Fixed Point and Common Fixed Point of Modified Enriched Hardy-Rogers Contractions in Some Banach Spaces
DOI:
https://doi.org/10.33003/fjs-2026-1014-5274Keywords:
Modified enriched Hardy-Rogers contractions, Modified enriched Jungck–Hardy–Rogers contractions, Common fixed point, real Banach spacesAbstract
The celebrated Banach contraction principle and its numerous extensions continue to play a central role in nonlinear analysis, fixed point theory, optimization, and differential equations. Among its notable generalizations are the Hardy–Rogers contraction, enriched contractions, and Jungck-type contractive mappings, each of which has significantly broadened the scope of applicability of fixed point methods. Motivated by recent developments on enriched Hardy–Rogers contractions and the growing interest in enriched contractive frameworks, we introduce a new class of operators called modified enriched Hardy–Rogers contractions in arbitrary real Banach spaces. The proposed contractive conditions are shown to be structurally different from the existing enriched Hardy–Rogers-type formulations and establish a new enriched framework. By employing direct iterative techniques and a careful analysis of the associated contractive inequalities, we establish existence and uniqueness results for fixed points of modified enriched Hardy–Rogers contractions in Banach spaces. Furthermore, we introduce the notion of modified enriched Jungck–Hardy–Rogers contractions and prove a unique common fixed point theorem for commuting pairs of mappings satisfying the proposed condition. The developed theory extends and unifies several classical results including those of Banach, Reich, Hardy–Rogers, Berinde–Păcurar, and Jungck. In addition, illustrative examples are presented to demonstrate the independence and properness of the newly introduced classes of mappings. The results obtained herein contribute to the ongoing development of enriched fixed point theory and provide a flexible framework for studying nonlinear operators beyond the reach of existing contraction principles.
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