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In this paper, we consider the existence of mild solutions and approximate controllability for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 &lt;  α  &lt; 2. As far as we know, there are few articles investigating on this issue. Firstly, the mild solutions to the equations are proved using Laplace transform of the Riemann-Liouville derivative. Moreover, the estimations of resolve operators involving the Riemann-Liouville fractional derivative of order 1 &lt;  α  &lt; 2 are given. Then, the existence results are obtained via the noncompact measurement strategy and the Mönch fixed point theorem. The approximate controllability of this nonlinear Riemann-Liouville fractional nonlocal stochastic systems of order 1 &lt;  α  &lt; 2 is concerned under the assumption that the associated linear system is approximately controllable. Finally, the approximate controllability results are obtained by using Lebesgue dominated convergence theorem.

fractional calculus and applied analysis

Approximate controllability and existence of mild solutions for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 &lt;  α  &lt; 2