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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 ${f}^{\text{'}}\left(x\right)>0$ for $x<{x}_{0}$, ${f}^{\text{'}}\left(x\right)<0$ for $x>{x}_{0}$, and $\left(Sf\right)\left(x\right)<0$ for all $x\ne {x}_{0}$. 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

${x}^{\text{'}}\left(t\right)=-ax\left(t\right)+f\left(x\left(t-\tau \right)\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $a\ge 0,$ $\tau >0,$ 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.

##### MSC:
 34K25 Asymptotic theory of functional-differential equations 34K19 Invariant manifolds (functional-differential equations) 34K12 Growth, boundedness, comparison of solutions of functional-differential equations 34D09 Dichotomy, trichotomy 34D45 Attractors