Meir-Keeler condensing operators and applications

Abstract

Motivated by the open question posed by H. K. XU in $\cite{H. K}$ (Question $2.8$), Belhadj, Ben Amar and Boumaiza introduced in $\cite{Maha}$ the concept of Meir-Keeler condensing operator for self-mappings in a Banach space via an arbitrary measure of weak noncompactness. In this paper, we introduce the concept of Meir-Keeler condensing operator for nonself-mappings in a Banach space via a measure of weak noncompactness  and we establish fixed point results under the condition of Leray-Schauder type. Some basic hybrid fixed point theorems involving the sum as well as the product of two operators are also presented. These results generalize the results on the lines of Krasnoselskii and Dhage. An application is given to nonlinear hybrid linearly perturbed integral equations and an example is also presented.