Positive periodic solutions of Hill's equations with singular nonlinear perturbations

Nonlinear Analysis, Theory, Methods and Applications

Volumn 69, Issue 1, 2008, Pages 276-286

  Chu, Jifeng ^a,b   Li, Ming ^b  

^a HOHAI UNIVERSITY   (China)

^b TSINGHUA UNIVERSITY   (China)

Author keywords

Fixed point theorem in cones; Hill equation; Leray Schauder alternative principle; Positive periodic solution; Strong singularity; Weak singularity

Indexed keywords

PERTURBATION TECHNIQUES; SINGULAR VALUE DECOMPOSITION; THEOREM PROVING; TIME DOMAIN ANALYSIS;

FIXED POINT THEOREM; HILL EQUATIONS; LERAY SCHAUDER ALTERNATIVE PRINCIPLES; POSITIVE PERIODIC SOLUTIONS; SINGULARITY;

NONLINEAR EQUATIONS;

EID: 43049173303     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2007.05.016     Document Type: Article

References (21)

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
2. Chu J., Lin X., Jiang D., O'Regan D., and Agarwal R.P. Multiplicity of positive solutions to second order differential equations. Bull. Austral. Math. Soc. 73 (2006) 175-182
3. De Coster C., and Habets P. Upper and lower solutions in the theory of ODE boundary value problems: Classical and recent results. In: Zanolin F. (Ed). Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations. CISM-ICMS vol. 371 (1996), Springer-Verlag, New York 1-78
4. del Pino M.A., Manásevich R.F., and Montero A. T-periodic solutions for some second order differential equations with singularities. Proc. Roy. Soc. Edinburgh Sect. A 120 (1992) 231-243
5. del Pino M.A., and Manásevich R.F. Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity. J. Differential Equations 103 (1993) 260-277
6. Franco D., and Webb J.R.L. Collisionless orbits of singular and nonsingular dynamical systems. Discrete Contin. Dyn. Syst. 15 (2006) 747-757
7. D. Franco, P.J. Torres, Periodic solutions of singular systems without the strong force condition, Proc. Amer. Math. Soc. (in press)
8. Gordon W.B. Conservative dynamical systems involving strong forces. Trans. Amer. Math. Soc. 204 (1975) 113-135
9. Habets P., and Sanchez L. Periodic solution of some Liénard equations with singularities. Proc. Amer. Math. Soc. 109 (1990) 1135-1144
10. Jiang D., Chu J., and Zhang M. Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. Differential Equations 211 (2005) 282-302
11. Krasnosel'skii M.A. Positive Solutions of Operator Equations (1964), Noordhoff, Groningen
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. Mawhin J. Topological degree and boundary value problems for nonlinear differential equations. In: Furi M., and Zecca P. (Eds). Topological Methods for Ordinary Differential Equations. Lecture Notes Math. vol. 1537 (1993), Springer, New York, Berlin 74-142
14. Rachunková I., Tvrdý M., and Vrkoc ̌ I. Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. J. Differential Equations 176 (2001) 445-469
15. 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
16. 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
17. Torres P.J. Weak singularities may help periodic solutions to exist. J. Differential Equations 232 (2007) 277-284
18. Yan P., and Zhang M. Higher order nonresonance for differential equations with singularities. Math. Methods Appl. Sci. 26 (2003) 1067-1074
19. 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
20. α norms. Proc. Amer. Math. Soc. 130 (2002) 3325-3333
21. Zhang M. Periodic solutions of equations of Ermakov-Pinney type. Adv. Nonlinear Stud. 6 (2006) 57-67