Rupali S. Jain, M. B. Dhakne

Transcription

1 DEMONSTRATIO MATHEMATICA Vol. XLVIII No Rupali S. Jain, M. B. Dhakne ON EXISTENCE OF SOLUTIONS OF IMPULSIVE NONLINEAR FUNCTIONAL NEUTRAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITION Communicated by E. Weber Abstract. In the present paper, we investigate the existence, uniqueness and continuous dependence of mild solutions of an impulsive neutral integro-differential equations with nonlocal condition in Banach spaces. We use Banach contraction principle and the theory of fractional power of operators to obtain our results. 1. Introduction Many evolution processes and phenomena experience abrupt changes of state through short term perturbations. Since the duration of the perturbations are negligible in comparison with the duration of each process, it is quite natural to assume that these perturbations act in terms of impulses. For more details see monographs [15], [18]. It is to be noted that the recent progress in the development of the qualitative theory of solutions of impulsive differential equations has been studied by many researchers, see [1] [3], [11] [16], [18] [21]. Also neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention in the last few decades, e.g. see [3], [11], [17], [2], [21]. On the other hand, the theory of functional differential equations with nonlocal conditions has been extensively studied in the literature, see [1], [4] [1], [12] [14], [19], as they have applications in physics and many other areas of applied mathematics. The nonlocal condition is more precise for describing natural phenomena than the classical condition because more 21 Mathematics Subject Classification: Primary: 45J5; Secondary: 45N5, 47H1, 47B38. Key words and phrases: impulsive, functional, neutral, integro-differential equation, fixed point, nonlocal condition. DOI: /dema c Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology

2 414 R. S. Jain, M. B. Dhakne information is taken into account, thereby decreasing the negative effects incurred by a possibly single measurement taken at initial time. In this paper, we study nonlinear impulsive neutral functional integrodifferential equation with non local condition of the type : (1) (2) (3) d dt rxptq upt, x tqs ` Axptq fpt, x t, kpt, sqhps, x s qdsq, t P p, T s, t τ k, k 1, 2,..., m, xptq ` pgpx t1,..., x tp qqptq φptq, r t, xpτ k ` q I k xpτ k q, k 1, 2,..., m, where ă t 1 ă t 2 ă ă t p T, p P N, A and I k pk 1, 2,..., kq are the linear operators acting in a Banach space X. The functions f, h, g, k and φ are given functions satisfying some assumptions. xpτ k q xpτ k ` q xpτ k q and the impulsive moments τ k are such that τ ă τ 1 ă τ 2 ă ă τ k ă T, k P N. For any continuous function x and for any t P r, T s, we denote by x t, the element of Cpr r, s, Xq defined by x t pθq xpt ` θq for θ P r r, s. Equations of the form (1) (3) or their special forms arise in some physical applications as a natural generalization of the classical initial value problems. The results for semilinear functional differential evolution nonlocal problem are extended for the case of impulsive effect. As usual, in the theory of impulsive differential equations at the points of discontinuity τ i of the solution t Ñ xptq, we assume that xpτ i q xpτ i q. It is clear that, in general, the derivatives x 1 pτ i q do not exist. On the other hand, according to (1), there exist the limits x 1 pτ i q. According to the above convention, we assume that x 1 pτ i q x 1 pτ i q. The aim of the present paper is to study the existence, uniqueness and continuous dependence of mild solution of nonlocal initial value problem for an impulsive neutral functional integro-differential equation. We are generalizing the results reported in [2], [11], [14]. The main tool used in our analysis is based on an application of the Banach contraction theorem and the theory of fractional power of operators. This paper is organized as follows. Section 2 presents the preliminaries and hypotheses. In Section 3, we prove existence and uniqueness of mild solution. Finally in Section 4, we prove continuous dependence of solutions on initial data. 2. Preliminaries and hypotheses Throughout this paper, X will be a Banach space with norm } }. Let Cpra, bs, Xq denote the set tx : ra, bs Ñ X xptq is continuous at t τ k,

3 Nonlocal impulsive integro-differential equations 415 left continuous at t τ k, and the right limit xpτ k ` q exists for k 1, 2,..., mu. Clearly, Cpra, bs, Xq is a Banach space with the supremum norm }x} Cpra,bs,Xq supt}xptq} : t P ra, bsztτ 1, τ 2,..., τ m uu. Let A : DpAq Ñ X be the infinitesimal generator of an analytic semigroup of linear operators tt ptqu t on X. It is well known that there exists a constant K such that }T ptq} K, t. Also assume that k : r, T s ˆ r, T s Ñ R is a continuous function and as the set r, T s ˆ r, T s is compact, there exists a constant L ą such that kpt, sq L, for s t T. If T is uniformly bounded analytic semigroup such that P ρpaq, then it is possible to define the fractional power p Aq α, for ă α 1, as a closed linear operator on its domain Dp Aq α. Furthermore, the subspace Dp Aq α is dense in X and the expression }x} α }p Aq α x} defines a norm in Dp Aq α. If X α represents the space Dp Aq α endowed with the norm }.}, then the following properties are well known. Lemma 2.1. ([17], p. 74) Let ă α β 1. Then the following properties hold. (i) X β is a Banach space and X β Ñ X α is continuous. (ii) The function S ÞÑ paq α T psq is continuous in the uniform operator topology on p, 8q and there exists a positive constant C α such that }p Aq α T ptq} Cα t α, for every t ą. The following inequality will be useful while proving our results. Lemma 2.2. ([18], p. 12) Let a nonnegative piecewise continuous function uptq satisfy, for t t, the inequality uptq C ` t vpsqupsqds ` t ăτ i ăt β i upτ i q, where C, β i, vptq ą, τ i are the first kind discontinuity points of the function uptq. Then the following estimate holds for the function uptq uptq C ź ˆ p1 ` β i q exp vpsqds. t t ăτ i ăt Definition 2.1. A function x P Cpr r, T s, Xq, T ą is called the mild solution of the problem (1) (3) if xptq ` pgpx t1,..., x tp qqptq φptq, r t, the restriction of xp.q to the interval r, T q is continuous and for each t ă T, the function AT pt squps, x s q, s P r, tq is integrable and the following integral equation

4 416 R. S. Jain, M. B. Dhakne xptq T ptqrφpq gpx t1,..., x tp qpq up, φpq gpx t1,..., x tp qpqqs ˆ ` upt, x t q ` T pt sqf s, x s, kps, τqhpτ, x τ qdτ ds is satisfied. ` ` AT pt squps, x s qds T pt τ k qi k xpτ k q, ăτ k ăt t P p, T s Now we introduce the following hypotheses. (H 1 ) There exists β P p, 1q. For the functions f : r, T s ˆ Cpr r, s, X β q ˆ X β Ñ X β, u, h : r, T s ˆ Cpr r, s, X β q Ñ X β, I k : X β Ñ X β, there exist positive constants F, U, H, L k such that }p Aq β upt, x t q p Aq β upt, y t q} U}x y} Cpr r,ts,xβ q, }fpt, x t, φq fpt, y t, ψq} F p}x y} Cpr r,ts,xβ q ` }φ ψ}q, }hpt, x t q hpt, y t q} H}x y} Cpr r,ts,xβ q, }I k pvq} Xβ L k }v} Xβ, φ, ψ, v P X β, k 1, 2,..., m. (H 2 ) For the function g : Cpr r, s, X β q P Ñ Cpr r, s, X β q, there exists a constant G ą such that }gpx t1,..., x tp qptq gpy t1,..., y tp qptq} G}x y} Cpr r,t s,xβ q. (H 3 ) Assume that φ P Cpr r, s, X β q. 3. Existence and uniqueness Theorem 3.1. Suppose that hypotheses (H 1 ) (H 3 ) are satisfied and Γ ă 1, where Γ KG ` K}p Aq β }UG ` }p Aq β }U ` C 1 β T β U ` KF r1 ` LHT st ` K L k, ăτ k ăt then the nonlocal impulsive Cauchy problem (1) (3) has a unique mild solution x on r r, T s. Proof. We introduce an operator F on a Banach space Cpr r, T s, X β q as follows:

5 Nonlocal impulsive integro-differential equations 417 (4) pfxqptq $ φptq pgpx t1,..., x tp qqptq, if r t, & T ptqrφpq gpx t1,..., x tp qpq up, φpq gpx t1,..., x tp qpqs ` upt, x t q ` şt AT pt squps, x sqds ` şt % T pt sqf `s, x s, ş s kps, τqhpτ, s τ qdτ ds ` řăτ k ăt T pt τ kqi k xpτ k q, if t P p, T s. It is easy to see that F : Cpr r, T s, X β q Ñ Cpr r, T s, X β q. Now, we will show that F is a contraction on Cpr r, T s, X β q. Let x, y P Cpr r, T s, X β q. Then for t P r r, s, (5) }pfxqptq pfyqptq} }gpx t1,..., x tp qptq gpy t1,..., y tp qptq} and for t P p, T s, G}x y} Cpr r,t s,xβ q (6) }pfxqptq pfyqptq} gpx T ptq t1,..., x tp qpq gpy t1,..., y tp qpq `up, φpq gpx t1,..., x tp qpq ı up, φpq gpy t1,..., y tp qpqq `upt, x t q upt, y t q ` AT pt sqrups, x s q ups, y s qsds ` T pt sqrfps, x s, kps, τqhpτ, x τ qdτq fps, y s, kps, τqhpτ, y τ qdτqsds` T pt τ k qri k xpτ k q I k ypτ k qs ăτ k ăt }T ptq}}gpx t1,..., x tp qpq gpy t1,..., y tp qpq} `}T ptq}}up, φpq gpx t1,..., x tp qpqq up, φpq gpy t1,..., y tp qpqq} `}upt, x t q upt, y t q}` }AT pt sq}}ups, x s q ups, y s q}ds ` }T pt sq} f s, x s, kps, τqhpτ, x τ qdτ f s, y s, kps, τqhpτ, y τ qdτ ds` }T pt τ k q}}ri k xpτ k q I k ypτ k qs} ăτ k ăt KG}x y} Cpr r,t s,xβ q`kj 1`J 2`J 3`J 4`J 5,

6 418 R. S. Jain, M. B. Dhakne where (7) J 1 }up, φpq gpx t1,..., x tp qpqq up, φpq gpy t1,..., y tp qpqq} }p Aq β }}p Aq β up, φpq gpx t1,..., x tp qpqq p Aq β up, φpq gpy t1,..., y tp qpqq} }p Aq β }U}φpq gpx t1,..., x tp qpq φpq`gpy t1,..., y tp qpq} (8) (9) (1) }p Aq β }UG}x y} Cpr r,t s,xβ q, J 2 }upt, x t q upt, y t q} }p Aq β }U}x y} Cpr r,t s,xβ q, J 3 J 4 }AT pt sq}}ups, x s q ups, y s q}ds } AT pt sq}}p Aq β }}p Aq β rups, x s q ups, y s qs}ds }p Aq 1 β T pt sq}u}x y} Cpr r,ss,xβ qds C 1 β pt sq 1 β U}x y} Cpr r,ss,x β qds C 1 β T 1 β U}x y} Cpr r,ss,x β qds C 1 β T 1 β U}x y} Cpr r,t s,x β qt C 1 β T β U}x y} Cpr r,t s,xβ q, K }T pt sq}}fps, x s, fps, y s, KF KF KF kps, τqhpτ, x τ qdτq kps, τqhpτ, y τ qdτq}ds F }x y} Cpr r,ss,xβ q` }x y} Cpr r,ss,xβ q`l j kps, τq }hpτ, x τ q hpτ, y τ q}dτ ds j H}x y} Cpr r,τs,xβ qdτ ds }x y} Cpr r,ss,xβ q`lht }x y} Cpr r,ss,xβ q r1`lht s}x y} Cpr r,ss,xβ qds j ds KF r1`lht st }x y} Cpr r,t s,xβ q,

7 (11) J 5 K K ăτ k ăt ăτ k ăt ăτ k ăt ăτ k ăt Nonlocal impulsive integro-differential equations 419 }T pt τ k q}}i k xpτ k q I k ypτ k q} Xβ K}I k xpτ k q I k ypτ k q} Xβ L k }xpτ k q ypτ k q} Xβ L k }x y} Cpr r,t s,xβ q. Using (7) (11), inequality (6) becomes (12) }pfxqptq pfyqptq} KG}x y} Cpr r,t s,xβ q ` `K}p Aq β }UG ` }p Aq β }U ` C 1 β T β U ` KF r1 ` LHT st ` K L k }x y}cpr r,t s,xβ q, t P r, T s. ăτ k ăt In view of inequality (5) and (12), we can say that inequality (12) holds good for t P r r, T s. Therefore, for t P r r, T s, }pfxqptq pfyqptq} KG}x y} Cpr r,t s,xβ q ` K}p Aq β }UG ` }p Aq β }U ` C 1 β T β U ` KF r1 ` LHT st ` K L k }x y} Cpr r,t s,xβ q which implies where ăτ k ăt KG ` K}p Aq β }UG ` }p Aq β }U ` C 1 β T β u β ` KF r1 ` LHT st ` K L k }x y} Cpr r,t s,xβ q, ăτ k ăt }Fx Fy} Cpr r,t s,xβ q Γ}x y} Cpr r,t s,xβ q, Γ KG ` K}p Aq β }UG ` }p Aq β }U ` C 1 β T β U ` KF r1 ` LHT st ` K L k. ăτ k ăt Since Γ ă 1, the operator F satisfies all assumptions of the Banach contraction theorem and therefore, F has a unique fixed point in the space Cpr r, T s, X β q, which is the mild solution of nonlocal neutral initial value problem (1) (3) with impulse effect. This completes the proof of the theorem.

8 42 R. S. Jain, M. B. Dhakne 4. Continuous dependence of a mild solution Theorem 4.1. Suppose that hypotheses (H 1 ) (H 3 ) are satisfied and Γ ă 1. Then for each φ 1, φ 2 P Cpr r, T s, X β q and for the corresponding mild solutions x 1, x 2 of the problems (13) (14) (15) d dt rxptq ` upt, x tqs Axptq ` f ˆ t, x t, kpt, sqhps, x s qds, t P p, T s, xpτ k ` q Q k xpτ k q xpτ k q ` I k xpτ k q, k 1, 2,..., m, xptq ` gpx t1,..., x tp qptq φ i ptq, i 1, 2, t P r r, s, the following inequality holds (16) }x 1 x 2 } Cpr r,t s,xβ q ś ăτ k ăt p1 ` KL kq exppkf T qrk ` KU}p Aq β s ś ˆ}φ 1 φ 2 } r1 Λ 1 p1 ` KL k q exppkf T qs Cpr r,s,xβ q, where ăτ k ăt Λ 1 GK ` K}p Aq β }UG ` }p Aq β }U ` C 1 β UT β ` KF LHT 2. Moreover, if U and G, the above inequality reduces to classical inequality (17) }x 1 x 2 } Cpr r,t s,xβ q K ś ăτ k ăt p1 ` KL kq exppkf T q r1 KF LHT 2 ś ăτ k ăt p1 ` KL kq exppkf T qs ˆ }φ 1 φ 2 } Cpr r,s,xβ q. Proof. Let φ 1, φ 2 P Cpr r, T s, X β q be arbitrary functions and let x 1, x 2 be the mild solutions of the problem(13) (15). Then we have (18) x 1 ptq x 2 ptq T ptqrφ 1 pq φ 2 pqs T ptqrgpx 1t1,..., x 1tp qpq gpx 2t1,..., x 2tp qpqs T ptqrup, φ 1 pq gpx 1t1,..., x 1tp qpqq up, φ 2 pq gpx 2t1,..., x 2tp qpqqs `upt, x 1t q upt, x 2t q ` p Aq 1 β T pt sq p Aq β ups, x q p Aq β 1s ups, x 2s q ds ` T pt sq fps, x 1s, kps, τqhps, x 1s qdτq j fps, x 2s, kps, τqhps, x 2s qdτq ds ` T pt τ k qpi k x 1 pτ k q I k x 2 pτ k qq, t P p, T s, ăτ k ăt

9 Nonlocal impulsive integro-differential equations 421 and for t P r r, s, (19) x 1 ptq x 2 ptq φ 1 ptq φ 2 ptq rgpx 1t1,..., x 1tp qptq gpx 2t1,..., x 2tp qptqs. From (18) and using hypothesis (H 1 ) (H 3 ), we get (2) }x 1 ptq x 2 ptq} K}φ 1 φ 2 } Cpr r,s,xβ q ` GK}x 1 x 2 } Cpr r,t s,xβ q ` KU}p Aq β } φ 1 φ 2 } Cpr r,s,xβ q ` K}p Aq β }UG}x 1 x 2 } Cpr r,t s,xβ q ` }p Aq β }U}x 1 x 2 } Cpr r,t s,xβ q ` ` K ` K C 1 β UT β 1 }x 1 x 2 } Cpr r,ss,xβ qds ` K F LHT }x 1 x 2 } Cpr r,t s,xβ qds L k }x 1 pτ k q x 2 pτ k q}, t T ăτ k ăt F }x 1 x 2 } Cpr r,ss,xβ qds rk ` KU}p Aq β s}φ 1 φ 2 } Cpr r,s,xβ q ` GK}x 1 x 2 } Cpr r,t s,xβ q ` }p Aq β }UG}x 1 x 2 } Cpr r,t s,xβ q ` }p Aq β }U}x 1 x 2 } Cpr r,t s,xβ q ` C 1 β UT β }x 1 x 2 } Cpr r,t s,xβ q ` ` KF LHT 2 }x 1 x 2 } Cpr r,t s,xβ q ` K L k }x 1 pτ k q x 2 pτ k q}, t T ăτ k ăt KF }x 1 x 2 } Cpr r,ss,xβ qds rk ` KU}p Aq β s}φ 1 φ 2 } Cpr r,s,xβ q ` Λ 1 }x 1 x 2 } Cpr r,t s,xβ q ` KF }x 1 x 2 } Cpr r,ss,xβ qds ` K L k }x 1 pτ k q x 2 pτ k q}, ă τ k t. ăτ k ăt Simultaneously, by (19) and hypothesis (H 3 ), we get (21) }x 1 ptq x 2 ptq} }φ 1 φ 2 } Cpr r,s,xβ q ` G}x 1 x 2 } Cpr r,t s,xβ q, t P r r, s. Since K 1, the inequalities (2) and (21) imply

10 422 R. S. Jain, M. B. Dhakne (22) }x 1 x 2 } Cpr r,ts,xβ q rk ` KU}p Aq β s}φ 1 φ 2 } Cpr r,s,xβ q ` Λ 1 }x 1 x 2 } Cpr r,t s,xβ q ` KF }x 1 x 2 } Cpr r,ss,xβ qds ` K L k }x 1 pτ k q x 2 pτ k q}, ă τ k t, t P r, T s. ăτ k ăt Now applying Lemma 2.2 to the inequality (22), we get }x 1 x 2 } Cpr r,ts,xβ q ź p1 ` KL k q exppkf T q ăτ k ăt hence, we get }x 1 x 2 } Cpr r,t s,xβ q ź ˆ `rk ` KU}p Aq β s}φ 1 φ 2 } Cpr r,s,xβ q ` Λ 1 }x 1 x 2 } Cpr r,t s,xβ q, ăτ k ăt p1 ` KL k q exppkf T q ˆ `rk ` KU}p Aq β s}φ 1 φ 2 } Cpr r,s,xβ q ` Λ 1 }x 1 x 2 } Cpr r,t s,xβ q. The inequalities given by (16) and (17) are easy consequences of the above inequality. This completes the proof. References [1] H. Akca, A. Boucherif, V. Covachev, Impulsive functional differential equations with nonlocal conditions, Int. J. Math. Math. Sci. 29(5) (22), [2] A. Anguraj, K. Karthikeyan, Existence of solutions for impulsive functional differential equations with nonlocal conditions, Nonlinear Anal. 7 (29), [3] M. Benchora, J. Henderson, Existence results for impulsive multivalued semilinear neutral functional differential inclusions in Banach spaces, J. Math. Anal. Appl. 263 (21), [4] K. Balachandran, J. Y. Park, Nonlocal Cauchy problem for Sobolev type functional integro-differential equations, Bull. Korean Math. Soc. 39(1) (22), [5] K. Balachandran, J. Y. Park, Existence of a mild solution of a functinal integrodifferential equations with nonlocal condition, Bull. Korean Math. Soc. 38(1) (21), [6] L. Byszewski, On mild solution of a semilinear functional differential evolution nonlocal problem, J. Appl. Math. Stochastic Anal. 1(3) (1997), [7] L. Byszewski, H. Akca, Existence of solutions of a semilinear functional evolution nonlocal problem, Nonlinear Anal. 34 (1998), [8] L. Byszewski, V. Lakshamikantham, Theorems about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal. 4 (199),

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