Abstract and Applied Analysis
Volume 2010 (2010), Article ID 481648, 11 pages
doi:10.1155/2010/481648
Research Article

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

He Yang

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 15 March 2010; Accepted 11 July 2010

Academic Editor: Paul Eloe

Copyright © 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.

Abstract

This paper deals with the existence and uniqueness of mild solutions for the initial value problems of abstract impulsive evolution equations in an ordered Banach space E:u′(t)+Au(t)=f(t,u(t),Gu(t)), t∈[0,a], t≠tk, Δu|t=tk=Ik(u(tk)), 0<t1<t2<⋯<tm<a, u(0)=u0, where A:D(A)⊂E→E is a closed linear operator, and f:[0,a]×E×E→E is a nonlinear mapping. Under wide monotone conditions and measure of noncompactness conditions of nonlinearity f, some existence and uniqueness results are obtained by using a monotone iterative technique in the presence of lower and upper solutions.