Open Access Research

New iterative methods for a common solution of fixed points for pseudo-contractive mappings and variational inequalities

Rabian Wangkeeree1 and Kamonrat Nammanee2*

Author Affiliations

1 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

2 Department of Mathematics, School of Science, University of Phayao, Phayao, 56000, Thailand

For all author emails, please log on.

Fixed Point Theory and Applications 2013, 2013:233  doi:10.1186/1687-1812-2013-233


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/233


Received:29 May 2013
Accepted:18 August 2013
Published:5 September 2013

© 2013 Wangkeeree and Nammanee; 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 three iterative methods for finding a common element of the set of fixed points for a continuous pseudo-contractive mapping and the solution set of a variational inequality problem governed by continuous monotone mappings. Strong convergence theorems for the proposed iterative methods are proved. Our results improve and extend some recent results in the literature.

MSC: 47H05, 47H09, 47J25, 65J15.

Keywords:
pseudo-contractive mapping; monotone mapping; strong convergence theorem

1 Introduction

The theory of variational inequalities represents, in fact, a very natural generalization of the theory of boundary value problems and allows us to consider new problems arising from many fields of applied mathematics, such as mechanics, physics, engineering, the theory of convex programming, and the theory of control. While the variational theory of boundary value problems has its starting point in the method of orthogonal projection, the theory of variational inequalities has its starting point in the projection on a convex set.

Let C be a nonempty closed and convex subset of a real Hilbert space H. The classical variational inequality problem is to find <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M1">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M2">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M3">View MathML</a>, where A is a nonlinear mapping. The set of solutions of the variational inequality is denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M4">View MathML</a>. The variational inequality problem has been extensively studied in the literature; see [1-9] and the references therein. In the context of the variational inequality problem, this implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M5">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M6">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M7">View MathML</a> is a metric projection of H into C.

Let A be a mapping from C to H, then A is called monotone if and only if for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M8">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M9">View MathML</a>

(1.1)

An operator A is said to be strongly positive on H if there exists a constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M10">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M11">View MathML</a>

A mapping A of C into itself is called L-Lipschitz continuous if there exits a positive number L such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M12">View MathML</a>

A mapping A of C into H is called α-inverse-strongly monotone if there exists a positive real number α such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M13">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M14">View MathML</a>; see [9-14]. If A is an α-inverse strongly monotone mapping of C into H, then it is obvious that A is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M15">View MathML</a>-Lipschitz continuous, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M16">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M8">View MathML</a>. Clearly, the class of monotone mappings includes the class of α-inverse strongly monotone mappings.

A mapping A of C into H is called <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M18">View MathML</a>-strongly monotone if there exists a positive real number <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M18">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M20">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M14">View MathML</a>; see [15]. Clearly, the class of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M18">View MathML</a>-strongly monotone mappings includes the class of strongly positive mappings.

Recall that a mapping T of C into H is called pseudo-contractive if for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M8">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M24">View MathML</a>

(1.2)

T is said to be a k-strict pseudo-contractive mapping if there exists a constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M25">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M26">View MathML</a>

A mapping T of C into itself is called nonexpansive if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M27">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M8">View MathML</a>. We denote by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M29">View MathML</a> the set of fixed points of T. Clearly, the class of pseudo-contractive mappings includes the class of nonexpansive and strict pseudo-contractive mappings.

For a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M30">View MathML</a> of real numbers in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M31">View MathML</a> and arbitrary <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M32">View MathML</a>, let the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a> in C be iteratively defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M34">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M35">View MathML</a>

(1.3)

where T is a nonexpansive mapping of C into itself. Halpern [16] was first to study the convergence of algorithm (1.3) in the framework of Hilbert spaces. Lions [17] and Wittmann [18] improved the result of Halpern by proving strong convergence of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a> to a fixed point of T if the real sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M30">View MathML</a> satisfies certain conditions. Reich [19], Shioji and Takahashi [20], and Zegeye and Shahzad [21] extended the result of Wittmann [18] to the case of a Banach space.

In 2000, Moudafi [22] introduced a viscosity approximation method and proved that if H is a real Hilbert space, for given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M34">View MathML</a>, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a> generated by the algorithm

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M40">View MathML</a>

(1.4)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M41">View MathML</a> is a contraction mapping and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M42">View MathML</a> satisfies certain conditions, converges strongly to the unique solution <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M43">View MathML</a> in C of the variational inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M44">View MathML</a>

(1.5)

Moudafi [22] generalized Halpern’s theorems in the direction of viscosity approximations. In [23], Zegeye et al. extended Moudafi’s result to the case of Lipschitz pseudo-contractive mappings in Banach spaces more general that Hilbert spaces.

In 2006, Marino and Xu [24] introduced the following general iterative method:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M45">View MathML</a>

(1.6)

They proved that if the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M30">View MathML</a> of parameters satisfies appropriate conditions, then the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a> generated by (1.6) converges strongly to the unique solution of the variational inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M48">View MathML</a>

(1.7)

which is the optimality condition for the minimization problem

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M49">View MathML</a>

where h is a potential function for γf (i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M50">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M51">View MathML</a>).

Recently, Zegeye and Shahzad [25] introduced an iterative method and proved that if C is a nonempty subset of a real Hilbert space H, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M52">View MathML</a> is a pseudo-contractive mapping and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M53">View MathML</a> is a continuous monotone mapping such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M54">View MathML</a>. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M55">View MathML</a> defined <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M56">View MathML</a> by the following: for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M58">View MathML</a>, define

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M59">View MathML</a>

(1.8)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M60">View MathML</a>

(1.9)

Then the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a> generated by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M62">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M63">View MathML</a>

(1.10)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M41">View MathML</a> is a contraction mapping and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M65">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M66">View MathML</a> satisfy certain conditions, converges strongly to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M67">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M68">View MathML</a>.

In this paper, motivated and inspired by the method of Marino and Xu [24] and the work of Zegeye and Shahzad [25], we introduce a viscosity approximation method for finding a common fixed point of a set of fixed points of continuous pseudo-contractive mappings more general than nonexpansive mappings and a solution set of the variational inequality problem for continuous monotone mappings more general than α-inverse strongly monotone mappings in a real Hilbert space. Our result extend and unify most of the results that have been proved for important classes of nonlinear operators.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M69">View MathML</a> be a continuous pseudo-contractive mapping and a continuous monotone mapping, respectively. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M55">View MathML</a>, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M72">View MathML</a> be defined by (1.8) and (1.9).

We consider the three iterative methods given as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M73">View MathML</a>

(1.11)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M74">View MathML</a>

(1.12)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M75">View MathML</a>

(1.13)

where A is a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M18">View MathML</a>-strongly monotone and L-Lipschitzian continuous operator and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M77">View MathML</a> is a contraction mapping. We prove in Section 3 that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M30">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M66">View MathML</a> of parameters satisfy appropriate conditions, then the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M80">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M81">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M82">View MathML</a> converge strongly to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M83">View MathML</a>.

2 Preliminaries

Let C be a closed and convex subset of a real Hilbert space H. For every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57">View MathML</a>, there exists a unique nearest point in C, denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M85">View MathML</a>, such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M86">View MathML</a>

(2.1)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M7">View MathML</a> is called the metric projection of H onto C. We know that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M7">View MathML</a> is a nonexpansive mapping of H onto C. In connection with metric projection, we have the following lemma.

Lemma 2.1Let H be a real Hilbert space. The following identity holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M89">View MathML</a>

Lemma 2.2LetCbe a nonempty closed convex subset of a Hilbert spaceH. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M91">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M92">View MathML</a>if and only if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M93">View MathML</a>

(2.2)

Lemma 2.3[26]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M94">View MathML</a>be a sequence of nonnegative real numbers such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M95">View MathML</a>

where

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M96">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M97">View MathML</a>and

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M98">View MathML</a>or<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M99">View MathML</a>.

Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M100">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M101">View MathML</a>.

Lemma 2.4[27]

LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M102">View MathML</a>be a continuous monotone mapping. Then, for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M103">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M105">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M106">View MathML</a>

(2.3)

Moreover, by a similar argument as in the proof of Lemmas 2.8 and 2.9 in[28], Zegeye[27]obtained the following lemmas.

Lemma 2.5[27]

LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M102">View MathML</a>be a continuous monotone mapping. For<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M108">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57">View MathML</a>, define a mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M110">View MathML</a>as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M111">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57">View MathML</a>. Then the following hold:

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M113">View MathML</a>is single-valued;

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M113">View MathML</a>is a firmly nonexpansive type mapping, i.e., for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M115">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M116">View MathML</a>

(3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M117">View MathML</a>;

(4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M118">View MathML</a>is closed and convex.

In the sequel, we shall make use of the following lemmas.

Lemma 2.6[27]

LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M119">View MathML</a>be a continuous pseudo-contractive mapping. Then, for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M103">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M105">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M123">View MathML</a>

(2.4)

Lemma 2.7[27]

LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M124">View MathML</a>be a continuous pseudo-contractive mapping. For<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M103">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M57">View MathML</a>, define a mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M127">View MathML</a>as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M128">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M129">View MathML</a>. Then the following hold:

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M130">View MathML</a>is single-valued;

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M130">View MathML</a>is a firmly nonexpansive type mapping, i.e., for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M115">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M133">View MathML</a>

(3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M134">View MathML</a>;

(4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M29">View MathML</a>is closed and convex.

Lemma 2.8[15]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M136">View MathML</a>and letfbe anα-contraction of a real Hilbert spaceHinto itself, and letAbe a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M18">View MathML</a>-strongly monotone andL-Lipschitzian continuous operator ofHinto itself with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M138">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M139">View MathML</a>. Takeμ, γto be real numbers as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M140">View MathML</a>

If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M141">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M142">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M143">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M144">View MathML</a>

3 Main results

Now, we prove our main theorems.

Theorem 3.1LetHbe a real Hilbert space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M145">View MathML</a>be a continuous pseudo-contractive mapping and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M146">View MathML</a>be a continuous monotone mapping such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M147">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M136">View MathML</a>and letfbe anα-contraction ofHinto itself, and letAbe a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M149">View MathML</a>-strongly monotone andL-Lipschitzian continuous operator ofCintoHwith<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M138">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M139">View MathML</a>. Takeμ, γto be real numbers as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M152">View MathML</a>

For<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M153">View MathML</a>, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a>be a sequence generated by (1.11), where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M155">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M156">View MathML</a>are such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M157">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M158">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M159">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M160">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M161">View MathML</a>. Then the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a>converges strongly to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M163">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M83">View MathML</a>.

Proof Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M165">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M101">View MathML</a>, we may assume, without loss of generality, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M167">View MathML</a> for all n. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M168">View MathML</a>, it implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M169">View MathML</a> is a contraction of H into itself. Since H is a real Hilbert space, there exists a unique element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M170">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M83">View MathML</a>.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M172">View MathML</a>, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M173">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M174">View MathML</a>. Then we have from Lemma (2.5) and (2.7) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M175">View MathML</a>

(3.1)

Moreover, from (1.11) and (3.1), we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M176">View MathML</a>

(3.2)

It follows from induction that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M177">View MathML</a>

(3.3)

Thus <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a> is bounded, and hence so are <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M179">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M180">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M181">View MathML</a>. Next, to show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M182">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M183">View MathML</a>

(3.4)

where .

Moreover, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M185">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M186">View MathML</a>, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M187">View MathML</a>

(3.5)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M188">View MathML</a>

(3.6)

Putting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M189">View MathML</a> in (3.5) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M190">View MathML</a> in (3.6), we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M191">View MathML</a>

(3.7)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M192">View MathML</a>

(3.8)

Adding (3.7) and (3.8), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M193">View MathML</a>

(3.9)

which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M194">View MathML</a>

(3.10)

Now, using the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M195">View MathML</a> is monotone, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M196">View MathML</a>

(3.11)

and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M197">View MathML</a>

(3.12)

Without loss of generality, let us assume that there exists a real number b such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M198">View MathML</a> for all . Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M200">View MathML</a>

(3.13)

and hence from (3.13) we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M201">View MathML</a>

(3.14)

where .

Furthermore, from (3.4) and (3.14) we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M203">View MathML</a>

(3.15)

Hence by Lemma 2.3, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M204">View MathML</a>

(3.16)

Consequently, from (3.14) and (3.16), we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M205">View MathML</a>

(3.17)

Moreover, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M206">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M207">View MathML</a>, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M208">View MathML</a>

(3.18)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M209">View MathML</a>

(3.19)

Putting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M210">View MathML</a> in (3.18) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M211">View MathML</a> in (3.19), we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M212">View MathML</a>

(3.20)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M213">View MathML</a>

(3.21)

Adding (3.20) and (3.21), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M214">View MathML</a>

(3.22)

which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M215">View MathML</a>

(3.23)

Now, using the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M216">View MathML</a> is pseudo-contractive, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M217">View MathML</a>

(3.24)

and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M218">View MathML</a>

(3.25)

Thus, using the method in (3.13) and (3.14), we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M219">View MathML</a>

(3.26)

where .

Therefore, from (3.16) and the property of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M66">View MathML</a>, we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M222">View MathML</a>

(3.27)

Furthermore, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M223">View MathML</a>, we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M224">View MathML</a>

(3.28)

From <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M165">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M226">View MathML</a>.

Now, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M172">View MathML</a>, using Lemma 2.5, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M228">View MathML</a>

(3.29)

and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M229">View MathML</a>

(3.30)

Therefore, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M230">View MathML</a>

(3.31)

Hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M231">View MathML</a>

(3.32)

So, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M232">View MathML</a>

(3.33)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M233">View MathML</a>, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M234">View MathML</a>

Next, we show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M235">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M83">View MathML</a>.

To show this equality, we choose a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M237">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M238">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M239">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M237">View MathML</a> is bounded, there exists a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M241">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M237">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M243">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M244">View MathML</a>. Without loss of generality, we may assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M245">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M246">View MathML</a> and H is closed and convex, we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M243">View MathML</a>. Moreover, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M248">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M249">View MathML</a>, we have that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M250">View MathML</a>.

Now, we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M251">View MathML</a>. Note that from the definition of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M252">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M253">View MathML</a>

(3.34)

Put <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M254">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M255">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M256">View MathML</a>. Consequently, we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M257">View MathML</a>. From (3.34) it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M258">View MathML</a>

From the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M259">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M101">View MathML</a>, we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M261">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M101">View MathML</a>.

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M195">View MathML</a> is monotone, we have that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M264">View MathML</a>. Thus, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M265">View MathML</a>

and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M266">View MathML</a>

(3.35)

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M267">View MathML</a> and using the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M195">View MathML</a> is continuous, we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M269">View MathML</a>

(3.36)

This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M270">View MathML</a>.

Furthermore, from the definition of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M271">View MathML</a> we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M272">View MathML</a>

(3.37)

Put <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M254">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M255">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M256">View MathML</a>. Consequently, we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M257">View MathML</a>. From (3.33) and pseudo-contractivity of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M216">View MathML</a>, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M278">View MathML</a>

(3.38)

Then, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M279">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M101">View MathML</a>, we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M281">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M282">View MathML</a>. Thus, as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M283">View MathML</a>, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M284">View MathML</a>

(3.39)

and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M285">View MathML</a>

(3.40)

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M267">View MathML</a> and using the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M216">View MathML</a> is continuous, we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M288">View MathML</a>

(3.41)

Now, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M289">View MathML</a>. Then we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M290">View MathML</a> and hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M291">View MathML</a>.

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M292">View MathML</a> and since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M83">View MathML</a>, by Lemma 2.2, implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M294">View MathML</a>

(3.42)

Now, we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M295">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M101">View MathML</a>. From <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M297">View MathML</a>, we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M298">View MathML</a>

(3.43)

This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M299">View MathML</a>

(3.44)

where , <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M301">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M302">View MathML</a>. We put <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M303">View MathML</a>. It is easy to see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M304">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M305">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M306">View MathML</a> by (3.42). Hence, by Lemma 2.3, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a> converges strongly to z. This completes the proof. □

Theorem 3.2LetHbe a real Hilbert space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M308">View MathML</a>be a continuous pseudo-contractive mapping and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M146">View MathML</a>be a continuous monotone mapping such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M147">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M136">View MathML</a>and letfbe anα-contraction ofHinto itself, and letAbe a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M149">View MathML</a>-strongly monotone andL-Lipschitzian continuous operator ofHinto itselfHwith<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M138">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M139">View MathML</a>. Takeμ, γto be real numbers as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M315">View MathML</a>

For<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M316">View MathML</a>, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M81">View MathML</a>be a sequence generated by (1.12), where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M155">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M156">View MathML</a>are such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M157">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M158">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M159">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M160">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M161">View MathML</a>. The sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M81">View MathML</a>converges strongly to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M163">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M83">View MathML</a>.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a> be the sequence given by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M329">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M330">View MathML</a>

From Theorem 3.1, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M295">View MathML</a>. We claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M332">View MathML</a>. Indeed, we estimate

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M333">View MathML</a>

(3.45)

It follows from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M158">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M335">View MathML</a> and Lemma 2.3 that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M336">View MathML</a>.

Consequently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M332">View MathML</a> as required. □

Theorem 3.3LetHbe a real Hilbert space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M308">View MathML</a>be a continuous pseudo-contractive mapping and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M146">View MathML</a>be a continuous monotone mapping such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M147">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M136">View MathML</a>and letfbe anα-contraction ofHinto itself, and letAbe a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M149">View MathML</a>-strongly monotone andL-Lipschitzian continuous operator ofHinto itselfHwith<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M138">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M139">View MathML</a>. Takeμ, γto be real numbers as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M345">View MathML</a>

For<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M346">View MathML</a>, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M82">View MathML</a>be a sequence generated by (1.13), where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M155">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M156">View MathML</a>are such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M157">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M158">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M159">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M160">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M161">View MathML</a>. The sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M82">View MathML</a>converges strongly to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M163">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M83">View MathML</a>.

Proof Define the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M81">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M359">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M360">View MathML</a>

Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M361">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M362">View MathML</a>

(3.46)

It follows from induction that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M363">View MathML</a>

(3.47)

Thus both <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M82">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M81">View MathML</a> are bounded. We observe that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M366">View MathML</a>

Thus Theorem 3.2 implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M81">View MathML</a> converges to some point z. In this case, we also have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M368">View MathML</a>

Hence the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M82">View MathML</a> converges to some point z. This completes the proof. □

Setting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M370">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M371">View MathML</a>, where I is the identity mapping in Theorem 3.1, we have the following result.

Corollary 3.4LetHbe a real Hilbert space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M372">View MathML</a>be a continuous pseudo-contractive mapping and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M146">View MathML</a>be a continuous monotone mapping such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M374">View MathML</a>. Letfbe a contraction ofHinto itself and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a>be a sequence generated by<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M153">View MathML</a>and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M377">View MathML</a>

(3.48)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M155">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M379">View MathML</a>are such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M157">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M381">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M382">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M160">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M384">View MathML</a>. The sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a>converges strongly to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M386">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M387">View MathML</a>.

In Theorem 3.1, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M370">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M371">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M390">View MathML</a> is a constant mapping, then we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M391">View MathML</a>. In fact, we have the following corollary.

Corollary 3.5LetHbe a real Hilbert space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M372">View MathML</a>be a continuous pseudo-contractive mapping and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M146">View MathML</a>be a continuous monotone mapping such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M374">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a>be a sequence generated by<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M396">View MathML</a>and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M397">View MathML</a>

(3.49)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M155">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M379">View MathML</a>are such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M157">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M381">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M382">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M160">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M384">View MathML</a>. The sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a>converges strongly to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M386">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M391">View MathML</a>.

In Theorem 3.1, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M370">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M409">View MathML</a>, where I is the identity mapping, then we have the following corollary.

Corollary 3.6LetHbe a real Hilbert space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M146">View MathML</a>be a continuous monotone mapping such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M411">View MathML</a>. Letfbe a contraction ofHinto itself, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a>be a sequence generated by<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M153">View MathML</a>and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M414">View MathML</a>

(3.50)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M155">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M379">View MathML</a>are such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M157">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M381">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M382">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M160">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M384">View MathML</a>. The sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M33">View MathML</a>converges strongly to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M423">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/233/mathml/M424">View MathML</a>.

Remark 3.7 Our results extend and unify most of the results that have been proved for these important classes of nonlinear operators. In particular, Theorem 3.1 extends Theorem 3.1 of Iiduka and Takahashi [12] and Zegeye et al.[25], Corollary 3.2 of Su et al.[29] in the sense that our convergence is for the more general class of continuous pseudo-contractive and continuous monotone mappings. Corollary 3.4 also extends Theorem 4.2 of Iiduka and Takahashi [12] in the sense that our convergence is for the more general class of continuous pseudo-contractive and continuous monotone mappings.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank Naresuan University and the university of Phayao. Moreover, the authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.

References

  1. Borwein, FE: Nonlinear monotone operators and convex sets in Banach spaces. Bull. Am. Math. Soc.. 71, 780–785 (1965). Publisher Full Text OpenURL

  2. Bruck, RE: On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J. Math. Anal. Appl.. 61, 159–164 (1977). Publisher Full Text OpenURL

  3. Noor, MA, Noor, KI, Al-Said, E: Iterative methods for solving general quasi-variational inequalities. Optim. Lett.. 4, 513–530 (2010). Publisher Full Text OpenURL

  4. Takahashi, W: Nonlinear complementarity problem and systems of convex inequalities. J. Optim. Theory Appl.. 24, 493–508 (1978)

  5. Yao, Y, Liou, YC, Yao, JC: An extragradient method for fixed point problems and variational inequality problems. J. Inequal. Appl. (2007). PubMed Abstract | Publisher Full Text OpenURL

  6. Yao, Y, Liou, YC, Yao, JC: An iterative algorithm for approximating convex minimization problem. Appl. Math. Comput.. 188, 648–656 (2007). Publisher Full Text OpenURL

  7. Yao, Y, Noor, MA, Liou, YC: Strong convergence of a modified extragradient method to the minimum-norm solution of variational inequalities. Abstr. Appl. Anal. (2012). Publisher Full Text OpenURL

  8. Yao, Y, Cho, YJ, Chen, R: An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems. Nonlinear Anal.. 71, 3363–3373 (2009). Publisher Full Text OpenURL

  9. Zegeye, H, Shahzad, N: Strong convergence for monotone mappings and relatively weak nonexpansive mappings. Nonlinear Anal.. 70, 2707–2716 (2009). Publisher Full Text OpenURL

  10. Borwein, JM: Fifty years of maximal monotonicity. Optim. Lett.. 4, 473–490 (2010). Publisher Full Text OpenURL

  11. Browder, FE, Petryshyn, WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl.. 20, 197–228 (1967). Publisher Full Text OpenURL

  12. Iiduka, H, Takahashi, W, Toyoda, M: Approximation of solutions of variational inequalities for monotone mappings. Panam. Math. J.. 14, 49–61 (2004)

  13. Liu, F, Nashed, MZ: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Anal.. 6, 313–344 (1998). Publisher Full Text OpenURL

  14. Nakajo, K, Takahashi, W: Strong and weak convergence theorems by an improved splitting method. Commun. Appl. Nonlinear Anal.. 9, 99–107 (2002)

  15. Lin, LJ, Takahashi, W: A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications. Positivity (2012). Publisher Full Text OpenURL

  16. Halpern, B: Fixed points of nonexpansive maps. Bull. Am. Math. Soc.. 73, 957–961 (1967). Publisher Full Text OpenURL

  17. Lions, PL: Approximation de points fixes de contractions. C. R. Acad. Sci. Paris Sér. A-B. 284, 1357–1359 (1977). PubMed Abstract OpenURL

  18. Wittmann, R: Approximation of fixed points of nonexpansive mappings. Arch. Math.. 58, 486–491 (1991)

  19. Reich, S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl.. 75, 287–292 (1980). Publisher Full Text OpenURL

  20. Shioji, N, Takahashi, W: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc.. 70, 45–57 (2009)

  21. Zegeye, H, Shahzad, N: Viscosity approximation methods for fixed-points problems. Appl. Math. Comput.. 191, 155–163 (2007). Publisher Full Text OpenURL

  22. Moudafi, A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl.. 241, 46–55 (2000). Publisher Full Text OpenURL

  23. Zegeye, H, Shahzad, N, Mekkonen, T: Viscosity approximation methods for pseudocontractive mappings in Banach spaces. Appl. Math. Comput.. 185, 538–546 (2007). Publisher Full Text OpenURL

  24. Marino, G, Xu, HK: A general iterative method for nonexpansive mapping in Hilbert spaces. J. Math. Anal. Appl.. 318, 43–52 (2006). Publisher Full Text OpenURL

  25. Zegeye, H, Shahzad, N: Strong convergence of an iterative method for pseudo-contractive and monotone mappings. J. Glob. Optim.. 54, 173–184 (2012). Publisher Full Text OpenURL

  26. Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc.. 66, 240–256 (2002). Publisher Full Text OpenURL

  27. Zegeye, H: An iterative approximation methods for a common fixed point of two pseudo-contractive mappings. ISRN Math. Anal.. 2011, Article ID 621901 (2011)

    Article ID 621901

    Publisher Full Text OpenURL

  28. Takahashi, W, Zembayashi, K: Strong and weak convergence theorems for equilibriums problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal.. 70, 45–57 (2009). Publisher Full Text OpenURL

  29. Su, Y, Shang, M, Qin, X: An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal.. 69, 2709–2719 (2008). Publisher Full Text OpenURL