{"title":"Existence of Solutions for a Fourth-Order Periodic Boundary Value Problem near Resonance","authors":"Xiaoxiao Su, Ruyun Ma, Mantang Ma","doi":"10.1134/s0001434624030325","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We show the existence and multiplicity of solutions for the fourth-order periodic boundary value problem </p><span>$$\\begin{cases} u''''(t)-\\lambda u(t)=f(t,u(t))-h(t), \\qquad t\\in [0,1],\\\\ u(0)=u(1),\\;u'(0)=u'(1),\\; u''(0)=u''(1),\\;u'''(0)=u'''(1), \\end{cases}$$</span><p> where <span>\\(\\lambda\\in\\mathbb{R}\\)</span> is a parameter, <span>\\(h\\in L^1(0,1)\\)</span>, and <span>\\(f:[0,1]\\times \\mathbb{R}\\rightarrow\\mathbb{R}\\)</span> is an <span>\\(L^1\\)</span>-Carathéodory function. Moreover, <span>\\(f\\)</span> is sublinear at <span>\\(+\\infty\\)</span> and nondecreasing with respect to the second variable. We obtain that if <span>\\(\\lambda\\)</span> is sufficiently close to <span>\\(0\\)</span> 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. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Notes","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0001434624030325","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We show the existence and multiplicity of solutions for the fourth-order periodic boundary value problem
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.
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。
期刊介绍：
Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.
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