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Dynamics of difference equation \(x_{n+1}=f( x_{nl},x_{nk})\)
Advances in Difference Equations volume 2018, Article number: 447 (2018)
Abstract
In this paper, we present the asymptotic behavior of the solutions for a general class of difference equations. We introduce general theorems in order to study the stability and periodicity of the solutions. Moreover, we use a new technique to study the existence of periodic solutions of this general equation. By using our general results, we can study many special cases that have not been studied previously and some problems that were raised previously. Some numerical examples are provided to illustrate the new results.
Introduction
Amid the most recent two decades, there has been an extraordinary research of the utilization of difference equations in the solution of numerous issues that emerge in economy, statistics, and engineering science. Likewise, difference equations have been utilized as approximations to ordinary and partial differential equations (ODEs and PDEs) because of the improvement of rapid advanced processing hardware. It tends to be said that difference equations identify with differential equations as discrete mathematics identifies with continuous mathematics. Any individual who has made an investigation of differential equations will realize that even elementary examples can be difficult to solve. By contrast, elementary difference equations are moderately simple to study. For many reasons, computer scientists take an interest difference equations. For instance, difference equations often emerge while determining the cost of an algorithm in bigO notation. In 1943, the difference equations were commonly used for solving partial differential equations. Problems involving timedependent fluid flows, neutron diffusion and transport, radiation flow, thermonuclear reactions, and problems involving the solution of several simultaneous partial differential equations are being solved by the use of difference equations. Other than the utilization of difference equations as approximations to ODEs and PDEs, they afford a powerful method for the analysis of electrical, mechanical, thermal, and other systems in which there is a recurrence of identical sections. By using the difference equations, the investigation of the conduct of electricwave filters, multistage amplifiers, magnetic amplifiers, insulator strings, continuous beams of equal span, crankshafts of multicylinder engines, acoustical filters, etc., is enormously facilitated. The standard techniques for solving such systems are generally very lengthy when the number of elements involved is large. The use of difference equations greatly reduces the complexity and labor in problems of this type.
As a result of the many applications of difference equations in various fields, many mathematicians are interested in the asymptotic behavior of different types of difference equations; see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]. Also, many powerful methods for studying qualitative behavior of difference equations have been established and developed; see [5, 20] and [30].
In particular, we review some difference equations that are special cases of the general studied equation. In [11], Devault et al. studied the recursive sequence
where A is a real number. Khuong in [23] investigated the behavior of the positive solutions of the difference equation
where l, k and α are positive integers , \(a>1\) and \(0\leq k< l\). In [33], Stevic investigated the behavior of the positive solutions of the difference equation (1.1) when a and α are positive real numbers, \(l=1\) and \(k=0\). The case \(\alpha =1\) has been considered in [10]. In [19], Elsayed studied the periodicity and the boundedness of the positive solutions of the difference equation
where a, b, c, d and e are positive real number. For further study of Eq. (1.2) with \(a=0\), see [12, 17, 22] and [26]. Elsayed in [20] and Moaaz in [29] studied the qualitative behavior of solutions of the equation
where a, b and c are real number.
Our aim in this paper is to investigate the qualitative behavior of the solutions of the difference equation
where l and k are positive integers, the function \(f ( u,v ) \) is a continuous real function and is homogeneous with degree α and the initial conditions \(x_{\mu }, x_{\mu +1}, \ldots, x_{0}\) are real numbers for \(\mu =\max \{ l,k \} \). In this paper, we study the local/global stability and periodicity character of solutions of the difference equation in a general form using a homogeneous function. We use a new and powerful method to study the prime period two solution of this equation. Moreover, we apply general results on some special cases. We can use our results to answer some of the problems raised earlier, as
Problem 1
(Kulenovic and Ladas [25])
Suppose that a, b, c and d are real numbers. Investigate the forbidden set of the difference equation
and investigate the asymptotic behavior and the periodic nature of its solution.
Existence of periodic solutions
The following theorems state a new necessary and sufficient condition that Eq. (E) has periodic solution of prime period two.
Theorem 2.1
Assume that l and k are odd or l and k are even. If \(\alpha \neq 1\), then Eq. (E) has no prime positive period two solution.
Proof
On the contrary, we assume that Eq. (E) has a prime period two distinct solution
If l and k are odd, then we have \(x_{nl}=x_{nk}=\rho \). From Eq. (E), we get
Thus, we obtain
This is a contradiction.
Next, we let l and k be even. Then we get \(x_{nl}=x_{nk}=\sigma \), and hence
Therefore
and so
Hence,
which is a contradiction. Thus, the proof is completed. □
Theorem 2.2
Assume that l is odd and k is even. Equation (E) has a prime period two solution \(\ldots, \rho , \sigma , \rho , \sigma , \ldots\) if and only if
where \(\tau =\rho /\sigma \).
Proof
We suppose without loss of generality that \(l>k\). Now, we assume that Eq. (E) has a prime period two solution
Since l is odd and k is even, we have \(x_{nl}=\rho \) and \(x_{nk}=\sigma \). From Eq. (E), we get
Then
On the other hand, we let (2.1) hold. If \(\alpha \neq 1\), then we choose
for \(\mu =0,1,\ldots, ( l1 ) /2\), where \(\tau \in \mathbb{R} \backslash \{ 1 \} \). Hence, we obtain
From (2.1), we have
Since f is homogeneous with degree α, we get
Also, we have
Hence, it is concluded by induction that
Therefore, Eq. (E) has a prime period two solution. If \(\alpha =1\), then we choose
where \(\tau \in \mathbb{R} \backslash \{ 1 \} \) and c arbitrary real number. Thus, we get
and
Then it is concluded by induction that \(x_{2n1}=c\tau \) and \(x _{2n}=c\) for all \(n>0\). Therefore, Eq. (E) has a prime period two solution and the proof is completed. □
Theorem 2.3
Assume that l is even and k is odd. Equation (E) has a prime period two solution \(\ldots, \rho , \sigma , \rho , \sigma , \ldots\) , if and only if
where \(\tau =\rho /\sigma \).
Proof
The proof is similar to that of proof of Theorem 2.2 and hence is omitted. □
Example 2.1
Let the difference equation
where α is an integer, a and b are positive real numbers and \(\vert \alpha \vert \neq 1\). From Theorem 2.3, Eq. (2.3) has a prime period two solution
if and only if
and so
We have
and
which with (2.4) gives \(b ( \alpha 1 ) >a ( \alpha +1 ) \). For example, for \(\alpha =2\), \(a=7\), \(b=2\), \(x_{1}=3.107\) and \(x_{0}=1.553\), the prime period two solution of (2.3) is shown in Fig. 1.
Stability of Equation (E)
In this section, we study the local stability and global attractivity of the equilibrium point of Eq. (E).
Lemma 3.1
If \(\alpha \neq 1\), then Eq. (E) has a positive equilibrium point
Also, if \(\alpha =1\) and \(f ( 1,1 ) \neq 1\), then Eq. (E) has only zero equilibrium point.
Proof
The equilibrium point of Eq. (E) is given by \(\overline{x}=f ( \overline{x},\overline{x} ) \). Since f is homogeneous with degree α, we obtain
If \(\alpha \neq 1\), then \(\overline{x}=0\) or \(\overline{x}=f^{1/ ( 1\alpha ) } ( 1,1 ) \). Otherwise, if \(\alpha =1\) and \(f ( 1,1 ) \neq 1\), then we have \(\overline{x}=0\). Thus, the proof is completed. □
Theorem 3.1
The zero equilibrium point of Eq. (E) is locally asymptotically stable if \(\alpha >1\), or
Proof
Since f homogeneous with degree α, we have \(f_{u}\) and \(f_{v}\) are homogeneous with degree \(\alpha 1\). Now, if \(\alpha >1\), then we get \(f_{u} ( \overline{x},\overline{x} ) = \overline{x}^{\alpha 1}f_{u} ( 1,1 ) =0\) and \(f_{v} ( \overline{x},\overline{x} ) =0\), for \(\overline{x}=0\). Hence, \(\overline{x}=0\) is locally asymptotically stable. Next, if \(\alpha =1\), then \(f_{u} ( \overline{x},\overline{x} ) =f _{u} ( 1,1 ) \) and \(f_{v} ( \overline{x},\overline{x} ) =f_{v} ( 1,1 ) \). By using Theorem 1.3.7 in [24], we see that Eq. (E) is locally stable if
which completes the proof. □
Theorem 3.2
The positive equilibrium point of Eq. (E) is locally asymptotically stable if
or
Proof
The linearized equation of (E) about x̅ is the linear difference equation
From Theorem 1.3.7 in [24], Eq. (3.4) is locally stable if
Using Corollary 2 in [7], we see that \(f_{u}\) and \(f_{v}\) are homogeneous with degree \(\alpha 1\). This implies
If we admit that \(f_{u}>0\) and \(f_{v}>0\), then we obtain
From Euler’s homogeneous function theorem, we deduce that
which with (3.5) gives \(0<\alpha <1\). Similarly, if \(f_{u}<0\) and \(f_{v}<0\), then we get \(1<\alpha <0\).
In the case where \(f_{u}>0\) and \(f_{v}<0\), we have
Combining (3.6) with (3.7), we get
Finally, if \(f_{u}<0\) and \(f_{v}>0\), then we find
Hence, the proof is completed. □
Example 3.1
Let the difference equation
where α, a and b are real numbers \(,~a>0\), \(b>0\) and \(\alpha \neq 1\). Since \(f ( u,v ) =au^{\alpha }+bv^{\alpha }\), we get
By using Theorem 3.2, the positive equilibrium point \(\overline{x}= ( a+b ) ^{1/ ( 1\alpha ) }\) of Eq. (3.8) is locally asymptotically stable if \(\vert \alpha \vert <1\). For example, for \(\alpha =0.6\), \(a=0.2\), \(b=0.7\), \(x_{1}=2.0\) and \(x_{0}=0.2\), the stable solution of (3.8) is shown in Fig. 2.
Remark 3.1
If \(\alpha =0\), then \(uf_{u}+vf_{v}=0\) and \(f_{u}f_{v}<0\). Thus, Eq. (3.8) is locally asymptotically stable if
Theorem 3.3
Assume that f has nonpositive partial derivatives. Then Eq. (E) has a unique positive equilibrium x̅ and every solution of Eq. (E) converges to x̅.
Proof
Let \(( m,M ) \) is a solution of the system
This implies
and so
Hence, we get \(m=M\). By Theorem 1.4.7 in [26], we see that every solution of Eq. (E) converges to x̅. Hence, the proof is completed. □
Remark 3.2
Assume that \(f_{u}>0\) and \(f_{v}<0\). By Theorem 1.4.5 in [26], if we were able to obtain a condition that ensures that
for all \(z\in ( 0,\infty ) \), then the equilibrium point x̅ would be a global attractor of Eq. (E).
Remark 3.3
Assume that \(f\in C ( [ 0,\infty ) \times [ 0, \infty ) , [ 0,\infty ) ) \), \(f_{u}f_{v}>0\), \(\vert \alpha \vert <1,~l=0\) and \(k=1\). Then, by Euler’s homogeneous function theorem, we see that
for all \(u,v\in ( 0,\infty ) \). Thus, by using Theorem 1.4.4 in [26], Eq. (E) has exactly one of the following three cases for all solutions (stability trichotomy):

(a)
\(\lim_{n\rightarrow \infty }x_{n} =\infty\) for \(x_{1}x_{0} \neq 0\).

(b)
\(\lim_{n\rightarrow \infty }x_{n} =0\) and Eq. (E) has only a zero equilibrium point.

(c)
\(\lim_{n\rightarrow \infty }x_{n} =\overline{x}\) for \(x_{1}x _{0}\neq 0\) and x̅ is the only positive equilibrium point.
Discussion and numerical examples
Corollary 4.1
Assume that l and k are odd or l and k are even. If \(f_{u}<0\) and \(f_{v}>0\), then Eq. (E) has a unique equilibrium x̅ and every solution of Eq. (E) converges to x̅.
Proof
From Theorem 2.1, if l and k are odd or l and k are even, then Eq. (E) has no prime period two solution. Thus, by Theorem 1.4.6 in [26], we see that every solution of Eq. (E) converges to x̅. □
Remark 4.1
Notice that equations that have been studied in [2, 3, 10,11,12,13,14,15,16,17,18,19,20,21,22,23, 26] and [31,32,33,34] are special cases of Eq. (E). For example, Elsayed in [19] investigated the stability character and the periodicity of solutions of Eq. (1.2). From Remark 3.1, the positive equilibrium point of Eq. (1.2) is locally asymptotically stable if
Next, if \(be>cd\), then we have \(f_{u}>0\) and \(f_{v}<0\). Note that the condition \(c\geq b\) ensures that
This implies
Hence, by Remark 3.2, the equilibrium point is a global attractor of (E) if \(be>cd\) and \(c\geq b\) (Theorem 5.2 in [19]). Finally, by using Theorem 2.2 and 2.3, we can obtain the results of Theorem 6.1 in [19].
In the following, two special cases are given to validate the asymptotic behavior of the proposed new class of difference equations.
Example 4.1
Consider the difference equation (1.3). We have
homogeneous with degree \(one\). Then the partial derivatives of f are
From Lemma 3.1, Eq. (1.3) has only zero equilibrium point if \(a+b\neq c+d\). By Theorem 3.1, the zero equilibrium point of Eq. (1.3) is locally asymptotically stable if one of the following cases holds:

(i)
\(bc>ad\) and \(a+b< c+d\),

(ii)
\(bc< ad\) and \(a ( c+3d ) +b ( dc ) < ( c+d) ^{2}\).
For the periodicity of solutions of Eq. (1.3), we assume that a, b, c and d are real numbers, \(\vert c\vert +\vert d\vert \neq 0\) and \(\vert a\vert +\vert d\vert \neq 0\). Using Theorem 2.3, we see that Eq. (1.3) has a prime period two solution if and only if
Then
and
Thus, we get \(a+d=0\) and
We define the function \(H ( \tau ) := ( 1+\tau^{2} ) /\tau \). Then we find
Therefore, Eq. (1.3) has a prime period two solution if \(a+d=0\) and one of the following conditions holds:

(a)
\(\frac{cb}{ad} >1\) for \(x_{1}x_{0}>0\),

(b)
\(\frac{cb}{ad} <1\) for \(x_{1}x_{0}<0\).
For a numerical example, we take \(a=b=1\), \(c=3.5\), \(d=1\), \(x_{1}=2\) and \(x_{0}=1\); see Fig. 3.
Example 4.2
Consider the difference equation
where α, a and b are real numbers, \(a>0\) and \(b>0\). We have
homogeneous with degree α. Then the partial derivatives of f are
If \(\alpha <0\), then \(f_{u}<0\) and \(f_{v}>0\). By Theorem 3.2, we see that the positive equilibrium point \(\overline{x}= ( ae^{b} ) ^{1/ ( 1\alpha ) }\) of Eq. (4.1) is locally asymptotically stable if \(b<\alpha <1\) or \(2b<1+\alpha \). For a numerical example, we take \(l=0\), \(k=1\), \(\alpha =0.5\), \(a=2\) and \(b=0.1\); see Fig. 4.
For the periodicity of solutions of Eq. (4.1), we assume that l is odd and k is even. By using Theorem 2.2, we see that Eq. (4.1) has a prime period two solution
if and only if
We have
which with (4.2) gives \(2b< ( \alpha 1 ) \). For a numerical example, we take \(l=1\), \(k=0\), \(\alpha =2\), \(a=1\), \(b=2\ln 2\), \(x_{1}=2\sqrt[3]{2}\) and \(x_{0}=\sqrt[3]{2}\); see Fig. 5.
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The author offers earnest thanks to the editors and two anonymous referees for the careful reading of the first original manuscript and valuable remarks that helped to improve the presentation of the results in this manuscript and accentuate important details.
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Moaaz, O. Dynamics of difference equation \(x_{n+1}=f( x_{nl},x_{nk})\). Adv Differ Equ 2018, 447 (2018). https://doi.org/10.1186/s1366201818960
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MSC
 39A10
 39A21
 39A23
 39A30
Keywords
 Difference equation
 Equilibrium points
 Local and global stability
 Prime period two