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WienerHopf 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 WienerHopf 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. 10:107109, 1997), Wu and Li (4th International Congress on Image and Signal Processing, pp. 28022805, 2011), and Noor and Huang (Appl. Math. Comput. 191:504510, 2007).
1 Introduction
Let H be a real Hilbert space, whose inner product and norm are denoted by \u3008\cdot ,\cdot \u3009 and \parallel \cdot \parallel, respectively. Let K be a nonempty closed convex subset of H. Let T,S:K\to K be nonlinear mappings. Let M:H\to {2}^{H} be a multivalued operator and let f:K\times K\to R be a bifunction, where R is the set of real numbers. Let {P}_{K} be the projection of H onto the closed convex set K and {Q}_{K}=I{P}_{K}, where the I is the identity operator.
In 1994, Blum and Oettli (see [1]) introduced the equilibrium problem (EP) which is to find \overline{x}\in K, such that
The set of solutions of (1.1) is denoted by EP(f). Let f(x,y)=\u3008Tx,yx\u3009 for all x,y\in K, then the EP reduces to the variational inequality problem (VIP) which is to find \overline{x}\in K 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 \parallel SxSy\parallel \le \parallel xy\parallel for all x,y\in K. Let us denote the set of fixed points of S and set of solutions of problem (1.2) by F(S), VI(K,T), 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: \u3008Tuf,vu\u3009\ge 0, \mathrm{\forall}v\in K and WienerHopf equation: (T{P}_{K}+{Q}_{K})v=f, where f\in H. Noor (see [4]) established the equivalence between the generalized VIP: \u3008Tu,g(v)g(u)\u3009\ge 0, \mathrm{\forall}g(v)\in K and the generalized WienerHopf equation: T{g}^{1}{P}_{K}z+{\rho}^{1}{Q}_{K}z=0 and introduced an iterative scheme based on this equivalence,
where {g}^{1} exists.
Afterwards, by using different generalized WienerHopf equation techniques, Verma and AlShemas et al. introduced several algorithms for solving generalized VIPs, respectively (see, for example, [5–7]). It has been shown that the WienerHopf 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 F(S), EP(f), VI(K,T).
For solving EP(f)\cap F(S):
Takahashi and Takahashi (see [8]) introduced the following iterative schemes based on the viscosity approximation method:
For solving EP(f)\cap VI(K,T):
Li and Su (see [9]) introduced the following iterative schemes:
For solving EP(f)\cap VIP(K,T)\cap F(S):
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 WienerHopf equation technique which is more flexible and general than projection methods has not been used for solving EP(f)\cap F(S), EP(f)\cap VI(K,T), and EP(f)\cap VI(K,T)\cap F(S).
Remark 1.1 It is worth mentioning that although there are some papers which have applied the WienerHopf equation technique to solve VIP and VI(K,T)\cap F(S), their research does not include EP(f). However, our idea is just to apply the WienerHopf equation technique to study the common element problem which is related to EP(f).
In this paper, motivated and inspired by the above analysis and ongoing research in this field, we combine the WienerHopf equation technique and auxiliary principle to introduce some iterative schemes for solving the common element problem which is related to EP(f). This paper is organized as follows: In Section 2, some preliminaries are presented. Section 3 is devoted to solving the EP(f)\cap VI(K,T). In Section 4, we consider a nonexpansive mapping S and obtain some iterative schemes and strong convergent results for solving EP(f)\cap VI(K,T)\cap F(S). 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 GVI(K,T)\cap EP(f) and GVI(K,T)\cap EP(f)\cap F(S), 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 \u3008\cdot ,\cdot \u3009 and \parallel \cdot \parallel, respectively. Let K be a nonempty closed convex subset of H. Let T,S:K\to K be nonlinear mapping. Let M:H\to {2}^{H} be a multivalued operator and let f:K\times K\to R be a bifunction, where R is the set of real numbers. Let {P}_{K} be the projection of H onto the closed convex set K and {Q}_{K}=I{P}_{K}, where I is the identity operator.
Definition 2.1 The operator T:K\to K is said to be:

(i)
μLipschitz continuous, if there exists a constant \mu >0 such that \parallel TxTy\parallel \le \mu \parallel xy\parallel for all x,y\in K;

(ii)
rstrongly monotone, if there exists a constant r>0 such that \u3008TxTy,xy\u3009\ge r{\parallel xy\parallel}^{2} for all x,y\in K;

(iii)
γcocoercive, if there exists a constant \gamma >0 such that \u3008TxTy,xy\u3009\ge \gamma {\parallel TxTy\parallel}^{2} for all x,y\in K;

(iv)
relaxed γcocoercive, if there exists a constant \gamma >0 such that \u3008TxTy,xy\u3009\ge \gamma {\parallel TxTy\parallel}^{2} for all x,y\in K;

(v)
relaxed (\gamma ,r)cocoercive, if there exist two constants \gamma ,r>0 such that \u3008TxTy,xy\u3009\ge \gamma {\parallel TxTy\parallel}^{2}+r{\parallel xy\parallel}^{2} for all x,y\in K.
Definition 2.2 The multivalued operator M:H\to {2}^{H} is said to be:

(i)
a relaxed monotone operator, if there exists a constant k>0 such that \u3008{w}_{1}{w}_{2},uv\u3009\ge k{\parallel uv\parallel}^{2}, \mathrm{\forall}{w}_{1}\in Mu, \mathrm{\forall}{w}_{2}\in Mv;

(ii)
Lipschitz continuous if there exists a constant \lambda >0 such that \parallel {w}_{1}{w}_{2}\parallel \le \lambda \parallel uv\parallel, \mathrm{\forall}{w}_{1}\in Mu, \mathrm{\forall}{w}_{2}\in Mv.
Lemma 2.1 (see [10])
Let the bifunction f:K\times K\to R satisfy the following conditions:

(i)
f(x,x)=0 for all x\in K;

(ii)
f is monotone, i.e. f(x,y)+f(y,x)\le 0 for all x,y\in K;

(iii)
for each x,y,z\in K, {lim}_{t\to 0}f(tz+(1t)x,y)\le f(x,y);

(iv)
for each x\in K, f(x,\cdot ) is convex and lower semicontinuous.
Then EP(f)\ne \mathrm{\varnothing}.
Lemma 2.2 (see [10])
Let r>0, x\in H, and f satisfy the conditions (i)(iv) in Lemma 2.1. Then there exists z\in K such that f(z,y)+\frac{1}{r}\u3008yz,zx\u3009\ge 0, \mathrm{\forall}y\in K.
Lemma 2.3 (see [10])
Let r>0, x\in H, and f satisfy the conditions (i)(iv) in Lemma 2.1. Define a mapping {T}_{r}:H\to K as {T}_{r}(x)=\{z\in K:f(z,y)+\frac{1}{r}\cdot \u3008yz,zx\u3009\ge 0,\mathrm{\forall}y\in K\}.
Then the following hold:

(a)
{T}_{r} is singlevalued;

(b)
{T}_{r} is firmly nonexpansive, i.e. \parallel {T}_{r}x{T}_{r}y\parallel \le \u3008{T}_{r}x{T}_{r}y,xy\u3009 for all x,y\in H;

(c)
EP(f)=F({T}_{r}), where F({T}_{r}) denotes the sets of fixed point of {T}_{r};

(d)
EP(f) is closed and convex.
In [4], Noor introduced the following generalized WienerHopf equation:
Here he assumed {g}^{1} exists, and note that if g=I, 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 WHE(T,g) and WHE(T), respectively.
Lemma 2.4 (see [3])
The variational inequality (1.2) has a solution \overline{x}\in H if and only if the WienerHopf equation (2.2) has a solution \overline{z}\in H, where \overline{x}={P}_{K}\overline{z}, \overline{z}=\overline{x}\rho T\overline{x}.
In 1988, Noor (see [19]) introduced the generalized variational inequality (GVIP) which is to find \overline{x}\in H such that g(\overline{x})\in H and
Denote the set of solutions of (2.3) by GVI(K,T). Clearly, if g=I, 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 \tilde{x}\in H if and only if the WienerHopf equation (2.1) has a solution \tilde{z}\in H, where g(\overline{x})={P}_{K}\tilde{z}, \tilde{z}=g(\overline{x})\rho T\overline{x}, and \rho >0 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 WienerHopf equation which included a nonexpansive mapping:
Furthermore the equivalence was established between the WienerHopf equation (2.4) and the variational inequality (1.2) as follows.
Lemma 2.6 (see [33])
The variational inequality (1.2) has a solution \tilde{x} if and only if the WienerHopf equation (2.4) has a solution \tilde{z}, where \tilde{z}=\tilde{x}\rho T\tilde{x}, \tilde{x}=S{P}_{K}\tilde{z}.
Wu and Li (see [29]) introduced the WienerHopf equation which includes a nonexpansive mapping S:
and established the equivalence between the WienerHopf equation (2.5) and the generalized variational inequality which is to find u\in K such that
Next, denote the sets of solutions of (2.5) by WHE(T,S).
Lemma 2.7 (see [29])
The variational inequality (2.6) has a solution \tilde{c}\in H if and only if the WienerHopf equation (2.5) has a solution \tilde{z}\in H, where \tilde{z}=\tilde{c}\rho (T\tilde{c}+w), \tilde{c}=S{P}_{K}\tilde{z}.
Lemma 2.8 (see [30])
Assume that \{{a}_{n}\} is a sequence of nonnegative real numbers such that
where {n}_{0} is some nonnegative integer, and \{{\lambda}_{n}\} is a sequence in [0,1] such that {\sum}_{n=1}^{\mathrm{\infty}}{\lambda}_{n}=\mathrm{\infty}, {b}_{n}=o({\lambda}_{n}). Then {lim}_{n\to \mathrm{\infty}}{a}_{n}=0.
Lemma 2.9 For given x,z\in K, if
for any y\in K. Then x=z.
Proof Assume x\ne z. Put y=x, then (2.7) reduces to 0\le \u3008xz,zx\u3009={\parallel xz\parallel}^{2}<0, which is a contradiction. □
3 Results for solving EP(f)\cap VI(K,T)
In this section, we firstly use Lemma 2.4 to introduce some iterative schemes and the convergence theorems for solving EP(f)\cap VI(K,T).
Algorithm 3.1 For a given {z}_{0}, compute the approximate solution {z}_{n+1} by the iterative schemes:
By an appropriate rearrangement, Algorithm 3.1 can be written in the following form.
Algorithm 3.2 For a given {z}_{0}, compute the approximate solution {z}_{n+1} by the iterative schemes:
If f(x,y)=0 for all x,y\in K, 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 {z}_{0}, compute the approximate solution {z}_{n+1} by the iterative schemes:
Theorem 3.1 Let K be the nonempty closed convex subset of H. The bifunction f:K\times K\to R satisfies the conditions (i)(iv) of Lemma 2.1. Let T:K\to K be a αstrongly monotone and βLipschitz continuous operator such that EP(f)\cap VI(K,T)\ne \mathrm{\varnothing}. Let \{{z}_{n}\}, \{{u}_{n}\}, \{{v}_{n}\} be the sequences generated by Algorithm 3.1, where \rho >0 is a constant and
Then \{{u}_{n}\}, \{{v}_{n}\} generated by Algorithm 3.1 converge to s\in EP(f)\cap VI(K,T), and \{{z}_{n}\} generated by Algorithm 3.1 converges to \tilde{z}\in WHE(T).
Proof Let \tilde{z}\in WHE(T) and s\in EP(f)\cap VI(K,T).
Step 1. Estimate \parallel {z}_{n+1}\tilde{z}\parallel.
From Lemma 2.4, we have
Hence
Here, we have used the αstrong monotonicity and μLipschitz continuity of T in (3.3) and (3.4).
Step 2. Estimate \parallel {v}_{n}s\parallel.
Since s\in EP(f)\cap VI(K,T), we have
Putting y={v}_{n} in (3.5) and y=s 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 \parallel {u}_{n}s\parallel and prove the strong convergence of sequences generated by Algorithm 3.1.
Due to the nonexpansivity of {P}_{K}, we find
By the above three steps, we have
which implies that
Since
we conclude
Therefore, from
we have
□
4 Results for solving EP(f)\cap VI(K,T)\cap F(S)
In this section, applying Lemma 2.6, we consider a nonexpansive mapping and analyze several iterative schemes and convergence theorems for solving EP(f)\cap VI(K,T)\cap F(S).
Algorithm 4.1 For a given {z}_{0}\in H, compute the approximate solution {z}_{n+1} by the iterative schemes
For S=I, the identity operator, Algorithm 4.1 collapses to the following iterative method.
Algorithm 4.2 For a given {z}_{0}\in H, compute the approximate solution {z}_{n+1} by the iterative schemes
For {\alpha}_{n}=1, S=I, the identity operator, Algorithm 4.1 collapses to the following iterative method.
Algorithm 4.3 For a given {z}_{0}\in H, compute the approximate solution {z}_{n+1} by the iterative schemes
For f=0 and via Lemma 2.9, Algorithm 4.1 reduces to the following iterative method for solving VI(K,T)\cap F(S).
Algorithm 4.4 For a given {z}_{0}\in H, compute the approximate solution {z}_{n+1} by the iterative schemes
For f=0, S=I, {\alpha}_{n}=1, and via Lemma 2.9, Algorithm 4.1 reduces to the following iterative method for solving VIP.
Algorithm 4.5 For a given {z}_{0}\in H, compute the approximate solution {z}_{n+1} 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 T:K\to K be relaxed (\gamma ,r)cocoercive and μLipschitz continuous, and let S:K\to K be a kstrictly pseudocontractive mapping such that EP(f)\cap VIP(K,T)\cap F(S)\ne \mathrm{\varnothing}, where
Then the sequences \{{u}_{n}\}, \{{v}_{n}\} generated by Algorithm 4.1 converge to \tilde{c}\in EP(f)\cap VI(K,T)\cap F(S) and the sequence \{{z}_{n}\} generated by Algorithm 4.1 converges to \tilde{z}\in WHE(T).
Proof Let \tilde{z}\in WHE(T) and \tilde{c}\in EP(f)\cap VI(K,T)\cap F(S).
Step 1. Estimate \parallel {z}_{n+1}\tilde{z}\parallel.
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 (\gamma ,r)cocoercivity and μLipschitz continuity of the operator T.
Step 2. Estimate \parallel {v}_{n}\tilde{c}\parallel.
Since \tilde{c}\in EP(f)\cap VI(K,T)\cap F(S), we get
Putting y={v}_{n} in (4.12) and y=\tilde{c} 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 \parallel {u}_{n}\tilde{c}\parallel and prove the strong convergence of sequences generated by Algorithm 4.1.
Since
we have
By the above three steps, we get
where \theta =\sqrt{(1+2\rho \gamma {\mu}^{2}2\rho r+{\rho}^{2}{\mu}^{2})}.
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 multivalued variational inequality, the extended general quasivariational inequalities and so on. Related to the variational inequality, the WienerHopf equation has also been extended, and the equivalence between generalized variational inequalities and generalized WienerHopf 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 {z}_{0}\in H, compute the approximate solution {z}_{n+1} by the iterative schemes
Theorem 5.1 Let K be the nonempty closed convex subset of H and the bifunction f:K\times K\to R satisfy the conditions (i)(iv) of Lemma 2.1. Let T:K\to K be a αstrongly monotone and βLipschitz continuous singlevalued operator and let g:K\to K be a σstrongly monotone and δLipschitz continuous singlevalued operator such that {g}^{1} exists and EP(f)\cap GVI(K,T)\ne \mathrm{\varnothing}, where
Then the sequences \{{u}_{n}\}, \{{v}_{n}\} generated by Algorithm 5.1 converge strongly to s\in EP(f)\cap GVI(K,T) and \{{z}_{n}\} generated by Algorithm 5.1 converges strongly to \tilde{z}\in WHE(T,g).
Proof Let s\in EP(f)\cap GVI(K,T), \tilde{z}\in WHE(T,g).
Step 1. Estimate \parallel {z}_{n+1}\tilde{z}\parallel.
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 k=\sqrt{12\sigma +{\delta}^{2}}\ne 1, t=\sqrt{12\rho \alpha +{\beta}^{2}{\rho}^{2}}.
Step 2. Estimate \parallel {v}_{n}s\parallel.
Employing the technique that we use in the proof of Theorem 3.1, we get
Step 3. Estimate \parallel {u}_{n}s\parallel 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 f(x,y)=0, \mathrm{\forall}x,y\in K, 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 {z}_{0}\in H, compute the approximate solution {z}_{n+1} by the iterative schemes
If S=I, Algorithm 5.2 reduces to the following.
Algorithm 5.3 For a given {z}_{0}\in H, compute the approximate solution {z}_{n+1} by the iterative schemes
If S=I and {\alpha}_{n}=1, Algorithm 5.2 reduces to the following.
Algorithm 5.4 For a given {z}_{0}\in H, compute the approximate solution {z}_{n+1} by the iterative schemes
If f=0, Algorithm 5.2 reduces to the following.
Algorithm 5.5 For a given {z}_{0}\in H, compute the approximate solution {z}_{n+1} by the iterative schemes
which was introduced in [8].
If f=0, S=I, Algorithm 5.2 reduces to the following.
Algorithm 5.6 For a given {z}_{0}\in H, compute the approximate solution {z}_{n+1} by the iterative schemes
which was introduced in [8].
If f=0, S=I, and {\alpha}_{n}=1, Algorithm 5.2 reduces to the following.
Algorithm 5.7 For a given {z}_{0}\in H, compute the approximate solution {z}_{n+1} 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 T:K\to K be a relaxed (\gamma ,r)cocoercive and μLipschitz continuous operator, and S:K\to K be kstrictly such that EP(f)\cap VI(K,T)\cap F(S)\ne \mathrm{\varnothing}. Let M:H\to {2}^{H} be a multivalued relaxed monotone and Lipschitz continuous operator with the corresponding constant k>0, m>0, where
Then the sequences \{{u}_{n}\}, \{{v}_{n}\} generated by Algorithm 5.2 converge strongly to \tilde{c}\in EP(f)\cap VI(K,T)\cap F(S), and \{{z}_{n}\} generated by Algorithm 5.2 converges strongly to \tilde{z}\in WHE(T,S).
Proof Let \tilde{c}\in EP(f)\cap VI(K,T)\cap F(S) and \tilde{z}\in WHE(T,S).
Step 1. Estimate \parallel {z}_{n+1}\tilde{z}\parallel and \parallel {u}_{n}\tilde{c}\parallel.
By the same technique in [29], we have
where \theta =\sqrt{1+2\rho (\gamma \mu r+k)+{\rho}^{2}{(\mu +m)}^{2}}.
Step 2. Estimate \parallel {v}_{n}\tilde{c}\parallel.
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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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. WienerHopf 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/10.1186/1029242X2014286
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DOI: https://doi.org/10.1186/1029242X2014286
Keywords
 equilibrium problems
 algorithms
 WienerHopf equation technique
 auxiliary principle