This article is part of the series Variational Analysis and Fixed Point Theory.

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Hybrid methods for a mixed equilibrium problem and fixed points of a countable family of multivalued nonexpansive mappings

Aunyarat Bunyawat and Suthep Suantai*

Author Affiliations

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

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Fixed Point Theory and Applications 2013, 2013:236  doi:10.1186/1687-1812-2013-236


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/236


Received:7 May 2013
Accepted:6 August 2013
Published:16 September 2013

© 2013 Bunyawat and Suantai; 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

In this paper, we prove a strong convergence theorem for a new hybrid method, using shrinking projection method introduced by Takahashi and a fixed point method for finding a common element of the set of solutions of mixed equilibrium problem and the set of common fixed points of a countable family of multivalued nonexpansive mappings in Hilbert spaces. We also apply our main result to the convex minimization problem and the fixed point problem of a countable family of multivalued nonexpansive mappings.

MSC: 47H09, 47H10.

Keywords:
multivalued nonexpansive mappings; mixed equilibrium problem; shrinking projection method

1 Introduction

The mixed equilibrium problem (MEP) includes several important problems arising in optimization, economics, physics, engineering, transportation, network, Nash equilibrium problems in noncooperative games, and others. Variational inequalities and mathematical programming problems are also viewed as the abstract equilibrium problems (EP) (e.g., [1,2]). Many authors have proposed several methods to solve the EP and MEP, see, for instance, [1-9] and the references therein.

Fixed point problems for multivalued mappings are more difficult than those of single-valued mappings and play very important role in applied science and economics. Recently, many authors have proposed their fixed point methods for finding a fixed point of both multivalued mapping and a family of multivalued mappings. All of those methods have only weak convergence.

It is known that Mann’s iterations have only weak convergence even in the Hilbert spaces. To overcome this problem, Takahashi [10] introduced a new method, known as shrinking projection method, which is a hybrid method of Mann’s iteration, and the projection method, and obtained strong convergence results of such method. In this paper, we use the shrinking projection method defined by Takahashi [10] and our new method to define a new hybrid method for MEP and a fixed point problem for a family of nonexpansive multivalued mappings.

An element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M1">View MathML</a> is called a fixed point of a single-valued mapping T if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M2">View MathML</a> and of a multivalued mapping T if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M3">View MathML</a>. The set of fixed points of T is denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M4">View MathML</a>.

Let X be a real Banach space. A subset K of X is called proximinal if for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M5">View MathML</a>, there exists an element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M6">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M7">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M8">View MathML</a> is the distance from the point x to the set K.

Let X be a uniformly convex real Banach space, and let K be a nonempty closed convex subset of X, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M9">View MathML</a> be a family of nonempty closed bounded subsets of K, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M10">View MathML</a> be a nonempty proximinal bounded subsets of K.

For multivalued mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M11">View MathML</a>, define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M12">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M13">View MathML</a>.

The Hausdorff metric on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M14">View MathML</a> is defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M15">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M16">View MathML</a>.

A multivalued mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M17">View MathML</a> is said to be nonexpansive if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M18">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M19">View MathML</a>.

Let H be a real Hilbert space with the inner product <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M20">View MathML</a> and the norm <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M21">View MathML</a>. Let D be a nonempty closed convex subset of H. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M22">View MathML</a> be a bifunction, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M23">View MathML</a> be a function such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M24">View MathML</a>, where ℝ is the set of real numbers and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M25">View MathML</a>.

Flores-Bazán [11] introduced the following mixed equilibrium problem:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M26">View MathML</a>

(1.1)

The set of solutions of (1.1) is denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M27">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M28">View MathML</a>, then the mixed equilibrium problem (1.1) reduces to the following equilibrium problem:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M29">View MathML</a>

(1.2)

The set of solutions of (1.2) is denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M30">View MathML</a> (see Combettes and Hirstoaga [12]).

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M31">View MathML</a>, then the mixed equilibrium problem (1.1) reduces to the following convex minimization problem:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M32">View MathML</a>

(1.3)

The set of solutions of (1.3) is denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M33">View MathML</a>.

In an infinite-dimensional Hilbert space, the Mann iteration algorithms have only a weak convergence. In 2003, Nakajo and Takahashi [13] introduced the method, called CQ method, to modify Mann’s iteration to obtain the strong convergence theorem for nonexpansive mapping in a Hilbert space. The CQ method has been studied extensively by many authors, for instance, Marino and Xu [14]; Zhou [15]; Zhang and Cheng [16].

In 2008, Takahashi et al.[10] introduced the following iteration scheme, which is usually called the shrinking projection method. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M34">View MathML</a> be a sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M35">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M36">View MathML</a>. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M37">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M38">View MathML</a>, define a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39">View MathML</a> of D as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M40">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M41">View MathML</a> is the metric projection of H onto <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M42">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M43">View MathML</a> is a family of nonexpansive mappings. They proved that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39">View MathML</a> converges strongly to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M45">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M46">View MathML</a>. The shrinking projection method has been studied widely by many authors, for example, Tada and Takahashi [17]; Aoyama et al.[18]; Yao et al.[19]; Kang et al.[20]; Cholamjiak and Suantai [21]; Ceng et al.[22]; Tang et al.[23]; Cai and Bu [24]; Kumam et al.[25]; Kimura et al.[26]; Shehu [27,28]; Wang et al.[29].

In 2009, Wangkeeree and Wangkeeree [30] proved a strong convergence theorem of an iterative algorithm based on extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of common fixed points of a family of infinitely nonexpansive mappings and the set of the variational inequality for a monotone Lipschitz continuous mapping in a Hilbert space.

In 2011, Rodjanadid [31] introduced another iterative method modified from an iterative scheme of Klin-eam and Suantai [32] for finding a common element of the set of solutions of mixed equilibrium problems and the set of common fixed points of countable family of nonexpansive mappings in real Hilbert spaces. The mixed equilibrium problems have been studied by many authors, for instance, Peng and Yao [33]; Zeng et al.[34]; Peng et al.[35]; Wangkeeree and Kamraksa [36]; Jaiboon and Kumam [37]; Chamnarnpan and Kumam [38]; Cholamjiak et al.[39].

Nadler [40] started to study fixed points of multivalued contractions and nonexpansive mapping by using the Hausdorff metric.

Sastry and Babu [41] defined Mann and Ishikawa iterates for a multivalued map T with a fixed point p, and proved that these iterates converge strongly to a fixed point q of T under the compact domain in a real Hilbert space. Moreover, they illustrated that fixed point q may be different from p.

Panyanak [42] generalized results of Sastry and Babu [41] to uniformly convex Banach spaces and proved a strong convergence theorem of Mann iterates for a mapping defined on a noncompact domain and satisfying some conditions. He also obtained a strong convergence result of Ishikawa iterates for a mapping defined on a compact domain.

Hussain and Khan [43], in 2003, introduced the best approximation operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M47">View MathML</a> to find fixed points of ∗-nonexpansive multivalued mapping and proved strong convergence of its iterates on a closed convex unbounded subset of a Hilbert space, which is not necessarily compact.

Hu et al.[44] obtained common fixed point of two nonexpansive multivalued mappings satisfying certain contractive conditions.

Cholamjiak and Suantai [45] proved strong convergence theorems of two new iterative procedures with errors for two quasi-nonexpansive multivalued mappings by using the best approximation operator and the end point condition in uniformly convex Banach spaces. Later, Cholamjiak et al.[46] introduced a modified Mann iteration and obtained weak and strong convergence theorems for a countable family of nonexpansive multivalued mappings by using the best approximation operator in a Banach space. They also gave some examples of multivalued mappings T such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M47">View MathML</a> are nonexpansive.

Later, Eslamian and Abkar [47] generalized and modified the iteration of Abbas et al.[48] from two mappings to the infinite family of multivalued mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M49">View MathML</a> such that each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M50">View MathML</a> satisfies the condition (C).

In this paper, we introduce a new hybrid method for finding a common element of the set of solutions of a mixed equilibrium problem and the set of common fixed points of a countable family of multivalued nonexpansive mappings in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by the proposed method without the assumption of compactness of the domain and other conditions imposing on the mappings.

In Section 2, we give some preliminaries and lemmas, which will be used in proving the main results. In Section 3, we introduce a new hybrid method and a fixed point method defined by (3.1) and prove strong convergence theorem for finding a common element of the set of solutions between mixed equilibrium problem and common fixed point problems of a countable family of multivalued nonexpansive mappings in Hilbert spaces. We also give examples of the control sequences satisfying the control conditions in main results. In Section 4, we summarize the main results of this paper.

2 Preliminaries

Let D be a closed convex subset of H. For every point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M51">View MathML</a>, there exists a unique nearest point in D, denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M52">View MathML</a>, such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M53">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M54">View MathML</a> is called the metric projection of H onto D. It is known that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M54">View MathML</a> is a nonexpansive mapping of H onto D. It is also know that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M54">View MathML</a> satisfies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M57">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M58">View MathML</a>. Moreover, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M52">View MathML</a> is characterized by the properties: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M60">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M61">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M62">View MathML</a>.

Lemma 2.1[13]

LetDbe a nonempty closed convex subset of a real Hilbert spaceHand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M63">View MathML</a>be the metric projection fromHontoD. Then the following inequality holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M64">View MathML</a>

Lemma 2.2[14]

LetHbe a real Hilbert space. Then the following equations hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M65">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M66">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M67">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M68">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M58">View MathML</a>.

Lemma 2.3[21]

LetHbe a real Hilbert space. Then for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M70">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M71">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M72">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M73">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M74">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M75">View MathML</a>.

Lemma 2.4[49]

LetDbe a nonempty closed and convex subset of a real Hilbert spaceH. Given<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M76">View MathML</a>and also given<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M77">View MathML</a>, the set

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M78">View MathML</a>

is convex and closed.

For solving the mixed equilibrium problem, we assume the bifunction F, φ and the set D satisfy the following conditions:

(A1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M79">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M80">View MathML</a>;

(A2) F is monotone, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M81">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M82">View MathML</a>;

(A3) for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M83">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M84">View MathML</a>;

(A4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M85">View MathML</a> is convex and lower semicontinuous for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M80">View MathML</a>;

(B1) for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M51">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M88">View MathML</a>, there exist a bounded subset <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M89">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M90">View MathML</a> such that for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M91">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M92">View MathML</a>

(B2) D is a bounded set.

Lemma 2.5[35]

LetDbe a nonempty closed and convex subset of a real Hilbert spaceH. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M22">View MathML</a>be a bifunction satisfying conditions (A1)-(A4) and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M94">View MathML</a>be a proper lower semicontinuous and convex function such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M95">View MathML</a>. For<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M88">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M80">View MathML</a>, define a mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M98">View MathML</a>as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M99">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M51">View MathML</a>. Assume that either (B1) or (B2) holds. Then the following conclusions hold:

(1) for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M51">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M102">View MathML</a>;

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M103">View MathML</a>is single-valued;

(3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M103">View MathML</a>is firmly nonexpansive, that is, for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M58">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M106">View MathML</a>

(4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M107">View MathML</a>;

(5) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M27">View MathML</a>is closed and convex.

As in ([21], Lemma 2.7), the following lemma holds true for multivalued mapping. To avoid repetition, we omit the details of proof.

Lemma 2.6LetDbe a closed and convex subset of a real Hilbert spaceH. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M109">View MathML</a>be a multivalued nonexpansive mapping with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M110">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M47">View MathML</a>is nonexpansive. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M4">View MathML</a>is a closed and convex subset ofD.

3 Main results

In the following theorem, we prove strong convergence of the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39">View MathML</a> defined by (3.1) to a common element of the set of solutions of a mixed equilibrium problem and the set of common fixed points of a countable family of multivalued nonexpansive mappings.

Theorem 3.1LetDbe a nonempty closed and convex subset of a real Hilbert spaceH. LetFbe a bifunction from<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M114">View MathML</a>tosatisfying (A1)-(A4), and letφbe a proper lower semicontinuous and convex function fromDto<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M115">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M116">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M117">View MathML</a>be multivalued nonexpansive mappings for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M119">View MathML</a>such that all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M50">View MathML</a>are nonexpansive. Assume that either (B1) or (B2) holds and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M121">View MathML</a>satisfies the condition<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M122">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M123">View MathML</a>. Define the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39">View MathML</a>as follows: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M125">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M126">View MathML</a>

(3.1)

where the sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M127">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M128">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M121">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M130">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M131">View MathML</a>for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118">View MathML</a>. Then the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39">View MathML</a>converges strongly to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M134">View MathML</a>.

Proof We split the proof into six steps.

Step 1. Show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M135">View MathML</a> is well defined for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M136">View MathML</a>.

By Lemmas 2.5-2.6, we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M27">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M138">View MathML</a> is a closed and convex subset of D. Hence Ω is a closed and convex subset of D. It follows from Lemma 2.4 that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M139">View MathML</a> is a closed and convex for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M140">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M141">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M142">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M144">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M145">View MathML</a>

for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M140">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M147">View MathML</a>

(3.2)

Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M148">View MathML</a>, so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M149">View MathML</a>. Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M135">View MathML</a> is well defined.

Step 2. Show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M151">View MathML</a> exists.

Since Ω is a nonempty closed convex subset of H, there exists a unique <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M141">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M153">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M154">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M155">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M156">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M157">View MathML</a>

On the other hand, as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M158">View MathML</a>, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M159">View MathML</a>

It follows that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39">View MathML</a> is bounded and nondecreasing. Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M151">View MathML</a> exists.

Step 3. Show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M162">View MathML</a>.

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M163">View MathML</a>, by the definition of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M42">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M165">View MathML</a>. By applying Lemma 2.1, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M166">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M151">View MathML</a> exists, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39">View MathML</a> is Cauchy. Hence there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M169">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M170">View MathML</a>.

Step 4. Show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M171">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M172">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118">View MathML</a>.

From <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M174">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M175">View MathML</a>

(3.3)

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M141">View MathML</a>, by Lemma 2.3 and (3.2), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M177">View MathML</a>

This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M178">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M179">View MathML</a>. By the given control condition on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M180">View MathML</a> and (3.3), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M181">View MathML</a>

By Lemma 2.5, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M182">View MathML</a>

Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M183">View MathML</a>. By Lemma 2.3, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M184">View MathML</a>

This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M185">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M179">View MathML</a>. From (3.3), we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M187">View MathML</a>. It follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M188">View MathML</a>

Step 5. Show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M189">View MathML</a>.

By <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M190">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M191">View MathML</a>

(3.4)

From <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M170">View MathML</a>, we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M193">View MathML</a>.

We will show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M194">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M195">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M196">View MathML</a>

It follows by (A2) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M197">View MathML</a>

Hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M198">View MathML</a>

It follows from (3.4), (A4) and the lower semicontinuous of φ that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M199">View MathML</a>

For t with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M200">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M62">View MathML</a>, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M202">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M203">View MathML</a> and D is convex, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M204">View MathML</a> and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M205">View MathML</a>

This implies by (A1), (A4) and the convexity of φ, that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M206">View MathML</a>

Dividing by t, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M207">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M208">View MathML</a>, it follows from the weakly semicontinuity of φ that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M209">View MathML</a>

Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M194">View MathML</a>. Next, we will show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M211">View MathML</a>. For each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M212">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M213">View MathML</a>

By Steps 3-4, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M214">View MathML</a>. Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M215">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M212">View MathML</a>.

Step 6. Show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M217">View MathML</a>.

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M154">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M219">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M220">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M221">View MathML</a>

Now, we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M217">View MathML</a>.

This completes the proof. □

Setting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M28">View MathML</a> in Theorem 3.1, we have the following result.

Corollary 3.2LetDbe a nonempty closed and convex subset of a real Hilbert spaceH. LetFbe a bifunction from<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M114">View MathML</a>tosatisfying (A1)-(A4). Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M117">View MathML</a>be multivalued nonexpansive mappings for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M227">View MathML</a>such that all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M50">View MathML</a>are nonexpansive. Assume that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M229">View MathML</a>satisfies the condition<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M122">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M123">View MathML</a>. Define the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39">View MathML</a>as follows: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M125">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M234">View MathML</a>

(3.5)

where the sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M127">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M128">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M121">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M130">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M131">View MathML</a>for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118">View MathML</a>. Then the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39">View MathML</a>converges strongly to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M134">View MathML</a>.

Setting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M31">View MathML</a> in Theorem 3.1, we have the following result.

Corollary 3.3LetDbe a nonempty closed and convex subset of a real Hilbert spaceH. Letφbe a proper lower semicontinuous and convex function fromDto<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M115">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M116">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M117">View MathML</a>be multivalued nonexpansive mappings for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M248">View MathML</a>such that all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M50">View MathML</a>are nonexpansive. Assume that either (B1) or (B2) holds, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M250">View MathML</a>satisfies the condition<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M251">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M123">View MathML</a>. Define the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39">View MathML</a>as follows: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M125">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M255">View MathML</a>

(3.6)

where the sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M127">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M128">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M121">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M130">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M131">View MathML</a>for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M118">View MathML</a>. Then the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M39">View MathML</a>converges strongly to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M134">View MathML</a>.

Remark 3.4

(i) Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M180">View MathML</a> be double sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M265">View MathML</a>. Let (a) and (b) be the following conditions: It is easy to see that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M180">View MathML</a> satisfies the condition (a), then it satisfies the condition (b). So Theorem 3.1 and Corollaries 3.2-3.3 hold true when the control double sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M180">View MathML</a> satisfies the condition (a).

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M266">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M123">View MathML</a>,

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M268">View MathML</a> exist and lie in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M265">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M270">View MathML</a> .

(ii) The following double sequences are examples of the control sequences in Theorem 3.1 and Corollaries 3.2-3.3:

(1)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M273">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M274">View MathML</a>

We see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M275">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M276">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M277">View MathML</a> .

(2)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M278">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M279">View MathML</a>

We see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M280">View MathML</a> does not exist and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M281">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/236/mathml/M277">View MathML</a> .

4 Conclusions

We use the shrinking projection method defined by Takahashi [10] together with our method for finding a common element of the set of solutions of mixed equilibrium problem and common fixed points of a countable family of multivalued nonexpansive mappings in Hilbert spaces. The main results of paper can be applied for solving convex minimization problems and fixed point problems.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

AB studied and researched a nonlinear analysis and also wrote this article. SS participated in the process of the study and helped to draft the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank Chiang Mai University, Chiang Mai, Thailand for the financial support of this paper.

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