Open Access

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq1_HTML.gif the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq2_HTML.gif -dimensional Euclidean space, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq3_HTML.gif its dual, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq4_HTML.gif the duality pairing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq5_HTML.gif ; moreover, we denote that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq6_HTML.gif and int  https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq7_HTML.gif is its interior. We also define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq8_HTML.gif (or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq9_HTML.gif ) in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq10_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq11_HTML.gif (or by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq12_HTML.gif int  https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq15_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq16_HTML.gif may not be equal to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq17_HTML.gif ), and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq19_HTML.gif to https://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., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq21_HTML.gif are https://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., https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq24_HTML.gif if we restrict https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq25_HTML.gif or https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq27_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq28_HTML.gif are replaced by set-valued maps https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq29_HTML.gif and https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq31_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq33_HTML.gif to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq35_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq36_HTML.gif be as above, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq37_HTML.gif be set-valued maps from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq38_HTML.gif to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq40_HTML.gif solves
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ3_HTML.gif
(1.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ4_HTML.gif
(1.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ5_HTML.gif
(1.5)
When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq41_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq43_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq44_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq46_HTML.gif . Indeed, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq47_HTML.gif is compact and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq48_HTML.gif is continuous, compact valued with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq49_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq50_HTML.gif is compact and there is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq51_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq52_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq53_HTML.gif . Hence https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq57_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq60_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq63_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq64_HTML.gif ?

Assumption.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq65_HTML.gif is a fixed point and https://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: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq67_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq68_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq70_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq71_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq72_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq73_HTML.gif ). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq74_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq75_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq76_HTML.gif be functions and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq78_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq79_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq80_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq82_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq83_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq84_HTML.gif as follows.

Definition 2.1.

(1) https://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  https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq86_HTML.gif is closed in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq87_HTML.gif . Moreover, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq89_HTML.gif and each neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq90_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq91_HTML.gif , there exists a neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq92_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq93_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq94_HTML.gif .

  1. (2)

    Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq95_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq96_HTML.gif ), and define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq97_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq98_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq99_HTML.gif ). Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq101_HTML.gif is u.s.c. on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq102_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq103_HTML.gif .

     

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

(4)

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

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

  1. (5)

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

     
Remark 2.2.
  1. (1)

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

     
  2. (2)
    For the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq157_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq158_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq159_HTML.gif exists if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq160_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq161_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq162_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq163_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq164_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq165_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq166_HTML.gif exists, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq167_HTML.gif for any https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq169_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq170_HTML.gif . Moreover, we have
    https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq171_HTML.gif is u.s.c., then https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq175_HTML.gif (the closure of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq176_HTML.gif ) is compact and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq177_HTML.gif is closed, then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq179_HTML.gif is compact and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq181_HTML.gif is compact in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq182_HTML.gif .

     
  4. (5)

    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq183_HTML.gif is convex (or compact) and https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq187_HTML.gif are convex, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq188_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq189_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq190_HTML.gif , then
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ7_HTML.gif
    (2.2)
     
Hence https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq192_HTML.gif is compact.

     

In case https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq194_HTML.gif , then by (2), (4), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq195_HTML.gif is closed and the range https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq196_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq197_HTML.gif is compact. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq198_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq199_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq200_HTML.gif ; hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq201_HTML.gif is u.s.c. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq202_HTML.gif , supposing that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq203_HTML.gif graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq204_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq205_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq206_HTML.gif graph  https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq207_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq208_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq209_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq210_HTML.gif . Hence, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq213_HTML.gif graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq214_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq215_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq216_HTML.gif and there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq217_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq218_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq219_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq220_HTML.gif By (4), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq221_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq222_HTML.gif are compact, so we can suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq223_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq224_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq225_HTML.gif . By (2), both https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq226_HTML.gif are closed, this implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq227_HTML.gif , and thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq228_HTML.gif is closed. Hence by (3), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq229_HTML.gif is u.s.c. because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq230_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq232_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq233_HTML.gif ]).

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

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

(2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq246_HTML.gif solves (1.2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq247_HTML.gif for https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq251_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq252_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq253_HTML.gif (resp., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq254_HTML.gif ) is finite for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq255_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq256_HTML.gif (resp., https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq258_HTML.gif .

     
  2. (2)

    (see [11]) If https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq260_HTML.gif is compact, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq261_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq262_HTML.gif (or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq263_HTML.gif defined by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq265_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq266_HTML.gif be convex compact, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq267_HTML.gif be defined on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq268_HTML.gif . If, for each https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq271_HTML.gif and there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq272_HTML.gif such that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq275_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq276_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq277_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq278_HTML.gif by
https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq279_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq280_HTML.gif ,

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

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

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq289_HTML.gif and https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq291_HTML.gif ,

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

(3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq296_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq297_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq298_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq300_HTML.gif solves (1.3) because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq301_HTML.gif and https://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 https://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 https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq305_HTML.gif (2) and (3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq306_HTML.gif (1).

(1) https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ9_HTML.gif
(3.2)
Then for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq308_HTML.gif and each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq309_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ10_HTML.gif
(3.3)

By taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq310_HTML.gif , it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq311_HTML.gif . Hence, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq313_HTML.gif is a nonempty extremal subset of https://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  https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq315_HTML.gif is nonempty with ext  https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq316_HTML.gif ext  https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq317_HTML.gif . So there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq318_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq319_HTML.gif ext  https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq320_HTML.gif . This implies by (3.3) that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq321_HTML.gif , and therefore (2) follows.

(3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq322_HTML.gif (1). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq323_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq324_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq325_HTML.gif be presented in (3). Since https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ11_HTML.gif
(3.4)

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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq330_HTML.gif implies that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq332_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq333_HTML.gif be defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq334_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq335_HTML.gif Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq337_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq338_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq339_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq340_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq341_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq342_HTML.gif and therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ12_HTML.gif
(3.5)

This implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq343_HTML.gif (or https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq348_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq349_HTML.gif , we define the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq350_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq351_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq352_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq353_HTML.gif by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq354_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq355_HTML.gif are finite with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq356_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq357_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq358_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq359_HTML.gif , so the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq360_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq362_HTML.gif (or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq363_HTML.gif ) the minimax value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq364_HTML.gif (or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq365_HTML.gif ) if it exists, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq366_HTML.gif (or https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq368_HTML.gif (or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq369_HTML.gif ) the restriction of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq370_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq371_HTML.gif (or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq372_HTML.gif to https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq374_HTML.gif exists if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq375_HTML.gif .

(ii)

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

(2)In particular, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq382_HTML.gif exists with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq383_HTML.gif , then for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq384_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq385_HTML.gif such that https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq387_HTML.gif ,

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

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

That is, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq402_HTML.gif is not the necessary condition of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq404_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq405_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq406_HTML.gif be defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq407_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq408_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq409_HTML.gif Then both https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq410_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq412_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq413_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq414_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq415_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq416_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq417_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq418_HTML.gif Therefore,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq419_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq420_HTML.gif solves (1.5) but https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq422_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq423_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq424_HTML.gif . Consider the auxiliary inequality system
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ15_HTML.gif
(4.3)
Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq426_HTML.gif solves (1.2) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq427_HTML.gif , and in particular, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq429_HTML.gif solves (1.4). Define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq430_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq431_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ16_HTML.gif
(4.4)

denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq432_HTML.gif the minimax value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq433_HTML.gif if it exists, and denote by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq435_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq436_HTML.gif exists and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq437_HTML.gif is nonempty. Moreover, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq439_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq440_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq441_HTML.gif .

     
  2. (2)

    The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq442_HTML.gif is continuous and strictly decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq443_HTML.gif with https://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), https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq446_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq447_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq448_HTML.gif , applying Lemma 2.4 to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq449_HTML.gif and substituting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq450_HTML.gif for https://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), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq452_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq454_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq455_HTML.gif is u.s.c. on https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ17_HTML.gif
(4.5)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq457_HTML.gif and https://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), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq459_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq460_HTML.gif are finite. Then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq461_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq462_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq463_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq464_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ19_HTML.gif
(4.7)
Hence for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq465_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq466_HTML.gif is continuous on https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq468_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq469_HTML.gif .

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

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

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

This implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq484_HTML.gif . Therefore, https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq486_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq487_HTML.gif are l.s.c. on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq488_HTML.gif .

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

(3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq492_HTML.gif exists if and only if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq494_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq495_HTML.gif , the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq496_HTML.gif is continuous on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq498_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq499_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq501_HTML.gif is a sequence with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq502_HTML.gif , then for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq503_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq504_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq505_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq506_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq507_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq508_HTML.gif are compact and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq509_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq510_HTML.gif , we may choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq511_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq512_HTML.gif such that
    https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq513_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq514_HTML.gif are closed. Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq515_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq516_HTML.gif , which in turn imply that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq517_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ24_HTML.gif
(4.12)
Combining this with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq518_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq519_HTML.gif , it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ25_HTML.gif
(4.13)
Hence by (4.1), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq520_HTML.gif is u.s.c. on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq521_HTML.gif , so is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq522_HTML.gif on https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq525_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq526_HTML.gif such that

     
https://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), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq527_HTML.gif . Hence https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq529_HTML.gif solves(1.4), then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq530_HTML.gif exists with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq531_HTML.gif .

     

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

Proof.
  1. (1)
    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq537_HTML.gif solves (1.4), then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq539_HTML.gif , and there is a unique https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq540_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq541_HTML.gif . Moreover, also from Lemma 4.5, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq544_HTML.gif . We will prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq545_HTML.gif exists with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq546_HTML.gif . Let
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ27_HTML.gif
    (4.15)
     

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

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq549_HTML.gif solves (4.3), there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq550_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq551_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq552_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq553_HTML.gif . Hence for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq554_HTML.gif As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq555_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq556_HTML.gif , it follows from (4.1) that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq557_HTML.gif and thus
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq558_HTML.gif we have https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq560_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq561_HTML.gif
https://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), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq562_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq563_HTML.gif solves (4.3). Since https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq566_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq567_HTML.gif exists and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq569_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq570_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq571_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq572_HTML.gif solves (1.4). Moreover, for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq573_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq574_HTML.gif such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq576_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq577_HTML.gif (i.e., (4.4) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq578_HTML.gif ), it follows that, for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq579_HTML.gif and each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq580_HTML.gif ,
https://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),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ33_HTML.gif
(4.21)
It follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq581_HTML.gif implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq582_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq583_HTML.gif and so
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq584_HTML.gif , we then conclude that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq585_HTML.gif solves (4.3). So there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq586_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq587_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq588_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq589_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq590_HTML.gif solves (1.4) because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq591_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq592_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq593_HTML.gif .

For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq594_HTML.gif . Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq596_HTML.gif on https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ35_HTML.gif
(4.23)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq598_HTML.gif is u.s.c. on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq599_HTML.gif , from (4.23) there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq600_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq601_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq602_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ36_HTML.gif
(4.24)

By taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq603_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq604_HTML.gif , and therefore https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq605_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq606_HTML.gif because https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq608_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq610_HTML.gif solves
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ37_HTML.gif
(4.25)
Define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq611_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq612_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ38_HTML.gif
(4.26)

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

Lemma 4.9.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq618_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq620_HTML.gif exists and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq621_HTML.gif is nonempty.

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

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

Proof.

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

Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq633_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq634_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq635_HTML.gif .

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq636_HTML.gif is convex and compact in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq637_HTML.gif , replacing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq638_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq639_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq640_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq641_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq642_HTML.gif by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq644_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq645_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq646_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ40_HTML.gif
(4.28)

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

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq650_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq651_HTML.gif , and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq654_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq655_HTML.gif . It follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq656_HTML.gif and thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq657_HTML.gif is strict decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq658_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq659_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq660_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq661_HTML.gif . This implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq662_HTML.gif with https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq664_HTML.gif exists if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq665_HTML.gif is nonempty, where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq667_HTML.gif solves (4.25) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq668_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq669_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq670_HTML.gif exists with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq671_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq672_HTML.gif is nonempty with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq673_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq674_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq675_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq676_HTML.gif .

     

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq677_HTML.gif is continuous on https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq680_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq681_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq682_HTML.gif is nonempty.)

(2)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq683_HTML.gif solves (4.25) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq684_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq686_HTML.gif exists with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq687_HTML.gif , and there is a unique https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq688_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq689_HTML.gif . In particular, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq691_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq693_HTML.gif exists with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq694_HTML.gif . (In fact, let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ42_HTML.gif
(4.30)

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

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq697_HTML.gif solves (4.25) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq698_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq699_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq700_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq701_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq702_HTML.gif . It follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq703_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq704_HTML.gif , hence https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq706_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq707_HTML.gif for any https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq709_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq710_HTML.gif which implies by (4.26) that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq711_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq712_HTML.gif . Hence from Lemma 4.9, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq713_HTML.gif solves (4.25), and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq714_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq715_HTML.gif exist with https://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 https://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 https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq719_HTML.gif and
    https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ45_HTML.gif
(4.33)
As https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq721_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq722_HTML.gif , we obtain that, for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq723_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ46_HTML.gif
(4.34)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq724_HTML.gif is u.s.c. on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq725_HTML.gif , from (4.34) there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq726_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ47_HTML.gif
(4.35)
Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq727_HTML.gif This implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq728_HTML.gif solves (4.25) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq729_HTML.gif and any https://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
https://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), https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq732_HTML.gif and each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq733_HTML.gif ,
https://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), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq734_HTML.gif . This implies that, for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq735_HTML.gif and there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq736_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq737_HTML.gif So we obtain
https://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) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq739_HTML.gif and there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq740_HTML.gif such that https://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 https://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 https://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), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq744_HTML.gif exists with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq745_HTML.gif .

Now we let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq746_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq747_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq748_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq749_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq750_HTML.gif , then we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq751_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq752_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq753_HTML.gif (where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_IEq754_HTML.gif This implies by (4.1) that
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543061/MediaObjects/13660_2010_Article_2183_Equ52_HTML.gif
(4.40)

Hence, https://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) https://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.