Volume 1, Issue 2

Linear boundary value problems for a system of ordinary differential equations are considered. The stability of the solution with respect to small perturbations of coefficients and boundary values is investigated.

A formula is obtained for the asymptotic representation of solutions of the basic equations of the couple-stress theory of elasticity. The formula is used in proving the uniqueness theorems of the external boundary value problems.

A theorem of the unique solvability of the first boundary value problem in the Sobolev weighted spaces is proved for higher-order strictly hyperbolic systems in the conic domain with special orientation.

This article deals with Chogoshvili cohomotopy functors which are defined by extending a cohomology functor given on some special auxiliary subcategories of the category of topological spaces. The question of choosing these subcategories is discussed. In particular, it is shown that in the singular case to define absolute groups it is sufficient that auxiliary subcategories should have as objects only spheres S n , Moore spaces P n ( t ) = S n –1 ∪ t e n , and one-point unions of these spaces.

By constructing the corresponding Green's function in a trapezoidal domain, we establish the existence of self-adjoint realizations of incorporating boundary conditions of the form u ( s , 0) = u ( s, T ) = 0. Such operators correspond to the historically important concept of a ”simultaneous crossing of the axis” for vibrating strings.

The concepts of polynomial stabilization, strong polynomial stabilization, and strong stabilization are introduced for a fundamental system of solutions of linear differential equations. Some criteria of such kinds of stabilizations and applications to the theory of existence and uniqueness of solutions of ordinary differential equations are given. An abstract scheme of the obtained results is presented for Banach spaces.

Limit distribution is studied for the integrated squared error of the projection regression estimator (2) constructed on the basis of independent observations (1). By means of the obtained limit theorems, a test is given for verifying the hypothesis on the regression, and the power of this test is calculated in the case of Pitman alternatives.

The two main theorems are cencerned with the approximation of (complex-valued) functions on the real plane by sums of Bernstein pseudoentire functions. They are formulated and proved in Section 4, after prior determination of the suitable integral operators. Analogous results for pseudopolynomial approximations were obtained by , Gonska, and Jetter ([Izv. Akad. Nauk SSSR, Ser. Mat. 34: 564-583, 1970],[J. Approx. Theory 48: 396-406, 1986]).