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# Global behavior of the solutions of some difference equations

- Elmetwally M Elabbasy
^{1}Email author, - Hamdy A El-Metwally
^{2, 4}and - Elsayed M Elsayed
^{3, 4}

**2011**:28

https://doi.org/10.1186/1687-1847-2011-28

© Elabbasy et al; licensee Springer. 2011

**Received: **5 April 2011

**Accepted: **23 August 2011

**Published: **23 August 2011

## Abstract

In this article we study the difference equation

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.

## Keywords

- Stability
- Solutions of the difference equations

## 1 Introduction

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.

*I*be some interval of real numbers and let

*x*

_{-k },

*x*

_{-k+1},...,

*x*

_{0}∈

*I*, the difference equation

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

That is, ${x}_{n}=\stackrel{\u0304}{x}$ for *n* ≥ 0, is a solution of Equation 2, or equivalently, $\stackrel{\u0304}{x}$ is a fixed point of *f*.

### Definition 1 *(Stability)*

- (i)The equilibrium point $\stackrel{\u0304}{x}$ 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$|{x}_{-k}-\stackrel{\u0304}{x}|+|{x}_{-k+1}-\stackrel{\u0304}{x}|+\cdots +|{x}_{0}-\stackrel{\u0304}{x}|<\delta ,$

- (ii)The equilibrium point $\stackrel{\u0304}{x}$ of Equation 2 is locally asymptotically stable if $\stackrel{\u0304}{x}$ 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$|{x}_{-k}-\stackrel{\u0304}{x}|+|{x}_{-k+1}-\stackrel{\u0304}{x}|+\dots +|{x}_{0}-\stackrel{\u0304}{x}|<\gamma ,$

- (iii)The equilibrium point $\stackrel{\u0304}{x}$ of Equation 2 is global attractor if for all
*x*_{ -k },*x*_{-k+1},...,*x*_{-1},*x*_{0}∈*I*, we have$\underset{n\to \infty}{lim}\phantom{\rule{0.3em}{0ex}}{x}_{n}=\stackrel{\u0304}{x}.$ - (iv)
The equilibrium point $\stackrel{\u0304}{x}$ of Equation 2 is globally asymptotically stable if $\stackrel{\u0304}{x}$ is locally stable and $\stackrel{\u0304}{x}$ is also a global attractor of Equation 2.

- (v)
The equilibrium point $\stackrel{\u0304}{x}$ of Equation 2 is unstable if $\stackrel{\u0304}{x}$ is not locally stable.

#### Theorem A [2]

*p*,

*q*∈

*R*and

*k*∈ {0, 1, 2,...}. Then

**Remark 1**

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

*where p*

_{1},

*p*

_{2},...,

*p*

_{ k }∈

*R and k*∈ {1, 2,...}.

*Then Equation*4 is asymptotically stable provided that

### 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 CO_{2} levels, a simple epidemics model and a model of waves of disease that can be analyzed by difference equations are shown in [3]. 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 [4]. In that paper they investigated a simple first order difference-delay equation that models the concentration of blood-level CO_{2}. 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.

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

For some related work see [1–29].

The article proceeds as follows. In Sect. 2 we show that when 2*a* |*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

if *a* ≠ *b-c, b* ≠ *c*, then the unique equilibrium point is $\stackrel{\u0304}{x}=0.$

*f*: (0,

*∞*)

^{4}→ (0, ∞) be a function defined by

### Theorem 1

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

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 [1].

### Theorem 2

*a*,

*b*] be an interval of real numbers and assume that

- (i)
*f*(*x*_{1},*x*_{2},....,*x*_{ k+1}) is non-increasing in one component (for example*x*_{ t }) for each*x*_{ r }(*r*≠*t*) in [*a*,*b*] and non-decreasing in the remaining components for each*x*_{ t }in [*a*,*b*]. - (ii)
If $\left(m,\phantom{\rule{2.77695pt}{0ex}}M\right)\in \left[a,\phantom{\rule{2.77695pt}{0ex}}b\right]\times \left[a,\phantom{\rule{2.77695pt}{0ex}}b\right]$ 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

Then Equation 2 has a unique equilibrium $\stackrel{\u0304}{x}\in \left[a,\phantom{\rule{2.77695pt}{0ex}}b\right]$ and every solution of Equation 2 converges to $\stackrel{\u0304}{x}$

**Proof:**Set

*i*= 1, 2,...set

*i*≥ 0,

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*).

from which the result follows.

### Theorem 3

*a*,

*b*] be an interval of real numbers and assume that

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

Then Equation 2 has a unique equilibrium $\stackrel{\u0304}{x}\in \left[a,\phantom{\rule{2.77695pt}{0ex}}b\right]$ and every solution of Equation 2 converges to $\stackrel{\u0304}{x}$

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

### Theorem 4

The equilibrium point $\stackrel{\u0304}{x}$ of Equation 1 is global attractor if *c* ≠ *a*.

**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*).

It follows by Theorem 2 that $\stackrel{\u0304}{x}$ 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*).

It follows by the Theorem 3 that $\stackrel{\u0304}{x}$ is a global attractor of Equation 1 and then the proof is complete.

## 4 Special cases of Equation 1

### 4.1 Case (1)

where the initial conditions *x*
_{-1}, *x*
_{0} are arbitrary positive real numbers.

#### Theorem 5

*n*= 0, 1,...

where *x*
_{-1} = *k*, *x*
_{0} = *h* and *F*
_{
n-1
}, *F*
_{
n-2}are 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;

Hence, the proof is completed.

*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)

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

### Theorem 6

*n*= 1, 2,...

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;

Hence, the proof is completed.

*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)

*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

### 4.4 Case (4)

*x*

_{-2}=

*r*,

*x*

_{-1}=

*k*,

*x*

_{0}=

*h*. Then the solution of the following difference equation

### 4.5 Case (5)

*x*

_{ - 2}=

*r*,

*x*

_{ - 1}=

*k*,

*x*

_{0}=

*h*. Then the solution of the following difference equation

### 4.6 Case (6)

*x*

_{ - 2}=

*r, x*

_{ - 1}=

*k, x*

_{0}=

*h*, Then the solution of the following difference equation

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.

### 4.7 Case (7)

Let *x*
_{-2} = *r, x*
_{-1} = *k, x*
_{0} = *h* and F_{
n-1}F_{, n-2}, *F*
_{
n
} are the Fibonacci terms.

## Declarations

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.