A time non-local problem for the fractional differential equation

A class of non-local problems, Dtρu(t) + Au(t) = f(t) (1 < ρ < 2, 0 < t ≤ T), αu(0) + βu(ξ) = ’; u0(0) = (α; β are constants and ξ 2 (0; T) is a fixed number), in an arbitrary separable Hilbert space H is considered. Here A is a strongly positive selfadjoint operator with the domain of definition D(A) and Dtρ is the Caputo derivative. The main goal of the work is to study the correctness of the problem depending on the pair of parameters (α; β) 2 R2. If jαj > jβj and ’; 2 D(A), then the solution of the problem exists and is unique. Inequalities of coercivity type are obtained.