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Stability in neutral nonlinear differential equations with functional delays using fixed-point theory. (English) Zbl 1083.34536

The paper deals with the stability of the zero solution of the scalar neutral differential equation

${x}^{\text{'}}\left(t\right)=-a\left(t\right)x\left(t\right)+c\left(t\right){x}^{\text{'}}\left(t-g\left(t\right)\right)+q\left(t,x\left(t\right),x\left(t-g\left(t\right)\right)\right),$

where $a$, $b$, $g$ and $q$ are continuous functions of their arguments. Noting that the construction of a Lyapunov functional solving this problem is an open problem (the difficulties that arise are illustrated by the case $q\equiv 0$), the author gets sufficient conditions for the stability of the zero solution on the base of the contraction mapping principle applied to the equivalent Volterra-type integral equation. Both bounded and unbounded delays are considered and the obtained results are illustrated by examples.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K40 Neutral functional-differential equations