On some nonlinear ordinary differential equations with advanced arguments. (Q1868020)

The authors consider the following nonlinear differential equation with advanced argument \[ y'(t)= [y(\beta t)]^{1/\beta},\tag{1} \] with \(t\geq 0\) and \(\beta> 1\). By making use of the technique of lower and upper solutions, they classify the solutions of (1) (those that satisfy the initial condition \(y(0)= y_0\) and a certain growth condition) with respect to their growth. They consider the analytic solutions of the Cauchy problem. More precisely, the authors prove that if \(y_0> 0\) and \(\beta> 1\) then there exist analytic solutions. They obtain a characterization of these solutions, too.