Georgian Mathematical Journal 1(1994), No. 3, 229-233

SOME OPEN PROBLEMS ABOUT THE SOLUTIONS OF THE DELAY DIFFERENCE EQUATION x_{n+1} = A/x_{n}^{2} + 1/x_{n−k}^{p}

M.ARCIERO, G.LADAS AND S.W.SCHULTZ

Abstract. We discuss the dynamics of the positive solutions of the delay difference equation in the title for some special values of the parameters A, p and k and we pose a conjecture and two open problems.

1. Introduction. Consider the difference equation

x_{n+1} = \frac{A}{x_{n}^{2}} + \frac{1}{\sqrt{x_{n−1}}} , n = 0, 1, …,  (1)

where A ∈ (0, ∞) and the initial conditions x_{−1} and x_{0} are arbitrary positive numbers. The following conjecture is predicted by computer simulations.

2. Conjecture. Let x̄ denote the unique positive equilibrium of Eq. (1). (a) Show that when

0 < A < \frac{15}{4}  (2)

the positive equilibrium of Eq. (1) is globally asymptotically stable. (b) Show that when

A > \frac{15}{4}  (3)

there exists a periodic cycle with period two which is asymptotically stable.

With the use of a computer one can easily experiment with difference equations and one can easily discover that such equations possess fascinating properties with a great deal of structure and regularity. Of course all computer observations and predictions must also be proven analytically. Therefore this is a fertile area of research, still in its infancy, with deep and important results which require our attention.

1991 Mathematics Subject Classification. 39A12.

For some developments on the global behavior of solutions of delay difference equations the reader is referred to the forthcoming monograph by Kocic and Ladas [2]. See also [1] and [3].

Although we are unable to establish the above conjecture, we have proven the following result.

Theorem 1. (a) Assume that (2) holds. Then the positive equilibrium x̄ of Eq. (1) is locally asymptotically stable.

(b) Assume that (3) holds. Then Eq. (1) has a periodic solution with period two.

Proof. (a) Set ϱ = \sqrt{x̄}. Then the linearized equation of Eq. (1) about x̄ is

y_{n+1} + \frac{2A}{ϱ^{6}}y_{n} + \frac{1}{2ϱ^{3}} y_{n−1} = 0, n = 0, 1, ….  (4)

From the well-known Schur-Cohn criterion, Eq. (4) is asymptotically stable provided that

\frac{2A}{ϱ^{6}} < 1 + \frac{1}{2ϱ^{3}} < 2.  (5)

Note that ϱ satisfies the equation

ϱ^{2} = \frac{A}{ϱ^{4}} + \frac{1}{ϱ}.  (6)

Hence ϱ > 1 and (5) is satisfied if and only if

2A < ϱ^{6} + \frac{1}{2}ϱ^{3} = A + \frac{3}{2}ϱ^{3},

that is,

ϱ > (\frac{2A}{3})^{1/3}.  (7)

Set f(t) = t^{6} − t^{3} − A and observe that f(t) < 0 if 0 < t < ϱ and f(t) > 0 if t > ϱ. Hence (7) is equivalent to f((\frac{2A}{3})^{1/3}) < 0; that is

A < \frac{15}{4}.

(b) Eq. (1) has a periodic solution of the form {p, q, p, q, … } or {q, p, q, p, … } if and only if

p = \frac{A}{q^{2}} + \frac{1}{\sqrt{p}} and q = \frac{A}{p^{2}} + \frac{1}{\sqrt{q}} .  (8)

Set x = \sqrt{p} and y = \sqrt{q}. Then the system of algebraic equations (8) is equivalent to

\begin{matrix}x^{2} & = \frac{A}{y^{4}} + \frac{1}{x} \\ y^{2} & = \frac{A}{x^{4}} + \frac{1}{y} \end{matrix}\} with x, y > 0.  (9)

Set ξ = x + y, η = xy and ζ = η^{3}. Then x and y are the roots of the quadratic equation λ^{2} − ξλ + η = 0 and these roots are real, positive, and distinct if and only if

ξ, η ∈ (0, ∞) and η < \frac{1}{4}ξ^{2}.  (10)

Cancel the denominators in (9), then multiply the first equation by x and the second by y, equate the terms x^{4}y^{4}, and divide by x − y. This leads to

Aξ = η(ξ^{2} − η).  (11)

Cancel the denominators in (9), subtract and then divide by x−y. This yields

η^{3} = −A + ξ(ξ^{2} − 2η).  (12)

Subtract from the first equation in (9), the second, and use (12) to obtain

ξ = \frac{(A − 1)η^{3} + A^{2}}{η^{4}} .  (13)

By substituting (13) into (11) we find

G(ζ) = ζ^{3} + (A − 1)ζ^{2} + A^{2}(2 − A)ζ − A^{4} = 0.  (14)

Note that

G(z) < 0 if z < ζ and G(z) > 0 if z > ζ.  (15)

In view of (10) and (13) we obtain

4ζ^{3} < (A − 1)^{2}ζ^{2} + 2A^{2}(A − 1)ζ + A^{4}

and so by using (14) we find

H(ζ) = (A + 3)(A − 1)ζ^{2} + 2A^{2}(3 − A)ζ − 3A^{4} > 0.

The positive root of this quadratic equation is ζ = 3A^{2}/(A + 3) and so H(ζ) > 0 if and only if G(3A^{2}/(A + 3)) < 0, that is

A > \frac{15}{4}.

The proof of the theorem is complete.

3. Open problems. A related difference equation is

x_{n+1} = \frac{a}{x_{n}^{2}} + \frac{1}{x_{n−1}} , n = 0, 1, …,  (16)

where a ∈ (0, ∞) and x_{−1}, x_{0} ∈ (0, ∞).

One can show that the following result holds.

Theorem 2. The following statements are true:

(a) The unique positive equilibrium x̄ of Eq. (16) is locally asymptotically stable if

a < 2\sqrt{3}  (17)

and unstable if

a > 2\sqrt{3}.  (18)

(b) When (18) holds, Eq. (16) has a periodic cycle with period two, {p, q, p, q, … }.

Furthermore

p = \frac{a + \sqrt{a^{2} + 2 − 2\sqrt{1 + 4a^{2}}}}{2} and q = \frac{a − \sqrt{a^{2} + 2 − 2\sqrt{1 + 4a^{2}}}}{2} .

Open problem 1. (a) For what values of a is the positive equilibrium x̄ of Eq. (16) globally asymptotically stable?

(b) For what values of a is the periodic cycle {p, q, p, q, … } of Eq. (16) asymptotically stable? What is its basin of attraction?

Eqs. (1) and (16) are special cases of the delay difference equation

x_{n+1} = \frac{A}{x_{n}^{2}} + \frac{1}{x_{n−k}^{p}} n = 0, 1, …  (19)

where A, p ∈ (0, ∞) and k ∈ {0, 1, … } and the initial conditions x_{−k}, …, x_{0} are arbitrary positive numbers.

Open problem 2. (a) Obtain conditions on A, p and k under which the positive equilibrium of Eq. (19) is globally asymptotically stable.

(b) Obtain conditions on A, p and k under which Eq. (19) has periodic cycles of period two. Under what conditions on A, p and k are these periodic cycles stable? What is the basin of attraction?

(c) Do there exist periodic cycles of period greater than two?

References

1. J.Hale and H.Kocak, Dynamics and Bifurcations. Texts in Applied Mathematics 3, Springer Verlag, New York, 1991.

2. V.L.Kocic and G.Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic Publishers, (to appear).

3. S.A.Kuruklis and G.Ladas, Oscillations and global attractivity in a discrete delay logistic model. Quart. Appl. Math., 50(1992), 227-233.

(Received 12.04.1993)

Authors’ addresses:

M.Arciero, G.Ladas Department of Mathematics The University of Rhode Island Kingston, RI 02881, U.S.A.

S.W.Schultz Department of Mathematics Providence College, Providence Rhode Island, 02918, U.S.A.