Bounded solutions to functional integro-differential equations arising from heat conduction in materials with memory

In this paper, we consider recurrent behavior of bounded solutions of a functional integro-differential equation arising in the theory of heat conduction in materials with memory. We give a new version of the theorem on the composition of measure pseudo-almost-automorphic functions involved in delay. Based on recently obtained results on the uniform exponential stability and the contraction mapping principle, we prove some existence and uniqueness theorems for the recurrence of bounded mild solutions of equations with infinite delay. Also, we present an example of a partial integro-differential equation appearing in the study of heat conduction.