Fixed Point Theory and Applications
Volume 2011 (2011), Article ID 748918, 17 pages
doi:10.1155/2011/748918
Research Article

An Implicit Hierarchical Fixed-Point Approach to General Variational Inequalities in Hilbert Spaces

L. C. Zeng,1,2 Ching-Feng Wen,3 and J. C. Yao3

1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
3Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Received 13 December 2010; Accepted 5 March 2011

Academic Editor: Jong Kim

Copyright © 2011 L. C. Zeng 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 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 . Let 𝐹 ∶ 𝐶 → 𝐻 be a 𝜅 -Lipschitzian and 𝜂 -strongly monotone operator with constants 𝜅 , 𝜂 > 0 , 𝑉 , 𝑇 ∶ 𝐶 → 𝐶 be nonexpansive mappings with F i x ( 𝑇 ) ≠ ∅ where F i x ( 𝑇 ) denotes the fixed-point set of 𝑇 , and 𝑓 ∶ 𝐶 → 𝐻 be a 𝜌 -contraction with coefficient 𝜌 ∈ [ 0 , 1 ) . Let 0 < 𝜇 < 2 𝜂 / 𝜅 2 and 0 < 𝛾 ≤ 𝜏 , where 𝜏 = 1 − √ 1 − 𝜇 ( 2 𝜂 − 𝜇 𝜅 2 ) . For each 𝑠 , 𝑡 ∈ ( 0 , 1 ) , let 𝑥 𝑠 , 𝑡 be a unique solution of the fixed-point equation 𝑥 𝑠 , 𝑡 = 𝑃 𝐶 [ 𝑠 𝛾 𝑓 ( 𝑥 𝑠 , 𝑡 ) + ( 𝐼 − 𝑠 𝜇 𝐹 ) ( 𝑡 𝑉 + ( 1 − 𝑡 ) 𝑇 ) 𝑥 𝑠 , 𝑡 ] . We derive the following conclusions on the behavior of the net { 𝑥 𝑠 , 𝑡 } along the curve 𝑡 = 𝑡 ( 𝑠 ) : (i) if 𝑡 ( 𝑠 ) = 𝑂 ( 𝑠 ) , as 𝑠 → 0 , then 𝑥 𝑠 , 𝑡 ( 𝑠 ) → 𝑧 ∞ strongly, which is the unique solution of the variational inequality of finding 𝑧 ∞ ∈ F i x ( 𝑇 ) such that ⟨ [ ( 𝜇 𝐹 − 𝛾 𝑓 ) + 𝑙 ( 𝐼 − 𝑉 ) ] 𝑧 ∞ , 𝑥 − 𝑧 ∞ ⟩ ≥ 0 , f o r a l l 𝑥 ∈ F i x ( 𝑇 ) and (ii) if 𝑡 ( 𝑠 ) / 𝑠 → ∞ , as 𝑠 → 0 , then 𝑥 𝑠 , 𝑡 ( 𝑠 ) → 𝑥 ∞ strongly, which is the unique solution of some hierarchical variational inequality problem.