General convergence analysis of projection methods for a system of variational inequalities in q-uniformly smooth Banach spaces

Abstract

In this paper, we introduce and consider a system of variational inequalities involving two different operators in q-uniformly 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. 18:1286-1292, 2005) and Yao, Liou and Kang (J. Glob. Optim., 2011, doi:10.1007/s10898-011-9804-0) 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

$$J_q(x)=\{x^{\ast }\in E^{\ast }:〈x,x^{\ast }〉={\parallel x\parallel }^q,\parallel x^{\ast }\parallel ={\parallel x\parallel }^{q-1}\}$$

for all x in E. In particular, $J=J_2$ is called the normalized duality mapping and $J_q(x)={\parallel x\parallel }^{q-2}{J}_2(x)$ 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 single-valued, which is denoted by $j_q$.

Recall the variational inequality problem of finding $x^{\ast }\in C$ such that

$$〈T{x}^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in C,$$

(1.1)

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, Wiener-Hopf 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 two-step 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

$$\{\begin{array}{cc}〈\rho T(y^{\ast })+x^{\ast }-y^{\ast },x-x^{\ast }〉\ge 0,\hfill & \mathrm{\forall }x\in C,\rho >0,\hfill \\ 〈\eta T(x^{\ast })+y^{\ast }-x^{\ast },x-y^{\ast }〉\ge 0,\hfill & \mathrm{\forall }x\in C,\eta >0.\hfill \end{array}$$

(1.2)

In order to solve problem (1.2), Verma [11] introduced the following projection method:

$$\{\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{P}_C[y_n-\rho T(y_n)],\hfill \\ y_n=(1-{\beta }_n)x_n+{\beta }_n{P}_C[x_n-\eta T(x_n)],n\ge 0,\hfill \end{array}$$

(1.3)

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 2-uniformly smooth Banach spaces: Find $(x^{\ast },y^{\ast })\in C\times C$ such that

$$\{\begin{array}{cc}〈\rho T_1(y^{\ast })+x^{\ast }-y^{\ast },j(x-x^{\ast })〉\ge 0,\hfill & \mathrm{\forall }x\in C,\rho >0,\hfill \\ 〈\eta T_2(x^{\ast })+y^{\ast }-x^{\ast },j(x-y^{\ast })〉\ge 0,\hfill & \mathrm{\forall }x\in C,\eta >0,\hfill \end{array}$$

(1.4)

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:

$$\{\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{\mathrm{\Pi }}_C[y_n-\rho T_1(y_n)],\hfill \\ y_n={\mathrm{\Pi }}_C[x_n-\eta T_2(x_n)],n\ge 0,\hfill \end{array}$$

(1.5)

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 2-uniformly smooth Banach spaces, or more generally, in q-uniformly 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 q-uniformly smooth Banach spaces: Find $(x^{\ast },y^{\ast })\in C\times C$ such that

$$\{\begin{array}{cc}〈\rho T_1(y^{\ast },x^{\ast })+x^{\ast }-y^{\ast },j_q(x-x^{\ast })〉\ge 0,\hfill & \mathrm{\forall }x\in C,\rho >0,\hfill \\ 〈\eta T_2(x^{\ast },y^{\ast })+y^{\ast }-x^{\ast },j_q(x-y^{\ast })〉\ge 0,\hfill & \mathrm{\forall }x\in C,\eta >0.\hfill \end{array}$$

(1.6)

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 q-uniformly 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

$$\underset{t\to 0}{lim}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$$

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

$${\psi }_E(t)=sup\{\frac{1}{2}(\parallel x+y\parallel +\parallel x-y\parallel )-1:x\in S(E),\parallel y\parallel \le t\}.$$

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 q-uniformly 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 q-uniformly 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 ${\mathrm{\Pi }}_C[tx+(1-t){\mathrm{\Pi }}_C]={\mathrm{\Pi }}_{C}x$;

• (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 μ-Lipschitz continuous in the first variable if there exists a constant $\mu >0$ such that

$$\parallel T(x,\cdot )-T(y,\cdot )\parallel \le \mu \parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

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) r-strongly accretive in the first variable if there exists a constant $r>0$ such that

  $$〈T(x,\cdot )-T(y,\cdot ),j_q(x-y)〉\ge r{\parallel x-y\parallel }^q,\mathrm{\forall }x,y\in C.$$

• (ii) γ-cocoercive in the first variable if there exists a constant $\gamma >0$ such that

  $$〈T(x,\cdot )-T(y,\cdot ),j_q(x-y)〉\ge -\gamma {\parallel T(x,\cdot )-T(y,\cdot )\parallel }^q,\mathrm{\forall }x,y\in C.$$

• (iii) relaxed $(\gamma ,r)$-cocoercive in the first variable if there exist constants $\gamma ,r>0$ such that

  $$〈T(x,\cdot )-T(y,\cdot ),j_q(x-y)〉\ge -\gamma {\parallel T(x,\cdot )-T(y,\cdot )\parallel }^q+r{\parallel x-y\parallel }^q,\mathrm{\forall }x,y\in C.$$

Remark 2.1 In Definition 2.2, the r-strongly accretive operator includes the r-strongly monotone and the r-strongly accretive ones defined in [11, 12] as special cases.

Remark 2.2 Obviously, an r-strongly 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 r-strongly accretive one.

Lemma 2.1 [15]

Let E be a real q-uniformly smooth Banach space, then there exists a constant $C_q>0$ such that

$${\parallel x+y\parallel }^q\le {\parallel x\parallel }^q+q〈y,j_q(x)〉+C_q{\parallel y\parallel }^q,\mathrm{\forall }x,y\in E.$$

In particular, if E is a real 2-uniformly smooth Banach space, then there exists the best smooth constant $K>0$ such that

$${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2〈y,j(x)〉+2{\parallel Ky\parallel }^2,\mathrm{\forall }x,y\in E.$$

Lemma 2.2 [16, 17]

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

$$〈u-{\mathrm{\Pi }}_C[u],J(y-{\mathrm{\Pi }}_C[u])〉\le 0,\mathrm{\forall }u\in E,y\in C.$$

Lemma 2.3 [18]

Assume that $\{a_n\}$ is a sequence of nonnegative real numbers such that

$$a_{n+1}\le (1-{\gamma }_n)a_n+{\sigma }_n,n\ge 0,$$

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

$$x^{\ast }={\mathrm{\Pi }}_C[y^{\ast }-\rho T_1(y^{\ast },x^{\ast })],$$

(3.1a)

$$y^{\ast }={\mathrm{\Pi }}_C[x^{\ast }-\eta T_2(x^{\ast },y^{\ast })].$$

(3.1b)

This alternative formula enables us to suggest and analyze a two-step 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 q-uniformly 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:

$$\{\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{\mathrm{\Pi }}_C[y_n-\rho T_1(y_n,x_n)],\hfill \\ y_{n+1}=(1-{\beta }_n)x_{n+1}+{\beta }_n{\mathrm{\Pi }}_C[x_{n+1}-\eta T_2(x_{n+1},y_n)],\hfill \end{array}$$

(3.2)

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 ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n(1-{\beta }_{n-1})<\mathrm{\infty }$;

• (ii) $0<\rho <{(\frac{q(r_1-{\gamma }_1{\mu }_1^q)}{C_q{\mu }_1^q})}^{\frac{1}{q-1}}$ and $0<\eta <{(\frac{q(r_2-{\gamma }_2{\mu }_2^q)}{C_q{\mu }_2^q})}^{\frac{1}{q-1}}$.

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

$$\begin{array}{rl}\parallel x_{n+1}-x^{\ast }\parallel & =\parallel (1-{\alpha }_n)x_n+{\alpha }_n{\mathrm{\Pi }}_C[y_n-\rho T_1(y_n,x_n)]-x^{\ast }\parallel \\ \le (1-{\alpha }_n)\parallel x_n-x^{\ast }\parallel +{\alpha }_n\parallel {\mathrm{\Pi }}_C[y_n-\rho T_1(y_n,x_n)]-{\mathrm{\Pi }}_C[y^{\ast }-\rho T_1(y^{\ast },x^{\ast })]\parallel \\ \le (1-{\alpha }_n)\parallel x_n-x^{\ast }\parallel +{\alpha }_n\parallel y_n-y^{\ast }-\rho (T_1(y_n,x_n)-T_1(y^{\ast },x^{\ast }))\parallel .\end{array}$$

(3.3)

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

$$\begin{array}{r}{\parallel y_n-y^{\ast }-\rho (T_1(y_n,x_n)-T_1(y^{\ast },x^{\ast }))\parallel }^q\\ ={\parallel y_n-y^{\ast }\parallel }^q-\rho q〈T_1(y_n,x_n)-T_1(y^{\ast },x^{\ast }),j_q(y_n-y^{\ast })〉\\ +C_q{\rho }^q{\parallel T_1(y_n,x_n)-T_1(y^{\ast },x^{\ast })\parallel }^q\\ \le {\parallel y_n-y^{\ast }\parallel }^q-\rho q(-{\gamma }_1{\parallel T_1(y_n,x_n)-T_1(y^{\ast },x^{\ast })\parallel }^q+r_1{\parallel y_n-y^{\ast }\parallel }^q)\\ +C_q{\rho }^q{\parallel T_1(y_n,x_n)-T_1(y^{\ast },x^{\ast })\parallel }^q\\ \le (1-\rho q{r}_1+\rho q{\gamma }_1{\mu }_1^q+C_q{\rho }^q{\mu }_1^q){\parallel y_n-y^{\ast }\parallel }^q,\end{array}$$

which implies that

$$\parallel y_n-y^{\ast }-\rho (T_1(y_n,x_n)-T_1(y^{\ast },x^{\ast }))\parallel \le {\theta }_1\parallel y_n-y^{\ast }\parallel ,$$

(3.4)

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

$$\parallel x_{n+1}-x^{\ast }\parallel \le (1-{\alpha }_n)\parallel x_n-x^{\ast }\parallel +{\alpha }_n{\theta }_1\parallel y_n-y^{\ast }\parallel .$$

(3.5)

Similarly, it follows from (3.1b) and (3.2) that

$$\begin{array}{r}\parallel y_{n+1}-y^{\ast }\parallel \\ =\parallel (1-{\beta }_n)x_{n+1}+{\beta }_n{\mathrm{\Pi }}_C[x_{n+1}-\eta T_2(x_{n+1},y_n)]-y^{\ast }\parallel \\ \le (1-{\beta }_n)\parallel x_{n+1}-y^{\ast }\parallel +{\beta }_n\parallel {\mathrm{\Pi }}_C[x_{n+1}-\eta T_2(x_{n+1},y_n)]-{\mathrm{\Pi }}_C[x^{\ast }-\eta T_2(x^{\ast },y^{\ast })]\parallel \\ \le (1-{\beta }_n)\parallel x_{n+1}-y^{\ast }\parallel +{\beta }_n\parallel x_{n+1}-x^{\ast }-\eta (T_2(x_{n+1},y_n)-T_2(x^{\ast },y^{\ast }))\parallel .\end{array}$$

(3.6)

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

$$\begin{array}{r}{\parallel x_{n+1}-x^{\ast }-\eta (T_2(x_{n+1},y_n)-T_2(x^{\ast },y^{\ast }))\parallel }^q\\ ={\parallel x_{n+1}-x^{\ast }\parallel }^q-\eta q〈T_2(x_{n+1},y_n)-T_2(x^{\ast },y^{\ast }),j_q(x_{n+1}-x^{\ast })〉\\ +C_q{\eta }^q{\parallel T_2(x_{n+1},y_n)-T_2(x^{\ast },y^{\ast })\parallel }^q\\ \le {\parallel x_{n+1}-x^{\ast }\parallel }^q-\eta q(-{\gamma }_2{\parallel T_2(x_{n+1},y_n)-T_2(x^{\ast },y^{\ast })\parallel }^q+r_2{\parallel x_{n+1}-x^{\ast }\parallel }^q)\\ +C_q{\eta }^q{\parallel T_2(x_{n+1},y_n)-T_2(x^{\ast },y^{\ast })\parallel }^q\\ \le (1-\eta q{r}_2+\eta q{\gamma }_2{\mu }_2^q+C_q{\eta }^q{\mu }_2^q){\parallel x_{n+1}-x^{\ast }\parallel }^q,\end{array}$$

which implies that

$$\parallel x_{n+1}-x^{\ast }-\eta (T_2(x_{n+1},y_n)-T_2(x^{\ast },y^{\ast }))\parallel \le {\theta }_2\parallel x_{n+1}-x^{\ast }\parallel ,$$

(3.7)

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

$$\begin{array}{rl}\parallel y_{n+1}-y^{\ast }\parallel & \le (1-{\beta }_n)\parallel x_{n+1}-y^{\ast }\parallel +{\beta }_n{\theta }_2\parallel x_{n+1}-x^{\ast }\parallel \\ \le (1-{\beta }_n)\parallel x_{n+1}-x^{\ast }\parallel +(1-{\beta }_n)\parallel x^{\ast }-y^{\ast }\parallel +{\beta }_n{\theta }_2\parallel x_{n+1}-x^{\ast }\parallel \\ =[1-(1-{\theta }_2){\beta }_n]\parallel x_{n+1}-x^{\ast }\parallel +(1-{\beta }_n)\parallel x^{\ast }-y^{\ast }\parallel \\ <\parallel x_{n+1}-x^{\ast }\parallel +(1-{\beta }_n)\parallel x^{\ast }-y^{\ast }\parallel ,\end{array}$$

(3.8)

that is,

$$\parallel y_n-y^{\ast }\parallel \le \parallel x_n-x^{\ast }\parallel +(1-{\beta }_{n-1})\parallel x^{\ast }-y^{\ast }\parallel .$$

(3.9)

It follows from (3.5) and (3.9) that

$$\begin{array}{rl}\parallel x_{n+1}-x^{\ast }\parallel & \le (1-{\alpha }_n)\parallel x_n-x^{\ast }\parallel +{\alpha }_n{\theta }_1(\parallel x_n-x^{\ast }\parallel +(1-{\beta }_{n-1})\parallel x^{\ast }-y^{\ast }\parallel )\\ \le [1-(1-{\theta }_1){\alpha }_n]\parallel x_n-x^{\ast }\parallel +{\alpha }_n(1-{\beta }_{n-1})\parallel x^{\ast }-y^{\ast }\parallel .\end{array}$$

(3.10)

Since $1-{\theta }_1>0$, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$ and ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n(1-{\beta }_{n-1})<\mathrm{\infty }$, we apply Lemma 2.3 to get

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-x^{\ast }\parallel =0.$$

(3.11)

Combining condition (ii), (3.9) and (3.11), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-y^{\ast }\parallel =0.$$

(3.12)

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 q-uniformly 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:

$$\{\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{\mathrm{\Pi }}_C[y_n-\rho T_1(y_n,x_n)],\hfill \\ y_{n+1}={\mathrm{\Pi }}_C[x_{n+1}-\eta T_2(x_{n+1},y_n)],\hfill \end{array}$$

(3.13)

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) $0<\rho <{(\frac{q{r}_1}{C_q{\mu }_1^q})}^{\frac{1}{q-1}}$ and $0<\eta <{(\frac{q{r}_2}{C_q{\mu }_2^q})}^{\frac{1}{q-1}}$.

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 r-strongly 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 2-uniformly smooth Banach space with constant $C_q=2K^2$, we obtain the following result.

Theorem 3.3 Let C be a nonempty closed convex subset of a 2-uniformly 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:

$$\{\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{\mathrm{\Pi }}_C[y_n-\rho T_1(y_n)],\hfill \\ y_{n+1}={\mathrm{\Pi }}_C[x_{n+1}-\eta T_2(x_{n+1})],\hfill \end{array}$$

(3.14)

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 2-uniformly 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:

$$\{\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{P}_C[y_n-\rho T_1(y_n)],\hfill \\ y_{n+1}=(1-{\beta }_n)x_{n+1}+{\beta }_n{P}_C[x_{n+1}-\eta T_2(x_{n+1})],\hfill \end{array}$$

(3.15)

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 ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n(1-{\beta }_{n-1})<\mathrm{\infty }$;

• (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 2-uniformly smooth Banach space to a q-uniformly 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 q-uniformly smooth Banach space. Moreover, the property defined on the underlying operator T is extended from r-strongly 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 one-step and two-step projection-type methods in a q-uniformly 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.