Subspace Almost Periodic $C$-Distribution Semigroups and $C$-Distribution Cosine Functions

TL;DR

This paper introduces and analyzes subspace almost periodicity concepts for $C$-distribution semigroups and cosine functions in Banach spaces, extending the understanding of their behavior in abstract Volterra equations and ill-posed problems.

Contribution

It develops new notions of subspace almost periodicity and weak almost periodicity for $C$-distribution semigroups and cosine functions, advancing the theoretical framework.

Findings

Defined subspace almost periodicity for $C$-distribution semigroups.

Extended almost periodicity concepts to $C$-distribution cosine functions.

Applied the theory to abstract Volterra integro-differential equations.

Abstract

The main aim of this paper is to introduce and analyze the notions of subspace almost periodicity and subspace weak almost periodicity for $C$-distribution semigroups and $C$-distribution cosine functions in Banach spaces. We continue our previous research study of almost periodicity of abstract Volterra integro-differential equations \cite{aot}, focusing our attention on the abstract ill-posed Cauchy problems of first order.