# An Optimal Lower Eigenvalue System

Authors: Yingfan Liu
Journal: Abstract and Applied Analysis
 (2011)

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## 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.


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## Body

## 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 byRk=(Rk,∥·∥) the real k-dimensional Euclidean space with the dual Rk*=Rk, R+k the set of all nonnegative vectors of Rk, and intR+k its interior. We also define y1≥(or>)y2 in Rk by y1-y2∈R+k (or ∈intR+k).Letλ∈R+=̂R+1, F⊆R+m, X⊆R+n, and T=̂(T1,…,Tm), S=̂(S1,…,Sm):X→intR+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)(a){λ>0:∃x∈Xs.t.Tx-λSx∈F+R+m,thatis,Tx-Sx≥cforsomec∈F,(b){0<λ⟶max⁡:∃x∈Xs.t.Tx-λSx∈F+R+m.
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 caseF={c} with c∈R+m, then (1.1) becomes(1.2)(a){λ>0:∃x∈Xs.t.Tx≥λSx+c,(b){0<λ⟶max⁡:∃x∈Xs.t.Tx≥λSx+c.
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).


### 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 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:(1.3)(a){∃x∈Xs.t.x-Ax=c,(b){∃x∈Xs.t.x-Sx∋c,
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→2R+n is the enterprise's single-valued or set-valued 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 15.1.4]) 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 thatF⊆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) IfT,S:X→intR+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)(a){x∈Xs.t.Tx-Sx∈F+R+m,(b){x∈Xs.t.Tx-Sx≥c.
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 Ti,-Si 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) IfT,S:X⊆R+n→intR+m are supposed to be the enterprise's output and input (or invest) maps, respectively, and set Λ=̂{λ>0:∃x∈Xs.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 λ0 can be selected to represent the enterprise's desired minimum growth no matter whether λ0∈Λ or not. By taking c=0 and restricting λ≥λ0, then from (1.2) we obtain the von-Neumann type growth and optimal growth factor problem:(1.5)(a){λ∈[λ0,+∞):∃x∈Xs.t.Tx≥λSx,(b){λ0≤λ⟶max⁡:∃x∈Xs.t.Tx≥λSx.
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) Ifm=n, X⊆R+n is the enterprise's admission output vector set, I the identity map from Rn to itself, and A=(aij)n×n, B=(bij)n×n∈R+n2 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:(1.6)(a){μ>1:∃x∈Xs.t.(I-A)x≥(μ-1)Bx,(b){1<μ⟶max⁡:∃x∈Xs.t.(I-A)x≥(μ-1)Bx.
Both (a) and (b) are just the static descriptions of the dynamic Leontief model (1.7)(a)μ>1or(b)1<μ⟶max⁡:∃x∈Xs.t.x(t)=μtxwith(I-A+B)x(t)≥Bx(t+1),t=1,2,….
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.


### 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).
Ifλ>0, 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 eigenvalueλ¯=λ¯(F) to (1.1) exist? How to describe it?Question 3 (Stability).
If the answer to the Question2 is positive, whether the corresponding map F↦λ¯=λ¯(F) 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.
X⊂R+n is nonempty, convex, and compact.Assumption 2.
For alli=1,2,…,m, Ti:X→intR+ is upper semicontinuous and concave, Si:X→intR+ is lower semicontinuous and convex.Assumption 3.
𝔹+m={F⊂R+m:F is nonempty, convex, and compact} and F∈𝔹+m.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.


## 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 byRk=(Rk,∥·∥) the real k-dimensional Euclidean space with the dual Rk*=Rk, R+k the set of all nonnegative vectors of Rk, and intR+k its interior. We also define y1≥(or>)y2 in Rk by y1-y2∈R+k (or ∈intR+k).Letλ∈R+=̂R+1, F⊆R+m, X⊆R+n, and T=̂(T1,…,Tm), S=̂(S1,…,Sm):X→intR+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)(a){λ>0:∃x∈Xs.t.Tx-λSx∈F+R+m,thatis,Tx-Sx≥cforsomec∈F,(b){0<λ⟶max⁡:∃x∈Xs.t.Tx-λSx∈F+R+m.
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 caseF={c} with c∈R+m, then (1.1) becomes(1.2)(a){λ>0:∃x∈Xs.t.Tx≥λSx+c,(b){0<λ⟶max⁡:∃x∈Xs.t.Tx≥λSx+c.
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).


## 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 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:(1.3)(a){∃x∈Xs.t.x-Ax=c,(b){∃x∈Xs.t.x-Sx∋c,
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→2R+n is the enterprise's single-valued or set-valued 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 15.1.4]) 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 thatF⊆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) IfT,S:X→intR+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)(a){x∈Xs.t.Tx-Sx∈F+R+m,(b){x∈Xs.t.Tx-Sx≥c.
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 Ti,-Si 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) IfT,S:X⊆R+n→intR+m are supposed to be the enterprise's output and input (or invest) maps, respectively, and set Λ=̂{λ>0:∃x∈Xs.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 λ0 can be selected to represent the enterprise's desired minimum growth no matter whether λ0∈Λ or not. By taking c=0 and restricting λ≥λ0, then from (1.2) we obtain the von-Neumann type growth and optimal growth factor problem:(1.5)(a){λ∈[λ0,+∞):∃x∈Xs.t.Tx≥λSx,(b){λ0≤λ⟶max⁡:∃x∈Xs.t.Tx≥λSx.
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) Ifm=n, X⊆R+n is the enterprise's admission output vector set, I the identity map from Rn to itself, and A=(aij)n×n, B=(bij)n×n∈R+n2 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:(1.6)(a){μ>1:∃x∈Xs.t.(I-A)x≥(μ-1)Bx,(b){1<μ⟶max⁡:∃x∈Xs.t.(I-A)x≥(μ-1)Bx.
Both (a) and (b) are just the static descriptions of the dynamic Leontief model (1.7)(a)μ>1or(b)1<μ⟶max⁡:∃x∈Xs.t.x(t)=μtxwith(I-A+B)x(t)≥Bx(t+1),t=1,2,….
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.


## 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).
Ifλ>0, 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 eigenvalueλ¯=λ¯(F) to (1.1) exist? How to describe it?Question 3 (Stability).
If the answer to the Question2 is positive, whether the corresponding map F↦λ¯=λ¯(F) 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.
X⊂R+n is nonempty, convex, and compact.Assumption 2.
For alli=1,2,…,m, Ti:X→intR+ is upper semicontinuous and concave, Si:X→intR+ is lower semicontinuous and convex.Assumption 3.
𝔹+m={F⊂R+m:F is nonempty, convex, and compact} and F∈𝔹+m.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.


## 2. Terminology

Letf,gα(α∈Λ):X⊂Rk→R and φ:P×X⊂Rm×Rn→R be functions. In the sections below, we need some well known concepts of f,gα(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.Definition 2.1.
(1) Ifinfp∈Psupx∈Xφ(p,x)=supx∈Xinfp∈Pφ(p,x)=̂v(φ), then we claim that the minimax value v(φ) (of φ) exists.
(2) If(p¯,x¯)∈P×X such that supx∈Xφ(p¯,x)=infp∈Pφ(p,x¯), then we call (p¯,x¯) a saddle point of φ, and denote by S(φ) the set of all saddle points.Remark 2.2.
From the definition, we can see that(1)
v(φ) exists if and only if infp∈Psupx∈Xφ(p,x)≤supx∈Xinfp∈Pφ(p,x),(2)
(p¯,x¯)∈S(φ) if and only if (p¯,x¯)∈P×X with supx∈Xφ(p¯,x)≤infp∈Pφ(p,x¯) if and only if (p¯,x¯)∈P×X such that φ(p¯,x)≤φ(p,x¯) for (p,x)∈P×X,(3)
ifS(φ)≠∅, then v(φ) exists and v(φ)=φ(p¯,x¯)=supx∈Xφ(p¯,x)=infp∈Pφ(p,x¯) for all (p¯,x¯)∈S(φ).Definition 2.3.
Letf be a function from Rk to R∪{+∞} with the domain dom(f)={x∈Rk:f(x)<+∞} and g a function from Rk*(=Rk) to R∪{+∞}. Then one has the following.(1)
f is said to be proper if dom(f)≠∅. The epigraph epi(f) of f is the subset of Rk×R defined by epi(f)={(x,a)∈Rk×R:f(x)≤a}.(2)
The conjugate functions off and g are the functions f*:Rk*→R∪{+∞} and g*:Rk→R∪{+∞} defined by f*(p)=supx∈Rk[〈p,x〉-f(x)] for p∈Rk* and g*(x)=supp∈Rk[〈p,x〉-g(p)] for x∈Rk, respectively. The biconjugate f** of f is therefore defined on Rk**(=Rk) by f**=(f*)*.(3)
Iff is a proper function from Rk to R∪{+∞} and x0∈dom(f), then the subdifferential of f at x0 is the (possibly empty) subset ∂f(x0) of Rk* defined by ∂f(x0)={p∈Rk*:f(x0)-f(x)≤〈p,x0-x〉 for all x∈Rk}.Remark 2.4.
Iff is a proper function from Rk to R∪{-∞}, then the domain of f should be defined by dom(f)={x∈Rk:f(x)>-∞}, and f is said to be proper if dom(f)≠∅.Definition 2.5.
Let𝔹(Rk) be the collection of all nonempty closed bounded subsets of Rk. Let x∈Rk and A,B∈𝔹(Rk). Then one has the following.(1)
The distanced(x,A) from x to A is defined by d(x,A)=infy∈Ad(x,y).(2)
Letρ(A,B)=supx∈Ad(x,B). Then the Hausdorff distance dH(A,B) between A and B is defined by dH(A,B)=max{ρ(A,B),ρ(B,A)}.The following lemmas are useful to prove the main theorems in the next section.Lemma 2.6 (see [9]).
(1) A proper functionf:Rk→R∪{+∞} is convex or l.s.c. if and only if its epigraph epi(f) is convex or closed in Rk×R.
(2) The upper envelopesupα∈Λfα(x) of proper convex (or l.s.c.) functions fα(x):Rk→R∪{+∞}(α∈Λ) is also proper convex (or l.s.c.) when the dom(supα∈Λfα)={x∈Rk:supα∈Λfα(x)<+∞} is nonempty.
(3) The lower envelopeinfα∈Λgα(x) of proper concave (or u.s.c.) functions gα(x):Rk→R∪{+∞}(α∈Λ) is also proper concave (or u.s.c.) when the dom(infα∈Λgα)={x∈Rk:infα∈Λgα(x)>-∞} is nonempty.Remark 2.7.
Sinceepi(supα∈Λfα)=⋂α∈Λepi(fα) thanks to Proposition 1.1.1 of [9], and a function f defined on Rk 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.Lemma 2.8 (see [9]).
LetX⊂Rn, Y be a compact subset of Rm, and let f:X×Y→R be l.s.c. (or, u.s.c.). Then h:X→R defined by h(x)=infy∈Yf(x,y) (or, k:X→R defined by k(x)=supy∈Yf(x,y)) is also l.s.c. (or, u.s.c.).Lemma 2.9 (see [8]).
LetP⊆Rm, X⊆Rn 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. Then infp∈Psupx∈Xφ(p,x)=supx∈Xinfp∈Pφ(p,x) and there exists (p¯,x¯)∈P×X such that φ(p¯,x¯)=supx∈Xφ(p¯,x)=infp∈Pφ(p,x¯).Lemma 2.10 (see [8]).
A proper functionf defined on Rk is convex and l.s.c. if and only if f=f**.Lemma 2.11 (see [8]).
Letf be a proper function defined on Rk, and p0∈Rk*. Then x0 minimizes x↦f(x)-〈p0,x〉 on U if and only if x0∈∂f*(p0) and f(x0)=f**(x0).Remark 2.12.
Iff is a finite function from X⊆Rk to R, define fX by fX(x)=f(x) if x∈X, or =+∞ if x∈Rk∖X, then we can use the preceding associated concepts and lemmas for f by identifying f with fX.


## 3. Solvability Results to (1.1)

### 3.1. Auxiliary Functions

In the sequel, we assume that(3.1)(1)Assumptions1–3inSection1aresatisfied,andλ∈R+,F∈B+m,(2)P⊆R+m∖{0}isaconvexcompactsubsetwithR+P=R+m.
Denote by 〈·,·〉 the duality paring on 〈Rm*,Rm〉, and for each λ∈R+ and F∈𝔹+m, define two auxiliary functions fλ,F(p,x) and gF(p,x) on P×X by(3.2)fλ,F(p,x)=sup⁡c∈F〈p,Tx-λSx-c〉=sup⁡(c1,c2,…,cm)∈F∑i=1mpi(Tix-λSix-ci),(3.3)gF(p,x)=sup⁡c∈F〈p,Tx-c〉〈p,Sx〉=sup⁡(c1,c2,…,cm)∈F∑i=1mpi(Tix-ci)∑i=1mpiSix.
Just as indicated by Definition 2.1, the minimax values and saddle point sets of φ(p,x)=fλ,F(p,x) and φ(p,x)=gF(p,x), if existed or nonempty, are denoted by v(fλ,F), v(gF), S(fλ,F), and S(gF), respectively.By (3.1)–(3.3), (p,x)↦〈p,Sx〉, and (p,x)↦〈p,Tx〉 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(3.4)0<ɛ0=inf⁡p∈P,x∈X〈p,Sx〉<+∞,0<ɛ1=sup⁡p∈P,x∈X〈p,Tx〉<+∞,
and both fλ,F(p,x) and gF(p,x) are finite for all λ∈R+,(p,x)∈P×X and F∈𝔹+m.We also define the extensionsx↦f̂λ,F(p,x) to x↦-fλ,F(p,x) (for each fixed p∈P) and p↦f̃λ,F(p,x) to p↦fλ,F(p,x) (for each fixed x∈X) by(3.5)f̂λ,F(p,x)={-fλ,F(p,x),x∈X,+∞,x∈Rn∖X,f̃λ,F(p,x)={fλ,F(p,x),p∈P,+∞,p∈Rm∖P.
According to Definition 2.3, the conjugate and biconjugate functions of x↦f̂λ,F(p,x) and p↦f̃λ,F(p,x) are then denoted by(3.6)q⟼f̂λ,F*(p,q),q∈Rn,x⟼f̂λ,F**(p,x),x∈Rn(foreachfixedp∈P),r⟼f̃λ,F*(r,x),r∈Rm,p⟼f̃λ,F**(p,x),p∈Rm(foreachfixedx∈X).
By Definition 2.5, the Hausdorff distance in 𝔹+m (see Assumption 3) is provided by(3.7)dH(F1,F2)=max⁡{sup⁡c1∈F1d(c1,F2),sup⁡c2∈F2d(c2,F1)}forF1,F2∈B+m.


### 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)v(fλ,F) exists and S(fλ,F) is a nonempty convex compact subset of P×X. Furthermore, λ↦v(fλ,F) is continuous and strictly decreasing on R+ with v(f+∞,F)=̂limλ→+∞v(fλ,F)=-∞.
(2)v(gF) exists if and only if S(gF)≠∅. Moreover, if v(f0,F)>0, then v(gF) exists and S(gF) is a nonempty compact subset of P×X.Theorem 3.2.
(1)λ is a lower eigenvalue to (1.1) and x its eigenvector if and only if infp∈Pfλ,F(p,x)≥0 if and only if infp∈PgF(p,x)≥λ.
(2)λ is a lower eigenvalue to (1.1) if and only if one of the following statements is true:
(a)
v(fλ,F)≥0,(b)
fλ,F(p̂,x̂)≥0 for (p̂,x̂)∈S(fλ,F),(c)
v(gF) exists with v(gF)≥λ,(d)
S(gF)≠∅ and gF(p̂,x̂)≥λ for (p̂,x̂)∈S(gF).
(3) The following statements are equivalent:
(a)
System (1.1) has at least one lower eigenvalue,(b)
v(f0,F)>0,(c)
v(gF) exists with v(gF)>0,(d)
S(gF)≠∅ and gF(p̂,x̂)>0 for (p̂,x̂)∈S(gF).Theorem 3.3.
(1)λ¯ exists if and only if one of the following statements is true:
(a)
v(f0,F)>0,(b)
f0,F(p̂,x̂)>0 for (p̂,x̂)∈S(f0,F),(c)
v(fλ¯,F)=0,(d)
v(gF) exists with v(gF)=λ¯,(e)
S(gF)≠∅ and gF(p¯,x¯)=λ¯ for (p¯,x¯)∈S(gF).
Where λ¯=λ¯(F)(>0) is the maximal lower eigenvalue to (1.1).
(2) Ifv(f0,F)>0, or equivalently, if v(gF) exists with v(gF)>0, then one has the following.
(a)
x¯ is an optimal eigenvector if and only if there exists p¯∈P with (p¯,x¯)∈S(gF) if and only if infp∈PgF(p,x¯)=λ¯.(b)
There existx̂∈X,ĉ∈F and i0∈{1,2,…,m} such that Tx̂≥λ¯Sx̂+ĉ and Ti0x̂=λ¯Si0x̂+ĉi0.(c)
λ¯=λ¯(F) is the maximal lower eigenvalue to (1.1) and (p¯,x¯)∈S(gF) if and only if λ¯>0 and (p¯,x¯)∈P×X satisfy x¯∈∂f̂λ¯,F*(p¯,0)and p¯∈∂f̃λ¯,F*(0,x¯). Where ∂f̂λ¯,F*(p¯,0) and ∂f̃λ¯,F*(0,x¯) are the subdifferentials of f̂λ¯,F*(p¯,q) at q=0 and f̃λ¯,F*(r,x¯) at r=0, respectively.(d)
The set of all lower eigenvalues to (1.1) coincides with the interval (0,v(gF)].
(3) Letℂ+m={F∈𝔹+m:v(f0,F)>0}, where 𝔹+m is defined as in Assumption 3. Then
(a)
ℂ+m≠∅, and for each F∈ℂ+m,λ¯=λ¯(F) exists with λ¯(F)=v(gF),(b)
for allF1,F2∈ℂ+m, |λ¯(F1)-λ¯(F2)|≤(supp∈P∥p∥/ɛ0)dH(F1,F2), where ɛ0 is defined by (3.4).
Thus, F↦λ¯(F) is Lipschitz on ℂ+m with the Hausdorff distance dH(·,·).Remark 3.4.
If we takeP=Σm-1={p∈R+m:∑i=1mpi=1}, then Σm-1 satisfies (3.1)(2), hence Theorems 3.1–3.3 are also true.


### 3.3. Proofs of the Main Theorems

In order to prove Theorems3.1–3.3, we need the following eight lemmas.Lemma 3.5.
Ifλ∈R+ is fixed, then one has the following.(1)
p↦fλ,F(p,x)(x∈X) and p↦supx∈Xfλ,F(p,x) are l.s.c. and convex on P.(2)
x↦fλ,F(p,x)(p∈P) and x↦infp∈Pfλ,F(p,x) are u.s.c. and concave on X.(3)
v(fλ,F) exists and S(fλ,F) is a nonempty convex compact subset of P×X.Proof.
By (3.1)–(3.3), it is easily to see that
(3.8)(a)∀x∈X,∀c∈F,p⟼〈p,Tx-λSx-c〉isconvexl.s.c.on⁡P,(b)∀p∈P,(x,c)⟼〈p,Tx-λSx-c〉isu.s.c.on⁡X×F.
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
(3.9)∀x∈X,p⟼fλ,F(p,x)isconvexl.s.c.on⁡Pandinf⁡p∈Pfλ,F(p,x)isfinite,∀p∈P,x⟼fλ,F(p,x)isu.s.c.on⁡Xandsup⁡x∈Xfλ,F(p,x)isfinite.
Ifxi∈X(i=1,2), then by (3.2), there exist ci=(c1i,c2i,…,cmi)∈F(i=1,2) such that fλ,F(p,xi)=〈p,Txi-λSxi-ci〉. Since Ti,-λSi(i=1,2,…,m) are concave, X,F are convex and p∈P is nonnegative, we have for each α∈[0,1],(3.10)fλ,F(p,αx1+(1-α)x2)≥〈p,T[αx1+(1-α)x2]-λS[αx1+(1-α)x2]-αc1-(1-α)c2〉≥α〈p,Tx1-λSx1-c1〉+(1-α)〈p,Tx2-λSx2-c2〉=αfλ,F(p,x1)+(1-α)fλ,F(p,x2),thatis,x⟼fλ,F(p,x)isconcaveon⁡X.
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(fλ,F) exists and S(fλ,F) is nonempty. It remains to verify that S(fλ,F) is convex and closed because P×X is convex and compact.
Ifα∈[0,1] and (pi,xi)∈S(fλ,F)(i=1,2), then supx∈Xfλ,F(pi,x)=infp∈Pfλ,F(p,xi) for i=1,2. By (1) and (2) (i.e., p↦supx∈Xfλ,F(p,x) is convex on P and x↦infp∈Pfλ,F(p,x) is concave on X), we have (3.11)sup⁡x∈Xfλ,F(αp1+(1-α)p2,x)≤αsup⁡x∈Xfλ,F(p1,x)+(1-α)sup⁡x∈Xfλ,F(p2,x)=αinf⁡p∈Pfλ,F(p,x1)+(1-α)inf⁡p∈Pfλ,F(p,x2)≤inf⁡p∈Pfλ,F(p,αx1+(1-α)x2).
This implies by Remark 2.2(2) that α(p1,x1)+(1-α)(p2,x2)∈S(fλ,F), and thus S(fλ,F) is convex.
If(pk,xk)∈S(fλ,F) with (pk,xk)→(p0,x0)∈P×X(k→∞), then supx∈Xfλ,F(pk,x)=infp∈Pfλ,F(p,xk) for all k=1,2,…. By taking k→∞, from (1) and (2) (that is, p↦supx∈Xfλ,F(p,x) is l.s.c. on P and x↦infp∈Pfλ,F(p,x) is u.s.c. on X), we obtain that (3.12)sup⁡x∈Xfλ,F(p0,x)≤lim⁡inf⁡k→∞sup⁡x∈Xfλ,F(pk,x)≤limsup⁡k→∞inf⁡p∈Pfλ,F(p,xk)≤inf⁡p∈Pfλ,F(p,x0).
Hence by Remark 2.2(2), (p0,x0)∈S(fλ,F) and S(fλ,F) is closed. Hence the first lemma follows.Lemma 3.6.
λ↦v(fλ,F) is continuous and strictly decreasing on R+ with v(f+∞,F)=̂limλ→+∞v(fλ,F)=-∞.Proof.
Since(λ,p)↦〈p,Tx-λSx-c〉 is continuous on R+×P for each c∈F and x∈X, (λ,x,c)↦〈p,Tx-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
(3.13)R+×P→R:(λ,p)⟼fλ,F(p,x)=sup⁡c∈F〈p,Tx-λSx-c〉isl.s.c.,R+×X→R:(λ,x)⟼fλ,F(p,x)=sup⁡c∈F〈p,Tx-λSx-c〉isu.s.c..
From Lemma 2.6(2)-(3), it follows that
(3.14)R+×P⟼R:(λ,p)⟼sup⁡x∈Xfλ,F(p,x)isl.s.c.,R+×X⟼R:(λ,x)⟼inf⁡p∈Pfλ,F(p,x)isu.s.c..
First applying Lemma 2.8 to both functions of (3.14), and then using Lemma 3.5(3), we further obtain that
(3.15)R+⟼R:λ⟼inf⁡p∈Psup⁡x∈Xfλ,F(p,x)isl.s.c.,R+⟼R:λ⟼sup⁡x∈Xinf⁡p∈Pfλ,F(p,x)isu.s.c.,
and thus λ↦v(fλ,F) is continuous on R+.
Suppose thatλ2>λ1≥0, then by (3.2), fλ1,F(p,x)=fλ2,F(p,x)+(λ2-λ1)〈p,Sx〉 for all (p,x)∈P×X. This implies by (3.4) that v(fλ1,F)≥v(fλ2,F)+(λ2-λ1)ɛ0, where ɛ0=infp∈P,x∈X〈p,Sx〉∈(0,+∞). Hence λ↦v(fλ,F) is strictly decreasing.
By Lemma3.5(3), Remark 2.2(3) and (3.2), it is easily to see that for each λ∈R+ and (p¯,x¯)∈S(fλ,F), (3.16)v(fλ,F)=sup⁡c∈F〈p¯,Tx¯-λSx¯-c〉≤sup⁡p∈P,x∈X〈p,Tx〉-λinf⁡p∈P,x∈X〈p,Sx〉=ɛ1-λɛ0.
Hence by (3.4), v(f+∞,F)=-∞ and the second lemma is proved.Lemma 3.7.
(1)λ is a lower eigenvalue to (1.1) and x its eigenvector if and only if infp∈Pfλ,F(p,x)≥0.
(2)λ is a lower eigenvalue to (1.1) if and only if v(fλ,F)≥0 if and only if fλ,F(p̂,x̂)≥0 for (p̂,x̂)∈S(fλ,F).Proof.
(1) Ifλ>0 and (x,c)∈X×F satisfy Tx≥λSx+c, then for each p∈P(⊆R+m), fλ,F(p,x)≥〈p,Tx-λSx-c〉≥0. Hence, infp∈Pfλ,F(p,x)≥0.
Ifλ>0 and x∈X satisfy infp∈Pfλ,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*∈Rm∖{0} such that -∞<supc∈F〈p*,Tx-λSx-c〉<infy∈R+m〈p*,y〉. Clearly, we have p*∈R+m∖{0} (or else, we obtain infy∈R+m〈p*,y〉=-∞, which is impossible), infy∈R+m〈p*,y〉=0 and thus supc∈F〈p*,Tx-λSx-c〉<0. Since R+P=R+m, there exist t>0 and p̂∈P with p̂=tp*. It follows that infp∈Pfλ,F(p,x)≤fλ,F(p̂,x)=tsupc∈F〈p*,Tx-λSx-c〉<0. This is a contradiction. So we can select c∈F such that Tx≥λSx+c.
(2) Ifλ>0 is a lower eigenvalue to (1.1), then there exists an eigenvector xλ∈X, which gives, by statement (1) and Lemma 3.5(3), v(fλ,F)≥infp∈Pfλ,F(p,xλ)≥0. If v(fλ,F)≥0, then Remark 2.2(3) and Lemma 3.5(3) imply that fλ,F(p̂,x̂)=v(fλ,F)≥0 for all (p̂,x̂)∈S(fλ,F). If (p̂,x̂)∈S(fλ,F) with fλ,F(p̂,x̂)≥0, then infp∈Pfλ,F(p,x̂)=fλ,F(p̂,x̂)≥0, which gives, by statement (1), that λ is a lower eigenvalue to (1.1) and x̂ 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)
v(f0,F)>0.(c)
f0,F(p̂,x̂)>0 for (p̂,x̂)∈S(f0,F).(d)
There is a uniqueλ̂>0 with v(fλ̂,F)=0.(e)
The maximal lower eigenvalueλ¯=λ¯(F) to (1.1) exists.
In particular, λ̂=λ¯ if either v(f0,F)>0 or one of the λ̂ and λ¯ exists.
(2) Ifv(f0,F)>0, then the set of all lower eigenvalues to (1.1) equals to (0,λ¯].Proof.
(1) Ifλ0(>0) is a lower eigenvalue to (1.1), then by Lemmas 3.6 and 3.7(2), v(f0,F)>v(fλ0,F)≥0. In view of Lemma 3.5(3) and Remark 2.2, we also see that v(f0,F)>0 if and only if f0,F(p̂,x̂)>0 for any (p̂,x̂)∈S(f0,F). If v(f0,F)>0, then also by Lemmas 3.6 and 3.7(2), there exists a unique λ̂>0 such that v(fλ̂,F)=0, and λ̂ 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 (1.1). Hence statement (1) follows.
(2) Statement (2) is obvious. Thus the lemma follows.Lemma 3.9.
IfF∈𝔹+m, then one has the following.(1)
p↦gF(p,x)(x∈X) and p↦supx∈XgF(p,x) are continuous on P.(2)
x↦gF(p,x)(p∈P) and x↦infp∈PgF(p,x) are u.s.c. on X.(3)
v(gF) exists if and only if S(gF)≠∅.Proof.
(1) Since for eachx∈X and c∈F, p↦〈p,Tx-c〉/〈p,Sx〉 is continuous on P, by (3.3), and Lemma 2.6(2), we see that p↦gF(p,x)=supc∈F(〈p,Tx-c〉/〈p,Sx〉)(x∈X) and p↦supx∈XgF(p,x) are l.s.c. on P. On the other hand, by Assumptions 1–3, we can verify that (p,x,c)↦〈p,Tx-c〉/〈p,Sx〉 is u.s.c. on P×X×F. It follows from Lemma 2.8 that both functions (p,x)↦gF(p,x)=supc∈F(〈p,Tx-c〉/〈p,Sx〉) on P×X and p↦supx∈XgF(p,x) on P are u.s.c., so is p↦gF(p,x). Hence (1) is true.
(2) As proved above, we know that for eachp∈P, x↦gF(p,x) is u.s.c. on X, so is x↦infp∈PgF(p,x) because of Lemma 2.6(3).
(3) By Remark2.2(3), we only need to prove the necessary part. Assume v(gF) exists, that is, infp∈Psupx∈XgF(p,x)=supx∈Xinfp∈PgF(p,x), then both (1) and (2) imply that there exist p¯∈P and x¯∈X with supx∈XgF(p¯,x)=infp∈PgF(p,x¯), which means that (p¯,x¯)∈S(gF) and S(gF) 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 infp∈PgF(p,x)≥λ.
(2)λ is a lower eigenvalue to (1.1) if and only if supx∈Xinfp∈PgF(p,x)≥λ.Proof.
(1) Supposeλ>0 and x∈X. Since for each p∈P, gF(p,x)=supc∈F(〈p,Tx-c〉/〈p,Sx〉)≥λ equals to fλ,F(p,x)=supc∈F〈p,Tx-λSx-c〉≥0, which implies that infp∈PgF(p,x)≥λ if and only if infp∈Pfλ,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. Ifsupx∈Xinfp∈PgF(p,x)≥λ(>0), then Lemma 3.9(2) shows that there exists xλ∈X with infp∈PgF(p,xλ)=supx∈Xinfp∈PgF(p,x)≥λ. Hence λ is a lower eigenvalue to (1.1) and xλ its eigenvector. This completes the proof.Lemma 3.11.
(1)v(f0,F)>0 if and only if v(gF) exists with v(gF)=λ¯ if and only if S(gF)≠∅ and gF(p¯,x¯)=λ¯ for (p¯,x¯)∈S(gF). 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(gF) exists with v(gF)≥λ if and only if S(gF)≠∅ and gF(p̂,x̂)≥λ for (p̂,x̂)∈S(gF).
(3)  System (1.1) has at least one lower eigenvalue if and only if v(gF) exist with v(gF)>0 if and only if S(gF)≠∅ and gF(p̂,x̂)>0 for (p̂,x̂)∈S(gF).Proof.
(1) We divide the proof of (1) into three steps.Step 1.
Ifv(f0,F)>0, then by Lemma 3.8(1), the maximal eigenvalue λ¯(>0) to (1.1) exists with v(fλ¯,F)=0. We will prove that v(gF) exists with v(gF)=λ¯. Let λ*=supx∈Xinfp∈PgF(p,x),λ*=infp∈Psupx∈XgF(p,x), then λ*≤λ*, and the left is to show λ*≤λ¯≤λ*.
By Lemma3.5(2), there exists x¯∈X such that infp∈Pfλ¯,F(p,x¯)=v(fλ¯,F)=0. This shows that supc∈F〈p,Tx¯-λ¯Sx¯-c〉=fλ¯,F(p,x¯)≥0 for any p∈P, that is, λ¯≤supc∈F(〈p,Tx¯-c〉/〈p,Sx¯〉)=gF(p,x¯)(p∈P). Hence, λ¯≤infp∈PgF(p,x¯)≤λ*. On the other hand, since for each p∈P, λ*≤supx∈XgF(p,x), by Lemma 3.9(2), there exists xp∈X such that λ*≤supx∈XgF(p,x)=gF(p,xp)=supc∈F(〈p,Txp-c〉/〈p,Sxp〉). It follows that supx∈Xfλ*,F(p,x)≥fλ*,F(p,xp)=supc∈F〈p,Txp-λ*Sxp-c〉≥0 for any p∈P. Hence by Lemma 3.5(3), v(fλ*,F)=infp∈Psupx∈Xfλ*,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(gF) exists with v(gF)=λ¯.
Step 2.
Ifv(gF) exists with v(gF)=λ¯(>0), then Lemma 3.9(3) and Remark 2.2(3) deduce that S(gF)≠∅ and gF(p¯,x¯)=v(gF)=λ¯>0 for (p¯,x¯)∈S(gF).Step 3.
IfS(gF)≠∅ and (p¯,x¯)∈S(gF) with gF(p¯,x¯)=λ¯(>0), then infp∈PgF(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(f0,F)>0.
(2) Ifλ>0 is a lower eigenvalue to (1.1), then Lemmas 3.8(1), 3.10(2) and statement (1) imply that v(f0,F)>0,v(gF) exists and v(gF)=supx∈Xinfp∈PgF(p,x)≥λ. If v(gF) exists with v(gF)≥λ, then from Lemma 3.9(3) and Remark 2.2(3), it follows that S(gF)≠∅ and gF(p̂,x̂)=v(gF)≥λ for (p̂,x̂)∈S(gF). If S(gF)≠∅ and gF(p̂,x̂)≥λ for (p̂,x̂)∈S(gF), then by Remark 2.2(3) and Lemma 3.10(1), we see that infp∈PgF(p,x̂)=gF(p̂,x̂)≥λ, and thus λ is a lower eigenvalue to (1.1) and x̂ its eigenvector.
(3) Statement (3) follows immediately from (1) and (2). This completes the proof.Lemma 3.12.
(1) Ifv(f0,F)>0, or equivalently, if v(gF) exists with v(gF)>0, then S(gF) is a nonempty compact subset of P×X.
(2) The first three statements of Theorem3.3(2) are true.
(3) Theorem3.3(3) is true.Proof.
(1) By Lemma3.11(1), S(gF) 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 (pk,xk)∈S(gF) such that (pk,xk)→(p0,x0)∈P×X as k→∞, then
(3.17)sup⁡x∈XgF(p0,x)=lim⁡inf⁡k→∞sup⁡x∈XgF(pk,x)≤limsup⁡k→∞inf⁡p∈PgF(p,xk)≤inf⁡p∈PgF(p,x0).
Hence, S(gF) is closed, and also compact.
(2) Now we prove the first three statements of Theorem3.3(2).
By the condition of Theorem3.3(2), Lemmas 3.8(1) and 3.11(1), we know that the maximal lower eigenvalue λ̅ to (1.1) and v(gF) exist with v(gF)=λ̅.
First we prove statement (a). Ifx̅∈X is an optimal eigenvector, then by Lemma 3.10(1), we have infp∈PgF(p,x̅)≥λ̅. On the other hand, by Lemma 3.9(1), there exists p̅∈P such that v(gF)=supx∈XgF(p̅,x). So we obtain that supx∈XgF(p̅,x)=λ̅≤infp∈PgF(p,x̅), and thus (p̅,x̅)∈S(gF). If p̅∈P such that (p̅,x̅)∈S(gF), then Remark 2.2(3) implies that infp∈PgF(p,x̅)=v(gF)=λ̅. If infp∈PgF(p,x̅)=λ̅, then Lemma 3.10(1) shows that x̅ is an optimal eigenvector. Hence, Theorem 3.3(2)(a) follows.
Next we prove statement (b). By Lemmas3.5(2) and 3.8(1), there exists x̂∈X with(3.18)0=v(fλ̅,F)=sup⁡x∈Xinf⁡p∈Pfλ̅,F(p,x)=inf⁡p∈Pfλ̅,F(p,x̂)=inf⁡p∈Psup⁡c∈F〈p,Tx̂-λ̅Sx̂-c〉.
By applying Lemma 2.9 to φ(p,c)=〈p,Tx̂-λ̅Sx̂-c〉 on P×F, this leads to
(3.19)sup⁡c∈Finf⁡p∈P〈p,Tx̂-λ̅Sx̂-c〉=inf⁡p∈Psup⁡c∈F〈p,Tx̂-λ̅Sx̂-c〉=0.
Since c↦infp∈P〈p,Tx̂-λ̅Sx̂-c〉 is u.s.c. on F and p↦〈p,Tx̂-λ̅Sx̂-c〉 is continuous on P, from (3.19), first there exists ĉ=(ĉ1,ĉ2,…,ĉm)∈F and then there exists p̂=(p̂1,p̂2,…,p̂m)∈P such that
(3.20)0=sup⁡c∈Finf⁡p∈P〈p,Tx̂-λ̅Sx̂-c〉=inf⁡p∈P〈p,Tx̂-λ̅Sx̂-ĉ〉=〈p̂,Tx̂-λ̅Sx̂-ĉ〉.
As R+P=R+m, for each i=1,2,…,m, there exists ti>0 with pi=(0,…,0,ti︷i,0,…,0︸m)∈P, which implies by (3.20) that for each i=1,2,…,m,
(3.21)ti(Tix̂-λ̅Six̂-ĉi)=〈pi,Tx̂-λ̅Sx̂-ĉ〉≥0,thatis,Tx̂≥λ̅Sx̂+ĉ.
On the other hand, we can see thatIp̂={i:p̂i>0} is nonempty because p̂∈P⊆R+m∖{0}. This gives, by (3.20) and (3.21), that for each i0∈Ip̂,(3.22)0≤p̂i0(Ti0x̂-λ̅Si0x̂-ĉi0)≤〈p̂,Tx̂-λ̅Sx̂-ĉ〉=0,thatis,Ti0x̂=λ̅Si0x̂+ĉi0.
Both (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 λ̅ is the maximal lower eigenvalue to (1.1) and (p̅,x̅)∈S(gF) if and only if λ̅>0 and (p̅,x̅)∈P×X satisfy gF(p̅,x)≤gF(p̅,x̅)=λ̅≤gF(p,x̅) for (p,x)∈P×X, which amounts to say(3.23)fλ̅,F(p̅,x)≤fλ̅,F(p̅,x̅)=0≤fλ̅,F(p,x̅),(p,x)∈P×X,
because for each (p,x)∈P×X,
(3.24)gF(p̅,x)=sup⁡c∈F〈p̅,Tx-c〉〈p̅,Sx〉≤λ̅ifffλ̅,F(p̅,x)=sup⁡c∈F〈p̅,Tx-λ̅Sx-c〉≤0,gF(p̅,x̅)=sup⁡c∈F〈p̅,Tx̅-c〉〈p̅,Sx̅〉=λ̅ifffλ̅,F(p̅,x̅)=sup⁡c∈F〈p̅,Tx̅-λ̅Sx̅-c〉=0,gF(p,x̅)=sup⁡c∈F〈p,Tx̅-c〉〈p,Sx̅〉≥λ̅ifffλ̅,F(p,x̅)=sup⁡c∈F〈p,Tx̅-λ̅Sx̅-c〉≥0.
In view of (3.5), we know that (3.23) is also equivalent to
(3.25)min⁡x∈Rn[f̂λ̅,F(p̅,x)]=min⁡x∈X[-fλ̅,F(p̅,x)]=0=-fλ̅,F(p̅,x̅)=f̂λ̅,F(p̅,x̅),min⁡p∈Rmf̃λ̅,F(p,x̅)=min⁡p∈Pfλ̅,F(p,x̅)=0=fλ̅,F(p̅,x̅)=f̃λ̅,F(p̅,x̅).
Also by (3.5), we have epif̂λ̅,F(p̅,·)={(x,a)∈X×R:-fλ̅,F(p̅,x)≤a} and epif̃λ̅,F(·,x̅)={(p,a)∈P×R:fλ̅,F(p,x̅)≤a}. Combining this with Lemma 3.5(1)-(2) and using the fact that X and P are convex compact, we can see that epif̂λ̅,F(p̅,·) (or epif̃λ̅,F(·,x̅)) is closed convex in Rn×R (or in Rm×R). Hence Lemmas 2.6(1) and 2.10 imply that both x↦f̂λ̅,F(p̅,x)(x∈Rn) and p↦f̃λ̅,F(p,x̅)(p∈Rm) are proper convex and l.s.c. with
(3.26)f̂λ̅,F(p̅,x)=f̂λ̅,F**(p̅,x)(x∈Rn),f̃λ̅,F(p,x̅)=f̃λ̅,F**(p,x̅)(p∈Rm).
Applying Lemma 2.11 to the functions x↦f̂λ̅,F(p̅,x)-〈q0,x〉 on Rn with q0=0 and p↦f̃λ̅,F(p,x̅)-〈r0,p〉 on Rm with r0=0, and using (3.25) and (3.26), we conclude that (3.23) holds if and only if x̅∈∂f̂λ̅,F*(p̅,0) and p̅∈∂f̃λ̅,F*(0,x̅). Hence Theorem 3.3(2)(c) is also true.
(3) Finally we prove Theorem3.3(3).
(a) By (3.1), we know that for each x0∈X,Tx0∈intR+m. So there exists c0∈R+m such that c0<Tx0 (that is, c0i<Tix0 for i=1,2,…,m). Take Fx0={c0}, then supx∈Xf0,Fx0(p,x)=supx∈X〈p,Tx-c0〉≥〈p,Tx0-c0〉>0 for any p∈P. Hence, v(f0,Fx0)=infp∈Psupx∈Xf0,Fx0(p,x)>0 because p↦supx∈Xf0,Fx0(p,x) is l.s.c. on P. This shows that ℂ+m={F∈𝔹+m:v(f0,F)>0} is nonempty. Moreover, Lemma 3.11(1) implies that λ̅(F) exists with λ̅(F)=v(gF) for any F∈ℂ+m. Hence statement (a) follows.
(b) LetFi∈ℂ+m(i=1,2), then we have λ(Fi)=v(gFi)(i=1,2). Suppose that (p,x)∈P×X. Since Fi(i=1,2) are compact, we can select ci∈Fi(i=1,2) such that gF1(p,x)=supc∈F1(〈p,Tx-c〉/〈p,Sx〉)=〈p,Tx-c1〉/〈p,Sx〉 and ∥c1-c2∥=d(c1,F2). This deduces that(3.27)gF1(p,x)=〈p,Tx-c2+(c2-c1)〉〈p,Sx〉≤gF2(p,x)+sup⁡p∈P⁡∥p∥ε0dH(F1,F2),
because 〈p,Tx-c2〉/〈p,Sx〉≤gF2(p,x), ∥c1-c2∥=d(c1,F2)≤dH(F1,F2) and ε0=infp∈P,x∈X〈p,Sx〉 is positive. By taking minimax values for both sides of (3.27), we have v(gF1)≤v(gF2)+(supp∈P∥p∥/ε0)dH(F1,F2). Therefore, |λ̅(F1)-λ̅(F2)|=|v(gF1)-v(gF2)|≤(supp∈P∥p∥/ε0)dH(F1,F2) because dH(F1,F2)=dH(F2,F1), and the last lemma follows.Proofs of Theorems3.1–3.3
Proof.
(i) For Theorem3.1. (1) follows from Lemmas 3.5(3) and 3.6, and (2) from Lemmas 3.9(3), 3.11(1), and 3.12(1).
(ii) For Theorem3.2. (1) can be deduced from Lemmas 3.7(1) and 3.10(1), (2) from Lemmas 3.7(2) and 3.11(2), while (3) from Lemmas 3.8(1) and 3.11(3).
(iii) For Theorem3.3. 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.


## 3.1. Auxiliary Functions

In the sequel, we assume that(3.1)(1)Assumptions1–3inSection1aresatisfied,andλ∈R+,F∈B+m,(2)P⊆R+m∖{0}isaconvexcompactsubsetwithR+P=R+m.
Denote by 〈·,·〉 the duality paring on 〈Rm*,Rm〉, and for each λ∈R+ and F∈𝔹+m, define two auxiliary functions fλ,F(p,x) and gF(p,x) on P×X by(3.2)fλ,F(p,x)=sup⁡c∈F〈p,Tx-λSx-c〉=sup⁡(c1,c2,…,cm)∈F∑i=1mpi(Tix-λSix-ci),(3.3)gF(p,x)=sup⁡c∈F〈p,Tx-c〉〈p,Sx〉=sup⁡(c1,c2,…,cm)∈F∑i=1mpi(Tix-ci)∑i=1mpiSix.
Just as indicated by Definition 2.1, the minimax values and saddle point sets of φ(p,x)=fλ,F(p,x) and φ(p,x)=gF(p,x), if existed or nonempty, are denoted by v(fλ,F), v(gF), S(fλ,F), and S(gF), respectively.By (3.1)–(3.3), (p,x)↦〈p,Sx〉, and (p,x)↦〈p,Tx〉 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(3.4)0<ɛ0=inf⁡p∈P,x∈X〈p,Sx〉<+∞,0<ɛ1=sup⁡p∈P,x∈X〈p,Tx〉<+∞,
and both fλ,F(p,x) and gF(p,x) are finite for all λ∈R+,(p,x)∈P×X and F∈𝔹+m.We also define the extensionsx↦f̂λ,F(p,x) to x↦-fλ,F(p,x) (for each fixed p∈P) and p↦f̃λ,F(p,x) to p↦fλ,F(p,x) (for each fixed x∈X) by(3.5)f̂λ,F(p,x)={-fλ,F(p,x),x∈X,+∞,x∈Rn∖X,f̃λ,F(p,x)={fλ,F(p,x),p∈P,+∞,p∈Rm∖P.
According to Definition 2.3, the conjugate and biconjugate functions of x↦f̂λ,F(p,x) and p↦f̃λ,F(p,x) are then denoted by(3.6)q⟼f̂λ,F*(p,q),q∈Rn,x⟼f̂λ,F**(p,x),x∈Rn(foreachfixedp∈P),r⟼f̃λ,F*(r,x),r∈Rm,p⟼f̃λ,F**(p,x),p∈Rm(foreachfixedx∈X).
By Definition 2.5, the Hausdorff distance in 𝔹+m (see Assumption 3) is provided by(3.7)dH(F1,F2)=max⁡{sup⁡c1∈F1d(c1,F2),sup⁡c2∈F2d(c2,F1)}forF1,F2∈B+m.


## 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)v(fλ,F) exists and S(fλ,F) is a nonempty convex compact subset of P×X. Furthermore, λ↦v(fλ,F) is continuous and strictly decreasing on R+ with v(f+∞,F)=̂limλ→+∞v(fλ,F)=-∞.
(2)v(gF) exists if and only if S(gF)≠∅. Moreover, if v(f0,F)>0, then v(gF) exists and S(gF) is a nonempty compact subset of P×X.Theorem 3.2.
(1)λ is a lower eigenvalue to (1.1) and x its eigenvector if and only if infp∈Pfλ,F(p,x)≥0 if and only if infp∈PgF(p,x)≥λ.
(2)λ is a lower eigenvalue to (1.1) if and only if one of the following statements is true:
(a)
v(fλ,F)≥0,(b)
fλ,F(p̂,x̂)≥0 for (p̂,x̂)∈S(fλ,F),(c)
v(gF) exists with v(gF)≥λ,(d)
S(gF)≠∅ and gF(p̂,x̂)≥λ for (p̂,x̂)∈S(gF).
(3) The following statements are equivalent:
(a)
System (1.1) has at least one lower eigenvalue,(b)
v(f0,F)>0,(c)
v(gF) exists with v(gF)>0,(d)
S(gF)≠∅ and gF(p̂,x̂)>0 for (p̂,x̂)∈S(gF).Theorem 3.3.
(1)λ¯ exists if and only if one of the following statements is true:
(a)
v(f0,F)>0,(b)
f0,F(p̂,x̂)>0 for (p̂,x̂)∈S(f0,F),(c)
v(fλ¯,F)=0,(d)
v(gF) exists with v(gF)=λ¯,(e)
S(gF)≠∅ and gF(p¯,x¯)=λ¯ for (p¯,x¯)∈S(gF).
Where λ¯=λ¯(F)(>0) is the maximal lower eigenvalue to (1.1).
(2) Ifv(f0,F)>0, or equivalently, if v(gF) exists with v(gF)>0, then one has the following.
(a)
x¯ is an optimal eigenvector if and only if there exists p¯∈P with (p¯,x¯)∈S(gF) if and only if infp∈PgF(p,x¯)=λ¯.(b)
There existx̂∈X,ĉ∈F and i0∈{1,2,…,m} such that Tx̂≥λ¯Sx̂+ĉ and Ti0x̂=λ¯Si0x̂+ĉi0.(c)
λ¯=λ¯(F) is the maximal lower eigenvalue to (1.1) and (p¯,x¯)∈S(gF) if and only if λ¯>0 and (p¯,x¯)∈P×X satisfy x¯∈∂f̂λ¯,F*(p¯,0)and p¯∈∂f̃λ¯,F*(0,x¯). Where ∂f̂λ¯,F*(p¯,0) and ∂f̃λ¯,F*(0,x¯) are the subdifferentials of f̂λ¯,F*(p¯,q) at q=0 and f̃λ¯,F*(r,x¯) at r=0, respectively.(d)
The set of all lower eigenvalues to (1.1) coincides with the interval (0,v(gF)].
(3) Letℂ+m={F∈𝔹+m:v(f0,F)>0}, where 𝔹+m is defined as in Assumption 3. Then
(a)
ℂ+m≠∅, and for each F∈ℂ+m,λ¯=λ¯(F) exists with λ¯(F)=v(gF),(b)
for allF1,F2∈ℂ+m, |λ¯(F1)-λ¯(F2)|≤(supp∈P∥p∥/ɛ0)dH(F1,F2), where ɛ0 is defined by (3.4).
Thus, F↦λ¯(F) is Lipschitz on ℂ+m with the Hausdorff distance dH(·,·).Remark 3.4.
If we takeP=Σm-1={p∈R+m:∑i=1mpi=1}, then Σm-1 satisfies (3.1)(2), hence Theorems 3.1–3.3 are also true.


## 3.3. Proofs of the Main Theorems

In order to prove Theorems3.1–3.3, we need the following eight lemmas.Lemma 3.5.
Ifλ∈R+ is fixed, then one has the following.(1)
p↦fλ,F(p,x)(x∈X) and p↦supx∈Xfλ,F(p,x) are l.s.c. and convex on P.(2)
x↦fλ,F(p,x)(p∈P) and x↦infp∈Pfλ,F(p,x) are u.s.c. and concave on X.(3)
v(fλ,F) exists and S(fλ,F) is a nonempty convex compact subset of P×X.Proof.
By (3.1)–(3.3), it is easily to see that
(3.8)(a)∀x∈X,∀c∈F,p⟼〈p,Tx-λSx-c〉isconvexl.s.c.on⁡P,(b)∀p∈P,(x,c)⟼〈p,Tx-λSx-c〉isu.s.c.on⁡X×F.
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
(3.9)∀x∈X,p⟼fλ,F(p,x)isconvexl.s.c.on⁡Pandinf⁡p∈Pfλ,F(p,x)isfinite,∀p∈P,x⟼fλ,F(p,x)isu.s.c.on⁡Xandsup⁡x∈Xfλ,F(p,x)isfinite.
Ifxi∈X(i=1,2), then by (3.2), there exist ci=(c1i,c2i,…,cmi)∈F(i=1,2) such that fλ,F(p,xi)=〈p,Txi-λSxi-ci〉. Since Ti,-λSi(i=1,2,…,m) are concave, X,F are convex and p∈P is nonnegative, we have for each α∈[0,1],(3.10)fλ,F(p,αx1+(1-α)x2)≥〈p,T[αx1+(1-α)x2]-λS[αx1+(1-α)x2]-αc1-(1-α)c2〉≥α〈p,Tx1-λSx1-c1〉+(1-α)〈p,Tx2-λSx2-c2〉=αfλ,F(p,x1)+(1-α)fλ,F(p,x2),thatis,x⟼fλ,F(p,x)isconcaveon⁡X.
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(fλ,F) exists and S(fλ,F) is nonempty. It remains to verify that S(fλ,F) is convex and closed because P×X is convex and compact.
Ifα∈[0,1] and (pi,xi)∈S(fλ,F)(i=1,2), then supx∈Xfλ,F(pi,x)=infp∈Pfλ,F(p,xi) for i=1,2. By (1) and (2) (i.e., p↦supx∈Xfλ,F(p,x) is convex on P and x↦infp∈Pfλ,F(p,x) is concave on X), we have (3.11)sup⁡x∈Xfλ,F(αp1+(1-α)p2,x)≤αsup⁡x∈Xfλ,F(p1,x)+(1-α)sup⁡x∈Xfλ,F(p2,x)=αinf⁡p∈Pfλ,F(p,x1)+(1-α)inf⁡p∈Pfλ,F(p,x2)≤inf⁡p∈Pfλ,F(p,αx1+(1-α)x2).
This implies by Remark 2.2(2) that α(p1,x1)+(1-α)(p2,x2)∈S(fλ,F), and thus S(fλ,F) is convex.
If(pk,xk)∈S(fλ,F) with (pk,xk)→(p0,x0)∈P×X(k→∞), then supx∈Xfλ,F(pk,x)=infp∈Pfλ,F(p,xk) for all k=1,2,…. By taking k→∞, from (1) and (2) (that is, p↦supx∈Xfλ,F(p,x) is l.s.c. on P and x↦infp∈Pfλ,F(p,x) is u.s.c. on X), we obtain that (3.12)sup⁡x∈Xfλ,F(p0,x)≤lim⁡inf⁡k→∞sup⁡x∈Xfλ,F(pk,x)≤limsup⁡k→∞inf⁡p∈Pfλ,F(p,xk)≤inf⁡p∈Pfλ,F(p,x0).
Hence by Remark 2.2(2), (p0,x0)∈S(fλ,F) and S(fλ,F) is closed. Hence the first lemma follows.Lemma 3.6.
λ↦v(fλ,F) is continuous and strictly decreasing on R+ with v(f+∞,F)=̂limλ→+∞v(fλ,F)=-∞.Proof.
Since(λ,p)↦〈p,Tx-λSx-c〉 is continuous on R+×P for each c∈F and x∈X, (λ,x,c)↦〈p,Tx-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
(3.13)R+×P→R:(λ,p)⟼fλ,F(p,x)=sup⁡c∈F〈p,Tx-λSx-c〉isl.s.c.,R+×X→R:(λ,x)⟼fλ,F(p,x)=sup⁡c∈F〈p,Tx-λSx-c〉isu.s.c..
From Lemma 2.6(2)-(3), it follows that
(3.14)R+×P⟼R:(λ,p)⟼sup⁡x∈Xfλ,F(p,x)isl.s.c.,R+×X⟼R:(λ,x)⟼inf⁡p∈Pfλ,F(p,x)isu.s.c..
First applying Lemma 2.8 to both functions of (3.14), and then using Lemma 3.5(3), we further obtain that
(3.15)R+⟼R:λ⟼inf⁡p∈Psup⁡x∈Xfλ,F(p,x)isl.s.c.,R+⟼R:λ⟼sup⁡x∈Xinf⁡p∈Pfλ,F(p,x)isu.s.c.,
and thus λ↦v(fλ,F) is continuous on R+.
Suppose thatλ2>λ1≥0, then by (3.2), fλ1,F(p,x)=fλ2,F(p,x)+(λ2-λ1)〈p,Sx〉 for all (p,x)∈P×X. This implies by (3.4) that v(fλ1,F)≥v(fλ2,F)+(λ2-λ1)ɛ0, where ɛ0=infp∈P,x∈X〈p,Sx〉∈(0,+∞). Hence λ↦v(fλ,F) is strictly decreasing.
By Lemma3.5(3), Remark 2.2(3) and (3.2), it is easily to see that for each λ∈R+ and (p¯,x¯)∈S(fλ,F), (3.16)v(fλ,F)=sup⁡c∈F〈p¯,Tx¯-λSx¯-c〉≤sup⁡p∈P,x∈X〈p,Tx〉-λinf⁡p∈P,x∈X〈p,Sx〉=ɛ1-λɛ0.
Hence by (3.4), v(f+∞,F)=-∞ and the second lemma is proved.Lemma 3.7.
(1)λ is a lower eigenvalue to (1.1) and x its eigenvector if and only if infp∈Pfλ,F(p,x)≥0.
(2)λ is a lower eigenvalue to (1.1) if and only if v(fλ,F)≥0 if and only if fλ,F(p̂,x̂)≥0 for (p̂,x̂)∈S(fλ,F).Proof.
(1) Ifλ>0 and (x,c)∈X×F satisfy Tx≥λSx+c, then for each p∈P(⊆R+m), fλ,F(p,x)≥〈p,Tx-λSx-c〉≥0. Hence, infp∈Pfλ,F(p,x)≥0.
Ifλ>0 and x∈X satisfy infp∈Pfλ,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*∈Rm∖{0} such that -∞<supc∈F〈p*,Tx-λSx-c〉<infy∈R+m〈p*,y〉. Clearly, we have p*∈R+m∖{0} (or else, we obtain infy∈R+m〈p*,y〉=-∞, which is impossible), infy∈R+m〈p*,y〉=0 and thus supc∈F〈p*,Tx-λSx-c〉<0. Since R+P=R+m, there exist t>0 and p̂∈P with p̂=tp*. It follows that infp∈Pfλ,F(p,x)≤fλ,F(p̂,x)=tsupc∈F〈p*,Tx-λSx-c〉<0. This is a contradiction. So we can select c∈F such that Tx≥λSx+c.
(2) Ifλ>0 is a lower eigenvalue to (1.1), then there exists an eigenvector xλ∈X, which gives, by statement (1) and Lemma 3.5(3), v(fλ,F)≥infp∈Pfλ,F(p,xλ)≥0. If v(fλ,F)≥0, then Remark 2.2(3) and Lemma 3.5(3) imply that fλ,F(p̂,x̂)=v(fλ,F)≥0 for all (p̂,x̂)∈S(fλ,F). If (p̂,x̂)∈S(fλ,F) with fλ,F(p̂,x̂)≥0, then infp∈Pfλ,F(p,x̂)=fλ,F(p̂,x̂)≥0, which gives, by statement (1), that λ is a lower eigenvalue to (1.1) and x̂ 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)
v(f0,F)>0.(c)
f0,F(p̂,x̂)>0 for (p̂,x̂)∈S(f0,F).(d)
There is a uniqueλ̂>0 with v(fλ̂,F)=0.(e)
The maximal lower eigenvalueλ¯=λ¯(F) to (1.1) exists.
In particular, λ̂=λ¯ if either v(f0,F)>0 or one of the λ̂ and λ¯ exists.
(2) Ifv(f0,F)>0, then the set of all lower eigenvalues to (1.1) equals to (0,λ¯].Proof.
(1) Ifλ0(>0) is a lower eigenvalue to (1.1), then by Lemmas 3.6 and 3.7(2), v(f0,F)>v(fλ0,F)≥0. In view of Lemma 3.5(3) and Remark 2.2, we also see that v(f0,F)>0 if and only if f0,F(p̂,x̂)>0 for any (p̂,x̂)∈S(f0,F). If v(f0,F)>0, then also by Lemmas 3.6 and 3.7(2), there exists a unique λ̂>0 such that v(fλ̂,F)=0, and λ̂ 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 (1.1). Hence statement (1) follows.
(2) Statement (2) is obvious. Thus the lemma follows.Lemma 3.9.
IfF∈𝔹+m, then one has the following.(1)
p↦gF(p,x)(x∈X) and p↦supx∈XgF(p,x) are continuous on P.(2)
x↦gF(p,x)(p∈P) and x↦infp∈PgF(p,x) are u.s.c. on X.(3)
v(gF) exists if and only if S(gF)≠∅.Proof.
(1) Since for eachx∈X and c∈F, p↦〈p,Tx-c〉/〈p,Sx〉 is continuous on P, by (3.3), and Lemma 2.6(2), we see that p↦gF(p,x)=supc∈F(〈p,Tx-c〉/〈p,Sx〉)(x∈X) and p↦supx∈XgF(p,x) are l.s.c. on P. On the other hand, by Assumptions 1–3, we can verify that (p,x,c)↦〈p,Tx-c〉/〈p,Sx〉 is u.s.c. on P×X×F. It follows from Lemma 2.8 that both functions (p,x)↦gF(p,x)=supc∈F(〈p,Tx-c〉/〈p,Sx〉) on P×X and p↦supx∈XgF(p,x) on P are u.s.c., so is p↦gF(p,x). Hence (1) is true.
(2) As proved above, we know that for eachp∈P, x↦gF(p,x) is u.s.c. on X, so is x↦infp∈PgF(p,x) because of Lemma 2.6(3).
(3) By Remark2.2(3), we only need to prove the necessary part. Assume v(gF) exists, that is, infp∈Psupx∈XgF(p,x)=supx∈Xinfp∈PgF(p,x), then both (1) and (2) imply that there exist p¯∈P and x¯∈X with supx∈XgF(p¯,x)=infp∈PgF(p,x¯), which means that (p¯,x¯)∈S(gF) and S(gF) 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 infp∈PgF(p,x)≥λ.
(2)λ is a lower eigenvalue to (1.1) if and only if supx∈Xinfp∈PgF(p,x)≥λ.Proof.
(1) Supposeλ>0 and x∈X. Since for each p∈P, gF(p,x)=supc∈F(〈p,Tx-c〉/〈p,Sx〉)≥λ equals to fλ,F(p,x)=supc∈F〈p,Tx-λSx-c〉≥0, which implies that infp∈PgF(p,x)≥λ if and only if infp∈Pfλ,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. Ifsupx∈Xinfp∈PgF(p,x)≥λ(>0), then Lemma 3.9(2) shows that there exists xλ∈X with infp∈PgF(p,xλ)=supx∈Xinfp∈PgF(p,x)≥λ. Hence λ is a lower eigenvalue to (1.1) and xλ its eigenvector. This completes the proof.Lemma 3.11.
(1)v(f0,F)>0 if and only if v(gF) exists with v(gF)=λ¯ if and only if S(gF)≠∅ and gF(p¯,x¯)=λ¯ for (p¯,x¯)∈S(gF). 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(gF) exists with v(gF)≥λ if and only if S(gF)≠∅ and gF(p̂,x̂)≥λ for (p̂,x̂)∈S(gF).
(3)  System (1.1) has at least one lower eigenvalue if and only if v(gF) exist with v(gF)>0 if and only if S(gF)≠∅ and gF(p̂,x̂)>0 for (p̂,x̂)∈S(gF).Proof.
(1) We divide the proof of (1) into three steps.Step 1.
Ifv(f0,F)>0, then by Lemma 3.8(1), the maximal eigenvalue λ¯(>0) to (1.1) exists with v(fλ¯,F)=0. We will prove that v(gF) exists with v(gF)=λ¯. Let λ*=supx∈Xinfp∈PgF(p,x),λ*=infp∈Psupx∈XgF(p,x), then λ*≤λ*, and the left is to show λ*≤λ¯≤λ*.
By Lemma3.5(2), there exists x¯∈X such that infp∈Pfλ¯,F(p,x¯)=v(fλ¯,F)=0. This shows that supc∈F〈p,Tx¯-λ¯Sx¯-c〉=fλ¯,F(p,x¯)≥0 for any p∈P, that is, λ¯≤supc∈F(〈p,Tx¯-c〉/〈p,Sx¯〉)=gF(p,x¯)(p∈P). Hence, λ¯≤infp∈PgF(p,x¯)≤λ*. On the other hand, since for each p∈P, λ*≤supx∈XgF(p,x), by Lemma 3.9(2), there exists xp∈X such that λ*≤supx∈XgF(p,x)=gF(p,xp)=supc∈F(〈p,Txp-c〉/〈p,Sxp〉). It follows that supx∈Xfλ*,F(p,x)≥fλ*,F(p,xp)=supc∈F〈p,Txp-λ*Sxp-c〉≥0 for any p∈P. Hence by Lemma 3.5(3), v(fλ*,F)=infp∈Psupx∈Xfλ*,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(gF) exists with v(gF)=λ¯.
Step 2.
Ifv(gF) exists with v(gF)=λ¯(>0), then Lemma 3.9(3) and Remark 2.2(3) deduce that S(gF)≠∅ and gF(p¯,x¯)=v(gF)=λ¯>0 for (p¯,x¯)∈S(gF).Step 3.
IfS(gF)≠∅ and (p¯,x¯)∈S(gF) with gF(p¯,x¯)=λ¯(>0), then infp∈PgF(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(f0,F)>0.
(2) Ifλ>0 is a lower eigenvalue to (1.1), then Lemmas 3.8(1), 3.10(2) and statement (1) imply that v(f0,F)>0,v(gF) exists and v(gF)=supx∈Xinfp∈PgF(p,x)≥λ. If v(gF) exists with v(gF)≥λ, then from Lemma 3.9(3) and Remark 2.2(3), it follows that S(gF)≠∅ and gF(p̂,x̂)=v(gF)≥λ for (p̂,x̂)∈S(gF). If S(gF)≠∅ and gF(p̂,x̂)≥λ for (p̂,x̂)∈S(gF), then by Remark 2.2(3) and Lemma 3.10(1), we see that infp∈PgF(p,x̂)=gF(p̂,x̂)≥λ, and thus λ is a lower eigenvalue to (1.1) and x̂ its eigenvector.
(3) Statement (3) follows immediately from (1) and (2). This completes the proof.Lemma 3.12.
(1) Ifv(f0,F)>0, or equivalently, if v(gF) exists with v(gF)>0, then S(gF) is a nonempty compact subset of P×X.
(2) The first three statements of Theorem3.3(2) are true.
(3) Theorem3.3(3) is true.Proof.
(1) By Lemma3.11(1), S(gF) 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 (pk,xk)∈S(gF) such that (pk,xk)→(p0,x0)∈P×X as k→∞, then
(3.17)sup⁡x∈XgF(p0,x)=lim⁡inf⁡k→∞sup⁡x∈XgF(pk,x)≤limsup⁡k→∞inf⁡p∈PgF(p,xk)≤inf⁡p∈PgF(p,x0).
Hence, S(gF) is closed, and also compact.
(2) Now we prove the first three statements of Theorem3.3(2).
By the condition of Theorem3.3(2), Lemmas 3.8(1) and 3.11(1), we know that the maximal lower eigenvalue λ̅ to (1.1) and v(gF) exist with v(gF)=λ̅.
First we prove statement (a). Ifx̅∈X is an optimal eigenvector, then by Lemma 3.10(1), we have infp∈PgF(p,x̅)≥λ̅. On the other hand, by Lemma 3.9(1), there exists p̅∈P such that v(gF)=supx∈XgF(p̅,x). So we obtain that supx∈XgF(p̅,x)=λ̅≤infp∈PgF(p,x̅), and thus (p̅,x̅)∈S(gF). If p̅∈P such that (p̅,x̅)∈S(gF), then Remark 2.2(3) implies that infp∈PgF(p,x̅)=v(gF)=λ̅. If infp∈PgF(p,x̅)=λ̅, then Lemma 3.10(1) shows that x̅ is an optimal eigenvector. Hence, Theorem 3.3(2)(a) follows.
Next we prove statement (b). By Lemmas3.5(2) and 3.8(1), there exists x̂∈X with(3.18)0=v(fλ̅,F)=sup⁡x∈Xinf⁡p∈Pfλ̅,F(p,x)=inf⁡p∈Pfλ̅,F(p,x̂)=inf⁡p∈Psup⁡c∈F〈p,Tx̂-λ̅Sx̂-c〉.
By applying Lemma 2.9 to φ(p,c)=〈p,Tx̂-λ̅Sx̂-c〉 on P×F, this leads to
(3.19)sup⁡c∈Finf⁡p∈P〈p,Tx̂-λ̅Sx̂-c〉=inf⁡p∈Psup⁡c∈F〈p,Tx̂-λ̅Sx̂-c〉=0.
Since c↦infp∈P〈p,Tx̂-λ̅Sx̂-c〉 is u.s.c. on F and p↦〈p,Tx̂-λ̅Sx̂-c〉 is continuous on P, from (3.19), first there exists ĉ=(ĉ1,ĉ2,…,ĉm)∈F and then there exists p̂=(p̂1,p̂2,…,p̂m)∈P such that
(3.20)0=sup⁡c∈Finf⁡p∈P〈p,Tx̂-λ̅Sx̂-c〉=inf⁡p∈P〈p,Tx̂-λ̅Sx̂-ĉ〉=〈p̂,Tx̂-λ̅Sx̂-ĉ〉.
As R+P=R+m, for each i=1,2,…,m, there exists ti>0 with pi=(0,…,0,ti︷i,0,…,0︸m)∈P, which implies by (3.20) that for each i=1,2,…,m,
(3.21)ti(Tix̂-λ̅Six̂-ĉi)=〈pi,Tx̂-λ̅Sx̂-ĉ〉≥0,thatis,Tx̂≥λ̅Sx̂+ĉ.
On the other hand, we can see thatIp̂={i:p̂i>0} is nonempty because p̂∈P⊆R+m∖{0}. This gives, by (3.20) and (3.21), that for each i0∈Ip̂,(3.22)0≤p̂i0(Ti0x̂-λ̅Si0x̂-ĉi0)≤〈p̂,Tx̂-λ̅Sx̂-ĉ〉=0,thatis,Ti0x̂=λ̅Si0x̂+ĉi0.
Both (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 λ̅ is the maximal lower eigenvalue to (1.1) and (p̅,x̅)∈S(gF) if and only if λ̅>0 and (p̅,x̅)∈P×X satisfy gF(p̅,x)≤gF(p̅,x̅)=λ̅≤gF(p,x̅) for (p,x)∈P×X, which amounts to say(3.23)fλ̅,F(p̅,x)≤fλ̅,F(p̅,x̅)=0≤fλ̅,F(p,x̅),(p,x)∈P×X,
because for each (p,x)∈P×X,
(3.24)gF(p̅,x)=sup⁡c∈F〈p̅,Tx-c〉〈p̅,Sx〉≤λ̅ifffλ̅,F(p̅,x)=sup⁡c∈F〈p̅,Tx-λ̅Sx-c〉≤0,gF(p̅,x̅)=sup⁡c∈F〈p̅,Tx̅-c〉〈p̅,Sx̅〉=λ̅ifffλ̅,F(p̅,x̅)=sup⁡c∈F〈p̅,Tx̅-λ̅Sx̅-c〉=0,gF(p,x̅)=sup⁡c∈F〈p,Tx̅-c〉〈p,Sx̅〉≥λ̅ifffλ̅,F(p,x̅)=sup⁡c∈F〈p,Tx̅-λ̅Sx̅-c〉≥0.
In view of (3.5), we know that (3.23) is also equivalent to
(3.25)min⁡x∈Rn[f̂λ̅,F(p̅,x)]=min⁡x∈X[-fλ̅,F(p̅,x)]=0=-fλ̅,F(p̅,x̅)=f̂λ̅,F(p̅,x̅),min⁡p∈Rmf̃λ̅,F(p,x̅)=min⁡p∈Pfλ̅,F(p,x̅)=0=fλ̅,F(p̅,x̅)=f̃λ̅,F(p̅,x̅).
Also by (3.5), we have epif̂λ̅,F(p̅,·)={(x,a)∈X×R:-fλ̅,F(p̅,x)≤a} and epif̃λ̅,F(·,x̅)={(p,a)∈P×R:fλ̅,F(p,x̅)≤a}. Combining this with Lemma 3.5(1)-(2) and using the fact that X and P are convex compact, we can see that epif̂λ̅,F(p̅,·) (or epif̃λ̅,F(·,x̅)) is closed convex in Rn×R (or in Rm×R). Hence Lemmas 2.6(1) and 2.10 imply that both x↦f̂λ̅,F(p̅,x)(x∈Rn) and p↦f̃λ̅,F(p,x̅)(p∈Rm) are proper convex and l.s.c. with
(3.26)f̂λ̅,F(p̅,x)=f̂λ̅,F**(p̅,x)(x∈Rn),f̃λ̅,F(p,x̅)=f̃λ̅,F**(p,x̅)(p∈Rm).
Applying Lemma 2.11 to the functions x↦f̂λ̅,F(p̅,x)-〈q0,x〉 on Rn with q0=0 and p↦f̃λ̅,F(p,x̅)-〈r0,p〉 on Rm with r0=0, and using (3.25) and (3.26), we conclude that (3.23) holds if and only if x̅∈∂f̂λ̅,F*(p̅,0) and p̅∈∂f̃λ̅,F*(0,x̅). Hence Theorem 3.3(2)(c) is also true.
(3) Finally we prove Theorem3.3(3).
(a) By (3.1), we know that for each x0∈X,Tx0∈intR+m. So there exists c0∈R+m such that c0<Tx0 (that is, c0i<Tix0 for i=1,2,…,m). Take Fx0={c0}, then supx∈Xf0,Fx0(p,x)=supx∈X〈p,Tx-c0〉≥〈p,Tx0-c0〉>0 for any p∈P. Hence, v(f0,Fx0)=infp∈Psupx∈Xf0,Fx0(p,x)>0 because p↦supx∈Xf0,Fx0(p,x) is l.s.c. on P. This shows that ℂ+m={F∈𝔹+m:v(f0,F)>0} is nonempty. Moreover, Lemma 3.11(1) implies that λ̅(F) exists with λ̅(F)=v(gF) for any F∈ℂ+m. Hence statement (a) follows.
(b) LetFi∈ℂ+m(i=1,2), then we have λ(Fi)=v(gFi)(i=1,2). Suppose that (p,x)∈P×X. Since Fi(i=1,2) are compact, we can select ci∈Fi(i=1,2) such that gF1(p,x)=supc∈F1(〈p,Tx-c〉/〈p,Sx〉)=〈p,Tx-c1〉/〈p,Sx〉 and ∥c1-c2∥=d(c1,F2). This deduces that(3.27)gF1(p,x)=〈p,Tx-c2+(c2-c1)〉〈p,Sx〉≤gF2(p,x)+sup⁡p∈P⁡∥p∥ε0dH(F1,F2),
because 〈p,Tx-c2〉/〈p,Sx〉≤gF2(p,x), ∥c1-c2∥=d(c1,F2)≤dH(F1,F2) and ε0=infp∈P,x∈X〈p,Sx〉 is positive. By taking minimax values for both sides of (3.27), we have v(gF1)≤v(gF2)+(supp∈P∥p∥/ε0)dH(F1,F2). Therefore, |λ̅(F1)-λ̅(F2)|=|v(gF1)-v(gF2)|≤(supp∈P∥p∥/ε0)dH(F1,F2) because dH(F1,F2)=dH(F2,F1), and the last lemma follows.Proofs of Theorems3.1–3.3
Proof.
(i) For Theorem3.1. (1) follows from Lemmas 3.5(3) and 3.6, and (2) from Lemmas 3.9(3), 3.11(1), and 3.12(1).
(ii) For Theorem3.2. (1) can be deduced from Lemmas 3.7(1) and 3.10(1), (2) from Lemmas 3.7(2) and 3.11(2), while (3) from Lemmas 3.8(1) and 3.11(3).
(iii) For Theorem3.3. 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.


## 4. Solvability Results to (1.2)

LetF={c}(c∈R+m), λ∈R+, fλ,F=fλ,c, gF=gc, and λ¯(F)=λ¯(c) (if exists), then(4.1)∀(p,x)∈P×X,fλ,c(p,x)=〈p,Tx-λSx-c〉,gc(p,x)=〈p,Tx-c〉〈p,Sx〉,
and the functions f̂λ,c(p,x),f̃λ,c(p,x),f̂λ,c*(p,q),f̂λ,c**(p,x),f̃λ,c*(r,x) and f̃λ,c**(p,x) can be obtained from (3.5) and (3.6) by replacing c for F, respectively. From Theorems 3.1–3.3, we immediately obtain the solvability results to (1.2) as follows.Theorem 4.1.
(1)v(fλ,c) exists and S(fλ,c) is a nonempty convex compact subset of P×X. Furthermore, λ↦v(fλ,c) is continuous and strictly decreasing on R+ with v(f+∞,c)=̂limλ→+∞v(fλ,c)=-∞.
(2)v(gc) exists if and only if S(gc)≠∅. Moreover, if v(f0,c)>0, then v(gc) exists and S(gc) is a nonempty compact subset of P×X.Theorem 4.2.
(1)λ is a lower eigenvalue to (1.2) and x its eigenvector if and only if infp∈Pfλ,c(p,x)≥0 if and only if infp∈Pgc(p,x)≥λ.
(2)λ is a lower eigenvalue to (1.2) if and only if one of the following statements is true.
(a)
v(fλ,c)≥0,(b)
fλ,c(p̂,x̂)≥0 for (p̂,x̂)∈S(fλ,c),(c)
v(gc) exists with v(gc)≥λ,(d)
S(gc)≠∅ and gc(p̂,x̂)≥λ for (p̂,x̂)∈S(gc).
(3) The following statements are equivalent.
(a)
System (1.2) has at least one lower eigenvalue,(b)
v(f0,c)>0,(c)
v(gc) exists with v(gc)>0,(d)
S(gc)≠∅ and gc(p̂,x̂)>0 for (p̂,x̂)∈S(gc).Theorem 4.3.
(1)λ¯ exists if and only if one of the following statements is true.
(a)
v(f0,c)>0.(b)
f0,c(p̂,x̂)>0 for (p̂,x̂)∈S(f0,c).(c)
v(fλ¯,c)=0.(d)
v(gc) exists with v(gc)=λ¯.(e)
S(gc)≠∅ and gc(p¯,x¯)=λ¯ for (p¯,x¯)∈S(gc).Where λ¯=λ¯(c)(>0) is the maximal lower eigenvalue to (1.2).
(2) Ifv(f0,c)>0, or equivalently, if v(gc) exists with v(gc)>0, then
(a)
x¯ is an optimal eigenvector if and only if there exists p¯∈P with (p¯,x¯)∈S(gc) if and only if infp∈Pgc(p,x¯)=λ¯.(b)
There existx̂∈X and i0∈{1,2,…,m} such that Tx̂≥λ¯Sx̂+c and Ti0x̂=λ¯Si0x̂+ci0.(c)
λ¯=λ¯(c) is the maximal lower eigenvalue to (1.2) and (p¯,x¯)∈S(gc) if and only if λ¯>0 and (p¯,x¯)∈P×X satisfy x¯∈∂f̂λ¯,c*(p¯,0) and p¯∈∂f̃λ¯,c*(0,x¯). Where ∂f̂λ¯,c*(p¯,0) and ∂f̃λ¯,c*(0,x¯) are the subdifferentials of f̂λ¯,c*(p¯,q) at q=0 and f̃λ¯,c*(r,x¯) at r=0, respectively.(d)
The set of all lower eigenvalues to (1.2) coincides with the interval (0,v(gc)].
(3) Letℂ+=̂{c∈R+m:v(f0,c)>0}. Then one has the following.
(a)
ℂ+≠∅, and for each c∈ℂ+,λ¯=λ¯(c) exists with λ¯(c)=v(gc).(b)
|λ¯(c1)-λ¯(c2)|≤(supp∈P∥p∥/ɛ0)∥c1-c2∥(c1,c2∈ℂ+), where ɛ0 is also defined by (3.4).Hence, c↦λ¯(c) is Lipschitz on ℂ+.


## 5. Solvability Results to (1.4)–(1.6)

We now use Theorems3.1–3.3 and 4.1–4.3 to study the solvability of (1.4)–(1.6). For convenience sake, we only present some essential results.


### 5.1. Solvability to (1.4)

SinceF∈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.Theorem 5.1.
(1) Inequality (1.4)(a) is solvable to F∈𝔹+m if and only if one of the following statements is true.
(a)
There existsλ¯≥1 with v(fλ¯,F)=0.(b)
v(gF) exists with v(gF)≥1.(c)
S(gF)≠∅ and gF(p¯,x¯)≥1 for (p¯,x¯)∈S(gF).
(2) Inequality (1.4)(b) is solvable to c∈R+m if and only if one of the following statements is true.
(a)
There existsλ¯≥1 with v(fλ¯,c)=0.(b)
v(gc) exists with v(gc)≥1.(c)
S(gc)≠∅ and gc(p¯,x¯)≥1 for (p¯,x¯)∈S(gc).


### 5.2. Solvability to (1.5)

By Theorem4.1, for each λ≥0, v(fλ,0) exists, S(fλ,0) is nonempty, and if (p¯,x¯)∈S(f0,0), then v(f0,0)=〈p¯,Tx¯〉>0. Hence v(g0) exists, S(g0) is nonempty, and the maximal lower eigenvalue λ¯=λ¯(0) to (1.2) exists with v(g0)=λ¯=g0(p¯,x¯) for (p¯,x¯)∈S(g0). 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≥λ0 with infp∈Pfλ,0(p,x)≥0 if and only if infp∈Pg0(p,x)≥λ≥λ0.
(2)λ is a growth factor to (1.5) if and only if λ≥λ0 with v(fλ,0)≥0 if and only if v(g0)≥λ≥λ0.
(3) Growth fact problem (1.5) is efficient if and only if there exists λ≥λ0 with v(fλ,0)≥0 if and only if v(g0)≥λ0.
(4)λ¯ is the optimal growth factor to (1.5) if and only if λ¯≥λ0 with v(fλ¯,0)=0 if and only if v(g0)=λ¯≥λ0.
(5) Ifv(g0)=λ¯≥λ0, then there exist x¯∈X and i0∈{1,2,…,m} such that Tx¯≥λ¯Sx¯ and Ti0x¯=λ¯Si0x¯.
(6)λ¯ is the optimal growth factor to (1.5) and (p¯,x¯)∈S(g0) if and only if λ¯≥λ0 and (p¯,x¯)∈P×X satisfy x¯∈∂f̂λ¯,0*(p¯,0) and p¯∈∂f̃λ¯,0*(0,x¯).


### 5.3. Solvability to (1.6)

To present the solvability results to (1.6), we assume that(5.1)(a)X⊂int⁡R+nisconvexcompact,(b)A,B∈R+n2with(I-A)X⊂int⁡R+n,BX⊂int⁡R+n,
and define fλ(p,x)=fλ,0(p,x) and g(p,x)=g0(p,x) on Σn-1×X by(5.2)fλ(p,x)=〈p,(I-A)x-λBx〉,(p,x)=〈p,(I-A)x〉〈p,Bx〉for(p,x)∈Σn-1×X,
where Σn-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.Theorem 5.3.
If (5.1) holds and fλ,g be defined by (5.1). Then one has the following.(1)
There existsλ¯>0 such that v(fλ¯)=0 and λ¯=v(g).(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.


## 5.1. Solvability to (1.4)

SinceF∈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.Theorem 5.1.
(1) Inequality (1.4)(a) is solvable to F∈𝔹+m if and only if one of the following statements is true.
(a)
There existsλ¯≥1 with v(fλ¯,F)=0.(b)
v(gF) exists with v(gF)≥1.(c)
S(gF)≠∅ and gF(p¯,x¯)≥1 for (p¯,x¯)∈S(gF).
(2) Inequality (1.4)(b) is solvable to c∈R+m if and only if one of the following statements is true.
(a)
There existsλ¯≥1 with v(fλ¯,c)=0.(b)
v(gc) exists with v(gc)≥1.(c)
S(gc)≠∅ and gc(p¯,x¯)≥1 for (p¯,x¯)∈S(gc).


## 5.2. Solvability to (1.5)

By Theorem4.1, for each λ≥0, v(fλ,0) exists, S(fλ,0) is nonempty, and if (p¯,x¯)∈S(f0,0), then v(f0,0)=〈p¯,Tx¯〉>0. Hence v(g0) exists, S(g0) is nonempty, and the maximal lower eigenvalue λ¯=λ¯(0) to (1.2) exists with v(g0)=λ¯=g0(p¯,x¯) for (p¯,x¯)∈S(g0). 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≥λ0 with infp∈Pfλ,0(p,x)≥0 if and only if infp∈Pg0(p,x)≥λ≥λ0.
(2)λ is a growth factor to (1.5) if and only if λ≥λ0 with v(fλ,0)≥0 if and only if v(g0)≥λ≥λ0.
(3) Growth fact problem (1.5) is efficient if and only if there exists λ≥λ0 with v(fλ,0)≥0 if and only if v(g0)≥λ0.
(4)λ¯ is the optimal growth factor to (1.5) if and only if λ¯≥λ0 with v(fλ¯,0)=0 if and only if v(g0)=λ¯≥λ0.
(5) Ifv(g0)=λ¯≥λ0, then there exist x¯∈X and i0∈{1,2,…,m} such that Tx¯≥λ¯Sx¯ and Ti0x¯=λ¯Si0x¯.
(6)λ¯ is the optimal growth factor to (1.5) and (p¯,x¯)∈S(g0) if and only if λ¯≥λ0 and (p¯,x¯)∈P×X satisfy x¯∈∂f̂λ¯,0*(p¯,0) and p¯∈∂f̃λ¯,0*(0,x¯).


## 5.3. Solvability to (1.6)

To present the solvability results to (1.6), we assume that(5.1)(a)X⊂int⁡R+nisconvexcompact,(b)A,B∈R+n2with(I-A)X⊂int⁡R+n,BX⊂int⁡R+n,
and define fλ(p,x)=fλ,0(p,x) and g(p,x)=g0(p,x) on Σn-1×X by(5.2)fλ(p,x)=〈p,(I-A)x-λBx〉,(p,x)=〈p,(I-A)x〉〈p,Bx〉for(p,x)∈Σn-1×X,
where Σn-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.Theorem 5.3.
If (5.1) holds and fλ,g be defined by (5.1). Then one has the following.(1)
There existsλ¯>0 such that v(fλ¯)=0 and λ¯=v(g).(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.


## 6. Conclusion

In this article, we have studied an optimal lower eigenvalue system (namely, (1.1)), and proved three solvability theorems (i.e., Theorems 3.1–3.3) including a series of necessary 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.



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*Source: 208624-2011-06-28.xml*