Fixed Point Results for Two Mappings on Closed Balls in Continuous Controlled Metric Spaces with Applications to Urysohn Integral Equations

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DOI: https://doi.org/10.28924/2291-8639-24-2026-279
Published: Sep 2, 2026
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Mahmood Akhtar Khan, Abdullah Shoaib, Thabet Abdeljawad, Aiman Mukheimer, Fixed Point Results for Two Mappings on Closed Balls in Continuous Controlled Metric Spaces with Applications to Urysohn Integral Equations, Int. J. Anal. Appl., 24 (2026), 279.

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Mahmood Akhtar Khan, Abdullah Shoaib, Thabet Abdeljawad, Aiman Mukheimer

Abstract

This manuscript investigates complete continuous controlled metric spaces (CCMSs). New fixed point theorems are established under Geraghty and Geraghty-Ćirić type contractive conditions, formulated on a closed ball of a CCMS rather than on the entire space. This localized approach extends the existing fixed point theory in controlled metric spaces, where such results on closed balls have not been previously reported. Concrete examples are provided to illustrate the effectiveness of the obtained theorems. Furthermore, the applicability of the theoretical results is demonstrated by proving the existence and uniqueness of solutions for a system of Urysohn-type integral equations.

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