Nonnegative periodic solutions for a totally nonlinear iterative differential equation with variable delay (Q6545023)

In this paper, the authors apply a fixed point theorem for the so-called large contractions due to \textit{T. A. Burton} [Proc. Am. Math. Soc. 124, No. 8, 2383--2390 (1996; Zbl 0873.45003)] to provide conditions for the existence of nonnegative periodic solutions of a nonlinear differential equation of the form \N\[\N\omega'(t)=-p(t)g(\omega(t))+q(t)\omega^m(t-r(t))f(t, \omega^{[1]}(t),\cdots,\omega^{[n]}(t)),\N\] \Nwhere \(\omega^{[i]}\) is the iterate \(\omega^{[i]}=\omega(\omega^{[i-1]}),\) \(i=1, 2, \cdots.\) Here \(p,q,r\) are positive continuous periodic functions with the same period, \(f\in C(\mathbb{R}\times\mathbb{R}^n,\mathbb{R}^n)\) and \(g\in C(\mathbb{R},\mathbb{R})\), with \(g(0)=0\) and \(m\geq 0.\) The key is to transform the problem into an integral equivalent equation and write it as the suitable sum of a compact continuous mapping and of a large contraction.