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A new iterative method for a common solution of fixed points for pseudo-contractive mappings and variational inequalities

Tanom Chamnarnpan and Poom Kumam*

Author Affiliations

Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand

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Fixed Point Theory and Applications 2012, 2012:67  doi:10.1186/1687-1812-2012-67


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/67


Received:26 October 2011
Accepted:24 April 2012
Published:24 April 2012

© 2012 Chamnarnpan and Kumam; 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 article, we introduce a new iterative scheme 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 for the proposed iterative scheme is proved. Our results improve and extend some recent results in the literature.

2000 Mathematics Subject Classification: 46C05; 47H09; 47H10.

Keywords:
monotone mapping; nonexpansive mapping; pseudo-contractive mappings; variational inequality

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 u C such that 〈v-u, Au〉 ≥ 0 for all v C, where A is a nonlinear mapping. The set of solutions of the variational inequality is denoted by VI(C, A). The variational inequality problem has been extensively studied in the literature, see [1-5] and the reference therein. In the context of the variational inequality problem, this implies that u VI(C, A) ⇔ u = PC(u - λAu), ∀λ > 0, where PC 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 x, y C,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M1">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/2012/1/67/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M2">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M3">View MathML</a>

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M4">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/2012/1/67/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M5">View MathML</a>

for all x, y C; see [2,6-10]. 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/2012/1/67/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M6">View MathML</a>-Lipschitz continuous, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M7">View MathML</a> for all x, y C. Clearly, the class of monotone mappings include the class of α-inverse strongly monotone mappings.

Recall that a mapping T of C into H is called pseudo-contractive if for each x, y C, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M8">View MathML</a>

(1.2)

T is said to be a k-strict pseudo-contractive mapping if there exists a constant 0 ≤ k ≤ 1 such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M9">View MathML</a>

A mapping T of C into itself is called nonexpansive if ∥Tx - Ty∥ ≤ ∥x - y∥, for all x, y C. We denote by F(T) the set of fixed points of T. Clearly, the class of pseudo-contractive mappings include the class of nonexpansive and strict pseudo-contractive mappings.

For finding an element of F(T), where T is a nonexpansive mapping of C into itself, Halpern [11] was the first to study the convergence of the following scheme:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M10">View MathML</a>

(1.3)

where u, x0 C and a sequence {αn} of real numbers in (0,1) in the framework of Hilbert spaces. Lions [12] improved the result of Halpern by proving strong convergence of {xn} to a fixed point of T provided that the real sequence {αn} satisfies certain mild conditions. In 2000, Moudafi [13] introduced viscosity approximation method and proved that if H is a real Hilbert space, for given x0 C, the sequence {xn} generated by the algorithm

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M11">View MathML</a>

(1.4)

where f : C C is a contraction mapping with a constant β ∈ (0,1) and {αn} ⊂ (0,1) satisfies certain conditions, converges strongly to fixed point of Moudafi [13] generalizes Halpern's theorems in the direction of viscosity approximations. In [14,15], Zegeye and Shahzad extended Moudafi's result to Banach spaces which more general than Hilbert spaces. For other related results, see [16-18]. Viscosity approximations are very important because they are applied to convex optimization, linear programming, monotone inclusion and elliptic differential equations. Marino and Xu [19], studied the viscosity approximation method for nonexpansive mappings and considered the following general iterative method:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M12">View MathML</a>

(1.5)

They proved that if the sequence {αn} of parameters satisfies appropriate conditions, then the sequence {xn} generated by (1.5) converges strongly to the unique solution of the variational inequality

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M13">View MathML</a>

which is the optimality condition for the minimization problem

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M14">View MathML</a>

where h is a potential function for γf (i.e., h'(x) = γf(x) for x H).

For finding an element of F(T) ∩ VI(C, A), where T is nonexpansive and A is α-inverse strongly monotone, Takahashi and Toyoda [20] introduced the following iterative scheme:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M15">View MathML</a>

(1.6)

where x0 C, {αn} is a sequence in (0,1), and {λn} is a sequence in (0, 2α), and obtained weak convergence theorem in a Hilbert space H. Iiduka and Takahashi [7] proposed a new iterative scheme x1 = x C and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M16">View MathML</a>

(1.7)

and obtained strong convergence theorem in a Hilbert space.

Motivated and inspired by the work mentioned above which combined from Equations (1.5) and (1.6), in this article, we introduced a new iterative scheme (3.1) below which converges strongly to common element of the set of fixed points of continuous pseudo-contractive mappings which more general than nonexpansive mappings and the solution set of the variational inequality problem of continuous monotone mappings which more general than α-inverse strongly monotone mappings. As a consequence, we provide an iterative scheme which converges strongly to a common element of set of fixed points of finite family continuous pseudo-contractive mappings and the solutions set of finite family of variational inequality problems for continuous monotone mappings. Our theorems extend and unify most the results that have been proved for these important class of nonlinear operators.

2. Preliminaries

Let H be a nonempty closed and convex subset of a real Hilbert space H. Let A be a mapping from C into H. For every point x H, there exists a unique nearest point in C, denoted by PCx, such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M17">View MathML</a>

PC is called the metric projection of H onto C. We know that PC is a nonexpansive mapping of H onto C.

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M18">View MathML</a>

Lemma 2.2.Let C be a closed convex subset of a Hilbert space H. Let x H and x0 C. Then x0 = PCx if and only if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M19">View MathML</a>

Lemma 2.3.[21]Let {an} be a sequence of nonnegative real numbers satisfying the following relation

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M20">View MathML</a>

where,

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M21">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M22">View MathML</a>.

Then, the sequence {an} → 0 as n → ∞.

Lemma 2.4.[22]Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A : C H be a continuous monotone mapping. Then, for r > 0 and x H, there exist z C such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M23">View MathML</a>

(2.1)

Moreover, by a similar argument of the proof of Lemmas 2.8 and 2.9 in[23], Zegeye[22]obtained the following lemmas:

Lemma 2.5.[22]Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A : C H be a continuous monotone mapping. For r > 0 and x H, define a mapping Fr : H C as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M24">View MathML</a>

for all x H. Then the following hold:

(1) Fr is single-valued;

(2) Fr is a firmly nonexpansive type mapping, i.e., for all x, y H,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M25">View MathML</a>

(3) F(Fr) = VI(C,A);

(4) VI(C, A) is closed and convex.

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

Lemma 2.6.[22]Let C be a nonempty closed and convex subset of a real Hilbert space H. Let T : C H be a continuous pseudo-contractive mapping. Then, for r > 0 and x H, there exist z C such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M26">View MathML</a>

(2.2)

Lemma 2.7.[22]Let C be a nonempty closed and convex subset of a real Hilbert space H. Let T : C C be a continuous pseudo-contractive mapping. For r > 0 and x H, define a mapping Tr : H C as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M27">View MathML</a>

for all x H. Then the following hold:

(1) Tr is single - valued;

(2) Tr is a firmly nonexpansive type mapping, i.e., for all x, y H,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M28">View MathML</a>

(3) F(Tr) = F(T);

(4) F(T) is closed and convex.

Lemma 2.8.[19]Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M2">View MathML</a>and 0 < ρ ≤ ∥A-1. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M29">View MathML</a>.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let T : C C be a continuous pseudo-contractive mapping and A : C H be a continuous monotone mapping. Then in what follows, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M30">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M31">View MathML</a> will be defined as follows: For x H and {rn} ⊂ (0, ∞), defined

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M32">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M33">View MathML</a>

3. Strong convergence theorems

In this section, we will prove a strong convergence theorem 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.

Theorem 3.1.Let C be a nonempty closed and convex subset of a real Hilbert space H. Let T : C C be a continuous pseudo-contractive mapping and A : C H be a continuous monotone mapping such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M34">View MathML</a>. Let f be a contraction of H into itself with a contraction constant β and let B : H H be a strongly positive linear bounded self-adjoint operator with coefficients <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M35">View MathML</a>and let {xn} be a sequence generated by x1 C and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M36">View MathML</a>

(3.1)

where {αn} ⊂ [0,1] and {rn} ⊂ (0, ∞) such that

(C1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M37">View MathML</a>;

(C2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M38">View MathML</a>;

(C3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M39">View MathML</a>.

Then, the sequence {xn} converges strongly to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M40">View MathML</a>, which is the unique solution of the variational inequality:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M41">View MathML</a>

(3.2)

Equivalently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M42">View MathML</a>, which is the optimality condition for the minimization problem

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M43">View MathML</a>

where h is a potential function for γf (i.e.,h'(z) = γf(z) for z H).

Remark: (1) The variational inequality (3.2) has the unique solution; (see [19]). (2) It follows from condition (C1) that (1 - δn)I - αnB is positive and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M44">View MathML</a> for all n ≥ 1; (see [24]).

Proof. We processed the proof with following four steps:

Step 1. First, we will prove that the sequence {xn} is bounded.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M45">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M46">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M47">View MathML</a>. Then, from Lemmas 2.5 and 2.7 that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M48">View MathML</a>

(3.3)

Moreover, from (3.1) and (3.2), we compute

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M49">View MathML</a>

Therefore, by the simple introduction, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M50">View MathML</a>

which show that {xn} is bounded, so {yn}, {un}, and {f(xn)} are bounded.

Step 2. We will show that ∥xn+1 - xn∥ → 0 and ∥un - yn∥ → 0 as n → ∞.

Notice that each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M30">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M31">View MathML</a> are firmly nonexpansive. Hence, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M51">View MathML</a>

From (3.1), we note that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M52">View MathML</a>

(3.4)

where K = ∥γf(xn-1) - Bxn-1∥ = 2 sup{∥f(xn) ∥ + ∥un∥:n N}. Moreover, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M47">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M53">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M54">View MathML</a>

(3.5)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M55">View MathML</a>

(3.6)

Putting y = yn+1 in (3.5) and y = yn in (3.6), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M56">View MathML</a>

(3.7)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M57">View MathML</a>

(3.8)

Adding (3.7) and (3.8), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M58">View MathML</a>

which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M59">View MathML</a>

Using the fact that A is monotone, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M60">View MathML</a>

and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M61">View MathML</a>

We observe that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M62">View MathML</a>

(3.9)

Without loss of generality, let k be a real number such that rn > k > 0 for all n N. Then, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M63">View MathML</a>

(3.10)

where M = sup{∥yn - xn∥: n N}. Furthermore, from (3.4) and (3.10), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M64">View MathML</a>

Using Lemma 2.3, and by the conditions (C1) and (C3), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M65">View MathML</a>

Consequently, from (3.10), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M66">View MathML</a>

(3.11)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M46">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M67">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M68">View MathML</a>

(3.12)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M69">View MathML</a>

(3.13)

Putting y := un+1 in (3.12) and y := un in (3.13), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M70">View MathML</a>

(3.14)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M71">View MathML</a>

(3.15)

Adding (3.14) and (3.15), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M72">View MathML</a>

Using the fact that T is pseudo-contractive, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M73">View MathML</a>

and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M74">View MathML</a>

Thus, using the methods in (3.9) and (3.10), we can obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M75">View MathML</a>

(3.16)

where M1 = sup{∥un - yn∥: n N}. Therefore, from (3.11) and property of {rn}, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M76">View MathML</a>

Furthermore, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M77">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M78">View MathML</a>

Thus, by (C1) and (C2), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M79">View MathML</a>

(3.17)

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M45">View MathML</a>, using Lemma 2.5, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M80">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M81">View MathML</a>

(3.18)

Therefore, from (3.1), the convexity of ∥·∥2, (3.2) and (3.18), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M82">View MathML</a>

and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M83">View MathML</a>

(3.19)

So, we have ∥yn - v∥ → 0 as n → ∞. Consequently, from (3.16) and (3.18), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M84">View MathML</a>

Step 3. We will show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M85">View MathML</a>

(3.20)

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M86">View MathML</a>, and since, Q(I - B + γf) is contraction on H into C (see also [[25], pp. 18]) and H is complete. Thus, by Banach Contraction Principle, then there exist a unique element z of H such that z = Q(I - B + γf)z.

We choose subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M87">View MathML</a> of {xn} such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M88">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M87">View MathML</a> is bounded, there exists a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M89">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M87">View MathML</a> and y C such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M90">View MathML</a>. Without loss of generality, we may assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M91">View MathML</a>. Since C is closed and convex it is weakly closed and hence y C. Since xn - yn → 0 as n → ∞ we have that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M92">View MathML</a>. Now, we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M93">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M94">View MathML</a>, Lemma 2.5 and using (3.5), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M95">View MathML</a>

(3.21)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M96">View MathML</a>

(3.22)

Set vt = tv + (1- t)y for all t ∈ (0,1] and v C. Consequently, we get vi C. From (3.22), it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M97">View MathML</a>

from the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M98">View MathML</a> as i → ∞, we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M99">View MathML</a> as i → ∞. Since A is monotone, we also have that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M100">View MathML</a>. Thus, if follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M101">View MathML</a>

and hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M102">View MathML</a>.

If t → 0, the continuity of A gives that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M103">View MathML</a>

This implies that y VI(C, A).

Furthermore, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M46">View MathML</a>, Lemma 2.5 and using (3.12), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M104">View MathML</a>

(3.23)

Put zt = t(v) + (1 - t)y for all t ∈ (0,1] and v C. Then, zt C and from (3.23) and pseudo-contractivity of T, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M105">View MathML</a>

Thus, since un - yn → 0, as n → ∞ we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M106">View MathML</a> as i → ∞. Therefore, as i → ∞, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M107">View MathML</a>

and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M108">View MathML</a>

Taking t → 0 and since T is continuous we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M109">View MathML</a>

Now, we get v = Ty. Then we obtain that y = Ty and hence y F(T). Therefore, y F(T) ∩ VI(C, A) and since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M42">View MathML</a>, Lemma 2.2 implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M110">View MathML</a>

(3.24)

Step 4. Finally, we will show that xn z as n → ∞, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M111">View MathML</a>.

From (3.1) and (3.2) we observe that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M112">View MathML</a>

which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M113">View MathML</a>

By the condition (C1), (3.24) and using Lemma 2.3, we see that limn→∞ xn - z∥ = 0. This complete to proof. □

If we take f(x) = u, ∀x H and γ = 1, then by Theorem 3.1, we have the following corollary:

Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C C be a continuous pseudo-contractive mapping and A : C H be a continuous monotone mapping such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M34">View MathML</a>. let B : H H be a strongly positive linear bounded self-adjoint operator with coefficients <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M35">View MathML</a>and let {xn} be a sequence generated by x1 H and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M114">View MathML</a>

(3.25)

where {αn} ⊂ [0,1] and {rn} ⊂ (0, ∞) such that

(C1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M115">View MathML</a>;

(C2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M116">View MathML</a>;

(C3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M117">View MathML</a>.

Then, the sequence {xn} converges strongly to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M40">View MathML</a>, which is the unique solution of the variational inequality:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M118">View MathML</a>

(3.26)

Equivalently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M119">View MathML</a>.

If we take T ≡ 0, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M120">View MathML</a> (the identity map on C). So by Theorem 3.1, we obtain the following corollary.

Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C H be a continuous monotone mapping such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M121">View MathML</a>. Let f be a contraction of H into itself and let B : H H be a strongly positive linear bounded self-adjoint operator with coefficients <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M35">View MathML</a>and let {xn} be a sequence generated by x1 H and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M122">View MathML</a>

(3.27)

where {αn} ⊂ [0,1] and {rn} ⊂ (0, ∞) such that

(C1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M115">View MathML</a>;

(C2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M116">View MathML</a>;

(C3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M117">View MathML</a>.

Then, the sequence {xn} converges strongly to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M40">View MathML</a>, which is the unique solution of the variational inequality:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M123">View MathML</a>

(3.28)

Equivalently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M42">View MathML</a>.

If we take A ≡ 0, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M124">View MathML</a> (the identity map on C). So by Theorem 3.1, we obtain the following corollary.

Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C C be a continuous pseudo-contractive mapping such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M125">View MathML</a>. Let f be a contraction of H into itself and let B : H H be a strongly positive linear bounded self-adjoint operator with coefficients <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M35">View MathML</a>and let {xn} be a sequence generated by x1 H and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M126">View MathML</a>

(3.29)

where {αn} ⊂ [0,1] and {rn} ⊂ (0, ∞) such that

(C1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M115">View MathML</a>;

(C2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M116">View MathML</a>;

(C3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M117">View MathML</a>.

Then, the sequence {xn}n≥1 converges strongly to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M40">View MathML</a>, which is the unique solution of the variational inequality:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M127">View MathML</a>

(3.30)

Equivalently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M42">View MathML</a>.

If we take C H in Theorem 3.1, then we obtain the following corollary.

Corollary 3.5. Let H be a real Hilbert space. Let Tn : H H be a continuous pseudo-contractive mapping and A : H H be a continuous monotone mapping such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M128">View MathML</a>. Let f be a contraction of C into itself and let B : H H be a strongly positive linear bounded self-adjoint operator with coefficients <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M35">View MathML</a>and let {xn} be a sequence generated by x1 H and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M129">View MathML</a>

(3.31)

where {αn} ⊂ [0,1] and {rn} ⊂ (0, ∞) such that

(C1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M115">View MathML</a>;

(C2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M116">View MathML</a>;

(C3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M117">View MathML</a>.

Then, the sequence {xn} converges strongly to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M40">View MathML</a>, which is the unique solution of the variational inequality:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M130">View MathML</a>

(3.32)

Equivalently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/67/mathml/M42">View MathML</a>.

Proof. Since D(A) = H, we note that VI(H, A) = A-1(0). So, by Theorem 3.1, we obtain the desired result. □

Remark 3.6. 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 [7] and Zegeye et al. [26], Corollary 3.2 of Su et al. [27] 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 [7] 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 article. All authors read and approved the final manuscript.

Acknowledgements

This study was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC No. 54000267).

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