New generalized systems of nonlinear ordered variational inclusions involving ⊕ operator in real ordered Hilbert spaces

Abstract

This manuscript deals with two general systems of nonlinear ordered variational inclusion problems. We also construct some new iterative algorithms for finding approximation solutions to the general systems of nonlinear ordered variational inclusions and prove the convergence of the sequences obtained by the schemes. The results presented in the manuscript are new and improve some well-known results in the literature.

Introduction

A lot of work has been added into the theory of variational inequalities since its seed was planted by Lions et al. [24]. On account of its wide applications in physics and applied sciences etc., the classical variational inequalities have been extensively studied by many researchers in different ways [1, 4, 5, 7–10].

A useful and important generalization of variational inequality problem is variational inclusion problem which was introduced and studied by Hasounni et al. [16]. Furthermore, they proposed a perturbed iterative algorithm for solving the variational inclusion problem.

Fang et al. [12] introduced and studied H-monotone operators, which was used to design a resolvent operator and to prove its Lipschitz continuity. Furthermore, they also introduced a class of variational inclusions in Hilbert space. Fang et al. [13] additionally presented another class of generalized monotone operators, called $(H,\eta )$-monotone operators, which generalize different classes of maximal monotone, maximal η-monotone and H-monotone operators.

Recently, Lan et al. [17] presented another idea of $(A,\eta )$-accretive mappings, which generalized the current monotone or accretive operators, and concentrated a few properties of mappings. They examined a class of variational inclusions using the resolvent operator related with $(A,\eta )$-accretive mappings.

Amann [6] studied the number of fixed points for a continuous operator $A:[x,y]\to [x,y]$ on a bounded order interval $[x,y]\subset \mathcal{E}$, an ordered Banach space. The nonlinear mapping fixed point theory and applications have been widely studied in ordered Banach spaces [4, 14, 15]. In this manner, it is essential that summed up nonlinear ordered variational inclusions (ordered equation) are contemplated.

Plenty of research concerned with the ordered equations and ordered variational inequalities in ordered Banach spaces has been done by Li et al.; see [21, 23]. Many problems concerning ordered variational inclusions are answered by the resolvent technique linked with RME set-valued mappings [19], $(\alpha ,\lambda )$-NODM set-valued mapping [20], $({\gamma }_G,\lambda )$-weak RRD mapping [2] and $(\alpha ,\lambda )$-weak ANODD set-valued map with strongly comparison mapping A [21] and many more see; e.g., [3, 22, 25, 26, 29] and the references therein.

In this work, we make use of the resolvent operator approach for the approximation solvability of solutions of implicit system of generalized nonlinear ordered variational inclusions in real ordered Hilbert spaces.

Preliminaries

In this part, we present some basic notions and results for the building up the manuscript.

Allow $\mathcal{E}$ to be a real ordered Hilbert space endowed with a norm $\parallel \cdot \parallel$, and an inner product $\langle \cdot ,\cdot \rangle$, d be a metric induced by the norm $\parallel \cdot \parallel$, $CB(\mathcal{E})$ be a collection of all closed and bounded subsets of $\mathcal{E}$ and $D(\cdot ,\cdot )$ be a Hausdorff metric on $CB(\mathcal{E})$ defined as

$$D(M,N)=max\{\underset{x\in M}{sup}d(x,N),\underset{y\in N}{sup}d(M,y)\},$$

where $M,N\in CB(\mathcal{E})$, $d(x,N)={inf}_{y\in N}d(x,y)$ and $d(M,y)={inf}_{x\in M}d(x,y)$.

Definition 2.1

Let $\mathfrak{C}$ be a nonvoid closed, convex subset of $\mathcal{E}$. Then $\mathfrak{C}$ is called a cone if

1. $x\in \mathfrak{C}$ and $\kappa >0$, $\kappa x\in \mathfrak{C}$;

2. x and $-x\in \mathfrak{C}$, then $x=\mathrm{\Theta }$.

Definition 2.2

([11])

A cone $\mathfrak{C}$ is said to be normal iff there exists ${\lambda }_{N_C}>0$ with $0\le x\le y$ implying $\parallel x\parallel \le {\lambda }_{N_C}\parallel y\parallel$, where ${\lambda }_{N_C}$ is called a normal constant of $\mathfrak{C}$.

Definition 2.3

A relation ≤ defined as $x\le y$ iff $y-x\in \mathfrak{C}$ for $x,y\in \mathcal{E}$ is known as a partial order relation expounded by $\mathfrak{C}$ in $\mathcal{E}$; then ($\mathcal{E},\le$) is called a real ordered Hilbert space.

Definition 2.4

([27])

Members $x,y\in \mathcal{E}$ having the relation $x\le y$ (or $y\le x$) are called comparable with each other.

Definition 2.5

([27])

For arbitrary elements $x,y\in \mathcal{E}$, $lub\{x,y\}$ and $glb\{x,y\}$ mean the least upper bound and the greatest upper bound of the set $\{x,y\}$. Suppose $lub\{x,y\}$ and $glb\{x,y\}$ exist; some binary relations are defined as follows:

1. $x\vee y=lub\{x,y\}$;

2. $x\wedge y=glb\{x,y\}$;

3. $x\oplus y=(x-y)\vee (y-x)$;

4. $x\odot y=(x-y)\wedge (y-x)$.

The operations ∨, ∧, ⊕ and ⊙ are called OR, AND, XOR and XNOR operations, respectively.

Proposition 1

([11])

For any positive integer n, if $x\propto y_n$ and $y_n\to y^{\ast }$ ($n\to \mathrm{\infty }$), then $x\propto y^{\ast }$.

Proposition 2

([11, 20])

Let $\mathbf{\text{XOR}},\mathbf{\text{XNOR}}$ be two operations on $\mathcal{E}$. Then the following hold:

1. $x\odot x=0,x\odot y=y\odot x=-(x\oplus y)=-(y\oplus x)$;

2. $(\lambda x)\oplus (\lambda y)=|\lambda |(x\oplus y)$;

3. $x\odot 0\le 0$, if $x\propto 0$;

4. $0\le x\oplus y$, if $x\propto y$;

5. if $x\propto y$, then $x\oplus y=0$ if and only if $x=y$;

6. $(x+y)\odot (u+v)\ge (x\odot u)+(y\odot v)$;

7. $(x+y)\odot (u+v)\ge (x\odot v)+(y\odot u)$;

8. $(\alpha x\oplus \beta x)=|\alpha -\beta |x$, if $x\propto 0$, $\mathrm{\forall }x,y,u,v\in \mathcal{E}$ and $\alpha ,\beta ,\lambda \in \mathbb{R}$.

Proposition 3

([11])

Let $\mathfrak{C}$ be a normal cone in $\mathcal{E}$ with normal constant ${\lambda }_{N_C}$, then, for each $x,y\in \mathcal{E}$, the following hold:

1. $\parallel 0+0\parallel =\parallel 0\parallel =0$;

2. $\parallel x\vee y\parallel \le \parallel x\parallel \vee \parallel y\parallel \le \parallel x\parallel +\parallel y\parallel$;

3. $\parallel x\oplus y\parallel \le \parallel x-y\parallel \le {\lambda }_{N_C}\parallel x\oplus y\parallel$;

4. if $x\propto y$, then $\parallel x\oplus y\parallel =\parallel x-y\parallel$.

Definition 2.6

([20])

Let $A:\mathcal{E}\to \mathcal{E}$ to be a single-valued map.

1. A is called a δ-order non-extended map, if there is a positive constant $\delta >0$ such that

   $$\delta (x\oplus y)\le A(x)\oplus A(y)\text{for all }x,y\in \mathcal{E};$$

2. A is called a strongly comparison map, if it is a comparison map and $A(x)\propto A(y)$ iff $x\propto y$, for all $x,y\in \mathcal{E}$.

Definition 2.7

([2])

A single-valued map $A:\mathcal{E}\to \mathcal{E}$ is termed a β-ordered compression, if it is comparison map and

$$A(x)\oplus A(y)\le \beta (x\oplus y),\text{for }0<\beta <1.$$

Definition 2.8

([18])

A map $A:\mathcal{E}\times \mathcal{E}\to \mathcal{E}$ is called $({\alpha }_1,{\alpha }_2)$-restricted-accretive map, if it is a comparison and ∃ constants $0\le {\alpha }_1,{\alpha }_2\le 1$ such that

$$(A(x,\cdot )+I(x))\oplus (A(y,\cdot )+I(y))\le {\alpha }_1(A(x,\cdot )\oplus A(y,\cdot ))+{\alpha }_2(x\oplus y),\text{for all }x,y\in \mathcal{E}$$

where I is the identity map on $\mathcal{E}$.

Lemma 2.1

([28])

Let $\theta \in (0,1)$ be a constant. Then the function $f(\lambda )=1-\lambda +\lambda \theta$ for $\lambda \in [0,1]$ is nonnegative and strictly decreases and $f(\lambda )\in [0,1]$. Furthermore, if $\lambda \ne 0$, then $f(\lambda )\in (0,1)$.

Lemma 2.2

([30])

Assume that $\{a_n\}$ and $\{b_n\}$ be two sequences of nonnegative real numbers such that

$$a_{n+1}\le \theta a_n+b_n,$$

where $\theta \in (0,1)$ and ${lim}_{n\to \mathrm{\infty }}{b}_n=0$. Then ${lim}_{n\to \mathrm{\infty }}{a}_n=0$.

Ordered weak-ARD mapping in ordered Hilbert spaces

Definition 3.1

Let $A:\mathcal{E}\to \mathcal{E}$ be a strong comparison and β-ordered compression mapping and $M:\mathcal{E}\to CB(\mathcal{E})$ be a set-valued mapping. Then

1. M is said to be a comparison mapping, if for any $v_x\in M(x)$, $x\propto v_x$ and if $x\propto y$, then, for any $v_x\in M(x)$ and any $v_y\in M(y)$, $v_x\propto v_y$, for all $x,y\in \mathcal{E}$;

2. a comparison mapping M is said to be ordered rectangular, if for each $x,y\in \mathcal{E}$, $v_x\in M(x)$ and $v_y\in M(y)$ such that

   $$\langle v_x\odot v_y,-(x\oplus y)\rangle =0;$$
3. a comparison mapping M is said to be a γ-ordered rectangular with respect to A, if there exists a constant ${\gamma }_A>0$ for any $x,y\in \mathcal{E}$, there exist $v_x\in M(A(x))$ and $v_y\in M(A(y))$ such that

   $$\langle v_x\odot v_y,-(A(x)\oplus A(y))\rangle \ge {\gamma }_A{\parallel A(x)\oplus A(y)\parallel }^2,$$

   holds, where $v_x$ and $v_y$ are said to be ${\gamma }_A$-elements, respectively;

4. M is said to be a weak comparison mapping with respect to A, if, for any $x,y\in \mathcal{E}$, $x\propto y$, there exist $v_x\in M(A(x))$ and $v_y\in M(A(y))$ such that $x\propto v_x$, $y\propto v_y$, where $v_x$ and $v_y$ are said to be weak comparison elements, respectively;

5. M is said to be a λ-weak ordered different comparison mapping with respect to A, if there exists a constant $\lambda >0$ such that, for any $x,y\in \mathcal{E}$, there exist $v_x\in M(A(x))$ and $v_y\in M(A(y))$, $\lambda (v_x-v_y)\propto (x-y)$ holds, where $v_x$ and $v_y$ are said to be λ-elements, respectively;

6. a weak comparison mapping M is said to be a $({\gamma }_A,\lambda )$-weak ARD mapping with respect to A, if M is a ${\gamma }_A$-ordered rectangular and λ-weak ordered different comparison mapping with respect to A and $(A+\lambda M)(\mathcal{E})=\mathcal{E}$, for $\lambda >0$ and there exist $v_x\in M(A(x))$ and $v_y\in M(A(y))$ such that $v_x$ and $v_y$ are $({\gamma }_A,\lambda )$-elements, respectively.

Definition 3.2

A set-valued mapping $A:\mathcal{E}\to CB(\mathcal{E})$ is said to be ${\delta }_A$-Lipschitz continuous, if for each $x,y\in \mathcal{E},x\propto y$, there exists a constant ${\delta }_A$ such that

$$D(A(x),A(y))\le {\delta }_A\parallel x\oplus y\parallel ,\mathrm{\forall }x,y\in \mathcal{E}.$$

Definition 3.3

Let $M:\mathcal{E}\to CB(\mathcal{E})$ be a set-valued mapping, $A:\mathcal{E}\to \mathcal{E}$ be a single-valued mapping and $I:\mathcal{E}\to \mathcal{E}$ be an identity mapping. Then a weak comparison mapping M is said to be a $({\gamma }^{\prime },\lambda )$-weak-ARD mapping with respect to $(I-A)$, if M is a ${\gamma }^{\prime }$-ordered rectangular and λ-weak ordered different comparison mapping with respect to $(I-A)$ and $[(I-A)+\lambda M](\mathcal{E})=\mathcal{E}$, for $\lambda >0$ and there exist $v_x\in M((I-A)(x))$ and $v_y\in M((I-A)(y))$ such that $v_x$ and $v_y$ are called $({\gamma }^{\prime },\lambda )$-elements, respectively.

Definition 3.4

Let $\mathfrak{C}$ be a normal cone with normal constant ${\lambda }_{N_C}$ and $M:\mathcal{E}\to CB(\mathcal{E})$ be a weak-ARD set-valued mapping. Let $I:\mathcal{E}\to \mathcal{E}$ be the identity mapping and $A:\mathcal{E}\to \mathcal{E}$ be a set-valued mapping and $A:\mathcal{E}\to \mathcal{E}$ be a single-valued mapping. The relaxed resolvent operator $R_{M,\lambda }^{(I-A)}:\mathcal{E}\to \mathcal{E}$ associated with I, A and M is defined by

$$R_{M,\lambda }^{(I-A)}(x)={[(I-A)+\lambda M]}^{-1}(x),\mathrm{\forall }x\in \mathcal{E}\text{ and }\lambda >0.$$ 1

The relaxed resolvent operator defined by (1) is single-valued, a comparison mapping and Lipschitz continuous.

Proposition 4

([2])

Let $A:\mathcal{E}\to \mathcal{E}$ be a β-ordered compression mapping and $M:\mathcal{E}\to CB(\mathcal{E})$ be the set-valued ordered rectangular mapping. Then the resolvent $R_{M,\lambda }^{(I-A)}:\mathcal{E}\to \mathcal{E}$ is single-valued, for all $\lambda >0$.

Proposition 5

([2])

Let $M:\mathcal{E}\to CB(\mathcal{E})$ be a $({\gamma }_A,\lambda )$-weak-ARD set-valued mapping with respect to $R_{M,\lambda }^{(I-A)}$. Let $A:\mathcal{E}\to \mathcal{E}$ be a strongly comparison mapping with respect to $R_{M,\lambda }^{(I-A)}$ and $I:\mathcal{E}\to \mathcal{E}$ be the identity mapping. Then the resolvent operator $R_{M,\lambda }^{(I-A)}:\mathcal{E}\to \mathcal{E}$ is a comparison mapping.

Proposition 6

([2])

Let $M:\mathcal{E}\to CB(\mathcal{E})$ be a $({\gamma }_A,\lambda )$-weak-ARD set-valued mapping with respect to $R_{M,\lambda }^{(I-A)}$. Let $A:\mathcal{E}\to \mathcal{E}$ be a strongly comparison and β-ordered compression mapping with respect to $R_{M,\lambda }^{(I-A)}$ with condition $\lambda {\gamma }_A>\beta +1$. Then the following condition survives:

$$\parallel R_{M,\lambda }^{(I-A)}(x)\oplus R_{M,\lambda }^{(I-A)}(y)\parallel \le \left(\frac{1}{\lambda {\gamma }_A-\beta -1}\right)\parallel x\oplus y\parallel .$$

Formulation of the problems

Let $F_i:{\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m\to {\mathcal{E}}_i$, $A_i:{\mathcal{E}}_i\to {\mathcal{E}}_i$ and $g_i:{\mathcal{E}}_i\to {\mathcal{E}}_i$ to be single-valued mappings, for $i,j=1,2,3,\dots ,m$. Let $U_{ij}:{\mathcal{E}}_j\to CB({\mathcal{E}}_j)$ be a set-valued map and $M_i:{\mathcal{E}}_i\to CB({\mathcal{E}}_i)$ be set-valued weak-ARD mapping. Then we have the problem:

Find $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })\in {\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m$ and $u_{ij}^{\ast }\in U_{ij}(x_j^{\ast })$, for $i,j=1,2,3,\dots m$, such that

$$0\in {\rho }_i{F}_i(u_{i1}^{\ast },u_{i2}^{\ast },\dots ,u_{im}^{\ast })\oplus {\lambda }_i{M}_i\left(g_i\left(x_i^{\ast }\right)\right),$$ 2

where ${\rho }_i$ and ${\lambda }_i$ are given positive constants. Problem (2) is called a generalized set-valued system of nonlinear ordered variational inclusions problem for weak-ARD mappings.

If $U_{ij}=T_{ij}$ is a single-valued mapping, then problem (2) becomes:

Find $x_j\in {\mathcal{E}}_j$, such that

$$0\in {\rho }_i{F}_i(T_{i1}{x}_1^{\ast },T_{i2}{x}_2^{\ast },\dots ,T_{im}{x}_m^{\ast })\oplus {\lambda }_i{M}_i\left(g_i\left(x_i^{\ast }\right)\right).$$ 3

This problem is known as a generalized system of nonlinear ordered variational inclusions problem involving weak-ARD mappings.

Remark

Here, we discuss special cases for our problem (2), which was encountered by Li et al.

Case 1.

For $i,j=1$, ${\rho }_i=1$, ${\lambda }_i=1$ and $U_{ij}=g_i=I$, then problem (2) is reduced to finding $x\in {\mathcal{E}}_1$ such that

$$0\in F_1(x)\oplus M_1(x).$$ 4

This problem was considered by Li et al. [23] and coined a general nonlinear mixed-order quasi-variational inclusion (GNMOQVI) involving the ⊕ operator in an ordered Banach space.

Case 2.

If $F=0$ (zero mapping), then problem (4) is reduced to finding $x\in \mathcal{E}$ such that

$$0\in M(x).$$ 5

This problem were considered by Li for ordered RME set-valued mappings [19] and $(\alpha ,\lambda )$-NODM set-valued mappings [20].

Lemma 4.1

Let $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })\in {\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m$ and $u_{ij}^{\ast }\in U_{ij}(x_j^{\ast })$ for $i,j=1,2,3,\dots ,m$. Then $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast },u_{11}^{\ast },u_{12}^{\ast },\dots ,u_{1m}^{\ast },\dots ,u_{m1}^{\ast },u_{m2}^{\ast },\dots ,u_{mm}^{\ast })$ is a solution of problem (2) if and only if it satisfies

$$g_i\left(x_i^{\ast }\right)=J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^{\ast }\right)\right)+{\rho }_i{F}_i(u_{i1}^{\ast },u_{i2}^{\ast },\dots ,u_{im}^{\ast })],$$ 6

where $J_{{\lambda }_i,M_i}^{I_i-A_i}(x)={[(I_i-A_i)+{\lambda }_i{M}_i]}^{-1}(x)$ and ${\rho }_i,{\lambda }_i>0$ for $i=1,2,\dots ,m$.

Proof

The proof follows from the definition of the relaxed resolvent operator. □

Design of the algorithms

Remark

If we choose $\lambda =1$ and $U_{ij}=T_{ij}$ for $i,j=1,2,\dots ,m$, is single-valued operator, then Algorithm 1 reduces to Algorithm 2 for problem (3).

Algorithm 1.

for the problem ( 2 ):

Algorithm 2.

for the problem ( 3 ):

Main results

Theorem 6.1

Let $A_i:{\mathcal{E}}_i\to {\mathcal{E}}_i$, $g_i:{\mathcal{E}}_i\to {\mathcal{E}}_i$ and $F_i:{\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m\to {\mathcal{E}}_i$ be the single-valued mappings such that $A_i$ be ${\lambda }_{A_i}$-ordered compression mapping, $g_i$ be ${\lambda }_{g_i}$-ordered compression, $({\alpha }_1^i,{\alpha }_2^i)$-ordered restricted-accretive mapping and $F_i$ be ${\lambda }_{ij}$-ordered compression mapping with respect to the jth argument. Let $U_{ij}:{\mathcal{E}}_j\to CB({\mathcal{E}}_j)$ be a $D_i$-${\delta }_{ij}$-ordered Lipschitz continuous set-valued mapping. Let $M_i:{\mathcal{E}}_i\to CB({\mathcal{E}}_i)$ be a $({\gamma }_{A_i},{\lambda }_i)$-weak rectangular different compression mapping with respect to $A_i$ and if $x_i\propto y_i$, $J_{{\lambda }_i,M_i}^{I_i-A_i}(x_i)\propto J_{{\lambda }_i,M_i}^{I_i-A_i}(y_i)$ and for all ${\lambda }_i$, ${\rho }_i>0$, then the following condition holds:

$${\theta }_j=\{{\alpha }_1^j+{\alpha }_2^j{\lambda }_{g_j}+L_j({\lambda }_{g_j}+{\lambda }_{A_j}{\lambda }_{g_j})+\sum _{i\ne j,i=1}^m{L}_i{\rho }_i{\lambda }_{F_{ij}}{\delta }_{D_{ij}}\}<1,$$ 9

for all $j=1,2,3,\dots ,m$, which in turn, implies that problem (2) admits a solution $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast },u_{11}^{\ast },u_{12}^{\ast },\dots ,u_{1m}^{\ast },\dots ,u_{m1}^{\ast },\dots ,u_{mm}^{\ast })$, where $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })\in {\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m$ and $u_{ij}^{\ast }\in U_{ij}(x_j^{\ast })$. Moreover, iterative sequences $\{x_j^n\}$ and $\{u_{ij}^n\}$ generated by Algorithm 1, converge strongly to $x_j^{\ast }$ and $u_{ij}^{\ast }$, for $i,j=1,2,\dots ,m$, respectively.

Proof

Using Algorithm 1 and Proposition 2, for $i=1,2,\dots ,m$, we have

$$\begin{array}{rl}{x}_i^{n+1}\oplus x_i^n=& ((1-\lambda )x_i^n+\lambda [x_i^n-g_i\left(x_i^n\right)+J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^n\right)\right)\\ & +{\rho }_i{F}_i(u_{i1}^n,u_{i2}^n,\dots ,u_{im}^n)\left]\right])\oplus ((1-\lambda )x_i^{n-1}\\ & +\lambda [x_i^{n-1}-g_i\left(x_i^{n-1}\right)+J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^{n-1}\right)\right)\\ & +{\rho }_i{F}_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{im}^{n-1})\left]\right])\\ \le & (1-\lambda )(x_i^n\oplus x_i^{n-1})+\lambda [(x_i^n-g_i\left(x_i^n\right))\oplus (x_i^{n-1}-g_i\left(x_i^{n-1}\right))]\\ & +\lambda \{J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^n\right)\right)+{\rho }_i{F}_i(u_{i1}^n,u_{i2}^n,\dots ,u_{im}^n)]\\ & \oplus J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^{n-1}\right)\right)+{\rho }_i{F}_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{im}^{n-1})]\}\\ \le & (1-\lambda )(x_i^n\oplus x_i^{n-1})+\lambda ({\alpha }_1^i(x_i^n\oplus x_i^{n-1})+{\alpha }_2^i(g_i\left(x_i^n\right)\oplus g_i\left(x_i^{n-1}\right))\\ & +\lambda \{J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^n\right)\right)+{\rho }_i{F}_i(u_{i1}^n,u_{i2}^n,\dots ,u_{im}^n)]\\ & \oplus J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^{n-1}\right)\right)+{\rho }_i{F}_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{im}^{n-1})]\}\\ \le & (1-\lambda )(x_i^n\oplus x_i^{n-1})+\lambda ({\alpha }_1^i+{\alpha }_2^i{\lambda }_{g_i})(x_i^n\oplus x_i^{n-1})\\ & +\lambda \{J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^n\right)\right)+{\rho }_i{F}_i(u_{i1}^n,u_{i2}^n,\dots ,u_{im}^n)]\\ & \oplus J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^{n-1}\right)\right)+{\rho }_i{F}_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{im}^{n-1})]\}.\end{array}$$ 10

Using Definition 2.2, Proposition 6 and Eq. (10), we get

$$\begin{array}{rl}\parallel x_i^{n+1}\oplus x_i^n\parallel \le & {\lambda }_{N_C}[(1-\lambda )+\lambda ({\alpha }_1^i+{\lambda }_{g_i}{\alpha }_2^i)]\parallel x_i^n\oplus x_i^{n-1}\parallel \\ & +\lambda {\lambda }_{N_C}{L}_i\parallel [(I_i-A_i)\left(g_i\left(x_i^n\right)\right)+{\rho }_i{F}_i(u_{i1}^n,u_{i2}^n,\dots ,u_{im}^n)]\\ & \oplus [(I_i-A_i)\left(g_i\left(x_i^{n-1}\right)\right)+{\rho }_i{F}_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{im}^{n-1})]\parallel \\ \le & {\lambda }_{N_C}[1-\lambda (1-({\alpha }_1^i+{\lambda }_{g_i}{\alpha }_2^i))]\parallel x_i^n\oplus x_i^{n-1}\parallel \\ & +\lambda {\lambda }_{N_C}{L}_i\left[\parallel (I_i-A_i)\left(g_i\left(x_i^n\right)\right)\oplus (I_i-A_i)\left(g_i\left(x_i^{n-1}\right)\right)\parallel \right]\\ & +\lambda {\lambda }_{N_C}{L}_i{\rho }_i\parallel F_i(u_{i1}^n,u_{i2}^n,\dots ,u_{im}^n)\oplus F_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{im}^{n-1})\parallel \\ \le & {\lambda }_{N_C}[1-\lambda (1-({\alpha }_1^i+{\lambda }_{g_i}{\alpha }_2^i))]\parallel x_i^n\oplus x_i^{n-1}\parallel \\ & +\lambda {\lambda }_{N_C}{L}_i[\parallel (g_i\left(x_i^n\right)\oplus g_i\left(x_i^n\right))\parallel +\parallel A_i\left(g_i\left(x_i^n\right)\right)\oplus A_i\left(g_i\left(x_i^{n-1}\right)\right)\parallel ]\\ & +\lambda {\lambda }_{N_C}{L}_i{\rho }_i\left[\parallel F_i(u_{i1}^n,u_{i2}^n,\dots ,u_{im}^n)\oplus F_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{im}^{n-1})\parallel \right]\\ \le & {\lambda }_{N_C}[1-\lambda (1-({\alpha }_1^i+{\lambda }_{g_i}{\alpha }_2^i))]\parallel x_i^n\oplus x_i^{n-1}\parallel \\ & +\lambda {\lambda }_{N_C}{L}_i[({\lambda }_{g_i}+{\lambda }_{A_i}{\lambda }_{g_i})\parallel x_i^n\oplus x_i^{n-1}\parallel ]\\ & +\lambda {\lambda }_{N_C}{L}_i{\rho }_i\left[\parallel F_i(u_{i1}^n,u_{i2}^n,\dots ,u_{im}^n)\oplus F_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{im}^{n-1})\parallel \right].\end{array}$$ 11

Now, from Eq. (11), we compute

$$\begin{array}{rl}& \parallel F_i(u_{i1}^n,u_{i2}^n,\dots ,u_{ii-1}^n,u_{ii}^n,u_{ii+1}^n,\dots ,u_{im}^n)\oplus F_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{ii-1}^{n-1},u_{ii}^{n-1},u_{ii+1}^{n-1},\dots ,u_{im}^{n-1})\parallel \\ & \le \parallel F_i(u_{i1}^n,u_{i2}^n,\dots ,u_{ii-1}^n,u_{ii}^n,u_{ii+1}^n,\dots ,u_{im}^n)\\ & \oplus F_i(u_{i1}^{n-1},u_{i2}^n,\dots ,u_{ii-1}^n,u_{ii}^n,u_{ii+1}^n,\dots ,u_{im}^n)\parallel \\ & +\parallel F_i(u_{i1}^{n-1},u_{i2}^n,\dots ,u_{ii-1}^n,u_{ii}^n,u_{ii+1}^n,\dots ,u_{im}^n)\\ & \oplus F_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{ii-1}^{n-1},u_{ii}^{n-1},u_{ii+1}^{n-1},\dots ,u_{im}^{n-1})\parallel +\cdots \\ & +\parallel F_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{ii-1}^{n-1},u_{ii}^{n-1},u_{im-1}^{n-1},u_{im}^n)\\ & \oplus F_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{ii-1}^{n-1},u_{ii}^{n-1},u_{ii+1}^{n-1},\dots ,u_{im}^{n-1})\parallel .\end{array}$$ 12

By the definition of $F_i$ as a ${\lambda }_{F_{ij}}$-ordered compression map with respect to the jth argument, we have

$$\begin{array}{rl}& \parallel F_i(u_{i1}^n,u_{i2}^n,\dots ,u_{ii-1}^n,u_{ii}^n,u_{ii+1}^n,\dots ,u_{im}^n)\oplus F_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{ii-1}^{n-1},u_{ii}^{n-1},u_{ii+1}^{n-1},\dots ,u_{im}^{n-1})\parallel \\ & \le {\lambda }_{F_{i1}}\parallel u_{i1}^n\oplus u_{i1}^{n-1}\parallel +{\lambda }_{F_{i2}}\parallel u_{i2}^n\oplus u_{i2}^{n-1}\parallel +\cdots +{\lambda }_{F_{im}}\parallel u_{im}^n\oplus u_{im}^{n-1}\parallel \\ & =\sum _{i\ne j,j=1}^m{\lambda }_{F_{ij}}\parallel u_{ij}^n\oplus u_{ij}^{n-1}\parallel \\ & \le \sum _{i\ne j,j=1}^m{\lambda }_{F_{ij}}(1+\frac{1}{(n+1)})D_j(U_{ij}\left(x_j^n\right),U_{ij}\left(x_j^{n-1}\right))\\ & \le (1+\frac{1}{(n+1)})\sum _{i\ne j,j=1}^m{\lambda }_{F_{ij}}{\delta }_{D_{ij}}\parallel x_j^n\oplus x_j^{n-1}\parallel .\end{array}$$ 13

Using Proposition 6 and Eq. (13) in Eq. (11), we obtain

$$\begin{array}{rl}\parallel x_i^{n+1}\oplus x_i^n\parallel =& \parallel x_i^{n+1}-x_i^n\parallel \\ \le & {\lambda }_{N_C}[1-\lambda (1-({\alpha }_1^i+{\lambda }_{g_i}{\alpha }_2^i))]\parallel x_i^n-x_i^{n-1}\parallel \\ & +\lambda {\lambda }_{N_C}{L}_i[({\lambda }_{g_i}+{\lambda }_{A_i}{\lambda }_{g_i})\parallel x_i^n-x_i^{n-1}\parallel ]\\ & +\lambda {\lambda }_{N_C}{L}_i{\rho }_i(1+\frac{1}{(n+1)})\sum _{i\ne j,j=1}^m{\lambda }_{F_{ij}}{\delta }_{D_{ij}}\parallel x_j^n-x_j^{n-1}\parallel \\ \le & \{{\lambda }_{N_C}[1-\lambda (1-({\alpha }_1^i+{\alpha }_2^i{\lambda }_{g_i}))]+\lambda {\lambda }_{N_C}{L}_i({\lambda }_{g_i}+{\lambda }_{A_i}{\lambda }_{g_i})\}\parallel x_i^n-x_i^{n-1}\parallel \\ & +\lambda {\lambda }_{N_C}{L}_i{\rho }_i(1+\frac{1}{(n+1)})\sum _{i\ne j,j=1}^m{\lambda }_{F_{ij}}{\delta }_{D_{ij}}\parallel x_j^n-x_j^{n-1}\parallel ,\end{array}$$

which implies that

$$\begin{array}{rl}& \sum _{j=1}^m\parallel x_j^{n+1}-x_j^n\parallel \\ & =\sum _{i=1}^m\parallel x_i^{n+1}-x_i^n\parallel \\ & \le \sum _{i=1}^m\{[{\lambda }_{N_C}[1-\lambda (1-({\alpha }_1^i+{\alpha }_2^i{\lambda }_{g_i}))]+\lambda {\lambda }_{N_C}{L}_i({\lambda }_{g_i}+{\lambda }_{A_i}{\lambda }_{g_i})]\parallel x_i^n-x_i^{n-1}\parallel \\ & +\lambda {\lambda }_{N_C}(1+\frac{1}{(n+1)})\sum _{i\ne j,j=1}^m{L}_j{\rho }_j{\lambda }_{F_{ij}}{\delta }_{D_{ij}}\parallel x_j^n-x_j^{n-1}\parallel \}\\ & =\sum _{i=1}^m{\lambda }_{N_C}[1-\lambda \{1-({\alpha }_1^i+{\lambda }_{g_i}{\alpha }_2^i)+L_i({\lambda }_{g_i}+{\lambda }_{A_i}{\lambda }_{g_i})\}]\parallel x_i^n-x_i^{n-1}\parallel \\ & +\lambda {\lambda }_{N_C}(1+\frac{1}{(n+1)})\sum _{i=1}^m\sum _{i\ne j,j=1}^m{L}_j{\rho }_j{\lambda }_{F_{ij}}{\delta }_{D_{ij}}\parallel x_j^n-x_j^{n-1}\parallel \\ & =\sum _{j=1}^m{\lambda }_{N_C}[1-\lambda \{1-({\alpha }_1^j+{\alpha }_2^j{\lambda }_{g_j})+L_j({\lambda }_{g_j}+{\lambda }_{A_j}{\lambda }_{g_j})\}]\parallel x_j^n-x_j^{n-1}\parallel \\ & +{\lambda }_{N_C}\lambda (1+\frac{1}{(n+1)})\sum _{j=1}^m\sum _{i\ne j,i=1}^m{L}_j{\rho }_j{\lambda }_{F_{ij}}{\delta }_{D_{ij}}\parallel x_j^n-x_j^{n-1}\parallel \\ & =\sum _{j=1}^m{\lambda }_{N_C}[1-\lambda +\lambda \{{\alpha }_1^j+{\alpha }_2^j{\lambda }_{g_j}+L_j({\lambda }_{g_j}+{\lambda }_{A_j}{\lambda }_{g_j})\\ & +(1+\frac{1}{(n+1)})\sum _{i\ne j,i=1}^m{L}_i{\rho }_i{\lambda }_{F_{ij}}{\delta }_{D_{ij}}\left\}\right]\parallel x_j^n-x_j^{n-1}\parallel \\ & =\sum _{j=1}^m{\lambda }_{N_C}(1-\lambda +\lambda {\theta }_j^n)\parallel x_j^n-x_j^{n-1}\parallel \\ & \le {\lambda }_{N_C}{f}_n(\lambda )\sum _{j=1}^m\parallel x_j^n-x_j^{n-1}\parallel ,\end{array}$$ 14

where

$${\theta }_j^n=\{{\alpha }_1^j+{\alpha }_2^j{\lambda }_{g_j}+L_j({\lambda }_{g_j}+{\lambda }_{A_j}{\lambda }_{g_j})+(1+\frac{1}{(n+1)})\sum _{i\ne j,i=1}^m{L}_i{\rho }_i{\lambda }_{F_{ij}}{\delta }_{D_{ij}}\}<1$$

and

$$f_n(\lambda )=\underset{1\le j\le m}{max}\{1-\lambda +\lambda {\theta }_j^n\}.$$

From Eq. (14), we know that the sequence $\{{\theta }_j^n\}$ is monotonic decreasing and ${\theta }_j^n\to {\theta }_j$ as $n\to \mathrm{\infty }$. Thus, $f(\lambda )={lim}_{n\to \mathrm{\infty }}{f}_n(\lambda )={max}_{1\le j\le m}\{1-\lambda +\lambda {\theta }_j\}$. Since $0<{\theta }_j<1$ for $j=1,2,\dots ,m$. We get $\theta ={max}_{1\le j\le m}\{{\theta }_j\}\in (0,1)$. By Lemma 2.1, we have $f(\lambda )=1-\lambda +\lambda \theta \in (0,1)$, from Eq. (14), it follows that $\{x_j^n\}$ is a Cauchy sequence and there exists $x_j^{\ast }\in {\mathcal{E}}_j$ such that $x_j^n\to x_j^{\ast }$ as $n\to \mathrm{\infty }$ for $j=1,2,\dots ,m$. Next, we show that $u_{ij}^n\to u_{ij}^{\ast }\in U_{ij}(x_j^{\ast })$ as $n\to \mathrm{\infty }$ for $i,j=1,2,\dots ,m$. It follows from Eq. (13) that the $\{u_{ij}^n\}$ are also Cauchy sequences. Hence, there exists $u_{ij}^{\ast }\in {\mathcal{E}}_j$ such that $u_{ij}^n\to u_{ij}^{\ast }$ as $n\to \mathrm{\infty }$ for $i,j=1,2,\dots ,m$. Furthermore,

$$\begin{array}{rcl}d(u_{ij}^{\ast },U_{ij}\left(x_j^{\ast }\right))& =& inf\{\parallel u_{ij}^{\ast }\oplus t\parallel :t\in U_{ij}\left(x_j^{\ast }\right)\}\\ & \le & \parallel u_{ij}^{\ast }\oplus u_{ij}^n\parallel +d(u_{ij}^n,U_{ij}\left(x_j^{\ast }\right))\\ & \le & \parallel u_{ij}^{\ast }\oplus u_{ij}^n\parallel +d(U_{ij}\left(x_j^n\right),U_{ij}\left(x_j^{\ast }\right))\\ & \le & \parallel u_{ij}^{\ast }\oplus u_{ij}^n\parallel +{\delta }_{D_{ij}}\parallel x_j^{\ast }\oplus x_j^n\parallel \\ & \le & \parallel u_{ij}^{\ast }-u_{ij}^n\parallel +{\delta }_{D_{ij}}\parallel x_j^{\ast }-x_j^n\parallel \to 0(n\to \mathrm{\infty }).\end{array}$$

Since $U_{ij}(x_j^{\ast })$ is closed for $i,j=1,2,\dots ,m$, we have $u_{ij}^{\ast }\in U_{ij}(x_j^{\ast })$ for $i,j=1,2,\dots ,m$. By using continuity $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })\in {\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m$ and $u_{ij}^{\ast }\in U_{ij}(x_j^{\ast })$ for $i,j=1,2,\dots ,m$ satisfy Eq. (6) and so by Lemma 4.1, problem (2) has a solution $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast },u_{11}^{\ast },u_{12}^{\ast },\dots ,u_{1m}^{\ast },\dots ,u_{m1}^{\ast },u_{m2}^{\ast },\dots ,u_{mm}^{\ast })$, where $u_{ij}^{\ast }\in U_{ij}(x_j^{\ast })$ for $i,j=1,2,\dots ,m$ and $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })\in {\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m$. This completes the proof. □

Theorem 6.2

Suppose that $A_i,g_i$ and $M_i$ are the same as in Theorem 6.1 for $i=1,2,\dots ,m$. Let $T_{ij}:{\mathcal{E}}_j\to {\mathcal{E}}_j$ be ${\gamma }_{ij}$-Lipschitz continuous and $F_i:{\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m\to {\mathcal{E}}_i$ be ${\lambda }_{F_{ij}}$-ordered compression mapping with respect to the jth argument. Let there be constants ${\lambda }_j>0$, for $j=1,2,\dots ,m$ such that

$${\theta }_j=[{\lambda }_{N_C}\{({\alpha }_1^j+{\alpha }_2^j{\lambda }_{g_i})+L_j({\lambda }_{g_j}+{\lambda }_{A_j}{\lambda }_{g_j})\}+{\lambda }_{N_C}\sum _{i\ne j,i=1}^m{L}_i{\rho }_i{\lambda }_{F_{ij}}{\gamma }_{ij}]<1.$$

Then problem (3) has a unique solution $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })\in {\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m$. Moreover, the iterative sequence $\{x_j^n\}$ generated by Algorithm 2 converges strongly to $x_j^{\ast }$ for $j=1,2,\dots ,m$.

Proof

Let us define a norm ${\parallel \cdot \parallel }_{\ast }$ on the product space ${\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m$ by

$${\parallel (x_1,x_2,\dots ,x_m)\parallel }_{\ast }=\sum _{i=1}^m\parallel x_j\parallel ,\mathrm{\forall }(x_1,x_2,\dots ,x_m)\in {\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m.$$ 15

Then it can easily be seen that $({\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m,{\parallel \cdot \parallel }_{\ast })$ is a Banach space.

Setting

$$\begin{array}{rcl}{y}_i& =& x_i-g_i(x_i)+J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)(g_i(x_i))\\ & & +{\rho }_i{F}_i(T_{i1}{x}_1,\dots ,T_{ii-1}{x}_{i-1},T_{ii}{x}_i,T_{ii+1}{x}_{i+1},\dots ,T_{im}{x}_m)].\end{array}$$

Define a mapping $Q:{\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m\to {\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m$ as

$$Q(x_1,x_2,\dots ,x_m)=(y_1,y_2,\dots ,y_m),\mathrm{\forall }(x_1,x_2,\dots ,x_m)\in {\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m.$$

For any $(x_1^1,x_2^1,\dots ,x_m^1)$, $(x_1^2,x_2^2,\dots ,x_m^2)\in {\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m$ we have

$$\begin{array}{rl}& {\parallel Q(x_1^1,x_2^1,\dots ,x_m^1)\oplus Q(x_1^2,x_2^2,\dots ,x_m^2)\parallel }_{\ast }\\ & \le {\parallel Q(x_1^1,x_2^1,\dots ,x_m^1)-Q(x_1^2,x_2^2,\dots ,x_m^2)\parallel }_{\ast }\\ & \le {\parallel (y_1^1,y_2^1,\dots ,y_m^1)-(y_1^2,y_2^2,\dots ,y_m^2)\parallel }_{\ast }\\ & \le \sum _{i=1}^m\parallel y_i^1-y_i^2\parallel .\end{array}$$ 16

First of all, we have to calculate $(y_i^1\oplus y_i^2)$ as follows:

$$\begin{array}{rcl}(y_i^1\oplus y_i^2)& =& (x_i^1-g_i\left(x_i^1\right)+J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^1\right)\right)\\ & & +{\rho }_i{F}_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\left]\right)\\ & & \oplus (x_i^2-g_i\left(x_i^2\right)+J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^2\right)\right)\\ & & +{\rho }_i{F}_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\left]\right)\\ & =& ((x_i^1-g_i\left(x_i^1\right))\oplus (x_i^2-g_i\left(x_i^2\right)))+\left(J_{{\lambda }_i,M_i}^{I_i-A_i}\right[(I_i-A_i)\left(g_i\left(x_i^1\right)\right)\\ & & +{\rho }_i{F}_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)]\\ & & \oplus J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^2\right)\right)\\ & & +{\rho }_i{F}_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\left]\right).\end{array}$$

From Definition 2.2 and Proposition 3, we have

$$\begin{array}{rcl}\parallel y_i^1\oplus y_i^2\parallel & \le & \parallel y_i^1-y_i^2\parallel \\ & \le & {\lambda }_{N_C}\parallel \left\{\right((x_i^1-g_i\left(x_i^1\right))\\ & & \oplus (x_i^2-g_i\left(x_i^2\right)))+(J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^1\right)\right)\\ & & +{\rho }_i{F}_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)]\\ & & \oplus J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^2\right)\right)\\ & & +{\rho }_i{F}_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\left]\right)\}\parallel \\ & \le & {\lambda }_{N_C}\{{\alpha }_1^j\parallel x_i^1-x_i^2\parallel +{\alpha }_2^j{\lambda }_{g_i}\parallel x_i^1-x_i^2\parallel \}\\ & & +{\lambda }_{N_C}\parallel \left(J_{{\lambda }_i,M_i}^{I_i-A_i}\right[(I_i-A_i)\left(g_i\left(x_i^1\right)\right)+{\rho }_i{F}_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,\\ & & T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1\left)\right]\oplus J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^2\right)\right)\\ & & +{\rho }_i{F}_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\left]\right)\parallel \\ & \le & {\lambda }_{N_C}({\alpha }_1^j+{\alpha }_2^j{\lambda }_{g_i})\parallel x_i^1-x_i^2\parallel +{\lambda }_{N_C}\parallel \left(J_{{\lambda }_i,M_i}^{I_i-A_i}\right[(I_i-A_i)\left(g_i\left(x_i^1\right)\right)\\ & & +{\rho }_i{F}_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)]\\ & & \oplus J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^2\right)\right)\\ & & +{\rho }_i{F}_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\left]\right)\parallel .\end{array}$$ 17

Further, we calculate

$$\begin{array}{rl}& \parallel J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^1\right)\right)+{\rho }_i{F}_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)]\\ & \oplus J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^2\right)\right)+{\rho }_i{F}_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)]\parallel \\ & \le L_i[\parallel \{(I_i-A_i)\left(g_i\left(x_i^1\right)\right)\oplus (I_i-A_i)\left(g_i\left(x_i^2\right)\right)\}\parallel \\ & +{\rho }_i\parallel F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\\ & \oplus F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\parallel ]\\ & \le L_i[\parallel (g_i\left(x_i^1\right)\oplus g_i\left(x_i^2\right))+(A_i\left(g_i\left(x_i^1\right)\right)\oplus A_i\left(g_i\left(x_i^2\right)\right))\parallel \\ & +{\rho }_i\parallel F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\\ & \oplus F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\parallel ]\\ & \le L_i[\{{\lambda }_{g_i}\parallel x_i^1\oplus x_i^2\parallel +{\lambda }_{A_i}{\lambda }_{g_i}\parallel x_i^1\oplus x_i^2\parallel \}\\ & +{\rho }_i\parallel F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\\ & \oplus F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\parallel ]\\ & \le L_i[({\lambda }_{g_i}+{\lambda }_{A_i}{\lambda }_{g_i})\parallel x_i^1\oplus x_i^2\parallel \\ & +{\rho }_i\parallel F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\\ & F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\parallel ].\end{array}$$ 18

Now we calculate the inner part estimate of the above expression with the help of the properties of the $F_i$-operator for $i=1,2,\dots ,m$. We have

$$\begin{array}{rl}& \parallel F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\\ & \oplus F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\parallel \\ & =\parallel F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\oplus F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^1,\\ & T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\oplus F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,\\ & T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\oplus F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\parallel \\ & \le \parallel F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\\ & \oplus F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\parallel +\cdots \\ & +\parallel F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\\ & \oplus F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\parallel \\ & +\parallel F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\\ & \oplus F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\parallel \\ & +\parallel F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\\ & \oplus F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^1)\parallel +\cdots \\ & +\parallel F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^1)\\ & \oplus F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\parallel \\ & \le {\lambda }_{F_{i1}}\parallel T_{i1}{x}_1^1\oplus T_{i1}{x}_1^2\parallel +{\lambda }_{F_{i2}}\parallel T_{i2}{x}_2^1\oplus T_{i2}{x}_2^2\parallel +\cdots \\ & +{\lambda }_{F_{ii-1}}\parallel T_{ii-1}{x}_{i-1}^1\oplus T_{ii-1}{x}_{i-1}^2\parallel +{\lambda }_{F_{ii}}\parallel T_{ii}{x}_i^1\oplus T_{ii}{x}_i^2\parallel \\ & +{\lambda }_{F_{ii+1}}\parallel T_{ii+1}{x}_{i+1}^1\oplus T_{ii+1}{x}_{i+1}^2\parallel +\cdots +{\lambda }_{F_{im}}\parallel T_{im}{x}_m^1\oplus T_{im}{x}_m^2\parallel .\end{array}$$ 19

By using the Lipschitz continuity of $T_{ij}$-operator in Eq. (19), we have

$$\begin{array}{rl}& \parallel F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\\ & \oplus F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\parallel \\ & \le \sum _{i\ne j,j=1}^m{\lambda }_{F_{ij}}{\gamma }_{ij}\parallel x_j^1\oplus x_j^2\parallel .\end{array}$$ 20

Using Eq. (20) in Eq. (18) and then use it in Eq. (17), we have

$$\begin{array}{rcl}\parallel y_i^1\oplus y_i^2\parallel & \le & \parallel y_i^1-y_i^2\parallel \\ & \le & {\lambda }_{N_C}({\alpha }_1^i+{\lambda }_{g_i}{\alpha }_2^i)\parallel x_i^1-x_i^2\parallel \\ & & +{\lambda }_{N_C}[L_i({\lambda }_{g_i}+{\lambda }_{A_i}{\lambda }_{g_i})\parallel x_i^1-x_i^2\parallel +L_i{\rho }_i\sum _{i\ne j,j=1}^m{\lambda }_{F_{ij}}{\gamma }_{ij}\parallel x_j^1-x_j^2\parallel ].\end{array}$$ 21

Now, Eq. (16) can be rewritten as

$$\begin{array}{rl}& {\parallel Q(x_1^1,x_2^1,\dots ,x_m^1)-Q(x_1^2,x_2^2,\dots ,x_m^2)\parallel }_{\ast }\\ & \le \sum _{i=1}^m\parallel y_i^1-y_i^2\parallel \\ & \le \sum _{i=1}^m\{{\lambda }_{N_C}[({\alpha }_1^i+{\lambda }_{g_i}{\alpha }_2^i)+L_i({\lambda }_{g_i}+{\lambda }_{A_i}{\lambda }_{g_i})]\parallel x_i^1-x_i^2\parallel \\ & +{\lambda }_{N_C}{L}_i{\rho }_i\sum _{i\ne j,j=1}^m{\lambda }_{F_{ij}}{\gamma }_{ij}\parallel x_j^1-x_j^2\parallel \}\\ & \le \sum _{i=1}^m\left\{{\lambda }_{N_C}[({\alpha }_1^i+{\lambda }_{g_i}{\alpha }_2^i)+L_i({\lambda }_{g_i}+{\lambda }_{A_i}{\lambda }_{g_i})]\right\}\parallel |x_i^1-x_i^2\parallel \\ & +\sum _{i=1}^m{\lambda }_{N_C}{L}_i{\rho }_i\sum _{i\ne j,j=1}^m{\lambda }_{F_{ij}}{\gamma }_{ij}\parallel x_j^1-x_j^2\parallel \\ & \le \sum _{j=1}^m[{\lambda }_{N_C}\{({\alpha }_1^j+{\alpha }_2^j{\lambda }_{g_i})+L_j({\lambda }_{g_j}+{\lambda }_{A_j}{\lambda }_{g_j})\}\\ & +{\lambda }_{N_C}\sum _{i\ne j,i=1}^m{L}_i{\rho }_i{\lambda }_{F_{ij}}{\gamma }_{ij}]\parallel x_j^1-x_j^2\parallel \\ & =\sum _{j=1}^m{\theta }_j\parallel x_j^1-x_j^2\parallel \\ & \le \theta \sum _{j=1}^m\parallel x_j^1-x_j^2\parallel \\ & \le \theta {\parallel (x_1^1,x_2^1,\dots ,x_m^1)-(x_1^2,x_2^2,\dots ,x_m^2)\parallel }_{\ast },\end{array}$$ 22

where $\theta ={max}_{1\le j\le m}{\theta }_j$. Finally, from Eq. (22), Eq. (16) can be written as

$$\begin{array}{rl}{\parallel Q(x_1^1,x_2^1,\dots ,x_m^1)-Q(x_1^2,x_2^2,\dots ,x_m^2)\parallel }_{\ast }& \le \theta \sum _{j=1}^m\parallel x_j^1-x_j^2\parallel \\ & =\theta {\parallel (x_1^1,x_2^1,\dots ,x_m^1)-(x_1^2,x_2^2,\dots ,x_m^2)\parallel }_{\ast }.\end{array}$$ 23

It follows from the condition (9) that $0<\theta <1$. This implies that $Q:{\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m\to {\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m$ is a contraction which in turn implies that there exists a unique $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })\in {\mathcal{E}}_1\times {\mathcal{E}}_2\times \cdots \times {\mathcal{E}}_m$ such that $Q(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })=(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })$. Thus, $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })$ is the unique solution of problem (3). Now, we prove that $x_i^n\to x_i^{\ast }$ as $n\to \mathrm{\infty }$ for $i=1,2,\dots ,m$. In fact, it follows from Eq. (8) and the Lipschitz continuity of the relaxed resolvent operator that

$$\begin{array}{rl}\parallel x_i^{n+1}\oplus x_i^{\ast }\parallel =& \parallel [x_i^n-g_i\left(x_i^n\right)+J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^n\right)\right)\\ & +{\rho }_i{F}_i(T_{i1}{x}_1^n,T_{i2}{x}_2^n,\dots ,T_{im}{x}_m^n)]+w_i^n\oplus [x_i^{\ast }-g_i\left(x_i^{\ast }\right)\\ & +J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^{\ast }\right)\right)\\ & +{\rho }_i{F}_i(T_{i1}{x}_1^{\ast },T_{i2}{x}_2^{\ast },\dots ,T_{im}{x}_m^{\ast })\left]\right]]\parallel \\ \le & \parallel (x_i^n-g_i\left(x_i^n\right))\oplus (x_i^{\ast }-g_i\left(x_i^{\ast }\right))\parallel \\ & +\parallel J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^n\right)\right)\\ & +{\rho }_i{F}_i(T_{i1}{x}_1^n,T_{i2}{x}_2^n,\dots ,T_{im}{x}_m^n)]\\ & \oplus J_{{\lambda }_i,M_i}^{I_i-A_i}[(I_i-A_i)\left(g_i\left(x_i^{\ast }\right)\right)\\ & +{\rho }_i{F}_i(T_{i1}{x}_1^{\ast },T_{i2}{x}_2^{\ast },\dots ,T_{im}{x}_m^{\ast })]\parallel +\parallel w_i^n\oplus 0\parallel .\end{array}$$ 24

From the previous calculations, we have

$$\begin{array}{rl}\sum _{i=1}^m\parallel x_i^{n+1}\oplus x_i^{\ast }\parallel =& \sum _{i=1}^m\parallel x_i^{n+1}-x_i^{\ast }\parallel \\ \le & [{\lambda }_{N_C}\{({\alpha }_1^j+{\alpha }_2^j{\lambda }_{g_i})+L_j({\lambda }_{g_j}+{\lambda }_{A_j}{\lambda }_{g_j})\}\\ & +{\lambda }_{N_C}\sum _{i\ne j,i=1}^m{L}_i{\rho }_i{\lambda }_{F_{ij}}{\gamma }_{ij}]\sum _{j=1}^m\parallel x_j^n-x_j^{\ast }\parallel +\sum _{j=1}^m\parallel w_j^n\parallel \\ =& \sum _{j=1}^m{\theta }_j\parallel x_j^n-x_j^{\ast }\parallel +\sum _{j=1}^m\parallel w_j^n\parallel ,\end{array}$$ 25

where $a_n={\sum }_{j=1}^m\parallel x_j^n-x_j^{\ast }\parallel$, $b_n={\sum }_{j=1}^m\parallel w_j^n\parallel$. Algorithm 2 yields ${lim}_{n\to \mathrm{\infty }}{b}_n=0$. Now, Lemma 2.2 implies that ${lim}_{n\to \mathrm{\infty }}{a}_n=0$, and so $x_j^n\to x_j^{\ast }$ as $n\to \mathrm{\infty }$ for $j=1,2,\dots ,m$. This completes the proof. □

Conclusion

Two of the most troublesome and imperative issues identified with inclusions are the foundation of generalized inclusions and the improvement of an iterative calculation. In this article, two systems of variational inclusions were presented and contemplated, which is a broader aim than the numerous current systems of generalized ordered variational inclusions in the literature. An iterative calculation is performed with a weak ARD mapping to an inexact solution of our systems, and the convergence criterion is likewise addressed.

We comment that our outcomes are new and valuable for additionally investigations. Considerably more work is required in every one of these regions to address utilizations of the system of general ordered variational inclusions in engineering and physical sciences.

Authors’ contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Funding

No funding available.

Competing interests

The authors declare that they have no competing interests.