ON A SOLVABLE p − DIMENSIONAL SYSTEM OF NONLINEAR DIFFERENCE EQUATIONS

. In this paper, we investigate the solutions of the following system of p − nonlinear difference equations where N 0 = N ∪ { 0 } the sequences (cid:16) a ( i ) (cid:17) , (cid:16) b ( i ) (cid:17) , (cid:16) c ( i ) (cid:17) , are non-zero real numbers and initial values x ( i ) − j , j ∈ { 0 , 1 , 2 } , i ∈ { 1 ,..., p } . Finally, we give some applications concerning aforementioned system of difference equations.


INTRODUCTION
In the recent years, there has been a lot of interest in studying nonlinear difference equations and systems. Not surprisingly therefore, several studies have been published on this topic (see, e.g., [1]- [28], and the related references therein). Besides their theoretical value, most of the recent applications have appeared in many scientific areas such as biology (population dynamics in particular), ecology, physics, engineering and economics (see, e.g. [8], [9], [12], [22]). It is very worthy to find systems belonging to solvable nonlinear difference equations systems in closed-form.
Since the paper by Brand [5], the following one-dimensional nonlinear difference equation of Riccati type, (1.1) x n+1 = ax n + d cx n + b , n ∈ N 0 , where the initial value x 0 is a real number or complex number and the parameters a, b, c and d are the real numbers with the restrictions c = 0 and ab = cd, have the most diverse and interesting properties, especially as regards the distribution of their cluster points. This finding, led Stević [24] to study the solutions of the Eq. (1.1).
In Abo-Zeid et al. [3] the authors presented the solutions of the one-dimensional system of nonlinear difference equations which reduced to the Riccati difference equation under appropriate transformations, Then, in [10] and [11], equations in (1.3) were generalized to the following equations where max {l, k, p, q} is nonnegative integer and a, b, c are positive constants. Moreover, in [4] the authors presented the solutions of the following two-dimensional four systems of nonlinear difference equation generalization of Eq. (1.2): (1.4) x n+1 = y n y n−2 x n−1 + y n−2 , y n+1 = x n x n−2 ±y n−1 ± x n−2 , n ∈ N 0 .
Its extension with constant coefficients and p−dimensional is a system of a huge interest. For this reason, another extension of system (1.6) is the following system of p−dimensional non- Now, we consider system (1.7) in the case when a a (i) = 0 for all i ∈ {1, ..., p} . Noticing that in this case, system (1.7) can be written in the form , for all i ∈ {1, ..., p}, we see that we may assume that a (i) = 1, for all i ∈ {1, ..., p}. Hence we consider, without loss of generality, the system , n ∈ N 0 , p ∈ N,i ∈ {1, ..., p} .
using the same notation for coefficients as in (1.7) except for the coefficients a (i) , assuming that a (i) = 1, for all i ∈ {1, ..., p}.
The remainder of the paper is organized as follows: In section 2, we study the solutions of the given system of the p−dimensional nonlinear rational difference equations by using convenient transformation. In the next section, we obtain well-known Fibonacci numbers and Pell numbers in the solutions of aforementioned system for some cases. Section 4 concludes.

EXPLICIT FORMULAS FOR THE SOLUTIONS OF SYSTEM
n , ..., x is not defined. Thus, for every well-defined solution of system (1.8), we get that . Note that the system (1.8) can be written in the form Next, by employing the change of variables Then system (2.1) can be written as .., p} , p ∈ N, are 2p−solutions to the following system of difference equations From now on, we assume that the sequence t To solve system (2.5) we need to use the following lemma.
Lemma 2.1. For a, b ∈ R, consider the homogeneous linear second-order difference equation where b = 0 and a 2 + 4b = 0, the general solution for equation (2.6) as follows: s −2 is calculated by using the following relations bs n−1 = s n+1 − as n for n = −1.
Proof. The proof follows essentially the same arguments as in Stević [24].
Remark 2.1. By taking a = 1 (resp. a = 2) and b = 1 in equation then the sequence (s n ) n≥−1 reduce to the well known Fibonacci (resp. Pell) sequence.
..,p} be the solution to system (2.5) such that s .., p} are calculated by using the following relations b (i) s i ∈ {1, ..., p} for m = −1. From the system in (2.4) and the system (2.7), it follows that Hence, for k ∈ {1, 2}, that is, Using the change of variables thus we have where l ∏ j=i y j = 1 if l < i. From all above mentioned we see that the following corollary holds.
be a solution of system (1.8). Then for n ≥ p, be a solution of system (1.8). Then for n ≥ p, if p is even, with (s (i) m ) m≥−1,i∈{1,...,p} be the solution to system (2.5) such that s Proof. By Corollary 2.1, we obtain Using (2.8), we get The rest of assertions are immediate.

SOME APPLICATIONS
In this section, we will give some applications for some special cases of the coefficients of the system (1.7). be a well-defined solution to the following system,  be a well-defined solution to the following system, for n ∈ N 0 , k ∈ {0, ..., 2p} , i ∈ {1, ..., 2p + 1} , p ∈ N, where m = p (n − 2l) − l + k 2 + n 2 + t ∨ t 2 − j − 1 and (P n ) n≥−1 is the solution to the following difference equation P n+1 = 2P n + P n−1 , n ∈ N 0 , satisfying the initial conditions P −1 = 0, P 0 = 1. The sequence (P n ) n≥−1 is called the Pell sequence in literature.

CONCLUSION
In this paper, we represented the general solutions of p−dimensional systems of nonlinear rational difference equations with constant coefficients using suitable transformation reducing to the equations in Riccati type. Secondly, the solutions of this system are related to both Fibanacci numbers and Pell numbers for some special cases. Finally, we will give the following important open problem for system of difference equations theory to researchers. The system (1.7) can extend to equations more general than that in (1.7). For example, the p−dimensional system of nonlinear rational difference equations of (max {m, k, l, s} + 1) −order,

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.