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PERIODIC SOLUTIONS OF SINGULAR DIFFERENTIAL EQUATIONS WITH SIGN-CHANGING POTENTIAL

  • JIFENG CHU (a1) and ZIHENG ZHANG (a2)
Abstract
Abstract

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.

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Corresponding author
For correspondence; e-mail: chujf05@mails.tsinghua.edu.cn
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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.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[4] J. Chu and M. Li , ‘Positive periodic solutions of Hill’s equations with singular nonlinear perturbations’, Nonlinear Anal. 69 (2008), 276286.

[6] J. Chu and J. J. Nieto , ‘Impulsive periodic solutions of first-order singular differential equations’, Bull. London Math. Soc. 40 (2008), 143150.

[7] J. Chu and J. J. Nieto , ‘Recent existence results for second-order singular periodic differential equations’, Bound. Value Probl. 540863 (2009), 120.

[8] J. Chu and P. J. Torres , ‘Applications of Schauder’s fixed point theorem to singular differential equations’, Bull. London Math. Soc. 39 (2007), 653660.

[9] J. Chu , P. J. Torres and M. Zhang , ‘Periodic solutions of second order non-autonomous singular dynamical systems’, J. Differential Equations 239 (2007), 196212.

[11] M. del Pino and R. Manásevich , ‘Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity’, J. Differential Equations 103 (1993), 260277.

[12] A. Fonda , R. Manásevich and F. Zanolin , ‘Subharmonic solutions for some second order differential equations with singularities’, SIAM J. Math. Anal. 24 (1993), 12941311.

[13] D. Franco and P. J. Torres , ‘Periodic solutions of singular systems without the strong force condition’, Proc. Amer. Math. Soc. 136 (2008), 12291236.

[15] W. B. Gordon , ‘Conservative dynamical systems involving strong forces’, Trans. Amer. Math. Soc. 204 (1975), 113135.

[16] P. Habets and L. Sanchez , ‘Periodic solution of some Liénard equations with singularities’, Proc. Amer. Math. Soc. 109 (1990), 11351144.

[17] P. Habets and F. Zanolin , ‘Upper and lower solutions for a generalized Emden–Fowler equation’, J. Math. Anal. Appl. 181 (1994), 684700.

[18] D. Jiang , J. Chu , D. O’Regan and R. P. Agarwal , ‘Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces’, J. Math. Anal. Appl. 286 (2003), 563576.

[19] D. Jiang , J. Chu and M. Zhang , ‘Multiplicity of positive periodic solutions to superlinear repulsive singular equations’, J. Differential Equations 211 (2005), 282302.

[20] A. C. Lazer and S. Solimini , ‘On periodic solutions of nonlinear differential equations with singularities’, Proc. Amer. Math. Soc. 99 (1987), 109114.

[21] I. Rachunková , M. Tvrdý and I. Vrkoc̆ , ‘Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems’, J. Differential Equations 176 (2001), 445469.

[22] P. J. Torres , ‘Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem’, J. Differential Equations 190 (2003), 643662.

[23] P. J. Torres , ‘Non-collision periodic solutions of forced dynamical systems with weak singularities’, Discrete Contin. Dyn. Syst. 11 (2004), 693698.

[24] P. J. Torres , ‘Weak singularities may help periodic solutions to exist’, J. Differential Equations 232 (2007), 277284.

[26] P. J. Torres and M. Zhang , ‘A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle’, Math. Nachr. 251 (2003), 101107.

[27] P. Yan and M. Zhang , ‘Higher order nonresonance for differential equations with singularities’, Math. Methods Appl. Sci. 26 (2003), 10671074.

[29] M. Zhang , ‘Periodic solutions of equations of Emarkov–Pinney type’, Adv. Nonlinear Stud. 6 (2006), 5767.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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