Uniform mean convergence theorems for hybrid mappings in Hilbert spaces

Aoyama Koji, Kohsaka, Kohsaka Fumiaki

Description

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. 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.

Sujets

• Ergodic theory
• Metric map

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 4
• Langue: English

Extrait

Aoyama and KohsakaFixed Point Theory and Applications2012,2012:193 http://www.fixedpointtheoryandapplications.com/content/2012/1/193

R E S E A R C HOpen Access Uniform mean convergence theorems for hybrid mappings in Hilbert spaces 1 2* Koji Aoyamaand Fumiaki Kohsaka

* Correspondence: [e-mail] 2 Department of Computer Science and Intelligent Systems, Oita University, Dannoharu, Oita-shi, Oita, 870-1192, Japan Full list of author information is available at the end of the article

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:Primary 47H25; 47H09; secondary 47H10; 40H05 Keywords:ergodic theorem; hybrid mapping; nonexpansive mapping; nonspreading mapping; uniform mean convergence theorem

1 Introduction ∞ Using the notion of asymptotically invariant sequences of means onl, we obtain a mean convergence theorem for pointwise convergent sequences of hybrid mappings in Hilbert spaces. By assuming the strong regularity on the sequences of means, we also obtain a uniform mean convergence theorem. In , Baillon [] established a nonlinear ergodic theorem for nonexpansive mappings in Hilbert spaces. Several results related to Baillon’s ergodic theorem have been obtained since then; see, for instance, [–] and the references therein. Especially, using the notion of asymptotically invariant nets of means on semitopological semigroups, Hirano, Kido, and Takahashi [] and Lau, Shioji, and Takahashi [] generalized Baillon’s ergodic theorem to commutative and noncommutative semigroups of nonexpansive mappings in Banach spaces, respectively. On the other hand, Akatsuka, Aoyama, and Takahashi [] obtained another general- ization 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. Recently, the au- thors [] generalized some results in [] for pointwise convergent sequences of hybrid mappings in the sense of []. The aim of the present paper is to obtain further generalizations of the results in [, ] ∞ by using a sequence{µ n }of means onl. In particular, by assuming the strong regularity on{µ n }, we prove a uniform mean convergence theorem (Theorem .) for pointwise convergent sequences of hybrid mappings in Hilbert spaces. Our paper is organized as follows. In Section , we recall some definitions and some pre- liminary results. In Section , we prove mean convergence theorems by using sequences ∞ of means onl; see Theorems . and .. In Section , we obtain some consequences of Theorem .; see Theorems ., ., and .. In Section , we give two applications of Theorem ..

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