For a differential equation of the form y"(t) - By(t) = 0, t is an element of(0, infinity), where B is a weakly positive linear operator in a Banach space B, the conditions on the operator B, under which this equation is uniformly or uniformly exponentially stable are given. As distinguished from earlier works dealing only with continuous at 0 solutions, in this paper no conditions on behavior of a solution near 0 are imposed.

For a C 0-semigroup \({\{U(t)\}_{t \geq 0}}\) of linear operators in a Banach space \({{\mathfrak{B}}}\) with generator A, we describe the set of elements \({x \in {\mathfrak{B}}}\) whose orbits U(t)x can be extended to entire \({{\mathfrak{B}}}\)-valued functions of a finite order and a finite type, and establish the conditions under which this set is dense in \({{\mathfrak{B}}}\). The Hille problem of finding vectors \({x \in {\mathfrak{B}}}\) such that there exists the limit \({\lim\limits_{n \to \infty}\left(I + \frac{tA}{n}\right)^{n}x}\) is also solved in the paper. We prove that this limit exists if and only if x is an entire vector of the operator A, and if this is the case, then it coincides with U(t)x.

For a differential equation of the form y'(t) + Ay (t) = 0, t epsilon (0, infinity), where A is the generating operator of a C-0-semigroup of linear operators on a Banach space B, we give conditions on the operator A, under which this equation is uniformly (uniformly exponentially) stable, that is, every its weak solution defined on the open semiaxis (0, infinity) tends (tends exponentially) to 0 as t -> infinity. As distinguished from the previous works dealing only with solutions continuous at 0, in this paper no conditions on the behavior of a solution near 0 are imposed. In the case where the equation is parabolic, there always exist weak solutions which have singularities of any order. The criterions below not only generalize, but make more precise a number of earlier results in this direction.

In the paper, we consider an abstract differential equation of the form (partial derivative(2)/partial derivative t(2) -B)(m) y(t) = 0, where B is a positive operator in a Banach space B. For solutions of this equation on (0, infinity), it is established the analogue of the Phragmen-Lindelof principle on the basis of which we show that the Dirichlet problem for the above equation is uniquely solvable in the class of vector-valued functions admitting an exponential estimate at infinity. The Dirichlet data may be both usual and generalized with respect to the operator B-1/2.The formula for the solution is given, and some applications to partial differential equations are adduced.

The aim of this work is to describe the weak solutions of a first-order differential equation on the interval (0, infinity) in a Banach space and their behavior when approaching to the ends of this interval.

We describe all classical solutions of an abstract m -harmonic equation on (0,∞) and study their properties on this interval and in the neighborhood of the singular point 0.

In trigonometric series terms all polyharmonic functions inside the unit disk are described. For such functions it is proved the existence of their boundary values on the unit circle in the space of hyperfunctions. The necessary and sufficient conditions are presented for the boundary value to belong to certain subspaces of the space of hyperfunctions.

We consider the equation Au = f, where A is a linear operator with compact inverse A−1 in a separable Hilbert space ℌ. For the approximate solution u n of this equation by the least squares method in a coordinate system {e k }k∈ℕ that is an orthonormal basis of eigenvectors of a self-adjoint operator B similar to A (\(\mathcal{D}\)(B) = \(\mathcal{D}\)(A)), we give a priori estimates for the asymptotic behavior of the expressions r n = ∥u n − u∥ and R n = ∥Au n − f∥ as n → ∞. A relationship between the order of smallness of these expressions and the degree of smoothness of u with respect to the operator B is established.

∞n=0 λ n A n x

We find conditions on a closed operator A in a Banach space that are necessary and sufficient for the existence of solutions of a differential equation y ′( t ) = Ay ( t ), t ∈[0,∞),in the classes of entire vector functions with given order of growth and type. We present criteria for the denseness of classes of this sort in the set of all solutions. These criteria enable one to prove the existence of a solution of the Cauchy problem for the equation under consideration in the class of analytic vector functions and to justify the convergence of the approximate method of power series. In the special case where A is a differential operator, the problem of applicability of this method was first formulated by Weierstrass. Conditions under which this method is applicable were found by Kovalevskaya.

For operator differential equations in a Banach space, we present the conditions for initial data which are necessary and sufficient for the Cauchy problem to have a solution in the class of analytic, entire, or exponential-type entire vector functions. In the case where an operator differential equation is a system of partial differential equations, the sufficient condition obtained coincides with the well-known Cauchy-Kovalevskaya theorem on the solvability of the Cauchy problem in the class of analytic functions.

We describe all weak solutions of a first-order differential equation in a Banach space on (0, ∞) and investigate their behavior in the neighborhood of zero. We use the results obtained to establish necessary and sufficient conditions for the essential maximal dissipativity of a dissipative operator in a Hilbert space.

The aim of this survey is to give a brief exposition of M. Krein's contribution to the extension theory of symmetric operators and the theory of entire operators, to describe the further development of his investigations in these fields and their application to the spectral theory of boundary value problems for differential equations.