Bielecki–Hyers–Ulam stability of non-linear fractional Volterra Fredholm Hammerstein integro-delay dynamic systems with instantaneous impulses on time scale

  • Syed Omar Shah orcid

    School of Mathematics, Southwest Jiaotong University, Chengdu 611756, China
  • Jun Zheng orcid

    School of Mathematics, Southwest Jiaotong University, Chengdu 611756, China; Department of Electrical Engineering, Polytechnique Montréal, Montreal, QC H3T 1J4, Canada
Article ID: 3353
DOI: https://doi.org/10.59400/adecp3353
Keywords: time scale, Volterra Fredholm Hammerstein integral, Bielecki–Hyers –Ulam stability, impulses, integro-dynamic system

Abstract

In this paper, we study the existence and uniqueness of solutions, Bielecki–Hyers–Ulam stability, and Bielecki–Hyers–Ulam–Rassias stability for non-linear fractional Volterra Fredholm Hammerstein integro-delay dynamic systems with instantaneous impulses on time scale. Such systems provide a unified framework that encompasses both continuous and discrete models, making them highly appropriate for describing complex real-world phenomena involving memory effects, hereditary properties, and sudden perturbations. Sufficient conditions are established for the existence and uniqueness of solutions to the considered systems. In particular, the Picard operator and the Banach fixed point theorem are utilized to prove the existence and uniqueness of solutions. Moreover, we analyze the qualitative behavior of solutions by proving Bielecki–Hyers–Ulam stability and Bielecki–Hyers–Ulam–Rassias stability. To obtain these stability results, Grönwall’s inequality on time scales is used as the main analytical tool. For our results, some suitable assumptions are imposed along with appropriate Lipschitz conditions on the nonlinear terms. By constructing appropriate contractive mappings in a suitably defined Bielecki-type normed space, we develop a unified and systematic framework to handle the combined effects of integral operators, fractional dynamics, delay arguments, and impulsive perturbations. Finally, an illustrative example is provided to demonstrate the effectiveness and applicability of the theoretical findings.

Published
2026-05-12
How to Cite
Shah, S. O., & Zheng, J. (2026). Bielecki–Hyers–Ulam stability of non-linear fractional Volterra Fredholm Hammerstein integro-delay dynamic systems with instantaneous impulses on time scale. Advances in Differential Equations and Control Processes, 33(2). https://doi.org/10.59400/adecp3353
Issue
Vol. 33 No. 2 (2026): In Progress
Section
Article

Copyright (c) 2026 Syed Omar Shah, Jun Zheng

Creative Commons License

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

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