Interval Oscillation Criteria for Self-adjoint Alpha-Fractional Matrix Differential Systems with Damping

In this paper, we are concerned with the oscillation criteria for self-adjoint alpha-fractional matrix differential system with damping term. By using the generalized Riccati technique and the averaging technique, some new oscillation criteria are obtained.


Introduction
Fractional calculus is a branch of mathematics, which is as old as calculus but the applications are rather recent. It deals with differentiation and integration of arbitrary orders. It merges and generalizes the ideas of integer-order differentiation and n-fold integration whereas the fractional order models capture phenomena and properties that integer order neglect. The fractional order differential equations have been used to model several physical phenomena emerging in various Physical sciences, Biological, Ecological, Economics and Financial mathematics. See, for example [1,8,10,11,14,20,21,[28][29][30]35] and the references cited therein.
The R-L and Caputo fractional derivatives are based on integral expressions and gamma functions which are nonlocal. In 2014, Khalil et al [19], introduced a new fractional derivative called the conformable derivative, using a limit definition analogous to that of standard derivative. The conformable derivative of Khalil was soon generalized by Katugampola which is reffered as katugampola fractional derivative or α-fractional derivative. See [2][3][4]7,17,18] and the references cited therein.
An important tool in the study of oscillatory behavior of solutions for the matrix systems and corresponding the scalar analogue is the averaging technique which goes back as far as the classical properties of Wintner [32] and Hartman [13] giving sufficient oscillation conditions for those equations. The results of Wintner was improved by Kamenev [16], and further extensions of Kamenev's criterion have been obtained by Philos [27] and for the corresponding matrix system by Erbe, Kong and Ruan [12], Meng, Wang and Zhang [25], Kumari and Umamaheswaram [23] and Wang [31].
To the best of the our knowledge, there exists no literature and the oscillation of αfractional matrix differential systems. Motivated by this gap, we proposed to initiate the following α-fractional matrix differential system of the form where A(t), B(t), X(t) are n × n real continuous matrix functions with A(t), B(t) symmetric and A solution of the system (1) is said to be nontrivial if det X(t) = 0 for atleast one t ∈ [t 0 , ∞), and a nontrivial solution X(t) of (1) is said to be prepared or self-conjugate if where for any matrix A, the transpose of A is denoted by A * . A prepared solution In this paper, by using generalized Riccati technique and the averaging technique and by considering the function H(t, s)k(s) which may not have a nonpositive partial derivative on D 0 = {(t, s) : t > s ≥ t 0 } with respect to the second variable, we obtain some new general oscillation criteria for the system (1), that is, criteria given by the behavior of (1) (or of A(t) and B(t)) only on a sequence of subintervals of [t 0 , ∞). By choosing appropriate functions H, k, ρ, we present a series of explicit oscillation criteria.
Hereafter we denote the trace of n × n matrix A by tr(A). Further, E n is the n × n identity matrix, and the eigenvalues of the n × n symmetric matrix A (an increasing order) are .

Preliminaries
In this section, we give some basic definitions of the katugampola α-fractional derivatives, integrals and lemmas which are useful throughout this paper. The α-fractional derivative satisfies the following properties.
Let α ∈ (0, 1] and f, g be α-differentiable at a point t > 0. Then . DOI : http://doi.org/10.26524/cm47 Definition 2.2 [17] Let a ≥ 0 and t ≥ a. Also, let y be a function defined on (a, t] and α ∈ R. Then, the α-fractional integral of y is given by if the Riemann improper integral exists.

Remark 2.3
Throughout the paper, we use the following notation. Further, if Also, if each X i,j (t) is differentiable, then and hence D α X(t) = t 1−α X (t).

Main Results
In this section, we study oscillatory behavior of solutions of the α-fractional matrix differential system (1).
Theorem 3.1 Suppose that the functions H ∈ C(D, R), h 1 , h 2 ∈ C(D 0 , R) and k, ρ ∈ C α ([t 0 , ∞), (0, ∞)) satisfy the following conditions: Assume also that for each sufficiently large T 0 ≥ t 0 , there exist a, b, c ∈ R with T 0 ≤ a < c < b such that Then the system (1) is oscillatory.
Proof: Assume that there exists a prepared solution X(t) of the system (1) which is not oscillatory. Without loss of generality, we may assume that det X(t) = 0 for t ≥ t 0 . Define By α-differentiating the matrix (3) and making use of (1) we find that W (t) satisfies Riccati equation for t ∈ [t 0 , ∞); Multiplying by t α−1 on both sides and apply (p 6 ), we get On multiplying (4) by H(t, s)k(s) and integrating with respect to s from c to t for t ∈ [c, b), we obtain  Since A(t) > 0, we can let Substituting V (t) into the above equation, we obtain Letting t → b − in (5) and dividing both sides by H(b, c), we obtain Similarly to the above proof, multiplying (4), with t replaced by s, by H(s, t)k(s) and integrating with respect to s from t to c for t ∈ (a, c], we obtain Since A(t) > 0, we can again let Substituting V (t) into the above equation, we get