Theory and Modern Applications

# Global behavior of the solutions of some difference equations

## Abstract

${x}_{n+1}=\frac{a{x}_{n-l}{x}_{n-k}}{b{x}_{n-p}-c{x}_{n-q}},\phantom{\rule{2.77695pt}{0ex}}n\phantom{\rule{2.77695pt}{0ex}}=0,1,\phantom{\rule{2.77695pt}{0ex}}\dots ,$

where the initial conditions x -r , x -r+1, x -r+2,..., x 0 are arbitrary positive real numbers, r = max{l, k, p, q} is nonnegative integer and a, b, c are positive constants: Also, we study some special cases of this equation.

## 1 Introduction

${x}_{n+1}=\frac{a{x}_{n-l}{x}_{n-k}}{b{x}_{n-p}-c{x}_{n-q}},\phantom{\rule{2.77695pt}{0ex}}n\phantom{\rule{2.77695pt}{0ex}}=0,1,\phantom{\rule{2.77695pt}{0ex}}\dots ,$
(1)

where the initial conditions x -r , x -r+1, x -r+2,..., x 0 are arbitrary positive real numbers, r = max{l, k, p, q} is nonnegative integer and a, b, c are positive constants: Moreover, we obtain the form of the solution of some special cases of Equation 1 and some numerical simulations to the equation are given to illustrate our results.

Let us introduce some basic definitions and some theorems that we need in the sequel.

Let I be some interval of real numbers and let

$f:{I}^{k+1}\to I,$

be a continuously differentiable function. Then for every set of initial conditions x -k , x -k+1,..., x 0 I, the difference equation

${x}_{n+1}=f\left({x}_{n},\phantom{\rule{2.77695pt}{0ex}}{x}_{n-1},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{x}_{n-k}\right),\phantom{\rule{2.77695pt}{0ex}}n=0,1,\phantom{\rule{2.77695pt}{0ex}}\dots ,$
(2)

has a unique solution ${\left\{{x}_{n}\right\}}_{n=-k}^{\infty }\left[1\right]$.

A point $\stackrel{̄}{x}\in I$ is called an equilibrium point of Equation 2 if

$\stackrel{̄}{x}=f\left(\stackrel{̄}{x},\phantom{\rule{2.77695pt}{0ex}}\stackrel{̄}{x},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}\stackrel{̄}{x}\right).$

That is, ${x}_{n}=\stackrel{̄}{x}$ for n ≥ 0, is a solution of Equation 2, or equivalently, $\stackrel{̄}{x}$ is a fixed point of f.

### Definition 1 (Stability)

1. (i)

The equilibrium point $\stackrel{̄}{x}$ of Equation 2 is locally stable if for every ε > 0, there exists δ > 0 such that for all x -k , x -k+1 ,..., x -1, x 0 I with

$|{x}_{-k}-\stackrel{̄}{x}|+|{x}_{-k+1}-\stackrel{̄}{x}|+\cdots +|{x}_{0}-\stackrel{̄}{x}|<\delta ,$

we have

1. (ii)

The equilibrium point $\stackrel{̄}{x}$ of Equation 2 is locally asymptotically stable if $\stackrel{̄}{x}$ is locally stable solution of Equation 2 and there exists γ > 0, such that for all x -k , x -k+1,..., x -1, x 0 I with

$|{x}_{-k}-\stackrel{̄}{x}|+|{x}_{-k+1}-\stackrel{̄}{x}|+\dots +|{x}_{0}-\stackrel{̄}{x}|<\gamma ,$

we have

$\underset{n\to \infty }{lim}\phantom{\rule{0.3em}{0ex}}{x}_{n}=\stackrel{̄}{x}.$
1. (iii)

The equilibrium point $\stackrel{̄}{x}$ of Equation 2 is global attractor if for all x -k , x -k+1,..., x -1, x 0 I, we have

$\underset{n\to \infty }{lim}\phantom{\rule{0.3em}{0ex}}{x}_{n}=\stackrel{̄}{x}.$
2. (iv)

The equilibrium point $\stackrel{̄}{x}$ of Equation 2 is globally asymptotically stable if $\stackrel{̄}{x}$ is locally stable and $\stackrel{̄}{x}$ is also a global attractor of Equation 2.

3. (v)

The equilibrium point $\stackrel{̄}{x}$ of Equation 2 is unstable if $\stackrel{̄}{x}$ is not locally stable.

The linearized equation of Equation 2 about the equilibrium $\stackrel{̄}{x}$ is the linear difference equation

${y}_{n+1}={\sum }_{i=0}^{k}\frac{\partial f\left(\stackrel{̄}{x},\stackrel{̄}{x},\dots ,\stackrel{̄}{x}\right)}{\partial {x}_{n-i}}{y}_{n-i}.$
(3)

#### Theorem A 

Assume that p, q R and k {0, 1, 2,...}. Then

$|p|+|q|<1,$

is a sufficient condition for the asymptotic stability of the difference equation

${x}_{n+1}+p{x}_{n}+q{x}_{n-k}=0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}n\phantom{\rule{2.77695pt}{0ex}}=0,1,\phantom{\rule{2.77695pt}{0ex}}\dots .$

Remark 1 Theorem A can be easily extended to a general linear equations of the form

${x}_{n+k}+{p}_{1}{x}_{n+k-1}+\dots +{p}_{k}{x}_{n}=0,\phantom{\rule{2.77695pt}{0ex}}n=0,1,\phantom{\rule{2.77695pt}{0ex}}\dots ,$
(4)

where p 1, p 2,..., p k R and k {1, 2,...}. Then Equation 4 is asymptotically stable provided that

$\sum _{i=1}^{k}|{p}_{i}|<1.$

### Definition 2

(Fibonacci Sequence) The sequence ${\left\{{F}_{m}\right\}}_{m=0}^{\infty }=\left\{1,2,3,5,8,13,\phantom{\rule{2.77695pt}{0ex}}\dots \right\}$ i.e. F m = F m-1 + F m-2, m ≥ 0, F -2 = 0, F -1 = 1 is called Fibonacci Sequence.

The nature of many biological systems naturally leads to their study by means of a discrete variable. Particular examples include population dynamics and genetics. Some elementary models of biological phenomena, including a single species population model, harvesting of fish, the production of red blood cells, ventilation volume and blood CO2 levels, a simple epidemics model and a model of waves of disease that can be analyzed by difference equations are shown in . Recently, there has been interest in so-called dynamical diseases, which correspond to physiological disorders for which a generally stable control system becomes unstable. One of the first papers on this subject was that of Mackey and Glass . In that paper they investigated a simple first order difference-delay equation that models the concentration of blood-level CO2. They also discussed models of a second class of diseases associated with the production of red cells, white cells, and platelets in the bone marrow.

The study of the nonlinear rational difference equations of a higher order is quite challenging and rewarding, and the results about these equations offer prototypes towards the development of the basic theory of the global behavior of nonlinear difference equations of a big order, recently, many researchers have investigated the behavior of the solution of difference equations for example: Elabbasy et al.  investigated the global stability, periodicity character and gave the solution of special case of the following recursive sequence

${x}_{n+1}=a{x}_{n}-\frac{b{x}_{n}}{c{x}_{n}-d{x}_{n-1}}.$

Elabbasy et al.  investigated the global stability, boundedness, periodicity character and gave the solution of some special cases of the difference equation

${x}_{n+1}=\frac{\alpha {x}_{n-k}}{\beta +\gamma {\prod }_{i=0}^{k}{x}_{n-i}}.$

Elabbasy et al.  investigated the global stability character, boundedness and the periodicity of solutions of the difference equation

${x}_{n+1}=\frac{\alpha {x}_{n}+\beta {x}_{n-1}+\gamma {x}_{n-2}}{A{x}_{n}+B{x}_{n-1}+C{x}_{n-2}}.$

El-Metwally et al.  investigated the asymptotic behavior of the population model:

${x}_{n+1}=\alpha \phantom{\rule{2.77695pt}{0ex}}+\beta {x}_{n-1}{e}^{-{x}_{n}},$

where α is the immigration rate and β is the population growth rate.

Yang et al.  investigated the invariant intervals, the global attractivity of equilibrium points and the asymptotic behavior of the solutions of the recursive sequence

${x}_{n+1}=\frac{a{x}_{n-1}+b{x}_{n-2}}{c+d{x}_{n-1}{x}_{n-2}}.$

Cinar [10, 11] has got the solutions of the following difference equations

${x}_{n+1}=\frac{{x}_{n-1}}{1+a{x}_{n}{x}_{n-1}},\phantom{\rule{2.77695pt}{0ex}}{x}_{n+1}=\frac{{x}_{n-1}}{-1+a{x}_{n}{x}_{n-1}}.$

Aloqeili  obtained the form of the solutions of the difference equation

${x}_{n+1}=\frac{{x}_{n-1}}{a-{x}_{n}{x}_{n-1}}\phantom{\rule{2.77695pt}{0ex}}.$

Yalçinkaya  studied the following nonlinear difference equation

${x}_{n+1}=\alpha \phantom{\rule{2.77695pt}{0ex}}+\frac{{x}_{n-m}}{{x}_{n}^{k}}.$

For some related work see .

The article proceeds as follows. In Sect. 2 we show that when 2a |b - c| + a(b + c) < (b - c)2, then the equilibrium point of Equation 1 is locally asymptotically stable. In Sect. 3 we prove that the equilibrium point of Equation 1 is global attractor. In Sect. 4 we give the solutions of some special cases of Equation 1 and give a numerical examples of each case and draw it by using Matlab 6.5.

## 2 Local stability of Equation 1

In this section we investigate the local stability character of the solutions of Equation 1. Equation 1 has a unique positive equilibrium point and is given by

$\overline{x}=\frac{a{\overline{x}}^{2}}{b\overline{x}-c\overline{x}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}},$

if ab-c, bc, then the unique equilibrium point is $\stackrel{̄}{x}=0.$

Let f : (0, )4 → (0, ∞) be a function defined by

$f\left(u,\phantom{\rule{2.77695pt}{0ex}}v,\phantom{\rule{2.77695pt}{0ex}}w,\phantom{\rule{2.77695pt}{0ex}}s\right)=\frac{auv}{bw-cs}.$
(5)

Therefore, it follows that

${f}_{u}\left(u,\phantom{\rule{2.77695pt}{0ex}}v,\phantom{\rule{2.77695pt}{0ex}}w,\phantom{\rule{2.77695pt}{0ex}}s\right)=\frac{av}{\left(bw-cs\right)},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{f}_{v}\left(u,\phantom{\rule{2.77695pt}{0ex}}v,\phantom{\rule{2.77695pt}{0ex}}w,\phantom{\rule{2.77695pt}{0ex}}s\right)=\frac{au}{\left(bw-cs\right)},$
${f}_{w}\left(u,\phantom{\rule{2.77695pt}{0ex}}v,\phantom{\rule{2.77695pt}{0ex}}w,\phantom{\rule{2.77695pt}{0ex}}s\right)=\frac{-bauv}{{\left(bw-cs\right)}^{2}},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{f}_{s}\left(u,\phantom{\rule{2.77695pt}{0ex}}v,\phantom{\rule{2.77695pt}{0ex}}w,\phantom{\rule{2.77695pt}{0ex}}s\right)=\frac{cauv}{{\left(bw-cs\right)}^{2}},$

we see that

${f}_{u}\left(\stackrel{̄}{x},\phantom{\rule{2.77695pt}{0ex}}\stackrel{̄}{x},\phantom{\rule{2.77695pt}{0ex}}\stackrel{̄}{x},\phantom{\rule{2.77695pt}{0ex}}\stackrel{̄}{x}\right)=\frac{a}{\left(b-c\right)},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{f}_{v}\left(\stackrel{̄}{x},\phantom{\rule{2.77695pt}{0ex}}\stackrel{̄}{x},\phantom{\rule{2.77695pt}{0ex}}\stackrel{̄}{x},\phantom{\rule{2.77695pt}{0ex}}\stackrel{̄}{x}\right)=\frac{a}{\left(b-c\right)},$
${f}_{w}\left(\stackrel{̄}{x},\phantom{\rule{2.77695pt}{0ex}}\stackrel{̄}{x},\phantom{\rule{2.77695pt}{0ex}}\stackrel{̄}{x},\phantom{\rule{2.77695pt}{0ex}}\stackrel{̄}{x}\right)=\frac{-ab}{{\left(b-c\right)}^{2}},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{f}_{s}\left(\stackrel{̄}{x},\phantom{\rule{2.77695pt}{0ex}}\stackrel{̄}{x},\phantom{\rule{2.77695pt}{0ex}}\stackrel{̄}{x},\phantom{\rule{2.77695pt}{0ex}}\stackrel{̄}{x}\right)=\frac{ac}{{\left(b-c\right)}^{2}}.$

The linearized equation of Equation 1 about $\stackrel{̄}{x}$ is

${y}_{n+1}+\frac{a}{\left(b-c\right)}{y}_{n-1}+\frac{a}{\left(b-c\right)}{y}_{n-k}-\frac{ab}{{\left(b-c\right)}^{2}}{y}_{n-p}+\frac{ac}{{\left(b-c\right)}^{2}}{y}_{n-q}=0.$
(6)

### Theorem 1

Assume that

$a\left(3\zeta -\eta \right)<{\left(b-c\right)}^{2}.$

where ζ = max{b, c}, η = min{b, c}. Then the equilibrium point of Equation 1 is locally asymptotically stable.

Proof: It is follows by Theorem A that Equation 6 is asymptotically stable if

$\left|\frac{a}{\left(b-c\right)}\right|+\left|\frac{a}{\left(b-c\right)}\right|+\left|\frac{ab}{{\left(b-c\right)}^{2}}\right|+\left|\frac{ab}{{\left(b-c\right)}^{2}}\right|<1,$

or

$\left|\frac{2a}{\left(b-c\right)}\right|+\left|\frac{a\left(b+c\right)}{{\left(b-c\right)}^{2}}\right|<1,$

and so

$2a|b-c|+a\left(b+c\right)<{\left(b-c\right)}^{2}.$

The proof is complete.

## 3 Global attractivity of the equilibrium point of Equation 1

In this section we investigate the global attractivity character of solutions of Equation 1.

We give the following two theorems which is a minor modification of Theorem A.0.2 in .

### Theorem 2

Let [a, b] be an interval of real numbers and assume that

$f:{\left[a,b\right]}^{k+1}\to \left[a,b\right],$

is a continuous function satisfying the following properties:

1. (i)

f(x 1, x 2,...., x k+1) is non-increasing in one component (for example x t ) for each x r (rt) in [a, b] and non-decreasing in the remaining components for each x t in [a, b].

2. (ii)

If $\left(m,\phantom{\rule{2.77695pt}{0ex}}M\right)\in \left[a,\phantom{\rule{2.77695pt}{0ex}}b\right]×\left[a,\phantom{\rule{2.77695pt}{0ex}}b\right]$ is a solution of the system

M = f(M, M,...,M, m, M,...,M, M) and m = f(m, m,...,m, M, m,...m, m) implies

$m=M.$

Then Equation 2 has a unique equilibrium $\stackrel{̄}{x}\in \left[a,\phantom{\rule{2.77695pt}{0ex}}b\right]$ and every solution of Equation 2 converges to $\stackrel{̄}{x}$

Proof: Set

${m}_{0}=a\phantom{\rule{0.3em}{0ex}}\mathsf{\text{and}}\phantom{\rule{0.3em}{0ex}}{M}_{0}=b,$

and for each i = 1, 2,...set

${m}_{i}=f\left({m}_{i-1},\phantom{\rule{2.77695pt}{0ex}}{m}_{i-1},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{m}_{i-1},\phantom{\rule{2.77695pt}{0ex}}{M}_{i-1},\phantom{\rule{2.77695pt}{0ex}}{m}_{i-1},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{m}_{i-1},\phantom{\rule{2.77695pt}{0ex}}{m}_{i-1}\right),$

and

${M}_{i}=f\left({M}_{i-1},\phantom{\rule{2.77695pt}{0ex}}{M}_{i-1},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{M}_{i-1},\phantom{\rule{2.77695pt}{0ex}}{m}_{i-1},\phantom{\rule{2.77695pt}{0ex}}{M}_{i-1},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{M}_{i-1},\phantom{\rule{2.77695pt}{0ex}}{M}_{i-1}\right).$

Now observe that for each i ≥ 0,

$a\phantom{\rule{2.77695pt}{0ex}}={m}_{0}\le {m}_{1}\le \dots \le {m}_{i}\le \dots \le {M}_{i}\le \dots \le {M}_{1}\le {M}_{0}=b,$

and

${m}_{i}\le {x}_{p}\le {M}_{i}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{for}}\phantom{\rule{0.3em}{0ex}}p\ge \left(k+1\right)i+1.$

Set

$m=\underset{x\to \infty }{lim}{m}_{i}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{and}}\phantom{\rule{0.3em}{0ex}}M=\underset{i\to \infty }{lim}{M}_{i}.$

Then

$M\ge \underset{i\to \infty }{lim}sup{x}_{i}\ge lim\phantom{\rule{2.77695pt}{0ex}}\underset{i\to \infty }{inf}{x}_{i}\ge m$

and by the continuity of f,

M = f(M, M,...,M, m, M,...,M, M) and m = f(m, m,...,m, M, m,...m, m).

In view of (ii),

$m=M=\stackrel{̄}{x},$

from which the result follows.

### Theorem 3

Let [a, b] be an interval of real numbers and assume that

$f:{\left[a,b\right]}^{k+1}\to \left[a,b\right],$

is a continuous function satisfying the following properties:

1. (i)

f(x 1, x 2,...,x k+1) is non-increasing in one component (for example x t ) for each x r (rt) in [a, b] and non-increasing in the remaining components for each x t in [a, b].

2. (ii)

If (m, M)[a, b] × [a, b] is a solution of the system

M = f(m, m,...,m, M, m,...m, m) and m = f(M, M,...,M, m, M,...,M, M)' implies

$m=M.$

Then Equation 2 has a unique equilibrium $\stackrel{̄}{x}\in \left[a,\phantom{\rule{2.77695pt}{0ex}}b\right]$ and every solution of Equation 2 converges to $\stackrel{̄}{x}$

Proof: As the proof of Theorem 2 and will be omitted.

### Theorem 4

The equilibrium point $\stackrel{̄}{x}$ of Equation 1 is global attractor if ca.

Proof: Let p, q are a real numbers and assume that $f:{\left[p,\phantom{\rule{2.77695pt}{0ex}}q\right]}^{4}\to \left[p,\phantom{\rule{2.77695pt}{0ex}}q\right]$ be a function defined by Equation 5, then we can easily see that the function f(u, v, w, s) increasing in s and decreasing in w.

Case (1) If bw-cs > 0, then we can easily see that the function f(u, v, w, s)

increasing in u, v, s and decreasing in w.

Suppose that (m, M) is a solution of the system

M = f(m, m, M, m) and M = f(M, M, m, M).

Then from Equation 1, we see that

$m=\frac{a{m}^{2}}{bM-cm},\phantom{\rule{1em}{0ex}}M=\frac{a{M}^{2}}{bm-cM},$
$bM\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}cm+am,\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}bm=cM+aM,$

then

$\left(M-m\right)\left(b+c+a\right)\phantom{\rule{2.77695pt}{0ex}}=0.$

Thus

$M=m.$

It follows by Theorem 2 that $\stackrel{̄}{x}$ is a global attractor of Equation 1 and then the proof is complete.

Case (2) If bw-cs < 0, then we can easily see that the function f(u, v, w, s) decreasing in u, v, w and increasing in s.

Suppose that (m, M) is a solution of the system

M = f(m, m, m, M) and m = f(M, M, M, m).

Then from Equation 1, we see that

$M\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}\frac{a{m}^{2}}{bm-cM},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}m=\frac{a{M}^{2}}{bM-cm},$
$bmM-c{M}^{2}\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}a{m}^{2},\phantom{\rule{2.77695pt}{0ex}}bmM-c{m}^{2}=a{M}^{2},$

then

$\left({M}^{2}-{m}^{2}\right)\left(c-a\right)\phantom{\rule{2.77695pt}{0ex}}=0,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}a\phantom{\rule{2.77695pt}{0ex}}\ne c.$

Thus,

$M=m.$

It follows by the Theorem 3 that $\stackrel{̄}{x}$ is a global attractor of Equation 1 and then the proof is complete.

## 4 Special cases of Equation 1

### 4.1 Case (1)

In this section we study the following special case of Equation 1

${x}_{n+1}=\frac{{x}_{n}{x}_{n-1}}{{x}_{n}-{x}_{n-1}},$
(7)

where the initial conditions x -1, x 0 are arbitrary positive real numbers.

#### Theorem 5

Let ${\left\{{x}_{n}\right\}}_{n=-1}^{\infty }$ be a solution of Equation 7. Then for n = 0, 1,...

${x}_{n}=\frac{{\left(-1\right)}^{n}hk}{{F}_{n-1}k-{F}_{n-2}h},$

where x -1 = k, x 0 = h and F n-1 , F n-2are the Fibonacci terms.

Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n-1, n-2. That is;

${x}_{n-2}=\frac{{\left(-1\right)}^{n-2}hk}{{F}_{n-3}k-{F}_{n-4}h},\phantom{\rule{2.77695pt}{0ex}}{x}_{n-1}=\frac{{\left(-1\right)}^{n-1}hk}{{F}_{n-2}k-{F}_{n-3}h}.$

Now, it follows from Equation 7 that

Hence, the proof is completed.

For confirming the results of this section, we consider numerical example for x -1 = 11, x 0 = 4 (see Figure 1), and for x -1 = 6, x 0 = 15 (see Figure 2), since the solutions take the forms {6, -12, 4, -3, 1.714286, -1.090909, .6666667, -.4137931, .2553191,....}, {-60, 10, -8.571428, 4.615385, -3, 1.818182, -1.132075, .6976744,...}.

### 4.2 Case (2)

In this section we study the following special case of Equation 1

${x}_{n+1}=\frac{{x}_{n-1}{x}_{n-2}}{{x}_{n-1}-{x}_{n-2}},$
(8)

where the initial conditions x -2, x -1, x 0 are arbitrary positive real numbers.

### Theorem 6

Let ${\left\{{x}_{n}\right\}}_{n=-2}^{\infty }$ be a solution of Equation 8. Then ${x}_{1}=\frac{rk}{k-r}$, for n = 1, 2,...

${x}_{n+1}=\frac{hkr}{{g}_{n-4}hk+{g}_{n-3}kr+{g}_{n-2}hr},$

where x -2 = r, x -1 = k, x 0 = h, ${\left\{{g}_{m}\right\}}_{m=0}^{\infty }=\left\{1,\phantom{\rule{2.77695pt}{0ex}}-2,0,3,\phantom{\rule{2.77695pt}{0ex}}-2,\phantom{\rule{2.77695pt}{0ex}}-3,\phantom{\rule{2.77695pt}{0ex}}\dots \right\},$ i.e., g m = g m-2+ g m-3, m ≥ 0, g -3 = 0, g -2 = -1, g -1 = 1.

Proof: For n = 1, 2 the result holds. Now suppose that n > 1 and that our assumption holds for n - 1, n - 2. That is;

${x}_{n-2}=\frac{hkr}{{g}_{n-7}hk+{g}_{n-6}kr+{g}_{n-5}hr},$ ${x}_{n-1}=\frac{hkr}{{g}_{n-6}hk+{g}_{n-5}kr+{g}_{n-4}hr}.$ Now, it follows from Equation 8 that

$\begin{array}{lll}\hfill {x}_{n+1}& =\frac{{x}_{n-1}{x}_{n-2}}{{x}_{n-1}-{x}_{n-2}}\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\frac{\left(\frac{hkr}{{g}_{n-6}hk+{g}_{n-5}kr+{g}_{n-4}hr}\right)\left(\frac{hkr}{{g}_{n-7}hk+{g}_{n-6}kr+{g}_{n-5}hr}\right)}{\left(\frac{hkr}{{g}_{n-6}hk+{g}_{n-5}kr+{g}_{n-4}hr}-\frac{hkr}{{g}_{n-7}hk+{g}_{n-6}kr+{g}_{n-5}hr}\right)}\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =\frac{hkr}{\left({g}_{n-7}hk+{g}_{n-6}kr+{g}_{n-5}hr-{g}_{n-6}hk+{g}_{n-5}kr+{g}_{n-4}hr\right)}\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ =\frac{hkr}{{g}_{n-4}hk+{g}_{n-3}kr+{g}_{n-2}hr}.\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$

Hence, the proof is completed.

Assume that x -2 = 8, x -1 = 15, x 0 = 7, then the solution will be {17.14286, -13.125, 11.83099, 7.433628, -6.222222, -20, 3.387097, -9.032259,...}(see Figure 3).

The proof of following cases can be treated similarly.

### 4.3 Case (3)

Let x -2 = r, x -1 = k, x 0 = h, $\prod _{i=0}^{-1}{A}_{i}=1$ and F 2i-1, F 2i , F 2i+1(where i = 0 to n) are the Fibonacci terms. Then the solution of the difference equation

${x}_{n+1}=\frac{{x}_{n-1}{x}_{n-2}}{{x}_{n}-{x}_{n-2}},$
(9)

is given by

${x}_{2n}=\frac{h\prod _{i=0}^{n-1}\left({F}_{2i-1}h-{F}_{2i}r\right)}{\prod _{i=0}^{n-1}\left({F}_{2i+1}r-{F}_{2i}h\right)},\phantom{\rule{0.3em}{0ex}}{x}_{2n+1}=\frac{kr\prod _{i=0}^{n-1}\left({F}_{2i+1}r-{F}_{2i}h\right)}{\prod _{i=0}^{n}\left({F}_{2i-1}h-{F}_{2i}r\right)},\phantom{\rule{0.3em}{0ex}}n=0,1,...\phantom{\rule{2.77695pt}{0ex}}.$

Figure 4 shows the solution when x -2 = 9, x -1 = 12, x 0 = 17.

### 4.4 Case (4)

Let x -2 = r, x -1 = k, x 0 = h. Then the solution of the following difference equation

${x}_{n+1}=\frac{{x}_{n-1}{x}_{n}}{{x}_{n}-{x}_{n-2}}$
(10)

is given by

${x}_{2n-1}={\left(\frac{h}{h-r}\right)}^{n}k,\phantom{\rule{1em}{0ex}}{x}_{2n}=\frac{{h}^{n+1}}{{r}^{n}},\phantom{\rule{2.77695pt}{0ex}}n\phantom{\rule{2.77695pt}{0ex}}=0,1,\phantom{\rule{2.77695pt}{0ex}}\dots \phantom{\rule{2.77695pt}{0ex}}.$

Figure 5 shows the solution when x - 2= 21, x - 1= 6, x 0 = 3.

### 4.5 Case (5)

Let x - 2= r, x - 1= k, x 0 = h. Then the solution of the following difference equation

${x}_{n+1}=\frac{{x}_{n-1}{x}_{n}}{{x}_{n-1}-{x}_{n-2}},$
(11)

is given by

${x}_{4n}=\frac{h{\left(hk\right)}^{2n}}{{\left(rk\left(h-k\right)\left(k-r\right)\right)}^{n}},\phantom{\rule{0.3em}{0ex}}{x}_{4n+1}=\frac{{\left(hk\right)}^{2n+1}}{{\left(rk\left(h-k\right)\right)}^{n}{\left(k-r\right)}^{n+1}},$
${x}_{4n+2}=\frac{h{\left(hk\right)}^{2n+1}}{{\left(\left(h-k\right)\left(k-r\right)\right)}^{n+1}{\left(rk\right)}^{n}},\phantom{\rule{0.3em}{0ex}}{x}_{4n+3}=\frac{{\left(hk\right)}^{2n}}{{\left(r\left(h-k\right)\left(k-r\right)\right)}^{n+1}{k}^{n}},n\phantom{\rule{2.77695pt}{0ex}}=0,1,\phantom{\rule{2.77695pt}{0ex}}\dots \phantom{\rule{2.77695pt}{0ex}}.$

Figure 6 shows the solution when x - 2= 9, x - 1= 5, x 0 = 4.

Figure 7 shows the solution when x - 2= .9, x - 1= 5, x 0 = .4.

### 4.6 Case (6)

Let x - 2= r, x - 1= k, x 0 = h, Then the solution of the following difference equation

${x}_{n+1}=\frac{{x}_{n-2}{x}_{n}}{{x}_{n}-{x}_{n-2}},$
(12)

is given by

${x}_{n}=\frac{hkr}{{u}_{n-3}hr+{u}_{n-2}hk+{u}_{n-1}kr},\phantom{\rule{2.77695pt}{0ex}}n\phantom{\rule{2.77695pt}{0ex}}=0,1,\phantom{\rule{2.77695pt}{0ex}}\dots \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}},$

Where ${\left\{{u}_{m}\right\}}_{m=0}^{\infty }=\left\{-1,1,0,\phantom{\rule{2.77695pt}{0ex}}-1,2,\phantom{\rule{2.77695pt}{0ex}}-2,1,1,\phantom{\rule{2.77695pt}{0ex}}-3,\phantom{\rule{2.77695pt}{0ex}}\dots \right\}$ i. e. u m = u m-1- u m-3, m ≥ 0, u -3 = 0, u -2 = 0, u -1 = 1.

Figure 8 shows the solution when x -2 = 11, x -1 = 6, x 0 = 17.

### 4.7 Case (7)

Let x -2 = r, x -1 = k, x 0 = h and F n-1F, n-2, F n are the Fibonacci terms.

Then the solution of the following difference equation

${x}_{n+1}=\frac{{x}_{n-2}{x}_{n}}{{x}_{n-1}-{x}_{n-2}},$
(13)

is given by

${x}_{2n}\phantom{\rule{2.77695pt}{0ex}}=\frac{hkr}{\left({F}_{n-2}k-{F}_{n-1}r\right)\left({F}_{n-2}h-{F}_{n-1}k\right)},$
${x}_{2n+1}=\frac{hkr}{\left({F}_{n-1}k-{F}_{n}r\right)\left({F}_{n-2}h-{F}_{n-1}k\right)},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}n\phantom{\rule{2.77695pt}{0ex}}=0,1,\phantom{\rule{2.77695pt}{0ex}}\dots \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}.$

Figure 9 shows the solution when x - 2= 8, x - 1= 5, x 0 = 0.9.

## References

1. Kulenovic MRS aa, Ladas G: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman & Hall/CRC Press, Boca Raton, FL 2001.

2. Kocic VL, Ladas G: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic Publishers, Dordrecht; 1993.

3. Mickens RE: Difference Equations. Van Nostrand Reinhold Comp, New York 1987.

4. Mackey MC, Glass L: Oscillation and chaos in physiological control system. Science 1977, 197: 287-289. 10.1126/science.267326

5. Elabbasy EM, El-Metwally H, Elsayed EM:On the difference equation ${x}_{n+1}=a{x}_{n}-\frac{b{x}_{n}}{c{x}_{n}-d{x}_{n-1}}.$ Adv Differ Equ 2006, 1-10. Article ID 82579

6. Elabbasy EM, El-Metwally H, Elsayed EM:On the difference equations ${x}_{n+1}=\frac{\alpha {x}_{n-k}}{\beta +\gamma {\Pi }_{i=0}^{k}{x}_{n-i}}.$ J Conc Appl Math 2007,5(2):101-113.

7. Elabbasy EM, El-Metwally H, Elsayed EM: Global attractivity and periodic character of a fractional difference equation of order three. Yoko-hama Math J 2007, 53: 89-100.

8. El-Metwally H, Grove EA, Ladas G, Levins R, Radin M:On the difference equation ${x}_{n+1}=\alpha +\beta {x}_{n-1}{e}^{-{x}_{n}}.$ Nonlinear Anal: Theory Methods Appl 2003,47(7):4623-4634.

9. Yang X, Su W, Chen B, Megson GM, Evans DJ:On the recursive Sequence ${x}_{n+1}=\frac{a{x}_{n-1}+b{x}_{n-2}}{c+d{x}_{n-1}{x}_{n-2}}.$ Appl Math Comput 2005, 162: 1485-1497. 10.1016/j.amc.2004.03.023

10. Cinar C:On the positive solutions of the difference equation ${x}_{n+1}=\frac{{x}_{n-1}}{1+a{x}_{n}{x}_{n-1}}.$ Appl Math Comput 2004,158(3):809-812. 10.1016/j.amc.2003.08.140

11. Cinar C:On the positive solutions of the difference equation ${x}_{n+1}=\frac{{x}_{n-1}}{-1+a{x}_{n}{x}_{n-1}}.$ Appl Math Comput 2004,158(3):793-797. 10.1016/j.amc.2003.08.139

12. Aloqeili M: Dynamics of a rational difference equation. Appl Math Comput 2006,176(2):768-774. 10.1016/j.amc.2005.10.024

13. Yalçinkaya I:On the difference equation ${x}_{n+1}=\alpha +\frac{{x}_{n-m}}{{x}_{n}^{k}}.$ Discrete Dyn Nat Soc 2008. Article ID 805460, 8

14. Agarwal R: Difference Equations and Inequalities. Theory, Methods and Applications. Marcel Dekker Inc., New York; 1992.

15. Agarwal RP, Elsayed EM: Periodicity and stability of solutions of higher order rational difference equation. Adv Stud Contemp Math 2008,17(2):181-201.

16. Agarwal RP, Zhang W: Periodic solutions of difference equations with general periodicity. Comput Math Appl 2001, 42: 719-727. 10.1016/S0898-1221(01)00191-2

17. Elabbasy EM, Elsayed EM: On the global attractivity of difference equation of higher order. Carpathian J Math 2008,24(2):45-53.

18. Elabbasy EM, Elsayed EM: Dynamics of a rational difference equation. Chin Ann Math Ser B 2009,30B(2):187-198.

19. El-Metwally H, Grove EA, Ladas G, McGrath LC: On the difference equation ${y}_{n+1}=\frac{{y}_{n-\left(2k+1\right)}+p}{{y}_{n-\left(2k+1\right)}+q{y}_{n-2l}}.$ Proceedings of the 6th ICDE, Taylor and Francis, London; 2004.

20. Elsayed EM: Qualitative behavior of difference equation of order three. Acta Sci Math (Szeged) 2009,75(1-2):113-129.

21. Elsayed EM: Qualitative behavior of s rational recursive sequence. Indagat Math 2008,19(2):189-201. 10.1016/S0019-3577(09)00004-4

22. Elsayed EM: On the Global attractivity and the solution of recursive sequence. Stud Sci Math Hung 2010,47(3):401-418.

23. Elsayed EM: Qualitative properties for a fourth order rational difference equation. Acta Appl Math 2010,110(2):589-604. 10.1007/s10440-009-9463-z

24. Wang C, Wang S, Yan X: Global asymptotic stability of 3-species mutualism models with diffusion and delay effects. Discrete Dyn Nat Sci 2009., 20: Article ID 317298

25. Wang C, Gong F, Wang S, Li L, Shi Q: Asymptotic behavior of equilibrium point for a class of nonlinear difference equation. Adv Diff Equ 2009, 8::. Article ID 214309

26. Yalçinkaya I, Iricanin BD, Cinar C: On a max-type difference equation. Discrete Dyn. Nat Soc 2007. Article ID 47264, 10

27. Yalçinkaya I: On the global asymptotic stability of a second-order system of difference equations. Discrete Dyn Nat Soc 2008. Article ID 860152, 12

28. Yalçinkaya I: On the global asymptotic behavior of a system of two nonlinear difference equations. ARS Combinatoria 2010, 95: 151-159.

29. Zayed EME, El-Moneam MA:On the rational recursive sequence ${x}_{n+1}=a{x}_{n}-\frac{b{x}_{n}}{c{x}_{n}-d{x}_{n-k}}.$ Commun Appl Nonlinear Anal 2008, 15: 47-57.

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Correspondence to Elmetwally M Elabbasy.

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The authors declare that they have no competing interests.

### Authors' contributions

EMEla investigated the behavior of the solutions, HAE-M found the solutions of the special cases and EMEls carried out the theoretical proof and gave the examples. All authors read and approved the final manuscript.

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Elabbasy, E.M., El-Metwally, H.A. & Elsayed, E.M. Global behavior of the solutions of some difference equations. Adv Differ Equ 2011, 28 (2011). https://doi.org/10.1186/1687-1847-2011-28 