Oscillation Results for a Class of Nonlinear Fractional Order Difference Equations with Damping Term

Abstract

The paper studies the oscillation of a class of nonlinear fractional order difference equations with damping term of the form where , stands for the fractional difference operator in Riemann-Liouville settings and of order , , and is a quotient of odd positive integers and . New oscillation results are established by the help of certain inequalities, features of fractional operators, and the generalized Riccati technique. We verify the theoretical outcomes by presenting two numerical examples.

1. Introduction and Background

The objective of this paper is to provide oscillation theorems for the equation where , is a quotient of odd positive integers, is the fractional difference operator in the sense of Riemann-Liouville (RL) and of order , , and

The fractional sum for , (see [1]) is defined by where the fractional sum is defined from to , is defined for and is defined for . The falling function is where is the Gamma function, given by for .

Let and be positive integers such that , namely, . Set . Then, fractional difference (see [2]) is defined as

The following conditions are assumed to hold throughout this work: (i)(H₁), and are positive sequences with (ii)(H₂)for and (iii)(H₃) is a nonnegative sequence on for some . There exists a such that for (iv)(H₄) is a monotone decreasing function satisfying such that

A nontrivial solution of Eq. (1) is called oscillatory if it has arbitrarily large zeros. Otherwise, it is called nonoscillatory. Eq. (1) is called oscillatory if all its solutions are oscillatory.

In the literature, Wang et al. [3] extended some oscillation results from integer-order differential equation to the fractional-order differential equation where denotes the standard RL differential operator of order with is a positive real-valued function, is a continuous function satisfying and denotes RL integral operator. Xiang et al. [4] investigated conditions for oscillation of fractional-order differential equation where , is a fractional-order, is a quotient of odd positive integers, and is the RL right-sided fractional derivative of order of defined by for .

In this paper and motivated by the above work, we intend to carry forward the oscillation results from fractional differential Eq. (12) to the fractional difference Eq. (1). Moreover, we consider Eq. (1) under a damping term.

Fractional difference calculus is evolving as a powerful tool for studying problems in many fields such as biology, mechanics, control systems, ecology, electrical networks, electrochemical of corrosion, chemical physics, optics, and signal processing, economics and so forth (see [5–12]) and the references therein. Basic definitions and properties of fractional difference calculus were presented by Goodrich and Peterson [1]. In addition, there are other research works dealing with fractional difference equations which have helped to build up some of the basic theory in this area (see [2, 13–15]).

Fractional difference equations are applied to model physical processes which vary with time and space and its nonlocal property enables to model systems with memory effect. The study of the qualitative analysis of these equations has gained momentum in recent years and thus numerous publications have been reviled [16–20].

Of late, the investigation of the oscillation of solutions for fractional order difference equations has accelerated with several articles (see [21–29]). In what follows, we state some lemmas and preliminaries that will contribute in proving our main results.

Definition 1 (see [2]). For any , the falling factorial is known as where is as in (5).

Lemma 2 (see [23]). Let be a solution of (1) and then

Lemma 3 (see [30]). The product and quotient rules of the difference operator are defined to be and where .

Lemma 4 (see [31]). Let be a quotient of two positive odd integers. If then

Lemma 5 (see [32]). Let with. Then, the inequality holds for all .

2. Main Results

Herein, new oscillation theorems for Eq. (1) are established by using mathematical inequalities, the properties of RL sum and difference operators, and the generalized Riccati technique.

Define the sequence

Then , and hence

Theorem 6. Assume that hold. If and then Eq. (1) is oscillatory.

Proof. Suppose that is a nonoscillatory solution of Eq. (1). Without loss of generality, we may assume that is an eventually positive solution of Eq. (1). Then, there exits such that , for , where is defined in Lemma 2. Considering the assumption and using Eq. (1), we get or where is defined in (2). Hence, we proceed from (22) and (17) to Then, is strictly decreasing on and is eventually of constant sign. Since , , and is a quotient of odd positive integers, we observe that is eventually of constant sign.
First, we show that If not, then there exits such that , and we obtain which implies for that is So we arrive at that on . Hence for , we get From (16), we have and summing from to , we obtain and hence we have Now, by letting , we get which contradicts that , . Therefore for .
Now, since and we have from (16), we obtain Define the generalized Riccati function It is clear that . Using (17) and (18) for , we have Since and , then ; it follows from that and by Lemma 4.
Using the fact that is strictly decreasing, we arrive at Also we have from (41).
Now substituting (45), (46), and (47) in (43), we get Using (48) and (21) with , , and we obtain and hence Summing up (51) from to , we have and hence Letting , which contradicts with (26).

Theorem 7. Assume that (25)and (26) hold and there exists a positive sequence such that (1) for ; for (2) for If then Eq. (1) is oscillatory, where is defined in Theorem 6 and

Proof. Suppose that is a nonoscillatory solution of (1). Without loss of generality, we may assume that is an eventually positive solution of (1). Then, there exists such that , for . Proceeding as in proof of Theorem 6 we arrive at (48). Multiplying (48) by and summing up from to , we obtain Using summation by parts formula, we get Hence, from (57) Now by using (21) with , and we obtain for . Then which yields Taking limit as , we get which contradicts with (55).
Next, we consider the condition which implies that (25) does not hold. Under this condition, we have the following result.

Theorem 8. Assume that , (26), and (65) hold. If then every solution is oscillatory or satisfies , where is defined in Lemma 2.

Proof. Assume that is a nonoscillatory solution of (1). Without loss of generality, assume that is eventually a positive solution of (1). Proceeding like in the proof of Theorem 6, we get that (28) holds. Then, there are two signs of . When is eventually positive, we conclude from the proof of Theorem 6 that equation (1) is oscillatory.
Next, assume that is eventually negative, then there exits such that for . Since , we have and hence On the other hand, Since holds, we get Now, taking the limit of the both sides of (68) as tends to , we get Since for , we have Claim that . If not, then for . Now we have by (29). Summing up from to , we have which yields and hence The last inequality above implies that Summing up from to , we get Taking the limit of the both sides of the above inequality as tends to , we end up with which contradicts to for . Therefore, we obtain , that is

3. Applications

Example 9. Consider the equation where , and that is a quotient of odd positive integers in .
Comparing with Eq. (1), we get , , , , , , , , , , and that where is a certain positive number. It is clear that assumptions hold.
Further from (22), we have Thus, conditions (25) and (26) are satisfied. Therefore, all solutions of (80) are oscillatory by Theorem 6.

Example 10. Consider the equation where , , , and that is a quotient of odd positive integers in . Comparing with (1), we have , , , , , , , , , , and that where is a certain positive number. It is clear that assumptions hold.
Further from (22), we have We define the double sequence as follows: (1) for (2) for (3)(4) for and Then for , and hence Therefore, condition (55) is satisfied. We deduce that all solutions of Eq. (84) are oscillatory by Theorem 7.

4. A Concluding Remark

In this paper, we obtained new oscillation theorems for a class of fractional difference equation. The main outcomes are proved via the means of mathematical inequalities, properties of fractional operators, and generalized Riccati technique. We claim that the concluded results have merit and considered as an extension for the corresponding fractional differential equations. Particular examples that are consistent to the main results are demonstrated at the end of the paper.

The consideration of Eq. (1) with forcing term of the form could be further investigated. We leave this for future consideration.

Data Availability

Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

Acknowledgments

J. Alzabut would like to thank Prince Sultan University for supporting this work through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.