On the global stability of some k-order difference equations

Yıl 2018, Cilt: 1 Sayı: 1, 13 - 18, 15.03.2018

Vasile Berinde , Hafiz Fukhar-ud-din , Mădălina Păcurar

Öz

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.

Anahtar Kelimeler

Difference equation;, Equilibrium point, Global attractor, Global asymptotic stability, Presic fixed point theorem

Kaynakça

• 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.