<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>In this paper, we prove the following generalization of the classical Darbo fixed point principle : Let <italic>X</italic> be a Banach space and <italic>µ</italic> be a montone measure of noncompactness on <italic>X</italic> which satisfies the generalized Cantor intersection property. Let <italic>C</italic> be a nonempty bounded closed convex subset of <italic>X</italic> and <italic>T : C → C</italic> be a continuous mapping such that for any countable set Ω ⊂ <italic>C</italic>, we have <italic>µ</italic>(<italic>T</italic>(Ω)) ≤ <italic>kµ</italic>(Ω), where <italic>k</italic> is a constant, 0 ≤ <italic>k</italic> &lt; 1. Then <italic>T</italic> has at least one fixed point in <italic>C</italic>. The proof is based on a combined use of topological methods and partial ordering techniques and relies on the Schauder and the Knaster-Tarski fixed point principles.</p>
</abstract>

In this paper, we prove the following generalization of the classical Darbo fixed point principle : Let X be a Banach space and µ be a montone measure of noncompactness on X which satisfies the generalized Cantor intersection property. Let C be a nonempty bounded closed convex subset of X and T : C → C be a continuous mapping such that for any countable set Ω ⊂ C, we have µ(T(Ω)) ≤ kµ(Ω), where k is a constant, 0 ≤ k < 1. Then T has at least one fixed point in C. The proof is based on a combined use of topological methods and partial ordering techniques and relies on the Schauder and the Knaster-Tarski fixed point principles.

On Darbo’s fixed point principle

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"On Darbo\u2019s fixed point principle"}

In this paper, we prove the following generalization of the classical Darbo fixed point principle : Let X be a Banach space and µ be a montone measure of...

On Darbo’s fixed point principle

Published Online: Sep 29, 2023

Page range: 304 - 310

Received: May 20, 2023

Accepted: Sep 12, 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0020

Keywords
Fixed point theorems, measure of noncompactness, Banach spaces

© 2023 Mohamed Aziz Taoudi, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On Darbo’s fixed point principle

Published Online: Sep 29, 2023

Page range: 304 - 310

Received: May 20, 2023

Accepted: Sep 12, 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0020

KeywordsFixed point theorems, measure of noncompactness, Banach spaces

© 2023 Mohamed Aziz Taoudi, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Fixed point theorems, measure of noncompactness, Banach spaces