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<p>This research aims to thoroughly investigate the existence and uniqueness of solutions to the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((\Phi , \varphi )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-order Caputo fractional integro-differential system involving a Kernel operator. This equation features an initial condition that involves the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {G}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">G</mi> </math></EquationSource> </InlineEquation>-Caputo derivative, which operates in two distinct orders. To accomplish this, we use the generalized Laplace transform method to find the solution, and then, we employ Krasnoselskii’s fixed point theorem as a foundational tool to explore and ascertain the existence of a solution. Following this, we apply Banach’s fixed point theorem to delve into the uniqueness of the solution, ensuring a comprehensive understanding of its properties. To further elucidate our findings, we present an illustrative example demonstrating the main results and their implications within fractional calculus.</p>

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An existence and unicity study of the \((\Phi , \varphi )\)-order Caputo fractional integro-differential system involving a Kernel operator

  • 期刊论文

  • 发表日期：

    2025年9月2日

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  • Asmaa Baihi,
  • Samira Zerbib,
  • Khalid Hilal,
  • Ahmed Kajouni

摘要

This research aims to thoroughly investigate the existence and uniqueness of solutions to the \((\Phi , \varphi )\) ( Φ , φ ) -order Caputo fractional integro-differential system involving a Kernel operator. This equation features an initial condition that involves the \(\mathbb {G}\) G -Caputo derivative, which operates in two distinct orders. To accomplish this, we use the generalized Laplace transform method to find the solution, and then, we employ Krasnoselskii’s fixed point theorem as a foundational tool to explore and ascertain the existence of a solution. Following this, we apply Banach’s fixed point theorem to delve into the uniqueness of the solution, ensuring a comprehensive understanding of its properties. To further elucidate our findings, we present an illustrative example demonstrating the main results and their implications within fractional calculus.