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Existence of Solutions for a Fourth-Order Periodic Boundary Value Problem near Resonance
Xiaoxiao Sua,
Ruyun Maab,
Mantang Mab a School of Mathematics and Statistics, Xidian University, Shannxi, P. R. China
b Department of Mathematics, Northwest Normal University, Lanzhou, P. R. China
Аннотация:
We show the existence and multiplicity of solutions for the fourth-order periodic boundary value problem
\begin{equation*} \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} \end{equation*}
where
$\lambda\in\mathbb{R}$ is a parameter,
$h\in L^1(0,1)$, and $f\colon[0,1]\times \mathbb{R}\rightarrow\mathbb{R}$ 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.
Ключевые слова:
fourth-order equation, periodic boundary value problem, near-resonance problem,
lower and upper solutions, bifurcation.
Поступило: 19.04.2022
Исправленный вариант: 19.04.2022
Язык публикации: английский