Inequalities and stability for a linear scalar functional differential equation
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 \[ | x(t)| = O\Biggl(\exp\biggl(\frac{1}{2}\int_{t_0}^{t-1/2} (a(s) + b(s+1)) \,ds\biggr)\Biggr), \] if \(-1/2 \leq a(t) + b(t+1) \leq -b^2(t+1)\) for all \(t\). Lower bounds are also derived. The proofs use Lyapunov functionals.
scientific article; zbMATH DE number 2117725