Volume 6, Issue 1

The existence and uniqueness of a solution of the first, the second and the third plane boundary value problem are considered for the basic homogeneous equations of statics in the theory of elastic mixtures. Applying the general Kolosov–Muskhelishvili representations from [Basheleishvili, Georgian Math. J. 4: 223–242, 1997], these problems can be splitted and reduced to the first and the second boundary value problem for an elliptic equation which structurally coincides with the equation of statics of an isotropic elastic body.

A formula of a radial derivative is obtained with the aid of derivatives with respect to θ and to φ of the functions closely connected with the spherical Poisson integral 𝑢 f ( r, θ, φ ) and the boundary values are determined for . The boundary values are also found for partial derivatives with respect to the Cartesian coordinates , and .

It is shown that the completion of the tensor product of two non-Archimedean weighted spaces of continuous functions is topologically isomorphic to another weighted space. Several applications of this result are given.

Sufficient conditions are established for the existence and uniqueness of an ω -periodic solution of the functional differential equation where f is a continuous operator acting from the space of n -dimensional ω -periodic continuous vector functions into the space of n -dimensional ω -periodic and summable on [0, ω ] vector functions.

The optimal sufficient conditions are found for weights, which guarantee the validity of two-weighted inequalities for singular integrals in the Lorentz spaces defined on homogeneous groups. In some particular case the found conditions are necessary for the corresponding inequalities to be valid. Also, the necessary and sufficient conditions are found for pairs of weights, which provide the validity of two-weighted inequalities for the generalized Hardy operator in the Lorentz spaces defined on homogeneous groups.

Theorems determining Weyl's multipliers for the summability almost everywhere by the | c , 1| method of the series with respect to block-orthonormal systems are proved. In particular, it is stated that if the sequence { ω ( n )} is the Weyl multiplier for the summability almost everywhere by the | c , 1| method of all orthogonal series, then there exists a sequence { N k } such that { ω ( n )} will be the Weyl multiplier for the summability almost everywhere by the | c , 1| method of all series with respect to the Δ k -orthonormal systems.

Let X and Y be two Hilbert spaces, and the space of bounded linear transformations from X into Y . Let { A n } ⊂ be a weakly periodic sequence of period T . Spectral theory of weakly periodic sequences in a Hilbert space is studied by H. L. Hurd and V. Mandrekar (Spectral theory of periodically and quasi-periodically stationary S α S sequences, University of North Carolina, Chapel Hill, 1991). In this work we proceed further to characterize { A n } by a positive measure μ and a number T of -valued functions a 0 , . . . , a T –1 ; in the spectral form , where and Φ is an -valued Borel set function on [0, 2 π ) such that (Φ(Δ) x , Φ(Δ′) x ′) Y = ( x , x ′) X μ (Δ ∩ Δ′).