NONLOCAL BOUNDARY VALUE PROBLEM FOR SECOND ORDER ANTI-PERIODIC NONLINEAR IMPULSIVE q k INTEGRODIFFERENCE EQUATION
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1 Euroean Journal of ahemac an Comuer Scence Vol No 7 ISSN NONLOCAL BOUNDARY VALUE PROBLE FOR SECOND ORDER ANTI-PERIODIC NONLINEAR IPULSIVE - INTEGRODIFFERENCE EQUATION Hao Wang Yuhang Zhang ngyang Shen & Yanheng He * Dearmen of ahemac Yanban Unvery Yanj PR CHINA ABSTRACT A econ orer an-eroc nonlnear mulve negrofference euaon whn he frame of - uanum calculu nvegae by alyng ung fxe on heorem The conon for exence an unuene of oluon are obane KEYWORDS: - Inegrofference Euaon - ervave - Value Problem INTRODUCTION negral Bounary The calculu wa nae n wene of he la cenury However ha gane conerable oulary an morance urng he la hree ecae or o Ther uy ha no only moran heorecal meanng bu alo we alcaon n conformal uanum mechanc hgh energy hyc ec We refer he reaer o recen arcle [-7] Recenly n [8] auhor reearch fr orer nonlocal bounary value roblem for nonlnear mulve negrofference euaon an n [9] auhor reearch exence of oluon for a cla of an-eroc bounary value roblem wh fraconal -fference euaon On h lne of hough n h aer we uy he exence an unuene of oluon for econ orer nonlnear negrofference euaon wh nonlocal bounary conon an mule: ' D u( f ( u( I g( u( J u( I ( u( D u( D u( L where ( ( u( ( - u( - u( D u( - D u( D I are ervave an negral ( reecvely f g C( J RR I ( RR h C J [] ' J ]\ u( u( u( [ Where u ( an u( enoe he rgh an he lef lm of u ( a ( reecvely Progreve Acaemc Publhng UK Page wwwublcaonorg
2 Euroean Journal of ahemac an Comuer Scence Vol No 7 ISSN PRELIINARIES Le u e J ] J ( ] J ( ] J ( ] an nrouce he ace: [ ( PC( J R u : J R u C( ] an x an x( ex wh x( x( ( An PC J R u PC J R D D ex an x( for where [] J noe ha R u u u D u ( PC J PC PC Defnon a funcon R oluon of ( f afe ( D PC J a Banach ace wh he norm u PC J wh ervave of econ orer exng on J a For convenence le u recall ome bac conce of calculu (J Tarboon e al For an J we efne he ervave of a real value connuou funcon f a D f ( lm D f ( f ( ( D f ( ( ( ( f ( Hgher orer ervave are gven by n n D K f ( f ( D f ( D D f ( n N J K (5 The negral of a funcon f efne by n n n : I f f f J (6 n Prove he ere converge If a ( an f efne on he nerval ( hen Oberve ha a f f f a D I f ( D f ( f ( I D f D f f f (8 a I D f ( D f ( f ( f ( a a a ( For he followng reverng orer of negraon hol J f ( r r r( Noe ha f an n ( an (6 Then where D f D f I f I f D an I are he well-nown ervave an negral of he funcon f ( efne by f ( r r (7 (9 Progreve Acaemc Publhng UK Page wwwublcaonorg
3 Euroean Journal of ahemac an Comuer Scence Vol No 7 ISSN f ( f ( D f ( ( ( n n I f ( f ( ( f ( n Lemma For gven y C J R he funcon R mulve negrofference euaon ' D u( y ( J D u( D u( ( ( L u u PC J a oluon of he u( I ( u( ( - u( - u( D u( - D ( u f an only f u afe he negral euaon + ( ( L u y y ( ( y J u ( L u y I u Lu [( ( ] y y ( ( y ( ( y J ( Proof Le u be a oluon of fference euaon ( For J alyng he oeraor I on boh e of D u( y ( we have D u( D u( I y D u y ( ( ( ( u u D u y Thu D u( D u( y ( ( u u D u y Smlarly for alyng he oeraor I on boh e of D ( u y hen J D u( D u( y ( (5 In vew of D u( D u( L ( u( an u( u( u( I ( u( hol ( Progreve Acaemc Publhng UK Page wwwublcaonorg
4 Euroean Journal of ahemac an Comuer Scence Vol No 7 ISSN D u ( D u ( L ( u ( y ( y ( (6 [( ( ] y ( [( ( ] y ( (7 Reeang he above roce we can ge An D u( D u( L ( u( y ( y ( (8 u( u( ( D u( I ( u( ( L ( u( (9 ( y ( [( ( ] y ( [( ( ] y ( Ung he bounary value conon gven n ( we can ge ( Converely aume ha u afe he mulve negral euaon (; alyng D on boh e of ( an ubung n ( hen ( hol Th comlee he roof AIN RESULTS Leng y ( f ( u( ( ( I g u n vew of Lemma we nrouce an oeraor R R Q : PC J PC J a ( Qu( ( ( f u( + g r u r r ( ( f u( + g r u r r [( ( ] f u( + g r u r r An f u( + g r u r r L u f u( + g r u r r I u Lu ( ( + ( ( ( ( ( ( ( ( ( ( ( u u D u I u L u y ( D Qu( f u g r u r r L u f u( + g r u r r ( Progreve Acaemc Publhng UK Page 5 wwwublcaonorg
5 Euroean Journal of ahemac an Comuer Scence Vol No 7 ISSN L u f u( + g r u r By reverng he orer of negraon we oban ( Qu( r ( ( ( [( ( ] ( f u g u ( ( f u( [( ( ] g u( [( ( ] ( ( f u g u f u( g u( L u f u g u I u Lu ( Then he mulve - negrofference euaon ( ha a oluon f an only f he oeraor euaon u Qu ha a fxe on In orer o rove he exence of oluon for ( we nee he followng nown reul (J X Sun 8 Lemma Le E be a Banach ace Aume ha T : E E a comleely connuou oeraor an he e V { x E x Tx } boune Then T ha a fxe on n E Theorem Aume he followng (H There ex nonnegave boune funcon ( ( uch ha f ( u ( ( u g( u ( ( u for any J u enoe u ( J (H There ex ove conan L L uch ha I ( u L L ( u L ( for any u L Then he roblem ( ha a lea one oluon rove u ( J Where ( ( ( ( Progreve Acaemc Publhng UK Page 6 wwwublcaonorg
6 Euroean Journal of ahemac an Comuer Scence Vol No 7 ISSN Proof Frly we rove he oeraor Q : PC ( J PC ( J comleely connuou Clearly connuy of he oeraor Q follow from he connuy of f g I Le PC ( J be boune hen J u There ex ove conan L uch ha f ( u L g( u L I ( u L L ( u L Thu ( Qu( ( ( ( [( ( ] ( f u g u ( ( f u( [( ( ] g u( [( ( ] f u g u f u( g u( L u f u g u I u Lu ( ( [( ( ] L L L L ( ( [( ( ] [( ( ] L L L L L L L L L ( L L L ( L ( L ( ( ( ( ( L L ( L L L ( ( L ( L L L L ( ( L ( L L + L [( L ( L L ] L L Progreve Acaemc Publhng UK Page 7 wwwublcaonorg
7 Euroean Journal of ahemac an Comuer Scence Vol No 7 ISSN L L ( P L L: = (conan (5 Th mle Qu An ( D Qu( ( [( ( ] ( f u g u ( [( ( ] ( f u g u L u L u f u g u L [( ( ] L L L L L ( ( L ( L L L ( L ( L L L [ ( L ( L L] L L ( L : (conan (6 Th mle D Qu Furhermore for any J ( L afyng we have ( Qu( ( Qu( ( ( ( [( ( ] ( f u g u f u( g u( L u f u g u [( ( ] ( ( f u( [( ( ] g u( f u( g u( L L Progreve Acaemc Publhng UK Page 8 wwwublcaonorg
8 Euroean Journal of ahemac an Comuer Scence Vol No 7 ISSN L u f u g u L u L u ( ( ( [( ( ] ( f u g u ( f u( [( ( ( ( ] g u( g ( ( L u ( f u( u( L u f u g u ( f u( ( g u( ( f u( [( ( ] g u( ( ( g ( f u u L u f u g u ( L [ L ( L ]( [( ] L ( ( L L L L L (7 An ( D Qu( ( D Qu( ( [( ( ] ( f u g u ( [( ( ] ( f u g u ( [( ( ] ( f u g u ( ( ( f u g u [ LL ( ]( L ( ( (8 A he rgh han e of he above neualy en o zero Thu Q( relavely comac A a coneuence of Arzela Acol heorem Q a comac oeraor Therefore Q a comleely connuou oeraor Defne he e W { u PC ( J u Qu } Nex we how W boune Le u W ; hen u Qu For any J by conon (H an (H we have L ( ( ( g u( Progreve Acaemc Publhng UK Page 9 wwwublcaonorg
9 Euroean Journal of ahemac an Comuer Scence Vol No 7 ISSN u( ( Qu( ( ( ( [( ( ] ( f u g u ( ( f u( [( ( ] g u( [( ( ] f u g u f u( g u( f u g u I u Lu Lu ( ( u( [( ( ]( u( ( ( u [( ( ] u [( ( ] u u u u u u L L L ( ( ( u ( ( u ( u P ( ( u L L ( ( u ( ( ( ( ( ( ( ( ( ( u u u u u ( ( u ( ( u ( u ( u ( u ( ( ( u ( ( u ( u + ( ( u ( ( u ( u L L Progreve Acaemc Publhng UK Page wwwublcaonorg
10 Euroean Journal of ahemac an Comuer Scence Vol No 7 ISSN u ( ( L L (9 u ( ( Smlarly for any J by conon (H an (H we have D u D u ( [( ( ] ( f u g u ( [( ( ] ( f u g u L u L u f u g u u( [( ( ]( u( ( [( ( ]( ( u u L u u ( ( ( u ( ( u ( u ( ( u ( ( u ( u [ ( ( u ( ( u ( u ] L L ( u ( ( P D L u ( : =conan ( I L Progreve Acaemc Publhng UK Page wwwublcaonorg
11 Euroean Journal of ahemac an Comuer Scence Vol No 7 ISSN P Where = ( u u PC J u Du PC PC I u PC So he e W boune Thu Lemma enure he mulve - negrofference euaon ( ha a lea one oluon Corollary Aume he followng (H There ex nonnegave conan L uch ha f ( u L g( u L I ( u L L ( u L ( for any J u L Then roblem ( ha a lea one oluon Theorem 5 Aume he followng (H There ex nonnegave boune funcon ( an N ( uch ha f ( u f ( v ( u v g( u g( v N( u v ( for J uv (H There ex ove conan K G X uch ha (H 6 I( u I( v K u v L( u L( v X u v (5 for uv an L P P K u ( X K J Then roblem ( ha a unue oluon Proof Clearly Q a connuou oeraor Denoe u ( u N( N For u v PC ( J by (H an (H 5 we have (6 ( ( f u( f ( v( [( ( ] g u( g( v( 5 ( Qu( ( Qv( ( ( f u( f ( v( [( ( ] g u( g( v( [( ( ] ( ( ( [( ( ] ( ( ( f u f v g u g v f u( f ( g u( g ( J J Progreve Acaemc Publhng UK Page wwwublcaonorg
12 Euroean Journal of ahemac an Comuer Scence Vol No 7 ISSN L u L f u f g u g I u I L u L ( ( [( ( ] N u [( ( ] N u N u N u K u X u X u ( ( ( ( ( N N N u ( ( ( ( ( N N N u ( ( N N N u ( ( ( [( ( ] N( ( u v( ( ( ( N N u + [( ( N N] u X K u N ( X K u K uv (7 A K by (H 6 Therefore Q a conracve ma Thu he concluon of he Theorem 5 follow by Banach conracon mang rncle EXAPLE Coner he followng econ orer an-eroc nonlnear wh mule D/( u( 8 ln 5 u( 5 n u( /( /( ( negrofference euaon Progreve Acaemc Publhng UK Page wwwublcaonorg
13 Euroean Journal of ahemac an Comuer Scence Vol No 7 ISSN D/( u D/( u n u u co u 6 u u D/ u( - D/9 u( (8 Obvouly / ( ( 6 / ( ( 6 f ( u 8 ln( 5 g( u n u I ( u co u L ( u n u By a mle calculaon we can ge f ( u 8 5 u( 5 g( u u 5 u( I ( u L ( u Tae ( 8 5 ( an L L Then 5 ( ( all conon of Theorem hol By Theorem econ orer an-eroc nonlnear mulve negrofference (8 ha a lea one oluon (9 REFERENCES [] Ferrera R A C Nonrval Soluon for Soluon for Fraconal -Dfference Bounary Value Problem [J] Elecon J Qual Theory Dffer Eu 7( : - [] F Ac an P W Eloe Inal value roblem n cree fraconal calculu Proceeng of he Amercan ahemacal Socey 9 vol 7 no [] F Ac an S Senguel oelng wh fraconal fference euaon Journal of ahemacal Analy an Alcaon vol 69 no 9 [] T Abeljawa D Baleanu F Jara an R P Agarwal Fraconal um an fference wh bnomal coeffcen Dcree Dynamc n Naure an Socey vol Arcle ID 7 6 age [5] G A Anaaou Prncle of ela fraconal calculu on me cale an neuale ahemacal an Comuer oellng vol 5 no [6] G C Wu an D Baleanu Dcree fraconal logc ma an chao Nonlnear Dynamc vol 75 no [7] T Holm The Theory of Dcree Fraconal Calculu: Develomen an Alcaon [PhD he] Unvery of NebraaLncoln Lncoln Nebraa [8] LHong Zhang Dumru Baleanu an Guoao Wang Nonlocal bounary value roblem for nonlnear mulve - negrofference euaon Abrac an Ale Analy vol Arcle ID age [9] SUN ngzhe HOU Chengmn Exence of oluon for a cla of an-eroc bounary value roblem wh fraconal -fference euaonjournal of Jln Unvery vol 5 no [] J Tarboon an S K Nouya Quanum calculu on fne nerval an alcaon o mulve fference euaon Avance n Dfference Euaon vol no 8 Progreve Acaemc Publhng UK Page wwwublcaonorg
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