Mixed Monotone Iterative Technique for Abstract Impulsive Evolution Equations in Banach Spaces
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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, [email protected] Received 29 December 2009; Revised 20 July 2010; Accepted 3 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.
1. 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-quasisolutions for the periodic boundary value problem PBVP of impulsive evolution equations u t Aut ft, ut, ut, Δu|ttk Ik utk , utk , u0 uω
a.e. t ∈ J,
k 1, 2, . . . , p,
1.1
2
Journal of Inequalities and Applications
in an ordered Banach space X, where A : DA ⊂ X → X is a closed linear operator and −A generates a C0 -semigroup T t t ≥ 0 in X; f : J × X × X → X only satisfies weak Carath´eodory condition, J 0, ω, ω > 0 is a constant; 0 t0 < t1 < t2 < · · · < tp < tp1 ω; Ik : X × X → X is an impulsive function, k 1, 2, . . . , p; Δu|ttk denotes the jump of ut at t tk , that is, Δu|ttk utk − ut−k , where utk and ut−k represent the right and left tk limits of ut at t tk , respectively. Let PCJ, X : {u : J → X | ut is continuous at t / and left continuous at t tk , and utk exists, k 1, 2, . . . , p}. Evidently, PCJ, X is a Banach space with the norm uPC supt∈J ut. Let J J \ {t1 , t2 , . . . , tp }, J J \ {0, t1 ,
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