LAUSR

In this paper we consider the following system of differential equations, y′ = f(x, y), y(x0) = y1 and z ′ = g(x, z), z(x0) = z1, where f, g are bounded L1 functions defined on a rectangle in R2. We give sufficient conditions for the existence of two functions φ and ψ, on an interval I containing x0, such that |y1 + ∫ x x0 f(t, φ(t))dt− φ(x)| ≤ |y1 − z1|, |z1 + ∫ x x0 g(t, ψ(t))dt− ψ(x)| ≤ |y1 − z1| 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.