Approximation of a bounded solution of a linear difference equation on Z2 by solutions of corresponding boundary-value problems in a Banach space

We study the problem of existence and uniqueness of solutions of boundary-value difference problems in a Banach space that correspond to a certain difference equation on Z². We prove a theorem on approximation of the unique bounded solution of the considered equation by solutions of the corresponding boundary-value problems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 535–541, April, 1995.     

Approximation of a bounded solution of a difference equation by solutions of corresponding boundary problems in Banach space

Translated from Matematicheskie Zametki, Vol. 51, No. 4, pp. 17–22, April, 1992.     

Existence of solutions of boundary-value problems corresponding to a certain difference equation in a Banach space

The problem of the existence and uniqueness of solutions is studied for boundary-value difference problems corresponding to a certain linear difference equation in a Banach space.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 431–435, March, 1993.     

On a nonautonomous difference equation with bounded coefficient

In this paper we investigate the boundedness, the persistence and the attractivity of the positive solutions of the nonautonomous difference equation

     

Some problems of the spectral theory of a second-order differential equation with a variable unbounded operator coefficient

A self-adjoint second-order differential expression with an unbounded operator coefficient is considered in the space of vector functions. The domain of definition of the minimum and maximum operators generated by this expression is investigated, it is shown that any generalized resolvent of the minimum operator is an integral operator, and expansion in eigenfunctions is performed.     

Basic boundary-value problems for one equation with fractional derivatives

We prove certain properties of solutions of the equation

On boundary-value problems for the laplacian in bounded and in unbounded domains with perforated boundaries

In this paper we consider boundary-value problems in domains with perforated boundaries. We use the classification of homogenized (limit) problems depending on the ratio of small parameters, which characterize the diameter of the holes and the distance between them. We study the analogue of the Helmholtz resonator for domains with a perforated boundary.     

On the solution of bounded and unbounded mixed complementarity problems

A reformulation of the bounded mixed complementarity problem is introduced. It is proved that the level sets of the objective function are bounded and, under reasonableassumptions, stationary points coincide with solutions of the original variationalinequality problem. Therefore, standard minimization algorithms applied to the new reformulation must succeed. This result is applied to the compactification of unboundedmixed complementarity problems     

On bounded and unbounded global solutions of a supercritical semilinear heat equation

We consider the Cauchy problem where u ₀ C ₀ ( ^N ), the space of all continuous functions on ^N that decay to zero at infinity, and p is supercritical in the sense that N11 and . We first examine the domain of attraction of steady states (and also of general solutions) in a class of admissible functions. In particular, we give a sharp condition on the initial function u ₀ so that the solution of the above problem converges to a given steady state. Then we consider the asymptotic behavior of global solutions bounded above and below by classical steady states (such solutions have compact trajectories in C ₀ ( ^N ), under the supremum norm). Our main result reveals an interesting possibility: the solution may approach a continuum of steady states, not settling down to any particular one of them. Finally, we prove the existence of global unbounded solutions, a phenomenon that does not occur for Sobolev-subcritical exponents. Mathematics Subject Classification (2000)Primary 35K15, Secondary 35B35, 35B40, 35B33     

On periodic and bounded solutions of the operator Riccati equation

Sufficient conditions for the existence of periodic and bounded solutions of the operator Riccati equation are presented.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 2, pp. 239–242, February, 1993.     

On analytic solutions of the heat equation with an operator coefficient

Let A be a bounded linear operator on a Banach space and let g a be vector-valued function that is analytic in a neighborhood of the origin of ℝ. We obtain conditions of the existence of analytic solutions for the Cauchy problem Moreover, we consider a representation of the solution of this problem as a Poisson integral and study the Cauchy problem for the corresponding inhomogeneous equation. Bibliography: 22 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 139–162.     

Existence of bounded and unbounded non-oscillatory solutions for partial difference equations

In this paper, the existence of bounded and unbounded non-oscillatory solutions for high-order neutral partial difference equations with variable coefficients and delays are studied. Some new sufficient conditions for bounded and unbounded non-oscillatory solutions are given.     

Approximation of unbounded operators by bounded operators in a Hilbert space

We determine the best approximation of an arbitrary power A ^k of an unbounded self-adjoint operator A in a Hilbert space H on the class {x ∈ D(A ^r ) : ∥A ^r x∥ ≤ 1}, k < r.