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RESEARCH ARTICLE

Revisiting Krasnoselskii’s fixed point theorem: Extensions and applications to operator equations

Abir Yakoub1 ,  Abdelaziz Mennouni1 ,  Ravi P. Agarwal2*
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1 Laboratory of Mathematical Techniques, Department of Mathematics, Faculty of Mathematics and Computer Science, University of Batna 2, Batna, Algeria
2 Department of Mathematics and Systems Engineering, College of Engineering and Science, Florida Institute of Technology, Melbourne, Florida, United States of America
IJOCTA, 025350147 https://doi.org/10.36922/IJOCTA025350147
Received: 26 August 2025 | Revised: 20 October 2025 | Accepted: 29 October 2025 | Published online: 12 December 2025
© 2025 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

Fixed point theory stands as a fundamental pillar in nonlinear functional analysis, being essential for proving the existence of solutions for nonlinear differential and integral equations. Krasnoselskii’s hybrid fixed point theorem, which combines the Banach contraction principle with Schauder’s theorem, is a pivotal contribution. Recent efforts have focused on refining and relaxing the conditions of this theorem. This study aims to extend the theoretical framework of Krasnoselskii-type fixed point theorems to address a broader and more general class of nonlinear operator equations within a Banach algebra setting. It also seeks to establish rigorous conditions for the existence (and uniqueness, where possible) of solutions. The approach involves developing local variants of the classic Krasnoselskii fixed point theorems. We performed a comparative analysis of previous studies, introduced modifications to the operator equations to relax restrictive assumptions, and theoretically generalized the theorems to accommodate a complex structure involving four operators. To validate the results, they were applied to a nonlinear functional integral equation within the Banach space C[0,1]. We successfully generalized existing results by incorporating the Hölder continuity condition, which is less restrictive than the standard Lipschitz condition. The unified theoretical framework led to the establishment of a comprehensive set of theorems and corollaries covering a wide class of operator equations such as: Ax(ρ2)Bxρ1 + Cx(ρ3)Dxρ1 = x. Our results provide less restrictive local existence conditions and wider applicability in the analysis of complex mathematical systems.

Keywords
Fixed point
Operator equations
Hölderian
Nonlinear functional
Lipschitzian
Integral equations
Funding
None.
Conflict of interest
Ravi P. Agarwal is an Editorial Board Member of this journal, but was not in any way involved in the editorial and peer-review process conducted for this study, directly or indirectly. Separately, other authors declared that they have no known competing financial interests or personal relationships that could have influenced the work reported in this study.
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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