A best proximity point approach to existence of solutions for a system of ordinary differential equations

Abstract

We establish the existence of a solution for the following system of differential equations \begin{equation*} \label{system}\begin{cases}x'(t) = f (t, x(t)), & x(t_0) = x^*,\\ y'(t) = g (t, y(t)), & y(t_0) = x^{**}, \end{cases} \end{equation*} in the space of all bounded and continuous real functions on $[0,+\infty[$. We use best proximity point methods and measure of noncompactness theory under suitable assumptions on $f$ and $g$. Some new best proximity point theorems play a key role in the above result.