×

The existence and uniqueness of periodic solutions for a kind of Duffing-type equation with two deviating arguments. (English) Zbl 1205.34084

The Duffing-type delay differential equation
\[ x''(t) + f(x(t-\tau(t))) + g(x(t-\gamma(t))) = e(t) \]
is considered, where \(\tau\), \(\gamma\) and \(e\) are real-valued continuous periodic functions with period \(T\), \(\tau'(t) <1\), \(\gamma'(t) < 1\), \(\int^T_0e(t)\,dt = 0\), and \(f\) and \(g\) are \(C^1\) functions satisfying \(f(c) + g(c) \not\equiv e(t)\) for all \(t\) and \(c\). Sufficient conditions are provided to ensure the existence, as well as uniqueness, of \(T\)-periodic solutions and it is claimed that these complement some known results.

MSC:

34K13 Periodic solutions to functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Torres, P. J., Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations, 190, 643-662 (2003) · Zbl 1032.34040
[2] Liu, B. W., Existence and uniqueness of periodic solutions for a kind of Liénard type \(p\)-Laplacian equation, Nonlinear Anal., 69, 724-729 (2008) · Zbl 1161.34022
[3] Liu, B. W.; Huang, L. H., Existence and uniqueness of periodic solutions for a kind of Liénard equation with a deviating argument, Appl. Math. Lett., 21, 56-62 (2008) · Zbl 1136.34331
[4] Liu, X. P.; Jia, M.; Ren, R., On the existence and uniqueness of periodic solutions to a type of Duffing equation with complex deviating argument, Acta Math. Sci., 27, 037-049 (2007), (in Chinese)
[5] Pournakia, M. R.; Razanib, A., On the existence of periodic solutions for a class of generalized forced Lienard equations, Appl. Math. Lett., 20, 248-254 (2007) · Zbl 1122.34313
[6] Walther, H. O., A periodic solution of a differential equation with state-dependent delay, J. Differential Equations, 244, 1910-1945 (2008) · Zbl 1146.34048
[7] Li, X. J.; Lu, S. P., Periodic solutions for a kind of high-order \(p\)-Laplacian differential equation with sign-changing coefficient ahead of nonlinear term, Nonlinear Anal., 70, 1011-1022 (2009) · Zbl 1161.34331
[8] Lu, S. P., Existence of periodic solutions to a \(p\)-Laplacian Liénard differential equation with a deviating argument, Nonlinear Anal., 68, 1453-1461 (2008) · Zbl 1139.34316
[9] Yang, Z. H.; Cao, J. D., Periodic solutions for general nonlinear state-dependent delay logistic equations, Nonlinear Anal., 66, 1378-1387 (2007) · Zbl 1119.34051
[10] Wang, Z. X.; Lu, S. P.; Cao, J. D., Existence of periodic solutions for a \(p\)-Laplacian neutral functional differential equation with multiple variable parameters, Nonlinear Anal., 72, 734-747 (2010) · Zbl 1182.34092
[11] Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer: Springer Berlin · Zbl 0326.34021
[12] Lu, S. P.; Ge, W. G., Periodic solutions for a kind of second order differential equation with multiple deviating arguments, Appl. Math. Comput., 146, 195-209 (2003) · Zbl 1037.34065
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.