Existence and uniqueness of pseudo-almost periodic solutions to semilinear differential equations and applications

This paper deals with the existence and uniqueness of pseudo-almost periodic solutions to the semilinear differential equations of the form equation(*)u′(t)=Au(t)+Bu(t)+f(t,u(t)),u′(t)=Au(t)+Bu(t)+f(t,u(t)), where A,BA,B are densely defined closed linear operators on a Hilbert space HH, and f:R×H↦Hf:R×H↦H is a jointly continuous function. Using both the so-called method of the invariant subspaces for unbounded linear operators and the classical Banach fixed-point principle, the existence of a pseudo-almost periodic solution to (*) is obtained under some suitable assumptions. As applications, we examine the existence and uniqueness of pseudo-almost periodic solutions to some second-order hyperbolic equations.