Anti-periodic solutions for nonlinear evolution equations

In this paper, we use the homotopy method to establish the existence and uniqueness of anti-periodic solutions for the nonlinear anti-periodic problem {(x) over dot + A(t,x) + Bx = f(t) a.e. t is an element of R, {x(t + T) = -x(t), where A(t,x) is a nonlinear map and B is a bounded linear operator from R-N to R-N. Sufficient conditions for the existence of the solution set are presented. Also, we consider the nonlinear evolution problems with a perturbation term which is multivalued. We show that, for this problem, the solution set is nonempty and weakly compact in W-1,W-2(I, R-N) for the case of convex valued perturbation and prove the existence theorems of anti-periodic solutions for the nonconvex case. All illustrative examples are provided.