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Global asymptotic stability for a higher order nonlinear rational difference equations. (English) Zbl 1110.39011

Consider the rational difference equation

${x}_{n+1}=\frac{a{x}_{n}+b{x}_{n-k}}{A+B{x}_{n}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{4pt}{0ex}}n=0,1,\cdots \phantom{\rule{2.em}{0ex}}\left(*\right)$

where $a\phantom{\rule{3.33333pt}{0ex}},b\phantom{\rule{3.33333pt}{0ex}},A,B$ are positive real numbers, $k\ge 1$ is a positive integer, and the initial conditions ${x}_{-k},\phantom{\rule{3.33333pt}{0ex}}\cdots ,\phantom{\rule{3.33333pt}{0ex}}{x}_{-1},\phantom{\rule{3.33333pt}{0ex}}{x}_{0}$ are nonnegative real numbers. The authors solve an open problem posed by M. R. S. Kulenović and G. Ladas [Dynamics of second order rational difference equations, Chapman & Hall / CRC, Boca Raton, FL (2002; Zbl 0981.39011), p. 129]. They prove the following

Theorem: (a) If $b\le A-a,$ then the zero equilibrium of Eq. ($*$) is globally asymptotically stable. (b) If $A-a then the positive equilibrium of Eq. ($*$) is globally asymptotically stable.

The boundedness, periodic character, invariant intervals of all nonnegative solutions of ($*$) are investigated.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Generalized difference equations