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A new iterative algorithm of pseudomonotone mappings for equilibrium problems in Hilbert spaces
Journal of Inequalities and Applications volume 2013, Article number: 128 (2013)
In this paper, we introduce a new algorithm for finding a common element of the set of fixed points of N strict pseudocontractions and the set of solutions of equilibrium problems with a pseudomonotone and Lipschitz-type continuous bifunction. The scheme is motivated by the idea of extragradient methods and fixed point iteration methods. We show that the iterative sequences generated by this algorithm converge strongly to the above mentioned common element under some suitable conditions on algorithm parameters in a real Hilbert space. And also, we consider the variational inequality problems as an application.
MSC:46H09, 47H10, 47J25, 65K10.
Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product and the norm , and let f be a bifunction from into R such that for all . We consider the equilibrium problem in the sense of Blum and Oettli : Find such that
for all .
We denote by the set of solutions of the equilibrium problem .
We know that the problem covers many important problems in optimization and nonlinear analysis. It has also found many applications in economics, transportation and engineering (see [1, 2] and the references quoted therein). Theory and methods for solving this problem have been developed by many authors [3–7]. Alternatively, the problem of finding a common fixed point of a sequence of finite self-mappings () is described as follows: Find such that
where is the set of fixed points of the mappings () on C. This problem has now become a mature subject in nonlinear analysis. The theory and solution methods of this problem can be found in many research papers and monographs (see [8–10]).
We are interested in the problem of finding a common element of the set of solutions of the equilibrium problem and the set of solutions of the fixed problem (FP), namely: Find such that
A special case of problem (1.1) is that , and this problem is reduced to finding a common element of the set of solutions of variational inequalities, i.e., find such that
In this paper, we introduce a new iterative scheme for solving problem (1.1). This method can be considered to be an improvement of the viscosity approximation method in [15, 18, 19] and the iterative method in  via an improvement of the extragradient methods [3, 4, 21–23].
The paper is organized as follows. Section 2 recalls some concepts in equilibrium problems and fixed point problems that are used in the sequel and an iterative algorithm for solving problem (1.1). In Section 3, we prove the convergence theorems for the algorithms which are defined in Section 2 as the main results of this paper. In Section 4, we consider the variational inequality problems as an application of the main theorem.
We first recall the following definitions that will be used for the main theorem.
Definition 2.1 Let C be a nonempty closed convex subset of a real Hilbert space H. A bifunction is said to be
monotone on C if , ;
pseudomonotone on C if implies , ;
Lipschitz-type continuous on C with two constants and if(2.1)
We know that every monotone bifunction f is pseudomonotone, but the converse is not true (see ).
Definition 2.2 Let C be a nonempty closed convex subset of a real Hilbert space H. A mapping is said to be a strict pseudocontraction if there exists a constant such that
where I is the identity mapping on H. If , then S is called nonexpansive on C.
Now, we define the projection on C, denoted by , i.e.,
And we use the symbols ⇀ and → to denote weak convergence and strong convergence, respectively. The following proposition gives some useful properties for strict pseudocontractions.
Proposition 2.3 
Let C be a nonempty closed convex subset of a real Hilbert space H, let be an L-strict pseudocontraction, and for each , let be an -strict pseudocontraction for some . Then we have the following.
S satisfies the following Lipschitz condition:
is demiclosed at zero. That is, if the sequence is in C such that and , then ;
The set is closed and convex;
If () and , then is an -strict pseudocontraction, where ;
If is the same as in (d) and has a common fixed point, then
Many authors studied the problem of finding a common fixed point of a finite family of mappings. For instance, Marino and Xu  constructed an iterative algorithm for finding a common fixed point of N strict pseudocontractions (). They defined the sequence starting from and taking
where the control sequence of parameters was made in order to get the guarantee for the convergence of the iterative sequence . And they proved that the sequence converges weakly to the point .
Recently, Chen et al.  introduced a new iterative scheme for finding a common element of the set of common fixed points of a sequence of strict pseudocontractions and the set of solutions of the equilibrium problem in a real Hilbert space H. Given a starting point , three iterative sequences , and are generated as the following scheme:
Here, two sequences and are given as control parameters. The authors proved that the sequences , and converged strongly to the same point , under certain conditions on and , such that
where S is a nonexpansive mapping of C into itself defined by
for all .
We need the following assumptions for the main theorems.
Assumption 2.4 The bifunction f satisfies the following conditions:
f is pseudomonotone and weakly continuous on C;
f is Lipschitz-type continuous on C;
for each , is convex and subdifferentiable on C.
Assumption 2.5 Every is an -strict pseudocontraction for some .
Assumption 2.6 The solution set of (1.1) is nonempty, i.e.,
Note that if , where is the set of relative interior points of the domain of , then Assumption 2.4(iii) is satisfied. Now we construct the new algorithms as follows.
Algorithm 2.7 Initialization: Choose positive sequences , , , and satisfying the following conditions:
Take an initial point and set .
Iteration k: Carry out three steps below continuously.
Step 1. Solve two strongly convex programs:
Step 2. Compute the iterations
Step 3. Set
Increase k by one and go back to Step 1.
3 Convergence of the algorithms
In this section, we study the convergence of Algorithm 2.7. We need the following useful lemmas for the main theorems.
Lemma 3.1 
Let C be a nonempty closed convex subset of a real Hilbert space H, and let be subdifferentiable on C. Then is a solution of the following convex problem:
if and only if
where denotes the subdifferential of g and is the (outward) normal cone of C at .
Lemma 3.2 
Let C be a nonempty closed convex subset of a real Hilbert space H and . Let be a bounded sequence such that every weakly cluster point of belongs to C and
Then converges strongly to as .
Now, we are in a position to prove the main theorem.
Theorem 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Suppose that Assumptions 2.4-2.6 are satisfied. Then the sequences , and generated by Algorithm 2.7 converge strongly to the same point , where
Proof The proof of this theorem is divided into several steps.
Step 1. Suppose that . Then we have
Since is convex on C for each , by Lemma 3.1, we see that
if and only if
where is the (outward) normal cone of C at .
Since is subdifferentiable on C, by the well-known Moreau-Rockafellar theorem (see ), there exists such that
Substituting into this inequality, we obtain
And also, it follows from (3.3) that , where and . By the definition of the normal cone , we have
Substituting into the last inequality, we obtain
Combining (3.4) and (3.6), we have
Since , for all , and f is pseudomonotone on C, we have . Hence, (3.7) implies that
From Lipschitz condition (2.1) for f with , and , we have
Combining (3.8) and (3.9), we get
Similarly, since is the unique solution of the strongly convex program
Substituting into the last inequality, we have
from (3.10), (3.11), we have
Hence, we have
The implies that the inequality (3.2) holds.
Step 2. Next, we show that
for all .
Using Step 1 and , we have
Let , using Proposition 2.3(d), (3.12) and the relation
and , we have
where . This means that . Hence
Step 3. Now, we have to prove that
for all .
We show this assertion by mathematical induction. For we have . Hence by Step 2, we obtain
Assume that for some ,
From it follows that
Using this and (3.14), we have
Hence we have
Then it follows from Step 2 that
Consequently, we have
Step 4. Next, we claim that
It follows from Step 2 and that
Hence, we get that is bounded. By Step 1, also the sequences and are bounded. Otherwise, we have
and hence . Using this and , we have
Therefore, there exists
Using , and the property of projections
Combining this and (3.16), we get
It follows from that , i.e.,
Then, by (3.17), we have
Step 2 and (3.16) imply that is bounded, and hence and are also bounded.
By (3.13), we have
From this and (3.18), we obtain
Using (3.13), we also have
Similarly, we have
Combining (3.20), (3.21) and , we have
This completes the proof of Step 4.
In Step 5 and Step 6 of this theorem, we consider weakly clusters of . It follows from (3.15) that the sequence is bounded, and hence there exists a subsequence converging weakly to as . By Step 4, also the sequences , and converge weakly to .
Step 5. Claim that .
For each , we suppose that converges as such that . Then we have
Since , from Step 4 and
we obtain that . By Proposition 2.3(b), we have
Then, it implies that from Proposition 2.3(e).
Step 6. Now we prove that if as , then we have .
Since is the unique strongly convex problem
from Lemma 3.1, we have
It follows that
where and . The definition of the normal cone implies that
On the other hand, since is subdifferentiable on C, by the Moreau-Rockafellar theorem , there exists such that
Combining this with (3.23), we have
Then, using , Step 2, as and weak continuity of f, we have
This means that .
Step 7. Finally, we claim that the sequences , , and converge strongly to the same point , where
From Step 5 and Step 6 it follows that for every weakly cluster point of the sequence ,
On the other hand, using the definition of , we have
Combining this with (3.15), we obtain
for all . For , we have
By Lemma 3.2, we know that the sequence converges strongly to as , where
We also have that as by Step 4. □
Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a function from C into H. In this section, we consider the variational inequality problem which is presented as follows:
Find such that
Let be defined by . Then problem can be written in . The set of solutions of is denoted by .
The function F is called
strongly monotone on C with if
monotone on C if
pseudomonotone on C if
Lipschitz continuous on C with constants if
from Algorithm 2.7, we obtain the algorithm for finding a common element of the set of fixed points of p strict pseudocontractions and the solution set of variational inequality problem .
Algorithm 4.1 Initialization: Choose positive sequences , , , and satisfying the conditions:
Find an initial point .
Iteration k: Perform the three steps below.
Step 1. Solve two strongly convex programs:
Step 2. Compute the iterations
Step 3. Set
Increase k by one and go back to Step 1.
Now, we can prove the following convergence theorem with respect to from Theorem 3.3.
Theorem 4.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a function from C into H such that F is pseudomonotone, weakly continuous and L-Lipschitz continuous on C. If each , is -strict pseudocontraction for some and
then the sequences , and generated by Algorithm 4.1 converge strongly to the same point , where
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This work was supported by the Kyungnam University Foundation Grant 2011.
The authors declare that they have no competing interests.
The main idea of this paper was proposed by JKK. JKK and WHL prepared the manuscript initially and performed all the steps of proof in this research. Both authors read and approved the final manuscript.
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Kim, J.K., Lim, W.H. A new iterative algorithm of pseudomonotone mappings for equilibrium problems in Hilbert spaces. J Inequal Appl 2013, 128 (2013). https://doi.org/10.1186/1029-242X-2013-128
- strict pseudocontractions
- Lipschitz-type continuous
- equilibrium problems
- fixed points