Existence of Solutions and Approximate Controllability of Fractional Nonlocal Stochastic Differential Equations of Order $$1<q\le 2$$ 1 < q ≤ 2 with Infinite Delay and Poisson Jumps

Abstract

In this paper, we investigate the existence of mild solutions and the approximate controllability of a class of nonlinear fractional stochastic differential equations of order \(1<q\le 2\) with infinite delay and Poisson jumps which satisfies the nonlocal conditions in Hilbert space. The existence of mild solutions is proved by using Sadovskii’s fixed point theorem. Also the approximate controllability of the nonlinear fractional nonlocal stochastic differential equations of order \(1<q\le 2\) with infinite delay and Poisson jumps is checked by using Lebesgue dominated convergence theorem. Finally an example is included to illustrate the results.