Solvability of a nonlinear second order m-point boundary value problem with p-Laplacian at resonance

IF 1.7 4区 数学 Q1 Mathematics Boundary Value Problems Pub Date : 2024-03-29 DOI:10.1186/s13661-024-01856-0
Meiyu Liu, Minghe Pei, Libo Wang
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Abstract

We study the existence of solutions of the nonlinear second order m-point boundary value problem with p-Laplacian at resonance $$ \textstyle\begin{cases} (\phi _{p}(x'))'=f(t,x,x'),\quad t\in [0,1],\\ x'(0)=0, \qquad x(1)=\sum_{i=1}^{m-2}a_{i}x(\xi _{i}), \end{cases} $$ where $\phi _{p}(s)=|s|^{p-2}s$ , $p>1$ , $f:[0,1]\times \mathbb{R}^{2}\to \mathbb{R}$ is a continuous function, $a_{i}>0$ ( $i=1,2,\ldots ,m-2$ ) with $\sum_{i=1}^{m-2}a_{i}=1$ , $0<\xi _{1}<\xi _{2}<\cdots <\xi _{m-2}<1$ . Based on the topological transversality method together with the barrier strip technique and the cut-off technique, we obtain new existence results of solutions of the above problem. Meanwhile some examples are also given to illustrate our main results.
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共振时具有 p-Laplacian 的非线性二阶 m 点边界值问题的可解性
我们研究了非线性二阶 m 点边界值问题的共振时 p-Laplacian 的解的存在性 $$ text\style\begin{cases} (\phi _{p}(x'))'=f(t. x,x'),\quad t\in [0,1],\qquad x'(0)=0、x,x'),quad t\in [0,1],\x'(0)=0, \qquad x(1)=sum_{i=1}^{m-2}a_{i}x(\xi _{i}), \end{cases} $$ 其中 $\phi _{p}(s)=|s|^{p-2}s$ , $p>1$ , $f:$f: [0,1]\times \mathbb{R}^{2}\to \mathbb{R}$ 是一个连续函数, $a_{i}>0$ ( $i=1,2,\ldots ,m-2$ ) with $\sum_{i=1}^{m-2}a_{i}=1$ , $0<\xi _{1}<\xi _{2}<\cdots <\xi _{m-2}<1$ 。基于拓扑横断性方法以及障带技术和截断技术,我们得到了上述问题解的新的存在性结果。同时,我们还给出了一些例子来说明我们的主要结果。
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来源期刊
Boundary Value Problems
Boundary Value Problems MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
3.00
自引率
5.90%
发文量
83
审稿时长
4 months
期刊介绍: The main aim of Boundary Value Problems is to provide a forum to promote, encourage, and bring together various disciplines which use the theory, methods, and applications of boundary value problems. Boundary Value Problems will publish very high quality research articles on boundary value problems for ordinary, functional, difference, elliptic, parabolic, and hyperbolic differential equations. Articles on singular, free, and ill-posed boundary value problems, and other areas of abstract and concrete analysis are welcome. In addition to regular research articles, Boundary Value Problems will publish review articles.
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