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This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Impulsive differential equations with nonlocal conditions in general Banach spaces Advances in Difference Equations 2012, 2012:10 doi:10.1186/1687-1847-2012-10 Lanping Zhu (lpzmath@yahoo.com.cn) Qixiang Dong (qxdongyz@yahoo.com.cn) Gang Li (gli@yzu.edu.cn) ISSN 1687-1847 Article type Research Submission date 7 August 2011 Acceptance date 14 February 2012 Publication date 14 February 2012 Article URL http://www.advancesindifferenceequations.com/content/2012/1/10 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). 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Impulsive differential equations with nonlocal conditions in general Banach spaces Lanping Zhu ∗ , Qixiang Dong and Gang Li School of Mathematics, Yangzhou University, Yangzhou 225002, China ∗ Corresponding author: lpzmath@yahoo.com.cn Email addresses: QD: qxdongyz@yahoo.com.cn GL: gli@yzu.edu.cn Abstract This article is concerned with impulsive semilinear differential equations with nonlocal initial conditions in Banach spaces. The approach used is fixed point theorem combined with the technique of operator transformation. Existence results are obtained when the nonlocal item is Lipschitz continuous. An example is also given to illustrate the obtained theorem. 1 AMS classification: 34G10; 47D06. Keywords: impulsive differential equations; measure of noncompactness; fixed point theorem; mild solutions. 1 Introduction In this article, we deal with the existence of mild solutions for the following impulsive semi- linear nonlocal problem                u  (t) = Au(t) + f(t, u(t)), t ∈ [0, T ], t = t i , u(0) = g(u), u(t i ) = I i (u(t i )), i = 1, 2, . . . , p, 0 < t 1 < t 2 < · · · < t p < T, (1. 1) where A : D(A) ⊆ X → X is the infinitesimal generator of strongly continuous semigroup S(t) for t > 0 in a real Banach space X, u(t i ) = u(t + i ) − u(t − i ) constitutes an impulsive condition. f and g are X−valued functions to be given later. In recent years, the theory of impulsive differential inclusions has become an important object of investigation because of its wide applicability in biology, medicine, mechanics, control and in more and more fields. The impulsive conditions are the appropriate model for describing some phenomena. For example, at certain moments, the system changes their state rapidly, which cannot be modeled by traditional initial value problems. For more detailed bibliography and exposition on this subject, we refer to [1–6]. Here we first recall the study of nonlocal semilinear initial value problems. It was first considered by Byszewski. Because it has better effect in the applications than the classi- 2 cal initial condition, more and more authors have studied the following type of semilinear differential equation under various conditions on S(t), f, and g,        u  (t) = Au(t) + f(t, u(t)), t ∈ [0, T ], u(0) = g(u). (1. 2) For instance, Byszewski and Lakshmikantham [7] proved the existence and uniqueness of mild solutions for nonlocal semilinear differential equations when f and g satisfy Lipschitz type conditions. In [8], Ntouyas and Tsamatos studied the case with compactness condi- tions. Byszewski and Akca [9] established the existence of solution to functional-differential equation when the semigroup is compact, and g is convex and compact on a given ball. Sub- sequently, Benchohra and Ntouyas [10] discussed second order differential equation under compact conditions. Recently, Dong and Li [11] study the semilinear differential inclusion when g is compact. By making full use of the measure of noncompactness, Obukhovski and Zecca [12] discuss the controllability for semilinear differential inclusions with a noncompact semigroup, Xue [13–15] established new existence theorems for semilinear and nonlinear nonlocal problem, respectively. Next, we focus on the study of impulsive problems. Liu [5] discuss the classic initial problem when f is Lipschitz continuous with respect to its second variable and the impul- sive functions I i are Lipschitz continuous. Cardinali and Rubbioni [3] study the multivalued impulsive semilinear differential equation by means of the Hausdorff measure of noncompact- ness. Liang et al. [16] investigate the nonlocal impulsive problems under the assumptions of g is compact, Lipschitz, and g is not compact and not Lipschitz, respectively. 3 The goal of this article is to make use of the Hausdorff measure of noncompactness and the fixed point theory to deal with the impulsive semilinear differential equation (1.1). We obtain the existence of mild solution of the nonlocal problem (1.1) when g is Lipschitz continuous. In particular, in our proof, we do not need the Lipschitz continuity of f. Thus the compactness of S(t) or f and the Lipschitz continuity of f are the special case of our results. This article is organized as follows. In Section 2, we recall some facts about the measure of noncompactness, fixed point theorem and semilinear differential equations. In Section 3, we give the existence result of the problem (1.1) when g is Lipschitz continuous. In Section 4, an example is given to illustrate our abstract results. 2 Preliminaries Let E be a real Banach space, we introduce the Hausdorff measure of noncompactness α defined on each bounded subset Ω of E by α(Ω) = inf{r > 0; there are finite points x 1 , x 2 , . . . , x n ∈ E with Ω ⊂  n i=1 B(x i , r)}. Now, we recall some basic properties of the Hausdorff measure of noncompactness. Lemma 2.1 [17]. For all bounded subsets Ω, Ω 1 , Ω 2 of E, the following properties are satisfied: (1) Ω is precompact if and only if α(Ω) = 0; (2) α(Ω) = α(Ω) = α(convΩ), where Ω and convΩ mean the closure and convex 4 hull of Ω, respectively; (3) α(Ω 1 ) ≤ α(Ω 2 ) when Ω 1 ⊂ Ω 2 ; (4) α(Ω 1  Ω 2 ) ≤ max{α (Ω 1 ), α(Ω 2 )}; (5) α(λΩ) = |λ|α(Ω), for any λ ∈ R; (6) α(Ω 1 + Ω 2 ) ≤ α (Ω 1 ) + α(Ω 2 ), where Ω 1 + Ω 2 = {x + y; x ∈ Ω 1 , y ∈ Ω 2 }; (7) if {W n } +∞ n=1 is a decreasing sequence of nonempty bounded closed subsets of E and lim n→∞ α(W n ) = 0, then ∩ +∞ n=1 W n is nonempty and compact in E. The map Q : D ⊂ E → E is said to be an α−contraction, if there exists a positive constant k < 1 such that α(QB) < kα(B) for every bounded closed subset B ⊂ D (see [18]). Lemma 2.2 ([17]: Darbo–Sadovskii). If D ⊂ E is bounded closed and convex, the contin- uous map Q : D → D is an α−contraction, then the map Q has at least one fixed point in D. Throughout this article, let (X, ||.||) be a real Banach space. We denote by C([0, T]; X) the Banach space of all continuous functions from [0, T] to X with the norm ||u|| = sup {||u(t)||, t ∈ [0, T ]} and by L 1 ([0, T ]; X) the Banach space of all X−valued Bochner integrable functions defined on [0, T ] with the norm ||u|| 1 =  T 0 ||u(t)||dt. Let PC([0, T ]; X) = {u : [0, T ] → X : u(t) be continuous at t = t i and left continuous at t = t i and the right limit u(t + i ) exists for i = 1, 2, . . . , p}. It is easy to check that PC([0, T ]; X) is a Banach space with the norm ||u|| P C = sup{||u(t)||, t ∈ [0, T]} and C([0, T ]; X) ⊆ PC([0, T ]; X) ⊆ L 1 ([0, T ]; X). Moreover, we denote β by the Hausdorff measure of noncompactness of X, denote β c by 5 the Hausdorff measure of noncompactness of C([0, T ]; X) and denote β pc by the Hausdorff measure of noncompactness of P C([0, T ]; X). C 0 −semigroup S(t) is said to be equicontinuous if {S(t)x : x ∈ B} is equicontinuous for t > 0 for all bounded set B ⊂ E. Consequently, the following lemma is easily verified. Lemma 2.3 If the semigroup S(t) is equicontinuous and w ∈ L 1 ([0, T ]; R + ), then the set {  t 0 S(t − s)u(s)ds, u(s) ≤ w(s) for a.e. s ∈ [0, T ] } is equicontinuous for t ∈ [0, T ]. Lemma 2.4 If W ⊆ PC([0, T ]; X) is bounded, then we have sup t∈[0,T ] β(W (t)) ≤ β pc (W ), where W (t) = {u(t); u ∈ W } ⊂ X. Proof. For arbitrary ε > 0, there exists W i ⊆ P C([0, T ]; X), i = 1, 2, . . . , n, such that W ⊆ ∪ n i=1 W i and diam(W i ) ≤ 2β pc (W ) + ε, i = 1, 2, . . . , n. Hence, for every t ∈ [0, T ], W (t) ⊆ ∪ n i=1 W i (t), and diam(W i (t)) ≤ diam(W i ), i = 1 , 2, . . . , n, that is 2β(W (t)) ≤ max 1≤i≤n diam(W i (t)) ≤ max 1≤i≤n diam(W i ) ≤ 2β pc (W ) + ε, and therefore, sup t∈[0,T ] β(W (t)) ≤ β pc (W ). To discuss the problem (1.1), we also need the following lemma. 6 Lemma 2.5 [4]. If W ⊆ C([0, T ]; X) is bounded, then for all t ∈ [0, T ], β(W (t)) ≤ β c (W ), where W(t) = {u(t); u ∈ W } ⊂ X. Furthermore, if W is equicontinuous on [0, T ], then β(W (t)) is continuous on [0, T] and β c (W ) = sup{β(W (t)) : t ∈ [0, T ]}. We will also use the sequential measure of noncompactness β 0 generated by β, that is, for any bounded subset Ω ⊂ X, we define β 0 (Ω) = sup{β({x n : n ≥ 1}) : {x n } +∞ n=1 is a sequence in Ω}. It follows that β 0 (Ω) ≤ β(Ω) ≤ 2β 0 (Ω). (2. 1) In addition, when X is separable, we have β 0 (Ω) = β(Ω). For the above related results on the sequential measure of noncompactness, we refer to [17]. Definition 2.1 A function u ∈ C([0, T ]; X) is said to be a mild solution of the nonlocal problem (1.2), if u(t) = S(t)g(u) + t  0 S(t − s)f(s, u(s))ds for all t ∈ [0, T ]. 7 Let Γf be the only mild solution of the following semilinear system u  (t) = Au(t) + f(t), a.e. t ∈ [0, T], u(0) = u 0 . Now, we give the following result about β−estimation of mild solutions (see [19]), similarly, see also [20,21]. Lemma 2.6 Let {f k } +∞ k=1 be a sequence of functions in L 1 ([0, T ]; X). Assume that there exists ϕ ∈ L 1 ([0, T ]; R + ) satisfying ||f k (t)|| ≤ ϕ(t) a.e. on [0, T ] for all k ≥ 1. Then for all t ∈ [0, T ], we have β({(Γf k )(t) : k ≥ 1}) ≤ 2M t  0 β({f k (s) : k ≥ 1})ds. Definition 2.2 A function u ∈ PC([0, T ]; X) is said to be a mild solution of the nonlocal problem (1.1), if it satisfies u(t) = S(t)g(u) + t  0 S(t − s)f(s, u(s))ds +  0<t i <t S(t − t i )I i (u(t i )), 0 ≤ t ≤ T. Since {S(t) : t ∈ [0, T ]} is a strongly continuous semigroup of bounded linear operators, we may assume ||S(t)|| ≤ M for all t ∈ [0, T ]. In addition, let r be a finite positive constant, and set B r := {x ∈ X : ||x|| ≤ r} and W r := {u ∈ PC([0, T ]; X) : u(t) ∈ B r , ∀ t ∈ [0, T ]}. 8 3 Main results In this section, by using the method and technique of operator transformation, Hausdorff measure of noncompactness and fixed point, we give the existence result for the nonlocal problem (1.1). First, we give the following hypotheses: (H A ) The C 0 semigroup S(t) generated by A is equicontinuous; (H f ) f : [0, T ] × X → X satisfies the following conditions: (1) f(·, x) : [0, T ] → X is measurable for all x ∈ X, (2) f(t, ·) : X → X is continuous for a.e. t ∈ [0, T ], (3) there exists l(t) ∈ L 1 (0, T ; R + ) such that β(f(t, D) ≤ l(t)β(D), for a.e. t ∈ [0, T] and every bounded subset D ⊂ X; (H I ) I i : X → X is Lipschitz continuous with Lipschitz constant k i , for i = 1, 2, . . . , p; (H g ) There exists a constant k ∈ (0, 1/M −  p i=1 k i ) such that ||g(u) − g(v)|| ≤ k||u − v||, for u, v ∈ P C([0, T ]; X); (H r ) M(||g(0)|| +  p i=1 ||I i (0)|| + T · sup t∈[0,T ],u∈W r ||f(t, u(t))||) ≤ (1 − M(k +  p i=1 k i ))r. Theorem 3.1 Assume that the conditions (H A ), (H f )(1)–(3), (H I ), (H g ), and (H r ) are satisfied. Then the nonlocal problem (1.1) has at least one mild solution on [0, T ] provided that M(4l 1 + k +  p i=1 k i ) < 1, where l 1 =  T 0 l(s)ds. 9 [...]... 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Impulsive differential equations with nonlocal conditions in general Banach spaces Advances in Difference Equations 2012, 2012:10 doi:10.1186/1687-1847-2012-10 Lanping Zhu (lpzmath@yahoo.com.cn) Qixiang. distribution, and reproduction in any medium, provided the original work is properly cited. Impulsive differential equations with nonlocal conditions in general Banach spaces Lanping Zhu ∗ , Qixiang Dong. equations with nonlocal initial conditions in Banach spaces. The approach used is fixed point theorem combined with the technique of operator transformation. Existence results are obtained when the nonlocal