Uniform mean convergence theorems for hybrid mappings in Hilbert spaces

Koji Aoyama,Fumiaki Kohsaka

doi:10.1186/1687-1812-2012-193

Koji Aoyama, Fumiaki Kohsaka

Open Access

https://doi.org/10.1186/1687-1812-2012-193

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Abstract

Using the notion of sequences of means on the Banach space of all bounded real sequences, we prove mean and uniform mean convergence theorems for pointwise convergent sequences of hybrid mappings in Hilbert spaces.MSC:47H25, 47H09, 47H10, 40H05.

Highlights

1 Introduction Using the notion of asymptotically invariant sequences of means on l∞, we obtain a mean convergence theorem for pointwise convergent sequences of hybrid mappings in Hilbert spaces
Akatsuka, Aoyama, and Takahashi [ ] obtained another generalization of Baillon’s ergodic theorem for pointwise convergent sequences of nonexpansive mappings in Hilbert spaces. Their result was applied to the problem of approximating common fixed points of countable families of nonexpansive mappings
In Section, we prove mean convergence theorems by using sequences of means on l∞; see Theorems . and

Summary

Introduction

Using the notion of asymptotically invariant sequences of means on l∞, we obtain a mean convergence theorem for pointwise convergent sequences of hybrid mappings in Hilbert spaces. The following hold: (i) if {Snx} is bounded, F(S) is nonempty and G({Skx}, μ) is a fixed point of S for each Banach limit μ; (ii) if F(S) is nonempty, {Snx} is bounded, {PF(S)Snx} is strongly convergent, and. As in the proof of [ , the corollary of Theorem ], we can show the following uniform mean convergence theorem in the case when the strong regularity of {μn} is assumed. (i) if {μn} is asymptotically invariant, the sequence {Gμn ({Skx})}n∈N converges weakly to the strong limit of {PF(S)Snx};. (ii) if {μn} is strongly regular, the sequence {Gμn ({Sk+px}k)}n,p∈N converges weakly to the strong limit of {PF(S)Snx} as n → ∞ uniformly in p ∈ N. In order to obtain our final result, we need the following theorem, which was originally shown in strictly convex Banach spaces. {Tn} a sequence of nonexpansive mappings of C into H such that is nonempty, and (γn) a sequence of ( , ) such that γk

Then the mapping

By and the definitions of and