Volume 7, Issue 3

Generalized Wiener classes are considered. For these classes the exact order of Fourier coefficients with respect to the trigonometrical system is established. The property of Fourier coefficient indices is investigated, at which the specified exact order is achieved.

This paper deals with nonclassical initial boundary value problems for string oscillation and telegraph equations. Algorithms for a direct construction of solutions are proposed. The existence and uniqueness of a solution of a nonlocal boundary value problem is proved in the case of a general one-dimensional hyperbolic equation.

The basic mixed plane boundary value problem of equations of statics of the elastic mixture theory is considered in a simply connected domain when the displacement vector is given on one part of the boundary and the stress vector on the remaining part. The problem is investigated using the general displacement vector and stress vector representations obtained in [Basheleishvili, Georgian Math. J. 4: 223–242, 1997]. These representations enable us to reduce the considered problem to a system of singular integral equations with discontinuous coefficients of special kind. The solvability of this system in a certain class is proved, which implies that the basic plane boundary value problem has a solution and this solution is unique.

Dirichlet- and Neumann-type boundary value problems of statics are considered in three-dimensional domains with cuspidal edges filled with a homogeneous anisotropic medium. Using the method of the theory of a potential and the theory of pseudodifferential equations on manifolds with boundary, we prove the existence and uniqueness theorems in Besov and Bessel-potential spaces, and study the smoothness and a complete asymptotics of solutions near the cuspidal edges.

In this paper a left adjoint is constructed for the restriction (or pullback) functor associated to a morphism of Lie algebras. Colimits in the category of crossed modules in Lie algebras and a new result on crossed modules induced by a morphism of Lie algebras ι : P → Q in the case where ι is the inclusion of an ideal, are obtained.

The boundedness of Cesáro maximal operators for multiple Walsh–Fourier series is studied from the martingale Hardy–Lorentz space H pq into the Lorentz space L pq . Supremum in the maximal operators is taken over a positive cone.

Sufficient conditions are found for the existence of an upper and a lower solutions of the boundary value problem where and are linear bounded operators, and and are continuous, generally speaking nonlinear, operators. Kamke type theorems are proved on functional differential inequalities.

A problem of bending of a plate is considered for an infinite doubly-connected domain bounded by two convex broken lines when the plate boundary is hinge-supported and normally bending moments are applied to the points at infinity. A similar reasoning is used to study a problem of bending of a plate for an infinite domain bounded by a convex polygon and a rectilinear cut or for an infinite domain with two rectilinear cuts.

A fixed point theorem is proved in a complete metrically convex metric space. Our result generalizes the theorems of Assad [Tamkang J. Math. 7: 91–94, 1976] and Chatterjea [C.R. Acad., Bulgare Sci. 25: 727–730, 1972].

The results of positive and negative character on growth order of integral means are presented. Estimates of growth order of multiple Fourier–Haar series and trigonometric Fourier series are implied.

A way of finding explicit exact formulas for the number of representations of positive integers by some diagonal positive quadratic forms in fourteen variables is suggested.

Statements connected with the ( C , α ) ( – 1 < α < 0)-summability almost everywhere of series with respect to block-orthonormal systems are given. In particular, it is stated that if , then the condition guarantees the ( C , α )-summability almost everywhere of the series with respect to an { ϕ n } – ( N k , N k +1 ]-orthonormal system.

Our concern is to consider delay differential equations of Euler type. Necessary and sufficient conditions for the oscillation of solutions are given. The results extend some famous facts about Euler differential equations without delay.

Upper and lower solutions method is used to obtain the existence of a solution for the following system of boundary value problems: where ũ i ( k ) = ( u i ( k ), Δ u i ( k ), … , Δ n–2 u i ( k )), 1 ≤ i ≤ m .