Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-18T17:37:41.315Z Has data issue: false hasContentIssue false
  • English
  • Français

MULTIPLICITY OF SOLUTIONS FOR A FOURTH-ORDER IMPULSIVE DIFFERENTIAL EQUATION VIA VARIATIONAL METHODS

Part of: Boundary value problems

Published online by Cambridge University Press:  13 October 2010

JUNTAO SUN ,
HAIBO CHEN  and
TIEJUN ZHOU
JUNTAO SUN
Affiliation:
Department of Mathematics, Central South University, Changsha, 410075 Hunan, PR China (email: [email protected])
HAIBO CHEN*
Affiliation:
Department of Mathematics, Central South University, Changsha, 410075 Hunan, PR China (email: [email protected])
TIEJUN ZHOU
Affiliation:
College of Science, Hunan Agricultural University, Changsha, 410128, Hunan, PR China (email: [email protected])
*
✉For correspondence; e-mail: math˙[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 deal with the multiplicity of solutions for a fourth-order impulsive differential equation with a parameter. Using variational methods and a ‘three critical points’ theorem, we give some new criteria to guarantee that the impulsive problem has at least three classical solutions. An example is also given in order to illustrate the main results.

Keywords

impulsive differential equationboundary value problemcritical pointsvariational methods

MSC classification

Secondary: 34B15: Nonlinear boundary value problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 3 , December 2010 , pp. 446 - 458
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The first author was supported by the Graduate Degree Thesis Innovation Foundation of Central South University (CX2009B023). The second author was supported by NFSC (10871206).

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]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
[3]Bonanno, G. and Bella, B. Di., ‘A boundary value problem for fourth-order elastic beam equations’, J. Math. Anal. Appl. 343 (2008), 1166–1176.CrossRefGoogle Scholar
[4]Bonanno, G. and Candito, P., ‘Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities’, J. Differential Equations 244 (2008), 3031–3059.CrossRefGoogle Scholar
[5]Bonanno, G. and Marano, S. A., ‘On the structure of the critical set of non-differentiable functions with a weak compactness condition’, Appl. Anal. 89 (2010), 1–10.CrossRefGoogle Scholar
[6]Bonanno, G. and Riccobono, G., ‘Multiplicity results for Sturm–Liouville boundary value problems’, Appl. Math. Comput. 210 (2009), 294–297.CrossRefGoogle Scholar
[7]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
[8]Chen, L., Tisdel, C. C. and Yuan, R., ‘On the solvability of periodic boundary value problems with impulse’, J. Math. Anal. Appl. 331(2) (2007), 233–244.CrossRefGoogle Scholar
[9]Chu, J. and Nieto, J. J., ‘Impulsive periodic solution of first-order singular differential equations’, Bull. London Math. Soc. 40 (2008), 143–150.CrossRefGoogle Scholar
[10]Du, Z., Lin, X. and Tisdel, C. C., ‘A multiplicity result for p-Laplacian boundary value problems via critical points theorem’, Appl. Math. Comput. 205 (2008), 231–237.CrossRefGoogle Scholar
[11]Hernandez, E., Henriquez, H. R. and McKibben, M. A., ‘Existence results for abstract impulsive second-order neutral functional differential equations’, Nonlinear Anal. 70 (2009), 2736–2751.CrossRefGoogle Scholar
[12]Karaca, I., ‘On positive solutions for fourth-order boundary value problem with impulse’, J. Comput. Appl. Math. 225 (2009), 356–364.CrossRefGoogle Scholar
[13]Lakshmikantham, V., Bainov, D. D. and Simeonov, P. S., Theory of Impulsive Differential Equations (World Scientific, Singapore, 1989).CrossRefGoogle Scholar
[14]Li, J., Nieto, J. J. and Shen, J., ‘Impulsive periodic boundary value problems of first-order differential equations’, J. Math. Anal. Appl. 325 (2007), 226–299.CrossRefGoogle Scholar
[15]Ma, T., ‘Positive solutions for a beam equation on a nonlinear elastic foundation’, Math. Comput. Modelling 39 (2004), 1195–1201.Google Scholar
[16]Mawhin, J. and Willem, M., Critical Point Theory and Hamiltonian Systems (Springer, Berlin, 1989).CrossRefGoogle Scholar
[17]Nieto, J. J. and O’Regan, D., ‘Variational approach to impulsive differential equations’, Nonlinear Anal.: Real World Appl. 10 (2009), 680–690.CrossRefGoogle Scholar
[18]Nieto, J. J. and Rodriguez-Lopez, R., ‘New comparison results for impulsive integro-differential equations and applications’, J. Math. Anal. Appl. 328 (2007), 1343–1368.CrossRefGoogle Scholar
[19]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
[20]Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65 (American Mathematical Society, Providence, RI, 1986).CrossRefGoogle Scholar
[21]Samoilenko, A. M. and Perestyuk, N. A., Impulsive Differential Equations (World Scientific, Singapore, 1995).CrossRefGoogle Scholar
[22]Struwe, M., Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 3rd edn (Springer, Berlin, 2000).CrossRefGoogle Scholar
[23]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.: TMA 72 (2010), 4575–4586.CrossRefGoogle Scholar
[24]Sun, J., Chen, H. and Yang, L., ‘The existence and multiplicity of solutions for an impulsive differential equation with two parameters via variational method’, Nonlinear Anal.: TMA 73 (2010), 440–449.CrossRefGoogle Scholar
[25]Tian, Y. and Ge, W., ‘Second-order Sturm–Liouville boundary value problem involving the one-dimensional p-Laplacian’, Rocky Mountain J. Math. 38 (2008), 309–327.CrossRefGoogle Scholar
[26]Timoshenko, S., Weaver, W. Jr and Young, D. H., Vibrations Problems in Engineering, 5th edn (John Wiley, New York, 1990).Google Scholar
[27]Zeidler, E., Nonlinear Functional Analysis and its Applications, Vol. 2 (Springer, Berlin, 1990).Google Scholar
[28]Zhang, L. and Ge, W., ‘Solvability of a kind of Sturm–Liouville boundary value problems with impulses via variational methods’, Acta Appl. Math. 110 (2010), 1237–1248.CrossRefGoogle Scholar
[29]Zhang, H. and Li, Z., ‘Variational approach to impulsive differential equations with periodic boundary conditions’, Nonlinear Anal.: Real World Appl. 11 (2010), 67–78.CrossRefGoogle Scholar