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## On the global stability of some k-order diﬀerence equations

#### Vasile BERİNDE [1] , Hafiz FUKHAR-UD-DİN [2] , Mădălina PĂCURAR [3]

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.

Diﬀerence equation;, Equilibrium point, Global attractor, Global asymptotic stability, Presic ﬁxed point theorem
• 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 = maxc, xp n xp n−1. Appl. Math. Lett. 21 (2008), no. 8, 791–796.
Primary Language en Mathematics Articles Author: Vasile BERİNDE (Primary Author)Institution: Technical University of Cluj-Napoca, Department of Mathematics and Computer, Baia MareCountry: Romania Author: Hafiz FUKHAR-UD-DİN Institution: King Fahd University of Petroleum And Minerals, Department of Mathematics and Statistics, DhahraCountry: Saudi Arabia Author: Mădălina PĂCURAR Institution: Babeş-Bolyai University, Department of Statistics, Forecast and Mathematics, Faculty of Economics and Business AdministrationCountry: Romania Publication Date : March 15, 2018
 Bibtex @research article { rna415423, journal = {Results in Nonlinear Analysis}, issn = {}, eissn = {2636-7556}, address = {}, publisher = {Erdal KARAPINAR}, year = {2018}, volume = {1}, pages = {13 - 18}, doi = {}, title = {On the global stability of some k-order diﬀerence equations}, key = {cite}, author = {Beri̇nde, Vasile and Fukhar-ud-di̇n, Hafiz and Păcurar, Mădălina} } APA Beri̇nde, V , Fukhar-ud-di̇n, H , Păcurar, M . (2018). On the global stability of some k-order diﬀerence equations . Results in Nonlinear Analysis , 1 (1) , 13-18 . Retrieved from https://dergipark.org.tr/en/pub/rna/issue/36561/415423 MLA Beri̇nde, V , Fukhar-ud-di̇n, H , Păcurar, M . "On the global stability of some k-order diﬀerence equations" . Results in Nonlinear Analysis 1 (2018 ): 13-18 Chicago Beri̇nde, V , Fukhar-ud-di̇n, H , Păcurar, M . "On the global stability of some k-order diﬀerence equations". Results in Nonlinear Analysis 1 (2018 ): 13-18 RIS TY - JOUR T1 - On the global stability of some k-order diﬀerence equations AU - Vasile Beri̇nde , Hafiz Fukhar-ud-di̇n , Mădălina Păcurar Y1 - 2018 PY - 2018 N1 - DO - T2 - Results in Nonlinear Analysis JF - Journal JO - JOR SP - 13 EP - 18 VL - 1 IS - 1 SN - -2636-7556 M3 - UR - Y2 - 2018 ER - EndNote %0 Results in Nonlinear Analysis On the global stability of some k-order diﬀerence equations %A Vasile Beri̇nde , Hafiz Fukhar-ud-di̇n , Mădălina Păcurar %T On the global stability of some k-order diﬀerence equations %D 2018 %J Results in Nonlinear Analysis %P -2636-7556 %V 1 %N 1 %R %U ISNAD Beri̇nde, Vasile , Fukhar-ud-di̇n, Hafiz , Păcurar, Mădălina . "On the global stability of some k-order diﬀerence equations". Results in Nonlinear Analysis 1 / 1 (March 2018): 13-18 . AMA Beri̇nde V , Fukhar-ud-di̇n H , Păcurar M . On the global stability of some k-order diﬀerence equations. RNA. 2018; 1(1): 13-18. Vancouver Beri̇nde V , Fukhar-ud-di̇n H , Păcurar M . On the global stability of some k-order diﬀerence equations. Results in Nonlinear Analysis. 2018; 1(1): 13-18.

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