Anti-periodic Solutions For Evolution Equations In Banach Spaces And Hilbert Spaces

The study of anti-periodic solutions for nonlinear evolution equations is originated from the study of periodic solutions, and it was initiated by Okochi[1]. She showed that the following equation x’(t)∈-(?)φ{x(t))+f(t), a. e. t∈R dose not have a periodic solution in general, so she added some assumptions. Anti-periodic problems have been studied by many authors, see [1-11] and references therein. In [1] Okochi showed that the following equation has a solution, where φ:D(φ)(?)Hâ†'H is an even lower semi-continuous convex function, (?)φ is the subdifferential of φ. f(t):Râ†'H satisfying f(t+T)=-f(t) and f(t)∈L2(0,T). Y. Q. Chen[7-11] studied anti-periodic solution for evolution equations associated with maximal monotone mapping, self-adjoint mapping, and the subdifferential of convex function. In [18] Y. Q. Chen showed that the following equation has a weak solution, where A:D(A)(?)Hâ†'H is a linear densely defined closed self-adjoint operator that only has point spectrum and f(t):Râ†'H satisfying f(t+T) =-f(t) and f(t)(?)L2(0,T).In this paper, we consider the following equation under different conditions on L(t,u). The purpose of this paper is to study this problem, we will show this equation has a solution under different conditions of L(t,u).Also we consider the following anti-periodic problem in Banach Space and Hilbert Space. under different conditions of L(t,u).