A global stability criterion in nonautonomous delay differential equations

Consider the following nonautonomous nonlinear delay differential equation:{dy(t)dt=−a(t)y(t)−∑i=0mai(t)gi(y(τi(t))),t⩾t0,y(t)=ϕ(t),t⩽t0, where we assume that there is a strictly monotone increasing function f(x)f(x) on (−∞,+∞)(−∞,+∞) such that{f(0)=0,0