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

We establish the existence of a solution for the following system of differential equations (y x ′′((t t ) ) = = g f ((t t ,y x ((t t )) )) ,y x ((t t 0 0) ) = = x x *** in the space of all bounded and continuous real functions on [0, +∞[. 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.