Geodesic
Existence and Stability of Periodic Solutions for Nonlinear Neutral Differential Equations with Variable Delay Using Fixed Point Technique
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 54 (2015) no. 1, pp. 95-108 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.].
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.].
Classification : 34K20, 34K30, 34K40, 45D05, 45J05, 47H10
Keywords: Fixed point theorem; contraction; compactness; neutral differential equation; integral equation; periodic solution; positive solution; stability 
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     title = {Existence and {Stability} of {Periodic} {Solutions} for {Nonlinear} {Neutral} {Differential} {Equations} with {Variable} {Delay} {Using} {Fixed} {Point} {Technique}},
     journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
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MESMOULI, Mouataz Billah; Ardjouni, Abdelouaheb; Djoudi, Ahcene. Existence and Stability of Periodic Solutions for Nonlinear Neutral Differential Equations with Variable Delay Using Fixed Point Technique. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 54 (2015) no. 1, pp. 95-108. http://geodesic.mathdoc.fr/item/AUPO_2015_54_1_a6/