Skip to main content

Coincidence theorems and minimax inequalities in abstract convex spaces

Abstract

In this paper, we deal with the notion of abstract convex spaces via minimal spaces as an extended version of other forms of convexity and establish some well-known results such as coincidence theorems for the classes m-KKM and ms-KKM of multimaps and Ky Fan's type minimax inequality.

Mathematics Subject Classification (2000)

26A51; 26B25; 54H25; 55M20; 47H10; 54A05.

1 Introduction

Many problems in nonlinear analysis can be solved by showing the nonemptyness of the intersection of certain family of subsets of an underlying set. Each point of the intersection can be a fixed point, a coincidence point, an equilibrium point, a saddle point and an optimal point or others of the corresponding equilibrium problem under consideration.

The first remarkable result on the nonempty intersection was the celebrated Knaster-Kuratowski-Mazurkiewicz theorem (simply, KKM principle) in 1929 [1], which concerns with certain types of multimaps called the KKM maps later. The KKM theory first called by Park in [2] and [3] was the study of KKM maps and their applications. At the beginning, the theory was mainly devoted to study on convex subsets of topological vector spaces. Later, it has been extended to convex spaces by Lassonde [4], to C-spaces (or H-spaces) by Horvath [58] and to others.

In 1993, Park and Kim [9] introduced the concept of generalized convex spaces, and the KKM theory is extended to generalized convex (G-convex) in a sequence of papers by many authors (for details, see [2, 3, 9] and [10]) and the references cited therein. Note that, in the KKM theory, there have appeared a number of coincidence theorems with many significant applications.

In 1996, since Chang and Yen [11] introduced the class KKM(X, Y) of multimaps, it was developed by many authors. Recently, Lin et al. [12] studied the class KKM(X, Y) in topological vector space and proved some KKM type coincidence and fixed point theorems. In all KKM type theorems, the convexity plays important role and so many efforts have done to establish KKM type theorems without convexity structure. For example, see KKM type theorems in minimal generalized convex spaces [1315], G-convex spaces [9], FC-spaces [16] and topological ordered spaces [17].

Recently, Zafarani [18] and Park [19] have introduced a new concept of abstract convex space and certain broad classes KC and KO of multimaps (having the KKM property). With this new concept, the KKM type maps were used to obtain matching theorems, coincidence theorems, fixed point theorems and others.

In this paper, by using the concept of abstract convexity and minimal spaces, the classes m- KKM(X, Y) and ms-KKM(X, Y, Z) as a generalization of KKM(X, Y) and s-KKM(X, Y, Z) are introduced. Some generalized KKM and s-KKM type theorems for minimal transfer closed valued multimaps are established. As applications, some new coincidence theorems and a new version of Ky Fan's minimax theorem are obtained.

2 Preliminaries

A multimap F : X Y is a function from a set X into the power set of Y. For any A X, set F(A) = xAF(x). Define the graph of F as G F = {(x, y) X × Y : y F(x)}. Furthermore, for any x X, define Fc (x) = {y Y : y F(x)} and, for any y Y, F-(y) = {x X : y F(x)} and F*(y) = {x X : y F(x)}. Note that F*(y) = X\F-(y) for any y Y. For any set D, put D as the family of all nonempty finite subsets of D.

Proposition 2.1. [20]Suppose F, G : X Y are two multimaps. Then we have the following.

  1. (1)

    for each x X, F(x) G(x) if and only if G*(y) F*(y) for each y Y,

  2. (2)

    y F(x) if and only if x F*(y),

  3. (3)

    for each x X, (F*)*(x) = F(x),

  4. (4)

    for each xX,F ( x ) if and only if y Y F * ( y ) =,

  5. (5)

    for each y Y, (Fc )*(y) = F -(y),

  6. (6)

    for each y Y, (F -) c (y) = F*(y).

A family MP ( X ) is said to be a minimal structure on , X if , X M . In this case, ( X , M ) is called a minimal space. For example, let (X, τ) be a topological space, then τ, SO(X), PO(X), αO(X) and βO(X) are minimal structures on X[21]. In a minimal space ( X , M ) , AP ( X ) is called an m-open set if AM and also BP ( X ) is called an m-closed set if B c M. Set m  - Int ( A ) = { U : U A , U M } and m  - Cl ( A ) = { B : A B , B c M } . Notice that, for any set A X, m-Cl(A) (resp., m-Int(A)) is not necessarily m-closed (resp., m-open).

It is not hard to see that there are many minimal spaces which are not topological space. Furthermore, in the following example, it is shown that there are some linear minimal spaces which are not topological vector space. Moreover in [13], there is a minimal G-convex space which is not G-convex space (see [9] and [13]).

Definition 2.2[22]. Let ( X , M ) and ( Y , N ) be two minimal spaces. A function f: ( X , M ) ( Y , N ) is called minimal continuous (briefly m-continuous) if f - 1 ( U ) M for any UN.

Example 2.3[13]. Consider the real field . Clearly M= { ( a , b ) : a , b { ± } } is a minimal structure on . We claim that M is a linear minimal structure on . For this, we must prove that, two operations + and · are m-continuous. Suppose (x0, y0) + -1(a, b) and so x0 + y0 (a, b). Put ε = min{x0 + y0 - a, b - (x0 + y0)} and so x 0 ( x 0 - ε 2 , x 0 + ε 2 ) and y 0 ( y 0 - ε 2 , y 0 + ε 2 ) . Hence,

x 0 + y 0 x 0 - ε 2 , x 0 + ε 2 + y 0 - ε 2 , y 0 + ε 2 ( a , b ) ;

which implies that + -1(a, b) is m-open in the minimal product space × ; that is + is m-continuous. Also, suppose (α0, x0) ·-1(a, b). Since α0x0 (a, b) and lims,t→0(α0 - s)(x0 - t) = α0x0, so one can find some 0 < δ for which |α0 - s| < δ and |x0 - t| < δ imply that a < (α0 - s)(x0 - t) < b. Therefore, (α0, x0) (α0 - δ, α0 + δ) · (x0 - δ, x0 + δ) (a, b); i.e., ·-1(a, b) is m-open in the minimal product space × , which implies that the operation · is m-continuous.

For the main results in this paper, we recall some basic definitions and results. More details can be found in [13, 15, 2124] and [22] and references therein.

Proposition 2.4. [21]For any two sets A and B,

  1. (1)

    m-Int(A) A and m-Int(A) = A if A is an m-open set.

  2. (2)

    A m-Cl(A) and A = m-Cl(A) if A is an m-closed set.

  3. (3)

    m-Int(A) m-Int(B) and m-Cl(A) m-Cl(B) if A B.

  4. (4)

    m-Int(m-Int(A)) = m-Int(A) and m-Cl(m-Cl(B)) = m-Cl(B),

  5. (5)

    (m-Cl(A)) c = m-Int(Ac ) and (m-Int(A)) c = m-Cl(Ac ).

Definition 2.5[14]. Let X be a nonempty set and Y be a minimal space. A multimap F : X Y is said to be

  1. (1)

    minimal transfer open valued if, for each x X and y F(x), there exists x 0 X such that y m-Int(F(x 0)).

  2. (2)

    minimal transfer closed valued if, for any x X and y F(x), there exists x 0 X such that y m-Cl(F(x 0)).

Theorem 2.6. Suppose that X is a nonempty set and Y is a minimal space. Then the following are equivalent.

  1. (1)

    The multimap F : X Y is minimal transfer closed valued,

  2. (2)

    xX F(x) = xX m-Cl(F(x)),

  3. (3)

    xX Fc (x) = xX m-Int(Fc (x)),

  4. (4)

    xX Fc (x) = xX(m-Cl(F(x))) c ,

  5. (5)

    Fc is minimal transfer open valued.

Proof. See theorems 2.1, 2.2 and 2.3 in [14].

Definition 2.7[22]. For a minimal space ( X , M ) ,

  1. (1)

    a family A= { A j : j J } of m-open sets in X is called an m-open cover of K if K jJ A j . Any subfamily of A which is also an m-open cover of K is called a subcover of A for K,

  2. (2)

    a subset K of X is m-compact whenever given any m-open cover of K has a finite subcover.

Definition 2.8. Suppose that X is a nonempty set and Y is a minimal space. A multimap T : X Y is called m-compact if m-Cl(T(X)) is an m-compact subset of Y.

Lemma 2.9. [15]Suppose that ( X , M ) is an m-compact minimal space and {A i : i I} is a family of subsets of X. If {m- Cl(A i ): i I} has the finite intersection property, then i I m - Cl ( A i ) . .

3 Coincidence theorems in abstract convex spaces

Now, we give some definitions for the results in this section as follows.

Definition 3.1[19]. An abstract convex space (X, D, Γ) consists of two nonempty sets X, D and a multimap Γ: D X. In case to emphasize X D, (X, D, Γ) will be denoted by (X D, Γ); and if X = D, then (X X; Γ) by (X, Γ). If D X and E X, then E is called abstract convex if Γ(A) E for each A DE. Obviously, for any B D, we can define CoΓ(B) = A | A B}.

Motivated by the results of Chang et al. [23], we introduce a new definition about the family of multimaps with the ms-KKMC (ms-KKMO) property in minimal abstract convex space as follows.

Definition 3.2. Let Y be a nonempty set, Z be a minimal space, s : YD be a function and (X, D, Γ) be an abstract convex space. Let T : X Z and F : Y Z be two multimaps. we say that F is generalized s-KKM with respect to T if

T ( Γ ( s ( A ) ) ) F ( A ) for any A Y .

The multimap T has ms- KKMC (resp. ms- KKMO) property if the following conditions.

  1. (1)

    for any y D, F(y) = m-Cl(A y ) (resp. F(y) = m-Int(A y )) for some A y Z,

  2. (2)

    F is generalized s-KKM with respect to T,

imply that the family {F(y): y D} has the finite intersection property.

Set

  1. (1)

    ms-KKMC(X, Y, Z) = {T : X Z : T has ms-KKMC property},

  2. (2)

    ms-KKMO(X, Y, Z) = {T : X Z : T has ms-KKMO property}.

Remark 3.3. Note that

  1. (1)

    If D = Y and the function s : YD is the identity map Id D , the class ms-KKMC(X, Y, Z) reduces to the class m-KKMC(X, Z) introduced and investigated in [1315].

  2. (2)

    The classes m-KKMC(X, Y) and ms-KKMC(X, Y, Z) generalize the classes KKM(X, Y) and s-KKM(X, Y, Z) in G-convex spaces and their subclasses (see [24]), in topological space.

Proposition 3.4. Suppose that X is a minimal space and A, B X such that B is m-compact and m- Cl(A) B. Then m- Cl(A) is m-compact.

Proof. Suppose that m-Cl(A) iIG i for any m-open cover {G i : i I}. Since B m-Cl(A) (m-Cl(A)) c , we have B m-Cl(A) m-Int(Ac ). From the definition of "m-Int", it follows that

B i I G i m  -  Int A c i I G i j J U j : U j M , U j A c .

Thus the compactness of B implies that

B i = 1 n G i j = 1 m { U j : U j M , U j A c } i = 1 n G i m  -  Int ( A c ) .

Therefore, we have B i = 1 n G i ( m  -  Cl ( A ) ) c , which implies that m- Cl ( A ) =Bm  -  Cl ( A) i = 1 n G i . Hence m-Cl(A) is m-compact. This completes the proof.

Theorem 3.5. Let (X, D, Γ) be an abstract convex space and Z be a minimal space. Suppose that s : YD is a mapping, T ms- KKMC(X, Y, Z) is an m-compact mapping such that F : Y Z is generalized s- KKM with respect to T. Then we have

m  -  Cl ( T ( C o Γ ( s ( Y ) ) ) ) y Y m  -  Cl ( F ( y ) ) .

Proof. Consider the multimap G : Y m-Cl(T(CoΓ(s(Y)))) defined by

G ( y ) = m  -  Cl ( m  -  Cl ( T ( C o Γ ( s ( Y ) ) ) ) F ( y ) )

for any y Y. It is easy to check that G is well defined. Since F is generalized s-KKM with respect to T, for any A Y, T(Γ(s(A))) F(A). Also, T(Γ(s(A))) m-Cl(T(CoΓ(s(Y)))) for each A Y. Hence T(Γ(s(A))) m-Cl(m-Cl(T(CoΓ(s(Y)))) ∩ F(A)) = G(A) and so G is generalized s-KKM with respect to T. Since T ms-KKMC(X, Y, Z), it follows that {G(y) = m-Cl(m-Cl(T(CoΓ(s(Y)))) ∩ F(Y)): y Y} has the finite intersection property in m-Cl(T(CoΓ(s(Y)))), which is an m-compact subset of Z by Proposition 3.4. Thus it follows from Lemma 2.9 that y Y m  - Cl ( m  - Cl ( T ( C o Γ ( s ( Y ) ) ) F ( y ) ) , which, from Proposition 2.4, we have y Y m  - Cl ( T ( C o Γ ( s ( Y ) ) ) m  - Cl( F ( y ) ) . Therefore, we have m  - Cl ( T ( C o Γ ( s ( Y ) ) ) y Y m  - Cl( F ( y ) ) . This completes the proof.

Theorem 3.6. Let (X, D, Γ) be an abstract convex space and Z be a minimal space. Suppose that s : YD is a mapping and T ms- KKMC(X, Y, Z) is m-compact. If a multimap F : Y Z satisfies the following conditions.

  1. (1)

    F is minimal transfer closed valued,

  2. (2)

    F is generalized s-KKM with respect to T.

Then m  - Cl ( T ( C o Γ ( s ( Y ) ) ) y Y F ( y ) .

Proof. It follows from Theorem 3.5 and the part (2) of Theorem 2.6.

Corollary 3.7. Let (X, D, Γ) is an abstract convex space and Y is a minimal space. Let T m- KKMC(X, Y) be m-compact. If a multimap F : D Y satisfies the following conditions.

  1. (1)

    F is minimal transfer closed valued,

  2. (2)

    F is generalized KKM with respect to T.

Thenm  - Cl ( T ( C o Γ ( D ) ) ) x D F ( x ) .

Proof. All conditions of Theorem 3.6 are satisfied for D instead of Y and for Y instead of Z, where s is the identity map on D. Therefore, one can deduce that m  - Cl ( T ( C o Γ ( D ) ) ) x D F ( x ) .

Remark 3.8. Note that

  1. (1)

    Theorem 3.5 generalizes Theorem 4.3 in [23] in convex space, theorems 13 and 14 in [25] in L-convex spaces, theorems 3.2 and 3.3 in [26], Theorem 3.2 in [27] in FC-spaces, Theorem 1 in [28] in G-convex spaces and Corollary 1.1 in [29] in abstract convex minimal spaces.

  2. (2)

    Theorem 3.6 generalizes Theorem 3.1 in [26] and [27], without the assumption "compactly" in FC-spaces and Corollary 3.1 [29] in abstract convex minimal spaces.

  3. (3)

    Corollary 3.7 generalizes Theorem 2.2 in [12] from convex spaces to abstract convex spaces and also from the class KKM(X, Y) in convex spaces to the class m-KKM(X, Y) in abstract convex spaces.

  4. (4)

    The compactness condition of multimap T in Theorem 3.5 and Theorem 3.6 can be replaced by the coercivity condition (see [24]) in minimal spaces. However, we use this condition in the following theorems.

Theorem 3.9. Let (X, D, Γ) be an abstract convex space and Y be a minimal space. Let T m- KKMC(X, Y) be m-compact. If F, G, H : D Y are three multimaps satisfying the following conditions.

  1. (1)

    F is minimal transfer closed valued,

  2. (2)

    F*(y) G*(y) and H*(y) T*(y) for any y Y,

  3. (3)

    CoΓ(G*(y)) H*(y) for any y Y.

Thenm  - Cl ( T ( C o Γ ( D ) ) ) x D F ( x ) .

Proof. We claim that G is generalized KKM with respect to T. To see this, suppose that there exists A D such that T(Γ(A)) xAG(x). Then there exist x ̄ Γ ( A ) and ȳT ( x ̄ ) such that ȳG ( x ) for all x A. Hence x G * ( ȳ ) for all x A and so A G * ( ȳ ) . From (3), we have x ̄ Γ ( A ) H * ( ȳ ) and then the condition (2) implies that x ̄ T * ( ȳ ) . Thus it follows that ȳT ( x ̄ ) , which is a contradiction. Therefore, G is generalized KKM with respect to T. Since G*(y) F*(y) for all y Y, from Proposition 2.1, F is generalized KKM with respect to T. Now, all conditions of Corollary 3.7 hold and hence m  - Cl ( T ( C o Γ ( D ) ) ) x D F ( x ) . This completes the proof.

Remark 3.10. According to the Remark 3.8, Theorem 3.9 generalizes the main theorem in [30] in FC-spaces, also it generalizes Theorem 2 in [31].

Theorem 3.11. Let (X, D, Γ) be an abstract convex space and Y be a minimal space. Let T m- KKMC(X, Y) be m-compact. If F : D Y and H, P : Y D are three multimaps satisfying the following conditions.

  1. (1)

    F is minimal transfer closed valued and DF -(y) for all y Y,

  2. (2)

    H*(x) F(x) for any x D,

  3. (3)

    CoΓ(H(y)) P(y) for all y Y.

Then there exists ( x ̄ , ȳ ) D×Ysuch thatȳT ( x ̄ ) and x ̄ P ( ȳ ) .

Proof. By the condition (2), it is easy to see that F*(y) (H*)*(y) for all y Y. Now, suppose that the conclusion does not hold, that is T ( x ) P - ( x ) = for all x D. Choose x P(y). Then y P-(x) and so y T(x), which gives x T*(y). Thus P(y) T*(y) for all y Y. The condition (3) implies that CoΓ((H*)*(y)) (P*)*(y) for all y Y. Hence all conditions of Theorem 3.9 are satisfied for P* and H*, which implies that m- Cl ( T ( Γ ( s ( Y ) ) ) ) x D F ( x ) and so there exists y0 m-Cl(T(CoΓ(D))) such that y0 F(x) for all x D. This means that D = F-(y0), which contradicts the condition (1). Therefore, there exists ( x ̄ , ȳ ) D×Y such that ȳT ( x ̄ ) and x ̄ P ( ȳ ) . This completes the proof.

Theorem 3.12. Let (X, D, Γ) be an abstract convex space and Y be a minimal space. Let T m- KKMC(X, Y) be m-compact. If H, P, Q : Y D are three multimaps satisfying the following conditions.

  1. (1)

    Q - is minimal transfer open valued and Q ( y ) for all y Y,

  2. (2)

    H*(x) Q*(x) for all x D,

  3. (3)

    CoΓ(H(y)) P(y) for all y Y.

Then there exists ( x ̄ , ȳ ) D×Ysuch thatȳT ( x ̄ ) and x ̄ P ( ȳ ) .

Proof. The condition (1) and Theorem 2.6 imply that Q* is minimal transfer closed valued. By the definition of Q*, (Q*)- = D\Q(y) and so D ≠ (Q*)-(y) for all y Y. By applying Theorem 3.11 with Q* instead of F, there exists ( x ̄ , ȳ ) D×Y such that ȳT ( x ̄ ) and x ̄ P ( ȳ ) . This completes the proof.

Remark 3.13. Theorem 3.12 generalizes Theorem 2.5 in [12] from convex spaces to abstract convex spaces and from the class KKM(X, Y) in convex spaces to the class m-KKM(X, Y) in abstract convex spaces.

Lemma 3.14. Let (X D, Γ) be an abstract convex space and Y be a minimal space. Let T m- KKMC(X, Y). If M is an abstract convex subset of X, then T/ M m- KKMC(M, Y).

Proof. Set D' = DM and let Γ': D' M be the multimap defined by Γ'(A) = Γ(A) for any A D'. Consider a multimap F : M Y defined by F(x) = m-Cl(A x ) for all x M. Then F is generalized KKM with respect to T| M . Now, define the multimap G : D Y by

G ( x ) = F ( x ) , i f x D , Y , i f x D \ D .

It is not hard to check that G is generalized KKM with respect to T. Since T m-KKMC(X, Y), the family {G(x): x D} has the finite intersection property and so {F(x): x D'} has the finite intersection property. Then T| M m-KKMC(M, Y). This completes the proof.

In the results on the KKM and coincidence type theorems mentioned above, it is assumed that T ms-KKMC(X, Y, Z) or T m-KKMC(X, Y) is an m-compact multimap, but, when it is not m-compact, we can use the coercivity conditions instead of the m-compactness condition of T as follows.

Proposition 3.15[14]. Let X and Y be two minimal spaces. Then the following statements for a multimap F : X Y are equivalent.

  1. (1)

    F- : Y X is minimal transfer open valued and F ( x ) for all x X.

  2. (2)

    X = {m-Int (F -(y)): y Y}.

Definition 3.16[32]. Let ( X , M ) be a minimal space and Y be a nonempty subset of X. The family M | Y = { U Y : U M } is called induced minimal structure by M on Y. ( Y , M | Y ) is called minimal subspace of ( X , M ) . Also, for any subset A of X we define m  -  Int Y ( A ) = { V : V M | Y and V A}.

Lemma 3.17. Let ( X , M ) be a minimal space and A be a nonempty subset of X. Then we have

m  -  Int ( A ) B m  -  Int B ( A B ) .

Proof. From Definition 3.16, we have

m  - Int A B = U : U M  and  U A B = { U B : U M  and  U A } { G : G M | B  and  G A B } = m  -  Int B ( A B ) .

Theorem 3.18. Let (X D, Γ) be an abstract convex space, Y be a minimal space and T m- KKMC(X, Y). If H, P, Q: Y D are three multimaps satisfying the following conditions.

  1. (1)

    Q - is minimal transfer open valued and Q ( y ) for all y Y,

  2. (2)

    H*(x) Q* (x) for all x D,

  3. (3)

    CoΓ (H(y)) P(y) for all y Y,

  4. (4)

    for each m-compact subset C D, m-Cl(T(C)) is m-compact in Y,

  5. (5)

    there is an m-compact subset K Y such that, for any A D, there is an m-compact abstract convex subset L A D containing A such that T ( L A ) \K x L A m  -  Int ( Q - ( x ) ) .

Then there exists ( x ̄ , ȳ ) D×Y such that ȳT ( x ̄ ) and x ̄ P ( ȳ ) .

Proof. The condition (1) and Proposition 3.15 imply that Y = xDm-Int(Q- (x)) and so we have

K = x D m  - Int ( Q - ( x ) ) K x D m  - Int ( Q - ( x ) ) = x D j J x U x , j ,

where U x, j is m-open and U x,j Q-(x). Since K is an m-compact subset of Y, K i = 1 n U x i , j i i = 1 n m  -  Int ( Q - ( x i ) ) - for j i J x i .

On the other hand, by the condition (5), there exists an m-compact abstract convex subset L A of D containing A such that T ( L A ) \K x L A m  -  Int ( Q - ( x ) ) . Thus we have

T ( L A ) x L A m  - Int ( Q - ( x ) ) K = x L A m  - Int ( Q - ( x ) ) i = 1 n m - Int ( Q - ( x i ) ) ,

where x i A L A . Then T ( L A ) x L A m  -  Int ( Q - ( x ) ) . By Lemma 3.17, it follows that

T ( L A ) = x L A m  - Int ( Q - ( x ) ) T ( L A ) x L A m  - In t T ( L A ) ( Q - ( x ) T ( L A ) ) T ( L A ) .

From the condition (4), it follows that m-Cl(T(L A )) is m-compact subset of Y. Moreover, since L A is an abstract convex subset of X, applying Lemma 3.14, T | L A m  - KKMC ( L A , Y ) .

Now, consider three multimaps P ̄ , Q ̄ , H ̄ :T ( L A ) L A defined by P ̄ ( y ) =P ( y ) L A , Q ̄ ( y ) =Q ( y ) L A and H ̄ ( y ) =H ( y ) L A for all y T(L A ). Then T | L A , P ̄ , Q ̄ and H ̄ satisfy in all conditions of Theorem 3.12 (notice Proposition 3.15) and hence there exist ȳT ( L A ) Y and x ̄ L A D such that ȳT | L A ( x ̄ ) =T ( x ̄ ) and x ̄ P ̄ ( ȳ ) P ( ȳ ) . This completes the proof.

Corollary 3.19. Let (X D, Γ) be an abstract convex space, Y be a minimal space and T m- KKMC(X, Y). If P, Q : Y D are two multimaps satisfying the following conditions.

  1. (1)

    Q- is minimal transfer open valued and Q ( y ) for all y Y,

  2. (2)

    CoΓ(Q(y)) P(y) for all y Y,

  3. (3)

    for each m-compact subset C D, m-Cl(T(C)) is m-compact in Y,

  4. (4)

    there exists an m-compact subset K Y such that, for any A D, there exists an m-compact abstract convex subset L A D containing A satisfying T ( L A ) \K x L A m  -  Int ( Q - ( x ) ) .

Then there exists ( x ̄ , ȳ ) D×Ysuch thatȳT ( x ̄ ) and x ̄ P ( ȳ ) .

Proof. Letting H* = Q* in Theorem 3.18, the conclusion follows.

Remark 3.20. Theorem 3.18 and Corollary 3.19 generalize Theorem 2.6 in [12] from the class of convex spaces to the class of abstract convex spaces under the weaker assumptions.

4 Applications to minimax inequalities

Fan's minimax inequality [33] has played very important roles in the study of modern nonlinear analysis and especially in mathematical economics. Moreover, some general minimax theorems and extensions of these inequalities have been obtained for the functions with KKM property under the various weaker conditions in many spaces with the convex structure (see [10] and [3439]).

In this section, we prove some results to obtain Ky Fan's type minimax theorem as follows.

Theorem 4.1. Let (X D, Γ) be an abstract convex space, Y be a minimal space and T : X Y is a multimap and T m- KKMC(X, Y). If G, H : D Y are two multimaps satisfying the following conditions.

  1. (1)

    G*(y) T*(y) for all y Y,

  2. (2)

    CoΓ (H*(y)) G* (y) for all y Y,

  3. (3)

    H is minimal transfer closed valued,

  4. (4)

    for any m-compact subset C D, m-Cl(T(C)) is m-compact in Y,

  5. (5)

    there exists an m-compact subset K Y such that, for any A D, there exists an m-compact abstract convex subset L A D containing A satisfying T ( L A ) x L A m  -  Cl ( H ( x ) ) K .

Then x D H ( x ) .

Proof. Suppose that x D H ( x ) =. For all y Y, there exists x D such that y H(x) and so x H*(y). Thus H* is nonempty valued. Since H is minimal transfer closed valued, from Lemma 2.6 and Proposition 2.1, it follows that Hc = (H*)- is minimal transfer open valued and m-Cl(H(x)) = (m-Int((H*)-(x))) c for all x D. Hence, from the condition (5), there exists an m-compact subset K Y such that, for any A D, there exists an m-compact abstract convex subset L A D containing A such that T ( L A ) \K x L A m  - Int ( H * ) - ( x ) ) . Applying Corollary 3.19 for (G*, H*) instead of (P, Q), there exists ( x ̄ , ȳ ) D×Y such that ȳT ( x ̄ ) and x ̄ G * ( ȳ ) . Thus x ̄ T * ( ȳ ) and x ̄ G * ( ȳ ) , which contradicts the condition (1). Therefore, x D H ( x ) .

Corollary 4.2. Let (X D, Γ) be an abstract convex space, Y be a minimal space and T m- KKMC(X, Y). If H : D Y is a multimap satisfying the following conditions.

  1. (1)

    CoΓ (H*(y)) T* (y) for all y Y,

  2. (2)

    H is minimal transfer closed valued,

  3. (3)

    for any m-compact subset C D, m-Cl(T(C)) is m-compact in Y,

  4. (4)

    there exists an m-compact subset K Y such that, for any A D, there exists an m-compact abstract convex subset L A D containing A satisfying T ( L A ) x L A m  -  Cl ( H ( x ) )K.

Then x D H ( x ) .

Proof. Letting G = T| D in Theorem 4.1, we can get the conclusion.

Lemma 4.3. Let (X, Γ) is an abstract convex space and Y be a nonempty set. If T, S : X Y are two multimaps, then the following statements are equivalent.

  1. (1)

    CoΓ (T*(y)) S* (y) for each y Y.

  2. (2)

    For any A X, S(Γ(A)) T(A).

Proof. (1) (2): Let A X and y S(Γ(A)). Then there exists x Γ(A) such that y S(x). Hence, from the definition of S - ,xΓ ( A ) S - ( y ) . So, it follows from Proposition 2.1 that Γ(A) S*(y). From (1), it follows that A T*(y) and so A T - ( y ) . Choose z AT-(y). Then y T(z) T(A). This means that S(Γ(A)) T(A) and so (1) (2) is proved.

  1. (2)

    (1): Choose y Y such that A T * ( y ) Γ ( A ) = C o Γ ( T * ( y ) ) S * ( y ) . Then there exist A T*(y) and x Γ(A) such that x S*(y), which implies that y S(x) S(Γ(A)). On the other hand, A T*(y) implies that A D \ T -(y) and so A T - ( y ) =. Thus it follows that y T (A) and hence S(Γ(A)) T(A), which this contradicts (2).

Remark 4.4. Note that

  1. (1)

    If X = D, then, by Lemma 4.3, in the condition (2) of Theorem 4.1, we can put G(Γ(A)) H(A) for all A D and so T(Γ(A)) H(A) for all A D instead of the condition (1) in Corollary 4.2.

  2. (2)

    Theorem 4.1 and Corollary 4.2 are generalizations of Theorem 3.3 and Corollary 3.2 in [40] respectively.

Definition 4.5. Let (X D, Γ) be an abstract convex space, Y be a nonempty set and f be a real-valued bifunction defined on X × Y. Then f is said to be

  1. (1)

    quasi-abstract convex in the first variable if, for all y Y and γ , the set (x X : f(x, y) < γ} is abstract convex.

  2. (2)

    quasi-abstract concave in the first variable if, for all y Y and γ , the set (x X : f(x, y) > γ} is abstract convex.

Definition 4.6. Let X, Y be minimal spaces and f be a real-valued bifunction defined on X × Y. Then f is said to be

  1. (1)

    strongly path minimal transfer lower semi-continuous in the first variable (shortly, spmt l.s.c.) if, for any (x, y) X × Y and ε > 0, there exists an m-open set N(x) containing x in X and there exists y' Y such that f(x, y) < f(x', y') + ε for any x' N(x).

  2. (2)

    strongly path minimal transfer upper semi-continuous in the first variable (shortly, spmt u.s.c.) if the function - f is l.s.c. in the first variable.

Theorem 4.7. Let (X D, Γ) be an abstract convex space, Y be a minimal space and T m- KKMC(X, Y). If f, g : D × Y are two functions and γ= inf ( x , y ) G T f ( x , y ) satisfying the following conditions.

  1. (1)

    for any m-compact subset C D, m-Cl(T(C)) is m-compact in Y,

  2. (2)

    f(x, y) ≤ g(x, y) for all (x, y) D × Y,

  3. (3)

    f is quasi-abstract convex in the first variable,

  4. (4)

    g is spmt u.s.c in the second variable,

  5. (5)

    there exists an m-compact subset K Y such that, for any A D, there exists an m-compact abstract convex subset L A D containing A such that, for all ȳT ( L A ) \K, there exists x L A such that ȳm  - Int { y : g ( x , y ) < γ } . Then

    inf ( x , y ) G T f ( x , y ) sup y Y inf x D g ( x , y ) .

Proof. Consider the multimap H : D Y defined by

H ( x ) = { y Y : g ( x , y ) γ }

for all x D. We claim that C o Γ ( H * ( ȳ ) ) T * ( ȳ ) for all y Y. Suppose that this is not the case, so there exists ȳY such that C o Γ ( H * ( ȳ ) ) T * ( ȳ ) or C o Γ ( D \ H - ( ȳ ) ) X\ T - ( ȳ ) . Thus there exist A D \ H - ( ȳ ) and x ̄ Γ ( A ) such that x ̄ X\ T - ( ȳ ) . Then ȳT ( x ̄ ) , which implies that ( x ̄ , ȳ ) G T .

On the other hand, from AD\ H - ( ȳ ) , we have A H - ( ȳ ) =. It follows that ȳH ( A ) and hence g ( x , ȳ ) <γ, for all x A. By the condition (2), we have f ( x , ȳ ) <γ for all x A. The condition (3) implies that f ( x , ȳ ) <γ for all x Γ(A). Therefore, f ( x ̄ , ȳ ) <γ, which contradicts γ= inf ( x , y ) G T f ( x , y ) . Thus it follows that CoΓ (H*(y)) T*(y) for all y Y.

To prove that H is minimal transfer closed valued, suppose that (x, y) D × Y and y H(x) and hence g(x, y) < γ. By the condition (4), setting ε = γ - g(x, y), there exist x' D and an m-open set N(y) containing y such that

- g ( x , y ) < - g ( x , y ) + ε = - g ( x , y ) + γ - g ( x , y )

for all y' N(y). Thus g(x', y') < γ for all y' N(y) and so N ( y ) H ( x ) =. It follows that N(y) (H(x')) c and so we have

yN ( y ) m  - Int ( ( H ( x ) ) c ) = ( m  - Cl ( H ( x ) ) ) c .

Therefore, y m-Cl(H(x')). From the condition (5), it follows that T ( L A ) \K x L A m  -  Int ( H c ( x ) ), which implies that T ( L A ) \K ( x L A m  - C l ( H c ( x ) ) or T ( L A ) x L A m  -  C ( H ( x ) )K. Thus the condition (5) is equivalent to the condition (4) in Corollary 4.2 for multimap H.

Now, applying Corollary 4.2, we have x D H ( x ) . There exists y' Y such that g(x, y') ≥ γ for all x D and so sup y Y inf x D g ( x , y ) γ. This implies that

inf ( x , y ) G T f ( x , y ) sup y Y inf x D g ( x , y ) .

Remark 4.8. Note that

  1. (1)

    The conditions (2) and (3) in Theorem 4.7 are satisfied if we assume that f is g-quasiconvex in the first variable, that is, for any y Y and A D, we have f ( x , y ) max x A g ( x , y ) for all x Γ(A).

  2. (2)

    Theorem 4.7 is a generalization of Theorem 1 in [41], Theorem 8 in [39] and Theorem 6.4 in [42].

References

  1. Knaster B, Kuratowski K, Mazurkiewicz S: Ein beweis des fixpunktsatsez für n -dimensionale simplexe. Fund Math 1929, 14: 132–137 1.

    Google Scholar 

  2. Park S: Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps. J Korean Math Soc 1994, 31: 493–519 1.

    MathSciNet  Google Scholar 

  3. Park S: Some coincidence theorems on acyclic multifunctions and applications to KKM theory. In Fixed Point Theory and Applications. Edited by: Tan K-K. World Scientific Publishers, River Edge; 1992:248–277. 1

    Google Scholar 

  4. Lassonde M: On the use of KKM multifunctions in fixed point theory and related topics. J Math Anal Appl 1983, 97: 151–201 1. 10.1016/0022-247X(83)90244-5

    MathSciNet  Article  Google Scholar 

  5. Horvath CD: Contractibility and generalized convexity. J Math Anal Appl 1991, 156: 341–357. 10.1016/0022-247X(91)90402-L

    MathSciNet  Article  Google Scholar 

  6. Horvath CD: Extension and selection theorems in topological spaces with a generalized convexity structure. Ann Fac Sci Toulouse Math 1993, 6: 253–269.

    Article  Google Scholar 

  7. Horvath CD: Convexite generalise et applications. In Sem Math Super. Volume 110. Press, University of Montre'al; 1990:79–99.

    Google Scholar 

  8. Horvath CD: Some results on multivalued mappings and inequalities without convexity. In Nonlinear and Convex Analysis (Proc. in honor of Ky Fan). Edited by: Lin BL, Simons S. Marcel Dekker, New York; 1987:99–106.

    Google Scholar 

  9. Park S, Kim H: Admissible classes of multifunctions on generalized convex spaces. Proc Coll Nature Sci SNU 1993, 18: 1–21. 1, 2

    Google Scholar 

  10. Park S: Coincidence, almost fixed point, and minimax theorems on generalized convex spaces. J Nonlinear Convex Anal 2003, 4: 151–164.

    MathSciNet  Google Scholar 

  11. Chang TH, Yen CL: KKM property and fixed point theorems. J Math Anal Appl 1996, 203: 224–235. 1 10.1006/jmaa.1996.0376

    MathSciNet  Article  Google Scholar 

  12. Lin LJ, Ansari QH, Wu JY: Geometric properties and coincidence theorems whit applications to generalized vector equilibrium problems. J Optim Theory Appl 2003, 117: 121–137. 1, 3.8, 3.13, 3.20 10.1023/A:1023656507786

    MathSciNet  Article  Google Scholar 

  13. Alimohammady M, Roohi M, Delavar MR: Knaster-Kuratowski-Mazurkiewicz theorem in minimal generalized convex spaces. Nonlinear Funct Anal Appl 2008, 13: 483–492. 2, 2.3, 2

    MathSciNet  Google Scholar 

  14. Alimohammady M, Roohi M, Delavar MR: Transfer closed and transfer open multimaps in minimal spaces. Chaos, Solitons Fractals 2009, 40: 1162–1168. 2.5, 2, 3.15 10.1016/j.chaos.2007.08.071

    MathSciNet  Article  Google Scholar 

  15. Alimohammady M, Roohi M, Delavar MR: Transfer closed multimaps and Fan-KKM principle. Nonlinear Funct Anal Appl 2008, 13: 597–611. 2, 2.9

    MathSciNet  Google Scholar 

  16. Ding XP: Maximal element theorems in product FC -spaces and generalized games. J Math Anal Appl 2005, 305: 29–42. 1 10.1016/j.jmaa.2004.10.060

    MathSciNet  Article  Google Scholar 

  17. Horvath CD, Llinares Ciscar JV: Maximal elements and fixed point for binary relations on topological ordered spaces. J Math Econ 1996, 25: 291–306. 1 10.1016/0304-4068(95)00732-6

    MathSciNet  Article  Google Scholar 

  18. Zafarani J: KKM property in topological spaces. Bull Soc R Liege 2004, 73: 171–185. 1

    MathSciNet  Google Scholar 

  19. Park S: On generalizations of the KKM principle on abstract convex spaces. Nonlinear Anal Forum 2006, 11: 67–77. 1, 3.1

    MathSciNet  Google Scholar 

  20. Lan KQ, Wu JH: A fixed-point theorem and applications to problems on sets with convex sections and to Nash equilibria. Math Comput Model 2002, 36: 139–145. 2.1 10.1016/S0895-7177(02)00110-3

    MathSciNet  Article  Google Scholar 

  21. Maki H: On generalizing semi-open sets and preopen sets. Proceedings of the 8th Meetings on Topolgical Spaces Theory and its Application 1996, 13–18. 2, 2.4

    Google Scholar 

  22. Popa V, Noiri T: On M -continuous functions. Anal. Univ. "Dunarea Jos-Galati''. Ser Mat Fiz Mec Teor Fasc II 2000,18(23):31–41. 2.2, 2, 2.7

    Google Scholar 

  23. Chang TH, Huang YY, Jeng JC, Kuo KW: On S -KKM property and related topics. J Math Anal Appl 1999, 229: 212–227. 3, 3.8 10.1006/jmaa.1998.6154

    MathSciNet  Article  Google Scholar 

  24. Park S: The KKM principle in abstract convex spaces: Equivalent formulations and applications. Nonlinear Anal 2010, 73: 1028–1042. 3.3, 3.8 10.1016/j.na.2010.04.029

    MathSciNet  Article  Google Scholar 

  25. Chen CM: R -KKM theorems on L -convex spaces and its applications. Sci Math Jpn 2007, 65: 195–207. 3.8

    MathSciNet  Google Scholar 

  26. Ding XP: Generalized KKM type theorems in FC-spaces with applications (I). J Glob Optim 2006, 36: 581–596. 3.8 10.1007/s10898-006-9028-x

    Article  Google Scholar 

  27. Ding XP, Ding TM: KKM type theorems and generalized vector equilibrium problems in noncompact FC-spaces. J Math Anal Appl 2007, 331: 1230–1245. 3.8 10.1016/j.jmaa.2006.09.059

    MathSciNet  Article  Google Scholar 

  28. Lin LJ: A KKM type theorem and it's applications. Bull Austral Math Soc 1999, 59: 481–493. 3.8 10.1017/S0004972700033189

    MathSciNet  Article  Google Scholar 

  29. Park S: Application of the KKM principle on abstract convex minimal space. Nonlinear Funct Anal Appl 2008, 13: 179–191. 3.8

    MathSciNet  Google Scholar 

  30. Ding XP: Generalized KKM type theorems in FC -spaces with applications (II). J Glob Optim 2007, 38: 367–385. 3.10 10.1007/s10898-006-9070-8

    Article  Google Scholar 

  31. Park S: Elements of the KKM theory on abstract convex spaces. J Korean Math Soc 2008, 45: 1–27. 3.10

    MathSciNet  Article  Google Scholar 

  32. Alimohammady M, Roohi M: Linear minimal space. Chaos, Solitons Fractals 2007, 33: 1348–1354. 3.16 10.1016/j.chaos.2006.01.100

    MathSciNet  Article  Google Scholar 

  33. Fan K: A minimax inequality and applications. In Inequalities. Edited by: Sisha O. Academic Press, San Diego; 1972:103–113. 4

  34. Ansari QH, Lin YC, Yao JC: General KKM theorem with applications to minimax and variational inequalities. J Optim Theory Appl 2000, 104: 41–57.

    MathSciNet  Article  Google Scholar 

  35. Chang SS, Lee BS, Wu X, Cho YJ, Lee M: The generalized quase-variational inequalities problems. J Math Anal Appl 1996, 203: 686–711. 10.1006/jmaa.1996.0406

    MathSciNet  Article  Google Scholar 

  36. Kuo TY, Jeng JC, Huang YY: Fixed point theorems for compact multimaps on almost Γ-convex sets in generalized convex spaces. Nonlinear Anal 2007, 66: 415–426. 10.1016/j.na.2005.11.036

    MathSciNet  Article  Google Scholar 

  37. Lan KQ: New fixed-point theorems for two maps and applications to problems on sets with convex section and minimax inequalities. Comput Math 2004, 47: 195–205.

    Google Scholar 

  38. Lin LJ: Applications of a fixed point theorem in G -convex spaces. Nonlinear Anal 2001, 46: 601–608. 10.1016/S0362-546X(99)00456-3

    MathSciNet  Article  Google Scholar 

  39. Zhang JH: Some minimax inequalities for mappping with noncompact domain. Appl Math Lett 2003, 17: 717–720. 4.8

    Article  Google Scholar 

  40. Lin LJ, Wan WP: KKM type theorems and coincidence theorems with applications to the existence of equilibria. J Optim Theory Appl 2004, 123: 105–122. 4.4

    MathSciNet  Article  Google Scholar 

  41. Ha CW: On a minimax of Ky Fan. Proc Amer Math Soc 1987, 99: 680–682. 4.8 10.1090/S0002-9939-1987-0877039-9

    MathSciNet  Article  Google Scholar 

  42. Park S: Equilibrium existence theorems in KKM spaces. Nonlinear Anal 2008, 69: 4352–5364. 4.8 10.1016/j.na.2007.10.058

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2011-0021821).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mehdi Roohi.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Je Cho, Y., Delavar, M.R., Mohammadzadeh, S.A. et al. Coincidence theorems and minimax inequalities in abstract convex spaces. J Inequal Appl 2011, 126 (2011). https://doi.org/10.1186/1029-242X-2011-126

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2011-126

Keywords

  • abstract convexity
  • abstract convex space
  • KKM theorem
  • coincidence theorem
  • minimax inequality