<p>In this paper, we consider the existence, nonexistence and multiplicity of positive solutions for a class of nonlocal differential equation with eigenvalue parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>. By using the fixed point index theory, we prove that the problem has at least two positive solutions for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mu \in \left( \mu _,+\infty \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>∈</mo> <mfenced close=")" open="("> <mmultiscripts> <mi>μ</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> <mo>,</mo> <mo>+</mo> <mi>∞</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, one nonnegative solution for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mu \in \left( \mu _,\mu ^\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>∈</mo> <mfenced close=")" open="("> <mmultiscripts> <mi>μ</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> <mo>,</mo> <msup> <mi>μ</mi> <mo>∗</mo> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, one positive solution for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu =\mu _{\text { or }} \mu ^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>=</mo> <mmultiscripts> <mi>μ</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> <mrow> <mspace width="0.333333em" /> <mtext>or</mtext> <mspace width="0.333333em" /> </mrow> <msup> <mi>μ</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> and no positive solution for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mu \in \left[ 0,\mu _\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>∈</mo> <mfenced close=")" open="["> <mn>0</mn> <mo>,</mo> <mmultiscripts> <mi>μ</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> </mfenced> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu _,\mu ^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>μ</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> <mo>,</mo> <msup> <mi>μ</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> are constants. Finally an example of the obtained results is given. The results of this study complement those of reference [Qingcong Song and Xinan Hao, Multiplicity of positive solutions for convolution equations with nonlocal boundary condition, Fixed Point Theory, 2025, doi: 10.24193/fpt-ro.2025.2.19].</p>

Study on the effects of eigenvalue parameter on the existence of positive solutions for nonlocal differential equation

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2026年4月15日

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Zhaocai Hao,
Qianqian Gu

摘要

In this paper, we consider the existence, nonexistence and multiplicity of positive solutions for a class of nonlocal differential equation with eigenvalue parameter $\mu $ $μ$ . By using the fixed point index theory, we prove that the problem has at least two positive solutions for $\mu \in \left( \mu _*,+\infty \right) $ $μ \in (μ_{*}, + \infty)$ , one nonnegative solution for $\mu \in \left( \mu _*,\mu ^*\right) $ $μ \in (μ_{*}, μ^{*})$ , one positive solution for $\mu =\mu _*{\text { or }} \mu ^*$ $μ = μ_{*} or μ^{*}$ and no positive solution for $\mu \in \left[ 0,\mu _*\right) $ $μ \in [0, μ_{*})$ where $\mu _*,\mu ^*$ $μ_{*}, μ^{*}$ are constants. Finally an example of the obtained results is given. The results of this study complement those of reference [Qingcong Song and Xinan Hao, Multiplicity of positive solutions for convolution equations with nonlocal boundary condition, Fixed Point Theory, 2025, doi: 10.24193/fpt-ro.2025.2.19].