Evolution of the Minimax Inequality of Ky Fan

There are quite a few generalizations or applications of the 1984 minimax inequality of Ky Fan compared with his original 1972 minimax inequality. In a certain sense, the relationship between the 1984 inequality and several hundreds of known generalizations of the original 1972 inequality has not been recognized for a long period. Hence, it would be necessary to seek such relationship. In this paper, we give several generalizations of the 1984 inequality and some known applications in order to clarify the close relationship among them. Some new types of minimax inequalities are added.


Introduction
The KKM theory is originated from the Knaster-Kuratowski-Mazurkiewicz (KKM for short) theorem of 1929 [1].Since then, it has been found a large number of results which are equivalent to the KKM theorem; see [2,3].Typical examples of the most remarkable and useful equivalent formulations are Ky Fan's KKM lemma of 1961 [4] and his minimax inequality of 1972 [5].The inequality and its various generalizations are very useful tools in various fields of mathematical sciences.
Since 1961, Ky Fan showed that the KKM theorem provides the foundation for many of the modern essential results in diverse areas of mathematical sciences.Actually, a milestone in the history of the KKM theory was erected by Fan in 1961 [4].His 1961 KKM Lemma (or the Fan-KKM theorem) extended the KKM theorem to arbitrary topological vector spaces and had been applied to various problems in his subsequent papers [5][6][7][8][9][10].
Recall that, at the beginning, the basic theorems in the KKM theory and their applications were established for convex subsets of topological vector spaces mainly by Fan in 1961Fan in -1984 [4-10] [4-10].A number of intersection theorems and their applications to various equilibrium problems followed.In our previous review [11], we recalled Fan's contributions to the KKM theory based on his celebrated 1961 KKM lemma, and introduced relatively recent applications of the lemma due to other authors in the twenty-first century.
Then, the KKM theory was extended to convex spaces by Lassonde in 1983 [12] and to -spaces (or H-spaces) by Horvath in 1983Horvath in -1993 [13-16] [13-16] and others.Since 1993, the theory has been extended to generalized convex (G-convex) spaces in a sequence of papers of the present author and others; see [2].Since 2006, the main theme of the theory has become abstract convex spaces in the sense of Park [17][18][19][20][21][22][23][24][25][26][27][28][29][30].The basic theorems in the theory have numerous applications to various equilibrium problems in nonlinear analysis and other fields.In our previous review [30], we recalled our versions of general KKM type theorems for abstract convex spaces and introduced relatively recent applications of various generalized KKM type theorems due to other authors in the twenty-first century.
While we were studying on [11,30], we noticed that there are quite a few generalizations and applications of the 1984 minimax inequality of Fan compared with his 1972 inequality.In a certain sense, the 1984 inequality is not connected with several hundreds of known generalizations of the original minimax inequality.Hence, it would be necessary to seek such relationship.In this paper, we introduce several generalizations of two minimax inequalities of Fan and some known direct applications in order to clarify the close relationship among them.

The Origin and Fan's Applications
The KKM theorem of Knaster, Kuratowski, and Mazurkiewicz of 1929 [1] was extended by Fan in 1961 [4] as follows.
The 1961 KKM Lemma of Fan.Let  be an arbitrary set in a Hausdorff topological vector space .To each  ∈ , let a closed set () in  be given such that the following two conditions are satisfied: This is usually known as the Fan-KKM lemma, the Fan-KKM theorem, or the KKMF theorem.Fan and his followers applied his KKM lemma to various problems in many fields in mathematics; see [2,29].
Five decades after the birth of the lemma, the above original form is still adopted by many authors in each year.But, it was found quite a long time ago by Lassonde [12] that the Hausdorffness is redundant.Moreover, note that  can be any convex subset of a topological vector space.
Recall that an extended real-valued function  :  → R, where  is a topological space, is lower semicontinuous For a convex set , a function  :  → R is said to be quasiconcave if { ∈  : () > } is convex for each  ∈ R.
Similarly, the upper semicontinuity (u.s.c.) and the quasiconvexity can be defined.
One of the most remarkable equivalent formulations of the KKM theorem is the minimax inequality established by Fan from his KKM lemma.The following is the original form given by Fan in 1972 [5].
The Fan Minimax Inequality.Let  be a compact convex set in a Hausdorff topological vector space.Let  be a real-valued function defined on  ×  such that (a) for each fixed  ∈ , (, ) is a lower semicontinuous function of  on ; (b) for each fixed  ∈ , (, ) is a quasiconcave function of  on .
Then, the minimax inequality In [10], Fan applied his inequality to the following: ( The inequality became a crucial tool in proving many existence problems in various fields of mathematical sciences, for example, nonlinear analysis, especially in fixed point theory, variational inequality problems, various equilibrium theory, mathematical programming, partial differential equations, game theory, impulsive control, and mathematical economics; see [21,26,31] and the references therein.
Theorem 1 (see [10]).Let  be a nonempty convex set in a Hausdorff topological vector space.Let  be a real-valued function defined on  ×  such that (a) for each fixed  ∈ , (, ) is a lower semicontinuous function of  on ; (b) for each fixed y ∈ X, (, ) is a quasi-concave function of  on ; (c) (, ) ≤ 0 for all  ∈ ; (d)  has a nonempty compact convex subset  0 such that the set { ∈  : (, ) ≤ 0    ∈  0 } is compact.
The following proof was given by Fan [10].
Theorem 1 ⇒ The Fan Minimax Inequality.Using condition (a), we see that in the case compact , condition (d) is fulfilled by taking  0 as any nonempty closed convex subset of .Thus, Theorem 1 reduces to the minimax inequality.
In [10], Theorem 1 was applied to a best approximation theorem, coincidence and fixed point theorems, a matching theorem for two closed covers of a convex set, another proof of the Brouwer fixed point theorem, and a generalization of Shapley's KKM theorem.
Recall that the original Nash equilibrium theorem was proved by the Brouwer or the Kakutani fixed point theorem; see [33,34].Later, Fan [7] proved it by applying his result on sets with convex sections.Nowadays, it is known to be one of the most important applications of the Fan minimax inequality; see [10,27].Note that, in a wide sense, the Brouwer theorem, the KKM theorem, the Kakutani theorem, the Nash theorem, Fan's theorem on sets with convex sections, the Fan inequality, the Fan-Browder fixed point theorem, and many others are mutually equivalent; see [2,3].

Evolution of the Minimax Inequality of Ky Fan
The Fan minimax inequality has been followed by a large number of generalizations and applications in the KKM theory on convex subsets of topological vector spaces, Lassonde type convex spaces, Horvath type H-spaces, generalized convex spaces due to Park, and other types of spaces.Furthermore, many authors generalized the lower semicontinuity and quasiconcavity in the inequality or replaced them by other requirements.Therefore, even for convex spaces, it is necessary to establish proper forms of the Fan minimax inequality which unify as many particular cases as possible.
Multimaps are also simply called maps.Let ⟨⟩ denote the set of all nonempty finite subsets of a set .
In this section, we show several major generalizations of the Fan minimax inequality in the chronological order.

Lassonde's Theorem.
The concept of convex sets in a topological vector space was extended to convex spaces by Lassonde in 1983 [12], and further to c-spaces by Horvath in 1983-1993 [13][14][15][16].A number of other authors also extended the concept of convexity for various purposes.Definition 2. Let  be a subset of a vector space and  a nonempty subset of .One calls (, ) a convex space if co  ⊂  and  has a topology that induces the Euclidean topology on the convex hulls of any  ∈ ⟨⟩ (see Park [35]).For a convex space (, ), a subset  of  is said to be convex if for each  ∈ ⟨⟩,  ⊂  implies co ⊂ .
If  =  is convex, then  = (, ) becomes a convex space in the sense of Lassonde [12].
A nonempty subset  of a convex space  is called a ccompact set [12] if for each finite subset  ⊂  there is a compact convex set   ⊂  such that  ∪  ⊂   .
Lassonde [12] presented a simple and unified treatment of a large variety of minimax and fixed point problems.He first noticed that the Hausdorffness in the 1961 Fan-KKM lemma is redundant.More specifically, he gave several KKM type theorems for convex spaces (, ) and proposed a systematic development of the method based on the KKM theorem; the principal topics treated by him may be listed as follows: Fixed point theory for multimaps; Minimax equalities;

Extensions of monotone sets;
Variational inequalities; Special best approximation problems.
Furthermore, assume that the following "coercivity" condition holds: (iv) there are a compact set  ⊂  and a c-compact set  ⊂  such that for each  ∈  \ , there is an  ∈  with (, ) + () > ().

Remark 4. (1)
As shown in the above proof, the Hausdorffness in the 1984 inequality is essential.However, it is redundant in the original 1972 minimax inequality.

Chang Type Coercivity.
In 1989, Chang [37] obtained a KKM theorem with a coercivity condition which eliminated the concept of c-compact sets.From a Fan-Browder type fixed point theorem equivalent to her KKM theorem, we noticed that the following holds [38].
Then, there exists a point  0 ∈  such that Moreover, the set of all such solutions  0 is a compact subset of .
Actually, Theorem 5 is equivalent to the following.(iv) There exist a nonempty compact subset  of  and, for each finite subset  of , a compact convex subset Therefore, all of the requirements of Theorem 5 are satisfied.Therefore, the conclusion of Theorem 3 follows from Theorem 5.

For H-Spaces.
In this subsection, we follow [39].Definition 6.A triple (, ; Γ) is called an H-space if  is a topological space,  is a nonempty subset of , and Γ = {Γ  } is a family of contractible (or, more generally, -connected) subsets of  indexed by  ∈ ⟨⟩ such that Γ  ⊂ Γ  whenever  ⊂  ∈ ⟨⟩.
If  = , we denote (; Γ) instead of (, ; Γ), which is called a -space by Horvath [13][14][15][16] or an H-space by Bardaro and Ceppitelli [40].Horvath noted that a torus, the Möbius band, or the Klein bottle can be regarded as c-spaces and are examples of compact c-spaces without having the fixed point property.
In the frame of H-spaces, we obtained several generalized minimax inequalities.The following is one of them in [39, Theorem 7].
Suppose that there exists a nonempty compact subset  of  such that either (i) there exists an  ∈ ⟨⟩ such that (ii) or for each  ∈ ⟨⟩, there exists a compact H-subspace   of  containing  such that for each  ∈   \  there exists an  ∈   ∩  satisfying (, ) > .
Note that Theorem 7 extends all of Theorems 1-5, 3  , and 5  and that Theorem 7 with the coercivity condition (i) generalizes the original minimax inequality of Fan.
For a G-convex space (, ; Γ), a subset  of  is said to be G-convex with respect some   ⊂  if for each  ∈ ⟨  ⟩, we have Γ  ⊂ .
There are lots of examples of G-convex spaces; see [3,41] and the references therein.So, the KKM theory was extended to the study of KKM maps on G-convex spaces.
The following minimax inequality for G-convex spaces originates from [41,42].Theorem 9. Let (, ; Γ) be a G-convex space, let  : × → R and  :  ×  → R be functions, and let  ∈ R such that Suppose that there exists a nonempty compact subset  of  such that either (i) there exists an  ∈ ⟨⟩ such that (ii) or for each  ∈ ⟨⟩, there exists a compact G-convex subset   of  relative to some   ⊂  such that  ⊂   , and for each  ∈   \  there exists a  ∈   satisfying (, ) > .
Then, there exists a ŷ ∈  such that Theorem 9 ⇒ Theorem 7. Any H-space is a G-convex space.
There are lots of examples of   -spaces; see [3,17,43,44] and the references therein.So, the KKM theory was extended to the study of KKM maps on   -spaces.
The following minimax inequality for   -spaces is new.
A subset  of  is called a Γ-convex subset of (, ; Γ) relative to   if for any  ∈ ⟨  ⟩, we have Γ  ⊂ ; that is, co Γ   ⊂ .
Definition 13.Let (, ; Γ) be an abstract convex space and  be a topological space.For a multimap  :  ⊸  with nonempty values, if a multimap  :  ⊸  satisfies then  is called a KKM map with respect to .A KKM map  :  ⊸  is a KKM map with respect to the identity map 1  .A multimap  : ⊸ is called a KC-map (resp., a KO-map) if, for any closed-valued (resp., open-valued) KKM map  : ⊸ with respect to , the family {()} ∈ has the finite intersection property.In this case, we denote  ∈ KC(, , ) (resp.,  ∈ KO(, , )).Definition 14.The partial KKM principle for an abstract convex space (, ; Γ) is the statement 1  ∈ KC(, , ); that is, for any closed-valued KKM map  :  ⊸ , the family {()} ∈ has the finite intersection property.The KKM principle is the statement that the same property also holds for any open-valued KKM map.
An abstract convex space is called a (partial) KKM space if it satisfies the (partial) KKM principle, respectively.
In our recent works [3,18,19], we studied the elements or foundations of the KKM theory on abstract convex spaces and noticed that many important results therein are related to the partial KKM principle.
Example 15.We give known examples of partial KKM spaces; see [3] and the references therein.
(2) A connected linearly ordered space (, ≤) can be made into a KKM space.
(6) A B-space due to Briec and Horvath is a KKM space.
The following whole intersection property for the mapvalues of a KKM map is a standard form of the KKM type theorems [20,23,24].
Further, if (3) ⋂ ∈ () is compact for some  ∈ ⟨⟩, then one has Since all of the spaces in Section 3 are   -spaces and hence KKM spaces, Theorem A can be applied to them, that is, Theorem 1-11 for the case (3).
Consider the following related four conditions for a map  :  ⊸ .(d)  is closed valued.
(2) We may assume that  is closed.Then, the closure notations in each coercivity condition can be eliminated.
Proof of Theorem 17.By Lemma,  is a KKM map.Therefore, all the requirements of Theorem B are satisfied.Hence, (a) follows from Theorem B. Note that (b) is a simple consequence of (a).
(2) In condition (1), transfer closedness can be replaced by mere closedness.For a long period, instead of closedness, some authors adopted the so-called compactly closedness, transfer compactly closedness, or even by the finitely closedness when  is a convex subset of a t.v.s.Moreover, (1) holds whenever  is l.s.c.This also can be replaced by compactly l.s.c., transfer compactly l.s.c, or finitely l.s.c.This kind of terminology is not essential and generalizes nothing important.

Remark 20. (1)
The above proof shows that Theorem 17 subsumes all of Theorems 1-11 and numerous variants or particular forms of them appeared in previous works.
Then there exists a point  * ∈  such that (,  * ) ≤  for all  ∈ .
Lin and Tian [31] proved Theorem 21 by applying the partition of unity argument [this is why Hausdorffness is assumed] and the Brouwer fixed point theorem.Moreover, they showed that Theorem 21 is equivalent to the Fan KKM lemma (where Hausdorffness is redundant).Note that Theorem 21 improves the 1984 inequality of Fan.
Here we introduce a new usage of   -spaces and a new minimax inequality.

5. 1 .
Generalizations of Quasiconcavity.The compactness, convexity, lower semicontinuity, and quasiconcavity in the inequality are extended or modified by a large number of authors.For example, the quasicocavity is extended to -DQCV by Zhou and Chen in 1988.Further, Lin and Tian in 1993[31, Theorem 3]  defined -DQCV in slightly more general form: