Solvability Criteria for Some Set-Valued Inequality Systems

Journal of Inequalities and Applications20102010:543061

DOI: 10.1155/2010/543061

Received: 23 May 2010

Accepted: 9 July 2010

Published: 28 July 2010

Abstract

Arising from studying some multivalued von Neumann model, three set-valued inequality systems are introduced, and two solvability questions are considered. By constructing some auxiliary functions and studying their minimax and saddle-point properties, solvability criteria composed of necessary and sufficient conditions regarding these inequality systems are obtained.

1. Introduction

Arising from considering some multivalued von Neumann model, this paper aims to study three set-valued inequality systems and try to find their solvability criteria. Before starting with this subject, we need to review some necessary backgrounds as follows.

We denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq1_HTML.gif the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq2_HTML.gif -dimensional Euclidean space, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq3_HTML.gif its dual, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq4_HTML.gif the duality pairing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq5_HTML.gif ; moreover, we denote that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq6_HTML.gif and int  http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq7_HTML.gif is its interior. We also define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq8_HTML.gif (or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq9_HTML.gif ) in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq10_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq11_HTML.gif (or by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq12_HTML.gif int  http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq13_HTML.gif ).

It is known that the generalized (linear or nonlinear) von Neumann model, which is composed of an inequality system and a growth factor problem described by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ1_HTML.gif
(1.1)
is one of the most important issues in the input-output analysis [13], where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq14_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq15_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq16_HTML.gif may not be equal to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq17_HTML.gif ), and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq18_HTML.gif are two nonnegative or positive maps from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq19_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq20_HTML.gif . A series of researches on (1.1) have been made by the authors of [15] for the linear case (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq21_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq22_HTML.gif matrices) and by the authors of [6, 7] for the nonlinear case (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq23_HTML.gif are some types of nonlinear maps). Since (a) or (b) of (1.1) is precisely a special example of the inequality http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq24_HTML.gif if we restrict http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq25_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq26_HTML.gif , it is enough for (1.1) to consider the inequality system. This idea can be extended to the set-valued version. Indeed, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq27_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq28_HTML.gif are replaced by set-valued maps http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq29_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq30_HTML.gif , respectively, then (1.1) yields a class of multivalued von Neumann model, and it solves a proper set-valued inequality system to study. With this idea, by [8] (as a set-valued extension to [6, 7]) we have considered the following multivalued inequality system:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ2_HTML.gif
(1.2)

and obtained several necessary and sufficient conditions for its solvability, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq31_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq32_HTML.gif is a class of set-valued maps from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq33_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq34_HTML.gif . Along the way, three further set-valued inequality systems that we will study in the sequel can be stated as follows.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq35_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq36_HTML.gif be as above, and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq37_HTML.gif be set-valued maps from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq38_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq39_HTML.gif , then we try to find the solvability criteria (i.e., the necessary and sufficient conditions) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq40_HTML.gif solves
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ3_HTML.gif
(1.3)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ4_HTML.gif
(1.4)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ5_HTML.gif
(1.5)
When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq42_HTML.gif are single-valued maps, then (1.3)–(1.5) return to the models of [6, 7]. When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq45_HTML.gif are set-valued maps, there are three troubles if we try to obtain some meaningful solvability criteria regarding (1.3)–(1.5) just like what we did in [8].
  1. (1)

    For (1.2) and (1.3), it is possible that only (1.2) has solution for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq46_HTML.gif . Indeed, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq47_HTML.gif is compact and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq48_HTML.gif is continuous, compact valued with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq49_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq50_HTML.gif is compact and there is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq51_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq52_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq53_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq54_HTML.gif solves (1.2) but does not solve (1.3).

     
  2. (2)

    It seems that the solvability criteria (namely, necessary and sufficient results concerning existence) to (1.4) can be obtained immediately by [8] with the replacement http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq55_HTML.gif . However, this type of result is trivial because it depends only on the property of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq56_HTML.gif but not on the respective information of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq57_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq58_HTML.gif . This opinion is also applicable to (1.3) and (1.5).

     
  3. (3)

    Clearly, (1.3) (or (1.5)) is more fine and more useful than (1.2) (or (1.4)). However, the method used for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq59_HTML.gif in [8] (or the possible idea for (1.4)) to obtain solvability criteria fails to be applied to find the similar characteristic results for (1.3) (or (1.5)) because there are some examples (see Examples 3.5 and 4.4) to show that, without any additional restrictions, no necessary and sufficient conditions concerning existence for them can be obtained. This is also a main cause that the author did not consider http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq61_HTML.gif in [8].

     

So some new methods should be introduced if we want to search out the solvability criteria to (1.3)–(1.5). In the sections below, we are devoted to study (1.3)–(1.5) by considering two questions under two assumptions as follows:

Question 1.

Whether there exist any criteria that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq62_HTML.gif solves (1.3) in some proper way?

Question 2.

Like Question 1, whether there exist any solvability criteria to (1.4) or (1.5) that depend on the respective information of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq63_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq64_HTML.gif ?

Assumption.

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq65_HTML.gif is a fixed point and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq66_HTML.gif is a convex compact subset.

Assumption.

Consider the following: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq67_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq68_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq69_HTML.gif are upper semicontinuous and convex set-valued maps with nonempty convex compact values.

By constructing some functions and studying their minimax properties, some progress concerning both questions has been made. The paper is arranged as follows. We review some concepts and known results in Section 2 and prove three Theorems composed of necessary and sufficient conditions regarding the solvability of (1.3)–(1.5) in Sections 3 and 4. Then we present the conclusion in Section 5.

2. Terminology

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq70_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq71_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq72_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq73_HTML.gif ). Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq74_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq75_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq76_HTML.gif be functions and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq77_HTML.gif a set-valued map. We need some concepts concerning http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq78_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq79_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq80_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq81_HTML.gif such as convex or concave and upper or lower semicontinuous (in short, u.s.c. or l.s.c.) and continuous (i.e., both u.s.c. and l.s.c.), whose definitions can be found in [911], therefore, the details are omitted here. We also need some further concepts to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq82_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq83_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq84_HTML.gif as follows.

Definition 2.1.

(1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq85_HTML.gif is said to be closed if its graph defined by graph  http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq86_HTML.gif is closed in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq87_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq88_HTML.gif is said to be upper semicontinuous (in short, u.s.c.) if, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq89_HTML.gif and each neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq90_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq91_HTML.gif , there exists a neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq92_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq93_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq94_HTML.gif .

  1. (2)

    Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq95_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq96_HTML.gif ), and define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq97_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq98_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq99_HTML.gif ). Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq100_HTML.gif is said to be upper hemicontinuous (in short, u.h.c.) if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq101_HTML.gif is u.s.c. on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq102_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq103_HTML.gif .

     

(3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq104_HTML.gif is said to be convex if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq105_HTML.gif is convex and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq106_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq107_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq108_HTML.gif .

(4)

(a)If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq109_HTML.gif , then one claims that the minimax equality of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq110_HTML.gif holds. Denoting by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq111_HTML.gif the value of the preceding equality, one also says that the minimax value http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq112_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq113_HTML.gif exists. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq114_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq115_HTML.gif , then one calls http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq116_HTML.gif a saddle point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq117_HTML.gif . Denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq118_HTML.gif the set of all saddle points of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq119_HTML.gif (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq120_HTML.gif ), and define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq121_HTML.gif , the restriction of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq122_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq123_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq124_HTML.gif is nonempty.

(b)Replacing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq125_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq126_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq127_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq128_HTML.gif , with the similar method one can also define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq129_HTML.gif (the minimax value of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq130_HTML.gif ), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq131_HTML.gif (the saddle-point set of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq132_HTML.gif ), and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq133_HTML.gif (the restriction of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq134_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq135_HTML.gif ).

  1. (5)

    If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq136_HTML.gif is a convex set and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq137_HTML.gif a subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq138_HTML.gif , one claims that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq139_HTML.gif is an extremal subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq140_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq141_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq142_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq143_HTML.gif entails http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq144_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq145_HTML.gif is an extremal point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq146_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq147_HTML.gif is an extremal subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq148_HTML.gif , and the set of all extremal points of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq149_HTML.gif is denoted by ext http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq150_HTML.gif .

     
Remark 2.2.
  1. (1)

    Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq151_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq152_HTML.gif , we can see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq153_HTML.gif is u.h.c. if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq154_HTML.gif is l.s.c. on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq155_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq156_HTML.gif .

     
  2. (2)
    For the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq157_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq158_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq159_HTML.gif exists if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq160_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq161_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq162_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq163_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq164_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq165_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq166_HTML.gif exists, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq167_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq168_HTML.gif . The same properties are also true for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq169_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq170_HTML.gif . Moreover, we have
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ6_HTML.gif
    (2.1)
     

We also need three known results as follows.

Lemma 2.3.
  1. (1)

    (see [9]) If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq171_HTML.gif is u.s.c., then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq172_HTML.gif is u.h.c.

     
  1. (2)

    (see [9]) If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq173_HTML.gif is u.s.c. with closed values, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq174_HTML.gif is closed.

     
  2. (3)

    (see [9]) If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq175_HTML.gif (the closure of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq176_HTML.gif ) is compact and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq177_HTML.gif is closed, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq178_HTML.gif is u.s.c.

     
  3. (4)

    If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq179_HTML.gif is compact and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq180_HTML.gif is u.s.c. with compact values, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq181_HTML.gif is compact in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq182_HTML.gif .

     
  4. (5)

    If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq183_HTML.gif is convex (or compact) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq184_HTML.gif are convex (or u.s.c. with compact values), then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq185_HTML.gif are also convex (or u.s.c.) for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq186_HTML.gif .

     

Proof.

We only need to prove (5).
  1. (a)
    If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq187_HTML.gif are convex, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq188_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq189_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq190_HTML.gif , then
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ7_HTML.gif
    (2.2)
     
Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq191_HTML.gif is convex.
  1. (b)

    Now we assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq192_HTML.gif is compact.

     

In case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq193_HTML.gif is u.s.c. with compact values and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq194_HTML.gif , then by (2), (4), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq195_HTML.gif is closed and the range http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq196_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq197_HTML.gif is compact. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq198_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq199_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq200_HTML.gif ; hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq201_HTML.gif is u.s.c. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq202_HTML.gif , supposing that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq203_HTML.gif graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq204_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq205_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq206_HTML.gif graph  http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq207_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq208_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq209_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq210_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq211_HTML.gif is closed and also u.s.c. because of (3).

In case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq212_HTML.gif are u.s.c. with compact values, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq213_HTML.gif graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq214_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq215_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq216_HTML.gif and there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq217_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq218_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq219_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq220_HTML.gif By (4), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq221_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq222_HTML.gif are compact, so we can suppose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq223_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq224_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq225_HTML.gif . By (2), both http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq226_HTML.gif are closed, this implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq227_HTML.gif , and thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq228_HTML.gif is closed. Hence by (3), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq229_HTML.gif is u.s.c. because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq230_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq231_HTML.gif is compact.

Lemma 2.4 (see [8, Theorems http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq232_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq233_HTML.gif ]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq234_HTML.gif be convex compact with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq235_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq236_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq237_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq238_HTML.gif is convex and u.s.c. with nonempty convex compact values, and define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq239_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq240_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq241_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq242_HTML.gif . Then

(1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq243_HTML.gif exists and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq244_HTML.gif is a convex compact subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq245_HTML.gif ,

(2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq246_HTML.gif solves (1.2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq247_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq248_HTML.gif .

In particular, both (1) and (2) are also true if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq249_HTML.gif .

Lemma 2.5.
  1. (1)

    (see [10, 11]) If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq250_HTML.gif is convex or l.s.c. (resp., concave or u.s.c.) on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq251_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq252_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq253_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq254_HTML.gif ) is finite for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq255_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq256_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq257_HTML.gif ) is also convex or l.s.c. (resp., concave or u.s.c.) on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq258_HTML.gif .

     
  2. (2)

    (see [11]) If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq259_HTML.gif is l.s.c. (or u.s.c.) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq260_HTML.gif is compact, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq261_HTML.gif defined by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq262_HTML.gif (or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq263_HTML.gif defined by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq264_HTML.gif ) is also l.s.c. (or u.s.c.).

     
  3. (3)

    (see [911], Minimax Theorem) Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq265_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq266_HTML.gif be convex compact, and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq267_HTML.gif be defined on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq268_HTML.gif . If, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq269_HTML.gif is convex and l.s.c. and, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq270_HTML.gif is concave and u.s.c., then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq271_HTML.gif and there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq272_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq273_HTML.gif .

     

3. Solvability Theorem to (1.3)

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq274_HTML.gif be introduced as in Lemma 2.4, and define the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq275_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq276_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq277_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq278_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ8_HTML.gif
(3.1)

Remark 3.1.

By both Assumptions in Section 1, Definition 2.1, and Lemmas 2.4 and 2.5, we can see that

(1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq279_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq280_HTML.gif ,

(2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq281_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq282_HTML.gif exist, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq283_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq284_HTML.gif are nonempty,

(3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq285_HTML.gif solves (1.2) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq286_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq287_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq288_HTML.gif .

Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq289_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq290_HTML.gif are nonempty. Moreover, we have the following.

Theorem 3.2.

For (1.3), the following three statements are equivalent to each other:

(1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq291_HTML.gif ,

(2)for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq292_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq293_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq294_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq295_HTML.gif ,

(3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq296_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq297_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq298_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq299_HTML.gif .

Remark 3.3.

Clearly, each of (2) and (3) implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq300_HTML.gif solves (1.3) because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq301_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq302_HTML.gif . So we conclude from Theorem 3.2 that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq303_HTML.gif solves (1.3) in the way of (2) or in the way of (3) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq304_HTML.gif .

Proof of Theorem 3.2.

We only need to prove (1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq305_HTML.gif (2) and (3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq306_HTML.gif (1).

(1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq307_HTML.gif (2). Assume that (1) holds. By (3.1) and Remarks 2.2 and 3.1, it is easy to see that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ9_HTML.gif
(3.2)
Then for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq308_HTML.gif and each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq309_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ10_HTML.gif
(3.3)

By taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq310_HTML.gif , it follows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq311_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq312_HTML.gif . On the other hand, it is easy to verify that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq313_HTML.gif is a nonempty extremal subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq314_HTML.gif . The Crain-Milmann Theorem (see [12]) shows that ext  http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq315_HTML.gif is nonempty with ext  http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq316_HTML.gif ext  http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq317_HTML.gif . So there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq318_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq319_HTML.gif ext  http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq320_HTML.gif . This implies by (3.3) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq321_HTML.gif , and therefore (2) follows.

(3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq322_HTML.gif (1). Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq323_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq324_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq325_HTML.gif be presented in (3). Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq326_HTML.gif solves (1.3) and also solves (1.2), by (3.1) and Remarks 2.2 and 3.1, we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ11_HTML.gif
(3.4)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq327_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq328_HTML.gif and the theorem follows.

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq329_HTML.gif

Remark 3.4.

From the Theorem, we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq330_HTML.gif implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq331_HTML.gif solves (1.3). However, without any additional restricting conditions, the inverse may not be true.

Example 3.5.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq332_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq333_HTML.gif be defined by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq334_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq335_HTML.gif Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq336_HTML.gif is an u.s.c. and convex set-valued map with convex compact values, and for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq337_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq338_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq339_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq340_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq341_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq342_HTML.gif and therefore,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ12_HTML.gif
(3.5)

This implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq343_HTML.gif (or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq344_HTML.gif ) may not be the necessary (or the sufficient) condition that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq345_HTML.gif solves (1.3).

4. Solvability Theorems to (1.4) and (1.5)

Taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq346_HTML.gif , then from both Assumptions, Lemma 2.4, and Theorem 3.2, we immediately obtain the necessary and sufficient conditions to the solvability of (1.4) and (1.5). However, just as indicated in Section 1, this type of result is only concerned with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq347_HTML.gif . To get some further solvability criteria to (1.4) and (1.5) depending on the respective information of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq348_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq349_HTML.gif , we define the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq350_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq351_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq352_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq353_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ13_HTML.gif
(4.1)

By both Assumptions, we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq354_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq355_HTML.gif are finite with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq356_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq357_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq358_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq359_HTML.gif , so the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq360_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq361_HTML.gif defined by (4.1) are well defined.

In view of Definition 2.1, we denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq362_HTML.gif (or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq363_HTML.gif ) the minimax value of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq364_HTML.gif (or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq365_HTML.gif ) if it exists, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq366_HTML.gif (or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq367_HTML.gif ) the saddle point set if it is nonempty, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq368_HTML.gif (or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq369_HTML.gif ) the restriction of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq370_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq371_HTML.gif (or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq372_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq373_HTML.gif ). Then we have the solvability result to (1.4) and (1.5) as follows.

Theorem 4.1.

(i) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq374_HTML.gif exists if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq375_HTML.gif .

(ii)

(1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq376_HTML.gif solves (1.4) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq377_HTML.gif exists with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq378_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq379_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq380_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq381_HTML.gif .

(2)In particular, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq382_HTML.gif exists with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq383_HTML.gif , then for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq384_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq385_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq386_HTML.gif

Theorem 4.2.

For (1.5), the following three statements are equivalent to each other:

(1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq387_HTML.gif ,

(2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq388_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq389_HTML.gif , and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq390_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq391_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq392_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq393_HTML.gif ,

(3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq394_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq395_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq396_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq397_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq398_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq399_HTML.gif

That is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq400_HTML.gif solves (1.5) in the way of (2) or in the way of (3) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq401_HTML.gif .

Remark 4.3.

It is also needed to point out that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq402_HTML.gif is not the necessary condition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq403_HTML.gif making (1.5) solvable without any other restricting conditions.

Example 4.4.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq404_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq405_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq406_HTML.gif be defined by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq407_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq408_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq409_HTML.gif Then both http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq410_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq411_HTML.gif are u.s.c. convex set-valued maps with convex compact values, and for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq412_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq413_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq414_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq415_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq416_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq417_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq418_HTML.gif Therefore,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ14_HTML.gif
(4.2)

This implies that, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq419_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq420_HTML.gif solves (1.5) but http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq421_HTML.gif .

The proof of both Theorems 4.1 and 4.2 can be divided into eight lemmas.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq422_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq423_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq424_HTML.gif . Consider the auxiliary inequality system
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ15_HTML.gif
(4.3)
Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq425_HTML.gif solves (4.3) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq426_HTML.gif solves (1.2) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq427_HTML.gif , and in particular, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq428_HTML.gif solves (4.3) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq429_HTML.gif solves (1.4). Define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq430_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq431_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ16_HTML.gif
(4.4)

denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq432_HTML.gif the minimax value of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq433_HTML.gif if it exists, and denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq434_HTML.gif the saddle point set if it is nonempty. Then we have the following.

Lemma 4.5.
  1. (1)

    For each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq435_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq436_HTML.gif exists and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq437_HTML.gif is nonempty. Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq438_HTML.gif solves (4.3) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq439_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq440_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq441_HTML.gif .

     
  2. (2)

    The function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq442_HTML.gif is continuous and strictly decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq443_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq444_HTML.gif .

     
Proof.
  1. (1)

    By both Assumptions and Lemmas 2.3(4) and 2.3(5), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq445_HTML.gif is convex and u.s.c. with nonempty convex compact values for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq446_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq447_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq448_HTML.gif , applying Lemma 2.4 to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq449_HTML.gif and substituting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq450_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq451_HTML.gif , we know that (1) is true.

     
  1. (2)

    We prove (2) in three steps as follows.

     
(a)By Lemma 2.3(1), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq452_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq453_HTML.gif are u.h.c., which implies by Definition 2.1(2) and Remark 2.2(1) that for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq454_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq455_HTML.gif is u.s.c. on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq456_HTML.gif . Then from Lemma 2.5(1)(2), we know that both functions
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ17_HTML.gif
(4.5)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq457_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq458_HTML.gif are compact by both Assumptions and Lemma 2.3(4), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq459_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq460_HTML.gif are finite. Then for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq461_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq462_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq463_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ18_HTML.gif
(4.6)
This implies that, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq464_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ19_HTML.gif
(4.7)
Hence for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq465_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq466_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq467_HTML.gif . Also from Lemmas 2.5(1) and 2.5(2), it follows that both functions
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ20_HTML.gif
(4.8)

So we conclude from (4.5), (4.8), and Statement (1) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq468_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq469_HTML.gif .

(b)Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq470_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq471_HTML.gif int  http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq472_HTML.gif is compact, it is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq473_HTML.gif . Thus for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq474_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ21_HTML.gif
(4.9)

which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq475_HTML.gif , and hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq476_HTML.gif is strict decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq477_HTML.gif .

(c)Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq478_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq479_HTML.gif . By both Assumptions, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq480_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq481_HTML.gif are finite. Thus for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq482_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq483_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ22_HTML.gif
(4.10)

This implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq484_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq485_HTML.gif . This completes the proof.

Lemma 4.6.

(1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq486_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq487_HTML.gif are l.s.c. on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq488_HTML.gif .

(2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq489_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq490_HTML.gif are u.s.c. on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq491_HTML.gif .

(3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq492_HTML.gif exists if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq493_HTML.gif is nonempty.

Proof.
  1. (1)

    Since, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq494_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq495_HTML.gif , the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq496_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq497_HTML.gif , from Lemma 2.5(1), we can see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq498_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq499_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq500_HTML.gif are l.s.c., hence (1) is true.

     
  1. (2)
    Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq501_HTML.gif is a sequence with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq502_HTML.gif , then for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq503_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq504_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq505_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq506_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq507_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq508_HTML.gif are compact and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq509_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq510_HTML.gif , we may choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq511_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq512_HTML.gif such that
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ23_HTML.gif
    (4.11)
     
By Lemma 2.3(2), both http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq513_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq514_HTML.gif are closed. Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq515_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq516_HTML.gif , which in turn imply that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq517_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ24_HTML.gif
(4.12)
Combining this with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq518_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq519_HTML.gif , it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ25_HTML.gif
(4.13)
Hence by (4.1), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq520_HTML.gif is u.s.c. on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq521_HTML.gif , so is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq522_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq523_HTML.gif thanks to Lemma 2.5(1).
  1. (3)

    Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq524_HTML.gif exists. By (1) and (2), there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq525_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq526_HTML.gif such that

     
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ26_HTML.gif
(4.14)

By Remark 2.2(2), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq527_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq528_HTML.gif is nonempty. The inverse is obvious. This completes the proof.

Lemma 4.7.
  1. (1)

    If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq529_HTML.gif solves(1.4), then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq530_HTML.gif exists with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq531_HTML.gif .

     

(2)If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq532_HTML.gif exists with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq533_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq534_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq535_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq536_HTML.gif .

Proof.
  1. (1)
    If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq537_HTML.gif solves (1.4), then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq538_HTML.gif solves (4.3). From Lemma 4.5, we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq539_HTML.gif , and there is a unique http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq540_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq541_HTML.gif . Moreover, also from Lemma 4.5, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq542_HTML.gif is the biggest number that makes (4.3) solvable, and thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq543_HTML.gif solves (4.3) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq544_HTML.gif . We will prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq545_HTML.gif exists with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq546_HTML.gif . Let
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ27_HTML.gif
    (4.15)
     

then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq547_HTML.gif . It is needed to show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq548_HTML.gif .

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq549_HTML.gif solves (4.3), there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq550_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq551_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq552_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq553_HTML.gif . Hence for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq554_HTML.gif As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq555_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq556_HTML.gif , it follows from (4.1) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq557_HTML.gif and thus
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ28_HTML.gif
(4.16)
On the other hand, by (4.15), for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq558_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq559_HTML.gif . By (4.1) and Lemma 4.6(2), there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq560_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ29_HTML.gif
(4.17)
It deduces from (4.4) that for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq561_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ30_HTML.gif
(4.18)
Hence by Lemma 4.5(1), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq562_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq563_HTML.gif solves (4.3). Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq564_HTML.gif is the biggest number that makes (4.3) solvable, we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq565_HTML.gif . Combining this with (4.16), we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq566_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq567_HTML.gif exists and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq568_HTML.gif
  1. (2)

    follows immediately from Lemma 4.6(3) and Remark 2.2(2). The third lemma follows.

     

Lemma 4.8.

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq569_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq570_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq571_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq572_HTML.gif solves (1.4). Moreover, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq573_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq574_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq575_HTML.gif

Proof.

By (4.1) and Remark 2.2(2), we know that, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq576_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ31_HTML.gif
(4.19)
Combining this with the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq577_HTML.gif (i.e., (4.4) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq578_HTML.gif ), it follows that, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq579_HTML.gif and each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq580_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ32_HTML.gif
(4.20)
Hence by Definition 2.1(4) and Remark 2.2(2),
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ33_HTML.gif
(4.21)
It follows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq581_HTML.gif implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq582_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq583_HTML.gif and so
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ34_HTML.gif
(4.22)

Applying Lemma 4.5(1) to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq584_HTML.gif , we then conclude that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq585_HTML.gif solves (4.3). So there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq586_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq587_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq588_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq589_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq590_HTML.gif solves (1.4) because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq591_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq592_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq593_HTML.gif .

For each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq594_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq595_HTML.gif by (4.4), applying Lemma 2.5(3) to the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq596_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq597_HTML.gif and associating with (4.22), we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ35_HTML.gif
(4.23)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq598_HTML.gif is u.s.c. on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq599_HTML.gif , from (4.23) there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq600_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq601_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq602_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ36_HTML.gif
(4.24)

By taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq603_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq604_HTML.gif , and therefore http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq605_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq606_HTML.gif because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq607_HTML.gif . This completes the proof.

Proof of Theorem 4.1.

By Lemmas 4.6(3), 4.7, and 4.8, we know that Theorem 4.1 is true.

To prove Theorem 4.2, besides using Lemmas 4.5–4.8, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq608_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq609_HTML.gif , we also need to study the condition that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq610_HTML.gif solves
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ37_HTML.gif
(4.25)
Define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq611_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq612_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ38_HTML.gif
(4.26)

We denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq613_HTML.gif the minimax values of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq614_HTML.gif if it exists, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq615_HTML.gif the saddle point set if it is nonempty, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq616_HTML.gif its restriction to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq617_HTML.gif .

Lemma 4.9.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq618_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq619_HTML.gif be fixed. Then one has the following.

(1)For each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq620_HTML.gif exists and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq621_HTML.gif is nonempty.

(2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq622_HTML.gif solves (4.25) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq623_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq624_HTML.gif implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq625_HTML.gif

(3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq626_HTML.gif is continuous and strict decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq627_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq628_HTML.gif

Proof.

Define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq629_HTML.gif from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq630_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq631_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ39_HTML.gif
(4.27)

Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq632_HTML.gif is a single-valued continuous map with the convex condition defined by Definition 2.1(3) because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq633_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq634_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq635_HTML.gif .

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq636_HTML.gif is convex and compact in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq637_HTML.gif , replacing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq638_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq639_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq640_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq641_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq642_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq643_HTML.gif , from Lemma 2.4, we know that both (1) and (2) are true. Moreover, with the same method as in proving Lemma 4.5(2), we can show that (3) is also true. (In fact, since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq644_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq645_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq646_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq647_HTML.gif are compact, by (4.26) and Lemmas 2.5(1) and 2.5(2), we can see that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ40_HTML.gif
(4.28)

Hence by (1), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq648_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq649_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq650_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq651_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq652_HTML.gif be defined as in the proof of Lemma 4.5(2).

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq653_HTML.gif , also by (4.26), we can see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq654_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq655_HTML.gif . It follows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq656_HTML.gif and thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq657_HTML.gif is strict decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq658_HTML.gif .

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq659_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq660_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq661_HTML.gif . This implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq662_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq663_HTML.gif .) Hence the fifth lemma follows.

Lemma 4.10.

(1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq664_HTML.gif exists if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq665_HTML.gif is nonempty, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq666_HTML.gif is defined by (4.1)(b).
  1. (2)

    If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq667_HTML.gif solves (4.25) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq668_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq669_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq670_HTML.gif exists with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq671_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq672_HTML.gif is nonempty with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq673_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq674_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq675_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq676_HTML.gif .

     

Proof.

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq677_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq678_HTML.gif , by Lemma 2.5(1), it is easy to see that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ41_HTML.gif
(4.29)

(1)By (4.29) and with the same method as in proving Lemma 4.6(3), we can show that (1) is true. (Indeed, we only need to prove the necessary part. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq679_HTML.gif exists, then by (4.29), there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq680_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq681_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq682_HTML.gif is nonempty.)

(2)If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq683_HTML.gif solves (4.25) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq684_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq685_HTML.gif , then from Lemma 4.9 we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq686_HTML.gif exists with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq687_HTML.gif , and there is a unique http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq688_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq689_HTML.gif . In particular, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq690_HTML.gif is the biggest number that makes (4.25) solvable for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq691_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq692_HTML.gif .

Applying the same method as in proving Lemma 4.7(1), we can show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq693_HTML.gif exists with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq694_HTML.gif . (In fact, let
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ42_HTML.gif
(4.30)

Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq695_HTML.gif . We need to show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq696_HTML.gif .

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq697_HTML.gif solves (4.25) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq698_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq699_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq700_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq701_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq702_HTML.gif . It follows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq703_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq704_HTML.gif , hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq705_HTML.gif On the other hand, by the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq706_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq707_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq708_HTML.gif . By (4.29)(a) and (4.1)(b), there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq709_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq710_HTML.gif which implies by (4.26) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq711_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq712_HTML.gif . Hence from Lemma 4.9, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq713_HTML.gif solves (4.25), and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq714_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq715_HTML.gif exist with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq716_HTML.gif .) So we conclude from (1) and Remark 2.2 that (2) is true. This completes the proof.

Lemma 4.11.

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq717_HTML.gif , then Theorem 4.2(2) is true.

Proof.
  1. (i)
    If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq718_HTML.gif , then by Lemma 4.6(3) and Remark 2.2, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq719_HTML.gif and
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ43_HTML.gif
    (4.31)
     
By the same proof of (4.21) we can show that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ44_HTML.gif
(4.32)
Combining this with Lemma 4.5(1) and using Remark 2.2(2), we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ45_HTML.gif
(4.33)
As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq720_HTML.gif by (4.4), applying Lemma 2.5(3) to the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq721_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq722_HTML.gif , we obtain that, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq723_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ46_HTML.gif
(4.34)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq724_HTML.gif is u.s.c. on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq725_HTML.gif , from (4.34) there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq726_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ47_HTML.gif
(4.35)
Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq727_HTML.gif This implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq728_HTML.gif solves (4.25) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq729_HTML.gif and any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq730_HTML.gif . So we conclude from Lemma 4.10 and Remark 2.2(2) that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ48_HTML.gif
(4.36)
On the other hand, by (4.1), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq731_HTML.gif . Combining this with (4.36), it follows that, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq732_HTML.gif and each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq733_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ49_HTML.gif
(4.37)
Hence, also by (4.1), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq734_HTML.gif . This implies that, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq735_HTML.gif and there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq736_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq737_HTML.gif So we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ50_HTML.gif
(4.38)

By using the same method as in proving (1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq738_HTML.gif (2) of Theorem 3.2, we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq739_HTML.gif and there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq740_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq741_HTML.gif . Hence Theorem 4.2(2) is true.

Lemma 4.12.

If Theorem 4.2(3) holds,then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq742_HTML.gif .

Proof.

If Theorem 4.2(3) holds, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq743_HTML.gif solves both (1.4) and (1.5), and by Lemma 4.7(1), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq744_HTML.gif exists with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq745_HTML.gif .

Now we let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq746_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq747_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq748_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq749_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq750_HTML.gif , then we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq751_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq752_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq753_HTML.gif (where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq754_HTML.gif This implies by (4.1) that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ51_HTML.gif
(4.39)
Combining this with the fact that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq755_HTML.gif and using Remark 2.2 and (4.1), we obtain that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ52_HTML.gif
(4.40)

Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq756_HTML.gif .

Proof of Theorem 4.2.

Since (2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq757_HTML.gif (3) of is clear, Theorem 4.2 follows immediately from Lemmas 4.11 and 4.12.

5. Conclusion

Based on the generalized and multivalued input-output inequality models, in this paper we have considered three types of set-valued inequality systems (namely, (1.3)–(1.5)) and two corresponding solvability questions. By constructing some auxiliary functions and studying their minimax and saddle point properties with the nonlinear analysis approaches, three solvability theorems (i.e., Theorems 3.2, 4.1, and 4.2) composed of necessary and sufficient conditions regarding these inequality systems have been obtained.

Authors’ Affiliations

(1)
Department of Mathematics, College of Science, Nanjing University of Posts and Telecommunications

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Copyright

© Yingfan Liu. 2010

This article is published under license to BioMed Central Ltd. 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.