Infinite first order differential systems with nonlocal initial conditions

TL;DR

This paper investigates the existence of solutions for infinite systems of first order differential equations with nonlocal initial conditions, using fixed point theorems in Fréchet spaces, and includes a finite case and example.

Contribution

It extends solvability results to infinite differential systems with nonlocal conditions using Schauder-Tychonoff fixed point theorem in Fréchet spaces.

Findings

Existence of solutions under sub-linear growth conditions

Application to finite systems as a special case

Provided illustrative example

Abstract

We discuss the solvability of an infinite system of first order ordinary differential equations on the half line, subject to nonlocal initial conditions. The main result states that if the nonlinearities possess a suitable "sub-linear" growth then the system has at least one solution. The approach relies on the application, in a suitable Fr\'echet space, of the classical Schauder-Tychonoff fixed point theorem. We show that, as a special case, our approach covers the case of a system of a finite number of differential equations. An illustrative example of application is also provided.