Nonlinear differential equations in abstract Banach subspace of $BC(\R)$

TL;DR

This paper establishes existence and uniqueness results for nonlinear differential equations within an abstract Banach subspace of bounded continuous functions, broadening the theoretical understanding of such equations in functional analysis.

Contribution

It introduces new existence and uniqueness theorems for nonlinear differential equations in Banach subspaces of $BC( )$, including vector-valued cases, with general conditions and illustrative examples.

Findings

Proved existence of solutions under broad conditions.

Established uniqueness of solutions in the abstract framework.

Provided multiple examples in various function spaces.

Abstract

We prove results of existence of a solution (resp. existence and uniqness of a solution) for nonlinear differential equations of type $x'(t) +G(x,t) x(t) = F(x,t),$ in an abstract Banach subspace $X$ of the space of bounded real-valued continuous functions, satisfying some general and natural property. In our work, the functions $F$ and $G$ jointly depend on the variables $(x,t)\in X\times \R$. Several examples will be given, in various function spaces, to illustrate our results. The vector-valued framework is also considered.