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

Jun Zhou, Jun Shen
2021 Discrete and continuous dynamical systems. Series B
<p style='text-indent:20px;'>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 <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 <inline-formula><tex-math id="M2">\begin{document}$n = 2$\end{document}</tex-math></inline-formula>, this equation can be regarded
more » ... s a mixed-type functional differential equation with state-dependence <inline-formula><tex-math id="M3">\begin{document}$\dot{x}(t) = f(t,x(t),x(T(t,x(t))))$\end{document}</tex-math></inline-formula> of a special form but, being a nonlinear operator, <inline-formula><tex-math id="M4">\begin{document}$n$\end{document}</tex-math></inline-formula>-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.</p>