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Infinitely many solutions for a class of fourth-order impulsive differential equations. (English) Zbl 1411.34043

In this paper, the authors consider the following fourth-order impulsive boundary value problem \[ \begin{aligned} u^{iv}(t)-(p(t)u'(t))'+q(t)u(t) & =\lambda f(t,u(t))+\mu g(t,u(t)),\quad t\neq t_j, \quad t\in (0,1),\\ \Delta(u''(t_j)) & =I_{1j}(u'(t_j)), \quad j=1,2,\ldots,m,\\ -\Delta(u'''(t_j)) & =I_{2j}(u(t)), \quad j=1,2,\ldots, m,\\ u(0) & =u(1)=u''(0)=u''(1)=0, \end{aligned} \] where \(p, q\in L^{\infty}([0,1]),\) \(f, g: [0,1]\times \mathbb{R}\to \mathbb{R}\) are \(L^1\)-Carathéodory functions, \(I_{1j}, I_{2j}\in C(\mathbb{R}, \mathbb{R})\) for \(1\leq j\leq m,\) \(0=t_0<t_1<t_2<\ldots<t_m<t_{m+1}=1,\) \(\Delta(u(t_j))=u(t_j^+)-u(t_j^-),\) \(u(t_j^+), (t_j^-)\) are the right and left limits, respectively, of \(u\) at \(t_j\) and \(\lambda>0\), \(\mu\geq 0\) are real parameters. Existence of infinitely many solutions are proved via critical point theory.

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
34A37 Ordinary differential equations with impulses
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