Strong Convergence of Iterative Algorithm for a New System of Generalized 𝐻(⋅,⋅)−𝜂-Cocoercive Operator Inclusions in Banach Spaces

Abstract

We introduce and study a new system of generalized 𝐻(⋅,⋅)−𝜂-cocoercive operator inclusions in Banach spaces. Using the resolvent operator technique associated with 𝐻(⋅,⋅)−𝜂-cocoercive operators, we suggest and analyze a new generalized algorithm of nonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method. We highlight the applicability of our results by examples in function spaces.

1. Introduction

The resolvent operator technique is a powerful tool to study the approximation solvability of nonlinear variational inequalities and variational inclusions, which have been applied widely to optimization and control, mechanics and physics, economics and transportation equilibrium, and engineering sciences, see, for example, [1–4] and the references therein.

In a series of papers [5–8], the authors investigated (𝐴,𝜂)-accretive and 𝐻(⋅,⋅)-accretive operators for solving variational inclusions in Banach spaces. Convergence and stability of iterative algorithms for the systems of (𝐴,𝜂)-accretive operators have been studied in [9, 10]. The notion of (𝐻,𝜙)−𝜂-monotone operators has been introduced and investigated by the authors in [11]. Generalized mixed variational inclusions involving (𝐻(⋅,⋅),𝜂)-monotone operators have been discussed in [12]. Some results on 𝐻((⋅,⋅),𝜂)-accretive operators and application for solving set-valued variational inclusions in Banach spaces have been proved in [7]. Some other related articles on the variational inclusion problems can be found in [13–22].

Very recently, Ahmad et al. [23] introduced a new 𝐻(⋅,⋅)−𝜂-cocoercive operator and its resolvent operator in the setting of Banach spaces. The authors proposed concrete examples in support of 𝐻(⋅,⋅)−𝜂-cocoercive operators and they also proved the Lipschitz continuity of resolvent operator associated with 𝐻(⋅,⋅)−𝜂-cocoercive operator. Motivated and inspired by the research works mentioned above, in this paper, we introduce and study a new system of 𝐻(⋅,⋅)−𝜂-cocoercive mapping inclusions in Banach spaces. Using the resolvent operator associated with 𝐻(⋅,⋅)−𝜂-cocoercive mapping, we suggest and analyze a new general algorithm and establish the existence and uniqueness of solutions for this system of 𝐻(⋅,⋅)−𝜂-cocoercive mappings.

2. Preliminaries

Throughout this paper, we denote the set of positive integers by ℕ. Let 𝑋 be a Banach space with the norm ‖⋅‖ and the dual space 𝑋^∗. For any 𝑥∈𝑋, we denote the value of 𝑥^∗∈𝑋^∗ at 𝑥 by ⟨𝑥,𝑥^∗⟩. When {𝑥_𝑛} is a sequence in 𝑋, we denote the strong convergence of {𝑥_𝑛} to 𝑥∈𝑋 by 𝑥_𝑛→𝑥 as 𝑛→∞. We denote by 2^𝑋 the family of all nonempty subsets of 𝑋. Let 𝐶𝐵(𝑋) be the family of all nonempty, closed, and bounded subsets of 𝑋. The Hausdörff metric on 𝐶𝐵(𝑋) [24] is defined by 𝐷(𝐴,𝐵)=maxsup_𝑥∈𝐴𝑑(𝑥,𝐵),sup_𝑦∈𝐵𝑑,(𝑋),𝐴,𝑦𝐴,𝐵∈𝐶𝐵 (1) where 𝑑(𝑥,𝐵)=inf_𝑏∈𝐵‖𝑥−𝑏‖ and 𝑑(𝐴,𝑦)=inf_𝑎∈𝐴‖𝑎−𝑦‖.

Definition 1 (see [25]). A continuous and strictly increasing function 𝜙∶[0,+∞)→[0,∞) such that 𝜙(0)=0 and lim_𝑡→∞𝜙(𝑡)=∞ is called a gauge function.

Definition 2 (see [25]). Let 𝑋 be a Banach space. Given a gauge function 𝜙, the mapping 𝐽_𝜙∶𝑋→2^𝑋∗ corresponding to 𝜙 defined by 𝐽_𝜙(𝑥)=𝑥^∗∈𝑋^∗∶𝑥,𝑥^∗=‖𝑥‖𝑥^∗,𝑥^∗=𝜙(‖𝑥‖),∀𝑥∈𝑋, (2) is called the duality mapping with gauge function 𝜙.
In particular, if 𝜙(𝑡)=𝑡, the duality map 𝐽=𝐽_𝜙 is called the normalized duality mapping.

Lemma 3 (see [26]). Let 𝑋 be a real Banach space and 𝐽∶𝑋→2^𝑋∗ be the normalized duality mapping. Then, for any 𝑥,𝑦∈𝑋, 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑗𝑥+𝑦, (3) for all 𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦).

Definition 4. Let 𝑋 be a Banach space. Let 𝐴∶𝑋→𝑋 and 𝜂∶𝑋×𝑋→𝑋 be two mappings and 𝐽∶𝑋→2^𝑋∗ be the normalized duality mapping. Then, 𝐴 is called(i)𝜂-cocoercive, if there exists a constant 𝜇₁>0 such that 𝐴𝑥−𝐴𝑦,𝑗𝜂𝑥,𝑦≥𝜇₁𝐴𝑥−𝐴𝑦²𝑗𝜂𝜂,,∀𝑥,𝑦∈𝑋,𝑥,𝑦∈𝐽𝑥,𝑦 (4)(ii) 𝜂-accretive, if 𝜂𝑗𝜂𝜂,𝐴𝑥−𝐴𝑦,𝑗𝑥,𝑦≥0,∀𝑥,𝑦∈𝑋,𝑥,𝑦∈𝐽𝑥,𝑦 (5)(iii) 𝜂-strongly accretive, if there exists a constant 𝛽₁>0 such that 𝐴𝑥−𝐴𝑦,𝑗𝜂𝑥,𝑦≥𝛽₁𝑥−𝑦²𝑗𝜂𝜂,,∀𝑥,𝑦∈𝑋,𝑥,𝑦∈𝐽𝑥,𝑦 (6)(iv) 𝜂-relaxed cocoercive, if there exists a constant 𝛾₁>0 such that 𝐴𝑥−𝐴𝑦,𝑗𝜂𝑥,𝑦≥−𝛾₁𝐴𝑥−𝐴𝑦²𝑗𝜂𝜂,,∀𝑥,𝑦∈𝑋,𝑥,𝑦∈𝐽𝑥,𝑦 (7)(v) Lipschitz continuous, if there exists a constant 𝜆_𝐴>0 such that 𝐴𝑥−𝐴𝑦≤𝜆_𝐴𝑥−𝑦,∀𝑥,𝑦∈𝑋, (8)(vi) 𝛼-expansive, if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦≥𝛼𝑥−𝑦,∀𝑥,𝑦∈𝑋, (9)(vii) 𝜂 is said to be Lipschitz continuous, if there exists a constant 𝜏>0 such that 𝜂𝑥,𝑦≤𝜏𝑥−𝑦,∀𝑥,𝑦∈𝑋. (10)

Definition 5. Let 𝑋 be a Banach space. Let 𝐴,𝐵∶𝑋→𝑋, 𝐻∶𝑋×𝑋→𝑋, 𝜂∶𝑋×𝑋→𝑋 be four single-valued mappings and 𝐽∶𝑋→2^𝑋∗ be the normalized duality mapping. Then,(i)𝐻(𝐴,⋅) is said to be 𝜂-cocoercive with respect to 𝐴, if there exists a constant 𝜇>0 such that 𝐻(𝐴𝑥,𝑢)−𝐻𝐴𝑦,𝑢,𝑗𝜂𝑥,𝑦≥𝜇𝐴𝑥−𝐴𝑦²𝑗𝜂𝜂,,∀𝑥,𝑦,𝑢∈𝑋,𝑥,𝑦∈𝐽𝑥,𝑦 (11)(ii) 𝐻(⋅,𝐵) is said to be 𝜂-relaxed cocoercive with respect to 𝐵, if there exists a constant 𝛾>0 such that 𝐻()𝜂≥𝑢,𝐵𝑥−𝐻𝑢,𝐵𝑦,𝑗𝑥,𝑦−𝛾𝐵𝑥−𝐵𝑦²𝑗𝜂𝜂,,∀𝑥,𝑦,𝑢∈𝑋,𝑥,𝑦∈𝐽𝑥,𝑦 (12)(iii) 𝐻(𝐴,⋅) is said to be 𝑟₁-Lipschitz continuous with respect to 𝐴, if there exists a constant 𝑟₁>0 such that 𝐻(𝐴𝑥,𝑢)−𝐻𝐴𝑦,𝑢≤𝑟₁𝑥−𝑦,∀𝑥,𝑦,𝑢∈𝑋, (13)(iv) 𝐻(⋅,𝐵) is said to be 𝑟₂-Lipschitz continuous with respect to 𝐵, if there exists a constant 𝑟₂>0 such that 𝐻(𝑢,𝐵𝑥)−𝐻𝑢,𝐵𝑦≤𝑟₂𝑥−𝑦,∀𝑥,𝑦,𝑢∈𝑋. (14)

Definition 6. Let 𝑋 be a Banach space. A set-valued mapping 𝑀∶𝑋→2^𝑋 is said to be 𝜂-cocoercive, if there exists a constant 𝜇₂>0 such that 𝑢−𝑣,𝑗𝜂𝑥,𝑦≥𝜇₂‖𝑢−𝑣‖²(𝑥)𝑦𝜂𝜂.,∀𝑥,𝑦∈𝑋,𝑢∈𝑀,𝑣∈𝑀,𝑗𝑥,𝑦∈𝐽𝑥,𝑦 (15)

Definition 7. Let 𝑋 be a Banach space. A mapping 𝑇∶𝑋→𝐶𝐵(𝑋) is said to be 𝒟-Lipschitz continuous, if there exists a constant 𝜆_𝑇>0 such that 𝒟𝑇𝑥,𝑇𝑦≤𝜆_𝑇𝑥−𝑦,𝑥,𝑦∈𝑋. (16)

Definition 8. Let 𝑋 be a Banach space. Let 𝑇,𝑄∶𝑋→𝐶𝐵(𝑋) be the mappings. A mapping 𝑁∶𝑋×𝑋→𝑋 is said to be(i) Lipschitz continuous in the first argument with respect to 𝑇, if there exists a constant 𝑡₁>0 such that 𝑁𝑤₁,⋅−𝑁𝑤₂,⋅≤𝑡₁𝑤₁−𝑤₂,∀𝑢₁,𝑢₂𝑤∈𝑋,₁∈𝑇𝑢₁,𝑤₂∈𝑇𝑢₂, (17)(ii) Lipschitz continuous in the second argument with respect to 𝑄, if there exists a constant 𝑡₂>0 such that 𝑁⋅,𝑣₁−𝑁⋅,𝑣₂≤𝑡₂𝑣₁−𝑣₂,∀𝑢₁,𝑢₂𝑣∈𝑋,₁∈𝑄𝑢₁,𝑣₂∈𝑄𝑢₂, (18)(iii) 𝜂-relaxed Lipschitz in the first argument with respect to 𝑇, if there exists a constant 𝜏₁>0 such that 𝑁𝑤₁,⋅−𝑁𝑤₂,⋅,𝑗𝜂𝑢₁,𝑢₂≤−𝜏₁𝑢₁−𝑢₂²,∀𝑢₁,𝑢₂𝑤∈𝑋,₁∈𝑇𝑢₁,𝑤₂∈𝑇𝑢₂,𝑗𝜂𝑢₁,𝑢₂∈𝐽𝜂𝑢₁,𝑢₂, (19) (iv) 𝜂-relaxed Lipschitz in the second argument with respect to 𝑄, if there exists a constant 𝜏₂>0 such that 𝑁⋅,𝑣₁−𝑁⋅,𝑣₂,𝑗𝜂𝑢₁,𝑢₂≤−𝜏₂𝑢₁−𝑢₂²,∀𝑢₁,𝑢₂∈𝑋,𝑣₁∈𝑄𝑢₁,𝑣₂∈𝑄𝑢₂,𝑗𝜂𝑢₁,𝑢₂∈𝐽𝜂𝑢₁,𝑢₂. (20)

Definition 9. Let 𝑋 be a Banach space. Let 𝐴,𝐵∶𝑋→𝑋, 𝐻∶𝑋×𝑋→𝑋, 𝜂∶𝑋×𝑋→𝑋 be four single-valued mappings. Let 𝑀∶𝑋→2^𝑋 be a set-valued mapping. 𝑀 is said to be 𝐻(⋅,⋅)−𝜂-cocoercive operator with respect to 𝐴 and 𝐵, if 𝑀 is 𝜂-cocoercive and (𝐻(𝐴,𝐵)+𝜆𝑀)(𝑋)=𝑋, for every 𝜆>0.

Example 10. Let 𝑋=ℝ×ℝ and 𝐴,𝐵∶𝑋→𝑋 be defined by 𝐴𝑥₁,𝑥₂=2𝑥₁−𝑥₂,𝑥₁−𝑥₂,𝐵𝑦₁,𝑦₂=−2𝑦₂,𝑦₁−𝑦₂,∀𝑥₁,𝑥₂,𝑦₁,𝑦₂∈𝑋. (21) Assume now that 𝐻(𝐴,𝐵), 𝜂∶𝑋×𝑋→𝑋 are defined by 𝐻𝐴𝑥,𝐵𝑦=𝐴𝑥+𝐵𝑦,𝜂𝑥,𝑦=𝑥−𝑦,∀𝑥,𝑦∈𝑋. (22)
Let 𝑀=𝐼, where 𝐼 is the identity mapping. Then, 𝑀 is 𝐻(⋅,⋅)−𝜂-cocoercive with respect to 𝐴 and 𝐵.

Example 11. Let 𝑋=𝐶[0,1], the space of all real valued continuous functions defined on closed interval [0,1] with the norm 𝑓=max_[]𝑡∈0,1𝑓(𝑡). (23) Let 𝐴,𝐵∶𝑋→𝑋 be defined by 𝐴𝑓=sin²𝑓,𝐵𝑔=cos²𝑔,∀𝑓,𝑔∈𝑋. (24) Let 𝐻(𝐴,𝐵)∶𝑋×𝑋→𝑋 be defined by 𝐻𝐴𝑓,𝐵𝑔=𝐴𝑓+𝐵𝑔,∀𝑓,𝑔∈𝑋. (25) Suppose that 𝑀(𝑓)=𝑓², where 𝑓²(𝑡)=𝑓(𝑡)𝑓(𝑡) for all 𝑡∈[0,1]. Then, for 𝜆=1, we conclude that (𝐻(𝐴,𝐵)+𝑀)𝑓=𝐴𝑓+𝐵𝑓+𝑓²=max_[]𝑡∈0,1sin²𝑓(𝑡)+cos²𝑓(𝑡)+𝑓²(𝑡)=1+𝑓²(𝑡)>0. (26) This proves that 0∉(𝐻(𝐴,𝐵)+𝑀)(𝑋) and 𝑀 is not 𝐻(𝐴,𝐵)−𝜂-cocoercive with respect to 𝐴 and 𝐵.

Proposition 12 (see [23]). Let 𝐻(𝐴,𝐵) be 𝜂-cocoercive with respect to 𝐴 with constant 𝜇>0 and 𝜂-relaxed cocoercive with respect to 𝐵 with constant 𝛾>0, 𝐴 be 𝛼-expansive and 𝐵 be 𝛽-Lipschitz continuous 𝜇>𝛾 and 𝛼>𝛽. Let 𝑀∶𝑋→2^𝑋 be 𝐻(𝐴,𝐵)−𝜂-cocoercive operator. Suppose that 𝜂()(𝑀),𝑗𝜂()𝜂().𝑥−𝑦,𝑗𝑢,𝑣≥0,∀𝑣,𝑦∈𝐺𝑟𝑎𝑝ℎ𝑢,𝑣∈𝐽𝑢,𝑣 (27) Then, 𝑥∈𝑀𝑢, where 𝐺𝑟𝑎𝑝ℎ(𝑀)={(𝑢,𝑥)∈𝑋×𝑋∶𝑥∈𝑀𝑢}.

Theorem 13 (see [23]). Let 𝐻(𝐴,𝐵) be 𝜂-cocoercive with respect to 𝐴 with constant 𝜇>0 and 𝜂-relaxed cocoercive with respect to 𝐵 with constant 𝛾>0, 𝐴 be 𝛼-expansive and 𝐵 be 𝛽-Lipschitz continuous, 𝜇>𝛾 and 𝛼>𝛽. Let 𝑀 be an 𝐻(⋅,⋅)−𝜂-cocoercive operator with respect to 𝐴 and 𝐵. Then, for each 𝜆>0, the operator (𝐻(𝐴,𝐵)+𝜆𝑀)⁻¹ is single-valued.

Definition 14. Let 𝑋 be a Banach space. Let 𝐻(𝐴,𝐵) be 𝜂-cocoercive with respect to 𝐴 with constant 𝜇>0 and 𝜂-relaxed cocoercive with respect to 𝐵 with constant 𝛾>0, 𝐴 be 𝛼-expansive 𝐵 be 𝛽-Lipschitz continuous and 𝜂 be 𝛽-Lipschitz continuous, 𝜇>𝛾, and 𝛼>𝛽. Let 𝑀 be a 𝐻(⋅,⋅)−𝜂-cocoercive operator with respect to 𝐴 and 𝐵. Then, the resolvent 𝑅^{𝐻(⋅,⋅)−𝜂}_𝜆,𝑀∶𝑋→𝑋 is defined by 𝑅^{𝐻(⋅,⋅)−𝜂}_𝜆,𝑀(𝑢)=(𝐻(𝐴,𝐵)+𝜆𝑀)⁻¹(𝑢),∀𝑢∈𝑋. (28)

Theorem 15 (see [23]). Let 𝑋 be a Banach space. Let 𝐻(𝐴,𝐵) be 𝜂-cocoercive with respect to 𝐴 with constant 𝜇>0 and 𝜂-relaxed cocoercive with respect to 𝐵 with constant 𝛾>0, 𝐴 be 𝛼-expansive 𝐵 be 𝛽-Lipschitz continuous, and 𝜂 be 𝜌-Lipschitz continuous; 𝜇>𝛾 and 𝛼>𝛽. Let 𝑀 be 𝐻(⋅,⋅)−𝜂-cocoercive operator with respect to 𝐴 and 𝐵. Then, the resolvent operator 𝑅^{𝐻(⋅,⋅)−𝜂}_𝜆,𝑀∶𝑋→𝑋 is 𝜌/(𝜇𝛼²−𝛾𝛽²)-Lipschitz continuous, that is, 𝑅^{𝐻(⋅,⋅)−𝜂}_𝜆,𝑀(𝑢)−𝑅^{𝐻(⋅,⋅)−𝜂}_𝜆,𝑀(𝑣≤𝜌)𝜇𝛼²−𝛾𝛽²‖𝑢−𝑣‖,∀𝑢,𝑣∈𝑋. (29)

3. Strong Convergence Theorem

In this section, using the resolvent operator technique associated with 𝐻(⋅,⋅)−𝜂-cocoercive operators, we propose a new generalized algorithm of nonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method.

For 𝑖=1,2, let 𝑋_𝑖 be real Banach spaces with the norm ‖⋅‖_𝑖. Let 𝐴_𝑖,𝐵_𝑖∶𝑋_𝑖→𝑋_𝑖, 𝐻_𝑖∶𝑋_𝑖×𝑋_𝑖→𝑋_𝑖, 𝜂_𝑖∶𝑋_𝑖×𝑋_𝑖→𝑋_𝑖, 𝐹∶𝑋₁×𝑋₂→𝑋₁, and 𝐺∶𝑋₁×𝑋₂→𝑋₂ be single-valued mappings, and 𝑇∶𝑋₁→𝐶𝐵(𝑋₁), 𝑄∶𝑋₂→𝐶𝐵(𝑋₂) be set-valued mappings. Let 𝑀∶𝑋₁×𝑋₁→2^𝑋1, 𝑁∶𝑋₂×𝑋₂→2^𝑋2 be 𝐻₁(⋅,⋅)−𝜂₁-cocoercive and 𝐻₂(⋅,⋅)−𝜂₂-cocoercive operators with respect to (𝐴₁,𝐵₁) and (𝐴₂,𝐵₂), respectively. We consider the following problem.

Find (𝑥,𝑦)∈𝑋₁×𝑋₂, 𝑤∈𝑇(𝑥), and 𝑣∈𝑄(𝑦) such that ()(),().0∈𝑀𝑥,𝑥+𝐹𝑤,𝑣0∈𝑁𝑦,𝑦+𝐺𝑤,𝑣 (30) We call problem (30) a system of generalized 𝐻(⋅,⋅)−𝜂-cocoercive operator inclusions.

Under the assumptions mentioned above, we have the following key and simple lemma.

Lemma 16. (𝑥,𝑦)∈𝑋₁×𝑋₂, 𝑤∈𝑇(𝑥), 𝑣∈𝑄(𝑦) is a solution of problem (30) if and only if 𝑥=𝑅^𝐻1_𝜆^{(⋅,⋅)−𝜂}_{1(),𝑀⋅,𝑥}𝐻₁𝐴₁𝑥,𝐵₁𝑥−𝜆₁𝐹(𝑤,𝑣),𝑦=𝑅^𝐻2_𝜆^{(⋅,⋅)−𝜂}_{2,𝑁⋅,𝑦}𝐻₂𝐴₂𝑦,𝐵₂𝑦−𝜆₂𝐺(𝑤,𝑣), (31) where 𝑅^𝐻1_𝜆^{(⋅,⋅)−𝜂}_{1,𝑀(⋅,𝑥)}=(𝐻₁(𝐴₁𝑥,𝐵₁𝑥)+𝜆₁𝑀(⋅,𝑥))⁻¹, 𝑅^𝐻2_𝜆^{(⋅,⋅)−𝜂}_{1,𝑁(⋅,𝑦)}=(𝐻₂(𝐴₂𝑦,𝐵₂𝑦)+𝜆₂𝑁(⋅,𝑦))⁻¹, and 𝜆₁,𝜆₂>0 are constants.

Proof . This is an easy and direct consequence of Definition 14.

Algorithm 17. For 𝑖=1,2, let 𝑋_𝑖 be real Banach spaces with the norm ‖⋅‖_𝑖. Let 𝐴_𝑖,𝐵_𝑖∶𝑋_𝑖→𝑋_𝑖, 𝐻_𝑖∶𝑋_𝑖×𝑋_𝑖→𝑋_𝑖, 𝜂_𝑖∶𝑋_𝑖×𝑋_𝑖→𝑋_𝑖, 𝐹∶𝑋₁×𝑋₂→𝑋₁, and 𝐺∶𝑋₁×𝑋₂→𝑋₂ be single-valued mappings, and 𝑇∶𝑋₁→𝐶𝐵(𝑋₁), 𝑄∶𝑋₂→𝐶𝐵(𝑋₂) be set-valued mappings. Let 𝑀∶𝑋₁×𝑋₁→→2^𝑋1, 𝑁∶𝑋₂×𝑋₂→2^𝑋2 be such that, for each fixed 𝑥∈𝑋₁, 𝑦∈𝑋₂, 𝑀(⋅,𝑥) and 𝑁(⋅,𝑦) are 𝐻₁(⋅,⋅)−𝜂₁-cocoercive and 𝐻₂(⋅,⋅)−𝜂₂-cocoercive operators with respect to (𝐴₁,𝐵₁) and (𝐴₂,𝐵₂), respectively. For any given constants 𝜆_𝑖>0(𝑖=1,2), define the mappings 𝑆₁∶𝑋₁×𝑋₂→𝑋₁ and 𝑆₂∶𝑋₁×𝑋₂→𝑋₂ by 𝑆₁=𝑥,𝑦_{(𝑥)𝑤∈𝑇𝑦𝑣∈𝑄}𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1(),𝑀⋅,𝑥}𝐻₁𝐴₁𝑥,𝐵₁𝑥−𝜆₁𝐹(,𝑆𝑤,𝑣)₂=𝑥,𝑦_{(𝑥)𝑤∈𝑇𝑦𝑣∈𝑄}𝑅^{𝐻2(⋅,⋅)−𝜂2}_{𝜆2,𝑁⋅,𝑦}𝐻₂𝐴₂𝑦,𝐵₂𝑦−𝜆₂𝐺(𝑤,𝑣). (32) For any given (𝑥₀,𝑦₀)∈𝑋₁×𝑋₂, 𝑤₀∈𝑇(𝑥₀), 𝑣₀∈𝑄(𝑦₀), let 𝑧₀=𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1,𝑀⋅,𝑥0}×𝐻₁𝐴₁𝑥₀,𝐵₁𝑥₀−𝜆₁𝐹𝑤₀,𝑣₀∈𝑆₁𝑥₀,𝑦₀,𝑢₀=𝑅^{𝐻2(⋅,⋅)−𝜂2}_{𝜆2,𝑁⋅,𝑦0}×𝐻₂𝐴₂𝑦₀,𝐵₂𝑦₀−𝜆₂𝐺𝑤₀,𝑣₀∈𝑆₂𝑥₀,𝑦₀. (33) Since 𝑤₀∈𝑇(𝑥₀)⊂𝐶𝐵(𝑋₁) and 𝑣₀∈𝑄(𝑦₀)⊂𝐶𝐵(𝑋₂), in view of Nadler's theorem [24], there exist 𝑤₁∈𝑇(𝑥₁) and 𝑣₁∈𝑄(𝑦₁) such that 𝑤₁−𝑤₀₁≤(1+1)𝐷𝑇𝑥₁,𝑇𝑥₀,𝑣₁−𝑣₀₂≤(1+1)𝐷𝑄𝑦₁,𝑄𝑦₀. (34) By induction, we define iterative sequences {𝑥_𝑛}, {𝑦_𝑛}, {𝑤_𝑛}, and {𝑣_𝑛} as follows: 𝑥_𝑛+1=𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1,𝑀⋅,𝑥𝑛}×𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛∈𝑆₁𝑥_𝑛,𝑦_𝑛,𝑦_𝑛+1=𝑅^{𝐻2(⋅,⋅)−𝜂2}_{𝜆2,𝑁⋅,𝑦𝑛}×𝐻₂𝐴₂𝑦_𝑛,𝐵₂𝑦_𝑛−𝜆₂𝐺𝑤_𝑛,𝑣_𝑛∈𝑆₂𝑥_𝑛,𝑦_𝑛,𝑤_𝑛∈𝑇𝑥_𝑛,𝑤_𝑛+1−𝑤_𝑛1≤11+𝐷𝑇𝑥𝑛+1^𝑛+1,𝑇𝑥_𝑛,𝑣_𝑛∈𝑄𝑦_𝑛,𝑣_𝑛+1−𝑣_𝑛2≤11+𝐷𝑄𝑦𝑛+1^𝑛+1,𝑄𝑦_𝑛, (35) where 𝑛=0,1,2,…, and 𝜆₁,𝜆₂>0 are constants.

Theorem 18. For 𝑖=1,2, let 𝑋_𝑖 be real Banach spaces with the norm ‖⋅‖_𝑖. Let 𝐴_𝑖,𝐵_𝑖∶𝑋_𝑖→𝑋_𝑖, 𝐻_𝑖∶𝑋_𝑖×𝑋_𝑖→𝑋_𝑖, 𝜂_𝑖∶𝑋_𝑖×𝑋_𝑖→𝑋_𝑖, 𝐹∶𝑋₁×𝑋₂→𝑋_1, and 𝐺∶𝑋₁×𝑋₂→𝑋₂ be single-valued mappings, and 𝑇∶𝑋₁→𝐶𝐵(𝑋₁), 𝑄∶𝑋₂→𝐶𝐵(𝑋₂) be set-valued mappings. Let 𝑀∶𝑋₁×𝑋₁→2^𝑋1 and 𝑁∶𝑋₂×𝑋₂→2^𝑋2 be such that, for each fixed 𝑥∈𝑋₁,𝑦∈𝑋₂, 𝑀(⋅,𝑥) and 𝑁(⋅,𝑦) are 𝐻₁(⋅,⋅)−𝜂₁-cocoercive and 𝐻₂(⋅,⋅)−𝜂₂-cocoercive operators with respect to (𝐴₁,𝐵₁) and (𝐴₂,𝐵₂), respectively. Suppose that the following conditions are satisfied.(i) 𝐻_𝑖(𝐴_𝑖,𝐵_𝑖) is 𝜂_𝑖-cocoercive with respect to 𝐴_𝑖 with constant 𝜇_𝑖 and 𝜂_𝑖-relaxed cocoercive with respect to 𝐵_𝑖 with constant 𝛾_𝑖, 𝑖=1,2.(ii) 𝐴_𝑖 is 𝛼_𝑖-expansive and 𝐵_𝑖 is 𝛽_𝑖-Lipschitz continuous, 𝑖=1,2.(iii) 𝐻_𝑖(𝐴_𝑖,𝐵_𝑖) is 𝑟_𝑖-Lipschitz continuous with respect to 𝐴_𝑖 and 𝑠_𝑖-Lipschitz continuous with respect to 𝐵_𝑖, 𝑖=1,2.(iv) 𝑇 is 𝐷-Lipschitz continuous with constant 𝜆_𝑇 and 𝑄 is 𝐷-Lipschitz continuous with constant 𝜆_𝑄.(v) 𝐹 is 𝑡₁-Lipschitz continuous with respect to 𝑇 in the first argument and 𝑡₂-Lipschitz continuous with respect to 𝑄 in the second argument.(vi) 𝐺 is 𝑙₁-Lipschitz continuous with respect to 𝑇 in the first argument and 𝑙₂-Lipschitz continuous with respect to 𝑄 in the second argument.(vii) 𝜂_𝑖 is 𝜌_𝑖-Lipschitz continuous, 𝑖=1,2.(viii)𝐹 is 𝜂₁-relaxed Lipschitz continuous with respect to 𝑇 in the first argument and 𝜂₁-relaxed Lipschitz continuous with respect to 𝑄 in the second argument with constants 𝜏₁ and 𝜏₂, respectively.(ix) 𝐺 is 𝜂₂-relaxed Lipschitz continuous with respect to 𝑇 in the first argument and 𝜂₂-relaxed Lipschitz continuous with respect to 𝑄 in the second argument with constants 𝜖₁ and 𝜖₂, respectively. Furthermore, assume that there exist constants 𝜎₁,𝜎₂>0 such that 𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1,𝑀⋅,𝑥1}(𝑥)−𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1,𝑀⋅,𝑥2}(𝑥)₁≤𝜎₁𝑥₁−𝑥₂₁,∀𝑥,𝑥₁,𝑥₂∈𝑋₁,𝑅^{𝐻2(⋅,⋅)−𝜂2}_{𝜆2,𝑁⋅,𝑦1}𝑦−𝑅^{𝐻2(⋅,⋅)−𝜂2}_{𝜆2,𝑁⋅,𝑦2}𝑦₂≤𝜎₂𝑦₁−𝑦₂₂,∀𝑦,𝑦₁,𝑦₂∈𝑋₂, (36)(37)and 𝜆₁,𝜆₂>0 are constants satisfying the following conditions: 𝑝₀=𝑟²₁+2𝜆₁𝑡₁𝜆_𝑇𝑟₁+𝜆₁𝑡₁𝜆_𝑇+𝜌₁−2𝜆₁𝜏₁,𝑞₀=𝑟²₂+2𝜆₂𝑙₂𝜆_𝑄𝑟₂+𝜆₂𝑙₂𝜆_𝑄+𝜌₂−2𝜆₂𝜖₂,𝑟²₁+2𝜆₁𝑡₁𝜆_𝑇𝑟₁+𝜆₁𝑡₁𝜆_𝑇+𝜌₁>2𝜆₁𝜏₁,𝑟²₂+2𝜆₂𝑙₂𝜆_𝑄𝑟₂+𝜆₂𝑙₂𝜆_𝑄+𝜌₂>2𝜆₂𝜖₂,𝜇_𝑖>𝛾_𝑖,𝛼_𝑖>𝛽_𝑖𝜃,𝑖=1,2,₁=𝜌₁𝜇₁𝛼²₁−𝛾₁𝛽²₁,𝜃₂=𝜌₂𝜇₂𝛼²₂−𝛾₂𝛽²₂,𝜎₁+𝜃₁𝑝₀+𝜃₁𝑠₁+𝜆₂𝜃₂𝑙₁𝜆_𝑇𝜎<1,₂+𝜃₂𝑞₀+𝜃₂𝑠₂+𝜆₁𝜃₁𝑡₂𝜆_𝑄<1. (38) Then, the iterative sequences {𝑥_𝑛}, {𝑦_𝑛}, {𝑤_𝑛}, and {𝑣_𝑛} generated by Algorithm 17 converge strongly to 𝑥, 𝑦, 𝑤, and 𝑣, respectively, and (𝑥,𝑦,𝑤,𝑣) is a solution of problem (30).

Proof. In view of Theorem 13, the resolvent operator 𝑅^{𝐻1(⋅,⋅)−𝜂1}_𝜆1,𝑀 is 𝜃₁-Lipschitz continuous. This, together with Algorithm 17 and (36), implies that𝑥_𝑛+1−𝑥_𝑛1=𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1,𝑀⋅,𝑥𝑛}𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1,𝑀⋅,𝑥𝑛−1}×𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛−1−𝜆₁𝐹𝑤_𝑛−1,𝑣_𝑛−11≤𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1,𝑀⋅,𝑥𝑛}𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1,𝑀⋅,𝑥𝑛−1}𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛1+𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1,𝑀⋅,𝑥𝑛−1}𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1,𝑀⋅,𝑥𝑛−1}×𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛−1−𝜆₁𝐹𝑤_𝑛−1,𝑣_𝑛−11≤𝜎₁𝑥_𝑛−𝑥_𝑛−11+𝜃₁𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛−1−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛−11≤𝜎₁𝑥_𝑛−𝑥_𝑛−11+𝜃₁𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛−1−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛1+𝜆₁𝜃₁𝐹𝑤_𝑛−1,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛−11≤𝜎₁𝑥_𝑛−𝑥_𝑛−11+𝜃₁𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛1+𝜃₁𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛−11+𝜆₁𝜃₁𝐹𝑤_𝑛−1,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛−11. (39) Since 𝐹 is 𝑡₁-Lipschitz continuous with respect to 𝑇 in the first argument and 𝑡₂-Lipschitz continuous in the second argument, 𝑇 is 𝜆_𝑇-Lipschitz continuous, and 𝑄 is 𝜆_𝑄-Lipschitz continuous, by Algorithm 17, we get 𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛1≤𝑡₁𝑤_𝑛−𝑤_𝑛−11≤𝑡₁11+𝑛𝐷𝑇𝑥_𝑛,𝑇𝑥_𝑛−1≤𝑡₁𝜆_𝑇11+𝑛𝑥_𝑛−𝑥_𝑛−11,𝐹𝑤_𝑛−1,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛−11≤𝑡₂𝑣_𝑛−𝑣_𝑛−12≤𝑡₂11+𝑛𝐷𝑄𝑦_𝑛,𝑄𝑦_𝑛−1≤𝑡₂𝜆_𝑄11+𝑛𝑦_𝑛−𝑦_𝑛−12. (40)(41) As 𝐻₁(⋅,⋅) is 𝑟₁-Lipschitz continuous with respect to 𝐴₁, we obtain 𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛1≤𝑟₁𝑥_𝑛−𝑥_𝑛−11. (42) Since 𝜂₁ is 𝜌₁-Lipschitz continuous, we conclude that 𝜂₁𝑥_𝑛,𝑥_𝑛−11≤𝜌₁𝑥_𝑛−𝑥_𝑛−11. (43)
Since 𝐻₁(⋅,⋅) is 𝜂₁-relaxed Lipschitz continuous with respect to 𝑇 and 𝜂₂-relaxed Lipschitz continuous with respect to 𝑄 in the first and second arguments with constants 𝜏₁ and 𝜏₂, respectively, we have𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛,𝑗𝜂₁𝑥_𝑛,𝑥_𝑛−1≤−𝜏₁𝑥_𝑛−𝑥_𝑛−1²₁. (44) Employing Lemma 3 and taking into account (39)–(44), we obtain 𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛²₁≤𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛²₁−2𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛,𝑗𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛=𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛²₁−2𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛,𝑗𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛+𝑗𝜂₁𝑥_𝑛,𝑥_𝑛−1+2𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛,𝑗𝜂₁𝑥_𝑛,𝑥_𝑛−1≤𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛²₁+2𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛1×𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛1+𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛1+𝜂₁𝑥_𝑛,𝑥_𝑛−11+2𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛,𝑗𝜂₁𝑥_𝑛,𝑥_𝑛−1≤𝑟²₁𝑥_𝑛−𝑥_𝑛−1²₁+2𝜆₁𝑡₁𝜆_𝑇11+𝑛𝑥_𝑛−𝑥_𝑛−11×𝑟₁𝑥_𝑛−𝑥_𝑛−11+𝜆₁𝑡₁𝜆_𝑇11+𝑛×𝑥_𝑛−𝑥_𝑛−11+𝜌₁𝑥_𝑛−𝑥_𝑛−11−2𝜆₁𝜏₁𝑥_𝑛−𝑥_𝑛−1²₁=𝑟²₁+2𝜆₁𝑡₁𝜆_𝑇11+𝑛×𝑟₁+𝜆₁𝑡₁𝜆_𝑇11+𝑛+𝜌₁−2𝜆₁𝜏₁𝑥_𝑛−𝑥_𝑛−1²₁. (45) This implies that 𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤_𝑛−1,𝑣_𝑛1≤𝑟²₁+2𝜆₁𝑡₁𝜆_𝑇11+𝑛𝑟₁+𝜆₁𝑡₁𝜆_𝑇11+𝑛+𝜌₁−2𝜆₁𝜏₁×𝑥_𝑛−𝑥_𝑛−11=𝑝_𝑛𝑥_𝑛−𝑥_𝑛−11, (46) where 𝑝_𝑛=𝑟²₁+2𝜆₁𝑡₁𝜆_𝑇11+𝑛𝑟₁+𝜆₁𝑡₁𝜆_𝑇11+𝑛+𝜌₁−2𝜆₁𝜏₁. (47)
Using 𝑠₁-Lipschitz continuity of 𝐻₁(⋅,⋅) with respect to 𝐵₁, we deduce that 𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥_𝑛−1,𝐵₁𝑥_𝑛−11≤𝑠₁𝑥_𝑛−𝑥_𝑛−11. (48) In view of (41), (46), (48), (39) becomes 𝑥_𝑛+1−𝑥_𝑛1≤𝜎₁𝑥_𝑛−𝑥_𝑛−11+𝜃₁𝑝_𝑛𝑥_𝑛−𝑥_𝑛−11+𝜃₁𝑠₁𝑥_𝑛−𝑥_𝑛−11+𝜆₁𝜃₁𝑡₂𝜆_𝑄11+𝑛𝑦_𝑛−𝑦_𝑛−12=𝜎₁+𝜃₁𝑝_𝑛+𝜃₁𝑠₁𝑥_𝑛−𝑥_𝑛−11+𝜆₁𝜃₁𝑡₂𝜆_𝑄11+𝑛𝑦_𝑛−𝑦_𝑛−12. (49) Similarly, we have 𝑦_𝑛+1−𝑦_𝑛2=𝑅^{𝐻2(⋅,⋅)−𝜂2}_{𝜆2,𝑁⋅,𝑦𝑛}𝐻₂𝐴₂𝑦_𝑛,𝐵₂𝑦_𝑛−𝜆₂𝐺𝑤_𝑛,𝑣_𝑛−𝑅^{𝐻2(⋅,⋅)−𝜂2}_{𝜆2,𝑁⋅,𝑦𝑛−1}𝐻₂𝐴₂𝑦_𝑛−1,𝐵₂𝑦_𝑛−1−𝜆₂𝐺𝑤_𝑛−1,𝑣_𝑛−12≤𝑅^{𝐻2(⋅,⋅)−𝜂2}_{𝜆2,𝑁⋅,𝑦𝑛}𝐻₂𝐴₂𝑦_𝑛,𝐵₂𝑦_𝑛−𝜆₂𝐺𝑤_𝑛,𝑣_𝑛−𝑅^{𝐻2(⋅,⋅)−𝜂2}_{𝜆2,𝑁⋅,𝑦𝑛−1}𝐻₂𝐴₂𝑦_𝑛,𝐵₂𝑦_𝑛−𝜆₂𝐺𝑤_𝑛,𝑣_𝑛2+𝑅^{𝐻2(⋅,⋅)−𝜂2}_{𝜆2,𝑁(⋅,𝑦𝑛−1)}𝐻₂𝐴₂𝑦_𝑛,𝐵₂𝑦_𝑛−𝜆₂𝐺𝑤_𝑛,𝑣_𝑛−𝑅^{𝐻2(⋅,⋅)−𝜂2}_{𝜆2,𝑁⋅,𝑦𝑛−1}𝐻₂𝐴₂𝑦_𝑛−1,𝐵₂𝑦_𝑛−1−𝜆₂𝐺𝑤_𝑛−1,𝑣_𝑛−12≤𝜎₂𝑦_𝑛−𝑦_𝑛−12+𝜃₂𝐻₂𝐴₂𝑦_𝑛,𝐵₂𝑦_𝑛−𝐻₂𝐴₂𝑦_𝑛−1,𝐵₂𝑦_𝑛−1−𝜆₂𝐺𝑤_𝑛,𝑣_𝑛−𝐺𝑤_𝑛−1,𝑣_𝑛−12≤𝜎₂𝑦_𝑛−𝑦_𝑛−12+𝜃₂𝐻₂𝐴₂𝑦_𝑛,𝐵₂𝑦_𝑛−𝐻₂𝐴₂𝑦_𝑛−1,𝐵₂𝑦_𝑛−1−𝜆₂𝐺𝑤_𝑛,𝑣_𝑛−𝐺𝑤_𝑛,𝑣_𝑛−12+𝜆₂𝜃₂𝐺𝑤_𝑛,𝑣_𝑛−1−𝐺𝑤_𝑛−1,𝑣_𝑛−12≤𝜎₂𝑦_𝑛−𝑦_𝑛−12+𝜃₂𝐻₂𝐴₂𝑦_𝑛,𝐵₂𝑦_𝑛−𝐻₂𝐴₂𝑦_𝑛−1,𝐵₂𝑦_𝑛−𝜆₂𝐺𝑤_𝑛,𝑣_𝑛−𝐺𝑤_𝑛,𝑣_𝑛−12+𝜃₂𝐻₂𝐴₂𝑦_𝑛−1,𝐵₂𝑦_𝑛−𝐻₂𝐴₂𝑦_𝑛−1,𝐵₂𝑦_𝑛−12+𝜆₂𝜃₂𝐺𝑤_𝑛,𝑣_𝑛−1−𝐺𝑤_𝑛−1,𝑣_𝑛−12≤𝜎₂𝑦_𝑛−𝑦_𝑛−12+𝜃₂𝑞_𝑛𝑦_𝑛−𝑦_𝑛−12+𝜃₂𝑠₂𝑦_𝑛−𝑦_𝑛−12+𝜆₂𝜃₂𝑡₁𝜆_𝑇11+𝑛𝑥_𝑛−𝑥_𝑛−11=𝜎₂+𝜃₂𝑞_𝑛+𝜃₂𝑠₂𝑦_𝑛−𝑦_𝑛−12+𝜆₂𝜃₂𝑙₁𝜆_𝑇11+𝑛𝑥_𝑛−𝑥_𝑛−11, (50) where 𝑞_𝑛=𝑟²₂+2𝜆₂𝑙₂𝜆_𝑄11+𝑛𝑟₂+𝜆₂𝑙₂𝜆_𝑄11+𝑛+𝜌₂−2𝜆₂𝜖₂. (51)
In view of (49) and (50), we obtain 𝑥_𝑛+1−𝑥_𝑛1+𝑦_𝑛+1−𝑦_𝑛2≤𝑐_𝑛𝑥_𝑛−𝑥_𝑛−11+𝑑_𝑛𝑦_𝑛−𝑦_𝑛−12≤𝑘_𝑛𝑥_𝑛−𝑥_𝑛−11+𝑦_𝑛−𝑦_𝑛−12, (52) where 𝑐_𝑛=𝜎₁+𝜃₁𝑝_𝑛+𝜃₁𝑠₁+𝜆₂𝜃₂𝑙₁𝜆_𝑇(1+1/𝑛), 𝑑_𝑛=𝜎₂+𝜃₂𝑞_𝑛+𝜃₂𝑠₂+𝜆₁𝜃₁𝑡₂𝜆_𝑄(1+1/𝑛), and 𝑘_𝑛=max{𝑐_𝑛,𝑑_𝑛}.
Letting 𝑛→∞, we obtain 𝑘_𝑛→𝑘, where 𝑘=max𝜎₁+𝜃₁𝑝₀+𝜃₁𝑠₁+𝜆₂𝜃₂𝑙₁𝜆_𝑇,𝜎₂+𝜃₂𝑞₀+𝜃₂𝑠₂+𝜆₁𝜃₁𝑡₂𝜆_𝑄. (53)
Next, we define the norm ‖⋅‖ on 𝑋₁×𝑋₂ by 𝑥,𝑦=‖𝑥‖₁+𝑦₂,𝑥,𝑦∈𝑋₁×𝑋₂. (54) One can easily check that (𝑋₁×𝑋₂,‖⋅‖) is a Banach space.
Define 𝑎_𝑛+1=(𝑥_𝑛+1,𝑦_𝑛+1). Then, we have 𝑎_𝑛+1−𝑎_𝑛=𝑥_𝑛+1−𝑥_𝑛1+𝑦_𝑛+1−𝑦_𝑛2. (55)
In view of (38), we conclude that 0<𝑘<1. This implies that there exist 𝑛₀∈ℕ and 𝑘₀∈(0,1) such that 𝑘_𝑛≤𝑘₀ for all 𝑛≥𝑛₀. It follows from (52) and (54) that 𝑎_𝑛+1−𝑎_𝑛≤𝑘₀𝑎_𝑛−𝑎_𝑛−1,∀𝑛≥𝑛₀. (56) In view of (56), we obtain 𝑎_𝑛+1−𝑎_𝑛≤𝑘^𝑛−𝑛0₀𝑎_𝑛0+1−𝑎_𝑛0,∀𝑛≥𝑛₀. (57) This implies that for any 𝑚≥𝑛≥𝑛₀, 𝑥_𝑚−𝑥_𝑛1≤𝑎_𝑚−𝑎_𝑛≤^𝑚−1_𝑖=𝑛𝑎_𝑖+1−𝑎_𝑖≤^𝑚−1_𝑖=𝑛𝑘^𝑖−𝑛0₀𝑎_𝑛0+1−𝑎_𝑛0. (58) Since 0<𝑘₀<1, it follows from (58) that ‖𝑥_𝑚−𝑥_𝑛‖₁→0 and 𝑛→∞. This proves that {𝑥_𝑛} is a Cauchy sequence in 𝑋₁. Similarly, we conclude that {𝑦_𝑛} is a Cauchy sequence in 𝑋₂. Thus, there exist 𝑥∈𝑋₁ and 𝑦∈𝑋₂ such that 𝑥_𝑛→𝑥 and 𝑦_𝑛→𝑦 as 𝑛→∞.
Next, we prove that 𝑤_𝑛→𝑤∈𝑇(𝑥) and 𝑣_𝑛→𝑣∈𝑄(𝑦). In view of Lipschitz continuity of 𝑇 and 𝑄 and Algorithm 17, we obtain 𝑤_𝑛−𝑤_𝑛−11≤11+𝑛𝜆_𝑇𝑥_𝑛−𝑥_𝑛−11,𝑣_𝑛−𝑣_𝑛−12≤11+𝑛𝜆_𝑄𝑦_𝑛−𝑦_𝑛−12. (59) From (59), we deduce that {𝑤_𝑛},{𝑣_𝑛} are Cauchy sequences in 𝑋₁ and 𝑋₂, respectively. Thus, there exist 𝑤∈𝑇(𝑥) and 𝑣∈𝑄(𝑦) such that 𝑤_𝑛→𝑤 and 𝑣_𝑛→𝑣 as 𝑛→∞. Since 𝑇 is 𝐷-Lipschitz continuous with constant 𝜆_𝑇, it is obvious that 𝑑(𝑤,𝑇(𝑥))≤𝑤−𝑤_𝑛1+𝑑𝑤_𝑛(𝑥)≤,𝑇𝑤−𝑤_𝑛−11+𝐷𝑇𝑥_𝑛(𝑥)≤,𝑇𝑤−𝑤_𝑛1+𝜆_𝑇‖𝑥_𝑛−𝑥‖₁⟶0(𝑛⟶∞). (60) By the closedness of 𝑇(𝑥), we conclude that 𝑤∈𝑇(𝑥). Similarly, we have 𝑣∈𝑄(𝑦).
Assume now that 𝑥₀=𝑅^𝐻1_𝜆^{(⋅,⋅)−𝜂}_{1(),𝑀⋅,𝑥}𝐻₁𝐴₁𝑥,𝐵₁𝑥−𝜆₁𝐹(,𝑦𝑤,𝑣)₀=𝑅^𝐻2_𝜆^{(⋅,⋅)−𝜂}_{2,𝑁⋅,𝑦}𝐻₂𝐴₂𝑦,𝐵₂𝑦−𝜆₂𝐺(𝑤,𝑣). (61) Then, we have 𝑥_𝑛+1−𝑥₀₁≤𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1,𝑀⋅,𝑥𝑛}𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1(),𝑀⋅,𝑥}𝐻₁𝐴₁𝑥,𝐵₁𝑥−𝜆₁𝐹(𝑤,𝑣)₁≤𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1,𝑀⋅.,𝑥𝑛}𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1(),𝑀⋅,𝑥}𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛1+𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1(),𝑀⋅,𝑥}𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝑅^{𝐻1(⋅,⋅)−𝜂1}_{𝜆1(),𝑀⋅,𝑥}𝐻₁𝐴₁𝑥,𝐵₁𝑥−𝜆₁𝐹(𝑤,𝑣)₁≤𝜎₁𝑥_𝑛−𝑥₁+𝜃₁𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐻₁𝐴₁𝑥,𝐵₁𝑥−𝜆₁𝐹(𝑤,𝑣)₁≤𝜎₁𝑥_𝑛−𝑥₁+𝜃₁𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥,𝐵₁𝑥₁+𝜆₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹(𝑤,𝑣)₁≤𝜎₁𝑥_𝑛−𝑥₁+𝜃₁𝐻₁𝐴₁𝑥_𝑛,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥,𝐵₁𝑥_𝑛1+𝐻₁𝐴₁𝑥,𝐵₁𝑥_𝑛−𝐻₁𝐴₁𝑥,𝐵₁𝑥₁+𝜆₁𝜃₁𝐹𝑤_𝑛,𝑣_𝑛−𝐹𝑤,𝑣_𝑛1+𝐹𝑤,𝑣_𝑛−𝐹(𝑤,𝑣)₁≤𝜎₁𝑥_𝑛−𝑥₁+𝜃₁𝑟₁+𝑠₁𝑥_𝑛−𝑥₁+𝜆₁𝜃₁𝑡₁𝑤_𝑛−𝑤₁+𝑡₂𝑣_𝑛−𝑣₂=𝜎₁+𝜃₁𝑟₁+𝜃₁𝑠₁𝑥_𝑛−𝑥₁+𝜆₁𝜃₁𝑡₁𝑤_𝑛−𝑤₁+𝑡₂𝑣_𝑛−𝑣₂. (62) Since 𝑥_𝑛→𝑥, 𝑤_𝑛→𝑤, and 𝑣_𝑛→𝑣 as 𝑛→∞, it follows from (62) that lim_𝑛→∞𝑥_𝑛−𝑥₀₁=0, (63) and hence 𝑥₀=𝑥.
A similar argument shows that 𝑦₀=𝑦. Therefore, 𝑥=𝑥₀=𝑅^𝐻1_𝜆^{(⋅,⋅)−𝜂}_{1(),𝑀⋅,𝑥}𝐻₁𝐴₁𝑥,𝐵₁𝑥−𝜆₁𝐹(𝑤,𝑣),𝑦=𝑦₀=𝑅^𝐻2_𝜆^{(⋅,⋅)−𝜂}_{2,𝑁⋅,𝑦}𝐻₂𝐴₂𝑦,𝐵₂𝑦−𝜆₂𝐺(𝑤,𝑣). (64) In view of Lemma 16, we conclude that (𝑥,𝑦,𝑤,𝑣) is a solution of problem (30), which completes the proof.

At the end of this paper, we include the following simple example in support of Theorem 18.

Example 19. Let 𝑋=ℝ² with the usual inner product. We define two mappings 𝐴,𝐵∶ℝ²→ℝ² by 𝐴(𝑥)1∶=4𝑥₁−𝑥₂,𝑥₁+14𝑥₂,𝐵(𝑥)−1∶=4𝑥₁+14𝑥₂1,−4𝑥₁−14𝑥₂,∀𝑥=𝑥₁,𝑥₂∈ℝ². (65) Let a mapping 𝐻∶ℝ²×ℝ²→ℝ² be defined by 𝐻𝐴𝑥,𝐵𝑦∶=𝐴𝑥+𝐵𝑦,∀𝑥,𝑦∈ℝ². (66) By similar arguments, as in Example 4.1 of [27], we can prove the following.(1) 𝐻(𝐴,𝐵) is 4/17-cocoercive with respect to 𝐴 and 1-relaxed cocoercive with respect to 𝐵.(2) 𝐴 is √17/𝑛-expansive, for 𝑛=4,5.(3) 𝐵 is 1/√𝑛-expansive, for 𝑛=1,2.(4) 𝐻(𝐴,𝐵) is √17/𝑛-Lipschitz continuous with constant √17/𝑛 with respect to 𝐴 and 𝐵, for 𝑛=1,2,…,15,16.(5) Let 𝑓,𝑔∶ℝ²→ℝ² be defined by 𝑓(𝑥)∶=8𝑥₁−85𝑥₂,85𝑥₁+8𝑥₂,𝑔(𝑥)∶=−178𝑥₁+58𝑥₂,58𝑥₁+178𝑥₂,∀𝑥=𝑥₁,𝑥₂∈ℝ². (67)(6) Now, we define a mapping 𝑀∶ℝ²×ℝ²→ℝ² by 𝑀𝑓𝑥,𝑔𝑦∶=𝑓𝑥−𝑔𝑦,∀𝑥,𝑦∈ℝ². (68)Let 𝑅,𝑆,𝑇∶ℝ²→ℝ² be the identity mappings. It is obvious that these mappings are 𝐷-Lipschitz continuous.(7)Assume that 𝐹,𝐺∶ℝ²→ℝ² are defined by 𝐹(𝑥)−1∶=4𝑥₁−18𝑥₂,18𝑥₁−14𝑥₂,𝐺(𝑥)−1∶=8𝑥₁+15𝑥₂1,−5𝑥₁−18𝑥₂,∀𝑥=𝑥₁,𝑥₂∈ℝ². (69) It could easily be seen that all the aspects of the hypotheses of Theorem 18 are satisfied, so we have the desired conclusion.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was funded by King Abdulaziz University, under Grant no. 23-130-1433-HiCi. The authors, therefore, acknowledge the technical and financial support of KAU. The authors would like to thank the referees for their sincere evaluation and constructive comments which improved the paper considerably.