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

In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for the general iterative functional differential equation \begin{document}$ \dot{x}(t) = f(t,x(t),x^{[2]}(t),...,x^{[n]}(t)). $\end{document} As \begin{document}$ n = 2 $\end{document} , this equation can be regarded as a mixed-type functional differential equation with state-dependence \begin{document}$ \dot{x}(t) = f(t,x(t),x(T(t,x(t)))) $\end{document} of a special form but, being a nonlinear operator, \begin{document}$ n $\end{document} -th order iteration makes more difficulties in estimation than usual state-dependence. Then we apply our results to the existence, uniqueness, boundedness, asymptotics and continuous dependence of solutions for the mixed-type functional differential equation. Finally, we present two concrete examples to show the boundedness and asymptotics of solutions to these two types of equations respectively.