Merve KARA, Yasin YAZLIK

3319

Solvability of a system of nonlinear difference equations of higher order

Solvability of a system of nonlinear difference equations of higher order

In this paper, we show that the following higher-order system of nonlinear difference equations, $x_{n=}frac{x_{n-kYn-k-l}}{Yn-l(a_n+b_nx_n-kYn-k-l)}$, $Y_{n=}frac{Y_{n-kxn-k-l}}{xn-l(a_n+b_nY_n-kxn-k-l)}$, $nin{mathbb{N}}_0$where k; l 2 N, (an)n2N0; (bn)n2N0; (n)n2N0; (n)n2N0and the initial values x?i; y?i , i = 1; k + l , are real numbers,can be solved and some results in the literature can be extended further. Also, by using these obtained formulas,we investigate the asymptotic behavior of well-defined solutions of the above difference equations system for the casek = 2; l = k .
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  • [1] Ahmed AM, Elsayed EM. The expressions of solutions and the periodicity of some rational difference equations systems. Journal of Applied Mathematics and Informatics 2016; 34 (1-2): 35-48.
  • [2] Alzahrani EO, El-Dessoky MM, Elsayed EM, Kuang Y. Solutions and properties of some degenerate systems of difference equations. Journal of Computational Analysis and Applications 2015; 18: 321-333.
  • [3] Andruch A, Migda SM. Further properties of the rational recursive sequence xn+1 = axn?1 b+cxnxn?1 . Opuscula Mathematica 2006; 26: 387-394.
  • [4] Cinar C. On the positive solutions of difference equation xn+1 = xn?1 1+xnxn?1 . Applied Mathematics and Computation 2004; 150 (1): 21-24.
  • [5] Dekkar I, Touafek N, Yazlik Y. Global stability of a third-order nonlinear system of difference equations with period-two coefficients. Revista de la Real Academia 2017; 111: 325-347.
  • [6] Din Q. On a system of fourth-order rational difference equations. Acta Universitatis Apulensis Mathematics Informatics 2014; 39: 137-150.
  • [7] Elaydi S. An Introduction to Difference Equations. New York, NY, USA: Springer, 1996.
  • [8] El-Dessoky MM, Elsayed EM, Alghamdi M. Solutions and periodicity for some systems of fourth order rational difference equations. Journal Computational Analysis and Applications 2015; 18 (1): 179-194.
  • [9] El-Metwally H, Elsayed EM. Qualitative study of solutions of some difference equations. Abstract and Applied Analysis 2012; 2012: 248291.
  • [10] El-Metwally H, Elsayed EM. Solution and behavior of a third rational difference equation. Utilitas Mathematica 2012; 88: 27-42.
  • [11] El-Metwally H, Elsayed EM. Qualitative behavior of some rational difference equations. Journal Compututational Analysis and Applications 2016; 20 (2): 226-236.
  • [12] Elsayed EM. Solution for systems of difference equations of rational form of order two. Computational and Applied Mathematics 2014; 33: 751-765.
  • [13] Elsayed EM. Expression and behavior of the solutions of some rational recursive sequences. Mathematical Methods in the Applied Sciences 2016; 39: 5682-5694.
  • [14] Elsayed EM, Alshabi KN, Alzahrani F. Qualitative study of solution of some higher order difference equations. Journal of Computational Analysis And Applications 2019; 26 (7): 1179-1191.
  • [15] Elsayed EM, El-Metwally H. On the solutions of some nonlinear systems of difference equations. Advances in Difference Equations 2013; 161 (1): 1-14.
  • [16] Grove EA, Ladas G. Periodicities in Nonlinear Difference Equations Advances in Discrete Mathematics and Applications. Vol. 4. London, UK: Chapman & Hall/CRC, London, 2005.
  • [17] Haddad N, Touafek N, Rabago, JFT. Well-defined solutions of a system of difference equations. Journal of Applied Mathematics and Computing 2018; 56 (1-2): 439-458.
  • [18] Halim Y, Touafek N, Yazlık Y. Dynamic behavior of a second-order nonlinear rational difference equation. Turkish Journal of Mathematics 2015; 39 (6): 1004-1018.
  • [19] Ibrahim TF. On the third order rational difference equation xn+1 = xnxn?2 xn?1(a+bxnxn?2) . International Journal of Contemporary Mathematical Sciences 2009; 4: 1321-1334.
  • [20] Ibrahim TF, Touafek N. On a third order rational difference equation with variable coefficients. Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms 2013; 20: 251-264.
  • [21] Khaliq A, Elsayed EM. The dynamics and solution of some difference equations. Journal of Nonlinear Science and Applications 2016; 9: 1052-1063.
  • [22] Khan AQ, Din Q, Qureshi MN, Ibrahim TF. Global behavior of an anti-competitive system of fourth-order rational difference equations. Computational Ecology and Software 2014; 4 (1): 35-46.
  • [23] Kulenovic MRS. Discrete Dynamical Systems and Difference Equations with Mathematica. London, UK: Chapman & Hall/CRC Press, 2002.
  • [24] Kurbanli AS, Cinar C, Yalcinkaya I. On the behavior of positive solutions of the system of rational difference equations xn+1 = xn?1 ynxn?1+1 ; yn+1 = yn?1 xnyn?1+1 . Mathematical and Computer Modelling 2011; 53: 1261-1267.
  • [25] Öcalan Ö. Oscillation of nonlinear difference equations with several coefficients. Communications in Mathematical Analysis 2008; 4 (1): 35–44.
  • [26] Okumus I, Soykan Y. Dynamical behavior of a system of three-dimensional nonlinear difference equations. Advances in Difference Equations 2018; 2018 (223): 1-15.
  • [27] Özban AY. On the positive solutions of the system of rational difference equations xn+1 = 1 yn?k ; yn+1 = yn xn?myn?m?k . Journal of Mathematical Analysis and Applications 2006; 323: 26-32.
  • [28] Papaschinopoulos G, Schinas CJ. On the behavior of the solutions of a system of two nonlinear difference equations. Communications on Applied Nonlinear Analysis 1998; 5 (2): 47-59.
  • [29] Papaschinopoulos G, Stefanidou G. Asymptotic behavior of the solutions of a class of rational difference equations. International Journal of Difference Equations 2010; 5 (2): 233-249.
  • [30] Rabago JFT, Bacani JB. On a nonlinear difference equations. Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 2017; 24: 375-394.
  • [31] Raouf A. Global behavior of the rational Riccati difference equation of order two: the general case. Journal of Difference Equations and Applications 2012; 18: 947-961.
  • [32] Stević S. On a solvable rational system of difference equations. Applied Mathematics and Computation 2012; 219: 2896-2908.
  • [33] Stević S. On the system xn+1 = ynxn?k yn?k+1(an+bnynxn?k) , yn+1 = xnyn?k xn?k+1(cn+dnxnyn?k) . Applied Mathematics and Computation 2013; 219: 4526-4534.
  • [34] Stević S, Alghamdi MA, Alotaibi A, Elsayed EM. On a class of solvable higher-order difference equations. Filomat 2017; 31 (2): 461-477.
  • [35] Stević S, Diblík J, Iričanin B, Šmarda Z. On some solvable difference equations and systems of difference equations. Abstract and Applied Analysis 2012; 2012: 541761.
  • [36] Stević S, Diblík J, Iričanin B, Šmarda Z. Solvability of nonlinear difference equations of fourth order. Electronic Journal of Differential Equations 2014; 2014 (264): 1-14.
  • [37] Stević S, Iričanin B, Šmarda Z. On a close to symmetric system of difference equations of second order. Advances in Difference Equations 2015; 2015 (264): 1-17.
  • [38] Tollu DT, Yazlik Y, Taşkara N. On the solutions of two special types of Riccati difference equation via Fibonacci numbers. Advances in Difference Equations 2013; 2013: 174.
  • [39] Tollu DT, Yazlik Y, Taşkara N. On fourteen solvable systems of difference equations. Applied Mathematics and Computation 2014; 233: 310-319.
  • [40] Tollu DT, Yazlık Y, Taşkara N. On a solvable nonlinear difference equation of higher order. Turkish Journal of Mathematics 2018; 42: 1765-1778.
  • [41] Touafek N, Elsayed EM. On the periodicity of some systems of nonlinear difference equations. Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie 2012; 55 (103): 217-224.
  • [42] Yalcinkaya I. On the difference equation xn+1 = + xn?m xk n . Discrete Dynamics in Nature and Society 2008; 2008: 805460.
  • [43] Yang X. On the system of rational difference equations xn = A + yn?1 xn?pyn?q ; yn = A + xn?1 xn?ryn?s . Journal of Mathematical Analysis and Applications 2006; 307: 305-311.
  • [44] Yazlik Y. On the solutions and behavior of rational difference equations. Journal of Computational Analysis and Applications 2014; 17 (3): 584-594.
  • [45] Yazlik Y, Elsayed EM, Taşkara N. On the behavior of the solutions of difference equation systems. Journal of Computational Analysis and Applications 2014; 16 (5): 932-941.
  • [46] Yazlik Y, Tollu DT, Taşkara N. On the behavior of solutions for some systems of difference equations. Journal of Computational Analysis and Applications 2015; 18 (1): 166-178.
  • [47] Yazlik Y, Tollu DT, Taşkara N. On the solutions of a max-type difference equation system. Mathematical Methods in the Applied Sciences 2015; 38 (17): 4388-4410.
  • [48] Yazlik Y, Tollu DT, Taşkara N. On the solutions of a three-dimensional system of difference equations. Kuwait Journal of Sciences 2016; 43 (1): 95-111.
  • [49] Zayed EME, El-Moneam MA. On the rational recursive sequence xn+1 = Axn+( xn+ xn?k) Bxn+Cxn?k . Communications on Applied Nonlinear Analysis 2009; 16: 91-106.
Turkish Journal of Mathematics-Cover
  • ISSN: 1300-0098
  • Yayın Aralığı: Yılda 6 Sayı
  • Yayıncı: TÜBİTAK
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