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Zbl 1064.34062
Wang, Tingxiu
Inequalities and stability for a linear scalar functional differential equation. (English)
[J] J. Math. Anal. Appl. 298, No. 1, 33-44 (2004). ISSN 0022-247X

The paper is concerned with the stability of the zero solution for the nonautonomous linear scalar functional-differential equation with one fixed delay $$x'(t) = a(t)x(t) + b(t)x(t-h) \quad\text{for }t > t_0,$$ addressing the cases where $a(t)$ may have variable sign or $b(t)$ may be unbounded. A typical result, assuming $h = 1$ without loss of generality, implies that $$\vert x(t)\vert = O\Biggl(\exp\biggl(\frac{1}{2}\int_{t_0}^{t-1/2} (a(s) + b(s+1)) \,ds\biggr)\Biggr),$$ if $-1/2 \le a(t) + b(t+1) \le -b^2(t+1)$ for all $t$. Lower bounds are also derived. The proofs use Lyapunov functionals.
[Hans Engler (Bonn)]

MSC 2000:: *34K20 Stability theory of functional-differential equations
34K06 Linear functional-differential equations
34K12 Properties of solutions of functional-differential equations

Keywords: functional-differential equation; stability; Lyapunov functional

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