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<p>In this paper, we examine the existence and uniqueness of solutions for nonlinear Langevin fractional boundary value integro-differential equations involving <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3178_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\( \psi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation>-Caputo derivative with two distinct variable-orders. We begin by establishing the equivalence between the original problem and an integral equation. Using Krasnoselskii’s fixed-point theorem, we investigate the existence of solutions, and we apply the Banach contraction principle to analyze the uniqueness of the solution. Finally, we provide an illustrative numerical example to demonstrate our main findings.</p>

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On the Langevin fractional boundary value integro-differential equations involving \( \psi \)-Caputo derivative with two distinct variable-orders

  • 期刊论文

  • 发表日期：

    2025年4月3日

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  • Samira Zerbib,
  • Khalid Hilal,
  • Ahmed Kajouni

摘要

In this paper, we examine the existence and uniqueness of solutions for nonlinear Langevin fractional boundary value integro-differential equations involving \( \psi \) ψ -Caputo derivative with two distinct variable-orders. We begin by establishing the equivalence between the original problem and an integral equation. Using Krasnoselskii’s fixed-point theorem, we investigate the existence of solutions, and we apply the Banach contraction principle to analyze the uniqueness of the solution. Finally, we provide an illustrative numerical example to demonstrate our main findings.