This paper is concerned with the existence and uniqueness of mild solution of the fractional differential equations with nonlocal conditions d q x ( t ) / d t q = − A x ( t ) + f ( t , x ( t ) , G x ( t ) ) ,   t ∈ [ 0 , T ] ,    and    x ( 0 ) + g ( x ) = x 0 , in a Banach space X , where 0 < q < 1 . General existence and uniqueness theorem, which extends many previous results, are given.