Approximation methods for solutions of generalized multi-valued mixed quasi-variational inclusion systems

Abstract

The purpose of this paper is to introduce new approximation methods for solutions of generalized non-accretive multi-valued mixed quasi-variational inclusion systems involving $(A,\eta )$-accretive mappings in q-uniformly smooth Banach spaces and, by using the new resolvent operator technique associated with $(A,\eta )$-accretive mappings, Nadler’s fixed point theorem and Liu’s inequality, we prove some existence theorems of solutions for our systems by constructing the new Mann iterative algorithm. Further, we study the stability of the iterative sequence generated by the perturbed iterative algorithms. The results presented in this paper improve and generalize the corresponding results of recent works given by some authors.

1 Introduction

It is well known that the ideas and techniques of the variational inequalities and variational inclusions are being applied in a variety of diverse fields of pure and applied sciences and proven to be productive and innovative. It has been shown that this theory provides the most natural, direct, simple, unified, and efficient framework for a general treatment of a wide class of linear and nonlinear problems. Correspondingly, the existence of solutions or the convergence and stability of a suitable iterative algorithm to the system of nonlinear variational inequalities or variational inclusions has also been studied by many authors, see [1–22] and the references therein.

Recently, Lan et al. [9] introduced a new concept of $(A,\eta )$-accretive mappings, which provides a unifying framework for maximal monotone operators, m-accretive operators, η-subdifferential operators, maximal η-monotone operators, H-monotone operators, generalized m-accretive mappings, H-accretive operators, $(H,\eta )$-monotone operators, A-monotone mappings. Further, we studied some properties of $(A,\eta )$-accretive mappings and defined the resolvent operators associated with $(A,\eta )$-accretive mappings which include the existing resolvent operators as special cases. By using the new resolvent operator technique, we also developed a new perturbed iterative algorithm with errors to solve a class of nonlinear relaxed cocoercive variational inclusions with $(A,\eta )$-accretive mappings in q-uniformly smooth Banach spaces and prove the convergence and stability of the iterative sequence generated by the perturbed iterative algorithm. For details, we can refer to [2–5, 7, 8, 10, 11, 13, 23].

On the other hand, some systems of variational inequalities, variational inclusions, complementarity problems and equilibrium problems have been studied by some authors in recent years because of their close relations to Nash equilibrium problems. Huang and Fang [6] introduced a system of order complementarity problems and established some existence results for the problems by using fixed point theory. Kassay and Kolumbán [8] introduced a system of variational inequalities and proved an existence theorem by using Ky Fan’s lemma. In [1], Cho et al. developed an iterative algorithm to approximate the solution of a system of nonlinear variational inequalities by using the classical resolvent operator technique. By using the resolvent operator technique associated with an $(H,\eta )$-monotone operator, Fang et al. [3] further studied the approximating solution of a system of variational inclusions in Hilbert spaces. Very recently, Guan and Hu [22] introduced and studied a system of generalized variational inclusions involving a new monotone mapping in Banach spaces. Furthermore, by using the concept of $(A,\eta )$-accretive mappings and the new resolvent operator technique associated with $(A,\eta )$-accretive mappings, Lan [15] introduced and studied a system of general mixed quasi-variational inclusions involving $(A,\eta )$-accretive mappings in Banach spaces, and construct a new perturbed iterative algorithm with mixed errors for this system of nonlinear $(A,\eta )$-accretive variational inclusions in q-uniformly smooth Banach spaces. Kazmi et al. [16] considered the convergence and stability of an iterative algorithm for a system of generalized implicit variational-like inclusions in Banach spaces. Suwannawit and Petrot [17] studied the existence of solutions and the stability of iterative algorithm for a system of random set-valued variational inclusion problems involving $(A,m,\eta )$-generalized monotone operators. Because stability is one of optimization theory, it is not surprising to see a number of papers dealing with the study of convergence and stability to investigate various important themes. For other related works, we refer to [10, 12, 18, 19, 24] and the references therein.

Motivated and inspired by the above works, in this paper, we consider the following system of generalized non-accretive multi-valued mixed quasi-variational inclusions:

Find $(x,y)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$, $u\in F(x)$, and $v\in G(y)$ such that

$$\{\begin{array}{c}0\in N_1(x,v)+M_1(x,x),\hfill \\ 0\in N_2(u,y)+M_2(y,y),\hfill \end{array}$$

(1.1)

where ${\mathbb{B}}_i$ is a real Banach space, $N_i:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_i$, $A_i:{\mathbb{B}}_i\to {\mathbb{B}}_i$, and ${\eta }_i:{\mathbb{B}}_i\times {\mathbb{B}}_i\to {\mathbb{B}}_i$ are single-valued mappings, and $F:{\mathbb{B}}_1\to 2^{{\mathbb{B}}_1}$ and $G:{\mathbb{B}}_2\to 2^{{\mathbb{B}}_2}$ are multi-valued mappings, $M_i:{\mathbb{B}}_i\times {\mathbb{B}}_i\to 2^{{\mathbb{B}}_i}$ is an any nonlinear mapping such that $M_i(\cdot ,t):{\mathbb{B}}_i\to 2^{{\mathbb{B}}_i}$ is an $(A_i,{\eta }_i)$-accretive mapping for all $t\in {\mathbb{B}}_i$ and $i=1,2$.

We remark that, for suitable choices of $N_i$, $M_i$, $A_i$, ${\eta }_i$, F, G, and ${\mathbb{B}}_i$ for $i=1,2$, it is easy to see that the problem (1.1) includes a number (systems) of quasi-variational inclusions, generalized quasi-variational inclusions, quasi-variational inequalities, implicit quasi-variational inequalities studied by many authors as special cases. See, for example, [1–22] and the following examples:

Example 1.1 If $F:{\mathbb{B}}_1\to {\mathbb{B}}_1$ and $G:{\mathbb{B}}_2\to {\mathbb{B}}_2$ are two single-valued mappings, then, from the problem (1.1), we have the following problem: Find $(x,y)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$ such that

$$\{\begin{array}{c}0\in N_1(x,G(y))+M_1(x,x),\hfill \\ 0\in N_2(F(x),y)+M_2(y,y).\hfill \end{array}$$

(1.2)

Example 1.2 In (1.2), for any $(a,b)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$, if $N_1(x,G(y))=E_1(f(x),y)-a$ and $N_2(x,G(y))=E_2(x,g(y))-b$, where $f:{\mathbb{B}}_1\to {\mathbb{B}}_1$ and $g:{\mathbb{B}}_2\to {\mathbb{B}}_2$ are two single-valued mappings, then the problem (1.2) reduces to finding $(x,y)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$ such that

$$\{\begin{array}{c}a\in E_1(f(x),y)+M_1(x,x),\hfill \\ b\in E_2(x,g(y))+M_2(y,y).\hfill \end{array}$$

(1.3)

The problem (1.3) is called a system of mixed quasi-variational inclusion problems, which was studied by Lan [15].

Example 1.3 If $F=G=I$, $M_i(\cdot ,t)=M_i(\cdot )$ for all $t\in {\mathbb{B}}_i$ and $M_2(\cdot ,t)=M_2(\cdot )$ for all $t\in {\mathbb{B}}_2$, then the problem (1.2) is equivalent to the problem of finding $(x,y)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$ such that

$$\{\begin{array}{c}0\in N_1(x,y)+M_1(x),\hfill \\ 0\in N_2(x,y)+M_2(y),\hfill \end{array}$$

(1.4)

which is studied by Fang et al. [3] and Verma [10] when $M_i$ is A-monotone and $(H,\eta )$-monotone for $i=1,2$, respectively. Some special cases of the problem (1.4) can be found in [1, 8, 12–14] and the references therein.

Moreover, in this paper, by using the new resolvent operator technique associated with $(A,\eta )$-accretive mappings, Nadler’s fixed point theorem and Liu’s inequality, we prove some existence theorems of solutions for our systems by constructing the new Mann iterative algorithm. Further, we study the stability of the iterative sequence generated by the perturbed iterative algorithms. The results presented in this paper improve and generalize the corresponding results of recent works given by some authors.

2 Preliminaries

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

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

for all $x\in \mathbb{B}$, where $q>1$ is a constant. In particular, $J_2$ is the usual normalized duality mapping. It is well known that, in general, $J_q(x)={\parallel x\parallel }^{q-2}{J}_2(x)$ for all $x\ne 0$ and $J_q$ is single-valued if $\mathbb{B}$ is strictly convex.

In the sequel, we always suppose that $\mathbb{B}$ is a real Banach space such that $J_q$ is single-valued and ℋ is a Hilbert space. If $\mathbb{B}=\mathcal{H}$, then $J_2$ becomes the identity mapping on ℋ.

The modulus of smoothness of $\mathbb{B}$ is the function ${\varrho }_{\mathbb{B}}:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ defined by

$${\varrho }_{\mathbb{B}}(t)=sup\{\frac{1}{2}(\parallel x+y\parallel +\parallel x-y\parallel )-1:\parallel x\parallel \le 1,\parallel y\parallel \le t\}.$$

• (1) A Banach space $\mathbb{B}$ is said to be uniformly smooth if

  $$\underset{t\to 0}{lim}\frac{{\varrho }_{\mathbb{B}}(t)}{t}=0.$$

• (2) $\mathbb{B}$ is said to be q-uniformly smooth if there exists a constant $c>0$ such that

  $${\varrho }_{\mathbb{B}}(t)\le c{t}^q$$

for all $q>1$.

Note that $J_q$ is single-valued if $\mathbb{B}$ is uniformly smooth. In the study of characteristic inequalities in q-uniformly smooth Banach spaces, Xu [25] proved the following result.

Lemma 2.1 Let $q>1$ be a given real number and $\mathbb{B}$ be a real uniformly smooth Banach space. Then $\mathbb{B}$ is q-uniformly smooth if and only if there exists a constant $c_q>0$ such that, for all $x,y\in \mathbb{B}$,

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

In the sequel, we give some concept and lemmas for our main results later.

Definition 2.1 Let $\mathbb{B}$ be a q-uniformly smooth Banach space and $f,A:\mathbb{B}\to \mathbb{B}$ be two single-valued mappings. T is said to be:

• (1) accretive if

  $$〈f(x)-f(y),J_q(x-y)〉\ge 0$$

  for all $x,y\in \mathbb{B}$;

• (2) strictly accretive if T is accretive and

  $$〈f(x)-f(y),J_q(x-y)〉=0$$

  if and only if $x=y$;

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

  $$〈f(x)-f(y),J_q(x-y)〉\ge r{\parallel x-y\parallel }^q$$

  for all $x,y\in \mathbb{B}$;

• (4) γ-strongly accretive with respect to A if there exists a constant $\gamma >0$ such that

  $$〈f(x)-f(y),J_q(A(x)-A(y))〉\ge \gamma {\parallel x-y\parallel }^q$$

  for all $x,y\in \mathbb{B}$;

• (5) m-relaxed cocoercive with respect to A if there exists a constant $m>0$ such that

  $$〈f(x)-f(y),J_q(A(x)-A(y))〉\ge -m{\parallel f(x)-f(y)\parallel }^q$$

  for all $x,y\in \mathbb{B}$;

• (6) $(\alpha ,\xi )$-relaxed cocoercive with respect to A if there exist constants $\alpha ,\xi >0$ such that

  $$〈f(x)-f(y),J_q(A(x)-A(y))〉\ge -\alpha {\parallel f(x)-f(y)\parallel }^q+\xi {\parallel x-y\parallel }^q$$

  for all $x,y\in \mathbb{B}$;

• (7) s-Lipschitz continuous if there exists a constant $s>0$ such that

  $$\parallel f(x)-f(y)\parallel \le s\parallel x-y\parallel$$

  for all $x,y\in \mathbb{B}$.

Remark 2.1 When $\mathbb{B}=\mathcal{H}$, (1)-(4) of Definition 2.1 reduce to the definitions of monotonicity, strict monotonicity, strong monotonicity, and strong monotonicity with respect to A, respectively (see [2, 3]).

Definition 2.2 A multi-valued mapping $F:\mathbb{B}\to 2^{\mathbb{B}}$ is said to be ζ-$\hat{\mathbf{H}}$-Lipschitz continuous if there exists a constant $\zeta >0$ such that

$$\hat{\mathbf{H}}(F(x),F(y))\le \zeta \parallel x-y\parallel$$

for all $x,y\in \mathbb{B}$, where $\hat{\mathbf{H}}:2^{\mathbb{B}}\times 2^{\mathbb{B}}\to (-\mathrm{\infty },+\mathrm{\infty })\cup \{+\mathrm{\infty }\}$ is the Hausdorff metric, i.e.,

$$\hat{\mathbf{H}}(A,B)=max\{\underset{x\in A}{sup}\underset{y\in B}{inf}\parallel x-y\parallel ,\underset{x\in B}{sup}\underset{y\in A}{inf}\parallel x-y\parallel \}$$

for all $A,B\in 2^{\mathbb{B}}$.

Definition 2.3 A single-valued mapping $\eta :\mathbb{B}\times \mathbb{B}\to \mathbb{B}$ is said to be τ-Lipschitz continuous if there exists a constant $\tau >0$ such that

$$\parallel \eta (x,y)\parallel \le \tau \parallel x-y\parallel$$

for all $x,y\in \mathbb{B}$.

Definition 2.4 Let $\mathbb{B}$ be a q-uniformly smooth Banach space, $\eta :\mathbb{B}\times \mathbb{B}\to \mathbb{B}$ and $H:\mathbb{B}\to \mathbb{B}$ be single-valued mappings. Then set-valued mapping $M:\mathbb{B}\to 2^{\mathbb{B}}$ is said to be:

• (1) accretive if

  $$〈u-v,J_q(x-y)〉\ge 0$$

  for all $x,y\in \mathbb{B}$, $u\in M(x)$, and $v\in M(y)$;

• (2) η-accretive if

  $$〈u-v,J_q(\eta (x,y))〉\ge 0$$

  for all $x,y\in \mathbb{B}$, $u\in M(x)$, and $v\in M(y)$;

• (3) strictly η-accretive if M is η-accretive and equality holds if and only if $x=y$;

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

  $$〈u-v,J_q(\eta (x,y))〉\ge r{\parallel x-y\parallel }^q$$

  for all $x,y\in \mathbb{B}$, $u\in M(x)$, and $v\in M(y)$;

• (5) α-relaxed η-accretive if there exists a constant $\alpha >0$ such that

  $$〈u-v,J_q(\eta (x,y))〉\ge -\alpha {\parallel x-y\parallel }^q$$

  for all $x,y\in \mathbb{B}$, $u\in M(x)$, and $v\in M(y)$;

• (6) m-accretive if M is accretive and $(I+\rho M)(\mathbb{B})=\mathbb{B}$ for all $\rho >0$, where I denotes the identity operator on $\mathbb{B}$;

• (7) generalized m-accretive if M is η-accretive and $(I+\rho M)(\mathbb{B})=\mathbb{B}$ for all $\rho >0$;

• (8) H-accretive if M is accretive and $(H+\rho M)(\mathbb{B})=\mathbb{B}$ for all $\rho >0$;

• (9) $(H,\eta )$-accretive if M is η-accretive and $(H+\rho M)(\mathbb{B})=\mathbb{B}$ for every $\rho >0$.

In a similar way, we can define strictly η-accretivity and strongly η-accretivity of the single-valued mapping $A:\mathbb{B}\to \mathbb{B}$.

Definition 2.5 The mapping $N:\mathbb{B}\times \mathbb{B}\to \mathbb{B}$ is said to be ϵ-Lipschitz continuous with respect to the first argument if there exists a constant $ϵ>0$ such that

$$\parallel N(x,\cdot )-N(y,\cdot )\parallel \le ϵ\parallel x-y\parallel$$

for all $x,y\in \mathbb{B}$.

In a similar way, we can define the Lipschitz continuity of the mapping $N(\cdot ,\cdot )$ with respect to the second argument.

Definition 2.6 Let $A:\mathbb{B}\to \mathbb{B}$, $\eta :\mathbb{B}\times \mathbb{B}\to \mathbb{B}$ be two single-valued mappings. Then a multi-valued mapping $M:\mathbb{B}\to 2^{\mathbb{B}}$ is said to be $(A,\eta )$-accretive if

• (1) M is m-relaxed η-accretive;

• (2) $(A+\rho M)(\mathbb{B})=\mathbb{B}$ for all $\rho >0$.

Lemma 2.2 ([9])

Let $\mathbb{B}$ be a q-uniformly smooth Banach space and $\eta :\mathbb{B}\times \mathbb{B}\to \mathbb{B}$ be τ-Lipschitz continuous, $A:\mathbb{B}\to \mathbb{B}$ be a r-strongly η-accretive mapping and $M:\mathbb{B}\to 2^{\mathbb{B}}$ be an $(A,\eta )$-accretive mapping. Then the resolvent operator $R_{\eta ,M}^{\rho ,A}:\mathbb{B}\to \mathbb{B}$ defined by

$$R_{\eta ,M}^{\rho ,A}(u)={(A+\rho M)}^{-1}(u)$$

for all $u\in \mathbb{B}$ is $\frac{{\tau }^{q-1}}{r-\rho m}$-Lipschitz continuous, i.e.,

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

for all $x,y\in \mathbb{B}$, where $\rho \in (0,\frac{r}{m})$ is a constant.

3 Approximation methods and main results

In this section, by using the resolvent operator technique associated with $(A,\eta )$-accretive mappings, we introduce the new Mann iterative algorithm with mixed errors for solving the system (1.1) of generalized nonlinear mixed quasi-variational inclusion in Banach spaces and prove the convergence and stability of the iterative sequence generated by the Mann iterative algorithm.

Definition 3.1 Let S be a self-mapping of $\mathbb{B}$, $x_0\in X$ and let $\{x_n\}$ be an iterative sequence in $\mathbb{B}$ defined by $x_{n+1}=h(S,x_n)$ for all $n\ge 0$. Suppose that $\{x\in \mathbb{B}:Sx=x\}\ne \mathrm{\varnothing }$ and $\{x_n\}$ converges to a fixed point $x^{\ast }$ of S. Let $\{{\upsilon }_n\}$ be a sequence in $\mathbb{B}$ and let ${ϵ}_n=\parallel {\upsilon }_{n+1}-h(S,{\upsilon }_n)\parallel$. If $lim{ϵ}_n=0$ implies that ${\upsilon }_n\to x^{\ast }$, then the iterative sequence $\{x_n\}$ defined by $x_{n+1}=h(S,x_n)$ for all $n\ge 0$ is said to be S-stable or stable with respect to S.

Lemma 3.1 ([26])

Let $\{a_n\}$, $\{b_n\}$, $\{c_n\}$ be three nonnegative real sequences satisfying the following condition: there exists a natural number $n_0$ such that

$$a_{n+1}\le (1-t_n)a_n+b_n{t}_n+c_n$$

for all $n\ge n_0$, where $t_n\in [0,1]$, ${\sum }_{n=0}^{\mathrm{\infty }}{t}_n=\mathrm{\infty }$, ${lim}_{n\to \mathrm{\infty }}{b}_n=0$ and ${\sum }_{n=0}^{\mathrm{\infty }}{c}_n<\mathrm{\infty }$. Then $a_n\to 0$ as $n\to \mathrm{\infty }$.

The solvability of the problem (1.1) depends on the equivalence between (1.1) and the problem of finding the fixed point of the associated generalized resolvent operator. From Definition 2.6, we can obtain the following.

Lemma 3.2 For $i=1,2$, let $A_i$, ${\eta }_i$, $M_i$, $N_i$, F, and G be the same as in the problem (1.1). Then the following statements are mutually equivalent:

• (1) An element $(x,y,u,v)\in {\mathbb{B}}_1\times {\mathbb{B}}_2\times {\mathbb{B}}_1\times {\mathbb{B}}_2$ is a solution to the problem (1.1).

• (2) There exist $(x,y)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$, $u\in F(x)$, and $v\in G(y)$ such that

  $$\begin{array}{r}x=R_{{\eta }_1,M_1(\cdot ,x)}^{\lambda ,A_1}[A_1(x)-\lambda N_1(x,v)],\\ y=R_{{\eta }_2,M_2(\cdot ,y)}^{\rho ,A_2}[A_2(y)-\rho N_2(u,y)],\end{array}$$

  (3.1)

  where $R_{{\eta }_1,M_1(\cdot ,x)}^{\lambda ,A_1}={(A_1+\lambda M_1(\cdot ,x))}^{-1}$, $R_{{\eta }_2,M_2(\cdot ,y)}^{\rho ,A_2}={(A_2+\rho M_2(\cdot ,y))}^{-1}$, and $\lambda >0$, $\rho >0$ are two constants.

• (3) For any $\lambda >0$ and $\rho >0$, the mapping $T_{\lambda ,\rho }:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_1\times {\mathbb{B}}_2$ defined by

  $$T_{\lambda ,\rho }(x,y)=(P_{\lambda }(x,y),Q_{\rho }(x,y))$$

  for all $(x,y)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$ has a fixed point $(x,y,u,v)\in {\mathbb{B}}_1\times {\mathbb{B}}_2\times {\mathbb{B}}_1\times {\mathbb{B}}_2$, where mappings $P_{\lambda }:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_1$ and $Q_{\rho }:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_2$ are defined by

  $$P_{\lambda }(x,y)=R_{{\eta }_1,M_1(\cdot ,x)}^{\lambda ,A_1}[A_1(x)-\lambda N_1(x,v)]$$

  for all $v\in G(y)$ and

  $$Q_{\rho }(x,y)=R_{{\eta }_2,M_2(\cdot ,y)}^{\rho ,A_2}[A_2(y)-\rho N_2(u,y)]$$

  for all $u\in F(x)$, respectively.

This fixed point formulation allows us to construct the following perturbed iterative algorithm with mixed errors.

Algorithm 3.1 Step 1. For any $(x_0,y_0)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$, define the iterative sequence $\{(x_n,y_n)\}$ by

$$\{\begin{array}{l}{x}_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{R}_{{\eta }_1,M_1(\cdot ,x_n)}^{\lambda ,A_1}(z_n)+{\alpha }_n{d}_n+e_n,\\ y_{n+1}=(1-{\alpha }_n)y_n+{\alpha }_n{R}_{{\eta }_2,M_2(\cdot ,y_n)}^{\rho ,A_2}(w_n)+{\alpha }_n{f}_n+h_n,\\ z_n=A_1(x_n)-\lambda N_1(x_n,v_n),\\ w_n=A_2(y_n)-\rho N_2(u_n,y_n)\end{array}$$

(3.2)

for all $n\ge 0$, where $u_n\in F(x_n)$, $v_n\in G(x_n)$, and $\lambda ,\rho >0$ are constants.

Step 2. Choose the sequences $\{{\alpha }_n\}$, $\{d_n\}$, $\{e_n\}$, $\{f_n\}$, and $\{h_n\}$ such that, for all $n\ge 0$, $\{{\alpha }_n\}$ is a sequence in $(0,1]$ with ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, $\{d_n\},\{e_n\}\subset {\mathbb{B}}_1$, and $\{f_n\},\{h_n\}\subset {\mathbb{B}}_2$ are the sequences of errors and satisfy the following conditions:

• (a) $d_n=d_n^{\prime }+d_n^{\prime \prime }$ and $f_n=f_n^{\prime }+f_n^{\prime \prime }$, where $\{d_n^{\prime }\},\{d_n^{\prime \prime }\}\subset {\mathbb{B}}_1$ and $\{f_n^{\prime }\},\{f_n^{\prime \prime }\}\subset {\mathbb{B}}_2$;

• (b) ${lim}_{n\to \mathrm{\infty }}\parallel d_n^{\prime }\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel f_n^{\prime }\parallel =0$;

• (c) ${\sum }_{n=0}^{\mathrm{\infty }}\parallel d_n^{\prime \prime }\parallel <\mathrm{\infty }$, ${\sum }_{n=0}^{\mathrm{\infty }}\parallel e_n\parallel <\mathrm{\infty }$, ${\sum }_{n=0}^{\mathrm{\infty }}\parallel f_n^{\prime \prime }\parallel <\mathrm{\infty }$ and ${\sum }_{n=0}^{\mathrm{\infty }}\parallel h_n\parallel <\mathrm{\infty }$.

Step 3. If the sequences $\{x_n\}$, $\{y_n\}$, $\{z_n\}$, $\{w_n\}$, $\{{\alpha }_n\}$, $\{d_n\}$, $\{e_n\}$, $\{f_n\}$, and $\{h_n\}$ satisfy (3.2) to sufficient accuracy, then go to Step 4. Otherwise, set $n:=n+1$ and return to Step 1.

Step 4. Let $\{({\phi }_n,{\psi }_n)\}$ be any sequence in ${\mathbb{B}}_1\times {\mathbb{B}}_2$ and define a sequence $\{({ϵ}_n,{\epsilon }_n)\}$ in $\mathbb{R}\times \mathbb{R}$ by

$$\{\begin{array}{l}{ϵ}_n=\parallel {\phi }_{n+1}-\{(1-{\alpha }_n){\phi }_n+{\alpha }_n{R}_{{\eta }_1,M_1(\cdot ,{\phi }_n)}^{\lambda ,A_1}(s_n)+{\alpha }_n{d}_n+e_n\}\parallel ,\\ {\epsilon }_n=\parallel {\psi }_{n+1}-\{(1-{\alpha }_n){\psi }_n+{\alpha }_n{R}_{{\eta }_2,M_2(\cdot ,{\psi }_n)}^{\rho ,A_2}(t_n)+{\alpha }_n{f}_n+h_n\}\parallel ,\\ s_n=A_1({\phi }_n)-\lambda N_1({\phi }_n,{\varpi }_n),\\ t_n=A_2({\psi }_n)-\rho N_2({\chi }_n,{\psi }_n),\end{array}$$

(3.3)

where ${\chi }_n\in F({\phi }_n)$ and ${\varpi }_n\in G({\psi }_n)$.

Step 5. If the sequences $\{{ϵ}_n\}$, $\{{\epsilon }_n\}$, $\{{\phi }_{n+1}\}$, $\{{\psi }_{n+1}\}$, $\{s_n\}$, $\{t_n\}$, $\{{\alpha }_n\}$, $\{d_n\}$, $\{e_n\}$, $\{f_n\}$, and $\{h_n\}$ satisfy (3.3) to sufficient accuracy, the stop here. otherwise, set $n:=n+1$ and return to Step 2.

Now, we show the existence of solutions of the problem (1.1) and prove the convergence and stability of Algorithm 3.1.

Theorem 3.1 For $i=1,2$, let ${\mathbb{B}}_i$ be a $q_i$-uniformly smooth Banach space with $q_i>1$, ${\eta }_i:{\mathbb{B}}_i\times {\mathbb{B}}_i\to {\mathbb{B}}_i$ be ${\tau }_i$-Lipschitz continuous, $A_i:{\mathbb{B}}_i\to {\mathbb{B}}_i$ be $r_i$-strongly ${\eta }_i$-accretive and ${\sigma }_i$-Lipschitz continuous, $F:{\mathbb{B}}_1\to CB({\mathbb{B}}_1)$ be k-$\hat{\mathbf{H}}$-Lipschitz continuous, $G:{\mathbb{B}}_2\to CB({\mathbb{B}}_2)$ be κ-$\hat{\mathbf{H}}$-Lipschitz continuous, $M_1:{\mathbb{B}}_1\times {\mathbb{B}}_1\to 2^{{\mathbb{B}}_1}$ be $(A_1,{\eta }_1)$-accretive in the first variable and $M_2:{\mathbb{B}}_2\times {\mathbb{B}}_2\to 2^{{\mathbb{B}}_2}$ be $(A_2,{\eta }_2)$-accretive in the first variable. Suppose that $N_1:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_1$ is $({\pi }_1,{\iota }_1)$-relaxed cocoercive with respect to $A_1$, ${\delta }_1$-Lipschitz continuous in the first argument, ${\beta }_2$-Lipschitz continuous in the second variable and $N_2:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_2$ is $({\pi }_2,{\iota }_2)$-relaxed cocoercive with respect to $A_2$, ${\delta }_2$-Lipschitz continuous in the second argument and ${\beta }_1$-Lipschitz continuous in the first variable. If

$$\begin{array}{r}\parallel R_{{\eta }_1,M_1(\cdot ,x)}^{\lambda ,A_1}(z)-R_{{\eta }_1,M_1(\cdot ,y)}^{\lambda ,A_1}(z)\parallel \le {\nu }_1\parallel x-y\parallel ,\\ \parallel R_{{\eta }_2,M_2(\cdot ,x)}^{\rho ,A_2}(z)-R_{{\eta }_2,M_2(\cdot ,y)}^{\rho ,A_2}(z)\parallel \le {\nu }_2\parallel x-y\parallel \end{array}$$

(3.4)

for all $(x,y,z)\in {\mathbb{B}}_2\times {\mathbb{B}}_2\times {\mathbb{B}}_2$ and there exist constants $\lambda \in (0,r_1/m_1)$ and $\rho \in (0,r_2/m_2)$ such that

$$\{\begin{array}{l}\frac{{\tau }_1^{q_1-1}\sqrt[q_1]{{\sigma }_1^{q_1}-q_1\lambda {\iota }_1+q_1\lambda {\pi }_1{\delta }_1^{q_1}+c_{q_1}{\lambda }^{q_1}{\delta }_1^{q_1}}}{r_1-\lambda m_1}+\frac{\rho k{\beta }_1{\tau }_2^{q_2-1}}{r_2-\rho m_2}+{\nu }_1<1,\\ \frac{{\tau }_2^{q_2-1}\sqrt[q_2]{{\sigma }_2^{q_2}-q_2\rho {\iota }_2+q_2\rho {\pi }_2{\delta }_2^{q_2}+c_{q_2}{\rho }^{q_2}{\delta }_2^{q_2}}}{r_2-\rho m_2}+\frac{\lambda \kappa {\beta }_2{\tau }_1^{q_1-1}}{r_1-\lambda m_1}+{\nu }_2<1,\end{array}$$

(3.5)

where $c_{q_1}$, $c_{q_2}$ are the constants as in Lemma  2.1, then

• (1) the problem (1.1) has a solution $(x^{\ast },y^{\ast },u^{\ast },v^{\ast })$;

• (2) the iterative sequence $\{(x_n,y_n,u_n,v_n)\}$ generated by Algorithm 3.1 converges strongly to the solution $(x^{\ast },y^{\ast },u^{\ast },v^{\ast })$;

• (3) if, in addition, there exists $\alpha >0$ such that ${\alpha }_n\ge \alpha$ for all $n\ge 0$, then

  $$\underset{n\to \mathrm{\infty }}{lim}({\phi }_n,{\psi }_n,{\chi }_n,{\varpi }_n)=(x^{\ast },y^{\ast },u^{\ast },v^{\ast })⟺\underset{n\to \mathrm{\infty }}{lim}({ϵ}_n,{\epsilon }_n)=(0,0),$$

  where $({ϵ}_n,{\epsilon }_n)$ is defined by (3.3).

Proof For any $\lambda >0$ and $\rho >0$, define $P_{\lambda }:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_1$ and $Q_{\rho }:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_2$ by

$$\{\begin{array}{c}{P}_{\lambda }(x,y)=R_{{\eta }_1,M_1(\cdot ,x)}^{\lambda ,A_1}[A_1(x)-\lambda N_1(x,v)],\hfill \\ Q_{\rho }(x,y)=R_{{\eta }_2,M_2(\cdot ,y)}^{\rho ,A_2}[A_2(y)-\rho N_2(u,y)]\hfill \end{array}$$

(3.6)

for all $(x,y)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$, $u\in F(x)$ and $v\in G(v)$. Now, define the norm ${\parallel \cdot \parallel }_{\ast }$ on ${\mathbb{B}}_1\times {\mathbb{B}}_2$ by

$${\parallel (x,y)\parallel }_{\ast }=\parallel x\parallel +\parallel y\parallel$$

for all $(x,y)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$. It is easy to see that $({\mathbb{B}}_1\times {\mathbb{B}}_2,{\parallel \cdot \parallel }_{\ast })$ is a Banach space (see [4]). By (3.6), for any $\lambda >0$ and $\rho >0$, define $T_{\lambda ,\rho }:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_1\times {\mathbb{B}}_2$ by

$$T_{\lambda ,\rho }(x,y)=(P_{\lambda }(x,y),Q_{\rho }(x,y))$$

for all $(x,y)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$.

Now, we prove that $T_{\lambda ,\rho }$ is a contractive mapping. In fact, for any $(x_i,y_i)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$ and $i=1,2$, there exist $u_1\in F(x_1)$ and $v_1\in G(y_1)$ such that

$$\{\begin{array}{c}{P}_{\lambda }(x_1,y_1)=R_{{\eta }_1,M_1(\cdot ,x_1)}^{\lambda ,A_1}[A_1(x_1)-\lambda N_1(x_1,v_1)],\hfill \\ Q_{\rho }(x_1,y_1)=R_{{\eta }_2,M_2(\cdot ,y_1)}^{\rho ,A_2}[A_2(y_1)-\rho N_2(u_1,y_1)].\hfill \end{array}$$

(3.7)

Since $F(x_2)\in CB({\mathbb{B}}_1)$ and $G(y_2)\in CB({\mathbb{B}}_2)$, it follows from Nadler’s result [27] that there exist $u_2\in F(x_2)$ and $v_2\in G(y_2)$ such that

$$\parallel u_1-u_2\parallel \le \hat{\mathbf{H}}\parallel F(x_1)-F(x_2)\parallel ,\parallel v_1-v_2\parallel \le \hat{\mathbf{H}}\parallel G(y_1)-G(y_2)\parallel .$$

(3.8)

Setting

$$\{\begin{array}{c}{P}_{\lambda }(x_2,y_2)=R_{{\eta }_1,M_1(\cdot ,x_2)}^{\lambda ,A_1}[A_1(x_2)-\lambda N_1(x_2,v_2)],\hfill \\ Q_{\rho }(x_2,y_2)=R_{{\eta }_2,M_2(\cdot ,y_2)}^{\rho ,A_2}[A_2(y_2)-\rho N_2(u_2,y_2)].\hfill \end{array}$$

(3.9)

Thus it follows from (3.4), (3.6), (3.9), and Lemma 2.2 that

$$\begin{array}{r}\parallel P_{\lambda }(x_1,y_1)-P_{\lambda }(x_2,y_2)\parallel \\ \le \parallel R_{{\eta }_1,M_1(\cdot ,x_1)}^{\lambda ,A_1}[A_1(x_1)-\lambda N_1(x_1,v_1)]-R_{{\eta }_1,M_1(\cdot ,x_2)}^{\lambda ,A_1}[A_1(x_1)-\lambda N_1(x_1,v_1)]\parallel \\ +\parallel R_{{\eta }_1,M_1(\cdot ,x_2)}^{\lambda ,A_1}[A_1(x_1)-\lambda N_1(x_1,v_1)]-R_{{\eta }_1,M_1(\cdot ,x_2)}^{\lambda ,A_1}[A_1(x_2)-\lambda N_1(x_2,v_2)]\parallel \\ \le \frac{{\tau }_1^{q_1-1}}{r_1-\lambda m_1}\parallel A_1(x_1)-A_1(x_2)-\lambda (N_1(x_1,v_1)-N_1(x_2,v_1))\parallel \\ +{\nu }_1\parallel x_1-x_2\parallel +\frac{\lambda {\tau }_1^{q_1-1}}{r_1-\lambda m_1}\parallel N_1(x_2,v_1)-N_1(x_2,v_2)\parallel \end{array}$$

(3.10)

and

$$\begin{array}{r}\parallel Q_{\rho }(x_1,y_1)-Q_{\rho }(x_2,y_2)\parallel \\ \le \frac{{\tau }_2^{q_2-1}}{r_2-\rho m_2}\parallel A_2(y_1)-A_2(y_2)-\rho (N_2(u_1,y_1)-N_2(u_1,y_2))\parallel \\ +{\nu }_2\parallel y_1-y_2\parallel +\frac{\rho {\tau }_2^{q_2-1}}{r_2-\rho m_2}\parallel N_2(u_1,y_2)-N_2(u_2,y_2)\parallel .\end{array}$$

(3.11)

By the assumptions, (3.8), and Lemma 2.1, we have

$$\begin{array}{r}{\parallel A_1(x_1)-A_1(x_2)-\lambda (N_1(x_1,v_1)-N_1(x_2,v_1))\parallel }^{q_1}\\ \le {\parallel A_1(x_1)-A_1(x_2)\parallel }^{q_1}-q_1\lambda 〈N_1(x_1,v_1)-N_1(x_2,v_1),J_{q_1}(A_1(x_1)-A_1(x_2))〉\\ +{\lambda }^{q_1}{c}_{q_1}{\parallel N_1(x_1,v_1)-N_1(x_2,v_1)\parallel }^{q_1}\\ \le ({\sigma }_1^{q_1}-q_1\lambda {\iota }_1+q_1\lambda {\pi }_1{\delta }_1^{q_1}+c_{q_1}{\lambda }^{q_1}{\delta }_1^{q_1}){\parallel x_1-x_2\parallel }^{q_1},\end{array}$$

(3.12)

$$\begin{array}{r}{\parallel A_2(y_1)-A_2(y_2)-\rho (N_2(u_1,y_1)-N_2(u_1,y_2))\parallel }^{q_2}\\ \le ({\sigma }_2^{q_2}-q_2\rho {\iota }_2+q_2\rho {\pi }_2{\delta }_2^{q_2}+c_{q_2}{\rho }^{q_2}{\delta }_2^{q_2}){\parallel y_1-y_2\parallel }^{q_2},\end{array}$$

(3.13)

$$\begin{array}{r}\parallel N_1(x_2,v_1)-N_1(x_2,v_2)\parallel \\ \le {\beta }_2\parallel v_1-v_2\parallel \le {\beta }_2\hat{\mathbf{H}}\parallel G(y_1)-G(y_2)\parallel \le \kappa {\beta }_2\parallel y_1-y_2\parallel ,\end{array}$$

(3.14)

$$\begin{array}{r}\parallel N_2(u_1,y_2)-N_2(u_2,y_2)\parallel \\ \le {\beta }_1\parallel u_1-u_2\parallel \le {\beta }_1\hat{\mathbf{H}}\parallel F(x_1)-F(x_2)\parallel \le k{\beta }_1\parallel x_1-x_2\parallel .\end{array}$$

(3.15)

Combining (3.10)-(3.15), we infer

$$\{\begin{array}{l}\parallel P_{\lambda }(x_1,y_1)-P_{\lambda }(x_2,y_2)\parallel \\ \le ({\nu }_1+\frac{{\tau }_1^{q_1-1}}{r_1-\lambda m_1}\sqrt[q_1]{{\sigma }_1^{q_1}-q_1\lambda {\iota }_1+q_1\lambda {\pi }_1{\delta }_1^{q_1}+c_{q_1}{\lambda }^{q_1}{\delta }_1^{q_1}})\parallel x_1-x_2\parallel \\ +\frac{\lambda \kappa {\beta }_2{\tau }_1^{q_1-1}}{r_1-\lambda m_1}\parallel y_1-y_2\parallel ,\\ \parallel Q_{\rho }(x_1,y_1)-Q_{\rho }(x_2,y_2)\parallel \\ \le ({\nu }_2+\frac{{\tau }_2^{q_2-1}}{r_2-\rho m_2}\sqrt[q_2]{{\sigma }_2^{q_2}-q_2\rho {\iota }_2+q_2\rho {\pi }_2{\delta }_2^{q_2}+c_{q_2}{\rho }^{q_2}{\delta }_2^{q_2}})\parallel y_1-y_2\parallel \\ +\frac{\rho k{\beta }_1{\tau }_2^{q_2-1}}{r_2-\rho m_2}\parallel x_1-x_2\parallel .\end{array}$$

(3.16)

It follows from (3.16) that

$$\begin{array}{r}\parallel P_{\lambda }(x_1,y_1)-P_{\lambda }(x_2,y_2)\parallel +\parallel Q_{\rho }(x_1,y_1)-Q_{\rho }(x_2,y_2)\parallel \\ \le \theta (\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel ),\end{array}$$

(3.17)

where

$$\begin{array}{rcl}\theta & =& max\{\frac{{\tau }_1^{q_1-1}}{r_1-\lambda m_1}\sqrt[q_1]{{\sigma }_1^{q_1}-q_1\lambda {\iota }_1+q_1\lambda {\pi }_1{\delta }_1^{q_1}+c_{q_1}{\lambda }^{q_1}{\delta }_1^{q_1}}+\frac{\rho k{\beta }_1{\tau }_2^{q_2-1}}{r_2-\rho m_2}+{\nu }_1,\\ \frac{{\tau }_2^{q_2-1}}{r_2-\rho m_2}\sqrt[q_2]{{\sigma }_2^{q_2}-q_2\rho {\iota }_2+q_2\rho {\pi }_2{\delta }_2^{q_2}+c_{q_2}{\rho }^{q_2}{\delta }_2^{q_2}}+\frac{\lambda \kappa {\beta }_2{\tau }_1^{q_1-1}}{r_1-\lambda m_1}+{\nu }_2\}.\end{array}$$

By (3.5), we know that $0\le \theta <1$ and it follows from (3.17) that

$${\parallel T_{\lambda ,\rho }(x_1,y_1)-T_{\lambda ,\rho }(x_2,y_2)\parallel }_{\ast }\le \theta {\parallel (x_1,y_1)-(x_2,y_2)\parallel }_{\ast }.$$

This proves that $T_{\lambda ,\rho }:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_1\times {\mathbb{B}}_2$ is a contraction mapping. Thus, from Nadler’s fixed point theorem [27], it follows that there exist $(x^{\ast },y^{\ast })\in {\mathbb{B}}_1\times {\mathbb{B}}_2$, $u^{\ast }\in F(x^{\ast })$ and $v^{\ast }\in G(y^{\ast })$ such that

$$T_{\lambda ,\rho }(x^{\ast },y^{\ast })=(x^{\ast },y^{\ast }),$$

that is,

$$x^{\ast }=R_{{\eta }_1,M_1(\cdot ,x^{\ast })}^{\lambda ,A_1}[A_1\left(x^{\ast }\right)-\lambda N_1(x^{\ast },v^{\ast })],y^{\ast }=R_{{\eta }_2,M_2(\cdot ,y^{\ast })}^{\rho ,A_2}[A_2\left(y^{\ast }\right)-\rho N_2(u^{\ast },y^{\ast })].$$

Hence, by Lemma 3.2, $(x^{\ast },y^{\ast })$ is a solution of the problem (1.1).

Next, for any $u^{\ast }\in F(x^{\ast })$ and $v^{\ast }\in G(y^{\ast })$, let

$$\begin{array}{c}{z}^{\ast }=A_1\left(x^{\ast }\right)-\lambda N_1(x^{\ast },v^{\ast }),w^{\ast }=A_2\left(y^{\ast }\right)-\rho N_2(u^{\ast },y^{\ast }),\hfill \\ x^{\ast }=R_{{\eta }_1,M_1(\cdot ,x^{\ast })}^{\lambda ,A_1}\left(z^{\ast }\right),y^{\ast }=R_{{\eta }_2,M_2(\cdot ,y^{\ast })}^{\rho ,A_2}\left(w^{\ast }\right).\hfill \end{array}$$

Then, by (3.2) and the proof of (3.16), it follows that

$$\begin{array}{c}\begin{array}{rl}\parallel z_n-z^{\ast }\parallel & =\parallel A_1(x_n)-A_1\left(x^{\ast }\right)-\lambda (N_1(x_n,v_n)-N_1(x^{\ast },v^{\ast }))\parallel \\ \le \sqrt[q_1]{{\sigma }_1^{q_1}-q_1\lambda {\iota }_1+q_1\lambda {\pi }_1{\delta }_1^{q_1}+c_{q_1}{\lambda }^{q_1}{\delta }_1^{q_1}}\parallel x_n-x^{\ast }\parallel +\lambda \kappa {\beta }_2\parallel y_n-y^{\ast }\parallel ,\end{array}\hfill \\ \begin{array}{rl}\parallel w_n-w^{\ast }\parallel & =\parallel A_2(y_n)-A_2\left(y^{\ast }\right)-\rho (N_2(x_n,y_n)-N_2(x^{\ast },y^{\ast }))\parallel \\ \le \sqrt[q_2]{{\sigma }_2^{q_2}-q_2\rho {\iota }_2+q_2\rho {\pi }_2{\delta }_2^{q_2}+c_{q_2}{\rho }^{q_2}{\delta }_2^{q_2}}\parallel y_n-y^{\ast }\parallel +\rho k{\beta }_1\parallel x_n-x^{\ast }\parallel ,\end{array}\hfill \\ \begin{array}{r}\parallel x_{n+1}-x^{\ast }\parallel \\ \le (1-{\alpha }_n)\parallel x_n-x^{\ast }\parallel +{\alpha }_n(\parallel d_n^{\prime }\parallel +\parallel d_n^{\prime \prime }\parallel )+\parallel e_n\parallel +{\alpha }_n\parallel R_{{\eta }_1,M_1(\cdot ,x_n)}^{\lambda ,A_1}(z_n)-x^{\ast }\parallel \\ \le (1-{\alpha }_n)\parallel x_n-x^{\ast }\parallel +{\alpha }_n(\parallel d_n^{\prime }\parallel +\parallel d_n^{\prime \prime }\parallel )+\parallel e_n\parallel \\ +{\alpha }_n\parallel R_{{\eta }_1,M_1(\cdot ,x_n)}^{\lambda ,A_1}(z_n)-R_{{\eta }_1,M_1(\cdot ,x^{\ast })}^{\lambda ,A_1}(z_n)\parallel +{\alpha }_n\parallel R_{{\eta }_1,M_1(\cdot ,x^{\ast })}^{\lambda ,A_1}(z_n)-R_{{\eta }_1,M_1(\cdot ,x^{\ast })}^{\lambda ,A_1}\left(z^{\ast }\right)\parallel \\ \le (1-{\alpha }_n)\parallel x_n-x^{\ast }\parallel +{\alpha }_n{\nu }_1\parallel x_n-x^{\ast }\parallel +{\alpha }_n\frac{{\tau }_1^{q_1-1}}{r_1-\lambda m_1}\parallel z_n-z^{\ast }\parallel \\ +{\alpha }_n\parallel d_n^{\prime }\parallel +(\parallel d_n^{\prime \prime }\parallel +\parallel e_n\parallel ),\end{array}\hfill \end{array}$$

and

$$\begin{array}{r}\parallel y_{n+1}-y^{\ast }\parallel \\ \le (1-{\alpha }_n)\parallel y_n-y^{\ast }\parallel +{\alpha }_n(\parallel f_n^{\prime }\parallel +\parallel f_n^{\prime \prime }\parallel )+\parallel h_n\parallel +{\alpha }_n\parallel R_{{\eta }_2,M_2(\cdot ,y_n)}^{\rho ,A_2}(w_n)-y^{\ast }\parallel \\ \le (1-{\alpha }_n)\parallel y_n-y^{\ast }\parallel +{\alpha }_n(\parallel f_n^{\prime }\parallel +\parallel f_n^{\prime \prime }\parallel )+\parallel h_n\parallel \\ +{\alpha }_n\parallel R_{{\eta }_2,M_2(\cdot ,y_n)}^{\rho ,A_2}(w_n)-R_{{\eta }_2,M_2(\cdot ,y^{\ast })}^{\rho ,A_2}(w_n)\parallel +{\alpha }_n\parallel R_{{\eta }_2,M_2(\cdot ,y^{\ast })}^{\rho ,A_2}(w_n)-R_{{\eta }_2,M_2(\cdot ,y^{\ast })}^{\rho ,A_2}\left(w^{\ast }\right)\parallel \\ \le (1-{\alpha }_n)\parallel y_n-y^{\ast }\parallel +{\alpha }_n{\nu }_2\parallel y_n-y^{\ast }\parallel +{\alpha }_n\frac{{\tau }_2^{q_2-1}}{r_2-\rho m_2}\parallel w_n-w^{\ast }\parallel \\ +{\alpha }_n\parallel f_n^{\prime }\parallel +(\parallel f_n^{\prime \prime }\parallel +\parallel h_n\parallel ).\end{array}$$

Thus we obtain

$$\begin{array}{r}\parallel x_{n+1}-x^{\ast }\parallel +\parallel y_{n+1}-y^{\ast }\parallel \\ \le (1-{\alpha }_n)(\parallel x_n-x^{\ast }\parallel +\parallel y_n-y^{\ast }\parallel )\\ +{\alpha }_n[{\nu }_1+\frac{\rho k{\beta }_1{\tau }_2^{q_2-1}}{r_2-\rho m_2}+\frac{{\tau }_1^{q_1-1}}{r_1-\lambda m_1}\sqrt[q_1]{{\sigma }_1^{q_1}-q_1\lambda {\iota }_1+q_1\lambda {\pi }_1{\delta }_1^{q_1}+c_{q_1}{\lambda }^{q_1}{\delta }_1^{q_1}}]\parallel x_n-x^{\ast }\parallel \\ +{\alpha }_n[{\nu }_2+\frac{\lambda \kappa {\beta }_2{\tau }_1^{q_1-1}}{r_1-\lambda m_1}+\frac{{\tau }_2^{q_2-1}}{r_2-\rho m_2}\sqrt[q_2]{{\sigma }_2^{q_2}-q_2\rho {\iota }_2+q_2\rho {\pi }_2{\delta }_2^{q_2}+c_{q_2}{\rho }^{q_2}{\delta }_2^{q_2}}]\\ \times \parallel y_n-y^{\ast }\parallel +{\alpha }_n(\parallel f_n^{\prime }\parallel +\parallel d_n^{\prime }\parallel )+(\parallel f_n^{\prime \prime }\parallel +\parallel h_n\parallel +\parallel d_n^{\prime \prime }\parallel +\parallel e_n\parallel )\\ \le [1-{\alpha }_n(1-\theta )](\parallel x_n-x^{\ast }\parallel +\parallel y_n-y^{\ast }\parallel )\\ +{\alpha }_n(1-\theta )\cdot \frac{1}{1-\theta }(\parallel f_n^{\prime }\parallel +\parallel d_n^{\prime }\parallel )+(\parallel f_n^{\prime \prime }\parallel +\parallel h_n\parallel +\parallel d_n^{\prime \prime }\parallel +\parallel e_n\parallel ).\end{array}$$

(3.18)

Since ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, it follows from Lemma 3.1, (3.5), and (3.18) that

$$\parallel x_n-x^{\ast }\parallel +\parallel y_n-y^{\ast }\parallel \to 0$$

as $n\to \mathrm{\infty }$. Further, by $u_n\in F(x_n)$, $u^{\ast }\in F(x^{\ast })$, $v_n\in G(y_n)$, $v^{\ast }\in G(y^{\ast })$, and the $\hat{\mathbf{H}}$-Lipschitz continuity of F and G, we obtain

$$\parallel u_n-u^{\ast }\parallel \le \hat{\mathbf{H}}\parallel F(x_n)-F\left(x^{\ast }\right)\parallel \le k\parallel x_n-x^{\ast }\parallel$$

and

$$\parallel v_n-v^{\ast }\parallel \le \hat{\mathbf{H}}\parallel G(y_n)-G\left(y^{\ast }\right)\parallel \le \kappa \parallel y_n-y^{\ast }\parallel .$$

Thus we know that the sequence $\{(x_n,y_n,u_n,v_n)\}$ converges to a solution $(x^{\ast },y^{\ast },u^{\ast },v^{\ast })$ of the problem (1.1).

Now, we prove the conclusion (3). By (3.3), we know

$$\{\begin{array}{c}\parallel {\phi }_{n+1}-x^{\ast }\parallel \le \parallel (1-{\alpha }_n){\phi }_n+{\alpha }_n{R}_{{\eta }_1,M_1(\cdot ,{\phi }_n)}^{\lambda ,A_1}(s_n)+{\alpha }_n{d}_n+e_n-x^{\ast }\parallel +{ϵ}_n,\hfill \\ \parallel {\psi }_{n+1}-y^{\ast }\parallel \le \parallel (1-{\alpha }_n){\psi }_n+{\alpha }_n{R}_{{\eta }_2,M_2(\cdot ,{\psi }_n)}^{\rho ,A_2}(t_n)+{\alpha }_n{f}_n+h_n-y^{\ast }\parallel +{\epsilon }_n.\hfill \end{array}$$

(3.19)

As in the proof of the inequality (3.18), we have

$$\begin{array}{r}\parallel (1-{\alpha }_n){\phi }_n+{\alpha }_n{R}_{{\eta }_1,M_1(\cdot ,{\phi }_n)}^{\lambda ,A_1}(s_n)+{\alpha }_n{d}_n+e_n-x^{\ast }\parallel \\ +\parallel (1-{\alpha }_n){\psi }_n+{\alpha }_n{R}_{{\eta }_2,M_2(\cdot ,{\psi }_n)}^{\rho ,A_2}(t_n)+{\alpha }_n{f}_n+h_n-y^{\ast }\parallel \\ \le [1-{\alpha }_n(1-\theta )](\parallel {\phi }_n-x^{\ast }\parallel +\parallel {\psi }_n-y^{\ast }\parallel )\\ +{\alpha }_n(1-\theta )\cdot \frac{1}{1-\theta }(\parallel d_n^{\prime }\parallel +\parallel f_n^{\prime }\parallel )+(\parallel d_n^{\prime \prime }\parallel +\parallel e_n\parallel +\parallel f_n^{\prime \prime }\parallel +\parallel h_n\parallel ).\end{array}$$

(3.20)

Since $0<\alpha \le {\alpha }_n$, it follows from (3.17) and (3.18) that

$$\begin{array}{r}\parallel {\phi }_{n+1}-x^{\ast }\parallel +\parallel {\psi }_{n+1}-y^{\ast }\parallel \\ \le [1-{\alpha }_n(1-\theta )](\parallel {\phi }_n-x^{\ast }\parallel +\parallel {\psi }_n-y^{\ast }\parallel )\\ +{\alpha }_n(1-\theta )\cdot \frac{1}{1-\theta }(\parallel d_n^{\prime }\parallel +\parallel f_n^{\prime }\parallel +\frac{{ϵ}_n+{\epsilon }_n}{\alpha })+(\parallel d_n^{\prime \prime }\parallel +\parallel f_n^{\prime \prime }\parallel +\parallel e_n\parallel +\parallel h_n\parallel ).\end{array}$$

Suppose that ${lim}_{n\to \mathrm{\infty }}({ϵ}_n,{\epsilon }_n)=(0,0)$. Then, from ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$ and Lemma 3.1, it follows that

$$\parallel {\phi }_n-x^{\ast }\parallel +\parallel {\psi }_n-y^{\ast }\parallel \to 0$$

as $n\to \mathrm{\infty }$. Further, from ${\chi }_n\in F({\phi }_n)$, $u^{\ast }\in F(x^{\ast })$, ${\varpi }_n\in G({\psi }_n)$, $v^{\ast }\in G(y^{\ast })$, and the $\hat{\mathbf{H}}$-Lipschitz continuity of F and G, we have

$$\parallel {\chi }_n-u^{\ast }\parallel \le \hat{\mathbf{H}}\parallel F({\phi }_n)-F\left(x^{\ast }\right)\parallel \le k\parallel {\phi }_n-x^{\ast }\parallel$$

and

$$\parallel {\varpi }_n-v^{\ast }\parallel \le \hat{\mathbf{H}}\parallel G({\psi }_n)-G\left(y^{\ast }\right)\parallel \le \kappa \parallel {\psi }_n-y^{\ast }\parallel .$$

Hence we know that ${lim}_{n\to \mathrm{\infty }}({\phi }_n,{\psi }_n,{\chi }_n,{\varpi }_n)=(x^{\ast },y^{\ast },u^{\ast },v^{\ast })$.

Conversely, if ${lim}_{n\to \mathrm{\infty }}({\phi }_n,{\psi }_n,{\chi }_n,{\varpi }_n)=(x^{\ast },y^{\ast },u^{\ast },v^{\ast })$, then we have

$$\begin{array}{c}\begin{array}{rl}{ϵ}_n& =\parallel {\phi }_{n+1}-\{(1-{\alpha }_n){\phi }_n+{\alpha }_n{R}_{{\eta }_1,M_1(\cdot ,{\phi }_n)}^{\lambda ,A_1}(s_n)+{\alpha }_n{d}_n+e_n\}\parallel \\ \le \parallel {\phi }_{n+1}-x^{\ast }\parallel +\parallel (1-{\alpha }_n){\phi }_n+{\alpha }_n{R}_{{\eta }_1,M_1(\cdot ,{\phi }_n)}^{\lambda ,A_1}(s_n)+{\alpha }_n{d}_n+e_n-x^{\ast }\parallel ,\end{array}\hfill \\ \begin{array}{rl}{\epsilon }_n& =\parallel {\psi }_{n+1}-\{(1-{\alpha }_n){\psi }_n+{\alpha }_n{R}_{{\eta }_2,M_2(\cdot ,{\psi }_n)}^{\rho ,A_2}(t_n)+{\alpha }_n{f}_n+h_n\}\parallel \\ \le \parallel {\psi }_{n+1}-y^{\ast }\parallel +\parallel (1-{\alpha }_n){\psi }_n+{\alpha }_n{R}_{{\eta }_2,M_2(\cdot ,{\psi }_n)}^{\rho ,A_2}(t_n)+{\alpha }_n{f}_n+h_n-y^{\ast }\parallel \end{array}\hfill \end{array}$$

and

$$\begin{array}{rcl}{ϵ}_n+{\epsilon }_n& \le & \parallel {\phi }_{n+1}-x^{\ast }\parallel +\parallel {\psi }_{n+1}-y^{\ast }\parallel +[1-{\alpha }_n(1-\theta )](\parallel {\phi }_n-x^{\ast }\parallel +\parallel {\psi }_n-y^{\ast }\parallel )\\ +{\alpha }_n(1-\theta )\cdot \frac{1}{1-\theta }(\parallel d_n^{\prime }\parallel +\parallel f_n^{\prime }\parallel )+(\parallel d_n^{\prime \prime }\parallel +\parallel f_n^{\prime \prime }\parallel +\parallel e_n\parallel +\parallel h_n\parallel )\to 0\end{array}$$

as $n\to \mathrm{\infty }$. This completes the proof. □

Remark 3.1 If ${\mathbb{B}}_1$ and ${\mathbb{B}}_2$ are both 2-uniformly smooth Banach space and $0<\lambda =\rho <min\{r_1/m_1,r_2/m_2\}$ is a constant such that

$$\{\begin{array}{l}|\rho -\frac{{\iota }_1+{\pi }_1{\delta }_1^2-r_1(1-{\nu }_1)(m_1(1-{\nu }_1)+k{\beta }_1{\tau }_1)}{c_2{\delta }_1^2{\tau }_1^2-{(m_1(1-{\nu }_1)+k{\beta }_1{\tau }_1)}^2}|\\ <\frac{\sqrt{{[{\iota }_1+{\pi }_1{\delta }_1^2-r_1(1-{\nu }_1)(m_1(1-{\nu }_1)+k{\beta }_1{\tau }_1)]}^2-[{\sigma }_1^2{\tau }_1^2-r_1^2(1-{\nu }_1^2)][c_2{\delta }_1^2{\tau }_1^2-{(m_1(1-{\nu }_1)+k{\beta }_1{\tau }_1)}^2]}}{c_2{\delta }_1^2{\tau }_1^2-{(m_1(1-{\nu }_1)+k{\beta }_1{\tau }_1)}^2},\\ |\rho -\frac{{\iota }_2+{\pi }_2{\delta }_2^2-r_2(1-{\nu }_2)(m_2(1-{\nu }_2)+\kappa {\beta }_2{\tau }_2}{c_2{\delta }_2^2{\tau }_2^2-{(m_2(1-{\nu }_2)+\kappa {\beta }_2{\tau }_2)}^2}|\\ <\frac{\sqrt{{[{\iota }_2+{\pi }_2{\delta }_2^2-r_2(1-{\nu }_2)(m_2(1-{\nu }_2)+\kappa {\beta }_2{\tau }_2)]}^2-[{\sigma }_2^2{\tau }_2^2-r_2^2(1-{\nu }_2^2)][c_2{\delta }_2^2{\tau }_2^2-{(m_2(1-{\nu }_2)+\kappa {\beta }_2{\tau }_2)}^2]}}{c_2{\delta }_2^2{\tau }_2^2-{(m_2(1-{\nu }_2)+\kappa {\beta }_2{\tau }_2)}^2},\\ {\iota }_1+{\pi }_1{\delta }_1^2-r_1(1-{\nu }_1)(m_1(1-{\nu }_1)+k{\beta }_1{\tau }_1)\\ >\sqrt{[{\sigma }_1^2{\tau }_1^2-r_1^2(1-{\nu }_1^2)][c_2{\delta }_1^2{\tau }_1^2-{(m_1(1-{\nu }_1)+k{\beta }_1{\tau }_1)}^2]},\\ {\iota }_2+{\pi }_2{\delta }_2^2-r_2(1-{\nu }_2)(m_2(1-{\nu }_2)+\kappa {\beta }_2{\tau }_2)\\ >\sqrt{[{\sigma }_2^2{\tau }_2^2-r_2^2(1-{\nu }_2^2)][c_2{\delta }_2^2{\tau }_2^2-{(m_2(1-{\nu }_2)+\kappa {\beta }_2{\tau }_2)}^2]},\\ c_2{\delta }_1^2{\tau }_1^2>{(m_1(1-{\nu }_1)+k{\beta }_1{\tau }_1)}^2,c_2{\delta }_2^2{\tau }_2^2>{(m_2(1-{\nu }_2)+\kappa {\beta }_2{\tau }_2)}^2,\end{array}$$

then (3.5) holds. We note that Hilbert space and $L_p$ (or $l_p$) ($2\le p<\mathrm{\infty }$) spaces are 2-uniformly smooth Banach spaces.

From Theorem 3.1, we have the following results.

Corollary 3.1 For $i=1,2$, let ${\eta }_i$, $A_i$, $M_i$, F, G, and ${\mathbb{B}}_i$ be the same as in Theorem  3.1. Suppose that $N_1:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_1$ is ${\iota }_1$-strong accretive with respect to $A_1$, ${\delta }_1$-Lipschitz continuous in the first argument, ${\beta }_2$-Lipschitz continuous in the second variable and $N_2:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_2$ is ${\iota }_2$- with respect to $A_2$, ${\delta }_2$-Lipschitz continuous in the second argument and ${\beta }_1$-Lipschitz continuous in the first variable. If condition (3.5) in Theorem  3.1 holds, and there exist constants $\lambda \in (0,r_1/m_1)$ and $\rho \in (0,r_2/m_2)$ such that

$$\{\begin{array}{l}\frac{{\tau }_1^{q_1-1}\sqrt[q_1]{{\sigma }_1^{q_1}-q_1\lambda {\iota }_1+c_{q_1}{\lambda }^{q_1}{\delta }_1^{q_1}}}{r_1-\lambda m_1}+\frac{\rho k{\beta }_1{\tau }_2^{q_2-1}}{r_2-\rho m_2}+{\nu }_1<1,\\ \frac{{\tau }_2^{q_2-1}\sqrt[q_2]{{\sigma }_2^{q_2}-q_2\rho {\iota }_2+c_{q_2}{\rho }^{q_2}{\delta }_2^{q_2}}}{r_2-\rho m_2}+\frac{\lambda \kappa {\beta }_2{\tau }_1^{q_1-1}}{r_1-\lambda m_1}+{\nu }_2<1,\end{array}$$

where $c_{q_1}$, $c_{q_2}$ are the constants as in Lemma  2.1, then the iterative sequence $\{(x_n,y_n,u_n,v_n)\}$ generated by Algorithm 3.1 converges strongly to a solution $(x^{\ast },y^{\ast },u^{\ast },v^{\ast })$ of the problem (1.1). Moreover, if, in addition, there exists $\alpha >0$ such that ${\alpha }_n\ge \alpha$ for all $n\ge 0$, then

$$\underset{n\to \mathrm{\infty }}{lim}({\phi }_n,{\psi }_n,{\chi }_n,{\varpi }_n)=(x^{\ast },y^{\ast },u^{\ast },v^{\ast })⟺\underset{n\to \mathrm{\infty }}{lim}({ϵ}_n,{\epsilon }_n)=(0,0),$$

where $({ϵ}_n,{\epsilon }_n)$ is defined by (3.3).

Corollary 3.2 For $i=1,2$, let ${\eta }_i$, $A_i$, $M_i$, $N_i$, and ${\mathbb{B}}_i$ be the same as in Theorem  3.1, and $F:{\mathbb{B}}_1\to {\mathbb{B}}_1$ be k-Lipschitz continuous and $G:{\mathbb{B}}_2\to {\mathbb{B}}_2$ be κ-Lipschitz continuous. Assume that for any $(x_0,y_0)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$, the iterative sequence $\{(x_n,y_n)\}$ is generated by

$$\{\begin{array}{l}{x}_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{R}_{{\eta }_1,M_1(\cdot ,x_n)}^{\lambda ,A_1}(z_n)+{\alpha }_n{d}_n+e_n,\\ y_{n+1}=(1-{\alpha }_n)y_n+{\alpha }_n{R}_{{\eta }_2,M_2(\cdot ,y_n)}^{\rho ,A_2}(w_n)+{\alpha }_n{f}_n+h_n,\\ z_n=A_1(x_n)-\lambda N_1(x_n,G(y_n)),\\ w_n=A_2(y_n)-\rho N_2(F(x_n),y_n),\end{array}$$

where $\lambda ,\rho >0$ are constants, and for all $n\ge 0$, the sequences $\{{\alpha }_n\}$ is a sequence in $(0,1]$ with ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, $\{d_n\},\{e_n\}\subset {\mathbb{B}}_1$, and $\{f_n\},\{h_n\}\subset {\mathbb{B}}_2$ are the sequences of errors and satisfy the following conditions:

• (1) $d_n=d_n^{\prime }+d_n^{\prime \prime }$ and $f_n=f_n^{\prime }+f_n^{\prime \prime }$, where $\{d_n^{\prime }\},\{d_n^{\prime \prime }\}\subset {\mathbb{B}}_1$ and $\{f_n^{\prime }\},\{f_n^{\prime \prime }\}\subset {\mathbb{B}}_2$;

• (2) ${lim}_{n\to \mathrm{\infty }}\parallel d_n^{\prime }\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel f_n^{\prime }\parallel =0$;

• (3) ${\sum }_{n=0}^{\mathrm{\infty }}\parallel d_n^{\prime \prime }\parallel <\mathrm{\infty }$, ${\sum }_{n=0}^{\mathrm{\infty }}\parallel e_n\parallel <\mathrm{\infty }$, ${\sum }_{n=0}^{\mathrm{\infty }}\parallel f_n^{\prime \prime }\parallel <\mathrm{\infty }$ and ${\sum }_{n=0}^{\mathrm{\infty }}\parallel h_n\parallel <\mathrm{\infty }$.

If conditions (3.4) and (3.5) in Theorem 3.1 hold, then the iterative sequence $\{(x_n,y_n)\}$ converges strongly to the unique solution $(x^{\ast },y^{\ast })$ of the problem (1.2). Further, if, in addition, there exists $\alpha >0$ such that ${\alpha }_n\ge \alpha$ for all $n\ge 0$, then

$$\underset{n\to \mathrm{\infty }}{lim}({\phi }_n,{\psi }_n)=(x^{\ast },y^{\ast })⟺\underset{n\to \mathrm{\infty }}{lim}({ϵ}_n,{\epsilon }_n)=(0,0),$$

where $({ϵ}_n,{\epsilon }_n)\in \mathbb{R}\times \mathbb{R}$ is defined by

$$\{\begin{array}{l}{ϵ}_n=\parallel {\phi }_{n+1}-\{(1-{\alpha }_n){\phi }_n+{\alpha }_n{R}_{{\eta }_1,M_1(\cdot ,{\phi }_n)}^{\lambda ,A_1}(s_n)+{\alpha }_n{d}_n+e_n\}\parallel ,\\ {\epsilon }_n=\parallel {\psi }_{n+1}-\{(1-{\alpha }_n){\psi }_n+{\alpha }_n{R}_{{\eta }_2,M_2(\cdot ,{\psi }_n)}^{\rho ,A_2}(t_n)+{\alpha }_n{f}_n+h_n\}\parallel ,\\ s_n=A_1({\phi }_n)-\lambda N_1({\phi }_n,G({\psi }_n)),\\ t_n=A_2({\psi }_n)-\rho N_2(F({\phi }_n),{\psi }_n)\end{array}$$

for any sequence $\{({\phi }_n,{\psi }_n)\}\subset {\mathbb{B}}_1\times {\mathbb{B}}_2$.

Proof For any $\lambda >0$ and $\rho >0$, define $P_{\lambda }:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_1$ and $Q_{\rho }:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_2$ by

$$\{\begin{array}{c}{P}_{\lambda }(x,y)=R_{{\eta }_1,M_1(\cdot ,x)}^{\lambda ,A_1}[A_1(x)-\lambda N_1(x,G(y))],\hfill \\ Q_{\rho }(x,y)=R_{{\eta }_2,M_2(\cdot ,y)}^{\rho ,A_2}[A_2(y)-\rho N_2(F(x),y)]\hfill \end{array}$$

(3.21)

for all $(x,y)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$. Now, define the norm ${\parallel \cdot \parallel }_{\ast }$ on ${\mathbb{B}}_1\times {\mathbb{B}}_2$ by

$${\parallel (x,y)\parallel }_{\ast }=\parallel x\parallel +\parallel y\parallel$$

for all $(x,y)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$. It is easy to see that $({\mathbb{B}}_1\times {\mathbb{B}}_2,{\parallel \cdot \parallel }_{\ast })$ is a Banach space (see [4]). By (3.21), for any $\lambda >0$ and $\rho >0$, define $T_{\lambda ,\rho }:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_1\times {\mathbb{B}}_2$ by

$$T_{\lambda ,\rho }(x,y)=(P_{\lambda }(x,y),Q_{\rho }(x,y))$$

for all $(x,y)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$.

Thus, $T_{\lambda ,\rho }$ is a contractive mapping. In fact, for any $(x_i,y_i)\in {\mathbb{B}}_1\times {\mathbb{B}}_2$ and $i=1,2$, it follows from (3.4) and (3.21) that

$$\begin{array}{c}{P}_{\lambda }(x_1,y_1)=R_{{\eta }_1,M_1(\cdot ,x_1)}^{\lambda ,A_1}[A_1(x_1)-\lambda N_1(x_1,G(y_1))],\hfill \\ Q_{\rho }(x_1,y_1)=R_{{\eta }_2,M_2(\cdot ,y_1)}^{\rho ,A_2}[A_2(y_1)-\rho N_2(F(x_1),y_1)],\hfill \\ P_{\lambda }(x_2,y_2)=R_{{\eta }_1,M_1(\cdot ,x_2)}^{\lambda ,A_1}[A_1(x_2)-\lambda N_1(x_2,G(y_2))],\hfill \\ Q_{\rho }(x_2,y_2)=R_{{\eta }_2,M_2(\cdot ,y_2)}^{\rho ,A_2}[A_2(y_2)-\rho N_2(F(x_2),y_2)],\hfill \end{array}$$

and

$${\parallel T_{\lambda ,\rho }(x_1,y_1)-T_{\lambda ,\rho }(x_2,y_2)\parallel }_{\ast }\le \vartheta {\parallel (x_1,y_1)-(x_2,y_2)\parallel }_{\ast },$$

where

$$\begin{array}{rcl}\vartheta & =& max\{\frac{{\tau }_1^{q_1-1}}{r_1-\lambda m_1}\sqrt[q_1]{{\sigma }_1^{q_1}-q_1\lambda {\iota }_1+q_1\lambda {\pi }_1{\delta }_1^{q_1}+c_{q_1}{\lambda }^{q_1}{\delta }_1^{q_1}}+\frac{\rho k{\beta }_1{\tau }_2^{q_2-1}}{r_2-\rho m_2}+{\nu }_1,\\ \frac{{\tau }_2^{q_2-1}}{r_2-\rho m_2}\sqrt[q_2]{{\sigma }_2^{q_2}-q_2\rho {\iota }_2+q_2\rho {\pi }_2{\delta }_2^{q_2}+c_{q_2}{\rho }^{q_2}{\delta }_2^{q_2}}+\frac{\lambda \kappa {\beta }_2{\tau }_1^{q_1-1}}{r_1-\lambda m_1}+{\nu }_2\}.\end{array}$$

From (3.5), now we know that $T_{\lambda ,\rho }:{\mathbb{B}}_1\times {\mathbb{B}}_2\to {\mathbb{B}}_1\times {\mathbb{B}}_2$ is a Banach contraction mapping. Hence, $(x^{\ast },y^{\ast })$ is unique solution of the problem (1.1). The rest of proof is similar to that of Theorem 3.1 and we omit the details. This completes the proof. □

Remark 3.2 If $d_n=0$ or $e_n=0$ or $f_n=0$ or $h_n=0$ for all $n\ge 0$ in Algorithm 3.1 and Corollary 3.2, then the conclusions of Theorem 3.1 also hold. The results of Theorem 3.1 improve and generalize the corresponding results of [3, 9, 10]. For other related works, we refer to [1–8, 11–14].

4 Conclusions

In this paper, we first introduced a system of generalized nonlinear mixed quasi-variational inclusions with $(A,\eta )$-accretive mappings in Banach spaces, which includes some systems of quasi-variational inclusions and variational inequality problems as special cases. Then, by using the new resolvent operator technique associated with $(A,\eta )$-accretive mappings, Nadler’s fixed point theorem, and Liu’s inequality, we constructed some new Mann iterative algorithms with mixed errors for the existence of solutions for generalized nonlinear variational inclusion systems in q-uniformly smooth Banach spaces. Furthermore, we proved the convergence and stability of the iterative sequences generated by the perturbed iterative algorithm. The results presented in this paper improve and generalize the corresponding results in the literature.

By similar methods to the ones this paper, we can study the existence of solutions and the convergence and stability to the following system of general nonlinear mixed quasi-variational inclusions:

Find $(x_1,x_2,\dots ,x_m)\in {\mathbb{B}}_1\times {\mathbb{B}}_2\times \cdots \times {\mathbb{B}}_m$ and $u_i\in F_i(x_i)$ ($i=1,2,\dots ,m$) such that

$$\{\begin{array}{l}0\in N_1(x_1,u_2,x_3,\dots ,x_m)+M_1(x_1,x_1),\\ 0\in N_2(x_1,x_2,u_3,\dots ,x_m)+M_2(x_2,x_2),\\ \cdots \\ 0\in N_m(u_1,x_2,x_3,\dots ,x_m)+M_m(x_m,x_m),\end{array}$$

where $N_i:{\mathbb{B}}_1\times {\mathbb{B}}_2\times \cdots \times {\mathbb{B}}_m\to {\mathbb{B}}_i$, $A_i:{\mathbb{B}}_i\to {\mathbb{B}}_i$, and ${\eta }_i:{\mathbb{B}}_i\times {\mathbb{B}}_i\to {\mathbb{B}}_i$ are single-valued mappings, $F_i:{\mathbb{B}}_i\to 2^{{\mathbb{B}}_i}$ is multi-valued mapping, $M_i:{\mathbb{B}}_i\times {\mathbb{B}}_i\to 2^{{\mathbb{B}}_i}$ is an any nonlinear mapping such that $M_i(\cdot ,t):{\mathbb{B}}_i\to 2^{{\mathbb{B}}_i}$ is an $(A_i,{\eta }_i)$-accretive mapping for all $t\in {\mathbb{B}}_i$ and $i=1,2$, which are still worthy of being studied in further research.