Taiwanese Journal of Mathematics

VARIATIONAL METHODS TO MIXED BOUNDARY VALUE PROBLEM FOR IMPULSIVE DIFFERENTIAL EQUATIONS WITH A PARAMETER

Yu Tian, Jun Wang, and Weigao Ge

Full-text: Open access

Abstract

In this paper, we study mixed boundary value problem for secondorder impulsive differential equations with a parameter. By using critical point theory, several new existence results are obtained. This is one of the first times that impulsive boundary value problems are studied by means of variational methods.

Article information

Source
Taiwanese J. Math., Volume 13, Number 4 (2009), 1353-1370.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405513

Digital Object Identifier
doi:10.11650/twjm/1500405513

Mathematical Reviews number (MathSciNet)
MR2543748

Zentralblatt MATH identifier
1189.34060

Subjects
Primary: 34B15: Nonlinear boundary value problems 34B37: Boundary value problems with impulses 58E30: Variational principles

Keywords
mixed boundary value problem impulsive effect critical point theory

Citation

Tian, Yu; Wang, Jun; Ge, Weigao. VARIATIONAL METHODS TO MIXED BOUNDARY VALUE PROBLEM FOR IMPULSIVE DIFFERENTIAL EQUATIONS WITH A PARAMETER. Taiwanese J. Math. 13 (2009), no. 4, 1353--1370. doi:10.11650/twjm/1500405513. https://projecteuclid.org/euclid.twjm/1500405513


Export citation

References

  • R. P. Agarwal, D. O'Regan, Multiple nonnegative solutions for second order impulsive differential equations, Appl. Math. Comput., 114 (2000), 51-59.
  • D. Averna, G. Bonanno, A three critical points theorem and its applications to the ordinary Dirichlet problem, Topol. Methods Nonlinear Anal., 22 (2003), 93-104.
  • D. Franco, J. J. Nieto, Maximum principle for periodic impulsive first order problems, J. Comput. Appl. Math., 88 (1998), 149-159.
  • Guo Dajun, Nonlinear Functional Analysis, Shandong science and technology Press, Shandong, China, 1985.
  • V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, Series Modern Appl. Math., vol. 6, World Scientific, Teaneck, NJ, 1989.
  • E. K. Lee, Y. H. Lee, Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equation, Appl. Math. Comput., 158 (2004), 745-759.
  • J. Li, J. J. Nieto, J. Shen, Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal. Appl., 325 (2007), 226-236.
  • Xiaoning Lin, Daqing Jiang, Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations, J. Math. Anal. Appl., 321 (2006), 501-514.
  • J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, Berlin, 1989.
  • J. J. Nieto, R. Rodriguez-Lopez, Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations, J. Math. Anal. Appl., 318 (2006), 593-610.
  • J. J. Nieto, R. Rodriguez-Lopez, New comparison results for impulsive integro-differential equations and applications, J. Math. Anal. Appl., 328 (2007), 1343-1368.
  • D. Qian, X. Li, Periodic solutions for ordinary differential equations with sublinear impulsive effects, J. Math. Anal. Appl., 303 (2005), 288-303.
  • P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applicatins to Differential Equations, in: CBMS Regional Conf. Ser. in Math., Vol. 65, American Mathematical Society, Providence, RI, 1986.
  • B. Ricceri, On a three critical points theorem, Arch. Math. $($Basel$)$, 75 (2000), 220-226.
  • B. Ricceri, A general multiplicity theorem for certain nonlinear equations in Hilbert spaces, Proc. Amer. Math. Soc., 133 (2005), 3255-3261.
  • Y. V. Rogovchenko, Impulsive evolution systems: Main results and new trends, Dynam. Contin. Discrete Impuls. Systems, 3 (1997), 57-88.
  • A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
  • Y. Tian, W. G. Ge, Periodic solutions of non-autonomous second-order systems with a p-Laplacian, Nonlinear Anal., 66 (2007), 192-203.
  • Y. Tian, W. G. Ge, Multiple positive solutions for a second-order Sturm-Liouville boundary value problem with a p-Laplacian via variational methods, Rocky Mountain J. Math., in press.
  • Y. Tian, W. G. Ge, Applications of Variational Methods to Boundary Value Problem for Impulsive Differential Equations, Proceedings of Edinburgh Mathematical Society, 51 (2008), 509-527.