×

Boundedness, periodicity, and convergence of solutions in a retarded Liénard equation. (English) Zbl 0803.34064

From authors’ introduction: “The paper deals with the retarded Liénard equation \[ x''+ f(x)x'+ g(x(t- h))= e(t),{(*)} \] where \(h\geq 0\) is the delay and \(e\) is a bounded function. With appropriate assumptions on \(f\) and \(g\) the authors obtain necessary and sufficient conditions for solutions of \((*)\) to be uniformly ultimately bounded. Thus by a fixed point theorem, those conditions imply that \((*)\) has a \(T\)-periodic solution whenever \(e\) is \(T\)-periodic. Also are given conditions under which all solutions of \((*)\) converge.
Reviewer: W.M.Oliva (Lisboa)

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C11 Growth and boundedness of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34D40 Ultimate boundedness (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Burton, T. A., The generalized Liénard equation, SLAM J. Control Ser. A, 3, 223-230 (1965) · Zbl 0135.30201
[2] Burton, T. A., Stability and Periodic Solutions of Ordinary and Functional Differential Equations (1985), Orlando, Florida: Academic Press, Orlando, Florida · Zbl 0635.34001
[3] Burton, T. A.; Hatvani, L., Stability theorems for nonautonomous functional differential equations by Liapunov functionals, Tôhoku Math. J., 41, 65-104 (1989) · Zbl 0677.34060
[4] Burton, T. A.; Townsend, C. G., On the generalized Liénard equation with forcing function, J. Diff. Eq., 4, 620-633 (1968) · Zbl 0174.13602
[5] Burton, T. A.; Zhang, Bo, Uniform ultimate boundedness and periodicity in functional differential equations, Tôhoku Math. J., 42, 93-100 (1990) · Zbl 0678.34073
[6] Graef, J. R., On the generalized Liénard equation with negative damping, J. Diff. Eq., 12, 34-62 (1972) · Zbl 0254.34038
[7] Hale, J. K.; Lopes, O., Fixed point theorems and dissipative processes, J. Diff. Eq., 13, 391-402 (1973) · Zbl 0256.34069
[8] Hara, T.; Yoneyama, T., On the global center of generalized Liénard equation and its application to stability problems, Funkeialaj Ekvacioj, 28, 171-192 (1985) · Zbl 0585.34038
[9] Krasovskii, N. N., Stability of Motion (1963), Stanford, California: Stanford University Press, Stanford, California · Zbl 0109.06001
[10] Murakami, S., Asymptotic behavior of solutions of some differential equations, J. Math. Anal. Appl., 109, 534-545 (1985) · Zbl 0594.34077
[11] Sansone, G.; Conti, R., Non-linear Differential Equations (1964), New York: MacMillan, New York · Zbl 0128.08403
[12] Somolinos, A., Periodic solutions of the sunflower equation, Quart. Appl. Math., 35, 465-478 (1978) · Zbl 0385.34017
[13] Sugie, J., On the generalized Liénard equation without the Signum condition, J. Math. Anal. Appl., 128, 80-91 (1987) · Zbl 0643.34043
[14] Sugie, J., On the boundedness of solutions of the generalized Liénard equation without the Signum condition, Nonlinear Analysis, 11, 1391-1397 (1987) · Zbl 0648.34036
[15] Villari, G., On the qualitative behaviour of solutions of Liénard equation, J. Diff. Eq., 67, 269-277 (1987) · Zbl 0613.34031
[16] Villari, G.; Zanolin, F., On a dynamical system in the Liénard plane. Necessary and sufficient conditions for the intersection with the vertical isocline and application, Funcialaj Ekvacioj, 33, 19-38 (1990) · Zbl 0731.34049
[17] Waltman, P.; Bridgland, T. F., On convergence of solutions of the forced Liénard equation, J. Math. Phys., 44, 284-287 (1965) · Zbl 0137.28301
[18] Yoshizawa, T., Asymptotic behavior of solutions of differential equations, Differential Equations qualitative Theory (Szeged, 1984), 1141-1172 (1984), Amsterdam: North-Holland, Amsterdam
[19] BoZhang,On the retarded Liénard equation, Proc. Amer. Math. Soc., in press. · Zbl 0756.34075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.