Global Attractivity of a Rational Difference Equation

Transcription

1 Math. Sci. Lett. 2, No. 3, (2013) 161 Mathematical Sciences Letters An International Journal Global Attractivity of a Rational Difference Equation Nouressadat Touafek 1 Yacine Halim 2, 1 LMAM Laboratory, Department of Mathematics, University of Jijel, Algeria 2 Institute of Science Technology, University Center of Mila, Algeria Received: 23 Feb. 2013, Revised: 3 May. 2013, Accepted: 4 May Published online: 1 Sep Abstract: In this paper, we investigate the global attractivity of positive solutions of the nonlinear difference equation x n+1 = ax3 n+ bx n xn cx2 nx n 1 + dxn 1 3 Axn+ 3 Bx n xn 1 2 +Cx2 nx n 1 + Dxn 1 3, n=0,1,... the parameters a, b, c, d, A, B, C, D are positive real numbers the initial values x 0, x 1 are arbitrary positive numbers. Keywords: Difference equations, global attractivity, periodicity. 1 Introduction preliminaries Difference equations appear naturally as discrete analogues as numerical solutions of differential delay differential equations which model phenomena in ecology, economy, automatic control theory so forth. Recently, there has been a great attention in studying nonlinear difference equations, see, for instance, [1], [3], [5], [10], [11], references cited therein, as well as in studying systems of difference equations (see, e.g. [6], [9]). Consider the following second-order difference equation x n+1 = ax3 n+ bx n xn cx2 nx n 1 + dxn 1 3 Axn+ 3 Bx n xn 1 2 +Cx2 nx n 1 + Dxn 1 3, n=0,1,... (1) the initial conditions x 0, x 1 (0, ) the parameters a, b, c, d, A, B, C, D (0, ). In this paper, we study the stability of the unique positive equilibrium point x= a+b+c+d, the boundedness the convergence of the solutions of the equation (1). Now, we review some definitions (see for example [7], [8]), which will be useful in the sequel. Let I be an interval of real numbers let F : I I I be a continuously differentiable function. Consider the difference equation with initial values x 1, x 0 I. x n+1 = F(x n,x n 1 ) (2) Definition 11 A point x I is called an equilibrium point of Eq.(2) if x=f(x,x). Definition 12 Let x be an equilibrium point of Eq.(2). (i) The equilibrium x is called locally stable if for every ε > 0, there exist δ > 0 such that for all x 1, x 0 I with x 1 x + x 0 x <δ, we have x n x <ε, for all n 1. (ii) The equilibrium x is called locally asymptotically stable if it is locally stable, if there exists γ > 0 such that if x 1, x 0 I x 1 x + x 0 x <γ then lim x n = x. (iii) The equilibrium x is called global attractor if for all x 1, x 0 I, we have lim x n = x. (iv) The equilibrium x is called global asymptotically stable if it is locally stable a global attractor. Corresponding author halyacine@yahoo.fr

2 162 N. Touafek, Y. Halim: Global Attractivity of a Rational... (v) The equilibrium x is called unstable if it is not stable. (Vi) Let p = F F x (x,x) q = y (x,x). Then the equation y n+1 = py n + qy n 1, n=0,1,... (3) is called the linearized equation of Eq.(2) about the equilibrium point x. The next result, which was given by Clark [2], provides a sufficient condition for the locally asymptotically stability of Eq. (3). Theorem 11 Consider the difference equation (3) with p, q R. Then, p + q <1 is a sufficient condition for the locally asymptotically stability. Definition 13 The difference equation (2) is said to be permanent, if there exist numbers α, β with 0<α β < such that for any initial x 1, x 0 I there exists a positive integer N wich depends on the initial conditions such that α x n β for all n N. 2 Main results Let f :(0,+ ) 2 (0,+ ) be the function defined by f(x,y)= ax3 + bxy 2 + cx 2 y+dy 3 Ax 3 + Bxy 2 +Cx 2 y+dy 3. In the sequel we need the following real numbers: r 1 = ab ba, r 2 = ac ca, r 3 = ad da, r 4 = cb bc, r 5 = bd db, r 6 = cd dc. Lemma Assume that A a max( B b, C c), D d min( B b, C c), 3r 3 + r 4 0. Then, f is nondecreasing in x for each y it is nonincreasing in y for each x. 2. Assume that A a min( B b, C c), D d max( B b, C c), 3r 3 + r 4 0. Then, f is nonincreasing in x for each y it is nondecreasing in y for each x. Proof. 1. We have, 3r 3 + r 4 0 it is easy to see that a A max( b B, c C ) d D max( b B, c C ) implies r 1,r 2,r 5,r 6 0. So the result follows from the two formulaes f x (x,y) = r 2x 4 y+2r 1 x 3 y 2 +(3r 3 + r 4 )x 2 y 3 (Ax 3 + Bxy 2 +Cx 2 y+dy 3 ) 2 + 2r 6 xy 4 + r 5 y 5 (Ax 3 + Bxy 2 +Cx 2 y+dy 3 ) 2, f y (x,y) = r 2x 5 2r 1 x 4 y (3r 3 + r 4 )x 3 y 2 (Ax 3 + Bxy 2 +Cx 2 y+dy 3 ) 2 + 2r 6 x 2 y 3 r 5 xy 4 (Ax 3 + Bxy 2 +Cx 2 y+dy 3 ) The proof of 2) is similar will be omitted. Theorem 21 Assume that 2 2r 1 + r 2 + 3r 3 + r 4 + r 5 + 2r 6 (a+b+c+d)() < 1. Then the positive equilibrium point x = a+b+c+d (1) is locally asymptotically stable. of Eq. Proof. The linearized equation of Eq. (1) about x= a+b+c+d is x n+1 = px n + qx n 1 p= (2r 1+ r 2 + 3r 3 + r 4 + r 5 + 2r 6 ) (a+b+c+d)(), q= (2r 1+ r 2 + 3r 3 + r 4 + r 5 + 2r 6 ) (a+b+c+d)(). So, by theorem (11) we get that x is locally asymptotically stable if 2 2r 1 + r 2 + 3r 3 + r 4 + r 5 + 2r 6 (a+b+c+d)() < 1. In the following theorem we prove the permanent of the difference equation (1). Theorem 22 Let {x n } + equation (1). 1. Assume that a A max( b B, c C ), d D min( b B, c C ), Then, 2. Assume that a A min( b B, c C ), d D max( b B, c C ), n= 1 be a positive solution of d D x n a A

3 Math. Sci. Lett. 2, No. 3, (2013) / Proof. Then, 1. We have x n+1 a A = x n+1 d D = a A x n d D r 2 x 2 nx n 1 r 1 x n x 2 n 1 r 3x 3 n 1 A(Ax 3 n+ Bx n x 2 n 1 +Cx2 nx n 1 + Dx 3 n 1 ), r 3 x 3 n+ r 6 x 2 nx n 1 + r 5 x n x 2 n 1 D(Ax 3 n+ Bx n x 2 n 1 +Cx2 nx n 1 + Dx 3 n 1 ). Now using the fact that r 1,r 2,r 3,r 5,r 6 0, it follows that d D x n a A 2. The proof of 2) is similar will be omitted. Here we study the global asymptotic stability of Eq.(1). Theorem 23 Let p 1 = Da+Ab+(A+C)d, p 2 = (B+D)a+(A+C)b+(A D)c+(A+B+C)d, p 3 = (A B C D)a+(A+B+C D)b + (A B+C D)c+(A+B+C+ D)d. Assume that A a max( B b, C c), D d min( B b, C c), 3r 3 + r 4 0, 2(2r 1+r 2 +3r 3 +r 4 +r 5 +2r 6 ) (a+b+c+d)() < 1, p 1, p 2, p 3 0. Then the equilibrium point x = a+b+c+d globally asymptotically stable. of Eq.(1) is Proof. Let{x n } + n= 1 be a positive solution of equation (1). In view of theorem (21), we need only to prove that x is global attractor. Let m= lim in f x n To prove that M = lim supx n. lim x n = x, it suffices to show that m=m. Let ε ]0,m[, then there exist n 0 N such that for all n n 0 we get m ε x n M+ ε. Thus by using lemma (21), Part 1; we get for all n n 0 +1 x n+1 f(m ε,m+ ε), So, Hence x n+1 f(m+ ε,m ε). Then, we get the following inequalities These inequalities yields m f(m ε,m+ ε), M f(m+ ε,m ε). m am3 + bmm 2 + cm 2 M+ dm 3 Am 3 + BmM 2 +Cm 2 M+ DM 3, M am3 + bm 2 M+ cmm 2 + dm 3 AM 3 + Bm 2 M+CmM 2 + Dm 3. mm M(am3 + bmm 2 + cm 2 M+ dm 3 ) Am 3 + BmM 2 +Cm 2 M+ DM 3, mm m(am3 + bm 2 M+ cmm 2 + dm 3 ) AM 3 + Bm 2 M+CmM 2 + Dm 3. M(am 3 + bmm 2 + cm 2 M+ dm 3 ) Am 3 + BmM 2 +Cm 2 M+ DM 3 m(am3 + bm 2 M+ cmm 2 + dm 3 ) AM 3 + Bm 2 M+CmM 2 + Dm 3 0, which can be written (M m)r(m,m)=0, R(m,M)= L(m,M) K(m,M), L(m,M) = da(m 6 + M 6 ) + p 1 mm(m 4 + M 4 ) + p 2 m 2 M 2 (m 2 + M 2 )+ p 3 m 3 M 3, K(m,M) = (Am 3 + BmM 2 + Cm 2 M + DM 3 )(AM 3 + BmM 2 +Cm 2 M+ Dm 3 ). Since we get So, m=m = x. R(m,M)>0 M m. By the same arguments, we can easily prove the following theorem. Theorem 24 Let q 1 = (B+D)a+Dc Ad, q 2 = (B+C+ D)a+( A+D)b+(B+D)c (A+C)d, q 3 = (A+B+C+ D)a+( A+B C+D)b + ( A+B+C+ D)c+( A C B+D)d. Assume that

4 164 N. Touafek, Y. Halim: Global Attractivity of a Rational... A a min( B b, C c), D d max( B b, C c), 3r 3 + r 4 0, 2(2r 1+r 2 +3r 3 +r 4 +r 5 +2r 6 ) (a+b+c+d)() < 1, q 1, q 2, q 3 0. Then the equilibrium point x = a+b+c+d globally asymptotically stable. of Eq.(1) is Theorem 25 Assume that q 1, q 2, q 3 0. Then Eq.(1) has no positive prime period two solution. Proof. For the sake of contradiction, assume that there exist distinct positive real numbers α β, such that..., α, β, α, β,... is a period two solution of Eq.(1). Then, α, β satisfy the system α = f(β,α), β = f(α,β). Thus, we have which gives β f(β,α)=αf(α,β), (β α)s(α,β)=0 S(α,β)= F(α,β) K(α,β) F(α,β) = ad(α 6 + β 6 ) + q 1 αβ(α 4 + β 4 ) + q 2 α 2 β 2 (α 2 + β 2 )+q 3 α 3 β 3 ; K(α,β) = (Aβ 3 + Bβα 2 + Cβ 2 α + Dα 3 )(Aα 3 + Bαβ 2 +Cα 2 β + Dβ 3 ). Since S(α,β)>0. We get β = α, which is a contradiction. 3 Numerical examples In order to illustrate support our theoretical discussions, we consider interesting numerical examples in this section. Example 31 Let (a,b,c,d,a,b,c,d) = (1, 3 2,1, 1 2,1,2,2, 3 2 ) (x 1,x 0 )=(0.5,0.7). Then all conditions of theorem (23) are satisfied we have the following results: Example 32 Let (a,b,c,d,a,b,c,d) = (1,2,2, 2 3,1, 3 2,1, 2 1) (x 1,x 0 )=(1.5,2.7). Then all conditions of theorem (24) are satisfied we have the following results: References [1] C. Cinar, On the positive solutions of the difference equation system x n+1 = 1/y n, y n+1 = y n /x n 1 y n 1, Appl. Math. Comput., 158, (2004). [2] C. W. Clark, A delayed recruitment of a population dynamics with an application to baleen whale populations, J. Math. Biol., 3, (1976). [3] E. M. Elabbasy E. M. Elsayed, On the global attractivity of difference equation of higher order, Carpathian J. Math., 24, (2008). [4] E. M. Elsayed, On the solution of some difference equations, European J. Pure Appl. Math., 4, (2011). [5] E. M. Elsayed, On the dynamics of a higher order rational recursive sequence, Commun. Math. Anal., 12, (2012). [6] B. D. Iricǎnin N. Touafek, On a second order max-type system of difference equations, Indian J. Math., 54, (2012). [7] V. L. Kocic G. Ladas, Global bahavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht, (1993). [8] M. R. S. Kulenovic G. Ladas, Dynamics of second order rational difference equations with open problems conjectures, Chapman Hall, CRC Press, (2001). [9] A. S. Kurbanli, C. Cinar I. Yalçinkaya, On the behavior of positive solutions of the system of rational difference equations, Math. Comput. Modelling, 53, (2011),

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