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On the asymptotics of the difference equation y n (1+y n-1 y n-k+1 )=y n-k . (English) Zbl 1220.39011

The existence of positive solutions of the difference equation

y n =y n-k 1+y n-1 y n-k+1 ,n 0 ,(*)

where k{1}, converging to zero is studied. The main result of this article is the following Theorem: For every k{1}, equation (*) has a positive solution with the following asymptotics

y n =k (k-1)n 1/(k-1) 1+alnn n+bln 2 n n 2 +oln 2 n n 2 ,

where for k=2p+1,

a=2p+1 8p 2 andb=(2p+1) 3 128p 4 ,

while for k=2p+2,

a=p+1 (2p+1) 2 andb=(p+1) 3 (2p+1) 4 ·

MSC:
39A20Generalized difference equations
39A22Growth, boundedness, comparison of solutions (difference equations)