Scholarly article on topic 'Minimax problems under hierarchical structures'

Minimax problems under hierarchical structures Academic research paper on "Mathematics"

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Academic research paper on topic "Minimax problems under hierarchical structures"

Lin Journal of Inequalities and Applications (2015) 2015:57 DOI 10.1186/s13660-015-0575-x

O Journal of Inequalities and Applications

a SpringerOpen Journal

RESEARCH

Open Access

Minimax problems under hierarchical structures

Yen-Cherng Lin*

*Correspondence: ydin@maiLamu.edu.tw Department of OccupationalSafety and Health, College of Public Health, China MedicalUniversity, Taichung, 40421, Taiwan

Abstract

We discuss the minimax problems for set-valued mappings with several hierarchical structures, and scalar hierarchical minimax theorems, hierarchical minimax theorems, hierarchical minimax inequalities for set-valued mappings, and the existence of cone-saddle points.

Keywords: minimax theorems; minimax inequalities; cone-convexities; cone-saddle points

ringer

1 Introduction and preliminaries

Let U, V be two nonempty sets in two Hausdorff topological vector spaces, respectively, W be a Hausdorff topological vector space, D c W a closed convex and pointed cone with apex at the origin and intD = 0. Let D* = {g e W* :g(c) > 0 for all c e D}, where W* is the set of all continuous linear functional on W. The scalar hierarchical minimax theorems are introduced and discussed by Lin [1] as follows: given three mappings A, B, C: U x V ^ R, under suitable conditions the following relation holds:

min y max y A(u, v) < max y min y C(u, v). (sH)

ueU veV veV ueU

In [1], the three versions (H1)-(H3) of minimax theorems with hierarchical structures are also discussed: given three mappings A, B, C: U x V ^ W, under suitable conditions the following relation holds:

Ma^y Minw y C(u, v) c Mi^c^y Max^yA(u, v)] + D, (Hi)

veV ueU ^ ueU veV '

Ma^y Minw y C(u, v) c Min y Maxw y A(u, v) + D, (H)

veV ueU ueU veV

Min y Max^ y A(u, v) c Max y Min^ C(u, v) + W \ (D \ {0}). (H3)

ueU veV veV ueU

In [2], given three mappings A, B, C: U x U ^ W, Lin et al. investigated the following two versions of minimax inequalities, the so-called hierarchical minimax inequalities:

Max y C(u, u) c Minicoi y MaxwyA(u, v)]) + D, (Hi)

ueU ueU veU

Max y C(u, u) c Min y Max^ y A(u, v) + D. ()

ueU ueU veU

© 2015 Lin; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

In this paper, we propose new hierarchical structures relative to several non-continuous set-valued mappings which obey one of the following relations: (sH), (Hi), (H2), (H3), (Hii), and (Hi2). As applications, the existence of saddle points for set-valued mappings is also discussed.

The fundamental concepts of maximal (minimal) point and weakly maximal (weakly minimal) point will be used in the sequel.

Definition 1 [3,4] Let L be a nonempty subset of W. A point w e L is called a

(a) minimal point of L if L n (w - D) = {w}; Min L denotes the set of all minimal points ofL;

(b) maximal point of L if L n (w + D) = {w}; Max L denotes the set of all maximal points ofL;

(c) weakly minimal point of L if L n (w - int D) = 0; MinwL denotes the set of all weakly minimal points of L;

(d) weakly maximal point of L if L n (w + int D) = 0; MaxwL denotes the set of all weakly maximal points of L.

Both Max and Maxw are denoted by max (both Min and Minw by min) in R since both Max and Maxw (both Min and Minw) are the same in R. We note that for a nonempty compact set L, both sets MaxL and MinL are nonempty. Furthermore, MinL c MinwL, MaxL c MaxwL, L c MinL + D, and L c MaxL - D.

Definition 2 [5, 6] Let U, V be two Hausdorff topological spaces. A set-valued mapping F: U ^ V with nonempty values is said to be

(a) upper semicontinuous on U if for any xo e U and for every open set N containing F(xo), there exists a neighborhood M of xo such that F(M) c N;

(b) lower semicontinuous on U if for any xo e U and any sequence {xn} c U such that xn ^ xo and any yo e F(xo), there exists a sequence yn e F(xn) such that yn ^ yo;

(c) continuous on U ifF is both upper semicontinuous and lower semicontinuous at any xo e U.

Definition 3 [4, 7] The Gerstewitzfunction pkw: W ^ R is defined by

Vkw(u) = min{t e R: u e w + tk - D}, where k e int D and w e W. Some properties of the scalarization function are as follows:

Proposition 1 [4, 7] The Gerstewitz function pkw : W ^ R has the following properties:

(a) Vkw(w) >r we w + rk - D;

(b) ^kw(w) > r we w + rk - int D;

(c) ^kw(-) is a convex function;

(d) ^kw(-) is an increasing function, that is, w2 - wi e int D ^ ^kw(wi) < ^kw(w2);

(e) ^kw(-) is a continuous function.

We also need the following cone-convexities for set-valued mappings.

Definition 4 [3] Let U be a nonempty convex subset of a topological vector space. A set-valued mapping F: U ^ W is said to be

(a) above-D-convex (respectively, above-D-concave) on W if for all ui, u2 e U and all a e [o,i],

F(aui + (i - a)u2) c aF(ui) + (i - a)F(u2) - D (respectively, aF(ui) + (i - a)F(u2) c F(aui + (i - a)u2) - D);

(b) above-naturally D-quasi-convex on W if for all ui, u2 e U and all a e [o, i],

F(aui + (i - a)u2) c co{F(ui) U F(u2)} - D,

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

(c) above-D-quasi-convex on W if for each w e W, the set {u e U: F(u) c w - D} is a convex subset of U.

By definition, the above-D-convex mapping is also an above-naturally D-quasi-convex on U. The following whole intersection theorem is a variant form of Ha [8].

Lemma 1 Let U be a nonempty convex subset of a real Hausdorff topological space, V be a nonempty compact convex subset of a real Hausdorff topological space. Let the three mappings L,M,N: U ^ V with L(u) c M(u) c N(u) for allu e U satisfy

(a) L(u), N(u) are open in V for each u e U, L-i(v), N-i(v) are convex in U for each ve V; and

(b) V \ M(u) is convex for each u e U, and M-i (v) is open in U for each v e V.

Then either there is an vo e V such that L-i(vo) is a empty set, or the whole intersection P|veVN-i(v) is nonempty.

In the sequel we also need the following proposition.

Proposition 2 Let U be a nonempty set, k e intD and w e W. Suppose that the set-valued mappings F, G: U ^ W come with nonempty compact values and, for some u e U, Maxw F(u) c Maxw G(u) - D. We have thefollowing two results:

(a) for any y e D*, the inequality

max yF(u) < max yG(u) holds;

(b) for the Gerstewitzfunction ykw: W ^ R, the inequality

maxykwF(u) < maxykwG(u) holds.

Proof For the proof of (a), we refer to Proposition 1.2 [9]. We omit the proof of (b) since it is quite similar to the proof of (a). □

2 Scalar hierarchical minimax theorems

We first establish the following scalar hierarchical minimax theorem.

Theorem 1 Let U be a nonempty convex subset of a real Hausdorff topological space, V be a nonempty compact convex subset of a real Hausdorff topological space. Suppose that the set-valued mappings A, B, C : U x V ^ R with nonempty compact values satisfy the following conditions:

(i) the mappings u ^ A(u, v) and u ^ C(u, v) are above-R+-quasi-convex on Ufor each v e V, and the mappings v ^ A(u, v) and v ^ C(u, v) are upper semicontinuous on V for each u e U;

(ii) the mapping u ^ B(u, v) is upper semicontinuous on U for each v e V, and the mapping v ^ B(u, v) is above-R+-concavefor each u e U; and

(iii) for all (u, v) e U x V, maxA(u, v) < maxB(u, v) < max C(u, v). Then, for each t e R, either there isv0 e V such that

C(u, vo) n (t + R+) = 0

for all u e U, or there is u0 e U such that

A(u0, v) c t - intR+

for all v e V.

Proof Give any t e R. Define three mappings L, M, N: U ^ V by

L(u) = {v e V: Vh e C(u, v), h < t}, M(u) = {v e V: Vg e B(u, v),g < t},

N(u) = {v e V: Vf e A(u, v),f < t}

for all u e U .By (iii), L(u) c M(u) c N(u) for all u e U.

Since the mapping u ^ C(u, v) is above-R+-quasi-convex on U for each v e V, the set L-1(v) is convex for each v e V. Similarly, the set N-1(v) is convex for each v e V. Next, we claim that the set L(u) is open in V, or the set V \ L(u) = {v e V: 3h e C(u, v), h > t} is closed for each u e U. For any net {vv }c V \ L(u) that converges to some point v0 e V, there exists hv e C(u, vv) suchthat hv > t. By the upper semicontinuity of H at v, C(u, v0) is compact. By Lemma 2.2 [10], there exist h0 e C(u, v0) and a subnet {hVa} that converges to h0. Since hVa > t, we have h0 > t, and hence v0 e V \ L(u). This proves that the set V \ L(u) is closed, and the set L(u) is open for each u e U. Similarly, by the upper semicontinuity of A and B, the sets M-1(v) and N(u) are open for each u e U and v e V.

Next, we claim that the set V \ M(u) is convex in V for each u e U. For each u e U, for any v1, v2 e V \ M(u) andanyr e [0,1]. There exist g1 e B(u, v1)withg1 > t and g2 e B(u, v2) with g2 > t, tg1 + (1 - t)g2 > t. By the above-R+-concavity of B,

Tg1 + (1 - T)g2 c B(u, Tv1 + (1 - t)v^ - R

Thus, there is agT e B(x, t vi + (1- t)v2) such that rgi + (1- t)g2 < gT. Hence, t v1 + (1- t) v2 e V \ M(u) and the set V \ M(u) is convex in V for each u e U.

Since all conditions of Lemma 1 hold, by Lemma 1, either there is an vo e V such that £-1(vo) is an empty set, or the whole intersection P|veVD-1(v) is nonempty. That is, for each t e R, either there is vo e V such that

C(u, vo) n (t + R+) = 0

for all u e U,or there is uo e U such that

A(uo, v) c t - int R+

for all v e V. □

Theorem 2 We work under the framework of Theorem 1, in addition, U is compact, for each (u, v) e U x V, the union U«eU C(u, v) is compact, and the mappings u ^ A(u, v) and v ^ C(u, v) are lower semicontinuous on U and V, respectively. If the following condition holds: for each v e V, there is anuv e U such that

max C(uv, v) < max ^J min ^J C(u, v), (L)

veV ueU

then (sH) is valid.

Proof For any t > max (JveV min (JueU C(u, v). From (L), we see that, for each v e V there is an uv e U such that

C(uv, v) n (t + R+) = 0.

Hence, by Theorem i, there is uo e U such that

A(uo, v) c t - intR+

for all v e V. This will suffice to show that (sH) holds. □

We note that Theorems i and 2 include some special cases as follows.

Corollary 1 If we replace (iii) of Theorem 2 by any one of the following conditions:

(i) for all (u, v) e U x V, A(u, v) = B(u, v) = C(u, v);

(ii) for all (u, v) e U x V, A(u, v) c B(u, v) = C(u, v);

(iii) for all (u, v) e U x V, A(u, v) = B(u, v) c C(u, v);

(iv) for all (u, v) e U x V, A(u, v) c B(u, v) c C(u, v);

(v) for all (u, v) e U x V, maxA(u, v) < maxB(u, v) < max C(u, v), but A(u, v) = B(u, v) = C(u, v);

(vi) for all (u, v) e U x V, maxA(u, v) < maxB(u, v) < max C(u, v), but A(u, v) c B(u, v) = C(u, v);

(vii) for all (u, v) e U x V, maxA(u, v) < maxB(u, v) < max C(u, v), but A(u, v) = B(u, v) c C(u, v),

then (sH) is valid.

We state the first one of Corollary 1 as follows.

Corollary 2 Let U, V be two nonempty compact convex subset of real Hausdorff topological spaces, respectively. Suppose that the set-valued mappings A : U x V ^ R come with nonempty compact values and satisfy the following conditions:

(i) the mapping u ^ A(u, v) is above-R+ -quasi-convex on Ufor each v e V, and the mapping v ^ A(u, v) is continuous on Vfor each u e U;

(ii) the mapping u ^ A(u, v) is continuous on Ufor each v e V, and the mapping v ^ A(u, v) is above-R+-concavefor each u e U.

If the following condition holds: for each v e V, there is anuv e U such that

maxA(uv, v) < max ^J min ^J A(u, v),

veV ueU

then (sH) with A = B = Cis valid.

From Proposition 3.12 [3], every above-naturally R+-quasi-convex is an above-R+-quasi-convex. We can see that Corollary 1 slightly generalizes Theorem 2.1 [4].

3 Hierarchical minimax theorems

In this section, we will discuss three versions of hierarchical minimax theorems. The first one is as follows.

Theorem 3 Let U, V be nonempty compact convex subsets of real Hausdorff topological spaces, respectively, W be a complete locally convex Hausdorff topological vector space. Suppose that the set-valued mappings A, B, C: U x V ^ W come with nonempty compact values and satisfy the following conditions:

(i) (u, v) ^ A(u, v) is upper semicontinuous on U x V, and u ^ A(u, v) is above-naturally D-quasi-convex and lower semicontinuous on U for each v e V;

(ii) u ^ B(u, v) is upper semicontinuous on U for each v e V, and v ^ B(u, v) is above-D-concave on V for each u e U;

(iii) (u, v) ^ C(u, v) is upper semicontinuous on U x V, u ^ C(u, v) is above-naturally D-quasi-convex on Ufor each v e V, and v ^ C(u, v) is continuous on V for each u e U;

(iv) for any y e C* and for each v e V, there is an uv e U such that

maxyC(uv, v) < max ^J min ^J yC(u, v);

veV ueU

(v) for each v e V,

Max y Minw y C(u, v) c Minw y C(u, v) + D; and

veV ueU ueU

(vi) for all (u, v) e U x V, Maxw A(u, v) c MaxwB(u, v) - D, and Maxw B(u, v) c Maxw C(u, v) - D.

Then (H1) is valid.

Proof We omit some parts of the proof since the techniques of the proof are similar to Theorem 3.1 [1]. Suppose that v £ co(Uu£U MaxwUv£VA(u, v)) + D. There is a nonzero continuous linear functional y : Z ^ R such that

y(v) < min ^J max ^J yA(u, v).

u£U v£V

Since u ^ A(u, v) and u ^ C(u, v) are above-naturally D-quasi-convex for each v £ V,by Proposition 3.13 [3], u ^ yA(u, v) and u ^ yC(u, v) are above-naturally R+-quasi-convex for each v £ V and y £ C*. Since v ^ B(u, v) is above-D-concave on V for each u £ U, by Proposition 3.9 [3], v ^ yB(u, v) is above-R+-concave on V for each u £ U and y £ C*. Since every y £ C* is continuous, all continuities of Theorem 2 are satisfied for the mappings yA, yB, yC. By Proposition 2 and (vi), yA(u, v) < yB(u, v) < yC(u, v) for all (u, v) £ U x V. Thus, all conditions of Theorem 2 hold for yA, yB, yC. Hence,

y(v) < max ^J min ^J yC(u, v).

v£V u£U

Since V is compact, there is a v' £ V such that y(v) < min ^J yC(u, v').

v £ y C(u, v') + D,

and hence,

v £ Min^C(u, v) + D. (2)

If v £ Max Uv£VMin^a£U C(u, v), then, by (v), v £ Minw y C(u, v') + D,

which contradicts (2). Hence, for every v £ MaxUv£V Minw u£ U C(u, v),

v £ col y Maxw y A(u, v)) + D.

u£U v£V

That is, (Hi) is valid. □

Corollary 3 If we replace (vi) of Theorem 3 by any one of the following conditions:

(i) for all (u, v) £ U x V, A(u, v) = B(u, v) = C(u, v);

(ii) for all (u, v) £ U x V, A(u, v) c B(u, v) = C(u, v);

(iii) for all (u, v) £ U x V, A(u, v) = B(u, v) c C(u, v);

(iv) for all (u, v) £ U x V, A(u, v) c B(u, v) c C(u, v);

(v) for all (u, v) e U x V, for all (u, v) e U x V, MaxwA(u, v) c MaxwB(u, v) -D, and MaxwB(u, v) c MaxwC(u, v)-D, but A(u, v) c B(u, v) c C(u, v);

(vi) for all (u, v) e U x V, for all (u, v) e U x V, MaxwA(u, v) c MaxwB(u, v) -D, and MaxwB(u, v) c MaxwC(u, v)-D, but A(u, v) c B(u, v) c C(u, v);

(vii) for all (u, v) e U x V, Maxw A(u, v) c MaxwB(u, v) - D, and MaxwB(u, v) c MaxwC(u, v)-D, but A(u, v) c B(u, v) c C(u, v),

then (H1) is valid.

The following example illustrates that Theorem 3 is true. Example 1 Let U = V = [0,1], D = R+, andf: U ^ R be defined by

f (v) =

[-1,0], v = 0, {0}, v =0.

Define A, B, C: U x V ^ R2 by

A(u, v)= {l -cos(un/2)} x f (v),

B(u, v) = {l + cos(un/2)} x [v - l,l],

C(u, v) = {2 + u2} x [v2 + l,2],

for all (u, v) e U x V.

We can easily see that the mappings A, B, C satisfy (vi) and all continuities in Theorem 3. For each v e V, the mapping u ^ A(u, v) is above-naturally Д-quasi-convex on U for each v e V since, for any a e [0, l] and ui, u2 e U,

A(aul + (l -a)u2, v)

= {l - cos((aul + (l - a)u2)п/2)} x f (v) С a{l -cos(ulп/2)} xf (v) + (l -a){l -cos(u2n/2)} xf (v)-D = cojA(ul, v) U A(u2, v)J - D.

We see that the mapping v ^ B(u, v) is above-D-concave on V for each u e U since, for any a e [0, l] and vl, v2 e V,

aB(u, vl) + (l - a)B(v2)

= a{l + cos(un/2)} x [vl - l, l] + (l -a){l + cos(un/2)} x [v2 - l, l] = {l + cos(un/2)} x [avl + (l - a)v2 - l,l] С B(u, avl) + (l - a)v2 - D.

We note that the mapping u ^ C(u, v) is above-D-convex on U for each v e V. Hence, by definition, u ^ C(u, v) is above-naturally D-quasi-convex on U for each v e V. Thus,

conditions (i)-(iii) of Theorem 3 are valid. Now we claim that condition (iv) holds. Indeed, for each v e V and p = p p2) e D*, we need to find an uv e U such that

maxpC(uv, v) = maxjpi(2 + u2) + p2t: v2 + 1 < t < 2} = pi(2 + u2) + 2p2 < 2pi + 2p2

= max y min y pC(u, v).

veV ueU

Hence, we choose uv by the following rule:

any point in [0,1], = 0, 0, ^i=0,

then (iv) of Theorem 3 holds. Next, we claim (v) of Theorem 3 is valid. Indeed, by a simple calculation, we get

Max y Minw y C(u, v)

veV ueU

= {(3,2)}

C ({2} x [y2 + 1,2]) U([2,3] x {y2 + 1}) + D = Minw y C(u, v)+ D

for each v e V. Thus, condition (v) of Theorem 3 holds. By Theorem 3, (Hi) is valid. Indeed,

Max ^J Minw [J C(u, v)

veV ueU

= {(3,2)} C {(0,-1)} + D

= Mini co y Maxw y A(u, v) ) + D,

ueU veV

and hence the conclusion of Theorem 3 is valid.

In the following result, we apply the Gerstewitz function pkw : W ^ R to introduce the second version of the hierarchical minimax theorems, where k e int D and w e W.

Theorem 4 Let U, V be nonempty compact convex subsets of real Hausdorff topological spaces, respectively, W be a real Hausdorff topological vector space. We work under the framework of Theorem 3 except (iv) and the concavity of B. If, in addition, the mapping v ^ pkwB(u, v) is above-R+-concave on V for each u e U, and for any Gerstewitz function

ykw and for each v e V, there is anuv e U such that

(iv)' maxykwC(uv, v) < max y min y ykwC(u, v);

veV ueU

then (H2) is valid.

Proof Using the same steps as in the proof of Theorem 3, we see that the set (J ueU Maxw x UveV A(u, v) is nonempty and compact. Suppose that v e UueU Maxw UveV A(u, v)+D. For any k e int D, there is a Gerstewitz function ykw: W ^ R such that

Vkw(u) > 0 (3)

for all u e (JueU MaxwUveVA(u, v). Then, for each u e U, there is v*u e Y and f (u, v£) e F(u, v£) withf (u, v*u) e Maxw |JveVA(u, v) such that

Vkw{f(u, v*u)) = max [J ykwA(u, v).

Choosing u = f (u, v*u) in (3), max y ykwA(u, v)> 0

for all u e U. Therefore,

min y max y ykwA(u, v) > 0.

ueU veV

By conditions (i)-(iii) and (iv'), we see that all conditions of Theorem 2 hold for the mappings ykwA, ykwB, ykwC, and hence, by (sH),

max y min y ykwC(u, v) > 0.

veV ueU

Since V is compact, there is a y' e Y such that min y pkwC(u, v') > 0.

v e y C(u, v') + D,

and hence

v e Minw U C(u, v) + D.

If v e MaxUveVMin^ UueUA(m, v), then, by (v), v e Minw y C(u, v') + D,

which contradicts (4). From this, we can deduce (H2). □

The third version of hierarchical minimax theorems is as follows. We remove condition (v) in Theorem 4 to deduce (H3).

Theorem 5 We work under the framework of Theorem 4 except condition (v). Equation (H3) is valid.

Proof Following the proof of Theorem 4. Fix any v e MinUueU Maxw |JveVA(u, v). Then ( (J Maxw U A(u, v) j \ {v} n (v - D) = 0.

\eU veV '

For any k e intD, there is a Gerstewitz function : W ^ R such that Vkw(u) > 0

for all u e (JueUMaxw |JveVA(u, v) \ {v}. For each u e U, max ^J ykwA(u, v) > 0,

min ^J max ^J ykwA(u, v) > 0.

ueU veV

Hence, by Theorem 2 for the mappings ykwA, ykwB, ykwC, max ^J min ^J ykwC(u, v) > 0.

veV ueU

Since U and V are compact, there are u0 e U, v0 e V, and h0 e C(u0, v0) such that ykw(ho) = min U ykwC(u, vo) > 0.

Applying Proposition 3.14 [3], h0 e MinwUueUC(u, v0). If h0 = v, v e h0 + (D \ {0}). If h0 = v, ykw(h0) > 0, and hence h0e v - D. Therefore, v e h0 + (D \ {0}). Thus, in any case, v e h0 + W \ (D \ {0}). This implies (H3). □

4 Hierarchical minimax inequalities

As an application of scalar hierarchical minimax theorems, we discuss minimax inequalities which were investigated by Lin et al. [2]. The following result as regards (Hi1) is different from [2] and holds under very different conditions.

Theorem 6 Let U be a nonempty compact convex subset of a real Hausdorff topological vector space, W be a complete locally convex Hausdorff topological vector space. Let the set-valued mappings A, B, C: U x U ^ W come with nonempty compact values and satisfy the following conditions:

(i) the mappings u ^ A(u, v) and u ^ C(u, v) are above-naturally D-quasi-convex on U for each v e U, the mappings (u, v) ^ A(u, v) and (u, v) ^ C(u, v) are upper semicontinuous on U for each ue U, and the mappings u ^ A(u, v) and

v ^ C(u, v) are lower semicontinuous on U;

(ii) u ^ B(u, v) is upper semicontinuous on U for each ve U, and the mapping v ^ B(u, v) is above-D-concave on U for each ue U;

(iii) for each v e U, for each y e D*, there is an uv e U such that

maxyC(uv, u) < max ^J min ^J yC(u, v);

veU u eU

(iv) for each veU,

Max y C(u, u) c Minw y C(u, v) + D; and

u U u U

(v) for all (u, v) e U x U,

MaxwA(u, v) c MaxwB(u, v)-D

Maxw B(u, v) c Maxw C(u, v) - D. Then (Hii) is valid.

Proof Suppose that v e co(UueUMaxwUveUA(u, v)) + D. With the help of technique in the proof of Theorems 2 and 3 for the mappings yA, yB, yC, we can see that

y(v) < max y min ^J yC(u, v).

y U u U

In a similar way to Theorem 3, there isa v' e V such that y(v) < min y yC(u, v').

Hence, v e Minw |JMeU C(u, v') + D. By condition (iv), we see that v e Max y C(u, u).

Therefore, (Hii) is valid. □

In the following example we modify Example 1, which serves to illustrate Theorem 6.

Example 2 Let X = [0,1], D = R+ and f : U ^ R be defined by f (v) =

[-1,0], y = 0, {0}, y =0.

Define A, B, C: U x U ^ R2 by

A(u, v)= {1 -cos(un/2)} x f (v), B(u, v) =1 + cos(un/2)} x [v -1,1], C(u, v) = {2 + u2} x [v2 + 1,2],

for all (u, v) e U x V.

We can easily see that the mappings A, B, C satisfy (v) and all continuities in Theorem 6. From the illustrations in Example 1, we see that the mapping u ^ A(u, v) is above-naturally D-quasi-convex on U for each v e V, the mapping v ^ B(u, v) is above-D-concave on V for each ue U, the mapping u ^ C(u, v) is above-naturally D-quasi-convex on U for each v e V. Furthermore, for each ve V and y = (pi, y2) e D*, by using the same choice of uv as in Example 1, (iii) of Theorem 6 holds. Next, we claim (iv) of Theorem 6 is valid. Indeed, by a simple calculation, we get

Max y C(u, u)

= Max ^J {2 + u2} x [u2 + 1,2]

= {(3,2)}

C ({2} x [v2 + 1,2]) U ([2,3] x {v2 + 1}) + D = Minw y C(u, v)+D

for each v g V. Thus, condition (iv) of Theorem 6 holds. By Theorem 6, (Hi1) is valid. Indeed,

Max y C(u, u)

= {(3,2)} C {(0,-1)} + D

= Mini co y Maxw y A(u, v) J + D,

u U v U

and hence the conclusion of Theorem 6 is valid.

Theorem 7 Let U be a nonempty compact convex subset of real Hausdorff topological vector space, W be a real Hausdorff topological vector space. We work under theframework of Theorem 6 except (iii) and the convexities ofB. If, in addition, the mapping v ^ ykwB(u, v)

is above-R+ -concave on Ufor each ue U, and for each v e U, there is anxve U such that for any Gerstewitzfunction ykw,

(iii)' maxykwC(uv, v) < max ^J min ^J ykwC(u, v),

veU u eU

then (Hi2) is valid.

Proof Suppose that v e UueU MaxwUveUA(u, v) + D. Using a similar technique to the proofs of Theorems 2 and 4 for the mappings pkwA, ykwB, ykwC, we can see that

max y min ^J ykwC(u, v) > 0.

v U u U

By the same technique as in Theorem 6 and condition (iv), we see that v e Max y C(u, u).

Hence, (Hi2) is valid. □

5 Saddle points

In this section, we list the existence of saddle points for set-valued mappings as applications of scalar hierarchical minimax theorems. The proofs of the following results can be deduced directly from Corollary 2, so we omit them. We refer the reader to [2, 3] for more details. Nevertheless, the conditions used in Theorems 8-10 are quite different from the ones used in the literature [2, 3].

Theorem 8 Under the framework of Corollary 2.2, we have max y A(u, v) = min ^J A(x, v) = A(u, v),

v V u U

which means: A has R+-saddlepoint (u, v).

Theorem 9 Let U, V be nonempty compact convex subsets of real Hausdorff topological spaces, respectively. W is a complete locally convex Hausdorff topological vector space. Suppose that the set-valued mappings F: U x V ^ W have nonempty compact values and satisfy the following conditions:

(i) (u, v) ^ A(u, v) is upper semicontinuous on U x V, and u ^ A(u, v) is above-naturally D-quasi-convex and lower semicontinuous on U for each ve V;

(ii) v ^ A(u, v) is above-D-concave on V for each ue U;

(iii) v ^ A(u, v) is continuous on V for each ue U; and

(iv) for any y eD* and for each ve V, there is an uv e U such that

maxyA(uv, v) < max ^J min ^J yA(u, v).

v V u U

A(u, v) n | Max^ ^J A( u, v) j n (Minw (J A(x, v ) J = 0,

^ v eV ' ^ Me U '

which means: A has a weakly D-saddle point (u, v).

Theorem 10 Let U, V be nonempty compact convex subsets of real Hausdorff topological spaces, respectively. W is a real Hausdorff topological vector space. We work under the framework of Theorem 9 except (iv) and the convexities of A. If, in addition, the mapping v ^ ykwA(u, v) is above-R+ -concave on V for each ueU, and for any Gerstewitz function ykw and for each v e V, there is anuv eU such that

(ivr) maxykwA(uv, v) < max min ykwA(u, v);

veV u eU

then A has a weakly D-saddle point (u, v).

Competing interests

The author declares that they have no competing interests. Acknowledgements

This work was supported by grant MOST103-2115-M-039-001 of the Ministry of Science and Technology of Taiwan (Republic of China).

Received: 4 October 2014 Accepted: 26 January 2015 Published online: 19 February 2015 References

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