Conditions of non-empty intersections of closed sets using non-self maps

Abstract

In this paper, we find a set of conditions under which two closed subsets of a metric space have nonempty intersection. We show that the existence of a non-self map from one of the sets to the other satisfying a weak contraction inequality along with some other conditions is sufficient to ensure the nonempty intersection. Further it is shown that the intersection contains the unique fixed point of the mapping. The result has a corollary and is illustrated with two examples. One of the examples show that the main theorem properly contains its corollary.