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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 $$\left\{\aligned &-u''(t)+g(t)u(t)=f(t,u(t)) \quad \text{a.e.}\,\, t\in [0,T]\\ &u(0)=u(T)=0,\\ & \Delta u'(t_j)=u'(t_j^+)-u'(t_j^-)=I_j(u(t_j)), \,\,\, j=1,2,\dots, p, \endaligned \right. $$ where $t_0=0<t_1<t_2<\dots<t_p<t_{p+1}=T,$ $g\in L^{\infty}[0,T],$ $f: [0,T]\times {\Bbb R}\to {\Bbb R}$ is continuous and $I_j: {\Bbb R}\to {\Bbb R},$ $j=1,2,\dots,p$ are continuous. Existence and multiplicity results are obtained via Lax-Milgram theorem and critical points theorems. The main results are illustrated by examples.

34B37Boundary value problems for ODE with impulses
58E05Abstract critical point theory
Full Text: DOI
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