On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

The nonlocal boundary value problem, d(t)(rho)u(t) + Au(t) = f (t) (0 < rho < 1, 0 < t <= T), u(xi) = alpha u(0) + ? (alpha is a constant and 0 < xi <= T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator d(t) on the left hand side of the equation expresses either the Caputo derivative or the Riemann-Liouville derivative; naturally, in the case of the Riemann-Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant alpha on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function ? in the boundary conditions are investigated.