In this paper we prove two general minimax theorems which gen-
eralize famous classical saddle point theorems of M. Sion [6] and J. von Neu-
mann. Our theorems also include some results of S. Park [3]-[5]. Results
of this type have many applications in the Game theory, because they gives
existence of solution of zero sums games

Aranđelović, Ivan D.

Mišić, Vesna

machinery

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTEISSN (p) 2303-4874, ISSN (o) 2303-4955www.imvibl.org /JOURNALS / BULLETINBull. Int. Math. Virtual Inst., Vol. 11(3)(2021), 413-418DOI: 10.7251/BIMVI2103413MFormerBULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKAISSN 0354-5792 (o), ISSN 1986-521X (p)EXTENSION OFTWO MINIMAX THEOREMS OF S. PARKVesna Mǐsić and Ivan D. ArandelovićAbstract. In this paper we prove two general minimax theorems which gen-eralize famous classical saddle point theorems of M. Sion [6] and J. von Neu-mann. Our theorems also include some results of S. Park [3]-[5]. Resultsof this type have many applications in the Game theory, because they givesexistence of solution of zero sums games.1. IntroductionToday KKM theory, which investigated generalizations of KKM principle andtheirs applications, is an important branch of nonlinear functional analysis (see [2],[7] and [8]).Many definitions of convexity on topological spaces without algebraic structuresexists. For convex fixed point theory and KKM theory, the most important ofthem are definition of Abstract convex spaces with KKM structure (shortly KKMspaces) introduced by S. Park (for details and references see [4], [5]). KKM spacesinclude earlier considered classes of topological spaces defined by of H. Komiya(Convex topological spaces), M. Lassonde (Convex spaces), C. Horvath (Pseudoconvex spaces and C - spaces), C. Bardaro, R. Ceppitelli (H - spaces), S. Park and H.Kim (Generalized convex spaces), H. Ben-El-Mechaiekh, S. Chebbi, M. Florenzanoand J.-V. Linares (L - spaces), J. Huang (spaces which satisfies property H) andX. P. Ding (FC - spaces).In this paper we prove two general minimax theorems which generalize famousclassical saddle point theorems of M. Sion [6] and J. von Neumann. Our theorems2010 Mathematics Subject Classification. 49K35, 49J35.Key words and phrases. Minimax theorem, saddle point, KKM space.Communicated by Ozgur EGE, Izmir, Turkey.413Extensions of two minimax theorems of S. Park 414also include some results of S. Park [3]-[5]. Results of this type have many appli-cations in the Game theory, because they give existence of solution of zero sumsgames.2. Preliminary notesLet X and Y be non-empty sets; we denote by 2X family of all non-empty sub-sets of X, F(X) a family of all non-empty finite subsets of X and P(X) a family ofall subsets of X. A set-valued function G from X into Y is a map G : X −→ P(Y ).We denote such set-valued function by G : X ( Y. A multifunction G from X intoY is a map G : X −→ 2Y i.e. multifunction is a set-valued function with non-emptyvalues. Let G : X ( Y be a set-valued function. We define G−1 : Ux∈XG(x) ( Xby G−1(y) = {x ∈ X : y ∈ G(x)}. If B ⊆ Y then set {x ∈ X : G(x) ∩ B ̸= ∅} isdenoted by G−1(B). Let X be a non-empty set and let F : X ( X be a set-valuedfunction. x0 ∈ X is a fixed point of set-valued function F if and only if x0 ∈ F (x0).Since statements x ∈ F (x) and x ∈ F−1(x) are equivalent, set-valued functions Fand F−1 have the same fixed points.Let S be a linearly ordered set. If its every subset has a least upper bound,then S is complete linearly ordered .Let S be a complete linearly ordered set. f : X −→ S is upper (lower) semi-continuous function on X if and only if set {x ∈ X : f(x) < r}({x ∈ X : f(x) > r})is open for each r ∈ S. If X is the compact topological space and f : X −→ Supper (lower) semi-continuous function on X, then f has maximum (minimum) onX.Let A be an arbitrary family of upper (lower) semi-continuous function de-fined on compact topological space X and let g be a function defined by g(x) =inff∈A f(x)(g(x) = supf∈A f(x)) for any x ∈ X. If g is finite on X then g is lower(upper) semi-continuous.A family of sets has the finite intersection property if and only if the intersec-tion of each its finite subfamily is non-empty.An abstract convex space (E,D; Γ) consists of a non-empty set E, a non-emptyset D and a multifunction Γ : F(D) ( X.Let (E,D; Γ) be an abstract convex space. For any A ⊆ D, the Γ-convex hullof A is denoted and defined byconvΓ(A) =∪B⊆AΓ(B).A subset X ⊆ E is called a Γ-convex subset of (E,D; Γ) relative to D′ ⊆ D, if forany A ⊆ D′, we have Γ(A) ⊆ X, that is convΓ(D′) ⊆ X.When D ⊆ E, the space is denoted by (E ⊇ D; Γ). In such case, a subset X ofE is said to be Γ-convex ifconvΓ(D ∩X) ⊆ X;in other words, X is Γ-convex relative to D ∩X.In case E = D, let (E; Γ) := (E,E; Γ).415 V. Mii and I. AraneloviLet (E,D; Γ) be an abstract convex space and Z be an arbitrary set, F : E ( Zand G : D ( Z be multifunctions which satisfiesF (Γ(A)) ⊂ G(A) :=∪y∈AG(y).Then G is called a KKM map with respect to F. A KKM map G : D ( E is aKKM map with respect to identity map IE .A multifunction F : E ( Z is said to have KKM property and called a K-mapif, for any KKM map G : D ( Z with respect to F, the family {G(y)} has thefinite intersection property. We denoteK(E,Z) := {F ( Z : F is a K −map}.When Z is a topological space then F : E ( Z is KC-map (KO-map) if, forany close-valued (open-valued) KKM map G : D ( Z with respect to F , thefamily {G(y)} has the finite intersection property. Sets KC(E,Z) and KO(E,Z)are defined similarly as K(E,Z). We have:K(E,Z) ⊆ KC(E,Z)∩KO(E,Z).Let (E,D; Γ) be an abstract convex topological space (this mean that E istopological space and (E,D; Γ) is abstract convex space) . The KKM principle isstatementI ∈ KC(E,E)∩KO(E,E).KKM space is an abstract convex topological space satisfying the KKM prin-ciple.Let S be a complete linearly ordered set, (E,D; Γ) an abstract convex spaceand f : X −→ S. f is said to be quasi-concave (quasi-convex) if for every t ∈ S set{x : f(x) > t} ({x : f(x) < t}) is convex.In [4],[5] S. Park proved the following fixed point theorem which generalizesfamous Fan-Browder theorem.Theorem 2.1 ([4],[5]). Let (E,D; Γ) be a KKM space, and F : E ( E,G :E ( D set-valued functions which satisfies:(1) for each x ∈ E, convΓ(G(x)) ⊆ F (x) i.e. F (x) is Γ-convex relative to G(x);(2) E = G−1(A) for some A ⊆ F(D);(3) G has open values.Then F has a fixed point.Next result was also proved by S. Park [4],[5].Theorem 2.2 ([4],[5]). Let {(Ei, Di; Γii)}i∈I be any family of abstract convexspaces. Let E :=∏i∈I Ei and D :=∏i∈I Di. For each i ∈ I let πi : D −→ Di bethe projection. For each A ∈ D define Γ(A) =∏i∈I Γi(πi(A)). Then (E,D; Γ) isan abstract convex space.Extensions of two minimax theorems of S. Park 4163. ResultsOur first result is the following coincidence theorem which generalize earlierresults of J. von Neumann and Ky Fan.Theorem 3.1. Let X ⊆ E and Y ⊆ F be non-empty convex compact sets inabstract convex spaces (E,ΓE) and (F,ΓF ), and A,B : X ( Y set-valued functionssuch that:a) A(x) is open and B(x) non-empty ΓF -convex set for any x ∈ X;b) B−1(y) open and A−1(y) non-empty ΓE-convex set for any y ∈ Y.If (E × F,ΓE×F ) is a KKM space, where ΓE×F is product convexity defined asabove (Theorem(2.2)), then there exists x0 ∈ X such thatA(x0)∩B(x0)B ̸= ∅.Proof. From Theorem (2.2) it follows that set Z = X×Y in KKM space E×Fis non-empty, Γ-convex and compact. Set-valued multifunction T : Z −→ P(Z)defined byT (x, y) = A−1(y)×B(x)satisfy:: 1) T (x, y) is non-empty Γ-convex set for each (x, y) ∈ Z;: 2) T−1(x, y) = B−1(y)×A(x) is open set for each (x, y) ∈ Z.By Theorem (2.1) we get that there exists fixed point of set-valued function T−1. Sothere exists (x0, y0) ∈ Z such that (x0, y0) ∈ T (x0, y0). From (x0, y0) ∈ A−1(y0)×B(x0) it follows that y0 ∈ A(x0) and y0 ∈ B(x0) which implies y0 ∈ A(x0)∩B(x0).Now we shall prove our main result.Theorem 3.2. Let (E,ΓE) and (F,ΓF ) be abstract convex spaces, S a completelinearly ordered set, X ⊆ E and Y ⊆ F compact Γ-convex sets and f, g : X×Y −→S. Suppose that:1) f(x, y) 6 g(x, y) for each x ∈ X and y ∈ Y ;2) there exists a subset T ⊆ S, such that a ∈ g(X,Y ) b ∈ f(X,Y ) with a < bimplies T ∩ (a, b) ̸= ∅;3) for each x ∈ X function y −→ f(x, y) is lower semi-continuous on Y andset {y ∈ Y |g(x, y) < t} is ΓF -convex for any t ∈ T ;4) for each y ∈ Y function x −→ g(x, y) is upper semi-continuous on X andset {x ∈ X|f(x, y) > t} is ΓE-convex for any t ∈ T.If (E × F,ΓE×F ) is a KKM space, where ΓE×F is product convexity defined asabove (Theorem (2.2)), thenminy∈Ysupx∈Xf(x, y) 6 maxx∈Xinfy∈Yg(x, y).Proof. Not that y −→ supx∈X f(x, y) is lower semi-continuous on Y andx −→ infy∈Y g(x, y) is upper semi-continuous on X. Therefore, the both sides of417 V. Mii and I. Araneloviinequality exists. Ifmaxx∈Xinfy∈Yg(x, y) < miny∈Ysupx∈Xf(x, y),then by 2) it follows that there exists t ∈ T such thatmaxx∈Xinfy∈Yg(x, y) < t < miny∈Ysupx∈Xf(x, y).Let A,B : X ( Y be set valued functions defined by:A(x) = {y ∈ Y : f(x, y) > r} and B(x) = y ∈ Y : g(x, y) < r}.By using Theorem (3.1) there exists x0 ∈ X such that A(x0)∩B(x0) ̸= ∅. Thisimplies that there exists y0 ∈ Y such that r < f(x0, y0) 6 f(x0, y0) < r which is acontradiction. If x −→ f(x, y) is quasi-concave on X, for each y ∈ Y, y −→ g(x, y) is quasi-convex on Y and S = R then Theorem (3.2) reduce to S. Parks theorem ([4]-Theorem 21; [5] -Theorem 8.2.) If f = g by using Lema 3.2 we obtained general-ization of famous theorems on existence of saddle points of J. von Neumann andM. Sion [6]. It also include result of first author [1] -Theorem 3.Now we present one result which extends one another minimax result of S. Park[3] - Theorem 3.Theorem 3.3. Let (E,ΓE) and (F,ΓF ) be KKM spaces, S a complete linearlyordered set, X ⊆ E be a Γ-convex set, Y ⊆ F a compact Γ-convex set, f : X×Y −→S. Suppose that:1) there exists a subset T ⊆ S, such that a, b ∈ f(X,Y ) with a < b impliesT ∩ (a, b) ̸= ∅;2) for each x ∈ X function y −→ f(x, y) is lower semi-continuous on Y andset {y ∈ Y |f(x, y) < t} is ΓF -convex for any t ∈ T ;3) for each y ∈ Y function x −→ f(x, y) is upper semi-continuous on X andset {x ∈ X|f(x, y) > t} is ΓE-convex for any t ∈ T.If (E × F,ΓE×F ) is a KKM space, where ΓE×F is product convexity defined asabove (Theorem (2.2)), thensupx∈Xminy∈Yf(x, y) = miny∈Ysupx∈Xf(x, y).Proof. Since f(x, y) is lower semi-continuous on Y, p(x) = miny∈Y f(x, y)exists for each x ∈ X. Since q(y) = supx∈X f(x, y) is lower semi-continuous foreach y ∈ Y, there exists q(y0) = minx∈X q(y). Note thatp(x) 6 min y ∈ Y f(x, y) 6 f(x, y) 6 maxx∈Xf(x, y) = q(y),for all x ∈ X and y ∈ Y. Suppose that the equality does not hold. Sosupx∈Xp(x) < min y ∈ Y g(y).This implies that there exists r ∈ T such that:supx∈Xp(x) < r < min y ∈ Y g(y),Extensions of two minimax theorems of S. Park 418for some r ∈ R.Let A,B : X ( Y be set-valued functions defined by:A(x) = {y ∈ Y : f(x, y) > r} and B(x) = {y ∈ Y : f(x, y) < r}.By using Theorem (3.1) there exists x0 ∈ X such that A(x0)∩B(x0) ̸= ∅. Hencethere exists y0 ∈ Y such that r < f(x0, y0) < r which is a contradiction. If E and F are convex spaces then Theorem (3.3) reduce to minimax result ofS. Park [3]. It also include earlier result of first author [1] -Theorem 3.Acknowledgement. The authors thank the reviewers for their suggestions.References[1] I. Arandelović. An extension of the Sions minimax theorem. Zb. Rad. Filoz. Fak. Nǐs, Ser.Mat., 6(1)(1992), 1-3.[2] M. Lassonde. On use of KKM-multifunctions in fixed point theory and relatedtopics. J.Math. Anal. Appl., 97(1)(1983), 151-201.[3] S. Park. Minimax theorems in convex spaces. Novi Sad J. Math., 28(2)(1998), 1-8.[4] S. Park. Elements of the KKM theory on abstract convex spaces. J. Korean Math. Soc.,45(1)(2008), 1-27.[5] S. Park. Equilibrium existence theorems in KKM spaces. Nonlinear Analysis, 69(1)(2008),4352-4364.[6] M. Sion. On general minimax theorems. Pacific J. Math., 8(1)(1958), 171-176.[7] S. Singh B. Watson P. Srivastava. Fixed point theory and best approximation: The KKM-map Principle. Kluwer Academic Publishers, Dordrecht, 1997.[8] G. X. Z. Yuan. KKM theory and applications in Nonlinear Analysis. Marcel Dekker, NewYork, 1999.Received by editors 05.11.2020; Revised version 12.03.2021; Available online 22.03.2021.V. Mǐsić: University of East Sarajevo - Faculty of Transport and Traffic Engi-neering, Vojvode Mǐsića 52, 53000 Doboj, Bosnia and HerzegovinaE-mail address: vesnasmisic@yahoo.comI. D. Arandelović: Univrsity of Belgrade, Faculty of Mechanical Engineering,11000 Beograd, Kraljice Marije 16, SerbiaE-mail address: iarandjelovic@mas.bg.ac.rs

Extension of two minimax theorems of S. Park

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Extension of two minimax theorems of S. Park

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