QUALITATIVE ANALYSIS OF A.P.A. SOLUTION FOR FRACTIONAL ORDER NEUTRAL STOCHASTIC EVOLUTION EQUATIONS DRIVEN BY G-BROWNIAN MOTION

where A(γ) : D(A(γ)) ⊂ LG(F )→ LG(F ) is densely closed linear operator and the functions D,φ ,φ and ψ : LG(F )→ LG(F ) are jointly continuous. We drive square mean almost pseudo automorphic mild solution for fractional order neutral stochastic evolution equations driven by G-Brownian motion is obtain by using evolution operator theorem and fixed point theorem. Moreover, we prove that this mild solution of equation (1) is unique.


INTRODUCTION
Some results on of existence and uniqueness of the square-mean almost pseudo almost automorphic mild solutions for fractional differential equation have been discussed by some authors which can be found in [1,2,3,4,5,6]. The aims of this article is to discussed the square mean almost pseudo automorphic mild solution for neutral stochastic evolution equations of fractional order driven by G-Brownian motion(G-NSEEF for short), which is given by equation (1). Now we will recall following definitions of fractional derivative, Riemann-Liouville definition [5,6]: For α ∈ [n − 1, n) the α -derivative of f is Caputo definition [5,6]:For α ∈ (n − 1, n) the α -derivative of f is   for any γ ≥ s and s ∈ R.
For our convenience and further use we consider the following assumptions.

MAIN RESULT
Theorem If the hypothesis (H1) − (H3) are satisfied, and then, the system (1) has a unique mild solution ℵ ∈ SPAA(R, L 2 G (F )) and this solution can be expressed as Proof Firstly we will discuss the Existence of the square mean almost pseudo automorphic mild solution for of equation ( 1).
Claim: For all γ ≥ s and at each s ∈ R, we will show that ℵ(γ) defined by (3) satisfies the equation (2) and hence ℵ(γ) will be a mild solution of (1).
Note that, for any γ ∈ R, B (γ) the difference B (γ + r n ) − B (r n ) has the same distribution with B (γ) and by using the Cauchy-Schwarz inequality again, we have Therefor, we have LetB(γ) = B(γ + r n ) − B(r n ) for each γ ∈ R, thenB(γ) is also a G-Brownian motion with the same distribution as B(γ), we obtain where the last estimate converges to zero as n → ∞.
Therefore, we can conclude that By the same arguments as above, we obtain, From the Steps 1 and 2, we conclude that, (Φ 1 ℵ)(γ) ∈ SAA(R, L 2 G (F )).

CONCLUSION
In this paper, we analysed square mean almost pseudo automorphic mild solution for fractional order neutral stochastic evolution equations driven by G-Brownian motion is obtain by using evolution operator theorem and fixed point theorem. Moreover, we proved this mild solution of equation (1) is unique.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.