The rise and fall of L-spaces, II

In 2005, Ben-El-Mechaiekh, Chebbi, and Florenzano obtained a generalization of Ky Fan's 1984 KKM theorem on the intersection of a family of closed sets on non-compact convex sets in a topological vector space. They also extended the Fan-Browder xed point theorem to multimaps on non-compact convex sets. Since then several groups of the L-space theorists introduced coercivity families and applied them to L-spaces, H-spaces, etc. In this article, we show that better forms of such works can be deduced from a general KKM theorem on abstract convex spaces in our previous works. Consequently, all of known KKM theoretic results on L-spaces related coercivity families are extended to corresponding better forms on abstract convex spaces. This article is a continuation of our [38] and a revised and extended version of [34].


Introduction
The KKM theory, rst named by the author in 1992, was originally devoted to convex subsets of topological vector spaces mainly by Ky Fan and Granas, and later to the so-called convex spaces by Lassonde, to c-spaces (or H-spaces) by Horvath and others, to generalized convex (G-convex) spaces mainly by the present author. Since 2006, we proposed new concepts of abstract convex spaces and (partial) KKM spaces which are proper generalizations of G-convex spaces and adequate to establish the KKM theory. Consequently we have obtained a large number of new results in such frame. topological vector spaces and was applied to various problems in his subsequent papers. Moreover, his lemma was extended in 1979 and 1984 [14,15] to his 1984 KKM theorem with a new coercivity (or compactness) condition for noncompact convex sets; see [21,25].
In 1993, we introduced generalized convex (G-convex) spaces (E, D; Γ) [39] and, in 1998, we derived new concept of them removing the original monotonicity restriction on them; see [21,25]. Motivated our original G-convex spaces in 1993, Ben-El-Mechaiekh, Chebbi, Florenzano, and Llinares [3,4] in 1998 introduced Lspaces (E, Γ) and claimed incorrectly that G-convex spaces are particular to their L-spaces. Since then a number of authors followed the misconception of [4] and published incorrect obsolete articles even after we established the KKM theory on abstract convex spaces in 20062010.
In 2005, Ben-El-Mechaiekh, Chebbi, and Florenzano [5] obtained a generalization of Ky Fan's 1984 KKM theorem [15] on the intersection of a family of closed sets on non-compact convex sets in a topological vector space. They also extended the Fan-Browder xed point theorem to multimaps on non-compact convex sets.
This type of studies also followed by Chebbi [7,9] and others. In 2011, Chebbi, Gourdel, and Hammami [11] introduced a generalized coercivity type condition for multimaps dened on topological spaces endowed with a generalized convex structure and extended Fan's KKM theorem.
In our previous work [34] in 2013, we showed that better forms of theorems in [5,7,8,9,11] can be deduced from a KKM theorem on abstract convex spaces in our sense [2327]. Moreover, in our recent work [38], we showed that our KKM theory on abstract convex spaces improves and extends all of typical results on L-spaces. Since such studies are beyond of L-spaces, we cordially claimed that now is the proper time to give up the useless study on L-spaces and their variants FC-spaces.
After we have published [34], we found similar articles by the L-space theorists such as Chebbi [8], Hammami [17], Gourdel-Hammami [16], Chebbi [10] and Altwaijry-Ounaies-Chebbi [1] in the chronological order. These authors introduce several coercing families and applied them to corresponding KKM theorems, Fan's matching theorems, Fan-Browder xed point theorems, minimax inequalities, etc. However, these seem to be of little use since their terminology and coercivity conditions are not practical.
In our previous work [38] in 2020, which will be called Part I, we recalled the history of relation between G-convex spaces and L-spaces, and showed that other authors' main works on L-spaces are consequences of our KKM theory on abstract convex spaces. In fact, the study on the KKM theoretic results related certain coercing families on particular types of KKM spaces like H-spaces, L-spaces, FC-spaces, etc. are not necessary, and hence, we concluded that now is the proper time to give up such useless study on L-spaces and their variants FC-spaces.
As a continuation of our [38], the present Part II is a revised and extended version of [34] and shows that some previous or new results generalize or improve corresponding ones in [1,8,10,16,17].
This article is organized as follows: Section 2 is a routine preliminary on terminology of abstract convex spaces. Here we introduce the KKM maps with respect to a multimap, KC-maps [resp. KO-maps], and (partial) KKM spaces. We add a typical general form of the KKM theorem for abstract convex spaces. In Section 3, we begin with a routine diagram showing typical subclasses of abstract convex spaces. Moreover, various subclasses and examples of abstract convex spaces are given for applications of all theorems in this article. Section 4 is to generalize the coercing families in [5, 8, 11, 16. 17]. We show that such coercing families can be unied by our condition [I]. In Sections 57, we show that better forms of main theorems in the works on L-spaces can be deduced from a KKM theorem on abstract convex spaces in the sense of [23][24][25][26][27]. Section 5 deals with the KKM type results, and Section 6 with the Fan-Browder type xed point theorems. Section 7 deals with improvements of minimax inequalities due to the L-space theorists.
Finally, in Section 8, some historical remarks related to L-spaces will be given.

Abstract convex spaces
Since 2006 we have introduced the concepts of abstract convex spaces, KKM spaces, and partial KKM spaces; see our recent works [27][28][29][30][31][32][33] and the references therein. Denition 2.1. An abstract convex space (E, D; Γ) consists of a topological space E, a nonempty set D, and a multimap Γ : D E with nonempty values Γ A := Γ(A) for A ∈ D , where D is the set of all nonempty nite subsets of D.
For any D ⊂ D, the Γ-convex hull of D is denoted and dened by Some remarks and examples on KC-maps and KO-maps can be seen in [23,24]. In this article, we need only the fact that any continuous function s : E → Z belongs to KC(E, Z). Denition 2.3. The partial KKM principle for an abstract convex space (E, D; Γ) is the statement 1 E ∈ KC(E, E); that is, for any closed-valued KKM map G : D E, the family {G(y)} y∈D has the nite intersection property. The KKM principle is the statement 1 E ∈ KC(E, E) ∩ KO(E, E); that is, the same property also holds for any open-valued KKM map.
An abstract convex space is called a (partial) KKM space if it satises the (partial) KKM principle, respectively. Example 2.1. Consider the following related four conditions for a multimap G : D Z from a set D into a topological space Z: From the denition of KC-maps, we have a whole intersection property of the Fan type under certaiǹ coercivity' conditions. The following is given in [28,[30][31][32][33]: Theorem 2.1. Let (E, D; Γ) be an abstract convex space, Z a topological space, F ∈ KC(E, Z), and G :

D
Z a multimap such that (1) G is a KKM map w.r.t. F ; and (2) there exists a nonempty compact subset K of Z such that either (i) K ⊃ {G(y) | y ∈ M } for some M ∈ D ; or (ii) for each N ∈ D , there exists a Γ-convex subset L N of E relative to some D ⊂ D such that N ⊂ D , F (L N ) is compact, and Remark 2.1. 1. The coercivity (ii) is originated from S. Y. Chang [6] in 1989.
2. Taking K instead of K, we may assume K is closed and the closure notations in (i) and (ii) can be erased.
3. In [33], we showed that a particular form of

Various subclasses and examples of abstract convex spaces
The following diagram for triples (E, D; Γ) on abstract convex spaces is well known: Example 3.1. We introduce typical subclasses of KKM spaces. Some details can be seen in [25] and the references therein.
(1) Simplex -The 1929 KKM theorem [19] was for the triple (∆ n , V ; co), where ∆ n is the standard n-simplex with vertices A = {e i } n i=0 , V the set of vertices and co : V ∆ n the convex hull operation.
(2) Convex subsets of a t.v.s. -The 1961 KKM lemma of Ky Fan (so called the KKMF theorem) [13] was for the triple (Y ⊃ X; co), where X is an arbitrary set in a topological vector space Y . Moreover, every nonempty convex subset X of a topological vector space is a convex space [in the following sense in (3)] with respect to any nonempty subset D of X, and the converse is not true.
(3) A convex space (X, D) = (X, D; Γ) is a triple where X is a subset of a vector space, D ⊂ X such that co D ⊂ X, and each Γ A is the convex hull of A ∈ D equipped with the Euclidean topology; see [25]. Note that (X, D) can be represented by (X, D; Γ) where Γ : D X is the convex hull operator. If X = D is convex, then X = (X, X) becomes a convex space in the sense of Lassonde [20].
(4) A triple (X, D; Γ) is called an H-space by Park in 1992 if X is a topological space, D a nonempty subset of X, and Γ = {Γ A } a family of contractible (or, more generally, ω-connected) subsets of X indexed by A ∈ D such that Γ A ⊂ Γ B whenever A ⊂ B ∈ D).
If D = X , we denote (X; Γ) instead of (X, X; Γ), which is called a c-space by Horvath [18] or an H-space by Bardaro and Ceppitelli in 1988.
(5) A generalized convex space or a G-convex space (X, D; Γ) is an abstract convex space such that for each A ∈ D with the cardinality |A| = n + 1, there exists a continuous function φ A : [21,25].
When X = D, a G-convex space is called an L-space; see [4].
consists of a topological space X, a nonempty set D, and a family of continuous functions φ A : ∆ n → X (that is, singular n-simplices) for A ∈ D with the cardinality |A| = n + 1. All theorems in this article can be applied to these spaces.
Let us begin with the following particular form of the condition (ii) in Theorem 2.1 with sG : D Z instead of G : D Z: [I] ([34]) Let (E, D; Γ) be an abstract convex space, G : D E a multimap, Z a topological space, and s : E → Z a continuous map such that (C) there exists a nonempty compact subset K of Z such that, for each N ∈ D , there exists a compact Γ-convex subset L N of E relative to some D ⊂ D such that N ⊂ D and Under the situation of [I], we have the following: Proof. Since L N is compact and s is continuous, s(L N ) is compact. Since L N is Γ-convex relative to some D ⊂ D such that N ⊂ D , for any A ∈ D , we have Γ(A) ⊂ L N and hence sΓ(N ) ⊂ s(L N ). Therefore, s(L N ) is sΓ-convex relative to D ⊂ D.
In 2011, Chebbi et al. [11] introduced the notion of coercing family in L-spaces for a given map as follows: [II] ([11]) Let D be an arbitrary set in an L-space (E, Γ), Z a topological space, and s : E → Z a continuous map. A family {(C a , K)} a∈E is said to be L-coercing for a map F : to the intention of the so-called L-space theorists.
Proof. Under the situation of [II], note that (E ⊃ D; Γ) is a G-convex space and hence a (partial) KKM space. Let G := s −1 F : D E and, for any N ∈ D , we have a compact Γ-convex subset L N of E containing N . Choose an x ∈ L N and let D : Motivated by [5], we dene the following: [III] Let (E, D; Γ) be an abstract convex space and Z a topological space. We say that a map F : D Z has a coercing family {(D i , K i )} i∈I if and only if (1), we have K i and D i such that, for each N ∈ D , there exists a compact Γ-convex subset L N : Since i was arbitrary, we may assume k = i and K Hence the coercivity condition [I] holds. [IV] ([5]) Consider a subset X of a Hausdor topological vector space E and a topological space Z. A family {(D i , K i )} i∈I of pairs of sets is said to be coercing for a map F : X Z if and only if: If I is a singleton, the family is called a single coercing family. Note that (E, X; co) is a G-convex space and that (ii) will be shown redundant.
Remark 4.1. In [5], it is noted that the condition (iii) holds if and only if the`dual' map Φ : In [5] [V] ([8]) Let (X, Γ) be an H-space and Y a topological space. A family {(C i , K i )} i∈I is said to be H-coercing for a correspondence F : X → Y if and only if: (i) For each i ∈ I, C i is an H-compact subset of X and K i is a compact subset of Y ; (ii) For each i, j ∈ I, there exists k ∈ I such that C i ∪ C j ⊆ C k ; (iii) For each i ∈ I, there exists k ∈ I such that: The following was adopted by Hammami [17] in 2007 and Gourdel-Hammami [16] in 2007, and originated from Chebbi-Gourdel-Hammami [11] in 2011: [VI] ([17]) Let Z be an arbitrary set of an L-space (X, Γ), Y a topological space, and s : X → Y a continuous map. A family {(C a , K)} a∈X is said to be L-coercing for a correspondence F : Z Y with respect to s if and only if: (i) K is a quasi-compact subset of Y , (ii) for each A ∈ Z , there exists a quasi-compact L-convex set D A in X containing A such that: Note that (X, Z; Γ) is a G-convex space.
Hammami [17] added: For more explanation of the L-coercivity and to see that this coercivity can't be compared to the coercivity in the sense of Ben-El-Mechaiekh, Chebbi and Florenzano in [5], see [11].
However, we note the following:  [17], and Gourdel- Hammami [16] are all ignorant of G-convex spaces. This is a proper evidence that they did not seriously read any works on G-convex spaces.

Generalization of the KKM principle
In this section, we deduce generalized better forms of KKM type theorems on the L-spaces in [1,5,7,8,10,11,16,17]  (2) the coercivity condition [I] holds for R instead of G.
Then we have K ∩ y∈D G(y) = ∅.
Proof. We apply Theorem 2.1 with F = s. Remark 5.1. From the above corollaries, we notice the following: 1. The quasi-compactly closed sets are compactly closed sets in modern usage and can be replaced by mere closed sets by adopting compactly generated extension (like k-spaces) of the original topology.
2. Our proofs are based on Theorem 2.1 and dierent from that of [5,7] We add up some more consequences of Theorem 5.1 As in [17], for any correspondence F : (a) for every x ∈ X, F (x) is strongly compactly closed, (b) for every x ∈ X, G(x) ⊂ F (x), (c) there exists a continuous function s : X → Y such that: 1. for every x ∈ X, s(x) ∈ G(x), 2. for every x ∈ X, S * (x) where S is dened by S(x) = s −1 (G(x)) is L-convex, 3. there exists an L-coercing family {(C x , K)} x∈X for F with respect to s.
Proof. By replacing the topology of X by the strongly compactly generated one, F has simply closed values. In order to apply Corollary 5.1, it suces to show that the correspondence R : X X dened by R(x) = s −1 (F (x)) is KKM. Let A ∈ X and z ∈ Γ(A), then by (c.1), s(z) ∈ G(Γ(A)). One can check that Condition (1) For every x ∈ X, F (x) is compactly closed in X.
(2) For some continuous map s : X → Y the correspondence G : X → X given by G(x) = s −1 (F (x)) is H-KKM. Proof. By switching the topology of X to the compactly generating one (as in k-spaces), all F (x) becomes simply closed in (1). Since H-spaces are KKM spaces, the closed-valued KKM map G in (2) has the nite intersection property for its values. The condition (3) is an abstract form of [IV] in Section 3. Since [IV] implies [I], The conclusion follows from Theorem 5.1.
The following [ [8], Theorem 3.2] is a simple consequences of the preceding one: Corollary 5.5. ([8]) Let (X, Γ) be an H-space and F, G : X → X two correspondences such that: (a) For every x ∈ X, G(x) is compactly closed and F (x) ⊂ G(x).
In [8] this was used to prove results on minimax inequalities for Riesz spaces. Our original G-convex space paper [38] was the origin of L-spaces, and is quoted in [8] for more details about generalized convexity. Corollary 5.6. ( [17]) Let Z be an arbitrary set in the L-space (X; Γ), Y an arbitrary topological space and F, G : Z Y two correspondences such that: (a) for every x ∈ Z, F (x) is strongly compactly closed, (b) for every x ∈ Z, G(x) ⊂ F (x), (c) there is a continuous function s : X → Y satisfying: 1. the correspondence R : Z X dened by R(x) = s −1 (F (x)) is L-KKM, 2. there exists an L-coercing family {(C a , K)} a∈X for G with respect to s, 3. for each quasi-compact L-convex set C in X: Proof. Note that (X ⊃ Z; Γ) is a G-convex space and hence an abstract convex space, and F is closed-valued by switching the topology of X to the strongly compactly generated topology (as for k-spaces). Moreover, R = s −1 F is KKM and Condition (c.2) implies the coercivity condition [I] for R = s −1 G by Proposition 4.3 Then by Theorem 5.2, we have the conclusion.
Comments on Corollary 5.6: (1) Corollary 5.6 is extremely articial and its usefulness is doubtful because of the following observations.
(2) Note that (X ⊃ Z; Γ) is an abstract convex space. More precisely it is a G-convex space, contrary to the routine claim of the L-space theorists. Traditionally L-spaces are pairs not triples; see our [38].
(3) Switching the topology of Y to compactly generated one (like k-spaces), each T (x) can be simply closed.
(4) Existence of L-coercing family is the same to [II] which implies the more reasonable ordinary coercivity  The following is rather articial KKMF theorem: Corollary 5.7. ( [11]) Let (X, Γ) be an L-space, Z a nonempty subset of X and F : Z X an L-KKM correspondence with strongly compactly closed values. Suppose that for some z ∈ Z, the correspondence F (z) is quasi-compact, then x∈Z F (x) = ∅.
Note that (X ⊃ Z; Γ) is a G-convex space.
In 2018, Altwaijry-Ounaies-Chebbi [1] claimed that the KKMF principle can be extended to L-spaces as follows: Corollary 5.8. ( [1]) Let (E, Γ) be an L-space, X a nonempty subset of E and F : X → E a set-valued map satisfying: Proof. This is a simple consequence of Theorem 2.1(i). Moreover, this is one of the simplest cases of Corollary Then the conclusion follows from Corollary 5.6 Remark 5.2. 1. The authors of [1] gave a proof using the KKM theorem and stated that this generalizes several well-known previous theorems. However, this is already known and they did not mention any of plenty of proper generalizations of this theorem.
2. Note that Theorems 5.1 and 5.2 can be applied to all subclasses of abstract convex spaces in Section 4. Recall that, for the last two decades, the L-space theorists have insisted that their L-spaces (X, Γ) generalize our generalized (G-convex) spaces (E, D; Γ); and could not recognize the existence of various types of extensions of their obsolete L-spaces such as G-convex spaces, φ A -spaces, KKM spaces, partial KKM spaces, and abstract convex spaces; see [38], where we declared the fall of L-spaces.

Generalizations of the Fan-Browder xed point theorems
In [34], we deduced the better form of the Fan-Browder xed point theorems in [5] and [11] from the KKM theorem 2.1 on abstract convex spaces. In this section, we add up more better forms of the theorems. (2) T (x) ⊃ co Γ S(x) for each x ∈ E. Suppose that there exists a nonempty compact subset K of E satisfying (3) for each N ∈ D , there exist D ⊂ D containing N and a compact Γ-convex subset L N of E relative to D such that Then either (a) S has a maximal element x 0 ∈ K, that is, S(x 0 ) = ∅; or (b) T has a xed point x 1 ∈ E, that is, x 1 ∈ T (x 1 ).
Proof. Suppose T has no xed point. Dene a map G : D E by Then G is closed-valued. Moreover, G is a KKM map. In fact, suppose on the contrary that there exists an N ∈ D such that Γ N ⊂ G(N ); that is, there exists an x ∈ Γ N such that x / ∈ G(y) for all y ∈ N . In other words, N ∈ D \ G − (x) and for all y ∈ N . Hence N ⊂ S(x) and, by (2), we have x ∈ Γ N ⊂ T (x). This is a contradiction. Note that (3) implies condition (ii) of Theorem 2.1 with E = D and s = 1 E ∈ KC(E, D, E) since (E, D; Γ) is a partial KKM space. Therefore, by Theorem 2.1, we have K ∩ y∈D G(y) = ∅.
Then we have an x 0 ∈ K and x 0 ∈ G(y) = E \ S − (y) or y / ∈ S(x 0 ) for all y ∈ D. Hence S has a maximal element x 0 ∈ K. Proof. We will use Theorem 6.1 with E = D = X, Γ = co, S = Φ, T = co(Φ). Since Φ has non-empty values, it does not have a maximal element. Now it suces to show that (iii) implies condition (3) of Theorem 6.1.
Suppose K is a compact subset of X and C is contained in a compact convex subset L of X. Let N ∈ X . Since X is a convex subset of a Hausdor topological vector space, there exists a compact convex subset L N of X containing D := L ∪ N . Note that Φ(x) ∩ D ⊃ Φ(x) ∩ C = ∅ for all x ∈ X \ K by (iii) , that is, Therefore, x / ∈ X \ K and hence x ∈ K. So condition (3) Suppose that there exists a nonempty compact subset K of E satisfying (3) for each N ∈ D , there exist D ⊂ D containing N and a compact Γ-convex subset L N of E relative to D such that Then either (a) G − s has a maximal element x 0 ∈ E, that is, G − s(x 0 ) = ∅; or (b) s −1 H has a xed point Proof. In view of Propositions 4.1 and 4.2, (Z, D; sΓ) is a partial KKM space and s(L N ) is a compact sΓ-convex subset relative to D . We apply Theorem 6.1 replacing (E, D; Γ) by (Z, D; sΓ), S := G − s and T := s −1 H.
Therefore, by Theorem6.1, the conclusion follows.  [11]) Let (X; Γ) be an L-space, Z an arbitrary topological space, s : X → Z a continuous map and S : X Z a multimap such that: (i) for each x ∈ X, S(x) is quasi-compactly open in Z; (ii) for each z ∈ Z, S −1 (z) is nonempty and L-convex; (iii) there exists an L-coercing family {(C x , K)} x∈X for the map Q(x) = Z \ S(x) with respect to s. Then there exists x 0 ∈ X such that s(x 0 ) ∈ S(x 0 ). In particular, if s is the identity map, then S has a xed point.
Proof. Put E = D = X and G = H = S in Theorem 6.2 As was noted in [5], Theorems in Sections 5 and 6 can be used to extend existing results on various equilibrium problems, solvability of complementarity problems, existence of zero on non-compact domains, and existence of equilibria for qualitative games and abstract economies. (iii) There exists an H-coercing family {(C i , K i )} i∈I for the correspondence F : X → Y dened by F (x) = Y \ S(x), ∀x ∈ X.
Then, for each continuous function s : X → Y , there exists an x 0 ∈ X such that s(x 0 ) ∈ S(x 0 ). In particular, S has a xed point.
The last conclusion should be In particular, if X = Y and s = 1 X , then S has a xed point.
Then g satises at least one of the following properties: (1) There existsŷ ∈ Y such that g(x,ŷ) / ∈ A for all x ∈ X. (2) There isx ∈ X such that g(x, s(x)) ∈ A. Corollary 6.5. ( [1]) Let X be a nonempty compact L-convex subset of an L-space (E, Γ), F, G : X X two set-valued maps satisfying the following: Then G admits a xed point, that is a pointx ∈ X such thatx ∈ G(x).
The authors of [1] gave a proof using their KKM theorem and stated that this generalizes several wellknown previous theorems. However, they did not mention any of so many proper generalizations of this theorem.
Then the authors continue: The concept of KF-majorization due to Borglin and Keiding is then easily extended to L-spaces and a result on the existence of a maximal element for such correspondence is deduced.
As an application, we prove an equilibrium existence result for qualitative games dened in an L-space and an equilibrium result for an abstract economy.
Note that such arguments were already given by many authors for much more general convexity structure than L-structures.

On general minimax inequalities and applications
It is well-known that any KKM type theorem can be reformulated equivalently to the Fan-Browder type xed point theorems, matching theorems, minimax inequalities, and so on.
In this section, we indicate that results of Chebbi in [7,9] and Gourdal-Hammami in [16] can be improved following our preceding arguments.
The following is a KKM type minimax inequality given as [ [32], Theorem 5.3]: Theorem 7.1. ([32]) Let (E, D; Γ) be a partial KKM space. Let f : E × D → R be an extended real-valued function and γ ∈ R such that (1) for each y ∈ D, {x ∈ E | f (x, y) ≤ γ} is intersectionally closed [resp. transfer closed]; (2) for each N ∈ D and x ∈ Γ N , min{f (x, y) | y ∈ N } ≤ γ; and Suppose that there exists a nonempty compact subset K of E satisfying (3) for each N ∈ D , there exist D ⊂ D containing N and a compact Γ-convex subset L N of E relative to D such that L N ∩ y∈D {x ∈ E | f (x, y) ≤ γ} ⊂ K. Corollary 7.1. (Chebbi [7]) Let X be a nonempty convex subset of a t.v.s. E, and f : X × X → R be a function satisfying (i) f is l.s.c. in the rst variable on each compact convex subsets of X; (ii) for each A ∈ X , sup x∈co A min y∈A f (x, y) ≤ 0; and (iii) the coercivity condition (IV) with X = Z and F (y) := {x ∈ X | f (x, y) ≤ 0} f or y ∈ X.
Then there exists an x 0 ∈ X such that f (x 0 , y) ≤ 0 for all y ∈ X.
Proof. Note that X can be regarded a convex space in the sense of Lassonde [20] and endowed the compactly generated extension of of its original topology. Then (i) becomes simply`f is l.s.c.' and hence, condition (1) of Theorem 7.1 is satised. Moreover, it is clear that (ii) implies (2) of Theorem 7.1. Further, (iii) implies the coercivity condition (I) in Section 4 with s = 1 E and G(y) := {x ∈ E | f (x, y) ≤ γ} for y ∈ D. Therefore, the conclusion of Corollary 7.1 follows from Theorem 7.1(a) with γ = 0.
Recall that Condition (ii) in [IV] is redundant.
Corollary 7.1 is applied to some equilibrium problems in [7] and to some quasi-variational inequalities in [9]. Note that Corollary 7.1 can be improved by adopting more general conditions [I][III] with s = 1 E and Z = E. Moreover, any interested reader can check that all results in [7] and [9] can be improved by applying Theorem 7.1 instead of Corollary 7.1.
In 2007, Gourdal-Hammami [16] applied a form of Ky Fan's matching theorem to minimax and variational inequalities. As usually some of the L-space theorists do, they listed our [38] in the references, but nothing was quoted from it.
For any subset A of R and every z ∈ R, A ≤ z denotes for all a ∈ A, a ≤ z and A ≤ z means that there exists a ∈ A such that a > z. Denition 7.1. ([16]) Let X be a topological space. A correspondence Q : X R is said to be weakly lower semi-continuous (weakly l.s.c) on X if for each p ∈ R, the set {x ∈ X | Q(x) ≤ p} is closed in X or equivalently, the set {x ∈ X | Q(x) ∩ ]p, +] = ∅} is open in X. Proposition 7.1. ([16]) If Q is a lower semi-continuous correspondence then it is weakly lower semicontinuous.
The Ky Fan minimax inequality can be extended in the following way: