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On the global stability of some k-order difference equations

Year 2018, Volume: 1 Issue: 1, 13 - 18, 15.03.2018

Abstract

We use two different techniques, one of them including fixed point tools, i.e., the Prešić type fixed point theorem, in order to study the asymptotic stability of some k-order difference equations for k = 1 and k = 2. In this way, we can study the global stability for more general initial value problems associated with particular forms of difference equations.

References

  • References
  • [1] Abu-Saris, R. M., DeVault, R., Global stability of yn+1 = A + yn yn−k . Appl. Math. Lett. 16 (2003), no. 2, 173–178.
  • [2] Aloqeili, M., On the difference equation xn+1 = α + xp n xp n−1 . J. Appl. Math. Comput. 25 (2007), no. 1-2, 375–382.
  • [3] Amleh, A. M., Grove, E. A., Ladas, G., Georgiou, D. A., On the recursive sequence xn+1 = α + xn−1/xn. J. Math. Anal. Appl. 233 (1999), no. 2, 790–798.
  • [4] Berinde, V., Exploring, Investigating and Discovering in Mathematics, Birkhäuser, Basel, 2004.
  • [5] Berinde, V., Iterative approximation of fixed points, Second edition. Lecture Notes in Mathematics, 1912. Springer, Berlin, 2007.
  • [6] Berinde, V., Păcurar, M., Stability of k-step fixed point iterative methods for some Prešić type contractive mappings. J. Inequal. Appl. 2014, 2014:149, 12 pp.
  • [7] Berinde, V., Păcurar, M., Two elementary applications of some Prešić type fixed point theorems. Creat. Math. Inform. 20 (2011), no. 1, 32–42
  • [8] Berinde, V., Păcurar, M., O metodă de tip punct fix pentru rezolvarea sistemelor ciclice, Gazeta Matematică, Seria B, 116 (2011), No. 3, 113-123
  • [9] Camouzis, E., DeVault, R., Ladas, G., On the recursive sequence xn+1 = −1 + (xn−1/xn). J. Differ. Equations Appl. 7 (2001), no. 3, 477–482.
  • [10] Cirić, L.B., Prešić, S.B., On Prešić type generalization of the Banach contraction mapping principle, Acta Math. Univ. Comenianae, 76 (2007), No. 2, 143-147
  • [11] Chen, Y.-Z., A Prešić type contractive condition and its applications, Nonlinear Anal. 71 (2009), no. 12, e2012–e2017
  • [12] DeVault, R., Ladas, G., Schultz, S. W., On the recursive sequence xn+1 = A/xn + 1/xn−2. Proc. Amer. Math. Soc. 126 (1998), no. 11, 3257–3261.
  • [13] Elabbasy, E. M., El-Metwally, H., Elsayed, E. M., On the difference equation xn+1 = axn − bxn/(cxn − dxn−1). Adv. Difference Equ. 2006, Art. ID 82579, 10 pp.
  • [14] El-Owaidy, H. M., Ahmed, A. M., Mousa, M. S., On asymptotic behaviour of the difference equation xn+1 = α+ xn−1p xnp . J. Appl. Math. Comput. 12 (2003), no. 1-2, 31–37.
  • [15] El-Owaidy, H. M., Ahmed, A. M., Mousa, M. S., On asymptotic behaviour of the difference equation xn+1 = α + xn−k xn . Appl. Math. Comput. 147 (2004), no. 1, 163–167.
  • [16] Kocić, V. L., Ladas, G., Global behavior of nonlinear difference equations of higher order with applications. Mathematics and its Applications, 256. Kluwer Academic Publishers Group, Dordrecht, 1993.
  • [17] Păcurar, M., Approximating common fixed points of Prešić-Kannan type operators by a multi-step iterative method, An. Ştiinţ,. Univ. "Ovidius" Constanţa Ser. Mat. 17 (2009), no. 1, 153–168
  • [18] Păcurar, M., Iterative Methods for Fixed Point Approximation, Risoprint, Cluj-Napoca, 2010
  • [19] Păcurar, M., A multi-step iterative method for approximating fixed points of Prešić-Kannan operators, Acta Math. Univ. Comen. New Ser., 79 (2010), No. 1, 77-88
  • [20] Păcurar, M., A multi-step iterative method for approximating common fixed points of Prešić-Rus type operators on metric spaces, Stud. Univ. Babeş-Bolyai Math. 55 (2010), no. 1, 149–162.
  • [21] Păcurar, M., Fixed points of almost Prešić operators by a k-step iterative method, An. Ştiint,. Univ. Al. I. Cuza Iaşi, Ser. Noua, Mat. 57 (2011), Supliment 199–210
  • [22] Păvăloiu, I., Rezolvarea ecuaţiilor prin interpolare, Editura Dacia, Cluj-Napoca, 1981
  • [23] Păvăloiu, I. and Pop, N., Interpolare şi aplicaţii, Risoprint, Cluj-Napoca, 2005
  • [24] Prešić, S.B., Sur une classe d’ inéquations aux différences finites et sur la convergence de certaines suites, Publ. Inst. Math. (Beograd)(N.S.), 5(19) (1965), 75–78
  • [25] Rus, I.A., An iterative method for the solution of the equation x = f(x,...,x), Rev. Anal. Numer. Théor. Approx., 10 (1981), No.1, 95–100
  • [26] Saleh, M.; Aloqeili, M., On the rational difference equation yn+1 = A + yn−k yn . Appl. Math. Comput. 171 (2005), no. 2, 862–869.
  • [27] Saleh, M., Aloqeili, M., On the rational difference equation yn+1 = A + yn yn−k . Appl. Math. Comput. 177 (2006), no. 1, 189–193.
  • [28] Stević, S., On the recursive sequence xn+1 = α + xp n−1 xp n . J. Appl. Math. Comput. 18 (2005), no. 1-2, 229–234.
  • [29] Stević, S., On the recursive sequence xn+1 = A + xp n/xp n−1. Discrete Dyn. Nat. Soc. 2007, Art. ID 34517, 9 pp.
  • [30] Stević, S., On the recursive sequence xn+1 = A + xp n/xr n−1. Discrete Dyn. Nat. Soc. 2007, Art. ID 40963, 9 pp.
  • [31] Stević, S., On the recursive sequence xn+1 = maxc, xp n xp n−1. Appl. Math. Lett. 21 (2008), no. 8, 791–796.
Year 2018, Volume: 1 Issue: 1, 13 - 18, 15.03.2018

Abstract

References

  • References
  • [1] Abu-Saris, R. M., DeVault, R., Global stability of yn+1 = A + yn yn−k . Appl. Math. Lett. 16 (2003), no. 2, 173–178.
  • [2] Aloqeili, M., On the difference equation xn+1 = α + xp n xp n−1 . J. Appl. Math. Comput. 25 (2007), no. 1-2, 375–382.
  • [3] Amleh, A. M., Grove, E. A., Ladas, G., Georgiou, D. A., On the recursive sequence xn+1 = α + xn−1/xn. J. Math. Anal. Appl. 233 (1999), no. 2, 790–798.
  • [4] Berinde, V., Exploring, Investigating and Discovering in Mathematics, Birkhäuser, Basel, 2004.
  • [5] Berinde, V., Iterative approximation of fixed points, Second edition. Lecture Notes in Mathematics, 1912. Springer, Berlin, 2007.
  • [6] Berinde, V., Păcurar, M., Stability of k-step fixed point iterative methods for some Prešić type contractive mappings. J. Inequal. Appl. 2014, 2014:149, 12 pp.
  • [7] Berinde, V., Păcurar, M., Two elementary applications of some Prešić type fixed point theorems. Creat. Math. Inform. 20 (2011), no. 1, 32–42
  • [8] Berinde, V., Păcurar, M., O metodă de tip punct fix pentru rezolvarea sistemelor ciclice, Gazeta Matematică, Seria B, 116 (2011), No. 3, 113-123
  • [9] Camouzis, E., DeVault, R., Ladas, G., On the recursive sequence xn+1 = −1 + (xn−1/xn). J. Differ. Equations Appl. 7 (2001), no. 3, 477–482.
  • [10] Cirić, L.B., Prešić, S.B., On Prešić type generalization of the Banach contraction mapping principle, Acta Math. Univ. Comenianae, 76 (2007), No. 2, 143-147
  • [11] Chen, Y.-Z., A Prešić type contractive condition and its applications, Nonlinear Anal. 71 (2009), no. 12, e2012–e2017
  • [12] DeVault, R., Ladas, G., Schultz, S. W., On the recursive sequence xn+1 = A/xn + 1/xn−2. Proc. Amer. Math. Soc. 126 (1998), no. 11, 3257–3261.
  • [13] Elabbasy, E. M., El-Metwally, H., Elsayed, E. M., On the difference equation xn+1 = axn − bxn/(cxn − dxn−1). Adv. Difference Equ. 2006, Art. ID 82579, 10 pp.
  • [14] El-Owaidy, H. M., Ahmed, A. M., Mousa, M. S., On asymptotic behaviour of the difference equation xn+1 = α+ xn−1p xnp . J. Appl. Math. Comput. 12 (2003), no. 1-2, 31–37.
  • [15] El-Owaidy, H. M., Ahmed, A. M., Mousa, M. S., On asymptotic behaviour of the difference equation xn+1 = α + xn−k xn . Appl. Math. Comput. 147 (2004), no. 1, 163–167.
  • [16] Kocić, V. L., Ladas, G., Global behavior of nonlinear difference equations of higher order with applications. Mathematics and its Applications, 256. Kluwer Academic Publishers Group, Dordrecht, 1993.
  • [17] Păcurar, M., Approximating common fixed points of Prešić-Kannan type operators by a multi-step iterative method, An. Ştiinţ,. Univ. "Ovidius" Constanţa Ser. Mat. 17 (2009), no. 1, 153–168
  • [18] Păcurar, M., Iterative Methods for Fixed Point Approximation, Risoprint, Cluj-Napoca, 2010
  • [19] Păcurar, M., A multi-step iterative method for approximating fixed points of Prešić-Kannan operators, Acta Math. Univ. Comen. New Ser., 79 (2010), No. 1, 77-88
  • [20] Păcurar, M., A multi-step iterative method for approximating common fixed points of Prešić-Rus type operators on metric spaces, Stud. Univ. Babeş-Bolyai Math. 55 (2010), no. 1, 149–162.
  • [21] Păcurar, M., Fixed points of almost Prešić operators by a k-step iterative method, An. Ştiint,. Univ. Al. I. Cuza Iaşi, Ser. Noua, Mat. 57 (2011), Supliment 199–210
  • [22] Păvăloiu, I., Rezolvarea ecuaţiilor prin interpolare, Editura Dacia, Cluj-Napoca, 1981
  • [23] Păvăloiu, I. and Pop, N., Interpolare şi aplicaţii, Risoprint, Cluj-Napoca, 2005
  • [24] Prešić, S.B., Sur une classe d’ inéquations aux différences finites et sur la convergence de certaines suites, Publ. Inst. Math. (Beograd)(N.S.), 5(19) (1965), 75–78
  • [25] Rus, I.A., An iterative method for the solution of the equation x = f(x,...,x), Rev. Anal. Numer. Théor. Approx., 10 (1981), No.1, 95–100
  • [26] Saleh, M.; Aloqeili, M., On the rational difference equation yn+1 = A + yn−k yn . Appl. Math. Comput. 171 (2005), no. 2, 862–869.
  • [27] Saleh, M., Aloqeili, M., On the rational difference equation yn+1 = A + yn yn−k . Appl. Math. Comput. 177 (2006), no. 1, 189–193.
  • [28] Stević, S., On the recursive sequence xn+1 = α + xp n−1 xp n . J. Appl. Math. Comput. 18 (2005), no. 1-2, 229–234.
  • [29] Stević, S., On the recursive sequence xn+1 = A + xp n/xp n−1. Discrete Dyn. Nat. Soc. 2007, Art. ID 34517, 9 pp.
  • [30] Stević, S., On the recursive sequence xn+1 = A + xp n/xr n−1. Discrete Dyn. Nat. Soc. 2007, Art. ID 40963, 9 pp.
  • [31] Stević, S., On the recursive sequence xn+1 = maxc, xp n xp n−1. Appl. Math. Lett. 21 (2008), no. 8, 791–796.
There are 32 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Vasile Berinde

Hafiz Fukhar-ud-din

Mădălina Păcurar This is me

Publication Date March 15, 2018
Published in Issue Year 2018 Volume: 1 Issue: 1

Cite

APA Berinde, V., Fukhar-ud-din, H., & Păcurar, M. (2018). On the global stability of some k-order difference equations. Results in Nonlinear Analysis, 1(1), 13-18.
AMA Berinde V, Fukhar-ud-din H, Păcurar M. On the global stability of some k-order difference equations. RNA. April 2018;1(1):13-18.
Chicago Berinde, Vasile, Hafiz Fukhar-ud-din, and Mădălina Păcurar. “On the Global Stability of Some K-Order difference Equations”. Results in Nonlinear Analysis 1, no. 1 (April 2018): 13-18.
EndNote Berinde V, Fukhar-ud-din H, Păcurar M (April 1, 2018) On the global stability of some k-order difference equations. Results in Nonlinear Analysis 1 1 13–18.
IEEE V. Berinde, H. Fukhar-ud-din, and M. Păcurar, “On the global stability of some k-order difference equations”, RNA, vol. 1, no. 1, pp. 13–18, 2018.
ISNAD Berinde, Vasile et al. “On the Global Stability of Some K-Order difference Equations”. Results in Nonlinear Analysis 1/1 (April 2018), 13-18.
JAMA Berinde V, Fukhar-ud-din H, Păcurar M. On the global stability of some k-order difference equations. RNA. 2018;1:13–18.
MLA Berinde, Vasile et al. “On the Global Stability of Some K-Order difference Equations”. Results in Nonlinear Analysis, vol. 1, no. 1, 2018, pp. 13-18.
Vancouver Berinde V, Fukhar-ud-din H, Păcurar M. On the global stability of some k-order difference equations. RNA. 2018;1(1):13-8.