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<p>This paper investigates the existence, non-existence, uniqueness, and multiplicity of positive solutions for a boundary value problem involving the Riemann-Liouville derivative of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \in (1,2]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. First, we establish existence and non-existence results for both sublinear and superlinear cases by characterizing the first eigenvalue <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\lambda _1(\alpha )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Second, we prove the uniqueness of positive solutions; especially, for the superlinear case, we employ an approach by leveraging the non-degeneracy of solutions near <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha =2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Finally, we demonstrate the existence of at least three positive solutions for Hénon-type problems.</p>

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Positive solutions for fractional-order boundary value problems with or without dependence of integer-order ones

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  • 发表日期：

    2026年1月26日

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  • Inbo Sim,
  • Satoshi Tanaka

摘要

This paper investigates the existence, non-existence, uniqueness, and multiplicity of positive solutions for a boundary value problem involving the Riemann-Liouville derivative of order \(\alpha \in (1,2]\) α ∈ ( 1 , 2 ] . First, we establish existence and non-existence results for both sublinear and superlinear cases by characterizing the first eigenvalue \(\lambda _1(\alpha )\) λ 1 ( α ) . Second, we prove the uniqueness of positive solutions; especially, for the superlinear case, we employ an approach by leveraging the non-degeneracy of solutions near \(\alpha =2\) α = 2 . Finally, we demonstrate the existence of at least three positive solutions for Hénon-type problems.