Weak singularities may help periodic solutions to exist

Journal of Differential Equations

Volumn 232, Issue 1, 2007, Pages 277-284

  Torres, Pedro J ^a  

^a UNIVERSITY OF GRANADA   (Spain)

Author keywords

Periodic solution; Schauder's fixed point theorem; Weak singularity

Indexed keywords

EID: 33750163929     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2006.08.006     Document Type: Article

References (28)

1. α. Differ. Equ. 8 (1972) 1609-1612
2. Bonheure D., and De Coster C. Forced singular oscillators and the method of upper and lower solutions. Topol. Methods Nonlinear Anal. 22 (2003) 297-317
3. Bonheure D., Fabry C., and Smets D. Periodic solutions of forced isochronous oscillators at resonance. Discrete Contin. Dyn. Syst. 8 4 (2002) 907-930
4. Ding T. A boundary value problem for the periodic Brillouin focusing system. Acta Sci. Natur. Univ. Pekinensis 11 (1965) 31-38 (in Chinese)
5. Fonda A. Periodic solutions of scalar second order differential equations with a singularity. Mém. Cl. Sciences Acad. R. Belguique (4) 8 (1993) 68-98
6. Fonda A., Manásevich R., and Zanolin F. Subharmonics solutions for some second order differential equations with singularities. SIAM J. Math. Anal. 24 (1993) 1294-1311
7. Franco D., and Webb J.K.L. Collisionless orbits of singular and nonsingular dynamical systems. Discrete Contin. Dyn. Syst. 15 (2006) 747-757
8. Gordon W.B. Conservative dynamical systems involving strong forces. Trans. Amer. Math. Soc. 204 (1975) 113-135
9. Gordon W.B. A minimizing property of Keplerian orbits. Amer. J. Math. 99 (1977) 961-971
10. Habets P., and Sanchez L. Periodic solutions of some Liénard equations with singularities. Proc. Amer. Math. Soc. 109 (1990) 1135-1144
11. Jiang D., Chu J., and Zhang M. Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. Differential Equations 211 2 (2005) 282-302
12. Lazer A.C., and Solimini S. On periodic solutions of nonlinear differential equations with singularities. Proc. Amer. Math. Soc. 99 (1987) 109-114
13. 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
14. del Pino M., Manásevich R., and Montero A. T-periodic solutions for some second order differential equations with singularities. Proc. Roy. Soc. Edinburgh Sect. A 120 3-4 (1992) 231-243
15. ″ + g (u) = s (1 + h (t)) using the Poincaré-Birkhoff theorem. J. Differential Equations 95 (1992) 240-258
16. Rachunková I., Tvrdý M., and Vrkoč I. Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. J. Differential Equations 176 (2001) 445-469
17. Rachunková I., Stanek S., and Tvrdý M. Singularities and Laplacians in Boundary Value Problems for Nonlinear Ordinary Differential Equations. Handbook of Differential Equations (Ordinary Differential Equations) vol. 3 (2006), Elsevier
18. Torres P.J. Bounded solutions in singular equations of repulsive type. Nonlinear Anal. 32 (1998) 117-125
19. Torres P.J. Existence and uniqueness of elliptic periodic solutions of the Brillouin electron beam focusing system. Math. Methods Appl. Sci. 23 (2000) 1139-1143
20. Torres P.J. Twist solutions of a Hill's equations with singular term. Adv. Nonlinear Stud. 2 (2002) 279-287
21. 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
22. P.J. Torres, Existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, in press
23. 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) 101-118
24. Torres P.J., and Zhang M. Twist periodic solutions of repulsive singular equations. Nonlinear Anal. 56 (2004) 591-599
25. Ye Y., and Wang X. Nonlinear differential equations in electron beam focusing theory. Acta Math. Appl. Sin. 1 (1978) 13-41 (in Chinese)
26. Zhang M. Periodic solutions of Liénard equations with singular forces of repulsive type. J. Math. Anal. Appl. 203 (1996) 254-269
27. 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
28. Zhang M. The best bound on the rotations in the stability of periodic solutions of a Newtonian equation. J. London Math. Soc. (2) 67 (2003) 137-148