This article considers a neutral type functional differential equation with two incommensurable delays at the derivative, with focusing on stability issues. The invertibility of the operator at the derivative in Lebesgue spaces Lp is investigated, and the location of the roots of its characteristic equation on the complex plane is analyzed.To study the invertibility of the operator at the derivative, the spectrum of the internal superposition operator S is defined, and its description is provided in terms of coefficients of the original equation. The resulting description of the spectrum allows formulating the conditions under which the operator at the derivative is invertible. Further, the invertibility of this operator facilitates the identification of criteria for exponential stability and instability.A connection between the coefficients of the operator S, the type of stability of the original equation, and the invertibility of the operator in arbitrary Lebesgue functional space, as well as the location of roots of the characteristic equation, is established.It has been demonstrated that the presence of roots of the characteristic equation to the right of the imaginary axis is equivalent to the instability of the original neutral type equation and the non-invertibility of the operator at the derivative. Conversely, if all the roots of the characteristic equation are located to the left of the imaginary axis and are separated from it, then the operator at the derivative is invertible, and the neutral type equation is exponentially stable. These conditions have been shown to be effectively verifiable in terms of the coefficients of the original equation. A “critical” case has also been described, when the roots of the characteristic equation lie to the left of the imaginary axis but are not separated from it; specifically, a vertical chain of roots approaches arbitrarily close to the imaginary axis. In this scenario, it is established that the operator at the derivative is non-invertible, and the neutral type equation cannot be exponentially stable.

difference equations, exponential stability, uniform stability, D-decomposition.