Partial approximate controllability of fractional systems with Riemann–Liouville derivatives and nonlocal conditions

In this work, we investigate the partial approximate controllability of nonlocal Riemann–Liouville fractional systems with integral initial conditions in Hilbert spaces without assuming the Lipschitz continuity of nonlinear function. We also exclude the conditions of Lipschitz continuity and compactness for the nonlocal function. The existence results are derived using Shauder fixed point theorem, then the partial approximate controllability result is proved by assuming that the associated linear system is partial approximately controllable for $$\varphi =0$$ , where $$\varphi$$ is nonlocal function. Lastly, an example is provided to apply our results.