zbMATH — the first resource for mathematics

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.

34B37 Boundary value problems with impulses for ordinary differential equations
58E30 Variational principles in infinite-dimensional spaces
Full Text: DOI
[1] Chipot, M., Elements of nonlinear analysis, (2000), Birkhauser Verlag Basel · Zbl 0977.35050
[2] Lax, P.D.; Milgram, A.N., (), 167-190
[3] Brattka, V.; Yoshikawa, A., Towards computability of elliptic boundary value problems in variational formulation, J. complexity, 22, 858-880, (2006) · Zbl 1126.03052
[4] Drivaliaris, D.; Yannakalis, N., Generalizations of the lax – milgram theorem, Bound. value probl., 2007, (2007), (Art. ID 87104) 9 pages · Zbl 1140.47303
[5] Nieto, J.J.; O’Regan, D., Variational approach to impulsive differential equations, Nonlinear anal. RWA, 10, 680-690, (2009) · Zbl 1167.34318
[6] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag Berlin · Zbl 0676.58017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.