Convergence theorems for finding zero points of maximal monotone operators and equilibrium problems in Banach spaces
© Saewan et al.; licensee Springer 2013
Received: 23 October 2012
Accepted: 30 April 2013
Published: 16 May 2013
In this paper, we construct a new hybrid projection method for approximating a common element of the set of zeroes of a finite family of maximal monotone operators and the set of common solutions to a system of generalized equilibrium problems in a uniformly smooth and strictly convex Banach space. We prove strong convergence theorems of the algorithm to a common element of these two sets. As application, we also apply our results to find common solutions of variational inequalities and zeroes of maximal monotone operators.
MSC:47H05, 47H09, 47H10.
exists for any . E is said to be uniformly smooth if the limit (1.1) is attained uniformly for in . A Banach space E is said to be strictly convex if for all with and (see  for more details).
One of the major problems in the theory of monotone operators is as follows.
where B is an operator from E into . Such is called a zero point of B. We denote the set of zeroes of the operator B by .
If E is a Hilbert space, then for all .
A point p in C is said to be a strongly asymptotic fixed point of T  if C contains a sequence which converges strongly to p such that . The set of strong asymptotic fixed points of T will be denoted by .
for all and ;
Kohasaka and Takahashi  proved that if E is a smooth strictly convex and reflexive Banach space and B is a continuous monotone operator with , then is a weak relatively nonexpansive mapping. By Takahashi , we know that is closed and convex, where is the set of fixed points of .
where and .
Also, Rockafellar  proved that the sequence defined by (1.5) converges weakly to an element of .
Let C be a nonempty closed convex subset of E and let ℝ be the set of real numbers. Let be a bifunction and be a nonlinear mapping for . The system of generalized equilibrium problems is as follows.
If and in (1.6), then from the problem (1.6) we have the following generalized equilibrium problem denoted by .
The generalized equilibrium problems include fixed point problems, optimization problem, monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, vector equilibrium problems, Nash equilibria in noncooperative games and equilibrium problems as special cases (see, for example, ). Also, some solution methods have been proposed to solve the equilibrium problem (see, for example, [7–14]) and numerous problems in physics, optimization and economics reduce to finding a solution of problem (1.7).
Recently, Li and Su  introduced the hybrid iterative scheme for approximating a common solution of the equilibrium problems and the variational inequality problems in a 2-uniformly convex real Banach space which is also uniformly smooth. In 2010, Zegeye and Shahzad  introduced the iterative process which converges strongly to a common solution of the variational inequality problems for two monotone mappings in Banach spaces.
Quite recently, Shehu  introduced an iterative scheme by the hybrid method for approximating a common element of the set of zeroes of a finite family of α-inverse-strongly monotone operators and the set of common solutions of a system of generalized mixed equilibrium problems in a 2-uniformly convex real Banach space which is also uniformly smooth.
Motivated by the results of Shehu , we prove some strong convergence theorems for finding a common zero of a finite family of continuous monotone mappings and a solution of the system of generalized equilibrium problems in a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property.
Throughout this paper, let E be a Banach space with its dual space . For a sequence of E and a point , the weak convergence of to x is denoted by and the strong convergence of to x is denoted by .
where denotes the duality pairing.
If E is an arbitrary Banach space, then J is monotone and bounded;
If E is a strictly convex, then J is strictly monotone;
If E is a smooth, then J is single-valued and semi-continuous;
If E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E;
If E is reflexive, smooth and strictly convex, then the normalized duality mapping J is single-valued, one-to-one and onto;
If E is a reflexive, strictly convex and smooth Banach space and J is the duality mapping from E into , then is also single-valued, bijective and is also the duality mapping from into E and thus and ;
If E is uniformly smooth, then E is smooth and reflexive;
E is uniformly smooth if and only if is uniformly convex.
where J is the normalized duality mapping from E to .
If E is a Hilbert space, then .
Remark 2.1 If E is a reflexive, strictly convex and smooth Banach space, then, for any , if and only if . It is sufficient to show that if , then . From (1.4) we have . This implies that . From the definition of J, one has . Therefore, we have (see [1, 17] for more details).
The existence and uniqueness of the operator follows from the properties of the functional and the strict monotonicity of the mapping J (see, for example, [1, 17–20]). If E is a Hilbert space, then becomes the metric projection of E onto C.
Example 2.2 (Qin et al. )
Let be the generalized projection from a smooth strictly convex and reflexive Banach space E onto a nonempty closed convex subset C of E. Then is a closed relatively quasi-nonexpansive mapping from E onto C with .
We also need the following lemmas for the proof of our main results.
Lemma 2.3 (Alber )
Lemma 2.4 (Alber )
is a single-valued mapping from E to . For any , the Yosida approximation of B is defined by for all . We know that for all and .
Lemma 2.5 (Kohsaka and Takahashi )
for all and ;
for all ;
Lemma 2.6 (Rockafellar )
Let E be a reflexive strictly convex and smooth Banach space. Then an operator is maximal monotone if and only if for all .
For solving the equilibrium problem for a bifunction , assume that f satisfies the following conditions:
(A1) for all ;
(A2) f is monotone, i.e., for all ;
(A4) for each , is convex and lower semi-continuous.
The following result is given in Blum and Oettli .
Lemma 2.8 (Takahashi and Zembayashi )
- (2)is a firmly nonexpansive-type mapping for all , that is,
is closed and convex.
Lemma 2.9 (Takahashi and Zembayashi )
Proof Define a bifunction by for all . We show that Θ satisfies the conditions (A1)-(A4).
the condition (A2) is satisfied.
So, the condition (A2) is satisfied.
The condition (A3) is satisfied.
Finally, we show that Θ satisfies the condition (A4) since is convex and continuous; that is, is convex and lower semi-continuous. Since is convex and lower semi-continuous, is convex and lower semi-continuous.
This completes the proof. □
- (2)is firmly nonexpansive, i.e., for all ,
is closed and convex;
for all and .
Thus, from Lemmas 2.8 and 2.9, we obtain the conclusion. This completes the proof. □
where for all and .
3 Main results
where for some for all and for all . Then the sequence converges strongly to a point , where .
Proof We split the proof into five steps as follows.
Thus is closed and convex for all .
This shows that , which implies that . Hence for all . This implies that the sequence is well defined.
This implies that is bounded and so exists. In particular, by (1.4), the sequence is bounded. This implies is also bounded. So, and are also bounded. Since E is reflexive and is closed and convex, without loss of generality, we may assume that there exists such that .
that is, for all and . This implies that for each . Therefore, .
Letting in the inequality above, we get . Since is maximal monotone for each , we obtain .
Therefore, by Lemma 2.3, we can conclude that and as . The proof is completed. □
If and , we have the following.
where for some and . Then the sequence converges strongly to a point , where .
In this section, we apply our result to find a common solution of the variational inequality problems and zeros of the maximal operators.
We need the following lemma for our result, which is a special case of Lemmas 2.8 and 2.9 of .
Lemma 4.1 (Zegeye and Shahzad )
- (2)is a firmly nonexpansive-type mapping, i.e., for any ,
is closed and convex;
for any .
where for some for all and for all . Then the sequence converges strongly to a point , where .
Proof Taking for all in Theorem 3.1, we can get the desired conclusion. □
By Theorem 4.2, if we set for each , we obtain the following.
where for some for each . Then the sequence converges strongly to a point , where .
Remark 4.4 Corollary 4.3 extends and improves the result of Zegeye and Shahzad  to a common solution of the variational inequality problems.
This work was supported by Thaksin University Research Fund and was also supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).
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