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DOI: 10.1515/ms-2017-0130 Math. Slovaca 68 (2018), No. 3, 625–638
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ON THE SOLUTIONS OF A SECOND-ORDER DIFFERENCE EQUATION IN TERMS OF GENERALIZED PADOVAN SEQUENCES Yacine Halim* — Julius Fergy T. Rabago** (Communicated by Michal Feˇ ckan )
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ABSTRACT. This paper deals with the solution, stability character and asymptotic behavior of the rational difference equation αxn−1 + β , n ∈ N0 , xn+1 = γxn xn−1 where N0 = N ∪ {0}, α, β, γ ∈ R+ , and the initial conditions x−1 and x0 are non zero real numbers such that their solutions are associated to generalized Padovan numbers. Also, we investigate the two-dimensional case of the this equation given by αxn−1 + β αyn−1 + β xn+1 = , yn+1 = , n ∈ N0 . γyn xn−1 γxn yn−1
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2018 Mathematical Institute Slovak Academy of Sciences
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1. Introduction
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The term difference equation refers to a specific type of recurrence relation, a mathematical relationship expressing xn as some combination of xi with i < n. These equations usually appear as discrete mathematical models of many biological and environmental phenomena such as population growth and predator-prey interactions (see, e.g., [8] and [18]), and so these equations are studied because of their rich and complex dynamics. Recently, the problem of finding closedform solutions of rational difference equations and systems of rational of difference equations have gained considerable interest from many mathematicians. In fact, countless papers have been published previously focusing on the aforementioned topic, see for example [5–7, 16, 20]. Interestingly, some of the solution forms of these equations are even expressible in terms of well-known integer sequences such as the Fibonacci numbers, Horadam numbers and Padovan numbers (see, e.g., [9, 11, 12, 14, 22, 24–27, 33]). It is well-known that linear recurrences with constant coefficients, such as the recurrence relation Fn+1 = Fn + Fn−1 defining the Fibonacci numbers, can be solved through various techniques (see, e.g., [17]). Finding the closed-form solutions of nonlinear types of difference equations, however, are far more interesting and challenging compared to those of linear types. In fact, as far as we know, there has no known general method to deal with different classes of difference equations 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 39A10; Secondary 40A05. K e y w o r d s: difference equations, general solution, stability, generalized Padovan numbers. This work was supported by LMAM Laboratory, Jijel University.
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YACINE HALIM — JULIUS FERGY T. RABAGO
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solvable in closed-forms. Nevertheless, numerous studies have recently dealt with finding appropriate techniques in solving closed-form solutions of some systems of difference equations (see, e.g., [2, 5–7, 15, 23]). Motivated by these aforementioned works, we investigate the rational difference equation αxn−1 + β , n ∈ N0 , . (1.1) xn+1 = γxn xn−1 Particularly, we seek to find its closed-form solution and examine the global stability of its positive solutions. We establish the solution form of equation (1.1) using appropriate transformation reducing the equation into a linear type difference equation. Also, we examine the solution form of the two-dimensional analogue of equation (1.1) given in the following more general form αyn−1 + β αxn−1 + β , yn+1 = , n ∈ N0 . (1.2) xn+1 = γyn xn−1 γxn yn−1
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The case α = β = γ = 1 has been studied by Tollu, Yazlik and Taskara in [33]. The authors in [33] established the solution form of system (1.2) (in the case α = β = γ = 1) through induction principle. The paper is organized as follows. In the next section (Section 2), we review some definitions and important results necessary for the success of our study, and this includes a brief discussion about generalized Padovan numbers. In Section 3 and 4, we established the respective solution forms of equations (1.1) and the system (1.2), and examine their respective stability properties. Finally, we end our paper with a short summary in Section 5.
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2. Preliminaries
2.1. Linearized stability of an equation
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Let I be an interval of real numbers and let F : I k+1 −→ I
be a continuously differentiable function. Consider the difference equation xn+1 = F (xn , xn−1 , . . . , xn−k )
(2.1)
with initial values x0 , x−1 , . . . x−k ∈ I.. 1. A point x ∈ I is called an equilibrium point of equation(2.1) if
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Definition
x = F (x, x, . . . , x).
2. Let x be an equilibrium point of equation(2.1). i) The equilibrium x is called locally stable if for every ε > 0, there exist δ > 0 such that for allx−k , x−k+1 , . . . x0 ∈ I with
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Definition
|x−k − x| + |x−k+1 − x| + · · · + |x0 − x| < δ,
we have |xn − x| < ε, for all n ≥ −k. ii) The equilibrium x is called locally asymptotically stable if it is locally stable, and if there exists γ > 0 such that if x−1 , x0 ∈ I and |x−k − x| + |x−k+1 − x| + · · · + |x0 − x| < γ, then lim xn = x.
n→+∞
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ON THE SOLUTIONS OF A SECOND-ORDER DIFFERENCE EQUATION
iii) The equilibrium x is called global attractor if for all x−k , x−k+1 , . . . x0 ∈ I, we have lim xn = x.
n→+∞
iv) The equilibrium x is called global asymptotically stable if it is locally stable and a global attractor. v) The equilibrium x is called unstable if it is not stable. vi) Let pi =
∂f ∂ui (x, x, . . . , x),
i = 0, 1, . . . , k. Then, the equation (2.2)
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yn+1 = p0 yn + p1 yn−1 + · · · + pk yn−k , is called the linearized equation of equation (2.1) about the equilibrium point x.
The next result, which was given by Clark [3], provides a sufficient condition for the locally asymptotically stability of equation (2.1). Theorem
2.1 ([3]). Consider the difference equation (2.2). Let pi ∈ R, then, |p0 | + |p1 | + · · · + |pk | < 1
is a sufficient condition for the locally asymptotically stability of equation (2.1).
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2.2. Linearized stability of the second-order systems Let f and g be two continuously differentiable functions: f : I 2 × J 2 −→ I,
g : I 2 × J 2 −→ J,
I, J ⊆ R
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where I, J are some interval of real numbers. For n ∈ N0 , consider the system of difference equations ( xn+1 = f (xn , xn−1 , yn , yn−1 ) (2.3) yn+1 = g (xn , xn−1 , yn , yn−1 )
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where (x−1 , x0 ) ∈ I 2 and (y−1 , y0 ) ∈ J 2 . Define the map H : I 2 × J 2 −→ I 2 × J 2 by H(W ) = (f0 (W ), f1 (W ), g0 (W ), g1 (W ))
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where W = (u0 , u1 , v0 , v1 ) , f0 (W ) = f (W ), f1 (W ) = u0 , g0 (W ) = g(W ), g1 (W ) = v0 . T Let Wn = [xn , xn−1 , yn , yn−1 ] . Then, we can easily see that system (2.3) is equivalent to the following system written in vector form Wn+1 = H(Wn ), n = 0, 1, . . . ,
(2.4)
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that is
xn+1 = f (xn , xn−1 , yn , yn−1 ) x =x n n . yn+1 = g (xn , xn−1 , yn , yn−1 ) yn = yn
Definition 3 (Equilibrium point). An equilibrium point (x, y) ∈ I ×J of system (2.3) is a solution of the system ( x = f (x, x, y, y) ,
y = g (x, x, y, y) . Furthermore, an equilibrium point W ∈ I 2 × J 2 of system (2.4) is a solution of the system W = H(W ). 627
YACINE HALIM — JULIUS FERGY T. RABAGO Definition 4 (Stability). Let W be an equilibrium point of system (2.4) and k . k be any norm (e.g. the Euclidean norm).
(1) The equilibrium point W is called stable (or locally stable) if for every ε > 0 exist δ such that kW0 − W k < δ implies kWn − W k < ε for n ≥ 0. (2) The equilibrium point W is called asymptotically stable (or locally asymptotically stable) if it is stable and there exist γ > 0 such that kW0 − W k < γ implies kWn − W k → 0,
n → +∞.
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(3) The equilibrium point W is said to be global attractor (respectively global attractor with basin of attraction a set G ⊆ I 2 × J 2 , if for every W0 (respectively for every W0 ∈ G) kWn − W k → 0,
n → +∞.
(4) The equilibrium point W is called globally asymptotically stable (respectively globally asymptotically stable relative to G) if it is asymptotically stable, and if for every W0 (respectively for every W0 ∈ G), kWn − W k → 0, n → +∞.
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(5) The equilibrium point W is called unstable if it is not stable.
Remark 1. Clearly, (x, y) ∈ I × J is an equilibrium point for system (2.3) if and only if W = (x, x, , y, y, ) ∈ I 2 × J 2 is an equilibrium point of system (2.4).
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From here on, by the stability of the equilibrium points of system (2.3), we mean the stability of the corresponding equilibrium points of the equivalent system (2.4).
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2.3. Generalized Padovan sequence
The integer sequence defined by the recurrence relation Pn+1 = Pn−1 + Pn−2 ,
n ∈ N,
(2.5)
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with the initial conditions P−2 = 1, P−1 = 0, P0 = 1 (so P0 = P1 = P2 = 1), is known as the Padovan numbers and was named after Richard Padovan. This is the same recurrence relation as for the Perrin sequence, but with different initial conditions (P0 = 3, P1 = 0, P2 = 2). The first few terms of the recurrence sequence are 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, . . . . The Binet’s formula for this recurrence sequence can easily be obtained and is given by
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(ρ − 1)(ρ − 1) n (σ − 1)(ρ − 1) n (σ − 1)(ρ − 1) n σ + ρ + ρ . (σ − ρ)(σ − ρ) (ρ − σ)(ρ − ρ) (σ − ρ)(ρ − ρ) p √ √ 2 3 +12 (the so-called plastic number), ρ = − σ2 + i 23 6r − 2r and r = 108 + 12 69. where σ = r 6r √ The plastic number corresponds to the golden number 1+2 5 associated with the equiangular spiral related to the conjoined squares in Fibonacci numbers, that is,
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Pn =
lim
n→∞
Pn+1 = σ. Pn
For more informations associated with Padovan sequence, see [4] and [19]. Here we define an extension of the Padovan sequence in the following way S−2 = 0,
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S−1 = 0,
S0 = 1,
Sn+1 = pSn−1 + qSn−2 ,
n ∈ N.
(2.6)
ON THE SOLUTIONS OF A SECOND-ORDER DIFFERENCE EQUATION
The Binet’s formula for this recurrence sequence is given by (ϕ − 1)(ϕ − 1) n (φ − 1)(ϕ − 1) n (φ − 1)(ϕ − 1) n φ + ϕ + ϕ . (φ − ϕ)(φ − ϕ) (ϕ − φ)(ϕ − ϕ) (φ − ϕ)(ϕ − ϕ) q √ p φ 2p R2 +12p 3 R where φ = 6R , ϕ = − 2 + i 2 6 − R and R = 3 108q + 12 −12p3 + 81q 2 . One can easily verify that Sn+1 lim = φ. n→∞ Sn
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Sn =
3. Closed-form solutions and stability of equation (1.1)
For the rest of our discussion we assume Sn , the n-th generalized Padovan number, to satisfy the recurrence equation Sn+1 = pSn−1 + qSn−2 , n ∈ N0 ,
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with initial conditions S−2 = 0, S−1 = 0, S0 = 1.
3.1. Closed-form solutions of equation (1.1)
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In this section, we derive the solution form of equation (1.1) through an analytical approach. β α We put q = and p = , hence we have the equation γ γ pxn−1 + q ; xn xn−1
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xn+1 =
n ∈ N0 .
(3.1)
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Consider the equivalent form of equation (3.1) given by q p + xn+1 = xn xn xn−1 which, upon the change of variable xn+1 = zn+1 /zn , transforms into zn+1 = pzn−1 + qzn−2 .
(3.2)
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Now, we iterate the right hand side of equation (3.2) as follows zn+1 = pzn−1 + qzn−2 = qzn−2 + p2 zn−3 + qpzn−4 = p2 zn−3 + 2pqzn−4 + q 2 zn−5 = 2pqzn−4 + (p3 + q 2 )zn−5 + qp2 zn−6 = (p3 + q 2 )zn−5 + 3p2 qzn−6 + 2pq 2 zn−7 = 3p2 qzn−6 + (p4 + 3pq 2 )zn−7 + (p3 + q 3 )zn−8 = (p4 + 3pq 2 )zn−7 + (q 3 + 4qp3 )zn−8 + 3p2 q 2 zn−9 .. . = Sn+1 z0 + Sn+2 z−1 + Sn qz−2 . 629
YACINE HALIM — JULIUS FERGY T. RABAGO
Hence, zn+1 Sn+1 z0 + Sn+2 z−1 + Sn qz−2 = zn Sn z0 + Sn+1 z−1 + Sn−1 qz−2 . =
= =
−2 0 Sn+1 zz−1 + Sn+2 + Sn−2 q zz−1 −2 0 + Sn+1 + Sn−1 q zz−1 Sn zz−1
1 Sn+1 x0 + Sn+2 + Sn q x−1 1 Sn x0 + Sn+1 + Sn−1 q x−1
Sn+1 x0 x−1 + Sn+2 x−1 + Sn q . Sn x0 x−1 + Sn+1 x−1 + Sn−1 q
The above computations prove the following result. Theorem
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xn+1 =
3.1. Let {xn }n≥−1 be a solution of (3.1). Then, for n = 1, 2, . . . , xn =
Sn+1 x−1 + Sn x0 x−1 + qSn−1 . Sn x−1 + Sn−1 x0 x−1 + qSn−2
(3.3)
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where the initial conditions x−1 , x0 ∈ R − F , with F is the Forbidden Set of equation (3.1) given by ∞ n o [ F = (x−1 , x0 ) : Sn x−1 + Sn−1 x0 x−1 + qSn−2 = 0 .
R
n=−1
If α = β = γ, then from (3.3) we get
Pn+1 x−1 + Pn x0 x−1 + qPn−1 . Pn x−1 + Pn−1 x0 x−1 + qPn−2
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xn =
Hence, for α = β = γ we have Sn = Pn , n ∈ N, and consequently we get the solution given in [33].
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3.2. Global stability of solutions of equation (1.1)
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In this section we study the global stability character of the solutions of equation (3.1). It is easy to show that (3.1) has a unique positive equilibrium point given by x = φ. Let I = (0, +∞), and consider the function f : I 2 −→ I defined by py + q . f (x, y) = xy Theorem
3.2. The equilibrium point x is locally asymptotically stable.
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P r o o f. The linearized equation of equation (3.1) about the equilibrium x is yn+1 = t1 yn + t2 yn−1
where
t1 =
∂f pR2 + 12p2 + 6qR (x, x) = − 3 ∂x R6 + pR2 + 12p2 + 48p2 R
and t2 =
∂f 6qR (x, x) = − 6 2 ∂y R + pR + 12p2 +
and the characteristic polynomial is λ2 + t1 λ + t2 = 0. 630
48p3 R2
ON THE SOLUTIONS OF A SECOND-ORDER DIFFERENCE EQUATION
Consider the two functions defined by a(λ) = λ2 ,
b(λ) = −(t1 λ + t2 ).
We have pR2 + 12p2 + 12qR < 1. R6 + pR2 + 12p2 + 48p23 R
|b(λ)| < |a(λ)| ,
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Then for all λ : |λ| = 1
Thus, by Rouche’s Theorem, all zeros of P (λ) = a(λ) − b(λ) = 0 lie in |λ| < 1. So, by Theorem 2.1 we get that x is locally asymptotically stable. Theorem
3.3. The equilibrium point x is globally asymptotically stable.
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P r o o f. Let {xn }n≥−k be a solution of equation (3.1). By Theorem (3.2) we need only to prove that x is global attractor, that is lim xn = φ.
n→∞
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it follows from Theorem 3.1 that
Sn+1 x−1 + Sn x0 x−1 + qSn−1 Sn x−1 + Sn−1 x0 x−1 + qSn−2 Sn−1 x + x x + q Sn SSn+1 −1 0 −1 Sn n = lim Sn−2 Sn−1 n→∞ Sn x−1 + Sn x0 x−1 + q Sn
lim xn = lim
n→∞
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n→∞
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Sn+1 Sn x−1
+ x0 x−1 + q SSn−1 n n→∞ ( q1 Sn+1 − pq Sn−1 ) Sn−1 x−1 + Sn x0 x−1 + q Sn φ x−1 + φ1 x0 x−1 + q φ1 = lim n→∞ x−1 + 1 x0 x−1 + φ + p φ φ
= lim
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Then
lim xn = φ.
n→∞
Example 1. For confirming results of this section, we consider the following numerical example. Let α = 2, β = 5 and γ = 4 in (1.1), then we obtain the equation xn+1 =
2xn−1 + 5 . 4xn xn−1
(3.4)
Assume x−1 = 3 and x0 = 0.2, (see Figure 1). 631
YACINE HALIM — JULIUS FERGY T. RABAGO
5 4.5 4
3 2.5 2 1.5 1 0.5
0
10
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30
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50
60
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n
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x(n)
3.5
Figure 1. This figure shows that the solution of the equation (3.4) is global attractor, that is, lim xn = φ.
n→∞
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4. Closed-form and stability of solutions of system (1.2)
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4.1. Closed-form solutions of system (1.2) In this section, we derive the respective solution form of system (1.2). We put q = pxn−1 + q , yn xn−1
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Hence, we have the system
xn+1 =
yn+1 =
pyn−1 + q , xn yn−1
n ∈ N0
α β and p = . γ γ (4.1)
The following theorem describes the form of the solutions of system (4.1). 4.1. Let {xn , yn }n≥−1 be a solution of (4.1). Then for n = 1, 2, . . . , S n+1 y−1 + Sn x0 y−1 + qSn−1 Sn y−1 + Sn−1 x0 y−1 + qSn−2 , if n is even, (4.2) xn = Sn+1 x−1 + Sn y0 x−1 + qSn−1 , if n is odd, Sn x−1 + Sn−1 y0 x−1 + qSn−2 S n+1 x−1 + Sn y0 x−1 + qSn−1 Sn x−1 + Sn−1 y0 x−1 + qSn−2 , if n is even, (4.3) yn = Sn+1 y−1 + Sn x0 y−1 + qSn−1 , if n is odd, Sn y−1 + Sn−1 x0 y−1 + qSn−2 where the initial conditions x−1 , x0 , y−1 and y0 ∈ R r (F1 ∪ F2 ), with F1 and F2 are the forbidden sets of system (4.1) given by ∞ n o [ F1 = (x−1 , x0 , y−1 , y0 ) : Sn x−1 + Sn−1 y0 x−1 + qSn−2 = 0 ,
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Theorem
n=−1
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ON THE SOLUTIONS OF A SECOND-ORDER DIFFERENCE EQUATION
and F2 =
∞ n o [ (x−1 , x0 , y−1 , y0 ) : Sn y−1 + Sn−1 x0 y−1 + qSn−2 = 0 . n=−1
P r o o f. The closed-form solution of (4.1) can be established through a similar approach we used in proving the one-dimensional case. However, for convenience, we shall prove the theorem by induction. For the basis step, we have px−1 + q y0 x−1
and y1 =
py−1 + q , x0 y−1
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x1 =
so the result clearly holds for n = 0. Suppose that n > 0 and that our assumption holds for n − 1. That is, S2n−1 y−1 + S2n−2 x0 y−1 + qS2n−3 , S2n−2 y−1 + S2n−3 x0 y−1 + qS2n−4
x2n−1 =
S2n x−1 + S2n−1 y0 x−1 + qS2n−2 , S2n−1 x−1 + S2n−2 y0 x−1 + qS2n−3
y2n−2 =
S2n−1 x−1 + S2n−2 y0 c−1 + qS2n−3 , S2n−2 x−1 + S2n−3 y0 x−1 + qS2n−4
y2n−1 =
S2n y−1 + S2n−1 x0 y−1 + qS2n−2 . S2n−1 y−1 + S2n−2 x0 y−1 + qS2n−3
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x2n−2 =
x2n =
px2n−2 + q y2n−1 x2n−2
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Now it follows from system (4.1) that
S2n−1 y−1 + S2n−2 x0 y−1 + qS2n−3 +q S2n−2 y−1 + S2n−3 x0 y−1 + qS2n−4 = S2n y−1 + S2n−1 x0 y−1 + qS2n−2 S2n−1 y−1 + S2n−2 x0 y−1 + qS2n−3 S2n−1 y−1 + S2n−2 x0 y−1 + qS2n−3 S2n−2 y−1 + S2n−3 x0 y−1 + qS2n−4
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p
p(S2n−1 y−1 + S2n−2 x0 y−1 + qS2n−3 ) + q(S2n−2 y−1 + S2n−3 x0 y−1 + qS2n−4 ) S2n y−1 + S2n−1 x0 y−1 + qS2n−2
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=
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So, we have
x2n =
S2n+1 y−1 + S2n x0 y−1 + qS2n−1 . S2n y−1 + S2n−1 x0 y−1 + qS2n−2
Also it follows from system (4.1) that y2n =
py2n−2 + q x2n−1 y2n−2
S2n−1 x−1 + S2n−2 y0 x−1 + qS2n−3 +q S2n−2 x−1 + S2n−3 y0 x−1 + qS2n−4 = S2n x−1 + S2n−1 y0 x−1 + qS2n−2 S2n−1 x−1 + S2n−2 y0 c−1 + qS2n−3 S2n−1 x−1 + S2n−2 y0 x−1 + qS2n−3 S2n−2 x−1 + S2n−3 y0 x−1 + qS2n−4 p
=
p(S2n−1 x−1 + S2n−2 y0 x−1 + qS2n−3 ) + q(S2n−2 x−1 + S2n−3 y0 x−1 + qS2n−4 ) . S2n x−1 + S2n−1 y0 x−1 + qS2n−2 633
YACINE HALIM — JULIUS FERGY T. RABAGO
Hence, we have y2n =
S2n+1 x−1 + S2n y0 c−1 + qS2n−1 . S2n x−1 + S2n−1 y0 x−1 + qS2n−2
Using the same argument it follows from system (4.1) that x2n+1 =
px2n−1 + q y2n x2n−1
S2n x−1 + S2n−1 y0 x−1 + qS2n−2 +q S2n−1 x−1 + S2n−2 y0 x−1 + qS2n−3 = S2n+1 x−1 + S2n y0 x−1 + qS2n−1 S2n x−1 + S2n−1 y0 x−1 + qS2n−2 S2n x−1 + S2n−1 y0 x−1 + qS2n−2 S2n−1 x−1 + S2n−2 y0 x−1 + qS2n−3 =
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p
p(S2n x−1 + S2n−1 y0 x−1 + qS2n−2 ) + q(S2n−1 x−1 + S2n−2 y0 x−1 + qS2n−3 ) . S2n+1 x−1 + S2n y0 x−1 + qS2n−1
This yields x2n+1 =
S2n+2 x−1 + S2n+1 y0 x−1 + qS2n . S2n+1 x−1 + S2n y0 c−1 + qS2n−1
y2n+1 =
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Moreover, we have py2n−1 + q x2n y2n−1
S2n y−1 + S2n−1 x0 y−1 + qS2n−2 +q S2n−1 y−1 + S2n−2 x0 y−1 + qS2n−3 = S2n+1 y−1 + S2n x0 y−1 + qS2n−1 S2n y−1 + S2n−1 x0 y−1 + qS2n−2 S2n y−1 + S2n−1 x0 y−1 + qS2n−2 S2n−1 y−1 + S2n−2 x0 y−1 + qS2n−3
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p(S2n y−1 + S2n−1 x0 y−1 + qS2n−2 ) + q(S2n−1 y−1 + S2n−2 x0 y−1 + qS2n−3 ) , S2n+1 y−1 + S2n x0 y−1 + qS2n−1
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=
R
p
and this implies that
y2n+1 =
S2n+2 y−1 + S2n+1 x0 y−1 + qS2n . S2n+1 y−1 + S2n x0 y−1 + qS2n−1
This completes the proof of the theorem.
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4.2. Global attractor of solutions of system (1.2)
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Our aim in this section is to study the asymptotic behavior of positive solutions of system (4.1). Let I = J = (0, +∞), and consider the functions f : I 2 × J 2 −→ I
and
g : I 2 × J 2 −→ J
pu1 + q v0 u1
and
g(u0 , u1 , v0 , v1 ) =
defined by f (u0 , u1 , v0 , v1 ) =
pv1 + q , u 0 v1
respectively. Lemma
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4.1. System (3.1) has a unique equilibrium point in I × J, namely 2 R + 12p R2 + 12p , . E= 6R 6R
ON THE SOLUTIONS OF A SECOND-ORDER DIFFERENCE EQUATION
P r o o f. Clearly the system py + q px + q , , y= yx xy has a unique solution in I 2 × J 2 which is 2 R + 12p R2 + 12p , . E= 6R 6R x=
4.2. The equilibrium point E is global attractor.
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Theorem
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P r o o f. Let {xn , yn }n≥0 be a solution of system (3.1). Let n → ∞ in Theorem 4.1. That is, we have Sn+1 y−1 + Sn x0 y−1 + qSn−1 lim x2n = lim n→∞ n→∞ Sn y−1 + Sn−1 x0 y−1 + qSn−2 Sn−1 y + x y + q Sn SSn+1 −1 0 −1 Sn n = lim Sn−2 Sn−1 n→∞ Sn y−1 + Sn x0 y−1 + q Sn Sn+1 Sn y−1
+ x0 y−1 + q SSn−1 n n→∞ ( q1 Sn+1 − pq Sn−1 ) Sn−1 y−1 + Sn x0 y−1 + q Sn φ y−1 + φ1 x0 y−1 + q φ1 = φ. = y−1 + φ1 x0 y−1 + φ + φp and
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Sn+1 x−1 + Sn y0 x−1 + qSn−1 Sn x−1 + Sn−1 y0 x−1 + qSn−2 Sn−1 x + y x + q Sn SSn+1 −1 0 −1 Sn n = lim Sn−2 Sn−1 n→∞ Sn x−1 + Sn y0 x−1 + q Sn
lim x2n+1 = lim
n→∞
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n→∞
Sn+1 Sn x−1
+ y0 x−1 + q SSn−1 n n→∞ ( q1 Sn+1 − pq Sn−1 ) Sn−1 x−1 + Sn y0 x−1 + q Sn φ x−1 + φ1 y0 x−1 + q φ1 = = φ. x−1 + φ1 y0 x−1 + φ + φp
= lim
Then lim xn = φ. Similarly, we obtain lim yn = φ. Thus, we have n→∞
n→∞
lim (xn , yn ) = E.
n→∞
Example 2. As an illustration of our results, we consider the following numerical example. Let α = 2, β = 3 and γ = 5 in system (1.2), then we obtain the system 2yn−1 + 3 2xn−1 + 3 , yn+1 = , n ∈ N0 (4.4) xn+1 = 5yn xn−1 5xn yn−1 635
YACINE HALIM — JULIUS FERGY T. RABAGO
Assume x−1 = 1.2, x0 = 3.6, y−1 = 2.3 and y0 = 0.8. (See Figure 2).
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x(n), y(n)
2.5
2
1.5
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0.5
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Figure 2. This figure shows that the solution of the system (4.4) is global attractor, that is lim xn = E.
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5. Summary and recommendations
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In this work, we have successfully established the closed-form solution of the rational difference equation αxn−1 + β xn+1 = γxn xn−1
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as well as the closed-form solutions of its corresponding two-dimensional case xn+1 =
αxn−1 + β , γyn xn−1
yn+1 =
αyn−1 + β . γxn yn−1
Also, we obtained stability results for the positive solutions of these systems. Particularly, we have shown that the positive solutions of each of these equations tends to a computable finite number, and is in fact expressible in terms of the well-known plastic number. Meanwhile, for future investigation, one could also derive the closed-form solution and examine the stability of solutions of the system xn+1 =
αxn−1 − β , γyn xn−1
This work we leave to the interested readers. 636
yn+1 =
αyn−1 ± β . γxn yn−1
ON THE SOLUTIONS OF A SECOND-ORDER DIFFERENCE EQUATION REFERENCES
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* Department of Mathematics and Computer Sciences Mila University Center BP26 RP Mila 43000 ALGERIA E-mail:
[email protected]
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** Department of Mathematics and Computer Sciences College of Science, University of the Philippines Gov. Pack Road, Baguio City 2600, Benguet PHILIPPINES E-mail:
[email protected]
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