In this paper, we study the existence of positive solutions of third order Riemann-Stieltjes integral boundary value problem: <Equation ID="Equ126"> <MediaObject ID="MO1"> <ImageObject Color="BlackWhite" FileRef="MediaObjects/9_2025_2915_Equ126_HTML.png" Format="PNG" Height="152" Rendition="HTML" Resolution="300" Type="Linedraw" Width="1066" /> </MediaObject> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2915_Article_IEq1.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">$\int _0^1y(t)\textrm{d}A(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </msubsup> <mi>y</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mtext>d</mtext> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2915_Article_IEq2.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">$\int _0^1y(t)\textrm{d}B(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </msubsup> <mi>y</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mtext>d</mtext> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote Riemann-Stieltjes integral of y with respect to bounded variation functions A, B with positive measures respectively. Based on the fixed point index, the existence of positive solutions is investigated under some conditions concerning the spectral radius corresponding to the relevant linear differential equation. Different from the existed studies, the relevant linear differential equation involves the unknown functions and its first-order derivative. Two examples are presented to illustrate the obtained results.

Study of Third Order Riemann–Stieltjes Integral Boundary Value Problem with Dependence on the First Order Derivative

期刊论文
发表日期：

2025年7月30日

查看全文

Yujun Cui,
Yue Du,
Yumei Zou

摘要

In this paper, we study the existence of positive solutions of third order Riemann-Stieltjes integral boundary value problem: where $\int _0^1y(t)\textrm{d}A(t)$ $\int_{0}^{1} y (t) d A (t)$ , $\int _0^1y(t)\textrm{d}B(t)$ $\int_{0}^{1} y (t) d B (t)$ denote Riemann-Stieltjes integral of y with respect to bounded variation functions A, B with positive measures respectively. Based on the fixed point index, the existence of positive solutions is investigated under some conditions concerning the spectral radius corresponding to the relevant linear differential equation. Different from the existed studies, the relevant linear differential equation involves the unknown functions and its first-order derivative. Two examples are presented to illustrate the obtained results.