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Existence, Unicite et Multiplicite de Solutions Periodiques D'Equations Differentielles de Duffing Non-Lineaires Avec Dissipation

Published online by Cambridge University Press:  20 November 2018

M. N. Nkashama
M. N. Nkashama*
Affiliation:
Università Degli Studi Delia Calabria, Calabria, Italia
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Nous nous intéressons à l'existence, l'unicité et la multiplicité de solutions de l'équation différentielle forcée:

1.1

vérifiant les conditions périodiques

1.2

où e ∊ L1(0, 2π), c ∊ R, c arbitraire, g:[0, 2π] × R → R satisfait les conditions de Carathéodory i.e.,g(·, x) est measurable pour tout x ∊ R, g(t, ·) est continu pour presque tout t ∊ [0, 2π].

Le problème (1.1)-(1.2) a été étudié par plusieurs auteurs. Nous mentionerons les travaux [22], [18], [21], [20], [19], [7], [29], [25] et la bibliographie contenue dans ces articles.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 3 , 01 June 1987 , pp. 583 - 602
Copyright
Copyright © Canadian Mathematical Society 1987

References

Bibliographie

1. Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review, 18 (1976), 620–709.Google Scholar
2. Amann, H. et Hess, P., A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edin. 84A (1979), 145–151.Google Scholar
3. Ambrosetti, A. et Prodi, G., On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. di Mat. Pura ed Appl. 93 (1972), 231–247.Google Scholar
4. Castro, A., A two point boundary value problem with jumping nonlinearities, Proc. AMS 79 (1980), 207–211.Google Scholar
5. Cronin, J., Fixed points and topological degree in nonlinear analysis, Mathematical Surveys 11 (A.M.S. Providence, R.I., 1964).Google Scholar
6. Cronin, J., Differential equations: introduction and qualitative theory (Marcel Dekker, Inc., New York and Basel, 1980).Google Scholar
7. Dancer, E. N., On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edin. 76A (1977), 283–300.Google Scholar
8. Dancer, E. N., On the range of certain weakly non-linear elliptic partial differential equations, J. Math. Pures et Appl. 57 (1978), 351–366.Google Scholar
9. Dancer, E. N., Non-uniqueness for nonlinear boundary-value problems, Rocky Mount. J. of Math. 13 (1983), 401–412.Google Scholar
10. de Figueiredo, D. G., Lectures on boundary value problem of the Ambrosetti-Prodi type, Atas do 12° Seminário Brasileiro de Análise, São Paolo (1980), 230–291.Google Scholar
11. Drabek, P., Nonlinear noncoercive equations and applications, Zeit. für Analysis and ihrer Anwendungen 1 (1983), 53–65.Google Scholar
12. Drabek, P. et Invernizzi, S., On the periodic bvpfor the forced Duffing equation with jumping nonlinearity, Nonlinear Analysis, Theory, Methods and Applications (To appear).Google Scholar
13. Fucik, S., Solvability of nonlinear equations and boundary value problems (Reidel, Dordrecht, 1980).Google Scholar
14. Iannacci, R. et Nkashama, M. N., On periodic solutions of forced second order differential equations with a deviating argument, Université de Louvain, Séminaire de Math. (N.S.), Rapport 57 (Cabay, août, 1984).Google Scholar
15. Kadzan, J. L. et Warner, F. W., Remarks on some quasilinear elliptic equations, Communications on Pure and Applied Math. 28 (1975), 567–597.Google Scholar
16. Kolesov, Ju. S., Positive periodic solutions of a class of differential equations of the second order, Soviet Math. Dokl. 8 (1967), 68–70.Google Scholar
17. Komlenko, Ju. V., Solvability conditions of some boundary value problems for an ordinary linear differential equation of second order, Soviet Math. Dokl. 8 (1967), 708–711.Google Scholar
18. Komlenko, Ju. V. et Tonkov, E. L., A periodic boundary value problem for an ordinary second-order differential equation, Soviet Math. Dokl. 9 (1968), 305–308.Google Scholar
19. Lasota, A. et Opial, Z., Sur les solutions périodiques des équations différentielles ordinaires, Annales Polonici Mathematici 16 (1964), 69–94.Google Scholar
20. Lazer, A. C. et Leach, D. E., Bounded perturbations of forced harmonie oscillators at resonance, Ann. di Mat. Pura ed Appl. 82 (1969), 49–68.Google Scholar
21. Leach, D. E., On Poincare's perturbation theorem and a theorem of W. S. Loud, Journal of Diff. Eq. 7 (1970), 34–53.Google Scholar
22. Loud, W. S., Periodic solutions of nonlinear differential equations of Duffing type, Proc. U.S. – Japan Semin. on Diff. and Funct. Equations (Benjamin, New York, 1967), 199–224.Google Scholar
23. Loud, W. S., Subharmonic solutions of second-order equations arising near harmonic solutions, J. Diff. Eq. 53 (1984), 192–212.Google Scholar
24. Mawhin, J., Contractive mappings and periodically perturbed conservative systems, Arch. Math. (Brno) 12 (1976), 67–74.Google Scholar
25. Mawhin, J., Compacité, monotonie et convexité dans l'étude de problèmes aux limites semi-linéaires, Sém. Anal. Moderne 19 (Université de Sherbrooke, Québec, 1981).Google Scholar
26. Mawhin, J., Topological degree methods in nonlinear boundary value problems, CBMS Regional Conference Series in Math. 40 (Amer. Math. Soc., Providence, RI, 1979).CrossRefGoogle Scholar
27. Mawhin, J. et Nkashama, M. N., Asymptotic conditions for periodic solutions of forced Liénard differential equations with jumping nonlinearities (preprint).Google Scholar
28. Protter, M. H. et Weinberger, H. F., Maximum principles in differential equations (Prentice Hall, Inc., Englewoods Cliffs, New Jersey, 1967).Google Scholar
29. Reissig, R., Contractive mappings and periodically perturbed non-conservative systems, Lincei-Rend. Se. fis. mat. e. nat. 58 (1975), 696–702.Google Scholar
30. Ruf, B., Remarks and generalizations related to a recent multiplicity result of A. Lazer and P. McKenna, Publications du Laboratoire d'Analyse numérique, Université Pierre et Marie Curie (Paris VI), Vol. 2, fasc. 7 (1983), (preprint).Google Scholar