We show that the next difference equations system x_{n+1}=((a_{n}x_{n-k+1}y_{n-k})/(y_{n}-α_{n}))+β_{n+1}, y_{n+1}=((b_{n}y_{n-k+1}x_{n-k})/(x_{n}-β_{n}))+α_{n+1}, n∈N₀, where N₀=N∪{0}, the sequences (a_{n})_{n∈N₀}, (b_{n})_{n∈N₀}, (α_{n})_{n∈N₀}, (β_{n})_{n∈N₀} are two periodic and the initial conditions x_{-i}, y_{-i} i∈{0,1,…,k}, are non-zero real numbers, can be solved. Also, we investigate the behavior of solutions of above mentioned system when a₀=b₁ and a₁=b₀.

In this study, we construct q-analog C(q) of Catalan matrix and study the sequence spaces c0(C(q)) and c(C(q)) defined as the domain of q-Catalan matrix C(q) in the spaces c0 and c, respectively. We exhibit some topological properties, obtain Schauder bases and determine ?-, ?-, and ?-duals of the spaces c0(C(q)) and c(C(q)). Finally, we characterize certain class of matrix mappings from the spaces c0(C(q)) and c(C(q)) to the space ? = {??, c0, c, ?1} and give the necessary and sufficient conditions for a matrix operator to be compact.

In this study, we investigate the boundedness, persistence of positive solutions, local and global stability of the unique positive equilibrium point and rate of convergence of positive solutions of the following difference equations systems of exponential forms: \begin{document}$ \begin{equation*} \Upsilon_{n+1} = \frac{\Gamma_{1}+\delta_{1}e^{-\Psi_{n-1}}}{\Theta_{1}+\Psi_{n}}, \ \Psi_{n+1} = \frac{\Gamma_{2}+\delta_{2}e^{-\Omega_{n-1}}}{\Theta_{2}+\Omega_{n}}, \ \Omega_{n+1} = \frac{\Gamma_{3}+\delta_{3}e^{-\Upsilon_{n-1}}}{\Theta_{3}+\Upsilon_{n}}, \end{equation*} $\end{document} \begin{document}$ \begin{equation*} \Upsilon_{n+1} = \frac{\Gamma_{1}+\delta_{1}e^{-\Psi_{n-1}}}{\Theta_{1}+\Upsilon_{n}}, \ \Psi_{n+1} = \frac{\Gamma_{2}+\delta_{2}e^{-\Omega_{n-1}}}{\Theta_{2}+\Psi_{n}}, \ \Omega_{n+1} = \frac{\Gamma_{3}+\delta_{3}e^{-\Upsilon_{n-1}}}{\Theta_{3}+\Omega_{n}}, \end{equation*} $\end{document} for $ n\in \mathbb{N}_{0} $, where the initial conditions $ \Upsilon_{-j} $, $ \Psi_{-j} $, $ \Omega_{-j} $, for $ j\in\{0, 1\} $ and the parameters $ \Gamma_{i} $, $ \delta_{i} $, $ \Theta_{i} $ for $ i\in\{1, 2, 3\} $ are positive constants.

In this paper we show that the system of difference equations\[ x_n= a y_{n-k}+\frac{dy_{n-k}x_{n-( k+l ) }}{b x_{n-(k+l)}+cy_{n-l}}=\alpha x_{n-k}+\frac{\delta x_{n-k}y_{n-(k+l)}}{\beta y_{n-(k+l)}}+\gamma x_{n-l}, \] where $n\in \mathbb{N}_{0},$ $k$ and $l$ are positive integers, the parameters $a$, $b$, $c$, $d$, $\alpha $, $\beta $, $\gamma $, $\delta $ are real numbers and the initial values $x_{-j}$, $y_{-j}$, $j=\overline{1,k+l}$, are real numbers, can be solved in the closed form. We also determine the asymptotic behavior of solutions for the case $l=1$ and describe the forbidden set of the initial values using the obtained formulas. Our obtained results significantly extend and develop some recent results in the literature.

In this work, we indicate three-dimensional system of difference equations xn = ayn?k + dyn?kxn?k?l/?bxn?k?l+?czn?l, yn = ?zn?k + ?zn?kyn?k?l/??yn?k?l + ??xn?l, zn = exn?k + hxn?kzn?k?l/?fzn?k?l +?g?yn?l , n ? N0, where k and l are positive integers, the parameters a,?b, ?c, d, ?, ??, ??, ?, e, ?f ,?g, h and the initial values x?j, y?j, z?j j = ?1,k+l, are non-zero real numbers, can be solved in closed form. In addition, we obtain explicit formulas for the well-defined solutions of the aforementioned system for the case l = 1. Also, the set of undefinable solutions of the system is found. Finally, an application about a three-dimensional system of difference equations is given.