Trichotomy of a system of two difference equations

J. Math. Anal. Appl. 289 (2004) 216 230 www.elsevier.com/locate/jmaa Trichotomy of a system of two difference equations G. Papaschinopoulos and G. Stefanidou 1 Democritus University of Thrace, Department of Electrical and Computer Engineering, 67100 Xanthi, Greece Received 11 April 2003 Submitted by R.P. Agarwal Abstract In this paper we study the boundedness and the asymptotic behavior of the positive solutions of the system of difference equations x n+1 = A + a i x k n pi c i y n pi m, y n+1 = B + b j y m, n qj d j x n qj where k,m {1, 2,...}, A,B,a i,c i,b j,d j, i {1,...,k}, j {1,...,m}, are positive constants, p i, q j, i {1,...,k}, j {1,...,m}, are positive integers such that p 1 <p 2 < <p k, q 1 <q 2 < <q m and the initial values x i,y i, i { π, π + 1,...,0}, π = max{p k,q m } are positive numbers. 2003 Elsevier Inc. All rights reserved. Keywords: Difference equations; Trichotomy; Boundedness; Persistence; Asymptotic behavior 1. Introduction The importance of difference equations mainly owed to the fast progress of computer science which uses mathematics models with exclusively discrete variables and not continuous and also to many applications in economics, biology, engineering etc. (see Agarwal [1], Elaydi [5], Kocic and Ladas [6]). * Corresponding author. E-mail address: gpapas@ee.duth.gr (G. Papaschinopoulos). 1 This work is a part of her Doctoral Thesis. 0022-247X/$ see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2003.09.046

G. Papaschinopoulos, G. Stefanidou / J. Math. Anal. Appl. 289 (2004) 216 230 217 In [2] Amleh, Grove, Ladas and Georgiou investigated the global stability, the boundedness and the periodic nature of the positive solutions of the equation x n+1 = a + x n 1, x n where A is a positive constant and x 1,x 0 are positive numbers. In [4] DeVault, Ladas and Schultz investigated the boundedness, the persistence, the oscillatory behavior and the global asymptotic stability of the positive solutions of the difference equation x n+1 = A + x n, n= 0, 1,..., x n 1 where A (0, ). In [7] Papaschinopoulos and Schinas studied the oscillatory behavior, the periodicity and the asymptotic behavior of the positive solutions of the system of the two nonlinear difference equations x n+1 = A + x n 1, y n+1 = A + y n 1, n= 0, 1,..., y n x n where A (0, ) and x 1,x 0,y 1,y 0 are positive numbers. In [8] Papaschinopoulos, Kiriakouli and Hatzifilippidis studied the difference equations k 1 s=0 x n+1 = A + c sx n s, n= 0, 1,..., x n k where A,c s, s {0, 1,...,k 1}, are positive constants and x k,...,x 0 (0, ). Finally, in [3] Camouzis and Papaschinopoulos investigated the global asymptotic behavior of the positive solutions of the system x n+1 = A + x n, y n+1 = B + y n, n= 0, 1,..., y n m x n m where the initial conditions x i,y i, i = m, m + 1,...,0, A,B are positive constants and m is a positive integer. In this paper we consider the system of difference equations a i x n pi x n+1 = A + m, y n+1 = B + b j y n qj c i y n pi m d j x n qj, (1) where k,m {1, 2,...}, A,B,a i,c i,b j,d j, i {1,...,k}, j {1,...,m}, are positive constants, p i,q j, i {1,...,k}, j {1,...,m}, are positive integers such that p 1 <p 2 < <p k, q 1 <q 2 < <q m and the initial values x i,y i, i { π, π + 1,...,0}, π = max{p k,q m } are positive numbers. We say that a positive solution (x n,y n ) of (1) is bounded and persists if there exist positive constants M,N such that M x n,y n N, n = 0, 1,...

218 G. Papaschinopoulos, G. Stefanidou / J. Math. Anal. Appl. 289 (2004) 216 230 In this paper we study the boundedness, persistence, the periodicity, the asymptotic behavior of the positive solutions of (1). In addition for k 2 and under some conditions on the parameters q 1,q 2,...,q m we prove that system (1) possesses the following trichotomy a k (i) If A<Dor B<Cor A>D, B = C or A = D, B>C, C = i c m, D = i b m, j d j then system (1) has unbounded solutions. (ii) If A = D and B = C then every positive solution of the system (1) converges to a period r solution of (1). (iii) If A>Dand B>Cthen there exists a unique positive equilibrium (x, y) of (1) and every positive solution of (1) tends to (x, y). Finally, we note that we are going to use the results concerning system (1) and obtained in this paper in order to study the corresponding fuzzy difference equation, as in the papers [9 11]. 2. Main results Firstly, we find conditions so that system (1) has unbounded solutions. Proposition 1. Consider system (1) where k,m {1, 2,...}, A,B,a i,c i,b j,d j, i {1,...,k}, j {1,...,m}, are positive constants and the initial values x i,y i, i { π, π + 1,...,0}, are positive numbers. Then the following statements are true: I. If one of the following conditions (i) A<D, B= C, (ii) A>D, B= C, (iii) A = D, B < C, (iv) A = D, B > C, (v) A<D, B>C, (vi) A>D, B<C, where (2) C = a i m b j, D= c i m d j is satisfied then every positive solution of (1) is unbounded. II. Moreover, if B<C (3) then system (1) has unbounded solutions (x n,y n ) such that lim x n =, and if lim y n = B (4) A<D (5) then system (1) has unbounded solutions (x n,y n ) such that lim x n = A, lim y n =. (6)

G. Papaschinopoulos, G. Stefanidou / J. Math. Anal. Appl. 289 (2004) 216 230 219 Proof. I. Let (x n,y n ) be an arbitrary solution of (1). Suppose that (x n,y n ) is bounded. Then we have l 1 = lim inf x n A, l 2 = lim inf y n B, x n <, L 2 = lim sup y n <. L 1 = lim sup Then relations (1) and (7) imply that L 1 (B C) (A D)l 2, L 2 (A D) (B C)l 1. (8) So from (7) and (8) it is obvious that relations (2) are not satisfied. This completes the proof of the statement I. II. Firstly, suppose that (3) is satisfied. We consider the difference equation a i v n pi v n+1 = A +, n= 0, 1,... (9) a i Then we can easily prove that v n = na k a i a i + k a i p i, n= p k, p k + 1,... is solution of (9) such that lim v n =. (10) Let (x n,y n ) be a positive solution of (1) with initial values satisfying x i CD C B, y i <C, i= π, π + 1,...,0. (11) Then in view of (1) and (11) we can prove that x 1 >A+ CD C B > CD C B, y 1 <C. (12) Using (1), (11), (12) and working inductively we can prove that x n > CD C B, y n <C, n= 1, 2,... (13) Therefore, relations (1) and (13) for n = 1, 2,...imply that a i x k n pi x n+1 >A+ C a i x n pi m = A + b k. (14) j a i Since v i 0fori = p k, p k + 1,...,0 then relation (14) implies that a i v pi x 1 >A+ = v 1 a i and working inductively we can prove that x n v n, n= 1, 2,... (15) (7)

220 G. Papaschinopoulos, G. Stefanidou / J. Math. Anal. Appl. 289 (2004) 216 230 Then in view of (10) and (15) we take that lim n =. (16) Moreover, from (1) and (13) we get c i B<y n <B+ C m, d j x n qj 1 n π + 2. (17) Therefore, in view of (16) and (17) we get lim y n = B. (18) Hence, from (16) and (18) we take relations (4). Finally, suppose that (5) holds. We consider the difference equation w n+1 = B + We can easily prove that w n = c i w n pi c i, n= 0, 1,... (19) nb k c i c i + k c i p i, n= p k, p k + 1,... is a solution of (19) such that lim w n =. (20) Let (x n,y n ) be a positive solution of (1) with initial values satisfying y i CD D A, x i <D, i= π, π + 1,...,0. Then form (20) and arguing as above we can prove that (6) are satisfied. This completes the proof of the statement II of the proposition. In the following proposition we study the periodicity of the positive solutions of system (1). Proposition 2. Consider system (1) where k,m {1, 2,...}, A,B,a i,c i,b j,d j, i {1,...,k}, j {1,...,m}, are positive constants and the initial values x i,y i, i { π, π + 1,...,0}, are positive numbers. Suppose that A = D and B = C. (21) Let r be a common divisor of the integers p i + 1, i = 1, 2,...,k.Ifr is also a common divisor of the positive integers q j + 1, j = 1, 2,...,m,thensystem(1) has periodic solutions of period r. Proof. From hypothesis there exist positive integers r i,s j, i = 1, 2,...,k, j = 1, 2,...,m, such that the following relations are satisfied:

G. Papaschinopoulos, G. Stefanidou / J. Math. Anal. Appl. 289 (2004) 216 230 221 p i + 1 = rr i, r i {1, 2,...}, i= 1, 2,...,k, (22) q j + 1 = rs j, s j {1, 2,...}, j= 1, 2,...,m. (23) Moreover, from (22) and (23) if φ = max{r k,s m } we get π + 1 = rφ. Let (x n,y n ) be a positive solution of (1) with initial values satisfying x rφ+rλ+θ = x r+θ, y rφ+rλ+θ = y r+θ, λ= 0, 1,...,φ 1, θ= 1, 2,...,r, x w >A, y w = Bx w, w= r + 1, r + 2,...,0. (24) x w A It is obvious that for i {1, 2,...,k}, j {1, 2,...,m}, θ {1, 2,...,r} where rr i + θ = rφ + (φ r i )r + θ, rs j + θ = rφ + (φ s j )r + θ (25) φ r i,φ s j {0, 1,...,φ 1}, Therefore, from relations (1), (21) (25) we get i = 1, 2,...,k, j = 1, 2,...,m. a i x rri +θ x θ = A + m b j y rsj +θ = A + Bx r+θ y r+θ = x r+θ. (26) c i y rri +θ y θ = B + m = B + Ay r+θ = y r+θ. (27) d j x rsj +θ x r+θ Let a v {1, 2,...}. Suppose that for all κ {1, 2,...,v 1} we have x κr+θ = x r+θ, y κr+θ = y r+θ, θ = 1, 2,...,r. (28) Relations (1), (22) and (23) imply that a i x rri +vr+θ x vr+θ = A + m. (29) b j y rsj +vr+θ Firstly, suppose that there exists at least one i {1, 2,...,k} such that r i v. Then there exists an ω {1, 2,...,k} such that r i v, i = 1, 2,...ω, r i v + 1, i = ω + 1,ω+ 2,...,k. (30) Then in view of (26), (28), (30) we have for i = 1, 2,...,ω x rri +vr+θ = x r(v ri )+θ = x r+θ. (31) Also, from (24) and (30) we have for i = ω + 1,ω+ 2,...,k x rri +vr+θ = x rφ+(φ ri +v)r+θ = x r+θ. (32) In addition, if for all i = 1, 2,...,k relations r i v + 1 hold it is obvious that (32) is true. Therefore, from (31) and (32) it is obvious that

222 G. Papaschinopoulos, G. Stefanidou / J. Math. Anal. Appl. 289 (2004) 216 230 a i x rri +vr+θ = x r+θ a i. (33) Similarly, we can prove that m m b j y rsj +vr+θ = y r+θ b j. (34) Therefore, in view of (21), (24), (29), (33) and (34) we have x vr+θ = x r+θ. (35) Similarly, we can prove that y vr+θ = y r+θ. (36) So from (26) (28), (35) and (36) the proof of the proposition is completed. In the following proposition we study the boundedness and persistence of the positive solutions of (1). Proposition 3. Consider system (1) where k,m {1, 2,...}, A,B,a i,c i,b j,d j, i {1,...,k}, j {1,...,m}, are positive constants and the initial values x i,y i, i { π, π + 1,...,0}, are positive numbers. Suppose that either relations (21) or A>D, B>C (37) are satisfied. Then every positive solution of (1) is bounded and persists. Proof. Let (x n,y n ) be a positive solution of (1). Firstly, suppose that (21) are satisfied. From (1) it is obvious that x n >A, y n >B, n= 1, 2,... (38) Therefore, from (38) there exist positive numbers L 1 > 1, L 2 > 1 such that AL 1 <x i < AL 1 L 1 1, BL 2 <y i < BL 2, i = 1, 2,...,π + 1. (39) L 2 1 Hence, from (39) if L = min{l 1,L 2 } we get x i [ AL, ], y i [ BL, AL L 1 BL L 1 Then in view of (1), (21) and (40) we take a i x π+1 pi ABL(L 1) x π+2 = A + m A + = AL, b j y π+1 qj BL x π+2 A + Similarly, we take ABL BL(L 1) = AL L 1. ], i = 1, 2,...,π + 1. (40) (41)

G. Papaschinopoulos, G. Stefanidou / J. Math. Anal. Appl. 289 (2004) 216 230 223 y π+2 BL, y π+2 BL L 1. (42) Then using (1), (21), (40) (42) and arguing as above we take [ ] [ ] x n AL,, y n BL,, n= 1, 2,... AL L 1 BL L 1 and so (x n,y n ) is bounded and persists if (21) holds. Suppose now that (37) hold. Then from (38) there exist positive numbers L 1,L 2,such that and C<L 1 B, D < L 2 A (43) [ x i A, ] [ AB, y i B, L 1 C Then using (1), (43), (44) and arguing as above we take [ ] [ x n A,, y n B, AB L 1 C ] AB, i {1, 2,...,π + 1}. (44) L 2 D AB L 2 D ], n= 1, 2,... and so we have that (x n,y n ) is bounded and persists if (37) is satisfied. This completes the proof of the proposition. In the following proposition we study the convergence of the positive solutions of (1). We need the following lemma. Lemma 1. Let r 1,r 2,...,r k, k = 2, 3,...be positive integers such that (r 1,r 2,...,r k ) = 1, (45) where (r 1,r 2,...,r k ) is the greatest common divisor of r 1,r 2,...,r k.lets 1,s 2,...,s m, m = 1, 2,..., be positive integers. Then for any positive integer σ there exist integers b wi = b wi (σ ), c wj = c wj (σ ), w = 1, 2,...,k+ m, i = 1, 2,...,k, j = 1, 2,...,m,such that where σ = m b wi (σ )r i + c wj (σ )s j, (46) b ii > 0, i = 1, 2,...,k, c k+j,j > 0, j = 1, 2,...,m, b wi < 0, i = 1, 2,...,k, w i, c wj < 0, j = 1, 2,...,m, w k + j, w = 1, 2,...,k+ m, and c wj, w= 1, 2,...,k+ m, j = 1, 2,...,m, (47) are even numbers.

224 G. Papaschinopoulos, G. Stefanidou / J. Math. Anal. Appl. 289 (2004) 216 230 Proof. Let σ be an arbitrary positive integer. Since (45) holds there exist integers β i, i = 1, 2,...,k, such that σ = β i r i. (48) For any ρ,ξ {1, 2,...,k}, τ,ψ {1, 2,...,m} we set ( m b ρρ = β ρ ω r i + s j ), b ρξ = β ξ + ωr ρ, ρ ξ,,i ρ b k+τ,ρ = β ρ + ωs τ, c ρτ = ωr ρ, c k+τ,ψ = ωs τ, τ ψ, (49) ( m c k+τ,τ = ω r i + s j ),,j τ where ω is an even integer, such that { ω<min B, B β, r 1 s k,i ρ }, 1 r i + (50) m s j B = max{β i,i= 1, 2,...,k}, β = min{β i,i = 1, 2,...,k}. Then from (48), (49) and (50) we get ( ) m m b ρi r i + c ρj s j = (β i + ωr ρ )r i + β ρ r ρ ωr ρ r i + s j and,i ρ m b k+τ,i r i + c k+τ,j s j =,i ρ m + ωr ρ s j = σ (51) ( (β i + ωs τ )r i s τ ω r i + + ωs τ m,j τ m,j τ s j ) s j = σ. (52) Therefore, in view of (49) (52) we take (46) and (47). This completes the proof of the lemma. Proposition 4. Consider system (1) where k,m {1, 2,...}, A,B, a i,c i,b j,d j, i {1,...,k}, j {1,...,m}, are positive constants and the initial values x i,y i, i { π, π + 1,...,0}, are positive numbers. Then the following statements are true: I. Suppose that (37) holds. Then (1) has a unique positive equilibrium (x, y) and every positive solution (x n,y n ) of (1) tends to the (x, y) as n.

G. Papaschinopoulos, G. Stefanidou / J. Math. Anal. Appl. 289 (2004) 216 230 225 II. Let r be the greatest common divisor of the positive integers p i + 1, i = 1, 2,...,k. Then if k 2, relations (21) are satisfied and r is a common divisor of q j + 1, j = 1, 2,...,m, every positive solution (x n,y n ) of (1) tends to a period r solution of (1). Proof. I. From (1) and (37) we can easily prove that (x, y) where AB DC AB DC x =, y = B C A D is the unique positive equilibrium of (1). Moreover, from Proposition 3 and relations (7) and (8) it is obvious that every positive solution (x n,y n ) of (1) tends to the (x, y) as n. This completes the proof of I. II. Let (x n,y n ) be an arbitrary positive solution of (1). We prove that there exist the lim x nr+i = ε i, i = 0, 1,...,r 1. (53) We fix a τ {0, 1,...,r 1}. From hypothesis there exist positive integers r i,s j, i = 1, 2,...,k, j = 1, 2,...,m, such that relations (22) and (23) are satisfied. Then in view of (1), (22) and (23) we obtain a i x k r(n ri )+τ c i y r(n ri )+τ x rn+τ = A + m, y rn+τ = B + b j y m. (54) r(n sj )+τ d j x r(n sj )+τ Since from Proposition 3 the solution (x n,y n ) is bounded and persists, we have lim inf x nr+τ = l τ A, lim inf y nr+τ = m τ B, lim sup x nr+τ = L τ <, lim sup y nr+τ = M τ <. Moreover, relations (54) and (55) imply that L τ A + CL τ, M τ B + DM τ, l τ A + Cl τ, m τ l τ M τ Therefore, from (21) and (56) we take (55) m τ B + Dm τ L τ. (56) m τ = BL τ L τ A, l τ = AM τ M τ B. (57) We prove that (53) is true for i = τ. Suppose on the contrary that l τ <L τ. Then from (55) there exists an ε>0 such that L τ >l τ + ε>a+ ε. (58) In view of (55) there exists a sequence n µ, µ = 1, 2,...,such that n µ >θ, n µ >ζ and lim x rn µ µ +τ = L τ, lim x r(n µ µ θ)+τ = T θτ L τ, lim y r(n µ µ ζ)+τ = S ζτ m τ, where (59)

226 G. Papaschinopoulos, G. Stefanidou / J. Math. Anal. Appl. 289 (2004) 216 230 θ = θ(β 1,β 2,...,β k,γ 1,γ 2,...,γ m ) = ζ = ζ(λ 1,λ 2,...,λ k,µ 1,µ 2,...,µ m ) = m β i r i + γ j s j, m λ i r i + µ j s j, β i,λ i {0, 1,...,δ i }, γ j,µ j {0, 1,...,ψ j }, δ i = max { b ii (σ ) b vi (σ ), v = 1, 2,...,k+ m, σ = 1, 2,...,φ }, ψ j = max { c k+j,j (σ ) c vj (σ ), v = 1, 2,...,k+ m, σ = 1, 2,...,φ }, b vi (σ ), c vj (σ ) are defined in (49) and m γ j (respectively m µ j ) is an even (respectively odd) number. In view of (21), (54), (55), (57) and (59) it is obvious that a i T ri,τ L τ = A + m b j S sj,τ and obviously, we have that A + BL τ m τ = L τ T ri,τ = L τ, S sj,τ = m τ, i = 1, 2,...,k, j = 1, 2,...,m. (60) Furthermore, using (21), (54), (57), (59) and the second relation of (60) it is obvious that for i = 1, 2,...,m c i S sj +r i,τ m τ = B + mv=1 d v T sj +s v,τ andsowetake B + Am τ L τ = m τ S sj +r i,τ = m τ, T sj +s v,τ = L τ, j,v {1, 2,...,m}, i= 1, 2,...,k. (61) Therefore, from relations (21), (54), (57), (59) (61) and arguing as above we can prove that lim µ x r(n µ θ)+τ = L τ, Let a σ {0, 1,...,φ}. Relations (54) imply that lim y r(n µ µ ζ)+τ = m τ. (62) a i x r(nµ +σ r i )+τ x r(nµ +σ)+τ = A + m, b j y r(nµ +σ s j )+τ c i y r(nµ +σ r i )+τ y r(nµ +σ)+τ = B + m. d j x r(nµ +σ s j )+τ (63) We claim that lim µ x r(n µ +σ)+τ = L τ, lim µ x r(n µ +σ s j )+τ = L τ, lim x r(n µ µ +σ r i )+τ = L τ, i = 1, 2,...,k, lim y r(n µ µ +σ s j )+τ = m τ, j = 1, 2,...,m. (64)

G. Papaschinopoulos, G. Stefanidou / J. Math. Anal. Appl. 289 (2004) 216 230 227 Using (54) we can easily prove that there exist positive integers p,q and a continuous function F : R R R R such that x r(nµ +σ)+τ = A + F(x ζnµ,1 +τ,...,x ζnµ,p +τ,y ξnµ,1 +τ,...,y ξnµ,q +τ ), (65) where ( m ζ nµ,l = r n µ + σ β li r i γ lj s j ), l = 1, 2,...,p, (66) ( ξ nµ,v = r n µ + σ m ε vi r i θ vj s j ), v= 1, 2,...,q, (67) m γ lj (respectively m θ vj ) is an even (respectively odd) number and for any l {1, 2,...,p} either there exists a i l {1, 2,...,k} such that β l,il = b il,i l, β li b ii, i = 1, 2,...,k, γ lj c k+j,j, j = 1, 2,...,m or there exists a j l {1, 2,...,m} such that γ l,jl = c k+jl,j l, γ lj c k+j,j, j = 1, 2,...,m, β li b ii, i = 1, 2,...,k. In addition, for any v {1, 2,...,q} either there exists an i v {1, 2,...,k} such that ε v,iv = b iv,i v, ε vi b ii, i = 1, 2,...,k, θ vj c k+j,j, j = 1, 2,...,m or there exists a j v {1, 2,...,m} such that (68) (69) (70) θ v,jv = c k+jv,j v, θ vj c k+j,j, j = 1, 2,...,m, (71) ε vi b ii, i = 1, 2,...,k. Let a l {1, 2,...,p}. Suppose that there exists an i l {1, 2,...,k} such that (68) is true. Since (45) holds, from Lemma 1 and relations (66) and (68) we get ( m ζ nµ,l = r n µ (β li b il,i)r i (γ lj c il,j )s j ), l = 1, 2,...,p. (72),i i l Then in view of (62), (68) and (72) we have for l = 1, 2,...,p lim µ x ζ nµ,l +τ = L τ. (73) We consider that there exists a j l such that (69) hold. Since (45) holds, from Lemma 1 and relations (66), (69) we get

228 G. Papaschinopoulos, G. Stefanidou / J. Math. Anal. Appl. 289 (2004) 216 230 ( ζ nµ,l = r n µ (β li b k+jl,i)r i ) m (γ lj c k+jl,j )s j,,j j l l = 1, 2,...,p. (74) Therefore, from (62) and (74) we take (73). Similarly, using (62), (67), (70) and (71) we can prove that for v = 1, 2,...,q lim y ξ µ nµ,v +τ = m τ. (75) Then in view of (57), (65), (73) and (75) we take lim x r(n µ µ +σ)+τ = A + F(L τ,...,l τ,m τ,...,m τ ) = A + BL τ = L τ. (76) m τ Similarly, we can prove that for σ {0, 1,...,φ} lim x r(n µ µ +σ r i )+τ = L τ, i = 1, 2,...,k. (77) Using (57), (63), (76) and (77) we take for σ {0, 1,...,φ} lim r(n µ µ +σ s j )+τ = m τ, j = 1, 2,...,m. (78) In addition, from (78) we take lim y rn µ µ +τ = m τ. (79) Furthermore, using (55) and without loss of generality we can suppose that lim µ r(n µ θ)+τ = T θτ L τ, lim µ r(n µ ζ)+τ = S ζτ m τ, (80) where θ = θ ( ) m β 1, β 2,..., β k, γ 1, γ 2,..., γ m = β i r i + γ j s j, ζ = ζ ( λ ) m 1, λ 2,..., λ k, µ 1, µ 2,..., µ m = λ i r i + µ j s j, β i, λ i {0, 1,...,δ i }, γ j, µ j {0, 1,...,ψ j }, m γ j (respectively m µ j ) is an odd (respectively even) number. Then using (21), (54), (57), (79), (80) and using the same argument to prove (61) we can easily take S ri,τ = m τ, T sj,τ = L τ, j = 1, 2,...,m, i= 1, 2,...,k. (81) Therefore, in view of (21), (54), (57), (80), (81) and using the same argument to prove (76) we can prove that lim x r(n µ µ +σ s j )+τ = L τ, j = 1, 2,...,m. (82)

G. Papaschinopoulos, G. Stefanidou / J. Math. Anal. Appl. 289 (2004) 216 230 229 So in view of (76), (77), (78) and (82) relation (64) is true. From (58) we can define the positive number δ as follows δ = ε(m τ B) L τ ε A. In view of (64) there exists a µ 0 {1, 2,...} such that x r(nµ0 +κ)+τ L τ ε, κ = φ, φ + 1,...,φ, y r(nµ0 +ν)+τ m τ + δ, ν = s m, s m + 1,...,φ s 1. In addition, from (1) we take a i x r(nµ0 x r(nµ0 +φ+1)+τ = A + +φ+1 r i)+τ m, b j y r(nµ0 +φ+1 s j )+τ c i y r(nµ0 y r(nµ0 +φ+1 s j )+τ = B + +φ+1 s j r i )+τ mv=1. d j x r(nµ0 +φ+1 s j s v )+τ (83) (84) Then from (21), (57), (83) and (84) we take y r(nµ0 +φ+1 s j )+τ B + A(m τ + δ) = m τ + δ. (85) L τ ε Hence relations (21), (57), (58), (83) (85) imply that x r(nµ0 +φ+1)+τ A + B(L τ ε) = L τ ε>l τ. m τ + δ Similarly, we can easily prove that x r(nµ0 +φ+m)+τ L τ ε>l τ, m= 2, 3,... which is a contradiction since lim inf x rn+τ = l τ. Therefore, since τ is an arbitrary number such that τ {0, 1,...,r 1} relations (53) are satisfied. In addition, from (53), (55) and (57) it is obvious that lim y nr+i = ξ i, i = 0, 1,...,r 1, (86) where ξ i, i = 0, 1,...,r 1, are positive numbers. Therefore, from relations (53) and (86) the proof of the Part II of the proposition is completed. Using Propositions 1 4 we take the last proposition concerning the following trichotomy for system (1). Proposition 5. Consider system (1) where k {2, 3,...}, m {1, 2,...}, A,B, a i,c i,b j,d j, i {1,...,k}, j {1,...,m}, are positive constants and the initial values x i,y i, i { π, π + 1,...,0}, are positive numbers. Let r be the greatest common divisor of the positive integers p i + 1, i = 1, 2,...,k.Thenifr is a common divisor of the positive integers q j + 1, j = 1, 2,...,m,system(1) possesses the following trichotomy: (i) If A<D or B<Cor A>D, B = C or A = D, B>Cthen system (1) has unbounded solutions.

230 G. Papaschinopoulos, G. Stefanidou / J. Math. Anal. Appl. 289 (2004) 216 230 (ii) If A = D and B = C then every positive solution of the system (1) converges to a period r solution of (1). (iii) If A>Dand B>Cthen there exists a unique positive equilibrium (x, y) of (1) and every positive solution of (1) tends to (x, y). References [1] R.P. Agarwal, Difference Equations and Inequalities, Dekker, New York, 1992. [2] A.M. Amleh, E.A. Grove, G. Ladas, D.A. Georgiou, On the recursive sequence x n+1 = a + x n 1 x n,j.math. Anal. Appl. 233 (2) (1999) 790 798. [3] E. Camouzis, G. Papaschinopoulos, Asymptotic behavior of positive solutions of rational difference equations, Appl. Math. Lett., submitted for publication. [4] R. DeVault, G. Ladas, S.W. Schultz, On the recursive sequence x n+1 = x A n + x 1, Proc. Amer. Math. n 2 Soc. 126 (11) (1998) 3257 3261. [5] S. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 1996. [6] V.L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order With Applications, Kluwer Academic, Dordrecht, 1993. [7] G. Papaschinopoulos, C.J. Schinas, On the system of two nonlinear difference equations x n+1 = A + x n 1 y n, y n+1 = A + y n 1 x n, Internat. J. Math. Math. Sci. 23 (12) (2000) 839 848. [8] G. Papaschinopoulos, P.Ch. Kiriakouli, V. Hatzifilippidis, Oscillation and asymptotic stability of a nonlinear difference equation, Nonlinear Anal. 47 (2001) 4885 4896. [9] G. Papaschinopoulos, B.K. Papadopoulos, On the fuzzy difference equation x n+1 = A + x xn n m, Fuzzy Sets and Systems 129 (2002) 73 81. [10] G. Papaschinopoulos, B.K. Papadopoulos, On the fuzzy difference equation x n+1 = A + x B n, Soft Computing 6 (2002) 456 461. [11] G. Papaschinopoulos, G. Stefanidou, Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation, Fuzzy Sets and Systems, submitted for publication.