Life Science Journal 2013;10(3) http://www.lifesciencesite.com http://www.lifesciencesite.com lifesciencej@gmail.com 361 Discussing the Existence of the Solutions and Their Dynamics of some Difference Equations H. El-Metwally 1,3 , R. Alsaedi 1 and E. M. Elsayed 2,3 1. Department of Mathematics, Rabigh College of Science and Art, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia. 2. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. 3. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E-mail: helmetwally2001@yahoo.com , ramzialsaedi@yahoo.co.uk , emmelsayed@yahoo.com Abstract: In this paper we care about the existence and study some qualitative properties of solutions to the following rational nonlinear difference equation 0,1,..., = , = 2) (3 1) (2 2) (3 1 n x x cx b x x k n k n k n k n n + − + − − + − + + where b and c are real numbers, k is a non-negative integer number and the initial conditions x -3k-2 , x -3k-1,…, x -1 , x 0 are arbitrary non- negative real numbers. Also, we derive the solutions of some special cases of the equation under consideration. [El-Metwally H, Alsaedi R, Elsayed, EM. Discussing the Existence of the Solutions and Their Dynamics of some Difference Equations. Life Sci J 2013;10(3):361-370] (ISSN:1097-8135). http://www.lifesciencesite.com . 55 Keywords: recursive sequence, stability, boundedness, periodicity, solutions of difference equations. Mathematics Subject Classification: 39A10. 1. Introduction Our aim in this paper is to investigate the dynamics of the solutions to the following difference equation 0,1,..., = , = 2) (3 1) (2 2) (3 1 n x x cx b x x k n k n k n k n n + − + − − + − + + (1) where b and c are real numbers, k is a non negative integer number and the initial conditions x -3k-2 , x -3k- 1,…, x -1 , x 0 are arbitrary non-negative real numbers. Also, we obtain the solutions of some special cases of Eq.(1). The study of Difference Equations has been growing continuously for the last decades. This is largely due to the fact that difference equations manifest themselves as mathematical models describing real life situations in probability theory, queuing theory, statistical problems, stochastic time series, combinatorial analysis, number theory, geometry, electrical network, quanta in radiation, genetics in biology, economics, psychology, sociology, etc. In fact, now it occupies a central position in applicable analysis and will no doubt continue to play an important role in mathematics as a whole. Recently there has been a lot of interest in studying the global attractivity, boundedness character, periodicity and the solution form of nonlinear rational difference equations. The study of this kind of equations is quite challenging and rewarding and is still in its infancy. The nonlinear rational difference equations are of paramount importance in their own right, and furthermore we believe that these results about such equations over prototypes for the development of the basic theory of the global behavior of nonlinear rational difference equations. Some results on rational difference equations and systems of difference equations can be found in refs. [1-23]. Let I be some intervals of real numbers and let , : 1 I I f k → + be a continuously differentiable function. Then for every set of initial conditions , ,..., , 0 1 I x x x k k ∈ + − − the difference equation 0,1,..., = ), ,..., , ( = 1 1 n x x x f x k n n n n − − + (2) has a unique solution ∞ −k n n x = } { . Definition 1. (Periodicity) A sequence ∞ −k n n x = } { is said to be periodic with period p if n p n x x = + for all . k n − ≥ The linearized equation of Eq.(2) about the equilibrium x is the linear difference equation . ) ,..., , ( = 0 = 1 i n i n k i n y x x x x f y − − + ∂ ∂ ∑ (3) Theorem A [15]: Assume that k i R p i 1,2,..., = , ∈ and } {0,1,2,... ∈ k . Then 1, < 1 = i k i p ∑ is a sufficient condition for the asymptotic stability of the difference equation