#### On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$

Yıl: 2020 Cilt: 8 Sayı: 2 Sayfa Aralığı: 155 - 163 Metin Dili: İngilizce İndeks Tarihi: 11-01-2022

On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$

Öz:
In this paper, given solutions fort he following difference equation $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$ , n ∈  where the initial conditions are positive real numbers. The initial conditions of the equation are arbitrary positive real numbers. We investigate periodic behavior of this equation. Also some numerical examples and graphs of solutions are given.
Anahtar Kelime:

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık
• [1]. Amleh A. M., Grove E. A., Ladas G., Georgiou D. A., On the recursive sequence 1 1 n n n x x x α − + = + , J. Math. Anal. Appl., 233, 2, (1999),790-798.
• [2]. Ari, M., Gelı̇ şken, A., Periodic and asymptotic behavior of a difference equation. Asian-European Journal of Mathematics, 12, 6, (2019), 2040004.
• [3]. Belhannache, F., Nouressadat T., and Raafat A., Dynamics of a third-order rational difference equation, Bull. Math. Soc. Sci. Math. Roumanie, 59, 1, (2016).
• [4]. Cinar C., On the positive solutions of the difference equation 1 1 1 1 n n n n x x ax x − + − = + , Appl. Math. Comp., 158, 3, (2004), 809–812.
• [5]. Cinar C., On the positive solutions of the difference equation 1 1 1 1 n n n n x x ax x − + − = − + , Appl. Math. Comp., 158, 3, (2004), 793–797.
• [6]. Cinar C., On the positive solutions of the difference equation 1 1 1 1 n n n n ax x bx x − + − = + , Appl. Math. Comp., 156, 3, (2004), 587–590.
• [7]. Cı̇ nar, G., Gelı̇ şken, A., Özkan, O., Well-defined solutions of the difference equation xn= xn− 3 kxn− 4 kxn− 5 kxn− kxn− 2 k (±1±xn− 3 kxn− 4 kxn− 5 k). Asian-European Journal of Mathematics, 12, 6, (2019), 2040016. [8]. DeVault R., Ladas G., Schultz W.S., On the recursive sequence 1 2 1 n n n A x x x + − = + , Proc. Amer. Math. Soc., 126, 11, (1998), 3257-3261.
• [9]. Elabbasy E. M., El-Metwally H., Elsayed E. M., On the difference equation 1 1 n n n n n bx x ax cx dx + − = − − , Advances in Difference Equation, (2006), 1-10.
• [10]. Elabbasy E. M., El-Metwally H., Elsayed E. M., Qualitative behavior of higher order difference equation, Soochow Journal of Mathematics, 33, 4, (2007), 861-873.
• [11]. Elabbasy E. M., El-Metwally H., Elsayed E. M., Global attractivity and periodic character of a fractional difference equation of order three, Yokohama Mathematical Journal, 53, (2007), 89-100.
• [12]. Elabbasy E. M., El-Metwally H., Elsayed E. M., On the difference equation 1 0 n k n k n i i x x x α β γ − + − = = + ∏ , J. Conc. Appl. Math., 5(2), (2007), 101-113.
• [13]. Elabbasy E. M. and Elsayed E. M., On the Global Attractivity of Difference Equation of Higher Order, Carpathian Journal of Mathematics, 24, 2, (2008), 45–53.
• [14]. Elsayed E. M., On the Solution of Recursive Sequence of Order Two, Fasciculi Mathematici, 40, (2008), 5–13.
• [15]. Elsayed E. M., Dynamics of a rational recursive sequences. International Journal of Difference Equations, 4, 2, 185–200, 2009.
• [16]. Elsayed E. M., Dynamics of a Recursive Sequence of Higher Order, Communications on Applied Nonlinear Analysis, 16, 2, (2009), 37–50.
• [17]. Elsayed E. M., Solution and atractivity for a rational recursive sequence, Discrete Dynamics in Nature and Society, (2011), 17.
• [18]. Elsayed E. M., On the solution of some difference equation, Europan Journal of Pure and Applied Mathematics, 4, 3, (2011), 287–303.
• [19]. Elsayed E. M., On the Dynamics of a higher order rational recursive sequence, Communications in Mathematical Analysis, 12, 1, (2012), 117–133.
• [20]. Elsayed E. M., Solution of rational difference system of order two, Mathematical and Computer Modelling, 55, (2012), 378–384.
• [21]. Gelisken A., On A System of Rational Difference Equations, J. Comput. Anal. Appl, 23, 4, (2017), 593-606.
• [22]. Gibbons C. H., Kulenović M. R. S. and Ladas G., On the recursive sequence 1 1 n n n x x x α β χ − + + = + , Math. Sci. Res. HotLine, 4, 2, (2000), 1-11.
• [23]. Ibrahim, T. F., Periodicity and analytic solution of a recursive sequence with numerical examples, Journal of Interdisciplinary Mathematics, 12, 5, (2009), 701-708.
• [24]. Ibrahim, T. F. On the third order rational difference equation, Int. J. Contemp. Math. Sciences 4, 27, (2009), 1321-1334.
• [25]. Ibrahim, T. F., and Touafek, N., On a third order rational difference equation with variable coefficients, DCDIS Series B: Applications & Algorithms 20, (2013), 251-264.
• [26]. Ibrahim T. F., Periodicity and Global Attractivity of Difference Equation of Higher Order, Journal of Computational Analysis & Applications, 16, 1, (2014).
• [27]. Khaliq A., Alzahrani F., and Elsayed E. M., Global attractivity of a rational difference equation of order ten, J. Nonlinear Sci. Appl, 9, 6, (2016), 4465-4477.
• [28]. Kulenović M.R.S., Ladas G., Sizer W.S., On the recursive sequence 1 1 1 n n n n n x x x x x α β χ δ − + − + = + Math. Sci. Res. Hot-Line, 2, 5, (1998), 1-16.
• [29]. Kulenovic, M. R. S., Moranjkic, S., and Nurkanovic, Z., Naimark-Sacker bifurcation of second order rational difference equation with quadratic terms., J. Nonlinear Sci. Appl, 10, (2017), 3477-3489.
• [30]. Stevic S., On the recursive sequence 1 1 ( ) n n n x x g x − + = , Taiwanese J. Math., 6, 3, (2002), 405-414.
• [31]. Simsek D., Cinar C. and Yalcinkaya I., On the recursive sequence 3 1 1 1 n n n x x x − + − = + , Int. J. Contemp. Math. Sci., 1, 9, 12, (2006), 475-480.
• [32]. Simsek D., Cinar C., Karatas R., Yalcinkaya I., On the recursive sequence 5 1 2 1 n n n x x x − + − = + , Int. J. Pure Appl. Math., 27, 4, (2006), 501-507.
• [33]. Simsek D., Cinar C., Karatas R., Yalcinkaya I., "n the recursive sequence 5 1 1 3 1 n n n n x x x x − + − − = + , Int. J. Pure Appl. Math., 28, 1, (2006), 117-124.
• [34]. Simsek D., Cinar C., Yalcinkaya I., On The Recursive Sequence x(n+1) = x[n-(5k+9)] / 1+x(n-4)x(n-9) x[n-(5k+4)], Taiwanese Journal of Mathematics, 12, 5, (2008), 1087-1098.
• [35]. Simsek D., Dogan A., On A Class of Recursive Sequence, Manas Journal of Engineering, 2, 1, (2014), 16-22.
• [36]. Simsek D., Eroz M., Solutions of The Rational Difference Equations 3 1 1 2 1 n n nn n x x xx x − + − − = + , Manas Journal of Engineering, 4, 1, (2016), 12-20.
• [37]. Simsek D., Ogul B., Solutions of The Rational Difference Equations (2 k 1) 1 1 n n n k x x x − + + − = + , Manas Journal of Engineering, 5, 3, (2017), 57-68.
• [38]. Simsek D., Ogul B., Abdullayev F., Solutions of The Rational Difference Equations 11 1 2 58 1 . n n n nn x x x xx − + − −− = + , AIP Conference Proceedings, 1880, 1, 040003, (2017).
• [39]. Simsek D., Abdullayev F., On The Recursive Sequence (4 3) 1 2 ( 1) 0 1 n k n n k tk t x x x − + + −+ − = = +∏ , Journal of Mathematics Sciences, 6, 222, (2017), 762-771.
• [40]. Simsek, D., Abdullayev, F. G., On the Recursive Sequence ( 1) 1 1 ... 1 n k n n n nk x x xx x − + + − − = + , Journal of Mathematical Sciences, 234, 1, (2018), 73-81.
• [41]. Simsek, D., Ogul, B., Cinar, C., Solution of the rational difference equation xn+ 1= xn-17/1+ xn-5• xn-11, Filomat, 33, 5, (2019), 1353-1359.
• [42]. Simsek, D., Ogul, B., On The Recursive Sequence x (n+ 1)= x (n-20)/[1+ x (n-2) x (n-5) x (n-8) x (n-11) x (n-14) x (n17)], MANAS Journal of Engineering, 7, 2, (2019), 147-156.
• [43]. Yalcinkaya, I., Cinar, C., Atalay, M., On the solutions of systems of difference equations, Advances in Difference Equations, 2008, (2008), 1-9.
• [44]. Yalcinkaya, I., Cinar, C., Simsek, D., Global asymptotic stability of a system of difference equations, Applicable Analysis, 87, 6, (2008), 677-687.
• [45]. Yalcinkaya, I., Cinar, C., Global asymptotic stability of a system of two nonlinear difference equations, Fasciculi Mathematici, 43, (2010), 171-180.
 APA Ogul B, ŞİMŞEK D (2020). On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$. , 155 - 163. Chicago Ogul Burak,ŞİMŞEK Dağıstan On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$. (2020): 155 - 163. MLA Ogul Burak,ŞİMŞEK Dağıstan On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$. , 2020, ss.155 - 163. AMA Ogul B,ŞİMŞEK D On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$. . 2020; 155 - 163. Vancouver Ogul B,ŞİMŞEK D On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$. . 2020; 155 - 163. IEEE Ogul B,ŞİMŞEK D "On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$." , ss.155 - 163, 2020. ISNAD Ogul, Burak - ŞİMŞEK, Dağıstan. "On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$". (2020), 155-163.
 APA Ogul B, ŞİMŞEK D (2020). On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$. Manas Journal of Engineering, 8(2), 155 - 163. Chicago Ogul Burak,ŞİMŞEK Dağıstan On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$. Manas Journal of Engineering 8, no.2 (2020): 155 - 163. MLA Ogul Burak,ŞİMŞEK Dağıstan On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$. Manas Journal of Engineering, vol.8, no.2, 2020, ss.155 - 163. AMA Ogul B,ŞİMŞEK D On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$. Manas Journal of Engineering. 2020; 8(2): 155 - 163. Vancouver Ogul B,ŞİMŞEK D On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$. Manas Journal of Engineering. 2020; 8(2): 155 - 163. IEEE Ogul B,ŞİMŞEK D "On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$." Manas Journal of Engineering, 8, ss.155 - 163, 2020. ISNAD Ogul, Burak - ŞİMŞEK, Dağıstan. "On the recursive sequence $x_{n+1}=frac{x_{n-14}}{1-x_{n-2} x_{n-5} x_{n-8} x_{n-11}}$". Manas Journal of Engineering 8/2 (2020), 155-163.