- Research
- Open Access
Oscillation criteria for a class of fractional delay differential equations
- Pengxian Zhu^{1, 2} and
- Qiaomin Xiang^{1, 2}Email author
https://doi.org/10.1186/s13662-018-1813-6
© The Author(s) 2018
- Received: 2 July 2018
- Accepted: 24 September 2018
- Published: 31 October 2018
Abstract
This paper is devoted to the oscillatory problem in the fractional-order delay differential equations. First, we prove the convergence of the Laplace transform of a fractional operator by the generalized Gronwall inequality with singularity and fractional calculus technique. Then we show that it exhibits oscillation dynamics if the corresponding characteristic equation has no real roots. We further provide other direct and effective criteria depending on the system parameters and fractional exponent. Finally, we carry out some numerical simulations to illustrate our results.
Keywords
- Oscillation
- Fractional differential equation
- Delay
MSC
- 39A21
- 34A08
1 Introduction
Fractional calculus has drawn much attention in various fields of science and engineering over the past few decades [1–3]. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. This main advantage makes them useful to model some neglected effects with classical integer-order models. The list of applications of fractional calculus has been growing, including viscoelastic materials and rheology, electrical engineering, electrochemistry, biology, biophysics and bioengineering, signal and image processing, mechanics, mechatronics, physics, and control theory [4, 5].
On the other hand, delay differential equations are adopted to represent systems with time delay. Such effects arise in many processes, such as chemical processes (behaviors in chemical kinetics), technical processes (electric, pneumatic, and hydraulic networks), biosciences (heredity in population dynamics), economics (dynamics of business cycles), and other branches. The basic qualitative theory of these delay differential equations is well established, especially in the linear case (for general references, see [6–9]).
With the combination of both fractional derivative and time delay, the topic of fractional-order delay differential equations (FDDEs) is enjoying growing interest among mathematicians and physicists. For instance, the general results on the existence of solutions of FDDEs were presented in [10]; the analytical stability bound for FDDEs was discussed in [11–13]; the finite time stability of robotic systems was studied in [14, 15], where time delay appears in PD^{α} fractional control system; the necessary and sufficient conditions for asymptotic stability of d-dimensional linear FDDEs are obtained by using the inverse Laplace transform method in [16]; the stability and asymptotic properties of FDDEs were analyzed in [17].
As is customary, a nontrivial solution of a differential equation is said to be oscillatory if eventually it is neither positive nor negative. Otherwise, the solution is called nonoscillatory. If all solutions of an equation are oscillatory, then this equation is said to be oscillatory.
2 Preliminaries
Let us first recall the necessary definitions of the fractional calculus.
Definition 2.1
Definition 2.2
Definition 2.3
Lemma 2.4
If \(x(t)\) is the solution of Eq. (1.1), then \(X(s)\) exists.
Proof
Combining (2.3) with (2.2), we can see that \(\vert x(t) \vert \le r(t)E_{\alpha} (pt^{\alpha} )\) for \(t \ge 0\). It is easy to show that \(X(s)\) exists for \(\operatorname{Re} (s) > p^{\frac{1}{\alpha}} \). The proof is complete. □
3 Oscillation criteria for FDDEs
Now we are in position to give our main results.
Theorem 3.1
Proof
Having the above result, we further obtain sufficient conditions of oscillation for Eq. (1.1).
Theorem 3.2
Proof
Corollary 3.3
Proof
4 Oscillation criteria for FDDEs with two constant coefficients
Proposition 4.1
Proof
The proof of this result is similar to that of Theorem 3.1, and thus we omit it. □
Theorem 4.2
Proof
Corollary 4.3
Proof
Corollary 4.4
Proof
5 Numerical simulation
To illustrate the effectiveness and the flexibility of our theoretical analysis, we give two numerical examples.
5.1 Numerical algorithm
The numerical methods used for solving ordinary differential equations cannot be used directly to solve fractional differential equations because of the nonlocal nature of fractional differential equations. Diethelm et al. [24] proposed a numerical algorithm for solving fractional differential equations. This scheme is a generalization of the Adams–Bashforth–Moulton method, that is, the predictor–corrector approach. Recently, Bhalekar and Daftardar-Gejji [25] have extended this algorithm to solve FDDEs.
5.2 Numerical results
Example 5.1
Example 5.2
6 Conclusion
In this paper, we studied the oscillatory behavior of solutions of FDDEs. We have proved that FDDEs exhibit oscillation dynamics if the corresponding characteristic equation has no negative real roots. We further give some sufficient conditions for the oscillation of FDDEs based on parameters and fractional exponent, which are very convenient for using in practice. Nevertheless, necessary conditions for the oscillation of FDDEs and oscillatory criteria for nonlinear FDDEs remain open. They will be the subjects of our further investigation.
Declarations
Acknowledgements
The authors would like to thank reviewers for their insightful and valuable comments.
Funding
Not applicable.
Authors’ contributions
The authors have achieved equal contributions. Both authors read and approved the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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