Applied Mathematics E-Notes, 1(2001), 31-33 \begin{matrix}c \\ ◯ \end{matrix} Available free at mirror sites of http://math2.math.nthu.edu.tw/∼amen/

Common Domain of Asymptotic Stability of a Family of Difference Equations ^{∗}

Yi-Zhong Lin^{†}

Received 26 August 2000

Abstract

A necessary and sufficient condition is obtained for each difference equation in a family to be asymptotically stable.

The following difference equation (see e.g. [1,2] for its importance)

u_{n} = au_{n−τ} + bu_{n−σ}, n = 0, 1, 2, ...  (1)

where a, b are nontrivial real numbers and τ, σ are positive integers, is said to be (globally) asymptotically stable if each of its solutions tends to zero.

When the delays τ and σ are given, whether the corresponding equation (1) is asymptotically stable clearly depends on the coefficients a and b. For this reason, we denote the set of all pairs (x, y) such that the equation

u_{n} = xu_{n−τ} + yu_{n−σ}, n = 0, 1, 2, ...  (2)

is asymptotically stable by D(x, y|τ, σ). It is well known that equation (1) is asymptotically stable if, and only if, all the roots of its characteristic equation

λ^{n} = aλ^{n−τ} + bλ^{n−σ},

are inside the open unit disk [3]. Since the latter statement holds if, and only if, all the roots of the equation

1 = aλ^{−τ} + bλ^{−σ}  (3)

are inside the open unit disk, the set D(x, y|τ, σ) is also the set of pairs (x, y) such that all the roots of

1 = xλ^{−τ} + bλ^{−σ}  (4)

has magnitude less than one.

By means of commercial software such as the MATLAB, it is not difficult to generate domains D(x, y|τ, σ) in the x, y-plane for different values of the delays τ and σ. It is interesting to observe that the set {(x, y)| |x| + |y| < 1} is included in all of these computer generated domains. This motivates the following theorem.

∗Mathematics Subject Classifications: 39A10

†Department of Mathematics, Fujian Normal University, Fuzhou, Fujian 350007, P. R. China

THEOREM 1. Let D(x, y|τ, σ) be the set of all pairs of the form (x, y) such that equation (2) is asymptotically stable. Then we have

⋂_{τ,σ∈N} D(x, y|τ, σ) = {(x, y)| |x| + |y| < 1} ,

where N is the set of all positive integers.

One part of the proof is easy. Let μ be a nonzero root of equation (3). If |a|+|b| < 1, then since

|a| + |b| < 1 ≤ |a| |μ|^{−τ} + |b| |μ|^{−σ} ,

we see that

|a| < |a| |μ|^{−τ}

or

|b| < |b| |μ|^{−σ} .

But then |μ|^{τ} < 1 or |μ|^{σ} < 1. In other words, |μ| < 1.

In order to complete our proof, we need the following preparatory lemma.

LEMMA 1 (cf. [4, Lemma 2.1]). Suppose a, b are real numbers such that |a|+|b| ≠ 0, and τ and σ are two positive integers. Then the equation

|a| x^{−τ} + |b| x^{−σ} = 1, x > 0

has a unique solution in (0, ∞).

PROOF. Consider the function

f(x) = |a| x^{−τ} + |b| x^{−σ}, x > 0.

Since f is continuous on (0, ∞), lim_{x→0^{+}} f(x) = ∞, lim_{x→∞} f(x) = 0 and

f′(x) = − (|a| τx^{−τ−1} + |b| σx^{−σ−1}) < 0, x > 0,

thus our proof follows from the intermediate value theorem.

Now if (a, b) belongs to ⋂_{τ,σ∈N} D(x, y|τ, σ), then for each pair (τ, σ) of integers, each root μ of equation (3) satisfies |μ| < 1. Let us write μ = re^{iθ} and write (3) in the form

ar^{−τ} cos τθ + br^{−σ} cos σθ = 1,  (5)
ar^{−τ} sin τθ + br^{−σ} sin σθ = 0.  (6)

There are several cases to consider: (i) a = 0 or b = 0; (ii) a > 0, b > 0; (iii) a < 0, b < 0; (iv) a < 0, b > 0; and (v) a > 0, b < 0. The first case is easily dealt with. In the second case, since the equation

ax^{−τ} + bx^{−σ} = 1

has a positive root ρ_{1}, thus (r, θ) = (ρ_{1}, 0) is a solution of equations (5)-(6). This implies that ρ_{1} =r = |μ| < 1. But then

1 = aρ_{1}^{−τ} + bρ_{1}^{−σ} > a + b = |a| + |b| .

In the third case, since the equation

−ax^{−τ} − bx^{−σ} = 1

has a positive root ρ_{2}, if we pick τ = 1 and σ = 3, then (r, θ) = (ρ_{2}, π) is a solution of equations (5)-(6). This implies ρ_{2} = |μ| < 1. But then

1 = −aρ_{2}^{−τ} − bρ_{2}^{−σ} > −a − b = |a| + |b| .

In the fourth case, since the equation

−ax^{−τ} + bx^{−σ} = 1

has a positive root ρ_{3}, if we pick τ = 1 and σ = 2, then (r, θ) = (ρ_{3}, π) is a solution of equations (5)-(6). This implies ρ_{3} = |μ| < 1. But then

1 = −aρ_{3}^{−τ} + bρ_{3}^{−σ} > −a + b = |a| + |b| .

In the final case, since the equation

ax^{−τ} − bx^{−σ} = 1

has a positive root ρ_{4}, if we pick τ = 2 and σ = 3, then (r, θ) = (ρ_{4}, π) is a solution of equations (5-6). This implies ρ_{4} < 1 and consequently

1 ≥ aρ_{4}^{−τ} − bρ_{4}^{−σ} > a − b = |a| + |b| .

The proof is complete.

References

[1] S. A. Levin and R. M. May, A note on difference-delay equations, Theoret. Popul. Biol., 9(1976), 178-187.

[2] S. A. Kuruklis, The asymptotic stability of x_{n+1} −ax_{n} +bx_{n−k} = 0, J. Math. Anal. Appl., 188(1994), 719-731.

[3] R. J. Duffin, Algorithms for classical stability problems, SIAM Review, 11(2)(1969), 196-213.

[4] L. A. V. Carvalho, An analysis of the characteristic equation of the scalar linear difference equation with two delays, Functional Differential Equations and Bifurcation, Lecture Notes in Mathematics, 799, Springer, Berlin, 1980.