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

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 formwith initial condition , where .

In [24], Secer and Adiguzel established the oscillation results for a class of nonlinear fractional difference equations of the formwhere 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 formwhere 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 bywhere 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 obtainSince is a quotient of two odd positive integers, thenSubstituting (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) andthen, 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 obtainTherefore, it follows from the definition of and the product rule (11) of the difference operator thatThen, 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 getthat is,So we can get on . From these terms, for , we haveBy (23), we obtain , which means, for some sufficiently large , on . By Lemma 3, we haveBy (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 haveSubstituting with in (32), a summation for (32) with respect to from to yieldswhich impliestherefore,Substituting with in (35), a summation for (35) with respect to from to yieldsnamely,therefore,that is,Substituting with in (39), a summation for (39) with respect to from to yieldsthenBy (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 obtainThen 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. Ifwhere 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 haveFrom and Lemma 3, we haveand ; then ; it follows from (B) thatand by (15).
Using Lemma 9 and the fact that is strictly decreasing, we haveNow substituting (49), (50), and (51) into (48), we obtainTaking , , and in (52), using Lemma 6, we obtainSubstituting with in (53), a summation for (53) with respect to from to yieldsSince , it is clear thattherefore,From (46) and (57), we obtain thatTaking in (54) as , we havewhich 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 getBy and , we haveand by Lemma 3; then ; from (B) and (16), we obtain thatProceeding the proof of Lemma 7, we havewhich implies is strictly increasing. By virtue of Lemma 10, , andwe havewhere and .
Combining (60), (61), (62), and (66), we can obtainTakingusing Lemma 6, we haveSubstituting with in (69), a summation for (69) with respect to from to yieldsTaking in (70) as , we havewhich 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 thatIfwhere 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 yieldUsing summation by parts formula, we obtainTherefore,Taking ,  , and in (76), using Lemma 6, we obtainwhich meansfor . Thenwhich meansSince , it is obvious thattherefore,By (73) and (83), we haveTaking in (80) as , we obtainwhich 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 havewhere is a certain positive number.
It is clear that (A) and (B) hold. Moreover, it follows from (21) thatFurthermore,which means (23), (24), and (25) hold.
For a sufficiently large , we haveFor ,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).