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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 $\left[ a,b\right] $ into itself is called an $SU$-map if it has a unique critical point $x_{0}$ such that $f^{\prime}\left( x\right) >0$ for $x<x_{0}$, $f^{\prime}\left( x\right) <0$ for $x>x_{0}$, and $\left( Sf\right) \left( x\right) <0$ for all $x\neq x_{0}$. 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 $$x^{\prime}\left( t\right) =-ax\left( t\right) +f\left( x\left( t-\tau\right) \right),\tag1$$ where $a\geq0,$ $\tau>0,$ 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.

34K25Asymptotic theory of functional-differential equations
34K19Invariant manifolds (functional-differential equations)
34K12Growth, boundedness, comparison of solutions of functional-differential equations
34D09Dichotomy, trichotomy
Full Text: DOI arXiv
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