Pseudo-Almost Automorphic 𝓒0-Solutions: Well-Posedness and Asymptotic Behaviour in Evolution Equations with Nonlocal Constraints

Abstract

This research establishes an innovative analytical framework for examining C0-solutions of pseudo-almost automorphic type in evolution systems governed by nonlocal initial conditions within Banach spaces. Our methodological approach is founded upon the compactness properties of the resolvent operator (I−A)−1 and the semigroup generated by A. By integrating semigroup theory with fixed-point methodologies, we develop a unified strategy that successfully overcomes challenges arising from nonlocal initial specifications. The practical applicability of our theoretical framework is verified through its implementation on a transport equation with nonlocal initial history, thereby demonstrating both the existence of solutions and their distinctive character.