A unified existence theory for evolution equations and systems under nonlocal conditions

TL;DR

This paper develops a comprehensive existence theory for evolution equations and systems with nonlocal conditions, analyzing how the support of these conditions affects the existence of solutions in Banach spaces.

Contribution

It introduces a unified framework that connects Volterra and Fredholm operators for nonlocal conditions, extending to systems with variable supports and independent nonlinearities.

Findings

Established existence results for a range of nonlocal conditions

Extended the theory to systems with variable nonlocal supports

Unified approach bridging integral operators in evolution equations

Abstract

We investigate the effect of nonlocal conditions expressed by linear continuous mappings over the hypotheses which guarantee the existence of global mild solutions for functional-differential equations in a Banach space. A progressive transition from the Volterra integral operator associated to the Cauchy problem, to Fredholm type operators appears when the support of the nonlocal condition increases from zero to the entire interval of the problem. The results are extended to systems of equations in a such way that the system nonlinearities behave independently as much as possible and the support of the nonlocal condition may differ from one variable to another.