Solution estimates and stability tests for linear neutral differential equations

Explicit exponential stability tests are obtained for the scalar neutral differential equation $$\dot{x}(t)-a(t)\dot{x}(g(t))=-\sum _{k=1}^m{b}_k(t)x(h_k(t)),$$ together with exponential estimates for its solutions. Estimates for solutions of a non-homogeneous neutral equation are also obtained, they are valid on every finite segment, thus describing both asymptotic and transient behavior. For neutral differential equations, exponential estimates are obtained here for the first time. Both the coefficients and the delays are assumed to be measurable, not necessarily continuous functions.