/pdf/a-convexity-in-metric-space-and-nonexpansive-mappings-i-1swzxbvu36.pdf

A convexity in metric space and nonexpansive mappings. I.

In previous papers we have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis. Here 'uniform' means independence from parameters in compact spaces. A recent case study in fixed point theory systematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result. Our metatheorems deal with general classes of spaces like metric spaces, hyperbolic spaces, CAT(0)-spaces, normed linear spaces, uniformly convex spaces, as well as inner product spaces.

Some logical metatheorems with applications in functional analysis

A quadratic rate of asymptotic regularity for CAT(0)-spaces

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A common fixed point theorem for a commuting family of nonexpansive mappings

11 Tx Ty 11 0) be a family of nonexpansive mappings of C into itself such that S(0) = I, S(t + s) = S(t)S(s) for all t, s E [0, oo) and S(t)x is continuous in t E [0, oo) for each x E C. Then, S is called a nonexpansive semigroup on C. The fixed point set F(S) of S is defined by

/pdf/a-nonlinear-ergodic-theorem-for-an-amenable-semigroup-of-2l5qz1btn6.pdf

A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space

Let K be a subset of a Banach space B. A mapping 5 of K into B is said to be nonexpansive if for each pair of elements x and y of K, we have \\\\sx—sy\\\\^\\\\x—y\\\\. Kakutani [5] and Markov [7] proved the following theorem: Let K be a compact convex subset of a locally convex linear topological space B and S be a commuting family of linear continuous mappings of K into itself. Then S has a common fixed point in K. Day [2] showed that this is true even if S is an amenable semigroup. On the other hand, de Marr [3] proved a fixed point theorem for a family of nonlinear mappings: Let K be a nonempty compact convex subset of a Banach space B. If S is a nonempty commutative family of nonexpansive mappings of K into itself, then the family S has a common fixed point in K. The question naturally arises as to whether this is true if one considers an amenable semigroup of nonexpansive mappings. In this paper, we shall show that the answer is affirmative.

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Fixed point theorem for amenable semigroup of nonexpansive mappings

/pdf/invariant-functions-for-amenable-semigroups-of-positive-398q0d8ol5.pdf

Invariant functions for amenable semigroups of positive contractions on $L^{1}$

/pdf/invariant-ideals-for-amenable-semigroups-of-markov-operators-fk16coew5c.pdf

Wataru Takahashi

Papers

A convexity in metric space and nonexpansive mappings. I.

Fixed point theorem for amenable semigroup of nonexpansive mappings

Invariant functions for amenable semigroups of positive contractions on $L^{1}$

Invariant ideals for amenable semigroups of Markov operators