Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: A. C. Lazer and P. J. McKenna
Journal: Trans. Amer. Math. Soc. 315 (1989), 721-739
MSC: Primary 34A10; Secondary 34C25, 34D05, 70K20, 70K40
DOI: https://doi.org/10.1090/S0002-9947-1989-0979963-1
MathSciNet review: 979963
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] W. A. Coppel, Stability and asymptotic behavior of differential equations, Heath Math. Monographs, Heath, Boston, Mass., 1965. MR 0190463 (32:7875)
  • [2] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321-340. MR 0288640 (44:5836)
  • [3] Th. Gallouët and O. Kavian, Résultats d'existence et de non-existence pour certains problèmes demi-linéaires á l'infini, Ann. Fac. Sci. Toulouse Math. (5) 3 (1981), 201-246. MR 658734 (83m:35058)
  • [4] -, Resonance for jumping non-linearities, Comm. Partial Differential Equations 7 (1982), 325-342. MR 646710 (84f:35054)
  • [5] J. Glover, A. C. Lazer and P. J. McKenna, Existence of stability of large scale nonlinear oscillations in suspension bridges, J. Appl. Math. Phys. (ZAMP) 40 (1989), 172-200. MR 990626 (90c:34037)
  • [6] J. K. Hale, Ordinary differential equations, Wiley-Interscience, New York, 1969. MR 0419901 (54:7918)
  • [7] M. W. Hirsh and S. Smale, Differential equations, dynamical systems, and linear algebra, Academic Press, New York, 1974. MR 0486784 (58:6484)
  • [8] A. C. Lazer, Applications of a lemma on bilinear forms to a problem in nonlinear oscillations, Proc. Amer. Math. Soc. 33 (1972), 89-94. MR 0293179 (45:2258)
  • [9] A. C. Lazer and P. J. McKenna, Critical point theory and boundary value problems with non-linearities crossing multiple eigenvalues. II, Comm. Partial Differential Equations 11 (1986), 1653-1676. MR 871108 (88c:35063)
  • [10] -, Large scale oscillatory behaviour in loaded asymmetric systems, Ann. Inst. H. Poincaré Sect. (N.S.) 4 (1987), 243-276. MR 898049 (88m:58028)
  • [11] -, A symmetry theorem and applications to nonlinear partial differential equations, J. Differential Equations (to appear). MR 929199 (89b:47087)
  • [12] D. E. Leach, On Poincaré's perturbation theorem and a theorem of W. S. Loud, J. Differential Equations 7 (1970), 34-53. MR 0251308 (40:4539)
  • [13] W. S. Loud, Periodic solutions of $ x'' + cx\prime + g(x) = \varepsilon f(t)$, Mem. Amer. Math. Soc. No. 31 (1959). MR 0107058 (21:5785)
  • [14] R. F. Manasevich, A non-variational version of a max-min principle, Nonlinear Anal. TMA 7 (1983), 565-570.
  • [15] J. L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J. 17 (1950), 457-475. MR 0040512 (12:705g)
  • [16] L. Nirenberg, Topics in nonlinear functional analysis, Courant Institute of Mathematical Sciences, 1974. MR 0488102 (58:7672)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34A10, 34C25, 34D05, 70K20, 70K40

Retrieve articles in all journals with MSC: 34A10, 34C25, 34D05, 70K20, 70K40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0979963-1
Keywords: Unique periodic solution, continuation argument, implicit function theorem
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society