A new iteration process for equilibrium, variational inequality, fixed point problems, and zeros of maximal monotone operators in a Banach space
© Saewan and Kumam; licensee Springer 2013
Received: 10 September 2012
Accepted: 27 December 2012
Published: 16 January 2013
In this article, a new iterative process is introduced to approximate a common element of a fixed point set, the solutions of equilibrium problems, the solution set of variational inequality problems, and the set of zeros of maximal monotone operators in a uniformly smooth and strictly convex Banach space by using a hybrid projection method. Also, we prove new strong convergence theorems for this proposed iterative precess in a Banach space.
MSC:47H05, 47H09, 47H10.
where and are the resolvent of A. Solovov and Svaitor  proposed a modified proximal point algorithm which converges strongly to a solution of the equation by using the projection method. Many problems in nonlinear analysis and optimization can be formulated by the proximal point algorithm (see [4–9]).
The equilibrium problem is very general in the sense that it includes, as special cases, optimization problems, variational inequality problems, min-max problems, saddle point problem, fixed point problem, Nash EP. In 2008, Takahashi and Zembayashi [10, 11] introduced iterative sequences for finding a common solution of an equilibrium problem and a fixed point problem.
The set of solutions of (1.4) is denoted by .
A point is a fixed point of T provided . Denote by the fixed point set of T; that is, . A point p in C is called an asymptotic fixed point of T  if C contains a sequence which converges weakly to p such that . The asymptotic fixed point set of T is denoted by .
T is said to be closed if for any sequence such that and , .
Remark 1.1 Every quasi-ϕ-nonexpansive mapping implies a quasi-ϕ-asymptotically nonexpansive mapping and a quasi-ϕ-asymptotically nonexpansive mapping implies a total quasi-ϕ-asymptotically nonexpansive mapping, but the converse is not true.
The existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping J. 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 .
Recently, Qin et al.  considered a pair of asymptotically quasi-ϕ-nonexpansive mappings. To be more precise, they proved the following results.
In 2008, Alber et al.  proved the strong convergence theorems to approximate a fixed point of a total asymptotically nonexpansive mapping in a Hilbert space. In 2011, Chang et al. [22, 23] proved the strong convergence theorems for finding the set of fixed points of a total quasi-ϕ-asymptotically nonexpansive mapping in the framework of Banach spaces.
Motivated and inspired by the work mentioned above, in this paper, we introduce a new hybrid projection algorithm for a pair of total quasi-ϕ-asymptotically nonexpansive mappings for finding a set of solutions of the equilibrium problem, a zero point of maximal monotone operators, and a set of solutions of the variation inequality in a uniformly smooth and strictly convex Banach space.
In this article, we denote the strong convergence and weak convergence of a sequence by and , respectively.
A Banach space E is uniformly convex if and only if for all . Let p be a fixed real number with . A Banach space E is said to be p-uniformly convex if there exists a constant such that for all . Observe that every p-uniformly convex is uniformly convex. One should note that no Banach space is p-uniformly convex for .
Remark 2.1 The basic properties of E, J, and are as follows (see ).
If E is an arbitrary Banach space, then J is monotone and bounded;
If E is strictly convex, then J is strictly monotone;
If E is 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;
If E is a reflexive and strictly convex Banach space, then is norm-weak∗-continuous.
Remark 2.2 If E is a reflexive strictly convex and smooth Banach space, then if and only if . It is sufficient to show that if , then . From (1.5), we have . This implies that . From the definition of J, one has . Therefore, we have (see [24–26] for more details).
Recall that a Banach space E has the Kadec-Klee property [24, 25, 27] if for any sequence and with and , then as . It is well known that if E is a uniformly convex Banach space, then E has the Kadec-Klee property.
The generalized projection  from E into C is defined by . The existence and uniqueness of the operator follow from the properties of the functional and the strict monotonicity of the mapping J (see, for example, [4, 15, 24, 25, 28]). If E is a Hilbert space, then and becomes the metric projection . If C is a nonempty closed and convex subset of a Hilbert space H, then is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. We also need the following lemmas for the proof of our main results.
Lemma 2.3 (Alber )
Lemma 2.4 (Alber )
Lemma 2.5 (Change et al. )
Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let be a closed and total quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences and with , as and a strictly increasing continuous function with . If , then the fixed point set is a closed convex subset of C.
For solving the equilibrium problem for a bifunction , let us 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 in Blum and Oettli .
Lemma 2.6 (Blum and Oettli )
Lemma 2.7 (Takahashi and Zembayashi )
- (2)is a firmly nonexpansive-type mapping , that is, for all ,
is closed and convex.
Lemma 2.8 (Takahashi and Zembayashi )
Lemma 2.9 
for all and with .
is a single-valued mapping from E to . For any , the Yosida approximation of A is defined by for all . We know that for all and .
Lemma 2.10 (Kohsaka and Takahashi )
for all and ;
for all ;
Lemma 2.11 (Rockafellar )
Let E be a reflexive strictly convex and smooth Banach space. Then an operator is maximal monotone if and only if for all .
3 Main result
where , , and are sequences in such that , for some , , for all , . If and , then converges strongly to .
That is, is convex for all . By the definition of , it is obvious that is closed for all .
where . This shows that , thus . Hence, F⊂ for all . This implies that the sequence is well defined.
We show that .
Thus . From (A3), we have for all . Hence, .
For any , it follows from the monotonicity of A that for all . Letting , we get . Therefore, since A is maximal monotone, we obtain .
Since and , it yields that , . From , we get , that is, . In view of the closeness of S, we have . This implies that . By the same way, we have that .
By Lemma 2.3, we can conclude that and as . The proof is completed. □
Let A be a continuous and monotone operator of C into . Then we can find a solution of in a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property by using the following lemma.
Lemma 3.2 (Zegeye and Shahzad )
is a closed and convex subset of C;
for all .
where , and are sequences in such that , for some , , for all , . If and , then converges strongly to .
Since are continuous and monotone mappings, we have . From (3.31) and (3.32), it follows that . Take the limit as and . We get for all . Therefore, for all . If , we have that for all . Hence, . The proof is completed. □
where , , and are sequences in such that , for some , , for all , . If and , then converges strongly to .
The authors would like to express their thanks to the reviewer for helpful suggestions and comments for the improvement of this paper. This work was supported by Thaksin University.
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