On some nonlinear ordinary differential equations with advanced arguments

The paper deals with ODEs with an advanced argument, in particular, with such equations
as y(t) = (y(t))1=, the simplest example is y(t) =

y(2t). The corresponding initial-value
problem with y(0) = y0¿0 has three types of solutions: (a) a unique analytic solution, (b)
solutions of subexponential growth, and (c) for every ¿0 a solution that behaves like et as
t →∞. In the case y0 = 0 there is (besides y ≡ 0) a positive solution, for some ¿1 it can
be analytic, but the above example  = 2 admits in9nitely many analytic solutions.
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