On the Dynamics of the Higher Order Nonlinear Rational Difference Equation

Transcription

1 Math. Sci. Lett. 3, No. 2, (214) 121 Mathematical Sciences Letters An International Journal On the Dynamics of the Higher Order Nonlinear Rational Difference Equation M. A. El-Moneam Mathematics Department, Faculty of Science Arts in Farasan, Jazan University, Kingdom of Saudi Araia Received: 26 Nov. 213, Revised: 3 Mar. 214, Accepted: 6 Mar. 214 Pulished online: 1 May 214 Astract: In this article, we study the periodicity, the oundedness the gloal staility of the positive solutions of the following nonlinear difference equation x x n+1 = Ax n + Bx n k +Cx n l + Dx n σ + n k [dx n k ex n l ], n=,1,2,..., where the coefficients A, B,C, D,, d, e (, ), while k, l σ are positive integers. The initial conditions x σ,...,x l,...,x k,...,x 1,x are aritrary positive real numers such k < l < σ. Some numerical examples will e given to illustrate our results. Keywords: Difference equations, prime period two solution, oundedness character, locally asymptotically stale, gloal attractor, gloal staility. 1 Introduction The topic of difference equations is interesting attractive to many mathematicians working in this field. The qualitative study of difference equations is a fertile research area increasingly attracts many mathematicians. This topic draws its importance from the fact many real life phenomena are modeled using difference equations. Examples from economy, iology, etc. can e found in [12,17,18]. It is known nonlinear difference equations are capale of producing a complicated ehavior regardless its order. This can e easily seen from the family x n+1 = g µ (x n ), µ >, n. This ehavior is ranging according to the value of µ, from the existence of a ounded numer of periodic solutions to chaos. There has een a great interest in studying the gloal attractivity, the oundedness character the periodicity nature of nonlinear difference equations. For example, in the articles [13 16, 19 26] closely related gloal convergence results were otained which can e applied to nonlinear difference equations in proving every solution of these equations converges to a period two solution. For other closely related results, (see [2] [11],[28] [31]) the references cited therein. The study of these equations is challenging rewarding is still in its infancy. We elieve the nonlinear rational difference equations are of paramount importance in their own right. Furthermore the results aout such equations offer prototypes for the development of the asic theory of the gloal ehavior of nonlinear difference equations. The ojective of this article is to investigate some qualitative ehavior of the solutions of the nonlinear difference equation x x n+1 = Ax n + Bx n k +Cx n l + Dx n σ + n k [dx n k ex n l ], n =,1,2,... (1) where the coefficients A,B,C,D,,d,e (, ), while k, l σ are positive integers. The initial conditions x σ,...,x l,...,x k,...,x 1,x are aritrary positive real numers such k < l < σ. Note the special cases of Eq.(1) have een studied in [1] when B = C = D =, k =,l = 1, is replaced y in [27] when B = C = D =, k =, is replaced y in [33] when B = C = D =, l = in [32] when A= C=D=, l =, is replaced y. Corresponding author madelmeneam214@yahoo.com Natural Sciences Pulishing Cor.

2 122 M. A. El-Moneam: On the Dynamics of the Higher Order Nonlinear... Our interest now is to study ehavior of the solutions of Eq.(1) in its general form. For the related work, (see [34] [39]). Let us now recall some well known results [12] which will e useful in the sequel. Definition 1.Consider a difference equation in the form x n+1 = F(x n,x n k,x n l,x n σ ), n=,1,2,... (2) where F is a continuous function, while k l are positive integers such k < l < σ. An equilirium point x of this equation is a point satisfies the condition x = F( x, x, x, x). That is, the constant sequence {x n } with x n = x f or all n k l σ is a solution of equation. Definition 2.Let x (, ) e an equilirium point of Eq.(2). Then we have (i) An equilirium point x of Eq.(2) is called locally stale if for every ε > there exists δ > such, if x σ,...,x l,...,x k,..., x 1, x (, ) with x σ x x l x x k x x 1 x + x x < δ, then x n x <ε for all n k l. (ii) An equilirium point x of Eq.(2) is called locally asymptotically stale if it is locally stale there exists γ > such, if x σ,...,x l,...,x k,..., x 1, x (, ) with x σ x x l x x k x x 1 x + x x <γ, then lim x n = x. n (iii) An equilirium point x of Eq.(2) is called a gloal attractor if for every x σ,...,x l,...,x k,..., x 1, x (, ) we have lim n x n = x. (iv) An equilirium point x of Eq.(2) is called gloally asymptotically stale if it is locally stale a gloal attractor. (v) An equilirium point x of Eq.(2) is called unstale if it is not locally stale. Definition 3.A sequence {x n } n= σ is said to e periodic with period r if x n+r = x n for all n σ. A sequence {x n } n= σ is said to e periodic with prime period r if r is the smallest positive integer having this property. Definition 4.Eq.(2) is called permanent ounded if there exists numers m M with <m<m < such for any initial conditions x σ,...,x l,...,x k,..., x 1, x (, ) there exists a positive integer N which depends on these initial conditions such m x n M f or all n N. Definition 5.The linearized equation of Eq.(2) aout the equilirium point x is defined y the equation where z n+1 = ρ z n + ρ 1 z n k + ρ 2 z n l + ρ 3 z n σ =, (3) ρ = F( x, x, x, x) x n ρ 3 = F( x, x, x, x) x n σ., ρ 1 = F( x, x, x, x), ρ 2 = F( x, x, x, x), x n k x n l The characteristic equation associated with Eq.(3) is ρ(λ)=λ σ+1 ρ λ σ ρ 1 λ σ k ρ 2 λ σ l ρ 3 =. (4) Theorem 1.[12]. Assume F is a C 1 function let x e an equilirium point of Eq.(2). Then the following statements are true. (i) If all roots of Eq.(4) lie in the open unit disk λ <1, then the equilirium point x is locally asymptotically stale. (ii) If at least one root of Eq.(4) has asolute value greater than one, then the equilirium point x is unstale. (iii) If all roots of Eq.(4) have asolute value greater than one, then the equilirium point x is a source. Theorem 2.[18]. Assume ρ,ρ 1,ρ 2 ρ 3 R. Then ρ + ρ 1 + ρ 2 + ρ 3 <1, (5) is a sufficient condition for the asymptotic staility of Eq.(2). Theorem 3.[12]. Consider the difference equation (2). Let x I e an equilirium point of Eq.(2). Suppose also (i) F is a nondecreasing function in each of its arguments. (ii) The function F satisfies the negative feedack property [F(x,x,x,x) x](x x)< f or all x I { x}, where I is an open interval of real numers. Then x is gloal attractor for all solutions of Eq.(2). 2 The local staility of the solutions The equilirium point x of Eq.(1) is the positive solution of the equation x=(a+b+c+ D) x+ x (d e) x, (6) Natural Sciences Pulishing Cor.

3 Math. Sci. Lett. 3, No. 2, (214) / where d e. If [(A+B+C+ D) 1](e d) >, then the only positive equilirium point x of Eq.(1) is given y x= [(A+B+C+ D) 1](e d). (7) Let us now introduce a continuous function F :(, ) 4 (, ) which is defined y F(u,u 1,u 2,u 3 )=Au + Bu 1 +Cu 2 + Du 3 + provided du 1 eu 2. Consequently, we get F( x, x, x, x) u = A=ρ, F( x, x, x, x) e [(A+B+C+D) 1] u = B 1 (e d) = ρ 1, F( x, x, x, x) u 2 = C+ e[(a+b+c+d) 1] (e d) = ρ 2, F( x, x, x, x) u 3 = D=ρ 3, u 1 (du 1 eu 2 ), where e d. Thus, the linearized equation of Eq.(1) aout x takes the form (8) (9) z n+1 ρ z n ρ 1 z n k ρ 2 z n l ρ 3 z n σ =, (1) where ρ,ρ 1,ρ 2 ρ 3 are given y (9). Theorem 4.Assume e d, A+B+C+ D 1 A(e d) + B(e d) e [(A+B+C+ D) 1] + C(e d)+e [(A+B+C+ D) 1] + D(e d) < e d, (11) then the equilirium point (7) of Eq.(1) is locally asymptotically stale. Proof.From (9) (11) we deduce ρ + ρ 1 + ρ 2 + ρ 3 < 1, hence the proof follows with the aid of Theorem 2. 3 Periodic solutions In this section, we study the existence of periodic solutions of Eq.(1). The following theorem states the necessary sufficient conditions the equation has periodic solutions of prime period two. Theorem 5.If k, l σ are oth even positive integers, then Eq.(1) has no prime period two solution. Proof.Assume there exists distinct positive solutions...,p,q,p,q,... of prime period two of Eq.(1). If k,l σ are oth even positive integers, then x n = x n k = x n l = x n σ. It follows from Eq.(1) P=(A+B+C+ D)Q Q=(A+B+C+ D)P By sutracting (13) from (12), we get (P Q)[A+B+C+ D + 1]=. (e d), (12) (e d). (13) Since A+B+ C + D+1, then P = Q. This is a contradiction. Thus, the proof is now completed. Theorem 6.If k, l σ are oth odd positive integers A+1 B+C+ D, then Eq.(1) has no prime period two solution. Proof.Following the proof of Theorem 5, we deduce if k,l σ are oth odd positive integers, then x n+1 = x n k = x n l = x n σ. It follows from Eq.(1) P=AQ+(B+C+ D)P Q=AP+(B+C+ D)Q By sutracting (15) from (14), we get (P Q)[(A+1) (B+C+ D)]=. (e d), (14) (e d). (15) Since (A+1) (B+C+ D), then P = Q. This is a contradiction. Thus, the proof is now completed. Theorem 7.If k,l are even σ is odd positive integers (A+B+C)+1 D, then Eq.(1) has no prime period two solution. Proof.Following the proof of Theorem 5, we deduce if k,l are even σ is odd positive integers, then x n = x n k = x n l x n+1 = x n σ. It follows from Eq.(1) P=(A+B+C)Q+DP Q=(A+B+C)P+DQ By sutracting (17) from (16), we get (P Q)[(A+B+C+ 1) D]=, (e d), (16) (e d). (17) Since (A+B+C+ 1) D, then P = Q. This is a contradiction. Thus, the proof is now completed. Natural Sciences Pulishing Cor.

4 124 M. A. El-Moneam: On the Dynamics of the Higher Order Nonlinear... Theorem 8.If σ is even k,l are odd positive integers (A + D + 1) (B +C), then Eq.(1) has no prime period two solution. Proof.Following the proof of Theorem 5, we deduce if σ is even k,l are odd positive integers, then x n = x n σ x n+1 = x n k = x n l. It follows from Eq.(1) P=(A+D)Q+(B+C)P Q=(A+D)P+(B+C)Q By sutracting (19) from (18), we get (P Q)[(A+D+1) (B+C)]=, (e d), (18) (e d). (19) Since (A+D+1) (B+C), then P = Q. This is a contradiction. Thus, the proof is now completed. Theorem 9.If k is even l,σ are odd positive integers, then Eq.(1) has prime period two solution if the condition (1 (C+ D))(3e d)<(e+d)(a+b), (2) is valid, provided (C + D) < 1 e (1 (C+ D)) d (A+B)>. Proof.If k is even l,σ are odd positive integers, then x n = x n k x n+1 = x n l = x n σ. It follows from Eq.(1) P=(A+B)Q+(C+ D)P Q=(A+B)P+(C+ D)Q Consequently, we get Q (ep dq), (21) P (e Q dp). (22) e P 2 dpq = e (A+B)PQ d (A+B)Q 2 + e(c+ D)P 2 (C+ D)dPQ Q, (23) e Q 2 dpq = e (A+B)PQ d (A+B)P 2 + e(c+ D)Q 2 (C+ D)dPQ P. (24) By sutracting (24) from (23), we get P+Q= [e (1 (C+ D)) d (A+B)], (25) where e (1 (C+ D)) d (A+B) >. By adding (23) (24), we otain e 2 (1 (C+ D)) PQ= (e+d)[k 1 +(A+B)][e K 1 d (A+B)] 2, (26) where K 1 = (1 (C+ D)), provided (C+ D) < 1. Assume P Q are two positive distinct real roots of the quadratic equation Thus, we deduce t 2 ( P+Q)t+ PQ=. (27) (P+Q) 2 > 4PQ. (28) Sustituting (25) (26) into (28), we get the condition (2). Thus, the proof is now completed. Theorem 1.If l is even k, σ are odd positive integers, then Eq.(1) has prime period two solution if the condition (A+C)(3e d)<(e+d)(1 (B+D)), (29) is valid, provided (B + D) < 1 d (1 (B+D)) e (A+C)>. Proof.If l is even k,σ are odd positive integers, then x n = x n l x n+1 = x n k = x n σ. It follows from Eq.(1) P=(A+C)Q+(B+D)P Q=(A+C)P+(B+D)Q Consequently, we get P+Q= P (e Q dp), (3) Q (e P dq). (31) [d (1 (B+D)) e (A+C)], (32) where d (1 (B+D)) e (A+C)>, e 2 (A+C) PQ= (e+d)[k 2 +(A+C)][d K 2 e (A+C)] 2, (33) where K 2 = (1 (B+D)), provided (B+D) < 1. Sustituting (32) (33) into (28), we get the condition (29). Thus, the proof is now completed. Theorem 11.If l,σ are even k is odd positive integers, then Eq.(1) has prime period two solution if the condition (A+C+ D)(3e d)<(e+d)(1 B), (34) is valid, provided B<1 d (1 B) e (A+C+ D)>. Proof.If l, σ are even k is odd positive integers, then x n = x n l = x n σ x n+1 = x n k. It follows from Eq.(1) P=(A+C+ D)Q+BP P (e Q dp), (35) Natural Sciences Pulishing Cor.

5 Math. Sci. Lett. 3, No. 2, (214) / Q=(A+C+ D)P+BQ Consequently, we get P+Q= where d (1 B) e (A+C+ D)>, Q (ep dq). (36) [d (1 B) e (A+C+ D)], (37) e 2 (A+C+ D) PQ= (e+d)[(1 B)+K 3 ][d (1 B) e K 3 ] 2, (38) where K 3 = (A+C+ D), provided B < 1. Sustituting (37) (38) into (28), we get the condition (34). Thus, the proof is now completed. Theorem 12.If k, σ are even l is odd positive integers, then Eq.(1) has prime period two solution if the condition (3e d) (1 C )<(e+d)(a+b+d), (39) is valid, provided C< 1 e (1 C ) d (A+B+D)>. Proof.If k,σ are even l is odd positive integers, then x n = x n k = x n σ x n+1 = x n l. It follows from Eq.(1) P=(A+B+D)Q+CP Q=(A+B+D)P+CQ Consequently, we get P+Q= Q (e P dq), (4) P (e Q dp). (41) [e (1 C ) d (A+B+D)], (42) where e (1 C ) d (A+B+D)>, e 2 (1 C ) PQ= (e+d)[(1 C)+K 4 ][e (1 C ) d K 4 ] 2, (43) where K 4 = (A+B+D), provided C < 1. Sustituting (42) (43) into (28), we get the condition (39). Thus, the proof is now completed. 4 Boundedness of the solutions In this section, we investigate the oundedness of positive solutions of Eq.(1). the Theorem 13.Let {x n } e a solution of Eq.(1). Then the following statements are true: (i) Suppose < d for some N, the intial conditions [ ] x N l+1,...,x N k+1,...,x N 1,x N d,1, are valid, then for e d 2 e, we have the inequality 2 d (A+B+C+ D)+ (d 2 e) x n (A+B+C+ D)+ ( e), (44) for all n N. (ii) Suppose > d for some N, the intial conditions [ x N l+1,...,x N k+1,...,x N 1,x N 1, ], d are valid, then for e d 2 e, we have the inequality (A+B+C+ D)+ ( e) x n d 2 (A+B+C+ D)+ (d 2 e), (45) for all n N. Proof.First of all, if for some N, d x N 1 e, we have x N k x N+1 = Ax N + Bx N k +Cx N l + Dx N σ + dx N k ex N l x N k A+B+C+ D+. (46) dx N k ex N l But, it is easy to see dx N k ex N l e, then for e, we get x N+1 A+B+C+ D+ e. (47) Similarly, we can show x N+1 x N k (A+B+C+ D)+. (48) d dx N k ex N l But, one can see dx N k ex N l d2 e d, then for d 2 e, we get x N+1 d (A+B+C+ D)+ 2 d 2 e. (49) From (47) (49) we deduce for all n N the inequality (44) is valid. Hence, the proof of part (i) is completed. Similarly, if 1 x N d, then we can prove part (ii) which is omitted here for convenience. Thus, the proof is now completed. Natural Sciences Pulishing Cor.

6 126 M. A. El-Moneam: On the Dynamics of the Higher Order Nonlinear... 5 Gloal staility plot of 14.5 x y(n+1)=(a*y(n)+b*y(n 2)+C*y(n 4)+D*y(n 6))+((*y(n 2))/(d*y(n 2) e*y(n 4))) 137 In this section we study the gloal asymptotic staility of the positive solutions of Eq.(1). Theorem 14.If < A+B+ C+ D < 1 e d, then the equilirium point x given y (7) of Eq.(1) is gloal attractor. Proof.We consider the following function F(x,y,z,w)=Ax+By+Cz+Dw+ y (dy ez), (5) where dy ez, provided B(dy ez) 2 >ez C(dy ez) 2 + ey>. It is easy to verify the condition (i) of Theorem 3. Let us now verify the condition (ii) of Theorem 3 as follows: { [F(x,x,x,x) x](x x) = K 5 x } e d x { } x [K 5 1](e d) { } x(e d)[k5 1] 2 = e d 1 [K 5 1], (51) where K 5 =(A+B+C+ D). Since <K 5 < 1 e d, then we deduce from (51) [F(x,x,x,x) x](x x)<. (52) According to Theorem 3, x is gloal attractor. Thus, the proof is now completed. On comining the two Theorems 4 14, we have the result. Theorem 15.The equilirium point x given y (7) of Eq.(1) is gloally asymptotically stale. Example 2. Figure 2, shows Eq.(1) has no prime period two solutions if k = 1, l = 3, σ = 5, x 5 = 1, x 4 = 2, x 3 = 3, x 2 = 4, x 1 = 5, x = 6, A=75, B=5, C=25, D=2, =5, d = 3, e=2. plot of 18 x y(n+1)=(a*y(n)+b*y(n 1)+C*y(n 3)+D*y(n 5))+((*y(n 1))/(d*y(n 1) e*y(n 3))) Example 3. Figure 3, shows Eq.(1) has no prime period two solutions if k = 2, l = 4, σ = 5, x 5 = 1, x 4 = 2, x 3 = 3, x 2 = 4, x 1 = 5, x = 6, A=1, B=2, C=3, D=4, =5, d = 3, e=2. 6 Numerical examples In order to illustrate the results of the previous section to support our theoretical discussions, we consider some numerical examples in this section. These examples represent different types of qualitative ehavior of solutions of Eq.(5). Example 1. Figure 1, shows Eq.(1) has no prime period two solutions if k = 2, l = 4, σ = 6, x 6 = 1, x 5 = 2, x 4 = 3, x 3 = 4, x 2 = 5, x 1 = 6, x = 7, A = 3, B = 2, C = 1, D = 75, = 5, d = 3, e=2. plot of 12 x y(n+1)=(a*y(n)+b*y(n 2)+C*y(n 4)+D*y(n 5))+((*y(n 2))/(d*y(n 2) e*y(n 4))) Natural Sciences Pulishing Cor.

7 Math. Sci. Lett. 3, No. 2, (214) / Example 4. Figure 4, shows Eq.(1) has no prime period two solutions if k = 1, l = 3, σ = 4, x 4 = 1, x 3 = 2, x 2 = 3, x 1 = 4, x = 5, A=1, B=5, C=25, D=2, =5, d = 3, e=2. plot of 2.5 x y(n+1)=(a*y(n)+b*y(n 1)+C*y(n 3)+D*y(n 4))+((*y(n 1))/(d*y(n 1) e*y(n 3))) numerical examples y using the mathematical programs Matla Mathematica to confirm the otained results. Note example 1 verifies Theorem 5 which shows if k,l σ are all even positive integers, then Eq.(1) has no prime period two solution example 2 verifies Theorem 6 which shows if k,l σ are all odd positive integers, then Eq.(1) has no prime period two solution example 3 verifies Theorem 7 which shows if k,l are even σ is odd positive integers, then Eq.(1) has no prime period two solution. But example 4 verifies Theorem 8 which shows if σ is even k,l are odd positive integers, then Eq.(1) has no prime period two solution, while example 5 verifies Theorem 15 which shows Eq.(1) is gloally asymptotically stale..5 References Example 5. shows Eq.(1) is gloally asymptotically stale if k = 1, l = 3, σ = 4, x 4 = 1, x 3 = 2, x 2 = 3, x 1 = 4, x = 5, A =.1, B =.2, C=.3, D=.4, =5, d = 3, e=2. plot of y(n+1)=(a*y(n)+b*y(n 1)+C*y(n 3)+D*y(n 4))+((*y(n 1))/(d*y(n 1) e*y(n 3))) Conclusion We have discussed some properties of the nonlinear rational difference equation (1), namely the periodicity, the oundedness the gloal staility of the positive solutions of this equation. We gave some figures to illustrate the ehavior of these solutions. The results in this article can e considered as a more generalization than the results otained in Refs. [1, 27, 32, 33]. We presented some numerical examples y giving some numerical values for the initial values the coefficients of each case. Some figures have een given to explain the ehavior of the otained solutions in the case of [1] E. M. Elaasy, H. El- Metwally E. M. Elsayed, On the difference equation x n+1 = ax n x n /(cx n dx n 1 ), Advances in Difference Equations, Volume 26, Article ID 82579, pages 1-1, doi: /26/ [2] H. El-Metwally E. M. Elsayed, Solution ehavior of a third rational difference equation, Utilitas Mathematica, 88 (212), [3] M. A. El-Moneam S.O. Alamoudy, On study of the asymptotic ehavior of some rational difference equations, DCDIS Series A: Mathematical Analysis (to appear). [4] M. A. El-Moneam E. M. E. Zayed, Dynamics of the rational difference equation, Inf. Sci. Lett. 3, No. 2, 1-9 (214). [5] E. M. Elsayed, Solution attractivity for a rational recursive sequence, Discrete Dynamics in Nature Society, Volume 211, Article ID 98239, 17 pages. [6] E. M. Elsayed, Solutions of rational difference system of order two, Mathematical Computer Modelling, 55(212), [7] E. M. Elsayed, Behavior expression of the solutions of some rational difference equations, J. Computational Analysis Applications, 15(213), [8] E. M. Elsayed M. M. El-Dessoky, Dynamics ehavior of a higher order rational recursive sequence, Advances in Difference Equations 212, 212:69 (28 May 212). [9] E. M. Elsayed H. A. El-Metwally, On the solutions of some nonlinear systems of difference equations, Advances in Difference Equations 213, 213:16, doi:1.1186/ , Pulished: 7 June 213. [1] M. E. Erdogan, C. Cinar I. Yalcinkaya, On the dynamics of the recursive sequence, Computers & Mathematics with Applications, 61(211), [11] M. E. Erdogan, C. Cinar I. Yalcinkaya, On the dynamics of the recursive sequence, Mathematical Computer Modelling, 54(211), [12] E. A. Grove G. Ladas, Periodicities in nonlinear difference equations, Vol.4, Chapman & Hall / CRC, 25. [13] T. F. Irahim, Solutions of a system of nonlinear fractional recursion equations, International Journal of Mathematical Education: Policy Practice, 2(212), Natural Sciences Pulishing Cor.

8 128 M. A. El-Moneam: On the Dynamics of the Higher Order Nonlinear... [14] T. F. Irahim, Periodicity solution of rational recurrence relation of order six, Applied Mathematics, 3(212), [15] T. F. Irahim, Three-dimensional max-type cyclic system of difference equations, International Journal of Physical Sciences, 8(213), [16] T. F. Irahim N. Touafek, On a third-order rational difference equation with variale coefficients, DCDIS Series B: Applications & Algorithms, 2(213), [17] V. L. Kocic G. Ladas, Gloal ehavior of nonlinear difference equations of higher order with applications, Kluwer Academic Pulishers, Dordrecht, [18] M. R. S. Kulenovic G. Ladas, Dynamics of second order rational difference equations with open prolems conjectures, Chapman & Hall / CRC, Florida, 21. [19] N. Touafek E. M. Elsayed, On the solutions of systems of rational difference equations, Mathematical Computer Modelling, 55(212), [2] N. Touafek E. M. Elsayed, On the periodicity of some systems of nonlinear difference equations, Bull. Math. Soc. Sci. Math. Roumanie, Tome 55(13), No. 2, (212), [21] I. Yalcinkaya C. Cinar, On the dynamics of the difference equation x n+1 = ax n k /(+cx p n), Fasc. Math., 42 (29), [22] E. M. E. Zayed, On the rational recursive sequence x n+1 = Ax n + Bx n k + (1+x n x n k )/(px n + x n k ),, Acta Math. Vietnamica, 37(212), [23] E. M. E. Zayed M. A. El-Moneam, On the rational recursive sequence x n+1 = (D+αx n + βx n 1 + γx n 2 )/(Ax n + Bx n 1 +Cx n 2 ), Comm. Appl. Nonl. Anal., 12(25), [24] E. M. E. Zayed M. A. El-Moneam, On the rational recursive sequence x n+1 =(αx n +βx n 1 +γx n 2 + δx n 3 )/(Ax n + Bx n 1 +Cx n 2 + Dx n 3 ), J. Appl. Math. & Computing, 22(26), [25] E. M. E. Zayed M. A. El-Moneam, On the rational recursive sequence x n+1 = ( A+ k i= α ix n i ) / k i= β i x n i, Math. Bohemica, 133(28), No.3, [26] E. M. E. Zayed M. A. El-Moneam, On the rational recursive sequence x n+1 = ( A+ k i= α i x n i ) / ( B+ k i= β i x n i ), Int. J. Math. & Math. Sci., Volume 27, Article ID 23618, 12 pages, doi: /27/ [27] E. M. E. Zayed M. A. El-Moneam, On the rational recursive sequence x n+1 = ax n x n /(cx n dx n k ), Comm. Appl. Nonlinear Analysis, 15(28), [28] E. M. E. Zayed M. A. El-Moneam, On the Rational Recursive Sequence x n+1 =(α+ βx n k )/(γ x n ), J. Appl. Math. & Computing, 31(29) [29] E. M. E. Zayed M. A. El-Moneam, On the rational recursive sequence x n+1 = Ax n + (βx n + γx n k )/(Cx n + Dx n k ), Comm. Appl. Nonl. Anal., 16(29), [3] E. M. E. Zayed M. A. El-Moneam, On the Rational Recursive Sequence x n+1 = γx n k + (ax n + x n k )/(cx n dx n k ), Bull. Iranian Math. Soc., 36(21) [31] E. M. E. Zayed M. A. El-Moneam, On the rational recursive sequence x n+1 = Ax n + Bx n k + (βx n + γx n k )/(Cx n + Dx n k ), Acta Appl. Math., 111(21), [32] E. M. E. Zayed M. A. El-Moneam, On the rational recursive two sequences x n+1 = ax n k + x n k /(cx n + δdx n k ), Acta Math. Vietnamica, 35(21), [33] E. M. E. Zayed M. A. El-Moneam, On the gloal attractivity of two nonlinear difference equations, J. Math. Sci., 177(211), [34] E. M. E. Zayed M. A. El-Moneam, On the rational recursive sequence x n+1 = (A+α x n + α 1 x n σ )/(B+β x n + β 1 x n τ ), Acta Math. Vietnamica, 36(211), [35] E. M. E. Zayed M. A. El-Moneam, On the rational recursive sequence x n+1 = (α x n + α 1 x n l + α 2 x n k )/(β x n + β 1 x n l + β 2 x n k ), Math. Bohemica, 135(21), [36] E. M. E. Zayed M. A. El-Moneam, On the gloal asymptotic staility for a rational recursive sequence, Iranian Journal of science Technology (A: siences), 35(A4) (211), [37] E. M. E. Zayed M. A. El-Moneam, On the rational recursive sequence x n+1 = α x n +α 1 x n l +α 2 x n m +α 3 x n k, β x n +β 1 x n l +β 2 x n m +β 3 x n k WSEAS Transactions Math., 11(212), [38] E. M. E. Zayed M. A. El-Moneam, On the qualitative study of the nonlinear difference equation x n+1 = αx n σ β+γx p n τ Fasciculi Mathematici, 5(213), [39] E. M. E. Zayed M. A. El-Moneam, Dynamics of the rational difference equation x n+1 = γx n + αx n l+βx n k Ax n l +Bx n k, Comm. Appl. Nonl. Anal., 21(214), , Natural Sciences Pulishing Cor.

9 Math. Sci. Lett. 3, No. 2, (214) / M. A. El-Moneam Assistant Professor of Mathematics (from 21 till now) at Mathematics Department, Faculty of Science Arts in Farasan, Jazan University, Kingdom of Saudi Araia. His interests are the nonlinear differential equations, Nonlinear difference equations. He Pulished many articles in famous International Journals around the world. He has reviewed many articles for many international Journals. He has got MSC degree in Mathematics from Faculty of Science, at Zagazig University, Egypt, supervised Prof. E. M. E. Zayed. He has got his PHD in Mathematics from Faculty of Science, at Zagazig University, Egypt, supervised Prof. E. M. E. Zayed. He has got the Man of the Year award, (211), y the American Biographical, Institute, U.S.A. He is a memer of International Society of Difference Equations (ISDE) in U.S.A. He is a memer of peer reviewer in AIP Conference Proceeding 21 Athena (21, 211, 212, 213). He is a memer of peer reviewer in the 2nd Scientific Conference for students of higher education in Jedda, Kingdom of Saudi Araia (1432 G.). He is a memer of peer reviewer in the 3rd Scientific Conference for students of higher education in Al Khoar, Kingdom of Saudi Araia He is a memer of peer reviewer in the 4th Scientific Conference for students of higher education in Makkah, Kingdom of Saudi Araia He is a memer of peer reviewer in the 5th Scientific Conference for students of higher education in Riyadh, Kingdom of Saudi Araia He is a peer reviewed to the Scientific research No. (164/662/1432), King Ad Al Aziz University, Kingdom of Saudi Araia (1432 H.). He edited reviewed the Scientific ook No. (28), Taif University, Kingdom of Saudi Araia, 213 G., (1434 H. ). He has attended the Scientific sessions of the 4th international Exhiition Conference on Higher Education (16-17/4/213 G.), Riyadh, Kingdom of Saudi Araia. He has attended the 1st Saudi Scientific Pulishing Conference (27-29/3/1435 H.)(28-3/1/214 G.) King Khalid University, Aha, Kingdom of Saudi Araia. He is a peer reviewed to the Scientific research No. (LGP-35-4), King Adulaziz City for Science Technology (KACST), Kingdom of Saudi Araia (1435 H.). Natural Sciences Pulishing Cor.