A Global Stability Criterion for Scalar Functional Differential Equations

We consider scalar delay differential equations $x'(t) = -\delta x(t) + f(t,x_t) \ (*)$ with nonlinear f satisfying a sort of negative feedback condition combined with a boundedness condition. The well-known Mackey--Glass-type equations, equations satisfying the Yorke condition, and equations with maxima all fall within our considerations. Here, we establish a criterion for the global asymptotical stability of a unique steady state to (*). As an example, we study Nicholson's blowflies equation, where our computations support the Smith conjecture about the equivalence between global and local asymptotical stabilities in this population model.