Optimum solutions for a system of differential equations via measure of noncompactness

New classes of mappings, called cyclic (noncyclic) condensing operators, are introduced and used to investigate the existence of best proximity points (best proximity pairs) with the help of a suitable measure of noncompactness. In this way, we obtain some real generalizations of Schauder and Darbo's fixed point theorems. In the last section, we apply such results to study the existence of optimum solutions to a system of differential equations.