A global stability criterion for scalar functional differential equations

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We consider scalar delay differential equations x'(t) = -δx(t) + f(t,xₜ) (*) 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, equations with maxima are kept 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 Smith's conjecture about the equivalence of global and local asymptotical stability in this population model.