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Báo cáo hóa học: "Research Article Impulsive Integrodifferential Equations Involving Nonlocal Initial Conditions"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Impulsive Integrodifferential Equations Involving Nonlocal Initial Conditions | Hindawi Publishing Corporation Advances in Difference Equations Volume 2011 Article ID 634701 18 pages doi 10.1155 2011 634701 Research Article Impulsive Integrodifferential Equations Involving Nonlocal Initial Conditions Rong-Nian Wang and Jun Xia Department of Mathematics NanChang University NanChang JiangXi 330031 China Correspondence should be addressed to Rong-Nian Wang rnwang@mail.ustc.edu.cn Received 23 November 2010 Revised 22 February 2011 Accepted 7 March 2011 Academic Editor Jin Liang Copyright 2011 R.-N. Wang and J. Xia. 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. We focus on a Cauchy problem for impulsive integrodifferential equations involving nonlocal initial conditions where the linear part is a generator of a solution operator on a complex Banach space. A suitable mild solution for the Cauchy problem is introduced. The existence and uniqueness of mild solutions for the Cauchy problem under various criterions are proved. In the last part of the paper we construct an example to illustrate the feasibility of our results. 1. Introduction Let X II II denote a complex Banach space and denote L X by the space of all bounded linear operators from X into X with the usual operator norm II L X . Let us recall the following definitions. Definition 1.1 see 1 . Let f R R be a continuous function and Y 1. Then the expression -X c1 t sf 2 . . Ff t r n f s ds 1.1 Jo r y 1 is called the Riemann-Liouville integral of order Y 1. Definition 1.2 see 2 . Let A be a linear and closed operator with domain D A defined on X. By a solution operator associated with A in X we mean a family Qa R L X of 2 Advances in Difference Equations strongly continuous operators satisfying 1 Aa Re A ỡ c p A and 2 Xa-1R Xa A x eAiQa t xdt Re A Q x e X 0 1.2 where Q e R is a constant and R X A XI - A 1 stands for the resolvent of