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In this paper, we make extensive use of the well-known Krasnoselskii fixed point theorem to obtain the existence of square-mean almost periodic solutions to some classes of hyperbolic stochastic evolution equations with infinite delay. Next, the existence of square-mean almost periodic solutions to not only the heat equation but also to a boundary value problem with infinite delay arising in control systems are studied. [ABSTRACT FROM AUTHOR]

This paper introduces and examines a new class of functions called Stepanov-like pseudo almost periodic functions (or Sp-pseudo almost periodic functions), which generalizes in a natural fashion the classical notion of pseudo almost periodicity. We then make extensive use of these new functions to study the existence and uniqueness of a pseudo almost periodic solution to the semilinear equation u′(t)=Au(t) + F(t,u(t)), where A : (D)(A) ⊂ X ↦ X is the infinistesimal generator of an exponentially stable C0-semigroup on a Banach space X and F: ℝ × X ↦ X is Sp-pseudo almost periodic for p > 1 and jointly continuous. [ABSTRACT FROM AUTHOR]