Qualitative analysis of A.P.A. solution for fractional order neutral stochastic evolution equations driven by G-Brownian motion

Abstract

In this paper, we will analyses the square mean almost pseudo automorphic mild solution for fractional order equation,

(1) ^c₀D^α_γ [ℵ(γ)−D(γ,ℵ(γ))] = [Aℵ(γ) +φ(γ,ℵ(γ))]dγ + ϕ(γ,ℵ(γ))d<B>(γ) + ψ(γ,ℵ(γ))dB(γ), γ∈R

where A(γ) : D(A(γ)) ⊂ L 2 G (F) → L 2 G (F) is densely closed linear operator and the functions D, φ, ϕ and ψ: L²_G(F) → L²_G(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.