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Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. (English) Zbl 1175.34035
The authors consider the following Dirichlet boundary value problem with impulses
\left\{\begin{aligned} &-u''(t)+g(t)u(t)=f(t,u(t)) \quad \text{a.e.}\,\, t\in [0,T]\\ &u(0)=u(T)=0,\\ & \Delta u'(t_j)=u'(t_j^+)-u'(t_j^-)=I_j(u(t_j)), \,\,\, j=1,2,\dots, p, \end{aligned} \right. where $$t_0=0<t_1<t_2<\dots<t_p<t_{p+1}=T,$$ $$g\in L^{\infty}[0,T],$$ $$f: [0,T]\times {\mathbb R}\to {\mathbb R}$$ is continuous and $$I_j: {\mathbb R}\to {\mathbb R},$$ $$j=1,2,\dots,p$$ are continuous. Existence and multiplicity results are obtained via Lax-Milgram theorem and critical points theorems. The main results are illustrated by examples.

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