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Zbl 1191.34039
Zhang, Ziheng; Yuan, Rong
An application of variational methods to Dirichlet boundary value problem with impulses. (English)
[J] Nonlinear Anal., Real World Appl. 11, No. 1, 155-162 (2010). ISSN 1468-1218

The authors consider the impulsive Dirichlet boundary value problem $$\aligned &-u''(t) + \lambda u(t) = f(t,u(t)) + p(t), \quad t \in [0,T], \\ &\triangle u'(t_j) = I_j(u(t_j)),\ j = 1,\dots,k, \\ &u(0) = u(T) = 0, \endaligned$$ where $0 < t_1 < \dots < t_k < T$ are impulse instants, the impulsive functions $I_j : {\Bbb R} \to {\Bbb R}$ and the right-hand side $f$ are continuous, $p \in L^2[0,T]$, $\lambda > -\pi^2/T^2$. Sufficient conditions for the existence of at least one and infinitely many weak solutions are found. The proofs are based on the critical points theory.
[Jan Tomeček (Olomouc)]

MSC 2000:: *34B37 Boundary value problems with impulses
58E05 Abstract critical point theory
47J30 Variational methods

Keywords: impulsive; critical points theory; Dirichlet boundary conditions

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