An Optimal Lower Eigenvalue System

and Applied Analysis 3 by Fujimoto 3 and Liu and Chen 4, 5 with the functional analysis approach. As for b , in case X is convex compact, and S 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 8–10 , in particular, using the classical RogalskiCornet 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 F ⊆ R or c ∈ R is an expected demand set or an expected demand of the market, and X ⊆ R some enterprise’s raw material bundle set. Then the von-Neumann type inequality problems including input-output inequalities, alongwith growth and optimal growth factors can be stated, respectively, as follows. 1 If T, S : X → intR are supposed to be the enterprise’s output or producing and consuming maps, respectively, by taking λ 1, then from both a of 1.1 and 1.2 , we obtain the von-Neumann type input-output inequalities:


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 R k R k , • the real k-dimensional Euclidean space with the dual R k * R k , R k the set of all nonnegative vectors of R k , and int R k its interior.We also define , and T T 1 , . . ., T m , S S 1 , . . ., S m : X → int R m be two single-valued maps, where m may not be equal to n.Then the optimal lower eigenvalue system that we will study and use to consider the preceding inequality problems can be described by λ, F, X, T , and S as follows:

1.1
We call λ > 0 a lower eigenvalue to 1.1 if it solves a , and its solution x the eigenvector, claim λ λ F > 0 the maximal lower eigenvalue to 1.1 if it maximizes b i.e., λ solves a , but μ not if μ > λ , and its solution x the optimal eigenvector.
In case F {c} with c ∈ R m , then 1.1 becomes a

1.2
All the concepts concerning 1.1 are reserved for 1.2 , and for convenience, the maximal lower eigenvalue λ λ {c} to 1.2 , if existed, is denoted by λ λ c .

Some Economic Backgrounds
As indicated above, the aim of this article is to consider some inequality problems in inputoutput 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 F, X, T , S, c, and λ 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 c ∈ R n is an expected demand of the market, X ⊂ R n some enterprise's admission output bundle set, and A : X → R n or S : X → 2 R n is the enterprise's single-valued or setvalued consuming map.The economic implication of a or b is whether there exists x ∈ X or there exist x ∈ X and y ∈ Sx such that the pure output x − Ax or x − y is precisely equal to the expected demand c.If X R n , and A is described by a nth 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 X is convex compact, and A 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 X is convex compact, and S 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 8-10 , in particular, using the classical Rogalski-Cornet Theorem see 8, Theorem 15.1.4and 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 F ⊆ R m or c ∈ R m is an expected demand set or an expected demand of the market, and X ⊆ R n 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 T, S : X → int R m are supposed to be the enterprise's output or producing and consuming maps, respectively, by taking λ 1, then from both a of 1.1 and 1.2 , we obtain the von-Neumann type input-output inequalities:

1.4
The economic implication of a or b is whether there exist x ∈ X and c ∈ F or there exists x ∈ X such that the pure output Tx−Sx satisfies sufficiently the expected demand c.If X R n , and T, S are described by two m × n 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 X is convex compact, and T, S are two nonlinear maps such that T i , −S i are upper semicontinuous concave for any i 1, . . ., m, 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 8-10 .Along the way, in case X is convex compact, and T , S 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 λ 1 makes 1.1 a or makes 1.2 a have solutions, and also, if and only if the maximal lower eigenvalue λ F to 1.1 exists with λ F ≥ 1 or the maximal lower eigenvalue λ c to 1.2 exists with λ c ≥ 1 , 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 T, S : X ⊆ R n → int R m are supposed to be the enterprise's output and input or invest maps, respectively, and set Λ {λ > 0 : ∃x ∈ X s.t.Tx ≥ λSx}, then Λ is nonempty, and in some degree, each λ ∈ Λ 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 λ 0 can be selected to represent the enterprise's desired minimum growth no matter whether 4 Abstract and Applied Analysis λ 0 ∈ Λ or not.By taking c 0 and restricting λ ≥ λ 0 , then from 1.2 we obtain the von-Neumann type growth and optimal growth factor problem:

1.5
We call λ a growth factor to 1.5 if it solves a , its solution x the intensity vector, and say that 1.5 is efficient if it has at least one growth factor.We also claim λ the optimal growth factor to 1.5 if it maximizes b , and its solution x the optimal intensity vector.If X R n , and S, T are described by two m × n 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 T, S 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 Tx ≥ λSx can be rewritten as Tx − λS x ≥ 0. 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 m n, X ⊆ R n is the enterprise's admission output vector set, I the identity map from R n to itself, and A a ij n×n , B b ij n×n ∈ R n 2 are two nth square matrixes used to describe the enterprise's consuming and reinvesting, respectively.Set λ μ − 1, S B, T I − A, and c 0, 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

1.7
This model also shows that why the Leontief model 1.6 should be restricted to the linear version.We call μ > 1 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 μ > 1 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 A and B, 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.

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.In order to analyse the preceding questions and obtain some meaningful results, we need three assumptions as follows.
Assumption 1. X ⊂ R n is nonempty, convex, and compact.Assumption 2. For all i 1, 2, . . ., m, T i : X −→ int R is upper semicontinuous and concave, S i : X −→ int R is lower semicontinuous and convex.By virtue of the nonlinear analysis methods attributed to 8-10 , 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.

Terminology
R be functions.In the sections below, we need some well known concepts of f, g α α ∈ Λ and ϕ 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 8-10 , 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.
2 The conjugate functions of f and g are the functions The following lemmas are useful to prove the main theorems in the next section.

Lemma 2.6 see 9 . 1 A proper function
3 The lower envelope inf α∈Λ g α x of proper concave (or u.s.c.) functions then we can use the preceding associated concepts and lemmas for f by identifying f with f X .

Auxiliary Functions
In the sequel, we assume that 1 Assumptions 1-3 in Section 1 are satisfied, and λ ∈ R , F ∈ B m ,

3.1
Denote by •, • the duality paring on R m * , R m , and for each λ ∈ R and F ∈ B m , define two auxiliary functions f λ,F p, x and g F p, x on P × X by

3.3
Just as indicated by Definition 2.1, the minimax values and saddle point sets of ϕ p, x f λ,F p, x and ϕ p, x g F p, x , if existed or nonempty, are denoted by v f λ,F , v g F , S f λ,F , and S g F , respectively.By 3.1 -3.3 , p, x → p, Sx , and p, x → p, T x are strictly positive on P × X, and the former is l.s.c.while the latter is u.s.c..So we can see that and both f λ,F p, x and g F p, x are finite for all λ ∈ R , p, x ∈ P × X and F ∈ B m .
We also define the extensions x → f λ,F p, x to x → −f λ,F p, x for each fixed p ∈ P and p → f λ,F p, x to p → f λ,F p, x for each fixed x ∈ X by

3.5
According to Definition 2.3, the conjugate and biconjugate functions of x → f λ,F p, x and p → f λ,F p, x are then denoted by

3.6
By Definition 2.5, the Hausdorff distance in B m see Assumption 3 is provided by

Main
3 The following statements are equivalent: d S g F / ∅ and g F p, x > 0 for p, x ∈ S g F .

Theorem 3.3. 1 λ exists if and only if one of the following statements is true:
e S g F / ∅ and g F p, x λ for p, x ∈ S g F .
Where λ λ F > 0 is the maximal lower eigenvalue to 1.1 .
p, q at q 0 and f * λ,F r, x at r 0, respectively.
d The set of all lower eigenvalues to 1.1 coincides with the interval 0, v g F .
, where B m is defined as in Assumption 3. Then a C m / ∅, and for each

Proofs of the Main Theorems
In order to prove Theorems 3.1-3.3,we need the following eight lemmas.Lemma 3.5.If λ ∈ R is fixed, then one has the following.
1 p → f λ,F p, x x ∈ X and p → sup x∈X f λ,F p, x are l.s.c. and convex on P .
2 x → f λ,F p, x p ∈ P and x → inf p∈P f λ,F p, x are u.s.c. and concave on X.
3 v f λ,F exists and S f λ,F is a nonempty convex compact subset of P × X.
Proof.By 3.1 -3.3 , it is easily to see that

3.8
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 F 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 ∀x ∈ X, p −→ f λ,F p, x is convex l.s.c. on P and inf . ., m are concave, X, F are convex and p ∈ P is nonnegative, we have for each α ∈ 0, 1 ,

3.10
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, v f λ,F exists and S f λ,F is nonempty.It remains to verify that S f λ,F is convex and closed because P × X is convex and compact.If α ∈ 0, 1 and p i , x i ∈ S f λ,F i 1, 2 , then sup x∈X f λ,F p i , x inf p∈P f λ,F p, x i for i 1, 2. By 1 and 2 i.e., p → sup x∈X f λ,F p, x is convex on P and x → inf p∈P f λ,F p, x is concave on X , we have
If p k , x k ∈ S f λ,F with p k , x k → p 0 , x 0 ∈ P ×X k → ∞ , then sup x∈X f λ,F p k , x inf p∈P f λ,F p, x k for all k 1, 2, . ... By taking k → ∞, from 1 and 2 that is, p → sup x∈X f λ,F p, x is l.s.c. on P and x → inf p∈P f λ,F p, x is u.s.c. on X , we obtain that

3.12
Hence by Remark 2.2 2 , p 0 , x 0 ∈ S f λ,F and S f λ,F is closed.Hence the first lemma follows.
Lemma 3.6.λ → v f λ,F is continuous and strictly decreasing on Proof.Since λ, p → p, T x − λSx − c is continuous on R × P for each c ∈ F and x ∈ X, λ, x, c → p, T x −λSx −c is u.s.c. on R ×X ×F for each p ∈ P , and F 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 λ → v f λ,F is continuous on R .Suppose that λ 2 > λ 1 ≥ 0, then by 3.2 , f λ 1 ,F p, x f λ 2 ,F p, x λ 2 − λ 1 p, Sx for all p, x ∈ P × X.This implies by 3.
If λ > 0 and x ∈ X satisfy inf p∈P f λ,F p, x ≥ 0, but no c ∈ F can be found such that Tx ≥ λSx c, then Tx − λSx − F ∩ R m ∅.Since Tx − λSx − F is convex compact and R m is closed convex, the Hahn-Banach separation theorem implies that there exists p * ∈ R m \ {0} such that −∞ < sup c∈F p * , Tx − λSx − c < inf y∈R m p * , y .Clearly, we have p * ∈ R m \ {0} or else, we obtain inf y∈R m p * , y −∞, which is impossible , inf y∈R m p * , y 0 and thus sup c∈F p * , Tx − λSx − c < 0. Since R P R m , there exist t > 0 and p ∈ P with p tp * .It follows that inf p∈P f λ,F p, x ≤ f λ,F p, x t sup c∈F p * , Tx−λSx−c < 0. This is a contradiction.So we can select c ∈ F such that Tx ≥ λSx c.
2 If λ > 0 is a lower eigenvalue to 1.1 , then there exists an eigenvector x λ ∈ X, which gives, by statement 1 and Lemma 3.5 3 , v f λ,F ≥ inf p∈P f λ,F p, x λ ≥ 0. If v f λ,F ≥ 0, then Remark 2.2 3 and Lemma 3.5 3 imply that f λ,F p, x v f λ,F ≥ 0 for all p, x ∈ S λ,F .If p, x ∈ S f λ,F with f λ,F p, x ≥ 0, then inf p∈P f λ,F p, x f λ,F p, x ≥ 0, which gives, by statement 1 , that λ is a lower eigenvalue to 1.1 and x its eigenvector.This completes the proof.
e The maximal lower eigenvalue λ λ F to 1.1 exists.
In particular, λ λ if either v f 0,F > 0 or one of the λ and λ exists.
2 If v f 0,F > 0, then the set of all lower eigenvalues to 1.1 equals to 0, λ .
Proof. 1 If λ 0 > 0 is a lower eigenvalue to 1.1 , then by Lemmas 3.6 and 3.7 2 , v f 0,F > v f λ 0 ,F ≥ 0. In view of Lemma 3.5 3 and Remark 2.2, we also see that v f 0,F > 0 if and only if f 0,F p, x > 0 for any p, x ∈ S f 0,F .If v f 0,F > 0, then also by Lemmas 3.6 and 3.7 2 , there exists a unique λ > 0 such that v f λ,F 0, and λ is precisely the maximal lower eigenvalue λ.If the maximal lower eigenvalue λ to 1.1 exists, then λ is also a lower eigenvalue to 1. Proof. 1 Suppose λ > 0 and x ∈ X.Since for each p ∈ P , g F p, x sup c∈F p, T x − c / p, Sx ≥ λ equals to f p, x sup c∈F p, T x − λSx − c ≥ 0, which implies that inf p∈P g F p, x ≥ λ if and only if inf p∈P f λ,F p, x ≥ 0. 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 sup x∈X inf p∈P g F p, x ≥ λ > 0 , then Lemma 3.9 2 shows that there exists x λ ∈ X with inf p∈P g F p, x λ sup x∈X inf p∈P g F p, x ≥ λ.Hence λ is a lower eigenvalue to 1.1 and x λ its eigenvector.This completes the proof.Lemma 3.11. 1 v f 0,F > 0 if and only if v g F exists with v g F λ if and only if S g F / ∅ and g F p, x λ for p, x ∈ S g F .Where λ λ F > 0 is the maximal lower eigenvalue to 1.1 . 2 λ is a lower eigenvalue to 1.1 if and only if v g F exists with v g F ≥ λ if and only if S g F / ∅ and g F p, x ≥ λ for p, x ∈ S g F .
3 System 1.1 has at least one lower eigenvalue if and only if v g F exist with v g F > 0 if and only if S g F / ∅ and g F p, x > 0 for p, x ∈ S g F .
Proof. 1 We divide the proof of 1 into three steps.
Step 1.If v f 0,F > 0, then by Lemma 3.8 1 , the maximal eigenvalue λ > 0 to 1.1 exists with v f λ,F 0. We will prove that v g F exists with v g F λ. Let λ * sup x∈X inf p∈P g F p, x , λ * inf p∈P sup x∈X g F p, x , then λ * ≤ λ * , and the left is to show λ * ≤ λ ≤ λ * .
By Lemma 3.5 2 , there exists x ∈ X such that inf p∈P f λ,F p, x v f λ,F 0. This shows that sup c∈F p, T x − λSx − c f λ,F p, x ≥ 0 for any p ∈ P , that is, λ ≤ sup c∈F p, T x − c / p, Sx g F p, x p ∈ P .Hence, λ ≤ inf p∈P g F p, x ≤ λ * .On the other hand, since for each p ∈ P , λ * ≤ sup x∈X g F p, x , by Lemma 3.9 2 , there exists x p ∈ X such that λ * ≤ sup x∈X g F p, x g F p, x p sup c∈F p, T x p − c / p, Sx p .It follows that sup x∈X f λ * ,F p, x ≥ f λ * ,F p, x p sup c∈F p, T x p − λ * Sx p − c ≥ 0 for any p ∈ P .Hence by Lemma 3.5 3 , v f λ * ,F inf p∈P sup x∈X f λ * ,F p, x ≥ 0. From Lemma 3.7 2 , this implies that λ * is a lower eigenvalue to 1.1 , and thus λ * ≤ λ.Therefore, v g F exists with v g F λ.
Step 2. If v g F exists with v g F λ > 0 , then Lemma 3.9 3 and Remark 2.2 3 deduce that S g F / ∅ and g F p, x v g F λ > 0 for p, x ∈ S g F .
Step 3. If S g F / ∅ and p, x ∈ S g F with g F p, x λ > 0 , then inf p∈P g F p, x λ > 0 .This implies by Lemmas 3.10 1 and 3.8 1 that λ is a lower eigenvalue to 1.1 , and thus v f 0,F > 0.
with v g F ≥ λ, then from Lemma 3.9 3 and Remark 2.2 3 , it follows that S g F / ∅ and g F p, x v g F ≥ λ for p, x ∈ S g F .If S g F / ∅ and g F p, x ≥ λ for p, x ∈ S g F , then by Remark 2.2 3 and Lemma 3.10 1 , we see that inf p∈P g F p, x g F p, x ≥ λ, and thus λ is a lower eigenvalue to 1.1 and x its eigenvector.
3 Statement 3 follows immediately from 1 and 2 .This completes the proof.
Lemma 3.12. 1 If v f 0,F > 0, or equivalently, if v g F exists with v g F > 0, then S g F is a nonempty compact subset of × X.
2 The first three statements of Theorem 3.3(2) are true.
Proof. 1 By Lemma 3.11 1 , S g F 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 Hence, S g F 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 λ to 1.1 and v g F exist with v g F λ. First we prove statement a .If x ∈ X is an optimal eigenvector, then by Lemma 3.10 1 , we have inf p∈P g F p, x ≥ λ.On the other hand, by Lemma 3.9 1 , there exists p ∈ P such that v g F sup x∈X g F p, x .So we obtain that sup x∈X g F p, x λ ≤ inf p∈P g F p, x , and thus p, x ∈ S g F .If p ∈ P such that p, x ∈ S g F , then Remark 2.2 3 implies that inf p∈P g F p, x v g F λ.If inf p∈P g F p, x λ, then Lemma 3.10 1 shows that x is an optimal eigenvector.Hence, Theorem 3.

3.21
On the other hand, we can see that I p {i : p i > 0} is nonempty because p ∈ P ⊆ R m \ {0}.This gives, by 3.20 and 3.21 , that for each i 0 ∈ I p ,

3.24
In view of 3.5 , we know that 3.23 is also equivalent to

3.25
Also by 3.5 , we have epi f λ,F p, • { x, a ∈ X × R : −f λ,F p, x ≤ a} and epi f λ,F •, x { p, a ∈ P × R : f λ,F p, x ≤ a}.Combining this with Lemma 3.5 1 -2 and using the fact that X and P are convex compact, we can see that epi f λ,F p, • or epi f λ,F •, x is closed convex in R n × R or in R m × R .Hence Lemmas 2.6 1 and 2.10 imply that both x → f λ,F p, x x ∈ R n and p → f λ,F p, x p ∈ R m are proper convex and l.s.c. with

3.26
Applying Lemma 2.11 to the functions x → f λ,F p, x − q 0 , x on R n with q 0 0 and p → f λ,F p, x − r 0 , p on R m with r 0 0, and using 3.25 and and the last lemma follows.
iii For Theorem 3.3.By Lemmas 3.5 3 , 3.8 1 and 3.11 1 , 1 is true.From Lemmas 3.8 2 and 3.12 2 , 2 is valid.Applying Lemma 3.12 3 , we obtain the last statement.b There exist x ∈ X and i 0 ∈ {1, 2, . . ., m} such that T x ≥ λS x c and T i 0 x λS i 0 x c i 0 .c λ λ c is the maximal lower eigenvalue to 1.2 and p, x ∈ S g c if and only if λ > 0 and p, x ∈ P × X satisfy x ∈ ∂ f *   c S g c / ∅ and g c p, x ≥ 1 for p, x ∈ S g c .

Solvability to 1.5
By Theorem 4.1, for each λ ≥ 0, v f λ,0 exists, S f λ,0 is nonempty, and if p, x ∈ S f 0,0 , then v f 0,0 p, T x > 0. Hence v g 0 exists, S g 0 is nonempty, and the maximal lower eigenvalue λ λ 0 to 1.2 exists with v g 0 λ g 0 p, x for p, x ∈ S g 0 .By Theorems 4.2 and 4.3 for c 0, we obtain the solvability results to 1.5 as follows.
Theorem 5.2. 1 λ is a growth factor to 1.5 and x its intensity vector if and only if λ ≥ λ 0 with inf p∈P f λ,0 p, x ≥ 0 if and only if inf p∈P g 0 p, x ≥ λ ≥ λ 0 .
2 λ is a growth factor to 1.5 if and only if λ ≥ λ 0 with v f λ,0 ≥ 0 if and only if v g 0 ≥ λ ≥ λ 0 .
3 Growth fact problem 1.5 is efficient if and only if there exists λ ≥ λ 0 with v f λ,0 ≥ 0 if and only if v g 0 ≥ λ 0 .
4 λ is the optimal growth factor to 1.5 if and only if λ ≥ λ 0 with v f λ,0 0 if and only if v g 0 λ ≥ λ 0 .
6 λ is the optimal growth factor to 1.5 and p, x ∈ S g 0 if and only if λ ≥ λ 0 and p, x ∈ P × X satisfy x ∈ ∂ f * λ,0 p, 0 and p ∈ ∂ f * λ,0 0, x .1.6 To present the solvability results to 1.6 , we assume that 1 There exists λ > 0 such that v f λ 0 and λ v g .

Solvability to
2 μ λ 1 is the optimal balanced growth factor to 1.6 .
3 Growth path problem 1.6 is efficient, and μ is a balanced growth factor to 1.6 if and only if μ ∈ 1, 1 λ .
Remark 5.4.Assumption 5.1 is only an essential condition to get the conclusions of Theorem 5.3.By applying Theorems 4.1-4.3 and using some analysis methods or matrix techniques, one may obtain some more solvability results to the Leontief-type balanced and optimal balanced growth path problem.

Conclusion
In this article, we have studied an optimal lower eigenvalue system namely, 1.1 , and proved three solvability theorems i.e., Theorems 3.1-3.3including a series of necessary 20 Abstract and Applied Analysis and sufficient conditions concerning existence and a Lipschitz continuity result concerning stability.With the theorems, we have also obtained some existence criteria namely, Theorems 5.1-5.3 to the von-Neumann type input-output inequalities, growth and optimal growth factors, as well as to the Leontief type balanced and optimal balanced growth path problems.

Assumption 3 .
B m {F ⊂ R m : F is nonempty, convex, and compact} and F ∈ B m .

Definition 2 . 5 .
then the domain of f should be defined by dom f {x ∈ R k : f x > −∞}, and f is said to be proper if dom f / ∅.Let B R k be the collection of all nonempty closed bounded subsets of R k .Let x ∈ R k and A, B ∈ B R k .Then one has the following.1 The distance d x, A from x to A is defined by d x, A inf y∈A d x, y . 2 Let ρ A, B sup x∈A d x, B .Then the Hausdorff distance d H A, B between A and B is defined by d H A, B max{ρ A, B , ρ B, A }.

Lemma 3 . 8 . 1
The following statements are equivalent.aSystem 1.1 has at least one lower eigenvalue.
3 2 a follows.Next we prove statement b .By Lemmas 3.5 2 and 3.8 1 , there exists x ∈ X with 0 v f λ,F sup x∈X inf p∈P f λ,F p, x inf p∈P f λ,F p, x inf p∈P sup c∈F p, T x − λS x − c .3.18 By applying Lemma 2.9 to ϕ p, c p, T x − λS x − c on P × F, this leads to sup c∈F inf p∈P p, T x − λS x − c inf p∈P sup c∈F p, T x − λS x − c 0. 3.19 Since c → inf p∈P p, T x−λS x−c is u.s.c. on F and p → p, T x−λS x−c is continuous on P , from 3.19 , first there exists c c 1 , c 2 , . . ., c m ∈ F and then there exists p p 1 , p 2 , . . ., p m ∈ P such that 0 sup c∈F inf p∈P p, T x − λS x − c inf p∈P p, T x − λS x − c p, T x − λS x − c .

d
are the subdifferentials of f * λ,c p, q at q 0 and f * λ,c r, x at r 0, respectively.The set of all lower eigenvalues to 1.2 coincides with the interval 0, v g c . 3 Let C {c ∈ R m : v f 0,c > 0}.Then one has the following.a C / ∅, and for each c ∈ C , λ λ c exists with λ c v g c .b |λ c 1 − λ c 2 | ≤ sup p∈P p /ε 0 c 1 − c 2 c 1 , c 2 ∈ C , where ε 0 is also defined by 3.4 .Hence, c → λ c is Lipschitz on C .

5. 1 .Theorem 5 . 1 . 1 a
Solvability to 1.4 Since F ∈ B m or c ∈ R m makes a or b of 1.4 solvable if and only if λ 1 is a lower eigenvalue to 1.1 or 1.2 if and only if the maximal lower eigenvalue λ λ F to 1.1 or λ λ c to 1.2 exists with λ F ≥ 1 or λ c ≥ 1 , by applying Theorems 3.3 and 4.3, we have the solvability results to 1.4 as follows.Inequality 1.4 (a) is solvable to F ∈ B m if and only if one of the following statements is true.a There exists λ ≥ 1 with v f λ,F 0. b v g F exists with v g F ≥ 1. c S g F / ∅ and g F p, x ≥ 1 for p, x ∈ S g F . 2 Inequality 1.4 (b) is solvable to c ∈ R m if and only if one of the following statements is true.There exists λ ≥ 1 with v f λ,c 0. b v g c exists with v g c ≥ 1.

5 . 1 andTheorem 5 . 3 .
define f λ p, x f λ,0 p, x and g p, x g 0 p, x on Σ n−1 × X byf λ p, x p, I − A x − λBx , p, x p, I − A x p, Bx for p, x ∈ Σ n−1 × X,5.2 where Σ n−1 is the n − 1 simplex.Applying Theorems 4.1-4.3 to S B, T I − A, c 0 and λ μ − 1, we obtain existence results to 1.6 as follows.If 5.1 holds and f λ , g be defined by 5.1 .Then one has the following.
Lemma 2.8 see 9 .Let X ⊂ R n , Y be a compact subset of R m , and let f : X × Y → R be l.s.c.(or, u.s.c.).Then h : X → R defined by h x inf Then inf p∈P sup x∈X ϕ p, x sup x∈X inf p∈P ϕ p, x and there exists p, x ∈ P × X such that ϕ p, x sup x∈X ϕ p, x inf p∈P ϕ p, x .Lemma 2.10 see 8 .A proper function f defined on R k is convex and l.s.c.if and only if f f * * .
is nonempty.Remark 2.7.Since epi sup α∈Λ f α α∈Λ epi f α thanks to Proposition 1.1.1 of 9 , and a function f defined on R k is concave or u.s.c.if and only if −f 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.y∈Y f x, y (or, k : X → R defined by k x sup y∈Y f x, y ) is also l.s.c.(or, u.s.c.).Lemma 2.9 see 8 .Let P ⊆ R m , X ⊆ R n be two convex compact subsets, and let ϕ : P × X → R be a function such that for all x ∈ X, p → ϕ p, x is l.s.c. and convex on P , and for all p ∈ P , x → ϕ p, x is u.s.c. and concave on X. Lemma 2.11 see 8 .Let f be a proper function defined on R k , and

Theorem 3.2. 1
λ is a lower eigenvalue to 1.1 and x its eigenvector if and only if inf p∈P f λ,F p, x ≥ 0 if and only if inf p∈P g F p, x ≥ λ.2 λ is a lower eigenvalue to 1.1 if and only if one of the following statements is true: Theorems to 1.1 With 3.1 -3.7 , we state the main solvability theorems to 1.1 as follows.Theorem 3.1.1vf λ,F exists and S f λ,F is a nonempty convex compact subset of P ×X.Furthermore, λ → v f λ,F is continuous and strictly decreasing on R withv f ∞,F lim λ → ∞ v f λ,F −∞.2 v g F exists if and only if S g F / ∅.Moreover, if v f 0,F > 0, then v g F exists and S g F is a nonempty compact subset of P × X.
or equivalently, if v g F exists with v g F > 0, then one has the following.a x is an optimal eigenvector if and only if there exists p ∈ P with p, x ∈ S g F if and only if inf p∈P g F p, x λ.
b There exist x ∈ X, c ∈ F and i 0 ∈ {1, 2, ..., m} such that T x ≥ λS x c andT i 0 x λS i 0 x c i 0 .cλ λ Fis the maximal lower eigenvalue to 1.1 and p, x ∈ S g F if and only if λ > 0 and p, x ∈ P × X satisfy x ∈ ∂ f * λ,F p, 0 and p ∈ ∂ f * λ,F 0, x .Where ∂ f * λ,F p, 0 and → sup x∈X g F p, x are l.s.c. on P .On the other hand, by Assumptions 1-3, we can verify that p, x, c → p, T x − c / p, Sx is u.s.c. on P × X × F. It follows from Lemma 2.8 that both functions p, x → g F p, x sup c∈F p, T x − c / p, Sx on P × X and p → sup x∈X g F p, x on P are u.s.c., so is p → g F p, x .Hence 1 is true.infp∈Pg F p, x because of Lemma 2.6 3 .3ByRemark 2.2 3 , we only need to prove the necessary part.Assume v g F exists, that is, inf p∈P sup x∈X g F p, x sup x∈X inf p∈P g F p, x , then both 1 and 2 imply that there exist p ∈ P and x ∈ X with sup x∈X g F p, x inf p∈P g F p, x , which means that p, x ∈ S g F and S g F is nonempty.Hence the lemma is true.Lemma 3.10. 1 λ is a lower eigenvalue to 1.1 and x its eigenvector if and only if inf p∈P g F p, x ≥ λ.2 λ is a lower eigenvalue to 1.1 if and only if sup x∈X inf p∈P g F p, x ≥ λ.
1 .Hence statement 1 follows.2Statement 2 is obvious.Thus the lemma follows.Lemma 3.9.If F ∈ B m , then one has the following.1 p → g F p, x x ∈ X and p → sup x∈X g F p,x are continuous on P .2x→g F p, x p ∈ P and x → inf p∈P g F p, x are u.s.c. on X.3 v g F exists if and only if S g F / ∅.Proof. 1 Since for each x ∈ X and c ∈ F, p → p, T x − c / p, Sx is continuous on P , 2 As proved above, we know that for each p ∈ P , x → g F p, x is u.s.c. on X, so is x → that is, T i 0 x λS i 0 x c i 0 .3.22Both 3.21 and 3.22 show that Theorem 3.3 2 b is true.Then we prove statement c .From 3.2 , 3.3 , and Lemmas 3.8 1 and 3.11 1 , as well as Remark 2.2 2 , we know that λ is the maximal lower eigenvalue to 1.1 and p, x ∈ S g F if and only if λ > 0 and p, x ∈ P ×X satisfy g F p, x ≤ g F p, x λ ≤ g F p, x for p, x ∈ P ×X, which amounts to say 1 , F 2 and ε 0 inf p∈P,x∈X p, Sx is positive.By taking minimax values for both sides of 3.27 , we have