Novel fixed point theorems for order-theoretic enriched contractions with applications to boundary value problems and matrix equations

  • 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; Research Center of Applied Mathematics, Khazar University, Baku AZ1096, Azerbaijan
  • Muhammad Din

    Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan
  • 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 Brasov, 500091 Brasov, Romania
Article ID: 4602
DOI: https://doi.org/10.59400/adecp4602
Keywords: fixed point; enriched contraction; binary relation; order-theoretic contraction; almost contraction; Krasnoselskii iteration; matrix equation; fractional differential equation

Abstract

This work puts forward a family of operators, which we term enriched ∇-preserving contractions, acting on a normed space carrying an arbitrary binary relation ∇. The family gathers several familiar classes under one roof: Banach contractions, order-theoretic contractions, and certain nonexpansive maps, among them averaged maps whose fixed points lie beyond the reach of ordinary contraction arguments; all arise through suitable choices of the constants and of the relation. Working in this relational framework, we show that any such operator admits a fixed point, even though its defining contractive estimate is demanded only on ∇-related pairs rather than across the whole space. A device used repeatedly is the symmetry of the norm: it forces ∇ and its symmetric closure to act compatibly with the operator, and this is what keeps the fixed-point conclusions meaningful. The fixed points themselves are located through the Krasnoselskii iteration, and a path criterion formulated in the symmetric closure of ∇ then yields their uniqueness; an almost contraction variant of the scheme is treated as well. By way of application, the theory settles the solvability, and under the stated hypotheses the uniqueness, of solutions to a Caputo fractional boundary value problem with an integral condition and to a nonlinear matrix equation. Numerical experiments over several groups of Lipschitz coefficients confirm the convergence threshold predicted by the theory, and the iteration is compared with Riemann-Hilbert analysis, Lie group integrators, physics-informed neural networks, and neural symbolic derivation tools, so that strengths and limits of the framework can be judged objectively.

Published
2026-08-28
How to Cite
Akram, M., Ishtiaq, U., Din, M., & Popa, I.-L. (2026). Novel fixed point theorems for order-theoretic enriched contractions with applications to boundary value problems and matrix equations. Advances in Differential Equations and Control Processes, 33(3). https://doi.org/10.59400/adecp4602
Issue
Vol. 33 No. 3 (2026): In Progress
Section
Articles

Copyright (c) 2026 Mohammad Akram, Umar Ishtiaq, Muhammad Din, Ioan-Lucian Popa

Creative Commons License

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

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