Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 RESEARCH Open Access Global behavior of the solutions of some difference equations Elmetwally M Elabbasy1*, Hamdy A El-Metwally2,4 and Elsayed M Elsayed3,4 * Correspondence: emelabbasy@mans.edu.eg Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Full list of author information is available at the end of the article Abstract In this article we study the difference equation xn+1 = axn−l xn−k , n = 0, 1, , bxn−p − cxn−q where the initial conditions x-r, x-r+1, x-r+2, , x0 are arbitrary positive real numbers, r = max{l, k, p, q} is nonnegative integer and a, b, c are positive constants: Also, we study some special cases of this equation Keywords: Stability, Solutions of the difference equations Introduction The purpose of this article is to investigate the global attractivity of the equilibrium point, and the asymptotic behavior of the solutions of the following difference equation xn+1 = axn−l xn−k , n = 0, 1, , bxn−p − cxn−q (1) where the initial conditions x-r, x-r+1, x-r+2, , x0 are arbitrary positive real numbers, r = max{l, k, p, q} is nonnegative integer and a, b, c are positive constants: Moreover, we obtain the form of the solution of some special cases of Equation and some numerical simulations to the equation are given to illustrate our results Let us introduce some basic definitions and some theorems that we need in the sequel Let I be some interval of real numbers and let f : Ik+1 → I, be a continuously differentiable function Then for every set of initial conditions x-k, x-k+1, , x0 Î I, the difference equation xn+1 = f (xn , xn−1 , , xn−k ), n = 0, 1, , (2) has a unique solution {xn }∞ [1] n=−k A point x ∈ I is called an equilibrium point of Equation if ¯ ¯ ¯ x = f (¯ , x, , x) x ¯ © 2011 Elabbasy et al; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 Page of 16 ¯ That is, xn = x for n ≥ 0, is a solution of Equation 2, or equivalently, x is a fixed point ¯ of f Definition (Stability) (i) The equilibrium point x of Equation is locally stable if for every ε >0, there exists ¯ δ >0 such that for all x-k, x-k+1, , x-1, x0 Ỵ I with ¯ ¯ ¯ |x−k − x| + |x−k+1 − x| + · · · + |x0 − x| < δ, we have ¯ |xn − x| < ε for all n ≥ −k (ii) The equilibrium point x of Equation is locally asymptotically stable if x is ¯ ¯ locally stable solution of Equation and there exists g >0, such that for all x-k, x-k+1, , x-1, x0 Ỵ I with ¯ ¯ ¯ |x−k − x| + |x−k+1 − x| + + |x0 − x| < γ , we have ¯ lim xn = x n→∞ (iii) The equilibrium point x of Equation is global attractor if for all x-k, x-k+1, , x-1, ¯ x0 Ỵ I, we have ¯ lim xn = x n→∞ (iv) The equilibrium point x of Equation is globally asymptotically stable if x is ¯ ¯ locally stable and x is also a global attractor of Equation ¯ (v) The equilibrium point x of Equation is unstable if x is not locally stable ¯ ¯ The linearized equation of Equation about the equilibrium x is the linear difference ¯ equation yn+1 = k i=0 ¯ ∂f (¯ , x, , x) x ¯ yn−i ∂xn−i (3) Theorem A [2] Assume that p, q Ỵ R and k Ỵ {0, 1, 2, } Then |p| + |q| < 1, is a sufficient condition for the asymptotic stability of the difference equation xn+1 + pxn + qxn−k = 0, n = 0, 1, Remark Theorem A can be easily extended to a general linear equations of the form xn+k + p1 xn+k−1 + + pk xn = 0, n = 0, 1, , (4) where p1, p2, , pk Î R and k Î {1, 2, } Then Equation is asymptotically stable provided that k |pi | < i=1 Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 Definition (Fibonacci Sequence) The sequence {Fm }∞ = {1, 2, 3, 5, 8, 13, }i.e Fm = Fm-1 + Fm-2, m=0 m ≥ 0, F-2 = 0, F-1 = is called Fibonacci Sequence The nature of many biological systems naturally leads to their study by means of a discrete variable Particular examples include population dynamics and genetics Some elementary models of biological phenomena, including a single species population model, harvesting of fish, the production of red blood cells, ventilation volume and blood CO2 levels, a simple epidemics model and a model of waves of disease that can be analyzed by difference equations are shown in [3] Recently, there has been interest in so-called dynamical diseases, which correspond to physiological disorders for which a generally stable control system becomes unstable One of the first papers on this subject was that of Mackey and Glass [4] In that paper they investigated a simple first order difference-delay equation that models the concentration of blood-level CO They also discussed models of a second class of diseases associated with the production of red cells, white cells, and platelets in the bone marrow The study of the nonlinear rational difference equations of a higher order is quite challenging and rewarding, and the results about these equations offer prototypes towards the development of the basic theory of the global behavior of nonlinear difference equations of a big order, recently, many researchers have investigated the behavior of the solution of difference equations for example: Elabbasy et al [5] investigated the global stability, periodicity character and gave the solution of special case of the following recursive sequence xn+1 = axn − bxn cxn − dxn−1 Elabbasy et al [6] investigated the global stability, boundedness, periodicity character and gave the solution of some special cases of the difference equation xn+1 = αxn−k β+γ k i=0 xn−i Elabbasy et al [7] investigated the global stability character, boundedness and the periodicity of solutions of the difference equation xn+1 = αxn + βxn−1 + γ xn−2 Axn + Bxn−1 + Cxn−2 El-Metwally et al [8] investigated the asymptotic behavior of the population model: xn+1 = α + βxn−1 e−xn , where a is the immigration rate and b is the population growth rate Yang et al [9] investigated the invariant intervals, the global attractivity of equilibrium points and the asymptotic behavior of the solutions of the recursive sequence xn+1 = axn−1 + bxn−2 c + dxn−1 xn−2 Page of 16 Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 Page of 16 Cinar [10,11] has got the solutions of the following difference equations xn−1 xn−1 , xn+1 = + axn xn−1 −1 + axn xn−1 xn+1 = Aloqeili [12] obtained the form of the solutions of the difference equation xn−1 a − xn xn−1 xn+1 = Yalỗinkaya [13] studied the following nonlinear difference equation xn+1 = α + xn−m xk n For some related work see [1-29] The article proceeds as follows In Sect we show that when 2a |b - c| + a(b + c) < (b - c)2, then the equilibrium point of Equation is locally asymptotically stable In Sect we prove that the equilibrium point of Equation is global attractor In Sect we give the solutions of some special cases of Equation and give a numerical examples of each case and draw it by using Matlab 6.5 Local stability of Equation In this section we investigate the local stability character of the solutions of Equation Equation has a unique positive equilibrium point and is given by x= ax2 , bx − cx if a ≠ b-c, b ≠ c, then the unique equilibrium point is x = ¯ Let f : (0, ∞)4 ® (0, ∞) be a function defined by f (u, v, w, s) = auv bw − cs (5) Therefore, it follows that fu (u, v, w, s) = fw (u, v, w, s) = av , (bw − cs) −bauv (bw − cs) fv (u, v, w, s) = , au , (bw − cs) fs (u, v, w, s) = cauv (bw − cs)2 , we see that fu (¯ , x, x, x) = x ¯ ¯ ¯ fw (¯ , x, x, x) = x ¯ ¯ ¯ a , (b − c) −ab (b − c) fv (¯ , x, x, x) = x ¯ ¯ ¯ , a , (b − c) fs (¯ , x, x, x) = x ¯ ¯ ¯ ac (b − c)2 The linearized equation of Equation about x is ¯ yn+1 + a a ab ac y + yn−q = yn−1 + yn−k − n−p (b − c) (b − c) (b − c) (b − c)2 (6) Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 Theorem Assume that a(3ζ − η) < (b − c)2 , where ζ = max{b, c}, h = min{b, c} Then the equilibrium point of Equation is locally asymptotically stable Proof: It is follows by Theorem A that Equation is asymptotically stable if a ab ab a + + + < 1, (b − c) (b − c) (b − c) (b − c)2 or a(b + c) 2a + < 1, (b − c) (b − c)2 and so 2a|b − c| + a(b + c) < (b − c)2 The proof is complete Global attractivity of the equilibrium point of Equation In this section we investigate the global attractivity character of solutions of Equation We give the following two theorems which is a minor modification of Theorem A.0.2 in [1] Theorem Let [a, b] be an interval of real numbers and assume that f : [a, b]k+1 → [a, b], is a continuous function satisfying the following properties: (i) f(x1, x2, , xk+1) is non-increasing in one component (for example xt) for each xr (r ≠ t) in [a, b] and non-decreasing in the remaining components for each xt in [a, b] (ii) If (m, M) ∈ [a, b] × [a, b] is a solution of the system M = f(M, M, ,M, m, M, ,M, M) and m = f(m, m, ,m, M, m, m, m) implies m = M ¯ Then Equation has a unique equilibrium x ∈ [a, b] and every solution of Equation converges to x ¯ Proof: Set m0 = a and M0 = b, and for each i = 1, 2, set mi = f (mi−1 , mi−1 , , mi−1 , Mi−1 , mi−1 , , mi−1 , mi−1 ), and Mi = f (Mi−1 , Mi−1 , , Mi−1 , mi−1 , Mi−1 , , Mi−1 , Mi−1 ) Page of 16 Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 Now observe that for each i ≥ 0, a = m0 ≤ m1 ≤ ≤ mi ≤ ≤ Mi ≤ ≤ M1 ≤ M0 = b, and mi ≤ xp ≤ Mi for p ≥ (k + 1)i + Set m = lim mi and M = lim Mi x→∞ i→∞ Then M ≥ lim sup xi ≥ lim inf xi ≥ m i→∞ i→∞ and by the continuity of f, M = f(M, M, ,M, m, M, ,M, M) and m = f(m, m, ,m, M, m, m, m) In view of (ii), ¯ m = M = x, from which the result follows Theorem Let [a, b] be an interval of real numbers and assume that f : [a, b]k+1 → [a, b], is a continuous function satisfying the following properties: (i) f(x1, x2, ,xk+1) is non-increasing in one component (for example xt) for each xr (r ≠ t) in [a, b] and non-increasing in the remaining components for each xt in [a, b] (ii) If (m, M)ẻ[a, b] ì [a, b] is a solution of the system M = f(m, m, ,m, M, m, m, m) and m = f(M, M, ,M, m, M, ,M, M)’ implies m = M ¯ Then Equation has a unique equilibrium x ∈ [a, b] and every solution of Equation converges to x ¯ Proof: As the proof of Theorem and will be omitted Theorem The equilibrium point x of Equation is global attractor if c ≠ a ¯ Proof: Let p, q are a real numbers and assume that f : [p, q]4 → [p, q] be a function defined by Equation 5, then we can easily see that the function f(u, v, w, s) increasing in s and decreasing in w Case (1) If bw-cs > 0, then we can easily see that the function f(u, v, w, s) increasing in u, v, s and decreasing in w Suppose that (m, M) is a solution of the system M = f(m, m, M, m) and M = f(M, M, m, M) Then from Equation 1, we see that m= am2 , bM − cm M= aM2 , bm − cM Page of 16 Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 bM = cm + am, Page of 16 bm = cM + aM, then (M − m)(b + c + a) = Thus M = m It follows by Theorem that x is a global attractor of Equation and then the proof ¯ is complete Case (2) If bw-cs < 0, then we can easily see that the function f(u, v, w, s) decreasing in u, v, w and increasing in s Suppose that (m, M) is a solution of the system M = f(m, m, m, M) and m = f(M, M, M, m) Then from Equation 1, we see that am2 , bm − cM M = m= aM2 , bM − cm bmM − cM2 = am2 , bmM − cm2 = aM2 , then (M2 − m2 )(c − a) = 0, a = c Thus, M = m It follows by the Theorem that x is a global attractor of Equation and then the ¯ proof is complete Special cases of Equation 4.1 Case (1) In this section we study the following special case of Equation xn+1 = xn xn−1 , xn − xn−1 (7) where the initial conditions x-1, x0 are arbitrary positive real numbers Theorem Let {xn }∞ be a solution of Equation Then for n = 0, 1, n=−1 xn = (−1)n hk , Fn−1 k − Fn−2 h where x-1 = k, x0 = h and Fn-1, Fn-2 are the Fibonacci terms Proof: For n = the result holds Now suppose that n > and that our assumption holds for n-1, n-2 That is; xn−2 = (−1)n−2 hk (−1)n−1 hk , xn−1 = Fn−3 k − Fn−4 h Fn−2 k − Fn−3 h Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 Page of 16 Now, it follows from Equation that xn = xn−1 xn−2 = xn−1 − xn−2 = (−1)n−1 hk Fn−2 k − Fn−3 h (−1)n−2 hk Fn−3 k − Fn−4 h (−1)n−1 hk (−1)n−2 hk − Fn−2 k − Fn−3 h Fn−3 k − Fn−4 h (−1)n−1 hk Fn−2 k − Fn−3 h −1 Fn−3 k − Fn−4 h 1 + Fn−2 k − Fn−3 h Fn−3 k − Fn−4 h (−1)n hk n = Fn−1 k − Fn−2 h = (−1)n hk (Fn−2 k − Fn−3 h + Fn−3 k − Fn−4 h) Hence, the proof is completed For confirming the results of this section, we consider numerical example for x-1 = 11, x0 = (see Figure 1), and for x-1 = 6, x0 = 15 (see Figure 2), since the solutions take the forms {6, -12, 4, -3, 1.714286, -1.090909, 6666667, -.4137931, 2553191, }, {-60, 10, -8.571428, 4.615385, -3, 1.818182, -1.132075, 6976744, } 4.2 Case (2) In this section we study the following special case of Equation xn+1 = xn−1 xn−2 , xn−1 − xn−2 (8) plot of x(n+1)= x(n)*x(n−1)/(x(n)−x(n−1)) 12 10 x(n) −2 −4 −6 −8 Figure This figure shows the solution of xn+1 10 12 14 16 n xn xn−1 = , where x-1 = 11, x0 = xn − xn−1 18 20 Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 Page of 16 plot of x(n+1)= x(n)*x(n−1)/(x(n)−x(n−1)) 15 10 x(n) −5 −10 −15 −20 −25 −30 Figure This figure shows the solution of xn+1 = 10 n 12 14 16 18 20 xn xn−1 , for x-1 = 6, x0 = 15 xn − xn−1 where the initial conditions x-2, x-1, x0 are arbitrary positive real numbers Theorem Let {xn }∞ be a solution of Equation Then x1 = n=−2 xn+1 = rk , for n = 1, 2, k−r hkr , gn−4 hk + gn−3 kr + gn−2 hr where x-2 = r, x-1 = k, x0 = h, {gm }∞ = {1, −2, 0, 3, −2, −3, }, i.e., gm = gm-2 + m=0 gm-3, m ≥ 0, g-3 = 0, g-2 = -1, g-1 = Proof: For n = 1, the result holds Now suppose that n > and that our assumption holds for n - 1, n - That is; hkr hkr xn−2 = ,xn−1 = Now, it follows gn−7 hk + gn−6 kr + gn−5 hr gn−6 hk + gn−5 kr + gn−4 hr from Equation that xn+1 = xn−1 xn−2 xn−1 − xn−2 hkr hkr gn−6 hk + gn−5 kr + gn−4 hr gn−7 hk + gn−6 kr + gn−5 hr = hkr hkr − gn−6 hk + gn−5 kr + gn−4 hr gn−7 hk + gn−6 kr + gn−5 hr hkr = (gn−7 hk + gn−6 kr + gn−5 hr − gn−6 hk + gn−5 kr + gn−4 hr) hkr = gn−4 hk + gn−3 kr + gn−2 hr Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 Page 10 of 16 Hence, the proof is completed Assume that x-2 = 8, x-1 = 15, x0 = 7, then the solution will be {17.14286, -13.125, 11.83099, 7.433628, -6.222222, -20, 3.387097, -9.032259, }(see Figure 3) The proof of following cases can be treated similarly 4.3 Case (3) −1 Let x-2 = r, x-1 = k, x0 = h, Ai = and F2i-1, F2i, F2i+1 (where i = to n) are the Fibo- i=0 nacci terms Then the solution of the difference equation xn−1 xn−2 , xn − xn−2 xn+1 = (9) is given by n−1 h x2n = i=0 n−1 n−1 (F2i−1 h − F2i r) kr , x2n+1 = (F2i+1 r − F2i h) i=0 n , n = 0, 1, (F2i−1 h − F2i r) (F2i+1 r − F2i h) i=0 i=0 Figure shows the solution when x-2 = 9, x-1 = 12, x0 = 17 plot of x(n+1)= x(n−1)*x(n−2)/(x(n−1)−x(n−2)) 20 15 10 x(n) −5 −10 −15 −20 10 15 Figure This figure shows the solution of xn+1 = 20 n 25 30 35 xn−1 xn−2 , where x-2 = 8, x-1 = 15, x0 = xn−1 − xn−2 40 Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 Page 11 of 16 plot of x(n+1)= x(n−1)*x(n−2)/(x(n)−x(n−2)) 150 100 50 x(n) −50 −100 −150 −200 10 15 n Figure This figure shows the solution of xn+1 = 20 25 30 xn−1 xn−2 , when x-2 = 9, x-1 = 12, x0 = 17 xn − xn−2 4.4 Case (4) Let x-2 = r, x-1 = k, x0 = h Then the solution of the following difference equation xn+1 = xn−1 xn xn − xn−2 (10) is given by x2n−1 = h h−r n k, x2n = hn+1 , n = 0, 1, rn Figure shows the solution when x-2 = 21, x-1 = 6, x0 = 4.5 Case (5) Let x-2 = r, x-1 = k, x0 = h Then the solution of the following difference equation xn+1 = xn−1 xn , xn−1 − xn−2 (11) is given by x4n = h(hk)2n (hk)2n+1 , n , x4n+1 = (rk(h − k)(k − r)) (rk(h − k))n (k − r)n+1 x4n+2 = h(hk)2n+1 ((h − k)(k − r))n+1 (rk) , x4n+3 = n (hk)2n (r(h − k)(k − r))n+1 kn Figure shows the solution when x-2 = 9, x-1= 5, x0 = , n = 0, 1, Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 Page 12 of 16 plot of x(n+1)= x(n)*x(n−1)/(x(n)−x(n−2)) 25 20 x(n) 15 10 −5 10 n Figure This figure shows the solution of xn+1 = xn−1 xn , where x-2 = 21, x-1 = 6, x0 = xn − xn−2 plot of x(n+1)= x(n)*x(n−1)/(x(n−1)−x(n−2)) x 10 3.5 2.5 x(n) 1.5 0.5 −0.5 −1 10 20 30 40 50 60 70 80 n Figure This figure shows the solution of xn+1 = xn−1 xn , for x-2 = 9, x-1 = 5, x0 = xn−1 − xn−2 90 Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 Page 13 of 16 Figure shows the solution when x-2= 9, x-1 = 5, x0 = 4.6 Case (6) Let x-2 = r, x-1 = k, x0 = h, Then the solution of the following difference equation xn+1 = xn−2 xn , xn − xn−2 (12) is given by xn = hkr , n = 0, 1, , un−3 hr + un−2 hk + un−1 kr Where {um }∞ = {−1, 1, 0, −1, 2, −2, 1, 1, −3, } i e um = um-1 - um-3, m ≥ 0, um=0 = 0, u-2 = 0, u-1 = Figure shows the solution when x-2 = 11, x-1 = 6, x0 = 17 4.7 Case (7) Let x-2 = r, x-1 = k, x0 = h and Fn-1F, n-2, Fn are the Fibonacci terms Then the solution of the following difference equation xn+1 = xn−2 xn , xn−1 − xn−2 (13) plot of x(n+1)= x(n)*x(n−1)/(x(n−1)−x(n−2)) x(n) −1 10 20 30 40 50 60 70 80 n Figure This figure shows the solution of xn+1 = xn−1 xn , when x-2 = 0.9, x-1 = 5, x0 = 0.4 xn−1 − xn−2 90 Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 Page 14 of 16 plot of x(n+1)= x(n)*x(n−2)/(x(n)−x(n−2)) 40 30 20 x(n) 10 −10 −20 −30 10 20 30 40 50 60 70 80 90 n Figure This figure shows the solution of xn+1 = xn−2 xn , where x-2 = 11, x-1 = 6, x0 = 17 xn − xn−2 plot of x(n+1)= x(n)*x(n−2)/(x(n−1)−x(n−2)) x(n) −2 −4 Figure This figure shows the solution of xn+1 = 10 n 12 14 16 18 xn−2 xn when x-2 = 8, x-1 = 5, x0 = 0.9 xn−1 − xn−2 20 Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 Page 15 of 16 is given by x2n = hkr , (Fn−2 k − Fn−1 r)(Fn−2 h − Fn−1 k) x2n+1 = hkr , n = 0, 1, (Fn−1 k − Fn r)(Fn−2 h − Fn−1 k) Figure shows the solution when x-2 = 8, x-1 = 5, x0 = 0.9 Author details Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt 2Department of Mathematics, Faculty of Science & Art in Rabigh, King AbdulAziz University, Rabigh 21911, Saudi Arabia 3Mathematics Department, Faculty of Science, King AbdulAziz University, P O Box 80203, Jeddah 21589, Saudi Arabia 4Permanent address: Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Authors’ contributions EMEla investigated the behavior of the solutions, HAE-M found the solutions of the special cases and EMEls carried out the theoretical proof and gave the examples All authors read and approved the final manuscript Competing interests The authors declare that they have no competing interests Received: April 2011 Accepted: 23 August 2011 Published: 23 August 2011 References Kulenovic MRS, aa, Ladas, G: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures Chapman & Hall/CRC Press, Boca Raton, FL (2001) Kocic, VL, Ladas, G: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications Kluwer Academic Publishers, Dordrecht (1993) Mickens, RE: Difference Equations Van Nostrand Reinhold Comp, New York (1987) Mackey, MC, Glass, L: Oscillation and chaos in physiological control system Science 197, 287–289 (1977) doi:10.1126/ science.267326 bxn Adv Differ Equ 1–10 Elabbasy, EM, El-Metwally, H, Elsayed, EM: On the difference equation xn+1 = axn − cxn − dxn−1 (2006) Article ID 82579 αxn−k J Conc Appl Math 5(2), Elabbasy, EM, El-Metwally, H, Elsayed, EM: On the difference equations xn+1 = β + γ k xn−i i=0 101–113 (2007) Elabbasy, EM, El-Metwally, H, Elsayed, EM: Global attractivity and periodic character of a fractional difference equation of order three Yoko-hama Math J 53, 89–100 (2007) El-Metwally, H, Grove, EA, Ladas, G, Levins, R, Radin, M: On the difference equation xn+1 = α + βxn−1 e−xn Nonlinear Anal: Theory Methods Appl 47(7), 4623–4634 (2003) ax + bxn−2 Appl Math Comput Yang, X, Su, W, Chen, B, Megson, GM, Evans, DJ: On the recursive Sequence xn+1 = n−1 c + dxn−1 xn−2 162, 1485–1497 (2005) doi:10.1016/j.amc.2004.03.023 xn−1 Appl Math Comput 158(3), 809–812 10 Cinar, C: On the positive solutions of the difference equation xn+1 = + axn xn−1 (2004) doi:10.1016/j.amc.2003.08.140 xn−1 Appl Math Comput 158(3), 793–797 11 Cinar, C: On the positive solutions of the difference equation xn+1 = −1 + axn xn−1 (2004) doi:10.1016/j.amc.2003.08.139 12 Aloqeili, M: Dynamics of a rational difference equation Appl Math Comput 176(2), 768774 (2006) doi:10.1016/j amc.2005.10.024 xnm 13 Yalỗinkaya, I: On the difference equation xn+1 = α + xk Discrete Dyn Nat Soc (2008) Article ID 805460, n 14 Agarwal, R: Difference Equations and Inequalities Theory, Methods and Applications Marcel Dekker Inc., New York (1992) 15 Agarwal, RP, Elsayed, EM: Periodicity and stability of solutions of higher order rational difference equation Adv Stud Contemp Math 17(2), 181–201 (2008) 16 Agarwal, RP, Zhang, W: Periodic solutions of difference equations with general periodicity Comput Math Appl 42, 719–727 (2001) doi:10.1016/S0898-1221(01)00191-2 17 Elabbasy, EM, Elsayed, EM: On the global attractivity of difference equation of higher order Carpathian J Math 24(2), 45–53 (2008) 18 Elabbasy, EM, Elsayed, EM: Dynamics of a rational difference equation Chin Ann Math Ser B 30B(2), 187–198 (2009) yn−(2k+1) + p Proceedings of 19 El-Metwally, H, Grove, EA, Ladas, G, McGrath, LC: On the difference equation yn+1 = yn−(2k+1) + qyn−2l the 6th ICDE, Taylor and Francis, London (2004) 20 Elsayed, EM: Qualitative behavior of difference equation of order three Acta Sci Math (Szeged) 75(1-2), 113–129 (2009) 21 Elsayed, EM: Qualitative behavior of s rational recursive sequence Indagat Math 19(2), 189–201 (2008) doi:10.1016/ S0019-3577(09)00004-4 22 Elsayed, EM: On the Global attractivity and the solution of recursive sequence Stud Sci Math Hung 47(3), 401–418 (2010) Elabbasy et al Advances in Difference Equations 2011, 2011:28 http://www.advancesindifferenceequations.com/content/2011/1/28 23 Elsayed, EM: Qualitative properties for a fourth order rational difference equation Acta Appl Math 110(2), 589–604 (2010) doi:10.1007/s10440-009-9463-z 24 Wang, C, Wang, S, Yan, X: Global asymptotic stability of 3-species mutualism models with diffusion and delay effects Discrete Dyn Nat Sci 20 (2009) Article ID 317298 25 Wang, C, Gong, F, Wang, S, Li, L, Shi, Q: Asymptotic behavior of equilibrium point for a class of nonlinear difference equation Adv Diff Equ 8, : (2009) Article ID 214309 26 Yalỗinkaya, I, Iricanin, BD, Cinar, C: On a max-type difference equation Discrete Dyn Nat Soc (2007) Article ID 47264, 10 27 Yalỗinkaya, I: On the global asymptotic stability of a second-order system of difference equations Discrete Dyn Nat Soc (2008) Article ID 860152, 12 28 Yalỗinkaya, I: On the global asymptotic behavior of a system of two nonlinear difference equations ARS Combinatoria 95, 151–159 (2010) bxn Commun Appl Nonlinear 29 Zayed, EME, El-Moneam, MA: On the rational recursive sequence xn+1 = axn − cxn − dxn−k Anal 15, 47–57 (2008) doi:10.1186/1687-1847-2011-28 Cite this article as: Elabbasy et al.: Global behavior of the solutions of some difference equations Advances in Difference Equations 2011 2011:28 Submit your manuscript to a journal and benefit from: Convenient online submission Rigorous peer review Immediate publication on acceptance Open access: articles freely available online High visibility within the field Retaining the copyright to your article Submit your next manuscript at springeropen.com Page 16 of 16 ... investigated the behavior of the solutions, HAE-M found the solutions of the special cases and EMEls carried out the theoretical proof and gave the examples All authors read and approved the final... Page of 16 Cinar [10,11] has got the solutions of the following difference equations xn−1 xn−1 , xn+1 = + axn xn−1 −1 + axn xn−1 xn+1 = Aloqeili [12] obtained the form of the solutions of the difference. .. and rewarding, and the results about these equations offer prototypes towards the development of the basic theory of the global behavior of nonlinear difference equations of a big order, recently,