This article is part of the series Recent Trends on Fixed Point Theory and its Applications.

Open Access Research

Strong convergence of a modified iterative algorithm for hierarchical fixed point problems and variational inequalities

Yuanheng Wang1* and Wei Xu2

Author Affiliations

1 Department of Mathematics, Zhejiang Normal University, Zhejiang, 321004, China

2 Department of Mathematics, Tongji Zhejiang College, Zhejiang, 314000, China

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Fixed Point Theory and Applications 2013, 2013:121  doi:10.1186/1687-1812-2013-121


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/121


Received:24 November 2012
Accepted:23 April 2013
Published:7 May 2013

© 2013 Wang and Xu; 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.

Abstract

This article aims to deal with a new modified iterative projection method for solving a hierarchical fixed point problem. It is shown that under certain approximate assumptions of the operators and parameters, the modified iterative sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1">View MathML</a> converges strongly to a fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M2">View MathML</a> of T, also the solution of a variational inequality. As a special case, this projection method solves some quadratic minimization problem. The results here improve and extend some recent corresponding results by other authors.

MSC: 47H10, 47J20, 47H09, 47H05.

Keywords:
hierarchical fixed point; nonexpansive mapping; Lipschitzian and strongly monotone mapping; quadratic minimization; modified iterative projection algorithm

1 Introduction

Let Ω be a nonempty closed convex subset of a real Hilbert space H with the inner product <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M3">View MathML</a> and the norm <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M4">View MathML</a>. Recall that a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M5">View MathML</a> is called L-Lipschitzian if there exits a constant L such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M6">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M7">View MathML</a>. In particular, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M8">View MathML</a>, then T is said to be a contraction; if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M9">View MathML</a>, then T is called a nonexpansive mapping. We denote by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M10">View MathML</a> the set of the fixed points of T, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M11">View MathML</a>.

A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M12">View MathML</a> is called η-strongly monotone if there exists a constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M13">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M14">View MathML</a>

In particular, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M15">View MathML</a>, then F is said to be monotone.

A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M16">View MathML</a> is called a metric projection if there exists a unique nearest point in Ω denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M17">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M18">View MathML</a>

Recently many authors investigated the fixed point problem of nonexpansive mappings, generalized nonexpansive mappings with C-conditions, a family of finite or infinite nonexpansive mappings and pseudo-contractions and obtained many useful results; see, for example, [1-12] and the references therein.

Now, we focus on the following problem.

To find a hierarchical fixed point of T with respect to another operator S is to find an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M19">View MathML</a> satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M20">View MathML</a>

(1)

which is equivalent to the following fixed point problem: to find an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M21">View MathML</a> that satisfies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M22">View MathML</a>. We know that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M10">View MathML</a> is closed and convex, so the metric projection <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M24">View MathML</a> is well defined.

It is well known that the iterative methods for finding hierarchical fixed points of nonexpansive mappings can also be used to solve a convex minimization problem; see, for example, [13,14] and the references therein. In 2006, Marino and Xu [15] considered the following general iterative method:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M25">View MathML</a>

(2)

where f is a contraction, T is a nonexpansive mapping, A is a bounded linear strongly positive operator: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M26">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M27">View MathML</a>, for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M28">View MathML</a>. And it is proved that if the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M29">View MathML</a> of parameters satisfies appropriate conditions, then the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1">View MathML</a> generated by (2) converges strongly to the unique solution of the variational inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M31">View MathML</a>

which is the optimality condition for the minimization problem

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M32">View MathML</a>

where h is a potential function for γf, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M33">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M34">View MathML</a>.

In 2010, Tian [16] introduced the general steepest-descent method

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M35">View MathML</a>

(3)

where F is an L-Lipschitzian and η-strongly monotone operator. Under certain approximate conditions, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1">View MathML</a> generated by (3) converges strongly to a fixed point of T, which solves the variational inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M37">View MathML</a>

Very recently, Ceng et al.[17] investigated the following iterative method:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M38">View MathML</a>

(4)

where U is a Lipschitzian (possibly non-self) mapping, and F is a Lipschitzian and strongly monotone mapping. They proved that under some approximate assumptions on the operators and parameters, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1">View MathML</a> generated by (4) converges strongly to the unique solution of the variational inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M40">View MathML</a>

(5)

On the other hand, in 2010, Yao et al.[18] investigated an iterative method for a hierarchical fixed point problem by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M41">View MathML</a>

(6)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M42">View MathML</a> is a nonexpansive mapping. Under some approximate assumptions of the parameters, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1">View MathML</a> generated by (6) converges strongly to the unique solution of the variational inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M44">View MathML</a>

Motivated and inspired by the above research work, we introduce the following modified iterative method for a hierarchical fixed point problem:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M45">View MathML</a>

(7)

where S, T are nonexpansive mappings with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M46">View MathML</a>, U is a γ-Lipschitzian (possibly non-self) mapping, F is an L-Lipschitzian and η-strongly monotone operator. We prove that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1">View MathML</a> generated by (7) converges strongly to the unique solution of the variational inequality (5) if the operators and parameters satisfy some approximate conditions. As a special case, this projection method also solves the quadratic minimization problem <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M48">View MathML</a>.

Obviously, (2), (3), (4) and (6) are some special cases of (7), respectively. So, our results improve and extend many recent corresponding results of other authors such as [5,13,15-19].

2 Preliminaries

This section contains some lemmas which will be used in the proofs of our main results in the following section.

Lemma 2.1[18]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M49">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M50">View MathML</a>be any points. The following results hold.

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M51">View MathML</a>is nonexpansive and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M52">View MathML</a>if and only if the following relation holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M53">View MathML</a>

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M52">View MathML</a>if and only if the following relation holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M55">View MathML</a>

Lemma 2.2[4]

LetHbe a real Hilbert space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M56">View MathML</a>, the following inequality holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M57">View MathML</a>

Lemma 2.3[17]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M58">View MathML</a>be aγ-Lipschitzian mapping with a constant<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M59">View MathML</a>and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M12">View MathML</a>be ak-Lipschitzian andη-strongly monotone mapping with constants<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M61">View MathML</a>, then for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M62">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M63">View MathML</a>

That is to say, the operator<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M64">View MathML</a>is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M65">View MathML</a>-strongly monotone.

Lemma 2.4[10] (Demiclosedness principle)

Let Ω be a nonempty closed convex subset of a real Hilbert spaceHand let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M66">View MathML</a>be a nonexpansive mapping with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M67">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1">View MathML</a>is a sequence in Ω weakly converging toxand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M69">View MathML</a>converges strongly toy, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M70">View MathML</a>. In particular, if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M71">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M72">View MathML</a>.

Lemma 2.5[19]

Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M73">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M74">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M12">View MathML</a>be anL-Lipschitzian andη-strongly monotone operator with constants<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M76">View MathML</a>. In association with a nonexpansive mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M77">View MathML</a>, define the mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M78">View MathML</a>by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M79">View MathML</a>

Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M80">View MathML</a>is a contraction provided<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M81">View MathML</a>, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M82">View MathML</a>

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M83">View MathML</a>.

Lemma 2.6[20]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M84">View MathML</a>be a sequence of nonnegative real numbers satisfying the following relation:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M85">View MathML</a>

with conditions

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M86">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M87">View MathML</a>;

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M88">View MathML</a>or<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M89">View MathML</a>.

Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M90">View MathML</a>.

3 Main results

Theorem 3.1Let Ω be a nonempty closed convex subset of a real Hilbert spaceHand let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M91">View MathML</a>be any given initial guess. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M92">View MathML</a>be nonexpansive mappings such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M46">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M12">View MathML</a>be anL-Lipschitzian andη-strongly monotone (possibly non-self) operator with coefficients<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M76">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M58">View MathML</a>be aγ-Lipschitzian (possibly non-self) mapping with a coefficient<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M59">View MathML</a>. Suppose the parameters satisfy<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M98">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M99">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M100">View MathML</a>. And suppose the sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M101">View MathML</a>satisfy the following conditions:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M102">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M103">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M104">View MathML</a>;

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M105">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M106">View MathML</a>.

Then the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1">View MathML</a>generated by (7) converges strongly to a fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M2">View MathML</a>ofT, which is the unique solution of the variational inequality (5). In particular, if we take<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M109">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M110">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M111">View MathML</a>defined by (7) converges in norm to the minimum norm fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M112">View MathML</a>of T, namely, the point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M112">View MathML</a>is the unique solution to the quadratic minimization problem<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M114">View MathML</a>.

Proof We divide the proof into six steps.

Step 1. We first show that the variational inequality (5) has only one solution. Observe that the constants satisfy <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M99">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M116">View MathML</a>

therefore the operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M64">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M118">View MathML</a>-strongly monotone, and we get the uniqueness of the solution of the variational inequality (5) and denote it by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M19">View MathML</a>.

Step 2. Then we get that the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M121">View MathML</a> are bounded. By condition (ii), without loss of generality, we may assume <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M122">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M123">View MathML</a>. Taking a fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M124">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M125">View MathML</a>

(8)

On the other hand, denoting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M126">View MathML</a>, from (7) we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M127">View MathML</a>

(9)

Together with (8) and (9), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M128">View MathML</a>

Hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M129">View MathML</a>

We get the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1">View MathML</a> is bounded, and so are <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M121">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M132">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M133">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M134">View MathML</a><a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M135">View MathML</a>.

Step 3. Next we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M136">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M137">View MathML</a>. Estimate <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M138">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M139">View MathML</a>

(10)

where M is a constant such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M140">View MathML</a>

(11)

Substituting (10) into (11), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M141">View MathML</a>

Notice the conditions (i) and (iii), by Lemma 2.6, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M142">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M137">View MathML</a>.

Step 4. Next we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M144">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M137">View MathML</a>.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M146">View MathML</a>

Notice that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M147">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M148">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M149">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M150">View MathML</a> are bounded, and we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M151">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M137">View MathML</a>.

Step 5. Now we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M153">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M2">View MathML</a> is the unique solution of the variational inequality. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1">View MathML</a> is bounded, we take a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M156">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M1">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M158">View MathML</a>

and we assume <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M159">View MathML</a>. By Lemma 2.4, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M160">View MathML</a>. Therefore

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M161">View MathML</a>

Hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M162">View MathML</a>

(12)

On the other hand, taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M163">View MathML</a> in (8), we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M164">View MathML</a>. Together with (12), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M165">View MathML</a>

which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M166">View MathML</a>

By the conditions (i) and (ii), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M167">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M168">View MathML</a>

According to Lemma 2.6, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M169">View MathML</a>.

Step 6. In particular, if we take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M109">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M110">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M172">View MathML</a>, which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M2">View MathML</a> is the minimum norm fixed point of T and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M2">View MathML</a> satisfies the variational inequality (5)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M175">View MathML</a>

So, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M176">View MathML</a>, we deduce <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M177">View MathML</a>, i.e., the point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M2">View MathML</a> is the unique solution to the quadratic minimization problem <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M114">View MathML</a>. This completes the proof. □

Remark 3.1 Prototypes for the iteration parameters in Theorem 3.1 are, for example, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M180">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M181">View MathML</a> (with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M182">View MathML</a>). It is not difficult to prove that the conditions (i)-(iii) are satisfied.

Remark 3.2 Our Theorem 3.1 improves and extends many recent corresponding main results of other authors (see, for example, [5,13,15-19]) in the following ways:

(a) Some self-mappings in other papers (see [15,16,19]) are extended to the cases of non-self-mappings. Such as the self-contraction mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M183">View MathML</a> in [15,16,19] is extended to the case of a Lipschitzian (possibly non-self-)mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M184">View MathML</a> on a nonempty closed convex subset C of H. The Lipschitzian and strongly monotone (self-)mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M185">View MathML</a> in [16] is extended to the case of a Lipschitzian and strongly monotone (possibly non-self-)mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M186">View MathML</a>.

(b) The contractive mapping f with a coefficient <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M187">View MathML</a> in other papers (see [15,16,18,19]) is extended to the cases of the Lipschitzian mapping U with a coefficient constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M188">View MathML</a>.

(c) The Mann-type iterative format in [15-17,19] has been extended to the Ishikawa-type iterative format (7) in our Theorem 3.1. So, their iterative formats (2), (3), (4) and (6) are some special cases of our iterative format (7), and some of their main results have been included in our Theorem 3.1, respectively.

(d) The iterative approximating fixed point of T in Theorem 3.1 is also the unique solution of the variational inequality (5). In fact, (5) is a hierarchical fixed point problem which closely relates to a convex minimization problem. In hierarchical fixed point problem (1), if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M189">View MathML</a>, then we can get the variational inequality (5). In (5), if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M190">View MathML</a>, then we get the variational inequality <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M191">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M176">View MathML</a>, which just is the variational inequality studied by Suzuki [19]. If the Lipschitzian mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M193">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M194">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M195">View MathML</a>, we get the variational inequality <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M196">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M176">View MathML</a>, which is the variational inequality studied by Yao et al.[18]. So, the results of Theorem 3.1 in this paper have many useful applications such as the quadratic minimization problem <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/121/mathml/M114">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in this research work. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the article. This study was supported by the National Natural Science Foundations of China (Grant Nos. 11271330, 11071169 ), the Natural Science Foundations of Zhejiang Province of China (Grant No. Y6100696).

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