New Results on the Existence of Periodic Solutions for Rayleigh Equation with State-Dependent Delay
DOI:
https://doi.org/10.5614/j.math.fund.sci.2013.45.2.5Keywords:
Rayleigh equation, non-linear, second order, state-dependent delay, periodic solution.Abstract
We give certain sufficient conditions for the existence of periodic solutions to a Rayleigh-type equation with state-dependent delay. With this work we extend and improve some known results in the literature.References
Lu, S.P., Ge, W.G. & Zheng, Z.X., Periodic Solutions for a Kind of Rayleigh Equation with A Deviating Argument, Appl. Math. Lett., 17, pp. 443-449, 2004.
Wang, G.Q. & Yan, J.R., Existence Theorem of Periodic Positive Solutions for The Rayleigh Equation of Retarded Type, Portugal. Math., 57(2), pp. 153-160, 2000.
Alzabut, J. & Tun, C., Existence of Periodic Solutions for A Type of Rayleigh Equation with State-Dependent Delay, Electron. J. Diff. Equ., 2012(77), pp. 1-8, 2012.
Deimling, K., Nonlinear Functional Analysis, Springer, Berlin, 1985.
Strutt, J.W. [Lord Rayleigh], Theory of Sound, 1, Dover Publications, New York (1877) re-issued 1945.
Wang, Y. & Zhang, L., Existence of Asymptotically Stable Periodic Solutions of A Rayleigh Type Equation, Nonlinear Anal., 71(5-6), pp.1728-1735, 2009.
Gains, R.E. & Mawhin, J.L., Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics, 568, Springer, Berlin, 1977.
Chen, F.D., Existence and Uniqueness of Almost Periodic Solutions for Forced Rayleigh Equations, Ann. Differential Equations 17(1), pp. 1-9, 2001.
Chen, F.D., Chen, X.X., Lin, F.X. & Shi, J.L., Periodic Solution and Global Attractivity of A Class of Differential Equations with Delays, Acta Math. Appl. Sinica, 28(1), pp.55-64, 2005.
Liu, F., On The Existence of The Periodic Solutions of Rayleigh Equation, Acta Math. Sinica, 37(5), pp. 639-644, 1994.
Lu, S.P. & Ge, W.G., Some New Results on the Existence of Periodic Solutions to A Kind of Rayleigh Equation with A Deviating Argument, Nonlinear Anal., 56, pp. 501-514, 2004.
Lu, S.P., Ge, W.G. & Zheng, Z.X., Periodic Solutions for A Kind of Raleigh Equation with A Deviating Argument, Acta Math. Sinica, 47(2), pp. 299-304, 2004.
Tun, C. & Sirma, A., Stability Analysis of A Class of Generalized Neutral Equations, J. Comput. Anal. Appl., 12(4), pp. 754-759, 2010.
Zhou, Y. & Tang, X., On Existence of Periodic Solutions of Rayleigh Equation of Retarded Type, J. Comput. Appl. Math., 203(1), pp. 1-5, 2007.
Wang, G.Q. & Cheng, S.S., A Priori Bounds for Periodic Solutions of A Delay Rayleigh Equation, Appl. Math. Lett., 12, pp. 41-44, 1999.
Wang, G.Q. & Yan, J.R., On Existence of Periodic Solutions of The Rayleigh Equation of Retarded Type, Internat. J. Math. Math. Sci., 23(1), pp. 65-68, 2000.
