Vol 54, No 3 (2018)

We prove the invertibility of second-order differential operators with constant operator coefficients acting on the Banach space of bounded continuous functions on the real line under the condition that they are uniformly injective (in particular, left invertible) or surjective (in particular, right invertible). We show that if these operators are considered on the space of periodic functions, then the unilateral invertibility does not imply the invertibility of such operators. We obtain criteria for the injectivity, surjectivity, and invertibility of differential operators on the space of periodic functions.

We study the Tricomi problem for equations of mixed type with multiple functional delay and advance. A general solution of the equations is obtained. The problem is uniquely solvable.

We prove that the mixed problem for the Klein–Gordon–Fock equation u_tt(x, t) − u_xx(x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle Q_T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ L_p[0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class L_p(Q_T) for p ≥ 1. We construct the solution in explicit analytic form.

We study the existence of a generalized solution of an initial–boundary value problem describing the process of unsteady filtration of a liquid in a bounded region of an n-dimensional space. We consider the case in which the Kirchhoff transformation used to determine the generalized solution takes the real axis into a semiaxis bounded below. An auxiliary problem is constructed. It is proved that any solution of the auxiliary problem is a solution of the problem under study. The solvability of the auxiliary problem is established by using the method of semidiscretization in time and the Galerkin method.

We carry out spectral analysis of operator functions that are the symbols of integro-differential equations with unbounded operator coefficients in a separable Hilbert space. The structure and localization of the spectrum of operator functions which are symbols of these equations play an important role in studies of the asymptotic behavior of their solutions.

We study the existence and uniqueness of a solution of the Bellman equation formalizing the operation of a manufacturing company under conditions of irregular demand and debt load. The class of functions where this equation has a unique solution is determined. The choice of a specific solution is justified.

The Gellerstedt eigenvalue problem with homogeneous boundary conditions on interior characteristics is studied. We prove that the eigenfunction system of this problem is a Riesz basis in the L₂ space in the elliptic domain.