A hybrid iterative method for common solutions of variational inequality problems and fixed point problems for single-valued and multi-valued mappings with applications
- Nawitcha Onjai-uea^{1} and
- Withun Phuengrattana^{1, 2}Email author
https://doi.org/10.1186/s13663-015-0265-x
© Onjai-uea and Phuengrattana; licensee Springer. 2015
Received: 27 October 2014
Accepted: 13 January 2015
Published: 1 February 2015
Abstract
In this article, we propose a new iterative method for approximating a common element of the set of common fixed points of a finite family of k-strictly pseudononspreading single-valued mappings, the set of common fixed points of a finite family of quasi-nonexpansive multi-valued mappings, and the set of common solutions of a finite family of variational inequality problems in Hilbert spaces. Furthermore, we prove that the proposed iterative method converges strongly to a common element of the above three sets, and we also apply our results to complementarity problems. Finally, we give two numerical examples to support our main result.
Keywords
fixed point strictly pseudononspreading mappings multi-valued mappings variational inequality problems Hilbert spacesMSC
46C05 47H09 47H101 Introduction
- (i)nonexpansive if$$H(Tx,Ty)\leq\|x-y\| \quad\mbox{for all }x,y\in D; $$
- (ii)quasi-nonexpansive if \(F(T)\ne\emptyset\) and$$H(Tx,Tp)\leq\|x-p\| \quad\mbox{for all }x\in D \mbox{ and }p\in F(T); $$
- (iii)L-Lipschitzian if there exists \(L > 0\) such that$$H(Tx,Ty)\leq L\|x-y\| \quad\mbox{for all }x,y\in D. $$
It is clear that every nonexpansive multi-valued mapping T with \(F(T)\ne\emptyset\) is quasi-nonexpansive. It is known that if T is a quasi-nonexpansive multi-valued mapping, then \(F(T)\) is closed. In general, the fixed point set of a quasi-nonexpansive multi-valued mapping T is not necessary to be convex. In the next lemma, we show that \(F(T)\) is convex under the assumption that \(Tp=\{p\}\) for all \(p\in F(T)\). The proof of this fact is very easy, therefore we omit it.
Lemma 1.1
Let D be a nonempty closed convex subset of a real Hilbert space X. Assume that \(T:D\to CB(D)\) is a quasi-nonexpansive multi-valued mapping. If \(Tp=\{p\}\) for all \(p\in F(T)\), then \(F(T)\) is convex.
The fixed point theory of multi-valued mappings is much more complicated and harder than the corresponding theory of single-valued mappings. However, some classical fixed point theorems for single-valued mappings have already been extended to multi-valued mappings; see [1, 2]. The recent fixed point results for multi-valued mappings can be found in [3–11] and the references cited therein.
Lemma 1.2
([15])
Let D be a nonempty closed convex subset of a real Hilbert space X, and let \(t : D\to D\) be a k-strictly pseudononspreading mapping. If \(F(t)\ne\emptyset\), then it is closed and convex.
Many researchers studied the existence and convergence theorems of those single-valued mappings in both Hilbert spaces and Banach spaces (e.g., see [16–23]).
Lemma 1.3
Let D be a nonempty closed convex subset of a strictly convex Banach space X. Let \(\{t_{i}\}_{i=1}^{N}\) be a finite family of nonexpansive mappings of D into itself with \(\bigcap_{i=1}^{N} F(t_{i})\ne\emptyset\), and let \(\delta_{j}=(\delta_{1}^{j}, \delta_{2}^{j}, \delta_{3}^{j})\in[0,1]\times[0,1]\times[0,1]\), \(j=1,2,\ldots,N\), where \(\delta_{1}^{j}+ \delta_{2}^{j}+ \delta_{3}^{j}=1\), \(\delta_{1}^{j}\in(0,1)\) for all \(j=1,2,\ldots,N-1\), \(\delta_{1}^{N}\in(0,1]\), and \(\delta_{2}^{j}, \delta_{3}^{j}\in[0,1)\) for all \(j=1,2,\ldots,N\). Let S be the S-mapping generated by \(t_{1}, t_{2},\ldots, t_{N}\) and \(\delta_{1},\delta_{2},\ldots,\delta_{N}\). Then S is a nonexpansive mapping and \(F(S) = \bigcap_{i=1}^{N} F(t_{i})\).
Applications of W-mappings and S-mappings for fixed point problems can be found in [26–31].
Variational inequality theory, which was first introduced by Stampacchia [33] in 1964, emerged as an interesting and fascinating branch of applicable mathematics with a wide range of applications in economics, industry, network analysis, optimizations, pure and applied sciences etc. In recent years, much attention has been given to developing efficient iterative methods for treating solution problems of variational inequalities (e.g., see [34–39]).
In 2003, Takahashi and Toyoda [40] introduced an iterative method for finding a common element of the set of fixed points of nonexpansive single-valued mappings and the set of solutions of variational inequalities for ϕ-inverse strongly monotone mappings in Hilbert spaces. Recently, by using the concept of S-mapping, Kangtunyakarn [41] introduced a new method for finding a common element of the set of fixed points of k-strictly pseudononspreading single-valued mappings and the set of solutions of variational inequality problems in Hilbert spaces.
Question A
How can we construct an iteration process for finding a common element of the set of common fixed points of a finite family of k-strictly pseudononspreading single-valued mappings, the set of common fixed points of a finite family of quasi-nonexpansive multi-valued mappings, and the set of common solutions of a finite family of variational inequality problems?
In the recent years, the problem of finding a common element of the set of fixed points of single-valued mappings and multi-valued mappings in the framework of Hilbert spaces and Banach spaces has been intensively studied by many researchers. However, no researchers have studied the problem of finding a common element of three sets, i.e., the set of common fixed points of a finite family of single-valued mappings, the set of common fixed points of a finite family of multi-valued mappings, and the set of common solutions of a finite family of variational inequality problems.
In this article, motivated by [41] and the research described above, we propose a new hybrid iterative method for finding a common element of the set of a common fixed point of a finite family of k-strictly pseudononspreading single-valued mappings, the set of common fixed points of a finite family of quasi-nonexpansive multi-valued mappings, and the set of common solutions of a finite family of variational inequality problems in Hilbert spaces and provide an affirmative answer to Question A.
2 Preliminaries
In this section, we give some useful lemmas for proving our main results. Let D be a nonempty closed convex subset of a real Hilbert space X. Let \(P_{D}\) be the metric projection of X onto D, i.e., for \(x\in X\), \(P_{D}x\) satisfies the property \(\|x-P_{D}x\|=\min_{y\in D}\|x-y\|\). It is well known that \(P_{D}\) is a nonexpansive mapping of X onto D.
Lemma 2.1
([42])
Lemma 2.2
([43])
Lemma 2.3
([44])
Lemma 2.4
([43])
Lemma 2.5
([45])
Lemma 2.6
([46])
Lemma 2.7
([42])
The following lemma obtained by Kangtunyakarn [41] is useful for our results.
Lemma 2.8
Let D be a nonempty closed convex subset of a Hilbert space X. Let \(t:D\to D\) be a k-strictly pseudononspreading mapping with \(F(t)\ne\emptyset\). Then \(F(t)=VI(D,I-t)\).
3 Main results
In this section, we prove a strong convergence theorem which solves the problem of finding a common element of the set of common fixed points of a finite family of k-strictly pseudononspreading single-valued mappings, the set of common fixed points of a finite family of quasi-nonexpansive multi-valued mappings, and the set of common solutions of a finite family of variational inequality problems in Hilbert spaces. Before starting the main theorem of this section, we need to prove the following useful lemma in Hilbert spaces.
Lemma 3.1
- (i)
W is quasi-nonexpansive;
- (ii)
\(F(W)=\bigcap_{i=1}^{N}F(t_{i})=\bigcap_{i=1}^{N}F(R_{i})\).
Proof
We now prove our main theorem.
Theorem 3.2
- (i)
\(\alpha_{n}^{(1)}+\alpha_{n}^{(2)}+\alpha_{n}^{(3)}=1\), \(\lim_{n\to \infty}\alpha_{n}^{(1)}=0\), and \(0< a\leq\alpha_{n}^{(2)},\alpha_{n}^{(3)}<1\);
- (ii)
\(0< b\leq\gamma_{n}^{(i)}<1\) for all \(i=0,1,\ldots,N\) and \(\sum_{i=0}^{N} \gamma_{n}^{(i)}=1\).
Proof
We shall divide our proof into 6 steps.
Step 1. We show that \(P_{C_{n+1}}x_{1}\) is well defined for every \(x_{1}\in X\).
Step 2. We show that \(\lim_{n\rightarrow\infty} x_{n}=q\) for some \(q\in D\).
Step 3. We show that \(q\in\bigcap_{i=1}^{N}F(T_{i})\).
Step 4. We show that \(q\in\bigcap^{N}_{i=1} VI(D,B_{i})\).
Step 5. We show that \(q\in\bigcap^{N}_{i=1}F(t_{i})\).
Step 6. Finally, we show that \(q=u=P_{\mathcal{F}}x_{1}\).
By Step 1 to Step 6, we conclude that \(\{x_{n}\}\), \(\{y_{n}\}\), and \(\{z_{n}\}\) converge strongly to \(u=P_{\mathcal{F}}x_{1}\). This completes the proof. □
As a direct consequence of Theorem 3.2, we have the following two corollaries.
Corollary 3.3
- (i)
\(\alpha_{n}^{(1)}+\alpha_{n}^{(2)}+\alpha_{n}^{(3)}=1\), \(\lim_{n\to \infty}\alpha_{n}^{(1)}=0\), and \(0< a\leq\alpha_{n}^{(2)},\alpha_{n}^{(3)}<1\);
- (ii)
\(0< b\leq\gamma_{n}^{(i)}<1\) for all \(i=0,1,\ldots,N\) and \(\sum_{i=0}^{N} \gamma_{n}^{(i)}=1\).
Proof
Let \(B_{i}x = 0\) for all \(x\in D\) and \(i=1,2,\ldots,N\) in Theorem 3.2. Then we obtain that \(Sx_{n}=x_{n}\) for all \(n\in \mathbb{N}\). Therefore the conclusion follows. □
Corollary 3.4
- (i)
\(\alpha_{n}^{(1)}+\alpha_{n}^{(2)}+\alpha_{n}^{(3)}=1\), \(\lim_{n\to \infty}\alpha_{n}^{(1)}=0\), and \(0< a\leq\alpha_{n}^{(2)},\alpha_{n}^{(3)}<1\);
- (ii)
\(0< b\leq\gamma_{n}^{(0)}, \gamma_{n}^{(1)}<1\) and \(\gamma _{n}^{(0)} + \gamma_{n}^{(1)}=1\).
4 Applications to complementarity problems
A complementarity problem is a special case of a variational inequality problem. The following lemma indicates the equivalence between the complementarity problem and the variational inequality problem. The proof of this fact can be found in [42]; for convenience of the readers, we include the details.
Lemma 4.1
Let D be a nonempty closed convex cone in a real Hilbert space X, and let \(D^{*}\) be the polar of D. Let B be a mapping of D into X. Then \(VI(D,B)=CP(D,B)\).
Proof
Theorem 4.2
- (i)
\(\alpha_{n}^{(1)}+\alpha_{n}^{(2)}+\alpha_{n}^{(3)}=1\), \(\lim_{n\to \infty}\alpha_{n}^{(1)}=0\), and \(0< a\leq\alpha_{n}^{(2)},\alpha_{n}^{(3)}<1\);
- (ii)
\(0< b\leq\gamma_{n}^{(i)}<1\) for all \(i=0,1,\ldots,N\) and \(\sum_{i=0}^{N} \gamma_{n}^{(i)}=1\).
Proof
5 Numerical results
In this section, we give two numerical examples to support our main result.
Example 5.1
The values of the sequences \(\pmb{\{x_{n}\}}\) , \(\pmb{\{y_{n}\}}\) , and \(\pmb{\{z_{n}\}}\) in Example 5.1
n | \(\boldsymbol {x_{n}}\) | \(\boldsymbol {y_{n}}\) | \(\boldsymbol {z_{n}}\) |
---|---|---|---|
1 | 4.5000000 | 3.7807316 | 1.3778666 |
2 | 2.5792991 | 2.1441268 | 0.7327468 |
3 | 1.4384368 | 1.1914900 | 0.3981720 |
4 | 0.7948310 | 0.6572003 | 0.2171369 |
5 | 0.4371686 | 0.3610815 | 0.1184806 |
⋮ | ⋮ | ⋮ | ⋮ |
16 | 0.0005599 | 0.0004611 | 0.0001484 |
⋮ | ⋮ | ⋮ | ⋮ |
23 | 0.0000079 | 0.0000065 | 0.0000021 |
24 | 0.0000043 | 0.0000035 | 0.0000011 |
25 | 0.0000023 | 0.0000019 | 0.0000006 |
26 | 0.0000013 | 0.0000010 | 0.0000003 |
27 | 0.0000007 | 0.0000006 | 0.0000002 |
From Table 1, we see that the sequences \(\{x_{n}\}\), \(\{y_{n}\}\), and \(\{z_{n}\}\) converge to 0. We observe that \(x_{27}=0.0000007\) is an approximation of the common element in \(\bigcap_{i=1}^{2}F(t_{i}) \cap\bigcap_{i=1}^{2}F(T_{i}) \cap\bigcap_{i=1}^{2}VI([0,5],B_{i})\) with accuracy at 6 significant digits.
Next, we give the numerical example to support our main theorem in a two-dimensional space of real numbers.
Example 5.2
The values of the sequences \(\pmb{\{\mathbf{x}_{n}\}}\) , \(\pmb{\{\mathbf{y}_{n}\}}\) , and \(\pmb{\{\mathbf{z}_{n}\}}\) with the initial point \(\pmb{\mathbf{x}_{1}=(7.5,9.1)}\) in Example 5.2
n | \(\boldsymbol {x_{n}}\) | \(\boldsymbol {y_{n}}\) | \(\boldsymbol {z_{n}}\) |
---|---|---|---|
1 | (7.5000000,9.1000000) | (6.3029396,7.6067234) | (3.0687437,3.7035235) |
2 | (4.6619648,5.6749079) | (3.8840555,4.7013476) | (1.7741233,2.1474385) |
3 | (2.8183693,3.4332496) | (2.3412722,2.8357215) | (1.0459308,1.2668192) |
4 | (1.6857031,2.0577916) | (1.3983068,1.6970922) | (0.6176583,0.7496374) |
5 | (1.0092457,1.2223240) | (0.8364468,1.0071576) | (0.3669562,0.4418484) |
6 | (0.5676300,0.7527995) | (0.4701681,0.6199094) | (0.2053231,0.2707153) |
7 | (0.7174567,0.1573218) | (0.5940218,0.1294942) | (0.2585586,0.0563647) |
8 | (0.3517952,0.4346565) | (0.2911794,0.3576574) | (0.1264280,0.1552922) |
9 | (0.4298726,0.0764960) | (0.3557172,0.0629290) | (0.1541523,0.0272707) |
⋮ | ⋮ | ⋮ | ⋮ |
45 | (0.0000963,0.0000292) | (0.0000795,0.0000240) | (0.0000340,0.0000103) |
⋮ | ⋮ | ⋮ | ⋮ |
55 | (0.0000088,0.0000042) | (0.0000072,0.0000034) | (0.0000031,0.0000015) |
56 | (0.0000033,0.0000065) | (0.0000027,0.0000053) | (0.0000012,0.0000023) |
57 | (0.0000054,0.0000020) | (0.0000044,0.0000017) | (0.0000019,0.0000007) |
58 | (0.0000022,0.0000036) | (0.0000018,0.0000030) | (0.0000008,0.0000013) |
59 | (0.0000033,0.0000009) | (0.0000030,0.0000008) | (0.0000012,0.0000003) |
60 | (0.0000015,0.0000020) | (0.0000012,0.0000017) | (0.0000005,0.0000007) |
61 | (0.0000020,0.0000004) | (0.0000016,0.0000003) | (0.0000007,0.0000001) |
62 | (0.0000010,0.0000011) | (0.0000008,0.0000009) | (0.0000004,0.0000004) |
The values of the sequences \(\pmb{\{\mathbf{x}_{n}\}}\) , \(\pmb{\{\mathbf{y}_{n}\}}\) , and \(\pmb{\{\mathbf{z}_{n}\}}\) with the initial point \(\pmb{\mathbf{x}_{1}=(0,75.6)}\) in Example 5.2
n | \(\boldsymbol {x_{n}}\) | \(\boldsymbol {y_{n}}\) | \(\boldsymbol {z_{n}}\) |
---|---|---|---|
1 | (0.0000000,75.6000000) | (0.0000000,63.1943178) | (0.0000000,30.7677335) |
2 | (0.0000000,46.9810256) | (0.0000000,38.9211840) | (0.0000000,17.7780617) |
3 | (0.0000000,28.3496228) | (0.0000000,23.4156103) | (0.0000000,10.4605987) |
4 | (0.0000000,16.9381045) | (0.0000000,13.9691137) | (0.0000000,6.1704195) |
5 | (0.0000000,10.0697666) | (0.0000000,8.2971795) | (0.0000000,3.6400418) |
6 | (0.0000000,5.9686106) | (0.0000000,4.9149845) | (0.0000000,2.1463806) |
7 | (0.0000000,3.5306826) | (0.0000000,2.9061647) | (0.0000000,1.2649601) |
8 | (0.0000000,2.0855624) | (0.0000000,1.7161061) | (0.0000000,0.7451208) |
9 | (0.0000000,1.2306135) | (0.0000000,1.0123560) | (0.0000000,0.4387110) |
⋮ | ⋮ | ⋮ | ⋮ |
15 | (0.0000000,0.0512698) | (0.0000000,0.0421428) | (0.0000000,0.0181501) |
⋮ | ⋮ | ⋮ | ⋮ |
30 | (0.0000000,0.0000175) | (0.0000000,0.0000144) | (0.0000000,0.0000062) |
31 | (0.0000000,0.0000103) | (0.0000000,0.0000084) | (0.0000000,0.0000036) |
32 | (0.0000000,0.0000060) | (0.0000000,0.0000049) | (0.0000000,0.0000021) |
33 | (0.0000000,0.0000035) | (0.0000000,0.0000029) | (0.0000000,0.0000012) |
34 | (0.0000000,0.0000021) | (0.0000000,0.0000017) | (0.0000000,0.0000007) |
35 | (0.0000000,0.0000012) | (0.0000000,0.0000010) | (0.0000000,0.0000004) |
36 | (0.0000000,0.0000007) | (0.0000000,0.0000006) | (0.0000000,0.0000003) |
37 | (0.0000000,0.0000004) | (0.0000000,0.0000003) | (0.0000000,0.0000001) |
Declarations
Acknowledgements
The authors would like to thank the referees for valuable suggestions on the research paper.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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