We study an initial value problem for an implicit fractional differential equation with the Liouville–Caputo fractional derivative. By using fixed point theory and an approximation method, we obtain some existence and uniqueness results.

This paper is concerned with the existence and continuous dependence of mild solutions to stochastic differential equations with non-instantaneous impulses driven by fractional Brownian motions. Our approach is based on a Banach fixed point theorem and Krasnoselski-Schaefer type fixed point theorem.

In this paper, we prove some random fixed point theorems in generalized Banach spaces. We establish a random version of a Krasnoselskii-type fixed point theorem for the sum of a contraction random operator and a compact operator. The results are used to prove the existence of solution for random differential equations with initial and boundary conditions. Finally, some examples are given to illustrate the results.