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# The extragradient-Armijo method for pseudomonotone equilibrium problems and strict pseudocontractions

Pham Ngoc Anh1* and Nguyen Duc Hien2

Author Affiliations

1 Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology, Hanoi, Vietnam

2 Department of Natural Sciences, Duy Tan University, Danang, Vietnam

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Fixed Point Theory and Applications 2012, 2012:82  doi:10.1186/1687-1812-2012-82

 Received: 2 January 2012 Accepted: 10 May 2012 Published: 10 May 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 present a new iteration method for finding a common element of the set of fixed points of p strict pseudocontractions and the set of solutions of equilibrium problems for pseudomonotone bifunctions without Lipschitz-type continuous conditions. The iterative process is based on the extragradient method and Armijo-type linesearch techniques. We obtain weak convergence theorems for the sequences generated by this process in a real Hilbert space.

AMS 2010 Mathematics Subject Classification: 65 K10; 65 K15; 90 C25; 90 C33.

##### Keywords:
equilibrium problems; pseudomonotone; extragradient method; strict pseudocontractions; fixed point; linesearch

### 1 Introduction

Let C be a nonempty closed convex subset of a real Hilbert space and f be a bifunction from C × C to . We consider the following equilibrium problems (shortly EP(f, C)):

The set of solutions of Problem EP(f, C) is denoted by Sol(f, C). These problems apprear frequently in many practical problems arising, for instance, physics, engineering, game theory, transportation, economics and network, and become an attractive field for many researchers both theory and applications (see [1-6]). The bifunction f is called

monotone if

pseudomonotone if

Lipschitz-type continuous with constants c1 > 0 and c2 > 0 if

It is clear that every monotone bifunction f is pseudomonotone.

Let C be a nonempty closed convex subset of . A self-mapping S : C C is called a strict pseudocontraction if there exists a constant 0 ≤ L < 1 such that

where I is the identity mapping on C. The set of fixed points of S is denoted by Fix(S). The following proposition lists some useful properties for strict pseudocontractions.

Proposition 1.1 [7]Let C be a nonempty closed convex subset of a real Hilbert space , S : C C be a L-strict pseudocontraction and for each i = 1, ..., p, Si : C C is a Li-strict pseudocontraction for some 0 ≤ Li < 1. Then,

(a) S satisfies the Lipschitz condition

(b) I - S is demiclosed at 0. That is, if {xn} is a sequence in C such that and (I - S)(xn) → 0, then ;

(c) the fixed point set Fix(S) is closed and convex;

(d) if λi > 0 and , then is a -strict pseudocontraction with ;

(e) if λi is given as in (d) and {Si : i = 1, ..., p} has a common fixed point, then

For finding a common fixed point of p strict pseudocontractions , Mastroeni [5] introduced an iterative algorithm in a real Hilbert space. Let sequences {xn} be defined by

Under appropriate assumptions on the sequence {λn,i}, the authors showed that the sequence {xn} converges weakly to the same point .

For obtaining a common element of set of solutions of Problem EP(f, C) and the set of fixed points of a nonexpansive mapping S in a real Hilbert space , Takahashi and Takahashi [8] first introduced an iterative scheme by the viscosity approximation method. The sequence {xn} is defined by

The authors showed that under certain conditions over {αn} and {rn}, sequences {xn} and {un} converge strongly to z = PrFix(T)∩Sol(f,C) (g(z)), where PrC is denoted the projection on C and g : C C is contractive, i.e., ||g(x) - g(y)|| ≤ δ||x - y|| for all x, y C.

Recently, for finding a common element of the set of common fixed points of a strict pseudocontraction sequence and the set of solutions of Problem EP (f, C), Chen et al. [9] proposed new iterative scheme in a real Hilbert space. Let sequences {xn}, {yn} and {zn} be defined by

Then, they showed that under certain appropriate conditions imposed on {αn} and {rn}, the sequences {xn}, {yn} and {zn} converge strongly to PrFix(S)∩Sol(f,C)(x0), where S is a mapping of C into itself defined by for all x C.

There exist some another solution methods for finding a common element of the set of solutions of Problem EP(f, C) and (see [3,10-19]). Most of these algorithms are based on solving approximation equilibrium problems for strongly monotone or monotone and Lipschitz-type continuous bifunctions on C. In this article, we introduce a new iteration method for finding a common element of the set of common fixed points of p strict pseudocontractions and the set of solutions of equilibrium problems for pseudomonotone bifunctions. The fundamental difference here is that at each iteration n, we only solve a strongly convex problem and perform a projection on C. The iterative process is based on the extragradient method and Armijo-type linesearch techniques. We obtain weak convergence theorems for sequences generated by this process in a real Hilbert space .

### 2 Preliminaries

Let C be a nonempty closed convex subset of a real Hilbert space . For each point , there exists the unique nearest point in C, denoted by PrC(x), such that

PrC is called the metric projection on C. We know that PrC is a nonexpansive mapping on C. It is also known that PrC is characterized by the following properties

(2.1)

for all , y C. In the context of the convex optimization, it is also known that if is convex and subdifferentiable on C, then is a solution to the following convex problem

if and only if , where is out normal cone at on C and ∂g(·) denotes the subdifferential of g (see [20]).

Now we are in a position to describe the extragradient-Armijo algorithm for finding a common element of .

Algorithm 2.1 Given a tolerance ε > 0. Choose x0 C, k = 0, γ ∈ (0, 1), and positive sequences {λn,i} and {αn} satisfy the conditions:

Step 1. Solve the strongly convex problem

If ||r(xn)|| ≠ 0 then go to Step 2. Otherwise, set wn = xn and go to Step 3.

Step 2. (Armijo-type linesearch techniques) Find the smallest positive integer number mn such that

(2.2)

Compute

where and , and go to Step 3.

Step 3. Compute

Increase n by 1 and go back to Step 1.

Remark 2.2 If ||r(xn)|| = 0 then xn is a solution to Problem EP(f, C) but it may be not a common fixed point of .

Indeed, ||r(xn)|| = 0, i.e., xn is the unique solution to

Then

Hence

Combining this inequality with f(xn, xn) = 0 and the convexity of f(xn, ·), i.e.,

we have f(xn, x) ≥ 0 for all x C. It means that xn is a solution to Problem EP(f, C).

### 3 Convergence results

In this section, we show the convergence of the sequences {xn}, {yn} and {wn} defined by Algorithm 2.1 is based on the extragradient method and Armijo-type linesearch techniques which solves the problem of finding a common element of two sets . To prove it's convergence, we need the following preparatory result.

Lemma 3.1 [21]Let C be a nonempty closed convex subset of a real Hilbert space . Suppose that, for all u C, the sequence {xn} satisfies

Then, the sequence {PrC(xn)} converges strongly to .

We now state and prove the convergence of the proposed iteration method.

Theorem 3.2 Let C be a nonempty closed convex subset of , Si : C C be a Li-Lipschitz pseudocontractions for all i = 1, ..., p and satisfy the following conditions:

(i) f(x, x) = 0 for all x C, f is pseudomonotone on C,

(ii) f is continuous on C,

(iii) For each x C, f(x, ·) is convex and subdifferentiable on C,

(iv) If the sequence{tn} is bounded then {vn} is also bounded, where vn ∈ ∂2f(tn, tn),

(v) .

Then the sequences {xn}, {yn} and {wn} generated by Algorithm 2.1 converge weakly to the point x*, where .

Proof. We divide the proof into several steps.

Step 1. If there exists n0 such that xn = yn for all n n0, then the sequences {xn}, {yn} and {wn} generated by Algorithm 2.1 converge weakly to .

Indeed, since xn = yn for all n n0, we have wn = xn and

This iteration process is originally introduced by Marino and Xu in a real Hilbert space (see [5]). Under assumptions of Algorithm 2.1 on the sequence {λn,i}, the author showed that the sequence {xn} converges weakly to the same point . Then, the sequence {xn} converges weakly to in . Consequently, the sequences {yn} and {wn} also converge weakly to as n → ∝. In this case, the sequences {zn} and {vn} might not converge weakly to the point .

Otherwise, we consider the following steps.

Step 2. If ||r(xn)|| ≠ 0, then there exists the smallest nonnegative integer mn such that

For ||r(xn)|| ≠ 0 and γ ∈ (0, 1), we suppose for contradiction that for every nonnegative integer m, we have

Passing to the limit above inequality as m → ∝, by continuity of f, we obtain

(3.1)

On the other hand, since yn is the unique solution of the strongly convex problem

we have

With y = xn, the last inequality implies

(3.2)

Combining (3.1) with (3.2), we obtain

Hence it must be either ||r(xn)|| = 0 or . The first case contradicts to ||r(xn)|| ≠ 0, while the second one contradicts to the fact .

Step 3. We claim that if ||r(xn)|| ≠ 0 then xn Hn.

From , it follows that

Then using (4.1) and the assumption f(x, x) = 0 for all x C, we have

Hence

This implies that xn Hn.

Step 4. We claim that if ||r(xn)|| ≠ 0 then , where .

For and ||w|| ≠ 0, we know that

Hence,

Otherwise, for every y C Hn there exists λ ∈ (0, 1) such that

where

From Step 2, it follows that xn C but xn Hn. Therefore, we have

(3.3)

because . Also we have

Using and the Pythagorean theorem, we can reduce that

(3.4)

From (3.3) and (3.4), we have

which means

Step 5. We claim that if ||r(xn)|| ≠ 0 then Sol(f, C) ⊆ C Hn.

Indeed, suppose x* ∈ Sol(f, C). Using the definition of x*, f(x*, x) ≥ 0 for all x C and f is pseudomonotone on C, we get

(3.5)

It follows from vn ∈ ∂2f (zn, zn) that

(3.6)

Combining (3.5) and (3.6), we have

By the definition of Hn, we have x* ∈ Hn. Thus Sol(f, C) ⊆ C Hn.

Step 6. We claim that if ||r(xn)|| ≠ 0 and the sequence {vn} is uniformly bounded by M > 0 then the sequence {||xn - x*||} is nonincreasing and hence convergent. Moreover, we have

(3.7)

where , and .

In the case ||r(xn)|| ≠ 0, by Step 4, we have , i.e.,

where . Substituting z = x* ∈ Sol(f, C) ⊆ C Hn by Step 5, then we have

which implies that

Hence

(3.8)

Since and

we have

(3.9)

It follows from vn ∈ ∂2f(zn, zn) that

(3.10)

Replacing y by yn and combining with assumptions f(zn, zn) = 0 and ,

we have

Combining this inequality with (4.1) and assumption γ ∈ (0, 1), we obtain

(3.11)

Substituting y = x* into (3.10) and using f(zn, zn) = 0, we have

(3.12)

Since f is pseudomonotone on C and f(x*, x) ≥ 0, ∀x C, we have

Combining this with (3.12), we get

(3.13)

Using (3.9), (3.11) and (3.13), we have

(3.14)

Combining (3.8) with (3.14), we obtain

(3.15)

Using and the equality

(3.16)

we have

(3.17)

In the case ||r(xn)|| = 0, by Algorithm 2.1 and (3.16), we have wn = xn and

Combining this and (3.17), we get

So the sequence {||xn - x*||} is nonincreasing and hence convergent. Since (3.17) and the sequence {vn} is uniformly bounded by M > 0, i.e.,

we obtain (3.7).

Step 7. We claim that there exists , where . Consequently, the sequences {xn}, {yn}, {zn}, {vn} and {wn} are bounded.

By Step 6, there exists

(3.18)

From and Step 6, it follows that

Hence

(3.19)

Using , we have

(3.20)

Hence

(3.21)

From (3.21) and (3.19), it follows that

Since yn is the unique solution to

we have

With y = xn C and f(xn, xn) = 0, we have

(3.22)

Since f (xn, ·) is convex and subdifferentiable on C, i.e.,

where un 2 f (xn, xn). Using y = yn, we have

Combining this and (3.22), we obtain

Hence

(3.23)

From the assumption (iv) and (3.18), it implies that the sequence {un} is bounded. Then, it follows from (3.23) that {yn} is bounded and hence is also bounded. Also the sequences {vn} and {wn} are bounded.

Step 8. We claim that there exists a subsequence of the sequence {xn} which converges weakly to and hence the whole sequence {xn} converges weakly to .

Suppose that is a subsequence of {xn} such that

By Step 7 and the assumption (iv), the sequence {vn} is bounded by M > 0. We show that

(3.24)

where , and . Indeed, if nk+1 = nk + 1 then it is clear from Step 6. Otherwise, we suppose that there exists a positive integer p such that nk + p + 1 = nk+1. Note that for all i = 0, 1, ..., p - 1. Using , (3.17) and Step 6, we have

This implies (3.24). Then, since {||xn - x*||} is convergent, it is easy to see that

The cases remaining to consider are the following.

Case 1. . This case must follow that . Since is bounded, there exists an accumulation point of . In other words, a subsequence converges weakly to some , as j → ∝ such that . Then by Remark 2.2, we have .

Case 2. . Since is convergent, there is the subsequence of which converges weakly to , as j → ∝. Since is the smallest nonnegative integer, does not satisfy (4.1). Hence, we have

Passing onto the limit, as j → ∝ and using the continuity of f, we have and

(3.25)

where . It follows from (3.2) that

Since f is continuous and passing onto the limit, as j → ∝, we obtain

Combining this with (3.25), we have

which implies , and hence . Thus every cluster point of the sequence is a solution to Problem EP(f, C).

Now we show that every cluster point of is a fixed point of p strict pseudocontractions . Suppose that there exists a subsequence of which converges weakly to , as j → ∝. By the above proof, we have . Then and converge weakly also to , as j → ∝. For each i = 1, ..., p, we suppose that converges as i → ∝ such that

Then, we have

For each , it follows from (3.20) that

Combining this and Step 6, we get

Then, using (a) of Proposition 1.1, we obtain

So . Then, it follows from (e) of Proposition 1.1 that . Thus letting and using Step 7, we have

We conclude that the whole sequence {xn} converges weakly to . Consequently, the sequences {yn} and {wn} also converge weakly to .

Step 9. We claim that the sequences {xn}, {yn} and {wn} converge weakly to , where .

By Step 8, we suppose that and as n → ∝. Using the definition of PrC(·), we have

(3.26)

It follows from Step 7 that

By Lemma 3.1, we have

(3.27)

Pass the limit in (3.26) and combining this with (3.27), we have

This means that and

It follows from Step 8 that the sequences {xn}, {yn} and {wn} converge weakly to , where

The proof is completed.

### 4 Application to variational inequalities

Let C be a nonempty closed convex subset of and F be a function from C into . In this section, we consider the variational inequalitiy problem which is presented as follows

Let be defined by f(x, y) = 〈F(x), y - x〉. Then problem P(f, C) can be written in V I(F, C). The set of solutions of V I(F, C) is denoted by Sol(F, C). Recall that the function F is called

monotone on C if

pseudomonotone on C if

Lipschitz continuous on C with constants L > 0 (shortly, L-Lipschitz continuous) if

Since

Algorithm 2.1, the convergence algorithm for finding a common element of the set of common fixed points of p strict pseudocontractions and the set of solutions of equilibrium problems for pseudomonotone bifunctions is presented as follows:

Algorithm 4.1 Give a tolerance ε > 0. Choose x0 C, k = 0, γ ∈ (0, 1), and positive sequences {λn,i} and {αn} satisfy the conditions:

Step 1. Compute

If ||r(xn)||≠ 0 then go to Step 2. Otherwise, set wn = xn and go to Step 3.

Step 2. (Armijo-type linesearch techniques) Find the smallest positive integer number mn such that

(4.1)

Compute

where and , and go to Step 3.

Step 3. Compute

Increase n by 1 and go back to Step 1.

Using Theorem 3.2, we also have the convergence of Algorithm 4.1 as the follows:

Theorem 4.2 Let C be a nonempty closed convex subset of . Let be continuous and pseudomonotone, and Si : C C be a Li-Lipschitz pseudocontractions for all i = 1, ..., p such that . Then the sequences {xn}, {yn} and {wn} generated by Algorithm 4.1 converge weakly to the point x*, where .

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

The main idea of this paper is proposed by PNA. All authors read and approved the final manuscript.

### Acknowledgements

The work was supported by the Vietnam National Foundation for Science Technology Development (NAFOSTED).

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