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<p>In this paper, we investigate the existence, uniqueness, and stability of solutions for a class of nonlinear fractional difference equations involving the Atangana–Baleanu–Caputo (ABC) operator, subject to prescribed initial conditions. A novel Gronwall-type inequality is established within the framework of ABC fractional differences, serving as a foundational analytical tool. By employing Picard’s iteration method alongside Schauder’s and Banach’s fixed-point theorems, we establish sufficient conditions that ensure the existence and uniqueness of solutions. Furthermore, attractive stability properties are explored, and Ulam–Hyers stability is analyzed using the newly established inequality. To demonstrate the practical relevance of the theoretical results, numerical approximations are presented using Newton’s iteration method.</p>

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Stability Analysis of Nonlinear Fractional Discrete Dynamical Systems

  • 期刊论文

  • 发表日期：

    2025年8月26日

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  • Anshul Sharma,
  • S. N. Mishra,
  • Anurag Shukla

摘要

In this paper, we investigate the existence, uniqueness, and stability of solutions for a class of nonlinear fractional difference equations involving the Atangana–Baleanu–Caputo (ABC) operator, subject to prescribed initial conditions. A novel Gronwall-type inequality is established within the framework of ABC fractional differences, serving as a foundational analytical tool. By employing Picard’s iteration method alongside Schauder’s and Banach’s fixed-point theorems, we establish sufficient conditions that ensure the existence and uniqueness of solutions. Furthermore, attractive stability properties are explored, and Ulam–Hyers stability is analyzed using the newly established inequality. To demonstrate the practical relevance of the theoretical results, numerical approximations are presented using Newton’s iteration method.