Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 5232147, 11 pages
https://doi.org/10.1155/2018/5232147

Research Article

The Asymptotic Behavior of Solutions for a Class of Nonlinear Fractional Difference Equations with Damping Term

Zhihong Bai and Run Xu

School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Correspondence should be addressed to Run Xu; xurun_2005@163.com

Received 20 October 2017; Accepted 2 January 2018; Published 29 January 2018

Academic Editor: Chris Goodrich

Copyright © 2018 Zhihong Bai and Run Xu. 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.

Based on generalized Riccati transformation and some inequalities, some oscillation results are established for a class of nonlinear fractional difference equations with damping term. An example is given to illustrate the validity of the established results.

1. Introduction

During the past two decades, fractional calculus arose in the field of viscoelasticity, electrochemistry, physics, control, porous media, electromagnetism, and so forth; see [1–11] and the references therein. Fractional differential equations and difference equations can be used to describe some complex systems more accurately, and we may get the corresponding equations from those phenomena. However, as we all know, it is usually difficult to find the exact solutions to fractional differential or difference equations. In recent years, the study on qualitative properties of solutions of fractional differential equations, such as the existence, uniqueness, boundedness, oscillation, and other asymptotic behaviors, attracted much attention and some excellent results were obtained; we refer the reader to see [12–29] and the references cited therein.

In [12], Marian et al. studied the oscillation of fractional nonlinear difference equations of the form

Sagayaraj et al. [13] and Selvam et al. [14] investigated the oscillation of the following nonlinear fractional difference equations:

where and is a quotient of odd positive integers.

In [15], Sagayaraj et al. studied oscillatory behavior of the following fractional difference equations:

where and is a quotient of odd positive integers.

In [16], Li investigated the oscillation of forced fractional difference equations with damping term of the form

with initial condition , where .

In [24], Secer and Adiguzel established the oscillation results for a class of nonlinear fractional difference equations of the form

where and and are the quotients of two odd positive numbers.

Motivated by the idea in [24], in this paper, we are concerned with the oscillation of a class of nonlinear fractional difference equations with damping term of the form

where is a quotient of two odd positive integers,   is a constant, denotes the Riemann-Liouville fractional difference operator of order , and .

By a solution of (6), we mean a real-valued sequence satisfying (6) for . A nontrivial solution of (6) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. Equation (6) is called oscillatory if all of its solutions are oscillatory.

Throughout this paper, we assume that the following conditions hold:

(A), , , and are positive sequences, .(B) is a monotone decreasing function satisfying ; for .

For convenience, in the rest of this paper, we set

2. Preliminaries and Lemmas

In this section, the definitions of the Riemann-Liouville fractional sum and difference are given; then some basic lemmas are presented, which will be used in the following proof.

Definition 1 (see [28]). The th fractional sum of , for , is defined by

where is defined for , is defined for , and . The fractional sum is a map from to , where .

Definition 2 (see [28]). Let and , where denotes a positive integer; . Set ; then th fractional difference is defined as

Lemma 3 (see [14]). Let be a solution of (6), , and then

Lemma 4 (see [29]). The product and quotient rules of the difference operator are as follows:

where .

Lemma 5. If is a quotient of two odd positive integers, then the following two inequalities are established:

Proof. Using the inequality (see [30])

we have the following results.
If , let , , and ; then ; it follows from (17) that = ≥ = , so
If , then ; let , , and ; then ; by (17), we obtain

Since is a quotient of two odd positive integers, then

Substituting (19) into (18), we have , which means .

Lemma 6. Let , ; then .

Proof.

We define the following sequence:

then

3. Main Results

Lemma 7. Assume that is an eventually positive solution of (6) and

then, there exists a sufficiently large such that on and one of the following two conditions holds: (i) on and (ii) on and .

Proof. Since is an eventually positive solution of (6), then there exists a sufficiently large such that So Noting assumption (B), from (6) we obtain

Therefore, it follows from the definition of and the product rule (11) of the difference operator that

Then, is strictly decreasing on , and thus is eventually of one sign. For is sufficiently large, we claim that on . Otherwise, assume that there exists a sufficiently large such that ; then, for , we get

that is,

So we can get on . From these terms, for , we have

By (23), we obtain , which means, for some sufficiently large , on . By Lemma 3, we have

By (24), we obtain , which contradicts . Therefore, . Thus, is eventually of one sign. There are two possibilities: (i) on and (ii) on , where is sufficiently large.
Now, we assume that , where is sufficiently large. Then, by Lemma 3, we have . Since , we have We claim that . Otherwise, assume that . Then, . By (27), we have

Substituting with in (32), a summation for (32) with respect to from to yields

which implies

therefore,

Substituting with in (35), a summation for (35) with respect to from to yields

namely,

therefore,

that is,

Substituting with in (39), a summation for (39) with respect to from to yields

then

By (25), it follows from (41) that , which contradicts . Then we get that , which is . This completes the proof of Lemma 7.

By the same proof as above, if is an eventually negative solution of (6), we can obtain and Lemma 8 holds.

Lemma 8. Assume that is an eventually negative solution of (6) and (23)–(25) hold. Then, there exists a sufficiently large such that on and one of the following two conditions holds: (i) on and (ii) on and .

Lemma 9. Assume that is an eventually positive solution of (6) such that , on , where is sufficiently large. Then

Proof. Assume that is an eventually positive solution of (6); then we obtain that is strictly decreasing on ; by (27) we have,

By Lemma 3, we obtain

Then the proof is complete.

With the same proof as that in Lemma 9, we can obtain the following.

Lemma 10. Assume that is an eventually negative solution of (6) such that , on , where is sufficiently large. Then

Theorem 11. Assume that (23)–(25) hold. If

where is sufficiently large, is defined as in (21), and , then (6) is oscillatory or satisfies .

Proof. Suppose to the contrary that (6) has a nonoscillatory solution ,  ; then is either eventually positive or eventually negative.
In the case when is eventually positive, we assume that on , where is sufficiently large; then . By Lemma 7, we obtain on , where is sufficiently large, and either on or .
If on , then the conclusion of Lemma 9 holds.
Define the generalized Riccati function as follows:

It is clear that . By the product rule (12) and the quotient rule (13), for , we have

From and Lemma 3, we have

and ; then ; it follows from (B) that

and by (15).
Using Lemma 9 and the fact that is strictly decreasing, we have

Now substituting (49), (50), and (51) into (48), we obtain

Taking , , and in (52), using Lemma 6, we obtain

Substituting with in (53), a summation for (53) with respect to from to yields

Since , it is clear that

therefore,

From (46) and (57), we obtain that

Taking in (54) as , we have

which contradicts (58).
If on , then, from Lemma 7, we get that
In the case when is eventually negative, we assume that on , where is sufficiently large; then . By Lemma 8, we obtain on , where is sufficiently large, and either on or .
If on , then the conclusion of Lemma 10 holds.
Define as in (47); since , , then .
Using the product rule (12) and the quotient rule (14), for , we get

By and , we have

and by Lemma 3; then ; from (B) and (16), we obtain that

Proceeding the proof of Lemma 7, we have

which implies is strictly increasing. By virtue of Lemma 10, , and

we have

where and .
Combining (60), (61), (62), and (66), we can obtain

Taking

using Lemma 6, we have

Substituting with in (69), a summation for (69) with respect to from to yields

Taking in (70) as , we have

which contradicts (46).
If on , then, from Lemma 8, we get that .
The proof of Theorem 11 is complete.

Theorem 12. Assume that (23)–(25) hold and there exists a positive sequence such that

If

where and and are the same as in Theorem 11, then (6) is oscillatory or satisfies .

Proof. Suppose on the contrary that is a nonoscillatory solution of (6); then is either eventually positive or eventually negative.
In the case when is eventually positive, we assume that on , where is sufficiently large. According to the proof of Theorem 11, if on , then (52) holds.
Substituting with in (52), multiplying both sides by , and then summing with respect to from to yield

Using summation by parts formula, we obtain

Therefore,

Taking ,  , and in (76), using Lemma 6, we obtain

which means

for . Then

which means

Since , it is obvious that

therefore,

By (73) and (83), we have

Taking in (80) as , we obtain

which contradicts (84).
If on , then, from Lemma 7, we have
In the case when is eventually negative, it can be proved similarly; here we omit it.
The proof of Theorem 12 is complete.

4. Applications

In this section, an example is shown to illustrate the validity of the established results above.

Example 13. Consider the following fractional difference equation:

where and . Comparing with (6), we have

where is a certain positive number.
It is clear that (A) and (B) hold. Moreover, it follows from (21) that

Furthermore,

which means (23), (24), and (25) hold.
For a sufficiently large , we have

For ,

Hence,

which implies that condition (46) is satisfied. Therefore, (86) is oscillatory or satisfies by virtue of Theorem 11.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

Zhihong Bai carried out the oscillation criteria and completed the corresponding proof. Run Xu participated in Section 4. All authors read and approved the final manuscript.

Acknowledgments

This research is supported by National Science Foundation of China (11671227).

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