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
HTML articles powered by AMS MathViewer

by Rafael Ortega and Antonio Tineo PDF

Proc. Amer. Math. Soc. 124 (1996), 2089-2096 Request permission

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.

References

Shair Ahmad, A nonstandard resonance problem for ordinary differential equations, Trans. Amer. Math. Soc. 323 (1991), no. 2, 857–875. MR 1010407, DOI 10.1090/S0002-9947-1991-1010407-9
W. A. Coppel, Dichotomies in stability theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. MR 0481196
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, DOI 10.1007/978-3-642-69409-7
R. Ortega, A boundedness result of Landesman - Lazer type, Differential and Integral Equations, 8 (1995), 729–734.
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
N. Rouche and J. Mawhin, Équations différentielles ordinaires, Masson et Cie, Éditeurs, Paris, 1973 (French). Tome I: Théorie générale. MR 0481181
A. Tineo, An iterative scheme for the N-competing species problem, J. Diff. Eq. 116 (1995), 1–15.
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, DOI 10.1090/S0002-9939-1981-0597653-2
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, DOI 10.1090/S0002-9947-1992-1066450-8
T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions, Applied Mathematical Sciences, Vol. 14, Springer-Verlag, New York-Heidelberg, 1975. MR 0466797, DOI 10.1007/978-1-4612-6376-0