Let I be any (finite or infinite) set of agents. A fuzzy abstract economy is de-fined as a family of order quadruples Ω = (Xα, Aα, Fα, Pα)α∈I such that for each α ∈ I, Xα is a topological space, Aα : X := Q
β∈IXβ −→ 2Xα is a constraint correspondence, Fα : X −→ 2Xα is a fuzzy constraint correspondence, and Pα : X × X −→ 2Xα is a preference correspondence. An equilibrium point of Ω is a point (ˆx, ˆy) ∈ X × X such that for each α ∈ I, ˆxα ∈ clAα(ˆx), ˆyα ∈ Fα(ˆx) and Aα(ˆx) ∩ Pα(ˆx, ˆy) = ∅. In a real market, any preference of a real agent would be unstable by the fuzziness of consumers’ behavior or market situations. Thus it is reasonable to introduce fuzzy constraint correspondences in defining an abstract economy.
For each α ∈ I, when Fα(x) = Xα for each x ∈ X, and Pα is independent of the vari-able of y, i.e., Pα : X −→ 2Xα, the above fuzzy abstract economy reduces to the following standard abstract economy: Ωs = (Xα, Aα, Pα)α∈I, in which an equilibrium point of Ωs is a point ˆx ∈ X such that for each α ∈ I, ˆxα ∈ clAα(ˆx) and Aα(ˆx) ∩ Pα(ˆx) = ∅. The purpose of this section is to obtain the equilibria for noncompact fuzzy abstract economies.
Let X, Y be two topological spaces, and T : X −→ 2Y be a set-valued mapping.
(1) T is said to be transfer open valued on X, if for each x ∈ X and y ∈ T (x), there exists some x0 ∈ X such that y ∈ intYT (x0).
(2) T is said to be transfer open inverse valued in Y , if T−1 is transfer open valued on Y ,
We first list and establish some results which we will use later.
Proposition 3.4.1. [17] Let X, Y be two topological spaces, and the set-valued mapping
S : X −→ 2Y be transfer open valued. Then (2) S is transfer open inverse valued in Y .
Then T has a continuous selection; i.e. there exists a continuous function g : X −→ Y such that g(x) ∈ T (x) for each x ∈ X mapping with closed H-convex values, then T :=Q
α∈ITα has a fixed point.
Proof. By Theorem 1.2.4, there exists a nonempty compact H-convex subset K :=
Q
β∈IKβ of X such that Tα(K) ⊆ Kα for each α ∈ I. Clearly, the restriction Tα : K −→
Kα is also upper semicontinuous and each Tα(x) is nonempty compact and H-convex for all x ∈ K. Since Tα(K) ⊆ Kα, we obtain
T (x) =Y
α∈I
Tα(x) ⊆Y
α∈I
Kα = K
for all x ∈ K, that is, T (K) ⊆ K. Applying Proposition 3.1.4, we note that T is also upper semicontinuous with nonempty H-convex values. Thus, it follows from Theorem
3.1.2 that T has a fixed point. 2
For convenience, we denote a class of set-valued mapping by H(X, Y ) as follows:
H(X, Y ) := {T : X −→ 2Y | T is u.s.c. with nonempty closed H-convex values.}
Now, we are able to establish our main existence result of equilibria.
Theorem 3.4.4. Let Ω = (Xα, Aα, Fα, Pα)α∈I be a fuzzy abstract economy and X = Q
β∈IXβ, where I is a set of agents such that for each α ∈ I, (1) Xα is an l.c.-space with precompact polytopes,
(2) both clAα and Fα are Qα-condensing, and clAα, Fα ∈H(X, Xα) (3) for each x, y ∈ X, xα ∈ H-coP/ α(x, y),
(4) Aα∩ (H-coPα) is transfer open inverse valued in Xα,
(5) Wα := {(x, y) ∈ X × X | Aα(x) ∩ (H-coPα(x, y)) 6= ∅} is paracompact.
Then Ω has an equilibrium point (ˆx, ˆy) ∈ X × X.
Proof. Fixed an arbitrary α ∈ I. Define φα: X × X −→ 2Xα by
φα(x, y) := Aα(x) ∩ (H-coPα(x, y)) for all (x, y) ∈ X × X.
Assume that Wα 6= ∅. By (4), φα is transfer open inverse valued in Xα. Note that Wα=S
yα∈Xαφ−1α (yα), so we have Wα is open in X × X by Proposition 3.4.1.
Note that φα|Wα : Wα −→ 2Xα is transfer open inverse valued in Xα with nonempty H-convex values. Therefore, by Proposition 3.4.2, there exists a continuous function fα : Wα −→ Xα such that fα(x, y) ∈ φα|Wα(x, y) for each (x, y) ∈ Wα.
Since clAα, Fα are Qα-condensing, by Theorem 1.2.4, there exist two nonempty com-pact H-convex subsets K := Q
β∈IKβ, K0 := Q
β∈IKβ0 of X such that clAα(K) ⊆ Kα, and Fα(K0) ⊆ Kα0, respectively. Now, we define Bα : K × K0 −→ 2Kα×Kα0 by
Bα(x, y) := cl[H-cofα(x, y)] × Fα(x) , if (x, y) ∈ (K × K0) ∩ Wα, clAα(x) × Fα(x) , if (x, y) ∈ (K × K0) \ Wα.
We shall show that Bα ∈ H(K × K0, Kα × Kα0). Let Vα be an open subset of Kα× Kα0. Since for each (x, y) ∈ Wα,
cl[H-cofα(x, y)] ⊆ cl[H-coφα(x, y)] ⊆ clAα(x), the set
{(x, y) ∈ K × K0 | Bα(x, y) ⊆ Vα}
= {(x, y) ∈ (K × K0) ∩ Wα | cl[H-cofα(x, y)] × Fα(x) ⊆ Vα} [{(x, y) ∈ (K × K0) \ Wα | clAα(x) × Fα(x) ⊆ Vα}
= {(x, y) ∈ (K × K0) ∩ Wα | cl[H-cofα(x, y)] × Fα(x) ⊆ Vα} [{(x, y) ∈ K × K0 | clAα(x) × Fα(x) ⊆ Vα}.
is open in K ×K0 by Proposition 3.1.5 and upper semicontinuity of clAαand Fα. Therefore Bα ∈H(K × K0, Kα× Kα0).
Finally, for each α ∈ I, we define a set-valued mapping Tα : K × K0 −→ 2Kα×Kα0 by Tα(x, y) := Bα(x, y) , if Wα 6= ∅,
clAα(x) × Fα(x) , if Wα = ∅.
Then each Tα is Qα-condensing and Tα ∈H(K ×K0, Kα× Kα0). Hence, by Theorem 3.4.4, there exists a point (ˆx, ˆy) ∈ K × K0 such that (ˆxα, ˆyα) ∈ Tα(ˆx, ˆy). It is easy to check that for each α ∈ I, ˆxα ∈ clAα(ˆx), ˆyα ∈ Fα(ˆx) and Aα(ˆx) ∩ Pα(ˆx, ˆy) = ∅. Thus, (ˆx, ˆy) is an
equilibrium point of Ω. 2
Remark that Theorem 3.4.4 generalizes and improves [17, Theorem 2] in which they deal with the case of locally convex topological vector spaces under some compactness con-ditions. Note that if X is metrizable, then Wαis also metrizable and hence is paracompact.
Therefore the assumption (5) of Theorem 3.4.4 is automatically satisfied. Furthermore, for each α ∈ I, when Fα(x) = Xα for all x ∈ X and Pα is independent of the variable y, we obtain the following generalization of [17, Corollary 1] to H-spaces.
Corollary 3.4.5. Let Ωs = (Xα, Aα, Pα)α∈I be an abstract economy, where I is a set of agents such that for each α ∈ I,
(1) Xα is a metrizable compact H-space, (2) clAα ∈H(X, Xα),
(3) for each x ∈ X, xα ∈ H-coP/ α(x),
(4) Aα∩ (H-coPα) is transfer open inverse valued in Xα. Then Ωs has an equilibrium point.
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