Solution for systems of difference equations of rational form of order two
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Solution for systems of difference equations of rational form of order two E. M. Elsayed
Received: 24 January 2013 / Accepted: 28 September 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract In this paper, we deal with the existence of solutions and the periodicity character of the following systems of rational difference equations with order two xn yn−1 yn xn−1 xn+1 = , yn+1 = , yn (±1 + xn yn−1 ) xn (±1 ± yn xn−1 ) with initial conditions x−1 , x0 , y−1 and y0 are nonzero real numbers. Keywords Difference equations · Recursive sequences · Stability · Periodic solution · System of difference equations Mathematics Subject Classification (2000)
39A10
1 Introduction This paper is devoted to study the form of the solution and the periodicity of the following systems of a rational difference equations of order two xn yn−1 yn xn−1 xn+1 = , yn+1 = , yn (±1 + xn yn−1 ) xn (±1 ± yn xn−1 ) with initial conditions x−1 , x0 , y−1 and y0 are nonzero real numbers.
Communicated by Jinyun Yuan. E. M. Elsayed Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia e-mail: [email protected] E. M. Elsayed (B) Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt e-mail: [email protected]
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E. M. Elsayed
Difference equations and systems of difference equations of rational form can be found in the following papers: Agarwal and Elsayed (2010), Battaloglu et al. (2010), Clark and Kulenovic (2002), Elabbasy et al. (2011), El-Metwally and Elsayed (2012), El-Metwally (2007), Elsayed (2010, 2011a,b, 2013), Elsayed and El-Metwally (2013), Elsayed (2012), Elsayed et al. (2012), Erdo˘gan et al. (2011), Grove et al. (2001), Grove and Ladas (2005). Moreover, as difference equations have many applications in applied sciences, there are many papers and books that can be found concerning the theory and applications of difference equations (for partial review of the theory of difference equations, systems of difference equations and their applications see Agarwal 2000; El-Metwally 2007; Kurbanli et al. 2011; Kurbanli 2011; Özban 2006 and the references cited therein). Recently, there has been great interest in studying difference equation systems; for example, the global asymptotic behavior of the positive solutions of the rational difference system xn yn x n+1 = 1 + , yn+1 = 1 + , yn−m xn−m has been studied by Camouzis and Papaschinopoulos (2004). The periodicity of the positive solutions of the rational difference equations systems xn+1 =
1 , yn
yn+1 =
yn , xn−1 yn−1
has been obtained by Cinar (2004). Elsayed (2012) has got the solutions of the following systems of the difference equations xn−1 yn−1 xn+1 = , yn+1 = . ±1 + xn−1 yn ∓1 + yn−1 xn Grove et al. (2001) has studied existence and behavior of solutions of the rational system xn+1 =
a b + , xn yn
yn+1 =
c d + . xn yn
The behavior of positive solutions of the following system xn−1 yn−1 xn+1 = , yn+1 = . 1 + xn−1 yn 1 + yn−1 xn has been
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