Oscillatory Behavior of Three Dimensional α-Fractional Delay Differential Systems

Adem Kilicman 1,2,* , Vadivel Sadhasivam 3 and Muthusamy Deepa 3 and Nagamanickam Nagajothi 3 1 Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400 UPM, Selangor, Malaysia 2 Department of Electrical and Electronic Engineering, Istanbul Gelisim University, Avcilar, Istanbul 34310, Turkey 3 Post Graduate and Research Department of Mathematics, Thiruvalluvar Government Arts College (Periyar University), Rasipuram 637 401, Namakkal Dt., India; ovsadha@gmail.com (V.S.); mdeepa.maths@gmail.com (M.D.); nagajothi006@gmail.com (N.N.) * Correspondence: akilic@upm.edu.my; Tel.: +603-89466813


Introduction
In the literature there are many advanced strategies in the expansion of ordinary and partial differential equations of fractional order and they have been used as excellent sources and tools in order to model many phenomena in the different fields of engineering, science and technology.Further, these tools are also used in fields such as chemical processes, polymer rheology, mathematical biology, industrial robotics, viscoelasticity, and many more, see the monographs [1][2][3][4][5][6][7].
At the end of the nineteenth century, Henry Poincare initiated the method and used the qualitative analysis of nonlinear systems of integer order differential equations.Since then, there has been significant development in the theory of oscillation of integer order differential systems [8][9][10][11][12][13][14][15][16][17][18].
In a study [19], Vreeke et al. applied the differential systems in the application of physics in order to solve the problem of a nuclear reactor which involved two temperature feedback.In the current literature there are many established results in the oscillation theory of classical differential systems (see [20][21][22][23][24]).However, in the nonlinear fractional differential system development is relatively slow due to the occurrence of nonlocal behavior of fractional derivatives that possess weakly singular kernels.
In 2014, Khalil et al. introduced the idea of conformable fractional derivative as a kind of local derivative with no memory (see [25][26][27]).By following the idea of Khalil, an interesting application of the conformable fractional derivative in physics was discussed and the action principle for particles under the frictional forces were formulated, see [28].
The idea of conformable fractional derivatives was generalized by Katugampola, and today it is known as the Katugampola fractional derivative.Nowadays, many researchers have interest in this type of derivative for their useful properties (see [29][30][31]).In this respect, we list the contributions of where 0 < α ≤ 1, D α denotes the α-fractional derivative respect to t.
A proper solution U(t) ∈ P for the system (1) is called oscillatory if all the components are oscillatory, otherwise it is nonoscillatory.Further, the system (1) is said to be oscillatory if all proper solutions oscillate.
The main goal of this paper is to establish some new oscillation criteria for the system (1) by making use of the generalized Riccati transformation and inequality technique.The study is structured as follows.In Section 2, we recall some preliminary concepts relative to the α-fractional derivative.In Section 3, some new conditions for the oscillatory behavior of the solutions of system (1) were presented.Illustrative examples are included in the final part of the paper in order to demonstrate the efficiency of new theorems.

Preliminaries
We begin this section with the following definition of the operator D α .Definition 1. [30] Let y : [0, ∞) → R, then α-fractional derivative of y is defined by If y is differentiable α-times in some (0, a) with a > 0, lim α-fractional derivative satisfies the following properties.[30] Let α ∈ (0, 1] and g, h be α-differentiable for t > 0. Then Definition 2. [30] Let a ≥ 0, t ≥ a and a function y defined on (a, t] with α ∈ R.Then, α-fractional integral as follows provided improper integral exists.

Main Results
In this section, the oscillatory behavior of solutions for the system (1) under certain conditions are established.Next we give the following lemmas that will be used in our further discussion.Lemma 1.If U(t) ∈ P is a nonoscillatory solution for (1), then the component function x(t) is always nonoscillatory.
Lemma 2. Suppose that (A 1 ) and (A 4 ) holds.Then there exists a t 1 ≥ t 0 such that either (I) Proof.Let u(t) be an eventually positive solution for (1) on [t 0 , ∞).Now, system (1) will be reduced to the following inequality which implies, From (5), Thus the proof completes on using the Lemma 3.2 in [34].
The following notations are employed in the sequel. where Theorem 1. Suppose that (A 1 ) − (A 4 ) hold.Assume also that there exists a positive function Then every solution of system (1) is oscillatory.
We now derive various oscillatory criteria on using the earlier results and we can generalize the Philos type kernel.Let us define a class of functions Ω.Consider Then each solution of system (1) is oscillatory.
Proof.As we proceed in the proof of Theorem 1 and from ( 16), we have the inequality Therefore assumption (18) is contradicted.Thus every solution of (1) oscillates.
We immediately obtain the following oscillation result for (1).
Theorem 3. Assume that (A 1 )-(A 4 ) hold.Also assume that there exists a function ρ Then every solution of system (1) is oscillatory.
Proof.Proceeding as in the proof of Theorem 1, multiplying inequality ( 17) by H(t, s) and integrating, we get H(t, s)w (s)ds ≤ H(t, t 1 )w(t 1 ).
Taking limsup as t → ∞, and hence lim sup which contradicts the hypothesis in (20).
The following theorem is to be proved using the techniques employed in the previous theorems.
Proof.We consider the Case (II) of Lemma 2, D α u(t) < 0, D α (a(t)D α u(t)) > 0 for t ≥ t By (A 4 ), we get Again integrating, we obtain Note: The decreasing condition imposed on q(t) and r(t) is only a sufficient condition, however it is not a necessary one.The following example ensures the oscillatory behavior of the system (34) even though q(t) and r(t) is nondecreasing.
Theorem 1 are satisfying the new conditions arriving at the solution for (34) is oscillatory and it is given as (u(t), v(t), w(t)) = (sin t, cos t, sin t).
Here further the condition (9) of the above Theorem 1 seems to be not satisfied, in view of the fact that (A 4 ) fails to hold, and hence the system (35) is not oscillatory.In fact, (u(t), v(t), w(t)) = (e t , e −t , e t ) it is a solution for (35), and nonoscillatory.

Conclusions
Through this article, we have derived some new oscillation results for a certain class of nonlinear three-dimensional α-fractional differential systems by using the Riccati transformation and inequality technique.This work extends and also improves some classical results in the literature [16,18,32] to the α-fractional systems and studied the oscillation criteria.Further, the present results are essentially new and, in order to illustrate the validity of the obtained results, we have provided three examples.