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On some nonlinear ordinary differential equations with advanced arguments. (English) Zbl 1069.34095
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.

MSC:
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
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[1] Błaz, J.; Walter, W., Über funktional-differentialgleichungen mit voreilendem argument, Monatsh. math., 82, 1-16, (1976) · Zbl 0339.34069
[2] Doss, S.; Nasr, S.K., On the functional equation dy/dx=f(x,y,y(x+h)), h>0, Amer. J. math., 75, 713-716, (1953) · Zbl 0053.06101
[3] Hale, J., Theory of functional differential equations, (1977), Springer New York, Heidelberg, Berlin
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