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Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities. (English) Zbl 0725.34042

Conditions for the existence, uniqueness and asymptotic stability of periodic solutions are obtained for 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, a necessary and sufficient condition is given.

MSC:

34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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[1] W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Co., Boston, Mass., 1965. · Zbl 0154.09301
[2] Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321 – 340. · Zbl 0219.46015
[3] 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). · Zbl 0495.35001
[4] Thierry Gallouët and Otared Kavian, Resonance for jumping nonlinearities, Comm. Partial Differential Equations 7 (1982), no. 3, 325 – 342. · Zbl 0497.35080 · doi:10.1080/03605308208820225
[5] 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. · Zbl 0677.73046 · doi:10.1007/BF00944997
[6] Jack K. Hale, Ordinary differential equations, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. Pure and Applied Mathematics, Vol. XXI. · Zbl 0433.34003
[7] Morris W. Hirsch and Stephen Smale, Differential equations, dynamical systems, and linear algebra, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Pure and Applied Mathematics, Vol. 60. · Zbl 0309.34001
[8] A. C. Lazer, Application of a lemma on bilinear forms to a problem in nonlinear oscillations, Proc. Amer. Math. Soc. 33 (1972), 89 – 94. · Zbl 0257.34041
[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. · Zbl 0654.35082 · doi:10.1080/03605308608820479
[10] 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). · Zbl 0633.34037
[11] 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. · Zbl 0666.47038 · doi:10.1016/0022-0396(88)90150-7
[12] D. E. Leach, On Poincaré’s perturbation theorem and a theorem of W. S. Loud, J. Differential Equations 7 (1970), 34 – 53. · Zbl 0186.15501 · doi:10.1016/0022-0396(70)90122-1
[13] W. S. Loud, Periodic solutions of \?”+\?\?’+\?(\?)=\?\?(\?), Mem. Amer. Math. Soc. no. 31 (1959), 58 pp. (1959). · Zbl 0091.26701
[14] R. F. Manasevich, A non-variational version of a max-min principle, Nonlinear Anal. TMA 7 (1983), 565-570. · Zbl 0527.49019
[15] José L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J. 17 (1950), 457 – 475. · Zbl 0038.25002
[16] 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.
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