A generalization of Suzuki's lemma

B. Panyanak; A. Cuntavepanit

doi:10.1155/2011/824718

A generalization of Suzuki's lemma

B. Panyanak
, A. Cuntavepanit

Chiang Mai University

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

Let {z_n}, {w_n}, and {v_n} be bounded sequences in a metric space of hyperbolic type (X, d), and let {α_n } be a sequence in [0,1] with 0 < lim inf_n α_n < lim sup_nα _n < 1. If z_n+1=α_nw_n (1-α _n)v _n for all n ∈ ℕ , lim_nd (z_n, v _n) = 0, and lim sup_n(d (w_n+1, w_n) - d (z_n+1, z_n)) ≤ 0, then lim_nd (w _n, z_n) = 0. This is a generalization of Lemma 2.2 in (T. Suzuki, 2005). As a consequence, we obtain strong convergence theorems for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces.

Original language	English
Article number	824718
Journal	Abstract and Applied Analysis
Volume	2011
DOIs	https://doi.org/10.1155/2011/824718
Publication status	Published - 2011
Externally published	Yes

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title = "A generalization of Suzuki's lemma",

abstract = "Let \{zn\}, \{wn\}, and \{vn\} be bounded sequences in a metric space of hyperbolic type (X, d), and let \{αn \} be a sequence in [0,1] with 0 < lim infn αn < lim supnα n < 1. If zn+1=αnwn (1-α n)v n for all n ∈ ℕ , limnd (zn, v n) = 0, and lim supn(d (wn+1, wn) - d (zn+1, zn)) ≤ 0, then limnd (w n, zn) = 0. This is a generalization of Lemma 2.2 in (T. Suzuki, 2005). As a consequence, we obtain strong convergence theorems for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces.",

author = "B. Panyanak and A. Cuntavepanit",

year = "2011",

doi = "10.1155/2011/824718",

language = "English",

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journal = "Abstract and Applied Analysis",

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T1 - A generalization of Suzuki's lemma

AU - Panyanak, B.

AU - Cuntavepanit, A.

PY - 2011

Y1 - 2011

N2 - Let {zn}, {wn}, and {vn} be bounded sequences in a metric space of hyperbolic type (X, d), and let {αn } be a sequence in [0,1] with 0 < lim infn αn < lim supnα n < 1. If zn+1=αnwn (1-α n)v n for all n ∈ ℕ , limnd (zn, v n) = 0, and lim supn(d (wn+1, wn) - d (zn+1, zn)) ≤ 0, then limnd (w n, zn) = 0. This is a generalization of Lemma 2.2 in (T. Suzuki, 2005). As a consequence, we obtain strong convergence theorems for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces.

AB - Let {zn}, {wn}, and {vn} be bounded sequences in a metric space of hyperbolic type (X, d), and let {αn } be a sequence in [0,1] with 0 < lim infn αn < lim supnα n < 1. If zn+1=αnwn (1-α n)v n for all n ∈ ℕ , limnd (zn, v n) = 0, and lim supn(d (wn+1, wn) - d (zn+1, zn)) ≤ 0, then limnd (w n, zn) = 0. This is a generalization of Lemma 2.2 in (T. Suzuki, 2005). As a consequence, we obtain strong convergence theorems for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces.

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DO - 10.1155/2011/824718

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AN - SCOPUS:80052686272

SN - 1085-3375

VL - 2011

JO - Abstract and Applied Analysis

JF - Abstract and Applied Analysis

M1 - 824718

ER -

A generalization of Suzuki's lemma

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