Scholarly article on topic 'Generalized hierarchical minimax theorems for set-valued mappings'

Generalized hierarchical minimax theorems for set-valued mappings Academic research paper on "Mathematics"

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Academic research paper on topic "Generalized hierarchical minimax theorems for set-valued mappings"

Wang Journal of Inequalities and Applications (2016) 2016:103 DOI 10.1186/s13660-016-1050-z

O Journal of Inequalities and Applications

a SpringerOpen Journal

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Generalized hierarchical minimax theorems for set-valued mappings

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Haijun Wang*

"Correspondence: wanghjshx@126.com Department of Mathematics, Taiyuan NormalUniversity, Taiyuan, 030619, P.R. China

Abstract

In this paper, we discuss generalized hierarchical minimax theorems with four set-valued mappings and we propose some scalar hierarchical minimax theorems and generalized hierarchical minimax theorems in topological spaces. Some examples are given to illustrate our results.

Keywords: minimax theorems; set-valued mappings; cone-convexities

£ Springer

1 Introduction

It is well known that minimax theorems are important in the areas of game theory, and mathematical economical and optimization theory (see [1-5]). Within recent years, many generalizations of minimax theorems have been successfully obtained. On the one hand, the minimax theorem of two functions has been studied based on the two-person nonzero-sum games (see [6, 7]); on the other hand, with the development of vector optimization, there are many authors paying their attention to minimax problems of vector-valued mappings (see [8-10]).

Since Kuroiwa [11] investigated minimax problems of set-valued mappings in 1996, many authors have devoted their efforts to the study of the minimax problems for set-valued mappings. Li et al. [12] proved some minimax theorems for set-valued by using section theorem and separation theorem. Some other minimax theorems for set-valued mappings can be found in [13-16]. Zhang etal. [17] established some minimax theorems for two set-valued mappings, which improved the corresponding results in [12,13]. Lin et al. [18, 19] investigated some bilevel minimax theorems and hierarchical minimax theorems for set-valued mappings by using nonlinear scalarization function.

Recently, Balaj [20] proposed some minimax theorems for four real-valued functions by using some new alternative principles. Inspired by [17-20] we shall study some generalized hierarchical minimax theorems for set-valued mappings. The imposed conditions involve four set-valued mappings. In the second section, we introduce some notions and preliminary results. In the third section, we prove the hierarchical minimax theorem for scalar set-valued mappings. In the fourth section, we show some hierarchical minimax theorems for set-valued mappings in Hausdorff topological vector spaces by using the results obtained in the previous section.

2 Preliminary

In this section, we recall some notations and some known facts.

© 2016 Wang. This article is distributed under the terms of the Creative Commons Attribution 4.0 InternationalLicense (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use,distribution, and reproduction in any medium, provided you give appropriate credit to the originalauthor(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Let X, Y be two nonempty sets in two local convex Hausdorff topological vector spaces, respectively, Z be a local convex Hausdorff topological vector space, S c Z be a closed convex pointed cone with intS = 0, and let Z* denote the topological dual space of Z. A set-valued mapping F: X ^ 2Z are associated with other two mappings F- : Z ^ 2X, the inverse of F and F*: Z ^ 2X the dual of F, defined as F-(z) = {x e X: z e F(x)} and F * (z) = X \ F -(z).

Definition 2.1 ([21]) Let A c Z be a nonempty subset.

(i) A point z e A is called a minimal point of A if A n (z - S) = {z}, and Min A denotes the set of all minimal points of A.

(ii) A point z e A is called a weakly minimal point of A if A n (z - int S) = 0, andMinwA denotes the set of all weakly minimal points of A.

(iii) A point z e A is called a maximal point of A if A n (z + S) = {z}, and Max A denotes the set of all maximal points of A.

(iv) A point z e A is called a weakly maximal point of A if A n (z + int S) = 0, and MaxwA denotes the set of all weakly maximal points of A.

For a nonempty compact subset A c Z, it follows from [12] that 0 = Min A c MinwA; A c Min A + S and 0 = Max A c MaxwA; A c Max A - S. We note that, when Z = R, Min A and Max A are equivalent to Minw A and Maxw A, respectively.

Definition 2.2 ([22]) Let F: X ^ 2Z be a set-valued mapping with nonempty values.

(i) F is said to be upper semicontinuous (shortly, u.s.c.) at xo e X, if for any neighborhood N(F(xo)) of F(xo), there exists a neighborhood N(xo) of xo such that F(x) c N(F(xo)), Vx e N(xo). F is u.s.c. on X if F is u.s.c. at any x e X.

(ii) F is said to be lower semicontinuous (shortly, l.s.c.) at xo e X, if for any open neighborhood N in Z satisfying F(xo) n N = 0, there exists a neighborhood N(xo) of xo such that F(x) n N = 0, Vx e N(xo). F is l.s.c. on X if F is l.s.c. at any x e X.

(iii) F is said to be continuous at xo e X,if F is both u.s.c. and l.s.c. at xo. F is continuous on X if F is continuous at any x e X.

(iv) F is said to be closed if the graph of F is closed subset of X x Z.

Definition 2.3 ([17]) Let X be a nonempty subset of a topological vector space, F: X ^ 2Z be a set-valued mapping.

(i) F is said to be S-concave (respectively, S-convex) on X, if for any x1, x2 e X and X e [o,1],

XF(xi) + (1 - X)F(x2) c F(Xxi + (1 - X)x2) - S (respectively, F(Xx1 + (1 - X)x2) c XF(x1) + (1 - X)F(x2) - S);

(ii) F is said to be properly S-quasiconcave (respectively, properly S-quasiconvex) on X, if for any x1, x2 e X and X e [o, 1],

either F(x1) c F(Xx1 + (1 - X)x2) - S or F(x2) c F(Xx1 + (1 - X)x2) - S (respectively, either F(Xx1 + (1 - X)x2) c F(x1) - S or F(Xx1 + (1 - X)x2) c F(x2) - s);

(iii) F is said to be naturally S-quasiconcave (respectively, naturally S-quasiconvex) on X, if for any xi, x2 e X and X e [0,1]

co(F(x1) U F(x2)) C F(Xx1 + (1 - X)x2) - S (respectively, F(Xx1 + (1 - X)x2) C co(F(x1) U F(x2)) - S).

Remark 2.1

(1) Obviously, any S-concave (S-convex) mapping F is naturally S-quasiconcave (naturally S-quasiconvex); any properly S-quasiconcave (properly S-quasiconvex) mapping F is naturally S-quasiconcave (naturally S-quasiconvex).

(2) One should note that the S-concave (respectively, S-convex, properly S-quasiconcave, properly S-quasiconvex, naturally S-quasiconcave, naturally S-quasiconvex) mapping is defined as above S-concave (respectively, above S-convex, above properly S-quasiconcave, above properly S-quasiconvex, above naturally S-quasiconcave, above naturally S-quasiconvex) mapping in [18,19].

Lemma 2.1 ([22]) Let F: X ^ 2Z be a set-valued mapping. IfX is compact and F is u.s.c. with compact values, then F(X) = IJxeX F(x) is compact.

Lemma 2.2 ([17]) Let F: X ^ 2Z be a continuous set-valued mapping with compact values. Then the set-valued mapping

r(x) = MaxwF (x)

is nonempty closed and upper semicontinuous.

In the sequel we need the following alternative theorem which is a variant form of Balaj [20].

Lemma 2.3 ([20]) Let X, Y be two nonempty compact convex subsets in two local convex Hausdorff topological vector spaces. The set-valued mappings Fi: X ^ Z, i = 1,2,3,4, satisfy the following conditions:

(i) for each x e X, co F1(x) C F2(x) C F3(x);

(ii) F3(co A) C F4(A) for any finite subset A C X;

(iii) F1 and are u.s.c.;

(iv) F2 and F have compact values.

Then at least one of the following assertions holds:

(a) There exists x0 e X such that F1(x0) = 0.

(b) HxeX F4(x)= 0.

3 Hierarchical minimax theorems for scalar set-valued mappings

In this section, we first establish the following hierarchical minimax theorems for scalar set-valued mappings.

Theorem 3.1 Let X, Y be two nonempty compact convex subsets of local convex Hausdorff topological vector spaces, respectively. Let Fi: X x Y ^ 2R, i = 1,2,3,4 be set-valued mappings such thatFi(x,y) C Fi+1(x,y) - R+. Assume that

(i) (x, y) ^ F1(x, y) is u.s.c. with nonempty closed values, and (x, y) ^ F4(x, y) is l.s.c.

(ii) y ^ F2(x, y) is naturally R+ -quasiconcave on Y for each x e X, and x ^ F3(x, y) is naturally R+-quasiconvex on X for each y e Y.

(iii) y ^ F2(x, y) is closed for all x e X, and x ^ F3(x, y) is l.s.c. for all y e Y.

Then either there is xo e X such that FL(xo,y) c (-to,a) for ally e Y or there is yo e Y such thatF4(x,yo) n [p, = 0 for allx e X.

Furthermore, assume that the sets UyeYF1(x, y) and UxeXF4(x, y) are compact for all y e Y and x e X, respectively. Assume the following condition holds:

(iv) for each w e Y, there exists xw e X such that

MaxF4(xw, w) < Max^J Min ^J F4(x,y). (1)

yeY xeX

Min U Ma^ Fl(x,y) < Ma^y Min U F4(x,y). (2)

xeX yeY yeY xeX

Proof For any real numbers a, p e R with a > p, we define the mappings Fi : X ^ 2Y, i = 1,2,3,4 by

Fi(x) = {y e Y: f e F1(x,y),f > a}, Fz(x) = {y e Y: f e F2(x,y),f > a}, F3(x) = {y e Y: f e F3(x,y),f > p}, F4(x) = {y e Y: f e F*(x,y),f > a}.

Then we can see that FL(x) c F2(x) c F3(x) c F4(x), Vx e X. For any x e X,if y e FL(x), there existsf e FL(x,y) such thatfL > a. Since FL(x,y) c F2(x,y)-R+, there aref2 e F2(x,y) and r e R+ such thatf2 = fL + r > a. Then y e F2(x), and so FL(x) c F2(x). Noticing that a > p, one can show F2(x) c F3(x) c F4(x) by using similar deduction.

For any x e X, we seethatF2(x) is convex valued. In fact, for any yL, y1 e F2(x), there exist f e F2(x,yL) andf e F2(x,y1) suchthatfL > a andf > a. Sincey ^ F2(x,y) is naturally R+-quasiconcave, we have XfL+ (1-Xf e XF2(x, yL) + (1-X)F2(x, y1) c co(F2(x, yL) U F2(x, y1)) c F2(x,XyL + (1 - X)y1) - R+, VX e [o, 1]. Then there existf e F2(x,XyL + (1 - X)y1) and r e R+ such thatf e XfL + (1 - X)f' + r > a. Therefore, XyL + (1 - X)y1 e F2(x), i.e. F2(x) is convex valued. Thus co FL(x) c co F2(x) = F2(x), Vx e X.

Let y e F3(co A) for a finite subset A c X. Without loss of generality, we suppose that y e F3(Xxl + (1 - X)x2) for some xL,x2 e A and X e [o, 1]. Then there exists f e F3(Xxl + (1 - X)x2,y) such thatf > p. Since x ^ F3(x,y) is naturally R+-quasiconvex for each y e Y, there existsf' e co(F3(xL,y) UF3(x2,y)) such thatf e f' -R+. Therefore, there exist ¡x e [o, 1] andfL,f2 e F3(xl,y) U F3(x2,y) and r e R+ such thatf = f - r = ¡f + (1 - ¡)f2 - r > p. Then at least one of the assertions f > p andf2 > p holds. Hence, y e (F3(xL) U F3(x2)) c F3(A). Therefore, F3(coA) c F3(A) c F4(A).

For any sequence (xn,yn) e graph = {(x,y): f e FL(x,y),f > a} with (xn,yn) ^ (x,y), there exist fn e FL(xn,yn) such that fn > a. We can take subsequence {f„k} such that limk^TOfnk = liminfn^TOfn = fo. Thenfo > a. Since Fl is u.s.c. with closed values, Then Fl is closed. Thusfo e F(xo,yo), and so (xo,yo) e graphFl. This implies that is closed. From compactness of Y it follows that is upper semicontinuous.

Now, we show that graph= {(x,y) : Vf e F4(x,y),f < fi} is closed. Let (xn,yn) e graphwith (xn,yn) ^ (x0,y0). From lower semicontinuity of F4, it follows that for anyf0 e F4(x0,y0), there exists fn e F4(xn,yn) such that fn ^f0. Thenf0 < fi. Therefore graph is closed. Noticing the compactness of Y, we see that is upper semicontinu-ous.

Since y ^ F2(x, y) is closed for all x e X, F2 is closed valued. In fact, for any sequence yn C F2(x) with yn ^ y0, there exists fn e F2(x,yn) such that fn > a. We can take subsequence {f„k} such that limk^TOfnk = liminfn^TOfn = f0. Thenf0 > a. It follows from the closedness of F(x, •) thatf0 e F(x,y0), and so F2 has closed values. Next, we claim that F3* has closed values. For any sequence xn C F|(y) that converges to some point x0 e X, we see that ye F3(xn). Then f < fi for any f e F3(xn, y). Since x ^ F3(x, y) is lower semicon-tinuous for ally e Y, for any y0 e F(x0,y) there existsf e F3(xn,y) such thatfn ^f0. Then f0 < fi and hence x0 e F|(y). This proves that has closed values. It follows from the compactness of X and Y that both F2 and have compact values.

Then from Lemma 2.3, it follows that either there is x0 e X such that F1(x0) = 0, or f|xeX F4(x) = 0. That is, for any real numbers a, fi e R with a > fi, either there is x0 e X such that F1(x0, y) C a)forall y e Y orthereis y0 e Y such that F4(x, y0) n [fi,+<») = 0 for all x e X.

Furthermore, the compactness of |JxeXF4(x,y) implies that MinUxeXF4(x,y) is nonempty for all y e Y. Since (x,y) ^ F4(x,y) is lower semicontinuous, it follows that y ^ UxeXF4(x,y) is lower semicontinuous. By the compactness of Y and the proof of Lemma 3.2 [12], the set UyeYMinUxeXF4(x,y) is nonempty and compact, and so MaxUyeY MinUxeXF4(x,y) = 0. Set any real numbers a, fi e R with a > fi > MaxUyeYMinUxeXF4(x,y). From (iv), we see that, for each w e Y, there exists xw e X such that F4(xw, w) n [fi, +cxj) = 0. Therefore, there is x0 e X such that F1(x0,y) C a) for all y e Y. Hence

Min ^J Max^J F1(x,y) < Max^J F1(x0,y) < a.

xeX yeY yeY

By the arbitrariness of a and fi, (2) holds. □

Example 3.1 Let X = Y = [0,1] C R. Define four mappings Fi: X x Y ^ 2R, i = 1,2,3,4, as

F1(x,y) = [x2 -1 +y,x]; F2(x,y) =

F3(x, y) = [x2 + y, x2 +1; F4(x, y) = [x2 + y, x +

We can see that Fi(x,y) C Fi+1(x,y) - R+ for all (x,y) e X x Y and conditions (i)-(iii) of Theorem 3.1 hold. It is obvious that UxeXF1(x,y) and |JyeYF4(x,y) are compact for all y e Y and x e X, respectively. Now, we show condition (iv) of Theorem 3.1 is true. One can calculate that Min UxeX F4(x,y) = {y}, Vy e Y, and Max UyeY Min UxeX F4(x,y) = 1. Taking x = 0, we have

x — + y, x + -2 * 2

MaxF4(0,y) < Ma^ Min y F4(x,y), Vy e Y.

yeY xeX

Then all of the conditions of Theorem 3.1 valid. So, the conclusion of Theorem 3.1 holds. In fact, MinUxeXMaxUyerFi(x,y) = o <1 = Max(JyerMin^L^ix,y).

When Fl(x,y) = F2(x,y) = F(x,y) and F3(x,y) = F4(x,y) = G(x,y) in Theorem 3.1, we state the special case of Theorem 3.1 as follows.

Theorem 3.2 LetX, Y be two nonempty compact convex subsets of local convex Hausdorff topological vector spaces, respectively. The set-valued mappings F, G : X x Y ^ 2R with F(x,y) c G(x,y) -R+. Assume that

(i) (x, y) ^ F(x, y) is u.s.c. with nonempty closed values, and (x, y) ^ G(x, y) is l.s.c.

(ii) y ^ F(x, y) is naturally R+-quasiconcave on Y for each x e X, and x ^ G(x, y) is naturally R+-quasiconvex on X for each y e Y.

Then either there is xo e X such that F(xo,y) c (-to,a) for ally e Y or there is yo e Y such that G(x,yo) n [p, +to) = 0 for all x e X.

Furthermore, assume that the sets UyeY F (x, y) andUxeX G(x, y) are compact for ally e Y and x e X, respectively. Assume the following condition holds:

(iii) for each w e Y, there exists xw e X such that

Max G(xw, w) < max ^J Min ^J F4(x,y).

yeY xeX

Min U Max U F(x,y) < Max U Min U G(x,y).

xe X ye Y ye Y xe X

Proof Since F is u.s.c. with nonempty closed values, it follows that y ^ F(x,y) is closed for all x e X by Proposition 7 in [22], p. 1io. From Theorem 3.1, it is easy to show the conclusion holds. □

Remark 3.1 It is obvious that F(x,y) c G(x,y) implies F(x,y) c G(x,y) - R+. So Theorem 3.2 generalizes Theorem 2.1 in [18].

It is well known that both sets UyeYF(x,y) and UxeX G(x,y) are compact for any y e Y and x e X whenever the mappings F and G are upper semicontinuous with nonempty compact values. Hence we can deduce the following result.

Corollary 3.1 LetX, Y be two nonempty compact convex subsets of local convex Hausdorff topological vector spaces, respectively. The set-valued mappings F, G : X x Y ^ 2R come with nonempty compact values and F(x,y) c G(x,y) - R+. Assume that

(i) (x, y) ^ F(x, y) is u.s.c., and (x, y) ^ G(x, y) is continuous.

(ii) y ^ F(x, y) is naturally R+-quasiconcave on Y for each x e X, and x ^ G(x, y) is naturally R+-quasiconvex on X for each y e Y.

(iii) For each w e Y, there exists xw e X such that

Max G(xw, w) < Max^J Min ^J G(x,y).

yeY xeX

Min U Max U F(x,y) < Ma^y Min U G(x,y).

xeX yeY yeY xeX

Remark 3.2 Corollary 3.1 generalizes Theorem 2.1 in [17] and weakens the continuity of F1 in Theorem 2.1 in [17]. It also generalizes Theorem 2.1 in [12] from one set-valued mapping to two set-valued mappings.

4 Generalized hierarchical minimax theorem

In this section, we will discuss some generalized hierarchical minimax theorems for set-valued mappings valued in a complete locally convex Hausdorff topological vector space.

Lemma 4.1 LetX, Y be two nonempty compact convex subsets of local convex Hausdorff topological vector spaces, respectively. The set-valued mapping F: X x Y ^ 2Z comes with nonempty compact values. If (x, y) ^ F(x, y) is u.s.c, andx ^ F(x, y) isl.s.c. for each y e Y, then the set-valued mapping

A(x) = Maxw ^J F (x, y)

is u.s.c. with nonempty compact values. Proof Define a set-valued mapping T: X ^ 2Z as T (x) = U F (x, y).

It follows from Lemma 2.4 in [16] that T is continuous. By Lemma 2.1 and compactness of Y, T is compact-valued. Then, by Lemma 2.2, we see that A is nonempty closed and u.s.c. on X. By compactness of X, it follows that A(x) is compact for each x e X. □

Theorem 4.1 LetX, Y be two nonempty compact convex subsets of local convex Hausdorff topological vector spaces, respectively, Zbea complete locally convex Hausdorff topological vector space. The set-valued mappings Fi: X x Y ^ 2Z, i = 1,2,3,4 come with nonempty compact values and Fi(x,y) c Fi+i(x,y) - S. Assume that

(i) (x,y) ^ F1(x,y) is u.s.c., x ^ F1(x,y) is l.s.c. for each y e Y, and (x,y) ^ F4(x,y) is continuous;

(ii) y ^ F2(x, y) is naturally S-quasiconcave on Y for each x e X, and x ^ F3(x, y) is naturally S-quasiconvex on X for each y e Y;

(iii) y ^ F2(x,y) is u.s.c. for all x e X, and x ^ F3(x,y) is l.s.c. for all y e Y;

(iv) for each w e Y, there exists xw e X such that

Max ^J Minw ^J F4(x, y) - F4(xw, w) c S;

yeY xeX

(v) for each w e Y

Max [J Minw U F4(x, y) C Minw ^J F4(x, w) + S.

yeY xeX xeX

Max U Minw U F4 (x, y) c Min i co U Maxw U Fl (x, y)) + 5. (3)

xeX yeY ^ yeY xeX '

Proof Let L(x) := Maxw UyeYFL(x,y). By Lemma 4.1, L(x) is u.s.c. with nonempty compact values. From Lemma 2.1, it follows that L(X) = (JxeXL(x) is compact, and so is co(L(X)). Then co(L(X)) + S is a closed set with nonempty interior. Suppose that v e Z and v e co(L(X)) + S. By the separation theorem, there exist f e Z* and aL, a2 e R such that

f (v) < a1 < a2< f (u + s), Vu e co(L(X)), Vs e S. (4)

By using a similar discussion to Theorem 3.1 in [17], we have f e S* and f (S) = R+. From assumptions (i) and (iii), it is easy to see that (x,y) ^ f (FL(x,y)) is u.s.c., (x,y) ^ f (F4(x,y)) is l.s.c., y ^ f (F2(x,y)) is closed for all x e X, and x ^ f (F3(x,y)) is l.s.c. for all y e Y. From condition (ii), applying Proposition 3.9 and Proposition 3.13 in [16], we see that y ^ f (F2(x,y)) is naturally R+-quasiconcave on Y for each x e X, and x ^ f (F3(x,y)) is naturally R+-quasiconvex on X for each y e Y. By the condition (iv), for each w e Y, there exists xw e X such that

Maxf (F4(xw, w)) < Max U Min U f (F4(x,y)).

yeY xeX

Since F1 and F4 are u.s.c. and come with compact values, we see that UxeX f (FL(x,y)) and UyeY f (F4(x,y)) are compact for all y e Y and x e X, respectively. Then for set-valued mappings f (Fi), i = 1,2,3,4, all conditions of Theorem 3.1 hold. Therefore we see that

Min U Max U f (Fl (x, y)) < Ma^ Min U f (F4 (x, y)). (5)

xeX yeY yeY xeX

Since Y is compact and F1 has nonempty compact values, for any x e X, there exist yx and f (x, yx) e Fl(x, yx) withf (x, yx) e L(x) such that

f (Fi(x, yx)) = MaxU f (Fi(x, y)).

From (4), choosing s = o and u = f (x,yx), it follows that f (v) < f (f (x,yx)) = Max U f (Fi(x,y))

for all x e X. Then

f (v) < Min U Ma^ f (Fi (x, y)).

xeX yeY

By (5),

f (v)<Ma^y Mi^ f (F4(x, y]).

yeY xeX

Since Y is compact, there exists y' e Y such that

f (v) < Min U f (F4 (x,/) ) = Ma^ Min U f (F4(x,y)).

yeY xeX

From f (5) > 0 for all 5 e S, it follows that v e UxeX (F4(x, y')) + S, and then v e Min„U (F4(x, y')) + S.

Combined with the assumption (v), we have v e Max^J Minw ^J F4(x,y).

yeY xeX

That is, for any v e Max |JyeY Minw |JxeX Fi(x,y),

v e co(i(x)) + S. Hence

Ma^ Min^ (F4(x,y)) c co(L(x)) + S.

y Y x X

Since co(L(X)) = co(UxeXL(x)) = co(UxeXMaxw UyeYFi(x,y)) is compact, we have coi (jMax„UFi(x,y)j c Minicoi (jMax„UFi(x,y) j j + S.

x X y Y x X y Y

xxeX yeY

Therefore, (3) holds.

Example 4.1 Let X = Y = [0,1], Z = R2, and S = R+. Define set-valued mappings Fi : X x Y ^ 2Z, i = 1,2,3,4, as

F1(x, y) = [x2 -1 + y, x] x {-1}, F2(x, y) =

x — + y, x + -2 * 2

x < x - ■

F3(x,y) = [x2 + y,x + 1 x jy^ 2 J, F4(x,y) = {x + 1}x [y + 1,2]. For all (x,y) e X x Y,we can see that the Fi(x,y), i = 1,2,3,4, are compact and Fi(x,y) c Fi+1(x,y) -S.

It is easy to show that the conditions (i)-(iii) hold in Theorem 4.1. We explain conditions (iv) and (v) are valid. We can calculate that

Minw U F4(x, y) = {1}x [y + 1,2] U [1,2] x{y + 1},

Max U Minw U F4(x,y) = {(2,2)}.

yeY xeX

For each w e Y, let xw = 0. Then

Ma^Min^F4(x, y)-Fi(xw, w)= {(2,2)} - {1} x [y + 1,2] c 5.

yeY xeX

The condition (iv) holds. We can see that Max^J Minw ^J F4(x, y)

yeY xeX

= {(2,2)} c{1}x [y + 1,2] U [1,2] x{y + 1} + 5 = Min^F4(x, y) + 5.

Then all of the assumptions of Theorem 4.1 are valid. So, the conclusion of Theorem 4.1 holds. In fact,

Min(coUMaxwUF1(x, y)) = {(0,-1)}.

^ xeX yeY '

Max^J Minw ^J Fj(x, y)

yeY xeX

= {(2,2)} c {(0,-1)} + 5 = Minw Minicoi (jMaxwUFi(x, y)] j + 5.

^ VeX yeY ' '

When Fl(x,y) = F2(x,y) = F(x,y) and F3(x,y) = F4(x,y) = G(x,y) in Theorem 4.1, we state the special case of Theorem 4.1 as follows.

Corollary 4.1 LetX, Y be two nonempty compact convex subsets of local convex Hausdorff topological vector spaces, respectively, Zbea complete locally convex Hausdorff topological space. The set-valued mappings F, G : X x Y ^ 2Z come with nonempty compact values and F(x,y) c G(x,y) - 5. Assume that

(i) (x, y) ^ F(x, y) is u.s.c., x ^ F(x, y) is l.s.c. for each y e Y, and (x, y) ^ G(x, y) is continuous;

(ii) y ^ F(x, y) is naturally 5-quasiconcave on Y for each x e X, and x ^ G(x, y) is naturally 5-quasiconvex on X for each y e Y;

(iii) for each w e Y, there exists xw e X such that

Max^J Minw U G(x, y) - G(xw, w) c 5;

yeY xeX

(iv) for each w e Y

Max^J Minw ^J G(x, y) c Minw ^J G(x, w) + 5.

yeY xeX xeX

Max F(x, y) I + S.

Remark 4.1

(1) Corollary 4.1 generalizes Theorem 3.1 in [17] and weakens the continuity if F.

(2) Corollary 4.1 also generalizes Theorem 3.1 in [18] since F(x, y) c G(x, y) implies F(x, y) c G(x, y) - S.

5 Concluding remarks

We have proven some hierarchical minimax theorems for scalar set-valued mappings and generalized hierarchical minimax theorems for set-valued mappings valued in a complete locally convex Hausdorff topological vector space. The imposed conditions involved four set-valued mappings. The main tools to prove our results have been an alternative principle and separation theorems. Some examples have been provided to illustrate our results.

Competing interests

The author declares to have no competing interests. Acknowledgements

The author would like to thank the editor and anonymous reviewers for their valuable comments and suggestions which helped to improve the paper. This work was supported by 'Department of Mathematics, Taiyuan Normal University, China, which is gratefully acknowledged.

Received: 17 November 2015 Accepted: 18 March 2016 Published online: 31 March 2016 References

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