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On the Asymptotic Behavior of Some Difference Equations by Using Invariants A. Aghajani1 , Z. Shouli
School of Mathematics, Iran University of Science and Technology, Tehran, Iran.
Abstract In this paper, we consider some classes of difference equations and investigate the asymptotic behavior of solutions by using invariants. Keywords: Difference Equation; Invariant; Boundedness; Forbidden set. c 2010 Published by Islamic Azad University-Karaj Branch.
1
Introduction
Difference equations have many applications in biology, economy, and other sciences. So it is important to study the properties of these equations and asymptotic behavior of their solutions. If for a difference equation no stability theorem applies, it is necessary to examine this difference equation directly. For many equations, however, it is neither obvious whether solutions are bounded or stable, nor it is trivial to prove such behavior. A useful way to prove boundedness is to find the invariant of difference equation. This theory is based on the existence of expressions that remain constant or invariant along solutions of a difference equation and which reveal the behavior of the solutions of the considered equation. The invariants of difference equations play an important role in understanding the qualitative behavior of their solutions. 1
Corresponding Author. E-mail Address:
[email protected]
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The invariant is not uniquely determined, but rather if one invariant is given, then any appropriate function (continuous, analytic) of this invariant is invariant itself. Furthermore, the difference equation can have several ”independent” invariants (e.g. two invariants neither of which can be derived from the other through homeomorphic changes of variable). For more details see [1-5] and the references cited therein. In section 2, we recall some definitions and preliminaries needed. In section 3, We consider a class of difference equations and find the corresponding invariant to investigate the boundedness of solutions. Finally, in section 4, we consider a difference equation and completely determine the asymptotic behavior of solutions.
2
Preliminaries
Let f : Dk → D be a continuous function for D ⊂ R . The equation xn+1 = f (xn , xn−1 , ..., xn−k+1 ),
n = 0, 1, ...,
(1)
is said a difference equation or recursive sequence of order k that for every set of initial values x−k+1 , x−k+2 , ..., x−1 , x0 ∈ D , equation (1) has a unique solution {xn }∞ n=−k+1 . Definition 2.1 A point x ¯ ∈ D is an equilibrium point for difference equation (1) or a fixed point for map f if x ¯ = f (¯ x, ..., x ¯). Definition 2.2 A non-constant continuous function I : Rk → R is an invariant for difference equation (1) if In+1 := I(xn+1 , xn , ..., xn−k+2 ) = I(xn , ..., xn−k+1 ) =: In for every solution {xn }∞ n=−k+1 and all n ≥ 0. Definition 2.3 The forbidden set of difference equation (1) is the set F of all initial conditions (x−k+1 , x−k+2 , ..., x−1 , x0 ) through which the equation (1) is not well defined for all n ≥ 0 . Hence the solution {xn }∞ n=−k+1 of (1) exists for all n ≥ 0 , if and only if, (x−k+1 , x−k+2 , ..., x−1 , x0 ) ∈ / F.
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In the following, we mention two examples of invariants from the literature: Consider the Lyness equation of order k + 2(see[1]): xn+1 =
a + xn + xn−1 + · · · + xn−k , xn−k−1
n = 0, 1, ...,
where a > 0 is a parameter and initial values are positive. This equation possesses the following invariant: 1 I(xn , xn−1 , ..., xn−k−1 ) = 1+ xn
1+
1
xn−1
1
· · · 1+
xn−k−1
(a+xn +xn−1 +· · · xn−k−1 ),
which implies that every solution is bounded from above and from below by positive constants. Also, consider the generalized Lyness equation (see[1,4]): xn+1 =
axn + b , (cxn + d)xn−1
n = 0, 1, ...,
where a, b, c, d are positive parameters and initial values are positive. This equation possesses the following invariant:
I(xn , xn−1 ) = (b + axn + axn−1 + dxn xn−1 ) c +
d d a , + + xn xn−1 xn xn−1
which implies that every solution is bounded and persists. Now, we will consider some difference equations and find their invariants to investigate the asymptotic behavior of solutions of these equations.
3
A Class of Difference Equations
In this section, we consider the difference equation: xn G(xn−1 , ..., xn−k ) + axn − H1 (xn ) + k−1 i=1 [Hi (xn−i ) − Hi+1 (xn−i )] + Hk (xn−k ) , G(xn , ..., xn−k+1 ) + a (2) P
xn+1 =
for n = 0, 1, ..., where a > 0 is a parameter, initial values are positive and functions G, Hi satisfy the following conditions:
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i) G ∈ C((0, +∞)k , [0, +∞)), ii) Hi ∈ C((0, +∞), [0, +∞)), for i = 1, ..., k iii) ax ≥ H1 (x) ≥ H2 (x) ≥ · · · ≥ Hk (x) ≥ 0, for every x ∈ (0, +∞). Now, we prove the main theorem of this section. Theorem 3.1 Equation (2) possesses the invariant I(xn , ..., xn−k ) = xn (G(xn−1 , ..., xn−k ) + a) +
k X
Hi (xn−i ),
(3)
i=1
and every solution with positive initial values is bounded. Proof. From (3) we have In+1 = I(xn+1 , xn , ..., xn−k+1 ) = xn+1 (G(xn , ..., xn−k+1 ) + a) +
k X
Hi (xn−i+1 )
i=1
xn G(xn−1 , ..., xn−k ) + axn − H1 (xn ) + k−1 i=1 [Hi (xn−i ) − Hi+1 (xn−i )] + Hk (xn−k ) = ( ) G(xn , ..., xn−k+1 ) + a P
× (G(xn , ..., xn−k+1 ) + a) +
k X
Hi (xn−i+1 )
i=1
= xn G(xn−1 , ..., xn−k ) + axn + = xn (G(xn−1 , ..., xn−k ) + a) +
k X
i=1 k X
Hi (xn−i ) −
k−1 X
Hi+1 (xn−i ) +
i=0
k X
Hi (xn−i+1 )
i=1
Hi (xn−i )
i=1
= I(xn , ..., xn−k ) = In , Since In+1 = In , I is an invariant of difference equation (2). To prove that every solution of equation (2) is bounded, suppose that there is a solution {xn }∞ n=−k of equation (2) which is unbounded. Then there exist a subsequence {xni } tends to ∞ , so I(xni , xni −1 , ..., xni −k ) → ∞ which contradicts being constant for all n ≥ 0. Therefore every solution of equation (2) is bounded. Remark. Every point in R is an equilibrium point of equation (2). To see this, notice
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that: x ¯G(¯ x, ..., x ¯) + a¯ x − H1 (¯ x) + k−1 x) − Hi+1 (¯ x)] + Hk (¯ x) i=1 [Hi (¯ G(¯ x, ..., x ¯) + a x ¯(G(¯ x, ..., x ¯) + a) =x ¯. G(¯ x, ..., x ¯) + a P
=
Now, we shall give some applications of theorem 3.1. Example 3.2 Consider the following equation: xn+1 =
xn x2n−1 + xn−2 , 1 + x2n
n = 0, 1, ...
(4)
If we let k = 2, a = 1, G(x, y) = x2 , H1 (x) = H2 (x) = x then equation (2) reduces to the equation (4). So by theorem 3.1, equation (4) possesses the invariant I(xn , xn−1 , xn−2 ) = xn (x2n−1 + 1) + xn−1 + xn−2 , and every solution of (4) is bounded. Example 3.3 Consider the following equation: xn+1 =
xn x2n−1 + β0 xn + β1 xn−1 + · · · + βk−1 xn−k+1 + βk xn−k , n = 0, 1, ..., x2n + a
where a > 0, βi ≥ 0 for i = 0, ..., k and a ≥ G(x1 , ..., xk ) = x21 , Hj (x) = (a −
Pj−1 i=0
Pj
i=0 βi
(5)
for j = 0, ..., k − 1. If we let
βi )x for j = 1, ..., k then equation (2) reduces to
the equation (5). So by theorem 3.1, equation (5) possesses the invariant I(xn , ..., xn−k ) = xn (x2n−1 + a) +
k X j=1
a−
j−1 X
βi xn−j ,
i=0
and every solution of (5) is bounded. Example 3.4 Consider the following equation: xn+1 =
x2n + xn xn−1 · · · xn−k+1 x2n−k , xn−k + xn xn−1 · · · xn−k
n = 0, 1, ...,
(6)
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If we let a = 1, G(x1 , ..., xk ) = 1/(x1 x2 ...xk ) , and Hi (x) = x for i = 1, ..., k then equation (2) reduces to the equation (6). So by theorem 3.1, equation (6) possesses the invariant I(xn , ..., xn−k ) =
xn + xn + xn−1 + · · · + xn−k , xn−1 · · · xn−k
(7)
and every solution of (6) is bounded. Now we find a bound for every solution of (6): Consider a positive solution {xn }∞ n=−k of (6). Let λ = I(x0 , ..., x−k ), then by (7) for every n ≥ 0 , we have λ = I(xn , ..., xn−k ) = s
≥ (k + 2)
k+2
xn + xn + xn−1 + · · · + xn−k xn−1 · · · xn−k
xn xn · · · xn−k xn−1 · · · xn−k
q
= (k + 2) k+2 x2n , so
xn ≤
λ k+2
k+2 2
for every n ≥ 0.
Hence, for every n ≥ −k , we have xn < max{x−k , ..., x0 , (λ/(k + 2))(k+2)/2 }.
4
The Difference Equation xn+1 =
x2n +x2n−1 −xn (xn−1 +xn−2 ) xn−1 −xn−2
In this section, we consider the difference equation: xn+1 =
x2n + x2n−1 − xn (xn−1 + xn−2 ) , xn−1 − xn−2
n = 0, 1, ...,
(8)
and investigate some properties of this equation. First we determine the forbidden set of the above equation. Theorem 4.1 The forbidden set for equation (8) is F = {(x−2 , x−1 , x0 ) : x−2 = x−1 or x−1 = x0 }.
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Proof. Clearly, if x−2 = x−1 or x−1 = x0 then equation (8) implies that such a solution does not exist. If x−2 6= x−1 and x−1 6= x0 , then xn−1 −xn−2 6= 0 for all n ≥ 0. Because from equation (8), we have xn+1 − xn =
(xn − xn−1 )2 , xn−1 − xn−2
n = 0, 1, . . . .
It is easy to see that equation (8) has the following invariant I(xn , xn−1 , xn−2 ) =
xn − xn−2 , xn − xn−1
(9)
which is useful for determine the character of the solutions of equation (8). So we have the following theorem: Theorem 4.2 Consider the equation (8) with initial values x−2 , x−1 , x0 such that x−2 6= x−1 and x−1 6= x0 . Let c = (x0 − x−2 )/(x0 − x−1 ) . Then we have the following statements: (a) If c = 0 , then {xn }∞ n=−2 is periodic of period 2. (b) If 0 < c < 1 or 1 < c ≤ 2, then {xn }∞ n=−2 is unbounded. (c) If c < 0 or c > 2 , then {xn }∞ n=−2 converges to ((c − 1)x−1 − x−2 )/(c − 2). Proof. If c = 0 then x0 = x−2 and by using the equation (8), we find that x1 = x−1 . So by induction, we obtain xn+1 = xn−1 for all n ≥ 0. Therefore {xn }∞ n=−2 is periodic of period 2. By (9), since I is constant along the solution, we have xn − xn−2 x0 − x−2 = = c, xn − xn−1 x0 − x−1
for all n ≥ 0,
which implies that xn =
c 1 xn−1 − xn−2 , c−1 c−1
for all n ≥ 0,
(10)
and note that c 6= 1 because x−2 6= x−1 . Equation (10) is a linear difference equation with constant coefficients and its characteristic equation is λ2 −
1 c λ+ = 0, c−1 c−1
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which has the following roots λ1 = 1, λ2 =
1 . c−1
So xn = αλn1 + βλn2 = α +
β , (c − 1)n
for c 6= 2 and n = 0, 1, ...,
(11)
and xn = αλn + βnλn = α + nβ,
for c = 2 and n = 0, 1, ..., (12)
where α, β are constants. From (10), (11) and (12), we obtain: xn =
x−2 − x−1 1 ((c − 1)x−1 − x−2 ) + , c−2 (c − 2)(c − 1)n+1
for c 6= 2 and n = 0, 1, ..., (13)
and xn = 2x−1 − x−2 + n(x−1 − x−2 ),
for c = 2 and n = 0, 1, .... (14)
If 0 < c < 1 or 1 < c < 2 , by (13), we have: xn → ∞ as n → ∞. If c = 2 , by (14), we also get: xn → ∞ as n → ∞. Thus, for 0 < c < 1 or 1 < c ≤ 2 , {xn }∞ n=−2 is unbounded. If c < 0 or c > 2 , by (13), we obtain: xn →
1 ((c − 1)x−1 − x−2 ) as n → ∞. c−2
So, for c < 0 or c > 2, {xn }∞ n=−2 converges to ((c − 1)x−1 − x−2 )/(c − 2).
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Conclusion
By using invariants we established some results on the asymptotic behavior of positive solutions of a large class of difference equations. Also we included some example to show how we can apply our results.
References [1] Kocic V.L., Ladas G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. [2] Kulenovic M.R.S., Merino O., Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman and Hall/CRC Press, Boca Raton, 2002. [3] Nesemann T. (2001) ”Invariants and liapunov functions for nonautonomous systems,” Comput. Math. Appl., 42, 385-392. [4] Schinas C.J. (1997) ”Invariants for some difference equations,” J. Math. Anal. Appl., 212, 281-291. [5] Schinas C.J. (1997) ”Invariants for difference equations and systems of difference equations of rational form,” J. Math. Anal. Appl., 216, 164-179.
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