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General convergence analysis of projection methods for a system of variational inequalities in quniformly smooth Banach spaces
Journal of Inequalities and Applicationsvolume 2013, Article number: 481 (2013)
Abstract
In this paper, we introduce and consider a system of variational inequalities involving two different operators in quniformly smooth Banach spaces. We suggest and analyze a new explicit projection method for solving the system under some more general conditions. Our results extend and unify the results of Verma (Appl. Math. Lett. 18:12861292, 2005) and Yao, Liou and Kang (J. Glob. Optim., 2011, doi:10.1007/s1089801198040) and some other previously known results.
MSC:47H05, 47H10, 47J25.
1 Introduction
Let E and ${E}^{\ast}$ be a real Banach space and the dual space of E, respectively. Let C be a subset of E and $q>1$ be a real number. The generalized duality mapping ${J}_{q}:E\to {E}^{\ast}$ is defined by
for all x in E. In particular, $J={J}_{2}$ is called the normalized duality mapping and ${J}_{q}(x)={\parallel x\parallel}^{q2}{J}_{2}(x)$ for $x\ne 0$. If E is a Hilbert space, then $J=I$, the identity mapping. It is well known that if E is smooth, then ${J}_{q}$ is singlevalued, which is denoted by ${j}_{q}$.
Recall the variational inequality problem of finding ${x}^{\ast}\in C$ such that
where $T:C\to H$ is a nonlinear mapping. Variational inequalities theory, which was introduced by Stampacchia [1], emerged as an interesting and fascinating branch of mathematical and engineering sciences. The ideas and techniques of variational inequalities have been applied in structural analysis, economics, optimization, operations research fields. It has been shown that variational inequalities provide the most natural, direct, simple and efficient framework for a general treatment of some unrelated problems arising in various fields of pure and applied sciences. In recent years, there have been considerable activities in the development of numerical techniques including projection methods, WienerHopf equations, auxiliary principle and descent framework for solving variational inequalities; see [1–14] and the references therein. These activities have motivated us to generalize and extend the variational inequalities and related optimization problems in several directions using novel techniques.
Recently, Verma [11] proved the strong convergence of twostep projection method for solving the following system of variational inequality problems in a Hilbert space: Find $({x}^{\ast},{y}^{\ast})\in C\times C$ such that
In order to solve problem (1.2), Verma [11] introduced the following projection method:
where ${P}_{C}$ is the projection of a Hilbert space H onto C. This method contains several previously known projection schemes as special cases, while some have been applied to problems arising, especially, from complementarity problems, convex quadratic programming and other variational problems; see [2–4, 11] and the references therein.
Very recently, Yao et al. [12] considered the following system of variational inequality problems in 2uniformly smooth Banach spaces: Find $({x}^{\ast},{y}^{\ast})\in C\times C$ such that
where ${T}_{1}$, ${T}_{2}$ are two different nonlinear operators. Moreover, they modified projection method to system (1.4) in Banach spaces and introduced the following iterative method:
where ${\mathrm{\Pi}}_{C}$ is a sunny nonexpansive retraction from E onto C.
One question arises naturally: Do Yao et al.’s new projection methods work for two bivariate nonlinear operators in 2uniformly smooth Banach spaces, or more generally, in quniformly smooth Banach spaces with $q>1$, under more general control conditions?
In order to give some affirmative answers to the question raised above, we introduce the following system of bivariate variational inequality problems in quniformly smooth Banach spaces: Find $({x}^{\ast},{y}^{\ast})\in C\times C$ such that
The purpose of this paper is not only to show that the projection technique can be extended to the system of bivariate variational inequality problems in quniformly smooth Banach spaces, but also to suggest and analyze a new explicit iterative method, which includes the previously known projection methods as special cases, and whose convergence analysis is proved under some more general conditions. Our results extend and unify the corresponding results of [5, 7, 11, 12] and many others.
2 Preliminaries
Let E and ${E}^{\ast}$ be a real Banach space and the dual space of E, respectively. Let C be a nonempty closed convex subset of E, and let $S(E)=\{x\in E:\parallel x\parallel =1\}$. Then the norm of E is said to be Gâteaux differentiable if the following limit
exists for each $x,y\in S(E)$. In this case, E is called smooth. The norm of E is said to be uniformly Gâteaux differentiable if for each $y\in S(E)$, the limit above is attained uniformly for $x\in S(E)$. The norm of E is called Fréchet differentiable if for each $x\in S(E)$, the limit above is attained uniformly for $y\in S(E)$. The norm of E is called uniformly Fréchet differentiable if the limit above is attained uniformly for $x,y\in S(E)$. It is well known that (uniform) Fréchet differentiability of the norm of E implies (uniform) Gâteaux differentiability of the norm of E.
Recall that ${\psi}_{E}:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$, the modulus of smoothness of E, is defined by
A Banach space E is said to be uniformly smooth if $\frac{{\psi}_{E}(t)}{t}$ as $t\to 0$. A Banach space E is said to be quniformly smooth, if there exists a fixed constant $c>0$ such that ${\psi}_{E}(t)\le c{t}^{q}$ ($q>1$). It is well known that E is uniformly smooth if and only if the norm of E is uniformly Fréchet differentiable. If E is quniformly smooth, then $q\le 2$ and E is uniformly smooth, and hence the norm of E is uniformly Fréchet differentiable, in particular, the norm of E is Fréchet differentiable. It is well known that Hilbert and Lebesgue ${L}^{p}$ ($p>1$) spaces are uniformly smooth. More precisely, ${L}^{p}$ is $min\{p,2\}$uniformly smooth for every $p>1$.
In order to prove our main results, we also need the following concepts and lemmas.
Let C be a nonempty closed and convex subset of a real Banach space E, and let K be a nonempty subset of C. Let ${\mathrm{\Pi}}_{C}:C\to K$ be a mapping, and ${\mathrm{\Pi}}_{C}$ is said to be:

(a)
sunny if for each $x\in C$ and $t\in [0,1]$, we have ${\mathrm{\Pi}}_{C}[tx+(1t){\mathrm{\Pi}}_{C}]={\mathrm{\Pi}}_{C}x$;

(b)
a retraction of C onto K if ${\mathrm{\Pi}}_{C}x=x$, $\mathrm{\forall}x\in K$;

(c)
a sunny nonexpansive retraction if ${\mathrm{\Pi}}_{C}$ is sunny, nonexpansive and retraction onto K.
Definition 2.1 An operator $T:C\to E$ is said to be μLipschitz continuous in the first variable if there exists a constant $\mu >0$ such that
Definition 2.2 Let C be a nonempty closed convex subset of a smooth Banach space E, and ${J}_{q}:E\to {E}^{\ast}$ is a generalized duality mapping. A bivariate operator $T:C\times C\to E$ is said to be:

(i)
rstrongly accretive in the first variable if there exists a constant $r>0$ such that
$$\u3008T(x,\cdot )T(y,\cdot ),{j}_{q}(xy)\u3009\ge r{\parallel xy\parallel}^{q},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in C.$$ 
(ii)
γcocoercive in the first variable if there exists a constant $\gamma >0$ such that
$$\u3008T(x,\cdot )T(y,\cdot ),{j}_{q}(xy)\u3009\ge \gamma {\parallel T(x,\cdot )T(y,\cdot )\parallel}^{q},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in C.$$ 
(iii)
relaxed $(\gamma ,r)$cocoercive in the first variable if there exist constants $\gamma ,r>0$ such that
$$\u3008T(x,\cdot )T(y,\cdot ),{j}_{q}(xy)\u3009\ge \gamma {\parallel T(x,\cdot )T(y,\cdot )\parallel}^{q}+r{\parallel xy\parallel}^{q},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in C.$$
Remark 2.1 In Definition 2.2, the rstrongly accretive operator includes the rstrongly monotone and the rstrongly accretive ones defined in [11, 12] as special cases.
Remark 2.2 Obviously, an rstrongly accretive operator must be a relaxed $(\gamma ,r)$cocoercive whenever $\gamma =0$, but the converse is not true. Therefore the relaxed $(\gamma ,r)$cocoercive operator is more general than rstrongly accretive one.
Lemma 2.1 [15]
Let E be a real quniformly smooth Banach space, then there exists a constant ${C}_{q}>0$ such that
In particular, if E is a real 2uniformly smooth Banach space, then there exists the best smooth constant $K>0$ such that
Let C be a nonempty subset of a smooth Banach space E, and let ${\mathrm{\Pi}}_{C}:E\to C$ be a retraction. Then ${\mathrm{\Pi}}_{C}$ is sunny and nonexpansive if and only if
Lemma 2.3 [18]
Assume that $\{{a}_{n}\}$ is a sequence of nonnegative real numbers such that
where $\{{\gamma}_{n}\}$ is a sequence in $(0,1)$ and $\{{\sigma}_{n}\}$ is a sequence in R such that

(i)
${\sum}_{n=0}^{\mathrm{\infty}}{\gamma}_{n}=\mathrm{\infty}$;

(ii)
${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\sigma}_{n}/{\gamma}_{n}\le 0$ or ${\sum}_{n=0}^{\mathrm{\infty}}{\sigma}_{n}<\mathrm{\infty}$.
Then ${lim}_{n\to \mathrm{\infty}}{a}_{n}=0$.
3 Main results
By Lemma 2.2, we establish the equivalence between the system of variational inequalities (1.6) and the fixed point problem with projection technique, that is, $({x}^{\ast},{y}^{\ast})\in C\times C$ is a solution of the system of variational inequalities (1.6) if and only if
This alternative formula enables us to suggest and analyze a twostep explicit projection method for solving system (1.6), and this is the main motivation of our next result.
Theorem 3.1 Let C be a nonempty closed convex subset of a quniformly smooth Banach space E. Let ${T}_{i}:C\times C\to E$ be relaxed $({\gamma}_{i},{r}_{i})$cocoercive and ${\mu}_{i}$Lipschitz continuous in the first variable, $i=1,2$. For arbitrarily chosen initial points $({x}_{0},{y}_{0})\in C\times C$, define sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ in the following manner:
where ${\mathrm{\Pi}}_{C}$ is a sunny nonexpansive retraction from E onto C, the following conditions are satisfied:

(i)
${\alpha}_{n},{\beta}_{n}\in [0,1]$, ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$ and ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}(1{\beta}_{n1})<\mathrm{\infty}$;

(ii)
$0<\rho <{(\frac{q({r}_{1}{\gamma}_{1}{\mu}_{1}^{q})}{{C}_{q}{\mu}_{1}^{q}})}^{\frac{1}{q1}}$ and $0<\eta <{(\frac{q({r}_{2}{\gamma}_{2}{\mu}_{2}^{q})}{{C}_{q}{\mu}_{2}^{q}})}^{\frac{1}{q1}}$.
Then the sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ converge to ${x}^{\ast}$ and ${y}^{\ast}$, respectively, where $({x}^{\ast},{y}^{\ast})$ is a solution of the system of variational inequalities (1.6).
Proof Let $({x}^{\ast},{y}^{\ast})\in C\times C$ be a solution of (1.6). From (3.1a) and (3.2), we have
Since the operator ${T}_{1}$ is relaxed $({\gamma}_{1},{r}_{1})$cocoercive and ${\mu}_{1}$Lipschitz continuous definition in the first variable, it follows from Lemma 2.1 that
which implies that
where ${\theta}_{1}=\sqrt[q]{1\rho q{r}_{1}+\rho q{\gamma}_{1}{\mu}_{1}^{q}+{C}_{q}{\rho}^{q}{\mu}_{1}^{q}}$. We can obtain ${\theta}_{1}\in (0,1)$ by condition (ii). Substituting (3.4) into (3.3), we have
Similarly, it follows from (3.1b) and (3.2) that
Since the operator ${T}_{2}$ is relaxed $({\gamma}_{2},{r}_{2})$cocoercive and ${\mu}_{2}$Lipschitz continuous definition in the first variable, it follows that
which implies that
where ${\theta}_{2}=\sqrt[q]{1\eta q{r}_{2}+\eta q{\gamma}_{2}{\mu}_{2}^{q}+{C}_{q}{\eta}^{q}{\mu}_{2}^{q}}$. We can obtain ${\theta}_{2}\in (0,1)$ from condition (ii). Substituting (3.7) into (3.6), we have
that is,
It follows from (3.5) and (3.9) that
Since $1{\theta}_{1}>0$, ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$ and ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}(1{\beta}_{n1})<\mathrm{\infty}$, we apply Lemma 2.3 to get
Combining condition (ii), (3.9) and (3.11), we have
It shows that ${lim}_{n\to \mathrm{\infty}}{x}_{n}={x}^{\ast}$, ${lim}_{n\to \mathrm{\infty}}{y}_{n}={y}^{\ast}$, respectively, satisfying the system of variational inequalities (1.6). This completes the proof. □
Theorem 3.2 Let C be a nonempty closed convex subset of a quniformly smooth Banach space E. Let ${T}_{i}:C\times C\to E$ be ${r}_{i}$strongly accretive and ${\mu}_{i}$Lipschitz continuous in the first variable, $i=1,2$. For arbitrarily chosen initial points $({x}_{0},{y}_{0})\in C\times C$, define sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ in the following manner:
where ${\mathrm{\Pi}}_{C}$ is a sunny nonexpansive retraction from E onto C, the following conditions are satisfied:

(i)
${\alpha}_{n}\in (0,1)$, ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$;

(ii)
$0<\rho <{(\frac{q{r}_{1}}{{C}_{q}{\mu}_{1}^{q}})}^{\frac{1}{q1}}$ and $0<\eta <{(\frac{q{r}_{2}}{{C}_{q}{\mu}_{2}^{q}})}^{\frac{1}{q1}}$.
Then the sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ converge to ${x}^{\ast}$ and ${y}^{\ast}$, respectively, where $({x}^{\ast},{y}^{\ast})$ is a solution of the system of variational inequalities (1.6).
Proof As $\gamma =0$ and ${\beta}_{n}=1$, from Remark 2.2, we know that a relaxed $(0,r)$cocoercive operator reduces to rstrongly accretive, and iterative algorithm (3.2) reduces to (3.13), respectively. Then the conclusion follows immediately from Theorem 3.1. This completes the proof. □
If ${T}_{1},{T}_{2}:C\to E$ are univariate operators, applying Theorem 3.1 to a 2uniformly smooth Banach space with constant ${C}_{q}=2{K}^{2}$, we obtain the following result.
Theorem 3.3 Let C be a nonempty closed convex subset of a 2uniformly smooth Banach space E. Let ${T}_{i}:C\to E$ be relaxed $({\gamma}_{i},{r}_{i})$cocoercive and ${\mu}_{i}$Lipschitz continuous, $i=1,2$. For arbitrarily chosen initial points ${x}_{0},{y}_{0}\in C$, define sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ in the following manner:
where ${\mathrm{\Pi}}_{C}$ is a sunny nonexpansive retraction from E onto C, the following conditions are satisfied:

(i)
${\alpha}_{n}\in [0,1]$, ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$;

(ii)
$0<\rho <\frac{{r}_{1}{\gamma}_{1}{\mu}_{1}^{2}}{{K}^{2}{\mu}_{1}^{2}}$ and $0<\eta <\frac{{r}_{2}{\gamma}_{2}{\mu}_{2}^{2}}{{K}^{2}{\mu}_{2}^{2}}$.
Then the sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ converge to ${x}^{\ast}$ and ${y}^{\ast}$, respectively, where $({x}^{\ast},{y}^{\ast})$ is a solution of the system of variational inequalities (1.4).
Since the Hilbert space H is a 2uniformly smooth Banach space with the best smooth constant $K=\frac{\sqrt{2}}{2}$, from Theorem 3.1 we obtain the following result.
Theorem 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let ${T}_{i}:C\to H$ be relaxed $({\gamma}_{i},{r}_{i})$cocoercive and ${\mu}_{i}$Lipschitz continuous, $i=1,2$. For arbitrarily chosen initial points ${x}_{0},{y}_{0}\in C$, define sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ in the following manner:
where ${P}_{C}$ is the projection from H onto C, the following conditions are satisfied:

(i)
${\alpha}_{n},{\beta}_{n}\in [0,1]$, ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$ and ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}(1{\beta}_{n1})<\mathrm{\infty}$;

(ii)
$0<\rho <\frac{2({r}_{1}{\gamma}_{1}{\mu}_{1}^{2})}{{\mu}_{1}^{2}}$ and $0<\eta <\frac{2({r}_{2}{\gamma}_{2}{\mu}_{2}^{2})}{{\mu}_{2}^{2}}$.
Then the sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ converge to ${x}^{\ast}$ and ${y}^{\ast}$, respectively, where $({x}^{\ast},{y}^{\ast})$ is a solution of the system of variational inequalities (1.2).
Remark 3.1 Theorems 3.1 and 3.3 extend Theorem 3.1 of [12] from a 2uniformly smooth Banach space to a quniformly smooth Banach space. Moreover, the underlying operator T is extended to a bivariate operator, and the property defined on T is more general than [12] in convergence analysis.
Remark 3.2 Theorems 3.1 and 3.4 extend Theorem 3.1 of [5, 7, 11] from a real Hilbert space to a quniformly smooth Banach space. Moreover, the property defined on the underlying operator T is extended from rstrongly monotone to relaxed $(\gamma ,r)$cocoercive, respectively.
Remark 3.3 Algorithm (3.2) includes the projection methods in [2, 5, 7, 11, 12] as special cases and unifies the previously known onestep and twostep projectiontype methods in a quniformly smooth Banach space. Furthermore, the computation workload of the present explicit projection method is much less than the implicit algorithm in [5] at each iteration step.
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Acknowledgements
Supported by the National Science Foundation of China (11001287), Natural Science Foundation Project of Chongqing (CSTC, 2012jjA00039) and Science and Technology Research Project of Chongqing Municipal Education Commission (KJ130712, KJ130731).
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Authors’ contributions
GQF carried out the primary studies for projection methods for a system of variational inequality, participated in its design and coordination. WDJ participated in the convergence analysis and drafted the manuscript. All authors read and approved the final manuscript.
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Keywords
 system of variational inequalities
 explicit projection methods
 relaxed $(\gamma ,r)$cocoercive operator
 μLipschitz continuous
 quniformly smooth Banach spaces