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.


34K20 Stability theory of functional-differential equations
34K40 Neutral functional-differential equations
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