Ishikawa-hybrid proximal point algorithm for NSVI system

Abstract

A nonlinear set-valued inclusions system framework for an Ishikawa-hybrid proximal point algorithm is developed and studied using the notion of an $(A,\eta )$-accretive mapping. Convergence analysis for the algorithm of solving the nonlinear set-valued inclusions system and existence analysis of solution for the system are explored along with some results on the resolvent operator corresponding to the $(A,\eta )$-accretive mapping in a Banach space. The result that the sequence generated by the algorithm converges linearly to a solution of the system with the convergence rate $\parallel \mathbf{\Psi }\parallel$ is proved.

MSC:49J40, 47H06.

1 Introduction

The nonlinear set-valued inclusions system, which was introduced and studied by Hassouni and Moudafi [1], is a useful and important extension of the variational inequality and variational inclusions system. In recent years, various variational inclusions systems and nonlinear set-valued inclusions systems have been intensively studied. For example, Kassay and Kolumbán [2], Chen, Deng and Tan [3], Yan, Fang and Huang [4], Fang, Huang and Thompson [5], Jin [6], Verma [7], Li, Xu and Jin [8], Kang, Cho and Liu [9]et al. introduced and studied various set-valued variational inclusions systems. For the past few years, many existence results and iterative algorithms for various variational inclusions systems have been studied. For details, please see [1–28] and the references therein.

Example 1.1 In 2001, Chen, Deng and Tan [3] have studied the problem associated with the following system of variational inequalities, which is finding $(x,y)\in H\times H$ (H, Hilbert space) such that

$$\{\begin{array}{c}〈\rho T(y)+x-y,w-x〉\ge \rho {\phi }_1(x)-\rho {\phi }_1(w),\hfill \\ 〈tS(x)+y-x,w-y〉\ge t{\phi }_2(y)-\rho {\phi }_2(w)(\mathrm{\forall }w\in H),\hfill \end{array}$$

(1)

where ${\phi }_i:H\to R\cup \mathrm{\infty }$ is a proper, convex, lower semicontinuous functional and $\partial {\phi }_i(\cdot )$ denotes the subdifferential operator of ${\phi }_i$ ($i=1,2$).

Example 1.2 Let X be a real q-uniformly smooth Banach space, and $S,T,M_1,M_2:X\to X$ be four single-valued mappings. Find $x,y\in X$ such that

$$\{\begin{array}{c}0\in x-y+\rho T(y)+\rho M_1(x),\hfill \\ 0\in y-x+tS(x)+t{M}_2(y),\hfill \end{array}$$

(2)

which is studied by Jin in [6].

Inspired and motivated by Examples 1.1-1.2 and recent research work in this field (see [7, 8]), in this paper, we will introduce and discuss the problem associated with the following class of new nonlinear set-valued inclusions systems (NSVI Systems), which is finding $(x,y)\in X\times X$ for any $f,g:X\to X$ such that $z\in S(x)$, $w\in T(y)$, and

$$\{\begin{array}{c}f(x)\in F(z,y)+M(y),\hfill \\ g(y)\in G(w,x)+N(x),\hfill \end{array}$$

(3)

where X is a real q-uniformly smooth Banach space, $A,B:X\to X$, ${\eta }_1,{\eta }_2:X\times X\to X$, and $F,G:X\times X\to X$ are single-valued mappings; $M:X\to 2^X$ is a set-valued $(A,{\eta }_1)$-accretive mapping and $N:X\to 2^X$ is a set-valued $(B,{\eta }_2)$-accretive mapping, and $S,T:X\to CB(X)$ are two set-valued mappings.

If $f(x)=-x$, $g(y)=-y$, $F(w,y)=\rho T(y)-y$, $G(y,x)=tS(x)-x$, $N=t{M}_2$ and $M=\rho M_1(\cdot )$, then the problem (3) reduces to Example 1.1. If $M_1=M_2=\partial \phi$, ${\phi }_i:H\to R\cup \mathrm{\infty }$ is a proper, convex, lower semicontinuous functional and $\partial {\phi }_i(\cdot )$ denotes the subdifferential operator of ${\phi }_i$ ($X=H$, Hilbert space, and $i=1,2$), then the problem (3) changes to Example 1.2.

If X is a real q-uniformly smooth Banach space, and $G(\cdot ,\cdot )=N(y,g(x))$, $f(u)=u$ and $S(u)=Q(u)(u\in X)$, then the problem (3) reduces to the problem associated with the following variational inclusions:

For any $u\in X$, find $x\in X$ and $y=Q(x)$ such that

$$u\in N(y,g(x))+M(y),$$

(4)

which is developed by Li in 2010 [8].

The main purpose of this paper is to introduce and study a generalized nonlinear set-valued inclusions system framework for an Ishikawa-hybrid proximal point algorithm using the notion of $(A,\eta )$-accretive due to Lan-Cho-Verma [10] in a Banach space, to analyse convergence for the algorithm of solving the system and existence of a solution for the system and to prove the result that sequence ${\{(x^n,y^n)\}}_{n=0}^{\mathrm{\infty }}$ generated by the algorithm converges linearly to a solution of the nonlinear set-valued inclusions system with the convergence rate $\parallel \mathbf{\Psi }\parallel$.

2 Preliminaries

Let X be a real q-uniformly smooth Banach space with a dual space $X^{\ast }$, $〈\cdot ,\cdot 〉$ be the dual pair between X and $X^{\ast }$, $2^X$ denote the family of all the nonempty subsets of X, and $CB(X)$ denote the family of all nonempty closed bounded subsets of X. The generalized duality mapping $J_q:X\to 2^{X^{\ast }}$ is defined by

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

where $q>1$ is a constant. Let us recall the following results and concepts.

Definition 2.1 A single-valued mapping $\eta :X\times X\to X$ is said to be τ-Lipschitz continuous if there exists a constant $\tau >0$ such that

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

Definition 2.2 A single-valued mapping $A:X\to X$ is said to be

• (i) accretive if

  $$〈A(x_1)-A(x_2),J_q(x_1-x_2)〉\ge 0,\mathrm{\forall }{x}_1,x_2\in X;$$

• (ii) strictly accretive if A is accretive and $〈A(x_1)-A(x_2),J_q(x_1-x_2)〉=0$ if and only if $x_1=x_2$, $\mathrm{\forall }{x}_1,x_2\in X$;

• (iii) r-strongly η-accretive if there exists a constant $r>0$ such that

  $$〈A(x_1)-A(x_2),J_q(\eta (x_1,x_2))〉\ge r{\parallel x_1-x_2\parallel }^q,\mathrm{\forall }{x}_1,x_2\in X;$$

• (iv) γ-Lipschitz continuous if there exists a constant $\gamma >0$ such that

  $$\parallel A(x_1)-A(x_2)\parallel \le \gamma \parallel x_1-x_2\parallel ,\mathrm{\forall }{x}_1,x_2\in X;$$

• (v) Let $f:X\to X$ be a single-valued mapping. A is said to be $(\sigma ,\phi )$-relaxed cocoercive with respect to f if for any $x_1,x_2\in X$, there exist two constants $\sigma ,\phi >0$ such that

  $$〈A(x_1)-A(x_2),J_q(f(x_1)-f(x_2))〉\ge -\sigma {\parallel A(x_1)-A(x_2)\parallel }^q+\phi {\parallel x_1-x_2\parallel }^q.$$

Definition 2.3 A set-valued mapping $S:X\to CB(X)$ is said to be

• (i) D-Lipschitz continuous if there exists a constant $\alpha >0$ such that

  $$D(S(x),S(y))\le \alpha \parallel x-y\parallel ,\mathrm{\forall }x,y\in X,$$

where $D(\cdot ,\cdot )$ is the Hausdorff metric on $CB(X)$.

• (ii) β-strongly η-accretive if there exists a constant $\beta >0$ such that

  $$〈u_1-u_2,J_q(\eta (x,y))〉\ge \beta {\parallel x-y\parallel }^q,\mathrm{\forall }x,y\in X,u_1\in S(x),u_2\in S(y).$$

Definition 2.4 Let $A:X\to X$ and $\eta :X\times X\to X$ be single-valued mappings. A set-valued mapping $M:X\to 2^X$ is said to be

• (i) accretive if

  $$〈u_1-u_2,J_q(x,y)〉\ge 0,\mathrm{\forall }x,y\in X,u_1\in M(x),u_2\in M(y);$$

• (ii) η-accretive if

  $$〈u_1-u_2,J_q(\eta (x,y))〉\ge 0,\mathrm{\forall }x,y\in X,u_1\in M(x),u_2\in M(y);$$

• (iii) m-relaxed η-accretive, if there exists a constant $m>0$ such that

  $$〈u_1-u_2,J_q(\eta (x,y))〉\ge -m{\parallel x-y\parallel }^q,\mathrm{\forall }x,y\in X,u_1\in M(x),u_2\in M(y);$$

• (iv) A-accretive if M is accretive and $(A+\rho M)(X)=X$ for all $\rho >0$;

• (v) $(A,\eta )$-accretive if M is m-relaxed η-accretive and $(A+\rho M)(X)=X$ for every $\rho >0$.

Based on [10], we can define the resolvent operator $R_{\rho ,M}^{A,\eta }$ as follows.

Lemma 2.5 ([10])

Let $\eta :X\times X\to X$ be a τ-Lipschitz continuous mapping, $A:X\to X$ be an r-strongly η-accretive mapping, and $M:X\to 2^X$ be a set-valued $(A,\eta )$-accretive mapping. Then the generalized resolvent operator $R_{\rho ,M}^{A,\eta }:X\to X$ is ${\tau }^{q-1}/(r-m\rho )$-Lipschitz continuous; that is,

$$\parallel R_{\rho ,M}^{A,\eta }(x)-R_{\rho ,M}^{A,\eta }(y)\parallel \le \frac{{\tau }^{q-1}}{r-m\rho }\parallel x-y\parallel \mathit{\text{for all}}x,y\in X,$$

where $\rho \in (0,r/m)$, $q>1$.

Remark 2.6 The $(A,\eta )$-accretive mappings are more general than $(H,\eta )$-monotone mappings, A-monotone operators and η-subdifferential operators in a Banach space or a Hilbert space, and the resolvent operators associated with $(A,\eta )$-accretive mappings include as special cases the corresponding resolvent operators associated with them, respectively [3–6, 9, 25].

In the study of characteristic inequalities in q-uniformly smooth Banach spaces X, Xu [14] proved the following result.

Lemma 2.7 ([14])

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 $x,y\in X$,

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

Lemma 2.8 ([8])

Let $a,b,c>0$ be real, for any real $q\ge 1$, if $a^q\le b^q+c^q$, then

$$a\le b+c.$$

3 Existence theorem of solutions

Let us study the existence theorem of solutions for the inclusions system (3).

Theorem 3.1 Let X be a Banach space, $f,g:X\to X$ be two single-valued mappings, $F:X\times X\to X$ be a $({\mu }_1,{\nu }_1)$-Lipschitz continuous mapping and $G:X\times X\to X$ be a $({\mu }_2,{\nu }_2)$-Lipschitz continuous mapping, ${\eta }_i:X\times X\to X$ be a ${\tau }_i$-Lipschitz continuous mapping ($i=1,2$), $A:X\to X$ be an $r_1$-strongly ${\eta }_1$-accretive mapping, $B:X\to X$ be an $r_2$-strongly ${\eta }_2$-accretive mapping, $M:X\to 2^X$ be a set-valued $(A,{\eta }_1)$-accretive mapping and $N:X\to 2^X$ be a set-valued $(B,{\eta }_2)$-accretive mapping. Then the following statements are mutually equivalent:

• (i) An element $(x,y)$ is a solution of the problem (3);

• (ii) For $(x,y)\in X\times X$, $z\in S(x)$ and $w\in T(y)$, the following relations hold:

  $$\{\begin{array}{c}x=R_{{\rho }_1,M}^{A,{\eta }_1}(A(x)+{\rho }_{1}f(x)-{\rho }_{1}F(z,y)),\hfill \\ y=R_{{\rho }_2,N}^{B,{\eta }_2}(B(y)+{\rho }_{2}g(y)-{\rho }_{2}G(w,x)),\hfill \end{array}$$

  (5)

where ${\rho }_i>0$ is a constant ($i=1,2$);

• (iii) For $(x,y)\in X\times X$, $z\in S(x)$, $w\in T(y)$, and any $1>\lambda >0$, the following relations hold:

  $$\{\begin{array}{c}x=(1-\lambda )x+\lambda R_{{\rho }_1,M}^{A,{\eta }_1}(A(x)+{\rho }_{1}f(x)-{\rho }_{1}F(z,y)),\hfill \\ y=(1-\lambda )y+\lambda R_{{\rho }_2,N}^{B,{\eta }_2}(B(y)+{\rho }_{2}g(y)-{\rho }_{2}G(w,x)),\hfill \end{array}$$

  (6)

where ${\rho }_i>0$ is a constant ($i=1,2$);

Proof This directly follows from the definition of $R_{{\rho }_1,M}^{A,{\eta }_1}$, $R_{{\rho }_2,N}^{B,{\eta }_2}$, and the problem (3) for $i=1,2$. □

Theorem 3.2 Let X be a q-uniformly smooth Banach space. Let $f,g:X\to X$ be two single-valued ${\kappa }_1$ or ${\kappa }_2$-Lipschitz continuous mappings, respectively, ${\eta }_i:X\times X\to X$ be a single-valued ${\tau }_i$-Lipschitz continuous mapping ($i=1,2$), $F,G:X\times X\to X$ be two single-valued $({\mu }_1,{\nu }_1)$ or $({\mu }_2,{\nu }_2)$-Lipschitz continuous mappings, respectively. Let $A:X\to X$ be single-valued $r_1$-strongly ${\eta }_1$-accretive, ${\omega }_1$-Lipschitz continuous, $({\sigma }_1,{\phi }_1)$-relaxed cocoercive with respect to f, and $B:X\to X$ be single-valued $r_2$-strongly ${\eta }_2$-accretive, ${\omega }_2$-Lipschitz continuous, $({\sigma }_2,{\phi }_2)$-relaxed cocoercive with respect to g. Let $S,T:X\to X$ be two set-valued ${\gamma }_1$ or ${\gamma }_2$-Lipschitz continuous mappings, respectively. If $M:X\to 2^X$ is a set-valued $(A,{\eta }_1)$-accretive mapping and $N:X\to 2^X$ is a set-valued $(B,{\eta }_2)$-accretive mapping, and the following condition holds:

$$\{\begin{array}{c}{\tau }^q({\rho }_1{\mu }_1{\gamma }_1+l_1)<\tau (r_1-m_1{\rho }_1),{\tau }^q({\rho }_2{\mu }_2{\gamma }_2+l_2)<\tau (r_2-m_2{\rho }_2),\hfill \\ l_1=\sqrt[q]{{\omega }_1^q+c_q{\rho }_1^q{\kappa }_1^q+q{\sigma }_1{\omega }_1^q-q{\phi }_1},l_2=\sqrt[q]{{\omega }_2^q+c_q{\rho }_2^q{\kappa }_2^q+q{\sigma }_2{\omega }_2^q-q{\phi }_2},\hfill \end{array}$$

(7)

where $c_q>0$ is the same as in Lemma  2.7 and ${\rho }_i\in (0,\frac{r_i}{m_i})$ ($i=1,2$), then the problem (3) has a solution $x^{\ast },y^{\ast }\in X$, $z^{\ast }\in S(x^{\ast })$, $w^{\ast }\in T(y^{\ast })$.

Proof Define two mappings $Q_1,Q_2:X\to X$ as follows:

$$\{\begin{array}{c}{Q}_1(x)=(1-\lambda )x+\lambda R_{{\rho }_1,M}^{A,{\eta }_1}(A(x)+{\rho }_{1}f(x)-{\rho }_{1}F(z,y)),\hfill \\ Q_2(y)=(1-\lambda )y+\lambda R_{{\rho }_2,N}^{B,{\eta }_2}(B(y)+{\rho }_{2}g(y)-{\rho }_{2}G(w,x))\hfill \\ (\mathrm{\forall }x,y\in X,z\in S(x),w\in T(y)).\hfill \end{array}$$

(8)

For elements $x_1,x_2,y_1,y_2\in X$, if letting

$${\mathrm{\Omega }}_i=A(x_i)-{\rho }_{1}f(x_i)-{\rho }_{1}F(z_i,y_i)(i=1,2),$$

then by (8), Lemma 2.5 and Lemma 2.7, we have

$$\begin{array}{rcl}\parallel Q_1(x_1)-Q_1(x_2)\parallel & =& \parallel (1-\lambda )x_1+\lambda R_{{\rho }_1,M}^{A,{\eta }_1}({\mathrm{\Omega }}_1)-(1-\lambda )x_2-\lambda R_{{\rho }_1,M}^{A,{\eta }_1}({\mathrm{\Omega }}_2)\parallel \\ \le & (1-\lambda )\parallel x_1-x_2\parallel +\lambda \parallel R_{{\rho }_1,M}^{A,{\eta }_1}({\mathrm{\Omega }}_1)-R_{{\rho }_1,M}^{A,{\eta }_1}({\mathrm{\Omega }}_2)\parallel \\ \le & (1-\lambda )\parallel x_1-x_2\parallel +\lambda \frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1}[{\rho }_1\left(\parallel F(z_2,y_2)-F(z_1,y_1)\parallel \right)\\ +\parallel A(x_1)-A(x_2)-{\rho }_1(f(x_1)-f(x_2))\parallel ],\end{array}$$

(9)

and by $({\mu }_1,{\nu }_1)$-Lipschitz continuity of $F(\cdot ,\cdot )$ and ${\gamma }_1$-Lipschitz continuity of S, we obtain

$$\begin{array}{rcl}\parallel F(z_2,y_2)-F(z_1,y_1)\parallel & \le & {\mu }_1\parallel z_2-z_1\parallel +{\nu }_1\parallel y_2-y_1\parallel \\ \le & {\mu }_1{\gamma }_1\parallel x_2-x_1\parallel +{\nu }_1\parallel y_2-y_1\parallel .\end{array}$$

(10)

Since A is ${\omega }_1$-Lipschitz continuous and $({\sigma }_1,{\phi }_1)$-relaxed cocoercive with respect to f, and f is ${\kappa }_1$-Lipschitz continuous so that for $z_1\in S(x_1)$, $z_2\in S(x_2)$, we have

(11)

Combining (9), (10) and (11), we can get

(12)

where

$${\theta }_1=\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1}({\rho }_1{\mu }_1{\gamma }_1+\sqrt[q]{{\omega }_1^q+c_q{\rho }_1^q{\kappa }_1^q+q{\sigma }_1{\omega }_1^q-q{\phi }_1}).$$

For elements $x_1,x_2,y_1,y_2\in X$, $z_i\in S(x_i)$, $y_i\in T(y_i)$ ($i=1,2$), if letting

$${\mathrm{\Theta }}_i=B(y_i)+{\rho }_{2}g(y_i)-{\rho }_{2}G(w_i,x_i)(i=1,2),$$

then by using the same method as the one used above,

$$\begin{array}{rcl}\parallel Q_2(y_1)-Q_2(y_2)\parallel & =& \parallel (1-\lambda )y_1+\lambda R_{{\rho }_2,N}^{B,{\eta }_2}({\mathrm{\Theta }}_1)-(1-\lambda )y_2-\lambda R_{{\rho }_2,N}^{B,{\eta }_2}({\mathrm{\Theta }}_2)\parallel \\ \le & (1-\lambda )\parallel y_1-y_2\parallel +\lambda \parallel R_{{\rho }_2,N}^{B,{\eta }_2}({\mathrm{\Theta }}_1)-R_{{\rho }_2,N}^{B,{\eta }_2}({\mathrm{\Theta }}_2)\parallel \\ \le & (1-\lambda )\parallel y_1-y_2\parallel +\lambda \frac{{\tau }^{q-1}}{r_2-m_2{\rho }_2}({\rho }_2\parallel G(w_2,x_2)-G(w_1,x_1)\parallel \\ +\parallel B(y_1)-B(y_2)-{\rho }_2(g(y_1)-g(y_2))\parallel )\\ \le & \lambda \frac{{\tau }^{q-1}}{r_2-m_2{\rho }_2}{\rho }_2{\nu }_2\parallel x_2-x_1\parallel +[(1-\lambda )+\lambda {\theta }_2]\parallel y_2-y_1\parallel \end{array}$$

(13)

hold, where

$${\theta }_2=\frac{{\tau }^{q-1}}{r_2-m_2{\rho }_2}({\rho }_2{\mu }_2{\gamma }_2+\sqrt[q]{{\omega }_2^q+c_q{\rho }_2^q{\kappa }_2^q+q{\sigma }_2{\omega }_2^q-q{\phi }_2}).$$

If setting

$$\begin{array}{r}{\mathrm{\Gamma }}_{11}={\theta }_1,{\mathrm{\Gamma }}_{12}=\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1}{\rho }_1{\nu }_1,\\ {\mathrm{\Gamma }}_{21}=\frac{{\tau }^{q-1}}{r_2-m_2{\rho }_2}{\rho }_2{\nu }_2,{\mathrm{\Gamma }}_{22}={\theta }_2,\end{array}$$

(14)

$\overrightarrow{a}={(\parallel Q_1(x_1)-Q_1(x_2)\parallel ,\parallel Q_2(y_1)-Q_2(y_2)\parallel )}^T$ and $\overrightarrow{b}={(\parallel x_1-x_2\parallel ,\parallel y_1-y_2\parallel )}^T$, then from (12), (13) and (14), we have $\overrightarrow{a}\le (1-\lambda )\mathbf{E}+\lambda \mathbf{\Psi }\overrightarrow{b}$, where

$$\mathbf{E}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),\mathbf{\Psi }=\left(\begin{array}{cc}{\mathrm{\Gamma }}_{11}& {\mathrm{\Gamma }}_{12}\\ {\mathrm{\Gamma }}_{21}& {\mathrm{\Gamma }}_{22}\end{array}\right),0<\lambda <1,$$

(15)

where Ψ is called the matrix for nonlinear set-valued inclusions system. By using [16], we have

$$\parallel \overrightarrow{a}\parallel \le (1-\lambda )+\lambda \parallel \mathbf{\Psi }\parallel \parallel \overrightarrow{b}\parallel .$$

(16)

Letting

$$\parallel \mathbf{\Psi }\parallel =max\{{\mathrm{\Gamma }}_{11},{\mathrm{\Gamma }}_{12},{\mathrm{\Gamma }}_{21},{\mathrm{\Gamma }}_{22}\}.$$

It follows from (16), the assumption of the condition (7) and $S(x),T(y)\in CB(X)$ that $0<\parallel \mathbf{\Psi }\parallel <1$, $(1-\lambda )+\lambda \parallel \mathbf{\Psi }\parallel <1$, and there exist $x^{\ast },y^{\ast }\in X$ and $z^{\ast }\in S(x^{\ast })$, $w^{\ast }\in T(y^{\ast })$ such that

$$\{\begin{array}{c}{Q}_1(x^{\ast })=x^{\ast },\hfill \\ Q_2(y^{\ast })=y^{\ast }.\hfill \end{array}$$

Therefore, the following relations hold for Theorem 3.1(ii)-(iii):

$$\{\begin{array}{c}{x}^{\ast }=R_{{\rho }_1,M}^{A,{\eta }_1}(A(x^{\ast })+{\rho }_{1}f(x^{\ast })-{\rho }_{1}F(z^{\ast },y^{\ast })),\hfill \\ y^{\ast }=R_{{\rho }_2,N}^{B,{\eta }_2}(B(y^{\ast })+{\rho }_{2}g(y^{\ast })-{\rho }_{2}G(w^{\ast },x^{\ast })),\hfill \end{array}$$

(17)

where ${\rho }_i>0$ is a constant ($i=1,2$). Thus, by Theorem 3.1, we know that $(x^{\ast },y^{\ast },z^{\ast },w^{\ast })$ is a solution of the problem (3). This completes the proof. □

4 Ishikawa-hybrid proximal algorithm

In 2008, Verma developed a hybrid version of the Eckstein-Bertsekas [11] proximal point algorithm, introduced the algorithm based on the $(A,\eta )$-maximal monotonicity framework [7] and studied convergence of the algorithm, and so did Li, Xu and Jin in [12]. Based on Theorem 3.1, we develop an Ishikawa-hybrid proximal point algorithm for finding an iterative sequence solving the problem (3) as follows.

Algorithm 4.1 Let X be a q-uniformly smooth Banach space. Let $f,g:X\to X$ be two single-valued ${\kappa }_1$ or ${\kappa }_2$-Lipschitz continuous mappings, respectively, ${\eta }_i:X\times X\to X$ be a single-valued ${\tau }_i$-Lipschitz continuous mapping ($i=1,2$), $F,G:X\times X\to X$ be two single-valued $({\mu }_1,{\nu }_1)$ or $({\mu }_2,{\nu }_2)$-Lipschitz continuous mappings, respectively. Let $A:X\to X$ be single-valued $r_1$-strongly ${\eta }_1$-accretive, ${\omega }_1$-Lipschitz continuous, $({\sigma }_1,{\phi }_1)$-relaxed cocoercive with respect to f, and $B:X\to X$ be single-valued $r_2$-strongly ${\eta }_2$-accretive, ${\omega }_2$-Lipschitz continuous, $({\sigma }_2,{\phi }_2)$-relaxed cocoercive with respect to g. Let $S,T:X\to X$ be two set-valued ${\gamma }_1$ or ${\gamma }_2$-Lipschitz continuous mappings, respectively, $M:X\to 2^X$ be a set-valued $(A,{\eta }_1)$-accretive mapping and $N:X\to 2^X$ be a set-valued $(B,{\eta }_2)$-accretive mapping. Suppose that ${\{{\alpha }_i^n\}}_{n=0}^{\mathrm{\infty }}$, ${\{{\beta }_i^n\}}_{n=0}^{\mathrm{\infty }}$, ${\{{\xi }_i^n\}}_{n=0}^{\mathrm{\infty }}$, ${\{{\zeta }_i^n\}}_{n=0}^{\mathrm{\infty }}$ and ${\{{\rho }_i^n\}}_{n=0}^{\mathrm{\infty }}$ ($i=1,2$) are ten nonnegative sequences such that

then we can get $x^1,y^1\in X$ and $z^1\in S(x^1)$, $w^1\in T(y^1)$ as follows.

Step 1: For arbitrarily chosen initial points $x^0\in X$, $y^0\in X$, we choose suitable $z^0\in S(x^0)$, $w^0\in T(y^0)$, setting

$$\{\begin{array}{c}{u}^0=(1-{\alpha }_1^0)x^0+{\alpha }_1^0{e}_1^0,\hfill \\ x^1=(1-{\beta }_1^0)x^0+{\beta }_1^0{d}_1^0,\hfill \end{array}$$

where $e_1^0$, $d_1^0$ satisfy

$$\{\begin{array}{c}\parallel e_1^0-R_{{\rho }_1^0,M}^{A,{\eta }_1}(A(x^0)+{\rho }_1^{0}f(x^0)+{\rho }_1^{0}F(z^0,y^0))\parallel \hfill \\ \le {\xi }_1^0\parallel e_1^0-x^0\parallel (z^0\in S(x^0)),\hfill \\ \parallel d_1^0-R_{{\rho }_1^0,M}^{A,{\eta }_1}(A(u^0)+{\rho }_1^{0}f(u^0)+{\rho }_1^{0}F(z_1^0,y^0))\parallel \hfill \\ \le {\zeta }_1^0\parallel d_1^0-u^0\parallel (z_1^0\in S(u^0)),\hfill \end{array}$$

and

$$\{\begin{array}{c}{v}^0=(1-{\alpha }_2^0)y^0+{\alpha }_2^0{e}_2^0,\hfill \\ y^1=(1-{\beta }_2^0)y^0+{\beta }_2^0{d}_2^0,\hfill \end{array}$$

where $e_2^0$, $d_2^0$ satisfy

$$\{\begin{array}{c}\parallel e_2^0-R_{{\rho }_2^0,N}^{B,{\eta }_2}(B(y^0)+{\rho }_2^{0}g(y^0)-{\rho }_2^{0}G(w^0,x^0))\parallel \hfill \\ \le {\xi }_2^0\parallel e_2^0-y^0\parallel (w^0\in T(y^0),\hfill \\ \parallel d_2^0-R_{{\rho }_2^0,N}^{B,{\eta }_2}(B(v^0)+{\rho }_2^{0}g(v^0)-{\rho }_2^{0}G(w_2^0,x^0))\parallel \hfill \\ \le {\zeta }_2^0\parallel d_2^0-v^0\parallel (w_2^0\in T(v^0)).\hfill \end{array}$$

By using Nadler [15], we can choose suitable $z^1\in S(x^1)$, $w^1\in T(y^1)$ such that

$$\{\begin{array}{c}\parallel z^0-z^1\parallel \le (1+\frac{1}{1})D(S(x^0),S(x^1)),\hfill \\ \parallel w^0-w^1\parallel \le (1+\frac{1}{1})D(T(y^0),T(y^1)),\hfill \\ \parallel z_1^0-z_1^1\parallel \le (1+\frac{1}{1})D(S(U^0),S(U^1)),\hfill \\ \parallel w_2^0-w_2^1\parallel \le (1+\frac{1}{1})D(T(V^0),T(V^1)).\hfill \end{array}$$

Therefore, we obtain $x^1,y^1\in X$ and $z^1\in S(x^1)$, $w^1\in T(y^1)$ and give the next step for generating sequences ${\{x^n\}}_{n=2}^{\mathrm{\infty }}$, ${\{y^n\}}_{n=2}^{\mathrm{\infty }}$, ${\{z^n\}}_{n=2}^{\mathrm{\infty }}$ and ${\{w^n\}}_{n=2}^{\mathrm{\infty }}$.

Step 2: From $x^1,y^1\in X$ and $z^1\in S(x^1)$, $w^1\in T(y^1)$, the sequences ${\{x^n\}}_{n=2}^{\mathrm{\infty }}$, ${\{y^n\}}_{n=2}^{\mathrm{\infty }}$, ${\{z^n\}}_{n=2}^{\mathrm{\infty }}$ and ${\{w^n\}}_{n=2}^{\mathrm{\infty }}$ are generated by the iterative procedure

$$\{\begin{array}{c}{u}^n=(1-{\alpha }_1^n)x^n+{\alpha }_1^n{e}_1^n,\hfill \\ x^{n+1}=(1-{\beta }_1^n)x^n+{\beta }_1^n{d}_1^n,\hfill \\ \parallel e_1^n-R_{{\rho }_1^n,M}^{A,{\eta }_1}(A(x^n)+{\rho }_1^{n}f(x^n)+{\rho }_1^{n}F(z^n,y^n))\parallel \hfill \\ \le {\xi }_1^n\parallel e_1^n-x^n\parallel (z^n\in S(x^n)),\hfill \\ \parallel d_1^n-R_{{\rho }_1^n,M}^{A,{\eta }_1}(A(u^n)+{\rho }_1^{n}f(u^n)-{\rho }_1^{n}F(z_1^n,y^n))\parallel \hfill \\ \le {\zeta }_1^n\parallel d_1^n-u^n\parallel (z_1^n\in S(u^n)),\hfill \end{array}$$

(18)

and

$$\{\begin{array}{c}{v}^n=(1-{\alpha }_2^n)y^n+{\alpha }_2^n{e}_2^n,\hfill \\ y^{n+1}=(1-{\beta }_2^n)y^n+{\beta }_2^n{d}_2^n,\hfill \\ \parallel e_2^n-R_{{\rho }_2^n,N}^{B,{\eta }_2}(B(y^n)+{\rho }_2^{n}g(y^n)-{\rho }_2^{n}G(w^n,x^n))\parallel \hfill \\ \le {\xi }_2^n\parallel e_2^n-y^n\parallel (w^n\in T(y^n)),\hfill \\ \parallel d_2^n-R_{{\rho }_2^n,N}^{B,{\eta }_2}(B(v^n)+{\rho }_2^{n}g(v^n)-{\rho }_2^{n}G(w_2^n,x^n))\parallel \hfill \\ \le {\zeta }_2^n\parallel d_2^n-v^n\parallel (w_2^n\in T(v^n)).\hfill \end{array}$$

(19)

By using Nadler [15], we can choose suitable $z^{n+1}\in S(x^{n+1})$, $w^{n+1}\in T(y^{n+1})$ such that

$$\{\begin{array}{c}\parallel z^n-z^{n+1}\parallel \le (1+\frac{1}{1+n})D(S(x^n),S(x^{n+1})),\hfill \\ \parallel w^n-w^{n+1}\parallel \le (1+\frac{1}{1+n})D(T(y^n),T(y^{n+1})),\hfill \end{array}$$

(20)

for $n=0,1,2,\dots$ .

Remark 4.2 If we choose some suitable operators A, B, ${\eta }_1$, ${\eta }_1$, F, G, S, T, M, N, f, g and a space X, then Algorithm 4.1 can degenerate to a number of known algorithms for solving the system of variational inequalities and variational inclusions (see [2–6, 8–10, 25]).

5 Convergence of Ishikawa-hybrid proximal Algorithm 4.1

In this section, we prove that ${\{(x^n,y^n,z^n,w^n)\}}_{n=0}^{\mathrm{\infty }}$ generated by Ishikawa-hybrid proximal Algorithm 4.1 converges linearly to a solution $(x^{\ast },y^{\ast },z^{\ast },w^{\ast })$ of the problem (3) as the convergence rate $\parallel \mathbf{\Psi }\parallel$.

Theorem 5.1 Let X be a q-uniformly smooth Banach space. Let $f,g:X\to X$ be two single-valued ${\kappa }_1$ or ${\kappa }_2$-Lipschitz continuous mappings, respectively, ${\eta }_i:X\times X\to X$ be a single-valued ${\tau }_i$-Lipschitz continuous mapping ($i=1,2$), $F,G:X\times X\to X$ be two single-two valued $({\mu }_1,{\nu }_1)$ or $({\mu }_2,{\nu }_2)$-Lipschitz continuous mappings, respectively. Let $A:X\to X$ be a single-valued $r_1$-strongly ${\eta }_1$-accretive and ${\omega }_1$-Lipschitz continuous mapping, and let $B:X\to X$ be a single-valued $r_2$-strongly ${\eta }_2$-accretive and ${\omega }_2$-Lipschitz continuous mapping. Let $S,T:X\to X$ be two set-valued ${\gamma }_1$ or ${\gamma }_2$-Lipschitz continuous mappings, respectively, A be $({\sigma }_1,{\phi }_1)$-relaxed cocoercive with respect to f and B be $({\sigma }_2,{\phi }_2)$-relaxed cocoercive with respect to g. Suppose that $M:X\to 2^X$ is a set-valued $(A,{\eta }_1)$-accretive mapping and $N:X\to 2^X$ is a set-valued $(B,{\eta }_2)$-accretive mapping, and the following conditions hold:

$$\{\begin{array}{c}max\{{\rho }_1{\mu }_1{\gamma }_1+l_1,{\alpha }_1{\beta }_1{\nu }_1{\theta }_1(1+{\rho }_1-{\beta }_1{\rho }_1),\hfill \\ {\beta }_2{\rho }_2{\nu }_2(1+2{\alpha }_2{\theta }_2),({\rho }_2{\mu }_2{\gamma }_2+l_2)\}<{\tau }^{1-q}(r_1-m_1{\rho }_1),\hfill \\ 2\frac{{\beta }_1(1-{\alpha }_1)}{{\alpha }_1+{\beta }_1}{\theta }_1+\frac{{\alpha }_1{\beta }_1}{{\alpha }_1+{\beta }_1}{\theta }_1^2<1-{\theta }_1,\hfill \\ (1-{\beta }_2+{\beta }_2{\theta }_2)+3(1-{\alpha }_2+{\alpha }_2{\theta }_2){\beta }_2{\theta }_2+{\beta }_2{\theta }_2<1,\hfill \\ l_1=\sqrt[q]{{\omega }_1^q+c_q{\rho }_1^q{\kappa }_1^q+q{\sigma }_1{\omega }_1^q-q{\phi }_1},l_2=\sqrt[q]{{\omega }_2^q+c_q{\rho }_2^q{\kappa }_2^q+q{\sigma }_2{\omega }_2^q-q{\phi }_2},\hfill \\ {\theta }_1=\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1}({\rho }_1{\mu }_1{\gamma }_1+l_1),{\theta }_2=\frac{{\tau }^{q-1}}{r_2-m_2{\rho }_2}({\rho }_2{\mu }_2{\gamma }_2+l_2),\hfill \end{array}$$

(21)

and eight nonnegative sequences ${\{{\alpha }_i^n\}}_{n=0}^{\mathrm{\infty }}$, ${\{{\beta }_i^n\}}_{n=0}^{\mathrm{\infty }}$, ${\{{\xi }_i^n\}}_{n=0}^{\mathrm{\infty }}$, ${\{{\zeta }_i^n\}}_{n=0}^{\mathrm{\infty }}$ and ${\{{\rho }_i^n\}}_{n=0}^{\mathrm{\infty }}$ ($i=1,2$) satisfy the following conditions:

(22)

(23)

Then the problem (3) has a solution $(x^{\ast },y^{\ast },z^{\ast },w^{\ast })z^{\ast }\in S(x^{\ast })$, $w^{\ast }\in T(y^{\ast })$, and the sequence ${\{x^n,y^n\}}_{n=0}^{\mathrm{\infty }}$ generated by Ishikawa-hybrid proximal Algorithm 4.1 converges linearly to a solution $(x^{\ast },y^{\ast })$ of the problem (3) as the convergence rate

$$\begin{array}{rcl}\parallel \mathbf{\Psi }\parallel & =& max\{1-({\alpha }_1+{\beta }_1)+({\alpha }_1+{\beta }_1){\theta }_1+(2-2{\alpha }_1+{\alpha }_1{\theta }_1){\beta }_1{\theta }_1,\\ {\alpha }_1{\beta }_1{\nu }_1{\theta }_1\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1}(1+{\rho }_1-{\beta }_1{\rho }_1),\\ 1-{\beta }_2+2{\beta }_2{\theta }_2(2-{\alpha }_2+{\alpha }_2{\theta }_2)+{\beta }_2{\theta }_2(1-{\alpha }_2+{\alpha }_2{\theta }_2),\\ (1-{\beta }_2+{\beta }_2{\theta }_2)+3(1-{\alpha }_2+{\alpha }_2{\theta }_2){\beta }_2{\theta }_2+{\beta }_2{\theta }_2\},\end{array}$$

(24)

where $c_q>0$ is the same as in Lemma  2.5, ${\rho }_i\in (0,\frac{r_i}{m_i})$ ($i=1,2$).

Proof Let $(x^{\ast },y^{\ast },z^{\ast },w^{\ast })$ ($z^{\ast }\in S(x^{\ast })$, $w^{\ast }\in T(y^{\ast })$) be the solution of the problem (3), then for any $\lambda >0$,

$$\{\begin{array}{c}{x}^{\ast }=(1-\lambda )x^{\ast }+\lambda R_{{\rho }_1,M}^{A,{\eta }_1}(A(x^{\ast })+{\rho }_{1}f(x^{\ast })-{\rho }_{1}F(z^{\ast },y^{\ast })),\hfill \\ y^{\ast }=(1-\lambda )y^{\ast }+\lambda R_{{\rho }_2,N}^{B,{\eta }_2}(B(y^{\ast })+{\rho }_{2}g(y^{\ast })-{\rho }_{2}G(w^{\ast },x^{\ast })).\hfill \end{array}$$

(25)

For $n\ge 0$, we write

$$\{\begin{array}{c}{s}_1^n=(1-{\alpha }_1^n)x^n+{\alpha }_1^n{R}_{{\rho }_1^n,M}^{A,{\eta }_1}(A(x^n)+{\rho }_1^{n}f(x^n)+{\rho }_1^{n}F(z^n,y^n)),\hfill \\ t_1^{n+1}=(1-{\beta }_1^n)x^n+{\beta }_1^n{R}_{{\rho }_1^n,M}^{A,{\eta }_1}(A(s_1^n)+{\rho }_1^{n}f(s_1^n)+{\rho }_1^{n}F(z_2^n,y^n))\hfill \\ (z_2^n\in S(s_1^n)).\hfill \end{array}$$

(26)

It follows from the hypotheses of the mappings A, f, F, S, M, ${\eta }_1$ and $R_{{\rho }_1^n,M}^{A,{\eta }_1}$ in Algorithm 4.1 that

that is,

(27)

where ${\theta }_1(n)=\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1^n}({\rho }_1^n{\mu }_1{\gamma }_1+\sqrt[q]{{\omega }_1^q+c_q{({\rho }_1^n)}^q{\kappa }_1^q+q{\sigma }_1{\omega }_1^q-q{\phi }_1})$, $z^n\in S(x^n)$ and $x^{\ast }\in S(x^{\ast })$.

From (24)-(27) and (13), we have

$$\begin{array}{rcl}\parallel t_1^{n+1}-x^{\ast }\parallel & \le & (1-{\beta }_1^n)\parallel x^n-x^{\ast }\parallel \\ +{\beta }_1^n\parallel R_{{\rho }_1^n,M}^{A,{\eta }_1}(A\left(x^{\ast }\right)+{\rho }_1^{n}f\left(x^{\ast }\right)-{\rho }_1^{n}F(z^{\ast },y^{\ast }))\\ -R_{{\rho }_1^n,M}^{A,{\eta }_1}(A\left(s_1^n\right)+{\rho }_1^{n}f\left(s_1^n\right)+{\rho }_1^{n}F(z_2^n,y^n))\parallel \\ \le & ((1-{\beta }_1^n)+{\beta }_1^n\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1^n}({\rho }_1^n{\mu }_1{\gamma }_1\\ +\sqrt[q]{{\omega }_1^q+c_q{\left({\rho }_1^n\right)}^q{\kappa }_1^q+q{\sigma }_1{\omega }_1^q-q{\phi }_1}\left)\right)\parallel s_1^n-x^{\ast }\parallel \\ +{\beta }_1^n\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1^n}{\nu }_1\parallel y^n-y^{\ast }\parallel \\ \le & ((1-{\beta }_1^n)+{\beta }_1^n{\theta }_1(n))\parallel s_1^n-x^{\ast }\parallel +{\beta }_1^n\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1^n}{\nu }_1\parallel y^n-y^{\ast }\parallel .\end{array}$$

(28)

By Algorithm 4.1, $x^{n+1}-x^n={\beta }_1^n(d_1^n-x^n)$ and $u^n-x^n={\alpha }_1^n(e_1^n-x^n)$, we have

$$\begin{array}{rcl}\parallel x^{n+1}-t_1^{n+1}\parallel & \le & \parallel (1-{\beta }_1^n)x^n+{\beta }_1^n{d}_1^n-(1-{\beta }_1^n)x^n\\ -{\beta }_1^n{R}_{{\rho }_1^n,M}^{A,{\eta }_1}(A\left(s_1^n\right)+{\rho }_1^{n}f\left(s_1^n\right)-{\rho }_1^{n}F(z_2^n,y^n))\parallel \\ \le & {\beta }_1^n\parallel d_1^n-R_{{\rho }_1^n,M}^{A,{\eta }_1}(A\left(s_1^n\right)+{\rho }_1^{n}f\left(s_1^n\right)-{\rho }_1^{n}F(z_2^n,y^n))\parallel \\ \le & {\beta }_1^n(\parallel d_1^n-R_{{\rho }_1^n,M}^{A,{\eta }_1}(A\left(u^n\right)+{\rho }_1^{n}f\left(u^n\right)-{\rho }_1^{n}F(z_1^n,y^n))\parallel \\ +\parallel R_{{\rho }_1^n,M}^{A,{\eta }_1}(A\left(u^n\right)+{\rho }_1^{n}f\left(u^n\right)-{\rho }_1^{n}F(z_1^n,y^n))\\ -R_{{\rho }_1^n,M}^{A,{\eta }_1}(A\left(s_1^n\right)+{\rho }_1^{n}f\left(s_1^n\right)-{\rho }_1^{n}F(z_2^n,y^n))\parallel )\\ \le & {\beta }_1^n{\zeta }_1^n\parallel d_1^n-u^n\parallel \\ +{\beta }_1^n\parallel R_{{\rho }_1^n,M}^{A,{\eta }_1}(A\left(u^n\right)+{\rho }_1^{n}f\left(u^n\right)-{\rho }_1^{n}F(z_1^n,y^n))\\ -R_{{\rho }_1^n,M}^{A,{\eta }_1}(A\left(s_1^n\right)+{\rho }_1^{n}f\left(s_1^n\right)-{\rho }_1^{n}F(z_2^n,y^n))\parallel \\ \le & {\beta }_1^n{\zeta }_1^n\parallel d_1^n-u^n\parallel \\ +{\beta }_1^n\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1^n}[{\rho }_1^n{\mu }_1{\gamma }_1\parallel s_1^n-u^n\parallel +{\nu }_1\parallel y^n-y^n\parallel \\ +\sqrt[q]{{\omega }_1^q+c_q{\left({\rho }_1^n\right)}^q{\kappa }_1^q+q{\sigma }_1{\omega }_1^q-q{\phi }_1}\parallel s_1^n-u^n\parallel ]\\ \le & {\zeta }_1^n\parallel x^{n+1}-x^n\parallel +{\beta }_1^n{\zeta }_1^n\parallel u^n-x^n\parallel \\ +{\beta }_1^n\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1^n}[{\rho }_1^n{\mu }_1{\gamma }_1\parallel s_1^n-u^n\parallel +{\nu }_1\parallel y^n-y^n\parallel \\ +\sqrt[q]{{\omega }_1^q+c_q{\left({\rho }_1^n\right)}^q{\kappa }_1^q+q{\sigma }_1{\omega }_1^q-q{\phi }_1}\parallel s_1^n-u^n\parallel ]\\ +{\beta }_1^n\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1^n}[{\rho }_1^n{\mu }_1{\gamma }_1+\sqrt[q]{{\omega }_1^q+c_q{\left({\rho }_1^n\right)}^q{\kappa }_1^q+q{\sigma }_1{\omega }_1^q-q{\phi }_1}]\parallel s_1^n-u^n\parallel \\ \le & {\zeta }_1^n\parallel x^{n+1}-x^{\ast }\parallel +{\zeta }_1^n\parallel x^n-x^{\ast }\parallel +{\beta }_1^n{\zeta }_1^n\parallel u^n-x^n\parallel \\ +{\beta }_1^n\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1^n}[{\rho }_1^n{\mu }_1{\gamma }_1+\sqrt[q]{{\omega }_1^q+c_q{\left({\rho }_1^n\right)}^q{\kappa }_1^q+q{\sigma }_1{\omega }_1^q-q{\phi }_1}]\\ \times (\parallel s_1^n-x^{\ast }\parallel +\parallel x^{\ast }-x^n\parallel +\parallel x^n-u^n\parallel )\\ \le & {\zeta }_1^n\parallel x^{n+1}-x^{\ast }\parallel +({\zeta }_1^n+{\beta }_1^n{\theta }_1(n))\parallel x^n-x^{\ast }\parallel \\ +{\beta }_1^n({\zeta }_1^n+{\theta }_1(n))(\parallel u^n-x^{\ast }\parallel +\parallel x^n-x^{\ast }\parallel )+{\beta }_1^n{\theta }_1\parallel s_1^n-x^{\ast }\parallel \\ \le & {\zeta }_1^n\parallel x^{n+1}-x^{\ast }\parallel +({\zeta }_1^n+2{\beta }_1^n{\theta }_1(n)+{\beta }_1^n{\zeta }_1^n)\parallel x^n-x^{\ast }\parallel \\ +{\beta }_1^n({\zeta }_1^n+{\theta }_1(n))\parallel u^n-x^{\ast }\parallel +{\beta }_1^n{\theta }_1(n)\parallel s_1^n-x^{\ast }\parallel .\end{array}$$

(29)

It follows from (26)-(29) that

$$\begin{array}{rcl}\parallel x^{n+1}-x^{\ast }\parallel & \le & \parallel x^{n+1}-t_1^{n+1}\parallel +\parallel t_1^{n+1}-x^{\ast }\parallel \\ \le & {\zeta }_1^n\parallel x^{n+1}-x^{\ast }\parallel +[{\zeta }_1^n+2{\beta }_1^n{\theta }_1(n)+{\beta }_1^n{\zeta }_1^n\\ +{\beta }_1^n({\zeta }_1^n+{\theta }_1(n))\frac{1}{1-{\xi }_1^n}(1-{\alpha }_1^n+{\xi }_1^n+{\alpha }_1^n{\theta }_1(n))\\ +(1-{\beta }_1^n)(1-{\alpha }_1^n+{\alpha }_1^n{\theta }_1(n))]\parallel x^n-x^{\ast }\parallel \\ +[{\beta }_1^n({\zeta }_1^n+{\theta }_1(n))\frac{1}{1-{\xi }_1^n}{\alpha }_1^n\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1^n}{\nu }_1\\ +(1-{\beta }_1^n){\beta }_1^n{\alpha }_1^n{\theta }_1(n){\rho }_1^n{\nu }_1\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1^n}]\parallel y^n-y^{\ast }\parallel ,\end{array}$$

and

$$\begin{array}{rcl}\parallel x^{n+1}-x^{\ast }\parallel & \le & \frac{1}{1-{\zeta }_1^n}[{\zeta }_1^n+2{\beta }_1^n{\theta }_1(n)+{\beta }_1^n{\zeta }_1^n\\ +{\beta }_1^n({\zeta }_1^n+{\theta }_1(n))\frac{1}{1-{\xi }_1^n}(1-{\alpha }_1^n+{\xi }_1^n+{\alpha }_1^n{\theta }_1(n))\\ +(1-{\beta }_1^n)(1-{\alpha }_1^n+{\alpha }_1^n{\theta }_1(n))]\parallel x^n-x^{\ast }\parallel \\ +\frac{1}{1-{\zeta }_1^n}[{\beta }_1^n({\zeta }_1^n+{\theta }_1(n))\frac{1}{1-{\xi }_1^n}{\alpha }_1^n\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1^n}{\nu }_1\\ +(1-{\beta }_1^n){\beta }_1^n{\alpha }_1^n{\theta }_1(n){\rho }_1^n{\nu }_1\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1^n}]\parallel y^n-y^{\ast }\parallel ,\end{array}$$

(30)

where

$${\theta }_1(n)=\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1^n}({\rho }_1^n{\mu }_1{\gamma }_1+\sqrt[q]{{\omega }_1^q+c_q{\left({\rho }_1^n\right)}^q{\kappa }_1^q+q{\sigma }_1{\omega }_1^q-q{\phi }_1}).$$

For $n\ge 0$, we write

$$\{\begin{array}{c}{s}_2^n=(1-{\alpha }_2^n)y^n+{\alpha }_2^n{R}_{{\rho }_2^n,N}^{B,{\eta }_2}(B(y^n)+{\rho }_2^{n}g(y^n)-{\rho }_2^{n}G(w^n,x^n))\hfill \\ (w^n\in T(y^n)),\hfill \\ t_2^{n+1}=(1-{\beta }_2^n)y^n+{\beta }_2^n{R}_{{\rho }_2^n,N}^{B,{\eta }_2}(B(s_2^n)+{\rho }_2^{n}g(s_2^n)-{\rho }_2^{n}G(w_2^n,x^n))\hfill \\ (w_2^n\in T(s_2^n)).\hfill \end{array}$$

(31)

By using the hypotheses of the mappings B, g, G, T, N, ${\eta }_2$ and $R_{{\rho }_2^n,N}^{B,{\eta }_2}$ in Theorem 5.1, and the same method as the one above, we can get

that is,

(32)

where ${\alpha }_2^n(e_2^n-y^n)=v^n-y^n$, and ${\theta }_2(n)=\frac{{\tau }^{q-1}}{r_2-m_2{\rho }_2^n}({\rho }_2^n{\mu }_2{\gamma }_2+\sqrt[q]{{\omega }_2^q+c_q{({\rho }_2^n)}^q{\kappa }_2^q+q{\sigma }_2{\omega }_2^q-q{\phi }_2})$.

Moreover, we have

$$\begin{array}{rcl}\parallel t_2^{n+1}-y^{\ast }\parallel & \le & \parallel (1-{\beta }_2^n)y^n+{\beta }_2^n{d}_2^n-(1-{\beta }_2^n)y^{\ast }\\ -{\beta }_2^n{R}_{{\rho }_2^n,N}^{B,{\eta }_2}(B\left(y^{\ast }\right)+{\rho }_2^{n}g\left(y^{\ast }\right)-{\rho }_2^{n}G(w^{\ast },x^{\ast }))\parallel \\ \le & (1-{\beta }_2^n)\parallel y^n-y^{\ast }\parallel +{\beta }_2^n\parallel d_2^n-R_{{\rho }_2^n,N}^{B,{\eta }_2}(B\left(v^n\right)+{\rho }_2^{n}g\left(v^n\right)-{\rho }_2^{n}G(w_2^n,x^n))\parallel \\ +{\beta }_2^n\parallel R_{{\rho }_2^n,N}^{B,{\eta }_2}(B\left(v^n\right)+{\rho }_2^{n}g\left(v^n\right)-{\rho }_2^{n}G(w_2^n,x^n))\\ -R_{{\rho }_2^n,N}^{B,{\eta }_2}(B\left(y^{\ast }\right)+{\rho }_2^{n}g\left(y^{\ast }\right)-{\rho }_2^{n}G(w^{\ast },x^{\ast }))\parallel \\ \le & (1-{\beta }_2^n)\parallel y^n-y^{\ast }\parallel +{\zeta }_2^n{\beta }_2^n\parallel d_2^n-y^n\parallel +{\beta }_2^n{\zeta }_2^n\parallel y^n-s_2^n\parallel \\ +{\beta }_2^n\frac{{\tau }^{q-1}}{r_2-m_2{\rho }_2^n}{\rho }_2^n{\nu }_2\parallel x^n-x^{\ast }\parallel +{\beta }_2^n{\theta }_2(n)\parallel v^n-y^{\ast }\parallel \\ \le & (1-{\beta }_2^n+{\zeta }_2^n+{\beta }_2^n{\zeta }_2^n)\parallel y^n-y^{\ast }\parallel +{\beta }_2^n\frac{{\tau }^{q-1}}{r_2-m_2{\rho }_2^n}{\rho }_2^n{\nu }_2\parallel x^n-x^{\ast }\parallel \\ +{\zeta }_2^n\parallel y^{n+1}-y^{\ast }\parallel +{\beta }_2^n{\zeta }_2^n\parallel s_2^n-y^{\ast }\parallel +{\beta }_2^n{\theta }_2(n)\parallel v^n-y^{\ast }\parallel ,\end{array}$$

for (19) $y^{n+1}-y^n={\beta }_2^n(d_2^n-y^n)$.

It follows from (26) that

$$\begin{array}{rcl}\parallel y^{n+1}-t_2^{n+1}\parallel & \le & \parallel (1-{\beta }_2^n)y^n+{\beta }_2^n{d}_2^n-(1-{\beta }_2^n)y^n\\ -{\beta }_2^n{R}_{{\rho }_2^n,N}^{B,{\eta }_2}(B\left(s_2^n\right)+{\rho }_2^{n}g\left(s_2^n\right)-{\rho }_2^{n}G(w_2^n,x^n))\parallel \\ \le & {\beta }_2^n\parallel d_2^n-R_{{\rho }_2^n,N}^{B,{\eta }_2}(B\left(s_2^n\right)+{\rho }_2^{n}g\left(s_2^n\right)-{\rho }_2^{n}G(w_2^n,x^n))\parallel \\ \le & {\beta }_2^n{\zeta }_2^n\parallel d_2^n-v^n\parallel +{\beta }_2^n\frac{{\tau }^{q-1}}{r_2-m_2{\rho }_2^n}{\rho }_2^n{\nu }_2\parallel x^n-x^n\parallel +{\beta }_2^n{\theta }_2(n)\parallel s_2^n-v^n\parallel \\ \le & {\zeta }_2^n(\parallel y^{n+1}-y^{\ast }\parallel +\parallel y^n-y^{\ast }\parallel )+{\zeta }_2^n{\beta }_2^n(\parallel y^n-y^{\ast }\parallel +\parallel v^n-y^{\ast }\parallel )\\ +{\beta }_2^n{\theta }_2(n)(\parallel s_2^n-y^{\ast }\parallel +\parallel y^{\ast }-v^n\parallel )\\ \le & {\zeta }_2^n\parallel y^{n+1}-y^{\ast }\parallel +({\zeta }_2^n+{\zeta }_2^n{\beta }_2^n)\parallel y^n-y^{\ast }\parallel \\ +({\zeta }_2^n{\beta }_2^n+{\beta }_2^n{\theta }_2(n))\parallel v^n-y^{\ast }\parallel +{\beta }_2^n{\theta }_2(n)\parallel s_2^n-y^{\ast }\parallel .\end{array}$$

(33)

Combining (30), (31), (32), (33) and (19), we have

$$\begin{array}{rcl}\parallel y^{n+1}-y^{\ast }\parallel & \le & \frac{1}{1-2{\zeta }_2^n}[(1-{\beta }_2^n+2{\beta }_2^n{\zeta }_2^n+2{\zeta }_2^n)\parallel y^n-y^{\ast }\parallel \\ +{\beta }_2^n\frac{{\tau }^{q-1}}{r_2-m_2{\rho }_2^n}{\rho }_2^n{\nu }_2\parallel x^n-x^{\ast }\parallel +(2{\beta }_2^n{\theta }_2(n)+{\zeta }_2^n{\beta }_2^n)\parallel v^n-y^{\ast }\parallel \\ +{\beta }_2^n({\zeta }_2^n+{\theta }_2(n))\parallel s_2^n-y^{\ast }\parallel ]\\ \le & \frac{1}{1-2{\zeta }_2^n}(1-{\beta }_2^n+2{\beta }_2^n{\zeta }_2^n+2{\zeta }_2^n\\ +(2{\beta }_2^n{\theta }_2(n)+{\zeta }_2^n{\beta }_2^n)\frac{1}{1-{\xi }_2^n}(2-{\alpha }_2^n+{\alpha }_2^n{\theta }_2(n))\\ +{\beta }_2^n({\zeta }_2^n+{\theta }_2(n))[(1-{\alpha }_2^n)+{\alpha }_2^n{\theta }_2(n)])\parallel y^n-y^{\ast }\parallel \\ +\frac{1}{1-2{\zeta }_2^n}({\beta }_2^n\frac{{\tau }^{q-1}}{r_2-m_2{\rho }_2^n}{\rho }_2^n{\nu }_2+{\beta }_2^n({\zeta }_2^n+{\theta }_2(n)){\alpha }_2^n\frac{{\tau }^{q-1}}{r_2-m_2{\rho }_2^n}{\rho }_2^n{\nu }_2\\ +(2{\beta }_2^n{\theta }_2(n)+{\zeta }_2^n{\beta }_2^n)\frac{1}{1-{\xi }_2^n}{\alpha }_2^n{\rho }_2^n{\nu }_2\frac{{\tau }^{q-1}}{r_2-m_2{\rho }_2^n})\parallel x^n-x^{\ast }\parallel .\end{array}$$

(34)

By using (22) and (23), let

(35)

where

Let $\overrightarrow{A}={(\parallel x^{n+1}-x^{\ast }\parallel ,\parallel y^{n+1}-y^{\ast }\parallel )}^T$ and $\overrightarrow{B}={(\parallel x^n-x^{\ast }\parallel ,\parallel y^n-y^{\ast }\parallel )}^T$, then from (33), (34) and (35), we have $\overrightarrow{a}\le \mathbf{\Psi }\overrightarrow{b}$, where

$$\mathbf{\Psi }=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right),$$

(36)

which is called the matrix for a nonlinear set-valued inclusions system involving $(A,\eta )$-accretive mappings. By using [16], we have

$$\parallel \overrightarrow{A}\parallel \le \parallel \mathbf{\Psi }\parallel \parallel \overrightarrow{B}\parallel .$$

(37)

Let

$$\parallel \mathbf{\Psi }\parallel =max\{a_{11},a_{12},a_{21},a_{22}\}.$$

It follows from (21)-(23), Theorem 3.1 and [15] that $0<\parallel \mathbf{\Psi }\parallel <1$ and there exist $x^{\ast },y^{\ast }\in X$ and $z^{\ast }\in S(x^{\ast })$, $w^{\ast }\in T(y^{\ast })$ [17] such that

$$\{\begin{array}{c}{Q}_1(x^{\ast })=x^{\ast },\hfill \\ Q_2(y^{\ast })=y^{\ast };\hfill \end{array}$$

and the sequence ${\{x^n,y^n\}}_{n=0}^{\mathrm{\infty }}$ generated by Ishikawa-hybrid proximal Algorithm 4.1 converges linearly to a solution $(x^{\ast },y^{\ast })$ of the problem (3) as the convergence rate

$$\begin{array}{rcl}\parallel \mathbf{\Psi }\parallel & =& max\{1-({\alpha }_1+{\beta }_1)+({\alpha }_1+{\beta }_1){\theta }_1+(2-2{\alpha }_1+{\alpha }_1{\theta }_1){\beta }_1{\theta }_1,\\ {\alpha }_1{\beta }_1{\nu }_1{\theta }_1\frac{{\tau }^{q-1}}{r_1-m_1{\rho }_1}(1+{\rho }_1-{\beta }_1{\rho }_1),\\ 1-{\beta }_2+2{\beta }_2{\theta }_2(2-{\alpha }_2+{\alpha }_2{\theta }_2)+{\beta }_2{\theta }_2(1-{\alpha }_2+{\alpha }_2{\theta }_2),\\ (1-{\beta }_2+{\beta }_2{\theta }_2)+3(1-{\alpha }_2+{\alpha }_2{\theta }_2){\beta }_2{\theta }_2+{\beta }_2{\theta }_2\},\end{array}$$

where $c_q>0$ is the same as in Lemma 2.7, ${\rho }_i\in (0,\frac{r_i}{m_i})$ ($i=1,2$). This completes the proof. □

Remark 5.2 For a suitable choice of the mappings A, B, ${\eta }_i$, F, G, M, N, S, T, f, g and X, we can obtain several known results in [7, 9, 11, 12] as special cases of Theorem 5.1.