## Annals of Functional Analysis

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

### 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 $\left\{T\right(s){\}}_{s>0}$ is a nonexpansive semigroup on $K$. By using the following implicit iteration sequence $\left\{{x}_{n}\right\}$ defined by $${x}_{n}=(1-{\alpha}_{n}){x}_{n-1}+{\alpha}_{n}\cdot \frac{1}{{t}_{n}}{\int}_{0}^{{t}_{n}}T\left(s\right){x}_{n}\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}\forall n\ge 1,$$ the main purpose of this paper is to establish a weak convergence theorem for the nonexpansive semigroup $\left\{T\right(s){\}}_{s>0}$ 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**

Received: 27 May 2016

Accepted: 26 October 2016

First available in Project Euclid: 22 April 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/20088752-0000018X

**Mathematical Reviews number (MathSciNet)**

MR3689997

**Zentralblatt MATH identifier**

06754361

**Subjects**

Primary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc.

Secondary: 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]

**Keywords**

nonexpansive semigroups fixed point implicit iteration scheme uniformly convex Banach spaces

#### 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