Solutions structures for some systems of fractional difference equations

1. Introduction

he theory of discrete dynamical systems of difference equations has been utilized to study natural phenomena which change over discrete time. A massive number of researchers investigated various real life problems that occur in population dynamics, genetics in biology, engineering, queuing problems, electrical networks, physics, economics, etc [1]. The long-term behaviours of such problems have been recently discussed by some scholars. For example, in [2] Asiri et al. explained the periodic solutions of the following system of difference equations: \begin{equation*} x_{n+1}=\frac{y_{n-2}}{1-y_{n-2}x_{n-1}y_{n}},\ \ \ y_{n+1}=\frac{x_{n-2}}{ \pm 1\pm x_{n-2}y_{n-1}x_{n}}. \end{equation*} Cinar [3] explored the periodicity of non-negative solutions of a system of fractional difference equations given by the form: \begin{equation*} x_{n+1}=\frac{1}{y_{n}},\ \ \ y_{n+1}=\frac{y_{n}}{x_{n-1}y_{n-1}}. \end{equation*} Din [4] analyzed and obtained the equilibrium points, local asymptotic stability and global behaviour of the fixed points of Lotka-Volterra model which is illustrated by the system: \begin{equation*} x_{n+1}=\frac{\alpha x_{n}-\beta x_{n}y_{n}}{1+\gamma x_{n}},\ \ \ y_{n+1}= \frac{\delta y_{n}+\epsilon x_{n}y_{n}}{1+\eta y_{n}}. \end{equation*} El-Metwally et al. [5] found the solutions and the periodicity of the following third order system of rational recursive equations: \begin{equation*} x_{n+1}=\frac{y_{n-2}}{-1-y_{n-2}x_{n-1}y_{n}},\ \ \ y_{n+1}=\frac{x_{n-2}}{ \pm 1\pm x_{n-2}y_{n-1}x_{n}}. \end{equation*} In [6], Elsayed presented the solutions of the following system of second order difference equations: \begin{equation*} x_{n+1}=\frac{x_{n-1}}{\pm 1\pm x_{n-1}y_{n}},\ \ \ y_{n+1}=\frac{y_{n-1}}{ \mp 1+y_{n-1}x_{n}}. \end{equation*} Elsayed and Alzubaidi [7] found the solutions of following systems of rational difference equations: \[ x_{n+1}=\frac{y_{n-8}}{1+y_{n-2}x_{n-5}y_{n-8}},\ \ \ \ \ y_{n+1} =\frac{x_{n-8}}{\pm1\pm x_{n-2}y_{n-5}x_{n-8}}. \] G\H{u}m\H{u}\c{s} and \H{O}calan [8] explored the positive solutions of the systems: \[ u_{n+1}=\frac{\alpha u_{n-1}}{\beta+\gamma v_{n}^{p}v_{n-2}^{q}} ,\ \ \ v_{n+1}=\frac{\alpha_{1}v_{n-1}}{\beta_{1}+\gamma_{1}u_{n}^{p_{1} }u_{n-2}^{q_{1}}}. \] Kurbanli et al. [9] obtained the solutions of the following system of difference equations: \begin{equation*} x_{n+1}=\frac{x_{n-1}}{y_{n}x_{n-1}-1},\ \ y_{n+1}=\frac{y_{n-1}}{ x_{n}y_{n-1}-1},\ \ z_{n+1}=\frac{x_{n}}{y_{n}z_{n-1}}. \end{equation*} In [10], Touafek et al. discovered the periodicity and solution of the system: \begin{equation*} x_{n+1}=\frac{x_{n-3}}{\pm 1\pm x_{n-3}y_{n-1}},\ \ \ y_{n+1}=\frac{y_{n-3}}{ \pm 1\pm y_{n-3}x_{n-1}}. \end{equation*} The author in [11] examined the dynamics of the following system of recursive equations: \begin{equation*} x_{n+1}=\frac{x_{n-2}}{B+y_{n}y_{n-1}y_{n-2}},\ \ \ y_{n+1}=\frac{y_{n-2}}{ A+x_{n}x_{n-1}x_{n-2}}. \end{equation*} For more information about basic theory and qualitative behaviour of dynamical systems of difference equations, one can see references [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 35, 26, 27].
This work aims to present the solutions of discrete dynamical systems of difference equations which are given by \begin{equation*} x_{n+1}=\frac{x_{n-1}y_{n-3}}{y_{n-1}\left( -1-x_{n-1}y_{n-3}\right) },\ \ \ y_{n+1}=\frac{y_{n-1}x_{n-3}}{x_{n-1}\left( \pm 1\pm y_{n-1}x_{n-3}\right) } ,\ \ \ n=0,1,\ldots, \end{equation*} where the initial values \(x_{-3},\ x_{-2},\ x_{-1},\ \)\ \(x_{0},\ y_{-3},\ y_{-2},\ y_{-1}\) and\(\ \ y_{0}\) are required to be real numbers.

2. Main Results

2.1. First system \(x_{n+1}=\frac{x_{n-1}y_{n-3}}{y_{n-1}\left( -1-x_{n-1}y_{n-3}\right) },\ y_{n+1}=\frac{y_{n-1}x_{n-3}}{x_{n-1}\left( 1+y_{n-1}x_{n-3}\right) }\)

This subsections is devoted for the solutions of the following system of recursive equations: The initial values of this system are required to be arbitrary real numbers.

Theorem Let \(\left\{ x_{n},y_{n}\right\} \) be a solution to system (1) and let \(x_{-3}=a,\ x_{-2}=b,\ x_{-1}=c,\ x_{0}=d,\ y_{-3}=\alpha ,\ y_{-2}=\beta ,\ y_{-1}=\gamma \) and \(y_{0}=\omega .\) Then, for \(n=0,\ 1,\ \ldots\), we have \begin{eqnarray*} x_{4n-3} &=&\frac{\left( -1\right) ^{n}c^{n}\alpha ^{n}\underset{i=0}{ \overset{n-1}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] }{ a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}},\ \ \ x_{4n-2}=\frac{\left( -1\right) ^{n}d^{n}\beta ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i\right) b\omega +1\right] }{b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n}}, \\ x_{4n-1} &=&\frac{c^{n+1}\alpha ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] }{a^{n}\gamma ^{n}},\ \ \ x_{4n}=\frac{ d^{n+1}\beta ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+1\right) b\omega +1\right] }{b^{n}\omega ^{n}}, \end{eqnarray*} and \begin{eqnarray*} y_{4n-3} &=&\frac{a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\underset{i=0}{\overset {n-1}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] },\ \ \ y_{4n-2}= \frac{b^{n}\omega ^{n}}{d^{n}\beta ^{n-1}\underset{i=0}{\overset{n-1}{\Pi }} \left[ \left( 2i+1\right) b\omega +1\right] }, \\ y_{4n-1} &=&\frac{\left( -1\right) ^{n}a^{n}\gamma ^{n+1}\left( c\alpha +1\right) ^{n}}{c^{n}\alpha ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+2\right) a\gamma +1\right] },\ \ \ y_{4n}=\frac{\left( -1\right) ^{n}b^{n}\omega ^{n+1}\left( d\beta +1\right) ^{n}}{d^{n}\beta ^{n}\underset{ i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+2\right) b\omega +1\right] }. \end{eqnarray*}

Proof. It is easy to see that the results hold for \(n=0\). Next, we assume that \(n>1\) and suppose that the solutions hold for \(n-1.\) That is \begin{eqnarray*} x_{4n-7} &=&\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] }{ a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}},\ \ \ x_{4n-6}=\frac{ \left( -1\right) ^{n-1}d^{n-1}\beta ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i\right) b\omega +1\right] }{b^{n-2}\omega ^{n-1}\left( d\beta +1\right) ^{n-1}}, \\ x_{4n-5} &=&\frac{c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] }{a^{n-1}\gamma ^{n-1}},\ \ \ x_{4n-4}= \frac{d^{n}\beta ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) b\omega +1\right] }{b^{n-1}\omega ^{n-1}} \end{eqnarray*} and \begin{eqnarray*} y_{4n-7} &=&\frac{a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] },\ \ \ y_{4n-6}=\frac{b^{n-1}\omega ^{n-1}}{d^{n-1}\beta ^{n-2}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i+1\right) b\omega +1\right] }, \\ y_{4n-5} &=&\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i+2\right) a\gamma +1\right] },\ \ \ y_{4n-4}=\frac{\left( -1\right) ^{n-1}b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n-1}}{ d^{n-1}\beta ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) b\omega +1\right] }. \end{eqnarray*} Following this, system (1) gives us \begin{eqnarray*} x_{4n-3} &=&\frac{x_{4n-5}y_{4n-7}}{y_{4n-5}\left( -1-x_{4n-5}y_{4n-7}\right) } \\ &=&\frac{\frac{c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] }{a^{n-1}\gamma ^{n-1}}\frac{ a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i+1\right) a\gamma +1\right] }}{\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1} \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma +1 \right] }\left[ -1-\frac{c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi } }\left[ \left( 2i+1\right) a\gamma +1\right] }{a^{n-1}\gamma ^{n-1}}\frac{ a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i+1\right) a\gamma +1\right] }\right] } \\ &=&\frac{-\left( -1\right) ^{-n+1}c^{n}\alpha ^{n}\underset{i=0}{\overset{n-2 }{\Pi }}\left[ \left( 2i+2\right) a\gamma +1\right] }{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}\left[ 1+c\alpha \right] }=\frac{\left( -1\right) ^{n}c^{n}\alpha ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] }{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}}. \end{eqnarray*} Moreover, one can observe from system (1) that \begin{eqnarray*} y_{4n-3} &=&\frac{y_{4n-5}x_{4n-7}}{x_{4n-5}\left[ 1+y_{4n-5}x_{4n-7}\right] } \\ &=&\frac{\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i+2\right) a\gamma +1\right] }\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] }{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}}{\frac{c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] }{a^{n-1}\gamma ^{n-1}}\left[ 1+\frac{ \left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{ c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma +1\right] }\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma +1 \right] }{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}\right] } \end{eqnarray*} \begin{eqnarray*} &=&\frac{\frac{a\gamma \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] }{\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma +1\right] }}{\frac{c^{n}\alpha ^{n-1}\underset{i=0 }{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] }{ a^{n-1}\gamma ^{n-1}}\left[ 1+\frac{a\gamma \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] }{\underset{i=0}{\overset{n-2}{ \Pi }}\left[ \left( 2i+2\right) a\gamma +1\right] }\right] } \\ &=&\frac{a\gamma \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] a^{n-1}\gamma ^{n-1}}{c^{n}\alpha ^{n-1}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] \left[ \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma +1 \right] +a\gamma \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma +1\right] \right] } \\ &=&\frac{a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-1}{ \Pi }}\left[ \left( 2i+1\right) a\gamma +1\right] }. \end{eqnarray*} Similarly, other results can be proved. The proof has been completed.

Second System \(x_{n+1}=\frac{x_{n-1}y_{n-3}}{y_{n-1}\left( -1-x_{n-1}y_{n-3}\right) },\ y_{n+1}=\frac{y_{n-1}x_{n-3}}{x_{n-1}\left( -1-y_{n-1}x_{n-3}\right) }\)

In this section, we obtained the solutions of the following system of differential equations: The initial values of system (2) are required to be arbitrary real numbers.

Theorem 2. Let \(\left\{ x_{n},y_{n}\right\} \) be a solution to system (2) and assume that \(x_{-3}=a,\ x_{-2}=b,\ x_{-1}=c,\ x_{0}=d,\ y_{-3}=\alpha ,\ y_{-2}=\beta ,\ y_{-1}=\gamma \) and \(y_{0}=\omega .\) Then, for \(n=0,\ 1,\ ...\) we have \begin{eqnarray*} x_{4n-3} &=&\frac{\left( -1\right) ^{n}c^{n}\alpha ^{n}}{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}},\ \ \ x_{4n-2}=\frac{\left( -1\right) ^{n}d^{n}\beta ^{n}}{b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n}}, \\ x_{4n-1} &=&\frac{\left( -1\right) ^{n}c^{n+1}\alpha ^{n}\left( a\gamma +1\right) ^{n}}{a^{n}\gamma ^{n}},\ \ \ x_{4n}=\frac{\left( -1\right) ^{n}d^{n+1}\beta ^{n}\left( b\omega +1\right) ^{n}}{b^{n}\omega ^{n}} \end{eqnarray*} and \begin{eqnarray*} y_{4n-3} &=&\frac{\left( -1\right) ^{n}a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\left( a\gamma +1\right) ^{n}},\ \ \ y_{4n-2}=\frac{\left( -1\right) ^{n}b^{n}\omega ^{n}}{d^{n}\beta ^{n-1}\left( b\omega +1\right) ^{n}}, \\ y_{4n-1} &=&\frac{\left( -1\right) ^{n}a^{n}\gamma ^{n+1}\left( c\alpha +1\right) ^{n}}{c^{n}\alpha ^{n}},\ \ \ y_{4n}=\frac{\left( -1\right) ^{n}b^{n}\omega ^{n+1}\left( d\beta +1\right) ^{n}}{d^{n}\beta ^{n}}. \end{eqnarray*}

Proof. The solutions are verified for \(n=0\). Next, we let \(n>1\) and assume that the results hold for \(n-1.\) That is \begin{eqnarray*} x_{4n-7} &=&\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}}{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}},\ \ \ x_{4n-6}=\frac{\left( -1\right) ^{n-1}d^{n-1}\beta ^{n-1}}{b^{n-2}\omega ^{n-1}\left( d\beta +1\right) ^{n-1} }, \\ x_{4n-5} &=&\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\left( a\gamma +1\right) ^{n-1}}{a^{n-1}\gamma ^{n-1}},\ \ \ x_{4n-4}=\frac{\left( -1\right) ^{n-1}d^{n}\beta ^{n-1}\left( b\omega +1\right) ^{n-1}}{ b^{n-1}\omega ^{n-1}} \end{eqnarray*} and \begin{eqnarray*} y_{4n-7} &=&\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\left( a\gamma +1\right) ^{n-1}},\ \ \ y_{4n-6}=\frac{\left( -1\right) ^{n-1}b^{n-1}\omega ^{n-1}}{d^{n-1}\beta ^{n-2}\left( b\omega +1\right) ^{n-1}}, \\ y_{4n-5} &=&\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}},\ \ \ y_{4n-4}=\frac{\left( -1\right) ^{n-1}b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n-1}}{ d^{n-1}\beta ^{n-1}}. \end{eqnarray*} Now, the first relation is given by \begin{eqnarray*} x_{4n-3} &=&\frac{x_{4n-5}y_{4n-7}}{y_{4n-5}\left( -1-x_{4n-5}y_{4n-7}\right) } \\ &=&\frac{\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\left( a\gamma +1\right) ^{n-1}}{a^{n-1}\gamma ^{n-1}}\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\left( a\gamma +1\right) ^{n-1}}}{\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}}\left[ -1-\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\left( a\gamma +1\right) ^{n-1}}{a^{n-1}\gamma ^{n-1} }\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\left( a\gamma +1\right) ^{n-1}}\right] } \\ &=&\frac{-\left( -1\right) ^{-n+1}c\alpha \ c^{n-1}\alpha ^{n-1}}{ a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}\left[ 1+c\alpha \right] }= \frac{\left( -1\right) ^{n}c^{n}\alpha ^{n}}{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}}. \end{eqnarray*} Similarly, system (2) leads \begin{eqnarray*} y_{4n-3} &=&\frac{y_{4n-5}x_{4n-7}}{x_{4n-5}\left( -1-y_{4n-5}x_{4n-7}\right) } \\ &=&\frac{\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}}\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}}{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}}{\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\left( a\gamma +1\right) ^{n-1}}{a^{n-1}\gamma ^{n-1}}\left[ -1-\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1} }\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}}{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}\right] } \\ &=&\frac{-\left( -1\right) ^{-n+1}a\gamma \ a^{n-1}\gamma ^{n-1}}{ c^{n}\alpha ^{n-1}\left( a\gamma +1\right) ^{n-1}\left[ 1+a\gamma \right] }= \frac{\left( -1\right) ^{n}a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\left( a\gamma +1\right) ^{n}}. \end{eqnarray*} Accordingly, the remaining relations of system (2) can be verified. Hence, this achieves the proof.

2.3. Third System \(x_{n+1}=\frac{x_{n-1}y_{n-3}}{y_{n-1}\left( -1-x_{n-1}y_{n-3}\right) },\ y_{n+1}=\frac{y_{n-1}x_{n-3}}{x_{n-1}\left( 1-y_{n-1}x_{n-3}\right) }\)

In this subsection, the solutions of the following dynamic discrete system will be formulated: where the initial values are required to be arbitrary real numbers.

Theorem 3. Assume that \(\left\{ x_{n},y_{n}\right\} \) is a solution to system (3) and suppose that \(x_{-3}=a,\ x_{-2}=b,\ x_{-1}=c,\ x_{0}=d,\ y_{-3}=\alpha ,\ y_{-2}=\beta ,\ y_{-1}=\gamma \) and \(y_{0}=\omega .\) Then, for \(n=0,\ 1,\ ...\) we have \begin{eqnarray*} x_{4n-3} &=&\frac{c^{n}\alpha ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i\right) a\gamma -1\right] }{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}},\ \ \ x_{4n-2}=\frac{d^{n}\beta ^{n}\underset{i=0}{\overset{ n-1}{\Pi }}\left[ \left( 2i\right) b\omega -1\right] }{b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n}}, \\ x_{4n-1} &=&\frac{\left( -1\right) ^{n}c^{n+1}\alpha ^{n}\underset{i=0}{ \overset{n-1}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }{ a^{n}\gamma ^{n}},\ \ \ x_{4n}=\frac{\left( -1\right) ^{n}d^{n+1}\beta ^{n} \underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+1\right) b\omega -1 \right] }{b^{n}\omega ^{n}} \end{eqnarray*} and \begin{eqnarray*} y_{4n-3} &=&\frac{\left( -1\right) ^{n}a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1} \underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1 \right] },\ \ \ y_{4n-2}=\frac{\left( -1\right) ^{n}b^{n}\omega ^{n}}{ d^{n}\beta ^{n-1}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+1\right) b\omega -1\right] }, \\ y_{4n-1} &=&\frac{a^{n}\gamma ^{n+1}\left( c\alpha +1\right) ^{n}}{ c^{n}\alpha ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+2\right) a\gamma -1\right] },\ \ \ y_{4n}=\frac{b^{n}\omega ^{n+1}\left( d\beta +1\right) ^{n}}{d^{n}\beta ^{n}\underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+2\right) b\omega -1\right] }. \end{eqnarray*}

Proof. The solutions hold for \(n=0\). Now, we suppose that \(n>1\) and assume that the solutions hold for \(n-1.\) That is \begin{eqnarray*} x_{4n-7} &=&\frac{c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i\right) a\gamma -1\right] }{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}},\ \ \ x_{4n-6}=\frac{d^{n-1}\beta ^{n-1}\underset{ i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) b\omega -1\right] }{ b^{n-2}\omega ^{n-1}\left( d\beta +1\right) ^{n-1}}, \end{eqnarray*} \begin{eqnarray*} x_{4n-5} &=&\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }{ a^{n-1}\gamma ^{n-1}},\ \ \ x_{4n-4}=\frac{\left( -1\right) ^{n-1}d^{n}\beta ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) b\omega -1 \right] }{b^{n-1}\omega ^{n-1}} \end{eqnarray*} and \begin{eqnarray*} y_{4n-7} &=&\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1 \right] },\ \ \ y_{4n-6}=\frac{\left( -1\right) ^{n-1}b^{n-1}\omega ^{n-1}}{ d^{n-1}\beta ^{n-2}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) b\omega -1\right] }, \\ y_{4n-5} &=&\frac{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{ c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma -1\right] },\ \ \ y_{4n-4}=\frac{b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n-1}}{d^{n-1}\beta ^{n-1}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i+2\right) b\omega -1\right] }. \end{eqnarray*} Next, it can be simply seen from system (3) that \begin{eqnarray*} x_{4n-3} &=&\frac{x_{4n-5}y_{4n-7}}{y_{4n-5}\left( -1-x_{4n-5}y_{4n-7}\right) } \\ &=&\frac{\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }{ a^{n-1}\gamma ^{n-1}}\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n-1}}{ c^{n-1}\alpha ^{n-2}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }}{\frac{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i+2\right) a\gamma -1\right] }\left[ -1-\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }{a^{n-1}\gamma ^{n-1}}\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }\right] } \\ &=&-\frac{c\alpha \ c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }} \left[ \left( 2i+2\right) a\gamma -1\right] }{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}\left[ 1+c\alpha \right] }=\frac{c^{n}\alpha ^{n} \underset{i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i\right) a\gamma -1\right] }{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}}. \end{eqnarray*} Furthermore, system (3) gives that \begin{eqnarray*} y_{4n-3} &=&\frac{y_{4n-5}x_{4n-7}}{x_{4n-5}\left( 1-y_{4n-5}x_{4n-7}\right) } \\ &=&\frac{\frac{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{ c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma -1\right] }\frac{c^{n-1}\alpha ^{n-1}\underset{i=0}{ \overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma -1\right] }{ a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}}{\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }{a^{n-1}\gamma ^{n-1}}\left[ 1-\frac{ a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1} \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma -1 \right] }\frac{c^{n-1}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma -1\right] }{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}\right] } \\ &=&\frac{\frac{a\gamma \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma -1\right] }{\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma -1\right] }}{\frac{\left( -1\right) ^{n-1}c^{n}\alpha ^{n-1}\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }{a^{n-1}\gamma ^{n-1}}\left[ 1-\frac{a\gamma \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma -1\right] }{\underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma -1 \right] }\right] } \\ &=&\frac{\left( -1\right) ^{-n+1}a^{n}\gamma ^{n}\underset{i=0}{\overset{n-2} {\Pi }}\left[ \left( 2i\right) a\gamma -1\right] }{c^{n}\alpha ^{n-1} \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1 \right] \left[ \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i+2\right) a\gamma -1\right] -a\gamma \underset{i=0}{\overset{n-2}{\Pi }}\left[ \left( 2i\right) a\gamma -1\right] \right] } \\ &=&\frac{\left( -1\right) ^{n}a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\underset{ i=0}{\overset{n-1}{\Pi }}\left[ \left( 2i+1\right) a\gamma -1\right] }. \end{eqnarray*} The remaining formulas can be shown in a similar way.

2.4 Fourth System \(x_{n+1}=\frac{x_{n-1}y_{n-3}}{y_{n-1}\left( -1-x_{n-1}y_{n-3}\right) },\ y_{n+1}=\frac{y_{n-1}x_{n-3}}{x_{n-1}\left( -1+y_{n-1}x_{n-3}\right) }\)

In the next theorem, we will discover the solutions of the following nonlinear system of difference equations: where the initial conditions are required to be arbitrary real numbers.

Theorem 4. Assume that \(\left\{ x_{n},y_{n}\right\} \) is a solution to system (4) and let \(x_{-3}=a,\ x_{-2}=b,\ x_{-1}=c,\ x_{0}=d,\ y_{-3}=\alpha ,\ y_{-2}=\beta ,\ y_{-1}=\gamma \) and \(y_{0}=\omega .\) Then, for \(n=0,\ 1,\ ... \) we have \begin{eqnarray*} x_{4n-3} &=&\frac{\left( -1\right) ^{n}c^{n}\alpha ^{n}}{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}},\ \ \ x_{4n-2}=\frac{\left( -1\right) ^{n}d^{n}\beta ^{n}}{b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n}}, \\ x_{4n-1} &=&\frac{c^{n+1}\alpha ^{n}\left( a\gamma -1\right) ^{n}}{ a^{n}\gamma ^{n}},\ \ \ x_{4n}=\frac{d^{n+1}\beta ^{n}\left( b\omega -1\right) ^{n}}{b^{n}\omega ^{n}} \end{eqnarray*} and \begin{eqnarray*} y_{4n-3} &=&\frac{a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\left( a\gamma -1\right) ^{n}},\ \ \ y_{4n-2}=\frac{b^{n}\omega ^{n}}{d^{n}\beta ^{n-1}\left( b\omega -1\right) ^{n}}, \\ y_{4n-1} &=&\frac{\left( -1\right) ^{n}a^{n}\gamma ^{n+1}\left( c\alpha +1\right) ^{n}}{c^{n}\alpha ^{n}},\ \ \ y_{4n}=\frac{\left( -1\right) ^{n}b^{n}\omega ^{n+1}\left( d\beta +1\right) ^{n}}{d^{n}\beta ^{n}}. \end{eqnarray*}

Proof. It is clear that the relations hold for \(n=0.\) Now, we suppose that \(n>1\) and assume that the solutions hold for \(n-1.\) That is \begin{eqnarray*} x_{4n-7} &=&\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}}{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}},\ \ \ x_{4n-6}=\frac{\left( -1\right) ^{n-1}d^{n-1}\beta ^{n-1}}{b^{n-2}\omega ^{n-1}\left( d\beta +1\right) ^{n-1} }, \\ x_{4n-5} &=&\frac{c^{n}\alpha ^{n-1}\left( a\gamma -1\right) ^{n-1}}{ a^{n-1}\gamma ^{n-1}},\ \ \ x_{4n-4}=\frac{d^{n}\beta ^{n-1}\left( b\omega -1\right) ^{n-1}}{b^{n-1}\omega ^{n-1}} \end{eqnarray*} and \begin{eqnarray*} y_{4n-7} &=&\frac{a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\left( a\gamma -1\right) ^{n-1}},\ \ \ y_{4n-6}=\frac{b^{n-1}\omega ^{n-1}}{d^{n-1}\beta ^{n-2}\left( b\omega -1\right) ^{n-1}}, \\ y_{4n-5} &=&\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}},\ \ \ y_{4n-4}=\frac{\left( -1\right) ^{n-1}b^{n-1}\omega ^{n}\left( d\beta +1\right) ^{n-1}}{ d^{n-1}\beta ^{n-1}}. \end{eqnarray*} Now, it can be obviously observed from system (4) that \begin{eqnarray*} x_{4n-3} &=&\frac{x_{4n-5}y_{4n-7}}{y_{4n-5}\left( -1-x_{4n-5}y_{4n-7}\right) } \\ &=&\frac{\frac{c^{n}\alpha ^{n-1}\left( a\gamma -1\right) ^{n-1}}{ a^{n-1}\gamma ^{n-1}}\frac{a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\left( a\gamma -1\right) ^{n-1}}}{\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}}\left[ -1-\frac{ c^{n}\alpha ^{n-1}\left( a\gamma -1\right) ^{n-1}}{a^{n-1}\gamma ^{n-1}} \frac{a^{n-1}\gamma ^{n-1}}{c^{n-1}\alpha ^{n-2}\left( a\gamma -1\right) ^{n-1}}\right] } \\ &=&\frac{\left( -1\right) ^{-n}c^{n}\alpha ^{n}}{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}\left[ 1+c\alpha \right] }=\frac{\left( -1\right) ^{n}c^{n}\alpha ^{n}}{a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n}}. \end{eqnarray*} Similarly, one can obtain from system (4) that \begin{eqnarray*} y_{4n-3} &=&\frac{y_{4n-5}x_{4n-7}}{x_{4n-5}\left( -1+y_{4n-5}x_{4n-7}\right) } \\ &=&\frac{\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}}\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}}{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}}{\frac{c^{n}\alpha ^{n-1}\left( a\gamma -1\right) ^{n-1}}{ a^{n-1}\gamma ^{n-1}}\left[ -1+\frac{\left( -1\right) ^{n-1}a^{n-1}\gamma ^{n}\left( c\alpha +1\right) ^{n-1}}{c^{n-1}\alpha ^{n-1}}\frac{\left( -1\right) ^{n-1}c^{n-1}\alpha ^{n-1}}{a^{n-2}\gamma ^{n-1}\left( c\alpha +1\right) ^{n-1}}\right] } \end{eqnarray*} \begin{eqnarray*} &=&\frac{a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\left( a\gamma -1\right) ^{n-1} \left[ -1+a\gamma \right] }=\frac{a^{n}\gamma ^{n}}{c^{n}\alpha ^{n-1}\left( a\gamma -1\right) ^{n}}. \end{eqnarray*} Other relations can be likewise proved. Therefore, this completes our proof.

2.5. Numerical Examples

This subsection is allocated to confirm our theoretical discussion by illustrating some numerical examples. These examples show the behaviour of the solutions of each system.

Example 1. In this example, we describe the behaviour of the solution of system (1). Our initial data has been taken as follows: \(x_{-3}=0.21,\ x_{-2}=-0.25,\ x_{-1}=-0.032,\ x_{0}=2,\ y_{-3}=1.06,\ y_{-2}=-0.4,\ y_{-1}=-1.55\) and \(\ y_{0}=0.082.\) See Figure 1.

Example 2. This example demonstrates the plot of system (2). The initial values are considered as follows: \( x_{-3}=0.22,\ x_{-2}=-0.25,\ x_{-1}=0.2,\ x_{0}=0.6,\ y_{-3}=1.03,\ y_{-2}=-0.43,\ y_{-1}=1.5\) and \(\ y_{0}=0.8,\) as depicted in Figure 2.

Example 3. Here, we plot the curves of solutions of system (3). Figure 3 presents this behaviour under the following initial values: \(x_{-3}=0.03,\ x_{-2}=-0.2,\ x_{-1}=10,\ x_{0}=-0.05,\ y_{-3}=2,\ y_{-2}=-4.1,\ y_{-1}=1.4\) and \(\ y_{0}=1.03.\)

Example 4. Figure \ref{fig4} illustrates the behaviour of the solution of system (4). The considered initial conditions in this example are given as follows: \(x_{-3}=0.02,\ x_{-2}=-0.2,\ x_{-1}=0.21,\ x_{0}=-0.03,\ y_{-3}=0.1,\ y_{-2}=-4.9,\ y_{-1}=1,\ y_{0}=-1.\)

3. Conclusion

This paper has been written to highlight the analytical and numerical solutions of four different systems of difference equations. In subsection 1, we have provided the solution of System 1 and illustrated its behaviour under specific conditions in Figure 1. Theorem 2 and Theorem 3 presented the exact solutions of System 2 and System 3, respectively. Finally, Figure 4 demonstrated the curve of the solution of System 4 which is given in Subsection 4.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.