Abstract

We introduce a new class of generalized accretive mappings, named (𝐻,𝜙)-𝜂-accretive mappings, in Banach spaces. We define a resolvent operator associated with (𝐻,𝜙)-𝜂-accretive mappings and show its Lipschitz continuity. We also introduce and study a new system of generalized variational inclusions with (𝐻,𝜙)-𝜂-accretive mappings in Banach spaces. By using the resolvent operator technique associated with (𝐻,𝜙)-𝜂-accretive mappings, we construct a new iterative algorithm for solving this system of generalized variational inclusions in Banach spaces. We also prove the existence of solutions for the generalized variational inclusions and the convergence of iterative sequences generated by algorithm. Our results improve and generalize many known corresponding results.

1. Introduction

Variational inequalities and variational inclusions are among the most interesting and important mathematical problems and have been studied intensively in the past years since they have wide applications in mechanics, physics, optimization and control, nonlinear programming, economics and transportation equilibrium, and engineering sciences, and so forth (see, e.g., [1–4]).

Recently Ding [5], Huang and Fang [6], Verma [7], Fang and Huang [8, 9], Huang and Fang [10], Fang et al. [11], Kazmi and Khan [12], and Lan et al. [13, 14] introduced the concepts of 𝜂-subdifferential operators, maximal 𝜂-monotone operators, A-monotone operators, and (𝐻,𝜂)-monotone operators in Hilbert spaces, H-accretive operators, generalized m-accretive mappings, (𝐻,𝜂)-accretive operators, P-𝜂-accretive operators, and (𝐴,𝜂)-accretive mappings in Banach spaces, and their resolvent operators, respectively. In [15], Luo and Huang introduced a new concept of (𝐻,𝜙)-𝜂-monotone mappings in Banach spaces and defined the proximal mapping associated with (𝐻,𝜙)-𝜂-monotone mappings.

Motivated and inspired by the research work going on this field, in this paper, we introduce a new concept of (𝐻,𝜙)-𝜂-accretive mappings and give the definition of its resolvent operator in Banach spaces. We also introduce and study a new system of generalized variational inclusions with (𝐻,𝜙)-𝜂-accretive mappings in Banach spaces, and we construct a new iterative algorithm for solving this system of generalized variational inclusions in Banach spaces. We also prove the existence of solutions for the generalized variational inclusions and the convergence of iterative sequences generated by algorithm. The results in this paper improve and extend some known results in the literature.

2. Preliminaries

Let 𝐸 be a real Banach space equipped with norm ‖⋅‖; let 𝐸∗ be the topological dual space of 𝐸; let ⟨⋅,⋅⟩ be the pair between 𝐸 and 𝐸∗; let 2𝐸 be the power set of 𝐸; let 𝐻(⋅,⋅) be the Hausdorff metric on CB(𝐸) defined by𝐻(𝐴,𝐵)=maxsup𝑥∈𝐴𝑑(𝑥,𝐵),sup𝑦∈𝐵𝑑(𝐴,𝑦),∀𝐴,𝐵∈CB(𝐸).(2.1)

Definition 2.1 (see [16, 17]). For 𝑞>1, a mapping 𝐽𝑞∶𝐸→2𝐸∗ is said to be generalized duality mapping if it is defined by 𝐽𝑞(𝑥)=𝑓∈𝐸∗∶⟨𝑥,𝑓⟩=‖𝑥‖𝑞,‖𝑥‖𝑞−1=‖𝑓‖,∀𝑥∈𝐸.(2.2)
In particular, 𝐽2 is the usual normalized duality mapping on 𝐸.
It is well known that 𝐽𝑞(𝑥)=‖𝑥‖𝑞−1𝐽2(𝑥),∀𝑥(≠0)∈𝐸.(2.3)
Note that if 𝐸=ℋ is a real Hilbert space, then 𝐽2 becomes the identity mapping on ℋ.

Definition 2.2 (see [18]). A Banach space 𝐸 is called smooth, if for every 𝑥∈𝐸 with ‖𝑥‖=1, there exists a unique 𝑓∈𝐸∗ such that ‖𝑓‖=𝑓(𝑥)=1. The modulus of 𝐸 is the function 𝜌𝐸∶[0,∞)→[0,∞), defined by 𝜌𝐸(𝜏)=sup‖𝑥+𝑦‖+‖𝑥−𝑦‖2.−1∶𝑥,𝑦∈𝐸,‖𝑥‖=1,‖𝑦‖=𝜏(2.4)

Definition 2.3 (see [17]). The Banach space 𝐸 is said to be(i)uniformly smooth, if lim𝜏→0𝜌𝐸(𝜏)𝜏=0;(2.5)(ii)q-uniformly smooth, for 𝑞>1, if there exists a constant 𝜏>0 such that 𝜌𝐸(𝜏)≤𝑐𝜏𝑞[,𝜏∈0,∞).(2.6) It is well known (see [16]) that𝐿𝑞or𝑙𝑞is𝑞-uniformlysmooth,if1<𝑞≤2,2-uniformlysmooth,if𝑞≥2.(2.7)
Note that if 𝐸 is uniformly smooth, 𝐽𝑞 becomes single-valued, and 𝐽𝑞 is single-valued if 𝐸∗ is strictly convex. In the sequel, unless otherwise specified, we always suppose that 𝐸 is a real Banach space such that 𝐽𝑞 is single-valued.

Lemma 2.4 (see [19]). Let 𝑞>1 be a real number and let 𝐸 be a smooth Banach space. Then 𝐸 is q-uniformly smooth if and only if there exists a constant 𝑐𝑞>0 such that for every 𝑥,𝑦∈𝐸, ‖𝑥+𝑦‖𝑞≤‖𝑥‖𝑞+𝑞𝑦,𝐽𝑞(𝑥)+𝑐𝑞‖𝑦‖𝑞.(2.8)

Definition 2.5 (see [9, 20]). Let 𝑃∶𝐸→𝐸 be a single-valued mapping. 𝑃 is said to be(i)accretive if 𝑃(𝑥)−𝑃(𝑦),𝐽𝑞(𝑥−𝑦)≥0,iff𝑥,𝑦∈𝐸;(2.9)(ii)strictly accretive if P is accretive and 𝑃(𝑥)−𝑃(𝑦),𝐽𝑞(𝑥−𝑦)=0if𝑥=𝑦;(2.10)(iii)r-strongly accretive if there exists a constant 𝑟>0 such that 𝑃(𝑥)−𝑃(𝑦),𝐽𝑞(𝑥−𝑦)≥𝑟‖𝑥−𝑦‖𝑞,∀𝑥,𝑦∈𝐸;(2.11)(iv)m-relaxed accretive if there exists a constant 𝑚>0 such that 𝑃(𝑥)−𝑃(𝑦),𝐽𝑞(𝑥−𝑦)≥−𝑚‖𝑥−𝑦‖𝑞,∀𝑥,𝑦∈𝐸;(2.12)(v)(𝛼,𝜉)-relaxed cocoercive if there exist constants 𝛼,𝜉>0 such that 𝑃(𝑥)−𝑃(𝑦),𝐽𝑞‖(𝑥−𝑦)≥−𝛼‖𝑃(𝑥)−𝑃(𝑦)𝑞+𝜉‖𝑥−𝑦‖𝑞,∀𝑥,𝑦∈𝐸;(2.13)(vi)s-Lipschitz continuous if there exists a constant 𝑠>0 such that ‖𝑃(𝑥)−𝑃(𝑦)‖≤𝑠‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐸.(2.14)

Definition 2.6 (see [12]). A mapping 𝜂∶𝐸×𝐸→𝐸 is said to be 𝜏-Lipschitz continuous if there exists a constant 𝜏>0 such that ‖𝜂(𝑥,𝑦)‖≤𝜏‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐸.(2.15)

Definition 2.7 (see [20]). Let 𝜂∶𝐸×𝐸→𝐸 and 𝑃∶𝐸→𝐸 be single-valued mappings. Then a multi-valued mapping 𝑇∶𝐸→2𝐸 is said to be(i)𝜂-accretive if 𝑢−𝑣,𝐽𝑞(𝜂(𝑥,𝑦))≥0,∀𝑥,𝑦∈𝐸,∀𝑢∈𝑇(𝑥),𝑣∈𝑇(𝑦);(2.16)(ii)strictly 𝜂-accretive if 𝑢−𝑣,𝐽𝑞(𝜂(𝑥,𝑦))≥0,∀𝑥,𝑦∈𝐸,∀𝑢∈𝑇(𝑥),𝑣∈𝑇(𝑦),(2.17) and equality holds if and only if 𝑥=𝑦;(iii)𝛾-strongly 𝜂-accretive if there exists a constant 𝛾>0 such that 𝑢−𝑣,𝐽𝑞(𝜂(𝑥,𝑦))≥𝛾‖𝑥−𝑦‖𝑞,∀𝑥,𝑦∈𝐸,∀𝑢∈𝑇(𝑥),𝑣∈𝑇(𝑦);(2.18)(iv)m-relaxed 𝜂-accretive if there exists a constant 𝑚>0 such that 𝑢−𝑣,𝐽𝑞(𝜂(𝑥,𝑦))≥−𝑚‖𝑥−𝑦‖𝑞,∀𝑥,𝑦∈𝐸,∀𝑢∈𝑇(𝑥),𝑣∈𝑇(𝑦);(2.19)(v)𝜂-𝑚-accretive if 𝑇 is 𝜂-accretive and (𝐼+𝜌𝑇)(𝐸)=𝐸 holds for all 𝜌>0;(vi)𝑃-𝜂-accretive if 𝑇 is 𝜂-accretive and (𝑃+𝜌𝑇)(𝐸)=𝐸 holds for all 𝜌>0.

Definition 2.8. Let 𝐸1,𝐸2,…,𝐸𝑛 be real Banach spaces, and, for 𝑖=1,2,…,𝑛, let 𝑁𝑖∶∏𝑛𝑗=1𝐸𝑗→𝐸𝑖 be a single-valued mapping. Then 𝑁𝑖 is said to be 𝜉𝑖𝑗-Lipschitz continuous in the 𝑗th argument, if there exists a constant 𝜉𝑖𝑗>0 such that ‖‖𝑁𝑖𝑥1,…,𝑥𝑗−1,𝑦𝑗1,𝑥𝑗+1,…,𝑥𝑛−𝑁𝑖𝑥1,…,𝑥𝑗−1,𝑦𝑗2,𝑥𝑗+1,…,𝑥𝑛‖‖≤𝜉𝑖𝑗‖‖𝑦𝑗1−𝑦𝑗2‖‖,∀𝑦𝑗1,𝑦𝑗2∈𝐸𝑗,𝑥𝑖∈𝐸𝑖(𝑖∈{1,2,…,𝑛},𝑖≠𝑗).(2.20)

Definition 2.9. Let 𝐴∶𝐸→CB(𝐸) be a set-valued mapping. 𝐴 is said to be 𝐻-Lipschitz continuous, if there exists a constant 𝑡>0 such that 𝐻(𝐴(𝑥),𝐵(𝑦))≤𝑡‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐸,(2.21) where 𝐻(⋅,⋅) denotes the Hausdorff metric on CB(𝐸).

Definition 2.10. Let ℋ be a Hilbert space, and let 𝐻∶ℋ→ℋ be a single-valued mapping. 𝐻 is said to be(i)coercive if lim‖𝑥‖→∞⟨𝐻(𝑥),𝑥⟩‖𝑥‖=+∞;(2.22)(ii)hemicontinuous if for any fixed 𝑥,𝑦,𝑧∈ℋ, the function 𝑡→⟨𝐻(𝑥+𝑡𝑦),𝑧⟩ is continuous at 0+.

3. (𝐻,𝜙)-𝜂-Accretive Mappings

In this section, we will introduce a new class of generalized accretive mappings, (𝐻,𝜙)-𝜂-accretive mappings, and discuss some properties of (𝐻,𝜙)-𝜂-accretive mappings.

Definition 3.1. Let 𝐸 be a Banach space and let 𝐻,𝜙∶𝐸→𝐸, 𝜂∶𝐸×𝐸→𝐸 be single-valued mappings and 𝑀∶𝐸→2𝐸 a multi-valued mapping. The mapping 𝑀 is said to be a (𝐻,𝜙)-𝜂-accretive mapping, if 𝜙∘𝑀 is 𝜂-accretive and (𝐻+𝜙∘𝑀)(𝐸)=𝐸.

Remark 3.2. (i) If 𝑀 is 𝜂-accretive and 𝜙(𝑥)=𝜆𝑥, for all 𝑥∈𝐸, 𝜆>0, then (𝐻,𝜙)-𝜂-accretive mapping reduces to the 𝑃-𝜂-accretive mapping studied by Kazmi and Khan [12].
(ii) If 𝜂(𝑥,𝑦)=𝑥−𝑦, for all 𝑥,𝑦∈𝐸, 𝑀 is 𝜂-accretive, and 𝜙(𝑥)=𝜆𝑥, for all 𝑥∈𝐸, 𝜆>0, then (𝐻,𝜙)-𝜂-accretive mapping reduces to the 𝐻-accretive mapping studied by Fang and Huang [9].
(iii) If 𝑀 is 𝑚-relaxed 𝜂-accretive and 𝜙(𝑥)=𝜆𝑥, for all 𝑥∈𝐸, 𝜆>0, then (𝐻,𝜙)-𝜂-accretive mapping reduces to the (𝐴,𝜂)-accretive mapping studied by Lan et al. [14].
Similarly, we give the following definition.

Definition 3.3. Let 𝐻,𝜙∶𝐸→𝐸 be single-valued mappings and 𝑀∶𝐸→2𝐸 a multi-valued mapping. The mapping 𝑀 is said to be a (𝐻,𝜙)-accretive mapping, if 𝜙∘𝑀 is accretive and (𝐻+𝜙∘𝑀)(𝐸)=𝐸.

Example 3.4. Let ℋ be a Hilbert space and for every 𝑥,𝑦∈ℋ, 𝜂(x,𝑦)=𝑥−𝑦, 𝜙(𝑥)=𝜆𝑥, where 𝜆>0 is a constant. Let 𝑀∶ℋ→2ℋ be a maximal monotone mapping and 𝐻∶ℋ→ℋ a bounded, coercive, hemicontinuous, and 𝛼-strongly 𝜂-accretive mapping. Then it follows from Corollary  32.26 of [21] that 𝑀 is (𝐻,𝜙)-𝜂-accretive mapping.

Theorem 3.5. Let 𝜙∶𝐸→𝐸, 𝜂∶𝐸×𝐸→𝐸 be single-valued mappings, 𝐻∶𝐸→𝐸 a strictly 𝜂-accretive mapping, and 𝑀∶𝐸→2𝐸 a (𝐻,𝜙)-𝜂-accretive mapping. Then (𝐻+𝜙∘𝑀)−1 is a single-valued mapping.

Proof. For any given 𝑧∈𝐸, let 𝑥,𝑦∈(𝐻+𝜙∘𝑀)−1(𝑧). It follows that 𝑧−𝐻(𝑥)∈𝜙∘𝑀(𝑥),𝑧−𝐻(𝑦)∈𝜙∘𝑀(𝑦).(3.1) Then 𝜂-accretivity of 𝜙∘𝑀 implies that 𝑧−𝐻(𝑥)−(𝑧−𝐻(𝑦)),𝐽𝑞=(𝜂(𝑥,𝑦))−𝐻(𝑥)+𝐻(𝑦),𝐽𝑞(𝜂(𝑥,𝑦))≥0.(3.2) This implies that 𝑥=𝑦 and so (𝐻+𝜙∘𝑀)−1 is a single-valued mapping. This completes the proof.

By Theorem 3.5, we can define the resolvent operator 𝐽𝐻,𝜂𝑀,𝜙 associated with an (𝐻,𝜙)-𝜂-accretive mapping 𝑀 as follows.

Definition 3.6. Let 𝜙∶𝐸→𝐸, 𝜂∶𝐸×𝐸→𝐸 be single-valued mappings, let 𝐻∶𝐸→𝐸 be a strictly 𝜂-accretive mapping, and let 𝑀∶𝐸→2𝐸 be a (𝐻,𝜙)-𝜂-accretive mapping. A resolvent operator 𝐽𝐻,𝜂𝑀,𝜙∶𝐸→𝐸 is defined by 𝐽𝐻,𝜂𝑀,𝜙(𝑥)=(𝐻+𝜙∘𝑀)−1(𝑥),∀𝑥∈𝐸.(3.3)

Remark 3.7. (i) If 𝑀 is 𝜂-accretive and 𝜙(𝑥)=𝜆𝑥, for all 𝑥∈𝐸, 𝜆>0, then the resolvent operator 𝐽𝐻,𝜂𝑀,𝜙 reduces to the 𝑃-𝜂-proximal point mapping 𝐽𝑀𝜂,𝜌 introduced by Kazmi and Khan [12].
(ii) If 𝜂(𝑥,𝑦)=𝑥−𝑦, for all 𝑥,𝑦∈𝐸, 𝑀 is 𝜂-accretive, and 𝜙(𝑥)=𝜆𝑥, for all 𝑥∈𝐸, 𝜆>0, then the resolvent operator 𝐽𝐻,𝜂𝑀,𝜙 reduces to the proximal-point mapping introduced by Fang and Huang [9].
(iii) If 𝑀 is 𝑚-relaxed 𝜂-accretive and 𝜙(𝑥)=𝜆𝑥, for all 𝑥∈𝐸, 𝜆>0, then the resolvent operator 𝐽𝐻,𝜂𝑀,𝜙 reduces to the resolvent operator 𝑅𝜌,𝐴𝜂,𝑀 introduced by Lan at el. [14].

Theorem 3.8. Let 𝜙∶𝐸→𝐸 be a single-valued mapping, let 𝜂∶𝐸×𝐸→𝐸 be a 𝜏-Lipschitz continuous mapping, let 𝐻∶𝐸→𝐸 be a 𝛾-strongly 𝜂-accretive mapping, and let 𝑀∶𝐸→2𝐸 be a (𝐻,𝜙)-𝜂-accretive mapping. Then the resolvent operator 𝐽𝐻,𝜂𝑀,𝜙 is Lipschitz continuous with constant 𝜏𝑞−1/𝛾, that is, ‖‖𝐽𝐻,𝜂𝑀,𝜙(𝑥)−𝐽𝐻,𝜂𝑀,𝜙‖‖≤𝜏(𝑦)𝑞−1𝛾‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐸.(3.4)

Proof. Let 𝑥,𝑦 be any given points in 𝐸, it follows from Definition 3.6 that 𝐽𝐻,𝜂𝑀,𝜙(𝑥)=(𝐻+𝜙∘𝑀)−1(𝑥),𝐽𝐻,𝜂𝑀,𝜙(𝑦)=(𝐻+𝜙∘𝑀)−1(𝑦).(3.5) This implies that 𝐽𝑥−𝐻𝐻,𝜂𝑀,𝜙𝐽(𝑥)∈𝜙∘𝑀𝐻,𝜂𝑀,𝜙,𝐽(𝑥)𝑦−𝐻𝐻,𝜂𝑀,𝜙𝐽(𝑦)∈𝜙∘𝑀𝐻,𝜂𝑀,𝜙.(𝑦)(3.6) Since 𝑀 is (𝐻,𝜙)-𝜂-accretive, we have 𝐽𝑥−𝐻𝐻,𝜂𝑀,𝜙−𝐽(𝑥)𝑦−𝐻𝐻,𝜂𝑀,𝜙(𝑦),𝐽𝑞𝜂𝐽𝐻,𝜂𝑀,𝜙(𝑥),𝐽𝐻,𝜂𝑀,𝜙=𝐻𝐽(𝑦)𝑥−𝑦−𝐻,𝜂𝑀,𝜙𝐽(𝑥)−𝐻𝐻,𝜂𝑀,𝜙(𝑦),𝐽𝑞𝜂𝐽𝐻,𝜂𝑀,𝜙(𝑥),𝐽𝐻,𝜂𝑀,𝜙(𝑦)≥0.(3.7) The inequality above implies that ‖‖𝜂𝐽‖𝑥−𝑦‖𝐻,𝜂𝑀,𝜙(𝑥),𝐽𝐻,𝜂𝑀,𝜙‖‖(𝑦)𝑞−1=‖‖𝐽‖𝑥−𝑦‖𝑞𝜂𝐽𝐻,𝜂𝑀,𝜙(𝑥),𝐽𝐻,𝜂𝑀,𝜙‖‖≥(𝑦)𝑥−𝑦,𝐽𝑞𝜂𝐽𝐻,𝜂𝑀,𝜙(𝑥),𝐽𝐻,𝜂𝑀,𝜙≥𝐻𝐽(𝑦)𝐻,𝜂𝑀,𝜙𝐽(𝑥)−𝐻𝐻,𝜂𝑀,𝜙(𝑦),𝐽𝑞𝜂𝐽𝐻,𝜂𝑀,𝜙(𝑥),𝐽𝐻,𝜂𝑀,𝜙‖‖𝐽(𝑦)≥𝛾𝐻,𝜂𝑀,𝜙(𝑥)−𝐽𝐻,𝜂𝑀,𝜙‖‖(𝑦)𝑞.(3.8) Since 𝜂 is Lipschitz continuous with a constant 𝜏, we have ‖‖𝜂𝐽𝐻,𝜂𝑀,𝜙(𝑥),𝐽𝐻,𝜂𝑀,𝜙‖‖‖‖𝐽(𝑦)≤𝜏𝐻,𝜂𝑀,𝜙(𝑥)−𝐽𝐻,𝜂𝑀,𝜙‖‖.(𝑦)(3.9) It follows from (3.8) and (3.9) that ‖𝑥−𝑦‖𝜏𝑞−1‖‖𝐽𝐻,𝜂𝑀,𝜙(𝑥)−𝐽𝐻,𝜂𝑀,𝜙‖‖(𝑦)𝑞−1‖‖𝐽≥𝛾𝐻,𝜂𝑀,𝜙(𝑥)−𝐽𝐻,𝜂𝑀,𝜙‖‖(𝑦)𝑞.(3.10) Hence, we get ‖‖𝐽𝐻,𝜂𝑀,𝜙(𝑥)−𝐽𝐻,𝜂𝑀,𝜙‖‖≤𝜏(𝑦)𝑞−1𝛾‖𝑥−𝑦‖.(3.11) This completes the proof.

4. A New System of Generalized Variational Inclusions

In this section, we will introduce a new system of generalized variational inclusions with (𝐻,𝜙)-𝜂-accretive mappings and construct a new iterative algorithm for solving this system of generalized variational inclusions. In what follows, for each 𝑖=1,2,3, suppose that 𝐸𝑖 is a Banach space, 𝐻𝑖,𝑓𝑖,𝜙𝑖,𝑝𝑖∶𝐸𝑖→𝐸𝑖, 𝜂𝑖∶𝐸𝑖×𝐸𝑖→𝐸𝑖, 𝐹𝑖,𝑁𝑖∶∏3𝑘=1𝐸𝑘→𝐸𝑖 are single-valued mappings, 𝑇𝑘𝑖∶𝐸𝑘→CB(𝐸𝑖) is a set-valued mapping, and 𝑀𝑖∶𝐸𝑖×𝐸𝑖→2𝐸𝑖 is a (𝐻𝑖,𝜙𝑖)-𝜂𝑖-accretive mapping in the second argument. Assume that 𝑓𝑖(𝐸𝑖)∩dom𝑀(𝑤𝑖,⋅)≠∅, for each 𝑤𝑖∈𝐸i. We consider the following system of generalized variational inclusions. Find (𝑥1,𝑥2,𝑥3,𝑢11,𝑢12,𝑢13,𝑢21,𝑢22,𝑢23,𝑢31,𝑢32,𝑢33) such that for each 𝑖=1,2,3, 𝑥𝑖∈𝐸𝑖, 𝑢1𝑖∈𝑇1𝑖(𝑥1), 𝑢2𝑖∈𝑇2𝑖(𝑥2), 𝑢3𝑖∈𝑇3𝑖(𝑥3), and0∈𝐹𝑖𝑥1−𝑝1𝑥1,𝑥2−𝑝2𝑥2,𝑥3−𝑝3𝑥3+𝑁𝑖𝑢𝑖1,𝑢𝑖2,𝑢𝑖3+𝑀𝑖𝑥𝑖,𝑓𝑖𝑥𝑖.(4.1) The following are some special cases of problem (4.1).

(i) If 𝐸1=ℋ1, 𝐸2=ℋ2, and 𝐸3=ℋ3 are three Hilbert spaces, and, for each 𝑖=1,2,3, 𝑝𝑖=0, 𝑁𝑖=0, 𝑀1(𝑥,𝑓1(𝑥))=𝜕𝜂1𝜑1(𝑥), for all 𝑥∈𝐸1, 𝑀2(𝑦,𝑓2(𝑦))=𝜕𝜂2𝜑2(𝑦), for all 𝑦∈𝐸2, and 𝑀3(𝑧,𝑓3(𝑧))=𝜕𝜂3𝜑3(𝑧), for all 𝑧∈𝐸3, where 𝜑𝑖∶𝐸𝑖→𝑅∪{∞} is a proper lower semicontinuous and 𝜂𝑖-subdifferential function, 𝜕𝜂1𝜑1(𝑥) is the 𝜂1-subdifferential of 𝜑1 at 𝑥, 𝜕𝜂2𝜑2(𝑦) is the 𝜂2-subdifferential of 𝜑2 at 𝑦, and 𝜕𝜂3𝜑3(𝑧) is the 𝜂3-subdifferential of 𝜑3 at 𝑧, then SGVIP (4.1) reduces to the following system of variational inequalities, which is to find (𝑥1,𝑥2,𝑥3)∈𝐸1×𝐸2×𝐸3 such that𝐹1𝑥1,𝑥2,𝑥3,𝜂1𝑎,𝑥1+𝜑1(𝑎)−𝜑1𝑥1≥0,∀𝑎∈𝐸1,𝐹2𝑥1,𝑥2,𝑥3,𝜂2𝑏,𝑥2+𝜑2(𝑏)−𝜑2𝑥2≥0,∀𝑏∈𝐸2,𝐹3𝑥1,𝑥2,𝑥3,𝜂3𝑐,𝑥3+𝜑3(𝑐)−𝜑3𝑥3≥0,∀𝑐∈𝐸3.(4.2)

If 𝜂1(𝑎,𝑥)=𝑎−𝑥, for all 𝑎,𝑥∈𝐸1, 𝜂2(𝑏,𝑦)=𝑏−𝑦, for all 𝑏,𝑦∈𝐸2, 𝜂3(𝑐,𝑧)=𝑐−𝑧, for all 𝑐,𝑧∈𝐸3, and 𝑀1(𝑥,𝑓1(𝑥))=𝜕𝜑1(𝑥) is the subdifferential of 𝜑1 at 𝑥, 𝑀2(𝑦,𝑓2(𝑦))=𝜕𝜑2(𝑦) is the subdifferential of 𝜑2 at 𝑦, and 𝑀3(𝑧,𝑓3(𝑧))=𝜕𝜑3(𝑧) is the subdifferential of 𝜑3 at 𝑧, then problem (4.2) reduces to the following system of variational inequalities, which is to find (𝑥1,𝑥2,𝑥3)∈𝐸1×𝐸2×𝐸3 such that𝐹1𝑥1,𝑥2,𝑥3,𝑎−𝑥1+𝜑1(𝑎)−𝜑1𝑥1≥0,∀𝑎∈𝐸1,𝐹2𝑥1,𝑥2,𝑥3,𝑏−𝑥2+𝜑2(𝑏)−𝜑2𝑥2≥0,∀𝑏∈𝐸2,𝐹3𝑥1,𝑥2,𝑥3,𝑐−𝑥3+𝜑3(𝑐)−𝜑3𝑥3≥0,∀𝑐∈𝐸3.(4.3)

If 𝑀1(𝑥,𝑓1(𝑥))=𝜕𝛿𝑘1(𝑥), 𝑀2(𝑦,𝑓2(𝑦))=𝜕𝛿𝑘2(𝑦), and 𝑀3(𝑧,𝑓3(𝑧))=𝜕𝛿𝑘3(𝑧), for all 𝑥∈𝐾1, 𝑦∈𝐾2, and 𝑧∈𝐾3, where 𝐾1⊂𝐸1, 𝐾2⊂𝐸2, and 𝐾3⊂𝐸3 are three nonempty, closed, and convex subsets, 𝛿𝑘1, 𝛿𝑘2, and 𝛿𝑘3 denote the indicator functions of 𝐾1, 𝐾2, and 𝐾3, respectively, then problem (4.3) reduces to the following system of variational inequalities, which is to find (𝑥1,𝑥2,𝑥3)∈𝐸1×𝐸2×𝐸3 such that𝐹1𝑥1,𝑥2,𝑥3,𝑎−𝑥1≥0,∀𝑎∈𝐾1,𝐹2𝑥1,𝑥2,𝑥3,𝑏−𝑥2≥0,∀𝑏∈𝐾2,𝐹3𝑥1,𝑥2,𝑥3,𝑐−𝑥3≥0,∀𝑐∈𝐾3.(4.4)

If 𝐸1=𝐸2=𝐸3=ℋ is a Hilbert space, 𝐾1=𝐾2=𝐾3=𝐾 is a nonempty, closed, and convex subset, 𝐹1(𝑥,𝑦,𝑧)=𝑠𝑇1(𝑦,𝑧,𝑥)+𝑥−𝑦, 𝐹2(𝑥,𝑦,𝑧)=𝑡𝑇2(𝑧,𝑥,𝑦)+𝑦−𝑧, and 𝐹3(𝑥,𝑦,𝑧)=𝑟𝑇3(𝑥,𝑦,𝑧)+𝑧−𝑥, for all 𝑥,𝑦,𝑧∈𝐾, where 𝑇1,𝑇2,𝑇3∶𝐾×𝐾×𝐾→ℋ are mappings on 𝐾×𝐾×𝐾, 𝑟,𝑠,𝑡>0 are three numbers, then problem (4.4) reduces to the following system of variational inequalities, which is to find 𝑥1,𝑥2,𝑥3∈𝐾 such that𝑠𝑇1𝑥2,𝑥3,𝑥1+𝑥1−𝑥2,𝑎−𝑥1≥0,∀𝑎∈𝐾,𝑡𝑇2𝑥3,𝑥1,𝑥2+𝑥2−𝑥3,𝑎−𝑥2≥0,∀𝑎∈𝐾,𝑟𝑇3𝑥1,𝑥2,𝑥3+𝑥3−𝑥1,𝑎−𝑥3≥0,∀𝑎∈𝐾.(4.5) Problem (4.5) was introduced and studied by Cho and Qin [22].

(ii) If 𝐸2=𝐸3, 𝐹1(⋅,⋅,⋅)=𝐹1(⋅,⋅), 𝐹2(⋅,⋅,⋅)=𝐹2(⋅,⋅), and 𝐹2=𝐹3 and for each 𝑖=1,2,3, 𝑝𝑖=0, 𝑁𝑖=0, 𝑀2(𝑦,𝑓2(𝑦))=𝑀3(𝑧,𝑓3(𝑧)), for all (𝑦,𝑧)∈𝐸2×𝐸3, then SGVIP (4.1) reduces to the following system of variational inclusions, which is to find (𝑥1,𝑥2)∈𝐸1×𝐸2 such that0∈𝐹1𝑥1,𝑥2+𝑀1𝑥1,𝑓1𝑥1,0∈𝐹2𝑥1,𝑥2+𝑀2𝑥2,𝑓2𝑥2.(4.6)

If 𝐸1=ℋ1 and 𝐸2=ℋ2 are two Hilbert spaces and 𝑀1(𝑥,𝑓1(𝑥))=𝜕𝜂1𝜑1(𝑥), for all 𝑥∈𝐸1, 𝑀2(𝑦,𝑓2(𝑦))=𝜕𝜂2𝜑2(𝑦), for all 𝑦∈𝐸2, then the problem (4.6) reduces to the following system of variational inequalities, which is to find (𝑥1,𝑥2)∈𝐸1×𝐸2 such that𝐹1𝑥1,𝑥2,𝜂1𝑎,𝑥1+𝜑1(𝑎)−𝜑1𝑥1≥0,∀𝑎∈𝐸1,𝐹2𝑥1,𝑥2,𝜂2𝑏,𝑥2+𝜑2(𝑏)−𝜑2𝑥2≥0,∀𝑏∈𝐸2.(4.7)

If 𝜂1(𝑎,𝑥)=𝑎−𝑥, for all 𝑎,𝑥∈𝐸1, 𝜂2(𝑏,𝑦)=𝑏−𝑦, for all 𝑏,𝑦∈𝐸2, 𝑀1(𝑥,𝑓1(𝑥))=𝜕𝜑1(𝑥) is the subdifferential of 𝜑1 at 𝑥, and 𝑀2(𝑦,𝑓2(𝑦))=𝜕𝜑2(𝑦) is the subdifferential of 𝜑2 at 𝑦, then the problem (4.7) reduces to the following system of variational inequalities, which is to find (𝑥1,𝑥2)∈𝐸1×𝐸2 such that𝐹1𝑥1,𝑥2,𝑎−𝑥1+𝜑1(𝑎)−𝜑1𝑥1≥0,∀𝑎∈𝐸1,𝐹2𝑥1,𝑥2,𝑏−𝑥2+𝜑2(𝑏)−𝜑2𝑥2≥0,∀𝑏∈𝐸2.(4.8) Problem (4.8) was introduced and studied by Cho et al. [23].

If 𝑀1(𝑥,𝑓1(𝑥))=𝜕𝛿𝑘1(𝑥), 𝑀2(𝑦,𝑓2(𝑦))=𝜕𝛿𝑘2(𝑦), for all 𝑥∈𝐾1, 𝑦∈𝐾2, where 𝐾1⊂𝐸1, 𝐾2⊂𝐸2 are two nonempty, closed, and convex subsets and 𝜕𝛿𝑘1 and 𝜕𝛿𝑘2 denote the indicator functions of 𝐾1, 𝐾2, respectively, then problem (4.8) reduces to the following system of variational inequalities, which is to find (𝑥1,𝑥2)∈𝐸1×𝐸2 such that𝐹1𝑥1,𝑥2,𝑎−𝑥1≥0,∀𝑎∈𝐾1,𝐹2𝑥1,𝑥2,𝑏−𝑥2≥0,∀𝑏∈𝐾2.(4.9) Problem (4.9) is just the problem [24] with 𝐹1 and 𝐹2 being single-valued.

If 𝐸1=𝐸2=ℋ is a Hilbert space and 𝐾1=𝐾2=𝐾 is a nonempty, closed, and convex subset, 𝐹1(𝑥,𝑦)=𝜌𝑇(𝑦,𝑥)+𝑥−𝑦 and 𝐹2(𝑥,𝑦)=𝜆𝑇(𝑥,𝑦)+𝑦−𝑥, for all 𝑥,𝑦∈𝐾, where 𝑇∶𝐾×𝐾→ℋ is a mapping on 𝐾×𝐾, 𝜌,𝜆>0 are two numbers, then problem (4.9) reduces to the following problem, which is to find 𝑥1,𝑥2∈𝐾 such that𝑥𝜌𝑇2,𝑥1+𝑥1−𝑥2,𝑎−𝑥1𝑥≥0,∀𝑎∈𝐾,𝜆𝑇1,𝑥2+𝑥2−𝑥1,𝑎−𝑥2≥0,∀𝑎∈𝐾.(4.10) Problem (4.10) was introduced and studied by Verma [25].

Lemma 4.1. Let, for 𝑖=1,2,3, 𝜙𝑖∶𝐸𝑖→𝐸𝑖 be a single-valued mapping satisfying 𝜙𝑖(𝑥+𝑦)=𝜙𝑖(𝑥)+𝜙𝑖(𝑦) and ker𝜙𝑖={0}, 𝜂𝑖∶𝐸𝑖×𝐸𝑖→𝐸𝑖 a single-valued mapping, 𝐻𝑖∶𝐸𝑖→𝐸𝑖 a strictly 𝜂𝑖-accretive mapping and 𝑀𝑖∶𝐸𝑖×𝐸𝑖→2𝐸𝑖 a (𝐻𝑖,𝜙𝑖)-𝜂𝑖-accretive mapping in the second argument. Then (𝑥1,𝑥2,𝑥3,𝑢11,𝑢12,𝑢13,𝑢21,𝑢22,𝑢23,𝑢31,𝑢32,𝑢33) in which 𝑥𝑖∈𝐸𝑖, 𝑢1𝑖∈𝑇1𝑖(𝑥1), 𝑢2𝑖∈𝑇2𝑖(𝑥2), 𝑢3𝑖∈𝑇3𝑖(𝑥3)(𝑖=1,2,3) is a solution of the problem (4.1) if and only if 𝑓𝑖𝑥𝑖=𝐽𝐻𝑖,𝜂𝑖𝑀𝑖𝑥𝑖,⋅,𝜙𝑖𝐻𝑖∘𝑓𝑖𝑥𝑖−𝜙𝑖∘𝐹𝑖𝑥1−𝑝1𝑥1,𝑥2−𝑝2𝑥2,𝑥3−𝑝3𝑥3−𝜙𝑖∘𝑁𝑖𝑢𝑖1,𝑢𝑖2,𝑢𝑖3,(4.11) where 𝐽𝐻𝑖,𝜂𝑖𝑀𝑖(𝑥𝑖,⋅),𝜙𝑖=(𝐻𝑖+𝜙𝑖∘𝑀𝑖(𝑥𝑖,⋅))−1.

Proof. The fact directly follows from Definition 3.6.

Algorithm 4.2. For any 𝑥0𝑖∈𝐸𝑖, take 𝑢01𝑖∈𝑇1𝑖(𝑥01), 𝑢02𝑖∈𝑇2𝑖(𝑥02), and 𝑢03𝑖∈𝑇3𝑖(𝑥03)(𝑖=1,2,3). For 𝑖=1,2,3, let 𝑥1𝑖=𝑥0𝑖−𝑓𝑖𝑥0𝑖+𝐽𝐻𝑖,𝜂𝑖𝑀𝑖𝑥0𝑖,⋅,𝜙𝑖𝐻𝑖∘𝑓𝑖𝑥0𝑖−𝜙𝑖∘𝐹𝑖𝑥01−𝑝1𝑥01,𝑥02−𝑝2𝑥02,𝑥03−𝑝3𝑥03−𝜙𝑖∘𝑁𝑖𝑢0𝑖1,𝑢0𝑖2,𝑢0𝑖3.(4.12) Since 𝑢01𝑖∈𝑇1𝑖(𝑥01), 𝑢02𝑖∈𝑇2𝑖(𝑥02), 𝑢03𝑖∈𝑇3𝑖(𝑥03)(𝑖=1,2,3), by Nodler's theorem [26], there exist 𝑢11𝑖∈𝑇1𝑖(𝑥11), 𝑢12𝑖∈𝑇2𝑖(𝑥12), 𝑢13𝑖∈𝑇3𝑖(𝑥13)(𝑖=1,2,3), such that, for each 𝑖=1,2,3, ‖‖𝑢11𝑖−𝑢01𝑖‖‖≤𝐻𝑇(1+1)1𝑖𝑥11,𝑇1𝑖𝑥01,‖‖𝑢12𝑖−𝑢02𝑖‖‖𝐻𝑇≤(1+1)2𝑖𝑥12,𝑇2𝑖𝑥02,‖‖𝑢13𝑖−𝑢03𝑖‖‖≤𝐻𝑇(1+1)3𝑖𝑥13,𝑇3𝑖𝑥03.(4.13) For 𝑖=1,2,3, let 𝑥2𝑖=𝑥1𝑖−𝑓𝑖𝑥1𝑖+𝐽𝐻𝑖,𝜂𝑖𝑀𝑖𝑥1𝑖,⋅,𝜙𝑖𝐻𝑖∘𝑓𝑖𝑥1𝑖−𝜙𝑖∘𝐹𝑖𝑥11−𝑝1𝑥11,𝑥12−𝑝2𝑥12,𝑥13−𝑝3𝑥13−𝜙𝑖∘𝑁𝑖𝑢1𝑖1,𝑢1𝑖2,𝑢1𝑖3.(4.14) Again by Nodler's theorem [26], there exist 𝑢21𝑖∈𝑇1𝑖(𝑥21), 𝑢22𝑖∈𝑇2𝑖(𝑥22), 𝑢23𝑖∈𝑇3𝑖(𝑥23)(𝑖=1,2,3), such that, for each 𝑖=1,2,3, ‖‖𝑢21𝑖−𝑢11𝑖‖‖≤11+2𝐻𝑇1𝑖𝑥21,𝑇1𝑖𝑥11,‖‖𝑢22𝑖−𝑢12𝑖‖‖≤11+2𝐻𝑇2𝑖𝑥22,𝑇2𝑖𝑥12,‖‖𝑢23𝑖−𝑢13𝑖‖‖≤11+2𝐻𝑇3𝑖𝑥23,𝑇3𝑖𝑥13.(4.15) By induction, we can compute the sequences 𝑥𝑛𝑖, 𝑢𝑛1𝑖, 𝑢𝑛2𝑖, 𝑢𝑛3𝑖(𝑖=1,2,3) by the following iterative schemes such that, for each 𝑖=1,2,3, 𝑥𝑖𝑛+1=𝑥𝑛𝑖−𝑓𝑖𝑥𝑛𝑖+𝐽𝐻𝑖,𝜂𝑖𝑀𝑖𝑥𝑛𝑖,⋅,𝜙𝑖𝐻𝑖∘𝑓𝑖𝑥𝑛𝑖−𝜙𝑖∘𝐹𝑖𝑥𝑛1−𝑝1𝑥𝑛1,𝑥𝑛2−𝑝2𝑥𝑛2,𝑥𝑛3−𝑝3𝑥𝑛3−𝜙𝑖∘𝑁𝑖𝑢𝑛𝑖1,𝑢𝑛𝑖2,𝑢𝑛𝑖3,𝑢𝑛1𝑖∈𝑇1𝑖𝑥𝑛1,‖‖𝑢𝑛1𝑖−𝑢𝑛−11𝑖‖‖≤11+𝑛𝐻𝑇1𝑖𝑥𝑛1,𝑇1𝑖𝑥1𝑛−1,𝑢𝑛2𝑖∈𝑇2𝑖𝑥𝑛2,‖‖𝑢𝑛2𝑖−𝑢𝑛−12𝑖‖‖≤11+𝑛𝐻𝑇2𝑖𝑥𝑛2,𝑇2𝑖𝑥2𝑛−1,𝑢𝑛3𝑖∈𝑇3𝑖𝑥𝑛3,‖‖𝑢𝑛3𝑖−𝑢𝑛−13𝑖‖‖≤11+𝑛𝐻𝑇3𝑖𝑥𝑛3,𝑇3𝑖𝑥3𝑛−1,(4.16) for all 𝑛=0,1,2,….

Now we prove the existence of solution of the SGVIP (4.1) and the convergence of Algorithm 4.2.

Theorem 4.3. Let, for 𝑖=1,2,3, 𝐸𝑖 be a q-uniformly smooth Banach space and let 𝜙𝑖∶𝐸𝑖→𝐸𝑖 be a 𝜃𝑖-Lipschitz continuous mapping satisfying 𝜙𝑖(𝑥+𝑦)=𝜙𝑖(𝑥)+𝜙𝑖(𝑦) and ker𝜙𝑖={0}. Let 𝜂𝑖∶𝐸𝑖×𝐸𝑖→𝐸𝑖 be 𝜏𝑖-Lipschitz continuous, let 𝐻𝑖∶𝐸𝑖→𝐸𝑖 be a 𝛾𝑖-strongly 𝜂𝑖-accretive and 𝑠𝑖-Lipschitz continuous mapping, 𝑓𝑖∶𝐸𝑖→𝐸𝑖 be (𝛼𝑖,𝜇𝑖)-relaxed cocoercive and 𝜉𝑖-Lipschitz continuous, and let 𝑝𝑖∶𝐸𝑖→𝐸𝑖 be a strongly accretive mapping with constant 𝛿𝑝𝑖 and Lipschitz continuous with constant 𝜆𝑝𝑖. Suppose that 𝐹𝑖∶∏3𝑘=1𝐸𝑘→𝐸𝑖 is 𝛽𝑖𝑘-Lipschitz continuous in the kth argument and 𝑁𝑖∶∏3𝑘=1𝐸𝑘→𝐸𝑖 be 𝜁𝑖𝑘-Lipschitz continuous in the kth argument for 𝑘=1,2,3, 𝑀𝑖∶𝐸𝑖×𝐸𝑖→2𝐸𝑖 is a (𝐻𝑖,𝜙𝑖)-𝜂𝑖-accretive mapping in the second argument, and set-valued mappings 𝑇1𝑖∶𝐸1→𝐶𝐵(𝐸𝑖), 𝑇2𝑖∶𝐸2→𝐶𝐵(𝐸𝑖), 𝑇3𝑖∶𝐸3→𝐶𝐵(𝐸𝑖) are 𝐻-Lipschitz continuous with constants 𝑡1𝑖>0, 𝑡2𝑖>0, 𝑡3𝑖>0, respectively. In addition if for all 𝑖=1,2,3, one has ‖‖‖𝐽𝐻𝑖,𝜂𝑖𝑀𝑖𝑥𝑖1,⋅,𝜙𝑖𝑧𝑖−𝐽𝐻𝑖,𝜂𝑖𝑀𝑖𝑥𝑖2,⋅,𝜙𝑖𝑧𝑖‖‖‖≤𝛿𝑖‖‖𝑥𝑖1−𝑥𝑖2‖‖,∀𝑥𝑖1,𝑥𝑖2,𝑧𝑖∈𝐸𝑖,(4.17)0<1+𝑞𝛼1𝜉𝑞1−𝑞𝜇1+𝑐𝑞𝜉𝑞11/𝑞+𝛿1+𝜏1𝑞−1𝛾1×𝑠𝑞1𝜉𝑞1+𝑞𝜃1𝛽11𝑠1𝑞−1𝜉1𝑞−11−𝑞𝛿𝑝1+𝑐𝑞𝜆𝑞𝑝11/𝑞+𝑐𝑞𝜃𝑞1𝛽𝑞111−𝑞𝛿𝑝1+𝑐𝑞𝜆𝑞𝑃11/𝑞+3𝑗=1𝜃1𝜁1𝑗𝑡1𝑗+𝜃2𝛽21𝜏2𝑞−1𝛾2+𝜃3𝛽31𝜏3𝑞−1𝛾31−𝑞𝛿𝑝1+𝑐𝑞𝜆𝑞𝑃11/𝑞<1,0<1+𝑞𝛼2𝜉𝑞2−𝑞𝜇2+𝑐𝑞𝜉𝑞21/𝑞+𝛿2+𝜏2𝑞−1𝛾2×𝑠𝑞2𝜉𝑞2+𝑞𝜃2𝛽22𝑠2𝑞−1𝜉2𝑞−11−𝑞𝛿𝑝2+𝑐𝑞𝜆𝑞𝑝21/𝑞+𝑐𝑞𝜃𝑞2𝛽𝑞221−𝑞𝛿𝑝2+𝑐𝑞𝜆𝑞𝑃21/𝑞+3𝑗=1𝜃2𝜁2𝑗𝑡2𝑗+𝜃1𝛽12𝜏1𝑞−1𝛾1+𝜃3𝛽32𝜏3𝑞−1𝛾31−𝑞𝛿𝑝2+𝑐𝑞𝜆𝑞𝑃21/𝑞<1,0<1+𝑞𝛼3𝜉𝑞3−𝑞𝜇3+𝑐𝑞𝜉𝑞31/𝑞+𝛿3+𝜏3𝑞−1𝛾3×𝑠𝑞3𝜉𝑞3+𝑞𝜃3𝛽33𝑠3𝑞−1𝜉3𝑞−11−𝑞𝛿𝑝3+𝑐𝑞𝜆𝑞𝑝31/𝑞+𝑐𝑞𝜃𝑞3𝛽𝑞331−𝑞𝛿𝑝3+𝑐𝑞𝜆𝑞𝑃31/𝑞+3𝑗=1𝜃3𝜁3𝑗𝑡3𝑗+𝜃1𝛽13𝜏1𝑞−1𝛾1+𝜃2𝛽23𝜏2𝑞−1𝛾21−𝑞𝛿𝑝3+𝑐𝑞𝜆𝑞𝑃31/𝑞<1.(4.18) Then the problem (4.1) admits a solution (𝑥1,𝑥2,𝑥3,𝑢11,𝑢12,𝑢13,𝑢21,𝑢22,𝑢23,𝑢31,𝑢32,𝑢33) and sequences 𝑥𝑛1,𝑥𝑛2,𝑥𝑛3,𝑢𝑛11,𝑢𝑛12,𝑢𝑛13,𝑢𝑛21,𝑢𝑛22,𝑢𝑛23,𝑢𝑛31,𝑢𝑛32,𝑢𝑛33 converge to 𝑥1,𝑥2,𝑥3, 𝑢11, 𝑢12, 𝑢13, 𝑢21, 𝑢22, 𝑢23, 𝑢31, 𝑢32, 𝑢33, respectively, where 𝑥𝑛𝑖,𝑢𝑛1𝑖,𝑢𝑛2𝑖,𝑢𝑛3𝑖(𝑖=1,2,3) are sequences generated by Algorithm 4.2.

Proof. For 𝑖=1,2,3, let 𝐷𝑛𝑖=𝐻𝑖∘𝑓𝑖𝑥𝑛𝑖−𝜙𝑖∘𝐹𝑖𝑥𝑛1−𝑝1𝑥𝑛1,𝑥𝑛2−𝑝2𝑥𝑛2,𝑥𝑛3−𝑝3𝑥𝑛3−𝜙𝑖∘𝑁𝑖𝑢𝑛𝑖1,𝑢𝑛𝑖2,𝑢𝑛𝑖3.(4.19) By Algorithm 4.2 and (4.17), we have ‖‖𝑥𝑖𝑛+1−𝑥𝑛𝑖‖‖=‖‖‖𝑥𝑛𝑖−𝑓𝑖𝑥𝑛𝑖+𝐽𝐻𝑖,𝜂𝑖𝑀𝑖𝑥𝑛𝑖,⋅,𝜙𝑖𝐻𝑖∘𝑓𝑖𝑥𝑛𝑖−𝜙𝑖∘𝐹𝑖𝑥𝑛1−𝑝1𝑥𝑛1,𝑥𝑛2−𝑝2𝑥𝑛2,𝑥𝑛3−𝑝3𝑥𝑛3−𝜙𝑖∘𝑁𝑖𝑢𝑛𝑖1,𝑢𝑛𝑖2,𝑢𝑛𝑖3−𝑥𝑖𝑛−1−𝑓𝑖𝑥𝑖𝑛−1+𝐽𝐻𝑖,𝜂𝑖𝑀𝑖𝑥𝑖𝑛−1,⋅,𝜙𝑖𝐻𝑖∘𝑓𝑖𝑥𝑖𝑛−1−𝜙𝑖∘𝐹𝑖𝑥1𝑛−1−𝑝1𝑥1𝑛−1,𝑥2𝑛−1−𝑝2𝑥2𝑛−1,𝑥3𝑛−1−𝑝3𝑥3𝑛−1−𝜙𝑖∘𝑁𝑖𝑢𝑛−1𝑖1,𝑢𝑛−1𝑖2,𝑢𝑛−1𝑖3‖‖‖≤‖‖𝑥𝑛𝑖−𝑥𝑖𝑛−1−𝑓𝑖𝑥𝑛𝑖−𝑓𝑖𝑥𝑖𝑛−1‖‖+‖‖‖𝐽𝐻𝑖,𝜂𝑖𝑀𝑖𝑥𝑛𝑖,⋅,𝜙𝑖𝐷𝑛𝑖−𝐽𝐻𝑖,𝜂𝑖𝑀𝑖𝑥𝑛𝑖,⋅,𝜙𝑖𝐷𝑖𝑛−1‖‖‖+‖‖‖𝐽𝐻𝑖,𝜂𝑖𝑀𝑖(𝑥𝑛𝑖,⋅),𝜙𝑖𝐷𝑖𝑛−1−𝐽𝐻𝑖,𝜂𝑖𝑀𝑖(𝑥𝑖𝑛−1,⋅),𝜙𝑖𝐷𝑖𝑛−1‖‖‖≤‖‖𝑥𝑛𝑖−𝑥𝑖𝑛−1−𝑓𝑖𝑥𝑛𝑖−𝑓𝑖𝑥𝑖𝑛−1‖‖+𝜏𝑖𝑞−1𝛾𝑖‖‖𝐷𝑛𝑖−𝐷𝑖𝑛−1‖‖+𝛿𝑖‖‖𝑥𝑛𝑖−𝑥𝑖𝑛−1‖‖.(4.20) Since 𝑓𝑖∶𝐸𝑖→𝐸𝑖 is (𝛼𝑖,𝜇𝑖)-relaxed cocoercive and 𝜉𝑖-Lipschitz continuous and 𝑝𝑖∶𝐸𝑖→𝐸𝑖 is 𝛿𝑝𝑖-strongly accretive and 𝜆𝑝𝑖-Lipschitz continuous, we have ‖‖𝑥𝑛𝑖−𝑥𝑖𝑛−1−𝑓𝑖𝑥𝑛𝑖−𝑓𝑖𝑥𝑖𝑛−1‖‖𝑞≤‖‖𝑥𝑛𝑖−𝑥𝑖𝑛−1‖‖𝑞𝑓−𝑞𝑖𝑥𝑛𝑖−𝑓𝑖𝑥𝑖𝑛−1,𝐽𝑞𝑥𝑛𝑖−𝑥𝑖𝑛−1+𝑐𝑞‖‖𝑓𝑖𝑥𝑛𝑖−𝑓𝑖𝑥𝑖𝑛−1‖‖𝑞≤1+𝑞𝛼𝑖𝜉𝑞𝑖−𝑞𝜇𝑖+𝑐𝑞𝜉𝑞𝑖‖‖𝑥𝑛𝑖−𝑥𝑖𝑛−1‖‖𝑞,‖‖𝑥𝑛𝑖−𝑥𝑖𝑛−1−𝑝𝑖𝑥𝑛𝑖−𝑝𝑖𝑥𝑖𝑛−1‖‖𝑞≤‖‖𝑥𝑛𝑖−𝑥𝑖𝑛−1‖‖𝑞𝑝−𝑞𝑖𝑥𝑛𝑖−𝑝𝑖𝑥𝑖𝑛−1,𝐽𝑞𝑥𝑛𝑖−𝑥𝑖𝑛−1+𝑐𝑞‖‖𝑝𝑖𝑥𝑛𝑖−𝑝𝑖𝑥𝑖𝑛−1‖‖𝑞≤1−𝑞𝛿𝑝𝑖+𝑐𝑞𝜆𝑞𝑝𝑖‖‖𝑥𝑛𝑖−𝑥𝑖𝑛−1‖‖𝑞.(4.21) From (4.19), we have ‖‖𝐷𝑛𝑖−𝐷𝑖𝑛−1‖‖=‖‖𝐻𝑖∘𝑓𝑖𝑥𝑛𝑖−𝜙𝑖∘𝐹𝑖𝑥𝑛1−𝑝1𝑥𝑛1,𝑥𝑛2−𝑝2𝑥𝑛2,𝑥𝑛3−𝑝3𝑥𝑛3−𝜙𝑖∘𝑁𝑖𝑢𝑛𝑖1,𝑢𝑛𝑖2,𝑢𝑛𝑖3−𝐻𝑖∘𝑓𝑖𝑥𝑖𝑛−1−𝜙𝑖∘𝐹𝑖𝑥1𝑛−1−𝑝1𝑥1𝑛−1,𝑥2𝑛−1−𝑝2𝑥2𝑛−1,𝑥3𝑛−1−𝑝3𝑥3𝑛−1−𝜙𝑖∘𝑁𝑖𝑢𝑛−1𝑖1,𝑢𝑛−1𝑖2,𝑢𝑛−1𝑖3‖‖≤‖‖𝐻𝑖∘𝑓𝑖𝑥𝑛𝑖−𝐻𝑖∘𝑓𝑖𝑥𝑖𝑛−1-𝜙𝑖∘𝐹𝑖𝑥𝑛1−𝑝1𝑥𝑛1,𝑥𝑛2−𝑝2𝑥𝑛2,𝑥𝑛3−𝑝3𝑥𝑛3−𝜙𝑖∘𝐹𝑖𝑥1𝑛−1−𝑝1𝑥1𝑛−1,𝑥2𝑛−1−𝑝2𝑥2𝑛−1,𝑥3𝑛−1−𝑝3𝑥3𝑛−1‖‖+‖‖𝜙𝑖∘𝑁𝑖𝑢𝑛𝑖1,𝑢𝑛𝑖2,𝑢𝑛𝑖3−𝜙𝑖∘𝑁𝑖𝑢𝑛−1𝑖1,𝑢𝑛−1𝑖2,𝑢𝑛−1𝑖3‖‖.(4.22) Since 𝐹𝑖∶∏3𝑘=1𝐸𝑘→𝐸𝑖 is 𝛽𝑖𝑘-Lipschitz continuous in the 𝑘th argument and by continuity of H𝑖, 𝑓𝑖, 𝜙𝑖(𝑖=1,2,3), we have ‖‖𝐻𝑖∘𝑓𝑖𝑥𝑛𝑖−𝐻𝑖∘𝑓𝑖𝑥𝑖𝑛−1−𝜙𝑖∘𝐹𝑖𝑥𝑛1−𝑝1𝑥𝑛1,𝑥𝑛2−𝑝2𝑥𝑛2,𝑥𝑛3−𝑝3𝑥𝑛3−𝜙𝑖∘𝐹𝑖𝑥1𝑛−1−𝑝1𝑥1𝑛−1,𝑥2𝑛−1−𝑝2𝑥2𝑛−1,𝑥3𝑛−1−𝑝3𝑥3𝑛−1‖‖≤𝑠𝑞𝑖𝜉𝑞𝑖+𝑞𝜃𝑖𝛽𝑖𝑖𝑠𝑖𝑞−1𝜉𝑖𝑞−11−𝑞𝛿𝑝𝑖+𝑐𝑞𝜆𝑞𝑝𝑖1/𝑞+𝑐𝑞𝜃𝑞𝑖𝛽𝑞𝑖𝑖1−𝑞𝛿𝑝𝑖+𝑐𝑞𝜆𝑞𝑃𝑖1/𝑞‖‖𝑥𝑛𝑖−𝑥𝑖𝑛−1‖‖+𝑖−1𝑗=1𝜃𝑖𝛽𝑖𝑗1−𝑞𝛿𝑝𝑗+𝑐𝑞𝜆𝑞𝑃𝑗1/𝑞‖‖𝑥𝑛𝑗−𝑥𝑗𝑛−1‖‖+3𝑗=𝑖+1𝜃𝑖𝛽𝑖𝑗1−𝑞𝛿𝑝𝑗+𝑐𝑞𝜆𝑞𝑃𝑗1/𝑞‖‖𝑥𝑛𝑗−𝑥𝑗𝑛−1‖‖.(4.23) It follows from the Lipschitz continuity of 𝑁𝑖 and the 𝐻-Lipschitz continuity of 𝑇𝑖𝑗 that ‖‖𝑁𝑖𝑢𝑛𝑖1,𝑢𝑛𝑖2,𝑢𝑛𝑖3−𝑁𝑖𝑢𝑛−1𝑖1,𝑢𝑛−1𝑖2,𝑢𝑛−1𝑖3‖‖≤‖‖𝑁𝑖𝑢𝑛𝑖1,𝑢𝑛𝑖2,𝑢𝑛𝑖3−𝑁𝑖𝑢𝑛−1𝑖1,𝑢𝑛𝑖2,𝑢𝑛𝑖3‖‖+‖‖𝑁𝑖𝑢𝑛−1𝑖1,𝑢𝑛𝑖2,𝑢𝑛𝑖3−𝑁𝑖𝑢𝑛−1𝑖1,𝑢𝑛−1𝑖2,𝑢𝑛𝑖3‖‖+‖‖𝑁𝑖𝑢𝑛−1𝑖1,𝑢𝑛−1𝑖2,𝑢𝑛𝑖3−𝑁𝑖𝑢𝑛−1𝑖1,𝑢𝑛−1𝑖2,𝑢𝑛−1𝑖3‖‖≤𝜁𝑖1‖‖𝑢𝑛𝑖1−𝑢𝑛−1𝑖1‖‖+𝜁𝑖2‖‖𝑢𝑛𝑖2−𝑢𝑛−1𝑖2‖‖+𝜁𝑖3‖‖𝑢𝑛𝑖3−𝑢𝑛−1𝑖3‖‖=3𝑗=1𝜁𝑖𝑗‖‖𝑢𝑛𝑖𝑗−𝑢𝑛−1𝑖𝑗‖‖≤3𝑗=1𝜁𝑖𝑗11+𝑛H𝑇𝑖𝑗𝑥𝑛𝑖,𝑇𝑖𝑗𝑥𝑖𝑛−1≤3𝑗=1𝜁𝑖𝑗𝑡𝑖𝑗11+𝑛‖‖𝑥𝑛𝑖−𝑥𝑖𝑛−1‖‖,𝑖=1,2,3.(4.24) It follows from (4.20)–(4.24) that, for each 𝑖=1,2,3, ‖‖𝑥𝑖𝑛+1−𝑥𝑛𝑖‖‖≤1+𝑞𝛼𝑖𝜉𝑞𝑖−𝑞𝜇𝑖+𝑐𝑞𝜉𝑞𝑖1/𝑞+𝛿𝑖+𝜏𝑖𝑞−1𝛾𝑖𝑠𝑞𝑖𝜉𝑞𝑖+𝑞𝜃𝑖𝛽𝑖𝑖𝑠𝑖𝑞−1𝜉𝑖𝑞−11−𝑞𝛿𝑝𝑖+𝑐𝑞𝜆𝑞𝑝𝑖1/𝑞+𝑐𝑞𝜃𝑞𝑖𝛽𝑞𝑖𝑖1−𝑞𝛿𝑝𝑖+𝑐𝑞𝜆𝑞𝑃𝑖1/𝑞+3𝑗=1𝜃𝑖𝜁𝑖𝑗𝑡𝑖𝑗11+𝑛×‖‖𝑥𝑛𝑖−𝑥𝑖𝑛−1‖‖+𝑖−1𝑗=1𝜃𝑖𝛽𝑖𝑗𝜏𝑖𝑞−1𝛾𝑖1−𝑞𝛿𝑝𝑗+𝑐𝑞𝜆𝑞𝑃𝑗1/𝑞‖‖𝑥𝑛𝑗−𝑥𝑗𝑛−1‖‖+3𝑗=𝑖+1𝜃𝑖𝛽𝑖𝑗𝜏𝑖𝑞−1𝛾𝑖1−𝑞𝛿𝑝𝑗+𝑐𝑞𝜆𝑞𝑃𝑗1/𝑞‖‖𝑥𝑛𝑗−𝑥𝑗𝑛−1‖‖.(4.25) Therefore, 3𝑖=1‖‖𝑥𝑖𝑛+1−𝑥𝑛𝑖‖‖≤3𝑖=11+𝑞𝛼𝑖𝜉𝑞𝑖−𝑞𝜇𝑖+𝑐𝑞𝜉𝑞𝑖1/𝑞+𝛿𝑖+𝜏𝑖𝑞−1𝛾𝑖𝑠𝑞𝑖𝜉𝑞𝑖+𝑞𝜃𝑖𝛽𝑖𝑖𝑠𝑖𝑞−1𝜉𝑖𝑞−11−𝑞𝛿𝑝𝑖+𝑐𝑞𝜆𝑞𝑝𝑖1/𝑞+𝑐𝑞𝜃𝑞𝑖𝛽𝑞𝑖𝑖×1−𝑞𝛿𝑝𝑖+𝑐𝑞𝜆𝑞𝑃𝑖1/𝑞+3𝑗=1𝜃𝑖𝜁𝑖𝑗𝑡𝑖𝑗11+𝑛×‖‖𝑥𝑛𝑖−𝑥𝑖𝑛−1‖‖+𝑖−1𝑗=1𝜃𝑖𝛽𝑖𝑗𝜏𝑖𝑞−1𝛾𝑖1−𝑞𝛿𝑝𝑗+𝑐𝑞𝜆𝑞𝑃𝑗1/𝑞‖‖𝑥𝑛𝑗−𝑥𝑗𝑛−1‖‖+3𝑗=𝑖+1𝜃𝑖𝛽𝑖𝑗𝜏𝑖𝑞−1𝛾𝑖1−𝑞𝛿𝑝𝑗+𝑐𝑞𝜆𝑞𝑃𝑗1/𝑞‖‖𝑥𝑛𝑗−𝑥𝑗𝑛−1‖‖=1+𝑞𝛼1𝜉𝑞1−𝑞𝜇1+𝑐𝑞𝜉𝑞11/𝑞+𝛿1+𝜏1𝑞−1𝛾1×⎛⎜⎜⎝𝑠𝑞1𝜉𝑞1+𝑞𝜃1𝛽11𝑠1𝑞−1𝜉1𝑞−11−𝑞𝛿𝑝1+𝑐𝑞𝜆𝑞𝑝11/𝑞+𝑐𝑞𝜃𝑞1𝛽𝑞111−𝑞𝛿𝑝1+𝑐𝑞𝜆𝑞𝑃11/𝑞+3𝑗=1𝜃1𝜁1𝑗𝑡1𝑗11+𝑛+𝜃2𝛽21𝜏2𝑞−1𝛾2+𝜃3𝛽31𝜏3𝑞−1𝛾31−𝑞𝛿𝑝1+𝑐𝑞𝜆𝑞𝑃11/𝑞×‖‖𝑥𝑛1−𝑥1𝑛−1‖‖+1+𝑞𝛼2𝜉𝑞2−𝑞𝜇2+𝑐𝑞𝜉𝑞21/𝑞+𝛿2+𝜏2𝑞−1𝛾2×𝑠𝑞2𝜉𝑞2+𝑞𝜃2𝛽22𝑠2𝑞−1𝜉2𝑞−11−𝑞𝛿𝑝2+𝑐𝑞𝜆𝑞𝑝21/𝑞+𝑐𝑞𝜃𝑞2𝛽𝑞221−𝑞𝛿𝑝2+𝑐𝑞𝜆𝑞𝑃21/𝑞+3𝑗=1𝜃2𝜁2𝑗𝑡2𝑗11+𝑛+𝜃1𝛽12𝜏1𝑞−1𝛾1+𝜃3𝛽32𝜏3𝑞−1𝛾31−𝑞𝛿𝑝2+𝑐𝑞𝜆𝑞𝑃21/𝑞×‖‖𝑥𝑛2−𝑥2𝑛−1‖‖+1+𝑞𝛼3𝜉𝑞3−𝑞𝜇3+𝑐𝑞𝜉𝑞31/𝑞+𝛿3+𝜏3𝑞−1𝛾3×𝑠𝑞3𝜉𝑞3+𝑞𝜃3𝛽33𝑠3𝑞−1𝜉3𝑞−11−𝑞𝛿𝑝3+𝑐𝑞𝜆𝑞𝑝31/𝑞+𝑐𝑞𝜃𝑞3𝛽𝑞331−𝑞𝛿𝑝3+𝑐𝑞𝜆𝑞𝑃31/𝑞+3𝑗=1𝜃3𝜁3𝑗𝑡3𝑗11+𝑛+𝜃1𝛽13𝜏1𝑞−1𝛾1+𝜃2𝛽23𝜏2𝑞−1𝛾21−𝑞𝛿𝑝3+𝑐𝑞𝜆𝑞𝑃31/𝑞×‖‖𝑥𝑛3−𝑥3𝑛−1‖‖≤𝜃𝑛𝑛𝑖=1‖‖𝑥𝑛𝑖−𝑥𝑖𝑛−1‖‖,(4.26) where 𝜃𝑛=max1+𝑞𝛼1𝜉𝑞1−𝑞𝜇1+𝑐𝑞𝜉𝑞11/𝑞+𝛿1+𝜏1𝑞−1𝛾1𝑠𝑞1𝜉𝑞1+𝑞𝜃1𝛽11𝑠1𝑞−1𝜉1𝑞−11−𝑞𝛿𝑝1+𝑐𝑞𝜆𝑞𝑝11/𝑞+𝑐𝑞𝜃𝑞1𝛽𝑞111−𝑞𝛿𝑝1+𝑐𝑞𝜆𝑞𝑃11/𝑞+3𝑗=1𝜃1𝜁1𝑗𝑡1𝑗11+𝑛+𝜃2𝛽21𝜏2𝑞−1𝛾2+𝜃3𝛽31𝜏3𝑞−1𝛾31−𝑞𝛿𝑝1+𝑐𝑞𝜆𝑞𝑃11/𝑞,1+𝑞𝛼2𝜉𝑞2−𝑞𝜇2+𝑐𝑞𝜉𝑞21/𝑞+𝛿2+𝜏2𝑞−1𝛾2𝑠𝑞2𝜉𝑞2+𝑞𝜃2𝛽22𝑠2𝑞−1𝜉2𝑞−11−𝑞𝛿𝑝2+𝑐𝑞𝜆𝑞𝑝21/𝑞+𝑐𝑞𝜃𝑞2𝛽𝑞221−𝑞𝛿𝑝2+𝑐𝑞𝜆𝑞𝑃21/𝑞+3𝑗=1𝜃2𝜁2𝑗𝑡2𝑗11+𝑛+𝜃1𝛽12𝜏1𝑞−1𝛾1+𝜃3𝛽32𝜏3𝑞−1𝛾31−𝑞𝛿𝑝2+𝑐𝑞𝜆𝑞𝑃21/𝑞,1+𝑞𝛼3𝜉𝑞3−𝑞𝜇3+𝑐𝑞𝜉𝑞31/𝑞+𝛿3+𝜏3𝑞−1𝛾3𝑠𝑞3𝜉𝑞3+𝑞𝜃3𝛽33𝑠3𝑞−1𝜉3𝑞−11−𝑞𝛿𝑝3+𝑐𝑞𝜆𝑞𝑝31/𝑞+𝑐𝑞𝜃𝑞3𝛽𝑞331−𝑞𝛿𝑝3+𝑐𝑞𝜆𝑞𝑃31/𝑞+3𝑗=1𝜃3𝜁3𝑗𝑡3𝑗11+𝑛+𝜃1𝛽13𝜏1𝑞−1𝛾1+𝜃2𝛽23𝜏2𝑞−1𝛾21−𝑞𝛿𝑝3+𝑐𝑞𝜆𝑞𝑃31/𝑞.(4.27) Let 𝜃=max1+𝑞𝛼1𝜉𝑞1−𝑞𝜇1+𝑐𝑞𝜉𝑞11/𝑞+𝛿1+𝜏1𝑞−1𝛾1×𝑠𝑞1𝜉𝑞1+𝑞𝜃1𝛽11𝑠1𝑞−1𝜉1𝑞−11−𝑞𝛿𝑝1+𝑐𝑞𝜆𝑞𝑝11/𝑞+𝑐𝑞𝜃𝑞1𝛽𝑞111−𝑞𝛿𝑝1+𝑐𝑞𝜆𝑞𝑃11/𝑞+3𝑗=1𝜃1𝜁1𝑗𝑡1𝑗+𝜃2𝛽21𝜏2𝑞−1𝛾2+𝜃3𝛽31𝜏3𝑞−1𝛾31−𝑞𝛿𝑝1+𝑐𝑞𝜆𝑞𝑃11/𝑞,1+𝑞𝛼2𝜉𝑞2−𝑞𝜇2+𝑐𝑞𝜉𝑞21/𝑞+𝛿2+𝜏2𝑞−1𝛾2×𝑠𝑞2𝜉𝑞2+𝑞𝜃2𝛽22𝑠2𝑞−1𝜉2𝑞−11−𝑞𝛿𝑝2+𝑐𝑞𝜆𝑞𝑝21/𝑞+𝑐𝑞𝜃𝑞2𝛽𝑞221−𝑞𝛿𝑝2+𝑐𝑞𝜆𝑞𝑃21/𝑞+3𝑗=1𝜃2𝜁2𝑗𝑡2𝑗+𝜃1𝛽12𝜏1𝑞−1𝛾1+𝜃3𝛽32𝜏3𝑞−1𝛾31−𝑞𝛿𝑝2+𝑐𝑞𝜆𝑞𝑃21/𝑞,1+𝑞𝛼3𝜉𝑞3−𝑞𝜇3+𝑐𝑞𝜉𝑞31/𝑞+𝛿3+𝜏3𝑞−1𝛾3×𝑠𝑞3𝜉𝑞3+𝑞𝜃3𝛽33𝑠3𝑞−1𝜉3𝑞−11−𝑞𝛿𝑝3+𝑐𝑞𝜆𝑞𝑝31/𝑞+𝑐𝑞𝜃𝑞3𝛽𝑞331−𝑞𝛿𝑝3+𝑐𝑞𝜆𝑞𝑃31/𝑞+3𝑗=1𝜃3𝜁3𝑗𝑡3𝑗+𝜃1𝛽13𝜏1𝑞−1𝛾1+𝜃2𝛽23𝜏2𝑞−1𝛾21−𝑞𝛿𝑝3+𝑐𝑞𝜆𝑞𝑃31/𝑞.(4.28) We know that 𝜃𝑛→𝜃 as 𝑛→∞.
Define ‖⋅‖∗ on ∏𝐸=3𝑘=1𝐸𝑘 by ‖‖𝑧1,𝑧2,𝑧3‖‖∗=‖‖𝑧1‖‖+‖‖𝑧2‖‖+‖‖𝑧3‖‖𝑧,∀1,𝑧2,𝑧3∈𝐸.(4.29) It is easy to see that (𝐸,‖⋅‖∗) is a Banach space. Define 𝑧𝑛+1=(𝑥1𝑛+1,𝑥2𝑛+1,𝑥3𝑛+1). Then we have ‖‖𝑧𝑛+1−𝑧𝑛‖‖∗=‖‖𝑥1𝑛+1−𝑥𝑛1‖‖+‖‖𝑥2𝑛+1−𝑥𝑛2‖‖+‖‖𝑥3𝑛+1−𝑥𝑛3‖‖.(4.30) It follows from (4.18) that 0<𝜃<1, and hence there exists an 𝑛0>0 and 𝜃0∈(0,1) such that 𝜃𝑛≤𝜃0, for all 𝑛≥𝑛0. Therefore, by (4.26) and (4.30), we have ‖‖𝑧𝑛+1−𝑧𝑛‖‖∗≤𝜃𝑛−𝑛𝑜0‖‖𝑧𝑛0+1−𝑧𝑛0‖‖∗.(4.31) Hence, for any 𝑚≥𝑛≥𝑛0, it follows that ‖‖𝑥𝑚1−𝑥𝑛1‖‖≤‖𝑧𝑚−𝑧𝑛‖∗≤𝑚−1𝑖=𝑛‖‖𝑧𝑖+1−𝑧𝑖‖‖∗≤𝑚−1𝑖=𝑛𝜃𝑖−𝑛00‖‖𝑧𝑛0+1−𝑧𝑛0‖‖∗.(4.32) Since 0<𝜃0<1, it follows from (4.32) that ‖𝑥𝑚1−𝑥𝑛1‖→0 as 𝑛→∞, and hence {𝑥𝑛1} is a Cauchy sequence in 𝐸1. By the same argument, we also have {𝑥𝑛𝑖} is a Cauchy sequence in 𝐸𝑖(𝑖=2,3). Thus, there exist 𝑥1∈𝐸1, 𝑥2∈𝐸2, 𝑥3∈𝐸3 such that 𝑥𝑛1→𝑥1, 𝑥𝑛2→𝑥2, 𝑥𝑛3→𝑥3 as 𝑛→∞.
Now we prove that 𝑢𝑛1𝑖→𝑢1𝑖∈𝑇1𝑖(𝑥1), 𝑢𝑛2𝑖→𝑢2𝑖∈𝑇2𝑖(𝑥2), 𝑢𝑛3𝑖→𝑢3𝑖∈𝑇3𝑖(𝑥3)(𝑖=1,2,3). In fact, it follows from the Lipschitz continuity of 𝑇1𝑖, 𝑇2𝑖, 𝑇3𝑖 that for, 𝑖=1,2,3, ‖‖𝑢𝑛1𝑖−𝑢𝑛−11𝑖‖‖≤11+𝑛𝑡1𝑖‖‖𝑥𝑛1−𝑥1𝑛−1‖‖,‖‖𝑢𝑛2𝑖−𝑢𝑛−12𝑖‖‖≤11+𝑛𝑡2𝑖‖‖𝑥𝑛2−𝑥2𝑛−1‖‖,‖‖𝑢𝑛3𝑖−𝑢𝑛−13𝑖‖‖≤11+𝑛𝑡3𝑖‖‖𝑥𝑛3−𝑥3𝑛−1‖‖.(4.33) From (4.33), we have that 𝑢𝑛1𝑖, 𝑢𝑛2𝑖, 𝑢𝑛3𝑖(𝑖=1,2,3) are also Cauchy sequences. Therefore, there exist 𝑢1𝑖∈𝐸𝑖, 𝑢2𝑖∈𝐸𝑖, 𝑢3𝑖∈𝐸𝑖 such that 𝑢𝑛1𝑖→𝑢1𝑖, 𝑢𝑛2𝑖→𝑢2𝑖, 𝑢𝑛3𝑖→𝑢3𝑖 as 𝑛→∞. Further, for 𝑖=1,2,3, 𝑑𝑢1𝑖,𝑇1𝑖𝑥1≤‖‖𝑢1𝑖−𝑢𝑛1𝑖‖‖𝑢+𝑑𝑛1𝑖,𝑇1𝑖𝑥1≤‖‖𝑢1𝑖−𝑢𝑛1𝑖‖‖+𝐻𝑇1𝑖𝑥𝑛1,𝑇1𝑖𝑥1≤‖‖𝑢1𝑖−𝑢𝑛1𝑖‖‖+𝑡1𝑖‖‖𝑥𝑛1−𝑥1‖‖⟶0.(4.34) Since 𝑇1𝑖(𝑥1) is closed, we have 𝑢1𝑖∈𝑇1𝑖(𝑥1)(𝑖=1,2,3). Similarly 𝑢2𝑖∈𝑇2𝑖(𝑥2), 𝑢3𝑖∈𝑇3𝑖(𝑥3)(𝑖=1,2,3). By continuity of 𝑓𝑖, 𝐻𝑖, 𝜙𝑖, 𝐹𝑖, 𝑁𝑖, 𝑇1𝑖, 𝑇2𝑖, 𝑇3𝑖, 𝐽𝐻𝑖,𝜂𝑖𝑀𝑖(𝑥𝑖,⋅),𝜙𝑖 and Algorithm 4.2, we know that 𝑥1,𝑥2,𝑥3, 𝑢11, 𝑢12, 𝑢13, 𝑢21, 𝑢22, 𝑢23, 𝑢31, 𝑢32, 𝑢33 satisfy the following relation: 𝑓𝑖𝑥𝑖=𝐽𝐻𝑖,𝜂𝑖𝑀𝑖𝑥𝑖,⋅,𝜙𝑖𝐻𝑖∘𝑓𝑖𝑥𝑖−𝜙𝑖∘𝐹𝑖𝑥1−𝑝1𝑥1,𝑥2−𝑝2𝑥2,𝑥3−𝑝3𝑥3−𝜙𝑖∘𝑁𝑖𝑢𝑖1,𝑢𝑖2,𝑢𝑖3,𝑖=1,2,3.(4.35) By Lemma 4.1, (𝑥1,𝑥2,𝑥3,𝑢11,𝑢12,𝑢13,𝑢21,𝑢22,𝑢23,𝑢31,𝑢32,𝑢33) is a solution of the SGVIP (4.1). This completes the proof.