Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations (Q2090317)

In this paper, the authors state some results on the global existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions of a general iterative functional differential equation \[ \dot x(t) = f (t, x(t), x^{[2]}(t), \dots, x^{[n]}(t)), \] where \(f : \mathbb{R}^{n+1}\to \mathbb{R}\) is a nonlinear function and \(x^{[n]}(t)\) represents the \(n\)-th order iteration of \(x(t)\), i.e. \((x^{[n]} (t) = x(x^{[n-1]}(t))\) for all \(n\in\mathbb{N}\) and \(x^{[0]}(t) = t\). To achieve their goal, they use Krasnoselskii fixed point theorem.