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Multiplicity of solutions for a fourth-order impulsive differential equation via variational methods. (English) Zbl 1213.34048

The authors consider the boundary value problem with impulses
\[ \begin{aligned} &u^{(4)}(t) = \lambda f(t,u(t))\quad \text{a.e. } t \in [0,1],\\ &\triangle u''(t_j) = I_j(u'(t_j)), \;\triangle u'''(t_j) = N_j(u(t_j)), \quad j=1,\ldots,l,\\ &u(0) = u'(0) = u''(1^-) = 0, \;u'''(1^-) = g(u(1)), \end{aligned} \]
where \(0 < t_1 < \ldots < t_l < 1\); \(\lambda > 0\); \(f\), \(g\), \(I_j\), \(N_j\) are continuous. Sufficient conditions for the existence of at least three solutions are given. The proofs are based on a variational approach.

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47J30 Variational methods involving nonlinear operators
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