The backward problem for time-fractional evolution equations

In this paper, we consider the backward problem for fractional in time evolution equations partial derivative(alpha)(t)u(t) = Au(t) with the Caputo derivative of order 0 < alpha <= 1, where A is a self-adjoint and bounded above operator on a Hilbert space H. First, we extend the logarithmic convexity technique to the fractional framework by analyzing the properties of the Mittag-Leffler functions. Then we prove conditional stability estimates of H & ouml;lder type for initial conditions under a weaker norm of the final data. Finally, we give several applications to show the applicability of our abstract results.