Solutions Form for Some Rational Systems of Difference Equations

with nonzero real numbers initial conditions and then investigate the obtained solutions. Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economy, physics, and so on. So, recently there has been an increasing interest in the study of qualitative analysis of scalar rational difference equations and systems of rational difference equations. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solutions. See [1–7] and the references cited therein. The periodicity of the positive solutions for the following system of rational difference equations


Introduction
Our aim in this paper is to find the solutions form for the following systems of rational difference equations: ,  = 0, 1, . . ., with nonzero real numbers initial conditions and then investigate the obtained solutions.Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economy, physics, and so on.So, recently there has been an increasing interest in the study of qualitative analysis of scalar rational difference equations and systems of rational difference equations.Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solutions.See [1][2][3][4][5][6][7] and the references cited therein.
The periodicity of the positive solutions for the following system of rational difference equations was studied by Cinar et al. [8].
Özban [9] has studied the positive solutions for the following system: The behavior of the positive solutions for the following system has been studied by Kurbanlı et al. [10].Touafek and Elsayed [11] studied the periodicity and gave the form of the solutions for the following systems: Yalcinkaya [12] investigated the sufficient condition for the global asymptotic stability for the following system of difference equations: Yang [13] investigated the positive solutions for the system Clark et al. [14,15] investigate the global asymptotic stability of the following difference equations: Camouzis and Papaschinopoulos [16] studied the global asymptotic behavior of the positive solutions of the system of rational difference equations as follows:

On the System: 𝑥
In this section, we study the existence of analytical forms of the solutions for the following system of difference equations: with nonzero real initials conditions  −2 ,  −1 ,  0 ,  −2 ,  −1 , and  0 .
In the sequel we assume that ∏ −1 =0     = 1, for any real numbers   and   .
Lemma 2. Every positive solution for system (10) is bounded, and Proof.It follows from system (10) that for  large, we see that Then the subsequences =0 are decreasing and so are bounded from above by , , , and , respectively, where  = max{ −1 ,  0 } and  = max{ −1 ,  0 }.
Then for  = 0, 1, 2, . .., Proof.For  = 0 the result holds.Now suppose that  > 0 and that our assumption holds for  − 1.That is, Now, it follows from system (23) that Similarly one can prove the other relations.The proof is complete.

On the System: 𝑥
In this section, we present the solutions form for the following system: with nonzero real numbers initial conditions where  −1 ̸ =  −2 and  0 ̸ =  −1 .The following theorems can be proved similarly to those in Sections 2 and 3.
Theorem 13.Suppose that {  ,   } is a solution for system (33).Assume that  −2 ,  −1 ,  0 ,  −2 ,  −1 , and  0 are arbitrary nonzero real numbers.Then Lemma 14.Every positive solution of the equation Theorem 15.Let {  ,   } be a solution for the system Theorem 16.The solution form for the following system Theorem 17.The following system has a solution form given by the following relations: where  −1 ̸ =  −2 and  0 ̸ =  −1 .

Other Systems
In this section, we give the solutions form for the following systems of difference equations: with nonzero real numbers initial conditions.