The exact boundaries of the stability domains of linear differential equations with distributed delay (Q735863)

This paper addresses the following linear differential equation with distributed varying delay \[ \dot{x}(t) + \int_{t-r(t)}^t x(s) ds = f(t), \quad t\in \mathbb{R}_+;\qquad x(\xi)=0, \text{ for } \xi<0. \tag{1} \] Stability criteria similar to criteria introduced by \textit{A.~D.~Myshkis} [Mat. Sbor. 28, No. 3, 641--658 (1951; Zbl 0054.03901)] for equations with concentrated varying delay are studied. The main result is the following theorem. Theorem. Let \(0<\inf_{t\in \mathbb{R}_+} r(t) \leq \sup_{t\in \mathbb{R}_+} r(t) <2\). Then equation (1) is exponentially stable. Stability criteria for some generalization (when integral part of equation (1) has a parameter) of equation (1) are also formulated.