Averaging principle for non-Lipschitz fractional stochastic evolution equations with random delays modulated by two-time-scale Markov switching processes

In the paper, we consider the averaging principle for a class of fractional stochastic evolution equations with random delays modulated by a two-time-scale continuous-time Markov chain under the non-Lipschitz coefficients, which extends the existing results: from Lipschitz to non-Lipschitz case, from classical to fractional equations, from constant to random delays. Using alpha$$ \alpha $$-order fractional resolvent operator theory and stopping time technique, a general theorem on the existence and uniqueness of mild solutions is obtained first; further, strong averaging principle for fractional stochastic delay evolution equation is investigated, which simplifies the original system.