Research ArticleOpen AccessGoogle Scholar indexed
Multiplicity Results for Second Order Impulsive Differential Equations via Variational Methods
School of Mathematics Statistics, Shandong Normal University, Jinan, China
School of Mathematics Statistics, Shandong Normal University, Jinan, China
School of Mathematics Statistics, Shandong Normal University, Jinan, China
- 1 School of Mathematics Statistics, Shandong Normal University, Jinan, China
- 2 School of Mathematics Statistics, Shandong Normal University, Jinan, China
- 3 School of Mathematics Statistics, Shandong Normal University, Jinan, China
Copy link · social · email
Abstract
In this paper we investigate a class of impulsive differential equations with Dirichlet boundary conditions. Firstly, we define new inner product of and prove that the norm which is deduced by the inner product is equivalent to the usual norm. Secondly, we construct the lower and upper solutions of (1.1). Thirdly, we obtain the existence of a positive solution, a negative solution and a sign-changing solution by using critical point theory and variational methods. Finally, an example is presented to illustrate the application of our main result.
KeywordsImpulsive Differential EquationSign-Changing SolutionCritical Point TheoryVariational Method
- Lakshmikantham, V., Bainov, D.D. and Simeonov, P.S. (1989) Theory of Impulsive Differential Equations. Vol. 6, World Scientific, Singapore. https://doi.org/10.1142/0906
- Samoilenko, A.M. and Perestyuk, N.A. (1995) Impulsive Differential Equations. Vol. 14, World Scientific, Singapore. https://doi.org/10.1142/2892
- Zavalishchin, S.T. and Sesekin, A.N. (1997) Dynamic Impulse Systems: Theory and Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8893-5
- Benchohra, M., Henderson, J. and Ntouyas, S.K. (2006) Impulsive Differential Equations and Inclusions. Vol. 2, Hindawi Publishing Corporation, New York.
- Hao, X.N., Liu, L.S. and Wu, Y.H. (2011) Positive Solutions for Second Order Impulsive Differential Equations with Integral Boundary Conditions. Communications in Nonlinear Science and Numerical Simulation, 16, 101-111. https://doi.org/10.1016/j.cnsns.2010.04.007
- Rachunkova, I. and Tomecek, J. (2014) Existence Principle for BVPS with State-Dependent Impulses. Topological Methods in Nonlinear Analysis, 44, 349-368. https://doi.org/10.12775/TMNA.2014.050
- Sun, H.X. and Chen, H.B. (2016) Multiplicity Results for a Class of Boundary Value Problems with Impulsive Effects. Mathematische Nachrichten, 289, 718-726. https://doi.org/10.1002/mana.201400341
- Liu, J. and Zhao, Z.Q. (2017) Multiple Solutions for Impulsive Problems with Non-Autonomous Perturbations. Applied Mathematics Letters, 64, 143-149. https://doi.org/10.1016/j.aml.2016.08.020
- Graef, J., Heidarkhani, S. and Kong, L. (2016) Nontrivial Solutions of a Dirichlet Boundary Value Problem with Impulsive Effects. Dynamic Systems and Applications, 25, 335-350.
- D’Agui, G., Di Bella, B. and Tersian, S. (2016) Multiplicity Results for Superlinear Boundary Value Problems with Impulsive Effects. Mathematical Methods in the Applied Sciences, 39, 1060-1068. https://doi.org/10.1002/mma.3545
- Xia, Y.H. (2011) Global Analysis of an Impulsive Delayed Lotka-Volterra Competition System. Communications in Nonlinear Science and Numerical Simulation, 16, 1597-1616. https://doi.org/10.1016/j.cnsns.2010.07.014
- Xiao, J., Nieto, J.J. and Luo, Z.G. (2012) Multiplicity of Solutions for Nonlinear Second Order Impulsive Differential Equations with Linear Derivative Dependence via Variational Methods. Communications in Nonlinear Science and Numerical Simulation, 17, 426-432. https://doi.org/10.1016/j.cnsns.2011.05.015