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International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI ...

International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.

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  • International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 2 Issue 4 || April. 2014 || PP-58-66 www.ijmsi.org 58 | P a g e Controllability of Neutral Integrodifferential Equations with Infinite Delay Jackreece P. C. Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Nigeria. ABSTRACT: In this paper sufficient conditions for the controllability of nonlinear neutral integrodifferential equations with infinite delay where established using the fixed point theorem due to Schaefer. KEYWORDS: Controllability, Infinite delay, Neutral Integrodifferential equation, Schaefer fixed point. I. INTRODUCTION In this paper, we establish a controllability result to the following nonlinear neutral integrodifferential equations with infinite delay:                 1.1 0,,,,, 0     x txtfdsxstgtButxtAxthtx dt d t t st where the state .x takes values in the Banach space X endowed with the norm . , the control function .u is given in the Banach space of admissible control function  UJL ,2 with U as a Banach space.     XADtA : is an infinitesimal generator of a strongly continuous semigroup of bounded linear operator   0, ttT in X .  is a bounded linear operator from U into X , where XJJg : , XJf : , and XJh : are given functions. The delay   Xxt  0,: defined by      txxt belongs to some abstract phase space  , which will be a linear space of functions mapping  0, into X endowed with the seminorm  . in . Controllability problems of linear and nonlinear systems represented by Ordinary differential Equations in finite dimensional space has been studies extensively [6, 15]. Several authors extended the controllability concept to infinite dimensional systems in abstract spaces with unbounded operators [1, 2, 8]. There are many systems that can be written as abstract neutral functional equations with infinite delays. In recent years, the theory of neutral functional differential equations with infinite delay in infinite dimension has received much attention [4, 7, 10]. Meanwhile, the controllability problem of such systems was also discussed by several scholars. Meili et al. [13] studied the controllability of neutral functional integrodifferential systems in abstract spaces using fractional power of operators and Sadovski fixed point theorem. Balachandran and Nandha [5] established sufficient conditions for the controllability of neutral functional integrodifferential systems with infinite delay in Banach spaces by means of Schaefer fixed point theorem. Li et al. [12] used Hausdoff measure of noncompactness and Kakutani’s fixed point theorem to establish sufficient conditions for the controllability of nonlinear integrodifferential systems with nonlocal condition in separable Banach space. The purpose of this paper is to establish a set of sufficient conditions for the controllability of neutral integrodifferential equations with infinite delay using Schaefer fixed point theorem. II. PRELIMINARIES In this section, we introduce notations, definitions and theorems which are used throughout this paper. In the study of equations with infinite delay such as Eq. (1.1), we need to introduce the phase space. In this paper we employ an axiomatic definition of the phase space first introduced by Hale and Kato [9] and widely discussed by Hino et al [11]. The phase space  is a linear space of functions mapping ]0,( into X endowed with the seminorm  . and assume that  satisfies the following axioms:
  • Controllability of Neutral Integrodifferential Equations with Infinite Delay www.ijmsi.org 59 | P a g e (A1) If   ,0,,:  bXbx is continuous on  b,0 and 0x , then for every  bt ,0 the Following conditions hold: (i) tx is in  (ii)     txHtx (iii)          00:sup xtMtssxtKxt where 0H is a constant; ),0[),0[:, MK , K is continuous and M is locally bounded, and MKH ,, are independent of .x . (A2) For the function .x in (A1), tx is a  -valued continuous function on ],0[ b . (A3) The space  is complete. We need the fixed point theorem as stated below, Schafer’s theorem [14]: Let S be a convex subset of a normed linear space E and S0 . Let SSF : be completely continuous operator and let     1,0:   someforxFxSxF Then either  F is unbounded or F has a fixed point. We assume the following hypothesis: (H1) A is the infinitesimal generator of a compact semigroup of bounded linear operators   0, ttT on X and their exists 1M and 01 M such that   MtT  and   1MtAT  . (H2) For XJh : is completely continuous and there exists constants 21,cc such that 1 ~ 1 cK and   21, ccth      ,,0 bt , where   JttKK  :max ~ . (H3) The function XJf : satisfies the following conditions (i). For each Jt  , the function XJf : is continuous. (ii) For each  , the function   XJf :,;  is strongly measurable. (iii). For each positive integer k, there exists    bLk ,0. 1  such that      eaJttkxxtf kt .,:,sup   . and    xJtxxtf ,,, where  ,0[),0[: is a continuous nondecreasing function.
  • Controllability of Neutral Integrodifferential Equations with Infinite Delay www.ijmsi.org 60 | P a g e (H4) XJJg : is continuous and there exists 0N , such that   Nstg ,, For every .0 bts  (H5) for each  ,     0 ,,lim)( a a dssstgtq  exists and it is continuous. Further there exists 01 N such that   .1Ntq  (H6) there exists a compact set XV  , such that             ,,,,, sgtTsButTsftT and     VsqtT  for all , , and .0 bts  (H7) The linear operator   XUJLW ,: 2 , defined by      b dssBusbTWu 0 has an inverse operator 1 W , which takes values in   WUJL ker/,2 and there exist positive constants 0, 43 MM such that 4 1 3 MWandMB   (H8)         b s ss ds dssm 0 ~ where      dsxMMNbbMN dscxcMcxcccMxMMN b s b st     0 1 0 21121211433 0     MNbMNMNbcMcccMKM cK c    1321121 1 0 ~~ ~ 1 1    JttMM  :max ~           11 11 ~ 1 ~ ,~ 1 ~ max~ cK MK cK cMK tm Definition 2.1: The system (1.1) is said to be controllable on the interval J iff, for every Xxx b ,0 , there exists a control  UJLu ,2  such that the mild solution  tx of (1.1) satisfies   00 xx  and   bxbx  . Definition 2.2: A function   Xbx  ,: is called a mild solution of the integrodifferential equation (1.1) on [0, b] iff ,                             1.2,0,,, ,,,00 0 0 0 btdsxsfdxsgsqsBustT dsxshstATxthhtTtx t s t s t tt             is satisfied. See [3,10]
  • Controllability of Neutral Integrodifferential Equations with Infinite Delay www.ijmsi.org 61 | P a g e III. MAIN RESULT Theorem 3.1: Assume that (H1)-(H8) holds. Then the system (1.1) is controllable. Proof: Consider the map defined by                           )1.3(,,, ,,,00 0 0 0           t s t s t st JtdsxsfdxsgsqsBustT dsxshstATxthhttx   and define the control function  tu as                              )2.3(,,, ,,,00 0 0 0 0 0 1         b b s b s b sbb tdsxsfsbTdsdxsgsbTdssqsbT dsxshsbATxbhhbTxWtu    First we show that there is a priori bound 0K such that JtKxt  , , where K depends only on b and on the function . . Then we show that the operator has a fixed point, which is then a solution to the system (1.1). Obviously,    1xbx  , which means that the control u steers the system from the initial function  to 1x in time b, provided that the nonlinear operator has a fixed point. Substituting (3.2) into (2.1) we obtain                 t st dsxshstATxthhtTtx 0 ,,,00                    t b sb dsxshsbATxbhhbTxBWtT 0 0 1 1 ,,,00                   b b s b s ddsxsfsbTdsdxsgsbTdssqsbT 0 0 0 0 ,,,                 t s s dsxsfdxsgsqstT 0 0 ,,,   Then, we have        MNbMNMNdscxcMcxcccMtx t st   13 0 21121210  dsxM t s  0 From which using Axiom (A1)(iii), it follows that       tMtssxtKxt  0:sup    MtssxK ~ 0:sup ~     MNbMNMNbcMcccMKM  13212210 ~~          t t sss dsxMKdsxcMKxcK 0 0 111 ~~ sup ~
  • Controllability of Neutral Integrodifferential Equations with Infinite Delay www.ijmsi.org 62 | P a g e Let    tsxt s  0:sup , then the function  t is continuous and nondecreasing and from Axiom (A2) we have      MNbMNMNbcMcccMKMt  13211210 ~~         tt dsMKdsscMKtcK 00 111 ~~~  From which it follows that             t t dss Kc MK dss cK cMK ct 0 0 11 11 1 ~ ~ 1 ~  Denoting the right hand side of the above inequality as  t , we have   c0 ,     Jttt  , and      s cK MK s cK cMK t       11 11 ~ 1 ~ ~ 1 ~    s cK MK s cK cMK       11 11 ~ 1 ~ ~ 1 ~       sstm    which implies that          Jt ss ds dssm ss dst t         0 0 0 ~ This inequality implies that there is a constant K such that   JtKt  , and hence,     KtKttxt  , , where K only depends on b and on the function . . We now rewrite the initial value problem (1.1) as follows: For  , define ˆ by                   atifhtT tiftht t 0,00 0,ˆ    If        atttytxandy ,,ˆ   , then it is easy to see that x satisfies (2.1) if and only if y satisfies,   0,00  tyty                                 b ssbb t t sstt dsyshsbATybhhbTxBWtT dsyshstATythhtTty 00 1 1 0 ˆ,ˆ,,00 ˆ,ˆ,,0       Jts cM M scMK cK           11 11 1 ~ ~ 1 1
  • Controllability of Neutral Integrodifferential Equations with Infinite Delay www.ijmsi.org 63 | P a g e                ddsysfsbTdsdysgsbTdssqsbT b s b ss b     0 0 00 ˆ,ˆ,,         JtdsysfdysgsqstT t s ss         ,ˆ,ˆ,, 0 0   We define the operator  0:,: 0000  yy by                             dsyshsbATybhhbTxBWtT dsyshstATythhtTty t b ssbb t sstt       0 0 1 1 0 ˆ,ˆ,,00 ˆ,ˆ,,0                  ddsysfsbTdsdysgsbTdssqsbT b b b ss     0 0 0 0 ˆ,ˆ,,                t s ss JtdsysfdysgsqstT 0 0 ,ˆ,ˆ,,   From the definition of an operator  defined on equation (3.1), it can be noted that the equation (2.1) can be written as      3.310,   tyty Now, we prove that  is completely continuous. For any kBy  , let btt  210 , then      21 tyty           2211 ˆ,ˆ,,0 2121 tttt ythythhtTtT               2 1 1 ˆ,ˆ, 2 0 21 t t ss t ss dsyshstATdsyshstTstTA                 1 0 1 1 21 ˆ,,00 t bbybhhbTxBWtTtT         dssqsbTyshsbAT b b ss   0 0 ˆ,             ddsysfsbTdsdysgsbT b b ss    0 0 0 ˆ,ˆ,,             2 1 ˆ,,001 1 2 t t bbybhhbTxBWtT            b b ss dssqsbTdsyshsbAT 0 0 ˆ,              ddsysfsbTdsdysgsbT b ss b    00 0 ˆ,ˆ,,                  1 0 0 21 ˆ,ˆ,, t s ss dsysfdysgsqstTstT                 2 1 0 2 ˆ,ˆ,, t t s ss dsysfdysgsqstT  
  • Controllability of Neutral Integrodifferential Equations with Infinite Delay www.ijmsi.org 64 | P a g e          2211 ˆ,ˆ,,0 2121 tttt ythythhtTtT              1 2 10 2122121 ˆˆt t t ssss dscycstATdscycstTstTA  (3.4)           1 0 12114321 ˆ,00 t bb bMNcychMxMMstTstT             b b t t ss hMxMMstTdsdsyMdsNM 0 0 1432 2 1 ,00ˆ           b b b bbbb dsdssMdsNdMbMNdscycMcyc 0 0 0 0 121121 ˆˆ                 1 2 10 12121 t t t dssNNstTdssNNstTstT  The right-hand side of equation (3.4) is independent of ky  and tends to zero as 012  tt , since g is completely continuous and the compactness of  tT for 0t implies the continuity in the uniform operator topology. Thus  maps k into an equicontinuous family of functions.                    t sstt dsyshstATythhtTty 0 ˆ,ˆ,,0                 t bbybhhbTxBWstT 0 1 1 ˆ,,00                b b b ssbs dsysfsbTdssqsbTdsyshsbAT 0 0 0 ˆ,ˆ,                      b t s ss s dsdysgysfsqstTdsdsdysgsbT 0 0 00 ˆ,,ˆ,ˆ,,                     t sstt yshtATTythhtT 0 ˆ,ˆ,,0                   t bbybhhbTxBWstTT 0 1 1 ˆ,,00               b b ss b bb dsysfsbTdssqsbTdsyshsbAT 0 00 ˆ,ˆ,         b s dsdsdysgsbT 0 0 ˆ,,             dsdysgysfsqstTT t s ss           0 0 ˆ,,ˆ, Since  tT is a compact operator, the set      kytytY  : is precompact in X for every ,
  • Controllability of Neutral Integrodifferential Equations with Infinite Delay www.ijmsi.org 65 | P a g e t 0 . Moreover, for every ky  we have                 t t sstttt dsyshstATythythtyty   ˆ,ˆ,ˆ,          t t bbybhhMxMMstT   ˆ,,00143               b ss b b ss dsysfstTdssqstTdsyshstAT 00 0 ˆ,ˆ,               dsdsysgysfsqstT dsdsdysgstT t t s ss b s                 0 0 0 ˆ,,ˆ, ˆ,, Clearly       0 tyty  as   0 . Therefore, there is a family of precompact sets which are arbitrarily close to the set    kyty  : . Hence, the set    kyty  : is precompact in X . Next we prove that the set 00 : bb  is continuous. Let   0 1 bnny  with yyn  in 0 b . We have                  t sssntttnn dsyshyshstATythythtyty st 0 ˆ,ˆ,ˆ,ˆ,          t t sssn dsyshyshMMMstT s0 0 143 ˆ,ˆ.              b s b sssnn dsdsysfysfMdsysgysgM s0 0 0 ˆ,ˆ,ˆ,,ˆ,,                   t s nsssn dsysgysgysfysfstT s0 0 ˆ,,ˆ,,ˆ,ˆ,    From the assumption (H1) and the Lebesgue Dominated Convergence theorem we conclude that yyn  as n in 0 b . Thus . is continuous, which completes the proof that . is completely continuous. Hence there exists a unique fixed point  tx for bon  . Obviously .x is a mild solution of the system (1.1) satisfying   1xbx  . IV. CONCLUTION Sufficient condition for the controllability of the nonlinear neutral integrodiffential equation was established using Schaefer’s fixed point theorem. First we show that there is a priori bound 0K such that JtKxt  , . Then we show that the operator has a fixed point, which is then a solution to the system (1.1). Obviously,    1xbx  , which means that the control u steers the system from the initial function  to 1x in time b, provided that the nonlinear operator has a fixed point.
  • Controllability of Neutral Integrodifferential Equations with Infinite Delay www.ijmsi.org 66 | P a g e REFERENCE [1]. Atmania R, Mazouzi S. Controllability of Semilinear Integrodifferential Equations with Nonlocal conditions, Electronic J of Diff Eq., 2005, 2005: 1-9. [2]. Balachandran K, Balasubramania P, Dauer J. P., Null Controllability of nonlinear Functional Differential Systems in Banach Space. J. Optim Theory Appl., 1996, 88:61-75. [3]. Balachandran K, Sakthivel , Existence of Solutions of Neutral Functional Integrodifferential Equations in Banach Spaces, Proceedings of the Indian Academy of Science (Mathematical Sciences) 109 (1999): 325-332. [4]. Balachandran K, Anandhi E. R, Controllability of neutral integrodifferential infinite delay systems ins Banach spaces, Taiwanese Journal of Mathematics, 2004, Vol. 8, No. 4: 689-702. [5]. Balachandran K, Nandha G, neutral integrodifferential control systems with infinite delay in Banach spaces, J. KSIAM, 2004, Vol. 8, No. 41: 41-51. [6]. Balachandran, K., Dauer, J.P., Controllability of nonlinear systems via fixed point theorems, J. Optim Appl. 1987, 53, 345-352. [7]. Bouzahor H., On Neutral Functional Differential Equations, Fixed Point Theory, 2005, 5: 11-21. [8]. Chulwu E. N, Lenhart S. M, Controllability questions for nonlinear systems in abstract spaces, J. Optim Theory Appl. 68(3), 1991: 437-462. [9]. Hale, J., Kato, J., Phase space for retarded Equations with Infinite Delay, Funlcial Ekvac, 21 (1978), 11-41. [10]. Hernandez E, Henriquez H. R, Existence results for partial neutral functional differential equations with unbounded delay, J Math. Anal Appl, 1998, 221: 452-475. [11]. Hino, Y., Murakami, S., Naito, T., Functional Differential Equations with Unbounded Delay, Lecture Notes in Mathematics, 1991, Vol. 1473, Springer-Verlag, Berlin. [12]. Li, G. C., Song, S. J., Zhang, B., Controllability of nonlinear integrodifferential systems in Banach space with nonlocal conditions, Dynamic Systems and Applications 16 (2007), 729-742. [13]. Meili L., Yongrui D., Xianlong F., and Miansen W., Controllability of neutral functional integrodifferential systems in abstract spaces. J. Appl. Math and Computing, Vol. 23(2007), no. 1-2: 102-112. [14]. Schaefer H., Uber die method der a priori schranken. Mathematische Amalen, 129, (1955), pp 415 – 416. [15]. Triggiani R., On the stabilizability problems in Banach space, J. Math. Anal. Appl. 52(3) (1975), 383-403.