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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 635149, 5 pages
http://dx.doi.org/10.1155/2013/635149
Research Article

Monotonicity of Eventually Positive Solutions for a Second Order Nonlinear Difference Equation

1School of Mathematics and Computer Sciences, Datong University, Datong, Shanxi 037009, China
2Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China

Received 7 October 2012; Revised 27 December 2012; Accepted 19 January 2013

Academic Editor: Tailei Zhang

Copyright © 2013 Huiqin Chen et al. 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.

Abstract

We derive several sufficient conditions for monotonicity of eventually positive solutions on a class of second order perturbed nonlinear difference equation. Furthermore, we obtain a few nonexistence criteria for eventually positive monotone solutions of this equation. Examples are provided to illustrate our main results.

1. Introduction

The theory of difference equations and their applications have received intensive attention. In the last few years, new research achievements kept emerging (see [17]). Among them, in [3], Saker considered the second order nonlinear delay difference equation

Saker used the Riccati transformation technique to obtain several sufficient conditions which guarantee that every solution of (1) oscillates or converges to zero. In [4], Rath et al. considered the more general second order equations

They found necessary conditions for the solutions of the above equations to be oscillatory or tend to zero. Following this trend, this paper is concerned with the second order perturbed nonlinear difference equation where is a positive sequence, are two continuous functions, and is the forward difference operator defined as .

In [8], Li and Cheng considered the special case of (3)

They got the sufficient conditions for asymptotically monotone solutions of (4). Enlightened by [8, 9], in this paper, we derive several sufficient conditions for monotonicity of eventually positive solutions on (3) and obtain a few nonexistence criteria for eventually positive monotone solutions of (3). Our results improve and generalize results in [8]. We also provide examples to illustrate our main results.

For convenience, these essential conditions used in main results are listed as follows: there exists a continuous function such that for all ;   is a derivable function and for ; there exist two sequences and , such that and for ;  , is a positive integral number,

where , and are all in (3).

2. Main Results

We first state a result which relates a positive sequence and a positive nondecreasing function. Its proof can be found in [8].

Lemma 1 (see [8]). Let be a positive nondecreasing function defined for . Let be a real sequence such that for . Then

Theorem 2. Suppose that conditions hold, and satisfy the following conditions:  ;  
for all . Then eventually positive solutions of (3) are eventually monotone increasing.

Proof. Suppose that is a positive solution of (3), say for . If conclusion cannot hold, without any loss of generality, assume , in view of (3) and conditions, we have by summing (6) from to , then
Making use of condition , we know for . Summing (3) and using , we have
By summing (8), we then see that which contradicts the fact . The proof is complete.

Example 3. Consider the difference equation where is any function of and . By taking , we have
So conditions of Theorem 2 hold. By Theorem 2, (10) has a positive monotone increasing solution .

Theorem 4. If conditions hold, there exist and   for such that
Then eventually positive solutions of (3) are eventually monotone increasing or .

Proof. Suppose is a positive solution of (3), there exists such that for . Let , and
If , then there exist and a number such that for ; in view of (7), we get
Summing (14) and making use of Lemma 1, we know
By , the right side of (15) tends to as , whereas the left side is finite. This contradiction completes our proof.

Example 5. Consider the difference equation
By taking , we have . So conditions of Theorem 4 hold. By Theorem 4, (16) has a positive monotone increasing solution .

Theorem 6. If conditions hold, and holds for all . Then eventually positive solutions of (3) are eventually monotone increasing or eventually monotone decreasing and .

Proof. Suppose is a positive solution of (3), there exists such that for . Let and , then there exists such that
From (7), we have
If , then there exists such that . There is no harm in assumption for . Summing (19), we obtain which is a contrary. The proof is complete.

Example 7. Consider the difference equation where is any function of and . By taking , we have
So conditions of Theorem 6 hold. By Theorem 6, (21) has a monotone decreasing positive solution .

Theorem 8. If conditions hold and  ; for all .
Then eventually positive solutions of (3) are eventually monotone increasing.

Proof. Suppose for , is a solution of (3), and . If the result does not hold, without any loss of generality, assume . In view of (7), we see that
Summing (23) and using Lemma 1, we know
This is a contradiction. The proof is complete.

Remark 9. In Theorems 2 and 4, condition is essential; that is, the series with positive terms is divergent, but it is not required in Theorems 6 and 8.

Remark 10. The eventually positive solutions in Theorems 4 and 8 are increasing it is not necessarily so in Theorem 6.
Next, we will derive several nonexistence criteria for eventually positive monotone solutions of (3).

Theorem 11. If conditions hold and
Then, (3) cannot have any eventually positive monotone increasing solutions.

Proof of Theorem 11 is obvious. If is an eventually positive increasing solution, by means of conditions, (7) is a contrary.

Theorem 12. If conditions hold, and there is a nonnegative and nondegenerate sequence such that holds for all . Then, (3) cannot have any eventually positive nondecreasing solutions.

Proof. Suppose that is a positive solution of (3), there exists such that and for . Multiplying (7) by , we have
So we obtain
This is contrary to our condition. The proof is complete.

Theorem 13. If and hold, is a nondecreasing sequence, is a nondecreasing function, and there is a nonnegative sequence , where is bounded, and   for all ;  , .
Then, (3) cannot have any eventually positive monotone increasing solutions.

Proof. Suppose that is a solution of (3), and there exists such that and for . Multiplying (3) by and summing from to again, we have
Namely,
As is a nondecreasing sequence, we get
Thus
From (30), we obtain using Lemma 1 and conditions, we have
By letting , we see that the left-hand side of (33) is bounded, this is contrary to our condition . The proof is complete.

By means of proof of Theorem 13, we get

Corollary 14. If , , and hold, is a nondecreasing sequence, is a nondecreasing function, and there is a nonnegative sequence , is bounded, and for . Then, (3) cannot have any eventually positive nondecreasing bounded solutions.

Corollary 15. Suppose , , and hold, is a nondecreasing sequence, is a nondecreasing function, and there is a nonnegative nonincreasing sequence . Then, (3) cannot have any eventually positive monotone increasing solutions.

Theorem 16. Suppose , , and hold, is a nondecreasing function, and   for all ;  .
Then, (3) cannot have any eventually positive nondecreasing solutions.

Proof. Assume to the contrary that there exists such that and for is a solution of (3). By means of (3) and , we get by summing (35) from to , thus
As is a nondecreasing function, we know , so
In view of Lemma 1, we see that
This contradiction establishes our assertion.

By means of proof of Theorem 16, we obtain the following.

Corollary 17. Suppose , , , and hold, is a nondecreasing function. Then (3) cannot have any eventually positive nondecreasing bounded solutions.

Acknowledgments

The authors are very grateful to the referee for her/his valuable suggestions. This work is supported by National Science Foundation of China (11271235), Shanxi Province (2008011002-1), Shanxi Datong University (2009-Y-15,2010-B-1), Sci-tech Research and Development Projects in Institutions of Higher Education of Shanxi Province (20111117, 20111020), and the Program for international cooperation of Shanxi Province (2010081005).

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