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The methods for variational inequality problems and fixed point of κ-strictly pseudononspreading mapping
Fixed Point Theory and Applications volume 2013, Article number: 171 (2013)
Abstract
In this paper, we introduce the methods for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and a finite family of the set of solutions of variational inequality problems. The strong convergence theorem of the proposed method is established under some suitable control conditions. Moreover, by using our main result, we prove interesting theorem involving an iterative scheme for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and a finite family of the set of fixed points of a -strictly pseudocontractive mappings.
1 Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that the mapping is said to be nonexpansive if for all . In 2008, Kohsaka and Takahashi [1] introduced the nonspreading mapping in Hilbert spaces H which is defined as follows: , . Following the terminology of Browder and Petryshyn [2], in 2011, Osilike and Isiogugu [3] introduced that the mapping is called a κ-strictly pseudononspreading mapping if there exists such that
for all . Clearly every nonspreading mapping is κ-strictly pseudononspreading; see, for example, [3]. A point is called a fixed point of T if . The set of fixed points of T is denoted by .
Let . The variational inequality problem is to find a point such that
for all . The set of solutions of (1.1) is denoted by .
The variational inequality has emerged as a fascinating and interesting branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, social, ecology, regional, pure and applied sciences; see, e.g., [4–7].
A mapping A of C into H is called α-inverse strongly monotone (see [8]) if there exists a positive real number α such that
for all .
In 2003, Takahashi and Toyoda [9] proved a convergence theorem for finding a common element of the set of fixed points of nonexpansive mappings and the set of solutions of variational inequalities for α-inverse strongly monotone mappings as follows.
Theorem 1.1 Let K be a closed convex subset of a real Hilbert space H. Let . Let A be an α-inverse strongly monotone mapping of K into H, and let S be a nonexpansive mapping of K into itself such that . Let be a sequence generated by and
for every , where for some and for some . Then converges weakly to , where .
Recently, Osilike and Isiogugu [3] proved strong convergence theorems for strictly pseudononspreading mappings as follows.
Theorem 1.2 Let C be a nonempty closed convex subset of a real Hilbert space and let be a κ-strictly pseudononspreading mapping with a nonempty fixed point set . Let and let be a real sequence in such that and . Let and let and be sequences in C generated from an arbitrary by
where . Then and converge strongly to , where is the metric projection of H onto .
Theorem 1.3 Let C be a nonempty closed convex subset of a real Hilbert space and let be a κ-strictly pseudononspreading mapping with a nonempty fixed point set . Let and let . Let be a real sequence in satisfying the conditions
-
(C1)
and
-
(C2)
.
Let be a fixed anchor in C and let be a sequence in C generated from an arbitrary by
Then converges strongly to a fixed point p of T.
Inspired and motivated by [3] and the research in the same direction, we prove a strong convergence theorem of κ-strictly pseudononspreading mappings and introduce the methods for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and a finite family of the set of solutions of variational inequality problems. Moreover, by using our main result, we prove an interesting theorem involving an iterative scheme for finding a common element of the set of fixed points of κ-strictly pseudononspreading mappings and a finite family of the set of fixed points of -strictly pseudocontractive mappings.
2 Preliminaries
We need the following lemmas to prove our main result. Let H be a real Hilbert space and let C be a nonempty closed convex subset of H, let be the metric projection of H onto C, i.e., for , satisfies the property
The following characterizes the projection .
Lemma 2.1 (See [10])
Given and . Then if and only if the following inequality holds:
Lemma 2.2 (See [10])
Let H be a Hilbert 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 2.3 (See [11])
Let be a sequence of nonnegative real numbers satisfying
where is a sequence in and is a sequence such that
-
(1)
,
-
(2)
or .
Then .
Lemma 2.4 (See [11])
Let be a sequence of nonnegative real numbers satisfying
where , satisfy the conditions
-
(1)
, ,
-
(2)
or .
Then .
Lemma 2.5 (See [12])
Let E be a uniformly convex Banach space, let C be a nonempty closed convex subset of E and let be a nonexpansive mapping. Then is demi-closed at zero.
In 2009, Kangtunykarn and Suantai [13] defined an S-mapping and proved their lemmas as follows.
Definition 2.1 Let C be a nonempty convex subset of a real Banach space. Let be a finite family of nonexpanxive mappings of C into itself. For each , let , where and . Define the mapping as follows:
This mapping is called an S-mapping generated by and .
Lemma 2.6 Let C be a nonempty closed convex subset of strictly convex Banach space. Let be a finite family of nonexpanxive mappings of C into itself with and let , , where , , for all , , for all . Let S be a mapping generated by and . Then .
Remark 2.7 It is easy to see that the mapping S is a nonexpansive mapping.
Lemma 2.8 Let C be a nonempty closed convex subset of H. Let be a κ-strictly pseudononspreading mapping with . Then .
Proof It is easy to see that . Put . Let and . Since , we have
Since T is a κ-strictly pseudononspreading mapping, we have
which implies that
Then we have . Therefore . Hence . □
Remark 2.9 From Lemmas 2.2 and 2.8, we have , .
Example 2.1 Let be defined by
To see that T is κ-strictly pseudononspreading, if for all , then we have and . From the definition of T, we have
and
From the above, then there exists such that
For every , we have , . From the definition of T, we have
and
From the above, then there exists such that
Finally, for every and , we have and . From the definition of T, we have
and
From the above, then there exists such that
Then, for all , we have
for some . Hence T is a κ-strictly pseudononspreading mapping. Observe that . From Lemma 2.8, we have .
3 Main results
Theorem 3.1 Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. For every , let be -inverse strongly monotone mappings and let be a κ-strictly pseudononspreading mapping for some . Let be defined by for every and for every , and let , , where , , for all , , for all . Let be the S-mapping generated by and . Assume that . For every , , let and be a sequence generated by
where such that , , and suppose the following conditions hold:
-
(i)
and ,
-
(ii)
,
-
(iii)
, , , .
Then the sequence converges strongly to .
Proof Let . First, we show that , where . From Remark 2.9, we have . From the nonexpansiveness of , we have
Since T is a κ-strictly pseudononspreading mapping and , we have
which implies that
From (3.3), we have
From (3.4) and (3.2), we can imply that
Next, we will show that the mapping is a nonexpansive mapping for every . Let . Since is -inverse strongly monotone and , for every , we have
Thus is a nonexpansive mapping for every . The proof of the above result can be also found in Imnang and Suantai [14]. From the definition of , we have are nonexpansive mappings for all . Since , by Lemma 2.2, we have
From Lemma 2.6, we have . Next, we will show that is bounded. From the definition of and (3.5), we have
Put . From (3.8) we can show by induction that , . This implies that is bounded and so are , . Next, we will show that
Since T is κ-strictly pseudononspreading, we have
which implies that
Putting and in (3.10), we have
which implies that
From (3.11) we have (3.9). Since , and (3.9), we have is bounded.
Next, we will show that
From the definition of , we have
where . From Lemma 2.3 and conditions (i)-(iii), we have (3.12). Next, we will show that
From the definition of and (3.5), we have
which implies that
From (3.15) and (3.12), we have
Since
from condition (i) and (3.16), we have
Since
from (3.12) and (3.17), we have (3.14). Since
from condition (i) (3.12) and (3.14), we have
Next, we will show that
where . To show this equality, take a subsequence of such that
Without loss of generality, we may assume that as where . We shall show that . From Remark 2.9, we have . Assume that . Since as , by Opial’s property, (3.14) and condition (ii), we have
This is a contradiction. Then . From (3.18), we have
From the nonexpansiveness of S, as and Lemma 2.5, we can imply that
Since for every and , by Lemma 2.2, we have
By Lemma 2.6, we have
From (3.21) and (3.22), we have . Hence . Since as and , we have
Finally, we show that converges strongly to . From the definition of and (3.5), we have
From (3.19) and Lemma 2.4, we have converges strongly to . This completes the proof. □
The following result can be obtained from Theorem 3.1. We, therefore, omit the proof.
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a δ-inverse strongly monotone mapping and let be a κ-strictly pseudononspreading mapping for some . Assume that . For every , let and be a sequence generated by
where such that , , , and , and suppose that the following conditions hold:
-
(i)
and ,
-
(ii)
,
-
(iii)
, , , .
Then the sequence converges strongly to .
4 Application
In this section, by using our main result, we prove strong a convergence theorem involving a strictly pseudononspreading mapping and a finite family of strictly pseudocontractive mappings. Before proving the next theorem, we need the following definition.
Definition 4.1 The mapping is said to be strictly pseudocontractive [2] with the coefficient if
Remark 4.1 If C is a nonempty closed convex subset of H and is a κ-strictly pseudocontractive mapping with , then . To show this, put . Let and . Since , , . Since is a κ-strictly pseudocontractive mapping, we have
It implies that
Then we have , therefore . Hence . It is easy to see that .
Remark 4.2 is a -inverse strongly monotone mapping. To show this, let , we have
Then we have
Theorem 4.3 Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let be -strictly pseudocontractive mappings for every , and let be a κ-strictly pseudononspreading mapping for some . Let be defined by for every and for every , and let , , where , , for all , , for all . Let be the S-mapping generated by and . Assume that . For every , , let and be a sequence generated by
where such that , , and suppose that the following conditions hold:
-
(i)
and ,
-
(ii)
,
-
(iii)
, , , .
Then the sequence converges strongly to .
Proof From Remark 4.2, we have is -inverse strongly monotone for every . From Remark 4.1 and Lemma 2.2, we have for every . Put and for every in Theorem 3.1. The conclusion of Theorem 4.3 can be obtained from Theorem 3.1 □
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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 methods for variational inequality problems and fixed point of κ-strictly pseudononspreading mapping. Fixed Point Theory Appl 2013, 171 (2013). https://doi.org/10.1186/1687-1812-2013-171
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DOI: https://doi.org/10.1186/1687-1812-2013-171