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The extragradient-Armijo method for pseudomonotone equilibrium problems and strict pseudocontractions
Fixed Point Theory and Applications volume 2012, Article number: 82 (2012)
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.
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, S i : C → C is a L i -strict pseudocontraction for some 0 ≤ L i < 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 thatand (I - S)(xn ) → 0, then;
(c) the fixed point set Fix(S) is closed and convex;
(d) if λ i > 0 and, thenis a-strict pseudocontraction with;
(e) if λ i is given as in (d) and {S i : 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 {α n } and {r n }, sequences {xn } and {un } converge strongly to z = PrFix(T)∩Sol(f,C)(g(z)), where Pr C 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 {α n } and {r n }, 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 Pr C (x), such that
Pr C is called the metric projection on C. We know that Pr C is a nonexpansive mapping on C. It is also known that Pr C is characterized by the following properties
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 {α n } 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 m n such that
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 {x n }, {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 {Pr C (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, S i : C → C be a L i -Lipschitz pseudocontractions for all i = 1, ..., p andsatisfy 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 m n 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
On the other hand, since yn is the unique solution of the strongly convex problem
we have
With y = xn , the last inequality implies
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∉ H n .
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 ∉ H n .
Step 4. We claim that if ||r(xn )|| ≠ 0 then , where .
For and ||w|| ≠ 0, we know that
Hence,
Otherwise, for every y ∈ C ∩ H n there exists λ ∈ (0, 1) such that
where
From Step 2, it follows that xn ∈ C but xn ∉ H n . Therefore, we have
because . Also we have
Using and the Pythagorean theorem, we can reduce that
From (3.3) and (3.4), we have
which means
Step 5. We claim that if ||r(xn )|| ≠ 0 then Sol(f, C) ⊆ C ∩ H n .
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
It follows from vn ∈ ∂2f (zn , zn ) that
Combining (3.5) and (3.6), we have
By the definition of H n , we have x* ∈ H n . Thus Sol(f, C) ⊆ C ∩ H n .
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
where , and .
In the case ||r(xn )|| ≠ 0, by Step 4, we have , i.e.,
where . Substituting z = x* ∈ Sol(f, C) ⊆ C ∩ H n by Step 5, then we have
which implies that
Hence
Since and
we have
It follows from vn ∈ ∂2f(zn , zn ) that
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
Substituting y = x* into (3.10) and using f(zn , zn ) = 0, we have
Since f is pseudomonotone on C and f(x*, x) ≥ 0, ∀x ∈ C, we have
Combining this with (3.12), we get
Using (3.9), (3.11) and (3.13), we have
Combining (3.8) with (3.14), we obtain
Using and the equality
we have
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
From and Step 6, it follows that
Hence
Using , we have
Hence
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
Since f (xn , ·) is convex and subdifferentiable on C, i.e.,
where un ∈ ∂2f (xn , xn ). Using y = yn , we have
Combining this and (3.22), we obtain
Hence
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
where , and . Indeed, if nk+1= n k + 1 then it is clear from Step 6. Otherwise, we suppose that there exists a positive integer p such that n k + 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
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 Pr C (·), we have
It follows from Step 7 that
By Lemma 3.1, we have
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
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monotone on C if
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pseudomonotone on C if
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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 {α n } 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 m n such that
Compute
whereand, 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 S i : C → C be a L i -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.
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The work was supported by the Vietnam National Foundation for Science Technology Development (NAFOSTED).
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The main idea of this paper is proposed by PNA. All authors read and approved the final manuscript.
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Anh, P.N., Hien, N.D. The extragradient-Armijo method for pseudomonotone equilibrium problems and strict pseudocontractions. Fixed Point Theory Appl 2012, 82 (2012). https://doi.org/10.1186/1687-1812-2012-82
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DOI: https://doi.org/10.1186/1687-1812-2012-82