Abstract
In this paper, we introduce a new iterative algorithm for finding a common element of the set of fixed points of a finite family of -strictly pseudo-contractive mappings and the set of solutions of new variational inequalities problems in Hilbert space. By using our main results, we obtain an interesting theorem involving a finite family of κ-strictly pseudo-contractive mappings and two sets of solutions of the variational inequalities problem.
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1 Introduction
Let H be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let C be a nonempty closed convex subset of H. A mapping is called nonexpansive if
for all .
A mapping S is called a κ-strictly pseudo-contractive mapping if there exists such that
for all .
It is easy to see that every noexpansive mapping is a κ-strictly pseudo-contractive mapping.
Let . The variational inequality problem is to find a point such that
for all . The set of solutions of (1.1) is denoted by .
Variational inequalities were initially studied by Kinderlehrer and Stampacchia [1] and Lions and Stampacchia [2]. Such a problem has been studied by many researchers, and it is connected with a wide range of applications in industry, finance, economics, social sciences, ecology, regional, pure and applied sciences; see, e.g., [3–9].
A mapping A of C into H is called α-inverse-strongly monotone, see [10], if there exists a positive real number α such that
for all .
Let be two mappings. In 2008, Ceng et al. [11] introduced a problem for finding such that
which is called a system of variational inequalities where . By a modification of (1.2), we consider the problem for finding such that
which is called a modification of system of variational inequalities, for every and . If , (1.3) reduce to (1.2).
In 2008, Ceng et al. [11] introduce and studied a relaxed extragradient method for finding solutions of a general system of variational inequalities with inverse-strongly monotone mappings in a real Hilbert space as follows.
Theorem 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let the mappings be α-inverse-strongly monotone and β-inverse-strongly monotone, respectively. Let be a nonexpansive mapping such that , where Ω is the set of fixed points of the mapping , defined by , for all . Suppose that and is generated by
where , and , , are three sequences in such that
Then converges strongly to and is a solution of problem (1.2), where .
In the last decade, many author studied the problem for finding an element of the set of fixed points of a nonlinear mapping; see, for instance, [12–14].
From the motivation of [11] and the research in the same direction, 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 and the set of solutions of a modified general system of variational inequalities problems. Moreover, in the last section, we prove an interesting theorem involving the set of a finite family of -strictly pseudo-contractive mappings and two sets of solutions of variational inequalities problems by using our main results.
2 Preliminaries
In this section, we collect and give some useful lemmas that will be used for our main result in the next section.
Let C be a closed convex subset of a real Hilbert space H, let be the metric projection of H onto C, i.e., for , satisfies the property
It is well known that is a nonexpansive mapping and satisfies
Obviously, this immediately implies that
The following characterizes the projection .
Lemma 2.1 (See [15])
Given and . Then if and only if the following inequality holds:
Lemma 2.2 (See [16])
Let be a sequence of nonnegative real numbers satisfying
where , satisfy the conditions
Then .
Lemma 2.3 (See [17])
Let and be bounded sequences in a Banach space X and let be a sequence in with . Suppose that
for all integer and
Then .
Definition 2.1 (See [18])
Let C be a nonempty convex subset of a real Hilbert space. Let be a finite family of -strict pseudo-contractions of C into itself. For each , let , where and . Define the mapping as follows:
This mapping is called S-mapping generated by and .
Lemma 2.4 (See [18])
Let C be a nonempty closed convex subset of a real Hilbert space. Let be a finite family of κ-strict pseudo-contractive mappings of C into C with and and let , , where , , for all and , , for all . Let S be a mapping generated by and . Then and S is a nonexpansive mapping.
Lemma 2.5 (See [19])
Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and let be a nonexpansive mapping. Then is demi-closed at zero.
Lemma 2.6 In a real Hilbert space H, the following inequality holds:
for all .
Lemma 2.7 Let C be a nonempty closed convex subset of a Hilbert space H and let be mappings. For every and , the following statements are equivalent:
-
(a)
is a solution of problem (1.3),
-
(b)
is a fixed point of the mapping , i.e., , defined by
where .
Proof (a) ⇒ (b) Let be a solution of problem (1.3). For every and , we have
From the properties of , we have
It implies that
Hence, we have , where .
(b) ⇒ (a) Let and . Then, we have
From the properties of , we have
Hence, we have is a solution of (1.3). □
3 Main results
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H and let be -inverse strongly monotone mappings, respectively. Define the mapping by for all , and . Let be a finite family of κ-strict pseudo-contractive mappings of C into C with and and let , , where , , for all and , , for all . Let S be a mapping generated by and . Suppose that and let be the sequence generated by
where , and , , are sequences in . Assume that the following conditions hold:
Then converges strongly to and is a solution of (1.3), where .
Proof First, we show that and are nonexpansive mappings for every , . Let . Since is -inverse strongly monotone and , we have
Thus is a nonexpansive mapping. By using the same method as (3.2), we have is a nonexpansive mapping. Hence, , are nonexpansive mappings. It is easy to see that the mapping G is a nonexpansive mapping. Let . Then we have and
Put and , we can rewrite (3.1) by
and .
From the definition of , we have
By induction we can conclude that for all . It implies that is bounded and so are and .
Next, we show that .
Let
where .
Since and (3.3), we have
It follows that
From conditions (ii) and (iii), we have
From Lemma 2.3 and (3.3) we have . Since , then we have
From the definition of , we have
From (3.4), we obtain
From the definition of , we have
From (3.4), conditions (ii) and (iii), we have
From the definition of , we have
From (3.4) and (3.7), we derive
From the nonexpansiveness of and , we have
It implies that
From (3.4), (3.9) conditions (ii) and (iii), we have
Put and . From the definition of , we have
which implies that
From (3.4), (3.11), conditions (ii) and (iii), we can conclude
Next, we show that
From the definition of , we have
It implies that
From the nonexpansiveness of and (3.14), we have
It follows that
From condition (ii), (3.4) and (3.10), we have
From the definition of , , , , we have
and
From the properties of , we have
From (3.6), (3.12) and (3.15), we have
Since
and (3.15), (3.16), then we have
From (3.6) and (3.17), we can conclude that
Next we show that
where . To show this inequality, take a subsequence of such that
Without loss of generality, we may assume that as , where . From (3.17), we have as . From Lemma 2.5 and (3.13), we have
From Lemma 2.4, we have . Then we obtain
From the nonexpansiveness of the mapping G and the definition of , we have
From (3.17), we have
From as , (3.19) and Lemma 2.5, we have
Hence, we can conclude that .
Since as and , we have
From the definition of and , we have
From condition (ii), (3.18) and Lemma 2.2, we can conclude that the sequence converges strongly to . This completes the proof. □
Remark 3.2 (1) If we take , then the iterative scheme (3.1) reduces to the following scheme:
which is an improvement to (1.4). From Theorem 3.1, we obtain that the sequence generated by (3.21) converges strongly to , where the mapping defined by for all and is a solution of (1.2) where .
(2) If we take , and , then the iterative scheme (3.1) reduces to the following scheme:
From Theorem 3.1, we obtain that the sequence generated by (3.22) converges strongly to , where the mapping defined by for all and is a solution of (1.3) where .
4 Applications
In this section we prove a strong convergence theorem involving variational inequalities problems by using our main result. We need the following lemmas to prove the desired results.
Lemma 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be nonexpansive mappings. Define a mapping by for every and . Then and is a nonexpansive mapping.
Proof It is easy to see that . Let and . By the definition of , we have
From (4.1), it implies that
Then we have , that is, . By the definition of , we have
It follows that . Then we have . Hence .
Next, we show that is a nonexpansive mapping. Let , since
Then we have is a nonexpansive mapping. □
Lemma 4.2 (See [15])
Let H be a real Hibert space, let C be a nonempty closed convex subset of H and let A be a mapping of C into H. Let . Then for ,
where is the metric projection of H onto C.
Lemma 4.3 Let C be a nonempty closed convex subset of a real Hilbert space H and let be -inverse strongly monotone mappings, respectively, which . Define a mapping as in Lemma 2.7 for every , and . Then .
Proof First, we show that , are nonexpansive. Let . Since is -inverse strongly monotone and , we have
Thus is nonexpansive. By using the same method as (4.3), we have is a nonexpansive mapping. Hence , are nonexpansive mappings. From
for every and Lemma 4.1, we have
From Lemma 4.2, we have
□
Theorem 4.4 Let C be a nonempty closed convex subset of a real Hilbert space H and let be -inverse strongly monotone mappings, respectively. Define the mapping by for all , and . Let be a finite family of κ-strict pseudo-contractive mappings of C into C with and and let , , where , , for all and , , for all . Let S be a mapping generated by and . Suppose that and let be a sequence generated by
where , and , , are sequences in . Assume that the following conditions hold:
Then converges strongly to .
Proof From Lemma 4.3 and 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. An iterative algorithm to approximate a common element of the set of common fixed points for a finite family of strict pseudo-contractions and of the set of solutions for a modified system of variational inequalities. Fixed Point Theory Appl 2013, 143 (2013). https://doi.org/10.1186/1687-1812-2013-143
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DOI: https://doi.org/10.1186/1687-1812-2013-143