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Weak convergence theorem for variational inequality problems with monotone mapping in Hilbert space
Journal of Inequalities and Applications volume 2016, Article number: 286 (2016)
Abstract
We know that variational inequality problem is very important in the nonlinear analysis. The main purpose of this paper is to propose an iterative method for finding an element of the set of solutions of a variational inequality problem with a monotone and Lipschitz continuous mapping in Hilbert space. This iterative method is based on the extragradient method. We get a weak convergence theorem. Using this result, we obtain three weak convergence theorems for the equilibrium problem, the constrained convex minimization problem, and the split feasibility problem.
Introduction
The variational inequality problem is a generalization of the nonlinear complementarity problem. It is widely used in economics, engineering, mechanics, signal processing, image processing, and so on. The variational inequality was first derived from the mechanics problems in the early 1960s. In 1964, the existence and uniqueness of solutions of variational inequalities were presented for the first time. Subsequently, some scientists have published a series of articles. In the 1970s, the variational inequality problem had been used in many fields. In the 1990s, the variational inequality problem became more important in nonlinear analysis.
Let \(\mathbb{R}\) be the set of real numbers. Let H be a real Hilbert space with the inner product \(\langle\cdot,\cdot\rangle \) and norm \(\\cdot\\) and let C be a nonempty closed convex subset of H. A mapping \(A:C\rightarrow H\) is called monotone if
A mapping \(A:C\rightarrow H\) is called Lipschitz continuous if there exists \(k\in\mathbb{R}\) with \(k>0\) such that
Such A is called kLipschitz continuous. If \(k=1\), such A is called a nonexpansive mapping. The variational inequality problem is to find \(x^{*}\in C\) such that
We denote the set of solutions of this variational inequality problem by \(\operatorname{VI}(C,A)\).
In 1976, Korpelevich [1] proposed the following socalled extragradient method for solving the variational inequality problem in the finitedimensional Euclidean space \(\mathbb{R}^{n}\).
Theorem 1.1
[1]
Let C be a nonempty closed convex subset of an ndimensional Euclidean space \(\mathbb{R}^{n}\). Let A be a monotone and kLipschitz continuous mapping of C into H. Assume that \(\operatorname{VI}(C,A)\) is nonempty. Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be sequences generated by \(x_{0}=x\in C\) and
for every \(n=0,1,2,\ldots\) , where \(\lambda\in(0,\frac{1}{k})\). Then the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) converge to the same point \(z\in \operatorname{VI}(C,A)\).
In this paper, based on the extragradient method, we introduce an iterative method for finding an element of the set of solutions of a variational inequality problem for a monotone and Lipschitz continuous mapping in Hilbert space. We obtain a weak convergence theorem. As applications, we can use this result to solve equilibrium problems, constrained convex minimization problems, and split feasibility problems.
Preliminaries
Let \(\mathbb{R}\) be the set of real numbers. Let H be a real Hilbert space with the inner product \(\langle\cdot,\cdot\rangle \) and norm \(\\cdot\\). Let \(\{x_{n}\}\) be a sequence in H, we denote the sequence \(\{x_{n}\}\) converging weakly to x by \(x_{n}\rightharpoonup x\) and the sequence \(\{x_{n}\}\) converging strongly to x by \(x_{n}\rightarrow x\). Let C be a nonempty closed convex subset of H. For each \(x\in H\), there exists a unique nearest point in C, denoted by \(P_{C}x\), such that
\(P_{C}\) is called the metric projection of H into C. We know that \(P_{C}\) is nonexpansive. A setvalued mapping \(T:H\rightarrow2^{H}\) is called monotone if
A monotone mapping \(T:H\rightarrow2^{H}\) is called maximal if its graph is not properly contained in the graph of any other monotone mapping on H. It is known that a monotone mapping T is maximal if and only if for \((x,u)\in H\times H\), \(\langle xy, uv\rangle\geq0\) for each \((y,v)\in G(T)\) implies \(u\in Tx\).
Lemma 2.1
[2]
Let C be a nonempty closed convex subset of a real Hilbert space H. Given \(x\in H\) and \(z\in C\). Then \(z=P_{C}x\) if and only if we have the inequality
Lemma 2.2
[2]
Let C be a nonempty closed convex subset of a real Hilbert space H. Given \(x\in H\) and \(z\in C\). Then \(z=P_{C}x\) if and only if we have the inequality
Lemma 2.3
[3]
Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and kLipschitz continuous mapping of C into H and let \(N_{C}v\) be the normal cone to C at \(v\in C\); i.e.,
Define
Then T is maximal monotone and \(0\in Tv\) if and only if \(v\in \operatorname{VI}(C,A)\).
Lemma 2.4
[4]
Let C be a nonempty closed convex subset of a real Hilbert space H. Let \(\{x_{n}\}\) be a sequence in H satisfying the properties:

(i)
\(\lim_{n\rightarrow\infty}\x_{n}u\\) exists for each \(u\in C\);

(ii)
\(\omega_{w}(x_{n})\subset C\).
Then \(\{x_{n}\}\) converges weakly to a point in C.
Lemma 2.5
[5]
Let C be a nonempty closed convex subset of a real Hilbert space H. Let \(\{x_{n}\}\) be a sequence in H. Suppose that
for every \(n=0,1,2,\ldots\) . Then the sequence \(\{P_{C}x_{n}\}\) converges strongly to a point in C.
Main results
The main task of this article is to find an element of the set of solutions of a variational inequality problem with a monotone and Lipschitz continuous mapping in Hilbert space. We obtain a weak convergence theorem.
Theorem 3.1
Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let A be a monotone and kLipschitz continuous mapping of C into H. Assume that \(\operatorname{VI}(C,A)\neq\emptyset\). Let the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) be generated by \(x_{0}=x\in C\) and
for every \(n=0,1,2,\ldots\) , where \(\{\lambda_{n}\}\subset[a,b]\) for some \(a,b\in(0,\frac{1}{k})\). Then the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) converge weakly to the same point \(z\in \operatorname{VI}(C,A)\), where \(z=\lim_{n\rightarrow\infty} P_{\operatorname{VI}(C,A)}x_{n}\).
Proof
For each \(u\in \operatorname{VI}(C,A)\). From Lemma 2.2, we have
Then, from Lemma 2.1, we obtain
So, we have
Therefore, there exists
and the sequence \(\{x_{n}\}\) is bounded. From (3.2), we also get
So, we obtain
Hence
On the other hand, we have
Hence
Since A is Lipschitz continuous, we get
From
we have
Since \(\{x_{n}\}\) is bounded, there is a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) that converges weakly to a point z. We prove that \(z\in \operatorname{VI}(C,A)\). From (3.5) and (3.9), we have \(y_{n_{i}}\rightharpoonup z\) and \(x_{n_{i}+1}\rightharpoonup z\).
Let
From Lemma 2.3, we know that T is maximal monotone and \(0\in Tv\) if and only if \(v\in \operatorname{VI}(C,A)\).
For each \((v,w)\in G(T)\), we have
Hence
So, we obtain
On the other hand, from \(v\in C\) and
we get
and hence
Therefore from (3.10) and (3.11), we obtain
As \(i\rightarrow\infty\), we have
Since T is maximal monotone, we have \(0\in Tz\) and hence \(z\in \operatorname{VI}(C,A)\).
From Lemma 2.4, we get
Since \(x_{n}y_{n}\rightarrow0\), we also have
From Lemma 2.5, we obtain
□
Application
In the applications of this method, they are useful in nonlinear analysis and optimization problems in Hilbert space. This section is concerned with three weak convergence theorems for the equilibrium problem, the constrained convex minimization problem, and the split feasibility problem by Theorem 3.1.
Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let F be a bifunction of \(C\times C\) into \(\mathbb{R}\). The equilibrium problem [6] is to find \(x^{*}\) such that
The set of solutions of problem (4.1) is denoted by \(\operatorname{EP}(F)\).
Lemma 4.1
Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction of \(C\times C\) into \(\mathbb{R}\) satisfying the properties:

(A1)
\(F(x,x)\)=0 for all \(x\in C\);

(A2)
for each \(x\in C\), \(y\mapsto F(x,y)\) is convex and differentiable.
Then \(z\in \operatorname{EP}(F)\) if and only if \(z\in \operatorname{VI}(C,S)\), where \(Sx= \nabla F_{y}(x,y)_{y=x}\).
Proof
Let \(z\in \operatorname{EP}(F)\). For each \(y\in C\), \(z+\lambda(yz)= \lambda y+(1\lambda)z\in C\), \(\forall\lambda\in(0,1)\). Since for each \(x\in C\), \(y\mapsto F(x,y)\) is differentiable. Then
Conversely. If \(z\in \operatorname{VI}(C,S)\); i.e., \(\langle\nabla F_{y}(z,y)_{y=z}, yz\rangle\geq0\), \(\forall y\in C\). Since for each \(x\in C\), \(y\mapsto F(x,y)\) is convex. Then \(F(z,y)\geq F(z,z)=0\). □
Applying Theorem 3.1 and Lemma 4.1, we obtain the following result.
Theorem 4.2
Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction of \(C\times C\) into \(\mathbb{R}\) satisfying (A1) and (A2). Assume that S is monotone and kLipschitz continuous and \(\operatorname{EP}(F)\neq\emptyset\). Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be sequences generated by \(x_{0}=x\in C\) and
for every \(n=0,1,2,\ldots\) , where \(S(x)=\nabla F_{y}(x,y)_{y=x}\) and \(\{\lambda_{n}\}\subset[a,b]\) for some \(a,b\in(0,\frac{1}{k})\). Then the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) converge weakly to the same point \(z\in \operatorname{EP}(F)\), where \(z=\lim_{n\rightarrow\infty} P_{\operatorname{EP}(F)}x_{n}\).
Proof
Putting \(A=S\) in Theorem 3.1, we get the desired result by Lemma 4.1. □
Consider the following constrained convex minimization problem [7]: Find \(x^{*}\in C\) such that
where C is a nonempty closed convex subset of a real Hilbert space H and f is a realvalued convex function.
Lemma 4.3
Let H is a real Hilbert space and let C be a nonempty closed convex subset of H. Let f be a convex function of H into \(\mathbb{R}\). If f is differentiable, then z is a solution of (4.3) if and only if \(z\in \operatorname{VI}(C,\nabla f)\).
Proof
Let z be a solution of (4.3). For each \(x\in C\), \(z+\lambda(xz)\in C\), \(\forall\lambda\in(0,1)\). Since f is differentiable, we have
Conversely, if \(z\in \operatorname{VI}(C,S)\), \(\langle\nabla f(z), xz\rangle\geq0\), \(\forall x\in C\). Since f is convex, we have
Hence z is a solution of (4.3). □
Applying Theorem 3.1 and Lemma 4.3, we obtain the following result.
Theorem 4.4
Let H is a real Hilbert space and let C be a nonempty closed convex subset of H. Let f be a function of H into \(\mathbb{R}\). Assume f is differentiable and we assume that the set of solutions of (4.3) is nonempty and ∇f is kLipschitz continuous. Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be sequences generated by \(x_{0}=x\in C\) and
for every \(n=0,1,2,\ldots\) , where \(\{\lambda_{n}\}\subset[a,b]\) for some \(a,b\in(0,\frac{1}{k})\). Then the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) converge weakly to the same point z, where z is a solution of (4.3).
Proof
Since f is convex, we see that ∇f is monotone. Putting \(A=\nabla f\) in Theorem 3.1, we obtain the desired result by Lemma 4.3. □
Very recently, the split feasibility problem (SFP) [8–11] has been proposed. It is very important in nonlinear analysis and optimization problems. The SFP is to find a point \(x^{*}\) such that
where C and Q are nonempty closed convex subsets of real Hilbert spaces \(H_{1}\) and \(H_{2}\) and B is a bounded linear operator of \(H_{1}\) into \(H_{2}\).
Lemma 4.5
[8]
Let \(H_{1}\) and \(H_{2}\) be real Hilbert spaces. Let C and Q be nonempty closed convex subsets of \(H_{1}\) and \(H_{2}\). Let B be a bounded linear operator of \(H_{1}\) into \(H_{2}\). Assume that \(C\cap B^{1}Q\) is nonempty. Let \(\lambda \geq0\). Then the following propositions are equivalent:

(i)
\(z\in \operatorname{VI}(C,B^{*}(IP_{Q})B)\);

(ii)
\(z=P_{C}(I\lambda B^{*}(IP_{Q})B)z\);

(iii)
\(z\in C\cap B^{1}Q\),
where \(B^{*}\) is the adjoint operator of B.
Lemma 4.6
Let \(H_{1}\) and \(H_{2}\) be real Hilbert spaces. Let B be a bounded linear operator of \(H_{1}\) into \(H_{2}\) such that \(B\neq0\). Let Q be a nonempty closed convex subset of \(H_{2}\). Then \(B^{*}(IP_{Q})B\) is monotone and \(\B\^{2}\)Lipschitz continuous.
Proof
Let \(x,y\in H_{1}\),
Hence
So, we obtain
On the other hand, we have
Then \(B^{*}(IP_{Q})B\) is monotone and \(\B\^{2}\)Lipschitz continuous. □
Applying Theorem 3.1 and Lemma 4.5, we obtain the following result.
Theorem 4.7
Let \(H_{1}\) and \(H_{2}\) be real Hilbert spaces. Let C and Q be nonempty closed convex subsets of \(H_{1}\) and \(H_{2}\). Let \(B:H_{1}\rightarrow H_{2}\) be a bounded linear operator such that \(B\neq0\). Assume that \(C\cap B^{1}Q\) is nonempty. Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be sequences generated by \(x_{0}=x\in C\) and
for every \(n=0,1,2,\ldots\) , where \(\{\lambda_{n}\}\subset[a,b]\) for some \(a,b\in(0,\frac{1}{\B\^{2}})\). Then the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) converge weakly to the same point \(z\in C\cap B^{1}Q\), where \(z=\lim_{n\rightarrow\infty}P_{C\cap B^{1}Q}x_{n}\).
Proof
By Lemma 4.6, we get \(B^{*}(IP_{Q})B\) is monotone and \(\B\^{2}\)Lipschitz continuous. Putting \(A=B^{*}(IP_{Q})B\) and \(k=\B\^{2}\) in Theorem 3.1, we get the desired result by Lemma 4.5. □
Numerical result
In this section, we use our iterative method to solve some specific practical numerical calculation problems. By using the algorithm in Theorem 4.4 and Theorem 4.7, we illustrate its convergence in solving constrained convex minimization problem and linear system of equations.
The first example is the constrained convex minimization problem of a function of one variable, which uses the algorithm in Theorem 4.4.
Example 1
In Theorem 4.4, we suppose that \(H=\mathbb{R}\) and \(C=[0,2]\). Consider the constrained convex minimization problem (4.3) and let the function
Then the problem (4.3) can be written as
It is easy to find a point \(x^{*}=1\) solving the problem (5.2). We can know that ∇f is monotone and 12Lipschitz continuous. Take \(k=12\) and \(\lambda_{n}=\frac{1}{36(n+1)}+\frac{1}{36}\).
Then by Theorem 4.4, the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) are generated by
As \(n\rightarrow\infty\), we have \(x_{n}\rightarrow x^{*}=1\).
From Table 1, we can see that with the increase of the number of iterations, \(\{x_{n}\}\) approaches the solution \(x^{*}\) and the errors gradually approach zero.
The second example is a \(3\times3\) linear system of equations, which use the algorithm in Theorem 4.7.
Example 2
In Theorem 4.7, we suppose that \(H_{1}=H_{2}=\mathbb{R}^{3}\). Take
Let \(B=A\), \(C=\mathbb{R}^{3}\) and \(Q=\{b\}\). Then the SFP (4.5) is transformed into the problem of system of linear equations. That is to say, \(x_{*}\) is the solution of linear system of equations \(Ax=b\) and
Then by Theorem 4.7, the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) are generated by
As \(n\rightarrow\infty\), we have \(x_{n}\rightarrow x^{*}\).
From Table 2, we can also see that with the increase of iterative number, \(x_{n}\) approaches the exact solution \(x^{*}\) and the errors gradually approach zero.
Conclusion
The variational inequality problem is a very important field of study in mathematics. It is not only playing an important role in optimization problems and nonlinear analysis, but also widely used in many fields, such as economics, mechanics, signal processing, etc. So, more and more scientists devote their efforts to the study of variational inequalities. For a variational inequalities, we mainly study the algorithm and its convergence, existence and uniqueness of the solutions. In a real Hilbert space, The gradientprojection method for solving the variational inequality problem for an inversestrongly monotone mapping has been studied. But this method will not be used if the inversestrongly monotone is changed to monotone in the condition. So we propose a new iterative method to solve it. In this paper, we introduce an iterative method for finding an element of the set of solutions of a variational inequality problem with a monotone and Lipschitz continuous mapping in Hilbert space. In particular, under certain conditions, equilibrium problem, constrained convex minimization problem and split feasibility problem are, respectively, equivalent to a variational inequality problem. Then the new weak convergence theorem are obtained. The algorithm in Theorem 3.1 improves and extends Korpelevich’s method [1] in the following ways:

(i)
The finitedimensional Euclidean space \(\mathbb{R}^{n}\) is extended to the case of an infinitedimensional Hilbert space H.

(ii)
The fixed coefficient λ is extended to the case of a sequence \(\{\lambda_{n}\}\).
Recently, the variational inequality problem has been further developed. This will attract more scholars interested in the study of the variational inequality problem. Many scholars will devote their efforts to its study. Then the variational inequality problem can be better developed in the future.
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Acknowledgements
The authors thank the referees for their helping comments, which notably improved the presentation of this paper. This paper was supported by Fundamental Research Funds for the Central Universities (Grant: 3122016L006). Ming Tian was supported by the Foundation of Tianjin Key Laboratory for Advanced Signal Processing.
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Tian, M., Jiang, BN. Weak convergence theorem for variational inequality problems with monotone mapping in Hilbert space. J Inequal Appl 2016, 286 (2016). https://doi.org/10.1186/s1366001612373
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DOI: https://doi.org/10.1186/s1366001612373
MSC
 58E35
 47H09
 65J15
Keywords
 iterative method
 extragradient method
 weak convergence
 variational inequality
 monotone mapping
 equilibrium problem
 constrained convex minimization problem
 split feasibility problem