Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 208624, 20 pages
http://dx.doi.org/10.1155/2011/208624
Research Article

## An Optimal Lower Eigenvalue System

Department of Mathematics, College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210046, China

Received 8 November 2010; Accepted 6 April 2011

Academic Editor: Nicholas D.Β Alikakos

Copyright Β© 2011 Yingfan Liu. 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.

#### Abstract

An optimal lower eigenvalue system is studied, and main theorems including a series of necessary and suffcient conditions concerning existence and a Lipschitz continuity result concerning stability are obtained. As applications, solvability results to some von-Neumann-type input-output inequalities, growth, and optimal growth factors, as well as Leontief-type balanced and optimal balanced growth paths, are also gotten.

#### 1. Introduction

##### 1.1. The Optimal Lower Eigenvalue System

Arising from considering some inequality problems in input-output analysis such as von-Neumann type input-output inequalities, growth and optimal growth factors, as well as Leontief type balanced and optimal balanced growth paths, we will study an optimal lower eigenvalue system.

To this end, we denote by the real -dimensional Euclidean space with the dual , the set of all nonnegative vectors of , and its interior. We also define in by (or ).

Let , , , and , be two single-valued maps, where may not be equal to . Then the optimal lower eigenvalue system that we will study and use to consider the preceding inequality problems can be described by , , , , and as follows: We call a lower eigenvalue to (1.1) if it solves (a), and its solution the eigenvector, claim the maximal lower eigenvalue to (1.1) if it maximizes (b) (i.e., solves (a), but not if ), and its solution the optimal eigenvector.

In case with , then (1.1) becomes All the concepts concerning (1.1) are reserved for (1.2), and for convenience, the maximal lower eigenvalue to (1.2), if existed, is denoted by .

##### 1.2. Some Economic Backgrounds

As indicated above, the aim of this article is to consider some inequality problems in input-output analysis by studying (1.1). So it is natural to know how many (or what types of) problems in input-output analysis can be deduced from (1.1) or (1.2) by supplying , , , , , and with some proper economic implications. Indeed, in the input-output analysis found by Leontief [1], there are two classes of important economic systems.

One is the Leontief type input-output equality problem composed of an equation and an inclusion as follows: where is an expected demand of the market, some enterprise's admission output bundle set, and or is the enterprise's single-valued or set-valued consuming map. The economic implication of (a) or (b) is whether there exists or there exist and such that the pure output or is precisely equal to the expected demand . If , and is described by a th square matrix, then (a) is precisely the classical Leontief input-output equation, which has been studied by Leontief [1] and Miller and Blair [2] with the matrix analysis method. If is convex compact, and is continuous, then (a) is a Leontief type input-output equation, which has been considered by Fujimoto [3] and Liu and Chen [4, 5] with the functional analysis approach. As for (b), in case is convex compact, and is convex compact-valued with and without the upper hemicontinuous condition, it has also been studied by Liu and Zhang [6, 7] with the nonlinear analysis methods attributed to [810], in particular, using the classical Rogalski-Cornet Theorem (see [8, Theorem 15.1.4]) and some Rogalski-Cornet type Theorems (see [6, Theorems 2.8, 2.9 and 2.12]). However, since the methods to tackle (1.3) are quite different from those to study (1.1), we do not consider it here.

Another is the von-Neumann type and Leontief type inequality problems which can be viewed as some special examples of (1.1) or (1.2).

(i) Assume that or is an expected demand set or an expected demand of the market, and some enterprise's raw material bundle set. Then the von-Neumann type inequality problems including input-output inequalities, along with growth and optimal growth factors can be stated, respectively, as follows.

(1) If are supposed to be the enterprise's output (or producing) and consuming maps, respectively, by taking , then from both (a) of (1.1) and (1.2), we obtain the von-Neumann type input-output inequalities: The economic implication of (a) or (b) is whether there exist and or there exists such that the pure output satisfies sufficiently the expected demand . If , and are described by two matrixes, then (b) returns to the classical von-Neumann input-output inequality, which has also been studied by Leontief [1] and Miller and Blair [2] with the matrix analysis method. If is convex compact, and are two nonlinear maps such that are upper semicontinuous concave for any , then (b) (as a nonlinear von-Neumann input-output inequality) has been handled by Liu [11] and Liu and Zhang [12] with the nonlinear analysis methods in [810]. Along the way, in case is convex compact, and , are replaced by two upper semicontinuous convex set-valued maps with convex compact values, then (b) (as a set-valued von-Neumann input-output inequality) has also been studied by Liu [13, 14]. However, (a) has not been considered up to now. Since (a) (or (b)) is solvable if and only if makes (1.1)(a) (or makes (1.2)(a)) have solutions, and also, if and only if the maximal lower eigenvalue to (1.1) exists with (or the maximal lower eigenvalue to (1.2) exists with ), we see that the lower eigenvalue approach yielded from studying (1.1) or (1.2) may be applied to obtain some new solvability results to (1.4).

(2) If are supposed to be the enterprise's output and input (or invest) maps, respectively, and set , then is nonempty, and in some degree, each can be used to describe the enterprise's growth behavior. Since the enterprise always hopes his growth as big as possible, a fixed positive number can be selected to represent the enterprise's desired minimum growth no matter whether or not. By taking and restricting , then from (1.2) we obtain the von-Neumann type growth and optimal growth factor problem: We call a growth factor to (1.5) if it solves (a), its solution the intensity vector, and say that (1.5) is efficient if it has at least one growth factor. We also claim the optimal growth factor to (1.5) if it maximizes (b), and its solution the optimal intensity vector. If , and are described by two matrixes, then (a) reduces to the classical von-Neumann growth model, and has been studied by Leontief [1], Miller and Blair [2], Medvegyev [15], and Bidard and Hosoda [16] with the matrix analysis method. Unfortunately, if are nonlinear maps, in my knowledge, no any references regarding (1.5) can be seen. Clearly, the matrix analysis method is useless to the nonlinear version. On the other hand, it seems that the methods of [11, 12] fit for (1.4)(b) may probably be applied to tackle (a) because can be rewritten as . However, since the most important issue regarding (1.5) is to find the optimal growth fact (or equivalently, to search out all the growth facts), which is much more difficult to be tackled than to determine a single growth fact, we suspect that it is impossible to solve both (a) and (b) completely only using the methods of [11, 12]. So a possible idea to deal with (1.5) for the nonlinear version is to study (1.2) and obtain some meaningful results.

(ii) If , is the enterprise's admission output vector set, the identity map from to itself, and , are two th square matrixes used to describe the enterprise's consuming and reinvesting, respectively. Set , , , and , then under the zero profit principle, from (1.2) we obtain the Leontief type balanced and optimal balanced growth path problem: Both (a) and (b) are just the static descriptions of the dynamic Leontief model This model also shows that why the Leontief model (1.6) should be restricted to the linear version. We call a balanced growth factor to (1.6) if it solves (a), (1.6) is efficient if it has at least one balanced growth factor, and claim the optimal balanced growth factor to (1.6) if it maximizes (b). It is also needed to stress that at least to my knowledge, only (1.6)(a) has been considered, that is to say, up to now we do not know under what conditions of and , the optimal balanced growth fact to (1.6) must exist, and how many possible balanced growth factors to (1.6) could be found. So we hope to consider (1.6) by studying (1.2), and obtain its solvability results.

##### 1.3. Questions and Assumptions

In the sequel, taking (1.2) and (1.4)–(1.6) as the special examples of (1.1), we will devote to study (1.1) by considering the following three solvability questions.

Question 1 (Existence). If , does it solve (1.1)(a)? Can we presentany sufficient conditions, or if possible, any necessary and sufficient conditions?

Question 2 (Existence). Does the maximal lower eigenvalue to (1.1) exist? How to describe it?

Question 3 (Stability). If the answer to the Question 2 is positive, whether the corresponding map is stable in any proper way?

In order to analyse the preceding questions and obtain some meaningful results, we need three assumptions as follows.

Assumption 1. is nonempty, convex, and compact.

Assumption 2. For all , is upper semicontinuous and concave, is lower semicontinuous and convex.

Assumption 3. is nonempty, convex, and compact} and .

By virtue of the nonlinear analysis methods attributed to [810], in particular, using the minimax, saddle point, and the subdifferential techniques, we have made some progress for the solvability questions to (1.1) including a series of necessary and sufficient conditions concerning existence and a Lipschitz continuity result concerning stability. The plan of this paper is as follows, we introduce some concepts and known lemmas in Section 2, prove the main (solvability) theorems concerning (1.1) in Section 3, list the solvability results concerning (1.2) in Section 4, followed by some applications to (1.4)–(1.6) in Section 5, then present the conclusion in Section 6.

#### 2. Terminology

Let and be functions. In the sections below, we need some well known concepts of and such as convex or concave, 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 [810], so the details are omitted here. In order to deal with the solvability questions to (1.1) stated in Section 1, we also need some further concepts as follows.

Definition 2.1. (1) If , then we claim that the minimax value (of ) exists.
(2) If such that , then we call a saddle point of , and denote by the set of all saddle points.

Remark 2.2. From the definition, we can see that(1) exists if and only if ,(2) if and only if with if and only if such that for ,(3)if , then exists and for all .

Definition 2.3. Let be a function from to with the domain and a function from to . Then one has the following.(1) is said to be proper if . The epigraph of is the subset of defined by =.(2)The conjugate functions of and are the functions and defined by for and for , respectively. The biconjugate of is therefore defined on by .(3)If is a proper function from to and , then the subdifferential of at is the (possibly empty) subset of defined by for all .

Remark 2.4. If is a proper function from to , then the domain of should be defined by , and is said to be proper if .

Definition 2.5. Let be the collection of all nonempty closed bounded subsets of . Let and . Then one has the following.(1)The distance from to is defined by .(2)Let . Then the Hausdorff distance between and is defined by .

The following lemmas are useful to prove the main theorems in the next section.

Lemma 2.6 (see [9]). (1) A proper function is convex or l.s.c. if and only if its epigraph is convex or closed in .
(2) The upper envelope of proper convex (or l.s.c.) functions is also proper convex (or l.s.c.) when the is nonempty.
(3) The lower envelope of proper concave (or u.s.c.) functions is also proper concave (or u.s.c.) when the is nonempty.

Remark 2.7. Since thanks to Proposition 1.1.1 of [9], and a function defined on is concave (or u.s.c.) if and only if is convex (or l.s.c.), it is easily to see that in Lemma 2.6, the proofs from (1) to (2) and (2) to (3) are simple.

Lemma 2.8 (see [9]). Let , be a compact subset of , and let be l.s.c. (or, u.s.c.). Then defined by (or, defined by ) is also l.s.c. (or, u.s.c.).

Lemma 2.9 (see [8]). Let , be two convex compact subsets, and let be a function such that for all is l.s.c. and convex on , and for all , is u.s.c. and concave on . Then and there exists such that .

Lemma 2.10 (see [8]). A proper function defined on is convex and l.s.c. if and only if .

Lemma 2.11 (see [8]). Let be a proper function defined on , and . Then minimizes on if and only if and .

Remark 2.12. If is a finite function from to , define by if , or if , then we can use the preceding associated concepts and lemmas for by identifying with .

#### 3. Solvability Results to (1.1)

##### 3.1. Auxiliary Functions

In the sequel, we assume that Denote by the duality paring on , and for each and , define two auxiliary functions and on by Just as indicated by Definition 2.1, the minimax values and saddle point sets of and , if existed or nonempty, are denoted by , , , and , respectively.

By (3.1)–(3.3), , and are strictly positive on , and the former is l.s.c. while the latter is u.s.c.. So we can see that and both and are finite for all and .

We also define the extensions to (for each fixed ) and to (for each fixed ) by According to Definition 2.3, the conjugate and biconjugate functions of and are then denoted by By Definition 2.5, the Hausdorff distance in (see Assumption 3) is provided by

##### 3.2. Main Theorems to (1.1)

With (3.1)–(3.7), we state the main solvability theorems to (1.1) as follows.

Theorem 3.1. (1)   exists and is a nonempty convex compact subset of . Furthermore, is continuous and strictly decreasing on with .
(2)   exists if and only if . Moreover, if , then exists and is a nonempty compact subset of .

Theorem 3.2. (1)   is a lower eigenvalue to (1.1) and its eigenvector if and only if if and only if .
(2)   is a lower eigenvalue to (1.1) if and only if one of the following statements is true:
(a),(b) for ,(c) exists with ,(d) and for .
(3) The following statements are equivalent:
(a)System (1.1) has at least one lower eigenvalue,(b),(c) exists with ,(d) and for .

Theorem 3.3. (1) exists if and only if one of the following statements is true:
(a),(b) for ,(c),(d) exists with ,(e) and for . Where is the maximal lower eigenvalue to (1.1).
(2) If , or equivalently, if exists with , then one has the following.
(a) is an optimal eigenvector if and only if there exists with if and only if .(b)There exist and such that and .(c) is the maximal lower eigenvalue to (1.1) and if and only if and satisfy and . Where and are the subdifferentials of at and at , respectively.(d)The set of all lower eigenvalues to (1.1) coincides with the interval .
(3) Let , where is defined as in Assumption 3. Then
(a), and for each exists with ,(b)for all , , where is defined by (3.4). Thus, is Lipschitz on with the Hausdorff distance .

Remark 3.4. If we take , then satisfies (3.1)(2), hence Theorems 3.13.3 are also true.

##### 3.3. Proofs of the Main Theorems

In order to prove Theorems 3.13.3, we need the following eight lemmas.

Lemma 3.5. If is fixed, then one has the following. (1) and are l.s.c. and convex on .(2) and are u.s.c. and concave on .(3) exists and is a nonempty convex compact subset of .

Proof. By (3.1)–(3.3), it is easily to see that Applying Lemma 2.6(2) (resp., Lemma 2.8) to the function of (3.8)(a) (resp., of (3.8)(b)), and using the fact that is compact, and any l.s.c. (or u.s.c.) function defined on a compact set attains its minimum (or its maximum), we obtain that
If , then by (3.2), there exist such that . Since are concave, are convex and is nonnegative, we have for each , Combining (3.9) with (3.10), and using Lemmas 2.6(2)(3) and 2.9, it follows that both statements (1) and (2) hold, exists and is nonempty. It remains to verify that is convex and closed because is convex and compact.
If and , then for . By (1) and (2) (i.e., is convex on and is concave on ), we have This implies by Remark 2.2(2) that , and thus is convex.
If with , then for all . By taking , from (1) and (2) (that is, is l.s.c. on and is u.s.c. on ), we obtain that Hence by Remark 2.2(2), and is closed. Hence the first lemma follows.

Lemma 3.6. is continuous and strictly decreasing on with .

Proof. Since is continuous on for each and , is u.s.c. on for each , and is compact, by Lemmas 2.6(2) and 2.8, we see that From Lemma 2.6(2)-(3), it follows that First applying Lemma 2.8 to both functions of (3.14), and then using Lemma 3.5(3), we further obtain that and thus is continuous on .
Suppose that , then by (3.2), for all . This implies by (3.4) that , where . Hence is strictly decreasing.
By Lemma 3.5(3), Remark 2.2(3) and (3.2), it is easily to see that for each and , Hence by (3.4), and the second lemma is proved.

Lemma 3.7. (1) is a lower eigenvalue to (1.1) and its eigenvector if and only if .
(2) is a lower eigenvalue to (1.1) if and only if if and only if for .

Proof. (1) If and satisfy , then for each , . Hence, .
If and satisfy , but no can be found such that , then . Since is convex compact and is closed convex, the Hahn-Banach separation theorem implies that there exists such that . Clearly, we have (or else, we obtain , which is impossible), and thus . Since , there exist and with . It follows that . This is a contradiction. So we can select such that .
(2) If is a lower eigenvalue to (1.1), then there exists an eigenvector , which gives, by statement (1) and Lemma 3.5(3), . If , then Remark 2.2(3) and Lemma 3.5(3) imply that for all . If with , then , which gives, by statement (1), that is a lower eigenvalue to (1.1) and its eigenvector. This completes the proof.

Lemma 3.8. (1) The following statements are equivalent.
(a)System (1.1) has at least one lower eigenvalue.(b). (c) for .(d)There is a unique with .(e)The maximal lower eigenvalue to (1.1) exists. In particular, if either or one of the and exists.
(2) If , then the set of all lower eigenvalues to (1.1) equals to .

Proof. (1) If is a lower eigenvalue to (1.1), then by Lemmas 3.6 and 3.7(2), . In view of Lemma 3.5(3) and Remark 2.2, we also see that if and only if for any . If , then also by Lemmas 3.6 and 3.7(2), there exists a unique such that , and is precisely the maximal lower eigenvalue . If the maximal lower eigenvalue to (1.1) exists, then is also a lower eigenvalue to (1.1). Hence statement (1) follows.
(2) Statement (2) is obvious. Thus the lemma follows.

Lemma 3.9. If , then one has the following. (1) and are continuous on .(2) and are u.s.c. on .(3) exists if and only if .

Proof. (1) Since for each and , is continuous on , by (3.3), and Lemma 2.6(2), we see that and are l.s.c. on . On the other hand, by Assumptions 13, we can verify that is u.s.c. on . It follows from Lemma 2.8 that both functions on and on are u.s.c., so is . Hence (1) is true.
(2) As proved above, we know that for each , is u.s.c. on , so is because of Lemma 2.6(3).
(3) By Remark 2.2(3), we only need to prove the necessary part. Assume exists, that is, , then both (1) and (2) imply that there exist and with , which means that and is nonempty. Hence the lemma is true.

Lemma 3.10. (1)   is a lower eigenvalue to (1.1) and its eigenvector if and only if .
(2)   is a lower eigenvalue to (1.1) if and only if .

Proof. (1) Suppose and . Since for each , equals to , which implies that if and only if . Combining this with Lemma 3.7(1), we know that (1) is true.
(2) By (1), it is enough to prove the sufficient part. If , then Lemma 3.9(2) shows that there exists with . Hence is a lower eigenvalue to (1.1) and its eigenvector. This completes the proof.

Lemma 3.11. (1)   if and only if exists with if and only if and for . Where is the maximal lower eigenvalue to (1.1).
(2)   is a lower eigenvalue to (1.1) if and only if exists with if and only if and for .
(3) System (1.1) has at least one lower eigenvalue if and only if exist with if and only if and for .

Proof. (1) We divide the proof of (1) into three steps.Step 1. If , then by Lemma 3.8(1), the maximal eigenvalue to (1.1) exists with . We will prove that exists with . Let , then , and the left is to show .
By Lemma 3.5(2), there exists such that . This shows that for any , that is, . Hence, . On the other hand, since for each , , by Lemma 3.9(2), there exists such that . It follows that for any . Hence by Lemma 3.5(3), . From Lemma 3.7(2), this implies that is a lower eigenvalue to (1.1), and thus . Therefore, exists with .
Step 2. If exists with , then Lemma 3.9(3) and Remark 2.2(3) deduce that and for .Step 3. If and with , then . This implies by Lemmas 3.10(1) and 3.8(1) that is a lower eigenvalue to (1.1), and thus .
(2) If is a lower eigenvalue to (1.1), then Lemmas 3.8(1), 3.10(2) and statement (1) imply that exists and . If exists with , then from Lemma 3.9(3) and Remark 2.2(3), it follows that and for . If and for , then by Remark 2.2(3) and Lemma 3.10(1), we see that , and thus is a lower eigenvalue to (1.1) and its eigenvector.
(3) Statement (3) follows immediately from (1) and (2). This completes the proof.

Lemma 3.12. (1) If , or equivalently, if exists with , then is a nonempty compact subset of .
(2) The first three statements of Theorem 3.3(2) are true.
(3) Theorem 3.3(3) is true.

Proof. (1) By Lemma 3.11(1), is nonempty. Furthermore, with the same procedure as in proving the last part of Lemma 3.5 and using Lemma 3.9(1)-(2), we can show that if such that as , then Hence, is closed, and also compact.
(2) Now we prove the first three statements of Theorem 3.3(2).
By the condition of Theorem 3.3(2), Lemmas 3.8(1) and 3.11(1), we know that the maximal lower eigenvalue to (1.1) and exist with .
First we prove statement (a). If is an optimal eigenvector, then by Lemma 3.10(1), we have . On the other hand, by Lemma 3.9(1), there exists such that . So we obtain that , and thus . If such that , then Remark 2.2(3) implies that . If , then Lemma 3.10(1) shows that is an optimal eigenvector. Hence, Theorem 3.3(2)(a) follows.
Next we prove statement (b). By Lemmas 3.5(2) and 3.8(1), there exists with By applying Lemma 2.9 to on , this leads to Since is u.s.c. on and is continuous on , from (3.19), first there exists and then there exists such that