article · International Journal of Analysis and Applications
Research examines implicit Kannan-type contractions situated in non-convex, rho-complete 2-modular spaces. This mathematical framework brings together modular Kannan-type contractions and specific generalised involutions governed by implicit contractive conditions. Working under the Delta-2 condition, specifically with a constant strictly between one and two that rules out modular convexity, the analysis proves that a unique fixed point exists. The solution is established through the implementation of a Krasnosel’skiĭ–Mann iterative process, demonstrating how iterative sequences can converge in these general, non-convex structures. Furthermore, the theoretical findings are applied directly to derive a coincidence point theorem specifically tailored for commuting involutions, extending classical fixed-point principles to broader and less restrictive analytical settings.
Fixed point theorems provide fundamental tools for determining whether mathematical systems possess stable, unique solutions. By establishing these results in generalised 2-modular spaces without requiring convexity, this work broadens the theoretical scope of iterative algorithms. It assists mathematicians in understanding convergence behaviour in abstract settings where standard geometric assumptions fail, supporting ongoing foundational developments across nonlinear analysis and optimization theory.
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In this paper, we study implicit Kannan-type contractions in non-convex and \(\rho\)-complete \(2\)-modular spaces. The class under consideration unifies modular Kannan-type contractions and certain generalized involutions satisfying an implicit contractive condition. Under the \(\Delta_{2}\)-condition with \(\Delta_{2}\)-constant \(\kappa\in(1,2)\), which is incompatible with the convexity of the \(2\)-modular, we establish the existence and uniqueness of a fixed point by means of a Krasnosel’skiĭ–Mann iterative scheme. As an application, we obtain a coincidence point theorem for commuting involutions.
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DOI: 10.28924/2291-8639-24-2026-265
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