Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T02:03:45.021Z Has data issue: false hasContentIssue false
  • English
  • Français

IMPULSIVE PERIODIC SOLUTIONS FOR SINGULAR PROBLEMS VIA VARIATIONAL METHODS

Part of: Boundary value problems

Published online by Cambridge University Press:  16 February 2012

JUNTAO SUN  and
DONAL O’REGAN
JUNTAO SUN*
Affiliation:
School of Science, Shandong University of Technology, Zibo, 255049 Shandong, China (email: [email protected])
DONAL O’REGAN
Affiliation:
Department of Mathematics, National University of Ireland, Galway, Ireland (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we study impulsive periodic solutions for second-order nonautonomous singular differential equations. Our proof is based on the mountain pass theorem. Some recent results in the literature are extended.

Keywords

Periodic solutionsimpulsesingular problemsvariational methodsmountain pass theorem

MSC classification

Secondary: 34B15: Nonlinear boundary value problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 2 , October 2012 , pp. 193 - 204
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Agarwal, R. P., Franco, D. and O’Regan, D., ‘Singular boundary value problems for first and second order impulsive differential equations’, Aequationes Math. 69 (2005), 83–96.CrossRefGoogle Scholar
[2]Agarwal, R. P. and O’Regan, D., ‘Existence criteria for singular boundary value problems with sign changing nonlinearities’, J. Differential Equations 183 (2002), 409–433.CrossRefGoogle Scholar
[3]Agarwal, R. P., Perera, K. and O’Regan, D., ‘Multiple positive solutions of singular problems by variational methods’, Proc. Amer. Math. Soc. 134 (2005), 817–824.CrossRefGoogle Scholar
[4]Ahmad, B. and Nieto, J. J., ‘Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions’, Nonlinear Anal. 69 (2008), 3291–3298.CrossRefGoogle Scholar
[5]Boucherif, A. and Daoudi-Merzagui, N., ‘Periodic solutions of singular nonautonomous second order differential equations’, Nonlinear Differ. Equ. Appl. 15 (2008), 147–158.CrossRefGoogle Scholar
[6]Chen, L. and Sun, J., ‘Nonlinear boundary value problem for first-order impulsive functional differential equations’, J. Math. Anal. Appl. 318 (2006), 726–741.CrossRefGoogle Scholar
[7]Chen, L., Tisdell, C. C. and Yuan, R., ‘On the solvability of periodic boundary value problems with impulse’, J. Math. Anal. Appl. 331 (2007), 233–244.CrossRefGoogle Scholar
[8]Choisy, M., Guégan, J. F. and Rohani, P., ‘Dynamics of infectious deseases and pulse vaccination: teasing apart the embedded resonance effects’, Physica D 223 (2006), 26–35.CrossRefGoogle Scholar
[9]Chu, J., Fan, N. and Torres, P. J., ‘Periodic solutions for second order singular damped differential equations’, J. Math. Anal. Appl. 388 (2012), 665–675.CrossRefGoogle Scholar
[10]Chu, J., Lin, X., Jiang, D., O’Regan, D. and Agarwal, P. R., ‘Multiplicity of positive solutions to second order differential equations’, Bull. Aust. Math. Soc. 73 (2006), 175–182.CrossRefGoogle Scholar
[11]Chu, J. and Nieto, J. J., ‘Impulsive periodic solution of first-order singular differential equations’, Bull. Lond. Math. Soc. 40 (2008), 143–150.CrossRefGoogle Scholar
[12]Chu, J. and O’Regan, D., ‘Multiplicity results for second order nonautonomous singular Dirichlet systems’, Acta Appl. Math. 105 (2009), 323–338.CrossRefGoogle Scholar
[13]Chu, J., Torres, P. J. and Zhang, M., ‘Periodic solutions of second order nonautonomous singular dynamical systems’, J. Differential Equations 239 (2007), 196–212.CrossRefGoogle Scholar
[14]Chu, J. and Zhang, Z., ‘Periodic solutions of singular differential equations with sign-changing potential’, Bull. Aust. Math. Soc. 82 (2010), 437–445.CrossRefGoogle Scholar
[15]Daoudi-Merzagui, N., ‘Periodic solutions of nonautonomous second order differential equations with a singularity’, Appl. Anal. 73 (1999), 449–462.CrossRefGoogle Scholar
[16]Gao, S., Chen, L., Nieto, J. J. and Torres, A., ‘Analysis of a delayed epidemic model with pulse vaccination and saturation incidence’, Vaccine 24 (2006), 6037–6045.CrossRefGoogle ScholarPubMed
[17]George, R. K., Nandakumaran, A. K. and Arapostathis, A., ‘A note on controllability of impulsive systems’, J. Math. Anal. Appl. 241 (2000), 276–283.CrossRefGoogle Scholar
[18]Lakshmikantham, V., Bainov, D. D. and Simeonov, P. S., Theory of Impulsive Differential Equations (World Scientific, Singapore, 1989).CrossRefGoogle Scholar
[19]Li, W., Chang, Y. and Nieto, J. J., ‘Solvability of impulsive neutral evolution differential inclusions with state-dependent delay’, Math. Comput. Modelling 49 (2009), 1920–1927.CrossRefGoogle Scholar
[20]Li, J. and Nieto, J. J., ‘Existence of positive solutions for multipoint boundary value problem on the half-line with impulses’, Bound. Value Probl. (2009), (Article ID 834158, 12 pages).CrossRefGoogle Scholar
[21]Nenov, S., ‘Impulsive controllability and optimization problems in population dynamics’, Nonlinear Anal. 36 (1999), 881–890.CrossRefGoogle Scholar
[22]Nieto, J. J. and O’Regan, D., ‘Variational approach to impulsive differential equations’, Nonlinear Anal. Real World Appl. 10 (2009), 680–690.CrossRefGoogle Scholar
[23]Mawhin, J. and Willem, M., Critical Point Theory and Hamiltonian Systems (Springer, Berlin, 1989).CrossRefGoogle Scholar
[24]Qian, D. and Li, X., ‘Periodic solutions for ordinary differential equations with sublinear impulsive effects’, J. Math. Anal. Appl. 303 (2005), 288–303.CrossRefGoogle Scholar
[25]Rachunková, I. and Tvrdý, M., ‘Existence results for impulsive second-order periodic problems’, Nonlinear Anal. 59 (2004), 133–146.CrossRefGoogle Scholar
[26]Sun, J., Chen, H., Nieto, J. J. and Otero-Novoa, M., ‘Multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects’, Nonlinear Anal. 72 (2010), 4575–4586.CrossRefGoogle Scholar
[27]Tian, Y. and Ge, W., ‘Applications of variational methods to boundary-value problem for impulsive differential equations’, Proc. Edinb. Math. Soc. (2) 51 (2008), 509–527.CrossRefGoogle Scholar