Viability for time fractional functional differential equations driven by the fractional Brownian motion

Let BH be a fractional Brownian motion with Hurst index 1/2< H< 1. In this paper, we consider the time fractional functional differential equation of the form \[ \{{array}{l@{10.0pt}l}{C}{Dₜγ}x(t)=f(t,{xₜ})+G(t,{xₜ})d/dt{BH}(t),{3.33333pt}{3.33333pt}{3.33333pt}{3.33333pt}{1em}& t∈ (0,T],\\ {} {x₀}(t)=η (t),{3.33333pt}{3.33333pt}{3.33333pt}{3.33333pt}{3.33333pt}{1em}& t∈ [-r,0],{array}.\] where 3/2-H< γ < 1, CDₜγ denotes the Caputo derivative, and xₜ∈ Cᵣ=C([-r,0],R) with xₜ(u)=x(t+u), u∈ [-r,0]. We prove the global existence and uniqueness of the solution of the equation and study its viability. As an application, we also discuss the existence of positive solutions.