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Variational formulation of a damped Dirichlet impulsive problem. (English) Zbl 1197.34041
Summary: We introduce the concept of a weak solution for a damped linear equation with Dirichlet boundary conditions and impulses. We use the classical Lax-Milgram Theorem to reveal the variational structure of the problem and get the existence and uniqueness of weak solutions as critical points. This will allow us in the future to deal with the corresponding nonlinear problems and look for solutions as critical points of weakly lower semicontinuous functionals.
MSC:
34B37Boundary value problems for ODE with impulses
58E30Variational principles on infinite-dimensional spaces
References:
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[3]Brattka, V.; Yoshikawa, A.: Towards computability of elliptic boundary value problems in variational formulation, J. complexity 22, 858-880 (2006) · Zbl 1126.03052 · doi:10.1016/j.jco.2006.04.007
[4]Drivaliaris, D.; Yannakalis, N.: Generalizations of the Lax–Milgram theorem, Bound. value probl. 2007 (2007) · Zbl 1140.47303 · doi:10.1155/2007/87104
[5]Nieto, J. J.; O’regan, D.: Variational approach to impulsive differential equations, Nonlinear anal. RWA 10, 680-690 (2009) · Zbl 1167.34318 · doi:10.1016/j.nonrwa.2007.10.022
[6]Mawhin, J.; Willem, M.: Critical point theory and Hamiltonian systems, (1989)