The Cauchy type problem for a differential equation with fractional derivative and self-adjoint operator in a Hilbert space is considered. The problem of parameter determination in equation by the value of the solution at a fixed point is presented. Theorems of existence and uniqueness of the solution are proved.

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The two inverse problems of determining an unknown parameter in a nonhomogeneous part of the equation for an abstract second-order elliptic equation in a Banach space with boundary conditions of Bitsadze-Samarski type are considered. For the first problem we use the conditions of Dirichlet, and for the second problem we use the conditions of Neumann. Theorems of existence and uniqueness of solutions for both direct and inverse problems are proved. Explicit formulas for the solutions are obtained.

We consider in a Banach space E the inverse problem(\mathbf{D}_{t}^{\alpha}u)(t)=Au(t)+\mathcal{F}(t)f,\quad t\in[0,T],u(0)=u^{0}% ,u(T)=u^{T},\,0<\alpha<1with operator A, which generates the analytic and compact α-times resolvent family {\{S_{\alpha}(t,A)\}_{t\geq 0}}, the function {\mathcal{F}(\,\cdot\,)\in C^{1}[0,T]} and {u^{0},u^{T}\in D(A)} are given and {f\in E} is an unknown element. Under natural conditions we have proved the Fredholm solvability of this problem. In the special case for a self-adjoint operator A, the existence and uniqueness theorems for the solution of the inverse problem are proved. The semidiscrete approximation theorem for this inverse problem is obtained.