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<p>This research focuses on a detailed investigation of the existence and uniqueness of mild solutions to initial value problems for semilinear fractional evolution systems involving a kernel operator and a variable-order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-Caputo derivative. Specifically, we derive the solution formula in terms of the semigroup generated by the resolvent and the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-function associated with the Caputo fractional derivative, using the generalized Laplace transform method along with the aid of certain probability density functions. To establish the existence of solutions, we utilize Krasnoselskii’s fixed point theorem, while Banach’s fixed point theorem is employed to confirm uniqueness. Finally, we illustrate the main results and their implications within the context of semigroups and variable-order fractional derivatives through a concrete example.</p>

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Analytical results for semilinear fractional evolution systems with variable-order \(\Phi \)-Caputo derivatives using generalized Laplace transforms

  • 期刊论文

  • 发表日期：

    2025年12月22日

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  • Samira Zerbib,
  • Khalid Hilal,
  • Ahmed Kajouni

摘要

This research focuses on a detailed investigation of the existence and uniqueness of mild solutions to initial value problems for semilinear fractional evolution systems involving a kernel operator and a variable-order \(\Phi \) Φ -Caputo derivative. Specifically, we derive the solution formula in terms of the semigroup generated by the resolvent and the \(\Phi \) Φ -function associated with the Caputo fractional derivative, using the generalized Laplace transform method along with the aid of certain probability density functions. To establish the existence of solutions, we utilize Krasnoselskii’s fixed point theorem, while Banach’s fixed point theorem is employed to confirm uniqueness. Finally, we illustrate the main results and their implications within the context of semigroups and variable-order fractional derivatives through a concrete example.