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RESEARCH ARTICLE

On different generalized interpolative proximal-type contractions in metric spaces with applications

Umar Ishtiaq1,2* ,  Fahad Jahangeer3 ,  Tayyab Kamran3,4 ,  Sina Etemad5,6 ,  Manuel De La Sen7
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1 Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore, Pakistan
2 Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan
3 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
4 Center for Theoretical Physics, Khazar University, 41 Mehseti Str., Baku, Azerbaijan
5 Institute of Graduate Studies and Research, Cyprus International University, Nicosia, Northern Cyprus, Turkey
6 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
7 University of the Basque Country Campus of Leioa (Bizkaia), Leioa, Spain
IJOCTA, 025480214 https://doi.org/10.36922/IJOCTA025480214
Received: 27 November 2025 | Revised: 6 January 2026 | Accepted: 12 January 2026 | Published online: 26 February 2026
© 2026 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

In this work, we establish the conditions for ensuring the existence and uniqueness of common best proximity points for non-self-mappings defined on the general metric spaces. A unified theoretical framework is formulated to cover a broad class of contraction mappings. We describe the required conditions on the real-valued functions (ℵ, Φ) : [0,∞) → R and verify that these secure the existence of common best proximity points for (ℵ, Φ)−interpolative contractions in complete metric spaces. The study further extends this concept by examining multiple forms of interpolative proximal-type contractions, such as proximal, Ćirić —Reich—Rus, Kannan, and Hardy-Rogers variants, through the use of the auxiliary functions (ℵ, Φ). Several illustrated examples are included to demonstrate the applicability of our findings. Finally, we conclude with an application involving a nonlinear fractional differential equation, showing that it fully satisfies the assumption of our main result.

Keywords
Interpolative contraction
Best proximity point
Common fixed point
Metric space
Funding
None.
Conflict of interest
The authors declare they have no competing interests.
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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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