# Convergence Results on a Second-Order Rational Difference Equation with Quadratic Terms

## Abstract

We investigate the global behavior of the second-order difference equation Open image in new window , where initial conditions and all coefficients are positive. We find conditions on Open image in new window under which the even and odd subsequences of a positive solution converge, one to zero and the other to a nonnegative number; as well as conditions where one of the subsequences diverges to infinity and the other either converges to a positive number or diverges to infinity. We also find initial conditions where the solution monotonically converges to zero and where it diverges to infinity.

## Keywords

Differential Equation Partial Differential Equation Ordinary Differential Equation Functional Analysis Functional Equation## 1. Introduction and Preliminaries

- (i)
Let Open image in new window and Open image in new window , then we have the following.

- (a)
There are infinitely many solutions, Open image in new window , such that for each, one of its subsequences, Open image in new window , Open image in new window , converges to zero and the other diverges to infinity.

- (b)
There exist solutions, Open image in new window , which

- (1)
converge to zero if Open image in new window ;

- (2)
diverge to infinity if Open image in new window ;

- (3)
are constant if Open image in new window .

- (1)

- (a)

- (i)
Let Open image in new window and Open image in new window . Then for each positive solution Open image in new window , one of the subsequences, Open image in new window , Open image in new window , diverges to infinity and the other to a positive number that can be arbitrarily large depending on initial values. Further there, are positive initial values for which the corresponding solution, Open image in new window , increases monotonically to infinity.

- (ii)
Let Open image in new window and Open image in new window . Then for each positive solution Open image in new window , one of the subsequences, Open image in new window , Open image in new window , converges to zero and the other to a nonnegative number. Further, there are positive initial values for which the corresponding solution, Open image in new window , decreases monotonically to zero.

We note that the following results address and solve the first five conjectures posed by Sedaghat in [10].

## 2. Results

Theorem 2.1.

- (i)
There are infinitely many solutions, Open image in new window , such that for each, one of its subsequences, Open image in new window , Open image in new window , converges to zero and the other to infinity.

- (ii)
There exist solutions, Open image in new window , which

- (a)
converge to zero if Open image in new window ;

- (b)
diverge to infinity if Open image in new window ;

- (c)
are constant if Open image in new window .

- (a)

Proof.

Starting with (2.1), let the function Open image in new window be defined as Open image in new window . Note that for Open image in new window , Open image in new window is a decreasing function since Open image in new window . Also note that Open image in new window and Open image in new window . Hence Open image in new window has a unique positive fixed point Open image in new window .

We next compute the expression Open image in new window and simplify, it including canceling the common factor Open image in new window from the numerator and denominator, thereby obtaining the following:

Note that since Open image in new window , Open image in new window and Open image in new window . Thus the numerator of Open image in new window has one and only one sign change. Therefore, by Descartes' rule of signs, the numerator of Open image in new window has exactly one positive root, Open image in new window .

In addition, we see that Open image in new window and so, given that Open image in new window is the only positive root of the numerator of Open image in new window , we have Open image in new window for Open image in new window . Thus, since Open image in new window and Open image in new window is continuous, we must have Open image in new window for Open image in new window . Therefore,

We consider two cases depending on the initial value Open image in new window for (2.1).

Case 1 ( Open image in new window ).

Thus, Open image in new window Since Open image in new window is the only positive fixed point of Open image in new window , then we must have Open image in new window and Open image in new window

Case 2 ( Open image in new window ).

The argument is similar to that in Case 1 in showing Open image in new window and Open image in new window In both cases, the solution, Open image in new window , of (2.1) is divided into even and odd subsequences, Open image in new window and Open image in new window , where one subsequence converges monotonically to zero and the other to infinity.

We now go back to (1.1) by inferring the behavior of Open image in new window from Open image in new window . To do this we first consider Open image in new window . Without loss of generality, we will assume that Open image in new window and so Open image in new window and Open image in new window .

Next, observe that

Hence, result (i) is true.

Now consider Open image in new window . Then Open image in new window for all Open image in new window , and so Open image in new window for all Open image in new window . Induction then gives us Open image in new window for all Open image in new window . We thus have one of the following:

- (1)
- (2)
- (3)
If Open image in new window ( Open image in new window ), then Open image in new window is a constant solution Open image in new window

Thus the result (ii) is true and this completes the proof.

- (P1)
Open image in new window , with Open image in new window undefined when Open image in new window .

- (P2)
- (P3)
- If
we consider the addition restriction that Open image in new window and Open image in new window , we also obtain

- (P4)
if Open image in new window , then Open image in new window , or Open image in new window .

Lemma 2.2.

- (1)
- (2)
- (3)
Open image in new window , and Open image in new window and Open image in new window are undefined; or if either Open image in new window or Open image in new window is not zero, then Open image in new window is a solution of (1.1).

- (4)

Proof.

where Statements 1 and 2 and the continuity of Open image in new window (Property (P1) hold. Finally, Statement 4 follows immediately from Statement 3 and Property (P4).

In the first three results, we characterize the convergence of the odd and even subsequences of solutions of (1.1).

Theorem 2.3.

Let Open image in new window and Open image in new window in (1.1). Then for each positive solution, Open image in new window , one of the subsequences, Open image in new window , Open image in new window , converges to zero and the other to a nonnegative number.

Proof.

Consider (1.1) with Open image in new window , Open image in new window , and Open image in new window . Then it follows from Lemma 2.2 that for each positive solution of (1.1), Open image in new window , one of the subsequences, Open image in new window , Open image in new window , converges to zero and the other to a nonnegative number.

Theorem 2.4.

Let Open image in new window and Open image in new window in (1.1). Then for each positive solution Open image in new window , one of the subsequences, Open image in new window , Open image in new window , diverges to infinity and the other to a positive number or diverges to infinity.

Proof.

Then Open image in new window , and so it follows from Lemma 2.2 that for each positive solution of (2.16), Open image in new window , one of the subsequences, Open image in new window , Open image in new window , converges to zero and the other to a nonnegative number. Hence, for each positive solution of (1.1), Open image in new window , one of the subsequences, Open image in new window , Open image in new window , diverges to infinity and the other to a positive number or diverges to infinity.

In the following results, we show the existence of monotonic solutions for (1.1). As with Theorem 2.1 we use the substitution Open image in new window

Theorem 2.5.

Let Open image in new window and Open image in new window in (1.1). Then there are positive initial values for which the corresponding solutions, Open image in new window , decrease monotonically to zero.

Proof.

Set Open image in new window Given Descartes' rule of signs, we have that there exists a unique positive equilibrium, Open image in new window , where Open image in new window and Open image in new window Recall that Open image in new window and let Open image in new window for all Open image in new window . Then Open image in new window for all Open image in new window . It follows from induction that Open image in new window for all Open image in new window . Since Open image in new window , Open image in new window , with Open image in new window , decreases monotonically to zero.

Theorem 2.6.

Let Open image in new window and Open image in new window in (1.1). Then there are positive initial values for which the corresponding solution, Open image in new window , increases monotonically to infinity.

Proof.

As in the previous proof, an equilibrium equation for (2.1) satisfies (2.17). Setting Open image in new window we obtain from Descartes' rule of signs, a unique positive equilibrium, Open image in new window , where Open image in new window and Open image in new window Recall that Open image in new window and let Open image in new window for all Open image in new window . Then Open image in new window for all Open image in new window . It follows from induction that Open image in new window for all Open image in new window . Since Open image in new window , Open image in new window , with Open image in new window , increases monotonically to infinity.

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