Abstract
In this paper, we use methods different from extragradient methods to prove a strong convergence theorem for the sets of fixed points of two finite families of nonexpansive and strictly pseudo-contractive mappings and the set of solutions of modification of a system of variational inequalities problems in a uniformly convex and 2-uniformly smooth Banach space. Applying the main result we obtain a strong convergence theorem involving two sets of solutions of variational inequalities problems introduced by Aoyama et al. (Fixed Point Theory Appl. 2006:35390, 2006, doi:10.1155/FPTA/2006/35390) in a uniformly convex and 2-uniformly smooth Banach space. We also give a numerical example to support our result.
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1 Introduction
Let E be a real 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 symbols ‘→’ and ‘⇀’ to denote strong and weak convergence, respectively. 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 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
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. Hilbert space, (or ) spaces, and the Sobolev spaces, , are q-uniformly smooth. Hilbert spaces are 2-uniformly smooth, while
Definition 1.3 A mapping J from E onto satisfying the condition
is called the normalized duality mapping of E. The duality pair represents for and .
It is well known that if E is smooth, then J is a single value, which we denote by j.
Definition 1.4 Let C be a nonempty subset of a Banach space E and be a self-mapping. T 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 .
Example 1.1 Let ℝ be a real line endowed with Euclidean norm and let the mapping defined by
for all . Then T is -strictly pseudo-contractive mapping.
Example 1.2 Let E be 2-uniformly smooth Banach space and let be λ-strictly pseudo-contractive mapping. Let K be the 2-uniformly smooth constant of E and , then is a nonexpansive mapping.
Definition 1.5 Let be closed convex and be a mapping of E onto C. The mapping is said to be sunny if for all and . A mapping is called retraction if . A subset C of E is called a sunny nonexpansive retract of E if there exists a sunny nonexpansive retraction of E onto C.
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 1.1 From (1.2) and (1.3), if T is an η-strictly pseudo-contractive mapping, then is an η-inverse strongly accretive.
In 2000, Ansari and Yao [1] introduced the system of generalized implicit variational inequalities and proved the existence of its solution. They derived the existence results for a solution of system of generalized variational inequalities and used their results as tools to establish the existence of a solution of system of optimization problems.
Ansari et al. [2] introduced the system of vector equilibrium problems and prove the existence of its solution. Moreover, they also applied their result to the system of vector variational inequalities. The results of [1] and [2] were used as tools to solve Nash problem for vector-value functions and (non)convex vector valued function.
Let be two nonlinear mappings. In 2010 Yao et al. [3] introduced the system of general variational inequalities problem for finding such that
They proved fixed points theorem by using modification of extragradient methods as follows.
Theorem 1.2 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E which admits a weakly sequentially continuous duality mapping. Let be the sunny nonexpansive retraction from X into C. Let the mappings be α-inverse strongly accretive with and β-inverse strongly accretive with , respectively. Define the mapping by for all and the set of fixed point of G denoted by . For given , let the sequence be generated by
where , , and are three sequences in . Suppose the sequences , , and satisfy the following conditions:
Then converges strongly to , where is the sunny nonexpansive retraction of C onto Ω.
In 2013, Cai and Bu [4] introduced the system of a general variational inequalities problem for finding such that
where . The set of solutions of (1.5) we denote by . If , then problem (1.5) reduces to (1.4). In Hilbert space (1.5) reduces to
which is introduced by Ceng et al. [5]. If , then (1.6) reduces to a problem for finding such that
which is introduced by Verma [6]. If , then problem (1.7) reduces to the variational inequality for finding such that
Variational inequality theory is one of very important mathematical tools for solving many problems in economic, engineering, physical, pure and applied science etc.
Many authors have studied the iterative scheme for finding the solutions of a variational inequality problem; see for example [7–10].
By using the extragradient methods, Cai and Bu [4] proved a strong convergence theorem for finding the solutions of (1.5) as follows.
Theorem 1.3 Let C be a nonempty closed convex subset of a 2-uniformly smooth and uniformly convex Banach space E such that . Let be the sunny nonexpansive retraction from E to C. Let the mapping be α-inverse strongly accretive and β-inverse strongly accretive, respectively. Let be an infinite family of nonexpansive mapping with . Let be a nonexpansive mapping and be a strongly positive linear bounded operator with the coefficient such that . For arbitrarily given , let the sequence be generated iteratively by
where and . Assume that , , and are three sequences in satisfying the following conditions:
Suppose that for any bounded subset of C there exists an increasing, continuous, and convex function from such that and . Let T be a mapping from C into C defined by for all and suppose that . Then converges strongly to , which also solves the following variational inequality:
For the research related to the extragradient methods, some additional references are [11–13].
Motivated by (1.4) and (1.5), we introduce the problem for finding such that
for all , and . This problem is called the modification of a system of variational inequalities problems in Banach space. If , then (1.8) reduces to (1.5).
Motivated by Theorems 1.2 and 1.3, we use the methods different from extragradient methods to prove a strong convergence theorem for finding the solutions of (1.8) and an element of the set of fixed points of two finite families of nonexpansive and strictly pseudo-contractive mappings in a uniformly convex and 2-uniformly smooth Banach space. Applying the main result, we obtain a strong convergence theorem involving two sets of solutions of variational inequalities problems introduced by Aoyama et al. [14] in a uniformly convex and 2-uniformly smooth Banach space. Moreover, we also give a numerical example to support our main results in the last section.
2 Preliminaries
The following lemmas and definitions are important tools to prove the results in the next sections.
Definition 2.1 ([15])
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 . Define the mapping as follows:
This mapping is called the -mapping generated by , , and .
Lemma 2.1 ([15])
Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth 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.
Lemma 2.2 ([16])
Let be a sequence of nonnegative real numbers satisfying
where is a sequence in and is a sequence such that
Then .
Lemma 2.3 ([17])
Let E be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:
for any .
Lemma 2.4 ([18])
Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E and let T be a nonexpansive mapping of C into itself with . Then is a sunny nonexpansive retract of C.
Lemma 2.5 ([19])
Let C be a nonempty closed convex subset of a smooth Banach space and be a retraction from E onto C. Then the following are equivalent:
-
(i)
is both sunny and nonexpansive;
-
(ii)
for all and .
It is obvious that if E is a Hilbert space, we find that a sunny nonexpansive retraction is coincident with the metric projection from E onto C. From Lemma 2.5, let and . Then we have if and only if , for all , where is a sunny nonexpansive retraction from E onto C.
Lemma 2.6 ([20])
Let E 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 2.7 ([21])
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 2.8 ([17])
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 ([22])
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 .
Lemma 2.10 Let C be a nonempty closed convex subset of a smooth Banach space E and let be mappings. Let be a sunny nonexpansive retraction of E onto C. For every and . The following are equivalent:
-
(a)
is a solution of (1.8);
-
(b)
is a fixed point of mapping , i.e., , defined by
where .
Proof First we show that (a) ⇒ (b). Let is a solution of (1.8), and we have
for all . From Lemma 2.5, we have
and .
It follows that
Then , where .
Next we claim that (b) ⇒ (a). Let and . Then
From Lemma 2.5, we have
for all . Then we find that is a solution of (1.8). □
Example 2.1 Let ℝ be a real line with the Euclidean norm and let defined by and for all . The mapping defined by
for all . Then and is a solution of (1.8).
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 and let be a sunny nonexpansive retraction of E onto C. Let be α- and β-inverse strongly accretive operators, respectively. Define the mapping by for all , , and , where K is the 2-uniformly smooth constant of E. 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 and with . Let , where , , , , and for all . Let be the -mapping generated by , , and . Assume that . Let the sequence be generated by and
where with . Suppose that the following conditions are satisfied:
Then the sequence converges strongly to and is a solution of (1.8), where .
Proof First, we show that and are nonexpansive mappings. Let ; we have
Then is a nonexpansive mapping. By using the same method we find that is a nonexpansive mapping. From the definition of G, we see that G is a nonexpansive mapping. Let . Put for all . From the definition of and Lemma 2.10, we have
Applying mathematical induction, we can conclude that the sequence is bounded and so is .
From the definition of , we have
Applying (3.2), the condition (iii), and Lemma 2.2, we have
From the definition of , we have
It follows that
From (3.3) and the conditions (i) and (ii), we have
From the property of g, we have
From the definition of , we have
From the condition (i) and (3.4), we obtain
From the definition of , we have
From the condition (i) and (3.4), we obtain
From the nonexpansiveness of , we have
From (3.5) and (3.6), we have
From the definition of , we have
From (3.3) and (3.5), we have
Define the mapping by for all and . From Lemmas 2.1 and 2.9, we have . From the definition of B, (3.7) and (3.8), we have
Since G and are nonexpansive mappings, we have B is a nonexpansive mapping. From Lemma 2.7, we have
where .
Finally, we show that the sequence converges strongly to . From the definition of , we have
Applying Lemma 2.2, the condition (i) and (3.10), we can conclude that the sequence converges strongly to and is a solution of (1.8), where . This completes the proof. □
The following corollary is a strong convergence theorem involving problem (1.5). This result is a direct proof from Theorem 3.1. We, therefore, omit the proof.
Corollary 3.2 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E and let be a sunny nonexpansive retraction of E onto C. Let be α- and β-inverse strongly accretive operators, respectively. Define the mapping by for all , , , where K is the 2-uniformly smooth constant of E. 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 and with . Let , where , , , , and for all . Let be the -mapping generated by , , and . Assume that . Let the sequence be generated by and
where with . Suppose that the following conditions are satisfied:
Then the sequence converges strongly to and is a solution of (1.5), where .
4 Applications
In this section, we prove a strong convergence theorem involving two sets of solutions of variational inequalities in Banach space. We give some useful lemmas and definitions to prove Theorem 4.4.
Let be a mapping. 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. [14]. The set of solutions of the variational inequality in a Banach space is denoted by , that is,
The variational inequalities problems have been studied by many authors; see, for example, [11, 23].
Lemma 4.1 ([14])
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 4.2 Let C be a nonempty closed convex subset of a uniformly convex Banach space E. Let be nonexpansive mappings with . Define the mapping by for all and . Then and is a nonexpansive mapping.
Proof It is easy to see that . Let and . From the definition of , we have
It follows that
Applying the property of g, we have , that is, . Since and , we have
It follows that . Hence . Applying (4.3), we have is a nonexpansive mapping. □
Lemma 4.3 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E and let be a sunny nonexpansive retraction from E onto C. Let be α- and β-inverse strongly accretive operators, respectively. Define a mapping G as in Lemma 2.10 and for every , and where K is 2-uniformly smooth constant. If , then .
Proof From Lemma 4.1, we have
Using the same method as Theorem 3.1, we find that and are nonexpansive mappings.
From the definition of G and Lemma 4.2, we have
□
From Theorem 3.1 and Lemma 4.3, we have the following theorem.
Theorem 4.4 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E and let be a sunny nonexpansive retraction of E onto C. Let be α- and β-inverse strongly accretive operators, 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 and with , where K is the 2-uniformly smooth constant of E. Let , where , , , , and for all . Let be the -mapping generated by , , and . Assume that . Let the sequence be generated by , and
where and with , , . Suppose that the following conditions are satisfied:
Then the sequence converges strongly to and is a solution of (1.8), where .
From Theorem 4.4, we have the following result.
Example 4.1 Let with norm define by . Define the mappings by and for all .
For every , define the mappings by and . Let be -mapping generated by , , and where for all and . Let the sequence be generated by and
where is a sunny nonexpansive retraction of onto . Then the sequence converges strongly to 0 and is a solution of (1.8).
Remark 4.5 If (), then Theorem 4.4 also holds.
5 Example and numerical results
In this section, we give a numerical example to support the main result.
Example 5.1 Let ℝ be the real line with Euclidean norm and let and be mappings defined by and for all . For every , define the mapping by and for all and .
Suppose that is the -mapping generated by , , and where and for all . Define the mapping by for all . Let the sequence be generated by (3.1), where , , and for all . Then converges strongly to 0 and is a solution of (1.8).
Solution. For every , it is easy to see that is a nonexpansive mapping and is -strictly pseudo-contractive mappings with . Then A is -inverse strongly accretive and B is -inverse strongly accretive. From the definition of G, we have and is a solution of (1.8). Then .
For every and , the mappings , , G, A, B and sequences , satisfy all conditions in Theorem 3.1. Since the sequence is generated by (3.1), from Theorem 3.1, we find that the sequence converges strongly to 0 and is a solution in (1.8).
Next, we will divide our iterations into two cases as follows:
-
(i)
, and ,
-
(ii)
, and .
Table 1 and Figure 1 show the values of sequence for both cases.
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. The modification of system of variational inequalities for fixed point theory in Banach spaces. Fixed Point Theory Appl 2014, 123 (2014). https://doi.org/10.1186/1687-1812-2014-123
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DOI: https://doi.org/10.1186/1687-1812-2014-123