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<p>Based on Krasnosel’skii fixed point theorem, k-set contractions fixed point theorem, Banach’s fixed point theorem, the existence and uniqueness of solutions is investigated for Mittag-Leffler kernel-type fractional differential equations under nonlocal delay and impulsive boundary value conditions, considering both compact and non-compact resolvent operators. However, a challenge arises due to the order of the derivative (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(0&lt;\vartheta &lt;1\)</EquationSource> </InlineEquation>), which complicates the proof of equicontinuity; one of the research objectives of this study is to solve this issue. In addition, another contribution in this process is that the constant <i>L</i> is generalized as an unbounded Lebesgue integrable function in case of non-compact measure conditions. Moreover, the Lipschitz conditions of nonlinear terms and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\hbar,\rho \)</EquationSource> </InlineEquation> are expanded from non-negative constants <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L,\hbar _{1},\rho _{1}\)</EquationSource> </InlineEquation> to unbounded Lebesgue integrable functions. Finally, three examples are provided to demonstrate the validity of the present work.</p>

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On Mild Solutions of the Mittag-Leffler Fractional-Order Equation Subjected to Nonlocal Delay and Impulsive Conditions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月12日

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  • Limin Guo,
  • Cheng Li,
  • Wanjie Zhu,
  • Shuang Li

摘要

Based on Krasnosel’skii fixed point theorem, k-set contractions fixed point theorem, Banach’s fixed point theorem, the existence and uniqueness of solutions is investigated for Mittag-Leffler kernel-type fractional differential equations under nonlocal delay and impulsive boundary value conditions, considering both compact and non-compact resolvent operators. However, a challenge arises due to the order of the derivative ( \(0<\vartheta <1\) ), which complicates the proof of equicontinuity; one of the research objectives of this study is to solve this issue. In addition, another contribution in this process is that the constant L is generalized as an unbounded Lebesgue integrable function in case of non-compact measure conditions. Moreover, the Lipschitz conditions of nonlinear terms and \(\hbar,\rho \) are expanded from non-negative constants \(L,\hbar _{1},\rho _{1}\) to unbounded Lebesgue integrable functions. Finally, three examples are provided to demonstrate the validity of the present work.