Existence and Uniqueness Results for Fractional Differential Equations With Nonlocal Conditions in Lp$$ {L}^p $$ Spaces

ABSTRACT This paper deals with nonlocal initial value problems (IVPs) for modified Caputo fractional differential equations (FDEs) of order . The novelty of this work is that the nonlinearity is considered in spaces, where , instead of the conventional space of continuous functions. In each scenario, we meticulously derive the equivalences between the FDEs and the corresponding integral equations in the spaces of interest. Several new existence and uniqueness results based on the Banach contraction principle are established, imposing weaker assumptions on the data as much as possible. To exemplify our main results, we present three constructive examples, together with their corresponding unique solution trajectories.