Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities

Lazer, A. C.; McKenna, P. J.

doi:10.1090/S0002-9947-1989-0979963-1

Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities
HTML articles powered by AMS MathViewer

by A. C. Lazer and P. J. McKenna PDF

Trans. Amer. Math. Soc. 315 (1989), 721-739 Request permission

Abstract:

We give conditions for the existence, uniqueness, and asymptotic stability of periodic solutions of a second-order differential equation with piecewise linear restoring and $2\pi$-periodic forcing where the range of the derivative of the restoring term possibly contains the square of an integer. With suitable restrictions on the restoring and forcing in the undamped case, we give a necessary and sufficient condition.

References

W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Company, Boston, Mass., 1965. MR 0190463
Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI 10.1016/0022-1236(71)90015-2
Thierry Gallouët and Otared Kavian, Résultats d’existence et de non-existence pour certains problèmes demi-linéaires à l’infini, Ann. Fac. Sci. Toulouse Math. (5) 3 (1981), no. 3-4, 201–246 (1982) (French, with English summary). MR 658734
Thierry Gallouët and Otared Kavian, Resonance for jumping nonlinearities, Comm. Partial Differential Equations 7 (1982), no. 3, 325–342. MR 646710, DOI 10.1080/03605308208820225
J. Glover, A. C. Lazer, and P. J. McKenna, Existence and stability of large scale nonlinear oscillations in suspension bridges, Z. Angew. Math. Phys. 40 (1989), no. 2, 172–200. MR 990626, DOI 10.1007/BF00944997
Jack K. Hale, Ordinary differential equations, Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. MR 0419901
Morris W. Hirsch and Stephen Smale, Differential equations, dynamical systems, and linear algebra, Pure and Applied Mathematics, Vol. 60, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0486784
A. C. Lazer, Application of a lemma on bilinear forms to a problem in nonlinear oscillations, Proc. Amer. Math. Soc. 33 (1972), 89–94. MR 293179, DOI 10.1090/S0002-9939-1972-0293179-9
A. C. Lazer and P. J. McKenna, Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues. II, Comm. Partial Differential Equations 11 (1986), no. 15, 1653–1676. MR 871108, DOI 10.1080/03605308608820479
A. C. Lazer and P. J. McKenna, Large scale oscillatory behaviour in loaded asymmetric systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), no. 3, 243–274 (English, with French summary). MR 898049
A. C. Lazer and P. J. McKenna, A symmetry theorem and applications to nonlinear partial differential equations, J. Differential Equations 72 (1988), no. 1, 95–106. MR 929199, DOI 10.1016/0022-0396(88)90150-7
D. E. Leach, On Poincaré’s perturbation theorem and a theorem of W. S. Loud, J. Differential Equations 7 (1970), 34–53. MR 251308, DOI 10.1016/0022-0396(70)90122-1
W. S. Loud, Periodic solutions of $x''+cx’+g(x)=\varepsilon f(t)$, Mem. Amer. Math. Soc. 31 (1959), 58 pp. (1959). MR 107058

A non-variational version of a max-min principle

7

José L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J. 17 (1950), 457–475. MR 40512
L. Nirenberg, Topics in nonlinear functional analysis, Courant Institute of Mathematical Sciences, New York University, New York, 1974. With a chapter by E. Zehnder; Notes by R. A. Artino; Lecture Notes, 1973–1974. MR 0488102