Journal of Applied Mathematics
Volume 2012 (2012), Article ID 417234, 13 pages
http://dx.doi.org/10.1155/2012/417234
Research Article

Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

Dan Zhang, Xiaolong Qin, and Feng Gu

Department of Mathematics, Institute of Applied Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

Received 14 October 2011; Accepted 11 December 2011

Academic Editor: Rudong Chen

Copyright © 2012 Dan Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝐻 be a real Hilbert space. Consider on 𝐻 a nonexpansive semigroup 𝑆 = { 𝑇 ( 𝑠 ) ∶ 0 ≤ 𝑠 < ∞ } with a common fixed point, a contraction 𝑓 with the coefficient 0 < 𝛼 < 1 , and a strongly positive linear bounded self-adjoint operator 𝐴 with the coefficient 𝛾 >  0. Let 0 < 𝛾 < 𝛾 / 𝛼 . It is proved that the sequence { 𝑥 𝑛 } generated by the iterative method 𝑥 0 ∈ 𝐻 , 𝑥 𝑛 + 1 = 𝛼 𝑛 𝛾 𝑓 ( 𝑥 𝑛 ) + 𝛽 𝑛 𝑥 𝑛 + ( ( 1 − 𝛽 𝑛 ) 𝐼 − 𝛼 𝑛 𝐴 ) ( 1 / 𝑠 𝑛 ) ∫ 𝑠 𝑛 0 𝑇 ( 𝑠 ) 𝑥 𝑛 𝑑 𝑠 , 𝑛 ≥ 0 converges strongly to a common fixed point 𝑥 ∗ ∈ 𝐹 ( 𝑆 ) , where 𝐹 ( 𝑆 ) denotes the common fixed point of the nonexpansive semigroup. The point 𝑥 ∗ solves the variational inequality ⟨ ( 𝛾 𝑓 − 𝐴 ) 𝑥 ∗ , 𝑥 − 𝑥 ∗ ⟩ ≤ 0 for all 𝑥 ∈ 𝐹 ( 𝑆 ) .