On Mild Solutions of the Mittag-Leffler Fractional-Order Equation Subjected to Nonlocal Delay and Impulsive Conditions

Based on Krasnosel’skii fixed point theorem, k-set contractions fixed point theorem, Banach’s fixed point theorem, the existence and uniqueness of solutions is investigated for Mittag-Leffler kernel-type fractional differential equations under nonlocal delay and impulsive boundary value conditions, considering both compact and non-compact resolvent operators. However, a challenge arises due to the order of the derivative ($$0<\vartheta <1$$), which complicates the proof of equicontinuity; one of the research objectives of this study is to solve this issue. In addition, another contribution in this process is that the constant L is generalized as an unbounded Lebesgue integrable function in case of non-compact measure conditions. Moreover, the Lipschitz conditions of nonlinear terms and $$\hbar,\rho $$ are expanded from non-negative constants $$L,\hbar _{1},\rho _{1}$$ to unbounded Lebesgue integrable functions. Finally, three examples are provided to demonstrate the validity of the present work.