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Strong convergence of the hybrid method for a finite family of nonspreading mappings and variational inequality problems
Fixed Point Theory and Applications volume 2012, Article number: 188 (2012)
Abstract
In this paper, we prove a strong convergence theorem by the hybrid method for finding a common element of the set of fixed points of a finite family of nonspreading mappings and the set of solutions of a finite family of variational inequality problems.
1 Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H. Then a mapping is said to be nonexpansive if for all . Recall that the mapping is said to be quasi-nonexpansive if , and , where denotes the set of fixed points of T. In 2008, Kohsaka and Takahashi [1] introduced the mapping T called the nonspreading mapping in Hilbert spaces H and defined it as follows: , .
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., [2–5].
A mapping A of C into H is called inverse-strongly monotone (see [6]) if there exists a positive real number α such that
for all . Throughout this paper, we will use the following notation:
-
1.
⇀ for weak convergence and → for strong convergence.
-
2.
denotes the weak ω-limit set of .
In 2008, Takahashi, Takeuchi and Kubota [7] proved the following strong convergence theorems by using the hybrid method for nonexpansive mappings in Hilbert spaces.
Theorem 1.1 Let H be a Hilbert space and C be a nonempty closed convex subset of H. Let T be a nonexpansive mapping of C into H such that and let . For and , define a sequence of C as follows:
where for all . Then converges strongly to .
In 2009, Iemoto and Takahashi [8] proved the convergence theorem of nonexpansive and nonspreading mappings as follows.
Theorem 1.2 Let H be a Hilbert space, and let C be a nonempty closed convex subset of H. Let S be a nonspreading mapping of C into itself, and let T be a nonexpansive mapping of C into itself such that . Define a sequence as follows.
for all , where . Then the following hold:
-
(i)
If and , then converges weakly to .
-
(ii)
If and , then converges weakly to .
-
(iii)
If and , then converges weakly to .
Inspired and motivated by these facts and the research in this direction, we prove the strong convergence theorem by the hybrid method for finding a common element of the set of fixed points of a finite family of nonspreading mappings and the set of solutions of a finite family of variational inequality problems.
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
The following characterizes the projection .
Lemma 2.1 (See [9])
Given and . Then if and only if the following inequality holds:
Lemma 2.2 (See [8])
Let C be a nonempty closed convex subset of H. Then a mapping is nonspreading if and only if
for all .
Example 2.3 Let ℛ denote the reals with the usual norm. Let be defined by
for all .
To see that T is a nonspreading mapping, if , then we have and . From the definition of the mapping T, we have
and
The above implies that
For every , we have and . From the definition of T, we have
and
From above, we have
Finally, for every and , we have and . From the definition of T, we have
and
From above, we have
Hence, for all , we have
Then T is a nonspreading mapping.
Lemma 2.4 (See [1])
Let H be a Hilbert space, let C be a nonempty closed convex subset of H, and let S be a nonspreading mapping of C into itself. Then is closed and convex.
Lemma 2.5 (See [9])
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 2.6 (See [10])
Let C be a closed convex subset of a strictly convex Banach space E. 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, nonexpansive and holds.
Lemma 2.7 (See [11])
Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E, and be a nonexpansive mapping. Then is demi-closed at zero.
Lemma 2.8 (See [12])
Let C be a closed convex subset of H. Let be a sequence in H and . Let . If is such that and satisfies the condition
then , as .
In 2009, Kangtunyakarn and Suantai [13] introduced an S-mapping generated by and as follows.
Definition 2.1 Let C be a nonempty convex subset of a real Banach space. Let be a finite family of (nonexpansive) 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 .
The next lemma is very useful for our consideration.
Lemma 2.9 Let C be a nonempty closed convex subset of a real Hilbert space. Let be a finite family of nonspreading mappings of C into C with , and let , , where , , for all and , , for all . Let S be the mapping generated by and . Then and S is a quasi-nonexpansive mapping.
Proof It easy to see that . Let and . Since is a finite family of nonspreading mappings of C into itself, for every , we have
This implies that
From the definition of S and (2.4),
From (2.5), we have
which implies that
Since for all and (2.6), we have . From and the definition of S, we have
From (2.5) and , we have
which implies that
Since for all and (2.7), we have . From the definition of S and , we have
By continuing in this way, we can show that and for all .
Finally, we shall show that .
Since
and , we obtain so that . Then we have . Hence, .
Next, we show that S is a quasi-nonexpansive mapping. Let and . From (2.5), we can imply that
Then we have the S-mapping is quasi-nonexpansive. □
Example 2.10 Let be a mapping defined by
for all .
Let be a mapping defined by
for all .
To see that is a nonspreading mapping, observe that if , we have and . Then we have
and
From above, we have
For every , we have and . From the definition of , we have
and
From above, we have
Finally, for every and , we have and . From the definition of , we have
and
From above, we have
Then for all , we have
Hence, we have is a nonspreading mapping.
Next, we show that is a nonspreading mapping. Let , then we have and . From the definition of , we have
and
From above, we have
For every , we have and . From the definition of , we have
and
From above, we have
Finally, for every and , we have and . From the definition of , we have
and
From above, we have
Then for every , we have
Hence, we have is a nonspreading mapping. Observe that . Let the mapping be the S-mapping generated by , and , , where and . From Lemma 2.9, we have .
3 Main result
Theorem 3.1 Let C be a nonempty closed convex subset of a Hilbert space H. For every , let be an -inverse strongly monotone mapping, and let be a finite family of nonspreading mappings with . For every , define the mapping by and . Let , , where , , for all and , for all , and let S be the S-mapping generated by and . Let be a sequence generated by and
where , and suppose the following conditions hold:
Then the sequence converges strongly to .
Proof First, we show that is a nonexpansive mapping for every . Let . Since A is an -inverse strongly monotone and , we have
Thus is a nonexpansive mapping for every . Since is a nonexpansive mapping, we have is a nonexpansive mapping for every . From Lemma 2.5, we have
From (3.2), is closed and convex. Let . From (3.2), we have for every . By nonexpansiveness of , we have
Next, we show that is closed and convex for every . It is obvious that is closed. In fact, we know that for ,
So, for every and , it follows that
then, we have is convex. Since is closed and convex for every , we have is closed and convex. From Lemma 2.4, we have is closed and convex. Hence, we have is closed and convex. This implies that is well defined. Next, we show that for every . Let , then we have
It follows that . Hence, we have for every . This implies that is well defined. Since , for every , we have
In particular, we have
By (3.4) we have is bounded, so are , for every , , and . Since and , we have
which implies that
Hence, we have exists. Since
it implies that
Since , we have
By (3.7) we have
Since
by (3.7) and (3.8), we have
Next, we will show that
For every , we have
From the definition of and (3.11), we have
It follows that
From conditions (i), (ii) and (3.9), it implies that
Since
it implies that
From the definition of and (3.13), we have
which implies that
From conditions (i), (ii), (3.9) and (3.12), we have
Since
from (3.14), we have
Since
from (3.9) and (3.15), we have
Next, we will show that
From the definition of , we have
From (3.17) and condition (ii), we have
Form (3.9), we have
By using the same method as (3.18), we can conclude that
Let be the set of all weakly ω-limit of . We shall show that . Since is bounded, then . Let , there exists a subsequence of which converges weakly to q.
Put defined by
Since is a nonexpansive mapping, for every , from Lemma 2.6 and 2.5, we have
Since
from the condition (i) and (3.15), we have
From (3.21), we have
From (3.19), it is easy to see that Q is a nonexpansive mapping. By Lemma 2.7 and as , we have From (3.2), we have
Next, we will show that . Assume that . From the Opial property, (3.10) and (3.16), we have
This is a contradiction. Then, we have . From Lemma 2.9, we have
From (3.22) and (3.23), we have . Hence, . Therefore, by (3.5) and Lemma 2.8, 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 Hilbert space H. For every , let be an -inverse strongly monotone mapping, and let be a nonspreading mapping with . For every , define the mapping by and . Let be a sequence generated by and
where , and suppose the following conditions hold:
Then the sequence converges strongly to .
Corollary 3.3 Let C be a nonempty closed convex subset of a Hilbert space H. Let be an α-inverse strongly monotone mapping, and let be a finite family of nonspreading mappings with . Let , , where , , for all and , , for all , and let S be the S-mapping generated by and . Let be a sequence generated by and
where , and . Then the sequence converges strongly to .
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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. Strong convergence of the hybrid method for a finite family of nonspreading mappings and variational inequality problems. Fixed Point Theory Appl 2012, 188 (2012). https://doi.org/10.1186/1687-1812-2012-188
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DOI: https://doi.org/10.1186/1687-1812-2012-188