Hindawi Publishing Corporation Advances in Difference Equations Volume 2008, Article ID 143943, 9 pages doi:10.1155/2008/143943 Research Article On the Solutions of Systems of Difference Equations ˙ Ibrahim Yalc¸inkaya, Cengiz C¸ inar, and Muhammet Atalay Mathematics Department, Faculty of Education, Selcuk University, 42099, Konya, Turkey Correspondence should be addressed to ˙ Ibrahim Yalc¸inkaya, iyalcinkaya1708@yahoo.com Received 19 March 2008; Accepted 19 May 2008 Recommended by Bing Zhang We show that every solution of the following system of difference equations x 1 n1 x 2 n /x 2 n - 1, x 2 n1 x 3 n /x 3 n - 1,...,x k n1 x 1 n /x 1 n - 1 as well as of the system x 1 n1 x k n /x k n - 1, x 2 n1 x 1 n /x 1 n - 1,...,x k n1 x k-1 n /x k-1 n - 1 is periodic with period 2k if k/ 0 mod2, and with period k if k 0 mod 2 where the initial values are nonzero real numbers for x 1 0 ,x 2 0 ,...,x k 0 / 1. Copyright q 2008 ˙ Ibrahim Yalc¸inkaya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economy, physics, and so on 1. So, recently there has been an increasing interest in the study of qualitative analysis of rational difference equations and systems of difference equations. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solutions. see 1–12 and the references cited therein. Papaschinopoulos and Schinas 9, 10 studied the behavior of the positive solutions of the system of two Lyness difference equations x n1 by n c x n-1 , y n1 dx n e y n-1 , n 0, 1, 2,..., 1.1 where b, c, d, e are positive constants and the initial values x -1 ,x 0 ,y -1 ,y 0 are positive. In 2 Camouzis and Papaschinopoulos studied the behavior of the positive solutions of the system of two difference equations x n1 1 x n y n-m , y n1 1 y n x n-m , n 0, 1, 2,..., 1.2