Feng, Meiqiang Periodic solutions and nontrivial periodic solutions for a class of Rayleigh-type equation with two deviating arguments. (English) Zbl 1292.34067 J. Funct. Spaces Appl. 2013, Article ID 414901, 7 p. (2013). Summary: The Rayleigh equation with two deviating arguments \[ x''(t)+f(x'(t))+g_1(t,x(t-\tau_1(t)))+g_2(t,x(t-\tau_2(t)))=e(t) \] is studied. By using the Leray-Schauder index theorem and the Leray-Schauder fixed point theorem, we obtain some new results on the existence of periodic solutions, especially for the existence of nontrivial periodic solutions to this equation. The results are illustrated with two examples, which cannot be handled using the existing results. Cited in 1 Document MSC: 34K13 Periodic solutions to functional-differential equations 47N20 Applications of operator theory to differential and integral equations Keywords:fixed point theorem; fixed point index PDFBibTeX XMLCite \textit{M. Feng}, J. Funct. Spaces Appl. 2013, Article ID 414901, 7 p. (2013; Zbl 1292.34067) Full Text: DOI References: [1] G. J. Ji, Z. X. Wang, and D. W. Lai, “Existence of periodic solutions to overvoltage models in electric power systems,” Acta Mathematica Scientia A, vol. 16, no. 1, pp. 99-104, 1996 (Chinese). · Zbl 0979.34514 [2] Z. Wang and D. Lai, “A delay differential equation appeared in the study of overvoltage,” in Report to the Italian Symposium, vol. 12, 1984. [3] C. H. Feng, “Existence and uniqueness of almost periodic solutions for some delayed differential equations appearing in a power system,” Acta Mathematica Sinica, vol. 46, no. 5, pp. 931-936, 2003 (Chinese). · Zbl 1048.34120 [4] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional-Differential Equations, vol. 178, Academic Press, Orlando, Fla, USA, 1985. · Zbl 0635.34001 [5] J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 1977. · Zbl 0352.34001 [6] T. Yashizaw, “Asymptotic behavior of solutions of differential equations,” in Differential Equation: Qualitative Theory (Szeged, 1984), vol. 47 of Colloquia mathematica Societatis János Bolyai, pp. 1141-1172, North-Holland, Amsterdam, 1987. [7] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, NY, USA, 1993. · Zbl 0777.34002 [8] L. Peng, B. Liu, Q. Zhou, and L. Huang, “Periodic solutions for a kind of Rayleigh equation with two deviating arguments,” Journal of the Franklin Institute, vol. 343, no. 7, pp. 676-687, 2006. · Zbl 1114.34051 [9] C. Huang, Y. He, L. Huang, and W. Tan, “New results on the periodic solutions for a kind of Rayleigh equation with two deviating arguments,” Mathematical and Computer Modelling, vol. 46, no. 5-6, pp. 604-611, 2007. · Zbl 1161.34345 [10] B. Liu, “Existence and uniqueness of periodic solutions for a kind of Rayleigh equation with two deviating arguments,” Computers & Mathematics with Applications, vol. 55, no. 9, pp. 2108-2117, 2008. · Zbl 1145.34038 [11] E. Zeidler, Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems, Springer, New York, NY, USA, 1986. · Zbl 0583.47050 [12] D. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Ji Nan, China, 1985. [13] S. Lu and W. Ge, “Periodic solutions for a kind of second order differential equations with multiple with deviating arguments,” Applied Mathematics and Computation, vol. 146, pp. 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.