Resonance and non-resonance in a problem of boundedness

Ortega, Rafael; Tineo, Antonio

doi:10.1090/S0002-9939-96-03457-0

Resonance and non-resonance
in a problem of boundedness

Authors: Rafael Ortega and Antonio Tineo
Journal: Proc. Amer. Math. Soc. 124 (1996), 2089-2096
MSC (1991): Primary 34B15, 34C11
DOI: https://doi.org/10.1090/S0002-9939-96-03457-0
MathSciNet review: 1342038
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies the existence of bounded solutions of a forced non-linear differential equation of arbitrary order. Necessary and sufficient conditions for the existence of such solutions are obtained. These results are inspired by classical results on the periodic problem, both in the resonant and non-resonant cases.

1. Shair Ahmad, A nonstandard resonance problem for ordinary differential equations, Trans. Amer. Math. Soc. 323 (1991), no. 2, 857–875. MR 1010407, https://doi.org/10.1090/S0002-9947-1991-1010407-9
2. W. A. Coppel, Dichotomies in stability theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. MR 0481196
3. M. A. Krasnosel′skiĭ and P. P. Zabreĭko, Geometrical methods of nonlinear analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 263, Springer-Verlag, Berlin, 1984. Translated from the Russian by Christian C. Fenske. MR 736839
4. R. Ortega, A boundedness result of Landesman - Lazer type, Differential and Integral Equations, 8 (1995), 729--734. CMP 95:05
5. G. Reuter, Boundedness theorems for nonlinear differential equations of the second order (II), J. London Math. Soc., 27 (1952), 48-58. MR 13:844b
6. N. Rouche and J. Mawhin, Équations différentielles ordinaires, Masson et Cie, Éditeurs, Paris, 1973 (French). Tome I: Théorie générale. MR 0481181
7. A. Tineo, An iterative scheme for the N-competing species problem, J. Diff. Eq. 116 (1995), 1--15.
8. James R. Ward Jr., Asymptotic conditions for periodic solutions of ordinary differential equations, Proc. Amer. Math. Soc. 81 (1981), no. 3, 415–420. MR 597653, https://doi.org/10.1090/S0002-9939-1981-0597653-2
9. James R. Ward Jr., A topological method for bounded solutions of nonautonomous ordinary differential equations, Trans. Amer. Math. Soc. 333 (1992), no. 2, 709–720. MR 1066450, https://doi.org/10.1090/S0002-9947-1992-1066450-8
10. T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions, Springer-Verlag, New York-Heidelberg, 1975. Applied Mathematical Sciences, Vol. 14. MR 0466797