ах, — bx,, Xp+1 n= 1, 2,.., (1.1) where a,b,c>0 and m21.. Now, the difference equations (as well as differential equations and delay differential equations) model various diverse Xn-k Xn+1 = (1) (2n-k)' + a where the initial values x_k+1 = a-k+! ,l = 0,1, 2, ...k. are nonzero real numbers and with (x-k+1)9-1 # -a for l = 0,1,2, ..k. . Moreover , we have studied the stability and periodicity of solutions for the generalized nonlinear rational difference equations in (1) and for some special cases . Xn+1 (2) (xn)? + a where (xo) + -a In order to do this we introduce the following notations: Let xo = a Proposition [22] Assume that p, q € R. Then \p| + lgl < 1 is a sufficient condition for the asymptotic stability of the difference equation In+1 - prn - qan-1 = 0, n= 0,1,.... Lemma 1. The equilibrium points of the difference equation (2) are 0 and t/1-a Proof. (7) +a (2)3 + az = T (2) + (a – 1)7 = 0 퍼((1)2 + (a - 1)} =D0 This means that the equilibrium points are 0 and ±V1- a. Remark 1. When a = 1, then the only equilibrium point of the difference equation (2) is 0. Theorem 7. 1) The equilibrium point E = 0 is locally asymptotically stable if lal >1. 2)The equilibrium points E=±V1– aare locally asymptotically stable if 0 < a <1. 3) All solutions of equation (2) at the equilibrium points 0 and ±T– aare unstable if a so -1< Proof. let f: (0, 00) → (0, oo) be a continuous function defined by f(u) = u? + a It is easy see that df (u) (u? + a)? -u? +a %3D du At the equilibrium point 7 = 0, we have df (u) \z=0 == =p du 1 The corresponding linearized equation about 7 = 0 is given by Yn+1 - Pyn = 0 This implies that the characteristic equation is Hence the equilibrium point 7= 0 is locally asymptotically stable if la| >1. Now we will prove the theorem at the equilibrium point z = +y1-a and the proof at the equilibrium point 7= -V1- a by the same way. At the equilibrium point i = V1-a we have df (u), du E-V-a = 2a – 1 = g The corresponding linearized equation about 7 = VT- a is given by Yn+1 - qYn = 0 This implies that the characteristic equation is А — (2а — 1) 3 0 Hence the equilibrium point 7 = v1- a is locally asymptotically stable if |2a – 1| <1. This means that 0< a < 1.