A Massera Type Criterion for a Partial Neutral Functional Differential Equation

where A is the infinitesimal generator of a compact analytic semigroup of linear operators, (T (t))t≥0, on a Banach space X; the history xt, xt(θ) = x(t+θ), belongs to an appropriate phase space D and G,F : R×D → X are continuous functions. The Massera Theorem [10] is a pioneer work in the study of the relations between the boundedness of solutions and the existence of periodic solutions. In Massera [10], this relation was explained for a two dimensional periodic ordinary differential equation. Subsequently, several authors considered similar relations, see for example Yoshizawa for a n-dimensional differential equation; Lopes and Hale for n-dimensional ordinary and functional equations with delay and Yong [14], for functional differential equations. Recently Ezzinbi [1], using a Massera type criterion, showed the existence of a periodic solution for a partial functional differential equation modeled in the form