# Dynamics of difference equation \(x_{n+1}=f( x_{n-l},x_{n-k})\)

## 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.

## Keywords

Difference equation Equilibrium points Local and global stability Prime period two## MSC

39A10 39A21 39A23 39A30## 1 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 big-O notation. In 1943, the difference equations were commonly used for solving partial differential equations. Problems involving time-dependent 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 electric-wave 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].

*A*is a real number. Khuong in [23] investigated the behavior of the positive solutions of the difference equation

*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

*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

*a*,

*b*and

*c*are real number.

*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])

*a*,

*b*,

*c*and

*d*are real numbers. Investigate the forbidden set of the difference equation

## 2 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

*l*and

*k*are odd, then we have \(x_{n-l}=x_{n-k}=\rho \). From Eq. (E), we get

*l*and

*k*be even. Then we get \(x_{n-l}=x_{n-k}=\sigma \), and hence

### 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

*l*is odd and

*k*is even, we have \(x_{n-l}=\rho \) and \(x_{n-k}=\sigma \). From Eq. (E), we get

*f*is homogeneous with degree

*α*, we get

*c*arbitrary real number. Thus, we get

### 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

*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

## 3 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

### Proof

*f*is homogeneous with degree

*α*, we obtain

### Theorem 3.1

*The zero equilibrium point of Eq*. (E)

*is locally asymptotically stable if*\(\alpha >1\),

*or*

### Proof

*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

### Theorem 3.2

*The positive equilibrium point of Eq*. (E)

*is locally asymptotically stable if*

*or*

### Proof

*x̅*is the linear difference equation

### Example 3.1

*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

### Remark 3.1

### Theorem 3.3

*Assume that**f**has non*-*positive partial derivatives*. *Then Eq*. (E) *has a unique positive equilibrium**x̅**and every solution of Eq*. (E) *converges to**x̅*.

### Proof

*x̅*. Hence, the proof is completed. □

### Remark 3.2

### Remark 3.3

- (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.

## 4 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

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

*f*are

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

- (ii)
\(bc< ad\) and \(a ( c+3d ) +b ( d-c ) < ( c+d) ^{2}\).

*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

- (a)
\(\frac{c-b}{a-d} >1\) for \(x_{-1}x_{0}>0\),

- (b)
\(\frac{c-b}{a-d} <-1\) for \(x_{-1}x_{0}<0\).

### Example 4.2

*α*,

*a*and

*b*are real numbers, \(a>0\) and \(b>0\). We have

*α*. Then the partial derivatives of

*f*are

*l*is odd and

*k*is even. By using Theorem 2.2, we see that Eq. (4.1) has a prime period two solution

## Notes

### Acknowledgements

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.

### Availability of data and materials

Data sharing not appropriate to this article as no datasets were produced down amid the current investigation.

### Authors’ contributions

The author wrote, read and approved the final manuscript.

### Funding

The author received no direct funding for this work.

### Competing interests

The author declares that they have no competing interests.

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