LAUSR

In this paper, a fractional order neutral differential equation with deviated argument is studied. The existence and uniqueness of mild solution are obtained by using Banach contraction principle without imposing additional assumptions such as compactness, analyticity, uniform boundedness etc. We also discuss the approximate controllability of a fractional neutral differential equation with deviated argument by assuming a simple range condition $$({ HR})$$(HR). Thereby, we remove the need to assume the invertibility of a controllability operator used by Tai (Appl. Math. Lett. 24: 2158–2161, 2011), which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by Sakthivel et al. (Rep. Math. Phys. 70: 291–311, 2012), which are practically difficult to verify and apply. An example is provided to illustrate the presented theory.