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A Review on the Solutions of Difference Equations via Integer Sequences such as Fibonacci Numbers and Tribonacci Numbers

Year 2019, Volume: 2 Issue: 4, 281 - 292, 29.12.2019

Abstract

In this review article, we study the recent investigations on the forms of solutions of systems difference equations and difference equations in terms of well-known integer sequences such as Fibonacci numbers, Padovan numbers. We focus on the papers given some interesting relationships both between the exact solutions of difference equations and the integer sequences and between the equilibrium points of difference equations and the golden ratio.

References

  • [1] D.T. Tollu, Y. Yazlik, N. Taskara, On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Adv. Differ. Equ., 2013 (2013), 174.
  • [2] J.F.T. Rabago, On the closed-form solution of a nonlinear difference equation and another proof to Sroysang’s conjecture, arXiv:1604.06659v1 [math.NT] (2016).
  • [3] Y. Yazlik, D.T. Tollu, N. Taskara, On the solutions of difference equation systems with Padovan numbers, Appl. Math., 4 (2013), 15-20.
  • [4] D.T. Tollu, Y. Yazlik, N. Taskara, The solutions of four Riccati difference equations associated with Fibonacci numbers, Balkan J. Math., 2 (2014), 163-172.
  • [5] D.T. Tollu, Y. Yazlik, N. Taskara, On fourteen solvable systems of difference equations, Appl. Math. Comput., 233 (2014), 310-319.
  • [6] Y. Halim, Global character of systems of rational difference equations, Elect. J. Mathe. Anal. Appl., 3(1) (2015), 204-214.
  • [7] J.B. Bacani, J.F.T. Rabago, On two nonlinear difference equations. Dynamics of continuous, Discrete Impul. Sys., (Serias A) to appear (2015).
  • [8] Y. Halim, M. Bayram, On the solutions of a higher-order difference equation in terms of generalized Fibonacci sequences, Mathe. Meth. Appl. Sci., 39 (2016), 2974-2982.
  • [9] Y. Halim, A System of difference equations with solutions associated to Fibonacci numbers, Int. J. Differ Equ., 11(1) (2016), 65-77.
  • [10] M. M. El-Dessoky, On the dynamics of higher order difference equations $x_{n+1}=ax_{n}+\frac{\alpha x_{n}x_{n-l}}{\beta x_{n}+\gamma x_{n-k}}$, J. Comput. Anal. Appl., 22(7) (2017), 1309-1322.
  • [11] Y. Halim, J. F. T. Rabago, On some solvable systems of difference equations with solutions associated to Fibonacci numbers, Elect. J. Mathe. Anal. Appl., 5(1) (2017), 166-178.
  • [12] Y. Halim, J. F. T. Rabago, On the solutions of a second-order difference equation in terms of generalized Padovan sequences, Math. Slovaca, 68(3) (2018), 625-638.
  • [13] S. Stevic, B. Iricanin, W. Kosmala, Z. Smarda, Representation of solutions of a solvable nonlinear difference equation of second order, Elect. J. Qualitative Theory of Dif. Equ., 95 (2018), 1-18.
  • [14] A. M. Alotaibi, M. S. M. Noorani, M. A. El-Moneam, On the solutions of a system of third-order rational difference equations, Discrete Dyn. Nat. Soc., 2018 (2018), Article ID 1743540, 11 pages.
  • [15] M. M. El-Dessoky, E. M. Elabbasy, A. Asiri, Dynamics and solutions of a fifth-order nonlinear difference equations, Discrete Dyn. Nat. Soc., 2018 (2018), Article ID 9129354, 21 pages.
  • [16] H. Matsunaga, R. Suzuki, Classification of global behavior of a system of rational difference equations, Appl. Math. Lett., 85 (2018), 57-63.
  • [17] Ö. Öcalan, O. Duman, On solutions of the recursive equations $x_{n+1}=x_{n-1}^{p}/x_{n}^{p}$ ($p>0$) via Fibonacci-type sequences, Elect. J. Math. Anal. Appl., 7(1) (2019), 102-115.
  • [18] Y. Akrour, N. Touafek, N, Y. Halim, On a system of difference equations of second order solved in a closed form, arXiv:1904.04476v1, [math.DS] (2019).
  • [19] İ. Okumuş, Y. Soykan, On the solutions of four rational difference equations associated to Tribonacci numbers, Preprints, (2019), preprints201906.0266.v1
  • [20] İ. Okumuş, Y. Soykan, On the solutions of four second-order nonlinear difference equations, Preprints, (2019), preprints201906.0265.v1.
  • [21] İ. Okumuş, Y. Soykan, On the solutions of systems of difference equations via Tribonacci numbers, arXiv preprint, (2019), arXiv:1906.09987v1 [math.DS].
  • [22] İ. Okumuş, Y. Soykan, On the dynamics of solutions of a rational difference equation via generalized Tribonacci numbers, arXiv preprint, (2019), arXiv:1906.11629v1 [math.DS].
  • [23] A. Asiri, E. M. Elsayed, Dynamics and solutions of some recursive sequences of higher order, J. Comput. Anal. Appl., 27(4) (2019), 656-670.
  • [24] A. Raouf, Global behaviour of the rational Riccati difference equation of order two: the general case, J. Differ. Equ. Appl., 18(6) (2012), 947–961.
  • [25] E. M. Elsayed, Solution for systems of difference equations of rational form of order two, Comput. Appl. Math., 33 (2014), 751-765.
  • [26] E. M. Elsayed, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacettepe J. Math. Sta., 44(6) (2015), 1361-1390.
  • [27] M. Dehghan, R. Mazrooei-Sebdani, H. Sedaghat, Global behaviour of the Riccati difference equation of order two, J. Differ. Equ. Appl., 17(4) (2011), 467–477.
  • [28] M. Göcen, A. Cebeci, On the periodic solutions of some systems of higher order difference equations, Rocky Mountain J. Math., 48(3) (2018), 845-858.
  • [29] M. Göcen, M. Güneysu, The global attractivity of some rational difference equations, J. Comput. Anal. Appl., 25(7) (2018), 1233-1243.
  • [30] Y. Halim, N. Touafek, Y. Yazlik, Dynamics behavior of a second-order nonlinear rational difference equation, Turkish J. Math., 39 (2015), 1004-1018.
  • [31] J. F. T. Rabago, Effective methods on determining the periodicity and form of solutions of some systems of nonlinear difference equations, arXiv preprints, (2015), arXiv:1512.01605v1 [math.DS].
  • [32] E. Taşdemir, Y. Soykan, Stability of negative equilibrium of a non-linear difference equation, J. Math. Sci. Adv. Appl., 49(1) (2018), 51-57.
  • [33] E. Taşdemir, Y. Soykan, Dynamical analysis of a non-linear difference equation, J. Comput. Anal. Appl., 26(2) (2019), 288-301.
  • [34] S. Stevic, More on a rational recurrence relation, Appl. Math. E-Notes, 4 (2004), 80-84.
  • [35] S. Stevic, On some solvable systems of difference equations, Appl. Math. Comput., 218 (2012), 5010-5018.
  • [36] S. Stevic, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Elect. J. Qualitative Theory Dif. Equ., 67 (2014), 1-15.
Year 2019, Volume: 2 Issue: 4, 281 - 292, 29.12.2019

Abstract

References

  • [1] D.T. Tollu, Y. Yazlik, N. Taskara, On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Adv. Differ. Equ., 2013 (2013), 174.
  • [2] J.F.T. Rabago, On the closed-form solution of a nonlinear difference equation and another proof to Sroysang’s conjecture, arXiv:1604.06659v1 [math.NT] (2016).
  • [3] Y. Yazlik, D.T. Tollu, N. Taskara, On the solutions of difference equation systems with Padovan numbers, Appl. Math., 4 (2013), 15-20.
  • [4] D.T. Tollu, Y. Yazlik, N. Taskara, The solutions of four Riccati difference equations associated with Fibonacci numbers, Balkan J. Math., 2 (2014), 163-172.
  • [5] D.T. Tollu, Y. Yazlik, N. Taskara, On fourteen solvable systems of difference equations, Appl. Math. Comput., 233 (2014), 310-319.
  • [6] Y. Halim, Global character of systems of rational difference equations, Elect. J. Mathe. Anal. Appl., 3(1) (2015), 204-214.
  • [7] J.B. Bacani, J.F.T. Rabago, On two nonlinear difference equations. Dynamics of continuous, Discrete Impul. Sys., (Serias A) to appear (2015).
  • [8] Y. Halim, M. Bayram, On the solutions of a higher-order difference equation in terms of generalized Fibonacci sequences, Mathe. Meth. Appl. Sci., 39 (2016), 2974-2982.
  • [9] Y. Halim, A System of difference equations with solutions associated to Fibonacci numbers, Int. J. Differ Equ., 11(1) (2016), 65-77.
  • [10] M. M. El-Dessoky, On the dynamics of higher order difference equations $x_{n+1}=ax_{n}+\frac{\alpha x_{n}x_{n-l}}{\beta x_{n}+\gamma x_{n-k}}$, J. Comput. Anal. Appl., 22(7) (2017), 1309-1322.
  • [11] Y. Halim, J. F. T. Rabago, On some solvable systems of difference equations with solutions associated to Fibonacci numbers, Elect. J. Mathe. Anal. Appl., 5(1) (2017), 166-178.
  • [12] Y. Halim, J. F. T. Rabago, On the solutions of a second-order difference equation in terms of generalized Padovan sequences, Math. Slovaca, 68(3) (2018), 625-638.
  • [13] S. Stevic, B. Iricanin, W. Kosmala, Z. Smarda, Representation of solutions of a solvable nonlinear difference equation of second order, Elect. J. Qualitative Theory of Dif. Equ., 95 (2018), 1-18.
  • [14] A. M. Alotaibi, M. S. M. Noorani, M. A. El-Moneam, On the solutions of a system of third-order rational difference equations, Discrete Dyn. Nat. Soc., 2018 (2018), Article ID 1743540, 11 pages.
  • [15] M. M. El-Dessoky, E. M. Elabbasy, A. Asiri, Dynamics and solutions of a fifth-order nonlinear difference equations, Discrete Dyn. Nat. Soc., 2018 (2018), Article ID 9129354, 21 pages.
  • [16] H. Matsunaga, R. Suzuki, Classification of global behavior of a system of rational difference equations, Appl. Math. Lett., 85 (2018), 57-63.
  • [17] Ö. Öcalan, O. Duman, On solutions of the recursive equations $x_{n+1}=x_{n-1}^{p}/x_{n}^{p}$ ($p>0$) via Fibonacci-type sequences, Elect. J. Math. Anal. Appl., 7(1) (2019), 102-115.
  • [18] Y. Akrour, N. Touafek, N, Y. Halim, On a system of difference equations of second order solved in a closed form, arXiv:1904.04476v1, [math.DS] (2019).
  • [19] İ. Okumuş, Y. Soykan, On the solutions of four rational difference equations associated to Tribonacci numbers, Preprints, (2019), preprints201906.0266.v1
  • [20] İ. Okumuş, Y. Soykan, On the solutions of four second-order nonlinear difference equations, Preprints, (2019), preprints201906.0265.v1.
  • [21] İ. Okumuş, Y. Soykan, On the solutions of systems of difference equations via Tribonacci numbers, arXiv preprint, (2019), arXiv:1906.09987v1 [math.DS].
  • [22] İ. Okumuş, Y. Soykan, On the dynamics of solutions of a rational difference equation via generalized Tribonacci numbers, arXiv preprint, (2019), arXiv:1906.11629v1 [math.DS].
  • [23] A. Asiri, E. M. Elsayed, Dynamics and solutions of some recursive sequences of higher order, J. Comput. Anal. Appl., 27(4) (2019), 656-670.
  • [24] A. Raouf, Global behaviour of the rational Riccati difference equation of order two: the general case, J. Differ. Equ. Appl., 18(6) (2012), 947–961.
  • [25] E. M. Elsayed, Solution for systems of difference equations of rational form of order two, Comput. Appl. Math., 33 (2014), 751-765.
  • [26] E. M. Elsayed, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacettepe J. Math. Sta., 44(6) (2015), 1361-1390.
  • [27] M. Dehghan, R. Mazrooei-Sebdani, H. Sedaghat, Global behaviour of the Riccati difference equation of order two, J. Differ. Equ. Appl., 17(4) (2011), 467–477.
  • [28] M. Göcen, A. Cebeci, On the periodic solutions of some systems of higher order difference equations, Rocky Mountain J. Math., 48(3) (2018), 845-858.
  • [29] M. Göcen, M. Güneysu, The global attractivity of some rational difference equations, J. Comput. Anal. Appl., 25(7) (2018), 1233-1243.
  • [30] Y. Halim, N. Touafek, Y. Yazlik, Dynamics behavior of a second-order nonlinear rational difference equation, Turkish J. Math., 39 (2015), 1004-1018.
  • [31] J. F. T. Rabago, Effective methods on determining the periodicity and form of solutions of some systems of nonlinear difference equations, arXiv preprints, (2015), arXiv:1512.01605v1 [math.DS].
  • [32] E. Taşdemir, Y. Soykan, Stability of negative equilibrium of a non-linear difference equation, J. Math. Sci. Adv. Appl., 49(1) (2018), 51-57.
  • [33] E. Taşdemir, Y. Soykan, Dynamical analysis of a non-linear difference equation, J. Comput. Anal. Appl., 26(2) (2019), 288-301.
  • [34] S. Stevic, More on a rational recurrence relation, Appl. Math. E-Notes, 4 (2004), 80-84.
  • [35] S. Stevic, On some solvable systems of difference equations, Appl. Math. Comput., 218 (2012), 5010-5018.
  • [36] S. Stevic, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Elect. J. Qualitative Theory Dif. Equ., 67 (2014), 1-15.
There are 36 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

İnci Okumuş 0000-0003-3711-8144

Yüksel Soykan 0000-0002-1895-211X

Publication Date December 29, 2019
Submission Date July 15, 2019
Acceptance Date October 1, 2019
Published in Issue Year 2019 Volume: 2 Issue: 4

Cite

APA Okumuş, İ., & Soykan, Y. (2019). A Review on the Solutions of Difference Equations via Integer Sequences such as Fibonacci Numbers and Tribonacci Numbers. Communications in Advanced Mathematical Sciences, 2(4), 281-292.
AMA Okumuş İ, Soykan Y. A Review on the Solutions of Difference Equations via Integer Sequences such as Fibonacci Numbers and Tribonacci Numbers. Communications in Advanced Mathematical Sciences. December 2019;2(4):281-292.
Chicago Okumuş, İnci, and Yüksel Soykan. “A Review on the Solutions of Difference Equations via Integer Sequences Such As Fibonacci Numbers and Tribonacci Numbers”. Communications in Advanced Mathematical Sciences 2, no. 4 (December 2019): 281-92.
EndNote Okumuş İ, Soykan Y (December 1, 2019) A Review on the Solutions of Difference Equations via Integer Sequences such as Fibonacci Numbers and Tribonacci Numbers. Communications in Advanced Mathematical Sciences 2 4 281–292.
IEEE İ. Okumuş and Y. Soykan, “A Review on the Solutions of Difference Equations via Integer Sequences such as Fibonacci Numbers and Tribonacci Numbers”, Communications in Advanced Mathematical Sciences, vol. 2, no. 4, pp. 281–292, 2019.
ISNAD Okumuş, İnci - Soykan, Yüksel. “A Review on the Solutions of Difference Equations via Integer Sequences Such As Fibonacci Numbers and Tribonacci Numbers”. Communications in Advanced Mathematical Sciences 2/4 (December 2019), 281-292.
JAMA Okumuş İ, Soykan Y. A Review on the Solutions of Difference Equations via Integer Sequences such as Fibonacci Numbers and Tribonacci Numbers. Communications in Advanced Mathematical Sciences. 2019;2:281–292.
MLA Okumuş, İnci and Yüksel Soykan. “A Review on the Solutions of Difference Equations via Integer Sequences Such As Fibonacci Numbers and Tribonacci Numbers”. Communications in Advanced Mathematical Sciences, vol. 2, no. 4, 2019, pp. 281-92.
Vancouver Okumuş İ, Soykan Y. A Review on the Solutions of Difference Equations via Integer Sequences such as Fibonacci Numbers and Tribonacci Numbers. Communications in Advanced Mathematical Sciences. 2019;2(4):281-92.

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