Mild Solutions to Time Fractional Stochastic 2D-Stokes Equations with Bounded and Unbounded Delay

In this paper, the well-posedness of stochastic time fractional 2D-Stokes equations of order α ∈ ( 0 , 1 ) containig finite or infinite delay with multiplicative noise is established, respectively, in the spaces C ( [ - h , 0 ] ; L 2 ( Ω ; L σ 2 ) ) and C ( ( - ∞ , 0 ] ; L 2 ( Ω ; L σ 2 ) ) . The existence and uniqueness of mild solution to such kind of equations are proved by using a fixed-point argument. Also the continuity with respect to initial data is shown. Finally, we conclude with several comments on future research concerning the challenging model: time fractional stochastic delay 2D-Navier–Stokes equations with multiplicative noise. Hence, this paper can be regarded as a first step to study this challenging topic.