Anti-periodic solutions for semilinear evolution equations

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.  2002 Elsevier Science (USA). All rights reserved.