- Open Access
Iterative scheme for a nonexpansive mapping, an η-strictly pseudo-contractive mapping and variational inequality problems in a uniformly convex and 2-uniformly smooth Banach space
© Kangtunyakarn; licensee Springer. 2013
Received: 21 July 2012
Accepted: 13 January 2013
Published: 1 February 2013
In this paper, we introduce an iterative scheme by the modification of Mann’s iteration process for finding a common element of the set of solutions of a finite family of variational inequality problems and the set of fixed points of an η-strictly pseudo-contractive mapping and a nonexpansive mapping. Moreover, we prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of -strictly pseudo-contractive mappings for every in uniformly convex and 2-uniformly smooth Banach spaces.
Let E be a Banach space with its dual space and let C be a nonempty closed convex subset of E. Throughout this paper, we denote the norm of E and by the same symbol . We use the symbol → to denote the strong convergence. Recall the following definition.
Definition 1.1 A Banach space E is said to be uniformly convex iff for any ϵ, , the inequalities , and imply there exists a such that .
Let . A Banach space E is said to be q-uniformly smooth if there exists a fixed constant such that . It is easy to see that if E is q-uniformly smooth, then and E is uniformly smooth.
is called the normalized duality mapping of E. The duality pair represents for and .
for all .
for every and for some .
Let C and D be nonempty subsets of a Banach space E such that C is nonempty closed convex and , then a mapping is sunny  provided for all and , whenever . The mapping is called a retraction if for all . Furthermore, P is a sunny nonexpansive retraction from C onto D if P is a retraction from C onto D which is also sunny and nonexpansive. The subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D.
Remark 1.1 From (1.1) and (1.2), if T is an η-strictly pseudo-contractive mapping, then is η-inverse strongly accretive.
where the sequence . If T is a nonexpansive mapping with a fixed point and the control sequence is chosen so that , then the sequence generated by normal Mann’s iterative process (1.5) converges weakly to a fixed point of T.
In 2008, Cho et al.  modified the normal Mann’s iterative process and proved strong convergence for a finite family of nonexpansive mappings in the framework of Banach spaces without any commutative assumption as follows.
where is the W-mapping generated by and . Then converges strongly to , where and is the unique sunny nonexpansive retraction from C onto F.
In 2008, Zhou  proved a strong convergence theorem for the modification of normal Mann’s iteration algorithm generated by a strict pseudo-contraction in a real 2-uniformly smooth Banach space as follows.
Then converges strongly to , where and is the unique sunny nonexpansive retraction from C onto .
In 2005, Aoyama et al.  proved a weak convergence theorem for finding a solution of problem (1.3) as follows.
for every , where is a sequence of positive real numbers and is a sequence in . If and are chosen so that for some and for some b, c with , then converges weakly to some element z of , where K is the 2-uniformly smoothness constant of E.
In this paper, motivated by Theorems 1.2, 1.3 and 1.4, we prove a strong convergence theorem for finding a common element of the set of solutions of a finite family of variational inequality problems and the set of fixed points of a nonexpansive mapping and an η-strictly pseudo-contractive mapping in uniformly convex and 2-uniformly smooth spaces. Moreover, by using our main result, we prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of -strictly pseudo-contractive mappings for every in uniformly convex and 2-uniformly smooth Banach spaces.
In this section, we collect and prove the following lemmas to use in our main result.
Lemma 2.1 (See )
for any .
Definition 2.1 (See )
Such a mapping K is called the K-mapping generated by and .
Lemma 2.2 (See )
Let C be a nonempty closed convex subset of a strictly convex Banach space. Let be a finite family of nonexpanxive mappings of C into itself with and let be real numbers such that for every and . Let K be the K-mapping generated by and . Then .
Remark 2.3 From Lemma 2.2, it is easy to see that the K mapping is a nonexpansive mapping.
Lemma 2.4 (See )
Lemma 2.5 (See )
for all and all with .
Lemma 2.6 (See )
Lemma 2.7 (See )
Let C be a closed convex subset of a strictly convex Banach space X. Let be a sequence of nonexpansive mappings on C. Suppose is nonempty. Let be a sequence of positive numbers with . Then a mapping S on C defined by for is well defined, non-expansive and holds.
Lemma 2.8 (See )
Let . If E is uniformly convex, then there exists a continuous, strictly increasing and convex function , such that for all and for any , we have .
Lemma 2.9 (See )
for all , .
Lemma 2.10 (See )
Lemma 2.11 (See )
Lemma 2.12 Let C be a nonempty closed convex subset of a 2-uniformly smooth Banach space E and let be a nonexpansive mapping and be an η-strictly pseudocontractive mapping with . Define a mapping by for all and , where K is the 2-uniformly smooth constant of E. Then .
Then we have , that is, .
Then we have . Therefore, . It follows that . Hence, . □
Remark 2.13 Applying (2.2), we have that the mapping is nonexpansive.
3 Main results
Proof First, we will show that is a nonexpansive mapping for every .
Next, we will show that the sequence is bounded.
By induction, we can conclude that the sequence is bounded and so are , , .
where and .
where and .
Define a mapping by for all and . From Lemma 2.7, 2.12 and (3.2), we have .
Then solves the fixed point equation .
From the condition (i) and Lemma 2.11, we can imply that converses strongly to . This completes the proof. □
The following results can be obtained from Theorem 3.1. We, therefore, omit the proof.
To prove the next theorem, we needed the following lemma.
Lemma 4.1 Let C be a nonempty closed convex subset of a Banach space E and let be an η-strictly pseudo-contractive mapping with . Then .
It implies that , that is, . Then we have . Hence, we have . □
for all .
Proof Since is an -strictly pseudo-contractive mapping, then we have is an -inverse strongly accretive mapping for every . For every , putting in Theorem 3.1, from Remark 4.2 and Theorem 3.1, we can conclude the desired results. □
Next corollaries are derived from Theorem 4.3. We, therefore, omit the proof.
This research was supported by the Research Administration Division of King Mongkut’s Institute of Technology Ladkrabang.
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