A Nonlinear System of Generalized Ordered XOR-Inclusion Problem in Hilbert Space with S-Iterative Algorithm

Abstract

In this article, a nonlinear system of generalized ordered XOR-inclusion problems in Hilbert space is introduced and studied. Initially, we define the resolvent operator related to the $(\alpha ,\lambda )$-XOR-weak-ANODD multivalued mapping. Using the fixed point technique, we demonstrate the results of existence. In order to make the suggested system more realistic, we create the S-iterative algorithm and demonstrate that the sequence generated through this technique strongly converges with the proposed system’s solution. One example is provided in support of the existence result as well.

1. Introduction

In recent decades, the variational inclusion problems have been extended and generalized in various directions by using novel and innovative techniques. One useful and important form of inclusion problem is the set-valued inclusion problem. Many problems related to control theory, game theory, optimization economics applied sciences, etc. can be constructed in terms of set-valued inclusion problem $0\in M\left(x\right)$, for a given set-valued mapping M in Hilbert space. For more details, we refer to [1,2,3,4,5] and references therein. A system of variational inequalities (inclusions), which is a natural generalization of variational inequalities, was taken into consideration and researched by many mathematicians as [6,7,8] and reference therein.

In 1972, Amann [9] introduced a technique for finding the number of solutions of nonlinear equations in ordered Hilbert spaces and Banach spaces. The applications of nonlinear equations involving nonlinear increasing operators were investigated by Du [10] and Srivastava et al. [11,12,13]. A large amount of literature is available related to ordered inequalities (inclusions) problems in arbitrarily ordered spaces. In 2008, Li and his coauthors [14] introduced and studied ordered variational inequalities (inclusion) with XOR-operations. Generally, XOR-operation is applicable in parity checks, neural networks, controlled inverters, binary to gray/gray to binary conversion, combinational logic circuits, etc. A lot of work related to ordered inclusions problem with XOR-operation can be found in [15,16,17,18,19,20,21,22].

In 2021, Ahmad and his coauthors introduced a nonlinear system of Mixed ordered variational inclusions with XOR-operation [23]. Inspired by the aforementioned work. In this article, we consider a system of nonlinear ordered XOR-inclusion problems in an ordered Hilbert space.

The paper is designed in the following way.

In Section 2, we have presented some prerequisites that we are needed to achieve our goal. In Section 3, we formulate our problem and prove an existence result by using the resolvent operator technique. In Section 4, we develop the S-iterative algorithm for the considered system and show the convergence of the sequence generated by S-iterative algorithm. In the last section, we have presented the conclusion of our work.

2. Prerequisites

Throughout this article, we assume $\mathcal{H}$ to be a real ordered positive Hilbert space with the norm ${\parallel .\parallel }_{\mathcal{H}}$ and inner product ${\langle ·,·\rangle }_{\mathcal{H}\times \mathcal{H}},$ the metric $\mathfrak{d}$ induced by the norm ${\parallel .\parallel }_{\mathcal{H}}$, $2^{\mathcal{H}}$ (respectively, $CB\left(\mathcal{H}\right)$) the nonempty collection of (respectively, closed and bounded) subsets of $\mathcal{H},$ and the Hausdorff metric $\mathbb{H}(.,.)$ on $CB\left(\mathcal{H}\right)$, is defined as:

$$\mathbb{H}(A,B)=max\{\underset{\mathfrak{s}\in A}{sup}\mathfrak{d}(\mathfrak{s},B),\underset{\mathfrak{t}\in B}{sup}\mathfrak{d}(A,\mathfrak{t})\},\forall A,B\in CB\left(\mathcal{H}\right),$$

where $\mathfrak{d}(\mathfrak{s},B)=\underset{\mathfrak{t}\in B}{inf}\{\parallel \mathfrak{s}-\mathfrak{t}\parallel \}$ and $\mathfrak{d}(A,\mathfrak{t})=\underset{\mathfrak{s}\in A}{inf}\{\parallel \mathfrak{s}-\mathfrak{t}\parallel \}$.

Definition 1

([22,24]). Let $\varphi \ne P\subseteq \mathcal{H}$ be closed convex subset of $\mathcal{H}$. Then P is called

(i)

a cone if for any $\mathfrak{s}\in P$ and any $\lambda >0,$  $\lambda \mathfrak{s}\in P,$

(ii)

pointed cone if $\mathfrak{s}\in P$ and $-\mathfrak{s}\in P,$ then $\mathfrak{s}=0.$

Definition 2

([22,24]). Let P be the cone, then

(i)

P is called a normal cone if ∃ a constant ${\lambda }_P>0$ such that $0\le \mathfrak{s}\le \mathfrak{t}$ implies ${\parallel \mathfrak{s}\parallel }_{\mathcal{H}}\le {\lambda }_P{\parallel \mathfrak{t}\parallel }_{\mathcal{H}},\forall \mathfrak{s},\mathfrak{t}\in \mathcal{H},$

(ii)

for any $\mathfrak{s},\mathfrak{t}\in \mathcal{H},$$\mathfrak{s}\le \mathfrak{t}$ if and only if $\mathfrak{t}-\mathfrak{s}\in P,$

(iii)

$\mathfrak{s}$ and $\mathfrak{t}$ are said to be comparative to each other if either $\mathfrak{s}\le \mathfrak{t}$ or $\mathfrak{t}\le \mathfrak{s}$ holds and is denoted by $\mathfrak{s}\propto \mathfrak{t}.$

Definition 3

([22,24]). For all $\mathfrak{s},\mathfrak{t}\in \mathcal{H},$ let $lub\{\mathfrak{s},\mathfrak{t}\}$ denotes least upper bound and $glb\{\mathfrak{s},\mathfrak{t}\}$ denotes greatest lower bound of the set $\{\mathfrak{s},\mathfrak{t}\}.$ Suppose $lub\{\mathfrak{s},\mathfrak{t}\}$ and $glb\{\mathfrak{s},\mathfrak{t}\}$ for the set $\{\mathfrak{s},\mathfrak{t}\}$ exists, then we define binary operations as below:

(i)

$\mathfrak{s}\vee \mathfrak{t}=lub\{\mathfrak{s},\mathfrak{t}\},$

(ii)

$\mathfrak{s}\wedge \mathfrak{t}=glb\{\mathfrak{s},\mathfrak{t}\},$

(iii)

$\mathfrak{s}\oplus \mathfrak{t}=(\mathfrak{s}-\mathfrak{t})\vee (\mathfrak{t}-\mathfrak{s}),$

(iv)

$\mathfrak{s}\odot \mathfrak{t}=(\mathfrak{s}-\mathfrak{t})\wedge (\mathfrak{t}-\mathfrak{s}).$

The operations$\vee ,\wedge ,$⊕ and ⊙ are called OR, AND, XOR and XNOR operations, respectively.

Proposition 1

([15,22]). Let ⊕ be an XOR-operation and ⊙ be an XNOR-operation. Then the following holds:

(i)

$\mathfrak{s}\odot \mathfrak{t}=0,$$\mathfrak{s}\odot \mathfrak{t}=\mathfrak{t}\odot \mathfrak{s}=-(\mathfrak{s}\oplus \mathfrak{t})=-(\mathfrak{t}\oplus \mathfrak{s}),$

(ii)

if $\mathfrak{s}\propto 0,$ then $-\mathfrak{s}\oplus 0\le \mathfrak{s}\le \mathfrak{s}\oplus 0,$

(iii)

$\left(\lambda \mathfrak{s}\right)\oplus \left(\lambda \mathfrak{t}\right)=\left|\lambda \right|(\mathfrak{s}\oplus \mathfrak{t}),$

(iv)

$0\le \mathfrak{s}\oplus \mathfrak{t},$ if $\mathfrak{s}\propto \mathfrak{t},$

(v)

if $\mathfrak{s}\propto \mathfrak{t},$ then $\mathfrak{s}\oplus \mathfrak{t}=0$ if and only if $\mathfrak{s}=\mathfrak{t},$

(vi)

$(\mathfrak{s}+\mathfrak{t})\odot (\mathfrak{u}+\mathfrak{v})\ge (\mathfrak{s}\odot \mathfrak{u})+(\mathfrak{t}\odot \mathfrak{v}),$

(vii)

$(\mathfrak{s}+\mathfrak{t})\odot (\mathfrak{u}+\mathfrak{v})\ge (\mathfrak{s}\odot \mathfrak{v})+(\mathfrak{t}\odot \mathfrak{u}),$

(viii)

if $\mathfrak{s},\mathfrak{t}$ and $\mathfrak{w}$ are comparable to each other, then $(\mathfrak{s}\oplus \mathfrak{t})\le (\mathfrak{s}\oplus \mathfrak{w})+(\mathfrak{w}\oplus \mathfrak{t}),$

(ix)

if $\mathfrak{s},\mathfrak{t}$ and $\mathfrak{w}$ are comparable to each other, then $\left[\right(\mathfrak{w}\oplus \mathfrak{s})\oplus (\mathfrak{w}\oplus \mathfrak{t}\left)\right]=(\mathfrak{s}\oplus \mathfrak{t}),$

(x)

$\alpha \mathfrak{s}\oplus \beta \mathfrak{s}=|\alpha -\beta |\mathfrak{s}=(\alpha \oplus \beta )\mathfrak{s},$ if $\mathfrak{s}\propto 0,$$\forall \mathfrak{s},\mathfrak{t},\mathfrak{u},\mathfrak{v},\mathfrak{w}\in \mathcal{H}$ and $\alpha ,\beta ,\lambda \in \mathbb{R}$.

Proposition 2

([15,22]). Let P be a normal cone with normal constant ${\lambda }_P$ in $\mathcal{H},$ then for each $\mathfrak{s},\mathfrak{t}\in \mathcal{H},$ the following hold:

(i)

$\parallel 0\oplus 0\parallel = \parallel 0\parallel =0,$

(ii)

$\parallel \mathfrak{s}\vee \mathfrak{t}\parallel \le \parallel \mathfrak{s}\parallel \vee \parallel \mathfrak{t}\parallel \le \parallel \mathfrak{s}\parallel +\parallel \mathfrak{t}\parallel ,$

(iii)

$\parallel \mathfrak{s}\oplus \mathfrak{t}\parallel \le \parallel \mathfrak{s}-\mathfrak{t}\parallel \le {\lambda }_P\parallel \mathfrak{s}\oplus \mathfrak{t}\parallel$,

(iv)

if $\mathfrak{s}\propto \mathfrak{t},$ then $\parallel \mathfrak{s}\oplus \mathfrak{t}\parallel = \parallel \mathfrak{s}-\mathfrak{t}\parallel .$

Definition 4

([15,22]). A single-valued mapping $A:\mathcal{H}\to \mathcal{H}$ is said to be

(i)

a comparison mapping if for each $\mathfrak{s},\mathfrak{t}\in \mathcal{H}$ and $\mathfrak{s}\propto \mathfrak{t}$ then $A\left(\mathfrak{s}\right)\propto A\left(\mathfrak{t}\right),$ $\mathfrak{s}\propto A\left(\mathfrak{s}\right)$ and $\mathfrak{t}\propto A\left(\mathfrak{t}\right),$

(ii)

strongly comparison mapping if A is a comparison mapping and $A\left(\mathfrak{s}\right)\propto A\left(\mathfrak{t}\right)$ if and only if $\mathfrak{s}\propto \mathfrak{t},$ for any $\mathfrak{s},\mathfrak{t}\in \mathcal{H},$

(iii)

β-ordered compression mapping if A is a comparison mapping and

$$A\left(\mathfrak{s}\right)\oplus A\left(\mathfrak{t}\right)\le \beta (\mathfrak{s}\oplus \mathfrak{t}),for\beta \in (0,1)and\mathfrak{s},\mathfrak{t}\in \mathcal{H}.$$

Definition 5.

Let $\mathfrak{f},\mathfrak{g}:\mathcal{H}\to \mathcal{H}$ be strongly comparison single-valued mappings. Then, the mapping $\mathcal{P}:\mathcal{H}\times \mathcal{H}\to \mathcal{H}$ is said to be

(i)

${\gamma }_{\mathcal{P}}^{\mathfrak{f}}$-ordered compression mapping in the first argument with respect to $\mathfrak{f}$ if there exists a constant ${\gamma }_{\mathcal{P}}^{\mathfrak{f}}>0$ such that

$$\mathcal{P}(\mathfrak{f}\left(\mathfrak{s}\right),.)\oplus \mathcal{P}(\mathfrak{f}\left(\mathfrak{t}\right),.)\le {\gamma }_{\mathcal{P}}^{\mathfrak{f}}(\mathfrak{s}\oplus \mathfrak{t}),$$

(ii)

${\xi }_{\mathcal{P}}^{\mathfrak{g}}$-ordered compression mapping in the second argument with respect to $\mathfrak{g}$ if there exists a constant ${\gamma }_{\mathcal{P}}^{\mathfrak{g}}>0$ such that

$$\mathcal{P}(.,\mathfrak{g}\left(\mathfrak{s}\right))\oplus \mathcal{P}(.,\mathfrak{g}\left(\mathfrak{t}\right))\le {\xi }_{\mathcal{P}}^{\mathfrak{g}}(\mathfrak{s}\oplus \mathfrak{t}).$$

Definition 6

([15,22]). Let $\mathcal{F}:\mathcal{H}\to 2^{\mathcal{H}}$ be a multi-valued mapping and $A:\mathcal{H}\to \mathcal{H}$ be a single-valued mapping. Then

(i)

$\mathcal{F}$ is said to be $\mathbb{H}$-Lipschitz-type-continuous if for any $\mathfrak{s},\mathfrak{t}\in \mathcal{H}$, $\mathfrak{s}\propto \mathfrak{t},$ there exists a constant ${\lambda }_{{\mathbb{H}}_{\mathcal{F}}}>0$ such that

$$\mathbb{H}(\mathcal{F}\left(\mathfrak{s}\right),\mathcal{F}\left(\mathfrak{t}\right))\le {\lambda }_{{\mathbb{H}}_{\mathcal{F}}}\parallel \mathfrak{s}\oplus \mathfrak{t}\parallel ,$$

(ii)

$\mathcal{F}$ is said to be a comparison mapping if for any $v_{\mathfrak{s}}\in \mathcal{F}\left(\mathfrak{t}\right),$$\mathfrak{s}\propto v_{\mathfrak{s}},$ and if $\mathfrak{s}\propto \mathfrak{t},$ then for $v_{\mathfrak{s}}\in \mathcal{F}\left(\mathfrak{s}\right)$ and $v_{\mathfrak{t}}\in \mathcal{F}\left(\mathfrak{t}\right),$$v_{\mathfrak{s}}\propto v_{\mathfrak{t}},\forall \mathfrak{s},\mathfrak{t}\in \mathcal{H},$

(iii)

a comparison mapping $\mathcal{F}$ is said to be ${\alpha }_A$-non-ordinary difference mapping with respect to A if for each $\mathfrak{s},\mathfrak{t}\in \mathcal{H},$$v_{\mathfrak{s}}\in \mathcal{F}\left(\mathfrak{s}\right)$ and $v_{\mathfrak{t}}\in \mathcal{F}\left(\mathfrak{t}\right)$ such that

$$(v_{\mathfrak{s}}\odot v_{\mathfrak{t}})\oplus {\alpha }_A(A\left(\mathfrak{s}\right)\oplus A\left(\mathfrak{t}\right))=0,$$

(iv)

a comparison mapping $\mathcal{F}$ is said to be λ-$XOR$-ordered different weak comparison mapping with respect to A, if $\mathfrak{s}\propto \mathfrak{t},$ then ∃ a constant $\lambda >0$ such that

$$\lambda (v_{\mathfrak{s}}\oplus v_{\mathfrak{t}})\ge \mathfrak{s}\oplus \mathfrak{t},\forall \mathfrak{s},\mathfrak{t}\in \mathcal{H},v_{\mathfrak{s}}\in \mathcal{F}\left(A\left(\mathfrak{s}\right)\right),v_{\mathfrak{t}}\in \mathcal{F}\left(A\left(\mathfrak{t}\right)\right),$$

(v)

$\mathcal{F}$ is said to be a $({\alpha }_A,\lambda )$-XOR-weak-ANODD mapping, if $\mathcal{F}$ is a ${\alpha }_A$-weak-non-ordinary difference mapping with respect to A, λ-XOR-ordered different weak comparison mapping with respect to A and $[A\oplus \lambda \mathcal{F}]\left(\mathcal{H}\right)=\mathcal{H}$.

Definition 7

([23]). Let $\mathcal{F}:\mathcal{H}\to 2^{\mathcal{H}}$ be a $({\alpha }_A,\lambda )$-XOR-weak-ANODD multi-valued mapping with respect to the strongly comparison and β-ordered compression mapping $A:\mathcal{H}\to \mathcal{H}$. Then the resolvent operator ${\mathcal{R}}_{\lambda ,\mathcal{F}}^A:\mathcal{H}\to \mathcal{H}$ associated with A and $\mathcal{F}$ is defined by

$${\mathcal{R}}_{\lambda ,\mathcal{F}}^A\left(\mathfrak{s}\right)={[A\oplus \lambda \mathcal{F}]}^{-1}\left(\mathfrak{s}\right),\forall \mathfrak{s}\in \mathcal{H},$$

(1)

where ${\alpha }_A,\lambda >0,and\beta \in (0,1).$

Proposition 3

([22,23]). Let $\mathcal{F}:\mathcal{H}\to 2^{\mathcal{H}}$ be a $({\alpha }_A,\lambda )$-XOR-weak-ANODD multi-valued mapping with respect to the strongly comparison and β-ordered compression mapping $A:\mathcal{H}\to \mathcal{H}$ such that ${\alpha }_A\lambda >\beta$. Then the resolvent operator ${\mathcal{R}}_{\lambda ,\mathcal{F}}^A:\mathcal{H}\to \mathcal{H}$ is well defined and single value for $\alpha \lambda >0$ and $\beta \in (0,1)$.

Proposition 4

([22,23]). Let $\mathcal{F}:\mathcal{H}\to 2^{\mathcal{H}}$ be a $({\alpha }_A,\lambda )$-XOR-weak-ANODD multi value mapping with respect to ${\mathcal{R}}_{\lambda ,\mathcal{F}}^A$. Let $A:\mathcal{H}\to \mathcal{H}$ be the strongly comparison and β-ordered compression mapping with respect to ${\mathcal{R}}_{\lambda ,\mathcal{F}}^A$ for $\mu \ge 1$ and ${\alpha }_A\lambda >\beta$. Then, ${\mathcal{R}}_{\lambda ,\mathcal{F}}^A$ is a comparison mapping and satisfies the condition

$${\mathcal{R}}_{\lambda ,\mathcal{F}}^A\left(\mathfrak{s}\right)\oplus {\mathcal{R}}_{\lambda ,\mathcal{F}}^A\left(\mathfrak{t}\right)\le \frac{\mu }{(\lambda {\alpha }_A\oplus \beta )}(\mathfrak{s}\oplus \mathfrak{t}),\forall \mathfrak{s},\mathfrak{t}\in \mathcal{H},$$

(2)

i.e., the resolvent operator ${\mathcal{R}}_{\lambda ,\mathcal{F}}^A$ is Lipschitz-type-continuous.

3. Problem Formulation Furthermore, Existence of Solution

Throughout, in rest portion of this article, we assume for $\kappa \in \mathbb{N}$, $i=1,2,\cdots ,\kappa$. Let $A_i,{\mathfrak{f}}_i,{\mathfrak{g}}_i:\mathcal{H}\to \mathcal{H}$ and ${\mathcal{P}}_i:\mathcal{H}\times \mathcal{H}\to \mathcal{H}$ be single valued mappings. Suppose ${\mathcal{F}}_i:\mathcal{H}\to 2^{\mathcal{H}}$ are $({\alpha }_{A_i},{\lambda }_i)$-XOR-weak-ANODD multi-valued mapping and ${\mathcal{M}}_i,{\mathcal{N}}_i:\mathcal{H}\to CB\left(\mathcal{H}\right)$ are closed and bounded multi-valued mappings. Then, the problem of finding $(x_1,\cdots ,x_{\kappa })\in \underset{\kappa -\mathrm{times}}{\underbrace{\mathcal{H}\times \cdots \times \mathcal{H}}}$, $(u_1,\cdots ,u_k)\in {\mathcal{M}}_1\left(x_1\right)\times \cdots \times {\mathcal{M}}_{\kappa }\left(x_{\kappa }\right)$ and $(v_1,\cdots ,v_k)\in {\mathcal{N}}_1\left(x_1\right)\times \cdots \times {\mathcal{N}}_{\kappa }\left(x_{\kappa }\right)$ such that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_1\in {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\oplus {\mathcal{F}}_1(u_1,v_2),\hfill \\ {\omega }_2\in {\mathcal{P}}_2({\mathfrak{f}}_2,\left(x_2\right),{\mathfrak{g}}_3\left(x_3\right))\oplus {\mathcal{F}}_2(u_2,v_3),\hfill \\ ⋮\hfill \\ {\omega }_{\kappa }\in {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }\right),{\mathfrak{g}}_1\left(x_1\right))\oplus {\mathcal{F}}_{\kappa }(u_{\kappa },v_1),\hfill \end{array}\right.\end{array}$$

(3)

holds, for some $\widehat{\omega }=({\omega }_1,\cdots {\omega }_{\kappa })\in \underset{\kappa -\mathrm{times}}{\underbrace{\mathcal{H}\times \cdots \times \mathcal{H}}},$ is called a nonlinear system of generalized ordered XOR-inclusion problems. By suitable assumptions of mappings involving in system (3), one can obtain many existing problems like [25]. Some of them are mentioned here.

Special Cases:

• For $i=1,2.$ If we define ${\mathcal{N}}_i={\mathfrak{g}}_i=I$ (the identity operator on $\mathcal{H}$), ${\mathcal{M}}_i={\mathfrak{h}}_i$ and ⊕ is replaced by +, then the System (3) reduces to the system of finding $(x,y)\in \mathcal{H}\times \mathcal{H}$ such that

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_1\in {\mathcal{P}}_1({\mathfrak{f}}_1\left(x\right),y)+{\mathcal{F}}_1({\mathfrak{h}}_1\left(x\right),y),\hfill \\ {\omega }_2\in {\mathcal{P}}_2({\mathfrak{f}}_2\left(y\right),x)+{\mathcal{F}}_2({\mathfrak{h}}_2\left(y\right),x),\hfill \end{array}\right.\end{array}$$

  (4)

  for some $({\omega }_1,{\omega }_2)\in \mathcal{H}\times \mathcal{H}$. System (4) is introduced and studied by [26].
• For $i=1,2$, if we define ${\mathcal{P}}_1({\mathfrak{f}}_1\left(x\right),y)=S(x-{\mathfrak{f}}_1\left(x\right),y)+C_{A,\lambda }^{\mathcal{M}}\left(x\right)$, ${\mathcal{P}}_2({\mathfrak{f}}_2\left(y\right),x)=T(x,y-{\mathfrak{f}}_2\left(y\right))+C_{A,\lambda }^{\mathcal{N}}\left(y\right)$, ${\mathcal{F}}_1(u_1\left(x\right),v_2\left(y\right))=\mathcal{M}\left(x\right)and{\mathcal{F}}_2(u_2\left(x\right),v_1\left(y\right))=\mathcal{N}\left(y\right)$, then for ${\omega }_i=0,\forall i,$ the system (3) reduces into the System of Multi-Valued Mixed Variational Inclusions with XOR-Operation; that is, the problem of finding $(x,y)\in \mathcal{H}\times \mathcal{H}$, $u\in \mathcal{M}\left(x\right)$ and $v\in \mathcal{N}$ such that

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}0\in S(x-{\mathfrak{f}}_1\left(x\right),y)+C_{A,\lambda }^{\mathcal{M}}\left(x\right)\oplus \mathcal{M}\left(x\right),\hfill \\ 0\in T(x,y-{\mathfrak{f}}_2\left(y\right))+C_{A,\lambda }^{\mathcal{N}}\left(y\right)\oplus \mathcal{N}\left(y\right),\hfill \end{array}\right.\end{array}$$

  (5)

  hold. The system (5) studied by [27].

The fixed point formulation for the nonlinear system of generalized ordered XOR -inclusion problems is the following lemma.

Lemma 1.

The triplets $(x,u,v)$, i.e., $x=(x_1,\cdots ,x_{\kappa })\in \mathcal{H}\times \cdots \times \mathcal{H},$$u=(u_1,\cdots ,u_k)\in {\mathcal{M}}_1\left(x_1\right)\times \cdots \times {\mathcal{M}}_{\kappa }\left(x_{\kappa }\right)$ and $v=(v_1,\cdots ,v_k)\in {\mathcal{N}}_1\left(x_1\right)\times \cdots \times {\mathcal{N}}_{\kappa }\left(x_{\kappa }\right)$ is a solution of system (3) if and only if the following equations hold:

$$\begin{array}{cc}\hfill x_1& ={\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1,v_2)}^{A_1}\left[A_1\left(x_1\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\}\right],\hfill \\ \hfill x_2& ={\mathfrak{R}}_{{\lambda }_2,{\mathcal{F}}_2(u_2,v_3)}^{A_2}\left[A_2\left(x_2\right)+{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(x_2\right),{\mathfrak{g}}_3\left(x_3\right))\}\right],\hfill \\ \hfill & ⋮\hfill \\ \hfill x_{\kappa }& ={\mathfrak{R}}_{{\lambda }_{\kappa },{\mathcal{F}}_{\kappa }(u_{\kappa },v_1)}^{A_{\kappa }}\left[A_{\kappa }\left(x_{\kappa }\right)+{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }\right),{\mathfrak{g}}_1\left(x_1\right))\}\right].\hfill \end{array}$$

(6)

hold, where ${\lambda }_i^{{}^{\prime }s}>0$ are constants.

Proof. 

Utilizing Definition 7 of the resolvent operator, the proof is simple and easily achieved.

Under some reasonable assumptions, the existence of the solution to system (3) is demonstrated by the following theorem.    □

Theorem 1.

For $ı=1,2,\cdots ,\kappa$, $\kappa \in \mathbb{N}and we set\kappa +1=1$, let $A_i,{\mathfrak{f}}_i,{\mathfrak{g}}_i:\mathcal{H}\to \mathcal{H}$ be single value mappings such that ${A_i}^{{}^{\prime }s}$ are ${\beta }^{A_i}$-ordered compression mapping and ${\mathfrak{f}}_i$, ${\mathfrak{g}}_i$ be strongly comprasion mappings. Assume that ${\mathcal{P}}_i:\mathcal{H}\times \mathcal{H}\to \mathcal{H}$ is single value mapping such that ${\mathcal{P}}_i$ is ${\gamma }_{{\mathcal{P}}_i}^{{\mathfrak{f}}_i}$-ordered compression in first argument with respect to ${\mathfrak{f}}_i$ and ${\xi }_{{\mathcal{P}}_i}^{{\mathfrak{g}}_i}$-ordered compression in second argument with respect to ${\mathfrak{g}}_i$, respectively. Suppose ${\mathcal{F}}_i:\mathcal{H}\to 2^{\mathcal{H}}$ is a $({\alpha }_{A_i},{\lambda }_i)$-XOR-weak-ANODD multi-value mapping and ${\mathcal{M}}_i,{\mathcal{N}}_i:\mathcal{H}\to CB\left(\mathcal{H}\right)$ are $\mathbb{H}$-Lipschitz continuous with constants ${\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_i}}$ and ${\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_i}}$, respectively.

Furthermore, if $x_i\propto x_{i+1}$, ${\mathfrak{f}}_i\left(x_i\right)\propto {\mathfrak{f}}_i\left(x_{i+1}\right)$, ${\mathfrak{g}}_i\left(x_i\right)\propto {\mathfrak{g}}_i\left(x_{i+1}\right)$, ${\mathfrak{R}}_{{\lambda }_i,{\mathcal{F}}_i(u_i,v_{i+1})}^{A_i}\left(x_i\right)\propto {\mathfrak{R}}_{{\lambda }_i,{\mathcal{F}}_i(u_i^{\prime },v_{i+1}^{\prime })}^{A_i}\left(y_i\right)$ and for all constants ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_{\kappa },{}_1{\mathsf{\Lambda }}_2,{}_2{\mathsf{\Lambda }}_3,\cdots ,{}_{\kappa }{\mathsf{\Lambda }}_1>0$, the following relations hold:

$${\mathfrak{R}}_{{\lambda }_i,{\mathcal{F}}_i(u_i,v_{i+1})}^{A_i}\left(x_i\right)\oplus {\mathfrak{R}}_{{\lambda }_i,{\mathcal{F}}_i(u_i^{\prime },v_{i+1}^{\prime })}^{A_i}\left(x_i\right)\le {}_i{\mathsf{\Lambda }}_{i+1}(u_i\oplus u_i^{\prime }+v_{i+1}\oplus v_{i+1}^{\prime }),foreachi,$$

(7)

and

$$\begin{array}{cc}\hfill {\mathsf{\Theta }}_1({\beta }^{A_1}+{\lambda }_1{\gamma }_{{\mathfrak{g}}_1}^{{\mathfrak{f}}_1})+{ϵ}_1({}_1{\mathsf{\Lambda }}_2{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_1}}+{}_1{\mathsf{\Lambda }}_2{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_2}})& <\frac{1}{{\lambda }_C}-{\mathsf{\Theta }}_{\kappa }{\lambda }_{\kappa }{\xi }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{g}}_1},\hfill \\ \hfill {\mathsf{\Theta }}_2({\beta }^{A_2}+{\lambda }_2{\gamma }_{{\mathfrak{g}}_2}^{{\mathfrak{f}}_2})+{ϵ}_2({}_2{\mathsf{\Lambda }}_3{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_2}}+{}_2{\mathsf{\Lambda }}_3{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_3}})& <\frac{1}{{\lambda }_C}-{\mathsf{\Theta }}_1{\lambda }_1{\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2},\hfill \\ \hfill & ⋮\hfill \\ \hfill {\mathsf{\Theta }}_{\kappa }({\beta }^{A_{\kappa }}+{\lambda }_{\kappa }{\gamma }_{{\mathfrak{g}}_{\kappa }}^{{\mathfrak{f}}_{\kappa }})+{ϵ}_{\kappa }({}_{\kappa }{\mathsf{\Lambda }}_1{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_{\kappa }}}+{}_{\kappa }{\mathsf{\Lambda }}_1{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_1}})& <\frac{1}{{\lambda }_C}-{\mathsf{\Theta }}_{\kappa -1}{\lambda }_{\kappa -1}{\xi }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{g}}_{\kappa }},\hfill \end{array}$$

(8)

where

$${\mathsf{\Theta }}_i=\frac{{\mu }_i}{{\lambda }_i{\alpha }_{A_i}\oplus {\beta }^{A_i}}.$$

Proof. 

We define the mapping $B:\underset{\kappa -\mathrm{times}}{\underbrace{\mathcal{H}\times \cdots \times \mathcal{H}}}⟶\underset{\kappa -\mathrm{times}}{\underbrace{\mathcal{H}\times \cdots \times \mathcal{H}}}$ by

$$\begin{array}{c}\hfill B(x_1,\cdots ,x_{\kappa }):=(B_1(x_1,x_2),\cdots ,B_{\kappa }(x_{\kappa },x_1)),\forall (x_1,\cdots ,x_{\kappa })\in \underset{\kappa -\mathrm{times}}{\underbrace{\mathcal{H}\times \cdots \times \mathcal{H}}},\end{array}$$

where $B_i:\mathcal{H}\times \mathcal{H}\to \mathcal{H}$ are mappings defined as

$$\begin{array}{c}\hfill B_1(x_1,x_2)={\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1,v_2)}^{A_1}\left[A_1\left(x_1\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\}\right],\\ \hfill B_2(x_2,x_3)={\mathfrak{R}}_{{\lambda }_2,{\mathcal{F}}_2(u_2,v_3)}^{A_2}\left[A_2\left(x_2\right)+{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(x_2\right),{\mathfrak{g}}_3\left(x_3\right))\}\right],\\ \hfill ⋮\\ \hfill B_{\kappa }(x_{\kappa },x_1)={\mathfrak{R}}_{{\lambda }_{\kappa },{\mathcal{F}}_{\kappa }(u_{\kappa },v_1)}^{A_{\kappa }}\left[A_{\kappa }\left(x_{\kappa }\right)+{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }\right),{\mathfrak{g}}_1\left(x_1\right))\}\right].\end{array}$$

For any $x_i,y_i\in \mathcal{H},u_i\in {\mathcal{M}}_i\left(x_i\right),v_i\in {\mathcal{N}}_i\left(x_i\right)$ such that $x_i\propto x_j,y_i\propto y_j,u_i\propto u_j$, $v_i\propto v_j$ and using Proposition 1, we have

$$\begin{array}{ccc}\hfill 0& \le & B_1(x_1,x_2)\oplus B_1(y_1,y_2)\hfill \\ & =& {\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1,v_2)}^{A_1}\left[A_1\left(x_1\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\}\right]\hfill \\ & & \oplus {\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1^{\prime },v_2^{\prime })}^{A_1}\left[A_1\left(y_1\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1\right),{\mathfrak{g}}_2\left(y_2\right))\}\right].\hfill \end{array}$$

Using the Proposition 1, Proposition 4,  (7) and the compression of $A_1$, we have

$$\begin{array}{ccc}\hfill 0& \le & B_1(x_1,x_2)\oplus B_1(y_1,y_2)\hfill \\ & \le & {\mathsf{\Theta }}_1[A_1\left(x_1\right)\oplus A_1\left(y_1\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\}\hfill \\ & & \oplus {\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1\right),{\mathfrak{g}}_2\left(y_2\right))\}]+{}_1{\mathsf{\Lambda }}_2(u_1\oplus u_1^{\prime }+v_2\oplus v_2^{\prime })\hfill \\ & \le & {\mathsf{\Theta }}_1\left[{\beta }^{A_1}(x_1\oplus y_1)+{\lambda }_1({\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\oplus P_1({\mathfrak{f}}_1\left(y_1\right),{\mathfrak{g}}_2\left(y_2\right)))\right]\hfill \\ & & +{}_1{\mathsf{\Lambda }}_2(u_1\oplus u_1^{\prime }+v_2\oplus v_2^{\prime }).\hfill \end{array}$$

(9)

By using the normal cone definition and (9), we have

$$\begin{array}{ccc}\hfill \parallel B_1(x_1,x_2)-B_1(y_1,y_2)\parallel & \le & {\lambda }_P\parallel B_1(x_1,x_2)\oplus B_1(y_1,y_2)\parallel \hfill \\ & \le & {\lambda }_P[{\mathsf{\Theta }}_1{\beta }^{A_1}\parallel x_1\oplus y_1\parallel +{\mathsf{\Theta }}_1{\lambda }_1\parallel {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\hfill \\ & & \oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1\right),{\mathfrak{g}}_2\left(y_2\right))\parallel +{}_1{\mathsf{\Lambda }}_2[\parallel u_1\oplus u_1^{\prime }\parallel +\parallel v_2\oplus v_2^{\prime }\parallel ]].\hfill \end{array}$$

(10)

Using $\left(viii\right)$ of Proposition 1, we calculate

$$\begin{array}{cc}\hfill \parallel {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\oplus P_1({\mathfrak{f}}_1\left(y_1\right),{\mathfrak{g}}_2\left(y_2\right))\parallel & \le \parallel {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(y_2\right))\parallel \hfill \\ \hfill & +\parallel {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(y_2\right))\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1\right),{\mathfrak{g}}_2\left(y_2\right))\parallel .\hfill \end{array}$$

(11)

Since ${\mathcal{P}}_1$ is ${\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1}$-ordered compression mapping in the first argument with respect to ${\mathfrak{f}}_1$ and ${\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}$-ordered compression mapping in the second argument with respect to ${\mathfrak{g}}_2$, we have

$$\begin{array}{c}\hfill \parallel {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1\right),{\mathfrak{g}}_2\left(y_2\right))\parallel \le {\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1}\parallel x_1\oplus y_1\parallel +{\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}\parallel x_2\oplus y_2\parallel .\end{array}$$

(12)

Since $u_1\in {\mathcal{M}}_1\left(x_1\right),$ $v_2\in {\mathcal{N}}_2\left(x_2\right)$, by Nadler [28], ∃$u_1^{\prime }\in {\mathcal{M}}_1\left(y_1\right),v_2^{\prime }\in {\mathcal{N}}_2\left(y_2\right)$ such that

$$\begin{array}{cc}\hfill {}_1{\mathsf{\Lambda }}_2\parallel u_1\oplus u_1{}^{\prime }\parallel +{}_1{\mathsf{\Lambda }}_2\parallel v_2\oplus v_2{}^{\prime }\parallel & \le {}_1{\mathsf{\Lambda }}_2{ϵ}_1\mathbb{H}({\mathcal{M}}_1\left(x_1\right),{\mathcal{M}}_1\left(y_1\right))\hfill \\ \hfill & +{}_1{\mathsf{\Lambda }}_2{ϵ}_2\mathbb{H}({\mathcal{N}}_2\left(x_2\right),{\mathcal{N}}_2\left(y_2\right)).\hfill \end{array}$$

(13)

Using $\mathbb{H}$ -Lipschitz continuity of ${\mathcal{M}}_1$ and ${\mathcal{N}}_2$, we obtain

$$\begin{array}{cc}\hfill {}_1{\mathsf{\Lambda }}_2\parallel u_1\oplus u_1{}^{\prime }\parallel +{}_1{\mathsf{\Lambda }}_2\parallel v_2\oplus v_2{}^{\prime }\parallel & \le {}_1{\mathsf{\Lambda }}_2[{ϵ}_1{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_1}}\parallel x_1\oplus y_1\parallel +{ϵ}_2{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_2}}\parallel x_2\oplus y_2\parallel ].\hfill \end{array}$$

(14)

Using (12), (14) and Proposition 1 in (10), we get

$$\begin{array}{ccc}\hfill \parallel B_1(x_1,x_2)-B_1(y_1,y_2)\parallel & \le & {\lambda }_P[{\mathsf{\Theta }}_1{\beta }^{A_1}\parallel x_1\oplus y_1\parallel +{\mathsf{\Theta }}_1{\lambda }_1[{\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1}\parallel x_1\oplus y_1\parallel +{\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}\parallel x_2\oplus y_2\parallel ]\hfill \\ & & +{}_1{\mathsf{\Lambda }}_2[{ϵ}_1{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_1}}\parallel x_1\oplus y_1\parallel +{ϵ}_2{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_2}}\parallel x_2\oplus y_2\parallel ]]\hfill \\ & =& {\lambda }_P[({\mathsf{\Theta }}_1{\beta }^{A_1}+{\mathsf{\Theta }}_1{\lambda }_1{\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1}+{}_1{\mathsf{\Lambda }}_2{ϵ}_1{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_1}})\parallel x_1-y_1\parallel \hfill \\ & & +({\mathsf{\Theta }}_1{\lambda }_1{\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}+{}_1{\mathsf{\Lambda }}_2{ϵ}_2{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_2}})\parallel x_2-y_2\parallel ].\hfill \end{array}$$

(15)

For any $x_i,y_i\in \mathcal{H},u_3\in {\mathcal{M}}_3\left(x_3\right),v_3\in {\mathcal{N}}_3\left(x_3\right)$ such that $x_i\propto x_j,y_i\propto y_j,u_i\propto u_j$,  $v_i\propto v_j$ and using Proposition 1, we have

$$\begin{array}{ccc}\hfill 0& \le & B_2(x_2,x_3)\oplus B_2(y_2,y_3)\hfill \\ & =& {\mathfrak{R}}_{{\lambda }_2,{\mathcal{F}}_2(u_2,v_3)}^{A_2}\left[A_2\left(x_2\right)+{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(x_2\right),{\mathfrak{g}}_3\left(x_3\right))\}\right]\hfill \\ & & \oplus {\mathfrak{R}}_{{\lambda }_2,{\mathcal{F}}_2(u_2^{\prime },v_3^{\prime })}^{A_2}\left[A_2\left(y_2\right)+{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(y_2\right),{\mathfrak{g}}_3\left(y_3\right))\}\right].\hfill \end{array}$$

Using the Proposition 1, Proposition 4,  (7) and the compression of $A_2$, we have

$$\begin{array}{ccc}\hfill 0& \le & B_2(x_2,x_3)\oplus B_2(y_2,y_3)\hfill \\ & \le & {\mathsf{\Theta }}_2[A_2\left(x_2\right)\oplus A_2\left(y_2\right)+{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(x_2\right),{\mathfrak{g}}_3\left(x_3\right))\}\hfill \\ & & \oplus {\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(y_2\right),{\mathfrak{g}}_3\left(y_3\right))\}]+{}_2{\mathsf{\Lambda }}_3(u_2\oplus u_2^{\prime }+v_3\oplus v_3^{\prime })\hfill \\ & \le & {\mathsf{\Theta }}_2\left[{\beta }^{A_2}(x_2\oplus y_2)+{\lambda }_2({\mathcal{P}}_2({\mathfrak{f}}_2\left(x_2\right),{\mathfrak{g}}_3\left(x_3\right))\oplus P_2({\mathfrak{f}}_2\left(y_2\right),{\mathfrak{g}}_3\left(y_3\right)))\right]\hfill \\ & & +{}_2{\mathsf{\Lambda }}_3(u_2\oplus u_2^{\prime }+v_3\oplus v_3^{\prime }).\hfill \end{array}$$

(16)

Using Proposition 2 and (16), we have

$$\begin{array}{ccc}\hfill \parallel B_2(x_2,x_3)-B_2(y_2,y_3)\parallel & \le & {\lambda }_P\parallel B_2(x_2,x_3)\oplus B_2(y_2,y_3)\parallel \hfill \\ & \le & {\lambda }_P[{\mathsf{\Theta }}_2{\beta }^{A_2}\parallel x_2\oplus y_2\parallel +{\mathsf{\Theta }}_2{\lambda }_2\parallel {\mathcal{P}}_2({\mathfrak{f}}_2\left(x_2\right),{\mathfrak{g}}_3\left(x_3\right))\hfill \\ & & \oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(y_2\right),{\mathfrak{g}}_3\left(y_3\right))\parallel +{}_2{\mathsf{\Lambda }}_3[\parallel u_2\oplus u_2^{\prime }\parallel +\parallel v_3\oplus v_3^{\prime }\parallel ]].\hfill \end{array}$$

(17)

Since ${\mathcal{P}}_2$ is ${\gamma }_{{\mathcal{P}}_2}^{{\mathfrak{f}}_2}$-ordered compression mapping in the first argument with respect to ${\mathfrak{f}}_2$ and ${\xi }_{{\mathcal{P}}_2}^{{\mathfrak{g}}_3}$-ordered compression mapping in the second argument with respect to ${\mathfrak{g}}_3$. Hence, importing the same logic as in (11) for (12), we have

$$\begin{array}{c}\hfill \parallel {\mathcal{P}}_2({\mathfrak{f}}_2\left(x_2\right),{\mathfrak{g}}_3\left(x_3\right))\oplus P_2({\mathfrak{f}}_2\left(y_2\right),{\mathfrak{g}}_3\left(y_3\right))\parallel \le {\gamma }_{{\mathcal{P}}_2}^{{\mathfrak{f}}_2}\parallel x_2\oplus y_2\parallel +{\xi }_{{\mathcal{P}}_2}^{{\mathfrak{g}}_3}\parallel x_3\oplus y_3\parallel .\end{array}$$

(18)

Similarly,

$$\begin{array}{ccc}\hfill {}_2{\mathsf{\Lambda }}_3\parallel u_2\oplus u_2{}^{\prime }\parallel +{}_2{\mathsf{\Lambda }}_3\parallel v_3\oplus v_3{}^{\prime }\parallel & \le & {}_2{\mathsf{\Lambda }}_3[{ϵ}_2{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_2}}\parallel x_2\oplus y_2\parallel +{ϵ}_3{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_3}}\parallel x_3\oplus y_3\parallel ].\hfill \end{array}$$

(19)

Using (18),  (19) and Proposition 1 in (17), we obtain

$$\begin{array}{ccc}\hfill \parallel B_2(x_2,x_3)-B_2(y_2,y_3)\parallel & \le & {\lambda }_P[{\mathsf{\Theta }}_2{\beta }^{A_2}\parallel x_2\oplus y_2\parallel +{\mathsf{\Theta }}_2{\lambda }_2[{\gamma }_{{\mathcal{P}}_2}^{{\mathfrak{f}}_2}\parallel x_2\oplus y_2\parallel +{\xi }_{{\mathcal{P}}_2}^{{\mathfrak{g}}_2}\parallel x_3\oplus y_3\parallel ]\hfill \\ & & +{}_2{\mathsf{\Lambda }}_3[{ϵ}_2{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_2}}\parallel x_2\oplus y_2\parallel +{ϵ}_3{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_3}}\parallel x_3\oplus y_3\parallel ]]\hfill \\ & =& {\lambda }_P[({\mathsf{\Theta }}_2{\beta }^{A_2}+{\mathsf{\Theta }}_2{\lambda }_2{\gamma }_{{\mathcal{P}}_2}^{{\mathfrak{f}}_2}+{}_2{\mathsf{\Lambda }}_3{ϵ}_2{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_2}})\parallel x_2-y_2\parallel \hfill \\ & & +({\mathsf{\Theta }}_2{\lambda }_2{\xi }_{{\mathcal{P}}_2}^{{\mathfrak{g}}_3}+{}_2{\mathsf{\Lambda }}_3{ϵ}_3{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_3}})\parallel x_3-y_3\parallel .\hfill \end{array}$$

(20)

Continuing in this way, for any $x_i,y_i\in \mathcal{H},u_{\kappa }\in {\mathcal{M}}_{\kappa }\left(x_{\kappa }\right),v_1\in {\mathcal{N}}_1\left(x_1\right)$ such that $x_i\propto x_j,y_i\propto y_j,u_i\propto u_j$,  $v_i\propto v_j$ and using Proposition 1, we have

$$\begin{array}{ccc}\hfill 0& \le & B_{\kappa }(x_{\kappa },x_1)\oplus B_{\kappa }(y_{\kappa },y_1)\hfill \\ & =& {\mathfrak{R}}_{{\lambda }_{\kappa },{\mathcal{F}}_{\kappa }(u_{\kappa },v_1)}^{A_{\kappa }}\left[A_{\kappa }\left(x_{\kappa }\right)+{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }\right),{\mathfrak{g}}_1\left(x_1\right))\}\right]\hfill \\ & & \oplus {\mathfrak{R}}_{{\lambda }_{\kappa },{\mathcal{F}}_{\kappa }(u_{\kappa }^{\prime },v_1^{\prime })}^{A_{\kappa }}\left[A_{\kappa }\left(y_{\kappa }\right)+{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(y_{\kappa }\right),{\mathfrak{g}}_1\left(y_1\right))\}\right].\hfill \end{array}$$

Using the Proposition 4,  (7) and the compression of $A_{\kappa }$, we have

$$\begin{array}{ccc}\hfill 0& \le & B_{\kappa }(x_{\kappa },x_1)\oplus B_{\kappa }(y_{\kappa },y_1)\hfill \\ & \le & {\mathsf{\Theta }}_{\kappa }[A_{\kappa }\left(x_{\kappa }\right)\oplus A_{\kappa }\left(y_{\kappa }\right)+{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }\right),{\mathfrak{g}}_1\left(x_1\right))\}\hfill \\ & & \oplus {\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(y_{\kappa }\right),{\mathfrak{g}}_1\left(y_1\right))\}]+{}_{\kappa }{\mathsf{\Lambda }}_1(u_{\kappa }\oplus u_{\kappa }^{\prime }+v_1\oplus v_1^{\prime })\hfill \\ & \le & {\mathsf{\Theta }}_{\kappa }\left[{\beta }^{A_{\kappa }}(x_{\kappa }\oplus y_{\kappa })+{\lambda }_{\kappa }({\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }\right),{\mathfrak{g}}_1\left(x_1\right))\oplus P_{\kappa }({\mathfrak{f}}_{\kappa }\left(y_{\kappa }\right),{\mathfrak{g}}_1\left(y_1\right)))\right]\hfill \\ & & +{}_{\kappa }{\mathsf{\Lambda }}_1(u_{\kappa }\oplus u_{\kappa }^{\prime }+v_1\oplus v_1^{\prime }).\hfill \end{array}$$

(21)

Using Proposition 2 and (21), we have

$$\begin{array}{ccc}\hfill \parallel B_{\kappa }(x_{\kappa },x_1)-B_{\kappa }(y_{\kappa },y_1)\parallel & \le & {\lambda }_P\parallel B_{\kappa }(x_{\kappa },x_1)\oplus B_{\kappa }(y_{\kappa },y_1)\parallel \hfill \\ & \le & {\lambda }_P[{\mathsf{\Theta }}_{\kappa }{\beta }^{A_{\kappa }}\parallel x_{\kappa }\oplus y_{\kappa }\parallel +{\mathsf{\Theta }}_{\kappa }{\lambda }_{\kappa }\parallel {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }\right),{\mathfrak{g}}_1\left(x_1\right))\hfill \\ & & \oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(y_{\kappa }\right),{\mathfrak{g}}_1\left(y_1\right))\parallel +{\delta }_1^{\kappa }\parallel u_{\kappa }\oplus u_{\kappa }^{\prime }\parallel +\parallel v_1\oplus v_1^{\prime }\parallel \left]\right].\hfill \end{array}$$

(22)

Importing the same logic as in (11) for (12), we obtain

$$\begin{array}{c}\hfill \parallel {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }\right),{\mathfrak{g}}_1\left(x_1\right))\oplus P_{\kappa }({\mathfrak{f}}_{\kappa }\left(y_{\kappa }\right),{\mathfrak{g}}_1\left(y_1\right))\parallel \le {\gamma }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{f}}_{\kappa }}\parallel x_{\kappa }\oplus y_{\kappa }\parallel +{\xi }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{g}}_1}\parallel x_1\oplus y_1\parallel .\end{array}$$

(23)

Using (23), Proposition 1 and $\mathbb{H}$-Lipschitz continuity of ${\mathcal{M}}_{\kappa }$ and ${\mathcal{N}}_1$ in (22), we obtain

$$\begin{array}{ccc}\hfill \parallel B_{\kappa }(x_{\kappa },x_1)-B_{\kappa }(y_{\kappa },y_1)\parallel & \le & {\lambda }_P[{\mathsf{\Theta }}_{\kappa }{\beta }^{A_{\kappa }}\parallel x_{\kappa }\oplus y_{\kappa }\parallel +{\mathsf{\Theta }}_{\kappa }{\lambda }_{\kappa }[{\gamma }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{f}}_{\kappa }}\parallel x_{\kappa }\oplus y_{\kappa }\parallel +{\xi }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{g}}_{\kappa }}\parallel x_1\oplus y_1\parallel ]\hfill \\ & & +{}_{\kappa }{\mathsf{\Lambda }}_1[{ϵ}_{\kappa }{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_{\kappa }}}\parallel x_{\kappa }\oplus y_{\kappa }\parallel +{ϵ}_1{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_1}}\parallel x_1\oplus y_1\parallel ]]\hfill \\ & =& {\lambda }_P[({\mathsf{\Theta }}_{\kappa }{\beta }^{A_{\kappa }}+{\mathsf{\Theta }}_{\kappa }{\lambda }_{\kappa }{\gamma }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{f}}_{\kappa }}+{}_{\kappa }{\mathsf{\Lambda }}_1{ϵ}_{\kappa }{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_{\kappa }}})\parallel x_{\kappa }-y_{\kappa }\parallel \hfill \\ & & +({\mathsf{\Theta }}_{\kappa }{\lambda }_{\kappa }{\xi }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{g}}_1}+{}_{\kappa }{\mathsf{\Lambda }}_1{ϵ}_1{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_1}})\parallel x_1-y_1\parallel .\hfill \end{array}$$

(24)

From (15), (20) and (24), we have

$$\begin{array}{cc}\hfill \parallel B_1(x_1,x_2)-B_1(y_1,y_2)\parallel & +\parallel B_2(x_2,x_3)-B_2(y_2,y_3)\parallel +\cdots +\parallel B_{\kappa }(x_{\kappa },x_1)-B_{\kappa }(y_{\kappa },y_1)\parallel \hfill \\ \hfill & \le {\lambda }_P[\{{\mathsf{\Theta }}_1({\beta }^{A_1}+{\lambda }_1{\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1})+{ϵ}_1({}_1{\mathsf{\Lambda }}_2{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_1}}+{}_{\kappa }{\mathsf{\Lambda }}_1{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_1}})+{\mathsf{\Theta }}_{\kappa }{\lambda }_{\kappa }{\xi }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{g}}_1}\}\parallel x_1-y_1\parallel \hfill \\ \hfill & +\{{\mathsf{\Theta }}_2({\beta }^{A_2}+{\lambda }_2{\gamma }_{{\mathcal{P}}_2}^{{\mathfrak{f}}_2})+{ϵ}_2({}_2{\mathsf{\Lambda }}_3{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_2}}+{}_1{\mathsf{\Lambda }}_2{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_2}})+{\mathsf{\Theta }}_1{\lambda }_1{\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}\}\parallel x_2-y_2\parallel \hfill \\ \hfill & +\cdots +\{{\mathsf{\Theta }}_{\kappa }({\beta }^{A_{\kappa }}+{\lambda }_{\kappa }{\gamma }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{f}}_{\kappa }})+{ϵ}_{\kappa }({}_{\kappa }{\mathsf{\Lambda }}_1{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_{\kappa }}}+{}_{\kappa -1}{\mathsf{\Lambda }}_{\kappa }{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_{\kappa }}})\hfill \\ & +{\mathsf{\Theta }}_{\kappa -1}{\lambda }_{\kappa -1}{\xi }_{{\mathcal{P}}_{\kappa -1}}^{{\mathfrak{g}}_{\kappa }}\}\parallel x_{\kappa }-y_{\kappa }\parallel ],\hfill \end{array}$$

that is,

$$\begin{array}{cc}\hfill \parallel B_1(x_1,x_2)-B_1(y_1,y_2)\parallel & +\parallel B_2(x_2,x_3)-B_2(y_2,y_3)\parallel +\cdots +\parallel B_{\kappa }(x_{\kappa },x_1)-B_{\kappa }(y_{\kappa },y_1)\parallel \hfill \\ \hfill & ={\varphi }_1\parallel x_1-y_1\parallel +{\varphi }_2\parallel x_2-y_2\parallel +\cdots +{\varphi }_{\kappa }\parallel x_{\kappa }-y_{\kappa }\parallel ,\hfill \end{array}$$

(25)

where

$$\begin{array}{cc}\hfill {\varphi }_1& ={\lambda }_P[{\mathsf{\Theta }}_1({\beta }^{A_1}+{\lambda }_1{\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1})+{ϵ}_1({}_1{\mathsf{\Lambda }}_2{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_1}}+{}_{\kappa }{\mathsf{\Lambda }}_1{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_1}})+{\mathsf{\Theta }}_{\kappa }{\lambda }_{\kappa }{\xi }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{g}}_1}]\hfill \\ \hfill {\varphi }_2& ={\lambda }_P[{\mathsf{\Theta }}_2({\beta }^{A_2}+{\lambda }_2{\gamma }_{{\mathcal{P}}_2}^{{\mathfrak{f}}_2})+{ϵ}_2({}_2{\mathsf{\Lambda }}_3{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_2}}+{}_1{\mathsf{\Lambda }}_2{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_2}})+{\mathsf{\Theta }}_1{\lambda }_1{\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}]\hfill \\ \hfill & ⋮\hfill \\ \hfill {\varphi }_{\kappa }& ={\lambda }_P[{\mathsf{\Theta }}_{\kappa }({\beta }^{A_{\kappa }}+{\lambda }_{\kappa }{\gamma }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{f}}_{\kappa }})+{ϵ}_{\kappa }({}_{\kappa }{\mathsf{\Lambda }}_1{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_{\kappa }}}+{}_{\kappa -1}{\mathsf{\Lambda }}_{\kappa }{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_{\kappa }}})+{\mathsf{\Theta }}_{\kappa -1}{\lambda }_{\kappa -1}{\xi }_{{\mathcal{P}}_{\kappa -1}}^{{\mathfrak{g}}_{\kappa }}].\hfill \end{array}$$

Now, we define $\parallel (x_1,\cdots ,x_{\kappa }){\parallel }_{1}on\mathcal{H}\times \cdots \times \mathcal{H}$   by

$$\begin{array}{c}\hfill \parallel (x_1,\cdots ,x_{\kappa }){\parallel }_1=\parallel x_1\parallel +\cdots +\parallel x_{\kappa }\parallel ,\forall (x_1,\cdots ,x_{\kappa })\in \mathcal{H}\times \cdots \times \mathcal{H}.\end{array}$$

(26)

It is easy to prove that $(\mathcal{H}\times \cdots \times \mathcal{H},\parallel .{\parallel }_1)$ is a Banach space. Hence, from definition of the mapping B, (25) and (26), we have

$$\begin{array}{cc}\hfill \parallel B(x_1,\cdots ,x_{\kappa })-B(y_1,\cdots ,y_{\kappa }){\parallel }_1& =\parallel B_1(x_1,x_2)-B_1(y_1,y_2)\parallel +\cdots +\parallel B_{\kappa }(x_{\kappa },x_1)-B_{\kappa }(y_{\kappa },y_1)\parallel \hfill \\ \hfill & \le max\left\{{\varphi }_1,{\varphi }_2,\cdots ,{\varphi }_{\kappa }\right\}(\parallel x_1-y_1\parallel +\parallel x_2-y_2\parallel \cdots +\parallel x_{\kappa }-y_{\kappa }\parallel ).\hfill \end{array}$$

(27)

From (8) we know that $max\left\{{\varphi }_1,\cdots ,{\varphi }_{\kappa }\right\}<1$. Therefore, the mapping B is a contraction mapping. Hence, there exists a unique fixed point $(x_1,\cdots ,x_{\kappa })\in \mathcal{H}\times \cdots \times \mathcal{H}$ of B (by Banach contraction principle); that is,

$$\begin{array}{c}\hfill B(x_1,\cdots ,x_{\kappa })=(x_1,\cdots ,x_{\kappa }).\end{array}$$

This leads to

$$\begin{array}{cc}\hfill x_1& ={\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1,v_2)}^{A_1}\left[A_1\left(x_1\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\}\right],\hfill \\ \hfill x_2& ={\mathfrak{R}}_{{\lambda }_2,{\mathcal{F}}_2(u_2,v_3)}^{A_2}\left[A_2\left(x_2\right)+{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(x_2\right),{\mathfrak{g}}_3\left(x_3\right))\}\right],\hfill \\ \hfill & ⋮\hfill \\ \hfill x_{\kappa }& ={\mathfrak{R}}_{{\lambda }_{\kappa },{\mathcal{F}}_{\kappa }(u_{\kappa },v_1)}^{A_{\kappa }}\left[A_{\kappa }\left(x_{\kappa }\right)+{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }\right),{\mathfrak{g}}_1\left(x_1\right))\}\right].\hfill \end{array}$$

(28)

It is determined by Lemma (1) that $(x,u,v)$ is a solution of system (3).    □

In support of Theorem 1, we give the following numerical example.

Example 1.

Let $ı=1,2$, $\mathcal{H}=P={\mathbb{R}}_{+}\cup \left\{0\right\}$, the mappings $A_{ı},f_{ı},g_{ı}:\mathcal{H}\to \mathcal{H}$, ${\mathcal{F}}_i:\mathcal{H}:\to 2^{\mathcal{H}}$, be defined by

$$f_i\left(x_i\right)=\frac{2x_i}{7(i+1)},g_i\left(x_i\right)=\frac{3x_i}{49i},A_i\left(x_i\right)=\frac{i}{14}{x}_{i}and{\mathcal{F}}_i\left(x_i\right)=\left\{\frac{x_i}{3i}\right\}.$$

Suppose that the mappings $P_i:\mathcal{H}\times \mathcal{H}\to \mathcal{H}$ is defined as

$$P_i(x_1,x_2)=\frac{x_1+x_2}{10+i}$$

Now,

$$\begin{array}{ccc}\hfill A_i\left(x_i\right)\oplus A_i\left(y_i\right)& =& \left(\frac{i}{14}{x}_i\oplus \frac{i}{14}{y}_i\right)\hfill \\ & \le & \frac{2i}{14}(x_i\oplus y_i)\hfill \end{array}$$

i.e., $A_i$ is $\frac{i}{7}$-ordered compression. Furthermore,

$$\begin{array}{cc}\hfill \parallel P_i\left(f_i\left(x_i\right),g_{i+1}\left(x_{i+1}\right)\right)\oplus P_i\left(f_i\left(y_i\right),g_{i+1}\left(y_{i+1}\right)\right)\parallel & =\left(\frac{1}{49(i+1)(i+10)}\right)\parallel (14x_i+3x_{i+1})\hfill \\ \hfill & \oplus (14y_i+3y_{i+1})\parallel \hfill \\ \hfill & \le \frac{21}{49(i+1)(i+10)}\parallel x_i+y_i\parallel \hfill \\ \hfill & +\frac{7}{49(i+1)(i+10)}\parallel x_{i+1}+y_{i+1}\parallel \hfill \end{array}$$

i.e., $P_i$ is $\frac{3}{7(i+1)(10+i)}$-ordered compression in first argument with respect to $f_i$ and $\frac{1}{7(i+1)(10+i)}$ in second argument with respect to $g_i$.

It is also trivial to verify that ${\mathcal{F}}_i$ are $\frac{14}{3i^2}$-weak-non-ordinary-difference mappings and ${\lambda }_i$-XOR-ordered different weak compression with respect to $A_i$ where ${\lambda }_i\ge \frac{42}{i}$. For ${\lambda }_i\ge \frac{42}{i}$, $[A_i+{\lambda }_i{F}_i]\left(\mathcal{H}\right)=\mathcal{H}$ which exhibits that ${\mathcal{F}}_i$ are $\left(\frac{14}{3i^2},\frac{42}{i}\right)$-XOR weak ANODD multi-mappings.

Hence the resolvent operator ${\mathfrak{R}}_{{\lambda }_i,{\mathcal{F}}_i}^{A_i}:\mathcal{H}\to \mathcal{H}$ assciated with $A_i$ and ${\mathcal{F}}_i$ are of the form

$${\mathfrak{R}}_{{\lambda }_i,{\mathcal{F}}_i}^{A_i}\left(x_i\right)=\frac{x_i}{\frac{i}{14}\oplus \frac{{\lambda }_i}{3i}}=\left(\frac{42i}{3i^2\oplus 14{\lambda }_i}\right)x_i,$$

which are single-valued and comparisons. Now,

$$\begin{array}{ccc}\hfill {\mathfrak{R}}_{{\lambda }_i,{\mathcal{F}}_i}^{A_i}\left(x_i\right)\oplus {\mathfrak{R}}_{{\lambda }_i,{\mathcal{F}}_i}^{A_i}\left(y_i\right)& =& \left(\frac{42i}{3i^2\oplus 14{\lambda }_i}\right)x_i\oplus \left(\frac{42i}{3i^2\oplus 14{\lambda }_i}\right)y_i\hfill \\ & =& \frac{42i}{3i^2\oplus 14{\lambda }_i}(x_i\oplus y_i)\hfill \\ & \le & \frac{7i}{{\lambda }_i}(x_i\oplus y_i),where{\lambda }_i\ge \frac{42}{i}.\hfill \end{array}$$

Therefore, the resolvent operators ${\mathfrak{R}}_{{\lambda }_i,{\mathcal{F}}_i}^{A_i}$ are continuous Lipschitz type with constants $\frac{7i}{{\lambda }_i}$. Clearly, Theorem 1’s requirements are all satisfied.

4. S-Iteration and Its Convergence

In this section, based on Lemma 1, we develop the S-iterative algorithm for finding the approximate solution of the new system of generalized XOR-inclusion problem (Algorithm 1). The convergence of the sequence generated by the S-iterative algorithm is shown under some suitable assumptions. For more details related to S-iterative algorithm, we refer to [29] and reference therein.

Algorithm 1 S-iterative algorithm

For $i=1,2,\cdots ,\kappa$, $\kappa \in \mathbb{N}and\kappa +1=1$, let $A_i,{\mathfrak{f}}_i,{\mathfrak{g}}_i:\mathcal{H}\to \mathcal{H}$ and ${\mathcal{P}}_i:\mathcal{H}\times \mathcal{H}\to \mathcal{H}$ be the single-valued mappings. Suppose ${\mathcal{M}}_i,{\mathcal{N}}_i:\mathcal{H}\to CB\left(\mathcal{H}\right)$ are $\mathbb{H}$-Lipschitz continuous mappings and ${\mathcal{F}}_i:\mathcal{H}\times \mathcal{H}\to 2^{\mathcal{H}}$ are $({\alpha }_{A_i},{\lambda }_i)$ XOR-weak-ANODD multi-valued mappings. Then, Initially: Choose $(x_1^0,\cdots ,x_k^0)\in \underset{\kappa -\mathrm{times}}{\underbrace{\mathcal{H}\times \cdots \times \mathcal{H}}}$, $(u_1^0,\cdots ,u_{\kappa }^0)\in {\mathcal{M}}_1\left(x_1^0\right)\times \cdots \times {\mathcal{M}}_{\kappa }\left(x_{\kappa }^0\right)$, and $(v_1^0,\cdots ,v_{\kappa }^0)\in {\mathcal{N}}_1\left(x_1^0\right)\times \cdots \times {\mathcal{N}}_{\kappa }\left(x_{\kappa }^0\right)$. Step: I  $Let{x}_i^{(n+1)}\propto x_i^{\left(n\right)},u_i^{\left(n\right)}\propto u_i^{\left(n\right)}and{v}_i^{\left(n\right)}\propto v_i^{\left(n\right)}.$ We define $$\begin{array}{cc}\hfill x_1^{(n+1)}& =(1-{\alpha }_n){\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1^{\left(n\right)},v_2^{\left(n\right)})}^{A_1}[A_1\left(x_1^{\left(n\right)}\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(x_2^{\left(n\right)}\right))\}]\hfill \\ \hfill & +{\alpha }_n{\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1^{\prime \left(n\right)},v_2^{\prime \left(n\right)})}^{A_1}[A_1\left(y_1^{\left(n\right)}\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(y_2^{\left(n\right)}\right))\}],\hfill \\ \hfill y_1^{\left(n\right)}& =(1-{\beta }_n)x_1^{\left(n\right)}+{\beta }_n{\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1^{\left(n\right)},v_2^{\left(n\right)})}^{A_1}[A_1\left(x_1^{\left(n\right)}\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(x_2^{\left(n\right)}\right))\}],\hfill \\ \hfill x_2^{(n+1)}& =(1-{\alpha }_n){\mathfrak{R}}_{{\lambda }_2,{\mathcal{F}}_2(u_2^{\left(n\right)},v_3^{\left(n\right)})}^{A_2}[A_2\left(x_2^{\left(n\right)}\right)+{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(x_2^{\left(n\right)}\right),{\mathfrak{g}}_3\left(x_3^{\left(n\right)}\right))\}]\hfill \\ \hfill & +{\alpha }_n{\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_2(u_2^{\prime \left(n\right)},v_3^{\prime \left(n\right)})}^{A_2}[A_2\left(y_2^{\left(n\right)}\right)+{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(y_2^{\left(n\right)}\right),{\mathfrak{g}}_3\left(y_3^{\left(n\right)}\right))\}],\hfill \\ \hfill y_2^{\left(n\right)}& =(1-{\beta }_n)x_2^{\left(n\right)}+{\beta }_n{\mathfrak{R}}_{{\lambda }_2,{\mathcal{F}}_2(u_2^{\left(n\right)},v_2^{\left(n\right)})}^{A_2}[A_2\left(x_2^{\left(n\right)}\right)+{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(x_2^{\left(n\right)}\right),{\mathfrak{g}}_3\left(x_3^{\left(n\right)}\right))\}],\hfill \\ \hfill & ⋮\hfill \\ \hfill x_{\kappa }^{(n+1)}& =(1-{\alpha }_n){\mathfrak{R}}_{{\lambda }_{\kappa },{\mathcal{F}}_{\kappa }(u_{\kappa }^{\left(n\right)},v_1^{\left(n\right)})}^{A_{\kappa }}[A_{\kappa }\left(x_{\kappa }^{\left(n\right)}\right)+{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }^{\left(n\right)}\right),{\mathfrak{g}}_1\left(x_1^{\left(n\right)}\right))\}]\hfill \\ \hfill & +{\alpha }_n{\mathfrak{R}}_{{\lambda }_{\kappa },{\mathcal{F}}_{\kappa }(u_{\kappa }^{\prime \left(n\right)},v_1^{\prime \left(n\right)})}^{A_{\kappa }}[A_{\kappa }\left(y_{\kappa }^{\left(n\right)}\right)+{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(y_{\kappa }^{\left(n\right)}\right),{\mathfrak{g}}_1\left(y_1^{\left(n\right)}\right))\}],\hfill \\ \hfill y_{\kappa }^{\left(n\right)}& =(1-{\beta }_n)x_{\kappa }^{\left(n\right)}+{\beta }_n{\mathfrak{R}}_{{\lambda }_{\kappa },{\mathcal{F}}_{\kappa }(u_{\kappa }^{\left(n\right)},v_1^{\left(n\right)})}^{A_{\kappa }}[A_{\kappa }\left(x_{\kappa }^{\left(n\right)}\right)+{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }^{\left(n\right)}\right),{\mathfrak{g}}_1\left(x_1^{\left(n\right)}\right))\}],\hfill \end{array}$$ for $n=0,1,2,\cdots ,$ where ${\lambda }_i>0$ constants and ${\alpha }_n,{\beta }_n$ are real sequences in (0,1) such that the following condition satisfies $$\begin{array}{c}\hfill \sum _{n=1}^{\infty }{\alpha }_n{\beta }_n(1-{\beta }_n)=\infty .\end{array}$$ Step: II  Choose $u_i^{(n+1)}\in {\mathcal{M}}_i\left(x_i^{(n+1)}\right)$ and $v_i^{(n+1)}\in {\mathcal{N}}_i\left(x_i^{(n+1)}\right)$ such that $$\begin{array}{ccc}\hfill u_i^{(n+1)}\oplus u_i^{\left(n\right)}& \le & (1+\frac{1}{n+1})\mathbb{H}({\mathcal{M}}_i\left(x_i^{(n+1)}\right),{\mathcal{M}}_i\left(x_i^{\left(n\right)}\right)),\hfill \end{array}$$ (29) $$\begin{array}{ccc}\hfill v_i^{(n+1)}\oplus v_i^{\left(n\right)}& \le & (1+\frac{1}{n+1})\mathbb{H}({\mathcal{N}}_i\left(x_i^{(n+1)}\right),{\mathcal{N}}_i\left(x_i^{\left(n\right)}\right)),\hfill \end{array}$$ (30) where $\mathbb{H}(.,.)$ is the Hausdorff metric on $\mathcal{H}$. Step: III  If $x_i^{(n+1)},u_i^{\left(n\right)}$ and $v_i^{\left(n\right)}$,  $\forall i$ satisfying step-I and the accuracy is satisfactory, quit; otherwise, set $n=n+1$ and go back to step-II.

Lemma 2.

Let $v_n$ and ${\eta }_n$ be the sequences in $[0,\infty )$ such that that the following are satisfy

(i)

$0\le {\eta }_n<1,n=0,1,2,\cdots$ and $limsu{p}_n{\eta }_n<1$;

(ii)

$v_{n+1}\le {\eta }_n{v}_n,n=0,1,2,3,\cdots .$

Then $v_n\to 0$ as $n\to \infty$.

Theorem 2.

With the exception of the condition (8), let all the mappings and conditions be the same as in Theorem 1. Moreover,

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_i^{\left(n\right)}:& ={\mathsf{\Theta }}_i{\beta }^{A_i}+{\mathsf{\Theta }}_i{\lambda }_i{\gamma }_{{\mathcal{P}}_i}^{{\mathfrak{f}}_i}+{}_i{\mathsf{\Lambda }}_{i+1}(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_i}},\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {\delta }_i^{\left(n\right)}:& ={\mathsf{\Theta }}_i{\lambda }_i{\xi }_{{\mathcal{P}}_i}^{{\mathfrak{g}}_{i+1}}+{}_i{\mathsf{\Lambda }}_{i+1}(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_{i+1}}},\hfill \end{array}$$

(32)

and

$$\begin{array}{cc}\hfill \underset{n\ge 1}{sup}\left\{{\alpha }_n{\beta }_n({\mathsf{\Omega }}_1^{\left(n\right)}(1-{\mathsf{\Omega }}_1^{\left(n\right)}-{\delta }_{\kappa }^{\left(n\right)})+{\delta }_{\kappa }^{\left(n\right)}(1-{\mathsf{\Omega }}_{\kappa }^{\left(n\right)}-{\delta }_{\kappa -1}^{\left(n\right)}))\right\}& <\underset{n\ge 1}{sup}({\mathsf{\Omega }}_1^{\left(n\right)}+{\delta }_{\kappa }^{\left(n\right)}),\hfill \\ \hfill \underset{n\ge 1}{sup}\left\{{\alpha }_n{\beta }_n({\mathsf{\Omega }}_2^{\left(n\right)}(1-{\mathsf{\Omega }}_2^{\left(n\right)}-{\delta }_1^{\left(n\right)})+{\delta }_1^{\left(n\right)}(1-{\mathsf{\Omega }}_1^{\left(n\right)}-{\delta }_{\kappa }^{\left(n\right)}))\right\}& <\underset{n\ge 1}{sup}({\mathsf{\Omega }}_2^{\left(n\right)}+{\delta }_1^{\left(n\right)}),\hfill \\ \hfill & ⋮\hfill \\ \hfill \underset{n\ge 1}{sup}\left\{{\alpha }_n{\beta }_n({\mathsf{\Omega }}_{\kappa }^{\left(n\right)}(1-{\mathsf{\Omega }}_{\kappa }^{\left(n\right)}-{\delta }_{\kappa -1}^{\left(n\right)})+{\delta }_{\kappa -1}^{\left(n\right)}(1-{\mathsf{\Omega }}_{\kappa -1}^{\left(n\right)}-{\delta }_{\kappa -2}^{\left(n\right)}))\right\}& <\underset{n\ge 1}{sup}({\mathsf{\Omega }}_{\kappa }^{\left(n\right)}+{\delta }_{\kappa -1}^{\left(n\right)}),\hfill \end{array}$$

(33)

hold, then the sequences $\{(x_i^{\left(n\right)},u_i^{\left(n\right)},v_i^{\left(n\right)}\}$ generated by Algorithm 1 converge strongly to the solution $\left\{(x_i,u_i,v_i)\right\}$ of system (3).

Proof. 

Using Proposition 1 and iterative Algorithm 1, we have

$$\begin{array}{cc}\hfill 0& \le x_1^{(n+1)}\oplus x_1\hfill \\ \hfill & =((1-{\alpha }_n){\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1^{\left(n\right)},v_2^{\left(n\right)})}^{A_1}[A_1\left(x_1^{\left(n\right)}\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(x_2^{\left(n\right)}\right))\}]\hfill \\ \hfill & +{\alpha }_n{\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1^{\prime \left(n\right)},v_2^{\prime \left(n\right)})}^{A_1}[A_1\left(y_1^{\left(n\right)}\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(y_2^{\left(n\right)}\right))\}])\hfill \\ \hfill & \oplus ((1-{\alpha }_n){\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1,v_2)}^{A_1}[A_1\left(x_1\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\}]\hfill \\ \hfill & +{\alpha }_n{\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1^{\prime },v_2^{\prime })}^{A_1}[A_1\left(y_1\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1\right),{\mathfrak{g}}_2\left(y_2\right))\}])\hfill \\ \hfill & \le (1-{\alpha }_n){\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1^{\left(n\right)},v_2^{\left(n\right)})}^{A_1}[A_1\left(x_1^{\left(n\right)}\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(x_2^{\left(n\right)}\right))\}]\hfill \\ \hfill & \oplus (1-{\alpha }_n){\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1,v_2)}^{A_1}[A_1\left(x_1\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\}]\hfill \\ \hfill & +{\alpha }_n{\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1^{\prime \left(n\right)},v_2^{\prime \left(n\right)})}^{A_1}[A_1\left(y_1^{\left(n\right)}\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(y_2^{\left(n\right)}\right))\}]\hfill \\ \hfill & \oplus {\alpha }_n{\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1^{\prime },v_2^{\prime })}^{A_1}[A_1\left(y_1\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1\right),{\mathfrak{g}}_2\left(y_2\right))\}].\hfill \end{array}$$

Applying (2) and (7), we get

$$\begin{array}{ccc}\hfill & \le & (1-{\alpha }_n)[{\mathsf{\Theta }}_1[A_1\left(x_1^{\left(n\right)}\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(x_2^{\left(n\right)}\right))\}]\hfill \\ & & \oplus \left[A_1\left(x_1\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\}\right]+{}_1{\mathsf{\Lambda }}_2(u_1^{\left(n\right)}\oplus u_1+v_2^{\left(n\right)}\oplus v_2)]\hfill \\ & & +{\alpha }_n[{\mathsf{\Theta }}_1[A_1\left(y_1^{\left(n\right)}\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(y_2^{\left(n\right)}\right))\}]\hfill \\ & & \oplus \left[A_1\left(y_1\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1\right),{\mathfrak{g}}_2\left(y_2\right))\}\right]+{}_1{\mathsf{\Lambda }}_2(u_1^{\prime \left(n\right)}\oplus u_1^{\prime }+v_2^{\prime \left(n\right)}\oplus v_2^{\prime })]\hfill \end{array}$$

(34)

By using Proposition 1 and ${\beta }^{A_1}$-ordered compression of $A_1$, we obtain

$$\begin{array}{cc}\hfill x_1^{(n+1)}\oplus x_1& \le (1-{\alpha }_n)[{\mathsf{\Theta }}_1(A_1\left(x_1^{\left(n\right)}\right)\oplus A_1\left(x_1\right))+{\mathsf{\Theta }}_1({\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(x_2^{\left(n\right)}\right))\}\hfill \\ \hfill & \oplus {\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\})+{}_1{\mathsf{\Lambda }}_2(u_1^{\left(n\right)}\oplus u_1+v_2^{\left(n\right)}\oplus v_2)]\hfill \\ \hfill & +{\alpha }_n[{\mathsf{\Theta }}_1(A_1\left(y_1^{\left(n\right)}\right)\oplus A_1\left(y_1\right))+{\mathsf{\Theta }}_1({\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(y_2^{\left(n\right)}\right))\}\hfill \\ \hfill & \oplus {\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1\right),{\mathfrak{g}}_2\left(y_2\right))\})+{}_1{\mathsf{\Lambda }}_2(u_1^{\prime \left(n\right)}\oplus u_1^{\prime }+v_2^{\prime \left(n\right)}\oplus v_2^{\prime })]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le (1-{\alpha }_n)[{\mathsf{\Theta }}_1{\beta }^{A_1}(x_1^{\left(n\right)}\oplus x_1)+{\mathsf{\Theta }}_1{\lambda }_1({\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(x_2^{\left(n\right)}\right))\hfill \\ \hfill & \oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right)))+{}_1{\mathsf{\Lambda }}_2(u_1^{\left(n\right)}\oplus u_1+v_2^{\left(n\right)}\oplus v_2)]\hfill \\ \hfill & +{\alpha }_n[{\mathsf{\Theta }}_1{\beta }^{A_1}(y_1^{\left(n\right)}\oplus y_1)+{\mathsf{\Theta }}_1{\lambda }_1\{{\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(y_2^{\left(n\right)}\right))\hfill \\ \hfill & \oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1\right),{\mathfrak{g}}_2\left(y_2\right))\}+{}_1{\mathsf{\Lambda }}_2(u_1^{\prime \left(n\right)}\oplus u_1^{\prime }+v_2^{\prime \left(n\right)}\oplus v_2^{\prime })],\hfill \end{array}$$

(35)

which implies that

$$\begin{array}{cc}\hfill \parallel x_1^{(n+1)}\oplus x_1\parallel & \le (1-{\alpha }_n)[{\mathsf{\Theta }}_1{\beta }^{A_1}\parallel x_1^{\left(n\right)}\oplus x_1\parallel +{\mathsf{\Theta }}_1{\lambda }_1\parallel {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(x_2^{\left(n\right)}\right))\hfill \\ \hfill & \oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\parallel +{}_1{\mathsf{\Lambda }}_2\parallel u_1^{\left(n\right)}\oplus u_1+v_2^{\left(n\right)}\oplus v_2\parallel ]\hfill \\ \hfill & +{\alpha }_n[{\mathsf{\Theta }}_1{\beta }^{A_1}\parallel y_1^{\left(n\right)}\oplus y_1\parallel +{\mathsf{\Theta }}_1{\lambda }_1\parallel {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(y_2^{\left(n\right)}\right))\hfill \\ \hfill & \oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(y_1\right),{\mathfrak{g}}_2\left(y_2\right))\parallel +{}_1{\mathsf{\Lambda }}_2\parallel u_1^{\prime \left(n\right)}\oplus u_1^{\prime }+v_2^{\prime \left(n\right)}\oplus v_2^{\prime }\parallel ].\hfill \end{array}$$

(36)

Since ${\mathcal{P}}_1$ is a ${\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1}$-ordered compression mapping in the first component with respect to ${\mathfrak{f}}_1$ and a ${\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}$-ordered compression mapping in the second component with respect to ${\mathfrak{g}}_2$, therefore, by applying the same logic as in (11) for (12), we obtain

$$\begin{array}{ccc}\hfill \parallel x_1^{(n+1)}\oplus x_1\parallel & \le & (1-{\alpha }_n)[{\mathsf{\Theta }}_1{\beta }^{A_1}\parallel x_1^{\left(n\right)}\oplus x_1\parallel +{\mathsf{\Theta }}_1{\lambda }_1\{{\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1}\parallel x_1^{\left(n\right)}\oplus x_1\parallel \hfill \\ & & +{\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}\parallel x_2^{\left(n\right)}\oplus x_2\parallel \}+{}_1{\mathsf{\Lambda }}_2\{\parallel u_1^{\left(n\right)}\oplus u_1\parallel +\parallel v_2^{\left(n\right)}\oplus v_2\parallel \left\}\right]\hfill \\ & & +{\alpha }_n[{\mathsf{\Theta }}_1{\beta }^{A_1}\parallel y_1^{\left(n\right)}\oplus y_1\parallel +{\mathsf{\Theta }}_1{\lambda }_1\{{\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1}\parallel y_1^{\left(n\right)}\oplus y_1\parallel \hfill \\ & & +{\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}\parallel y_2^{\left(n\right)}\oplus y_2\parallel \}+{}_1{\mathsf{\Lambda }}_2\{\parallel u_1^{\prime \left(n\right)}\oplus u_1^{\prime }\parallel +\parallel v_2^{\prime \left(n\right)}\oplus v_2^{\prime }\parallel \left\}\right].\hfill \end{array}$$

(37)

Using (29), (30) and $\mathbb{H}$-Lipschitz continuity of ${\mathcal{M}}_1$ and ${\mathcal{N}}_2$, respectively, there will

$$\begin{array}{cc}\hfill \parallel x_1^{(n+1)}\oplus x_1\parallel & \le (1-{\alpha }_n)[{\mathsf{\Theta }}_1{\beta }^{A_1}\parallel x_1^{\left(n\right)}\oplus x_1\parallel +{\mathsf{\Theta }}_1{\lambda }_1\{{\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1}\parallel x_1^{\left(n\right)}\oplus x_1\parallel \hfill \\ \hfill & +{\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}\parallel x_2^{\left(n\right)}\oplus x_2\parallel \}+{}_1{\mathsf{\Lambda }}_2\{(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_1}}\parallel x_1^{\left(n\right)}\oplus x_1\parallel \hfill \\ \hfill & +(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_2}}\parallel x_2^{\left(n\right)}\oplus x_2\parallel \left\}\right]\hfill \\ \hfill & +{\alpha }_n[{\mathsf{\Theta }}_1{\beta }^{A_1}\parallel y_1^{\left(n\right)}\oplus y_1\parallel +{\mathsf{\Theta }}_1{\lambda }_1\{{\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1}\parallel y_1^{\left(n\right)}\oplus y_1\parallel \hfill \\ \hfill & +{\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}\parallel y_2^{\left(n\right)}\oplus y_2\parallel \}+{}_1{\mathsf{\Lambda }}_2\{(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_1}}\parallel y_1^{\left(n\right)}\oplus y_1\parallel \hfill \\ \hfill & +(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_2}}\parallel y_2^{\left(n\right)}\oplus y_2\parallel \left\}\right]\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & =(1-{\alpha }_n)[\{{\mathsf{\Theta }}_1{\beta }^{A_1}+{\mathsf{\Theta }}_1{\lambda }_1{\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1}+{}_1{\mathsf{\Lambda }}_2(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_1}}\}\parallel x_1^{\left(n\right)}\oplus x_1\parallel \hfill \\ \hfill & +\{{\mathsf{\Theta }}_1{\lambda }_1{\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}+{}_1{\mathsf{\Lambda }}_2(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_2}}\}\parallel x_2^{\left(n\right)}\oplus x_2\parallel ]\hfill \\ \hfill & +{\alpha }_n[\{{\mathsf{\Theta }}_1{\beta }^{A_1}+{\mathsf{\Theta }}_1{\lambda }_1{\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1}+{}_1{\mathsf{\Lambda }}_2(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_1}}\}\parallel y_1^{\left(n\right)}\oplus y_1\parallel \hfill \\ \hfill & +\{{\mathsf{\Theta }}_1{\lambda }_1{\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}+{}_1{\mathsf{\Lambda }}_2(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_2}}\}\parallel y_2^{\left(n\right)}\oplus y_2\parallel ].\hfill \end{array}$$

(39)

Using (31) and (32), we have

$$\begin{array}{cc}\hfill \parallel x_1^{(n+1)}\oplus x_1\parallel & \le |(1-{\alpha }_n)\left|\right[{\mathsf{\Omega }}_1^{\left(n\right)}\parallel x_1^{\left(n\right)}\oplus x_1\parallel +{\delta }_1^{\left(n\right)}\parallel x_2^{\left(n\right)}\oplus x_2\parallel ]\\ & +|{\alpha }_n^{\left(n\right)}\left|\right[{\mathsf{\Omega }}_1\parallel y_1^{\left(n\right)}\oplus y_1\parallel +{\delta }_1^{\left(n\right)}\parallel y_2^{\left(n\right)}\oplus y_2\parallel ].\end{array}$$

(40)

Furthermore, by using algorithm (1), we calculate

$$\begin{array}{cc}\hfill \parallel y_1^{\left(n\right)}\oplus y_1\parallel & \le \parallel \{(1-{\beta }_n)x_1^{\left(n\right)}+{\beta }_n{\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1^{\left(n\right)},v_2^{\left(n\right)})}^{A_1}[A_1\left(x_1^{\left(n\right)}\right)\hfill \\ \hfill & +{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(x_2^{\left(n\right)}\right))\}\left]\right\}\hfill \\ \hfill & \oplus \{(1-{\beta }_n)x_1+{\beta }_n{\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1,v_2)}^{A_1}[A_1\left(x_1\right)\hfill \\ \hfill & +{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\}\left]\right\}\parallel .\hfill \end{array}$$

(41)

Using Proposition 1, Lipschitz-type-continuity of ${\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1^{\left(n\right)},v_2^{\left(n\right)})}^{A_1}$ and condition (7), we have

$$\begin{array}{cc}\hfill \parallel y_1^{\left(n\right)}\oplus y_1\parallel & \le (1-{\beta }_n)\parallel x_1^{\left(n\right)}\oplus x_1\parallel +{\beta }_n\parallel {\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1^{\left(n\right)},v_2^{\left(n\right)})}^{A_1}[A_1\left(x_1^{\left(n\right)}\right)\hfill \\ \hfill & +{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(x_2^{\left(n\right)}\right))\}]\hfill \\ \hfill & \oplus {\mathfrak{R}}_{{\lambda }_1,{\mathcal{F}}_1(u_1,v_2)}^{A_1}[A_1\left(x_1\right)+{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\}]\parallel \hfill \\ \hfill & \le (1-{\beta }_n)\parallel x_1^{\left(n\right)}\oplus x_1\parallel +{\beta }_n[{\mathsf{\Theta }}_1\parallel (A_1\left(x_1^{\left(n\right)}\right)\oplus A_1\left(x_1\right))\hfill \\ \hfill & +{\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(x_2^{\left(n\right)}\right))\}\oplus {\lambda }_1\{{\omega }_1\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\}\parallel \hfill \\ \hfill & +{}_1{\mathsf{\Lambda }}_2\{\parallel u_1^{\left(n\right)}\oplus u_1+v_2^{\left(n\right)}\oplus v_2\parallel ]\hfill \\ \hfill & \le (1-{\beta }_n)\parallel x_1^{\left(n\right)}\oplus x_1\parallel +{\beta }_n[{\mathsf{\Theta }}_1{\beta }^{A_1}\parallel x_1^{\left(n\right)}\oplus x_1\parallel \hfill \\ \hfill & +{\mathsf{\Theta }}_1{\lambda }_1\parallel {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1^{\left(n\right)}\right),{\mathfrak{g}}_2\left(x_2^{\left(n\right)}\right))\oplus {\mathcal{P}}_1({\mathfrak{f}}_1\left(x_1\right),{\mathfrak{g}}_2\left(x_2\right))\parallel \hfill \\ \hfill & +{}_1{\mathsf{\Lambda }}_2\{\parallel u_1^{\left(n\right)}\oplus u_1\parallel +\parallel v_2^{\left(n\right)}\oplus v_2\parallel ].\hfill \end{array}$$

(42)

Using (12) and $\mathbb{H}$-Lipschitz continuity of ${\mathcal{M}}_1$ and ${\mathcal{N}}_2$, we have

$$\begin{array}{cc}\hfill \parallel y_1^{\left(n\right)}\oplus y_1\parallel & \le |(1-{\beta }_n)|\parallel x_1^{\left(n\right)}\oplus x_1\parallel +|{\beta }_n\left|\right[{\mathsf{\Theta }}_1{\beta }^{A_1}\parallel x_1^{\left(n\right)}\oplus x_1\parallel \hfill \\ \hfill & +{\mathsf{\Theta }}_1{\lambda }_1({\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1}\parallel x_1^{\left(n\right)}\oplus x_1\parallel +{\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}\parallel x_2^{\left(n\right)}\oplus x_2\parallel )\hfill \\ \hfill & +{}_1{\mathsf{\Lambda }}_2\{(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_1}}\parallel x_1^{\left(n\right)}\oplus x_1\parallel \hfill \\ \hfill & +(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_2}}\parallel x_2^{\left(n\right)}\oplus x_2\parallel \left\}\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le (1-{\beta }_n)\parallel x_1^{\left(n\right)}\oplus x_1\parallel +|{\beta }_n\left|\right[\{{\mathsf{\Theta }}_1{\beta }^{A_1}+{\mathsf{\Theta }}_1{\lambda }_1{\gamma }_{{\mathcal{P}}_1}^{{\mathfrak{f}}_1}\hfill \\ \hfill & +{}_1{\mathsf{\Lambda }}_2(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_1}}\}\parallel x_1^{\left(n\right)}\oplus x_1\parallel \hfill \\ \hfill & +\{{\mathsf{\Theta }}_1{\lambda }_1{\xi }_{{\mathcal{P}}_1}^{{\mathfrak{g}}_2}+{}_1{\mathsf{\Lambda }}_2(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_2}}\}\parallel x_2^{\left(n\right)}\oplus x_2\parallel ].\hfill \end{array}$$

(43)

Using (31) and (32), we have

$$\parallel y_1^{\left(n\right)}\oplus y_1\parallel \le (1-{\beta }_n)\parallel x_1^{\left(n\right)}\oplus x_1\parallel +{\beta }_n[{\mathsf{\Omega }}_1^{\left(n\right)}\parallel x_1^{\left(n\right)}\oplus x_1\parallel +{\delta }_1^{\left(n\right)}\parallel x_2^{\left(n\right)}\oplus x_2\parallel ].$$

(44)

Similarly, we may obtain

$$\begin{array}{cc}\hfill \parallel y_2^{\left(n\right)}\oplus y_2\parallel & \le (1-{\beta }_n)\parallel x_2^{\left(n\right)}\oplus x_2\parallel +{\beta }_n\left[\right\{{\mathsf{\Theta }}_2{\beta }^{A_2}+{\mathsf{\Theta }}_2{\lambda }_2{\gamma }_{{\mathcal{P}}_2}^{{\mathfrak{f}}_2}\hfill \\ \hfill & +{}_2{\mathsf{\Lambda }}_3(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_2}}\}\parallel x_2^{\left(n\right)}\oplus x_2\parallel \hfill \\ \hfill & +\{{\mathsf{\Theta }}_2{\lambda }_2{\xi }_{{\mathcal{P}}_2}^{{\mathfrak{g}}_3}+{}_2{\mathsf{\Lambda }}_3(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_3}}\}\parallel x_3^{\left(n\right)}\oplus x_3\parallel ].\hfill \end{array}$$

(45)

That is,

$$\parallel y_2^{\left(n\right)}\oplus y_2\parallel \le (1-{\beta }_n)\parallel x_2^{\left(n\right)}\oplus x_2\parallel +{\beta }_n[{\mathsf{\Omega }}_2^{\left(n\right)}\parallel x_2^{\left(n\right)}\oplus x_2\parallel +{\delta }_2^{\left(n\right)}\parallel x_3^{\left(n\right)}\oplus x_3\parallel ].$$

(46)

Using (44) and (46) in (40), we get

$$\begin{array}{cc}\hfill \parallel x_1^{(n+1)}\oplus x_1\parallel & \le {\mathsf{\Omega }}_1^{\left(n\right)}(1-{\alpha }_n{\beta }_n+{\alpha }_n{\beta }_n{\mathsf{\Omega }}_1^{\left(n\right)})\parallel x_1^{\left(n\right)}\oplus x_1\parallel \hfill \\ \hfill & +{\delta }_1^{\left(n\right)}(1-{\alpha }_n{\beta }_n+{\alpha }_n{\beta }_n{\mathsf{\Omega }}_1^{\left(n\right)}+{\alpha }_n{\beta }_n{\mathsf{\Omega }}_2^{\left(n\right)})\parallel x_2^{\left(n\right)}\oplus x_2\parallel \hfill \\ \hfill & +{\alpha }_n{\beta }_n{\delta }_1^{\left(n\right)}{\delta }_2^{\left(n\right)}\parallel x_3^{\left(n\right)}\oplus x_3\parallel .\hfill \end{array}$$

(47)

Since the cone P is normal, by Proposition 1, we have

$$\begin{array}{cc}\hfill \parallel x_1^{(n+1)}-x_1\parallel & \le {\lambda }_P\parallel x_1^{(n+1)}\oplus x_1\parallel \hfill \\ \hfill & \le {\lambda }_P[{\mathsf{\Omega }}_1^{\left(n\right)}(1-{\alpha }_n{\beta }_n+{\alpha }_n{\beta }_n{\mathsf{\Omega }}_1^{\left(n\right)})\parallel x_1^{\left(n\right)}-x_1\parallel \hfill \\ \hfill & +{\delta }_1^{\left(n\right)}(1-{\alpha }_n{\beta }_n+{\alpha }_n{\beta }_n{\mathsf{\Omega }}_1^{\left(n\right)}+{\alpha }_n{\beta }_n{\mathsf{\Omega }}_2^{\left(n\right)})\parallel x_2^{\left(n\right)}-x_2\parallel \hfill \\ \hfill & +{\alpha }_n{\beta }_n{\delta }_1^{\left(n\right)}{\delta }_2^{\left(n\right)}\parallel x_3^{\left(n\right)}-x_3\parallel ].\hfill \end{array}$$

(48)

Using the Proposition 1 and Algorithm 1, we calculate

$$\begin{array}{cc}\hfill 0& \le x_2^{(n+1)}\oplus x_2\hfill \\ \hfill & =((1-{\alpha }_n){\mathfrak{R}}_{{\lambda }_2,{\mathcal{F}}_2(u_2^{\left(n\right)},v_2^{\left(n\right)})}^{A_2}[A_2\left(x_2^{\left(n\right)}\right)+{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(x_2^{\left(n\right)}\right),{\mathfrak{g}}_3\left(x_3^{\left(n\right)}\right))\}]\hfill \\ \hfill & +{\alpha }_n{\mathfrak{R}}_{{\lambda }_2,{\mathcal{F}}_2(u_2^{\prime \left(n\right)},v_2^{\prime \left(n\right)})}^{A_2}[A_2\left(y_2^{\left(n\right)}\right)+{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(y_2^{\left(n\right)}\right),{\mathfrak{g}}_3\left(y_3^{\left(n\right)}\right))\}])\hfill \\ \hfill & \oplus ((1-{\alpha }_n){\mathfrak{R}}_{{\lambda }_2,{\mathcal{F}}_2(u_2,v_2)}^{A_2}[A_2\left(x_2\right)+{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(x_2\right),{\mathfrak{g}}_3\left(x_3\right))\}]\hfill \\ \hfill & +{\alpha }_n{\mathfrak{R}}_{{\lambda }_2,{\mathcal{F}}_2(u_2^{\prime },v_3^{\prime })}^{A_2}\left[A_2\left(y_2\right)+{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(y_2\right),{\mathfrak{g}}_3\left(y_3\right))\}\right])\hfill \end{array}$$

(49)

Applying the same logic as in (35) for (38), we obtain

$$\begin{array}{cc}\hfill \parallel x_2^{(n+1)}\oplus x_2\parallel & =(1-{\alpha }_n)[\{{\mathsf{\Theta }}_2{\beta }^{A_2}+{\mathsf{\Theta }}_2{\lambda }_2{\gamma }_{{\mathcal{P}}_2}^{{\mathfrak{f}}_2}+{}_2{\mathsf{\Lambda }}_3(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_2}}\}\parallel x_2^{\left(n\right)}\oplus x_2\parallel \hfill \\ \hfill & +\{{\mathsf{\Theta }}_2{\lambda }_2{\xi }_{{\mathcal{P}}_2}^{{\mathfrak{g}}_3}+{}_2{\mathsf{\Lambda }}_3(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_3}}\}\parallel x_3^{\left(n\right)}\oplus x_3\parallel ]\hfill \\ \hfill & +{\alpha }_n[\{{\mathsf{\Theta }}_1{\beta }^{A_2}+{\mathsf{\Theta }}_2{\lambda }_2{\gamma }_{{\mathcal{P}}_2}^{{\mathfrak{f}}_2}+{}_2{\mathsf{\Lambda }}_3(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_2}}\}\parallel y_2^{\left(n\right)}\oplus y_2\parallel \hfill \\ \hfill & +\{{\mathsf{\Theta }}_2{\lambda }_2{\xi }_{{\mathcal{P}}_2}^{{\mathfrak{g}}_3}+{}_2{\mathsf{\Lambda }}_3(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_3}}\}\parallel y_3^{\left(n\right)}\oplus y_3\parallel ].\hfill \end{array}$$

(50)

Using (31) and (32), we have

$$\begin{array}{cc}\hfill \parallel x_2^{(n+1)}\oplus x_2\parallel & \le (1-{\alpha }_n)[{\mathsf{\Omega }}_2^{\left(n\right)}\parallel x_2^{\left(n\right)}\oplus x_2\parallel +{\delta }_2^{\left(n\right)}\parallel x_3^{\left(n\right)}\oplus x_3\parallel ]\\ & +{\alpha }_n[{\mathsf{\Omega }}_2^{\left(n\right)}\parallel y_2^{\left(n\right)}\oplus y_2\parallel +{\delta }_2^{\left(n\right)}\parallel y_3^{\left(n\right)}\oplus y_3\parallel ].\end{array}$$

(51)

Furthermore, by iterative Algorithm (1), we calculate

$$\begin{array}{cc}\hfill \parallel y_2^{\left(n\right)}\oplus y_2\parallel & \le \parallel \{(1-{\beta }_n)x_2^{\left(n\right)}+{\beta }_n{\mathfrak{R}}_{{\lambda }_2,{\mathcal{F}}_2(u_2^{\left(n\right)},v_3^{\left(n\right)})}^{A_2}[A_2\left(x_2^{\left(n\right)}\right)\hfill \\ \hfill & +{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(x_2^{\left(n\right)}\right),{\mathfrak{g}}_3\left(x_3^{\left(n\right)}\right))\}\left]\right\}\hfill \\ \hfill & \oplus \{(1-{\beta }_n)x_2+{\beta }_n{\mathfrak{R}}_{{\lambda }_2,{\mathcal{F}}_2(u_2,v_3)}^{A_2}[A_2\left(x_2\right)\hfill \\ \hfill & +{\lambda }_2\{{\omega }_2\oplus {\mathcal{P}}_2({\mathfrak{f}}_2\left(x_2\right),{\mathfrak{g}}_3\left(x_3\right))\}\left]\right\}\parallel .\hfill \end{array}$$

(52)

Importing the same logic as in (42) for (43), we have

$$\begin{array}{cc}\hfill \parallel y_2^{\left(n\right)}\oplus y_2\parallel & \le (1-{\beta }_n)\parallel x_2^{\left(n\right)}\oplus x_2\parallel +{\beta }_n\left[\right\{{\mathsf{\Theta }}_2{\beta }^{A_2}+{\mathsf{\Theta }}_2{\lambda }_2{\gamma }_{{\mathcal{P}}_2}^{{\mathfrak{f}}_2}\hfill \\ \hfill & +{}_2{\mathsf{\Lambda }}_3(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_2}}\}\parallel x_2^{\left(n\right)}\oplus x_2\parallel \hfill \\ \hfill & +\{{\mathsf{\Theta }}_2{\lambda }_2{\xi }_{{\mathcal{P}}_2}^{{\mathfrak{g}}_3}+{}_2{\mathsf{\Lambda }}_3(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_3}}\}\parallel x_3^{\left(n\right)}\oplus x_3\parallel ].\hfill \end{array}$$

(53)

Using (31) and (32), we have

$$\parallel y_2^{\left(n\right)}\oplus y_2\parallel \le (1-{\beta }_n)\parallel x_2^{\left(n\right)}\oplus x_2\parallel +{\beta }_n[{\mathsf{\Omega }}_2^{\left(n\right)}\parallel x_2^{\left(n\right)}\oplus x_2\parallel +{\delta }_2^{\left(n\right)}\parallel x_3^{\left(n\right)}\oplus x_3\parallel ].$$

(54)

Similarly, we get

$$\begin{array}{cc}\hfill \parallel y_3^{\left(n\right)}\oplus y_3\parallel & \le (1-{\beta }_n)\parallel x_3^{\left(n\right)}\oplus x_3\parallel +{\beta }_n\left[\right\{{\mathsf{\Theta }}_3{\beta }^{A_3}+{\mathsf{\Theta }}_3{\lambda }_3{\gamma }_{{\mathcal{P}}_3}^{{\mathfrak{f}}_3}\hfill \\ \hfill & +{}_3{\mathsf{\Lambda }}_4(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_3}}\}\parallel x_3^{\left(n\right)}\oplus x_3\parallel \hfill \\ \hfill & +\{{\mathsf{\Theta }}_3{\lambda }_3{\xi }_{{\mathcal{P}}_3}^{{\mathfrak{g}}_4}+{}_3{\mathsf{\Lambda }}_4(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_4}}\}\parallel x_4^{\left(n\right)}\oplus x_4\parallel ].\hfill \end{array}$$

(55)

That is,

$$\parallel y_3^{\left(n\right)}\oplus y_3\parallel \le (1-{\beta }_n)\parallel x_3^{\left(n\right)}\oplus x_3\parallel +{\beta }_n[{\mathsf{\Omega }}_3^{\left(n\right)}\parallel x_2^{\left(n\right)}\oplus x_3\parallel +{\delta }_3^{\left(n\right)}\parallel x_4^{\left(n\right)}\oplus x_4\parallel ].$$

(56)

By applying the definition of normal cone, (54) and (56) in Equation (51), we have

$$\begin{array}{cc}\hfill \parallel x_2^{(n+1)}-x_2\parallel & \le {\lambda }_P[{\mathsf{\Omega }}_2^{\left(n\right)}(1-{\alpha }_n{\beta }_n+{\alpha }_n{\beta }_n{\mathsf{\Omega }}_2^{\left(n\right)})\parallel x_2^{\left(n\right)}-x_2\parallel \hfill \\ \hfill & +{\delta }_2^{\left(n\right)}(1-{\alpha }_n{\beta }_n+{\alpha }_n{\beta }_n{\mathsf{\Omega }}_2^{\left(n\right)}+{\alpha }_n{\beta }_n{\mathsf{\Omega }}_3^{\left(n\right)})\parallel x_3^{\left(n\right)}-x_3\parallel \hfill \\ \hfill & +{\alpha }_n{\beta }_n{\delta }_2^{\left(n\right)}{\delta }_3^{\left(n\right)}\parallel x_4^{\left(n\right)}-x_4\parallel ].\hfill \end{array}$$

(57)

Continuing in this way, we have from Algorithm 1 that

$$\begin{array}{cc}\hfill 0& \le x_{\kappa }^{(n+1)}\oplus x_{\kappa }\hfill \\ \hfill & =((1-{\alpha }_n){\mathfrak{R}}_{{\lambda }_{\kappa },{\mathcal{F}}_{\kappa }(u_{\kappa }^{\left(n\right)},v_{\kappa }^{\left(n\right)})}^{A_{\kappa }}[A_{\kappa }\left(x_{\kappa }^{\left(n\right)}\right)+{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }^{\left(n\right)}\right),{\mathfrak{g}}_1\left(x_1^{\left(n\right)}\right))\}]\hfill \\ \hfill & +{\alpha }_n{\mathfrak{R}}_{{\lambda }_{\kappa },{\mathcal{F}}_{\kappa }(u_{\kappa }^{\prime \left(n\right)},v_{\kappa }^{\prime \left(n\right)})}^{A_{\kappa }}[A_{\kappa }\left(y_{\kappa }^{\left(n\right)}\right)+{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(y_{\kappa }^{\left(n\right)}\right),{\mathfrak{g}}_1\left(y_1^{\left(n\right)}\right))\}])\hfill \\ \hfill & \oplus ((1-{\alpha }_n){\mathfrak{R}}_{{\lambda }_{\kappa },{\mathcal{F}}_{\kappa }(u_{\kappa },v_{\kappa })}^{A_{\kappa }}[A_{\kappa }\left(x_{\kappa }\right)+{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }\right),{\mathfrak{g}}_1\left(x_1\right))\}]\hfill \\ \hfill & +{\alpha }_n{\mathfrak{R}}_{{\lambda }_{\kappa },{\mathcal{F}}_{\kappa }(u_{\kappa }^{\prime },v_1^{\prime })}^{A_{\kappa }}[A_{\kappa }\left(y_{\kappa }\right)+{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(y_{\kappa }\right),{\mathfrak{g}}_1\left(y_1\right))\}]).\hfill \end{array}$$

(58)

Applying the same logic as in (34) for (38), we obtain

$$\begin{array}{cc}\hfill \parallel x_{\kappa }^{(n+1)}\oplus x_{\kappa }\parallel & \le (1-{\alpha }_n)[\{{\mathsf{\Theta }}_{\kappa }{\beta }^{A_{\kappa }}+{\mathsf{\Theta }}_{\kappa }{\lambda }_{\kappa }{\gamma }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{f}}_{\kappa }}+{}_{\kappa }{\mathsf{\Lambda }}_1(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_{\kappa }}}\}\parallel x_{\kappa }^{\left(n\right)}\oplus x_{\kappa }\parallel \hfill \\ \hfill & +\{{\mathsf{\Theta }}_{\kappa }{\lambda }_{\kappa }{\xi }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{g}}_1}+{}_{\kappa }{\mathsf{\Lambda }}_1(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_1}}\}\parallel x_1^{\left(n\right)}\oplus x_1\parallel ]\hfill \\ \hfill & +{\alpha }_n[\{{\mathsf{\Theta }}_{\kappa }{\beta }^{A_{\kappa }}+{\mathsf{\Theta }}_{\kappa }{\lambda }_{\kappa }{\gamma }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{f}}_{\kappa }}+{}_{\kappa }{\mathsf{\Lambda }}_1(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_{\kappa }}}\}\parallel y_{\kappa }^{\left(n\right)}\oplus y_{\kappa }\parallel \hfill \\ \hfill & +\{{\mathsf{\Theta }}_{\kappa }{\lambda }_{\kappa }{\xi }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{g}}_1}+{}_{\kappa }{\mathsf{\Lambda }}_1(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_1}}\}\parallel y_1^{\left(n\right)}\oplus y_1\parallel ].\hfill \end{array}$$

(59)

Using (31) and (32), we have

$$\begin{array}{cc}\hfill \parallel x_{\kappa }^{(n+1)}\oplus x_{\kappa }\parallel & \le (1-{\alpha }_n)[{\mathsf{\Omega }}_{\kappa }^{\left(n\right)}\parallel x_{\kappa }^{\left(n\right)}\oplus x_{\kappa }\parallel +{\delta }_{\kappa }^{\left(n\right)}\parallel x_1^{\left(n\right)}\oplus x_1\parallel ]\\ & +{\alpha }_n[{\mathsf{\Omega }}_{\kappa }^{\left(n\right)}\parallel y_{\kappa }^{\left(n\right)}\oplus y_{\kappa }\parallel +{\delta }_{\kappa }^{\left(n\right)}\parallel y_1^{\left(n\right)}\oplus y_1\parallel ].\end{array}$$

(60)

Furthermore, by using Algorithm 1, we calculate

$$\begin{array}{cc}\hfill \parallel y_{\kappa }^{\left(n\right)}\oplus y_{\kappa }\parallel & \le \parallel \{(1-{\beta }_n)x_{\kappa }^{\left(n\right)}+{\beta }_n{\mathfrak{R}}_{{\lambda }_{\kappa },{\mathcal{F}}_{\kappa }(u_{\kappa }^{\left(n\right)},v_1^{\left(n\right)})}^{A_{\kappa }}[A_{\kappa }\left(x_{\kappa }^{\left(n\right)}\right)\hfill \\ \hfill & +{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }^{\left(n\right)}\right),{\mathfrak{g}}_1\left(x_1^{\left(n\right)}\right))\}\left]\right\}\hfill \\ \hfill & \oplus \{(1-{\beta }_n)x_{\kappa }+{\beta }_n{\mathfrak{R}}_{{\lambda }_{\kappa },{\mathcal{F}}_{\kappa }(u_{\kappa },v_1)}^{A_{\kappa }}[A_{\kappa }\left(x_{\kappa }\right)\hfill \\ \hfill & +{\lambda }_{\kappa }\{{\omega }_{\kappa }\oplus {\mathcal{P}}_{\kappa }({\mathfrak{f}}_{\kappa }\left(x_{\kappa }\right),{\mathfrak{g}}_1\left(x_1\right))\}\left]\right\}\parallel .\hfill \end{array}$$

(61)

Importing the same logic as in (42) for (43), we have

$$\begin{array}{cc}\hfill \parallel y_{\kappa }^{\left(n\right)}\oplus y_{\kappa }\parallel & \le (1-{\beta }_n)\parallel x_{\kappa }^{\left(n\right)}\oplus x_{\kappa }\parallel +|{\beta }_n\left|\right[\{{\mathsf{\Theta }}_{\kappa }{\beta }^{A_{\kappa }}+{\mathsf{\Theta }}_{\kappa }{\lambda }_{\kappa }{\gamma }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{f}}_{\kappa }}\hfill \\ \hfill & +{}_{\kappa }{\mathsf{\Lambda }}_1(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_{\kappa }}}\}\parallel x_{\kappa }^{\left(n\right)}\oplus x_{\kappa }\parallel \hfill \\ \hfill & +\{{\mathsf{\Theta }}_{\kappa }{\lambda }_{\kappa }{\xi }_{{\mathcal{P}}_{\kappa }}^{{\mathfrak{g}}_1}+{}_{\kappa }{\mathsf{\Lambda }}_1(1+\frac{1}{n+1}){\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_1}}\}\parallel x_1^{\left(n\right)}\oplus x_1\parallel ].\hfill \end{array}$$

(62)

By (31) and (32), we have

$$\parallel y_{\kappa }^{\left(n\right)}\oplus y_{\kappa }\parallel \le (1-{\beta }_n)\parallel x_{\kappa }^{\left(n\right)}\oplus x_{\kappa }\parallel +|{\beta }_n\left|\right[{\mathsf{\Omega }}_{\kappa }^{\left(n\right)}\parallel x_{\kappa }^{\left(n\right)}\oplus x_{\kappa }\parallel +{\delta }_{\kappa }^{\left(n\right)}\parallel x_1^{\left(n\right)}\oplus x_1\parallel ].$$

(63)

By using definition of normal cone, Equations (63) and (44) in Equation (60), we get

$$\begin{array}{cc}\hfill \parallel x_{\kappa }^{(n+1)}-x_{\kappa }\parallel & \le {\lambda }_P[{\mathsf{\Omega }}_{\kappa }^{\left(n\right)}(1-{\alpha }_n{\beta }_n+{\alpha }_n{\beta }_n{\mathsf{\Omega }}_{\kappa }^{\left(n\right)})\parallel x_{\kappa }^{\left(n\right)}-x_{\kappa }\parallel \hfill \\ \hfill & +{\delta }_{\kappa }^{\left(n\right)}(1-{\alpha }_n{\beta }_n+{\alpha }_n{\beta }_n{\mathsf{\Omega }}_{\kappa }^{\left(n\right)}+{\alpha }_n{\beta }_n{\mathsf{\Omega }}_1^{\left(n\right)})\parallel x_1^{\left(n\right)}-x_1\parallel \hfill \\ \hfill & +{\alpha }_n{\beta }_n{\delta }_{\kappa }^{\left(n\right)}{\delta }_1^{\left(n\right)}\parallel x_2^{\left(n\right)}-x_2\parallel ].\hfill \end{array}$$

(64)

From (48), (57) and (64), we have

$$\begin{array}{cc}\hfill & \parallel x_1^{(n+1)}-x_1\parallel +\parallel x_2^{(n+1)}-x_2\parallel +\cdots +\parallel x_{\kappa }^{(n+1)}-x_{\kappa }\parallel \hfill \\ \hfill & \le {\lambda }_P[({\mathsf{\Omega }}_1^{\left(n\right)}+{\delta }_{\kappa }^{\left(n\right)})-{\alpha }_n{\beta }_n\{{\mathsf{\Omega }}_1^{\left(n\right)}(1-{\mathsf{\Omega }}_1^{\left(n\right)}-{\delta }_{\kappa }^{\left(n\right)})\hfill \\ \hfill & +{\delta }_{\kappa }^{\left(n\right)}(1-{\mathsf{\Omega }}_{\kappa }^{\left(n\right)}-{\delta }_{\kappa -1}^{\left(n\right)})\left\}\right]\parallel x_1^{\left(n\right)}-x_1\parallel \hfill \\ \hfill & +{\lambda }_P[({\mathsf{\Omega }}_2^{\left(n\right)}+{\delta }_1^{\left(n\right)})-{\alpha }_n{\beta }_n\{{\mathsf{\Omega }}_2^{\left(n\right)}(1-{\mathsf{\Omega }}_2^{\left(n\right)}-{\delta }_1^{\left(n\right)})\hfill \\ \hfill & +{\delta }_1^{\left(n\right)}(1-{\mathsf{\Omega }}_1^{\left(n\right)}-{\delta }_{\kappa }^{\left(n\right)})\left\}\right]\parallel x_2^{\left(n\right)}-x_2\parallel \hfill \\ \hfill & +\cdots +\hfill \\ \hfill & +{\lambda }_P[({\mathsf{\Omega }}_{\kappa }^{\left(n\right)}+{\delta }_{\kappa -1}^{\left(n\right)})-{\alpha }_n{\beta }_n\{{\mathsf{\Omega }}_{\kappa }^{\left(n\right)}(1-{\mathsf{\Omega }}_{\kappa }^{\left(n\right)}-{\delta }_{\kappa -1}^{\left(n\right)})\hfill \\ \hfill & +{\delta }_{\kappa -1}^{\left(n\right)}(1-{\mathsf{\Omega }}_{\kappa -1}^{\left(n\right)}-{\delta }_{\kappa -2}^{\left(n\right)})\left\}\right]\parallel x_{\kappa }^{\left(n\right)}-x_{\kappa }\parallel \hfill \\ \hfill & ={\lambda }_P[{▵}_1^{\left(n\right)}\parallel x_1^{\left(n\right)}-x_1\parallel +{▵}_2^{\left(n\right)}\parallel x_2^{\left(n\right)}-x_2\parallel +\cdots +{▵}_{\kappa }^{\left(n\right)}\parallel x_{\kappa }^{\left(n\right)}-x_{\kappa }\parallel ],\hfill \end{array}$$

(65)

where

$$\begin{array}{cc}\hfill {▵}_1^{\left(n\right)}& =[({\mathsf{\Omega }}_1^{\left(n\right)}+{\delta }_{\kappa }^{\left(n\right)})-{\alpha }_n{\beta }_n\{{\mathsf{\Omega }}_1^{\left(n\right)}(1-{\mathsf{\Omega }}_1^{\left(n\right)}-{\delta }_{\kappa }^{\left(n\right)})+{\delta }_{\kappa }^{\left(n\right)}(1-{\mathsf{\Omega }}_{\kappa }^{\left(n\right)}-{\delta }_{\kappa -1}^{\left(n\right)})\}]\hfill \\ \hfill {▵}_2^{\left(n\right)}& =[({\mathsf{\Omega }}_2^{\left(n\right)}+{\delta }_1^{\left(n\right)})-{\alpha }_n{\beta }_n\{{\mathsf{\Omega }}_2^{\left(n\right)}(1-{\mathsf{\Omega }}_2^{\left(n\right)}-{\delta }_1^{\left(n\right)})+{\delta }_1^{\left(n\right)}(1-{\mathsf{\Omega }}_1^{\left(n\right)}-{\delta }_{\kappa }^{\left(n\right)})\}]\hfill \\ \hfill & ⋮\hfill \\ \hfill {▵}_{\kappa }^{\left(n\right)}& =[({\mathsf{\Omega }}_{\kappa }^{\left(n\right)}+{\delta }_{\kappa -1}^{\left(n\right)})-{\alpha }_n{\beta }_n\{{\mathsf{\Omega }}_{\kappa }^{\left(n\right)}(1-{\mathsf{\Omega }}_{\kappa }^{\left(n\right)}-{\delta }_{\kappa -1}^{\left(n\right)})+{\delta }_{\kappa -1}^{\left(n\right)}(1-{\mathsf{\Omega }}_{\kappa -1}^{\left(n\right)}-{\delta }_{\kappa -2}^{\left(n\right)})\}].\hfill \end{array}$$

(66)

Let ${\mathsf{\Phi }}^{\left(n\right)}=\frac{1}{{\lambda }_P}[max\{{▵}_1^{\left(n\right)},{▵}_2^{\left(n\right)},\cdots ,{▵}_{\kappa }^{\left(n\right)}\}]$. From (31) and (32), ${\mathsf{\Omega }}_i^{\left(n\right)}\to {\mathsf{\Omega }}_i$ and ${\delta }_i^{\left(n\right)}\to {\delta }_i$, $\forall i$ as $n\to \infty$. Where

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_i& ={\mathsf{\Theta }}_i{\beta }^{A_i}+{\mathsf{\Theta }}_i{\lambda }_i{\gamma }_{{\mathcal{P}}_i}^{{\mathfrak{f}}_i}+{}_i{\mathsf{\Lambda }}_{i+1}{\lambda }_{{\mathbb{H}}_{{\mathcal{M}}_i}}\hfill \\ \hfill {\delta }_i& ={\mathsf{\Theta }}_i{\lambda }_i{\xi }_{{\mathcal{P}}_i}^{{\mathfrak{g}}_{i+1}}+{}_i{\mathsf{\Lambda }}_{i+1}{\lambda }_{{\mathbb{H}}_{{\mathcal{N}}_{i+1}}}.\hfill \end{array}$$

By algebra of convergence of sequences ${\mathsf{\Omega }}_i^{\left(n\right)}$ and ${\delta }_i^{\left(n\right)}$, we may say that $\exists {▵}_i$ such that ${▵}_i^{\left(n\right)}\to {▵}_i$ as $n\to \infty$. Therefore, ${\mathsf{\Phi }}^{\left(n\right)}\to \mathsf{\Phi }$, as $n\to \infty$, where $\mathsf{\Phi }=max\left\{{▵}_1,{▵}_2,\cdots ,{▵}_{\kappa }\right\}$. Condition (33) implies that $\mathsf{\Phi }<1$, so ${\mathsf{\Phi }}^{\left(n\right)}<1,$ for sufficiently large n.

Let $v_n=(\parallel x_1^{(n+1)}-x_1\parallel +\parallel x_2^{(n+1)}-x_2\parallel +\dots +\parallel x_{\kappa }^{(n+1)}-x_{\kappa }\parallel )$. Then, (65) can be written as

$$\begin{array}{ccc}\hfill v_{n+1}& \le & {\mathsf{\Phi }}^{\left(n\right)}{v}_n,n=0,1,2,\cdots .\hfill \end{array}$$

Choosing ${\mathsf{\Phi }}^{\left(n\right)}<1$ such that $lim{sup}_{n\ge 1}{\mathsf{\Phi }}^{\left(n\right)}<1.$ As a result of Lemma 2 that $0\le {\mathsf{\Phi }}_n<1.$ Therefore, $\left\{(x_i^{\left(n\right)},u_i^{\left(n\right)},v_i^{\left(n\right)})\right\}$ converge strongly to the solution $\left\{(x_i,u_i,v_i)\right\}$ of system (3). □

5. Conclusions

In this article, a nonlinear system of generalized ordered XOR- inclusion problem is studied in Hilbert space, which is more general than the problems studied in [21,25]. We analyze suitable conditions to prove the existence and convergence result of the considered system by using the theory of strong compression mappings, the resolvent operator, the Banach contraction theorem and the theory of converging sequences. Our results are new and easy to understand.