id sid tid token lemma pos ejpam-891 1 1 13_xxx_jonalgedda.dvi 13_xxx_jonalgedda.dvi NUM ejpam-891 1 2 european european PROPN ejpam-891 1 3 journal journal PROPN ejpam-891 1 4 of of ADP ejpam-891 1 5 pure pure ADJ ejpam-891 1 6 and and CCONJ ejpam-891 1 7 applied apply VERB ejpam-891 1 8 mathematics mathematic NOUN ejpam-891 1 9 vol vol NOUN ejpam-891 1 10 . . PUNCT ejpam-891 2 1 3 3 NUM ejpam-891 2 2 , , PUNCT ejpam-891 2 3 no no INTJ ejpam-891 2 4 . . NOUN ejpam-891 2 5 4 4 NUM ejpam-891 2 6 , , PUNCT ejpam-891 2 7 2010 2010 NUM ejpam-891 2 8 , , PUNCT ejpam-891 2 9 737 737 NUM ejpam-891 2 10 - - SYM ejpam-891 2 11 747 747 NUM ejpam-891 2 12 issn issn PROPN ejpam-891 2 13 1307 1307 NUM ejpam-891 2 14 - - SYM ejpam-891 2 15 5543 5543 NUM ejpam-891 2 16 – – PUNCT ejpam-891 2 17 www.ejpam.com www.ejpam.com X ejpam-891 2 18 existence existence NOUN ejpam-891 2 19 , , PUNCT ejpam-891 2 20 uniqueness uniqueness NOUN ejpam-891 2 21 of of ADP ejpam-891 2 22 solutions solution NOUN ejpam-891 2 23 for for ADP ejpam-891 2 24 set set ADJ ejpam-891 2 25 differential differential ADJ ejpam-891 2 26 equations equation NOUN ejpam-891 2 27 involving involve VERB ejpam-891 2 28 causal causal ADJ ejpam-891 2 29 operators operator NOUN ejpam-891 2 30 with with ADP ejpam-891 2 31 memory memory NOUN ejpam-891 2 32 j. j. PROPN ejpam-891 2 33 vasundhara vasundhara PROPN ejpam-891 2 34 devi devi PROPN ejpam-891 2 35 gvp-prof.v.lakshmikantham gvp-prof.v.lakshmikantham AUX ejpam-891 2 36 institute institute VERB ejpam-891 2 37 for for ADP ejpam-891 2 38 advanced advanced ADJ ejpam-891 2 39 studies study NOUN ejpam-891 2 40 , , PUNCT ejpam-891 2 41 department department NOUN ejpam-891 2 42 of of ADP ejpam-891 2 43 mathematics mathematic NOUN ejpam-891 2 44 , , PUNCT ejpam-891 2 45 gvp gvp PROPN ejpam-891 2 46 college college PROPN ejpam-891 2 47 of of ADP ejpam-891 2 48 engineering engineering PROPN ejpam-891 2 49 , , PUNCT ejpam-891 2 50 visakhapatnam visakhapatnam PROPN ejpam-891 2 51 , , PUNCT ejpam-891 2 52 ap ap PROPN ejpam-891 2 53 , , PUNCT ejpam-891 2 54 india india PROPN ejpam-891 2 55 . . PUNCT ejpam-891 3 1 abstract abstract PROPN ejpam-891 3 2 . . PUNCT ejpam-891 4 1 in in ADP ejpam-891 4 2 this this DET ejpam-891 4 3 paper paper NOUN ejpam-891 4 4 , , PUNCT ejpam-891 4 5 we we PRON ejpam-891 4 6 obtain obtain VERB ejpam-891 4 7 existence existence NOUN ejpam-891 4 8 and and CCONJ ejpam-891 4 9 uniqueness uniqueness NOUN ejpam-891 4 10 results result NOUN ejpam-891 4 11 of of ADP ejpam-891 4 12 ivp ivp NOUN ejpam-891 4 13 for for ADP ejpam-891 4 14 set set VERB ejpam-891 4 15 differential differential ADJ ejpam-891 4 16 equations equation NOUN ejpam-891 4 17 involving involve VERB ejpam-891 4 18 causal causal ADJ ejpam-891 4 19 operators operator NOUN ejpam-891 4 20 with with ADP ejpam-891 4 21 memory memory NOUN ejpam-891 4 22 . . PUNCT ejpam-891 5 1 this this DET ejpam-891 5 2 paper paper NOUN ejpam-891 5 3 is be AUX ejpam-891 5 4 2nd 2nd ADJ ejpam-891 5 5 in in ADP ejpam-891 5 6 sequel sequel NOUN ejpam-891 5 7 . . PUNCT ejpam-891 6 1 in in ADP ejpam-891 6 2 the the DET ejpam-891 6 3 first first ADJ ejpam-891 6 4 one one NUM ejpam-891 6 5 we we PRON ejpam-891 6 6 obtained obtain VERB ejpam-891 6 7 inequality inequality NOUN ejpam-891 6 8 results result NOUN ejpam-891 6 9 and and CCONJ ejpam-891 6 10 existence existence NOUN ejpam-891 6 11 results result VERB ejpam-891 6 12 for for ADP ejpam-891 6 13 set set VERB ejpam-891 6 14 causal causal NOUN ejpam-891 6 15 operators operator NOUN ejpam-891 6 16 involving involve VERB ejpam-891 6 17 memory memory NOUN ejpam-891 6 18 . . PUNCT ejpam-891 7 1 2000 2000 NUM ejpam-891 7 2 mathematics mathematic NOUN ejpam-891 7 3 subject subject NOUN ejpam-891 7 4 classifications classification NOUN ejpam-891 7 5 : : PUNCT ejpam-891 7 6 34a12 34a12 NUM ejpam-891 7 7 key key ADJ ejpam-891 7 8 words word NOUN ejpam-891 7 9 and and CCONJ ejpam-891 7 10 phrases phrase NOUN ejpam-891 7 11 : : PUNCT ejpam-891 7 12 set set VERB ejpam-891 7 13 differential differential ADJ ejpam-891 7 14 equations equation NOUN ejpam-891 7 15 , , PUNCT ejpam-891 7 16 existence existence NOUN ejpam-891 7 17 , , PUNCT ejpam-891 7 18 uniqueness uniqueness NOUN ejpam-891 7 19 and and CCONJ ejpam-891 7 20 comparison comparison NOUN ejpam-891 7 21 theorems theorem NOUN ejpam-891 7 22 and and CCONJ ejpam-891 7 23 extremal extremal ADJ ejpam-891 7 24 solutions solution NOUN ejpam-891 7 25 . . PUNCT ejpam-891 8 1 1 1 X ejpam-891 8 2 . . X ejpam-891 8 3 introduction introduction NOUN ejpam-891 8 4 owing owe VERB ejpam-891 8 5 to to ADP ejpam-891 8 6 the the DET ejpam-891 8 7 generalization generalization NOUN ejpam-891 8 8 encompassed encompass VERB ejpam-891 8 9 in in ADP ejpam-891 8 10 both both DET ejpam-891 8 11 set set VERB ejpam-891 8 12 differential differential ADJ ejpam-891 8 13 equations equation NOUN ejpam-891 8 14 and and CCONJ ejpam-891 8 15 the the DET ejpam-891 8 16 causal causal PROPN ejpam-891 8 17 operators operator NOUN ejpam-891 8 18 , , PUNCT ejpam-891 8 19 the the DET ejpam-891 8 20 study study NOUN ejpam-891 8 21 of of ADP ejpam-891 8 22 set set VERB ejpam-891 8 23 differential differential ADJ ejpam-891 8 24 equations equation NOUN ejpam-891 8 25 involving involve VERB ejpam-891 8 26 causal causal ADJ ejpam-891 8 27 operators operator NOUN ejpam-891 8 28 with with ADP ejpam-891 8 29 memory memory NOUN ejpam-891 8 30 has have AUX ejpam-891 8 31 been be AUX ejpam-891 8 32 initiated initiate VERB ejpam-891 8 33 in in ADP ejpam-891 8 34 [ [ X ejpam-891 8 35 6 6 NUM ejpam-891 8 36 ] ] PUNCT ejpam-891 8 37 . . PUNCT ejpam-891 9 1 the the DET ejpam-891 9 2 basic basic ADJ ejpam-891 9 3 differential differential ADJ ejpam-891 9 4 inequality inequality NOUN ejpam-891 9 5 and and CCONJ ejpam-891 9 6 existence existence NOUN ejpam-891 9 7 results result NOUN ejpam-891 9 8 have have AUX ejpam-891 9 9 been be AUX ejpam-891 9 10 developed develop VERB ejpam-891 9 11 for for ADP ejpam-891 9 12 both both PRON ejpam-891 9 13 set set VERB ejpam-891 9 14 causal causal ADJ ejpam-891 9 15 operators operator NOUN ejpam-891 9 16 with with ADP ejpam-891 9 17 memory memory NOUN ejpam-891 9 18 and and CCONJ ejpam-891 9 19 set set VERB ejpam-891 9 20 differential differential ADJ ejpam-891 9 21 equations equation NOUN ejpam-891 9 22 involving involve VERB ejpam-891 9 23 causal causal ADJ ejpam-891 9 24 operators operator NOUN ejpam-891 9 25 with with ADP ejpam-891 9 26 memory memory NOUN ejpam-891 9 27 . . PUNCT ejpam-891 10 1 the the DET ejpam-891 10 2 set set VERB ejpam-891 10 3 differential differential ADJ ejpam-891 10 4 equations equation NOUN ejpam-891 10 5 [ [ X ejpam-891 10 6 4 4 X ejpam-891 10 7 ] ] PUNCT ejpam-891 10 8 have have VERB ejpam-891 10 9 certain certain ADJ ejpam-891 10 10 advantages advantage NOUN ejpam-891 10 11 that that PRON ejpam-891 10 12 dictate dictate VERB ejpam-891 10 13 the the DET ejpam-891 10 14 continued continue VERB ejpam-891 10 15 interest interest NOUN ejpam-891 10 16 in in ADP ejpam-891 10 17 them they PRON ejpam-891 10 18 . . PUNCT ejpam-891 11 1 they they PRON ejpam-891 11 2 are be AUX ejpam-891 11 3 useful useful ADJ ejpam-891 11 4 to to PART ejpam-891 11 5 study study VERB ejpam-891 11 6 multivalued multivalue VERB ejpam-891 11 7 differential differential ADJ ejpam-891 11 8 inclusions inclusion NOUN ejpam-891 11 9 or or CCONJ ejpam-891 11 10 multivalued multivalue VERB ejpam-891 11 11 differential differential ADJ ejpam-891 11 12 equations equation NOUN ejpam-891 11 13 . . PUNCT ejpam-891 12 1 moreover moreover ADV ejpam-891 12 2 , , PUNCT ejpam-891 12 3 they they PRON ejpam-891 12 4 include include VERB ejpam-891 12 5 the the DET ejpam-891 12 6 theory theory NOUN ejpam-891 12 7 of of ADP ejpam-891 12 8 ordinary ordinary ADJ ejpam-891 12 9 differential differential ADJ ejpam-891 12 10 equations equation NOUN ejpam-891 12 11 and and CCONJ ejpam-891 12 12 ordinary ordinary ADJ ejpam-891 12 13 differential differential ADJ ejpam-891 12 14 systems system NOUN ejpam-891 12 15 as as ADP ejpam-891 12 16 special special ADJ ejpam-891 12 17 cases case NOUN ejpam-891 12 18 . . PUNCT ejpam-891 13 1 this this PRON ejpam-891 13 2 yields yield VERB ejpam-891 13 3 the the DET ejpam-891 13 4 theory theory NOUN ejpam-891 13 5 of of ADP ejpam-891 13 6 ordinary ordinary ADJ ejpam-891 13 7 differential differential ADJ ejpam-891 13 8 equations equation NOUN ejpam-891 13 9 and and CCONJ ejpam-891 13 10 that that PRON ejpam-891 13 11 of of ADP ejpam-891 13 12 systems system NOUN ejpam-891 13 13 in in ADP ejpam-891 13 14 a a DET ejpam-891 13 15 semilinear semilinear ADJ ejpam-891 13 16 metric metric ADJ ejpam-891 13 17 space space NOUN ejpam-891 13 18 instead instead ADV ejpam-891 13 19 of of ADP ejpam-891 13 20 a a DET ejpam-891 13 21 linear linear ADJ ejpam-891 13 22 metric metric ADJ ejpam-891 13 23 space space NOUN ejpam-891 13 24 which which PRON ejpam-891 13 25 is be AUX ejpam-891 13 26 an an DET ejpam-891 13 27 additional additional ADJ ejpam-891 13 28 benefit benefit NOUN ejpam-891 13 29 . . PUNCT ejpam-891 14 1 a a DET ejpam-891 14 2 causal causal ADJ ejpam-891 14 3 operator operator NOUN ejpam-891 14 4 [ [ X ejpam-891 14 5 1,3 1,3 X ejpam-891 14 6 ] ] PUNCT ejpam-891 14 7 or or CCONJ ejpam-891 14 8 a a DET ejpam-891 14 9 non non ADJ ejpam-891 14 10 anticipative anticipative NOUN ejpam-891 14 11 operator operator NOUN ejpam-891 14 12 is be AUX ejpam-891 14 13 a a DET ejpam-891 14 14 term term NOUN ejpam-891 14 15 adopted adopt VERB ejpam-891 14 16 from from ADP ejpam-891 14 17 engineering engineering NOUN ejpam-891 14 18 literature literature NOUN ejpam-891 14 19 . . PUNCT ejpam-891 15 1 the the DET ejpam-891 15 2 study study NOUN ejpam-891 15 3 of of ADP ejpam-891 15 4 causal causal NOUN ejpam-891 15 5 or or CCONJ ejpam-891 15 6 volterra volterra NOUN ejpam-891 15 7 operators operator NOUN ejpam-891 15 8 envelopes envelop VERB ejpam-891 15 9 the the DET ejpam-891 15 10 study study NOUN ejpam-891 15 11 of of ADP ejpam-891 15 12 several several ADJ ejpam-891 15 13 dynamic dynamic ADJ ejpam-891 15 14 systems system NOUN ejpam-891 15 15 such such ADJ ejpam-891 15 16 as as ADP ejpam-891 15 17 ordinary ordinary ADJ ejpam-891 15 18 differential differential ADJ ejpam-891 15 19 equations equation NOUN ejpam-891 15 20 [ [ X ejpam-891 15 21 2 2 NUM ejpam-891 15 22 ] ] PUNCT ejpam-891 15 23 , , PUNCT ejpam-891 15 24 delay delay NOUN ejpam-891 15 25 differential differential VERB ejpam-891 15 26 equations[2 equations[2 PROPN ejpam-891 15 27 ] ] PUNCT ejpam-891 15 28 , , PUNCT ejpam-891 15 29 integro integro PROPN ejpam-891 15 30 email email NOUN ejpam-891 15 31 address address NOUN ejpam-891 15 32 : : PUNCT ejpam-891 15 33 jvdevi jvdevi ADJ ejpam-891 15 34 � � PROPN ejpam-891 15 35 gmail gmail NOUN ejpam-891 15 36 . . PUNCT ejpam-891 16 1 om om PROPN ejpam-891 16 2 http://www.ejpam.com http://www.ejpam.com X ejpam-891 17 1 737 737 NUM ejpam-891 18 1 c c NOUN ejpam-891 18 2 © © NOUN ejpam-891 18 3 2010 2010 NUM ejpam-891 18 4 ejpam ejpam NOUN ejpam-891 18 5 all all DET ejpam-891 18 6 rights right NOUN ejpam-891 18 7 reserved reserve VERB ejpam-891 18 8 . . PUNCT ejpam-891 19 1 j. j. PROPN ejpam-891 19 2 devi devi PROPN ejpam-891 19 3 / / SYM ejpam-891 19 4 eur eur PROPN ejpam-891 19 5 . . PUNCT ejpam-891 20 1 j. j. PROPN ejpam-891 20 2 pure pure PROPN ejpam-891 20 3 appl appl PROPN ejpam-891 20 4 . . PROPN ejpam-891 20 5 math math PROPN ejpam-891 20 6 , , PUNCT ejpam-891 20 7 3 3 NUM ejpam-891 20 8 ( ( PUNCT ejpam-891 20 9 2010 2010 NUM ejpam-891 20 10 ) ) PUNCT ejpam-891 20 11 , , PUNCT ejpam-891 20 12 737 737 NUM ejpam-891 20 13 - - SYM ejpam-891 20 14 747 747 NUM ejpam-891 20 15 738 738 NUM ejpam-891 20 16 differential differential NOUN ejpam-891 20 17 equations[5 equations[5 NOUN ejpam-891 20 18 ] ] PUNCT ejpam-891 20 19 and and CCONJ ejpam-891 20 20 integral integral ADJ ejpam-891 20 21 equations equation NOUN ejpam-891 20 22 to to PART ejpam-891 20 23 name name VERB ejpam-891 20 24 a a DET ejpam-891 20 25 few few ADJ ejpam-891 20 26 . . PUNCT ejpam-891 21 1 in in ADP ejpam-891 21 2 this this DET ejpam-891 21 3 paper paper NOUN ejpam-891 21 4 we we PRON ejpam-891 21 5 continue continue VERB ejpam-891 21 6 to to PART ejpam-891 21 7 combine combine VERB ejpam-891 21 8 these these DET ejpam-891 21 9 two two NUM ejpam-891 21 10 areas area NOUN ejpam-891 21 11 and and CCONJ ejpam-891 21 12 study study VERB ejpam-891 21 13 set set VERB ejpam-891 21 14 differential differential ADJ ejpam-891 21 15 equations equation NOUN ejpam-891 21 16 involving involve VERB ejpam-891 21 17 causal causal ADJ ejpam-891 21 18 operators operator NOUN ejpam-891 21 19 with with ADP ejpam-891 21 20 memory memory NOUN ejpam-891 21 21 . . PUNCT ejpam-891 22 1 this this PRON ejpam-891 22 2 will will AUX ejpam-891 22 3 provide provide VERB ejpam-891 22 4 a a DET ejpam-891 22 5 unified unified ADJ ejpam-891 22 6 treatment treatment NOUN ejpam-891 22 7 of of ADP ejpam-891 22 8 the the DET ejpam-891 22 9 basic basic ADJ ejpam-891 22 10 theory theory NOUN ejpam-891 22 11 of of ADP ejpam-891 22 12 set set VERB ejpam-891 22 13 differential differential ADJ ejpam-891 22 14 equations equation NOUN ejpam-891 22 15 ( ( PUNCT ejpam-891 22 16 sde sde PROPN ejpam-891 22 17 ’s ’s PART ejpam-891 22 18 ) ) PUNCT ejpam-891 22 19 , , PUNCT ejpam-891 22 20 sde sde PROPN ejpam-891 22 21 ’s ’ VERB ejpam-891 22 22 with with ADP ejpam-891 22 23 delay delay NOUN ejpam-891 22 24 and and CCONJ ejpam-891 22 25 set set VERB ejpam-891 22 26 integro integro PROPN ejpam-891 22 27 differential differential ADJ ejpam-891 22 28 equations equation NOUN ejpam-891 22 29 which which PRON ejpam-891 22 30 in in ADP ejpam-891 22 31 turn turn NOUN ejpam-891 22 32 include include VERB ejpam-891 22 33 ordinary ordinary ADJ ejpam-891 22 34 dynamic dynamic ADJ ejpam-891 22 35 systems system NOUN ejpam-891 22 36 of of ADP ejpam-891 22 37 the the DET ejpam-891 22 38 corresponding corresponding ADJ ejpam-891 22 39 type type NOUN ejpam-891 22 40 . . PUNCT ejpam-891 23 1 2 2 X ejpam-891 23 2 . . X ejpam-891 23 3 preliminaries preliminary NOUN ejpam-891 23 4 we we PRON ejpam-891 23 5 begin begin VERB ejpam-891 23 6 with with ADP ejpam-891 23 7 the the DET ejpam-891 23 8 definitions definition NOUN ejpam-891 23 9 of of ADP ejpam-891 23 10 kc(r kc(r NOUN ejpam-891 23 11 n n CCONJ ejpam-891 23 12 ) ) PUNCT ejpam-891 23 13 , , PUNCT ejpam-891 23 14 the the DET ejpam-891 23 15 semilinear semilinear ADJ ejpam-891 23 16 space space NOUN ejpam-891 23 17 in in ADP ejpam-891 23 18 which which PRON ejpam-891 23 19 we we PRON ejpam-891 23 20 work work VERB ejpam-891 23 21 . . PUNCT ejpam-891 24 1 we we PRON ejpam-891 24 2 next next ADV ejpam-891 24 3 define define VERB ejpam-891 24 4 the the DET ejpam-891 24 5 hausdorff hausdorff NOUN ejpam-891 24 6 metric metric NOUN ejpam-891 24 7 , , PUNCT ejpam-891 24 8 the the DET ejpam-891 24 9 hukuhara hukuhara ADJ ejpam-891 24 10 difference difference NOUN ejpam-891 24 11 , , PUNCT ejpam-891 24 12 the the DET ejpam-891 24 13 hukuhara hukuhara ADV ejpam-891 24 14 derivative derivative NOUN ejpam-891 24 15 and and CCONJ ejpam-891 24 16 the the DET ejpam-891 24 17 hukuhara hukuhara ADV ejpam-891 24 18 integral integral ADJ ejpam-891 24 19 . . PUNCT ejpam-891 25 1 we we PRON ejpam-891 25 2 also also ADV ejpam-891 25 3 state state VERB ejpam-891 25 4 all all DET ejpam-891 25 5 the the DET ejpam-891 25 6 important important ADJ ejpam-891 25 7 properties property NOUN ejpam-891 25 8 that that PRON ejpam-891 25 9 are be AUX ejpam-891 25 10 useful useful ADJ ejpam-891 25 11 in in ADP ejpam-891 25 12 this this DET ejpam-891 25 13 paper paper NOUN ejpam-891 25 14 . . PUNCT ejpam-891 26 1 we we PRON ejpam-891 26 2 further far ADV ejpam-891 26 3 define define VERB ejpam-891 26 4 a a DET ejpam-891 26 5 partial partial ADJ ejpam-891 26 6 order order NOUN ejpam-891 26 7 in in ADP ejpam-891 26 8 kc(r kc(r NOUN ejpam-891 26 9 n n CCONJ ejpam-891 26 10 ) ) PUNCT ejpam-891 26 11 . . PUNCT ejpam-891 27 1 we we PRON ejpam-891 27 2 also also ADV ejpam-891 27 3 state state VERB ejpam-891 27 4 all all DET ejpam-891 27 5 the the DET ejpam-891 27 6 required require VERB ejpam-891 27 7 results result NOUN ejpam-891 27 8 developed develop VERB ejpam-891 27 9 in in ADP ejpam-891 27 10 [ [ X ejpam-891 27 11 6 6 NUM ejpam-891 27 12 ] ] PUNCT ejpam-891 27 13 that that PRON ejpam-891 27 14 will will AUX ejpam-891 27 15 be be AUX ejpam-891 27 16 used use VERB ejpam-891 27 17 in in ADP ejpam-891 27 18 this this DET ejpam-891 27 19 paper paper NOUN ejpam-891 27 20 . . PUNCT ejpam-891 28 1 let let VERB ejpam-891 28 2 kc(r kc(r NOUN ejpam-891 28 3 n n CCONJ ejpam-891 28 4 ) ) PUNCT ejpam-891 28 5 denote denote VERB ejpam-891 28 6 the the DET ejpam-891 28 7 collection collection NOUN ejpam-891 28 8 of of ADP ejpam-891 28 9 all all DET ejpam-891 28 10 nonempty nonempty ADJ ejpam-891 28 11 , , PUNCT ejpam-891 28 12 compact compact ADJ ejpam-891 28 13 and and CCONJ ejpam-891 28 14 convex convex ADJ ejpam-891 28 15 subsets subset NOUN ejpam-891 28 16 of of ADP ejpam-891 28 17 rn rn PROPN ejpam-891 28 18 . . PROPN ejpam-891 28 19 define define VERB ejpam-891 28 20 the the DET ejpam-891 28 21 hausdorff hausdorff NOUN ejpam-891 28 22 metric metric ADJ ejpam-891 28 23 by by ADP ejpam-891 28 24 d[a d[a PROPN ejpam-891 28 25 , , PUNCT ejpam-891 28 26 b b X ejpam-891 28 27 ] ] X ejpam-891 28 28 = = NOUN ejpam-891 28 29 max[sup max[sup NOUN ejpam-891 28 30 x∈b x∈b NOUN ejpam-891 28 31 d(x d(x PROPN ejpam-891 28 32 , , PUNCT ejpam-891 28 33 a a X ejpam-891 28 34 ) ) PUNCT ejpam-891 28 35 , , PUNCT ejpam-891 28 36 sup sup NOUN ejpam-891 28 37 y∈a y∈a NOUN ejpam-891 28 38 d(y d(y PROPN ejpam-891 28 39 , , PUNCT ejpam-891 28 40 b b NOUN ejpam-891 28 41 ) ) PUNCT ejpam-891 28 42 ] ] PUNCT ejpam-891 28 43 , , PUNCT ejpam-891 28 44 ( ( PUNCT ejpam-891 28 45 1 1 X ejpam-891 28 46 ) ) PUNCT ejpam-891 29 1 where where SCONJ ejpam-891 29 2 d(x d(x NOUN ejpam-891 29 3 , , PUNCT ejpam-891 29 4 a a X ejpam-891 29 5 ) ) PUNCT ejpam-891 29 6 = = SYM ejpam-891 29 7 inf[d(x inf[d(x ADP ejpam-891 29 8 , , PUNCT ejpam-891 29 9 y y PROPN ejpam-891 29 10 ) ) PUNCT ejpam-891 29 11 : : PUNCT ejpam-891 29 12 y y PROPN ejpam-891 29 13 ∈ ∈ PROPN ejpam-891 29 14 a a X ejpam-891 29 15 ] ] X ejpam-891 29 16 , , PUNCT ejpam-891 29 17 a a DET ejpam-891 29 18 , , PUNCT ejpam-891 29 19 b b PROPN ejpam-891 29 20 are be AUX ejpam-891 29 21 bounded bound VERB ejpam-891 29 22 sets set NOUN ejpam-891 29 23 in in ADP ejpam-891 29 24 rn rn PROPN ejpam-891 29 25 . . PUNCT ejpam-891 30 1 we we PRON ejpam-891 30 2 note note VERB ejpam-891 30 3 that that SCONJ ejpam-891 30 4 kc(r kc(r NOUN ejpam-891 30 5 n n CCONJ ejpam-891 30 6 ) ) PUNCT ejpam-891 30 7 with with ADP ejpam-891 30 8 this this DET ejpam-891 30 9 metric metric NOUN ejpam-891 30 10 is be AUX ejpam-891 30 11 a a DET ejpam-891 30 12 complete complete ADJ ejpam-891 30 13 metric metric ADJ ejpam-891 30 14 space space NOUN ejpam-891 30 15 . . PUNCT ejpam-891 31 1 it it PRON ejpam-891 31 2 is be AUX ejpam-891 31 3 known know VERB ejpam-891 31 4 that that SCONJ ejpam-891 31 5 if if SCONJ ejpam-891 31 6 the the DET ejpam-891 31 7 space space NOUN ejpam-891 31 8 kc(r kc(r X ejpam-891 31 9 n n CCONJ ejpam-891 31 10 ) ) PUNCT ejpam-891 31 11 is be AUX ejpam-891 31 12 equipped equip VERB ejpam-891 31 13 with with ADP ejpam-891 31 14 the the DET ejpam-891 31 15 natural natural ADJ ejpam-891 31 16 algebraic algebraic ADJ ejpam-891 31 17 operations operation NOUN ejpam-891 31 18 of of ADP ejpam-891 31 19 addition addition NOUN ejpam-891 31 20 and and CCONJ ejpam-891 31 21 non non ADJ ejpam-891 31 22 - - ADJ ejpam-891 31 23 negative negative ADJ ejpam-891 31 24 scalar scalar ADJ ejpam-891 31 25 multiplication multiplication NOUN ejpam-891 31 26 , , PUNCT ejpam-891 31 27 then then ADV ejpam-891 31 28 kc(r kc(r NOUN ejpam-891 31 29 n n CCONJ ejpam-891 31 30 ) ) PUNCT ejpam-891 31 31 becomes become VERB ejpam-891 31 32 a a DET ejpam-891 31 33 semilinear semilinear ADJ ejpam-891 31 34 metric metric ADJ ejpam-891 31 35 space space NOUN ejpam-891 31 36 which which PRON ejpam-891 31 37 can can AUX ejpam-891 31 38 be be AUX ejpam-891 31 39 embedded embed VERB ejpam-891 31 40 as as ADP ejpam-891 31 41 a a DET ejpam-891 31 42 complete complete ADJ ejpam-891 31 43 cone cone NOUN ejpam-891 31 44 into into ADP ejpam-891 31 45 a a DET ejpam-891 31 46 corresponding corresponding ADJ ejpam-891 31 47 banach banach NOUN ejpam-891 31 48 space space NOUN ejpam-891 31 49 . . PUNCT ejpam-891 32 1 the the DET ejpam-891 32 2 hausdorff hausdorff PROPN ejpam-891 32 3 metric metric NOUN ejpam-891 32 4 ( ( PUNCT ejpam-891 32 5 1 1 NUM ejpam-891 32 6 ) ) PUNCT ejpam-891 32 7 satisfies satisfy VERB ejpam-891 32 8 the the DET ejpam-891 32 9 following follow VERB ejpam-891 32 10 properties property NOUN ejpam-891 32 11 : : PUNCT ejpam-891 32 12 d[a+ d[a+ PROPN ejpam-891 32 13 c c NOUN ejpam-891 32 14 , , PUNCT ejpam-891 32 15 b+ b+ X ejpam-891 32 16 c c X ejpam-891 32 17 ] ] X ejpam-891 32 18 = = SYM ejpam-891 32 19 d[a d[a ADJ ejpam-891 32 20 , , PUNCT ejpam-891 32 21 b b NOUN ejpam-891 32 22 ] ] PUNCT ejpam-891 32 23 and and CCONJ ejpam-891 32 24 d[a d[a ADJ ejpam-891 32 25 , , PUNCT ejpam-891 32 26 b b X ejpam-891 32 27 ] ] X ejpam-891 32 28 = = SYM ejpam-891 32 29 d[b d[b PROPN ejpam-891 32 30 , , PUNCT ejpam-891 32 31 a a PRON ejpam-891 32 32 ] ] X ejpam-891 32 33 , , PUNCT ejpam-891 32 34 ( ( PUNCT ejpam-891 32 35 2 2 X ejpam-891 32 36 ) ) PUNCT ejpam-891 32 37 d[λa d[λa NOUN ejpam-891 32 38 , , PUNCT ejpam-891 32 39 λb λb ADP ejpam-891 32 40 ] ] X ejpam-891 33 1 = = SYM ejpam-891 33 2 λd[a λd[a PROPN ejpam-891 33 3 , , PUNCT ejpam-891 33 4 b b NOUN ejpam-891 33 5 ] ] X ejpam-891 33 6 , , PUNCT ejpam-891 33 7 ( ( PUNCT ejpam-891 33 8 3 3 X ejpam-891 33 9 ) ) PUNCT ejpam-891 33 10 d[a d[a ADJ ejpam-891 33 11 , , PUNCT ejpam-891 33 12 b b X ejpam-891 33 13 ] ] PUNCT ejpam-891 33 14 ≤ ≤ NUM ejpam-891 33 15 d[a d[a PROPN ejpam-891 33 16 , , PUNCT ejpam-891 33 17 c c X ejpam-891 33 18 ] ] X ejpam-891 34 1 + + CCONJ ejpam-891 34 2 d[c d[c ADJ ejpam-891 34 3 , , PUNCT ejpam-891 34 4 b b NOUN ejpam-891 34 5 ] ] X ejpam-891 34 6 , , PUNCT ejpam-891 34 7 ( ( PUNCT ejpam-891 34 8 4 4 X ejpam-891 34 9 ) ) PUNCT ejpam-891 34 10 for for ADP ejpam-891 34 11 all all DET ejpam-891 34 12 a a DET ejpam-891 34 13 , , PUNCT ejpam-891 34 14 b b NOUN ejpam-891 34 15 , , PUNCT ejpam-891 34 16 c c PROPN ejpam-891 34 17 ∈ ∈ PROPN ejpam-891 34 18 kc(r kc(r X ejpam-891 34 19 n n CCONJ ejpam-891 34 20 ) ) PUNCT ejpam-891 34 21 and and CCONJ ejpam-891 34 22 λ λ PROPN ejpam-891 34 23 ∈ ∈ PROPN ejpam-891 34 24 r+ r+ X ejpam-891 34 25 . . PUNCT ejpam-891 35 1 let let VERB ejpam-891 35 2 a a DET ejpam-891 35 3 , , PUNCT ejpam-891 35 4 b b PROPN ejpam-891 35 5 ∈ ∈ PROPN ejpam-891 35 6 kc(r kc(r NOUN ejpam-891 35 7 n n CCONJ ejpam-891 35 8 ) ) PUNCT ejpam-891 35 9 . . PUNCT ejpam-891 36 1 the the DET ejpam-891 36 2 set set NOUN ejpam-891 36 3 c c PROPN ejpam-891 36 4 ∈ ∈ PROPN ejpam-891 36 5 kc(r kc(r X ejpam-891 36 6 n n CCONJ ejpam-891 36 7 ) ) PUNCT ejpam-891 36 8 satisfying satisfy VERB ejpam-891 36 9 a a DET ejpam-891 36 10 = = SYM ejpam-891 36 11 b b PROPN ejpam-891 36 12 + + CCONJ ejpam-891 36 13 c c NOUN ejpam-891 36 14 is be AUX ejpam-891 36 15 known know VERB ejpam-891 36 16 as as ADP ejpam-891 36 17 the the DET ejpam-891 36 18 hukuhara hukuhara ADJ ejpam-891 36 19 difference difference NOUN ejpam-891 36 20 of of ADP ejpam-891 36 21 the the DET ejpam-891 36 22 sets set NOUN ejpam-891 36 23 a a PRON ejpam-891 36 24 and and CCONJ ejpam-891 36 25 b b NOUN ejpam-891 36 26 and and CCONJ ejpam-891 36 27 is be AUX ejpam-891 36 28 denoted denote VERB ejpam-891 36 29 by by ADP ejpam-891 36 30 the the DET ejpam-891 36 31 symbol symbol NOUN ejpam-891 36 32 a− a− PROPN ejpam-891 36 33 b. b. PROPN ejpam-891 37 1 we we PRON ejpam-891 37 2 say say VERB ejpam-891 37 3 that that SCONJ ejpam-891 37 4 the the DET ejpam-891 37 5 mapping mapping NOUN ejpam-891 37 6 f f X ejpam-891 37 7 : : PUNCT ejpam-891 37 8 i i PROPN ejpam-891 37 9 → → SYM ejpam-891 37 10 kc(r kc(r X ejpam-891 37 11 n n CCONJ ejpam-891 37 12 ) ) PUNCT ejpam-891 37 13 has have VERB ejpam-891 37 14 a a DET ejpam-891 37 15 hukuhara hukuhara ADJ ejpam-891 37 16 derivative derivative ADJ ejpam-891 37 17 dh dh NOUN ejpam-891 37 18 f(t0 f(t0 NOUN ejpam-891 37 19 ) ) PUNCT ejpam-891 37 20 at at ADP ejpam-891 37 21 a a DET ejpam-891 37 22 point point NOUN ejpam-891 37 23 t0 t0 X ejpam-891 37 24 ∈ ∈ PROPN ejpam-891 38 1 i i PRON ejpam-891 38 2 , , PUNCT ejpam-891 38 3 if if SCONJ ejpam-891 38 4 lim lim PROPN ejpam-891 38 5 h→0 h→0 NOUN ejpam-891 38 6 + + NUM ejpam-891 38 7 f(t0 f(t0 NOUN ejpam-891 38 8 + + CCONJ ejpam-891 38 9 h)− h)− PROPN ejpam-891 38 10 f(t0 f(t0 NOUN ejpam-891 38 11 ) ) PUNCT ejpam-891 38 12 h h NOUN ejpam-891 38 13 and and CCONJ ejpam-891 38 14 lim lim PROPN ejpam-891 38 15 h→0 h→0 PROPN ejpam-891 38 16 + + CCONJ ejpam-891 38 17 f(t0)− f(t0)− PROPN ejpam-891 38 18 f(t0 f(t0 NOUN ejpam-891 38 19 − − PROPN ejpam-891 38 20 h h NOUN ejpam-891 38 21 ) ) PUNCT ejpam-891 38 22 h h NOUN ejpam-891 38 23 exist exist VERB ejpam-891 38 24 in in ADP ejpam-891 38 25 the the DET ejpam-891 38 26 topology topology NOUN ejpam-891 38 27 of of ADP ejpam-891 38 28 kc(r kc(r NOUN ejpam-891 38 29 n n CCONJ ejpam-891 38 30 ) ) PUNCT ejpam-891 38 31 and and CCONJ ejpam-891 38 32 are be AUX ejpam-891 38 33 equal equal ADJ ejpam-891 38 34 to to ADP ejpam-891 38 35 dh dh PROPN ejpam-891 38 36 f(t0 f(t0 NOUN ejpam-891 38 37 ) ) PUNCT ejpam-891 38 38 . . PUNCT ejpam-891 39 1 here here ADV ejpam-891 39 2 i i PRON ejpam-891 39 3 is be AUX ejpam-891 39 4 any any DET ejpam-891 39 5 interval interval NOUN ejpam-891 39 6 in in ADP ejpam-891 39 7 r. r. PROPN ejpam-891 39 8 with with ADP ejpam-891 39 9 these these DET ejpam-891 39 10 preliminaries preliminary NOUN ejpam-891 39 11 , , PUNCT ejpam-891 39 12 we we PRON ejpam-891 39 13 consider consider VERB ejpam-891 39 14 the the DET ejpam-891 39 15 set set ADJ ejpam-891 39 16 differential differential ADJ ejpam-891 39 17 equation equation NOUN ejpam-891 39 18 dh dh PROPN ejpam-891 39 19 u u PROPN ejpam-891 39 20 = = PROPN ejpam-891 39 21 f(t f(t NOUN ejpam-891 39 22 , , PUNCT ejpam-891 39 23 u u NOUN ejpam-891 39 24 ) ) PUNCT ejpam-891 39 25 , , PUNCT ejpam-891 39 26 u(t0 u(t0 NOUN ejpam-891 39 27 ) ) PUNCT ejpam-891 39 28 = = PUNCT ejpam-891 39 29 u0 u0 PROPN ejpam-891 39 30 ∈ ∈ PROPN ejpam-891 39 31 kc(r kc(r NOUN ejpam-891 39 32 n n CCONJ ejpam-891 39 33 ) ) PUNCT ejpam-891 39 34 , , PUNCT ejpam-891 39 35 t0 t0 PROPN ejpam-891 39 36 ≥ ≥ NUM ejpam-891 39 37 0 0 NUM ejpam-891 39 38 , , PUNCT ejpam-891 39 39 ( ( PUNCT ejpam-891 39 40 5 5 X ejpam-891 39 41 ) ) PUNCT ejpam-891 39 42 j. j. PROPN ejpam-891 39 43 devi devi PROPN ejpam-891 39 44 / / SYM ejpam-891 39 45 eur eur PROPN ejpam-891 39 46 . . PUNCT ejpam-891 40 1 j. j. PROPN ejpam-891 40 2 pure pure PROPN ejpam-891 40 3 appl appl PROPN ejpam-891 40 4 . . PROPN ejpam-891 40 5 math math PROPN ejpam-891 40 6 , , PUNCT ejpam-891 40 7 3 3 NUM ejpam-891 40 8 ( ( PUNCT ejpam-891 40 9 2010 2010 NUM ejpam-891 40 10 ) ) PUNCT ejpam-891 40 11 , , PUNCT ejpam-891 40 12 737 737 NUM ejpam-891 40 13 - - SYM ejpam-891 40 14 747 747 NUM ejpam-891 40 15 739 739 NUM ejpam-891 40 16 where where SCONJ ejpam-891 40 17 f f PROPN ejpam-891 40 18 ∈ ∈ PROPN ejpam-891 40 19 c[r+ c[r+ NOUN ejpam-891 40 20 × × NOUN ejpam-891 40 21 kc(r kc(r X ejpam-891 40 22 n n CCONJ ejpam-891 40 23 ) ) PUNCT ejpam-891 40 24 , , PUNCT ejpam-891 40 25 kc(r kc(r NOUN ejpam-891 40 26 n n CCONJ ejpam-891 40 27 ) ) PUNCT ejpam-891 40 28 ] ] PUNCT ejpam-891 40 29 . . PUNCT ejpam-891 41 1 the the DET ejpam-891 41 2 mapping mapping NOUN ejpam-891 41 3 u u NOUN ejpam-891 41 4 ∈ ∈ PROPN ejpam-891 41 5 c1[j c1[j NOUN ejpam-891 41 6 , , PUNCT ejpam-891 41 7 kc(r kc(r NOUN ejpam-891 41 8 n n CCONJ ejpam-891 41 9 ) ) PUNCT ejpam-891 41 10 ] ] PUNCT ejpam-891 41 11 , , PUNCT ejpam-891 41 12 j j X ejpam-891 41 13 = = PUNCT ejpam-891 42 1 [ [ X ejpam-891 42 2 t0 t0 PROPN ejpam-891 42 3 , , PUNCT ejpam-891 42 4 t0 t0 PROPN ejpam-891 42 5 + + CCONJ ejpam-891 42 6 a a PRON ejpam-891 42 7 ] ] X ejpam-891 42 8 is be AUX ejpam-891 42 9 said say VERB ejpam-891 42 10 to to PART ejpam-891 42 11 be be AUX ejpam-891 42 12 a a DET ejpam-891 42 13 solution solution NOUN ejpam-891 42 14 of of ADP ejpam-891 42 15 ( ( PUNCT ejpam-891 42 16 5 5 NUM ejpam-891 42 17 ) ) PUNCT ejpam-891 42 18 on on ADP ejpam-891 42 19 j j PROPN ejpam-891 42 20 if if SCONJ ejpam-891 42 21 it it PRON ejpam-891 42 22 satisfies satisfy VERB ejpam-891 42 23 ( ( PUNCT ejpam-891 42 24 5 5 NUM ejpam-891 42 25 ) ) PUNCT ejpam-891 42 26 on on ADP ejpam-891 42 27 j j PROPN ejpam-891 42 28 . . PUNCT ejpam-891 43 1 since since SCONJ ejpam-891 43 2 u(t u(t NOUN ejpam-891 43 3 ) ) PUNCT ejpam-891 43 4 is be AUX ejpam-891 43 5 continuously continuously ADV ejpam-891 43 6 differentiable differentiable ADJ ejpam-891 43 7 , , PUNCT ejpam-891 43 8 we we PRON ejpam-891 43 9 have have VERB ejpam-891 43 10 u(t u(t NOUN ejpam-891 43 11 ) ) PUNCT ejpam-891 43 12 = = PUNCT ejpam-891 44 1 u0 u0 PROPN ejpam-891 45 1 + + X ejpam-891 45 2 ∫ ∫ PROPN ejpam-891 45 3 t t PROPN ejpam-891 45 4 t0 t0 PROPN ejpam-891 45 5 dh dh PROPN ejpam-891 45 6 u(s)ds u(s)ds PROPN ejpam-891 45 7 , , PUNCT ejpam-891 45 8 t t PROPN ejpam-891 45 9 ∈ ∈ PROPN ejpam-891 45 10 j j PROPN ejpam-891 45 11 . . PUNCT ejpam-891 46 1 ( ( PUNCT ejpam-891 46 2 6 6 NUM ejpam-891 46 3 ) ) PUNCT ejpam-891 46 4 hence hence ADV ejpam-891 46 5 , , PUNCT ejpam-891 46 6 we we PRON ejpam-891 46 7 can can AUX ejpam-891 46 8 associate associate VERB ejpam-891 46 9 with with ADP ejpam-891 46 10 the the DET ejpam-891 46 11 ivp ivp NOUN ejpam-891 46 12 ( ( PUNCT ejpam-891 46 13 5 5 NUM ejpam-891 46 14 ) ) PUNCT ejpam-891 46 15 the the DET ejpam-891 46 16 hukuhara hukuhara ADV ejpam-891 46 17 integral integral ADJ ejpam-891 46 18 u(t u(t NOUN ejpam-891 46 19 ) ) PUNCT ejpam-891 46 20 = = PUNCT ejpam-891 47 1 u0 u0 PROPN ejpam-891 48 1 + + X ejpam-891 48 2 ∫ ∫ PROPN ejpam-891 48 3 t t PROPN ejpam-891 48 4 t0 t0 PROPN ejpam-891 48 5 f(s f(s PROPN ejpam-891 48 6 , , PUNCT ejpam-891 48 7 u(s))ds u(s))ds PROPN ejpam-891 48 8 , , PUNCT ejpam-891 48 9 t t PROPN ejpam-891 48 10 ∈ ∈ PROPN ejpam-891 48 11 j j PROPN ejpam-891 48 12 . . PUNCT ejpam-891 49 1 ( ( PUNCT ejpam-891 49 2 7 7 X ejpam-891 49 3 ) ) PUNCT ejpam-891 49 4 where where SCONJ ejpam-891 49 5 the the DET ejpam-891 49 6 integral integral ADJ ejpam-891 49 7 is be AUX ejpam-891 49 8 the the DET ejpam-891 49 9 hukuhara hukuhara ADV ejpam-891 49 10 integral integral ADJ ejpam-891 49 11 which which PRON ejpam-891 49 12 is be AUX ejpam-891 49 13 defined define VERB ejpam-891 49 14 as as ADP ejpam-891 49 15 , , PUNCT ejpam-891 49 16 intf(s)ds intf(s)d NOUN ejpam-891 49 17 = = SYM ejpam-891 49 18 { { PUNCT ejpam-891 49 19 ∫ ∫ PROPN ejpam-891 49 20 f f X ejpam-891 49 21 ( ( PUNCT ejpam-891 49 22 s)ds s)ds PROPN ejpam-891 49 23 : : PUNCT ejpam-891 49 24 f f PROPN ejpam-891 49 25 is be AUX ejpam-891 49 26 any any DET ejpam-891 49 27 continuous continuous ADJ ejpam-891 49 28 selector selector NOUN ejpam-891 49 29 of of ADP ejpam-891 49 30 f f PROPN ejpam-891 49 31 } } PUNCT ejpam-891 49 32 observe observe VERB ejpam-891 49 33 that that SCONJ ejpam-891 49 34 u(t u(t NOUN ejpam-891 49 35 ) ) PUNCT ejpam-891 49 36 is be AUX ejpam-891 49 37 a a DET ejpam-891 49 38 solution solution NOUN ejpam-891 49 39 of of ADP ejpam-891 49 40 ( ( PUNCT ejpam-891 49 41 5 5 NUM ejpam-891 49 42 ) ) PUNCT ejpam-891 49 43 on on ADP ejpam-891 49 44 j j PROPN ejpam-891 49 45 iff iff PROPN ejpam-891 49 46 it it PRON ejpam-891 49 47 satisfies satisfy VERB ejpam-891 49 48 ( ( PUNCT ejpam-891 49 49 7 7 NUM ejpam-891 49 50 ) ) PUNCT ejpam-891 49 51 on on ADP ejpam-891 49 52 j j PROPN ejpam-891 49 53 . . PUNCT ejpam-891 50 1 we we PRON ejpam-891 50 2 now now ADV ejpam-891 50 3 define define VERB ejpam-891 50 4 a a DET ejpam-891 50 5 partial partial ADJ ejpam-891 50 6 order order NOUN ejpam-891 50 7 in in ADP ejpam-891 50 8 the the DET ejpam-891 50 9 metric metric ADJ ejpam-891 50 10 space space NOUN ejpam-891 50 11 kc(r kc(r NOUN ejpam-891 50 12 n n CCONJ ejpam-891 50 13 ) ) PUNCT ejpam-891 50 14 . . PUNCT ejpam-891 51 1 to to PART ejpam-891 51 2 do do VERB ejpam-891 51 3 so so ADV ejpam-891 51 4 , , PUNCT ejpam-891 51 5 we we PRON ejpam-891 51 6 need need VERB ejpam-891 51 7 the the DET ejpam-891 51 8 definition definition NOUN ejpam-891 51 9 of of ADP ejpam-891 51 10 a a DET ejpam-891 51 11 cone cone NOUN ejpam-891 51 12 in in ADP ejpam-891 51 13 kc(r kc(r NOUN ejpam-891 51 14 n n CCONJ ejpam-891 51 15 ) ) PUNCT ejpam-891 51 16 , , PUNCT ejpam-891 51 17 which which PRON ejpam-891 51 18 is be AUX ejpam-891 51 19 given give VERB ejpam-891 51 20 below below ADV ejpam-891 51 21 . . PUNCT ejpam-891 52 1 let let AUX ejpam-891 52 2 k(k0 k(k0 NOUN ejpam-891 52 3 ) ) PUNCT ejpam-891 52 4 be be AUX ejpam-891 52 5 the the DET ejpam-891 52 6 subfamily subfamily NOUN ejpam-891 52 7 of of ADP ejpam-891 52 8 kc(r kc(r NOUN ejpam-891 52 9 n n CCONJ ejpam-891 52 10 ) ) PUNCT ejpam-891 52 11 consisting consist VERB ejpam-891 52 12 of of ADP ejpam-891 52 13 sets set NOUN ejpam-891 52 14 u u PROPN ejpam-891 52 15 ∈ ∈ PROPN ejpam-891 52 16 kc(r kc(r X ejpam-891 52 17 n n CCONJ ejpam-891 52 18 ) ) PUNCT ejpam-891 52 19 such such ADJ ejpam-891 52 20 that that SCONJ ejpam-891 52 21 any any DET ejpam-891 52 22 u u PROPN ejpam-891 52 23 ∈ ∈ PROPN ejpam-891 52 24 u u NOUN ejpam-891 52 25 is be AUX ejpam-891 52 26 a a DET ejpam-891 52 27 nonnegative nonnegative ADJ ejpam-891 52 28 ( ( PUNCT ejpam-891 52 29 positive positive ADJ ejpam-891 52 30 ) ) PUNCT ejpam-891 52 31 vector vector NOUN ejpam-891 52 32 of of ADP ejpam-891 52 33 n n CCONJ ejpam-891 52 34 - - PUNCT ejpam-891 52 35 components component NOUN ejpam-891 52 36 satisfying satisfy VERB ejpam-891 52 37 ui ui PROPN ejpam-891 52 38 ≥ ≥ NUM ejpam-891 52 39 0(ui 0(ui X ejpam-891 52 40 > > X ejpam-891 52 41 0 0 NUM ejpam-891 52 42 ) ) PUNCT ejpam-891 52 43 for for ADP ejpam-891 52 44 i i PRON ejpam-891 52 45 = = NOUN ejpam-891 52 46 1,2,3 1,2,3 NUM ejpam-891 52 47 , , PUNCT ejpam-891 52 48 . . PUNCT ejpam-891 52 49 . . PUNCT ejpam-891 53 1 . . PUNCT ejpam-891 54 1 , , PUNCT ejpam-891 54 2 n. n. PROPN ejpam-891 54 3 then then ADV ejpam-891 54 4 k k PROPN ejpam-891 54 5 is be AUX ejpam-891 54 6 a a DET ejpam-891 54 7 cone cone NOUN ejpam-891 54 8 in in ADP ejpam-891 54 9 kc(r kc(r NOUN ejpam-891 54 10 n n CCONJ ejpam-891 54 11 ) ) PUNCT ejpam-891 54 12 and and CCONJ ejpam-891 54 13 k0 k0 PROPN ejpam-891 54 14 is be AUX ejpam-891 54 15 the the DET ejpam-891 54 16 nonempty nonempty ADJ ejpam-891 54 17 interior interior NOUN ejpam-891 54 18 of of ADP ejpam-891 54 19 k. k. PROPN ejpam-891 54 20 definition definition PROPN ejpam-891 54 21 1 1 NUM ejpam-891 54 22 . . PUNCT ejpam-891 55 1 for for ADP ejpam-891 55 2 any any DET ejpam-891 55 3 u u NOUN ejpam-891 55 4 and and CCONJ ejpam-891 55 5 v v NOUN ejpam-891 55 6 ∈ ∈ PROPN ejpam-891 55 7 kc(r kc(r NOUN ejpam-891 55 8 n n CCONJ ejpam-891 55 9 ) ) PUNCT ejpam-891 55 10 , , PUNCT ejpam-891 55 11 if if SCONJ ejpam-891 55 12 there there PRON ejpam-891 55 13 exists exist VERB ejpam-891 55 14 a a DET ejpam-891 55 15 z z PROPN ejpam-891 55 16 ∈ ∈ PROPN ejpam-891 55 17 kc(r kc(r X ejpam-891 55 18 n n CCONJ ejpam-891 55 19 ) ) PUNCT ejpam-891 55 20 such such ADJ ejpam-891 55 21 that that SCONJ ejpam-891 55 22 z z PROPN ejpam-891 55 23 ∈ ∈ PROPN ejpam-891 55 24 k(k0 k(k0 NOUN ejpam-891 55 25 ) ) PUNCT ejpam-891 55 26 and and CCONJ ejpam-891 55 27 u u X ejpam-891 55 28 = = NOUN ejpam-891 55 29 v v PROPN ejpam-891 55 30 + + NOUN ejpam-891 55 31 z z NOUN ejpam-891 55 32 then then ADV ejpam-891 55 33 we we PRON ejpam-891 55 34 say say VERB ejpam-891 55 35 that that SCONJ ejpam-891 55 36 u u PROPN ejpam-891 55 37 ≥ ≥ NUM ejpam-891 55 38 v v NOUN ejpam-891 55 39 ( ( PUNCT ejpam-891 55 40 u u NOUN ejpam-891 55 41 > > X ejpam-891 55 42 v v NOUN ejpam-891 55 43 ) ) PUNCT ejpam-891 55 44 . . PUNCT ejpam-891 56 1 similarly similarly ADV ejpam-891 56 2 we we PRON ejpam-891 56 3 can can AUX ejpam-891 56 4 define define VERB ejpam-891 56 5 u u NOUN ejpam-891 56 6 ≤ ≤ X ejpam-891 56 7 v v NOUN ejpam-891 56 8 ( ( PUNCT ejpam-891 56 9 u u NOUN ejpam-891 56 10 < < X ejpam-891 56 11 v v NOUN ejpam-891 56 12 ) ) PUNCT ejpam-891 56 13 . . PUNCT ejpam-891 57 1 to to PART ejpam-891 57 2 define define VERB ejpam-891 57 3 the the DET ejpam-891 57 4 causal causal ADJ ejpam-891 57 5 operator operator NOUN ejpam-891 57 6 we we PRON ejpam-891 57 7 introduce introduce VERB ejpam-891 57 8 the the DET ejpam-891 57 9 following following ADJ ejpam-891 57 10 notation notation NOUN ejpam-891 57 11 . . PUNCT ejpam-891 58 1 let let VERB ejpam-891 58 2 e e NOUN ejpam-891 58 3 = = SYM ejpam-891 58 4 c[[t0 c[[t0 PROPN ejpam-891 58 5 , , PUNCT ejpam-891 58 6 t t PROPN ejpam-891 58 7 ] ] PUNCT ejpam-891 58 8 , , PUNCT ejpam-891 58 9 kc(r kc(r NOUN ejpam-891 58 10 n n CCONJ ejpam-891 58 11 ) ) PUNCT ejpam-891 58 12 ] ] PUNCT ejpam-891 58 13 and and CCONJ ejpam-891 59 1 e0 e0 PROPN ejpam-891 59 2 = = SYM ejpam-891 59 3 c[[t0 c[[t0 PROPN ejpam-891 59 4 − − PROPN ejpam-891 59 5 h1 h1 PROPN ejpam-891 59 6 , , PUNCT ejpam-891 59 7 t t PROPN ejpam-891 59 8 ] ] PUNCT ejpam-891 59 9 , , PUNCT ejpam-891 59 10 kc(r kc(r NOUN ejpam-891 59 11 n n CCONJ ejpam-891 59 12 ) ) PUNCT ejpam-891 59 13 ] ] PUNCT ejpam-891 59 14 , , PUNCT ejpam-891 59 15 where where SCONJ ejpam-891 59 16 u u PROPN ejpam-891 59 17 ∈ ∈ PROPN ejpam-891 59 18 e0 e0 PROPN ejpam-891 59 19 implies imply VERB ejpam-891 59 20 u(t u(t NOUN ejpam-891 59 21 ) ) PUNCT ejpam-891 59 22 = = PUNCT ejpam-891 59 23 φ0(t φ0(t PROPN ejpam-891 59 24 ) ) PUNCT ejpam-891 59 25 , , PUNCT ejpam-891 59 26 t0 t0 PROPN ejpam-891 59 27 − − PROPN ejpam-891 59 28 h1 h1 VERB ejpam-891 59 29 ≤ ≤ PROPN ejpam-891 59 30 t t NOUN ejpam-891 59 31 ≤ ≤ NUM ejpam-891 59 32 t0 t0 PROPN ejpam-891 59 33 and and CCONJ ejpam-891 59 34 u(t u(t NOUN ejpam-891 59 35 ) ) PUNCT ejpam-891 59 36 is be AUX ejpam-891 59 37 any any DET ejpam-891 59 38 arbitrarily arbitrarily ADV ejpam-891 59 39 continuous continuous ADJ ejpam-891 59 40 function function NOUN ejpam-891 59 41 on on ADP ejpam-891 59 42 [ [ X ejpam-891 59 43 t0 t0 PROPN ejpam-891 59 44 , , PUNCT ejpam-891 59 45 t t PROPN ejpam-891 59 46 ] ] PUNCT ejpam-891 59 47 . . PUNCT ejpam-891 60 1 we we PRON ejpam-891 60 2 define define VERB ejpam-891 60 3 a a DET ejpam-891 60 4 norm norm NOUN ejpam-891 60 5 on on ADP ejpam-891 60 6 e e PROPN ejpam-891 60 7 as as SCONJ ejpam-891 60 8 follows follow VERB ejpam-891 60 9 : : PUNCT ejpam-891 60 10 for for ADP ejpam-891 60 11 u u PROPN ejpam-891 60 12 , , PUNCT ejpam-891 60 13 v v NOUN ejpam-891 60 14 ∈ ∈ PROPN ejpam-891 60 15 e e NOUN ejpam-891 60 16 d0[u d0[u NOUN ejpam-891 60 17 , , PUNCT ejpam-891 60 18 v v X ejpam-891 60 19 ] ] X ejpam-891 60 20 = = SYM ejpam-891 60 21 supt0≤t≤t supt0≤t≤t NUM ejpam-891 60 22 d[u(t d[u(t NOUN ejpam-891 60 23 ) ) PUNCT ejpam-891 60 24 , , PUNCT ejpam-891 60 25 v v X ejpam-891 60 26 ( ( PUNCT ejpam-891 60 27 t t PROPN ejpam-891 60 28 ) ) PUNCT ejpam-891 60 29 ] ] PUNCT ejpam-891 60 30 where where SCONJ ejpam-891 60 31 d d PROPN ejpam-891 60 32 denotes denote VERB ejpam-891 60 33 the the DET ejpam-891 60 34 hausdorff hausdorff PROPN ejpam-891 60 35 metric metric PROPN ejpam-891 60 36 . . PUNCT ejpam-891 61 1 definition definition NOUN ejpam-891 61 2 2 2 NUM ejpam-891 61 3 . . PUNCT ejpam-891 61 4 by by ADP ejpam-891 61 5 a a DET ejpam-891 61 6 causal causal ADJ ejpam-891 61 7 operator operator NOUN ejpam-891 61 8 or or CCONJ ejpam-891 61 9 a a DET ejpam-891 61 10 volterra volterra NOUN ejpam-891 61 11 operator operator NOUN ejpam-891 61 12 or or CCONJ ejpam-891 61 13 a a DET ejpam-891 61 14 nonanticipative nonanticipative ADJ ejpam-891 61 15 operator operator NOUN ejpam-891 61 16 we we PRON ejpam-891 61 17 mean mean VERB ejpam-891 61 18 a a DET ejpam-891 61 19 mapping mapping NOUN ejpam-891 61 20 q q NOUN ejpam-891 61 21 : : PUNCT ejpam-891 61 22 e e X ejpam-891 61 23 → → SYM ejpam-891 61 24 e e X ejpam-891 61 25 satisfying satisfy VERB ejpam-891 61 26 the the DET ejpam-891 61 27 property property NOUN ejpam-891 62 1 that that PRON ejpam-891 62 2 if if SCONJ ejpam-891 62 3 u(s u(s ADJ ejpam-891 62 4 ) ) PUNCT ejpam-891 62 5 = = SYM ejpam-891 62 6 v v NOUN ejpam-891 62 7 ( ( PUNCT ejpam-891 62 8 s),t0 s),t0 NOUN ejpam-891 62 9 ≤ ≤ PROPN ejpam-891 62 10 s s PART ejpam-891 62 11 ≤ ≤ NOUN ejpam-891 62 12 t t PROPN ejpam-891 62 13 < < X ejpam-891 62 14 t t PROPN ejpam-891 62 15 then then ADV ejpam-891 62 16 ( ( PUNCT ejpam-891 62 17 qu)(s qu)(s PROPN ejpam-891 62 18 ) ) PUNCT ejpam-891 62 19 = = PUNCT ejpam-891 62 20 ( ( PUNCT ejpam-891 62 21 qv qv INTJ ejpam-891 62 22 ) ) PUNCT ejpam-891 62 23 ( ( PUNCT ejpam-891 62 24 s),t0 s),t0 VERB ejpam-891 62 25 ≤ ≤ NUM ejpam-891 62 26 s s PART ejpam-891 62 27 ≤ ≤ NOUN ejpam-891 62 28 t t NOUN ejpam-891 62 29 < < X ejpam-891 62 30 t. t. NOUN ejpam-891 62 31 by by ADP ejpam-891 62 32 a a DET ejpam-891 62 33 causal causal ADJ ejpam-891 62 34 operator operator NOUN ejpam-891 62 35 with with ADP ejpam-891 62 36 memory memory NOUN ejpam-891 62 37 we we PRON ejpam-891 62 38 mean mean VERB ejpam-891 62 39 a a DET ejpam-891 62 40 mapping mapping NOUN ejpam-891 62 41 q q NOUN ejpam-891 62 42 : : PUNCT ejpam-891 62 43 e0 e0 PROPN ejpam-891 62 44 → → SYM ejpam-891 62 45 e e X ejpam-891 62 46 such such ADJ ejpam-891 62 47 that that PRON ejpam-891 62 48 for for ADP ejpam-891 62 49 u(s u(s PROPN ejpam-891 62 50 ) ) PUNCT ejpam-891 62 51 = = SYM ejpam-891 62 52 v v NOUN ejpam-891 62 53 ( ( PUNCT ejpam-891 62 54 s),t0 s),t0 NOUN ejpam-891 62 55 ≤ ≤ PROPN ejpam-891 62 56 s s PART ejpam-891 62 57 ≤ ≤ PROPN ejpam-891 62 58 t t PROPN ejpam-891 62 59 < < X ejpam-891 62 60 t t PROPN ejpam-891 62 61 , , PUNCT ejpam-891 62 62 q(u q(u NOUN ejpam-891 62 63 , , PUNCT ejpam-891 62 64 φ0)(s φ0)(s PROPN ejpam-891 62 65 ) ) PUNCT ejpam-891 63 1 = = SYM ejpam-891 63 2 q(v q(v NOUN ejpam-891 63 3 , , PUNCT ejpam-891 63 4 φ0)(s φ0)(s PROPN ejpam-891 63 5 ) ) PUNCT ejpam-891 63 6 , , PUNCT ejpam-891 63 7 t0 t0 PROPN ejpam-891 63 8 ≤ ≤ PROPN ejpam-891 63 9 s s PART ejpam-891 63 10 ≤ ≤ NOUN ejpam-891 63 11 t t PROPN ejpam-891 63 12 < < X ejpam-891 63 13 t t PROPN ejpam-891 63 14 and and CCONJ ejpam-891 63 15 φ0 φ0 PROPN ejpam-891 63 16 ∈ ∈ PROPN ejpam-891 63 17 c1 c1 NOUN ejpam-891 64 1 = = PROPN ejpam-891 64 2 c[[t0 c[[t0 PROPN ejpam-891 64 3 − − PROPN ejpam-891 64 4 h1 h1 PROPN ejpam-891 64 5 , , PUNCT ejpam-891 64 6 t0 t0 PROPN ejpam-891 64 7 ] ] PUNCT ejpam-891 64 8 , , PUNCT ejpam-891 64 9 kc(r kc(r NOUN ejpam-891 64 10 n n CCONJ ejpam-891 64 11 ) ) PUNCT ejpam-891 64 12 ] ] PUNCT ejpam-891 64 13 . . PUNCT ejpam-891 65 1 j. j. PROPN ejpam-891 65 2 devi devi PROPN ejpam-891 65 3 / / SYM ejpam-891 65 4 eur eur PROPN ejpam-891 65 5 . . PUNCT ejpam-891 66 1 j. j. PROPN ejpam-891 66 2 pure pure PROPN ejpam-891 66 3 appl appl PROPN ejpam-891 66 4 . . PROPN ejpam-891 66 5 math math PROPN ejpam-891 66 6 , , PUNCT ejpam-891 66 7 3 3 NUM ejpam-891 66 8 ( ( PUNCT ejpam-891 66 9 2010 2010 NUM ejpam-891 66 10 ) ) PUNCT ejpam-891 66 11 , , PUNCT ejpam-891 66 12 737 737 NUM ejpam-891 66 13 - - SYM ejpam-891 66 14 747 747 NUM ejpam-891 66 15 740 740 NUM ejpam-891 66 16 we we PRON ejpam-891 66 17 now now ADV ejpam-891 66 18 state state VERB ejpam-891 66 19 the the DET ejpam-891 66 20 following follow VERB ejpam-891 66 21 theorems theorem NOUN ejpam-891 66 22 from from ADP ejpam-891 66 23 [ [ X ejpam-891 66 24 6 6 NUM ejpam-891 66 25 ] ] PUNCT ejpam-891 66 26 which which PRON ejpam-891 66 27 are be AUX ejpam-891 66 28 needed need VERB ejpam-891 66 29 to to PART ejpam-891 66 30 prove prove VERB ejpam-891 66 31 results result NOUN ejpam-891 66 32 in in ADP ejpam-891 66 33 the the DET ejpam-891 66 34 next next ADJ ejpam-891 66 35 sections section NOUN ejpam-891 66 36 . . PUNCT ejpam-891 67 1 before before ADP ejpam-891 67 2 proceeding proceed VERB ejpam-891 67 3 further far ADV ejpam-891 67 4 , , PUNCT ejpam-891 67 5 we we PRON ejpam-891 67 6 set set VERB ejpam-891 67 7 d0[u d0[u NOUN ejpam-891 67 8 , , PUNCT ejpam-891 67 9 v v NOUN ejpam-891 67 10 ] ] X ejpam-891 67 11 = = PUNCT ejpam-891 67 12 sup sup NOUN ejpam-891 67 13 t0≤s≤t t0≤s≤t NOUN ejpam-891 67 14 d[u(s d[u(s PROPN ejpam-891 67 15 ) ) PUNCT ejpam-891 67 16 , , PUNCT ejpam-891 67 17 v v X ejpam-891 67 18 ( ( PUNCT ejpam-891 67 19 s s NOUN ejpam-891 67 20 ) ) PUNCT ejpam-891 67 21 ] ] PUNCT ejpam-891 67 22 theorem theorem NOUN ejpam-891 67 23 1 1 X ejpam-891 67 24 . . PUNCT ejpam-891 67 25 assume assume VERB ejpam-891 67 26 that that SCONJ ejpam-891 67 27 ( ( PUNCT ejpam-891 67 28 i i NOUN ejpam-891 67 29 ) ) PUNCT ejpam-891 67 30 q q X ejpam-891 67 31 is be AUX ejpam-891 67 32 nondecreasing nondecrease VERB ejpam-891 67 33 in in ADP ejpam-891 67 34 u u NOUN ejpam-891 67 35 for for ADP ejpam-891 67 36 each each DET ejpam-891 67 37 t t NOUN ejpam-891 67 38 ∈ ∈ PROPN ejpam-891 68 1 i i PRON ejpam-891 68 2 = = PUNCT ejpam-891 69 1 [ [ X ejpam-891 69 2 t0 t0 PROPN ejpam-891 69 3 , , PUNCT ejpam-891 69 4 t t PROPN ejpam-891 69 5 ] ] PUNCT ejpam-891 69 6 . . PUNCT ejpam-891 70 1 ( ( PUNCT ejpam-891 70 2 ii ii NOUN ejpam-891 70 3 ) ) PUNCT ejpam-891 70 4 dh dh NOUN ejpam-891 70 5 v v ADP ejpam-891 70 6 ( ( PUNCT ejpam-891 70 7 t t NOUN ejpam-891 70 8 ) ) PUNCT ejpam-891 70 9 ≤ ≤ NOUN ejpam-891 70 10 ( ( PUNCT ejpam-891 70 11 qv qv INTJ ejpam-891 70 12 ) ) PUNCT ejpam-891 70 13 ( ( PUNCT ejpam-891 70 14 t t PROPN ejpam-891 70 15 ) ) PUNCT ejpam-891 70 16 dhw dhw PROPN ejpam-891 70 17 ( ( PUNCT ejpam-891 70 18 t t PROPN ejpam-891 70 19 ) ) PUNCT ejpam-891 70 20 ≥ ≥ NOUN ejpam-891 70 21 ( ( PUNCT ejpam-891 70 22 qw qw NOUN ejpam-891 70 23 ) ) PUNCT ejpam-891 70 24 ( ( PUNCT ejpam-891 70 25 t t PROPN ejpam-891 70 26 ) ) PUNCT ejpam-891 70 27 where where SCONJ ejpam-891 70 28 v v NOUN ejpam-891 70 29 , , PUNCT ejpam-891 70 30 w w PROPN ejpam-891 70 31 ∈ ∈ PROPN ejpam-891 70 32 c1[i c1[i NOUN ejpam-891 70 33 , , PUNCT ejpam-891 70 34 kc(r kc(r X ejpam-891 70 35 n n CCONJ ejpam-891 70 36 ) ) PUNCT ejpam-891 70 37 ] ] PUNCT ejpam-891 70 38 and and CCONJ ejpam-891 70 39 ( ( PUNCT ejpam-891 70 40 iii iii NOUN ejpam-891 70 41 ) ) PUNCT ejpam-891 70 42 v v NOUN ejpam-891 70 43 ( ( PUNCT ejpam-891 70 44 t0) > X ejpam-891 87 48 0 0 PROPN ejpam-891 87 49 , , PUNCT ejpam-891 87 50 t t PROPN ejpam-891 87 51 ∈ ∈ PROPN ejpam-891 88 1 i i PRON ejpam-891 88 2 where where SCONJ ejpam-891 88 3 ω ω X ejpam-891 88 4 = = SYM ejpam-891 88 5 { { PUNCT ejpam-891 88 6 u u NOUN ejpam-891 88 7 , , PUNCT ejpam-891 88 8 v v PROPN ejpam-891 88 9 ∈ ∈ PROPN ejpam-891 88 10 e0 e0 PROPN ejpam-891 88 11 : : PUNCT ejpam-891 88 12 max max PROPN ejpam-891 88 13 s∈[t0−h1,t s∈[t0−h1,t PROPN ejpam-891 88 14 ] ] PUNCT ejpam-891 88 15 d[u(s d[u(s PROPN ejpam-891 88 16 ) ) PUNCT ejpam-891 88 17 , , PUNCT ejpam-891 88 18 v v X ejpam-891 88 19 ( ( PUNCT ejpam-891 88 20 s s NOUN ejpam-891 88 21 ) ) PUNCT ejpam-891 88 22 ] ] PUNCT ejpam-891 88 23 = = PUNCT ejpam-891 88 24 d[u(t d[u(t PROPN ejpam-891 88 25 ) ) PUNCT ejpam-891 88 26 , , PUNCT ejpam-891 88 27 v v X ejpam-891 88 28 ( ( PUNCT ejpam-891 88 29 t)]t t)]t NOUN ejpam-891 88 30 ∈ ∈ PROPN ejpam-891 89 1 i i NOUN ejpam-891 89 2 } } PUNCT ejpam-891 89 3 then then ADV ejpam-891 89 4 there there PRON ejpam-891 89 5 exists exist VERB ejpam-891 89 6 a a DET ejpam-891 89 7 unique unique ADJ ejpam-891 89 8 solution solution NOUN ejpam-891 89 9 u(t u(t NOUN ejpam-891 89 10 ) ) PUNCT ejpam-891 89 11 of of ADP ejpam-891 89 12 the the DET ejpam-891 89 13 ivp(8 ivp(8 NOUN ejpam-891 89 14 ) ) PUNCT ejpam-891 89 15 and and CCONJ ejpam-891 89 16 ( ( PUNCT ejpam-891 89 17 9 9 X ejpam-891 89 18 ) ) PUNCT ejpam-891 89 19 provided provide VERB ejpam-891 89 20 t t PROPN ejpam-891 89 21 − − PROPN ejpam-891 89 22 t0 t0 PROPN ejpam-891 89 23 < < X ejpam-891 89 24 1 1 NUM ejpam-891 89 25 l l NOUN ejpam-891 89 26 . . PUNCT ejpam-891 90 1 proof proof NOUN ejpam-891 90 2 . . PUNCT ejpam-891 91 1 define define VERB ejpam-891 91 2 d0[u d0[u NOUN ejpam-891 91 3 , , PUNCT ejpam-891 91 4 v v X ejpam-891 91 5 ] ] X ejpam-891 91 6 ( ( PUNCT ejpam-891 91 7 t t NOUN ejpam-891 91 8 ) ) PUNCT ejpam-891 92 1 = = SYM ejpam-891 92 2 max max PROPN ejpam-891 92 3 s∈[t0−h1,t s∈[t0−h1,t PROPN ejpam-891 92 4 ] ] PUNCT ejpam-891 92 5 d[u(s d[u(s PROPN ejpam-891 92 6 ) ) PUNCT ejpam-891 92 7 , , PUNCT ejpam-891 92 8 v v X ejpam-891 92 9 ( ( PUNCT ejpam-891 92 10 s s NOUN ejpam-891 92 11 ) ) PUNCT ejpam-891 92 12 ] ] PUNCT ejpam-891 92 13 for for ADP ejpam-891 92 14 any any DET ejpam-891 92 15 u u PROPN ejpam-891 92 16 ∈ ∈ PROPN ejpam-891 92 17 e0 e0 NOUN ejpam-891 92 18 , , PUNCT ejpam-891 92 19 define define VERB ejpam-891 92 20 the the DET ejpam-891 92 21 hukuhara hukuhara ADJ ejpam-891 92 22 integral integral ADJ ejpam-891 92 23 operator operator NOUN ejpam-891 92 24 t t PROPN ejpam-891 92 25 on on ADP ejpam-891 92 26 i i PRON ejpam-891 92 27 by by ADP ejpam-891 92 28 ( ( PUNCT ejpam-891 92 29 t t NOUN ejpam-891 92 30 u)(t u)(t ADJ ejpam-891 92 31 ) ) PUNCT ejpam-891 92 32 = = SYM ejpam-891 92 33 φ0(t0 φ0(t0 PROPN ejpam-891 92 34 ) ) PUNCT ejpam-891 93 1 + + CCONJ ejpam-891 93 2 ∫ ∫ PROPN ejpam-891 93 3 t t PROPN ejpam-891 93 4 t0 t0 PROPN ejpam-891 93 5 q(u q(u NOUN ejpam-891 93 6 , , PUNCT ejpam-891 93 7 φ0)(s)ds φ0)(s)ds X ejpam-891 93 8 ( ( PUNCT ejpam-891 93 9 12 12 NUM ejpam-891 93 10 ) ) PUNCT ejpam-891 93 11 ut0 ut0 NOUN ejpam-891 93 12 = = NOUN ejpam-891 93 13 φ0 φ0 PROPN ejpam-891 93 14 ∈ ∈ PROPN ejpam-891 93 15 c1 c1 PROPN ejpam-891 93 16 on on ADP ejpam-891 93 17 [ [ X ejpam-891 93 18 t0 t0 PROPN ejpam-891 93 19 − − PROPN ejpam-891 93 20 h1 h1 PROPN ejpam-891 93 21 , , PUNCT ejpam-891 93 22 t0 t0 PROPN ejpam-891 93 23 ] ] PUNCT ejpam-891 93 24 ( ( PUNCT ejpam-891 93 25 13 13 NUM ejpam-891 93 26 ) ) PUNCT ejpam-891 93 27 now now ADV ejpam-891 93 28 for for ADP ejpam-891 93 29 u u PROPN ejpam-891 93 30 , , PUNCT ejpam-891 93 31 v v PROPN ejpam-891 93 32 ∈ ∈ PROPN ejpam-891 93 33 ω ω NOUN ejpam-891 93 34 , , PUNCT ejpam-891 93 35 using use VERB ejpam-891 93 36 the the DET ejpam-891 93 37 properties property NOUN ejpam-891 93 38 of of ADP ejpam-891 93 39 hausdorff hausdorff NOUN ejpam-891 93 40 metric metric ADJ ejpam-891 93 41 and and CCONJ ejpam-891 93 42 hypothesis hypothesis NOUN ejpam-891 93 43 of of ADP ejpam-891 93 44 the the DET ejpam-891 93 45 theorem theorem NOUN ejpam-891 93 46 , , PUNCT ejpam-891 93 47 d[t d[t VERB ejpam-891 93 48 u(t u(t NOUN ejpam-891 93 49 ) ) PUNCT ejpam-891 93 50 , , PUNCT ejpam-891 93 51 t t PROPN ejpam-891 93 52 v v PROPN ejpam-891 93 53 ( ( PUNCT ejpam-891 93 54 t t PROPN ejpam-891 93 55 ) ) PUNCT ejpam-891 93 56 ] ] PUNCT ejpam-891 94 1 = = PUNCT ejpam-891 94 2 d[φ0(t0 d[φ0(t0 NOUN ejpam-891 94 3 ) ) PUNCT ejpam-891 95 1 + + CCONJ ejpam-891 95 2 ∫ ∫ PROPN ejpam-891 95 3 t t PROPN ejpam-891 95 4 t0 t0 PROPN ejpam-891 95 5 q(u q(u NOUN ejpam-891 95 6 , , PUNCT ejpam-891 95 7 φ0)(s)ds φ0)(s)ds PROPN ejpam-891 95 8 , , PUNCT ejpam-891 95 9 φ0(t0 φ0(t0 NOUN ejpam-891 95 10 ) ) PUNCT ejpam-891 96 1 + + CCONJ ejpam-891 97 1 ∫ ∫ PROPN ejpam-891 97 2 t t PROPN ejpam-891 97 3 t0 t0 PROPN ejpam-891 97 4 q(v q(v PROPN ejpam-891 97 5 , , PUNCT ejpam-891 97 6 φ0)(s)ds φ0)(s)ds PROPN ejpam-891 97 7 ] ] PUNCT ejpam-891 97 8 = = SYM ejpam-891 98 1 d d PROPN ejpam-891 98 2 [ [ PUNCT ejpam-891 98 3 ∫ ∫ PROPN ejpam-891 98 4 t t PROPN ejpam-891 98 5 t0 t0 PROPN ejpam-891 98 6 q(u q(u NOUN ejpam-891 98 7 , , PUNCT ejpam-891 98 8 φ0)(s)ds φ0)(s)ds PROPN ejpam-891 98 9 , , PUNCT ejpam-891 98 10 ∫ ∫ PROPN ejpam-891 98 11 t t PROPN ejpam-891 98 12 t0 t0 PROPN ejpam-891 98 13 q(v q(v PROPN ejpam-891 98 14 , , PUNCT ejpam-891 98 15 φ0)(s)ds φ0)(s)ds PROPN ejpam-891 98 16 ] ] PUNCT ejpam-891 98 17 ≤ ≤ NUM ejpam-891 98 18 ∫ ∫ PROPN ejpam-891 98 19 t t PROPN ejpam-891 98 20 t0 t0 PROPN ejpam-891 98 21 d[q(u d[q(u X ejpam-891 98 22 , , PUNCT ejpam-891 98 23 φ0)(s),q(v φ0)(s),q(v PROPN ejpam-891 98 24 , , PUNCT ejpam-891 98 25 φ0)(s)]ds φ0)(s)]ds PROPN ejpam-891 98 26 ≤ ≤ NUM ejpam-891 98 27 ∫ ∫ PROPN ejpam-891 98 28 t t PROPN ejpam-891 98 29 t0 t0 PROPN ejpam-891 98 30 lmax lmax VERB ejpam-891 98 31 t0≤s≤t t0≤s≤t PROPN ejpam-891 98 32 d[u(s d[u(s PROPN ejpam-891 98 33 ) ) PUNCT ejpam-891 98 34 , , PUNCT ejpam-891 98 35 v v NOUN ejpam-891 98 36 ( ( PUNCT ejpam-891 98 37 s)]ds s)]ds NOUN ejpam-891 98 38 = = PUNCT ejpam-891 98 39 l l NOUN ejpam-891 98 40 ∫ ∫ PROPN ejpam-891 98 41 t t PROPN ejpam-891 98 42 t0 t0 PROPN ejpam-891 98 43 d[u(t d[u(t PROPN ejpam-891 98 44 ) ) PUNCT ejpam-891 98 45 , , PUNCT ejpam-891 98 46 v v X ejpam-891 98 47 ( ( PUNCT ejpam-891 99 1 t)]ds t)]ds ADV ejpam-891 99 2 ≤ ≤ ADJ ejpam-891 99 3 ld0[u ld0[u NOUN ejpam-891 99 4 , , PUNCT ejpam-891 99 5 v v NOUN ejpam-891 99 6 ] ] X ejpam-891 99 7 ( ( PUNCT ejpam-891 99 8 t t PROPN ejpam-891 99 9 − − PROPN ejpam-891 99 10 t0 t0 PROPN ejpam-891 99 11 ) ) PUNCT ejpam-891 99 12 ≤ ≤ NUM ejpam-891 99 13 l(t l(t PROPN ejpam-891 99 14 − − PUNCT ejpam-891 100 1 t0)d0[u t0)d0[u NOUN ejpam-891 100 2 , , PUNCT ejpam-891 100 3 v v NOUN ejpam-891 100 4 ] ] PUNCT ejpam-891 100 5 which which PRON ejpam-891 100 6 is be AUX ejpam-891 100 7 a a DET ejpam-891 100 8 contraction contraction NOUN ejpam-891 100 9 , , PUNCT ejpam-891 100 10 when when SCONJ ejpam-891 100 11 l(t l(t PROPN ejpam-891 100 12 − − PROPN ejpam-891 100 13 t0 t0 PROPN ejpam-891 100 14 ) ) PUNCT ejpam-891 100 15 < < X ejpam-891 100 16 1 1 NUM ejpam-891 100 17 or or CCONJ ejpam-891 100 18 ( ( PUNCT ejpam-891 100 19 t t PROPN ejpam-891 100 20 − − PROPN ejpam-891 100 21 t0 t0 PROPN ejpam-891 100 22 ) ) PUNCT ejpam-891 100 23 < < X ejpam-891 100 24 1 1 NUM ejpam-891 100 25 l l NOUN ejpam-891 100 26 . . PUNCT ejpam-891 101 1 thus thus ADV ejpam-891 101 2 , , PUNCT ejpam-891 101 3 since since SCONJ ejpam-891 101 4 ( ( PUNCT ejpam-891 101 5 t− t− PROPN ejpam-891 101 6 t0 t0 PROPN ejpam-891 101 7 ) ) PUNCT ejpam-891 101 8 < < X ejpam-891 101 9 1 1 NUM ejpam-891 101 10 l l NOUN ejpam-891 101 11 , , PUNCT ejpam-891 101 12 t t PROPN ejpam-891 101 13 is be AUX ejpam-891 101 14 contraction contraction NOUN ejpam-891 101 15 from from ADP ejpam-891 101 16 e e NOUN ejpam-891 101 17 to to ADP ejpam-891 101 18 e e NOUN ejpam-891 101 19 and and CCONJ ejpam-891 101 20 hence hence ADV ejpam-891 101 21 by by ADP ejpam-891 101 22 contraction contraction NOUN ejpam-891 101 23 mapping mapping NOUN ejpam-891 101 24 theorem theorem NOUN ejpam-891 101 25 there there PRON ejpam-891 101 26 exists exist VERB ejpam-891 101 27 a a DET ejpam-891 101 28 u u NOUN ejpam-891 101 29 ∈ ∈ NOUN ejpam-891 101 30 e e NOUN ejpam-891 101 31 such such ADJ ejpam-891 102 1 that that SCONJ ejpam-891 102 2 t t PROPN ejpam-891 102 3 u u NOUN ejpam-891 102 4 = = SYM ejpam-891 102 5 u u PROPN ejpam-891 102 6 , , PUNCT ejpam-891 102 7 ut0 ut0 PROPN ejpam-891 102 8 = = SYM ejpam-891 103 1 φ0 φ0 PROPN ejpam-891 103 2 we we PRON ejpam-891 103 3 get get VERB ejpam-891 103 4 that that SCONJ ejpam-891 103 5 u u NOUN ejpam-891 103 6 is be AUX ejpam-891 103 7 unique unique ADJ ejpam-891 103 8 solution solution NOUN ejpam-891 103 9 for for ADP ejpam-891 103 10 the the DET ejpam-891 103 11 ivp ivp NOUN ejpam-891 103 12 ( ( PUNCT ejpam-891 103 13 8) 8) NUM ejpam-891 103 14 and and CCONJ ejpam-891 103 15 ( ( PUNCT ejpam-891 103 16 9 9 X ejpam-891 103 17 ) ) PUNCT ejpam-891 103 18 whenever whenever SCONJ ejpam-891 103 19 ( ( PUNCT ejpam-891 103 20 t t PROPN ejpam-891 103 21 − − PROPN ejpam-891 103 22 t0 t0 PROPN ejpam-891 103 23 ) ) PUNCT ejpam-891 103 24 < < X ejpam-891 103 25 1 1 NUM ejpam-891 103 26 l l NOUN ejpam-891 103 27 . . PUNCT ejpam-891 104 1 hence hence ADV ejpam-891 104 2 the the DET ejpam-891 104 3 proof proof NOUN ejpam-891 104 4 is be AUX ejpam-891 104 5 complete complete ADJ ejpam-891 104 6 . . PUNCT ejpam-891 105 1 j. j. PROPN ejpam-891 105 2 devi devi PROPN ejpam-891 105 3 / / SYM ejpam-891 105 4 eur eur PROPN ejpam-891 105 5 . . PUNCT ejpam-891 106 1 j. j. PROPN ejpam-891 106 2 pure pure PROPN ejpam-891 106 3 appl appl PROPN ejpam-891 106 4 . . PROPN ejpam-891 106 5 math math PROPN ejpam-891 106 6 , , PUNCT ejpam-891 106 7 3 3 NUM ejpam-891 106 8 ( ( PUNCT ejpam-891 106 9 2010 2010 NUM ejpam-891 106 10 ) ) PUNCT ejpam-891 106 11 , , PUNCT ejpam-891 106 12 737 737 NUM ejpam-891 106 13 - - SYM ejpam-891 106 14 747 747 NUM ejpam-891 106 15 742 742 NUM ejpam-891 106 16 remark remark NOUN ejpam-891 106 17 1 1 NUM ejpam-891 106 18 . . PUNCT ejpam-891 107 1 the the DET ejpam-891 107 2 restriction restriction NOUN ejpam-891 107 3 ( ( PUNCT ejpam-891 107 4 t t PROPN ejpam-891 107 5 − − PROPN ejpam-891 107 6 t0 t0 PROPN ejpam-891 107 7 ) ) PUNCT ejpam-891 107 8 < < X ejpam-891 107 9 1 1 NUM ejpam-891 107 10 l l NOUN ejpam-891 107 11 can can AUX ejpam-891 107 12 be be AUX ejpam-891 107 13 avoided avoid VERB ejpam-891 107 14 by by ADP ejpam-891 107 15 using use VERB ejpam-891 107 16 a a DET ejpam-891 107 17 weighted weight VERB ejpam-891 107 18 norm norm NOUN ejpam-891 107 19 . . PUNCT ejpam-891 108 1 we we PRON ejpam-891 108 2 define define VERB ejpam-891 108 3 d0[u d0[u NOUN ejpam-891 108 4 , , PUNCT ejpam-891 108 5 v v NOUN ejpam-891 108 6 ] ] X ejpam-891 108 7 = = SYM ejpam-891 108 8 maxs∈[t0,t]d[u(s maxs∈[t0,t]d[u(s PROPN ejpam-891 108 9 ) ) PUNCT ejpam-891 108 10 , , PUNCT ejpam-891 108 11 v v NOUN ejpam-891 108 12 ( ( PUNCT ejpam-891 108 13 s)]e−λt s)]e−λt NOUN ejpam-891 108 14 . . PUNCT ejpam-891 109 1 where where SCONJ ejpam-891 109 2 λ λ PROPN ejpam-891 109 3 is be AUX ejpam-891 109 4 to to PART ejpam-891 109 5 be be AUX ejpam-891 109 6 chosen choose VERB ejpam-891 109 7 suitably suitably ADV ejpam-891 109 8 . . PUNCT ejpam-891 110 1 now now ADV ejpam-891 110 2 using use VERB ejpam-891 110 3 the the DET ejpam-891 110 4 relation relation NOUN ejpam-891 110 5 ( ( PUNCT ejpam-891 110 6 12 12 NUM ejpam-891 110 7 ) ) PUNCT ejpam-891 110 8 , , PUNCT ejpam-891 110 9 and and CCONJ ejpam-891 110 10 using use VERB ejpam-891 110 11 the the DET ejpam-891 110 12 properties property NOUN ejpam-891 110 13 of of ADP ejpam-891 110 14 hausdorff hausdorff NOUN ejpam-891 110 15 metric metric NOUN ejpam-891 110 16 we we PRON ejpam-891 110 17 arrive arrive VERB ejpam-891 110 18 at at ADP ejpam-891 110 19 , , PUNCT ejpam-891 110 20 d[(t d[(t NOUN ejpam-891 110 21 u)(t u)(t NOUN ejpam-891 110 22 ) ) PUNCT ejpam-891 110 23 , , PUNCT ejpam-891 110 24 ( ( PUNCT ejpam-891 110 25 t t PROPN ejpam-891 110 26 v v NOUN ejpam-891 110 27 ) ) PUNCT ejpam-891 110 28 ( ( PUNCT ejpam-891 110 29 t t PROPN ejpam-891 110 30 ) ) PUNCT ejpam-891 110 31 ] ] PUNCT ejpam-891 111 1 = = PUNCT ejpam-891 111 2 d[φ0(t0 d[φ0(t0 NOUN ejpam-891 111 3 ) ) PUNCT ejpam-891 112 1 + + CCONJ ejpam-891 112 2 ∫ ∫ PROPN ejpam-891 112 3 t t PROPN ejpam-891 112 4 t0 t0 PROPN ejpam-891 112 5 q(u q(u NOUN ejpam-891 112 6 , , PUNCT ejpam-891 112 7 φ0)(s)ds φ0)(s)ds PROPN ejpam-891 112 8 , , PUNCT ejpam-891 112 9 φ0(t0 φ0(t0 NOUN ejpam-891 112 10 ) ) PUNCT ejpam-891 113 1 + + CCONJ ejpam-891 114 1 ∫ ∫ PROPN ejpam-891 114 2 t t PROPN ejpam-891 114 3 t0 t0 PROPN ejpam-891 114 4 q(v q(v PROPN ejpam-891 114 5 , , PUNCT ejpam-891 114 6 φ0)(s)ds φ0)(s)ds PROPN ejpam-891 114 7 ] ] PUNCT ejpam-891 114 8 = = SYM ejpam-891 115 1 d d PROPN ejpam-891 115 2 [ [ PUNCT ejpam-891 115 3 ∫ ∫ PROPN ejpam-891 115 4 t t PROPN ejpam-891 115 5 t0 t0 PROPN ejpam-891 115 6 q(u q(u NOUN ejpam-891 115 7 , , PUNCT ejpam-891 115 8 φ0)(s)ds φ0)(s)ds PROPN ejpam-891 115 9 , , PUNCT ejpam-891 115 10 ∫ ∫ PROPN ejpam-891 115 11 t t PROPN ejpam-891 115 12 t0 t0 PROPN ejpam-891 115 13 q(v q(v PROPN ejpam-891 115 14 , , PUNCT ejpam-891 115 15 φ0)(s)ds φ0)(s)ds PROPN ejpam-891 115 16 ] ] PUNCT ejpam-891 115 17 ≤ ≤ NUM ejpam-891 115 18 ∫ ∫ PROPN ejpam-891 115 19 t t PROPN ejpam-891 115 20 t0 t0 PROPN ejpam-891 115 21 d[q(u d[q(u X ejpam-891 115 22 , , PUNCT ejpam-891 115 23 φ0)(s),q(v φ0)(s),q(v PROPN ejpam-891 115 24 , , PUNCT ejpam-891 115 25 φ0)(s)]ds φ0)(s)]ds PROPN ejpam-891 115 26 ≤ ≤ NUM ejpam-891 115 27 ∫ ∫ PROPN ejpam-891 115 28 t t PROPN ejpam-891 115 29 t0 t0 PROPN ejpam-891 115 30 l l PROPN ejpam-891 115 31 maxs∈[t0,t]d[u(s maxs∈[t0,t]d[u(s PROPN ejpam-891 115 32 ) ) PUNCT ejpam-891 115 33 , , PUNCT ejpam-891 115 34 v v X ejpam-891 115 35 ( ( PUNCT ejpam-891 115 36 s)]e−λseλsds s)]e−λseλsds ADJ ejpam-891 115 37 = = PUNCT ejpam-891 115 38 d0[u d0[u NOUN ejpam-891 115 39 , , PUNCT ejpam-891 115 40 v v X ejpam-891 115 41 ] ] X ejpam-891 115 42 l l NOUN ejpam-891 115 43 ∫ ∫ PROPN ejpam-891 115 44 t t PROPN ejpam-891 115 45 t0 t0 PROPN ejpam-891 115 46 eλsds eλsds NOUN ejpam-891 116 1 = = PUNCT ejpam-891 116 2 ld0[u ld0[u PROPN ejpam-891 116 3 , , PUNCT ejpam-891 116 4 v v NOUN ejpam-891 116 5 ] ] PUNCT ejpam-891 116 6 ∫ ∫ PROPN ejpam-891 117 1 t t PROPN ejpam-891 117 2 t0 t0 PROPN ejpam-891 117 3 eλsds eλsds NOUN ejpam-891 118 1 = = PUNCT ejpam-891 118 2 l l NOUN ejpam-891 118 3 λ λ X ejpam-891 118 4 d0[u d0[u NOUN ejpam-891 118 5 , , PUNCT ejpam-891 118 6 v v X ejpam-891 118 7 ] ] X ejpam-891 119 1 [ [ X ejpam-891 119 2 eλt eλt PROPN ejpam-891 119 3 − − PROPN ejpam-891 119 4 eλt0 eλt0 PROPN ejpam-891 119 5 ] ] PUNCT ejpam-891 119 6 ≤ ≤ NUM ejpam-891 119 7 l l NOUN ejpam-891 119 8 λ λ PROPN ejpam-891 119 9 d0[u d0[u NOUN ejpam-891 119 10 , , PUNCT ejpam-891 119 11 v v X ejpam-891 119 12 ] ] X ejpam-891 119 13 eλt eλt PROPN ejpam-891 120 1 this this PRON ejpam-891 120 2 gives give VERB ejpam-891 120 3 e−λt e−λt NOUN ejpam-891 120 4 d[t d[t VERB ejpam-891 120 5 u(t u(t NOUN ejpam-891 120 6 ) ) PUNCT ejpam-891 120 7 , , PUNCT ejpam-891 120 8 t t PROPN ejpam-891 120 9 v v PROPN ejpam-891 120 10 ( ( PUNCT ejpam-891 120 11 t t PROPN ejpam-891 120 12 ) ) PUNCT ejpam-891 120 13 ] ] PUNCT ejpam-891 120 14 ≤ ≤ NUM ejpam-891 120 15 l l NOUN ejpam-891 120 16 λ λ PROPN ejpam-891 120 17 d0[u d0[u NOUN ejpam-891 120 18 , , PUNCT ejpam-891 120 19 v v X ejpam-891 120 20 ] ] PUNCT ejpam-891 120 21 thus thus ADV ejpam-891 120 22 , , PUNCT ejpam-891 120 23 d0[t d0[t PROPN ejpam-891 120 24 u u PROPN ejpam-891 120 25 , , PUNCT ejpam-891 120 26 t t PROPN ejpam-891 120 27 v v X ejpam-891 120 28 ] ] PUNCT ejpam-891 120 29 ≤ ≤ NUM ejpam-891 120 30 l l NOUN ejpam-891 120 31 λ λ PROPN ejpam-891 120 32 d0[u d0[u NOUN ejpam-891 120 33 , , PUNCT ejpam-891 120 34 v v X ejpam-891 120 35 ] ] PUNCT ejpam-891 120 36 now now ADV ejpam-891 120 37 to to PART ejpam-891 120 38 choose choose VERB ejpam-891 120 39 λ λ PROPN ejpam-891 120 40 , , PUNCT ejpam-891 120 41 we we PRON ejpam-891 120 42 observe observe VERB ejpam-891 120 43 that that SCONJ ejpam-891 120 44 l l NOUN ejpam-891 120 45 λ λ X ejpam-891 120 46 < < X ejpam-891 120 47 1 1 NUM ejpam-891 120 48 2 2 NUM ejpam-891 120 49 , , PUNCT ejpam-891 120 50 yields yield NOUN ejpam-891 120 51 that that PRON ejpam-891 120 52 t t PROPN ejpam-891 120 53 is be AUX ejpam-891 120 54 a a DET ejpam-891 120 55 contraction contraction NOUN ejpam-891 120 56 , , PUNCT ejpam-891 120 57 so so SCONJ ejpam-891 120 58 we we PRON ejpam-891 120 59 can can AUX ejpam-891 120 60 choose choose VERB ejpam-891 120 61 λ λ X ejpam-891 120 62 < < X ejpam-891 120 63 2 2 NUM ejpam-891 120 64 l l NOUN ejpam-891 120 65 . . PUNCT ejpam-891 121 1 hence hence ADV ejpam-891 121 2 we we PRON ejpam-891 121 3 get get VERB ejpam-891 121 4 that that DET ejpam-891 121 5 t t NOUN ejpam-891 121 6 is be AUX ejpam-891 121 7 a a DET ejpam-891 121 8 contraction contraction NOUN ejpam-891 121 9 and and CCONJ ejpam-891 121 10 that that SCONJ ejpam-891 121 11 there there PRON ejpam-891 121 12 exists exist VERB ejpam-891 121 13 a a DET ejpam-891 121 14 unique unique ADJ ejpam-891 121 15 u u NOUN ejpam-891 121 16 ∈ ∈ PROPN ejpam-891 121 17 e0 e0 NOUN ejpam-891 121 18 such such ADJ ejpam-891 121 19 that that SCONJ ejpam-891 121 20 u u PROPN ejpam-891 121 21 is be AUX ejpam-891 121 22 a a DET ejpam-891 121 23 solution solution NOUN ejpam-891 121 24 of of ADP ejpam-891 121 25 the the DET ejpam-891 121 26 ivp ivp NOUN ejpam-891 121 27 ( ( PUNCT ejpam-891 121 28 8) 8) NUM ejpam-891 121 29 and and CCONJ ejpam-891 121 30 ( ( PUNCT ejpam-891 121 31 9 9 NUM ejpam-891 121 32 ) ) PUNCT ejpam-891 121 33 . . PUNCT ejpam-891 122 1 in in ADP ejpam-891 122 2 order order NOUN ejpam-891 122 3 to to PART ejpam-891 122 4 establish establish VERB ejpam-891 122 5 existence existence NOUN ejpam-891 122 6 and and CCONJ ejpam-891 122 7 uniqueness uniqueness NOUN ejpam-891 122 8 result result NOUN ejpam-891 122 9 using use VERB ejpam-891 122 10 generalized generalized ADJ ejpam-891 122 11 lipschitz lipschitz NOUN ejpam-891 122 12 condition condition NOUN ejpam-891 122 13 . . PUNCT ejpam-891 123 1 we we PRON ejpam-891 123 2 need need VERB ejpam-891 123 3 the the DET ejpam-891 123 4 following follow VERB ejpam-891 123 5 comparison comparison NOUN ejpam-891 123 6 theorem theorem VERB ejpam-891 123 7 on on ADP ejpam-891 123 8 r+ r+ NOUN ejpam-891 123 9 from from ADP ejpam-891 123 10 [ [ X ejpam-891 123 11 4 4 NUM ejpam-891 123 12 ] ] PUNCT ejpam-891 123 13 . . PUNCT ejpam-891 124 1 theorem theorem ADJ ejpam-891 124 2 4 4 NUM ejpam-891 124 3 . . PUNCT ejpam-891 124 4 assume assume VERB ejpam-891 124 5 that that SCONJ ejpam-891 124 6 m m PROPN ejpam-891 124 7 ∈ ∈ NOUN ejpam-891 124 8 c[i c[i NOUN ejpam-891 124 9 , , PUNCT ejpam-891 124 10 r+ r+ X ejpam-891 124 11 ] ] PUNCT ejpam-891 124 12 , , PUNCT ejpam-891 124 13 g g PROPN ejpam-891 124 14 ∈ ∈ PROPN ejpam-891 124 15 c[i c[i NUM ejpam-891 124 16 ×r+,r+ ×r+,r+ NOUN ejpam-891 124 17 ] ] X ejpam-891 124 18 and and CCONJ ejpam-891 124 19 for for ADP ejpam-891 124 20 t t PROPN ejpam-891 124 21 ∈ ∈ PROPN ejpam-891 125 1 i i PRON ejpam-891 125 2 , , PUNCT ejpam-891 125 3 d−m(t d−m(t NOUN ejpam-891 125 4 ) ) PUNCT ejpam-891 125 5 ≤ ≤ PUNCT ejpam-891 125 6 g[t| g[t| NOUN ejpam-891 125 7 m m PRON ejpam-891 125 8 |0(t |0(t NOUN ejpam-891 125 9 ) ) PUNCT ejpam-891 125 10 ] ] PUNCT ejpam-891 125 11 , , PUNCT ejpam-891 125 12 ( ( PUNCT ejpam-891 125 13 14 14 NUM ejpam-891 125 14 ) ) PUNCT ejpam-891 125 15 where where SCONJ ejpam-891 125 16 | | ADV ejpam-891 125 17 m m VERB ejpam-891 125 18 |0(t |0(t NOUN ejpam-891 125 19 ) ) PUNCT ejpam-891 125 20 = = SYM ejpam-891 125 21 supt0≤s≤t supt0≤s≤t NUM ejpam-891 125 22 | | ADV ejpam-891 125 23 m(s m(s NOUN ejpam-891 125 24 ) ) PUNCT ejpam-891 125 25 | | ADV ejpam-891 125 26 . . PUNCT ejpam-891 126 1 j. j. PROPN ejpam-891 126 2 devi devi PROPN ejpam-891 126 3 / / SYM ejpam-891 126 4 eur eur PROPN ejpam-891 126 5 . . PUNCT ejpam-891 127 1 j. j. PROPN ejpam-891 127 2 pure pure PROPN ejpam-891 127 3 appl appl PROPN ejpam-891 127 4 . . PROPN ejpam-891 127 5 math math PROPN ejpam-891 127 6 , , PUNCT ejpam-891 127 7 3 3 NUM ejpam-891 127 8 ( ( PUNCT ejpam-891 127 9 2010 2010 NUM ejpam-891 127 10 ) ) PUNCT ejpam-891 127 11 , , PUNCT ejpam-891 127 12 737 737 NUM ejpam-891 127 13 - - SYM ejpam-891 127 14 747 747 NUM ejpam-891 127 15 743 743 NUM ejpam-891 127 16 suppose suppose VERB ejpam-891 127 17 that that SCONJ ejpam-891 127 18 r(t r(t NOUN ejpam-891 127 19 ) ) PUNCT ejpam-891 127 20 = = SYM ejpam-891 127 21 r(t r(t NOUN ejpam-891 127 22 , , PUNCT ejpam-891 127 23 t0 t0 PROPN ejpam-891 127 24 , , PUNCT ejpam-891 127 25 w0 w0 PROPN ejpam-891 127 26 ) ) PUNCT ejpam-891 127 27 is be AUX ejpam-891 127 28 the the DET ejpam-891 127 29 maximal maximal ADJ ejpam-891 127 30 solution solution NOUN ejpam-891 127 31 of of ADP ejpam-891 127 32 the the DET ejpam-891 127 33 scalar scalar ADJ ejpam-891 127 34 differential differential ADJ ejpam-891 127 35 equation equation NOUN ejpam-891 128 1 w′ w′ NOUN ejpam-891 128 2 = = SYM ejpam-891 128 3 g(t g(t PROPN ejpam-891 128 4 , , PUNCT ejpam-891 128 5 w w NOUN ejpam-891 128 6 ) ) PUNCT ejpam-891 128 7 , , PUNCT ejpam-891 128 8 w(t0 w(t0 NOUN ejpam-891 128 9 ) ) PUNCT ejpam-891 128 10 = = PUNCT ejpam-891 128 11 w0 w0 PROPN ejpam-891 128 12 ≥ ≥ NOUN ejpam-891 128 13 0 0 NUM ejpam-891 128 14 ( ( PUNCT ejpam-891 128 15 15 15 NUM ejpam-891 128 16 ) ) PUNCT ejpam-891 128 17 existing exist VERB ejpam-891 128 18 on on ADP ejpam-891 128 19 i. i. NOUN ejpam-891 128 20 then then ADV ejpam-891 128 21 m(t0)≤ m(t0)≤ PROPN ejpam-891 128 22 w0 w0 PROPN ejpam-891 128 23 implies imply VERB ejpam-891 128 24 m(t m(t NOUN ejpam-891 128 25 ) ) PUNCT ejpam-891 128 26 ≤ ≤ NOUN ejpam-891 128 27 r(t r(t NOUN ejpam-891 128 28 ) ) PUNCT ejpam-891 128 29 , , PUNCT ejpam-891 128 30 t t PROPN ejpam-891 128 31 ∈ ∈ PROPN ejpam-891 129 1 i i PRON ejpam-891 129 2 . . PUNCT ejpam-891 130 1 we we PRON ejpam-891 130 2 now now ADV ejpam-891 130 3 state state VERB ejpam-891 130 4 a a DET ejpam-891 130 5 comparison comparison NOUN ejpam-891 130 6 theorem theorem VERB ejpam-891 130 7 that that PRON ejpam-891 130 8 connects connect VERB ejpam-891 130 9 an an DET ejpam-891 130 10 estimate estimate NOUN ejpam-891 130 11 on on ADP ejpam-891 130 12 the the DET ejpam-891 130 13 solution solution NOUN ejpam-891 130 14 of of ADP ejpam-891 130 15 the the DET ejpam-891 130 16 ivp ivp NOUN ejpam-891 130 17 ( ( PUNCT ejpam-891 130 18 8) 8) NUM ejpam-891 130 19 and and CCONJ ejpam-891 130 20 ( ( PUNCT ejpam-891 130 21 9 9 NUM ejpam-891 130 22 ) ) PUNCT ejpam-891 130 23 with with ADP ejpam-891 130 24 the the DET ejpam-891 130 25 maximal maximal ADJ ejpam-891 130 26 solution solution NOUN ejpam-891 130 27 of of ADP ejpam-891 130 28 the the DET ejpam-891 130 29 initial initial ADJ ejpam-891 130 30 value value NOUN ejpam-891 130 31 problem problem NOUN ejpam-891 130 32 ( ( PUNCT ejpam-891 130 33 15 15 NUM ejpam-891 130 34 ) ) PUNCT ejpam-891 130 35 . . PUNCT ejpam-891 131 1 theorem theorem NOUN ejpam-891 131 2 5 5 NUM ejpam-891 131 3 . . PUNCT ejpam-891 132 1 let let VERB ejpam-891 132 2 q q PROPN ejpam-891 132 3 ∈ ∈ PROPN ejpam-891 132 4 c[e0 c[e0 NOUN ejpam-891 132 5 , , PUNCT ejpam-891 132 6 e e X ejpam-891 132 7 ] ] X ejpam-891 132 8 be be AUX ejpam-891 132 9 a a DET ejpam-891 132 10 causal causal ADJ ejpam-891 132 11 map map NOUN ejpam-891 132 12 such such ADJ ejpam-891 132 13 that that PRON ejpam-891 132 14 for for ADP ejpam-891 132 15 t t PROPN ejpam-891 132 16 ∈ ∈ PROPN ejpam-891 133 1 i i PRON ejpam-891 133 2 , , PUNCT ejpam-891 133 3 d[(qu)(t d[(qu)(t PROPN ejpam-891 133 4 ) ) PUNCT ejpam-891 133 5 , , PUNCT ejpam-891 133 6 ( ( PUNCT ejpam-891 133 7 qv qv INTJ ejpam-891 133 8 ) ) PUNCT ejpam-891 133 9 ( ( PUNCT ejpam-891 133 10 t)]≤ t)]≤ PROPN ejpam-891 133 11 g(t g(t PROPN ejpam-891 133 12 , , PUNCT ejpam-891 133 13 d0[u d0[u PROPN ejpam-891 133 14 , , PUNCT ejpam-891 133 15 v v X ejpam-891 133 16 ] ] X ejpam-891 133 17 ( ( PUNCT ejpam-891 133 18 t t PROPN ejpam-891 133 19 ) ) PUNCT ejpam-891 133 20 ] ] PUNCT ejpam-891 133 21 , , PUNCT ejpam-891 133 22 where where SCONJ ejpam-891 133 23 g g PROPN ejpam-891 133 24 ∈ ∈ PROPN ejpam-891 133 25 c[i c[i NUM ejpam-891 133 26 ×r+,r+ ×r+,r+ NOUN ejpam-891 133 27 ] ] PUNCT ejpam-891 133 28 . . PUNCT ejpam-891 133 29 suppose suppose VERB ejpam-891 133 30 further far ADV ejpam-891 133 31 that that SCONJ ejpam-891 133 32 the the DET ejpam-891 133 33 maximal maximal ADJ ejpam-891 133 34 solution solution NOUN ejpam-891 133 35 r(t r(t NOUN ejpam-891 133 36 , , PUNCT ejpam-891 133 37 t0 t0 PROPN ejpam-891 133 38 , , PUNCT ejpam-891 133 39 w0 w0 PROPN ejpam-891 133 40 ) ) PUNCT ejpam-891 133 41 of of ADP ejpam-891 133 42 the the DET ejpam-891 133 43 scalar scalar ADJ ejpam-891 133 44 differential differential ADJ ejpam-891 133 45 equation equation NOUN ejpam-891 133 46 ( ( PUNCT ejpam-891 133 47 15 15 NUM ejpam-891 133 48 ) ) PUNCT ejpam-891 133 49 exists exist VERB ejpam-891 133 50 on on ADP ejpam-891 133 51 i. i. PROPN ejpam-891 133 52 then then ADV ejpam-891 133 53 , , PUNCT ejpam-891 133 54 if if SCONJ ejpam-891 133 55 u(t u(t NOUN ejpam-891 133 56 ) ) PUNCT ejpam-891 133 57 , , PUNCT ejpam-891 133 58 v v X ejpam-891 133 59 ( ( PUNCT ejpam-891 133 60 t t NOUN ejpam-891 133 61 ) ) PUNCT ejpam-891 133 62 are be AUX ejpam-891 133 63 any any DET ejpam-891 133 64 two two NUM ejpam-891 133 65 solutions solution NOUN ejpam-891 133 66 of of ADP ejpam-891 133 67 ( ( PUNCT ejpam-891 133 68 8) 8) NUM ejpam-891 133 69 and and CCONJ ejpam-891 133 70 ( ( PUNCT ejpam-891 133 71 9 9 NUM ejpam-891 133 72 ) ) PUNCT ejpam-891 133 73 with with ADP ejpam-891 133 74 initial initial ADJ ejpam-891 133 75 function function NOUN ejpam-891 133 76 ut0 ut0 NOUN ejpam-891 133 77 = = PUNCT ejpam-891 133 78 vt0 vt0 NOUN ejpam-891 133 79 = = PUNCT ejpam-891 133 80 φ0 φ0 PROPN ejpam-891 133 81 ∈ ∈ PROPN ejpam-891 133 82 c1 c1 NOUN ejpam-891 133 83 , , PUNCT ejpam-891 133 84 then then ADV ejpam-891 133 85 we we PRON ejpam-891 133 86 have have VERB ejpam-891 133 87 d[u(t d[u(t NOUN ejpam-891 133 88 ) ) PUNCT ejpam-891 133 89 , , PUNCT ejpam-891 133 90 v v X ejpam-891 133 91 ( ( PUNCT ejpam-891 133 92 t t PROPN ejpam-891 133 93 ) ) PUNCT ejpam-891 133 94 ] ] PUNCT ejpam-891 133 95 ≤ ≤ NUM ejpam-891 133 96 r(t r(t NOUN ejpam-891 133 97 , , PUNCT ejpam-891 133 98 t0 t0 PROPN ejpam-891 133 99 , , PUNCT ejpam-891 133 100 w0 w0 PROPN ejpam-891 133 101 ) ) PUNCT ejpam-891 133 102 , , PUNCT ejpam-891 133 103 t t PROPN ejpam-891 133 104 ∈ ∈ PROPN ejpam-891 133 105 i i PRON ejpam-891 133 106 proof proof VERB ejpam-891 133 107 . . PUNCT ejpam-891 134 1 we we PRON ejpam-891 134 2 first first ADV ejpam-891 134 3 observe observe VERB ejpam-891 134 4 that that SCONJ ejpam-891 134 5 d0[ut0 d0[ut0 ADV ejpam-891 134 6 , , PUNCT ejpam-891 134 7 vt0 vt0 PROPN ejpam-891 134 8 ] ] PUNCT ejpam-891 134 9 ≤ ≤ NUM ejpam-891 134 10 w0 w0 PROPN ejpam-891 134 11 is be AUX ejpam-891 134 12 satisfied satisfied ADJ ejpam-891 134 13 automatically automatically ADV ejpam-891 134 14 . . PUNCT ejpam-891 135 1 the the DET ejpam-891 135 2 proof proof NOUN ejpam-891 135 3 of of ADP ejpam-891 135 4 the the DET ejpam-891 135 5 theorem theorem NOUN ejpam-891 135 6 is be AUX ejpam-891 135 7 exactly exactly ADV ejpam-891 135 8 same same ADJ ejpam-891 135 9 as as ADP ejpam-891 135 10 that that PRON ejpam-891 135 11 of of ADP ejpam-891 135 12 theorem theorem NOUN ejpam-891 135 13 5.7.2 5.7.2 NUM ejpam-891 135 14 in in ADP ejpam-891 135 15 [ [ X ejpam-891 135 16 4 4 NUM ejpam-891 135 17 ] ] PUNCT ejpam-891 135 18 . . PUNCT ejpam-891 136 1 hence hence ADV ejpam-891 136 2 we we PRON ejpam-891 136 3 avoid avoid VERB ejpam-891 136 4 the the DET ejpam-891 136 5 proof proof NOUN ejpam-891 136 6 . . PUNCT ejpam-891 137 1 we we PRON ejpam-891 137 2 are be AUX ejpam-891 137 3 now now ADV ejpam-891 137 4 in in ADP ejpam-891 137 5 a a DET ejpam-891 137 6 position position NOUN ejpam-891 137 7 to to PART ejpam-891 137 8 state state VERB ejpam-891 137 9 the the DET ejpam-891 137 10 existence existence NOUN ejpam-891 137 11 and and CCONJ ejpam-891 137 12 uniqueness uniqueness NOUN ejpam-891 137 13 result result NOUN ejpam-891 137 14 using use VERB ejpam-891 137 15 successive successive ADJ ejpam-891 137 16 approximations approximation NOUN ejpam-891 137 17 and and CCONJ ejpam-891 137 18 generalized generalized ADJ ejpam-891 137 19 lipschitz lipschitz NOUN ejpam-891 137 20 condition condition NOUN ejpam-891 137 21 . . PUNCT ejpam-891 138 1 once once ADV ejpam-891 138 2 again again ADV ejpam-891 138 3 the the DET ejpam-891 138 4 proof proof NOUN ejpam-891 138 5 is be AUX ejpam-891 138 6 very very ADV ejpam-891 138 7 much much ADV ejpam-891 138 8 similar similar ADJ ejpam-891 138 9 to to ADP ejpam-891 138 10 the the DET ejpam-891 138 11 corresponding corresponding ADJ ejpam-891 138 12 theorem theorem NOUN ejpam-891 138 13 , , PUNCT ejpam-891 138 14 theorem theorem VERB ejpam-891 138 15 5.7.3 5.7.3 NUM ejpam-891 138 16 in in ADP ejpam-891 138 17 [ [ X ejpam-891 138 18 4 4 NUM ejpam-891 138 19 ] ] PUNCT ejpam-891 138 20 . . PUNCT ejpam-891 139 1 hence hence ADV ejpam-891 139 2 we we PRON ejpam-891 139 3 omit omit VERB ejpam-891 139 4 it it PRON ejpam-891 139 5 . . PUNCT ejpam-891 140 1 observe observe VERB ejpam-891 140 2 that that SCONJ ejpam-891 140 3 the the DET ejpam-891 140 4 only only ADJ ejpam-891 140 5 difference difference NOUN ejpam-891 140 6 between between ADP ejpam-891 140 7 the the DET ejpam-891 140 8 two two NUM ejpam-891 140 9 results result NOUN ejpam-891 140 10 is be AUX ejpam-891 140 11 that that SCONJ ejpam-891 140 12 the the DET ejpam-891 140 13 following follow VERB ejpam-891 140 14 theorem theorem NOUN ejpam-891 140 15 has have VERB ejpam-891 140 16 memory memory NOUN ejpam-891 140 17 included include VERB ejpam-891 140 18 in in ADP ejpam-891 140 19 its its PRON ejpam-891 140 20 set set NOUN ejpam-891 140 21 up up ADP ejpam-891 140 22 . . PUNCT ejpam-891 141 1 theorem theorem VERB ejpam-891 141 2 6 6 NUM ejpam-891 141 3 . . PUNCT ejpam-891 142 1 suppose suppose VERB ejpam-891 142 2 that that SCONJ ejpam-891 142 3 ( ( PUNCT ejpam-891 142 4 i i NOUN ejpam-891 142 5 ) ) PUNCT ejpam-891 142 6 q q PROPN ejpam-891 142 7 ∈ ∈ PROPN ejpam-891 142 8 c[b c[b PROPN ejpam-891 142 9 , , PUNCT ejpam-891 142 10 e e X ejpam-891 142 11 ] ] PUNCT ejpam-891 142 12 be be AUX ejpam-891 142 13 a a DET ejpam-891 142 14 causal causal ADJ ejpam-891 142 15 map map NOUN ejpam-891 142 16 , , PUNCT ejpam-891 142 17 where where SCONJ ejpam-891 142 18 b b PROPN ejpam-891 142 19 ⊆ ⊆ NUM ejpam-891 142 20 e0 e0 PROPN ejpam-891 142 21 with with ADP ejpam-891 142 22 b b NOUN ejpam-891 142 23 = = SYM ejpam-891 142 24 { { PUNCT ejpam-891 142 25 u u NOUN ejpam-891 142 26 ∈ ∈ PROPN ejpam-891 142 27 e0 e0 PROPN ejpam-891 142 28 : : PUNCT ejpam-891 142 29 d0[u d0[u NOUN ejpam-891 142 30 , , PUNCT ejpam-891 142 31 φ0(t0)]≤ φ0(t0)]≤ NOUN ejpam-891 142 32 b b NOUN ejpam-891 142 33 , , PUNCT ejpam-891 142 34 d0[ut0 d0[ut0 INTJ ejpam-891 142 35 , , PUNCT ejpam-891 142 36 φ0 φ0 PROPN ejpam-891 142 37 ] ] X ejpam-891 142 38 = = SYM ejpam-891 142 39 0 0 NUM ejpam-891 142 40 , , PUNCT ejpam-891 142 41 t t PROPN ejpam-891 142 42 ∈ ∈ PROPN ejpam-891 143 1 i i X ejpam-891 143 2 } } PUNCT ejpam-891 143 3 and and CCONJ ejpam-891 143 4 d0[q(u d0[q(u PROPN ejpam-891 143 5 , , PUNCT ejpam-891 143 6 φ0),θ φ0),θ NOUN ejpam-891 143 7 ] ] PUNCT ejpam-891 143 8 ≤ ≤ NUM ejpam-891 143 9 m1 m1 NOUN ejpam-891 143 10 on on ADP ejpam-891 143 11 b. b. PROPN ejpam-891 143 12 ( ( PUNCT ejpam-891 143 13 ii ii PROPN ejpam-891 143 14 ) ) PUNCT ejpam-891 143 15 g g PROPN ejpam-891 143 16 ∈ ∈ PROPN ejpam-891 143 17 c[i c[i PROPN ejpam-891 143 18 ×r+,r+]g(t ×r+,r+]g(t NOUN ejpam-891 143 19 , , PUNCT ejpam-891 143 20 u u NOUN ejpam-891 143 21 ) ) PUNCT ejpam-891 143 22 ≤ ≤ NOUN ejpam-891 143 23 m2 m2 PROPN ejpam-891 143 24 on on ADP ejpam-891 143 25 i i PRON ejpam-891 143 26 × × NOUN ejpam-891 144 1 [ [ X ejpam-891 144 2 0,2b 0,2b X ejpam-891 144 3 ] ] X ejpam-891 144 4 , , PUNCT ejpam-891 144 5 g(t g(t PROPN ejpam-891 144 6 , , PUNCT ejpam-891 144 7 0 0 NUM ejpam-891 144 8 ) ) PUNCT ejpam-891 144 9 = = SYM ejpam-891 144 10 0 0 NUM ejpam-891 144 11 , , PUNCT ejpam-891 144 12 g(t g(t PROPN ejpam-891 144 13 , , PUNCT ejpam-891 144 14 u u NOUN ejpam-891 144 15 ) ) PUNCT ejpam-891 144 16 be be AUX ejpam-891 144 17 nondecreasing nondecrease VERB ejpam-891 144 18 in in ADP ejpam-891 144 19 u u NOUN ejpam-891 144 20 for for ADP ejpam-891 144 21 each each DET ejpam-891 144 22 t t NOUN ejpam-891 144 23 ∈ ∈ PROPN ejpam-891 145 1 i i PRON ejpam-891 145 2 and and CCONJ ejpam-891 145 3 w(t w(t PROPN ejpam-891 145 4 ) ) PUNCT ejpam-891 145 5 ≡ ≡ PROPN ejpam-891 145 6 0 0 NUM ejpam-891 145 7 is be AUX ejpam-891 145 8 the the DET ejpam-891 145 9 only only ADJ ejpam-891 145 10 solution solution NOUN ejpam-891 145 11 of of ADP ejpam-891 145 12 w′ w′ PROPN ejpam-891 145 13 = = SYM ejpam-891 145 14 g(t g(t PROPN ejpam-891 145 15 , , PUNCT ejpam-891 145 16 w w NOUN ejpam-891 145 17 ) ) PUNCT ejpam-891 145 18 , , PUNCT ejpam-891 145 19 w(t0 w(t0 PROPN ejpam-891 145 20 ) ) PUNCT ejpam-891 146 1 = = SYM ejpam-891 146 2 0 0 NUM ejpam-891 147 1 on on ADP ejpam-891 147 2 i i PRON ejpam-891 147 3 ( ( PUNCT ejpam-891 147 4 16 16 NUM ejpam-891 147 5 ) ) PUNCT ejpam-891 147 6 ( ( PUNCT ejpam-891 147 7 iii iii X ejpam-891 147 8 ) ) PUNCT ejpam-891 147 9 d[q(u d[q(u PROPN ejpam-891 147 10 , , PUNCT ejpam-891 147 11 φ0)(t),q(v φ0)(t),q(v PROPN ejpam-891 147 12 , , PUNCT ejpam-891 147 13 φ0)(t φ0)(t PROPN ejpam-891 147 14 ) ) PUNCT ejpam-891 147 15 ] ] PUNCT ejpam-891 147 16 ≤ ≤ NUM ejpam-891 147 17 g(t g(t PROPN ejpam-891 147 18 , , PUNCT ejpam-891 147 19 d0[u d0[u PROPN ejpam-891 147 20 , , PUNCT ejpam-891 147 21 v v X ejpam-891 147 22 ] ] X ejpam-891 147 23 ( ( PUNCT ejpam-891 147 24 t t PROPN ejpam-891 147 25 ) ) PUNCT ejpam-891 147 26 ] ] PUNCT ejpam-891 147 27 , , PUNCT ejpam-891 147 28 on on ADP ejpam-891 147 29 b b NOUN ejpam-891 147 30 then then ADV ejpam-891 147 31 the the DET ejpam-891 147 32 successive successive ADJ ejpam-891 147 33 approximations approximation NOUN ejpam-891 147 34 defined define VERB ejpam-891 147 35 by by ADP ejpam-891 147 36 un+1(t un+1(t PROPN ejpam-891 147 37 ) ) PUNCT ejpam-891 147 38 = = SYM ejpam-891 147 39 φ0(t0 φ0(t0 PROPN ejpam-891 147 40 ) ) PUNCT ejpam-891 148 1 + + CCONJ ejpam-891 149 1 ∫ ∫ PROPN ejpam-891 149 2 t t PROPN ejpam-891 149 3 t0 t0 PROPN ejpam-891 149 4 q(un q(un PROPN ejpam-891 149 5 , , PUNCT ejpam-891 149 6 φ0)(s)ds φ0)(s)ds PROPN ejpam-891 149 7 . . PUNCT ejpam-891 150 1 un+1t0 un+1t0 PROPN ejpam-891 150 2 = = PUNCT ejpam-891 150 3 φ0 φ0 PROPN ejpam-891 150 4 ∈ ∈ PROPN ejpam-891 150 5 c1 c1 NOUN ejpam-891 150 6 , , PUNCT ejpam-891 150 7 n= n= ADJ ejpam-891 150 8 0,1,2,3 0,1,2,3 NUM ejpam-891 150 9 . . PUNCT ejpam-891 150 10 . . PUNCT ejpam-891 150 11 . . PUNCT ejpam-891 151 1 exist exist VERB ejpam-891 151 2 on on ADP ejpam-891 151 3 i0 i0 PROPN ejpam-891 151 4 = = PUNCT ejpam-891 152 1 [ [ X ejpam-891 152 2 t0 t0 PROPN ejpam-891 152 3 , , PUNCT ejpam-891 152 4 t0 t0 PROPN ejpam-891 152 5 + + CCONJ ejpam-891 152 6 η η PROPN ejpam-891 152 7 ] ] X ejpam-891 152 8 where where SCONJ ejpam-891 152 9 η η PROPN ejpam-891 152 10 = = PROPN ejpam-891 152 11 min[t min[t PROPN ejpam-891 152 12 − − PROPN ejpam-891 152 13 t0 t0 PROPN ejpam-891 152 14 , , PUNCT ejpam-891 152 15 b b PROPN ejpam-891 152 16 2 2 NUM ejpam-891 152 17 m m NOUN ejpam-891 152 18 ] ] X ejpam-891 152 19 m m VERB ejpam-891 152 20 = = SYM ejpam-891 152 21 max[m1 max[m1 PROPN ejpam-891 152 22 , , PUNCT ejpam-891 152 23 m2 m2 PROPN ejpam-891 152 24 ] ] PUNCT ejpam-891 152 25 and and CCONJ ejpam-891 152 26 converge converge VERB ejpam-891 152 27 uniformly uniformly ADV ejpam-891 152 28 to to ADP ejpam-891 152 29 a a DET ejpam-891 152 30 unique unique ADJ ejpam-891 152 31 solution solution NOUN ejpam-891 152 32 u(t u(t NOUN ejpam-891 152 33 ) ) PUNCT ejpam-891 152 34 of of ADP ejpam-891 152 35 ( ( PUNCT ejpam-891 152 36 8) 8) NUM ejpam-891 152 37 and and CCONJ ejpam-891 152 38 ( ( PUNCT ejpam-891 152 39 9 9 NUM ejpam-891 152 40 ) ) PUNCT ejpam-891 152 41 . . PUNCT ejpam-891 153 1 j. j. PROPN ejpam-891 153 2 devi devi PROPN ejpam-891 153 3 / / SYM ejpam-891 153 4 eur eur PROPN ejpam-891 153 5 . . PUNCT ejpam-891 154 1 j. j. PROPN ejpam-891 154 2 pure pure PROPN ejpam-891 154 3 appl appl PROPN ejpam-891 154 4 . . PROPN ejpam-891 154 5 math math PROPN ejpam-891 154 6 , , PUNCT ejpam-891 154 7 3 3 NUM ejpam-891 154 8 ( ( PUNCT ejpam-891 154 9 2010 2010 NUM ejpam-891 154 10 ) ) PUNCT ejpam-891 154 11 , , PUNCT ejpam-891 154 12 737 737 NUM ejpam-891 154 13 - - SYM ejpam-891 154 14 747 747 NUM ejpam-891 154 15 744 744 NUM ejpam-891 154 16 4 4 NUM ejpam-891 154 17 . . PUNCT ejpam-891 154 18 global global ADJ ejpam-891 154 19 existence existence NOUN ejpam-891 154 20 let let AUX ejpam-891 154 21 be be AUX ejpam-891 154 22 = = PUNCT ejpam-891 154 23 c[[t0,∞ c[[t0,∞ VERB ejpam-891 154 24 ] ] X ejpam-891 154 25 , , PUNCT ejpam-891 154 26 kc(r kc(r NOUN ejpam-891 154 27 n n CCONJ ejpam-891 154 28 ) ) PUNCT ejpam-891 154 29 ] ] PUNCT ejpam-891 154 30 and and CCONJ ejpam-891 154 31 ce0 ce0 VERB ejpam-891 154 32 = = SYM ejpam-891 154 33 c[[t0 c[[t0 PROPN ejpam-891 154 34 − − PROPN ejpam-891 154 35 h1,∞ h1,∞ PROPN ejpam-891 154 36 ] ] PUNCT ejpam-891 154 37 , , PUNCT ejpam-891 154 38 kc(r kc(r NOUN ejpam-891 154 39 n n CCONJ ejpam-891 154 40 ] ] PUNCT ejpam-891 154 41 . . PUNCT ejpam-891 155 1 we we PRON ejpam-891 155 2 now now ADV ejpam-891 155 3 state state VERB ejpam-891 155 4 and and CCONJ ejpam-891 155 5 prove prove VERB ejpam-891 155 6 a a DET ejpam-891 155 7 global global ADJ ejpam-891 155 8 existence existence NOUN ejpam-891 155 9 result result NOUN ejpam-891 155 10 . . PUNCT ejpam-891 156 1 theorem theorem ADJ ejpam-891 156 2 7 7 NUM ejpam-891 156 3 . . PUNCT ejpam-891 156 4 assume assume VERB ejpam-891 156 5 that that SCONJ ejpam-891 156 6 q q PROPN ejpam-891 156 7 ∈ ∈ PROPN ejpam-891 156 8 c[ce0 c[ce0 PROPN ejpam-891 156 9 , , PUNCT ejpam-891 156 10 be be AUX ejpam-891 156 11 ] ] PUNCT ejpam-891 156 12 and and CCONJ ejpam-891 156 13 is be AUX ejpam-891 156 14 smooth smooth ADJ ejpam-891 156 15 enough enough ADV ejpam-891 156 16 to to PART ejpam-891 156 17 guarantee guarantee VERB ejpam-891 156 18 local local ADJ ejpam-891 156 19 existence existence NOUN ejpam-891 156 20 of of ADP ejpam-891 156 21 solutions solution NOUN ejpam-891 156 22 of of ADP ejpam-891 156 23 ivp ivp NOUN ejpam-891 156 24 ( ( PUNCT ejpam-891 156 25 8) 8) NUM ejpam-891 156 26 and and CCONJ ejpam-891 156 27 ( ( PUNCT ejpam-891 156 28 9 9 NUM ejpam-891 156 29 ) ) PUNCT ejpam-891 156 30 for for ADP ejpam-891 156 31 any any DET ejpam-891 156 32 ( ( PUNCT ejpam-891 156 33 t0,φ0(t0 t0,φ0(t0 NOUN ejpam-891 156 34 ) ) PUNCT ejpam-891 156 35 ) ) PUNCT ejpam-891 157 1 ∈ ∈ PROPN ejpam-891 157 2 r+× r+× PROPN ejpam-891 157 3 kc(r kc(r X ejpam-891 157 4 n n CCONJ ejpam-891 157 5 ) ) PUNCT ejpam-891 157 6 . . PUNCT ejpam-891 158 1 further far ADV ejpam-891 158 2 q(u q(u NOUN ejpam-891 158 3 , , PUNCT ejpam-891 158 4 φ0 φ0 PROPN ejpam-891 158 5 ) ) PUNCT ejpam-891 158 6 be be AUX ejpam-891 158 7 such such ADJ ejpam-891 158 8 that that SCONJ ejpam-891 158 9 d[q(u d[q(u PROPN ejpam-891 158 10 , , PUNCT ejpam-891 158 11 φ0)(t),θ φ0)(t),θ NOUN ejpam-891 158 12 ] ] PUNCT ejpam-891 158 13 ≤ ≤ NUM ejpam-891 158 14 g(t g(t PROPN ejpam-891 158 15 , , PUNCT ejpam-891 158 16 d0[u d0[u NOUN ejpam-891 158 17 , , PUNCT ejpam-891 158 18 θ](t θ](t ADJ ejpam-891 158 19 ) ) PUNCT ejpam-891 158 20 ] ] PUNCT ejpam-891 158 21 ( ( PUNCT ejpam-891 158 22 17 17 NUM ejpam-891 158 23 ) ) PUNCT ejpam-891 158 24 where where SCONJ ejpam-891 158 25 g g PROPN ejpam-891 158 26 ∈ ∈ PROPN ejpam-891 158 27 c[r+ c[r+ NOUN ejpam-891 158 28 2,r 2,r NUM ejpam-891 158 29 ] ] X ejpam-891 158 30 , , PUNCT ejpam-891 158 31 g(t g(t PROPN ejpam-891 158 32 , , PUNCT ejpam-891 158 33 w w NOUN ejpam-891 158 34 ) ) PUNCT ejpam-891 158 35 is be AUX ejpam-891 158 36 nondecreasing nondecrease VERB ejpam-891 158 37 in in ADP ejpam-891 158 38 w w NOUN ejpam-891 158 39 for for ADP ejpam-891 158 40 each each DET ejpam-891 158 41 t t NOUN ejpam-891 158 42 ∈ ∈ PROPN ejpam-891 158 43 r+ r+ NOUN ejpam-891 158 44 , , PUNCT ejpam-891 158 45 and and CCONJ ejpam-891 158 46 the the DET ejpam-891 158 47 maximal maximal ADJ ejpam-891 158 48 solution solution NOUN ejpam-891 158 49 r(t r(t NOUN ejpam-891 158 50 ) ) PUNCT ejpam-891 158 51 = = SYM ejpam-891 158 52 r(t r(t NOUN ejpam-891 158 53 , , PUNCT ejpam-891 158 54 t0 t0 PROPN ejpam-891 158 55 , , PUNCT ejpam-891 158 56 w0 w0 PROPN ejpam-891 158 57 ) ) PUNCT ejpam-891 158 58 of of ADP ejpam-891 158 59 scalar scalar ADJ ejpam-891 158 60 ivp ivp X ejpam-891 158 61 ( ( PUNCT ejpam-891 158 62 15 15 NUM ejpam-891 158 63 ) ) PUNCT ejpam-891 158 64 exists exist VERB ejpam-891 158 65 on on ADP ejpam-891 158 66 [ [ X ejpam-891 158 67 t0,∞ t0,∞ NUM ejpam-891 158 68 ) ) PUNCT ejpam-891 158 69 . . PUNCT ejpam-891 159 1 then then ADV ejpam-891 159 2 the the DET ejpam-891 159 3 largest large ADJ ejpam-891 159 4 interval interval NOUN ejpam-891 159 5 of of ADP ejpam-891 159 6 existence existence NOUN ejpam-891 159 7 for for ADP ejpam-891 159 8 any any DET ejpam-891 159 9 solution solution NOUN ejpam-891 159 10 u(t u(t NOUN ejpam-891 159 11 ) ) PUNCT ejpam-891 159 12 of of ADP ejpam-891 159 13 ( ( PUNCT ejpam-891 159 14 8) 8) NUM ejpam-891 159 15 and and CCONJ ejpam-891 159 16 ( ( PUNCT ejpam-891 159 17 9 9 NUM ejpam-891 159 18 ) ) PUNCT ejpam-891 159 19 is be AUX ejpam-891 159 20 [ [ X ejpam-891 159 21 t0,∞ t0,∞ NUM ejpam-891 159 22 ) ) PUNCT ejpam-891 159 23 , , PUNCT ejpam-891 159 24 whenever whenever SCONJ ejpam-891 159 25 d0[φ0,θ d0[φ0,θ PROPN ejpam-891 159 26 ] ] X ejpam-891 159 27 ≤ ≤ ADJ ejpam-891 159 28 w0 w0 PROPN ejpam-891 159 29 proof proof NOUN ejpam-891 159 30 . . PUNCT ejpam-891 160 1 suppose suppose VERB ejpam-891 160 2 that that SCONJ ejpam-891 160 3 u(t u(t NOUN ejpam-891 160 4 ) ) PUNCT ejpam-891 160 5 = = SYM ejpam-891 160 6 u(t u(t NOUN ejpam-891 160 7 , , PUNCT ejpam-891 160 8 t0,φ0(t0 t0,φ0(t0 NOUN ejpam-891 160 9 ) ) PUNCT ejpam-891 160 10 ) ) PUNCT ejpam-891 160 11 with with ADP ejpam-891 160 12 ut0 ut0 PROPN ejpam-891 160 13 = = SYM ejpam-891 160 14 φ0 φ0 PROPN ejpam-891 160 15 be be AUX ejpam-891 160 16 any any DET ejpam-891 160 17 solution solution NOUN ejpam-891 160 18 of of ADP ejpam-891 160 19 ( ( PUNCT ejpam-891 160 20 8) 8) NUM ejpam-891 160 21 and and CCONJ ejpam-891 160 22 ( ( PUNCT ejpam-891 160 23 9 9 X ejpam-891 160 24 ) ) PUNCT ejpam-891 160 25 existing exist VERB ejpam-891 160 26 on on ADP ejpam-891 160 27 [ [ X ejpam-891 160 28 t0,β t0,β NUM ejpam-891 160 29 ) ) PUNCT ejpam-891 160 30 , , PUNCT ejpam-891 160 31 t0 t0 PROPN ejpam-891 160 32 < < X ejpam-891 160 33 β β X ejpam-891 160 34 < < X ejpam-891 160 35 ∞ ∞ PROPN ejpam-891 160 36 , , PUNCT ejpam-891 160 37 with with ADP ejpam-891 160 38 d0[φ0,θ d0[φ0,θ NOUN ejpam-891 160 39 ] ] X ejpam-891 160 40 ≤ ≤ NUM ejpam-891 160 41 w0 w0 PROPN ejpam-891 160 42 and and CCONJ ejpam-891 160 43 the the DET ejpam-891 160 44 value value NOUN ejpam-891 160 45 of of ADP ejpam-891 160 46 β β X ejpam-891 160 47 can can AUX ejpam-891 160 48 not not PART ejpam-891 160 49 be be AUX ejpam-891 160 50 increased increase VERB ejpam-891 160 51 . . PUNCT ejpam-891 161 1 set set VERB ejpam-891 161 2 m(t m(t NOUN ejpam-891 161 3 ) ) PUNCT ejpam-891 161 4 = = PUNCT ejpam-891 162 1 d[u(t),θ d[u(t),θ PROPN ejpam-891 162 2 ] ] PUNCT ejpam-891 162 3 then then ADV ejpam-891 162 4 m(t0 m(t0 PROPN ejpam-891 162 5 ) ) PUNCT ejpam-891 162 6 = = PUNCT ejpam-891 162 7 d[u(t0),θ d[u(t0),θ NOUN ejpam-891 162 8 ] ] PUNCT ejpam-891 162 9 = = PUNCT ejpam-891 162 10 d[φ0(t0),θ]≤ d[φ0(t0),θ]≤ NUM ejpam-891 162 11 d0[φ0,θ d0[φ0,θ X ejpam-891 162 12 ] ] X ejpam-891 162 13 ≤ ≤ NUM ejpam-891 162 14 w0 w0 PROPN ejpam-891 162 15 consider consider VERB ejpam-891 162 16 d+m(t d+m(t NOUN ejpam-891 162 17 ) ) PUNCT ejpam-891 162 18 ≤ ≤ NOUN ejpam-891 162 19 d[dh d[dh ADP ejpam-891 162 20 u(t),θ u(t),θ NOUN ejpam-891 162 21 ] ] PUNCT ejpam-891 162 22 ≤ ≤ NUM ejpam-891 162 23 d[q(u d[q(u NUM ejpam-891 162 24 , , PUNCT ejpam-891 162 25 φ0)(t),θ φ0)(t),θ PROPN ejpam-891 162 26 ] ] PUNCT ejpam-891 162 27 ≤ ≤ NUM ejpam-891 162 28 g(t g(t PROPN ejpam-891 162 29 , , PUNCT ejpam-891 162 30 d0[u d0[u NOUN ejpam-891 162 31 , , PUNCT ejpam-891 162 32 θ](t θ](t ADJ ejpam-891 162 33 ) ) PUNCT ejpam-891 162 34 ) ) PUNCT ejpam-891 162 35 . . PUNCT ejpam-891 163 1 now now ADV ejpam-891 163 2 using use VERB ejpam-891 163 3 the the DET ejpam-891 163 4 comparison comparison NOUN ejpam-891 163 5 theorem theorem VERB ejpam-891 163 6 , , PUNCT ejpam-891 163 7 theorem theorem VERB ejpam-891 163 8 4 4 NUM ejpam-891 163 9 , , PUNCT ejpam-891 163 10 we we PRON ejpam-891 163 11 obtain obtain VERB ejpam-891 163 12 that that DET ejpam-891 163 13 m(t m(t NOUN ejpam-891 163 14 ) ) PUNCT ejpam-891 163 15 ≤ ≤ NOUN ejpam-891 163 16 r(t r(t NOUN ejpam-891 163 17 ) ) PUNCT ejpam-891 163 18 , , PUNCT ejpam-891 163 19 t0 t0 PROPN ejpam-891 163 20 ≤ ≤ PROPN ejpam-891 163 21 t t PROPN ejpam-891 163 22 ≤ ≤ NOUN ejpam-891 163 23 β β X ejpam-891 163 24 . . PUNCT ejpam-891 164 1 for for ADP ejpam-891 164 2 any any DET ejpam-891 164 3 t1 t1 NOUN ejpam-891 164 4 , , PUNCT ejpam-891 164 5 t2 t2 NOUN ejpam-891 164 6 such such ADJ ejpam-891 164 7 that that SCONJ ejpam-891 164 8 t0 t0 PROPN ejpam-891 164 9 < < X ejpam-891 164 10 t1 t1 X ejpam-891 164 11 < < X ejpam-891 164 12 t2 t2 PROPN ejpam-891 164 13 < < X ejpam-891 164 14 β β X ejpam-891 164 15 , , PUNCT ejpam-891 164 16 we we PRON ejpam-891 164 17 obtain obtain VERB ejpam-891 164 18 , , PUNCT ejpam-891 164 19 using use VERB ejpam-891 164 20 the the DET ejpam-891 164 21 properties property NOUN ejpam-891 164 22 of of ADP ejpam-891 164 23 the the DET ejpam-891 164 24 hausdorff hausdorff NOUN ejpam-891 164 25 metric metric NOUN ejpam-891 164 26 , , PUNCT ejpam-891 164 27 and and CCONJ ejpam-891 164 28 relation relation NOUN ejpam-891 164 29 ( ( PUNCT ejpam-891 164 30 17 17 NUM ejpam-891 164 31 ) ) PUNCT ejpam-891 164 32 , , PUNCT ejpam-891 164 33 d[u(t1 d[u(t1 PROPN ejpam-891 164 34 ) ) PUNCT ejpam-891 164 35 , , PUNCT ejpam-891 164 36 u(t2 u(t2 NOUN ejpam-891 164 37 ) ) PUNCT ejpam-891 164 38 ] ] PUNCT ejpam-891 165 1 = = PUNCT ejpam-891 165 2 d d X ejpam-891 165 3 [ [ PUNCT ejpam-891 165 4 ∫ ∫ PROPN ejpam-891 165 5 t t PROPN ejpam-891 165 6 t0 t0 PROPN ejpam-891 165 7 q(u q(u NOUN ejpam-891 165 8 , , PUNCT ejpam-891 165 9 φ0)(s)ds φ0)(s)ds PROPN ejpam-891 165 10 , , PUNCT ejpam-891 165 11 ∫ ∫ PROPN ejpam-891 165 12 t2 t2 PROPN ejpam-891 165 13 t0 t0 PROPN ejpam-891 165 14 q(u q(u NOUN ejpam-891 165 15 , , PUNCT ejpam-891 165 16 φ0)(s)ds φ0)(s)ds PROPN ejpam-891 165 17 ] ] PUNCT ejpam-891 165 18 ≤ ≤ NUM ejpam-891 165 19 ∫ ∫ PROPN ejpam-891 165 20 t2 t2 PROPN ejpam-891 165 21 t1 t1 PROPN ejpam-891 165 22 d[q(u d[q(u PROPN ejpam-891 165 23 , , PUNCT ejpam-891 165 24 φ0)(s),θ]ds φ0)(s),θ]ds ADJ ejpam-891 165 25 ≤ ≤ NUM ejpam-891 165 26 ∫ ∫ PROPN ejpam-891 165 27 t2 t2 PROPN ejpam-891 165 28 t1 t1 PROPN ejpam-891 165 29 g(s g(s PROPN ejpam-891 165 30 , , PUNCT ejpam-891 165 31 d0[u d0[u NOUN ejpam-891 165 32 , , PUNCT ejpam-891 165 33 θ])(s)ds θ])(s)ds PROPN ejpam-891 165 34 = = SYM ejpam-891 165 35 ∫ ∫ PROPN ejpam-891 165 36 t2 t2 PROPN ejpam-891 165 37 t1 t1 PROPN ejpam-891 165 38 g(s g(s PROPN ejpam-891 165 39 , , PUNCT ejpam-891 165 40 | | ADV ejpam-891 165 41 m m PROPN ejpam-891 165 42 |0 |0 X ejpam-891 165 43 ( ( PUNCT ejpam-891 165 44 s)])ds s)])d NOUN ejpam-891 165 45 . . PUNCT ejpam-891 165 46 using use VERB ejpam-891 165 47 the the DET ejpam-891 165 48 fact fact NOUN ejpam-891 165 49 that that SCONJ ejpam-891 165 50 m(t m(t NOUN ejpam-891 165 51 ) ) PUNCT ejpam-891 165 52 = = PUNCT ejpam-891 166 1 d[u(t),θ d[u(t),θ ADP ejpam-891 166 2 ] ] PUNCT ejpam-891 166 3 ≤ ≤ NUM ejpam-891 166 4 r(t r(t NOUN ejpam-891 166 5 ) ) PUNCT ejpam-891 166 6 and and CCONJ ejpam-891 166 7 g(t g(t PROPN ejpam-891 166 8 , , PUNCT ejpam-891 166 9 u u NOUN ejpam-891 166 10 ) ) PUNCT ejpam-891 166 11 is be AUX ejpam-891 166 12 nondecreasing nondecrease VERB ejpam-891 166 13 in in ADP ejpam-891 166 14 u u NOUN ejpam-891 166 15 for for ADP ejpam-891 166 16 each each DET ejpam-891 166 17 t t PROPN ejpam-891 166 18 , , PUNCT ejpam-891 166 19 we we PRON ejpam-891 166 20 get get VERB ejpam-891 166 21 d[u(t1 d[u(t1 NOUN ejpam-891 166 22 ) ) PUNCT ejpam-891 166 23 , , PUNCT ejpam-891 166 24 u(t2)]≤ u(t2)]≤ PROPN ejpam-891 166 25 r(t2)− r(t2)− ADJ ejpam-891 166 26 r(t1 r(t1 NOUN ejpam-891 166 27 ) ) PUNCT ejpam-891 167 1 since since SCONJ ejpam-891 167 2 lim lim PROPN ejpam-891 167 3 t→β t→β NUM ejpam-891 167 4 r(t r(t PROPN ejpam-891 167 5 , , PUNCT ejpam-891 167 6 t0 t0 PROPN ejpam-891 167 7 , , PUNCT ejpam-891 167 8 w0 w0 PROPN ejpam-891 167 9 ) ) PUNCT ejpam-891 167 10 j. j. PROPN ejpam-891 167 11 devi devi PROPN ejpam-891 167 12 / / SYM ejpam-891 167 13 eur eur PROPN ejpam-891 167 14 . . PUNCT ejpam-891 168 1 j. j. PROPN ejpam-891 168 2 pure pure PROPN ejpam-891 168 3 appl appl PROPN ejpam-891 168 4 . . PROPN ejpam-891 168 5 math math PROPN ejpam-891 168 6 , , PUNCT ejpam-891 168 7 3 3 NUM ejpam-891 168 8 ( ( PUNCT ejpam-891 168 9 2010 2010 NUM ejpam-891 168 10 ) ) PUNCT ejpam-891 168 11 , , PUNCT ejpam-891 168 12 737 737 NUM ejpam-891 168 13 - - SYM ejpam-891 168 14 747 747 NUM ejpam-891 168 15 745 745 NUM ejpam-891 168 16 exists exist NOUN ejpam-891 168 17 , , PUNCT ejpam-891 168 18 taking take VERB ejpam-891 168 19 the the DET ejpam-891 168 20 limit limit NOUN ejpam-891 168 21 as as ADP ejpam-891 168 22 t1 t1 NOUN ejpam-891 168 23 , , PUNCT ejpam-891 168 24 t2 t2 PROPN ejpam-891 168 25 → → SYM ejpam-891 168 26 β β PROPN ejpam-891 168 27 − − PROPN ejpam-891 168 28 , , PUNCT ejpam-891 168 29 we we PRON ejpam-891 168 30 conclude conclude VERB ejpam-891 168 31 that that SCONJ ejpam-891 168 32 { { PUNCT ejpam-891 168 33 u(tk u(tk NOUN ejpam-891 168 34 ) ) PUNCT ejpam-891 168 35 } } PUNCT ejpam-891 168 36 is be AUX ejpam-891 168 37 a a DET ejpam-891 168 38 cauchy cauchy ADJ ejpam-891 168 39 sequence sequence NOUN ejpam-891 168 40 and and CCONJ ejpam-891 168 41 therefore therefore ADV ejpam-891 168 42 the the DET ejpam-891 168 43 l l NOUN ejpam-891 168 44 imt→β−u(t imt→β−u(t NOUN ejpam-891 168 45 , , PUNCT ejpam-891 168 46 t0,φ0 t0,φ0 PROPN ejpam-891 168 47 ) ) PUNCT ejpam-891 168 48 = = SYM ejpam-891 169 1 uβ uβ NOUN ejpam-891 169 2 exists exist VERB ejpam-891 169 3 . . PUNCT ejpam-891 170 1 now now ADV ejpam-891 170 2 define define VERB ejpam-891 170 3 φβ φβ PRON ejpam-891 170 4 ( ( PUNCT ejpam-891 170 5 t t NOUN ejpam-891 170 6 ) ) PUNCT ejpam-891 170 7 = = PUNCT ejpam-891 171 1   PROPN ejpam-891 171 2   X ejpam-891 171 3   NOUN ejpam-891 171 4 φ0(t φ0(t PROPN ejpam-891 171 5 ) ) PUNCT ejpam-891 171 6 , , PUNCT ejpam-891 171 7 t0 t0 PROPN ejpam-891 171 8 − − PROPN ejpam-891 171 9 h1 h1 VERB ejpam-891 171 10 ≤ ≤ PROPN ejpam-891 171 11 t t NOUN ejpam-891 171 12 ≤ ≤ NUM ejpam-891 171 13 t0 t0 PROPN ejpam-891 171 14 u(t u(t PROPN ejpam-891 171 15 , , PUNCT ejpam-891 171 16 t0,φ0 t0,φ0 PROPN ejpam-891 171 17 ) ) PUNCT ejpam-891 171 18 , , PUNCT ejpam-891 171 19 t0 t0 PROPN ejpam-891 171 20 ≤ ≤ PROPN ejpam-891 171 21 t t PROPN ejpam-891 171 22 < < X ejpam-891 171 23 β β X ejpam-891 171 24 uβ uβ PROPN ejpam-891 171 25 , , PUNCT ejpam-891 171 26 t t PROPN ejpam-891 171 27 = = PUNCT ejpam-891 171 28 β β NOUN ejpam-891 171 29 and and CCONJ ejpam-891 171 30 consider consider VERB ejpam-891 171 31 the the DET ejpam-891 171 32 ivp ivp ADJ ejpam-891 171 33 dh dh NOUN ejpam-891 171 34 u(t u(t PROPN ejpam-891 171 35 ) ) PUNCT ejpam-891 171 36 = = PUNCT ejpam-891 172 1 q(u q(u NOUN ejpam-891 172 2 , , PUNCT ejpam-891 172 3 φ0)(t φ0)(t PROPN ejpam-891 172 4 ) ) PUNCT ejpam-891 172 5 , , PUNCT ejpam-891 172 6 t t PROPN ejpam-891 172 7 ≥ ≥ NUM ejpam-891 172 8 β β X ejpam-891 172 9 , , PUNCT ejpam-891 172 10 uβ uβ NOUN ejpam-891 172 11 = = NOUN ejpam-891 172 12 φβ φβ NOUN ejpam-891 172 13 on on ADP ejpam-891 172 14 [ [ X ejpam-891 172 15 t0 t0 X ejpam-891 172 16 − − ADP ejpam-891 172 17 h1,β h1,β PROPN ejpam-891 172 18 ] ] PUNCT ejpam-891 172 19 , , PUNCT ejpam-891 172 20 t0 t0 PROPN ejpam-891 172 21 ≥ ≥ NUM ejpam-891 172 22 0 0 NUM ejpam-891 172 23 ( ( PUNCT ejpam-891 172 24 18 18 NUM ejpam-891 172 25 ) ) PUNCT ejpam-891 172 26 setce0 setce0 NOUN ejpam-891 173 1 = = SYM ejpam-891 173 2 c[[t0 c[[t0 NOUN ejpam-891 173 3 − − NOUN ejpam-891 173 4 h1,β h1,β PROPN ejpam-891 173 5 + + CCONJ ejpam-891 173 6 a a DET ejpam-891 173 7 ] ] X ejpam-891 173 8 , , PUNCT ejpam-891 173 9 kc(r kc(r NOUN ejpam-891 173 10 n n CCONJ ejpam-891 173 11 ) ) PUNCT ejpam-891 173 12 ] ] PUNCT ejpam-891 173 13 and and CCONJ ejpam-891 173 14 be be AUX ejpam-891 173 15 = = PUNCT ejpam-891 173 16 c[[t0,β c[[t0,β NOUN ejpam-891 173 17 + + CCONJ ejpam-891 173 18 a a DET ejpam-891 173 19 ] ] X ejpam-891 173 20 , , PUNCT ejpam-891 173 21 kc(r kc(r NOUN ejpam-891 173 22 n n CCONJ ejpam-891 173 23 ) ) PUNCT ejpam-891 173 24 ] ] PUNCT ejpam-891 173 25 , , PUNCT ejpam-891 173 26 a a PRON ejpam-891 173 27 > > X ejpam-891 173 28 0 0 NUM ejpam-891 173 29 , , PUNCT ejpam-891 173 30 bb bb NOUN ejpam-891 173 31 ⊂ce0 ⊂ce0 PROPN ejpam-891 173 32 where where SCONJ ejpam-891 173 33 bb bb NOUN ejpam-891 173 34 = = PUNCT ejpam-891 173 35 { { PUNCT ejpam-891 173 36 u u NOUN ejpam-891 173 37 ∈ce0 ∈ce0 VERB ejpam-891 173 38 : : PUNCT ejpam-891 173 39 d0[u d0[u NOUN ejpam-891 173 40 , , PUNCT ejpam-891 173 41 φ0(t0)]≤ φ0(t0)]≤ PROPN ejpam-891 173 42 b b NOUN ejpam-891 173 43 , , PUNCT ejpam-891 173 44 d0[u(t0),φ0(t0 d0[u(t0),φ0(t0 PROPN ejpam-891 173 45 ) ) PUNCT ejpam-891 173 46 ] ] PUNCT ejpam-891 174 1 = = PUNCT ejpam-891 174 2 0 0 NUM ejpam-891 174 3 , , PUNCT ejpam-891 174 4 t t PROPN ejpam-891 174 5 ∈ ∈ PROPN ejpam-891 174 6 j j PROPN ejpam-891 174 7 } } PUNCT ejpam-891 174 8 then then ADV ejpam-891 174 9 q q NOUN ejpam-891 174 10 : : PUNCT ejpam-891 174 11 bb→ bb→ VERB ejpam-891 174 12 be be AUX ejpam-891 174 13 is be AUX ejpam-891 174 14 a a DET ejpam-891 174 15 causal causal ADJ ejpam-891 174 16 map map NOUN ejpam-891 174 17 such such ADJ ejpam-891 174 18 that that SCONJ ejpam-891 174 19 it it PRON ejpam-891 174 20 guarantees guarantee VERB ejpam-891 174 21 the the DET ejpam-891 174 22 local local ADJ ejpam-891 174 23 existence existence NOUN ejpam-891 174 24 of of ADP ejpam-891 174 25 a a DET ejpam-891 174 26 solution solution NOUN ejpam-891 174 27 , , PUNCT ejpam-891 174 28 hence hence ADV ejpam-891 174 29 there there PRON ejpam-891 174 30 exists exist VERB ejpam-891 174 31 u(t u(t NOUN ejpam-891 174 32 , , PUNCT ejpam-891 174 33 β β X ejpam-891 174 34 , , PUNCT ejpam-891 174 35 uβ uβ PROPN ejpam-891 174 36 ) ) PUNCT ejpam-891 174 37 satisfying satisfy VERB ejpam-891 174 38 ( ( PUNCT ejpam-891 174 39 17 17 NUM ejpam-891 174 40 ) ) PUNCT ejpam-891 174 41 on on ADP ejpam-891 174 42 some some DET ejpam-891 174 43 interval interval NOUN ejpam-891 174 44 [ [ X ejpam-891 174 45 β β X ejpam-891 174 46 , , PUNCT ejpam-891 174 47 β β PROPN ejpam-891 174 48 + + NOUN ejpam-891 174 49 α α NOUN ejpam-891 174 50 ] ] X ejpam-891 174 51 , , PUNCT ejpam-891 174 52 0 0 NUM ejpam-891 174 53 < < X ejpam-891 174 54 α α X ejpam-891 174 55 < < X ejpam-891 174 56 a. a. NOUN ejpam-891 174 57 thus thus ADV ejpam-891 174 58 u(t u(t NOUN ejpam-891 174 59 , , PUNCT ejpam-891 174 60 t0,φ0 t0,φ0 PROPN ejpam-891 174 61 ) ) PUNCT ejpam-891 174 62 can can AUX ejpam-891 174 63 be be AUX ejpam-891 174 64 extended extend VERB ejpam-891 174 65 beyond beyond ADP ejpam-891 174 66 β β NOUN ejpam-891 174 67 , , PUNCT ejpam-891 174 68 contradicting contradict VERB ejpam-891 174 69 our our PRON ejpam-891 174 70 assumption assumption NOUN ejpam-891 174 71 that that SCONJ ejpam-891 174 72 β β PROPN ejpam-891 174 73 can can AUX ejpam-891 174 74 not not PART ejpam-891 174 75 be be AUX ejpam-891 174 76 increased increase VERB ejpam-891 174 77 . . PUNCT ejpam-891 175 1 thus thus ADV ejpam-891 175 2 every every DET ejpam-891 175 3 solution solution NOUN ejpam-891 175 4 u(t u(t NOUN ejpam-891 175 5 , , PUNCT ejpam-891 175 6 t0,φ0 t0,φ0 PROPN ejpam-891 175 7 ) ) PUNCT ejpam-891 175 8 of of ADP ejpam-891 175 9 ( ( PUNCT ejpam-891 175 10 8) 8) NUM ejpam-891 175 11 , , PUNCT ejpam-891 175 12 ( ( PUNCT ejpam-891 175 13 9 9 X ejpam-891 175 14 ) ) PUNCT ejpam-891 175 15 such such ADJ ejpam-891 175 16 that that PRON ejpam-891 175 17 d0[φ0,θ d0[φ0,θ X ejpam-891 175 18 ] ] X ejpam-891 175 19 ≤ ≤ NUM ejpam-891 175 20 w0 w0 PROPN ejpam-891 175 21 exists exist VERB ejpam-891 175 22 globally globally ADV ejpam-891 175 23 on on ADP ejpam-891 175 24 [ [ X ejpam-891 175 25 t0 t0 NOUN ejpam-891 175 26 − − PROPN ejpam-891 175 27 h1,∞ h1,∞ PROPN ejpam-891 175 28 ) ) PUNCT ejpam-891 175 29 . . PUNCT ejpam-891 176 1 hence hence ADV ejpam-891 176 2 the the DET ejpam-891 176 3 proof proof NOUN ejpam-891 176 4 is be AUX ejpam-891 176 5 complete complete ADJ ejpam-891 176 6 . . PUNCT ejpam-891 177 1 theorem theorem ADJ ejpam-891 177 2 8 8 NUM ejpam-891 177 3 . . PUNCT ejpam-891 178 1 let let VERB ejpam-891 178 2 q q PROPN ejpam-891 178 3 ∈ ∈ PROPN ejpam-891 178 4 c1[ce0 c1[ce0 VERB ejpam-891 178 5 , , PUNCT ejpam-891 178 6 be be AUX ejpam-891 178 7 ] ] PUNCT ejpam-891 178 8 and and CCONJ ejpam-891 178 9 satisfy satisfy VERB ejpam-891 178 10 the the DET ejpam-891 178 11 estimate estimate NOUN ejpam-891 178 12 . . PUNCT ejpam-891 179 1 d[q(u d[q(u X ejpam-891 179 2 , , PUNCT ejpam-891 179 3 φ0)(t),θ φ0)(t),θ NOUN ejpam-891 179 4 ] ] PUNCT ejpam-891 179 5 ≤ ≤ NUM ejpam-891 179 6 g(t g(t PROPN ejpam-891 179 7 , , PUNCT ejpam-891 179 8 d[u(t),θ d[u(t),θ ADP ejpam-891 179 9 ] ] X ejpam-891 179 10 ] ] PUNCT ejpam-891 179 11 , , PUNCT ejpam-891 179 12 u u PROPN ejpam-891 179 13 ∈ ∈ PROPN ejpam-891 179 14 ω ω X ejpam-891 179 15 ( ( PUNCT ejpam-891 179 16 19 19 NUM ejpam-891 179 17 ) ) PUNCT ejpam-891 179 18 where where SCONJ ejpam-891 179 19 ω ω NOUN ejpam-891 179 20 = = PRON ejpam-891 179 21 { { PUNCT ejpam-891 179 22 u u NOUN ejpam-891 179 23 ∈ ∈ PROPN ejpam-891 179 24 e0 e0 PROPN ejpam-891 179 25 : : PUNCT ejpam-891 179 26 max max PROPN ejpam-891 179 27 t0−h1≤s≤t t0−h1≤s≤t PROPN ejpam-891 179 28 d[u(s),θ d[u(s),θ NOUN ejpam-891 179 29 ] ] X ejpam-891 179 30 = = PUNCT ejpam-891 179 31 d[u(t),θ d[u(t),θ PROPN ejpam-891 179 32 ] ] PUNCT ejpam-891 179 33 , , PUNCT ejpam-891 179 34 t t PROPN ejpam-891 179 35 ∈ ∈ PROPN ejpam-891 179 36 i i X ejpam-891 179 37 } } PUNCT ejpam-891 179 38 and and CCONJ ejpam-891 179 39 g g PROPN ejpam-891 179 40 ∈ ∈ PROPN ejpam-891 180 1 c[[t0,∞)×r+,r+ c[[t0,∞)×r+,r+ PROPN ejpam-891 180 2 ] ] X ejpam-891 180 3 , , PUNCT ejpam-891 180 4 g(t g(t PROPN ejpam-891 180 5 , , PUNCT ejpam-891 180 6 u u NOUN ejpam-891 180 7 ) ) PUNCT ejpam-891 180 8 is be AUX ejpam-891 180 9 monotone monotone ADJ ejpam-891 180 10 nondecreasing nondecrease VERB ejpam-891 180 11 in in ADP ejpam-891 180 12 u u NOUN ejpam-891 180 13 for for ADP ejpam-891 180 14 each each DET ejpam-891 180 15 t t NOUN ejpam-891 180 16 ∈ ∈ PROPN ejpam-891 181 1 [ [ X ejpam-891 181 2 t0,∞ t0,∞ NUM ejpam-891 181 3 ) ) PUNCT ejpam-891 181 4 . . PUNCT ejpam-891 182 1 assume assume VERB ejpam-891 182 2 that that SCONJ ejpam-891 182 3 for for ADP ejpam-891 182 4 every every DET ejpam-891 182 5 t0 t0 PROPN ejpam-891 182 6 > > X ejpam-891 182 7 0 0 PROPN ejpam-891 182 8 , , PUNCT ejpam-891 182 9 the the DET ejpam-891 182 10 scalar scalar ADJ ejpam-891 182 11 differential differential ADJ ejpam-891 182 12 equation equation NOUN ejpam-891 182 13 u′ u′ PROPN ejpam-891 182 14 = = SYM ejpam-891 182 15 g(t g(t PROPN ejpam-891 182 16 , , PUNCT ejpam-891 182 17 u),u(t0 u),u(t0 NOUN ejpam-891 182 18 ) ) PUNCT ejpam-891 182 19 = = PUNCT ejpam-891 182 20 u0 u0 ADJ ejpam-891 182 21 ≥ ≥ PROPN ejpam-891 182 22 o o NOUN ejpam-891 182 23 , , PUNCT ejpam-891 182 24 ( ( PUNCT ejpam-891 182 25 20 20 NUM ejpam-891 182 26 ) ) PUNCT ejpam-891 182 27 has have VERB ejpam-891 182 28 a a DET ejpam-891 182 29 solution solution NOUN ejpam-891 182 30 u(t u(t NOUN ejpam-891 182 31 ) ) PUNCT ejpam-891 182 32 existing exist VERB ejpam-891 182 33 on on ADP ejpam-891 182 34 [ [ X ejpam-891 182 35 t0,∞ t0,∞ NUM ejpam-891 182 36 ) ) PUNCT ejpam-891 182 37 . . PUNCT ejpam-891 183 1 then then ADV ejpam-891 183 2 for for ADP ejpam-891 183 3 φ0 φ0 PROPN ejpam-891 183 4 ∈ ∈ PROPN ejpam-891 183 5 c1 c1 NOUN ejpam-891 183 6 such such ADJ ejpam-891 183 7 that that SCONJ ejpam-891 183 8 d0[φ0,θ d0[φ0,θ X ejpam-891 183 9 ] ] X ejpam-891 183 10 ≤ ≤ ADJ ejpam-891 183 11 u0 u0 NOUN ejpam-891 183 12 there there PRON ejpam-891 183 13 exists exist VERB ejpam-891 183 14 a a DET ejpam-891 183 15 solution solution NOUN ejpam-891 183 16 u(t u(t NOUN ejpam-891 183 17 ) ) PUNCT ejpam-891 183 18 of of ADP ejpam-891 183 19 ( ( PUNCT ejpam-891 183 20 8) 8) NUM ejpam-891 183 21 , , PUNCT ejpam-891 183 22 ( ( PUNCT ejpam-891 183 23 9 9 NUM ejpam-891 183 24 ) ) PUNCT ejpam-891 183 25 on on ADP ejpam-891 183 26 [ [ X ejpam-891 183 27 t0,∞ t0,∞ NOUN ejpam-891 183 28 ) ) PUNCT ejpam-891 183 29 satisfying satisfy VERB ejpam-891 183 30 d[u(t),θ d[u(t),θ NOUN ejpam-891 183 31 ] ] PUNCT ejpam-891 183 32 ≤ ≤ ADJ ejpam-891 183 33 u(t u(t NOUN ejpam-891 183 34 ) ) PUNCT ejpam-891 183 35 , , PUNCT ejpam-891 183 36 t t PROPN ejpam-891 183 37 ∈ ∈ PROPN ejpam-891 184 1 [ [ X ejpam-891 184 2 t0,∞ t0,∞ NUM ejpam-891 184 3 ) ) PUNCT ejpam-891 184 4 ( ( PUNCT ejpam-891 184 5 21 21 NUM ejpam-891 184 6 ) ) PUNCT ejpam-891 184 7 proof proof NOUN ejpam-891 184 8 . . PUNCT ejpam-891 185 1 consider consider VERB ejpam-891 185 2 the the DET ejpam-891 185 3 space space NOUN ejpam-891 185 4 be be AUX ejpam-891 185 5 , , PUNCT ejpam-891 185 6 of of ADP ejpam-891 185 7 all all DET ejpam-891 185 8 continuous continuous ADJ ejpam-891 185 9 functions function NOUN ejpam-891 185 10 from from ADP ejpam-891 185 11 [ [ X ejpam-891 185 12 t0,∞ t0,∞ NUM ejpam-891 185 13 ) ) PUNCT ejpam-891 185 14 to to ADP ejpam-891 185 15 kc(r kc(r NOUN ejpam-891 185 16 n n CCONJ ejpam-891 185 17 ) ) PUNCT ejpam-891 185 18 , , PUNCT ejpam-891 185 19 and and CCONJ ejpam-891 185 20 a a DET ejpam-891 185 21 family family NOUN ejpam-891 185 22 of of ADP ejpam-891 185 23 pseudonorms pseudonorm NOUN ejpam-891 185 24 { { PUNCT ejpam-891 185 25 pn(u pn(u NOUN ejpam-891 185 26 ) ) PUNCT ejpam-891 185 27 } } PUNCT ejpam-891 185 28 ∞ ∞ NUM ejpam-891 185 29 n=1 n=1 PROPN ejpam-891 185 30 be be AUX ejpam-891 185 31 defined define VERB ejpam-891 185 32 for for SCONJ ejpam-891 185 33 u u PROPN ejpam-891 185 34 ∈ ∈ PROPN ejpam-891 185 35 be be AUX ejpam-891 185 36 , , PUNCT ejpam-891 185 37 pn(u pn(u NOUN ejpam-891 185 38 ) ) PUNCT ejpam-891 185 39 = = SYM ejpam-891 185 40 sup sup NOUN ejpam-891 185 41 t0≤t≤n t0≤t≤n VERB ejpam-891 185 42 d[u(t),θ d[u(t),θ ADP ejpam-891 185 43 ] ] PUNCT ejpam-891 185 44 . . PUNCT ejpam-891 186 1 j. j. PROPN ejpam-891 186 2 devi devi PROPN ejpam-891 186 3 / / SYM ejpam-891 186 4 eur eur PROPN ejpam-891 186 5 . . PUNCT ejpam-891 187 1 j. j. PROPN ejpam-891 187 2 pure pure PROPN ejpam-891 187 3 appl appl PROPN ejpam-891 187 4 . . PROPN ejpam-891 187 5 math math PROPN ejpam-891 187 6 , , PUNCT ejpam-891 187 7 3 3 NUM ejpam-891 187 8 ( ( PUNCT ejpam-891 187 9 2010 2010 NUM ejpam-891 187 10 ) ) PUNCT ejpam-891 187 11 , , PUNCT ejpam-891 187 12 737 737 NUM ejpam-891 187 13 - - SYM ejpam-891 187 14 747 747 NUM ejpam-891 187 15 746 746 NUM ejpam-891 187 16 let let VERB ejpam-891 187 17 the the DET ejpam-891 187 18 topology topology NOUN ejpam-891 187 19 on on ADP ejpam-891 187 20 be be AUX ejpam-891 187 21 be be AUX ejpam-891 187 22 generated generate VERB ejpam-891 187 23 by by ADP ejpam-891 187 24 this this DET ejpam-891 187 25 family family NOUN ejpam-891 187 26 . . PUNCT ejpam-891 188 1 a a DET ejpam-891 188 2 fundamental fundamental ADJ ejpam-891 188 3 system system NOUN ejpam-891 188 4 of of ADP ejpam-891 188 5 neighborhoods neighborhood NOUN ejpam-891 188 6 is be AUX ejpam-891 188 7 then then ADV ejpam-891 188 8 given give VERB ejpam-891 188 9 by by ADP ejpam-891 188 10 { { PUNCT ejpam-891 188 11 vn(u vn(u NOUN ejpam-891 188 12 } } PUNCT ejpam-891 188 13 ∞ ∞ PROPN ejpam-891 188 14 n=1 n=1 PROPN ejpam-891 188 15 , , PUNCT ejpam-891 188 16 where where SCONJ ejpam-891 188 17 vn(u vn(u NOUN ejpam-891 188 18 ) ) PUNCT ejpam-891 188 19 = = PUNCT ejpam-891 189 1 { { PUNCT ejpam-891 189 2 u u NOUN ejpam-891 189 3 ∈ ∈ PROPN ejpam-891 189 4 be be VERB ejpam-891 189 5 : : PUNCT ejpam-891 189 6 pn(u)≤ pn(u)≤ NOUN ejpam-891 189 7 1 1 NUM ejpam-891 189 8 } } PUNCT ejpam-891 189 9 under under ADP ejpam-891 189 10 this this DET ejpam-891 189 11 topology topology NOUN ejpam-891 189 12 , , PUNCT ejpam-891 189 13 be be AUX ejpam-891 189 14 becomes become VERB ejpam-891 189 15 a a DET ejpam-891 189 16 complete complete ADJ ejpam-891 189 17 , , PUNCT ejpam-891 189 18 locally locally ADV ejpam-891 189 19 convex convex ADJ ejpam-891 189 20 linear linear ADJ ejpam-891 189 21 space space NOUN ejpam-891 189 22 . . PUNCT ejpam-891 190 1 now now ADV ejpam-891 190 2 define define VERB ejpam-891 190 3 a a DET ejpam-891 190 4 subset subset NOUN ejpam-891 190 5 e e AUX ejpam-891 190 6 ⊂ ⊂ PROPN ejpam-891 190 7 be be AUX ejpam-891 190 8 as as SCONJ ejpam-891 190 9 follows follow VERB ejpam-891 190 10 . . PUNCT ejpam-891 191 1 e e X ejpam-891 191 2 = = PRON ejpam-891 191 3 { { PUNCT ejpam-891 191 4 u u NOUN ejpam-891 191 5 ∈ ∈ PROPN ejpam-891 191 6 ω ω NOUN ejpam-891 191 7 : : PUNCT ejpam-891 191 8 d[u(t),θ d[u(t),θ PROPN ejpam-891 191 9 ] ] PUNCT ejpam-891 191 10 ≤ ≤ ADJ ejpam-891 191 11 u(t u(t NOUN ejpam-891 191 12 ) ) PUNCT ejpam-891 191 13 , , PUNCT ejpam-891 191 14 t t PROPN ejpam-891 191 15 ≥ ≥ PROPN ejpam-891 191 16 t0 t0 PROPN ejpam-891 191 17 } } PUNCT ejpam-891 191 18 where where SCONJ ejpam-891 191 19 u(t u(t NOUN ejpam-891 191 20 ) ) PUNCT ejpam-891 191 21 is be AUX ejpam-891 191 22 a a DET ejpam-891 191 23 solution solution NOUN ejpam-891 191 24 of of ADP ejpam-891 191 25 ( ( PUNCT ejpam-891 191 26 20 20 NUM ejpam-891 191 27 ) ) PUNCT ejpam-891 191 28 existing exist VERB ejpam-891 191 29 on on ADP ejpam-891 191 30 [ [ X ejpam-891 191 31 t0,∞ t0,∞ NUM ejpam-891 191 32 ) ) PUNCT ejpam-891 191 33 . . PUNCT ejpam-891 192 1 then then ADV ejpam-891 192 2 under under ADP ejpam-891 192 3 the the DET ejpam-891 192 4 topology topology NOUN ejpam-891 192 5 of of ADP ejpam-891 192 6 be be AUX ejpam-891 192 7 , , PUNCT ejpam-891 192 8 e e NOUN ejpam-891 192 9 is be AUX ejpam-891 192 10 closed close VERB ejpam-891 192 11 convex convex NOUN ejpam-891 192 12 and and CCONJ ejpam-891 192 13 bounded bound VERB ejpam-891 192 14 . . PUNCT ejpam-891 193 1 consider consider VERB ejpam-891 193 2 the the DET ejpam-891 193 3 integral integral ADJ ejpam-891 193 4 operator operator NOUN ejpam-891 193 5 defined define VERB ejpam-891 193 6 by by ADP ejpam-891 193 7 ( ( PUNCT ejpam-891 193 8 t t NOUN ejpam-891 193 9 u)(t u)(t ADJ ejpam-891 193 10 ) ) PUNCT ejpam-891 194 1 = = SYM ejpam-891 194 2 φ0(t0 φ0(t0 PROPN ejpam-891 194 3 ) ) PUNCT ejpam-891 195 1 + + CCONJ ejpam-891 195 2 ∫ ∫ PROPN ejpam-891 195 3 t t PROPN ejpam-891 195 4 t0 t0 PROPN ejpam-891 195 5 q(u q(u NOUN ejpam-891 195 6 , , PUNCT ejpam-891 195 7 φ0)(s)ds φ0)(s)ds PROPN ejpam-891 195 8 and and CCONJ ejpam-891 195 9 ut0 ut0 PRON ejpam-891 195 10 = = NOUN ejpam-891 195 11 φ0 φ0 PROPN ejpam-891 195 12 ∈ ∈ PROPN ejpam-891 195 13 c1 c1 NOUN ejpam-891 195 14 it it PRON ejpam-891 195 15 is be AUX ejpam-891 195 16 obvious obvious ADJ ejpam-891 195 17 that that SCONJ ejpam-891 195 18 a a DET ejpam-891 195 19 fixed fix VERB ejpam-891 195 20 point point NOUN ejpam-891 195 21 of of ADP ejpam-891 195 22 t t PROPN ejpam-891 195 23 will will AUX ejpam-891 195 24 be be AUX ejpam-891 195 25 a a DET ejpam-891 195 26 solution solution NOUN ejpam-891 195 27 of of ADP ejpam-891 195 28 the the DET ejpam-891 195 29 ivp ivp X ejpam-891 195 30 ( ( PUNCT ejpam-891 195 31 8),(9 8),(9 NOUN ejpam-891 195 32 ) ) PUNCT ejpam-891 195 33 . . PUNCT ejpam-891 196 1 the the DET ejpam-891 196 2 operator operator NOUN ejpam-891 196 3 t t NOUN ejpam-891 196 4 is be AUX ejpam-891 196 5 compact compact ADJ ejpam-891 196 6 in in ADP ejpam-891 196 7 the the DET ejpam-891 196 8 topology topology NOUN ejpam-891 196 9 of of ADP ejpam-891 196 10 be be AUX ejpam-891 196 11 and and CCONJ ejpam-891 196 12 therefore therefore ADV ejpam-891 196 13 closure closure NOUN ejpam-891 196 14 of of ADP ejpam-891 196 15 t t PROPN ejpam-891 196 16 e e NOUN ejpam-891 196 17 is be AUX ejpam-891 196 18 compact compact ADJ ejpam-891 196 19 since since SCONJ ejpam-891 196 20 be be AUX ejpam-891 196 21 is be AUX ejpam-891 196 22 bounded bound VERB ejpam-891 196 23 . . PUNCT ejpam-891 197 1 the the DET ejpam-891 197 2 proof proof NOUN ejpam-891 197 3 of of ADP ejpam-891 197 4 the the DET ejpam-891 197 5 theorem theorem NOUN ejpam-891 197 6 is be AUX ejpam-891 197 7 complete complete ADJ ejpam-891 197 8 , , PUNCT ejpam-891 197 9 if if SCONJ ejpam-891 197 10 we we PRON ejpam-891 197 11 show show VERB ejpam-891 197 12 that that SCONJ ejpam-891 197 13 t t NOUN ejpam-891 197 14 e e PROPN ejpam-891 197 15 ⊆ ⊆ NUM ejpam-891 197 16 e. e. PROPN ejpam-891 197 17 hence hence ADV ejpam-891 197 18 consider consider VERB ejpam-891 197 19 u u PRON ejpam-891 197 20 ∈ ∈ PROPN ejpam-891 197 21 e. e. PROPN ejpam-891 197 22 then then ADV ejpam-891 197 23 d[(t d[(t NOUN ejpam-891 197 24 u)(t),θ u)(t),θ PROPN ejpam-891 197 25 ] ] X ejpam-891 197 26 = = PUNCT ejpam-891 197 27 d[φ0(t0 d[φ0(t0 NOUN ejpam-891 197 28 ) ) PUNCT ejpam-891 198 1 + + CCONJ ejpam-891 198 2 ∫ ∫ PROPN ejpam-891 198 3 t t PROPN ejpam-891 198 4 t0 t0 PROPN ejpam-891 198 5 q(u q(u NOUN ejpam-891 198 6 , , PUNCT ejpam-891 198 7 φ0)(s)ds φ0)(s)ds PROPN ejpam-891 198 8 , , PUNCT ejpam-891 198 9 θ θ NOUN ejpam-891 198 10 ] ] PUNCT ejpam-891 198 11 ≤ ≤ X ejpam-891 198 12 d0[φ0,θ d0[φ0,θ X ejpam-891 198 13 ] ] X ejpam-891 199 1 + + NUM ejpam-891 199 2 ∫ ∫ PROPN ejpam-891 199 3 t t PROPN ejpam-891 199 4 t0 t0 PROPN ejpam-891 199 5 d[q(u d[q(u X ejpam-891 199 6 , , PUNCT ejpam-891 199 7 φ)(t),θ φ)(t),θ PROPN ejpam-891 199 8 ] ] PUNCT ejpam-891 199 9 ≤ ≤ NUM ejpam-891 199 10 d0[φ0,θ d0[φ0,θ X ejpam-891 199 11 ] ] X ejpam-891 200 1 + + NUM ejpam-891 200 2 ∫ ∫ PROPN ejpam-891 200 3 t t PROPN ejpam-891 200 4 t0 t0 PROPN ejpam-891 200 5 g(t g(t PROPN ejpam-891 200 6 , , PUNCT ejpam-891 200 7 d[u(t),θ])ds d[u(t),θ])ds PROPN ejpam-891 200 8 ≤ ≤ NUM ejpam-891 200 9 d0[φ0,θ d0[φ0,θ X ejpam-891 200 10 ] ] X ejpam-891 201 1 + + NUM ejpam-891 201 2 ∫ ∫ PROPN ejpam-891 201 3 t t PROPN ejpam-891 201 4 t0 t0 PROPN ejpam-891 201 5 g(t g(t PROPN ejpam-891 201 6 , , PUNCT ejpam-891 201 7 u(s))ds u(s))ds PROPN ejpam-891 201 8 because because SCONJ ejpam-891 201 9 of of ADP ejpam-891 201 10 the the DET ejpam-891 201 11 relation relation NOUN ejpam-891 201 12 ( ( PUNCT ejpam-891 201 13 18),(20 18),(20 NUM ejpam-891 201 14 ) ) PUNCT ejpam-891 201 15 and and CCONJ ejpam-891 201 16 ( ( PUNCT ejpam-891 201 17 21 21 NUM ejpam-891 201 18 ) ) PUNCT ejpam-891 201 19 and and CCONJ ejpam-891 201 20 the the DET ejpam-891 201 21 monotonic monotonic ADJ ejpam-891 201 22 nature nature NOUN ejpam-891 201 23 of of ADP ejpam-891 201 24 g g PROPN ejpam-891 201 25 , , PUNCT ejpam-891 201 26 the the DET ejpam-891 201 27 definition definition NOUN ejpam-891 201 28 of of ADP ejpam-891 201 29 the the DET ejpam-891 201 30 set set NOUN ejpam-891 201 31 e e NOUN ejpam-891 201 32 and and CCONJ ejpam-891 201 33 the the DET ejpam-891 201 34 fact fact NOUN ejpam-891 201 35 that that SCONJ ejpam-891 201 36 u(t u(t NOUN ejpam-891 201 37 ) ) PUNCT ejpam-891 201 38 is be AUX ejpam-891 201 39 a a DET ejpam-891 201 40 solution solution NOUN ejpam-891 201 41 of of ADP ejpam-891 201 42 ( ( PUNCT ejpam-891 201 43 20 20 NUM ejpam-891 201 44 ) ) PUNCT ejpam-891 201 45 , , PUNCT ejpam-891 201 46 with with ADP ejpam-891 201 47 d0[φ0,θ d0[φ0,θ NOUN ejpam-891 201 48 ] ] X ejpam-891 201 49 ≤ ≤ X ejpam-891 201 50 u0 u0 PROPN ejpam-891 201 51 . . PUNCT ejpam-891 202 1 this this DET ejpam-891 202 2 yields yield NOUN ejpam-891 202 3 d[(t d[(t NOUN ejpam-891 202 4 u)(t),θ u)(t),θ NOUN ejpam-891 202 5 ] ] PUNCT ejpam-891 202 6 ≤ ≤ NUM ejpam-891 202 7 u0 u0 PROPN ejpam-891 202 8 + + X ejpam-891 202 9 ∫ ∫ PROPN ejpam-891 202 10 t t PROPN ejpam-891 202 11 t0 t0 PROPN ejpam-891 202 12 g(s g(s PROPN ejpam-891 202 13 , , PUNCT ejpam-891 202 14 u(s))ds u(s))ds PROPN ejpam-891 202 15 = = SYM ejpam-891 202 16 u(t u(t NOUN ejpam-891 202 17 ) ) PUNCT ejpam-891 203 1 hence hence ADV ejpam-891 203 2 t t X ejpam-891 203 3 u u X ejpam-891 203 4 ∈ ∈ PROPN ejpam-891 203 5 e e NOUN ejpam-891 203 6 or or CCONJ ejpam-891 203 7 t t PROPN ejpam-891 203 8 e e PROPN ejpam-891 203 9 ⊆ ⊆ NUM ejpam-891 203 10 e. e. PROPN ejpam-891 203 11 thus thus ADV ejpam-891 203 12 the the DET ejpam-891 203 13 proof proof NOUN ejpam-891 203 14 is be AUX ejpam-891 203 15 complete complete ADJ ejpam-891 203 16 . . PUNCT ejpam-891 204 1 acknowledgements acknowledgement NOUN ejpam-891 204 2 this this DET ejpam-891 204 3 work work NOUN ejpam-891 204 4 has have AUX ejpam-891 204 5 been be AUX ejpam-891 204 6 done do VERB ejpam-891 204 7 under under ADP ejpam-891 204 8 the the DET ejpam-891 204 9 project project NOUN ejpam-891 204 10 no no INTJ ejpam-891 204 11 . . PUNCT ejpam-891 205 1 sr sr PROPN ejpam-891 205 2 / / SYM ejpam-891 205 3 s4 s4 PROPN ejpam-891 205 4 / / SYM ejpam-891 205 5 ms ms NOUN ejpam-891 205 6 : : PUNCT ejpam-891 205 7 491/07 491/07 NUM ejpam-891 205 8 sanctioned sanction VERB ejpam-891 205 9 by by ADP ejpam-891 205 10 department department PROPN ejpam-891 205 11 of of ADP ejpam-891 205 12 science science NOUN ejpam-891 205 13 and and CCONJ ejpam-891 205 14 technology technology NOUN ejpam-891 205 15 , , PUNCT ejpam-891 205 16 government government NOUN ejpam-891 205 17 of of ADP ejpam-891 205 18 india india PROPN ejpam-891 205 19 . . PUNCT ejpam-891 206 1 the the DET ejpam-891 206 2 author author NOUN ejpam-891 206 3 acknowledges acknowledge VERB ejpam-891 206 4 their their PRON ejpam-891 206 5 support support NOUN ejpam-891 206 6 references reference NOUN ejpam-891 206 7 747 747 NUM ejpam-891 206 8 references reference NOUN ejpam-891 206 9 [ [ X ejpam-891 206 10 1 1 NUM ejpam-891 206 11 ] ] PUNCT ejpam-891 206 12 c.corduneanu c.corduneanu NOUN ejpam-891 206 13 , , PUNCT ejpam-891 206 14 functional functional ADJ ejpam-891 206 15 equations equation NOUN ejpam-891 206 16 with with ADP ejpam-891 206 17 causal causal ADJ ejpam-891 206 18 operators operator NOUN ejpam-891 206 19 , , PUNCT ejpam-891 206 20 taylor taylor PROPN ejpam-891 206 21 and and CCONJ ejpam-891 206 22 francis francis PROPN ejpam-891 206 23 , , PUNCT ejpam-891 206 24 newyork newyork PROPN ejpam-891 206 25 ( ( PUNCT ejpam-891 206 26 2003 2003 NUM ejpam-891 206 27 ) ) PUNCT ejpam-891 206 28 . . PUNCT ejpam-891 207 1 [ [ X ejpam-891 207 2 2 2 NUM ejpam-891 207 3 ] ] SYM ejpam-891 207 4 v.lakshmikantham v.lakshmikantham PROPN ejpam-891 207 5 and and CCONJ ejpam-891 207 6 s.leela s.leela PROPN ejpam-891 207 7 , , PUNCT ejpam-891 207 8 differential differential ADJ ejpam-891 207 9 and and CCONJ ejpam-891 207 10 integral integral ADJ ejpam-891 207 11 inequalities inequality NOUN ejpam-891 207 12 , , PUNCT ejpam-891 207 13 vol.i vol.i PROPN ejpam-891 207 14 and and CCONJ ejpam-891 207 15 ii ii PROPN ejpam-891 207 16 , , PUNCT ejpam-891 207 17 academic academic ADJ ejpam-891 207 18 press press NOUN ejpam-891 207 19 , , PUNCT ejpam-891 207 20 newyork newyork NOUN ejpam-891 207 21 , , PUNCT ejpam-891 207 22 ( ( PUNCT ejpam-891 207 23 1969 1969 NUM ejpam-891 207 24 ) ) PUNCT ejpam-891 207 25 . . PUNCT ejpam-891 208 1 [ [ X ejpam-891 208 2 3 3 X ejpam-891 208 3 ] ] SYM ejpam-891 208 4 v.lakshmikantham v.lakshmikantham PROPN ejpam-891 208 5 and and CCONJ ejpam-891 208 6 s.leela s.leela PRON ejpam-891 208 7 , , PUNCT ejpam-891 208 8 z.drici z.drici PROPN ejpam-891 208 9 and and CCONJ ejpam-891 208 10 mcrae mcrae PROPN ejpam-891 208 11 fa fa PROPN ejpam-891 208 12 , , PUNCT ejpam-891 208 13 theory theory NOUN ejpam-891 208 14 of of ADP ejpam-891 208 15 causal causal ADJ ejpam-891 208 16 differential differential ADJ ejpam-891 208 17 equations equation NOUN ejpam-891 208 18 , , PUNCT ejpam-891 208 19 atlantis atlantis PROPN ejpam-891 208 20 press press PROPN ejpam-891 208 21 and and CCONJ ejpam-891 208 22 world world NOUN ejpam-891 208 23 scientific scientific ADJ ejpam-891 208 24 , , PUNCT ejpam-891 208 25 ( ( PUNCT ejpam-891 208 26 2009 2009 NUM ejpam-891 208 27 ) ) PUNCT ejpam-891 208 28 . . PUNCT ejpam-891 209 1 [ [ X ejpam-891 209 2 4 4 X ejpam-891 209 3 ] ] SYM ejpam-891 209 4 v.lakshmikantham v.lakshmikantham NOUN ejpam-891 209 5 , , PUNCT ejpam-891 209 6 t.gnanabhaskar t.gnanabhaskar NOUN ejpam-891 209 7 and and CCONJ ejpam-891 209 8 j.vasundhara j.vasundhara ADJ ejpam-891 209 9 devi devi PROPN ejpam-891 209 10 , , PUNCT ejpam-891 209 11 theory theory NOUN ejpam-891 209 12 of of ADP ejpam-891 209 13 set set VERB ejpam-891 209 14 differential differential ADJ ejpam-891 209 15 equations equation NOUN ejpam-891 209 16 in in ADP ejpam-891 209 17 metric metric ADJ ejpam-891 209 18 spaces space NOUN ejpam-891 209 19 , , PUNCT ejpam-891 209 20 cambridge cambridge NOUN ejpam-891 209 21 scientific scientific ADJ ejpam-891 209 22 publishers publisher NOUN ejpam-891 209 23 , , PUNCT ejpam-891 209 24 2(006 2(006 NUM ejpam-891 209 25 ) ) PUNCT ejpam-891 209 26 . . PUNCT ejpam-891 210 1 [ [ X ejpam-891 210 2 5 5 NUM ejpam-891 210 3 ] ] SYM ejpam-891 210 4 v.lakshmikantham v.lakshmikantham NUM ejpam-891 210 5 and and CCONJ ejpam-891 210 6 m.rama m.rama PROPN ejpam-891 210 7 mohan mohan PROPN ejpam-891 210 8 rao rao PROPN ejpam-891 210 9 , , PUNCT ejpam-891 210 10 theory theory NOUN ejpam-891 210 11 of of ADP ejpam-891 210 12 integro integro PROPN ejpam-891 210 13 differential differential ADJ ejpam-891 210 14 equations equation NOUN ejpam-891 210 15 , , PUNCT ejpam-891 210 16 gordan gordan PROPN ejpam-891 210 17 and and CCONJ ejpam-891 210 18 breach breach VERB ejpam-891 210 19 science science NOUN ejpam-891 210 20 publishers publisher NOUN ejpam-891 210 21 , , PUNCT ejpam-891 210 22 amsterdam amsterdam PROPN ejpam-891 210 23 , , PUNCT ejpam-891 210 24 ( ( PUNCT ejpam-891 210 25 1995 1995 NUM ejpam-891 210 26 ) ) PUNCT ejpam-891 210 27 . . PUNCT ejpam-891 211 1 [ [ X ejpam-891 211 2 6 6 NUM ejpam-891 211 3 ] ] PUNCT ejpam-891 211 4 j.vasundhra j.vasundhra NOUN ejpam-891 211 5 devi devi PROPN ejpam-891 211 6 , , PUNCT ejpam-891 211 7 comparison comparison NOUN ejpam-891 211 8 theorems theorem NOUN ejpam-891 211 9 and and CCONJ ejpam-891 211 10 existence existence NOUN ejpam-891 211 11 results result VERB ejpam-891 211 12 for for ADP ejpam-891 211 13 set set VERB ejpam-891 211 14 causal causal NOUN ejpam-891 211 15 operators operator NOUN ejpam-891 211 16 with with ADP ejpam-891 211 17 memory memory NOUN ejpam-891 211 18 submitted submit VERB ejpam-891 211 19 to to ADP ejpam-891 211 20 nonlinear nonlinear ADJ ejpam-891 211 21 analysis analysis NOUN ejpam-891 211 22 , , PUNCT ejpam-891 211 23 tma tma PROPN ejpam-891 211 24 . . PUNCT