Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 293410, 15 pages doi:10.1155/2010/293410 Research Article Mixed Monotone Iterative Technique for Abstract Impulsive Evolution Equations in Banach Spaces He Yang Department of Mathematics, Northwest Normal University, Lanzhou 730070, China Correspondence should be addressed to He Yang, yanghe256@163.com Received 29 December 2009; Revised 20 July 2010; Accepted September 2010 Academic Editor: Alberto Cabada Copyright q 2010 He Yang This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited By constructing a mixed monotone iterative technique under a new concept of upper and lower solutions, some existence theorems of mild ω-periodic L-quasi solutions for abstract impulsive evolution equations are obtained in ordered Banach spaces These results partially generalize and extend the relevant results in ordinary differential equations and partial differential equations Introduction and Main Result Impulsive differential equations are a basic tool for studying evolution processes of real life phenomena that are subjected to sudden changes at certain instants In view of multiple applications of the impulsive differential equations, it is necessary to develop the methods for their solvability Unfortunately, a comparatively small class of impulsive differential equations can be solved analytically Therefore, it is necessary to establish approximation methods for finding solutions The monotone iterative technique of Lakshmikantham et al see 1–3 is such a method which can be applied in practice easily This technique combines the idea of method of upper and lower solutions with appropriate monotone conditions Recent results by means of monotone iterative method are obtained in 4–7 and the references therein In this paper, by using a mixed monotone iterative technique in the presence of coupled lower and upper L-quasisolutions, we consider the existence of mild ωperiodic L-quasi solutions for the periodic boundary value problem PBVP of impulsive evolution equations u t Δu|t Au t tk f t, u t , u t , I k u t k , u tk , u uω k a.e t ∈ J, 1, 2, , p, 1.1 Journal of Inequalities and Applications in an ordered Banach space X, where A : D A ⊂ X → X is a closed linear operator and −A generates a C0 -semigroup T t t ≥ in X; f : J × X × X → X only satisfies weak Carath´ odory condition, J e 0, ω , ω > is a constant; t0 < t1 < t2 < · · · < < ω; 1, 2, , p; Δu|t tk denotes the jump of u t Ik : X × X → X is an impulsive function, k u tk − u t− , where u tk and u t− represent the right and left at t tk , that is, Δu|t tk k k limits of u t at t tk , respectively Let PC J, X : {u : J → X | u t is continuous at t / tk and left continuous at t tk , and u tk exists, k 1, 2, , p} Evidently, PC J, X is a Banach J \ {t1 , t2 , , }, J J \ {0, t1 , t2 , , } space with the norm u PC supt∈J u t Let J · A · An Denote by X1 the Banach space generated by D A with the norm · abstract function u ∈ PC J, X ∩ C1 J , X ∩ C J , X1 is called a solution of the PBVP 1.1 if u t satisfies all the equalities of 1.1 Let X be an ordered Banach space with the norm · and the partial order “≤”, whose positive cone K : {u ∈ X | u ≥ 0} is normal with a normal constant N Let L ≥ If functions v0 , w0 ∈ PC J, X ∩ C1 J , X ∩ C J , X1 satisfy v0 t Av0 t ≤ f t, v0 t , w0 t Δv0 |t tk L v0 t − w0 t , ≤ Ik v0 tk , w0 tk , k t∈J, 1.2 1, 2, , p, v0 ≤ v0 ω , w0 t Aw0 t ≥ f t, w0 t , v0 t Δw0 |t L w0 t − v0 t , ≥ Ik w0 tk , v0 tk , tk k 1, 2, , p, t∈J, 1.3 w0 ≥ w ω , we call v0 , w0 coupled lower and upper L-quasisolutions of the PBVP 1.1 Only choosing “ ” in 1.2 and 1.3 , we call v0 , w0 coupled ω-periodic L-quasisolution pair of the PBVP 1.1 Furthermore, if u0 : v0 w0 , we call u0 an ω-periodic solution of the PBVP 1.1 Definition 1.1 Abstract functions u, v ∈ PC J, X are called a coupled mild ω-periodic Lquasisolution pair of the PBVP 1.1 if u t and v t satisfy the following integral equations: t ut T t B1 u, v T t−s G1 u, v s ds T t − t k I k u tk , v tk , t ∈ J, 0 and δ ∈ R such that T t ≤ Ceδt , t ≥ 2.1 Definition 2.1 A C0 -semigroup T t t ≥ is said to be exponentially stable in X if there exist constants C ≥ and δ > such that T t ≤ Ce−δt , t ≥ 2.2 Let I0 t0 , T Denote by C I0 , X the Banach space of all continuous X-value functions on interval I0 with the norm u C maxt∈I0 u t It is well-known 12, Chapter 4, Theorem 2.9 that for any x0 ∈ D A and h ∈ C1 I0 , X , the initial value problem IVP of linear evolution equation u t Au t u t0 h t , x0 t ∈ I0 , 2.3 Journal of Inequalities and Applications has a unique classical solution u ∈ C1 I0 , X ∩ C I0 , X1 expressed by t T t − t0 x0 u t T t − s h s ds, t ∈ I0 2.4 t0 If x0 ∈ X and h ∈ C I0 , X , the function u given by 2.4 belongs to C I0 , X We call it a mild solution of the IVP 2.3 To prove Theorem 1.2, for any h ∈ PC J, X , we consider the periodic boundary value problem PBVP of linear impulsive evolution equation in X u t Au t Δu|t yk , tk k u where yk ∈ X, k t ∈ J, t / tk , h t, 1, 2, , p, 2.5 u ω , 1, 2, , p Lemma 2.2 Let T t t ≥ be an exponentially stable C0 -semigroup in X Then for any h ∈ PC J, X and yk ∈ X, k 1, 2, , p, the linear PBVP 2.5 has a unique mild solution u ∈ PC J, X given by t u t T t B h T t − s h s ds T t − tk yk , where B h I−T ω −1 ω t ∈ J, 2.6 0