Abstract

We introduce and study a class of a system of random set-valued variational inclusion problems. Some conditions for the existence of solutions of such problems are provided, when the operators are contained in the classes of generalized monotone operators, so-called (𝐴,π‘š,πœ‚)-monotone operator. Further, the stability of the iterative algorithm for finding a solution of the considered problem is also discussed.

1. Introduction

It is well known that the ideas and techniques of the variational inequalities are being applied in a variety of diverse fields of pure and applied sciences and proven to be productive and innovative. It has been shown that this theory provides the most natural, direct, simple, unified, and efficient framework for a general treatment of a wide class of linear and nonlinear problems. The development of variational inequality theory can be viewed as the simultaneous pursuit of two different lines of research. On the one hand, it reveals the fundamental facts on the qualitative aspects of the solutions to important classes of problems. On the other hand, it also enables us to develop highly efficient and powerful new numerical methods for solving, for example, obstacle, unilateral, free, moving, and complex equilibrium problems. Of course, the concept of variational inequality has been extended and generalized in several directions, and it is worth to noticed that, an important and useful generalization of variational inequality problem is the concept of variational inclusion. Many efficient ways have been studied to find solutions for variational inclusions and a related technique, as resolvent operator technique, was of great concern.

In 2006, Jin [1] investigated the approximation solvability of a type of set-valued variational inclusions based on the convergence of (𝐻,πœ‚)-resolvent operator technique, while the convergence analysis for approximate solutions much depends on the existence of Cauchy sequences generated by a proposed iterative algorithm. In the same year, Lan [2] first introduced a concept of (𝐴,πœ‚)-monotone operators, which contains the class of (𝐻,πœ‚)-monotonicity, 𝐴-monotonicity (see [3–5]), and other existing monotone operators as special cases. In such paper, he studied some properties of (𝐴,πœ‚)-monotone operators and defined resolvent operators associated with (𝐴,πœ‚)-monotone operators. Then, by using this new resolvent operator, he constructed some iterative algorithms to approximate the solutions of a new class of nonlinear (𝐴,πœ‚)-monotone operator inclusion problems with relaxed cocoercive mappings in Hilbert spaces. After that, Verma [5] explored sensitivity analysis for strongly monotone variational inclusions using (𝐴,πœ‚)-resolvent operator technique in a Hilbert space setting. For more examples, ones may consult [6–11].

Meanwhile, in 2001, Verma [12] introduced and studied some systems of variational inequalities and developed some iterative algorithms for approximating the solutions of such those problems. Furthermore, in 2004, Fang and Huang [13] introduced and studied some new systems of variational inclusions involving 𝐻-monotone operators. By Using the resolvent operator associated with 𝐻-monotone operators, they proved the existence and uniqueness of solutions for the such considered problem, and also some new algorithms for approximating the solutions are provided. Consequently, in 2007, Lan et al. [14] introduced and studied another system of nonlinear 𝐴-monotone multivalued variational inclusions in Hilbert spaces. Recently, based on the generalized (𝐴,πœ‚)-resolvent operator method, Argarwal and Verma [15] considered the existence and approximation of solutions for a general system of nonlinear set-valued variational inclusions involving relaxed cocoercive mappings in Hilbert spaces. Notice that, the concept of a system of variational inequality is very interesting since it is well-known that a variety of equilibrium models, for example, the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium problem, and the general equilibrium programming problem, can be uniformly modelled as a system of variational inequalities. Additional researches on the approximate solvability of a system of nonlinear variational inequalities are problems; ones may see Cho et al. [16], Cho and Petrot [17], Noor [18], Petrot [19], Suantai and Petrot [20], and others.

On the other hand, the systematic study of random equations employing the techniques of functional analysis was first introduced by Špaček [21] and Hanő [22], and it has received considerable attention from numerous authors. It is well known that the theory of randomness leads to several new questions like measurability of solutions, probabilistic and statistical aspects of random solutions estimate for the difference between the mean value of the solutions of the random equations and deterministic solutions of the averaged equations. The main question concerning random operator equations is essentially the same as those of deterministic operator equations, that is, a question of existence, uniqueness, characterization, contraction, and approximation of solutions. Of course, random variational inequality theory is an important part of random function analysis. This topic has attracted many scholars and experts due to the extensive applications of the random problems. For the examples of research works in these fascinating areas, ones may see Ahmad and BazÑn [23], Huang [24], Huang et al. [25], Khan et al. [26], Lan [27], and Noor and Elsanousi [28].

In this paper, inspired by the works going on these fields, we introduce a system of set-valued random variational inclusion problems and provide the sufficient conditions for the existence of solutions and the algorithm for finding a solution of proposed problems, involving a class of generalized monotone operators by using the resolvent operator technique. Furthermore, the stability of the constructed iterative algorithm is also discussed.

2. Preliminaries

Let β„‹ be a real Hilbert space equipped with norm β€–β‹…β€– and inner product βŸ¨β‹…,β‹…βŸ©, and let 2β„‹ and CB(β„‹) denote for the family of all the nonempty subsets of β„‹ and the family of all the nonempty closed bounded subsets of β„‹, respectively. As usual, we will define 𝐷∢CB(β„‹)Γ—CB(β„‹)β†’[0,∞), the Hausdorff metric on CB(β„‹), by 𝐷(𝐴,𝐡)=maxsupπ‘₯∈𝐴infπ‘¦βˆˆπ΅β€–π‘₯βˆ’π‘¦β€–,supπ‘¦βˆˆπ΅infπ‘₯βˆˆπ΄β€–ξƒ°π‘₯βˆ’π‘¦β€–,βˆ€π΄,𝐡∈CB(β„‹).(2.1)

Let (Ξ©,Ξ£,πœ‡) be a complete 𝜎-finite measure space and ℬ(β„‹) the class of Borel 𝜎-fields in β„‹. A mapping π‘₯βˆΆΞ©β†’β„‹ is said to be measurable if {π‘‘βˆˆΞ©βˆΆπ‘₯(𝑑)∈𝐡}∈Σ, for all π΅βˆˆβ„¬(β„‹). We will denote by β„³β„‹ a set of all measurable mappings on β„‹, that is, β„³β„‹={π‘₯βˆΆΞ©β†’β„‹|π‘₯isameasurablemapping}.

Let β„‹1 and β„‹2 be two real Hilbert spaces. Let πΉβˆΆΞ©Γ—β„‹1Γ—β„‹2β†’β„‹1 and πΊβˆΆΞ©Γ—β„‹1Γ—β„‹2β†’β„‹2 be single-valued mappings. Let π‘ˆβˆΆΞ©Γ—β„‹1β†’CB(β„‹1),π‘‰βˆΆΞ©Γ—β„‹2β†’CB(β„‹2), and π‘€π‘–βˆΆΞ©Γ—β„‹π‘–β†’2ℋ𝑖 be set-valued mappings, for 𝑖=1,2. In this paper, we will consider the following problem: find measurable mappings π‘Ž,π‘’βˆΆΞ©β†’β„‹1 and 𝑏,π‘£βˆΆΞ©β†’β„‹2 such that 𝑒(𝑑)βˆˆπ‘ˆ(𝑑,π‘Ž(𝑑)),𝑣(𝑑)βˆˆπ‘‰(𝑑,𝑏(𝑑)) and0∈𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))+𝑀1(𝑑,π‘Ž(𝑑)),0∈𝐺(𝑑,𝑒(𝑑),𝑏(𝑑))+𝑀2(𝑑,𝑏(𝑑)),βˆ€π‘‘βˆˆΞ©.(2.2) The problem of type (2.2) is called the system of random set-valued variational inclusion problem. If π‘Ž,π‘’βˆΆΞ©β†’β„‹1 and 𝑏,π‘£βˆΆΞ©β†’β„‹2 are solutions of problem (2.2), we will denote by (π‘Ž,𝑒,𝑏,𝑣)∈SRSVI(𝑀1,𝑀2)(𝐹,𝐺,π‘ˆ,𝑉).

Notice that, if π‘ˆβˆΆΞ©Γ—β„‹1β†’β„‹1 and π‘‰βˆΆΞ©Γ—β„‹2β†’β„‹2 are two single-valued mappings, then the problem (2.2) reduces to the following problem: find π‘ŽβˆΆΞ©β†’β„‹1 and π‘βˆΆΞ©β†’β„‹2 such that0∈𝐹(𝑑,π‘Ž(𝑑),𝑉(𝑑,𝑏(𝑑)))+𝑀1(𝑑,π‘Ž(𝑑)),0∈𝐺(𝑑,π‘ˆ(𝑑,π‘Ž(𝑑)),𝑏(𝑑))+𝑀2(𝑑,𝑏(𝑑)),βˆ€π‘‘βˆˆΞ©.(2.3) In this case, we will denote by (π‘Ž,𝑏)∈SRSI(𝑀1,𝑀2)(𝐹,𝐺,π‘ˆ,𝑉). Other special cases of the problem (2.2) are presented the following. (I)If 𝑀1(𝑑,π‘Ž(𝑑))=πœ•πœ‘(𝑑,π‘Ž(𝑑)) and 𝑀2(𝑑,𝑏(𝑑))=πœ•πœ™(𝑑,𝑏(𝑑)), where πœ‘βˆΆΞ©Γ—β„‹1→ℝβˆͺ{+∞} and πœ™βˆΆΞ©Γ—β„‹2→ℝβˆͺ{+∞} are two proper convex and lower semicontinuous functions and πœ•πœ‘ and πœ•πœ™ denoted for the subdifferential operators of πœ‘ and πœ™, respectively, then (2.2) reduces to the following problem: find π‘Ž,π‘’βˆΆΞ©β†’β„‹1 and 𝑏,π‘£βˆΆΞ©β†’β„‹2 such that 𝑒(𝑑)βˆˆπ‘ˆ(𝑑,π‘Ž(𝑑)),𝑣(𝑑)βˆˆπ‘‰(𝑑,𝑏(𝑑)) and ⟨𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑)),π‘₯(𝑑)βˆ’π‘Ž(𝑑)⟩+πœ‘(π‘₯(𝑑))βˆ’πœ‘(π‘Ž(𝑑))β‰₯0,βˆ€π‘₯βˆˆβ„³β„‹1,⟨𝐺(𝑑,𝑒(𝑑),𝑏(𝑑)),𝑦(𝑑)βˆ’π‘(𝑑)⟩+πœ™(𝑦(𝑑))βˆ’πœ™(𝑏(𝑑))β‰₯0,βˆ€π‘¦βˆˆβ„³β„‹2,(2.4) for all π‘‘βˆˆΞ©. The problem (2.4) is called a system of random set-valued mixed variational inequalities. A special of problem (2.4) was studied in by Agarwal and Verma [15]. (II)Let 𝐾1βŠ†β„‹1,𝐾2βŠ†β„‹2 be two nonempty closed and convex subsets and 𝛿𝐾𝑖 the indicator functions of 𝐾𝑖 for 𝑖=1,2. If 𝑀1(𝑑,π‘₯(𝑑))=πœ•π›ΏπΎ1(π‘₯(𝑑)) and 𝑀2(𝑑,𝑦(𝑑))=πœ•π›ΏπΎ2(𝑦(𝑑)) for all π‘₯βˆˆβ„³πΎ1 and π‘¦βˆˆβ„³πΎ2. Then the problem (2.2) reduces to the following problem: find π‘Ž,π‘’βˆΆΞ©β†’β„‹1 and 𝑏,π‘£βˆΆΞ©β†’β„‹2 such that 𝑒(𝑑)βˆˆπ‘ˆ(𝑑,π‘Ž(𝑑)),𝑣(𝑑)βˆˆπ‘‰(𝑑,𝑏(𝑑)) and ⟨𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑)),π‘₯(𝑑)βˆ’π‘Ž(𝑑)⟩β‰₯0,βˆ€π‘₯βˆˆβ„³π’¦1,⟨𝐺(𝑑,𝑒(𝑑),𝑏(𝑑)),𝑦(𝑑)βˆ’π‘(𝑑)⟩β‰₯0,βˆ€π‘¦βˆˆβ„³π’¦2,(2.5) for all π‘‘βˆˆΞ©.(III)If β„‹1=β„‹2=β„‹ and 𝑀1(𝑑,π‘Ž(𝑑))=𝑀2(𝑑,𝑏(𝑑))=πœ•πœ‘(𝑑,π‘Ž(𝑑)), where πœ‘βˆΆΞ©Γ—β„‹β†’β„βˆͺ{+∞} is proper convex and lower semicontinuous function and πœ•πœ‘ is denoted for the subdifferential operators of πœ‘. Let π‘”βˆΆβ„‹β†’β„‹ be a nonlinear mapping and 𝜌,πœ‚>0. If we set 𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))=πœŒπ‘£(𝑑)+π‘Ž(𝑑)βˆ’π‘”(𝑏(𝑑)), and 𝐺(𝑑,𝑒(𝑑),𝑏(𝑑))=πœ‚π‘’(𝑑)+𝑏(𝑑)βˆ’π‘”(π‘Ž(𝑑)) where 𝑒(𝑑)βˆˆπ‘ˆ(𝑑,π‘Ž(𝑑)),𝑣(𝑑)βˆˆπ‘‰(𝑑,𝑏(𝑑)), then problem (2.2) reduces to the following system of variational inequalities: find π‘Ž,π‘βˆΆΞ©β†’β„‹,𝑒(𝑑)βˆˆπ‘ˆ(𝑑,π‘Ž(𝑑)) and 𝑣(𝑑)βˆˆπ‘‰(𝑑,𝑏(𝑑)) such that βŸ¨πœŒπ‘£(𝑑)+π‘Ž(𝑑)βˆ’π‘”(𝑏(𝑑)),𝑔(π‘₯(𝑑))βˆ’π‘Ž(𝑑)⟩+πœ‘(𝑔(π‘₯(𝑑)))βˆ’πœ‘(π‘Ž(𝑑))β‰₯0,βŸ¨πœ‚π‘’(𝑑)+𝑏(𝑑)βˆ’π‘”(π‘Ž(𝑑)),𝑔(π‘₯(𝑑))βˆ’π‘(𝑑)⟩+πœ‘(𝑔(π‘₯(𝑑)))βˆ’πœ‘(𝑏(𝑑))β‰₯0,(2.6) for all π‘‘βˆˆΞ© and 𝑔(π‘₯(𝑑))βˆˆβ„³β„‹. A special of problem (2.6) was studied by Argarwal et al. [29]. (IV)Let π‘‡βˆΆπΎβ†’β„‹ be a nonlinear mapping and 𝜌,πœ‚>0 two fixed constants. If β„‹1=β„‹2=β„‹,𝐾1=𝐾2=𝐾,𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))=πœŒπ‘‡(𝑣(𝑑))+π‘Ž(𝑑)βˆ’π‘£(𝑑), and 𝐺(𝑑,𝑒(𝑑),𝑏(𝑑))=πœ‚π‘‡(𝑒(𝑑))+𝑏(𝑑)βˆ’π‘’(𝑑). Then (2.5) reduces to the following system of variational inequalities: find π‘Ž,𝑒,𝑏,π‘£βˆΆΞ©β†’β„‹ such that 𝑒(𝑑)βˆˆπ‘ˆ(𝑑,π‘Ž(𝑑)),𝑣(𝑑)βˆˆπ‘‰(𝑑,𝑏(𝑑)) and βŸ¨πœŒπ‘‡(𝑣(𝑑))+π‘Ž(𝑑)βˆ’π‘£(𝑑),π‘₯(𝑑)βˆ’π‘Ž(𝑑)⟩β‰₯0,βŸ¨πœ‚π‘‡(𝑒(𝑑))+𝑏(𝑑)βˆ’π‘’(𝑑),𝑦(𝑑)βˆ’π‘(𝑑)⟩β‰₯0,(2.7) for all π‘₯,π‘¦βˆˆβ„³β„‹ and π‘‘βˆˆΞ©. Notice that, if π‘ˆ=𝑉=𝐼, then (2.5), (2.7) are studied by Kim and Kim [30].

We now recall important basic concepts and definitions, which will be used in this work.

Definition 2.1. A mapping π‘“βˆΆΞ©Γ—β„‹β†’β„‹ is called a random single-valued mapping if for any π‘₯βˆˆβ„‹, the mapping 𝑓(β‹…,π‘₯)βˆΆΞ©β†’β„‹ is measurable.

Definition 2.2. A set-valued mapping πΊβˆΆΞ©β†’2β„‹ is said to be measurable if πΊβˆ’1(𝐡)={π‘‘βˆˆΞ©βˆΆπΊ(𝑑)βˆ©π΅β‰ βˆ…}∈Σ, for all π΅βˆˆβ„¬(β„‹).

Definition 2.3. A set-valued mapping πΉβˆΆΞ©Γ—β„‹β†’2β„‹ is called a random set-valued mapping if for any π‘₯βˆˆβ„‹, the set-valued mapping 𝐹(β‹…,π‘₯)βˆΆΞ©β†’2β„‹ is measurable.

Definition 2.4. A single-valued mapping πœ‚βˆΆΞ©Γ—β„‹Γ—β„‹β†’β„‹ is said to be random 𝜏-Lipschitz continuous if there exists a measurable function πœβˆΆΞ©β†’(0,∞) such that β€–πœ‚(𝑑,π‘₯(𝑑),𝑦(𝑑))β€–β‰€πœ(𝑑)β€–π‘₯(𝑑)βˆ’π‘¦(𝑑)β€–,(2.8) for all π‘₯,π‘¦βˆˆβ„³β„‹,π‘‘βˆˆΞ©.

Definition 2.5. A set-valued mapping π‘ˆβˆΆΞ©Γ—β„‹β†’CB(β„‹) is said to be random πœ™-𝐷-Lipschitz continuous if there exists a measurable function πœ™βˆΆΞ©β†’(0,∞) such that 𝐷(π‘ˆ(𝑑,π‘₯(𝑑)),π‘ˆ(𝑑,𝑦(𝑑)))β‰€πœ™(𝑑)β€–π‘₯(𝑑)βˆ’π‘¦(𝑑)β€–,(2.9) for all π‘₯,π‘¦βˆˆβ„³β„‹ and π‘‘βˆˆΞ©, where 𝐷(β‹…,β‹…) is the Hausdorff metric on CB(β„‹).

Definition 2.6. A set-valued mapping πΉβˆΆΞ©Γ—β„‹β†’CB(β„‹) is said to be 𝐷-continuous if, for any π‘‘βˆˆΞ©, the mapping 𝐹(𝑑,β‹…)βˆΆβ„‹β†’CB(β„‹) is continuous in 𝐷(β‹…,β‹…), where 𝐷(β‹…,β‹…) is the Hausdorff metric on CB(β„‹).

Definition 2.7. Let π΄βˆΆΞ©Γ—β„‹β†’β„‹ and πœ‚βˆΆΞ©Γ—β„‹Γ—β„‹β†’β„‹ be two random single-valued mappings. Then 𝐴 is said to be (i)random 𝛽-Lipschitz continuous if there exists a measurable function π›½βˆΆΞ©β†’(0,∞) such that ‖𝐴(𝑑,π‘₯(𝑑))βˆ’π΄(𝑑,𝑦(𝑑))‖≀𝛽(𝑑)β€–π‘₯(𝑑)βˆ’π‘¦(𝑑)β€–,(2.10) for all π‘₯,π‘¦βˆˆβ„³β„‹,π‘‘βˆˆΞ©;(ii)random πœ‚-monotone if ⟨𝐴(𝑑,π‘₯(𝑑))βˆ’π΄(𝑑,𝑦(𝑑)),πœ‚(𝑑,π‘₯(𝑑),𝑦(𝑑))⟩β‰₯0,(2.11) for all π‘₯,π‘¦βˆˆβ„³β„‹,π‘‘βˆˆΞ©;(iii)random strictly πœ‚-monotone if, 𝐴 is a random πœ‚-monotone and ⟨𝐴(𝑑,π‘₯(𝑑))βˆ’π΄(𝑑,𝑦(𝑑)),πœ‚(𝑑,π‘₯(𝑑),𝑦(𝑑))⟩=0iffπ‘₯(𝑑)=𝑦(𝑑),(2.12) for all π‘₯,π‘¦βˆˆβ„³β„‹,π‘‘βˆˆΞ©;(iv)random (π‘Ÿ,πœ‚)-strongly monotone if there exists a measurable function π‘ŸβˆΆΞ©β†’(0,∞) such that ⟨𝐴(𝑑,π‘₯(𝑑))βˆ’π΄(𝑑,𝑦(𝑑)),πœ‚(𝑑,π‘₯(𝑑),𝑦(𝑑))⟩β‰₯π‘Ÿ(𝑑)β€–π‘₯(𝑑)βˆ’π‘¦(𝑑)β€–2,(2.13) for all π‘₯,π‘¦βˆˆβ„³β„‹,π‘‘βˆˆΞ©.

Definition 2.8. Let π΄βˆΆΞ©Γ—β„‹β†’β„‹ be a random single-valued mapping. A single-valued mapping πΉβˆΆΞ©Γ—β„‹Γ—β„‹β†’β„‹ is said to be (i)random (𝑐,πœ‡)-relaxed cocoercive with respect to 𝐴 in the second argument if there exist measurable functions 𝑐,πœ‡βˆΆΞ©β†’(0,∞) such that ⟨𝐹(𝑑,β‹…,π‘₯(𝑑))βˆ’πΉ(𝑑,β‹…,𝑦(𝑑)),𝐴(𝑑,π‘₯(𝑑))βˆ’π΄(𝑑,𝑦(𝑑))⟩β‰₯βˆ’π‘(𝑑)‖𝐹(𝑑,β‹…,π‘₯(𝑑))βˆ’πΉ(𝑑,β‹…,𝑦(𝑑))β€–2+πœ‡(𝑑)β€–π‘₯(𝑑)βˆ’π‘¦(𝑑)β€–2,(2.14) for all π‘₯,π‘¦βˆˆβ„³β„‹ and π‘‘βˆˆΞ©;(ii)random 𝛼-Lipschitz continuous in the second argument if there exists a measurable function π›ΌβˆΆΞ©β†’(0,∞) such that ‖𝐹(𝑑,β‹…,π‘₯(𝑑))βˆ’πΉ(𝑑,β‹…,𝑦(𝑑))‖≀𝛼(𝑑)β€–π‘₯(𝑑)βˆ’π‘¦(𝑑)β€–,(2.15) for all π‘₯,π‘¦βˆˆβ„³β„‹,π‘‘βˆˆΞ©.

Notice that, in a similar way, we can define the concepts of relaxed cocoercive and Lipschitz continuous in the third argument.

Definition 2.9. Let πœ‚βˆΆΞ©Γ—β„‹Γ—β„‹β†’β„‹ and π΄βˆΆΞ©Γ—β„‹β†’β„‹ be two random single-valued mappings. Then a set-valued mapping π‘€βˆΆΞ©Γ—β„‹β†’2β„‹ is said to be (i)random (π‘š,πœ‚)-relaxed monotone if there exists a measurable function π‘šβˆΆΞ©β†’(0,∞) such that βŸ¨π‘’(𝑑)βˆ’π‘£(𝑑),πœ‚(𝑑,π‘₯(𝑑),𝑦(𝑑))⟩β‰₯βˆ’π‘š(𝑑)β€–π‘₯(𝑑)βˆ’π‘¦(𝑑)β€–2,(2.16) for all π‘₯,π‘¦βˆˆβ„³β„‹,𝑒(𝑑)βˆˆπ‘€(𝑑,π‘₯(𝑑)),𝑣(𝑑)βˆˆπ‘€(𝑑,𝑦(𝑑)),π‘‘βˆˆΞ©;(ii)random (𝐴,π‘š,πœ‚)-monotone if 𝑀 is a random (π‘š,πœ‚)-relaxed monotone and (𝐴𝑑+𝜌(𝑑)𝑀𝑑)(β„‹)=β„‹ for all measurable function πœŒβˆΆΞ©β†’(0,∞) and π‘‘βˆˆΞ©, where 𝐴𝑑(π‘₯)=𝐴(𝑑,π‘₯(𝑑)),𝑀𝑑(π‘₯)=𝑀(𝑑,π‘₯(𝑑)).

Definition 2.10. Let π΄βˆΆΞ©Γ—β„‹β†’β„‹ be a random single-valued mapping and π‘€βˆΆΞ©Γ—β„‹β†’2β„‹ a random (𝐴,π‘š,πœ‚)-monotone mapping. For each measurable function πœŒβˆΆΞ©β†’(0,∞), the corresponding random (𝐴,π‘š,πœ‚)-resolvent operator π½πœ‚,π‘€πœŒ,π΄βˆΆΞ©Γ—β„‹β†’β„‹ is defined by π½πœ‚π‘‘,π‘€π‘‘πœŒ(𝑑),𝐴𝑑𝐴(π‘₯)=𝑑+𝜌(𝑑)π‘€π‘‘ξ€Έβˆ’1(π‘₯),βˆ€π‘₯βˆˆβ„³β„‹,π‘‘βˆˆΞ©,(2.17) where 𝐴𝑑(π‘₯)=𝐴(𝑑,π‘₯(𝑑)),𝑀𝑑(π‘₯)=𝑀(𝑑,π‘₯(𝑑)), and π½πœ‚π‘‘,π‘€π‘‘πœŒ(𝑑),𝐴𝑑(π‘₯)=π½πœ‚,π‘€πœŒ,𝐴(𝑑,π‘₯(𝑑)).

The following lemma, which related to π½πœ‚,π‘€πœŒ,𝐴 operator, is very useful in order to prove our results.

Lemma 2.11. Let πœ‚βˆΆΞ©Γ—β„‹Γ—β„‹β†’β„‹ be a random single-valued mapping, π΄βˆΆΞ©Γ—β„‹β†’β„‹ a random (π‘Ÿ,πœ‚)-strongly monotone mapping, and π‘€βˆΆΞ©Γ—β„‹β†’2β„‹ a random (𝐴,π‘š,πœ‚)-monotone mapping. If πœŒβˆΆΞ©β†’(0,∞) is a measurable function with 𝜌(𝑑)∈(0,π‘Ÿ(𝑑)/π‘š(𝑑)) for all π‘‘βˆˆΞ©, then the following are true. (i)The corresponding random (𝐴,π‘š,πœ‚)-resolvent operator π½πœ‚,π‘€πœŒ,𝐴 is a random single-valued mapping. (ii)If πœ‚βˆΆΞ©Γ—β„‹Γ—β„‹β†’β„‹ is a random 𝜏-Lipschitz continuous mapping, then the corresponding random (𝐴,π‘š,πœ‚)-resolvent operator π½πœ‚,π‘€πœŒ,𝐴 is a random 𝜏/(π‘Ÿβˆ’πœŒπ‘š)-Lipschitz continuous.

Proof. The proof is similar to Proposition 3.9 in [2].

In order to prove our main results, we also need the following well known facts.

Lemma 2.12 (see [31]). Let β„‹ be a separable real Hilbert space and π‘ˆβˆΆΞ©Γ—β„‹β†’CB(β„‹) be a 𝐷-continuous random set-valued mapping. Then for any measurable mapping π‘€βˆΆΞ©β†’β„‹, the set-valued mapping π‘ˆ(β‹…,𝑀(β‹…))βˆΆΞ©β†’CB(β„‹) is measurable.

Lemma 2.13 (see [31]). Let β„‹ be a separable real Hilbert space and π‘ˆ,π‘‰βˆΆΞ©β†’CB(β„‹) two measurable set-valued mappings; πœ€>0 be a constant and π‘’βˆΆΞ©β†’β„‹ a measurable selection of π‘ˆ. Then there exists a measurable selection π‘£βˆΆΞ©β†’β„‹ of 𝑉 such that ‖𝑒(𝑑)βˆ’π‘£(𝑑)‖≀(1+πœ€)𝐷(π‘ˆ(𝑑),𝑉(𝑑)),βˆ€π‘‘βˆˆΞ©.(2.18)

Lemma 2.14 (see [32]). Let {𝛾𝑛} be a nonnegative real sequence, and let {πœ†π‘›} be a real sequence in [0,1] such that Ξ£βˆžπ‘›=0πœ†π‘›=∞. If there exists a positive integer 𝑛1 such that 𝛾𝑛+1≀1βˆ’πœ†π‘›ξ€Έπ›Ύπ‘›+πœ†π‘›πœŽπ‘›,βˆ€π‘›β‰₯𝑛1,(2.19) where πœŽπ‘›β‰₯0 for all 𝑛β‰₯0 and πœŽπ‘›β†’0 as π‘›β†’βˆž, then limπ‘›β†’βˆžπ›Ύπ‘›=0.

3. Existence Theorems

In this section, we will provide sufficient conditions for the existence solutions of the problem (2.2). To do this, we will begin with a useful lemma.

Lemma 3.1. Let β„‹1 and β„‹2 be two real Hilbert spaces. Let πΉβˆΆΞ©Γ—β„‹1Γ—β„‹2β†’β„‹1 and πΊβˆΆΞ©Γ—β„‹1Γ—β„‹2β†’β„‹2 be single-valued mappings. Let π‘ˆβˆΆΞ©Γ—β„‹1β†’CB(β„‹1),π‘‰βˆΆΞ©Γ—β„‹2β†’CB(β„‹2), and π‘€π‘–βˆΆΞ©Γ—β„‹π‘–β†’2ℋ𝑖 be a set-valued mappings for 𝑖=1,2. Assume that 𝑀𝑖 are random (𝐴𝑖,π‘šπ‘–,πœ‚π‘–)-monotone mappings and π΄π‘–βˆΆΞ©Γ—β„‹π‘–β†’β„‹π‘– random (π‘Ÿπ‘–,πœ‚π‘–)-strongly monotone mappings, for 𝑖=1,2. Then we have the following statements: (i)if (π‘Ž,𝑒,𝑏,𝑣)∈SRSVI(𝑀1,𝑀2)(𝐹,𝐺,π‘ˆ,𝑉), then for any measurable functions 𝜌1,𝜌2βˆΆΞ©β†’(0,∞) we have π‘Ž(𝑑)=π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1ξ€»,(𝑑)𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))𝑏(𝑑)=π½πœ‚2𝑑,𝑀2π‘‘πœŒ2(𝑑),𝐴2𝑑𝐴2(𝑑,𝑏(𝑑))βˆ’πœŒ2ξ€»(𝑑)𝐺(𝑑,𝑒(𝑑),𝑏(𝑑)),βˆ€π‘‘βˆˆΞ©;(3.1)(ii)if there exist two measurable functions 𝜌1,𝜌2βˆΆΞ©β†’(0,∞) such that π‘Ž(𝑑)=π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1ξ€»,(𝑑)𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))𝑏(𝑑)=π½πœ‚2𝑑,𝑀2π‘‘πœŒ2(𝑑),𝐴2𝑑𝐴2(𝑑,𝑏(𝑑))βˆ’πœŒ2ξ€»,(𝑑)𝐺(𝑑,𝑒(𝑑),𝑏(𝑑))(3.2) for all π‘‘βˆˆΞ©, then (π‘Ž,𝑒,𝑏,𝑣)∈SRSVI(𝑀1,𝑀2)(𝐹,𝐺,π‘ˆ,𝑉).

Proof. (i) Let 𝜌1,𝜌2βˆΆΞ©β†’(0,∞) be any measurable functions. Since (π‘Ž,𝑒,𝑏,𝑣)∈SRSVI(𝑀1,𝑀2)(𝐹,𝐺,π‘ˆ,𝑉), we have 0∈𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))+𝑀1(𝑑,π‘Ž(𝑑)),0∈𝐺(𝑑,𝑒(𝑑),𝑏(𝑑))+𝑀2(𝑑,𝑏(𝑑)),βˆ€π‘‘βˆˆΞ©.(3.3) Let π‘‘βˆˆΞ© be fixed. By 0∈𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))+𝑀1(𝑑,π‘Ž(𝑑)), we obtain 𝐴1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1(𝑑)𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))∈𝐴1(𝑑,π‘Ž(𝑑))+𝜌1(𝑑)𝑀1(𝑑,π‘Ž(𝑑)).(3.4) This means 𝐴1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1𝐴(𝑑)𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))∈1𝑑+𝜌1(𝑑)𝑀1𝑑(π‘Ž(𝑑)).(3.5) Thus π‘Ž(𝑑)=π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1ξ€»,(𝑑)𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))(3.6) where (𝐴1𝑑+𝜌1(𝑑)𝑀1𝑑)βˆ’1=π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑.
Similarly, if 0∈𝐺(𝑑,𝑒(𝑑),𝑏(𝑑))+𝑀2(𝑑,𝑏(𝑑)), we can show that𝑏(𝑑)=π½πœ‚2𝑑,𝑀2π‘‘πœŒ2(𝑑),𝐴2𝑑[𝐴2(𝑑,𝑏(𝑑))βˆ’πœŒ2(𝑑)𝐺(𝑑,𝑒(𝑑),𝑏(𝑑))],where(𝐴2𝑑+𝜌2(𝑑)𝑀2𝑑)βˆ’1=π½πœ‚2𝑑,𝑀2π‘‘πœŒ2(𝑑),𝐴2𝑑. Hence (i) is proved.
(ii) Assume that there exist two measurable functions 𝜌1,𝜌2βˆΆΞ©β†’(0,∞) such that π‘Ž(𝑑)=π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1ξ€»,(𝑑)𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))𝑏(𝑑)=π½πœ‚2𝑑,𝑀2π‘‘πœŒ2(𝑑),𝐴2𝑑𝐴2(𝑑,𝑏(𝑑))βˆ’πœŒ2ξ€»,(𝑑)𝐺(𝑑,𝑒(𝑑),𝑏(𝑑))(3.7) for all π‘‘βˆˆΞ©. Let π‘‘βˆˆΞ© be fixed. Since π‘Ž(𝑑)=π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑[𝐴1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1(𝑑)𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))], then by the definition of π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑, we see that ξ€·π΄π‘Ž(𝑑)=1𝑑+𝜌1(𝑑)𝑀1π‘‘ξ€Έβˆ’1𝐴1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1ξ€».(𝑑)𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))(3.8) This implies that βˆ’πΉ(𝑑,π‘Ž(𝑑),𝑣(𝑑))βˆˆπ‘€1(𝑑,π‘Ž(𝑑)).(3.9) That is, 0∈𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))+𝑀1(𝑑,π‘Ž(𝑑)).(3.10)
Similarly, if 𝑏(𝑑)=π½πœ‚2𝑑,𝑀2π‘‘πœŒ2(𝑑),𝐴2𝑑[𝐴2(𝑑,𝑏(𝑑))βˆ’πœŒ2(𝑑)𝐺(𝑑,𝑒(𝑑),𝑏(𝑑))] we can show that 0∈𝐺(𝑑,𝑒(𝑑),𝑏(𝑑))+𝑀2(𝑑,𝑏(𝑑)). This completes the proof.

Due to Lemma 3.1, in order to prove our main theorems, the following assumptions should be needed.

Assumptionπ’œ
π’œ(π‘Ž)β„‹1 and β„‹2 are separable real Hilbert spaces. π’œ(𝑏)πœ‚π‘–βˆΆΞ©Γ—β„‹π‘–Γ—β„‹π‘–β†’β„‹π‘– are random πœπ‘–-Lipschitz continuous single-valued mappings, for 𝑖=1,2.π’œ(𝑐)π΄π‘–βˆΆΞ©Γ—β„‹π‘–β†’β„‹π‘– are random (π‘Ÿπ‘–,πœ‚π‘–)-strongly monotone and random 𝛽𝑖-Lipschitz continuous single-valued mappings, for 𝑖=1,2.π’œ(𝑑)π‘€π‘–βˆΆΞ©Γ—β„‹π‘–β†’2ℋ𝑖 are random (𝐴𝑖,π‘šπ‘–,πœ‚π‘–)-monotone set-valued mappings, for 𝑖=1,2.π’œ(𝑒)π‘ˆβˆΆΞ©Γ—β„‹1β†’CB(β„‹1) is a random πœ™1-𝐷-Lipschitz continuous set-valued mapping and π‘‰βˆΆΞ©Γ—β„‹2β†’CB(β„‹2) is a random πœ™2-𝐷-Lipschitz continuous set-valued mapping. π’œ(𝑓)πΉβˆΆΞ©Γ—β„‹1Γ—β„‹2β†’β„‹1 is a random single-valued mapping, which has the following conditions: (i)𝐹 is a random (𝑐1,πœ‡1)-relaxed cocoercive with respect to 𝐴1 in the third argument and a random 𝛼1-Lipschitz continuous in the third argument, (ii)𝐹 is a random 𝜁1-Lipschitz continuous in the second argument. π’œ(𝑔)πΊβˆΆΞ©Γ—β„‹1Γ—β„‹2β†’β„‹2 is a random single-valued mapping, which has the following conditions: (i)𝐺 is a random (𝑐2,πœ‡2)-relaxed cocoercive with respect to 𝐴2 in the second argument and a random 𝛼2-Lipschitz continuous in the second argument; (ii)𝐺 is a random 𝜁2-Lipschitz continuous in the third argument.

Now, we are in position to present our main results.

Theorem 3.2. Assume that Assumption (π’œ) holds and there exist two measurable functions 𝜌1,𝜌2βˆΆΞ©β†’(0,∞) such that πœŒπ‘–(𝑑)∈(0,π‘Ÿπ‘–(𝑑)/π‘šπ‘–(𝑑)), for each 𝑖=1,2 and 𝜏1(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1(𝑑)𝛽21(𝑑)βˆ’2𝜌1(𝑑)πœ‡1(𝑑)+2𝜌1(𝑑)𝛼21(𝑑)𝑐1(𝑑)+𝜌21(𝑑)𝛼21𝜏(𝑑)<1βˆ’2(𝑑)𝜌2(𝑑)𝜁2(𝑑)πœ™1(𝑑)π‘Ÿ2(𝑑)βˆ’πœŒ2(𝑑)π‘š2(,πœπ‘‘)2(𝑑)π‘Ÿ2(𝑑)βˆ’πœŒ2(𝑑)π‘š2(𝑑)𝛽22(𝑑)βˆ’2𝜌2(𝑑)πœ‡2(𝑑)+2𝜌2(𝑑)𝛼22(𝑑)𝑐2(𝑑)+𝜌22(𝑑)𝛼22(πœπ‘‘)<1βˆ’1(𝑑)𝜌1(𝑑)𝜁1(𝑑)πœ™2(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1,(𝑑)(3.11) for all π‘‘βˆˆΞ©. Then the problem (2.2) has a solution.

Proof. Let {πœ€π‘›} be a null sequence of positive real numbers. Starting with measurable mappings π‘Ž0βˆΆΞ©β†’β„‹1 and 𝑏0βˆΆΞ©β†’β„‹2. By Lemma 2.12, we know that the set-valued mappings π‘ˆ(β‹…,π‘Ž0(β‹…))βˆΆΞ©β†’CB(β„‹1) and 𝑉(β‹…,𝑏0(β‹…))βˆΆΞ©β†’CB(β„‹2) are measurable mappings. Consequently, by Himmelberg [33], there exist measurable selections 𝑒0βˆΆΞ©β†’β„‹1 of π‘ˆ(β‹…,π‘Ž0(β‹…)) and 𝑣0βˆΆΞ©β†’β„‹2 of 𝑉(β‹…,𝑏0(β‹…)). We define now the measurable mappings π‘Ž1βˆΆΞ©β†’β„‹1 and 𝑏1βˆΆΞ©β†’β„‹2 by π‘Ž1(𝑑)=π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1𝑑,π‘Ž0ξ€Έ(𝑑)βˆ’πœŒ1ξ€·(𝑑)𝐹𝑑,π‘Ž0(𝑑),𝑣0,𝑏(𝑑)ξ€Έξ€»1(𝑑)=π½πœ‚2𝑑,𝑀2π‘‘πœŒ2(𝑑),𝐴2𝑑𝐴2𝑑,𝑏0ξ€Έ(𝑑)βˆ’πœŒ2ξ€·(𝑑)𝐺𝑑,𝑒0(𝑑),𝑏0,(𝑑)ξ€Έξ€»(3.12) where π½πœ‚π‘–π‘‘,π‘€π‘–π‘‘πœŒπ‘–(𝑑),𝐴𝑖𝑑(π‘₯)=(𝐴𝑖𝑑+πœŒπ‘–(𝑑)𝑀𝑖𝑑)βˆ’1(π‘₯), for all π‘₯βˆˆβ„³β„‹,π‘‘βˆˆΞ©, and 𝑖=1,2. Further, by Lemma 2.12, the set-valued mappings π‘ˆ(β‹…,π‘Ž1(β‹…))βˆΆΞ©β†’CB(β„‹1),𝑉(β‹…,𝑏1(β‹…))βˆΆΞ©β†’CB(β„‹2) are measurable. Again, by Himmelberg [33] and Lemma 2.13, there exist measurable selections 𝑒1βˆΆΞ©β†’β„‹1 of π‘ˆ(β‹…,π‘Ž1(β‹…)) and 𝑣1βˆΆΞ©β†’β„‹2 of 𝑉(β‹…,𝑏1(β‹…)) such that ‖‖𝑒0(𝑑)βˆ’π‘’1‖‖≀(𝑑)1+πœ€1ξ€Έπ·ξ€·π‘ˆξ€·π‘‘,π‘Ž0ξ€Έξ€·(𝑑),π‘ˆπ‘‘,π‘Ž1,‖‖𝑣(𝑑)ξ€Έξ€Έ0(𝑑)βˆ’π‘£1(‖‖≀𝑑)1+πœ€1𝐷𝑉𝑑,𝑏0(𝑑),𝑉𝑑,𝑏1(,𝑑)ξ€Έξ€Έ(3.13) for all π‘‘βˆˆΞ©. Define measurable mappings π‘Ž2βˆΆΞ©β†’β„‹1 and 𝑏2βˆΆΞ©β†’β„‹2 as follows: π‘Ž2(𝑑)=π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1𝑑,π‘Ž1ξ€Έ(𝑑)βˆ’πœŒ1ξ€·(𝑑)𝐹𝑑,π‘Ž1(𝑑),𝑣1,𝑏(𝑑)ξ€Έξ€»2(𝑑)=π½πœ‚2𝑑,𝑀2π‘‘πœŒ2(𝑑),𝐴2𝑑𝐴2𝑑,𝑏1ξ€Έ(𝑑)βˆ’πœŒ2ξ€·(𝑑)𝐺𝑑,𝑒1(𝑑),𝑏1,(𝑑)ξ€Έξ€»(3.14) for all π‘‘βˆˆΞ©. Continuing this process, inductively, we obtain the sequences {π‘Žπ‘›},{𝑏𝑛},{𝑒𝑛}, and {𝑣𝑛} of measurable mappings satisfy the following: π‘Žπ‘›+1(𝑑)=π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1𝑑,π‘Žπ‘›ξ€Έ(𝑑)βˆ’πœŒ1ξ€·(𝑑)𝐹𝑑,π‘Žπ‘›(𝑑),𝑣𝑛,𝑏(𝑑)𝑛+1(𝑑)=π½πœ‚2𝑑,𝑀2π‘‘πœŒ2(𝑑),𝐴2𝑑𝐴2𝑑,𝑏𝑛(𝑑)βˆ’πœŒ2ξ€·(𝑑)𝐺𝑑,𝑒𝑛(𝑑),𝑏𝑛,‖‖𝑒(𝑑)𝑛(𝑑)βˆ’π‘’π‘›+1‖‖≀(𝑑)1+πœ€π‘›+1ξ€Έπ·ξ€·π‘ˆξ€·π‘‘,π‘Žπ‘›ξ€Έξ€·(𝑑),π‘ˆπ‘‘,π‘Žπ‘›+1,‖‖𝑣(𝑑)𝑛(𝑑)βˆ’π‘£π‘›+1‖‖≀(𝑑)1+πœ€π‘›+1𝐷𝑉𝑑,𝑏𝑛(𝑑),𝑉𝑑,𝑏𝑛+1,(𝑑)ξ€Έξ€Έ(3.15) where 𝑒𝑛(𝑑)βˆˆπ‘ˆ(𝑑,π‘Žπ‘›(𝑑)),𝑣𝑛(𝑑)βˆˆπ‘‰(𝑑,𝑏𝑛(𝑑)) and for all π‘‘βˆˆΞ©,𝑛=0,1,2,….
Now, since π½πœ‚1,𝑀1𝜌1,𝐴1 is a random 𝜏1/(π‘Ÿ1βˆ’πœŒ1π‘š1)-Lipschitz continuous mapping, we have β€–β€–π‘Žπ‘›+1(𝑑)βˆ’π‘Žπ‘›β€–β€–=‖‖𝐽(𝑑)πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1𝑑,π‘Žπ‘›ξ€Έ(𝑑)βˆ’πœŒ1ξ€·(𝑑)𝐹𝑑,π‘Žπ‘›(𝑑),𝑣𝑛(𝑑)ξ€Έξ€»βˆ’π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1𝑑,π‘Žπ‘›βˆ’1ξ€Έ(𝑑)βˆ’πœŒ1ξ€·(𝑑)𝐹t,π‘Žπ‘›βˆ’1(𝑑),π‘£π‘›βˆ’1β€–β€–β‰€πœ(𝑑)ξ€Έξ€»1(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1‖‖𝐴(𝑑)1𝑑,π‘Žπ‘›ξ€Έ(𝑑)βˆ’π΄1𝑑,π‘Žπ‘›βˆ’1ξ€Έ(𝑑)βˆ’πœŒ1𝐹(𝑑)𝑑,π‘Žπ‘›(𝑑),𝑣𝑛(𝑑)βˆ’πΉπ‘‘,π‘Žπ‘›βˆ’1(𝑑),π‘£π‘›βˆ’1β€–β€–β‰€πœ(𝑑)ξ€Έξ€»1(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1‖‖𝐴(𝑑)1𝑑,π‘Žπ‘›ξ€Έ(𝑑)βˆ’π΄1𝑑,π‘Žπ‘›βˆ’1ξ€Έ(𝑑)βˆ’πœŒ1𝐹(𝑑)𝑑,π‘Žπ‘›(𝑑),𝑣𝑛(𝑑)βˆ’πΉπ‘‘,π‘Žπ‘›βˆ’1(𝑑),𝑣𝑛‖‖+𝜌(𝑑)ξ€Έξ€»1(𝑑)𝜏1(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1‖‖𝐹(𝑑)𝑑,π‘Žπ‘›βˆ’1(𝑑),𝑣𝑛(𝑑)βˆ’πΉπ‘‘,π‘Žπ‘›βˆ’1(𝑑),π‘£π‘›βˆ’1ξ€Έβ€–β€–,(𝑑)(3.16) for all π‘‘βˆˆΞ©. On the other hand, by Assumptions π’œ(𝑐) and π’œ(𝑓), we see that ‖‖𝐴1(𝑑,π‘Žπ‘›(𝑑))βˆ’π΄1(𝑑,π‘Žπ‘›βˆ’1(𝑑))βˆ’πœŒ1ξ€Ί(𝑑)𝐹(𝑑,π‘Žπ‘›(𝑑),𝑣𝑛(𝑑))βˆ’πΉ(𝑑,π‘Žπ‘›βˆ’1(𝑑),𝑣𝑛‖‖(𝑑))2=‖‖𝐴1(𝑑,π‘Žπ‘›(𝑑))βˆ’π΄1(𝑑,π‘Žπ‘›βˆ’1β€–β€–(𝑑))2βˆ’2𝜌1(𝐹𝑑)𝑑,π‘Žπ‘›(𝑑),𝑣𝑛(𝑑)βˆ’πΉπ‘‘,π‘Žπ‘›βˆ’1(𝑑),𝑣𝑛(𝑑),𝐴1𝑑,π‘Žπ‘›(𝑑)βˆ’π΄1𝑑,π‘Žπ‘›βˆ’1(𝑑)+𝜌21(‖‖𝑑)𝐹(𝑑,π‘Žπ‘›(𝑑),𝑣𝑛(𝑑))βˆ’πΉ(𝑑,π‘Žπ‘›βˆ’1(𝑑),𝑣𝑛‖‖(𝑑))2≀𝛽21β€–β€–π‘Ž(𝑑)𝑛(𝑑)βˆ’π‘Žπ‘›βˆ’1β€–β€–(𝑑)2+2𝜌1(𝑑)𝑐1β€–β€–(𝑑)𝐹(𝑑,π‘Žπ‘›(𝑑),𝑣𝑛(𝑑))βˆ’πΉ(𝑑,π‘Žπ‘›βˆ’1(𝑑),𝑣𝑛‖‖(𝑑))2βˆ’2𝜌1(𝑑)πœ‡1β€–β€–π‘Ž(𝑑)𝑛(𝑑)βˆ’π‘Žπ‘›βˆ’1β€–β€–(𝑑)2+𝜌21β€–β€–(𝑑)𝐹(𝑑,π‘Žπ‘›(𝑑),𝑣𝑛(𝑑))βˆ’πΉ(𝑑,π‘Žπ‘›βˆ’1(𝑑),𝑣𝑛‖‖(𝑑))2=𝛽21(𝑑)βˆ’2𝜌1(𝑑)πœ‡1ξ€»β€–β€–π‘Ž(𝑑)𝑛(𝑑)βˆ’π‘Žπ‘›βˆ’1β€–β€–(𝑑)2+ξ€Ί2𝜌1(𝑑)𝑐1(𝑑)+𝜌21ξ€»Γ—β€–β€–(𝑑)𝐹(𝑑,π‘Žπ‘›(𝑑),𝑣𝑛(𝑑))βˆ’πΉ(𝑑,π‘Žπ‘›βˆ’1(𝑑),𝑣𝑛‖‖(𝑑))2≀𝛽21(𝑑)βˆ’2𝜌1(𝑑)πœ‡1ξ€»β€–β€–π‘Ž(𝑑)𝑛(𝑑)βˆ’π‘Žπ‘›βˆ’1β€–β€–(𝑑)2+ξ€Ί2𝜌1(𝑑)𝑐1(𝑑)+𝜌21𝛼(𝑑)21β€–β€–π‘Ž(𝑑)𝑛(𝑑)βˆ’π‘Žπ‘›βˆ’1β€–β€–(𝑑)2≀𝛽21(𝑑)βˆ’2𝜌1(𝑑)πœ‡1(𝑑)+2𝜌1(𝑑)𝑐1(𝑑)𝛼21(𝑑)+𝜌21(𝑑)𝛼21(ξ€»β€–β€–π‘Žπ‘‘)𝑛(𝑑)βˆ’π‘Žπ‘›βˆ’1β€–β€–(𝑑)2,(3.17) for all π‘‘βˆˆΞ©. This gives ‖‖𝐴1𝑑,π‘Žπ‘›ξ€Έ(𝑑)βˆ’π΄1𝑑,π‘Žπ‘›βˆ’1ξ€Έ(𝑑)βˆ’πœŒ1𝐹(𝑑)𝑑,π‘Žπ‘›(𝑑),𝑣𝑛(𝑑)βˆ’πΉπ‘‘,π‘Žπ‘›βˆ’1(𝑑),𝑣𝑛‖‖≀(𝑑)𝛽21(𝑑)βˆ’2𝜌1(𝑑)πœ‡1(𝑑)+2𝜌1(𝑑)𝑐1(𝑑)𝛼21(𝑑)+𝜌21(𝑑)𝛼21β€–β€–π‘Ž(𝑑)𝑛(𝑑)βˆ’π‘Žπ‘›βˆ’1β€–β€–,(𝑑)(3.18) for all π‘‘βˆˆΞ©.
Meanwhile, since 𝐹 is a random 𝜁1-Lipschitz continuous mapping in the second argument, we get ‖‖𝐹𝑑,π‘Žπ‘›βˆ’1(𝑑),𝑣𝑛(𝑑)βˆ’πΉπ‘‘,π‘Žπ‘›βˆ’1(𝑑),π‘£π‘›βˆ’1ξ€Έβ€–β€–(𝑑)β‰€πœ1‖‖𝑣(𝑑)𝑛(𝑑)βˆ’π‘£π‘›βˆ’1β€–β€–,(𝑑)(3.19) for all π‘‘βˆˆΞ©. From (3.16), (3.18), and (3.19), we obtain that β€–β€–π‘Žπ‘›+1(𝑑)βˆ’π‘Žπ‘›β€–β€–β‰€πœ(𝑑)1β€–β€–π‘Ž(𝑑)𝑛(𝑑)βˆ’π‘Žπ‘›βˆ’1β€–β€–(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1(𝑑)𝛽21(𝑑)βˆ’2𝜌1(𝑑)πœ‡1(𝑑)+2𝜌1(𝑑)𝑐1(𝑑)𝛼21(𝑑)+𝜌21(𝑑)𝛼21+𝜏(𝑑)1(𝑑)𝜌1(𝑑)𝜁1(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1‖‖𝑣(𝑑)𝑛(𝑑)βˆ’π‘£π‘›βˆ’1β€–β€–(𝑑)=Ξ”1β€–β€–π‘Ž(𝑑)𝑛(𝑑)βˆ’π‘Žπ‘›βˆ’1β€–β€–+𝜏(𝑑)1(𝑑)𝜌1(𝑑)𝜁1(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1‖‖𝑣(𝑑)𝑛(𝑑)βˆ’π‘£π‘›βˆ’1β€–β€–,(𝑑)(3.20) where Ξ”1𝜏(𝑑)=1(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1(𝑑)𝛽21(𝑑)βˆ’2𝜌1(𝑑)πœ‡1(𝑑)+2𝜌1(𝑑)𝑐1(𝑑)𝛼21(𝑑)+𝜌21(𝑑)𝛼21(𝑑)(3.21) for all π‘‘βˆˆΞ©.
Similarly, by using Assumptions π’œ(𝑐) and π’œ(𝑔), we know that ‖‖𝑏𝑛+1(𝑑)βˆ’π‘π‘›β€–β€–(𝑑)≀Δ2‖‖𝑏(𝑑)𝑛(𝑑)βˆ’π‘π‘›βˆ’1β€–β€–+𝜏(𝑑)2(𝑑)𝜌2(𝑑)𝜁2(𝑑)π‘Ÿ2(𝑑)βˆ’πœŒ2(𝑑)π‘š2(‖‖𝑒𝑑)𝑛(𝑑)βˆ’π‘’π‘›βˆ’1β€–β€–,(𝑑)(3.22) where Ξ”2𝜏(𝑑)=2(𝑑)π‘Ÿ2(𝑑)βˆ’πœŒ2(𝑑)π‘š2(𝑑)𝛽22(𝑑)βˆ’2𝜌2(𝑑)πœ‡2(𝑑)+2𝜌2(𝑑)𝑐2(𝑑)𝛼22(𝑑)+𝜌22(𝑑)𝛼22(𝑑)(3.23) for all π‘‘βˆˆΞ©.
Next, since π‘ˆ is a random πœ™1-𝐷-Lipschitz continuous mapping and 𝑉 is a random πœ™2-𝐷-Lipschitz continuous mapping, by the choices of {𝑒𝑛} and {𝑣𝑛}, we have ‖‖𝑣𝑛(𝑑)βˆ’π‘£π‘›βˆ’1‖‖≀(𝑑)1+πœ€π‘›ξ€Έπ·ξ€·π‘‰ξ€·π‘‘,𝑏𝑛(𝑑),𝑉𝑑,π‘π‘›βˆ’1≀(𝑑)ξ€Έξ€Έ1+πœ€π‘›ξ€Έπœ™2(‖‖𝑏𝑑)𝑛(𝑑)βˆ’π‘π‘›βˆ’1(β€–β€–,‖‖𝑒𝑑)𝑛(𝑑)βˆ’π‘’π‘›βˆ’1‖‖≀(𝑑)1+πœ€π‘›ξ€Έπ·ξ€·π‘ˆξ€·π‘‘,π‘Žπ‘›ξ€Έξ€·(𝑑),π‘ˆπ‘‘,π‘Žπ‘›βˆ’1≀(𝑑)ξ€Έξ€Έ1+πœ€π‘›ξ€Έπœ™1(β€–β€–π‘Žπ‘‘)𝑛(𝑑)βˆ’π‘Žπ‘›βˆ’1(β€–β€–,𝑑)(3.24) for all π‘‘βˆˆΞ©. Now, by (3.20), (3.22), and (3.24), we obtain that β€–β€–π‘Žπ‘›+1(𝑑)βˆ’π‘Žπ‘›β€–β€–+‖‖𝑏(𝑑)𝑛+1(𝑑)βˆ’π‘π‘›β€–β€–β‰€ξ‚΅Ξ”(𝑑)1ξ€·(𝑑)+1+πœ€π‘›ξ€Έπœ2(𝑑)𝜌2(𝑑)𝜁2(𝑑)πœ™1(𝑑)π‘Ÿ2(𝑑)βˆ’πœŒ2(𝑑)π‘š2ξ‚Άβ€–β€–π‘Ž(𝑑)𝑛(𝑑)βˆ’π‘Žπ‘›βˆ’1β€–β€–+ξ‚΅Ξ”(𝑑)2ξ€·(𝑑)+1+πœ€π‘›ξ€Έπœ1(𝑑)𝜌1(𝑑)𝜁1(𝑑)πœ™2(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1(×‖‖𝑏𝑑)𝑛(𝑑)βˆ’π‘π‘›βˆ’1β€–β€–,(𝑑)(3.25) for all π‘‘βˆˆΞ©. This implies that β€–β€–π‘Žπ‘›+1(𝑑)βˆ’π‘Žπ‘›β€–β€–+‖‖𝑏(𝑑)𝑛+1(𝑑)βˆ’π‘π‘›β€–β€–(𝑑)β‰€πœƒπ‘›ξ€·β€–β€–π‘Ž(𝑑)𝑛(𝑑)βˆ’π‘Žπ‘›βˆ’1β€–β€–+‖‖𝑏(𝑑)𝑛(𝑑)βˆ’π‘π‘›βˆ’1β€–β€–ξ€Έ,(𝑑)(3.26) where πœƒπ‘›ξ‚»Ξ”(𝑑)=max1ξ€·(𝑑)+1+πœ€π‘›ξ€Έπœ2(𝑑)𝜌2(𝑑)𝜁2(𝑑)πœ™1(𝑑)π‘Ÿ2(𝑑)βˆ’πœŒ2(𝑑)π‘š2(𝑑),Ξ”2ξ€·(𝑑)+1+πœ€π‘›ξ€Έπœ1(𝑑)𝜌1(𝑑)𝜁1(𝑑)πœ™2(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1ξ‚Ό,(𝑑)(3.27) for all π‘‘βˆˆΞ©.
Next, let us define a norm β€–β‹…β€–+ on β„‹1Γ—β„‹2 by β€–β€–(π‘₯,𝑦)+=β€–π‘₯β€–+‖𝑦‖,βˆ€(π‘₯,𝑦)βˆˆβ„‹1Γ—β„‹2.(3.28) It is well known that (β„‹1Γ—β„‹2,β€–β‹…β€–+) is a Hilbert space. Moreover, for each π‘›βˆˆβ„•, we have β€–β€–(π‘Žπ‘›+1(𝑑),𝑏𝑛+1(𝑑))βˆ’(π‘Žπ‘›(𝑑),𝑏𝑛‖‖(𝑑))+β‰€πœƒπ‘›β€–β€–(𝑑)(π‘Žπ‘›(𝑑),𝑏𝑛(𝑑))βˆ’(π‘Žπ‘›βˆ’1(𝑑),π‘π‘›βˆ’1β€–β€–(𝑑))+,(3.29) for all π‘‘βˆˆΞ©.
Let ξ‚»Ξ”πœƒ(𝑑)=max1𝜏(𝑑)+2(𝑑)𝜌2(𝑑)𝜁2(𝑑)πœ™1(𝑑)π‘Ÿ2(𝑑)βˆ’πœŒ2(𝑑)π‘š2(𝑑),Ξ”2𝜏(𝑑)+1(𝑑)𝜌1(𝑑)𝜁1(𝑑)πœ™2(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1ξ‚Ό,(𝑑)foreachπ‘‘βˆˆΞ©.(3.30) We see that πœƒπ‘›(𝑑)β†“πœƒ(𝑑) as π‘›β†’βˆž. Moreover, condition (3.11) yields that 0<πœƒ(𝑑)<1 for all π‘‘βˆˆΞ©. This allows us to choose πœ—βˆˆ(πœƒ(𝑑),1) and a natural number 𝑁 such that πœƒπ‘›(𝑑)<πœ— for all 𝑛β‰₯𝑁. Using this one together with (3.30), we get β€–β€–(π‘Žπ‘›+1(𝑑),𝑏𝑛+1(𝑑))βˆ’(π‘Žπ‘›(𝑑),𝑏𝑛‖‖(𝑑))+β€–β€–β‰€πœ—(π‘Žπ‘›(𝑑),𝑏𝑛(𝑑))βˆ’(π‘Žπ‘›βˆ’1(𝑑),π‘π‘›βˆ’1β€–β€–(𝑑))+,(3.31) for all π‘‘βˆˆΞ© and 𝑛β‰₯𝑁. Thus, for each 𝑛>𝑁, we obtain β€–β€–(π‘Žπ‘›+1(𝑑),𝑏𝑛+1(𝑑))βˆ’(π‘Žπ‘›(𝑑),𝑏𝑛‖‖(𝑑))+β‰€πœ—π‘›βˆ’π‘β€–β€–(π‘Žπ‘+1(𝑑),𝑏𝑁+1(𝑑))βˆ’(π‘Žπ‘(𝑑),𝑏𝑁‖‖(𝑑))+,(3.32) for all π‘‘βˆˆΞ©. So, for any π‘šβ‰₯𝑛>𝑁, we have β€–β€–(π‘Žπ‘š(𝑑),π‘π‘š(𝑑))βˆ’(π‘Žπ‘›(𝑑),𝑏𝑛‖‖(𝑑))+β‰€Ξ£π‘šβˆ’1𝑖=𝑛‖‖(π‘Žπ‘–+1(𝑑),𝑏𝑖+1(𝑑))βˆ’(π‘Žπ‘–(𝑑),𝑏𝑖‖‖(𝑑))+β‰€Ξ£π‘šβˆ’1𝑖=π‘›πœ—π‘–βˆ’π‘β€–β€–(π‘Žπ‘+1(𝑑),𝑏𝑁+1(𝑑))βˆ’(π‘Žπ‘(𝑑),𝑏𝑁‖‖(𝑑))+β‰€πœ—π‘›πœ—π‘β€–β€–(1βˆ’πœ—)(π‘Žπ‘+1(𝑑),𝑏𝑁+1(𝑑))βˆ’(π‘Žπ‘(𝑑),𝑏𝑁‖‖(𝑑))+,(3.33) for all π‘‘βˆˆΞ©. Since πœ—βˆˆ(0,1), it follows that {πœ—π‘›}βˆžπ‘›=𝑁+1 converges to 0, as π‘›β†’βˆž. This means that {(π‘Žπ‘›(𝑑),𝑏𝑛(𝑑))} is a Cauchy sequence, for each π‘‘βˆˆΞ©. Thus, there are π‘Ž(𝑑)βˆˆβ„‹1 and 𝑏(𝑑)βˆˆβ„‹2 such that π‘Žπ‘›(𝑑)β†’π‘Ž(𝑑) and 𝑏𝑛(𝑑)→𝑏(𝑑) as π‘›β†’βˆž, for each π‘‘βˆˆΞ©.
Next, we will show that {𝑒𝑛(𝑑)} and {𝑣𝑛(𝑑)} converge to an element of π‘ˆ(𝑑,π‘Ž(𝑑)) and 𝑉(𝑑,𝑏(𝑑)), for all π‘‘βˆˆΞ©. Indeed, for π‘šβ‰₯𝑛>𝑁, we have from (3.24) and (3.33) that β€–β€–(π‘’π‘š(𝑑),π‘£π‘š(𝑑))βˆ’(𝑒𝑛(𝑑),𝑣𝑛‖‖(𝑑))+=β€–β€–π‘’π‘š(𝑑)βˆ’π‘’π‘›β€–β€–+‖‖𝑣(𝑑)π‘š(𝑑)βˆ’π‘£π‘›β€–β€–(𝑑)β‰€Ξ£π‘šβˆ’1𝑖=𝑛‖‖𝑒𝑖+1(𝑑)βˆ’π‘’π‘–β€–β€–(𝑑)+Ξ£π‘šβˆ’1𝑖=𝑛‖‖𝑣𝑖+1(𝑑)βˆ’π‘£π‘–β€–β€–(𝑑)β‰€Ξ£π‘šβˆ’1𝑖=𝑛1+πœ€π‘–+1ξ€Έπœ™2β€–β€–π‘Ž(𝑑)𝑖+1(𝑑)βˆ’π‘Žπ‘–β€–β€–(𝑑)+Ξ£π‘šβˆ’1𝑖=𝑛1+πœ€π‘–+1ξ€Έπœ™1‖‖𝑏(𝑑)𝑖+1(𝑑)βˆ’π‘π‘–β€–β€–(𝑑)≀2πœ™(𝑑)Ξ£π‘šβˆ’1𝑖=𝑛‖‖(π‘Žπ‘–+1(𝑑),𝑏𝑖+1(𝑑))βˆ’(π‘Žπ‘–(𝑑),𝑏𝑖‖‖(𝑑))+=2πœ™(𝑑)πœ—π‘›πœ—π‘β€–β€–(1βˆ’πœ—)(π‘Žπ‘+1(𝑑),𝑏𝑁+1(𝑑))βˆ’(π‘Žπ‘(𝑑),𝑏𝑁‖‖(𝑑))+,(3.34) where πœ™(𝑑)=max{πœ™1(𝑑),πœ™2(𝑑)}, for each π‘‘βˆˆΞ©. This implies that {(𝑒𝑛(𝑑),𝑣𝑛(𝑑))} is a Cauchy sequence in (β„‹1Γ—β„‹2,β€–β‹…β€–+), for all π‘‘βˆˆΞ©. Therefore, there exist 𝑒(𝑑)βˆˆβ„‹1 and 𝑣(𝑑)βˆˆβ„‹2 such that 𝑒𝑛(𝑑)→𝑒(𝑑) and 𝑣𝑛(𝑑)→𝑣(𝑑) as π‘›β†’βˆž, for each π‘‘βˆˆΞ©. Furthermore, β€–β€–inf{‖𝑒(𝑑)βˆ’π‘’β€²(𝑑)β€–βˆΆπ‘’β€²(𝑑)βˆˆπ‘ˆ(𝑑,π‘Ž(𝑑))}≀𝑒(𝑑)βˆ’π‘’π‘›β€–β€–(𝑑)+inf𝑒(𝑑)βˆˆπ‘ˆ(𝑑,π‘Ž(𝑑))‖‖𝑒𝑛‖‖≀‖‖(𝑑)βˆ’π‘’(𝑑)𝑒(𝑑)βˆ’π‘’π‘›β€–β€–ξ€·π‘ˆξ€·(𝑑)+𝐷𝑑,π‘Žπ‘›ξ€Έξ€Έβ‰€β€–β€–(𝑑),π‘ˆ(𝑑,π‘Ž(𝑑))𝑒(𝑑)βˆ’π‘’π‘›β€–β€–(𝑑)+πœ™1β€–β€–π‘Ž(𝑑)𝑛‖‖.(𝑑)βˆ’π‘Ž(𝑑)(3.35) Since 𝑒𝑛(𝑑)→𝑒(𝑑) and π‘Žπ‘›(𝑑)β†’π‘Ž(𝑑) as π‘›β†’βˆž, we have from the closedness property of π‘ˆ(𝑑,π‘Ž(𝑑)) and (3.35) that 𝑒(𝑑)βˆˆπ‘ˆ(𝑑,π‘Ž(𝑑)), for all π‘‘βˆˆΞ©. Similarly, we can show that 𝑣(𝑑)βˆˆπ‘‰(𝑑,𝑏(𝑑)), for all π‘‘βˆˆΞ©.
Finally, in view of (3.15) and applying the continuity of 𝐴𝑖,𝐹,𝐺 and π½πœ‚π‘–,π‘€π‘–πœŒπ‘–,𝐴𝑖, for 𝑖=1,2, we see that π‘Ž(𝑑)=π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1ξ€»,(𝑑)𝐹(𝑑,π‘Ž(𝑑),𝑣(𝑑))𝑏(𝑑)=π½πœ‚2𝑑,𝑀2π‘‘πœŒ2(𝑑),𝐴2𝑑𝐴2(𝑑,𝑏(𝑑))βˆ’πœŒ2ξ€»,(𝑑)𝐺(𝑑,𝑒(𝑑),𝑏(𝑑))(3.36) for all π‘‘βˆˆΞ©. Thus Lemma 3.1(ii) implies that (π‘Ž,𝑏,𝑒,𝑣) is a solution to problem (2.2). This completes the proof.

In particular, we have the following result.

Theorem 3.3. Let π‘ˆβˆΆΞ©Γ—β„‹1β†’β„‹1 and π‘‰βˆΆΞ©Γ—β„‹2β†’β„‹2 be two random single-valued mappings. Assume that Assumption π’œ holds and there exist measurable functions 𝜌1,𝜌2 satisfing (3.11). Then problem (2.3) has a unique solution.

Proof. From Theorem 3.2, we know that the problem (2.3) has a solution. So it remains to prove that, in fact, it has the unique solution. Assume that π‘Ž,π‘Žβˆ—βˆΆΞ©β†’β„‹1 and 𝑏,π‘βˆ—βˆΆΞ©β†’β„‹2 such that (π‘Ž,𝑏),(π‘Žβˆ—,π‘βˆ—) are solutions of the problem (2.3). Using the same lines as obtaining (3.20) and (3.22), by replacing π‘Žπ‘› with π‘Ž and π‘Žπ‘›+1 with π‘Žβˆ—, we have β€–π‘Ž(𝑑)βˆ’π‘Žβˆ—(𝑑)‖≀Δ1(𝑑)β€–π‘Ž(𝑑)βˆ’π‘Žβˆ—πœ(𝑑)β€–+1(𝑑)𝜌1(𝑑)𝜁1(𝑑)πœ™2(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1(𝑑)‖𝑏(𝑑)βˆ’π‘βˆ—(𝑑)β€–,βˆ€π‘‘βˆˆΞ©,(3.37) and, by replacing 𝑏𝑛 with 𝑏 and 𝑏𝑛+1 with π‘βˆ—, we obtain that ‖𝑏(𝑑)βˆ’π‘βˆ—(𝑑)‖≀Δ2(𝑑)‖𝑏(𝑑)βˆ’π‘βˆ—πœ(𝑑)β€–+2(𝑑)𝜌2(𝑑)𝜁2(𝑑)πœ™1(𝑑)π‘Ÿ2(𝑑)βˆ’πœŒ2(𝑑)π‘š2(𝑑)β€–π‘Ž(𝑑)βˆ’π‘Žβˆ—(𝑑)β€–,βˆ€π‘‘βˆˆΞ©,(3.38) where Ξ”1(𝑑) and Ξ”2(𝑑) are defined as in (3.21) and (3.23), respectively. From (3.37) and (3.38), we get β€–(π‘Ž(𝑑),𝑏(𝑑))βˆ’(π‘Žβˆ—(𝑑),π‘βˆ—β€–(𝑑))+≀Δ1𝜏(𝑑)+2(𝑑)𝜌2(𝑑)𝜁2(𝑑)πœ™1(𝑑)π‘Ÿ2(𝑑)βˆ’πœŒ2(𝑑)π‘š2ξ‚Ή(𝑑)β€–π‘Ž(𝑑)βˆ’π‘Žβˆ—β€–+ξ‚ΈΞ”(𝑑)2𝜏(𝑑)+1(𝑑)𝜌1(𝑑)𝜁1(𝑑)πœ™2(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1(𝑑)‖𝑏(𝑑)βˆ’π‘βˆ—β€–β€–β€–ξ€·π‘Ž(𝑑)β‰€πœƒ(𝑑)(π‘Ž(𝑑),𝑏(𝑑))βˆ’βˆ—(𝑑),π‘βˆ—ξ€Έβ€–β€–(𝑑)+,βˆ€π‘‘βˆˆΞ©,(3.39) where πœƒ(𝑑) is defined as in (3.30). Since 0<πœƒ(𝑑)<1, it follows that (π‘Ž(𝑑),𝑏(𝑑))=(π‘Žβˆ—(𝑑),π‘βˆ—(𝑑)), for all π‘‘βˆˆΞ©. This completes the proof.

4. Stability Analysis

In the proof of Theorem 3.3, in fact, we have constructed a sequence of measurable mappings {(π‘Žπ‘›,𝑏𝑛)} and show that its limit point is nothing but the unique element of SRSI(𝑀1,𝑀2)(𝐹,𝐺,π‘ˆ,𝑉). In this section, we will consider the stability of such a constructed sequence.

We start with a definition for stability analysis.

Definition 4.1. Let β„‹1,β„‹2 be real Hilbert spaces. Let π‘„βˆΆΞ©Γ—β„‹1Γ—β„‹2β†’β„‹1Γ—β„‹2,(π‘Ž0(𝑑),𝑏0(𝑑))βˆˆβ„‹1Γ—β„‹2, and let (π‘Žπ‘›+1(𝑑),𝑏𝑛+1(𝑑))=β„Ž(𝑄,π‘Žπ‘›(𝑑),𝑏𝑛(𝑑)) define an iterative procedure which yields a sequence of points {(π‘Žπ‘›(𝑑),𝑏𝑛(𝑑))} in β„‹1Γ—β„‹2, where β„Ž is an iterative procedure involving the mapping 𝑄. Let 𝐹(𝑄)={(π‘Ž,𝑏)βˆˆβ„³β„‹1Γ—β„³β„‹2βˆΆπ‘„(𝑑,π‘Ž(𝑑),𝑏(𝑑))=(π‘Ž(𝑑),𝑏(𝑑)),forallπ‘‘βˆˆΞ©}β‰ βˆ… and that {(π‘Žπ‘›,𝑏𝑛)} converges to a random fixed point (π‘Ž,𝑏) of 𝑄. Let {(π‘₯𝑛,𝑦𝑛)} be an arbitrary sequence in β„³β„‹1Γ—β„³β„‹2 and let 𝛿𝑛(𝑑)=β€–(π‘₯𝑛+1(𝑑),𝑦𝑛+1(𝑑))βˆ’β„Ž(𝑄,π‘₯𝑛(𝑑),𝑦𝑛(𝑑))β€–, for each 𝑛β‰₯0 and π‘‘βˆˆΞ©. For each π‘‘βˆˆΞ©, if limπ‘›β†’βˆžπ›Ώπ‘›(𝑑)=0 implies that limπ‘›β†’βˆž(π‘₯𝑛(𝑑),𝑦𝑛(𝑑))β†’(π‘Ž(𝑑),𝑏(𝑑)), then the iteration procedure defined by (π‘Žπ‘›+1(𝑑),𝑏𝑛+1(𝑑))=β„Ž(𝑄,π‘Žπ‘›(𝑑),𝑏𝑛(𝑑)) is said to be 𝑄-stable or stable with respect to 𝑄.

Let 𝐹,𝐺,𝑀𝑖,πœ‚π‘–,𝐴𝑖, and πœŒπ‘–, for 𝑖=1,2, be random mappings defined as in Theorem 3.2. Now, for each π‘‘βˆˆΞ©, if {(π‘₯𝑛(𝑑),𝑦𝑛(𝑑))} is any sequence in β„‹1Γ—β„‹2. We will consider the sequence {(𝑆𝑛(𝑑),𝑇𝑛(𝑑))}, which is defined by𝑆𝑛(𝑑)=π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1𝑑,π‘₯𝑛(𝑑)βˆ’πœŒ1ξ€·(𝑑)𝐹𝑑,π‘₯𝑛(𝑑),𝑉𝑑,𝑦𝑛,𝑇(𝑑)𝑛(𝑑)=π½πœ‚2𝑑,𝑀2π‘‘πœŒ2(𝑑),𝐴2𝑑𝐴2𝑑,𝑦𝑛(𝑑)βˆ’πœŒ2ξ€·ξ€·(𝑑)𝐺𝑑,π‘ˆπ‘‘,π‘₯𝑛(t)ξ€Έ,𝑦𝑛,(𝑑)ξ€Έξ€»(4.1) where π‘ˆβˆΆΞ©Γ—β„‹1β†’β„‹1 and π‘‰βˆΆΞ©Γ—β„‹2β†’β„‹2 and π‘‘βˆˆΞ©. Consequently, we put𝛿𝑛‖‖(𝑑)=(π‘₯𝑛+1(𝑑),𝑦𝑛+1(𝑑))βˆ’(𝑆𝑛(𝑑),𝑇𝑛‖‖(𝑑))+.(4.2)

Meanwhile, let π‘„βˆΆΞ©Γ—β„‹1Γ—β„‹2β†’β„‹1Γ—β„‹2 be defined by𝐽𝑄(𝑑,π‘Ž(𝑑),𝑏(𝑑))=πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1ξ€»,𝐽(𝑑)𝐹(𝑑,π‘Ž(𝑑),𝑏(𝑑))πœ‚2𝑑,𝑀2π‘‘πœŒ2(𝑑),𝐴2𝑑𝐴2(𝑑,𝑏(𝑑))βˆ’πœŒ2(𝑑)𝐺(𝑑,π‘Ž(t),𝑏(t))(4.3) for all π‘Žβˆˆβ„³β„‹1,π‘βˆˆβ„³β„‹2,π‘‘βˆˆΞ©. In view of Lemma 3.1, we see that (π‘Ž,𝑏)∈SRSI(𝑀1,𝑀2)(𝐹,𝐺,π‘ˆ,𝑉) if and only if (π‘Ž,𝑏)∈𝐹(𝑄).

Now, we prove the stability of the sequence {(π‘Žπ‘›,𝑏𝑛)} with respect to mapping 𝑄, defined by (4.3).

Theorem 4.2. Assume that Assumption π’œ holds and there exist 𝜌1,𝜌2 satisfing (3.11). Then for each π‘‘βˆˆΞ©, we have limπ‘›β†’βˆžπ›Ώπ‘›(𝑑)=0 if and only if limπ‘›β†’βˆž(π‘₯𝑛(𝑑),𝑦𝑛(𝑑))=(π‘Ž(𝑑),𝑏(𝑑)), where 𝛿𝑛(𝑑) are defined by (4.2) and (π‘Ž(𝑑),𝑏(𝑑))∈𝐹(𝑄).

Proof. According to Theorem 3.3, the solution set SRSI(𝑀1,𝑀2)(𝐹,𝐺,π‘ˆ,𝑉) of problem (2.3) is a singleton set, that is, SRSI(𝑀1,𝑀2)(𝐹,𝐺,π‘ˆ,𝑉)={(π‘Ž,𝑏)}. For each π‘‘βˆˆΞ©, let {(π‘₯𝑛(𝑑),𝑦𝑛(𝑑))} be any sequence in β„‹1Γ—β„‹2. By (4.1) and (4.2), we have β€–β€–(π‘₯𝑛+1(𝑑),𝑦𝑛+1β€–β€–(𝑑))βˆ’(π‘Ž(𝑑),𝑏(𝑑))+≀‖‖(π‘₯𝑛+1(𝑑),𝑦𝑛+1(𝑑))βˆ’(𝑆𝑛(𝑑),𝑇𝑛‖‖(𝑑))++β€–β€–(𝑆𝑛(𝑑),𝑇𝑛‖‖(𝑑))βˆ’(π‘Ž(𝑑),𝑏(𝑑))+=β€–β€–(𝑆𝑛(𝑑),𝑇𝑛‖‖(𝑑))βˆ’(π‘Ž(𝑑),𝑏(𝑑))++𝛿𝑛=‖‖𝐽(𝑑)πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1𝑑,π‘₯𝑛(𝑑)βˆ’πœŒ1ξ€·(𝑑)𝐹𝑑,π‘₯𝑛(𝑑),𝑉𝑑,𝑦𝑛‖‖+‖‖𝐽(𝑑)ξ€Έξ€Έξ€»βˆ’π‘Ž(𝑑)πœ‚2𝑑,𝑀2π‘‘πœŒ2(𝑑),𝐴2𝑑𝐴2𝑑,𝑦𝑛(𝑑)βˆ’πœŒ2ξ€·ξ€·(𝑑)𝐺𝑑,π‘ˆπ‘‘,π‘₯𝑛(𝑑),𝑦𝑛‖‖(𝑑)ξ€Έξ€»βˆ’π‘(𝑑)+𝛿𝑛(𝑑).(4.4) Since π½πœ‚1,𝑀1𝜌1,𝐴1 is a random 𝜏1/(π‘Ÿ1βˆ’πœŒ1π‘š1)-Lipschitz continuous mapping, by Assumptions π’œ(𝑐),π’œ(𝑓) and Lemma 3.1(i), we get β€–β€–π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1𝑑,π‘₯𝑛(𝑑)βˆ’πœŒ1ξ€·(𝑑)𝐹𝑑,π‘₯𝑛(𝑑),𝑉𝑑,𝑦𝑛‖‖=‖‖𝐽(𝑑)ξ€Έξ€Έξ€»βˆ’π‘Ž(𝑑)πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1𝑑,π‘₯𝑛(𝑑)βˆ’πœŒ1ξ€·(𝑑)𝐹𝑑,π‘₯𝑛(𝑑),𝑉𝑑,𝑦𝑛(𝑑)ξ€Έξ€Έξ€»βˆ’π½πœ‚1𝑑,𝑀1π‘‘πœŒ2(𝑑),𝐴1𝑑𝐴1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1ξ€»β€–β€–β‰€πœ(𝑑)𝐹(𝑑,π‘Ž(𝑑),𝑉(𝑑,𝑏(𝑑)))1(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1(×‖‖𝐴𝑑)1𝑑,π‘₯𝑛(𝑑)βˆ’π΄1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1𝐹(𝑑)𝑑,π‘₯𝑛(𝑑),𝑉𝑑,π‘¦π‘›ξ€»β€–β€–β‰€πœ(𝑑)ξ€Έξ€Έβˆ’πΉ(𝑑,π‘Ž(𝑑),𝑉(𝑑,𝑏(𝑑)))1(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1×‖‖𝐴(𝑑)1𝑑,π‘₯𝑛(𝑑)βˆ’π΄1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1𝐹(𝑑)𝑑,π‘₯𝑛(𝑑),𝑉𝑑,𝑦𝑛(𝑑)ξ€Έξ€Έβˆ’πΉπ‘‘,π‘Ž(𝑑),𝑉𝑑,𝑦𝑛‖‖+𝜌(𝑑)ξ€Έξ€Έξ€»1(𝑑)𝜏1(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1‖‖𝐹(𝑑)𝑑,π‘Ž(𝑑),𝑉𝑑,𝑦𝑛‖‖.(𝑑)ξ€Έξ€Έβˆ’πΉ(𝑑,π‘Ž(𝑑),𝑉(𝑑,𝑏(𝑑)))(4.5)
On the other hand, by Assumptions π’œ(𝑐) and π’œ(𝑓), we see that ‖‖𝐴1(𝑑,π‘₯𝑛(𝑑))βˆ’π΄1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1ξ€Ί(𝑑)𝐹(𝑑,π‘₯𝑛(𝑑),𝑉(𝑑,𝑦𝑛(𝑑)))βˆ’πΉ(𝑑,π‘Ž(𝑑),𝑉(𝑑,𝑦𝑛‖‖(𝑑)))2=‖‖𝐴1(𝑑,π‘₯𝑛(𝑑))βˆ’π΄1β€–β€–(𝑑,π‘Ž(𝑑))2βˆ’2𝜌1(𝐹𝑑)𝑑,π‘₯𝑛(𝑑),𝑉𝑑,𝑦𝑛(𝑑)ξ€Έξ€Έβˆ’πΉπ‘‘,π‘Ž(𝑑),𝑉𝑑,𝑦𝑛(𝑑)ξ€Έξ€Έ,𝐴1𝑑,π‘₯𝑛(𝑑)βˆ’π΄1(𝑑,π‘Ž(𝑑))+𝜌21(‖‖𝑑)𝐹(𝑑,π‘₯𝑛(𝑑),𝑉(𝑑,𝑦𝑛(𝑑)))βˆ’πΉ(𝑑,π‘Ž(𝑑),𝑉(𝑑,𝑦𝑛‖‖(𝑑)))2≀𝛽21β€–β€–π‘₯(𝑑)𝑛‖‖(𝑑)βˆ’π‘Ž(𝑑)2+2𝜌1(𝑑)𝑐1β€–β€–(𝑑)𝐹(𝑑,π‘₯𝑛(𝑑),𝑉(𝑑,𝑦𝑛(𝑑)))βˆ’πΉ(𝑑,π‘Ž(𝑑),𝑉(𝑑,𝑦𝑛‖‖(𝑑)))2βˆ’2𝜌1(𝑑)πœ‡1β€–β€–π‘₯(𝑑)𝑛‖‖(𝑑)βˆ’π‘Ž(𝑑)2+𝜌21(‖‖𝑑)𝐹(𝑑,π‘₯𝑛(𝑑),𝑉(𝑑,𝑦𝑛(𝑑)))βˆ’πΉ(𝑑,π‘Ž(𝑑),𝑉(𝑑,𝑦𝑛‖‖(𝑑)))2=𝛽21(𝑑)βˆ’2𝜌1(𝑑)πœ‡1ξ€»β€–β€–π‘₯(𝑑)𝑛‖‖(𝑑)βˆ’π‘Ž(𝑑)2+ξ€Ί2𝜌1(𝑑)𝑐1(𝑑)+𝜌21ξ€»Γ—β€–β€–(𝑑)𝐹(𝑑,π‘₯𝑛(𝑑),𝑉(𝑑,𝑦𝑛(𝑑)))βˆ’πΉ(𝑑,π‘Ž(𝑑),𝑉(𝑑,𝑦𝑛‖‖(𝑑)))2≀𝛽21(𝑑)βˆ’2𝜌1(𝑑)πœ‡1ξ€»β€–β€–π‘₯(𝑑)𝑛‖‖(𝑑)βˆ’π‘Ž(𝑑)2+ξ€Ί2𝜌1(𝑑)𝑐1(𝑑)+𝜌21𝛼(𝑑)21β€–β€–π‘₯(𝑑)𝑛‖‖(𝑑)βˆ’π‘Ž(𝑑)2≀𝛽21(𝑑)βˆ’2𝜌1(𝑑)πœ‡1(𝑑)+2𝜌1(𝑑)𝑐1(𝑑)𝛼21(𝑑)+𝜌21(𝑑)𝛼21ξ€»β€–β€–π‘₯(𝑑)𝑛‖‖(𝑑)βˆ’π‘Ž(𝑑)2.(4.6)
This gives ‖‖𝐴1𝑑,π‘₯𝑛(𝑑)βˆ’π΄1(𝑑,π‘Ž(𝑑))βˆ’πœŒ1𝐹(𝑑)𝑑,π‘₯𝑛(𝑑),𝑉𝑑,𝑦𝑛(𝑑)ξ€Έξ€Έβˆ’πΉπ‘‘,π‘Ž(𝑑),𝑉𝑑,𝑦𝑛‖‖≀(𝑑)𝛽21(𝑑)βˆ’2𝜌1(𝑑)πœ‡1(𝑑)+2𝜌1(𝑑)𝑐1(𝑑)𝛼21(𝑑)+𝜌21(𝑑)𝛼21β€–β€–π‘₯(𝑑)𝑛‖‖.(𝑑)βˆ’π‘Ž(𝑑)(4.7)
Meanwhile, since 𝐹 is a random 𝜁1-Lipschitz continuous mapping in the second argument, we get ‖‖𝐹𝑑,π‘Ž(𝑑),𝑉𝑑,𝑦𝑛‖‖(𝑑)ξ€Έξ€Έβˆ’πΉ(𝑑,π‘Ž(𝑑),𝑉(𝑑,𝑏(𝑑)))β‰€πœ1(𝑑)πœ™2‖‖𝑦(𝑑)𝑛‖‖.(𝑑)βˆ’π‘(𝑑)(4.8)
From (4.5)–(4.8), we obtain that β€–β€–π½πœ‚1𝑑,𝑀1π‘‘πœŒ1(𝑑),𝐴1𝑑𝐴1𝑑,π‘₯𝑛(𝑑)βˆ’πœŒ1ξ€·(𝑑)𝐹𝑑,π‘₯𝑛(𝑑),𝑉𝑑,π‘¦π‘›β€–β€–β‰€πœ(𝑑)ξ€Έξ€Έξ€»βˆ’π‘Ž(𝑑)1(𝑑)𝛽21(𝑑)βˆ’2𝜌1(𝑑)πœ‡1(𝑑)+2𝜌1(𝑑)𝑐1(𝑑)𝛼21(𝑑)+𝜌21(𝑑)𝛼21(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1β€–β€–π‘₯(𝑑)𝑛‖‖+𝜌(𝑑)βˆ’π‘Ž(𝑑)1(𝑑)𝜏1(𝑑)𝜁1(𝑑)πœ™2(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1(‖‖𝑦𝑑)𝑛‖‖,(𝑑)βˆ’π‘(𝑑)(4.9) where Ξ”1𝜏(𝑑)=1(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1(𝑑)𝛽21(𝑑)βˆ’2𝜌1(𝑑)πœ‡1(𝑑)+2𝜌1(𝑑)𝑐1(𝑑)𝛼21(𝑑)+𝜌21(𝑑)𝛼21(𝑑).(4.10)
Similarly, since π½πœ‚2,𝑀2𝜌2,𝐴2 is a random 𝜏2/(π‘Ÿ2βˆ’πœŒ2π‘š2)-Lipschitz continuous mapping, by Assumption π’œ(𝑐),π’œ(𝑔), and Lemma 3.1, we obtain that β€–β€–π½πœ‚2𝑑,𝑀2π‘‘πœŒ2(𝑑),𝐴2𝑑𝐴2𝑑,𝑦𝑛(𝑑)βˆ’πœŒ2ξ€·ξ€·(𝑑)𝐺𝑑,π‘ˆπ‘‘,π‘₯𝑛(𝑑),𝑦𝑛‖‖(𝑑)ξ€Έξ€»βˆ’π‘(𝑑)≀Δ2(‖‖𝑦𝑑)𝑛(β€–β€–+πœπ‘‘)βˆ’π‘(𝑑)2(𝑑)𝜌2(𝑑)𝜁2(𝑑)πœ™1(𝑑)π‘Ÿ2(𝑑)βˆ’πœŒ2(𝑑)π‘š2β€–β€–π‘₯(𝑑)𝑛(β€–β€–,𝑑)βˆ’π‘Ž(𝑑)(4.11) where Ξ”2𝜏(𝑑)=2(𝑑)π‘Ÿ2(𝑑)βˆ’πœŒ2(𝑑)π‘š2(𝑑)𝛽22(𝑑)βˆ’2𝜌2(𝑑)πœ‡2(𝑑)+2𝜌2(𝑑)𝑐2(𝑑)𝛼22(𝑑)+𝜌22(𝑑)𝛼22(𝑑).(4.12)
Thus β€–β€–(π‘₯𝑛+1(𝑑),𝑦𝑛+1β€–β€–(𝑑))βˆ’(π‘Ž(𝑑),𝑏(𝑑))+≀Δ1𝜏(𝑑)+2(𝑑)𝜌2(𝑑)𝜁2(𝑑)πœ™1(𝑑)π‘Ÿ2(𝑑)βˆ’πœŒ2(𝑑)π‘š2ξ‚Ήβ€–β€–π‘₯(𝑑)𝑛‖‖+ξ‚ΈΞ”(𝑑)βˆ’π‘Ž(𝑑)2𝜏(𝑑)+1(𝑑)𝜌1(𝑑)𝜁1(𝑑)πœ™2(𝑑)π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1(‖‖𝑦𝑑)𝑛‖‖‖‖(𝑑)βˆ’π‘(𝑑)β‰€πœƒ(𝑑)(π‘₯𝑛(𝑑),𝑦𝑛‖‖(𝑑))βˆ’(π‘Ž(𝑑),𝑏(𝑑))++𝛿𝑛=β€–β€–(𝑑)(1βˆ’(1βˆ’πœƒ(𝑑)))(π‘₯𝑛(𝑑),𝑦𝑛‖‖(𝑑))βˆ’(π‘Ž(𝑑),𝑏(𝑑))++𝛿𝑛(𝑑),(4.13) where πœƒ(𝑑)=max{Ξ”1(𝑑)+𝜏2(𝑑)𝜌2(𝑑)𝜁2(𝑑)πœ™1(𝑑)/(π‘Ÿ2(𝑑)βˆ’πœŒ2(𝑑)π‘š2(𝑑)),Ξ”2(𝑑)+𝜏1(𝑑)𝜌1(𝑑)𝜁1(𝑑)πœ™2(𝑑)/(π‘Ÿ1(𝑑)βˆ’πœŒ1(𝑑)π‘š1(𝑑))}, for all π‘‘βˆˆΞ©.
So β€–β€–(π‘₯𝑛+1(𝑑),𝑦𝑛+1β€–β€–(𝑑))βˆ’(π‘Ž(𝑑),𝑏(𝑑))+≀‖‖(1βˆ’(1βˆ’πœƒ(𝑑)))(π‘₯𝑛(𝑑),𝑦𝑛‖‖(𝑑))βˆ’(π‘Ž(𝑑),𝑏(𝑑))++𝛿(1βˆ’πœƒ(𝑑))⋅𝑛(𝑑).(1βˆ’πœƒ(𝑑))(4.14) In view of (4.14), if limπ‘›β†’βˆžπ›Ώπ‘›(𝑑)=0, we see that Lemma 2.14 implies limπ‘›β†’βˆžξ€·π‘₯𝑛(𝑑),𝑦𝑛=(𝑑)(π‘Ž(𝑑),𝑏(𝑑)).(4.15)
On the other hand, by using (4.5) and (4.11), we see that 𝛿𝑛‖‖(𝑑)≀(π‘₯𝑛+1(𝑑),𝑦𝑛+1(𝑑))βˆ’(𝑆𝑛(𝑑),𝑇𝑛‖‖(𝑑))+≀‖‖(π‘₯𝑛+1(𝑑),𝑦𝑛+1β€–β€–(𝑑))βˆ’(π‘Ž(𝑑),𝑏(𝑑))++β€–β€–(π‘Ž(𝑑),𝑏(𝑑))βˆ’(𝑆𝑛(𝑑),𝑇𝑛‖‖(𝑑))+≀‖‖(π‘₯𝑛+1(𝑑),𝑦𝑛+1β€–β€–(𝑑))βˆ’(π‘Ž(𝑑),𝑏(𝑑))+β€–β€–+πœƒ(𝑑)(π‘₯𝑛(𝑑),𝑦𝑛‖‖(𝑑))βˆ’(π‘Ž(𝑑),𝑏(𝑑))+(4.16) for all π‘‘βˆˆΞ©. Consequently, if for each π‘‘βˆˆΞ© we assume limπ‘›β†’βˆž(π‘₯𝑛(𝑑),𝑦𝑛(𝑑))=(π‘Ž(𝑑),𝑏(𝑑)), we will have limπ‘›β†’βˆžπ›Ώπ‘›(𝑑)=0. This completes the proof.

Remark 4.3. Theorem 4.2 shows that the iterative sequence {(π‘Žπ‘›,𝑏𝑛)}, which has constructed in Theorem 3.3, is 𝑄-stable.

5. Conclusion

We have introduced a new system of set-valued random variational inclusions involving (𝐴,π‘š,πœ‚)-monotone operator and random relaxed cocoercive operators in Hilbert space. By using the resolvent operator technique, we have constructed an iterative algorithm and then the approximation solvability of a aforesaid problem is examined. Moreover, we have considered the stability of such iterative algorithm. It is worth noting that for a suitable and appropriate choice of the operators, as 𝐹,𝐺,𝑀,πœ‚,𝐴, one can obtain a large number of various classes of variational inequalities; this means that problem (2.2) is quite general and unifying. Consequently, the results presented in this paper are very interesting and improve some known corresponding results in the literature.

Acknowledgment

J. Suwannawit is supported by the Centre of Excellence in Mathematics, Commission on Higher Education, Thailand. N. Petrot is supported by the National Research Council of Thailand (Project no. R2554B102).