Monotone Iterative Technique for the Initial Value Problems of Impulsive Evolution Equations in Ordered Banach Spaces

and Applied Analysis 3 The purpose of this paper is to improve and extend the above mentioned results. We will delete the Lipschitz condition 1.5 for impulsive function Ik and the restriction condition 1.6 and improve condition 1.4 for nonlinear term f . Our main results are as follows. Theorem 1.1. Let E be an ordered Banach space, whose positive cone P is normal, A : D A ⊂ E → E be a closed linear operator, −A generate a positive C0-semigroup T t t ≥ 0 , f ∈ C J ×E×E, E , and Ik ∈ C E, E , k 1, 2, . . . , m. If IVP 1.1 has a lower solution v0 and an upper solutionw0 with v0 ≤ w0 and the following conditions are satisfied: H1 there exists a positive constant C such that f ( t, x2, y2 ) − f(t, x1, y1) ≥ −C x2 − x1 , 1.7 for any t ∈ J , v0 t ≤ x1 ≤ x2 ≤ w0 t , and Gv0 t ≤ y1 ≤ y2 ≤ Gw0 t , H2 for any x1, x2 ∈ E with v0 tk ≤ x1 ≤ x2 ≤ w0 tk , k 1, 2, . . . , m, one has Ik x1 ≤ Ik x2 , 1.8 H3 there exists a positive constant L such that α ({ f ( t, xn, yn )}) ≤ L(α {xn} α({yn})), 1.9 for any t ∈ J , and increasing or decreasing monotonic sequences {xn} ⊂ v0 t , w0 t and {yn} ∈ Gv0 t , Gw0 t . Then IVP 1.1 has minimal and maximal mild solutions between v0 and w0, which can be obtained by a monotone iterative procedure starting from v0 and w0, respectively. Clearly, condition H3 greatly improves the measure of noncompactness condition in 8 . Therefore, Theorem 1.1 greatly improves the main results in 6–8 . In Theorem 1.1, if Banach space E is weakly sequentially complete, condition H3 holds automatically; see 9, Theorem 2.2 . Hence, from Theorem 1.1, we have the following. Corollary 1.2. Let E be an ordered and weakly sequentially complete Banach space, whose positive cone P is normal, A : D A ⊂ E → E be a closed linear operator, −A generate a positive C0-semigroup T t t ≥ 0 , f ∈ C J × E × E, E , and Ik ∈ C E, E , k 1, 2, . . . , m. If IVP 1.1 has a lower solution v0 and an upper solution w0 with v0 ≤ w0, and the conditions H1 and H2 are satisfied, then IVP 1.1 has minimal and maximal mild solutions between v0 and w0, which can be obtained by a monotone iterative procedure starting from v0 and w0, respectively. The proof of Theorem 1.1 will be shown in the next section. In Section 2, we also discuss the uniqueness of mild solutions for IVP 1.1 between the lower solution and upper solution see Theorem 2.4 . 4 Abstract and Applied Analysis 2. Proof of the Main Results Let C J, E denote the Banach space of all continuous E-value functions on interval J with norm ‖u‖C maxt∈J‖u t ‖ and let C1 J, E denote the Banach space of all continuously differentiable E-value functions on interval J with norm ‖u‖C1 max{‖u‖C, ‖u‖C}. Consider the initial value problem IVP of linear evolution equation without impulse u′ t Au t h t , t ∈ J, u 0 u0. 2.1 It is well-known 10, chapter 4, Theorem 2.9 , when u0 ∈ D A and h ∈ C1 J, E , IVP 2.1 has a classical solution u ∈ C1 J, E ∩ C J, E1 expressed by u t T t u0 ∫ t 0 T t − s h s ds, t ∈ J. 2.2 Generally, when u0 ∈ E and h ∈ C J, E , the function u given by 2.2 belongs to C J, E and it is called a mild solution of IVP 2.1 . Let us start by defining what we mean by a mild solution of problem u′ t Au t h t , t ∈ J, t / tk, Δu|t tk Ik u tk , k 1, 2, . . . , m, u 0 u0. 2.3 Definition 2.1. A function u ∈ PC J, E is called a mild solution of IVP 2.3 , if u is a solution of integral equation u t T t u0 ∫ t 0 T t − s h s ds ∑ 0<tk<t T t − tk Ik u tk , t ∈ J. 2.4 To prove Theorem 1.1, for any h ∈ PC J, E , we consider the linear initial value problem LIVP of impulsive evolution equation u′ t Au t Cu t h t , t ∈ J, t / tk, Δu|t tk yk, k 1, 2, . . . , m, u 0 x, 2.5 where C ≥ 0, x ∈ E, and yk ∈ E, k 1, 2, . . . , m. Abstract and Applied Analysis 5 Lemma 2.2. For any h ∈ PC J, E , x ∈ E, and yk ∈ E, k 1, 2, . . . , m, LIVP 2.5 has a unique mild solution u ∈ PC J, E given byand Applied Analysis 5 Lemma 2.2. For any h ∈ PC J, E , x ∈ E, and yk ∈ E, k 1, 2, . . . , m, LIVP 2.5 has a unique mild solution u ∈ PC J, E given by u t S t x ∫ t 0 S t − s h s ds ∑ 0<tk<t S t − tk yk, t ∈ J, 2.6 where S t e−CtT t t ≥ 0 is a C0-semigroup generated by − A CI . Proof. Let y0 0. If u ∈ PC J, E is a mild solution of LIVP 2.5 , then the restriction of u on tk−1, tk satisfies the initial value problem of linear evolution equation without impulse u′ t Au t Cu t h t , tk−1 < t ≤ tk, u ( t k−1 ) u tk−1 yk−1. 2.7 Hence, on tk−1, tk , u t can be expressed by u t S t − tk−1 u tk−1 S t − tk−1 yk−1 ∫ t tk−1 S t − s h s ds. 2.8 Iterating successively in the above equality with u tj for j k − 1, k − 2, . . . , 1, 0, we see that u satisfies 2.6 . Inversely, we can verify directly that the function u ∈ PC J, E defined by 2.6 satisfies all the equalities of LIVP 2.5 . Let α · denote the Kuratowskii measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see 11 . For any B ⊂ C J, E and t ∈ J , set B t {u t | u ∈ B} ⊂ E. If B is bounded in C J, E , then B t is bounded in E, and α B t ≤ α B . In the proof of Theorem 1.1 we need the following lemma. Lemma 2.3. Let B {un} ⊂ PC J, E be a bounded and countable set. Then α B t is the Lebesgue integrable on J , and