New Generalized R-KKM Type Theorems in General Topological

Acta Mathematica Sinica, English Series Oct., 2007, Vol. 23, No. 10, pp. 1869–1880 Published online: Dec. 11, 2006 DOI: 10.1007/s10114-005-0876-y Http://www.ActaMath.com

New Generalized R-KKM Type Theorems in General Topological Spaces and Applications Xie Ping DING College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, P. R. China E-mail: xieping [email protected] Abstract In this paper, some new generalized R-KKM type theorems for generalized R-KKM mappings with finitely closed values and with finitely open values are established in noncompact topological spaces without any convexity structure under much weaker assumptions. As applications, some new minimax inequalities, saddle point theorem and equilibrium existence theorem for equilibrium problems with lower and upper bounds are established in general noncompact topological spaces. These theorems unify and generalize many known results in the literature. Keywords generalized R-KKM mapping, finitely closed-valued (open-valued), generalized R-KKM type theorem, minimax inequality, saddle point, equilibrium problem MR(2000) Subject Classification 49J27, 49J35, 91B50

1

Introduction

For a set X, we denote by 2X and X the family of all subsets of X and the family of all nonempty finite subsets of X, respectively. If X is a topological vector space and A is a subset of X, we denote by co(A) and clX (A) the convex hull of A and the closure of A in X, respectively. Chowdhury and Tan [1] generalized the celebrated Ky Fan’s KKM Lemma [2] as follows: Theorem 1.1 Let X be a nonempty convex subset of a topological vector space E and F : X → 2X \ {∅} be a mapping such that (i) clX (F (x0 )) is compact for some x0 ∈ X;  (ii) For each A ∈ X with x0 ∈ A and each x ∈ co(A), F (x) co(A) is closed in co(A);  (iii) For each A ∈ X, co(A) ⊂ x∈A F (x); (iv) For each A ∈ X with x0 ∈ A,          clX F (x) co(A) = F (x) co(A). Then

 x∈X

x∈co(A)

x∈co(A)

F (x) = ∅.

The Example 1 in [1, pp. 915–916] shows that Theorem 1.1 also generalizes Lemma 1 of Brezis, Nirenberg and Stampacchia [3]. Ding and Tarafdar [4] obtain a further generalization of Theorem 1.1 by relaxing the coercivity condition (i) and gave some applications for generalized variational-like inequalities. Received November 25, 2005, Accepted March 1, 2006 This project is supported by Natural Science Foundation of Sichuan Education Department of China (2003A081) and SZD0406

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Recently, Ding [5] further generalized Theorem 1.1 and Lemma 2.3 of Ding and Tarafdar [4] to G-convex spaces without any linear structure as follows: Theorem 1.2 Let X be a nonempty set and Y be a nonempty G-convex subset of a G-convex space (E, D; Γ) such that, for each M ∈ Y , G-co(M ) is compact in Y . Let F : X → 2Y be a  set-valued mapping and s : X → Y D be a single-valued mapping such that  (i) clY ( x∈A F (x)) is compact for some A ∈ X;

(ii) F is a generalized s-KKM mapping such that, for each M ∈ Y  with A ⊂ s−1 (M )  and for each x ∈ s−1 (G-co(M )), F (x) G-co(M ) is closed in G-co(M ); (iii) For each M = {y0 , . . . , yn } ∈ Y  with A ⊂ s−1 (M ),        F (x) G − co(M ) = clY

Then



x∈s−1 (G−co(M ))

x∈X

F (x)



G − co(M ).

x∈s−1 (G−co(M ))

F (x) = ∅.

Remark 1.1 Theorem 1.2 generalizes Lemma 2 of Chowdhury and Tan [1] (see Theorem 1.1), Lemma 2.3 of Ding and Tarafdar [4], Lemma 1 of Brezis, Nirenberg and Stampacchia [3], Lemma 1 of Ky Fan [2] and Theorem 3.2 of Verma [6] from topological vector space and G-Hspace to G-convex space without linear structure under weaker coercivity condition. In most of the known KKM type theorems, the convexity assumptions play a crucial role which strictly restricts the applicable area of these KKM type theorems. Hence Deng and Xia [7], Ding [8], and Ding, et al. [9] established some generalized R-KKM type theorems for generalized R-KKM mappings with compactly closed values and with compactly open values in general topological spaces without any convexity structure, respectively. These theorems include most of the known generalized KKM type theorems as special cases. The main purpose of this paper is to establish some new generalized R-KKM type theorems for generalized R-KKM mappings with finitely closed values and with finitely open values in noncompact topological spaces without any convexity structure under much weaker assumptions. As applications, some new minimax inequalities, saddle point theorem and equilibrium existence theorem for equilibrium problems with lower and upper bounds are established in general noncompact topological spaces. These theorems unify and generalize many known results in the literature. 2

Preliminaries

Let Δn be the standard n-dimensional simplex with vertices e0 , e1 , . . . , en . If J is a nonempty subset of {0, 1, . . . , n}, we denote by ΔJ the convex hull of the vertices {ej : j ∈ J}. The following notion was introduced by Deng and Xia [7]: Definition 2.1

Let X be a nonempty set and Y a topological space. A set-valued mapping

Y

G : X → 2 is said to be a generalized R-KKM mapping if, for any N = {x0 , x1 , . . . , xn } ∈ X, there exists a continuous mapping ϕN : Δn → Y such that, for each {ei0 , ei1 , . . . , eik ) ⊂ k (e0 , e1 , . . . , en }, ϕN (Δk ) ⊂ j=0 G(xij ), where Δk = co({ei0 , ei1 , . . . , eik }). The above class of generalized R-KKM mappings include those classes of KKM mappings, H-KKM mappings, generalized H-KKM mappings, G-KKM mappings, generalized

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G-KKM mappings, generalized S-KKM mappings, GLKKM mappings and GM KKM mappings defined in topological vector spaces, H-spaces, G-convex spaces, G-H-spaces, Lconvex spaces and hyperconvex metric spaces, respectively, as true subclasses, see [6, 7, 10–22]. Definition 2.2 Let X be a nonempty set and Y be a topological space. A set-valued mapping G : X → 2Y is said to be compactly closed-valued (resp., compactly open-valued) if, for each  x ∈ X and each compact subset K of Y , G(x) K is closed (resp., open) in K. Definition 2.3 Let X be a nonempty set and Y be a topological space. A generalized RKKM mapping G : X → 2Y is said to be finitely closed-valued (resp., finitely open-valued) if  for each N = {x0 , . . . , xn } ∈ X and each x ∈ N , ϕN (Δn ) G(x) is closed (resp., open) in ϕN (Δn ), where ϕN : Δn → Y is the continuous mapping in touch with N in Definition 2.1 of a generalized R-KKM mapping. Clearly, each generalized R-KKM mapping with compactly closed (resp., compactly open) values must be finitely closed-valued (resp., finitely open-valued) since, for each N = {x0 , . . . , xn } ∈ X, ϕN (Δn ) is compact in Y . The converse is not true in general. 3

Generalized R-KKM Type Theorems

In this section, we first prove some new generalized R-KKM type theorems for generalized RKKM mappings with finitely closed values and with finitely open values in general topological spaces. Theorem 3.1 Let X be a nonempty set, Y be a topological space and G : X → 2Y be a generalized R-KKM mapping with nonempty finitely closed values. Then for each N =

 n {x0 , x1 , . . . , xn } ∈ X, ϕN (Δn ) i=0 G(xi ) = ∅, where ϕN : Δn → Y is the continuous mapping in touch with N in Definition 2.1 of a generalized R-KKM mapping. Proof Since G is a generalized R-KKM mapping, for each N = {x0 , . . . , xn } ∈ X, there exists a continuous mapping ϕN : Δn → Y such that, for any {ei0 , . . . , eik } ⊂ {e0 , . . . , en }, ϕN (Δk ) ⊂

k 

G(xij ), j=0  ϕ−1 ϕN (Δn )) N (G(xij )

where Δk = co({ei0 , . . . , eik }). Let Eij = for j = 1, . . . k. For each z ∈ Δk , we have ϕN (z) ∈ ϕN (Δk ) ⊂ ϕN (Δn ). On the other hand, we have ϕN (z) ∈ ϕN (Δk ) ⊂ k  ϕN (Δn ), i.e., j=0 G(xij ). Hence there exists r ∈ {0, . . . , k} such that ϕN (z) ∈ G(xir )  −1 z ∈ ϕN (G(xir ) ϕN (Δn )). It follows that Δk = co({ei0 , . . . , eik }) ⊂

k  j=0

ϕ−1 A (G(xij )



ϕN (Δn )) =

k 

E ij .

j=0

 Since G is finitely closed-valued, G(xij ) ϕN (Δn ) is closed in ϕN (Δn ) for each j = 0, . . . , k. It follows from the continuity of ϕN that Eij is closed in Δn . By the classical KKM theorem, n  n i=0 Eij = ∅. It follows that ϕN (Δn ) ( i=0 G(xi )) = ∅. Remark 3.1 Theorem 3.1 generalizes Theorem 3.1 of Deng and Xia [7], Theorem 3.1 of Ding [8] and Theorem 2.1 of Ding, et al. [9], by weakening the assumption that G is compactly closed-valued. Theorem 3.1, in turn, generalizes Theorem 3.1 of Ding [10, 11], Theorem 2.1 of

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Ding [12], Lemma 2.1 of Verma [18], Theorem 2.3 of Tan [16], Theorem 3 of Khamsi [13] and Theorem 2.1 of Yuan [22] under much weaker assumptions. Theorem 3.2 Let X be a nonempty set, Y be a topological space and G : X → 2Y be a generalized R-KKM mapping with nonempty finitely open values. Then, for each N =  n {x0 , x1 , . . . , xn } ∈ X, ϕN (Δn ) ( i=0 G(xi )) = ∅, where ϕN : Δn → Y is the continuous mapping in touch with N in Definition 2.1 of a generalized R-KKM mapping. Suppose the conclusion is not true; then there exists N = {x0 , x1 , . . . , xn } ∈ X such  n that ϕN (Δn ) ( i=0 G(xi )) = ∅, where ϕN : Δn → Y is the continuous mapping in touch with N in Definition 2.1 of generalized R-KKM mapping. It follows that    n n  

ϕN (Δn ) = ϕN (Δn ) \ ϕN (Δn ) \ (ϕN (Δn ) G(xi )) . (ϕN (Δn ) G(xi )) = Proof

i=1

Since G is finitely open-valued, ϕN (Δn ) z ∈ Δn , let



i=1

G(x) is open in ϕN (Δn ) for each x ∈ N . For each

I(z) = {i ∈ {0, . . . , n} : ϕN (z) ∈ / G(xi )} and S(z) = co({ei : i ∈ I(z)}). If, for some z ∈ Δn , I(z) = ∅, then we have ϕN (z) ∈ G(xi ) for all i ∈ {0, . . . , n}, which   contradicts the assumption ϕN (Δn ) ( ni=0 G(xi )) = ∅. Therefore we can assume that I(z) = ∅ for each z ∈ Δn and hence S(z) is a nonempty compact convex subset of Δn for each z ∈ Δn .   Since i∈I(z) [ϕN (Δn ) \ (ϕN (Δn ) G(xi ))] is closed in ϕN (Δn ) and ϕN is continuous, we have /     −1 U = Δn \ ϕN [ϕN (Δn ) \ (ϕN (Δn ) G(xi )) i∈I(z) /

is an open neighborhood of z in Δn . For each z  ∈ U , we have ϕN (z  ) ∈ G(xi ) for all i ∈ / I(z) and    hence I(z ) ⊂ I(z). It follows that S(z ) ⊂ S(z) for all z ∈ U . This shows that S : Δn → 2Δn is a upper semicontinuous set-valued mapping with nonempty compact, convex values. By the Kakutani fixed point theorem, there exists a z0 ∈ Δn such that z0 ∈ S(z0 ). Note that G is an  R-KKM mapping; it follows that ϕN (z0 ) ∈ ϕN (S(z0 )) ⊂ i∈I(z0 ) G(xi ). Hence there exists an i0 ∈ I(z0 ) such that ϕN (z0 ) ∈ G(xi0 ). By the definition of I(z0 ), we have / G(xi ), ∀ i ∈ I(z0 ), ϕN (z0 ) ∈  n ( i=1 G(xi )) = ∅. which is a contradiction. Therefore ϕN (Δn ) Remark 3.2 Theorem 3.2 generalizes Theorem 2.2 of Ding, et al. [9] by weakening the assumption that G is compactly open-valued. Theorem 3.2 also generalizes Theorem 3.2 of Ding [5] and Theorem 2.3 of Ding [12] from G-convex spaces and L-convex spaces to general topological spaces without any convexity structure under much weaker assumptions. Theorem 3.2 also, in turn, generalizes Theorem 2.2 of Yuan [22] and Theorem 2.11.18 of Yuan [21] in the following aspects: (1) from hyperconvex metric space to general topological space; (2) from the class of GM KKM mappings to the class of R-KKM mappings; (3) from G being finitely metrically open-valued to G being finitely open-valued; (4) for each N ∈ X, the compactness assumption of co(N ) being dropped. Theorem 3.3 Let X be a nonempty set, Y be a topological space and G : X → 2Y be a generalized R-KKM mapping with nonempty finitely closed values such that

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 (i) clY ( x∈A G(x) ) is compact for some A ∈ X; (ii) For each N = {x0 , . . . , xn } ∈ X with A ⊂ N ,          clY G(x) G(x) ϕN (Δn ) = ϕN (Δn ), x∈N

x∈N

where ϕN : Δn → Y is the continuous mapping in touch with N in Definition 2.1 of a generalized R-KKM mapping.  Then x∈X G(x) = ∅. Proof Let {Ni }i∈I denote the family of all elements in X satisfying A ⊂ Ni for all i ∈ I and be partially ordered by the inclusion relation ⊂, i.e., for i, j ∈ I, i ≤ j if and only if Ni ⊂ Nj . For each i ∈ I and Ni = {xi,0 , . . . , xi,ni } ∈ X, define a set-valued mapping Gi : Ni → 2ϕNi (Δni ) by  Gi (x) = G(x) ϕNi (Δni ), ∀ x ∈ Ni , where ϕNi : Δni → Y is the continuous mapping in touch with Ni in Definition 2.1 of a generalized R-KKM mapping. Since G is finitely closed-valued, we have Gi (x) is closed in ϕNi (Δni ) for each x ∈ Ni . We show that Gi is also a generalized R-KKM mapping. Since G is a generalized R-KKM mapping, for each {xi,i0 , . . . , xi,ik } ∈ Ni , we have k  G(xi,ij ). ϕNi (Δk ) ⊂ j=0

Note k ≤ ni ; we have ϕNi (Δk ) ⊂ ϕNi (Δni ). Hence we have k k  

 ϕNi (Δk ) ⊂ G(xi,ij ) ϕNi (Δni ) = Gi (xi,ij ). j=0

j=0

This shows that Gi is also a generalized R-KKM mapping with closed values in Ni . By Theorem 3.1,  

G(x) ϕNi (Δni ) = ∅. x∈Ni

  Take ui ∈ x∈Ni (G(x) ϕNi (Δni )) for each i ∈ I and let Φi = {uj : i ≤ j and j ∈ I}. Clearly, we have (1) {Φi : i ∈ I} has the finite intersection property;  (2) Φi ⊂ x∈A G(x), ∀ i ∈ I, since A ⊂ Ni for all i ∈ I.  It follows that clY Φi ⊂clY ( x∈A G(x)) for each i ∈ I. By the condition (i) and the prop  erties (1) and (2), i∈I clY Φi = ∅. Take any yˆ ∈ i∈I clY Φi . Note that, for any i ∈ I and for any j ∈ I with i ≤ j,     



G(x) ϕNj (Δnj ) ⊂ G(x) ϕNj (Δnj ) ⊂ G(x). uj ∈ x∈N

x∈N

x∈N

i i  j Therefore Φi ⊂ x∈Ni G(x), ∀ i ∈ I. Clearly, for each x ∈ X, there exists i1 ∈ I such that x ∈ Ni1 and there exists i2 ∈ I such that yˆ ∈ Ni2 . Hence there exists i3 ∈ I with i1 ≤ i3 and i2 ≤ i3 such that x, yˆ ∈ Ni3 for all i ≥ i3 . It follows that, for all i ≥ i3 ,      G(z) ϕNi (Δni ) yˆ ∈ clY Φi ϕNi (Δni ) ⊂ clY

=

  z∈Ni

G(z)



z∈Ni

ϕNi (Δni )

(by the condition (iii))

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G(z) ϕNi (Δni ) ⊂ G(x). = z∈Ni

Thus yˆ ∈ G(x) for each x ∈ X and

 x∈X

G(x) = ∅.

Remark 3.3 Theorem 3.3 improves and generalizes Theorems 3.3 and 3.4 of Ding [5] in the following way: 1) from G-convex spaces to general topological spaces; (2) from s-KKM  mappings to generalized R-KKM mappings; (3) for each N ∈ Y D, the compactness assumption of G-co(N ) being dropped. Theorem 3.3, in turn, generalizes Theorem 3.2 of Verma [6] in the following way: (1) from G-H-space in the sense of Verma [6] to general topological space; (2) G may be a generalized R-KKM mapping; (3) the coercivity condition (i) being weaker than the condition (i) in Theorem 3.2 of Verma [6]. Corollary 3.1 Let X be a nonempty set, Y be a topological space and G : X → 2Y be a  generalized R-KKM mapping with closed values such that x∈A G(x) is nonempty compact for  some A ∈ X. Then x∈X G(x) = ∅. Proof Since G(x) is closed in Y for each x ∈ X, the condition (ii) of Theorem 3.3 is satisfied   trivially and clY ( x∈A G(x)) = x∈A G(x) is nonempty compact for some A ∈ X. Therefore the conclusion of Corollary 3.1 holds from Theorem 3.3. Corollary 3.2 Let X be a nonempty set and Y be a nonempty convex subset of a topological vector space E. Let G : X → 2Y be a set-valued mapping such that  (i) clY ( x∈A G(x) ) is nonempty compact for some A ∈ X; (ii) For each N = {x0 , . . . , xn } ∈ X with A ⊂ N , there exists N1 = {y0 , . . . yn } ∈ Y such  that, for each x ∈ N , G(x) co({y0 , . . . , yn }) is closed in co({y0 , . . . , yn }); (iii) For each N = {x0 , . . . , xn } ∈ X, there exists N1 = {y0 , . . . yn } ∈ Y such that n co({y0 , . . . , yn }) ⊂ i=0 G(xi ); (iv) For each N = {x0 , . . . , xn } ∈ X with A ⊂ N , there exists N1 = {y0 , . . . yn } ∈ Y such that          clY G(x) G(x) co(N1 ) = co(N1 ). Then

 x∈X

x∈N

x∈N

G(x) = ∅.

Proof By (iii), for each N = {x0 , . . . , xn } ∈ X there exists N1 = {y1 , . . . , yn } such that  n co({y0 , . . . , yn }) ⊂ ni=0 G(xi ). Define a mapping ϕN : Δn → Y by ϕN (α) = i=0 αi yi , n n ∀ α = i=0 αi ei ∈ Δn , where αi ≥ 0 and i=0 αi = 1. Then ϕN is continuous and n ϕN (Δn ) = co({y0 , . . . , yn } ⊂ i=0 G(xi ). By (iii), for any {ei0 , . . . , eik } ⊂ {e0 , . . . , en }, we  have co({yi0 , . . . , yik }) ⊂ kj=0 G(xij ), and for each u ∈ Δk = co({ei0 , . . . , eik }), there exist k k αij ≥ 0, j = 1, . . . , k with j=0 αij = 1 such that ϕN (u) = j=0 αij yij ∈ co({yi0 , . . . , yik }). Hence we have k  ϕN (Δk ) ⊂ co({yi0 , . . . , yik }) ⊂ G(xij ). j=0

It follows that G is a generalized R-KKM mapping. The condition (ii) implies G is finitely closed-valued. By the definition of ϕN , the condition (iv) implies that the condition (ii) of  Theorem 3.3 is satisfied. By Theorem 3.3, we have x∈X G(x) = ∅.

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From Corollary 3.2 with Y = X, we easily obtain the following result: Corollary 3.3 Let X be a nonempty convex subset of a topological vector space E. Let G : X → 2X be a set-valued mapping such that  (i) clY ( x∈A G(x)) is compact for some A ∈ X;  (ii) For each N = {x0 , . . . , xn } ∈ X with A ⊂ N and for each x ∈ N , G(x) co({x0 , . . . , xn }) is closed in co({x0 , . . . , xn }); n (iii) For each N = {x0 , . . . , xn } ∈ X, co(N ) ⊂ i=0 G(xi ); (iv) For each N = {x0 , . . . , xn } ∈ X with A ⊂ N ,          clY G(x) co(N ) = G(x) co(N ). Then



x∈N

x∈N

x∈X G(x) = ∅.

Remark 3.4 Corollaries 3.2 and 3.3 generalize Lemma 2 of Chowdhury and Tan [1] (see Theorem 1.1), Lemma 2.3 of Ding and Tarafdar [4], Lemma 1 of Brezis, Nirenberg and Stampacchia [3] and Lemma 1 of Ky Fan [2] under weaker assumptions. Hence Theorem 3.3 further generalizes the above results from topological vector space to general topological spaces without convexity structure under much weaker assumptions. 3

Applications

In this section, by applying the R-KKM type theorems obtained in the above section, we shall establish some new Ky Fan type minimax inequalities, saddle point theorem and equilibrium existence theorem for equilibrium problems with lower and upper bounds. Definition 4.1 Let X be a nonempty set and Y be a topological space. For some γ ∈ R, a  function f : X × Y → R {±∞} is said to be generalized γ-R-diagonally quasiconcave (resp., generalized γ-R-diagonally quasiconvex) in x if, for each N = {x0 , . . . , xn } ∈ X, there exists a continuous mapping ϕN : Δn → Y such that, for any {ei0 , . . . , eik } ⊂ {e0 , . . . , en } and any y ∈ ϕN (Δk ), min0≤j≤k f (xij , y) ≤ γ (resp., max0≤j≤k f (xij , y) ≥ γ), where Δk = co({ei0 , . . . , eik }). Definition 4.2 Let X be a nonempty set, Y be a topological space, and f : X × Y →  R {±∞}. For some α, β ∈ R with α ≤ β, f (x, y) is said to be generalized α-β-R-diagonally quasiconcave in x if, for each N = {x0 , . . . , xn } ∈ X, there exists a continuous mapping ϕN : Δn → Y such that, for any {ei0 , . . . , eik } ⊂ {e0 , . . . , en } and any y¯ ∈ ϕN (Δk ), there is an r ∈ {0, . . . , k} satisfying α ≤ f (xir , y¯) ≤ β, where Δk = co({ei0 , . . . , eik }). If α = −∞, then the notion in Definition 4.2 reduces to the corresponding notion in Definition 4.1. Theorem 4.1 Let X be a nonempty set and Y be a topological space. Let γ ∈ R be a constant  and f, g : X × Y → R {±∞} be two functions such that (i) For each (x, y) ∈ X × Y , g(x, y) ≤ f (x, y); (ii) f (x, y) is generalized γ-R-diagonally quasiconcave in x; (iii) For each N = {x0 , . . . , xn } ∈ X and each x ∈ N , the function y → g(x, y) is lower semicontinuous on ϕN (Δn ), where ϕN : Δn → Y is the continuous mapping in touch with N

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in Definition 4.1 of f (x, y) being generalized γ-R-diagonally quasiconcave in x;  (iv) There exists A ∈ X such that the set clY ( x∈A ({y ∈ Y : g(x, y) ≤ γ})) is nonempty compact; (v) For each N ∈ X with A ⊂ N , each x ∈ N and for any net {yλ }λ∈Λ in Y converging to y ∈ ϕN (Δn ), we have that g(x, yλ ) ≤ γ for all λ ∈ Λ implies g(x, y) ≤ γ. Then there exists a point yˆ ∈ Y such that g(x, yˆ) ≤ γ, ∀ x ∈ X. Proof

Define two set-valued mappings G, F : X → 2Y by F (x) = {y ∈ Y : f (x, y) ≤ γ} and

G(x) = {y ∈ Y : g(x, y) ≤ γ}, ∀ x ∈ X.

By (i), we have that F (x) ⊂ G(x) for all x ∈ X. By (ii) and Definition 4.1, for each N = {x0 , . . . , xn } ∈ X, there exists a continuous mapping ϕN (Δn ) → Y such that, for any {ei0 , . . . , eik } ⊂ {e0 , . . . , en } and for any y ∈ ϕN (Δk ), min1≤j≤k f (xij , y) ≤ γ. Hence there   exists r ∈ {0, . . . , k} such that f (xir , y) ≤ γ, i.e., y ∈ F (xir ) ⊂ kj=0 F (xij ) ⊂ kj=0 G(xij ). Since y ∈ ϕN (Δk ) is arbitrary, we have k  ϕN (Δk ) ⊂ G(xij ). j=0

Hence G is a generalized R-KKM mapping. The condition (iii) implies that G is finitely  closed-valued. The condition (iv) implies that there exists A ∈ X such that clY ( x∈A G(x))   is nonempty compact. For each N ∈ X with A ⊂ N , if y ∈ (clY ( x∈N G(x))) ϕN (Δn ),  then there exists a net {yλ }λ∈Λ ⊂ x∈N G(x) such that yλ → y ∈ ϕN (ΔN ) and hence, for each x ∈ N , g(x, yλ ) ≤ γ for all λ ∈ Λ. By the condition (v), we have g(x, y) ≤ γ for all x ∈ N . It   follows that y ∈ ( x∈N G(x)) ϕN (Δn ). Therefore we have          clY G(x) G(x) ϕN (Δn ) = ϕN (Δn ). x∈N

x∈N

All conditions of Theorem 3.3 are satisfied. By Theorem 3.3,  yˆ ∈ x∈X G(x), we obtain g(x, yˆ) ≤ γ, ∀ x ∈ X. Remark 4.1

 x∈X

G(x) = ∅. Taking any

Theorem 4.1 generalizes Theorem 4.1 of Ding [5] in the following way: (1) from

G-convex space to general topological space without any convexity structure; (2) from s-KKM mappings to generalized R-KKM mappings; (3) the compactness assumption that, for each  M ∈ Y D, G-co(M ) is compact being removed. Theorem 4.1, in turn, generalizes Theorem 3.3 of Verma [6] and Theorems 2.1–2.4 of Verma [19] to general topological space under much weaker assumptions. Theorem 4.2 Let X be a nonempty set and Y be a topological space. Let γ ∈ R be a constant  and f, g : X × Y → R {±∞} be two functions such that (i) For each (x, y) ∈ X × Y , g(x, y) ≤ f (x, y); (ii) f (x, y) is generalized γ-R-diagonally quasiconcave in x; (iii) For each N ∈ X and each x ∈ N , the function y → g(x, y) is lower semicontinuous on ϕN (Δn ), where ϕN : Δn → Y is the continuous mapping in touch with N in Definition 4.1 of f (x, y) being generalized γ-R-diagonally quasiconcave in x; (iv) There exist A ∈ X and a closed, compact subset K of Y such that, for each y ∈ X \K, there is a point x ∈ A satisfying g(x, y) > γ;

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(v) For each N ∈ X with A ⊂ N , each x ∈ N and for any net {yλ }λ∈Λ in Y converging to y ∈ ϕN (Δn ), we have that g(x, yλ ) ≤ γ for all λ ∈ Λ implies g(x, y) ≤ γ. Then there exists yˆ ∈ K such that g(x, yˆ) ≤ γ, ∀ x ∈ X.

 Proof From the condition (iv) it follows that the set x∈A ({y ∈ Y : g(x, y) ≤ γ}) ⊂ K is  nonempty and hence clY ( x∈A ({y ∈ Y : g(x, y) ≤ γ})) is nonempty compact. The condition (iv) of Theorem 4.1 holds. By Theorem 4.1 there exists yˆ ∈ Y such that g(x, yˆ) ≤ γ, ∀ x ∈ X. The condition (iv) implies yˆ ∈ K. Remark 4.2 Theorem 4.2 generalizes Theorem 4.2 of Ding [5] in several aspects. Theorem 4.2, in turn, generalizes Theorems 2.1–2.4 of Verma [19] in the following way: (1) from generalized H-space in the sense of Verma [19] to general topological spaces; (2) the conditions (ii)–(iv) being weaker than the conditions (6)–(8) in Theorem 2.1 of Verma [19]. Theorem 4.2 with γ = 0 generalizes Theorem 2 of Chowdhury and Tan [1], Theorem 7 of Arni [23] and Theorem 2.1 of Chowhury, Tarafdar and Tan [24] in the following way: (1) from topological vector spaces to general topological spaces without any convexity structure; (2) the coercivity condition (iv) being weaker than the condition (d) of Theorem 2 in [1] and the condition (d) of Theorem 7 in [23]. Hence, Theorem 4.2 generalizes the above results in [1, 5, 12, 19, 23, 24] under much weaker assumptions.  Corollary 4.1 Let X be a topological space and g : X × X → R {±∞} be a function with γ = supx∈X g(x, x) such that (i) g(x, y) is generalized γ-R-diagonally quasiconcave in x; (ii) For each N ∈ X and for each x ∈ N , the function y → g(x, y) is lower semicontinuous on ϕN (Δn ), where ϕN : Δn → Y is the continuous mapping in touch with N in Definition 4.1 of f (x, y) being generalized γ-R-diagonally quasiconcave in x; (iii) There exist A ∈ X and a closed, compact subset K of X such that, for each y ∈ X \K, there is a point x ∈ A satisfying ϕ(x, y) > supx∈X g(x, x) whenever supx∈X ϕ(x, x) < ∞; (iv) For each N ∈ X with A ⊂ N , each x ∈ N and for any net {yλ }λ∈Λ in X converging y ∈ ϕN (Δn ), we have that g(x, yλ ) ≤ γ for all λ ∈ Λ implies g(x, y) ≤ γ.

Then the following minimax inequality holds : inf

y∈K

sup g(x, y) ≤ sup g(x, x).

x∈X

x∈X

Proof If γ = supx∈X g(x, x) = ∞, then it is clear that the conclusion of Corollary 4.1 holds. Hence we may assume γ < ∞. By Theorem 4.2 with X = Y and f = g, there exists yˆ ∈ K such that supx∈X g(x, yˆ) ≤ γ = supx∈X ϕ(x, x). Hence we have inf y∈K supx∈X ϕ(x, y) ≤ supx∈X ϕ(x, x). Remark 4.3 Corollary 4.1 generalizes Corollary 4.3 of Ding [5], Theorem 3 of Chowdhury and Tan [1] and Theorem 1 of Ky Fan [25] from G-convex space and topological vector space to general topological spaces without any convexity structure under much weaker assumptions.  Theorem 4.3 Let X and Y be two topological spaces and f : X × Y → R {±∞} be a function such that

Ding X. P.

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(i) f (x, y) is generalized 0-R-diagonally quasiconcave in x and generalized 0-R-diagonally quasiconvex in y; (ii) For each N = {x0 , . . . , xn } ∈ X and each x ∈ N , the function y → f (x, y) is lower semicontinuous on ϕN (Δn ); (iii) For each M = {y0 , . . . , ym } ∈ Y  and each y ∈ M , the function x → f (x, y) is lower semicontinuous on ϕM (Δm );  (iv) There exist A ∈ X and B ∈ Y  such that the sets clY ( x∈A ({y ∈ Y : f (x, y) ≤ 0}))  and clX ( y∈B ({x ∈ X : f (x, y) ≥ 0})) are both nonempty compact; (v) For each N ∈ X with A ⊂ N , each x ∈ N and for any net {yλ }λ∈Λ in Y converging to y ∈ ϕN (Δn ), we have that f (x, yλ ) ≤ 0 for all λ ∈ Λ implies f (x, y) ≤ 0; (vi) For each M ∈ Y  with B ⊂ M , each y ∈ M and for any net {xα }α∈I in X converging to x ∈ ϕM (Δm ), we have that f (xα , y) ≥ 0 for all α ∈ I implies f (x, y) ≥ 0. Then f has a saddle point (ˆ x, yˆ) ∈ X × Y ; i.e., f (x, yˆ) ≤ f (ˆ x, yˆ) ≤ f (ˆ x, y), ∀ (x, y) ∈ X × Y. In particular, we have inf

y∈Y

sup f (x, y) = sup inf f (x, y) = 0.

x∈X

x∈X

y∈Y

Proof By Theorem 4.1 with γ = 0 and g ≡ f , there exists a point yˆ ∈ Y such that f (x, yˆ) ≤ 0 for all x ∈ X. Let g(y, x) = −f (x, y) for all (y, x) ∈ Y × X. Then, by using Theorem 4.1 with γ = 0 again, there exists a point x ˆ ∈ X such that g(y, x ˆ) ≤ 0 for each y ∈ Y . It follows that f (x, yˆ) ≤ 0 ≤ f (ˆ x, y) for all (x, y) ∈ X × Y . Hence we have f (ˆ x, yˆ) = 0 and f (x, yˆ) ≤ f (ˆ x, yˆ) ≤ f (ˆ x, y),

∀ (x, y) ∈ X × Y,

which implies inf

y∈Y

sup f (x, y) ≤ f (ˆ x, yˆ) ≤ sup inf f (x, y).

x∈X

x∈X

y∈Y

Since inf y∈Y supx∈X f (x, y) ≥ supx∈X inf y∈Y f (x, y) is always true, we obtain inf

y∈Y

sup f (x, y) = sup inf f (x, y) = 0.

x∈X

x∈X

y∈Y

Remark 4.4 Theorem 4.3 improves and generalizes Theorem 4.3 of Ding [5] and Theorem 4.2 of Tan [16] in several aspects. Theorem 4.4 Let X be a nonempty set, Y be a topological space and α, β be two real numbers  with α ≤ β. Let f, g : X × Y → R {±∞} be two functions such that (i) For each (x, y) ∈ X × Y , α ≤ f (x, y) ≤ β implies α ≤ g(x, y) ≤ β; (ii) f (x, y) is generalized α-β-R-diagonally concave in x;  (iii) There exists A ∈ X such that the set clY ( x∈A ({y ∈ Y : α ≤ g(x, y) ≤ β})) is non-empty compact; (iv) For each N = {x0 , . . . , xn } ∈ X and each x ∈ N , the set {y ∈ ϕN (Δn ) : α ≤ g(x, y) ≤ β} is closed in ϕN (Δn ), where ϕN : Δn → Y is the continuous mapping in touch with N in Definition 4.2 of f (x, y) being a generalized α-β-R-diagonally concave in x; (v) For each N ∈ X with A ⊂ N , each x ∈ N and for any net {yλ }λ∈Λ in Y converging y ∈ ϕN (Δn ), we have that α ≤ g(x, yλ ) ≤ β for all λ ∈ Λ implies α ≤ g(x, y) ≤ β.

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Then there exists yˆ ∈ Y such that α ≤ f (x, yˆ) ≤ β, ∀ x ∈ X. Proof

Define two mappings F, G : X → 2Y by F (x) = {y ∈ Y : α ≤ f (x, y) ≤ β},

G(x) = {y ∈ Y : α ≤ g(x, y) ≤ β}, ∀ x ∈ X.

Then, by (i), we have F (x) ⊂ G(x) for all x ∈ X. By (ii), for each N = {x0 , . . . , xn } ∈ X, there exists a continuous mapping ϕN : Δn → Y such that, for any {ei0 , . . . , eik } ⊂ {e0 , . . . , en } and any y¯ ∈ ϕN (Δk ), there is an r ∈ {0, . . . , k} satisfying α ≤ f (xir , y¯) ≤ β. It follows that k  G(xij ). y¯ ∈ {y ∈ Y : α ≤ f (xir , y) ≤ β} = F (xir ) ⊂ G(xir ) ⊂ j=0

 Since y¯ ∈ ϕN (Δk ) is arbitrary, we have ϕN (Δk ) ⊂ kj=0 G(xij ). This shows that G is a  generalized R-KKM mapping. By (iii), there exists A ∈ X such that the set clY ( x∈A G(x)) is nonempty compact. The condition (iv) implies that G is finitely closed-valued. For each   N ∈ X with A ⊂ N , if y ∈ (clY ( x∈N G(x))) ϕN (Δn ), then there exists a net {yλ }λ∈Λ ⊂  G(x) such that y → y ∈ ϕ (Δ ) and hence, for each x ∈ N , α ≤ g(x, yλ ) ≤ β for λ N N x∈N all λ ∈ Λ. By the condition (v), we have α ≤ g(x, y) ≤ β for all x ∈ N . It follows that   y ∈ ( x∈N G(x)) ϕN (Δn ). Therefore we have          clY G(x) ϕN (Δn ) = G(x) ϕN (Δn ). x∈N

x∈N

All conditions of Theorem 3.3 are satisfied. By Theorem 3.3,  ˆ ∈ Y and x∈X G(x), we get y

 x∈X

G(x) = ∅. Taking yˆ ∈

α ≤ f (x, yˆ) ≤ β, ∀ x ∈ X. Remark 4.5 Theorem 4.4 generalizes Theorem 4.5 of Ding [8] and Theorem 3.1 of Li [26] in several aspects. References [1] Chowdhury, M. S. R., Tan, K. K.: Generalization of Ky Fan’s minimax inequality with applications to generalized variational inequalities for pseudo-monotone operators and fixed point theorems. J Math. Anal. Appl., 204(3), 910–929 (1996) [2] Fan, Ky.: A generalization of Tychonoff’s fixed point theorem. Math. Ann., 142, 305–310 (1961) [3] Br´ ezis, H., Nirenberg, L., Stampacchia, G.: A remark on Ky Fan’s minimax principle. Boll Un Mat Ital, 6(4), 293–300 (1972) [4] Ding, X. P., Tarafdar, E.: Generalized variational-like inequalities with pseudo-monotone set-valued mappings. Archiv. Math., 74(4), 302–313 (2000) [5] Ding, X. P.: KKM type theorems, minimax inequalities and saddle point theorems in generalized convex spaces. Acta Mathematica Sinica, Chinese Series, 47(4), 711–722 (2004) [6] Verma, R. U.: Relative KKM type selection theorems and their applications. Appl. Math. Lett., 12(7), 39–43 (1999) [7] Deng, L., Xia, X.: Generalized R-KKM theorems in topolgical space and their applications. J. Math. Anal. Appl., 285, 679–690 (2003) [8] Ding, X. P.: Generalized R-KKM type theorems in topological spaces and applications. J. Sichuan Normal Univ., 28(5), 505–513 (2005) [9] Ding, X. P., Liou, Y. C., Yao, J. C.: Generalized R-KKM type theorems in topological spaces with applications. Appl. Math. Lett., 18(12), 1345–1350 (2005) [10] Ding, X. P.: Coincidence theorems and equilibria of generalized games. Indian J. Pure Appl. Math., 27(11), 1057–1071 (1996) [11] Ding, X. P.: Generalized G-KKM theorems in generalized convex spaces and their applications. J. Math. Anal. Appl., 266(1), 21–37 (2002) [12] Ding, X. P.: Generalized L-KKM type theorems in L-convex spaces with applications. Comput. Math. Appl., 43, 1249–1256 (2002)

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