Existence and Uniqueness Results for Random Impulsive Integro-Differential Equation

Transcription

1 Global Journal of Pure and Appled Mahemacs. ISSN Volume 4, Number 6 (8), pp Research Inda Publcaons hp:// Exsence and Unqueness Resuls for Random Impulsve Inegro-Dfferenal Equaon K. Malar Deparmen of Mahemacs, Erode Ars and Scence College, Erode, aml Nadu. Inda Absrac In hs arcle we concerned wh he exsence and unqueness of mld soluons for random mpulsve negro-dfferenal equaon hrough fxed pon echnque. Moreover, Lpschz condon as o be relaxed on he mpulsve erms n he dervng resuls. I s nvesgaed under suffcen condons. he resuls are obaned by usng he Banach fxed pon heorem. Keywords: random mpulsve negro- dfferenal equaon; mld soluon; fxed pon... INRODUCION Impulsve dfferenal sysems have been hghly recognzed and appled n he felds as dverse as physcs, populaon dynamcs, aeronaucs, economcs, elecommuncaons and engneerng are characerzed by he fac ha hey undergo abrup change of sae a ceran momens of me beween nervals of connuous evoluon. he duraon of hese changes are ofen neglgble compared o he oal duraon of process ac nsananeously n he form of mpulses see [6,7,8]. he mpulses may be deermnsc or random. here are lo of papers whch nvesgae he qualave properes of deermnsc mpulses see [,3,5] and he references heren. When he mpulses are exs a random pons, he soluons of he dfferenal sysems are sochasc processes. I s very dfferen from deermnsc mpulsve dfferenal sysems and also s dfferen from sochasc dfferenal equaons. hus he random mpulsve sysems gve more realsc han deermnsc mpulsve sysems. he sudy of random mpulsve dfferenal equaons s a new research area. Acual mpulses do no always happen a fxed pons bu usually a random pons,.e., mpulsve momens are random varable, see [,5,9-6] Recenly n [5], A.Vnodumar el. Suded he exsence and unqueness and sably of random mpulsve fraconal dfferenal equaon. In [4], he exsence and

2 8 K. Malar unqueness of random mpulsve dfferenal sysem by relaxng he lnear growh condons, suffcen condons for sably hrough connuous dependence on nal condons and exponenal sably va fxed pon heory. Moreover, Lpschz condon has o be relaxed on he mpulsve erms n he dervng resuls.. PRELIMINARIES Le X be a real separable Hlber space and a nonempy se. Assume ha s a random varable defned from o D def.(, d ) for all,,..., where d. Furhermore, assume ha and j are ndependen wh each oher as j for, j,,... Le be a consan. For he sae of smplcy, we denoe [, ]. We consder he negro-dfferenal equaons wh random mpulses of he form x '( ) Ax( ) f (, x( )) f (, x( )) d,,, x( ) b ( ) x( ),,,... (.) x x, where A s he nfnesmal generaor of a srongly connuous sem group of bounded lnear operaors () doman D(A) n X. f : X X, b : D for each,,...; [, ] and for,,..., here s arbrary real number. Obvously,......; x( ) lm x( ) accordng o her pahs wh he norm x sup x( s) for each sasfyng. Le us s denoe { B, } be he smple counng process generaed by{ n } { B n} { }, and denoe he alg ebra generaed by{ B, } n ha s,. hen (, P,{ }) s a probably space. Le L L (,{ }, X ) denoe he Hlber Space of all { }- measurable square negrable random varables wh values n X. Le denoe Banach space ([,],L ), he famly of all { }- measurable random varables wh he norm sup E ( ) [, ] DEFINIION.. Consder he nhomogeneous problem where f :[, ] X. x'( ) Ax( ) f ( ) x() x

3 Exsence and Unqueness Resuls for Random Impulsve Inegro-Dfferenal 8 Le A be he nfnesmal generaor of a C sem group (). Le x f L (, ; X ). hen he funcon x C([, ]; X ) s gven by x( ) ( ) x ( s) f ( s) ds, s he mld soluon of he above nal value problem for [, ]. DEFINIION.. A semgroup { ( ), } s sad o be unformly bounded f here exss a consan M such ha ( ) M, for DEFINIION.3. For a gven (, ), a sochasc process { x( ), } s called a mld soluon o equaon (.) n (, P,{ }), f ( ) x( ) X s adaped ; x( ) b ( ) ( ) x bj ( j ) ( s) f ( s, x( s)) ds ( s) f ( s, x( s)) ds j where, n jm b ( ) ( s) f (, x( s )) dds j j j X ( s) f (, x( s )) dds I[, ) ( ) [, ], (.) as m n, b ( ) b ( ) b ( )... b ( ), and I (.) s hendex funcon,. e., j j j A and I A, f A, (), f A. 3. MAIN RESULS In he secon, we dscuss he exsence and unqueness of he mld soluon for he sysem (.). Before provng he man resuls, we nroduce he followng hypohess whch s used n our resuls. (H) he funcon f :[, ] C X sasfes he Lpschz condon. ha s, for, X and here exss a consans L L ( ) and L L ( ) such ha

4 8 K. Malar E f (, ) f (, ) L E, E f (, ) L ( E ), E f (,), where s a (H) he condon C such ha consan. max bj( j) s unformly bounded f, here s a consan, j E max bj ( j ) C for all j Dj, j,,3,..., j heorem 3. Le he hypohess (H) (H) be hold. If he followng nequaly max{, }( ), (.) M C L s sasfed, hen he sysem (.) has a unque mld soluon n. Proof. Le be an arbrary number. Frs we defne he nonlnear operaor : as follows ( x)( ) b ( ) ( ) x bj ( j ) ( s) f ( s, x) ds ( s) f ( s, x) ds j b ( ) ( s) f (, x( s )) dds j j j ( s) f (, x( s )) dds I[, ) ( ) [, ], I s easy o prove he connuy of S. Now, we have o show ha S maps B no self. ( x)( ) b ( ) ( ) x bj( j) j ( s) f ( s, x( s)) ds ( s) f ( s, x( s)) ds

5 Exsence and Unqueness Resuls for Random Impulsve Inegro-Dfferenal 83 bj( j) ( s) f(, x( s )) dds j ( s) f (, x( s )) dds I[, ) ( ) b ( ) ( ) x I ( ) [, ) bj( j) ( s) f ( s, x( s)) ds j ( s) f ( s, x( s)) ds I[, ) ( ) bj( j) ( s) f(, x( s )) dds j ( s) f (, x( s )) dds I[, ) ( ) M max b ( ) x M max, b ( ) f ( s, x( s)) ds I ( ) j j [, ), j M max, b ( ) f (, x( s )) d ds I ( ) j j [, ], j s M C () M max{, C }( f ( s, x ) ds)

6 84 K. Malar M max{, C } f (, x( s )) dds M C x M max{, C }( ) f ( s, x( s)) ds M max{, C }( ) f(, x( s )) dds ( x)( ) M C x M max{, C }( ) f ( s, x( s)) ds M max{, C }( ) ( ) x( s ) dds E ( x)( ) M C x M max{, C }( ) E f ( s, x( s)) ds M max{, C }( )( ) ( ) E x( s ) dds M C x 4M max{, C }( ) L 4M max{, C }( ) L E x( s) ds 4M max{, C }( ) L hus, [, ] 4M max{, C }( ) L E x( s ) dds sup E ( x)( ) M C x 4M max{, C }( ) L s[, ] 4M max{, C }( ) L sup E x( s) ds 4M max{, C }( ) L

7 Exsence and Unqueness Resuls for Random Impulsve Inegro-Dfferenal 85 s[, ] 4M max{, C }( ) L sup E x( s ) dds M C x 4M max{, C }( ) L 4M max{, C }( ) L sup E x 4M max{, C }( ) L [, ] 4M max{, C }( ) L sup E x( ) d [, ] for all [, ], herefore maps no self Now, we have o show s a conracon mappng x y b s f s x s f s y s ds ( )( ) ( )( ) j( ) j ( ) (, ( )) (, ( )) j ( s) f ( s, x( s)) f ( s, y( s)) ds I[, ) ( ) bj( ) j ( s) f(, x( s )) f (, y( s )) dds j ( s) f (, x( s )) f (, y( s )) dds I[, ) ( ) M bj j f s x s f s y s ds I[, ), max, ( ) (, ( )) (, ( )) ( ) j M bj j f x s f y s dds I[, ), max, ( ) (, ( )) (, ( )) ( ) j max{, }( ) (, ( )) (, ( )) M C f s x s f s y s ds

8 86 K. Malar max{, }( ) (, ( )) (, ( )) M C f x s f y s d ds E ( x)( ) ( y)( ) M max{, C }( ) E f ( s, x( s)) f ( s, y( s)) ds max{, }( )( ) (, ( ) (, ( )) M C E f x s f y s d ds max{, }( ) ( ) ( ) M C L E x s y s ds max{, }( )( ) ( ) ( ) M C L E x s y s d ds ang supremum over, we ge, ( x) ( y) M max{, C }( ) L x y [, ] M max{, C }( ) ( ) L E( sup x( ) y( ) ) d ( sup ( ) ( ) ) [, ] x y E x y d By applyng he Gronwall nequaly n he above equaon, follows ha ( x) ( y) x y Snce. hs shows ha he operaor sasfes he conracon mappng prncple and herefore, has a unque fxed pon whch s he mld soluon of he sysem (.). hs complees he proof. REFERENCES [] A.Anguraj and A.Vnodumar, Exsence, unqueness and sably resuls of random mpulsve semlnear dfferenal sysems, Nonlnear Analyss Hybrd sysems, 3(), [] A.Anguraj, S.Wu and A.Vnodumar, Exsence and exponenal sably of semlnear funconal dfferenal equaons wh random mpulses under non unqueness, Nonlnear Analyss: heory, mehods & Applcaons, 74(),

9 Exsence and Unqueness Resuls for Random Impulsve Inegro-Dfferenal 87 [3] A.Anguraj and A.Vnodumar, Exsence and unqueness of neural funconal dfferenal equaons wh random mpulses, Inernaonal Journal of nonlnear scence, 8(4)(9) [4] A.Vnodumar, Exsence, unqueness of soluons for random mpulsve dfferenal equaon, Malaya Journal of Maema ()()8-3. [5] K.Malar and A.Vnodumar, Exsence, Unqueness and sably of random mpulsve fraconal dfferenal equaons, Aca Mahemaca Scena 6, 36B(): [6] K. Karheyan, A. Anguraj, K. Malar and Juan J. rujlo, Exsence of Mld and Classcal Soluons for Nonlocal Impulsve Inegrodfferenal Equaons n Banach Spaces wh Measure of Noncompacness, Hndaw Publshng Corporaon Inernaonal Journal of Dfferenal Equaons, Volume 4, Arcle ID 395, pages. [7] V.Lashmanhan, D.D.Banov and P.S.Smeonov, heory of Impulsve Dfferenal Equaons, world scenfc, Sngapore, 989. [8] A.M.Samoleno and N.A.Peresyu, Impulsve Dfferenal Equaons, World Scenfc, Sngapore, 995. [9] J.M.Sanz-Serna and A.M.Suar, Ergodcy of dsspave dfferenal equaons subjec o random mpulses, J.Dfferenal Equaons, 55(999),6-84. [] A.Vnodumar, Exsence resuls on random mpulsve semlnear funconal dfferenal nclusons wh delays, Ann. Func. Anal., 3(), [] A.Vnodumar and A.Anguraj, Exsence of random mpulsve absrac neural non-auonomous dfferenal nclusons wh delays, Nonlnear Anal. Hybrd Sysems, 5(), [] A.Vnodumar, Sably resuls of Random mpulsve semlnear dfferenal equaons, Scence drec, Aca Mahemaca Scenca 4, 34(B):55-7. [3] S.J.Wu and X.Z.Meng, Boundedness of nonlnear dfferenal sysems wh mpulsve effec on random momens, Aca Mah. Appl., Sn., ()(4), [4] S.J.Wu and Y.R.Duan, Oscllaon, sably, and boundedness of secondorder dfferenal sysems wh random mpulses, Compu. Mah. Appl., 49(9- )(5), [5] S.J.Wu,X.L.Guo and S.Q.Ln, Exsence and unqueness of soluons o random mpulsve dfferenal sysems, Aca Mah.Appl. Sn., (4)(6), [6] S.J.Wu, X.L. Guo and Y.Zhou, p-momen sably of funconal dfferenal equaons wh random mpulses, Compu. Mah. Appl., 5(6),

10 88 K. Malar