CC BY
0Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 190821, 8 pages http://dx.doi.org/10.1155/2014/190821
Research Article
The Hierarchical Minimax Inequalities for Set-Valued Mappings
Yen-Cherng Lin1 and Chin-Tzong Pang2
1 Department of Occupational Safety and Health, College of Public Health, China Medical University, Taichung 40421, Taiwan
2 Department of Information Management, and Innovation Center for Big Data and Digital Convergence, Yuan Ze University, Chung-Li 32003, Taiwan
Correspondence should be addressed to Chin-Tzong Pang; imctpang@saturn.yzu.edu.tw
Received 16 December 2013; Accepted 14 March 2014; Published 10 April 2014
Academic Editor: Ngai-Ching Wong
Copyright © 2014 Y.-C. Lin and C.-T. Pang. 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.
We study the minimax inequalities for set-valued mappings with hierarchical process and propose two versions of minimax inequalities in topological vector spaces settings. As applications, we discuss the existent results of solutions for set equilibrium problems. Some examples are given to illustrate the established results.
1. Introduction and Preliminaries
Let X be a nonempty set in a Hausdorff topological vector space, Z a Hausdorff topological vector space, and C c Z a closed convex and pointed cone with apex at the origin with int C = 0; that is, C is properly closed with int C = 0 and satisfies AC c C, for all A > 0; C + CcC;and Cn (-C) = {0}. The scalar hierarchical minimax inequalities are stated as follows: for given mappings F, G : X x X R, under some suitable conditions, the following inequality holds:
min y max y F (x, y) < max Jg (*,*).
(s-Hi)
For given mappings F, G : X x X =3 Z, the first version of hierarchical minimax theorems states that under some suitable conditions, the following inequality holds:
Max y F (x, x) c Min ( co (y Max y F (x, y))) + C.
xeX \ W yeX ) )
(Hi-1)
The second version of hierarchical minimax theorems states that under some suitable conditions, the following inequality holds:
Max y G (x, x) c Min y Max y F (*, y) + C. (Hi-2)
X£X X£X y£X
These versions, (Hi-1) and (Hi-2), arise naturally from some minimax theorems in the vector or real-valued settings. We refer to [1-4] and the references therein.
The notations we use in the above relations are as follows.
Definition 1 (see [1, 3]). Let A be a nonempty subset of Z. A point z e A is called a
(a) minimalpoint of A if An(z-C) = {z};MinA denotes the set of all minimal points of A;
(b) maximal point of A if An(z+C) = {z};MaxA denotes the set of all maximal points of A;
(c) weakly minimal point of A if An(z-int C) = 0;Min^A denotes the set of all weakly minimal points of A;
(d) weakly maximal point of A if A n (z + int C) = 0; Maxw A denotes the set of all weakly maximal points of A.
We note that, for a nonempty compact set A, both sets Max A and Min A are nonempty. Furthermore, Min A c Min^A, Max A c Max^A, A c Min A + C, and A c Max A - C. Following [3], we denote both Max and Max^ by max (both Min and Minm by min) in R since both Max and Maxw (both Min and Minm) are the same in R.
We present some fundamental concepts which will be used in the following.
Definition 2 (see [5, 6]). Let U, V be Hausdorff topological spaces. A set-valued map F : U =3 V with nonempty values is said to be
(a) upper semicontinuous at x0 e U if for every x0 e U and for every open set N containing F(x0) there exists a neighborhood M of x0 such that F(M) c N;
(b) lower semicontinuous at x0 e U ifforanynet [xv] c U, xv ^ x0, y0 e T(x0) implies that there exists net yv e T(xv) such that yv ^ yo;
(c) continuous at x0 eU if F is upper semicontinuous as well as lower semicontinuous at x0.
We note that if T is upper semicontinuous at x0 and T(x0) is compact, then for any net [xv] cU, xv ^ x0, and for any net yv e T(xv) for each v there exists y0 e T(x0) and a subnet [yv } such that yv ^ y0. We refer to [5, 6] for more details.
Definition 3 (see [3, 7]). Let k e intC and v e Z. The Gerstewitzfunction ^kv : Z ^ R is defined by
%kv (u) = min [t e R : u e v + tk-C}.
Some fundamental properties for the Gerstewitz function are as follows.
Proposition 4 (see [3, 7]). Let k e int C and v e Z. The Gerstewitz function ^kv : Z ^ R has the following properties:
(a) £kv(u) > r & u i V + rk - C;
(b) %kv(u) >r & u i V + rk- int C;
(c) ^kv(-) is a convex, continuous, and increasing function.
We also need the following different kinds of cone-convexities for set-valued mappings.
Definition 5 (see [1]). Let X be a nonempty convex subset of a topological vector space. A set-valued mapping F : X =3 Z is said to be
(a) above-C-convex (resp., above-C-concave) on X if, for all x1 ,x2 e X and all X e [0,1],
F (Xx1 + (1 - A) x2) c IF (x1) + (1-X)F (x2) - C
(resp., XF (x1) + (1-X)F(x2) c F (Xx1 + (1 - X) x2)-C);
(b) above-naturally C-quasiconvex on X if, for all X and all X e [0,1],
F (Xx1 +(1-X)x2) c co [F (x1) U F (x2)} - C, (3)
where co A denotes the convex hull of a set A;
(c) above-C-convex-like (resp., above-C-concave-like) on X (X is not necessary convex) if, for all x1,x2 e X and all A e [0,1], there is an x e X such that
F (x) cXF(x1) + (1-X)F (x2) - C (resp., XF (x1) + (1-X)F (x2) c F (x') - c) .
We note that whenever F is a scalar function and C = R+, the mappings in Definition 5 reduce to the classical ones. The following theorem is a special case of the scalar hierarchical minimax theorem by Lin [8].
Theorem 6. Let X be a nonempty compact convex subset of real Hausdorff topological vector space. Let the set-valued mappings F, G, H : X x X =3 R such that F(x, y) c G(x, y) c H(x,y) for all (x,y) e XxX; \Jy£X F(x,y) and H(x,y) are compact for each x e X and for each y e X and satisfy the following conditions:
(i) x ^ F(x, y) is lower semicontinuous on X for each y e X and y ^ F(x, y) is above-R+-concave on X for each x e X;
(ii) x ^ G(x,y) is above-naturally R+-quasiconvex for each y e X, and y ^ G(x, y) is lower semicontinuous on X for each x e X;
(iii) x ^ H(x,y) is lower semicontinuous on X for each y e X, y ^ H(x,y) is above-R+-concave on X for each x e X, and y ^ H(x, y) is lower semicontinuous for each x e X.
Then one has
min y max y F (x, y) < max y min y H (x, y). (5)
yeX yeX
Lemma 7. Let F : X =i R be such that max Uxex F(x), max Uxex max F(x), and max F(x) exist for all x e X. Then
= max y max F (x).
Proof. By using the similar technique of Lemma 3.3 [9], we can show that the conclusion is valid. □
2. Scalar Hierarchical Minimax Inequalities
We first state the following scalar hierarchical minimax inequalities.
Theorem 8. Let X be a nonempty compact (not necessarily convex) subset of a real Hausdorff topological space. Let the set-valued mappings F,S,T,G : X x X =3 R with nonempty compact values such that
(i) (x, y) ^ F(x, y) and (x, y) ^ G(x, y) are upper semicontinuous on X x X;
(ii) x ^ max S(x,y) is convex-like for each y e X, and y ^ max T(x, y) is concave-like on Y for each x e X;
(iii) for all (x,y) e XxX, max F(x,y) < max S(x,y) < max T(x,y) < maxG(x,y).
Then the relation (s-Hi) holds.
Proof. From (i), we know that both sides of (s-Hi) exist. For any r e R,
r > max y G (x, x).
Define M : X =lX by
M(x) = {yeX: max F (x, y) > r} (8)
for all x e X.By (i), the set M(x) is closed for all xeX.We claim that the whole intersection
is empty. Indeed, if not, there exists y0 e Hxex M(x) such that, for all x e X, maxF(x,y0) > r. In particular, we choose x = y0; then maxF(y0, y0) > r which, with the aid of condition (iii), contradicts the choice of r. Hence, by the compactness of X, there exist x1,x2,... ,xm e X such that
XcQ(X\M(x,)).
N(x) = {yeX: max T (x, y) > r} (11)
for all x e X. Then, by (iii), we have
Xc^(X\N(x,)).
This implies that, for each y e X, there is xio e [x1,x2,..., xm} such that
max T (xi0, y) < r. (13)
Define two sets as follows:
L1 : = co {(max T(x1,y), max T(x2,y),
..., max T(xm,y)) e Rm : y e X], L 2 := {(z1,z2'...,zm) e K™ : >r
By the concave-like property of T, we can see that these two sets are disjoint. For each y e X,by the separation theorem, there exists nonzero vector (t1,t2, ..., Tm) e Rm such that
max T(xi,y) < Y?izi>
¡=1 ¡=1
for all (z1, z2,..., zm) e L 2. Then, tt > 0 and Tj > 0 for all i = \,2,...,m. Let St = tj Y==1 ri for all i = 1,2,... ,m. Then we have
Y,8*max T(xi,y) <
For each i = 1,2,... ,m, by taking zi = r and noting maxS(Xj,y) < max T(xt, y),wehave
YSj max S (xj, y) < r,
for all y e X. Since the mapping x ^ max S(x, y) is convex-like for each y e Y, there is x0 e X such that
max S (x0, y) < r.
Since max F(x,y) < max S(x,y) for all (x,y) e X x X, we have
max F (x0, y) < r, (19)
for all y e X.By Lemma 7, we know that
min y max y F (x, y) < max y F (x0, y)
x£X yeY yeX
= max y maxF (x0, y) < r.
Therefore, the relation (s-Hi) holds.
Theorem 9. Let X be a nonempty compact convex subset of a real Hausdorff topological vector space. Let the set-valued mappings F,G : X x X R with nonempty compact values such that
(i) (x, y) ^ F(x, y) and (x, y) ^ G(x, y) are upper semicontinuous on X x X;
(ii) y ^ max F(x, y) is quasiconcave for each x e X; that is, for each x e X, the set [y e X : max F(x, y) > r} is convex in X;
(iii) for all (x,y) eXxX, max F(x,y) < max G(x,y). Then the relation (s-Hi) holds.
Proof. By (i), we know that both sides of (s-Hi) exist. Choose any re R satisfies
r > max y G (x, x). (21)
Define W: X X by
W(x) = [yeX: max F(x,y)>r], (22)
for all x e X.By (ii), the set W(x) is convex for all x e X.By (iii), we have
maxF (x, x) < maxG (x, x) < max y G (x, x) < r. (23)
Hence,
x$W(x),
for all x e X.By the upper semicontinuity of F, we know that the mapping x ^ max F(x, y) is upper semicontinuous for each x e X. Thus, for each x e X, W(x) is closed; hence it is compact. In order to claim that the mapping x ^ W(x) is upper semicontinuous on X, we only need to show that the mapping x ^ W(x) has a closed graph. Since, for any net [(xa, ya)} c Graph(W) we have the net [(xa, ya)}. converges to some point (x0, y0). Then, for each a,
maxF(xa, ya) > r. Since the mapping (x, y) ^ max F(x, y) is upper semicontinuous, we have
max F (x0, y0) > lim sup max F (xa, ya) > r.
Thus, (x0, y0) e Graph(W). Suppose that W(x) = 0 for all x e X. Then, by Kakutani fixed point theorem, the mapping W has a fixed point which is a contradiction to (24). Hence, there is an x0 e X such that W(x0) = 0. From this, we know that
r > min У max У F (x, y).
xeX yiY
This implies that the relation (s-Hi) holds. □
The following examples illustrate Theorems 8 and 9.
Example 10. Let X = {0} U {1/n : n e N} and f(x) = X2, g(y) = 1-y2 for all x,yeX. Define F,S,T,G : X x X =i R by F(x,y) = [0,f(x)g(y)], S(x,y) = [-1, f(x)g(y) + 1], T(x,y) = [2, f(x)g(y) + 2], and G(x,y) = [3,f(x)g(y) + 3]. Obviously, all conditions of Theorem 8 hold. Hence the relation (s-Hi) holds. Indeed, by simple calculation, we can see that
min У max У F (x, y) = 0,
хчХ учХ
max У G(x, x) = —.
Example 11. Let X = [0,1]. The mappings f,g,F, and G are the same as in Example 10. Then, all conditions of Theorem 9 hold. We can see that both values of min Uxex max Uy^x F(x,y) and max Ц*6х G(x,x) are the same as those in Example 10. Hence the relation (s-Hi) holds.
3. Hierarchical Minimax Inequalities
In this section, we will present two versions of hierarchical minimax inequalities. The following theorem is the first result satisfies the relation (Hi-1).
Theorem 12. Let X be a nonempty compact convex subset of a real Hausdorff topological vector space. Let the set-valued mappings F,G, H : XxX =t Z with nonempty compact values such that F(x,y) с G(x,y) с H(x,y) for all (x,y) e XxY satisfy the following conditions:
(i) (x, у) ^ F(x, y) is upper semicontinuous, у ^ F(x, y) is above-C-concave on Y for each x e X, and x ^ F(x,y) is lower semicontinuous on X for each у e Y;
(ii) x ^ G(x,y) is above-naturally C-quasiconvexfor each у eY, and у ^ G(x, y) is lower semicontinuous on Y for each x e X;
(iii) у ^ H(x, y) is lower semicontinuous and above-C-concave on Y for each x e X, and x ^ H(x,y) is lower semicontinuous on X for each у e Y;
(iv) for each у eY,
Max\JF (x, x) с Min\ H (x, y) + C. (28)
хчХ хчХ
Then the relation (Hi-1) is valid.
Proof. Let Г(х) := Maxw \JyeY F(x,y) for all x e X. From Lemma 2.4 and Proposition 3.5 in [1], the mapping x ^ Г(х) is upper semicontinuous with nonempty compact values on X.Hence Uxex П*) is compact and so is co(UxeX T(x)).Then co(Uxex П*)) + С is a closed convex set with nonempty interior. Suppose that v i co(Uxex П*)) + C. By separation theorem, there is a к e R, e > 0, and a nonzero continuous linear functional Ç : Z ^ R such that
%(v) <k- e < к <^(u + c),
for all u e co(\JxeX T(x)) and c e C.Fromthiswecanseethat Ç e C*, where C* = [g : Z ^ R : g(c) > e C}, and ç(v) < Ç(u) for all u e co(UxeX T(x)). By Proposition 3.14 of [1], for any x e X, there isa y* eY and f(x, y*) e F(x, y*) with f(x, y*x) e r(x) such that
tf (х'У*х) = max \$Р(х'У)-
Let us choose с = 0 and и = f(x, yX) in (29); we have
t(v) < t(f(x,y*)) = max У^(х,у) Ш)
for all x e X. Therefore,
(v) < min У max У <;F (x, y).
xeX yeY
From conditions (i)—(iii), by applying Proposition 3.9 and Proposition 3.13 in [1], all conditions of Theorem 6 hold. Hence, we have
<; (v) < max \ min \ <;H (x, y).
yiY xeX
Since Y is compact, there is y eY such that $(v) < min \J$H(x,y').
and, hence,
v i Ун(х,у') + С,
v i MinУн(х,у') + С.
Therefore,
MinУн(х,у') + Cc co (Уг(х)) + С. (37)
W xeX \xeX J
By taking into account condition (iv), we know that Thus,
MaxJF(x,x)c Mm JH{x,y')+ C. (38)
Hence, the relation (Hi-1) is valid. □
The following example illustrates that Theorem 12 is valid. Example 13. Let X = [0,1], C = R+,and f: X R define
[-1,0], y = 0, {0}, y = 0,
f(y) =
andF,G,H:XxX=l R2define
F{x,y) = {sin {^T)}xf(y)
G(x,y) =
H(x,y) = for all (x,y) eXxX.
. xn 0, sin I —
, xn 0, sin | —
x[y-1,0], (40) x [y2 - 1,0],
We can easily see that F(x, y) c G(x, y) c H(x, y) for all (x,y) e X xY and conditions (i)-(iii) of Theorem 12 are valid. Now we claim that condition (iv) holds. Indeed,
\jF(x,x) = ( y {sin(y)}x{0})U«0}x[-1,0])
xeX \xe(0,1] /
= (([0,1]x{0})U({0}x[-1,0])).
Hence, Maxw F(x,x) = {0} x [0,1]. On the other hand, U^xH(x,y) = [0,1] x [y2 - 1,0]. Hence,
Min\jH(x,y) = ({0} x [0,1]) y ({1} x [y2 - 1, 0]) .
Thus, condition (iv) of Theorem 12 holds. By Theorem 12, the relation (Hi-1) is valid. Indeed,
Max jF(x,x) = {(1,0)},
Max jF(x,y) = jF(x,y) = {sin (^r)} X [-1,0]'
yeX yeX
Hence,
co y Max yF(x,y)= y Max y F (x, y)
x£X y£X x£X y£X
= [0,1] x [-1,0].
Min ( co J Max jF(x,y) ) = {(0,-1)}, (45)
\ xeX W yeX J
and hence the conclusion of Theorem 12 is valid.
Theorem 14. Let X be a nonempty compact convex subset of real Hausdorff topological vector space. Let the set-valued mappings F,G, H : X x X =1 Z such that F(x, y) c G(x, y) c H(x, y) for all (x,y) e X x X and satisfy the following conditions:
(i) (x, y) ^ F(x, y) is continuous with nonempty compact values, and y ^ %kvF(x,y) is above-R+-concave on X for each x e X and any Gerstewitz function ^kv;
(ii) x ^ G(x,y) is above-naturally C-quasiconvexfor each y e X, and y ^ G(x, y) is lower semicontinuous on X for each x e X;
(iii) (x,y) ^ H(x,y) is upper semicontinuous with nonempty compact values, y ^ ^kvH(x, y) is above-R+-concave on X for each x e X, and x ^ H(x,y) is lower semicontinuous on X for each y e X and any Gerstewitz function ^kv;
(iv) for each y eY,
Max J G ^ x) c Min J H ^ y) + C. (46)
X£X X£X
Then the relation (Hi-2) is valid.
Proof. Let r(x) be defined the same as that in Theorem 12 for all x e X. From the process in the proof of Theorem 12, we know that the set JxgX H*) is nonempty compact. Suppose that v i J^x r(x)+C.Foranyfc e int C,thereisaGerstewitz function ^kv : Z ^ R such that
4v (u) > 0,
for all u e JxgX r(x). Then, for each x e X, there is y* e X and f(x,y*) e F(x,fx) with f(x,y*) e Maxw Jy6X F(x,y) such that
ïkv (f(^ fx)) = max y tkvF (^ y).
Choosing u = f(x, y*x) in (47), we have
max J 4vF (x, y) > 0,
for all x e X. Therefore,
min J max J ^F (x, y) > 0.
X£X y£X
By conditions (i)—(iii), we know that all conditions of Theorem 6 hold for the mappings ÇkvF(x, y), i;kvG(x, y), and ÇkvH(x, y), and, hence, we have
max y min y ÇkvH (x, y) > 0.
y£X X£X
Since X is compact, there is a y e X such that
min \J$kvH(x,y')>0. (52)
v t \H(x,y')+C, (53)
and, hence,
V t Mjn\H{X,y') + C. (54)
If v e Max \JyeX Min^ U*eX F(x> y), then, by (iv), we have v e Min\H{x,y') + C, (55)
which contradicts (54). From this, we can deduce that the relation (Hi-2) is valid. □
4. Strong and Weak Solutions for SEP
In our previous work [10], we establish existence of solutions for set equilibrium problems (SEP, for short). Let Y be a Hausdorff topological vector space, and let K be a nonempty compact convex subset of a Hausdorff topological vector space. For a given mapping T : K =3 Y and a trimapping F : TK x K x K =i Z,a weak solution for (SEP)F is a point x e K such that
F(s,x,y)t- int C, (56)
for all y e K and for some s e T(x). A strong solution for (SEP)f is a point x e K with some s e T(x) such that
F(s,x,y)t- int C, (57)
for all y e K. A strong solution is obviously a weak solution for (SEP) for the same mapping.
We recall that a set-valued mapping Q : X ^ Z is called a KKM mapping if cof^,..., xn] c \i=1 Q(xi) for each finite subset {x1,..., xn] c X.
Fan Lemma (see [11]). Let Q : X =i Z be a KKM mapping with nonempty closed values. If there exists an x0 e X such that Q(x0) is a compact set of Z, then Pl*^ Q(x) = 0.
We first state that the existent result of weak solution for (SEP) is as follows.
Theorem 15. Let Z be a finite dimensional space and the set-valued mappings F and T are two upper semicontinuous mappings with nonempty compact values such that,
(i) for each x e K, there is s e T(x) such that F(s, x, x) C
- int C;
(ii) for each x e K, the sets {(s, y) e TK x K : F(s, x, y) c
- int C} and T(x) are convex.
Then (SEP)f has a weak solution. Proof. Define Q : K =i K by
Q (y) := [x e K : F (s, x,y) C - int C for some s e T (%)},
for all y e K. By (i), y e Q(y) for all y e K. Hence the set Q(y) is nonempty for all y e K. Next, we claim that the set Q(y) is closed for all y e K. Let a net {xa} c Q converge to some point x0. Then there are sa e Txa and za e F(xa, xa, y) such that za e Z\(- int C). By the upper semicontinuities of F and T, the sets TK and F(TKxKxK) are compact. Hence, there is a convergent subnet {za } of {za} that converges to some point z0. Furthermore, the net {sa } has a convergent subnet {sa } which converges to some point s0. Again, by the
upper semicontinuities of F and T, we have z0 e T(x0) and z0 e F(s0,z0,y). Since the set Z \ (- int C) is closed, z0 e Z\(- int C). Hence, x0 e Q(y), and, thus, Q(y) is closed for all y e K. We next claim that the mapping Q : K =3 K is a KKM mapping. Indeed, if not, there exist x1,x2,... ,xn e K and x0 such that
X0 e co [{Xi,X2,...,Xn} c\Q(x). (59)
Then there is x0 = Yn=1 where ^i=1 =1 and Xt > 0 for all i = 1,2,...,n.
Since x0 t \\n=1 Q(xi), for all i e {1,2,..., n}, choose any st e T(x0); we have
F(si,x0,xi)c- int C. (60)
By(ii),
S0 eT (X0),
(s0,x0) e {(s,%) e TK x K : F (s,x0, x) c - int C}. This implies that
F (s0, x0, x0) c - int C, (62)
which contradicts (i). Thus, the mapping Q : K =3 K is a KKM mapping. By the Fan lemma, the whole intersection
r^OO (63)
is nonempty. Any point in the whole intersection is a weak solution for (SEP)f. □
For the existence of strong solution for (SEP), we propose the following results.
Theorem 16. Under the framework of Theorem 15, in addition, the mappings A,B,G : TK x K x K =3 Z with nonempty compact values such that
(i) the mapping s ^ G(s, x, y) is upper semicontinuous mappings for each x,ye K;
(ii) both sets UseT(x) x, y) and UysK G(s, x, y) are compact for x, y e K,s e T(x);
(iii) the mapping s ^ max B(s,x,y) is concave-like for each x,y e K, and the mapping y ^ max A(s, x, y) is convex-like for each x, y e K,s e T(x);
(iv) for each x,y e K,s e T(x), max F(s,x,y) < max A(s,x,y) < max B(s,x,y) < max G(s, x,y);
(v) for each y e K, there is an x e K with ~sy e T(x) such that
max G (ly, x, y) < max y min y G (s, x, y). (g^)
seT(x) yeK
Then (SEP)g has a strong solution.
Proof. According to Theorem 15, we know that (SEP)F has a weak solution. That is, there is an x e K such that
F (sx, x,x)<t- int C,
for all x e K and for some sx e T(x). For any k e int C, from Proposition 4, the Gerstewitz function %kQ satisfies
ÇkQ F(~sx,x,x)>0. (66)
Hence, there is x e K such that, for each x e K,
max y <;k0F (s, x, x) > 0.
S£T(x)
Thus, we have
min y max y Çk0F (s, x, x) > 0. (68)
xeK seTÇx)
By conditions (i)-(v), all conditions of Theorem 6 hold; hence we have
max y min y %kClG (s, x, x) > 0.
seT(x) xeK
Since T(x) is compact, there is s e T(x) such that min y <;k0G (s, x, x) > 0.
This implies that
ÇkQG (s, x,x)>0
G (s, x,x)t- int C, (72)
for all x e K. Therefore, (SEP)G has a strong solution. □
Finally, we give the following example to illustrate that Theorems 15 and 16 are valid.
Example 17. Let K = [1,2], C = R+, Z= R, and T : K =i R be defined by T(x) = [0,2x] for all x e K. Then we define F,A,B,G:TKxKxK =i R which are defined by
F (s, x, y) = {x (y - Çsx) : 0, e A (s, x, y) = {x (y - Çsx) + 1 : Ç e B (s, x, y) = {x (y - Çsx) +2 : Ç e G (s, x,y) = \x(y - Çsx) + 3 : 0, e
for all (x, y) eKxK.
Then, the set-valued mappings F and T are two upper semicontinuous mappings with nonempty compact. We can easily see that F(s, x,x) £ - int C for all x e K and if we choose any s e T(x) n [0,1]. So, condition (i) of Theorem 15 holds. It is obvious that condition (ii) of Theorem 15 holds since the mapping
(s,y) ^F(s,x,y)
is linear. Hence (SEP)F has a weak solution by Theorem 15. Indeed, x = 2 is a weak solution for (SEP)F where we can choose s = 1/2.
Next, we claim that (SEP)G has a strong solution. We can easily deduce that conditions (i), (iii), and (iv) hold. The condition (ii) is valid since
\jF(s,x,y)= y |x(7-Çs*):Çe
seT(x) se[0,2x]
= [xy - 2x3, xy] is compact for x,yeK and so is
yG(s,*,^)= y {*fr-Çs*) + 3:Çe
ycK yi[l,2]
2 , 0 ,
x- sx +3,2x---+ 3
for x e K,s e T(x). Finally, condition (v) of Theorem 16 is valid, since, for each y e K, we can choose an x e K with x > yy - 1 and Iy = 2x such that
iG{sy,x,y) = x(y--y)+3
< x + 3
= max y min \ G (s, x, y).
seT(x) yeK
Indeed, x = 3/2 with s = 2 is a strong solution for (SEP)G.
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
Acknowledgments
In this research, the first author was supported by Grant no.
NSC102-2115-M-039-001- of the National Science Council of
Taiwan (Taiwan). The second author was supported partly by
National Science Council of the Republic of China.
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