Anti-periodic solutions for semilinear evolution equations

In this paper, we study the existence problem of anti-periodic solutions for the following first order evolution equation:{u′(t)+Au(t)+∂Gu(t)+F(t,u(t))=0,a.e. t∈R;u(t+T)=−u(t),t∈R, in a separable Hilbert space H, where A is a self-adjoint operator, ∂G is the gradient of G and F   is a nonlinear mapping. An existence result is obtained under the assumptions that D(A)D(A) is compactly embedded into H, ∂G is a continuous bounded mapping in H and F   is a continuous mapping bounded by a L2L2 function, which extends some known results in [Y.Q. Chen et al., Anti-periodic solutions for semilinear evolution equations, J. Math. Anal. Appl. 273 (2002) 627–636] and [A. Haraux, Anti-periodic solutions of some nonlinear evolution equations, Manuscripta Math. 63 (1989) 479–505].