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Periodic solutions of some forced Liénard differential equations at resonance. (English) Zbl 0537.34037
L’A. examine, dans les conditions convenables, l’équation de Liénard $(1)\quad x''(t)+f(x(t)),x'(t)+g(t,x(t))=e(t),\quad t\in [0,2\pi],$ lorsque $$x(0)-x(2\pi)=x'(0)-x'(2\pi)=0$$. On obtient, comme des cas particuliers, certains résultats de Lazer, Reissig, Martelli, Chang, Gupta etc.
Puis, avec le deuxième théorème, on minimise le nombre des conditions suffisantes, par rapport à l’existence des solutions de l’équation (1) - dans les relations aux limites, notées ci-dessus.
Reviewer: S.Manolov

##### MSC:
 34C25 Periodic solutions to ordinary differential equations
##### Keywords:
forced Liénard differential equations; resonance
Full Text:
##### References:
 [1] S. H. Chang, Periodic solutions of certain second order nonlinear differential equations. J. Math. Anal. Appl.49, 263-266 (1975). · Zbl 0294.34032 [2] C. P.Gupta, On functional equations of Fredholm and Hammerstein type with applications to existence of periodic solutions of certain ordinary differential equations. J. Integral Equations, to appear. · Zbl 0457.34040 [3] A. C. Lazer, On Schauder’s fixed point theorem and forced second order nonlinear oscillations. J. Math. Anal. Appl.21, 421-425 (1968). · Zbl 0155.14001 [4] M. Martelli, On forced nonlinear oscillations. J. Math. Anal. Appl.69, 456-504 (1979). · Zbl 0413.34040 [5] M. Martelli andJ. D. Schuur, Periodic solutions of Liénard type second-order ordinary differential equations. Tohoku Math. J.32, 201-207 (1980). · Zbl 0438.34033 [6] J. Mawhin, An extension of a theorem of A. C. Lazer on forced nonlinear equations. J. Math. Anal. Appl.40, 20-29 (1972). · Zbl 0245.34035 [7] J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems. CBMS Conference in Math., Vol.40, Amer. Math. Soc., Providence, R. I., 1970. · Zbl 0203.46201 [8] J.Mawhin and J. R.Ward, Jr., Nonuniform nonresonance conditions at the first two eigenvalues for periodic solutions of forced Liénard and Duffing equations. To appear. [9] R. Reissig, Schwingungssätze für die verallgemeinerte Liénardsche Differentialgleichung. Abh. Math. Sem. Univ. Hamburg44, 45-51 (1975). · Zbl 0323.34033 [10] R. Reissig, Continua of periodic solutions of the Liénard equation, in ?Constructive Methods for Nonlinear Boundary Value Problems and Nonlinear Oscillations?. ISNM48, 126-133, Basel (1979). [11] R. Reissig, Periodic solutions of a second order differential equations including a onesided restoring term. Arch. Math.33, 85-90 (1979). · Zbl 0421.34045 [12] K. Schmitt, Periodic solutions of a forced nonlinear oscillator involving a one sided restoring force. Arch. Math.31, 70-73 (1978). · Zbl 0399.34035 [13] J. R. Ward, Jr., Asymptotic conditions for periodic solutions of ordinary differential equations. Proc. Amer. Math. Soc.81, 415-420 (1981). · Zbl 0461.34029 [14] J. R. Ward, Jr., Periodic solutions for systems of second order ordinary differential equations. J. Math. Anal. Appl.81, 92-98 (1981). · Zbl 0462.34023
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