuous continuous ADJ cana-697 45 11 if if SCONJ cana-697 45 12 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 45 13 , , PUNCT cana-697 45 14 π‘₯ π‘₯ NOUN cana-697 45 15 ) ) PUNCT cana-697 45 16 β†’ β†’ SYM cana-697 45 17 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 45 18 , , PUNCT cana-697 45 19 π‘₯ π‘₯ NOUN cana-697 45 20 ) ) PUNCT cana-697 45 21 then then ADV cana-697 45 22 for for ADP cana-697 45 23 𝑛 𝑛 PROPN cana-697 45 24 β†’ β†’ SYM cana-697 45 25 ∞ ∞ NUM cana-697 45 26 we we PRON cana-697 45 27 have have VERB cana-697 45 28 𝑝(𝑓(π‘₯𝑛 𝑝(𝑓(π‘₯𝑛 NOUN cana-697 45 29 ) ) PUNCT cana-697 45 30 , , PUNCT cana-697 45 31 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 45 32 ) ) PUNCT cana-697 45 33 ) ) PUNCT cana-697 46 1 β†’ β†’ PUNCT cana-697 46 2 𝑝(𝑓(π‘₯ 𝑝(𝑓(π‘₯ NOUN cana-697 46 3 ) ) PUNCT cana-697 46 4 , , PUNCT cana-697 46 5 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 46 6 ) ) PUNCT cana-697 46 7 ) ) PUNCT cana-697 46 8 . . PUNCT cana-697 47 1 lemma lemma PROPN cana-697 47 2 2.6 2.6 NUM cana-697 47 3 . . PUNCT cana-697 47 4 suppose suppose VERB cana-697 47 5 that that SCONJ cana-697 47 6 ( ( PUNCT cana-697 47 7 𝑋 𝑋 PROPN cana-697 47 8 , , PUNCT cana-697 47 9 𝑝 𝑝 NOUN cana-697 47 10 ) ) PUNCT cana-697 47 11 be be AUX cana-697 47 12 a a DET cana-697 47 13 partial partial ADJ cana-697 47 14 metric metric ADJ cana-697 47 15 space space NOUN cana-697 47 16 , , PUNCT cana-697 47 17 then then ADV cana-697 47 18 1 1 X cana-697 47 19 . . PUNCT cana-697 48 1 if if SCONJ cana-697 48 2 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 48 3 , , PUNCT cana-697 48 4 𝑦 𝑦 NOUN cana-697 48 5 ) ) PUNCT cana-697 48 6 = = SYM cana-697 48 7 0 0 PUNCT cana-697 49 1 then then ADV cana-697 49 2 π‘₯ π‘₯ X cana-697 49 3 = = SYM cana-697 49 4 𝑦 𝑦 SYM cana-697 49 5 2 2 NUM cana-697 49 6 . . PUNCT cana-697 50 1 if if SCONJ cana-697 50 2 π‘₯ π‘₯ PROPN cana-697 50 3 β‰  β‰  PROPN cana-697 50 4 𝑦 𝑦 NOUN cana-697 50 5 then then ADV cana-697 50 6 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 50 7 , , PUNCT cana-697 50 8 𝑦 𝑦 NOUN cana-697 50 9 ) ) PUNCT cana-697 50 10 > > X cana-697 50 11 0 0 PUNCT cana-697 51 1 lemma lemma PROPN cana-697 51 2 2.7 2.7 NUM cana-697 51 3 . . PUNCT cana-697 52 1 [ [ X cana-697 52 2 16 16 NUM cana-697 52 3 ] ] X cana-697 52 4 if if SCONJ cana-697 52 5 ( ( PUNCT cana-697 52 6 π‘Žπ‘› π‘Žπ‘› NOUN cana-697 52 7 ) ) PUNCT cana-697 52 8 is be AUX cana-697 52 9 nonincreasing nonincrease VERB cana-697 52 10 sequences sequence NOUN cana-697 52 11 of of ADP cana-697 52 12 nonnegative nonnegative ADJ cana-697 52 13 real real ADJ cana-697 52 14 numbers number NOUN cana-697 52 15 , , PUNCT cana-697 52 16 then then ADV cana-697 52 17 the the DET cana-697 52 18 sequence sequence NOUN cana-697 52 19 communications communication NOUN cana-697 52 20 on on ADP cana-697 52 21 applied apply VERB cana-697 52 22 nonlinear nonlinear ADJ cana-697 52 23 analysis analysis NOUN cana-697 52 24 issn issn NOUN cana-697 52 25 : : PUNCT cana-697 52 26 1074 1074 NUM cana-697 52 27 - - PUNCT cana-697 52 28 133x 133x NUM cana-697 52 29 vol vol NOUN cana-697 52 30 31 31 NUM cana-697 52 31 no no NOUN cana-697 52 32 . . PUNCT cana-697 53 1 2s 2s NUM cana-697 53 2 ( ( PUNCT cana-697 53 3 2024 2024 NUM cana-697 53 4 ) ) PUNCT cana-697 53 5 678 678 NUM cana-697 53 6 https://internationalpubls.com https://internationalpubls.com X cana-697 53 7 ( ( PUNCT cana-697 53 8 π‘Žπ‘› π‘Žπ‘› NOUN cana-697 53 9 + + CCONJ cana-697 53 10 π‘Žπ‘›+1 π‘Žπ‘›+1 VERB cana-697 53 11 π‘Žπ‘› π‘Žπ‘› NOUN cana-697 53 12 + + CCONJ cana-697 53 13 π‘Žπ‘›+1 π‘Žπ‘›+1 NUM cana-697 53 14 + + CCONJ cana-697 53 15 1 1 NUM cana-697 53 16 ) ) PUNCT cana-697 53 17 is be AUX cana-697 53 18 nonincreasing nonincrease VERB cana-697 53 19 too too ADV cana-697 53 20 . . PUNCT cana-697 54 1 we we PRON cana-697 54 2 extend extend VERB cana-697 54 3 corollary corollary ADJ cana-697 54 4 2.5 2.5 NUM cana-697 54 5 in in ADP cana-697 54 6 [ [ X cana-697 54 7 16 16 NUM cana-697 54 8 ] ] PUNCT cana-697 54 9 to to ADP cana-697 54 10 partial partial ADJ cana-697 54 11 metric metric ADJ cana-697 54 12 spaces space NOUN cana-697 54 13 . . PUNCT cana-697 55 1 corollary corollary ADJ cana-697 55 2 2.8 2.8 NUM cana-697 55 3 . . PUNCT cana-697 56 1 let let AUX cana-697 56 2 ( ( PUNCT cana-697 56 3 𝑋 𝑋 PROPN cana-697 56 4 , , PUNCT cana-697 56 5 𝑝 𝑝 NOUN cana-697 56 6 ) ) PUNCT cana-697 56 7 be be AUX cana-697 56 8 a a DET cana-697 56 9 partial partial ADJ cana-697 56 10 metric metric ADJ cana-697 56 11 space space NOUN cana-697 56 12 . . PUNCT cana-697 57 1 suppose suppose VERB cana-697 57 2 that that SCONJ cana-697 57 3 𝑓 𝑓 X cana-697 57 4 : : PUNCT cana-697 57 5 𝑋 𝑋 PROPN cana-697 57 6 β†’ β†’ SYM cana-697 57 7 𝑋 𝑋 PROPN cana-697 57 8 be be VERB cana-697 57 9 a a DET cana-697 57 10 nonexpansive nonexpansive ADJ cana-697 57 11 mapping mapping NOUN cana-697 57 12 and and CCONJ cana-697 57 13 π‘₯0 π‘₯0 PROPN cana-697 57 14 ∈ ∈ PROPN cana-697 57 15 𝑋. 𝑋. PROPN cana-697 57 16 if if SCONJ cana-697 57 17 ( ( PUNCT cana-697 57 18 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 57 19 ) ) PUNCT cana-697 57 20 is be AUX cana-697 57 21 a a DET cana-697 57 22 picard picard NOUN cana-697 57 23 sequence sequence NOUN cana-697 57 24 of of ADP cana-697 57 25 initial initial ADJ cana-697 57 26 point point NOUN cana-697 57 27 π‘₯0 π‘₯0 NOUN cana-697 57 28 , , PUNCT cana-697 57 29 then then ADV cana-697 57 30 the the DET cana-697 57 31 sequence sequence NOUN cana-697 57 32 ( ( PUNCT cana-697 57 33 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 57 34 , , PUNCT cana-697 57 35 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 57 36 ) ) PUNCT cana-697 58 1 + + CCONJ cana-697 58 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 58 3 , , PUNCT cana-697 58 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 58 5 ) ) PUNCT cana-697 58 6 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 58 7 , , PUNCT cana-697 58 8 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 58 9 ) ) PUNCT cana-697 59 1 + + CCONJ cana-697 59 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 59 3 , , PUNCT cana-697 59 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 59 5 ) ) PUNCT cana-697 59 6 + + CCONJ cana-697 59 7 1 1 X cana-697 59 8 ) ) PUNCT cana-697 59 9 ( ( PUNCT cana-697 59 10 2.1 2.1 NUM cana-697 59 11 ) ) PUNCT cana-697 59 12 is be AUX cana-697 59 13 nonincreasing nonincrease VERB cana-697 59 14 . . PUNCT cana-697 60 1 proof proof NOUN cana-697 60 2 : : PUNCT cana-697 60 3 since since SCONJ cana-697 60 4 ( ( PUNCT cana-697 60 5 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 60 6 ) ) PUNCT cana-697 60 7 is be AUX cana-697 60 8 a a DET cana-697 60 9 picard picard NOUN cana-697 60 10 sequence sequence NOUN cana-697 60 11 of of ADP cana-697 60 12 initial initial ADJ cana-697 60 13 point point NOUN cana-697 60 14 π‘₯0 π‘₯0 NOUN cana-697 60 15 then then ADV cana-697 60 16 π‘₯𝑛 π‘₯𝑛 VERB cana-697 60 17 = = SYM cana-697 60 18 𝑓(π‘₯π‘›βˆ’1 𝑓(π‘₯π‘›βˆ’1 PROPN cana-697 60 19 ) ) PUNCT cana-697 60 20 = = PUNCT cana-697 61 1 𝑓𝑛(π‘₯0 𝑓𝑛(π‘₯0 X cana-697 61 2 ) ) PUNCT cana-697 61 3 for for ADP cana-697 61 4 all all DET cana-697 61 5 𝑛 𝑛 DET cana-697 61 6 ∈ ∈ PROPN cana-697 61 7 β„•. β„•. PROPN cana-697 61 8 furthermore furthermore ADV cana-697 61 9 , , PUNCT cana-697 61 10 since since SCONJ cana-697 61 11 𝑓 𝑓 PRON cana-697 61 12 is be AUX cana-697 61 13 nonexpansive nonexpansive ADJ cana-697 61 14 mapping mapping NOUN cana-697 61 15 then then ADV cana-697 61 16 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 61 17 , , PUNCT cana-697 61 18 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 61 19 ) ) PUNCT cana-697 61 20 = = SYM cana-697 61 21 𝑝(𝑓(π‘₯π‘›βˆ’1 𝑝(𝑓(π‘₯π‘›βˆ’1 PROPN cana-697 61 22 ) ) PUNCT cana-697 61 23 , , PUNCT cana-697 61 24 𝑓(π‘₯𝑛 𝑓(π‘₯𝑛 NOUN cana-697 61 25 ) ) PUNCT cana-697 61 26 ) ) PUNCT cana-697 61 27 ≀ ≀ NOUN cana-697 62 1 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 62 2 , , PUNCT cana-697 62 3 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 62 4 ) ) PUNCT cana-697 62 5 for for ADP cana-697 62 6 all all DET cana-697 62 7 𝑛 𝑛 DET cana-697 62 8 ∈ ∈ NOUN cana-697 62 9 β„•. β„•. PROPN cana-697 62 10 hence hence ADV cana-697 62 11 by by ADP cana-697 62 12 lemma lemma PROPN cana-697 62 13 2.7 2.7 NUM cana-697 62 14 , , PUNCT cana-697 62 15 the the DET cana-697 62 16 statement statement NOUN cana-697 62 17 ( ( PUNCT cana-697 62 18 2.1 2.1 NUM cana-697 62 19 ) ) PUNCT cana-697 62 20 holds hold VERB cana-697 62 21 . . PUNCT cana-697 63 1 this this PRON cana-697 63 2 completes complete VERB cana-697 63 3 the the DET cana-697 63 4 proof proof NOUN cana-697 63 5 . . PUNCT cana-697 64 1 3 3 X cana-697 64 2 . . X cana-697 64 3 main main ADJ cana-697 64 4 results result NOUN cana-697 64 5 theorem theorem VERB cana-697 64 6 3.1 3.1 NUM cana-697 64 7 . . PUNCT cana-697 65 1 let let AUX cana-697 65 2 ( ( PUNCT cana-697 65 3 𝑋 𝑋 PROPN cana-697 65 4 , , PUNCT cana-697 65 5 𝑝 𝑝 NOUN cana-697 65 6 ) ) PUNCT cana-697 65 7 be be AUX cana-697 65 8 a a DET cana-697 65 9 complete complete ADJ cana-697 65 10 partial partial ADJ cana-697 65 11 metric metric ADJ cana-697 65 12 spaces space NOUN cana-697 65 13 endowed endow VERB cana-697 65 14 with with ADP cana-697 65 15 a a DET cana-697 65 16 binary binary ADJ cana-697 65 17 relation relation NOUN cana-697 65 18 β„œ β„œ PROPN cana-697 65 19 on on ADP cana-697 65 20 𝑋. 𝑋. PROPN cana-697 65 21 suppose suppose VERB cana-697 65 22 that that SCONJ cana-697 65 23 𝑓 𝑓 X cana-697 65 24 : : PUNCT cana-697 65 25 𝑋 𝑋 PROPN cana-697 65 26 β†’ β†’ SYM cana-697 65 27 𝑋 𝑋 PROPN cana-697 65 28 be be VERB cana-697 65 29 a a DET cana-697 65 30 nonexpansive nonexpansive ADJ cana-697 65 31 mappings mapping NOUN cana-697 65 32 such such ADJ cana-697 65 33 that that SCONJ cana-697 65 34 𝑝(𝑓(π‘₯ 𝑝(𝑓(π‘₯ NOUN cana-697 65 35 ) ) PUNCT cana-697 65 36 , , PUNCT cana-697 65 37 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 65 38 ) ) PUNCT cana-697 65 39 ) ) PUNCT cana-697 66 1 ≀ ≀ NOUN cana-697 66 2 ( ( PUNCT cana-697 66 3 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 66 4 , , PUNCT cana-697 66 5 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 66 6 ) ) PUNCT cana-697 66 7 ) ) PUNCT cana-697 67 1 + + CCONJ cana-697 67 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 67 3 , , PUNCT cana-697 67 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 67 5 ) ) PUNCT cana-697 67 6 ) ) PUNCT cana-697 67 7 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 67 8 , , PUNCT cana-697 67 9 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 67 10 ) ) PUNCT cana-697 67 11 ) ) PUNCT cana-697 68 1 + + CCONJ cana-697 68 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 68 3 , , PUNCT cana-697 68 4 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 68 5 ) ) PUNCT cana-697 68 6 ) ) PUNCT cana-697 69 1 + + CCONJ cana-697 69 2 1 1 NUM cana-697 69 3 + + NUM cana-697 69 4 π‘˜ π‘˜ X cana-697 69 5 ) ) PUNCT cana-697 69 6 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 69 7 , , PUNCT cana-697 69 8 𝑦 𝑦 NOUN cana-697 69 9 ) ) PUNCT cana-697 69 10 ( ( PUNCT cana-697 69 11 3.1 3.1 NUM cana-697 69 12 ) ) PUNCT cana-697 69 13 for for ADP cana-697 69 14 each each DET cana-697 69 15 π‘₯ π‘₯ PROPN cana-697 69 16 , , PUNCT cana-697 69 17 𝑦 𝑦 NOUN cana-697 69 18 ∈ ∈ PROPN cana-697 69 19 β„œ β„œ PROPN cana-697 69 20 , , PUNCT cana-697 69 21 where where SCONJ cana-697 69 22 π‘˜ π‘˜ PRON cana-697 69 23 ∈ ∈ PROPN cana-697 69 24 [ [ X cana-697 69 25 0,1 0,1 NUM cana-697 69 26 ) ) PUNCT cana-697 69 27 . . PUNCT cana-697 70 1 assume assume VERB cana-697 70 2 that that SCONJ cana-697 70 3 : : PUNCT cana-697 70 4 1 1 X cana-697 70 5 . . X cana-697 70 6 𝑓 𝑓 PRON cana-697 70 7 is be AUX cana-697 70 8 preserving preserve VERB cana-697 70 9 mapping mapping NOUN cana-697 70 10 2 2 NUM cana-697 70 11 . . PUNCT cana-697 71 1 𝑓 𝑓 PRON cana-697 71 2 is be AUX cana-697 71 3 continuous continuous ADJ cana-697 71 4 mapping mapping NOUN cana-697 71 5 3 3 NUM cana-697 71 6 . . PUNCT cana-697 72 1 𝐹𝑖π‘₯ 𝐹𝑖π‘₯ PROPN cana-697 72 2 ( ( PUNCT cana-697 72 3 𝑓 𝑓 X cana-697 72 4 ) ) PUNCT cana-697 72 5 is be AUX cana-697 72 6 well well ADV cana-697 72 7 ordered order VERB cana-697 72 8 with with ADP cana-697 72 9 respect respect NOUN cana-697 72 10 to to ADP cana-697 72 11 β„œ. β„œ. PROPN cana-697 72 12 if if SCONJ cana-697 72 13 there there PRON cana-697 72 14 exist exist VERB cana-697 72 15 π‘₯0 π‘₯0 NOUN cana-697 72 16 ∈ ∈ NOUN cana-697 72 17 𝑋 𝑋 NOUN cana-697 72 18 such such ADJ cana-697 72 19 that that SCONJ cana-697 72 20 ( ( PUNCT cana-697 72 21 π‘₯0 π‘₯0 NOUN cana-697 72 22 , , PUNCT cana-697 72 23 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 72 24 ) ) PUNCT cana-697 72 25 ) ) PUNCT cana-697 72 26 ∈ ∈ PROPN cana-697 72 27 β„œ β„œ PROPN cana-697 72 28 and and CCONJ cana-697 72 29 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 72 30 , , PUNCT cana-697 72 31 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 72 32 ) ) PUNCT cana-697 72 33 ) ) PUNCT cana-697 73 1 + + CCONJ cana-697 73 2 𝑝(𝑓(π‘₯0 𝑝(𝑓(π‘₯0 ADJ cana-697 73 3 ) ) PUNCT cana-697 73 4 , , PUNCT cana-697 73 5 𝑓2(π‘₯0 𝑓2(π‘₯0 NUM cana-697 73 6 ) ) PUNCT cana-697 73 7 ) ) PUNCT cana-697 73 8 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 73 9 , , PUNCT cana-697 73 10 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 73 11 ) ) PUNCT cana-697 73 12 ) ) PUNCT cana-697 74 1 + + CCONJ cana-697 74 2 𝑝(𝑓(π‘₯0 𝑝(𝑓(π‘₯0 ADJ cana-697 74 3 ) ) PUNCT cana-697 74 4 , , PUNCT cana-697 74 5 𝑓2(π‘₯0 𝑓2(π‘₯0 NUM cana-697 74 6 ) ) PUNCT cana-697 74 7 ) ) PUNCT cana-697 75 1 + + CCONJ cana-697 75 2 1 1 NUM cana-697 75 3 + + NUM cana-697 75 4 π‘˜ π‘˜ X cana-697 75 5 < < X cana-697 75 6 1 1 NUM cana-697 75 7 ( ( PUNCT cana-697 75 8 3.2 3.2 NUM cana-697 75 9 ) ) PUNCT cana-697 75 10 then then ADV cana-697 75 11 there there PRON cana-697 75 12 exist exist VERB cana-697 75 13 𝑧 𝑧 DET cana-697 75 14 ∈ ∈ NOUN cana-697 75 15 𝑋 𝑋 NOUN cana-697 75 16 such such ADJ cana-697 75 17 that that PRON cana-697 75 18 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 75 19 , , PUNCT cana-697 75 20 𝑧 𝑧 NOUN cana-697 75 21 ) ) PUNCT cana-697 75 22 = = SYM cana-697 75 23 0 0 X cana-697 75 24 . . PUNCT cana-697 76 1 furthermore furthermore ADV cana-697 76 2 a. a. NOUN cana-697 76 3 there there PRON cana-697 76 4 exists exist VERB cana-697 76 5 𝑧 𝑧 PRON cana-697 76 6 ∈ ∈ PROPN cana-697 76 7 𝑋 𝑋 NOUN cana-697 76 8 fixed fix VERB cana-697 76 9 point point NOUN cana-697 76 10 of of ADP cana-697 76 11 𝑓 𝑓 DET cana-697 76 12 b. b. NOUN cana-697 76 13 the the DET cana-697 76 14 picard picard PROPN cana-697 76 15 sequences sequence NOUN cana-697 76 16 of of ADP cana-697 76 17 initial initial ADJ cana-697 76 18 point point NOUN cana-697 76 19 π‘₯0 π‘₯0 NOUN cana-697 76 20 ∈ ∈ NOUN cana-697 76 21 𝑋 𝑋 NOUN cana-697 76 22 converges converge VERB cana-697 76 23 to to ADP cana-697 76 24 fixed fix VERB cana-697 76 25 point point NOUN cana-697 76 26 of of ADP cana-697 76 27 𝑓 𝑓 DET cana-697 76 28 c. c. NOUN cana-697 76 29 if if SCONJ cana-697 76 30 𝑧 𝑧 PROPN cana-697 76 31 and and CCONJ cana-697 76 32 𝑀 𝑀 PROPN cana-697 76 33 are be AUX cana-697 76 34 fixed fix VERB cana-697 76 35 point point NOUN cana-697 76 36 of of ADP cana-697 76 37 𝑓 𝑓 PRON cana-697 76 38 where where SCONJ cana-697 76 39 𝑧 𝑧 DET cana-697 76 40 β‰  β‰  PROPN cana-697 76 41 𝑀 𝑀 PROPN cana-697 76 42 then then ADV cana-697 76 43 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 76 44 , , PUNCT cana-697 76 45 𝑀 𝑀 X cana-697 76 46 ) ) PUNCT cana-697 76 47 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 76 48 , , PUNCT cana-697 76 49 𝑧 𝑧 NOUN cana-697 76 50 ) ) PUNCT cana-697 76 51 + + CCONJ cana-697 76 52 𝑝(𝑀 𝑝(𝑀 PROPN cana-697 76 53 , , PUNCT cana-697 76 54 𝑀 𝑀 ADP cana-697 76 55 ) ) PUNCT cana-697 76 56 + + CCONJ cana-697 76 57 1 1 NUM cana-697 76 58 β‰₯ β‰₯ NOUN cana-697 76 59 1 1 NUM cana-697 76 60 βˆ’ βˆ’ NOUN cana-697 76 61 π‘˜ π‘˜ SYM cana-697 76 62 2 2 NUM cana-697 76 63 communications communication NOUN cana-697 76 64 on on ADP cana-697 76 65 applied apply VERB cana-697 76 66 nonlinear nonlinear ADJ cana-697 76 67 analysis analysis NOUN cana-697 76 68 issn issn NOUN cana-697 76 69 : : PUNCT cana-697 76 70 1074 1074 NUM cana-697 76 71 - - PUNCT cana-697 76 72 133x 133x NUM cana-697 76 73 vol vol NOUN cana-697 76 74 31 31 NUM cana-697 76 75 no no NOUN cana-697 76 76 . . PUNCT cana-697 77 1 2s 2s NUM cana-697 77 2 ( ( PUNCT cana-697 77 3 2024 2024 NUM cana-697 77 4 ) ) PUNCT cana-697 77 5 679 679 NUM cana-697 77 6 https://internationalpubls.com https://internationalpubls.com X cana-697 77 7 proof proof NOUN cana-697 77 8 : : PUNCT cana-697 77 9 let let VERB cana-697 77 10 π‘₯0 π‘₯0 PROPN cana-697 77 11 ∈ ∈ PROPN cana-697 77 12 𝑋 𝑋 NOUN cana-697 77 13 be be VERB cana-697 77 14 such such ADJ cana-697 77 15 that that SCONJ cana-697 77 16 ( ( PUNCT cana-697 77 17 π‘₯0 π‘₯0 NOUN cana-697 77 18 , , PUNCT cana-697 77 19 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 77 20 ) ) PUNCT cana-697 77 21 ) ) PUNCT cana-697 77 22 ∈ ∈ PROPN cana-697 77 23 β„œ β„œ PROPN cana-697 77 24 and and CCONJ cana-697 77 25 ( ( PUNCT cana-697 77 26 3.2 3.2 NUM cana-697 77 27 ) ) PUNCT cana-697 77 28 holds hold VERB cana-697 77 29 . . PUNCT cana-697 77 30 suppose suppose VERB cana-697 77 31 that that SCONJ cana-697 77 32 ( ( PUNCT cana-697 77 33 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 77 34 ) ) PUNCT cana-697 77 35 be be AUX cana-697 77 36 a a DET cana-697 77 37 picard picard NOUN cana-697 77 38 sequence sequence NOUN cana-697 77 39 of of ADP cana-697 77 40 initial initial ADJ cana-697 77 41 point point NOUN cana-697 77 42 π‘₯0 π‘₯0 NOUN cana-697 77 43 such such ADJ cana-697 77 44 that that DET cana-697 77 45 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 77 46 = = SYM cana-697 77 47 𝑓(π‘₯π‘›βˆ’1 𝑓(π‘₯π‘›βˆ’1 PROPN cana-697 77 48 ) ) PUNCT cana-697 77 49 = = PUNCT cana-697 78 1 𝑓𝑛(π‘₯0 𝑓𝑛(π‘₯0 X cana-697 78 2 ) ) PUNCT cana-697 78 3 for for ADP cana-697 78 4 all all DET cana-697 78 5 𝑛 𝑛 DET cana-697 78 6 ∈ ∈ PROPN cana-697 78 7 β„•. β„•. PROPN cana-697 78 8 let let VERB cana-697 78 9 we we PRON cana-697 78 10 consider consider VERB cana-697 78 11 the the DET cana-697 78 12 following follow VERB cana-697 78 13 cases case NOUN cana-697 78 14 : : PUNCT cana-697 78 15 case case NOUN cana-697 78 16 1 1 NUM cana-697 78 17 . . PUNCT cana-697 78 18 suppose suppose VERB cana-697 78 19 that that SCONJ cana-697 78 20 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 NOUN cana-697 78 21 = = PUNCT cana-697 78 22 π‘₯𝑛 π‘₯𝑛 VERB cana-697 78 23 for for ADP cana-697 78 24 some some DET cana-697 78 25 𝑛 𝑛 PRON cana-697 78 26 ∈ ∈ PROPN cana-697 78 27 β„• β„• PROPN cana-697 78 28 then then ADV cana-697 78 29 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 PROPN cana-697 78 30 = = SYM cana-697 78 31 𝑓(π‘₯π‘›βˆ’1 𝑓(π‘₯π‘›βˆ’1 PROPN cana-697 78 32 ) ) PUNCT cana-697 78 33 it it PRON cana-697 78 34 means mean VERB cana-697 78 35 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 NOUN cana-697 78 36 is be AUX cana-697 78 37 fixed fix VERB cana-697 78 38 point point NOUN cana-697 78 39 of of ADP cana-697 78 40 𝑓. 𝑓. NOUN cana-697 78 41 hence hence ADV cana-697 78 42 , , PUNCT cana-697 78 43 the the DET cana-697 78 44 existence existence NOUN cana-697 78 45 of of ADP cana-697 78 46 a a DET cana-697 78 47 fixed fix VERB cana-697 78 48 point point NOUN cana-697 78 49 of of ADP cana-697 78 50 𝑓 𝑓 PRON cana-697 78 51 is be AUX cana-697 78 52 proved prove VERB cana-697 78 53 . . PUNCT cana-697 79 1 case case NOUN cana-697 79 2 2 2 X cana-697 79 3 . . PUNCT cana-697 79 4 suppose suppose VERB cana-697 79 5 that that SCONJ cana-697 79 6 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 PROPN cana-697 79 7 β‰  β‰  PROPN cana-697 79 8 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 79 9 for for ADP cana-697 79 10 all all DET cana-697 79 11 𝑛 𝑛 DET cana-697 79 12 ∈ ∈ PROPN cana-697 79 13 β„•. β„•. PROPN cana-697 79 14 let let VERB cana-697 79 15 we we PRON cana-697 79 16 consider consider VERB cana-697 79 17 that that PRON cana-697 79 18 ( ( PUNCT cana-697 79 19 π‘₯0 π‘₯0 NOUN cana-697 79 20 , , PUNCT cana-697 79 21 π‘₯1 π‘₯1 NOUN cana-697 79 22 ) ) PUNCT cana-697 79 23 = = PUNCT cana-697 79 24 ( ( PUNCT cana-697 79 25 π‘₯0 π‘₯0 PROPN cana-697 79 26 , , PUNCT cana-697 79 27 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 79 28 ) ) PUNCT cana-697 79 29 ) ) PUNCT cana-697 80 1 ∈ ∈ PROPN cana-697 80 2 β„œ. β„œ. PROPN cana-697 80 3 since since SCONJ cana-697 80 4 𝑓 𝑓 PRON cana-697 80 5 is be AUX cana-697 80 6 preserving preserve VERB cana-697 80 7 mapping mapping NOUN cana-697 80 8 then then ADV cana-697 80 9 we we PRON cana-697 80 10 have have VERB cana-697 80 11 ( ( PUNCT cana-697 80 12 𝑓(π‘₯0 𝑓(π‘₯0 NOUN cana-697 80 13 ) ) PUNCT cana-697 80 14 , , PUNCT cana-697 80 15 𝑓2(π‘₯0 𝑓2(π‘₯0 NUM cana-697 80 16 ) ) PUNCT cana-697 80 17 ) ) PUNCT cana-697 81 1 ∈ ∈ PROPN cana-697 81 2 β„œ β„œ PROPN cana-697 81 3 by by ADP cana-697 81 4 induction induction NOUN cana-697 81 5 , , PUNCT cana-697 81 6 we we PRON cana-697 81 7 have have VERB cana-697 81 8 ( ( PUNCT cana-697 81 9 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 NOUN cana-697 81 10 , , PUNCT cana-697 81 11 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 81 12 ) ) PUNCT cana-697 81 13 = = SYM cana-697 81 14 ( ( PUNCT cana-697 81 15 π‘“π‘›βˆ’1(π‘₯0 π‘“π‘›βˆ’1(π‘₯0 PROPN cana-697 81 16 ) ) PUNCT cana-697 81 17 , , PUNCT cana-697 81 18 𝑓𝑛(π‘₯0 𝑓𝑛(π‘₯0 NOUN cana-697 81 19 ) ) PUNCT cana-697 81 20 ) ) PUNCT cana-697 82 1 ∈ ∈ PROPN cana-697 82 2 β„œ β„œ PROPN cana-697 82 3 for for ADP cana-697 82 4 all all PRON cana-697 82 5 𝑛 𝑛 DET cana-697 82 6 ∈ ∈ PROPN cana-697 82 7 β„•. β„•. PROPN cana-697 82 8 since since SCONJ cana-697 82 9 ( ( PUNCT cana-697 82 10 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 NOUN cana-697 82 11 , , PUNCT cana-697 82 12 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 82 13 ) ) PUNCT cana-697 82 14 ∈ ∈ PROPN cana-697 82 15 β„œ β„œ PROPN cana-697 82 16 then then ADV cana-697 82 17 by by ADP cana-697 82 18 ( ( PUNCT cana-697 82 19 3.1 3.1 NUM cana-697 82 20 ) ) PUNCT cana-697 82 21 we we PRON cana-697 82 22 obtain obtain VERB cana-697 82 23 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 82 24 , , PUNCT cana-697 82 25 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 82 26 ) ) PUNCT cana-697 82 27 = = SYM cana-697 82 28 𝑝(𝑓(π‘₯π‘›βˆ’1 𝑝(𝑓(π‘₯π‘›βˆ’1 PROPN cana-697 82 29 ) ) PUNCT cana-697 82 30 , , PUNCT cana-697 82 31 𝑓(π‘₯𝑛 𝑓(π‘₯𝑛 NOUN cana-697 82 32 ) ) PUNCT cana-697 82 33 ) ) PUNCT cana-697 83 1 ⬚ ⬚ PROPN cana-697 83 2 ≀ ≀ NUM cana-697 83 3 ( ( PUNCT cana-697 83 4 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 83 5 , , PUNCT cana-697 83 6 𝑓(π‘₯𝑛 𝑓(π‘₯𝑛 NOUN cana-697 83 7 ) ) PUNCT cana-697 83 8 ) ) PUNCT cana-697 84 1 + + CCONJ cana-697 84 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 84 3 , , PUNCT cana-697 84 4 𝑓(π‘₯π‘›βˆ’1 𝑓(π‘₯π‘›βˆ’1 NOUN cana-697 84 5 ) ) PUNCT cana-697 84 6 ) ) PUNCT cana-697 84 7 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 PROPN cana-697 84 8 , , PUNCT cana-697 84 9 𝑓(π‘₯π‘›βˆ’1 𝑓(π‘₯π‘›βˆ’1 NOUN cana-697 84 10 ) ) PUNCT cana-697 84 11 ) ) PUNCT cana-697 85 1 + + CCONJ cana-697 85 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 85 3 , , PUNCT cana-697 85 4 𝑓(π‘₯𝑛 𝑓(π‘₯𝑛 NOUN cana-697 85 5 ) ) PUNCT cana-697 85 6 ) ) PUNCT cana-697 86 1 + + CCONJ cana-697 86 2 1 1 NUM cana-697 86 3 + + NUM cana-697 86 4 π‘˜ π‘˜ NOUN cana-697 86 5 ) ) PUNCT cana-697 86 6 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 86 7 , , PUNCT cana-697 86 8 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 86 9 ) ) PUNCT cana-697 86 10 ⬚ ⬚ PROPN cana-697 86 11 = = SYM cana-697 86 12 ( ( PUNCT cana-697 86 13 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 86 14 , , PUNCT cana-697 86 15 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 86 16 ) ) PUNCT cana-697 86 17 + + CCONJ cana-697 86 18 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 86 19 , , PUNCT cana-697 86 20 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 86 21 ) ) PUNCT cana-697 86 22 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 86 23 , , PUNCT cana-697 86 24 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 86 25 ) ) PUNCT cana-697 87 1 + + CCONJ cana-697 87 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 87 3 , , PUNCT cana-697 87 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 87 5 ) ) PUNCT cana-697 87 6 + + CCONJ cana-697 87 7 1 1 NUM cana-697 87 8 + + NUM cana-697 87 9 π‘˜ π‘˜ NOUN cana-697 87 10 ) ) PUNCT cana-697 87 11 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 87 12 , , PUNCT cana-697 87 13 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 87 14 ) ) PUNCT cana-697 87 15 ⬚ ⬚ PROPN cana-697 87 16 ≀ ≀ NUM cana-697 87 17 ( ( PUNCT cana-697 87 18 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 87 19 , , PUNCT cana-697 87 20 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 87 21 ) ) PUNCT cana-697 88 1 + + CCONJ cana-697 88 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 88 3 , , PUNCT cana-697 88 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 88 5 ) ) PUNCT cana-697 88 6 βˆ’ βˆ’ PROPN cana-697 89 1 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 89 2 , , PUNCT cana-697 89 3 π‘₯𝑛 π‘₯𝑛 PRON cana-697 89 4 ) ) PUNCT cana-697 90 1 + + CCONJ cana-697 90 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 90 3 , , PUNCT cana-697 90 4 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 90 5 ) ) PUNCT cana-697 90 6 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 90 7 , , PUNCT cana-697 90 8 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 90 9 ) ) PUNCT cana-697 91 1 + + CCONJ cana-697 91 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 91 3 , , PUNCT cana-697 91 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 91 5 ) ) PUNCT cana-697 91 6 + + CCONJ cana-697 92 1 1 1 NUM cana-697 92 2 + + NUM cana-697 92 3 π‘˜ π‘˜ NOUN cana-697 92 4 ) ) PUNCT cana-697 92 5 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 92 6 , , PUNCT cana-697 92 7 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 92 8 ) ) PUNCT cana-697 92 9 ⬚ ⬚ PROPN cana-697 92 10 = = SYM cana-697 92 11 ( ( PUNCT cana-697 92 12 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 92 13 , , PUNCT cana-697 92 14 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 92 15 ) ) PUNCT cana-697 93 1 + + CCONJ cana-697 93 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 93 3 , , PUNCT cana-697 93 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 93 5 ) ) PUNCT cana-697 93 6 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 93 7 , , PUNCT cana-697 93 8 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 93 9 ) ) PUNCT cana-697 94 1 + + CCONJ cana-697 94 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 94 3 , , PUNCT cana-697 94 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 94 5 ) ) PUNCT cana-697 94 6 + + CCONJ cana-697 95 1 1 1 NUM cana-697 95 2 + + NUM cana-697 95 3 π‘˜ π‘˜ NOUN cana-697 95 4 ) ) PUNCT cana-697 95 5 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 95 6 , , PUNCT cana-697 95 7 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 95 8 ) ) PUNCT cana-697 95 9 since since SCONJ cana-697 95 10 𝑓 𝑓 PRON cana-697 95 11 in in ADP cana-697 95 12 nonexpansive nonexpansive ADJ cana-697 95 13 mapping mapping NOUN cana-697 95 14 , , PUNCT cana-697 95 15 then then ADV cana-697 95 16 by by ADP cana-697 95 17 using use VERB cana-697 95 18 corollary corollary ADJ cana-697 95 19 2.8 2.8 NUM cana-697 95 20 we we PRON cana-697 95 21 obtain obtain VERB cana-697 95 22 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 95 23 , , PUNCT cana-697 95 24 π‘₯𝑛+1 π‘₯𝑛+1 NOUN cana-697 95 25 ) ) PUNCT cana-697 95 26 ≀ ≀ NOUN cana-697 95 27 ( ( PUNCT cana-697 95 28 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 95 29 , , PUNCT cana-697 95 30 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 95 31 ) ) PUNCT cana-697 96 1 + + CCONJ cana-697 96 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 96 3 , , PUNCT cana-697 96 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 96 5 ) ) PUNCT cana-697 96 6 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 96 7 , , PUNCT cana-697 96 8 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 96 9 ) ) PUNCT cana-697 97 1 + + CCONJ cana-697 97 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 97 3 , , PUNCT cana-697 97 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 97 5 ) ) PUNCT cana-697 97 6 + + CCONJ cana-697 97 7 1 1 NUM cana-697 97 8 + + NUM cana-697 97 9 π‘˜ π‘˜ NOUN cana-697 97 10 ) ) PUNCT cana-697 97 11 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 97 12 , , PUNCT cana-697 97 13 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 97 14 ) ) PUNCT cana-697 97 15 ⬚ ⬚ PROPN cana-697 97 16 ≀ ≀ NUM cana-697 97 17 ( ( PUNCT cana-697 97 18 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 97 19 , , PUNCT cana-697 97 20 π‘₯1 π‘₯1 PROPN cana-697 97 21 ) ) PUNCT cana-697 97 22 + + CCONJ cana-697 97 23 𝑝(π‘₯1 𝑝(π‘₯1 ADJ cana-697 97 24 , , PUNCT cana-697 97 25 π‘₯2 π‘₯2 NOUN cana-697 97 26 ) ) PUNCT cana-697 97 27 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 97 28 , , PUNCT cana-697 97 29 π‘₯1 π‘₯1 PROPN cana-697 97 30 ) ) PUNCT cana-697 97 31 + + CCONJ cana-697 97 32 𝑝(π‘₯1 𝑝(π‘₯1 ADJ cana-697 97 33 , , PUNCT cana-697 97 34 π‘₯2 π‘₯2 NOUN cana-697 97 35 ) ) PUNCT cana-697 97 36 + + CCONJ cana-697 97 37 1 1 NUM cana-697 97 38 + + NUM cana-697 97 39 π‘˜ π‘˜ NOUN cana-697 97 40 ) ) PUNCT cana-697 97 41 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 97 42 , , PUNCT cana-697 97 43 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 97 44 ) ) PUNCT cana-697 97 45 by by ADP cana-697 97 46 ( ( PUNCT cana-697 97 47 3.2 3.2 NUM cana-697 97 48 ) ) PUNCT cana-697 97 49 then then ADV cana-697 97 50 we we PRON cana-697 97 51 have have VERB cana-697 97 52 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 97 53 , , PUNCT cana-697 97 54 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 97 55 ) ) PUNCT cana-697 97 56 ≀ ≀ NOUN cana-697 97 57 ( ( PUNCT cana-697 97 58 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 97 59 , , PUNCT cana-697 97 60 π‘₯1 π‘₯1 PROPN cana-697 97 61 ) ) PUNCT cana-697 97 62 + + CCONJ cana-697 97 63 𝑝(π‘₯1 𝑝(π‘₯1 ADJ cana-697 97 64 , , PUNCT cana-697 97 65 π‘₯2 π‘₯2 NOUN cana-697 97 66 ) ) PUNCT cana-697 97 67 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 97 68 , , PUNCT cana-697 97 69 π‘₯1 π‘₯1 PROPN cana-697 97 70 ) ) PUNCT cana-697 97 71 + + CCONJ cana-697 97 72 𝑝(π‘₯1 𝑝(π‘₯1 ADJ cana-697 97 73 , , PUNCT cana-697 97 74 π‘₯2 π‘₯2 NOUN cana-697 97 75 ) ) PUNCT cana-697 97 76 + + CCONJ cana-697 97 77 1 1 NUM cana-697 97 78 + + NUM cana-697 97 79 π‘˜ π‘˜ NOUN cana-697 97 80 ) ) PUNCT cana-697 97 81 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 97 82 , , PUNCT cana-697 97 83 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 97 84 ) ) PUNCT cana-697 97 85 ⬚ ⬚ PROPN cana-697 97 86 ≀ ≀ NUM cana-697 97 87 ( ( PUNCT cana-697 97 88 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 97 89 , , PUNCT cana-697 97 90 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 97 91 ) ) PUNCT cana-697 97 92 ) ) PUNCT cana-697 98 1 + + CCONJ cana-697 98 2 𝑝(𝑓(π‘₯0 𝑝(𝑓(π‘₯0 ADJ cana-697 98 3 ) ) PUNCT cana-697 98 4 , , PUNCT cana-697 98 5 𝑓2(π‘₯0 𝑓2(π‘₯0 NUM cana-697 98 6 ) ) PUNCT cana-697 98 7 ) ) PUNCT cana-697 98 8 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 98 9 , , PUNCT cana-697 98 10 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 98 11 ) ) PUNCT cana-697 98 12 ) ) PUNCT cana-697 99 1 + + CCONJ cana-697 99 2 𝑝(𝑓(π‘₯0 𝑝(𝑓(π‘₯0 ADJ cana-697 99 3 ) ) PUNCT cana-697 99 4 , , PUNCT cana-697 99 5 𝑓2(π‘₯0 𝑓2(π‘₯0 NUM cana-697 99 6 ) ) PUNCT cana-697 99 7 ) ) PUNCT cana-697 100 1 + + CCONJ cana-697 100 2 1 1 NUM cana-697 100 3 + + NUM cana-697 100 4 π‘˜ π‘˜ NOUN cana-697 100 5 ) ) PUNCT cana-697 100 6 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 100 7 , , PUNCT cana-697 100 8 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 100 9 ) ) PUNCT cana-697 100 10 ⬚ ⬚ PROPN cana-697 100 11 = = SYM cana-697 100 12 𝛼𝑝(π‘₯π‘›βˆ’1 𝛼𝑝(π‘₯π‘›βˆ’1 PROPN cana-697 100 13 , , PUNCT cana-697 100 14 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 100 15 ) ) PUNCT cana-697 100 16 communications communication NOUN cana-697 100 17 on on ADP cana-697 100 18 applied apply VERB cana-697 100 19 nonlinear nonlinear ADJ cana-697 100 20 analysis analysis NOUN cana-697 100 21 issn issn NOUN cana-697 100 22 : : PUNCT cana-697 100 23 1074 1074 NUM cana-697 100 24 - - PUNCT cana-697 100 25 133x 133x NUM cana-697 100 26 vol vol NOUN cana-697 100 27 31 31 NUM cana-697 100 28 no no NOUN cana-697 100 29 . . PUNCT cana-697 101 1 2s 2s NUM cana-697 101 2 ( ( PUNCT cana-697 101 3 2024 2024 NUM cana-697 101 4 ) ) PUNCT cana-697 101 5 680 680 NUM cana-697 101 6 https://internationalpubls.com https://internationalpubls.com X cana-697 101 7 where where SCONJ cana-697 101 8 𝛼 𝛼 X cana-697 101 9 < < X cana-697 101 10 1 1 NUM cana-697 101 11 . . PUNCT cana-697 102 1 hence hence ADV cana-697 102 2 ( ( PUNCT cana-697 102 3 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 102 4 ) ) PUNCT cana-697 102 5 is be AUX cana-697 102 6 cauchy cauchy ADJ cana-697 102 7 sequences sequence NOUN cana-697 102 8 in in ADP cana-697 102 9 𝑋 𝑋 NOUN cana-697 102 10 since since SCONJ cana-697 102 11 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 PROPN cana-697 102 12 , , PUNCT cana-697 102 13 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 102 14 ) ) PUNCT cana-697 102 15 ≀ ≀ NOUN cana-697 102 16 𝛼𝑝(π‘₯π‘›βˆ’1 𝛼𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 102 17 , , PUNCT cana-697 102 18 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 102 19 ) ) PUNCT cana-697 102 20 , , PUNCT cana-697 102 21 𝛼 𝛼 X cana-697 102 22 < < X cana-697 102 23 1 1 NUM cana-697 102 24 . . PUNCT cana-697 103 1 therefor therefor SCONJ cana-697 103 2 we we PRON cana-697 103 3 have have VERB cana-697 103 4 lim lim NOUN cana-697 103 5 𝑛,π‘šβ†’βˆž 𝑛,π‘šβ†’βˆž NOUN cana-697 103 6 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 103 7 , , PUNCT cana-697 103 8 π‘₯π‘š π‘₯π‘š ADJ cana-697 103 9 ) ) PUNCT cana-697 103 10 = = SYM cana-697 103 11 0 0 PUNCT cana-697 103 12 since since SCONJ cana-697 103 13 𝑋 𝑋 PROPN cana-697 103 14 is be AUX cana-697 103 15 complete complete ADJ cana-697 103 16 partial partial ADJ cana-697 103 17 metric metric ADJ cana-697 103 18 spaces space NOUN cana-697 103 19 , , PUNCT cana-697 103 20 then then ADV cana-697 103 21 sequences sequence NOUN cana-697 103 22 ( ( PUNCT cana-697 103 23 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 103 24 ) ) PUNCT cana-697 103 25 is be AUX cana-697 103 26 converges converge NOUN cana-697 103 27 , , PUNCT cana-697 103 28 namely namely ADV cana-697 103 29 π‘₯𝑛 π‘₯𝑛 VERB cana-697 103 30 β†’ β†’ SYM cana-697 103 31 𝑧 𝑧 DET cana-697 103 32 ∈ ∈ PROPN cana-697 103 33 𝑋. 𝑋. PROPN cana-697 103 34 it it PRON cana-697 103 35 implies imply VERB cana-697 103 36 lim lim PROPN cana-697 103 37 π‘›β†’βˆž π‘›β†’βˆž NUM cana-697 103 38 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 PROPN cana-697 103 39 , , PUNCT cana-697 103 40 𝑧 𝑧 NOUN cana-697 103 41 ) ) PUNCT cana-697 103 42 = = SYM cana-697 103 43 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 103 44 , , PUNCT cana-697 103 45 𝑧 𝑧 NOUN cana-697 103 46 ) ) PUNCT cana-697 104 1 = = SYM cana-697 104 2 lim lim PROPN cana-697 104 3 𝑛,π‘šβ†’βˆž 𝑛,π‘šβ†’βˆž PROPN cana-697 104 4 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 104 5 , , PUNCT cana-697 104 6 π‘₯π‘š π‘₯π‘š ADJ cana-697 104 7 ) ) PUNCT cana-697 104 8 = = SYM cana-697 105 1 0 0 X cana-697 105 2 . . PUNCT cana-697 106 1 therefor therefor PROPN cana-697 106 2 , , PUNCT cana-697 106 3 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 106 4 , , PUNCT cana-697 106 5 𝑧 𝑧 NOUN cana-697 106 6 ) ) PUNCT cana-697 106 7 = = SYM cana-697 106 8 0 0 X cana-697 106 9 . . PUNCT cana-697 107 1 furthermore furthermore ADV cana-697 107 2 , , PUNCT cana-697 107 3 we we PRON cana-697 107 4 will will AUX cana-697 107 5 prove prove VERB cana-697 107 6 that that SCONJ cana-697 107 7 𝑧 𝑧 PROPN cana-697 107 8 is be AUX cana-697 107 9 fixed fix VERB cana-697 107 10 point point NOUN cana-697 107 11 of of ADP cana-697 107 12 𝑓. 𝑓. NOUN cana-697 107 13 since since SCONJ cana-697 107 14 𝑓 𝑓 PRON cana-697 107 15 is be AUX cana-697 107 16 nonexpansive nonexpansive ADJ cana-697 107 17 mapping mapping NOUN cana-697 107 18 then then ADV cana-697 107 19 𝑝(𝑓(𝑧 𝑝(𝑓(𝑧 PROPN cana-697 107 20 ) ) PUNCT cana-697 107 21 , , PUNCT cana-697 107 22 𝑓(𝑧 𝑓(𝑧 NUM cana-697 107 23 ) ) PUNCT cana-697 107 24 ) ) PUNCT cana-697 107 25 ≀ ≀ PROPN cana-697 107 26 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 107 27 , , PUNCT cana-697 107 28 𝑧 𝑧 NOUN cana-697 107 29 ) ) PUNCT cana-697 107 30 = = SYM cana-697 107 31 0 0 PUNCT cana-697 107 32 hence hence ADV cana-697 107 33 𝑝(𝑓(𝑧 𝑝(𝑓(𝑧 PROPN cana-697 107 34 ) ) PUNCT cana-697 107 35 , , PUNCT cana-697 107 36 𝑓(𝑧 𝑓(𝑧 NUM cana-697 107 37 ) ) PUNCT cana-697 107 38 ) ) PUNCT cana-697 107 39 = = PUNCT cana-697 108 1 0 0 X cana-697 108 2 . . PUNCT cana-697 108 3 by by ADP cana-697 108 4 hypothesis hypothesis NOUN cana-697 108 5 ( ( PUNCT cana-697 108 6 2 2 NUM cana-697 108 7 ) ) PUNCT cana-697 108 8 , , PUNCT cana-697 108 9 the the DET cana-697 108 10 continuity continuity NOUN cana-697 108 11 of of ADP cana-697 108 12 𝑓 𝑓 PRON cana-697 108 13 then then ADV cana-697 108 14 by by ADP cana-697 108 15 ( ( PUNCT cana-697 108 16 3.1 3.1 NUM cana-697 108 17 ) ) PUNCT cana-697 108 18 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 108 19 , , PUNCT cana-697 108 20 𝑓(𝑧 𝑓(𝑧 NUM cana-697 108 21 ) ) PUNCT cana-697 108 22 ) ) PUNCT cana-697 109 1 = = SYM cana-697 109 2 lim lim NOUN cana-697 109 3 π‘›β†’βˆž π‘›β†’βˆž NUM cana-697 109 4 𝑝(π‘₯𝑛+1 𝑝(π‘₯𝑛+1 PROPN cana-697 109 5 , , PUNCT cana-697 109 6 𝑓(𝑧 𝑓(𝑧 NUM cana-697 109 7 ) ) PUNCT cana-697 109 8 ) ) PUNCT cana-697 110 1 ⬚ ⬚ PROPN cana-697 110 2 = = SYM cana-697 110 3 lim lim PROPN cana-697 110 4 π‘›β†’βˆž π‘›β†’βˆž NUM cana-697 110 5 𝑝(𝑓(π‘₯𝑛 𝑝(𝑓(π‘₯𝑛 NOUN cana-697 110 6 ) ) PUNCT cana-697 110 7 , , PUNCT cana-697 110 8 𝑓(𝑧 𝑓(𝑧 NUM cana-697 110 9 ) ) PUNCT cana-697 110 10 ) ) PUNCT cana-697 110 11 ⬚ ⬚ PROPN cana-697 110 12 = = SYM cana-697 110 13 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 110 14 , , PUNCT cana-697 110 15 𝑧 𝑧 PART cana-697 110 16 ) ) PUNCT cana-697 110 17 since since SCONJ cana-697 110 18 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 110 19 , , PUNCT cana-697 110 20 𝑧 𝑧 NOUN cana-697 110 21 ) ) PUNCT cana-697 110 22 = = SYM cana-697 110 23 0 0 NUM cana-697 110 24 , , PUNCT cana-697 110 25 it it PRON cana-697 110 26 implies imply VERB cana-697 110 27 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 110 28 , , PUNCT cana-697 110 29 𝑓(𝑧 𝑓(𝑧 NUM cana-697 110 30 ) ) PUNCT cana-697 110 31 ) ) PUNCT cana-697 111 1 = = SYM cana-697 111 2 0 0 NUM cana-697 111 3 , , PUNCT cana-697 111 4 i.e. i.e. X cana-697 111 5 𝑧 𝑧 DET cana-697 111 6 ∈ ∈ NOUN cana-697 111 7 𝑓(𝑧 𝑓(𝑧 PROPN cana-697 111 8 ) ) PUNCT cana-697 111 9 . . PUNCT cana-697 112 1 it it PRON cana-697 112 2 means mean VERB cana-697 112 3 𝑧 𝑧 PROPN cana-697 112 4 is be AUX cana-697 112 5 fixed fix VERB cana-697 112 6 point point NOUN cana-697 112 7 of of ADP cana-697 112 8 𝑓. 𝑓. NOUN cana-697 112 9 suppose suppose VERB cana-697 112 10 that that SCONJ cana-697 112 11 𝑀 𝑀 VERB cana-697 112 12 is be AUX cana-697 112 13 another another DET cana-697 112 14 fixed fix VERB cana-697 112 15 point point NOUN cana-697 112 16 of of ADP cana-697 112 17 𝑓 𝑓 PRON cana-697 112 18 where where SCONJ cana-697 112 19 𝑧 𝑧 DET cana-697 112 20 β‰  β‰  PROPN cana-697 112 21 𝑀. 𝑀. NOUN cana-697 112 22 by by ADP cana-697 112 23 hypothesis hypothesis NOUN cana-697 112 24 ( ( PUNCT cana-697 112 25 3 3 X cana-697 112 26 ) ) PUNCT cana-697 112 27 we we PRON cana-697 112 28 obtain obtain VERB cana-697 112 29 ( ( PUNCT cana-697 112 30 𝑧 𝑧 NOUN cana-697 112 31 , , PUNCT cana-697 112 32 𝑀 𝑀 ADJ cana-697 112 33 ) ) PUNCT cana-697 112 34 ∈ ∈ PROPN cana-697 112 35 β„œ β„œ PROPN cana-697 112 36 , , PUNCT cana-697 112 37 then then ADV cana-697 112 38 ( ( PUNCT cana-697 112 39 3.1 3.1 NUM cana-697 112 40 ) ) PUNCT cana-697 112 41 we we PRON cana-697 112 42 have have VERB cana-697 112 43 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 112 44 , , PUNCT cana-697 112 45 𝑀 𝑀 ADP cana-697 112 46 ) ) PUNCT cana-697 112 47 = = SYM cana-697 112 48 𝑝(𝑓(𝑧 𝑝(𝑓(𝑧 PROPN cana-697 112 49 ) ) PUNCT cana-697 112 50 , , PUNCT cana-697 112 51 𝑓(𝑀 𝑓(𝑀 PROPN cana-697 112 52 ) ) PUNCT cana-697 112 53 ) ) PUNCT cana-697 113 1 ⬚ ⬚ PROPN cana-697 113 2 ≀ ≀ NUM cana-697 113 3 ( ( PUNCT cana-697 113 4 𝑝(𝑧 𝑝(𝑧 NOUN cana-697 113 5 , , PUNCT cana-697 113 6 𝑓(𝑀 𝑓(𝑀 PROPN cana-697 113 7 ) ) PUNCT cana-697 113 8 ) ) PUNCT cana-697 114 1 + + CCONJ cana-697 114 2 𝑝(𝑀 𝑝(𝑀 PROPN cana-697 114 3 , , PUNCT cana-697 114 4 𝑓(𝑧 𝑓(𝑧 NUM cana-697 114 5 ) ) PUNCT cana-697 114 6 ) ) PUNCT cana-697 114 7 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 114 8 , , PUNCT cana-697 114 9 𝑓(𝑧 𝑓(𝑧 NUM cana-697 114 10 ) ) PUNCT cana-697 114 11 ) ) PUNCT cana-697 115 1 + + CCONJ cana-697 115 2 𝑝(𝑀 𝑝(𝑀 PROPN cana-697 115 3 , , PUNCT cana-697 115 4 𝑓(𝑀 𝑓(𝑀 PROPN cana-697 115 5 ) ) PUNCT cana-697 115 6 ) ) PUNCT cana-697 116 1 + + CCONJ cana-697 116 2 1 1 NUM cana-697 116 3 + + NUM cana-697 116 4 π‘˜ π‘˜ X cana-697 116 5 ) ) PUNCT cana-697 116 6 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 116 7 , , PUNCT cana-697 116 8 𝑀 𝑀 X cana-697 116 9 ) ) PUNCT cana-697 116 10 ⬚ ⬚ PROPN cana-697 116 11 = = SYM cana-697 116 12 ( ( PUNCT cana-697 116 13 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 116 14 , , PUNCT cana-697 116 15 𝑀 𝑀 ADP cana-697 116 16 ) ) PUNCT cana-697 116 17 + + CCONJ cana-697 116 18 𝑝(𝑀 𝑝(𝑀 PROPN cana-697 116 19 , , PUNCT cana-697 116 20 𝑧 𝑧 NOUN cana-697 116 21 ) ) PUNCT cana-697 116 22 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 116 23 , , PUNCT cana-697 116 24 𝑧 𝑧 NOUN cana-697 116 25 ) ) PUNCT cana-697 116 26 + + CCONJ cana-697 116 27 𝑝(𝑀 𝑝(𝑀 PROPN cana-697 116 28 , , PUNCT cana-697 116 29 𝑀 𝑀 ADP cana-697 116 30 ) ) PUNCT cana-697 116 31 + + CCONJ cana-697 116 32 1 1 NUM cana-697 116 33 + + NUM cana-697 116 34 π‘˜ π‘˜ X cana-697 116 35 ) ) PUNCT cana-697 116 36 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 116 37 , , PUNCT cana-697 116 38 𝑀 𝑀 X cana-697 116 39 ) ) PUNCT cana-697 116 40 ⬚ ⬚ PROPN cana-697 116 41 = = SYM cana-697 116 42 ( ( PUNCT cana-697 116 43 2𝑝(𝑧 2𝑝(𝑧 NUM cana-697 116 44 , , PUNCT cana-697 116 45 𝑀 𝑀 PROPN cana-697 116 46 ) ) PUNCT cana-697 116 47 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 116 48 , , PUNCT cana-697 116 49 𝑧 𝑧 NOUN cana-697 116 50 ) ) PUNCT cana-697 116 51 + + CCONJ cana-697 116 52 𝑝(𝑀 𝑝(𝑀 PROPN cana-697 116 53 , , PUNCT cana-697 116 54 𝑀 𝑀 ADP cana-697 116 55 ) ) PUNCT cana-697 116 56 + + CCONJ cana-697 116 57 1 1 NUM cana-697 116 58 + + NUM cana-697 116 59 π‘˜ π‘˜ X cana-697 116 60 ) ) PUNCT cana-697 116 61 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 116 62 , , PUNCT cana-697 116 63 𝑀 𝑀 X cana-697 116 64 ) ) PUNCT cana-697 116 65 furthermore furthermore ADV cana-697 116 66 , , PUNCT cana-697 116 67 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 116 68 , , PUNCT cana-697 116 69 𝑀 𝑀 X cana-697 116 70 ) ) PUNCT cana-697 116 71 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 116 72 , , PUNCT cana-697 116 73 𝑧 𝑧 NOUN cana-697 116 74 ) ) PUNCT cana-697 116 75 + + CCONJ cana-697 116 76 𝑝(𝑀 𝑝(𝑀 PROPN cana-697 116 77 , , PUNCT cana-697 116 78 𝑀 𝑀 ADP cana-697 116 79 ) ) PUNCT cana-697 116 80 + + CCONJ cana-697 116 81 1 1 NUM cana-697 116 82 β‰₯ β‰₯ NOUN cana-697 116 83 1 1 NUM cana-697 116 84 βˆ’ βˆ’ NOUN cana-697 116 85 π‘˜ π‘˜ PRON cana-697 116 86 2 2 NUM cana-697 116 87 this this PRON cana-697 116 88 completes complete VERB cana-697 116 89 the the DET cana-697 116 90 proof proof NOUN cana-697 116 91 . . PUNCT cana-697 117 1 we we PRON cana-697 117 2 can can AUX cana-697 117 3 replace replace VERB cana-697 117 4 the the DET cana-697 117 5 continuity continuity NOUN cana-697 117 6 hypothesis hypothesis NOUN cana-697 117 7 in in ADP cana-697 117 8 theorem theorem NOUN cana-697 117 9 3.1 3.1 NUM cana-697 117 10 with with ADP cana-697 117 11 the the DET cana-697 117 12 convergence convergence NOUN cana-697 117 13 hypothesis hypothesis NOUN cana-697 117 14 as as ADP cana-697 117 15 the the DET cana-697 117 16 following follow VERB cana-697 117 17 theorem theorem NOUN cana-697 117 18 . . PUNCT cana-697 117 19 theorem theorem PROPN cana-697 117 20 3.2 3.2 NUM cana-697 117 21 . . PUNCT cana-697 118 1 let let AUX cana-697 118 2 ( ( PUNCT cana-697 118 3 𝑋 𝑋 PROPN cana-697 118 4 , , PUNCT cana-697 118 5 𝑝 𝑝 NOUN cana-697 118 6 ) ) PUNCT cana-697 118 7 be be AUX cana-697 118 8 a a DET cana-697 118 9 complete complete ADJ cana-697 118 10 partial partial ADJ cana-697 118 11 metric metric ADJ cana-697 118 12 spaces space NOUN cana-697 118 13 endowed endow VERB cana-697 118 14 with with ADP cana-697 118 15 a a DET cana-697 118 16 binary binary ADJ cana-697 118 17 relation relation NOUN cana-697 118 18 β„œ β„œ PROPN cana-697 118 19 on on ADP cana-697 118 20 𝑋. 𝑋. PROPN cana-697 118 21 suppose suppose VERB cana-697 118 22 that that SCONJ cana-697 118 23 𝑓 𝑓 X cana-697 118 24 : : PUNCT cana-697 118 25 𝑋 𝑋 PROPN cana-697 118 26 β†’ β†’ SYM cana-697 118 27 𝑋 𝑋 PROPN cana-697 118 28 be be VERB cana-697 118 29 a a DET cana-697 118 30 nonexpansive nonexpansive ADJ cana-697 118 31 mappings mapping NOUN cana-697 118 32 such such ADJ cana-697 118 33 that that SCONJ cana-697 118 34 𝑝(𝑓(π‘₯ 𝑝(𝑓(π‘₯ NOUN cana-697 118 35 ) ) PUNCT cana-697 118 36 , , PUNCT cana-697 118 37 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 118 38 ) ) PUNCT cana-697 118 39 ) ) PUNCT cana-697 119 1 ≀ ≀ NOUN cana-697 119 2 ( ( PUNCT cana-697 119 3 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 119 4 , , PUNCT cana-697 119 5 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 119 6 ) ) PUNCT cana-697 119 7 ) ) PUNCT cana-697 120 1 + + CCONJ cana-697 120 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 120 3 , , PUNCT cana-697 120 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 120 5 ) ) PUNCT cana-697 120 6 ) ) PUNCT cana-697 120 7 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 120 8 , , PUNCT cana-697 120 9 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 120 10 ) ) PUNCT cana-697 120 11 ) ) PUNCT cana-697 121 1 + + CCONJ cana-697 121 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 121 3 , , PUNCT cana-697 121 4 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 121 5 ) ) PUNCT cana-697 121 6 ) ) PUNCT cana-697 122 1 + + CCONJ cana-697 122 2 1 1 NUM cana-697 122 3 + + NUM cana-697 122 4 π‘˜ π‘˜ X cana-697 122 5 ) ) PUNCT cana-697 122 6 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 122 7 , , PUNCT cana-697 122 8 𝑦 𝑦 NOUN cana-697 122 9 ) ) PUNCT cana-697 122 10 ( ( PUNCT cana-697 122 11 3.3 3.3 NUM cana-697 122 12 ) ) PUNCT cana-697 122 13 communications communication NOUN cana-697 122 14 on on ADP cana-697 122 15 applied apply VERB cana-697 122 16 nonlinear nonlinear ADJ cana-697 122 17 analysis analysis NOUN cana-697 122 18 issn issn NOUN cana-697 122 19 : : PUNCT cana-697 122 20 1074 1074 NUM cana-697 122 21 - - PUNCT cana-697 122 22 133x 133x NUM cana-697 122 23 vol vol NOUN cana-697 122 24 31 31 NUM cana-697 122 25 no no NOUN cana-697 122 26 . . PUNCT cana-697 123 1 2s 2s NUM cana-697 123 2 ( ( PUNCT cana-697 123 3 2024 2024 NUM cana-697 123 4 ) ) PUNCT cana-697 123 5 681 681 NUM cana-697 123 6 https://internationalpubls.com https://internationalpubls.com X cana-697 123 7 for for ADP cana-697 123 8 each each DET cana-697 123 9 π‘₯ π‘₯ PROPN cana-697 123 10 , , PUNCT cana-697 123 11 𝑦 𝑦 NOUN cana-697 123 12 ∈ ∈ PROPN cana-697 123 13 β„œ β„œ PROPN cana-697 123 14 , , PUNCT cana-697 123 15 where where SCONJ cana-697 123 16 π‘˜ π‘˜ PRON cana-697 123 17 ∈ ∈ PROPN cana-697 123 18 [ [ X cana-697 123 19 0,1 0,1 NUM cana-697 123 20 ) ) PUNCT cana-697 123 21 . . PUNCT cana-697 123 22 assume assume VERB cana-697 123 23 that that SCONJ cana-697 123 24 : : PUNCT cana-697 123 25 1 1 X cana-697 123 26 . . X cana-697 123 27 𝑓 𝑓 PRON cana-697 123 28 is be AUX cana-697 123 29 preserving preserve VERB cana-697 123 30 mapping mapping NOUN cana-697 123 31 2 2 NUM cana-697 123 32 . . PUNCT cana-697 124 1 if if SCONJ cana-697 124 2 π‘₯𝑛 π‘₯𝑛 VERB cana-697 124 3 sequences sequence NOUN cana-697 124 4 in in ADP cana-697 124 5 𝑋 𝑋 PROPN cana-697 124 6 such such ADJ cana-697 124 7 that that SCONJ cana-697 124 8 ( ( PUNCT cana-697 124 9 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 NOUN cana-697 124 10 , , PUNCT cana-697 124 11 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 124 12 ) ) PUNCT cana-697 124 13 ∈ ∈ PROPN cana-697 124 14 β„œ β„œ PROPN cana-697 124 15 for for ADP cana-697 124 16 each each DET cana-697 124 17 𝑛 𝑛 PRON cana-697 124 18 ∈ ∈ PROPN cana-697 124 19 β„• β„• PROPN cana-697 124 20 and and CCONJ cana-697 124 21 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 124 22 β†’ β†’ SYM cana-697 124 23 𝑧 𝑧 PRON cana-697 124 24 ∈ ∈ NOUN cana-697 124 25 𝑋 𝑋 NOUN cana-697 124 26 as as ADP cana-697 124 27 𝑛 𝑛 PROPN cana-697 124 28 β†’ β†’ SYM cana-697 124 29 ∞ ∞ PROPN cana-697 124 30 , , PUNCT cana-697 124 31 then then ADV cana-697 124 32 ( ( PUNCT cana-697 124 33 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 NOUN cana-697 124 34 , , PUNCT cana-697 124 35 𝑧 𝑧 NOUN cana-697 124 36 ) ) PUNCT cana-697 124 37 ∈ ∈ PROPN cana-697 124 38 β„œ β„œ PROPN cana-697 124 39 for for ADP cana-697 124 40 all all PRON cana-697 124 41 𝑛 𝑛 DET cana-697 124 42 ∈ ∈ PROPN cana-697 124 43 β„• β„• PROPN cana-697 124 44 3 3 NUM cana-697 124 45 . . PUNCT cana-697 125 1 𝐹𝑖π‘₯ 𝐹𝑖π‘₯ PROPN cana-697 125 2 ( ( PUNCT cana-697 125 3 𝑓 𝑓 X cana-697 125 4 ) ) PUNCT cana-697 125 5 is be AUX cana-697 125 6 well well ADV cana-697 125 7 ordered order VERB cana-697 125 8 with with ADP cana-697 125 9 respect respect NOUN cana-697 125 10 to to ADP cana-697 125 11 β„œ. β„œ. PROPN cana-697 125 12 if if SCONJ cana-697 125 13 there there PRON cana-697 125 14 exist exist VERB cana-697 125 15 π‘₯0 π‘₯0 NOUN cana-697 125 16 ∈ ∈ NOUN cana-697 125 17 𝑋 𝑋 NOUN cana-697 125 18 such such ADJ cana-697 125 19 that that SCONJ cana-697 125 20 ( ( PUNCT cana-697 125 21 π‘₯0 π‘₯0 NOUN cana-697 125 22 , , PUNCT cana-697 125 23 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 125 24 ) ) PUNCT cana-697 125 25 ) ) PUNCT cana-697 125 26 ∈ ∈ PROPN cana-697 125 27 β„œ β„œ PROPN cana-697 125 28 and and CCONJ cana-697 125 29 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 125 30 , , PUNCT cana-697 125 31 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 125 32 ) ) PUNCT cana-697 125 33 ) ) PUNCT cana-697 126 1 + + CCONJ cana-697 126 2 𝑝(𝑓(π‘₯0 𝑝(𝑓(π‘₯0 ADJ cana-697 126 3 ) ) PUNCT cana-697 126 4 , , PUNCT cana-697 126 5 𝑓2(π‘₯0 𝑓2(π‘₯0 NUM cana-697 126 6 ) ) PUNCT cana-697 126 7 ) ) PUNCT cana-697 126 8 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 126 9 , , PUNCT cana-697 126 10 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 126 11 ) ) PUNCT cana-697 126 12 ) ) PUNCT cana-697 127 1 + + CCONJ cana-697 127 2 𝑝(𝑓(π‘₯0 𝑝(𝑓(π‘₯0 ADJ cana-697 127 3 ) ) PUNCT cana-697 127 4 , , PUNCT cana-697 127 5 𝑓2(π‘₯0 𝑓2(π‘₯0 NUM cana-697 127 6 ) ) PUNCT cana-697 127 7 ) ) PUNCT cana-697 128 1 + + CCONJ cana-697 128 2 1 1 NUM cana-697 128 3 + + NUM cana-697 128 4 π‘˜ π‘˜ X cana-697 128 5 < < X cana-697 128 6 1 1 NUM cana-697 128 7 ( ( PUNCT cana-697 128 8 3.4 3.4 NUM cana-697 128 9 ) ) PUNCT cana-697 128 10 then then ADV cana-697 128 11 there there PRON cana-697 128 12 exist exist VERB cana-697 128 13 𝑧 𝑧 DET cana-697 128 14 ∈ ∈ NOUN cana-697 128 15 𝑋 𝑋 NOUN cana-697 128 16 such such ADJ cana-697 128 17 that that PRON cana-697 128 18 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 128 19 , , PUNCT cana-697 128 20 𝑧 𝑧 NOUN cana-697 128 21 ) ) PUNCT cana-697 128 22 = = SYM cana-697 128 23 0 0 X cana-697 128 24 . . PUNCT cana-697 129 1 furthermore furthermore ADV cana-697 129 2 a. a. NOUN cana-697 129 3 there there PRON cana-697 129 4 exists exist VERB cana-697 129 5 𝑧 𝑧 PRON cana-697 129 6 ∈ ∈ PROPN cana-697 129 7 𝑋 𝑋 NOUN cana-697 129 8 fixed fix VERB cana-697 129 9 point point NOUN cana-697 129 10 of of ADP cana-697 129 11 𝑓 𝑓 DET cana-697 129 12 b. b. NOUN cana-697 129 13 the the DET cana-697 129 14 picard picard PROPN cana-697 129 15 sequences sequence NOUN cana-697 129 16 of of ADP cana-697 129 17 initial initial ADJ cana-697 129 18 point point NOUN cana-697 129 19 π‘₯0 π‘₯0 NOUN cana-697 129 20 ∈ ∈ NOUN cana-697 129 21 𝑋 𝑋 NOUN cana-697 129 22 converges converge VERB cana-697 129 23 to to ADP cana-697 129 24 fixed fix VERB cana-697 129 25 point point NOUN cana-697 129 26 of of ADP cana-697 129 27 𝑓 𝑓 DET cana-697 129 28 c. c. NOUN cana-697 129 29 if if SCONJ cana-697 129 30 𝑧 𝑧 PROPN cana-697 129 31 and and CCONJ cana-697 129 32 𝑀 𝑀 PROPN cana-697 129 33 are be AUX cana-697 129 34 fixed fix VERB cana-697 129 35 point point NOUN cana-697 129 36 of of ADP cana-697 129 37 𝑓 𝑓 PRON cana-697 129 38 where where SCONJ cana-697 129 39 𝑧 𝑧 DET cana-697 129 40 β‰  β‰  PROPN cana-697 129 41 𝑀 𝑀 PROPN cana-697 129 42 then then ADV cana-697 129 43 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 129 44 , , PUNCT cana-697 129 45 𝑀 𝑀 X cana-697 129 46 ) ) PUNCT cana-697 129 47 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 129 48 , , PUNCT cana-697 129 49 𝑧 𝑧 NOUN cana-697 129 50 ) ) PUNCT cana-697 129 51 + + CCONJ cana-697 129 52 𝑝(𝑀 𝑝(𝑀 PROPN cana-697 129 53 , , PUNCT cana-697 129 54 𝑀 𝑀 ADP cana-697 129 55 ) ) PUNCT cana-697 129 56 + + CCONJ cana-697 129 57 1 1 NUM cana-697 129 58 β‰₯ β‰₯ NOUN cana-697 129 59 1 1 NUM cana-697 129 60 βˆ’ βˆ’ NOUN cana-697 129 61 π‘˜ π‘˜ DET cana-697 129 62 2 2 NUM cana-697 129 63 proof proof NOUN cana-697 129 64 : : PUNCT cana-697 129 65 the the DET cana-697 129 66 proof proof NOUN cana-697 129 67 is be AUX cana-697 129 68 following follow VERB cana-697 129 69 theorem theorem VERB cana-697 129 70 3.1 3.1 NUM cana-697 129 71 ’s ’s PART cana-697 129 72 proof proof NOUN cana-697 129 73 . . PUNCT cana-697 130 1 we we PRON cana-697 130 2 only only ADV cana-697 130 3 have have VERB cana-697 130 4 to to PART cana-697 130 5 check check VERB cana-697 130 6 that that SCONJ cana-697 130 7 𝑧 𝑧 PROPN cana-697 130 8 is be AUX cana-697 130 9 fixed fix VERB cana-697 130 10 point point NOUN cana-697 130 11 of of ADP cana-697 130 12 𝑓. 𝑓. NOUN cana-697 130 13 by by ADP cana-697 130 14 hypothesis hypothesis NOUN cana-697 130 15 ( ( PUNCT cana-697 130 16 2 2 X cana-697 130 17 ) ) PUNCT cana-697 130 18 we we PRON cana-697 130 19 have have VERB cana-697 130 20 ( ( PUNCT cana-697 130 21 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 130 22 , , PUNCT cana-697 130 23 𝑧 𝑧 NOUN cana-697 130 24 ) ) PUNCT cana-697 130 25 ∈ ∈ PROPN cana-697 130 26 β„œ β„œ PROPN cana-697 130 27 for for ADP cana-697 130 28 all all PRON cana-697 130 29 𝑛 𝑛 DET cana-697 130 30 ∈ ∈ PROPN cana-697 130 31 β„•. β„•. PROPN cana-697 130 32 therefor therefor ADP cana-697 130 33 by by ADP cana-697 130 34 ( ( PUNCT cana-697 130 35 3.3 3.3 NUM cana-697 130 36 ) ) PUNCT cana-697 130 37 we we PRON cana-697 130 38 obtain obtain VERB cana-697 130 39 𝑝(π‘₯𝑛+1 𝑝(π‘₯𝑛+1 ADJ cana-697 130 40 , , PUNCT cana-697 130 41 𝑓(𝑧 𝑓(𝑧 NUM cana-697 130 42 ) ) PUNCT cana-697 130 43 ) ) PUNCT cana-697 131 1 = = SYM cana-697 131 2 𝑝(𝑓(π‘₯𝑛 𝑝(𝑓(π‘₯𝑛 NOUN cana-697 131 3 ) ) PUNCT cana-697 131 4 , , PUNCT cana-697 131 5 𝑓(𝑧 𝑓(𝑧 NUM cana-697 131 6 ) ) PUNCT cana-697 131 7 ) ) PUNCT cana-697 132 1 ⬚ ⬚ PROPN cana-697 132 2 ≀ ≀ NUM cana-697 132 3 ( ( PUNCT cana-697 132 4 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 132 5 , , PUNCT cana-697 132 6 𝑓(𝑧 𝑓(𝑧 NUM cana-697 132 7 ) ) PUNCT cana-697 132 8 ) ) PUNCT cana-697 133 1 + + CCONJ cana-697 133 2 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 133 3 , , PUNCT cana-697 133 4 𝑓(π‘₯𝑛 𝑓(π‘₯𝑛 NOUN cana-697 133 5 ) ) PUNCT cana-697 133 6 ) ) PUNCT cana-697 133 7 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 133 8 , , PUNCT cana-697 133 9 𝑓(π‘₯𝑛 𝑓(π‘₯𝑛 NOUN cana-697 133 10 ) ) PUNCT cana-697 133 11 ) ) PUNCT cana-697 134 1 + + CCONJ cana-697 134 2 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 134 3 , , PUNCT cana-697 134 4 𝑓(𝑧 𝑓(𝑧 NUM cana-697 134 5 ) ) PUNCT cana-697 134 6 ) ) PUNCT cana-697 135 1 + + CCONJ cana-697 136 1 1 1 NUM cana-697 136 2 + + NUM cana-697 136 3 π‘˜ π‘˜ X cana-697 136 4 ) ) PUNCT cana-697 136 5 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 PROPN cana-697 136 6 , , PUNCT cana-697 136 7 𝑧 𝑧 NOUN cana-697 136 8 ) ) PUNCT cana-697 136 9 ⬚ ⬚ NOUN cana-697 136 10 = = SYM cana-697 136 11 ( ( PUNCT cana-697 136 12 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 136 13 , , PUNCT cana-697 136 14 𝑓(𝑧 𝑓(𝑧 NUM cana-697 136 15 ) ) PUNCT cana-697 136 16 ) ) PUNCT cana-697 137 1 + + CCONJ cana-697 137 2 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 137 3 , , PUNCT cana-697 137 4 π‘₯𝑛+1 π‘₯𝑛+1 NOUN cana-697 137 5 ) ) PUNCT cana-697 137 6 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 137 7 , , PUNCT cana-697 137 8 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 137 9 ) ) PUNCT cana-697 137 10 + + CCONJ cana-697 137 11 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 137 12 , , PUNCT cana-697 137 13 𝑓(𝑧 𝑓(𝑧 NUM cana-697 137 14 ) ) PUNCT cana-697 137 15 ) ) PUNCT cana-697 138 1 + + CCONJ cana-697 138 2 1 1 NUM cana-697 138 3 + + NUM cana-697 138 4 π‘˜ π‘˜ X cana-697 138 5 ) ) PUNCT cana-697 138 6 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 PROPN cana-697 138 7 , , PUNCT cana-697 138 8 𝑧 𝑧 NOUN cana-697 138 9 ) ) PUNCT cana-697 138 10 taking take VERB cana-697 138 11 𝑛 𝑛 PRON cana-697 138 12 β†’ β†’ SYM cana-697 138 13 ∞ ∞ NUM cana-697 138 14 then then ADV cana-697 138 15 we we PRON cana-697 138 16 obtain obtain VERB cana-697 138 17 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 138 18 , , PUNCT cana-697 138 19 𝑓(𝑧 𝑓(𝑧 NUM cana-697 138 20 ) ) PUNCT cana-697 138 21 ) ) PUNCT cana-697 138 22 ≀ ≀ NOUN cana-697 138 23 ( ( PUNCT cana-697 138 24 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 138 25 , , PUNCT cana-697 138 26 𝑓(𝑧 𝑓(𝑧 PROPN cana-697 138 27 ) ) PUNCT cana-697 138 28 ) ) PUNCT cana-697 139 1 + + CCONJ cana-697 139 2 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 139 3 , , PUNCT cana-697 139 4 𝑧 𝑧 NOUN cana-697 139 5 ) ) PUNCT cana-697 139 6 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 139 7 , , PUNCT cana-697 139 8 𝑧 𝑧 NOUN cana-697 139 9 ) ) PUNCT cana-697 139 10 + + CCONJ cana-697 139 11 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 139 12 , , PUNCT cana-697 139 13 𝑓(𝑧 𝑓(𝑧 NUM cana-697 139 14 ) ) PUNCT cana-697 139 15 ) ) PUNCT cana-697 140 1 + + CCONJ cana-697 140 2 1 1 NUM cana-697 140 3 + + NUM cana-697 140 4 π‘˜ π‘˜ X cana-697 140 5 ) ) PUNCT cana-697 140 6 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 140 7 , , PUNCT cana-697 140 8 𝑧 𝑧 X cana-697 140 9 ) ) PUNCT cana-697 140 10 since since SCONJ cana-697 140 11 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 140 12 , , PUNCT cana-697 140 13 𝑧 𝑧 NOUN cana-697 140 14 ) ) PUNCT cana-697 140 15 = = SYM cana-697 140 16 0 0 NUM cana-697 140 17 , , PUNCT cana-697 140 18 it it PRON cana-697 140 19 implies imply VERB cana-697 140 20 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 140 21 , , PUNCT cana-697 140 22 𝑓(𝑧 𝑓(𝑧 PROPN cana-697 140 23 ) ) PUNCT cana-697 140 24 ) ) PUNCT cana-697 140 25 ≀ ≀ NUM cana-697 140 26 0 0 X cana-697 140 27 . . PUNCT cana-697 141 1 hence hence ADV cana-697 141 2 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 141 3 , , PUNCT cana-697 141 4 𝑓(𝑧 𝑓(𝑧 NUM cana-697 141 5 ) ) PUNCT cana-697 141 6 ) ) PUNCT cana-697 141 7 = = SYM cana-697 141 8 0 0 NUM cana-697 141 9 , , PUNCT cana-697 141 10 i.e. i.e. X cana-697 141 11 𝑧 𝑧 DET cana-697 141 12 ∈ ∈ NOUN cana-697 141 13 𝑓(𝑧 𝑓(𝑧 PROPN cana-697 141 14 ) ) PUNCT cana-697 141 15 . . PUNCT cana-697 142 1 it it PRON cana-697 142 2 means mean VERB cana-697 142 3 𝑧 𝑧 PROPN cana-697 142 4 is be AUX cana-697 142 5 fixed fix VERB cana-697 142 6 point point NOUN cana-697 142 7 of of ADP cana-697 142 8 𝑓. 𝑓. NOUN cana-697 142 9 for for ADP cana-697 142 10 the the DET cana-697 142 11 following follow VERB cana-697 142 12 proof proof NOUN cana-697 142 13 on on ADP cana-697 142 14 the the DET cana-697 142 15 similar similar ADJ cana-697 142 16 line line NOUN cana-697 142 17 with with ADP cana-697 142 18 theorem theorem ADJ cana-697 142 19 3.1 3.1 NUM cana-697 142 20 ’s ’s PART cana-697 142 21 proof proof NOUN cana-697 142 22 . . PUNCT cana-697 143 1 this this PRON cana-697 143 2 completes complete VERB cana-697 143 3 the the DET cana-697 143 4 proof proof NOUN cana-697 143 5 . . PUNCT cana-697 144 1 let let VERB cana-697 144 2 we we PRON cana-697 144 3 consider consider VERB cana-697 144 4 , , PUNCT cana-697 144 5 if if SCONJ cana-697 144 6 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 144 7 , , PUNCT cana-697 144 8 𝑧 𝑧 NOUN cana-697 144 9 ) ) PUNCT cana-697 144 10 and and CCONJ cana-697 144 11 𝑝(𝑀 𝑝(𝑀 PROPN cana-697 144 12 , , PUNCT cana-697 144 13 𝑀 𝑀 ADP cana-697 144 14 ) ) PUNCT cana-697 144 15 in in ADP cana-697 144 16 theorem theorem ADJ cana-697 144 17 3.1 3.1 NUM cana-697 144 18 and and CCONJ cana-697 144 19 3.2 3.2 NUM cana-697 144 20 are be AUX cana-697 144 21 zero zero NUM cana-697 144 22 , , PUNCT cana-697 144 23 then then ADV cana-697 144 24 we we PRON cana-697 144 25 have have VERB cana-697 144 26 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 144 27 , , PUNCT cana-697 144 28 𝑀 𝑀 X cana-697 144 29 ) ) PUNCT cana-697 144 30 β‰₯ β‰₯ NOUN cana-697 144 31 1 1 NUM cana-697 144 32 βˆ’ βˆ’ NOUN cana-697 144 33 π‘˜ π‘˜ DET cana-697 144 34 2 2 NUM cana-697 144 35 furthermore furthermore ADV cana-697 144 36 , , PUNCT cana-697 144 37 we we PRON cana-697 144 38 provide provide VERB cana-697 144 39 the the DET cana-697 144 40 following follow VERB cana-697 144 41 example example NOUN cana-697 144 42 motivated motivate VERB cana-697 144 43 by by ADP cana-697 144 44 aydi aydi VERB cana-697 144 45 [ [ X cana-697 144 46 1 1 NUM cana-697 144 47 ] ] PUNCT cana-697 144 48 . . PUNCT cana-697 145 1 example example NOUN cana-697 145 2 3.3 3.3 NUM cana-697 145 3 . . PUNCT cana-697 146 1 let let VERB cana-697 146 2 𝑋 𝑋 NOUN cana-697 146 3 = = PUNCT cana-697 147 1 [ [ X cana-697 147 2 0,1 0,1 NUM cana-697 147 3 ] ] PUNCT cana-697 147 4 and and CCONJ cana-697 147 5 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 147 6 , , PUNCT cana-697 147 7 𝑦 𝑦 NOUN cana-697 147 8 ) ) PUNCT cana-697 147 9 = = SYM cana-697 147 10 max max X cana-697 147 11 { { PUNCT cana-697 147 12 π‘₯ π‘₯ PROPN cana-697 147 13 , , PUNCT cana-697 147 14 𝑦 𝑦 NOUN cana-697 147 15 } } PUNCT cana-697 147 16 . . PUNCT cana-697 148 1 define define VERB cana-697 148 2 𝑓 𝑓 PRON cana-697 148 3 : : PUNCT cana-697 148 4 𝑋 𝑋 PROPN cana-697 148 5 β†’ β†’ SYM cana-697 148 6 𝑋 𝑋 NOUN cana-697 148 7 by by ADP cana-697 148 8 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 148 9 ) ) PUNCT cana-697 148 10 = = SYM cana-697 148 11 π‘₯2 π‘₯2 NOUN cana-697 148 12 for for ADP cana-697 148 13 all all DET cana-697 148 14 π‘₯ π‘₯ DET cana-697 148 15 ∈ ∈ PROPN cana-697 148 16 𝑋. 𝑋. PROPN cana-697 148 17 clearly clearly ADV cana-697 148 18 , , PUNCT cana-697 148 19 ( ( PUNCT cana-697 148 20 𝑋 𝑋 PROPN cana-697 148 21 , , PUNCT cana-697 148 22 𝑝 𝑝 NOUN cana-697 148 23 ) ) PUNCT cana-697 148 24 is be AUX cana-697 148 25 complete complete ADJ cana-697 148 26 partial partial ADJ cana-697 148 27 metric metric ADJ cana-697 148 28 spaces space NOUN cana-697 148 29 and and CCONJ cana-697 148 30 𝑓 𝑓 PRON cana-697 148 31 is be AUX cana-697 148 32 continuous continuous ADJ cana-697 148 33 mappings mapping NOUN cana-697 148 34 . . PUNCT cana-697 149 1 suppose suppose VERB cana-697 149 2 that that SCONJ cana-697 149 3 partial partial ADJ cana-697 149 4 metric metric ADJ cana-697 149 5 spaces space NOUN cana-697 149 6 ( ( PUNCT cana-697 149 7 𝑋 𝑋 PROPN cana-697 149 8 , , PUNCT cana-697 149 9 𝑝 𝑝 NOUN cana-697 149 10 ) ) PUNCT cana-697 149 11 is be AUX cana-697 149 12 endowed endow VERB cana-697 149 13 with with ADP cana-697 149 14 binary binary ADJ cana-697 149 15 relations relation NOUN cana-697 149 16 β„œ β„œ PROPN cana-697 149 17 where where SCONJ cana-697 149 18 ( ( PUNCT cana-697 149 19 π‘₯ π‘₯ NOUN cana-697 149 20 , , PUNCT cana-697 149 21 𝑦 𝑦 NOUN cana-697 149 22 ) ) PUNCT cana-697 149 23 ∈ ∈ PROPN cana-697 149 24 β„œ β„œ PROPN cana-697 149 25 it it PRON cana-697 149 26 means mean VERB cana-697 149 27 π‘₯ π‘₯ DET cana-697 149 28 β‰₯ β‰₯ PROPN cana-697 149 29 𝑦. 𝑦. PROPN cana-697 149 30 communications communication NOUN cana-697 149 31 on on ADP cana-697 149 32 applied apply VERB cana-697 149 33 nonlinear nonlinear ADJ cana-697 149 34 analysis analysis NOUN cana-697 149 35 issn issn NOUN cana-697 149 36 : : PUNCT cana-697 149 37 1074 1074 NUM cana-697 149 38 - - PUNCT cana-697 149 39 133x 133x NUM cana-697 149 40 vol vol NOUN cana-697 149 41 31 31 NUM cana-697 149 42 no no NOUN cana-697 149 43 . . PUNCT cana-697 150 1 2s 2s NUM cana-697 150 2 ( ( PUNCT cana-697 150 3 2024 2024 NUM cana-697 150 4 ) ) PUNCT cana-697 150 5 682 682 NUM cana-697 150 6 https://internationalpubls.com https://internationalpubls.com X cana-697 150 7 without without ADP cana-697 150 8 loss loss NOUN cana-697 150 9 of of ADP cana-697 150 10 generality generality NOUN cana-697 150 11 , , PUNCT cana-697 150 12 take take VERB cana-697 150 13 π‘₯ π‘₯ PRON cana-697 150 14 β‰₯ β‰₯ NOUN cana-697 150 15 𝑦 𝑦 X cana-697 150 16 > > X cana-697 150 17 0 0 X cana-697 150 18 . . PUNCT cana-697 151 1 let let VERB cana-697 151 2 we we PRON cana-697 151 3 consider consider VERB cana-697 151 4 that that SCONJ cana-697 151 5 𝑓 𝑓 NOUN cana-697 151 6 is be AUX cana-697 151 7 nonexpasive nonexpasive ADJ cana-697 151 8 mapping mapping NOUN cana-697 151 9 since since SCONJ cana-697 151 10 𝑝(𝑓(π‘₯ 𝑝(𝑓(π‘₯ NOUN cana-697 151 11 ) ) PUNCT cana-697 151 12 , , PUNCT cana-697 151 13 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 151 14 ) ) PUNCT cana-697 151 15 ) ) PUNCT cana-697 152 1 = = SYM cana-697 152 2 max{π‘₯2 max{π‘₯2 NOUN cana-697 152 3 , , PUNCT cana-697 152 4 𝑦2 𝑦2 NOUN cana-697 152 5 } } PUNCT cana-697 152 6 = = SYM cana-697 152 7 π‘₯2 π‘₯2 NOUN cana-697 152 8 ≀ ≀ NUM cana-697 152 9 π‘₯ π‘₯ NOUN cana-697 152 10 = = SYM cana-697 152 11 max{π‘₯ max{π‘₯ PROPN cana-697 152 12 , , PUNCT cana-697 152 13 𝑦 𝑦 NOUN cana-697 152 14 } } PUNCT cana-697 152 15 = = SYM cana-697 152 16 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 152 17 , , PUNCT cana-697 152 18 𝑦 𝑦 NOUN cana-697 152 19 ) ) PUNCT cana-697 152 20 . . PUNCT cana-697 153 1 take take VERB cana-697 153 2 π‘˜ π‘˜ NOUN cana-697 153 3 = = NOUN cana-697 153 4 1 1 NUM cana-697 153 5 2 2 NUM cana-697 153 6 and and CCONJ cana-697 153 7 let let VERB cana-697 153 8 we we PRON cana-697 153 9 consider consider VERB cana-697 153 10 the the DET cana-697 153 11 following follow VERB cana-697 153 12 cases case NOUN cana-697 153 13 : : PUNCT cana-697 153 14 case case NOUN cana-697 153 15 1 1 NUM cana-697 153 16 : : PUNCT cana-697 153 17 for for ADP cana-697 153 18 π‘₯2 π‘₯2 PROPN cana-697 153 19 β‰₯ β‰₯ PUNCT cana-697 153 20 𝑦 𝑦 NOUN cana-697 153 21 we we PRON cana-697 153 22 have have VERB cana-697 153 23 𝑝(𝑓(π‘₯ 𝑝(𝑓(π‘₯ NOUN cana-697 153 24 ) ) PUNCT cana-697 153 25 , , PUNCT cana-697 153 26 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 153 27 ) ) PUNCT cana-697 153 28 ) ) PUNCT cana-697 154 1 = = SYM cana-697 154 2 π‘₯2 π‘₯2 NOUN cana-697 154 3 , , PUNCT cana-697 154 4 and and CCONJ cana-697 154 5 ( ( PUNCT cana-697 154 6 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 154 7 , , PUNCT cana-697 154 8 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 154 9 ) ) PUNCT cana-697 154 10 ) ) PUNCT cana-697 155 1 + + CCONJ cana-697 155 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 155 3 , , PUNCT cana-697 155 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 155 5 ) ) PUNCT cana-697 155 6 ) ) PUNCT cana-697 155 7 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 155 8 , , PUNCT cana-697 155 9 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 155 10 ) ) PUNCT cana-697 155 11 ) ) PUNCT cana-697 156 1 + + CCONJ cana-697 156 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 156 3 , , PUNCT cana-697 156 4 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 156 5 ) ) PUNCT cana-697 156 6 ) ) PUNCT cana-697 157 1 + + CCONJ cana-697 157 2 1 1 NUM cana-697 157 3 + + NUM cana-697 157 4 π‘˜ π‘˜ X cana-697 157 5 ) ) PUNCT cana-697 157 6 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 157 7 , , PUNCT cana-697 157 8 𝑦 𝑦 NOUN cana-697 157 9 ) ) PUNCT cana-697 157 10 = = SYM cana-697 157 11 ( ( PUNCT cana-697 157 12 π‘₯ π‘₯ X cana-697 157 13 + + NUM cana-697 157 14 π‘₯2 π‘₯2 NOUN cana-697 157 15 π‘₯ π‘₯ NOUN cana-697 157 16 + + NUM cana-697 157 17 𝑦 𝑦 SYM cana-697 157 18 + + CCONJ cana-697 157 19 1 1 NUM cana-697 157 20 + + CCONJ cana-697 157 21 1 1 NUM cana-697 157 22 2 2 NUM cana-697 157 23 ) ) PUNCT cana-697 157 24 π‘₯ π‘₯ PRON cana-697 157 25 ⬚ ⬚ PROPN cana-697 157 26 = = SYM cana-697 157 27 2π‘₯2 2π‘₯2 NUM cana-697 157 28 + + NUM cana-697 157 29 2π‘₯3 2π‘₯3 NUM cana-697 157 30 + + CCONJ cana-697 157 31 ( ( PUNCT cana-697 157 32 1 1 NUM cana-697 157 33 + + NUM cana-697 157 34 π‘₯ π‘₯ PROPN cana-697 157 35 + + CCONJ cana-697 157 36 𝑦)π‘₯ 𝑦)π‘₯ X cana-697 157 37 2(π‘₯ 2(π‘₯ NUM cana-697 157 38 + + CCONJ cana-697 157 39 𝑦 𝑦 PRON cana-697 157 40 + + ADJ cana-697 157 41 1 1 NUM cana-697 157 42 ) ) PUNCT cana-697 157 43 ⬚ ⬚ NOUN cana-697 157 44 = = SYM cana-697 157 45 2π‘₯2(1 2π‘₯2(1 NUM cana-697 157 46 + + CCONJ cana-697 157 47 π‘₯ π‘₯ NOUN cana-697 157 48 ) ) PUNCT cana-697 158 1 + + CCONJ cana-697 158 2 2π‘₯2𝑦 2π‘₯2𝑦 NUM cana-697 158 3 βˆ’ βˆ’ NOUN cana-697 158 4 2π‘₯2𝑦 2π‘₯2𝑦 NUM cana-697 158 5 + + CCONJ cana-697 158 6 ( ( PUNCT cana-697 158 7 1 1 NUM cana-697 158 8 + + NUM cana-697 158 9 π‘₯ π‘₯ PROPN cana-697 158 10 + + CCONJ cana-697 158 11 𝑦)π‘₯ 𝑦)π‘₯ X cana-697 158 12 2(π‘₯ 2(π‘₯ NUM cana-697 158 13 + + CCONJ cana-697 158 14 𝑦 𝑦 PRON cana-697 158 15 + + ADJ cana-697 158 16 1 1 NUM cana-697 158 17 ) ) PUNCT cana-697 158 18 ⬚ ⬚ PROPN cana-697 158 19 = = SYM cana-697 158 20 π‘₯2 π‘₯2 NOUN cana-697 158 21 + + CCONJ cana-697 158 22 ( ( PUNCT cana-697 158 23 1 1 NUM cana-697 158 24 + + NUM cana-697 158 25 π‘₯ π‘₯ X cana-697 158 26 + + NUM cana-697 158 27 𝑦 𝑦 NOUN cana-697 158 28 βˆ’ βˆ’ PROPN cana-697 158 29 2π‘₯𝑦)π‘₯ 2π‘₯𝑦)π‘₯ NUM cana-697 158 30 2(π‘₯ 2(π‘₯ NUM cana-697 158 31 + + CCONJ cana-697 158 32 𝑦 𝑦 PRON cana-697 158 33 + + NOUN cana-697 158 34 1 1 NUM cana-697 158 35 ) ) PUNCT cana-697 158 36 since since SCONJ cana-697 158 37 2π‘₯𝑦 2π‘₯𝑦 ADJ cana-697 158 38 ≀ ≀ NUM cana-697 158 39 π‘₯2 π‘₯2 NOUN cana-697 158 40 + + CCONJ cana-697 158 41 𝑦2 𝑦2 PROPN cana-697 158 42 ≀ ≀ NUM cana-697 158 43 π‘₯ π‘₯ PROPN cana-697 159 1 + + NUM cana-697 159 2 𝑦 𝑦 NOUN cana-697 159 3 then then ADV cana-697 159 4 ( ( PUNCT cana-697 159 5 1 1 NUM cana-697 159 6 + + CCONJ cana-697 159 7 π‘₯ π‘₯ X cana-697 159 8 + + NUM cana-697 159 9 𝑦 𝑦 NOUN cana-697 159 10 βˆ’ βˆ’ PROPN cana-697 159 11 2π‘₯𝑦)π‘₯ 2π‘₯𝑦)π‘₯ NUM cana-697 159 12 2(π‘₯ 2(π‘₯ NUM cana-697 159 13 + + CCONJ cana-697 159 14 𝑦 𝑦 PRON cana-697 159 15 + + NOUN cana-697 159 16 1 1 NUM cana-697 159 17 ) ) PUNCT cana-697 159 18 > > X cana-697 159 19 0 0 PUNCT cana-697 160 1 it it PRON cana-697 160 2 implies imply VERB cana-697 160 3 ( ( PUNCT cana-697 160 4 𝑝(π‘₯ 𝑝(π‘₯ X cana-697 160 5 , , PUNCT cana-697 160 6 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 160 7 ) ) PUNCT cana-697 160 8 ) ) PUNCT cana-697 161 1 + + CCONJ cana-697 161 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 161 3 , , PUNCT cana-697 161 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 161 5 ) ) PUNCT cana-697 161 6 ) ) PUNCT cana-697 161 7 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 161 8 , , PUNCT cana-697 161 9 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 161 10 ) ) PUNCT cana-697 161 11 ) ) PUNCT cana-697 162 1 + + CCONJ cana-697 162 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 162 3 , , PUNCT cana-697 162 4 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 162 5 ) ) PUNCT cana-697 162 6 ) ) PUNCT cana-697 163 1 + + CCONJ cana-697 163 2 1 1 NUM cana-697 163 3 + + NUM cana-697 163 4 π‘˜ π‘˜ X cana-697 163 5 ) ) PUNCT cana-697 163 6 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 163 7 , , PUNCT cana-697 163 8 𝑦 𝑦 NOUN cana-697 163 9 ) ) PUNCT cana-697 163 10 β‰₯ β‰₯ NOUN cana-697 163 11 π‘₯2 π‘₯2 NOUN cana-697 163 12 = = SYM cana-697 163 13 𝑝(𝑓(π‘₯ 𝑝(𝑓(π‘₯ NOUN cana-697 163 14 ) ) PUNCT cana-697 163 15 , , PUNCT cana-697 163 16 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 163 17 ) ) PUNCT cana-697 163 18 ) ) PUNCT cana-697 164 1 = = SYM cana-697 164 2 π‘₯2 π‘₯2 NOUN cana-697 164 3 case case NOUN cana-697 164 4 2 2 NUM cana-697 164 5 : : PUNCT cana-697 164 6 for for ADP cana-697 164 7 π‘₯2 π‘₯2 PROPN cana-697 164 8 < < X cana-697 164 9 𝑦 𝑦 NOUN cana-697 164 10 we we PRON cana-697 164 11 have have VERB cana-697 164 12 𝑝(𝑓(π‘₯ 𝑝(𝑓(π‘₯ NOUN cana-697 164 13 ) ) PUNCT cana-697 164 14 , , PUNCT cana-697 164 15 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 164 16 ) ) PUNCT cana-697 164 17 ) ) PUNCT cana-697 165 1 = = SYM cana-697 165 2 π‘₯2 π‘₯2 NOUN cana-697 165 3 , , PUNCT cana-697 165 4 and and CCONJ cana-697 165 5 ( ( PUNCT cana-697 165 6 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 165 7 , , PUNCT cana-697 165 8 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 165 9 ) ) PUNCT cana-697 165 10 ) ) PUNCT cana-697 166 1 + + CCONJ cana-697 166 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 166 3 , , PUNCT cana-697 166 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 166 5 ) ) PUNCT cana-697 166 6 ) ) PUNCT cana-697 166 7 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 166 8 , , PUNCT cana-697 166 9 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 166 10 ) ) PUNCT cana-697 166 11 ) ) PUNCT cana-697 167 1 + + CCONJ cana-697 167 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 167 3 , , PUNCT cana-697 167 4 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 167 5 ) ) PUNCT cana-697 167 6 ) ) PUNCT cana-697 168 1 + + CCONJ cana-697 168 2 1 1 NUM cana-697 168 3 + + NUM cana-697 168 4 π‘˜ π‘˜ X cana-697 168 5 ) ) PUNCT cana-697 168 6 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 168 7 , , PUNCT cana-697 168 8 𝑦 𝑦 NOUN cana-697 168 9 ) ) PUNCT cana-697 168 10 = = SYM cana-697 168 11 ( ( PUNCT cana-697 168 12 π‘₯ π‘₯ X cana-697 168 13 + + PUNCT cana-697 168 14 𝑦 𝑦 NOUN cana-697 168 15 π‘₯ π‘₯ NOUN cana-697 168 16 + + NUM cana-697 168 17 𝑦 𝑦 SYM cana-697 168 18 + + CCONJ cana-697 168 19 1 1 NUM cana-697 168 20 + + CCONJ cana-697 168 21 1 1 NUM cana-697 168 22 2 2 NUM cana-697 168 23 ) ) PUNCT cana-697 168 24 π‘₯ π‘₯ PRON cana-697 168 25 ⬚ ⬚ PROPN cana-697 168 26 = = SYM cana-697 168 27 2π‘₯2 2π‘₯2 NUM cana-697 168 28 + + CCONJ cana-697 168 29 2π‘₯𝑦 2π‘₯𝑦 ADJ cana-697 168 30 + + CCONJ cana-697 168 31 ( ( PUNCT cana-697 168 32 1 1 NUM cana-697 168 33 + + NUM cana-697 168 34 π‘₯ π‘₯ PROPN cana-697 168 35 + + CCONJ cana-697 168 36 𝑦)π‘₯ 𝑦)π‘₯ X cana-697 168 37 2(π‘₯ 2(π‘₯ NUM cana-697 168 38 + + CCONJ cana-697 168 39 𝑦 𝑦 PRON cana-697 168 40 + + ADJ cana-697 168 41 1 1 NUM cana-697 168 42 ) ) PUNCT cana-697 168 43 ⬚ ⬚ NOUN cana-697 168 44 = = SYM cana-697 168 45 2π‘₯2(1 2π‘₯2(1 NUM cana-697 168 46 + + SYM cana-697 168 47 𝑦 𝑦 X cana-697 168 48 ) ) PUNCT cana-697 168 49 + + NUM cana-697 168 50 2π‘₯3 2π‘₯3 NUM cana-697 168 51 βˆ’ βˆ’ PROPN cana-697 168 52 2π‘₯3 2π‘₯3 NUM cana-697 168 53 + + CCONJ cana-697 168 54 ( ( PUNCT cana-697 168 55 1 1 NUM cana-697 168 56 + + NUM cana-697 168 57 π‘₯ π‘₯ PROPN cana-697 168 58 + + CCONJ cana-697 168 59 𝑦)π‘₯ 𝑦)π‘₯ X cana-697 168 60 2(π‘₯ 2(π‘₯ NUM cana-697 168 61 + + CCONJ cana-697 168 62 𝑦 𝑦 PRON cana-697 168 63 + + ADJ cana-697 168 64 1 1 NUM cana-697 168 65 ) ) PUNCT cana-697 168 66 ⬚ ⬚ PROPN cana-697 168 67 = = SYM cana-697 168 68 π‘₯2 π‘₯2 NOUN cana-697 168 69 + + CCONJ cana-697 168 70 ( ( PUNCT cana-697 168 71 1 1 NUM cana-697 168 72 + + NUM cana-697 168 73 π‘₯ π‘₯ X cana-697 168 74 + + NUM cana-697 168 75 𝑦 𝑦 NOUN cana-697 168 76 βˆ’ βˆ’ NOUN cana-697 168 77 2π‘₯2)π‘₯ 2π‘₯2)π‘₯ NUM cana-697 168 78 2(π‘₯ 2(π‘₯ NUM cana-697 168 79 + + CCONJ cana-697 168 80 𝑦 𝑦 NOUN cana-697 168 81 + + NOUN cana-697 168 82 1 1 NUM cana-697 168 83 ) ) PUNCT cana-697 168 84 since since SCONJ cana-697 168 85 π‘₯ π‘₯ PROPN cana-697 168 86 + + SYM cana-697 168 87 𝑦 𝑦 NOUN cana-697 168 88 ≀ ≀ NUM cana-697 168 89 π‘₯2 π‘₯2 NOUN cana-697 168 90 + + CCONJ cana-697 168 91 𝑦2 𝑦2 PROPN cana-697 168 92 ≀ ≀ NUM cana-697 168 93 2π‘₯2 2π‘₯2 NUM cana-697 168 94 then then ADV cana-697 168 95 ( ( PUNCT cana-697 168 96 1 1 NUM cana-697 168 97 + + CCONJ cana-697 168 98 π‘₯ π‘₯ X cana-697 168 99 + + NUM cana-697 168 100 𝑦 𝑦 NOUN cana-697 168 101 βˆ’ βˆ’ NOUN cana-697 168 102 2π‘₯2)π‘₯ 2π‘₯2)π‘₯ NUM cana-697 168 103 2(π‘₯ 2(π‘₯ NUM cana-697 168 104 + + CCONJ cana-697 168 105 𝑦 𝑦 NOUN cana-697 168 106 + + NOUN cana-697 168 107 1 1 NUM cana-697 168 108 ) ) PUNCT cana-697 168 109 > > X cana-697 168 110 0 0 NUM cana-697 168 111 communications communication NOUN cana-697 168 112 on on ADP cana-697 168 113 applied apply VERB cana-697 168 114 nonlinear nonlinear ADJ cana-697 168 115 analysis analysis NOUN cana-697 168 116 issn issn NOUN cana-697 168 117 : : PUNCT cana-697 168 118 1074 1074 NUM cana-697 168 119 - - PUNCT cana-697 168 120 133x 133x NUM cana-697 168 121 vol vol NOUN cana-697 168 122 31 31 NUM cana-697 168 123 no no NOUN cana-697 168 124 . . PUNCT cana-697 169 1 2s 2s NUM cana-697 169 2 ( ( PUNCT cana-697 169 3 2024 2024 NUM cana-697 169 4 ) ) PUNCT cana-697 169 5 683 683 NUM cana-697 169 6 https://internationalpubls.com https://internationalpubls.com X cana-697 169 7 hence hence ADV cana-697 169 8 ( ( PUNCT cana-697 169 9 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 169 10 , , PUNCT cana-697 169 11 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 169 12 ) ) PUNCT cana-697 169 13 ) ) PUNCT cana-697 170 1 + + CCONJ cana-697 170 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 170 3 , , PUNCT cana-697 170 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 170 5 ) ) PUNCT cana-697 170 6 ) ) PUNCT cana-697 170 7 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 170 8 , , PUNCT cana-697 170 9 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 170 10 ) ) PUNCT cana-697 170 11 ) ) PUNCT cana-697 171 1 + + CCONJ cana-697 171 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 171 3 , , PUNCT cana-697 171 4 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 171 5 ) ) PUNCT cana-697 171 6 ) ) PUNCT cana-697 172 1 + + CCONJ cana-697 172 2 1 1 NUM cana-697 172 3 + + NUM cana-697 172 4 π‘˜ π‘˜ X cana-697 172 5 ) ) PUNCT cana-697 172 6 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 172 7 , , PUNCT cana-697 172 8 𝑦 𝑦 NOUN cana-697 172 9 ) ) PUNCT cana-697 172 10 β‰₯ β‰₯ NOUN cana-697 172 11 π‘₯2 π‘₯2 NOUN cana-697 172 12 = = SYM cana-697 172 13 𝑝(𝑓(π‘₯ 𝑝(𝑓(π‘₯ NOUN cana-697 172 14 ) ) PUNCT cana-697 172 15 , , PUNCT cana-697 172 16 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 172 17 ) ) PUNCT cana-697 172 18 ) ) PUNCT cana-697 173 1 this this DET cana-697 173 2 condition condition NOUN cana-697 173 3 shows show VERB cana-697 173 4 that that SCONJ cana-697 173 5 ( ( PUNCT cana-697 173 6 3.1 3.1 NUM cana-697 173 7 ) ) PUNCT cana-697 173 8 in in ADP cana-697 173 9 theorem theorem ADJ cana-697 173 10 3.1 3.1 NUM cana-697 173 11 ( ( PUNCT cana-697 173 12 resp resp NOUN cana-697 173 13 . . PUNCT cana-697 174 1 ( ( PUNCT cana-697 174 2 3.3 3.3 NUM cana-697 174 3 ) ) PUNCT cana-697 174 4 in in ADP cana-697 174 5 theorem theorem NOUN cana-697 174 6 3.2 3.2 NUM cana-697 174 7 ) ) PUNCT cana-697 174 8 is be AUX cana-697 174 9 verified verify VERB cana-697 174 10 for for ADP cana-697 174 11 all all PRON cana-697 174 12 π‘₯ π‘₯ PROPN cana-697 174 13 , , PUNCT cana-697 174 14 𝑦 𝑦 PRON cana-697 174 15 ∈ ∈ PROPN cana-697 174 16 𝑋. 𝑋. PROPN cana-697 174 17 furthermore furthermore ADV cana-697 174 18 , , PUNCT cana-697 174 19 𝑓 𝑓 PRON cana-697 174 20 is be AUX cana-697 174 21 preserving preserve VERB cana-697 174 22 mappings mapping NOUN cana-697 174 23 since since SCONJ cana-697 174 24 ( ( PUNCT cana-697 174 25 π‘₯ π‘₯ NOUN cana-697 174 26 , , PUNCT cana-697 174 27 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 174 28 ) ) PUNCT cana-697 174 29 ) ) PUNCT cana-697 175 1 ∈ ∈ PROPN cana-697 175 2 β„œ β„œ PROPN cana-697 175 3 , , PUNCT cana-697 175 4 i.e. i.e. X cana-697 175 5 π‘₯ π‘₯ DET cana-697 175 6 β‰₯ β‰₯ NOUN cana-697 175 7 π‘₯2 π‘₯2 NOUN cana-697 175 8 then then ADV cana-697 175 9 we we PRON cana-697 175 10 have have VERB cana-697 175 11 π‘₯2 π‘₯2 PROPN cana-697 175 12 β‰₯ β‰₯ NOUN cana-697 175 13 π‘₯3 π‘₯3 NOUN cana-697 175 14 or or CCONJ cana-697 175 15 ( ( PUNCT cana-697 175 16 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 175 17 ) ) PUNCT cana-697 175 18 , , PUNCT cana-697 175 19 𝑓2(π‘₯ 𝑓2(π‘₯ X cana-697 175 20 ) ) PUNCT cana-697 175 21 ) ) PUNCT cana-697 176 1 ∈ ∈ PROPN cana-697 176 2 β„œ. β„œ. ADJ cana-697 176 3 for for ADP cana-697 176 4 π‘₯0 π‘₯0 NOUN cana-697 176 5 = = SYM cana-697 176 6 1 1 NUM cana-697 176 7 2 2 NUM cana-697 176 8 we we PRON cana-697 176 9 have have VERB cana-697 176 10 ( ( PUNCT cana-697 176 11 π‘₯0 π‘₯0 NOUN cana-697 176 12 , , PUNCT cana-697 176 13 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 176 14 ) ) PUNCT cana-697 176 15 ) ) PUNCT cana-697 177 1 ∈ ∈ PROPN cana-697 177 2 β„œ β„œ PROPN cana-697 177 3 since since SCONJ cana-697 177 4 1 1 NUM cana-697 177 5 2 2 NUM cana-697 177 6 > > SYM cana-697 177 7 1 1 NUM cana-697 177 8 4 4 NUM cana-697 177 9 = = SYM cana-697 177 10 ( ( PUNCT cana-697 177 11 1 1 NUM cana-697 177 12 2 2 NUM cana-697 177 13 ) ) PUNCT cana-697 177 14 2 2 NUM cana-697 177 15 and and CCONJ cana-697 177 16 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 177 17 , , PUNCT cana-697 177 18 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 177 19 ) ) PUNCT cana-697 177 20 ) ) PUNCT cana-697 178 1 + + CCONJ cana-697 178 2 𝑝(𝑓(π‘₯0 𝑝(𝑓(π‘₯0 ADJ cana-697 178 3 ) ) PUNCT cana-697 178 4 , , PUNCT cana-697 178 5 𝑓2(π‘₯0 𝑓2(π‘₯0 NUM cana-697 178 6 ) ) PUNCT cana-697 178 7 ) ) PUNCT cana-697 178 8 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 178 9 , , PUNCT cana-697 178 10 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 178 11 ) ) PUNCT cana-697 178 12 ) ) PUNCT cana-697 179 1 + + CCONJ cana-697 179 2 𝑝(𝑓(π‘₯0 𝑝(𝑓(π‘₯0 ADJ cana-697 179 3 ) ) PUNCT cana-697 179 4 , , PUNCT cana-697 179 5 𝑓2(π‘₯0 𝑓2(π‘₯0 NUM cana-697 179 6 ) ) PUNCT cana-697 179 7 ) ) PUNCT cana-697 180 1 + + CCONJ cana-697 180 2 1 1 NUM cana-697 180 3 + + NUM cana-697 180 4 π‘˜ π‘˜ NOUN cana-697 180 5 = = SYM cana-697 180 6 1 1 NUM cana-697 180 7 2 2 NUM cana-697 180 8 + + CCONJ cana-697 180 9 1 1 NUM cana-697 180 10 4 4 NUM cana-697 180 11 1 1 NUM cana-697 180 12 2 2 NUM cana-697 180 13 + + CCONJ cana-697 180 14 1 1 NUM cana-697 180 15 4 4 NUM cana-697 180 16 + + SYM cana-697 180 17 1 1 NUM cana-697 180 18 + + SYM cana-697 180 19 1 1 NUM cana-697 180 20 2 2 NUM cana-697 180 21 = = SYM cana-697 180 22 13 13 NUM cana-697 180 23 14 14 NUM cana-697 180 24 < < X cana-697 180 25 1 1 NUM cana-697 180 26 . . PUNCT cana-697 180 27 therefore therefore ADV cana-697 180 28 , , PUNCT cana-697 180 29 all all DET cana-697 180 30 hypotheses hypothesis NOUN cana-697 180 31 of of ADP cana-697 180 32 theorem theorem ADJ cana-697 180 33 3.1 3.1 NUM cana-697 180 34 are be AUX cana-697 180 35 satisfied satisfied ADJ cana-697 180 36 . . PUNCT cana-697 181 1 in in ADP cana-697 181 2 this this DET cana-697 181 3 case case NOUN cana-697 181 4 𝑓 𝑓 PRON cana-697 181 5 has have VERB cana-697 181 6 two two NUM cana-697 181 7 fixed fix VERB cana-697 181 8 points point NOUN cana-697 181 9 which which PRON cana-697 181 10 are be AUX cana-697 181 11 𝑧 𝑧 PRON cana-697 181 12 = = X cana-697 181 13 0 0 NUM cana-697 181 14 and and CCONJ cana-697 181 15 𝑀 𝑀 ADP cana-697 181 16 = = SYM cana-697 181 17 1 1 NUM cana-697 181 18 , , PUNCT cana-697 181 19 and and CCONJ cana-697 181 20 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 181 21 , , PUNCT cana-697 181 22 𝑀 𝑀 X cana-697 181 23 ) ) PUNCT cana-697 181 24 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 181 25 , , PUNCT cana-697 181 26 𝑧 𝑧 NOUN cana-697 181 27 ) ) PUNCT cana-697 182 1 + + CCONJ cana-697 182 2 𝑝(𝑀 𝑝(𝑀 PROPN cana-697 182 3 , , PUNCT cana-697 182 4 𝑀 𝑀 ADP cana-697 182 5 ) ) PUNCT cana-697 182 6 + + CCONJ cana-697 182 7 1 1 NUM cana-697 182 8 = = SYM cana-697 182 9 𝑝(0,1 𝑝(0,1 NOUN cana-697 182 10 ) ) PUNCT cana-697 182 11 𝑝(0,0 𝑝(0,0 NOUN cana-697 182 12 ) ) PUNCT cana-697 182 13 + + SYM cana-697 182 14 𝑝(1,1 𝑝(1,1 X cana-697 182 15 ) ) PUNCT cana-697 182 16 + + CCONJ cana-697 182 17 1 1 NUM cana-697 182 18 = = SYM cana-697 182 19 1 1 NUM cana-697 182 20 2 2 NUM cana-697 182 21 > > SYM cana-697 182 22 1 1 NUM cana-697 182 23 4 4 NUM cana-697 182 24 = = SYM cana-697 182 25 1 1 NUM cana-697 182 26 βˆ’ βˆ’ NOUN cana-697 182 27 π‘˜ π‘˜ PRON cana-697 182 28 2 2 NUM cana-697 182 29 . . PUNCT cana-697 183 1 in in ADP cana-697 183 2 the the DET cana-697 183 3 next next ADJ cana-697 183 4 results result NOUN cana-697 183 5 , , PUNCT cana-697 183 6 we we PRON cana-697 183 7 replace replace VERB cana-697 183 8 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 183 9 , , PUNCT cana-697 183 10 𝑦 𝑦 NOUN cana-697 183 11 ) ) PUNCT cana-697 183 12 in in ADP cana-697 183 13 ( ( PUNCT cana-697 183 14 3.1 3.1 NUM cana-697 183 15 ) ) PUNCT cana-697 183 16 by by ADP cana-697 183 17 𝒩(π‘₯ 𝒩(π‘₯ ADP cana-697 183 18 , , PUNCT cana-697 183 19 𝑦 𝑦 NOUN cana-697 183 20 ) ) PUNCT cana-697 183 21 as as ADP cana-697 183 22 on on ADP cana-697 183 23 the the DET cana-697 183 24 following following NOUN cana-697 183 25 theorem theorem VERB cana-697 183 26 . . PUNCT cana-697 184 1 this this DET cana-697 184 2 result result NOUN cana-697 184 3 generalizes generalize VERB cana-697 184 4 theorem theorem VERB cana-697 184 5 3.1 3.1 NUM cana-697 184 6 . . PUNCT cana-697 185 1 theorem theorem VERB cana-697 185 2 3.4 3.4 NUM cana-697 185 3 . . PUNCT cana-697 186 1 let let AUX cana-697 186 2 ( ( PUNCT cana-697 186 3 𝑋 𝑋 PROPN cana-697 186 4 , , PUNCT cana-697 186 5 𝑝 𝑝 NOUN cana-697 186 6 ) ) PUNCT cana-697 186 7 be be AUX cana-697 186 8 a a DET cana-697 186 9 complete complete ADJ cana-697 186 10 partial partial ADJ cana-697 186 11 metric metric ADJ cana-697 186 12 spaces space NOUN cana-697 186 13 endowed endow VERB cana-697 186 14 with with ADP cana-697 186 15 a a DET cana-697 186 16 binary binary ADJ cana-697 186 17 relation relation NOUN cana-697 186 18 β„œ β„œ PROPN cana-697 186 19 on on ADP cana-697 186 20 𝑋. 𝑋. PROPN cana-697 186 21 suppose suppose VERB cana-697 186 22 that that SCONJ cana-697 186 23 𝑓 𝑓 X cana-697 186 24 : : PUNCT cana-697 186 25 𝑋 𝑋 PROPN cana-697 186 26 β†’ β†’ SYM cana-697 186 27 𝑋 𝑋 PROPN cana-697 186 28 be be VERB cana-697 186 29 a a DET cana-697 186 30 nonexpansive nonexpansive ADJ cana-697 186 31 mappings mapping NOUN cana-697 186 32 such such ADJ cana-697 186 33 that that SCONJ cana-697 186 34 𝑝(𝑓(π‘₯ 𝑝(𝑓(π‘₯ NOUN cana-697 186 35 ) ) PUNCT cana-697 186 36 , , PUNCT cana-697 186 37 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 186 38 ) ) PUNCT cana-697 186 39 ) ) PUNCT cana-697 187 1 ≀ ≀ ADV cana-697 187 2 1 1 NUM cana-697 187 3 2 2 NUM cana-697 187 4 ( ( PUNCT cana-697 187 5 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 187 6 , , PUNCT cana-697 187 7 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 187 8 ) ) PUNCT cana-697 187 9 ) ) PUNCT cana-697 188 1 + + CCONJ cana-697 188 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 188 3 , , PUNCT cana-697 188 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 188 5 ) ) PUNCT cana-697 188 6 ) ) PUNCT cana-697 188 7 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 188 8 , , PUNCT cana-697 188 9 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 188 10 ) ) PUNCT cana-697 188 11 ) ) PUNCT cana-697 189 1 + + CCONJ cana-697 189 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 189 3 , , PUNCT cana-697 189 4 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 189 5 ) ) PUNCT cana-697 189 6 ) ) PUNCT cana-697 190 1 + + CCONJ cana-697 190 2 1 1 NUM cana-697 190 3 + + NUM cana-697 190 4 π‘˜ π‘˜ X cana-697 190 5 ) ) PUNCT cana-697 190 6 𝒩(π‘₯ 𝒩(π‘₯ ADP cana-697 190 7 , , PUNCT cana-697 190 8 𝑦 𝑦 NOUN cana-697 190 9 ) ) PUNCT cana-697 190 10 ( ( PUNCT cana-697 190 11 3.5 3.5 NUM cana-697 190 12 ) ) PUNCT cana-697 190 13 for for ADP cana-697 190 14 each each DET cana-697 190 15 π‘₯ π‘₯ PROPN cana-697 190 16 , , PUNCT cana-697 190 17 𝑦 𝑦 NOUN cana-697 190 18 ∈ ∈ PROPN cana-697 190 19 β„œ β„œ PROPN cana-697 190 20 , , PUNCT cana-697 190 21 where where SCONJ cana-697 190 22 π‘˜ π‘˜ PRON cana-697 190 23 ∈ ∈ PROPN cana-697 190 24 [ [ X cana-697 190 25 0,1 0,1 NUM cana-697 190 26 ) ) PUNCT cana-697 190 27 and and CCONJ cana-697 190 28 𝒩(π‘₯ 𝒩(π‘₯ ADP cana-697 190 29 , , PUNCT cana-697 190 30 𝑦 𝑦 NOUN cana-697 190 31 ) ) PUNCT cana-697 190 32 = = SYM cana-697 190 33 max max PROPN cana-697 190 34 { { PUNCT cana-697 190 35 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 190 36 , , PUNCT cana-697 190 37 𝑦 𝑦 NOUN cana-697 190 38 ) ) PUNCT cana-697 190 39 , , PUNCT cana-697 190 40 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 190 41 , , PUNCT cana-697 190 42 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 190 43 ) ) PUNCT cana-697 190 44 ) ) PUNCT cana-697 190 45 , , PUNCT cana-697 190 46 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 190 47 , , PUNCT cana-697 190 48 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 190 49 ) ) PUNCT cana-697 190 50 ) ) PUNCT cana-697 190 51 , , PUNCT cana-697 190 52 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 190 53 , , PUNCT cana-697 190 54 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 190 55 ) ) PUNCT cana-697 190 56 ) ) PUNCT cana-697 190 57 , , PUNCT cana-697 190 58 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 190 59 , , PUNCT cana-697 190 60 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 190 61 ) ) PUNCT cana-697 190 62 ) ) PUNCT cana-697 190 63 } } PUNCT cana-697 190 64 ( ( PUNCT cana-697 190 65 3.6 3.6 NUM cana-697 190 66 ) ) PUNCT cana-697 190 67 assume assume VERB cana-697 190 68 that that SCONJ cana-697 190 69 : : PUNCT cana-697 190 70 1 1 X cana-697 190 71 . . X cana-697 190 72 𝑓 𝑓 PRON cana-697 190 73 is be AUX cana-697 190 74 preserving preserve VERB cana-697 190 75 mapping mapping NOUN cana-697 190 76 2 2 NUM cana-697 190 77 . . PUNCT cana-697 191 1 𝑓 𝑓 PRON cana-697 191 2 is be AUX cana-697 191 3 continuous continuous ADJ cana-697 191 4 mapping mapping NOUN cana-697 191 5 3 3 NUM cana-697 191 6 . . PUNCT cana-697 192 1 𝐹𝑖π‘₯ 𝐹𝑖π‘₯ PROPN cana-697 192 2 ( ( PUNCT cana-697 192 3 𝑓 𝑓 X cana-697 192 4 ) ) PUNCT cana-697 192 5 is be AUX cana-697 192 6 well well ADV cana-697 192 7 ordered order VERB cana-697 192 8 with with ADP cana-697 192 9 respect respect NOUN cana-697 192 10 to to ADP cana-697 192 11 β„œ. β„œ. PROPN cana-697 192 12 if if SCONJ cana-697 192 13 there there PRON cana-697 192 14 exist exist VERB cana-697 192 15 π‘₯0 π‘₯0 NOUN cana-697 192 16 ∈ ∈ NOUN cana-697 192 17 𝑋 𝑋 NOUN cana-697 192 18 such such ADJ cana-697 192 19 that that SCONJ cana-697 192 20 ( ( PUNCT cana-697 192 21 π‘₯0 π‘₯0 NOUN cana-697 192 22 , , PUNCT cana-697 192 23 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 192 24 ) ) PUNCT cana-697 192 25 ) ) PUNCT cana-697 192 26 ∈ ∈ PROPN cana-697 192 27 β„œ β„œ PROPN cana-697 192 28 and and CCONJ cana-697 192 29 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 192 30 , , PUNCT cana-697 192 31 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 192 32 ) ) PUNCT cana-697 192 33 ) ) PUNCT cana-697 193 1 + + CCONJ cana-697 193 2 𝑝(𝑓(π‘₯0 𝑝(𝑓(π‘₯0 ADJ cana-697 193 3 ) ) PUNCT cana-697 193 4 , , PUNCT cana-697 193 5 𝑓2(π‘₯0 𝑓2(π‘₯0 NUM cana-697 193 6 ) ) PUNCT cana-697 193 7 ) ) PUNCT cana-697 193 8 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 193 9 , , PUNCT cana-697 193 10 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 193 11 ) ) PUNCT cana-697 193 12 ) ) PUNCT cana-697 194 1 + + CCONJ cana-697 194 2 𝑝(𝑓(π‘₯0 𝑝(𝑓(π‘₯0 ADJ cana-697 194 3 ) ) PUNCT cana-697 194 4 , , PUNCT cana-697 194 5 𝑓2(π‘₯0 𝑓2(π‘₯0 NUM cana-697 194 6 ) ) PUNCT cana-697 194 7 ) ) PUNCT cana-697 195 1 + + CCONJ cana-697 195 2 1 1 NUM cana-697 195 3 + + NUM cana-697 195 4 π‘˜ π‘˜ PRON cana-697 195 5 < < X cana-697 195 6 1 1 NUM cana-697 195 7 ( ( PUNCT cana-697 195 8 3.7 3.7 NUM cana-697 195 9 ) ) PUNCT cana-697 195 10 then then ADV cana-697 195 11 there there PRON cana-697 195 12 exist exist VERB cana-697 195 13 𝑧 𝑧 DET cana-697 195 14 ∈ ∈ NOUN cana-697 195 15 𝑋 𝑋 NOUN cana-697 195 16 such such ADJ cana-697 195 17 that that PRON cana-697 195 18 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 195 19 , , PUNCT cana-697 195 20 𝑧 𝑧 NOUN cana-697 195 21 ) ) PUNCT cana-697 195 22 = = SYM cana-697 195 23 0 0 X cana-697 195 24 . . PUNCT cana-697 196 1 furthermore furthermore ADV cana-697 196 2 a. a. NOUN cana-697 196 3 there there PRON cana-697 196 4 exists exist VERB cana-697 196 5 𝑧 𝑧 PRON cana-697 196 6 ∈ ∈ PROPN cana-697 196 7 𝑋 𝑋 NOUN cana-697 196 8 fixed fix VERB cana-697 196 9 point point NOUN cana-697 196 10 of of ADP cana-697 196 11 𝑓 𝑓 DET cana-697 196 12 b. b. NOUN cana-697 196 13 the the DET cana-697 196 14 picard picard PROPN cana-697 196 15 sequences sequence NOUN cana-697 196 16 of of ADP cana-697 196 17 initial initial ADJ cana-697 196 18 point point NOUN cana-697 196 19 π‘₯0 π‘₯0 NOUN cana-697 196 20 ∈ ∈ NOUN cana-697 196 21 𝑋 𝑋 NOUN cana-697 196 22 converges converge VERB cana-697 196 23 to to ADP cana-697 196 24 fixed fix VERB cana-697 196 25 point point NOUN cana-697 196 26 of of ADP cana-697 196 27 𝑓 𝑓 DET cana-697 196 28 communications communication NOUN cana-697 196 29 on on ADP cana-697 196 30 applied apply VERB cana-697 196 31 nonlinear nonlinear ADJ cana-697 196 32 analysis analysis NOUN cana-697 196 33 issn issn NOUN cana-697 196 34 : : PUNCT cana-697 196 35 1074 1074 NUM cana-697 196 36 - - PUNCT cana-697 196 37 133x 133x NUM cana-697 196 38 vol vol NOUN cana-697 196 39 31 31 NUM cana-697 196 40 no no NOUN cana-697 196 41 . . PUNCT cana-697 197 1 2s 2s NUM cana-697 197 2 ( ( PUNCT cana-697 197 3 2024 2024 NUM cana-697 197 4 ) ) PUNCT cana-697 197 5 684 684 NUM cana-697 197 6 https://internationalpubls.com https://internationalpubls.com X cana-697 197 7 c. c. NOUN cana-697 197 8 if if SCONJ cana-697 197 9 𝑧 𝑧 PROPN cana-697 197 10 and and CCONJ cana-697 197 11 𝑀 𝑀 PROPN cana-697 197 12 are be AUX cana-697 197 13 fixed fix VERB cana-697 197 14 point point NOUN cana-697 197 15 of of ADP cana-697 197 16 𝑓 𝑓 PRON cana-697 197 17 where where SCONJ cana-697 197 18 𝑧 𝑧 DET cana-697 197 19 β‰  β‰  PROPN cana-697 197 20 𝑀 𝑀 PROPN cana-697 197 21 then then ADV cana-697 197 22 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 197 23 , , PUNCT cana-697 197 24 𝑀 𝑀 X cana-697 197 25 ) ) PUNCT cana-697 197 26 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 197 27 , , PUNCT cana-697 197 28 𝑧 𝑧 NOUN cana-697 197 29 ) ) PUNCT cana-697 198 1 + + CCONJ cana-697 198 2 𝑝(𝑀 𝑝(𝑀 PROPN cana-697 198 3 , , PUNCT cana-697 198 4 𝑀 𝑀 ADP cana-697 198 5 ) ) PUNCT cana-697 198 6 + + CCONJ cana-697 198 7 1 1 NUM cana-697 198 8 β‰₯ β‰₯ NOUN cana-697 198 9 1 1 NUM cana-697 198 10 βˆ’ βˆ’ NOUN cana-697 198 11 π‘˜ π‘˜ DET cana-697 198 12 2 2 NUM cana-697 198 13 proof proof NOUN cana-697 198 14 : : PUNCT cana-697 198 15 suppose suppose VERB cana-697 198 16 that that SCONJ cana-697 198 17 π‘₯0 π‘₯0 PROPN cana-697 198 18 ∈ ∈ PROPN cana-697 198 19 𝑋 𝑋 NOUN cana-697 198 20 such such ADJ cana-697 198 21 that that SCONJ cana-697 198 22 ( ( PUNCT cana-697 198 23 π‘₯0 π‘₯0 NOUN cana-697 198 24 , , PUNCT cana-697 198 25 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 198 26 ) ) PUNCT cana-697 198 27 ) ) PUNCT cana-697 198 28 ∈ ∈ PROPN cana-697 198 29 β„œ β„œ PROPN cana-697 198 30 and and CCONJ cana-697 198 31 ( ( PUNCT cana-697 198 32 3.5 3.5 NUM cana-697 198 33 ) ) PUNCT cana-697 198 34 holds hold VERB cana-697 198 35 . . PUNCT cana-697 199 1 let let AUX cana-697 199 2 ( ( PUNCT cana-697 199 3 π‘₯𝑛 π‘₯𝑛 AUX cana-697 199 4 ) ) PUNCT cana-697 199 5 be be AUX cana-697 199 6 a a DET cana-697 199 7 picard picard NOUN cana-697 199 8 sequence sequence NOUN cana-697 199 9 of of ADP cana-697 199 10 initial initial ADJ cana-697 199 11 point point NOUN cana-697 199 12 π‘₯0 π‘₯0 NOUN cana-697 199 13 such such ADJ cana-697 199 14 that that DET cana-697 199 15 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 199 16 = = SYM cana-697 199 17 𝑓(π‘₯π‘›βˆ’1 𝑓(π‘₯π‘›βˆ’1 PROPN cana-697 199 18 ) ) PUNCT cana-697 199 19 = = PUNCT cana-697 200 1 𝑓𝑛(π‘₯0 𝑓𝑛(π‘₯0 X cana-697 200 2 ) ) PUNCT cana-697 200 3 for for ADP cana-697 200 4 all all DET cana-697 200 5 𝑛 𝑛 DET cana-697 200 6 ∈ ∈ PROPN cana-697 200 7 β„•. β„•. PROPN cana-697 200 8 let let VERB cana-697 200 9 we we PRON cana-697 200 10 consider consider VERB cana-697 200 11 , , PUNCT cana-697 200 12 if if SCONJ cana-697 200 13 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 NOUN cana-697 200 14 = = PUNCT cana-697 200 15 π‘₯𝑛 π‘₯𝑛 VERB cana-697 200 16 for for ADP cana-697 200 17 some some DET cana-697 200 18 𝑛 𝑛 PRON cana-697 200 19 ∈ ∈ PROPN cana-697 200 20 β„• β„• PROPN cana-697 200 21 then then ADV cana-697 200 22 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 PROPN cana-697 200 23 = = SYM cana-697 200 24 𝑓(π‘₯π‘›βˆ’1 𝑓(π‘₯π‘›βˆ’1 PROPN cana-697 200 25 ) ) PUNCT cana-697 200 26 . . PUNCT cana-697 201 1 it it PRON cana-697 201 2 means mean VERB cana-697 201 3 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 NOUN cana-697 201 4 is be AUX cana-697 201 5 fixed fix VERB cana-697 201 6 point point NOUN cana-697 201 7 of of ADP cana-697 201 8 𝑓. 𝑓. NOUN cana-697 201 9 hence hence ADV cana-697 201 10 , , PUNCT cana-697 201 11 the the DET cana-697 201 12 existence existence NOUN cana-697 201 13 of of ADP cana-697 201 14 a a DET cana-697 201 15 fixed fix VERB cana-697 201 16 point point NOUN cana-697 201 17 of of ADP cana-697 201 18 𝑓 𝑓 PRON cana-697 201 19 is be AUX cana-697 201 20 proved prove VERB cana-697 201 21 . . PUNCT cana-697 202 1 another another DET cana-697 202 2 condition condition NOUN cana-697 202 3 , , PUNCT cana-697 202 4 suppose suppose VERB cana-697 202 5 that that SCONJ cana-697 202 6 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 PROPN cana-697 202 7 β‰  β‰  PROPN cana-697 202 8 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 202 9 for for ADP cana-697 202 10 all all DET cana-697 202 11 𝑛 𝑛 DET cana-697 202 12 ∈ ∈ PROPN cana-697 202 13 β„•. β„•. PROPN cana-697 202 14 let let VERB cana-697 202 15 we we PRON cana-697 202 16 consider consider VERB cana-697 202 17 that that PRON cana-697 202 18 ( ( PUNCT cana-697 202 19 π‘₯0 π‘₯0 NOUN cana-697 202 20 , , PUNCT cana-697 202 21 π‘₯1 π‘₯1 NOUN cana-697 202 22 ) ) PUNCT cana-697 202 23 = = PUNCT cana-697 202 24 ( ( PUNCT cana-697 202 25 π‘₯0 π‘₯0 PROPN cana-697 202 26 , , PUNCT cana-697 202 27 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 202 28 ) ) PUNCT cana-697 202 29 ) ) PUNCT cana-697 203 1 ∈ ∈ PROPN cana-697 203 2 β„œ. β„œ. PROPN cana-697 203 3 since since SCONJ cana-697 203 4 𝑓 𝑓 PRON cana-697 203 5 is be AUX cana-697 203 6 preserving preserve VERB cana-697 203 7 mapping mapping NOUN cana-697 203 8 then then ADV cana-697 203 9 we we PRON cana-697 203 10 have have VERB cana-697 203 11 ( ( PUNCT cana-697 203 12 𝑓(π‘₯0 𝑓(π‘₯0 NOUN cana-697 203 13 ) ) PUNCT cana-697 203 14 , , PUNCT cana-697 203 15 𝑓2(π‘₯0 𝑓2(π‘₯0 NUM cana-697 203 16 ) ) PUNCT cana-697 203 17 ) ) PUNCT cana-697 204 1 ∈ ∈ PROPN cana-697 204 2 β„œ β„œ PROPN cana-697 204 3 by by ADP cana-697 204 4 induction induction NOUN cana-697 204 5 , , PUNCT cana-697 204 6 we we PRON cana-697 204 7 have have VERB cana-697 204 8 ( ( PUNCT cana-697 204 9 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 NOUN cana-697 204 10 , , PUNCT cana-697 204 11 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 204 12 ) ) PUNCT cana-697 204 13 = = SYM cana-697 204 14 ( ( PUNCT cana-697 204 15 π‘“π‘›βˆ’1(π‘₯0 π‘“π‘›βˆ’1(π‘₯0 PROPN cana-697 204 16 ) ) PUNCT cana-697 204 17 , , PUNCT cana-697 204 18 𝑓𝑛(π‘₯0 𝑓𝑛(π‘₯0 NOUN cana-697 204 19 ) ) PUNCT cana-697 204 20 ) ) PUNCT cana-697 205 1 ∈ ∈ PROPN cana-697 205 2 β„œ β„œ PROPN cana-697 205 3 for for ADP cana-697 205 4 all all PRON cana-697 205 5 𝑛 𝑛 DET cana-697 205 6 ∈ ∈ PROPN cana-697 205 7 β„•. β„•. PROPN cana-697 205 8 since since SCONJ cana-697 205 9 ( ( PUNCT cana-697 205 10 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 NOUN cana-697 205 11 , , PUNCT cana-697 205 12 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 205 13 ) ) PUNCT cana-697 205 14 ∈ ∈ PROPN cana-697 205 15 β„œ β„œ PROPN cana-697 205 16 then then ADV cana-697 205 17 by by ADP cana-697 205 18 ( ( PUNCT cana-697 205 19 3.5 3.5 NUM cana-697 205 20 ) ) PUNCT cana-697 205 21 we we PRON cana-697 205 22 obtain obtain VERB cana-697 205 23 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 205 24 , , PUNCT cana-697 205 25 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 205 26 ) ) PUNCT cana-697 205 27 = = SYM cana-697 205 28 𝑝(𝑓(π‘₯π‘›βˆ’1 𝑝(𝑓(π‘₯π‘›βˆ’1 PROPN cana-697 205 29 ) ) PUNCT cana-697 205 30 , , PUNCT cana-697 205 31 𝑓(π‘₯𝑛 𝑓(π‘₯𝑛 NOUN cana-697 205 32 ) ) PUNCT cana-697 205 33 ) ) PUNCT cana-697 206 1 ⬚ ⬚ PROPN cana-697 206 2 ≀ ≀ NUM cana-697 206 3 1 1 NUM cana-697 206 4 2 2 NUM cana-697 206 5 ( ( PUNCT cana-697 206 6 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 206 7 , , PUNCT cana-697 206 8 𝑓(π‘₯𝑛 𝑓(π‘₯𝑛 NOUN cana-697 206 9 ) ) PUNCT cana-697 206 10 ) ) PUNCT cana-697 207 1 + + CCONJ cana-697 207 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 207 3 , , PUNCT cana-697 207 4 𝑓(π‘₯π‘›βˆ’1 𝑓(π‘₯π‘›βˆ’1 NOUN cana-697 207 5 ) ) PUNCT cana-697 207 6 ) ) PUNCT cana-697 207 7 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 PROPN cana-697 207 8 , , PUNCT cana-697 207 9 𝑓(π‘₯π‘›βˆ’1 𝑓(π‘₯π‘›βˆ’1 NOUN cana-697 207 10 ) ) PUNCT cana-697 207 11 ) ) PUNCT cana-697 208 1 + + CCONJ cana-697 208 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 208 3 , , PUNCT cana-697 208 4 𝑓(π‘₯𝑛 𝑓(π‘₯𝑛 NOUN cana-697 208 5 ) ) PUNCT cana-697 208 6 ) ) PUNCT cana-697 209 1 + + CCONJ cana-697 209 2 1 1 NUM cana-697 209 3 + + NUM cana-697 209 4 π‘˜ π‘˜ X cana-697 209 5 ) ) PUNCT cana-697 209 6 𝒩(π‘₯π‘›βˆ’1 𝒩(π‘₯π‘›βˆ’1 NOUN cana-697 209 7 , , PUNCT cana-697 209 8 π‘₯𝑛 π‘₯𝑛 PRON cana-697 209 9 ) ) PUNCT cana-697 209 10 let let VERB cana-697 209 11 we we PRON cana-697 209 12 consider consider VERB cana-697 209 13 𝒩(π‘₯π‘›βˆ’1 𝒩(π‘₯π‘›βˆ’1 NOUN cana-697 209 14 , , PUNCT cana-697 209 15 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 209 16 ) ) PUNCT cana-697 209 17 = = SYM cana-697 209 18 max max PROPN cana-697 209 19 { { PUNCT cana-697 209 20 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 209 21 , , PUNCT cana-697 209 22 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 209 23 ) ) PUNCT cana-697 209 24 , , PUNCT cana-697 209 25 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 209 26 , , PUNCT cana-697 209 27 𝑓(π‘₯π‘›βˆ’1 𝑓(π‘₯π‘›βˆ’1 NOUN cana-697 209 28 ) ) PUNCT cana-697 209 29 ) ) PUNCT cana-697 209 30 , , PUNCT cana-697 209 31 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NOUN cana-697 209 32 , , PUNCT cana-697 209 33 𝑓(π‘₯𝑛 𝑓(π‘₯𝑛 NOUN cana-697 209 34 ) ) PUNCT cana-697 209 35 ) ) PUNCT cana-697 209 36 , , PUNCT cana-697 209 37 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 209 38 , , PUNCT cana-697 209 39 𝑓(π‘₯𝑛 𝑓(π‘₯𝑛 NOUN cana-697 209 40 ) ) PUNCT cana-697 209 41 ) ) PUNCT cana-697 209 42 , , PUNCT cana-697 209 43 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 209 44 , , PUNCT cana-697 209 45 𝑓(π‘₯π‘›βˆ’1 𝑓(π‘₯π‘›βˆ’1 NOUN cana-697 209 46 ) ) PUNCT cana-697 209 47 ) ) PUNCT cana-697 209 48 } } PUNCT cana-697 210 1 = = SYM cana-697 210 2 max{𝑝(π‘₯π‘›βˆ’1 max{𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 210 3 , , PUNCT cana-697 210 4 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 210 5 ) ) PUNCT cana-697 210 6 , , PUNCT cana-697 210 7 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 PROPN cana-697 210 8 , , PUNCT cana-697 210 9 π‘₯𝑛+1 π‘₯𝑛+1 NOUN cana-697 210 10 ) ) PUNCT cana-697 210 11 , , PUNCT cana-697 210 12 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 210 13 , , PUNCT cana-697 210 14 π‘₯𝑛+1 π‘₯𝑛+1 NOUN cana-697 210 15 ) ) PUNCT cana-697 210 16 , , PUNCT cana-697 210 17 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 210 18 , , PUNCT cana-697 210 19 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 210 20 ) ) PUNCT cana-697 210 21 } } PUNCT cana-697 210 22 = = SYM cana-697 210 23 max{𝑝(π‘₯π‘›βˆ’1 max{𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 210 24 , , PUNCT cana-697 210 25 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 210 26 ) ) PUNCT cana-697 210 27 , , PUNCT cana-697 210 28 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 PROPN cana-697 210 29 , , PUNCT cana-697 210 30 π‘₯𝑛+1 π‘₯𝑛+1 NOUN cana-697 210 31 ) ) PUNCT cana-697 210 32 , , PUNCT cana-697 210 33 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 210 34 , , PUNCT cana-697 210 35 π‘₯𝑛+1 π‘₯𝑛+1 NOUN cana-697 210 36 ) ) PUNCT cana-697 210 37 } } PUNCT cana-697 210 38 since since SCONJ cana-697 210 39 𝑓 𝑓 PRON cana-697 210 40 is be AUX cana-697 210 41 nonexpansive nonexpansive ADJ cana-697 210 42 mapping mapping NOUN cana-697 210 43 then then ADV cana-697 210 44 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 210 45 , , PUNCT cana-697 210 46 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 210 47 ) ) PUNCT cana-697 210 48 = = SYM cana-697 210 49 𝑝(𝑓(π‘₯𝑛 𝑝(𝑓(π‘₯𝑛 NOUN cana-697 210 50 ) ) PUNCT cana-697 210 51 , , PUNCT cana-697 210 52 𝑓(π‘₯𝑛+1 𝑓(π‘₯𝑛+1 NOUN cana-697 210 53 ) ) PUNCT cana-697 210 54 ) ) PUNCT cana-697 211 1 ≀ ≀ NOUN cana-697 211 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 211 3 , , PUNCT cana-697 211 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 211 5 ) ) PUNCT cana-697 211 6 therefore therefore ADV cana-697 211 7 𝒩(π‘₯π‘›βˆ’1 𝒩(π‘₯π‘›βˆ’1 PROPN cana-697 211 8 , , PUNCT cana-697 211 9 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 211 10 ) ) PUNCT cana-697 211 11 = = SYM cana-697 211 12 max max PROPN cana-697 211 13 { { PUNCT cana-697 211 14 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 211 15 , , PUNCT cana-697 211 16 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 211 17 ) ) PUNCT cana-697 211 18 , , PUNCT cana-697 211 19 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 211 20 , , PUNCT cana-697 211 21 π‘₯𝑛+1 π‘₯𝑛+1 NOUN cana-697 211 22 ) ) PUNCT cana-697 211 23 } } PUNCT cana-697 211 24 let let VERB cana-697 211 25 we we PRON cana-697 211 26 consider consider VERB cana-697 211 27 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 211 28 , , PUNCT cana-697 211 29 π‘₯𝑛+1 π‘₯𝑛+1 NOUN cana-697 211 30 ) ) PUNCT cana-697 211 31 ≀ ≀ NOUN cana-697 211 32 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 211 33 , , PUNCT cana-697 211 34 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 211 35 ) ) PUNCT cana-697 212 1 + + CCONJ cana-697 212 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 212 3 , , PUNCT cana-697 212 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 212 5 ) ) PUNCT cana-697 212 6 βˆ’ βˆ’ PROPN cana-697 213 1 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 213 2 , , PUNCT cana-697 213 3 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 213 4 ) ) PUNCT cana-697 213 5 ⬚ ⬚ PROPN cana-697 213 6 ≀ ≀ PROPN cana-697 213 7 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 213 8 , , PUNCT cana-697 213 9 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 213 10 ) ) PUNCT cana-697 214 1 + + CCONJ cana-697 214 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 214 3 , , PUNCT cana-697 214 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 214 5 ) ) PUNCT cana-697 214 6 ⬚ ⬚ PROPN cana-697 214 7 ≀ ≀ PROPN cana-697 214 8 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 214 9 , , PUNCT cana-697 214 10 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 214 11 ) ) PUNCT cana-697 214 12 + + CCONJ cana-697 214 13 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 214 14 , , PUNCT cana-697 214 15 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 214 16 ) ) PUNCT cana-697 214 17 ⬚ ⬚ PROPN cana-697 214 18 = = SYM cana-697 214 19 2𝑝(π‘₯π‘›βˆ’1 2𝑝(π‘₯π‘›βˆ’1 NUM cana-697 214 20 , , PUNCT cana-697 214 21 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 214 22 ) ) PUNCT cana-697 214 23 thus thus ADV cana-697 214 24 𝒩(π‘₯π‘›βˆ’1 𝒩(π‘₯π‘›βˆ’1 NOUN cana-697 214 25 , , PUNCT cana-697 214 26 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 214 27 ) ) PUNCT cana-697 214 28 ≀ ≀ NOUN cana-697 215 1 max{𝑝(π‘₯π‘›βˆ’1 max{𝑝(π‘₯π‘›βˆ’1 PROPN cana-697 215 2 , , PUNCT cana-697 215 3 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 215 4 ) ) PUNCT cana-697 215 5 , , PUNCT cana-697 215 6 2𝑝(π‘₯π‘›βˆ’1 2𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 215 7 , , PUNCT cana-697 215 8 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 215 9 ) ) PUNCT cana-697 215 10 } } PUNCT cana-697 215 11 = = SYM cana-697 215 12 2𝑝(π‘₯π‘›βˆ’1 2𝑝(π‘₯π‘›βˆ’1 NUM cana-697 215 13 , , PUNCT cana-697 215 14 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 215 15 ) ) PUNCT cana-697 215 16 communications communication NOUN cana-697 215 17 on on ADP cana-697 215 18 applied apply VERB cana-697 215 19 nonlinear nonlinear ADJ cana-697 215 20 analysis analysis NOUN cana-697 215 21 issn issn NOUN cana-697 215 22 : : PUNCT cana-697 215 23 1074 1074 NUM cana-697 215 24 - - PUNCT cana-697 215 25 133x 133x NUM cana-697 215 26 vol vol NOUN cana-697 215 27 31 31 NUM cana-697 215 28 no no NOUN cana-697 215 29 . . PUNCT cana-697 216 1 2s 2s NUM cana-697 216 2 ( ( PUNCT cana-697 216 3 2024 2024 NUM cana-697 216 4 ) ) PUNCT cana-697 216 5 685 685 NUM cana-697 216 6 https://internationalpubls.com https://internationalpubls.com X cana-697 216 7 hence hence ADV cana-697 216 8 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 216 9 , , PUNCT cana-697 216 10 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 216 11 ) ) PUNCT cana-697 216 12 = = SYM cana-697 216 13 𝑝(𝑓(π‘₯π‘›βˆ’1 𝑝(𝑓(π‘₯π‘›βˆ’1 PROPN cana-697 216 14 ) ) PUNCT cana-697 216 15 , , PUNCT cana-697 216 16 𝑓(π‘₯𝑛 𝑓(π‘₯𝑛 NOUN cana-697 216 17 ) ) PUNCT cana-697 216 18 ) ) PUNCT cana-697 217 1 ⬚ ⬚ PROPN cana-697 217 2 ≀ ≀ NUM cana-697 217 3 1 1 NUM cana-697 217 4 2 2 NUM cana-697 217 5 ( ( PUNCT cana-697 217 6 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 217 7 , , PUNCT cana-697 217 8 𝑓(π‘₯𝑛 𝑓(π‘₯𝑛 NOUN cana-697 217 9 ) ) PUNCT cana-697 217 10 ) ) PUNCT cana-697 218 1 + + CCONJ cana-697 218 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 218 3 , , PUNCT cana-697 218 4 𝑓(π‘₯π‘›βˆ’1 𝑓(π‘₯π‘›βˆ’1 NOUN cana-697 218 5 ) ) PUNCT cana-697 218 6 ) ) PUNCT cana-697 218 7 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 PROPN cana-697 218 8 , , PUNCT cana-697 218 9 𝑓(π‘₯π‘›βˆ’1 𝑓(π‘₯π‘›βˆ’1 NOUN cana-697 218 10 ) ) PUNCT cana-697 218 11 ) ) PUNCT cana-697 219 1 + + CCONJ cana-697 219 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 219 3 , , PUNCT cana-697 219 4 𝑓(π‘₯𝑛 𝑓(π‘₯𝑛 NOUN cana-697 219 5 ) ) PUNCT cana-697 219 6 ) ) PUNCT cana-697 220 1 + + CCONJ cana-697 220 2 1 1 NUM cana-697 220 3 + + NUM cana-697 220 4 π‘˜ π‘˜ X cana-697 220 5 ) ) PUNCT cana-697 220 6 2𝑝(π‘₯π‘›βˆ’1 2𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 220 7 , , PUNCT cana-697 220 8 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 220 9 ) ) PUNCT cana-697 220 10 ⬚ ⬚ NOUN cana-697 220 11 = = SYM cana-697 220 12 1 1 NUM cana-697 220 13 2 2 NUM cana-697 220 14 ( ( PUNCT cana-697 220 15 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 220 16 , , PUNCT cana-697 220 17 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 220 18 ) ) PUNCT cana-697 221 1 + + CCONJ cana-697 221 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 221 3 , , PUNCT cana-697 221 4 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 221 5 ) ) PUNCT cana-697 221 6 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 221 7 , , PUNCT cana-697 221 8 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 221 9 ) ) PUNCT cana-697 222 1 + + CCONJ cana-697 222 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 222 3 , , PUNCT cana-697 222 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 222 5 ) ) PUNCT cana-697 222 6 + + CCONJ cana-697 222 7 1 1 NUM cana-697 222 8 + + NUM cana-697 222 9 π‘˜ π‘˜ X cana-697 222 10 ) ) PUNCT cana-697 222 11 2𝑝(π‘₯π‘›βˆ’1 2𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 222 12 , , PUNCT cana-697 222 13 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 222 14 ) ) PUNCT cana-697 222 15 ⬚ ⬚ PROPN cana-697 222 16 ≀ ≀ NUM cana-697 222 17 1 1 NUM cana-697 222 18 2 2 NUM cana-697 222 19 ( ( PUNCT cana-697 222 20 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 222 21 , , PUNCT cana-697 222 22 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 222 23 ) ) PUNCT cana-697 223 1 + + CCONJ cana-697 223 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 223 3 , , PUNCT cana-697 223 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 223 5 ) ) PUNCT cana-697 223 6 βˆ’ βˆ’ PROPN cana-697 224 1 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 224 2 , , PUNCT cana-697 224 3 π‘₯𝑛 π‘₯𝑛 PRON cana-697 224 4 ) ) PUNCT cana-697 225 1 + + CCONJ cana-697 225 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 225 3 , , PUNCT cana-697 225 4 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 225 5 ) ) PUNCT cana-697 225 6 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 225 7 , , PUNCT cana-697 225 8 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 225 9 ) ) PUNCT cana-697 226 1 + + CCONJ cana-697 226 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 226 3 , , PUNCT cana-697 226 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 226 5 ) ) PUNCT cana-697 226 6 + + CCONJ cana-697 226 7 1 1 NUM cana-697 226 8 + + NUM cana-697 226 9 π‘˜ π‘˜ X cana-697 226 10 ) ) PUNCT cana-697 226 11 2𝑝(π‘₯π‘›βˆ’1 2𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 226 12 , , PUNCT cana-697 226 13 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 226 14 ) ) PUNCT cana-697 226 15 ⬚ ⬚ NOUN cana-697 226 16 = = SYM cana-697 226 17 1 1 NUM cana-697 226 18 2 2 NUM cana-697 226 19 ( ( PUNCT cana-697 226 20 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 226 21 , , PUNCT cana-697 226 22 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 226 23 ) ) PUNCT cana-697 227 1 + + CCONJ cana-697 227 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 227 3 , , PUNCT cana-697 227 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 227 5 ) ) PUNCT cana-697 227 6 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 227 7 , , PUNCT cana-697 227 8 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 227 9 ) ) PUNCT cana-697 228 1 + + CCONJ cana-697 228 2 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 NUM cana-697 228 3 , , PUNCT cana-697 228 4 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 228 5 ) ) PUNCT cana-697 228 6 + + CCONJ cana-697 228 7 1 1 NUM cana-697 228 8 + + NUM cana-697 228 9 π‘˜ π‘˜ X cana-697 228 10 ) ) PUNCT cana-697 228 11 2𝑝(π‘₯π‘›βˆ’1 2𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 228 12 , , PUNCT cana-697 228 13 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 228 14 ) ) PUNCT cana-697 228 15 ⬚ ⬚ PROPN cana-697 228 16 ≀ ≀ NUM cana-697 228 17 1 1 NUM cana-697 228 18 2 2 NUM cana-697 228 19 ( ( PUNCT cana-697 228 20 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 228 21 , , PUNCT cana-697 228 22 π‘₯1 π‘₯1 PROPN cana-697 228 23 ) ) PUNCT cana-697 228 24 + + CCONJ cana-697 228 25 𝑝(π‘₯1 𝑝(π‘₯1 ADJ cana-697 228 26 , , PUNCT cana-697 228 27 π‘₯2 π‘₯2 NOUN cana-697 228 28 ) ) PUNCT cana-697 228 29 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 228 30 , , PUNCT cana-697 228 31 π‘₯1 π‘₯1 PROPN cana-697 228 32 ) ) PUNCT cana-697 228 33 + + CCONJ cana-697 228 34 𝑝(π‘₯1 𝑝(π‘₯1 ADJ cana-697 228 35 , , PUNCT cana-697 228 36 π‘₯2 π‘₯2 NOUN cana-697 228 37 ) ) PUNCT cana-697 229 1 + + CCONJ cana-697 229 2 1 1 NUM cana-697 229 3 + + NUM cana-697 229 4 π‘˜ π‘˜ X cana-697 229 5 ) ) PUNCT cana-697 229 6 2𝑝(π‘₯π‘›βˆ’1 2𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 229 7 , , PUNCT cana-697 229 8 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 229 9 ) ) PUNCT cana-697 230 1 ⬚ ⬚ PROPN cana-697 230 2 = = SYM cana-697 230 3 ( ( PUNCT cana-697 230 4 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 230 5 , , PUNCT cana-697 230 6 π‘₯1 π‘₯1 PROPN cana-697 230 7 ) ) PUNCT cana-697 230 8 + + CCONJ cana-697 230 9 𝑝(π‘₯1 𝑝(π‘₯1 ADJ cana-697 230 10 , , PUNCT cana-697 230 11 π‘₯2 π‘₯2 NOUN cana-697 230 12 ) ) PUNCT cana-697 230 13 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 230 14 , , PUNCT cana-697 230 15 π‘₯1 π‘₯1 PROPN cana-697 230 16 ) ) PUNCT cana-697 230 17 + + CCONJ cana-697 230 18 𝑝(π‘₯1 𝑝(π‘₯1 ADJ cana-697 230 19 , , PUNCT cana-697 230 20 π‘₯2 π‘₯2 NOUN cana-697 230 21 ) ) PUNCT cana-697 231 1 + + CCONJ cana-697 231 2 1 1 NUM cana-697 231 3 + + NUM cana-697 231 4 π‘˜ π‘˜ NOUN cana-697 231 5 ) ) PUNCT cana-697 231 6 𝑝(π‘₯π‘›βˆ’1 𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 231 7 , , PUNCT cana-697 231 8 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 231 9 ) ) PUNCT cana-697 231 10 ⬚ ⬚ PROPN cana-697 231 11 ≀ ≀ PROPN cana-697 231 12 𝛼𝑝(π‘₯π‘›βˆ’1 𝛼𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 231 13 , , PUNCT cana-697 231 14 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 231 15 ) ) PUNCT cana-697 231 16 where where SCONJ cana-697 231 17 𝛼 𝛼 X cana-697 231 18 < < X cana-697 231 19 1 1 NUM cana-697 231 20 . . PUNCT cana-697 232 1 hence hence ADV cana-697 232 2 ( ( PUNCT cana-697 232 3 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 232 4 ) ) PUNCT cana-697 232 5 is be AUX cana-697 232 6 cauchy cauchy ADJ cana-697 232 7 sequences sequence NOUN cana-697 232 8 in in ADP cana-697 232 9 𝑋 𝑋 NOUN cana-697 232 10 since since SCONJ cana-697 232 11 𝑝(π‘₯𝑛 𝑝(π‘₯𝑛 PROPN cana-697 232 12 , , PUNCT cana-697 232 13 π‘₯𝑛+1 π‘₯𝑛+1 ADJ cana-697 232 14 ) ) PUNCT cana-697 232 15 ≀ ≀ NOUN cana-697 232 16 𝛼𝑝(π‘₯π‘›βˆ’1 𝛼𝑝(π‘₯π‘›βˆ’1 NOUN cana-697 232 17 , , PUNCT cana-697 232 18 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 232 19 ) ) PUNCT cana-697 232 20 , , PUNCT cana-697 232 21 𝛼 𝛼 X cana-697 232 22 < < X cana-697 232 23 1 1 NUM cana-697 232 24 . . PUNCT cana-697 233 1 the the DET cana-697 233 2 next next ADJ cana-697 233 3 proof proof NOUN cana-697 233 4 is be AUX cana-697 233 5 the the DET cana-697 233 6 same same ADJ cana-697 233 7 as as ADP cana-697 233 8 the the DET cana-697 233 9 proof proof NOUN cana-697 233 10 of of ADP cana-697 233 11 theorem theorem NOUN cana-697 233 12 3.1 3.1 NUM cana-697 233 13 . . PUNCT cana-697 234 1 this this PRON cana-697 234 2 completes complete VERB cana-697 234 3 the the DET cana-697 234 4 proof proof NOUN cana-697 234 5 . . PUNCT cana-697 235 1 analogous analogous ADJ cana-697 235 2 to to PART cana-697 235 3 theorem theorem VERB cana-697 235 4 3.4 3.4 NUM cana-697 235 5 , , PUNCT cana-697 235 6 we we PRON cana-697 235 7 can can AUX cana-697 235 8 generalize generalize VERB cana-697 235 9 theorem theorem VERB cana-697 235 10 3.2 3.2 NUM cana-697 235 11 as as SCONJ cana-697 235 12 follows follow VERB cana-697 235 13 . . PUNCT cana-697 236 1 theorem theorem ADJ cana-697 236 2 3.5 3.5 NUM cana-697 236 3 . . PUNCT cana-697 237 1 let let AUX cana-697 237 2 ( ( PUNCT cana-697 237 3 𝑋 𝑋 PROPN cana-697 237 4 , , PUNCT cana-697 237 5 𝑝 𝑝 NOUN cana-697 237 6 ) ) PUNCT cana-697 237 7 be be AUX cana-697 237 8 a a DET cana-697 237 9 complete complete ADJ cana-697 237 10 partial partial ADJ cana-697 237 11 metric metric ADJ cana-697 237 12 spaces space NOUN cana-697 237 13 endowed endow VERB cana-697 237 14 with with ADP cana-697 237 15 a a DET cana-697 237 16 binary binary ADJ cana-697 237 17 relation relation NOUN cana-697 237 18 β„œ β„œ PROPN cana-697 237 19 on on ADP cana-697 237 20 𝑋. 𝑋. PROPN cana-697 237 21 suppose suppose VERB cana-697 237 22 that that SCONJ cana-697 237 23 𝑓 𝑓 X cana-697 237 24 : : PUNCT cana-697 237 25 𝑋 𝑋 PROPN cana-697 237 26 β†’ β†’ SYM cana-697 237 27 𝑋 𝑋 PROPN cana-697 237 28 be be VERB cana-697 237 29 a a DET cana-697 237 30 nonexpansive nonexpansive ADJ cana-697 237 31 mappings mapping NOUN cana-697 237 32 such such ADJ cana-697 237 33 that that SCONJ cana-697 237 34 𝑝(𝑓(π‘₯ 𝑝(𝑓(π‘₯ NOUN cana-697 237 35 ) ) PUNCT cana-697 237 36 , , PUNCT cana-697 237 37 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 237 38 ) ) PUNCT cana-697 237 39 ) ) PUNCT cana-697 238 1 ≀ ≀ ADV cana-697 238 2 1 1 NUM cana-697 238 3 2 2 NUM cana-697 238 4 ( ( PUNCT cana-697 238 5 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 238 6 , , PUNCT cana-697 238 7 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 238 8 ) ) PUNCT cana-697 238 9 ) ) PUNCT cana-697 239 1 + + CCONJ cana-697 239 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 239 3 , , PUNCT cana-697 239 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 239 5 ) ) PUNCT cana-697 239 6 ) ) PUNCT cana-697 239 7 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 239 8 , , PUNCT cana-697 239 9 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 239 10 ) ) PUNCT cana-697 239 11 ) ) PUNCT cana-697 240 1 + + CCONJ cana-697 240 2 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 240 3 , , PUNCT cana-697 240 4 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 240 5 ) ) PUNCT cana-697 240 6 ) ) PUNCT cana-697 241 1 + + CCONJ cana-697 241 2 1 1 NUM cana-697 241 3 + + NUM cana-697 241 4 π‘˜ π‘˜ X cana-697 241 5 ) ) PUNCT cana-697 241 6 𝒩(π‘₯ 𝒩(π‘₯ ADP cana-697 241 7 , , PUNCT cana-697 241 8 𝑦 𝑦 NOUN cana-697 241 9 ) ) PUNCT cana-697 241 10 for for ADP cana-697 241 11 each each DET cana-697 241 12 π‘₯ π‘₯ PROPN cana-697 241 13 , , PUNCT cana-697 241 14 𝑦 𝑦 NOUN cana-697 241 15 ∈ ∈ PROPN cana-697 241 16 β„œ β„œ PROPN cana-697 241 17 , , PUNCT cana-697 241 18 where where SCONJ cana-697 241 19 π‘˜ π‘˜ PRON cana-697 241 20 ∈ ∈ PROPN cana-697 241 21 [ [ X cana-697 241 22 0,1 0,1 NUM cana-697 241 23 ) ) PUNCT cana-697 241 24 and and CCONJ cana-697 241 25 𝒩(π‘₯ 𝒩(π‘₯ ADP cana-697 241 26 , , PUNCT cana-697 241 27 𝑦 𝑦 NOUN cana-697 241 28 ) ) PUNCT cana-697 241 29 = = SYM cana-697 241 30 max max PROPN cana-697 241 31 { { PUNCT cana-697 241 32 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 241 33 , , PUNCT cana-697 241 34 𝑦 𝑦 NOUN cana-697 241 35 ) ) PUNCT cana-697 241 36 , , PUNCT cana-697 241 37 𝑝(π‘₯ 𝑝(π‘₯ NOUN cana-697 241 38 , , PUNCT cana-697 241 39 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 241 40 ) ) PUNCT cana-697 241 41 ) ) PUNCT cana-697 241 42 , , PUNCT cana-697 241 43 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 241 44 , , PUNCT cana-697 241 45 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 241 46 ) ) PUNCT cana-697 241 47 ) ) PUNCT cana-697 241 48 , , PUNCT cana-697 241 49 𝑝(π‘₯ 𝑝(π‘₯ PROPN cana-697 241 50 , , PUNCT cana-697 241 51 𝑓(𝑦 𝑓(𝑦 PROPN cana-697 241 52 ) ) PUNCT cana-697 241 53 ) ) PUNCT cana-697 241 54 , , PUNCT cana-697 241 55 𝑝(𝑦 𝑝(𝑦 PROPN cana-697 241 56 , , PUNCT cana-697 241 57 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-697 241 58 ) ) PUNCT cana-697 241 59 ) ) PUNCT cana-697 241 60 } } PUNCT cana-697 241 61 assume assume VERB cana-697 241 62 that that SCONJ cana-697 241 63 : : PUNCT cana-697 241 64 1 1 X cana-697 241 65 . . X cana-697 241 66 𝑓 𝑓 PRON cana-697 241 67 is be AUX cana-697 241 68 preserving preserve VERB cana-697 241 69 mapping mapping NOUN cana-697 241 70 2 2 NUM cana-697 241 71 . . PUNCT cana-697 242 1 if if SCONJ cana-697 242 2 π‘₯𝑛 π‘₯𝑛 VERB cana-697 242 3 sequences sequence NOUN cana-697 242 4 in in ADP cana-697 242 5 𝑋 𝑋 PROPN cana-697 242 6 such such ADJ cana-697 242 7 that that SCONJ cana-697 242 8 ( ( PUNCT cana-697 242 9 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 NOUN cana-697 242 10 , , PUNCT cana-697 242 11 π‘₯𝑛 π‘₯𝑛 NOUN cana-697 242 12 ) ) PUNCT cana-697 242 13 ∈ ∈ PROPN cana-697 242 14 β„œ β„œ PROPN cana-697 242 15 for for ADP cana-697 242 16 each each DET cana-697 242 17 𝑛 𝑛 PRON cana-697 242 18 ∈ ∈ PROPN cana-697 242 19 β„• β„• PROPN cana-697 242 20 and and CCONJ cana-697 242 21 π‘₯𝑛 π‘₯𝑛 PROPN cana-697 242 22 β†’ β†’ SYM cana-697 242 23 𝑧 𝑧 PRON cana-697 242 24 ∈ ∈ NOUN cana-697 242 25 𝑋 𝑋 NOUN cana-697 242 26 as as ADP cana-697 242 27 𝑛 𝑛 PROPN cana-697 242 28 β†’ β†’ SYM cana-697 242 29 ∞ ∞ PROPN cana-697 242 30 , , PUNCT cana-697 242 31 then then ADV cana-697 242 32 ( ( PUNCT cana-697 242 33 π‘₯π‘›βˆ’1 π‘₯π‘›βˆ’1 NOUN cana-697 242 34 , , PUNCT cana-697 242 35 𝑧 𝑧 NOUN cana-697 242 36 ) ) PUNCT cana-697 242 37 ∈ ∈ PROPN cana-697 242 38 β„œ β„œ PROPN cana-697 242 39 for for ADP cana-697 242 40 all all PRON cana-697 242 41 𝑛 𝑛 DET cana-697 242 42 ∈ ∈ PROPN cana-697 242 43 β„• β„• PROPN cana-697 242 44 3 3 NUM cana-697 242 45 . . PUNCT cana-697 243 1 𝐹𝑖π‘₯ 𝐹𝑖π‘₯ PROPN cana-697 243 2 ( ( PUNCT cana-697 243 3 𝑓 𝑓 X cana-697 243 4 ) ) PUNCT cana-697 243 5 is be AUX cana-697 243 6 well well ADV cana-697 243 7 ordered order VERB cana-697 243 8 with with ADP cana-697 243 9 respect respect NOUN cana-697 243 10 to to ADP cana-697 243 11 β„œ. β„œ. PROPN cana-697 243 12 if if SCONJ cana-697 243 13 there there PRON cana-697 243 14 exist exist VERB cana-697 243 15 π‘₯0 π‘₯0 NOUN cana-697 243 16 ∈ ∈ NOUN cana-697 243 17 𝑋 𝑋 NOUN cana-697 243 18 such such ADJ cana-697 243 19 that that SCONJ cana-697 243 20 ( ( PUNCT cana-697 243 21 π‘₯0 π‘₯0 NOUN cana-697 243 22 , , PUNCT cana-697 243 23 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 243 24 ) ) PUNCT cana-697 243 25 ) ) PUNCT cana-697 243 26 ∈ ∈ PROPN cana-697 243 27 β„œ β„œ PROPN cana-697 243 28 and and CCONJ cana-697 243 29 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 243 30 , , PUNCT cana-697 243 31 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 243 32 ) ) PUNCT cana-697 243 33 ) ) PUNCT cana-697 244 1 + + CCONJ cana-697 244 2 𝑝(𝑓(π‘₯0 𝑝(𝑓(π‘₯0 ADJ cana-697 244 3 ) ) PUNCT cana-697 244 4 , , PUNCT cana-697 244 5 𝑓2(π‘₯0 𝑓2(π‘₯0 NUM cana-697 244 6 ) ) PUNCT cana-697 244 7 ) ) PUNCT cana-697 244 8 𝑝(π‘₯0 𝑝(π‘₯0 ADJ cana-697 244 9 , , PUNCT cana-697 244 10 𝑓(π‘₯0 𝑓(π‘₯0 NUM cana-697 244 11 ) ) PUNCT cana-697 244 12 ) ) PUNCT cana-697 245 1 + + CCONJ cana-697 245 2 𝑝(𝑓(π‘₯0 𝑝(𝑓(π‘₯0 ADJ cana-697 245 3 ) ) PUNCT cana-697 245 4 , , PUNCT cana-697 245 5 𝑓2(π‘₯0 𝑓2(π‘₯0 NUM cana-697 245 6 ) ) PUNCT cana-697 245 7 ) ) PUNCT cana-697 246 1 + + CCONJ cana-697 246 2 1 1 NUM cana-697 246 3 + + NUM cana-697 246 4 π‘˜ π‘˜ X cana-697 246 5 < < X cana-697 246 6 1 1 NUM cana-697 246 7 then then ADV cana-697 246 8 there there PRON cana-697 246 9 exist exist VERB cana-697 246 10 𝑧 𝑧 DET cana-697 246 11 ∈ ∈ NOUN cana-697 246 12 𝑋 𝑋 NOUN cana-697 246 13 such such ADJ cana-697 246 14 that that PRON cana-697 246 15 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 246 16 , , PUNCT cana-697 246 17 𝑧 𝑧 NOUN cana-697 246 18 ) ) PUNCT cana-697 246 19 = = SYM cana-697 246 20 0 0 X cana-697 246 21 . . PUNCT cana-697 247 1 furthermore furthermore ADV cana-697 247 2 a. a. NOUN cana-697 247 3 there there PRON cana-697 247 4 exists exist VERB cana-697 247 5 𝑧 𝑧 PRON cana-697 247 6 ∈ ∈ PROPN cana-697 247 7 𝑋 𝑋 NOUN cana-697 247 8 fixed fix VERB cana-697 247 9 point point NOUN cana-697 247 10 of of ADP cana-697 247 11 𝑓 𝑓 DET cana-697 247 12 b. b. NOUN cana-697 247 13 the the DET cana-697 247 14 picard picard PROPN cana-697 247 15 sequences sequence NOUN cana-697 247 16 of of ADP cana-697 247 17 initial initial ADJ cana-697 247 18 point point NOUN cana-697 247 19 π‘₯0 π‘₯0 NOUN cana-697 247 20 ∈ ∈ NOUN cana-697 247 21 𝑋 𝑋 NOUN cana-697 247 22 converges converge VERB cana-697 247 23 to to ADP cana-697 247 24 fixed fix VERB cana-697 247 25 point point NOUN cana-697 247 26 of of ADP cana-697 247 27 𝑓 𝑓 DET cana-697 247 28 communications communication NOUN cana-697 247 29 on on ADP cana-697 247 30 applied apply VERB cana-697 247 31 nonlinear nonlinear ADJ cana-697 247 32 analysis analysis NOUN cana-697 247 33 issn issn NOUN cana-697 247 34 : : PUNCT cana-697 247 35 1074 1074 NUM cana-697 247 36 - - PUNCT cana-697 247 37 133x 133x NUM cana-697 247 38 vol vol NOUN cana-697 247 39 31 31 NUM cana-697 247 40 no no NOUN cana-697 247 41 . . PUNCT cana-697 248 1 2s 2s NUM cana-697 248 2 ( ( PUNCT cana-697 248 3 2024 2024 NUM cana-697 248 4 ) ) PUNCT cana-697 248 5 686 686 NUM cana-697 248 6 https://internationalpubls.com https://internationalpubls.com X cana-697 248 7 c. c. NOUN cana-697 248 8 if if SCONJ cana-697 248 9 𝑧 𝑧 PROPN cana-697 248 10 and and CCONJ cana-697 248 11 𝑀 𝑀 PROPN cana-697 248 12 are be AUX cana-697 248 13 fixed fix VERB cana-697 248 14 point point NOUN cana-697 248 15 of of ADP cana-697 248 16 𝑓 𝑓 PRON cana-697 248 17 where where SCONJ cana-697 248 18 𝑧 𝑧 DET cana-697 248 19 β‰  β‰  PROPN cana-697 248 20 𝑀 𝑀 PROPN cana-697 248 21 then then ADV cana-697 248 22 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 248 23 , , PUNCT cana-697 248 24 𝑀 𝑀 X cana-697 248 25 ) ) PUNCT cana-697 248 26 𝑝(𝑧 𝑝(𝑧 PROPN cana-697 248 27 , , PUNCT cana-697 248 28 𝑧 𝑧 NOUN cana-697 248 29 ) ) PUNCT cana-697 249 1 + + CCONJ cana-697 249 2 𝑝(𝑀 𝑝(𝑀 PROPN cana-697 249 3 , , PUNCT cana-697 249 4 𝑀 𝑀 ADP cana-697 249 5 ) ) PUNCT cana-697 249 6 + + CCONJ cana-697 249 7 1 1 NUM cana-697 249 8 β‰₯ β‰₯ NOUN cana-697 249 9 1 1 NUM cana-697 249 10 βˆ’ βˆ’ NOUN cana-697 249 11 π‘˜ π‘˜ DET cana-697 249 12 2 2 NUM cana-697 249 13 proof proof NOUN cana-697 249 14 : : PUNCT cana-697 249 15 the the DET cana-697 249 16 proof proof NOUN cana-697 249 17 of of ADP cana-697 249 18 this this DET cana-697 249 19 theorem theorem NOUN cana-697 249 20 is be AUX cana-697 249 21 following follow VERB cana-697 249 22 theorem theorem VERB cana-697 249 23 3.4 3.4 NUM cana-697 249 24 and and CCONJ cana-697 249 25 theorem theorem VERB cana-697 249 26 3.2 3.2 NUM cana-697 249 27 ’s ’s PART cana-697 249 28 proof proof NOUN cana-697 249 29 . . PUNCT cana-697 250 1 4 4 X cana-697 250 2 . . X cana-697 250 3 conclusions conclusion NOUN cana-697 250 4 in in ADP cana-697 250 5 this this DET cana-697 250 6 article article NOUN cana-697 250 7 , , PUNCT cana-697 250 8 we we PRON cana-697 250 9 have have AUX cana-697 250 10 extended extend VERB cana-697 250 11 the the DET cana-697 250 12 fixed fix VERB cana-697 250 13 point point NOUN cana-697 250 14 theorem theorem VERB cana-697 250 15 to to ADP cana-697 250 16 non non ADJ cana-697 250 17 - - ADJ cana-697 250 18 expansive expansive ADJ cana-697 250 19 mappings mapping NOUN cana-697 250 20 resulting result VERB cana-697 250 21 from from ADP cana-697 250 22 vetro vetro NOUN cana-697 250 23 in in ADP cana-697 250 24 the the DET cana-697 250 25 setting setting NOUN cana-697 250 26 of of ADP cana-697 250 27 partial partial ADJ cana-697 250 28 metric metric ADJ cana-697 250 29 spaces space NOUN cana-697 250 30 . . PUNCT cana-697 251 1 these these DET cana-697 251 2 results result NOUN cana-697 251 3 also also ADV cana-697 251 4 provide provide VERB cana-697 251 5 proof proof NOUN cana-697 251 6 of of ADP cana-697 251 7 one one NUM cana-697 251 8 of of ADP cana-697 251 9 aydi aydi NOUN cana-697 251 10 's 's PART cana-697 251 11 results result NOUN cana-697 251 12 . . PUNCT cana-697 252 1 furthermore furthermore ADV cana-697 252 2 , , PUNCT cana-697 252 3 we we PRON cana-697 252 4 generalize generalize VERB cana-697 252 5 our our PRON cana-697 252 6 previous previous ADJ cana-697 252 7 result result NOUN cana-697 252 8 by by ADP cana-697 252 9 using use VERB cana-697 252 10 the the DET cana-697 252 11 𝒩 𝒩 PROPN cana-697 252 12 function function NOUN cana-697 252 13 . . PUNCT cana-697 253 1 we we PRON cana-697 253 2 also also ADV cana-697 253 3 give give VERB cana-697 253 4 examples example NOUN cana-697 253 5 to to PART cana-697 253 6 illustrate illustrate VERB cana-697 253 7 our our PRON cana-697 253 8 results result NOUN cana-697 253 9 . . PUNCT cana-697 254 1 5 5 X cana-697 254 2 . . X cana-697 254 3 acknowledgement acknowledgement NOUN cana-697 254 4 this this DET cana-697 254 5 research research NOUN cana-697 254 6 was be AUX cana-697 254 7 funded fund VERB cana-697 254 8 by by ADP cana-697 254 9 the the DET cana-697 254 10 institute institute NOUN cana-697 254 11 for for ADP cana-697 254 12 research research NOUN cana-697 254 13 and and CCONJ cana-697 254 14 community community NOUN cana-697 254 15 service service NOUN cana-697 254 16 , , PUNCT cana-697 254 17 universitas universita NOUN cana-697 254 18 muhammadiyah muhammadiyah ADP cana-697 254 19 ponorogo ponorogo PROPN cana-697 254 20 , , PUNCT cana-697 254 21 indonesia indonesia PROPN cana-697 254 22 . . PUNCT cana-697 255 1 references reference NOUN cana-697 255 2 [ [ X cana-697 255 3 1 1 NUM cana-697 255 4 ] ] PUNCT cana-697 255 5 aydi aydi ADJ cana-697 255 6 , , PUNCT cana-697 255 7 h. h. PROPN cana-697 255 8 and and CCONJ cana-697 255 9 felhi felhi PROPN cana-697 255 10 , , PUNCT cana-697 255 11 a. a. NOUN cana-697 255 12 2017 2017 NUM cana-697 255 13 . . PUNCT cana-697 256 1 some some DET cana-697 256 2 fixed fix VERB cana-697 256 3 point point NOUN cana-697 256 4 results result NOUN cana-697 256 5 for for ADP cana-697 256 6 𝛼-nonexpansive 𝛼-nonexpansive ADJ cana-697 256 7 maps map NOUN cana-697 256 8 on on ADP cana-697 256 9 partial partial ADJ cana-697 256 10 metric metric ADJ cana-697 256 11 spaces space NOUN cana-697 256 12 . . PUNCT cana-697 257 1 j. j. PROPN cana-697 257 2 nonlinear nonlinear PROPN cana-697 257 3 sci sci PROPN cana-697 257 4 . . PUNCT cana-697 257 5 appl appl PROPN cana-697 257 6 . . PROPN cana-697 257 7 , , PUNCT cana-697 257 8 10 10 NUM cana-697 257 9 : : SYM cana-697 257 10 5509 5509 NUM cana-697 257 11 - - SYM cana-697 257 12 5527 5527 NUM cana-697 257 13 [ [ X cana-697 257 14 2 2 NUM cana-697 257 15 ] ] PUNCT cana-697 257 16 banach banach NOUN cana-697 257 17 , , PUNCT cana-697 257 18 s. s. PROPN cana-697 257 19 1992 1992 NUM cana-697 257 20 . . PUNCT cana-697 258 1 sur sur PROPN cana-697 258 2 les les PROPN cana-697 258 3 opΓ©rations opΓ©ration NOUN cana-697 258 4 dans dan NOUN cana-697 258 5 ensembles ensemble NOUN cana-697 258 6 abstraisn abstraisn PROPN cana-697 258 7 et et PROPN cana-697 258 8 leur leur X cana-697 258 9 application application PROPN cana-697 258 10 aux aux PROPN cana-697 258 11 Γ©quations Γ©quations PROPN cana-697 258 12 intΓ©grales intΓ©grale NOUN cana-697 258 13 , , PUNCT cana-697 258 14 fund fund PROPN cana-697 258 15 . . PUNCT cana-697 259 1 math math NOUN cana-697 259 2 . . PUNCT cana-697 260 1 3 3 NUM cana-697 260 2 : : SYM cana-697 260 3 157 157 NUM cana-697 260 4 - - SYM cana-697 260 5 165 165 NUM cana-697 260 6 . . PUNCT cana-697 260 7 http://eudml.org/doc/213289 http://eudml.org/doc/213289 NOUN cana-697 261 1 [ [ X cana-697 261 2 3 3 NUM cana-697 261 3 ] ] X cana-697 261 4 beg beg PROPN cana-697 261 5 , , PUNCT cana-697 261 6 i. i. NOUN cana-697 261 7 and and CCONJ cana-697 261 8 butt butt PROPN cana-697 261 9 , , PUNCT cana-697 261 10 a. a. NOUN cana-697 261 11 2013 2013 NUM cana-697 261 12 . . PUNCT cana-697 262 1 fixed fix VERB cana-697 262 2 point point NOUN cana-697 262 3 theorems theorem NOUN cana-697 262 4 for for ADP cana-697 262 5 set set ADJ cana-697 262 6 valued value VERB cana-697 262 7 mappings mapping NOUN cana-697 262 8 in in ADP cana-697 262 9 partially partially ADV cana-697 262 10 ordered order VERB cana-697 262 11 metric metric ADJ cana-697 262 12 spaces space NOUN cana-697 262 13 , , PUNCT cana-697 262 14 international international ADJ cana-697 262 15 journal journal NOUN cana-697 262 16 of of ADP cana-697 262 17 mathematical mathematical ADJ cana-697 262 18 sciences science NOUN cana-697 262 19 . . PUNCT cana-697 263 1 7 7 NUM cana-697 263 2 : : SYM cana-697 263 3 66 66 NUM cana-697 263 4 - - SYM cana-697 263 5 68 68 NUM cana-697 263 6 . . PUNCT cana-697 264 1 [ [ X cana-697 264 2 4 4 NUM cana-697 264 3 ] ] X cana-697 264 4 edelstein edelstein PROPN cana-697 264 5 , , PUNCT cana-697 264 6 m. m. NOUN cana-697 264 7 1964 1964 NUM cana-697 264 8 . . PUNCT cana-697 265 1 on on ADP cana-697 265 2 nonexpansive nonexpansive ADJ cana-697 265 3 mappings mapping NOUN cana-697 265 4 , , PUNCT cana-697 265 5 proceedings proceeding NOUN cana-697 265 6 of of ADP cana-697 265 7 the the DET cana-697 265 8 american american PROPN cana-697 265 9 mathematical mathematical ADJ cana-697 265 10 society society NOUN cana-697 265 11 15:689–695 15:689–695 NUM cana-697 265 12 [ [ X cana-697 265 13 5 5 NUM cana-697 265 14 ] ] X cana-697 265 15 eke eke ADJ cana-697 265 16 , , PUNCT cana-697 265 17 k. k. PROPN cana-697 265 18 s. s. PROPN cana-697 265 19 and and CCONJ cana-697 265 20 oghonyon oghonyon PROPN cana-697 265 21 , , PUNCT cana-697 265 22 j. j. PROPN cana-697 265 23 2018 2018 NUM cana-697 265 24 . . PUNCT cana-697 266 1 some some DET cana-697 266 2 fixed fix VERB cana-697 266 3 point point NOUN cana-697 266 4 theorems theorem NOUN cana-697 266 5 in in ADP cana-697 266 6 ordered order VERB cana-697 266 7 partial partial ADJ cana-697 266 8 metric metric ADJ cana-697 266 9 spaces space NOUN cana-697 266 10 with with ADP cana-697 266 11 applications application NOUN cana-697 266 12 , , PUNCT cana-697 266 13 cogent cogent NOUN cana-697 266 14 mathematics mathematic NOUN cana-697 266 15 and and CCONJ cana-697 266 16 statistics statistic NOUN cana-697 266 17 : : PUNCT cana-697 266 18 5 5 NUM cana-697 266 19 [ [ SYM cana-697 266 20 6 6 NUM cana-697 266 21 ] ] SYM cana-697 266 22 gΓΌlyaz gΓΌlyaz NOUN cana-697 266 23 , , PUNCT cana-697 266 24 s. s. PROPN cana-697 266 25 and and CCONJ cana-697 266 26 karapinar karapinar PROPN cana-697 266 27 , , PUNCT cana-697 266 28 e. e. PROPN cana-697 266 29 2013 2013 NUM cana-697 266 30 . . PUNCT cana-697 267 1 a a DET cana-697 267 2 coupled couple VERB cana-697 267 3 fixed fix VERB cana-697 267 4 point point NOUN cana-697 267 5 result result NOUN cana-697 267 6 in in ADP cana-697 267 7 partially partially ADV cana-697 267 8 ordered order VERB cana-697 267 9 partial partial ADJ cana-697 267 10 metric metric ADJ cana-697 267 11 space space NOUN cana-697 267 12 through through ADP cana-697 267 13 implicit implicit ADJ cana-697 267 14 function function NOUN cana-697 267 15 , , PUNCT cana-697 267 16 hacettepe hacettepe ADJ cana-697 267 17 journal journal NOUN cana-697 267 18 of of ADP cana-697 267 19 mathematics mathematic NOUN cana-697 267 20 and and CCONJ cana-697 267 21 statistics statistic NOUN cana-697 267 22 42 42 NUM cana-697 267 23 , , PUNCT cana-697 267 24 347–357 347–357 NUM cana-697 267 25 . . PUNCT cana-697 268 1 [ [ X cana-697 268 2 7 7 NUM cana-697 268 3 ] ] X cana-697 268 4 karapinar karapinar NOUN cana-697 268 5 , , PUNCT cana-697 268 6 e. e. PROPN cana-697 268 7 2011 2011 NUM cana-697 268 8 . . PUNCT cana-697 269 1 weak weak ADJ cana-697 269 2 Ο† Ο† NOUN cana-697 269 3 - - NOUN cana-697 269 4 contraction contraction NOUN cana-697 269 5 on on ADP cana-697 269 6 partial partial ADJ cana-697 269 7 metric metric ADJ cana-697 269 8 spaces space NOUN cana-697 269 9 and and CCONJ cana-697 269 10 existence existence NOUN cana-697 269 11 of of ADP cana-697 269 12 fixed fix VERB cana-697 269 13 points point NOUN cana-697 269 14 in in ADP cana-697 269 15 partially partially ADV cana-697 269 16 ordered order VERB cana-697 269 17 sets set NOUN cana-697 269 18 , , PUNCT cana-697 269 19 mathematica mathematica PROPN cana-697 269 20 aeterna aeterna PROPN cana-697 269 21 1 1 NUM cana-697 269 22 , , PUNCT cana-697 269 23 237–244 237–244 NUM cana-697 269 24 . . PUNCT cana-697 270 1 [ [ X cana-697 270 2 8 8 NUM cana-697 270 3 ] ] X cana-697 270 4 khamsi khamsi PROPN cana-697 270 5 , , PUNCT cana-697 270 6 m.a m.a PROPN cana-697 270 7 . . PROPN cana-697 270 8 and and CCONJ cana-697 270 9 reich reich PROPN cana-697 270 10 , , PUNCT cana-697 270 11 s. s. PROPN cana-697 270 12 1990 1990 NUM cana-697 270 13 . . PUNCT cana-697 271 1 nonexpansive nonexpansive ADJ cana-697 271 2 mappings mapping NOUN cana-697 271 3 and and CCONJ cana-697 271 4 semigroups semigroup NOUN cana-697 271 5 in in ADP cana-697 271 6 hyperconvex hyperconvex ADJ cana-697 271 7 spaces space NOUN cana-697 271 8 , , PUNCT cana-697 271 9 mathematica mathematica PROPN cana-697 271 10 japonica japonica PROPN cana-697 271 11 35 35 NUM cana-697 271 12 : : PUNCT cana-697 271 13 467–471 467–471 NUM cana-697 272 1 [ [ X cana-697 272 2 9 9 NUM cana-697 272 3 ] ] X cana-697 272 4 matthews matthew NOUN cana-697 272 5 , , PUNCT cana-697 272 6 s. s. PROPN cana-697 272 7 1992 1992 NUM cana-697 272 8 . . PUNCT cana-697 273 1 partial partial ADJ cana-697 273 2 metric metric ADJ cana-697 273 3 topology topology NOUN cana-697 273 4 . . PUNCT cana-697 274 1 university university NOUN cana-697 274 2 of of ADP cana-697 274 3 warwick warwick PROPN cana-697 274 4 . . PUNCT cana-697 275 1 [ [ X cana-697 275 2 10 10 NUM cana-697 275 3 ] ] X cana-697 275 4 matthews matthew NOUN cana-697 275 5 , , PUNCT cana-697 275 6 s. s. PROPN cana-697 275 7 1994 1994 NUM cana-697 275 8 . . PUNCT cana-697 276 1 partial partial ADJ cana-697 276 2 metric metric ADJ cana-697 276 3 topology topology NOUN cana-697 276 4 , , PUNCT cana-697 276 5 annals annal NOUN cana-697 276 6 of of ADP cana-697 276 7 the the DET cana-697 276 8 new new PROPN cana-697 276 9 york york PROPN cana-697 276 10 academy academy PROPN cana-697 276 11 of of ADP cana-697 276 12 sciences sciences PROPN cana-697 276 13 806 806 NUM cana-697 276 14 . . PUNCT cana-697 276 15 general general ADJ cana-697 276 16 topology topology NOUN cana-697 276 17 and and CCONJ cana-697 276 18 applications application NOUN cana-697 276 19 , , PUNCT cana-697 276 20 304–315 304–315 NUM cana-697 276 21 . . PUNCT cana-697 277 1 [ [ X cana-697 277 2 11 11 NUM cana-697 277 3 ] ] X cana-697 277 4 o’regan o’regan PROPN cana-697 277 5 , , PUNCT cana-697 277 6 d. d. PROPN cana-697 277 7 and and CCONJ cana-697 277 8 petruşel petruşel NOUN cana-697 277 9 , , PUNCT cana-697 277 10 a. a. NOUN cana-697 277 11 2008 2008 NUM cana-697 277 12 . . PUNCT cana-697 278 1 fixed fix VERB cana-697 278 2 point point NOUN cana-697 278 3 theorem theorem VERB cana-697 278 4 for for ADP cana-697 278 5 generalized generalized ADJ cana-697 278 6 contractions contraction NOUN cana-697 278 7 in in ADP cana-697 278 8 ordered order VERB cana-697 278 9 metric metric ADJ cana-697 278 10 spaces space NOUN cana-697 278 11 , , PUNCT cana-697 278 12 j. j. PROPN cana-697 278 13 math math PROPN cana-697 278 14 . . PUNCT cana-697 279 1 anal anal PROPN cana-697 279 2 . . PUNCT cana-697 279 3 appl appl PROPN cana-697 279 4 . . PUNCT cana-697 280 1 341 341 NUM cana-697 280 2 : : PUNCT cana-697 280 3 1241 1241 NUM cana-697 280 4 - - SYM cana-697 280 5 1252 1252 NUM cana-697 280 6 [ [ X cana-697 280 7 12 12 NUM cana-697 280 8 ] ] PUNCT cana-697 280 9 ran ran NOUN cana-697 280 10 , , PUNCT cana-697 280 11 a.c.m a.c.m NOUN cana-697 280 12 . . PUNCT cana-697 280 13 and and CCONJ cana-697 280 14 reurings reuring NOUN cana-697 280 15 , , PUNCT cana-697 280 16 m.c.b m.c.b NOUN cana-697 280 17 . . PUNCT cana-697 281 1 2004 2004 NUM cana-697 281 2 . . PUNCT cana-697 282 1 a a DET cana-697 282 2 fixed fix VERB cana-697 282 3 point point NOUN cana-697 282 4 theorem theorem VERB cana-697 282 5 in in ADP cana-697 282 6 partially partially ADV cana-697 282 7 ordered order VERB cana-697 282 8 sets set NOUN cana-697 282 9 and and CCONJ cana-697 282 10 some some DET cana-697 282 11 applications application NOUN cana-697 282 12 to to PART cana-697 282 13 matrix matrix VERB cana-697 282 14 equations equation NOUN cana-697 282 15 , , PUNCT cana-697 282 16 proc proc NOUN cana-697 282 17 . . PUNCT cana-697 283 1 amer amer PROPN cana-697 283 2 . . PUNCT cana-697 283 3 math math PROPN cana-697 283 4 . . PUNCT cana-697 284 1 soc soc PROPN cana-697 284 2 . . PUNCT cana-697 285 1 132 132 NUM cana-697 285 2 : : PUNCT cana-697 285 3 1435 1435 NUM cana-697 285 4 - - SYM cana-697 285 5 1443 1443 NUM cana-697 285 6 [ [ PUNCT cana-697 285 7 13 13 NUM cana-697 285 8 ] ] SYM cana-697 285 9 reich reich PROPN cana-697 285 10 , , PUNCT cana-697 285 11 s. s. PROPN cana-697 285 12 and and CCONJ cana-697 285 13 shafrir shafrir PROPN cana-697 285 14 , , PUNCT cana-697 285 15 i. i. PROPN cana-697 285 16 1987 1987 NUM cana-697 285 17 . . PUNCT cana-697 286 1 the the DET cana-697 286 2 asymptotic asymptotic ADJ cana-697 286 3 behavior behavior NOUN cana-697 286 4 of of ADP cana-697 286 5 firmly firmly ADV cana-697 286 6 nonexpansive nonexpansive ADJ cana-697 286 7 mappings mapping NOUN cana-697 286 8 , , PUNCT cana-697 286 9 proceedings proceeding NOUN cana-697 286 10 of of ADP cana-697 286 11 the the DET cana-697 286 12 american american PROPN cana-697 286 13 mathematical mathematical PROPN cana-697 286 14 society society NOUN cana-697 286 15 101:246–250 101:246–250 NUM cana-697 287 1 [ [ X cana-697 287 2 14 14 NUM cana-697 287 3 ] ] X cana-697 287 4 rhoades rhoade NOUN cana-697 287 5 , , PUNCT cana-697 287 6 b.e b.e PROPN cana-697 287 7 . . PROPN cana-697 287 8 1977 1977 NUM cana-697 287 9 . . PUNCT cana-697 288 1 a a DET cana-697 288 2 comparison comparison NOUN cana-697 288 3 of of ADP cana-697 288 4 various various ADJ cana-697 288 5 definitions definition NOUN cana-697 288 6 of of ADP cana-697 288 7 contractive contractive ADJ cana-697 288 8 mappings mapping NOUN cana-697 288 9 , , PUNCT cana-697 288 10 trans trans PROPN cana-697 288 11 . . PROPN cana-697 289 1 amer amer PROPN cana-697 289 2 . . PUNCT cana-697 289 3 math math PROPN cana-697 289 4 . . PUNCT cana-697 290 1 soc soc PROPN cana-697 290 2 . . PUNCT cana-697 291 1 226 226 NUM cana-697 291 2 : : PUNCT cana-697 291 3 257 257 NUM cana-697 291 4 - - SYM cana-697 291 5 290 290 NUM cana-697 291 6 [ [ SYM cana-697 291 7 15 15 NUM cana-697 291 8 ] ] X cana-697 291 9 suzuki suzuki PROPN cana-697 291 10 , , PUNCT cana-697 291 11 t. t. PROPN cana-697 291 12 2008 2008 NUM cana-697 291 13 . . PUNCT cana-697 292 1 fixed fix VERB cana-697 292 2 point point NOUN cana-697 292 3 theorems theorem NOUN cana-697 292 4 and and CCONJ cana-697 292 5 convergence convergence NOUN cana-697 292 6 theorems theorem NOUN cana-697 292 7 for for ADP cana-697 292 8 some some DET cana-697 292 9 generalized generalize VERB cana-697 292 10 nonexpansive nonexpansive ADJ cana-697 292 11 mappings mapping NOUN cana-697 292 12 , , PUNCT cana-697 292 13 journal journal NOUN cana-697 292 14 of of ADP cana-697 292 15 mathematical mathematical ADJ cana-697 292 16 analysis analysis NOUN cana-697 292 17 and and CCONJ cana-697 292 18 applications application NOUN cana-697 292 19 340 340 NUM cana-697 292 20 : : SYM cana-697 292 21 1088–1095 1088–1095 NUM cana-697 292 22 [ [ X cana-697 292 23 16 16 NUM cana-697 292 24 ] ] PUNCT cana-697 292 25 vetro vetro X cana-697 292 26 , , PUNCT cana-697 292 27 f. f. PROPN cana-697 292 28 2015 2015 NUM cana-697 292 29 . . PUNCT cana-697 293 1 fixed fix VERB cana-697 293 2 point point NOUN cana-697 293 3 result result NOUN cana-697 293 4 for for ADP cana-697 293 5 nonexpansive nonexpansive ADJ cana-697 293 6 mappings mapping NOUN cana-697 293 7 on on ADP cana-697 293 8 metric metric ADJ cana-697 293 9 spaces space NOUN cana-697 293 10 , , PUNCT cana-697 293 11 filomat filomat PROPN cana-697 293 12 29(9 29(9 NOUN cana-697 293 13 ): ): PUNCT cana-697 293 14 2011 2011 NUM cana-697 293 15 - - SYM cana-697 293 16 2020 2020 NUM