An algorithm for treating asymptotically strict pseudocontractions and monotone operators
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An algorithm for treating asymptotically strict pseudocontractions and monotone operators Mingliang Zhang* * Correspondence: [email protected] School of Mathematics and Information Science, Henan University, Kaifeng, 475000, China
Abstract In this paper, an algorithm for treating asymptotically κ -strict pseudocontractions and monotone operators is proposed. Convergence analysis of the algorithm is investigated in the framework of Hilbert spaces. Keywords: asymptotically κ -strict pseudocontraction; inverse-strongly monotone mapping; maximal monotone operator; fixed point
1 Introduction-preliminaries In this paper, we are concerned with the problem of finding a common element in the intersection F(T) ∩ (A + B)– (), where F(T) denotes the fixed point set of the mapping T and (A + B)– () denotes the zero point set of the sum of the operator A and the operator B. The motivation for the common element problem is mainly due to its possible applications to mathematical modeling of concrete complex problems. The common element problems include mini-max problems, complementarity problems, equilibrium problems, common fixed point problems and variational inequalities as special cases; see, for example, [–] and the references therein. Throughout the article, we always assume that H is a real Hilbert space with the inner product · , · and the norm · , respectively. Let C be a nonempty closed convex subset of H, and let ProjC be the metric projection from H onto C. Let A : C → H be a mapping. A– () stands for the zero point set of A; that is, A– () := {x ∈ C : Ax = }. Recall that A is said to be monotone iff Ax – Ay, x – y ≥ ,
∀x, y ∈ C.
A is said to be α-strongly monotone iff there exists a constant α > such that Ax – Ay, x – y ≥ αx – y ,
∀x, y ∈ C.
A is said to be α-inverse-strongly monotone iff there exists a constant α > such that Ax – Ay, x – y ≥ αAx – Ay ,
∀x, y ∈ C.
©2014 Zhang; 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.
Zhang Fixed Point Theory and Applications 2014, 2014:52 http://www.fixedpointtheoryandapplications.com/content/2014/1/52
Page 2 of 14
It is not hard to see that α-inverse-strongly monotone mappings are Lipschitz continuous. Indeed, we have αAx – Ay ≤ Ax – Ay, x – y ≤ Ax – Ayx – y. This shows that Ax – Ay ≤ α x – y. Recall that the classical variational inequality, denoted by VI(C, A), is to find u ∈ C such that Au, v – u ≥ ,
∀v ∈ C.
(.)
One can see that the variational inequality (.) is equivalent to a fixed point problem of the mapping ProjC (I – rA), where I is the identity and r is some positive real number. The element u ∈ C is a solution of the variational inequality (.) iff u ∈ C satisfies the equation u = PC (u – rAu). This alternative equivalent formulation has played
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