## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2012, Special Issue (2012), Article ID 417234, 13 pages.

### Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

#### Abstract

Let $H$ be a real Hilbert space. Consider on $H$ a nonexpansive semigroup $S=\{T(s):0\le s<\infty \}$ with a common fixed point, a contraction $f$ with the coefficient $0<\alpha <1$, and a strongly positive linear bounded self-adjoint operator $A$ with the coefficient $\overline{\gamma }$>  0. Let $0<\gamma <\overline{\gamma }$/$\alpha$. It is proved that the sequence $\{{x}_{n}\}$ generated by the iterative method ${x}_{0}\in H, {x}_{n+1}={\alpha }_{n}\gamma f({x}_{n})+{\beta }_{n}{x}_{n}+((1-{\beta }_{n})I-{\alpha }_{n}A)(1/{s}_{n}){\int }_{0}^{{s}_{n}}T(s){x}_{n}ds, n\ge 0$ converges strongly to a common fixed point ${x}^{*}\in F(S)$, where $F(S)$ denotes the common fixed point of the nonexpansive semigroup. The point ${x}^{*}$ solves the variational inequality $〈(\gamma f-A){x}^{*},x-{x}^{*}〉\le 0$ for all $x\in F(S)$.

#### Article information

Source
J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 417234, 13 pages.

Dates
First available in Project Euclid: 3 January 2013

https://projecteuclid.org/euclid.jam/1357180293

Digital Object Identifier
doi:10.1155/2012/417234

Mathematical Reviews number (MathSciNet)
MR2880849

Zentralblatt MATH identifier
1250.47079

#### Citation

Zhang, Dan; Qin, Xiaolong; Gu, Feng. Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces. J. Appl. Math. 2012, Special Issue (2012), Article ID 417234, 13 pages. doi:10.1155/2012/417234. https://projecteuclid.org/euclid.jam/1357180293

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