Explicit stability tests for linear neutral delay equations using infinite series

Abstract

We obtain new explicit exponential stability conditions for the linear scalar neutral equation with two bounded delays $(x(t)-a(t)x(g(t)){)}^{\prime }+b(t)x(h(t))=0,$ where $|a(t)|\le A_0<1$, $0<b_0\le b(t)\le B_0$, assuming that all parameters of the equation are measurable functions. To analyze exponential stability, we apply the Bohl-Perron theorem and a reduction of a neutral equation to an equation with an infinite number of non-neutral delay terms. This method has never been used before for this neutral equation; its application allowed to omit a usual restriction $|a(t)|<\frac{1}{2}$ in known asymptotic stability tests and consider variable delays.

11 pages, one figure. Accepted to Rocky Mountain Journal of Mathematics