Abstract
In this paper, interval oscillation criteria for the nonlinear damped dynamic equations with forcing terms on time scales within conformable fractional derivatives are established. Our approach is determined from the implementation of generalized Riccati transformation, some properties of conformable time-scale fractional calculus, and certain mathematical inequalities. Also, we extend the study of oscillation to conformable fractional Euler-type dynamic equation. Examples are presented to emphasize the validity of the main theorems\enleadertwodots.
1. Introduction
The consideration of dynamic equations on time scales has attracted many researchers because of its wide applications in the field of science and engineering. The theory of time scales was presented by Hilger [1] to unify the discrete and continuous analysis. It not only unifies the continuous and discrete cases but also gives new areas in between such as -calculus [2]. The qualitative analysis of solutions of dynamic equations on different time scales has received considerable notice. In particular, the investigation of the oscillation of solutions to dynamic equations [3–5], dynamic equations with damping term [6–8], and dynamic equations on various time scales [9–11] has gained extensive attention. Fractional calculus is a generalization of integration and differentiation to any order. Recently, it has been realized that the fractional calculus has numerous applications in engineering, signal processing, economics and finance, probability and statistics, neural networks, and thermodynamics; see for illustrations [12–16] and the citations therein. In recent times, the importance has been given to fractional order calculus rather than integer order due to its applications in engineering such as neural networks, electrical and mechanical engineering, and population dynamics. The fractional dynamic equations on time scales have been studied by only few authors [17–19].
In [19], Feng and Meng established the asymptotic and oscillatory behavior of the following dynamic equation of fractional order on time scales using the generalized Riccati transformation technique:
Alzabut et al. [17] considered the following nonlinear damped dynamic equation with a conformable fractional derivative:
Here, the authors established the oscillation of the above equation when the nonlinear function is increasing and nonincreasing. Besides, the results are carried out in light of the following two cases: and . Motivated by the above discussion, in this work, we established the oscillation results of nonlinear conformable fractional dynamic equations with a forced term.
In the recent papers [18, 20], the conformable time-scale fractional calculus has been introduced. Applications of the obtained results demonstrate that the newly defined calculus will be applied to investigate oscillation for both fractional differential and fractional difference equations at the same time. Therefore, the determination of oscillation of solutions of conformable fractional dynamic equations has become a promising topic for researchers. To the best of our observation, papers [17, 19] are the only research that has studied the oscillation of conformable fractional dynamic equations.
2. Problem Formulation and Preliminaries
In this article, we are concerned with a class of nonlinear conformable fractional dynamic equations with damped and forced terms on time scales of the kind:where denotes an arbitrary time scale, , is the conformable fractional dynamic operator of order , , and such that and the function , where is in . We constitute new interval criteria for oscillation of the solutions of equation (3) when the nonlinear function f is increasing and nonincreasing and extend the results to the Euler-type fractional dynamic equations.
By a solution of (3), we insist a nontrivial function fulfilling (3) for . If a solution of (3) is neither eventually positive (EP) nor eventually negative (EN), then it is called oscillatory. Or else, it is said to be nonoscillatory. If all solutions of (3) are oscillatory, then (3) is called oscillatory.
Before we proceed to the main results, we present essential preliminaries on conformable time-scale fractional calculus that will be used to justify further discussion. Terms and definitions are adopted from the papers [2, 18].
Definition 1 (see [2]). On any time scale , and are defined as the forward and backward jump operators, respectively.
A point is known as right-dense if , left-dense if , right-scattered if , and left scattered if .
The graininess function of the time scale is given by .
The set .
Definition 2 (see [2]). A real-valued function defined on is known as rd-continuous if at all left-dense points, a finite left limit of exists, and if it is continuous at each right-dense point.
Definition 3 (see [2]). A function is known as regressive if . is the collection that consists of all rd-continuous regressive functions . We define .
Definition 4 (see [2]). If , then the function defined byis called exponential, where .
Also, is a nonzero real function and .
Definition 5 (see [18]). For , and , the conformable fractional derivative of order for at is (if it exists) so that, for every positive , there is a -neighborhood fulfilling
Definition 6 (see [18]). If , then is said to be an -order antiderivative of and the integral of given byis named as Cauchy -fractional.
Theorem 1 (see [18]). By the definition of - order conformable fractional derivative, . Also, is decreasing (increasing) for , if for .
Theorem 2 (see [19]). Let . If , then, for the IVP and , the unique solution is the exponential function for fixed .
Theorem 3 (see [18]). Let be real-valued differentiable functions defined on at a point in . Then,(i)(ii)(iii)
Theorem 4 (see [18]). Assume ; be rd-continuous. Then,for .
Definition 7. Assume . Then, class is a collection of functions so that for ; for and has continuous -partial fractional derivatives and with respect to and , respectively, such that and , where .
For simplicity, we use the notion as follows:
3. Main Results
This part supplies the main theorems of the work. We will present the results in two folds based on the monotonicity of and extend the results for fractional Euler-type dynamic equation.
Lemma 1. Suppose that and hold.(i)If (3) has an solution y, then there is a sufficiently large so that(ii)If (3) has an solution y, then there is a sufficiently large so that
Proof. (i)Since is , there is an such that on . Now, we have By the assumptions, we take such that on with and . Then, on the interval , we have Since , we acquire that and is decreasing on . Therefore, by the assumption that is positive and , we get that is either negative or positive. Assume that on for sufficiently large . Thus, Letting , we get . This leads to a contradiction that is . Therefore, is .(ii)Since is , there is a sufficiently large so that on . Now, we haveBy the assumptions, we can choose such that on with and . Then, on , we attainSince , we get that and is increasing on . Therefore, by the assumption of and , we get that is either negative or positive.
Assume that on for sufficiently large . Therefore,Letting , we get , and this leads to a contradiction that is . Therefore, is .
3.1. Oscillation Criteria for (3) When Is Not Necessarily Increasing
To establish oscillation criteria, we make use of the following assumption: (H1) and for some .
Theorem 5. Let . If and holds. If there exists , a function on and for sufficiently large such thatthen (3) is oscillatory.
Proof. Assume that (3) has a nonoscillatory solution . Then, is either or for . Define a generalized Riccati function as follows:By Lemma 1, clearly andBy assumptions, we take for so that on with and is , or on with and is . On the intervals and , the above inequality implies thatBy multiplying and taking - fractional integral for (20) from to , we haveLetting , we getBy multiplying and taking integration of -order from to for (20), we haveLetting , we getDividing (33) by and (24) by , we attainAdding the above two inequalities, we get a contradiction to hypothesis.
Remark 1. Assume that the condition of Theorem 5 holds for every and function for sufficiently large . If there exist some so thatfor , then (3) is oscillatory.
Let . Then, , and we use for them. By using this in Theorem 5, we have the corollary as follows.
Corollary 1. Suppose that all the assumptions of Theorem 5 hold and for every , there is a so that . If there is a such thatthen (3) is oscillatory.
3.2. Oscillation Criteria for (3) When Is Increasing
To establish oscillation criteria, we make use of the following assumption:(H2) exists and for all and for some
Theorem 6. Let . If holds. If there exists , a function on and for sufficiently large such thatthen (3) is oscillatory.
Proof. Assume that (3) has a nonoscillatory solution . Define a generalized Riccati function as follows:Clearly, andBy the hypothesis, we take for such that on with and is , or on with and is . On the intervals and , we haveBy multiplying and taking - fractional integral for (31) from to , we haveLetting , we getBy multiplying and taking - fractional integral for (31) from to , we getLetting , we getDivide (22) by and (35) by , we obtainAdding the above two inequalities, imply a contradiction to hypothesis.
3.3. Oscillation Criteria for Fractional Euler Equation
We consider fractional Euler-type dynamic equation:
Theorem 7. Assume that . If there exists , a function on and , for sufficiently large such thatthen (37) is oscillatory.
Proof. The proof is the same as that of Theorem 6.
Remark 2. Suppose the conditions in Theorem 5 are replaced by the assumption given as follows:then Theorem 5 is accomplished.
Remark 3. In this article, we introduced the interval oscillation criteria for conformable fractional dynamic equations with Riccati transformation. So far, there are many results concerning the fractional model with oscillation criteria (see [19]). However, to the best of our cognizance, there are no results on the oscillation of fractional dynamic equations with forcing term.
4. Examples
In this part, in view of the conditions acquired in Section 3, we introduce some examples to represent the adequacy of the suggested nonlinear fractional model and the merits of our methodology.
Example 1. We take the nonlinear conformable fractional damped dynamic equation:Here, ; ; ; ; ; ; . Clearly, , , , ,Let , , , , , and . Then, we getHence, by Theorem 5, (40) is oscillatory.
Example 2. We take the nonlinear conformable fractional damped dynamic equation:Here, ; ; ; ; ; ; ; .
Therefore, we get , , , , which implies . Also,Let and . Then, all the constraints of Theorem 6 are satisfied with , and (40) is oscillatory.
Example 3. We take the following conformable fractional Euler-type dynamic equationswhere , , , and . Then, and since . If we let , , , and , then all the constraints of Theorem 7 are fulfilled with for suitable choice of and . Hence, we conclude that equation (45) is oscillatory.
5. Conclusion
In this article, we established the interval oscillation criteria for conformable fractional damped dynamic equations with forcing term. The obtained results are improved in the sense which provides sufficient criteria for the forced oscillation of the considered equation with various conditions. Finally, numerical examples are also presented to validate the theoretical results of this study, and in addition, the developed methodology has been tested on the forced oscillation of a fractional Euler-type dynamic equation with an example.
It would be interesting to extend the results proposed in this article to more complex systems, including fractional dynamic equations with time delay. This will be examined in our future works.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The fourth author was financially supported by Thailand Science Research and Innovation (TSRI) Grant Fund No. 64A146000001, and the second author was supported by DST-FIST (Grant No. SR/FST/MSI-115/2016).