A detailed study on a solvable system related to the linear fractional difference equation

1.   Introduction and preliminaries

Nonlinear difference equations have long interested both mathematics and other sciences. Since these equations play a key role in many applications such as the natural model of a discrete process, they appear in many disciplines such as population biology, optics, economics, probability theory, genetics, psychology. See e.g., ^{[1,2,3,4,5,6]} and the references therein. For the last two decades, there has been interest in studying the global attractivity, the boundedness character and the periodic nature of nonlinear difference equations. For some recent results see, for example, ^{[7,8,9,10,11,12,13,14,15,16]}. However, for the last decade, some researchers have focused on the solvability of nonlinear difference equations and their systems. For some recent results see, for example, ^{[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]}.

In this paper, we consider the following system of second-order nonlinear difference equations

where the parameters $a$, $b$, and the initial values $x_{-1}$, $x_{0}$, $y_{-1}$, $y_{0}$ are arbitrary real numbers such that the solution $\left\{ \left(\ x_{n}, y_{n}\right) \right\} _{n\geq -1}$ exists. System (1.1) can be obtained systematically as follows. First, we consider the following difference equation

where the parameters $a$, $b$, $c$, $d$, and the initial value $x_{0}$ are arbitrary such that $x_{n}$ are defined. Equation (1.2) is called a first order linear fractional difference equation. Equation (1.2) is solvable by virtue of several changes of variables. The most common method is to transform Eq (1.2) into the second order linear equation by using the change of variables $c+dx_{n} = z_{n}/z_{n-1}$. For a detailed background on Eq (1.2), see e.g, ^[34,35]. Also, for other equations related to Eq (1.2), see ^{[36,37,38,39]}. A different case occurs when we get $b = 0$. This case yields the following difference equation

Equation (1.3) can also be transformed into the second order linear equation by using the change of variables $x_{n} = z_{n-1}/z_{n}$ and so is solvable. Some generalizations of Eq (1.3) can inherit its solvability property. For example, the following difference equation

where the parameters $a$, $c$, $d$, and the initial values $x_{-1}$, $x_{0}$ are arbitrary such that $x_{n}$ are defined, is also solvable by using the change of variables $x_{n} = z_{n-1}/z_{n}$. Hence, the general solutions of (1.2)–(1.4) follow from the general solutions of the associated linear equations and the corresponding changes of variables. Note that both Eq (1.2) and Eq (1.3) can be reduced equations with one parameter or two parameters. If we choose $x_{n} = \frac{c}{d}u_{n}$, $\frac{ad }{c^{2}} = \alpha$, and $x_{n} = \sqrt{\frac{c}{d}}v_{n}$, $\frac{a}{c}\sqrt{ \frac{d}{c}} = \beta$, then they are reduced equations with one parameter in $u_{n}$ and $v_{n}$, respectively. Therefore, we can take $c = d = 1$ under favorable conditions.

Based on the above considerations, we investigate a two-dimensional generalization that maintains the solvability characteristic of Eq (1.4). So, we get a further generalization of (1.4), that is, the system given in (1.1). System (1.1) can also be transformed into a system of third-order linear equations by using the changes of variables $x_{n} = u_{n-1}/v_{n}$, $y_{n} = v_{n-1}/u_{n}$, and so can be solved. But, we will use a more practical method introduced firstly in ^[40] to solve the system.

We need to the following two results in the sequel of our study.

Lemma 1.1. ^[41] Consider the cubic equation

Equation (1.5) has the discriminant

Then, the following statements are true:

(i) If $\Delta < 0$, then the polynomial $P$ has three distinct real zeros $\rho _{1}$, $\rho _{2}$, $\rho _{3}$.

(ii) If $\Delta = 0$, then there are two subcases:

(a) if $\beta = \frac{-\alpha ^{2}}{3}\ $and$\ \gamma = \frac{\alpha ^{3}}{27}$ , then the polynomial $P$ has the triple root $\rho = \frac{\alpha }{3}$,

(b) if $\beta \neq -\frac{\alpha ^{2}}{3}\ $or$\ \gamma \neq \frac{\alpha ^{3}}{27}$, then the polynomial $P$ has the double root $r$ and the simple root $\rho$.

(iii) If $\Delta > 0$, then the polynomial $P$ has one real root $p$ and two complex roots $re^{\pm i\theta }, \; \theta \in \left(0, \pi \right)$.

Theorem 1.1 (Kronecker's theorem). ^[42] If $\theta$ is irrational, the set of points $u_{n} = n\theta - \left[n\theta \right]$ is dense in the interval $\left(0, 1\right)$.

In the above theorem, $n$ is an integer and $\left[n\theta \right]$ is greatest integer function of $n\theta$.

2.   Main results

This section, which contains our main results, is examined in three subsections.

2.1.   Representation forms of the general solution

In this subsection, by using an interesting and practical method, we solve system (1.1). If we take $a = 0$ in system (1.1), then we have $x_{n} = 0$ for every $n\geq 1$ and $y_{n} = b$ for every $n\geq 2$. If we take $b = 0$ in system (1.1), then we have $x_{n} = a$ for every $n\geq 2$ and $y_{n} = 0$ for every $n\geq 1$. So, to enable the use of the method, we suppose $ab\neq 0$ in the sequel of our study.

We start by writing system (1.1) in the following

for every $n\geq 0$. By multiplying (2.1), (2.2), (2.3) and (2.4) by

we have the followings

for every $n\geq 0$, respectively. In fact, the equalities (2.9)–(2.12) constitute a linear system with respect to (2.5)–(2.8). Hence, we should solve (2.9)–(2.12). By using (2.9) in (2.12), (2.12) in (2.9) and similarly by using (2.10) in (2.11), (2.11) in (2.10), we have the following statements

and

for every $n\geq 2$. Note that the equations in (2.13)–(2.16) are linear with respect to (2.5), (2.8), (2.6) and (2.7), respectively, and they can be represented by the following third-order difference equation

whose characteristic equation is the following equation

By Lemma 1.1, we see that there are three cases to be considered.

2.1.1.   The case $4/ab < -27$

In this case $P$ has three real distinct zeros denoted by $\rho _{1}$, $\rho _{2}$, $\rho _{3}$, respectively. Hence, from (2.13) and (2.17), we can write

from which it follows that

where $C_{1}$, $C_{2}$, $C_{3}$ are arbitrary real constants given by

for every $n\geq 0$. By using (2.20) in the first equation of system (1.1), we have

for every $n\geq -1$. On the other hand, the first equation of system (1.1) can be written as follows

for every $n\geq 0$. By using (2.21) and its backward shifted one from $n$ to $n-1$ in (2.22), we have

for every $n\geq 0$. Now, we consider Eq (2.16), which is linear with respect to (2.7) and (2.17). Hence, we have

from which it follows that

where $C_{1}^{\prime }$, $C_{2}^{\prime }$, $C_{3}^{\prime }$ are arbitrary real constants given by

for every $n\geq 0$. By using (2.25) in the second equation of system (1.1), we have

for every $n\geq -1$. On the other hand, the second equation of system (1.1) can be written as follows

for every $n\geq 0$. By using (2.26) and its backward shifted one from $n$ to $n-1$ in (2.27), we have

for every $n\geq 0$. Consequently, in the case $\frac{4}{ab} < -27$, the representation forms of the general solution of system (1.1) are given by (2.21), (2.23), (2.26) and (2.28).

2.1.2.   Case $4/ab=-27$

In this case $P\left(\lambda \right)$ has the simple root $\rho$ and the double root $r$. Moreover, since $ab = -\frac{4}{27}$, we have $\rho = -3/4$ and $r = -3$. Hence, from (2.13) and (2.17), we have

from which it follows that

where $C_{1}$, $C_{2}$, $C_{3}$ are arbitrary real constants given by

for every $n\geq 0$. By substituting (2.30) in the first equation of system (1.1), we have

for every $n\geq -1$. On the other hand, by using (2.31) and its backward shifted one from $n$ to $n-1$ in (2.22), we have

for every $n\geq 0$. Now, by considering Eqs (2.16) and (2.17). we have

from which it follows that

where $C_{1}^{\prime }$, $C_{2}^{\prime }$, $C_{3}^{\prime }$ are arbitrary real constants given by

for every $n\geq 0$. By substituting (2.34) in the second equation of system (1.1), we have

for every $n\geq -1$. On the other hand, by using (2.35) and its backward shifted one from $n$ to $n-1$ in (2.27), it follows that

for every $n\geq 0$. Consequently, in the case $\frac{4}{ab} = -27$, the representation forms of the general solution of system (1.1) are given by (2.31), (2.32), (2.35) and (2.36).

2.1.3.   Case $4/ab > -27$

In this case $P\left(\lambda \right)$ has one real root and two complex roots denoted by $\rho$ and $re^{\pm i\theta }, \; \theta \in \left(0, \pi \right)$, respectively. Hence, from (2.13) and (2.17), we have

from which it follows that

where $C_{1}$, $C_{2}$, $C_{3}$ are arbitrary real constants given by

for every $n\geq 0$. By using (2.38) in the first equation of system (1.1), we have

where $C_{4} = C_{2}\left(r+\cos \theta \right) -C_{3}\sin \theta$, $C_{5} = C_{3}\left(r+\cos \theta \right) +C_{2}\sin \theta$, for every $n\geq -1$. On the other hand, by using (2.39) and its backward shifted one from $n$ to $n-1$ in (2.22), we have

for every $n\geq 0$. Now, by considering Eqs (2.16) and (2.17), we have

from which it follows that

where $C_{1}^{\prime }$, $C_{2}^{\prime }$, $C_{3}^{\prime }$ are arbitrary real constants given by

for every $n\geq 0$. By using (2.38) in the second equation of system (1.1), we have

where $C_{4}^{\prime } = C_{2}^{\prime }\left(r+\cos \theta \right) -C_{3}^{\prime }\sin \theta$, $C_{5}^{\prime } = C_{3}^{\prime }\left(r+\cos \theta \right) +C_{2}^{\prime }\sin \theta$, for every $n\geq -1$. On the other hand, by using (2.43) and its backward shifted one from $n$ to $n-1$ in (2.27), we have

for every $n\geq 0$. Consequently, in the case $\frac{4}{ab} > -27$, the representation forms of the general solution of system (1.1) are given by (2.39), (2.40), (2.43) and (2.44).

2.2.   Forbidden set and well-defined solutions

The representation forms given in the previous subsection are valid where the denominators are not zero. That is, we can obtain the set of initial values that make the solutions of the system undefined from the forms by equating their denominators to zero. This operation enables us to obtain a set of initial values that produce the well-defined solutions of system (1.1). In the following we give a theorem that helps us characterize such solutions.

Theorem 2.1. Consider system (1.1). Then, the following statements are true:

(a) If $\frac{4}{ab} < -27$, then the forbidden set of system (1.1) is given by

where

(b) If $\frac{4}{ab} = -27$, then the forbidden set of system (1.1) is given by

where

and $\rho = -3/4$, $r = -3$.

(c) If $\frac{4}{ab} > -27$, then the forbidden set of system (1.1) is given by

where

Proof. The proof is simple and follows by equalizing denominators of the representation forms obtained in the previous section to zero.

By considering this theorem, we say that a well-defined solution of system (1.1) is a solution $\left\{ \left(x_{n}, y_{n}\right) \right\} _{n\geq -1}$ obtained using the initial values such that $\left(x_{-1}, x_{0}, y_{-1}, y_{0}\right)\in \mathbb{R}^4 \backslash F$.

2.3.   Long-term behavior of the solutions

In this subsection we study the long-term behavior of the solutions of system (1.1) by using the representation forms obtained in the first subsection. We analyze the solutions in the following cases of the parameter $ab$:

i) Case $\frac{4}{ab} < -27$: in this case we have $ab\in \left(-\frac{4}{27}, 0\right)$.

ii) Case $\frac{4}{ab} = -27$: in this case we have $ab = \frac{-4}{27}$.

iii) Case $\frac{4}{ab} > -27$: in this case we have $ab\in \left(-\infty, - \frac{4}{27}\right) \cup \left(0, +\infty \right)$.

2.3.1.   Case $4/ab < -27$

This case yields the following result.

Theorem 2.2. Let $\left\{ \left(x_{n}, y_{n}\right) \right\} _{n\geq -1}$ be a well-defined solution of system (1.1). Suppose that $\frac{4}{ab} < -27$. Then, the following statements are true:

(a) If $C_{i}\neq 0$ for $i\in \left\{ 1, 2, 3\right\}$ and $\left\vert \rho \right\vert = \max \left\{ \left\vert \rho _{1}\right\vert, \left\vert \rho _{2}\right\vert, \left\vert \rho _{3}\right\vert \right\}$, then $x_{2n+1}\rightarrow \frac{a\rho }{\rho +1}$ and $y_{2n}\rightarrow \frac{ b\rho }{\rho +1}$ as $n\rightarrow \infty$.

(b) If $C_{i}^{\prime }\neq 0$ for $i\in \left\{ 1, 2, 3\right\}$ and $\left\vert \rho \right\vert = \max \left\{ \left\vert \rho _{1}\right\vert, \left\vert \rho _{2}\right\vert, \left\vert \rho _{3}\right\vert \right\}$ , then $x_{2n}\rightarrow \frac{a\rho }{\rho +1}$ and $y_{2n+1}\rightarrow \frac{b\rho }{\rho +1}$ as $n\rightarrow \infty$.

(c) If $C_{i} = 0$ and $C_{j}C_{k}\neq 0$ for $i, j, k\in \left\{ 1, 2, 3\right\}$ with $i\neq j\neq k$ and $\left\vert \rho \right\vert = \max \left\{ \left\vert \rho _{j}\right\vert, \left\vert \rho _{k}\right\vert \right\}$, then $x_{2n+1}\rightarrow \frac{a\rho }{\rho +1}$ and $y_{2n}\rightarrow \frac{b\rho }{\rho +1}$ as $n\rightarrow \infty$.

(d) If $C_{i}^{\prime } = 0$ and $C_{j}^{\prime }C_{k}^{\prime }\neq 0$ for $i, j, k\in \left\{ 1, 2, 3\right\}$ with $i\neq j\neq k$ and $\left\vert \rho \right\vert = \max \left\{ \left\vert \rho _{j}\right\vert, \left\vert \rho _{k}\right\vert \right\}$, then $x_{2n}\rightarrow \frac{a\rho }{\rho +1}$ and $y_{2n+1}\rightarrow \frac{b\rho }{\rho +1}$ as $n\rightarrow \infty$.

Proof. (a)–(b) Let us assume without losing generality that $\left\vert \rho _{1}\right\vert = \max \left\{ \left\vert \rho _{1}\right\vert, \left\vert \rho _{2}\right\vert, \left\vert \rho _{3}\right\vert \right\}$. Then, we have the following limits

Since $\rho _{1}$ is a zero of the polynomial $P$, we have the relation

from (2.18). Hence, we have

and

The proofs of other cases are similar and so they will be omitted.

Remark 2.1. Note that the cases $\left\vert C_{1}\right\vert +\left\vert C_{2}\right\vert +\left\vert C_{3}\right\vert = \left\vert C_{i}\right\vert$ and $\left\vert C_{1}^{\prime }\right\vert +\left\vert C_{2}^{\prime }\right\vert +\left\vert C_{3}^{\prime }\right\vert = \left\vert C_{j}\right\vert$, $i, j\in \left\{ 1, 2, 3\right\}$ are impossible. Because, for example, if $C_{1} = C_{2} = 0$, then we need to the common solution of the system

This case requires that $\rho _{1} = \rho _{2}$ which is a contradiction.

Corollary 2.1. Suppose that $\frac{4}{ab} < -27$. Then, every well-defined solution of system (1.1) has a finite limit point.

2.3.2.   Case $4/ab=-27$

This case yields the following result.

Theorem 2.3. Let $\left\{ \left(x_{n}, y_{n}\right) \right\} _{n\geq -1}$ be a well-defined solution of system (1.1). Suppose that $\frac{4}{ab} = -27$. Then, the following statements are true:

(a) If $\left\vert C_{2}\right\vert +\left\vert C_{3}\right\vert \neq 0$, then $x_{2n+1}\rightarrow \frac{3a}{2}$and $y_{2n}\rightarrow \frac{3b}{2}$ as $n\rightarrow \infty$.

(b) If $\left\vert C_{2}^{\prime }\right\vert +\left\vert C_{3}^{\prime }\right\vert \neq 0$, then $x_{2n}\rightarrow \frac{3a}{2}$ and $y_{2n+1}\rightarrow \frac{3b}{2}$ as $n\rightarrow \infty$.

(c) If $C_{2} = C_{3} = 0$ and $C_{1}\neq 0$, then $x_{2n+1}\rightarrow -3a$ and $y_{2n}\rightarrow -3b$ as $n\rightarrow \infty$.

(d) If $C_{2}^{\prime } = C_{3}^{\prime } = 0$ and $C_{1}^{\prime }\neq 0$, then $x_{2n}\rightarrow -3a$, and $y_{2n+1}\rightarrow -3b$ as $n\rightarrow \infty$.

Proof. (a)–(b) Since $\rho = -3/4$ and $r = -3$, we have $\left\vert \rho \right\vert < \left\vert r\right\vert$. So, from (2.31), (2.32), (2.35) and (2.36), we have the following limits

The proofs of (c) and (d) are clear from the forms in (2.31), (2.32), (2.35) and (2.36).

Corollary 2.2. Suppose that $\frac{4}{ab} = -27$. Then, every well-defined solution of system (1.1) has a finite limit point.

2.3.3.   Case $4/ab > -27$

For this case we first prove the following lemma.

Lemma 2.1. Suppose that $\frac{4}{ab} > -27$ and the zeros of the polynomial $P\left(\lambda \right)$ are $\rho$ and $re^{\pm i\theta }$, $\; r > 0$, $\theta \in \left(0, \pi \right)$. Then, the following statements are true:

(a) If $ab\in \left(0, +\infty \right)$, then $r < \rho$

(b) If $ab\in \left(-\infty, -\frac{4}{27}\right)$, then $r > \left\vert \rho \right\vert$

Proof. Let $\frac{4}{ab} > -27$. Then, since $\rho$ and $re^{\pm i\theta }$ are the zeros of the polynomial $P\left(\lambda \right)$, the relations

are satisfied. We conclude from these relations that $ab\rho > 0$. This implies that if $ab < 0$, then $\rho < 0$ and if $ab > 0$, then $\rho > 0$. Also, from (2.18), we have

which implies

We must consider the following two cases:

(a) If $ab\in \left(0, +\infty \right)$, then $\rho > 0$ and so, from (2.45), we have

(b) If $ab\in \left(-\infty, -\frac{4}{27}\right) \,$, then we see from (2.45) that $\rho < -1$. So, from (2.45), we have

So, the proof is completed.

Remark 2.2. Note that the equality $r = \left\vert \rho \right\vert$ is impossible. Because, in this case we have $\frac{1}{ab} = \rho ^{3}$ which yields $ab =$ $- \frac{1}{8}\notin \left(-\infty, -\frac{4}{27}\right)$.

Theorem 2.4. Let $\left\{ \left(x_{n}, y_{n}\right) \right\} _{n\geq -1}$ be a well-defined solution of system (1.1). Suppose that $\frac{4}{ab} > -27$. Then, the following statements are true:

(a) If $ab\in \left(0, +\infty \right)$ and $C_{1}\neq 0$, then $x_{2n+1}\rightarrow \frac{a\rho }{\rho +1}$ and $y_{2n}\rightarrow \frac{ b\rho }{\rho +1}$ as $n\rightarrow \infty$.

(b) If $ab\in \left(0, +\infty \right)$ and $C_{1}^{\prime }\neq 0$, then $x_{2n}\rightarrow \frac{a\rho }{\rho +1}$ and $y_{2n+1}\rightarrow \frac{ b\rho }{\rho +1}$ as $n\rightarrow \infty$.

(c) If $ab\in \left(-\infty, -\frac{4}{27}\right)$ and $\left\vert C_{2}\right\vert +\left\vert C_{3}\right\vert \neq 0$, then both $x_{2n+1}$ and $y_{2n}$ are periodic or converge to a periodic solution or dense in $\mathbb{R}$.

(d) If $ab\in \left(-\infty, -\frac{4}{27}\right)$ and $\left\vert C_{2}^{\prime }\right\vert +\left\vert C_{3}^{\prime }\right\vert \neq 0$, then both $x_{2n}$ and $y_{2n+1}$ are periodic or converge to a periodic solution or dense in $\mathbb{R}$.

Proof. (a)–(b) The proof follows from the formulas given in (2.39), (2.40), (2.43), (2.44) and Lemma 2.1 by taking the limit.

(c)–(d) Since the proof is similar for the forms given in (2.39), (2.40), (2.43), (2.44), we only prove for (2.44). Suppose that $C_{1}^{\prime } = 0$. Then, from (2.44), we have

where $C_{4}^{\prime } = C_{2}^{\prime }\left(r+\cos \theta \right) -C_{3}^{\prime }\sin \theta$, $C_{5}^{\prime } = C_{3}^{\prime }\left(r+\cos \theta \right) +C_{2}^{\prime }\sin \theta$. Also, we see that the form given in (2.46) can be written as

where $\gamma$ is an arbitrary constant, which corresponds to the arbitrary constants $C_{2}^{\prime }$, $C_{3}^{\prime }$, and satisfies the equality

Now, we consider the following two cases:

(i) If $\theta = \frac{p}{q}\pi$ such that $p, q$ are co-prime integers, we have

which implies $x_{2n+q} = x_{2n}$. Suppose that $C_{1}^{\prime }\neq 0$. Then, since the inequality $r > \left\vert \rho \right\vert$ holds, from (2.44), we have

which leads to

for large enough values of $n$. Hence, the sequence $\left(x_{2n}\right) _{n\geq 0}$ converges to a periodic solution obtained in the case $C_{1}^{\prime } = 0$.

(ii) If $\theta = t\pi$ such that $t$ is irrational, then we have by virtue of the Kronecker's Theorem that the set $\left\{ \left(nt-\left[nt\right] \right) \pi :n\in \mathbb{N} _{0}\text{ and }t\ \text{is irrational}\right\}$ is dense in the interval $\left(0, \pi \right)$. Hence, we have

which implies the sequence $\left(x_{2n}\right) _{n\geq 0}$ is dense in $\left(-\infty, +\infty \right)$.

3.   Conclusions

In this paper we conducted a detailed analysis on all solutions of system (1.1). To do this analysis, we obtained the representation forms of general solution of the system by using a practical method. By using these forms, we characterized the well-defined solutions of system (1.1). Finally, we studied the long-term behavior of the well-defined solutions. We can summarize our results as follows:

Consider system (1.1). Then,

(a) If $ab\in \left(-\frac{4}{27}, 0\right)$, then every well-defined solution of system (1.1) has a finite limit point.

(b) If $ab\in \left(0, +\infty \right)$ and $C_{1}C_{1}^{\prime }\neq 0$, then every well-defined solution of system (1.1) has a finite limit point.

(c) If $ab\in \left(-\infty, -\frac{4}{27}\right)$ and $\left(\left\vert C_{2}\right\vert +\left\vert C_{3}\right\vert \right) \left(\left\vert C_{2}^{\prime }\right\vert +\left\vert C_{3}^{\prime }\right\vert \right) \neq 0$, then every well-defined solution of system (1.1) is periodic or converges to a periodic solution or dense in $\mathbb{R}^2$.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 71601072), Key Scientific Research Project of Higher Education Institutions in Henan Province of China (No. 20B110006) and the Fundamental Research Funds for the Universities of Henan Province. The authors thank the participants of this study for their valuable contributions.

Conflict of interest

The authors declare that there are no conflict of interest associated with this publication.