Generalized Halanay inequalities for dissipativity of Volterra functional differential equations (Q936581)

Let \(a> b> 0\), let \(y(t)\) satisfy \[ y'(t)\leq- ay(t)+ b \sup_{t-\tau\leq s\leq t} y(s),\quad t\geq t_0,\;\tau> 0. \] Then there exist \(\gamma> 0\), \(k> 0\) such that \(y(t)\leq ke^{-\gamma(t- t_0)}\), \(t\geq t_0\). The authors extend this well-known result to various cases which allow for \(t\)-dependent \(a\), \(b\). Applications to the asymptotics of various equations of Volterra type are given.