Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations

<p style='text-indent:20px;'>In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for general iterative functional differential equation <inline-formula><tex-math id="M1">\begin{document}$ \dot{x}(t) = f(t,x(t),x^{[2]}(t),...,x^{[n]}(t)). $\end{document}</tex-math></inline-formula> As id="M2">\begin{document}$ n 2 $\end{document}</tex-math></inline-formula>, can be regarded as a mixed-type with state-dependence id="M3">\begin{document}$ f(t,x(t),x(T(t,x(t)))) special form but, being nonlinear operator, id="M4">\begin{document}$ $\end{document}</tex-math></inline-formula>-th order iteration makes more difficulties in estimation than usual state-dependence. Then apply our results to boundedness, asymptotics equation. Finally, present two concrete examples show these types equations respectively.</p>