Abstract
The purpose of this research is to modify the variational inclusion problems and prove a strong convergence theorem for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problems in Hilbert space. By using our main result, we prove a strong convergence theorem involving a κ-quasi-strictly pseudo-contractive mapping in Hilbert space. We give a numerical example to support some of our results.
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1 Introduction
Throughout this article, let H be a real Hilbert space with inner product and norm . Let C be a nonempty closed convex subset of H. Let be a nonlinear mapping. A point is called a fixed point of if . The set of fixed points of is denoted by . A mapping is called nonexpansive if
In 2008, Kohsaka and Takahashi [1] introduced the nonspreading mapping in Hilbert space H as follows:
It is shown in [2] that (1.1) is equivalent to
The mapping is called a κ-strictly pseudononspreading mapping if there exists such that
This mapping was introduced by Osilike and Isiogugu [3] in 2011. Clearly every nonspreading mapping is a κ-strictly pseudononspreading mapping.
Remark 1.1 Let C be a nonempty closed convex subset of H. Then a mapping is a κ-strictly pseudononspreading if and only if
for all .
Proof Let and be a κ-strictly pseudononspreading mapping, then there exists such that
Since
then we have
From (1.3) and (1.5), we have
It follows that
Then
On the other hand, let and
Then
It follows that
From (1.4), we have
From (1.6) and (1.7), we have
Then
□
Example 1.2 Let be defined by
Then is a κ-strictly pseudononspreading mapping where .
Example 1.3 Let be defined by
Then is a -strictly pseudononspreading mapping.
The mapping is called α-inverse strongly monotone if there exists a positive real number α such that
for all .
Let be a mapping and be a multi-valued mapping. The variational inclusion problem is to find such that
where θ is a zero vector in H. The set of the solution of (1.8) is denoted by . It is well known that the variational inclusion problems are widely studied in mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, and game theory, etc. Many authors have increasingly investigated such a problem (1.8); see for instance [4–7] and references therein.
Let be a multi-valued maximal monotone mapping, then the single-valued mapping defined by
is called the resolvent operator associated with M, where λ is any positive number and I is an identity mapping; see [7].
Let be a bifunction. The equilibrium problem for Ψ is to determine its equilibrium point. The set of solution of equilibrium problem is denoted by
Finding a solution of an equilibrium problem can be applied to many problems in physics, optimization, and economics. Many researchers have proposed some methods to solve the equilibrium problem; see, for example, [8, 9] and the references therein.
In 2008, Zhang et al. [7] introduced an iterative scheme for finding a common element of the set of solutions of the variational inclusion problem with multi-valued maximal monotone mapping and inverse strongly monotone mappings and the set of fixed points of nonexpansive mappings in Hilbert space as follows:
and they proved a strong convergence theorem of the sequence under suitable conditions of the parameters and λ.
In 2013, Kangtunyakarn [10] introduced an iterative algorithm for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and the set of solutions of a finite family of variational inequality problems as follows:
and proved a strong convergence theorem of the sequence under suitable conditions of the parameters , , , and .
Very recently, Suwannaut and Kangtunyakarn [11] have modified (1.9) as follows:
where is for bifunctions and with for every . It is obvious that (1.10) reduces to (1.9), if , for all . They also introduced an iterative method for finding a common element of the set of fixed points of an infinite family of -strictly pseudo-contractive mappings and the set of solutions of a finite family of an equilibrium problem and a variational inequalities problem as follows:
Under some appropriate conditions, they proved a strong convergence theorem of the sequence converging to an element of a set where is a strongly positive linear bounded operator for every .
For , let be a single-valued mapping and let be a multi-valued mapping. From the concept of (1.8), we introduce the problem of finding such that
for all with and θ is a zero vector. This problem is called the modified variational inclusion. The set of solutions of (1.11) is denoted by . If for all , then (1.11) reduces to (1.8).
In this paper, motivated by the research described above, we prove fixed point theory involving the modified variational inclusion and introduce iterative scheme for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problem. By using the same method as our main theorem, we prove a strong convergence theorem for finding a common element of the set of fixed points of a κ-quasi-strictly pseudo-contractive mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problem in Hilbert space. Applying such a problem, we have a convergence theorem associated with a nonspreading mapping. In the last section, we also give numerical examples to support some of our results.
2 Preliminaries
In this paper, we denote weak and strong convergence by the notations ‘⇀’ and ‘→’, respectively. In a real Hilbert space H, recall that the (nearest point) projection from H onto C assigns to each the unique point satisfying the property
For a proof of the main theorem, we will use the following lemmas.
Lemma 2.1 ([12])
Given and , then if and only if we have the inequality
Lemma 2.2 ([13])
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 2.3 Let H be a real Hilbert space. Then
for all .
Lemma 2.4 ([14])
Let H be a Hilbert space. Then for all and for such that the following equality holds:
For finding solutions of the equilibrium problem, assume a bifunction to satisfy the following conditions:
-
(A1)
for all ;
-
(A2)
Ψ is monotone, i.e., for all ;
-
(A3)
for each ,
-
(A4)
for each , is convex and lower semicontinuous.
Lemma 2.5 ([15])
Let C be a nonempty closed convex subset of H and let Ψ be a bifunction of into ℝ satisfying (A1)-(A4). Let and . Then there exists such that
Lemma 2.6 ([8])
Assume that satisfies (A1)-(A4). For , define a mapping as follows:
for all . Then the following hold:
-
(i)
is single-valued;
-
(ii)
is firmly nonexpansive, i.e., for any ,
-
(iii)
;
-
(iv)
is closed and convex.
Lemma 2.7 ([11])
Let C be a nonempty closed convex subset of a real Hilbert space H. For , let be bifunctions satisfying (A1)-(A4) with . Then
where for every and .
Remark 2.8 ([11])
From Lemma 2.7,
where , for each , and .
Lemma 2.9 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be bifunctions satisfying (A1)-(A4) where , for each and . For every , let with as . Then as for all .
Proof For every , let with as , from which it follows that . For every , by Lemma 2.6, we have
and
In particular, we have
and
Summing up (2.1) and (2.2) and using (A2), we have
It follows that
This implies that
It follows that
Then we have
where . Since as , we have
□
Remark 2.10 Let be a κ-strictly pseudononspreading mapping with . Define by , where . Then the following hold:
-
(i)
;
-
(ii)
for every and ,
Proof (i) It is easy to see that .
(ii) Next, we show that . For every and , we have
□
Lemma 2.11 ([7])
is a solution of variational inclusion (1.8) if and only if , , i.e.,
Further, if , then is a closed convex subset in H.
Lemma 2.12 ([7])
The resolvent operator associated with M is single-valued, nonexpansive for all and 1-inverse strongly monotone.
Lemma 2.13 Let H be a real Hilbert space and let be a multi-valued maximal monotone mapping. For every , let be -inverse strongly monotone mapping with and . Then
where , and for every . Moreover, is a nonexpansive mapping, for all .
Proof Clearly .
Let and let . From Lemma 2.11, we have
Since , we have . From Lemma 2.11, we have
From the nonexpansiveness of , we have
This implies that
Then
Since , we have
From , we have
From (2.5) and (2.6), we have
From (2.4) and (2.7), we have
Since and we have (2.4) and (2.8),
for all . It implies that .
Hence
Applying (2.3), we can conclude that is a nonexpansive mapping for all . □
3 Main result
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a multi-valued maximal monotone mapping. For every , let be a bifunction satisfying (A1)-(A4) and be -inverse strongly monotone mapping with . Let be a κ-strictly pseudononspreading mapping. Assume . Let the sequences and be generated by and
where and with , , and , for every , , , for all . Suppose the following conditions hold:
-
(i)
and ,
-
(ii)
,
-
(iii)
, where ,
-
(iv)
,
-
(v)
, , , , , .
Then the sequences and converge strongly to .
Proof The proof of Theorem 3.1 will be divided into five steps:
Step 1. We show that the sequence is bounded.
Since satisfies (A1)-(A4), and
by Lemma 2.6 and Remark 2.8, we have and .
Let . From Lemma 2.11 and Lemma 2.13, we have
From the nonexpansiveness of , we have
From Remark 2.10, we have
From the definition of , (3.2), and (3.3), we have
By mathematical induction, we have , . It implies that is bounded and so is .
By continuing in the same direction as in Step 1 of Theorem 3.1 in [10], we have
From (3.4), we can conclude that is bounded.
Step 2. Put and . We will show that . From the definition of , we have
By continuing in the same direction as in Step 2 of Theorem 3.1 in [11], we have
By substituting (3.6) into (3.5), we obtain
Applying Lemma 2.2, (3.7), and the conditions (i), (ii), (v), we have
Step 3. We show that . By the definition of , (3.2), and (3.3), we have
It implies that
From the condition (i) and (3.8), we have
By continuing in the same direction as (3.9), we have
From the definition of , we have
From the condition (i), (3.8), (3.9), and (3.10), we have
Step 4. We will show that , where .
To show this, choose a subsequence of such that
Without loss of generality, we can assume that as . From (3.9), we obtain as .
First, we will show that . Assume that . By Lemmas 2.11 and 2.13, . Then , where . By the nonexpansiveness of , (3.10), and Opial’s condition, we obtain
This is a contradiction. Then we have
Next, we will show that . Assume that . From Remark 2.10(i), we get . Then . From the condition (ii), (3.11), and Opial’s condition, we obtain
This is a contradiction. Then we have
Since , , then we have as with . Applying Lemma 2.9, we have as . Next, we will show that . Assume that . From Remark 2.8, we have . By Opial’s condition and (3.9), we have
This is a contradiction. Then we have
From (3.13), (3.14), and (3.15), we can conclude that .
Since as and . By (3.12) and Lemma 2.1, we have
Step 5. Finally, we will show that , where . From the definition of , we have
From the condition (i), (3.16), and Lemma 2.2, we can conclude that the sequence converges strongly to . By (3.9), we find that converges strongly to . This completes the proof. □
As a direct proof of Theorem 3.1, we obtain the following results.
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a multi-valued maximal monotone mapping. For every , let be a bifunction satisfying (A1)-(A4) and let be an α-inverse strongly monotone mapping. Let be a κ-strictly pseudononspreading mapping. Assume . Let the sequences and be generated by and
where and with , , and , for every , , , for all . Suppose the following conditions hold:
-
(i)
and ,
-
(ii)
,
-
(iii)
,
-
(iv)
,
-
(v)
, , , , , .
Then the sequences and converge strongly to .
Proof Put for all in Theorem 3.1. So, from Theorem 3.1, we obtain the desired result. □
Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a multi-valued maximal monotone mapping. Let be a bifunction satisfying (A1)-(A4). For every , be -inverse strongly monotone mapping with . Let be a κ-strictly pseudononspreading mapping. Assume . Let the sequences and be generated by and
where and with , , and , for every , , , for all . Suppose the following conditions hold:
-
(i)
and ,
-
(ii)
,
-
(iii)
, where ,
-
(iv)
,
-
(v)
, , , , , .
Then the sequences and converge strongly to .
Proof Take , . By Theorem 3.1, we obtain the desired conclusion. □
4 Applications
In this section, we utilize our main theorem to prove a strong convergence theorem for finding a common element of the set of fixed points of a κ-quasi-strictly pseudo-contractive mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problem in Hilbert space. To obtain this result, we recall some definitions, lemmas, and remarks as follows.
Definition 4.1 Let C be a subset of a real Hilbert space H and let be a mapping. Then is said to be κ-quasi-strictly pseudo-contractive if there exists a constant such that
is said to be quasi-nonexpansive if
The class of κ-quasi-strictly pseudo-contractions includes the class of quasi-nonexpansive mappings.
Remark 4.1 If be a κ-strictly pseudononspreading mapping with , then is a κ-quasi-strictly pseudo-contractive mapping.
Example 4.2 Let be defined by
Then is a κ-strictly pseudononspreading mapping where . Since , is also κ-quasi-strictly pseudo-contractive mapping.
Next, we give the example to show that the converse of Remark 4.1 is not true.
Example 4.3 Let be defined by
First, show that is a κ-quasi-strictly pseudo-contractive mapping for all .
Observe that . Let , we have
and
Then is a -quasi-strictly pseudo-contractive mapping. Next, we show that is not a -strictly pseudononspreading mapping.
Choose and , we have
and
Then we have
By changing from being a κ-strictly pseudononspreading mapping with into a κ-quasi-strictly pseudo-contractive mapping, we obtain the same result as in Remark 2.10.
Remark 4.4 Let be a κ-quasi-strictly pseudo-contractive mapping with . Define by , where . Then the following hold:
-
(i)
;
-
(ii)
for every and ,
In 2009, Kangtunyakarn and Suantai [16] introduced the S-mapping generated by and as follows.
Definition 4.2 ([16])
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 the S-mapping generated by and .
Lemma 4.5 ([17])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a finite family of nonspreading mappings of C into itself with and let where , , for all and , , for all . Let S be the S-mapping generated by and . Then and S is a quasi-nonexpansive mapping.
Remark 4.6 From Lemma 4.5 it still holds if .
Theorem 4.7 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a multi-valued maximal monotone mapping. For every , let be a bifunction satisfying (A1)-(A4) and be -inverse strongly monotone mapping with . Let be a κ-quasi-strictly pseudo-contractive mapping. Assume . Let the sequences and be generated by and
where , and with , and , for every , , , for all . Suppose the following conditions hold:
-
(i)
and ,
-
(ii)
,
-
(iii)
, where ,
-
(iv)
,
-
(v)
, , , , , .
Then the sequences and converge strongly to .
Proof Using Remark 4.4 and the same method of proof in Theorem 3.1, we have the desired conclusion. □
Theorem 4.8 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a multi-valued maximal monotone mapping. For every , let be a bifunction satisfying (A1)-(A4), and let be -inverse strongly monotone mapping with . Let , for 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 the sequences and be generated by and
where and with , and , for every , , , for all . Suppose the following conditions hold:
-
(i)
and ,
-
(ii)
,
-
(iii)
, where ,
-
(iv)
,
-
(v)
, , , , , .
Then the sequences and converge strongly to .
Proof From Theorem 4.7 and Remark 4.6, we obtain the desired conclusion. □
5 Numerical results
The purpose of this section we give a numerical example to support our some result. The following example is given for supporting Theorem 3.1.
Example 5.1 Let ℝ be the set of real numbers. For every , let , be defined by
for all and let be defined by
For every , suppose that , , , . Let and be generated by (3.1), where , , , , , , and for every . Then the sequences and converge strongly to 0.
Solution. It is easy to see that is a κ-strictly pseudononspreading mapping. Since , we obtain
It is easy to check that satisfies all the conditions of Theorem 3.1 and . Then we have
Put , then we have
Let . is a quadratic function of y with coefficient , , and . Determine the discriminant Δ of G as follows:
We know that , . If it has at most one solution in ℝ, then , so we obtain
where .
Since and ,
From (5.1) and the definition of , we have
For every , , , , , , , and . Then the sequences , , , , , , and satisfy all the conditions of Theorem 3.1. For every , from (5.2), we rewrite (3.1) as follows:
Using the algorithm (5.4) and choosing with and , we have the numerical results in Table 1.
Conclusion
-
1.
The sequences and converge to 0 as shown in Table 1 and Figure 1.
-
2.
From Theorem 3.1, we can conclude that the sequences and , in Example 5.1, converge to 0.
Next, we give the numerical example to support our some result in a three dimensional space of real numbers.
Example 5.2 Let an inner product be defined by and a usual norm defined by for all . For every , let , be defined by
for all , and let be defined by
For every , suppose that , , , . Let and be generated by (3.1), where , , , , , , and for every . Then the sequences and converge strongly to 0 where .
Solution. It is easy to see that is a -strictly pseudononspreading mapping. Since , we obtain
for all , . It is easy to check that satisfies the condition of Theorem 3.1 and . Then we have
Put , we have
where , and . Then , , and are a quadratic function of y with coefficients , , , , , , , and , respectively. Determine the discriminant of as follows:
From (5.6), if , and it has most one solution in ℝ, then , so we obtain
Next, determine the discriminant of as follows:
From (5.6), if , and it has at most one solution in ℝ, then , so we obtain
Next, determine the discriminant of as follows:
From (5.6), if , and it has at most one solution in ℝ, then , so we obtain
Since and , then
From (5.5) and the definition of , we have
For every , , , , , , , and . Then the sequences , , , , , , and satisfy all the conditions of Theorem 3.1. For every , from (5.7), (5.8), and (5.9), we rewrite (3.1) as follows:
where and .
Using the algorithm (5.11), choose , , , and . The numerical results for the sequences and are shown in Table 2 and Figure 2.
Conclusion
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The authors appreciated the referees for providing valuable comments improving the content of this research paper. This research was supported by the Research Administration Division of King Mongkut’s Institute of Technology Ladkrabang.
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Khuangsatung, W., Kangtunyakarn, A. Algorithm of a new variational inclusion problem and strictly pseudononspreading mapping with application. Fixed Point Theory Appl 2014, 209 (2014). https://doi.org/10.1186/1687-1812-2014-209
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DOI: https://doi.org/10.1186/1687-1812-2014-209