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A hybrid projection method for solving a common solution of a system of equilibrium problems and fixed point problems for asymptotically strict pseudocontractions in the intermediate sense in Hilbert spaces

Chatchawan Watchararuangwit1, Pongrus Phuangphoo12 and Poom Kumam1*

Author Affiliations

1 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thung Kru, Bangkok, 10140, Thailand

2 Department of Mathematics, Faculty of Education, Bansomdejchaopraya Rajabhat University (BSRU), Thonburi, Bangkok, 10600, Thailand

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Journal of Inequalities and Applications 2012, 2012:252  doi:10.1186/1029-242X-2012-252

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2012/1/252


Received:15 June 2012
Accepted:15 October 2012
Published:30 October 2012

© 2012 Watchararuangwit et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we introduce a new iterative algorithm which is constructed by using the hybrid projection method for finding a common solution of a system of equilibrium problems of bifunctions satisfying certain conditions and a common solution of fixed point problems of a family of uniformly Lipschitz continuous and asymptotically <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M1">View MathML</a>-strict pseudocontractive mappings in the intermediate sense. We prove the strong convergence theorem for a new iterative algorithm under some mild conditions in Hilbert spaces. Finally, we also give a numerical example which supports our results.

MSC: 47H05, 47H09, 47H10.

Keywords:
asymptotically strict pseudocontraction in the intermediate sense; hybrid projection method; system of equilibrium problems; fixed point problems

1 Introduction

Let C be a closed and convex subset of a real Hilbert space H with the inner product <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M2">View MathML</a> and the norm <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M3">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M4">View MathML</a> be a family of bifunctions from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M5">View MathML</a> into ℝ, where ℝ is the set of real numbers and Γ is an arbitrary index set. The system of equilibrium problems is to find <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M6">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M7">View MathML</a>

(1.1)

The set of solutions of (1.1) is denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M8">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M9">View MathML</a>, that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M10">View MathML</a>

(1.2)

If Γ is a singleton, then the problem (1.1) is reduced to the equilibrium problem of finding <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M11">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M12">View MathML</a>

(1.3)

The set of solutions of (1.3) is denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M13">View MathML</a>.

Recall the following definitions.

A mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M14">View MathML</a> is called monotone if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M15">View MathML</a>

(1.4)

A mapping A is called α-inverse-strongly monotone[1,2], if there exists a positive real number α such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M16">View MathML</a>

(1.5)

Clearly, if A is α-inverse-strongly monotone, then A is monotone.

A mapping A is called β-strongly monotone if there exists a positive real number β such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M17">View MathML</a>

(1.6)

A mapping A is called L-Lipschitz continuous if there exists a positive real number L such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M18">View MathML</a>

(1.7)

It is easy to see that if A is an α-inverse-strongly monotone mapping from C into H, then A is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M19">View MathML</a>-Lipschitz continuous.

In 2009, Qin et al.[3] introduced the following algorithm for a finite family of asymptotically <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M1">View MathML</a>-strictly pseudocontractions.

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M21">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M22">View MathML</a> be a sequence in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M23">View MathML</a>. The sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a> is as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M25">View MathML</a>

(1.8)

It is called the explicit iterative sequence of a finite family of asymptotically <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M1">View MathML</a>-strictly pseudocontractions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M27">View MathML</a>. Since for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M28">View MathML</a>, it can be written as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M29">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M30">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M31">View MathML</a> is a positive integer and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M32">View MathML</a>, as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M33">View MathML</a>, we can rewrite the above table in the following compact form:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M34">View MathML</a>

Next, Sahu et al.[4] introduced new iterative schemes for asymptotically strictly pseudocontractive mappings in the intermediate sense. To be more precise, they proved the following theorem.

Theorem (SXY)LetCbe a nonempty closed and convex subset of a real Hilbert spaceHand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M35">View MathML</a>be a uniformly continuous asymptoticallyκ-strictly pseudocontractive mapping in the intermediate sense with a sequence<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M36">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M37">View MathML</a>is nonempty and bounded. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M38">View MathML</a>be a sequence in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M39">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M40">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M41">View MathML</a>. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M42">View MathML</a>be a sequence generated by the following (CQ) algorithm:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M43">View MathML</a>

(1.9)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M44">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M45">View MathML</a>. Then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>converges strongly to<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M47">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M48">View MathML</a>is a metric projection fromHinto<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M37">View MathML</a>.

In 2010, Hu and Cai [5] considered the asymptotically strictly pseudocontractive mappings in the intermediate sense concerning the equilibrium problem. They obtained the following result in a real Hilbert space. Next, Ceng et al.[6] introduced the viscosity approximation method for a modified Mann iteration process for asymptotically strict pseudocontractive mappings in the intermediate sense and they proved the strong convergence of a general CQ-algorithm and extended the concept of asymptotically strictly pseudocontractive mappings in the intermediate sense to the Banach space setting called nearly asymptotically strictly pseudocontractive mappings in the intermediate sense. Finally, they established a weak convergence theorem for a fixed point of nearly asymptotically strictly pseudocontractive mappings in the intermediate sense which are not necessarily Lipschitz continuous mappings.

Theorem (HC)LetCbe a nonempty closed and convex subset of a real Hilbert spaceHand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M50">View MathML</a>be an integer, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M51">View MathML</a>be a bifunction satisfying (A1)-(A4), and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M14">View MathML</a>be anα-inverse-strongly monotone mapping. Let for each<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M53">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M54">View MathML</a>be a uniformly continuous<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M55">View MathML</a>-strictly asymptotically pseudocontractive mapping in the intermediate sense for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M56">View MathML</a>with sequences<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M57">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M58">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M59">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M60">View MathML</a>. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M61">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M62">View MathML</a>, and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M63">View MathML</a>. Assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M64">View MathML</a>is nonempty and bounded. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M38">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M66">View MathML</a>be sequences in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M39">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M68">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M69">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M41">View MathML</a>, and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M71">View MathML</a>.

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M73">View MathML</a>be sequences generated by the following algorithm:

(1.10)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M75">View MathML</a>, as<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M76">View MathML</a>, and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M77">View MathML</a>. Then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>converges strongly to<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M79">View MathML</a>.

In 2011, Duan and Zhao [7] introduced new iterative schemes for finding a common solution set of a system of equilibrium problems and a solution of a fixed point set of asymptotically strict pseudocontractions in the intermediate sense and they proved these schemes converge strongly.

In 2012, Shui Ge [8] introduced a new hybrid algorithm with variable coefficients for a fixed point problem of a uniformly Lipschitz continuous mapping and asymptotically pseudocontractive mapping in the intermediate sense on unbounded domains and he proved strong convergence in a real Hilbert space.

Theorem (Ge)LetCbe a nonempty, closed, and convex subset of a real Hilbert spaceH, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M35">View MathML</a>be a uniformlyL-Lipschitz continuous mapping and asymptotically pseudocontractive mapping in the intermediate sense with sequences<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M81">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M82">View MathML</a>. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M83">View MathML</a>for each<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M84">View MathML</a>. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>be the sequence generated by the following hybrid algorithm with variable coefficients:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M86">View MathML</a>

(1.11)

where

Assume that the positive real number<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M88">View MathML</a>is chosen so that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M89">View MathML</a>and that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M38">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M66">View MathML</a>are sequences in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M23">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M93">View MathML</a>for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M94">View MathML</a>and for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M95">View MathML</a>.

Then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>converges strongly to a fixed point ofT.

In this paper, motivated and inspired by the previously mentioned above results, we introduce a new iterative algorithm by the hybrid projection method for finding a common solution of a system of equilibrium problems of bifunctions satisfying certain conditions and a common solution of fixed point problems of a family of uniformly Lipschitz continuous and asymptotically <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M1">View MathML</a>-strict pseudocontractive mappings in the intermediate sense in a real Hilbert space. Then, we prove a strong convergence theorem of the iterative algorithm generated by this conditions. Finally, we also give a numerical example which supports our results. The results obtained in this paper extend and improve several recent results in this area.

2 Preliminaries

Let H be a real Hilbert space with the inner product <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M98">View MathML</a> and the norm <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M3">View MathML</a>. Let C be a closed and convex subset of H. For any point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M100">View MathML</a>, there exists a unique nearest point in C, denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M101">View MathML</a>, such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M102">View MathML</a>

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M103">View MathML</a> is called the metric projection of H onto C defined by the following:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M104">View MathML</a>

We know that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M103">View MathML</a> is a nonexpansive mapping H onto C. It is also known that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M103">View MathML</a> satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M107">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M108">View MathML</a>

We will adopt the following notations:

(1) → for strong convergence and ⇀ for weak convergence.

(2) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M109">View MathML</a> denotes the weak w-limit set of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>.

(3) A nonlinear mapping S : <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M111">View MathML</a> is a self-mapping in C. We denote the set of fixed points of S by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M112">View MathML</a> (i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M113">View MathML</a>). Recall the following definitions.

Definition 2.1 Let S be a mapping from C to C. Then

(1) S is said to be nonexpansive if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M114">View MathML</a>

(2.1)

(2) S is said to be uniformly Lipschitz continuous if there exists a constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M115">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M116">View MathML</a>

(2.2)

(3) S is said to be asymptotically nonexpansive if there exists a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M117">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M118">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M119">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M120">View MathML</a>

(2.3)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk (see [9]) in 1972. It is known that if C is a nonempty, bounded, closed, and convex subset of a real Hilbert space H, then every asymptotically nonexpansive self-mapping has a fixed point. Further, the set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M112">View MathML</a> of fixed points of S is closed and convex.

(4) S is said to be asymptotically nonexpansive in the intermediate sense [10,11] if it is continuous and the following inequality holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M122">View MathML</a>

(2.4)

Putting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M123">View MathML</a>, we see that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M124">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M119">View MathML</a>. Then (2.4) is reduced to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M126">View MathML</a>

The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Kirk and Bruck et al. (see [10,11]) as a generalization of the class of asymptotically nonexpansive mappings. It is known that if C is a nonempty, bounded, closed, and convex subset of a real Hilbert space H, then every asymptotically nonexpansive self-mapping in the intermediate sense has a fixed point (see [12]).

(5) S is said to be contractive if there exists a coefficient <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M127">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M128">View MathML</a>

(2.5)

(6) S is said to be a λ-strict pseudocontraction if there exists a coefficient <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M129">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M130">View MathML</a>

(2.6)

The class of strict pseudocontractions was introduced by Brower and Petryshyn (see [1]) in 1967. Clearly, if S is a nonexpansive mapping, then S is a strict pseudocontraction with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M131">View MathML</a>. We also remark that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M132">View MathML</a>, then S is called a pseudocontractive mapping.

(7) S is said to be an asymptoticallyλ-strict pseudocontraction with the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M133">View MathML</a> (see also [13]) if there exists a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M134">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M135">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M119">View MathML</a> and a constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M137">View MathML</a> such that

(2.7)

The class of asymptotically strict pseudocontractions was introduced by Qihou [14] in 1996. Clearly, if S is an asymptotically nonexpansive mapping, then S is an asymptotically strict pseudocontraction with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M131">View MathML</a>. We also remark that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M140">View MathML</a>, then S is said to be an asymptotically pseudocontractive mapping which was introduced by Schu [15] in 1991.

(8) S is said to be an asymptoticallyλ-strict pseudocontraction in the intermediate sense with the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M133">View MathML</a>[4,5] if there exists a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M142">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M143">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M119">View MathML</a> and a constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M129">View MathML</a> such that

(2.8)

Putting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M147">View MathML</a>, we see that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M148">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M119">View MathML</a>. Then (2.8) is reduced to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M150">View MathML</a>

The class of asymptotically strict pseudocontractions in the intermediate sense was introduced by Sahu, Xu, and Yao [4] as a generalization of a class of asymptotically strict pseudocontractions.

For solving the equilibrium problem, let us give the following assumptions for the bifunction F and the set C:

(A1) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M151">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M11">View MathML</a>;

(A2) F is monotone, i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M153">View MathML</a>, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M154">View MathML</a>;

(A3) for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M155">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M156">View MathML</a>;

(A4) for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M11">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M158">View MathML</a> is convex and lower semicontinuous.

Lemma 2.2 ([16])

LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. For any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M159">View MathML</a>and given also a real number<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M160">View MathML</a>, the set

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M161">View MathML</a>

is closed and convex.

Lemma 2.3 ([17])

LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M162">View MathML</a>satisfy (A1)-(A4), and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M163">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M100">View MathML</a>. Then there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M165">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M166">View MathML</a>

Lemma 2.4 ([18])

Assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M167">View MathML</a>satisfies (A1)-(A4). For<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M168">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M169">View MathML</a>, define a mapping<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M170">View MathML</a>as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M171">View MathML</a>

(2.9)

Then the following hold:

(1) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M172">View MathML</a>is single-valued;

(2) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M172">View MathML</a>is firmly nonexpansive, i.e., for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M174">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M175">View MathML</a>

(2.10)

(3) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M176">View MathML</a>; and

(4) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M13">View MathML</a>is closed and convex.

Lemma 2.5 ([7,19])

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

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M178">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M179">View MathML</a>.

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M180">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M181">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M179">View MathML</a>.

(iii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M183">View MathML</a>.

Lemma 2.6 ([4])

LetCbe a nonempty closed and convex subset of a real Hilbert spaceH, and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M184">View MathML</a>be a uniformlyL-Lipschitz continuous and asymptoticallyλ-strict pseudocontraction in the intermediate sense. Then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M112">View MathML</a>is closed and convex.

Lemma 2.7 ([4])

LetCbe a nonempty closed and convex subset of a real Hilbert spaceHand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M184">View MathML</a>be a uniformlyL-Lipschitz continuous and asymptoticallyλ-strict pseudocontraction in the intermediate sense. Then the mapping<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M187">View MathML</a>is demiclosed at zero, that is, if the sequence<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>inCis such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M189">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M190">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M191">View MathML</a>.

Lemma 2.8 ([20])

LetCbe a nonempty closed and convex subset of a real Hilbert space H. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>be a sequence inHand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M193">View MathML</a>, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M194">View MathML</a>. Suppose that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>is such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M196">View MathML</a>and satisfies the condition

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M197">View MathML</a>

Then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M198">View MathML</a>.

Lemma 2.9 ([4])

LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M199">View MathML</a>be an asymptoticallyλ-strict pseudocontractive mapping in the intermediate sense with the sequence<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M200">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M201">View MathML</a>

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M202">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M84">View MathML</a>.

3 Main results

In this section, we prove a strong convergence theorem which solves the problem of finding a common solution of a system of equilibrium problems and a common solution of fixed point problems in Hilbert spaces.

Theorem 3.1LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M204">View MathML</a>be a positive integer. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M205">View MathML</a>be a bifunction satisfying (A1)-(A4). Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M206">View MathML</a>be a uniformly Lipschitz continuous and asymptotically<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M1">View MathML</a>-strict pseudocontractive mapping in the intermediate sense for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M208">View MathML</a>with the sequences<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M209">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M60">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M211">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M212">View MathML</a>. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M213">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M63">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M215">View MathML</a>. Assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M216">View MathML</a>is nonempty and bounded. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M38">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M66">View MathML</a>be sequences in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M39">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M220">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M221">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M222">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M223">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M224">View MathML</a>be a sequence in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M225">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M226">View MathML</a>.

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>be a sequence generated by the following algorithm:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M228">View MathML</a>

(3.1)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M229">View MathML</a>, as<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M76">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M231">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M232">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M233">View MathML</a>. Then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>converges strongly to some point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M235">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M236">View MathML</a>.

Proof The proof is split into seven steps.

Step 1. We will show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M237">View MathML</a> is well defined.

From Lemma 2.4, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M238">View MathML</a> is closed and convex. From the assumption of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M239">View MathML</a> and Lemma 2.6, it follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M240">View MathML</a> is closed and convex.

Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M241">View MathML</a> is closed and convex. Hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M237">View MathML</a> is well defined.

Step 2. We will show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M243">View MathML</a> is closed and convex for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M28">View MathML</a>.

By the assumption of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M245">View MathML</a>, it is easy to see that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M243">View MathML</a> is closed for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M28">View MathML</a>. We only show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M243">View MathML</a> is convex for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M249">View MathML</a>.

Note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M250">View MathML</a> is convex. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M251">View MathML</a> is convex for some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M252">View MathML</a>. Next, we show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M253">View MathML</a> is convex for the same k. For each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M254">View MathML</a>, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M255">View MathML</a>

is equivalent to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M256">View MathML</a>

(3.2)

Taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M257">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M258">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M253">View MathML</a> and putting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M260">View MathML</a>, it follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M261">View MathML</a>, and so

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M262">View MathML</a>

(3.3)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M263">View MathML</a>

(3.4)

Combining (3.3) with (3.4), we obtain that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M264">View MathML</a>

That is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M265">View MathML</a>

In view of the convexity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M251">View MathML</a>, we see that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M267">View MathML</a>. This implies that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M268">View MathML</a>. Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M253">View MathML</a> is convex. Hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M243">View MathML</a> is closed and convex for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M28">View MathML</a>.

Step 3. We will show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M272">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M28">View MathML</a>.

Put <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M274">View MathML</a> for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M275">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M276">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M84">View MathML</a>. Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M278">View MathML</a>. It is obvious that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M279">View MathML</a>. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M280">View MathML</a> for some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M252">View MathML</a>.

Next, we show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M282">View MathML</a> for the same k. Taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M283">View MathML</a> and for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M284">View MathML</a>, we see that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M285">View MathML</a> is nonexpansive and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M286">View MathML</a>. We note that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M287">View MathML</a>

(3.5)

We observe that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M288">View MathML</a>

(3.6)

By virtue of convexity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M289">View MathML</a>, one has

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M290">View MathML</a>

(3.7)

Substituting (3.5) and (3.6) into (3.7), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M291">View MathML</a>

(3.8)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M292">View MathML</a>

(3.9)

Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M293">View MathML</a>, and so <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M272">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M295">View MathML</a>. Hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a> is well defined.

Step 4. We will show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a> is bounded.

Since Ω is a nonempty closed and convex subset of H, there exists a unique <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M298">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M299">View MathML</a>. By the assumption, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M300">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M301">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M302">View MathML</a>

This implies that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a> is bounded. Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M73">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M305">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M306">View MathML</a> are also bounded.

Step 5. We will show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M307">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M308">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M119">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M310">View MathML</a>.

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M300">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M312">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M313">View MathML</a>

(3.10)

Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M314">View MathML</a>, and so

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M315">View MathML</a>

(3.11)

Thus, the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M316">View MathML</a> is nondecreasing. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a> is bounded, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M318">View MathML</a> exists. On the other hand, from (3.10), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M319">View MathML</a>

(3.12)

The fact that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M318">View MathML</a> exists implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M321">View MathML</a>

(3.13)

It is easy to see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M322">View MathML</a>

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M323">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M324">View MathML</a>

It follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M325">View MathML</a>

(3.14)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M326">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M119">View MathML</a> and from (3.13), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M328">View MathML</a>

(3.15)

For each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M329">View MathML</a>, it follows from the firmly nonexpansive <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M285">View MathML</a> that for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M331">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M332">View MathML</a>

Thus, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M333">View MathML</a>

(3.16)

This implies that for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M284">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M335">View MathML</a>

Therefore, by the convexity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M289">View MathML</a> and (3.8) and the nonexpansivity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M285">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M338">View MathML</a>

It follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M339">View MathML</a>

(3.17)

From (3.15) and (3.17), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M340">View MathML</a>

(3.18)

Then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M341">View MathML</a>

Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M342">View MathML</a>

(3.19)

From (3.13) and (3.19), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M343">View MathML</a>

(3.20)

It follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M344">View MathML</a>

(3.21)

Since for any positive integer <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M345">View MathML</a>, we can write <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M346">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M233">View MathML</a>, note that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M348">View MathML</a>

(3.22)

From the conditions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M349">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M350">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M351">View MathML</a>

From (3.15) and (3.19), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M352">View MathML</a>

(3.23)

It is obvious that the relations <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M353">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M354">View MathML</a> hold.

Therefore, we compute

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M355">View MathML</a>

Applying Lemma 2.9 and (3.21), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M356">View MathML</a>

(3.24)

From (3.22) and (3.24), it follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M357">View MathML</a>

(3.25)

Since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M358">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M359">View MathML</a>, which gives that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M360">View MathML</a>

(3.26)

Moreover, for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M359">View MathML</a>, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M362">View MathML</a>

This implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M363">View MathML</a>

(3.27)

Step 6. We will show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M364">View MathML</a>.

(6.1) We will show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M365">View MathML</a>.

We take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M366">View MathML</a> and assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M367">View MathML</a> for some subsequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M368">View MathML</a> of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>.

Note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M370">View MathML</a> is uniformly Lipschitz continuous and (3.27), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M371">View MathML</a>

(3.28)

It follows from Lemma 2.7 that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M372">View MathML</a>

(3.29)

(6.2) We will show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M373">View MathML</a>.

By Lemma 2.3, for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M284">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M375">View MathML</a>

From (A2), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M376">View MathML</a>

Taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M377">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M378">View MathML</a>

From (3.18), we obtain that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M379">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M380">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M275">View MathML</a> (especially <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M382">View MathML</a>). Considering this together with (3.18) and (A4), we have for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M284">View MathML</a> that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M384">View MathML</a>

For any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M385">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M386">View MathML</a>, we let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M387">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M386">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M389">View MathML</a>, we obtain that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M390">View MathML</a>, and so <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M391">View MathML</a>. It follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M392">View MathML</a>

Dividing by t, for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M284">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M394">View MathML</a>

Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M395">View MathML</a>, from (A3), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M396">View MathML</a>

Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M397">View MathML</a>, and so <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M398">View MathML</a>.

Step 7. We will show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a> converges strongly to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M400">View MathML</a>.

Set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M401">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M402">View MathML</a>

Since Ω is a nonempty closed and convex subset of H, there exists a unique <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M398">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M401">View MathML</a>. It follows from Lemma 2.8 that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M405">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M401">View MathML</a>. This completes proof. □

4 Deduced theorems

If we take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M407">View MathML</a> in Theorem 3.1, then we obtain the following result.

Theorem 4.1LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M204">View MathML</a>be a positive integer. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M409">View MathML</a>be a bifunction satisfying (A1)-(A4). Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M206">View MathML</a>be a uniformly Lipschitz continuous and asymptotically<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M1">View MathML</a>-strict pseudocontractive mapping in the intermediate sense for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M412">View MathML</a>with the sequences<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M209">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M60">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M415">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M416">View MathML</a>. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M213">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M418">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M419">View MathML</a>. Assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M420">View MathML</a>is nonempty and bounded. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M38">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M66">View MathML</a>be sequences in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M39">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M220">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M221">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M222">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M223">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M224">View MathML</a>be a sequence in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M429">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M226">View MathML</a>.

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>be a sequence generated by the following algorithm:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M432">View MathML</a>

(4.1)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M229">View MathML</a>, as<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M76">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M231">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M232">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M233">View MathML</a>. Then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>converges strongly to some point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M235">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M236">View MathML</a>.

Remark 4.2 Theorem 4.1 improves and extends the theorem of Tada and Takahashi [21] and the corollary of Duan and Zhao [7].

If we set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M441">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M442">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M443">View MathML</a> in Theorem 3.1, then we obtain the following result.

Theorem 4.3LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M204">View MathML</a>be a positive integer. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M445">View MathML</a>be a uniformly Lipschitz continuous and asymptotically<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M1">View MathML</a>-strict pseudocontractive mapping in the intermediate sense for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M412">View MathML</a>with the sequences<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M448">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M60">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M211">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M212">View MathML</a>. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M213">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M63">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M215">View MathML</a>. Assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M455">View MathML</a>is nonempty and bounded. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M38">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M66">View MathML</a>be sequences in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M39">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M459">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M350">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M222">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M462">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M224">View MathML</a>be a sequence in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M429">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M226">View MathML</a>.

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>be a sequence generated by the following algorithm:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M467">View MathML</a>

(4.2)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M229">View MathML</a>, as<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M76">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M231">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M232">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M233">View MathML</a>. Then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M24">View MathML</a>converges strongly to some point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M235">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M236">View MathML</a>.

Remark 4.4 Theorem 4.1 improves and extends the theorem of Sahu, Xu, and Yao [4], the theorem of Qin, Cho, Kang, and Shang [3] and the corollary of Duan and Zhao [7].

5 Numerical examples

In this section, in order to demonstrate the effectiveness, realization and convergence of algorithm of Theorem 3.1, we consider the following simple example that was presented in reference [4].

Example 5.1 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M476">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M477">View MathML</a>. For each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M11">View MathML</a>, we define

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M479">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M480">View MathML</a>.

It is easy to see that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M199">View MathML</a> is discontinuous at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M482">View MathML</a> and S is not Lipschitz continuous.

Set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M483">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M484">View MathML</a>.

For each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M485">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M486">View MathML</a>

For each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M487">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M488">View MathML</a>

For each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M489">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M490">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M491">View MathML</a>

It follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M492">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M493">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M84">View MathML</a> and for some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M495">View MathML</a>.

Therefore, S is an asymptotically k-strict pseudocontractive mapping in the intermediate sense.

In Theorem 3.1, we set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M496">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M441">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M498">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M499">View MathML</a>. We apply it to find the fixed point of S of Example 5.1.

Under the above assumption in Theorem 3.1 is simplified as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M500">View MathML</a>

(5.1)

In fact, in one-dimensional case, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M245">View MathML</a> is a closed interval. If we set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M502">View MathML</a>, then the projection point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M503">View MathML</a> of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M504">View MathML</a> onto <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M245">View MathML</a> can be expressed as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M506">View MathML</a>

The numerical results for an initial guess <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M507">View MathML</a> are shown in Table 1. From the table, we see that the iterations converge to 0 which is the unique fixed point of S. The convergence of each iteration is also shown in Figure 1 for comparison.

thumbnailFigure 1. The convergence comparison of different initial values<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M508">View MathML</a>.

Table 1. The numerical results for an initial guess<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/252/mathml/M507">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. All authors read and approved the final manuscript.

Acknowledgements

This research was supported by the Faculty of Science, KMUTT Research Fund 2553-2554.

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