Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 547828, 9 pages
doi:10.1155/2010/547828
Research Article

Convergence of Paths for Perturbed Maximal Monotone Mappings in Hilbert Spaces

Yuan Qing,1 Xiaolong Qin,1 Haiyun Zhou,2 and Shin Min Kang3

1Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
2Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China
3Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 16 July 2010; Revised 30 November 2010; Accepted 20 December 2010

Academic Editor: Ljubomir B. Ciric

Copyright © 2010 Yuan Qing 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 Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻 . Let 𝐴 ∶ 𝐶 → 𝐻 be a maximal monotone mapping and 𝑓 ∶ 𝐶 → 𝐶 a bounded demicontinuous strong pseudocontraction. Let { 𝑥 𝑡 } be the unique solution to the equation 𝑓 ( 𝑥 ) = 𝑥 + 𝑡 𝐴 𝑥 . Then { 𝑥 𝑡 } is bounded if and only if { 𝑥 𝑡 } converges strongly to a zero point of A as 𝑡 → ∞ which is the unique solution in 𝐴 − 1 ( 0 ) , where 𝐴 − 1 ( 0 ) denotes the zero set of 𝐴 , to the following variational inequality ⟨ 𝑓 ( 𝑝 ) − 𝑝 , 𝑦 − 𝑝 ⟩ ≤ 0 , for all 𝑦 ∈ 𝐴 − 1 ( 0 ) .