1 Introduction
The present paper is devoted to presenting a more general regularization method for a class of monotone variational inequality problems (MVIPs) with a monotone and hemicontinuous operator F over a nonempty, closed, and convex subset C of a real Hilbert space H. The so-called MVIP is to find a point \(x^{*}\in C\) such that
The set of solutions for MVIP (1.1) is denoted by \(VI(C,F)\).
$$ \bigl\langle {F{x^{*}},x - {x^{*}}} \bigr\rangle \geq0, \quad \mbox{for all }x\in C. $$
(1.1)
The monotone 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 [1‐23]).
Anzeige
Over the past five decades years or so, the researchers designed various iterative algorithms for solving MVIP (1.1); see [1‐23] and the references therein. An early and typical iterative algorithm for solving MVIP (1.1) seems to be the projected gradient method (PGM), see, for instance, [4, 8], which generates a sequence \(\{x_{n}\}\) by the recursive procedure
where \(P_{C}\) stands for the metric projection from H onto C, I is the identity mapping on H, and μ is an arbitrarily positive number.
$$ x_{1}\in C \quad\mbox{and}\quad {x_{n + 1}} = {P_{C}} \bigl[(I - \mu F){x_{n}} \bigr],\quad n\ge1, $$
(1.2)
It is well known that if F is k-Lipschitz continuous and η-strongly monotone, then MVIP (1.1) has a unique solution. Moreover, if one chooses \(\mu\in (0,\frac{2\eta}{k^{2}})\), then the sequence \(\{x_{n}\}\) generated by PGM (1.2) converges in norm to the unique solution to MVIP (1.1); see, for instance, [4, 8].
However, if F is simply k-Lipschitz continuous and monotone, but not η-strongly monotone, then MVIP (1.1) may fail to have a solution, which can be seen from the following counterexample.
Example 1.1
Let \(H=C=\mathbb{R}=(-\infty,\infty)\) and consider the function \(F:C\to H\) defined by \(F(x)=\arctan(x)-\frac{\pi }{2}\), \(x\in C\). Then F is 1-Lipschitz continuous and monotone. It is clear that the equation \(F(x)=0\) has no solutions in H, and hence MVIP (1.1) has no solutions.
Anzeige
For a k-Lipschitz continuous and monotone operator F, even though MVIP (1.1) has a solution, the PGM (1.2) associated with F does not yet necessarily converge to the solution of the MVIP (1.1).
Example 1.2
Let \(H=\mathbb{R}^{2}\), \(C=B=\{x\in H:\|x\|\le1\}\), \(B_{1}=\{x\in B:\|x\|\le\frac{1}{2}\}\), and \(B_{2}=\{x\in B:\frac {1}{2}\le\|x\|\le1\}\). If \(x=(a,b)\in H\), we write \(x^{\bot}=(b,-a)\in H\). Define \(F:C\to H\) by
Then, from Chidume and Mutangadura [17], we know that \(F:C\to H\) is a 6-Lipschitz continuous and monotone operator with the unique zero point \(\theta=(0,0)\in C\), but PGM (1.2) does not converge to θ for any \(\mu\in(0,1)\).
$$Fx= \left \{ \begin{array}{@{}l@{\quad}l} -x^{\bot}, & x \in B_{1},\\ 2x-\frac{x}{\|x\|}-x^{\bot},& x\in B_{2}. \end{array} \right . $$
The Example 1.2 tells us that PGM (1.2) is invalid for a k-Lipschitz continuous and monotone operator F, and therefore further modifications to PGM (1.2) are needed. In this regard, we pick up the following several known results.
In 1976, Korpelevich [18] made a kind of modification to PGM (1.2) by introducing the following extragradient method (EM):
for all \(n\geq1\), where \(\lambda\in(0,\frac{1}{k})\), C is a nonempty, closed, and convex subset of \(R^{N}\) and F is a k-Lipschitz continuous and monotone operator of C into \(R^{N}\), where N is a positive integer. He proved that if \(VI(C, F)\) is nonempty, then the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\), generated by EM (1.3), converge to the same point \(p\in VI(C, F)\) which is a solution to MVIP (1.1).
$$ \left \{ \begin{array}{@{}l} {x_{1}} \in C,\\ {y_{n}} = {P_{C}}[{x_{n}} - {\lambda}F{x_{n}}],\\ {x_{n + 1}} = {P_{C}}[{x_{n}} - {\lambda}F{y_{n}}], \end{array} \right . $$
(1.3)
EM (1.3) contains two metric projections and it is indeed a composite of two classical projected gradient methods. We also remark in passing that in an infinite-dimensional Hilbert space, Korpelevich’s EM (1.3) has only weak convergence, in general, moreover, it cannot be used to seek the minimum-norm solution of MVIP (1.1).
Recently, Xu and Xu [22] provided a general regularization method for solving MVIP (1.1), where the regularizer is a Lipschitz continuous and strongly monotone operator. They also introduced an iterative method as discretization of the regularization method. They proved that both regularization and iterative methods converge in norm to a solution to MVIP (1.1) under some conditions.
Very recently, Iemoto et al. [23] studied a variational inequality for a hemicontinuous and monotone operator over the fixed point set of a strongly nonexpansive mapping in a real Hilbert space. They proposed an iterative algorithm and analyzed the weak convergence of the proposed algorithm.
On the other hand, the construction of fixed points for pseudocontractive mappings has been studied extensively by several authors since 1974. A good number of results was reported recently; see, for instance, [24‐30].
Question
Can the Lipschitz continuity assumptions be removed or weakened in the results mentioned above?
The purpose of this paper is to answer the question mentioned above. In order to realize this objective, we first establish a new existence and uniqueness theorem for MVIP (1.1), where \(F:C\to H\) is a hemicontinuous and strongly monotone operator. By using the established existence and uniqueness theorem, we then introduce an implicit method for seeking a solution of MVIP (1.1). We also introduce an explicit iterative method. We prove that both the implicit and the explicit iterative methods converge in norm to the same solution of MVIP (1.1). Some applications are also included.
The rest of the paper is organized as follows. Section 2 contains some necessary concepts and useful facts. The new existence and uniqueness theorem for MVIP (1.1) and several convergence results of the proposed algorithms are established in Section 3. Finally, in Section 4, we provide some applications to the constrained minimization problem for a convex and continuously Fréchet differentiable functional and the minimum-norm fixed point problem for a generalized Lipschitz continuous and pseudocontractive mapping.
2 Preliminaries
Throughout this paper, we will assume that H is a real Hilbert space with inner product \(\langle\cdot,\cdot\rangle\) and its induced norm \(\| \cdot\|\). Let C be a nonempty, closed, and convex subset of H. We denote the strong convergence and weak convergence of \(\{x_{n}\}\) to \(x\in H\) by \(x_{n}\to x\) and \(x_{n}\rightharpoonup x\), respectively. We use ℝ to denote the set of real numbers. Let \(T:C\to H\) be a mapping. We use \(\operatorname{Fix}(T)\) to denote the set of fixed points of T. We also denote by \(\mathfrak{D}(T)\) and \(\mathfrak{R}(T)\) the domain and range of T, respectively. The letter I stands for the identity mapping on H.
In what follows, we shall collect some important concepts, facts, and tools, which will be used in Section 3.
It is well known that the following equalities hold:
for all \(x,y\in H\).
$$ \|x\pm y\|^{2}=\|x\|^{2}\pm2\langle x,y\rangle+\|y \|^{2} $$
(2.1)
Recall that the metric projection from a real Hilbert space H onto a nonempty, closed, and convex subset C of H is defined as follows: for each \(x\in H\), there exists a unique element \(P_{C}x\in C\) with the property
that is, for any \(x\in H\), \(\bar{x}=P_{C}x\) if and only if
$$ \|x-P_{C}x\|\le\|x-y\|,\quad\forall y\in C, $$
(2.2)
$$ \bar{x}\in C \quad\mbox{and}\quad \|x-\bar{x}\|=\inf\bigl\{ \|x-y\|:y\in C\bigr\} . $$
(2.3)
Lemma 2.1
(see [11])
Let
\(P_{C}\)
be the metric projection from
H
onto a nonempty, bounded, and closed and convex subset
C
of a real Hilbert space
H. Then the following conclusions hold true:
(C1)
Given
\(x\in H\)
and
\(z\in C\).Then
\(z=P_{C}x\)
if and only if the inequality
holds.
$$ \langle x-z,y-z\rangle\le0,\quad \forall y\in C $$
(2.4)
(C2)
in particular, one has
$$ \|P_{C}x-P_{C}y\|^{2}\le\langle P_{C}x-P_{C}y,x-y\rangle,\quad \forall x,y\in H, $$
(2.5)
$$ \|P_{C}x-P_{C}y\|\le\|x-y\|,\quad \forall x,y\in H. $$
(2.6)
(C3)
in particular, one has
$$ \bigl\| (I-P_{C})x-(I-P_{C})y \bigr\| ^{2}\le \bigl\langle (I-P_{C})x-(I-P_{C})y, x-y \bigr\rangle , \quad \forall x,y \in H, $$
(2.7)
$$ \bigl\| (I-P_{C})x-(I-P_{C})y \bigr\| \le\|x-y\|,\quad \forall x,y\in H. $$
(2.8)
Recall that an operator \(F:C\to H\) is said to be
(i) monotone if
for all \(x,y\in C\);
$$ \langle{Fx - Fy,x - y} \rangle\geq0 $$
(2.9)
(ii) η-strongly monotone if there exists an \(\eta>0\) such that
for all \(x,y\in C\);
$$ \langle{Fx - Fy,x - y} \rangle\ge\eta{ \Vert {x - y} \Vert ^{2}} $$
(2.10)
(iii) k-Lipschitz continuous if there exists a positive number k such that
for all \(x,y\in C\);
$$ \Vert {Fx - Fy} \Vert \le k\Vert {x - y} \Vert $$
(2.11)
(iv) generalized Lipschitz continuous if there exists a positive number c such that
for all \(x,y\in C\);
$$ \Vert {Fx - Fy} \Vert \le c \bigl(1+\Vert {x - y} \Vert \bigr) $$
(2.12)
(v) demicontinuous if \(Fx_{n}\rightharpoonup Fx\) as \(n\to\infty\), whenever \(x_{n}\to x\) for any \(\{x_{n}\}\subset C\) and \(x\in C\);
(vi) hemicontinuous if for any \(x,y\in C\) and \(z\in H\), the function
of \([0,1]\) into ℝ is continuous;
$$t\rightarrowtail \bigl\langle z,F \bigl(tx+(1-t)y \bigr) \bigr\rangle $$
(vii) a mapping \(T:C\to H\) is said to be pseudocontractive if
for all \(x,y\in C\);
$$ \langle Tx-Ty,x-y\rangle\le\|x-y\|^{2} $$
(2.13)
(viii) an operator \(A\subset H\times H\) is called maximal monotone if it is monotone and it is not properly contained in any other monotone. We denote by \(\mathfrak{G}(A)\) the graph of A.
It is well known that T is pseudocontractive if and only if \(F=I-T\) is monotone.
Example 2.1
Let \(f:H\to\mathbb{R}\) be a convex and continuously Fréchet differentiable function. Then the gradient ∇f of f is maximal monotone and hemicontinuous.
It is clear that the following implication relation holds:
\(F:C\to H\) is k-Lipschitz continuous ⇒ F is continuous ⇒ F is demicontinuous ⇒ F is hemicontinuous, however, the converse relation does not hold true, which can be seen from the following counterexample.
Example 2.2
Consider a function \(\varphi:\mathbb{R}^{2}\to \mathbb{R}\) of two variables \(\varphi(x,y)=xy^{2}(x^{2}+y^{4})^{-1}\) for \((x,y)\in\mathbb{R}^{2}\backslash\{(0,0)\}\) and \(\varphi(0,0)=0\). Then φ is hemicontinuous but demicontinuous.
If \(F:\mathfrak{D}(F)=H\to H\) is monotone, then F is demicontinuous if and only if it is hemicontinuous; see, for instance, [9].
We remark that if an operator \(F:C\to H\) is either k-Lipschitz continuous or has a bounded range \(\mathfrak{R}(F)\), then it is generalized Lipschitz continuous. However, a generalized Lipschitz continuous mapping may be not continuous.
Example 2.3
Consider the sign function \(F:\mathbb{R}\to \mathbb{R}\) defined by
Then F is generalized Lipschitz continuous but continuous.
$$Fx= \left \{ \begin{array}{@{}l@{\quad}l} 1,& x>0,\\ 0,& x=0,\\ -1,& x< 0. \end{array} \right . $$
Lemma 2.2
(see [7])
Let
\(A\subset H\times H\)
be a maximal monotone operator and let
\((x,y)\in H\times H\). If
for all
\((a,b)\in\mathfrak{G}(A)\), then
\((x,y)\in\mathfrak{G}(A)\).
$$ \langle a-x,b-y\rangle\ge0 $$
(2.14)
Lemma 2.3
(see [7])
Let
\(A\subset H\times H\)
be a monotone operator. Then
A
is maximal monotone if and only if
for all
\(r>0\).
$$ R(I+rA)=H $$
(2.15)
Let
C
be a nonempty, closed, and convex subset of
H
and
\(x\in C\). Then we define set
\(N_{C}(x)\)
of
H
by
Such a set
\(N_{C}(x)\)
is called the normal cone of
C
at
x.
$$ N_{C}(x)= \bigl\{ z\in H:\langle u-x,z\rangle\le0, \forall u\in C \bigr\} . $$
(2.16)
Lemma 2.4
(see [7])
Let
\(F:C\to H\)
be monotone and hemicontinuous and let
\(N_{C}(x)\)
denote the normal cone of
C
at
\(x\in C\). Define
Then, \(T\subset H\times H\)
is maximal monotone, moreover, \(T^{-1}0=VI(C,F)\).
$$ Tx= \left \{ \begin{array}{@{}l@{\quad}l} Fx+N_{C}(x),& x \in C,\\ \emptyset,& x\notin C. \end{array} \right . $$
(2.17)
Lemma 2.5
Let
\(F:C\rightarrow H\)
be a hemicontinuous monotone operator and
\(x^{*}\in C\). Then, the following variational inequalities are equivalent:
(i)
\(\langle Fx,x - {x^{*}} \rangle\ge0\), \(\forall x \in C\);
(ii)
\(\langle F{x^{*}},x - {x^{*}} \rangle\ge0\), \(\forall x \in C\).
From Lemma 2.5, we also deduce that \(VI(C,F)\) is closed convex.
Lemma 2.6
(see [31])
Let
\(\{a_{n}\}\)
be a sequence of nonnegative real numbers satisfying
where
\(\{ {\gamma_{n}}\} \subset(0,1)\)
and
\(\{ {\sigma_{n}}\} \)
satisfy
Then
\(\lim_{n\to\infty}a_{n}=0\).
$${a_{n + 1}} \le(1 - {\gamma_{n}}){a_{n}} + { \gamma_{n}} {\sigma_{n}},\quad n\geq0, $$
(i)
\(\sum_{n = 0}^{\infty}{\gamma_{n}} = \infty\);
(ii)
either
\(\lim{\sup_{n \to\infty}}{\sigma_{n}} \le0\)
or
\(\sum_{n = 0}^{\infty}|{\gamma_{n}}{\sigma_{n}}| < \infty\).
Now we are in a proposition to state and prove the main results in this paper.
3 Main results
In this section we first establish a new existence and uniqueness theorem to MVIP (1.1) with F being a hemicontinuous and η-strongly monotone operator. We then introduce two kinds of algorithms (one implicit and the other explicit) for solving MVIP (1.1).
Theorem 3.1
Let
C
be a nonempty, closed, and convex subset of a real Hilbert space
H. Let
\(F:C\to H\)
be a hemicontinuous and
η-strongly monotone operator. Let
\(y\in H\)
be an arbitrary but fixed element. Then the monotone variational inequality
has a unique solution.
$$ \bigl\langle F{x^{*}}-y,x - {x^{*}} \bigr\rangle \ge0,\quad \forall x \in C, $$
(3.1)
Proof
Let T be defined by (2.17). Then \(T\subset H\times H\) is maximal monotone by Lemma 2.4. By taking \(r_{n}>0\) with \(r_{n}\to0\) as \(n\to\infty\), in view of Lemma 2.3, we have \(R(r_{n}I+T)=H\) for all \(n\ge1\). Consequently, for any \(y\in H\), there exist \(x_{n}\in C\) such that
for all \(n\ge1\). We plan to prove that \(\{x_{n}\}\) is bounded. To end this, using (2.17) and (3.2), we get
for all \(n\ge1\), in particular, we have
Then we can write them by
$$ y\in r_{n}x_{n}+Tx_{n} $$
(3.2)
$$ y\in r_{n}x_{n}+Fx_{n}+N_{C}(x_{n}) $$
(3.3)
$$ y\in r_{1}x_{1}+Fx_{1}+N_{C}(x_{1}). $$
(3.4)
$$\begin{aligned}& y=r_{n}x_{n}+Fx_{n}+z_{n}, \quad \mbox{and } z_{n}\in N_{C}(x_{n}), \end{aligned}$$
(3.5)
$$\begin{aligned}& y=r_{1}x_{1}+Fx_{1}+z_{1},\quad \mbox{and } z_{1}\in N_{C}(x_{1}). \end{aligned}$$
(3.6)
Since F is η-strongly monotone, \(z_{n}\in N_{C}(x_{n})\) and \(z_{1}\in N_{C}(x_{1})\), from (3.5) and (3.6) we have
from which it turns out that
for all \(n\ge1\). Equation (3.7) shows that \(\{x_{n}\}\) is bounded. Without loss of generality, we may assume that \(x_{n}\rightharpoonup x^{*}\) as \(n\to\infty\). Since T is monotone, and \(y-r_{n}x_{n}\in Tx_{n}\), we have
for all \((a,b)\in\mathfrak{G}(T)\) and all \(n\ge1\). Letting \(n\to\infty\) and taking the limit in (3.8) yield
for all \((a,b)\in\mathfrak{G}(T)\). By Lemma 2.2, we conclude that \((x^{*},y)\in\mathfrak{G}(T)\), that is, \(y\in Tx^{*}\), and hence we have \(y-Fx^{*}\in N_{C}(x^{*})\) by (2.17), i.e., \(\langle Fx^{*}-y, x-x^{*}\rangle\ge0\), \(\forall x\in C\), that is, \(x^{*}\) is a solution of (3.1). We have proven that the monotone variational inequality (3.1) has a solution. We next prove that the monotone variational inequality (3.1) has a unique solution. Suppose \(x^{**}\) is another solution of (3.1), then \(\langle Fx^{**}-y, x-x^{**}\rangle\ge0\), \(\forall x\in C\). Thus we have \(\langle Fx^{*}-y, x^{**}-x^{*}\rangle\ge0\) and \(\langle Fx^{**}-y, x^{*}-x^{**}\rangle\ge0\). Adding up these two inequalities yields \(\langle Fx^{*}-Fx^{**}, x^{**}-x^{*}\rangle\ge0\), from which one derives that \(x^{**}=x^{*}\) by using the strong monotonicity of F. This completes the proof. □
$$\begin{aligned}[b] 0&=\langle r_{n}x_{n}-r_{1}x_{1}, x_{n}-x_{1}\rangle+\langle Fx_{n}-Fx_{1}, x_{n}-x_{1}\rangle+\langle z_{n}-z_{1},x_{n}-x_{1} \rangle \\ &\ge r_{n}\|x_{n}-x_{1}\|^{2}-|r_{n}-r_{1}| \|x_{1}\|\|x_{n}-x_{1}\|+\eta \|x_{n}-x_{1} \| ^{2} \\ &=(r_{n}+\eta)\|x_{n}-x_{1}\|^{2}-|r_{n}-r_{1}| \|x_{1}\|\|x_{n}x_{\|}, \end{aligned} $$
$$ \|x_{n}-x_{1}\|\le\frac{|r_{n}-r_{1}|\|x_{1}\|}{r_{n}+\eta} $$
(3.7)
$$ \langle a-x_{n}, b-y+r_{n}x_{n}\rangle \ge0 $$
(3.8)
$$\bigl\langle a-x^{*}, b-y \bigr\rangle \ge0 $$
From Theorem 3.1, we deduce immediately the following important result.
Corollary 3.1
Let
C
be a nonempty, closed, and convex subset of a real Hilbert space
H. Let
\(F:C\to H\)
be a hemicontinuous and
η-strongly monotone operator. Then the MVIP (1.1) has a unique solution.
Let \(F:C\to H\) be hemicontinuous and monotone and \(R:H\to H\) be hemicontinuous and η-strongly monotone. Choose arbitrarily a point \(u\in H\) and a sequence of positive numbers \(\{\gamma_{n}\}\) with \(\gamma_{n}\to0\) as \(n\to\infty\). Then \(\gamma _{n}(R-u)+F\) are hemicontinuous and \(\eta\gamma_{n}\)-strongly monotone for all \(n\ge1\). By using Theorem 3.1, we conclude that the variational inequality problem
has a unique solution \(\{y_{n}\}\subset C\) for every fixed \(n\ge 1\).
$$ \bigl\langle \gamma_{n}(Ry-u)+Fy, x-y \bigr\rangle \ge0,\quad \forall x \in C, $$
(3.9)
Take \(\gamma_{n}=\frac{{{\alpha_{n}}}}{{{\beta_{n}}}}\) in (3.9). Then (3.9) yields
and hence
It follows from Lemma 2.1(C1) that
$$ \bigl\langle \alpha_{n}(Ry_{n}-u)+\beta_{n} Fy_{n}, x-y_{n} \bigr\rangle \geq0,\quad \forall x\in C, $$
(3.10)
$$ \bigl\langle y_{n}-\alpha_{n}(Ry_{n}-u)- \beta_{n} Fy_{n}-y_{n}, x-y_{n} \bigr\rangle \leq0,\quad \forall x\in C. $$
(3.11)
$$ y_{n}=P_{C} \bigl[(I-\alpha_{n}R)y_{n}+ \alpha_{n} u-\beta_{n} Fy_{n} \bigr],\quad n \geq1. $$
(3.12)
Theorem 3.2
Let
C
be a nonempty, closed, and convex subset of a real Hilbert space
H. Let
\(F:C\to H\)
be a hemicontinuous monotone operator and
\(R:H\to H\)
be a hemicontinuous and
η-strongly monotone operator. Let
\(\{\alpha_{n}\}\)
and
\(\{\beta_{n}\}\)
be two sequences in
\((0,1]\)
that satisfy the following condition:
$$\frac{{{\alpha_{n}}}}{{{\beta_{n}}}} \to0 \quad\textit{as }n\to\infty. $$
Assume that
\(VI(C,F)\neq\emptyset\). Then, the sequence
\(\{y_{n}\}\)
generated by (3.12) converges in norm to
\({x^{*}} = {P_{VI(C,F)}}((I-R)x^{*}+u) \), in particular, if we take
\(R=I\)
and
\(u=0\)
in (3.12), then the sequence
\(\{y_{n}\}\)
generated by (3.12) converges in norm to the minimum-norm solution to MVIP (1.1).
Proof
Since \(VI(C,F)\) is nonempty, we see that \(VI(C,F)\) is nonempty, closed, and convex, and hence \(P_{VI(C,F)}h\) is well defined for any \(h\in H\). Let \(\{y_{n}\}\) be defined by (3.12). Then (3.12) is equivalent to (3.10). By Lemma 2.5, we have
for all \(n\ge1\). \(\forall x^{*}\in VI(C,F)\), we have
By Lemma 2.5, we have
Taking \(x=y_{n}\) in (3.15), we have
for all \(n\ge1\).
$$ \langle\alpha_{n}Rx-\alpha_{n}u+\beta_{n}Fx, x-y_{n}\rangle\ge0, \quad\forall x\in C $$
(3.13)
$$ \bigl\langle Fx^{*}, x-x^{*} \bigr\rangle \ge0,\quad \forall x\in C. $$
(3.14)
$$ \bigl\langle Fx,x-x^{*} \bigr\rangle \ge0,\quad \forall x\in C. $$
(3.15)
$$ \bigl\langle Fy_{n},y_{n}-x^{*} \bigr\rangle \ge0, $$
(3.16)
Taking \(x=x^{*}\) in (3.10), we have
for all \(n\ge1\).
$$ \bigl\langle \alpha_{n}Ry_{n}-\alpha_{n}u+ \beta_{n}Fy_{n}, y_{n}-x^{*} \bigr\rangle \le 0, $$
(3.17)
By using (3.16), (3.17), and the strong monotonicity of F, we get
from which it turns out that
in particular,
for all \(n\ge1\).
$$\begin{aligned} 0\ge{}& \bigl\langle \alpha_{n}Ry_{n}-\alpha_{n}u+ \beta_{n}Fy_{n}, y_{n}-x^{*} \bigr\rangle \\ ={}& \alpha_{n} \bigl\langle Ry_{n}-Rx^{*},y_{n}-x^{*} \bigr\rangle +\alpha_{n} \bigl\langle Rx^{*}-u, y_{n}-x^{*} \bigr\rangle \\ &{}+\beta_{n} \bigl\langle Fy_{n},y_{n}-x^{*} \bigr\rangle \\ \ge{}&\eta\alpha_{n} \|y_{n}-x^{*}\|^{2}+ \alpha_{n} \bigl\langle Rx^{*}-u, y_{n}-x^{*} \bigr\rangle , \end{aligned}$$
$$ \bigl\| y_{n}-x^{*}\bigr\| ^{2}\le\frac{1}{\eta} \bigl\langle Rx^{*}-u, x^{*}-y_{n} \bigr\rangle , \quad\forall x^{*}\in VI(C,F), $$
(3.18)
$$ \bigl\| y_{n}-x^{*}\bigr\| \le\frac{1}{\eta}\bigl\| Rx^{*}-u\bigr\| ,\quad \forall x^{*}\in VI(C,F), $$
(3.19)
Equation (3.19) shows that \(\{y_{n}\}\) is bounded. Without loss of generality, we may assume that \(y_{n}\rightharpoonup\hat{y}\) as \(n\to\infty\); then \(\hat{y}\in VI(C,F)\). Indeed, using (3.13), we have
Letting \(n\to\infty\) and taking the limit in (3.20) yield
$$ \biggl\langle \frac{\alpha_{n}}{\beta_{n}}(Rx-u)+Fx,x-y_{n} \biggr\rangle \ge0, \quad \forall x\in C. $$
(3.20)
$$ \langle Fx,x-\hat{y}\rangle\ge0, \quad\forall x\in C. $$
(3.21)
It follows from (3.21) and Lemma 2.5 that
that is, \(\hat{y}\in VI(C,F)\).
$$\langle F\hat{y},x-\hat{y}\rangle\ge0,\quad \forall x\in C. $$
Replace \(x^{*}\) by \(\hat{y}\) in (3.18) to yield
for all \(n\ge1\).
$$ \|y_{n}-\hat{y}\|^{2}\le\frac{1}{\eta}\langle R\hat{y}-u, \hat {y}-y_{n}\rangle, $$
(3.22)
Use \(y_{n}\rightharpoonup\hat{y}\) and (3.22) to conclude that \(y_{n}\to \hat{y}\) as \(n\to\infty\).
In view of (3.18), we have
for all \(n\ge1\).
$$ \bigl\langle Rx^{*}-u, x^{*}-y_{n} \bigr\rangle \ge0,\quad \forall x^{*}\in VI(C,F), $$
(3.23)
Letting \(n\to\infty\) and taking the limit in (3.23) yield
which implies that
$$\bigl\langle Rx^{*}-u, x^{*}-\hat{y} \bigr\rangle \ge0,\quad \forall x^{*}\in VI(C,F), $$
$$ \bigl\langle R\hat{y}-u, x^{*}-\hat{y} \bigr\rangle \ge0,\quad \forall x^{*}\in VI(C,F). $$
(3.24)
Theorem 3.1 tells us that \(\hat{y}\) is the unique solution of (3.24), which ensures that the whole sequence \(\{y_{n}\}\) converges in norm to \(\hat{y}\) as \(n\to \infty\). Moreover, it follows from Lemma 2.1(C1) and (3.24) that \(\hat {y}=P_{VI(C,F)}[(I-R)\hat{y}+u]\). This completes the proof. □
We next introduce an explicit iterative method for solving MVIP (1.1).
It is natural to consider the following iteration method that generates a sequence \(\{x_{n}\}\) according to the recursion:
where the initial guess \(x_{1}\in C\) and \(u\in H\) are selected arbitrarily, while \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are two sequences of positive numbers in \((0, 1]\).
$$ {x_{n+1}} = {P_{C}} \bigl[(I-\alpha_{n}R){x_{n}}+ \alpha_{n}u-{\beta _{n}}F{x_{n}} \bigr],\quad n \ge1, $$
(3.25)
Theorem 3.3
Let
\(F:C\to H\)
be a hemicontinuous, generalized Lipschitz continuous and monotone operator. Let
\(R:H\to H\)
be a hemicontinuous, generalized Lipschitz continuous, and
η-strongly monotone operator. 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
\(n \to\infty\).
Assume that
\(VI(C,F)\neq\emptyset\). Then, the sequence
\(\{x_{n}\}\)
generated by (3.25) converges in norm to
\({x^{*}} = {P_{VI(C,F)}}((I-R)x^{*}+u)\).
Proof
Let \(\{y_{n}\}\) be defined by (3.12). By using Theorem 3.2, we know that \(\{y_{n}\}\) converges in norm to \({x^{*}} = {P_{VI(C,F)}}((I-R)x^{*}+u) \). It is sufficient to show that \(x_{n+1}-y_{n}\to0\) as \(n\to\infty\). To end this, we first show that \(\{x_{n}\}\) is bounded.
For any \(p\in VI(C,F)\), we have
By using (3.25), (3.26), (2.1), and Lemma 2.1(C2), we have
Observe that
Since both F and R are generalized Lipschitz continuous, we have
and
Observe also that
In view of conditions (i) and (ii), without loss of generality, we may assume that
for all \(n\ge1\).
$$ p=P_{C} \bigl[(1-\alpha_{n})p+\alpha_{n}p- \beta_{n}Fp \bigr]. $$
(3.26)
$$\begin{aligned} \|x_{n+1}-p\|^{2}={}& \bigl\| P_{C} \bigl[(I-\alpha_{n}R)x_{n}+ \alpha_{n}u-\beta _{n}Fx_{n} \bigr]-P_{C} \bigl[(1- \alpha_{n})p+\alpha_{n}p-\beta_{n}Fp \bigr] \bigr\| ^{2} \\ \leq{}&\bigl\| x_{n}-p-\alpha_{n}(Rx_{n}-p)+ \alpha_{n}(u-p)-\beta_{n}(Fx_{n}-Fp)\bigr\| ^{2} \\ ={}&\|x_{n}-p\|^{2}-2\alpha_{n}\langle Rx_{n}-p, x_{n}-p\rangle \\ &{}+2\alpha_{n}\langle u-p, x_{n}-p\rangle-2 \beta_{n}\langle Fx_{n}-Fp, x_{n}-p\rangle \\ &{}+\bigl\| \alpha_{n}(Rx_{n}-u)+\beta_{n}(Fx_{n}-Fp) \bigr\| ^{2} \\ ={}&\|x_{n}-p\|^{2}-2\alpha_{n}\langle Rx_{n}-Rp, x_{n}-p\rangle-2\alpha_{n}\langle Rp-p,x_{n}-p\rangle \\ &{}+2\alpha_{n}\langle u-p, x_{n}-p\rangle-2 \beta_{n}\langle Fx_{n}-Fp, x_{n}-p\rangle \\ &{}+\bigl\| \alpha_{n}(Rx_{n}-u)+\beta_{n}(Fx_{n}-Fp) \bigr\| ^{2} \\ \leq{}&(1-2\eta\alpha_{n})\|x_{n}-p\|^{2}-2 \alpha_{n}\langle Rp-u,x_{n}-p\rangle \\ &{}+2\alpha_{n}^{2}\|Rx_{n}-u\|^{2}+2 \beta_{n}^{2}\|Fx_{n}-Fp\|^{2}. \end{aligned}$$
(3.27)
$$ 2\bigl|\langle Rp-u, x_{n}-p\rangle\bigr|\le2\eta\frac{\|Rp-u\|}{\eta} \|x_{n}-p\|\le \eta \biggl(\frac{\|Rp-u\|^{2}}{\eta^{2}}+\|x_{n}-p \|^{2} \biggr). $$
(3.28)
$$\begin{aligned} \|Fx_{n}-Fp\|^{2} &\le c^{2} \bigl(1+ \|x_{n}-p\| \bigr)^{2} \\ &\le 2c^{2} \bigl(1+\|x_{n}-p\|^{2} \bigr) \end{aligned}$$
(3.29)
$$\begin{aligned} \|Rx_{n}-Rp\|^{2} \le c^{2}\bigl(1+ \|x_{n}-p\|\bigr)^{2} \le 2c^{2} \bigl(1+\|x_{n}-p\|^{2} \bigr). \end{aligned}$$
(3.30)
$$ \|Rx_{n}-u\|^{2}\le2 \bigl(\|Rx_{n}-Rp \|^{2}+\|Rp-u\|^{2} \bigr). $$
(3.31)
$$ 8c^{2}\alpha_{n}^{2}+4c^{2}\beta \beta_{n}^{2}\le\frac{1}{2}\eta\alpha_{n}, $$
(3.32)
Substitute (3.28)-(3.32) into (3.27) and simplify to yield
for all \(n\ge1\). We have shown that \(\{x_{n}\}\) is bounded. Next, we shall prove that \(x_{n+1}-y_{n}\to0\) as \(n\to\infty\).
$$\begin{aligned} \|x_{n+1}-p\|^{2}&\leq \bigl(1-\eta \alpha_{n}+8c^{2} \alpha_{n}^{2}+4c^{2} \beta_{n}^{2} \bigr)\| x_{n}-p\|^{2} +\frac{1}{2}\eta\alpha_{n} \biggl(\frac{2}{\eta^{2}}+4+ \frac{c^{2}}{\eta} \biggr)\|Rp-u\| ^{2} \\ &\leq \biggl(1-\frac{1}{2}\eta\alpha_{n} \biggr) \|x_{n}-p\|^{2}+\frac{1}{2}\eta\alpha _{n} \biggl(\frac{2}{\eta^{2}}+4+\frac{c^{2}}{\eta} \biggr)\|Rp-u\|^{2} \\ &\leq \max \biggl\{ \|x_{1}-p\|^{2}, \biggl( \frac{2}{\eta^{2}}+4+\frac{c^{2}}{\eta} \biggr)\|Rp-u\| ^{2} \biggr\} =M, \end{aligned}$$
Since R is η-strongly monotone and F is monotone, we have
and
for all \(n\ge1\). Since both R and F are generalized Lipschitz continuous, we have
and
for all \(n\ge1\).
$$ \langle Rx_{n}-Ry_{n}, x_{n}-y_{n} \rangle\ge\eta\|x_{n}-y_{n}\|^{2} $$
(3.33)
$$ \langle Fx_{n}-Fy_{n}, x_{n}-y_{n} \rangle\ge0, $$
(3.34)
$$ \|Rx_{n}-Ry_{n}\|^{2}\le2c^{2} \bigl(1+ \|x_{n}-y_{n}\|^{2} \bigr) $$
(3.35)
$$ \|Fx_{n}-Fy_{n}\|^{2}\le2c^{2} \bigl(1+ \|x_{n}-y_{n}\|^{2} \bigr), $$
(3.36)
By using (3.12), Lemma 2.1(C2), the strong monotonicity of R, and the monotonicity of F, we have
which implies that
where M is a positive constant such that \(M\ge\max\{\|Ry_{n-1}\|+\| u\|, \|Fy_{n-1}\|\}\).
$$\begin{aligned} &\|y_{n}-y_{n-1}\|^{2} \\ &\quad\le \bigl\langle y_{n}-y_{n-1}, (I-\alpha_{n}R)y_{n}-(I- \alpha _{n-1}R)y_{n-1}+(\alpha_{n}- \alpha_{n-1})u-\beta_{n}Fy_{n}-\beta _{n-1}Fy_{n-1} \bigr\rangle \\ &\quad= \|y_{n}-y_{n-1}\|^{2}-\alpha_{n} \langle Ry_{n}-Ry_{n-1},y_{n}-y_{n-1}\rangle \\ &\qquad{}-(\alpha_{n}-\alpha_{n-1})\langle Ry_{n-1}-u,y_{n}-y_{n-1} \rangle \\ &\qquad{}-\beta_{n}\langle Fy_{n}-Fy_{n-1}, y_{n}-y_{n-1}\rangle-(\beta_{n}-\beta _{n-1})\langle Fy_{n-1}, y_{n}-y_{n-1} \rangle \\ &\quad\le (1-\eta\alpha_{n})\|y_{n}-y_{n-1} \|^{2}+|\alpha_{n}-\alpha_{n-1}|\| Ry_{n-1} \|\|y_{n}-y_{n-1}\| \\ &\qquad{}+|\alpha_{n}-\alpha_{n-1}|\|u\|\|y_{n}-y_{n-1} \|+|\beta_{n}-\beta_{n-1}|\| Fy_{n-1}\| \|y_{n}-y_{n-1}\|, \end{aligned}$$
(3.37)
$$ \|y_{n}-y_{n-1}\|\le\frac{M}{\eta}\frac{|\alpha_{n}-\alpha_{n-1}|+|\beta _{n}-\beta_{n-1}|}{\alpha_{n}}, $$
(3.38)
In view of conditions (i) and (ii), without loss of generality, we may assume that
for all \(n\ge1\).
$$ \alpha_{n}^{2}+\beta_{n}^{2}\le \frac{\eta}{4c^{2}}\alpha_{n}, $$
(3.39)
By using (3.12), (3.25), (2.1), Lemma 2.1(C2), and (3.33)-(3.39), we get
where we have used conditions (i)-(iii) and \(M_{1}\) is a fixed positive number.
$$\begin{aligned} {\Vert {{x_{n + 1}} - {y_{n}}} \Vert ^{2}} \le{}& { \bigl\Vert x_{n}-y_{n}- { \alpha_{n}}(R{x_{n}} -R{y_{n}}) - { \beta_{n}}(F{x_{n}} - F{y_{n}}) \bigr\Vert ^{2}} \\ ={}& \Vert x_{n}-y_{n} \Vert ^{2}-2 \alpha_{n} \langle Rx_{n}-Ry_{n}, x_{n}-y_{n}\rangle -2\beta_{n}\langle Fx_{n}-Fy_{n}, x_{n}-y_{n}\rangle \\ &{}+\bigl\| \alpha_{n}(Rx_{n}-Ry_{n})+ \beta_{n} (Fx_{n}-Fy_{n})\bigr\| ^{2} \\ \le{}&(1-2\eta\alpha_{n})\|x_{n}-y_{n} \|^{2}+2\alpha_{n}^{2}\|Rx_{n}-Ry_{n} \|^{2}+2\beta _{n}^{2}\|Fx_{n}-Fy_{n} \|^{2} \\ \le{}& (1 - 2\eta{\alpha_{n}}){\Vert {{x_{n}} - {y_{n}}} \Vert ^{2}}+4c^{2}\alpha_{n}^{2} \| \bigl(1+\|x_{n}-y_{n}\|^{2} \bigr) + 4c^{2}\beta_{n}^{2} \bigl(1+\|x_{n}-y_{n} \| ^{2} \bigr) \\ ={}& \bigl(1 - 2\eta{\alpha_{n}}+4c^{2} \bigl( \alpha_{n}^{2}+\beta_{n}^{2} \bigr) \bigr){ \Vert {{x_{n}} - {y_{n}}} \Vert ^{2}}+ 4c^{2} \bigl(\alpha_{n}^{2}+\beta_{n}^{2} \bigr) \\ \le{}&(1 -\eta\alpha_{n}) \bigl({\Vert {{x_{n}} - {y_{n-1}}} \Vert ^{2}}+2\| x_{n}-y_{n-1} \|\|y_{n}-y_{n-1}\| \\ &{}+\|y_{n}-y_{n-1}\|^{2} \bigr)+4c^{2} \bigl(\alpha_{n}^{2}+\beta_{n}^{2} \bigr) \\ \le{}&(1-\eta\alpha_{n})\|x_{n}-y_{n-1} \|^{2}+M_{1}\frac{|\alpha_{n}-\alpha _{n-1}|+|\beta_{n}-\beta_{n-1}|}{\alpha_{n}}+4c^{2} \bigl( \alpha_{n}^{2}+\beta_{n}^{2} \bigr) \\ \le{}& (1 -\eta\alpha_{n}){\Vert {{x_{n}} - {y_{n-1}}} \Vert ^{2}}+\circ (\eta\alpha_{n}), \end{aligned}$$
By Lemma 2.6, we conclude that \(\Vert {{x_{n + 1}} - {y_{n}}} \Vert \to0\), as \(n \to\infty\), which means that \(\{ {x_{n}}\} \) converges in norm to \({x^{*}} = {P_{VI(C,F)}}((I-R)x^{*}+u)\). This completes the proof. □
Remark 3.1
Remark 3.2
Choose the sequences \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) such that
where \(a<\frac{b+1}{2}\), \(0< b<a\) and \(a<2b\). Then it is clear that conditions (i)-(iii) of Theorem 3.3 are satisfied.
$$\alpha_{n}=\frac{1}{n^{a}} \quad\mbox{and}\quad \beta_{n}= \frac{1}{n^{b}},\quad n\ge1, $$
4 Applications
In this section, we give some applications of the results established in Section 3.
Consider the constrained convex minimization problem:
where C is a closed convex subset of a real Hilbert space H and \(\varphi:H\to\mathbb{R}\) is a real-valued convex function. Assume that φ is continuously Fréchet differentiable with a generalized Lipschitz continuous gradient:
for all \(x,y\in H\), where c is a positive constant.
$$ \min \bigl\{ \varphi(x):x\in C \bigr\} , $$
(4.1)
$$ \bigl\| \nabla\varphi(x)-\nabla\varphi(y)\bigr\| \le c\bigl(1+\|x-y\|\bigr), $$
(4.2)
It is well known that the minimization problem (4.1) is equivalent to the following variational inequality problem:
From Example 2.1, we know that \(\nabla\varphi:H\to H\) is maximal monotone and hemicontinuous.
$$ x^{*}\in C, \quad \bigl\langle \nabla\varphi \bigl(x^{*} \bigr),x-x^{*} \bigr\rangle \ge0,\quad \forall x\in C. $$
(4.3)
By virtue of Theorem 3.3, we can deduce the following convergence result.
Theorem 4.1
Assume that (4.1) has a solution. 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. Then
\(\{x_{n}\}\)
converges in norm to the minimum-norm solution of the constrained minimization problem (4.1).
$$ \forall{x_{1}} \in H,\quad {x_{n + 1}} = {P_{C}} \bigl[(1 - {\alpha_{n}}){x_{n}} - {\beta_{n}}\nabla \varphi({x_{n}}) \bigr],\quad n\ge1, $$
(4.4)
Proof
Apply Theorem 3.3 to the case where \(F=\nabla\varphi\), \(R=I\), and \(u=0\) to get the conclusion. □
Finally, we apply our results to the minimum-norm fixed point problem for pseudocontractive mappings.
Theorem 4.2
Let
C
be a nonempty, closed, and convex subset of a real Hilbert space
H. Let
\(T:C\to C\)
be a generalized Lipschitz continuous, hemicontinuous, and 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
\({x^{*}} = {P_{\operatorname{Fix}(T)}}u\), in particular, if we take
\(u=0\)
in (4.5), then the sequence
\(\{x_{n}\}\)
generated by (4.5) converges in norm to the minimum-norm fixed point of
T.
$$ x_{1}\in C, u\in H,\quad {x_{n + 1}} = {P_{C}} \bigl[ \bigl(1 - ({\alpha _{n}}+\beta_{n}) \bigr){x_{n}}+ \alpha_{n}u+{\beta_{n}}T{x_{n}} \bigr],\quad n \ge1, $$
(4.5)
Proof
Write \(F=I-T\). Then (4.5) reduces to the following iterative algorithm:
$$ x_{1}\in C, u\in H,\quad {x_{n + 1}} = {P_{C}} \bigl[(1 - \alpha _{n}){x_{n}}+\alpha_{n}u-{ \beta_{n}}F{x_{n}} \bigr],\quad n \ge1. $$
(4.6)
Since \(T:C\to C\) is a generalized Lipschitz continuous, hemicontinuous, and pseudocontractive mapping, we deduce that F is a generalized Lipschitz continuous, hemecontinuous, and monotone operator. By Lemma 2.1(C1), noting that \(T:C\to C\) is a self-mapping, we see that \(VI(C,F)=\operatorname{Fix}(P_{C}T)=\operatorname{Fix}(T)\ne\emptyset\) by our assumption. Apply Theorem 3.3 to the case where \(F=I-T\) and \(R=I\) to derive the desired conclusion. □
Corollary 4.1
Let
C
be a nonempty, bounded, and closed convex subset of a real Hilbert space
H. Let
\(T:C\to C\)
be a hemicontinuous and pseudocontractive mapping. Let
\(\{\alpha_{n}\}\)
and
\(\{\beta_{n}\}\)
be the same as in Theorem
4.2. Then the sequence
\(\{x_{n}\}\)
generated by (4.5) converges in norm to
\({x^{*}} = {P_{\operatorname{Fix}(T)}}u\).
Proof
Write \(F=I-T\). Then \(F:C\to H\) is a generalized Lipschitz continuous, hemicontinuous, and monotone operator. Indeed, since C is bounded, we see that there exists a positive number c such that
for all \(x,y\in C\). Since \(T:C\to C\) is a hemicontinuous and pseudocontractive mapping, we also know that F is a hemicontinuous and monotone operator. By Minty [15], we know that \(VI(C,F)\ne \emptyset\). Since \(VI(C,F)=\operatorname{Fix}(T)\), we have \(\operatorname{Fix}(T)\ne\emptyset\). Apply Theorem 4.2 to derive the desired conclusion. This completes the proof. □
$$\|Fx-Fy\|\le c\bigl(1+\|x-y\|\bigr) $$
Remark 4.1
5 Conclusion
We studied a class of monotone variational inequality problems (MVIPs) for generalized Lipschitz continuous, hemicontinuous, and monotone operators defined on a nonempty, closed, and convex subset of a real Hilbert space. Firstly, by using the maximal monotone theory, we established a new existence and uniqueness theorem for a variational inequality problem with a hemicontinuous and strongly monotone operator. Then, by virtue of the existence and uniqueness theorem, we introduced an implicit method and analyzed its strong convergence. We also introduced an explicit iterative method as discretization of the implicit method. We proved that both the implicit and the explicit methods converge in norm to the same solutions to the MVIPs. Finally, we applied our main results to the constrained minimization problems and the minimum-norm fixed point problems for generalized Lipschitz continuous, hemicontinuous, and pseudocontractive mappings.
Acknowledgements
This research was supported by the National Natural Science Foundation of China (11071053).
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.