Solvability Criteria for Some Set-Valued Inequality Systems
© Yingfan Liu. 2010
Received: 23 May 2010
Accepted: 9 July 2010
Published: 28 July 2010
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.
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.
and obtained several necessary and sufficient conditions for its solvability, where and is a class of set-valued maps from to . Along the way, three further set-valued inequality systems that we will study in the sequel can be stated as follows.
For (1.2) and (1.3), it is possible that only (1.2) has solution for some . Indeed, if is compact and is continuous, compact valued with , then is compact and there is with for . Hence solves (1.2) but does not solve (1.3).
It seems that the solvability criteria (namely, necessary and sufficient results concerning existence) to (1.4) can be obtained immediately by  with the replacement . However, this type of result is trivial because it depends only on the property of but not on the respective information of and . This opinion is also applicable to (1.3) and (1.5).
Clearly, (1.3) (or (1.5)) is more fine and more useful than (1.2) (or (1.4)). However, the method used for in  (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 and in .
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:
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.
Let , , and ( ). Let , , and be functions and a set-valued map. We need some concepts concerning , and and 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 [9–11], therefore, the details are omitted here. We also need some further concepts to , , and as follows.
(1) is said to be closed if its graph defined by graph is closed in . Moreover, is said to be upper semicontinuous (in short, u.s.c.) if, for each and each neighborhood of , there exists a neighborhood of such that .
(a)If , then one claims that the minimax equality of holds. Denoting by the value of the preceding equality, one also says that the minimax value of exists. If such that , then one calls a saddle point of . Denote by the set of all saddle points of (i.e., ), and define , the restriction of to if is nonempty.
We also need three known results as follows.
(see ) If is u.s.c., then is u.h.c.
In case is u.s.c. with compact values and , then by (2), (4), is closed and the range of is compact. If , then for any ; hence, is u.s.c. If , supposing that graph with , then graph such that as , which implies that . Hence, is closed and also u.s.c. because of (3).
In case are u.s.c. with compact values, if graph with , then and there exist , such that for all By (4), and are compact, so we can suppose and as . By (2), both are closed, this implies that , and thus is closed. Hence by (3), is u.s.c. because and is compact.
Lemma 2.4 (see [8, Theorems and ]).
(see ) If is l.s.c. (or u.s.c.) and is compact, then defined by (or defined by ) is also l.s.c. (or u.s.c.).
3. Solvability Theorem to (1.3)
By both Assumptions in Section 1, Definition 2.1, and Lemmas 2.4 and 2.5, we can see that
For (1.3), the following three statements are equivalent to each other:
Proof of Theorem 3.2.
By taking , it follows that . Hence, . On the other hand, it is easy to verify that is a nonempty extremal subset of . The Crain-Milmann Theorem (see ) shows that ext is nonempty with ext ext . So there exists such that ext . This implies by (3.3) that , and therefore (2) follows.
4. Solvability Theorems to (1.4) and (1.5)
In view of Definition 2.1, we denote by (or ) the minimax value of (or ) if it exists, (or ) the saddle point set if it is nonempty, and (or ) the restriction of to (or to ). Then we have the solvability result to (1.4) and (1.5) as follows.
For (1.5), the following three statements are equivalent to each other:
The proof of both Theorems 4.1 and 4.2 can be divided into eight lemmas.
We prove (2) in three steps as follows.
follows immediately from Lemma 4.6(3) and Remark 2.2(2). The third lemma follows.
Proof of Theorem 4.1.
By Lemmas 4.6(3), 4.7, and 4.8, we know that Theorem 4.1 is true.
(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 exists, then by (4.29), there exists such that . Hence is nonempty.)
Since solves (4.25) for and , there exist and such that . It follows that for any , hence On the other hand, by the definition of , we have for any . By (4.29)(a) and (4.1)(b), there exists such that which implies by (4.26) that for any . Hence from Lemma 4.9, solves (4.25), and . Therefore, exist with .) So we conclude from (1) and Remark 2.2 that (2) is true. This completes the proof.
Proof of Theorem 4.2.
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.
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