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The system of extended ordered XOR-inclusion problems (in short, SEOXORIP) involving generalized Cayley and Yosida operators is introduced and studied in this paper. The solution is obtained in a real ordered Banach space using a fixed-point approach. First, we develop the fixed-point lemma for the solution of SEOXORIP. By using the fixed-point lemma, we develop a three-step iterative scheme for obtaining the approximate solution of SEOXORIP. Under the Lipschitz continuous assumptions of the cost mappings, the strong convergence of the scheme is demonstrated. Lastly, we provide a numerical example with a convergence graph generated using MATLAB 2018a to verify the convergence of the sequence generated by the proposed scheme.

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1. Introduction

It is widely acknowledged that convex minimization problems, variational inequalities, equilibrium challenges, and feasibility dilemmas represent particular instances of variational inclusion problems. In the context of a single-valued mapping $A : X \to X$ and a multivalued (potentially nonlinear) operator $B : X \to 2^{X}$ , the objective is to determine $x \in X$ such that

(1) $0 \in (A + B) x .$

which is designated as a variational inclusion problem, exhibiting applications in signal processing [1], image processing [2], machine learning [3], and applications to multiple sets split feasibility problems [4].

The inability of projection methods to effectively address variational inclusion problems necessitated the emergence of resolvent operator methods, which have proven to be efficient in solving these problems. In their seminal works, Fang and Huang [5,6] put forth the concepts of H-monotone operators and H-accretive mappings, and by establishing the resolvent operators that correspond to these concepts, they investigated a specific category of variational inclusions within the context of Hilbert and Banach spaces. Following this, Xia and Huang [7] proposed the notion of general H-monotone operators as an extension of J-proximal mapping [8] and articulated a proximal mapping that is associated with general H-monotone operators, which diverges from the resolvent operator linked with the H-accretive mapping [6].

In 1972, a multitude of solutions pertaining to nonlinear equations were presented and analyzed by Amann [9]. In the recent past, fixed-point theory alongside its applications has undergone rigorous examination within the context of real ordered Banach spaces. Consequently, it is of paramount significance and a natural progression for generalized nonlinear ordered variational inequalities (ordered equations) to be meticulously investigated and deliberated. In 2008, Li [10] delineated the framework of generalized nonlinear ordered variational inequalities and proposed a computational algorithm aimed at approximating solutions for a specific class of generalized nonlinear ordered variational inequalities (ordered equations) within real ordered Banach spaces. Over the past two to three years, various forms of ordered variational inequalities and ordered inclusion problems have been extensively scrutinized, as evidenced in works such as [11,12,13] and the associated references therein.

On the contrary, Glowinski, in the year 1989 [14], and Noor, between the years 2000 and 2001 [15,16], formulated a three-step iterative algorithm aimed at addressing various categories of variational inequalities through the application of the Lagrangian multiplier and auxiliary principle methodologies. Consequently, it can be inferred that three-step iterative algorithms occupy a pivotal and consequential role in solutions to diverse challenges encountered in both pure and applied scientific disciplines. Glowinski et al. [14] and Noor [17] demonstrated that the three-step schemes yielded superior numerical outcomes in comparison to the two-step and one-step approximation iterations. In the year 2018, Iqbal et al. [18] introduced a three-step iterative framework for generalized mixed ordered quasi-variational inclusion that incorporates the XOR operator, while more recently, Ali et al. [19] examined the convergence and stability of the three-step iterative algorithm applied to the Extended Cayley–Yosida Inclusion Problem within the context of 2-Uniformly Smooth Banach Spaces.

Remark 1.

It is imperative to acknowledge that the selection of the governing sequences can significantly influence the efficacy of iterative algorithms.

Motivated by the preceding research while adhering to Remark 1, this manuscript presents a framework for extended ordered XOR-inclusion problems that incorporate generalized Cayley and Yosida operators. The solution is attained within a real Banach space utilizing a fixed-point methodology. Initially, we established the fixed-point lemma pertinent to the solutions of the proposed system. By applying the fixed-point lemma, we formulated a three-step iterative scheme aimed at deriving an approximate solution for SEOXORIP. Under the assumptions of Lipschitz continuity pertaining to the cost mappings, we elucidate the strong convergence of the iterative scheme. Finally, we furnish a numerical example accompanied by a convergence graph generated through MATLAB 2018a to substantiate the convergence of the sequence produced by the proposed scheme.

2. Prerequisites and Formulation of SEOXORIP

Throughout this manuscript, we consider $X$ to be a real ordered Banach space equipped with the norm ${∥ . ∥}_{X}$ and the inner product ${〈 \cdot, \cdot 〉}_{X \times X}$ , the metric $d$ generated by the norm ${∥ . ∥}_{X}$ , $2^{X}$ (correspondingly, $C B (X)$ ) representing the nonempty collection of (closed and bounded) subsets of $X$ , and the Hausdorff metric $D (., .)$ on $C B (X)$ , which is defined as follows:

$D (A, B) = max {sup_{s \in A} d (s, B), sup_{t \in B} d (A, t)}, \forall A, B \in C B (X),$

where

d (s, B) = inf_{t \in B} {∥ s - t ∥}

and

d (A, t) = inf_{s \in A} {∥ s - t ∥}

A cone is a $ϕ \neq C \subseteq X$ convex and closed subset of $X$ that satisfies, for any $x \in C$ and scalar $λ > 0$ , $λ x \in C$ . $C$ is called pointed cone if $C \cap {- C} = ϕ .$ For any $x, y \in X$ , we define an ordering $\leq_{C}$ such that $x \leq_{C} y i f a n d o n l y i f y - x \in C$ . If either $x \leq_{C} y$ or $y \leq_{C} x$ , then x and y are called comparable elements, and this is denoted by $x \propto y$ . The real Banach space $X$ equipped with ordering $\leq_{C}$ is said to be an ordered Banach space.

Let $l u b {x, y}$ and $g l b {x, y}$ denote the least upper bound and greatest lower bound of the set ${x, y}$ , respectively, for all $x, y \in X .$ Suppose that $l u b {x, y}$ and $g l b {x, y}$ for the set ${x, y}$ exist; then, we define binary operations:(i)

$x \lor y = l u b {x, y},$

(ii)

$x \land y = g l b {x, y},$

(iii)

$x \oplus y = (x - y) \lor (x - y),$

(iv)

$x ⊙ y = (x - y) \land (x - y) .$

The operations $\lor, \land,$ ⊕, and ⊙ are called OR, AND, XOR, and XNOR operations, respectively. For more details, refer to [20].

Proposition 1

([11,13]). Let ⊕ be an XOR operation and ⊙ be an XNOR operation. Then, the following holds:

(i)

$x ⊙ x = 0,$ $x ⊙ y = y ⊙ x = - (x \oplus y) = - (y \oplus x);$

(ii)

If $x \propto 0,$ then $- x \oplus 0 \leq x \leq x \oplus 0;$

(iii)

$(λ x) \oplus (λ y) = | λ | (x \oplus y);$

(iv)

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

(v)

If $x, y$ , and w are comparable to each other, then $(x \oplus y) \leq (x \oplus w) + (w \oplus y);$

(ix)

If $x, y$ , and w are comparable to each other, then $[(w \oplus x) \oplus (w \oplus y)] = (x \oplus y);$

(vi)

$∥ x \oplus y ∥ \leq ∥ x - y ∥ \leq λ_{P} ∥ x \oplus y ∥$ ;

(vii)

If $x \propto y,$ then $∥ x \oplus y ∥ = ∥ x - y ∥ .$

Definition 1

([11,13]). A single-valued mapping $f : X \to X$ is called

(i)

A comparison mapping if, for every $x, y \in X$ and $x \propto y$ , then $f (x) \propto f (y),$ $x \propto f (x)$ and $y \propto f (y);$

(ii)

A strong comparison mapping if f is a comparison mapping and $f (x) \propto f (y)$ if and only if $x \propto y$ for any $x, y \in X;$

(iii)

A $℘^{f}$ -ordered compression mapping if f is a comparison mapping and

$f (x) \oplus f (y) \leq ℘^{f} (x \oplus y), f o r ℘^{f} \in (0, 1) a n d x, y \in X .$

Definition 2.

Let $f, g : X \to X$ be strong comparison single-valued mappings. Then, the mapping $P : X \times X \to X$ is called

(i)

A $℘_{P}^{f}$ -ordered compression mapping associated with f in the first component if there exists a constant $℘_{P}^{f} > 0$ such that

$P (f (x), .) \oplus P (f (y), .) \leq ℘_{P}^{f} (x \oplus y),$

(ii)

A $℘_{P}^{g}$ -ordered compression mapping associated with g in the second component if there exists a constant $℘_{P}^{g} > 0$ such that

$P (., g (x)) \oplus P (., g (y)) \leq ℘_{P}^{g} (x \oplus y) .$

Definition 3

([21]). Let $(X, d)$ be a metric space and $D$ be a Hausdorff metric on $C B (X)$ . Then, the multivalued map $S : X \times X \to 2^{X}$ is called a multivalued Lipschitz-type mapping with respect to $N$ if there is a constant $λ_{N_{D}} > 0$ such that

$D (S (h, u_{x}), S (h, u_{y})) \leq λ_{N_{D}} ∥ x - y ∥, u_{x} \in N (x) a n d u_{y} \in N (y), \forall x, y \in X,$

where $h : X \to X$ be single-valued mapping.

Definition 4

([22]). Let $h : X \to X$ be single-valued mapping, $N : X \to C B (X)$ and $S : X \times X \to 2^{X}$ be a multivalued mappings. Then, $S (h (\cdot), z)$ is said to be α-strongly accretive with respect to h if there is a constant $α > 0$ such that

$〈 u - v, x - y 〉 \geq {α ∥ x - y ∥}^{2}, \forall x, y \in X, u \in S (h (x), z), v \in S (h (y), z) .$

Proposition 2

([22]). Let $h : X \to X$ be a single-valued mapping, $N : X \to C B (X)$ be a multivalued mapping, and $S : X \times X \to 2^{X}$ be α-strongly accretive with respect to h. Then, the mapping ${[I + λ S (h (\cdot), z)]}^{- 1}$ is single-valued for $λ \geq 0$ and $z \in N (x) .$

Definition 5

([22]). Let $h : X \to X$ be a single-valued mapping, $N : X \to C B (X)$ be a multivalued mapping, and $S : X \times X \to 2^{X}$ be α-strongly accretive with respect to h. Then, $S$ is said to be generalized α-maximal monotone with respect to h if

$[I + λ S (h, z)] X = X, f o r λ > 0 a n d z \in N (x) .$

Definition 6.

Let $h : X \to X$ be a single-valued mapping and $N : X \to C B (X)$ be a multivalued mapping. Let $S : X \times X \to 2^{X}$ be generalized α-strongly accretive with respect to h. Then, the generalized proximal-point mapping $ℜ_{I, λ}^{S (h (\cdot), z)} : X \to X$ associated with h and $N$ is defined by

(2) $ℜ_{I, λ}^{S (h (\cdot), z)} (x) = {[I + λ S (h (\cdot), z)]}^{- 1} (x), \forall x \in X .$

Proposition 3

([11,22]). Let $h : X \to X$ be a single-valued mapping and $N : X \to C B (X)$ be a multivalued mapping. Let $S : X \times X \to 2^{X}$ be generalized α-maximal monotone with respect to h. Then, the generalized proximal-point mapping $ℜ_{I, λ}^{S (h (\cdot), z)}$ is $L^{ℜ}$ -Lipschitz continuous. That is,

$∥ℜ_{I, λ}^{S (h (\cdot), z)} (x) - ℜ_{I, λ}^{S (h (\cdot), z)} (y)∥ \leq L^{ℜ} ∥ x - y ∥, \forall x, y \in X,$

where $L^{ℜ} = (\frac{1}{α λ + 1}) .$

Definition 7

([19]). The generalized Yosida approximation operator $Y_{I, λ}^{S (h (\cdot), z)}$ associated with the proximal-point mapping $ℜ_{I, λ}^{S (h (\cdot), z)}$ is a single-valued mapping; that is, $Y_{I, λ}^{S (h (\cdot), z)} : X \to X$ is defined by

(3) $Y_{I, λ}^{S (h (\cdot), z)} (x) = \frac{1}{λ} [I - ℜ_{I, λ}^{S (h (\cdot), z)}] (x), \forall x \in X .$

Definition 8

([19]). The generalized Cayley operator $C_{I, λ}^{S (h (\cdot), z)}$ associated with the proximal-point mapping $ℜ_{I, λ}^{S (h (\cdot), z)}$ is a single-valued mapping; that is, $C_{I, λ}^{S (h (\cdot), z)} : X \to X$ is defined by

(4) $C_{I, λ}^{S (h (\cdot), z)} (x) = [2 ℜ_{I, λ}^{S (h (\cdot), z)} - I] (x), \forall x \in X .$

Remark 2.

(i)

It is well known that the generalized Yosida approximation operator is Lipschitz-type continuous with constant $Θ_{Y} = \frac{1}{λ} (1 + L^{ℜ}) .$

(ii)

Similarly, the generalized Cayley operator is Lipschitz-type continuous with constant $Θ_{C} = (2 L^{ℜ} + 1) .$

SEOXORIP and the Existence of Its Solution

We assume that, for $k \in N$ , $i = 1, 2, \dots, k$ . Let $f_{i}, g_{i}, h_{i} : X \to X$ and $P_{i} : X \times X \to X$ be single-valued mappings. Suppose that $S_{i} : X \times X \to 2^{X}$ is a generalized $α_{i}$ -strongly accretive multivalued mapping with respect to $h_{i}$ , and $N_{i} : X \to C B (X)$ is a closed and bounded multivalued mapping. Then, our interest is in finding $(x_{1}, \dots, x_{k}) \in X^{k}$ and $(v_{1}, \dots, v_{k}) \in}} N_{1} (x_{1}) \times \dots \times}} N_{k} (x_{k})$ such that

(5) $\begin{matrix} \{\begin{matrix} w_{1} \in P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (x_{1}), v_{2})} (x_{1}), Y_{I, λ_{2}}^{S_{2} (h_{2} (x_{2}), v_{3})} (x_{2})) + f_{1} (x_{1}) \oplus g_{2} (x_{2}) + S_{1} (h_{1} (x_{1}), v_{2}), \\ w_{2} \in P_{2} (C_{I, λ_{2}}^{S_{2} (h_{2} (x_{2}), v_{3})} (x_{2}), Y_{I, λ_{3}}^{S_{3} (h_{3} (x_{3}), v_{4})} (x_{3})) + f_{2} (x_{2}) \oplus g_{3} (x_{3}) + S_{2} (h_{2} (x_{2}), v_{3}), \\ ⋮ \\ w_{k} \in P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (x_{k}), v_{1})} (x_{k}), Y_{I, λ_{1}}^{S_{1} (h_{1} (x_{1}), v_{2})} (x_{1})) + f_{k} (x_{k}) \oplus g_{1} (x_{1}) + S_{k} (h_{k} (x_{k}), v_{1}), \end{matrix} \end{matrix}$

holds, for some

\hat{w} = (w_{1}, \dots w_{k}) \in X^{k} .

This system is called an extended nonlinear system of ordered XOR-inclusion problems involving generalized Cayley–Yosida operators (in short, SEOXORIP).

Moreover, to highlight the level of generalization of our problem, we present several special cases below.

Special Cases:

If we define $C_{I, λ_{i}}^{S_{i} (h_{i} (x_{i}), v_{i + 1})} (x_{i}) = f_{i} (x_{i}),$ $Y_{I, λ_{i}}^{S_{i} (h_{i} (x_{i}), v_{i + 1})} (x_{i}) = x_{i}, \forall x_{i}$ and $f_{i} = g_{i + 1} \forall i$ , then SEOXORIP (5) becomes a system for finding $(x_{1}, x_{2}, \dots, x_{k}) \in X^{k}$ such that

(6) $\begin{matrix} \{\begin{matrix} w_{1} \in P_{1} (f_{1} (x_{1}), x_{2}) + S_{1} (h_{1} (x_{1}), v_{2}), \\ w_{2} \in P_{2} (f_{2} (x_{2}), x_{3}) + S_{2} (h_{2} (x_{2}), v_{3}), \\ ⋮ \\ w_{k} \in P_{k} (f_{k} (x_{k}), x_{1}) + S_{k} (h_{k} (x_{k}), v_{1}), \end{matrix} \end{matrix}$

for some

(w_{1}, w_{2}, \dots, w_{k}) \in X^{k}

. System (6) was introduced and studied by [23].

Let $i = 1, 2$ . If we define $v_{i} = {g_{i} (x_{i})}$ and $C_{I, λ_{1}}^{S_{i} (h_{i} (x_{i}), v_{i + 1})} (x_{i}) = Y_{I, λ_{i}}^{S_{i} (h_{i} (x_{i}), v_{i + 1})} (x_{i}) = x_{i}, \forall x_{i}$ , then for $ω_{i} = 0, \forall i,$ system (5) reduces to the following System of Generalized Variational Inclusions (SGVI), that is, a problem where $(x, y) \in X_{1} \times X_{2}$ must be found, such that

(7) $\begin{matrix} \{\begin{matrix} θ_{1} \in P_{1} (x, y) + S_{1} (h_{1} (x), g_{2} (y)), \\ θ_{2} \in P_{2} (x, y) + S_{2} (h_{2} (x), g_{1} (y)), \end{matrix} \end{matrix}$

hold, where

θ_{1} = 0

and

θ_{2} = 0

are the zeros of

X_{1}

and

X_{2}

, respectively. System (7) was studied by Kazmi et al. [22].

By using the proximal-point mapping, we can characterize the solution of system (5) in terms of fixed-point equations.

Lemma 1.

SEOXORIP (5) has the solution $(x, v)$ , i.e., $x = (x_{1}, \dots, x_{k}) \in X^{k}$ and $v = (v_{1}, \dots, v_{k}) \in N_{1} (x_{1}) \times \dots \times N_{k} (x_{k}),$ if and only if the following equations hold:

(8) $\begin{matrix} x_{1} & = ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} [x_{1} - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2})) + f_{1} (x_{1}) \oplus g_{2} (x_{2}) - w_{1}}], \\ x_{2} & = ℜ_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} [x_{2} - λ_{2} {P_{2} (C_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}), Y_{I, λ_{3}}^{S_{3} (h_{3} (\cdot), v_{4})} (x_{3})) + f_{2} (x_{2}) \oplus g_{3} (x_{3}) - w_{2}}], \\ ⋮ \\ x_{k} & = ℜ_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} [x_{k} - λ_{k} {P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (x_{k}), Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1})) + f_{k} (x_{k}) \oplus g_{1} (x_{1}) - w_{k}}], \end{matrix}$

where $λ_{i}^{' s} > 0$ are non-negative real numbers.

Proof.

The proof follows directly from Definition 6 of proximal-point mapping.

Now, we have the following theorem for the existence of the solution to system (5). □

Theorem 1.

Let $i = 1, 2, \dots, k$ , $k \in N, a n d w e s e t k + 1 = 1$ . Let $f_{i}, g_{i}, h_{i} : X \to X$ be $℘^{f_{i}}$ , $℘^{g_{i}}$ , and $℘^{h_{i}}$ strongly ordered comparison mappings. Assume that $P_{i} : X \times X \to X$ is a $℘_{P_{i}}^{C_{i}}$ -ordered compression mapping associated with $C_{I, λ_{i}}^{S_{i} (h_{i} (\cdot), z)}$ in the first component and $℘_{P_{i}}^{Y_{i}}$ -ordered compression mapping associated with $Y_{I, λ_{i}}^{S_{i} (h_{i} (\cdot), z)}$ in the second component. Let $S_{i} : X \times X \to 2^{X}$ be generalized $α_{i}$ -strongly accretive with respect to $h_{i}$ and $N_{i} : X \to C B (X)$ be a closed and bounded multivalued mapping. Suppose $ℜ_{I, λ_{i}}^{S_{i} (h_{i} (\cdot), z)}$ , $Y_{I, λ_{i}}^{S_{i} (h_{i} (\cdot), z)}$ , and $C_{I, λ_{i}}^{S_{i} (h_{i} (\cdot), z)}$ are Lipschitz continuous with constants $L^{ℜ_{i}}$ , $Θ_{Y_{i}}$ , and $Θ_{C_{i}}$ , respectively.

Additionally, let $x_{i} \propto x_{i + 1}$ , $y_{i} \propto y_{i + 1}$ $f_{i} (x_{i}) \propto f_{i} (y_{i})$ , $g_{i} (x_{i}) \propto g_{i} (y_{i})$ , and $ℜ_{I, λ_{i}}^{S_{i} (h_{i} (\cdot), z)} (x_{i}) \propto ℜ_{I, λ_{i}}^{S_{i} (h_{i} (\cdot), z)} (y_{i})$ , and for all constants $λ_{1}, λ_{2}, \dots, λ_{k} > 0$ , the following relations hold:

(9) $\begin{matrix} λ_{1} (℘_{P_{1}}^{C_{1}} Θ_{C_{1}} + ℘^{f_{1}}) + λ_{k} (℘_{P_{k}}^{Y_{1}} Θ_{Y_{1}} + ℘^{g_{1}}) & < (\frac{1}{L^{ℜ^{1}}} - 1), \\ λ_{2} (℘_{P_{2}}^{C_{2}} Θ_{C_{2}} + ℘^{f_{2}}) + λ_{1} (℘_{P_{1}}^{Y_{2}} Θ_{Y_{2}} + ℘^{g_{2}}) & < (\frac{1}{L^{ℜ^{2}}} - 1), \\ ⋮ \\ λ_{k} (℘_{P_{k}}^{C_{k}} Θ_{C_{k}} + ℘^{f_{k}}) + λ_{k - 1} (℘_{P_{k - 1}}^{Y_{k}} Θ_{Y_{k}} + ℘^{g_{k}}) & < (\frac{1}{L^{ℜ^{k}}} - 1), \end{matrix}$

where

$L^{ℜ^{i}} = (\frac{1}{α_{i} λ_{i} + 1}), Θ_{Y_{i}} = \frac{1}{λ_{i}} (L^{ℜ^{i}} - 1) a n d Θ_{C_{i}} = 2 L^{ℜ^{i}} - 1 .$

Then, SEOXORIP (5) admits a solution $(x_{i}, v_{i}) .$

Proof.

We define the mapping $Q : X^{k} ⟶ X^{k}$ by

$\begin{matrix} Q (x_{1}, \dots, x_{k}) : = (Q_{1} (x_{1}, x_{2}), \dots, Q_{κ} (x_{k}, x_{1})), \forall (x_{1}, \dots, x_{k}) \in X^{k}, \end{matrix}$

where the mappings

Q_{i} : X \times X \to X

are defined as

$\begin{matrix} Q_{1} (x_{1}, x_{2}) & = & ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} [x_{1} - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2})) + f_{1} (x_{1}) \oplus g_{2} (x_{2}) - w_{1}}], \\ Q_{2} (x_{2}, x_{3}) & = & ℜ_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} [x_{2} - λ_{2} {P_{2} (C_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}), Y_{I, λ_{3}}^{S_{3} (h_{3} (\cdot), v_{4})} (x_{3})) + f_{2} (x_{2}) \oplus g_{3} (x_{3}) - w_{2}}], \\ ⋮ \\ Q_{k} (x_{κ}, x_{1}) & = & ℜ_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} [x_{k} - λ_{k} {P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (x_{k}), Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1})) + f_{k} (x_{k}) \oplus g_{1} (x_{1}) - w_{k}}] . \end{matrix}$

Let

x_{i}, y_{i} \in X, v_{i} \in N_{i} (x_{i})

such that

x_{i} \propto x_{j}, y_{i} \propto y_{j}, v_{i} \propto v_{j}

, and by applying Proposition 3, we have

(10) $\begin{matrix} ∥ Q_{1} (x_{1}, x_{2}) \oplus Q_{1} (y_{1}, y_{2}) ∥ \\ = & ∥ ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} [x_{1} - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2})) + f_{1} (x_{1}) \oplus g_{2} (x_{2}) - w_{1}}] \\ \oplus ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} [y_{1} - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (y_{1}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (y_{2})) + f_{1} (y_{1}) \oplus g_{2} (y_{2}) - w_{1}}] ∥ \\ \leq & L^{ℜ_{1}} {∥ x_{1} \oplus y_{1} ∥ + λ_{1} ∥ P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2})) \oplus P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (y_{1}), \\ Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (y_{2})) ∥ + λ_{1} ∥ f_{1} (x_{1}) \oplus g_{2} (x_{2}) \oplus f_{1} (y_{1}) \oplus g_{2} (y_{2}) ∥} . \end{matrix}$

By applying Proposition 1 and the ordered compressionness of

f_{1}

and

g_{2}

, we have

(11) $\begin{matrix} ∥ f_{1} (x_{1}) \oplus g_{2} (x_{2}) \oplus f_{1} (y_{1}) \oplus g_{2} (y_{2}) ∥ & = & ∥ f_{1} (x_{1}) \oplus f_{1} (y_{1}) \oplus g_{2} (x_{2}) \oplus g_{2} (y_{2}) ∥ \\ \leq & ∥ f_{1} (x_{1}) \oplus f_{1} (y_{1}) ∥ + ∥ g_{2} (x_{2}) \oplus g_{2} (y_{2}) ∥ \\ \leq & ℘^{f_{1}} ∥ x_{1} \oplus y_{1} ∥ + ℘^{g_{2}} ∥ x_{2} \oplus y_{2} ∥ . \end{matrix}$

Since

P_{1}

is a

℘_{P_{1}}^{C_{1}}

-ordered compression mapping in the first component with respect to

C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})}

and a

℘_{P_{1}}^{Y_{1}}

-ordered compression mapping in the second component with respect to

Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})}

, we have

(12) $\begin{matrix} ∥ P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2})) \oplus P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (y_{1}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (y_{2})) ∥ \\ \leq ∥P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2})) \oplus P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (y_{1}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}))∥ \\ + ∥P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (y_{1}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2})) \oplus P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (y_{1}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (y_{2}))∥ \\ \leq ℘_{P_{1}}^{C_{1}} ∥C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}) \oplus C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (y_{1})∥ + ℘_{P_{1}}^{Y_{1}} ∥Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}) \oplus Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (y_{2})∥ \\ \leq ℘_{P_{1}}^{C_{1}} Θ_{C_{1}} ∥ x_{1} \oplus y_{1} ∥ + ℘_{P_{1}}^{Y_{1}} Θ_{Y_{1}} ∥ x_{2} \oplus y_{2} ∥ \\ \leq ℘_{P_{1}}^{C_{1}} Θ_{C_{1}} ∥ x_{1} - y_{1} ∥ + ℘_{P_{1}}^{Y_{1}} Θ_{Y_{1}} ∥ x_{2} - y_{2} ∥ . \end{matrix}$

Inserting (11) and (12) into (10) and applying Proposition 1, we get

(13) $\begin{matrix} ∥ Q_{1} (x_{1}, x_{2}) \oplus Q_{1} (y_{1}, y_{2}) ∥ = ∥ Q_{1} (x_{1}, x_{2}) - Q_{1} (y_{1}, y_{2}) ∥ \\ \leq & L^{ℜ_{1}} {∥ x_{1} - y_{1} ∥ + λ_{1} (℘_{P_{1}}^{C_{1}} Θ_{C_{1}} ∥ x_{1} - y_{1} ∥ + ℘_{P_{1}}^{Y_{1}} Θ_{Y_{1}} ∥ x_{2} - y_{2} ∥ \\ + ℘^{f_{1}} ∥ x_{1} - y_{1} ∥ + ℘^{g_{2}} ∥ x_{2} - y_{2} ∥)} \\ = & L^{ℜ_{1}} ([1 + λ_{1} (℘_{P_{1}}^{C_{1}} Θ_{C_{1}} + ℘^{f_{1}})] ∥ x_{1} - y_{1} ∥ + λ_{1} (℘_{P_{1}}^{Y_{1}} Θ_{Y_{1}} + ℘^{g_{2}}) ∥ x_{2} - y_{2} ∥) . \end{matrix}$

Continuing in similar fashion, let

x_{i}, y_{i} \in X, v_{i} \in N_{i} (x_{i})

such that

x_{i} \propto x_{j}, y_{i} \propto y_{j},

and

v_{i} \propto v_{j}

; applying Proposition 3, we have

(14) $\begin{matrix} ∥ Q_{k} (x_{k}, x_{1}) \oplus Q_{k} (y_{k}, y_{1}) ∥ \\ = & ∥ ℜ_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} [x_{k} - λ_{k} {P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (x_{k}), Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1})) + f_{k} (x_{k}) \oplus g_{1} (x_{1}) - w_{k}}] \\ - ℜ_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} [y_{k} - λ_{k} {P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (y_{k}), Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (y_{1})) + f_{k} (y_{k}) \oplus g_{1} (y_{1}) - w_{k}}] ∥ \\ \leq & L^{ℜ_{k}} {∥ x_{k} \oplus y_{k} ∥ + λ_{k} ∥ P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (x_{k}), Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1})) \oplus P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (y_{k}), \\ Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (y_{1})) ∥ + λ_{k} ∥ f_{k} (x_{k}) \oplus g_{1} (x_{1}) \oplus f_{k} (y_{k}) \oplus g_{1} (y_{1}) ∥} . \end{matrix}$

By applying Proposition 1 and the ordered compressionness of

f_{k}

and

g_{1}

, we have

(15) $\begin{matrix} ∥ f_{k} (x_{k}) \oplus g_{1} (x_{1}) \oplus f_{k} (y_{k}) \oplus g_{1} (y_{1}) ∥ & = & ∥ f_{k} (x_{k}) \oplus f_{k} (y_{k}) \oplus g_{1} (x_{1}) \oplus g_{1} (y_{1}) ∥ \\ \leq & ∥ f_{k} (x_{k}) \oplus f_{k} (y_{k}) ∥ + ∥ g_{1} (x_{1}) \oplus g_{1} (y_{1}) ∥ \\ \leq & ℘^{f_{k}} ∥ x_{k} - y_{k} ∥ + ℘^{g_{1}} ∥ x_{1} - y_{1} ∥ . \end{matrix}$

Since

P_{k}

is a

℘_{P_{k}}^{C_{k}}

-ordered compression mapping associated with

C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})}

in the first component and a

℘_{P_{k}}^{Y_{1}}

-ordered compression mapping associated with

Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})}

in the second component, we have

(16) $\begin{matrix} ∥ P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (x_{k}), Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1})) \oplus P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (y_{k}), Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (y_{1})) ∥ \\ \leq ∥P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (x_{k}), Y_{I, λ_{k}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1})) \oplus P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (y_{k}), Y_{I, λ_{1}}^{S_{1} (h_{2} (\cdot), v_{2})} (x_{1}))∥ \\ + ∥P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (y_{k}), Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1})) \oplus P_{k} (C_{I, λ_{1}}^{S_{k} (h_{k} (\cdot), v_{1})} (y_{1}), Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (y_{1}))∥ \\ \leq ℘_{P_{k}}^{C_{k}} ∥C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (x_{k}) \oplus C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (y_{k})∥ + ℘_{P_{k}}^{Y_{k}} ∥Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}) \oplus Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (y_{1})∥ \\ \leq ℘_{P_{k}}^{C_{k}} Θ_{C_{k}} ∥ x_{k} \oplus y_{k} ∥ + ℘_{P_{k}}^{Y_{k}} Θ_{Y_{k}} ∥ x_{1} \oplus y_{1} ∥ \\ \leq ℘_{P_{k}}^{C_{k}} Θ_{C_{k}} ∥ x_{k} - y_{k} ∥ + ℘_{P_{k}}^{Y_{k}} Θ_{Y_{k}} ∥ x_{1} - y_{1} ∥ . \end{matrix}$

Inserting (15) and (16) into (14) and applying Proposition 1, we get

(17) $\begin{matrix} ∥ Q_{k} (x_{k}, x_{1}) \oplus Q_{k} (y_{k}, y_{1}) ∥ = ∥ Q_{k} (x_{k}, x_{1}) - Q_{k} (y_{k}, y_{1}) ∥ \\ \leq & L^{ℜ_{k}} {∥ x_{k} - y_{k} ∥ + λ_{k} (℘_{P_{k}}^{C_{k}} Θ_{C_{k}} ∥ x_{k} - y_{k} ∥ + ℘_{P_{k}}^{Y_{k}} Θ_{Y_{1}} ∥ x_{1} - y_{1} ∥ \\ + ℘^{f_{k}} ∥ x_{k} - y_{k} ∥ + ℘^{g_{1}} ∥ x_{1} - y_{1} ∥)} \\ = & L^{ℜ_{k}} ([1 + λ_{k} (℘_{P_{k}}^{C_{k}} Θ_{C_{k}} + ℘^{f_{k}})] ∥ x_{k} - y_{k} ∥ + λ_{k} (℘_{P_{k}}^{Y_{1}} Θ_{Y_{1}} + ℘^{g_{1}}) ∥ x_{1} - y_{1} ∥) . \end{matrix}$

Based on (13) and (17), we have

$\begin{matrix} ∥ Q_{1} (x_{1}, x_{2}) - Q_{1} (y_{1}, y_{2}) ∥ & + \dots + ∥ Q_{k} (x_{k}, x_{1}) - Q_{k} (y_{k}, y_{1}) ∥ \\ \leq L^{ℜ_{1}} [1 + λ_{1} (℘_{P_{1}}^{C_{1}} Θ_{C_{1}} + ℘^{f_{1}}) + λ_{k} (℘_{P_{k}}^{Y_{1}} Θ_{Y_{1}} + ℘^{g_{1}})] ∥ x_{1} - y_{1} ∥ \\ + L^{ℜ_{2}} [1 + λ_{2} (℘_{P_{2}}^{C_{2}} Θ_{C_{2}} + ℘^{f_{2}}) + λ_{1} (℘_{P_{1}}^{Y_{2}} Θ_{Y_{2}} + ℘^{g_{2}})] ∥ x_{2} - y_{2} ∥ \\ + \dots + \\ + L^{ℜ_{k}} [1 + λ_{k} (℘_{P_{k}}^{C_{k}} Θ_{C_{k}} + ℘^{f_{k}}) + λ_{k -} (℘_{P_{k - 1}}^{Y_{k}} Θ_{Y_{k}} + ℘^{g_{k}})] ∥ x_{k} - y_{k} ∥, \end{matrix}$

that is,

(18) $\begin{matrix} \sum_{i = 1}^{k} ∥ Q_{i} (x_{i}, x_{i + 1}) - Q_{i} (y_{i}, y_{i + 1}) ∥ \leq \sum_{i = 1}^{k} Ξ_{i}^{k + i - 1} L^{ℜ^{i}} ∥ x_{i} - y_{i} ∥, \end{matrix}$

where

$\begin{matrix} Ξ_{1}^{k} & = L^{ℜ_{1}} [1 + λ_{1} (℘_{P_{1}}^{C_{1}} Θ_{C_{1}} + ℘^{f_{1}}) + λ_{k} (℘_{P_{k}}^{Y_{1}} Θ_{Y_{1}} + ℘^{g_{1}})], \\ Ξ_{2}^{1} & = L^{ℜ_{2}} [1 + λ_{2} (℘_{P_{2}}^{C_{2}} Θ_{C_{2}} + ℘^{f_{2}}) + λ_{1} (℘_{P_{1}}^{Y_{2}} Θ_{Y_{2}} + ℘^{g_{2}})], \\ ⋮ \\ Ξ_{k}^{k - 1} & = L^{ℜ_{k}} [1 + λ_{k} (℘_{P_{k}}^{C_{k}} Θ_{C_{k}} + ℘^{f_{k}}) + λ_{k -} (℘_{P_{k - 1}}^{Y_{k}} Θ_{Y_{k}} + ℘^{g_{k}})] . \end{matrix}$

The norm $∥ (x_{1}, \dots, x_{k}) ∥_{1} o n X^{k}$ is defined as

(19) $\begin{matrix} ∥ (x_{1}, \dots, x_{κ}) ∥_{1} = ∥ x_{1} ∥ + \dots + ∥ x_{κ} ∥, \forall (x_{1}, \dots, x_{k}) \in X^{k} . \end{matrix}$

Clearly, the structure

(X^{k}, ∥ . ∥_{1})

forms a Banach space with respect to the norm (19). Thus, the definition of the mapping

Q

, (18) and (19), implies that

(20) $\begin{matrix} ∥ Q (x_{1}, \dots, x_{κ}) - Q (y_{1}, \dots, y_{κ}) ∥_{1} & = ∥ Q_{1} (x_{1}, x_{2}) - Q_{1} (y_{1}, y_{2}) ∥ + \dots + ∥ Q_{k} (x_{k}, x_{1}) - Q_{k} (y_{k}, y_{1}) ∥ \\ \leq max \{Ξ_{1}^{k}, Ξ_{2}^{1}, \dots, Ξ_{k}^{k - 1}\} (∥ x_{1} - y_{1} ∥ + ∥ x_{2} - y_{2} ∥ \dots + ∥ x_{k} - y_{k} ∥) . \end{matrix}$

From (9), we conclude that

max \{Ξ_{1}^{k}, Ξ_{2}^{1}, \dots, Ξ_{k}^{k - 1}\} < 1

. Therefore, there exists a unique fixed point

(x_{1}, \dots, x_{k}) \in X^{k}

of the mapping

Q

. That is,

$\begin{matrix} Q (x_{1}, \dots, x_{k}) = (x_{1}, \dots, x_{k}) . \end{matrix}$

This leads to

(21) $\begin{matrix} x_{1} & = ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} [x_{1} - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2})) + f_{1} (x_{1}) \oplus g_{2} (x_{2}) - w_{1}}], \\ x_{2} & = ℜ_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} [x_{2} - λ_{2} {P_{2} (C_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}), Y_{I, λ_{3}}^{S_{3} (h_{3} (\cdot), v_{4})} (x_{3})) + f_{2} (x_{2}) \oplus g_{3} (x_{3}) - w_{2}}], \\ ⋮ \\ x_{k} & = ℜ_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} [x_{k} - λ_{k} {P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (x_{k}), Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1})) + f_{k} (x_{k}) \oplus g_{1} (x_{1}) - w_{k}}] . \end{matrix}$

Hence, Lemma (1) ensures that $(x, v)$ is a solution of system (5). □

3. Three-Step Iterative Scheme and Its Convergence

In this section, leveraging Lemma 1, we formulate a three-step iterative algorithm aimed at determining the approximate solution for the newly established system of ordered XOR-inclusion problems involving Cayley and Yosida operators (Algorithm 1). The convergence properties of the sequence produced by the algorithm are demonstrated under certain appropriate assumptions.

Algorithm 1: Three-Step Iterative Algorithm for the Approximate Solution of SEOXORIP

Let

i = 1, 2, \dots, k

k \in N a n d s e t k + 1 = 1

; let

f_{i}, g_{i}, h_{i} : X \to X

and

P_{i} : X \times X \to X

be single-valued mappings. Let

N_{i} : X \to C B (X)

be a

D

-Lipschitz continuous mapping with constants

λ_{N_{D}}

and let

S_{i} : X \times X \to 2^{X}

be a generalized

α_{i}

-strongly accretive mapping with respect to

h_{i}

. Then, Initially: Choose

(x_{1}^{0}, \dots, x_{k}^{0}) \in X^{k}

and

(v_{1}^{0}, \dots, v_{k}^{0}) \in N_{1} (x_{1}^{0}) \times \dots \times N_{k} (x_{k}^{0})

. Step I: Let

x_{i}^{(n + 1)} \propto x_{i}^{(n)}

and

v_{i}^{(n)} \propto v_{j}^{(n)} .

We define

(22) $\begin{matrix} x_{1}^{(n + 1)} & = α_{1}^{n} ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} [y_{1}^{n} - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (y_{1}^{n}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (y_{2}^{n})) \\ + f_{1} (y_{1}^{n}) \oplus g_{2} (y_{2}^{n}) - w_{1}}] + (1 - α_{1}^{n}) x_{1}^{n} + α_{1}^{n} δ_{1}^{n} \\ y_{1}^{(n)} & = β_{1}^{n} ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} [z_{1}^{n} - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (z_{1}^{n}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (z_{2}^{n})) \\ + f_{1} (z_{1}^{n}) \oplus g_{2} (z_{2}^{n}) - w_{1}}] + (1 - β_{1}^{n}) x_{1}^{(n)} + β_{1}^{n} δ_{2}^{n} \\ z_{1}^{(n)} & = γ_{1}^{n} ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} [x_{1}^{n} - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}^{n}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}^{n})) \\ + f_{1} (x_{1}^{n}) \oplus g_{2} (x_{2}^{n}) - w_{1}}] + (1 - γ_{1}^{n}) x_{1}^{(n)} + γ_{1}^{n} δ_{3}^{n} \\ ⋮ \\ x_{k}^{(n + 1)} & = α_{k}^{n} ℜ_{I, λ_{1}}^{S_{k} (h_{k} (\cdot), v_{1})} [y_{k}^{n} - λ_{k} {P_{1} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (y_{k}^{n}), Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (y_{1}^{n})) \\ + f_{k} (y_{k}^{n}) \oplus g_{1} (y_{1}^{n}) - w_{k}}] + (1 - α_{k}^{n}) x_{k}^{n} + α_{k}^{n} δ_{k}^{n} \\ y_{k}^{(n)} & = β_{k}^{n} ℜ_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} [z_{k}^{n} - λ_{k} {P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (z_{k}^{n}), Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (z_{1}^{n})) \\ + f_{k} (z_{k}^{n}) \oplus g_{1} (z_{1}^{n}) - w_{k}}] + (1 - β_{k}^{n}) x_{k}^{(n)} + β_{k}^{n} δ_{k + 1}^{n} \\ z_{k}^{(n)} & = γ_{k}^{n} ℜ_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} [x_{k}^{n} - λ_{k} {P_{k} (C_{I, λ_{k}}^{S_{k} (h_{k} (\cdot), v_{1})} (x_{k}^{n}), Y_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}^{n})) \\ + f_{k} (x_{k}^{n}) \oplus g_{1} (x_{1}^{n}) - w_{k}}] + (1 - γ_{k}^{n}) x_{k}^{(n)} + γ_{k}^{n} δ_{k + 2}^{n} . \end{matrix}$

for

n = 0, 1, 2, \dots,

where

λ_{i} > 0

is a constant and

α_{i}^{n}, β_{i}^{n}

, and

γ_{i}^{n}

are real sequences in (0,1) such that

$\begin{matrix} \sum_{n = 1}^{\infty} α_{i}^{n} = \infty, \forall i . \end{matrix}$

Step II: Choose

v_{i}^{(n + 1)} \in N_{i} (x_{i}^{(n + 1)})

such that

(23) $\begin{matrix} ∥ v_{i}^{(n + 1)} \oplus v_{i}^{(n)} ∥ \leq (1 + \frac{1}{n + 1}) D (N_{i} (x_{i}^{(n + 1)}), N_{i} (x_{i}^{(n)})), \end{matrix}$

where

D (., .)

is the Hausdorff metric on

X

. Step III: If the accuracy is satisfactory and

x_{i}^{(n + 1)}

and

v_{i}^{(n)}

\forall i

satisfy step I, then stop; if not, set

n = n + 1

and return to step I.

Remark 3.

Algorithm 1 becomes a two-step iterative algorithm (Ishikawa-type) if $γ_{i}^{n} = 0$ for all $n \geq 0$ , and it reduces to a one-step iterative scheme (Mann-type) when choosing $β_{i}^{n} = γ_{i}^{n} = 0$ for all $n \geq 0 .$ We also observe that if we define the appropriate operators for Algorithm 1, we can easily acquire many more methods that have been studied by several authors for addressing ordered variational inclusions, see, e.g., [24,25,26].

Lemma 2.

The sequence $v_{n} \to 0$ as $n \to \infty$ if $v_{n}$ and $η_{n}$ are sequences in $[0, \infty)$ such that

(i)

$0 \leq η_{n} < 1, n = 0, 1, 2, \dots$ and $l i m s u p_{n} η_{n} < 1$ ;

(ii)

$v_{n + 1} \leq η_{n} v_{n}, n = 0, 1, 2, 3, \dots,$ hold.

Theorem 2.

Let all the mappings and conditions be the same as in Theorem 1, except for condition (9). Additionally, if

(24) $\begin{matrix} ∥ ℜ_{I, λ_{i}}^{S_{i} (h (\cdot), u)} (x) \oplus ℜ_{I, λ_{i}}^{S_{i} (h (\cdot), v)} (x) ∥ \leq Λ_{i} ∥ u \oplus v ∥, \end{matrix}$

(25) $\begin{matrix} sup_{n \geq 1} {α_{i}^{n} β_{i}^{n} ν_{i}^{2} {γ_{i}^{n} ν_{i} + 1} + α_{i}^{n} ν_{i} + 1} < sup_{n \geq 1} {α_{i}^{n} β_{i}^{n} ν_{i}^{2} γ_{i}^{n} ν_{i} + α_{i}^{n} ν_{i} β_{i}^{n} + α_{i}^{n}}, \\ sup_{n \geq 1} {α_{i}^{n} β_{i}^{n} ν_{i} (ν_{i} γ_{i}^{n} + 1) (δ_{i + 1} + μ_{i + 1}) + α_{i}^{n} β_{i}^{n} ν_{i} δ_{i + 1} γ_{i + 1}^{n} ν_{i + 1} \\ + α_{i}^{n} ν_{i + 1} β_{i + 1}^{n} δ_{i + 1} ν_{i + 1} γ_{i + 1}^{n} + α_{i}^{n} δ_{i + 1}}, \\ < sup_{n \geq 1} {α_{i}^{n} β_{i}^{n} ν_{i} δ_{i + 1} γ_{i + 1}^{n} + α_{i}^{n} ν_{i + 1} β_{i + 1}^{n} δ_{i + 1} γ_{i + 1}^{n} + 1 + α_{i}^{n} δ_{i + 1} β_{i + 1}^{n}}, \\ sup_{n \geq 1} {α_{i}^{n} γ_{i + 1}^{n} δ_{i + 1} (β_{i}^{n} ν_{i} + β_{i + 1}^{n} ν_{i + 1}) (δ_{i + 2} + μ_{i + 2}) + α_{i}^{n} β_{i + 2}^{n} τ_{i + 2} ν_{i + 2} (γ_{i + 2}^{n} ν_{i + 2} + 1) \\ + α_{i}^{n} β_{i}^{n} ν_{i} τ_{i + 2} (ν_{i} γ_{i}^{n} + γ_{i + 2}^{n} ν_{i + 2}) + α_{i}^{n} β_{i + 1}^{n} δ_{i + 1} [γ_{i + 1}^{n} δ_{i + 2} ν_{i + 2} + δ_{i + 2} + μ_{i + 2}]} \\ < sup_{n \geq 1} \{α_{i}^{n} β_{i + 2}^{n} τ_{i + 2} ν_{i + 2} γ_{i + 2}^{n} + α_{i}^{n} β_{i}^{n} ν_{i} τ_{i + 2} γ_{i + 2}^{n} + 1 + α_{i}^{n} β_{i + 1}^{n} δ_{i + 1} γ_{i + 1}^{n} δ_{i + 2} + β_{i + 1}^{n}\}, \\ sup_{n \geq 1} {α_{i}^{n} β_{i}^{n} ν_{i} [γ_{i + 1}^{n} δ_{i + 1} δ_{i + 3} + γ_{i + 2}^{n} τ_{i + 2} (δ_{i + 3} + μ_{i + 3})] + α_{i}^{n} β_{i + 1}^{n} δ_{i + 1} [γ_{i + 1}^{n} ν_{i + 1} δ_{i + 3} \\ + γ_{i + 2}^{n} δ_{i + 2} δ_{i + 3} + γ_{i + 2}^{n} δ_{i + 2} μ_{i + 3} + τ_{i + 3} + γ_{i + 3}^{n} τ_{i + 3} ν_{i + 3}] \\ + α_{i}^{n} β_{i + 2}^{n} τ_{i + 2} [γ_{i + 2}^{n} ν_{i + 2} (δ_{i + 3} + μ_{i + 3}) + γ_{i + 3}^{n} δ_{i + 3} ν_{i + 3} + δ_{i + 3} + τ_{i + 3}]} \\ < sup_{n \geq 1} {1 + γ_{i + 3}^{n} τ_{i + 3} + α_{i}^{n} β_{i + 2}^{n} τ_{i + 2} γ_{i + 3}^{n} δ_{i + 3}}, \\ sup_{n \geq 1} {α_{i}^{n} β_{i}^{n} ν_{i} τ_{i + 2} τ_{i + 4} γ_{i + 2}^{n} + α_{i}^{n} β_{i + 1}^{n} δ_{i + 1} [δ_{i + 2} γ_{i + 2}^{n} τ_{i + 4} + γ_{i + 3}^{n} τ_{i + 3} (δ_{i + 4} + μ_{i + 4})] \\ + α_{i}^{n} β_{i + 2}^{n} [τ_{i + 2} τ_{i + 4} (γ_{i + 2}^{n} ν_{i + 2} + γ_{i + 4}^{n} ν_{i + 4} + 1) + γ_{i + 3}^{n} δ_{i + 3} (τ_{i + 3} δ_{i + 4} + τ_{i + 2} μ_{i + 4})]} \\ < sup_{n \geq 1} {1 + α_{i}^{n} β_{i + 2}^{n} τ_{i + 2} τ_{i + 4} γ_{i + 4}^{n}}, \\ sup_{n \geq 1} {α_{i}^{n} β_{i + 1}^{n} γ_{i + 3}^{n} δ_{i + 1} τ_{i + 3} τ_{i + 5} + α_{i}^{n} β_{i + 2}^{n} τ_{i + 2} [γ_{i + 3}^{n} δ_{i + 3} τ_{i + 5} + γ_{i + 4}^{n} τ_{i + 4} (δ_{i + 5} + μ_{i + 5})]} < 1, \\ sup_{n \geq 1} {α_{i}^{n} β_{i + 2}^{n} γ_{i + 4}^{n} τ_{i + 2} τ_{i + 4} τ_{i + 6}} < 1, \end{matrix}$

and

$lim_{n \to \infty} ∥ δ_{i}^{n} \lor (- δ_{i}^{n}) ∥ = 0, \forall, i = 1, 2, \dots, k .$

hold, then the sequences ${(x_{i}^{(n)}, v_{i}^{(n)})}$ generated by Algorithm 1 converge strongly to the solution $(x_{i}, v_{i})$ of system (5).

Proof of Convergence.

Theorem 1 guarantees that System (5) admits the solution $(x_{i}, v_{i})$ . Let us assume that $x^{*} = (x_{1}^{*}, x_{2}^{*}, \dots, x_{k}^{*})$ is a unique solution of SEOXORIP (5). Then, we have

(26) $\begin{matrix} x_{i}^{*} & = [α_{i}^{n} ℜ_{I, λ_{i}}^{S_{i} (h_{i} (\cdot), v_{i + 1})} (x_{i}^{*} - λ_{i} {P_{i} (C_{I, λ_{i}}^{S_{i} (h_{i} (\cdot), v_{i + 1})} (x_{i}^{*}), Y_{I, λ_{i + 1}}^{S_{i + 1} (h_{i + 1} (\cdot), v_{i + 2})} (x_{i + 1}^{*})) \\ + f_{i} {(x_{i})}^{*} \oplus g_{i + 1} (x_{i + 1}^{*}) - w_{i}}) + (1 - α_{i}^{n}) x_{i}^{*}] \\ = [β_{i}^{n} ℜ_{I, λ_{i}}^{S_{i} (h_{i} (\cdot), v_{i + 1})} [x_{i}^{*} - λ_{i} {P_{i} (C_{I, λ_{i}}^{S_{i} (h_{i} (\cdot), v_{i + 1})} (x_{i}^{*}), Y_{I, λ_{i + 1}}^{S_{i + 1} (h_{i + 1} (\cdot), v_{i + 2})} (x_{i + 1}^{*})) \\ + f_{i} (x_{i}^{*}) \oplus g_{i + 1} (x_{i + 1}^{*}) - w_{i}}] + (1 - β_{i}^{n}) x_{i}^{*}] \\ = [γ_{i}^{n} ℜ_{I, λ_{i}}^{S_{i} (h_{i} (\cdot), v_{i + 1})} [x_{i}^{*} - λ_{i} {P_{i} (C_{I, λ_{i}}^{S_{i} (h_{i} (\cdot), v_{i + 1})} (x_{i}^{*}), Y_{I, λ_{i + 1}}^{S_{i + 1} (h_{i + 1} (\cdot), v_{i + 2})} (x_{i + 1}^{*})) \\ + f_{i} (x_{i}^{*}) \oplus g_{i + 1} (x_{i + 1}^{*}) - w_{i}}] + (1 - γ_{i}^{n}) x_{i}^{*}] . \end{matrix}$

From (22), (26), and Proposition 3, we obtain

$\begin{matrix} ∥ x_{1}^{(n + 1)} \oplus x_{1}^{*} ∥ \\ = ∥ [α_{1}^{n} ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2}^{n})} (y_{1}^{n} - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2}^{n})} (y_{1}^{n}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3}^{n})} (y_{2}^{n})) \\ + f_{1} (y_{1}^{n}) \oplus g_{2} (y_{2}^{n}) - w_{1}}) + (1 - α_{1}^{n}) x_{1}^{n} + α_{1}^{n} δ_{1}^{n}] \\ \oplus [α_{1}^{n} ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}^{*} - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}^{*}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}^{*})) \\ + f_{1} (x_{1}^{*}) \oplus g_{2} (x_{2}^{*}) - w_{1}}) + (1 - α_{1}^{n}) x_{1}^{*}] \\ \leq α_{1}^{n} ∥ ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2}^{n})} (y_{1}^{n} - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2}^{n})} (y_{1}^{n}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3}^{n})} (y_{2}^{n})) \\ + f_{1} (y_{1}^{n}) \oplus g_{2} (y_{2}^{n}) - w_{1}}) \\ \oplus ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}^{*} - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}^{*}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}^{*})) \\ + f_{1} (x_{1}^{*}) \oplus g_{2} (x_{2}^{*}) - w_{1}}) ∥ + (1 - α_{1}^{n}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + ∥ α_{1}^{n} δ_{1}^{n} ∥ . \end{matrix}$

In similar way to (10), using (24), we have

(27) $\begin{matrix} ∥ x_{1}^{(n + 1)} \oplus x_{1}^{*} ∥ \\ \leq & α_{1}^{n} [Λ_{2} ∥ v_{2}^{n} \oplus v_{2} ∥ + L^{ℜ_{1}} {∥ y_{1}^{n} \oplus x_{1}^{*} ∥ + λ_{1} ∥ P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2}^{n})} (y_{1}^{n}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (y_{2}^{n})) \\ \oplus P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2}^{n})} (x_{1}^{*}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}^{*})) ∥ + λ_{1} ∥ f_{1} (y_{1}^{n}) \oplus g_{2} (y_{2}^{n}) \\ - f_{1} (x_{1}^{*}) \oplus g_{2} (x_{2}^{*}) ∥}] + (1 - α_{1}^{n}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + α_{1}^{n} ∥ δ_{1}^{n} \oplus 0 ∥ . \end{matrix}$

From Definitions 7 and 8 and (12), we have

(28) $\begin{matrix} ∥ P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2}^{n})} (y_{1}^{n}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (y_{2}^{n})) \oplus P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}^{*}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}^{*})) ∥ \\ \leq & ℘_{P_{1}}^{C_{1}} ∥C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2}^{n})} (y_{1}^{n}) \oplus C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}^{*})∥ + ℘_{P_{1}}^{Y_{1}} ∥Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3}^{n})} (y_{2}^{n}) \oplus Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}^{*})∥ \\ \leq & 3 ℘_{P_{1}}^{C_{1}} ∥ y_{1}^{n} \oplus x_{1}^{*} ∥ + ℘_{P_{1}}^{C_{1}} Λ_{2} (1 + \frac{1}{1 + n}) λ_{D_{N_{2}}} ∥ y_{2}^{n} \oplus x_{2}^{*} ∥ + \frac{℘_{P_{1}}^{Y_{1}}}{λ_{2}} (2 ∥ y_{2}^{n} \oplus x_{2}^{*} ∥ \\ + Λ_{3} (1 + \frac{1}{1 + n}) λ_{D_{N_{3}}} ∥ y_{3}^{n} \oplus x_{3}^{*} ∥) . \end{matrix}$

Since

v_{2}^{n} \in N_{2} (x_{2}^{n})

, according to Nadler [27], ∃

v_{2} \in N_{2} (x_{2}^{*})

such that

(29) $\begin{matrix} ∥ v_{2}^{n} \oplus v_{2} ∥ & \leq & (1 + \frac{1}{n + 1}) D (N_{2} (x_{2}^{n}), N_{2} (x_{2}^{*})) . \end{matrix}$

Using the

D

-Lipschitz continuity of

N_{2}

, we obtain

(30) $\begin{matrix} ∥ v_{2}^{n} \oplus v_{2} ∥ & \leq & (1 + \frac{1}{n + 1}) λ_{D_{N_{2}}} ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ . \end{matrix}$

Using (28), (30), and Proposition 3 in (27), we get

(31) $\begin{matrix} ∥ x_{1}^{(n + 1)} \oplus x_{1}^{*} ∥ \\ \leq & α_{1}^{n} [Λ_{2} (1 + \frac{1}{n + 1}) λ_{D_{N_{2}}} ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ + L^{ℜ_{1}} {∥ y_{1}^{n} \oplus x_{1}^{*} ∥ + λ_{1} (3 ℘_{P_{1}}^{C_{1}} ∥ y_{1}^{n} \oplus x_{1}^{*} ∥ \\ + ℘_{P_{1}}^{C_{1}} Λ_{2} (1 + \frac{1}{1 + n}) λ_{D_{N_{2}}} ∥ y_{2}^{n} \oplus x_{2}^{*} ∥ + \frac{℘_{P_{1}}^{Y_{1}}}{λ_{2}} (2 ∥ y_{2}^{n} \oplus x_{2}^{*} ∥ \\ + Λ_{3} (1 + \frac{1}{1 + n}) λ_{D_{N_{3}}} ∥ y_{3}^{n} \oplus x_{3}^{*} ∥) + λ_{1} (℘^{f_{1}} ∥ y_{1}^{n} \oplus x_{1}^{*} ∥ + ℘^{g_{2}} ∥ y_{2}^{n} \oplus x_{2}^{*} ∥)}] \\ + (1 - α_{1}^{n}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + α_{1}^{n} ∥ δ_{1}^{n} \oplus 0 ∥ . \end{matrix}$

(v i i)

of Proposition 1, we have

(32) $\begin{matrix} ∥ x_{1}^{(n + 1)} - x_{1}^{*} ∥ & = & ∥ x_{1}^{(n + 1)} \oplus x_{1}^{*} ∥ \\ = & α_{1}^{n} L^{ℜ_{1}} (1 + λ_{1} ℘^{f_{1}}) ∥ y_{1}^{n} - x_{1}^{*} ∥ + (1 - α_{1}^{n}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ \\ + α_{1}^{n} Λ_{2} (1 + \frac{1}{1 + n}) λ_{D_{N_{2}}} L^{ℜ_{1}} (℘_{P_{1}}^{C_{1}} + \frac{2 ℘_{P_{1}}^{Y_{1}}}{λ_{2}} + λ_{1} ℘^{g_{2}}) ∥ y_{2}^{n} \oplus x_{2}^{*} ∥ \\ + α_{1}^{n} Λ_{2} (1 + \frac{1}{1 + n}) λ_{D_{N_{2}}} ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ \\ + α_{1}^{n} \frac{L^{ℜ_{1}} ℘_{P_{1}}^{Y_{1}}}{λ_{2}} Λ_{3} (1 + \frac{1}{1 + n}) λ_{D_{N_{3}}} ∥ y_{3}^{n} \oplus x_{3}^{*} ∥ + α_{1}^{n} ∥ δ_{1}^{n} \oplus 0 ∥ . \end{matrix}$

Let

$ν_{1} = L^{ℜ_{1}} (1 + λ_{1} ℘^{f_{1}}), δ_{2} = Λ_{2} (1 + \frac{1}{1 + n}) λ_{D_{N_{2}}} L^{ℜ_{1}} (℘_{P_{1}}^{C_{1}} + \frac{2 ℘_{P_{1}}^{Y_{1}}}{λ_{2}} + λ_{1} ℘^{g_{2}}),$

$μ_{2} = Λ_{2} (1 + \frac{1}{1 + n}) λ_{D_{N_{2}}} a n d τ_{3} = \frac{L^{ℜ_{1}} ℘_{P_{1}}^{Y_{1}}}{λ_{2}} Λ_{3} (1 + \frac{1}{1 + n}) λ_{D_{N_{3}}} .$

(33) $\begin{matrix} ∥ x_{1}^{(n + 1)} \oplus x_{1}^{*} ∥ & = & α_{1}^{n} ν_{1} ∥ y_{1}^{n} \oplus x_{1}^{*} ∥ + α_{1}^{n} δ_{2} ∥ y_{2}^{n} \oplus x_{2}^{*} ∥ + α_{1}^{n} μ_{2} ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ \\ + (1 - α_{1}^{n}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + α_{1}^{n} τ_{3} ∥ y_{3}^{n} \oplus x_{3}^{*} ∥ + α_{1}^{n} ∥ δ_{1}^{n} \oplus 0 ∥ . \end{matrix}$

Similarly, we calculate

$\begin{matrix} ∥ y_{1}^{(n)} \oplus x_{1}^{*} ∥ \\ \leq ∥ (β_{1}^{n} ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} [z_{1}^{n} - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (z_{1}^{n}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (z_{2}^{n})) \\ + f_{1} (z_{1}^{n}) \oplus g_{2} (z_{2}^{n}) - w_{1}}] + (1 - β_{1}^{n}) x_{1}^{(n)} + β_{1}^{n} δ_{2}^{n}) \oplus (β_{1}^{n} ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} [x_{1}^{*} \\ - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}^{*}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}^{*})) + f_{1} (x_{1}^{*}) \oplus g_{2} (x_{2}^{*}) - w_{1}}] + (1 - β_{1}^{n}) x_{1}^{*}) ∥ . \end{matrix}$

(34) $\begin{matrix} ∥ y_{1}^{(n)} \oplus x_{1}^{*} ∥ \\ \leq β_{1}^{n} [Λ_{2} ∥ v_{2}^{n} \oplus v_{2} ∥ + L^{ℜ_{1}} {∥ z_{1}^{n} \oplus x_{1}^{*} ∥ + λ_{1} ∥ P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2}^{n})} (z_{1}^{n}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (z_{2}^{n})) \\ \oplus P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2}^{n})} (x_{1}^{*}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}^{*})) ∥ + λ_{1} ∥ f_{1} (z_{1}^{n}) \oplus g_{2} (z_{2}^{n}) \oplus f_{1} (x_{1}^{*}) \oplus g_{2} (x_{2}^{*}) ∥}] \\ + (1 - β_{1}^{n}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + β_{1}^{n} ∥ δ_{2}^{n} \oplus 0 ∥ \\ \leq β_{1}^{n} [Λ_{2} (1 + \frac{1}{n + 1}) λ_{D_{N_{2}}} ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ + L^{ℜ_{1}} {∥ z_{1}^{n} \oplus x_{1}^{*} ∥ + λ_{1} (3 ℘_{P_{1}}^{C_{1}} ∥ z_{1}^{n} \oplus x_{1}^{*} ∥ \\ + ℘_{P_{1}}^{C_{1}} Λ_{2} (1 + \frac{1}{1 + n}) λ_{D_{N_{2}}} ∥ z_{2}^{n} \oplus x_{2}^{*} ∥ + \frac{℘_{P_{1}}^{Y_{1}}}{λ_{2}} (2 ∥ z_{2}^{n} \oplus x_{2}^{*} ∥ \\ + Λ_{3} (1 + \frac{1}{1 + n}) λ_{D_{N_{3}}} ∥ z_{3}^{n} \oplus x_{3}^{*} ∥) + λ_{1} (℘^{f_{1}} ∥ z_{1}^{n} \oplus x_{1}^{*} ∥ + ℘^{g_{2}} ∥ y_{2}^{n} \oplus x_{2}^{*} ∥)}]] \\ + (1 - β_{1}^{n}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + β_{1}^{n} ∥ δ_{2}^{n} \oplus 0 ∥ \\ = β_{1}^{n} L^{ℜ_{1}} (1 + λ_{1} ℘^{f_{1}}) ∥ z_{1}^{n} \oplus x_{1}^{*} ∥ + (1 - β_{1}^{n}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ \\ + β_{1}^{n} Λ_{2} (1 + \frac{1}{1 + n}) λ_{D_{N_{2}}} L^{ℜ_{1}} (℘_{P_{1}}^{C_{1}} + \frac{2 ℘_{P_{1}}^{Y_{1}}}{λ_{2}} + λ_{1} ℘^{g_{2}}) ∥ z_{2}^{n} \oplus x_{2}^{*} ∥ \\ + β_{1}^{n} Λ_{2} (1 + \frac{1}{1 + n}) λ_{D_{N_{2}}} ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ \\ + β_{1}^{n} \frac{L^{ℜ_{1}} ℘_{P_{1}}^{Y_{1}}}{λ_{2}} Λ_{3} (1 + \frac{1}{1 + n}) λ_{D_{N_{3}}} ∥ z_{3}^{n} \oplus x_{3}^{*} ∥ + β_{1}^{n} ∥ δ_{2}^{n} \oplus 0 ∥ \\ = β_{1}^{n} ν_{1} ∥ z_{1}^{n} \oplus x_{1}^{*} ∥ + β_{1}^{n} δ_{2} ∥ z_{2}^{n} \oplus x_{2}^{*} ∥ + β_{1}^{n} μ_{2} ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ \\ + (1 - β_{1}^{n}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + β_{1}^{n} τ_{3} ∥ z_{3}^{n} \oplus x_{3}^{*} ∥ + β_{1}^{n} ∥ δ_{2}^{n} \oplus 0 ∥ . \end{matrix}$

In a similar way, by using the definition of

z_{1}^{n}

in Algorithm 1, we obtain

$\begin{matrix} ∥ z_{1}^{(n)} \oplus x_{1}^{*} ∥ \\ \leq ∥ (γ_{1}^{n} ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} [x_{1}^{n} - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}^{n}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}^{n})) \\ + f_{1} (x_{1}^{n}) \oplus g_{2} (x_{2}^{n}) - w_{1}}] + (1 - γ_{1}^{n}) x_{1}^{(n)} + γ_{1}^{n} δ_{2}^{n}) \oplus (γ_{1}^{n} ℜ_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} [x_{1}^{*} \\ - λ_{1} {P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2})} (x_{1}^{*}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}^{*})) + f_{1} (x_{1}^{*}) \oplus g_{2} (x_{2}^{*}) - w_{1}}] + (1 - γ_{1}^{n}) x_{1}^{*}) ∥ \\ \leq γ_{1}^{n} [Λ_{2} ∥ v_{2}^{n} \oplus v_{2} ∥ + L^{ℜ_{1}} {∥ x_{1}^{n} \oplus x_{1} ∥ + λ_{1} ∥ P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2}^{n})} (x_{1}^{n}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}^{n})) \\ \oplus P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (\cdot), v_{2}^{n})} (x_{1}^{*}), Y_{I, λ_{2}}^{S_{2} (h_{2} (\cdot), v_{3})} (x_{2}^{*})) ∥ + λ_{1} ∥ f_{1} (x_{1}^{n}) \oplus g_{2} (x_{2}^{n}) - f_{1} (x_{1}^{*}) \oplus g_{2} (x_{2}^{*}) ∥}] \\ + (1 - γ_{1}^{n}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + γ_{1}^{n} ∥ δ_{3}^{n} \oplus 0 ∥ \\ \leq γ_{1}^{n} [Λ_{2} (1 + \frac{1}{n + 1}) λ_{D_{N_{2}}} ∥ x_{2}^{n} \oplus x_{2} ∥ + L^{ℜ_{1}} {∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + λ_{1} (3 ℘_{P_{1}}^{C_{1}} ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ \\ + ℘_{P_{1}}^{C_{1}} Λ_{2} (1 + \frac{1}{1 + n}) λ_{D_{N_{2}}} ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ + \frac{℘_{P_{1}}^{Y_{1}}}{λ_{2}} (2 ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ \\ + Λ_{3} (1 + \frac{1}{1 + n}) λ_{D_{N_{3}}} ∥ x_{3}^{n} \oplus x_{3}^{*} ∥) + λ_{1} (℘^{f_{1}} ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + ℘^{g_{2}} ∥ x_{2}^{n} \oplus x_{2}^{*} ∥)}]] \\ + (1 - γ_{1}^{n}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + γ_{1}^{n} ∥ δ_{3}^{n} \oplus 0 ∥ . \end{matrix}$

(35) $\begin{matrix} ∥ z_{1}^{(n)} \oplus x_{1}^{*} ∥ \\ = γ_{1}^{n} L^{ℜ_{1}} (1 + λ_{1} ℘^{f_{1}}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + (1 - γ_{1}^{n}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ \\ + γ_{1}^{n} Λ_{2} (1 + \frac{1}{1 + n}) λ_{D_{N_{2}}} L^{ℜ_{1}} (℘_{P_{1}}^{C_{1}} + \frac{2 ℘_{P_{1}}^{Y_{1}}}{λ_{2}} + λ_{1} ℘^{g_{2}}) ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ \\ + γ_{1}^{n} Λ_{2} (1 + \frac{1}{1 + n}) λ_{D_{N_{2}}} ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ \\ + γ_{1}^{n} \frac{L^{ℜ_{1}} ℘_{P_{1}}^{Y_{1}}}{λ_{2}} Λ_{3} (1 + \frac{1}{1 + n}) λ_{D_{N_{3}}} ∥ x_{3}^{n} \oplus x_{3}^{*} ∥ + γ_{1}^{n} ∥ δ_{3}^{n} \oplus 0 ∥ \\ = (γ_{1}^{n} ν_{1} + (1 - γ_{1}^{n})) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + (γ_{1}^{n} δ_{2} + γ_{1}^{n} μ_{2}) ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ \\ + γ_{1}^{n} τ_{3} ∥ x_{3}^{n} \oplus x_{3}^{*} ∥ + γ_{1}^{n} ∥ δ_{3}^{n} \oplus 0 ∥ . \end{matrix}$

Using (34) and (35) in (33), we have

(36) $\begin{matrix} ∥ x_{1}^{n + 1} \oplus x_{1}^{*} ∥ \\ \leq α_{1}^{n} ν_{1} {β_{1}^{n} ν_{1} ∥ z_{1}^{n} \oplus x_{1}^{*} ∥ + β_{1}^{n} δ_{2} ∥ z_{2}^{n} \oplus x_{2}^{*} ∥ + β_{1}^{n} μ_{2} ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ \\ + (1 - β_{1}^{n}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + β_{1}^{n} τ_{3} ∥ z_{3}^{n} \oplus x_{3}^{*} ∥ + β_{1}^{n} ∥ δ_{2}^{n} \oplus 0 ∥} \\ + α_{1}^{n} δ_{2} {β_{2}^{n} ν_{2} ∥ z_{2}^{n} \oplus x_{2}^{*} ∥ + β_{2}^{n} δ_{3} ∥ z_{3}^{n} \oplus x_{3}^{*} ∥ + β_{2}^{n} μ_{3} ∥ x_{3}^{n} \oplus x_{3}^{*} ∥ \\ + (1 - β_{2}^{n}) ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ + β_{2}^{n} τ_{4} ∥ z_{4}^{n} \oplus x_{4}^{*} ∥ + β_{2}^{n} ∥ δ_{3}^{n} \oplus 0 ∥} \\ + α_{1}^{n} μ_{2} ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ + (1 - α_{1}^{n}) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + α_{1}^{n} ∥ δ_{1}^{n} \oplus 0 ∥ \\ + α_{1}^{n} τ_{3} {β_{3}^{n} ν_{3} ∥ z_{3}^{n} \oplus x_{3}^{*} ∥ + β_{3}^{n} δ_{4} ∥ z_{4}^{n} \oplus x_{4}^{*} ∥ + β_{3}^{n} μ_{4} ∥ x_{4}^{n} \oplus x_{4}^{*} ∥ \\ + (1 - β_{3}^{n}) ∥ x_{3}^{n} \oplus x_{3}^{*} ∥ + β_{3}^{n} τ_{5} ∥ z_{5}^{n} \oplus x_{5}^{*} ∥ + β_{3}^{n} ∥ δ_{4}^{n} \oplus 0 ∥} \\ \leq α_{1}^{n} β_{1}^{n} ν_{1}^{2} ∥ z_{1}^{n} \oplus x_{1}^{*} ∥ + (α_{1}^{n} β_{1}^{n} ν_{1} δ_{2} + α_{1}^{n} ν_{2} β_{2}^{n} δ_{2}) ∥ z_{2}^{n} \oplus x_{2}^{*} ∥ \\ + (α_{1}^{n} β_{1}^{n} ν_{1} τ_{3} + α_{1}^{n} β_{2}^{n} δ_{2} δ_{3} + α_{1}^{n} β_{3}^{n} τ_{3} ν_{3}) ∥ z_{3}^{n} \oplus z_{3}^{*} ∥ + (α_{1}^{n} β_{2}^{n} δ_{2} τ_{4} + α_{1}^{n} β_{3}^{n} τ_{3} δ_{4}) ∥ z_{4}^{n} \oplus x_{4}^{*} ∥ \\ + α_{1}^{n} β_{3}^{n} τ_{3} τ_{5} ∥ z_{5} \oplus x_{5}^{*} ∥ + ((1 - α_{1}^{n}) + α_{1}^{n} ν_{1} (1 - β_{1}^{n})) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ \\ + (α_{1}^{n} β_{1}^{n} ν_{1} μ_{2} + α_{1}^{n} (1 - β_{2}^{n}) δ_{2} + α_{1}^{n} μ_{2}) ∥ x_{2}^{*} \oplus x_{2} ∥ + ((1 - β_{3}^{n}) + α_{1}^{n} β_{2}^{n} δ_{2} μ_{3}) ∥ x_{3} \oplus x_{3}^{*} ∥ \\ + α_{1}^{n} β_{3}^{n} τ_{3} τ_{4} ∥ x_{4}^{n} \oplus x_{4}^{*} ∥ + α_{1}^{n} ν_{1} β_{1}^{n} ∥ δ_{2}^{n} \oplus 0 ∥ + α_{1}^{n} δ_{2} β_{2}^{n} ∥ δ_{3}^{n} \oplus 0 ∥ + α_{1}^{n} τ_{3} β_{3}^{n} ∥ δ_{4}^{n} \oplus 0 ∥ \\ + α_{1}^{n} ∥ δ_{1}^{n} \oplus 0 ∥ \\ \leq α_{1}^{n} β_{1}^{n} ν_{1}^{2} {(γ_{1}^{n} ν_{1} + (1 - γ_{1}^{n})) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + (γ_{1}^{n} δ_{2} + γ_{1}^{n} μ_{2}) ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ + γ_{1}^{n} τ_{3} ∥ x_{3}^{n} \oplus x_{3}^{*} ∥ \\ + γ_{1}^{n} ∥ δ_{3}^{n} \oplus 0 ∥} + (α_{1}^{n} β_{1}^{n} ν_{1} δ_{2} + α_{1}^{n} ν_{2} β_{2}^{n} δ_{2}) {(γ_{2}^{n} ν_{2} + (1 - γ_{2}^{n})) ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ \\ + (γ_{2}^{n} δ_{3} + γ_{2}^{n} μ_{3}) ∥ x_{3}^{n} \oplus x_{3}^{*} ∥ + γ_{2}^{n} τ_{4} ∥ x_{4}^{n} \oplus x_{4}^{*} ∥ + γ_{2}^{n} ∥ δ_{4}^{n} \oplus 0 ∥} + (α_{1}^{n} β_{1}^{n} ν_{1} τ_{3} + α_{1}^{n} β_{2}^{n} δ_{2} δ_{3} \\ + α_{1}^{n} β_{3}^{n} τ_{3} ν_{3}) {(γ_{3}^{n} ν_{3} + (1 - γ_{3}^{n})) ∥ x_{3}^{n} \oplus x_{3}^{*} ∥ + (γ_{3}^{n} δ_{4} + γ_{3}^{n} μ_{4}) ∥ x_{4}^{n} \oplus x_{4}^{*} ∥ + γ_{3}^{n} τ_{5} ∥ x_{5}^{n} \oplus x_{5}^{*} ∥ \\ + γ_{3}^{n} ∥ δ_{4}^{n} \oplus 0 ∥} + (α_{1}^{n} β_{2}^{n} δ_{2} τ_{4} + α_{1}^{n} β_{3}^{n} τ_{3} δ_{4}) {(γ_{4}^{n} ν_{4} + (1 - γ_{4}^{n})) ∥ x_{4}^{n} \oplus x_{4}^{*} ∥ \\ + (γ_{4}^{n} δ_{5} + γ_{4}^{n} μ_{5}) ∥ x_{5}^{n} \oplus x_{5}^{*} ∥ + γ_{4}^{n} τ_{6} ∥ x_{6}^{n} \oplus x_{6}^{*} ∥ + γ_{4}^{n} ∥ δ_{5}^{n} \oplus 0 ∥} + α_{1}^{n} β_{3}^{n} τ_{3} τ_{5} {(γ_{5}^{n} ν_{5} \\ + (1 - γ_{5}^{n})) ∥ x_{5}^{n} \oplus x_{5}^{*} ∥ + (γ_{5}^{n} δ_{6} + γ_{5}^{n} μ_{6}) ∥ x_{6}^{n} \oplus x_{6}^{*} ∥ + γ_{5}^{n} τ_{7} ∥ x_{7}^{n} \oplus x_{7}^{*} ∥ + γ_{5}^{n} ∥ δ_{7}^{n} \oplus 0 ∥} \\ + ((1 - α_{1}^{n}) + α_{1}^{n} ν_{1} (1 - β_{1}^{n})) ∥ x_{1}^{n} \oplus x_{1}^{*} ∥ + (α_{1}^{n} β_{1}^{n} ν_{1} μ_{2} + α_{1}^{n} (1 - β_{2}^{n}) δ_{2} + α_{1}^{n} μ_{2}) ∥ x_{2}^{n} \oplus x_{2}^{*} ∥ \\ + ((1 - β_{3}^{n}) + α_{1}^{n} β_{2}^{n} δ_{2} μ_{3}) ∥ x_{3}^{n} \oplus x_{3}^{*} ∥ + α_{1}^{n} β_{3}^{n} τ_{3} τ_{4} ∥ x_{4}^{n} \oplus x_{4} ∥ + α_{1}^{n} ν_{1} β_{1}^{n} ∥ δ_{2}^{n} \oplus 0 ∥ \\ + α_{1}^{n} δ_{2} β_{2}^{n} ∥ δ_{3}^{n} \oplus 0 ∥ + α_{1}^{n} τ_{3} β_{3}^{n} ∥ δ_{4}^{n} \oplus 0 ∥ + α_{1}^{n} ∥ δ_{1}^{n} \oplus 0 ∥ . \end{matrix}$

After simplification and by Proposition 1, we obtain

(37) $\begin{matrix} ∥ x_{1}^{n + 1} - x_{1}^{*} ∥ \\ \leq [α_{1}^{n} β_{1}^{n} ν_{1}^{2} {γ_{1}^{n} (ν_{1} - 1) + 1} + α_{1}^{n} {ν_{1} (1 - β_{1}^{n}) - 1} + 1] ∥ x_{1}^{n} - x_{1}^{*} ∥ \\ + [α_{1}^{n} β_{1}^{n} ν_{1} (ν_{1} γ_{1}^{n} + 1) (δ_{2} + μ_{2}) + α_{1}^{n} β_{1}^{n} ν_{1} δ_{2} γ_{2}^{n} (ν_{2} - 1) \\ + α_{1}^{n} ν_{2} β_{2}^{n} δ_{2} (ν_{2} γ_{2}^{n} - γ_{2}^{n} + 1) + α_{1}^{n} δ_{2} (1 - β_{2}^{n})] ∥ x_{2}^{n} - x_{2}^{*} ∥ \\ + [α_{1}^{n} γ_{2}^{n} δ_{2} (β_{1}^{n} ν_{1} + β_{2}^{n} ν_{2}) (δ_{3} + μ_{3}) + α_{1}^{n} β_{3}^{n} τ_{3} ν_{3} (γ_{3}^{n} ν_{3} + 1 - γ_{3}^{n}) + α_{1}^{n} β_{1}^{n} ν_{1} τ_{3} (ν_{1} γ_{1}^{n} \\ + γ_{3}^{n} ν_{3} - γ_{3}^{n} + 1) + α_{1}^{n} β_{2}^{n} δ_{2} [γ_{3}^{n} δ_{3} (ν_{3} - 1) + δ_{3} + μ_{3}] - β_{3}^{n} + 1] ∥ x_{3}^{n} - x_{3}^{*} ∥ \\ + [α_{1}^{n} β_{1}^{n} ν_{1} [γ_{2}^{n} δ_{2} δ_{4} + γ_{3}^{n} τ_{3} (δ_{4} + μ_{4})] + α_{1}^{n} β_{2}^{n} δ_{2} [γ_{2}^{n} ν_{2} δ_{4} + γ_{3}^{n} δ_{3} δ_{4} + γ_{3}^{n} δ_{3} μ_{4} \\ + τ_{4} + γ_{4}^{n} τ_{4} (ν_{4} - 1)] + α_{1}^{n} β_{3}^{n} τ_{3} [γ_{3}^{n} ν_{3} (δ_{4} + μ_{4}) + γ_{4}^{n} δ_{4} (ν_{4} - 1) + δ_{4} + τ_{4}]] ∥ x_{4}^{n} - x_{4}^{*} ∥ \\ + [α_{1}^{n} β_{1}^{n} ν_{1} τ_{3} τ_{5} γ_{3}^{n} + α_{1}^{n} β_{2}^{n} δ_{2} [δ_{3} γ_{3}^{n} τ_{5} + γ_{4}^{n} τ_{4} (δ_{5} + μ_{5})] + α_{1}^{n} β_{3}^{n} [τ_{3} τ_{5} (γ_{3}^{n} ν_{3} + γ_{5}^{n} ν_{5} - γ_{5}^{n} + 1) \\ + γ_{4}^{n} δ_{4} (τ_{4} δ_{5} + τ_{3} μ_{5})]] ∥ x_{5}^{n} - x_{5}^{*} ∥ + [α_{1}^{n} β_{2}^{n} γ_{4}^{n} δ_{2} τ_{4} τ_{6} + α_{1}^{n} β_{3}^{n} τ_{3} [γ_{4}^{n} δ_{4} τ_{6} \\ + γ_{5}^{n} τ_{5} (δ_{6} + μ_{6})]] ∥ x_{6}^{n} - x_{6}^{*} ∥ + α_{1}^{n} β_{3}^{n} γ_{5}^{n} τ_{3} τ_{5} τ_{7} ∥ x_{7}^{n} - x_{7}^{*} ∥ + α_{1}^{n} β_{1}^{n} ν_{1}^{2} γ_{1}^{n} ∥ δ_{3}^{n} \oplus 0 ∥ \\ + (α_{1}^{n} β_{1}^{n} ν_{1} δ_{2} + α_{1}^{n} ν_{2} β_{2}^{n} δ_{2}) γ_{2}^{n} ∥ δ_{4}^{n} \oplus 0 ∥ + (α_{1}^{n} β_{1}^{n} ν_{1} τ_{3} + α_{1}^{n} β_{2}^{n} δ_{2} δ_{3} + α_{1}^{n} β_{3}^{n} τ_{3} ν_{3}) γ_{3}^{n} ∥ δ_{4}^{n} \oplus 0 ∥ \\ + (α_{1}^{n} β_{2}^{n} δ_{2} τ_{4} + α_{1}^{n} β_{3}^{n} τ_{3} δ_{4}) γ_{4}^{n} ∥ δ_{5}^{n} \oplus 0 ∥ + α_{1}^{n} β_{3}^{n} τ_{3} τ_{5} γ_{5}^{n} ∥ δ_{7}^{n} \oplus 0 ∥ + α_{1}^{n} ν_{1} β_{1}^{n} ∥ δ_{2}^{n} \oplus 0 ∥ \\ + α_{1}^{n} δ_{2} β_{2}^{n} ∥ δ_{3}^{n} \oplus 0 ∥ + α_{1}^{n} τ_{3} β_{3}^{n} ∥ δ_{4}^{n} \oplus 0 ∥ + α_{1}^{n} ∥ δ_{1}^{n} \oplus 0 ∥ . \end{matrix}$

By applying the same logic as above, we have

(38) $\begin{matrix} ∥ x_{2}^{n + 1} - x_{2}^{*} ∥ \\ \leq [α_{2}^{n} β_{2}^{n} ν_{2}^{2} {γ_{2}^{n} (ν_{2} - 1) + 1} + α_{2}^{n} {ν_{2} (1 - β_{2}^{n}) - 1} + 1] ∥ x_{2}^{n} - x_{2}^{*} ∥ \\ + [α_{2}^{n} β_{2}^{n} ν_{2} (ν_{2} γ_{2}^{n} + 1) (δ_{3} + μ_{3}) + α_{2}^{n} β_{2}^{n} ν_{2} δ_{3} γ_{3}^{n} (ν_{3} - 1) + α_{2}^{n} ν_{3} β_{3}^{n} δ_{3} (ν_{3} γ_{3}^{n} - γ_{3}^{n} + 1) \\ + α_{2}^{n} δ_{3} (1 - β_{3}^{n})] ∥ x_{3}^{n} - x_{3}^{*} ∥ + [α_{2}^{n} γ_{3}^{n} δ_{3} (β_{2}^{n} ν_{2} + β_{3}^{n} ν_{3}) (δ_{4} + μ_{4}) \\ + α_{2}^{n} β_{4}^{n} τ_{4} ν_{4} (γ_{4}^{n} ν_{4} + 1 - γ_{4}^{n}) + α_{2}^{n} β_{2}^{n} ν_{2} τ_{4} (ν_{2} γ_{2}^{n} + γ_{4}^{n} ν_{4} - γ_{4}^{n} + 1) \\ + α_{2}^{n} β_{3}^{n} δ_{3} [γ_{4}^{n} δ_{4} (ν_{4} - 1) + δ_{4} + μ_{4}] - β_{4}^{n} + 1] ∥ x_{4}^{n} - x_{4}^{*} ∥ + [α_{2}^{n} β_{2}^{n} ν_{2} [γ_{3}^{n} δ_{3} δ_{5} \\ + γ_{4}^{n} τ_{4} (δ_{5} + μ_{5})] + α_{2}^{n} β_{3}^{n} δ_{3} [γ_{3}^{n} ν_{3} δ_{5} + γ_{4}^{n} δ_{4} δ_{5} + γ_{4}^{n} δ_{4} μ_{5} + τ_{5} + γ_{5}^{n} τ_{5} (ν_{5} - 1)] \\ + α_{2}^{n} β_{4}^{n} τ_{4} [γ_{4}^{n} ν_{4} (δ_{5} + μ_{5}) + γ_{5}^{n} δ_{5} (ν_{5} - 1) + δ_{5} + τ_{5}]] ∥ x_{5}^{n} - x_{5}^{*} ∥ + [α_{2}^{n} β_{2}^{n} ν_{2} τ_{4} τ_{6} γ_{4}^{n} \\ + α_{2}^{n} β_{3}^{n} δ_{3} [δ_{4} γ_{4}^{n} τ_{6} + γ_{5}^{n} τ_{5} (δ_{6} + μ_{6})] + α_{2}^{n} β_{4}^{n} [τ_{4} τ_{6} (γ_{4}^{n} ν_{4} + γ_{6}^{n} ν_{6} - γ_{6}^{n} + 1) + γ_{5}^{n} δ_{5} (τ_{5} δ_{6} \\ + τ_{4} μ_{6})]] ∥ x_{6}^{n} - x_{6}^{*} ∥ + [α_{2}^{n} β_{3}^{n} γ_{5}^{n} δ_{3} τ_{5} τ_{7} + α_{2}^{n} β_{4}^{n} τ_{4} [γ_{5}^{n} δ_{5} τ_{7} + γ_{6}^{n} τ_{6} (δ_{7} + μ_{7})]] ∥ x_{7}^{n} - x_{7}^{*} ∥ \\ + α_{2}^{n} β_{4}^{n} γ_{6}^{n} τ_{4} τ_{6} τ_{8} ∥ x_{8}^{n} - x_{8}^{*} ∥ + α_{2}^{n} β_{2}^{n} ν_{2}^{2} γ_{2}^{n} ∥ δ_{4}^{n} \oplus 0 ∥ + (α_{2}^{n} β_{2}^{n} ν_{2} δ_{3} + α_{2}^{n} ν_{3} β_{3}^{n} δ_{3}) γ_{3}^{n} ∥ δ_{5}^{n} \oplus 0 ∥ \\ + (α_{2}^{n} β_{2}^{n} ν_{2} τ_{4} + α_{2}^{n} β_{3}^{n} δ_{3} δ_{4} + α_{2}^{n} β_{4}^{n} τ_{4} ν_{4}) γ_{4}^{n} ∥ δ_{5}^{n} \oplus 0 ∥ + (α_{2}^{n} β_{3}^{n} δ_{3} τ_{5} + α_{2}^{n} β_{4}^{n} τ_{4} δ_{5}) γ_{5}^{n} ∥ δ_{6}^{n} \oplus 0 ∥ \\ + α_{2}^{n} β_{4}^{n} τ_{4} τ_{6} γ_{6}^{n} ∥ δ_{8}^{n} \oplus 0 ∥ + α_{2}^{n} ν_{2} β_{2}^{n} ∥ δ_{3}^{n} \oplus 0 ∥ + α_{2}^{n} δ_{3} β_{3}^{n} ∥ δ_{4}^{n} \oplus 0 ∥ \\ + α_{2}^{n} τ_{4} β_{4}^{n} ∥ δ_{5}^{n} \oplus 0 ∥ + α_{2}^{n} ∥ δ_{2}^{n} \oplus 0 ∥ . \end{matrix}$

Continuing in a similar way, we have

(39) $\begin{matrix} ∥ x_{k}^{n + 1} - x_{k}^{*} ∥ \\ \leq [α_{k}^{n} β_{k}^{n} ν_{k}^{2} {γ_{k}^{n} (ν_{k} - 1) + 1} + α_{k}^{n} {ν_{k} (1 - β_{k}^{n}) - 1} + 1] ∥ x_{k}^{n} - x_{k}^{*} ∥ \\ + [α_{k}^{n} β_{k}^{n} ν_{k} (ν_{k} γ_{k}^{n} + 1) (δ_{1} + μ_{1}) + α_{k}^{n} β_{k}^{n} ν_{k} δ_{1} γ_{1}^{n} (ν_{1} - 1) + α_{k}^{n} ν_{1} β_{1}^{n} δ_{1} (ν_{1} γ_{1}^{n} - γ_{1}^{n} + 1) \\ + α_{k}^{n} δ_{1} (1 - β_{1}^{n})] ∥ x_{1}^{n} - x_{1}^{*} ∥ + [α_{k}^{n} γ_{1}^{n} δ_{1} (β_{k}^{n} ν_{k} + β_{1}^{n} ν_{1}) (δ_{2} + μ_{2}) \\ + α_{k}^{n} β_{2}^{n} τ_{2} ν_{2} (γ_{2}^{n} ν_{2} + 1 - γ_{2}^{n}) + α_{k}^{n} β_{k}^{n} ν_{k} τ_{2} (ν_{k} γ_{k}^{n} + γ_{2}^{n} ν_{2} - γ_{2}^{n} + 1) \\ + α_{k}^{n} β_{1}^{n} δ_{1} [γ_{2}^{n} δ_{2} (ν_{2} - 1) + δ_{2} + μ_{2}] - β_{2}^{n} + 1] ∥ x_{2}^{n} - x_{2}^{*} ∥ \\ + [α_{k}^{n} β_{k}^{n} ν_{k} [γ_{1}^{n} δ_{1} δ_{3} + γ_{2}^{n} τ_{2} (δ_{3} + μ_{3})] + α_{k}^{n} β_{1}^{n} δ_{1} [γ_{1}^{n} ν_{1} δ_{3} + γ_{2}^{n} δ_{2} δ_{3} + γ_{2}^{n} δ_{2} μ_{3} \\ + τ_{3} + γ_{3}^{n} τ_{3} (ν_{3} - 1)] + α_{k}^{n} β_{2}^{n} τ_{2} [γ_{2}^{n} ν_{2} (δ_{3} + μ_{3}) + γ_{3}^{n} δ_{3} (ν_{3} - 1) + δ_{3} + τ_{3}]] ∥ x_{3}^{n} - x_{3}^{*} ∥ \\ + [α_{k}^{n} β_{k}^{n} ν_{k} τ_{2} τ_{4} γ_{2}^{n} + α_{k}^{n} β_{1}^{n} δ_{1} [δ_{2} γ_{2}^{n} τ_{4} + γ_{3}^{n} τ_{3} (δ_{4} + μ_{4})] \\ + α_{k}^{n} β_{2}^{n} [τ_{2} τ_{4} (γ_{2}^{n} ν_{2} + γ_{4}^{n} ν_{4} - γ_{4}^{n} + 1) + γ_{3}^{n} δ_{3} (τ_{3} δ_{4} + τ_{2} μ_{4})]] ∥ x_{4}^{n} - x_{4}^{*} ∥ \\ + [α_{k}^{n} β_{1}^{n} γ_{3}^{n} δ_{1} τ_{3} τ_{5} + α_{k}^{n} β_{2}^{n} τ_{2} [γ_{3}^{n} δ_{3} τ_{5} + γ_{4}^{n} τ_{4} (δ_{5} + μ_{5})]] ∥ x_{5}^{n} - x_{5}^{*} ∥ \\ + α_{k}^{n} β_{2}^{n} γ_{4}^{n} τ_{2} τ_{4} τ_{6} ∥ x_{6}^{n} - x_{6}^{*} ∥ + α_{k}^{n} β_{k}^{n} ν_{k}^{2} γ_{k}^{n} ∥ δ_{2}^{n} \oplus 0 ∥ + (α_{k}^{n} β_{k}^{n} ν_{k} δ_{1} + α_{k}^{n} ν_{1} β_{1}^{n} δ_{1}) γ_{1}^{n} ∥ δ_{3}^{n} \oplus 0 ∥ \\ + (α_{k}^{n} β_{k}^{n} ν_{k} τ_{2} + α_{k}^{n} β_{1}^{n} δ_{1} δ_{2} + α_{k}^{n} β_{2}^{n} τ_{2} ν_{2}) γ_{2}^{n} ∥ δ_{3}^{n} \oplus 0 ∥ + (α_{k}^{n} β_{1}^{n} δ_{1} τ_{3} + α_{k}^{n} β_{2}^{n} τ_{2} δ_{3}) γ_{3}^{n} ∥ δ_{4}^{n} \oplus 0 ∥ \\ + α_{k}^{n} β_{2}^{n} τ_{2} τ_{4} γ_{4}^{n} ∥ δ_{6}^{n} \oplus 0 ∥ + α_{k}^{n} ν_{k} β_{k}^{n} ∥ δ_{1}^{n} \oplus 0 ∥ + α_{k}^{n} δ_{1} β_{1}^{n} ∥ δ_{2}^{n} \oplus 0 ∥ \\ + α_{k}^{n} τ_{2} β_{2}^{n} ∥ δ_{3}^{n} \oplus 0 ∥ + α_{k}^{n} ∥ δ_{k}^{n} \oplus 0 ∥ . \end{matrix}$

From (37)–(39), we have

(40) $\begin{matrix} ∥ x_{1}^{(n + 1)} - x_{1} ∥ + ∥ x_{2}^{(n + 1)} - x_{2} ∥ + \dots + ∥ x_{κ}^{(n + 1)} - x_{κ} ∥ \\ \leq \sum_{i = 1}^{n} ({}_{1}Ω_{i}^{(n)} +_{2} Ω_{i}^{(n)} +_{3} Ω_{i}^{(n)} +_{4} Ω_{i}^{(n)} +_{5} Ω_{i}^{(n)} +_{6} Ω_{i}^{(n)} +_{7} Ω_{i}^{(n)}) ∥ x_{i}^{(n)} - x_{i} ∥ \\ + \sum_{i = 1}^{k} α_{i}^{n} β_{i}^{n} ν_{i}^{2} γ_{i}^{n} ∥ δ_{i + 2}^{n} \lor (- δ_{i + 2}^{n}) ∥ + \sum_{i = 1}^{k} (α_{i}^{n} β_{i}^{n} ν_{i} δ_{i + 1} + α_{i}^{n} ν_{i + 1} β_{i + 1}^{n} δ_{i + 1}) γ_{i + 1}^{n} ∥ δ_{i + 3}^{n} \lor (- δ_{i + 3}^{n}) ∥ \\ + \sum_{i = 1}^{k} (α_{i}^{n} β_{i}^{n} ν_{i} τ_{i + 2} + α_{i}^{n} β_{i + 1}^{n} δ_{i + 1} δ_{i + 2} + α_{i}^{n} β_{i + 2}^{n} τ_{i + 2} ν_{i + 2}) γ_{i + 2}^{n} ∥ δ_{i + 3}^{n} \lor (- δ_{i + 3}^{n}) ∥ \\ + \sum_{i = 1}^{k} (α_{i}^{n} β_{i + 1}^{n} δ_{i + 1} τ_{i + 3} + α_{i}^{n} β_{i + 2}^{n} τ_{i + 2} δ_{i + 3}) γ_{i + 3}^{n} ∥ δ_{i + 4}^{n} \lor (- δ_{i + 4}^{n}) ∥ \\ + \sum_{i = 1}^{k} α_{i}^{n} β_{i + 2}^{n} τ_{i + 2} τ_{i + 4} γ_{i + 4}^{n} ∥ δ_{i + 6}^{n} \lor (- δ_{i + 6}^{n}) ∥ + \sum_{i = 1}^{k} α_{i}^{n} ν_{i} β_{i}^{n} ∥ δ_{i + 1}^{n} \lor (- δ_{i + 1}^{n}) ∥ \\ + \sum_{i = 1}^{k} α_{i}^{n} δ_{i + 1} β_{i + 1}^{n} ∥ δ_{i + 2}^{n} \lor (- δ_{i + 2}^{n}) ∥ + \sum_{i = 1}^{k} α_{i}^{n} τ_{i + 2} β_{i + 2}^{n} ∥ δ_{i + 3}^{n} \lor (- δ_{i + 3}^{n}) ∥ \\ + \sum_{i = 1}^{k} α_{i}^{n} ∥ δ_{i}^{n} \lor (- δ_{i}^{n}) ∥ . \end{matrix}$

where

$\begin{matrix} {}_{1}Ω_{i}^{(n)} & = [α_{i}^{n} β_{i}^{n} ν_{i}^{2} {γ_{i}^{n} (ν_{i} - 1) + 1} + α_{i}^{n} {ν_{i} (1 - β_{i}^{n}) - 1} + 1], \\ {}_{2}Ω_{i}^{(n)} & = [α_{i}^{n} β_{i}^{n} ν_{i} (ν_{i} γ_{i}^{n} + 1) (δ_{i + 1} + μ_{i + 1}) + α_{i}^{n} β_{i}^{n} ν_{i} δ_{i + 1} γ_{i + 1}^{n} (ν_{i + 1} - 1) \\ + α_{i}^{n} ν_{i + 1} β_{i + 1}^{n} δ_{i + 1} (ν_{i + 1} γ_{i + 1}^{n} - γ_{i + 1}^{n} + 1) + α_{i}^{n} δ_{i + 1} (1 - β_{i + 1}^{n})], \\ {}_{3}Ω_{i}^{(n)} & = [α_{i}^{n} γ_{i + 1}^{n} δ_{i + 1} (β_{i}^{n} ν_{i} + β_{i + 1}^{n} ν_{i + 1}) (δ_{i + 2} + μ_{i + 2}) + α_{i}^{n} β_{i + 2}^{n} τ_{i + 2} ν_{i + 2} (γ_{i + 2}^{n} ν_{i + 2} \\ + 1 - γ_{i + 2}^{n}) + α_{i}^{n} β_{i}^{n} ν_{i} τ_{i + 2} (ν_{i} γ_{i}^{n} + γ_{i + 2}^{n} ν_{i + 2} - γ_{i + 2}^{n} + 1) \\ + α_{i}^{n} β_{i + 1}^{n} δ_{i + 1} [γ_{i + 1}^{n} δ_{i + 2} (ν_{i + 2} - 1) + δ_{i + 2} + μ_{i + 2}] - β_{i + 1}^{n} + 1], \end{matrix}$

$\begin{matrix} {}_{4}Ω_{i}^{(n)} & = [α_{i}^{n} β_{i}^{n} ν_{i} [γ_{i + 1}^{n} δ_{i + 1} δ_{i + 3} + γ_{i + 2}^{n} τ_{i + 2} (δ_{i + 3} + μ_{i + 3})] + α_{i}^{n} β_{i + 1}^{n} δ_{i + 1} [γ_{i + 1}^{n} ν_{i + 1} δ_{i + 3} \\ + γ_{i + 2}^{n} δ_{i + 2} δ_{i + 3} + γ_{i + 2}^{n} δ_{i + 2} μ_{i + 3} + τ_{i + 3} + γ_{i + 3}^{n} τ_{i + 3} (ν_{i + 3} - 1)] \\ + α_{i}^{n} β_{i + 2}^{n} τ_{i + 2} [γ_{i + 2}^{n} ν_{i + 2} (δ_{i + 3} + μ_{i + 3}) + γ_{i + 3}^{n} δ_{i + 3} (ν_{i + 3} - 1) + δ_{i + 3} + τ_{i + 3}]], \\ {}_{5}Ω_{i}^{(n)} & = [α_{i}^{n} β_{i}^{n} ν_{i} τ_{i + 2} τ_{i + 4} γ_{i + 2}^{n} + α_{i}^{n} β_{i + 1}^{n} δ_{i + 1} [δ_{i + 2} γ_{i + 2}^{n} τ_{i + 4} + γ_{i + 3}^{n} τ_{i + 3} (δ_{i + 4} + μ_{i + 4})] \\ + α_{i}^{n} β_{i + 2}^{n} [τ_{i + 2} τ_{i + 4} (γ_{i + 2}^{n} ν_{i + 2} + γ_{i + 4}^{n} ν_{i + 4} - γ_{i + 4}^{n} + 1) \\ + γ_{i + 3}^{n} δ_{i + 3} (τ_{i + 3} δ_{i + 4} + τ_{i + 2} μ_{i + 4})]], \\ {}_{6}Ω_{i}^{(n)} & = [α_{i}^{n} β_{i + 1}^{n} γ_{i + 3}^{n} δ_{i + 1} τ_{i + 3} τ_{i + 5} + α_{i}^{n} β_{i + 2}^{n} τ_{i + 2} [γ_{i + 3}^{n} δ_{i + 3} τ_{i + 5} + γ_{i + 4}^{n} τ_{i + 4} (δ_{i + 5} + μ_{i + 5})]], \\ {}_{7}Ω_{i}^{(n)} & = α_{i}^{n} β_{i + 2}^{n} γ_{i + 4}^{n} τ_{i + 2} τ_{i + 4} τ_{i + 6} . \end{matrix}$

Let

Φ^{(n)} = max ({}_{1}Ω_{i}^{(n)}, {}_{2}Ω_{i}^{(n)}, {}_{3}Ω_{i}^{(n)}, {}_{4}Ω_{i}^{(n)}, {}_{5}Ω_{i}^{(n)}, {}_{6}Ω_{i}^{(n)}, {}_{7}Ω_{i}^{(n)})

. From (25) and by algebra of convergence of sequences

{}_{k}Ω_{i}^{(n)}

and

1 \leq k \leq 7

, we may say that

Φ^{(n)} \to Φ

n \to \infty

, where

Φ = max ({}_{1}Ω_{i}, {}_{2}Ω_{i}, {}_{3}Ω_{i}, {}_{4}Ω_{i}, {}_{5}Ω_{i}, {}_{6}Ω_{i}, {}_{7}Ω_{i})

. Condition (25) implies that

Φ < 1

, so

Φ^{(n)} < 1

for sufficiently large n.

Let $v_{n + 1} = (∥ x_{1}^{(n + 1)} - x_{1} ∥ + ∥ x_{2}^{(n + 1)} - x_{2} ∥ + \dots + ∥ x_{k}^{(n + 1)} - x_{k} ∥)$ . Then, (40) can be written as

$\begin{matrix} v_{n + 1} & \leq & Φ^{(n)} v_{n}, n = 1, 2, \dots . \end{matrix}$

Clearly, $l i m {sup}_{n \geq 1} Φ^{(n)} < 1$ for $Φ^{(n)} < 1 .$ With the help of Lemma 2, we may claim that $0 \leq Φ_{n} < 1 .$ Hence, ${(x_{i}^{(n)}, v_{i}^{(n)})}$ converge strongly to the solution $(x_{i}, v_{i})$ of system (5). □

A numerical example is provided with a convergence graph.

4. Numerical Example

Let $k = 2$ , $X = R$ , $x = (x_{1}, x_{2}) \in R \times R$ and $(v_{1}, v_{2}) \in N_{1} (x_{1}) \times N_{2} (x_{2})$ such that

(41) $\begin{matrix} \{\begin{matrix} w_{1} \in P_{1} (C_{I, λ_{1}}^{S_{1} (h_{1} (x_{1}), v_{2})} (x_{1}), Y_{I, λ_{2}}^{S_{2} (h_{2} (x_{2}), v_{1})} (x_{2})) + f_{1} (x_{1}) \oplus g_{2} (x_{2}) + S_{1} (h_{1} (x_{1}), v_{2}), \\ w_{2} \in P_{2} (C_{I, λ_{2}}^{S_{2} (h_{2} (x_{2}), v_{1})} (x_{2}), Y_{I, λ_{3}}^{S_{2} (h_{2} (x_{2}), v_{1})} (x_{1})) + f_{2} (x_{2}) \oplus g_{3} (x_{1}) + S_{2} (h_{2} (x_{2}), v_{1}), \end{matrix} \end{matrix}$

for some

\hat{w} = (w_{1}, w_{2}) \in R \times R

Let the mappings $f_{i}, g_{i}, h_{i} : R \to R$ , $S_{i} : R \times R : \to 2^{R}$ , $N_{i} : R \times R : \to C B (R)$ be defined by

$f_{i} (x_{i}) = \frac{2 x_{i}}{7 (i + 1)}, g_{i} (x_{i}) = \frac{3 x_{i}}{49 i}, h_{i} (x_{i}) = 2 x_{i}, S_{i} (h_{i} (x_{i}), v_{i + 1}) = \{\frac{h_{i} (x_{i}) + 3 v_{i + 1}}{2 (i + 1)}\}$

and

N_{i} (x_{i}) = \{\frac{5 x_{i}}{16 i}\} .

Suppose that the mapping $P_{i} : R \times R \to R$ is defined as

$P_{i} (x_{i}, x_{i + 1}) = (\frac{x_{i} + x_{i + 1}}{7 + i}) .$

Now, we define the associated resolvent operator,

(42) $\begin{matrix} ℜ_{I, λ_{i}}^{S_{1} (h_{i}, v_{i + 1})} (x_{i}) = (\frac{2 i (i + 1) x_{i} - 3 v_{i + 1}}{2 (i^{2} + i + 1)}), \end{matrix}$

the Yosida approximation operator,

(43) $\begin{matrix} Y_{I, λ_{i}}^{S_{1} (h_{i}, v_{i + 1})} (x_{i}) = (\frac{(2 x_{i} + 3 v_{i + 1}) i}{2 (i^{2} + i + 1)}), \end{matrix}$

and the Cayley operator,

(44) $\begin{matrix} C_{I, λ_{i}}^{S_{1} (h_{i}, v_{i + 1})} (x_{i}) = (\frac{(i^{2} + i - 1) x_{i} - 3 v_{i + 1}}{i^{2} + i + 1}) . \end{matrix}$

Let us choose the controlling sequences

{α_{i}^{n}} = \{\frac{i}{6 n}\}, \{β_{i}^{n}\} = \{\frac{(7 n - 3) i}{12 n}\}, {γ_{i}^{n}} = \{\frac{(3 n + 1) i}{12 n}\}

, and

{δ_{i}^{n}} = \{\frac{i}{6 n + 3}\}

. Clearly,

{α_{i}^{n}}

{β_{i}^{n}}

{γ_{i}^{n}}

, and

{δ_{i}^{n}}

satisfy the conditions of Algorithm 1.

The sequences for the approximate solution of (41) are obtained by using Algorithm 1 in the following way:

(45) $\begin{matrix} x_{i}^{n + 1} & = \frac{i}{6 n} (\frac{160 i^{2} (i + 1)}{160 i^{2} (i + 1) + 171}) [y_{i}^{n} - \frac{1}{2 (7 + i) (i^{2} + i + 1)} {2 (i^{2} + i - 1) y_{i}^{n} \\ + 2 i y_{i + 1}^{n} (3 i - 6) v_{i + 1} + w_{i}^{n}}] + (\frac{6 n - i}{6 n}) x_{i}^{n} + \frac{i^{2}}{6 n (6 n + 3)}, \end{matrix}$

(46) $\begin{matrix} y_{i}^{n} & = (\frac{7 n - 3}{12 n}) (\frac{160 i^{3} (i + 1)}{160 i^{2} (i + 1) + 171}) [z_{i}^{n} - \frac{1}{2 (7 + i) (i^{2} + i + 1)} {2 (i^{2} + i - 1) z_{i}^{n} \\ + 2 i z_{i + 1}^{n} (3 i - 6) v_{i + 1} + w_{i}^{n}}] + (1 - \frac{(7 n - 3) i}{12 n}) x_{i}^{n} + \frac{(7 n - 3) (i^{2} + i)}{12 n (6 n + 3)}, \end{matrix}$

and

(47) $\begin{matrix} z_{i}^{n} & = (\frac{3 n + 1}{12 n}) (\frac{160 i^{3} (i + 1)}{160 i^{2} (i + 1) + 171}) [x_{i}^{n} - \frac{1}{2 (7 + i) (i^{2} + i + 1)} {2 (i^{2} + i - 1) x_{i}^{n} \\ + 2 i x_{i + 1}^{n} (3 i - 6) v_{i + 1} + w_{i}^{n}}] + (1 - \frac{(3 n + 1) i}{12 n}) x_{i}^{n} + \frac{(3 n + 1) (i^{2} + 2 i)}{12 n (6 n + 3)} . \end{matrix}$

Also, it is verified that all conditions of Theorem 2 are satisfied. Clearly, for

\hat{w} = (1, 1),

the sequence

x^{n} = (x_{1}^{n}, x_{2}^{n})

converges strongly to the solution

x = (- 0.39, 0.60)

of system (41). In this regard the convergence graph (Figure 1) and computational table (Table 1) are shown below. Also, the convergence of each sequence individually involved in Algorithm 1 is shown in Figure 2.

Remark 4.

We select identical operators to those in the above numerical illustration and compare our innovative three-step iterative Algorithm 1 against the two-step iterative algorithm (Ishikawa-style) and the one-step iterative algorithm (Mann-style). By setting $γ_{i}^{n} = 0$ , we derive the sequences ${x_{i}^{n}}$ and ${y_{i}^{n}}$ as detailed below:

(48) $\begin{matrix} x_{i}^{n + 1} & = \frac{i}{6 n} (\frac{160 i^{2} (i + 1)}{160 i^{2} (i + 1) + 171}) [y_{i}^{n} - \frac{1}{2 (7 + i) (i^{2} + i + 1)} {2 (i^{2} + i - 1) y_{i}^{n} \\ + 2 i y_{i + 1}^{n} (3 i - 6) v_{i + 1} + w_{i}^{n}}] + (\frac{6 n - i}{6 n}) x_{i}^{n} + \frac{i^{2}}{6 n (6 n + 3)}, \end{matrix}$

(49) $\begin{matrix} y_{i}^{n} & = (\frac{7 n - 3}{12 n}) (\frac{160 i^{3} (i + 1)}{160 i^{2} (i + 1) + 171}) [z_{i}^{n} - \frac{1}{2 (7 + i) (i^{2} + i + 1)} {2 (i^{2} + i - 1) z_{i}^{n} \\ + 2 i z_{i + 1}^{n} (3 i - 6) v_{i + 1} + w_{i}^{n}}] + (1 - \frac{(7 n - 3) i}{12 n}) x_{i}^{n} + \frac{(7 n - 3) (i^{2} + i)}{12 n (6 n + 3)} . \end{matrix}$

Moreover, by taking $β_{i}^{n} = γ_{i}^{n} = 0,$ for all $n \geq 0$ , we may approximate by using the Mann-style iterative scheme as follows:

(50) $\begin{matrix} x_{i}^{n + 1} & = \frac{i}{6 n} (\frac{160 i^{2} (i + 1)}{160 i^{2} (i + 1) + 171}) [y_{i}^{n} - \frac{1}{2 (7 + i) (i^{2} + i + 1)} {2 (i^{2} + i - 1) y_{i}^{n} \\ + 2 i y_{i + 1}^{n} (3 i - 6) v_{i + 1} + w_{i}^{n}}] + (\frac{6 n - i}{6 n}) x_{i}^{n} + \frac{i^{2}}{6 n (6 n + 3)} . \end{matrix}$

The iterative methods that we are employing will stop their operations once the stopping criterion $∥ x_{i}^{(n + 1)} - x_{i}^{n} ∥ \leq 10^{- 6}$ has been satisfactorily met. In the accompanying Table 2 and the illustrative Figure 3, we present a comprehensive comparison showcasing the performance of our innovative three-step Algorithm 1, alongside the well-established Ishikawa-type Equations (48) and (49), in conjunction with the Mann-type Algorithm (50), all of which are initiated with the chosen starting point of $(- 0.5, 1)$ .

The numerical findings that are meticulously detailed in Table 2 and the graphical representation provided in Figure 3 strongly suggest that our newly proposed three-step Algorithm 1 demonstrates impressive performance and appears to possess a significant competitive edge over the other methods. Based on this analysis, we can confidently assert that our algorithm exhibits remarkable speed and efficiency, typically requiring an average of about 10 to 12 iterations to achieve convergence successfully.

5. Conclusions

In this manuscript, the framework of extended ordered XOR-inclusion problems incorporating Cayley and Yosida operators is presented and examined within the context of real ordered Banach spaces, which encompasses a broader scope than the problems addressed in [11,19,28]. We investigated the existence of solutions to SEOXORIP by utilizing proximal-point mappings under specific conditions deemed appropriate. A three-step iterative methodology is proposed for deriving approximate solutions to SEOXORIP, and an analysis of the convergence of the proposed methodology was conducted. Ultimately, we provided a numerical example accompanied by a convergence graph generated using MATLAB 2018a to substantiate the convergence of the sequence produced by the proposed methodology.

Author Contributions

Conceptualization, D.F., I.A., and F.A.K.; methodology, I.A., N.H.E.E., and E.A.; formal analysis, M.S.A.; investigation, N.H.E.E.; resources, D.F., I.A., and F.A.K.; writing—original draft, I.A., N.H.E.E., and F.A.K.; writing—review and editing, D.F., E.A., and M.S.A.; supervision, F.A.K.; funding acquisition, D.F., E.A., and M.S.A. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding authors.

Acknowledgments

The authors are grateful to the anonymous reviewers for their valuable remarks which improved the results and presentation of this paper.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1 Convergence of $x^{n} = (x_{1}^{n}, x_{2}^{n})$ with initial value $(- 1, 2)$ .

Figure 2 Convergence of $x^{n} = (x_{1}^{n}, x_{2}^{n})$ , $y^{n} = (y_{1}^{n}, y_{2}^{n})$ , and $z^{n} = (z_{1}^{n}, z_{2}^{n})$ with initial values $(- 1, 2), (3, - 2), a n d (4, - 1)$ , respectively.

Figure 3 Convergence of $x^{n} = (x_{1}^{n}, x_{2}^{n})$ with initial value $(- 0.5, 1)$ .

Table 1

The computational table of ${x^{n}}$ , ${y^{n}}$ , and ${z^{n}}$ starting with $(x_{1}^{0}, x_{2}^{0}) = (- 1, 2)$ $(y_{1}^{0}, y_{2}^{0}) = (3, - 2)$ , and $(z_{1}^{0}, z_{2}^{0}) = (4, - 1)$ .

No. of	For $x^{n} = (- 1, 2)$	For $y^{n} = (3, - 2)$	For $z^{n} = (4, - 1)$
Iterations	$x^{n} = (x_{1}^{n}, x_{2}^{n})$	$y^{n} = (y_{1}^{n}, y_{2}^{n})$	$z^{n} = (z_{1}^{n}, z_{2}^{n})$
n = 1	(−1,2)	(3,−2)	(4,−1)
n = 2	(−0.49930, 0.81207)	(0.23862, 0.24273)	(−0.72659, 2.10466)
n = 3	(−0.45165, 0.71122)	(−0.44464,1.89461)	(−0.43555, 0.89865)
n = 4	(−0.46018, 0.80844)	(−0.33075, 0.90799)	(−0.43376, 0.89865)
n = 5	(−0.45581, 0.80425)	(−0.33841, 0.75610)	(−0.46637, 0.82427)
n = 10	(−0.44173, 0.76545)	(−0.36189, 0.69389)	(−0.4845, 0.74925)
n = 15	(−0.43366, 0.73856)	(−0.36339, 0.63498)	(−0.48451, 0.71074)
n = 20	(−0.42794, 0.71821)	(−0.36241, 0.60042)	(−0.48211, 0.68513)
n = 25	(−0.42341, 0.70210)	(−0.36087, 0.57680)	(−0.47936, 0.66615)
n = 30	(−0.41972, 0.68884)	(−0.35921, 0.55919)	(−0.47668, 0.65118)
n = 35	(−0.41659, 0.67762)	(−0.35758, 0.54531)	(−0.47417, 0.63887)
n = 40	(−0.41389, 0.66791)	(−0.35604, 0.53394)	(−0.47186, 0.62845)
n = 45	(−0.41149, 0.65938)	(−0.35458, 0.52436)	(−0.46972, 0.61943)
n = 55	(−0.40742, 0.64494)	(−0.35194, 0.50891)	(−0.46591 0.60444)
n = 70	(−0.40252, 0.62779)	(−0.34853, 0.49157)	(−0.46113, 0.58701)
n = 80	(−0.39981, 0.61841)	(−0.34656, 0.48248)	(−0.45835 0.57760)
n = 90	(−0.39742, 0.61020)	(−0.34479, 0.47473)	(−0.45588, 0.56946)
n = 100	(−0.39523, 0.60290)	(−0.34318, 0.46796)	(−0.45365, 0.56228)

Table 2

The values of $x^{n} = (x_{1}^{n}, x_{2}^{n})$ with the initial value $(- 0.5, 1)$ .

No. ofIterations	Three-Step Iterative Algorithm $x^{n} = (x_{1}^{n}, x_{2}^{n})$	Two-Step Iterative Algorithm $x^{n} = (x_{1}^{n}, x_{2}^{n})$	One Step Iterative Algorithm $x^{n} = (x_{1}^{n}, x_{2}^{n})$
n = 1	(−0.5,1)	(−0.5,1)	(−0.5,1)
n = 2	(0.12570, −0.25747)	(0.02153, −0.08605)	(0.03702, −0.37147)
n = 3	(0.20262, −0.30925)	(0.11337, −0.14096)	(0.18311, −0.41916)
n = 4	(0.16102, −0.13137)	(0.11288, −0.0982)	(0.22754, −0.34455)
n = 5	(0.11547, −0.05666)	(0.09339, −0.06932)	(0.2329, −0.27319)
n = 10	(0.01631, 0.00114)	(0.02223, −0.01653)	(0.15164, −0.11691)
n = 15	(0.00225, 0.00301)	(0.00459, −0.00439)	(0.09413, −0.07328)
n = 20	(0.00017, 0.00172)	(0.00071 −0.00100)	(0.06543, −0.05345)
n = 25	(0, 0.00093)	(0.00013, −0.0001)	(0.04982, −0.04209)
n = 30	(0, 0.00052)	(0, 0.00016)	(0.04026, −0.03471)
n = 35	(0, 0.00029)	(0, 0.00013)	(0.03382, −0.02954)
n = 40	(0, 0.00016)	(0, 0.00009)	(0.02918, −0.02571)
n = 45	(0, 0.00009)	(0, 0.00006)	(0.02566, −0.02276)
n = 55	(0, 0)	(0, 0.00003)	(0.02069, 0)
n = 70	(0, 0)	(0, 0)	(0.01604, 0)
n = 80	(0, 0)	(0, 0)	(0.01395, −0.01262)
n = 90	(0, 0)	(0, 0)	(0.01234, −0.0112)
n = 100	(0, 0)	(0, 0)	(0.01107, −0.01006)

References

1. Adamu, A.; Abass, H.H.; Ibrahim, A.I.; Kilicman, A. An accelerated Halpern-type algorithm for solving variational inclusion problems with applications. Bangmod J. Math. Comput. Sci.; 2022; 8, pp. 37-55. [DOI: https://dx.doi.org/10.58715/bangmodjmcs.2022.8.4]

2. Adamu, A.; Deepho, J.; Ibrahim, A.H.; Abubakar, A.B. Approximation of zeros of sum of Monotone Mappings with Applications to Variational Inequalities Problems and image processing. Nonlinear Fun. Anal. Appl.; 2021; 262, pp. 411-432.

3. Taiwo, A.; Reich, S.; Agarwal, R.P. Tseng-Type Algorithms for Solving Variational Inequalities Over the Solution Sets of Split Variational Inclusion Problems with An Application to A Bilevel Optimization Problem. J. Appl. Numer. Optim.; 2024; 6, pp. 41-57.

4. Lin, L.-J.; Chen, Y.-D.; Chuang, C.-S. Solutions for a variational inclusion problem with applications to multiple sets split feasibility problems. Fixed Point Theory Appl.; 2013; 2013, 333. [DOI: https://dx.doi.org/10.1186/1687-1812-2013-333]

5. Fang, Y.-P.; Huang, N.-J. H-monotone operator and resolvent operator technique for variational inclusions. Appl. Math. Comput.; 2003; 145, pp. 795-803. [DOI: https://dx.doi.org/10.1016/S0096-3003(03)00275-3]

6. Fang, Y.-P.; Huang, N.-J. H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces. Appl. Math. Lett.; 2004; 17, pp. 647-653. [DOI: https://dx.doi.org/10.1016/S0893-9659(04)90099-7]

7. Xia, F.-Q.; Huang, N.-J. Variational inclusions with a general H-monotone operator in Banach spaces. Comput. Math. Appl.; 2007; 54, pp. 24-30. [DOI: https://dx.doi.org/10.1016/j.camwa.2006.10.028]

8. Ding, X.P.; Xia, F.Q. A new class of completely generalized quasivariational inclusions in Banach spaces. J. Comput. Appl. Math.; 2002; 147, pp. 369-383. [DOI: https://dx.doi.org/10.1016/S0377-0427(02)00443-0]

9. Amann, H. On the number of solutions of nonlinear equations in ordered Banach spaces. J. Funct. Anal.; 1972; 11, pp. 346-384. [DOI: https://dx.doi.org/10.1016/0022-1236(72)90074-2]

10. Li, H.G. Approximation solution for general nonlinear ordered variational inequalities and ordered equations in ordered Banach space. Nonlinear Anal. Forum; 2008; 13, pp. 205-214.

11. Ali, I.; Ahmad, R.; Wen, C.F. Cayley Inclusion problem involving XOR-operation. Mathematics; 2019; 7, 302. [DOI: https://dx.doi.org/10.3390/math7030302]

12. Li, H.G.; Li, L.P.; Jin, M.M. A class of nonlinear mixed ordered ininclusion problems for ordered (α_A,λ)-ANODM set-valued mappings with strong compression mapping. Fixed Point Theory Appl.; 2014; 2014, 79. [DOI: https://dx.doi.org/10.1186/1687-1812-2014-79]

13. Li, H.G. A nonlinear inclusion problem involving (α,λ)-NODM set-valued mappings in ordered Hilbert space. Appl. Math. Lett.; 2012; 25, pp. 1384-1388. [DOI: https://dx.doi.org/10.1016/j.aml.2011.12.007]

14. Glowinski, G.; Le Tallec, P. Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics; SIAM: Philadelphia, PA, USA, 1989.

15. Noor, M.A. New approximation schemes for general variational inequalities. J. Math. Anal. Appl.; 2000; 251, pp. 217-229. [DOI: https://dx.doi.org/10.1006/jmaa.2000.7042]

16. Noor, M.A. A predictor-corrector algorithm for general variational inequalities. Appl. Math. Lett.; 2001; 14, pp. 53-58. [DOI: https://dx.doi.org/10.1016/S0893-9659(00)00112-9]

17. Noor, M.A. Three-step iterative algorithms for multivaled quasi-variational inclusions. J. Math. Anal. Appl.; 2001; 255, pp. 589-604. [DOI: https://dx.doi.org/10.1006/jmaa.2000.7298]

18. Ahmad, I.; Rahaman, M.; Ahmad, R.; Ali, I. Convergence analysis and stability of perturbed three-step iterative algorithm for generalized mixed ordered quasi-variational inclusion involving XOR operator. Optimization; 2020; 69, pp. 821-845. [DOI: https://dx.doi.org/10.1080/02331934.2019.1652910]

19. Ali, I.; Wang, Y.; Ahmad, R. Convergence and Stability of a Three-Step Iterative Algorithm for the Extended Cayley–Yosida Inclusion Problem in 2-Uniformly Smooth Banach Spaces: Convergence and Stability Analysis. Mathematics; 2024; 12, 1977. [DOI: https://dx.doi.org/10.3390/math12131977]

20. Schaefer, H.H. Banach Lattices and Positive Operators; Springer: Berlin/Heidelberg, Germany, 1974.

21. Aubin, J.P.; Cellina, A. Differential Inclusions; Springer: Berlin, Germany, 1984.

22. Kazmi, K.R.; Khan, F.A.; Shahzad, M. A system of generalized variational inclusions involving generalized H(·,·)-accretive mapping in real q-uniformly smooth Banach spaces. Appl. Math. Comput.; 2011; 217, pp. 9679-9688. [DOI: https://dx.doi.org/10.1016/j.amc.2011.04.052]

23. Salahuddin,. System of Generalized Mixed Nonlinear Ordered Variational Inclusions. Numer. Algebra Control. Optim.; 2019; 9, pp. 445-460. [DOI: https://dx.doi.org/10.3934/naco.2019026]

24. Du, Y.H. Fixed points of increasing operators in ordered Banach spaces and applications. Appl. Anal.; 1990; 38, pp. 1-20. [DOI: https://dx.doi.org/10.1080/00036819008839957]

25. Ahmad, I.; Ahmad, R.; Iqbal, J. A resolvent approach for solving a set-valued variational inclusion problem using weak-RRD set-valued mapping. Korean J. Math.; 2016; 16, pp. 199-213. [DOI: https://dx.doi.org/10.11568/kjm.2016.24.2.199]

26. Osilike, M.O. Stability results for the Ishikawa fixed point iteration procedure. Ind. J. Pure Appl. Math.; 1995; 26, pp. 937-945.

27. Nadler, S.B., Jr. Multi-valued contraction mappings. Pac. J. Math.; 1969; 30, pp. 475-488. [DOI: https://dx.doi.org/10.2140/pjm.1969.30.475]

28. Arifuzzaman,; Irfan, S.S.; Ahmad, I. Convergence Analysis for Cayley Variational Inclusion Problem Involving XOR and XNOR Operations. Axioms; 2025; 14, 149. [DOI: https://dx.doi.org/10.3390/axioms14030149]

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Three-Step Iterative Methodology for the Solution of Extended Ordered XOR-Inclusion Problems Incorporating Generalized Cayley–Yosida Operators

Content area

Abstract

Full text

1. Introduction

2. Prerequisites and Formulation of SEOXORIP

3. Three-Step Iterative Scheme and Its Convergence

4. Numerical Example

5. Conclusions