On the Solvability of an Inverse Fractional Abstract Cauchy Problem

This note is devoted to study an inverse Cauchy problem in a Hilbert space H for fractional abstract differential equations of the form; ), ( ) ( ) ( = ) ( t g t f t u A dt t u d    with the initial condition H u u = (0) 0 and the overdetermination condition: ), ( = ) ), ( ( t w v t u where (.,.) is the inner product in H , f is a real unknown function w is a given real function, 0 u , v are given elements in H , g is a given abstract function with values in H , 1 < 0   , u is unknown, and A is a linear closed operator defined on a dense subset of H . It is supposed that A generates a semigroup. An application is given to study an inverse problem in a suitable Sobolev space for general fractional parabolic partial differential equations with unknown source functions.