Perron type integral on compact zero-dimensional Abelian groups

Perron and Henstock type integrals defined directly on a compact zero-dimensional Abelian group are studied. It is proved that the considered Perron type integral defined by continuous majorants and minorants is equivalent to the integral defined in the same way, but without assumption on continuity of majorants and minorants.     

Estimates of Deviations from Exact Solutions of Elliptic Variational Inequalities

In this paper, we present a method of deriving majorants of the difference between exact solutions of elliptic type variational inequalities and functions lying in the admissible functional class of the problem under consideration. We analyze three classical problems associated with stationary variational inequalities: the problem with two obstacles, the elastoplastic torsion problem and the problem with friction type boundary conditions. The majorants are obtained by a modification of the duality technique earlier used by the author for variational problems with uniformly convex functionals. These majorants naturally reflects properties of exact solutions and possess necessary continuity conditions. Bibliography: 15 titles.     

Interpolation and harmonic majorants in big Hardy-Orlicz spaces

Free interpolation in Hardy spaces is characterized by the well-known Carleson condition. The result extends to Hardy-Orlicz spaces contained in the scale of classical Hardy spaces H ^p, p > 0. For the Smirnov and the Nevanlinna classes, interpolating sequences have been characterized in a recent paper in terms of the existence of harmonic majorants (quasi-bounded in the case of the Smirnov class). Since the Smirnov class can be regarded as the union over all Hardy-Orlicz spaces associated with a so-called strongly convex function, it is natural to ask how the condition changes from the Carleson condition in classical Hardy spaces to harmonic majorants in the Smirnov class. The aim of this paper is to narrow down this gap from the Smirnov class to “big” Hardy-Orlicz spaces. More precisely, we characterize interpolating sequences for a class of Hardy-Orlicz spaces that carry an algebraic structure and are strictly bigger than ⋃ _p>0 H ^p . It turns out that the interpolating sequences are again characterized by the existence of quasi-bounded majorants, but now the functions defining these quasi-bounded majorants have to be in suitable Orlicz spaces. The existence of harmonic majorants defined by functions in such Orlicz spaces is also discussed in the general situation. We finish the paper with a class of examples of separated Blaschke sequences which are interpolating for certain Hardy-Orlicz spaces without being interpolating for slightly smaller ones.     

基于凸优函数的Newton类方法的收敛定理

本文研究了求解Banach空间上非线性算子方程f(x)=0的Newton类方法的收敛性.利用优函数原理,在A(x0)1f满足关于某一凸优函数的广义Lipschitz条件下,得到了Newton类方法的一个半局部收敛定理.同时,当f和A(x)及初始点x0给定时,针对广义Lipschitz条件构造了相应的优函数,推广了Newton类方法的相关结果.     

Majorants and minorants for the classes of graphs with fixed diameter and number of vertices

We study majorants (minorants) for the classes of n-vertex diameter d graphs, that is, the extremal graphs on which the sharp upper (lower) bounds are attained for the number of distinct radius i balls for every i ≥ 0. We explicitly describe the minorants for all values of n and d, determine when the class of n-vertex diameter d graphs contains majorants, and describe these extremal graphs.     

On the Two-step Newton Method for the Solution of Nonlinear Operator Equations

Building on the method of Kantorovich majorants, we give convergence results and error estimates for the two-step Newton method for the approximate solution of a nonlinear operator equation.     

Ball convergence theorems for Halley’s method in Banach space

We provide local convergence results for Halley??s method in order to approximate a locally unique zero of an operator in a Banach space setting using convex majorants. Kantorovich-type and Smale-type results are considered as applications and special cases.     

The method of Kantorovich majorants to nonlinear singular integral equation with shift

The paper is concerned with the applicability of the method of Kantorovich majorants to nonlinear singular integral equation with shift. The result is illustrated in the generalized Hölder space.     

Nonlinear Cauchy problems with small analytic data

We study the lifespan of solutions to fully nonlinear Cauchy problems with small real- or complex-analytic data. Our proofs are based on the method of majorants and the fixed point theorem for a contraction mapping.

     

A Bilogarithmic Criterion for the Existence of a Regular Minorant that Does Not Satisfy the Bang Condition

Mathematical Notes - Problems of constructing regular majorants for sequences $$\mu=\{\mu_n\}_{n=0}^{\infty}$$ of numbers $$\mu_n\ge0$$ that are the Taylor coefficients of integer transcendental...     

Some properties of majorants of Lyapunov exponents for systems with unbounded coefficients

A number of well-known results on majorants of Lyapunov exponents of linear differential systems with bounded coefficients on the half-line (such as their simultaneous attainability in the class of infinitesimal perturbations, belonging to the second Baire class in the compact-open topology, etc.) are generalized to systems with unbounded coefficients.     

Polynomials with curved majorants on two segments

In this paper we prove covering theorems for analytic functions related to polynomials that have curved majorants on two symmetric intervals. These theorems contain and complement some new and classical results for polynomials with restrictions on one and two intervals.     

On a problem on minorants of Lyapunov exponents

We construct examples of families of linear systems whose coefficients continuously depend on the parameter but whose sets of points of lower semicontinuity of the Lyapunov exponents, as well as of their majorants and minorants, treated as functions of the parameter are empty.     

Generalized Ostrowski–Grüss-type Inequalities

The paper is devoted to inequalities of Ostrowski–Grüss type. The upper bounds of our inequalities involve least concave majorants of the first order moduli of continuity or differences of upper and lower bounds of first order derivatives. Several inequalities available in the literature are generalized, improved or modified.     

On majorants in the hardy-littlewood theorem for higher derivatives

We establish conditions for majorants under which the classical Hardy-Littlewood theorem for the class of functions analytic in a disk is true in terms of derivatives of arbitrary fixed order.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 11, pp. 1574–1576, November, 1995.     

Covering theorems for polynomials with curved majorants on two intervals

In this paper, covering theorems for analytic functions related to polynomials that have curved majorants on two symmetric intervals are proved. These theorems contain and complement some new and classical results for polynomials with constraints on one and two intervals. Bibliography: 15 titles.     

On the practical application of majorants for nonlinear matrix iterations

We introduce a majorant concept with minor deviations from that of Schröder [3]. However, we emphasize the constructional aspects of our majorants. In particular we establish some simple composition rules. We demonstrate some applications including a problem of a priori choice between different iteration functions.     

A unifying theorem on newton's method

In this paper a general theorem is proved about convergence and error estimates for Newton's method. It contains as special cases a theorem of Kantorowitsch-type, a theorem with nonlinear majorants and componentwise error estimates as well as a theorem of monotone type.     

Domination, discretization, and scalarization

This is an overview of a few possibilities that are open by model theory in applied mathematics. The most attention is paid to the present state and frontiers of the Cauchy method of majorants, approximation of operator equations with finite-dimensional analogs, and the Lagrange multiplier principle in multiobjective decision making.