On (θ, α)-metric space and fixed point results with application

  • Mohammad Akram orcid

    Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
  • Umar Ishtiaq orcid

    Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore 54770, Pakistan; Software Development Department, Biruni University, 34010 Istanbul, Turkey
  • Zainab Bibi

    Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan
  • Tayyab Kamran

    Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan; Center for Theoretical Physics, Khazar University, 41 Mehseti Street, Baku AZ1096, Azerbaijan
  • Ioan-Lucian Popa

    Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania; Faculty of Mathematics and Computer Science, Transilvania University of Braşov, 500091 Braşov, Romania
Article ID: 4603
DOI: https://doi.org/10.59400/adecp4603
Keywords: fixed point; contraction mapping; controlled metric space; extended b-metric space; (θ, α)-metric space; Caputo fractional boundary value problem; existence and uniqueness of solutions

Abstract

The notion of a double controlled metric type space was introduced by Abdeljawad et al. as a broadening of the controlled metric type spaces of Mlaiki et al., and shortly afterwards Aydi et al. formulated the new extended b-metric spaces to widen the extended b-metric spaces of Kamran et al. Building on these developments, the present work advances the class of (θ, α)-metric spaces, a structure that simultaneously subsumes b-metric spaces, new extended b-metric spaces, controlled metric type spaces and double controlled metric type spaces. This is accomplished by attaching two control functions θ(ζ, µ, s) and α(ζ, µ, s) to the terms situated on the right-hand side of the triangle inequality in the axioms defining a (θ, α)-metric space. We first prove, through explicit propositions, that each of the aforementioned structures is recovered as a special case under an appropriate choice of θ and α, and we supply non-trivial examples confirming that the new class is strictly broader than a b-metric space. Within this setting, a collection of fixed point theorems is derived for φ-contractive and Reich–Ćirić–Rus–type operators, together with their corollaries. Finally, two applications are presented: the principal theorem is first used to guarantee that a nonlinear algebraic equation possesses a unique solution, and it is then applied to establish the existence and uniqueness of a solution to a nonlinear Caputo fractional boundary value problem, thereby strengthening the connection of the theory with differential equations.

Published
2026-08-28
How to Cite
Akram, M., Ishtiaq, U., Bibi, Z., Kamran, T., & Popa, I.-L. (2026). On (θ, α)-metric space and fixed point results with application. Advances in Differential Equations and Control Processes, 33(3). https://doi.org/10.59400/adecp4603
Issue
Vol. 33 No. 3 (2026): In Progress
Section
Article

Copyright (c) 2026 Mohammad Akram, Umar Ishtiaq, Zainab Bibi, Tayyab Kamran, Ioan-Lucian Popa

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

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