Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2020, Cilt: 33 Sayı: 3, 767 - 778, 01.09.2020
https://doi.org/10.35378/gujs.641441

Öz

Kaynakça

  • [1] Cinar, C., “On the positive solutions of the difference equation ” , Appl. Math. Comput., 150(1): 21-24, (2004).
  • [2] Cinar, C., Toufik, M. and Yalcinkaya, I., “On the difference equation of higher order”, Util. Math., 92: 161-166, (2013).
  • [3] Din, Q., “Global behavior of a rational difference equation”, Acta Univ. Apulensis., 34: 35-49, (2013).
  • [4] Elabbasy, EM., El-Metwally, H. and Elsayed, EM., “Qualitative behavior of higher order difference equation”, Soochow J. Math., 33: 861-873, (2007).
  • [5] Elaydi, S., An Introduction to Difference Equations 3 nd ed., Springer, New York, (1996).
  • [6] Elmetwally, ME. and Elsayed, EM., “Dynamics of a rational difference equation”, Chin. Ann.Math. Ser. B., 30B(2): 187-198, (2009).
  • [7] Elmetwally, H., Yalcinkaya, I. and Cinar, C., “On the dynamics of a recursive sequence”, Electron. J. Math. Anal. Appl., 5(1): 196-201, (2017).
  • [8] Elsayed, EM., “On the solutions and periodic nature of some systems of difference equations”, Int. J. Biomath., 7(6): 1-26, (2014).
  • [9] Elsayed, EM., “Solution for systems of difference equations of rational form of order two”, Int. J. Comput. Appl. Math., 33: 751-765, (2014).
  • [10] Elsayed, EM., “Expression and behavior of the solutions of some rational recursive sequences”, Math. Methods Appl. Sci., 39: 5682-5694, (2016).
  • [11] Gelisken, A., Cinar, C. and Yalcinkaya, I., “On a max-type difference equation”, Adv. Difference Equ., 2010(1): 584890, (2010).
  • [12] Gelisken, A., Cinar, C. and Yalcinkaya, I., “On the periodicity of a difference equation with maximum”, Discrete Dyn. Nat. Soc., 2008: (2008).
  • [13] Gümüs, M. and Soykan, Y., “Global character of a six-dimensional nonlinear system of difference equations”, Discrete Dyn. Nat. Soc., 2016: (2016).
  • [14] Haddad, N., Touafek, N. and Rabago, JFT., “Well-defined solutions of a system of difference equations”, Journal of Appl. Math. and Comput., 56(1-2): 439-458, (2018).
  • [15] Kara, M. and Yazlik, Y., “Solvability of a system of non-linear difference equations of higher order”, Turk. J. Math., 43(3): 1533-1565, (2019).
  • [16] Kara, M., Tollu, DT. and Yazlik, Y., “Representation of solutions of some systems of difference equations via Padovan numbers”. (to appear).
  • [17] Kurbanli, AS., Yalcinkaya, I. and Gelisken, A., “On the behavior of the solutions of the system of rational difference equations”, International Journal of Physical Sciences, 8(2): 51-56, (2013).
  • [18] Kurbanli, AS., Cinar, C. and Yalcinkaya, I., “On the behavior of positive solutions of the system of rational difference equations ” , Math. Comput. Modelling., 53: 1261-1267, (2011).
  • [19] Öcalan, Ö., “Oscillation of nonlinear difference equations with several coefficients”, Commun. Math. Anal., 4 (1): 35-44 (2008).
  • [20] Okumus, I. and Soykan, Y., “Dynamical behavior of a system of three-dimensional nonlinear difference equations”, Adv. Difference Equ., 2018(223): 1-15, (2018).
  • [21] Papaschinopoulos, G., Schinas, CJ. and Ellina, G., “On the dynamics of the solutions of a biological model”, J. Difference Equ. Appl., 20(56): 694-705, (2014).
  • [22] Rabago, JFT., “On second-order linear recurrent homogeneous differential equations with period k”, Hacet. J. Math. Stat., 43(6): 923-933, (2014).
  • [23] Rabago, JFT., “On second-order linear recurrent functions with period k and proofs to two conjectures of Sroysang”, Hacet. J. Math. Stat., 45(2): 429-446, (2016).
  • [24] Rabago, JFT. and Bacani, JB., “On a nonlinear difference equations”, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 24: 375-394, (2017).
  • [25] Tollu, DT., Yazlik, Y. and Taskara, N., “On fourteen solvable systems of difference equations”, Appl. Math. Comput., 233: 310-319, (2014).
  • [26] Tollu, DT., Yazlik, Y. and Taskara, N., “Behavior of positive solutions of a difference equation”, J. Appl. Math. Informatics., 35: 217-230, (2017).
  • [27] Touafek, N., “On a second order rational difference equation”, Hacet. J. Math. Stat., 41: 867-874, (2012).
  • [28] Touafek, N. and Elsayed, EM., “On the periodicity of some systems of nonlinear difference equations”, Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie, 55(103): 217-224, (2012).
  • [29] Touafek, N. and Elsayed, EM., “On a second order rational systems of difference equations”, Hokkaido Math. J., 44: 29-45, (2015).
  • [30] Yalcinkaya, I. and Cinar, C., “Global asymptotic stability of a system of two nonlinear difference equations”, Fasc. Math., 43: 171-180, (2010).
  • [31] Yalcinkaya, I. and Tollu, DT., “Global behavior of a second order system of difference equations”, Adv. Stud. Contemp. Math., 26(4): 653-667, (2016).
  • [32] Yang, K. and Cushing, JM., “Global stability in a nonlinear difference-delay equation model of flour beetle population growth”, J. Difference Equ. Appl., 2(1): 31-37, (1996).
  • [33] Yazlik, Y., “On the solutions and behavior of rational difference equations”, J. Comput. Anal. Appl., 17(3): 584-594, (2014).
  • [34] Yazlik, Y., Elsayed, EM. and Taskara, N., “On the behaviour of the solutions the solutions of difference equation system”, J. Comput. Anal. Appl., 16(5): 932-941, (2014).
  • [35] Yazlik, Y., Tollu, DT. and Taskara, N., “On the behaviour of solutions for some systems of difference equations”, J. Comput. Anal. Appl., 18(1): 166-178, (2015).
  • [36] Yazlik, Y., Tollu, DT. and Taskara, N., "On the solutions of a three-dimensional system of difference equations", Kuwait J. Sci., 43(1): 95-111, (2015).
  • [37] Yazlik, Y. and Kara, M., “On a solvable system of difference equations of higher-order with period two coefficients”, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 68(2): 1675-1693, (2019).
  • [38] Yazlik, Y. and Kara, M., “Beşinci mertebeden fark denklem sisteminin çözülebilirliği üzerine”, Eskisehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B- Teorik Bilimler, 7(1): 29-45, (2019).
  • [39] Wang, X. and Li, Z., “Global asymptotic stability for two kinds of higher order recursive sequences”, J. Difference Equ. Appl., 22(10): 1542-1553, (2016).

Well-Defined Solutions of a Three-Dimensional System of Difference Equations

Yıl 2020, Cilt: 33 Sayı: 3, 767 - 778, 01.09.2020
https://doi.org/10.35378/gujs.641441

Öz

We show that the
three-dimensional system of difference equations

x_{n+1}=\frac{ax_{n}z_{n-1}}{z_{n}-\beta}+\gamma,  y_{n+1}=\frac{by_{n}x_{n-1}}{x_{n}-\gamma}+\alpha, z_{n+1}=\frac{cz_{n}y_{n-1}}{y_{n}-\alpha}+\beta,





where the parameters a,b,x, \alpha, \beta, \gamma  and the initial
conditions x_{-i}, y_{-i}, i\in\{0,1\}  are non-zero real
numbers, can be solved. Using the obtained formulas, we determine the asymptotic
behavior of solutions and give conditions for which periodic solutions exists.
Some numerical examples are given to demonstrate the theoretical results.

Kaynakça

  • [1] Cinar, C., “On the positive solutions of the difference equation ” , Appl. Math. Comput., 150(1): 21-24, (2004).
  • [2] Cinar, C., Toufik, M. and Yalcinkaya, I., “On the difference equation of higher order”, Util. Math., 92: 161-166, (2013).
  • [3] Din, Q., “Global behavior of a rational difference equation”, Acta Univ. Apulensis., 34: 35-49, (2013).
  • [4] Elabbasy, EM., El-Metwally, H. and Elsayed, EM., “Qualitative behavior of higher order difference equation”, Soochow J. Math., 33: 861-873, (2007).
  • [5] Elaydi, S., An Introduction to Difference Equations 3 nd ed., Springer, New York, (1996).
  • [6] Elmetwally, ME. and Elsayed, EM., “Dynamics of a rational difference equation”, Chin. Ann.Math. Ser. B., 30B(2): 187-198, (2009).
  • [7] Elmetwally, H., Yalcinkaya, I. and Cinar, C., “On the dynamics of a recursive sequence”, Electron. J. Math. Anal. Appl., 5(1): 196-201, (2017).
  • [8] Elsayed, EM., “On the solutions and periodic nature of some systems of difference equations”, Int. J. Biomath., 7(6): 1-26, (2014).
  • [9] Elsayed, EM., “Solution for systems of difference equations of rational form of order two”, Int. J. Comput. Appl. Math., 33: 751-765, (2014).
  • [10] Elsayed, EM., “Expression and behavior of the solutions of some rational recursive sequences”, Math. Methods Appl. Sci., 39: 5682-5694, (2016).
  • [11] Gelisken, A., Cinar, C. and Yalcinkaya, I., “On a max-type difference equation”, Adv. Difference Equ., 2010(1): 584890, (2010).
  • [12] Gelisken, A., Cinar, C. and Yalcinkaya, I., “On the periodicity of a difference equation with maximum”, Discrete Dyn. Nat. Soc., 2008: (2008).
  • [13] Gümüs, M. and Soykan, Y., “Global character of a six-dimensional nonlinear system of difference equations”, Discrete Dyn. Nat. Soc., 2016: (2016).
  • [14] Haddad, N., Touafek, N. and Rabago, JFT., “Well-defined solutions of a system of difference equations”, Journal of Appl. Math. and Comput., 56(1-2): 439-458, (2018).
  • [15] Kara, M. and Yazlik, Y., “Solvability of a system of non-linear difference equations of higher order”, Turk. J. Math., 43(3): 1533-1565, (2019).
  • [16] Kara, M., Tollu, DT. and Yazlik, Y., “Representation of solutions of some systems of difference equations via Padovan numbers”. (to appear).
  • [17] Kurbanli, AS., Yalcinkaya, I. and Gelisken, A., “On the behavior of the solutions of the system of rational difference equations”, International Journal of Physical Sciences, 8(2): 51-56, (2013).
  • [18] Kurbanli, AS., Cinar, C. and Yalcinkaya, I., “On the behavior of positive solutions of the system of rational difference equations ” , Math. Comput. Modelling., 53: 1261-1267, (2011).
  • [19] Öcalan, Ö., “Oscillation of nonlinear difference equations with several coefficients”, Commun. Math. Anal., 4 (1): 35-44 (2008).
  • [20] Okumus, I. and Soykan, Y., “Dynamical behavior of a system of three-dimensional nonlinear difference equations”, Adv. Difference Equ., 2018(223): 1-15, (2018).
  • [21] Papaschinopoulos, G., Schinas, CJ. and Ellina, G., “On the dynamics of the solutions of a biological model”, J. Difference Equ. Appl., 20(56): 694-705, (2014).
  • [22] Rabago, JFT., “On second-order linear recurrent homogeneous differential equations with period k”, Hacet. J. Math. Stat., 43(6): 923-933, (2014).
  • [23] Rabago, JFT., “On second-order linear recurrent functions with period k and proofs to two conjectures of Sroysang”, Hacet. J. Math. Stat., 45(2): 429-446, (2016).
  • [24] Rabago, JFT. and Bacani, JB., “On a nonlinear difference equations”, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 24: 375-394, (2017).
  • [25] Tollu, DT., Yazlik, Y. and Taskara, N., “On fourteen solvable systems of difference equations”, Appl. Math. Comput., 233: 310-319, (2014).
  • [26] Tollu, DT., Yazlik, Y. and Taskara, N., “Behavior of positive solutions of a difference equation”, J. Appl. Math. Informatics., 35: 217-230, (2017).
  • [27] Touafek, N., “On a second order rational difference equation”, Hacet. J. Math. Stat., 41: 867-874, (2012).
  • [28] Touafek, N. and Elsayed, EM., “On the periodicity of some systems of nonlinear difference equations”, Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie, 55(103): 217-224, (2012).
  • [29] Touafek, N. and Elsayed, EM., “On a second order rational systems of difference equations”, Hokkaido Math. J., 44: 29-45, (2015).
  • [30] Yalcinkaya, I. and Cinar, C., “Global asymptotic stability of a system of two nonlinear difference equations”, Fasc. Math., 43: 171-180, (2010).
  • [31] Yalcinkaya, I. and Tollu, DT., “Global behavior of a second order system of difference equations”, Adv. Stud. Contemp. Math., 26(4): 653-667, (2016).
  • [32] Yang, K. and Cushing, JM., “Global stability in a nonlinear difference-delay equation model of flour beetle population growth”, J. Difference Equ. Appl., 2(1): 31-37, (1996).
  • [33] Yazlik, Y., “On the solutions and behavior of rational difference equations”, J. Comput. Anal. Appl., 17(3): 584-594, (2014).
  • [34] Yazlik, Y., Elsayed, EM. and Taskara, N., “On the behaviour of the solutions the solutions of difference equation system”, J. Comput. Anal. Appl., 16(5): 932-941, (2014).
  • [35] Yazlik, Y., Tollu, DT. and Taskara, N., “On the behaviour of solutions for some systems of difference equations”, J. Comput. Anal. Appl., 18(1): 166-178, (2015).
  • [36] Yazlik, Y., Tollu, DT. and Taskara, N., "On the solutions of a three-dimensional system of difference equations", Kuwait J. Sci., 43(1): 95-111, (2015).
  • [37] Yazlik, Y. and Kara, M., “On a solvable system of difference equations of higher-order with period two coefficients”, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 68(2): 1675-1693, (2019).
  • [38] Yazlik, Y. and Kara, M., “Beşinci mertebeden fark denklem sisteminin çözülebilirliği üzerine”, Eskisehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B- Teorik Bilimler, 7(1): 29-45, (2019).
  • [39] Wang, X. and Li, Z., “Global asymptotic stability for two kinds of higher order recursive sequences”, J. Difference Equ. Appl., 22(10): 1542-1553, (2016).
Toplam 39 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Mathematics
Yazarlar

Merve Kara 0000-0001-8081-0254

Nouressedat Touafek Bu kişi benim 0000-0001-7079-6794

Yasin Yazlik 0000-0001-6369-540X

Yayımlanma Tarihi 1 Eylül 2020
Yayımlandığı Sayı Yıl 2020 Cilt: 33 Sayı: 3

Kaynak Göster

APA Kara, M., Touafek, N., & Yazlik, Y. (2020). Well-Defined Solutions of a Three-Dimensional System of Difference Equations. Gazi University Journal of Science, 33(3), 767-778. https://doi.org/10.35378/gujs.641441
AMA Kara M, Touafek N, Yazlik Y. Well-Defined Solutions of a Three-Dimensional System of Difference Equations. Gazi University Journal of Science. Eylül 2020;33(3):767-778. doi:10.35378/gujs.641441
Chicago Kara, Merve, Nouressedat Touafek, ve Yasin Yazlik. “Well-Defined Solutions of a Three-Dimensional System of Difference Equations”. Gazi University Journal of Science 33, sy. 3 (Eylül 2020): 767-78. https://doi.org/10.35378/gujs.641441.
EndNote Kara M, Touafek N, Yazlik Y (01 Eylül 2020) Well-Defined Solutions of a Three-Dimensional System of Difference Equations. Gazi University Journal of Science 33 3 767–778.
IEEE M. Kara, N. Touafek, ve Y. Yazlik, “Well-Defined Solutions of a Three-Dimensional System of Difference Equations”, Gazi University Journal of Science, c. 33, sy. 3, ss. 767–778, 2020, doi: 10.35378/gujs.641441.
ISNAD Kara, Merve vd. “Well-Defined Solutions of a Three-Dimensional System of Difference Equations”. Gazi University Journal of Science 33/3 (Eylül 2020), 767-778. https://doi.org/10.35378/gujs.641441.
JAMA Kara M, Touafek N, Yazlik Y. Well-Defined Solutions of a Three-Dimensional System of Difference Equations. Gazi University Journal of Science. 2020;33:767–778.
MLA Kara, Merve vd. “Well-Defined Solutions of a Three-Dimensional System of Difference Equations”. Gazi University Journal of Science, c. 33, sy. 3, 2020, ss. 767-78, doi:10.35378/gujs.641441.
Vancouver Kara M, Touafek N, Yazlik Y. Well-Defined Solutions of a Three-Dimensional System of Difference Equations. Gazi University Journal of Science. 2020;33(3):767-78.