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Abstract <p>In this paper, we derive a necessary condition for the existence of solutions to a boundary value problem involving the Riesz fractional derivative, a two-sided fractional operator. The problem is transformed into a singular integral equation within a weighted Lebesgue space. By applying the Kolmogorov compactness criterion and using the boundedness properties of the Hilbert transform, we establish the compactness and boundedness conditions necessary for the operator to satisfy the requirements of Krasnoselskii’s fixed point theorem. This allows us to prove the existence of solutions in the Lebesgue space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^{p}\)</EquationSource> <!--LobJMat2560745Guezanelakoud-m1--> </InlineEquation>. Additionally, we derive Lyapunov-type inequalities in a weighted Lebesgue space.</p>

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On a Riesz Fractional Boundary Value Problem in a Weighted Lebesgue Space

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • R. Khaldi,
  • A. Guezane-Lakoud

摘要

Abstract

In this paper, we derive a necessary condition for the existence of solutions to a boundary value problem involving the Riesz fractional derivative, a two-sided fractional operator. The problem is transformed into a singular integral equation within a weighted Lebesgue space. By applying the Kolmogorov compactness criterion and using the boundedness properties of the Hilbert transform, we establish the compactness and boundedness conditions necessary for the operator to satisfy the requirements of Krasnoselskii’s fixed point theorem. This allows us to prove the existence of solutions in the Lebesgue space \(L^{p}\) . Additionally, we derive Lyapunov-type inequalities in a weighted Lebesgue space.