r-function

The concept of measure of noncompactness (MNC) allows us to select an important class of mappings which are more general than compact operators. The proposed work exploits the axiomatic definitio of MNC and introduces the notion of relatively nonexpansive cyclic (noncyclic) SR-condensing operators along with the aid of SR-functions. The first phase of the paper concentrates on establishing the best proximity point (pair) theorems for such operators. The main results in this manuscript extend and generalize several state of art literature on Darbo type fixed point results. In the second phase, proposed results are applied to show the actuality of optimum solutions for system of second order differential equations with two initial conditions.

The basic purpose of this article is to introduce the concept of R - λ - G-contraction by using R-functions, lower semi-continuous functions and digraphs and discuss the existence and uniqueness of points of coincidence and common fixed points for a pair of self-mappings satisfying such contractions in the setting of metric spaces endowed with a graph. As some consequences of our results, we obtain several recent results in metric spaces and partial metric spaces.