On the resolvent existence and the separability of a hyperbolic operator with fast growing coefficients in L2(R2)

This paper studies the question of the resolvent existence, as well as, the smoothness of elements from the definition domain (separability) of a class of hyperbolic differential operators defined in an unbounded domain with greatly increasing coefficients after a closure in the space L2(R2). Such a problem was previously put forward by I.M. Gelfand for elliptic operators. Here, we note that a detailed analysis shows that when studying the spectral properties of differential operators specified in an unbounded domain, the behavior of the coefficients at infinity plays an important role.

Shadowing and structural stability for operators

A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by Palis and Pugh around 1968. We will exhibit a class of examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expansive and has the shadowing property. Moreover, we will show that if a structurally stable operator is expansive, then it must be uniformly expansive. Finally, we will characterize the weighted shifts on the spaces $c_{0}(\mathbb{Z})$ and $\ell _{p}(\mathbb{Z})$ ( $1\leq p<\infty$ ) that satisfy the shadowing property.

Initial problem for a nonlinear integro-differential equation with a higher-order hyperbolic operator and with reflection of the argument

In this paper it is studied the questions of one value solvability of initial value problem for nonlinear integro-differential equation with hyperbolic operator of the higher order, with degenerate kernel and reflective argument for regular values of spectral parameter. It is expressed the partial differential operator on the left-hand side of equation of higher order by the superposition of first-order partial differential operators. This is allowed us to present the considering integro-differential equation as an integral equation, describing the change of the unknown function along the characteristic. Further is applied the method of degenerate kernel. In proof of the theorem on one-value solvability of initial value problem is applied the method of successive approximations. Also is proved the stability of this solution with respect to the initial functions.

One-dimensional Schrodinger operator with a negative parameter and its applications to the study of the approximation numbers of a singular hyperbolic operator

In this paper we use the one-dimensional Schr?dinger operator with a negative parameter to the study of the approximation numbers of a hyperbolic type singular operator. Estimates for the distribution function of the approximation numbers are obtained.