Antiperiodic solutions for semilinear evolution equations

In this paper, we study the existence problem of antiperiodic solutions for the following first-order semilinear evolution equation: u ʹ ( t ) + A u ( t ) + ∂ G u ( t ) + f ( t ) = 0 , t ∈ R ; u ( t + T ) = - u ( t ) , t ∈ R , in a Hilbert space H, where A is a self-adjoint operator, ∂G is the gradient of G. Existence results are obtained under assumptions that D(A) is compactly embedded into H and ∂G is continuous or G is a convex function, which extend some known results in [1,2].