Paper's Title:

Applications of Relations and Relators in the Extensions of Stability Theorems for Homogeneous and Additive Functions

Author(s):

Árpád Száz

Abstract:

By working out an appropriate technique of relations and relators and extending the ideas of the direct methods of Z. Gajda and R. Ger, we prove some generalizations of the stability theorems of D. H. Hyers, T. Aoki, Th. M. Rassias and P. Găvruţă in terms of the existence and unicity of 2-homogeneous and additive approximate selections of generalized subadditive relations of semigroups to vector relator spaces. Thus, we obtain generalizations not only of the selection theorems of Z. Gajda and R. Ger, but also those of the present author.

Paper's Title:

Some Moduli and Inequalities Related to Birkhoff Orthogonality in Banach Spaces

Author(s):

Dandan Du and Yongjin Li

Department of Mathematics,
Sun Yat-sen University,
Guangzhou, 510275,
P.R. China.
E-mail: dudd5@mail2.sysu.edu.cn

 
Department of Mathematics,
Sun Yat-sen University,
Guangzhou, 510275,
P.R. China.
E-mail: stslyj@mail.sysu.edu.cn 

Abstract:

In this paper, we shall consider two new constants δ_B(X) and ρ_B(X), which are the modulus of convexity and the modulus of smoothness related to Birkhoff orthogonality, respectively. The connections between these two constants and other well-known constants are established by some equalities and inequalities. Meanwhile, we obtain two characterizations of Hilbert spaces in terms of these two constants, study the relationships between the constants δ_B(X), ρ_B(X) and the fixed point property for nonexpansive mappings. Furthermore, we also give a characterization of the Radon plane with affine regular hexagonal unit sphere.

Paper's Title:

On the Convergence in Law of Iterates of Random-Valued Functions

Author(s):

Karol Baron

Uniwersytet Śląski, Instytut Matematyki
 Bankowa 14, PL--40--007 Katowice,
Poland
baron@us.edu.pl

Abstract:

Given a probability space (Ω, A, P) a separable and complete metric space X with the σ-algebra B of all its Borel subsets and a B A -measurable f : X * Ω → X we consider its iterates fⁿ, n N, defined on X * Ω^N by f¹(x,ω) = f(x,ω₁)  and fⁿ⁺¹(x,ω)=f(fⁿ(x,ω),ω_n+1), provide a simple criterion for the convergence in law of fⁿ(x,·)) _{n N}, to a random variable independent of x X , and apply this criterion to linear functional equations in a single variable.

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