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Inequalities and stability for a linear scalar functional differential equation. (English) Zbl 1064.34062
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.

MSC:
34K20Stability theory of functional-differential equations
34K06Linear functional-differential equations
34K12Growth, boundedness, comparison of solutions of functional-differential equations
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References:
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