Existence of Solutions for a Fourth-Order Periodic Boundary Value Problem near Resonance

Pub Date : 2024-07-05 DOI:10.1134/s0001434624030325

Xiaoxiao Su, Ruyun Ma, Mantang Ma

引用次数: 0

Abstract

We show the existence and multiplicity of solutions for the fourth-order periodic boundary value problem

$$\begin{cases} u''''(t)-\lambda u(t)=f(t,u(t))-h(t), \qquad t\in [0,1],\\ u(0)=u(1),\;u'(0)=u'(1),\; u''(0)=u''(1),\;u'''(0)=u'''(1), \end{cases}$$

where $\lambda\in\mathbb{R}$ is a parameter, $h\in L^1(0,1)$, and $f:[0,1]\times \mathbb{R}\rightarrow\mathbb{R}$ is an $L^1$-Carathéodory function. Moreover, $f$ is sublinear at $+\infty$ and nondecreasing with respect to the second variable. We obtain that if $\lambda$ is sufficiently close to $0$ from the left or right, then the problem has at least one or two solutions, respectively. The proof of main results is based on bifurcation theory and the method of lower and upper solutions.

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共振附近的四阶周期性边界值问题的解的存在性

Abstract We show the existence and multiplicity of solutions for the fourth-order periodic boundary value problem $$\begin{cases} u''''(t)-\lambda u(t)=f(t,u(t))-h(t), \qquad t\in [0,1],\ u(0)=u(1),\；u''(0)=u''(1),； u'''(0)=u'''(1),； u''''(0)=u''''(1), end{cases}$$ 其中$\lambda\in\mathbb{R}$ 是一个参数，$h\in L^1(0,1)$, and$f：(f：[0,1]/times\mathbb{R}\rightarrow\mathbb{R}$ 是一个 $L^1$-Carathéodory 函数。此外，$f$在$+\infty$处是次线性的，并且相对于第二个变量是非递减的。我们得到，如果$\lambda$从左边或右边足够接近$0$，那么问题至少有一个或两个解。主要结果的证明基于分岔理论和上下解法。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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