Yıl 2018, Cilt 47, Sayı 5, Sayfalar 1128 - 1143 2018-10-16

Qualitative study of a higher order rational difference equation

Abdul Khaliq [1] , E.M. Elsayed [2]

14 23

In this paper we study the behavior of the difference equation

                     $x_{n+1}$ = $\dfrac{\alpha x_nx_{n-l}}{\beta x_{n-m}+\gamma x_{n-l}}$,$\quad n=0,1,$ $\cdots$

where the initial conditions $x_{-r}$, $x_{-r+1}$, $\cdots$ ,$x_0$ are arbitrary non zero real numbers where $r=\max\{l,m\}$ is a non-negative integer and  $\alpha$, $\beta$ and $\gamma$ are constants: Also, we obtain the solutions of some special cases of this equation. At the end we present some numerical examples to support our theoretical discussion.
difference equation, stability, boundedness, global attractivity
  • Abo-Zeid, R. and Cinar, C. Global behavior of the difference equation $x_{n+1}=\dfrac{A x_{n-1}}{B-Cx_nx_{n-2}}$, Bol. Soc. Paran. Mat. (3s.) 31 (1), 43-49, 2013.
  • Ahmed, A. M. and Eshtewy, N. A. Basin of attraction of the recursive sequence $x_{n+1}=\dfrac{\alpha+\beta x_n+\gamma x_{n-1} }{A+B x_n+C x_{n-1}}$, J. Fract. Calc. Appl. 5 (10), 1-8, 2014.
  • Amleh, A. M. Kirk, V. and Ladas, G. On the dynamics of $x_{n+1}=\dfrac{a+b x_{n-1}}{A+B x_{n-2}}$, Math. Sci. Res. Hot-Line 5, 1-15, 2001.
  • Bedford, E. and Kim, K. Dynamics of rational surface automorphisms: linear fractional recurrences, J. Geo. Anal. 19 (3), 553-583, 2009.
  • Belhannache, F. Touafek, N. and Abo-Zeid, R. Dynamics of a third-order rational difference equation, Bull. Math. Soc. Sci. Math. Roumanie, Tome 59 (107) (1), 13-22, 2016.
  • Beverton, R. J. and Holt, S. J. On the Dynamics of Exploited Fish Populations, vol.19, Fish Invest., London, 1957.
  • Cinar, C. On the positive solutions of the difference equation $x_{n+1}=\dfrac{a x_{n-1}}{1+b x_nx_{n-1}}$, Appl. Math. Comp. 156, 587-590, 2004.
  • Dehghan, M. and Rastegar, N. Stability and periodic character of a third order difference equation, Math. Comput. Modelling 54 (11-12), 2560-2564, 2011.
  • Din, Q. On a System of fourth-order rational difference equations, Acta Univ. Apulensis 39, 137-150, 2014.
  • Din, Q. Global character of a rational difference equation, Thai J. Math. 12 (1), 55-70, 2014.
  • Elabbasy, E. M. Barsoum, M. Y. and Alshawee, H. S. Behavior of solutions of a class of nonlinear rational difference equation $x_{n+1}=\alpha x_{n-k}+\dfrac{\beta x_{n-l}^\delta }{\gamma x_{n-s}^\delta}$, Electron. J. Math. Anal. Appl. 4 (2), 78-87, 2016.
  • Elabbasy, E. M. El-Metwally, H. and Elsayed, E. M. Global behavior of the solutions of some difference equation, Advances in difference equation 2011, 89-100, 2011.
  • El-Metwally H. and Elsayed, E. M. Solution and behavior of a third rational difference equation, Utilitas Mathematica 88, 27-42, 2012.
  • EI-Metwally, H. Grove, E. A. Ladas, G. Levins, R. and Radin, M. On the difference equation $x_{n+1}=\alpha+\beta x_{n-1}e^{-x_n} $, Nonlinear Anal: Theory, Methods & Applications 47 (7), 4623- 4634, 2003.
  • El-Moneam, M. A. and Alamoudy, S. O. On study of the asymptotic behavior of some rational difference equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A: Math. Anal. 22, 157-176, 2015.
  • Elsayed, E. M. Solution and attractivity for a rational recursive sequence, Discrete Dyn. Nat. Soc. 2011, Article ID 982309, 17 pages, 2011.
  • Elsayed, E.M. Solutions of rational difference system of order two, Math. Comput. Modelling 55, 378-384, 2012.
  • Elsayed, E. M. Behavior and expression of the solutions of some rational difference equa- tions, J. Comput. Anal. Appl. 15 (1), 73-81, 2013.
  • Elsayed, E. M. Solution for systems of difference equations of rational Form of order two, Comput. Appl. Math. 33 (3), 751-765, 2014.
  • Elsayed, E. M. On the solutions and periodic nature of some systems of difference equations, Int. J. Biomath. 7 (6), 1450067, 26 pages, 2014.
  • Elsayed, E. M. and Ahmed, A. M. Dynamics of a three-dimensional systems of rational difference equations, Math. Methods Appl. Sci. 39 (5), 1026-1038, 2016.
  • Elsayed, E. M. and Alghamdi, A. Dynamics and global stability of higher order nonlinear difference equation, J. Comput. Anal. Appl. 21 (3), 493-503, 2016.
  • Elsayed, E. M. and El-Dessoky, M. M. Dynamics and global behavior for a fourth-order rational difference equation, Hacet. J. Math. Stat. 42 (5), 479494, 2013.
  • Elsayed E. M. and El-Metwally, H. Stability and solutions for rational recursive sequence of order three, J. Comput. Anal. Appl. 17 (2), 305-315, 2014.
  • Elsayed E. M. and El-Metwally, H. Global behavior and periodicity of some difference equations, J. Comput. Anal. Appl. 19 (2), 298-309, 2015.
  • Elsayed, E. M. and Ibrahim, T. F. Solutions and periodicity of a rational recursive sequences of order five, Bull. Malays. Math. Sci. Society 38 (1), 95-112, 2015.
  • Elsayed, E. M. and Ibrahim, T. F. Periodicity and solutions for some systems of nonlinear rational difference equations, Hacet. J. Math. Stat. 44 (6), 13611390, 2015.
  • Elsayed, E. M. and Khaliq, A. Global attractivity and periodicity behavior of a recursive sequence, J. Comput. Anal. Appl. 22 (2), 369-379, 2017.
  • Erdogan, M. E. and Cinar, C. On the dynamics of the recursive sequence, Fasciculi Math. 50, 59-66, 2013.
  • Halim, Y. Global character of systems of rational difference equations, Electron. J. Math. Anal. Appl. 3 (1), 204-214, 2015.
  • Hassan, Sk. and Chatterjee, E. Dynamics of the equation in the complex plane, Cogent Math. 2, 1-12, 2015.
  • Ibrahim, T. F. Periodicity and Solution of Rational Recurrence Relation of Order Six, Appl. Math. 3 , 729-733, 2012.
  • Ibrahim, T. F. Periodicity and Global Attractivity of Difference Equation of Higher Order, J. Comput. Anal. Appl. 16 , 552-564, 2014.
  • Jana, D. and Elsayed, E. M. Interplay between strong Allee effect, harvesting and hydra effect of a single population discrete - time system, Int. J. Biomath. 9 (1), 1650004, 25 pages, 2016.
  • Khaliq, A. and Elsayed, E. M. The dynamics and solution of some difference equations, J. Nonlinear Sci. Appl. 9 (3), 1052-1063, 2016.
  • Khan, A. Q. Din, Q. Qureshi, M. N. and Ibrahim, T. F. Global behavior of an anti- competitive system of fourth-order rational difference equations, Computational Ecology and Software 4 (1), 35-46, 2014.
  • Kocic, V. L. and Ladas, G. Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
  • Kulenovic M. R. S. and Ladas, G. Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001.
  • Kulenovic, M. R. S. Ladas, G. and Sizer, W. S. On the rational recursive sequence $x_{n+1}=\dfrac{\alpha x_n+\beta x_{n-1}}{\gamma x_n+\delta x_{n-1}}$, Math. Sci Res Hot-Line 2 (5), 1-16, 1996.
  • Qureshi, M. N. and Khan, A. Q. Local stability of an open-access anchovy shery model, Comput. Ecology and Software 5 (1), 48-62, 2015.
  • Simsek, D. Cinar C. and Yalcinkaya, I. On the recursive sequence $x_{n+1}=\dfrac{x_{n-3}}{1+x_{n-1}}$, Int. J. Contemp. Math. Sci. 1 (10), 475-480, 2006.
  • Su, Y. H. and Li, W. T. Global asymptotic stability of a second-order nonlinear difference equation, Appl. Math. Comput. 168, 981-989, 2005.
  • Tollu, D. Yazlik, Y. and Taskara, N. The Solutions of Four Riccati Difference Equations Associated with Fibonacci Numbers, Balkan J. Math. 2, 163-172, 2014.
  • Touafek, N. On a second order rational difference equation, Hacet. J. Math. Stat. 41 (6), 867-874, 2012.
  • Touafek, N. and Elsayed, E. M. On the periodicity of some systems of nonlinear difference equations, Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie Tome 55 (103) (2), 217-224, 2012.
  • Touafek, N. and Elsayed, E. M. On the solutions of systems of rational difference equations, Math. Comput. Model. 55, 1987-1997, 2012.
  • Touafek, N. and Haddad, N. On a mixed max-type rational system of difference equations, Electron. J. Math. Anal. Appl. 3 (1), 164 - 169, 2015.
  • Wang, C. and Hu, M. On the solutions of a rational recursive sequence, J. Math. Inform. 1 (14), 25-33, 2013.
  • Yalcinkaya, I. and Cinar C. On the dynamics of difference equation $x_{n+1}=\dfrac{a x_{n-k}}{b+c x_n^p} $, Fasciculi Math. 42, 141-148, 2009.
  • Yalçınkaya, I. Cinar, C. and Atalay, M. On the solutions of systems of difference equations, Adv. Difference Equ. Vol. 2008, Article ID 143943, 9 pages, 2008.
  • Yan, X. and Li , W. Global attractivity in the recursive sequence $x_{n+1}=\dfrac{\alpha-\beta x_n}{\gamma-x_{n-1}}$, Appl. Math. Comp. 138 (2-3), 415-423, 2003.
  • Yang, X. On the global asymptotic stability of the difference equation $ x_{n+1}=\dfrac{x_{n-1}x_{n-2}+x_{n-3}+a}{x_{n-1}+x_{n-2}x_{n-3}+a}$ , Appl. Math. Comp. 171 (2), 857-861, 2005.
  • Yazlik, Y., Elsayed, E. M. and Taskara, N. On the behaviour of the solutions of difference equation systems, J. Comput. Anal. Appl. 16 (5), 932-941, 2014.
  • Zayed, E. M. E. Qualitative behavior of the rational recursive sequence $x_{n+1}=A x_n+B x_{n-k}+\dfrac{p+x_{n-k}}{q x_n+x_{n-k}} $, Int. J. Adv. Math. 1 (1), 44-55, 2014.
  • Zayed, E. M. E. and El-Moneam, M. A. On the rational recursive sequence $x_{n+1}=a x_n-\dfrac{b x_n}{c x_n-d x_{n-k}} $, Comm. Appl. Nonlinear Anal. 15, 47-57 2008.
  • Zhang, Q., Zhang, W., Liu, J. and Shao, Y. On a fuzzy logistic difference equation, WSEAS Trans. Math.13, 282-290, 2014.
Birincil Dil en
Konular Matematik
Dergi Bölümü Matematik
Yazarlar

Yazar: Abdul Khaliq (Sorumlu Yazar)
Kurum: RIPHAH INSTITUTE OF COMPUTING AND APPLIED SCIENCES, RIPHAH INTERNATIONAL UNIVERSITY
Ülke: Pakistan


Yazar: E.M. Elsayed
Kurum: MATHEMATICS DEPARTMENT, FACULTY OF SCIENCE, KING ABDULAZIZ UNIVERSITY
Ülke: Egypt


Bibtex @araştırma makalesi { hujms471054, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {Hacettepe Üniversitesi}, year = {2018}, volume = {47}, pages = {1128 - 1143}, doi = {}, title = {Qualitative study of a higher order rational difference equation}, key = {cite}, author = {Elsayed, E.M. and Khaliq, Abdul} }
APA Khaliq, A , Elsayed, E . (2018). Qualitative study of a higher order rational difference equation. Hacettepe Journal of Mathematics and Statistics, 47 (5), 1128-1143. Retrieved from http://dergipark.gov.tr/hujms/issue/39860/471054
MLA Khaliq, A , Elsayed, E . "Qualitative study of a higher order rational difference equation". Hacettepe Journal of Mathematics and Statistics 47 (2018): 1128-1143 <http://dergipark.gov.tr/hujms/issue/39860/471054>
Chicago Khaliq, A , Elsayed, E . "Qualitative study of a higher order rational difference equation". Hacettepe Journal of Mathematics and Statistics 47 (2018): 1128-1143
RIS TY - JOUR T1 - Qualitative study of a higher order rational difference equation AU - Abdul Khaliq , E.M. Elsayed Y1 - 2018 PY - 2018 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1128 EP - 1143 VL - 47 IS - 5 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2017 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Qualitative study of a higher order rational difference equation %A Abdul Khaliq , E.M. Elsayed %T Qualitative study of a higher order rational difference equation %D 2018 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 47 %N 5 %R %U
ISNAD Khaliq, Abdul , Elsayed, E.M. . "Qualitative study of a higher order rational difference equation". Hacettepe Journal of Mathematics and Statistics 47 / 5 (Ekim 2018): 1128-1143.