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On some systems of difference equations. (English) Zbl 1243.39009

The authors consider the following three systems of nonlinear difference equations

${u}_{n+1}=\frac{{v}_{n}}{1+{v}_{n}},\phantom{\rule{1.em}{0ex}}{v}_{n+1}=\frac{{u}_{n}}{1+{u}_{n}},$
${u}_{n+1}=\frac{{v}_{n}}{1+{u}_{n}},\phantom{\rule{1.em}{0ex}}{v}_{n+1}=\frac{{u}_{n}}{1+{v}_{n}},$

and

${u}_{n+1}=\frac{{u}_{n}}{1+{v}_{n}},\phantom{\rule{1.em}{0ex}}{v}_{n+1}=\frac{{v}_{n}}{1+{u}_{n}},$

where $n\in {ℕ}_{0}$ and the initial values ${u}_{0}$ and ${v}_{0}$ are given complex numbers. By help of the auxiliary Riccati equation

${x}_{n+1}=\frac{{x}_{n}}{a+b{x}_{n}},\phantom{\rule{1.em}{0ex}}a,\phantom{\rule{0.166667em}{0ex}}b\in ℂ,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}n\in {ℕ}_{0},$

the general solution of each system is explicitly given. As a consequence, the asymptotic behaviour of the solutions is analysed in detail as $n\to \infty$.

##### MSC:
 39A20 Generalized difference equations 39A22 Growth, boundedness, comparison of solutions (difference equations) 39A23 Periodic solutions (difference equations) 39A30 Stability theory (difference equations)
##### References:
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