Family of mappings with an equicontractive-type condition

Dariusz Wardowski

doi:10.1007/s11784-020-00789-2

Dariusz Wardowski

Open Access

https://doi.org/10.1007/s11784-020-00789-2

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Abstract

In a real Banach space X and a complete metric space M, we consider a compact mapping C defined on a closed and bounded subset A of X with values in M and the operator T:Atimes C(A) rightarrow X. Using a new type of equicontractive condition for a certain family of mappings and beta -condensing operators defined by the Hausdorff measure of noncompactness we prove that the operator xmapsto T(x,C(x)) has a fixed point. The obtained results are applied to the initial value problem.

Highlights

Introduction and preliminariesThe investigations concerning compact operators together with contractive mappings have their origin in the famous Krasnosel’skiı’s result [7]
Using a new type of equicontractive condition for a certain family of mappings and β-condensing operators defined by the Hausdorff measure of noncompactness we prove that the operator x → T (x, C(x)) has a fixed point
Using Krasnosel’skiı-Schaefer type method, Vetro and Wardowski [12] have recently proved an existence theorem producing a periodic solution of a nonlinear integral equation

Summary

Introduction and preliminaries

The investigations concerning compact operators together with contractive mappings have their origin in the famous Krasnosel’skiı’s result [7] This known theorem states that if M is a nonempty closed convex and bounded subset of the given Banach space X and there are given two mappings: a contraction A : M → X and a compact operator B : M → X satisfying A(M ) + B(M ) ⊂ M A + B has a fixed point. In [4], the authors merged the concepts due to Krasnosel’skiı with the Schaefer’s result [11]. In [9], Reich considered condensing mappings with bounded ranges and applied them to obtain the Schaefer’s alternative and a Krasnosel’skiı type fixed point theorem. Przeradzki in his work [8], using a concept of Hausdorff measure of noncompactness, relaxed a strong condition: A(M ) + B(M ) ⊂ M, 55 Page 2 of 9

Wardowski

Results

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Initial value problem

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