Nonresonance Conditions for Asymptotically Positively Homogeneous Differential Systems: The Fučik Spectrum and Its Generalization

Journal of Differential Equations

Volumn 145, Issue 2, 1998, Pages 332-366

  Zhang, Meirong ^a,b  

^a TSINGHUA UNIVERSITY   (China)

^b GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0011566211     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1997.3403     Document Type: Article

References (35)

1. Boccardo L., Dràbek P., Giachetti D., Kučera M. Generalization of Fredholm alternative for nonlinear differential operators. Nonlinear Anal. 10:1986;1083-1103.
2. Capietto A., Mawhin J., Zanolin F. A continuation approach to superlinear periodic boundary value problems. J. Differential Equations. 88:1990;347-395.
3. Capietto A., Mawhin J., Zanolin F. Continuation theorems for periodic perturbations of autonomous systems. Trans. Amer. Math. Soc. 329:1992;41-72.
4. Cuesta M. A nonresonance result with respect to the Fučik spectrum for a second order differential equation. Partial Differential Equations (Han-sur-Lesse, 1993). Math. Res. 82:1994;Akademie-Verlag, Berlin. p. 101-119.
5. Cuesta M., Gossez J.-P. A variational approach to nonresonance with respect to the Fuik spectrum. Nonlinear Anal. 19:1992;487-500.
6. Dancer E. N. Boundary value problems for weakly nonlinear ordinary differential equations. Bull. Austral. Math. Soc. 15:1976;321-328.
7. Dancer E. N. On the Dirichlet problem for weakly nonlinear elliptical partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A. 76:1977;283-300.
8. puftuuuTp. J. Differential Equations. 80:1989;1-13.
9. Ding T., Iannacci R., Zanolin F. Existence and multiplicity results for periodic solutions of semilinear Duffing equations. J. Differential Equations. 105:1993;364-409.
10. Ding T., Zanolin F. Time-maps for the solvability of periodically perturbed nonlinear Duffing equations. Nonlinear Anal. 17:1991;635-653.
11. Drábek P., Invernizzi S. On the periodic BVP for the forced Duffing equation with jumping nonlinearities. Nonlinear Anal. 10:1986;643-650.
12. O. C. Ezeilo, M. N. Nkashama, Resonant and nonresonant oscillations for some third order nonlinear differential equations, Trieste, 1986.
13. Fabry C. Landesman-Lazer conditions for periodic boundary value problems with asymmetric nonlinearities. J. Differential Equations. 116:1995;405-418.
14. Fabry C., Fayyad D. Periodic solutions of second order differential equations with ap. Rend. Istit. Mat. Univ. Trieste. 24:1992;207-227.
15. Fonda A., Habets P. Periodic solutions of asymptotically positively homogeneous differential equations. J. Differential Equations. 81:1989;68-97.
16. Fonda A., Zanolin F. Periodic solutions to second order differential equations of Liénard type with jumping nonlinearities. Comm. Math. Univ. Carolinae. 28:1987;33-41.
17. Fučik S. Boundary value problems with jumping nonlinearities. Časopis Pěst. Mat. 101:1976;69-87.
18. Fučik S. Solvability of Nonlinear Equations and Boundary Value Problems. 1980;Reidel, Dordrecht.
19. Garcia-Huidobro M. Jumping nonlinearities for a nonlinear equation with ann. Differential Integral Equations. 4:1991;1281-1291.
20. Gaudenzi M., Habets P. Fučik spectrum for third order equations. J. Differential Equations. 128:1996;556-593.
21. Gossez J.-P., Omari P. Nonresonance with respect to the Fučik spectrum for periodic solutions of second order ordinary differential equations. Nonlinear Anal. 14:1990;1079-1104.
22. Habets P., Metzen G. Existence of periodic solutions of Duffing equations. J. Differential Equations. 78:1989;1-32.
23. Habets P., Omari P., Zanolin F. Nonresonance conditions on the potential with respect to the Fučik spectrum for the periodic boundary value problem. Rocky Mountain Math. J. 25:1995;1305-1340.
24. Invernizzi S. A note on nonuniform nonresonance for jumping nonlinearities. Comm. Math. Univ. Carolinae. 27:1986;285-291.
25. Krejči P. On the solvability of equations of the 4th order with jumping nonlinearities. Časopis Pěst. Mat. 108:1983;29-39.
26. Lazer A. C., McKenna P. J. A semi-Fredholm principle for periodically forced systems with homogeneous nonlinearities. Proc. Amer. Math. Soc. 106:1989;119-125.
27. Lazer A. C., McKenna P. J. Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities. Trans. Amer. Math. Soc. 315:1989;721-739.
28. Lazer A. C., McKenna P. J. Nonresonance conditions for the existence, uniqueness and stability of periodic solutions of differential equations with asymmetric nonlinearities. Differential Integral Equations. 4:1991;719-730.
29. Mawhin J. Topological Degree Methods in Nonlinear Boundary Value Problems. CBMS. 40:1979;Amer. Math. Soc. Providence.
30. Mawhin J. Topological degree and boundary value problems for nonlinear differential equations. Topological Methods for Ordinary Differential Equations. Lecture Notes in Math. 1537:1993;Springer-Verlag, New York/Berlin. p. 74-142.
31. Mawhin J., Ward J. R. Nonuniform nonresonance conditions at the first two eigenvalues for periodic solutions of forced Liénard and Duffing equations. Rocky Mountain Math. J. 12:1982;643-654.
32. Omari P., Zanolin F. On forced nonlinear oscillations in n th order differential systems with geometric conditions. Nonlinear Anal. 8:1984;723-748.
33. Wang D. On the existence of 2πxgxpt. Chinese Ann. Math. Ser. B. 5:1984;61-72.
34. Zhang M. Nonuniform nonresonance at the first eigenvalue of thep. Nonlinear Anal. 29:1997;41-51.
35. Zhang M. An abstract result on asymptotically positively homogeneous differential equations. J. Math. Anal. Appl. 209:1997;291-298.