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Keywords:
Fixed point; stability; delay; stability; nonlinear neutral equation; large contraction mapping; integral equation
Fixed point; stability; delay; stability; nonlinear neutral equation; large contraction mapping; integral equation
Summary:
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 $G$.
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