Annulus arguments in the stability theory for functional differential equations.(English)Zbl 0897.34060

The paper is concerned with stability properties of functional-differential equations. The technique of the proofs is based on the method of Lyapunov functionals. This method is combined with the annulus argument – the method which can detect that a curve in $$\mathbb{R}^n$$ crosses an annulus around the origin infinitely many times. The author modifies this method and gives annulus arguments not requiring the boundedness of the derivatives of the functions involved. Using these results, the author establishes several Lyapunov type theorems for the attractivity, asymptotic $$D$$-stability and partial stability of the zero solution of a nonautonomous functional-differential equation $$x'(t)=f(t,x_t)$$. The right-hand side of this equation is not assumed to be bounded with respect to $$t$$. The results are applied to the scalar equation with one constant delay $x'(t)=-c(t)x(t)+b(t)x(t-h) \qquad (c(t)\geq 0),$ to the scalar equation with several constant delays $x'(t)=-c(t)x(t)+\sum _{i=1}^{n} b_i(t)x(t-h_i) \qquad (c(t)\geq 0)$ and to the system $x'(t)=B(t)x(t-h)-C(t)x(t),$ where $$B(t)$$ and $$C(t)$$ are continuous matrix functions. The results generalize and improve several results of S. N. Busenberg and K. L. Cooke [Q. Appl. Math. 42, 295-306 (1984; Zbl 0558.34059)].
Reviewer: J.Kalas (Brno)

MSC:

 34K20 Stability theory of functional-differential equations 34D20 Stability of solutions to ordinary differential equations

Zbl 0558.34059