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.


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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