We find the criteria for the solvability of the operator equation $AX-XB=C$, where $A,B$, and $C$ are unbounded operators, and use the result to show existence and regularity of solutions of nonhomogeneous Cauchy problems.

For stationary Schrödinger equation in ${ℝ}^n$ with the finite potential the singular pseudopotential is constructed in the form allowing us to find wave functions. The method does not require the knowledge of the explicit form of a potential and assumes only knowledge of the scattering amplitude for fixed level of energy.

Solodkĭ (1998) applied the modified projection scheme of Pereverzev (1995) for obtaining error estimates for a class of regularization methods for solving ill-posed operator equations. But, no a posteriori procedure for choosing the regularization parameter is discussed. In this paper, we consider Arcangeli′s type discrepancy principles for such a general class of regularization methods with modified projection scheme.

We investigate the local exact controllability of a linear age and space population dynamics model where the birth process is nonlocal. The methods we use combine the Carleman estimates for the backward adjoint system, some estimates in the theory of parabolic boundary value problems in $L^k$ and the Banach fixed point theorem.

A fixed point theorem for condensing maps due to Martelli is used to investigate the existence of solutions to second-order impulsive initial value problem for functional differential inclusions in Banach spaces.