Some generalizations of Darbo’s theorem and applications to fractional integral equations

Mohamed Jleli,Donal O’Regan,Erdal Karapinar,Bessem Samet

doi:10.1186/s13663-016-0497-4

Abstract

AbstractIn this paper, some generalizations of Darbo’s fixed point theorem are presented. An existence result for a class of fractional integral equations is given as an application of the obtained results.

Highlights

Introduction and preliminaries LetE be a Banach space over R with respect to a certain norm ·
For any subsets X and Y of E, we have the following notations: X denotes the closure of X; conv(X) denotes the convex hull of X; P(X) denotes the set of nonempty subsets of X; X + Y and λX (λ ∈ R) stand for algebraic operations on sets X and Y
If X is a nonempty subset of E and T : X → X is a given operator, we denote by Fix(T) the set of fixed points of T, that is, Fix(T) = {x ∈ X : Tx = x}

Summary

Then for

In [ ], Dhage introduced the following axiomatic definition of the measure of noncompactness. We say that σ is a D-measure of noncompactness (in the sense of Dhage) on E if the following conditions are satisfied:. In this paper, using the axiomatic definition of the measure of noncompactness given by Dhage, we obtain new generalizations of Theorem. The following lemma can be proved using a similar argument as in the proof of Theorem . Let IX be the set of mappings D : X → X such that (I ) D is continuous; (I ) there exists some φ ∈ such that σ (DW ).

Let γε

Thus we have