Abstract and Applied Analysis

Abstract and Applied Analysis / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 418273 | https://doi.org/10.1155/2010/418273

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
Received20 May 2010
Accepted02 Jun 2010
Published04 Jul 2010

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, [140] 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, 3438] 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, 3537]. 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, 511, 1820, 26, 30, 3335, 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 © 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.


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