Geodesic
Study of Stability in Nonlinear Neutral Differential Equations with Variable Delay Using Krasnoselskii–Burton’s Fixed Point
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 55 (2016) no. 2, pp. 129-142.

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In this paper, we use a modification of Krasnoselskii’s fixed point theorem introduced by Burton (see [Burton, T. A.: Liapunov functionals, fixed points and stability by Krasnoseskii’s theorem. Nonlinear Stud., 9 (2002), 181–190.] Theorem 3) to obtain stability results of the zero solution of the totally nonlinear neutral differential equation with variable delay \[ x^{\prime }\left( t\right) =-a\left( t\right) h\left( x\left( t\right) \right) +\frac{d}{dt}Q\left( t,x\left( t-\tau \left( t\right) \right) \right) +G\left( t,x\left( t\right) ,x\left( t-\tau \left( t\right) \right) \right) . \] The stability of the zero solution of this eqution provided that $h\left(0\right) =Q\left( t,0\right) =G\left( t,0,0\right) =0$. The Caratheodory condition is used for the functions $Q$ and $G$.
Classification : 34K20, 34K30, 34K40, 47H10
Keywords: Fixed point; stability; delay; stability; nonlinear neutral equation; large contraction mapping; integral equation 
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MESMOULI, Mouataz Billah; Ardjouni, Abdelouaheb; Djoudi, Ahcene. Study of Stability in Nonlinear Neutral Differential Equations with Variable Delay Using Krasnoselskii–Burton’s Fixed Point. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 55 (2016) no. 2, pp. 129-142. http://geodesic.mathdoc.fr/item/AUPO_2016__55_2_a10/