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<p>In this paper, we investigate the existence of multiple weak solutions for a class of weighted nonlocal problems involving the fractional <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((p(x), \psi )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Hilfer operator. As a preliminary step, we establish new continuous and compact embeddings of the associated functional spaces into the appropriate space of continuous functions. The main tool in our existence analysis is a three critical points theorem due to Ricceri, which allows us to guarantee the existence of at least three distinct weak solutions. Our results contribute to the growing literature on nonlocal problems with variable exponent structure and generalized fractional operators, and offer new insights into the analytical treatment of such equations.</p>

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On a New Weighted \((p(x), \psi )\)-Hilfer Fractional Problem

  • 期刊论文

  • 发表日期：

    2026年5月20日

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  • A. Kasmi,
  • M. Shimi,
  • E. Azroul

摘要

In this paper, we investigate the existence of multiple weak solutions for a class of weighted nonlocal problems involving the fractional \((p(x), \psi )\) ( p ( x ) , ψ ) -Hilfer operator. As a preliminary step, we establish new continuous and compact embeddings of the associated functional spaces into the appropriate space of continuous functions. The main tool in our existence analysis is a three critical points theorem due to Ricceri, which allows us to guarantee the existence of at least three distinct weak solutions. Our results contribute to the growing literature on nonlocal problems with variable exponent structure and generalized fractional operators, and offer new insights into the analytical treatment of such equations.