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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)fort>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

|x(t)|=Oexp 1 2 t 0 t-1/2 (a(s)+b(s+1)) d s,

if -1/2a(t)+b(t+1)-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