<p>In this paper, we consider the nonlocal differential equation possessing two different nonlocal elements of the form <Equation ID="Equ18"> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{aligned}&-M\left( \left( b*\left( g\circ u\right) \right) (t) \right) u''(t)= f(t,u(t),u'(t)),\quad t\in (0,1),\\&u(0) =\beta [u],\quad u'(1) =0. \end{aligned}\right. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd /> <mtd columnalign="left"> <mrow> <mo>-</mo> <mi>M</mi> <mfenced close=")" open="("> <mfenced close=")" open="("> <mi>b</mi> <mrow /> <mo>∗</mo> <mfenced close=")" open="("> <mi>g</mi> <mo>∘</mo> <mi>u</mi> </mfenced> </mfenced> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <msup> <mi>u</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>β</mi> <mrow> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Using a recently obtained Gronwall-type inequality and topological fixed point index theory, we prove the existence of at least one positive solution of the second-order nonlocal boundary value problems with nonlinear term having derivative dependence. An example is given to illustrate the main result.</p>

Positive solutions for doubly nonlocal boundary value problem with time-varying convolution coefficients

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发表日期：

2025年9月24日

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Shunze Chu,
Xinan Hao

摘要

In this paper, we consider the nonlocal differential equation possessing two different nonlocal elements of the form $\begin{aligned} \left\{ \begin{aligned}&-M\left( \left( b*\left( g\circ u\right) \right) (t) \right) u''(t)= f(t,u(t),u'(t)),\quad t\in (0,1),\\&u(0) =\beta [u],\quad u'(1) =0. \end{aligned}\right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} - M ((b * (g \circ u)), (t)) u^{''} (t) = f (t, u (t), u^{'} (t)), t \in (0, 1), \\ u (0) = β [u], u^{'} (1) = 0 . \end{matrix}) \end{matrix}$ Using a recently obtained Gronwall-type inequality and topological fixed point index theory, we prove the existence of at least one positive solution of the second-order nonlocal boundary value problems with nonlinear term having derivative dependence. An example is given to illustrate the main result.