Existence and uniqueness solution of an inverse problems for fractional evolution equations

In this paper we concerned with study an inverse problem in a Hilbert space H for fractional abstract differential equation of the form du(t) dtα = Au(t) + γ(t)u(t), (1) with the initial condition u(0) = u0 ∈ H (2) and the overdetermination condition (u(t), v) = w(t) (3) where (.,.) is the inner product in H, γ is a real an unknown function,w is a given real function, u0 and v are given elements in H, 0 < α ≤ 1,u is unknown,and A is a linear closed operator defined on a dense subset D(A) in H into H. It assumed that A generates an analytic semigroup Q(t).This condition implies ‖Q(t)‖ ≤ β for all t ≥ 0, β is a positive constant.