Existence of Solutions for a System of Integral Equations Using a Generalization of Darbo’s Fixed Point Theorem

Babak Mohammadi,Manuel De La Sen,Vahid Parvaneh,Ali Asghar Shole Haghighi,Maryam Khorshidi

doi:10.3390/math8040492

Abstract

In this paper, an extension of Darbo’s fixed point theorem via θ -F-contractions in a Banach space has been presented. Measure of noncompactness approach is the main tool in the presentation of our proofs. As an application, we study the existence of solutions for a system of integral equations. Finally, we present a concrete example to support the effectiveness of our results.

Highlights

Introduction and PreliminariesIntegral equations are equations in which an unknown function emerges under an integral sign
The existence of solutions for nonlinear integral equations have been perused in many papers via applying the measures of noncompactness approach which was initiated by Kuratowski [4]
The concepts of α-ψ and β-ψ condensing operators have been defined and using them some new fixed point results via the technique of measure of noncompactness have been presented

Summary

Introduction and Preliminaries

Integral equations are equations in which an unknown function emerges under an integral sign. The concepts of α-ψ and β-ψ condensing operators have been defined and using them some new fixed point results via the technique of measure of noncompactness have been presented. Let ∆ be the following subfamily of Γ consists of all functions W : R+ → R so that (W1 ) W is a continuous and strictly increasing mapping;. Let Ω be a nonempty, bounded, closed and convex subset of a Banach space E. Each continuous and compact mapping W : Ω → Ω possesses at least one fixed point in the set Ω. Let Ω be a nonempty, bounded, closed and convexsubset of a Banach space E and let Υ : Ω → Ω be a continuous mapping. Υ admits at least a fixed point in Ω

Main Results

Coupled Fixed Point

Application

Example