On the Oscillation of Fractional Order Emden-Fowler q-Difference Equations

In this article, we study the oscillatory behavior of fractional order Emden Fowler qdifference equations of the form Dq [ r(t) (cDα q z(t))] + φ(t) |x(σ(t))| x(σ(t)) = 0, t ≥ t0, where z(t) = x(t) + p(t)x(t− τ),D q denotes the Caputo q-fractional derivative of order α,0 < α ≤ 1. Using the generalized Riccati technique, new oscillation criteria are established.


Introduction
Fractional differential equations can be found in extensive range of many different subject areas. There are different concepts of fractional derivatives such as Riemann -Liouville and Caputo fractional derivatives are widely used. The Caputo fractional derivatives are based on integral expressions and gamma functions which are nonlocal. Fractional theory and its applications are mentioned many papers and monographs, we refer [1,12,15,17,24,26,27,28,29,33].
In [6], the researchers investigated the oscillatory behavior of second-order Emden-Fowler neutral delay differential equations of the form . In this paper, we investigate the following fractional order Emden-Fowler neutral delay q-difference equation where z(t) = x(t) + p(t)x(t − τ ), c D α q denotes the Caputo q-fractional derivative of order α, 0 < α ≤ 1. We assume the following conditions throughout this paper without mentioning that (A 1 ) γ ∈ R, where R is the set of all ratios of odd positive integers ; By a solution of (1) we mean a nontrivial function x satisfying (1) for t ≥ t x ≥ t 0 . in the sequel, we assume that solutions of (1) exist and can be continued indefinitely to the right. A solution of (1) is called oscillatory if it has arbitrarily on [t x , ∞); Otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillory. There is no work done on the oscillation of q-fractional Emden-Fowler equation. Our main aim of this paper is to establish new oscillation criteria for (1) by using generalized Riccati technique method.

Preliminaries
We state some definitions and fundamental results on quantum fractional calculus, see [9,23,30] and the references cited therein.

Definition 2.2
The Riemann-Liouville type of the fractional q-derivative of the of order ν ≥ 0 is defined by (D 0 q g)(t) = g(t) and is the smallest integer greater than or equal to ν.

Definition 2.3
The Caputo type of the fractional q-derivative of order ν ≥ 0 is defined by is the smallest integer greater than or equal to ν.

Definition 2.4
For any x, y > 0 is called q-beta function and we recall the relation Lemma 2.5 Assume ν, γ ≥ 0 and let g be a function defined on the interval [0,1]. Then Lemma 2.6 Let ν > 0. Then, the following result holds: Lemma 2.7 Assume ν ≥ 0 and n ∈ N. Then, the following equality holds: , the following is valid: For ρ = 0, a = 0, applying q-integration by parts, we get

Main Results
In this section, we establish some new oscillation criteria for (1). In the following for convenience. We denote If for all sufficiently large t 1 , and for all constants M > 0, L > 0 one has, then (1) is oscillatory.
Proof: Suppose to the contrary that x is a nonoscillatory solution of (1).Without loss of generality we may assume that x(t) > 0 for all large t. The case of x(t) < 0 can be considered by the same method.
From (1) we can easily obtain that there exists a t 1 ≥ t 0 such that Case I: Case II: In case I holds. We have that σ(t) ≤ t From the definition of Z, we have Define, Then W (t) > 0 for t ≥ t 1 . From (1), (7) and (8), we obtain By (6), (9) and c D α q (σ(t)) > 0, we get since x(t) is positive and increasing, we see that x(t) > x(qt) and this implies that .
Letting t → ∞ in (13) we get a contradiction with (2). In case II holds. We define the function V by Then and integrating it from ρ(t) to l we obtain, That is, Thus, So by (−r(t) c D α q Z(t)) > 0 and (14) we have, Where L = −r(t) c D α q Z(t). Differentiating (14), we get .
If for all sufficiently large t 1 , and for all constants M > 0 such that (2) holds and ∞ φ(t) then (1) is oscillatory.
Proof: Suppose to the contrary that x is a nonoscillatory solution of (1). Without loss of generality we may assume that x(t) > 0 for all large t. The case of x(t) < 0 can be considered by the same method. From (1) we can easily obtain that there exists a t 1 ≥ t 0 such that (4) or (5) holds. If (4) holds. Proceeding as in the proof of Theorem (3.1). We obtain a contradiction with (2). If case II holds. We proceed as in the proof of Theorem (3.1) then we get (16) and (18). Multiplying (18) by δ γ (t), and integrating from t 1 to t implies that In view of (16) we have, Therefore by the above inequality letting t → ∞ in (21). We obtain ∞ φ(t) which contradiction (20). Theorem 3.3 Assume that c D α q (p(t)) ≥ 0, and there exists ρ ∈ c 1 q ([t 0 , ∞), R), such that ρ(t) ≥ t, c D α q (ρ(t)) > 0, σ(t) = ρ(t) − τ . If for all sufficiently large t 1 , and for all constants M > 0 such that (2) holds and then (1) is oscillatory.
Proof: Suppose to the contrary that u is a nonoscillatory solution of (1).Without loss of generality we may assume that x(t) > 0 for all large t. The case of x(t) < 0 can be considered by the same method.
From (1) we can easily obtain that there exists a t 1 ≥ t 0 such that (4) or (5) holds.
If (4) holds. Procedding as in the proof of Theorem (3.1). We obtain a contradiction with (2). If case II holds. We proceed as in the proof of Theorem (3.1) then we get (15). Dividing (15) by r(s) c D α q Z(s) ≤ r(t)cD α q Z(t) r(s) .
and integrating it from ρ(t) to l, letting l → ∞ yields.

Conclusion
In this paper, we have obtained some oscillation results for the fractional order Emden-Fowler quantum difference equation using generalized Riccati technique. In this paper is q-analog of [6]. Our results are new.