Graph convergence for the $H(\cdot ,\cdot )$-mixed mappingwith an application for solving the system of generalized variationalinclusions

Abstract

In this paper, we investigate a class of accretive mappings called the$H(\cdot ,\cdot )$-mixed mappingsin Banach spaces. We prove that the proximal-point mapping associated with the$H(\cdot ,\cdot )$-mixed mapping issingle-valued and Lipschitz continuous. Some examples are given to justify thedefinition of $H(\cdot ,\cdot )$-mixed mapping.Further, a concept of graph convergence concerned with the$H(\cdot ,\cdot )$-mixed mapping isintroduced in Banach spaces and some equivalence theorems betweengraph-convergence and proximal-point mapping convergence for the$H(\cdot ,\cdot )$-mixed mappingssequence are proved. As an application, we consider a system of generalizedvariational inclusions involving $H(\cdot ,\cdot )$-mixed mappingsin real q-uniformly smooth Banach spaces. Using the proximal-pointmapping method, we prove the existence and uniqueness of solution and suggest aniterative algorithm for the system of generalized variational inclusions.Furthermore, we discuss the convergence criteria for the iterative algorithmunder some suitable conditions.

MSC: 47J19, 49J40, 49J53.

1 Introduction

Variational inclusions, as the generalization of variational inequalities, have beenwidely studied in recent years. Some of the most interesting and important problemsin the theory of variational inclusions include variational, quasi-variational,variational-like inequalities as special cases. For applications of variationalinclusions, we refer to [1]. Various kinds ofiterative methods have been studied to solve the variational inclusions. Among thesemethods, the proximal-point mapping technique for the study of variationalinclusions has been widely used by many authors. For details, we refer to[2–20].

In 2001, Huang and Fang [5] were the first tointroduce the generalized m-accretive mapping and give the definition ofthe proximal-point mapping for the generalized m-accretive mapping inBanach spaces. Since then a number of researchers have investigated several classesof generalized m-accretive mappings such as H-accretive,$H,\eta$-accretive,$(P,\eta )$-proximal-point,$(P,\eta )$-accretive,A-maximal relaxed accretive, $(A,\eta )$-accretive mappings.For details, we refer to [2, 3, 6, 7, 11, 14, 16, 18].

Recently, Zou and Huang [19, 20] introduced and studied $H(\cdot ,\cdot )$-accretive mappings;Kazmi et al.[8–10] introduced and studied generalized $H(\cdot ,\cdot )$-accretive mappings,$H(\cdot ,\cdot )$-η-proximal-point mappings. Veryrecently, Li and Huang [12] studied thegraph convergence for the $H(\cdot ,\cdot )$-accretive mappingand showed the equivalence between graph convergence and proximal-point mappingconvergence for the $H(\cdot ,\cdot )$-accretive mappingsequence in a Banach space, and Verma [17]studied the graph convergence for an A-maximal relaxed monotone mapping andgave the equivalence between the graph convergence and the proximal-point mappingconvergence for the A-maximal relaxed monotone mapping sequence in aHilbert space. They extended the concept of graph convergence introduced andconsidered by Attouch [21].

Motivated by the research work going on in this direction, we consider a class ofaccretive mappings called $H(\cdot ,\cdot )$-mixed mappings, anatural generalization of accretive (monotone) mappings in Banach spaces. Forrelated work, we refer to [2–4, 11, 14, 16, 18–20]. We prove that theproximal-point mapping of the $H(\cdot ,\cdot )$-mixed mapping issingle-valued and Lipschitz continuous and extends the concept of proximal-pointmappings associated with the $H(\cdot ,\cdot )$-accretive mappingsto the $H(\cdot ,\cdot )$-mixed mappings.Further, we study the graph convergence for the $H(\cdot ,\cdot )$-mixed mappings. Wepresent an equivalence theorem between graph convergence and proximal-point mappingconvergence for the $H(\cdot ,\cdot )$-mixed mappingsequence in Banach spaces. As an application, we consider a system of generalizedvariational inclusions involving the $H(\cdot ,\cdot )$-mixed mappings inreal q-uniformly smooth Banach spaces. Using the proximal-point mappingmethod, we prove the existence and uniqueness of solution and suggest an iterativealgorithm for the system of generalized variational inclusions. Furthermore, wediscuss the convergence criteria of the iterative algorithm under some suitableconditions. Our results can be viewed as a generalization of some known resultsgiven in [12, 17, 19–21].

2 Preliminaries

Let X be a real Banach space equipped with the norm $\parallel \cdot \parallel$, and let $X^{\ast }$be the topological dual space of X. Let $〈\cdot ,\cdot 〉$ be the dualpair between X and $X^{\ast }$,and let $2^X$be the power set of X.

Definition 2.1[22]

For $q>1$,a mapping $J_q:X\to 2^{X^{\ast }}$is said to be a generalized duality mapping if it is defined by

$$J_q(x)=\{f^{\ast }\in X^{\ast }:〈x,f^{\ast }〉={\parallel x\parallel }^q,\parallel f^{\ast }\parallel ={\parallel x\parallel }^{q-1}\},\mathrm{\forall }x\in X.$$

In particular, $J_2$is the usual normalized duality mapping on X. It is known that, ingeneral,

$$J_q(x)={\parallel x\parallel }^{q-1}{J}_2(x)\mathrm{\forall }x(\ne 0)\in X.$$

If $X\equiv H$a real Hilbert space, then $J_2$ becomesan identity mapping on H.

Definition 2.2[22]

A Banach space X is called smooth if, for every$x\in X$with $\parallel x\parallel =1$,there exists a unique $f\in X^{\ast }$such that $\parallel f\parallel =f(x)=1$.

The modulus of smoothness of X is a function${\rho }_X:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$defined by

$${\rho }_X(t)=sup\{\frac{1}{2}(\parallel x+y\parallel +\parallel x-y\parallel )-1:\parallel x\parallel \le 1,\parallel y\parallel \le t\}.$$

Definition 2.3[22]

A Banach space X is called

• (i) uniformly smooth if

  $$\underset{t\to 0}{lim}\frac{{\rho }_X(t)}{t}=0;$$

• (ii) q-uniformly smooth, for $q>1$, if there exists a constant $c>0$ such that

  $${\rho }_X(t)\le c{t}^q,t\in [0,\mathrm{\infty }).$$

Note that $J_q$is single-valued if X is uniformly smooth. Concerned with thecharacteristic inequalities in q-uniformly smooth Banach spaces, Xu[22] proved the following result.

Lemma 2.4 Let X be a real uniformly smooth Banach space. Then X is q-uniformly smooth if and only if there exists a constant$c_q>0$such that, for all$x,y\in X$,

$${\parallel x+y\parallel }^q\le {\parallel x\parallel }^q+q〈y,J_q(x)〉+c_q{\parallel y\parallel }^q.$$

From Lemma 2 of Liu [13], it is easy tohave the following lemma.

Lemma 2.5 Let $\{a_n\}$ and $\{b_n\}$ be two nonnegative real sequences satisfying

$$a_{n+1}\le k{a}_n+b_n$$

with$0<k<1$and$b_n\to 0$. Then${lim}_{n\to \mathrm{\infty }}{a}_n=0$.

Definition 2.6 Let $G:X\to X$be a single-valued mapping. Then

• (i) G is said to be accretive if

  $$〈G(x)-G(y),J_q(x-y)〉\ge 0,\mathrm{\forall }x,y\in X;$$

• (ii) G is said to be ξ-strongly accretive if there exists a constant $\xi >0$ such that

  $$〈G(x)-G(y),J_q(x-y)〉\ge \xi {\parallel x-y\parallel }^q,\mathrm{\forall }x,y\in X;$$

• (iii) G is said to be μ-cocoercive if there exists a constant $\mu >0$ such that

  $$〈G(x)-G(y),J_q(x-y)〉\ge \mu {\parallel G(x)-G(y)\parallel }^q,\mathrm{\forall }x,y\in X;$$

• (iv) G is said to be ${\lambda }_G$-Lipschitz continuous if there exists a constant ${\lambda }_G>0$ such that

  $$\parallel G(x)-G(y)\parallel \le {\lambda }_G\parallel x-y\parallel ,\mathrm{\forall }x,y\in X;$$

• (v) G is said to be α-expansive if there exists a constant $\alpha >0$ such that

  $$\parallel G(x)-G(y)\parallel \ge \alpha \parallel x-y\parallel ,\mathrm{\forall }x,y\in X;$$

if $\alpha =1$,then it is expansive.

Definition 2.7 Let $H:X\times X\to X$and $A,B:X\to X$be three single mappings. Then

• (i) $H(A,\cdot )$ is said to be μ-cocoercive with respect to A if there exists a constant $\mu >0$ such that

  $$〈H(Ax,u)-H(Ay,u),J_q(x-y)〉\ge \mu {\parallel Ax-Ay\parallel }^q,\mathrm{\forall }x,y,u\in X;$$

• (ii) $H(\cdot ,B)$ is said to be γ-relaxed accretive with respect to B if there exists a constant $\gamma >0$ such that

  $$〈H(u,Bx)-H(u,By),J_q(x-y)〉\ge (-\gamma ){\parallel x-y\parallel }^q,\mathrm{\forall }x,y,u\in X;$$

• (iii) $H(A,\cdot )$ is said to be $r_1$-Lipschitz continuous with respect to A if there exists a constant $r_1>0$ such that

  $$\parallel H(Ax,\cdot )-H(Ay,\cdot )\parallel \le r_1\parallel x-y\parallel ,\mathrm{\forall }x,y\in X;$$

• (iv) $H(\cdot ,B)$ is said to be $r_2$-Lipschitz continuous with respect to B if there exists a constant $r_2>0$ such that

  $$\parallel H(\cdot ,Bx)-H(\cdot ,By)\parallel \le r_2\parallel x-y\parallel ,\mathrm{\forall }x,y\in X.$$

Example 2.8 Let us consider the 2-uniformly smooth Banach space$X={\mathbb{R}}^2$ with the usual innerproduct. Let $A,B:{\mathbb{R}}^2\to {\mathbb{R}}^2$ bedefined by

$$Ax=\left(\begin{array}{c}m{x}_1-m{x}_2\\ -m{x}_1+2m{x}_2\end{array}\right),By=\left(\begin{array}{c}-m{y}_1+m{y}_2\\ -m{y}_1-m{y}_2\end{array}\right)$$

for all scalers $m\in \mathbb{R}$ and for all $x=(x_1,x_2),y=(y_1,y_2)\in {\mathbb{R}}^2$.

Suppose that $H:{\mathbb{R}}^2\times {\mathbb{R}}^2\to {\mathbb{R}}^2$ isdefined by $H(Ax,By)=Ax+By$,then $H(A,B)$ is$\frac{1}{3m}$-cocoercivewith respect to A and m-relaxed accretive with respect toB, and $\sqrt{5}m$-Lipschitzcontinuous with respect to A and $\sqrt{2}m$-Lipschitzcontinuous with respect to B.

Indeed, let for any $u\in X$,

$$\begin{array}{c}〈H(Ax,u)-H(Ay,u),x-y〉\hfill \\ =〈Ax-Ay,x-y〉\hfill \\ =〈(m{x}_1-m{x}_2,-m{x}_1+2m{x}_2)-(m{y}_1-m{y}_2,-m{y}_1+2m{y}_2),\hfill \\ (x_1-y_1,x_2-y_2)〉\hfill \\ =〈(m(x_1-y_1)-m(x_2-y_2),-m(x_1-y_1)+2m(x_2-y_2)),\hfill \\ (x_1-y_1,x_2-y_2)〉\hfill \\ =m{(x_1-y_1)}^2-2m(x_1-y_1)(x_2-y_2)+2m{(x_2-y_2)}^2,\hfill \\ {\parallel Ax-Ay\parallel }^2=〈Ax-Ay,Ax-Ay〉\hfill \\ \phantom{{\parallel Ax-Ay\parallel }^2}=〈((m{x}_1-m{x}_2,-m{x}_1+2m{x}_2)-(m{y}_1-m{y}_2,-m{y}_1+2m{y}_2)),\hfill \\ \phantom{{\parallel Ax-Ay\parallel }^2=}((m{x}_1-m{x}_2,-m{x}_1+2m{x}_2)-(m{y}_1-m{y}_2,-m{y}_1+2m{y}_2))〉\hfill \\ \phantom{{\parallel Ax-Ay\parallel }^2}=2m^2{(x_1-y_1)}^2-6m^2(x_1-y_1)(x_2-y_2)+5m^2{(x_2-y_2)}^2\hfill \\ \phantom{{\parallel Ax-Ay\parallel }^2}\le 3m^2{(x_1-y_1)}^2-6m^2(x_1-y_1)(x_2-y_2)+6m^2{(x_2-y_2)}^2\hfill \\ \phantom{{\parallel Ax-Ay\parallel }^2}=3m\{m{(x_1-y_1)}^2-2m(x_1-y_1)(x_2-y_2)+2m{(x_2-y_2)}^2\}\hfill \\ \phantom{{\parallel Ax-Ay\parallel }^2}=3m\left\{〈H(Ax,u)-H(Ay,u),x-y〉\right\},\hfill \end{array}$$

which implies that

$$〈H(Ax,u)-H(Ay,u),x-y〉\ge \frac{1}{3m}{\parallel Ax-Ay\parallel }^2,$$

that is, $H(A,B)$ is$\frac{1}{3m}$-cocoercivewith respect to A.

$$\begin{array}{c}〈H(u,Bx)-H(u,By),x-y〉\hfill \\ =〈Bx-By,x-y〉\hfill \\ =〈Bx-By,x-y〉\hfill \\ =〈(-m{x}_1+m{x}_2,-m{x}_1-m{x}_2)-(-m{y}_1+m{y}_2,-m{y}_1-m{y}_2),\hfill \\ (x_1-y_1,x_2-y_2)〉\hfill \\ =〈(-m(x_1-y_1)+m(x_2-y_2),-m(x_1-y_1)-m(x_2-y_2)),\hfill \\ (x_1-y_1,x_2-y_2)〉\hfill \\ =-m{(x_1-y_1)}^2-m{(x_2-y_2)}^2\hfill \\ =-m\{{(x_1-y_1)}^2+{(x_2-y_2)}^2\}\hfill \\ \ge -m{\parallel x-y\parallel }^2,\hfill \end{array}$$

which implies that

$$〈H(u,Bx)-H(u,By),x-y〉\ge (-m){\parallel x-y\parallel }^2,$$

that is, $H(A,B)$ ism-relaxed accretive with respect to B.

$$\begin{array}{rcl}{\parallel H(Ax,u)-H(Ay,u)\parallel }^2& =& {\parallel Ax-Ay\parallel }^2=〈Ax-Ay,Ax-Ay〉\\ =& 〈((m{x}_1-m{x}_2,-m{x}_1+2m{x}_2)-(m{y}_1-m{y}_2,-m{y}_1+2m{y}_2)),\\ ((m{x}_1-m{x}_2,-m{x}_1+2m{x}_2)-(m{y}_1-m{y}_2,-m{y}_1+2m{y}_2))〉\\ =& 2m^2{(x_1-y_1)}^2-6m^2(x_1-y_1)(x_2-y_2)+5m^2{(x_2-y_2)}^2\\ \le & 5m^2{(x_1-y_1)}^2+5m^2{(x_2-y_2)}^2,\end{array}$$

which implies that

$$\parallel H(Ax,u)-H(Ay,u)\parallel \le \sqrt{5}m\parallel x-y\parallel ,$$

that is, $H(A,B)$ is$\sqrt{5}m$-Lipschitzcontinuous with respect to A.

$$\begin{array}{rcl}{\parallel H(u,Bx)-H(u,By)\parallel }^2& =& {\parallel Bx-By\parallel }^2=〈Bx-By,Bx-By〉\\ =& 〈((-m{x}_1+m{x}_2,-m{x}_1-m{x}_2)-(-m{y}_1+m{y}_2,-m{y}_1-m{y}_2)),\\ ((-m{x}_1+m{x}_2,-m{x}_1-m{x}_2)-(-m{y}_1+m{y}_2,-m{y}_1-m{y}_2))〉\\ =& 2m^2{(x_1-y_1)}^2+2m^2{(x_2-y_2)}^2\end{array}$$

which implies that

$$\parallel H(u,Bx)-H(u,By)\parallel \le \sqrt{2}m\parallel x-y\parallel ,$$

that is, $H(A,B)$ is$\sqrt{2}m$-Lipschitzcontinuous with respect to B.

Definition 2.9 Let $\eta :X\times X\to X$and $H,A,B:X\to X$be mappings. Let $M:X\to 2^X$be a set-valued mapping. Then

• (i) η is said to be τ-Lipschitz continuous if there exists a constant $\tau >0$ such that

  $$\parallel \eta (x,y)\parallel \le \tau \parallel x-y\parallel ,\mathrm{\forall }x,y\in X;$$

• (ii) M is said to be accretive if

  $$〈u-v,J_q(x-y)〉\ge 0,\mathrm{\forall }x,y\in X,u\in Mx,v\in My;$$

• (iii) M is said to be ${\mu }^{\prime }$-strongly accretive if there exists a constant ${\mu }^{\prime }>0$ such that

  $$〈u-v,J_q(x-y)〉\ge {\mu }^{\prime }{\parallel x-y\parallel }^q,\mathrm{\forall }x,y\in X,u\in Mx,v\in My;$$

• (iv) M is said to be m-relaxed accretive if there exists a constant $m>0$ such that

  $$〈u-v,J_q(x-y)〉\ge -m{\parallel x-y\parallel }^q,\mathrm{\forall }x,y\in X,u\in Mx,v\in My;$$

• (v) M is said to be η-accretive if

  $$〈u-v,J_q(\eta (x,y))〉\ge 0,\mathrm{\forall }x,y\in X,u\in Mx,v\in My;$$

• (vi) M is said to be strictly η-accretive if M is η-accretive and equality holds if and only if $x=y$;

• (vii) M is said to be γ-strongly η-accretive if there exists a constant $\gamma >0$ such that

  $$〈u-v,J_q(\eta (x,y))〉\ge \gamma {\parallel x-y\parallel }^q,\mathrm{\forall }x,y\in X,u\in Mx,v\in My;$$

• (viii) M is said to be α-relaxed η-accretive if there exists a constant $\alpha >0$ such that

  $$〈u-v,J_q(\eta (x,y))〉\ge (-\alpha ){\parallel x-y\parallel }^q,\mathrm{\forall }x,y\in X,u\in Mx,v\in My;$$

• (ix) M is said to be m-accretive if M is accretive and $(I+\rho M)(X)=X$ for all $\rho >0$, where I denotes the identity operator on X;

• (x) M is said to be generalized m-accretive if M is η-accretive and $(I+\rho M)(X)=X$ for all $\rho >0$;

• (xi) M is said to be H-accretive if M is accretive and $(H+\rho M)(X)=X$ for all $\rho >0$;

• (xii) M is said to be $(H,\eta )$-accretive if M is η-accretive and $(H+\rho M)(X)=X$ for all $\rho >0$;

• (xiii) M is said to be $(A,\eta )$-accretive if M is m-relaxed η-accretive and $(A+\rho M)(X)=X$ for all $\rho >0$.

Definition 2.10[19]

Let $A,B:X\to X$,$H:X\times X\to X$be three single-valued mappings. Let $M:X\to 2^X$be a set-valued mapping. Then M is said to be $H(\cdot ,\cdot )$-accretivewith respect to A and B if M is accretive and$(H(\cdot ,\cdot )+\rho M)(X)=X$ for all$\rho >0$.

3 $H(\cdot ,\cdot )$-mixedmappings

In this section, we introduce the $H(\cdot ,\cdot )$-mixed mapping andshow some of its properties.

Definition 3.1 Let $H:X\times X\to X$,$A,B:X\to X$be three single-valued mappings. Let $H(A,B)$ beμ-cocoercive with respect to A, γ-relaxedaccretive with respect to B. Then the set-valued mapping$M:X\to 2^X$is said be $H(\cdot ,\cdot )$-mixed withrespect to mappings A and B if

• (i) M is m-relaxed accretive;

• (ii) $(H(A,B)+\rho M)(X)=X$ for all $\rho >0$.

Example 3.2 Let X, H, A, B be the same asin Example 2.8, and $M:{\mathbb{R}}^2\to {\mathbb{R}}^2$ bedefined by $M(x)=(-3\pi ,-3x_2)$,$\mathrm{\forall }x=(x_1,x_2)\in {\mathbb{R}}^2$.

We claim that M is a 3-relaxed accretive mapping. Indeed, for any$x=(x_1,x_2),y=(y_1,y_2)\in {\mathbb{R}}^2$

$$\begin{array}{c}〈Mx-My,x-y〉=〈(-3\pi ,-3x_2)-(-3\pi ,-3y_2),((x_1-y_1),(x_2-y_2))〉\hfill \\ \phantom{〈Mx-My,x-y〉}=〈(0,-3(x_2-y_2)),((x_1-y_1),(x_2-y_2))〉\hfill \\ \phantom{〈Mx-My,x-y〉}=-3{(x_2-y_2)}^2\hfill \\ \phantom{〈Mx-My,x-y〉}\ge -3\{{(x_1-y_1)}^2+{(x_2-y_2)}^2\}\hfill \\ \phantom{〈Mx-My,x-y〉}\ge -3{\parallel x-y\parallel }^2,\hfill \\ 〈Mx-My,x-y〉\ge (-3){\parallel x-y\parallel }^2.\hfill \end{array}$$

Furthermore, M is also an $H(\cdot ,\cdot )$-mixed mapping since$(H(A,B)+\rho M)({\mathbb{R}}^2)={\mathbb{R}}^2$ for any$\rho >0$.

Proposition 3.3 Let the set-valued mapping$M:X\to 2^X$be an$H(\cdot ,\cdot )$-mixed mappingwith respect to mappings A and B. If A is α-expansive and$\mu >\gamma$with$r=\mu {\alpha }^q-\gamma >m$,then the following inequality holds:

$$〈x-y,J_q(u-v)〉\ge 0,\mathrm{\forall }(v,y)\in graph(M),\mathit{\text{implies}}x\in Mu.$$

Proof Suppose on contrary that there exists $(u_0,x_0)\notin graphM$such that

$$〈x_0-y,J_q(u_0-v)〉\ge 0,\mathrm{\forall }(v,y)\in graph(M).$$

(3.1)

Since M is an $H(\cdot ,\cdot )$-mixed mapping, weknow that $(H(A,B)+\rho M)(X)=X$ holdsfor every $\rho >0$,and so there exists $(u_1,x_1)\in graph(M)$ such that

$$H(A{u}_0,B{u}_0)+\rho x_0=H(A{u}_1,B{u}_1)+\rho x_1\in X.$$

(3.2)

Now

$$\begin{array}{c}\rho x-\rho x_1=H(A{u}_1,B{u}_1)-H(Au,Bu)\in X,\hfill \\ 〈\rho x-\rho x_1,J_q(u-u_1)〉=〈H(A{u}_1,B{u}_1)-H(Au,B{u}_0),J_q(u-u_1)〉.\hfill \end{array}$$

Setting $(v,y)=(u_1,x_1)$ in (3.1) and thenfrom the resultant (3.2) and m-relaxed accretivity of M, we obtain

$$\begin{array}{rcl}-m{\parallel u_0-u_1\parallel }^q& \le & \rho 〈x_0-x_1,J_q(u_0-u_1)〉\\ =& -〈H(A{u}_0,B{u}_0)-H(A{u}_1,B{u}_1),J_q(u_0-u_1)〉\\ =& -〈H(A{u}_0,B{u}_0)-H(A{u}_1,B{u}_0),J_q(u_0-u_1)〉\\ -〈H(A{u}_1,B{u}_0)-H(A{u}_1,B{u}_1),J_q(u_0-u_1)〉.\end{array}$$

(3.3)

Since $H(A,B)$ isμ-cocoercive with respect to A and γ-relaxedaccretive with respect to B, and A is α-expansive,thus (3.3) becomes

$$\begin{array}{rcl}-m{\parallel u_0-u_1\parallel }^q& \le & -\mu {\parallel Au-A{u}_1\parallel }^q+\gamma {\parallel u-u_1\parallel }^q\\ \le & -\mu {\alpha }^2{\parallel u_0-u_1\parallel }^q+\gamma {\parallel u-u_1\parallel }^q\\ \le & -(\mu {\alpha }^q-\gamma ){\parallel u_0-u_1\parallel }^q\\ =& -r{\parallel u_0-u_1\parallel }^q\le 0,\text{where }r=\mu {\alpha }^q-\gamma \\ \le & -(r-m){\parallel u_0-u_1\parallel }^q\le 0.\end{array}$$

It implies that $u_0=u_1$since $r>m$.By (3.1), we have $x_0=x_1$,a contradiction. This completes the proof. □

Theorem 3.4 Let the set-valued mapping$M:X\to 2^X$be an$H(\cdot ,\cdot )$-mixed mappingwith respect to mappings A and B. If A is α-expansive and$\mu >\gamma$with$r=\mu {\alpha }^q-\gamma >\rho m$,then${(H(A,B)+\rho M)}^{-1}$is single-valued.

Proof For any given $u\in X$, let$x,y\in {(H(A,B)+\rho M)}^{-1}(u)$. It follows that

$$\begin{array}{c}-H(Ax,Bx)+u\in \rho Mx,\hfill \\ -H(Ay,By)+u\in \rho My.\hfill \end{array}$$

Since M is m-relaxed accretive, we have

$$\begin{array}{c}-m{\parallel x-y\parallel }^q\le \frac{1}{\rho }〈-H(Ax,Bx)+u-(-H(Ay,By)+u),J_q(x-y)〉,\hfill \\ -m\rho {\parallel x-y\parallel }^q=-〈H(Ax,Bx)-H(Ay,Bx),J_q(x-y)〉\hfill \\ \phantom{-m\rho {\parallel x-y\parallel }^q=}-〈H(Ay,Bx)-H(Ay,By),J_q(x-y)〉,\hfill \end{array}$$

which is like (3.3). Hence it follows that $\parallel x-y\parallel \le 0$.This implies that $x=y$ and so${(H(A,B)+\rho M)}^{-1}$is single-valued. □

Definition 3.5 Let the set-valued mapping $M:X\to 2^X$be an $H(\cdot ,\cdot )$-mixed mapping withrespect to mappings A and B. If A isα-expansive and $\mu >\gamma$with $r=\mu {\alpha }^q-\gamma >\rho m$,then the proximal-point mapping$R_{\rho ,M}^{H(\cdot ,\cdot )}:X\to X$is defined by

$$R_{\rho ,M}^{H(\cdot ,\cdot )}(u)={(H(A,B)+\rho M)}^{-1}(u),\mathrm{\forall }u\in X.$$

(3.4)

Now we prove that the proximal-point mapping defined by (3.4) is Lipschitzcontinuous.

Theorem 3.6 Let the set-valued mapping$M:X\to 2^X$be an$H(\cdot ,\cdot )$-mixed mappingwith respect to mappings A and B. If A is α-expansive and$\mu >\gamma$with$r=\mu {\alpha }^q-\gamma >\rho m$,then the proximal-point mapping$R_{\rho ,M}^{H(\cdot ,\cdot )}:X\to X$is$\frac{1}{r-\rho m}$-Lipschitzcontinuous, that is,

$$\parallel R_{\rho ,M}^{H(\cdot ,\cdot )}(u)-R_{\rho ,M}^{H(\cdot ,\cdot )}(v)\parallel \le \frac{1}{r-\rho m}\parallel u-v\parallel ,\mathrm{\forall }u,v\in X.$$

Proof Let $u\text{ and }v\in X$be any given points in X. It follows from (3.2) that

$$\begin{array}{c}\{\begin{array}{c}{R}_{\rho ,M}^{H(\cdot ,\cdot )}(u)={H((A,B)+\rho M))}^{-1}(u),\hfill \\ R_{\rho ,M}^{H(\cdot ,\cdot )}(v)={H((A,B)+\rho M))}^{-1}(v),\hfill \end{array}\hfill \\ \frac{1}{\rho }(u-H(A(R_{\rho ,M}^{H(\cdot ,\cdot )}(u)),B(R_{\rho ,M}^{H(\cdot ,\cdot )}(u)))\in M(R_{\rho ,M}^{H(\cdot ,\cdot )}(u)),\hfill \\ \frac{1}{\rho }(v-H(A(R_{\rho ,M}^{H(\cdot ,\cdot )}(v)),B(R_{\rho ,M}^{H(\cdot ,\cdot )}(v)))\in M(R_{\rho ,M}^{H(\cdot ,\cdot )}(v)).\hfill \end{array}$$

Let $z_1=R_{\rho ,M}^{H(\cdot ,\cdot )}(u)$ and $z_2=R_{\rho ,M}^{H(\cdot ,\cdot )}(v)$.

Since M is m-relaxed accretive, we have

$$\begin{array}{c}\frac{1}{\rho }〈(u-H(A(z_1),B(z_1))-(v-H(A(z_2),B(z_2))),J_q(z_1-z_2)〉\ge -m{\parallel z_1-z_2\parallel }^q,\hfill \\ 〈u-v-(H(A(z_1),B(z_1))-H(A(z_2),B(z_2))),J_q(z_1-z_2)〉\ge -\rho m{\parallel z_1-z_2\parallel }^q,\hfill \end{array}$$

which implies that

$$\begin{array}{rcl}\parallel u-v\parallel {\parallel z_1-z_2\parallel }^{q-1}& \ge & 〈u-v,J_q(z_1-z_2)〉\\ \ge & 〈H(A(z_1),B(z_1))-H(A(z_2),B(z_2)),J_q(z_1-z_2)〉-\rho m{\parallel z_1-z_2\parallel }^q\\ \ge & 〈H(A(z_1),B(z_1))-H(A(z_2),B(z_1)),J_q(z_1-z_2)〉\\ -〈H(A(z_2),B(z_1))-H(A(z_2),B(z_2)),J_q(z_1-z_2)〉-\rho m{\parallel z_1-z_2\parallel }^q\\ \ge & \mu {\parallel A(z_1)-A(z_2)\parallel }^q-\gamma {\parallel z_1-z_2\parallel }^q-\rho m{\parallel z_1-z_2\parallel }^q\\ \ge & \mu {\alpha }^q{\parallel z_1-z_2\parallel }^q-\gamma {\parallel z_1-z_2\parallel }^q-\rho m{\parallel z_1-z_2\parallel }^q\\ =& (\mu {\alpha }^q-\gamma -\rho m){\parallel z_1-z_2\parallel }^q,\\ =& (r-\rho m){\parallel z_1-z_2\parallel }^q,\text{where }r=\mu {\alpha }^q-\gamma ,\end{array}$$

and hence

$$\parallel u-v\parallel {\parallel z_1-z_2\parallel }^{q-1}\ge (r-\rho m){\parallel z_1-z_2\parallel }^q,$$

that is,

$$\parallel R_{\rho ,M}^{H(\cdot ,\cdot )}(u)-R_{\rho ,M}^{H(\cdot ,\cdot )}(v)\parallel \le \frac{1}{r-\rho m}\parallel u-v\parallel ,\mathrm{\forall }u,v\in X.$$

This completes the proof. □

4 Graph convergence for an $H(\cdot ,\cdot )$-mixedmapping

Let $M:X\to 2^X$be a set-valued mapping. The graph of the map M is defined by

$$graph(M)=\{(x,y)\in X\times X:y\in M(X)\}.$$

In this section we shall introduce the graph convergence for the$H(\cdot ,\cdot )$-mixed mapping.

Definition 4.1 Let $M_n,M:X\to 2^X$be the set-valued mappings such that M, $M_n$ are$H(\cdot ,\cdot )$-mixed mappings withrespect to the mappings A and B for $n=0,1,2,\dots$ .The sequence $\{M_n\}$ is said to begraph convergent to M, denoted by $M_n\stackrel{G}{⟶}M$,if for every $(x,y)\in graph(M)$, there exists a sequence$(x_n,y_n)\in graph(M_n)$ such that

$$x_n\to x,y_n\to y\text{as }n\to \mathrm{\infty }.$$

Theorem 4.2 Let$M_n,M:X\to 2^X$be the set-valued mappings such that M, $M_n$are$H(\cdot ,\cdot )$-mixed mappingswith respect to the mappings A and B for$n=0,1,2,\dots$ .Let$H(A,B)$be s-Lipschitz continuous with respect to A and t-Lipschitz continuous with respect to B. If A is α-expansive and$\mu >\gamma$with$r=\mu {\alpha }^q-\gamma >\rho m$,then$M_n\stackrel{G}{⟶}M$if and only if

$$R_{\rho ,M_n}^{H(\cdot ,\cdot )}(u)\to R_{\rho ,M}^{H(\cdot ,\cdot )}(u),\mathrm{\forall }u\in X,\rho >0,$$

where

$$R_{\rho ,M_n}^{H(\cdot ,\cdot )}(u)={(H(A,B)+\rho M_n)}^{-1}(u),R_{\rho ,M}^{H(\cdot ,\cdot )}(u)={(H(A,B)+\rho M)}^{-1}(u).$$

Proof It follows from Theorem 3.6 that $R_{\rho ,M_n}^{H(\cdot ,\cdot )}$ and$R_{\rho ,M}^{H(\cdot ,\cdot )}$ are both$\frac{1}{r-\rho m}$-Lipschitzcontinuous.

If part: Suppose that $M_n\stackrel{G}{⟶}M$.For any given $x\in X$,let

$$z_n=R_{\rho ,M_n}^{H(\cdot ,\cdot )}(x),z=R_{\rho ,M}^{H(\cdot ,\cdot )}(x).$$

Then

$$\frac{1}{\rho }[x-H(Az,Bz)]\in M(z),\text{that is}(z,\frac{1}{\rho }[x-H(Az,Bz)])\in graph(M).$$

In the light of Definition 4.1, we know that there exists a sequence$(z_n^{\prime },y_n^{\prime })\in graph(M_n)$ such that

$$z_n^{\prime }\to z,y_n^{\prime }\to \frac{1}{\rho }[x-H(Az,Bz)]\text{as }n\to \mathrm{\infty }.$$

(4.1)

Since $y_n^{\prime }\in M_n(z_n^{\prime })$, we have

$$H(A{z}_n^{\prime },B{z}_n^{\prime })+\rho y_n^{\prime }\in [H(A,B)+\rho M_n]\left(z_n^{\prime }\right)$$

and so

$$z_n^{\prime }=[H(A{z}_n^{\prime },B{z}_n^{\prime })+\rho y_n^{\prime }].$$

From the Lipschitz continuity of $M_n$, we get

$$\begin{array}{rcl}\parallel z_n-z\parallel & \le & \parallel z_n-z_n^{\prime }\parallel +\parallel z_n^{\prime }-z\parallel \\ =& \parallel R_{\rho ,M_n}^{H(\cdot ,\cdot )}(x)-R_{\rho ,M_n}^{H(\cdot ,\cdot )}[H(A{z}_n^{\prime },B{z}_n^{\prime })+\rho y_n^{\prime }]\parallel +\parallel z_n^{\prime }-z\parallel \\ \le & \frac{1}{r-\rho m}\parallel x-H(A{z}_n^{\prime },B{z}_n^{\prime })-\rho y_n^{\prime }\parallel +\parallel z_n^{\prime }-z\parallel \\ \le & \frac{1}{r-\rho m}[\parallel x-H(Az,Bz)-\rho y_n^{\prime }\parallel \\ +\parallel H(Az,Bz)-H(A{z}_n^{\prime },B{z}_n^{\prime })\parallel ]+\parallel z_n^{\prime }-z\parallel .\end{array}$$

(4.2)

From the Lipschitz continuity of $H(A,B)$, we have

$$\begin{array}{r}\parallel H(Az,Bz)-H(A{z}_n^{\prime },B{z}_n^{\prime })\parallel \\ \le \parallel H(Az,Bz)-H(Az,B{z}_n^{\prime })\parallel +\parallel H(Az,B{z}_n^{\prime })-H(A{z}_n^{\prime },B{z}_n^{\prime })\parallel \\ \le (s+t)\parallel z_n^{\prime }-z\parallel .\end{array}$$

(4.3)

It follows from (4.2) and (4.3) that

$$\parallel z_n-z\parallel \le \frac{1}{r-\rho m}\parallel x-H(Az,Bz)-\rho y_n^{\prime }\parallel +[1+\frac{1}{r-\rho m}(s+t)]\parallel z_n^{\prime }-z\parallel .$$

By (4.1), we have

$$\parallel z_n^{\prime }-z\parallel \to 0,\parallel \frac{1}{\rho }[x-H(Az,Bz)-y_n^{\prime }]\parallel \to 0$$

and so

$$\parallel z_n-z\parallel \to 0\text{as }n\to \mathrm{\infty }.$$

Only if part: Suppose that

$$R_{\rho ,M_n}^{H(\cdot ,\cdot )}\to R_{\rho ,M}^{H(\cdot ,\cdot )},\mathrm{\forall }u\in X,\rho >0.$$

For any given $(x,y)\in graph(M)$, we have

$$H(Ax,Bx)+\rho y\in [H(A,B)+\rho M](x)$$

and so

$$x=R_{\rho ,M}^{H(\cdot ,\cdot )}[H(Ax,Bx)+\rho y].$$

Let

$$x_n=R_{\rho ,M_n}^{H(\cdot ,\cdot )}[H(Ax,Bx)+\rho y].$$

Then

$$\frac{1}{\rho }[H(Ax,Bx)-H(A{x}_n,B{x}_n)+\rho y]\in M_n(x_n).$$

Let

$$y_n=\frac{1}{\rho }[H(Ax,Bx)-H(A{x}_n,B{x}_n)+\rho y].$$

It follows from (4.3) that

$$\begin{array}{rl}\parallel y_n-y\parallel & \le \parallel \frac{1}{\rho }[H(Ax,Bx)-H(A{x}_n,B{x}_n)+\rho y]-y\parallel =\frac{1}{\rho }\parallel H(Ax,Bx)-H(A{x}_n,B{x}_n)\parallel \\ \le \frac{1}{\rho }(s+t)\parallel x_n-x\parallel .\end{array}$$

(4.4)

Since $R_{\rho ,M_n}^{H(\cdot ,\cdot )}\to R_{\rho ,M}^{H(\cdot ,\cdot )}$ for any$u\in X$,we know that $\parallel x_n-x\parallel \to 0$. Now (4.4)implies that

$$y_n\to y\text{as }n\to \mathrm{\infty },$$

and so $M_n\stackrel{G}{⟶}M$.This completes the proof. □

5 An application of the $H(\cdot ,\cdot )$-mixed mappingfor solving the system of generalized variational inclusions

Throughout the rest of the paper, unless otherwise stated, we assume that for each$i=1,2$,$E_i$is a $q_i$-uniformlysmooth Banach space with the norm ${\parallel \cdot \parallel }_i$.

Let $A_1,B_1:X_1\to X_1$,$A_2,B_2:X_2\to X_2$,$N_1,H_1:X_1\times X_2\to X_1$and $N_2,H_2:X_1\times X_2\to X_2$be nonlinear mappings. Let $M_1:X_1\to 2^{X_1}$be $H_1(\cdot ,\cdot )$-mixed and$M_2:X_2\to 2^{X_2}$be $H_2(\cdot ,\cdot )$-mixed mappings,respectively. We consider the following system of generalized variational inclusions(SGVI): Find $(x,y)\in X_1\times X_2$such that

$$\{\begin{array}{c}{\theta }_1\in N_1(x,y)+M_1(x);\hfill \\ {\theta }_2\in N_2(x,y)+M_2(y),\hfill \end{array}$$

(5.1)

where ${\theta }_1$,${\theta }_2$are zero vectors of $X_1$ and$X_2$,respectively. The problem of type (5.1) was studied by Zou and Huang [20].

Definition 5.1 Let $A:X_1\to X_1$.A mapping $N:X_1\times X_2\to X_1$is said to be:

• (i) κ-strongly accretive in the first argument with respect to A if there exists a constant $\kappa >0$ such that

  $${〈N(x_1,y)-N(x_2,y),J_q(A(x_1)-A(x_2))〉}_1\ge \kappa {\parallel x-y\parallel }_1^{q_1},\mathrm{\forall }{x}_1,x_2\in X_1,y\in X_2;$$

• (ii) $L_N$-Lipschitz continuous in the first argument if there exists a constant $L_N>0$ such that

  $${\parallel N(x_1,y)-N(x_2,y)\parallel }_1\le L_N{\parallel x_1-x_2\parallel }_1^{q_1},\mathrm{\forall }{x}_1,x_2\in X_1,y\in X_2;$$

• (iii) $l_N$-Lipschitz continuous in the second argument if there exists a constant $l_N>0$ such that

  $${\parallel N(x,y_1)-N(x,y_2)\parallel }_1\le l_N{\parallel y_1-y_2\parallel }_2^{q_2},\mathrm{\forall }x\in X_1,y_1,y_2\in X_2.$$

The following lemma, which will be used in the sequel, is an immediate consequence ofthe definitions of $R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}$,$R_{{\rho }_2,M_2}^{H_2(\cdot ,\cdot )}$.

Lemma 5.2 For any given$(x,y)\in X_1\times X_2$,$(x,y)$is a solution of (SGVI) (5.1) if and only if$(x,y)$satisfies

$$x=R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x)-{\rho }_1{N}_1(x,y)],$$

(5.2)

$$y=R_{{\rho }_2,M_2}^{H_2(\cdot ,\cdot )}[H_2(A_2,B_2)(y)-{\rho }_2{N}_2(x,y)],$$

(5.3)

where$R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}={(H_1(A_1,B_1)+{\rho }_1{M}_1)}^{-1}$and$R_{{\rho }_2,M_2}^{H_2(\cdot ,\cdot )}={(H_2(A_2,B_2)+{\rho }_2{M}_2)}^{-1}$,and${\rho }_1,{\rho }_2>0$are constants.

Proof Consider first that an element $(x,y)\in X_1\times X_2$is a solution to (5.1). Then it follows that

$$\begin{array}{r}{\theta }_1\in N_1(x,y)+M_1(x)\\ \Rightarrow H_1(A_{1}x,B_{1}x)\in H_1(A_{1}x,B_{1}x)+{\rho }_1{N}_1(x,y)+{\rho }_1{M}_1(x)\\ \Rightarrow H_1(A_{1}x,B_{1}x)-{\rho }_1{N}_1(x,y)\in H_1(Ax,Bx)+{\rho }_1{M}_1(x)\\ \Rightarrow x=R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}[H_1(A_{1}x,B_{1}x)-{\rho }_1{N}_1(x,y)].\end{array}$$

In a similar way, we can show that

$$y=R_{{\rho }_2,M_2}^{H_2(\cdot ,\cdot )}[H_2(A_{2}y,B_{2}y)-{\rho }_2{N}_2(x,y)].$$

 □

A similar proof follows for the converse part:

$$\begin{array}{c}x=R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x)-{\rho }_1{N}_1(x,y)]\hfill \\ \Rightarrow x={(H_1(A_1,B_1)+{\rho }_1{M}_1)}^{-1}[H_1(A_1,B_1)(x)-{\rho }_1{N}_1(x,y)]\hfill \\ \Rightarrow (H_1(A_1,B_1)+{\rho }_1{M}_1)(x)\ni (H_1(A_{1}x,B_{1}x))-{\rho }_1{N}_1(x,y)\hfill \\ \Rightarrow {\theta }_1\in N_1(x,y)+M_1(x).\hfill \end{array}$$

In a similar way, we can show that

$${\theta }_2\in N_2(x,y)+M_2(y).$$

Theorem 5.3 For each$i=1,2$,let$X_i$be$q_i$-uniformlysmooth Banach spaces, let$A_i,B_i:X_i\to X_i$be single-valued mappings. Let the set-valuedmappings$M_i:X_i\to 2^{X_i}$be such that$M_i$are$H_i(\cdot ,\cdot )$-mixed mappingswith respect to mappings$A_i$and$B_i$,and$A_i$are${\alpha }_i$-expansiveand${\mu }_i>{\gamma }_i$with$r_i={\mu }_i{\alpha }_i^{q_i}-{\gamma }_i>{\rho }_i{m}_i$.Let$H_i:X_1\times X_2\to X_i$be$s_i$-Lipschitzcontinuous with respect to$A_i$and$t_i$-Lipschitzcontinuous with respect to$B_i$,and let$N_i:X_1\times X_2\to X_i$be a${\kappa }_i$-stronglyaccretive mapping in the ith argument, $L_{N_i}$-Lipschitzcontinuous in the first argument and$l_{N_i}$-Lipschitzcontinuous in the second argument. Suppose that there are twoconstants${\rho }_1,{\rho }_2>0$satisfying the following conditions:

$$\{\begin{array}{c}{\tau }_1=a_1+{\rho }_2{L}_2{L}_{N_2}<1;\hfill \\ {\tau }_2=a_2+{\rho }_2{L}_1{l}_{N_1}<1,\hfill \end{array}$$

(5.4)

where

$$\begin{array}{c}{a}_1=L_1[{(1-2q_1{r}_1+c_{q_1}{(s_1+t_1)}^{q_1})}^{\frac{1}{q_1}}+{(1-2{\rho }_1{q}_1{\kappa }_1+c_{q_1}{\rho }_1^{q_1}{L}_1^{q_1})}^{\frac{1}{q_1}}];\hfill \\ a_2=L_2[{(1-2q_2{r}_2+c_{q_2}{(s_2+t_2)}^{q_2})}^{\frac{1}{q_2}}+{(1-2{\rho }_2{q}_2{\kappa }_2+c_{q_2}{\rho }_2^{q_2}{L}_2^{q_2})}^{\frac{1}{q_2}}];\hfill \\ L_1=\frac{1}{r_1-{\rho }_1{m}_1};\hfill \\ L_2=\frac{1}{r_2-{\rho }_2{m}_2}.\hfill \end{array}$$

Then SGVI (5.1) has a unique solution$(x,y)\in X_1\times X_2$.

Proof For $i=1,2$,it follows that for $(x,y)\in X_1\times X_2$,the proximal-point mappings $R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}$ and$R_{{\rho }_2,M_2}^{H_2(\cdot ,\cdot )}$ are$\frac{1}{r_1-{\rho }_1{m}_1}$-Lipschitzcontinuous and $\frac{1}{r_2-{\rho }_2{m}_2}$-Lipschitzcontinuous, respectively.

Let $R:X_1\times X_2\to X_1\times X_2$be defined as follows:

$$R(x,y)=(P(x,y),Q(x,y)),\mathrm{\forall }(x,y)\in X_1\times X_2,$$

(5.5)

where $P:X_1\times X_2\to X_1$and $Q:X_1\times X_2\to X_2$are defined by

$$P(x,y)=R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x)-{\rho }_1{N}_1(x,y)]$$

(5.6)

and

$$Q(x,y)=R_{{\rho }_2,M_2}^{H_2(\cdot ,\cdot )}[H_2(A_2,B_2)(x)-{\rho }_2{N}_2(x,y)]$$

(5.7)

for ${\rho }_1,{\rho }_2>0$,respectively.

For any $(x_1,y_1),(x_2,y_2)\in X_1\times X_2$,it follows from (5.6) and (5.7) and the Lipschitz continuity of$R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}$ and$R_{{\rho }_2,M_2}^{H_2(\cdot ,\cdot )}$ that

$$\begin{array}{c}{\parallel P(x_1,y_1)-P(x_2,y_2)\parallel }_1\hfill \\ =\parallel R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x_1)-\rho N_1(x_1,y_1)]\hfill \\ -R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x_2)-\rho N_1(x_2,y_2)]{\parallel }_1\hfill \\ \le {\mathrm{Ł}}_1[{\parallel H_1(A_1,B_1)(x_1)-H_1(A_1,B_1)(x_2)-{\rho }_1(N_1(x_1,y_1)-N_1(x_2,y_1))\parallel }_1\hfill \\ +{\rho }_1{\parallel N_1(x_2,y_1)-N_1(x_2,y_2)\parallel }_1]\hfill \end{array}$$

(5.8)

and

$$\begin{array}{c}{\parallel Q(x_1,y_1)-Q(x_2,y_2)\parallel }_2\hfill \\ \le {\mathrm{Ł}}_2[{\parallel H_2(A_2,B_2)(y_1)-H_2(A_2,B_2)(y_2)-{\rho }_2(N_2(x_1,y_1)-N_2(x_1,y_2))\parallel }_2\hfill \\ +{\rho }_2{\parallel N_2(x_1,y_2)-N_2(x_2,y_2)\parallel }_2].\hfill \end{array}$$

(5.9)

Now

$$\begin{array}{c}{\parallel H_1(A_1,B_1)(x_1)-H_1(A_1,B_1)(x_2)-{\rho }_1(N_1(x_1,y_1)-N_1(x_2,y_1))\parallel }_1\hfill \\ ={\parallel H_1(A_1,B_1)(x_1)-H_1(A_1,B_1)(x_2)-(x_1-x_2)\parallel }_1\hfill \\ +{\parallel x_1-x_2-{\rho }_1(N_1(x_1,y_1)-N_1(x_2,y_1))\parallel }_1.\hfill \end{array}$$

(5.10)

Also,

$$\begin{array}{c}{\parallel H_2(A_2,B_2)(y_1)-H_2(A_2,B_2)(y_2)-(y_1-y_2)\parallel }_2\hfill \\ ={\parallel H_2(A_2,B_2)(y_1)-H_2(A_2,B_2)(y_2)-(y_1-y_2)\parallel }_2\hfill \\ +{\parallel y_1-y_2-{\rho }_2(N_2(x_1,y_1)-N_1(x_1,y_2))\parallel }_2.\hfill \end{array}$$

(5.11)

Now

$$\begin{array}{c}{\parallel H_1(A_1,B_1)(x_1)-H_1(A_1,B_1)(x_2)-(x_1-x_2)\parallel }_1^{q_1}\hfill \\ \le {\parallel x_1-x_2\parallel }_1^{q_1}-2q_1{〈H_1(A_1,B_1)(x_1)-H_1(A_1,B_1)(x_2)-J_{q_1}(x_1-x_2)〉}_1\hfill \\ +c_{q_1}{\parallel H_1(A_1,B_1)(x_1)-H_1(A_1,B_1)(x_2)\parallel }_1^{q_1}.\hfill \end{array}$$

(5.12)

Since $M_1$is an $H(\cdot ,\cdot )$-mixed mapping, then$H_1(A_1,B_1)$ is${\mu }_1$-cocoercivewith respect to $A_1$and ${\gamma }_1$-relaxedaccretive with respect to $B_1$, andfrom the fact that $A_1$ is${\alpha }_1$-expansive,we can obtain

$$\begin{array}{c}{〈H_1(A_1,B_1)(x_1)-H_1(A_2,B_2)(x_2)-J_{q_1}(x_1-x_2)〉}_1\hfill \\ ={〈H_1(A_1,B_1)(x_1)-H_1(A_2,B_1)(x_2)-J_{q_1}(x_1-x_2)〉}_1\hfill \\ +{〈H_1(A_2,B_1)(x_1)-H_1(A_2,B_2)(x_2)-J_{q_1}(x_1-x_2)〉}_1\hfill \\ \le {\mu }_1{\parallel A_1(x_1)-A_1(x_2)\parallel }^{q_1}-{\gamma }_1{\parallel x_1-x_2\parallel }^{q_1}\hfill \\ \le {\mu }_1{\alpha }_1^{q_1}{\parallel x_1-x_2\parallel }^{q_1}-{\gamma }_1{\parallel x_1-x_2\parallel }^{q_1}\hfill \\ =({\mu }_1{\alpha }_1^{q_1}-{\gamma }_1){\parallel x_1-x_2\parallel }^{q_1}.\hfill \end{array}$$

(5.13)

Since $H_1(A_1,B_1)$ is$s_1$-Lipschitzcontinuous with respect to $A_1$ and$t_1$-Lipschitzcontinuous with respect to $B_1$, wehave

$$\begin{array}{c}{\parallel H_1(A_1,B_1)(x_1)-H_1(A_1,B_1)(x_2)-(x_1-x_2)\parallel }_1\hfill \\ \le {\parallel H_1(A_1,B_1)(x_1)-H_1(A_2,B_1)(x_2)\parallel }_1+{\parallel H_1(A_2,B_1)(x_1)-H_1(A_2,B_2)(x_2)\parallel }_1\hfill \\ \le (s_1+t_1){\parallel x_1-x_2\parallel }_1.\hfill \end{array}$$

(5.14)

Using (5.12), (5.13) and (5.14), we have

$$\begin{array}{c}{\parallel H_1(A_1,B_1)(x_1)-H_1(A_1,B_1)(x_2)-(x_1-x_2)\parallel }_1^{q_1}\hfill \\ \le {\parallel x_1-x_2\parallel }_1^{q_1}-2q_1{〈H_1(A_1,B_1)(x_1)-H_1(A_1,B_1)(x_2)-J_{q_1}(x_1-x_2)〉}_1\hfill \\ \le [1-2q_1{r}_1+c_{q_1}{(s_1+t_1)}^{q_1}]{\parallel x_1-x_2\parallel }_1^{q_1},\hfill \end{array}$$

which implies that

$$\begin{array}{c}{\parallel H_1(A_1,B_1)(x_1)-H_1(A_1,B_1)(x_2)-(x_1-x_2)\parallel }_1\hfill \\ \le {[1-2q_1{r}_1+c_{q_1}{(s_1+t_1)}^{q_1}]}^{\frac{1}{q_1}}{\parallel x_1-x_2\parallel }_1,\text{where }{r}_1={\mu }_1{\alpha }_1^{q_1}-{\gamma }_1.\hfill \end{array}$$

(5.15)

In the light of (5.15), we can obtain

$$\begin{array}{c}{\parallel H_2(A_2,B_2)(y_1)-H_2(A_2,B_2)(y_2)-(y_1-y_2)\parallel }_2\hfill \\ \le {[1-2q_2{r}_2+c_{q_2}{(s_2+t_2)}^{q_2}]}^{\frac{1}{q_2}}{\parallel y_1-y_2\parallel }_2,\text{where }{r}_2={\mu }_2{\alpha }_2^{q_2}-{\gamma }_2.\hfill \end{array}$$

(5.16)

Again, since $N_i$is ${\kappa }_i$-stronglyaccretive in the first argument and $L_{N_i}$-Lipschitzcontinuous in the first argument and $l_{N_i}$-Lipschitzcontinuous in the second argument, then using Lemma 2.4, we have

$$\begin{array}{c}{\parallel x_1-x_2-{\rho }_1(N_1(x_1,y_1)-N_1(x_2,y_1))\parallel }_1^{q_1}\hfill \\ \le {\parallel x_1-x_2\parallel }_1^{q_1}-2{\rho }_1{q}_1{〈N_1(x_1,y_1)-N_1(x_2,y_1),J_{q_1}(x_1-x_2)〉}_1\hfill \\ +{\rho }_1^{q_1}{c}_{q_1}{\parallel N_1(x_1,y_1)-N_1(x_2,y_1)\parallel }_1^{q_1}\hfill \\ \le (1-2{\rho }_1{q}_1{\kappa }_1+c_{q_1}{\rho }_1^{q_1}{L}_1^{q_1}){\parallel x_1-x_2\parallel }_1^{q_1},\hfill \end{array}$$

which implies that

$${\parallel x_1-x_2-{\rho }_1(N_1(x_1,y_1)-N_1(x_2,y_1))\parallel }_1\le {(1-2{\rho }_1{q}_1{\kappa }_1+c_{q_1}{\rho }_1^{q_1}{L}_1^{q_1})}^{\frac{1}{q_1}}{\parallel x_1-x_2\parallel }_1.$$

(5.17)

In the light of (5.17), we have

$${\parallel y_1-y_2-{\rho }_2(N_2(x_1,y_1)-N_2(x_1,y_2))\parallel }_2\le {(1-2{\rho }_2{q}_2{\kappa }_2+c_{q_2}{\rho }_2^{q_2}{L}_2^{q_2})}^{\frac{1}{q_2}}{\parallel y_1-y_2\parallel }_2.$$

(5.18)

Using (5.8), (5.15) and (5.17), we have

$$\begin{array}{c}{\parallel P(x_1,y_1)-P(x_2,y_2)\parallel }_1\hfill \\ \le \left(L_1[{(1-2q_1{r}_1+c_{q_1}{(s_1+t_1)}^{q_1})}^{\frac{1}{q_1}}+{(1-2{\rho }_1{q}_1{\kappa }_1+c_{q_1}{\rho }_1^{q_1}{L}_1^{q_1})}^{\frac{1}{q_1}}]\right){\parallel x_1-x_2\parallel }_1\hfill \\ +L_1{\rho }_1{l}_{N_1}{\parallel y_1-y_2\parallel }_2.\hfill \end{array}$$

(5.19)

Using (5.9), (5.16) and (5.18), we have

$$\begin{array}{c}{\parallel Q(x_1,y_1)-Q(x_2,y_2)\parallel }_2\hfill \\ \le \left(L_2[{(1-2q_2{r}_2+c_{q_2}{(s_2+t_2)}^{q_2})}^{\frac{1}{q_2}}+{(1-2{\rho }_2{q}_2{\kappa }_2+c_{q_2}{\rho }_2^{q_2}{L}_2^{q_2})}^{\frac{1}{q_2}}]\right){\parallel y_1-y_2\parallel }_2\hfill \\ +L_2{\rho }_2{l}_{N_2}{\parallel x_1-x_2\parallel }_1.\hfill \end{array}$$

(5.20)

From (5.19) and (5.20), we have

$$\begin{array}{r}{\parallel P(x_1,y_1)-P(x_2,y_2)\parallel }_1+{\parallel Q(x_1,y_1)-Q(x_2,y_2)\parallel }_2\\ \le {\tau }_1{\parallel x_1-x_2\parallel }_1+{\tau }_2{\parallel y_1-y_2\parallel }_2\\ \le max\{{\tau }_1,{\tau }_2\}({\parallel x_1-x_2\parallel }_1+{\parallel y_1-y_2\parallel }_2),\end{array}$$

(5.21)

where

$$\{\begin{array}{c}{\tau }_1=a_1+{\rho }_2{L}_2{L}_{N_2};\hfill \\ {\tau }_2=a_2+{\rho }_2{L}_1{l}_{N_1},\hfill \end{array}$$

(5.22)

and

$$\begin{array}{c}{a}_1=L_1[{(1-2q_1{r}_1+c_{q_1}{(s_1+t_1)}^{q_1})}^{\frac{1}{q_1}}+{(1-2{\rho }_1{q}_1{\kappa }_1+c_{q_1}{\rho }_1^{q_1}{L}_1^{q_1})}^{\frac{1}{q_1}}];\hfill \\ a_2=L_2[{(1-2q_2{r}_2+c_{q_2}{(s_2+t_2)}^{q_2})}^{\frac{1}{q_2}}+{(1-2{\rho }_2{q}_2{\kappa }_2+c_{q_2}{\rho }_2^{q_2}{L}_2^{q_2})}^{\frac{1}{q_2}}];\hfill \\ L_1=\frac{1}{r_1-{\rho }_1{m}_1};\hfill \\ L_2=\frac{1}{r_2-{\rho }_2{m}_2}.\hfill \end{array}$$

Now define the norm ${\parallel \cdot \parallel }_{\star }$on $X_1\times X_2$by

$${\parallel (x,y)\parallel }_{\star }={\parallel x\parallel }_1+{\parallel y\parallel }_2,\mathrm{\forall }(x,y)\in X_1\times X_2.$$

(5.23)

We observe that $(X_1\times X_2,{\parallel \cdot \parallel }_{\star })$ is a Banach space.Hence it follows from (5.5), (5.21) and (5.23) that

$${\parallel R(x_1,y_1)-R(x_2,y_2)\parallel }_{\star }=max\{{\tau }_1,{\tau }_2\}{\parallel (x_1,y_1)-(x_2,y_2)\parallel }_{\star }.$$

(5.24)

Since $max\{{\tau }_1,{\tau }_2\}<1$ by (5.2), itfollows from (5.24) that R is a contraction mapping. Hence, by the Banachcontraction principle, there exists a unique point $(x,y)\in X_1\times X_2$such that

$$R(x,y)=(x,y),$$

which implies that

$$\begin{array}{c}x=R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x)-{\rho }_1{N}_1(x,y)],\hfill \\ y=R_{{\rho }_2,M_2}^{H_2(\cdot ,\cdot )}[H_2(A_2,B_2)(y)-{\rho }_2{N}_2(x,y)].\hfill \end{array}$$

It follows from Lemma 5.2 that $(x,y)$ is a unique solutionof SGVI (5.1). This completes the proof. □

6 Convergence of an iterative algorithm for SGVI (5.1)

Based on Lemma 5.2, we suggest and analyze the following iterative algorithm forfinding an approximate solution for SGVI (5.1).

Algorithm 6.1 For any given $(x_0,y_0)\in X_1\times X_2$,$(x_n,y_n)\in X_1\times X_2$by an iterative scheme

$$x_{n+1}=R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x_n)-{\rho }_1{N}_1(x_n,y_n)],$$

(6.1)

$$y_{n+1}=R_{{\rho }_2,M_2}^{H_2(\cdot ,\cdot )}[H_2(A_2,B_2)(y_n)-{\rho }_2{N}_2(x_n,y_n)],$$

(6.2)

where $n=0,1,2,\dots$and ${\rho }_1,{\rho }_2>0$are constants.

Theorem 6.2 For each$i=1,2$,let$X_i$be$q_i$-uniformlysmooth Banach spaces, let$A_i,B_i:X_i\to X_i$be single-valued mappings. Let the set-valuedmappings$M_{n_i},M_i:X_i\to 2^{X_i}$be such that$M_{n_i}$,$M_i$are$H_i(\cdot ,\cdot )$-mixed mappingswith respect to mappings$A_i$and$B_i$such that$M_{n_i}\stackrel{G}{⟶}{M}_i$for$n=0,1,2,\dots$ ,and$A_i$is${\alpha }_i$-expansiveand${\mu }_i>{\gamma }_i$with$r_i={\mu }_i{\alpha }_i^{q_i}-{\gamma }_i>{\rho }_i{m}_i$.Let$H_i:X_1\times X_2\to X_i$be$s_i$-Lipschitzcontinuous with respect to$A_i$and$t_i$-Lipschitzcontinuous with respect to$B_i$,and let$N_i:X_1\times X_2\to X_i$be a${\kappa }_i$-stronglyaccretive mapping in the ith argument, $L_{N_i}$-Lipschitzcontinuous in the first argument and$l_{N_i}$-Lipschitzcontinuous in the second argument. Suppose that there are twoconstants${\rho }_1,{\rho }_2>0$satisfying the following conditions:

$$\{\begin{array}{c}{\tau }_1=a_1+{\rho }_2{L}_2{L}_{N_2}<1;\hfill \\ {\tau }_2=a_2+{\rho }_2{L}_1{l}_{N_1}<1,\hfill \end{array}$$

(6.3)

where

$$\begin{array}{c}{a}_1=L_1[{(1-2q_1{r}_1+c_{q_1}{(s_1+t_1)}^{q_1})}^{\frac{1}{q_1}}+{(1-2{\rho }_1{q}_1{\kappa }_1+c_{q_1}{\rho }_1^{q_1}{L}_1^{q_1})}^{\frac{1}{q_1}}];\hfill \\ a_2=L_2[{(1-2q_2{r}_2+c_{q_2}{(s_2+t_2)}^{q_2})}^{\frac{1}{q_2}}+{(1-2{\rho }_2{q}_2{\kappa }_2+c_{q_2}{\rho }_2^{q_2}{L}_2^{q_2})}^{\frac{1}{q_2}}];\hfill \\ L_1=\frac{1}{r_1-{\rho }_1{m}_1};\hfill \\ L_2=\frac{1}{r_2-{\rho }_2{m}_2}.\hfill \end{array}$$

Then the approximate solution$(x_n,y_n)$generated by Algorithm  6.1 converges strongly to the uniquesolution$(x,y)$of SGVI (5.1).

Proof By Theorem 5.3, there exists a unique solution$(x,y)\in X_1\times X_2$of SGVI (5.1). It follows from Algorithm 6.1 and Theorem 3.6 that

$$\begin{array}{rcl}\parallel x_{n+1}-x\parallel & =& \parallel R_{{\rho }_1,M_{n_1}}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x_n)-{\rho }_1{N}_1(x_n,y_n)]\\ -R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x)-{\rho }_1{N}_1(x,y)]{\parallel }_1\\ \le & \parallel R_{{\rho }_1,M_{n_1}}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x_n)-{\rho }_1{N}_1(x_n,y_n)]\\ -R_{{\rho }_1,M_{n_1}}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x)-{\rho }_1{N}_1(x,y)]{\parallel }_1\\ +\parallel R_{{\rho }_1,M_{n_1}}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x)-{\rho }_1{N}_1(x,y)]\\ -R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x)-{\rho }_1{N}_1(x,y)]{\parallel }_1\end{array}$$

(6.4)

and

$$\begin{array}{rcl}\parallel y_{n+1}-y\parallel & \le & \parallel R_{{\rho }_2,M_{n_2}}^{H_2(\cdot ,\cdot )}[H_2(A_2,B_2)(y_n)-{\rho }_2{N}_2(x_n,y_n)]\\ -R_{{\rho }_2,M_{n_2}}^{H_2(\cdot ,\cdot )}[H_2(A_2,B_2)(y)-{\rho }_2{N}_2(x,y)]{\parallel }_2\\ +\parallel R_{{\rho }_2,M_{n_2}}^{H_2(\cdot ,\cdot )}[H_2(A_2,B_2)(y)-{\rho }_2{N}_2(x,y)]\\ -R_{{\rho }_2,M_{n_2}}^{H_2(\cdot ,\cdot )}[H_2(A_2,B_2)(y)-{\rho }_2{N}_2(x,y)]{\parallel }_2.\end{array}$$

(6.5)

By (5.8) and (5.19), we have

$$\begin{array}{c}\parallel R_{{\rho }_1,M_{n_1}}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x_n)-{\rho }_1{N}_1(x_n,y_n)]\hfill \\ -R_{{\rho }_1,M_{n_1}}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x)-{\rho }_1{N}_1(x,y)]{\parallel }_1\hfill \\ \le {\mathrm{Ł}}_1[{\parallel H_1(A_1,B_1)(x_n)-H_1(A_1,B_1)(x)-{\rho }_1(N_1(x_n,y_n)-N_1(x,y_n))\parallel }_1\hfill \\ +{\rho }_1{\parallel N_1(x,y_n)-N_1(x,y)\parallel }_1]\hfill \\ \le \left(L_1[{(1-2q_1{r}_1+c_{q_1}{(s_1+t_1)}^{q_1})}^{\frac{1}{q_1}}+{(1-2{\rho }_1{q}_1{\kappa }_1+c_{q_1}{\rho }_1^{q_1}{L}_1^{q_1})}^{\frac{1}{q_1}}]\right)\hfill \\ \times {\parallel x_n-x\parallel }_1+L_1{\rho }_1{l}_{N_1}{\parallel y_n-y\parallel }_2\hfill \end{array}$$

(6.6)

and

$$\begin{array}{c}{\parallel R_{{\rho }_2,M_{n_2}}^{H_2(\cdot ,\cdot )}[H_2(A_2,B_2)(y_n)-{\rho }_2{N}_2(x_n,y_n)]-R_{{\rho }_2,M_{n_2}}^{H_2(\cdot ,\cdot )}[H_2(A_2,B_2)(y)-{\rho }_2{N}_2(x,y)]\parallel }_2\hfill \\ \le {\mathrm{Ł}}_2[{\parallel H_2(A_2,B_2)(y_n)-H_2(A_2,B_2)(y)-{\rho }_2(N_2(x_n,y_n)-N_1(x_n,y))\parallel }_2\hfill \\ +{\rho }_2{\parallel N_2(x_n,y)-N_2(x,y)\parallel }_2]\hfill \\ \le \left(L_2[{(1-2q_2{r}_2+c_{q_2}{(s_2+t_2)}^{q_2})}^{\frac{1}{q_2}}+{(1-2{\rho }_2{q}_2{\kappa }_2+c_{q_2}{\rho }_2^{q_2}{L}_2^{q_2})}^{\frac{1}{q_2}}]\right)\hfill \\ \times {\parallel y_n-y\parallel }_2+L_2{\rho }_2{l}_{N_2}{\parallel x_n-x\parallel }_1.\hfill \end{array}$$

(6.7)

By Theorem 4.2, we have

$$R_{{\rho }_1,M_{n_1}}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x)-{\rho }_1{N}_1(x,y)]\to R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x)-{\rho }_1{N}_1(x,y)],$$

(6.8)

$$R_{{\rho }_2,M_{n_2}}^{H_2(\cdot ,\cdot )}[H_2(A_2,B_2)(y)-{\rho }_2{N}_2(x,y)]\to R_{{\rho }_2,M_2}^{H_2(\cdot ,\cdot )}[H_1(A_2,B_2)(y)-{\rho }_2{N}_2(x,y)].$$

(6.9)

Let

$$\begin{array}{c}{b}_{n_1}=\parallel R_{{\rho }_1,M_{n_1}}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x)-{\rho }_1{N}_1(x,y)]\hfill \\ \phantom{b_{n_1}=}-R_{{\rho }_1,M_1}^{H_1(\cdot ,\cdot )}[H_1(A_1,B_1)(x)-{\rho }_1{N}_1(x,y)]{\parallel }_1,\hfill \end{array}$$

(6.10)

$$\begin{array}{c}{b}_{n_2}=\parallel R_{{\rho }_2,M_{n_2}}^{H_2(\cdot ,\cdot )}[H_2(A_2,B_2)(y)-{\rho }_2{N}_2(x,y)]\hfill \\ \phantom{b_{n_2}=}-R_{{\rho }_2,M_2}^{H_2(\cdot ,\cdot )}[H_2(A_2,B_2)(y)-{\rho }_2{N}_2(x,y)]{\parallel }_2.\hfill \end{array}$$

(6.11)

From (6.4)-(6.11), we have

$${\parallel x_{n+1}-x\parallel }_1\le {\tau }_1{\parallel x_n-x\parallel }_1+b_{n_1},$$

(6.12)

$${\parallel y_{n+1}-y\parallel }_1\le {\tau }_2{\parallel y_n-y\parallel }_1+b_{n_2}.$$

(6.13)

From (6.12) and (6.13), we have

$${\parallel x_{n+1}-x\parallel }_1+{\parallel y_{n+1}-y\parallel }_2\le max\{{\tau }_1,{\tau }_2\}\{{\parallel x_n-x\parallel }_1+{\parallel y_n-y\parallel }_2\}+\{b_{n_1}+b_{n_2}\}.$$

(6.14)

Since $(X_1\times X_2,{\parallel \cdot \parallel }_{\star })$ is a Banach spacewith the norm ${\parallel \cdot \parallel }_{\star }$defined by (5.23), it follows from (5.5), (5.23) and (6.14) that

$$\begin{array}{rl}{\parallel (x_{n+1},y_{n+1})-(x,y)\parallel }_{\star }& ={\parallel x_{n+1}-x\parallel }_1+{\parallel y_{n+1}-y\parallel }_2\\ \le max\{{\tau }_1,{\tau }_2\}\parallel (x_n,y_n)-(x,y)\parallel +\{b_{n_1}+b_{n_2}\}.\end{array}$$

(6.15)

By condition (6.3), it follows that $max\{{\tau }_1,{\tau }_2\}<1$ andLemma 2.5 implies that ${\parallel (x_{n+1},y_{n+1})-(x,y)\parallel }_{\star }\to 0$as $n\to \mathrm{\infty }$.

Thus $\{(x_n,y_n)\}$ converges strongly to the unique solution$(x,y)$ of SGVI (5.1). Thiscompletes the proof. □