Convergence of Bregman Projection Methods for Solving Consistent Convex Feasibility Problems in Reflexive Banach Spaces

The problem that we consider is whether or under what conditions sequences generated in reflexive Banach spaces by cyclic Bregman projections on finitely many closed convex subsets Q _i with nonempty intersection converge to common points of the given sets.     

Iterative Methods of Solving Stochastic Convex Feasibility Problems and Applications

The stochastic convex feasibility problem (SCFP) is the problem of finding almost common points of measurable families of closed convex subsets in reflexive and separable Banach spaces. In this paper we prove convergence criteria for two iterative algorithms devised to solve SCFPs. To do that, we first analyze the concepts of Bregman projection and Bregman function with emphasis on the properties of their local moduli of convexity. The areas of applicability of the algorithms we present include optimization problems, linear operator equations, inverse problems, etc., which can be represented as SCFPs and solved as such. Examples showing how these algorithms can be implemented are also given.     

Downward Sets and their separation and approximation properties

We develop a theory of downward subsets of the space ^I, where I is a finite index set. Downward sets arise as the set of all solutions of a system of inequalities x^I,f_t(x)0 (tT), where T is an arbitrary index set and each f _t (tT) is an increasing function defined on ^I. These sets play an important role in some parts of mathematical economics and game theory. We examine some functions related to a downward set (the distance to this set and the plus-Minkowski gauge of this set, which we introduce here) and study lattices of closed downward sets and of corresponding distance functions. We discuss two kinds of duality for downward sets, based on multiplicative and additive min-type functions, respectively, and corresponding separation properties, and we give some characterizations of best approximations by downward sets. Some links between the multiplicative and additive cases are established.     

Some convexity properties of Dirichlet series with positive terms

Some basic results for Dirichlet series ψ with positive terms via log‐convexity properties are pointed out. Applications for Zeta, Lambda and Eta functions are considered. The concavity of the function 1/ψ is explored and, as a main result, it is proved that the function 1/ζ is concave on (ζ^–1(e), ∞). As a consequence of this fundamental result it is noted that Zeta at the odd positive integers is bounded above by the harmonic mean of its immediate even Zeta values which are known explicitly (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Converses of the Jensen inequality derived from the Green functions with applications in information theory

Jensen integral inequality has got much importance regarding their applications in different fields of mathematics. In this paper, two converses of Jensen integral inequality for convex function are obtained. The results are applied to establish converses of Hölder and Hermite-Hadamard inequalities as well. At the end, some useful applications in information theory of the obtained results are given. The idea used in this paper may inculcate further research.     

Mean value properties of nondifferentiation functions and their application in nonsmooth analysis

A calculus for (radial) upper convex approximations is developed and mean value properties of nondifferentiable functions are presented. The mean value theorems are formulated with radial upper convex approximations and are of equality type. Some applications in nonsmooth analysis are studied.     

Generalized derivatives of distance functions and the existence of nearest points

The relationships between the generalized directional derivative of the distance function and the existence of nearest points as well as some geometry properties in Banach spaces are studied. It is proved in the present paper that the condition that for each closed subset G$G$ of X$X$ and x∈X?G$x\in X?G$, the Clarke, Michel-Penot, Dini or modified Dini directional derivative of the distance function is 1 or −1 implying the existence of the nearest points to x$x$ from G$G$ is equivalent to X$X$ being compactly locally uniformly convex. Similar results for uniqueness of the nearest point are also established.     

Geometric properties of the functions Γ and 1/Γ

In this paper we determine the radius of convexity and the radius of starlikeness of the functions Γ and The basic tools of our work are the developments of the functions in function series.     

Continuity of optimal control in differential games and certain properties of weakly and strongly convex functions

In the first part of the paper, the equivalence of Lipschitzian differentiability of a function and a set of conditions of weak convexity and weak concavity of this function is proved, as well as sufficient conditions for the continuous dependence of the saddle point on a strongly convex-concave function of a parameter are given. In the second part, it is proved that the value function of a game is smooth and the optimal positional and programmed strategies of the players are continuous in zero-sum nonlinear differential games with strongly convex-concave Lagrangian. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 816–839 December, 1999.