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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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##### References:
 [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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