Wojciech M. Kozlowski

COMMON FIXED POINTS FOR SEMIGROUPS OF POINTWISE LIPSCHITZIAN MAPPINGS IN BANACH SPACES

We investigate the existence of common fixed points for pointwise Lipschitzian semigroups of nonlinear mappings acting in a bounded, closed, convex subset of a uniformly convex Banach space. We also demonstrate how the asymptotic aspect of the pointwise Lipschitzian semigroups can be expressed in terms of the respective Frechet derivatives. doi:10.1017/S0004972711002668

Fixed point iteration processes for asymptotic pointwise nonexpansive mapping in modular function spaces

Let Lρ be a uniformly convex modular function space with a strong Opial property. Let T:C→C be an asymptotic pointwise nonexpansive mapping, where C is a ρ-a.e. compact convex subset of Lρ. In this paper, we prove that the generalized Mann and Ishikawa processes converge almost everywhere to a fixed point of T. In addition, we prove that if C is compact in the strong sense, then both processes converge strongly to a fixed point.MSC:47H09, 47H10.

Modular Metric Spaces

The concept of a metric space is closely related to our intuitive understanding of space is the 3-dimensional Euclidean space. In fact, the notion of metric is a generalization of the Euclidean metric arising from the basic long known properties of the Euclidean distance. Maurice Frechet1is credited as the mathematician who introduced the abstract definition of a metric space. Metric spaces are seen as a nonlinear version of vector spaces endowed with a norm. Following the same direction, one may think of a nonlinear version of modular function spaces [74]. Indeed throughout this book we have seen that a modular function space is a vector space endowed with a modular function. Therefore it is natural to consider a nonlinear version of function modular spaces. The first to consider such generalization was V. Chistyakov [46,47]. Informally speaking, whereas a metric on a set represents nonnegative finite distances between any two points of the set, a modular on a set attributes a nonnegative (possibly, infinite valued) “field of (generalized) velocities” to each “time” \(\lambda> 0\) (the absolute value of) an average velocity \(w_\lambda(x,y)\) which is associated in such a way that in order to cover the “distance” between points \(x, y \in X\) it takes time λ to move from x to y with velocity \(w_\lambda(x,y)\). The nonlinear approach to modular function spaces was initiated in [1,2,3]

Fixed Point Existence Theorems in Modular Function Spaces

This chapter presents a series of fixed point existence theorems for nonlinear mappings acting in modular functions spaces. We cover a range of different types of mappings including ρ-contractions and their pointwise asymptotic versions and ρ-nonexpansive mappings and pointwise asymptotic ρ-nonexpansive mappings, under various assumptions on the function modular ρ and on the sets these mappings are defined in. In addition we will discuss some examples and applications to the theory of differential equations.

Fixed Point Theory in Metric Spaces: An Introduction

This chapter introduces the general concepts and results of the metric fixed point theory. It sets the foundation for the coming chapters.

Geometry of Modular Function Spaces

This chapter introduces general notions related to the geometry of modular function spaces. We define the modular version of uniform convexity and property (R) which will equip us with powerful tools for proving the fixed point theorems in modular function spaces. The geometrical theory also provides a set of powerful techniques for proving existence of common fixed points for commutative families of mappings acting in modular function spaces, and for investigating the topological properties of the set of common fixed points.

Semigroups of Nonlinear Mappings in Modular Function Spaces

Let us recall that a family \(\{T_t\}_{t \geq 0}\) of mappings forms a semigroup if \(T_0(x)=x\), and \(T_{s+t}=T_s(T_t(x))\). Such a situation is quite typical in mathematics and applications. For instance, in the theory of dynamical systems, the modular function space \(L_{\rho}\) would define the state space and the mapping \((t,x)\rightarrow T_t(x)\) would represent the evolution function of a dynamical system. The question about the existence of common fixed points, and about the structure of the set of common fixed points, can be interpreted as a question whether there exist points that are fixed during the state space transformation T t at any given point of time t, and if yes - what the structure of a set of such points may look like. In the setting of this chapter, the state space may be an infinite dimensional. Therefore, it is natural to apply these result to not only to deterministic dynamical systems but also to stochastic dynamical systems.

Fixed Point Construction Processes

Assume \(\rho \in \Re\) is \((UUC1)\). Let C be a ρ-closed ρ-bounded convex nonempty subset of \(L_{\rho}\). Let \(T: C\rightarrow C\) be a pointwise asymptotically nonexpansive mapping. According to Theorem 5.7 the mapping T has a fixed point. The proof of this important theorem is of the existential nature and does not describe any algorithm for constructing a fixed point of an asymptotic pointwise ρ-nonexpansive mapping. This chapter aims at filling this gap. Therefore, we will define iterative processes for the fixed point construction in modular function spaces and we will prove their convergence. These algorithms will be based on classical iterative methods introduced originally by Mann in [161] and Ishikawa [97], see also Section 2.6 of this book. The results of the current section draw mostly on the research exposed in [54].

On the convergence of iteration processes for semigroups of nonlinear mappings in modular function spaces

Let C be a ρ-bounded, ρ-closed, convexsubset of a modular function space . We investigate the problem of constructingcommon fixed points for asymptotic pointwise nonexpansive semigroups of mappings, i.e. a family such that, , and , where , for every .MSC: 47H09, 46B20, 47H10, 47E10.

Common fixed points for pointwise Lipschitzian semigroups in modular function spaces

AbstractLet C be a ρ-bounded, ρ-closed, convex subset of a modular function space Lρ. We investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups of nonlinear mappings Tt:C→C, i.e. a family such that T0(f)=f, Ts+t(f)=Ts∘Tt(f) and ρ(T(f)−T(g))≤αt(f)ρ(f−g), where lim supt→∞αt(f)≤1 for every f∈C. In particular, we prove that if Lρ is uniformly convex, then the common fixed point is nonempty ρ-closed and convex.MSC:47H09, 46B20, 47H10, 47E10.