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 evolution equation: { u ′ ( t ) + A u ( t ) + ∂ G u ( 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 ) is compactly embedded into H, ∂ G is a continuous bounded mapping in H and F is a continuous mapping bounded by a L 2 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].