On a criterion for the stability of a scalar equation with constant delay and periodical coefficient (Q1278061)

The author deals with the stability of the scalar equation \[ \dot x(t)-a(t)x(t-\omega)=f(t),\quad t>0,\qquad x(\xi)=0,\quad \xi<0,\quad x(0)=\nu, \] with a constant delay \(\omega>0\) and an \(m\omega\)-periodic coefficient \(a(\cdot)\). The stability is understood in the sense that the solutions to the considered scalar equation are bounded for all bounded and continuous functions \(f\). It is shown by \textit{Z. I. Rekhlitskij} [Izv. Akad. Nauk SSSR 30, No. 5, 971-974 (1966; Zbl 0207.39703)], that this equation is stable if and only if all complex roots \(z_\theta\) for all \(\theta\in[0,\omega]\) of certain equation \(\Delta_m(\theta,z)=0\) lie outside the unit circle. Without writing the explicit expression of \(\Delta_m(\theta,z)\) (given in Rekhlitskij's paper) it is to be noted that \(\Delta_m(\theta,z)\) is a determinant of order \(m\) whose elements are entire functions. Therefore the verification of the Rekhlitskij criterion for a concrete equation is a complicated task. A related criterion for the stability in terms of the Cauchy function \(C(t,s)\) for the respective homogeneous equation gives \textit{V. V. Malygina} and \textit{V. A. Sokolov} [Boundary value problems, Interuniv. Collect. sci. Works, Perm' 1983, 28-32 (1983; Zbl 0598.34056)]. The present paper gives in the cases \(m=2\) and \(m=3\) an explicit expression for \(\Delta_m(z,s)\), shows that in fact this function does not depend on \(s\), and reproves Rekhlitskij and Malygina criteria.