Stability in neutral nonlinear differential equations with functional delays using fixed-point theory.

*(English)*Zbl 1083.34536The paper deals with the stability of the zero solution of the scalar neutral differential equation

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

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.

Reviewer: Ivan Ginchev (Varna)