Hostname: page-component-7dd5485656-gs9qr Total loading time: 0 Render date: 2025-10-31T05:27:14.446Z Has data issue: false hasContentIssue false
  • English
  • Français

PERIODIC SOLUTIONS OF SINGULAR DIFFERENTIAL EQUATIONS WITH SIGN-CHANGING POTENTIAL

Part of: Qualitative theory

Published online by Cambridge University Press:  14 September 2010

JIFENG CHU  and
ZIHENG ZHANG
JIFENG CHU*
Affiliation:
Department of Mathematics, College of Science, Hohai University, Nanjing 210098, PR China (email: chujf05@mails.tsinghua.edu.cn)
ZIHENG ZHANG
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, PR China (email: zhzh@mail.bnu.edu.cn)
*
✉For correspondence; e-mail: chujf05@mails.tsinghua.edu.cn
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 the existence of positive periodic solutions to second-order singular differential equations with the sign-changing potential. Both the repulsive case and the attractive case are studied. The proof is based on Schauder’s fixed point theorem. Recent results in the literature are generalized and significantly improved.

Keywords

periodic solutionsingular differential equationSchauder’s fixed point theoremsign-changing potential

MSC classification

Secondary: 34C25: Periodic solutions

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 3 , December 2010 , pp. 437 - 445
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

Jifeng Chu is supported by the National Natural Science Foundation of China (Grant No. 10801044), Jiangsu Natural Science Foundation (Grant No. BK2008356), the Program for New Century Excellent Talents in University (Grant No. NCET-10-0325) and the Fundamental Research Funds for the Central Universities.

References

[1]Bonheure, D. and De Coster, C., ‘Forced singular oscillators and the method of lower and upper solutions’, Topol. Methods Nonlinear Anal. 22 (2003), 297–317.CrossRefGoogle Scholar
[2]Cabada, A. and Cid, J. A., ‘On the sign of the Green’s function associated to Hill’s equation with an indefinite potential’, Appl. Math. Comput. 205 (2008), 303–308.Google Scholar
[3]Chu, J. and Lee, Y. H., ‘The sign of Green’s functions and a uniqueness result for a weak singular equation’, Preprint.Google Scholar
[4]Chu, J. and Li, M., ‘Positive periodic solutions of Hill’s equations with singular nonlinear perturbations’, Nonlinear Anal. 69 (2008), 276–286.CrossRefGoogle Scholar
[5]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
[6]Chu, J. and Nieto, J. J., ‘Impulsive periodic solutions of first-order singular differential equations’, Bull. London Math. Soc. 40 (2008), 143–150.CrossRefGoogle Scholar
[7]Chu, J. and Nieto, J. J., ‘Recent existence results for second-order singular periodic differential equations’, Bound. Value Probl. 540863 (2009), 1–20.Google Scholar
[8]Chu, J. and Torres, P. J., ‘Applications of Schauder’s fixed point theorem to singular differential equations’, Bull. London Math. Soc. 39 (2007), 653–660.CrossRefGoogle Scholar
[9]Chu, J., Torres, P. J. and Zhang, M., ‘Periodic solutions of second order non-autonomous singular dynamical systems’, J. Differential Equations 239 (2007), 196–212.CrossRefGoogle Scholar
[10]Chu, J. and Zhang, M., ‘Rotation numbers and Lyapunov stability of elliptic periodic solutions’, Discrete Contin. Dyn. Syst. 21 (2008), 1071–1094.CrossRefGoogle Scholar
[11]del Pino, M. and Manásevich, R., ‘Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity’, J. Differential Equations 103 (1993), 260–277.CrossRefGoogle Scholar
[12]Fonda, A., Manásevich, R. and Zanolin, F., ‘Subharmonic solutions for some second order differential equations with singularities’, SIAM J. Math. Anal. 24 (1993), 1294–1311.CrossRefGoogle Scholar
[13]Franco, D. and Torres, P. J., ‘Periodic solutions of singular systems without the strong force condition’, Proc. Amer. Math. Soc. 136 (2008), 1229–1236.Google Scholar
[14]Franco, D. and Webb, J. R. L., ‘Collisionless orbits of singular and nonsingular dynamical systems’, Discrete Contin. Dyn. Syst. 15 (2006), 747–757.CrossRefGoogle Scholar
[15]Gordon, W. B., ‘Conservative dynamical systems involving strong forces’, Trans. Amer. Math. Soc. 204 (1975), 113–135.CrossRefGoogle Scholar
[16]Habets, P. and Sanchez, L., ‘Periodic solution of some Liénard equations with singularities’, Proc. Amer. Math. Soc. 109 (1990), 1135–1144.Google Scholar
[17]Habets, P. and Zanolin, F., ‘Upper and lower solutions for a generalized Emden–Fowler equation’, J. Math. Anal. Appl. 181 (1994), 684–700.CrossRefGoogle Scholar
[18]Jiang, D., Chu, J., O’Regan, D. and Agarwal, R. P., ‘Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces’, J. Math. Anal. Appl. 286 (2003), 563–576.CrossRefGoogle Scholar
[19]Jiang, D., Chu, J. and Zhang, M., ‘Multiplicity of positive periodic solutions to superlinear repulsive singular equations’, J. Differential Equations 211 (2005), 282–302.CrossRefGoogle Scholar
[20]Lazer, A. C. and Solimini, S., ‘On periodic solutions of nonlinear differential equations with singularities’, Proc. Amer. Math. Soc. 99 (1987), 109–114.Google Scholar
[21]Rachunková, I., Tvrdý, M. and Vrkoc̆, I., ‘Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems’, J. Differential Equations 176 (2001), 445–469.CrossRefGoogle Scholar
[22]Torres, P. J., ‘Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem’, J. Differential Equations 190 (2003), 643–662.CrossRefGoogle Scholar
[23]Torres, P. J., ‘Non-collision periodic solutions of forced dynamical systems with weak singularities’, Discrete Contin. Dyn. Syst. 11 (2004), 693–698.CrossRefGoogle Scholar
[24]Torres, P. J., ‘Weak singularities may help periodic solutions to exist’, J. Differential Equations 232 (2007), 277–284.CrossRefGoogle Scholar
[25]Torres, P. J., ‘Existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity’, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 195–201.CrossRefGoogle Scholar
[26]Torres, P. J. and Zhang, M., ‘A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle’, Math. Nachr. 251 (2003), 101–107.CrossRefGoogle Scholar
[27]Yan, P. and Zhang, M., ‘Higher order nonresonance for differential equations with singularities’, Math. Methods Appl. Sci. 26 (2003), 1067–1074.CrossRefGoogle Scholar
[28]Zhang, M., ‘A relationship between the periodic and the Dirichlet BVPs of singular differential equations’, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 1099–1114.CrossRefGoogle Scholar
[29]Zhang, M., ‘Periodic solutions of equations of Emarkov–Pinney type’, Adv. Nonlinear Stud. 6 (2006), 57–67.CrossRefGoogle Scholar
[30]Zhang, M. and Li, W., ‘A Lyapunov-type stability criterion using L α norms’, Proc. Amer. Math. Soc. 130 (2002), 3325–3333.Google Scholar