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A new mapping for finding a common element of the sets of fixed points of two finite families of nonexpansive and strictly pseudo-contractive mappings and two sets of variational inequalities in uniformly convex and 2-smooth Banach spaces
Fixed Point Theory and Applications volume 2013, Article number: 157 (2013)
Abstract
In this paper we introduce a new mapping in a uniformly convex and 2-smooth Banach space to prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of nonexpansive mappings and the set of fixed points of a finite family of strictly pseudo-contractive mappings and two sets of solutions of variational inequality problems. Moreover, we also obtain a strong convergence theorem for a finite family of the set of solutions of variational inequality problems and the set of fixed points of a finite family of strictly pseudo-contractive mappings by using our main result.
1 Introduction
Throughout this paper, we use E and to denote a real Banach space and a dual space of E, respectively. For any pair and , instead of . The duality mapping is defined by for all . It is well known that if E is a Hilbert space, then , where I is the identity mapping. Recall the following definitions.
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 .
Definition 1.2 A Banach space E is said to be smooth if for each , there exists a unique functional such that and .
It is obvious that if E is smooth, then J is single-valued which is denoted by j.
Definition 1.3 Let E be a Banach space. Then a function is said to be the modulus of smoothness of E if
A Banach space E is said to be uniformly smooth if
It is well known that every uniformly smooth Banach space is smooth.
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.
A mapping is called a nonexpansive mapping if
for all .
T is called an η-strictly pseudo-contractive mapping if there exists a constant such that
for every and for some . It is clear that (1.1) is equivalent to the following:
for every and for some . We give some examples for a strictly pseudo-contractive mapping as follows.
Example 1.1 Let ℝ be a real line endowed with the Euclidean norm and let . Define the mapping by
Then T is a -strictly pseudo-contractive mapping.
Example 1.2 (See [1])
Let ℝ be a real line endowed with the Euclidean norm. Let and let be defined by
Then T is a λ-strictly pseudo-contractive mapping where and , .
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 [2] provided for all and , whenever . A 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.
Subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D.
An operator A of C into E is said to be accretive if there exists such that
A mapping is said to be α-inverse strongly accretive if there exist and such that
Remark 1.3 From (1.1) and (1.2), if T is an η-strictly pseudo-contractive mapping, then is η-inverse strongly accretive.
The variational inequality problem in a Banach space is to find a point such that for some ,
This problem was considered by Aoyama et al. [3]. The set of solutions of the variational inequality in a Banach space is denoted by , that is,
Several problems in pure and applied science, numerous problems in physics and economics reduce to finding an element in (1.4); see, for instance, [4–6].
Recall that normal Mann’s iterative process was introduced by Mann [7] in 1953. The normal Mann’s iterative process generates a sequence in the following manner:
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 1967, Halpern has introduced the iteration method guaranteeing the strong convergence as follows:
where . Such an iteration is called Halpern iteration if T is a nonexpansive mapping with a fixed point. He also pointed out that the conditions and are necessary for the strong convergence of to a fixed point of T.
Many authors have modified the iteration (1.6) for a strong convergence theorem; see, for instance, [8–10].
In 2008, Zhou [11] 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.
Theorem 1.4 Let C be a closed convex subset of a real 2-uniformly smooth Banach space E and let be a λ-strict pseudo-contraction such that . Given and sequences , , and in , the following control conditions are satisfied:
Let a sequence be generated by
Then converges strongly to , where and is the unique sunny nonexpansive retraction from C onto .
In 2006, Aoyama et al. introduced a Halpern-type iterative sequence and proved that such a sequence converges strongly to a common fixed point of nonexpansive mappings as follows.
Theorem 1.5 Let E be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and let C be a nonempty closed convex subset of E. Let be a sequence of nonexpansive mappings of C into itself such that is nonempty and let be a sequence of such that and . Let be a sequence of C defined as follows: and
for every . Suppose that for any bounded subset B of C. Let T be a mapping of C into itself defined by for all and suppose that . If either
then converges strongly to Qx, where Q is the sunny nonexpansive retraction of E onto .
In 2005, Aoyama et al. [3] proved a weak convergence theorem for finding a solution of problem (1.3) as follows.
Theorem 1.6 Let E be a uniformly convex and 2-uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let be a sunny nonexpansive retraction from E onto C, let and let A be an α-inverse strongly accretive operator of C into E with . Suppose that and is given by
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 2009, Kangtunykarn and Suantai [12] introduced the S-mapping generated by a finite family of mappings and real numbers as follows.
Definition 1.4 Let C be a nonempty convex subset of a real Banach space. Let be a finite family of mappings of C into itself. For each , let , where and . Define the mapping as follows:
This mapping is called the S-mapping generated by and .
For every , put in (1.7), then the S-mapping generated by and reduces to the K-mapping generated by and , which is defined by Kangtunyakarn and Suantai [13].
Recently, Kangtunyakarn [14] introduced 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 as follows.
Theorem 1.7 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let be the sunny nonexpansive retraction from E onto C. For every , let be an -inverse strongly accretive mapping. Define a mapping by for all and , where , K is the 2-uniformly smooth constant of E. Let be the K-mapping generated by and , where , and . Let be a nonexpansive mapping and be an η-strictly pseudo-contractive mapping with . Define a mapping by , and . Let be a sequence generated by and
where is a contractive mapping and , and satisfy the following conditions:
Then the sequence converses strongly to , which solves the following variational inequality:
Question How can we prove a strong convergence theorem for the set of fixed points of a finite family of nonexpansive mappings and the set of fixed points of a finite family of strictly pseudo-contractive mappings and the set of solutions of variational inequality problems in a uniformly convex and 2-uniformly smooth Banach space?
Motivated by the S-mapping, we define a new mapping in the next section to answer the above question, and from Theorems 1.4, 1.5, 1.6 and 1.7 we modify the Halpern iteration for finding a common element of two sets of solutions of (1.3) and the set of fixed points of a finite family of nonexpansive mappings and the set of fixed points of a finite family of strictly pseudo-contractive mappings in a uniformly convex and 2-uniformly smooth Banach space. Moreover, by using our main result, we also obtain a strong convergence theorem for a finite family of the set of solutions of (1.3) and the set of fixed points of a finite family of strictly pseudo-contractive mappings.
2 Preliminaries
In this section we collect and prove the following lemmas to use in our main result.
Lemma 2.1 (See [15])
Let E be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:
for any .
Lemma 2.2 (See [16])
Let X be a uniformly convex Banach space and , . Then there exists a continuous, strictly increasing and convex function , such that
for all and all with .
Lemma 2.3 (See [3])
Let C be a nonempty closed convex subset of a smooth Banach space E. Let be a sunny nonexpansive retraction from E onto C and let A be an accretive operator of C into E. Then, for all ,
Lemma 2.4 (See [15])
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 2.5 (See [17])
Let C be a closed and convex subset of a real uniformly smooth Banach space E and let be a nonexpansive mapping with a nonempty fixed point . If is a bounded sequence such that . Then there exists a unique sunny nonexpansive retraction such that
for any given .
Lemma 2.6 (See [18])
Let be a sequence of nonnegative real numbers satisfying
where is a sequence in and is a sequence such that
Then .
From the S-mapping, we define the mapping generated by two sets of finite families of the mappings and real numbers as follows.
Definition 2.1 Let C be a nonempty convex subset of a Banach space. Let and be two finite families of mappings of C into itself. For each , let , where and . We define the mapping as follows:
This mapping is called the -mapping generated by , and .
Lemma 2.7 Let C be a nonempty closed convex subset of a 2-uniformly smooth and uniformly convex Banach space. Let be a finite family of -strict pseudo-contractions of C into itself and let be a finite family of nonexpansive mappings of C into itself with and with , where K is the 2-uniformly smooth constant of E. Let , where , , , and for all . Let be the -mapping generated by , and . Then and is a nonexpansive mapping.
Proof Let and , we have
For every and (2.2), we have
For every and (2.2) we have
which implies that
for every .
From (2.2), it implies that , that is, . From the definition of , we have
From (2.2) and , we have
It implies that .
From the definition of and , we have
From the definition of and (2.4), we have
From (2.2), (2.6), (2.7) and E is uniformly convex, we have
It implies that
Assume that , then we have . From the properties of , we have
This is a contradiction. Then we have . From (2.6), we have .
From the definition of , we have
By using the same method as above, we have
Continuing on this way, we can conclude that
for every . From (2.2) and (2.10), we have
It implies that
From the definition of and (2.10), we have
Then we have
Since for every and , then we have
for every . From (2.11), it implies that
From (2.12) and (2.13), we have
Hence, . It is easy to see that .
Applying (2.2), we have that the mapping is nonexpansive. □
Lemma 2.8 [19]
Let C be a closed convex subset of a strictly convex Banach space E. Let and be two nonexpansive mappings from C into itself with . Define a mapping S by
where λ is a constant in . Then S is nonexpansive and .
Applying Lemma 2.8, we have the following lemma.
Lemma 2.9 Let C be a closed convex subset of a strictly convex Banach space E. Let , and be three nonexpansive mappings from C into itself with . Define a mapping S by
where α, β, γ is a constant in and . Then S is nonexpansive and .
Proof For every and the definition of the mapping S, we have
where . From Lemma 2.8, we have and is a nonexpansive mapping. From Lemma 2.8 and (2.15), we have and S is a nonexpansive mapping. Hence we have . □
3 Main results
Theorem 3.1 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let be a sunny nonexpansive retraction from E onto C and let A, B be α- and β-inverse strongly accretive mappings of C into E, respectively. Let be a finite family of -strict pseudo-contractions of C into itself and let be a finite family of nonexpansive mappings of C into itself with and with , where K is the 2-uniformly smooth constant of E. Let , where , , , and for all . Let be the -mapping generated by , and . Let be the sequence generated by and
where and and satisfy the following conditions:
Then converges strongly to , where is the sunny nonexpansive retraction of C onto ℱ.
Proof First we show that and are nonexpansive mappings. Let , we have
Then we have is a nonexpansive mapping. By using the same methods as (3.2), we have is a nonexpansive mapping.
Let . From Lemma 2.3, we have and . By the definition of , we have
By induction, we have . We can imply that the sequence is bounded and so are , and .
Next, we show that . From the definition of , we have
Applying Lemma 2.6, we have
Next, we show that
From the definition of , we have
where and .
From Lemma 2.2, we have
which implies that
From (3.3) and condition (i), we obtain
From the property of , we have
By using the same method as (3.7), we can imply that
Define for all and . From Lemma 2.9, we have . From Lemmas 2.3 and 2.7, we have . By the definition of G, we obtain
From (3.4), we have
From Lemma 2.5 and (3.8), we have
where . Finally, we prove strong convergence of the sequence to . From the definition of , we have
Applying Lemma 2.6 and condition (i), we have . This completes the proof. □
4 Applications
From our main results, we obtain strong convergence theorems in a Banach space. Before proving these theorem, we need the following lemma which is the result from Lemma 2.7 and Definition 1.4. Therefore, we omit the proof.
Lemma 4.1 Let C be a nonempty closed convex subset of a 2-uniformly smooth and uniformly convex Banach space. Let be a finite family of -strict pseudo-contractions of C into itself with and with , where K is the 2-uniformly smooth constant of E. Let , where , , , and for all . Let S be the S-mapping generated by and . Then and S is a nonexpansive mapping.
Theorem 4.2 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let be a sunny nonexpansive retraction from E onto C and let A, B be α- and β-inverse strongly accretive mappings of C into E, respectively. Let be a finite family of -strict pseudo-contractions of C into itself with and with , where K is the 2-uniformly smooth constant of E. Let , where , , , and for all . Let S be the S-mapping generated by and . Let be the sequence generated by and
where and and satisfy the following conditions:
Then converges strongly to , where is the sunny nonexpansive retraction of C onto ℱ.
Proof Put in Theorem 3.1. From Lemma 4.1 and Theorem 3.1 we can conclude the desired result. □
Theorem 4.3 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let be a sunny nonexpansive retraction from E onto C. For every , let , A, B be -, α- and β-inverse strongly accretive mappings of C into E, respectively. Define a mapping by , where , K is the 2-uniformly smooth constant of E, for all and . Let be a finite family of -strict pseudo-contractions of C into itself and with and with . Let , where , , , and for all . Let be the -mapping generated by , and . Let be the sequence generated by and
where and and satisfy the following conditions:
Then converges strongly to , where is the sunny nonexpansive retraction of C onto ℱ.
Proof By using the same method as (3.2), we can conclude that is a nonexpansive mapping. From Lemma 2.3, we have for all . From Theorem 3.1 we can conclude the desired conclusion. □
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Acknowledgements
This research was supported by the Research Administration Division of King Mongkut’s Institute of Technology Ladkrabang.
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Kangtunyakarn, A. A new mapping for finding a common element of the sets of fixed points of two finite families of nonexpansive and strictly pseudo-contractive mappings and two sets of variational inequalities in uniformly convex and 2-smooth Banach spaces. Fixed Point Theory Appl 2013, 157 (2013). https://doi.org/10.1186/1687-1812-2013-157
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DOI: https://doi.org/10.1186/1687-1812-2013-157