In this article, we use fixed-point methods and MEASURE of non-compactness theory to focus on the problem of establishing the existence of at least a solution for the following functional integral equation...

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.

We propose a new notion of contraction mappings for two class of functions involving MEASURE of NONCOMPACTNESS in Banach space and derive some basic Darbo type fixed and coupled fixed point results. This work includes and extends the results of Falset and Latrach[ Falset, J. G., Latrach, K.: On Darbo-Sadovskii's fixed point theorems type for abstract MEASUREs of (weak) NONCOMPACTNESS, Bull. Belg. Math. Soc. Simon Stevin 22 (2015), 797-812. ] The results are also correlated with the classical generalized Banach fixed point theorems. Also we show the applicability of obtained results to the Volterra integral equations in Banach algebras.

In this paper, first we introduce a MEASURE of NONCOMPACTNESS in the Sobolev space Wk; 1ð XÞ and then, as an application, we study the existence of solutions for a class of the functional integral-differential equations using Darbo’ s fixed point theorem associated with this new MEASURE of NONCOMPACTNESS. Further, two examples are presented to verify the effectiveness and applicability of our main results.

A fixed point theorem is proved using a newly constructed contraction operator in this article, and the solvability of a more general type of fractional integrals based here on the proportional derivative is analyzed. We also use suitable examples to illustrate our findings.