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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 596216, 13 pages
A Rogalski-Cornet Type Inclusion Theorem Based on Two Hausdorff Locally Convex Vector Spaces
Department of Mathematics, College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210046, China
Received 30 December 2011; Accepted 21 February 2012
Academic Editor: Muhammad Aslam Noor
Copyright © 2012 Yingfan Liu and Youguo Wang. 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.
A Rogalski-Cornet type inclusion theorem based on two Hausdorff locally convex vector spaces is proved and composed of two parts. An example is presented to show that the associated set-valued map in the first part does not need any conventional continuity conditions including upper hemicontinuous. As an application, solvability results regarding an abstract von Neumann inclusion system are obtained.
The Rogalski-Cornet inclusion theorem, based on a Hausdorff locally convex vector space and as an useful tool to deal with some inclusion problems described by an upper hemicontinuous set-valued map, can be stated as follows.
Theorem 1.1 (see ). Let be a convex compact subset of a Hausdorff locally convex vector space , and an upper hemicontinuous set-valued map from to with nonempty closed convex values. If is outward, that is, with , then .
Two comments are in order.
(1) Theorem 1.1 can be used to deal with those inclusion problems described by a set-valued map whose domain and range are contained in a same space. For example, if denotes the set of all nonnegative vectors of the dimensional Euclidean space , is an expected demand of the market, some enterprise’s admission output bundle set, and or is the enterprise’s consuming map or correspondence (namely, set-valued map) from to . Then a class of Leontief type input-output inclusion system, as a nonlinear extension to the classical input-output equation [2–4], is composed of and has been studied by Sandberg  and Fujimoto  with the nonlinear analysis methods. Moreover, some primary extensions to Theorem 1.1, also based on a Hausdorff locally convex vector space, have been made by Liu and Zhang  such that the associated correspondence no longer needs the upper hemicontinuous condition, and the obtained results (see ) also have been used to deal with the solvability of (1.1)(b) (see ).
(2) However, if a problem is concerned with two different spaces, then Theorem 1.1 is generally useless even if this problem can be changed into an inclusion system. For example, assume that is an expected demand of the market, is some enterprise’s raw material bundle set, and (or ) are the enterprise’s output and input maps (or correspondences) from to , respectively. If the semiordering in is defined by iff , then the following von Neumann type input-output inequality system, composed of a single-valued inequality and a set-valued inequality, has been studied by Liu and Zhang , and Liu [10, 11] with the nonlinear analysis methods including the minimax and saddle point techniques attributed to [1, 12, 13]. We claim that (1.2) also includes some economic growth problems. For example, if we restrict (where may be viewed as the minimal growth fact of the output regarding input accepted by the enterprise) and replace by , respectively, then (1.2)(a) reduces to a single-valued von Neumann economic growth model ,, which has been studied by Medvegyev , and Bidard and Hosoda . Some other research regarding economic growth has also been made by Jones [16, 17], and Jones, Williams . It is easy to see that (1.2)(b) yields a set-valued economic growth problem. However, up to now, no any corresponding references can be seen. This shows that to study (1.2)(b) is also useful.
Returning to (1.2), if we set and for , then (1.2) equals to the following inclusion system: which is difficult to be handled by Theorem 1.1 except for . Moreover, the following (as an inclusion system described by the preceding or , which is clearly more practical than (1.1) and more fine than (1.2)) can also hardly be tackled by Theorem 1.1 even along with the minimax method.
Now let , and be a set-valued map, then (1.2)–(1.4) (equivalently, (1.3)-(1.4)) can be viewed as the special examples of the first inclusion in the following von Neumann type inclusion system We claim (1.5) an abstract von Neumann inclusion system if and are replaced by two Hausdorff locally convex vector spaces and , respectively.
In the sequel, we attempt to extend Theorem 1.1 to a new situation, that is, to present a Rogalski-Cornet type theorem composed of existence and continuity parts (as the main result proved in Section 3), such that the abstract von Neumann inclusion system (1.5) could be tackled, which means that the domain and range of may probably be contained in two different Hausdorff locally convex vector spaces and , respectively. Since Theorem 15.1.9 in  and Theorem 6.4.10 in  are such type of results while their correspondences need the upper hemicontinuous assumption, and a new paper completed by Lignola  provides some Ky Fan inequalities and Nash equilibrium points without semicontinuity and compactness, we concentrate our attention to the instance such that the associated correspondence no longer needs any conventional continuity conditions including upper hemicontinuous.
The paper is arranged as follows. We introduce some necessary concepts in the next section. In Section 3, we first prove a Rogalski-Cornet type inclusion theorem based on two different Hausdorff locally convex vector spaces, followed by an example to show that the first part of this theorem does not need any conventional continuity conditions such as upper semicontinuous, lower semicontinuous, and upper hemicontinuous. Then, as an application of this theorem, we provide some solvability results for (1.5). Finally we present the conclusion in Section 4.
In the sections below, without any special explanations, we always assume are two Hausdorff locally convex vector spaces, their duals and the duality paring on or . We need some concepts with respect to a function defined on or such as convex or concave, upper semicontinuous or lower semicontinuous (in short, u.s.c. or l.s.c.) and continuous, and concepts to a set-valued map from a subset to including closed, upper, and lower semicontinuous (in short, u.s.c. and l.s.c.) and continuous, whose definitions can be consulted in [1, 12, 13], so the details are omitted here.
Let and . Denote
We also need the following.
3. Main Theorem and Application to (1.5)
In this section, we always assume that
Under some additional assumptions, we first prove a Rogalski-Cornet type inclusion theorem for a set-valued map from to with and without u.h.c. condition, then present an application of this theorem to the abstract von Neumann inclusion system (1.5).
3.1. Main Theorem and a Counterexample
The Rogalski-Cornet type inclusion theorem we will prove is the following.
Theorem 3.1. (A) Results without u.h.c. Condition.
(i)Existence. (1) Assume that then , that is, there exist and such that . (2) If is the set of all such that then , that is, for each , there exists such that .(ii)Continuity. Assume that , the set of all such that is nonempty, then . Moreover, is compact, and the inverse of restricting to defined by is a u.s.c. and u.h.c. set-valued map.(B) Results with u.h.c. Condition.
Assume that is u.h.c., then the following are true. (i)If (3.2)(b) holds, then is nonempty and compact, and the inverse of restricting to defined by is a u.s.c. and u.h.c. set-valued map.(ii)If , the set of all satisfying (3.3)(b), is nonempty, then is compact with , and the inverse of restricting to defined by (3.5) is also a u.s.c and u.h.c. set-valued map.
To prove this theorem, we need some known results and state them in lemmas as follows.
Lemma 3.2 (see ). Let be a convex compact subset of and let satisfy that is lower semicontinuous, is quasi-concave, and . Then there exists such that .
Lemma 3.3 (see ). Let be a Hausdorff topological space, a compact Hausdorff topological space, and a closed set-valued map from to . Then is u.s.c.
Lemma 3.4 (see ). Let be a Hausdorff topological space, supplied with the weak topology , and a u.s.c. set-valued map from to . Then is u.h.c.
Remark 3.5. Since the original vector topology on is stronger than the weak topology , this Lemma is also true if is supplied with the original topology .
Lemma 3.6 (see ). Let be a Hausdorff topological space, supplied with the weak topology, and let be a u.h.c. set-valued map from to with nonempty closed convex values. Then the graph of denoted by graph is closed, that is, is closed.
With these lemmas, we proceed to prove Theorem 3.1.
Proof. Proof of Part A. (i) First we prove (i).
(1) Under the assumptions (3.1) and (3.2), we will prove by contradiction. If for all , then by (3.1), we see that for each , is closed convex with . So the Hahn-Banach separation theorem implies that
Setting for , from (3.7) we have Since is compact and , by (3.2)(a) we know that forms an open covering of . Therefore, holds for some finite subset , and there exists a continuous partition of unit associated to this finite covering of such that where is the closure of the set .
Define on by Clearly, satisfies the assumptions of Lemma 3.2. Indeed, it is easy to see that for any , is continuous, for any , is affine, and . So there exists such that . Take and , then from (3.8) and (3.9) we know that and . This further implies Associating this with (3.2)(b), we obtain that This is a contradiction. Therefore, .
(2) Statement (2) immediately follows from (1) because for each , is compact, and (3.2) holds for because of the assumption (3.3).(ii) Then we prove (ii).
By (3.4) and statement (i)(2), it is easy to see that , so the left is to show that is compact and defined by (3.5) is u.s.c. and u.h.c.
(1) Since is compact and , it is sufficient to verify that is closed, that is, to verify that if is a generalized sequence such that , then . To this end, for each and each , we define and will prove that which implies by (3.4) and (3.12) that , and thus is closed.
(a) First we prove (3.13)(a). Suppose that are fixed, and is a generalized sequence such that . By (3.12)(a) we have Since , for each there exists such that Associating this with (3.14), we obtain As , from (3.4)(a), (3.12)(a), and (3.16), we conclude that and because is closed. Hence, , which together with (3.15) yields
By taking , from (3.17) we obtain that . Hence, and (3.13)(a) follows.
(b) Then we prove (3.13)(b). Suppose that satisfy , and (3.15) holds for this and some . Since , by (3.4)(b), we have . In view of (3.15), we get Thus, because is arbitrary. This shows that and (3.13) is also true. Therefore, .
(2) To prove the continuity of , by Lemmas 3.3 and 3.4 and Remark 3.5, it is sufficient to verify that is closed because is compact. Assume that satisfy , then . This implies that Let be fixed. Since , for any , there exists such that , which together with (3.19) yields Therefore, . This implies because and thus is closed. Hence, . By letting , it follows that holds for any fixed . Since is closed convex by assumption (3.1), we have (also thanks to the Hahn-Banach separation theorem). Combing this with the fact , we conclude that graph . Therefore, defined by (3.5) is closed, and statement (ii) follows.
Proof of Part B. Since is u.h.c., by Definition 2.1 we know that for each , is lower semicontinuous. Hence the lower sections and of the function are closed in for all and all . This implies that all the conditions from (3.2) to (3.4) are satisfied, and thus all the statements of part (A) are true. So the remaining is to show that both and graph (where is defined by (3.6)) are closed also thanks to Lemmas 3.3 and 3.4 and Remark 3.5.
(1) Assume that is a generalized sequence of with , then (because is compact), and for each there exists such that . As is compact, choosing a generalized subsequence if necessary, we may assume . On the other hand, it is easy to see that in Definition 2.1, if denotes the original vector topology on , then can be supplied with any compatible topologies of including the weak topology because . Associating this with (3.1)(b) and using Hahn-Banach’s separation theorem, we conclude that is also a u.h.c. correspondence with nonempty closed convex values when is supplied with the weak topology , which implies by Lemma 3.6 that is closed. Therefore, and is closed.
(2) Assume that is a generalized sequence such that , then (because is closed), and . Thus, by the closeness of , and is closed. This completes the proof.
Remark 3.7. We claim that the part (A) of Theorem 3.1 does not need any conventional continuity conditions such as u.s.c., l.s.c. and u.h.c. See the following counterexample.
Example 3.8. Let , , , and let be the set of all rational numbers of , . Assume that is a fixed number, a single-valued map from to , and a set-valued map from to defined by
Then we have the following (i) It is easy to see that (i.e., is a continuous linear map from to ), is a nonempty convex compact subset of for each , and is not u.s.c. or l.s.c. at any point of . It can also be shown that is not u.h.c. on . Indeed, if with , then is not u.s.c. at any points of because
(ii) Now we verify that (3.2)–(3.4) hold for this example.
(1) It is easy to see that , and , we have Since and for and , from (3.23) and using the fact that for any , we obtain that , and , Both (3.24) and (3.25) imply that Therefore, all (a) of (3.2)–(3.4) are satisfied.
(2) As , , where and for , then for any and any , we see that From (3.26) and (3.27), it follows that and with , Both (3.28) and (3.29) show that , we have Hence, all (b) of (3.2)-(3.4) are satisfied. Therefore, Theorem 3.1(A) holds for this example. Indeed, we have .
Remark 3.10. From the proof of Theorem 3.1(A), we see that the Ky-Fan inequality is very important to the existence part. Regarding this inequality, some extensions have been made by Lignola , Lin and Simons , Alzer , as well as S. J. Li and X. B. Li , we think that probably in the future, these new Ky-Fan inequalities could also be used to obtain some new useful Rogalski-Cornet type theorems.
3.2. Application to (1.5)
(i) If , and (3.2) holds for , then , that is, (1.5)(a) is solvable.
(ii) Assume that Then (1) if In particular, (a) Equation (1.5)(a) is solvable (i.e., ) when ,(b) Equation (1.5)(b) has a solution (i.e., ) when ,(c) Equation (1.5)(c) is solvable (i.e., ) when . (2) The inverse of restricting to defined by is u.s.c. and u.h.c. if
(iii) If is u.h.c., then the assumptions (3.2)(a), (3.32), and (3.33) in statements (i) and (ii) can be removed.
Proof. (i) Applying Theorem 3.1(A)(i)(1) to , we see that the statement (i) is true.
(ii) Set . Since is a convex compact subset of , and a continuous linear map from to , we see that is a convex compact subset of . Moreover, for each , as equals to , from (3.31), we obtain that This means that the assumption (3.3)(b) holds for any .
(1)Associating (3.32) with (3.34) we know that (3.3) holds for all , which implies by Theorem 3.1(A)(i)(2), for , that , and thus all the statements (a), (b), and (c) of (ii)(1) are true.(2)From (3.33) and (3.34), we see that defined by (3.4) is precisely equal to . Hence by Theorem 3.1(A)(ii), the statement (ii)(2) follows.
(iii) The conclusion of (iii) is clear. This completes the proof.
In this paper, we have proved a Rogalski-Cornet type theorem (namely, Theorem 3.1) based on two Hausdorff locally convex vector spaces, and presented a counterexample (i.e., Example 3.8) to show that the first part of this theorem does not need any conventional continuity conditions such as upper semicontinuous, lower semicontinuous and upper hemicontinuous conditions. Applying this theorem, by Theorem 3.11 we have also provided the solvability results for a class of abstract von Neumann input-output inclusion system (namely, (1.5)).
This work is supported by the National Science Foundation of China (no. 61179027).
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