Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequality prob
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Hierarchical convergence of an implicit doublenet algorithm for nonexpansive semigroups and variational inequality problems Yonghong Yao1, Yeol Je Cho2* and Yeong-Cheng Liou3 * Correspondence: [email protected]. kr 2 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea Full list of author information is available at the end of the article
Abstract In this paper, we show the hierarchical convergence of the following implicit doublenet algorithm: 1 xs,t = s[tf (xs,t ) + (1 − t)(xs,t − μAxs,t )] + (1 − s) λs
λs
T(v)xs,t dν,
∀s, t ∈ (0, 1),
0
where f is a r-contraction on a real Hilbert space H, A : H ® H is an a-inverse strongly monotone mapping and S = {T(s)}s ≥ 0: H ® H is a nonexpansive semigroup with the common fixed points set Fix(S) ≠ ∅, where Fix(S) denotes the set of fixed points of the mapping S, and, for each fixed t Î (0, 1), the net {xs, t} converges in norm as s ® 0 to a common fixed point xt Î Fix(S) of {T(s)}s ≥ 0and, as t ® 0, the net {xt} converges in norm to the solution x* of the following variational inequality: ∗ x ∈ Fix(S); Ax∗ , x − x∗ ≥ 0, ∀x ∈ Fix(S). MSC(2000): 49J40; 47J20; 47H09; 65J15. Keywords: fixed point, variational inequality, double-net algorithm, hierarchical convergence, Hilbert space
1 Introduction In nonlinear analysis, a common approach to solving a problem with multiple solutions is to replace it by a family of perturbed problems admitting a unique solution and to obtain a particular solution as the limit of these perturbed solutions when the perturbation vanishes. In this paper, we introduce a more general approach which consists in finding a particular part of the solution set of a given fixed point problem, i.e., fixed points which solve a variational inequality. More precisely, the goal of this paper is to present a method for finding hierarchically a fixed point of a nonexpansive semigroup S = {T(s)}s ≥ 0 with respect to another monotone operator A, namely, Find x* Î Fix(S) such that Ax∗ , x − x∗ ≥ 0,
∀x ∈ Fix(S).
(1:1)
© 2011 Yao et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Yao et al. Fixed Point Theory and Applications 2011, 2011:101 http://www.fixedpointtheoryandapplications.com/content/2011/1/101
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This is an interesting topic due to the fact that it is closely related to convex programming problems. For the related works, refer to [1-19]. This paper is devoted to solve the problem (1.1). For this purpose, we propose a double-net algorithm which generates a net {xs,t} and prove that the net {xs,t} hierarchically converges to the solution of the problem (1.1), that is, for each fixed t Î (0, 1), the net {xs,t} converges in norm as s ® 0 to a common fixed point xt Î Fix(S) of the nonexpansive semigroup {T(
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