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Equilibrium of Bayesian fuzzy economies and quasivariational inequalities with random fuzzy mappings
Journal of Inequalities and Applications volume 2013, Article number: 374 (2013)
Abstract
In this paper, we introduce a Bayesian abstract fuzzy economy model, and we prove the existence of Bayesian fuzzy equilibrium. As applications, we prove the existence of the solutions for two types of random quasivariational inequalities with random fuzzy mappings, and we also obtain random fixed point theorems.
MSC:58E35, 47H10, 91B50, 91A44.
1 Introduction
The study of fuzzy games began with the paper written by Kim and Lee in 1998 [1]. This type of games is a generalization of classical abstract economies. For an overview of results concerning this topic, the reader is referred to [2]. However, the existence of random fuzzy equilibrium has not been studied so far. We introduce the new model of Bayesian abstract fuzzy economy, and we explore the existence of the Bayesian fuzzy equilibrium. Our model is characterized by a private information set, an action (strategy) fuzzy mapping, a random fuzzy constraint one and a random fuzzy preference mapping. The Bayesian fuzzy equilibrium concept is an extension of the deterministic equilibrium. We generalize the former deterministic models introduced by Debreu [3], Shafer and Sonnenschein [4], Yannelis and Prabhakar [5] or Patriche [2], and we search for applications.
Since Fichera and Stampacchia introduced the variational inequalities (in 1960s), this domain has been extensively studied. For recent results we refer the reader to [6–11] and the bibliography therein. Noor and Elsanousi [12] introduced the notion of a random variational inequality. The existence of solutions of the random variational inequality and random quasivariational inequality problems has been proved, for instance, in [13–21].
In this paper, we first define the model of the Bayesian abstract fuzzy economy and we prove a theorem of Bayesian fuzzy equilibrium existence. Then, we apply it in order to prove the existence of solutions for the two types of random quasivariational inequalities with random fuzzy mappings. We generalize some results obtained by Yuan in [22]. As a consequence, we obtain random fixed point theorems.
The paper is organized as follows. In the next section, some notational and terminological conventions are given. We also present, for the reader’s convenience, some results on Bochner integration. In Section 3, the model of differential information abstract fuzzy economy is introduced, and the main result is also stated. Section 4 contains existence results for solutions of random quasivariational inequalities with random fuzzy mappings.
2 Notation and definition
Throughout this paper, we shall use the following notation.
{\mathbb{R}}_{++} denotes the set of strictly positive reals. coD denotes the convex hull of the set D. \overline{co}D denotes the closed convex hull of the set D. {2}^{D} denotes the set of all nonempty subsets of the set D. If D\subset Y, where Y is a topological space, clD denotes the closure of D.
For the reader’s convenience, we review a few basic definitions and results from continuity and measurability of correspondences and Bochner integrable functions.
Let Z and Y be sets. Let Z, Y be topological spaces and P:Z\to {2}^{Y} be a correspondence. P is said to be upper semicontinuous if for each z\in Z and each open set V in Y with P(z)\subset V, there exists an open neighborhood U of z in Z such that P(y)\subset V for each y\in U. P is said to be lower semicontinuous if for each z\in Z and each open set V in Y with P(z)\cap V\ne \mathrm{\varnothing}, there exists an open neighborhood U of z in Z such that P(y)\cap V\ne \mathrm{\varnothing} for each y\in U.
Lemma 1 [22]
Let Z and Y be two topological spaces, and let D be an open subset of Z. Suppose {P}_{1}:Z\to {2}^{Y}, {P}_{2}:Z\to {2}^{Y} are upper semicontinuous correspondences such that {P}_{2}(z)\subset {P}_{1}(z) for all z\in D. Then the correspondence P:Z\to {2}^{Y} defined by
is also upper semicontinuous.
Let E be a topological vector space, and let {E}^{\mathrm{\prime}} be the dual space of E, which consists of all continuous linear functionals on E. The real part of pairing between {E}^{\mathrm{\prime}} and E is denoted by Re\u3008w,x\u3009 for each w\in {E}^{\mathrm{\prime}} and x\in E. The operator P:E\to {2}^{{E}^{\mathrm{\prime}}} is called monotone if Re\u3008uv,yx\u3009\ge 0 for all u\in P(y) and v\in P(x) and x,y\in E.
Let now (\mathrm{\Omega},\mathcal{F},\mu ) be a complete, finite measure space, and Y be a topological space. The correspondence P:\mathrm{\Omega}\to {2}^{Y} is said to have a measurable graph if {G}_{P}\in \mathcal{F}\otimes \beta (Y), where \beta (Y) denotes the Borel σalgebra on Y and ⊗ denotes the product σalgebra. The correspondence T:\mathrm{\Omega}\to {2}^{Y} is said to be lower measurable if for every open subset V of Y, the set \{\omega \in \mathrm{\Omega}:T(\omega )\cap V\ne \mathrm{\varnothing}\} is an element of ℱ. This notion of measurability is also called in the literature weak measurability or just measurability, in comparison with the strong measurability: the correspondence T:\mathrm{\Omega}\to {2}^{Y} is said to be strong measurable if for every closed subset V of Y, the set \{\omega \in \mathrm{\Omega}:T(\omega )\cap V\ne \mathrm{\varnothing}\} is an element of ℱ. In the framework we shall deal with (complete finite measure spaces), the two notions coincide (see [23]).
Recall (see Debreu [24], p.359) that if T:\mathrm{\Omega}\to {2}^{Y} has a measurable graph, then T is lower measurable. Furthermore, if T(\cdot ) is closedvalued and lower measurable, then T:\mathrm{\Omega}\to {2}^{Y} has a measurable graph.
Lemma 2 [25]
Let {P}_{n}:\mathrm{\Omega}\to {2}^{Y}, n=1,2\dots be a sequence of correspondences with measurable graphs. Then the correspondences {\bigcup}_{n}{P}_{n}, {\bigcap}_{n}{P}_{n} and Y\setminus {P}_{n} have measurable graphs.
Let (\mathrm{\Omega},\mathcal{F},\mu ) be a measure space, and let Y be a Banach space.
It is known (see [25], Theorem 2, p.45) that if x:\mathrm{\Omega}\to Y is a μmeasurable function, then x is the Bochner integrable if and only if {\int}_{\mathrm{\Omega}}\parallel x(\omega )\parallel \phantom{\rule{0.2em}{0ex}}d\mu (\omega )<\mathrm{\infty}.
It is denoted by {L}_{1}(\mu ,Y), the space of equivalence classes of Yvalued Bochner integrable functions x:\mathrm{\Omega}\to Y normed by \parallel x\parallel ={\int}_{\mathrm{\Omega}}\parallel x(\omega )\parallel \phantom{\rule{0.2em}{0ex}}d\mu (\omega ). Also, it is known (see [24], p.50) that {L}_{1}(\mu ,Y) is a Banach space.
The correspondence P:\mathrm{\Omega}\to {2}^{Y} is said to be integrably bounded if there exists a map h\in {L}_{1}(\mu ,R) such that sup\{\parallel x\parallel :x\in P(\omega )\}\le h(\omega ) μa.e.
We denote by {S}_{P} the set of all selections of the correspondence P:\mathrm{\Omega}\to {2}^{Y} that belong to the space {L}_{1}(\mu ,Y), i.e.,
We will find the conditions under which {S}_{P} is nonempty and weakly compact in {L}_{1}(\mu ,Y) by applying Aumann measurable selection theorem (see Appendix) and Diestel’s theorem (see Appendix).
Zadeh initiated the theory of fuzzy sets [26] as a framework for phenomena, which can not be characterized precisely. We present below several notions concerning the fuzzy sets and the fuzzy mappings.
Definition 1 (Chang [27])
If Y is a topological space, then a function A from Y into [0;1] is called a fuzzy set on Y. The family of all fuzzy sets on Y is denoted by \mathcal{F}(Y).

(2)
If X and Y are topological spaces, then a mapping P:X\to \mathcal{F}(Y) is called a fuzzy mapping.

(3)
If P is a fuzzy mapping, then, for each x\in X, P(x) is a fuzzy set in Y and P(x)(y)\in [0,1], y\in Y is called the degree of membership of y in P(x).

(4)
Let A\in \mathcal{F}(Y), a\in [0,1], then the set {(A)}_{a}=\{y\in Y:A(y)>a\} is called a strong acut set of the fuzzy set A.
The random fuzzy mappings have been defined in order to model random mechanisms generating impreciselyvalued data which can be properly described by using fuzzy sets.
Let Y be a topological space, let \mathcal{F}(Y) be a collection of all fuzzy sets over Y, and let (\mathrm{\Omega},\mathcal{F}) be a measurable space.
Definition 2 (See [28])
A fuzzy mapping P:\mathrm{\Omega}\to \mathcal{F}(Y) is said to be measurable if for any given a\in [0,1], {(P(\cdot ))}_{a}:\mathrm{\Omega}\to {2}^{Y} is a measurable setvalued mapping.

(2)
We say that a fuzzy mapping P:\mathrm{\Omega}\to \mathcal{F}(Y) is said to have a measurable graph if for any given a\in [0,1], the setvalued mapping {(P(\cdot ))}_{a}:\mathrm{\Omega}\to {2}^{Y} has a measurable graph.

(3)
A fuzzy mapping P:\mathrm{\Omega}\times X\to \mathcal{F}(Y) is called a random fuzzy mapping if, for any given x\in X, P(\cdot ,x):\mathrm{\Omega}\to \mathcal{F}(Y) is a measurable fuzzy mapping.
3 Bayesian fuzzy equilibrium existence for Bayesian abstract fuzzy economies
3.1 The model of a Bayesian abstract fuzzy economy
The framework of fuzziness became part of the language of applied mathematics. The uncertainties characterize the individual feature of the decisions of the agents involved in different economic activities, and they can be described by using random fuzzy mappings. In the fuzzy model of the abstract economy, which we will define below, for each agent i, the action choice is modelled by the measurable fuzzy mapping {X}_{i}, and the constraints and the preferences are modelled by the random fuzzy mappings {A}_{i} and, respectively, {P}_{i}. In the state of the world \omega \in \mathrm{\Omega}, the interpretation of the number {P}_{i}(\omega ,\tilde{x})(y)\in [0,1], associated with ({\tilde{x}}_{i}(\omega ),y), can be the degree of intensity, with which y is preferred to {\tilde{x}}_{i}(\omega ), or the degree of truth that y is preferred to {\tilde{x}}_{i}(\omega ). We can also see the value {A}_{i}(\omega ,\tilde{x})(y)\in [0,1] associated with ({\tilde{x}}_{i}(\omega ),y), as the belief of the player i that in the state ω, if the other players choose {({\tilde{x}}_{j}(\omega ))}_{j\ne i}, he can choose y\in Y. The element {z}_{i} is the action level in each state of the world, {a}_{i}(\tilde{x}) expresses the perceived degree of feasibility of the strategy \tilde{x}, and {p}_{i}(\tilde{x}) represents the preference level of the strategy \tilde{x}.
We now define the next model of the Bayesian abstract fuzzy economy, which generalizes the model in [29].
Let (\mathrm{\Omega},\mathcal{F},\mu ) be a complete finite measure space, where Ω denotes the set of states of nature of the world, and the σalgebra ℱ denotes the set of events. Let Y denote the strategy or commodity space, where Y is a separable Banach space.
Let I be a countable or uncountable set (the set of agents). For each i\in I, let {X}_{i}:\mathrm{\Omega}\to \mathcal{F}(Y) be a fuzzy mapping, and let {z}_{i}\in (0,1].
Let {L}_{{X}_{i}}=\{{x}_{i}\in {S}_{{({X}_{i}(\cdot ))}_{{z}_{i}}}:{x}_{i}\text{is}{\mathcal{F}}_{i}\text{measurable}\}. Denote by {L}_{X}={\prod}_{i\in I}{L}_{{X}_{i}} and by {L}_{{X}_{i}} the set {\prod}_{j\ne i}{L}_{{X}_{j}}. An element {x}_{i} of {L}_{{X}_{i}} is called a strategy for agent i. The typical element of {L}_{{X}_{i}} is denoted by {\tilde{x}}_{i} and that of {({X}_{i}(\omega ))}_{{z}_{i}} by {x}_{i}(\omega ) (or {x}_{i}).
Definition 3 A general Bayesian abstract fuzzy economy is a family G=\{(\mathrm{\Omega},\mathcal{F},\mu ),{({X}_{i},{\mathcal{F}}_{i},{A}_{i},{P}_{i},{a}_{i},{b}_{i},{z}_{i})}_{i\in I}\}, where

(a)
{X}_{i}:\mathrm{\Omega}\to \mathcal{F}(Y) is the action (strategy) fuzzy mapping of agent i;

(b)
{\mathcal{F}}_{i} is a sub σalgebra of ℱ, which denotes the private information of agent i;

(c)
for each \omega \in \mathrm{\Omega}, {A}_{i}(\omega ,\cdot ):{L}_{X}\to \mathcal{F}(Y) is the random fuzzy constraint mapping of agent i;

(d)
for each \omega \in \mathrm{\Omega}, {P}_{i}(\omega ,\cdot ):{L}_{X}\to \mathcal{F}(Y) is the random fuzzy preference mapping of agent i;

(e)
{a}_{i}:{L}_{X}\to (0,1] is a random fuzzy constraint function, and {p}_{i}:{L}_{X}\to (0,1] is a random fuzzy preference function of agent i;

(f)
{z}_{i}\in (0,1] is such that for all (\omega ,x)\in \mathrm{\Omega}\times {L}_{X}, {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\subset {({X}_{i}(\omega ))}_{{z}_{i}} and {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}\subset {({X}_{i}(\omega ))}_{{z}_{i}}.
Definition 4 A Bayesian fuzzy equilibrium for G is a strategy profile {\tilde{x}}^{\ast}\in {L}_{X} such that for all i\in I,

(i)
{\tilde{x}}_{i}^{\ast}(\omega )\in cl{({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})} μa.e.;

(ii)
{({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})}\cap {({P}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{p}_{i}({\tilde{x}}^{\ast})}=\mathrm{\varnothing} μa.e.
Remark 1 If the correspondences from the model above are constant with respect to Ω, we obtain the abstract fuzzy economy model.
3.2 Existence of the Bayesian fuzzy equilibrium
This is our first theorem. The constraint and preference correspondences, derived from the constraint and preference fuzzy mappings, verify the assumptions of measurable graph and weakly open lower sections. Our result is a generalization of Theorem 3 in [29].
Theorem 1 Let I be a countable or uncountable set. Let the family G=\{(\mathrm{\Omega},\mathcal{F},\mu ),{({X}_{i},{\mathcal{F}}_{i},{A}_{i},{P}_{i},{a}_{i},{b}_{i},{z}_{i})}_{i\in I}\} be a general Bayesian abstract economy satisfying (A.1)(A.4). Then there exists a Bayesian fuzzy equilibrium for G.
For each i\in I:
(A.1)

(a)
{X}_{i}:\mathrm{\Omega}\to \mathcal{F}(Y) is such that \omega \to {X}_{i}{(\omega )}_{{z}_{i}}:\mathrm{\Omega}\to {2}^{Y} is a nonempty convex weakly compactvalued and integrably bounded correspondence;

(b)
{X}_{i}:\mathrm{\Omega}\to \mathcal{F}(Y) is such that \omega \to {({X}_{i}(\omega ))}_{{z}_{i}}:\mathrm{\Omega}\to {2}^{Y} is {\mathcal{F}}_{i}lower measurable;
(A.2)

(a)
For each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}, {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})} is convex and has a nonempty interior in the relative norm topology of {({X}_{i}(\omega ))}_{{z}_{i}};

(b)
the correspondence (\omega ,\tilde{x})\to {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:\mathrm{\Omega}\times {L}_{X}\to {2}^{Y} has a measurable graph, i.e., \{(\omega ,\tilde{x},y)\in \mathrm{\Omega}\times {L}_{X}\times Y:y\in {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\}\in \mathcal{F}\otimes {\mathit{\text{\xdf}}}_{w}({L}_{X})\otimes \mathit{\text{\xdf}}(Y), where {\mathit{\text{\xdf}}}_{w}({L}_{X}) is the Borel σalgebra for the weak topology on {L}_{X} and \mathit{\text{\xdf}}(Y) is the Borel σalgebra for the norm topology on Y;

(c)
the correspondence (\omega ,\tilde{x})\to {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})} has weakly open lower sections, i.e., for each \omega \in \mathrm{\Omega} and for each y\in Y, the set {({({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})})}^{1}(\omega ,y)=\{\tilde{x}\in {L}_{X}:y\in {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\} is weakly open in {L}_{X};

(d)
For each \omega \in \mathrm{\Omega}, \tilde{x}\to cl{({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:{L}_{X}\to {2}^{Y} is upper semicontinuous in the sense that the set \{\tilde{x}\in {L}_{X}:cl{({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\subset V\} is weakly open in {L}_{X} for every norm open subset V of Y;
(A.3)

(a)
the correspondence (\omega ,\tilde{x})\to {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}:\mathrm{\Omega}\times {L}_{X}\to {2}^{Y} has open convex values such that {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}\subset {(X(\omega ))}_{{z}_{i}} for each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X};

(b)
the correspondence (\omega ,\tilde{x})\to {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}:\mathrm{\Omega}\times {L}_{X}\to {2}^{Y} has a measurable graph;

(c)
the correspondence (\omega ,\tilde{x})\to {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}:\mathrm{\Omega}\times {L}_{X}\to {2}^{Y} has weakly open lower sections, i.e., for each \omega \in \mathrm{\Omega} and for each y\in Y, the set {({({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})})}^{1}(\omega ,y)=\{\tilde{x}\in {L}_{X}:y\in {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}\} is weakly open in {L}_{X};
(A.4)

(a)
For each {\tilde{x}}_{i}\in {L}_{{X}_{i}}, for each \omega \in \mathrm{\Omega}, {\tilde{x}}_{i}(\omega )\notin {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\cap {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}.
Proof For each i\in I, let us define {\mathrm{\Phi}}_{i}:\mathrm{\Omega}\times {L}_{X}\to {2}^{Y} by {\mathrm{\Phi}}_{i}(\omega ,\tilde{x})={({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\cap {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}. We prove first that {L}_{X} is a nonempty convex weakly compact subset in {L}_{1}(\mu ,Y).
Since (\mathrm{\Omega},\mathcal{F},\mu ) is a complete finite measure space, Y is a separable Banach space, and {X}_{i}:\mathrm{\Omega}\to {2}^{Y} has a measurable graph, by Aumann’s selection theorem (see Appendix), it follows that there exists a {\mathcal{F}}_{i}measurable function {f}_{i}:\mathrm{\Omega}\to Y such that {f}_{i}(\omega )\in {X}_{i}(\omega ) μa.e. Since {X}_{i} is integrably bounded, we have that {f}_{i}\in {L}_{1}(\mu ,Y), hence {L}_{{X}_{i}} is nonempty and {L}_{X}={\prod}_{i\in I}{L}_{{X}_{i}} is nonempty. Obviously, {L}_{{X}_{i}} is convex and {L}_{X} is also convex. Since {X}_{i}:\mathrm{\Omega}\to {2}^{Y} is integrably bounded and it has convex weakly compact values, by Diestel’s theorem (see Appendix), it follows that {L}_{{X}_{i}} is a weakly compact subset of {L}_{1}(\mu ,Y). More over, {L}_{X} is weakly compact. {L}_{1}(\mu ,Y) equipped with the weak topology is a locally convex topological vector space.
The correspondence {\mathrm{\Phi}}_{i} is convex valued, by Lemma 2, it has a measurable graph, and for each \omega \in \mathrm{\Omega}, {\mathrm{\Phi}}_{i}(\omega ,\cdot ) has weakly open lower sections. Let {U}_{i}=\{(\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}:{\mathrm{\Phi}}_{i}(\omega ,\tilde{x})\ne \mathrm{\varnothing}\}. For each \tilde{x}\in {L}_{X}, let {U}_{i}^{\tilde{x}}=\{\omega \in \mathrm{\Omega}:{\mathrm{\Phi}}_{i}(\omega ,\tilde{x})\ne \mathrm{\varnothing}\} and for each \omega \in \mathrm{\Omega}, let {U}_{i}^{\omega}=\{\tilde{x}\in {L}_{X}:{\mathrm{\Phi}}_{i}(\omega ,\tilde{x})\ne \mathrm{\varnothing}\}. The values of {\mathrm{\Phi}}_{i/{U}_{i}} have nonempty interiors in the relative norm topology of {X}_{i}(\omega ). By the Caratheodorytype selection theorem (see Appendix), there exists a function {f}_{i}:{U}_{i}\to Y such that {f}_{i}(\omega ,\tilde{x})\in {\mathrm{\Phi}}_{i}(\omega ,\tilde{x}) for all (\omega ,\tilde{x})\in {U}_{i}, for each \tilde{x}\in {L}_{X}, {f}_{i}(\cdot ,\tilde{x}) is measurable on {U}_{i}^{\tilde{x}}, for each \omega \in \mathrm{\Omega}, {f}_{i}(\omega ,\cdot ) is continuous on {U}_{i}^{\omega} and, moreover {f}_{i}(\cdot ,\cdot ) is jointly measurable.
Define {G}_{i}:\mathrm{\Omega}\times {L}_{X}\to {2}^{Y} by {G}_{i}(\omega ,\tilde{x})=\{\begin{array}{cc}\{{f}_{i}(\omega ,\tilde{x})\}\hfill & \text{if}(\omega ,\tilde{x})\in {U}_{i};\hfill \\ cl{({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\hfill & \text{if}(\omega ,\tilde{x})\notin {U}_{i}.\hfill \end{array}
Define {G}_{i}^{\mathrm{\prime}}:{L}_{X}\to {2}^{{L}_{{X}_{i}}}, by {G}_{i}^{{}^{\mathrm{\prime}}}(\tilde{x})=\{{y}_{i}\in {L}_{{X}_{i}}:{y}_{i}(\omega )\in {G}_{i}(\omega ,\tilde{x})\phantom{\rule{0.25em}{0ex}}\mu \text{a.e.}\} and {G}^{{}^{\mathrm{\prime}}}:{L}_{X}\to {2}^{{L}_{X}} by {G}^{\mathrm{\prime}}(\tilde{x}):={\prod}_{i\in I}{G}_{i}^{\mathrm{\prime}}(\tilde{x}) for each \tilde{x}\in {L}_{X}. We shall prove that {G}^{\mathrm{\prime}} is an upper semicontinuous correspondence with respect to the weakly topology of {L}_{X} and has nonempty convex closed values. By applying FanGlicksberg’s fixedpoint Theorem [30] to {G}^{\mathrm{\prime}}, we obtain a fixed point, which is the equilibrium point for the abstract economy.
It follows by Theorem III.40 in [31] and the projection theorem that for each \tilde{x}\in {L}_{X}, the correspondence \tilde{x}\to cl{({A}_{i}(\cdot ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:\mathrm{\Omega}\to {2}^{Y} has a measurable graph. For each \tilde{x}\in {L}_{X}, the correspondence {G}_{i}(\cdot ,\tilde{x}) has a measurable graph. Since {\mathrm{\Phi}}_{i}(\omega ,\cdot ) has weakly open lower sections for each \omega \in \mathrm{\Omega}, it follows that {U}_{i}^{\omega} is weakly open in {L}_{X}. By Lemma 1, for each \omega \in \mathrm{\Omega}, {G}_{i}(\omega ,\cdot ):{L}_{X}\to {2}^{Y} is upper semicontinuous in the sense that the set \{\tilde{x}\in {L}_{X}:{G}_{i}(\omega ,\tilde{x})\}\subset V is weakly open in {L}_{X} for every norm open subset V of Y. Moreover, {G}_{i} is convex and nonemptyvalued.
{G}_{i} is nonemptyvalued, and for each \tilde{x}\in {L}_{X}, {G}_{i}(\cdot ,\tilde{x}) has a measurable graph. Hence, according to the Aumann measurable selection theorem for each fixed \tilde{x}\in {L}_{X}, there exists an {\mathcal{F}}_{i}measurable function {y}_{i}:\mathrm{\Omega}\to Y such that {y}_{i}(\omega )\in {G}_{i}(\omega ,\tilde{x}) μa.e. Since for each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}, {G}_{i}(\omega ,\tilde{x}) is contained in the values of the integrably bounded correspondence {X}_{i}(\cdot ), then {y}_{i}\in {L}_{{X}_{i}}, and we conclude that {y}_{i}\in {G}_{i}^{\mathrm{\prime}}(\tilde{x}) for each \tilde{x}\in {L}_{X}. Thus, {G}_{i}^{\mathrm{\prime}} is nonemptyvalued.
Since for each \tilde{x}\in {L}_{X}, {G}_{i}(\cdot ,\tilde{x}) has a measurable graph and for each \omega \in \mathrm{\Omega}, {G}_{i}(\omega ,\cdot ):{L}_{X}\to {2}^{Y} is upper semicontinuous and {G}_{i}(\omega ,\tilde{x})\subset {({X}_{i}(\omega ))}_{{z}_{i}} for each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}, by u.s.c. lifting theorem (see Appendix), it follows that {G}_{i}^{\mathrm{\prime}} is weakly upper semicontinuous. {G}_{i}^{\mathrm{\prime}} is convexvalued since {G}_{i} is such.
{G}^{\mathrm{\prime}} is a weakly upper semicontinuous correspondence, and it also has nonempty convex closed values.
The set {L}_{X} is weakly compact and convex, and then, by FanGlicksberg’s fixedpoint theorem in [30], there exists {\tilde{x}}^{\ast}\in {L}_{X} such that {\tilde{x}}^{\ast}\in {G}^{\mathrm{\prime}}({\tilde{x}}^{\ast}), i.e., for each i\in I, {\tilde{x}}_{i}^{\ast}\in {G}_{i}^{\mathrm{\prime}}({\tilde{x}}^{\ast}).
Then, {\tilde{x}}_{i}^{\ast}\in {L}_{{X}_{i}} and {\tilde{x}}_{i}^{\ast}(\omega )\in {G}_{i}(\omega ,{\tilde{x}}^{\ast}) μa.e. Since {\tilde{x}}_{i}^{\ast}(\omega )\notin {({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})}\cap {({P}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{p}_{i}({\tilde{x}}^{\ast})} μa.e., it follows that (\omega ,{\tilde{x}}^{\ast})\notin {U}_{i} for each i\in I and {\tilde{x}}_{i}^{\ast}\in cl{({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})} μa.e. We also have that {({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})}\cap {({P}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{p}_{i}({\tilde{x}}^{\ast})}=\mathrm{\varnothing}. □
Example 1 Let (\mathrm{\Omega},\mathcal{F},\mu ) be the measure space, where \mathrm{\Omega}=[0,1], \mathcal{F}=\mathit{\text{\xdf}}([0,1]) is the σalgebra of the Borel measurable subsets in [0,1] and μ is the Lebesgue measure.
Let Y=\mathbb{R} and I=\{1,2,\dots ,n\}.
For each i\in I, let us define the following.
The random fuzzy constraint function {z}_{i}:[0,1]\to (0,1] is defined by
The random fuzzy mapping {X}_{i}(\cdot ):[0,1]\to \mathcal{F}(\mathbb{R}) is defined by
Then, the correspondence {X}_{i}:[0,1]\to {2}^{\mathbb{R}} is defined by
It is a nonempty convex compact valued and integrably bounded correspondence. It is also {\mathcal{F}}_{i}lower measurable.
Let {L}_{{X}_{i}}=\{{x}_{i}\in {S}_{{({X}_{i}(\cdot ))}_{{z}_{i}}}:{x}_{i}\text{is}{\mathcal{F}}_{i}\text{measurable}\} and {L}_{X}={\prod}_{i\in I}{L}_{{X}_{i}}.
The random fuzzy constraint function {a}_{i}:{L}_{X}\to (0,1] is defined by
For each \omega \in [0,1], the random fuzzy constraint mapping of agent i, {A}_{i}(\omega ,\cdot ):{L}_{X}\to \mathcal{F}(\mathbb{R}) is defined by
Then, the correspondence (\omega ,\tilde{x})\to {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:[0,1]\times {L}_{X}\to {2}^{[0,1]} is defined by
For each \omega \in [0,1], it has weakly open lower sections in {L}_{X}, and it has a measurable graph.
For each (\omega ,\tilde{x})\in [0,1]\times {L}_{X}, {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})} is convex and with nonempty interior in [0,1].
For each \omega \in [0,1], the correspondence \tilde{x}\to cl{({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:{L}_{X}\to {2}^{[0,1]}, defined by cl{({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}=[0,\frac{9}{10}] for each \tilde{x}\in {L}_{X} is upper semicontinuous and nonemptyvalued.
The random fuzzy preference mapping {p}_{i}:{L}_{X}\to (0,1] is defined by
Let us define {D}_{i}={\prod}_{j\ne i}{L}_{{X}_{j}}\times \{{\tilde{x}}_{i}:[0,1]\to [0,1],{\tilde{x}}_{i}(\omega )={k}_{{\tilde{x}}_{i}}{\omega}^{i},\omega \in [0,1],{k}_{{\tilde{x}}_{i}}\in [0,1]\}. {D}_{i} is weakly closed in {L}_{X}.
For each \omega \in [0,1], the random fuzzy preference mapping of agent i, {P}_{i}(\omega ,\cdot ):{L}_{X}\to \mathcal{F}(\mathbb{R}) is defined by
Then, for each \omega \in [0,1], the correspondence \tilde{x}\to {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}:{L}_{X}\to {2}^{[0,1]} is defined by
For each \omega \in \mathrm{\Omega} and for each y\in Y, the set {({({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})})}^{1}(\omega ,y)=\{\tilde{x}\in {D}_{i}:0\le {\tilde{x}}_{i}(\omega )<5y4\} is weakly open in {D}_{i}, then it is weakly open in {L}_{X}. Therefore, the correspondence (\omega ,\tilde{x})\to {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})} has weakly open lower sections. It also has open convex values and a measurable graph.
For each i\in I, {\tilde{x}}_{i}(\omega )\notin {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}, for each \omega \in [0,1] and \tilde{x}\in {L}_{X}.
All the assumptions of Theorem 1 are fulfilled, then an equilibrium exists.
For example, \tilde{{x}^{\ast}}\in {L}_{X} such that for each i\in I, {\tilde{x}}_{i}^{\ast}(\omega )=\frac{3}{4}{\omega}^{i}, \omega \in [0,1] is an equilibrium for the abstract fuzzy economy, that is, for each i\in I and μa.e.:
4 Random quasivariational inequalities
In this section, we are establishing new random quasivariational inequalities with random fuzzy mappings and random fixed point theorems. The proofs rely on the theorem of Bayesian fuzzy equilibrium existence for the Bayesian abstract fuzzy economy.
This is our first theorem.
Theorem 2 Let I be a countable or uncountable set. Let (\mathrm{\Omega},\mathcal{F},\mu ) be a complete finite separable measure space, and let Y be a separable Banach space. Suppose that the following conditions are satisfied.
For each i\in I:
(A.1)

(a)
{X}_{i}:\mathrm{\Omega}\to \mathcal{F}(Y) is such that \omega \to {({X}_{i}(\omega ))}_{{z}_{i}}:\mathrm{\Omega}\to {2}^{Y} is a nonempty convex weakly compactvalued and integrably bounded correspondence;

(b)
{X}_{i}:\mathrm{\Omega}\to \mathcal{F}(Y) is such that \omega \to {({X}_{i}(\omega ))}_{{z}_{i}}:\mathrm{\Omega}\to {2}^{Y} is {\mathcal{F}}_{i}lower measurable;
(A.2)

(a)
For each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}, {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})} is convex and has a nonempty interior in the relative norm topology of {({X}_{i}(\omega ))}_{{z}_{i}};

(b)
the correspondence (\omega ,\tilde{x})\to {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:\mathrm{\Omega}\times {L}_{X}\to {2}^{Y} has a measurable graph, i.e., \{(\omega ,\tilde{x},y)\in \mathrm{\Omega}\times {L}_{X}\times Y:y\in {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\}\in \mathcal{F}\otimes {\mathit{\text{\xdf}}}_{w}({L}_{X})\otimes \mathit{\text{\xdf}}(Y), where {\mathit{\text{\xdf}}}_{w}({L}_{X}) is the Borel σalgebra for the weak topology on {L}_{X} and \mathit{\text{\xdf}}(Y) is the Borel σalgebra for the norm topology on Y;

(c)
the correspondence (\omega ,\tilde{x})\to {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})} has weakly open lower sections, i.e., for each \omega \in \mathrm{\Omega} and for each y\in Y, the set {({({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})})}^{1}(\omega ,y)=\{\tilde{x}\in {L}_{X}:y\in {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\} is weakly open in {L}_{X};

(d)
For each \omega \in \mathrm{\Omega}, \tilde{x}\to cl{({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:{L}_{X}\to {2}^{Y} is upper semicontinuous in the sense that the set \{\tilde{x}\in {L}_{X}:cl{({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\subset V\} is weakly open in {L}_{X} for every norm open subset V of Y;
(A.3) {\psi}_{i}:\mathrm{\Omega}\times {L}_{X}\times Y\to \mathbb{R}\cup \{\mathrm{\infty},+\mathrm{\infty}\} is such that:

(a)
\tilde{x}\to {\psi}_{i}(\omega ,\tilde{x},y) is lower semicontinuous on {L}_{X} for each fixed (\omega ,y)\in \mathrm{\Omega}\times Y;

(b)
\tilde{{x}_{i}}(\omega )\notin \{y\in Y:{\psi}_{i}(\omega ,\tilde{x},y)>0\} for each fixed (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X};

(c)
for each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}, {\psi}_{i}(\omega ,\tilde{x},\cdot ) is quasiconcave;

(d)
for each \omega \in \mathrm{\Omega}, \{\tilde{x}\in {L}_{X}:{\alpha}_{i}(\omega ,\tilde{x})>0\} is weakly open in {L}_{X}, where {\alpha}_{i}:\mathrm{\Omega}\times {L}_{X}\to R is defined by {\alpha}_{i}(\omega ,\tilde{x})={sup}_{y\in {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}}{\psi}_{i}(\omega ,\tilde{x},y) for each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X};

(e)
\{(\omega ,\tilde{x}):{\alpha}_{i}(\omega ,\tilde{x})>0\}\in {\mathcal{F}}_{i}\otimes B({L}_{X}).
Then, there exists {\tilde{x}}^{\ast}\in {L}_{X} such that for every i\in I and μa.e.:

(i)
{\tilde{{x}_{i}}}^{\ast}(\omega )\in cl{({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})};

(ii)
{sup}_{y\in {({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})}}{\psi}_{i}(\omega ,{\tilde{x}}^{\ast},y)\le 0.
Proof For every i\in I, let {P}_{i}:\mathrm{\Omega}\times {S}_{X}^{1}\to \mathcal{F}(Y), and let {p}_{i}:{L}_{X}\to (0,1] such that {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}=\{y\in Y:{\psi}_{i}(\omega ,\tilde{x},y)>0\} for each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}.
We shall show that the abstract economy G=\{(\mathrm{\Omega},\mathcal{F},\mu ),{({X}_{i},{\mathcal{F}}_{i},{A}_{i},{P}_{i}{a}_{i},{p}_{i},{z}_{z})}_{i\in I}\} satisfies all hypotheses of Theorem 1.
Suppose \omega \in \mathrm{\Omega}.
According to (A.3)(a), we have that \tilde{x}\to {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}:\mathrm{\Omega}\to {2}^{Y} has open lower sections, nonempty compact values and according to (A.3)(b), \tilde{{x}_{i}}(\omega )\notin {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})} for each \tilde{x}\in {L}_{X}. Assumption (A.3)(c) implies that \tilde{x}\to {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}:\mathrm{\Omega}\to {2}^{Y} has convex values.
By the definition of {\alpha}_{i}, we note that \{\tilde{x}\in {L}_{X}:{({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\cap {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}\ne \mathrm{\varnothing}\}=\{\tilde{x}\in {L}_{X}:{\alpha}_{i}(\omega ,\tilde{x})>0\} so that \{\tilde{x}\in {L}_{X}:{({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\cap {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}\ne \mathrm{\varnothing}\} is weakly open in {L}_{X} by (A.3)(d).
According to (A.2)(b) and (A.3)(e), it follows that the correspondences (\omega ,\tilde{x})\to {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:\mathrm{\Omega}\times {L}_{X}\to {2}^{Y} and (\omega ,\tilde{x})\to {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}:\mathrm{\Omega}\times {L}_{X}\to {2}^{Y} have measurable graphs.
Thus, the Bayesian abstract fuzzy economy G=\{(\mathrm{\Omega},\mathcal{F},\mu ),{({X}_{i},{\mathcal{F}}_{i},{A}_{i},{P}_{i},{a}_{i},{b}_{i},{z}_{i})}_{i\in I}\} satisfies all hypotheses of Theorem 1. Therefore, there exists {\tilde{x}}^{\ast}\in {L}_{X} such that for every i\in I:
that is, there exists {\tilde{x}}^{\ast}\in {L}_{X} such that for every i\in I and μa.e.:

(i)
{\tilde{{x}_{i}}}^{\ast}(\omega )\in cl{({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})};

(ii)
{sup}_{y\in {({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})}}{\psi}_{i}(\omega ,{\tilde{x}}^{\ast},y)\le 0.
□
Example 2 Let \mathrm{\Omega}=[0,1], \mathcal{F}=\mathit{\text{\xdf}}([0,1],\mu ), Y=\mathbb{R}, I=\{1,2,\dots ,n\}, and for each i\in I, let {\mathcal{F}}_{i}, {X}_{i}, {a}_{i} and {D}_{i} be defined as in Example 1.
Let us define {\psi}_{i}:[0,1]\times {L}_{X}\times \mathbb{R}\to \mathbb{R} as follows: if \tilde{{x}_{i}}(\omega )\in [0,1), then,
and if \tilde{{x}_{i}}(\omega )=1, {\psi}_{i}(\omega ,\tilde{x},y)=0 for each (\omega ,\tilde{x},y)\in [0,1]\times {L}_{X}\times \mathbb{R}.
For each i\in I, let {P}_{i}:\mathrm{\Omega}\times {L}_{X}\to \mathcal{F}(Y), and let {p}_{i}:{L}_{X}\to (0,1] as in Example 1, and then, {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}=\{y\in Y:{\psi}_{i}(\omega ,\tilde{x},y)>0\}=\{\begin{array}{cc}(\frac{\tilde{{x}_{i}}(\omega )+4}{5},1)\hfill & \text{if}\tilde{x}\in {D}_{i};\hfill \\ \mathrm{\varnothing}\hfill & \text{if}\tilde{x}\notin {D}_{i}.\hfill \end{array}
From the Example 1, we have that for every i\in I and for each \omega \in [0,1], the correspondence \tilde{x}\to {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}:{L}_{X}\to {2}^{[0,1]} has weakly open lower sections, open convex values and \tilde{{x}_{i}}(\omega )\notin {({P}_{i}(\omega ,\tilde{x}))}_{{p}_{i}(\tilde{x})}.
By the definition of {\alpha}_{i}, we note that for each \omega \in \mathrm{\Omega}, {N}_{i}(\omega )=\{\tilde{x}\in {L}_{X}:{\alpha}_{i}(\omega ,\tilde{x})>0\}=\{\tilde{x}\in {D}_{i}:{\tilde{x}}_{i}(\omega )\in [0,\frac{1}{2})\} is weakly open in {L}_{X\phantom{\rule{0.25em}{0ex}}} and {N}_{i}\in {\mathcal{F}}_{i}\otimes B({L}_{X}).
In Example 1, we proved that the correspondence (\omega ,\tilde{x})\to {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:\mathrm{\Omega}\times {L}_{X}\to {2}^{[0,1]} has a measurable graph.
Therefore, there exists {\tilde{x}}^{\ast}\in {L}_{X} such that, for each i\in I and μa.e.:

(i)
{\tilde{{x}_{i}}}^{\ast}(\omega )\in cl{({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})};

(ii)
{sup}_{y\in {({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})}}{\psi}_{i}(\omega ,{\tilde{x}}^{\ast},y)\le 0.
For instance, {\tilde{x}}^{\ast} is a solution for the variational inequality, where {\tilde{x}}^{\ast} is defined by {\tilde{x}}_{i}^{\ast}(\omega )=\frac{3}{4}{\omega}^{i} for each i\in \{1,2,\dots ,n\} and \omega \in \mathrm{\Omega}.
If \mathrm{I}=1, we obtain the following corollary of Theorem 2.
Corollary 1 Let (\mathrm{\Omega},\mathcal{F},\mu ) be a complete finite separable measure space, and let Y be a separable Banach space. Suppose that the following conditions are satisfied:
(A.1)

(a)
X:\mathrm{\Omega}\to \mathcal{F}(Y) is such that \omega \to {(X(\omega ))}_{z}:\mathrm{\Omega}\to {2}^{Y} is a nonempty convex weakly compactvalued and integrably bounded correspondence;

(b)
X:\mathrm{\Omega}\to \mathcal{F}(Y) is such that \omega \to {(X(\omega ))}_{z}:\mathrm{\Omega}\to {2}^{Y} is ℱlower measurable;
(A.2)

(a)
For each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}, {(A(\omega ,\tilde{x}))}_{a(\tilde{x})} is convex and has a nonempty interior in the relative norm topology of {(X(\omega ))}_{z};

(b)
the correspondence (\omega ,\tilde{x})\to {(A(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:\mathrm{\Omega}\times {L}_{X}\to {2}^{Y} has a measurable graph, i.e., \{(\omega ,\tilde{x},y)\in \mathrm{\Omega}\times {L}_{X}\times Y:y\in {(A(\omega ,\tilde{x}))}_{a(\tilde{x})}\}\in \mathcal{F}\otimes {\mathit{\text{\xdf}}}_{w}({L}_{X})\otimes \mathit{\text{\xdf}}(Y) where {\mathit{\text{\xdf}}}_{w}({L}_{X}) is the Borel σalgebra for the weak topology on {L}_{X} and \mathit{\text{\xdf}}(Y) is the Borel σalgebra for the norm topology on Y;

(c)
the correspondence (\omega ,\tilde{x})\to {(A(\omega ,\tilde{x}))}_{a(\tilde{x})} has weakly open lower sections, i.e., for each \omega \in \mathrm{\Omega} and for each y\in Y, the set {({(A(\omega ,\tilde{x}))}_{a(\tilde{x})})}^{1}(\omega ,y)=\{\tilde{x}\in {L}_{X}:y\in {(A(\omega ,\tilde{x}))}_{a(\tilde{x})}\} is weakly open in {L}_{X};

(d)
For each \omega \in \mathrm{\Omega}, \tilde{x}\to cl{(A(\omega ,\tilde{x}))}_{a(\tilde{x})}:{L}_{X}\to {2}^{Y} is upper semicontinuous in the sense that the set \{\tilde{x}\in {L}_{X}:cl{(A(\omega ,\tilde{x}))}_{a(\tilde{x})}\subset V\} is weakly open in {L}_{X} for every norm open subset V of Y;
(A.3) \psi :\mathrm{\Omega}\times {L}_{X}\times Y\to R\cup \{\mathrm{\infty},+\mathrm{\infty}\} is such that:

(a)
\tilde{x}\to \psi (\omega ,\tilde{x},y) is lower semicontinuous on {L}_{X} for each fixed (\omega ,y)\in \mathrm{\Omega}\times Y;

(b)
\tilde{x}(\omega )\notin \{y\in Y:\psi (\omega ,\tilde{x},y)>0\} for each fixed (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X};

(c)
for each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}, \psi (\omega ,\tilde{x},\cdot ) is quasiconcave;

(d)
for each \omega \in \mathrm{\Omega}, \{\tilde{x}\in {L}_{X}:\alpha (\omega ,\tilde{x})>0\} is weakly open in {L}_{X}, where \alpha :\mathrm{\Omega}\times {L}_{X}\to R is defined by \alpha (\omega ,\tilde{x})={sup}_{y\in {(A(\omega ,\tilde{x}))}_{a(\tilde{x})}}\psi (\omega ,\tilde{x},y) for each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X};

(e)
\{(\omega ,\tilde{x}):\alpha (\omega ,\tilde{x})>0\}\in \mathcal{F}\otimes B({L}_{X});
Then, there exists {\tilde{x}}^{\ast}\in {L}_{X} such that μa.e.:

(i)
{\tilde{x}}^{\ast}(\omega )\in cl{(A(\omega ,{\tilde{x}}^{\ast}))}_{a({\tilde{x}}^{\ast})};

(ii)
{sup}_{y\in {(A(\omega ,{\tilde{x}}^{\ast}))}_{a({\tilde{x}}^{\ast})}}\psi (\omega ,{\tilde{x}}^{\ast},y)\le 0.
As a consequence of Theorem 2, we prove the following Tan and Yuantype [22] random quasivariational inequality with random fuzzy mappings.
Theorem 3 Let (\mathrm{\Omega},\mathcal{F},\mu ) be a complete finite separable measure space, and let Y be a separable Banach space. Suppose that the following conditions are satisfied:
For each i\in I:
(A.1)

(a)
{X}_{i}:\mathrm{\Omega}\to \mathcal{F}(Y) is such that \omega \to {({X}_{i}(\omega ))}_{{z}_{i}}:\mathrm{\Omega}\to {2}^{Y} is a nonempty convex weakly compactvalued and integrably bounded correspondence;

(b)
{X}_{i}:\mathrm{\Omega}\to \mathcal{F}(Y) is such that \omega \to {({X}_{i}(\omega ))}_{{z}_{i}}:\mathrm{\Omega}\to {2}^{Y} is {\mathcal{F}}_{i}lower measurable;
A.2)

(a)
For each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}, {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})} is convex and has a nonempty interior in the relative norm topology of {({X}_{i}(\omega ))}_{{z}_{i}};

(b)
the correspondence (\omega ,\tilde{x})\to {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:\mathrm{\Omega}\times {L}_{X}\to {2}^{Y} has a measurable graph, i.e., \{(\omega ,\tilde{x},y)\in \mathrm{\Omega}\times {L}_{X}\times Y:y\in {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\}\in \mathcal{F}\otimes {\mathit{\text{\xdf}}}_{w}({L}_{X})\otimes \mathit{\text{\xdf}}(Y) where {\mathit{\text{\xdf}}}_{w}({L}_{X}) is the Borel σalgebra for the weak topology on {L}_{X}and \mathit{\text{\xdf}}(Y) is the Borel σalgebra for the norm topology on Y;

(c)
the correspondence (\omega ,\tilde{x})\to {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})} has weakly open lower sections, i.e., for each \omega \in \mathrm{\Omega} and for each y\in Y, the set {({({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})})}^{1}(\omega ,y)=\{\tilde{x}\in {L}_{X}:y\in {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\} is weakly open in {L}_{X};

(d)
For each \omega \in \mathrm{\Omega}, \tilde{x}\to cl{({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:{L}_{X}\to {2}^{Y} is upper semicontinuous in the sense that the set \{\tilde{x}\in {L}_{X}:cl{({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\subset V\} is weakly open in {L}_{X} for every norm open subset V of Y;
(A.3) {G}_{i}:\mathrm{\Omega}\times Y\to \mathcal{F}({Y}^{\mathrm{\prime}}) and {g}_{i}:Y\to (0,1] are such that

(a)
for each \omega \in \mathrm{\Omega}, y\to {({G}_{i}(\omega ,y))}_{{g}_{i}(y)}:Y\to {2}^{{Y}^{\mathrm{\prime}}} is monotone (that is Re\u3008uv,yx\u3009\ge 0 for all u\in {({G}_{i}(\omega ,y))}_{{g}_{i}(y)} and v\in {({G}_{i}(\omega ,x))}_{{g}_{i}(x)} and x,y\in Y) with nonempty values;

(b)
for each \omega \in \mathrm{\Omega}, y\to {({G}_{i}(\omega ,y))}_{{g}_{i}(y)}:L\cap Y\to {2}^{{Y}^{\mathrm{\prime}}} is lower semicontinuous from the relative topology of Y into the weak^{∗}topology \sigma ({Y}^{\mathrm{\prime}},Y) of {Y}^{\mathrm{\prime}} for each onedimensional flat L\subset Y;
(A.4)

(a)
for each fixed \omega \in \mathrm{\Omega}, the set \{\tilde{x}\in {S}_{X}^{1}:{sup}_{y\in {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}}[{sup}_{u\in {({G}_{i}(\omega ,y))}_{{g}_{i}(y)}}Re\u3008u,\tilde{x}y\u3009]>0\}\phantom{\rule{0.25em}{0ex}}\mathit{\text{is weakly open in}}\phantom{\rule{0.25em}{0ex}}{L}_{X};

(b)
\{(\omega ,\tilde{x}):{sup}_{u\in {({G}_{i}(\omega ,y))}_{{g}_{i}(y)}}Re\u3008u,\tilde{{x}_{i}}(\omega )y\u3009>0\}\in \mathcal{F}\otimes B({L}_{X}).
Then, there exists {\tilde{x}}^{\ast}\in {L}_{X} such that for every i\in I and μa.e.:

(i)
{\tilde{{x}_{i}}}^{\ast}(\omega )\in cl{({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})};

(ii)
{sup}_{u\in {({G}_{i}(\omega ,{\tilde{x}}^{\ast}(\omega )))}_{{g}_{i}({\tilde{x}}^{\ast}(\omega ))}}Re\u3008u,{\tilde{{x}_{i}}}^{\ast}(\omega )y\u3009\le 0 for all y\in ({({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})}).
Proof Let us define {\psi}_{i}:\mathrm{\Omega}\times {L}_{X}\times Y\to R\cup \{\mathrm{\infty},+\mathrm{\infty}\} by
We have that \tilde{x}\to {\psi}_{i}(\omega ,\tilde{x},y) is lower semicontinuous on {L}_{X} for each fixed (\omega ,y)\in \mathrm{\Omega}\times Y and \tilde{{x}_{i}}(\omega )\notin \{y\in Y:{\psi}_{i}(\omega ,\tilde{x},y)>0\} for each fixed (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}.
We also know that for each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}, {\psi}_{i}(\omega ,\tilde{x},\cdot ) is concave. This fact is a consequence of assumption (A.3)(a).
All the hypotheses of Theorem 2 are satisfied. According to Theorem 2, there exists {\tilde{x}}^{\ast}\in {L}_{X} such that {\tilde{{x}_{i}}}^{\ast}(\omega )\in cl{({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})} for every i\in I and

(1)
{sup}_{y\in {({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})}}{sup}_{u\in {({G}_{i}(\omega ,y))}_{{g}_{i}(y)}}[Re\u3008u,{\tilde{{x}_{i}}}^{\ast}(\omega )y\u3009]\le 0 for every i\in I.
Finally, we will prove that {sup}_{y\in {({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})}}{sup}_{u\in {({G}_{i}(\omega ,{\tilde{x}}^{\ast}(\omega )))}_{{g}_{i}({\tilde{x}}^{\ast}(\omega ))}}[Re\u3008u,{\tilde{{x}_{i}}}^{\ast}(\omega )y\u3009]\le 0 for every i\in I.
In order to do that, let us consider i\in I and the fixed point \omega \in \mathrm{\Omega}.
Let y\in {({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})}, \lambda \in [0,1] and {z}_{\lambda}^{i}(\omega ):=\lambda y+(1\lambda ){\tilde{{x}_{i}}}^{\ast}(\omega ). According to assumption (A.2)(a), {z}_{\lambda}^{i}(\omega )\in {A}_{i}(\omega ,{\tilde{x}}^{\ast}).
According to (1), we have {sup}_{u\in {({G}_{i}(\omega ,{z}_{\lambda}^{i}(\omega )))}_{{g}_{i}({z}_{\lambda}^{i}(\omega ))}}[Re\u3008u,{\tilde{{x}_{i}}}^{\ast}(\omega ){z}_{\lambda}(\omega )\u3009]\le 0 for each \lambda \in [0,1].
Therefore, for each \lambda \in [0,1], we have that
It follows that for each \lambda \in [0,1],

(2)
{sup}_{u\in {({G}_{i}(\omega ,{z}_{\lambda}^{i}(\omega )))}_{{g}_{i}({z}_{\lambda}^{i}(\omega ))}}[Re\u3008u,{\tilde{{x}_{i}}}^{\ast}(\omega )y\u3009]\le 0.
Now, we are using the lower semicontinuity of y\to {({G}_{i}(\omega ,y))}_{{g}_{i}(y)}:L\cap Y\to {2}^{{Y}^{\mathrm{\prime}}} in order to show the conclusion. For each {z}_{0}\in {({G}_{i}(\omega ,{\tilde{{x}_{i}}}^{\ast}(\omega )))}_{{g}_{i}({\tilde{{x}_{i}}}^{\ast}(\omega ))} and e>0 let us consider {U}_{{z}_{0}}^{i}, the neighborhood of {z}_{0} in the topology \sigma ({Y}^{\mathrm{\prime}},Y), defined by {U}_{{z}_{0}}^{i}:=\{z\in {Y}^{\mathrm{\prime}}:Re\u3008{z}_{0}z,{\tilde{{x}_{i}}}^{\ast}(\omega )y\u3009<e\}. As y\to {({G}_{i}(\omega ,y))}_{{g}_{i}(y)}:L\cap Y\to {2}^{{Y}^{\mathrm{\prime}}} is lower semicontinuous, where L=\{{z}_{\lambda}^{i}(\omega ):\lambda \in [0,1]\} and {U}_{{z}_{0}}^{i}\cap {({G}_{i}(\omega ,{\tilde{{x}_{i}}}^{\ast}(\omega )))}_{{g}_{i}({\tilde{{x}_{i}}}^{\ast}(\omega ))}\ne \mathrm{\varnothing}, there exists a nonempty neighborhood N({\tilde{{x}_{i}}}^{\ast}(\omega )) of {\tilde{{x}_{i}}}^{\ast}(\omega ) in L such that for each z\in N({\tilde{{x}_{i}}}^{\ast}(\omega )), we have that {U}_{{z}_{0}}^{i}\cap {({G}_{i}(\omega ,z))}_{{g}_{i}(z)}\ne \mathrm{\varnothing}. Then there exists \delta \in (0,1], t\in (0,\delta ) and u\in {({G}_{i}(\omega ,{z}_{\lambda}^{i}(\omega )))}_{{g}_{i}({z}_{\lambda}^{i}(\omega ))}\cap {U}_{{z}_{0}}^{i}\ne \mathrm{\varnothing} such that Re\u3008{z}_{0}u,{\tilde{{x}_{i}}}^{\ast}(\omega )y\u3009<e. Therefore, Re\u3008{z}_{0},{\tilde{{x}_{i}}}^{\ast}(\omega )y\u3009<Re\u3008{u}_{i},{\tilde{{x}_{i}}}^{\ast}(\omega )y\u3009+e.
It follows that
The last inequality comes from (2). Since e>0 and {z}_{0}\in {({G}_{i}(\omega ,{\tilde{{x}_{i}}}^{\ast}(\omega )))}_{{g}_{i}({\tilde{{x}_{i}}}^{\ast}(\omega ))} have been chosen arbitrarily, the next relation holds:
Hence, for each i\in I, we have that {sup}_{u\in {({G}_{i}(\omega ,{\tilde{x}}^{\ast}(\omega )))}_{{g}_{i}({\tilde{x}}^{\ast}(\omega ))}}[Re\u3008{z}_{0},{\tilde{{x}_{i}}}^{\ast}(\omega )y\u3009]\le 0 for every y\in cl{({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})}. □
Example 3 Let (\mathrm{\Omega},\mathcal{F},\mu ) be the measure space, where \mathrm{\Omega}=[0,1], \mathcal{F}=\mathit{\text{\xdf}}([0,1]) is the σalgebra of the Borel measurable subsets in [0,1], and μ is the Lebesgue measure.
Let Y=\mathbb{R} and I=\{1,2,\dots ,n\}.
For each i\in I, let us define the following
The correspondence {X}_{i} is as in Example 1, that is, {({X}_{i}(\omega ))}_{{z}_{i}}=[0,1] for each \omega \in [0,1].
For each \omega \in [0,1], {({X}_{i}(\omega ))}_{{z}_{i}} is a nonempty convex weakly compactly valued and integrably bounded correspondence. It is also {\mathcal{F}}_{i}lower measurable.
Let {L}_{{X}_{i}}=\{{x}_{i}\in {S}_{{({X}_{i}(\cdot ))}_{{z}_{i}}}:{x}_{i}\text{is}{\mathcal{F}}_{i}\text{measurable}\}, and let {L}_{X}={\prod}_{i\in I}{L}_{{X}_{i}}.
Let us define {D}_{i}={\prod}_{j\ne i}{L}_{{X}_{j}}\times \{{\tilde{x}}_{i}:[0,1]\to [0,1],{\tilde{x}}_{i}(\omega )={k}_{{\tilde{x}}_{i}},\omega \in [0,1],{k}_{{\tilde{x}}_{i}}\in [0,1]\}. {D}_{i} is weakly closed in {L}_{X}.
The random fuzzy constraint function {a}_{i}:{L}_{X}\to (0,1] is defined by
For each \omega \in [0,1], the random fuzzy constraint mapping of agent i, {A}_{i}(\omega ,\cdot ):{L}_{X}\to \mathcal{F}(\mathbb{R}) is defined by
Then, the correspondence (\omega ,\tilde{x})\to {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:[0,1]\times {L}_{X}\to {2}^{[0,1]} is defined by
For each \omega \in [0,1], it has weakly open lower sections in {L}_{X}, and it has a measurable graph.
For each (\omega ,\tilde{x})\in [0,1]\times {L}_{X}, {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})} is convex and with nonempty interior in [0,1].
For each \omega \in [0,1], the correspondence \tilde{x}\to cl{({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:{L}_{X}\to {2}^{[0,1]} is upper semicontinuous and nonemptyvalued.
For each \omega \in [0,1], let {G}_{i}(\omega ,\cdot ):\mathbb{R}\to \mathcal{F}(\mathbb{R}), and let {g}_{i}:\mathbb{R}\to (0,1] be such that
Then, for each \omega \in [0,1], {({G}_{i}(\omega ,\cdot ))}_{{g}_{i}(\cdot )}:\mathbb{R}\to {2}^{\mathbb{R}} is defined by
For each \omega \in [0,1], {({G}_{i}(\omega ,\cdot ))}_{{g}_{i}(\cdot )}:\mathbb{R}\to {2}^{\mathbb{R}} is monotone with nonempty values and lower semicontinuous.
For each fixed \omega \in \mathrm{\Omega}, let {m}_{i}(\omega )={sup}_{y\in {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}}[{sup}_{u\in {({G}_{i}(\omega ,y))}_{{g}_{i}(y)}}u(\tilde{{x}_{i}}(\omega )y)].
If \tilde{{x}_{i}}(\omega )\ge \frac{9}{10}, {m}_{i}(\omega )>0.
If \tilde{{x}_{i}}(\omega )<\frac{9}{10}, {m}_{i}(\omega )=\{\begin{array}{cc}{sup}_{y\in (\frac{1}{2},\frac{9}{10})}(y+1)(\tilde{{x}_{i}}(\omega )y)\hfill & \text{if}\frac{1}{2}y\tilde{{x}_{i}}(\omega );\hfill \\ 0\hfill & \text{if}y\in [\tilde{{x}_{i}}(\omega ),\frac{9}{10})\hfill \end{array}
Therefore, if \frac{1}{2}<\tilde{{x}_{i}}(\omega )<\frac{9}{10}, {m}_{i}(\omega )={sup}_{y\in (\frac{1}{2},\tilde{{x}_{i}}(\omega ))}(y+1)(\tilde{{x}_{i}}(\omega )y)>0 and if 0<\tilde{{x}_{i}}(\omega )\le \frac{1}{2}, {m}_{i}(\omega )=0.
Consequently, {m}_{i}(\omega )>0 for each \tilde{{x}_{i}}(\omega )\in (\frac{1}{2},1].
Then, for each \omega \in [0,1], the set
is weakly open in {L}_{X} and {M}_{i}\in \mathcal{F}\otimes B({L}_{X}).
There exists {\tilde{x}}^{\ast}\in {L}_{X} such that, for each i\in I, {\tilde{{x}_{i}}}^{\ast}(\omega )=0 for each \omega \in [0,1] and μa.e.

(i)
{\tilde{x}}_{i}^{\ast}(\omega )\in cl({({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})});

(ii)
{sup}_{u\in {({G}_{i}(\omega ,{\tilde{x}}^{\ast}(\omega )))}_{{g}_{i}({\tilde{x}}^{\ast}(\omega ))}}Re\u3008u,{\tilde{{x}_{i}}}^{\ast}(\omega )y\u3009\le 0 for all y\in {({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})}.
If \mathrm{I}=1, we obtain the following corollary of Theorem 3.
Corollary 2 Let (\mathrm{\Omega},\mathcal{F},\mu ) be a complete finite separable measure space, and let Y be a separable Banach space. Suppose that the following conditions are satisfied.
(A.1)

(a)
X:\mathrm{\Omega}\to \mathcal{F}(Y) is such that \omega \to {(X(\omega ))}_{z}:\mathrm{\Omega}\to {2}^{Y} is a nonempty convex weakly compactvalued and integrably bounded correspondence;

(b)
X:\mathrm{\Omega}\to \mathcal{F}(Y) is such that \omega \to {(X(\omega ))}_{z}:\mathrm{\Omega}\to {2}^{Y} is ℱlower measurable;
(A.2)

(a)
For each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}, {(A(\omega ,\tilde{x}))}_{a(\tilde{x})} is convex and has a nonempty interior in the relative norm topology of {(X(\omega ))}_{z};

(b)
the correspondence (\omega ,\tilde{x})\to {(A(\omega ,\tilde{x}))}_{a(\tilde{x})}:\mathrm{\Omega}\times {L}_{X}\to {2}^{Y} has a measurable graph, i.e., \{(\omega ,\tilde{x},y)\in \mathrm{\Omega}\times {L}_{X}\times Y:y\in {(A(\omega ,\tilde{x}))}_{a(\tilde{x})}\}\in \mathcal{F}\otimes {\mathit{\text{\xdf}}}_{w}({L}_{X})\otimes \mathit{\text{\xdf}}(Y), where {\mathit{\text{\xdf}}}_{w}({L}_{X}) is the Borel σalgebra for the weak topology on {L}_{X} and \mathit{\text{\xdf}}(Y) is the Borel σalgebra for the norm topology on Y;

(c)
the correspondence (\omega ,\tilde{x})\to {(A(\omega ,\tilde{x}))}_{a(\tilde{x})} has weakly open lower sections, i.e., for each \omega \in \mathrm{\Omega} and for each y\in Y, the set {({(A(\omega ,\tilde{x}))}_{a(\tilde{x})})}^{1}(\omega ,y)=\{\tilde{x}\in {L}_{X}:y\in {(A(\omega ,\tilde{x}))}_{a(\tilde{x})}\} is weakly open in {L}_{X};

(d)
For each \omega \in \mathrm{\Omega}, \tilde{x}\to cl{(A(\omega ,\tilde{x}))}_{a(\tilde{x})}:{L}_{X}\to {2}^{Y} is upper semicontinuous in the sense that the set \{\tilde{x}\in {L}_{X}:cl{(A(\omega ,\tilde{x}))}_{a(\tilde{x})}\subset V\} is weakly open in {L}_{X} for every norm open subset V of Y;
(A.3) G:\mathrm{\Omega}\times Y\to \mathcal{F}({Y}^{\mathrm{\prime}}) and g:Y\to (0,1] are such that:

(a)
for each \omega \in \mathrm{\Omega}, y\to {(G(\omega ,y))}_{g(y)}:Y\to {2}^{{Y}^{\mathrm{\prime}}} is monotone with nonempty values;

(b)
for each \omega \in \mathrm{\Omega}, y\to {(G(\omega ,y))}_{g(y)}:L\cap Y\to {2}^{{Y}^{\mathrm{\prime}}} is lower semicontinuous from the relative topology of Y into the weak ^{∗}topology \sigma ({Y}^{\mathrm{\prime}},Y) of {Y}^{\mathrm{\prime}} for each onedimensional flat L\subset Y;
(A.4)

(a)
for each fixed \omega \in \mathrm{\Omega}, the set \{\tilde{x}\in {S}_{X}^{1}:{sup}_{y\in {(A(\omega ,\tilde{x}))}_{a(\tilde{x})}}[{sup}_{u\in {(G(\omega ,y))}_{g(y)}}Re\u3008u,\tilde{x}y\u3009]>0\} is weakly open in {L}_{X};

(b)
\{(\omega ,\tilde{x}):{sup}_{u\in {(G(\omega ,y))}_{g(y)}}Re\u3008u,\tilde{x}y\u3009>0\}\in \mathcal{F}\otimes B({L}_{X}).
Then, there exists {\tilde{x}}^{\ast}\in {L}_{X} such that μa.e.:

(i)
{\tilde{x}}^{\ast}(\omega )\in cl{(A(\omega ,{\tilde{x}}^{\ast}))}_{a({\tilde{x}}^{\ast})};

(ii)
{sup}_{u\in {(G(\omega ,{\tilde{x}}^{\ast}(\omega )))}_{g({\tilde{x}}^{\ast}(\omega ))}}Re\u3008u,{\tilde{x}}^{\ast}(\omega )y\u3009\le 0 for all y\in ({(A(\omega ,{\tilde{x}}^{\ast}))}_{a({\tilde{x}}^{\ast})}).
We obtain the following random fixed point theorem by using a similar kind of proof as in the case of Theorem 1. This result is a generalization of Browder fixed point theorem [32].
Theorem 4 Let (\mathrm{\Omega},\mathcal{F},\mu ) be a complete finite separable measure space, and let Y be a separable Banach space. Suppose that the following conditions are satisfied.
For each i\in I:
(A.1)

(a)
{X}_{i}:\mathrm{\Omega}\to \mathcal{F}(Y) is such that \omega \to {({X}_{i}(\omega ))}_{{z}_{i}}:\mathrm{\Omega}\to {2}^{Y} is a nonempty convex weakly compactvalued and integrably bounded correspondence;

(b)
{X}_{i}:\mathrm{\Omega}\to \mathcal{F}(Y) is such that \omega \to {({X}_{i}(\omega ))}_{{z}_{i}}:\mathrm{\Omega}\to {2}^{Y} is {\mathcal{F}}_{i}lower measurable;
(A.2)

(a)
For each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}, {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})} is convex and has a nonempty interior in the relative norm topology of {({X}_{i}(\omega ))}_{z};

(b)
the correspondence (\omega ,\tilde{x})\to {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:\mathrm{\Omega}\times {L}_{X}\to {2}^{Y} has a measurable graph, i.e., \{(\omega ,\tilde{x},y)\in \mathrm{\Omega}\times {L}_{X}\times Y:y\in {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\}\in \mathcal{F}\otimes {\mathit{\text{\xdf}}}_{w}({L}_{X})\otimes \mathit{\text{\xdf}}(Y), where {\mathit{\text{\xdf}}}_{w}({L}_{X}) is the Borel σalgebra for the weak topology on {L}_{X} and \mathit{\text{\xdf}}(Y) is the Borel σalgebra for the norm topology on Y;

(c)
the correspondence (\omega ,\tilde{x})\to {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})} has weakly open lower sections, i.e., for each \omega \in \mathrm{\Omega} and for each y\in Y, the set {({({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})})}^{1}(\omega ,y)=\{\tilde{x}\in {L}_{X}:y\in {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\} is weakly open in {L}_{X};
Then, there exists {\tilde{x}}^{\ast}\in {L}_{X} such that for every i\in I and μa.e., {\tilde{{x}_{i}}}^{\ast}(\omega )\in {({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})}.
If \mathrm{I}=1, we obtain the following result.
Theorem 5 Let (\mathrm{\Omega},\mathcal{F},\mu ) be a complete finite separable measure space, and let Y be a separable Banach space. Suppose that the following conditions are satisfied.
(A.1)

(a)
X:\mathrm{\Omega}\to \mathcal{F}(Y) is such that \omega \to {(X(\omega ))}_{z}:\mathrm{\Omega}\to {2}^{Y} is a nonempty convex weakly compactvalued and integrably bounded correspondence;

(b)
X:\mathrm{\Omega}\to \mathcal{F}(Y) is such that \omega \to {(X(\omega ))}_{z}:\mathrm{\Omega}\to {2}^{Y} is ℱlower measurable;
(A.2)

(a)
For each (\omega ,\tilde{x})\in \mathrm{\Omega}\times {L}_{X}, {(A(\omega ,\tilde{x}))}_{a(\tilde{x})} is convex and has a nonempty interior in the relative norm topology of {(X(\omega ))}_{z};

(b)
the correspondence (\omega ,\tilde{x})\to {(A(\omega ,\tilde{x}))}_{a(\tilde{x})}:\mathrm{\Omega}\times {L}_{X}\to {2}^{Y} has a measurable graph, i.e., \{(\omega ,\tilde{x},y)\in \mathrm{\Omega}\times {L}_{X}\times Y:y\in {(A(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\}\in \mathcal{F}\otimes {\mathit{\text{\xdf}}}_{w}({L}_{X})\otimes \mathit{\text{\xdf}}(Y), where {\mathit{\text{\xdf}}}_{w}({L}_{X}) is the Borel σalgebra for the weak topology on {L}_{X} and \mathit{\text{\xdf}}(Y) is the Borel σalgebra for the norm topology on Y;

(c)
the correspondence (\omega ,\tilde{x})\to {(A(\omega ,\tilde{x}))}_{a(\tilde{x})} has weakly open lower sections, i.e., for each \omega \in \mathrm{\Omega} and for each y\in Y, the set {({(A(\omega ,\tilde{x}))}_{a(\tilde{x})})}^{1}(\omega ,y)=\{\tilde{x}\in {L}_{X}:y\in {(A(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}\} is weakly open in {L}_{X};
Then, there exists {\tilde{x}}^{\ast}\in {L}_{X} such that {\tilde{x}}^{\ast}(\omega )\in {(A(\omega ,{\tilde{x}}^{\ast}))}_{a({\tilde{x}}^{\ast})} μa.e.
Example 4 Let \mathrm{\Omega}=[0,1], \mathcal{F}=\mathit{\text{\xdf}}([0,1],\mu ), Y=\mathbb{R}, I=\{1,2,\dots ,n\}.
For each i\in \{1,2,\dots ,n\} let us define the following mathematical objects.
Let {X}_{i}, {L}_{{X}_{i}} and {L}_{X} be defined as in Example 1. {\mathcal{M}}_{i}=\{[0,\frac{1}{2}),[\frac{1}{2},\frac{2}{3}),\dots ,[\frac{i1}{i},1]\} and {\mathcal{F}}_{i}=\sigma ({\mathcal{M}}_{i}).
{C}_{i}=\{{\tilde{x}}_{i}:[0,1]\to [0,1]:{\tilde{x}}_{i}(\omega )=\{\begin{array}{cc}{c}_{{\tilde{x}}_{i}}\hfill & \text{if}\omega \in [\frac{i1}{i},1];\hfill \\ 0\hfill & \text{otherwise},\hfill \end{array}\text{where}{c}_{{\tilde{x}}_{i}}\in [0,1]\text{is constant}\} and {D}_{i}={\prod}_{j\ne i}{L}_{{X}_{j}}\times {C}_{i}.
We notice that if {x}_{i}\in {C}_{i}, then it is {\mathcal{F}}_{i}measurable and μintegrable, then {C}_{i}\subset {L}_{{X}_{i}} and {C}_{i} is weakly closed in {L}_{{X}_{i}}.
The random fuzzy constraint function {a}_{i}:{L}_{X}\to (0,1] is defined by
For each \omega \in [0,1], the random fuzzy constraint mapping of agent i, {A}_{i}(\omega ,\cdot ):{L}_{X}\to \mathcal{F}(\mathbb{R}) is defined by
Then, the correspondence (\omega ,\tilde{x})\to {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})}:[0,1]\times {L}_{X}\to {2}^{[0,1]} is defined by
It has weakly open lower sections in {L}_{X}, and it has a measurable graph.
For each (\omega ,\tilde{x})\in [0,1]\times {L}_{X}, {({A}_{i}(\omega ,\tilde{x}))}_{{a}_{i}(\tilde{x})} is convex and with a nonempty interior in [0,1].
There exists {\tilde{x}}^{\ast}\in {L}_{X} such that for every i\in I, {\tilde{{x}_{i}}}^{\ast}(\omega )\in {({A}_{i}(\omega ,{\tilde{x}}^{\ast}))}_{{a}_{i}({\tilde{x}}^{\ast})}. For instance, let {\tilde{x}}^{\ast} such that {\tilde{{x}_{1}}}^{\ast}(\omega )=\frac{1}{3} if \omega \in [0,1] and {\tilde{{x}_{i}}}^{\ast}(\omega )=\{\begin{array}{cc}\frac{1}{i+1}\hfill & \text{if}\omega \in [\frac{i1}{i},1];\hfill \\ 0\hfill & \text{otherwise.}\hfill \end{array} for each i\in \{2,\dots ,n\}.
Appendix
The results below have been used in the proof of our theorems. For more details and further references see the paper quoted.
Theorem 6 (Projection theorem)
Let (\mathrm{\Omega},\mathcal{F},\mu ) be a complete, finite measure space, and let Y be a complete separable metric space. If H belongs to \mathcal{F}\otimes \mathit{\text{\xdf}}(Y), its projection {\mathit{Proj}}_{\mathrm{\Omega}}(H) belongs to ℱ.
Theorem 7 (Aumann measurable selection theorem [33])
Let (\mathrm{\Omega},\mathcal{F},\mu ) be a complete finite measure space, let Y be a complete, separable metric space, and let T:\mathrm{\Omega}\to {2}^{Y} be a nonempty valued correspondence with a measurable graph, i.e., {G}_{T}\in \mathcal{F}\otimes \beta (Y). Then there is a measurable function f:\mathrm{\Omega}\to Y such that f(\omega )\in T(\omega ) μa.e.
Theorem 8 (Diestel’s theorem [34], Theorem 3.1)
Let (\mathrm{\Omega},\mathcal{F},\mu ) be a complete finite measure space, let Y be a separable Banach space, and let T:\mathrm{\Omega}\to {2}^{Y} be an integrably bounded, convex, weakly compact and a nonempty valued correspondence. Then {S}_{T}=\{x\in {L}_{1}(\mu ,Y):x(\omega )\in T(\omega )\phantom{\rule{0.25em}{0ex}}\mu \text{}\mathit{\text{a.e.}}\} is weakly compact in {L}_{1}(\mu ,Y).
Theorem 9 (Carathéodorytype selection theorem [25])
Let (\mathrm{\Omega},\mathcal{F},\mu ) be a complete measure space, let Z be a complete separable metric space, and let Y be a separable Banach space. Let X:\mathrm{\Omega}\to {2}^{Y} be a correspondence with a measurable graph, i.e., {G}_{X}\in \mathcal{F}\otimes \mathit{\text{\xdf}}(Y) and let T:\mathrm{\Omega}\times Z\to {2}^{Y} be a convexvalued correspondence (possibly empty) with a measurable graph, i.e., {G}_{T}\in \mathcal{F}\otimes \mathit{\text{\xdf}}(Z)\otimes \mathit{\text{\xdf}}(Y), where \mathit{\text{\xdf}}(Y) and \mathit{\text{\xdf}}(Z) are the Borel σalgebras of Y and Z , respectively.
Suppose that

(a)
for each \omega \in \mathrm{\Omega}, T(\omega ,x)\subset X(\omega ) for all x\in Z.

(b)
for each \omega \in \mathrm{\Omega}, T(\omega ,\cdot ) has open lower sections in Z, i.e., for each \omega \in \mathrm{\Omega} and y\in Y, {T}^{1}(\omega ,y)=\{x\in Z:y\in T(\omega ,x)\} is open in Z.

(c)
for each (\omega