Enriched polynomial contractions and a neural fixed-point solver for fractional boundary value problems

  • 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: 4605
DOI: https://doi.org/10.59400/adecp4605
Keywords: enriched polynomial contraction; almost enriched polynomial contraction; Banach space; fixed point; Krasnoselskii iteration; Caputo fractional differential equation; neural fixed-point solver

Abstract

We introduce two classes of operators on normed spaces, the enriched polynomial contractions and the almost enriched polynomial contractions. Both classes contain the enriched contractions of Berinde and Păcurar and the polynomial-type contractions of Jleli, Pacurar and Samet as special cases. In Banach spaces, we prove existence, uniqueness, and convergence theorems for the fixed points, first under continuity of the operator and then under the weaker assumption of Picard continuity, with the fixed points approximated by Krasnoselskii-type iteration. Several classical results, including the Banach contraction principle and the enriched contraction theorem, are recovered as special cases, and several worked examples are given. As the main application, we consider a nonlinear Caputo fractional boundary value problem of order 2 < N ≤ 3. Writing it as a fixed-point equation and assuming an explicit Lipschitz condition, we prove that it has a unique solution. We then examine the constructive side of this result numerically: on a manufactured problem with a known exact solution, we run the associated fractional-quadrature Krasnoselskii iteration, check that its measured geometric rate stays below the certified contraction factor, and recover the same solution with a neural fixed-point solver, a network trained to satisfy the discretized fixed-point equation of the same operator, and we relate the accuracy of both solvers to the underlying quadrature error. The numerical results are consistent with the theory and illustrate its use on a class of nonlinear fractional models of this form.

Published
2026-08-28
How to Cite
Akram, M., Ishtiaq, U., Din, M., & Popa, I.-L. (2026). Enriched polynomial contractions and a neural fixed-point solver for fractional boundary value problems. Advances in Differential Equations and Control Processes, 33(3). https://doi.org/10.59400/adecp4605
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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