THE SOLUTION OF A SYSTEM OF HIGHER-ORDER DIFFERENCE EQUATIONS IN TERMS OF BALANCING NUMBERS

A BSTRACT . In this paper, we are interested in the closed-form solution of the following system of nonlinear difference equations of higher order,


INTRODUCTION
Many researchers have interested in different types of difference equations, and we mention but are not limited to the homogeneous linear difference equation of the 2nd-order, u n+1 = αu n + βu n−1 , n ≥ 1, where α, β ∈ R or C such that β = 0, in particular, we give information about Balancing (resp. Pell) sequence that establishes a significant part of our study, defined as follows B n+1 = 6B n − B n−1 , (resp. P n+1 = 2P n + P n−1 ), n ≥ 1, with initial conditions B 0 = P 0 = 0 and B 1 = P 1 = 1. The following Binet formula of the Balancing (resp. Pell) numbers gives, B n = a n − b n a − b , P n = 2B 2n , (see., [11]) where a = 3 + 2 √ 2 and b = 3 − 2 √ 2. The search for solutions in the closed form of difference equations or systems has attracted the attention of many mathematicians (see., [1]- [26]). So, in this paper, we seek to provide a class of system of nonlinear difference equations which can be solved in explicit form, but the solutions are expressed by Balancing numbers, is the following system of difference equations, with initial conditions ω 0 , ω −1 ∈ R. Then, where (B n , n ≥ 0) is the Balancing sequence and (P n , n ≥ 0) is the Pell sequence.
Proof. Difference equation (2.0) is ordinarily solved by using the following characteristic polynomial, λ 2 − 34λ + 1 = 0, roots of this equation are These roots are linked to the roots of the Balancing number sequence. Then the closed form of the general solution of the equation (2.0) is where ω 0 , ω −1 are initial values such that and we have after some calculations, we get The lemma is proved.

Lemma 2.2. Consider the homogeneous linear difference equation with constant coefficients
with initial conditions θ 0 , θ −1 ∈ R. Then, where (B n , n ≥ 0) is the Balancing sequence and (P n , n ≥ 0) is the Pell sequence.
Proof. The difference equation (2.1) is ordinarily solved by using the following characteristic polynomial, λ 2 + 34λ + 1 = 0, roots of this equation are These roots are linked to the roots of the Balancing number sequence. Then the closed form of the general solution of the equation (2.1) is and we have after some calculations, we get The lemma is proved. Lemma 2.3. Consider the following system of rational difference equations Then, Proof. From system (2.3), we get the following system Using the change of variables ω n = x n + y n and θ n = x n − y n , we can write (2.4) as hence, the closed form of general solution of the system , n ≥ 0. The lemma is proved.
2.1. On the system (2.3). In this subsection, we consider the following system of difference equations of 1st-order, To find the closed form of the solutions of the system (2.3) we consider the following change variables where (B n , n ≥ 0) is the Balancing sequence and (P n , n ≥ 0) is the Pell sequence.
Proof. Straightforward and hence omitted.
2.2. On the system (1.0). In this paper, we study the System (1.0), which is an extension of System (2.3). Therefore, the System (1.0) can be written as follows for t ∈ {0, 1, ..., m} and n ∈ N. Now, using the following notation, we can get (m + 1) −systems similar to System (2.3), for t ∈ {0, 1, ..., m} . Through the above discussion, we can introduce the following Theorem where (B n , n ≥ 0) is the Balancing sequence and (P n , n ≥ 0) is the Pell sequence.
where (P n , n ≥ 0) is the Pell sequence and (Q n , n ≥ 0) is the Pell-Lucas sequence.
Proof. We see that it suffices to remark B n = a n 1 − b n 1 a 1 − b 1 a n 1 + b n 1 a 1 + b 1 = 1 2 P n Q n (see., [11]).
where (C n , n ≥ 0) is the Lucas-Balancing sequence.

GLOBAL STABILITY OF POSITIVE SOLUTIONS OF (1.0)
In the following, we will study the global stability character of the solutions of system (1.0). Obviously, the positive equilibriums of system (1.0) are 1) .
Let the functions f 1 , f 2 : (0, +∞) 2(m+1) → (0, +∞) defined by where z 0:m = (z 0 , z 1 , ..., z m ) . Now, it is usually useful to linearized system (1.0) around the equilibrium point U 2 in order to facilitate its study. For this purpose, introducing the vectors X n := X n , Y n where X n = (x n , x n−1 , ..., x n−m ) and Y n = (y n , y n−1 , ..., y n−m ) . With these notations, we obtain the following representation with O (k,l) denotes the matrix of order k × l whose entries are zeros, for simplicity, we set O (k) := O (k,k) and O (k) := O (k,1) and I (m) is the m × m identity matrix. We summarize the above discussion in the following theorem Theorem 3.1. The positive equilibrium point U 2 is locally asymptotically stable.
The following result is an immediate consequence of Theorem 3.1 and Corollary 3.1.

Corollary 3.2.
The unique positive equilibrium point U 2 is globally asymptotically stable.

NUMERICAL EXAMPLES
In order to clarify and shore theoretical results of the previous section, we consider some interesting numerical examples in this section.   Table 1. The initial conditions. The plot of the solutions is shown in Figure 2.  Table 1.  Table 2. The initial conditions. The plot of the solutions is shown in Figure 3.  Table 2.
In these examples, we show that the solutions of the system (1.0) for some cases are globally asymptotically stable.

Conflicts of Interest
The corresponding author declares no conflict of interest.