Mathematics

The Solvability of a System of Nonlinear Integral Equations of Hammerstein Type on the Whole Line

Authors: Khachatryan K. A.

In recent years, the interest has grown in nonlinear integral equations of convolution type in connection with their application in various fields of mathematical physics, inparticular, inthep-adic theory of an open-closed string, kinetic theory of gases, in the theory of radiation transfer in spectral lines. The paper is devoted to the questions of construction of nontrivial solutions and the study of their asymptotic behavior for one system of nonlinear integral equations of convolution type with a symmetric kernel on the whole axis.

Harmonic Analysis of Operator Semigroups Slowly Varying at Infinity

Authors: Strukov V. E., Strukova I. I.

The article focuses on studying of strongly continuous bounded operator semigroups. In the space of uniformly continuous functions with values inacomplex Banach space weconsider the subspace of integrally vanishing at infinity functions. This subspace includes the subspace of vanishing at infinity functions, but it is wider. We study the properties of the subspace under consideration.

Multiple Completeness of the Root Functions of the Pencils of Differential Operators with Constant Coefficients and Splitting Boundary Conditions

Authors: Rykhlov V. S.

In the space of square summable functions on the main segment [0,1], the class of polynomial pencils of ordinary differential operators of the n-th order is considered. The coefficients of the differential expression are assumed to be constants. The boundary conditions are assumed to be splitting and two-point at the ends 0 and 1 (l of boundary conditions is taken only at the point 0, and the remaining n − l is taken at the point 1). The differential expression and the boundary forms are assumed to be homogeneous, that is, they contain only main parts.

Martingale Inequalities in Symmetric Spaces with Semimultiplicative Weight

Authors: Volosivets S. S., Zaitsev N. N.

Let (Ω,Σ,P) be a complete probability space, F = {F n } ∞ n=0 be an increasing sequence of σ- algebras such that ∪ ∞ n=0 F n generates Σ. If f = {f n } ∞ n=0 is a martingale with respect to F and E n is the conditional expectation with respect to F n , then one can introduce a maximal function M(f) = sup n>0 |f n | and a square function S(f) =?∞P i=0|f i − f i−1 | 2 ¶ 1/2 , f −1 = 0. In the case of uniformly integrable martingales there exists g ∈ L 1 (Ω) such that E n g = f n and we consider a sharp maximal function f ♯ = sup n>0 E n |g − f n−1 |.