- Research
- Open Access
Dynamics of difference equation \(x_{n+1}=f( x_{n-l},x_{n-k})\)
- Osama Moaaz^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-018-1896-0
© The Author(s) 2018
- Received: 3 October 2018
- Accepted: 19 November 2018
- Published: 4 December 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.
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 [1–36]. Also, many powerful methods for studying qualitative behavior of difference equations have been established and developed; see [5, 20] and [30].
Problem 1
(Kulenovic and Ladas [25])
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
Theorem 2.2
Proof
Theorem 2.3
Proof
The proof is similar to that of proof of Theorem 2.2 and hence is omitted. □
Example 2.1
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
Theorem 3.1
Proof
Theorem 3.2
Proof
Example 3.1
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
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
- (i)
\(bc>ad\) and \(a+b< c+d\),
- (ii)
\(bc< ad\) and \(a ( c+3d ) +b ( d-c ) < ( c+d) ^{2}\).
- (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
Declarations
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.
Funding
The author received no direct funding for this work.
Authors’ contributions
The author wrote, read and approved the final manuscript.
Competing interests
The author declares that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Abdelrahman, M.A.E., Chatzarakis, G.E., Li, T., Moaaz, O.: On the difference equation \(x_{n+1}=ax_{n-l}+bx_{n-k}+f( x_{n-l},x_{n-k}) \). Adv. Differ. Equ. (2018). https://doi.org/10.1186/s13662-018-1880-8 View ArticleGoogle Scholar
- Abu-Saris, R.M., DeVault, R.: Global stability of \(y_{n+1}=A+y_{n}/y _{n-k}\). Appl. Math. Lett. 16, 173–178 (2003) MathSciNetView ArticleGoogle Scholar
- Amleh, A.M., Grove, E.A., Georgiou, D.A., Ladas, G.: On the recursive sequence \(x_{n+1}=\alpha +x_{n-1}/x_{n}\). J. Math. Anal. Appl. 233, 790–798 (1999) MathSciNetView ArticleGoogle Scholar
- Berenhaut, K.S., Foley, J.D., Stevic, S.: The global attractivity of the rational difference equation \(y_{n}=1+y_{n-k}/y_{n-m}\). Proc. Am. Math. Soc. 135, 1133–1140 (2007) View ArticleGoogle Scholar
- Berenhaut, K.S., Stevic, S.: The behaviour of the positive solutions of the difference equation \(x_{n}=A+ ( x_{n-2}/x_{n-1} ) ^{p}\). J. Differ. Equ. Appl. 12(9), 909–918 (2006) View ArticleGoogle Scholar
- Bohner, M., Hassan, T.S., Li, T.: Fite–Hille–Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments. Indag. Math. 29, 548–560 (2018) MathSciNetView ArticleGoogle Scholar
- Border, K.C.: Euler’s theorem for homogeneous functions. Caltech Div. Hum. Soc. Sci. 2017, 16–34 (2017) Google Scholar
- Chatzarakis, G.E., Li, T.: Oscillations of differential equations generated by several deviating arguments. Adv. Differ. Equ. 2017, Article ID 292 (2017). https://doi.org/10.1186/s13662-017-1353-5 MathSciNetView ArticleGoogle Scholar
- Chatzarakis, G.E., Li, T.: Oscillation criteria for delay and advanced differential equations with nonmonotone arguments. Complexity 2018, Article ID 8237634 (2018). https://doi.org/10.1155/2018/8237634 View ArticleGoogle Scholar
- Devault, R., Kent, C., Kosmala, W.: On the recursive sequence \(x_{n+1}=p+x_{n-k}/x_{n}\). J. Differ. Equ. Appl. 9(8), 721–730 (2003) View ArticleGoogle Scholar
- Devault, R., Ladas, G., Schultz, S.W.: On the recursive sequence \(x_{n+1}=A/x_{n}+1/x_{n-2}\). Proc. Am. Math. Soc. 126, 3257–3261 (1998) View ArticleGoogle Scholar
- Devault, R., Schultz, S.W.: On the dynamics of \(x_{n+1}=(\beta x_{n}+ \gamma x_{n-1})/(Bx_{n}+Dx_{n-2})\). Commun. Appl. Nonlinear Anal. 12, 35–40 (2005) MATHGoogle Scholar
- El-Dessoky, M.M.: On the periodicity of solutions of max-type difference equation. Math. Methods Appl. Sci. 38, 3295–3307 (2015) MathSciNetView ArticleGoogle Scholar
- El-Dessoky, M.M.: The form of solutions and periodicity for some systems of third-order rational difference equations. Math. Methods Appl. Sci. 39, 1076–1092 (2016) MathSciNetView ArticleGoogle Scholar
- El-Dessoky, M.M., Elsayed, E.M., Alghamdi, M.: Solutions and periodicity for some systems of fourth order rational difference equations. J. Comput. Anal. Appl. 18(1), 179–194 (2015) MathSciNetMATHGoogle Scholar
- El-Owaidy, H., Ahmed, A., Mousa, M.: On asymptotic behaviour of the difference equation \(x_{n+1}=\alpha +\) \(x_{n-k}/\) \(x_{n}\). Appl. Math. Comput. 147, 163–167 (2004) MathSciNetMATHGoogle Scholar
- Elabbasy, E.M., El-Metwally, H., Elsayed, E.M.: On the difference equation \(x_{n+1}=(\alpha x_{n-l}+\beta x_{n-k})/(Ax_{n-l}+Bx_{n-k})\). Acta Math. Vietnam. 33(1), 85–94 (2008) MathSciNetMATHGoogle Scholar
- Elabbasy, E.M., Hassan, T.S., Moaaz, O.: Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments. Opusc. Math. 32, 719–730 (2012) MathSciNetView ArticleGoogle Scholar
- Elsayed, E.M.: Dynamics and behavior of a higher order rational difference equation. J. Nonlinear Sci. Appl. 9, 1463–1474 (2015) MathSciNetView ArticleGoogle Scholar
- Elsayed, E.M.: New method to obtain periodic solutions of period two and three of a rational difference equation. Nonlinear Dyn. 79, 241–250 (2015) MathSciNetView ArticleGoogle Scholar
- Hamza, A.E.: On the recursive sequence \(x_{n+1}=\alpha +x_{n-1}/x_{n}\). J. Math. Anal. Appl. 322, 668–674 (2006) MathSciNetView ArticleGoogle Scholar
- Kalabusic, S., Kulenovic, M.R.S.: On the recursive sequence \(x_{n+1}=(\gamma x_{n-1}+\delta x_{n-2})/\) \((Cx_{n-1}+Dx_{n-2})\). J. Differ. Equ. Appl. 9(8), 701–720 (2003) View ArticleGoogle Scholar
- Khuong, V.V.: On the positive nonoscillatory solution of the difference equations \(x_{n+1}=\alpha + ( x_{n-k}/x_{n-m} ) ^{p}\). Appl. Math. J. Chin. Univ. Ser. A 24, 45–48 (2008) MathSciNetGoogle Scholar
- Kocic, V.L., Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic, Dordrecht (1993) View ArticleGoogle Scholar
- Kulenovic, M.R.S., Ladas, G.: Dynamics of Second Order Rational Difference Equations. Chapman & Hall, London (2002) MATHGoogle Scholar
- Kulenovic, M.R.S., Ladas, G., Sizer, W.S.: On the dynamics of \(x_{n+1}=(\alpha x_{n}+\beta x_{n-1})/(\gamma x_{n}+\delta x_{n-1})\). Math. Sci. Res. Hot-Line 2(5), 1–16 (1998) MathSciNetMATHGoogle Scholar
- Li, T., Rogovchenko, Y.V.: On asymptotic behavior of solutions to higher-order sublinear Emden–Fowler delay differential equations. Appl. Math. Lett. 67, 53–59 (2017) MathSciNetView ArticleGoogle Scholar
- Li, T., Rogovchenko, Y.V.: Oscillation criteria for second-order superlinear Emden–Fowler neutral differential equations. Monatshefte Math. 184, 489–500 (2017) MathSciNetView ArticleGoogle Scholar
- Moaaz, O.: Comment on “New method to obtain periodic solutions of period two and three of a rational difference equation [Nonlinear Dyn. 79:241–250]”. Nonlinear Dyn. 88, 1043–1049 (2017) View ArticleGoogle Scholar
- Moaaz, O., Elabbasy, E.M., Bazighifan, O.: On the asymptotic behavior of fourth-order functional differential equations. Adv. Differ. Equ. 2017, Article ID 261 (2017). https://doi.org/10.1186/s13662-017-1312-1 MathSciNetView ArticleGoogle Scholar
- Ocalan, O.: Dynamics of the difference equation \(x_{n+1}=p+x_{n-k}/x _{n}\) with a period-two coefficient. Appl. Math. Comput. 228, 31–37 (2014) MathSciNetGoogle Scholar
- Saleh, M., Aloqeili, M.: On the rational difference equation \(x_{n+1}=A+x_{n-k}/x_{n}\). Appl. Math. Comput. 171(2), 862–869 (2005) MathSciNetMATHGoogle Scholar
- Stevic, S.: On the recursive sequence \(x_{n+1}=\alpha +x_{n-1}^{p}/x _{n}^{p}\). J. Appl. Math. Comput. 18, 229–234 (2005) MathSciNetView ArticleGoogle Scholar
- Stevic, S., Kent, C., Berenaut, S.: A note on positive nonoscillatory solutions of the differential equation \(x_{n+1}=\alpha +x_{n-1}^{p}/x _{n}^{p}\). J. Differ. Equ. Appl. 12, 495–499 (2006) View ArticleGoogle Scholar
- Sun, T., Xi, H.: On convergence of the solutions of the difference equation \(x_{n+1}=1+x_{n-1}/x_{n}\). J. Math. Anal. Appl. 325(2), 1491–1494 (2007) MathSciNetView ArticleGoogle Scholar
- Yan, X., Li, W.T., Zhao, Z.: On the recursive sequence \(x_{n+1}= \alpha -( x_{n}/x_{n-1}) \). J. Appl. Math. Comput. 17(1), 269–282 (2005) MathSciNetView ArticleGoogle Scholar