Boundedness character of a max-type system of difference equations of second order

The boundedness character of positive solutions of the next max-type system of difference equations $$x_{n+1}=\max\left\{A,\frac{y_n^p}{x_{n-1}^q}\right\},\quad y_{n+1}=\max\left\{A,\frac{x_n^p}{y_{n-1}^q}\right\},\quad n\in\mathbb{N}_0,$$ with $\min\{A, p, q\}>0$, is characterized

Note on the binomial partial difference equation

Some formulas for the "general solution" to the binomial partial difference equation $$c_{m,n}=c_{m-1,n}+c_{m-1,n-1},$$ are known in the literature. However, it seems that there is no such a formula on the most natural domain connected to the equation, that is, on the set $D=\big\{(m,n)\in\mathbb{N}^2_0 : 0\le n\le m\big\}.$ By using a connection with the scalar linear first order difference equation we show that the equation on the domain $D\setminus\{(0,0)\}$, can be solved in closed form by presenting a formula for the solution in terms of the "side" values $c_{k,0}$, $c_{k,k}$, $k\in\mathbb{N}$

Product-type system of difference equations of second-order solvable in closed form

This paper presents solutions of the following second-order system of difference equations $$x_{n+1}=\frac{y_n^a}{z_{n-1}^b},\qquad y_{n+1}=\frac{z_n^c}{x_{n-1}^d},\qquad z_{n+1}=\frac{x_n^f}{y_{n-1}^g},\qquad n\in N_0,$$ where $a,b,c,d,f,g\in Z$, and $x_{-i}, y_{-i}, z_{-i}\in C\setminus\{0\},$ $i\in\{0,1\}$, in closed form

Global behavior of a max-type system of difference equations of the second order with four variables and period-two parameters

In this paper, we study global behavior of the following max-type system of difference equations of the second order with four variables and period-two parameters $\left\{\begin{array}{ll}x_{n} = \max\Big\{A_n , \frac{z_{n-1}}{y_{n-2}}\Big\}, \ y_{n} = \max \Big\{B_n, \frac{w_{n-1}}{x_{n-2}}\Big\}, \ z_{n} = \max\Big\{C_n , \frac{x_{n-1}}{w_{n-2}}\Big\}, \ w_{n} = \max \Big\{D_n, \frac{y_{n-1}}{z_{n-2}}\Big\}, \ \end{array}\right. \ \ n\in \{0, 1, 2, \cdots\},$ where $A_n, B_n, C_n, D_n\in (0, +\infty)$ are periodic sequences with period 2 and the initial values $x_{-i}, y_{-i}, z_{-i}, w_{-i}\in (0, +\infty)\ (1\leq i\leq 2)$. We show that if \min\{A_0C_1, B_0D_1, A_1C_0, B_1D_0\} < 1 , then this system has unbounded solutions. Also, if $\min\{A_0C_1, B_0D_1, A_1C_0, B_1D_0\}\geq 1$, then every solution of this system is eventually periodic with period $4$.</p

On a higher-order system of difference equations

Here we study the following system of difference equations xn = f −1 cnf(xn−2k) ∏k an + bn i=1 g(y) n−(2i−1))f(xn−2i) yn = g −1 γng(yn−2k) ∏k αn + βn i=1 f(x) n−(2i−1))g(yn−2i) n ∈ N0, where f and g are increasing real functions such that f(0) = g(0) = 0, and coefficients an, bn, cn, αn, βn, γn, n ∈ N0, and initial values x−i, y−i, i ∈ {1, 2,..., 2k} are real numbers. We show that the system is solvable in closed form, and study asymptotic behavior of its solutions

New solvable class of product-type systems of difference equations on the complex domain and a new method for proving the solvability

This paper continues the investigation of solvability of product-type systems of difference equations, by studying the following system with two variables: $$z_n=\alpha z_{n-1}^aw_{n-2}^b,\quad w_n=\beta w_{n-3}^cz_{n-2}^d,\quad n\in\mathbb{N}_0,$$ where $a,b,c,d\in\mathbb{Z}$, $\alpha,\beta\in\mathbb{C}\setminus\{0\}$, $w_{-3}, w_{-2}, w_{-1}, z_{-2}, z_{-1}\in\mathbb{C}\setminus\{0\}$. It is shown that there are some important cases such that the system cannot be solved by using our previous methods. Hence, we also present a method different from the previous ones by which the solvability of the system is shown also in the cases

On a practically solvable product-type system of difference equations of second order

The problem of solvability of the following second order system of difference equations z(n+1) = alpha Z(n)(a)w(n)(b), w(n+1) = beta w(n)(c)z(n-1)(d), n is an element of N-0, where a, b, c, d is an element of Z, alpha, beta is an element of C \ {0}, z(-1), z(0), w(0) is an element of C \ {0}, is studied in detail

Asymptotic representation of intermediate solutions to a cyclic systems of second-order difference equations with regularly varying coefficients

The cyclic system of second-order difference equations \begin{equation*} \Delta(p_i(n)|\Delta x_i(n)|^{\alpha_i-1}\Delta x_i(n)) = q_i(n)|x_{i+1}(n+1)|^{\beta_i-1}x_{i+1}(n+1), \end{equation*} for $i=\overline{1,N}$ where $x_{N+1}=x_1,$ is analysed in the framework of discrete regular variation. Under the assumption that $\alpha_i$ and $\beta_i$ are positive constants such that $\alpha_1\alpha_2\cdots\alpha_N>\beta_1\beta_2\cdots\beta_N$ and $p_i$ and $q_i$ are regularly varying sequences it is shown that the situation in which this system possesses regularly varying intermediate solutions can be completely characterized. Besides, precise information can be acquired about the asymptotic behavior at infinity of these solutions