A New System of Random Generalized Variational Inclusions with Random Fuzzy Mappings and Random (𝐻(⋅,⋅),𝜙)-𝜂-Accretive Mappings in Banach Spaces

Abstract

We introduce a new notion of random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mappings and prove the Lipschitz continuity of the random resolvent operator associated with the random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mappings. We introduce and study a new system of random generalized variational inclusions with random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mappings and random fuzzy mappings in Banach spaces. By using the random resolvent operator, an iterative algorithm for solving such system of random generalized variational inclusions is constructed in Banach spaces. Under some suitable conditions, we prove the convergence of the iterative sequences generated by the algorithm.

1. Introduction

Variational inclusions and variational inequalities have wide applications to many fields including, for example, economics, optimization and control theory, operators research, transportation network modeling, and mathematical programming. For these reasons, various variational inclusions and variational inequalities have been intensively studied in recent years. For details, we refer the reader to [1–3] and references therein.

Recently, Wang and Ding [4], and Nazemi [5] introduced the cocepts of (𝐻(⋅,⋅),𝜂)-accretive mappings, and (𝐻,𝜙)-𝜂-accretive mappings, respectively, which generalize the notion of 𝐻-accretive mappings, (𝐻,𝜂)-accretive mappings, 𝐻(⋅,⋅)-accretive mappings, (𝐴,𝜂)-accretive mappings and other existing accretive mappings as special cases. They also defined the resolvent operators associated with this mappings, and shown their Lipschitz continuity.

On the other hand, it is well known that the study of the random equations involving the random operators in view of their need for dealing with probabilistic models in applied sciences is very important. In 1989, Chang and Zhu [6] and Chang [7] first introduced the concepts of variational inequalities for fuzzy mappings. Since then several classes of variational inequalities for fuzzy mappings were considered by Chang and Huang [8], Noor [9, 10], Noor and Al-Said [11], Ding [12], Ding and Park [13], and Park and Jeong [14, 15] in the setting of Hilbert spaces. Recently, the random variational inequalities and random variational inclusion problems have been introduced and studied by Dai [16], Cho and Lan [17], Lan [18], Ahmad and Bazán [19], Chang and Huang [20, 21], Zang and Zhu [22], Huang [23, 24], Husain et al. [25], etc.

Motivated and inspired by the recent research works in this fascinating area, in this paper, we introduce a new class of random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mappings and study a new system of random generalized variational inclusions with random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mappings and random fuzzy mappings in Banach spaces. An iterative algorithm is defined to compute the approximate solutions of system of random generalized variational inclusions. The convergence of iterative sequences generated by the algorithm is also shown. Our results improve and generalize many known corresponding results.

2. Preliminaries

Let 𝐸 be a real Banach space with dual space 𝐸^∗ and ⟨⋅,⋅⟩ the dual pair between 𝐸^∗ and 𝐸. Let 𝐹(𝐸) be a collection of all fuzzy sets over 𝐸. A mapping 𝐹 from 𝐸 into 𝐹(𝐸) is called a fuzzy mapping on 𝐸. If 𝐹 is a fuzzy mapping on 𝐸, then 𝐹(𝑥) (denote it by 𝐹_𝑥 in the sequel) is a fuzzy set on 𝐸 and 𝐹_𝑥(𝑦) is the membership function of 𝑦 in 𝐹_𝑥. Let 𝑁∈𝐹(𝐸), 𝑞∈[0,1], then the set (𝑁)_𝑞={𝑥∈𝐸∶𝑁(𝑥)≥𝑞}(2.1) is called a 𝑞-cut set of 𝑁.

Let (Ω,Σ) be a measurable space, where Ω is a set and Σ is a 𝜎-algebra of subsets of Ω. Let 𝛽(𝐸), 2^𝐸, CB(𝐸), and 𝐻(⋅,⋅) be the class of Borel 𝜎-fields in 𝐸, the family of all nonempty subsets of 𝐸, the family of all nonempty, closed and bounded subsets of 𝐸, and the Hausdorff metric on CB(𝐸), respectively. Recall that the generalized duality mapping 𝐽_𝑞∶𝐸→2^𝐸∗ is defined by𝐽_𝑞(𝑥)=𝑓∈𝐸^∗∶⟨𝑥,𝑓⟩=‖𝑥‖^𝑞,‖𝑥‖^𝑞−1=‖𝑓‖,∀𝑥∈𝐸,(2.2) where 𝑞>1 is a constant. In particular, 𝐽₂ is the usual normalized duality mapping. It is known that, in general, 𝐽_𝑞(𝑥)=‖𝑥‖^𝑞−1𝐽₂(𝑥) for all 𝑥≠0, and 𝐽_𝑞 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 𝐽_𝑞 is single-valued. If 𝐸=ℋ is a Hilbert space, then 𝐽₂ becomes the identity mapping on 𝐻.

The modulus of smoothness of 𝐸 is the function 𝜌_𝐸∶[0,∞)→[0,∞) defined by𝜌_𝐸(𝑡)=sup12(‖𝑥+𝑦‖+‖𝑥−𝑦‖)−1∶‖𝑥‖≤1,‖𝑦‖≤𝑡.(2.3) A Banach space 𝐸 is called uniformly smooth iflim_𝑡→0𝜌_𝐸(𝑡)𝑡=0.(2.4)𝐸 is called 𝑞-uniformly smooth if there exists a constant 𝑐>0 such that𝜌_𝐸(𝑡)≤𝑐𝑡^𝑞,𝑞>1.(2.5) Note that 𝐽_𝑞 is single-valued if 𝐸 is uniformly smooth. In the following, we give an important inequality due to Xu [26] in 𝑞-uniformly smooth Banach spaces, which will play a crucial role in constructing the iterative algorithm to approximate the solution of system of variational inclusions.

Lemma 2.1 (see [26]). Let 𝐸 be a real uniformly smooth Banach space. Then 𝐸 is 𝑞-uniformly smooth if and only if there exists a constant 𝑐_𝑞>0 such that ‖𝑥+𝑦‖^𝑞≤‖𝑥‖^𝑞+𝑞𝑦,𝐽_𝑞(𝑥)+𝑐_𝑞‖𝑦‖^𝑞.(2.6)

Definition 2.2. A mapping 𝑥∶Ω→𝐸 is said to be measurable if for any 𝐵∈𝛽(𝐸), {𝑡∈Ω∶𝑥(𝑡)∈𝐵}∈Σ.

Definition 2.3. A mapping 𝑓∶Ω×𝐸→𝐸 is called a random single-valued mapping if for any 𝑥∈𝐸, 𝑓(𝑡,𝑥)=𝑦(𝑡) is measurable. A random mapping 𝑓 is said to be continuous if for any 𝑡∈Ω, the mapping 𝑓(𝑡,⋅)∶𝐸→𝐸 is continuous. Similarly, we can define a random mapping 𝑏∶Ω×𝐸×𝐸→𝐸. We will write 𝑓_𝑡(𝑥)=𝑓(𝑡,𝑥(𝑡)) and 𝑏_𝑡(𝑥,𝑦)=𝑏(𝑡,𝑥(𝑡),𝑦(𝑡)) for all 𝑡∈Ω and 𝑥(𝑡),𝑦(𝑡)∈𝐸.
It is well known that a measurable mapping is necessarily a random mapping.

Definition 2.4. A multivalued mapping 𝐺∶Ω→2^𝐸 is said to be measurable if for any 𝐵∈𝛽(𝐸), 𝐺⁻¹(𝐵)={𝑡∈Ω∶𝐺(𝑡)∩𝐵≠∅}∈Σ.

Definition 2.5. A mapping 𝑢∶Ω→𝐸 is called a measurable section of a multivalued mapping Γ∶Ω→2^𝐸 if 𝑢 is measurable and for any 𝑡∈Ω, 𝑢(𝑡)∈Γ(𝑡).

Definition 2.6. A mapping 𝑇∶Ω×𝐸→2^𝐸 is called a random multivalued mapping if for any 𝑥∈𝐸, 𝑇(⋅,𝑥) is measurable. A random multivalued mapping 𝑇∶Ω×𝐸→CB(𝐸) is said to be 𝐻-continuous if for any 𝑡∈Ω, 𝑇(𝑡,⋅) is continuous in 𝐻(⋅,⋅), where 𝐻(⋅,⋅) is the Hausdorff metric on CB(𝐸) defined as follows: for any given 𝐴,𝐵∈CB(𝐸), 𝐻(𝐴,𝐵)=maxsup_𝑥∈𝐴inf_𝑦∈𝐵‖𝑥−𝑦‖,sup_𝑦∈𝐵inf_𝑥∈𝐴‖𝑥−𝑦‖.(2.7)

Definition 2.7. A fuzzy mapping 𝐹∶Ω→𝐹(𝐸) is called measurable if for any 𝛼∈(0,1], (𝐹(⋅))_𝛼∶Ω→2^𝐸 is a measurable multivalued mapping.

Definition 2.8. A fuzzy mapping 𝐹∶Ω×𝐸→𝐹(𝐸) is called a random fuzzy mapping if for any 𝑥∈𝐸, 𝐹(⋅,𝑥)∶Ω→𝐹(𝐸) is a measurable fuzzy mapping.

Lemma 2.9 (see [27]). Let 𝑇∶Ω×𝐸→CB(𝐸) be a 𝐻-continuous random multivalued mapping. Then for any measurable mapping 𝑤∶Ω→𝐸 the multivalued mapping 𝑇(⋅,𝑤(⋅))∶Ω→CB(𝐸) is measurable.

Lemma 2.10 (see [27]). Let 𝑆,𝑇∶Ω→CB(𝐸) be two measurable multivalued mappings, 𝜖>0 a constant, and 𝑣∶Ω→𝐸 a measurable selection of 𝑆. Then there exists a measurable selection 𝑤∶Ω→𝐸 of T such that for all 𝑡∈Ω, ‖𝑣(𝑡)−𝑤(𝑡)‖≤(1+𝜖)𝐻(𝑆(𝑡),𝑇(𝑡)).(2.8)

Definition 2.11. A random mapping 𝑃∶Ω×𝐸→𝐸 is said to be(i)accretive if𝑃_𝑡𝑥₁−𝑃_𝑡𝑥₂,𝐽_𝑞𝑥₁(𝑡)−𝑥₂(𝑡)≥0,∀𝑥₁(𝑡),𝑥₂(𝑡)∈𝐸,𝑡∈Ω;(2.9)(ii)𝑟-strongly accretive if there exists a measurable function 𝑟∶Ω→(0,∞) such that𝑃≥_𝑡𝑟𝑥(₁𝑡)‖‖−𝑥𝑃_𝑡1𝑥₂,𝐽_𝑞𝑥₁(𝑡)−𝑥₂(𝑡)(𝑡)−𝑥₂(𝑡)‖‖^𝑞,∀𝑥₁(𝑡),𝑥₂(𝑡)∈𝐸,𝑡∈Ω;(2.10)(iii)𝑚-relaxed accretive if there exists a measurable function 𝑚∶Ω→(0,∞) such that𝑃≥_𝑡𝑥−₁𝑚−(𝑡)𝑃‖‖_𝑡𝑥𝑥₂₁,𝐽_𝑞𝑥₁(𝑡)−𝑥₂(𝑡)(𝑡)−𝑥₂(𝑡)‖‖^𝑞,∀𝑥₁(𝑡),𝑥₂(𝑡)∈𝐸,𝑡∈Ω;(2.11)(iv)(𝛼,𝜉)-relaxed cocoercive if there exist measurable functions 𝛼∶Ω→(0,∞), 𝜉∶Ω→(0,∞) such that≥𝑃_𝑡−𝑥𝛼₁(𝑡)−‖‖𝑃𝑃_𝑡𝑡𝑥₂,𝐽_𝑞𝑥₁(𝑡)−𝑥₂(𝑡)𝑥₁−∀𝑃𝑥_𝑡₁𝑥₂‖‖^𝑞+𝜉(𝑡)‖‖𝑥₁(𝑡)−𝑥₂(𝑡)‖‖^𝑞,(𝑡),𝑥₂(𝑡)∈𝐸,𝑡∈Ω;(2.12)(v)𝜆-Lipschitz continuous if there exists a measurable function 𝜆∶Ω→(0,∞) such that‖𝑝_𝑡𝑥₁−𝑝_𝑡𝑥₂‖≤𝜆(𝑡)‖𝑥₁(𝑡)−𝑥₂(𝑡)‖,∀𝑥₁(𝑡),𝑥₂(𝑡)∈𝐸,𝑡∈Ω.(2.13)

Definition 2.12. A random multivalued mapping 𝑀∶Ω×𝐸→2^𝐸 is said to be(i)accretive if𝑢≥(𝑡)0,−𝑣∀(𝑢𝑡)(𝑡,)𝐽∈_𝑞𝑀𝑥_1𝑡(𝑡)−𝑥₂(𝑡)𝑥₁,𝑣(𝑡)∈𝑀_𝑡𝑥₂,𝑥₁(𝑡),𝑥₂(𝑡)∈𝐸,𝑡∈Ω;(2.14)(ii)𝑚-accretive if 𝑀 is accretive and (𝐼_𝑡+𝜌(𝑡)𝑀_𝑡)(𝐸)=𝐸 holds for all 𝑡∈Ω and 𝜌(𝑡)>0.

Remark 2.13. If 𝐸=ℋ is a Hilbert space, then we can obtain corresponding definitions of monotonicity, strongly monotonicity, relaxed monotonicity, and maximal monotonicity from Definitions 2.11 and 2.12.

Definition 2.14. A random mapping 𝜂∶Ω×𝐸×𝐸→𝐸 is said to be 𝜏-Lipschitz continuous if there exists a measurable function 𝜏∶Ω→(0,∞) such that ‖𝜂_𝑡𝑥₁,𝑥₂‖≤𝜏(𝑡)‖𝑥₁(𝑡)−𝑥₂(𝑡)‖,∀𝑥₁(𝑡),𝑥₂(𝑡)∈𝐸,𝑡∈Ω.(2.15)

Definition 2.15. Let 𝐻,𝐴∶Ω×𝐸→𝐸 and 𝜂∶Ω×𝐸×𝐸→𝐸 be random single-valued mappings. Then a random multivalued mapping 𝑇∶Ω×𝐸→2^𝐸 is said to be(i)𝜂-accretive if𝑢≥(𝑡)0,−𝑣∀(𝑢𝑡)(𝑡,)𝐽∈_𝑞𝑇𝜂_𝑡𝑡𝑥₁,𝑥₂𝑥₁,𝑣(𝑡)∈𝑇_𝑡𝑥₂,𝑥₁(𝑡),𝑥₂(𝑡)∈𝐸,𝑡∈Ω;(2.16)(ii)strictly 𝜂-accretive if𝑢≥(𝑡)0,−𝑣∀(𝑢𝑡)(𝑡,)𝐽∈_𝑞𝑇𝜂_𝑡𝑡𝑥₁,𝑥₂𝑥₁,𝑣(𝑡)∈𝑇_𝑡𝑥₂,𝑥₁(𝑡),𝑥₂(𝑡)∈𝐸,𝑡∈Ω,(2.17) and equality holds if and only if 𝑥₁(𝑡)=𝑥₂(𝑡);(iii)𝛾-strongly 𝜂-accretive if there exists a measurable function 𝛾∶Ω→(0,∞) such that𝑢≥(𝑡)𝛾−(𝑡)𝑣‖‖(𝑥𝑡)₁,𝐽_𝑞𝜂_𝑡𝑥₁,𝑥₂(𝑡)−𝑥₂(𝑡)‖‖^𝑞,∀𝑢(𝑡)∈𝑇_𝑡𝑥₁,𝑣(𝑡)∈𝑇_𝑡𝑥₂,𝑥₁(𝑡),𝑥₂(𝑡)∈𝐸,𝑡∈Ω;(2.18)(iv)𝑚-relaxed 𝜂-accretive if there exists a measurable function 𝑚∶Ω→(0,∞) such that𝑢≥(𝑡)−−𝑚𝑣((𝑡𝑡))‖‖,𝑥𝐽_𝑞1𝜂_𝑡𝑥₁,𝑥₂(𝑡)−𝑥₂(𝑡)‖‖^𝑞,∀𝑢(𝑡)∈𝑇_𝑡𝑥₁,𝑣(𝑡)∈𝑇_𝑡𝑥₂,𝑥₁(𝑡),𝑥₂(𝑡)∈𝐸,𝑡∈Ω;(2.19)(v)𝜂-𝑚-accretive if 𝑇 is 𝜂-accretive and (𝐼_𝑡+𝜌(𝑡)𝑇_𝑡)(𝐸)=𝐸 holds for all 𝑡∈Ω and 𝜌(𝑡)>0;(vi)𝐻-accretive if 𝑇 is accretive and (𝐻_𝑡+𝜌(𝑡)𝑇_𝑡)(𝐸)=𝐸 holds for all 𝑡∈Ω and 𝜌(𝑡)>0;(vii)(𝐻,𝜂)-accretive if 𝑇 is 𝜂-accretive and (𝐻_𝑡+𝜌(𝑡)𝑇_𝑡)(𝐸)=𝐸 holds for all 𝑡∈Ω and 𝜌(𝑡)>0;(viii)(𝐴,𝜂)-accretive if 𝑇 is 𝑚-relaxed 𝜂-accretive and (𝐴_𝑡+𝜌(𝑡)𝑇_𝑡)(𝐸)=𝐸 holds for all 𝑡∈Ω and 𝜌(𝑡)>0.

Definition 2.16. Let 𝐻∶𝐸×𝐸→𝐸 be a single-valued mapping and 𝜂∶Ω×𝐸×𝐸→𝐸 and 𝐴,𝐵∶Ω×𝐸→𝐸 random single-valued mappings. Then(i)𝐻(𝐴,⋅) is said to be 𝛼-strongly 𝜂-accretive with respect to 𝐴 if there exists a measurable function 𝛼∶Ω→(0,∞) such that 𝐻≥𝛼𝐴(_𝑡𝑡)‖‖𝑥𝑥₁₁,𝑢(𝑡)−𝐻𝐴_𝑡𝑥₂,𝑢(𝑡),𝐽_𝑞𝜂_𝑡𝑥₁,𝑥₂(𝑡)−𝑥₂(𝑡)‖‖^𝑞,∀𝑥₁(𝑡),𝑥₂(𝑡),𝑢(𝑡)∈𝐸,𝑡∈Ω;(2.20)(ii)𝐻(⋅,𝐵) is said to be 𝛽-relaxed 𝜂-accretive with respect to 𝐵 if there exists a measurable function 𝛽∶Ω→(0,∞) such that 𝐻≥−𝑢(𝛽𝑡)(,𝑡)𝐵‖‖_𝑡𝑥₁𝑥₁−𝐻𝑢(𝑡),𝐵_𝑡𝑥₂,𝐽_𝑞𝜂_𝑡𝑥₁,𝑥₂(𝑡)−𝑥₂(𝑡)‖‖^𝑞,∀𝑥₁(𝑡),𝑥₂(𝑡),𝑢(𝑡)∈𝐸,𝑡∈Ω;(2.21)(iii)𝐻 is said to be 𝜆-Lipschitz continuous with respect to 𝐴 in the first argument if there exists a measurable function 𝜆∶Ω→(0,∞) such that ‖𝐻≤𝜆𝐴(_𝑡𝑡)𝑥‖₁𝑥₁,𝑢(𝑡)−𝐻𝐴_𝑡𝑥₂,𝑢(𝑡)‖(𝑡)−𝑥₂(𝑡)‖,∀𝑥₁(𝑡),𝑥₂(𝑡),𝑢(𝑡)∈𝐸,𝑡∈Ω.(2.22)

Example 2.17. Let ℋ be a Hilbert space and 𝑆∶Ω×ℋ→ℋ a 𝑠-Lipschitz continuous and accretive mapping, 𝜂_𝑡(𝑥,𝑦)=𝑥(𝑡)−𝑦(𝑡), 𝐻(𝑥(𝑡),𝑦(𝑡))=𝑥(𝑡)+𝑦(𝑡), 𝐴_𝑡(𝑥)=(1+𝑐_𝑘/16)𝑥(𝑡), and 𝐵_𝑡(𝑥)=−𝑐_𝑘𝑆_𝑡(𝑥)−(𝑐_𝑘/8)𝑥(𝑡), for all (𝑥(𝑡),𝑦(𝑡))∈ℋ×ℋ, 𝑡∈Ω and 𝑐_𝑘>0. Then 𝐻(𝐴,⋅) is (1+𝑐_𝑘/16)-strongly 𝜂-accretive with respect to 𝐴 and 𝐻(⋅,𝐵) is 𝑐_𝑘(𝑠+1/8)-relaxed 𝜂-accretive with respect to 𝐵.

Definition 2.18. Let 𝐸₁,𝐸₂,…,𝐸_𝑘 be real Banach spaces and for each 𝑖=1,2,…,𝑘, let 𝑁_𝑖∶Ω×∏^𝑘𝑗=1𝐸_𝑗→𝐸_𝑖 be a random single-valued mapping. Then 𝑁_𝑖 is said to be 𝜉_𝑖𝑗-Lipschitz continuous in the (𝑗+1)th argument if there exists a measurable function 𝜉_𝑖𝑗∶Ω→(0,∞) such that ‖𝑁_𝑖𝑡≤𝑥𝜉₁,_𝑖𝑗…,𝑥_𝑗−1,𝑦_𝑗1,𝑥_𝑗+1,…,𝑥_𝑘−𝑁_𝑖𝑡𝑥₁,…,𝑥_𝑗−1,𝑦_𝑗2,𝑥_𝑗+1,…,𝑥_𝑘‖(𝑡)‖𝑦_𝑗1(𝑡)−𝑦_𝑗2(𝑡)‖𝑡,∈∀Ω𝑦_𝑗1(𝑖(,𝑡𝑗),∈𝑦_𝑗2{1(𝑡,)2∈,…𝐸_𝑗,,𝑘}𝑥_𝑖,(𝑖𝑡≠)∈𝑗)𝐸._𝑖,(2.23)

Definition 2.19. Let ℋ be a Hilbert space, 𝐻∶ℋ×ℋ→ℋ a single-valued mapping, and 𝐴,𝐵∶Ω×ℋ→ℋ random single-valued mappings. 𝐻 is said to be coercive with respect to 𝐴 and 𝐵 if lim_{‖𝑥(𝑡)‖→∞}𝐻𝐴_𝑡(𝑥),𝐵_𝑡(𝑥),𝑥(𝑡)‖𝑥(𝑡)‖=+∞.(2.24)

Definition 2.20. Let ℋ be a Hilbert space, 𝐻∶ℋ×ℋ→ℋ a single-valued mapping and 𝐴,𝐵∶Ω×ℋ→ℋ random single-valued mappings. 𝐻 is said to be bounded with respect to 𝐴 and 𝐵 if 𝐻(𝐴_𝑡(𝑃),𝐵_𝑡(𝑃)) is bounded for every bounded subset 𝑃 of ℋ and 𝑡∈Ω. 𝐻 is said to be hemicontinuous with respect to 𝐴 and 𝐵 if for any fixed 𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)∈ℋ, the function 𝑘↦⟨𝐻(𝐴_𝑡(𝑥+𝑘𝑦),𝐵_𝑡(𝑥+𝑘𝑦)),𝑧(𝑡)⟩ is continuous at 0⁺.

3. Random (𝐻(⋅,⋅),𝜙)-𝜂-Accretive Mappings

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

Definition 3.1. Let 𝐻∶𝐸×𝐸→𝐸 be a single-valued mapping and 𝜂∶Ω×𝐸×𝐸→𝐸 and 𝜙,𝐴,𝐵∶Ω×𝐸→𝐸 random single-valued mappings. Then the random multivalued mapping 𝑀∶Ω×𝐸→2^𝐸 is said to be (𝐻(⋅,⋅),𝜙)-𝜂-accretive with respect to 𝐴 and 𝐵 if 𝜙∘𝑀 is 𝑚-relaxed 𝜂-accretive and 𝐻𝐴_𝑡,𝐵_𝑡+𝜙_𝑡∘𝑀_𝑡(𝐸)=𝐸,∀𝑡∈Ω.(3.1)

Remark 3.2. (i) If 𝐻(𝐴_𝑡,𝐵_𝑡)=𝐻_𝑡 and 𝜙∘𝑀 is 𝜂-accretive, then the random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mapping reduces to the random (𝐻,𝜙)-𝜂-accretive mapping.
(ii) If 𝜙_𝑡(𝑥)=𝜆(𝑡)𝑥(𝑡), for all 𝑥(𝑡)∈𝐸, 𝜆(𝑡)>0, then the random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mapping reduces to the random (𝐻(⋅,⋅),𝜂)-accretive mapping.

Remark 3.3. (i) If 𝑀 is 𝜂-accretive, 𝐻(𝐴_𝑡,𝐵_𝑡)=𝐻_𝑡, and 𝜙_𝑡(𝑥)=𝜆(𝑡)𝑥(𝑡), for all 𝑥(𝑡)∈𝐸, 𝜆(𝑡)>0, then the random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mapping reduces to the random (𝐻,𝜂)-accretive mapping studied by Uea and Kumam [28].
(ii) If 𝑀 is 𝑚-relaxed 𝜂-accretive, 𝐻(𝐴_𝑡,𝐵_𝑡)=𝐻_𝑡, and 𝜙_𝑡(𝑥)=𝜆(𝑡)𝑥(𝑡), for all 𝑥(𝑡)∈𝐸, 𝜆(𝑡)>0, then the random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mapping reduces to the random (𝐴,𝜂)-accretive mapping studied by Cho and Lan [17].
(iii) If 𝑀 is 𝜂-accretive, 𝜂_𝑡(𝑥,𝑦)=𝑥(𝑡)−𝑦(𝑡), 𝜙_𝑡(𝑥)=𝜆(𝑡)𝑥(𝑡), for all 𝑥(𝑡),𝑦(𝑡)∈𝐸, 𝜆(𝑡)>0, then the random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mapping reduces to the random 𝐻(⋅,⋅)-accretive mapping studied by Zhang [29].

Example 3.4. Let ℋ be a Hilbert space, 𝜆∶Ω→(0,∞) a measurable function; for every (𝑥(𝑡),𝑦(𝑡))∈ℋ×ℋ, 𝜂_𝑡(𝑥,𝑦)=𝑥(𝑡)−𝑦(𝑡), 𝜙_𝑡(𝑥)=𝜆(𝑡)𝑥(𝑡); 𝑀∶Ω×ℋ→2^𝐻 a random maximal monotone mapping, 𝐻∶ℋ×ℋ→ℋ a bounded, coercive, and hemicontinuous mapping, with respect to 𝐴 and 𝐵, and 𝛼-strongly 𝜂-accretive with respect to 𝐴 in the first argument and 𝛽-relaxed 𝜂-accretive with respect to 𝐵 in second argument. Then 𝑀 is a random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mapping with respect to 𝐴 and 𝐵.

Proof. For every 𝑡∈Ω, 𝜆(𝑡)𝑀 is maximal monotone since 𝑀 is maximal monotone. Since 𝐻 is bounded, coercive, and hemicontinuous with respect to 𝐴 and 𝐵,𝛼-strongly 𝜂-accretive with respect to 𝐴 in the first argument and 𝛽-relaxed 𝜂-accretive with respect to 𝐵 in the second argument, it follows from Corollary 32.26 of [30] that (𝐻(𝐴_𝑡,𝐵_𝑡)+𝜙_𝑡∘𝑀_𝑡)(ℋ)=ℋ holds for every 𝑡∈Ω. Thus 𝑀 is a random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mapping with respect to 𝐴 and 𝐵.

Example 3.5. Let ℋ=ℛ=(−∞,+∞), 𝑀_𝑡(𝑥)=1/(1+𝑥(𝑡)²), 𝐴_𝑡(𝑥)=2𝑥(𝑡), 𝐵_𝑡(𝑦)=−𝑦(𝑡), 𝐻(𝐴_𝑡(𝑥),𝐵_𝑡(𝑦))=𝐴_𝑡(𝑥)+𝐵_𝑡(𝑦), 𝜂_𝑡(𝑥,𝑦)=arctan(𝑥(𝑡))−arctan(𝑦(𝑡)), and 𝜙_𝑡∘𝑀_𝑡(𝑥)=∫^{𝑥(𝑡)−∞}𝑀_𝑡(𝑠)𝑑𝑠, for all 𝑥(𝑡),𝑦(𝑡)∈ℛ. Then ⟨=𝜙_𝑡∘𝑀ar_𝑡c(t𝑥an)−(𝑥𝜙(_𝑡𝑡)∘)𝑀+_𝑡𝜋(𝑦),𝜂_𝑡(𝑥,𝑦)⟩=(arctan(𝑥(𝑡))−arc2tan−(𝑦(𝑡a)r)c)t₂an(𝑦(𝑡))+𝜋2,arctan(𝑥(𝑡))−arctan(𝑦(𝑡))𝐻𝐴≥0,_𝑡,𝐵_𝑡+𝜙_𝑡∘𝑀_𝑡(𝑥)=𝑥(𝑡)+arctan(𝑥(𝑡))+𝜋2.(3.2) This implies that (𝐻(𝐴_𝑡,𝐵_𝑡)+𝜙_𝑡∘𝑀_𝑡) is surjective. Thus 𝑀 is a random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mapping with respect to 𝐴 and 𝐵.

Definition 3.6. Let 𝜂∶Ω×𝐸×𝐸→𝐸, 𝜙∶Ω×𝐸→𝐸 be random single-valued mappings, 𝐻(𝐴,𝐵) be 𝛼-strongly 𝜂-accretive with respect to 𝐴 and 𝛽-relaxed 𝜂-accretive with respect to 𝐵 and 𝑀∶Ω×𝐸→2^𝐸 be a random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mapping with respect to 𝐴 and 𝐵. Then the general resolvent operator 𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}∶Ω×𝐸→2^𝐸 is defined by 𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑢)=𝐻𝐴_𝑡,𝐵_𝑡+𝜙_𝑡∘𝑀_𝑡⁻¹(𝑢),∀𝑢(𝑡)∈𝐸,𝑡∈Ω.(3.3)

Remark 3.7. (i) If 𝑀 is 𝜂-accretive, 𝐻(𝐴_𝑡,𝐵_𝑡)=𝐻_𝑡, and 𝜙_𝑡(𝑥)=𝜆(𝑡)𝑥(𝑡), for all 𝑥(𝑡)∈𝐸, 𝜆(𝑡)>0, then the resolvent operator 𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡} reduces to the resolvent operator 𝐽^{𝜆(𝑡),𝑀𝑡}_𝐻𝑡 introduced by Uea and Kumam [28].
(ii) If 𝑀 is 𝑚-relaxed 𝜂-accretive, 𝐻(𝐴_𝑡,𝐵_𝑡)=𝐻_𝑡 and 𝜙_𝑡(𝑥)=𝜆(𝑡)𝑥(𝑡), for all 𝑥(𝑡)∈𝐸, 𝜆(𝑡)>0, then the resolvent operator 𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡} reduces to the resolvent operator 𝑅^{𝜆(𝑡),𝐴𝑡}_{𝜂𝑡,𝑀𝑡} introduced by Cho and Lan [17].
(iii) If 𝑀 is 𝜂-accretive, 𝜂_𝑡(𝑥,𝑦)=𝑥(𝑡)−𝑦(𝑡), 𝜙_𝑡(𝑥)=𝜆(𝑡)𝑥(𝑡), for all 𝑥(𝑡),𝑦(𝑡)∈𝐸, 𝜆(𝑡)>0, then the resolvent operator 𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡} reduces to the resolvent operator 𝑅^{𝐻(⋅,⋅)𝑀,𝜆} introduced by Zhang [29].

Theorem 3.8. Let 𝜙∶Ω×𝐸→𝐸 be a random single-valued mapping, 𝜂∶Ω×𝐸×𝐸→𝐸 be 𝜏-Lipschitz continuous and 𝐻(𝐴,𝐵) be 𝛼-strongly 𝜂-accretive with respect to 𝐴 and 𝛽-relaxed 𝜂-accretive with respect to 𝐵 and 𝛼(𝑡)≥𝛽(𝑡), ∀𝑡∈Ω. Let 𝑀∶Ω×𝐸→2^𝐸 be a random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mapping with respect to 𝐴 and 𝐵. Then the resolvent operator 𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡} is 𝜏(𝑡)^𝑞−1/(𝑟(𝑡)−𝑚(𝑡))-Lipschitz continuous for 𝑟(𝑡)>𝑚(𝑡), where 𝑟(𝑡)=𝛼(𝑡)−𝛽(𝑡), ∀𝑡∈Ω, that is, ‖𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑢)−𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑣)‖≤𝜏(𝑡)^𝑞−1𝑟(𝑡)−𝑚(𝑡)‖𝑢(𝑡)−𝑣(𝑡)‖,∀𝑢(𝑡),𝑣(𝑡)∈𝐸,𝑡∈Ω.(3.4)

Proof. Let 𝑢(𝑡),𝑣(𝑡)∈𝐸 be any given points; it follows from Definition 3.6 that 𝑅𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑢)=𝐻𝐴_𝑡,𝐵_𝑡+𝜙_𝑡∘𝑀_𝑡⁻¹(𝑢),_{𝐻(⋅,⋅),𝜂𝑡𝑀𝑡,𝜙𝑡}(𝑣)=𝐻𝐴_𝑡,𝐵_𝑡+𝜙_𝑡∘𝑀_𝑡⁻¹(𝑣).(3.5) This implies that 𝑣𝑢((𝑡𝑡))−−𝐻𝐻𝐴𝐴^𝑡_𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑢),𝐵_𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑢)∈𝜙_𝑡∘𝑀_𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑢),𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑣),𝐵_𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑣)∈𝜙_𝑡∘𝑀_𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑣).(3.6) Since 𝜙∘𝑀 is 𝑚-relaxed 𝜂-accretive, we get −𝑚(𝑡)‖‖𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑢)−𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑣)‖‖^𝑞≤−𝑢(𝑡𝑣)−(𝑡)𝐻−𝐻𝐴_𝑡𝐴𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡𝑡(𝑢),𝐵_𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑢)=𝑢(𝑡)−𝑣(𝑡)−𝐻𝑅𝐴^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡𝑡}(𝑣),𝐵_𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑣),𝐽_𝑞𝜂^𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑢),𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑣)−𝐻𝐴_𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑢),𝐵_𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑢)𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑣),𝐵_𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑣),𝐽_𝑞𝜂^𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑢),𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑣).(3.7) From (3.7) and Lipschitz continuity of 𝜂, we have 𝜏(𝑡)^𝑞−1‖𝑢(𝑡)−𝑣(𝑡)‖‖‖𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑢)−𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑣)‖‖^𝑞−1≥‖𝑢(𝑡)−𝑣(𝑡)‖‖‖𝜂_𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑢),𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑣)‖‖^𝑞−1≥≥𝑢𝐻(𝑡)𝐴−𝑣(𝑡),𝐽_𝑞𝜂^𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑢),𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑣)_𝑡𝐽𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑢),𝐵_𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑢)−𝐻𝐴_𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑣),𝐵_𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑣)_𝑞,𝜂^𝑡𝑅^{𝐻(⋅,⋅),𝜂𝑡𝑀}_𝑡^,𝜙_𝑡(𝑢),𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑣)−𝑚(𝑡)‖‖𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑢)−𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑣)‖‖^𝑞≥(𝛼(𝑡)−𝛽(𝑡)−𝑚(𝑡))‖‖𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑢)−𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑣)‖‖^𝑞=(𝑟(𝑡)−𝑚(𝑡))‖‖𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑢)−𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑣)‖‖^𝑞.(3.8) Hence ‖𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑢)−𝑅^{𝐻(⋅,⋅),𝜂𝑡}_{𝑀𝑡,𝜙𝑡}(𝑣)‖≤𝜏(𝑡)^𝑞−1𝑟(𝑡)−𝑚(𝑡)‖𝑢(𝑡)−𝑣(𝑡)‖,∀𝑢(𝑡),𝑣(𝑡)∈𝐸,𝑡∈Ω.(3.9) This completes the proof.

4. A New System of Random Generalized Variational Inclusions

In this section, we will introduce a new system of random generalized variational inclusions with random (𝐻(⋅,⋅),𝜙)-𝜂-accretive mappings and construct an iterative algorithm for solving this system of random generalized variational inclusions.

Let for 𝑖,𝑗=1,2,…,𝑘, 𝑇_𝑗𝑖∶Ω×𝐸_𝑗→𝖥(𝐸_𝑖), and 𝑆_𝑖∶Ω×𝐸_𝑖→𝖥(𝐸_𝑖) be random fuzzy mappings satisfying the following condition (C).(C)For 𝑖,𝑗=1,2,…,𝑘, there exist mappings 𝑎_𝑗∶𝐸_𝑗→[0,1], 𝑏_𝑖∶𝐸_𝑖→[0,1], such that𝑇𝑆_{𝑗𝑖𝑗𝑡,𝑦}_{𝑎𝑗(𝑦𝑗)}∈CB𝐸_𝑖,∀𝑡,𝑦_𝑗∈Ω×𝐸_𝑗(𝑖,𝑗=1,2,…,𝑘),_{𝑖𝑖𝑡,𝑧}_{𝑏𝑖(𝑧𝑖)}∈CB𝐸_𝑖,∀𝑡,𝑧_𝑖∈Ω×𝐸_𝑖(𝑖=1,2,…,𝑘).(4.1)

By using the random fuzzy mappings 𝑇_𝑗𝑖, 𝑆_𝑖(𝑖,𝑗=1,2,…,𝑘), we can define random multivalued mappings 𝑇_𝑗𝑖, 𝑆_𝑖 as follows.𝑇_𝑗𝑖∶Ω×𝐸_𝑗⟶CB𝐸_𝑖,𝑦_𝑗⟼𝑇_{𝑗𝑖𝑗𝑡,𝑦}𝑆_{𝑎𝑗(𝑦𝑗)}(𝑖,𝑗=1,2,…,𝑘),_𝑖∶Ω×𝐸_𝑖⟶CB𝐸_𝑖,𝑧_𝑖⟼𝑆_{𝑖𝑖𝑡,𝑧}_{𝑏𝑖(𝑧𝑖)}(𝑖=1,2,…,𝑘),(4.2) where (𝑆_{𝑖𝑖𝑡,𝑧})_{𝑏𝑖(𝑧𝑖)}=𝑆_𝑖(𝑡,𝑧_𝑖(𝑡)).

In the sequel, for 𝑖,𝑗=1,2,…,𝑘, 𝑇_𝑗𝑖 and 𝑆_𝑖 are called the random multivalued mappings induced by the random fuzzy mappings 𝑇_𝑗𝑖 and 𝑆_𝑖, respectively.

For 𝑖=1,2,…,𝑘, given mappings 𝑎_𝑖∶𝐸_𝑖→[0,1], 𝑏_𝑖∶𝐸_𝑖→[0,1], let 𝑇_1𝑖∶Ω×𝐸₁→𝖥(𝐸_𝑖), 𝑇_2𝑖∶Ω×𝐸₂→𝖥(𝐸_𝑖),…,𝑇_𝑘𝑖∶Ω×𝐸_𝑘→𝖥(𝐸_𝑖), 𝑆_𝑖∶Ω×𝐸_𝑖→𝖥(𝐸_𝑖) be random fuzzy mappings, 𝐻_𝑖∶𝐸_𝑖×𝐸_𝑖→𝐸_𝑖 be a single-valued mapping and 𝜙_𝑖,𝐴_𝑖,𝐵_𝑖,𝑓_𝑖,𝑝_𝑖∶Ω×𝐸_𝑖→𝐸_𝑖, 𝜂_𝑖∶Ω×𝐸_𝑖×𝐸_𝑖→𝐸_𝑖, 𝐹_𝑖,𝑁_𝑖∶Ω×∏^𝑘𝑗=1𝐸_𝑗→𝐸_𝑖 be random single-valued mappings and 𝑀_𝑖∶Ω×𝐸_𝑖×𝐸_𝑖→2^𝐸𝑖 be a random (𝐻_𝑖(⋅,⋅),𝜙_𝑖)-𝜂_𝑖-accretive mapping with respect to 𝐴_𝑖 and 𝐵_𝑖 in the second argument with Im𝑓_𝑖∩dom𝑀_𝑖𝑡(⋅,𝑤_𝑖)≠∅, for each 𝑤_𝑖(𝑡)∈𝐸_𝑖 and 𝑡∈Ω. Now we consider the following system of random generalized variational inclusions:

For 𝑖=1,2,…,𝑛, find measurable mappings 𝑥_𝑖,𝑢_1𝑖,𝑢_2𝑖,…,𝑢_𝑘𝑖,𝑣_𝑖∶Ω→𝐸_𝑖 such that for all 𝑡∈Ω and for each 𝑖=1,2,…,𝑘,𝑥_𝑖(𝑡)∈𝐸_𝑖, 𝑇_{1𝑖1𝑡,𝑥(𝑡)}(𝑢_1𝑖(𝑡))≥𝑎₁(𝑥₁(𝑡)), 𝑇_{2𝑖2𝑡,𝑥(𝑡)}(𝑢_2𝑖(𝑡))≥𝑎₂(𝑥₂(𝑡)), …, 𝑇_{𝑘𝑖𝑘𝑡,𝑥(𝑡)}(𝑢_𝑘𝑖(𝑡))≥𝑎_𝑘(𝑥_𝑘(𝑡)), 𝑆_{𝑖𝑖𝑡,𝑥(𝑡)}(𝑣_𝑖(𝑡))≥𝑏_𝑖(𝑥_𝑖(𝑡)) such that0∈𝐹_𝑖𝑡𝑥₁−𝑝_1𝑡𝑥₁,𝑥₂−𝑝_2𝑡𝑥₂,…,𝑥_𝑘−𝑝_𝑘𝑡𝑥_𝑘+𝑁_𝑖𝑡𝑢_𝑖1,𝑢_𝑖2,…,𝑢_𝑖𝑘+𝑀_𝑖𝑡𝑓_𝑖𝑡𝑥_𝑖,𝑣_𝑖.(4.3)

If for 𝑖=1,2,…,𝑘, 𝐸_𝑖=ℋ_𝑖 is a Hilbert space and 𝑀_𝑖𝑡(⋅,𝑣_𝑖)=𝜕𝜑_𝑖𝑡(⋅), for all 𝑡∈Ω, 𝑥_𝑖(𝑡)∈ℋ_𝑖 and 𝑣_𝑖(𝑡)∈𝑆_𝑖𝑡(𝑥_𝑖), where 𝜕𝜑_𝑖 denote the subdifferential of a proper, convex, and lower semicontinuous function 𝜑_𝑖∶Ω×ℋ_𝑖→𝑅∪{+∞}, then system of random generalized variational inclusions (4.3) reduces to the following system of random generalized mixed variational inequalities.

For 𝑖=1,2,…,𝑘, find measurable mappings 𝑥_𝑖,𝑢_1𝑖,𝑢_2𝑖,…,𝑢_𝑘𝑖∶Ω→ℋ_𝑖 such that for all 𝑡∈Ω and for each 𝑖=1,2,…,𝑘, 𝑥_𝑖(𝑡)∈ℋ_𝑖, 𝑇_{1𝑖1𝑡,𝑥(𝑡)}(𝑢_1𝑖(𝑡))≥𝑎₁(𝑥₁(𝑡)), 𝑇_{2𝑖2𝑡,𝑥(𝑡)}(𝑢_2𝑖(𝑡))≥𝑎₂(𝑥₂(𝑡)), …, 𝑇_{𝑘𝑖𝑘𝑡,𝑥(𝑡)}(𝑢_𝑘𝑖(𝑡))≥𝑎_𝑘(𝑥_𝑘(𝑡)) such that𝐹_𝑖𝑡≥𝑥𝜑₁−_𝑖𝑡𝑝_1𝑡𝑥₁,𝑥₂−𝑝_2𝑡𝑥₂,…,𝑥_𝑘−𝑝_𝑘𝑡𝑥_𝑘+𝑁_𝑖𝑡𝑢_𝑖1,𝑢_𝑖2,…,𝑢_𝑖𝑘,𝑦_𝑖(𝑡)−𝑓_𝑖𝑡𝑥_𝑖𝑓_𝑖𝑡𝑥_𝑖−𝜑_𝑖𝑡𝑦_𝑖,∀𝑡∈Ω,𝑦_𝑖(𝑡)∈ℋ_𝑖.(4.4)

We remark that system of random generalized variational inclusions (4.3) and system of random generalized mixed variational inequalities (4.4) include as special cases, many kinds of random variational inequalities and random variational inclusions of [17, 19–21], etc.

Lemma 4.1. For 𝑖=1,2,…,𝑘, let 𝐻_𝑖∶𝐸_𝑖×𝐸_𝑖→𝐸_𝑖 be a single-valued mapping and 𝜂_𝑖∶Ω×𝐸_𝑖×𝐸_𝑖→𝐸_𝑖, 𝜙_𝑖,𝐴_𝑖,𝐵_𝑖∶Ω×𝐸_𝑖→𝐸_𝑖 random single-valued mappings such that 𝐻_𝑖(𝐴_𝑖,𝐵_𝑖) be 𝛼_𝑖-strongly 𝜂_𝑖-accretive with respect to 𝐴_𝑖 and 𝛽_𝑖-relaxed 𝜂_𝑖-accretive with respect to 𝐵_𝑖. Then the set of measurable mappings 𝑥_𝑖,𝑢_1𝑖,𝑢_2𝑖,…,𝑢_𝑘𝑖∶Ω→𝐸_𝑖(𝑖=1,2,…,𝑘) is a random solution of problem (4.3) if and only if for all 𝑡∈Ω, 𝑥_𝑖(𝑡)∈𝐸_𝑖, 𝑢_1𝑖(𝑡)∈𝑇_1𝑖t(𝑥₁), 𝑢_2𝑖(𝑡)∈𝑇_2𝑖t(𝑥₂),…,𝑢_𝑘𝑖(𝑡)∈𝑇_𝑘𝑖t(𝑥_𝑘), 𝑣_𝑖(𝑡)∈𝑆_𝑖t(𝑥_𝑖), and 𝑓_𝑖𝑡𝑥_𝑖=𝑅−^{𝐻𝑖(⋅,⋅),𝜂𝑖𝑡}_{𝑀𝑖𝑡⋅,𝑣𝑖,𝜙𝑖𝑡}𝜙𝐻_𝑖𝑖𝑡𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥_𝑖,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥_𝑖−𝜙_𝑖𝑡∘𝐹_𝑖𝑡𝑥₁−𝑝_1𝑡𝑥₁,𝑥₂−𝑝_2𝑡𝑥₂,…,𝑥_𝑘−𝑝_𝑘𝑡𝑥_𝑘∘𝑁_𝑖𝑡𝑢_𝑖1,𝑢_𝑖2,…,𝑢_𝑖𝑘,(4.5) where 𝑅^{𝐻𝑖(⋅,⋅),𝜂𝑖𝑡}_{𝑀𝑖𝑡(⋅,𝑣𝑖),𝜙𝑖𝑡}=(𝐻_𝑖(𝐴_𝑖𝑡,𝐵_𝑖𝑡)+𝜙_𝑖𝑡∘𝑀_𝑖𝑡(⋅,𝑣_𝑖))⁻¹.

Algorithm 4.2. Suppose that for 𝑖,𝑗=1,2,…,𝑘, 𝑇_𝑗𝑖∶Ω×𝐸_𝑗→𝖥(𝐸_𝑖), 𝑆_𝑖∶Ω×𝐸_𝑖→𝖥(𝐸_𝑖) be random fuzzy mappings satisfying the condition (C). Let for 𝑖=1,2,…,𝑘, 𝑇_1𝑖∶Ω×𝐸₁→CB(𝐸_𝑖), 𝑇_2𝑖∶Ω×𝐸₂→CB(𝐸_𝑖),…,𝑇_𝑘𝑖∶Ω×𝐸_𝑘→CB(𝐸_𝑖), 𝑆_𝑖∶Ω×𝐸_𝑖→CB(𝐸_𝑖) be 𝐻-continuous random multivalued mappings induced by 𝑇_1𝑖,𝑇_2𝑖,…,𝑇_𝑘𝑖 and 𝑆_𝑖, respectively. Let 𝐻_𝑖∶𝐸_𝑖×𝐸_𝑖→𝐸_𝑖 be a single-valued mapping; 𝜂_𝑖∶Ω×𝐸_𝑖×𝐸_𝑖→𝐸_𝑖, 𝜙_𝑖,𝐴_𝑖,𝐵_𝑖∶Ω×𝐸_𝑖→𝐸_𝑖, 𝐹_𝑖,𝑁_𝑖∶Ω×∏^𝑘𝑗=1𝐸_𝑗→𝐸_𝑖 random single-valued mappings; the random multivalued mapping 𝑀_𝑖∶Ω×𝐸_𝑖×𝐸_𝑖→2^𝐸𝑖 be (𝐻_𝑖(⋅,⋅),𝜙_𝑖)-𝜂_𝑖-accretive with respect to 𝐴_𝑖 and 𝐵_𝑖 in the second argument. For any given measurable mapping 𝑥_𝑖∶Ω→𝐸_𝑖(𝑖=1,2,…,𝑘), the multivalued mappings 𝑇_1𝑖(⋅,𝑥⁰¹(⋅)), 𝑇_2𝑖(⋅,𝑥⁰²(⋅)),…,𝑇_𝑘𝑖(⋅,𝑥^0𝑘(⋅)), 𝑆_𝑖(⋅,𝑥^0𝑖(⋅))(𝑖=1,2,…,𝑘) are measurable by Lemma 2.9. Hence for 𝑖=1,2,…,𝑘, there exist measurable selections 𝑢^01𝑖∶Ω→𝐸_𝑖 of 𝑇_1𝑖(⋅,𝑥⁰¹(⋅)), 𝑢^02𝑖∶Ω→𝐸_𝑖 of 𝑇_2𝑖(⋅,𝑥⁰²(⋅)),…,𝑢^0𝑘𝑖∶Ω→𝐸_𝑖 of 𝑇_𝑘𝑖(⋅,𝑥^0𝑘(⋅)) and 𝑣^0𝑖∶Ω→𝐸_𝑖 of 𝑆_𝑖(⋅,𝑥^0𝑖(⋅)), by Himmelberg [31]. Let for 𝑖=1,2,…,𝑘, 𝑥=^1𝑖(𝑥𝑡)_0𝑖(𝑡)−𝑓_𝑖𝑡𝑥^0𝑖+𝑅−𝜙^{𝐻𝑖(⋅,⋅),𝜂𝑖𝑡}_{𝑀𝑖𝑡⋅,𝑣0𝑖,𝜙𝑖𝑡𝑖𝑡}𝐻_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^0𝑖,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^0𝑖−𝜙_𝑖𝑡∘𝐹_𝑖𝑡𝑥⁰¹−𝑝_1𝑡𝑥⁰¹,𝑥⁰²−𝑝_2𝑡𝑥⁰²,…,𝑥^0𝑘−𝑝_𝑘𝑡𝑥^0𝑘∘𝑁_𝑖𝑡𝑢^0𝑖1,𝑢^0𝑖2,…,𝑢^0𝑖𝑘.(4.6) It is easy to see that for 𝑖=1,2,…,𝑘, 𝑥^1𝑖∶Ω→𝐸_𝑖 is measurable. By Lemma 2.10, there exist measurable selections 𝑢^11𝑖∶Ω→𝐸_𝑖 of 𝑇_1𝑖(⋅,𝑥¹¹(⋅)), 𝑢^12𝑖∶Ω→𝐸_𝑖 of 𝑇_2𝑖(⋅,𝑥¹²(⋅)),…,𝑢^1𝑘𝑖∶Ω→𝐸_𝑖 of 𝑇_𝑘𝑖(⋅,𝑥^1𝑘(⋅)) and 𝑣^1𝑖∶Ω→𝐸_𝑖 of 𝑆_𝑖(⋅,𝑥^1𝑖(⋅))(𝑖=1,2,…,𝑘) such that for all 𝑡∈Ω and 𝑖=1,2,…,𝑘, ‖𝑢‖𝑢^01𝑖_02𝑖(𝑡)−𝑢^11𝑖(𝑡)‖≤(1+1)𝐻𝑇_1𝑖𝑡𝑥⁰¹,𝑇_1𝑖𝑡𝑥¹¹,‖𝑢(_0𝑘𝑖𝑡)−𝑢^12𝑖(𝑡)‖≤⋮(1+1)𝐻𝑇_2𝑖𝑡𝑥⁰²,𝑇_2𝑖𝑡𝑥¹²,‖𝑣_0𝑖(𝑡)−𝑢^1𝑘𝑖(𝑡)‖≤(1+1)𝐻𝑇_𝑘𝑖𝑡𝑥^0𝑘,𝑇_𝑘𝑖𝑡𝑥^1𝑘,(𝑡)−𝑣^1𝑖(𝑡)‖≤(1+1)𝐻𝑆_𝑖𝑡𝑥^0𝑖,𝑆_𝑖𝑡𝑥^1𝑖.(4.7) Let for 𝑖=1,2,…,𝑘, 𝑥^2𝑖(𝑡)=𝑥^1𝑖(𝑡)−𝑓_𝑖𝑡𝑥^1𝑖+𝑅−𝜙^{𝐻𝑖(⋅,⋅),𝜂𝑖𝑡}_{𝑀𝑖𝑡(⋅,𝑣1𝑖),𝜙𝑖𝑡𝑖𝑡}𝐻_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^1𝑖,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^1𝑖−𝜙_𝑖𝑡∘𝐹_𝑖𝑡𝑥¹¹−𝑝_1𝑡𝑥¹¹,𝑥¹²−𝑝_2𝑡𝑥¹²,…,𝑥^1𝑘−𝑝_𝑘𝑡𝑥^1𝑘∘𝑁_𝑖𝑡𝑢^1𝑖1,𝑢^1𝑖2,…,𝑢^1𝑖𝑘,(4.8) then 𝑥^2𝑖∶Ω→𝐸_𝑖 is measurable. Continuing the above process inductively, we can define the following random iterative sequences {𝑥^𝑛𝑖(𝑡)}, {𝑢^𝑛1𝑖(𝑡)}, {𝑢^𝑛2𝑖(𝑡)},…,{𝑢^𝑛𝑘𝑖(𝑡)} and {𝑣^𝑛𝑖(𝑡)}(𝑖=1,2,…,𝑘) for solving problem (4.3) as follows: 𝑥=𝑥^𝑖𝑛+1_𝑛𝑖(𝑡)(𝑡)−𝑓_𝑖𝑡𝑥^𝑛𝑖+𝑅−𝜙^{𝐻𝑖(⋅,⋅),𝜂𝑖𝑡}_{𝑀𝑖𝑡(⋅,𝑣}^𝑛𝑖_{),𝜙𝑖𝑡𝑖𝑡}𝐻_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖−𝜙_𝑖𝑡∘𝐹_𝑖𝑡𝑥^𝑛1−𝑝_1𝑡𝑥^𝑛1,𝑥^𝑛2−𝑝_2𝑡𝑥^𝑛2,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘𝑢∘𝑁_𝑖𝑡𝑢^𝑛𝑖1,𝑢^𝑛𝑖2,…,𝑢^𝑛𝑖𝑘_𝑛1𝑖,(𝑡)∈𝑇_1𝑖𝑡‖𝑢𝑥^𝑛1_𝑛1𝑖,𝑢^𝑛2𝑖(𝑡)∈𝑇_2𝑖t𝑥^𝑛2,…,𝑢^𝑛𝑘𝑖(𝑡)∈𝑇_𝑘𝑖𝑡𝑥^𝑛𝑘,𝑣^𝑛𝑖(𝑡)∈𝑆_𝑖𝑡𝑥^𝑛𝑖,(𝑡)−𝑢^𝑛+11𝑖(𝑡)‖≤1+1‖𝑢_𝑛2𝑖𝑛+1𝐻𝑇^1𝑖𝑡𝑥^𝑛1,𝑇_1𝑖𝑡𝑥^1𝑛+1,(𝑡)−𝑢^𝑛+12𝑖(𝑡)‖≤1+1‖𝑢_𝑛𝑘𝑖⋮𝑛+1𝐻𝑇^2𝑖𝑡𝑥^𝑛2,𝑇_2𝑖𝑡𝑥^2𝑛+1,(𝑡)−𝑢^{𝑛+1𝑘𝑖}(𝑡)‖≤1+1‖𝑣_𝑛𝑖𝑛+1𝐻𝑇^𝑘𝑖t𝑥^𝑛𝑘,𝑇_𝑘𝑖𝑡𝑥^𝑘𝑛+1,(𝑡)−𝑣^𝑖𝑛+1(𝑡)‖≤1+1𝑛+1𝐻𝑆^𝑖𝑡𝑥^𝑛𝑖,𝑆_𝑖𝑡𝑥^𝑖𝑛+1,(4.9) for any 𝑡∈Ω, 𝑖=1,2,…,𝑘 and 𝑛=0,1,2,….

Theorem 4.3. For 𝑖=1,2,…,𝑘, let 𝐸_𝑖 be a 𝑞-uniformly smooth Banach space and 𝜙_𝑖∶Ω×𝐸_𝑖→𝐸_𝑖 be a 𝜚_𝑖-Lipschitz continuous mapping satisfying 𝜙_𝑖𝑡(𝑥+𝑦)=𝜙_𝑖𝑡(𝑥)+𝜙_𝑖𝑡(𝑦) and ker𝜙_𝑖𝑡={0}, for all 𝑡∈Ω. Let 𝜂_𝑖∶Ω×𝐸_𝑖×𝐸_𝑖→𝐸_𝑖 be 𝜏_𝑖-Lipschitz continuous, 𝐻_𝑖∶𝐸_𝑖×𝐸_𝑖→𝐸_𝑖 be a single-valued mapping, and 𝐴_𝑖,𝐵_𝑖∶Ω×𝐸_𝑖→𝐸_𝑖 random single-valued mappings such that 𝐻_𝑖(𝐴_𝑖,𝐵_𝑖) be 𝛼_𝑖-strongly 𝜂_𝑖-accretive with respect to 𝐴_𝑖 and 𝛽_𝑖-relaxed 𝜂_𝑖-accretive with respect to 𝐵_𝑖 and 𝐻_𝑖(𝐴_𝑖,𝐵_𝑖) be 𝛿_𝑖1-Lipschitz continuous with respect to 𝐴_𝑖 and 𝛿_𝑖2-Lipschitz continuous with respect to 𝐵_𝑖. Let 𝑓_𝑖∶Ω×𝐸_𝑖→𝐸_𝑖 be (𝜖_𝑖,𝜇_𝑖)-relaxed cocoercive and 𝜉_𝑖-Lipschitz continuous, and 𝑝_𝑖∶Ω×𝐸_𝑖→𝐸_𝑖 be 𝛾_𝑝𝑖-strongly accretive and 𝜆_𝑝𝑖-Lipschitz continuous. Suppose that 𝐹_𝑖∶Ω×∏^𝑘𝑗=1𝐸_𝑗→𝐸_𝑖 be 𝜁_𝑖𝑗-Lipschitz continuous in the (𝑗+1) th argument for 𝑗=1,2,…,𝑘, 𝑁_𝑖∶Ω×∏^𝑘𝑗=1𝐸_𝑗→𝐸_𝑖 be 𝜈_𝑖𝑗-Lipschitz continuous in the (𝑗+1) th argument for 𝑗=1,2,…,𝑘. Let 𝑀_𝑖∶Ω×𝐸_𝑖×𝐸_𝑖→2^𝐸𝑖 be a random multivalued mapping such that 𝑀_𝑖 is a (𝐻_𝑖(⋅,⋅),𝜙_𝑖)-𝜂_𝑖-accretive mapping with respect to 𝐴_𝑖 and 𝐵_𝑖 in the second argument. Let 𝑇_1𝑖∶Ω×𝐸₁→𝖥(𝐸_𝑖), 𝑇_2𝑖∶Ω×𝐸₂→𝖥(𝐸_𝑖),…,𝑇_𝑘𝑖∶Ω×𝐸_𝑘→𝖥(𝐸_𝑖), 𝑆_𝑖∶Ω×𝐸_𝑖→𝖥(𝐸_𝑖) be random fuzzy mappings satisfying the condition (C) and 𝑇_1𝑖∶Ω×𝐸₁→CB(𝐸_𝑖), 𝑇_2𝑖∶Ω×𝐸₂→CB(𝐸_𝑖),…,𝑇_𝑘𝑖∶Ω×𝐸_𝑘→CB(𝐸_𝑖), and 𝑆_𝑖∶Ω×𝐸_𝑖→CB(𝐸_𝑖) random multivalued mappings induced by 𝑇_1𝑖,𝑇_2𝑖,…,𝑇_𝑘𝑖 and 𝑆_𝑖, respectively. Suppose that 𝑇_1𝑖, 𝑇_2𝑖,…,𝑇_𝑘𝑖, 𝑆_𝑖 are 𝐻-continuous with constants 𝑡_1𝑖, 𝑡_2𝑖, …, 𝑡_𝑘𝑖, and 𝜆_𝑠𝑖, respectively.In addition, if ‖𝑅^{𝐻𝑖(⋅,⋅),𝜂𝑖𝑡}_{𝑀𝑖𝑡⋅,𝑦𝑖1,𝜙𝑖𝑡}𝑧_𝑖∀𝑡−∈𝑅Ω,𝑦^{𝐻𝑖(⋅,⋅),𝜂𝑖𝑡}_{𝑀𝑖𝑡(⋅,𝑦𝑖2),𝜙𝑖𝑡𝑖1}𝑧_𝑖‖≤𝜆_𝑖(𝑡)‖𝑦_𝑖1(𝑡)−𝑦_𝑖2(𝑡)‖,(𝑡),𝑦_𝑖2(𝑡),𝑧_𝑖(𝑡)∈𝐸_𝑖(𝑖=1,2,…,𝑘),(4.10)0<1+𝑞𝜖₁(𝑡)𝜉₁(𝑡)^𝑞−𝑞𝜇₁(𝑡)+𝑐_𝑞𝜉₁(𝑡)^𝑞^1/𝑞+𝜆₁(𝑡)𝜆_𝑠1(𝑡)+𝜏¹(𝑡)^𝑞−1×𝛿₁₁𝑟₁(𝑡)−𝑚₁(𝑡)(𝑡)+𝛿₁₂(𝑡)^𝑞𝜉₁(𝑡)^𝑞+𝑞𝜚₁(𝑡)1−𝑞𝛾_𝑝1(𝑡)+𝑐_𝑞𝜆_𝑝1(𝑡)^𝑞^1/𝑞𝜁₁₁(𝑡)𝛿₁₁(𝑡)+𝛿₁₂(𝑡)^𝑞−1𝜉₁(𝑡)^𝑞−1+𝑐_𝑞𝜚₁(𝑡)^𝑞1−𝑞𝛾_𝑝1(𝑡)+𝑐_𝑞𝜆_𝑝1(𝑡)^𝑞𝜁₁₁(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=1+𝜚₁(𝑡)𝜈_1𝑗(𝑡)𝑡_1𝑗(𝑡)_𝑘_𝑗=2𝜏_𝑗(𝑡)^𝑞−1𝑟_𝑗(𝑡)−𝑚_𝑗(𝑡)𝜚^𝑗0<1+𝑞𝜖₂(𝑡)𝜁_𝑗1(𝑡)1−𝑞𝛾_𝑝1(𝑡)+𝑐_𝑞𝜆_𝑝1(𝑡)^𝑞^1/𝑞<1,(𝑡)𝜉₂(𝑡)^𝑞−𝑞𝜇₂(𝑡)+𝑐_𝑞𝜉₂(𝑡)^𝑞^1/𝑞+𝜆₂(𝑡)𝜆_𝑠2(𝑡)+𝜏²(𝑡)^𝑞−1×𝛿₂₁𝑟₂(𝑡)−𝑚₂(𝑡)(𝑡)+𝛿₂₂(𝑡)^𝑞𝜉₂(𝑡)^𝑞+𝑞𝜚₂(𝑡)1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞^1/𝑞𝜁₂₂(𝑡)𝛿₂₁(𝑡)+𝛿₂₂(𝑡)^𝑞−1𝜉₂(𝑡)^𝑞−1+𝑐_𝑞𝜚₂(𝑡)^𝑞1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞𝜁₂₂(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=1+𝜏𝜚₂(𝑡)𝜈_2𝑗(𝑡)𝑡_2𝑗(𝑡)₁(𝑡)^𝑞−1𝑟₁(𝑡)−𝑚₁(𝑡)𝜚¹(𝑡)𝜁₁₂(𝑡)1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=3𝜏_𝑗(𝑡)^𝑞−1𝑟_𝑗(𝑡)−𝑚_𝑗(𝑡)𝜚^𝑗0<1+𝑞𝜖_𝑘(𝑡)𝜁_𝑗2(𝑡)1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞^1/𝑞<1,…,(𝑡)𝜉_𝑘(𝑡)^𝑞−𝑞𝜇_𝑘(𝑡)+𝑐_𝑞𝜉_𝑘(𝑡)^𝑞^1/𝑞+𝜆_𝑘(𝑡)𝜆_𝑠𝑘(𝑡)+𝜏^𝑘(𝑡)^𝑞−1×𝛿_𝑘1𝑟_𝑘(𝑡)−𝑚_𝑘(𝑡)(𝑡)+𝛿_𝑘2(𝑡)^𝑞𝜉_𝑘(𝑡)^𝑞+𝑞𝜚_𝑘(𝑡)1−𝑞𝛾_𝑝𝑘(𝑡)+𝑐_𝑞𝜆_𝑝𝑘(𝑡)^𝑞^1/𝑞𝜁_𝑘𝑘(𝑡)𝛿_𝑘1(𝑡)+𝛿_𝑘2(𝑡)^𝑞−1𝜉_𝑘(𝑡)^𝑞−1+𝑐_𝑞𝜚_𝑘(𝑡)^𝑞1−𝑞𝛾_𝑝𝑘(𝑡)+𝑐_𝑞𝜆_𝑝𝑘(𝑡)^𝑞𝜁_𝑘𝑘(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=1+𝜚_𝑘(𝑡)𝜈_𝑘𝑗(𝑡)𝑡_𝑘𝑗(𝑡)_𝑘−1_𝑗=1𝜏_𝑗(𝑡)^𝑞−1𝑟_𝑗(𝑡)−𝑚_𝑗(𝑡)𝜚^𝑗(𝑡)𝜁_𝑗𝑘(𝑡)1−𝑞𝛾_𝑝𝑘(𝑡)+𝑐_𝑞𝜆_𝑝𝑘(𝑡)^𝑞^1/𝑞<1,(4.11) then there exist measurable mappings 𝑥_𝑖, 𝑢_1𝑖,𝑢_2𝑖,…,𝑢_𝑘𝑖,𝑣_𝑖∶Ω→𝐸_𝑖 such that (4.3) holds. Moreover for all 𝑡∈Ω,𝑥^𝑛𝑖(𝑡)→𝑥_𝑖(𝑡),𝑢^𝑛1𝑖(𝑡)→𝑢_1𝑖(𝑡), 𝑢^𝑛2𝑖(𝑡)→𝑢_2𝑖(𝑡),…,𝑢^𝑛𝑘𝑖(𝑡)→𝑢_𝑘𝑖(𝑡), 𝑣^𝑛𝑖(𝑡)→𝑣_𝑖(𝑡), where {𝑥^𝑛𝑖(𝑡)}, and {𝑢^𝑛1𝑖(𝑡)},…,{𝑢^𝑛𝑘𝑖(𝑡)}, {𝑣^𝑛𝑖(𝑡)} are random sequences obtained by Algorithm 4.2.

Proof. For 𝑖=1,2,…,𝑘, from (4.10), Lemma 4.1, and Algorithm 4.2, we have ‖=𝑥‖‖‖𝑥^𝑖𝑛+1(𝑡)−𝑥^𝑛𝑖(𝑡)‖_𝑛𝑖(𝑡)−𝑓_𝑖𝑡𝑥^𝑛𝑖+𝑅−𝜙^{𝐻𝑖(⋅,⋅),𝜂𝑖𝑡}_{𝑀𝑖𝑡⋅,𝑣𝑛𝑖,𝜙𝑖𝑡𝑖𝑡}𝐻_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖−𝜙_𝑖𝑡∘𝐹_𝑖𝑡𝑥^𝑛1−𝑝_1𝑡𝑥^𝑛1,𝑥^𝑛2−𝑝_2𝑡𝑥^𝑛2,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘−𝑥∘𝑁_𝑖𝑡𝑢^𝑛𝑖1,𝑢^𝑛𝑖2,…,𝑢^𝑛𝑖𝑘_𝑖𝑛−1+𝑅(𝑡)−𝑓_{𝑖𝑡𝐻𝑖(⋅,⋅),𝜂𝑖𝑡𝑀𝑖𝑡⋅,𝑣𝑖𝑛−1,𝜙𝑖𝑡}𝑥^𝑖𝑛−1−𝜙𝐻_𝑖𝑖𝑡𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1−𝜙_𝑖𝑡∘𝐹_𝑖𝑡𝑥^1𝑛−1−𝑝_1𝑡𝑥^1𝑛−1,𝑥^2𝑛−1−𝑝_2𝑡𝑥^2𝑛−1,…,𝑥^𝑘𝑛−1−𝑝_𝑘𝑡𝑥^𝑘𝑛−1≤‖‖𝑥∘𝑁_𝑖𝑡𝑢^{𝑛−1𝑖1},𝑢^{𝑛−1𝑖2},…,𝑢^{𝑛−1𝑖𝑘}_𝑛𝑖‖‖+‖‖‖𝑅(𝑡)−𝑥^𝑖𝑛−1(𝑡)−𝑓_𝑖𝑡𝑥^𝑛𝑖−𝑓_𝑖𝑡𝑥^𝑖𝑛−1_{𝐻𝑖(⋅,⋅),𝜂𝑖𝑡𝑀𝑖𝑡⋅,𝑣𝑛𝑖,𝜙𝑖𝑡}‖‖−𝐻𝜙_𝑖_𝑖𝑡𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖−𝜙_𝑖𝑡∘𝐹_𝑖𝑡𝑥^𝑛1−𝑝_1𝑡𝑥^𝑛1,𝑥^𝑛2−𝑝_2𝑡𝑥^𝑛2,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘−𝑅_{𝐻𝑖(⋅,⋅),𝜂𝑖𝑡𝑀𝑖𝑡⋅,𝑣𝑛𝑖,𝜙𝑖𝑡}∘𝑁_𝑖𝑡𝑢^𝑛𝑖1,𝑢^𝑛𝑖2,…,𝑢^𝑛𝑖𝑘−𝜙𝐻_𝑖𝑖𝑡𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1−𝜙_𝑖𝑡∘𝐹_𝑖𝑡𝑥^1𝑛−1−𝑝_1𝑡𝑥^1𝑛−1,𝑥^2𝑛−1−𝑝_2𝑡𝑥^2𝑛−1,…,𝑥^𝑘𝑛−1−𝑝_𝑘𝑡𝑥^𝑘𝑛−1+‖‖‖𝑅∘𝑁_𝑖𝑡𝑢^{𝑛−1𝑖1},𝑢^{𝑛−1𝑖2},…,𝑢^{𝑛−1𝑖𝑘}_{𝐻𝑖(⋅,⋅),𝜂𝑖𝑡𝑀𝑖𝑡⋅,𝑣𝑛𝑖,𝜙𝑖𝑡}‖‖‖−𝜙𝐻_𝑖𝑖𝑡𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1−𝜙_𝑖𝑡∘𝐹_𝑖𝑡𝑥^1𝑛−1−𝑝_1𝑡𝑥^1𝑛−1,𝑥^2𝑛−1−𝑝_2𝑡𝑥^2𝑛−1,…,𝑥^𝑘𝑛−1−𝑝_𝑘𝑡𝑥^𝑘𝑛−1−𝑅∘_{𝐻𝑖(⋅,⋅),𝜂𝑖𝑡𝑀𝑖𝑡⋅,𝑣𝑖𝑛−1,𝜙𝑖𝑡}𝑁_𝑖𝑡𝑢^{𝑛−1𝑖1},𝑢^{𝑛−1𝑖2},…,𝑢^{𝑛−1𝑖𝑘}−𝐻𝜙_𝑖𝑖𝑡𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1−𝜙_𝑖𝑡∘𝐹_𝑖𝑡𝑥^1𝑛−1−𝑝_1𝑡𝑥^1𝑛−1,𝑥^2𝑛−1−𝑝_2𝑡𝑥^2𝑛−1,…,𝑥^𝑘𝑛−1−𝑝_𝑘𝑡𝑥^𝑘𝑛−1≤‖‖𝑥∘𝑁_𝑖𝑡𝑢^{𝑛−1𝑖1},𝑢^{𝑛−1𝑖2},…,𝑢^{𝑛−1𝑖𝑘}_𝑛𝑖‖‖+(𝜏𝑡)−𝑥^𝑖𝑛−1(𝑡)−𝑓_𝑖𝑡𝑥^𝑛𝑖−𝑓_𝑖𝑡𝑥^𝑖𝑛−1_𝑖‖‖(𝑡)^𝑞−1𝑟_𝑖(𝑡)−𝑚_𝑖(𝑡)‖‖−𝐻𝐻_𝑖_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖−𝜙𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1_𝑖𝑡−𝜙_𝑖𝑡∘𝐹_𝑖𝑡𝑥^𝑛1−𝑝_1𝑡𝑥^𝑛1,𝑥^𝑛2−𝑝_2𝑡𝑥^𝑛2,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘+𝜏∘𝐹_𝑖𝑡𝑥^1𝑛−1−𝑝_1𝑡𝑥^1𝑛−1,𝑥^2𝑛−1−𝑝_2𝑡𝑥^2𝑛−1,…,𝑥^𝑘𝑛−1−𝑝_𝑘𝑡𝑥^𝑘𝑛−1_𝑖‖‖(𝑡)^𝑞−1𝑟_𝑖(𝑡)−𝑚_𝑖(𝑡)𝜚^𝑖+𝜆_𝑖(𝑡)‖‖𝑁_𝑖𝑡𝑢^𝑛𝑖1,𝑢^𝑛𝑖2,…,𝑢^𝑛𝑖𝑘−𝑁_𝑖𝑡𝑢^{𝑛−1𝑖1},𝑢^{𝑛−1𝑖2},…,𝑢^{𝑛−1𝑖𝑘}‖‖(𝑡)‖‖𝑣^𝑛𝑖(𝑡)−𝑣^𝑖𝑛−1(𝑡)‖‖.(4.12) Since 𝑓_𝑖∶Ω×𝐸_𝑖→𝐸_𝑖 be (𝜖_𝑖,𝜇_𝑖)-relaxed cocoercive and 𝜉_𝑖-Lipschitz continuous, we have ‖‖𝑥^𝑛𝑖(𝑡)−𝑥^𝑖𝑛−1(𝑡)−𝑓_𝑖𝑡𝑥^𝑛𝑖−𝑓_𝑖𝑡𝑥^𝑖𝑛−1‖‖^𝑞≤+‖‖𝑐𝑥^𝑛𝑖_𝑞(𝑡)−𝑥^𝑖𝑛−1(𝑡)‖‖^𝑞−𝑞𝑓_𝑖𝑡𝑥^𝑛𝑖−𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐽_𝑞𝑥^𝑛𝑖(𝑡)−𝑥^𝑖𝑛−1(𝑡)‖‖𝑓_𝑖𝑡𝑥^𝑛𝑖−𝑓_𝑖𝑡𝑥^𝑖𝑛−1‖‖^𝑞≤1+𝑞𝜖_𝑖(𝑡)𝜉_𝑖(𝑡)^𝑞−𝑞𝜇_𝑖(𝑡)+𝑐_𝑞𝜉_𝑖(𝑡)^𝑞‖‖𝑥^𝑛𝑖(𝑡)−𝑥^𝑖𝑛−1(𝑡)‖‖^𝑞.(4.13) Using 𝛾_𝑝𝑖-strongly accretivity and 𝜆_𝑝𝑖-Lipschitz continuity of 𝑝_𝑖, 𝜚_𝑖-Lipschitz continuity of 𝜑_𝑖 we have ‖‖𝑥^𝑛𝑖(𝑡)−𝑥^𝑖𝑛−1(𝑡)−𝑝_𝑖𝑡𝑥^𝑛𝑖−𝑝_𝑖𝑡𝑥^𝑖𝑛−1‖‖^𝑞≤+‖‖𝑥𝑐^𝑛𝑖_𝑞(𝑡)−𝑥^𝑖𝑛−1(𝑡)‖‖^𝑞−𝑞𝑝_𝑖𝑡𝑥^𝑛𝑖−𝑝_𝑖𝑡𝑥^𝑖𝑛−1,𝐽_𝑞𝑥^𝑛𝑖(𝑡)−𝑥^𝑖𝑛−1(𝑡)‖‖𝑝_𝑖𝑡𝑥^𝑛𝑖−𝑝_𝑖𝑡𝑥^𝑖𝑛−1‖‖^𝑞‖‖𝐻≤_𝑖1−𝑞𝛾_𝑝𝑖(𝑡)+𝑐_𝑞𝜆_𝑝𝑖(𝑡)^𝑞‖‖𝑥^𝑛𝑖(𝑡)−𝑥^𝑖𝑛−1(𝑡)‖‖^𝑞,−𝐴𝜙_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖−𝐻_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1_𝑖𝑡−𝜙∘_𝑖𝑡𝐹_𝑖𝑡𝑥^𝑛1−𝑝_1𝑡𝑥^𝑛1,𝑥^𝑛2−𝑝_2𝑡𝑥^𝑛2,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘≤‖‖𝐻∘𝐹_𝑖𝑡𝑥^1𝑛−1−𝑝_1𝑡𝑥^1𝑛−1,𝑥^2𝑛−1−𝑝_2𝑡𝑥^2𝑛−1,…,𝑥^𝑘𝑛−1−𝑝_𝑘𝑡𝑥^𝑘𝑛−1_𝑖‖‖−𝜙𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖−𝐻_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1_𝑖𝑡∘𝐹𝑥_𝑖𝑡𝑥^𝑛1_𝑛𝑖+1−𝑝_1𝑡𝑥^𝑛1,…,𝑥^𝑛𝑖−1−𝑝_𝑖−1𝑡𝑥^𝑛𝑖−1,𝑥^𝑛𝑖−𝑝_𝑖𝑡𝑥^𝑛𝑖,−𝜙_𝑖𝑡−𝑝_𝑖+1𝑡𝑥^𝑛𝑖+1,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘∘𝐹𝑥_𝑖𝑡𝑥^𝑛1_𝑛𝑖+1−𝑝_1𝑡𝑥^𝑛1,…,𝑥^𝑛𝑖−1−𝑝_𝑖−1𝑡𝑥^𝑛𝑖−1,𝑥^𝑖𝑛−1−𝑝_𝑖𝑡𝑥^𝑖𝑛−1,+𝜚_𝑖−𝑝_𝑖+1𝑡𝑥^𝑛𝑖+1,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘‖‖(𝑡)‖‖𝐹𝑥_𝑖𝑡𝑥^𝑛1_𝑛𝑖+1−𝑝_1𝑡𝑥^𝑛1,…,𝑥^𝑛𝑖−1−𝑝_𝑖−1𝑡𝑥^𝑛𝑖−1,𝑥^𝑖𝑛−1−𝑝_𝑖𝑡𝑥^𝑖𝑛−1,−𝐹_𝑖𝑡−𝑝_𝑖+1𝑡𝑥^𝑛𝑖+1,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘𝑥𝑥^1𝑛−1_{𝑛−1𝑖+1}−𝑝_1𝑡𝑥^1𝑛−1,…,𝑥^{𝑛−1𝑖−1}−𝑝_𝑖−1𝑡𝑥^{𝑛−1𝑖−1},𝑥^𝑖𝑛−1−𝑝_𝑖𝑡𝑥^𝑖𝑛−1,−𝑝_𝑖+1𝑡𝑥^{𝑛−1𝑖+1},…,𝑥^𝑘𝑛−1−𝑝_𝑘𝑡𝑥^𝑘𝑛−1‖‖.(4.14) Since 𝐻_𝑖(𝐴_𝑖,𝐵_𝑖) is 𝛿_𝑖1-Lipschitz continuous with respect to 𝐴_𝑖 and 𝛿_𝑖2-Lipschitz continuous with respect to 𝐵_𝑖, we have ‖‖𝐻≤_𝑖‖‖𝐴𝐻_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖−𝐻_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1_𝑖‖‖+‖‖𝐻𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖−𝐻_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖_𝑖‖‖≤𝛿𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖−𝐻_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1_𝑖1‖‖≤𝛿_𝑖1(𝑡)+𝛿_𝑖2(𝑡)‖‖𝑓_𝑖𝑡𝑥^𝑛𝑖−𝑓_𝑖𝑡𝑥^𝑖𝑛−1‖‖(𝑡)+𝛿_𝑖2(𝑡)𝜉_𝑖(𝑡)‖‖𝑥^𝑛𝑖(𝑡)−𝑥^𝑖𝑛−1(𝑡)‖‖.(4.15) Since 𝐹_𝑖∶Ω×∏^𝑘𝑗=1𝐸_𝑗→𝐸_𝑖 is 𝜁_𝑖𝑖-Lipschitz continuous in the (𝑖+1)th argument, by (4.15), we have ‖‖𝐻−_𝑖𝜙𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖−𝐻_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1_𝑖𝑡∘𝐹_𝑖𝑡𝑥𝑥^𝑛1_𝑛𝑖+1−𝑝_1𝑡𝑥^𝑛1,…,𝑥^𝑛𝑖−1−𝑝_𝑖−1𝑡𝑥^𝑛𝑖−1,𝑥^𝑛𝑖−𝑝_𝑖𝑡𝑥^𝑛𝑖,−𝜙_𝑖𝑡−𝑝_𝑖+1𝑡𝑥^𝑛𝑖+1,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘∘𝐹_𝑖𝑡𝑥𝑥^𝑛1_𝑛𝑖+1−𝑝_1𝑡𝑥^𝑛1,…,𝑥^𝑛𝑖−1−𝑝_𝑖−1𝑡𝑥^𝑛𝑖−1,𝑥^𝑖𝑛−1𝑝_𝑖𝑡𝑥^𝑖𝑛−1,−𝑝_𝑖+1𝑡𝑥^𝑛𝑖+1,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘‖‖^𝑞≤‖‖𝐻_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖−𝐻_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1‖‖^𝑞−𝑞𝜙_𝑖𝑡∘𝐹𝑥_𝑖𝑡_𝑛𝑖+1𝑥^𝑛1−𝑝_1𝑡𝑥^𝑛1,…,𝑥^𝑛𝑖−1−𝑝_𝑖−1𝑡𝑥^𝑛𝑖−1,𝑥^𝑛𝑖−𝑝_𝑖𝑡𝑥^𝑛𝑖,−𝜙_𝑖𝑡−𝑝_𝑖+1𝑡𝑥^𝑛𝑖+1,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘∘𝐹_𝑖𝑡𝑥𝑥^𝑛1_𝑛𝑖+1−𝑝_1𝑡𝑥^𝑛1,…,𝑥^𝑛𝑖−1−𝑝_𝑖−1𝑡𝑥^𝑛𝑖−1,𝑥^𝑖𝑛−1−𝑝_𝑖𝑡𝑥^𝑖𝑛−1,𝐽−𝑝_𝑖+1𝑡𝑥^𝑛𝑖+1,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘_𝑞,+𝑐_𝑞𝐻_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑛𝑖−𝐻_𝑖𝐴_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1,𝐵_𝑖𝑡𝑓_𝑖𝑡𝑥^𝑖𝑛−1𝜚_𝑖(𝑡)^𝑞‖‖𝐹𝑥_𝑖𝑡𝑥^𝑛1_𝑛𝑖+1−𝑝_1𝑡𝑥^𝑛1,…,𝑥^𝑛𝑖−1−𝑝_𝑖−1𝑡𝑥^𝑛𝑖−1,𝑥^𝑛𝑖−𝑝_𝑖𝑡𝑥^𝑛𝑖,−𝐹_𝑖𝑡−𝑝_𝑖+1𝑡𝑥^𝑛𝑖+1,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘𝑥𝑥^𝑛1_𝑛𝑖+1−𝑝_1𝑡𝑥^𝑛1,…,𝑥^𝑛𝑖−1−𝑝_𝑖−1𝑡𝑥^𝑛𝑖−1,𝑥^𝑖𝑛−1−𝑝_𝑖𝑡𝑥^𝑖𝑛−1,−𝑝_𝑖+1𝑡𝑥^𝑛𝑖+1,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘‖‖^𝑞≤𝛿_𝑖1(𝑡)+𝛿_𝑖2(𝑡)^𝑞𝜉_𝑖(𝑡)^𝑞+𝑞𝜚_𝑖(𝑡)1−𝑞𝛾_𝑝𝑖(𝑡)+𝑐_𝑞𝜆_𝑝𝑖(𝑡)^𝑞^1/𝑞𝜁_𝑖𝑖(𝑡)𝛿_𝑖1(𝑡)+𝛿_𝑖2(𝑡)^𝑞−1𝜉_𝑖(𝑡)^𝑞−1+𝑐_𝑞𝜚_𝑖(𝑡)^𝑞1−𝑞𝛾_𝑝𝑖(𝑡)+𝑐_𝑞𝜆_𝑝𝑖(𝑡)^𝑞𝜁_𝑖𝑖(𝑡)^𝑞‖‖𝑥^𝑛𝑖(𝑡)−𝑥^𝑖𝑛−1(𝑡)‖‖^𝑞,𝑖=1,2,…,𝑘.(4.16) Since 𝐹_𝑖∶Ω×∏^𝑘𝑗=1𝐸_𝑗→𝐸_𝑖 is 𝜁_𝑖𝑗-Lipschitz continuous in the (𝑗+1)th argument, for 𝑗=1,…,𝑖−1,𝑖+1,…,𝑘, we have ‖‖𝐹_𝑖𝑡𝑥𝑥^𝑛1−𝑝_1𝑡𝑥^𝑛1,…,𝑥^𝑛𝑖−1−𝑝_𝑖−1𝑡𝑥^𝑛𝑖−1,𝑥^𝑖𝑛−1−𝑝_𝑖𝑡𝑥^𝑖𝑛−1,𝑥^𝑛𝑖+1−𝑝_𝑖+1𝑡𝑥^𝑛𝑖+1,…,_𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘−𝑥𝐹_{𝑖𝑡𝑛−1𝑖+1}𝑥^1𝑛−1−𝑝_1𝑡𝑥^1𝑛−1,…,𝑥^{𝑛−1𝑖−1}−𝑝_𝑖−1𝑡𝑥^{𝑛−1𝑖−1},𝑥^𝑖𝑛−1−𝑝_𝑖𝑡𝑥^𝑖𝑛−1,≤‖‖𝐹−𝑝_𝑖+1𝑡𝑥^{𝑛−1𝑖+1},…,𝑥^𝑘𝑛−1−𝑝_𝑘𝑡𝑥^𝑘𝑛−1_𝑖𝑡‖‖𝑥𝑥^𝑛1_𝑖𝑛−1−𝑝_1𝑡𝑥^𝑛1,𝑥^𝑛2−𝑝_2𝑡𝑥^𝑛2,…,𝑥^𝑛𝑖−1−𝑝_𝑖−1𝑡𝑥^𝑛𝑖−1,−𝐹_𝑖𝑡−𝑝_𝑖𝑡𝑥^𝑖𝑛−1,𝑥^𝑛𝑖+1−𝑝_𝑖+1𝑡𝑥^𝑛𝑖+1,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘𝑥𝑥^1𝑛−1_𝑖𝑛−1−𝑝_1𝑡𝑥^1𝑛−1,𝑥^𝑛2−𝑝_2𝑡𝑥^𝑛2,…,𝑥^𝑛𝑖−1−𝑝_𝑖−1𝑡𝑥^𝑛𝑖−1,+‖‖𝐹−𝑝_𝑖𝑡𝑥^𝑖𝑛−1,𝑥^𝑛𝑖+1−𝑝_𝑖+1𝑡𝑥^𝑛𝑖+1,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘_𝑖𝑡‖‖𝑥𝑥^1𝑛−1_𝑖𝑛−1−𝑝_1𝑡𝑥^1𝑛−1,𝑥^𝑛2−𝑝_2𝑡𝑥^𝑛2,𝑥^𝑛3−𝑝_3𝑡𝑥^𝑛3,…,𝑥^𝑛𝑖−1−𝑝_𝑖−1𝑡𝑥^𝑛𝑖−1,−𝐹_𝑖𝑡−𝑝_𝑖𝑡𝑥^𝑖𝑛−1,𝑥^𝑛𝑖+1−𝑝_𝑖+1𝑡𝑥^𝑛𝑖+1,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘𝑥𝑥^1𝑛−1_𝑖𝑛−1−𝑝_1𝑡𝑥^1𝑛−1,𝑥^2𝑛−1−𝑝_2𝑡𝑥^2𝑛−1,𝑥^𝑛3−𝑝_3𝑡𝑥^𝑛3,…,𝑥^𝑛𝑖−1−𝑝_𝑖−1𝑡𝑥^𝑛𝑖−1,+⋯+‖‖𝐹_𝑖𝑡−𝑝_𝑖𝑡𝑥^𝑖𝑛−1,𝑥^𝑛𝑖+1−𝑝_𝑖+1𝑡𝑥^𝑛𝑖+1,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘‖‖𝑥𝑥^1𝑛−1−𝑝_1𝑡𝑥^1𝑛−1,𝑥^2𝑛−1−𝑝_2𝑡𝑥^2𝑛−1,𝑥^3𝑛−1−𝑝_3𝑡𝑥^3𝑛−1,…,_{𝑛−1𝑖−1}𝑥−𝑝_𝑖−1𝑡𝑥^{𝑛−1𝑖−1},𝑥^𝑖𝑛−1−𝑝_𝑖𝑡𝑥^𝑖𝑛−1,𝑥^{𝑛−1𝑖+1}−𝑝_𝑖+1𝑡𝑥^{𝑛−1𝑖+1},…,_{𝑛−1𝑘−1}−𝐹_𝑖𝑡−𝑝_𝑘−1𝑡𝑥^{𝑛−1𝑘−1},𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘𝑥𝑥^1𝑛−1−𝑝_1𝑡𝑥^1𝑛−1,𝑥^2𝑛−1−𝑝_2𝑡𝑥^2𝑛−1,𝑥^3𝑛−1−𝑝_3𝑡𝑥^3𝑛−1,…,_{𝑛−1𝑖−1}𝑥−𝑝_𝑖−1𝑡𝑥^{𝑛−1𝑖−1},𝑥^𝑖𝑛−1−𝑝_𝑖𝑡𝑥^𝑖𝑛−1,𝑥^{𝑛−1𝑖+1}−𝑝_𝑖+1𝑡𝑥^{𝑛−1𝑖+1},…,_{𝑛−1𝑘−1}≤𝜁_𝑖1−𝑝_𝑘−1𝑡𝑥^{𝑛−1𝑘−1},𝑥^𝑘𝑛−1−𝑝_𝑘𝑡𝑥^𝑘𝑛−1‖‖+𝜁(_𝑖2𝑡)1−𝑞𝛾_𝑝1(𝑡)+𝑐_𝑞𝜆_𝑝1(𝑡)^𝑞^1/𝑞‖‖𝑥^𝑛1(𝑡)−𝑥^1𝑛−1(𝑡)‖‖+⋯(+𝑡)𝜁1−_{𝑖,𝑖−1}𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞^1/𝑞‖‖𝑥^𝑛2(𝑡)−𝑥^2𝑛−1(𝑡)‖‖+𝜁_𝑖,𝑖+1(𝑡)1−𝑞𝛾_𝑝𝑖−1(𝑡)+𝑐_𝑞𝜆_𝑝𝑖−1(𝑡)^𝑞^1/𝑞‖‖𝑥^𝑛𝑖−1(𝑡)−𝑥^{𝑛−1𝑖−1}(𝑡)‖‖+⋯+(𝜁𝑡)_𝑖𝑘1−𝑞𝛾_𝑝𝑖+1(𝑡)+𝑐_𝑞𝜆_𝑝𝑖+1(𝑡)^𝑞^1/𝑞‖‖𝑥^𝑛𝑖+1(𝑡)−𝑥^{𝑛−1𝑖+1}(𝑡)‖‖=(𝑡)1−𝑞𝛾_𝑝𝑘(𝑡)+𝑐_𝑞𝜆_𝑝𝑘(𝑡)^𝑞^1/𝑞‖‖𝑥^𝑛𝑘(𝑡)−𝑥^𝑘𝑛−1(𝑡)_𝑖−1‖‖_𝑗=1+𝜁_𝑖𝑗(𝑡)1−𝑞𝛾_𝑝𝑗(𝑡)+𝑐_𝑞𝜆_𝑝𝑗(𝑡)^𝑞^1/𝑞‖‖𝑥^𝑛𝑗(𝑡)−𝑥^𝑗𝑛−1(𝑡)_𝑘‖‖_𝑗_=𝑖+1𝜁_𝑖𝑗(𝑡)1−𝑞𝛾_𝑝𝑗(𝑡)+𝑐_𝑞𝜆_𝑝𝑗(𝑡)^𝑞^1/𝑞‖‖𝑥^𝑛𝑗(𝑡)−𝑥^𝑗𝑛−1(𝑡)‖‖,𝑖=1,2,…,𝑘.(4.17) It follows from the Lipschitz continuity of 𝑁_𝑖, the 𝐻-continuity of 𝑇_1𝑖,𝑇_2𝑖,…,𝑇_𝑘𝑖(𝑖=1,2,…,𝑘) that ‖‖𝑁≤_𝑖𝑡‖‖𝑢𝑁^𝑛𝑖1,_𝑖𝑡𝑢^𝑛𝑖2,…,𝑢^𝑛𝑖𝑘−𝑁_𝑖𝑡𝑢^{𝑛−1𝑖1},𝑢^{𝑛−1𝑖2},…,𝑢^{𝑛−1𝑖𝑘}‖‖+‖‖𝑁_𝑖𝑡𝑢^𝑛𝑖1,𝑢^𝑛𝑖2,…,𝑢^𝑛𝑖𝑘−𝑁_𝑖𝑡𝑢^{𝑛−1𝑖1},𝑢^𝑛𝑖2,…,𝑢^𝑛𝑖𝑘‖‖+⋯+‖‖𝑢𝑁^{𝑛−1𝑖1}_𝑖𝑡,𝑢^𝑛𝑖2,𝑢^𝑛𝑖3,…,𝑢^𝑛𝑖𝑘−𝑁_𝑖𝑡𝑢^{𝑛−1𝑖1},𝑢^{𝑛−1𝑖2},𝑢^𝑛𝑖3,…,𝑢^𝑛𝑖𝑘‖‖≤_𝑘𝑢^{𝑛−1𝑖1},𝑢^{𝑛−1𝑖2},𝑢^{𝑛−1𝑖3},…,𝑢^{𝑛−1𝑖,𝑘−1},𝑢^𝑛𝑖𝑘−𝑁^𝑖_𝑡𝑢^{𝑛−1𝑖1},𝑢^{𝑛−1𝑖2},𝑢^{𝑛−1𝑖3},…,𝑢^{𝑛−1𝑖,𝑘−1},𝑢^{𝑛−1𝑖𝑘}‖‖_𝑗=1𝜈_𝑖𝑗(𝑡)‖‖𝑢^𝑛𝑖𝑗(𝑡)−𝑢^{𝑛−1𝑖𝑗}(𝑡)‖‖≤^𝑘_𝑗=1𝜈_𝑖𝑗(𝑡)1+1≤_𝑘𝑛𝐻𝑇^𝑖𝑗𝑡𝑥^𝑛𝑖,𝑇_𝑖𝑗𝑡𝑥^𝑖𝑛−1_𝑗=1𝜈_𝑖𝑗(𝑡)1+1𝑛𝑡^𝑖𝑗(𝑡)‖‖𝑥^𝑛𝑖(𝑡)−𝑥^𝑖𝑛−1(𝑡)‖‖,𝑖=1,2,…,𝑘.(4.18) It follows from (4.12)–(4.18) that for each 𝑖=1,2,…,𝑘, ‖𝑥≤^𝑖𝑛+1(𝑡)1−+𝑥𝑞^𝑛𝑖𝜖(_𝑖𝑡)‖(𝑡)𝜉_𝑖(𝑡)^𝑞−𝑞𝜇_𝑖(𝑡)+𝑐_𝑞𝜉_𝑖(𝑡)^𝑞^1/𝑞+𝜆_𝑖(𝑡)𝜆_𝑠𝑖(𝑡)1+1𝑛+𝜏^𝑖(𝑡)^𝑞−1×𝛿_𝑖1𝑟_𝑖(𝑡)−𝑚_𝑖(𝑡)(𝑡)+𝛿_𝑖2(𝑡)^𝑞𝜉_𝑖(𝑡)^𝑞+𝑞𝜚_𝑖(𝑡)1−𝑞𝛾_𝑝𝑖(𝑡)+𝑐_𝑞𝜆_𝑝𝑖(𝑡)^𝑞^1/𝑞𝜁_𝑖𝑖(𝑡)𝛿_𝑖1(𝑡)+𝛿_𝑖2(𝑡)^𝑞−1𝜉_𝑖(𝑡)^𝑞−1+𝑐_𝑞𝜚_𝑖(𝑡)^𝑞1−𝑞𝛾_𝑝𝑖(𝑡)+𝑐_𝑞𝜆_𝑝𝑖(𝑡)^𝑞𝜁_𝑖𝑖(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=1𝜚_𝑖(𝑡)𝜈_𝑖𝑗(𝑡)1+1𝑛𝑡^𝑖𝑗+(𝑡)‖𝑥^𝑛𝑖(𝑡)−𝑥^𝑖𝑛−1(𝑡)‖_𝑖−1_𝑗=1𝜏_𝑖(𝑡)^𝑞−1𝑟_𝑖(𝑡)−𝑚_𝑖(𝑡)𝜚^𝑖+(𝑡)𝜁_𝑖𝑗(𝑡)1−𝑞𝛾_𝑝𝑗(𝑡)+𝑐_𝑞𝜆_𝑝𝑗(𝑡)^𝑞^1/𝑞‖𝑥^𝑛𝑗(𝑡)−𝑥^𝑗𝑛−1(𝑡)‖_𝑘𝑗_=𝑖+1𝜏_𝑖(𝑡)^𝑞−1𝑟_𝑖(𝑡)−𝑚_𝑖(𝑡)𝜚^𝑖(𝑡)𝜁_𝑖𝑗(𝑡)1−𝑞𝛾_𝑝𝑗(𝑡)+𝑐_𝑞𝜆_𝑝𝑗(𝑡)^𝑞^1/𝑞‖𝑥^𝑛𝑗(𝑡)−𝑥^𝑗𝑛−1(𝑡)‖.(4.19) Therefore, ^𝑘_𝑖=1≤‖‖𝑥^𝑖𝑛+1(𝑡)−𝑥^𝑛𝑖(𝑡)_𝑘‖‖_𝑖=11+𝑞𝜖^𝑖(𝑡)𝜉_𝑖(𝑡)^𝑞−𝑞𝜇_𝑖(𝑡)+𝑐_𝑞𝜉_𝑖(𝑡)^𝑞^1/𝑞+𝜆_𝑖(𝑡)𝜆_𝑠𝑖(𝑡)1+1𝑛+𝜏^𝑖(𝑡)^𝑞−1×𝛿𝑟_𝑖(𝑡)−𝑚_𝑖(𝑡)_𝑖1(𝑡)+𝛿_𝑖2(𝑡)^𝑞𝜉_𝑖(𝑡)^𝑞+𝑞𝜚_𝑖(𝑡)1−𝑞𝛾_𝑝𝑖(𝑡)+𝑐_𝑞𝜆_𝑝𝑖(𝑡)^𝑞^1/𝑞×𝜁_𝑖𝑖(𝑡)𝛿_𝑖1(𝑡)+𝛿_𝑖2(𝑡)^𝑞−1𝜉_𝑖(𝑡)^𝑞−1+𝑐_𝑞𝜚_𝑖(𝑡)^𝑞1−𝑞𝛾_𝑝𝑖(𝑡)+𝑐_𝑞𝜆_𝑝𝑖(𝑡)^𝑞𝜁_𝑖𝑖(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=1𝜚_𝑖(𝑡)𝜈_𝑖𝑗(𝑡)1+1𝑛𝑡^𝑖𝑗+(𝑡)‖‖𝑥^𝑛𝑖(𝑡)−𝑥^𝑖𝑛−1(𝑡)_𝑖−1‖‖_𝑗=1𝜏_𝑖(𝑡)^𝑞−1𝑟_𝑖(𝑡)−𝑚_𝑖(𝑡)𝜚^𝑖+(𝑡)𝜁_𝑖𝑗(𝑡)1−𝑞𝛾_𝑝𝑗(𝑡)+𝑐_𝑞𝜆_𝑝𝑗(𝑡)^𝑞^1/𝑞‖‖𝑥^𝑛𝑗(𝑡)−𝑥^𝑗𝑛−1(𝑡)_𝑘‖‖_𝑗_=𝑖+1𝜏_𝑖(𝑡)^𝑞−1𝑟_𝑖(𝑡)−𝑚_𝑖(𝑡)𝜚^𝑖=1+𝑞𝜖₁(𝑡)𝜁_𝑖𝑗(𝑡)1−𝑞𝛾_𝑝𝑗(𝑡)+𝑐_𝑞𝜆_𝑝𝑗(𝑡)^𝑞^1/𝑞‖𝑥^𝑛𝑗(𝑡)−𝑥^𝑗𝑛−1(𝑡)‖(𝑡)𝜉₁(𝑡)^𝑞−𝑞𝜇₁(𝑡)+𝑐_𝑞𝜉₁(𝑡)^𝑞^1/𝑞+𝜆₁(𝑡)𝜆_𝑠1(𝑡)1+1𝑛+𝜏¹(𝑡)^𝑞−1×𝛿₁₁𝑟₁(𝑡)−𝑚₁(𝑡)(𝑡)+𝛿₁₂(𝑡)^𝑞𝜉₁(𝑡)^𝑞+𝑞𝜚₁(𝑡)1−𝑞𝛾_𝑝1(𝑡)+𝑐_𝑞𝜆_𝑝1(𝑡)^𝑞^1/𝑞𝜁₁₁(𝑡)𝛿₁₁(𝑡)+𝛿₁₂(𝑡)^𝑞−1𝜉₁(𝑡)^𝑞−1+𝑐_𝑞𝜚₁(𝑡)^𝑞1−𝑞𝛾_𝑝1(𝑡)+𝑐_𝑞𝜆_𝑝1(𝑡)^𝑞𝜁₁₁(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=1𝜚₁(𝑡)𝜈_1𝑗(𝑡)1+1𝑛𝑡^1𝑗+(𝑡)_𝑘_𝑗=2𝜏_𝑗(𝑡)^𝑞−1𝑟_𝑗(𝑡)−𝑚_𝑗(𝑡)𝜚^𝑗(𝑡)𝜁_𝑗1(𝑡)1−𝑞𝛾_𝑝1(𝑡)+𝑐_𝑞𝜆_𝑝1(𝑡)^𝑞^1/𝑞‖‖𝑥^𝑛1+1+𝑞𝜖₂(𝑡)−𝑥^1𝑛−1(𝑡)‖‖(𝑡)𝜉₂(𝑡)^𝑞−𝑞𝜇₂(𝑡)+𝑐_𝑞𝜉₂(𝑡)^𝑞^1/𝑞+𝜆₂(𝑡)𝜆_𝑠2(𝑡)1+1𝑛+𝜏²(𝑡)^𝑞−1×𝛿₂₁𝑟₂(𝑡)−𝑚₂(𝑡)(𝑡)+𝛿₂₂(𝑡)^𝑞𝜉₂(𝑡)^𝑞+𝑞𝜚₂(𝑡)1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞^1/𝑞𝜁₂₂(𝑡)𝛿₂₁(𝑡)+𝛿₂₂(𝑡)^𝑞−1𝜉₂(𝑡)^𝑞−1+𝑐_𝑞𝜚₂(𝑡)^𝑞1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞𝜁₂₂(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=1𝜚₂(𝑡)𝜈_2𝑗(𝑡)1+1𝑛𝑡^2𝑗+𝜏(𝑡)₁(𝑡)^𝑞−1𝑟₁(𝑡)−𝑚₁(𝑡)𝜚¹(𝑡)𝜁₁₂(𝑡)1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=3𝜏_𝑗(𝑡)^𝑞−1𝑟_𝑗(𝑡)−𝑚_𝑗(𝑡)𝜚^𝑗(𝑡)𝜁_𝑗2(𝑡)1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞^1/𝑞‖‖𝑥^𝑛2+⋯+1+𝑞𝜖_𝑘(𝑡)−𝑥^2𝑛−1(𝑡)‖‖(𝑡)𝜉_𝑘(𝑡)^𝑞−𝑞𝜇_𝑘(𝑡)+𝑐_𝑞𝜉_𝑘(𝑡)^𝑞^1/𝑞+𝜆_𝑘(𝑡)𝜆_𝑠𝑘(𝑡)1+1𝑛+𝜏^𝑘(𝑡)^𝑞−1×𝛿_𝑘1𝑟_𝑘(𝑡)−𝑚_𝑘(𝑡)(𝑡)+𝛿_𝑘2(𝑡)^𝑞𝜉_𝑘(𝑡)^𝑞+𝑞𝜚_𝑘(𝑡)1−𝑞𝛾_𝑝𝑘(𝑡)+𝑐_𝑞𝜆_𝑝𝑘(𝑡)^𝑞^1/𝑞𝜁_𝑘𝑘(𝑡)𝛿_𝑘1(𝑡)+𝛿_𝑘2(𝑡)^𝑞−1𝜉_𝑘(𝑡)^𝑞−1+𝑐_𝑞𝜚_𝑘(𝑡)^𝑞1−𝑞𝛾_𝑝𝑘(𝑡)+𝑐_𝑞𝜆_𝑝𝑘(𝑡)^𝑞𝜁_𝑘𝑘(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=1𝜚_𝑘(𝑡)𝜈_𝑘𝑗(𝑡)1+1𝑛𝑡^𝑘𝑗+(𝑡)_𝑘−1_𝑗=1𝜏_𝑗(𝑡)^𝑞−1𝑟_𝑗(𝑡)−𝑚_𝑗(𝑡)𝜚^𝑗(𝑡)𝜁_𝑗𝑘(𝑡)1−𝑞𝛾_𝑝𝑘(𝑡)+𝑐_𝑞𝜆_𝑝𝑘(𝑡)^𝑞^1/𝑞‖‖𝑥^𝑛𝑘≤𝜃_𝑛(𝑡)−𝑥^𝑘𝑛−1(𝑡)‖‖(𝑡)^𝑘_𝑖=1‖‖𝑥^𝑛𝑖(𝑡)−𝑥^𝑖𝑛−1(𝑡)‖‖,(4.20) where𝜃=_𝑛(m𝑡)ax1+𝑞𝜖₁(𝑡)𝜉₁(𝑡)^𝑞−𝑞𝜇₁(𝑡)+𝑐_𝑞𝜉₁(𝑡)^𝑞^1/𝑞+𝜆₁(𝑡)𝜆_𝑠1(𝑡)1+1𝑛+𝜏¹(𝑡)^𝑞−1×𝛿₁₁𝑟₁(𝑡)−𝑚₁(𝑡)(𝑡)+𝛿₁₂(𝑡)^𝑞𝜉₁(𝑡)^𝑞+𝑞𝜚₁(𝑡)1−𝑞𝛾_𝑝1(𝑡)+𝑐_𝑞𝜆_𝑝1(𝑡)^𝑞^1/𝑞𝜁₁₁(𝑡)𝛿₁₁(𝑡)+𝛿₁₂(𝑡)^𝑞−1𝜉₁(𝑡)^𝑞−1+𝑐_𝑞𝜚₁(𝑡)^𝑞1−𝑞𝛾_𝑝1(𝑡)+𝑐_𝑞𝜆_𝑝1(𝑡)^𝑞𝜁₁₁(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=1𝜚₁(𝑡)𝜈_1𝑗(𝑡)1+1𝑛𝑡^1𝑗+(𝑡)_𝑘_𝑗=2𝜏_𝑗(𝑡)^𝑞−1𝑟_𝑗(𝑡)−𝑚_𝑗(𝑡)𝜚^𝑗1+𝑞𝜖₂(𝑡)𝜁_𝑗1(𝑡)1−𝑞𝛾_𝑝1(𝑡)+𝑐_𝑞𝜆_𝑝1(𝑡)^𝑞^1/𝑞,(𝑡)𝜉₂(𝑡)^𝑞−𝑞𝜇₂(𝑡)+𝑐_𝑞𝜉₂(𝑡)^𝑞^1/𝑞+𝜆₂(𝑡)𝜆_𝑠2(𝑡)1+1𝑛+𝜏²(𝑡)^𝑞−1×𝛿₂₁𝑟₂(𝑡)−𝑚₂(𝑡)(𝑡)+𝛿₂₂(𝑡)^𝑞𝜉₂(𝑡)^𝑞+𝑞𝜚₂(𝑡)1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞^1/𝑞𝜁₂₂(𝑡)𝛿₂₁(𝑡)+𝛿₂₂(𝑡)^𝑞−1𝜉₂(𝑡)^𝑞−1+𝑐_𝑞𝜚₂(𝑡)^𝑞1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞𝜁₂₂(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=1𝜚₂(𝑡)𝜈_2𝑗(𝑡)1+1𝑛𝑡^2𝑗+𝜏(𝑡)₁(𝑡)^𝑞−1𝑟₁(𝑡)−𝑚₁(𝑡)𝜚¹(𝑡)𝜁₁₂(𝑡)1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=3𝜏_𝑗(𝑡)^𝑞−1𝑟_𝑗(𝑡)−𝑚_𝑗(𝑡)𝜚^𝑗1+𝑞𝜖_𝑘(𝑡)𝜁_𝑗2(𝑡)1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞^1/𝑞,…,(𝑡)𝜉_𝑘(𝑡)^𝑞−𝑞𝜇_𝑘(𝑡)+𝑐_𝑞𝜉_𝑘(𝑡)^𝑞^1/𝑞+𝜆_𝑘(𝑡)𝜆_𝑠𝑘(𝑡)1+1𝑛+𝜏^𝑘(𝑡)^𝑞−1×𝛿_𝑘1𝑟_𝑘(𝑡)−𝑚_𝑘(𝑡)(𝑡)+𝛿_𝑘2(𝑡)^𝑞𝜉_𝑘(𝑡)^𝑞+𝑞𝜚_𝑘(𝑡)1−𝑞𝛾_𝑝𝑘(𝑡)+𝑐_𝑞𝜆_𝑝𝑘(𝑡)^𝑞^1/𝑞𝜁_𝑘𝑘(𝑡)𝛿_𝑘1(𝑡)+𝛿_𝑘2(𝑡)^𝑞−1𝜉_𝑘(𝑡)^𝑞−1+𝑐_𝑞𝜚_𝑘(𝑡)^𝑞1−𝑞𝛾_𝑝𝑘(𝑡)+𝑐_𝑞𝜆_𝑝𝑘(𝑡)^𝑞𝜁_𝑘𝑘(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=1𝜚_𝑘(𝑡)𝜈_𝑘𝑗(𝑡)1+1𝑛𝑡^𝑘𝑗+(𝑡)_𝑘−1_𝑗=1𝜏_𝑗(𝑡)^𝑞−1𝑟_𝑗(𝑡)−𝑚_𝑗(𝑡)𝜚^𝑗(𝑡)𝜁_𝑗𝑘(𝑡)1−𝑞𝛾_𝑝𝑘(𝑡)+𝑐_𝑞𝜆_𝑝𝑘(𝑡)^𝑞^1/𝑞.(4.21) Let 𝜃=(𝑡)max1+𝑞𝜖¹(𝑡)𝜉₁(𝑡)^𝑞−𝑞𝜇₁(𝑡)+𝑐_𝑞𝜉₁(𝑡)^𝑞^1/𝑞+𝜆₁(𝑡)𝜆_𝑠1(𝑡)+𝜏¹(𝑡)^𝑞−1×𝛿₁₁𝑟₁(𝑡)−𝑚₁(𝑡)(𝑡)+𝛿₁₂(𝑡)^𝑞𝜉₁(𝑡)^𝑞+𝑞𝜚₁(𝑡)1−𝑞𝛾_𝑝1(𝑡)+𝑐_𝑞𝜆_𝑝1(𝑡)^𝑞^1/𝑞𝜁₁₁(𝑡)𝛿₁₁(𝑡)+𝛿₁₂(𝑡)^𝑞−1𝜉₁(𝑡)^𝑞−1+𝑐_𝑞𝜚₁(𝑡)^𝑞1−𝑞𝛾_𝑝1(𝑡)+𝑐_𝑞𝜆_𝑝1(𝑡)^𝑞𝜁₁₁(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=1+𝜚₁(𝑡)𝜈_1𝑗(𝑡)𝑡_1𝑗(𝑡)_𝑘_𝑗=2𝜏_𝑗(𝑡)^𝑞−1𝑟_𝑗(𝑡)−𝑚_𝑗(𝑡)𝜚^𝑗1+𝑞𝜖₂(𝑡)𝜁_𝑗1(𝑡)1−𝑞𝛾_𝑝1(𝑡)+𝑐_𝑞𝜆_𝑝1(𝑡)^𝑞^1/𝑞,(𝑡)𝜉₂(𝑡)^𝑞−𝑞𝜇₂(𝑡)+𝑐_𝑞𝜉₂(𝑡)^𝑞^1/𝑞+𝜆₂(𝑡)𝜆_𝑠2(𝑡)+𝜏²(𝑡)^𝑞−1×𝛿₂₁𝑟₂(𝑡)−𝑚₂(𝑡)(𝑡)+𝛿₂₂(𝑡)^𝑞𝜉₂(𝑡)^𝑞+𝑞𝜚₂(𝑡)1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞^1/𝑞𝜁₂₂(𝑡)𝛿₂₁(𝑡)+𝛿₂₂(𝑡)^𝑞−1𝜉₂(𝑡)^𝑞−1+𝑐_𝑞𝜚₂(𝑡)^𝑞(1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞𝜁₂₂(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=1+𝜏𝜚₂(𝑡)𝜈_2𝑗(𝑡)𝑡_2𝑗(𝑡)₁(𝑡)^𝑞−1𝑟₁(𝑡)−𝑚₁(𝑡)𝜚¹(𝑡)𝜁₁₂(𝑡)1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=3𝜏_𝑗(𝑡)^𝑞−1𝑟_𝑗(𝑡)−𝑚_𝑗(𝑡)𝜚^𝑗1+𝑞𝜖_𝑘(𝑡)𝜁_𝑗2(𝑡)1−𝑞𝛾_𝑝2(𝑡)+𝑐_𝑞𝜆_𝑝2(𝑡)^𝑞^1/𝑞,…,(𝑡)𝜉_𝑘(𝑡)^𝑞−𝑞𝜇_𝑘(𝑡)+𝑐_𝑞𝜉_𝑘(𝑡)^𝑞^1/𝑞+𝜆_𝑘(𝑡)𝜆_𝑠𝑘(𝑡)+𝜏^𝑘(𝑡)^𝑞−1×𝛿_𝑘1𝑟_𝑘(𝑡)−𝑚_𝑘(𝑡)(𝑡)+𝛿_𝑘2(𝑡)^𝑞𝜉_𝑘(𝑡)^𝑞+𝑞𝜚_𝑘(𝑡)1−𝑞𝛾_𝑝k(𝑡)+𝑐_𝑞𝜆_𝑝k(𝑡)^𝑞^1/𝑞𝜁_𝑘𝑘(𝑡)𝛿_𝑘1(𝑡)+𝛿_𝑘2(𝑡)^𝑞−1𝜉_𝑘(𝑡)^𝑞−1+𝑐_𝑞𝜚_𝑘(𝑡)^𝑞1−𝑞𝛾_𝑝𝑘(𝑡)+𝑐_𝑞𝜆_𝑝𝑘(𝑡)^𝑞𝜁_𝑘𝑘(𝑡)^𝑞^1/𝑞+^𝑘_𝑗=1+𝜚_𝑘(𝑡)𝜈_𝑘𝑗(𝑡)𝑡_𝑘𝑗(𝑡)_𝑘−1_𝑗=1𝜏_𝑗(𝑡)^𝑞−1𝑟_𝑗(𝑡)−𝑚_𝑗(𝑡)𝜚^𝑗(𝑡)𝜁_𝑗𝑘(𝑡)1−𝑞𝛾_𝑝𝑘(𝑡)+𝑐_𝑞𝜆_𝑝𝑘(𝑡)^𝑞^1/𝑞.(4.22) Then 𝜃_𝑛(𝑡)→𝜃(𝑡), for all 𝑡∈Ω.
Define ‖⋅‖_∗ on 𝐸=∏^𝑘𝑖=1𝐸_𝑖 by ‖‖𝑧₁(𝑡),𝑧₂(𝑡),…,𝑧_𝑘(𝑡)‖‖_∗=‖𝑧₁∀(𝑡)𝑧‖+‖𝑧₂(𝑡)‖+⋯+‖𝑧_𝑘(𝑡)‖,₁(𝑡),𝑧₂(𝑡),…,𝑧_𝑘(𝑡)∈𝐸.(4.23) It is easy to see that (𝐸,‖⋅‖_∗) is a Banach space.
Define 𝑧^𝑛+1(𝑡)=(𝑥^1𝑛+1(𝑡),𝑥^2𝑛+1(𝑡),…,𝑥^𝑘𝑛+1(𝑡)). Then, we have ‖‖𝑧^𝑛+1(𝑡)−𝑧^𝑛(𝑡)‖‖_∗=‖𝑥^1𝑛+1(𝑡)−𝑥^𝑛1(𝑡)‖+‖𝑥^2𝑛+1(𝑡)−𝑥^𝑛2(𝑡)‖+⋯+‖𝑥^𝑘𝑛+1(𝑡)−𝑥^𝑛𝑘(𝑡)‖.(4.24) From the condition (4.11), we know that for all 𝑡∈Ω, 0<𝜃(𝑡)<1, and hence there exists an 𝑛₀>0 and 𝜃₀(𝑡)∈(0,1) such that 𝜃_𝑛(𝑡)≤𝜃₀(𝑡), for all 𝑛≥𝑛₀. Therefore, by (4.20) and (4.24), we have ‖‖𝑧^𝑛+1(𝑡)−𝑧^𝑛(𝑡)‖‖_∗≤𝜃₀(𝑡)‖‖𝑧^𝑛(𝑡)−𝑧^𝑛−1(𝑡)‖‖_∗,∀𝑛≥𝑛₀.(4.25) It follows from (4.25) that ‖‖𝑧^𝑛+1(𝑡)−𝑧^𝑛(𝑡)‖‖_∗≤𝜃₀(𝑡)^𝑛−𝑛0‖‖𝑧^𝑛0+1(𝑡)−𝑧^𝑛0(𝑡)‖‖_∗.(4.26) Hence, for any 𝑚≥𝑛≥𝑛₀, it follows that ‖‖𝑥^𝑚1(𝑡)−𝑥^𝑛1(𝑡)‖‖≤‖𝑧^𝑚(𝑡)−𝑧^𝑛(𝑡)‖_∗≤^𝑚−1_𝑖=𝑛‖‖𝑧^𝑖+1(𝑡)−𝑧^𝑖(𝑡)‖‖_∗≤^𝑚−1_𝑖=𝑛𝜃₀(𝑡)^𝑖−𝑛0‖‖𝑧^𝑛0+1(𝑡)−𝑧^𝑛0(𝑡)‖‖_∗.(4.27) Since 0<𝜃₀(𝑡)<1, for all 𝑡∈Ω, it follows from (4.27) that ‖𝑥^𝑚1(𝑡)−𝑥^𝑛1(𝑡)‖→0 as 𝑛→∞, and hence {𝑥^𝑛1(𝑡)} is a Cauchy sequence in 𝐸₁. By the same argument, we also have that {𝑥^𝑛𝑖(𝑡)} is a Cauchy sequence in 𝐸_𝑖(𝑖=2,3,…,𝑘). Thus there exist measurable mappings 𝑥₁∶Ω→𝐸₁, 𝑥₂∶Ω→𝐸₂,…,𝑥_𝑘∶Ω→𝐸_𝑘 such that for all 𝑡∈Ω, 𝑥^𝑛1(𝑡)→𝑥₁(𝑡), 𝑥^𝑛2(𝑡)→𝑥₂(𝑡),…,𝑥^𝑛𝑘(𝑡)→𝑥_𝑘(𝑡) as 𝑛→∞.
Now we prove that 𝑢^𝑛1𝑖(𝑡)→𝑢_1𝑖(𝑡)∈𝑇_1𝑖t(𝑥₁), 𝑢^𝑛2𝑖(𝑡)→𝑢_2𝑖(𝑡)∈𝑇_2𝑖t(𝑥₂),…,𝑢^𝑛𝑘𝑖(𝑡)→𝑢_𝑘𝑖(𝑡)∈𝑇_𝑘𝑖t(𝑥_𝑘), 𝑣^𝑛𝑖(𝑡)→𝑣_𝑖(𝑡)∈𝑆_𝑖t(𝑥_𝑖)(𝑖=1,2,…,𝑘). In fact, it follows from the Lipschitz continuity of 𝑇_1𝑖, 𝑇_2𝑖,…,𝑇_𝑘𝑖, 𝑆_𝑖 and Algorithm 4.2 that for 𝑖=1,2,…,𝑘, ‖‖𝑢^𝑛1𝑖(𝑡)−𝑢^{𝑛−11𝑖}(𝑡)‖‖≤1+1𝑛𝑡^1𝑖‖‖𝑢(𝑡)‖‖𝑥^𝑛1(𝑡)−𝑥^1𝑛−1(𝑡)_𝑛2𝑖‖‖,(𝑡)−𝑢^{𝑛−12𝑖}(𝑡)‖‖≤1+1𝑛𝑡^2𝑖‖‖𝑢⋮(𝑡)‖‖𝑥^𝑛2(𝑡)−𝑥^2𝑛−1(𝑡)_𝑛𝑘𝑖‖‖,(𝑡)−𝑢^{𝑛−1𝑘𝑖}(𝑡)‖‖≤1+1𝑛𝑡^𝑘𝑖‖‖𝑣(𝑡)‖‖𝑥^𝑛𝑘(𝑡)−𝑥^𝑘𝑛−1(𝑡)_𝑛𝑖‖‖,(𝑡)−𝑣^𝑖𝑛−1(𝑡)‖‖≤1+1𝑛𝜆^𝑠𝑖(𝑡)‖‖𝑥^𝑛𝑖(𝑡)−𝑥^𝑖𝑛−1(𝑡)‖‖.(4.28) From (4.28), we know that {𝑢^𝑛1𝑖(𝑡)}, {𝑢^𝑛2𝑖(𝑡)},…,{𝑢^𝑛𝑘𝑖(𝑡)}, {𝑣^𝑛𝑖(𝑡)}(𝑖=1,2,…,𝑘) are also Cauchy sequences. Therefore, there exist 𝑢_1𝑖(𝑡)∈𝐸_𝑖, 𝑢_2𝑖(𝑡)∈𝐸_𝑖,…,𝑢_𝑘𝑖(𝑡)∈𝐸_𝑖, 𝑣_𝑖(𝑡)∈𝐸_𝑖(𝑖=1,2,…,𝑘) such that 𝑢^𝑛1𝑖(𝑡)→𝑢_1𝑖(𝑡), 𝑢^𝑛2𝑖(𝑡)→𝑢_2𝑖(𝑡),…,𝑢^𝑛𝑘𝑖(𝑡)→𝑢_𝑘𝑖(𝑡) as 𝑛→∞. Further, for 𝑖=1,2,…,𝑘, 𝑑𝑢_1𝑖(𝑡),𝑇_1𝑖𝑡𝑥₁≤≤‖‖‖‖𝑢𝑢_1𝑖1𝑖(𝑡)−𝑢^𝑛1𝑖(𝑡)‖‖+𝑑𝑢^𝑛1𝑖(𝑡),𝑇_1𝑖𝑡𝑥₁≤‖‖𝑢_1𝑖(𝑡)−𝑢^𝑛1𝑖(𝑡)‖‖+𝐻𝑇_1𝑖𝑡𝑥^𝑛1,𝑇_1𝑖𝑡𝑥₁(𝑡)−𝑢^𝑛1𝑖(𝑡)‖‖+𝑡_1𝑖(𝑡)‖𝑥^𝑛1(𝑡)−𝑥₁(𝑡)‖⟶0,∀𝑡∈Ω.(4.29) Since 𝑇_1𝑖t(𝑥₁) is closed, we have 𝑢_1𝑖(𝑡)∈𝑇_1𝑖𝑡(𝑥₁)(𝑖=1,2,…,𝑘). Similarly, 𝑢_2𝑖(𝑡)∈𝑇_2𝑖𝑡(𝑥₂),…,𝑢_𝑘𝑖(𝑡)∈𝑇_𝑘𝑖𝑡(𝑥_𝑘), 𝑣_𝑖(𝑡)∈𝑆_𝑖𝑡(𝑥_𝑖)(𝑖=1,2,…,𝑘). This completes the proof.

Now we show the existence of solutions for problem (4.4).

Lemma 4.4. Problem (4.4) is equivalent to find measurable mappings 𝑥_𝑖,𝑢_1𝑖,𝑢_2𝑖,…,𝑢_𝑘𝑖∶Ω→ℋ_𝑖 such that for all 𝑡∈Ω and 𝑖=1,2,…,𝑘, 𝑥_𝑖(𝑡)∈ℋ_𝑖, 𝑇_{1𝑖1𝑡,𝑥(𝑡)}(𝑢_1𝑖(𝑡))≥𝑎₁(𝑥₁(𝑡)), 𝑇_{2𝑖2𝑡,𝑥(𝑡)}(𝑢_2𝑖(𝑡))≥𝑎₂(𝑥₂(𝑡)),…,𝑇_{𝑘𝑖𝑘𝑡,𝑥(𝑡)}(𝑢_𝑘𝑖(𝑡))≥𝑎_𝑘(𝑥_𝑘(𝑡)) such that 𝑓=_𝑖𝑡𝐽𝑥_𝑖_{𝜑𝑖𝑡𝜌𝑖(𝑡)}𝑓_𝑖𝑡𝑥_𝑖−𝜌_𝑖(𝑡)𝐹_𝑖𝑡𝑥₁−𝑝_1𝑡𝑥₁,𝑥₂−𝑝_2𝑡𝑥₂,…,𝑥_𝑘−𝑝_𝑘𝑡𝑥_𝑘+𝑁_𝑖𝑡𝑢_𝑖1,𝑢_𝑖2,…,𝑢_𝑖𝑘,(4.30) where 𝜌_𝑖∶Ω→(0,∞) is a measurable function and 𝐽^𝜑𝑖𝑡_{𝜌𝑖(𝑡)}=(𝐼_𝑡+𝜌_𝑖(𝑡)𝜕𝜑_𝑖𝑡)⁻¹.
If for 𝑖=1,2,…,𝑘, 𝐸_𝑖=ℋ_𝑖 is a Hilbert space and 𝑀_𝑖𝑡(⋅,𝑣_𝑖)=𝜕𝜑_𝑖𝑡(⋅), for all 𝑡∈Ω, 𝑥_𝑖(𝑡)∈ℋ_𝑖 and 𝑣_𝑖(𝑡)∈𝑆_𝑖t(𝑥_𝑖), then Algorithm 4.2 reduces to Algorithm 4.5.

Algorithm 4.5. For any given measurable mapping 𝑥^0𝑖∶Ω→ℋ_𝑖(𝑖=1,2,…,𝑘), compute the following random iterative sequences {𝑥^𝑛𝑖(𝑡)}, {𝑢^𝑛1𝑖(𝑡)}, {𝑢^𝑛2𝑖(𝑡)}, …, {𝑢^𝑛𝑘𝑖(𝑡)}(𝑖=1,2,…,𝑘) for solving problem (4.4) as follows: 𝑥=𝑥^𝑖𝑛+1_𝑛𝑖(𝑡)(𝑡)−𝑓_𝑖𝑡𝑥^𝑛𝑖+𝐽^𝜑𝑖𝑡_{𝜌𝑖(𝑡)}𝑓_𝑖𝑡𝑥^𝑛𝑖−𝜌_𝑖𝑢(𝑡)𝐹_𝑖𝑡𝑥^𝑛1−𝑝_1𝑡𝑥^𝑛1,𝑥^𝑛2−𝑝_2𝑡𝑥^𝑛2,…,𝑥^𝑛𝑘−𝑝_𝑘𝑡𝑥^𝑛𝑘+𝑁_𝑖𝑡𝑢^𝑛𝑖1,𝑢^𝑛𝑖2,…,𝑢^𝑛𝑖𝑘_𝑛1𝑖,‖𝑢_𝑛1𝑖(𝑡)∈𝑇_1𝑖𝑡𝑥^𝑛1,𝑢^𝑛2𝑖(𝑡)∈𝑇_2𝑖𝑡𝑥^𝑛2,…,𝑢^𝑛𝑘𝑖(𝑡)∈𝑇_𝑘𝑖𝑡𝑥^𝑛𝑘,(𝑡)−𝑢^𝑛+11𝑖(𝑡)‖≤1+1‖𝑢_𝑛2𝑖𝑛+1𝐻𝑇^1𝑖𝑡𝑥^𝑛1,𝑇_1𝑖𝑡𝑥^1𝑛+1,(𝑡)−𝑢^𝑛+12𝑖(𝑡)‖≤1+1‖𝑢_𝑛𝑘𝑖⋮𝑛+1𝐻𝑇^2𝑖𝑡𝑥^𝑛2,𝑇_2𝑖𝑡𝑥^2𝑛+1,(𝑡)−𝑢^{𝑛+1𝑘𝑖}(𝑡)‖≤1+1𝑛+1𝐻𝑇^𝑘𝑖𝑡𝑥^𝑛𝑘,𝑇_𝑘𝑖𝑡𝑥^𝑘𝑛+1,(4.31) for any 𝑡∈Ω, 𝑖=1,2,…,𝑘, 𝑛=0,1,2,….

From Theorem 4.3, we have the following Theorem.

Theorem 4.6. For 𝑖=1,2,…,𝑘, let ℋ_𝑖 be a Hilbert space, 𝑓_𝑖∶Ω×ℋ_𝑖→ℋ_𝑖 be (𝜖_𝑖,𝜇_𝑖)-relaxed cocoercive and 𝜉_𝑖-Lipschitz continuous, and 𝑝_𝑖∶Ω×ℋ_𝑖→ℋ_𝑖 be 𝛾_𝑝𝑖-strongly accretive and 𝜆_𝑝𝑖-Lipschitz continuous. Let 𝐹_𝑖∶Ω×∏^𝑘𝑗=1ℋ_𝑗→ℋ_𝑖 be 𝜁_𝑖𝑗-Lipschitz continuous in the (𝑗+1)th argument for 𝑗=1,2,…,𝑘, and 𝑁_𝑖∶Ω×∏^𝑘𝑗=1ℋ_𝑗→ℋ_𝑖 be 𝜈_𝑖𝑗-Lipschitz continuous in the (𝑗+1)th argument for 𝑗=1,2,…,𝑘. Let 𝑇_1𝑖∶Ω×ℋ₁→𝖥(ℋ_𝑖), 𝑇_2𝑖∶Ω×ℋ₂→𝖥(ℋ_𝑖),…,𝑇_𝑘𝑖∶Ω×ℋ_𝑘→𝖥(ℋ_𝑖) be random fuzzy mappings satisfying the condition (C); 𝑇_1𝑖∶Ω×ℋ₁→CB(ℋ_𝑖), 𝑇_2𝑖∶Ω×ℋ₂→CB(ℋ_𝑖),…,𝑇_𝑘𝑖∶Ω×ℋ_𝑘→CB(ℋ_𝑖) be random multivalued mappings induced by 𝑇_1𝑖,𝑇_2𝑖,…,𝑇_𝑘𝑖, respectively. Suppose that 𝑇_1𝑖, 𝑇_2𝑖,…,𝑇_𝑘𝑖 are 𝐻-continuous with constants 𝑡_1𝑖, 𝑡_2𝑖,…,𝑡_𝑘𝑖, respectively. If there exists a constant 𝜌_𝑖(𝑡)>0(𝑖=1,2,…,𝑘) such that 0<1+2𝜖₁(𝑡)𝜉₁(𝑡)²−2𝜇₁(𝑡)+𝜉₁(𝑡)²^1/2++𝜉₁𝜌(𝑡₁)²+2𝜌₁(𝑡)1−2𝛾_𝑝1(𝑡)+𝜆_𝑝1(𝑡)²^1/2𝜁₁₁(𝑡)𝜉₁(𝑡)(𝑡)²1−2𝛾_𝑝1(𝑡)+𝜆_𝑝1(𝑡)²𝜁₁₁(𝑡)²^1/2+^𝑘_𝑗=1𝜌₁(𝑡)𝜈_1𝑗(𝑡)𝑡_1𝑗(𝑡)+^𝑘_𝑗=20<1+2𝜖₂𝜌_𝑗(𝑡)𝜁_𝑗1(𝑡)1−2𝛾_𝑝1(𝑡)+𝜆_𝑝1(𝑡)²^1/2<1,(𝑡)𝜉₂(𝑡)²−2𝜇₂(𝑡)+𝜉₂(𝑡)²^1/2+𝜉+₂(𝜌𝑡)²₂+2𝜌₂(𝑡)1−2𝛾_𝑝2(𝑡)+𝜆_𝑝2(𝑡)²^1/2𝜁₂₂(𝑡)𝜉₂(𝑡)(𝑡)²1−2𝛾_𝑝2(𝑡)+𝜆_𝑝2(𝑡)²𝜁₂₂(𝑡)²^1/2+^𝑘_𝑗=1𝜌₂(𝑡)𝜈_2𝑗(𝑡)𝑡_2𝑗(𝑡)+𝜌₁(𝑡)𝜁₁₂(𝑡)1−2𝛾_𝑝2(𝑡)+𝜆_𝑝2(𝑡)²^1/2+^𝑘_𝑗=30<1+𝜌2_𝑗𝜖(_𝑘𝑡)𝜁_𝑗2(𝑡)1−2𝛾_𝑝2(𝑡)+𝜆_𝑝2(𝑡)²^1/2<1,…,(𝑡)𝜉_𝑘(𝑡)²−2𝜇_𝑘(𝑡)+𝜉_𝑘(𝑡)²^1/2++𝜉𝜌_𝑘𝑘(𝑡)²+2𝜌_𝑘(𝑡)1−2𝛾_𝑝𝑘(𝑡)+𝜆_𝑝𝑘(𝑡)²^1/2𝜁_𝑘𝑘(𝑡)𝜉_𝑘(𝑡)(𝑡)²1−2𝛾_𝑝𝑘(𝑡)+𝜆_𝑝𝑘(𝑡)²𝜁_𝑘𝑘(𝑡)²^1/2+^𝑘_𝑗=1+𝜌_𝑘(𝑡)𝜈_𝑘𝑗(𝑡)𝑡_𝑘𝑗(𝑡)_𝑘−1_𝑗=1𝜌_𝑗(𝑡)𝜁_𝑗𝑘(𝑡)1−2𝛾_𝑝𝑘(𝑡)+𝜆_𝑝𝑘(𝑡)²^1/2<1.(4.32) Then there exist measurable mappings 𝑥_𝑖, 𝑢_1𝑖,𝑢_2𝑖,…,𝑢_𝑘𝑖∶Ω→ℋ_𝑖 such that (4.4) holds. Moreover, for all 𝑡∈Ω, 𝑥^𝑛𝑖(𝑡)→𝑥_𝑖(𝑡), 𝑢^𝑛1𝑖(𝑡)→𝑢_1𝑖(𝑡), 𝑢^𝑛2𝑖(𝑡)→𝑢_2𝑖(𝑡),…,𝑢^𝑛𝑘𝑖(𝑡)→𝑢_𝑘𝑖(𝑡), where {𝑥^𝑛𝑖(𝑡)}, {𝑢^𝑛1𝑖(𝑡)},…,{𝑢^𝑛𝑘𝑖(𝑡)} are random sequences obtained by Algorithm 4.5.