# 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

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

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,

(1.1)

An operator A is said to be strongly positive on H if there exists a constant such that

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

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

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 -Lipschitz continuous, that is, 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

(1.2)

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

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:

(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

(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:

(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

which is the optimality condition for the minimization problem

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:

(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

(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

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:

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

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

where,

(i) ;

(ii) .

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

(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:

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,

(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

(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:

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,

(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 and 0 < ρ ≤ ∥A-1. Then .

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, and will be defined as follows: For x H and {rn} ⊂ (0, ∞), defined

and

### 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 . 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 and let {xn} be a sequence generated by x1 C and

(3.1)

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

(C1) ;

(C2) ;

(C3) .

Then, the sequence {xn} converges strongly to , which is the unique solution of the variational inequality:

(3.2)

Equivalently, , which is the optimality condition for the minimization problem

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 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 and let and . Then, from Lemmas 2.5 and 2.7 that

(3.3)

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

Therefore, by the simple introduction, we have

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 and are firmly nonexpansive. Hence, we have

From (3.1), we note that

(3.4)

where K = ∥γf(xn-1) - Bxn-1∥ = 2 sup{∥f(xn) ∥ + ∥un∥:n N}. Moreover, since and , we get

(3.5)

and

(3.6)

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

(3.7)

and

(3.8)

Adding (3.7) and (3.8), we have

which implies that

Using the fact that A is monotone, we get

and hence

We observe that

(3.9)

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

(3.10)

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

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

Consequently, from (3.10), we obtain

(3.11)

Since and , we have

(3.12)

and

(3.13)

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

(3.14)

and

(3.15)

Adding (3.14) and (3.15), we have

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

and hence

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

(3.16)

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

Furthermore, since , we have

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

(3.17)

For , using Lemma 2.5, we obtain

and

(3.18)

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

and hence

(3.19)

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

Step 3. We will show that

(3.20)

Let , 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 of {xn} such that

Since is bounded, there exists a sequence of and y C such that . Without loss of generality, we may assume that . Since C is closed and convex it is weakly closed and hence y C. Since xn - yn → 0 as n → ∞ we have that . Now, we show that . Since , Lemma 2.5 and using (3.5), we get

(3.21)

and

(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

from the fact that as i → ∞, we obtain that as i → ∞. Since A is monotone, we also have that . Thus, if follows that

and hence .

If t → 0, the continuity of A gives that

This implies that y VI(C, A).

Furthermore, since , Lemma 2.5 and using (3.12), we get

(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

Thus, since un - yn → 0, as n → ∞ we obtain that as i → ∞. Therefore, as i → ∞, it follows that

and hence

Taking t → 0 and since T is continuous we obtain

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 , Lemma 2.2 implies that

(3.24)

Step 4. Finally, we will show that xn z as n → ∞, where .

From (3.1) and (3.2) we observe that

which implies that

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 . let B : H H be a strongly positive linear bounded self-adjoint operator with coefficients and let {xn} be a sequence generated by x1 H and

(3.25)

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

(C1) ;

(C2) ;

(C3) .

Then, the sequence {xn} converges strongly to , which is the unique solution of the variational inequality:

(3.26)

Equivalently, .

If we take T ≡ 0, then (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 . 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 and let {xn} be a sequence generated by x1 H and

(3.27)

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

(C1) ;

(C2) ;

(C3) .

Then, the sequence {xn} converges strongly to , which is the unique solution of the variational inequality:

(3.28)

Equivalently, .

If we take A ≡ 0, then (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 . 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 and let {xn} be a sequence generated by x1 H and

(3.29)

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

(C1) ;

(C2) ;

(C3) .

Then, the sequence {xn}n≥1 converges strongly to , which is the unique solution of the variational inequality:

(3.30)

Equivalently, .

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 . 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 and let {xn} be a sequence generated by x1 H and

(3.31)

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

(C1) ;

(C2) ;

(C3) .

Then, the sequence {xn} converges strongly to , which is the unique solution of the variational inequality:

(3.32)

Equivalently, .

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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