# Article

 Title: Existence and Stability of Periodic Solutions for Nonlinear Neutral Differential Equations with Variable Delay Using Fixed Point Technique (English) Author: MESMOULI, Mouataz Billah Author: Ardjouni, Abdelouaheb Author: Djoudi, Ahcene Language: English Journal: Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica ISSN: 0231-9721 Volume: 54 Issue: 1 Year: 2015 Pages: 95-108 Summary lang: English . Category: math . Summary: Our paper deals with the following nonlinear neutral differential equation with variable delay $\frac{d}{dt}Du_{t}(t) =p (t)-a(t)u (t)-a(t) g(u(t-\tau (t))) -h (u(t) ,u (t-\tau (t))) .$ By using Krasnoselskii’s fixed point theorem we obtain the existence of periodic solution and by contraction mapping principle we obtain the uniqueness. A sufficient condition is established for the positivity of the above equation. Stability results of this equation are analyzed. Our results extend and complement some results obtained in the work [Yuan, Y., Guo, Z.: On the existence and stability of periodic solutions for a nonlinear neutral functional differential equation Abstract and Applied Analysis 2013, ID 175479 (2013), 1–8.]. (English) Keyword: Fixed point theorem Keyword: contraction Keyword: compactness Keyword: neutral differential equation Keyword: integral equation Keyword: periodic solution Keyword: positive solution Keyword: stability MSC: 34K20 MSC: 34K30 MSC: 34K40 MSC: 45D05 MSC: 45J05 MSC: 47H10 idZBL: Zbl 1347.34108 idMR: MR3468603 . Date available: 2015-09-01T09:02:29Z Last updated: 2018-01-10 Stable URL: http://hdl.handle.net/10338.dmlcz/144370 . Reference: [1] Ardjouni, A., Djoudi, A.: Fixed points and stability in neutral nonlinear differential equations with variable delays. Nonlinear Anal. 74 (2011), 2062–2070. MR 2781737, 10.1016/j.na.2010.10.050 Reference: [2] Ardjouni, A., Djoudi, A.: Existence of positive periodic solutions for a nonlinear neutral differential equation with variable delay. Applied Mathematics E-Notes 2012 (2012), 94–101. Zbl 1254.34098, MR 2988223 Reference: [3] Burton, T. A.: Liapunov functionals, fixed points, and stability by Krasnoselskii’s theorem. Nonlinear Studies 9 (2001), 181–190. MR 1898587 Reference: [4] Burton, T. A.: Stability by fixed point theory or Liapunov’s theory: A comparison. Fixed Point Theory 4 (2003), 15–32. MR 2031819 Reference: [5] Burton, T. A.: Fixed points and stability of a nonconvolution equation. Proc. Amer. Math. Soc. 132 (2004), 3679–3687. Zbl 1050.34110, MR 2084091, 10.1090/S0002-9939-04-07497-0 Reference: [6] Burton, T. A.: Stability by Fixed Point Theory for Functional Differential Equations. Dover Publications, New York, 2006. Zbl 1160.34001, MR 2281958 Reference: [7] Ding, L., Li, Z.: Periodicity and stability in neutral equations by Krasnoselskii’s fixed point theorem. Nonlinear Analysis: Real World Applications 11, 3 (2010), 1220–1228. Zbl 1206.34091, MR 2646539 Reference: [8] Hatvani, L.: Annulus arguments in the stability theory for functional differential equations. Differential and Integral Equations 10 (1997), 975–1002. Zbl 0897.34060, MR 1741762 Reference: [9] Kolmanovskii, V. B., Nosov, V. R.: Stability of functional differential equations. Mathematics in Science and Engineering 180, Academic Press, London, 1986. MR 0860947 Reference: [10] Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Mathematics in Science and Engineering, 191, Academic Press, Boston, Mass, 1993. Zbl 0777.34002, MR 1218880 Reference: [11] Liu, Z., Li, X., Kang, S., Kwun, Y. C.: Positive periodic solutions for first-order neutral functional differential equations with periodic delays. Abstract and Applied Analysis 2012, ID 185692 (2012), 1–12. Zbl 1245.34073, MR 2922961 Reference: [12] Smart, D. R.: Fixed Points Theorems. Cambridge University Press, Cambridge, 1980. Reference: [13] Yuan, Y., Guo, Z.: On the existence and stability of periodic solutions for a nonlinear neutral functional differential equation. Abstract and Applied Analysis 2013, ID 175479 (2013), 1–8. Zbl 1279.34083, MR 3039158 Reference: [14] Zhang, B.: Fixed points and stability in differential equations with variable delays. Nonlinear Anal. 63 (2005), 233–242. Zbl 1159.34348, 10.1016/j.na.2005.02.081 .

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