On best proximity pairs with application to differential equations

Abstract

In this paper we consider the following system of differential equations, y = f (x, y), y(x 0) = y 1 and z = g(x, z), z(x 0) = z 1 , where f, g are bounded L 1 functions defined on a rectangle in R 2. We give sufficient conditions for the existence of two functions φ and ψ, on an interval I containing x 0 , such that |y 1 + x x 0 f (t, φ(t))dt − φ(x)| ≤ |y 1 − z 1 |, |z 1 + x x 0 g(t, ψ(t))dt − ψ(x)| ≤ |y 1 − z 1 | for all x ∈ I. To establish the same, we introduce a notation of c-cyclic contractive mapping and prove the existence of best proximity pairs for such a mapping.

On Proximal contractions via implicit relations and Best proximity points

“…In many cases of sciences and engineering, instead of solutions of certain kinds of integro-differential equations, one need optimal solutions of the same. On the other hand, if an integral or differential equation does not possess any solution, then we often show interest on the best proximity solutions of the equation (see [21], [6]). In the aforementioned two cases, the desired optimal and best proximity solutions can be dealt by best proximity point results only.…”