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Wiener-Hopf equation technique for solving equilibrium problems and variational inequalities and fixed points of a nonexpansive mapping
Journal of Inequalities and Applications volume 2014, Article number: 286 (2014)
Abstract
In this paper, we introduce some new iterative schemes based on the Wiener-Hopf equation technique and auxiliary principle for finding common elements of the set of solutions of equilibrium problems, the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality. Several strong convergence results for the sequences generated by these iterative schemes are established in Hilbert spaces. As the generation, we also consider two generalized variational inequalities, and obtain some iterative schemes and the proposed strong convergence theorems for solving these generalized variational inequalities, equilibrium problems, and a nonexpansive mapping. Our results and proof are new, and they extend the corresponding results of Verma (Appl. Math. Lett. 10:107-109, 1997), Wu and Li (4th International Congress on Image and Signal Processing, pp. 2802-2805, 2011), and Noor and Huang (Appl. Math. Comput. 191:504-510, 2007).
1 Introduction
Let H be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let K be a nonempty closed convex subset of H. Let be nonlinear mappings. Let be a multi-valued operator and let be a bifunction, where R is the set of real numbers. Let be the projection of H onto the closed convex set K and , where the I is the identity operator.
In 1994, Blum and Oettli (see [1]) introduced the equilibrium problem (EP) which is to find , such that
The set of solutions of (1.1) is denoted by . Let for all , then the EP reduces to the variational inequality problem (VIP) which is to find such that
This problem was introduced by Stampacchia (see [2]) in 1964. Related to EP and VIP, fixed point problems (FP) of a nonexpansive mapping are also considered by many authors. Recall that a mapping S is nonexpansive if for all . Let us denote the set of fixed points of S and set of solutions of problem (1.2) by , , respectively.
Recently, for solving the EP, VIP, and FP, many authors have introduced and extended lots of iterative schemes.
For solving variational inequality problems:
In 1991, Shi (see [3]) demonstrated the equivalence between the VIP: , and Wiener-Hopf equation: , where . Noor (see [4]) established the equivalence between the generalized VIP: , and the generalized Wiener-Hopf equation: and introduced an iterative scheme based on this equivalence,
where exists.
Afterwards, by using different generalized Wiener-Hopf equation techniques, Verma and Al-Shemas et al. introduced several algorithms for solving generalized VIPs, respectively (see, for example, [5–7]). It has been shown that the Wiener-Hopf equation techniques are more flexible and general than projection methods.
On the other hand, for getting the unified approach to solve EP, VIP and FP, many authors also suggest and analyze lots of iterative schemes for common elements of , , .
For solving :
Takahashi and Takahashi (see [8]) introduced the following iterative schemes based on the viscosity approximation method:
For solving :
Li and Su (see [9]) introduced the following iterative schemes:
For solving :
Plubtieng and Punpaeng (see [10]) introduced the following iterative schemes:
For more algorithms, please see, for instance, [1–33] and the references therein.
Summarizing the above algorithms, we know that these papers have mainly used the auxiliary principle, the projection technique and iterative schemes of fixed points. However, the Wiener-Hopf equation technique which is more flexible and general than projection methods has not been used for solving , , and .
Remark 1.1 It is worth mentioning that although there are some papers which have applied the Wiener-Hopf equation technique to solve VIP and , their research does not include . However, our idea is just to apply the Wiener-Hopf equation technique to study the common element problem which is related to .
In this paper, motivated and inspired by the above analysis and ongoing research in this field, we combine the Wiener-Hopf equation technique and auxiliary principle to introduce some iterative schemes for solving the common element problem which is related to . This paper is organized as follows: In Section 2, some preliminaries are presented. Section 3 is devoted to solving the . In Section 4, we consider a nonexpansive mapping S and obtain some iterative schemes and strong convergent results for solving . In Section 5, we extend the VIPs in Section 3 and Section 4, and we get some iterative schemes and strong convergent theorems for solving and , respectively. Our results extend the corresponding results of Verma (see [6]), Wu and Li (see [29]), and Noor and Huang (see [33]).
2 Preliminaries
In the rest of this paper, let H be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let K be a nonempty closed convex subset of H. Let be nonlinear mapping. Let be a multi-valued operator and let be a bifunction, where R is the set of real numbers. Let be the projection of H onto the closed convex set K and , where I is the identity operator.
Definition 2.1 The operator is said to be:
-
(i)
μ-Lipschitz continuous, if there exists a constant such that for all ;
-
(ii)
r-strongly monotone, if there exists a constant such that for all ;
-
(iii)
γ-co-coercive, if there exists a constant such that for all ;
-
(iv)
relaxed γ-co-coercive, if there exists a constant such that for all ;
-
(v)
relaxed -co-coercive, if there exist two constants such that for all .
Definition 2.2 The multi-valued operator is said to be:
-
(i)
a relaxed monotone operator, if there exists a constant such that , , ;
-
(ii)
Lipschitz continuous if there exists a constant such that , , .
Lemma 2.1 (see [10])
Let the bifunction satisfy the following conditions:
-
(i)
for all ;
-
(ii)
f is monotone, i.e. for all ;
-
(iii)
for each , ;
-
(iv)
for each , is convex and lower semicontinuous.
Then .
Lemma 2.2 (see [10])
Let , , and f satisfy the conditions (i)-(iv) in Lemma 2.1. Then there exists such that , .
Lemma 2.3 (see [10])
Let , , and f satisfy the conditions (i)-(iv) in Lemma 2.1. Define a mapping as .
Then the following hold:
-
(a)
is single-valued;
-
(b)
is firmly nonexpansive, i.e. for all ;
-
(c)
, where denotes the sets of fixed point of ;
-
(d)
is closed and convex.
In [4], Noor introduced the following generalized Wiener-Hopf equation:
Here he assumed exists, and note that if , the identity operator, then (2.1) reduces to
which was introduced by Shi (see [3]). Denote the sets of solutions of (2.1) and (2.2) by and , respectively.
Lemma 2.4 (see [3])
The variational inequality (1.2) has a solution if and only if the Wiener-Hopf equation (2.2) has a solution , where , .
In 1988, Noor (see [19]) introduced the generalized variational inequality (GVIP) which is to find such that and
Denote the set of solutions of (2.3) by . Clearly, if , the identity operator, the GVIP (2.3) reduces to VIP (1.2).
Lemma 2.5 (see [4])
The variational inequality (2.3) has a solution if and only if the Wiener-Hopf equation (2.1) has a solution , where , , and is a constant.
For finding the common element of the set of fixed points of a nonexpansive mapping and the set of solution of the variational inequality, Noor and Huang [33] introduced the Wiener-Hopf equation which included a nonexpansive mapping:
Furthermore the equivalence was established between the Wiener-Hopf equation (2.4) and the variational inequality (1.2) as follows.
Lemma 2.6 (see [33])
The variational inequality (1.2) has a solution if and only if the Wiener-Hopf equation (2.4) has a solution , where , .
Wu and Li (see [29]) introduced the Wiener-Hopf equation which includes a nonexpansive mapping S:
and established the equivalence between the Wiener-Hopf equation (2.5) and the generalized variational inequality which is to find such that
Next, denote the sets of solutions of (2.5) by .
Lemma 2.7 (see [29])
The variational inequality (2.6) has a solution if and only if the Wiener-Hopf equation (2.5) has a solution , where , .
Lemma 2.8 (see [30])
Assume that is a sequence of nonnegative real numbers such that
where is some nonnegative integer, and is a sequence in such that , . Then .
Lemma 2.9 For given , if
for any . Then .
Proof Assume . Put , then (2.7) reduces to , which is a contradiction. □
3 Results for solving
In this section, we firstly use Lemma 2.4 to introduce some iterative schemes and the convergence theorems for solving .
Algorithm 3.1 For a given , compute the approximate solution by the iterative schemes:
By an appropriate rearrangement, Algorithm 3.1 can be written in the following form.
Algorithm 3.2 For a given , compute the approximate solution by the iterative schemes:
If for all , Algorithm 3.1 collapses to the following iterative method for solving variational inequalities (1.2), which is mainly due to Shi [3].
Algorithm 3.3 For a given , compute the approximate solution by the iterative schemes:
Theorem 3.1 Let K be the nonempty closed convex subset of H. The bifunction satisfies the conditions (i)-(iv) of Lemma 2.1. Let be a α-strongly monotone and β-Lipschitz continuous operator such that . Let , , be the sequences generated by Algorithm 3.1, where is a constant and
Then , generated by Algorithm 3.1 converge to , and generated by Algorithm 3.1 converges to .
Proof Let and .
Step 1. Estimate .
From Lemma 2.4, we have
Hence
Here, we have used the α-strong monotonicity and μ-Lipschitz continuity of T in (3.3) and (3.4).
Step 2. Estimate .
Since , we have
Putting in (3.5) and in Algorithm 3.1, we have
From the monotonicity of f, we get
Combining (3.6) and (3.7), we obtain
It follows that
Step 3. Estimate and prove the strong convergence of sequences generated by Algorithm 3.1.
Due to the nonexpansivity of , we find
By the above three steps, we have
which implies that
Since
we conclude
Therefore, from
we have
□
4 Results for solving
In this section, applying Lemma 2.6, we consider a nonexpansive mapping and analyze several iterative schemes and convergence theorems for solving .
Algorithm 4.1 For a given , compute the approximate solution by the iterative schemes
For , the identity operator, Algorithm 4.1 collapses to the following iterative method.
Algorithm 4.2 For a given , compute the approximate solution by the iterative schemes
For , , the identity operator, Algorithm 4.1 collapses to the following iterative method.
Algorithm 4.3 For a given , compute the approximate solution by the iterative schemes
For and via Lemma 2.9, Algorithm 4.1 reduces to the following iterative method for solving .
Algorithm 4.4 For a given , compute the approximate solution by the iterative schemes
For , , , and via Lemma 2.9, Algorithm 4.1 reduces to the following iterative method for solving VIP.
Algorithm 4.5 For a given , compute the approximate solution by the iterative schemes
Theorem 4.1 Let K be the nonempty closed convex subset of H. The bifunction f satisfies the conditions (i)-(iv) of Lemma 2.1. Let be relaxed -co-coercive and μ-Lipschitz continuous, and let be a k-strictly pseudocontractive mapping such that , where
Then the sequences , generated by Algorithm 4.1 converge to and the sequence generated by Algorithm 4.1 converges to .
Proof Let and .
Step 1. Estimate .
From Lemma 2.6, we have
This implies that
Hence
In (4.8), (4.10), (4.11) of the above induction, we have used the -co-coercivity and μ-Lipschitz continuity of the operator T.
Step 2. Estimate .
Since , we get
Putting in (4.12) and in Algorithm 4.1, respectively, we have
From the monotonicity of f, we obtain
Combining (4.13) and (4.14), we know
It follows that
Step 3. Estimate and prove the strong convergence of sequences generated by Algorithm 4.1.
Since
we have
By the above three steps, we get
where .
From Lemma 2.8, it is easy to see that
and from
we have
□
5 Generation
Ever since the classical variational inequality was introduced, it has been extended to many forms in different directions, such as the nonconvex variational inequality, the multi-valued variational inequality, the extended general quasi-variational inequalities and so on. Related to the variational inequality, the Wiener-Hopf equation has also been extended, and the equivalence between generalized variational inequalities and generalized Wiener-Hopf equations has been studied.
In this section, we consider the two generalized variational inequalities (2.3) and (2.6), respectively. Furthermore, applying Lemma 2.5, we firstly suggest and give some iterative schemes for solving the variational inequality (2.3) and the equilibrium problem; secondly, via Lemma 2.7, we also analyze and introduce several iterative schemes for solving the variational inequality (2.6), the equilibrium problem, and a nonexpansive mapping.
5.1 Results for solving variational inequality (2.3) and equilibrium problem
In this subsection, we use a similar technique to Section 3, and the proposed results in this subsection can be considered as the generation of Section 3.
Algorithm 5.1 For a given , compute the approximate solution by the iterative schemes
Theorem 5.1 Let K be the nonempty closed convex subset of H and the bifunction satisfy the conditions (i)-(iv) of Lemma 2.1. Let be a α-strongly monotone and β-Lipschitz continuous single-valued operator and let be a σ-strongly monotone and δ-Lipschitz continuous single-valued operator such that exists and , where
Then the sequences , generated by Algorithm 5.1 converge strongly to and generated by Algorithm 5.1 converges strongly to .
Proof Let , .
Step 1. Estimate .
From the Lemma 2.5, we have
This implies that
Note that we use the strong monotonicity and Lipschitz continuity of T, g in (5.8), where , .
Step 2. Estimate .
Employing the technique that we use in the proof of Theorem 3.1, we get
Step 3. Estimate and prove the strong convergence of sequences generated by Algorithm 5.1.
From (5.3), we obtain
It follows that
Combining the above three steps, we have
Since
we get
□
Remark 5.1 Clearly, if g is the identity mapping, Theorem 5.1 reduces to Theorem 3.1.
Remark 5.2 Via Lemma 2.9, it is easy to see that if the bifunction , , Algorithm 5.1 reduces to
This algorithm was introduced by Noor in [4], which implies that Algorithm 5.1 in this paper extends the results in [4].
Remark 5.3 It is easy to show the condition (5.1) can be satisfied, for instance
5.2 Results for solving variational inequality (2.6), an equilibrium problem, and a nonexpansive mapping
In this subsection, we use a similar technique to Section 4, and the proposed results in this subsection can be considered as the generation of Section 4.
Algorithm 5.2 For a given , compute the approximate solution by the iterative schemes
If , Algorithm 5.2 reduces to the following.
Algorithm 5.3 For a given , compute the approximate solution by the iterative schemes
If and , Algorithm 5.2 reduces to the following.
Algorithm 5.4 For a given , compute the approximate solution by the iterative schemes
If , Algorithm 5.2 reduces to the following.
Algorithm 5.5 For a given , compute the approximate solution by the iterative schemes
which was introduced in [8].
If , , Algorithm 5.2 reduces to the following.
Algorithm 5.6 For a given , compute the approximate solution by the iterative schemes
which was introduced in [8].
If , , and , Algorithm 5.2 reduces to the following.
Algorithm 5.7 For a given , compute the approximate solution by the iterative schemes
which was introduced in [6].
Theorem 5.2 Let K be the nonempty closed convex subset of H and the bifunction f satisfy the conditions (i)-(iv) of Lemma 2.1. Let be a relaxed -co-coercive and μ-Lipschitz continuous operator, and be k-strictly such that . Let be a multi-valued relaxed monotone and Lipschitz continuous operator with the corresponding constant , , where
Then the sequences , generated by Algorithm 5.2 converge strongly to , and generated by Algorithm 5.2 converges strongly to .
Proof Let and .
Step 1. Estimate and .
By the same technique in [29], we have
where .
Step 2. Estimate .
Applying the technique in Theorem 3.2 of this paper, we get
Combining the above two steps, we can obtain
From Lemma 2.8, it follows that
Note that
We conclude that
□
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Acknowledgements
The authors would like to thank the referees for the valuable suggestions, which helped to improve this manuscript. This project is supported by the National Natural Science Foundation of China (11071109).
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Wang, Y., Zhang, C. Wiener-Hopf equation technique for solving equilibrium problems and variational inequalities and fixed points of a nonexpansive mapping. J Inequal Appl 2014, 286 (2014). https://doi.org/10.1186/1029-242X-2014-286
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DOI: https://doi.org/10.1186/1029-242X-2014-286