## Annals of Functional Analysis

### On the weak convergence theorem for nonexpansive semigroups in Banach spaces

#### Abstract

Assume that $K$ is a closed convex subset of a uniformly convex Banach space $E$, and assume that $\{T(s)\}_{s\gt 0}$ is a nonexpansive semigroup on $K$. By using the following implicit iteration sequence $\{x_{n}\}$ defined by $$x_{n}=(1-\alpha_{n})x_{n-1}+\alpha_{n}\cdot\frac{1}{t_{n}}\int _{0}^{t_{n}}T(s)x_{n}\,ds,\quad\forall n\geq1,$$ the main purpose of this paper is to establish a weak convergence theorem for the nonexpansive semigroup $\{T(s)\}_{s\gt 0}$ in uniformly convex Banach spaces without the Opial property. Our results are different from some recently announced results.

#### Article information

Source
Ann. Funct. Anal., Volume 8, Number 3 (2017), 341-349.

Dates
Accepted: 26 October 2016
First available in Project Euclid: 22 April 2017

https://projecteuclid.org/euclid.afa/1492826604

Digital Object Identifier
doi:10.1215/20088752-0000018X

Mathematical Reviews number (MathSciNet)
MR3689997

Zentralblatt MATH identifier
06754361

#### Citation

Yao, Rongjie; Yang, Liping. On the weak convergence theorem for nonexpansive semigroups in Banach spaces. Ann. Funct. Anal. 8 (2017), no. 3, 341--349. doi:10.1215/20088752-0000018X. https://projecteuclid.org/euclid.afa/1492826604

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