- Research
- Open Access
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 spaces
MSC
- 46C05
- 47H09
- 47H10
1 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
References
- Markin, JT: Continuous dependence of fixed point sets. Proc. Am. Math. Soc. 38, 545-547 (1973) View ArticleMATHMathSciNetGoogle Scholar
- Nadler, SB Jr.: Multivalued contraction mappings. Pac. J. Math. 30, 475-488 (1969) View ArticleMATHMathSciNetGoogle Scholar
- Assad, NA, Kirk, WA: Fixed point theorems for set-valued mappings of contractive type. Pac. J. Math. 43, 553-562 (1972) View ArticleMathSciNetGoogle Scholar
- Blasi, FS, Myjak, J, Reich, S, Zaslavski, AJ: Generic existence and approximation of fixed points for nonexpansive set-valued maps. Set-Valued Var. Anal. 17(1), 97-112 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Cholamjiak, W, Suantai, S: A hybrid method for a countable family of multivalued maps, equilibrium problems, and variational inequality problems. Discrete Dyn. Nat. Soc. 2010, Article ID 349158 (2010) View ArticleMathSciNetGoogle Scholar
- Dominguez Benavides, T, Gavira, B: A fixed point property for multivalued nonexpansive mappings. J. Math. Anal. Appl. 328, 1471-1483 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Downing, D, Kirk, WA: Fixed point theorems for set-valued mappings in metric and Banach spaces. Math. Jpn. 22, 99-112 (1977) MATHMathSciNetGoogle Scholar
- Geobel, K: On a fixed point theorem for multivalued nonexpansive mappings. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 29, 70-72 (1975) Google Scholar
- Lim, TC: A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space. Bull. Am. Math. Soc. 80, 1123-1126 (1974) View ArticleMATHGoogle Scholar
- Markin, JT: A fixed point theorem for set valued mappings. Bull. Am. Math. Soc. 74, 639-640 (1968) View ArticleMATHMathSciNetGoogle Scholar
- Xu, HK: Multivalued nonexpansive mappings in Banach spaces. Nonlinear Anal. 43, 693-706 (2001) View ArticleMATHMathSciNetGoogle Scholar
- Iemoto, S, Takahashi, W: Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space. Nonlinear Anal. 71, 2080-2089 (2009) View ArticleMathSciNetGoogle Scholar
- Kohsaka, F, Takahashi, W: Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch. Math. (Basel) 91, 166-177 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Kurokawa, Y, Takahashi, W: Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces. Nonlinear Anal. 73, 1562-1568 (2010) View ArticleMATHMathSciNetGoogle Scholar
- Osilike, MO, Isiogugu, FO: Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces. Nonlinear Anal. 74, 1814-1822 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Aoyama, K, Kohsaka, F: Fixed point theorem for α-nonexpansive mappings in Banach spaces. Nonlinear Anal. 74, 4387-4391 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Cao, S: The split common fixed point problem for ϱ-strictly pseudononspreading mappings. J. Appl. Math. 2013, Article ID 241789 (2013) Google Scholar
- Deng, BC, Chen, T, Li, ZF: Viscosity iteration algorithm for a ϱ-strictly pseudononspreading mapping in a Hilbert space. J. Inequal. Appl. 2013, Article ID 80 (2013) View ArticleMathSciNetGoogle Scholar
- Isiogugu, FO, Osilike, MO: Fixed point and convergence theorems for certain classes of mappings. J. Niger. Math. Soc. 31, 147-165 (2012) MATHMathSciNetGoogle Scholar
- Kocourek, P, Takahashi, W, Yao, JC: Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces. Taiwan. J. Math. 14, 2497-2511 (2010) MATHMathSciNetGoogle Scholar
- Lin, LJ, Chuang, CS, Yu, ZT: Fixed point theorems for some new nonlinear mappings in Hilbert spaces. Fixed Point Theory Appl. 2011, Article ID 51 (2011) View ArticleGoogle Scholar
- Takahashi, W: Fixed point theorems for new nonlinear mappings in a Hilbert space. J. Nonlinear Convex Anal. 11, 79-88 (2010) MATHMathSciNetGoogle Scholar
- Takahashi, W, Yao, JC: Weak convergence theorems for generalized hybrid mappings in Banach spaces. J. Nonlinear Anal. Optim. 2, 147-158 (2011) MathSciNetGoogle Scholar
- Atsushiba, S, Takahashi, W: Strong convergence theorems for a finite family of nonexpansive mappings and applications. B. N. Prasad birth centenary commemoration volume. Indian J. Math. 41, 435-453 (1999) MATHMathSciNetGoogle Scholar
- Kangtunyakarn, A, Suantai, S: Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings. Nonlinear Anal. Hybrid Syst. 3, 296-309 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Ceng, LC, Cubiotti, P, Yao, JC: Strong convergence theorems for finitely many nonexpansive mappings and applications. Nonlinear Anal. 67, 1464-1473 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Colao, V, Marino, G, Xu, H-K: An iterative method for finding common solutions of equilibrium and fixed point problems. J. Math. Anal. Appl. 344, 340-352 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Nakajo, K, Shimoji, K, Takahashi, W: On strong convergence by the hybrid method for families of mappings in Hilbert spaces. Nonlinear Anal. 71, 112-119 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Takahashi, W: Weak and strong convergence theorems for families of nonexpansive mappings and their applications. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 51, 277-292 (1997) MATHGoogle Scholar
- Takahashi, W, Shimoji, K: Convergence theorems for nonexpansive mappings and feasibility problems. Math. Comput. Model. 32, 1463-1471 (2000) View ArticleMATHMathSciNetGoogle Scholar
- Yao, Y: A general iterative method for a finite family of nonexpansive mappings. Nonlinear Anal. 66, 2676-2687 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Iiduka, H, Takahashi, W: Weak convergence theorem by Cesáro means for nonexpansive mappings and inverse-strongly monotone mappings. J. Nonlinear Convex Anal. 7, 105-113 (2006) MATHMathSciNetGoogle Scholar
- Stampacchia, G: Formes bilinéaires coercitives sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413-4416 (1964) MATHMathSciNetGoogle Scholar
- Ansari, QH, Yao, JC: Systems of generalized variational inequalities and their applications. Appl. Anal. 76(3-4), 203-217 (2000) View ArticleMATHMathSciNetGoogle Scholar
- Ceng, LC, Ansari, QH, Schaible, S, Yao, JC: Iterative methods for generalized equilibrium problems, systems of general generalized equilibrium problems and fixed point problems for nonexpansive mappings in Hilbert spaces. Fixed Point Theory 12(2), 293-308 (2011) MATHMathSciNetGoogle Scholar
- Latif, A, Ceng, LC, Ansari, QH: Multi-step hybrid viscosity method for systems of variational inequalities defined over sets of solutions of equilibrium problem and fixed point problems. Fixed Point Theory Appl. 2012, Article ID 186 (2012) View ArticleMathSciNetGoogle Scholar
- Ceng, LC, Latif, A, Yao, JC: On solutions of system of variational inequalities and fixed point problems in Banach spaces. Fixed Point Theory Appl. 2013, Article ID 176 (2013) View ArticleMathSciNetGoogle Scholar
- Latif, A, Luc, DT: Variational relation problems: existence of solutions and fixed points of set-valued contraction mappings. Fixed Point Theory Appl. 2013, Article ID 315 (2013) View ArticleMathSciNetGoogle Scholar
- Ceng, LC, Latif, A, Ansari, QH, Yao, YC: Hybrid extragradient method for hierarchical variational inequalities. Fixed Point Theory Appl. 2014, Article ID 222 (2014) View ArticleMathSciNetGoogle Scholar
- Takahashi, W, Toyoda, M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417-428 (2003) View ArticleMATHMathSciNetGoogle Scholar
- Kangtunyakarn, A: The methods for variational inequality problems and fixed point of κ-strictly pseudononspreading mapping. Fixed Point Theory Appl. 2013, Article ID 171 (2013) View ArticleGoogle Scholar
- Takahashi, W: Nonlinear Function Analysis. Yokohama Publishers, Yokohama (2000) Google Scholar
- Marino, G, Xu, HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336-346 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Nakajo, K, Takahashi, W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372-379 (2003) View ArticleMATHMathSciNetGoogle Scholar
- Zegeye, H, Shahzad, N: Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 62, 4007-4014 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Martinez-Yanesa, C, Xu, HK: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal. 64, 2400-2411 (2006) View ArticleMathSciNetGoogle Scholar