Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps

Let be a nonempty closed convex subset of a real Banach space and be a Lipschitz pseudocontractive self-map of with . An iterative sequence is constructed for which as . If, in addition, is assumed to be bounded, this conclusion still holds without the requirement that Moreover, if, in addition, has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of has the fixed point property for nonexpansive self-mappings, then the sequence converges strongly to a fixed point of . Our iteration method is of independent interest.

     

The loss of tightness of time distributions for homeomorphisms of the circle

For a minimal circle homeomorphism we study convergence in law of rescaled hitting time point process of an interval of length 0$">. Although the point process in the natural time scale never converges in law, we study all possible limits under a subsequence. The new feature is the fact that, for rotation numbers of unbounded type, there is a sequence going to zero exhibiting coexistence of two non-trivial asymptotic limit point processes depending on the choice of time scales used when rescaling the point process. The phenomenon of loss of tightness of the first hitting time distribution is an indication of this coexistence behaviour. Moreover, tightness occurs if and only if the rotation number is of bounded type. Therefore tightness of time distributions is an intrinsic property of badly approximable irrational rotation numbers.

     

Asymptotic properties of two interacting maps

In this paper we consider a system whose state x changes to (x) if a perturbation occurs at the time t, for . Moreover, the state x changes to the new state (x) at time t, for . It is assumed that the number of perturbations in an interval (0, t) is a Poisson process. Here and are measurable maps from a measure space into itself. We give conditions for the existence of a stationary distribution of the system when the maps and commute, and we prove that any stationary distribution is an invariant measure of these maps.     

On a Theorem of Holický and Zelený Concerning Borel Maps Without σ-Compact Fibers

The paper is concerned with a recent very interesting theorem obtained by Holický and Zelený. We provide an alternative proof avoiding games used by Holický and Zelený and give some generalizations to the case of set-valued mappings.     

On Vector Quasi-Equilibrium Problems with Set-Valued Maps

In this paper, we introduce a new class of vector quasi-equilibrium problems with set-valued maps. Almost all the vector equilibrium models of the Blum-Oettli type in the literature are special cases of our new class of equilibrium problems under consideration. Moreover, a number of C-diagonal quasiconvexity properties are proposed for set-valued maps, which are natural generalizations of the -diagonal quasiconvexity for real functions. Together with an application of continuous selection and fixed-point theorems, these conditions enable us to prove unified existence results of solutions for such vector equilibrium problems.     

Sheaves on Involutive Quantaloids

We extend the well-known notions of a singleton, complete -set, presheaf and sheaf over a complete Heyting algebra or a right-sided idempotent quantale to arbitrary involutive quantaloids. We show that sheaves on and complete -sets come to the same thing. This paper can be considered as a symmetric version of an earlier work of the author.     

Random Logistic Maps. I

Let {C _i} ₀ be a sequence of independent and identically distributed random variables with vales in [0, 4]. Let {X _n} ₀ be a sequence of random variables with values in [0, 1] defined recursively by X _n+1=C _n+1 X _n(1–X _n). It is shown here that: (i) E ln C ₁<0X _n0 w.p.1. (ii) E ln C ₁=0X _n0 in probability (iii) E ln C ₁>0, E |ln(4–C ₁)| such that (0, 1)=1 and is invariant for {X _n}. (iv) If there exits an invariant probability measure such that {0}=0, then E ln C ₁>0 and – ln(1–x) (dx)=E ln C ₁. (v) E ln C ₁>0, E |ln(4–C ₁)|< and {X _n} is Harris irreducible implies that the probability distribution of X _n converges in the Cesaro sense to a unique probability distribution on (0, 1) for all X ₀0.     

Existence and uniqueness of rectilinear slit maps

We consider a generalization of the parallel slit uniformization in which the angle of inclination of each image slit is assigned independently. Koebe proved that for domains of finite connectivity there is, up to a normalization, a unique rectilinear slit map achieving any given angle assignment. Koebe's theorem is partially extended to domains of infinite connectivity. A uniqueness result is shown for domains of countable connectivity and arbitrary angle assignments, and an existence result is proved for arbitrary domains under the assumption that the angle assignment is continuous and has finite range. In order to prove the existence result a new extremal length tool, called the crossing-module, is introduced. The crossing-module allows greater freedom in the family of admissible arcs than the classical module. Several results known for the module are extended to the crossing-module. A generalization of Jenkins' module condition for the parallel slit problem is given for the rectilinear slit problem in terms of the crossing-module and it is shown that rectilinear slit maps satisfying this crossing-module condition exist.

     

Syndetically proximal pairs

For continuous self-maps of compact metric spaces, we study the syndetically proximal relation, and in particular we identify certain sufficient conditions for the syndetically proximal cell of each point to be small. We show that any interval map f with positive topological entropy has a syndetically scrambled Cantor set, and an uncountable syndetically scrambled set invariant under some power of f. In the process of proving this, we improve a classical result about interval maps and establish that if f is an interval map with positive topological entropy and m?2, then there is n∈N such that the one-sided full shift on m symbols is topologically conjugate to a subsystem of fn² (the classical result gives only semi-conjugacy).     

Affine Near-Semirings Over Brandt Semigroups

In order to study the structure of A ⁺(B _n )—the affine near-semiring over a Brandt semigroup—this work completely characterizes the Green's classes of its semigroup reducts. In this connection, this work classifies the elements of A ⁺(B _n ) and reports the size of A ⁺(B _n ). Further, idempotents and regular elements of the semigroup reducts of A ⁺(B _n ) have also been characterized and studied some relevant semigroups in A ⁺(B _n ).