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Dichotomy results for delay differential equations with negative Schwarzian derivative. (English) Zbl 1207.34093

A C 3 -map f of a closed interval a,b into itself is called an SU-map if it has a unique critical point x 0 such that f ' x>0 for x<x 0 , f ' x<0 for x>x 0 , and Sfx<0 for all xx 0 . The authors establish a dichotomy result for SU-maps with negative Schwarzian derivative which is then applied to several classes of functional differential equations including Wright and Mackey-Glass delay differential equations. In particular, easily computable bounds for the global attractor of a delay differential equation

x ' t=-axt+fxt-τ,(1)

where a0, τ>0, and f is a continuous function, are obtained. This nice paper concludes with an interesting conjecture for (1), a discussion of related results and open problems.

MSC:
34K25Asymptotic theory of functional-differential equations
34K19Invariant manifolds (functional-differential equations)
34K12Growth, boundedness, comparison of solutions of functional-differential equations
34D09Dichotomy, trichotomy
34D45Attractors
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