Anti-periodic Problems for Semilinear Partial Neutral Evolution Equations†

We study the anti-periodic problem for the semilinear partial neutral evolution equation in the form d dt [u(t) + h(t, u(t))] + Au(t) = f(t, u(t)), t ∈ R in a Banach space X, where h, f are given X-valued functions, and −A : D(A) ⊆ X → X is the infinitesimal generator of a compact analytic semigroup. Some new theorems concerning the existence of anti-periodic mild solutions for the problem are established. The theorems formulated are essential extensions of those given previously for the anti-periodic problems for evolution equations in Banach spaces. The main tools in our study are the analytic semigroup theory of linear operators, fractional powers of closed operators, and a fixed point theorem due to Krasnoselskii. Furthermore, we provide an illustrative example to justify the practical usefulness of the obtained abstract results.