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Iterative methods for finding the minimumnorm solution of the standard monotone variational inequality problems with applications in Hilbert spaces
Journal of Inequalities and Applications volume 2015, Article number: 135 (2015)
Abstract
In this paper, we introduce two kinds of iterative methods for finding the minimumnorm solution to the standard monotone variational inequality problems in a real Hilbert space. We then prove that the proposed iterative methods converge strongly to the minimumnorm solution of the variational inequality. Finally, we apply our results to the constrained minimization problem and the split feasibility problem as well as the minimumnorm fixed point problem for pseudocontractive mappings.
Introduction
Let C be a nonempty, closed, and convex subset of a real Hilbert space H with the inner product\(\langle\cdot,\cdot\rangle\) and induced norm \(\\cdot\\). A mapping F is said to be monotone if
for all \(x,y\in C\).
The variational inequality problem (VIP) with respect to F and C is to find a point \(x^{*}\in C\) such that
Variational inequalities were initially investigated by Kinderlehrer and Stampacchia in [1], and have been widely studied by many authors ever since, due to the fact that they cover as diverse disciplines as partial differential equations, optimization, optimal control, mathematical programming, mechanics and finance (see [2–4]).
It is well known that if F is a kLipschitz continuous and ηstrongly monotone mapping, i.e., the following inequalities hold:
for all \(x,y\in C\), where k and η are fixed positive numbers, then (1.2) has a unique solution.
A mapping F is said to be hemicontinuous if for any sequence \(\{x_{n}\} \) converging to \(x_{0}\in H\) along a line implies \(T{x_{n}} \rightharpoonup T{x_{0}}\), i.e., \(T{x_{n}} = T({x_{0}} + {t_{n}}x)\rightharpoonup Tx_{0}\) as \(t_{n}\rightarrow0\) for all \(x\in H\).
Theorem 1.1
Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H. Let F be a monotone and hemicontinuous mapping of C into H. Then there exists \(x_{0}\in C\) such that
It is also well known that (1.2) is equivalent to the fixed point equation
where \(P_{C}\) stands for the metric projection from H onto C and μ is an arbitrarily positive number. Consequently, the wellknown iterative procedure, the projected gradient method (PGM), can be used to solve (1.2). PGM generates an iterative sequence by the recursion
When F is a kLipschitz continuous and ηstrongly monotone mapping, as \(\mu \in(0,\frac{{2\eta}}{{k^{2}}})\), the sequence \(\{ x_{n}\}\) generated by (1.4) converges strongly to a unique solution of (1.2).
However, if F fails to be Lipschitz continuous or strongly monotone, then the result above is false in general. We will assume that F is a hemicontinuous and general monotone mapping. Thus, VIP (1.2) is illposed and regularization is needed; moreover, a solution is often sought through iteration methods.
In 1976, Korpelevich [5] introduced the following socalled extragradient method:
for all \(n\geq0\), where \(\lambda\in(0,\frac{1}{k})\), C is a nonempty, closed, and convex subset of \(R^{n}\) and F is a monotone kLipschitz mapping of C into \(R^{n}\). He proved that if \(\operatorname{VI}(C, F)\) is nonempty, then the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\), generated by (EM), converge weakly to the same point \(p\in \operatorname{VI}(C, F)\), which is a solution of (1.2).
Recently Chen et al. [6] introduced the following iterative method:
where \(\gamma\in(0,\frac{{2\eta}}{{{k^{2}}}})\) is fixed, T is a nonexpansive mapping and \(If\) is a Lipschitz \((1\rho)\)strongly monotone mapping. Then the iterative sequence \(x_{n}\) converges strongly to the unique solution \(x^{*}\) of (VI) below:
Very recently Yao et al. [7] constructed the minimumnorm fixed points of pseudocontractions in Hilbert spaces by the following iterative algorithm:
where T is a LLipschitzian and pseudocontractive with \(\operatorname {Fix}(T)\neq\emptyset\).
Questions

1.
Can one modify extragradient method for general monotone operator variational inequality so that strong convergence of the modified algorithm is desirable?

2.
If F is a hemicontinuous and strongly monotone mapping, the solution of VIP (1.2) is unique or not?
The purpose of this paper is to solve the questions above. We introduce implicit and explicit iterative methods for construction of the solution of the monotone variational inequality problem and prove that our algorithms converge strongly to the minimumnorm solution of variational inequality problem (1.2). Finally, we apply our results to the constrained minimization problem and the split feasibility problem as well as the minimumnorm fixed point problem for pseudocontractive mappings.
Preliminaries
For our main results, we shall make use of the following lemmas.
Lemma 2.1
(see [8])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let \(A:C\rightarrow H\) be a hemicontinuous monotone operator. Then, for a fixed element \(x^{*}\in C\), the following variational inequalities are equivalent:

(i)
\(\langle Ax,x  {x^{*}}\rangle \ge0\), \(\forall x \in C\);

(ii)
\(\langle A{x^{*}},x  {x^{*}}\rangle \ge0\), \(\forall x \in C\).
Lemma 2.2
(see [9])
Let X be a reflexive Banach space and K is a unbounded closed convex subset of X with \(\theta\in K\). Let \(A:K\rightarrow X^{*}\) be a hemicontinuous monotone coercively operator, i.e., \(\forall u\in K\),
Then \(\forall w^{*}\in X^{*}\), there exists a \(u_{0}\in K\) such that
In Lemma 2.2, \(\theta\in K\) is needed. Indeed, if \(A:K \to{X^{*}}\) is a hemicontinuous ηstrongly monotone operator, then the restriction that \(\theta\in K\) can be omitted. To prove this, we give the following lemma.
Lemma 2.3
Let K be a unbounded, closed, and convex subset of reflexive Banach space X. Let \(A:K\to H\) be a hemicontinuous ηstrongly monotone operator. Then \(\forall w^{*}\in X^{*}\), there exists a \(u_{0}^{*}\in K\) such that the VI (2.1) holds.
Proof
Let \(\tilde{K} = K  {x_{0}}\), where \(x_{0}\) is a fixed element of K. Define \(\tilde{A}x = A(x + {x_{0}})\). Then we see that \(\tilde{A}\) is hemicontinuous ηstrongly monotone. For any \(x, y\in\tilde{K}\) we have
Since \(\langle\tilde{A}x  A{x_{0}},x\rangle \ge\eta\ x\^{2}\), we have
Then we get
By Lemma 2.2, \(\forall{w^{*}} \in{X^{*}}\), there exists a \({u_{0}} \in \tilde{K}\) such that
Putting \({u_{0}}^{*} = {u_{0}} + {x_{0}}\), then we have
Therefore, \({u_{0}}^{*}\) is a solution of VIP (2.1). □
Lemma 2.4
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let \(A:C\to H\) be a hemicontinuous ηstrongly monotone operator. Then variational inequality
has a unique solution.
Proof
Let \(\operatorname{VI}(C,A)\) be the solution set of VI (2.2). From Lemma 2.3, we know that \(\operatorname{VI}(C,A)\) is nonempty. Next, we show that \(\operatorname{VI}(C,A)\) has a unique element. Assume that \(x^{*}, y^{*}\in \operatorname{VI}(C,A)\). Then we have
and
Combining (2.3) and (2.4), we get
Since A is ηstrongly monotone, from (2.5) it follows that
Therefore, \(x^{*}=y^{*}\). This completes the proof. □
Lemma 2.5
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let \(A:C\to H\) be a hemicontinuous monotone operator and \({\gamma_{n}} > 0\) be a sequence of real numbers. Then \({\gamma_{n}}I + A\) are \({\gamma_{n}}\)strongly monotone.
Proof
\(\forall x, y\in C\), we have
So, \({\gamma_{n}}I + A\) are \({\gamma_{n}}\)strongly monotone. □
Lemma 2.6
(see [10])
Let \(\{\alpha_{n}\}\) be a sequence of nonnegative real numbers satisfying
where \(\{ {\gamma_{n}}\} \subset(0,1)\) and \(\{ {\sigma_{n}}\} \) satisfy

(i)
\(\sum_{n = 0}^{\infty}{\gamma_{n}} = \infty\);

(ii)
either \(\lim{\sup_{n \to\infty}}{\sigma_{n}} \le0\) or \(\sum_{n = 0}^{\infty}{\gamma_{n}}{\sigma_{n}} < \infty\).
Then \(\lim_{n\to\infty}\alpha_{n}=0\).
Lemma 2.7
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let \(T:C\to H\) be a mapping and write \(A:=IT\). Then \(\operatorname{VI}(C,A)=\operatorname{Fix}(P_{C} T)\). In particular, if \(T:C\to C\) is a selfmapping, then \(\operatorname{VI}(C,A)=\operatorname{Fix}(T)\).
Proof
Indeed,
If \(T:C\to C\) is a selfmapping, then we have
This completes the proof. □
Now we are in a proposition to state and prove the main results in this paper.
Main results
In this section we will introduce two iterative methods (one implicit and the other explicit). First, we introduce the implicit one. In what follows, we assume that \(A:C\to H\) is hemicontinuous and monotone.
For given \(\gamma_{n}>0\), we consider the sequences of operators \(\{A_{n}\} \) which are defined by
for all \(n\geq1\).
From Lemma 2.5, we know that \(A_{n}:C\to H\) are hemicontinuous and \(\gamma _{n}\)strongly monotone for all \(n\geq1\). It follows from Lemma 2.4 that the variational inequality
has a unique solution \(y_{n}\in C\) for every fixed \(n\geq1\).
Substitute (3.1) into (3.2) to obtain
Take \(\gamma_{n}=\frac{\alpha_{n}}{\beta_{n}}\). Then (3.3) yields
and hence
It turns out that
By virtue of the property of \(P_{C}\), we conclude
Theorem 3.1
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let A be a hemicontinuous monotone operator. Let \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) be two sequences in \([0,1]\) that satisfy the following condition:
Assume that \(\operatorname{VI}(C,A)\neq\emptyset\). Then the sequence \(\{y_{n}\}\) generated by (3.7) converges in norm to \({x^{*}} = {P_{\operatorname{VI}(C,A)}}\theta\) which is the minimumnorm solution of VIP (2.2).
Proof
Put \({z_{n}} = (1  {\alpha_{n}}){y_{n}}  {\beta_{n}}A{y_{n}}\). \(\forall p \in \operatorname{VI}(C,A)\), we have
By using (3.7) and (3.8), we get
It follows from the property of \(P_{C}\) that
which simplifies to
and then
Setting \({\gamma_{n}} = \frac{\alpha_{n}}{\beta_{n}}\), then we have
Since A is a monotone operator and \(p \in \operatorname{VI}(C,A)\), we know
and
Substitute (3.13) and (3.14) into (3.12) to obtain
Then we have
from which it turns out that
Therefore, \(\{y_{n}\}\) is bounded. Then we know that \(\{y_{n}\}\) has a subsequence \(\{y_{n_{j}}\}\) such that \(y_{n_{j}}\rightharpoonup x^{*}\) as \(j\to\infty\).
Furthermore, without loss of generality, we may assume that \(\{y_{n}\}\) converges weakly to a point \({x^{*}} \in C\).
We show that \({x^{*}}\) is a solution to VIP (2.2). For any \(x\in C\), by Lemma 2.5 we have
Combining (3.17) and (3.3), we get
Taking the limit as \(n\to\infty\) in (3.18) yields
By Lemma 2.1, we get
that is, \({x^{*}} \in \operatorname{VI}(C,A)\).
Therefore, we can substitute p by \(x^{*}\) in (3.16) to obtain
Since \({y_{n}} \rightharpoonup{x^{*}}\) as \(n\to\infty\), by (3.19) we get \({y_{n}} \to{x^{*}}\) as \(n\to\infty\).
Moreover, from (3.15) we get
By virtue of the property of the projection, we claim
So, the sequence \(\{y_{n}\}\) generated by (3.7) converges in norm to \(x^{*}=P_{\operatorname{VI}(C,A)}\theta\) as \(n\to\infty\).
Furthermore, it follows from (3.20) that
from which we know that \(x^{*}\) is the minimumnorm solution of VIP (2.2). This completes the proof. □
Now, we introduce an explicit method and establish its strongly convergence analysis.
From the implicit method, it is natural to consider the following iteration method that generates a sequence \(\{x_{n}\}\) according to the recursion
where the initial guess \(x_{1}\in C\) is selected arbitrarily and \(\{\alpha _{n}\}\) and \(\{\beta_{n}\}\) are two sequences of positive numbers in \((0, 1)\).
Theorem 3.2
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let A be a hemicontinuous monotone operator. Let \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) be two sequences in \([0,1]\) that satisfy the following conditions:

(i)
\(\frac{{{\alpha_{n}}}}{{{\beta_{n}}}} \to0\), \(\frac{{\beta _{n}^{2}}}{{{\alpha_{n}}}} \to0\) as \(n \to\infty\);

(ii)
\({\alpha_{n}} \to0\) as \(n\to\infty\), \(\sum_{n = 1}^{\infty}{{\alpha_{n}}} = \infty\);

(iii)
\(\frac{{\alpha_{n}}  {\alpha_{n  1}} + {\beta_{n}}  {\beta _{n  1}}}{\alpha_{n}^{2}} \to0\) as \(n \to\infty\).
Assume that both \(\{Ax_{n}\}\) and \(\{Ay_{n}\}\) are bounded and that \(\operatorname{VI}(C,A)\neq\emptyset\). Then the iterative sequence \(\{x_{n}\}\) generated by (3.23) converges in norm to \({x^{*}} = {P_{\operatorname{VI}(C,A)}}\theta\), which is the minimumnorm solution to VIP (2.2).
Proof
By using Theorem 3.1, we know that \(\{y_{n}\}\) converges in norm to \({x^{*}} = {P_{\operatorname{VI}(C,A)}}\theta\).
For any \(p\in \operatorname{VI}(C,A)\), from the property of \(P_{C}\) we know
By Lemma 2.1 we know
Substitute (3.25) into (3.24) to get
Since \(\{Ax_{n}\}\) is bounded, by condition (i), we see that there exists some positive constant \(M_{0}=\max\{\x_{1}p\, \p\^{2}+\beta_{n}\p\\ Ax_{n}\+\frac{\beta_{n}^{2}}{\alpha_{n}}\Ax_{n}\^{2}, n\geq1\}\) such that
for all \(n\ge1\), which implies that \(\{x_{n}\}\) is bounded.
By using (3.7) and (3.23), we get
Since both \(\{y_{n}\}\) and \(\{Ay_{n}\}\) are bounded, we get
Write \(M_{1}= \max\{ \y_{n  1}\, \A{y_{n  1}}\ \}\), \(n\geq1\). Then we have
From conditions (i) and (iii) we know that \(\frac{{\vert {{\alpha_{n}}  {\alpha_{n  1}}} \vert + \vert {{\beta_{n}}  {\beta_{n  1}}} \vert }}{{{\alpha_{n}}}} = o({\alpha_{n}})\) and \(\beta_{n}^{2}=o(\alpha_{n})\).
Putting \(M_{2}=\max\{M_{1}, 2\x_{n}y_{n1}\, \Ax_{n}Ay_{n}\, \ y_{n}y_{n1}\}\), then (3.27) turns out to be
By Lemma 2.6 and condition (ii), we have \(\Vert {{x_{n + 1}}  {y_{n}}} \Vert \to0\), as \(n \to\infty\). It follows that \(\{ {x_{n}}\} \) converges strongly to \({x^{*}} = {P_{\operatorname{VI}(C,A)}}\theta\). This completes the proof. □
If \(A:C\to H\) is a kLipschitz continuous and monotone, we have the following convergence result.
Theorem 3.3
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let A be a kLipschitz continuous and monotone operator. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) be two sequences in \([0,1]\) that satisfy the following conditions:

(i)
\(\frac{{{\alpha_{n}}}}{{{\beta_{n}}}} \to0\), \(\frac{{\beta _{n}^{2}}}{{{\alpha_{n}}}} \to0\) as \(n \to\infty\);

(ii)
\({\alpha_{n}} \to0\) as \(n\to\infty\), \(\sum_{n = 1}^{\infty}{{\alpha_{n}}} = \infty\);

(iii)
\(\frac{{{\alpha_{n}}  {\alpha_{n  1}} + {\beta_{n}}  {\beta _{n  1}}}}{{\alpha_{n}^{2}}} \to0\) as \(n \to\infty\).
Assume that \(\operatorname{VI}(C,A)\neq\emptyset\). Then the iterative sequence \(\{ x_{n}\}\) generated by (3.23) converges in norm to \({x^{*}} = {P_{\operatorname{VI}(C,A)}}\theta\), which is the minimumnorm solution of VIP (2.2).
Proof
From Theorem 3.1, we know that \(\{y_{n}\}\) converges in norm to \({x^{*}} = {P_{\operatorname{VI}(C,A)}}\theta\). Therefore, it is sufficient to show that \(x_{n+1}y_{n}\to0\) as \(n\to\infty\).
In view of condition (i), without loss of generality, we may assume that
for all \(n\ge1\). By using (3.7), (3.23), and (3.30), we get
From (3.29), (3.31), and condition (iii), we obtain
By condition (ii) and Lemma 2.6, we deduce that \(x_{n+1}y_{n}\to0\) as \(n\to\infty\). This completes the proof. □
Remark 3.1
Comparing our algorithm (3.23) with (EM), we find that algorithm (3.23) enjoys the following merits:
 (1)

(2)
The recursion (3.23) has the strong convergence property; while (EM) has only the weak convergence property in general.

(3)
The choice of the iterative parametric sequences \(\{\alpha_{n}\} \) and \(\{\beta_{n}\}\) in (3.23) does not depend on the Lipschitz constant of A, thus, (3.23) is also efficient even in the case where the Lipschitz constant of A is unknown.
Remark 3.2
Choose the sequences \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) such that
where \(a<\frac{b+1}{2}\), \(0< b<a\) and \(a<2b\) or \(b>\frac{1}{2}\). Then it is clear that conditions (i)(iii) of Theorems 3.2 and 3.3 are satisfied.
Applications
In this section, we give some applications of our results.
Problem 4.1
Let B be a bounded linear operator on a real Hilbert space H and \(b\in H\) be a fixed vector. Find the least square solutions with the minimum norm for the following class of operator equation:
It is well known that the above problem is equivalent to the following minimization problem:
We denote by \({S_{B}}\) the solution set of Problem 4.1. We consider the functional \(f(x) = \frac{1}{2}{\Vert {Bx  b} \Vert ^{2}}\). Then \(\nabla f(x) = {B^{*}}(Bx  b)\). It is easy to verify that \({S_{B}} = \operatorname{VI}(C, \nabla f)\) and \(x^{*}\) solves Problem 4.1 if and only if \({x^{*}} = {P_{{S_{B}}}}\theta\). Let \(\{ {x_{n}}\} \) be generated by the following recursion:
where \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are two sequences in \([0,1]\) that satisfy conditions (i)(iii) in Theorem 3.3.
By virtue of Theorem 3.3, we can deduce the following convergence result.
Theorem 4.1
Assume that \(S_{B}\ne\emptyset\) and \(\{x_{n}\}\) is generated by (4.3), then \(\{x_{n}\}\) converges in norm to \(x^{*}\).
Proof
Notice that
and
we see that ∇f is \({\Vert B \Vert ^{2}}\)Lipschitz continuous and monotone. By Theorem 3.3 we conclude that \(\{ {x_{n}}\} \) converges in norm to \({x^{*}} = {P_{\operatorname{VI}(C,\nabla f)}}\theta\). This completes the proof. □
Next, we turn to consider the split feasibility problem (SFP).
Problem 4.2
Let C and Q be nonempty, closed, and convex subsets in Hilbert spaces \(H_{1}\) and \(H_{2}\), respectively. The SFP is formulated as finding a point \(x\in C\) with the property:
where \(B: C\subset H_{1}\to H_{2} \) is a bounded linear operator.
We denote by Γ the solution set of Problem 4.2. Consider the functional
It is well known that if Problem 4.2 is consistent, i.e., \(\Gamma\ne\emptyset\), then Problem 4.2 is equivalent to the following minimization problem:
We know that \(x^{*}\) is a solution of the minimization problem (4.5) if and only if \(x^{*}\) is a solution of the following variational inequality:
Therefore, we have \(\Gamma=\operatorname{VI}(C,\nabla g)\) provided that \(\Gamma\ne \emptyset\). Let \(\{ {x_{n}}\} \) be generated by the following recursion:
where \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are two sequences in \([0,1]\) that satisfy conditions (i)(iii) in Theorem 3.3. By using Theorem 3.3, we have the following convergence result.
Theorem 4.2
Assume \(\Gamma\ne\emptyset\) and \(\{x_{n}\}\) is generated by (4.7), then \(\{ {x_{n}}\} \) converges in norm to \({x^{*}} = {P_{\operatorname{VI}(C,\nabla g)}}\theta\).
Proof
Note that \(\nabla g(x)=B^{*}(IP_{Q})Bx\). It is clear that ∇g is \({\Vert B \Vert ^{2}}\)Lipschitz continuous and monotone, by Theorem 3.3 we conclude that \(\{ {x_{n}}\} \) converges in norm to \({x^{*}} = {P_{\operatorname{VI}(C,\nabla g)}}\theta=P_{\Gamma}\theta\). This completes the proof. □
Finally, we apply our results to the minimumnorm fixed point problem for pseudocontractive mappings.
Theorem 4.3
Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H. Let \(T:C\to C\) be a hemicontinuous pseudocontractive mapping with \(\operatorname{Fix}(T)\neq\emptyset\). Assume that \(\{ \alpha_{n}\}\) and \(\{\beta_{n}\}\) are two sequences in \([0,1]\) that satisfy the following conditions:

(i)
\(\frac{{{\alpha_{n}}}}{{{\beta_{n}}}} \to0\), \(\frac{{\beta _{n}^{2}}}{{{\alpha_{n}}}} \to0\) as \(n \to\infty\);

(ii)
\({\alpha_{n}} \to0\) as \(n\to\infty\), \(\sum_{n = 1}^{\infty}{{\alpha_{n}}} = \infty\);

(iii)
\(\frac{{{\alpha_{n}}  {\alpha_{n  1}} + {\beta_{n}}  {\beta _{n  1}}}}{{\alpha_{n}^{2}}} \to0\) as \(n \to\infty\).
Then the sequence \(\{x_{n}\}\) generated by
converges in norm to \({x^{*}} = {P_{\operatorname{Fix}(T)}}\theta\).
Proof
Put \(A=IT\). Since \(T:C\to C\) is a hemicontinuous pseudocontractive mapping, then A is a hemicontinuous monotone operator. It follows from Theorem 1.1 that \(\operatorname{VI}(C,A)\ne\emptyset \). From the boundedness of C, we know that \(\{Ax_{n}\}\) and \(\{Ay_{n}\}\) are bounded. By Theorem 3.2, the iterative sequence \(\{x_{n}\}\) converges strongly to \({x^{*}} = {P_{\operatorname{VI}(C,A)}}\theta\). By Lemma 2.7 and noting that T is a selfmapping, we know that \(\operatorname{VI}(C,A)=\operatorname{Fix}(T)\). This completes the proof. □
Theorem 4.4
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let \(T:C\to C\) be a kLipschitz continuous pseudocontractive mapping with \(\operatorname{Fix}(T)\neq\emptyset\). Assume that \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are two sequences in \([0,1]\) that satisfy the following conditions:

(i)
\(\frac{{{\alpha_{n}}}}{{{\beta_{n}}}} \to0\), \(\frac{{\beta _{n}^{2}}}{{{\alpha_{n}}}} \to0\) as \(n \to\infty\);

(ii)
\({\alpha_{n}} \to0\) as \(n\to\infty\), \(\sum_{n = 1}^{\infty}{{\alpha_{n}}} = \infty\);

(iii)
\(\frac{{{\alpha_{n}}  {\alpha_{n  1}} + {\beta_{n}}  {\beta _{n  1}}}}{{\alpha_{n}^{2}}} \to0\) as \(n \to\infty\).
Then the sequence \(\{x_{n}\}\) generated by
converges in norm to \({x^{*}} = {P_{\operatorname{Fix}(T)}}\theta\).
Proof
Put \(A=IT\). Since \(T:C\to C\) is a kLipschitz continuous pseudocontractive mapping, A is a \((k+1)\)Lipschitz continuous monotone operator. By Lemma 2.7 and our assumption, we see that \(\operatorname{VI}(C,A)=\operatorname{Fix}(T)\ne\emptyset\). By Theorem 3.3, the iterative sequence \(\{x_{n}\}\) generated by (4.9) converges strongly to \({x^{*}} = {P_{\operatorname{VI}(C,A)}}\theta=P_{\operatorname{Fix}(T)}\theta\). This completes the proof. □
Remark 4.1
Theorem 4.2 improves some related results of [10] and [11] in the sense that the iterative parametric sequences do not depend on the norm of operator A. Theorem 4.3 seems to be a new result. Theorem 4.4 is similar to Theorem 3.2 of [7] with a different condition (iii) and different arguments.
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Acknowledgements
This research was supported by the National Natural Science Foundation of China (11071053).
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Zhou, Y., Zhou, H. & Wang, P. Iterative methods for finding the minimumnorm solution of the standard monotone variational inequality problems with applications in Hilbert spaces. J Inequal Appl 2015, 135 (2015). https://doi.org/10.1186/s1366001506597
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MSC
 41A65
 47H17
 47J20
Keywords
 standard monotone variational inequality problem
 minimumnorm solution
 iterative method
 strong convergence
 Hilbert space