On almost automorphic type solutions of abstract Integral equations, a Bohr-Neugebauer type property and some applications

TL;DR

This paper establishes conditions for the existence and uniqueness of almost automorphic solutions to abstract integral and integro-differential equations in Banach spaces, including applications to heat conduction and parabolic equations.

Contribution

It introduces the notion of λ-bounded functions, develops an abstract theory for almost automorphic solutions, and extends Bohr-Neugebauer type results to these equations.

Findings

Unique almost automorphic solutions exist under certain conditions.

Application to heat conduction models with memory.

Existence of solutions for delayed parabolic equations.

Abstract

In the present work we give some sufficient conditions to obtain a unique almost automorphic solution to abstract nonlinear integral equations which are simultaneously of advanced and delayed type and also a unique asymptotically almost automorphic mild solution to abstract integro-differential equations with nonlocal initial conditions, both situations are posed on Banach spaces. Also, we develop a Bohr-Neugebauer type result for the abstract integral equations. Before that, we introduce the notion of $\lambda$-bounded functions, develop the appropriate abstract theory and discuss the almost periodic situation. As applications, we study the existence of an asymptotically almost automorphic solution to integro-differential equations modeling heat conduction in materials with memory and also the existence of the almost automorphic solution to semilinear parabolic evolution equations with finite delay.