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We extract quantitative information (specifically, a rate of metastability in the sense of Terence Tao) from a proof due to Kazuo Kobayasi and Isao Miyadera, which shows strong convergence for Cesàro means of non-expansive maps on Banach spaces.

In [6] and [7], Krasnosel'skiĭ proved several fundamental fixed point principles for operators leaving invariant a cone in a Banach space. In [9], Nussbaum extended one of the results, the theorem about compression and expansion of a cone, to k-setcontraction maps, k < 1. Other versions for completely continuous maps were given by Fournier-Peitgen [2] and G. Fournier [1].

The purpose of this paper is to generalise some of these results to upper semi continuous multivalued maps which are K-set contractions, k < 1, and differentiable at the origin and infinity.

We first prove a nonlinear ergodic theorem for nonexpansive semigroups without convexity in a Hilbert space. Further we prove a fixed point theorem for non-expansive semigroups without convexity which generalizes simultaneously fixed point theorems for left amenable semigroups and left reversible semigroups.

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Bounds for a nonlinear ergodic theorem for Banach spaces

Fixed Point Principles for Cones of a Banach Space for the Multivalued Maps Differentiable at the Origin and Infinity

Fixed Point Theorem and Nonlinear Ergodic Theorem for Nonexpansive Semigroups Without Convexity

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3 results

Bounds for a nonlinear ergodic theorem for Banach spaces

Fixed Point Principles for Cones of a Banach Space for the Multivalued Maps Differentiable at the Origin and Infinity

Fixed Point Theorem and Nonlinear Ergodic Theorem for Nonexpansive Semigroups Without Convexity