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<p>This paper investigates Gronwall’s inequality within the framework of interval-valued functions under Caputo gH-differentiability. Employing interval-valued Caputo fractional derivatives and associated interval-valued Riemann–Liouville integrals, we develop a generalized Gronwall-type inequality adapted to fractional-order systems with interval uncertainty. The approach treats the interval endpoints component-wise under generalized Hukuhara <i>gH</i>-differentiability. For the case of constant kernel functions, explicit upper bounds are derived using the Mittag-Leffler function, leveraging Laplace transform techniques and structural properties of the interval-valued Caputo derivative. In the case of non-constant kernel functions, we utilize the convolution theorem in conjunction with the proposed fractional Gronwall inequality to obtain integral bounds for the interval-valued solutions. These results extend classical Gronwall inequalities to interval-valued fractional differential equations, offering a rigorous tool for estimating solutions in the presence of uncertainty and enhancing the analysis of interval-valued fractional dynamic systems.</p>

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Caputo gH-differentiability for Gronwalls inequality: insight, new results and application

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月18日

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  • Fatima Khan,
  • Anum Zehra,
  • Muhammad Farman,
  • Kottakkaran Sooppy Nisar,
  • Aceng Sambas,
  • Mohamed Hafez

摘要

This paper investigates Gronwall’s inequality within the framework of interval-valued functions under Caputo gH-differentiability. Employing interval-valued Caputo fractional derivatives and associated interval-valued Riemann–Liouville integrals, we develop a generalized Gronwall-type inequality adapted to fractional-order systems with interval uncertainty. The approach treats the interval endpoints component-wise under generalized Hukuhara gH-differentiability. For the case of constant kernel functions, explicit upper bounds are derived using the Mittag-Leffler function, leveraging Laplace transform techniques and structural properties of the interval-valued Caputo derivative. In the case of non-constant kernel functions, we utilize the convolution theorem in conjunction with the proposed fractional Gronwall inequality to obtain integral bounds for the interval-valued solutions. These results extend classical Gronwall inequalities to interval-valued fractional differential equations, offering a rigorous tool for estimating solutions in the presence of uncertainty and enhancing the analysis of interval-valued fractional dynamic systems.