Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities

Abstract

The main aim of this paper is twofold. Our first objective is to study a new system of generalized multivalued variational-like inequalities in Banach spaces and to establish its equivalence with a system of fixed point problems utilizing the concept of P-$\eta$-proximal mapping. The obtained alternative equivalent formulation is used and a new iterative algorithm for finding its approximate solution is suggested. Under some appropriate assumptions imposed on the mappings and parameters involved in the system of generalized multivalued variational-like inequalities, the existence of solution for the system mentioned above is proved and the convergence analysis of the sequences generated by our proposed iterative algorithm is discussed. The second objective of this work is to investigate and analyze the notion M-$\eta$-proximal mapping defined in the literature. Taking into account of the assumptions considered for such a mapping, we prove that every M-$\eta$-proximal mapping is actually P-$\eta$-proximal and is not a new one. At the same time, some comments relating to some existing results are pointed out.

1. Introduction

Variational inequality theory, which is mainly attributed to Stampacchia [1] and Fichera [2] provides very powerful techniques for studying a wide class of nonlinear problems arising in many diverse fields of pure mathematics and applied sciences, such as mechanics, optimization and control, physics, nonlinear programming, economics and transportation equilibrium, contact problems in elasticity, engineering sciences and other branches of mathematics, etc., see, e.g., refs. [3,4,5,6] and the relevant references therein.

In light of its wide applications in the study of a large variety of problems arising in the above-mentioned fields, during the last four decades, much attention has been given to extend and generalize variational inequalities in different directions by using novel and innovative techniques. An important and useful generalization of variational inequalities are generalized variational-like (quasi-variational-like) inequalities including a nonlinear term. In addition to the study of various classes of variational inequalities, many authors have also proceeded to construct and develop several quite different methods such as projection method and its variant forms, linear approximation, descent and Newton’s methods, and the methods based on auxiliary principle technique, for computing approximate solutions; see, for example, refs. [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] and the references therein. Considering the fact that to find the projection except in a very special case is not easy and depends on the inner product property of Hilbert spaces, the applicability of the projection method is limited. In fact, due to the presence of nonlinear term, the projection method cannot be applied to suggest any iterative algorithms for solving generalized variational-like (quasi-variational-like) inequalities. To overcome this drawback, in the last decades, Ding [16,29] and Ding and Luo [17] introduced the concepts of $\eta$-subdifferential and $\eta$-proximal point mapping of a proper functional and developed some iterative algorithms for solving some classes of generalized variational-like inequalities in the Hilbert space context.

Ding and Xia [19] introduced the notion of P-proximal (also referred to as J-proximal) mapping for a nonconvex lower semicontinuous subdifferentiable proper functional on Banach space. They established the existence and Lipschitz continuity of P-proximal mapping of a lower semicontinuous subdifferentiable proper functional under some appropriate assumptions in a reflexive Banach space setting. Utilizing the concept of P-proximal mapping and a similar technique of resolvent operator in Hilbert spaces, they suggested an iterative algorithm for approximating the solution of a class of completely generalized quasi-variational inequalities with nonconvex functional in the setting of reflexive Banach spaces. They also discussed the convergence analysis of the sequences generated by their proposed iterative algorithm under some suitable conditions.

Subsequently, Ahmad et al. [7] and Kazmi and Bhat [22] introduced independently the notion of P-$\eta$-proximal (as referred to as $J^{\eta }$-proximal) mapping, as a generalization of the concept of P-proximal mapping defined in [19], for a nonconvex lower semicontinuous $\eta$-subdifferentiable proper functional on Banach space and proved its existence and Lipschitz continuity under some suitable hypotheses. They also suggested some iterative algorithms for solving some classes of generalized multivalued nonlinear variational-like inequalities in a Banach space setting.

Recently, Kazmi et al. [30] defined the notion of M-$\eta$-proximal mapping, as an extension of P-$\eta$-proximal mapping introduced in [7,22], for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space and verified its existence and Lipschitz continuity under some appropriate conditions. They considered a system of generalized implicit variational-like inequalities (in short, $\mathrm{SGIVLI}$) in the Banach space context and using the notion of M-$\eta$-proximal mapping, proved its equivalence with a system of implicit equations. With the help of the obtained equivalence, they demonstrated the existence of a solution for the $\mathrm{SGIVLI}$ and proposed an iterative algorithm for approximating the solution of the $\mathrm{SGIVLI}$. They also discussed the convergence and stability analysis of the sequences generated by their suggested iterative algorithm.

On the other hand, Attouch [31] was the first to introduce and study the notion of the graph convergence for functions and operators involving the classical resolvent of multivalued mappings in a Hilbert space setting. Based on the equivalence between graph convergence and resolvent operator convergence considered in [31], many authors proceeded to construct some perturbed iterative algorithms for solving various kinds of variational inequality problems. They also studied the convergence analysis of the sequences generated by their proposed iterative algorithms under some appropriate assumptions, see, for instance, refs. [32,33,34,35,36,37,38].

Inspired and motivated by these results, the main purpose of this article is to study a new system of generalized multivalued variational-like inequalities in Banach spaces and to establish its equivalence with a system of fixed point problems utilizing the concept of P-$\eta$-proximal mapping. Under suitable conditions, some convergence analysis of the sequences generated by our proposed iterative algorithm is discussed. In our article, we also investigate and analyze the notion M-$\eta$-proximal mapping defined in [30], we prove that every M-$\eta$-proximal mapping is actually P-$\eta$-proximal mapping. At the same time, some comments relating to the results appearing in [30] are included.

The rest of the paper is organized as follows. Section 2 recalls the basic definitions and preliminaries concerning P-$\eta$-proximal mappings that are broadly used throughout the whole paper. At the end of this section, the notions of graph convergence and P-$\eta$-proximal mapping are employed and a new equivalence relationship between the graph convergence of a sequence of lower semicontinous and ${\eta }_n$-subdifferentiable proper functionals and their associated $P_n$-${\eta }_n$-proximal mappings, respectively, to a given lower semicontinuous and $\eta$-subdifferentiable proper functional and its associated P-$\eta$-proximal mapping is established.

In Section 3, we focus on a new system of generalized multivalued variational-like inequalities (in short, $\mathrm{SGMVLI}$) in Banach spaces and using the notion of P-$\eta$-proximal mapping, we establish its equivalence with a system of fixed point problems. In Section 4, with the goal of finding an approximate solution of the $\mathrm{SGMVLI}$, the notion of P-$\eta$-proximal mapping and Nadler’s technique [39] are used and a new iterative algorithm is suggested. We also discuss the existence of solution for the $\mathrm{SGMVLI}$ and study the convergence analysis of the sequences generated by our suggested iterative algorithm under some appropriate hypotheses imposed on the parameters and mappings involved in the $\mathrm{SGMVLI}$. The final section is devoted to the investigation and analysis of the notion of M-$\eta$-proximal mapping defined in [30]. We prove that under the assumptions considered in [30], every M-$\eta$-proximal mapping is actually P-$\eta$-proximal and is not a new one. By detecting some fatal errors in the iterative algorithm and results of [30], we point out that the main results appeared in [30] are not valid. Meanwhile, the correct versions of the corresponding algorithm and results given in [30] along with some important comments are provided.

2. Notation, Basic Definitions and Properties

Throughout, E will denote a real Banach space with a norm $\parallel ·\parallel$ and $E^{*}$ its continuous dual space with a norm ${\parallel ·\parallel }_{*}$. As usual, $w^{*}$ will stand for the weak star topology in $E^{*}$. Furthermore, the paring between E and $E^{*}$ is designated by $\langle .,.\rangle$. We denote by $CB(E)$ (resp. $C(E)$) the family of all the nonempty closed and bounded (resp. compact) subsets of E. The effective domain of an extended real-valued function f on E is the set of vectors for which the function takes a finite value and is denoted $\mathrm{dom}(f)$. An extended real-valued function $f:E\to ℝ\cup \{\pm \infty \}$ is called proper if it does not take the value $-\infty$ and $\mathrm{dom}(f)\ne \varnothing$.

The function $f:E\to ℝ\cup \{+\infty \}$ is said to be lower semicontinuous at $x\in E$ if for any sequence $\left\{x_n\right\}$ in E converging strongly to x we have

$$\begin{array}{c}\hfill f(x)\le \underset{n\to \infty }{lim\; inf}f(x_n).\end{array}$$

The function f is called lower semicontinuous if it is lower semicontinuous at every point of its domain.

Definition 1.

The function $f:E\times E\to ℝ\cup \{+\infty \}$ is called lower semicontinuous in the second argument on E if for each $x\in E$, the function $f(x,.):E\to ℝ\cup \{+\infty \}$ is lower semicontinuous on E.

Similarly, one can define the lower semicontinuity of the function f in the first argument. Recall that the function $f:E\to ℝ\cup \{+\infty \}$ is called convex if and only if the inequality

$$\begin{array}{c}\hfill f(\lambda x+(1-\lambda )y)\le \lambda f(x)+(1-\lambda )f(y),\end{array}$$

holds for every $\lambda \in [0,1]$ and all $x,y\in E$, for which the right-hand side is meaningful.

Definition 2 ([28]).

An extended real-valued functional $f:(x,y)\in E\times E\to f(x,y)\in ℝ\cup \{\pm \infty \}$ is said to be 0-diagonally quasi-concave (in short, $\mathrm{0-DQCV}$)

(i)

in the first argument (or with respect to x), if for any finite subset $\{x_1,x_2,\cdots ,x_n\}$ of X and any $\widehat{x}\in \mathrm{Co}(\{x_1,x_2,\cdots ,x_n\})$, we have

$$\begin{array}{c}\hfill \underset{1\le i\le n}{min}f(x_i,\widehat{x})\le 0,\end{array}$$

where for any given set $A\subset E$, $\mathrm{Co}(A)$ denotes the closed convex hull of A consisting of all vectors of the form ${\displaystyle \sum _{i=1}^n}{\lambda }_i{u}_i$ with $u_i\in A_i$, ${\lambda }_i\in {ℝ}_{+}=[0,+\infty )$ and ${\displaystyle \sum _{i=1}^n}{\lambda }_i=1$;

(ii)

in the second argument (or with respect to y), if for any finite subset $\{y_1,y_2,\cdots ,y_n\}$ of X and any $\widehat{y}\in \mathrm{Co}(\{y_1,y_2,\cdots ,y_n\})$, we have

$$\begin{array}{c}\hfill \underset{1\le i\le n}{min}f(\widehat{y},y_i)\le 0.\end{array}$$

Lemma 1 ([18]).

Let D be a nonempty convex subset of a topological vector space and let $f:D\times D\to ℝ\cup \{\pm \infty \}$ be an extended real-valued functional such that

(i)

f is lower semicontinuous in the second argument on every nonempty compact subset of D;

(ii)

f is $\mathrm{0-DQCV}$ in the first argument;

(iii)

there exists a nonempty compact convex subset $D_0$ of D and a nonempty compact subset K of D such that for each $y\in D\backslash K$, there is an $x\in \mathrm{Co}(D_0\cup \left\{y\right\})$ satisfying $f(x,y)>0$.

Then, there exists $\widehat{y}\in K$ such that $f(x,\widehat{y})\le 0$ for all $x\in D$.

The study of the notion of $\eta$-subdifferential, in a more general setting than that given in [40], was first initiated by Lee et al. [41] and Ding and Luo [17] independently as follows.

Definition 3 ([17,41]).

Let $\eta :E\times E\to E$ be a vector-valued mapping. A proper functional $\varphi :E\to ℝ\cup \{+\infty \}$ is called η-subdifferentiable at a point $x\in E$ if there exists a point $x^{*}\in E^{*}$ such that

$$\begin{array}{c}\hfill \langle x^{*},\eta (y,x)\rangle \le \varphi (y)-\varphi (x),\forall y\in E.\end{array}$$

Such a point $x^{*}$ is called η-subgradient of ϕ at x. The set of all η-subgradients of ϕ at x is denoted by ${\partial }_{\eta }\varphi (x)$. Associate with each ϕ, one can define the η-subdifferential mapping ${\partial }_{\eta }\varphi$ as follows:

$${\partial }_{\eta }\varphi (x)=\left\{\begin{array}{cc}\{x^{*}\in E^{*}:\langle x^{*},\eta (y,x)\rangle \le \varphi (y)-\varphi (x),\forall y\in E\},\hfill & x\in \mathrm{dom}\varphi ,\hfill \\ \varnothing ,\hfill & x\notin \mathrm{dom}\varphi .\hfill \end{array}\right.$$

For $x\in \mathrm{dom}\varphi$, ${\partial }_{\eta }\varphi (x)$ is called the η-subdifferential of ϕ at x.

It is worth highlighting that in the definition of $\eta$-subdifferential in the sense of Yang and Craven [40], the function $\varphi$ needs to be local Lipschitz and cannot take the value $+\infty$. In other words, the notion of $\eta$-subdifferential introduced in [17,41] is more general than that given in [40]. The following illustrates this fact.

Example 1.

Let E be the set of all real numbers endowed with the Euclidean norm $\parallel .\parallel =|.|$ and let the mappings $\varphi :E\to ℝ\cup \{+\infty \}$ and $\eta :E\times E\to E$ be defined, respectively, by

$$\varphi (x)=\left\{\begin{array}{cc}\varrho {\displaystyle \sum _{n=1}^{\frac{m+1}{2}}}{x}^{2n-1}\left|x\right|+\gamma ,\hfill & x\le 0,\hfill \\ +\infty ,\hfill & x>0,\hfill \end{array}\right.$$

and

$$\begin{array}{c}\hfill \eta (x,y)=\alpha \sum _{n=1}^{\frac{m+1}{2}}{x}^{2n-1}\left|x\right|+\beta \sum _{n=1}^{\frac{m+1}{2}}{y}^{2n-1}\left|y\right|,\forall x,y\in E,\end{array}$$

where m is an arbitrary but fixed odd natural number, and $\alpha ,\beta ,\varrho >0$ and $\gamma \in ℝ$ are arbitrary constants. We now show that for given $x\in \mathrm{dom}\varphi$, ${\partial }_{\eta }\varphi (x)=[\frac{\varrho }{\alpha },+\infty )$. For this purpose, take $x\in \mathrm{dom}\varphi$ arbitrarily but fixed. Then, we have

$$\begin{array}{c}\hfill \varphi (x)=\varrho \sum _{n=1}^{\frac{m+1}{2}}{x}^{2n-1}\left|x\right|+\gamma andx\le 0.\end{array}$$

If $\omega \in {\partial }_{\eta }\varphi (x)$, then we obtain

$$\begin{array}{c}\hfill \omega \left(\alpha \sum _{n=1}^{\frac{m+1}{2}}{y}^{2n-1}\left|y\right|+\beta \sum _{n=1}^{\frac{m+1}{2}}{x}^{2n-1}\left|x\right|\right)\le \varphi (y)-\varrho \sum _{n=1}^{\frac{m+1}{2}}{x}^{2n-1}\left|x\right|-\gamma ,\forall y\in E.\end{array}$$

In view of the fact that $\varphi (y)=+\infty$ for all $y>0$, we infer that

$$\begin{array}{c}\hfill \omega \left(\alpha \sum _{n=1}^{\frac{m+1}{2}}{y}^{2n-1}\left|y\right|+\beta \sum _{n=1}^{\frac{m+1}{2}}{x}^{2n-1}\left|x\right|\right)\le \varrho \sum _{n=1}^{\frac{m+1}{2}}\left(y^{2n-1}\left|y\right|-x^{2n-1}\left|x\right|\right),\forall y\le 0.\end{array}$$

(1)

If $x=0$, then from (1) it follows that

$$\begin{array}{c}\hfill \omega \alpha \sum _{n=1}^{\frac{m+1}{2}}{y}^{2n-1}\left|y\right|\le \varrho \sum _{n=1}^{\frac{m+1}{2}}{y}^{2n-1}\left|y\right|,\forall y\le 0,\end{array}$$

from which we deduce that $\omega \ge \frac{\varrho }{\alpha }$. For the case when $x<0$, owing to the fact that

$$\begin{array}{c}\hfill \alpha \sum _{n=1}^{\frac{m+1}{2}}{y}^{2n-1}\left|y\right|+\beta \sum _{n=1}^{\frac{m+1}{2}}{x}^{2n-1}\left|x\right|<0,\end{array}$$

using (1), we derive that

$$\begin{array}{c}\hfill \omega \ge \frac{\varrho {\displaystyle \sum _{n=1}^{\frac{m+1}{2}}}\left(y^{2n-1}\left|y\right|-x^{2n-1}\left|x\right|\right)}{\alpha {\displaystyle \sum _{n=1}^{\frac{m+1}{2}}}{y}^{2n-1}\left|y\right|+\beta {\displaystyle \sum _{n=1}^{\frac{m+1}{2}}}{x}^{2n-1}\left|x\right|},\forall y\le 0.\end{array}$$

(2)

Passing to the limit for $y\to -\infty$ in (2), we conclude that $\omega \ge \frac{\varrho }{\alpha }$. Therefore, in any case, we note that $\omega \ge \frac{\varrho }{\alpha }$ and so ${\partial }_{\eta }\varphi (x)\subseteq [\frac{\varrho }{\alpha },+\infty )$ for all $x\le 0$. To prove ${\partial }_{\eta }\varphi (x)=[\frac{\varrho }{\alpha },+\infty )$ for all $x\le 0$, it is sufficient to show that $[\frac{\varrho }{\alpha },+\infty )\subseteq {\partial }_{\eta }\varphi (x)$ for all $x\le 0$. Take $\omega \in [\frac{\varrho }{\alpha },+\infty )$ arbitrarily but fixed and suppose, on the contrary, that $\omega \notin {\partial }_{\eta }\varphi (x_0)$ for some $x_0\le 0$. Then, there exists $y_0\le 0$ such that

$$\begin{array}{c}\hfill \omega \left(\alpha \sum _{n=1}^{\frac{m+1}{2}}{y}_0^{2n-1}|y_0|+\beta \sum _{n=1}^{\frac{m+1}{2}}{x}_0^{2n-1}\left|x_0\right|\right)>\varrho \sum _{n=1}^{\frac{m+1}{2}}\left(y_0^{2n-1}|y_0|-x_0^{2n-1}\left|x_0\right|\right).\end{array}$$

(3)

Clearly, the case where $x_0=y_0=0$ cannot happen. If $x_0,y_0<0$, then in virtue of the fact that $\alpha {\displaystyle \sum _{n=1}^{\frac{m+1}{2}}}{y}_0^{2n-1}|y_0|+\beta {\displaystyle \sum _{n=1}^{\frac{m+1}{2}}}{x}_0^{2n-1}|x_0|<0$, making use of (3), yields

$$\begin{array}{c}\hfill \frac{\varrho }{\alpha }\le \omega <\frac{\varrho {\displaystyle \sum _{n=1}^{\frac{m+1}{2}}}\left(y_0^{2n-1}|y_0|-x_0^{2n-1}|x_0|\right)}{\alpha {\displaystyle \sum _{n=1}^{\frac{m+1}{2}}}{y}_0^{2n-1}|y_0|+\beta {\displaystyle \sum _{n=1}^{\frac{m+1}{2}}}{x}_0^{2n-1}|x_0|},\end{array}$$

from which it follows that $\varrho (\alpha +\beta ){\displaystyle \sum _{n=1}^{\frac{m+1}{2}}}{x}_0^{2n-1}\left|x_0\right|>0$. Since $\alpha ,\beta ,\varrho >0$ and m is an odd natural number, the last inequality ensures that $x_0>0$ which is a contradiction. If $x_0<0$ and $y_0=0$, then using (3), we conclude that $\omega <-\frac{\varrho }{\beta }$, which leads to a contradiction. Finally, for the case when $x_0=0$ and $y_0<0$, making use of (3), it follows that $\omega <\frac{\varrho }{\alpha }$ which is also a contradiction. These facts imply that $[\frac{\varrho }{\alpha },+\infty )\subseteq {\partial }_{\eta }\varphi (x)$, for all $x\le 0$. Accordingly, ${\partial }_{\eta }\varphi (x)=[\frac{\varrho }{\alpha },+\infty )$ for all $x\le 0$.

Definition 4 ([7,22]).

Let $\eta :E\times E\to E$ be a vector-valued mapping, $\varphi :E\to ℝ\cup \{+\infty \}$ be a proper η-subdifferentiable (not necessarily convex) functional and $P:X\to E^{*}$ be a single-valued mapping. If for any given point $x^{*}\in E^{*}$ and $\lambda >0$, there exists a unique point $x\in E$ satisfying

$$\begin{array}{c}\hfill \langle P(x)-x^{*},\eta (y,x)\rangle +\lambda \varphi (y)-\lambda \varphi (x)\ge 0,\forall y\in E,\end{array}$$

then the mapping $x^{*}\to x$, denoted by $R_{\lambda ,P}^{{\partial }_{\eta }\varphi }$, is called P-η-proximal mapping of ϕ. Evidently, in light of Definition 3, we have $x^{*}-P(x)\in \lambda {\partial }_{\eta }\varphi (x)$ and then it follows that $x=R_{\lambda ,P}^{{\partial }_{\eta }\varphi }(x^{*})={(P+\lambda {\partial }_{\eta }\varphi )}^{-1}(x^{*})$.

Definition 5.

Let $P:E\to E^{*}$ and $\eta :E\times E\to E$ be two vector-valued mappings. We say that the mapping P is

(i)

r-strongly η-monotone if there exists a constant $r>0$ such that

$$\begin{array}{c}\hfill \langle P(x)-P(y),\eta (x,y)\rangle \ge {r\parallel x-y\parallel }^2,\forall x,y\in E;\end{array}$$

(ii)

ϱ-Lipschitz continuous if there exists a constant $\varrho >0$ such that

$$\begin{array}{c}\hfill \parallel P(x)-P(y)\parallel \le \varrho \parallel x-y\parallel ,\forall x,y\in E.\end{array}$$

Definition 6.

The vector-valued mapping $\eta :X\times X\to X$ is said to be τ-Lipschitz continuous if there exists a constant $\tau >0$ such that $\parallel \eta (x,y)\parallel \le \tau \parallel x-y\parallel$, for all $x,y\in X$.

Taking into consideration the above-mentioned argument, an appropriate question then is whether for given vector-valued mappings $\eta :E\times E\to E$, $P:E\to E^{*}$, a lower semicontinuous $\eta$-subdifferentiable (not necessarily convex) proper functional $\varphi :E\to ℝ\cup \{+\infty \}$ and an arbitrary real constant $\lambda >0$, the P-$\eta$-proximal mapping associated with $\varphi ,P,\eta$ and $\lambda >0$ is well defined necessarily? The next theorem gives an affirmative answer to this question under some suitable conditions.

Theorem 1 ([7,22]).

Let E be a reflexive Banach space, $\eta :E\times E\to E$ be a τ-Lipschitz continuous mapping such that $\eta (x,y)+\eta (y,x)=0$ for all $x,y\in E$, and $P:E\to E^{*}$ be a γ-strongly η-monotone continuous mapping. Suppose that for any given $x^{*}\in E^{*}$, the function $h:(y,x)\in E\times E\to h(y,x)=\langle x^{*}-P(x),\eta (y,x)\rangle \in ℝ\cup \{+\infty \}$ is $\mathrm{0-DQCV}$ in the first argument. Furthermore, let $\varphi :E\to ℝ\cup \{+\infty \}$ be a lower semicontinuous η-subdifferentiable proper functional on E, which may not be convex. Then, for any given $\lambda >0$ and $x^{*}\in E^{*}$, there exists a unique point $x\in E$ such that

$$\begin{array}{c}\hfill \langle P(x)-x^{*},\eta (y,x)\rangle \ge \lambda \varphi (x)-\lambda \varphi (y),\forall y\in E,\end{array}$$

that is, $x=R_{\lambda ,\eta }^{\partial \varphi ,P}(x^{*})$ and so the P-η-proximal mapping associated with $\varphi ,P,\eta$ and $\lambda >0$ is well defined.

It should be pointed out that by a careful reading the proof of Theorem 3.1 in [22] and its comparing with the hypotheses mentioned in its context, we found that the mapping $\eta$ must be $\tau$-Lipschitz continuous. Indeed, in the context of ([22], Theorem 3.1), the continuity condition of the mapping $\eta$ must be replaced by the $\tau$-Lipschitz continuity condition, as we have done in the context of Theorem 4.

In the next theorem by Ahmad et al. [7] and Kazmi and Bhat [22], the appropriate conditions under which the P-$\eta$-proximal mapping $R_{\lambda ,\eta }^{\partial \varphi ,P}$ associated with the mappings $\varphi ,P,\eta$ and the constant $\lambda >0$ is Lipschitz continuous are stated and an estimate of its Lipschitz constant is also computed.

Theorem 2 ([7,22]).

Let E be a reflexive Banach space with the dual space $E^{*}$, $\eta :E\times E\to E$ be a τ-Lipschitz continuous mapping such that $\eta (x,y)+\eta (y,x)=0$ for all $x,y\in E$, and let $P:E\to E^{*}$ be a γ-strongly η-monotone continuous mapping. Suppose that for any given $x^{*}\in E^{*}$, the function $h:(y,x)\in E\times E\to h(y,x)=\langle x^{*}-P(x),\eta (y,x)\rangle \in ℝ\cup \{+\infty \}$ is $\mathrm{0-DQCV}$ in the first argument, $\varphi :E\to ℝ\cup \{+\infty \}$ is a lower semicontinuous η-subdifferentiable proper functional on E and $\lambda >0$ is an arbitrary real constant. Then, the P-η-proximal mapping $R_{\lambda ,\eta }^{\partial \varphi ,P}:E^{*}\to E$ associated with $\varphi ,P,\eta$ and $\lambda >0$ is $\frac{\tau }{\gamma }$-Lipschitz continuous, i.e.,

$$\begin{array}{c}\hfill \parallel R_{\lambda ,\eta }^{\partial \varphi ,P}(x^{*})-R_{\lambda ,\eta }^{\partial \varphi ,P}(y^{*})\parallel \le \frac{\tau }{\gamma }{\parallel x^{*}-y^{*}\parallel }_{*},\forall x^{*},y^{*}\in E^{*}.\end{array}$$

Definition 7 ([33]).

Given set-valued mappings $M_n,M:E\to 2^E$ $(n\ge 0)$, the sequence ${\left\{M_n\right\}}_{n=0}^{\infty }$ is said to be graph-convergent to M, denoted by $M_n\stackrel{G}{⟶}M$, if for every point $(x,u)\in Graph(M)$, there exists a sequence of points $(x_n,u_n)\in Graph(M_n)$ such that $x_n\to x$ and $u_n\to u$ as $n\to \infty$, where $Graph(M)$ is defined as follows:

$$\begin{array}{c}\hfill Graph(M)=\{(x,u)\in E\times E:u\in M(x)\}.\end{array}$$

We end this section with the following theorem in which a new equivalence relationship between the graph convergence of a sequence of lower semicontinuous ${\eta }_n$-subdifferentiable proper functionals and their associated $J_n^{{\eta }_n}$-proximal mappings, respectively, to a given lower semicontinuous $\eta$-subdifferentiable proper functional and its associated $J^{\eta }$-proximal mapping is established.

Theorem 3.

Let E be a real reflexive Banach space with the dual space $E^{*}$; ${\eta }^n:E\times E\to E$ be a ${\tau }^n$-Lipschitz continuous mapping for each $n\ge 0$ and $\eta :E\times E\to E$ be a τ-Lipschitz continuous mapping such that ${\eta }^n(x,y)+{\eta }^n(y,x)=\eta (x,y)+\eta (y,x)=0$, for all $n\ge 0$ and $x,y\in E$. Assume that $\widehat{P}:E\to E^{*}$ is a γ-strongly η-monotone and ζ-Lipschitz continuous mapping, and for each $n\ge 0$, ${\widehat{P}}^n:E\to E^{*}$ is a ${\gamma }^n$-strongly ${\eta }^n$-monotone and ${\zeta }^n$-Lipschitz continuous mapping. Let $\underset{n\to \infty }{lim}{\widehat{P}}^n(x)=\widehat{P}(x)$ for any $x\in E$, and the sequences ${\left\{\frac{1}{{\gamma }^n}\right\}}_{n=0}^{\infty }$, ${\left\{{\tau }^n\right\}}_{n=0}^{\infty }$, ${\left\{{\zeta }^n\right\}}_{n=0}^{\infty }$ be bounded. Suppose that for each $n\ge 0$ and for any given $x^{*}\in E^{*}$, the functions $h^n:(y,x)\in E\times E\to h^n(y,x)=\langle x^{*}-{\widehat{P}}^n(x),{\eta }^n(y,x)\rangle \in ℝ\cup \{+\infty \}$ and $h(y,x):\in E\times E\to h(y,x)=\langle x^{*}-\widehat{P}(x),\eta (y,x)\rangle \in ℝ\cup \{+\infty \}$ are $\mathrm{0-DQCV}$ in the first argument. Let for each $n\ge 0$, ${\varphi }^n:E\to ℝ\cup \{+\infty \}$ be a lower semicontinuous and ${\eta }^n$-subdifferentiable proper functional, and $\varphi :E\to ℝ\cup \{+\infty \}$ be a lower semicontinuous and η-subdifferentiable proper functional on E. Suppose further that ${\left\{{\lambda }_n\right\}}_{n=0}^{\infty }$ is a sequence of positive real constants convergent to a positive real constant λ. Then, $\partial {\varphi }^n\stackrel{G}{⟶}\partial \varphi$ if and only if

$$\begin{array}{c}\hfill R_{{\lambda }^n,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}(x^{*})\to R_{\lambda ,\eta }^{\partial \varphi ,\widehat{P}}(x^{*}),for all x^{*}\in E^{*},\end{array}$$

where for each $n\ge 0$, $R_{{\lambda }^n,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}={({\widehat{P}}^n+{\lambda }^n\partial {\varphi }^n)}^{-1}$ and $R_{\lambda ,\eta }^{\partial \varphi ,\widehat{P}}={(\widehat{P}+\lambda \partial \varphi )}^{-1}$.

Proof. 

Suppose first that $\partial {\varphi }^n\stackrel{G}{⟶}\partial \varphi$, and let $x^{*}\in E^{*}$ be arbitrary but fixed. In light of the assumptions and from Theorem 1, it follows that $\widehat{P}+\lambda \partial \varphi :E\to E^{*}$ is an injective and surjective function and so $(\widehat{P}+\lambda \partial \varphi )(E)=E^{*}$. Accordingly, there exists $(x,u^{*})\in \mathrm{Graph}(\partial \varphi )$ such that $x^{*}=\widehat{P}(x)+\lambda u^{*}$. According to Definition 7, there exists a sequence ${\left\{(x_n,u_n^{*})\right\}}_{n=0}^{\infty }\subseteq Graph(\partial {\varphi }^n)$ such that $x_n\to x$ and $u_n^{*}\to u^{*}$, as $n\to \infty$. Taking into consideration the facts that $(x,u^{*})\in Graph(\partial \varphi )$ and $(x_n,u_n^{*})\in Graph(\partial {\varphi }^n)$, we have

$$\begin{array}{c}\hfill x=R_{\lambda ,\eta }^{\partial \varphi ,\widehat{P}}[\widehat{P}(x)+\lambda u^{*}]\mathrm{and}{x}_n=R_{{\lambda }^n,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}[{\widehat{P}}^n(x_n)+{\lambda }^n{u}_n^{*}],\forall n\ge 0.\end{array}$$

(4)

Letting $x_n^{*}={\widehat{P}}^n(x_n)+{\lambda }^n{u}_n^{*}$, for each $n\ge 0$, by utilizing Theorem 2, (4) and with the help of our hypotheses, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& \parallel R_{{\lambda }^n,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}(x^{*})-R_{\lambda ,\eta }^{\partial \varphi ,\widehat{P}}(x^{*})\parallel \hfill \\ & \le \parallel R_{{\lambda }^n,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}(x^{*})-R_{{\lambda }^n,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}(x_n^{*})\parallel +\parallel R_{{\lambda }^n,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}(x_n^{*})-R_{\lambda ,\eta }^{\partial \varphi ,\widehat{P}}(x^{*})\parallel \hfill \\ & \le \frac{{\tau }^n}{{\gamma }^n}\parallel x_n^{*}-x^{*}{\parallel }_{*}+\parallel R_{{\lambda }^n,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}[{\widehat{P}}^n(x_n)+{\lambda }^n{u}_n^{*}]-R_{\lambda ,\eta }^{\partial \varphi ,\widehat{P}}[\widehat{P}(x)+\lambda u^{*}]\parallel \hfill \\ & \le \frac{{\tau }^n}{{\gamma }^n}\parallel x_n^{*}-x^{*}{\parallel }_{*}+\parallel x_n-x\parallel \hfill \\ & =\frac{{\tau }^n}{{\gamma }^n}\parallel {\widehat{P}}^n(x_n)+{\lambda }^n{u}_n^{*}-\widehat{P}(x)-\lambda u^{*}{\parallel }_{*}+\parallel x_n-x\parallel \hfill \\ & \le \frac{{\tau }^n}{{\gamma }^n}(\parallel {\widehat{P}}^n(x_n)-\widehat{P}(x){\parallel }_{*}+\parallel {\lambda }^n{u}_n^{*}-\lambda u^{*}{\parallel }_{*})+\parallel x_n-x\parallel \hfill \\ & \le \frac{{\tau }^n}{{\gamma }^n}(\parallel {\widehat{P}}^n(x_n)-{\widehat{P}}^n(x){\parallel }_{*}+{\parallel {\widehat{P}}^n(x)-\widehat{P}(x)\parallel }_{*}\hfill \\ & +\parallel {\lambda }^n{u}_n^{*}-{\lambda }^n{u}^{*}{\parallel }_{*}+\parallel {\lambda }^n{u}^{*}-\lambda u^{*}{\parallel }_{*})+\parallel x_n-x\parallel \hfill \\ & \le (1+\frac{{\zeta }^n{\tau }^n}{{\gamma }^n})\parallel x_n-x\parallel +\frac{{\tau }^n}{{\gamma }^n}{\parallel {\widehat{P}}^n(x)-\widehat{P}(x)\parallel }_{*}\hfill \\ & +\frac{{\lambda }^n{\tau }^n}{{\gamma }^n}\parallel u_n^{*}-u^{*}{\parallel }_{*}+\frac{|{\lambda }^n-\lambda |{\tau }^n}{{\gamma }^n}{\parallel u^{*}\parallel }_{*}.\hfill \end{array}\end{array}$$

Since $\underset{n\to \infty }{lim}{\lambda }_n=\lambda$ and the sequences ${\left\{\frac{1}{{\gamma }^n}\right\}}_{n=0}^{\infty }$ and ${\left\{{\tau }^n\right\}}_{n=0}^{\infty }$ are bounded, it follows that the sequence ${\left\{\frac{{\lambda }^n{\tau }^n}{{\gamma }^n}\right\}}_{n=0}^{\infty }$ is also bounded. In view of the assumptions, the right-hand side of the above inequality tends to zero, as $n\to \infty$, which implies that $R_{{\lambda }^n,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}(x^{*})\to R_{\lambda ,\eta }^{\partial \varphi ,\widehat{P}}(x^{*})$, as $n\to \infty$.

Converse, assume that for all $x^{*}\in E^{*}$, we have $R_{{\lambda }^n,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}(x^{*})\to R_{\lambda ,\eta }^{\partial \varphi ,\widehat{P}}(x^{*})$, as $n\to \infty$. Then, for any $u^{*}\in \partial \varphi (x)$, we have $x=R_{\lambda ,\eta }^{\partial \varphi ,\widehat{P}}[\widehat{P}(x)+\lambda u^{*}]$ and so $R_{{\lambda }^n,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}[\widehat{P}(x)+\lambda u^{*}]\to x$, as $n\to \infty$. Picking $x_n=R_{{\lambda }^n,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}[\widehat{P}(x)+\lambda u^{*}]$ for each $n\ge 0$, we conclude that $\widehat{P}(x)+\lambda u^{*}\in ({\widehat{P}}^n+{\lambda }^n\partial {\varphi }^n)(x_n)$. Consequently, for each $n\ge 0$, there exists $u_n^{*}\in \partial {\varphi }^n(x_n)$ such that $\widehat{P}(x)+\lambda u^{*}={\widehat{P}}^n(x_n)+{\lambda }^n{u}_n^{*}$. Then, for each $n\ge 0$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel {\lambda }_n{u}_n^{*}-\lambda u^{*}{\parallel }_{*}={\parallel {\widehat{P}}^n(x_n)-\widehat{P}(x)\parallel }_{*}& \le \parallel {\widehat{P}}^n(x_n)-{\widehat{P}}^n(x){\parallel }_{*}+{\parallel {\widehat{P}}^n(x)-\widehat{P}(x)\parallel }_{*}\hfill \\ & \le {\zeta }^n\parallel x_n-x\parallel +\parallel {\widehat{P}}^n(x_n)-\widehat{P}(x){\parallel }_{*}.\hfill \end{array}\end{array}$$

Since ${\left\{{\zeta }^n\right\}}_{n=0}^{\infty }$ is a bounded sequence, $x_n\to x$ and ${\widehat{P}}^n(x)\to \widehat{P}(x)$, as $n\to \infty$, it follows that ${\lambda }_n{u}_n\to \lambda u$, as $n\to \infty$. In the meantime, for each $n\ge 0$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \lambda \parallel u_n^{*}-u^{*}{\parallel }_{*}={\parallel \lambda u_n^{*}-\lambda u^{*}\parallel }_{*}& \le \parallel {\lambda }_n{u}_n^{*}-\lambda u_n^{*}{\parallel }_{*}+{\parallel {\lambda }_n{u}_n^{*}-\lambda u^{*}\parallel }_{*}\hfill \\ & = |{\lambda }_n-\lambda |\parallel u_n^{*}{\parallel }_{*}+{\parallel {\lambda }_n{u}_n^{*}-\lambda u^{*}\parallel }_{*}.\hfill \end{array}\end{array}$$

Taking into account that ${\lambda }_n\to \lambda$ and ${\lambda }_n{u}_n^{*}\to \lambda u^{*}$, as $n\to \infty$, we deduce that the right-hand side of the preceding inequality approaches zero, as $n\to \infty$. Thus, $u_n^{*}\to u^{*}$, as $n\to \infty$ and so invoking Definition 7, $\partial {\varphi }^n\stackrel{G}{⟶}\partial \varphi$. This completes the proof.    □

Corollary 1.

Let $E,{\widehat{P}}^n,{\widehat{\eta }}^n,\partial {\varphi }^n$ ($n\ge 0$), $\widehat{P},\widehat{\eta }$ and $\partial \varphi$ be the same as in Theorem 3 and let all the conditions of Theorem 3 hold. Then, $\partial {\varphi }^n\stackrel{G}{⟶}\partial \varphi$ if and only if

$$\begin{array}{c}\hfill R_{\lambda ,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}(x^{*})\to R_{\lambda ,\eta }^{\partial \varphi ,\widehat{P}}(x^{*}),\mathrm{for} \mathrm{all} x^{*}\in E^{*},\end{array}$$

where $\lambda >0$ is a constant; for each $n\ge 0$, $R_{\lambda ,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}={({\widehat{P}}^n+\lambda \partial {\varphi }^n)}^{-1}$ and $R_{\lambda ,\eta }^{\partial \varphi ,\widehat{P}}={(\widehat{P}+\lambda \partial \varphi )}^{-1}$.

3. Formulation of the Problem and Existence Result

For each $i\in \{1,2\}$, let $E_i$ be a real Banach space with a norm ${\parallel ·\parallel }_i$, $E_i^{*}$ be its dual space with a norm ${\parallel ·\parallel }_{*i}$, and let ${\langle .,.\rangle }_i$ be the dual pair between $E_i$ and $E_i^{*}$. Suppose that for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, $f_i:E_i\to E_i^{*}$, $g_i:E_i\to E_i$, $W_i:E_i^{*}\times E_i^{*}\times E_j^{*}\to E_i^{*}$, $P_i:E_j^{*}\times E_i^{*}\to E_i^{*}$, $Q_i:E_i^{*}\times E_j^{*}\to E_i^{*}$ and ${\eta }_i:E_i\times E_i\to E_i$ are single-valued mappings, and $K_i:E_i\to CB(E_i)$, $S_i:E_i\to CB(E_i^{*})$, $T_i:E_j\to CB(E_j^{*})$, $C_i,G_i:E_j\to CB(E_j^{*})$ and $D_i,F_i:E_i\to CB(E_i^{*})$ are multi-valued mappings. Furthermore, for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, let ${\varphi }_i:E_i\times E_i\to ℝ\cup \{+\infty \}$ be an extended real-valued bifunction such that for each fixed $t_i\in E_i$, ${\varphi }_i:(.,t_i):E_i\to ℝ\cup \{+\infty \}$ is a proper, lower semicontinuous and ${\eta }_i$-subdifferentiable on $E_i$ with $g_i(E_i)\cap \mathrm{dom}\partial {\varphi }_i(.,t_i)\ne \varnothing$. We consider the problem of finding $(x_1,x_2)\in E_1\times E_2$, $({\pi }_1,{\pi }_2)\in S_1(x_1)\times S_2(x_2)$, $({\iota }_1,{\iota }_2)\in T_1(x_2)\times T_2(x_1)$, $(u_1,u_2)\in C_1(x_2)\times C_2(x_1)$, $(v_1,v_2)\in D_1(x_1)\times D_2(x_2)$, $(w_1,w_2)\in F_1(x_1)\times F_2(x_2)$, $(z_1,z_2)\in G_1(x_2)\times G_2(x_1)$, $(t_1,t_2)\in K_1(x_1)\times K_2(x_2)$ such that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{\langle W_1(f_1(x_1),{\pi }_1,{\iota }_1)+{\rho }_1[P_1(u_1,v_1)+Q_1(w_1,z_1)],{\eta }_1(y_1,g_1(x_1))\rangle }_1\hfill \\ \ge {\rho }_1[{\varphi }_1(g_1(x_1),t_1)-{\varphi }_1(y_1,t_1)],\forall y_1\in E_1,\hfill \\ {\langle W_2(f_2(x_2),{\pi }_2,{\iota }_2)+{\rho }_2[P_2(u_2,v_2)+Q_2(w_2,z_2)],{\eta }_2(y_2,g_2(x_2))\rangle }_2\hfill \\ \ge {\rho }_2[{\varphi }_2(g_2(x_2),t_2)-{\varphi }_2(y_2,t_2)],\forall y_2\in E_2,\hfill \end{array}\right.\end{array}\end{array}$$

(5)

where ${\rho }_1,{\rho }_2>0$ are two arbitrary positive real constants. The problem (5) is called a system of generalized multi-valued variational-like inequalities (in short, $\mathrm{SGMVLI}$).

If for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, $W_i(u_i,v_i,w_j)=u_i+N_i(v_i,w_j)$, for all $(u_i,v_i,w_j)\in E_i^{*}\times E_i^{*}\times E_j^{*}$, where $N_i:E_i^{*}\times E_j^{*}\to E_i^{*}$ is a bifunction, $f_i=K_i\equiv 0$, $S_i:E_i\to E_i^{*}$ and $T_i:E_j\to E_j^{*}$ are single-valued mappings, $C_i,G_i:E_j\to C(E_j^{*})$ and $D_i,F_i:E_i\to C(E_i^{*})$ are multi-valued mappings, and ${\varphi }_i:E_i\to ℝ\cup \{+\infty \}$ is a proper, lower semicontinuous and ${\eta }_i$-subdifferentiable on $E_i$ with $g_i(E_i)\cap \mathrm{dom}\partial {\varphi }_i(.)\ne \varnothing$, then the $\mathrm{SGMVLI}$ (5) reduces to the problem of finding $(x_1,x_2,u_1,u_2,v_1,v_2,w_1,w_2,z_1,z_2)$, where for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, $x_i\in E_i$, $u_i\in C_i(x_j)$, $v_i\in D_i(x_i)$, $w_i\in F_i(x_i)$ and $z_i\in G_i(x_j)$ such that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{\langle N_1(S_1(x_1),T_1(x_2))+{\rho }_1[P_1(u_1,v_1)+Q_1(w_1,z_1)],{\eta }_1(y_1,g_1(x_1))\rangle }_1\hfill \\ \ge {\rho }_1[{\varphi }_1(g_1(x_1))-{\varphi }_1(y_1)],\forall y_1\in E_1,\hfill \\ {\langle N_2(S_2(x_2),T_2(x_1))+{\rho }_2[P_2(u_2,v_2)+Q_2(w_2,z_2)],{\eta }_2(y_2,g_2(x_2))\rangle }_2\hfill \\ \ge {\rho }_2[{\varphi }_2(g_2(x_2))-{\varphi }_2(y_2)],\forall y_2\in E_2,\hfill \end{array}\right.\end{array}\end{array}$$

(6)

where ${\rho }_1,{\rho }_2>0$ are two arbitrary positive real constants. The problem (6) is called a system of generalized implicit variational-like inequalities (in short, $\mathrm{SGIVLI}$) and was introduced and studied by Kazmi et al. [30].

Remark 1.

It should be pointed out that for appropriate and suitable choices of the mappings $W_i,P_i,Q_i,S_i,T_i,C_i,D_i,F_i,G_i,K_i,{\eta }_i,{\varphi }_i,f_i,g_i$, the constants ${\rho }_i$ and the underlying spaces $E_i$ $(i=1,2)$, one can obtain many known classes of variational inequalities, complementarity problems and their generalizations, studied previously by many authors, as special cases of the $\mathrm{SGMVLI}$ (5), see for example [7,16,17,19,27,30,42] and the references therein.

In order to provide a characterization of the solution of the $\mathrm{SGMVLI}$ (5), by the same arguments used in Lemma 3.1 of [43] and by utilizing the definition of ${\widehat{P}}_i$-proximal mapping $R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i),{\widehat{P}}_i}$ of the functional ${\varphi }_i(.,t_i)$ $(i=1,2)$, we give the following result whose proof is omitted.

Lemma 2.

For each $i\in \{1,2\}$, let $E_i$ be a real reflexive Banach space with a norm ${\parallel ·\parallel }_i$, $E_i^{*}$ be its dual space with a norm ${\parallel ·\parallel }_{*i}$, and let $f_i,g_i,W_i,P_i,Q_i,K_i,S_i,T_i,C_i,D_i,G_i,F_i,{\varphi }_i$ and ${\rho }_i>0$ be the same as in the $\mathrm{SGMVLI}$ (5). Suppose that for each $i\in \{1,2\}$, ${\eta }_i:E_i\times E_i\to E_i$ is a vector-valued continuous mapping such that ${\eta }_i({\widehat{x}}_i,y_i)+{\eta }_i(y_i,{\widehat{x}}_i)=0$ for all ${\widehat{x}}_i,y_i\in E_i$ and let ${\widehat{P}}_i:E_i\to E_i^{*}$ be a ${\gamma }_i$-strongly ${\eta }_i$-monotone continuous mapping such that $g_i(E_i)\subseteq \mathrm{dom}({\widehat{P}}_i)$. Let for each $i\in \{1,2\}$ and for any given $x_i^{*}\in E_i^{*}$, the function

$$\begin{array}{c}\hfill h_i:(y_i,{\widehat{x}}_i)\in E_i\times E_i\to h_i(y_i,{\widehat{x}}_i)={\langle x_i^{*}-{\widehat{P}}_i({\widehat{x}}_i),{\eta }_i(y_i,{\widehat{x}}_i)\rangle }_i\in ℝ\cup \{+\infty \}\end{array}$$

be $\mathrm{0-DQCV}$ in the first argument. Then, the following assertions are equivalent:

(i)

$(x_1,x_2,{\pi }_1,{\pi }_2,{\iota }_1,{\iota }_2,u_1,u_2,v_1,v_2,w_1,w_2,z_1,z_2,t_1,t_2)\in E_1\times E_2\times S_1(x_1)\times S_2(x_2)\times T_1(x_2)\times T_2(x_1)\times C_1(x_2)\times C_2(x_1)\times D_1(x_1)\times D_2(x_2)\times F_1(x_1)\times F_2(x_2)\times G_1(x_2)\times G_2(x_1)\times K_1(x_1)\times K_2(x_2)$ is a solution of the $\mathrm{SGMVLI}$ (5);

(ii)

for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$,

$$\begin{array}{c}\hfill g_i(x_i)=R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)(x_i)-{\lambda }_i(W_i(f_i(x_i),{\pi }_i,{\iota }_i)+{\rho }_i(P_i(u_i,v_i)+Q_i(w_i,z_i))],\end{array}$$

where for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, $R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i),{\widehat{P}}_i}={({\widehat{P}}_i+{\lambda }_i\partial {\varphi }_i(.,t_i))}^{-1}$ is ${\widehat{P}}_i$-${\eta }_i$-proximal mapping of ${\varphi }_i(.,t_i)$, ${\widehat{P}}_i\circ g_i$ denotes ${\widehat{P}}_i$ composition $g_i$, and ${\lambda }_i>0$ is a constant;

(iii)

$(x_1,x_2)\in E_1\times E_2$ is a fixed point of the multi-valued mapping ${\Phi }:E_1\times E_2\to 2^{E_1\times E_2}$ defined as follows:

$$\begin{array}{c}\hfill {\Phi }({\widehat{x}}_1,{\widehat{x}}_2)=({{\Psi }}_1({\widehat{x}}_1,{\widehat{x}}_2),{{\Psi }}_2({\widehat{x}}_1,{\widehat{x}}_2)),\forall ({\widehat{x}}_1,{\widehat{x}}_2)\in E_1\times E_2,\end{array}$$

(7)

where for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, ${{\Psi }}_i:E_1\times E_2\to 2^{E_i}$ is defined, for any $({\widehat{x}}_1,{\widehat{x}}_2)\in E_1\times E_2$, by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {{\Psi }}_i({\widehat{x}}_1,{\widehat{x}}_2)& ={\widehat{x}}_i-g_i({\widehat{x}}_i)+\bigcup _{\begin{array}{c}{\widehat{\pi }}_i\in S_i({\widehat{x}}_i)\\ {\widehat{\iota }}_i\in T_i({\widehat{x}}_j)\\ {\widehat{u}}_i\in C_i({\widehat{x}}_j)\end{array}}\bigcup _{\begin{array}{c}{\widehat{v}}_i\in D_i({\widehat{x}}_i)\\ {\widehat{w}}_i\in F_i({\widehat{x}}_i)\\ {\widehat{z}}_i\in G_i({\widehat{x}}_j)\\ {\widehat{t}}_i\in K_i({\widehat{x}}_i)\end{array}}{R}_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,{\widehat{t}}_i),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)\hfill \\ & -{\lambda }_i(W_i(f_i({\widehat{x}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)+{\rho }_i(P_i({\widehat{u}}_i,{\widehat{v}}_i)+Q_i({\widehat{w}}_i,{\widehat{z}}_i))],\hfill \end{array}\end{array}$$

and ${\lambda }_1,{\lambda }_2>0$ are two constants.

As an immediate result of the last assertion, we obtain a characterization of the solution of the $\mathrm{SGIVLI}$ (6) as follows.

Lemma 3.

For each $i\in \{1,2\}$, suppose that $E_i$ is a real reflexive Banach space with a norm ${\parallel ·\parallel }_i$, $E_i^{*}$ is its dual space with a norm ${\parallel ·\parallel }_{*i}$, and let the mappings $g_i,N_i,P_i,Q_i,S_i,T_i,C_i,D_i,G_i,F_i,{\varphi }_i$ and the constants ${\rho }_i>0$ be the same as in the $\mathrm{SGIVLI}$ (6). Assume that for each $i\in \{1,2\}$, ${\eta }_i:E_i\times E_i\to E_i$ is a vector-valued continuous mapping such that ${\eta }_i({\widehat{x}}_i,y_i)+{\eta }_i(y_i,{\widehat{x}}_i)=0$ for all ${\widehat{x}}_i,y_i\in E_i$ and let ${\widehat{P}}_i:E_i\to E_i^{*}$ be a ${\gamma }_i$-strongly ${\eta }_i$-monotone continuous mapping such that $g_i(E_i)\subseteq \mathrm{dom}({\widehat{P}}_i)$. Let for each $i\in \{1,2\}$ and for any given $x_i^{*}\in E_i^{*}$, the function

$$\begin{array}{c}\hfill h_i:(y_i,{\widehat{x}}_i)\in E_i\times E_i\to h_i(y_i,{\widehat{x}}_i)={\langle x_i^{*}-{\widehat{P}}_i({\widehat{x}}_i),{\eta }_i(y_i,{\widehat{x}}_i)\rangle }_i\in ℝ\cup \{+\infty \}\end{array}$$

be $\mathrm{0-DQCV}$ in the first argument. Then, the following conclusions are equivalent:

(i)

$(x_1,x_2,u_1,u_2,v_1,v_2,w_1,w_2,z_1,z_2)\in E_1\times E_2\times C_1(x_2)\times C_2(x_1)\times D_1(x_1)\times D_2(x_2)\times F_1(x_1)\times F_2(x_2)\times G_1(x_2)\times G_2(x_1)$ is a solution of the $\mathrm{SGIVLI}$ (6);

(ii)

for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$,

$$\begin{array}{c}\hfill g_i(x_i)=R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i,{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)(x_i)-{\lambda }_i(N_i(S_i(x_i),T_i(x_j))+{\rho }_i(P_i(u_i,v_i)+Q_i(w_i,z_i)))],\end{array}$$

where for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, $R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i,{\widehat{P}}_i}={({\widehat{P}}_i+{\lambda }_i\partial {\varphi }_i)}^{-1}$ is ${\widehat{P}}_i$-${\eta }_i$-proximal mapping of ${\varphi }_i$, ${\widehat{P}}_i\circ g_i$ denotes ${\widehat{P}}_i$ composition $g_i$, and ${\lambda }_i>0$ is a constant;

(iii)

$(x_1,x_2)\in E_1\times E_2$ is a fixed point of the multi-valued mapping $U:E_1\times E_2\to 2^{E_1\times E_2}$ defined as follows:

$$\begin{array}{c}\hfill U({\widehat{x}}_1,{\widehat{x}}_2)=(V_1({\widehat{x}}_1,{\widehat{x}}_2),V_2({\widehat{x}}_1,{\widehat{x}}_2)),\forall ({\widehat{x}}_1,{\widehat{x}}_2)\in E_1\times E_2,\end{array}$$

where for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, $V_i:E_1\times E_2\to 2^{E_i}$ is defined, for any $({\widehat{x}}_1,{\widehat{x}}_2)\in E_1\times E_2$, by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_i({\widehat{x}}_1,{\widehat{x}}_2)& ={\widehat{x}}_i-g_i({\widehat{x}}_i)+\bigcup _{\begin{array}{c}{\widehat{u}}_i\in C_i({\widehat{x}}_j)\\ {\widehat{v}}_i\in D_i({\widehat{x}}_i)\\ {\widehat{w}}_i\in F_i({\widehat{x}}_i)\\ {\widehat{z}}_i\in G_i({\widehat{x}}_j)\end{array}}{R}_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i,{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-{\lambda }_i(N_i(S_i({\widehat{x}}_i),T_i({\widehat{x}}_j))\hfill \\ & +{\rho }_i(P_i({\widehat{u}}_i,{\widehat{v}}_i)+Q_i({\widehat{w}}_i,{\widehat{z}}_i)))],\hfill \end{array}\end{array}$$

and ${\lambda }_1,{\lambda }_2>0$ are two constants.

Before turning to the main results of this section, let us recall the following notions and significant lemmas which will be used in the sequel.

Recall that a mapping $J:E\to 2^{E^{*}}$ satisfying the condition

$$\begin{array}{c}\hfill J(x)=\left\{x^{*}\in E^{*}:\langle x,x^{*}\rangle ={\parallel x\parallel }^2={\parallel x^{*}\parallel }_{*}^2\right\},\forall x\in E,\end{array}$$

where $2^{E^{*}}$ is the family of all the nonempty subsets of E, is called the normalized duality mapping on E. The Hahn–Banach theorem guarantees that $J(x)\ne \varnothing$ for every $x\in E$. A Banach space E is called smooth ([44], p. 21) if, for every $x\in E$ with $\parallel x\parallel =1$, there exists a unique $x^{*}\in E^{*}$ such that $\parallel x^{*}{\parallel }_{*}=x^{*}(x)=1$. It is well known that if E is smooth, then J is single-valued and if $E=\mathcal{H}$, a Hilbert space, then J is the identity mapping on $\mathcal{H}$.

The modulus of smoothness of E is the function ${\rho }_E:[0,\infty )\to [0,\infty )$ defined by

$$\begin{array}{c}\hfill {\rho }_E(t)=sup\left\{\frac{1}{2}(\parallel x+y\parallel +\parallel x-y\parallel )-1:\parallel x\parallel \le 1,\parallel y\parallel \le t\right\}.\end{array}$$

A Banach space E is called uniformly smooth [44,45], if $\underset{t\to 0}{lim}\frac{{\rho }_E(t)}{t}=0$.

It should be noted that every uniformly smooth Banach space is reflexive.

Definition 8.

A multi-valued mapping $T:E\to CB(E)$ is said to be ξ-H-Lipschitz continuous if there exists a constant $\xi >0$ such that

$$\begin{array}{c}\hfill H(T(x),T(y))\le \xi \parallel x-y\parallel ,\forall x,y\in E,\end{array}$$

where H is the Hausdorff metric on $CB(E)$ defined by

$$\begin{array}{c}\hfill H(A,B)=max\left\{\underset{x\in A}{sup}\underset{y\in B}{inf}\parallel x-y\parallel ,\underset{y\in B}{sup}\underset{x\in A}{inf}\parallel x-y\parallel \right\},\forall A,B\in CB(E).\end{array}$$

Definition 9.

Let E be a real smooth Banach space with the dual space $E^{*}$. A mapping $g:E\to E$ is said to be κ-strongly accretive, if there exists a constant $\kappa >0$ such that for any $x,y\in E$,

$$\begin{array}{c}\hfill \langle g(x)-g(y),J(x-y)\rangle \ge {\kappa \parallel x-y\parallel }^2,\end{array}$$

where J is the normalized duality mapping from E into $E^{*}$.

Definition 10.

Let E be a real Banach space. A vector-valued mapping $N:E\times E\times E\to E$ is said to be $({\varsigma }_1,{\varsigma }_2,{\varsigma }_3)$-mixed Lipschitz continuous if there exist constants ${\varsigma }_1,{\varsigma }_2,{\varsigma }_3>0$ such that

$$\begin{array}{c}\hfill \parallel N(x,y,z)-N(\widehat{x},\widehat{y},\widehat{z})\parallel \le {\varsigma }_1\parallel x-\widehat{x}\parallel +{\varsigma }_2\parallel y-\widehat{y}\parallel +{\varsigma }_3\parallel z-\widehat{z}\parallel ,\forall x,\widehat{x},y,\widehat{y},z,\widehat{z}\in E.\end{array}$$

Clearly, if N is $({\varsigma }_1,{\varsigma }_2,{\varsigma }_3)$-mixed Lipschitz continuous, then N is ${\varsigma }_1$-Lipschitz, ${\varsigma }_2$-Lipschitz, and ${\varsigma }_3$-Lipschitz continuous in the first, second and third argument, respectively. Similarly, we say that a vector-valued mapping $S:E\times E\to E$ is $(\delta ,\nu )$-mixed Lipschitz continuous if there exist constants $\delta ,\nu >0$ such that

$$\begin{array}{c}\hfill \parallel S(x,y)-S(\widehat{x},\widehat{y})\parallel \le \delta \parallel x-\widehat{x}\parallel +\nu \parallel y-\widehat{y}\parallel ,\forall x,\widehat{x},y,\widehat{y}\in E.\end{array}$$

Lemma 4.

Let E be a real uniformly smooth Banach space with the dual space $E^{*}$ and J be the normalized duality mapping from E into $E^{*}$. Then for all $x,y\in E$, we have

(i)

${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2\langle y,J(x+y)\rangle$;

(ii)

$\langle x-y,J(x)-J(y)\rangle \le 2d^2(x,y){\rho }_E\left(\frac{4\parallel x-y\parallel }{d(x,y)}\right)$, where $d(x,y)=\sqrt{\frac{{\parallel x\parallel }^2+{\parallel y\parallel }^2}{2}}$.

Remark 2.

It should be pointed out that in the original version of the above conclusion, d is used instead of $d(x,y)$. But, taking into consideration the fact that for any given $x,y\in E$, $d=\sqrt{\frac{{\parallel x\parallel }^2+{\parallel y\parallel }^2}{2}}$ depends on the choice of x and y, hence d must be replaced by $d(x,y)$, as we have done in Lemma 4(ii).

Lemma 5 ([39]).

Let $(X,d)$ be a complete metric space and $T:X\to CB(X)$ be a multi-valued mapping satisfying

$$\begin{array}{c}\hfill H(T(x),T(y))\le \kappa d(x,y),\forall x,y,\in X,\end{array}$$

where $\kappa \in (0,1)$ is a constant. Then, the mapping T has a fixed point in X.

Lemma 6 ([39]).

Let $(X,d)$ be a complete metric space and $T:X\to CB(X)$ be a multi-valued mapping. Then, for any $ϵ>0$ and for any given $x,y\in X$, $u\in T(x)$, there exists $v\in T(y)$ such that

$$\begin{array}{c}\hfill d(u,v)\le (1+ϵ)H(T(x),T(y)).\end{array}$$

Lemma 7 ([39]).

Let $(X,d)$ be a complete metric space and $T:X\to C(X)$ be a multi-valued mapping. Then, for any given $x,y\in X$, $u\in T(x)$, there exists $v\in T(y)$ such that

$$\begin{array}{c}\hfill d(u,v)\le H(T(x),T(y)).\end{array}$$

(8)

We are now in a position to give some sufficient conditions which guarantee the existence of a solution for the $\mathrm{SGMVLI}$ (5).

Theorem 4.

For each $i\in \{1,2\}$, let $E_i$ be a real uniformly smooth Banach space with ${\rho }_{E_i}(t)\le c_i{t}^2$ for some $c_i>0$ and for all $t\in [0,+\infty )$. Suppose that for each $i\in \{1,2\}$, ${\eta }_i:E_i\times E_i\to E_i$ is a ${\tau }_i$-Lipschitz continuous mapping such that ${\eta }_i(y_i,y_i^{\prime })+{\eta }_i(y_i^{\prime },y_i)=0$ for all $y_i,y_i^{\prime }\in E_i$, $g_i:E_i\to E_i$ is a $s_i$-strongly accretive and $L_{g_i}$-Lipschitz continuous mapping, ${\widehat{P}}_i:E_i\to E_i^{*}$ is a ${\gamma }_i$-strongly ${\eta }_i$-monotone and $L_{{\widehat{P}}_i}$-Lipschitz continuous mapping such that $g_i(E_i)\subseteq \mathrm{dom}({\widehat{P}}_i)$. Let for each $i\in \{1,2\}$ and any given $x_i^{*}\in E_i^{*}$, the function $h_i:(y_i,x_i^{\prime })\in E_i\times E_i\to h_i(y_i,x_i^{\prime })={\langle x_i^{*}-{\widehat{P}}_i(x_i^{\prime }),{\eta }_i(y_i,x_i^{\prime })\rangle }_i\in ℝ\cup \{+\infty \}$ be $\mathrm{0-DQCV}$ in the first argument. For each $i\in \{1,2\}$, let $f_i:E_i\to E_i^{*}$ be $L_{f_i}$-Lipschitz continuous, $K_i:E_i\to C(E_i)$ be $L_{K_i}$-$H_i$-Lipschitz continuous, $S_i:E_i\to C(E_i^{*})$ be $L_{S_i}$-$H_{*i}$-Lipschitz continuous, $T_i:E_j\to C(E_j^{*})$ $(j\in \{1,2\}\backslash \{i\})$ be $L_{T_i}$-$H_{*j}$-Lipschitz continuous, and $D_i,F_i:E_i\to C(E_i^{*})$ be $L_{D_i}$-$H_{*i}$-Lipschitz and $L_{F_i}$-$H_{*i}$-Lipschitz continuous, respectively. Assume that for each $i\in \{1,2\}$, $W_i:E_i^{*}\times E_i^{*}\times E_j^{*}\to E_i^{*}$ $(j\in \{1,2\}\backslash \{i\})$ is a $({\varsigma }_i,{\delta }_i,{\nu }_i)$-mixed Lipschitz continuous mapping and for any ${\tilde{x}}_i\in E_i$, ${\tilde{x}}_j\in E_j$, ${\tilde{\pi }}_i\in S_i({\tilde{x}}_i)$ and ${\tilde{\iota }}_i\in T_i({\tilde{x}}_j)$, $W_i(f_i(.),{\tilde{\pi }}_i,{\tilde{\iota }}_i)$ is ${\varpi }_i$-strongly accretive with respect to ${\widehat{P}}_i\circ g_i$. For each $i\in \{1,2\}$, let $C_i,G_i:E_j\to C(E_j^{*})$ $(j\in \{1,2\}\backslash \{i\})$ be $L_{C_i}$-$H_{*j}$-Lipschitz and $L_{G_i}$-$H_{*j}$-Lipschitz continuous, respectively, and $P_i:E_j^{*}\times E_i^{*}\to E_i^{*}$ and $Q_i:E_i^{*}\times E_j^{*}\to E_i^{*}$ be $(L_{(P_i,j)},L_{(P_i,i)})$-mixed Lipschitz and $(L_{(Q_i,i)},L_{(Q_i,j)})$-mixed Lipschitz continuous, respectively. Furthermore, let for each $i\in \{1,2\}$, ${\varphi }_i:E_i\times E_i\to ℝ\cup \{+\infty \}$ be an extended real-valued bifunction such that for each fixed ${\widehat{t}}_i\in E_i$, ${\varphi }_i(.,{\widehat{t}}_i):E_i\to ℝ\cup \{+\infty \}$ is a proper, lower semicontinuous and ${\eta }_i$-subdifferentiable on $E_i$ with $g_i(E_i)\cap \mathrm{dom}\partial {\varphi }_i(.,{\widehat{t}}_i)\ne \varnothing$. If there exist constants ${\mu }_i,{\lambda }_i>0$ $(i=1,2)$ such that for each $i\in \{1,2\}$,

$$\begin{array}{c}\hfill \parallel R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i^{\prime }),{\widehat{P}}_i}(x_i^{*})-R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i^{″}),{\widehat{P}}_i}(x_i^{*}){\parallel }_i\le {\mu }_i\parallel t_i^{\prime }-t_i^{″}\parallel ,\forall t_i^{\prime },t_i^{″}\in E_i,x_i^{*}\in E_i^{*}\end{array}$$

(9)

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}\sqrt{1-2s_i+64c_i{L}_{g_i}^2}+{\mu }_i{L}_{K_i}+\frac{{\tau }_i}{{\gamma }_i}(\sqrt{L_{{\widehat{P}}_i}^2{L}_{g_i}^2-2{\lambda }_i{\varpi }_i+64c_i{\lambda }_i^2{\varsigma }_i^2{L}_{f_i}^2}+{\lambda }_i({\delta }_i{L}_{S_i}\hfill \\ +{\rho }_i(L_{(P_i,i)}{L}_{D_i}+L_{(Q_i,i)}{L}_{F_i})))+\frac{{\tau }_j{\lambda }_j}{{\gamma }_j}({\nu }_j{L}_{T_j}+{\rho }_j(L_{(P_j,i)}{L}_{C_j}+L_{(Q_j,i)}{L}_{G_j}))<1,\hfill \\ 2s_i<1+64c_i{L}_{g_i}^2,2{\lambda }_i{\varpi }_i<L_{{\widehat{P}}_i}^2{L}_{g_i}^2+64c_i{\lambda }_i^2{\varsigma }_i^2{L}_{f_i}^2,\hfill \end{array}\right.\end{array}\end{array}$$

(10)

where $j\in \{1,2\}\backslash \left\{i\right\}$, then the $\mathrm{SGMVLI}$ (5) admits a solution.

Proof. 

Let us consider the multi-valued mapping ${\Phi }:E_1\times E_2\to 2^{E_1\times E_2}$ defined by (7). For any $({\widehat{x}}_1,{\widehat{x}}_2)\in E_1\times E_2$, due to the fact that for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, $S_i({\widehat{x}}_i),D_i({\widehat{x}}_i),F_i({\widehat{x}}_i)\in C(E_i^{*})$, $T_i({\widehat{x}}_j),C_i({\widehat{x}}_j),G_i({\widehat{x}}_j)\in C(E_j^{*})$, $K_i({\widehat{x}}_i)\in C(E_i)$ and ${\widehat{P}}_i,{\widehat{g}}_i,W_i,f_i,P_i,Q_i,R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i^{\prime }),{\widehat{P}}_i}$, for any $t_i^{\prime }\in K_i({\widehat{x}}_i)$ $(i=1,2)$ are continuous, for each $i\in \{1,2\}$, we have ${{\Psi }}_i({\widehat{x}}_1,{\widehat{x}}_2)\in C(E_i)$ and so ${\Phi }({\widehat{x}}_1,{\widehat{x}}_2)\in C(E_1)\times C(E_2)=C(E_1\times E_2)$. We now prove that ${\Phi }$ is a multi-valued contraction mapping. Indeed, for any $({\widehat{x}}_1,{\widehat{x}}_2),({\widehat{y}}_1,{\widehat{y}}_2)\in E_1\times E_2$ and $(q_1,q_2)\in {\Phi }({\widehat{x}}_1,{\widehat{x}}_2)$, by the definition ${\Phi }$, for each $i\in \{1,2\}$ there exist ${\widehat{\pi }}_i\in S_i({\widehat{x}}_i)$, ${\widehat{\iota }}_i\in T_i({\widehat{x}}_j)$, ${\widehat{u}}_i\in C_i({\widehat{x}}_j)$, ${\widehat{v}}_i\in D_i({\widehat{x}}_i)$, ${\widehat{w}}_i\in F_i({\widehat{x}}_i)$, ${\widehat{z}}_i\in G_i({\widehat{x}}_j)$ and ${\widehat{t}}_i\in K_i({\widehat{x}}_i)$ $(j\in \{1,2\}\backslash \{i\})$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill q_i& ={\widehat{x}}_i-g_i({\widehat{x}}_i)+R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,{\widehat{t}}_i),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-{\lambda }_i(W_i(f_i(\widehat{x_i}),{\widehat{\pi }}_i,{\widehat{\iota }}_i)\hfill \\ & +{\rho }_i(P_i({\widehat{u}}_i,{\widehat{v}}_i)+Q_i({\widehat{w}}_i,{\widehat{z}}_i)))].\hfill \end{array}\end{array}$$

(11)

By virtue of Lemma 7, for each $i\in \{1,2\}$ there exist ${\widehat{\pi }}_i^{\prime }\in S_i({\widehat{y}}_i)$, ${\widehat{\iota }}_i^{\prime }\in T_i({\widehat{y}}_j)$, ${\widehat{u}}_i^{\prime }\in C_i({\widehat{y}}_j)$, ${\widehat{v}}_i^{\prime }\in D_i({\widehat{y}}_j)$, ${\widehat{w}}_i^{\prime }\in F_i({\widehat{y}}_i)$, ${\widehat{z}}_i^{\prime }\in G_i({\widehat{y}}_j)$ and ${\widehat{t}}_i^{\prime }\in K_i({\widehat{y}}_i)$ $(j\in \{1,2\}\backslash \{i\})$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}& \parallel {\widehat{\pi }}_i-{\widehat{\pi }}_i^{\prime }{\parallel }_{*i}\le H_{*i}(S_i({\widehat{x}}_i),S_i({\widehat{y}}_i)),{\parallel {\widehat{\iota }}_i-{\widehat{\iota }}_i^{\prime }\parallel }_{*j}\le H_{*j}(T_i({\widehat{x}}_j),T_i({\widehat{y}}_j)),\hfill \\ & \parallel {\widehat{u}}_i-{\widehat{u}}_i^{\prime }{\parallel }_{*j}\le H_{*j}(C_i({\widehat{x}}_j),C_i({\widehat{y}}_j)),{\parallel {\widehat{v}}_i-{\widehat{v}}_i^{\prime }\parallel }_{*i}\le H_{*i}(D_i({\widehat{x}}_i),D_i({\widehat{y}}_i)),\hfill \\ & \parallel {\widehat{w}}_i-{\widehat{w}}_i^{\prime }{\parallel }_{*i}\le H_{*i}(F_i({\widehat{x}}_i),F_i({\widehat{y}}_i)),{\parallel {\widehat{z}}_i-{\widehat{z}}_i^{\prime }\parallel }_{*j}\le H_{*j}(G_i({\widehat{x}}_j),G_i({\widehat{y}}_j)),\hfill \\ & \parallel {\widehat{t}}_i-{\widehat{t}}_i^{\prime }{\parallel }_i\le H_i(K_i({\widehat{x}}_i),K_i({\widehat{y}}_i)).\hfill \end{array}\end{array}$$

(12)

Letting

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\widehat{q}}_i& ={\widehat{y}}_i-g_i({\widehat{y}}_i)+R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,{\widehat{t}}_i^{\prime }),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)({\widehat{y}}_i)-{\lambda }_i(W_i(f_i(\widehat{y_i}),{\widehat{\pi }}_i^{\prime },{\widehat{\iota }}_i^{\prime })\hfill \\ & +{\rho }_i(P_i({\widehat{u}}_i^{\prime },{\widehat{v}}_i^{\prime })+Q_i({\widehat{w}}_i^{\prime },{\widehat{z}}_i^{\prime })))]\hfill \end{array}\end{array}$$

(13)

for each $i\in \{1,2\}$, we have ${\widehat{q}}_i\in {{\Psi }}_i({\widehat{x}}_1,{\widehat{x}}_2)$ and so $({\widehat{q}}_1,{\widehat{q}}_2)\in {\Phi }({\widehat{x}}_1,{\widehat{x}}_2)$. From (9), (11), (13) and utilizing Theorem 2, for each $i\in \{1,2\}$, it follows that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel q_i-{\widehat{q}}_i{\parallel }_i& \le \parallel {\widehat{x}}_i-{\widehat{y}}_i-(g_i({\widehat{x}}_i)-g_i({\widehat{y}}_i)){\parallel }_i+\parallel R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,{\widehat{t}}_i),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)\hfill \\ & -{\lambda }_i(W_i(f_i(\widehat{x_i}),{\widehat{\pi }}_i,{\widehat{\iota }}_i)+{\rho }_i(P_i({\widehat{u}}_i,{\widehat{v}}_i)+Q_i({\widehat{w}}_i,{\widehat{z}}_i)))]\hfill \\ & -R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,{\widehat{t}}_i^{\prime }),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)({\widehat{y}}_i)-{\lambda }_i(W_i(f_i(\widehat{y_i}),{\widehat{\pi }}_i^{\prime },{\widehat{\iota }}_i^{\prime })\hfill \\ & +{\rho }_i(P_i({\widehat{u}}_i^{\prime },{\widehat{v}}_i^{\prime })+Q_i({\widehat{w}}_i^{\prime },{\widehat{z}}_i^{\prime })){)]\parallel }_i\hfill \\ & \le \parallel {\widehat{x}}_i-{\widehat{y}}_i-(g_i({\widehat{x}}_i)-g_i({\widehat{y}}_i)){\parallel }_i+\parallel R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,{\widehat{t}}_i),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)\hfill \\ & -{\lambda }_i(W_i(f_i(\widehat{x_i}),{\widehat{\pi }}_i,{\widehat{\iota }}_i)+{\rho }_i(P_i({\widehat{u}}_i,{\widehat{v}}_i)+Q_i({\widehat{w}}_i,{\widehat{z}}_i)))]\hfill \\ & -R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,{\widehat{t}}_i^{\prime }),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-{\lambda }_i(W_i(f_i(\widehat{x_i}),{\widehat{\pi }}_i,{\widehat{\iota }}_i)\hfill \\ & +{\rho }_i(P_i({\widehat{u}}_i,{\widehat{v}}_i)+Q_i({\widehat{w}}_i,{\widehat{z}}_i)){)]\parallel }_i\hfill \\ & +\parallel R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,{\widehat{t}}_i^{\prime }),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-{\lambda }_i(W_i(f_i(\widehat{x_i}),{\widehat{\pi }}_i,{\widehat{\iota }}_i)\hfill \\ & +{\rho }_i(P_i({\widehat{u}}_i,{\widehat{v}}_i)+Q_i({\widehat{w}}_i,{\widehat{z}}_i)))]\hfill \\ & -R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,{\widehat{t}}_i^{\prime }),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)({\widehat{y}}_i)-{\lambda }_i(W_i(f_i(\widehat{y_i}),{\widehat{\pi }}_i^{\prime },{\widehat{\iota }}_i^{\prime })\hfill \\ & +{\rho }_i(P_i({\widehat{u}}_i^{\prime },{\widehat{v}}_i^{\prime })+Q_i({\widehat{w}}_i^{\prime },{\widehat{z}}_i^{\prime })){)]\parallel }_i\hfill \\ & \le \parallel {\widehat{x}}_i-{\widehat{y}}_i-(g_i({\widehat{x}}_i)-g_i({\widehat{y}}_i)){\parallel }_i+{\mu }_i{\parallel {\widehat{t}}_i-{\widehat{t}}_i^{\prime }\parallel }_i\hfill \\ & +\frac{{\tau }_i}{{\gamma }_i}(\parallel ({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-({\widehat{P}}_i\circ g_i)({\widehat{y}}_i)\hfill \\ & -{\lambda }_i(W_i(f_i({\widehat{x}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)-W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)){\parallel }_{*i}\hfill \\ & +{\lambda }_i({\parallel W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)-W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i^{\prime },{\widehat{\iota }}_i^{\prime })\parallel }_{*i}\hfill \\ & +{\rho }_i{\parallel P_i({\widehat{u}}_i,{\widehat{v}}_i)-P_i({\widehat{u}}_i^{\prime },{\widehat{v}}_i^{\prime })\parallel }_{*i}\hfill \\ & +{\rho }_i{\parallel Q_i({\widehat{w}}_i,{\widehat{z}}_i)-Q_i({\widehat{w}}_i^{\prime },{\widehat{z}}_i^{\prime })\parallel }_{*i})).\hfill \end{array}\end{array}$$

(14)

Using Lemma 4 and taking into consideration the fact that that for each $i\in \{1,2\}$, the mapping $g_i$ is $s_i$-strongly accretive and $L_{g_i}$-Lipschitz continuous, we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}& \parallel {\widehat{x}}_i-{\widehat{y}}_i-(g_i({\widehat{x}}_i)-g_i(\widehat{y_i})){\parallel }_i^2\hfill \\ & \le \parallel {\widehat{x}}_i-{\widehat{y}}_i{\parallel }_i^2+2{\langle -(g_i({\widehat{x}}_i)-g_i({\widehat{y}}_i)),J_i({\widehat{x}}_i-\widehat{y_i}-(g_i({\widehat{x}}_i)-g_i({\widehat{y}}_i)))\rangle }_i\hfill \\ & \le \parallel {\widehat{x}}_i-{\widehat{y}}_i{\parallel }_i^2-2{\langle g_i(\widehat{x})-g_i({\widehat{y}}_i),J_i({\widehat{x}}_i-{\widehat{y}}_i)\rangle }_i\hfill \\ & +2{\langle g_i({\widehat{x}}_i)-g_i({\widehat{y}}_i),J_i({\widehat{x}}_i-{\widehat{y}}_i)-J_i({\widehat{x}}_i-{\widehat{y}}_i-(g_i({\widehat{x}}_i)-g_i({\widehat{y}}_i)))\rangle }_i\hfill \\ & \le \parallel {\widehat{x}}_i-{\widehat{y}}_i{\parallel }_i^2-2s_i\parallel {\widehat{x}}_i-{\widehat{y}}_i{\parallel }_i^2+64c_i{L}_{g_i}^2{\parallel {\widehat{x}}_i-{\widehat{y}}_i\parallel }_i^2\hfill \\ & =(1-2s_i+64c_i{L}_{g_i}^2){\parallel {\widehat{x}}_i-{\widehat{y}}_i\parallel }_i^2,\hfill \end{array}\end{array}$$

where for each $i\in \{1,2\}$, $J_i$ is the normalized duality mapping from $E_i$ into $E_i^{*}$. Then, the preceding inequality implies that for each $i\in \{1,2\}$,

$$\begin{array}{c}\hfill \parallel {\widehat{x}}_i-{\widehat{y}}_i-(g_i({\widehat{x}}_i)-g_i(\widehat{y_i})){\parallel }_i\le \sqrt{1-2s_i+64c_i{L}_{g_i}^2}{\parallel {\widehat{x}}_i-{\widehat{y}}_i\parallel }_i.\end{array}$$

(15)

From (12) and taking into account that for each $i\in \{1,2\}$, the mapping $K_i$ is $L_{K_i}$-$H_i$-Lipschitz continuous, we deduce that for each $i\in \{1,2\}$,

$$\begin{array}{c}\hfill \parallel {\widehat{t}}_i-{\widehat{t}}_i^{\prime }{\parallel }_i\le H_i(K_i({\widehat{x}}_i),K_i({\widehat{y}}_i))\le L_{K_i}{\parallel {\widehat{x}}_i-{\widehat{y}}_i\parallel }_i.\end{array}$$

(16)

Again employing Lemma 4(i), for each $i\in \{1,2\}$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& \parallel ({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-({\widehat{P}}_i\circ g_i)({\widehat{y}}_i)-{\lambda }_i(W_i(f_i({\widehat{x}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)-W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)){\parallel }_{*i}^2\hfill \\ & \le \parallel ({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-({\widehat{P}}_i\circ g_i)({\widehat{y}}_i){\parallel }_{*i}^2+2\langle -{\lambda }_i(W_i(f_i({\widehat{x}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)\hfill \\ & -W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)),J_i^{*}(({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-({\widehat{P}}_i\circ g_i)({\widehat{y}}_i))\hfill \\ & -J_i^{*}(({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-({\widehat{P}}_i\circ g_i)({\widehat{y}}_i))+J_i^{*}(({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-({\widehat{P}}_i\circ g_i)({\widehat{y}}_i)\hfill \\ & -{\lambda }_i(W_i(f_i({\widehat{x}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)-W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)){)\rangle }_i\hfill \\ & =\parallel ({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-({\widehat{P}}_i\circ g_i)({\widehat{y}}_i){\parallel }_{*i}^2-2{\lambda }_i\langle W_i(f_i({\widehat{x}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)-W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i),\hfill \\ & J_i^{*}(({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-({\widehat{P}}_i\circ g_i)({\widehat{y}}_i)){\rangle }_i+2\langle {\lambda }_i(W_i(f_i({\widehat{x}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)-W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)),\hfill \\ & J_i^{*}(({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-({\widehat{P}}_i\circ g_i)({\widehat{y}}_i)-{\lambda }_i(W_i(f_i({\widehat{x}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)-W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i))){\rangle }_i,\hfill \end{array}\end{array}$$

(17)

where for each $i\in \{1,2\}$, $J_i^{*}$ is the normalized duality mapping from $E_i^{*}$ into $E_i$. Considering the fact that for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, $W_i(f_i(.),{\tilde{\pi }}_i,{\tilde{\iota }}_i)$ is ${\varpi }_i$-strongly accretive with respect to ${\widehat{P}}_i\circ g_i$, for all ${\tilde{x}}_i\in E_i$, ${\tilde{x}}_j\in E_j$, ${\tilde{\pi }}_i\in S_i({\tilde{x}}_i)$, ${\tilde{\iota }}_i\in T_i({\tilde{x}}_j)$, it follows that

$$\begin{array}{c}\hfill \begin{array}{cc}& {\langle W_i(f_i({\widehat{x}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)-W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i),J_i^{*}(({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-({\widehat{P}}_i\circ g_i)({\widehat{y}}_i))\rangle }_i\hfill \\ & \ge {\varpi }_i{\parallel {\widehat{x}}_i-{\widehat{y}}_i\parallel }_i^2.\hfill \end{array}\end{array}$$

(18)

Thanks to Lemma 4(ii) and the facts that for each $i\in \{1,2\}$, $W_i$ is ${\varsigma }_i$-Lipschitz continuous in the first argument and $f_i$ is $L_{f_i}$-Lipschitz continuous, we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}& \langle {\lambda }_i(W_i(f_i({\widehat{x}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)-W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)),J_i^{*}(({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-({\widehat{P}}_i\circ g_i)({\widehat{y}}_i)\hfill \\ & -{\lambda }_i(W_i(f_i({\widehat{x}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)-W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)){)\rangle }_i\hfill \\ & \le 32c_i{\parallel {\lambda }_i(W_i(f_i({\widehat{x}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)-W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i))\parallel }_{*i}^2\hfill \\ & \le 32c_i{\lambda }_i^2{\varsigma }_i^2{\parallel f_i({\widehat{x}}_i)-f_i({\widehat{y}}_i)\parallel }_{*i}^2\hfill \\ & \le 32c_i{\lambda }_i^2{\varsigma }_i^2{L}_{f_i}^2{\parallel {\widehat{x}}_i-{\widehat{y}}_i\parallel }_i^2.\hfill \end{array}\end{array}$$

(19)

Since for each $i\in \{1,2\}$, ${\widehat{P}}_i$ and $g_i$ are $L_{\widehat{P_i}}$-Lipschitz and $L_{g_i}$-Lipschitz continuous mappings, respectively, we have

$$\begin{array}{c}\hfill \parallel ({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-({\widehat{P}}_i\circ g_i)({\widehat{y}}_i){\parallel }_{*i}\le L_{{\widehat{P}}_i}{L}_{g_i}{\parallel {\widehat{x}}_i-{\widehat{y}}_i\parallel }_i.\end{array}$$

(20)

Combining (17)–(20), for each $i\in \{1,2\}$ we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& \parallel ({\widehat{P}}_i\circ g_i)({\widehat{x}}_i)-({\widehat{P}}_i\circ g_i)({\widehat{y}}_i)-{\lambda }_i{(W_i(f_i({\widehat{x}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)-W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)\parallel }_{*i}\hfill \\ & \le \sqrt{L_{{\widehat{P}}_i}^2{L}_{g_i}^2-2{\lambda }_i{\varpi }_i+64c_i{\lambda }_i^2{\varsigma }_i^2{L}_{f_i}^2}{\parallel {\widehat{x}}_i-{\widehat{y}}_i\parallel }_i.\hfill \end{array}\end{array}$$

(21)

Since for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, $W_i$ is $({\varsigma }_i,{\delta }_i,{\nu }_i)$-mixed Lipschitz continuous, $S_i$ is $L_{S_i}$-$H_{*i}$-Lipschitz continuous, and $T_i$ is $L_{T_i}$-$H_{*j}$-Lipschitz continuous, using (12), it follows that

$$\begin{array}{c}\hfill \begin{array}{cc}& \parallel W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i,{\widehat{\iota }}_i)-W_i(f_i({\widehat{y}}_i),{\widehat{\pi }}_i^{\prime },{\widehat{\iota }}_i^{\prime }){\parallel }_{*i}\hfill \\ & \le {\delta }_i\parallel {\widehat{\pi }}_i-{\widehat{\pi }}_i^{\prime }{\parallel }_{*i}+{\nu }_i{\parallel {\widehat{\iota }}_i-{\widehat{\iota }}_i^{\prime }\parallel }_{*j}\hfill \\ & \le {\delta }_i{H}_{*i}(S_i({\widehat{x}}_i),S_i({\widehat{y}}_i))+{\nu }_i{H}_{*j}(T_i({\widehat{x}}_j),T_i({\widehat{y}}_j))\hfill \\ & \le {\delta }_i{L}_{S_i}\parallel {\widehat{x}}_i-{\widehat{y}}_i{\parallel }_i+{\nu }_i{L}_{T_i}{\parallel {\widehat{x}}_j-{\widehat{y}}_j\parallel }_j.\hfill \end{array}\end{array}$$

(22)

In light of the fact that for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, $P_i$ is $(L_{(P_i,j)},L_{(P_i,i)})$-mixed Lipschitz, $Q_i$ is $(L_{(Q_i,i)},L_{(Q_i,j)})$-mixed Lipschitz continuous, $C_i$ is $L_{C_i}$-$H_{*j}$-Lipschitz continuous, $D_i$ is $L_{D_i}$-$H_{*i}$-Lipschitz continuous, $F_i$ is $L_{F_i}$-$H_{*i}$-Lipschitz continuous, and $G_i$ is $L_{G_i}$-$H_{*j}$-Lipschitz continuous, one has

$$\begin{array}{c}\hfill \begin{array}{cc}& \parallel P_i({\widehat{u}}_i,{\widehat{v}}_i)-P_i({\widehat{u}}_i^{\prime },{\widehat{v}}_i^{\prime }){\parallel }_{*i}\hfill \\ & \le L_{(P_i,j)}\parallel {\widehat{u}}_i-{\widehat{u}}_i^{\prime }{\parallel }_{*j}+L_{(P_i,i)}{\parallel {\widehat{v}}_i-{\widehat{v}}_i^{\prime }\parallel }_{*i}\hfill \\ & \le L_{(P_i,j)}{H}_{*j}(C_i({\widehat{x}}_j),C_i({\widehat{y}}_j))+L_{(P_i,i)}{H}_{*i}(D_i({\widehat{x}}_i),D_i({\widehat{y}}_i))\hfill \\ & \le L_{(P_i,j)}{L}_{C_i}\parallel {\widehat{x}}_j-{\widehat{y}}_j{\parallel }_j+L_{(P_i,i)}{L}_{D_i}{\parallel {\widehat{x}}_i-{\widehat{y}}_i\parallel }_i\hfill \end{array}\end{array}$$

(23)

and

$$\begin{array}{c}\hfill \begin{array}{cc}& \parallel Q_i({\widehat{w}}_i,{\widehat{z}}_i)-Q_i({\widehat{w}}_i^{\prime },{\widehat{z}}_i^{\prime }){\parallel }_{*i}\hfill \\ & \le L_{(Q_i,i)}\parallel {\widehat{w}}_i-{\widehat{w}}_i^{\prime }{\parallel }_{*i}+L_{(Q_i,j)}{\parallel {\widehat{z}}_i-{\widehat{z}}_i^{\prime }\parallel }_{*j}\hfill \\ & \le L_{(Q_i,j)}{H}_{*i}(F_i({\widehat{x}}_i),F_i({\widehat{y}}_i))+L_{(Q_i,j)}{H}_{*j}(G_i({\widehat{x}}_j),G_i({\widehat{y}}_j))\hfill \\ & \le L_{(Q_i,i)}{L}_{F_i}\parallel {\widehat{x}}_i-{\widehat{y}}_i{\parallel }_i+L_{(Q_i,j)}{L}_{G_i}{\parallel {\widehat{x}}_j-{\widehat{y}}_j\parallel }_j.\hfill \end{array}\end{array}$$

(24)

Substituting (15), (16) and (21)–(24) into (14), for each $i\in \{1,2\}$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel q_i-{\widehat{q}}_i{\parallel }_i& \le \sqrt{1-2s_i+64c_i{L}_{g_i}^2}\parallel {\widehat{x}}_i-{\widehat{y}}_i{\parallel }_i+{\mu }_i{L}_{K_i}{\parallel {\widehat{x}}_i-{\widehat{y}}_i\parallel }_i\hfill \\ & +\frac{{\tau }_i}{{\gamma }_i}(\sqrt{L_{{\widehat{P}}_i}^2{L}_{g_i}^2-2{\lambda }_i{\varpi }_i+64c_i{\lambda }_i^2{\varsigma }_i^2{L}_{f_i}^2}{\parallel {\widehat{x}}_i-{\widehat{y}}_i\parallel }_i\hfill \\ & +{\lambda }_i({\delta }_i{L}_{S_i}\parallel {\widehat{x}}_i-{\widehat{y}}_i{\parallel }_i+{\nu }_i{L}_{T_i}{\parallel {\widehat{x}}_j-{\widehat{y}}_j\parallel }_j\hfill \\ & +{\rho }_i(L_{(P_i,j)}{L}_{C_i}\parallel {\widehat{x}}_j-{\widehat{y}}_j{\parallel }_j+L_{(P_i,i)}{L}_{D_i}{\parallel {\widehat{x}}_i-{\widehat{y}}_i\parallel }_i\hfill \\ & +L_{(Q_i,i)}{L}_{F_i}\parallel {\widehat{x}}_i-{\widehat{y}}_i{\parallel }_i+L_{(Q_i,j)}{L}_{G_i}\parallel {\widehat{x}}_j-{\widehat{y}}_j{\parallel }_j)))\hfill \\ & =a_i\parallel {\widehat{x}}_i-{\widehat{y}}_i{\parallel }_i+b_i{\parallel {\widehat{x}}_j-{\widehat{y}}_j\parallel }_j,\hfill \end{array}\end{array}$$

(25)

where $j\in \{1,2\}\backslash \left\{i\right\}$ and for each $i\in \{1,2\}$,

$$\begin{array}{c}\hfill \begin{array}{cc}& a_i=\sqrt{1-2s_i+64c_i{L}_{g_i}^2}+{\mu }_i{L}_{K_i}+\frac{{\tau }_i}{{\gamma }_i}(\sqrt{L_{{\widehat{P}}_i}^2{L}_{g_i}^2-2{\lambda }_i{\varpi }_i+64c_i{\lambda }_i^2{\varsigma }_i^2{L}_{f_i}^2}\hfill \\ & +{\lambda }_i\left({\delta }_i{L}_{S_i}+{\rho }_i(L_{(P_i,i)}{L}_{D_i}+L_{(Q_i,i)}{L}_{F_i})\right)),\hfill \\ & b_i=\frac{{\tau }_i{\lambda }_i}{{\gamma }_i}({\nu }_i{L}_{T_i}+{\rho }_i(L_{(P_i,j)}{L}_{C_i}+L_{(Q_i,j)}{L}_{G_i})).\hfill \end{array}\end{array}$$

Let us now define a norm ${\parallel ·\parallel }_{\diamond }$ on $E_1\times E_2$ by

$$\begin{array}{c}\hfill \parallel (x_1,x_2){\parallel }_{\diamond }=\parallel x_1{\parallel }_1+{\parallel x\parallel }_2,\forall (x_1,x_2)\in E_1\times E_2.\end{array}$$

It is easy to see that $(E_1\times E_2,\parallel ·{\parallel }_{\diamond })$ is a Banach space. Then, using (24), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel (q_1,q_2)-({\widehat{q}}_1,{\widehat{q}}_2){\parallel }_{\diamond }& =\parallel q_1-{\widehat{q}}_1{\parallel }_1+{\parallel q_2-{\widehat{q}}_2\parallel }_2\hfill \\ & \le a_1\parallel {\widehat{x}}_1-{\widehat{y}}_1{\parallel }_1+b_1{\parallel {\widehat{x}}_2-{\widehat{y}}_2\parallel }_2\hfill \\ & +a_2\parallel {\widehat{x}}_2-{\widehat{y}}_2{\parallel }_2+b_2{\parallel {\widehat{x}}_1-{\widehat{y}}_1\parallel }_1\hfill \\ & =(a_1+b_2)\parallel {\widehat{x}}_1-{\widehat{y}}_1{\parallel }_1+(a_2+b_1){\parallel {\widehat{x}}_2-{\widehat{y}}_2\parallel }_2\hfill \\ & \le max\{k_1,k_2\}{\parallel ({\widehat{x}}_1,{\widehat{x}}_2)-({\widehat{y}}_1,{\widehat{y}}_2)\parallel }_{\diamond },\hfill \end{array}\end{array}$$

(26)

where $k_1=a_1+b_2$ and $k_2=a_2+b_1$. It follows from (10) that $0<max\{k_1,k_2\}<1$. Hence, from (26), we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d((q_1,q_2),{\Phi }({\widehat{y}}_1,{\widehat{y}}_2))& =\underset{({\widehat{q}}_1,{\widehat{q}}_2)\in {\Phi }({\widehat{y}}_1,{\widehat{y}}_2)}{inf}{\parallel (q_1,q_2)-({\widehat{q}}_1,{\widehat{q}}_2)\parallel }_{\diamond }\hfill \\ & \le max\{k_1,k_2\}{\parallel ({\widehat{x}}_1,{\widehat{x}}_2)-({\widehat{y}}_1,{\widehat{y}}_2)\parallel }_{\diamond }.\hfill \end{array}\end{array}$$

Taking into account the arbitrariness in the choice of $(\widehat{q_1},\widehat{q_2})\in {\Phi }({\widehat{y}}_1,{\widehat{y}}_2)$, we conclude that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \underset{(q_1,q_2)\in {\Phi }({\widehat{x}}_1,{\widehat{x}}_2)}{sup}d((q_1,q_2),{\Phi }({\widehat{y}}_1,{\widehat{y}}_2))\le max\{k_1,k_2\}{\parallel ({\widehat{x}}_1,{\widehat{x}}_2)-({\widehat{y}}_1,{\widehat{y}}_2)\parallel }_{\diamond }.\end{array}\end{array}$$

(27)

In a similar fashion to the preceding analysis, one can prove that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \underset{({\widehat{q}}_1,{\widehat{q}}_2)\in {\Phi }({\widehat{y}}_1,{\widehat{y}}_2)}{sup}d({\Phi }({\widehat{x}}_1,{\widehat{x}}_2),({\widehat{q}}_1,{\widehat{q}}_2))\le max\{k_1,k_2\}{\parallel ({\widehat{x}}_1,{\widehat{x}}_2)-({\widehat{y}}_1,{\widehat{y}}_2)\parallel }_{\diamond }.\end{array}\end{array}$$

(28)

With the help of (27) and (28) and by the definition of Hausdorff metric H on $C(E_1\times E_2)$, we obtain that for all $({\widehat{x}}_1,{\widehat{x}}_2),({\widehat{y}}_1,{\widehat{y}}_2)\in E_1\times E_2$,

$$\begin{array}{c}\hfill H({\Phi }({\widehat{x}}_1,{\widehat{x}}_2),{\Phi }({\widehat{y}}_1,{\widehat{y}}_2))\le max\{k_1,k_2\}{\parallel ({\widehat{x}}_1,{\widehat{x}}_2)-({\widehat{y}}_1,{\widehat{y}}_2)\parallel }_{\diamond },\end{array}$$

which means that ${\Phi }$ is a multi-valued contraction mapping. Invoking Lemma 4, ${\Phi }$ has a fixed point $(x_1,x_1)\in E_1\times E_2$, i.e., $(x_1,x_2)\in {\Phi }(x_1,x_2)$. By the definition of ${\Phi }$, we have $x_1\in {{\Psi }}_1(x_1,x_2)$ and $x_2\in {{\Psi }}_2(x_1,x_2)$. Then, in light of the definition of ${{\Psi }}_i$ $(i=1,2)$, for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, there exist ${\pi }_i\in S_i(x_i)$, ${\iota }_i\in T_i(x_j)$, $u_i\in C_i(x_j)$, $v_i\in D_i(x_i)$, $w_i\in F_i(x_i)$, $z_i\in G_i(x_j)$ and $t_i\in K_i(x_i)$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {{\Psi }}_i(x_1,x_2)& =x_i-g_i(x_i)+R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)(x_i)-{\lambda }_i(W_i(f_i(x_i),{\pi }_i,{\iota }_i)\hfill \\ & +{\rho }_i(P_i(u_i,v_i)+Q_i(w_i,z_i))].\hfill \end{array}\end{array}$$

In accordance with Lemma 2, $(x_1,x_2,{\pi }_1,{\pi }_2,{\iota }_1,{\iota }_2,u_1,u_2,v_1,v_2,w_1,w_2,z_1,z_2,t_1,t_2)$ is a solution of the $\mathrm{SGMVLI}$ (5). This completes the proof.    □

As an immediate corollary of the previous result, one can prove the existence of a solution for the $\mathrm{SGIVLI}$ (6) and we omit its proof.

Corollary 2.

Suppose that for each $i\in \{1,2\}$, $E_i$ is a real uniformly smooth Banach space with ${\rho }_{E_i}(t)\le c_i{t}^2$ for some $c_i>0$ and for all $t\in [0,+\infty )$. For each $i\in \{1,2\}$, let ${\eta }_i:E_i\times E_i\to E_i$ be a ${\tau }_i$-Lipschitz continuous mapping such that ${\eta }_i(y_i,{\widehat{y}}_i)+{\eta }_i({\widehat{y}}_i,y_i)=0$ for all $y_i,{\widehat{y}}_i\in E_i$, $g_i:E_i\to E_i$ be a $s_i$-strongly accretive and $L_{g_i}$-Lipschitz continuous mapping, ${\widehat{P}}_i:E_i\to E_i^{*}$ be a ${\gamma }_i$-strongly ${\eta }_i$-monotone and $L_{{\widehat{P}}_i}$-Lipschitz continuous mapping such that $g_i(E_i)\subseteq \mathrm{dom}({\widehat{P}}_i)$. Assume that for each $i\in \{1,2\}$ and any given $x_i^{*}\in E_i^{*}$, the function $h_i:(y_i,x_i^{\prime })\in E_i\times E_i\to h_i(y_i,x_i^{\prime })={\langle x_i^{*}-{\widehat{P}}_i(x_i^{\prime }),{\eta }_i(y_i,x_i^{\prime })\rangle }_i\in ℝ\cup \{+\infty \}$ is $\mathrm{0-DQCV}$ in the first argument. For each $i\in \{1,2\}$, let $S_i:E_i\to E_i^{*}$ be $L_{S_i}$-Lipschitz continuous, $T_i:E_j\to E_j^{*}$ be $L_{T_i}$-Lipschitz continuous $(j\in \{1,2\}\backslash \{i\})$, and $D_i,F_i:E_i\to C(E_i^{*})$ be $L_{D_i}$-$H_{*i}$-Lipschitz and $L_{F_i}$-$H_{*i}$-Lipschitz continuous, respectively. Suppose that for each $i\in \{1,2\}$, $N_i:E_i^{*}\times E_j^{*}\to E_i^{*}$ $(j\in \{1,2\}\backslash \{i\})$ is a $({\delta }_i,{\nu }_i)$-mixed Lipschitz continuous mapping and for any ${\tilde{x}}_j\in E_j$, $N_i(S_i(.),T_i({\tilde{x}}_j))$ is ${\epsilon }_i$-strongly accretive with respect to ${\widehat{P}}_i\circ g_i$. For each $i\in \{1,2\}$, let $C_i,G_i:E_j\to C(E_j^{*})$ $(j\in \{1,2\}\backslash \{i\})$ be $L_{C_i}$-$H_{*j}$-Lipschitz and $L_{G_i}$-$H_{*j}$-Lipschitz continuous, respectively, and $P_i:E_j^{*}\times E_i^{*}\to E_i^{*}$ and $Q_i:E_i^{*}\times E_j^{*}\to E_i^{*}$ be $(L_{(P_i,j)},L_{(P_i,i)})$-mixed Lipschitz and $(L_{(Q_i,i)},L_{(Q_i,j)})$-mixed Lipschitz continuous, respectively. Moreover, let for each $i\in \{1,2\}$, ${\varphi }_i:E_i\to ℝ\cup \{+\infty \}$ be a proper, lower semicontinuous and ${\eta }_i$-subdifferentiable on $E_i$ with $g_i(E_i)\cap \mathrm{dom}\partial {\varphi }_i\ne \varnothing$. If there exist constants ${\lambda }_i>0$ $(i=1,2)$ such that for $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}\sqrt{1-2s_i+64c_i{L}_{g_i}^2}+\frac{{\tau }_i}{{\gamma }_i}(\sqrt{L_{{\widehat{P}}_i}^2{L}_{g_i}^2-2{\lambda }_i{\epsilon }_i+64c_i{\lambda }_i^2{\delta }_i^2{L}_{S_i}^2}+{\lambda }_i{\rho }_i(L_{(P_i,i)}{L}_{D_i}\hfill \\ +L_{(Q_i,i)}{L}_{F_i}))+\frac{{\tau }_j{\lambda }_j}{{\gamma }_j}({\nu }_j{L}_{T_j}+{\rho }_j(L_{(P_j,i)}{L}_{C_j}+L_{(Q_j,i)}{L}_{G_j}))<1,\hfill \\ 2s_i<1+64c_i{L}_{g_i}^2,2{\lambda }_i{\epsilon }_i<L_{{\widehat{P}}_i}^2{L}_{g_i}^2+64c_i{\lambda }_i^2{\delta }_i^2{L}_{S_i}^2,\hfill \end{array}\right.\end{array}\end{array}$$

(29)

then the $\mathrm{SGIVLI}$ (6) has a solution.

4. Iterative Algorithms and Convergence Analysis

The equivalence formulation given in Lemma 2 and Nadler technique [39] enable us to suggest the following iterative algorithm for finding the approximate solution of the $\mathrm{SGMVLI}$ (5).

In the next theorem, under some appropriate conditions imposed on the parameters and mappings involved in the $\mathrm{SGMVLI}$ (5), the convergence analysis of the sequences generated by Algorithm 1 is studied.

Algorithm 1 Iterative algorithm for finding the approximate solution of the SGIVLI (6)

Let $E_i,E_i^{*},C_i,D_i,F_i,G_i,P_i,Q_i,K_i,S_i,T_i,{\widehat{P}}_i,W_i,{\eta }_i,{\varphi }_i,f_i,g_i,h_i$ $(i=1,2)$ be the same as in Lemma 2. For any given $(x_1^0,x_2^0,{\pi }_1^0,{\pi }_2^0,{\iota }_1^0,{\iota }_2^0,u_1^0,u_2^0,v_1^0,v_2^0,w_1^0,w_2^0,z_1^0,z_2^0,t_1^0,t_2^0)$, where for each $i\in \{1,2\}$, ${\pi }_i^0\in S_i(x_i^0)$, ${\iota }_i^0\in T_i(x_j^0)$, $u_i^0\in C_i(x_j^0)$, $v_i^0\in D_i(x_i^0)$, $w_i^0\in F_i(x_i^0)$, $z_i^0\in G_i(x_j^0)$, $t_i^0\in K_i(x_i^0)$ $(j\in \{1,2\}\backslash \{i\})$, define the iterative sequences ${\left\{x_i^n\right\}}_{n=0}^{\infty }$, ${\left\{{\pi }_i^n\right\}}_{n=0}^{\infty }$, ${\left\{{\iota }_i^n\right\}}_{n=0}^{\infty }$, ${\left\{u_i^n\right\}}_{n=0}^{\infty }$, ${\left\{v_i^n\right\}}_{n=0}^{\infty }$, ${\left\{w_i^n\right\}}_{n=0}^{\infty }$, ${\left\{z_i^n\right\}}_{n=0}^{\infty }$ and ${\left\{t_i^n\right\}}_{n=0}^{\infty }$ $(i=1,2)$, in the following way: $$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{x}_i^{n+1}=(1-\mu )x_i^n+\mu \{x_i^n-g_i(x_i^n)+R_{{\lambda }_i,{\eta }_i}^{\partial {{\Phi }}_i(.,t_i^n),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)(x_i^n)\hfill \\ -{\lambda }_i(W_i(f_i(x_i),{\pi }_i^n,{\iota }_i^n)+{\rho }_i(P_i(u_i^n,v_i^n)+Q_i(w_i^n,z_i^n))]\}+\mu e_i^n+r_i^n,\hfill \\ {\pi }_i^n\in S_i(x_i^n):{\parallel {\pi }_i^{n+1}-{\pi }_i^n\parallel }_{*i}\le (1+{(1+n)}^{-1})H_{*i}(S_i(x_i^{n+1}),S_i(x_i^n)),\hfill \\ {\iota }_i^n\in T_i(x_j^n):{\parallel {\iota }_i^{n+1}-{\iota }_i^n\parallel }_{*j}\le (1+{(1+n)}^{-1})H_{*j}(T_i(x_j^{n+1}),T_i(x_j^n)),\hfill \\ u_i^n\in C_i(x_j^n):{\parallel u_i^{n+1}-u_i^n\parallel }_{*j}\le (1+{(1+n)}^{-1})H_{*j}(C_i(x_j^{n+1}),C_i(x_j^n)),\hfill \\ v_i^n\in D_i(x_i^n):{\parallel v_i^{n+1}-v_i^n\parallel }_{*i}\le (1+{(1+n)}^{-1})H_{*i}(D_i(x_i^{n+1}),D_i(x_i^n)),\hfill \\ w_i^n\in F_i(x_i^n):{\parallel w_i^{n+1}-w_i^n\parallel }_{*i}\le (1+{(1+n)}^{-1})H_{*i}(F_i(x_i^{n+1}),F_i(x_i^n)),\hfill \\ z_i^n\in G_i(x_j^n):{\parallel z_i^{n+1}-z_i^n\parallel }_{*j}\le (1+{(1+n)}^{-1})H_{*j}(G_i(x_j^{n+1}),G_i(x_j^n)),\hfill \\ t_i^n\in K_i(x_i^n):{\parallel t_i^{n+1}-t_i^n\parallel }_{*i}\le (1+{(1+n)}^{-1})H_{*i}(K_i(x_i^{n+1}),K_i(x_i^n)),\hfill \end{array}\right.\end{array}\end{array}$$ (30) where $n=0,1,2,\cdots$; ${\lambda }_i>0$ $(i=1,2)$ are constants, $\mu \in (0,1]$ is a relaxation parameter, $H_i$ and $H_{*i}$ are Hausdorff metric on $CB(E_i)$ and $CB(E_i^{*})$, respectively, and for each $i\in \{1,2\}$, ${\left\{e_i^n\right\}}_{n=0}^{\infty }$ and ${\left\{r_i^n\right\}}_{n=0}^{\infty }$ are two sequences in $E_i$ to take into account a possible inexact computation of the $P_i$-${\eta }_i$-proximal mapping points satisfying the following conditions: $$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}\underset{n\to \infty }{lim}\parallel e_i^n{\parallel }_i=\underset{n\to \infty }{lim}{\parallel r_i^n\parallel }_i=0,\hfill \\ {\displaystyle \sum _{n=1}^{\infty }}{\parallel e_i^n-e_i^{n-1}\parallel }_i<\infty ,\hfill \\ {\displaystyle \sum _{n=1}^{\infty }}{\parallel r_i^n-r_i^{n-1}\parallel }_i<\infty .\hfill \end{array}\right.\end{array}\end{array}$$ (31)

Theorem 5.

Let $E_i,E_i^{*},P_i,Q_i,{\widehat{P}}_i,W_i,{\eta }_i,{\varphi }_i,f_i,g_i,h_i$ $(i=1,2)$ be the same as in Theorem 4. Let for each $i\in \{1,2\}$, $K_i:E_i\to CB(E_i)$ be $L_{K_i}$-$H_i$-Lipschitz continuous, $S_i:E_i\to CB(E_i^{*})$ be $L_{S_i}$-$H_{*i}$-Lipschitz continuous and $D_i,F_i:E_i\to CB(E_i^{*})$ be $L_{D_i}$-$H_{*i}$-Lipschitz and $L_{F_i}$-$H_{*i}$-Lipschitz continuous, respectively. Suppose further that for each $i\in \{1,2\}$, $T_i:E_j\to CB(E_j^{*})$ $(j\in \{1,2\}\backslash \{i\})$ is $L_{T_i}$-$H_{*j}$-Lipschitz continuous, and $C_i,G_i:E_j\to CB(E_j^{*})$ is $L_{C_i}$-$H_{*j}$-Lipschitz and $L_{G_i}$-$H_{*j}$-Lipschitz continuous, respectively. If there exist constants ${\mu }_i,{\lambda }_i>0$ $(i=1,2)$ satisfying (9) and (10), then the sequences ${\left\{x_i^n\right\}}_{n=0}^{\infty }$, ${\left\{{\pi }_i^n\right\}}_{n=0}^{\infty }$, ${\left\{{\iota }_i^n\right\}}_{n=0}^{\infty }$, ${\left\{u_i^n\right\}}_{n=0}^{\infty }$, ${\left\{v_i^n\right\}}_{n=0}^{\infty }$, ${\left\{w_i^n\right\}}_{n=0}^{\infty }$, ${\left\{z_i^n\right\}}_{n=0}^{\infty }$ and ${\left\{t_i^n\right\}}_{n=0}^{\infty }$ $(i=1,2)$ generated by Algorithm 1 converge strongly to $x_i$, ${\pi }_i$, ${\iota }_i$, $u_i$, $v_i$, $w_i$, $z_i$ and $t_i$, respectively, and $(x_1,x_2,{\pi }_1,{\pi }_2,{\iota }_1,{\iota }_2,u_1,u_2,v_1,v_2,w_1,w_2,z_1,z_2,t_1,t_2)$ is a solution of the $\mathrm{SGMVLI}$ (5).

Proof. 

Making use of (9) and (30), Theorem 2 and the hypotheses mentioned in the context of the theorem, by an argument analogous to that in the proof of Theorem 4, for each $i\in \{1,2\}$ and $n\ge 0$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel x_i^{n+1}-x_i^n{\parallel }_i& \le (1-\mu )\parallel x_i^n-x_i^{n-1}{\parallel }_i+\mu ({\parallel x_i^n-x_i^{n-1}-(g_i(x_i^n)-g_i(x_i^{n-1}))\parallel }_i\hfill \\ & +\parallel R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i^n)}[({\widehat{P}}_i\circ g_i)(x_i^n)-{\lambda }_i(W_i(f_i(x_i^n),{\pi }_i^n,{\iota }_i^n)\hfill \\ & +{\rho }_i(P_i(u_i^n,v_i^n)+Q_i(w_i^n,z_i^n))]-R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i^{n-1})}[({\widehat{P}}_i\circ g_i)(x_i^{n-1})\hfill \\ & -{\lambda }_i(W_i(f_i(x_i^{n-1}),{\pi }_i^{n-1},{\iota }_i^{n-1})+{\rho }_i(P_i(u_i^{n-1},v_i^{n-1})\hfill \\ & +Q_i(w_i^{n-1},z_i^{n-1}){)]\parallel }_i)+\mu \parallel e_i^n-e_i^{n-1}{\parallel }_i+{\parallel r_i^n-r_i^{n-1}\parallel }_i\hfill \\ & \le (1-\mu )\parallel x_i^n-x_i^{n-1}{\parallel }_i+\mu ({\parallel x_i^n-x_i^{n-1}-(g_i(x_i^n)-g_i(x_i^{n-1}))\parallel }_i\hfill \\ & +\parallel R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i^n)}[({\widehat{P}}_i\circ g_i)(x_i^n)-{\lambda }_i(W_i(f_i(x_i^n),{\pi }_i^n,{\iota }_i^n)\hfill \\ & +{\rho }_i(P_i(u_i^n,v_i^n)+Q_i(w_i^n,z_i^n))]-R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i^{n-1})}[({\widehat{P}}_i\circ g_i)(x_i^n)\hfill \\ & -{\lambda }_i(W_i(f_i(x_i^n),{\pi }_i^n,{\iota }_i^n)+{\rho }_i(P_i(u_i^n,v_i^n)+Q_i(w_i^n,z_i^n))]{\parallel }_i\hfill \\ & +\parallel R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i^{n-1})}[({\widehat{P}}_i\circ g_i)(x_i^n)-{\lambda }_i(W_i(f_i(x_i^n),{\pi }_i^n,{\iota }_i^n)\hfill \\ & +{\rho }_i(P_i(u_i^n,v_i^n)+Q_i(w_i^n,z_i^n))]-R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i^{n-1})}[({\widehat{P}}_i\circ g_i)(x_i^{n-1})\hfill \\ & -{\lambda }_i(W_i(f_i(x_i^{n-1}),{\pi }_i^{n-1},{\iota }_i^{n-1})+{\rho }_i(P_i(u_i^{n-1},v_i^{n-1})\hfill \\ & +Q_i(w_i^{n-1},z_i^{n-1}){)]\parallel }_i)+\mu \parallel e_i^n-e_i^{n-1}{\parallel }_i+{\parallel r_i^n-r_i^{n-1}\parallel }_i\hfill \\ & \le (1-\mu )\parallel x_i^n-x_i^{n-1}{\parallel }_i+\mu ({\parallel x_i^n-x_i^{n-1}-(g_i(x_i^n)-g_i(x_i^{n-1}))\parallel }_i\hfill \\ & +{\mu }_i\parallel t_i^n-t_i^{n-1}{\parallel }_i+\frac{{\tau }_i}{{\gamma }_i}\parallel ({\widehat{P}}_i\circ g_i)(x_i^n)-({\widehat{P}}_i\circ g_i)(x_i^{n-1})\hfill \\ & -{\lambda }_i(W_i(f_i(x_i^n),{\pi }_i^n,{\iota }_i^n)+{\rho }_i(P_i(u_i^n,v_i^n)+Q_i(w_i^n,z_i^n)))\hfill \\ & +{\lambda }_i(W_i(f_i(x_i^{n-1}),{\pi }_i^{n-1},{\iota }_i^{n-1})+{\rho }_i(P_i(u_i^{n-1},v_i^{n-1})\hfill \\ & +Q_i(w_i^{n-1},z_i^{n-1}){))\parallel }_{*i})+\mu \parallel e_i^n-e_i^{n-1}{\parallel }_i+{\parallel r_i^n-r_i^{n-1}\parallel }_i\hfill \\ & \le (1-\mu )\parallel x_i^n-x_i^{n-1}{\parallel }_i+\mu ({\parallel x_i^n-x_i^{n-1}-(g_i(x_i^n)-g_i(x_i^{n-1}))\parallel }_i\hfill \\ & +{\mu }_i\parallel t_i^n-t_i^{n-1}{\parallel }_i+\frac{{\tau }_i}{{\gamma }_i}\parallel ({\widehat{P}}_i\circ g_i)(x_i^n)-({\widehat{P}}_i\circ g_i)(x_i^{n-1})\hfill \\ & -{\lambda }_i(W_i(f_i(x_i^n),{\pi }_i^n,{\iota }_i^n)-W_i(f_i(x_i^{n-1}),{\pi }_i^n,{\iota }_i^n)){\parallel }_i\hfill \\ & +{\lambda }_i({\parallel W_i(f_i(x_i^{n-1}),{\pi }_i^n,{\iota }_i^n)-W_i(f_i(x_i^{n-1}),{\pi }_i^{n-1},{\iota }_i^{n-1})\parallel }_{*i}\hfill \\ & +{\rho }_i(\parallel P_i(u_i^n,v_i^n)-P_i(u_i^{n-1},v_i^{n-1}){\parallel }_{*i}\hfill \\ & +\parallel Q_i(w_i^n,z_i^n)-Q_i(w_i^{n-1},z_i^{n-1}){\parallel }_{*i})))\hfill \\ & +\mu \parallel e_i^n-e_i^{n-1}{\parallel }_i+{\parallel r_i^n-r_i^{n-1}\parallel }_i\hfill \\ & \le (1-\mu )\parallel x_i^n-x_i^{n-1}{\parallel }_i+\mu (a_i^n\parallel x_i^n-x_i^{n-1}{\parallel }_i+b_i^n\parallel x_j^n-x_j^{n-1}{\parallel }_j)\hfill \\ & +\mu \parallel e_i^n-e_i^{n-1}{\parallel }_i+{\parallel r_i^n-r_i^{n-1}\parallel }_i,\hfill \end{array}\end{array}$$

(32)

where for $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$,

$$\begin{array}{c}\hfill \begin{array}{cc}& a_i^n=\sqrt{1-2s_i+64c_i{L}_{g_i}^2}+{\mu }_i{L}_{K_i}(1+n^{-1})+\frac{{\tau }_i}{{\gamma }_i}(\sqrt{L_{{\widehat{P}}_i}^2{L}_{g_i}^2-2{\lambda }_i{\varpi }_i+64c_i{\lambda }_i{\varsigma }_i^2{L}_{f_i}^2}\hfill \\ & +{\lambda }_i\left({\delta }_i{L}_{S_i}+{\rho }_i(L_{(P_i,i)L_{D_i}}+L_{(Q_i,i)L_{F_i}})\right)(1+n^{-1})),\hfill \\ & b_i^n=\frac{{\tau }_i{\lambda }_i}{{\gamma }_i}\left({\nu }_i{L}_{T_i}+{\rho }_i(L_{(P_i,j)}{L}_{C_i}+L_{(Q_i,j)}{L}_{G_i})\right)(1+n^{-1}).\hfill \end{array}\end{array}$$

From (32), it follows that for each $n\in N$,

$$\begin{array}{c}\hfill \begin{array}{cc}& \parallel (x_1^{n+1},x_2^{n+1})-(x_1^n,x_2^n){\parallel }_{\diamond }\hfill \\ & =\parallel x_1^{n+1}-x_1^n{\parallel }_1+{\parallel x_2^{n+1}-x_2^n\parallel }_2\hfill \\ & \le (1-\mu )\parallel (x_1^n,x_2^n)-(x_1^{n-1},x_2^{n-1}){\parallel }_{\diamond }\hfill \\ & +\mu ((a_1^n+b_2^n)\parallel x_1^n-x_1^{n-1}{\parallel }_1+(a_2^n+b_1^n)\parallel x_2^n-x_2^{n-1}{\parallel }_2)\hfill \\ & +\mu \parallel (e_1^n,e_2^n)-(e_1^{n-1},e_2^{n-1}){\parallel }_{\diamond }+{\parallel (r_1^n,r_2^n)-(r_1^{n-1},r_2^{n-1})\parallel }_{\diamond }\hfill \\ & \le (1-\mu )\parallel (x_1^n,x_2^n)-(x_1^{n-1},x_2^{n-1}){\parallel }_{\diamond }\hfill \\ & +\mu max\{k_1^n,k_2^n\}{\parallel (x_1^n,x_2^n)-(x_1^{n-1},x_2^{n-1})\parallel }_{\diamond }\hfill \\ & +\mu \parallel (e_1^n,e_2^n)-(e_1^{n-1},e_2^{n-1}){\parallel }_{\diamond }+{\parallel (r_1^n,r_2^n)-(r_1^{n-1},r_2^{n-1})\parallel }_{\diamond }\hfill \\ & =(1-\mu (1-max\{k_1^n,k_2^n\})){\parallel (x_1^n,x_2^n)-(x_1^{n-1},x_2^{n-1})\parallel }_{\diamond }\hfill \\ & +\mu \parallel (e_1^n,e_2^n)-(e_1^{n-1},e_2^{n-1}){\parallel }_{\diamond }+{\parallel (r_1^n,r_2^n)-(r_1^{n-1},r_2^{n-1})\parallel }_{\diamond },\hfill \end{array}\end{array}$$

(33)

where for each $n\in \mathbb{N}$, $k_1^n=a_1^n+b_2^n$ and $k_2^n=a_2^n+b_1^n$. Letting $\vartheta (n)=1-\mu (1-max\{k_1^n,k_2^n\})$, for each $n\in \mathbb{N}$, we infer that $\vartheta (n)\to \vartheta$ as $n\to \infty$, where $\vartheta =1-\mu (1-max\{k_1,k_2\})$. By (10), we know that $max\{k_1,k_2\}\in (0,1)$, and so $\vartheta \in (0,1)$. Therefore, there exist $\widehat{\vartheta }\in (0,1)$ (take $\widehat{\vartheta }=\frac{\vartheta +1}{2}\in (\vartheta ,1))$ and $n_0\in \mathbb{N}$ such that $\vartheta (n)\le \widehat{\vartheta }$, for all $n\ge n_0$. Employing (33), for all $n>n_0$, we deduce that

$$\begin{array}{c}\hfill \begin{array}{cc}& \parallel (x_1^{n+1},x_2^{n+1})-(x_1^n,x_2^n){\parallel }_{\diamond }\hfill \\ & \le \widehat{\vartheta }\parallel (x_1^n,x_2^n)-(x_1^{n-1},x_2^{n-1}){\parallel }_{\diamond }+\mu {\parallel (e_1^n,e_2^n)-(e_1^{n-1},e_2^{n-1})\parallel }_{\diamond }\hfill \\ & +\parallel (r_1^n,r_2^n)-(r_1^{n-1},r_2^{n-1}){\parallel }_{\diamond }\hfill \\ & \le \widehat{\vartheta }[\widehat{\vartheta }\parallel (x_1^{n-1},x_2^{n-1})-(x_1^{n-2},x_2^{n-2}){\parallel }_{\diamond }+\mu {\parallel (e_1^{n-1},e_2^{n-1})-(e_1^{n-2},e_2^{n-2})\parallel }_{\diamond }\hfill \\ & +\parallel (r_1^{n-1},r_2^{n-1})-(r_1^{n-2},r_2^{n-2}){\parallel }_{\diamond }]+\mu {\parallel (e_1^n,e_2^n)-(e_1^{n-1},e_2^{n-1})\parallel }_{\diamond }\hfill \\ & +\parallel (r_1^n,r_2^n)-(r_1^{n-1},r_2^{n-1}){\parallel }_{\diamond }\hfill \\ & ={\widehat{\vartheta }}^2\parallel (x_1^{n-1},x_2^{n-1})-(x_1^{n-2},x_2^{n-2}){\parallel }_{\diamond }+\mu (\widehat{\vartheta }{\parallel (e_1^{n-1},e_2^{n-1})-(e_1^{n-2},e_2^{n-2})\parallel }_{\diamond }\hfill \\ & +\parallel (e_1^n,e_2^n)-(e_1^{n-1},e_2^{n-1}){\parallel }_{\diamond })+\widehat{\vartheta }{\parallel (r_1^{n-1},r_2^{n-1})-(r_1^{n-2},r_2^{n-2})\parallel }_{\diamond }\hfill \\ & +\parallel (r_1^n,r_2^n)-(r_1^{n-1},r_2^{n-1}){\parallel }_{\diamond }\hfill \\ & \le \cdots \hfill \\ & \le {\widehat{\vartheta }}^{n-n_0}{\parallel (x_1^{n_0+1},x_2^{n_0+1})-(x_1^{n_0},x_2^{n_0})\parallel }_{\diamond }\hfill \\ & +\mu \sum _{q=1}^{n-n_0}{\widehat{\vartheta }}^{q-1}{\parallel (e_1^{n-(q-1)},e_2^{n-(q-1)})-(e_1^{n-q},e_2^{n-q})\parallel }_{\diamond }\hfill \\ & +\sum _{q=1}^{n-n_0}{\widehat{\vartheta }}^{q-1}{\parallel (r_1^{n-(q-1)},r_2^{n-(q-1)})-(r_1^{n-q},r_2^{n-q})\parallel }_{\diamond }.\hfill \end{array}\end{array}$$

(34)

Since $\widehat{\vartheta }\in (0,1)$, relying on (31) and (35), we conclude that $\parallel (x_1^{n+1},x_2^{n+1})-(x_1^n,x_2^n){\parallel }_{\diamond }\to 0$, as $n\to \infty$, and so for each $i\in \{1,2\}$, $\left\{x_i^n\right\}$ is a Cauchy sequence in $E_i$. In view of the completeness of $E_i$, we infer that for each $i\in \{1,2\}$ there exists $x_i\in E_i$ such that $x_i^n\to x_i$ as $n\to \infty$. Since for each $i\in \{1,2\}$, $S_i$ is $L_{S_i}$-$H_{*i}$-Lipschitz continuous, it follows from (30) that for each $i\in \{1,2\}$ and $n\ge 0$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel {\pi }_i^{n+1}-{\pi }_i^n{\parallel }_{*i}& \le (1+{(1+n)}^{-1})H_{*i}(S_i(x_i^{n+1}),S_i(x_i^n))\hfill \\ & \le (1+{(1+n)}^{-1})L_{S_i}{\parallel x_i^{n+1}-x_i^n\parallel }_i.\hfill \end{array}\end{array}$$

The preceding inequality guarantees that for each $i\in \{1,2\}$, $\left\{{\pi }_i^n\right\}$ is also a Cauchy sequence in $E_i^{*}$ and so for each $i\in \{1,2\}$ there exists ${\pi }_i\in E_i^{*}$ such that ${\pi }_i^n\to {\pi }_i$ as $n\to \infty$. Using (30) and $L_{S_i}$-$H_{*i}$-Lipshitz continuity of $S_i$ for each $i=1,2$, we conclude that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d_{*i}({\pi }_i,S_i(x_i))& =inf\{\parallel {\pi }_i-\pi {\parallel }_{*i}:\pi \in S_i(x_i)\}\hfill \\ & \le \parallel {\pi }_i-{\pi }_i^n{\parallel }_{*i}+d_{*i}({\pi }_i^n,S_i(x_i))\hfill \\ & \le \parallel {\pi }_i-{\pi }_i^n{\parallel }_{*i}+H_{*i}(S_i(x_i^n),S_i(x_i))\hfill \\ & \le \parallel {\pi }_i-{\pi }_i^n{\parallel }_{*i}+L_{S_i}{\parallel x_i^n-x_i\parallel }_i.\hfill \end{array}\end{array}$$

The right-hand side of the above inequality approaches zero, as $n\to \infty$. Taking into consideration the fact that for each $i\in \{1,2\}$, $S_i(x_i)$ is close, we deduce that ${\pi }_i\in S_i(x_i)$. By arguments analogous to the previous one, for each $i\in \{1,2\}$, ${\left\{v_i^n\right\}}_{n=0}^{\infty }$, ${\left\{w_i^n\right\}}_{n=0}^{\infty }$, ${\left\{t_i^n\right\}}_{n=0}^{\infty }$ and ${\left\{{\iota }_i^n\right\}}_{n=0}^{\infty }$, ${\left\{u_i^n\right\}}_{n=0}^{\infty }$, ${\left\{z_i^n\right\}}_{n=0}^{\infty }$ are also Cauchy sequences in $E_i^{*}$ and $E_j^{*}$ $(j\in \{1,2\}\backslash \{i\})$, respectively, and for each $i\in \{1,2\}$, $j\in \{1,2\}\backslash \left\{i\right\}$, there exist ${\iota }_i\in T_i(x_j)$, $u_i\in C_i(x_j)$, $v_i\in D_i(x_i)$, $w_i\in F_i(x_i)$, $z_i\in G_i(x_j)$ and $t_i\in K_i(x_i)$ such that ${\iota }_i^n\to {\iota }_i$, $u_i^n\to u_i$, $v_i^n\to v_i$, $w_i^n\to w_i$, $z_i^n\to z_i$ and $t_i^n\to t_i$, as $n\to \infty$. In a similar manner, using (9), Theorem 2 and the assumptions, for each $i\in \{1,2\}$ and $n\ge 0$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& \parallel R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i^n),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)(x_i^n)-{\lambda }_i(W_i(f_i(x_i^n),{\pi }_i^n,{\iota }_i^n)+{\rho }_i(P_i(u_i^n,v_i^n)+Q_i(w_i^n,z_i^n))]\hfill \\ & -R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i),{\widehat{P}}_i}{[({\widehat{P}}_i\circ g_i)(x_i)-{\lambda }_i(W_i(f_i(x_i),{\pi }_i,{\iota }_i)+{\rho }_i(P_i(u_i,v_i)+Q_i(w_i,z_i))]\parallel }_i\hfill \\ & \le \parallel R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i^n),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)(x_i^n)-{\lambda }_i(W_i(f_i(x_i^n),{\pi }_i^n,{\iota }_i^n)+{\rho }_i(P_i(u_i^n,v_i^n)+Q_i(w_i^n,z_i^n))]\hfill \\ & -R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i),{\widehat{P}}_i}{[({\widehat{P}}_i\circ g_i)(x_i^n)-{\lambda }_i(W_i(f_i(x_i^n),{\pi }_i^n,{\iota }_i^n)+{\rho }_i(P_i(u_i^n,v_i^n)+Q_i(w_i^n,z_i^n))]\parallel }_i\hfill \\ & +\parallel R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)(x_i^n)-{\lambda }_i(W_i(f_i(x_i^n),{\pi }_i^n,{\iota }_i^n)+{\rho }_i(P_i(u_i^n,v_i^n)+Q_i(w_i^n,z_i^n))]\hfill \\ & -R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i),{\widehat{P}}_i}{[({\widehat{P}}_i\circ g_i)(x_i)-{\lambda }_i(W_i(f_i(x_i),{\pi }_i,{\iota }_i)+{\rho }_i(P_i(u_i,v_i)+Q_i(w_i,z_i))]\parallel }_i\hfill \\ & \le L_{K_i}\parallel t_i^n-t_i{\parallel }_i+\frac{{\tau }_i}{{\gamma }_i}(\sqrt{L_{{\widehat{P}}_i}^2{L}_{g_i}^2-2{\lambda }_i{\varpi }_i+64c_i{\lambda }_i{\varsigma }_i^2{L}_{f_i}^2}+{\lambda }_i({\delta }_i{\parallel {\pi }_i^n-{\pi }_i\parallel }_i\hfill \\ & +{\rho }_i(L_{(P_i,i)}\parallel v_i^n-v_i{\parallel }_i+L_{(Q_i,i)}\parallel w_i^n-w_i{\parallel }_i)))\parallel x_i^n-x_i{\parallel }_i+\frac{{\tau }_i{\lambda }_i}{{\gamma }_i}({\nu }_i{\parallel {\iota }_i^n-{\iota }_i\parallel }_i\hfill \\ & +{\rho }_i(L_{(P_i,j)}\parallel u_i^n-u_i{\parallel }_i+L_{(Q_i,j)}\parallel z_i^n-z_i{\parallel }_i))\parallel x_j^n-x_j{\parallel }_j.\hfill \end{array}\end{array}$$

(35)

where $j\in \{1,2\}\backslash \left\{i\right\}$. Owing to the facts that for each $i\in \{1,2\}$, $x_i^n\to x_i$, $u_i^n\to u_i$, $v_i^n\to v_i$, $w_i^n\to w_i$, ${\pi }_i^n\to {\pi }_i$, ${\iota }_i^n\to {\iota }_i$, $z_i^n\to z_i$ and $t_i^n\to t_i$, as $n\to \infty$, it follows that the right-hand side of (35) tends to zero, as $n\to \infty$, and so its left-hand side approaches zero, as $n\to \infty$. Passing to the limit in the iterative scheme (30), it follows that for each $i\in \{1,2\}$,

$$\begin{array}{c}\hfill g_i(x_i)=R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i(.,t_i),{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)(x_i)-{\lambda }_i(W_i(f_i(x_i),{\pi }_i,{\iota }_i)+{\rho }_i(P_i(u_i,v_i)+Q_i(w_i,z_i))].\end{array}$$

Now, Lemma 2 ensures that $(x_1,x_2,{\pi }_1,{\pi }_2,{\iota }_1,{\iota }_2,u_1,u_2,v_1,v_2,w_1,w_2,z_1,z_2,t_1,t_2)$ is a solution of the $\mathrm{SGMVLI}$ (5). The proof is completed.    □

We now employ the equivalence formulation presented in Lemma 3.2 and Nadler technique [39], and construct the following iterative algorithm as a special case of Algorithm 1 for finding the approximate solution of the $\mathrm{SGIVLI}$ (6).

As an immediate consequence of Theorem 5, we obtain the following corollary in which the strong convergence of the sequences generated by Algorithm 2 to a solution of the $\mathrm{SGIVLI}$ (6) is proved.

Algorithm 2 Iterative algorithm for finding the approximate solution of the SGIVLI (5)

Suppose that $E_i,E_i^{*},C_i,D_i,F_i,N_i,P_i,Q_i,S_i,T_i,{\widehat{P}}_i,{\eta }_i,{\varphi }_i,g_i,h_i$ $(i=1,2)$ are the same as in Lemma 3. For any given $(x_1^0,x_2^0,u_1^0,u_2^0,v_1^0,v_2^0,w_1^0,w_2^0,z_1^0,z_2^0)$, where for each $i\in \{1,2\}$, $u_i^0\in C_i(x_j^0)$, $v_i^0\in D_i(x_i^0)$, $w_i^0\in F_i(x_i^0)$, $z_i^0\in G_i(x_j^0)$ $(j\in \{1,2\}\backslash \{i\})$, compute the sequences ${\left\{x_i^n\right\}}_{n=0}^{\infty }$, ${\left\{u_i^n\right\}}_{n=0}^{\infty }$, ${\left\{v_i^n\right\}}_{n=0}^{\infty }$, ${\left\{w_i^n\right\}}_{n=0}^{\infty }$ and ${\left\{z_i^n\right\}}_{n=0}^{\infty }$ by the iterative schemes $$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{x}_i^{n+1}=(1-\mu )x_i^n+\mu \{x_i^n-g_i(x_i^n)+R_{{\lambda }_i,{\eta }_i}^{\partial {{\Phi }}_i,{\widehat{P}}_i}[({\widehat{P}}_i\circ g_i)(x_i^n)\hfill \\ -{\lambda }_i(N_i(S_i(x_i^n),T_i(x_j^n))+{\rho }_i(P_i(u_i^n,v_i^n)+Q_i(w_i^n,z_i^n))]\},\hfill \\ u_i^n\in C_i(x_j^n):{\parallel u_i^{n+1}-u_i^n\parallel }_{*j}\le (1+{(1+n)}^{-1})H_{*j}(C_i(x_j^{n+1}),C_i(x_j^n)),\hfill \\ v_i^n\in D_i(x_i^n):{\parallel v_i^{n+1}-v_i^n\parallel }_{*i}\le (1+{(1+n)}^{-1})H_{*i}(D_i(x_i^{n+1}),D_i(x_i^n)),\hfill \\ w_i^n\in F_i(x_i^n):{\parallel w_i^{n+1}-w_i^n\parallel }_{*i}\le (1+{(1+n)}^{-1})H_{*i}(F_i(x_i^{n+1}),F_i(x_i^n)),\hfill \\ z_i^n\in G_i(x_j^n):{\parallel z_i^{n+1}-z_i^n\parallel }_{*j}\le (1+{(1+n)}^{-1})H_{*j}(G_i(x_j^{n+1}),G_i(x_j^n)),\hfill \end{array}\right.\end{array}\end{array}$$ (36) where $n=0,1,2,\cdots$; ${\lambda }_i>0$ $(i=1,2)$ are constants, $\mu \in (0,1]$ is a relaxation parameter, $H_i$ and $H_{*i}$ are Hausdorff metric on $CB(E_i)$ and $CB(E_i^{*})$, respectively.

Corollary 3.

Suppose that $E_i,E_i^{*},P_i,Q_i,{\widehat{P}}_i,N_i,S_i,T_i,{\eta }_i,{\varphi }_i,g_i,h_i$ $(i=1,2)$ are the same as in Corollary 2. Let further that for each $i\in \{1,2\}$, $D_i,F_i:E_i\to CB(E_i^{*})$ be $L_{D_i}$-$H_{*i}$-Lipschitz and $L_{F_i}$-$H_{*i}$-Lipschitz continuous, respectively, and $C_i,G_i:E_j\to CB(E_j^{*})$ $(j\in \{1,2\}\backslash \{i\})$ be $L_{C_i}$-$H_{*j}$-Lipschitz and $L_{G_i}$-$H_{*j}$-Lipschitz continuous, respectively. If there exist constants ${\lambda }_i>0$ $(i=1,2)$ satisfying (29), then the iterative sequences ${\left\{x_i^n\right\}}_{n=0}^{\infty }$, ${\left\{u_i^n\right\}}_{n=0}^{\infty }$, ${\left\{v_i^n\right\}}_{n=0}^{\infty }$, ${\left\{w_i^n\right\}}_{n=0}^{\infty }$ and ${\left\{z_i^n\right\}}_{n=0}^{\infty }$ $(i=1,2)$ generated by Algorithm 2 converge strongly to $x_i$, $u_i$, $v_i$, $w_i$ and $z_i$, respectively, and $(x_1,x_2,u_1,u_2,v_1,v_2,w_1,w_2,z_1,z_2)$ is a solution of the $\mathrm{SGIVLI}$ (6).

5. Some Remarks on $\mathit{M}$-$\mathbf{\eta }$-Proximal Mapping

This section is concerned with the investigation and analysis of the concept M-$\eta$-proximal mapping defined in [30] and pointing out some comments concerning it and the related results appeared in [30]. Some fatal errors in the main results of [30] are detected and their correct versions are given. We also point out that the modified versions of the corresponding results presented in [30] can be proved using the results of the previous sections of this paper.

Definition 11 ([30], Definition 3.1).

Let $\eta :E\times E\to E$, $A,B:E\to E$ be single-valued mappings and let $M:E\times E\to E^{*}$ be a nonlinear mapping. Then

(i)

$M(A,.)$ is said to be α-strongly η-monotone with respect to A if there exists a constant $\alpha >0$ satisfying

$$\begin{array}{c}\hfill \langle M(Ay,u)-M(Ax,u),\eta (y,x)\rangle \ge {\alpha \parallel x-y\parallel }^2,\forall x,y,u\in E;\end{array}$$

(ii)

$M(.,B)$ is said to be β-relaxed η-monotone with respect to B if there exists a constant $\beta >0$ satisfying

$$\begin{array}{c}\hfill \langle M(u,By)-M(u,Bx),\eta (y,x)\rangle \ge {-\beta \parallel x-y\parallel }^2,\forall x,y,u\in E;\end{array}$$

(iii)

$M(.,.)$ is said to be $\alpha \beta$-symmetric η-monotone with respect to A and B if $M(A,.)$ is α-strongly η-monotone with respect to A and $M(.,B)$ is β-relaxed η-monotone with respect to B with $\alpha >\beta$ and $\alpha =\beta$ if and only if $x=y$, $\forall x,y\in E$.

Remark 3.

Comparing ([30] Definition 3.1) with the proofs of two Theorems 3.1 and 3.2 in [30] (see lines 7 and 8 from the bottom on page 9211 and line 14 from the bottom on page 9212 in [30]), we found that there are two small mistakes in parts (i) and (ii) of Definition 3.1 of [30]. In fact, in parts (i) and (ii) of ([30], Definition 3.1), $M(Ax,u)-M(Ay,u)$ and $M(u,Bx)-M(u,By)$ must be replaced by $M(Ay,u)-M(Ax,u)$ and $M(u,By)-M(u,Bx)$, respectively, as we have done in Definition 11. At the same time, invoking parts (i) and (ii) of Definition 3.1 in [30], there also is a small mistake in Definition 3.1(iii) of [30]. Indeed, in ([30] Definition 3.1(iii)), $\alpha$-strongly monotone with respect to A and $\beta$-relaxed monotone with respect to B must be replaced by $\alpha$-strongly $\eta$-monotone with respect to A and $\beta$-relaxed $\eta$-monotone with respect to B, respectively, as we have done in Definition 11(iii).

The next proposition illustrates that every symmetric $\eta$-monotone mapping is actually strongly $\eta$-monotone.

Proposition 1.

Let $A,B:E\to E$, $\eta :E\times E\to E$ and $M:E\times E\to E^{*}$ be vector-valued mappings and suppose further that the mapping $P:E\to E^{*}$ is defined by $P(x):=M(Ax,Bx)$, for all $x\in E$. If $M(.,.)$ is $\alpha \beta$-symmetric η-monotone with respect to A and B, then P is $(\alpha -\beta )$-strongly monotone.

Proof. 

Since $M(.,.)$ is $\alpha \beta$-symmetric $\eta$-monotone with respect to A and B, it follows that for all $x,y\in E$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \langle P(y)-P(x),\eta (y,x)\rangle & =\langle M(Ay,By)-M(Ax,Bx),\eta (y,x)\rangle \hfill \\ & =\langle M(Ay,By)-M(Ax,By),\eta (y,x)\rangle \hfill \\ & +\langle M(Ax,By)-M(Ax,Bx),\eta (y,x)\rangle \hfill \\ & \ge {\alpha \parallel x-y\parallel }^2-\beta {\parallel x-y\parallel }^2\hfill \\ & ={(\alpha -\beta )\parallel x-y\parallel }^2.\hfill \end{array}\end{array}$$

Taking into consideration the fact that $\alpha \ge \beta$, and $\alpha =\beta$ if and only if $x=y$, for all $x,y\in E$, the preceding inequality implies that P is an $(\alpha -\beta )$-strongly $\eta$-monotone mapping. This gives the desired result.    □

Remark 4.

In view of the above-mentioned argument, it should be noticed that the two classes of symmetric η-monotone and strongly η-monotone mappings are the same. In fact, Definition 11(iii) is actually the same definition of $r=(\alpha -\beta )$-strong $\eta$-monotonicity of the mapping $P:=M(A,B)$ presented in Definition 3(i) and is not a new one.

Kazmi et al. [30] defined the notion of M-$\eta$-proximal mapping associated with a proper and $\eta$-subdifferentiable (not necessarily convex) functional as follows.

Definition 12 ([30], Definition 3.2).

Let $\eta :E\times E\to E$ and $A,B:E\to E$ be vector-valued mappings. Let $\varphi :E\to ℝ\cup \{+\infty \}$ be a proper and subdifferentiable (may not be convex) functional and $M:E\times E\to E^{*}$ be a nonlinear mapping. If for any given point $x^{*}\in E^{*}$ and $\rho >0$, there exists a unique point $x\in E$ satisfying

$$\begin{array}{c}\hfill \langle M(Ax,Bx)-x^{*},\eta (y,x)\rangle +\rho \varphi (y)-\rho \varphi (x)\ge 0,\forall y\in E,\end{array}$$

then the mapping $x^{*}\to x$, denoted by $R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}(x^{*})$, is called M-η-proximal mapping of ϕ. Clearly, we have $x^{*}-M(Ax,Bx)\in \rho \partial \varphi (x)$ and then it follows that

$$\begin{array}{c}\hfill R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}(x^{*})={(M(A,B)+\rho \partial \varphi )}^{-1}(x^{*}).\end{array}$$

Remark 5.

(i) It should be pointed out that the M-η-proximal mapping associated with a proper and η-subdifferentiable functional ϕ is denoted by $R_{\rho ,\eta }^{\partial \varphi }$ in [30]. For the sake of convenience for our discussion, throughout this section, we use the notation $R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}$ instead of $R_{\rho ,\eta }^{\partial \varphi }$, as it is used in Definition 12.

(ii) Defining $P:E\to E^{*}$ by $P(x):=M(Ax,Bx)$, for all $x\in E$, and taking $\lambda =\rho$, we observe that Definition 12 becomes actually the same Definition 5 and is not a new one.

The well definedness of the M-$\eta$-proximal mapping associated with a lower semicontinuous and subdifferentiable proper functional $\varphi :E\to ℝ\cup \{+\infty \}$ is proved in [30] under some appropriate conditions as follows.

Theorem 6 ([30], Theorem 3.1).

Let E be a reflexive Banach space. Let $\eta :E\times E\to E$ be a continuous mapping such that $\eta (y,y^{\prime })+\eta (y^{\prime },y)=0$ for all $y,y^{\prime }\in E$; let $M:E\times E\to E^{*}$ be $\alpha \beta$-symmetric η-monotone continuous with respect to A and B; let for any given $x^{*}\in E^{*}$, the function $h(y,x)=\langle x^{*}-M(Ax,Bx),\eta (y,x)\rangle$ be $\mathrm{0-DQCV}$ in y and let $\varphi :E\to ℝ\cup \{+\infty \}$ be a proper, lower semicontinuous and η-subdifferentiable functional which may not be convex. Then, for any given constant $\rho >0$ and $x^{*}\in E^{*}$, there exists a unique $x\in E$ such that

$$\begin{array}{c}\hfill \langle M(Ax,Bx)-x^{*},\eta (y,x)\rangle \ge \rho \varphi (x)-\rho \varphi (y),\forall y\in E,\end{array}$$

that is, $x=R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}(x^{*})$.

Proof. 

Let the mapping $P:E\to E^{*}$ be defined by $P(x):=M(Ax,Bx)$ for all $x\in E$. Since M is an $\alpha \beta$-symmetric $\eta$-monotone mapping, in virtue of Proposition 1, P is $(\alpha -\beta )$-strongly $\eta$-monotone. Taking $\gamma =\alpha -\beta$, all the conditions of Theorem 1 are satisfied. Invoking Theorem 1, for any constant $\rho >0$ and $x^{*}\in E^{*}$, there exists a unique $x\in E$ such that

$$\begin{array}{c}\hfill \langle P(x)-x^{*},\eta (y,x)\rangle =\langle M(Ax,Bx)-x^{*},\eta (y,x)\rangle \ge \rho \varphi (x)-\rho \varphi (y),\forall y\in E,\end{array}$$

that is, $x=R_{\rho ,\eta }^{\partial \varphi ,P}(x^{*})=R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}(x^{*})$ and so $P=M$-$\eta$-proximal mapping of $\varphi$ is well defined.    □

By a careful reading of the proof of Theorem 3.1 in [30], we found that there is a small mistake on page 9211, line 12 from the bottom. The authors used proof by contraction and proved that the functional $f:E\times E\to ℝ\cup \{+\infty \}$ defined by

$$\begin{array}{c}\hfill f(y,x)=\langle x^{*}-M(Ax,Bx),\eta (y,x)\rangle +\rho \varphi (x)-\rho \varphi (y),\forall x,y\in E,\end{array}$$

for any given $M:E\times E\to E^{*}$, $\rho >0$, $x^{*}\in E^{*}$ and $\varphi :E\to ℝ\cup \{+\infty \}$, satisfies Lemma 1(ii). However, there is an error in the process of achieving a contradiction. Indeed, on the contrary, they assumed that there exists a finite set $\{y_1,y_2,\cdots ,y_m\}\subset E$ and $x_0={\sum }_{i=1}^m{\lambda }_i{y}_i$ with ${\lambda }_i\ge 0$ and ${\sum }_{i=1}^m{\lambda }_i=1$ such that

$$\begin{array}{c}\hfill \langle x^{*}-M(A{x}_0,B{x}_0),\eta (y_i,x_0)\rangle +\rho \varphi (x_0)-\rho \varphi (y_i)>0,\forall i=1,2,\cdots ,m.\end{array}$$

(37)

Using the property of $\eta$-subdifferentiability of $\varphi$ at $x_0\in E$, they deduced the existence of a point $f^{*}\in E^{*}$ satisfying the following inequality:

$$\begin{array}{c}\hfill \rho \varphi (y_i)-\rho \varphi (x_0)\ge \rho \langle f^{*},\eta (y_i,x_0)\rangle ,\forall i=1,2,\cdots ,m.\end{array}$$

(38)

Then, employing (37) and (38), they concluded that

$$\begin{array}{c}\hfill \langle x^{*}-M(A{x}_0,B{x}_0)-\rho f^{*},\eta (y_i,x_0)\rangle >0,\forall i=1,2,\cdots ,m.\end{array}$$

(39)

Finally, making use of (39) and taking into account that the functional

$$\begin{array}{c}\hfill h:(y,x_0)\in E\times E\to h(y,x_0)=\langle x^{*}-M(A{x}_0,B{x}_0)-\rho f^{*},\eta (y,x_0)\rangle \in ℝ\cup \{+\infty \}\end{array}$$

is $\mathrm{0-DQCV}$ in y (in the first argument) and $\eta (x_0,x_0)=0$, they claimed that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 0& <\sum _{i=1}^m{\lambda }_i\langle x^{*}-M(A{x}_0,B{x}_0)-\rho f^{*},\eta (y_i,x_0)\rangle \hfill \\ & =\langle x^{*}-M(A{x}_0,B{x}_0)-\rho f^{*},\eta (x_0,x_0)\rangle =0,\hfill \end{array}\end{array}$$

(40)

which is a contradiction. In view of (40), they deduced the required contradiction in light of the fact that ${\displaystyle \sum _{i=1}^m}\eta (y_i,x_0)=\eta ({\displaystyle \sum _{i=1}^m}{y}_i,x_0)=\eta (x_0,x_0)=0$. But, thanks to the assumptions mentioned in the context of Theorem 3.1 of [30], ${\displaystyle \sum _{i=1}^m}\eta (y_i,x_0)=\eta ({\displaystyle \sum _{i=1}^m}{y}_i,x_0)$ does not hold necessarily. We now resolve this problem and present the correct proof. Since the functional h is $\mathrm{0-DQCV}$ in the first argument, we have

$$\begin{array}{c}\hfill \underset{1\le i\le m}{min}\langle x^{*}-M(A{x}_0,B{x}_0)-\rho f^{*},\eta (y_i,x_0)\rangle \le 0,\end{array}$$

(41)

which contradicts (39). Considering the above-mentioned argument, in order to achieve a contradiction in the proof of Theorem 3.1 of [30], line 12 from the bottom on page 9211, (40) must be replaced by (41).

Under some appropriate conditions, Kazmi et al. [30] verified the Lipschitz continuity of the M-$\eta$-proximal mapping $R_{\rho ,\eta }^{\partial \varphi ,M}:E^{*}\to E$ associated with a proper, lower semicontinuous $\eta$-subdifferentiable functional $\varphi$ and any constant $\rho >0$ as follows.

Theorem 7 ([30], Theorem 3.2).

Let $\eta :E\times E\to E$ be τ-Lipschitz continuous such that $\eta (y,y^{\prime })+\eta (y^{\prime },y)=0$ for all $y,y^{\prime }\in E$; let $M:E\times E\to E^{*}$ be $\alpha \beta$-symmetric η-monotone continuous with respect to A and B; let for any given $x^{*}\in E^{*}$, the function $h(y,x)=\langle x^{*}-M(Ax,Bx),\eta (y,x)\rangle$ be $\mathrm{0-DQCV}$ in y; let $\varphi :E\to ℝ\cup \{+\infty \}$ be a proper, lower semicontinuous and η-subdifferentiable functional and let $\rho >0$ be any given constant. Then, the M-η-proximal mapping $R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}$ of ϕ is $\frac{\tau }{\alpha -\beta }$-Lipschitz continuous, i.e., for any $x_1^{*},x_2^{*}\in E^{*}$,

$$\begin{array}{c}\hfill \parallel R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}(x_1^{*})-R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}(x_2^{*})\parallel \le \frac{\tau }{\alpha -\beta }\parallel x_1^{*}-x_2^{*}\parallel .\end{array}$$

Proof. 

Consider the mapping $P:E\to E^{*}$ defined by $P(x):=M(Ax,Bx)$, for all $x\in E$. Relying on the fact that M is $\alpha \beta$-symmetric $\eta$-monotone, by means of Proposition 1, it follows that P is $(\alpha -\beta )$-strongly $\eta$-monotone. Picking $\gamma =\alpha -\beta$, we note that all the conditions of Theorem 2 are satisfied. Letting $\lambda =\rho$ and invoking Theorem 2, the $P=M$-$\eta$-proximal mapping $R_{\rho ,\eta }^{\partial \varphi ,P}=R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}:E^{*}\to E$ associated with $\varphi ,M,\rho$ and $\eta$ is $\frac{\tau }{\gamma }=\frac{\tau }{\alpha -\beta }$-Lipschitz continuous, i.e.,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel R_{\rho ,\eta }^{\partial \varphi ,P}(x_1^{*})-R_{\rho ,\eta }^{\partial \varphi ,P}(x_2^{*})\parallel & =\parallel R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}(x_1^{*})-R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}(x_2^{*})\parallel \hfill \\ & \le \frac{\tau }{\gamma }\parallel x_1^{*}-x_2^{*}\parallel \hfill \\ & =\frac{\tau }{\alpha -\beta }\parallel x_1^{*}-x_2^{*}\parallel ,\forall x_1^{*},x_2^{*}\in E^{*}.\hfill \end{array}\end{array}$$

This gives the desired result.    □

For each $i\in \{1,2\}$, let $E_i,g_i,{\eta }_i,N_i,Q_i,P_i,S_i,T_i,C_i,G_i,D_i,F_i,{\varphi }_i$ and ${\rho }_i>0$ be the same as in the $\mathrm{SGIVLI}$ (6) and let $M_i:E_i\times E_i\to E_i^{*}$ and $A_i,B_i:E_i\to E_i$ be single-valued mappings. With the goal of concluding main results in [30], the authors presented Assumption 4.1 in [30] in which for each $i\in \{1,2\}$, the single-valued mappings $M_i,A_i,B_i$ and $g_i$ satisfy the following two conditions:

$$\begin{array}{c}\hfill \mathrm{dom}(M_i(A_i,B_i))\cap g_i(E_i)=\varnothing \end{array}$$

(42)

and

$$\begin{array}{c}\hfill \mathrm{dom}(M_i(A_i,B_i))\cap R_{{\rho }_i,{\eta }_i}^{\partial {\varphi }_i,M_i}(E_i^{*})=\varnothing .\end{array}$$

(43)

By careful checking, we found that (42) must be replaced by

$$\begin{array}{c}\hfill \mathrm{dom}(M_i(A_i,B_i))\cap g_i(E_i)\ne \varnothing \end{array}$$

and (44) is extra and must be removed.

By the argument similar to that of Lemma 3.1 in [43], and using the concept of M-$\eta$-proximal mapping, Kazmi et al. [30] provided a characterization of the solution of the $\mathrm{SGIVLI}$ (6) as follows.

Lemma 8 ([30], Lemma 4.1).

For each $i\in \{1,2\}$, let $E_i$ be a reflexive Banach space; let ${\eta }_i:E_i\times E_i\to E_i$ be a continuous mapping such that ${\eta }_i(y_i,y_i^{\prime })+{\eta }_i(y_i^{\prime },y_i)=0$, for all $y_i,y_i\in E_i$; let $A_i,B_i:E_i\to E_i$ be nonlinear mappings; let the mappings $M_i:E_i\times E_i\to E_i^{*}$ be ${\alpha }_i{\beta }_i$-symmetric ${\eta }_i$-monotone continuous with respect to $A_i$ and $B_i$; let for any given $x_i^{*}\in E_i^{*}$, the function $h_i(y_i,{\widehat{x}}_i)=\langle x_i^{*}-M_i(A_i{\widehat{x}}_i,B_i{\widehat{x}}_i),{\eta }_i(y_i,{\widehat{x}}_i)\rangle$ be $\mathrm{0-DQCV}$ in $y_i$ and let ${\varphi }_i:E_i\to ℝ\cup \{+\infty \}$ be a proper, lower semicontinuous and ${\eta }_i$-subdifferentiable functional. Then, the following are equivalent:

(i)

$(x_1,x_2,u_1,u_2,v_1,v_2,w_1,w_2,z_1,z_2)$ where $x_i\in E_i$, $u_i\in C_i(x_j)$, $v_i\in D_i(x_i)$, $w_i\in F_i(x_i)$, $z_i\in G_i(x_j)$ $(i\in \{1,2\}$, $j\in \{1,2\}\backslash \left\{i\right\})$ is a solution of the $\mathrm{SGIVLI}$ (6);

(ii)

For $i\in \{1,2\}$, $j\in \{1,2\}\backslash \left\{i\right\}$, $(x_1,x_2,u_1,u_2,v_1,v_2,w_1,w_2,z_1,z_2)$ satisfies the relation

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill g_i(x_i)& =R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i,M_i(A_i,B_i)}[(M_i(A_i,B_i)\circ g_i)(x_i)-{\lambda }_i(N_i(S_i(x_i),T_i(x_j))\hfill \\ & +{\rho }_i(P_i(u_i,v_i)+Q_i(w_i,z_i)))],\hfill \end{array}\end{array}$$

where $M_i(A_i,B_i)\circ g_i$ denotes $M_i(A_i,B_i)$ composition $g_i$ and ${\lambda }_i>0$ is a constant;

(iii)

$(x_1,x_2)\in E_1\times E_2$ is a fixed point of the multi-valued mapping $U:E_1\times E_2\to 2^{E_1\times E_2}$ defined as follows:

$$\begin{array}{c}\hfill U(\widehat{x_1},{\widehat{x}}_2)=(V_1(\widehat{x_1},{\widehat{x}}_2),V_2(\widehat{x_1},{\widehat{x}}_2)),\forall (\widehat{x_1},{\widehat{x}}_2)\in E_1\times E_2,\end{array}$$

where, for each $i\in \{1,2\}$, $V_i:E_1\times E_2\to 2^{E_i}$ is defined by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_i({\widehat{x}}_1,{\widehat{x}}_2)& ={\widehat{x}}_i-g_i({\widehat{x}}_i)+\bigcup _{\begin{array}{c}{\widehat{u}}_i\in C_i({\widehat{x}}_j)\\ {\widehat{v}}_i\in D_i({\widehat{x}}_i)\\ {\widehat{w}}_i\in F_i({\widehat{x}}_i)\\ {\widehat{z}}_i\in G_i({\widehat{x}}_j)\end{array}}{R}_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i,M_i(A_i,B_i)}[(M_i(A_i,B_i)\circ g_i)({\widehat{x}}_i)\hfill \\ & -{\lambda }_i(N_i(S_i({\widehat{x}}_i),T_i({\widehat{x}}_j))+{\rho }_i(P_i({\widehat{u}}_i,{\widehat{v}}_i)+Q_i({\widehat{w}}_i,{\widehat{z}}_i))],\hfill \end{array}\end{array}$$

$j\in \{1,2\}\backslash \left\{i\right\}$ and ${\lambda }_i>0$ is a constant.

Proof. 

Let us define for each $i\in \{1,2\}$, the mapping ${\widehat{P}}_i:E_i\to E_i^{*}$ as ${\widehat{P}}_i(x_i):=M_i(A_i{x}_i,B_i{x}_i)$, for all $x_i\in E_i$. From Proposition 1, we conclude that for each $i\in \{1,2\}$, ${\widehat{P}}_i$ is an $({\alpha }_i-{\beta }_i)$-strongly ${\eta }_i$-monotone continuous mapping. Meanwhile, in view of the fact that for each $i\in \{1,2\}$, $\mathrm{dom}(M_i(A_i,B_i))\cap g_i(E_i)\ne \varnothing$, it follows that for each $i\in \{1,2\}$, $\mathrm{dom}({\widehat{P}}_i)\cap g_i(E_i)\ne \varnothing$. Taking ${\gamma }_i={\alpha }_i-{\beta }_i$ for each $i\in \{1,2\}$, we now observe that all the conditions of Lemma 3 are satisfied. Consequently, according to Lemma 3, the assertions (i)–(iii) are equivalent.    □

It should be remarked that there are misprints in the context of Lemma 4.1 in [30]. In fact, in parts (ii) and (iii) of ([30], Lemma 4.1), $R_{{\rho }_i,{\eta }_i}^{\partial {\varphi }_i}$ must be replaced by $R_{{\rho }_i,{\eta }_i}^{\partial {\varphi }_i,M_i(A_i,B_i)}$, as we have done in Lemma 8. Furthermore, in ([30], Lemma 4.1(iii)), $U(x_1,x_2)$, $V_1(x_1,x_2)$ and $V_2(x_1,x_2)$ must be replaced by $U({\widehat{x}}_1,{\widehat{x}}_2)$, $V_1({\widehat{x}}_1,{\widehat{x}}_2)$ and $V_2({\widehat{x}}_1,{\widehat{x}}_2)$, respectively, where for each $i\in \{1,2\}$, $V_i({\widehat{x}}_1,{\widehat{x}}_2)$ is defined as Lemma 8(iii).

Kazmi et al. [30] derived the following conclusion in which under some appropriate conditions, the existence of a solution for the $\mathrm{SGIVLI}$ (6) is proved.

Theorem 8 ([30], Theorem 4.1).

For each $i\in \{1,2\}$, $j\in \{1,2\}\backslash \left\{i\right\}$, let $E_i$ be a uniformly smooth Banach space with ${\rho }_{E_i}(t)\le c_i{t}^2$ for some $c_i>0$; let ${\eta }_i:E_i\times E_i\to E_i$ be a continuous mapping such that ${\eta }_i(y_i,y_i^{\prime })+{\eta }_i(y_i^{\prime },y_i)=0$, for all $y_i,y_i^{\prime }\in E_i$; let $A_i,B_i:E_i\to E_i$ be nonlinear mappings; let the mapping $M_i:E_i\times E\to E_i^{*}$ be ${\alpha }_i{\beta }_i$-symmetric ${\eta }_i$-monotone continuous with respect to $A_i$ and $B_i$; let for any given $x_i^{*}\in E_i^{*}$, the function $h_i(y_i,x_i)={\langle x_i^{*}-M_i(A_i{x}_i,B_i{x}_i),{\eta }_i(y_i,x_i)\rangle }_i$ be $\mathrm{0-DQCV}$ in $y_i$. Let ${\varphi }_i:E_i\to ℝ\cup \{+\infty \}$ be a proper, lower semicontinuous and ${\eta }_i$-subdifferentiable functional; let $N_i,Q_i:E_i^{*}\times E_j^{*}\to E_i^{*}$, $P_i:E_j^{*}\times E_i^{*}\to E_i^{*}$ be $({\delta }_i,{\nu }_i)$, $(L_{(Q_i,i)},L_{(Q_i,j)})$ and $(L_{(P_i,i)},L_{(P_i,j)})$-mixed Lipcshitz continuous, respectively; let $S_i:E_i\to E_i^{*}$, $T_i:E_i\to E_j^{*}$ be $L_{S_i}$- and $L_{T_i}$-Lipschitz continuous, respectively, let, $C_i,G_i:E_j\to C(E_j^{*})$, $D_i,F_i:E_i\to C(E_i^{*})$ be such that $C_i$ is $L_{C_i}$-$H_{*j}$-Lipschitz continuous, $D_i$ is $L_{D_i}$-$H_{*i}$-Lipschitz continuous, $F_i$ is $L_{F_i}$-$H_{*i}$-Lipschitz continuous and $G_i$ is $L_{G_i}$-$H_{*j}$-Lipschitz continuous; let $g_i:E_i\to E_i$ be $s_i$-strongly and $L_{g_i}$-Lipschitz continuous; let $N_i(S_i(.),T_i(x_j))$ be ${\epsilon }_i$-strongly accretive with respect to $M_i(A_i,B_i)\circ g_i$ and let $M_i(A_i,B_i)\circ g_i$ be $L_{M_i}$-Lipschitz continuous. Suppose that there are constants ${\rho }_i>0$ $(i=1,2)$ such that for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$,

$$\begin{array}{c}\hfill \begin{array}{cc}& {(1-2s_i+64c_i{L}_{g_i}^2)}^{\frac{1}{2}}+\frac{{\tau }_i}{{\alpha }_i-{\beta }_i}[{(L_{M_i}^2-2{\lambda }_i{\epsilon }_i+64c_i{\lambda }_i^2{\delta }_i^2)}^{\frac{1}{2}}+{\rho }_i(L_{(P_i,i)}{L}_{D_i}\hfill \\ & +L_{(Q_i,i)}{L}_{F_i})]+\frac{{\tau }_i}{{\alpha }_i-{\beta }_i}[{\lambda }_i{\nu }_i{L}_{T_i}+{\rho }_i(L_{(P_i,j)}{L}_{C_i}+L_{(Q_i,j)}{L}_{G_i})]<1,\hfill \\ \end{array}\end{array}$$

(44)

then the $\mathrm{SGIVLI}$ (6)(that is, ([30], the$\mathrm{SGIVLI}$(4.1)–(4.2))) has a solution.

By a careful reading, we found that there are some errors in the proof of ([30], Theorem 4.1) which must be resolved. In order to provide the correct version of ([30], Theorem 4.1), we now investigate and analyze its proof.

On page 9214 of [30], using the assumptions, the authors obtained the two equalities (4.7) and (4.9) in [30] for each $i\in \{1,2\}$ as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill t_i& =x_i-g_i(x_i)+R_{{\rho }_i,{\eta }_i}^{\partial {\varphi }_i,M_i(A_i,B_i)}[(M_i(A_i,B_i)\circ g_i)(x_i)-{\lambda }_i(N_i(S_i(x_i),T_i(x_j))\hfill \\ & +{\rho }_i(P_i(u_i,v_i)+Q_i(w_i,z_i)))]\hfill \end{array}\end{array}$$

(45)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill t_i^{\prime }& =x_i^{\prime }-g_i(x_i^{\prime })+R_{{\rho }_i,{\eta }_i}^{\partial {\varphi }_i,M_i(A_i,B_i)}[(M_i(A_i,B_i)\circ g_i)(x_i^{\prime })-{\lambda }_i(N_i(S_i(x_i^{\prime }),T_i(x_j^{\prime }))\hfill \\ & +{\rho }_i(P_i(u_i^{\prime },v_i^{\prime })+Q_i(w_i^{\prime },z_i^{\prime })))],\hfill \end{array}\end{array}$$

(46)

where for each $i\in \{1,2\}$, ${\lambda }_i>0$ is a constant.

Utilizing (45) and (46), Theorem 7 and the assumptions, they deduced the inequalities (4.10)–(4.12) in [30] for each $i\in \{1,2\}$ as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel t_i-t_i^{\prime }{\parallel }_i& \le \parallel x_i-x_i^{\prime }-(g_i(x_i)-g_i(x_i^{\prime })){\parallel }_i+\frac{{\tau }_i}{{\alpha }_i-{\beta }_i}[\parallel (M_i(A_i,B_i)\circ g_i)(x_i)\hfill \\ & -(M_i(A_i,B_i)\circ g_i)(x_i^{\prime })-{\lambda }_i(N_i(S_i(x_i),T_i(x_j)))\hfill \\ & -N_i(S_i(x_i^{\prime }),T_i(x_j^{\prime })){\parallel }_{*i}+{\lambda }_i{\parallel N_i(S_i(x_i^{\prime }),T_i(x_j))-N_i(S_i(x_i^{\prime }),T_i(x_j^{\prime }))\parallel }_{*i}\hfill \\ & +{\rho }_i\parallel P_i(u_i,v_i)-P_i(u_i^{\prime },v_i^{\prime }){\parallel }_{*i}+{\rho }_i\parallel Q_i(w_i,z_i)-Q_i(w_i^{\prime },z_i^{\prime }){\parallel }_{*i}],\hfill \end{array}\end{array}$$

(47)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \parallel x_i-x_i^{\prime }-(g_i(x_i)-g_i(x_i^{\prime })){\parallel }_i\le (1-2s_i+64c_i{L}_{g_i}^2){\parallel x_i-x_i^{\prime }\parallel }_i^2\end{array}\end{array}$$

(48)

and

$$\begin{array}{c}\hfill \begin{array}{cc}& \parallel (M_i(A_i,B_i)\circ g_i)(x_i)-(M_i(A_i,B_i)\circ g_i)(x_i^{\prime })-{\lambda }_i(N_i(S_i(x_i),T_i(x_j))\hfill \\ & -N_i(S_i(x_i^{\prime }),T_i(x_j^{\prime })){\parallel }_{*i}^2\le (L_{M_i}^2-2{\lambda }_i{\epsilon }_i+64c_i{\lambda }_i^2{\delta }_i^2){\parallel x_i-x_i^{\prime }\parallel }_i^2.\hfill \end{array}\end{array}$$

(49)

Applying (4.10)–(4.15) in [30], for each $i\in \{1,2\}$, they obtained (4.16) in [30] as an estimate of $\parallel t_i-t_i^{\prime }{\parallel }_i$ as follows:

$$\begin{array}{c}\hfill \parallel t_i-t_i^{\prime }{\parallel }_i\le a_i\parallel x_i-x_i^{\prime }{\parallel }_i+b_i{\parallel x_j-x_j^{\prime }\parallel }_j,\end{array}$$

(50)

where for each $i\in \{1,2\}$,

$$\begin{array}{c}\hfill \begin{array}{cc}& a_i:={(1-2s_i+64c_i{L}_{g_i}^2)}^{\frac{1}{2}}+\frac{{\tau }_i}{{\alpha }_i-{\beta }_i}[{(L_{M_i}^2-2{\lambda }_i{\epsilon }_i+64c_i{\lambda }_i^2{\delta }_i^2)}^{\frac{1}{2}}\hfill \\ & +{\rho }_i(L_{(P_i,i)}{L}_{D_i}+L_{(Q_i,i)}{L}_{F_i})],\hfill \\ & b_i:=\frac{{\tau }_i}{{\alpha }_i-{\beta }_i}[{\lambda }_i{\nu }_i{L}_{T_i}+{\rho }_i(L_{(P_i,j)}{L}_{C_i}+L_{(Q_i,j)}{L}_{G_i})],\hfill \end{array}\end{array}$$

(51)

and $j\in \{1,2\}\backslash \left\{i\right\}$. With the help of the hypotheses of Theorem 8, for any arbitrary elements $(x_1,x_2),(x_1^{\prime },x_2^{\prime })\in E_1\times E_2$, for each $i\in \{1,2\}$, $t_i\in V_i(x_1,x_2)$ and $t_i^{\prime }\in V_i(x_1^{\prime },x_2^{\prime })$, we conclude that there exist $u_i\in C_i(x_j)$, $u_i^{\prime }\in C_i(x_j^{\prime })$, $v_i\in D_i(x_i)$, $v_i^{\prime }\in D_i(x_i^{\prime })$, $w_i\in F_i(x_i)$, $w_i^{\prime }\in F_i(x_i^{\prime })$, $z_i\in G_i(x_j)$ and $z_i^{\prime }\in G_i(x_j^{\prime })$ $(j\in \{1,2\}\backslash \{i\})$, such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill t_i& =x_i-g_i(x_i)+R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i,M_i(A_i,B_i)}[(M_i(A_i,B_i)\circ g_i)(x_i)-{\lambda }_i(N_i(S_i(x_i),T_i(x_j))\hfill \\ & +{\rho }_i(P_i(u_i,v_i)+Q_i(w_i,z_i)))]\hfill \end{array}\end{array}$$

(52)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill t_i^{\prime }& =x_i^{\prime }-g_i(x_i^{\prime })+R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i,M_i(A_i,B_i)}[(M_i(A_i,B_i)\circ g_i)(x_i^{\prime })-{\lambda }_i(N_i(S_i(x_i^{\prime }),T_i(x_j^{\prime }))\hfill \\ & +{\rho }_i(P_i(u_i^{\prime },v_i^{\prime })+Q_i(w_i^{\prime },z_i^{\prime })))],\hfill \end{array}\end{array}$$

(53)

where for each $i\in \{1,2\}$, ${\lambda }_i>0$ is a constant. In fact, in two Equation (45) and (46), $R_{{\rho }_i,{\eta }_i}^{\partial {\varphi }_i,M_i}$ must be replaced by $R_{{\lambda }_i,{\eta }_i}^{\partial {\varphi }_i,M_i}$.

Making use of (52) and (53), Theorem 7 and the assumptions, what we can obtain for each $i\in \{1,2\}$ is the following:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel t_i-t_i^{\prime }{\parallel }_i& \le \parallel x_i-x_i^{\prime }-(g_i(x_i)-g_i(x_i^{\prime })){\parallel }_i+\frac{{\tau }_i}{{\alpha }_i-{\beta }_i}[\parallel (M_i(A_i,B_i)\circ g_i)(x_i)\hfill \\ & -(M_i(A_i,B_i)\circ g_i)(x_i^{\prime })-{\lambda }_i(N_i(S_i(x_i),T_i(x_j))\hfill \\ & -N_i(S_i(x_i^{\prime }),T_i(x_j)){)\parallel }_{*i}+{\lambda }_i(\parallel N_i(S_i(x_i^{\prime }),T_i(x_j))-N_i(S_i(x_i^{\prime }),T_i(x_j^{\prime })){\parallel }_{*i}\hfill \\ & +{\rho }_i\parallel P_i(u_i,v_i)-P_i(u_i^{\prime },v_i^{\prime }){\parallel }_{*i}+{\rho }_i\parallel Q_i(w_i,z_i)-Q_i(w_i^{\prime },z_i^{\prime }){\parallel }_{*i})],\hfill \end{array}\end{array}$$

(54)

not (47). Using $s_i$-strong accretiveness and $L_{g_i}$-Lipschitz continuity of $g_i$ $(i\in \{1,2\})$, what we can obtain as an estimate of $\parallel x_i-x_i^{\prime }-(g_i(x_i)-g_i(x_i^{\prime })){\parallel }_i$ is the following:

$$\begin{array}{c}\hfill \parallel x_i-x_i^{\prime }-(g_i(x_i)-g_i(x_i^{\prime })){\parallel }_i\le \sqrt{1-2s_i+64c_i{L}_{g_i}^2}{\parallel x_i-x_i^{\prime }\parallel }_i^2,\end{array}$$

(55)

not (48). In light of the assumptions of Theorem 8, for each $i\in \{1,2\}$, $g_i$ is $L_{g_i}$-Lipschitz continuous and $M_i$ is continuous. If we replace the continuity condition of $M_i$ by the $L_{M_i(A_i,B_i)}$-Lipschitz continuity condition, then for each $i\in \{1,2\}$, it follows that $M_i(A_i,B_i)\circ g_i$ is $L_{M_i(A_i,B_i)}{L}_{g_i}$-Lipschitz continuous. Consequently, we remove the $L_{M_i}$-Lipschitz continuity condition of the mapping $M_{(}{A}_i,B_i)\circ g_i$ from the context of Theorem 4.1 in [30] and in return replace the continuity condition of $M_i$ by the $L_{M_i(A_i,B_i)}$-Lipschitz continuity condition. Then, in virtue of (54) and taking into account of the assumptions, for each $i\in \{1,2\}$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& \parallel (M_i(A_i,B_i)\circ g_i)(x_i)-(M_i(A_i,B_i)\circ g_i)(x_i^{\prime })\hfill \\ & -{\lambda }_i(N_i(S_i(x_i),T_i(x_j))-N_i(S_i(x_i^{\prime }),T_i(x_j))){\parallel }_{*i}^2\hfill \\ & \le L_{M_i(A_i,B_i)}^2{L}_{g_i}^2\parallel x_i-x_i^{\prime }{\parallel }_i^2-2{\lambda }_i{\epsilon }_i{\parallel x_i-x_i^{\prime }\parallel }_i^2\hfill \\ & +64c_i{\lambda }_i^2{\delta }_i^2{L}_{S_i}^2{\parallel x_i-x_i^{\prime }\parallel }_i^2\hfill \\ & =(L_{M_i(A_i,B_i)}^2{L}_{g_i}^2-2{\lambda }_i{\epsilon }_i+64c_i{\lambda }_i^2{\delta }_i^2{L}_{S_i}^2){\parallel x_i-x_i^{\prime }\parallel }_i^2.\hfill \end{array}\end{array}$$

(56)

Hence, using the assumptions, what we can obtain as an estimate of $\parallel (M_i(A_i,B_i)\circ g_i)(x_i)-(M_i(A_i,B_i)\circ g_i)(x_i^{\prime })-{\lambda }_i(N_i(S_i(x_i),T_i(x_j))-N_i(S_i(x_i^{\prime }),T_i(x_j))){\parallel }_{*i}^2$ is (56) not (49). Finally, utilizing (4.13)–(4.15) in [30] and (54)–(56), what we can obtain as an estimate of $\parallel t_i-t_i^{\prime }{\parallel }_i$ for each $i\in \{1,2\}$ is (50), where for each $i\in \{1,2\}$, $a_i$ and $b_i$ are as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}& a_i:={(1-2s_i+64c_i{L}_{g_i}^2)}^{\frac{1}{2}}+\frac{{\tau }_i}{{\alpha }_i-{\beta }_i}[{(L_{M_i(A_i,B_i)}^2-2{\lambda }_i{\epsilon }_i+64c_i{\lambda }_i^2{\delta }_i^2{L}_{S_i}^2)}^{\frac{1}{2}}\hfill \\ & +{\lambda }_i{\rho }_i(L_{(P_i,i)}{L}_{D_i}+L_{(Q_i,i)}{L}_{F_i})],\hfill \\ & b_i:=\frac{{\tau }_i}{{\alpha }_i-{\beta }_i}[{\lambda }_i({\nu }_i{L}_{T_i}+{\rho }_i(L_{(P_i,j)}{L}_{C_i}+L_{(Q_i,j)}{L}_{G_i}))],\hfill \end{array}\end{array}$$

not (51). Taking into consideration the above-mentioned arguments, we note that under the conditions appeared in the context of ([30], Theorem 4.1), its assertion need not be true. Indeed, the condition (44) (that is, (4.5) in [30]) and some hypotheses given in the context of Theorem 8 must be edited.

Taking into consideration the above-mentioned facts, we now present the correct version of Theorem 4.1 in [30] and prove it by using a conclusion of Section 3.

Theorem 9.

For each $i\in \{1,2\}$, let $E_i$ be a real uniformly smooth Banach space with ${\rho }_{E_i}(t)\le c_i{t}^2$ for some $c_i>0$ and for all $t\in [0,+\infty )$. Suppose that for each $i\in \{1,2\}$, ${\eta }_i:E_i\times E_i\to E_i$ is a ${\tau }_i$-Lipschitz continuous mapping such that ${\eta }_i(y_i,y_i^{\prime })+{\eta }_i(y_i^{\prime },y_i)=0$ for all $y_i,y_i^{\prime }\in E_i$, $g_i:E_i\to E_i$ is a $s_i$-strongly accretive and $L_{g_i}$-Lipschitz continuous mapping, $M_i:E_i\times E_i\to E_i^{*}$ is ${\alpha }_i{\beta }_i$-symmetric ${\eta }_i$-monotone and $L_{M_i(A_i,B_i)}$-Lipschitz continuous with respect to $A_i$ and $B_i$ such that $g_i(E_i)\subseteq \mathrm{dom}(M_i(A_i,B_i))$. For each $i\in \{1,2\}$ and any given $x_i^{*}\in E_i^{*}$, let the function $h_i:(y_i,x_i^{\prime })\in E_i\times E_i\to h_i(y_i,x_i^{\prime })=\langle x_i^{*}-M_i(A_i{x}_i^{\prime },B_i{x}_i^{\prime }),{\eta }_i(y_i,x_i^{\prime })\rangle \in ℝ\cup \{+\infty \}$ be $\mathrm{0-DQCV}$ in the first argument. Assume that for each $i\in \{1,2\}$, $S_i:E_i\to E_i^{*}$ is $L_{S_i}$-Lipschitz continuous, $T_i:E_j\to E_j^{*}$ $(j\in \{1,2\}\backslash \{i\})$ is $L_{T_i}$-Lipschitz continuous, and $D_i,F_i:E_i\to C(E_i^{*})$ are $L_{D_i}$-$H_{*i}$-Lipschitz and $L_{F_i}$-$H_{*i}$-Lipschitz continuous, respectively. Suppose that for each $i\in \{1,2\}$, $N_i:E_i^{*}\times E_j^{*}\to E_i^{*}$ $(j\in \{1,2\}\backslash \{i\})$ is $({\delta }_i,{\nu }_i)$-mixed Lipschitz continuous and for all ${\tilde{x}}_j\in E_j$, $N_i(S_i(.),T_i({\tilde{x}}_j))$ is ${\epsilon }_i$-strongly accretive with respect to $M_i(A_i,B_i)\circ g_i$. Let, for each $i\in \{1,2\}$, $C_i,G_i:E_j\to C(E_j^{*})$ $(j\in \{1,2\}\backslash \{i\})$ be $L_{C_i}$-$H_{*j}$-Lipschitz and $L_{G_i}$-$H_{*j}$-Lipschitz continuous, respectively, and $P_i:E_j^{*}\times E_i^{*}\to E_i^{*}$ and $Q_i:E_i^{*}\times E_j^{*}\to E_i^{*}$ be $(L_{(P_i,j)},L_{(P_i,i)})$-mixed Lipschitz and $(L_{(Q_i,i)},L_{(Q_i,j)})$-mixed Lipschitz continuous, respectively. In the meanwhile, let for each $i\in \{1,2\}$, ${\varphi }_i:E_i\to ℝ\cup \{+\infty \}$ be a proper, lower semicontinuous and ${\eta }_i$-subdifferentiable on $E_i$ with $g_i(E_i)\cap \mathrm{dom}\partial {\varphi }_i\ne \varnothing$. If there exist constants ${\lambda }_i>0$ $(i=1,2)$ such that for $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$,

$$\begin{array}{c}\hfill \begin{array}{cc}& {(1-2s_i+64c_i{L}_{g_i}^2)}^{\frac{1}{2}}+\frac{{\tau }_i}{{\alpha }_i-{\beta }_i}[{(L_{M_i(A_i,B_i)}^2{L}_{g_i}^2-2{\lambda }_i{\epsilon }_i+64c_i{\lambda }_i^2{\delta }_i^2{L}_{S_i}^2)}^{\frac{1}{2}}\hfill \\ & +{\lambda }_i{\rho }_i(L_{(P_i,i)}{L}_{D_i}+L_{(Q_i,i)}{L}_{F_i})]+\frac{{\tau }_j{\lambda }_j}{{\alpha }_j-{\beta }_j}[{\nu }_j{L}_{T_j}+{\rho }_j(L_{(P_j,i)}{L}_{C_j}\hfill \\ & +L_{(Q_j,i)}{L}_{G_j})]<1,\hfill \\ & 2s_i<1+64c_i{L}_{g_i}^2,2{\lambda }_i{\epsilon }_i<L_{M_i(A_i,B_i)}^2{L}_{g_i}^2+64c_i{\lambda }_i^2{\delta }_i^2{L}_{S_i}^2,\hfill \end{array}\end{array}$$

(57)

then the $\mathrm{SGIVLI}$ (6) admits a solution.

Proof. 

Let for each $i\in \{1,2\}$, ${\widehat{P}}_i:E_i\to E_i^{*}$ be defined by ${\widehat{P}}_i(x_i)=M_i(A_i{x}_i,B_i{x}_i)$, for all $x_i\in E_i$. Utilizing Proposition 1, it follows that for each $i\in \{1,2\}$, ${\widehat{P}}_i$ is an $({\alpha }_i-{\beta }_i)$-strongly ${\eta }_i$-monotone and $L_{{\widehat{P}}_i}=L_{M_i(A_i,B_i)}$-Lipschitz continuous mapping. Considering the fact that for each $i\in \{1,2\}$, $g_i(E_i)\subseteq \mathrm{dom}(M_i(A_i,B_i))$, we conclude that for each $i\in \{1,2\}$, $g_i(E_i)\subseteq \mathrm{dom}({\widehat{P}}_i)$. Picking ${\gamma }_i={\alpha }_i-{\beta }_i$ for each $i\in \{1,2\}$, we observe that (57) becomes the same condition (28) in Corollary 2. Then, all the conditions of Corollary 2 are satisfied. Now, Corollary 2 guarantees the existence of a solution for the $\mathrm{SGIVLI}$ (6) and we have completed the proof.    □

With the aim of suggesting an iterative algorithm for finding the approximate solution of the $\mathrm{SGIVLI}$ (6) and demonstrating the strong convergence of the sequences generated by it to the solution of the $\mathrm{SGIVLI}$ (6), they also imposed the following two conditions on the mappings $M^n(A^n,B^n),M(A,B),{\eta }^n,\eta ,{\varphi }^n,\varphi ,h^n,h$ $(n\in \mathbb{N}\cup \{0\})$.

Condition 1 ([30], Condition 5.1).

Let for each $n\ge 0$, ${\eta }^n,\eta :E\times E\to E$ be ${\tau }^n$-Lipschitz continuous such that ${\eta }^n(y,y^{\prime })+{\eta }^n(y^{\prime },y)=0$ and $\tau$-Lipschitz continuous such that $\eta (y,y^{\prime })+\eta (y^{\prime },y)=0$, for all $y^{\prime },y\in E$, respectively; let $M^n(A^n,B^n):E\to E^{*}$ be ${\alpha }^n{\beta }^n$-symmetric ${\eta }^n$-monotone and ${\nu }^n$-Lipschitz continuous with respect to $A^n$ and $B^n$; let $M(A,B):E\to E^{*}$ be $\alpha \beta$-symmetric $\eta$-monotone and $\nu$-Lipschitz continuous with respect to A and B; let for any given $x^{*}\in E^{*}$, the functions $h^n(y,x)=\langle x^{*}-M^n(A^{n}x,B^{n}x),{\eta }^n(y,x)\rangle$ and $h(y,x)=\langle x^{*}-M(Ax,Bx),\eta (y,x)\rangle$ be $\mathrm{0-DQCV}$ in y; let ${\varphi }^n:E\to ℝ\cup \{+\infty \}$ be a proper, lower semicontinuous and ${\eta }^n$-subdifferentiable function, and let $\varphi :E\to ℝ\cup \{+\infty \}$ be a proper, lower semicontinuous and ${\eta }^n$-subdifferentiable functional.

Condition 2 ([30], Condition 5.2).

Let Condition 1 (that is, ([30], Condition 5.1)) holds. The sequence ${\{\partial {\varphi }^n\}}_{n=0}^{\infty }$ which approximates $\partial \varphi$ in the following sense:

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{R}_{\rho ,{\eta }^n}^{\partial {\varphi }^n,M^n(A^n,B^n)}(x^{*})=R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}(x^{*}),\forall x^{*}\in E^{*}.\end{array}$$

Condition 2 tells that the sequence $M^n$-${\eta }^n$-proximal mappings ${\left\{R_{\rho ,{\eta }^n}^{\partial {\varphi }^n,M^n(A^n,B^n)}\right\}}_{n=0}^{\infty }$ converges to M-$\eta$-proximal mapping $R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}$. An appropriate question then is whether or not we can deduce Condition 2 under the assumptions mentioned in Condition 2? Fortunately, the answer of this question is affirmative if, in addition to the assumptions mentioned in Condition 2, we impose some other additional hypotheses on the mappings $\partial {\varphi }^n$, $M^n(A^n,B^n)$ $(n\ge 0)$, and the sequences ${\left\{\frac{1}{{\alpha }^n-{\beta }^n}\right\}}_{n=0}^{\infty }$, ${\left\{{\tau }^n\right\}}_{n=0}^{\infty }$, ${\left\{{\nu }^n\right\}}_{n=0}^{\infty }$.

In the next proposition, the conditions under which Condition 2 holds necessarily are stated.

Proposition 2.

Let $A^n,B^n,{\eta }^n,M^n(A^n,B^n),h^n,{\varphi }^n$ $(n\ge 0)$, $A,B,\eta ,M(A,B),h$ and ϕ be the same as in Condition 1 such that $\underset{n\to \infty }{lim}{M}^n(A^{n}x,B^{n}x)=M(Ax,Bx)$ for any $x\in E$. Suppose further that the sequences ${\left\{\frac{1}{{\alpha }^n-{\beta }^n}\right\}}_{n=0}^{\infty }$, ${\left\{{\tau }^n\right\}}_{n=0}^{\infty }$ and ${\left\{{\nu }^n\right\}}_{n=0}^{\infty }$ are bounded. Then, $\partial {\varphi }^n\stackrel{G}{⟶}\partial \varphi$ if and only if

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{R}_{\rho ,{\eta }^n}^{\partial {\varphi }^n,M^n(A^n,B^n)}(x^{*})=R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}(x^{*}),\forall x^{*}\in E^{*},\end{array}$$

where $\rho >0$ is constant; $R_{\rho ,{\eta }^n}^{\partial {\varphi }^n,M^n(A^n,B^n)}={(M^n(A^n,B^n)+\rho \partial {\varphi }^n)}^{-1}$ for all $(n\ge 0$, and $R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}={(M(A,B)+\rho \partial \varphi )}^{-1}$.

Proof. 

Let us define for each $n\ge 0$, the mappings ${\widehat{P}}^n,\widehat{P}:E\to E^{*}$ by ${\widehat{P}}^n(x):=M^n(A^{n}x,B^{n}x)$ and $\widehat{P}(x):=M(Ax,Bx)$ for all $x\in E$. Proposition 1 implies that for each $n\ge 0$, ${\widehat{P}}^n$ is an $({\alpha }^n-{\beta }^n)$-strongly ${\eta }^n$-monotone mapping. We note that all the conditions of Corollary 3 are satisfied and so according to Corollary 3,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{R}_{\rho ,{\eta }^n}^{\partial {\varphi }^n,M^n(A^n,B^n)}(x^{*})=\underset{n\to \infty }{lim}{R}_{\rho ,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}(x^{*})=R_{\rho ,\eta }^{\partial \varphi ,\widehat{P}}(x^{*})=R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}(x^{*}),\end{array}$$

for all $x^{*}\in E^{*}$, where $\rho >0$ is a constant; for all $n\ge 0$, $R_{\rho ,{\eta }^n}^{\partial {\varphi }^n,M^n(A^n,B^n)}=R_{\rho ,{\eta }^n}^{\partial {\varphi }^n,{\widehat{P}}^n}={({\widehat{P}}^n+\rho \partial {\varphi }^n)}^{-1}={(M^n(A^n,B^n)+\rho \partial {\varphi }^n)}^{-1}$ and $R_{\rho ,\eta }^{\partial \varphi ,M(A,B)}=R_{\rho ,\eta }^{\partial \varphi ,\widehat{P}}={(\widehat{P}+\rho \partial \varphi )}^{-1}={(M(A,B)+\rho \partial \varphi )}^{-1}$.    □

In light of the last conclusion, under some suitable conditions, Condition 2 is equivalent to the graph convergence of the sequence $\{\partial {\varphi }^n\}$ to $\partial \varphi$. Hence, under the assumptions given in Proposition 1, Condition 2 can be omitted by imposing the hypothesis $\partial {\varphi }^n\stackrel{G}{⟶}\partial \varphi$.

Assumption 1.

Let $E_i,E_i^{*},g_i,N_i,P_i,Q_i,S_i,T_i,C_i,D_i,G_i,F_i$ and ${\rho }_i$ $(i=1,2)$ be the same in the $\mathrm{SGIVLI}$ (6). Suppose that for each $i\in \{1,2\}$ and $n\ge 0$, ${\eta }_i^n:E_i\times E_i\to E_i$ is a continuous mapping such that ${\eta }_i^n(y_i,y_i^{\prime })+{\eta }_i^n(y_i^{\prime },y_i)=0$ for all $y_i,y_i^{\prime }\in E_i$, $A_i^n,B_i^n:E_i\to E_i$ are single-valued nonlinear mappings, and $M_i^n(A_i^n,B_i^n):E_i\times E_i\to E_i^{*}$ is ${\alpha }_i^n{\beta }_i^n$-symmetric ${\eta }_i^n$-monotone continuous with respect to $A_i^n$ and $B_i^n$ such that $g_i(E_i)\cap \mathrm{dom}(M_i^n(A_i^n,B_i^n))\ne \varnothing$. Let for each $i\in \{1,2\}$ and $n\ge 0$, and for any given $x_i^{*}\in E_i^{*}$, the functions $h_i^n:(y_i,x_i)\in E_i\times E_i\to h_i^n(y_i,x_i)={\langle x_i^{*}-M_i^n(A_i^n{x}_i,B_i^n{x}_i),{\eta }_i^n(y_i,x_i)\rangle }_i$ be $\mathrm{0-DQCV}$ in the first argument(in $y_i$), and ${\varphi }_i^n:E_i\to ℝ\cap \{+\infty \}$ be a proper, lower semicontinuous and ${\eta }_i^n$-subdifferentiable functional on $E_i$ with $g_i(E_i)\cap \mathrm{dom}\partial {\varphi }_i^n\ne \varnothing$.

Using Lemma 3, Corollary 3 and Nadler’s technique [39], Kazmi et al. [30] proposed an iterative algorithm for finding the approximate solution of the $\mathrm{SGIVLI}$ (6) (that is, ([30], the $\mathrm{SGIVLI}$ (4.1)–(4.2))) as follows.

They discussed the convergence analysis of the sequences generated by Algorithm 3 as follows.

Algorithm 3 [30] Iterative Algorithm 5.1

For $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, given $(x_1^0,x_2^0,u_1^0,$ $u_2^0,v_1^0,v_2^0,w_1^0,w_2^0,z_1^0,z_2^0)$, where $x_i^0\in E_i$, $u_i^0\in C_i(x_j^0)$, $v_i^0\in D_i(x_i^0)$, $w_i^0\in F_i(x_i^0)$, $z_i^0\in G_i(x_j^0)$, compute the sequences ${\left\{x_i^n\right\}}_{n=0}^{\infty }$, ${\left\{u_i^n\right\}}_{n=0}^{\infty }$, ${\left\{v_i^n\right\}}_{n=0}^{\infty }$, ${\left\{w_i^n\right\}}_{n=0}^{\infty }$, ${\left\{z_i^n\right\}}_{n=0}^{\infty }$ defined by the iterative schemes: $$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{x}_i^{n+1}=(1-{\mu }^n)x_i^n+{\mu }^n\{x_i^n-g_i(x_i^n)+R_{{\rho }_i,{\eta }_i^n}^{\partial {\varphi }_i^n,M_i^n(A_i^n,B_i^n)}[(M_i(A_i^n,B_i^n)\circ g_i)(x_i^n)\hfill \\ -{\lambda }_i(N_i(S_i(x_i^n),T_i(x_j^n))+{\rho }_i(P_i(u_i^n,v_i^n)+Q_i(w_i^n,z_i^n)))]\},\hfill \\ u_i^n\in C_i(x_j^n):{\parallel u_i^{n+1}-u_i^n\parallel }_{*j}\le H_{*j}(C_i(x_j^{n+1}),C_i(x_j^n)),\hfill \\ v_i^n\in D_i(x_i^n):{\parallel v_i^{n+1}-v_i^n\parallel }_{*i}\le H_{*i}(D_i(x_i^{n+1}),D_i(x_i^n)),\hfill \\ w_i^n\in F_i(x_i^n):{\parallel w_i^{n+1}-w_i^n\parallel }_{*i}\le H_{*i}(F_i(x_i^{n+1}),F_i(x_i^n)),\hfill \\ z_i^n\in G_i(x_j^n):{\parallel z_i^{n+1}-z_i^n\parallel }_{*j}\le H_{*j}(G_i(x_j^{n+1}),G_i(x_j^n)),\hfill \end{array}\right.\end{array}\end{array}$$ (58) where $n=0,1,2,\cdots$; ${\rho }_i,{\lambda }_i>0$ are constants; and $0<{\mu }^n<1$ are relaxation constants such that ${\displaystyle \sum _{n=0}^{\infty }}{\mu }^n=\infty$.

Theorem 10 ([30], Theorem 5.1).

For $i\in \{1,2\}$, let $E_i$ be a uniformly smooth Banach space with ${\rho }_{E_i}(t)\le c_i{t}^2$ for some $c_i>0$. For $i\in \{1,2\}$, $j\in \{1,2\}\backslash \left\{i\right\}$, let the mappings ${\eta }_i^n,{\eta }_i:E_i\times E_i\to E_i$, $A_i^n,B_i^n,A_i,B_i:E_i\to E_i$, $M_i^n(A_i^n,B_i^n)$, $M_i(A_i,B_i):E_i\to E_i^{*}$, ${\varphi }_i^n,{\varphi }_i:E_i\to ℝ\cup \{+\infty \}$, and for any given $x_i^{*}\in E_i^{*}$, the functions $h_i^n(y_i,x_i)={\langle x_i^{*}-M_i^n(A_i^n{x}_i,B_i^n{x}_i),{\eta }_i^n(y_i,x_i)\rangle }_i$ and $h_i(y_i,x_i)={\langle x_i^{*}-M_i(A_i{x}_i,B_i{x}_i),{\eta }_i(y_i,x_i)\rangle }_i$ satisfy Conditions 1 and 2. Let $N_i,Q_i:E_i^{*}\times E_j^{*}\to E_i^{*}$, $P_i:E_j^{*}\times E_i^{*}\to E_i^{*}$ be $({\delta }_i,{\nu }_i)$, $(L_{(Q_i,i)},L_{(Q_i,j)})$ and $(L_{(P_i,i)},L_{(P_i,j)})$-mixed Lipschitz continuous, respectively; let $S_i:E_i\to E_i^{*}$, $T_i:E_i\to E_j^{*}$ be $L_{S_i}$ and $L_{T_i}$-Lipschitz continuous; let $C_i,G_i:E_j\to C(E_j^{*})$, $D_i,F_i:E_i\to C(E_i^{*})$ be $L_{C_i}$-$H_{*j}$-Lipschitz continuous, $L_{G_i}$-$H_{*j}$-Lipschitz continuous, $L_{D_i}$-$H_{*i}$-Lipschitz continuous and $L_{F_i}$-$H_{*i}$-Lipschitz continuous, respectively; let $g_i:E_i\to E_i$ be $s_i$-strongly and $L_{g_i}$-Lipschitz continuous; let $N_i(S_i(.),T_i(x_j))$ be ${\epsilon }_i$-strongly accretive with respect to $M_i(A_i,B_i)\circ g_i$ and let $M_i(A_i,B_i)\circ g_i$ be $L_{M_i}$-Lipschitz continuous. Suppose that there is a constant ${\rho }_i>0$, $(i=1,2)$ such that for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$,

$$\begin{array}{c}\hfill \begin{array}{cc}& {(1-2s_i+64c_i{L}_{g_i}^2)}^{\frac{1}{2}}+\frac{{\tau }_i^n}{{\alpha }_i^n-{\beta }_i^n}[{(L_{M_i}^2-2{\lambda }_i{\epsilon }_i+64c_i{\lambda }_i^2{\delta }_i^2)}^{\frac{1}{2}}\hfill \\ & +{\rho }_i(L_{(P_i,j)}{L}_{D_i}+L_{(Q_i,i)}{L}_{F_i})]+\frac{{\tau }_i^n}{{\alpha }_i^n-{\beta }_i^n}[{\lambda }_i{\nu }_i{L}_{T_i}+{\rho }_i(L_{(P_i,j)}{L}_{C_i}\hfill \\ & +L_{(Q_i,j)}{L}_{G_i})]<1.\hfill \end{array}\end{array}$$

(59)

Then, for each $i\in \{1,2\}$, the sequences ${\left\{x_i^n\right\}}_{n=0}^{\infty }$, ${\left\{u_i^n\right\}}_{n=0}^{\infty }$, ${\left\{v_i^n\right\}}_{n=0}^{\infty }$, ${\left\{w_i^n\right\}}_{n=0}^{\infty }$, ${\left\{z_i^n\right\}}_{n=0}^{\infty }$ generated by Algorithm 3(that is, ([30], Iterative Algorithm 5.1)) converges strongly to, respectively, $x_i,u_i,v_i,w_i,z_i$, where $(x_1,x_2,u_1,u_2,v_1,v_2,w_1,w_2,z_1,z_2)$ is a solution of the $\mathrm{SGIVLI}$ (6) (that is, ([30], the $\mathrm{SGIVLI}$ (4.1)–(4.2))).

By a careful reading the proof of Theorem 5.1 in [30], some fatal errors are detected. On page 9216, line 10 from the bottom in [30], the authors concluded the existence of a solution $(x_1,x_2,u_1,u_2,v_1,v_2,w_1,w_2,z_1,z_2)$ for ([30], the $\mathrm{SGIVLI}$ (4.1)–(4.2)) according to ([30], Theorem 4.1), where for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, $x_i\in E_i$, $u_i\in C_i(x_j)$, $v_i\in D_i(x_i)$, $w_i\in F_i(x_i)$, $z_i\in G_i(x_j)$. However, as was pointed out, the assertion of ([30], Theorem 4.1) is not true necessarily. Hence, the existence of a solution for the $\mathrm{SGIVLI}$ (6) cannot be deduced by ([30], Theorem 4.1). Even without considering this fact, we discovered that there is a fatal error in the proof of the above-mentioned theorem. Indeed, on page 9217, lines 7–10, the authors estimated $\parallel x_i^{n+1}-x_i{\parallel }_i$ for all $n\ge 0$ and $i\in \{1,2\}$, with the help of the same arguments used in the proof of ([30], Theorem 4.1), by (5.5) in [30] as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel x_i^{n+1}-x_i{\parallel }_i& \le (1-{\mu }^n)\parallel x_i^n-x_i{\parallel }_i+{\mu }^n{(1-2s_i+64c_i{L}_{g_i}^2)}^{\frac{1}{2}}{\parallel x_i^n-x_i\parallel }_i\hfill \\ & +\frac{{\tau }_i^n{\mu }_i}{{\alpha }_i^n-{\beta }_i^n}[{(L_{M_i}^2-2{\lambda }_i{ϵ}_i+64c_i{\lambda }_i^2{\delta }_i^2)}^{\frac{1}{2}}{\parallel x_i^n-x_i\parallel }_i\hfill \\ & +{\lambda }_i{\nu }_i{L}_{T_i}\parallel x_j^n-x_j{\parallel }_j+{\rho }_i\{L_{(P_i,j)}{L}_{C_i}{\parallel x_j^n-x_j\parallel }_j\hfill \\ & +L_{(P_i,i)}{L}_{D_i}\parallel x_i^n-x_i{\parallel }_i+L_{(Q_i,i)}{L}_{F_i}{\parallel x_i^n-x_i\parallel }_i\hfill \\ & +L_{(Q_i,i)}{L}_{G_i}\parallel x_j^n-x_j{\parallel }_j\}]+{\mu }^n{\varphi }^n{\parallel x_i^{n+1}-x_i\parallel }_i\hfill \\ & \le (1-{\mu }^n)\parallel x_i^n-x_i{\parallel }_i+{\mu }^n(a_i^n\parallel x_i^n-x_i{\parallel }_i+b_i^n\parallel x_i^n-x_i{\parallel }_i)+{\mu }^n{\varphi }^n,\hfill \end{array}\end{array}$$

(60)

where for each $i\in \{1,2\}$ and $n\ge 0$,

$$\begin{array}{c}{a}_i^n:={(1-2s_i+64c_i{L}_{g_i}^2)}^{\frac{1}{2}}+\frac{{\tau }_i^n}{{\alpha }_i^n-{\beta }_i^n}[{(L_{M_i}^2-2{\lambda }_i{ϵ}_i+64c_i{\lambda }_i^2{\delta }_i^2)}^{\frac{1}{2}}\hfill \\ +{\rho }_i(L_{(P_i,i)}{L}_{D_i}+L_{(Q_i,i)}{L}_{F_i})],\hfill \end{array}$$

(61)

$$b_i^n:=\frac{{\tau }_i^n}{{\alpha }_i^n-{\beta }_i^n}[{\lambda }_i{v}_i{L}_{T_i}+{\rho }_i(L_{(P_i,j)}{L}_{C_i}+L_{(Q_i,j)}{L}_{G_i})]$$

(62)

There is a fatal error in obtaining an estimate of $\parallel x_i^{n+1}-x_i{\parallel }_i$ by (60). In fact, utilizing Lemma 3, (58) and taking into account that for each $i\in \{1,2\}$, $u_i\in C_i(x_j)$, $v_i\in D_i(x_i)$, $w_i\in F_i(x_i)$ and $z_i\in G_i(x_j)$ $(j\in \{1,2\}\backslash \{i\})$, they used the following inequalities in computing (60):

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}\parallel u_i^n-u_i{\parallel }_{*j}\le H_{*j}(C_i(x_j^n),C_i(x_j)),\hfill \\ \parallel v_i^n-v_i{\parallel }_{*i}\le H_{*i}(D_i(x_i^n),D_i(x_i)),\hfill \\ \parallel w_i^n-w_i{\parallel }_{*i}\le H_{*i}(F_i(x_i^n),F_i(x_i)),\hfill \\ \parallel z_i^n-z_i{\parallel }_{*j}\le H_{*j}(G_i(x_j^n),G_i(x_j)),\hfill \end{array}\right.\end{array}\end{array}$$

(63)

However, relying on Lemma 3, the preceding inequalities in (63) cannot be deduced. Indeed, for any given $x,y\in E$, $u\in T(x)$ and $v\in T(y)$, the inequality (8) in Lemma 3 does not hold necessarily. This fact is illustrated in the next example.

Example 2.

Consider $E=l^{\infty }(\mathbb{Z})=\{x={\left\{x_n\right\}}_{n=-\infty }^{\infty }:\underset{n\in \mathbb{Z}}{sup}|x_n|<\infty ,x_n\in ℝ\}$ the real Banach space consisting of all bounded real sequences $x={\left\{x_n\right\}}_{n=-\infty }^{\infty }$ with the supremum norm ${\parallel x\parallel }_{\infty }=\underset{n\in \mathbb{Z}}{sup}\left|x_n\right|$. Any element $x={\left\{x_n\right\}}_{n=-\infty }^{\infty }\in l^{\infty }(\mathbb{Z})$ can be written as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill x={\left\{x_n\right\}}_{n=-\infty }^{\infty }& =(\cdots ,x_{-2},x_{-1},x_0,x_1,x_2,\cdots )\hfill \\ & =\sum _{\sigma =-\infty }^{\infty }(\cdots ,0,0,\cdots ,0,x_{\sigma \delta +1},x_{\sigma \delta +2},\cdots ,x_{(\sigma +1)\delta },0,0,\cdots ),\hfill \end{array}\end{array}$$

where $\delta \ge 2$ is an arbitrary but fixed even natural number.

For each $\sigma \in \mathbb{Z}$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& (\cdots ,0,0,\cdots ,0,x_{\sigma \delta +1},x_{\sigma \delta +2},\cdots ,x_{(\sigma +1)\delta },0,0,\cdots )\hfill \\ & =(\cdots ,0,0,\cdots ,0,x_{\sigma \delta +1},0,0,\cdots ,0,x_{(\sigma +1)\delta },0,0,\cdots )\hfill \\ & +(\cdots ,0,0,\cdots ,0,x_{\sigma \delta +2},0,0,\cdots ,0,x_{(\sigma +1)\delta -1},0,0,\cdots )\hfill \\ & +\cdots +(\cdots ,0,0,\cdots ,0,x_{\frac{(2\sigma +1)\delta }{2}},x_{\frac{(2\sigma +1)\delta }{2}+1},0,0,\cdots )\hfill \\ & =\sum _{m=\sigma \delta +1}^{\frac{(2\sigma +1)\delta }{2}}(\cdots ,0,0,\cdots ,0,x_m,0,0,\cdots ,0,x_{(2\sigma +1)\delta -m+1},0,0,\cdots ).\hfill \end{array}\end{array}$$

For each $\sigma \in \mathbb{Z}$ and $m\in \{\sigma \delta +1,\sigma \delta +2,\cdots ,\frac{(2\sigma +1)\delta }{2}\}$, there are two real numbers $y_m$ and $y_{(2\sigma +1)\delta -m+1}$ such that

$$\begin{array}{c}\hfill \begin{array}{cc}& (\cdots ,0,0,\cdots ,0,x_m,0,0,\cdots ,0,x_{(2\sigma +1)\delta -m+1},0,0,\cdots )\hfill \\ & =(\cdots ,0,0,\cdots ,0,y_m+y_{(2\sigma +1)\delta -m+1},0,0,\cdots ,0,y_m-y_{(2\sigma +1)\delta -m+1},0,0,\cdots ).\hfill \end{array}\end{array}$$

Therefore, for any $x={\left\{x_n\right\}}_{n=-\infty }^{\infty }\in l^{\infty }(\mathbb{Z})$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill x& =\sum _{\sigma =-\infty }^{\infty }(\cdots ,0,0,\cdots ,0,x_{\sigma \delta +1},x_{\sigma \delta +2},\cdots ,x_{(\sigma +1)\delta },0,0,\cdots )\hfill \\ & =\sum _{\sigma =-\infty }^{\infty }\sum _{m=\sigma \delta +1}^{\frac{(2\sigma +1)\delta }{2}}(\cdots ,0,0,\cdots ,0,x_m,0,0,\cdots ,0,x_{(2\sigma +1)\delta -m+1},0,0,\cdots )\hfill \\ & =\sum _{\sigma =-\infty }^{\infty }\sum _{m=\sigma \delta +1}^{\frac{(2\sigma +1)\delta }{2}}(\cdots ,0,0,\cdots ,0,y_m+y_{(2\sigma +1)\delta -m+1},0,0,\cdots ,0,y_m-y_{(2\sigma +1)\delta -m+1},0,0,\cdots )\hfill \\ & =\sum _{\sigma =-\infty }^{\infty }\sum _{m=\sigma \delta +1}^{\frac{(2\sigma +1)\delta }{2}}[y_m(\cdots ,0,0,\cdots ,0,1_m,0,0,\cdots ,0,1_{(2\sigma +1)\delta -m+1},0,0,\cdots )\hfill \\ & +y_{(2\sigma +1)\sigma -m+1}(\cdots ,0,0,\cdots ,0,1_m,0,0,\cdots ,0,-1_{(2\sigma +1)\delta -m+1},0,0,\cdots )]\hfill \\ & =\sum _{\sigma =-\infty }^{\infty }\sum _{m=\sigma \delta +1}^{\frac{(2\sigma +1)\delta }{2}}(y_m{\mu }_{m,(2\sigma +1)\delta -m+1}+y_{(2\sigma +1)\delta -m+1}{\mu }_{m,(2\sigma +1)\delta -m+1}^{\prime }),\hfill \end{array}\end{array}$$

where for each $\sigma \in \mathbb{Z}$ and $m\in \{\sigma \delta +1,\sigma \delta +2,\cdots ,\frac{(2\sigma +1)\delta }{2}\}$,

$$\begin{array}{c}\hfill {\mu }_{m,(2\sigma +1)\delta -m+1}=(\cdots ,0,0,\cdots ,0,1_m,0,0,\cdots ,0,1_{(2\sigma +1)\delta -m+1},0,0,\cdots ),\end{array}$$

1 at the mth and $((2\sigma +1)\delta -m+1)$th coordinates and all other coordinates are zero, and

$$\begin{array}{c}\hfill {\mu }_{m,(2\sigma +1)\delta -m+1}^{\prime }=(\cdots ,0,0,\cdots ,0,1_m,0,0,\cdots ,0,-1_{(2\sigma +1)\delta -m+1},0,0,\cdots ),\end{array}$$

1 and $-1$ at the mth and $((2\sigma +1)\delta -m+1)$th coordinates, respectively, and all other coordinates are zero. Thus, the set

$$\begin{array}{c}\hfill \mathfrak{B}=\left\{{\mu }_{m,(2\sigma +1)\delta -m+1},{\mu }_{m,(2\sigma +1)\delta -m+1}^{\prime }:\sigma \in \mathbb{Z};m=\sigma \delta +1,\sigma \delta +2,\cdots ,\frac{(2\sigma +1)\delta }{2}\right\}\end{array}$$

spans the real Banach space $l^{\infty }(\mathbb{Z})$. It can be easily seen that the set $\mathfrak{B}$ is linearly independent and so it is a Schauder basis for the real Banach space $l^{\infty }(\mathbb{Z})$. Define the set-valued mapping $T:l^{\infty }(\mathbb{Z})\to CB(l^{\infty }(\mathbb{Z}))$ by

$$T(x)=\left\{\begin{array}{cc}{\psi }_1,\hfill & x\ne {\mu }_{r,(2\sigma +1)\delta -r+1}^{\prime },\hfill \\ {\psi }_2,\hfill & x={\mu }_{r,(2\sigma +1)\delta -r+1}^{\prime },\hfill \end{array}\right.$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}& {\psi }_1=\left\{{\left\{\frac{\beta }{{\gamma }^{n^2}{n}^{\theta }!}\right\}}_{n=-\infty }^{\infty },{\mu }_{m,(2\sigma +1)\delta -m+1}^{\prime }:\sigma \in \mathbb{Z};m=\sigma \delta +1,\sigma \delta +2,\cdots ,\frac{(2\sigma +1)\delta }{2}\right\},\hfill \\ & {\psi }_2=\left\{{\mu }_{m,(2\sigma +1)\delta -m+1}:\sigma \in \mathbb{Z};m=\sigma \delta +1,\sigma \delta +2,\cdots ,\frac{(2\sigma +1)\delta }{2}\right\},\hfill \end{array}\end{array}$$

$\beta \in [-1,0)$ and $\gamma >1$ are arbitrary but fixed real numbers, θ is an arbitrary but fixed even natural number, and $r=s\delta +t$ for an arbitrary but fixed $s\in \mathbb{Z}$ and $t\in \{1,2,\cdots ,\frac{\delta }{2}\}$. Take ${\mu }_{r,(2s+1)\delta -r+1}^{\prime }\ne x\in E$ arbitrarily but fixed, $y={\mu }_{r,(2s+1)\delta -r+1}^{\prime }$ and $u={\left\{\frac{\beta }{{\gamma }^{n^2}{n}^{\theta }!}\right\}}_{n=-\infty }^{\infty }$.

If $a={\left\{\frac{\beta }{{\gamma }^{n^2}{n}^{\theta }!}\right\}}_{n=-\infty }^{\infty }$, then thanks to the fact that $\beta <0$ for each $\sigma \in \mathbb{Z}$ and $m\in \{\sigma \delta +1,\sigma \delta +2,\cdots ,\frac{(2\sigma +1)\delta }{2}\}$, we infer that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d(a,{\mu }_{m,(2\sigma +1)\delta -m+1})& =\parallel {\left\{\frac{\beta }{{\gamma }^{n^2}{n}^{\theta }!}\right\}}_{n=-\infty }^{\infty }-{\mu }_{m,(2\sigma +1)\delta -m+1}{\parallel }_{\infty }\hfill \\ & =sup\left\{\right|\frac{\beta }{{\gamma }^{n^2}{n}^{\theta }!}|,|\frac{\beta }{{\gamma }^{m^2}{m}^{\theta }!}-1|,\hfill \\ & |\frac{\beta }{{\gamma }^{{((2\sigma +1)\delta -m+1)}^2}{((2\sigma +1)\delta -m+1)}^{\theta }!}-1|:\hfill \\ & n\in \mathbb{Z};n\ne m,(2\sigma +1)\delta -m+1\}\hfill \\ & =|\frac{\beta }{{\gamma }^{m^2}{m}^{\theta }!}-1|\hfill \\ & =1-\frac{\beta }{{\gamma }^{m^2}{m}^{\theta }!},\hfill \end{array}\end{array}$$

from which we deduce that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill d(a,T(y))=\underset{b\in T(y)}{inf}d(a,b)=inf\left\{1-\frac{\beta }{{\gamma }^{m^2}{m}^{\theta }!}:\sigma \in \mathbb{Z};m=\sigma \delta +1,\sigma \delta +2,\cdots ,\frac{(2\sigma +1)\delta }{2}\right\}=1.\end{array}\end{array}$$

If $a={\mu }_{k,(2p+1)\delta -k+1}^{\prime }$, for some $p\in \mathbb{Z}$ and $k\in \{p\delta +1,p\delta +2,\cdots ,\frac{(2p+1)\delta }{2}\}$, then for each $\sigma \in \mathbb{Z}$ and $m\in \{\sigma \delta +1,\sigma \delta +2,\cdots ,\frac{(2\sigma +1)\delta }{2}\}$, yields

$$d(a,{\mu }_{m,(2\sigma +1)\delta -m+1})=\left\{\begin{array}{cc}\parallel {\mu }_{k,(2p+1)\delta -k+1}^{\prime }-{\mu }_{k,(2p+1)\delta -k+1}{\parallel }_{\infty },\hfill & m=k,\hfill \\ \parallel {\mu }_{k,(2p+1)\delta -k+1}^{\prime }-{\mu }_{m,(2\sigma +1)\delta -m+1}{\parallel }_{\infty },\hfill & m\ne k,\hfill \end{array}\right.$$

$$=\left\{\begin{array}{cc}2\hfill & m=k,\hfill \\ 1\hfill & m\ne k,\hfill \end{array}\right.$$

and so

$$\begin{array}{c}\hfill d(a,T(y))=\underset{b\in T(y)}{inf}d(a,b)=1.\end{array}$$

Hence,

$$\begin{array}{c}\hfill \underset{a\in T(x)}{sup}d(a,T(y))=1.\end{array}$$

If $b={\mu }_{\varrho ,(2j+1)\delta -\varrho +1}$ for some $j\in \mathbb{Z}$ and $\delta \in \{j\delta +1,j\delta +2,\cdots ,\frac{(2j+1)\delta }{2}\}$, in view of the fact that $\beta <0$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d({\left\{\frac{\beta }{{\gamma }^{n^2}{n}^{\theta }!}\right\}}_{n=-\infty }^{\infty },{\mu }_{\varrho ,(2j+1)\delta -\varrho +1})& =\parallel {\left\{\frac{\beta }{{\gamma }^{n^2}{n}^{\theta }!}\right\}}_{n=-\infty }^{\infty }-{\mu }_{\varrho ,(2j+1)\delta -\varrho +1}{\parallel }_{\infty }\hfill \\ & =sup\left\{\right|\frac{\beta }{{\gamma }^{n^2}{n}^{\theta }!}|,|\frac{\beta }{{\gamma }^{{\varrho }^2}{\varrho }^{\theta }!}-1|,\hfill \\ & |\frac{\beta }{{\gamma }^{{((2j+1)\delta -\varrho +1)}^2}{((2j+1)\delta -\varrho +1)}^{\theta }!}-1|:\hfill \\ & n\in \mathbb{Z};n\ne \varrho ,(2j+1)\delta -\varrho +1\}\hfill \\ & =|\frac{\beta }{{\gamma }^{{\varrho }^2}{\varrho }^{\theta }!}-1|\hfill \\ & =1-\frac{\beta }{{\gamma }^{{\varrho }^2}{\varrho }^{\theta }!},\hfill \end{array}\end{array}$$

and for each $\sigma \in \mathbb{Z}$ and $m\in \{\sigma \delta +1,\sigma \delta +2,\cdots ,\frac{(2\sigma +1)\delta }{2}\}$,

$$d({\mu }_{m,(2\sigma +1)\delta -m+1}^{\prime },{\mu }_{\varrho ,(2j+1)\delta -\varrho +1})=\left\{\begin{array}{cc}\parallel {\mu }_{\varrho ,(2j+1)\delta -\varrho +1}^{\prime }-{\mu }_{\varrho ,(2j+1)\delta -\varrho +1}{\parallel }_{\infty },\hfill & m=\varrho ,\hfill \\ \parallel {\mu }_{m,(2\sigma +1)\delta -m+1}^{\prime }-{\mu }_{\varrho ,(2j+1)\delta -\varrho +1}{\parallel }_{\infty },\hfill & m\ne \varrho ,\hfill \end{array}\right.$$

$$=\left\{\begin{array}{cc}2\hfill & m=\varrho ,\hfill \\ 1\hfill & m\ne \varrho .\hfill \end{array}\right.$$

Since $\beta <0$, it follows that

$$\begin{array}{c}\hfill d(T(x),b)=\underset{a\in T(x)}{inf}d(a,b)=1,\end{array}$$

and so

$$\begin{array}{c}\hfill \underset{b\in T(y)}{sup}d(T(x),b)=1.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill D(T(x),T(y))=max\left\{\underset{a\in T(x)}{sup}d(a,T(y)),\underset{b\in T(y)}{sup}d(T(x),b)\right\}=1.\end{array}$$

Considering the fact that $\beta \in [-1,0)$, we deduce that for all $\sigma \in \mathbb{Z}$ and $m\in \{\sigma \delta +1,\sigma \delta +2,\cdots ,\frac{(2\sigma +1)\delta }{2}\}$,

$$\begin{array}{c}\hfill \parallel {\left\{\frac{\beta }{{\gamma }^{n^2}{n}^{\theta }!}\right\}}_{n=-\infty }^{\infty }-{\mu }_{m,(2\sigma +1)\delta -m+1}{\parallel }_{\infty }=1-\frac{\beta }{{\gamma }^{n^2}{n}^{\theta }!}>1,\end{array}$$

which implies that for any $v\in T(y)$,

$$\begin{array}{c}\hfill d(u,v)={\parallel u-v\parallel }_{\infty }>D(T(x),T(y)).\end{array}$$

Considering the above-mentioned arguments, it is essential to mention that to study the convergence analysis of the iterative sequences ${\left\{x_i^n\right\}}_{n=0}^{\infty }$, ${\left\{u_i^n\right\}}_{n=0}^{\infty }$, ${\left\{v_i^n\right\}}_{n=0}^{\infty }$, ${\left\{w_i^n\right\}}_{n=0}^{\infty }$ and ${\left\{z_i^n\right\}}_{n=0}^{\infty }$ generated by Algorithm 3, that is, to provide the correct version of Theorem 10, the strong convergence of the aforesaid sequences, respectively, to some $x_i\in E_i$, $u_i\in C_i(x_j)$, $v_i\in D_i(x_i)$, $w_i\in F_i(x_i)$ and $z_i\in G_i(x_j)$ $(i\in \{1,2\},j\in \{1,2\}\backslash \{i\})$ must first be proved and then the fact that $(x_1,x_2,u_1,u_2,v_1,v_2,w_1,w_2,z_1,z_2)$ is a solution of the $\mathrm{SGIVLI}$ (6) must be shown. Hence, contrary to the argument used in the proof of ([30], Theorem 5.1), and in the same way as the proof of Theorem 5, an estimate of $\parallel x_i^{n+1}-x_i^n{\parallel }_i$ must be computed for each $i\in \{1,2\}$ and $n\in \mathbb{N}$ instead of $\parallel x_i^{n+1}-x_i{\parallel }_i$ for each $i\in \{1,2\}$ and $n\ge 0$. However, there are two difficulties which must be resolved. Firstly, to overcome the difficulty of finding an estimate of $\parallel x_i^{n+1}-x_i^n{\parallel }_i$, relaxation parameters ${\mu }^n\in (0,1)$ $(n\ge 0)$ in Algorithm 3 must be replaced by a relaxation parameter $\mu \in (0,1)$. Secondly, to show the strong convergence of the sequences ${\left\{x_i^n\right\}}_{n=0}^{\infty }$ to some $x_i\in E_i$ for $i=1,2$, we need to prove $\parallel x_i^{n+1}-x_i^n{\parallel }_i\to 0$ as $n\to \infty$, for each $i\in \{1,2\}$. For this purpose, it can be easily observed that we need for each $i\in \{1,2\}$, $(M_i^n(A_i^n,B_i^n)\circ g_i)(x_i^{n-1})\to (M_i^{n-1}(A_i^{n-1},B_i^{n-1})\circ g_i)(x_i^{n-1})$, as $n\to \infty$, which does not hold in general. In fact, it is significant to emphasize that, for given variational inequality (inclusion) problems involving multivalued mappings, i.e., the multivalued variational inequality (inclusion) problems, the notion of graph convergence can be used in no way to study the convergence analysis of the iterative sequences generated by the suggested algorithms. To overcome this difficulty, in Algorithm 3, for each $i\in \{1,2\}$, the mappings $\partial {\varphi }_i^n$, $M_i^n(A_i^n,B_i^n)$ and ${\eta }_i^n$ $(n\ge 0)$ must be replaced by $\partial {\varphi }_i$, $M_i(A_i,B_i)$ and ${\eta }_i$, respectively. Thereby, in view of the facts mentioned above, Iterative Algorithm 5.1 in [30] does not generate the sequences converging to a solution of the $\mathrm{SGIVLI}$ (6) and must be edited. The correct version of it can be written as follows.

Remark 6.

Defining ${\widehat{P}}_i:E_i\to E_i^{*}$ for each $i\in \{1,2\}$ by ${\widehat{P}}_i(x_i):=M_i(A_i(x_i),B_i(x_i))$, for all $x_i\in E_i$, the iterative schemes (64) in Algorithm 4 become the same iterative schemes (36) in Algorithm 2. On the other hand, Proposition 1 implies that for each $i\in \{1,2\}$, ${\widehat{P}}_i$ is an $({\alpha }_i-{\beta }_i)$-strongly ${\eta }_i$-monotone continuous mapping. At the same time, in light of the fact that $g_i(E_i)\subseteq \mathrm{dom}(M_i(A_i,B_i))$ for each $i\in \{1,2\}$, it follows that $g_i(E_i)\subseteq \mathrm{dom}({\widehat{P}}_i)$ for each $i\in \{1,2\}$. Now, picking ${\alpha }_i-{\beta }_i={\gamma }_i$ for each $i\in \{1,2\}$, we observe that Algorithm 4 is actually the same Algorithm 2 and is not a new one.

We are now in a position to present the correct version of ([30], Theorem 5.1).

Theorem 11.

For each $i\in \{1,2\}$, let $E_i$ be a real uniformly smooth Banach space with ${\rho }_{E_i}(t)\le c_i{t}^2$ for some $c_i>0$ and for all $t\in [0,+\infty )$. For each $i\in \{1,2\}$, let ${\eta }_i:E_i\times E_i\to E_i$ be a ${\tau }_i$-Lipschitz continuous mapping such that ${\eta }_i(y_i,y_i^{\prime })+{\eta }_i(y_i^{\prime },y_i)=0$ for all $y_i,y_i^{\prime }\in E_i$. Suppose that for each $i\in \{1,2\}$, $g_i:E_i\to E_i$ is a $s_i$-strongly accretive and $L_{g_i}$-Lipschitz continuous mapping, $M_i(A_i,B_i):E_i\to E_i^{*}$ is an ${\alpha }_i{\beta }_i$-symmetric ${\eta }_i$-monotone and $L_{M_i(A_i,B_i)}$-Lipschitz continuous with respect to the mappings $A_i,B_i:E_i\to E_i$ such that $g_i(E_i)\subseteq \mathrm{dom}(M_i(A_i,B_i))$. Let for each $i\in \{1,2\}$ and for any given $x_i^{*}\in E_i^{*}$, the function $h_i:(y_i,{\widehat{x}}_i)\in E_i\times E_i\to h_i(y_i,{\widehat{x}}_i)={\langle x_i^{*}-M_i(A_i{\widehat{x}}_i,B_i{\widehat{x}}_i),{\eta }_i(y_i,{\widehat{x}}_i)\rangle }_i\in ℝ\cup \{+\infty \}$ be $\mathrm{0-DQCV}$ in the first argument(in $y_i$). Assume that for each $i\in \{1,2\}$, $S_i:E_i\to E_i^{*}$ is $L_{S_i}$-Lipschitz continuous, $T_i:E_j\to E_j^{*}$ $(j\in \{1,2\}\backslash \{i\})$ is $L_{T_i}$-Lipschitz continuous, and $D_i,F_i:E_i\to CB(E_i^{*})$ are $L_{D_i}$-$H_{*i}$-Lipschitz and $L_{F_i}$-$H_{*i}$-Lipschitz continuous, respectively. Let for each $i\in \{1,2\}$, $N_i:E_i^{*}\times E_j^{*}\to E_i^{*}$ $(j\in \{1,2\}\backslash \{i\})$ be a $({\delta }_i,{\nu }_i)$-mixed Lipschitz continuous mapping and for any ${\tilde{x}}_j\in E_j$, $N_i(S_i(.),T_i({\tilde{x}}_j))$ be ${\epsilon }_i$-strongly accretive with respect to $M_i(A_i,B_i)$. For each $i\in \{1,2\}$, let $C_i,G_i:E_j\to CB(E_j^{*})$ $(j\in \{1,2\}\backslash \{i\})$ be $L_{C_i}$-$H_{*j}$-Lipschitz and $L_{G_i}$-$H_{*j}$-Lipschitz continuous, respectively, and $P_i:E_j^{*}\times E_i^{*}\to E_i^{*}$ and $Q_i:E_i^{*}\times E_j^{*}\to E_i^{*}$ be $(L_{(P_i,j)},L_{(P_i,i)})$-mixed Lipschitz and $(L_{(Q_i,i)},L_{(Q_i,j)})$-mixed Lipschitz continuous, respectively. Assume that for each $i\in \{1,2\}$, ${\varphi }_i:E_i\to ℝ\cup \{+\infty \}$ is a lower semicontinuous and ${\eta }_i$-subdifferentiable proper functional on $E_i$ with $g_i(E_i)\cap \mathrm{dom}\partial {\varphi }_i\ne \varnothing$. If for $i=1,2$, there exist constants ${\lambda }_i>0$ satisfying (29), then the iterative sequences ${\left\{x_i^n\right\}}_{n=0}^{\infty }$, ${\left\{u_i^n\right\}}_{n=0}^{\infty }$, ${\left\{v_i^n\right\}}_{n=0}^{\infty }$, ${\left\{w_i^n\right\}}_{n=0}^{\infty }$, ${\left\{z_i^n\right\}}_{n=0}^{\infty }$ $(i=1,2)$ generated by Algorithm 4 converge strongly to $x_i,u_i,v_i,w_i,z_i$, respectively, and $(x_1,x_2,u_1,u_2,v_1,v_2,w_1,w_2,z_1,z_2)$ is a solution of the $\mathrm{SGIVLI}$ (6).

Algorithm 4 Actually it is the same as Algorithm 2 and it is not a new one

For each $i\in \{1,2\}$, let $E_i$ be a real reflexive Banach space with a norm ${\parallel .\parallel }_i$ and the dual space $E_i^{*}$. Suppose that for $i=1,2$, the mappings $N_i,P_i,Q_i,S_i,T_i,C_i,D_i,F_i,G_i$ and the constants ${\rho }_i>0$ are the same as in the $\mathrm{SGIVLI}$ (6). Let for each $i\in \{1,2\}$, ${\eta }_i:E_i\times E_i\to E_i$ is a ${\tau }_i$-Lipschitz continuous mapping such that ${\eta }_i(y_i,{\widehat{y}}_i)+{\eta }_i({\widehat{y}}_i,y_i)=0$ for all $y_i,{\widehat{y}}_i\in E_i$. Assume that for each $i\in \{1,2\}$, $M_i(A_i,B_i):E_i\to E_i^{*}$ is an ${\alpha }_i{\beta }_i$-symmetric ${\eta }_i$-monotone continuous mapping with respect to $A_i,B_i:E_i\to E_i$ such that $g_i(E_i)\subseteq \mathrm{dom}(M_i(A_i,B_i))$. Let for each $i\in \{1,2\}$ and for any given $x_i^{*}\in E_i^{*}$, the function $h_i:(y_i,{\widehat{x}}_i)\in E_i\times E_i\to h_i(y_i,{\widehat{x}}_i)={\langle x_i^{*}-M_i(A_i{\widehat{x}}_i,B_i{\widehat{x}}_i),{\eta }_i(y_i,{\widehat{x}}_i)\rangle }_i\in ℝ\cup \{+\infty \}$ be $\mathrm{0-DQCV}$ in the first argument. Furthermore, let for each $i\in \{1,2\}$, ${\varphi }_i:E_i\to ℝ\cup \{+\infty \}$ be a lower semicontinuous and ${\eta }_i$-subdifferentiable proper functional on $E_i$ with $g_i(E_i)\cap \mathrm{dom}\partial {\varphi }_i\ne \varnothing$. For any given $(x_1^0,x_2^0,u_1^0,u_2^0,v_1^0,v_2^0,w_1^0,w_2^0,z_1^0,z_2^0)$, where $x_i^0\in E_i$, $u_i^0\in C_i(x_j^0)$, $v_i^0\in D_i(x_i^0)$, $w_i^0\in F_i(x_i^0)$, $z_i^0\in G_i(x_j^0)$ for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, compute the sequences ${\left\{x_i^n\right\}}_{n=0}^{\infty }$, ${\left\{u_i^n\right\}}_{n=0}^{\infty }$, ${\left\{v_i^n\right\}}_{n=0}^{\infty }$, ${\left\{w_i^n\right\}}_{n=0}^{\infty }$, ${\left\{z_i^n\right\}}_{n=0}^{\infty }$ by the iterative schemes $$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{x}_i^{n+1}=(1-\mu )x_i^n+\mu \{x_i^n-g_i(x_i^n)+R_{{\rho }_i,{\eta }_i}^{\partial {\varphi }_i,M_i(A_i,B_i)}[(M_i(A_i,B_i)\circ g_i)(x_i^n)\hfill \\ -{\lambda }_i(N_i(S_i(x_i^n),T_i(x_j^n))+{\rho }_i(P_i(u_i^n,v_i^n)+Q_i(w_i^n,z_i^n)))]\},\hfill \\ u_i^n\in C_i(x_j^n):{\parallel u_i^{n+1}-u_i^n\parallel }_{*j}\le H_{*j}(C_i(x_j^{n+1}),C_i(x_j^n)),\hfill \\ v_i^n\in D_i(x_i^n):{\parallel v_i^{n+1}-v_i^n\parallel }_{*i}\le H_{*i}(D_i(x_i^{n+1}),D_i(x_i^n)),\hfill \\ w_i^n\in F_i(x_i^n):{\parallel w_i^{n+1}-w_i^n\parallel }_{*i}\le H_{*i}(F_i(x_i^{n+1}),F_i(x_i^n)),\hfill \\ z_i^n\in G_i(x_j^n):{\parallel z_i^{n+1}-z_i^n\parallel }_{*j}\le H_{*j}(G_i(x_j^{n+1}),G_i(x_j^n)),\hfill \end{array}\right.\end{array}\end{array}$$ (64) where $n=0,1,2,\cdots$; ${\lambda }_i>0$ $(i=1,2)$ are constants; $\mu \in (0,1]$ is a relaxation parameter; and $H_i$ and $H_{*i}$ are Hausdorff metric on $CB(E_i)$ and $CB(E_i^{*})$, respectively.

Proof. 

Let us define the mapping ${\widehat{P}}_i:E_i\to E_i^{*}$ for each $i\in \{1,2\}$ by ${\widehat{P}}_i(x_i):=M_i(A_i(x_i),$ $B_i(x_i))$ for all $x_i\in E_i$. In accordance with Proposition 1, ${\widehat{P}}_i$ is an $({\alpha }_i-{\beta }_i)$-strongly ${\eta }_i$-monotone mapping for each $i\in \{1,2\}$. Since for each $i\in \{1,2\}$, $g_i(E_i)\subseteq \mathrm{dom}(M_i(A_i,B_i))$, it follows that for each $i\in \{1,2\}$, $g_i(E_i)\subseteq \mathrm{dom}({\widehat{P}}_i)$. Taking ${\gamma }_i={\alpha }_i-{\beta }_i$ and $L_{{\widehat{P}}_i}=L_{M_i(A_i,B_i)}$ for each $i\in \{1,2\}$, we observe that all the conditions of Corollary 3 are satisfied. Now, the statement yields by Corollary 3 immediately.   □

Definition 13 ([30], Definition 2.5).

Let $E_1$, $E_2$ be real Banach spaces; let $T:E_1\times E_2\to E_1$ and $S:E_1\times E_2\to E_2$. Let $Q:E_1\times E_2\to E_1\times E_2$ be defined as $Q(x,y)=(T(x,y),S(x,y))$ for any $(x,y)\in E_1\times E_2$, and let $(x^0,y^0)\in E_1\times E_2$. Assume that $(x^{n+1},y^{n+1})=f(Q,x^n,y^n)=(g(T,x^n,y^n),g(S,x^n,y^n))$ define an iteration procedure which yields a sequence of points ${\left\{(x^n,y^n)\right\}}_{n=0}^{\infty }$ in $E_1\times E_2$. Suppose that $\mathrm{F}(Q)=\{(x,y)\in E_1\times E_2:(x,y)=Q(x,y)\}\ne \varnothing$ and ${\left\{(x^n,y^n)\right\}}_{n=0}^{\infty }$ converges to some $(p,q)\in \mathrm{F}(Q)$. Let ${\left\{(u^n,v^n)\right\}}_{n=0}^{\infty }$ be an arbitrary sequence in $E_1\times E_2$ and ${ϵ}_n=\parallel (u^{n+1},v^{n+1})-f(Q,x^n,y^n)\parallel$, $\forall n\ge 0$. If $\underset{n\to \infty }{lim}{ϵ}_n=0$ implies that $\underset{n\to \infty }{lim}(u^n,v^n)=(p,q)$, then the iteration procedure defined by $(x^{n+1},y^{n+1})=f(Q,x^n,y^n)$ is said to be Q-stable or stable with respect to Q. If ${\displaystyle \sum _{n=0}^{\infty }}{ϵ}^n<+\infty$ implies that $\underset{n\to \infty }{lim}(u^n,v^n)=(p,q)$, then the iterative procedure ${\left\{(x^n,y^n)\right\}}_{n=0}^{\infty }$ is said to be almost Q-stable.

We now end this section with the last result in [30], in which the authors studied the stability analysis of the sequences generated by Iterative Algorithm 5.1 in [30].

Theorem 12 ([30], Theorem 5.2).

Let for $i\in \{1,2\}$, $j\in \{1,2\}\backslash \left\{i\right\}$, $E_i$, ${\eta }_i^n$, ${\eta }_i$, $A_i^n$, $B_i^n$, $A_i$, $B_i$, $M_i^n(A_i^n,B_i^n)$, $M_i(A_i,B_i)$, ${\varphi }_i^n$, ${\varphi }_i$, $h_i^n$, $h_i$, $N_i$, $Q_i$, $P_i$, $S_i$, $T_i$, $C_i$, $G_i$, $D_i$, $F_i$, $g_i$, $N_i(S_i(.),T_i(x_j))$, $M_i(A_i,B_i)\circ g_i$ be the same as in Theorem 10 (that is, ([30], Theorem 5.1)) and let the condition (59) of Theorem 10 (that is, ([30], the Condition (5.2) of Theorem 5.1)) holds. Let Conditions 1 and 2 hold and let ${\left\{{\overline{x}}_1^n\right\}}_{n=0}^{\infty }$, ${\left\{{\overline{x}}_2^n\right\}}_{n=0}^{\infty }$ be the sequences in $E_1$ and $E_2$, respectively. Define $\left\{{ϵ}^n\right\}\subseteq [0,\infty )$ by

$$\begin{array}{c}\hfill {ϵ}^n=\sum _{i=1}^2{\parallel {\overline{x}}_i^{n+1}-A_i\parallel }_i,\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}& A_i=(1-{\mu }^n){\overline{x}}_i^n+{\mu }^n\{{\overline{x}}_i^n-g_i({\overline{x}}_i^n)+R_{{\rho }_i,{\eta }_i^n}^{\partial {\varphi }_i^n,M_i^n(A_i^n,B_i^n)}[(M_i(A_i^n,B_i^n)\circ g_i)({\overline{x}}_i^n)\hfill \\ & -{\lambda }_i(N_i(S_i({\overline{x}}_i^n),T_i({\overline{x}}_j^n))+{\rho }_i(P_i({\overline{u}}_i^n,{\overline{v}}_i^n)+Q_i({\overline{w}}_i^n,{\overline{z}}_i^n))]\},\hfill \\ & {\overline{u}}_i^n\in C_i({\overline{x}}_j^n):{\parallel {\overline{u}}_i^{n+1}-{\overline{u}}_i^n\parallel }_{*j}\le H_{*j}(C_i({\overline{x}}_j^{n+1}),C_i({\overline{x}}_j^n)),\hfill \\ & {\overline{v}}_i^n\in D_i({\overline{x}}_i^n):{\parallel {\overline{v}}_i^{n+1}-{\overline{v}}_i^n\parallel }_{*i}\le H_{*i}(D_i({\overline{x}}_i^{n+1}),D_i({\overline{x}}_i^n)),\hfill \\ & {\overline{w}}_i^n\in F_i({\overline{x}}_i^n):{\parallel {\overline{w}}_i^{n+1}-{\overline{w}}_i^n\parallel }_{*i}\le H_{*i}(F_i({\overline{x}}_i^{n+1}),F_i({\overline{x}}_i^n)),\hfill \\ & {\overline{z}}_i^n\in G_i({\overline{x}}_j^n):{\parallel {\overline{z}}_i^{n+1}-{\overline{z}}_i^n\parallel }_{*j}\le H_{*j}(G_i({\overline{x}}_j^{n+1}),G_i({\overline{x}}_j^n)),\hfill \end{array}\end{array}$$

(65)

$n=0,1,2,\cdots$; ${\rho }_i,{\lambda }_i>0$ are constants; and $0<{\mu }^n<1$ are relaxation constants such that ${\displaystyle \sum _{n=0}^{\infty }}{\mu }^n=\infty$.

Then

(a)

If ${\displaystyle \sum _{n=0}^{\infty }}{ϵ}^n<\infty$, then $\underset{n\to \infty }{lim}({\overline{x}}_i^n,{\overline{u}}_i^n,{\overline{v}}_i^n,{\overline{w}}_i^n,{\overline{z}}_i^n)=(x_i,u_i,v_i,w_i,z_i)$;

(b)

If $\underset{n\to \infty }{lim}({\overline{x}}_i^n,{\overline{u}}_i^n,{\overline{v}}_i^n,{\overline{w}}_i^n,{\overline{z}}_i^n)=(x_i,u_i,v_i,w_i,z_i)$ then $\underset{n\to \infty }{lim}{ϵ}^n=0$, where $(x_i,u_i,v_i,w_i,z_i)$ is a solution of the $\mathrm{SGIVLI}$ (6) (that is, ([30], the $\mathrm{SGIVLI}$ (4.1)–(4.2))).

By an easy checking, we found that the proof of Theorem 12 has inherited the fatal errors existing in the proof of Theorem 10. Indeed, on page 9218, the authors deduced the existence of a solution $(x_1,x_2,u_1,u_2,v_1,v_2,w_1,w_2,z_1,z_2)$ for the $\mathrm{SGIVLI}$ (6) (that is, ([30], the $\mathrm{SGIVLI}$ (4.1)–(4.2))) utilizing Theorem 7. However, as it was shown, the statement of Theorem 10 is not true necessarily. Even without considering this fact, they derived (5.13) in [30] as an estimate of $\parallel A_i-x_i{\parallel }_i$ for each $i\in \{1,2\}$, using the same arguments used in obtaining (60)–(62) ((5.5)–(5.7) in [30]) as follows:

$$\begin{array}{c}\hfill \parallel A_i-x_i{\parallel }_i\le (1-{\mu }^n)\parallel {\overline{x}}_i^n-x_i{\parallel }_i+{\mu }^n[a_i^n\parallel {\overline{x}}_i^n-x_i{\parallel }_i+b_i\parallel {\overline{x}}_i^n-x_i{\parallel }_i]+{\mu }^n{\varphi }^n.\end{array}$$

(66)

We discovered that with the help of (65) and in light of the fact that for each $i\in \{1,2\}$ and $j\in \{1,2\}\backslash \left\{i\right\}$, $u_i\in C_i(x_j)$, $v_i\in D_i(x_i)$, $w_i\in F_i(x_i)$ and $z_i\in G_i(x_j)$, they concluded the following inequalities by using Lemma 3 and used them in computing (66):

$$\begin{array}{c}\hfill \begin{array}{cc}& \parallel {\overline{u}}_i^n-u_i{\parallel }_{*j}\le H_{*j}(C_i({\overline{x}}_j^n),C_i(x_j)),\hfill \\ & \parallel {\overline{v}}_i^n-v_i{\parallel }_{*i}\le H_{*i}(D_i({\overline{x}}_i^n),D_i(x_i)),\hfill \\ & \parallel {\overline{w}}_i^n-w_i{\parallel }_{*i}\le H_{*i}(F_i({\overline{x}}_i^n),F_i(x_i)),\hfill \\ & \parallel {\overline{z}}_i^n-z_i{\parallel }_{*j}\le H_{*j}(G_i({\overline{x}}_j^n),G_i(x_j)).\hfill \end{array}\end{array}$$

(67)

At the same time, on page 9218 of [30], lines 2 and 3 from the bottom, the authors obtained an estimate of ${ϵ}^n$ by employing the last inequalities in (67) and deduced that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {ϵ}^n& \le \parallel ({\overline{x}}_1^{n+1},{\overline{x}}_2^{n+1})-(x_1,x_2){\parallel }_{*}+{\parallel (A_1,A_2)-(x_1,x_2)\parallel }_{*}\hfill \\ & \le \parallel ({\overline{x}}_1^{n+1},{\overline{x}}_2^{n+1})-(x_1,x_2){\parallel }_{*}+[1-(1-max\{k_1,k_2\}){\mu }^n]{\parallel ({\overline{x}}_1^n,{\overline{x}}_2^n)-(x_1,x_2)\parallel }_{*}\to 0,\hfill \end{array}\end{array}$$

as $n\to \infty$. But, as it was pointed out, Example 2 shows that the previous inequalities in (67) do not hold necessarily.

6. Concluding Remark

With the goal of solving some classes of generalized multivalued nonlinear variational-like inequalities in the setting of Banach spaces, the notion of $J^{\eta }$-proximal (also referred to as P-$\eta$-proximal) mapping for a nonconvex lower semicontinuous $\eta$-subdifferentiable proper functional in the setting of reflexive Banach spaces was initially introduced, independently, by Ahmad et al. [7] and Kazmi and Bhat [22]. In this paper, we have pursued two purposes. We have first studied a new system of generalized multivalued variational-like inequalities in Banach spaces and proved its equivalence with a system of fixed point problems using the concept of P-$\eta$-proximal mapping. Under some appropriate assumptions imposed on the mappings and parameters involved in the system of generalized multivalued variational-like inequalities, we have proved the existence of solutions for the system mentioned above and discussed the convergence analysis of the sequences generated by our proposed iterative algorithm. We have also investigated and analyzed the notion M-$\eta$–proximal mapping defined in [30], and demonstrated that every M-$\eta$-proximal mapping is actually P-$\eta$-proximal mapping, and it is not a new one. At the same time, we have pointed out some comments relating to the results appearing in [30].