## 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 $x'(t)=-a(t) x(t)+c(t) x'(t-g(t))+q(t,x(t),x(t-g(t))),$ 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
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### References:

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