Existence of the solution to a nonlocal-in-time evolutional problem

TL;DR

This paper investigates the existence of solutions for a nonlocal-in-time evolution problem in Banach spaces, introducing advanced analysis techniques to establish necessary and sufficient conditions for solutions.

Contribution

It provides a novel approach using Dunford-Cauchy formula to reduce the existence problem to locating zeros of an entire function, extending existing results.

Findings

Established necessary and sufficient conditions for solution existence

Validated the solution representation via Dunford-Cauchy formula

Presented new sufficient conditions generalizing prior results

Abstract

This work is devoted to the study of a nonlocal-in-time evolutional problem for the first order differential equation in Banach space. Our primary approach, although stems from the convenient technique based on the reduction of a nonlocal problem to its classical initial value analogue, uses more advanced analysis. That is a validation of the correctness in definition of the general solution representation via the Dunford-Cauchy formula. Such approach allows us to reduce the given existence problem to the problem of locating zeros of a certain entire function. It results in the necessary and sufficient conditions for the existence of a generalized (mild) solution to the given nonlocal problem. Aside of that we also present new sufficient conditions which in the majority of cases generalize existing results.