Stochastic fractional diffusion equations containing finite and infinite delays with multiplicative noise

In this work, we investigate stochastic fractional diffusion equations with Caputo–Fabrizio fractional derivatives and multiplicative noise, involving finite and infinite delays. Initially, the existence and uniqueness of mild solution in the spaces C p ( [ − a , b ] ; L q ( Ω , H ˙ r ) ) ) and C δ ( ( − ∞ , b ] ; L q ( Ω , H ˙ r ) ) ) are established. Next, besides investigating the regularity properties, we show the continuity of mild solutions with respect to the initial functions and the order of the fractional derivative for both cases of delay separately.