Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 310818,26 pages doi:10.1155/2012/310818

Research Article

Minimax Theorems for Set-Valued Mappings under Cone-Convexities

Yen-Cherng Lin,1 Qamrul Hasan Ansari,2,3 and Hang-Chin Lai4

1 Department of Occupational Safety and Health, College of Public Health, China Medical University, Taichung 404, Taiwan

2 Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India

3 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

4 Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan

Correspondence should be addressed to Yen-Cherng Lin, yclin@mail.cmu.edu.tw Received 7 September 2012; Accepted 27 October 2012 Academic Editor: Ondrej Dosly

Copyright © 2012 Yen-Cherng Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The aim of this paper is to study the minimax theorems for set-valued mappings with or without linear structure. We define several kinds of cone-convexities for set-valued mappings, give some examples of such set-valued mappings, and study the relationships among these cone-convexities. By using our minimax theorems, we derive some existence results for saddle points of set-valued mappings. Some examples to illustrate our results are also given.

1. Introduction

The minimax theorems for real-valued functions were introduced by Fan [1, 2] in the early fifties. Since then, these were extended and generalized in many different directions because of their applications in variational analysis, game theory, mathematical economics, fixed-point theory, and so forth (see, for example, [3-11] and the references therein). The minimax theorems for vector-valued functions have been studied in [4, 9, 10] with applications to vector saddle point problems. However, the minimax theorems for set-valued bifunctions have been studied only in few papers, namely, [4-8] and the references therein.

In this paper, we establish some new minimax theorems for set-valued mappings. Section 2 deals with preliminaries which will be used in rest of the paper. Section 3 denotes the cone-convexities of set-valued mappings. In Section 4, we establish some minimax theorems by using separation theorems, Fan-Browder fixed-point theorem. In the last

section, we discuss some existence results for different kinds of saddle points for set-valued mappings.

2. Preliminaries

Throughout the paper, unless otherwise specified, we assume that X, Y are two nonempty subsets, and Z is a real Hausdorff topological vector space, C is a closed convex pointed cone in Z with int C / 0. Let Z* be the topological dual space of Z, and let

C* = {g eZ*: g(c) > 0Vc e C}. (2.1)

We present some fundamental concepts which will be used in the sequel. Definition 2.1 (see [3, 4, 8]). Let A be a nonempty subset of Z. A point z e A is called a

(a) minimal point of A if A n (z - C) = {z}; Min A denotes the set of all minimal points of A;

(b) maximal point of A if A n (z + C) = {z};Max A denotes the set of all maximal points of A;

(c) weakly minimal point of A if A n (z - int C) = 0; Minro A denotes the set of all weakly minimal points of A;

(d) weakly maximal point of A if A n (z + int C) = 0; MaxroA denotes the set of all weakly maximal points of A.

It can be easily seen that Min A c MinroA and Max A c MaxroA.

Lemma 2.2 (see [3, 4]). Let A be a nonempty compact subset of Z. Then,

(a) Min A / 0;

(b) A c Min A + C;

(c) Max A / 0;

(d) A c Max A - C.

Following [6], we denote both Max and Maxro by max (both Min and Minro by min) in r since both Max and Maxro (both Min and Minro) are the same in r.

Definition 2.3. Let X, Y be Hausdorff topological spaces. A set-valued map F : X ^ Y with nonempty values is said to be

(a) upper semicontinuous at x0 e X if for every x0 e X and for every open set N containing F (x0), there exists a neighborhood M of x0 such that F (M) c N;

(b) lower semi-continuous at x0 eX if for any sequence {x„} c X such that xn ^ x0 and any y0 e F(x0), there exists a sequence yn e F(xn) such that yn ^ y0;

(c) continuous at x0 eX if F is upper semi-continuous as well as lower semi-continuous at x0.

We present the following fundamental lemmas which will be used in the sequel.

Lemma 2.4 (see [9, Lemma 3.1]). Let X, Y, and Z be three topological spaces. Let Y be compact, F : Xx Y ^ Z a set-valued mapping, and the set-valued mapping T : X ^ Z defined by

T(x) = U F (x,y), Vx eX. (22)

(a) IfF is upper semi-continuous on XxY, then T is upper semi-continuous on X.

(b) IfF is lower semi-continuous on X, so is T.

Lemma 2.5 (see [9, Lemma 3.2]). Let Z be a Hausdorff topological vector space, F : Z ^ r a set-valued mapping with nonempty compact values, and the functions p, q : Z ^ r defined by p(z) = max F(z) and q(z) = min F(z).

(a) If F is upper semi-continuous, so is p.

(b) If F is lower semi-continuous, so is p.

(c) If F is continuous, so are p and q.

We shall use the following nonlinear scalarization function to establish our results.

Definition 2.6 (see [6,10]). Let k e int C and v e Z. The Gerstewitz function £kv : Z ^ r is defined by

lkv(u)= min {t e r : u e v + tk - C}. (2.3)

We present some fundamental properties of the scalarization function.

Proposition 2.7 (see [6,10]). Let k e int C and v eZ. The Gerstewitz function £kv : Z ^ r has the following properties:

(a) £kv(u) <r & u e v + rk - int C;

(b) £kv(u) < r & u e v + rk - C;

(c) £kv(u) = 0 & u e v - dC, where dC is the topological boundary of C;

(d) £kv(u) > r & u / v + rk - C;

(e) £kv(u) > r & u / v + rk - int C;

(f) £kv (■) is a convex function;

(g) £kv (■) is an increasing function, that is, u2 - u e int S ^ ¿,kv(u\) < ¿,kv(u2);

(h) £kv(-) is a continuous function.

Theorem 2.8 ( Fan-Browder fixed-point theorem (see [12])). Let X be a nonempty compact convex subset of a Hausdorff topological vector space and let T : X ^ X be a set-valued mapping with nonempty convex values and open fibers, that is, T-1(y) = {x e X : y e T(x)} is open for all y e X. Then, T has a fixed point.

3. Cone-Convexities

In this section, we present different kinds of cone-convexities for set-valued mappings and give some relations among them. Some examples of such set-valued mappings are also given.

Definition 3.1. Let X be a nonempty convex subset of a topological vector space W. A set-valued mapping F : X ^ Z is said to be

(a) above -C-convex [4] (resp., above-C-concave [5]) on X if for all x1,x2 e X and all 1 e [0,1],

F(1x1 + (1 - 1)x2) c 1F(xx) + (1 - 1)F(x2) - C,

(resp., 1F(xi) + (1 - 1)F(x2) c F(1x1 + (1 - 1)x2) - C);

(b) below-C-convex [13] (resp., below-C-concave [9,13]) on X if for all x1,x2 e X and all 1 e [0,1],

1F(x1) + (1 - 1)F(x2) c F(1x1 + (1 - 1)x2) + C

(resp., F(1x1 + (1 - 1)x2) c 1F(x1) + (1 - 1)F(x2) + C);

(c) above-C-quasi-convex (resp., below-C-quasiconcave) [7, Definition 2.3] on X if the set

LevF<(z) := {x e X : F(x) c z - C}

(resp., LevF>(z) := {x e X : F(x) c z + C}),

is convex for all z eZ;

(d) above-properly C-quasiconvex (resp., above-properly C-quasiconcave [6]) on X if for all x1,x2 e X and all 1 e [0,1], either

F(1x1 + (1 - 1)x2) c F(x1) - C

(resp., F(x1) c F(1x1 + (1 - 1)x2) - C)

F(1x1 + (1 - 1)x2) c F(x2) - C

(resp., F(x2) c F(1x1 + (1 - 1)x2) - C);

(e) below-properly C-quasiconvex [7] (resp., below-properly C-quasiconcave) on X if for all x1,x2 e X and all 1 e [0,1], either

F(x1) c F(1x1 + (1 - 1)x2) + C

(resp., F(1x1 + (1 - 1)x2) c F(x1) + C)

F(x2) c F(Xx1 + (1 - X)x2) + C (resp., F(Xx1 + (1 - X)x2) c F(x2) + C);

(f) above-naturally C-quasiconvex [6] on X if for all x1,x2 e X and all 1 e [0,1],

F(Xx1 + (1 - X)x2) c co{F(x1) U F(x2)} - C,

where co A denotes the convex hull of a set A;

(g) above -C-convex-like (resp., above-C-concave-like) on X (X is not necessarily convex) if for all x1,x2 e X and all 1 e [0,1], there is an x' e X such that

(h) below -C-convex-like [13] (resp., below -C-concave-like) on X (X is not necessarily convex) if for all x1,x2 e X and all 1 e [0,1], there is an x' e X such that

It is obvious that every above-C-convex set-valued mapping or above-properly C-quasi-convex set-valued mapping is an above-naturally C-quasi-convex set-valued mapping, and every above-C-convex (above-C-concave) set-valued mapping is an above-C-convex-like ( above-C -concave-like ) set-valued mapping. Similar relations hold for cases below.

Remark 3.2. The definition of above-properly C-quasi-convex (above-properly C-quasi-concave) set-valued mapping is different from the one mentioned in [7, Definition 2.3] or [5, 6]. The following Examples 3.3 and 3.4 illustrate the reason why they are different from the one mentioned in [5-7]. However, if F is a vector-valued mapping or a single-valued mapping, both mappings reduce to the ordinary definition of a properly C-quasi-convex mapping for vector-valued functions [7]. The above-C-convexity in Definition 3.1 is also different from the below-C-convexity used in [5, 9].

Example 3.3. Consider C = {(s,t) e r2 : s > 0, t > 0}.LetF : [x1,x2] c r ^ r2 be a set-valued mapping defined by

F(x1) := {(s, t) e r2 : (s - 2)2 + (t - 4)2 = 1,2 < s < 3,4 < t < 5} |J{(s,5) : -1 < s < 2}, F(x2) := {(s,t) e r2 : (s - 6)2 + (t + 2)2 = 1,6 < s < 7,-2 < t <-1},

F(x') c XF(x1) + (1 - X)F(x2) - C (resp., XF(x1) + (1 - X)F(x2) c F(x') - C);

XF(x1) + (1 - X)F(x2) c F(x') + C (resp.,F(x1) c XF(x1) + (1 - X)F(xi) + C).

(3.10)

and for all 1 e (0,1),

F(1x1 + (1 - 1)x2) := {(s, t) e r2 : (s - 2)2 + (t - 2)2 = 4,0 < s < 2,0 < t < 2}. (3.12)

Then F is an above-properly C-quasi-convex set-valued mapping, but it is not below-properly C-quasi-convex.

On the other hand, let G : [x^x2] c r ^ r2 be a set-valued mapping defined by G(xx) := {(s, t) e r2 : (s - 1)2 + (t - 4)2 = 1,1 < s < 2,4 < t < 5},

(3.13)

G(x2) := {(s,t) e r2 : (s - 6)2 + (t + 2)2 = 1,6 < s < 7,-2 < t <-1},

and for all 1 e (0,1),

G(1x1 + (1 - 1)x2) : = f (s, t) e r2 : (s - 2)2 + (t - 2)2 = 4,0 < s < 2,0 < t < 2}

(3.14)

U{(s,0) : 2 < s < 3}.

Then, G is a below-properly C-quasi-convex set-valued mapping, but it is not above-properly C-quasi-convex.

Example 3.4. Let C = {(s,t) : s > 0,t > 0}. Define F : [-1,1] ^ r2 by

F(x) = {(x,t) : 1 - x2 < t < 1}, Vx e [-1,1]. (3.15)

Then F is continuous, above-C-quasi-convex, below-C-quasi-concave, above-properly C-quasi-convex, and above-properly C-quasi-concave, but it is not below-properly C-quasi-conconvex.

Proposition 3.5. Let X be a nonempty set (not necessarily convex) and for a given set-valued mapping F : X ^ Z with nonempty compact values, define a set-valued mapping M : X ^ Z as

M(x) = MaxwF(x), Vx e X. (3.16)

(a) If Fis above-C-convex-like, then M is so.

(b) If X is a topological space and F is a continuous mapping, then M is upper semicontinuous with nonempty compact values on X.

Proof. (a) Let F be above-C-convex-like, and let x1,x2 e X be arbitrary. Since F is above-C-convex-like, for any a e [0,1], there exists x' e X such that

F{x') c aF(x1) + (1 - a)F(x2) - C.

(3.17)

Abstract and Applied Analysis By Lemma 2.2,

MaxwF(x') c aMaxwF(x1) + (1 - a)MaxwF(x2) - C. (3.18)

Therefore, x ^ MaxwF(x) is above-C-convex-like.

(b) The upper semicontinuity of M was deduced in [4, Lemma 2]. □

Proposition 3.6. Let X be a nonempty convex set, and let F : X ^ Z be a set-valued mapping with nonempty compact values. Then, the set-valued mapping M : X ^ Z defined by

M(x) = MaxwF(x), Vx e X, (3.19)

is above-C-quasiconvex if F is so.

The following result can be easily derived, and therefore, we omit the proof.

Proposition 3.7. Let X be a nonempty convex set and F : X ^ r be above-R+-concave. Then the set-valued mapping x ^ maxF(x) is above-R+-concave and below-R+-quasiconcave. Furthermore, if F : X ^ r is above-properly R+-quasiconcave, then the set-valued mapping x ^ max F (x) is also above-properly r+-quasiconcave and below-R+-quasiconcave.

Let & e C* and F : X ^ Z be a set-valued mapping. Then, the composition mapping & o F : X ^ r is defined by

(& o F)(x) = &(F(x)) = U vx e X. (3.20)

yeF(x)

Clearly, the composition mapping & o F : X ^ r is also a set-valued mapping. Proposition 3.8. Let X be a nonempty set, F : X ^ Z a set-valued mapping, and & e C*.

(a) If F is above-C-convex-like, then & o F is above-R+-convex-like.

(b) If F is below-C-concave-like, then & o F is below-R+-concave-like.

(c) If X is a topological space and F is upper semi-continuous, then so is & o F.

Proof. (a) By the definition of above-C-convex-like set-valued mapping F : X ^ Z, for any xi ,x2 e X and all 1 e [0,1], there exists x' e X such that F (x') c XF (xi) + (1 - X)F (x2) - C. For any y e F(x'), there exist y1 e F(x1), y2 e F(x2) such that

Xyi + (1 - X)y2 e f - C. (3.21)

For any £ e C, we have £(y) < 1£(y0 + (1 - 1)£(y2). Hence, £(F(x')) c 1£(F(x1)) + (1 -1)£(F(x2)) - r+. Thus, £ o F is above-r+-convex-like.

The proof of (b) and (c) is easy, and therefore, we omit it. □

Proposition 3.9. Let X be a nonempty convex set and £ e C*.

(a) If F : X ^ Z is above-C-concave (above-properly C-quasi-concave), then £ o F : X ^ r is above-R+-concave (above-properly R+-quasi-concave).

(b) If F : X ^ Z is above-properly C-quasi-convex, then £ o F : X ^ r is above-R+-quasi-convex and above-properly R+-quasi-convex.

(c) If F : X ^ Z is above-C-convex, then £ o F : X ^ r is above-R+-convex and above-R+-quasi-convex.

Lemma 3.10. Let Z be a real Hausdorff topological vector space and C a closed convex pointed cone in Z with int C / 0. Let X be a nonempty compact subset of a topological space X, and let F : X ^ Z be an upper semi-continuous set-valued mapping with nonempty compact values. Then, for any £ e C*, there exists y e Maxw F(X) such that £(y) = maxUxeX £(F(x)).

Proof. For any given £ e C*, the mapping x ^ £(F(x)) is upper semi-continuous by Proposition 3.8 (c). By the compactness of X, there exist xo e X and y0 e F(xo) such that £(y0) = maxUxeX£(F(x)). By Lemma 2.2, there exists y e Maxw{JxeXF(x) such that y0 - y e -C, and hence £(y) > £(y0). On the other hand, y e Max^UxexF(x) c F(X), we know that £(y) e £(F(X)), and then £(y) < ma^UxeX£(F(x)) = £(y0). Therefore, the conclusion holds. □

Proposition 3.11. Let X be a nonempty convex set. If F : X ^ Z is above-properly C-quasi-convex, then it is above-C-quasi-convex.

Proof. For any z eZ, let x1, x2 e LevF<(z). Then, F(xi) and F(x2) are subsets of z - C. Since F is above-properly C-quasi-convex, for any X e [0,1], F(Xx1 + (1 - l)x2) is contained in either F (x1) - C or F(x2) - C, and hence, in z - C. Thus, the set LevF< (z) is convex, and therefore, F is above-C-quasi-convex. □

Proposition 3.12. Let X be a nonempty convex set. If F : X ^ Z is above-naturally C-quasi-convex, then it is above-C-quasi-convex.

Proof. Let z, x1, and x2 be the same as given as in Proposition 3.11. Then, co{F(x1 ) U F(x2)} c z - C since z - C is convex. By the above-naturally C-quasi-convexity, F(Xx1 + (1 - X)x2)} c z - C for all X e [0,1]. Thus, the set LevF<(z) is convex, and therefore, F is above-C-quasi-convex. □

Proposition 3.13. Let X be a nonempty convex set. If F : X ^ Z is above-naturally C-quasi-convex, then £ o F is above-naturally r+-quasi-convex for any £ e C*.

Proof. Let £ e C* be given. From the above-naturally C-quasi-convexity of F, for any x1,x2 e X and any X e [0,1],

F(Xx\ + (1 - l)x2) c co(F(xi) U F(x2)} - C.

(3.22)

For any y e F(axi + (1 - a)x2), there is a w e co{F(xi) U F(x2)} such that y e w - C. Then there exist yi e F(x1) U F(x2) and l e [0,1], 1 < i < n such that w = ^"=i Ijyj. Hence,

£(w) = En=1 ^(y^ and

£(y) e £(w) - r+ = £ li^yO - r+ c co{£(F(x1)) u ¿(Ffo))} - r+. (3.23)

Therefore, £ o F is a above-naturally r+-quasi-convex. □

Proposition 3.14. Let F : X ^ Z be a set-valued mapping with nonempty compact values. For any £ e C*,

(a) if £(d) = min UxeX £(F(x)) for some d eZ, then d e Minw UxeX F(x);

(b) if 1(e) = maxUxeX£(F(x)) for some e e Z, then e e Maxw UxeXF(x). Proof. Let £(d) = min UxeX £(F(x)). Suppose that d / Minw UxeX F(x). Then

\jF(x))f](d - int C)/ 0. (3.24)

Then, there exists w e UxeX F (x) and w e d - int C. Therefore, there exists s e X such that w e F (s) and d - w e int C. Since £ e C*, 1(d) > £(w) and £(w) > minU xeX£(F (x)). This implies that 1(d) > min |JxeX £(F(x)), which is a contradiction. This proves (a).

Analogously, we can prove (b), so we omit it. □

Remark 3.15. Propositions 3.8 and 3.9, Lemma 3.10, and Propositions 3.13 and 3.14 are always true except Proposition 3.8 (b) if we replace £ by any Gerstewitz function.

4. Minimax Theorems for Set-Valued Mappings

In this section, we establish some minimax theorems for set-valued mappings with or without linear structure.

Theorem 4.1. Let X, Y be two nonempty compact subsets (not necessarily convex) of real Hausdorff topological spaces X and Y, respectively. Let the set-valued mapping F : X x Y ^ r be lower semi-continuous on X and upper semi-continuous on Y such that for all (x,y) e X x Y, F(x,y) is nonempty compact and satisfies the following conditions:

(i) for each x e X,y ^ F(x, y) is below-R+-concave-like on Y;

(ii) for each y e Y,x ^ F(x, y) is above-R+-convex-like on X.

max y min (J F(x, y) = min (J max (J F(x,y).

yeY xeX xeX yeY

Proof. Since

max (J min (J F(x,y) < min (J max (J F(x,y),

yeY xeX xeX yeY

it is sufficient to prove that

max (J min (J F(x,y) > min (J max (J F(x,y).

yeY xeX xeX yeY

Choose any a e r such that a < min uxeX max uyeY F (x, y). For any y e Y, let

LevF< (y; a) = {x e X : F(x,y) c a - r+}.

Then, by the lower semi-continuity of the set-valued mapping x ^ F (x,y), the set LevF< (y; a) is closed, hence it is compact for all y e Y .By the choice of a, we have

P| LevF< (y; a) = 0.

Since X is compact and the collection {X \ LevF<(y;a) : y e Y} covers X, there exist finite number of points y1,y2,...,ym in Y such that

X cU(X \ LevF<(yt; a)) i=1

P) LevF^ yi; a) = 0.

This implies that

max (J F (x,y^ > a, Vx e X, i=1

and therefore,

min (J ma^y > a.

xeX i=1

Following the idea of Borwein and Zhuang [14], let

E := |(z,r) e rm+1 : there is x e X,F(x,yi) c r + zi - r+,i = 1,2,...,m},

(4.10)

where z = (z1,z2,...,zm). Then the set Eis convex, so is int E. We note that the interior int E of E is nonempty since

0,1 + max|JF(x,yi)) e int E, Vx e X. (4.11)

Since (0,a) / E, by separation hyperplane theorem [15, Theorem 14.2], there is a (S,e) = 0 x { 0} such that

((3, e), (z, r) > > ((3, e), (0, a) >, V(z, r) e E, (4.12)

where 3 = (11,12,..., 1m), that is,

3z + er > ea, V(z, r) e E. (4.13)

By (4.11), (4.13), and the choice of a, we have that e> 0. Furthermore, from the fact

~[(F(x,yi) + r) x{-r} c E, (4.14)

we have

{Ux,1 + r,nx,2 + r,...,nxrm + r,-r) e E, Vnx,i e F(x,. (4.15)

Hence, by (4.13), we have

X 1i(nx,i + r) + e(-r) > ea (4.16)

(1)nxii ^^mr11 - 0r > a, Vx e X, r e r. (4.17)

Thus, we have Xim1(1i/e) = 1. Hence, by (4.17), we have

m /1-\

2i( — )F(x,yi) c a + r+. (4.18)

i=1 ^ e '

Since F(x,y) is below-r+-concave-like in y, there is y' e Y such that

m /1 \

F(x,y') c ^(~)F(x,yi) + r+, Vx e X. (4.19)

i=1 ^ e /

Therefore,

1j F (.x,y') c a + r+

(4-20)

and hence,

max (J min (J F(x,y) > a. (4 21)

yeY xeX ( - )

This completes the proof. □

Remark 4.2. Theorem 4.1 is a modification of [14, Theorem A]. If F is a real-valued function, then Theorem 4.1 reduces to the well-known minimax theorem due to Fan [2].

We next establish a minimax theorem for set-valued mappings defined on the sets with linear structure.

Theorem 4.3. Let X, Y be two nonempty compact convex subsets of real Hausdorff topological vector spaces X and Y, respectively. Let the set-valued mapping F : X x Y ^ r be lower semi-continuous on X and upper semi-continuous on Y such that for all (x, y) e X x Y, F(x, y) is nonempty compact, and satisfies the following conditions:

(i) for each y e Y, x ^ F(x, y) is above-R+-quasi-convex on X;

(ii) for each x e X, y ^ F(x,y) is above-R+-concave, or above-properly R+-quasi-concave on Y;

(iii) for each y e Y, there is a xy e Y such that

max F(xy, y) < max (J min (J F(x, y). (4 22)

yeY xeX ( .

min (J max (J F(x,y) = max (J min (J F(x,y). (4 23)

xeX yeY yeY xeX ( . )

Proof. We only need to prove that

max (J min (J F(x,y) < min (J max (J F(x,y) (4 24)

yeY xeX xeX yeY

is impossible, since it is always true that

max (J min (J F(x,y) < min (J max (J F(x,y).

yeY xeX xeX yeY

Suppose that there is an a e r such that

max y min (J F(x,y) < a < min (J max (J F(x, y). (4 26)

yeY xeX xeX yeY

Define G : X x Y ^ X x Y by

G(x,y) = {s e X : maxF(s,y) < a} x {t e Y : maxF(x,t) > a}. (4.27)

For each x e X, maxUyeYF (x, y) > minUxeX maxUyeYF(x,y) > a. Since Y is compact and the set-valued mapping y ^ max F (x,y) is upper semi-continuous, there is a t e Y such that maxF(x,t) = maxUyeYF(x,y) > a.

On the other hand, from the condition (iii), for each y e Y, there is a xv e Y such that max F(xy,y) < a. Hence, for each (x,y) e X x Y, G(x,y)/ 0. By (i) and Proposition 3.6, the mapping x ^ maxF(x,y) is above-r+-quasi-convex on X. By (ii) and Proposition 3.7, the mapping y ^ max F (x,y) is below-r+-quasi-concave on y. Hence, for each (x, y) e X x Y, the set G(x,y) is convex. From the lower semi-continuities on X and upper semi-continuity on Y of F, the set

G~\s,t) = {x e X : max F(x,t) > a} x {y e Y : max F(s,y) < a} (4.28)

is open in X x Y. By Fan-Browder fixed-point Theorem 2.8, there exists (x,y) e X x Y such that

{x,y) e G{x,y), (4.29)

that is,

max Fix, y) > a> max F(x,y), (4.30)

which is a contradiction. This completes the proof. □

Remark 4.4. [5, Propositions 2.7 and 2.1] can be deduced from Theorem 4.3. Indeed, in [5, Proposition 2.1], the above-naturally C-quasi-convexity is used. By Proposition 3.12, the condition (i) of Theorem 4.3 holds. Hence the conclusion of Proposition 2.1 in [5] holds. We also note that, in Theorem 4.3, the mapping F need not be continuous on X x Y. Hence Theorem 4.3 is a slight generalization of [7, Theorem 3.1].

Theorem 4.5. Let X and Y be nonempty compact (not necessarily convex) subsets of real Hausdorff topological vector spaces X and Y, respectively. Let the mapping F : X x Y ^ Z be upper semi-continuous with nonempty compact values and lower semi-continuous on X such that

(i) for each x e X, y ^ F(x,y) is below-C-concave-like on Y;

(ii) for each y e Y, x ^ F(x,y) is above-C-convex-like on X;

(iii) for every y e Y,

Max U Min^ F(x, y) c Minw U F(x, y) + C. (4 31)

yeY xeX xeX \ ■ /

Then for any

Z1 e Max y Minw y F(x,y), (432)

yeY xeX

there is a

Z2 e Min ( cJ y Maxw y F(x, yU ) (4.33)

\ [ xeX yeY j /

such that

zi e Z2 + C, (4.34)

that is,

Max y Minw y F(x, y) c Min ^co { y Maxw y F(x, y) \\ + C. (4.35)

yeY xeX \ [xeX yeY j J

Proof. Let r(x) := Maxro UyeYF(x, y) for all x e X. From Lemma 2.4 and Proposition 3.5, the set-valued mapping x ^ r(x) is upper semi-continuous with nonempty compact values on X. Hence the set r(X) is compact, and so is co{T(X)}. Then co{T(X)}+C is a closed convex set with nonempty interior. Suppose that v / co{T(X)} + C. By separation hyperplane theorem [15, Theorem 14.2], there exist k e r, e > 0 and a nonzero continuous linear functional ¿: Z ^ r such that

¿(v) < k - e<k < ¿(u + c), for every u e co{r(X)}, c e C. (4.36)

Therefore,

¿(c) > ¿(v - u), for every u e co{r(X)}, c e C. (4.37)

This implies that £ e C* and ¿(v) < ¿(u) for all u e co{r(X)}.

Let g := ¿F : X x Y ^ r. From Lemma 3.10, for each fixed x e X, there exist y*x e Y

and f (x,yx) e F(x,y*x) with f (x,y*x) e r(x) such that ¿(f (x,y*x)) = maxUyeY ¿(F(x, y)).

Choosing c = 0 and u = f (x,y*x) in (4.36), we have

max y ¿(F(x,y)) = ¿f(x,y*x) > k>k - e > ¿(v), Vx e X.

Abstract and Applied Analysis 15 Therefore,

min U max U t(F(x,y)) > l{v). (439)

xeX yeY

By the conditions (i), (ii) and Proposition 3.8, the set-valued mapping y ^ ¿(F(x,y)) is below-r+-concave-like on Y for all x e X, and the set-valued mapping x ^ ¿(F(x,y)) is above-r+-convex-like on X for all y e Y. From Theorem 4.1, we have

maxUminU l(F(x,y)) > l(v). (440)

yeY xeX

Since Y is compact, there is an y' e Y such that minUxe_Xl(F(x,y')) > ¿(v). For any x e X and all g(x, y') e F (x, y'), we have

¿(g(x,y')) >№, (4.41)

that is,

è(g(x,y') - v) > 0, Vx e X, g(x,y') e F (x,y'). (4.42)

Thus, v / UxeX F(x,y') + C, and hence,

v / Minro y F(x,y') + C. (4.43)

If v e Max UyeY Minro UxeX F(x, y), by the condition (iii), v e Minro UxeX F(x, y') + C which contradicts (4.43). Hence, for every v e MaxUyeY Minro UxeX F(x,y),

v e co | y Maxro y F (x, y) | + C, (4.44)

[ xeX yeY j

that is,

Max y Min^ y F (x, y) c co ^ y Maxro y F(x,y)\ + C (4.45)

yeY xeX [xeX yeY j

Max y Minro y F (x, y) c Mini co^ y Maxro y F (x,y) ^

yeY xeX \ [xeX yeY j

The following examples illustrate Theorem 4.5.

+ C. (4.46) □

Example 4.6. Let X = Y = {0} u {1/n : n e n}, C = r+ and

F(x,y) = |(s, t) e r2: s = x2,t = 1 - y2}, V(x,y) e X x Y. (4.47)

It is obviously that F is below-r+-concave-like on Y and above-r+-convex-like on X. We now verify the condition (iii) of Theorem 4.5. Indeed, for any y e Y,

U F(x,y) = ({0} u {n~2 : n e n}) x {1 - y2},

xeX ^ ^ n J '

Min^ F(x,y) = ({0} u {"2" : n e x {1 - y2}.

= Min^ F(x,y) + C,

and the condition (iii) of Theorem 4.5 holds. Furthermore, for any x e X,

[J F (x, y) = {x2} x ^{1} u j 1 - -32 : n e n Maxro U F(x,y) = {x2} x ({1} u [ 1 - n e n

U Maxro U F(x,y) = ({0} u {1 : n e n}) x ({1} u { 1 - -1 : n e n ^x ^Y V yn J / \ Ln

co{ U Maxro U F (x, y) ^ = [0,1] x [0,1].

^ xeX yeY j

(4.48)

U Minro U F (x, y) = ({0} u {1 : n e n]) x ({1} u {1 - 1 : n e n}),

yeY xeX n2 n2

Max U Minro U F(x,y) = {(1,1)}.

yeY xeX

Thus, for every y e Y,

Max U Minro U F(x,y) c ({0} u f1 : n e nX) x {1 - y2} + C

y Y x X n2

(4.49)

(4.50)

(4.51)

Mini coJ U Max™ U F{X'Vn ) = {(0,0)},

\ LxeX yer J /

Max U Minro U F{x,y) = {(1,1)} c Mini cJ U Maxro U F(x,y)\ ) + c

\ IxeX yeY J /

(4.53)

yeY xeX

Hence, the conclusion of Theorem 4.5 holds.

Example 4.7. Let X = [0,1], Y = [-1,0], C = r+, and G : Y ^ Y be defined by

G(y) = i[-1,0]' » = (4.54)

w ll°), y/0.

Let F (x,y) = (x2}x G(y) for all (x,y) e X x Y. Then G is upper semi-continuous, but not lower semi-continuous on r, and F is not continuous but is upper semi-continuous on X x Y. Moreover, F has nonempty compact values and is lower semi-continuous on X. It is easy to see that F is below-C-concave-like on Y and is above-C-convex-like on X. We verify the condition (iii) of Theorem 4.5. Indeed, for all y e Y, UxeX F(x,y) = [0,1] x G(y).

Min. U F(x,y) = i[0,1] X{0}, » = 0, (4.55)

xeX V ' !({0}x [-1,0]) U ([0,1] x{-1}), y = 0.

U Min. U F(x,y) = ({0} x [-1,0]) U ([0,1] x {-1}) U ([0,1] x {0}),

yeY xeX

Max U Min. U F(x,y) = {(1,0)} c Min. U F(x,y) + C.

yeY xeX xeX

Therefore, the condition (iii) of Theorem 4.5 holds. Since

(4.56)

{x2} x [-1,0], y = 0, {x2} x{0}, yf 0,

F{x,y) = \ ^ (4.57)

for all (x, y) e X x Y, and Ma^Uyer Minro UxeXF(x, y) = {(1,0)}, for each y e Y, we can

choose xy = 0 e X such that

Max y Minro y F(x,y) c F(xy,y) + C.

xy,ync- (4.58)

yeY xeX

Furthermore,

U F(x,y) = {x2} X ( U G(y)j

yeY \yeY /

= {x2} x ([-1,0] U{0}) = {x2} x [-1,0], U Maxro U F(x,y) = [0,1] x [-1,0].

xeX yeY

(4.59)

Therefore,

Max U Minw U F(x,y) = {(1,0)} c {(0,-1)} + C

yeY xeX

= Min y Maxro y F(x,y) + C.

(4.60)

xeX yeY

Hence, the conclusion of Theorem 4.5 holds.

Remark 4.8. Theorem 3.1 in [5] Theorem 3.1 in [6], or Theorem 4.2 in [7] cannot be applied to Examples 4.6 and 4.7 because of the following reasons:

(i) the two sets X and Y are not convex in Example 4.6;

(ii) F is not continuous on X x Y in Examples 4.6 and 4.7.

Theorem 4.9. Let X, Y be two nonempty compact convex subsets of real Hausdorff topological vector spaces X and Y, respectively. Suppose that the set-valued mapping F : X x Y ^ Z has nonempty compact values, and it is continuous on Y and lower semi-continuous on X such that

(i) for each y e Y, x ^ F(x, y) is above-naturally C-quasi-convex on X;

(ii) for each x e X, y ^ F (x,y) is above-C-concave or above-properly C-quasi-concave on Y;

(iii) for every y e Y,

(iv) for any continuous increasing function h and for each y e Y, there exists xy e X such

Max y Minw y F(x,y) c Minw y F(x,y) + C;

(4.61)

yeY xeX xeX

max h(F{xy,y)) < max y min y h(F{x,y)).

yeY xeX

Then, for any z1 e Max UyeY Min. UxeX F(x, y), there is a

Z2 e Min U Max^ y F(x, y) (4 63)

xeX yeY V ' '

such that zi e z2 + C, that is,

Max y Minœ y F(x, y) c Min y Maxœ y F(x, y) + C.

yeY xeX xeX yeY

(4.64)

Proof. Let r(x) be defined as the same as in the proof of Theorem 4.5. Following the same perspective as in the proof of Theorem 4.5, suppose that v / UxeX Max. UyeY F(x, y) + C. For any k e int C and Gerstewitz function ¿kv : Z ^ r. By Proposition 2.7(d), we have

lkv(u) > 0, for every u e r(X). (4.65)

Let g := Ikv ◦ F : X x Y ^ R. From Lemma 3.10, for the mapping ¿kv and Remark 3.15, for each x e X, there exist y*x e Yand f (x, y*x) e F(x, y*x) with f (x, y*x) e Max. UyeY F(x, y) such that lkvf (x,y*x) = maxUyeY&kv(F(x,y)). Choosing u = f (x,y*x) in (4.65), we have

max y г,кv(F(x,y))> ° vx e x. (4.66)

Therefore,

min U max y lkv(F(x,y))> 0. (4 67)

xeX yeY

By conditions (i), (ii) and Remark3.15, the set-valued mapping y ^ Ikv(F(x,y)) is upper semi-continuous, and either above-r+-concave or above-properly r+-quasi-concave on Y, and the set-valued mapping x ^ ¿,kv(F(x,y)) is lower semi-continuous and above-r+-quasi-convex on X. From Theorem 4.3, we have

max y min y t,kv(F(x,y)) > (4 68)

yeY xeX

Since the set-valued mapping y ^ F(x,y) is lower semi-continuous on Y, by Lemma 2.4 (b) and Lemma 2.5 (b), the set-valued mapping y ^ minUxeX&kv(F(x,y)) is upper semi-continuous on Y. By the compactness of Y, there exists y' e Y such that minUxeXlkv(F(x,y')) > 0. For all x e X and all g(x,y') e F(x,y'),we have lkv(g(x,y')) > 0. Thus, v / UxeX F(x,y') + C, and hence,

v e Min. y F{x,y') + C. (4.69)

If v e Max UyeY Minro Uxex F(x, y), by the condition (iii), v e Minro UxeX F(x, y') + C which contradicts (4.69). Hence, for every v e MaxUyeY Minro UxeX F(x,y),

v e Min y Maxro y F(x,y) + C, (4 70)

xeX yeY ^ ' '

that is,

Max y Minro y F(x,y) c Min y Max^ y F(x,y) + C. (4.71)

yeY xeX xeX yeY

This completes the proof. □

The following example illustrates Theorem 4.9. Example 4.10. Let X = Y = [0,1], C = R+ and G : X ^ Y be a set-valued mapping defined as

G(x)H[°,1], x * 0, (4.72)

' {0}, x = 0.

Let F (x,y) = G(x) x {-y2} for all (x, y) e X x Y. Then G is lower semi-continuous, but not upper semi-continuous on r, and F is continuous on Y, and F has nonempty compact values and is lower semi-continuous on X. It is easy to see that F is above-C-concave or above-properly C-quasi-concave on Y and is above-naturally C-quasi-convex on X.

We verify the condition (iii) of Theorem 4.9. Indeed, for all y e Y, UxeXF(x,y) = [0,1] x {-y2} and Minro UxeXF(x,y) = [0,1] x {-y2}. Hence,

y Minro y F(x,y) = [0,1] x [-1,0],

yeY xeX

(4.73)

Max y Minro y F(x,y) = {(1,0)} c Min^ F(x,y) + C.

yeY xeX xeX

Therefore, the condition (iii) of Theorem 4.9 holds.

Since Max UyeY Minro UxeX F(x, y) = {(1,0)} for any y e Y, we can choose xy = 0 e X such that

F {xy, y) = { (0, -y2) } c Max y Minro y F{x,y) - C. (4.74)

yeY xeX

For any continuous increasing function h, the condition (iv) of Theorem 4.9 holds.

Furthermore, since for each x e X,

\jF(x,y) = G(x) x [-1,0],

Maxw U F(x,y) = i{0}X [-1,0], X = 0, ¥Yy v y \({1}x [-1,0])U([0,1] X{0}), x/0,

(4.75)

we have

U Max^ U F(x,y) = ({0} X [-1,0]) U([0,1] x {0}) U({1} x [-1,0]),

xeX yeY

Min U Maxw U F(x,y) = {(0,-1)}.

xeX yeY

Ma^ Minro U F(x,y) = {(1,0)} c {(0,-1)} + C

yeY xeX

= Min y Maxw y F(x,y) + C.

xeX yeY

(4.76)

(4.77)

Therefore, the conclusion of Theorem 4.9 holds.

Remark 4.11. Theorem 3.1 in [5], Theorem 3.1 in [6], or Theorem 4.2 in [7] cannot be applied to Example 4.10 as F is not continuous on X x Y.

If we choose Z = r and C = r+ in Theorems 4.5 and 4.9, we always have C* = r+ and for every y e Y,

max U min U F(x,y) > min U F(x,y). (4.78)

yeY xeX xeX

Hence, the condition (iii) of Theorem 4.5 holds. Thus, we have the following corollaries.

Corollary 4.12. Let X, Y be nonempty compact (not necessarily convex) subsets of real Hausdorff topological vector space X and Y, respectively. Suppose that the set-valued mapping F : X x Y ^ r has nonempty compact values such that it is lower semi-continuous on X and is upper semi-continuous on X x Y. Assume that the following conditions hold:

(i) for each x e X, y ^ F (x,y) is below-R+-concave-like on Y;

(ii) for each y e Y, x ^ F (x,y) is above-R+-convex-like on X;

(iii) for every y e Y,

max U min U F(x,y) > min U F(x,y). (4.79)

yeY xeX xeX

Then, for any

z1 e max (J min (J F(x,y),

yeY xeX

(4.80)

there is a

z2 e mini co< y max (J F(x,y) >

\ [xeX yeY J

(4.8i)

such that

Zi > Z2,

(4.82)

that is,

max y min y F(x, y) > min

yeY xeX

y max y F(x,y)

xeX yeY

(4.83)

Corollary 4.13. Under the framework of Corollary 4.12, in addition, let X, Y be two convex subsets, and let F be upper semi-continuous on X x Y. Then,

max y min y F(x,y) = min y max y F(x,y).

yeY xeX xeX yeY

(4.84)

Proof. By Corollary 4.12, we have

max y min y F(x, y) > min ( co < y max y F(x,y) > ).

yeY xeX I xeX yeY J

(4.85)

Since the set-valued mapping F is upper semi-continuous on X x Y and Y is compact, by Lemmas 2.4 and 2.5, the set-valued mapping x ^ ma^UyeY F(x, y) is upper semi-continuous on X. Since X is convex, it is connected. By [16, Theorem 3.1],

y max y F(x,y)

xeX yeY

(4.86)

is connected in r, and hence, it is convex. From (4.85),

max y min y F(x,y) > mini y max y F(x,y) ).

yeY xeX xeX yeY

(4.87)

This completes the proof. □

When Z = r and C = r+, from Theorem 4.9, we deduce the following corollary.

Corollary 4.14. Let X, Y be two nonempty compact convex subsets in real Hausdorff topological vector spaces X and Y, respectively. Suppose that the set-valued mapping F : X x Y ^ r has nonempty compact values such that it is continuous on Y and is lower semi-continuous on X. Assume that the following conditions hold:

(i) for each y e Y, x ^ F(x,y) is above-naturally r+-quasi-convex on X;

(ii) for each x e X, y ^ F(x, y) is above-R+-concave or above-properly R+-quasi-concave on Y;

(iii) for each y e Y, there exists xy e X such that

max F(xy,y) < max (J min (J F(x,y). (4 88)

yeY xeX ( .

max U F(x,y) = ma^ F(x,y). /4 89)

yeY xeX xeX yeY V ' '

Remark 4.15. Corollary 4.14 includes Proposition 2.1 in [5].

5. Saddle Points for Set-Valued Mappings

In this section, we discuss the existence of several kinds of saddle points for set-valued mappings including the C-loose saddle points, weak C-saddle points, r+-loose saddle points, and R+-saddle points of F on X x Y.

Definition 5.1. Let F:X x Y ^ Z be a set-valued mapping. A point (x,y) e X x Y is said to be a

(a) C-loose saddle point [7] of F on X x Y if

F (x, y) Ma^U F(x,y) ) / 0,

y 7 (5.1)

F(x,y)^( Min U F (x,y)\ / 0;

(b) weak C-saddle point [7] of F on X x Y if

F(x,y) H ( Max^ J F{x, y) ) Q ( Min^ F(x,y) ) / 0; (5.2)

yeY xeX

(c) R+-loose saddle point of F on X x Y if Z = R and

F(x,y) =

min (J F(x,y), max (J F(x,y)

xeX yeY

(d) R+-saddle point of F on X x Y if Z = r and

max U F{x,y) = min U Fix,y) = F(x,y)- (54)

yeY xeX

It is obvious that every weak C-saddle point is a C-loose saddle point and every r+-saddle point is a r+-loose saddle point.

Theorem 5.2. Under the framework of Theorem 4.1, F has R+-saddle point if the set-valued mapping y ^ F(x,y) is continuous.

Proof. By Lemmas 2.4 and 2.5, we attained the max and min in Theorem 4.1. By the compactness of X and Y and the lower semi-continuity of F on X and Y, respectively, there exists (x,y) e X x Y such that

max (J min (J F(x,y) = min (J F(x,y),

yeY xeX xeX

min (J max (J F(x, y) = max (J F(x, y) •

xeX yeY yeY

Combining this with Theorem 4.1, we have

max U F(x,y) = min U F(x,y) = F(x,y), (5 6)

yeY xeX

and hence, F has r+-saddle point. □

Theorem 5.3. Under the framework of Theorem 4.3, F has R+-saddle point if the set-valued mapping y ^ F(x,y) is continuous.

Theorem 5.4. Under the framework of Theorem 4.5 or Theorem 4.9, F has weak C-saddle point if the set-valued mapping y ^ F(x,y) is continuous.

Proof. For any £ e C*, the set-valued mapping £ o F satisfies all the conditions of Theorem 5.2 or Theorem 5.3. Hence, £ o F has r+-saddle point, that is, there exists (x,y) e X x Y such that

max U '^(xy)) = mi^ £(F(x,y) = £(F(x,y)). (5 7)

yeY xeX

Then, for any z e F (x,y),

¿(z) e mi^U ¿(F(x,y)),

¿(z) e ma^U l(F(x,y)).

Thus, by Proposition 3.14,

z e Minro y F(x,y)f)Maxw (J F(x,y),

and (x,y) is a weak C-saddle point of F.

Acknowledgments

In this paper, the first author was partially supported by Grant NSC101-2115-M-039-001- from

the National Science Council of Taiwan. The research part of the second author was done

during his visit to King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia.

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