Anti-periodic solutions for semilinear evolution equations

Abstract

In this paper, we study the existence problem of anti-periodic solutions for the following first-order nonlinear evolution equation: u′(t)+Au(t)+F(t,u(t))=0,t∈R,u(t+T)=−u(t),t∈R, in a Hilbert space H, where A is a self-adjoint operator and F is a continuous nonlinear operator. An existence result is obtained under assumptions that D(A) is compactly embedded into H and F is anti-periodic and bounded by a L2 function. Furthermore, anti-periodic solutions for second-order equations are also studied.