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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access October 12, 2012

On variational impulsive boundary value problems

Marek Galewski
From the journal Open Mathematics


Using the variational approach, we investigate the existence of solutions and their dependence on functional parameters for classical solutions to the second order impulsive boundary value Dirichlet problems with L1 right hand side.

MSC: 34B37; 47J30

[1] Appell J., Zabrejko P.P., Nonlinear Superposition Operators, Cambridge Tracts in Math., 95, Cambridge University Press, Cambridge, 1990 in Google Scholar

[2] Chen H., Li J., Variational approach to impulsive differential equations with Dirichlet boundary conditions, Bound. Value Probl., 2010, #325415 10.1155/2010/325415Search in Google Scholar

[3] Feng M., Xie D., Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations, J. Comput. Appl. Math., 2009, 223(1), 438–448 in Google Scholar

[4] Idczak D., Rogowski A., On a generalization of Krasnoselskii’s theorem, J. Aust. Math. Soc., 2002, 72(3), 389–394 10.1017/S1446788700150001Search in Google Scholar

[5] Jankowski T., Positive solutions to second order four-point boundary value problems for impulsive differential equations, Appl. Math. Comput., 2008, 202(2), 550–561 in Google Scholar

[6] Lakshmikantham V., Baĭnov D.D., Simeonov P.S., Theory of Impulsive Differential Equations, Ser. Modern Appl. Math., 6, World Scientific, Teaneck, 1989 in Google Scholar

[7] Ledzewicz U., Schättler H., Walczak S., Optimal control systems governed by second-order ODEs with Dirichlet boundary data and variable parameters, Illinois J. Math., 2003, 47(4), 1189–1206 10.1215/ijm/1258138099Search in Google Scholar

[8] Mawhin J., Problèmes de Dirichlet Variationnels non Linéaires, Sem. Math. Sup., 104, Presses de l’Université de Montréal, Montréal, 1987 Search in Google Scholar

[9] Nieto J.J., Variational formulation of a damped Dirichlet impulsive problem, Appl. Math. Lett., 2010, 23(8), 940–942 in Google Scholar

[10] Nieto J.J., O’Regan D., Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl., 2009, 10(2), 680–690 in Google Scholar

[11] Samoilenko A.M., Perestyuk N.A., Impulsive Differential Equations, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, 14, World Scientific, Singapore, 1995 10.1142/2892Search in Google Scholar

[12] Sun J., Chen H., Multiplicity of solutions for a class of impulsive differential equations with Dirichlet boundary conditions via variant fountain theorems, Nonlinear Anal. Real World Appl., 2010, 11(5), 4062–4071 in Google Scholar

[13] Tian Y., Ge W., Applications of variational methods to boundary-value problem for impulsive differential equations, Proc. Edinb. Math. Soc., 2008, 51(2), 509–527 in Google Scholar

[14] Xiao J., Nieto J.J., Variational approach to some damped Dirichlet nonlinear impulsive differential equations, J. Franklin Inst., 2011, 48(2), 369–377 in Google Scholar

[15] Zavalishchin S.T., Sesekin A.N., Dynamic Impulse Systems, Math. Appl., 394, Kluwer, Dordrecht, 1997 10.1007/978-94-015-8893-5Search in Google Scholar

[16] Zhang H., Li Z., Variational approach to impulsive differential equations with periodic boundary conditions, Nonlinear Anal. Real World Appl., 2010, 11(1), 67–78 in Google Scholar

[17] Zhang Z., Yuan R., An application of variational methods to Dirichlet boundary value problem with impulses, Nonlinear Anal. Real World Appl., 2010, 11(1), 155–162 in Google Scholar

Published Online: 2012-10-12
Published in Print: 2012-12-1

© 2012 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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