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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.