- Research Article
- Open Access
New Iterative Scheme for Finite Families of Equilibrium, Variational Inequality, and Fixed Point Problems in Banach Spaces
© Shenghua Wang and Caili Zhou. 2011
- Received: 6 December 2010
- Accepted: 30 January 2011
- Published: 24 February 2011
We introduced a new iterative scheme for finding a common element in the set of common fixed points of a finite family of quasi-ϕ-nonexpansive mappings, the set of common solutions of a finite family of equilibrium problems, and the set of common solutions of a finite family of variational inequality problems in Banach spaces. The proof method for the main result is simplified under some new assumptions on the bifunctions.
- Hilbert Space
- Banach Space
- Convex Subset
- Equilibrium Problem
- Nonexpansive Mapping
Let be a mapping from into itself. We denote the set of fixed points of by . is called to be nonexpansive if for all and quasi-nonexpansive if and for all and . A point is called to be an asymptotic fixed point of  if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of is denoted by . The mapping is said to be relatively nonexpansive [6–8] if and for all and . The mapping is said to be -nonexpansive if for all . is called to be quasi- -nonexpansive  if and for all and .
where is the duality mapping on , is a relatively nonexpansive mapping from into itself, and is a sequence of real numbers such that and and proved that the sequence generated by (1.5) converges strongly to , where is the generalized projection from onto .
For studying the equilibrium problem, is usually assumed to satisfy the following conditions:
(A1) for all ;
(A2) is monotone, that is, for all ;
(A3)for each , ;
(A4)for each , is convex and lower semicontinuous.
where is a closed quasi- -nonexpansive mapping for each , are real sequences in satisfying for each and for each and is a real sequence in with . Then the authors proved that converges strongly to , where .
where , such that . Further, they proved that converges strongly to an element of , where .
where are the real numbers in satisfying and for each , are the real numbers in satisfying . We will prove that the sequence generated by (1.13) converges strongly to an element in . In this paper, in order to simplify the proof, we will replace the condition (A3) with (A3'): for each fixed , is continuous.
Obviously, the condition (A3') implies (A3). Under the condition (A3'), we will show that each (as well as , , ) is closed which is such that the proof for the main result of this paper is simplified.
The space is said to be smooth if , , and is called uniformly smooth if and only if .
A Banach space is said to be strictly convex if for all with and . It is said to be uniformly convex if for any two sequences and in such that and . It is known that if a Banach space is uniformly smooth, then its dual space is uniformly convex.
A Banach space is called to have the Kadec-Klee property if for any sequence and with , where denotes the weak convergence, and , then as , where denotes the strong convergence. It is well known that every uniformly convex Banach space has the Kadec-Klee property. For more details on the Kadec-Klee property, the reader is referred to [3, 4].
Let be a nonempty closed and convex subset of a Banach space . A mapping is said to be closed if for any sequence such that and , .
The solution set of the above variational inequality problem is denoted by .
Next we state some lemmas which will be used later.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Let be a strictly convex and smooth Banach space, a nonempty closed and convex subset of , and a quasi- -nonexpansive mapping. Then is a closed and convex subset of .
Since the condition (A3') implies (A3), the following lemma is a natural result of [22, Lemmas 2.8 and 2.9].
there exists such that
define a mapping by
Then the following conclusions hold:
(1) is single-valued;
(2) is firmly nonexpansive, that is, for all ,
(4) is quasi- -nonexpansive;
(5) is closed and convex;
then satisfies all the conclusions in Lemma 2.4. See [25, Lemma 2.4].
Lemma 2.6 (see ).
for all and , where .
The following lemma can be obtained from Lemma 2.6 immediately; also see [20, Lemma 1.9].
Lemma 2.7 (see ).
for all and such that .
Let be a closed and convex subset of a uniformly smooth and strictly convex Banach space . Let be a bifunction satisfying (A1), (A2), (A3'), and (A4). Let and be a mapping defined by (2.7). Then Tr is closed.
which implies that . This completes the proof.
where and are defined by (1.11) and (1.12), are the real numbers in satisfying and for each , are the real numbers in satisfying . Then the sequence converges strongly to , where is the generalized projection from onto .
First we prove that is closed and convex for each . From the definition of , it is obvious that is closed. Moreover, since is equivalent to , it follows that is convex for each . By the definition of , we can conclude that is closed and convex for each .
for each . This shows that the sequence is bounded. It follows from (1.4) that the sequence is also bounded.
Since each is closed, by (3.30) and (3.31) we conclude that , that is, , . On the other hand, Lemma 2.4, Remark 2.5, and Lemma 2.8 show that and are closed. So, by (3.32) and (3.33) we have and . Now, it follows from Lemma 2.4 and Remark 2.5 that ( ) and ( ). Hence, ( ) and ( ). Therefore, .
In view of Lemma 2.1, we can obtain that . This completes the proof.
Obviously, the proof process of is simple since we replace the condition (A3) with (A3') which is such that and are closed. In fact, although the condition (A3') is stronger than (A3), it is not easier to verify the condition (A3) than verify the condition (A3'). Hence, from this point, the condition (A3') is acceptable. On the other hand, the definition of is of some interest.
If for each , for each and for each , then Theorem 3.1 reduces to the following result.
where and are defined by (1.11) and (1.12) with and , are the real numbers in satisfying . Then the sequence converges strongly to , where is the generalized projection from onto .
where and are defined by (1.11) and (1.12) with ( is the identity mapping), are the real numbers in satisfying and for each , are the real numbers in satisfying . Then the sequence converges strongly to , where is the projection from onto .
Since each is closed, we can conclude that , . Note that in a Hilbert space, a firmly-nonexpansive mapping is also nonexpansive. Hence, and are nonexpansive for each and . By demiclosed principle, we can conclude that and for each and . That is, . Then by the final part of proof of Theorem 3.1, we have . This completes the proof.
for all . Take . Then we have . That is, . This shows that , which implies that . So, . Based this, we have following result.
are the real numbers in satisfying and for each , are the real numbers in satisfying . Then the sequence converges strongly to , where is the projection from onto .
If , , and for each , , and , then Corollary 3.5 reduced the following result.
where is defined by (1.11) with and , is defined by (3.44) , and are the real numbers in satisfying . Then the sequence converges strongly to , where is the projection from onto .
This work was supported by the Natural Science Foundation of Hebei Province (A2010001482).
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