Research Article | Open Access

Bratislav D. Iričanin, "On a Higher-Order Nonlinear Difference Equation", *Abstract and Applied Analysis*, vol. 2010, Article ID 418273, 8 pages, 2010. https://doi.org/10.1155/2010/418273

# On a Higher-Order Nonlinear Difference Equation

**Academic Editor:**Stevo Stević

#### Abstract

This paper shows that all positive solutions of a higher-order nonlinear difference equation are bounded, extending some recent results in the literature.

#### 1. Introduction

There is a considerable interest in studying nonlinear difference equations nowadays; see, for example, [1–40] and numerous references listed therein.

The investigation of the higher-order nonlinear difference equation where and , and , , was suggested by Stević at numerous talks and in papers (see, e.g., [20, 28, 30, 34–38] and the related references therein).

In this paper we show that under some conditions on parameters , and all positive solutions of the difference equation where , are bounded. To do this we modify some methods and ideas from Stević's papers [30, 35–37]. Our motivation stems from these four papers.

The reader can find results for some particular cases of (1.2), as well as on some closely related equations treated in, for example, [1, 2, 5–11, 18–20, 26, 30, 33–35, 38, 40].

#### 2. Main Result

Here we investigate the boundedness of the positive solutions to (1.2) for the case The following result completely describes the boundedness of positive solutions to (1.2) in this case. The result is an extension of one of the main results in [35].

Theorem 2.1. *Assume, and Then every positive solution of (1.2) is bounded if
*

*Proof. *First note that from (1.2) it directly follows that
Using (1.2), it follows that
After steps we obtain the following formula

Two subcases can be considered now.*Case 1 (). * If , then by (2.2) equality (2.4) implies that
for This means that is a bounded sequence.*Case 2 (). * In this case we have

From (2.4) and (1.2) we further obtain
for each and every where the sequences satisfy the system
and the initial values are given by

Note that implies that Assume for every .

By a direct calculation it follows that , , which, along with (2.8) implies that are strictly increasing sequences.

From system (2.8), we have,

If it were , , then there was
Clearly is a solution of the equation

Since
and
we see that the function attains its maximum at the point

Further, by assumption (2.1) we get
which along with (2.13) implies that (2.12) does not have solutions on arriving at a contradiction.

This implies that there is a fixed index such that
From this, inequality (2.2), and identity (2.7) with , it follows that
for .

From (2.17) the boundedness of the sequence directly follows, as desired.

#### Acknowledgment

The research was partially supported by the Serbian Ministry of Science, through The Mathematical Institute of SASA, Belgrade, Project no. 144013.

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#### Copyright

Copyright © 2010 Bratislav D. Iričanin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.