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We study a second-order difference equation of the form

Second-order difference equations of the form

Among the important aspects of solutions of a difference equation are boundedness and global stability. On several occasions, the question of boundedness of all solutions of a particular difference equation was settled by finding invariant curves. Invariant curves of a second-order difference equation are plane curves on which forward orbits that start on a curve remain on the curve. Finding invariant curves, studying their properties and their relation with Liapunov functions is an active area of research [

In recent years, several results that give sufficient conditions for global stability or global attractivity of equilibrium solutions of difference equations have been established. Most of the results rely on the monotonicity of the function defined by the difference equation under investigation; see, for instance, [

In this paper, we focus on (

In this section, we consider (

Consider

Merino used this crucial lemma to show that the positive equilibrium of (

For typographical reasons, we define

We consider

(a) clarifies the proof of Case 1 while (b) clarifies the proof of Case 2. The scale on the axes is intentionally missing because the graphs represent the general situation.

In this section, we consider (

The two equilibrium points

This figure illustrates the magnitude of the eigenvalues associated with

Before we address the issue of boundedness, let us have a look at the possibility of periodic solutions. We always use period to denote the prime period. We start with period-two solutions by considering the equations

Let

Next, by algebraic manipulations of the equations

Finally, as a consequence of the boundedness and oscillation results that we establish later on, no periodic solutions of period higher than four exist. Thus, our discussion about the existence of periodic solutions ends by investigating the existence of period-four solutions. In fact, algebraic manipulations show that positive period-four solutions do not exist within the range of our parameters.

In this section, we prove that the only positive solution of (

This figure illustrates the regions

Now, write

Now, the following two basic results will be used in the sequel.

Let

If

If

(i) Because

(ii) Because we have

Now,

Let

It is obvious that

Next, we proceed to show that either

Consider (

Let

A solution

Recall that a point

Next, we choose a suitable value for

Now, we give the following result.

Let

Suppose that

Positive solutions

Let

Let

Since

After establishing an invariant region

When extrema of consecutive positive (and negative) semicycles form monotonic sequences, we obtain subsequences of the orbit that aids us in characterizing the orbit. This approach has been widely used to prove attractivity of equilibria [

Instead of bounding an orbit with the extrema of its semicycles, here we develop a technique that bounds the elements of semicycles by a monotonic sequence that does not form a subsequence of the orbit itself, and then we use the monotonic sequence to show the attractivity of

Consider the function

If

Parts (i) and (ii) are obvious. To verify part (iii), observe that

The next result about the attractivity of the small fixed point of

Consider

Since

Because our goal is to be able to let

Next, from the invariant region

The next proposition gives the feasible region for the inequalities in H1, H2, and H3. The proof is just algebraic manipulations of the inequalities.

The feasible region for the inequalities in H1, H2, and H3 is the region

Now, we established enough tools to give the following result.

Consider (

Based on Lemma

We close this section by the following remark.

It is worth mentioning that the method of this section can be developed under more general settings to handle a wider class of maps

In this paper, we considered the difference equation

For (the new range of parameters)

Several aspects of solutions of (

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank Nasser Al-Salti for the stimulating discussion they had while working on the problem.