An Optimal Lower Eigenvalue System

Abstract

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

1. Introduction

1.1. The Optimal Lower Eigenvalue System

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

To this end, we denote by 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 int𝑅^𝑘₊ its interior. We also define 𝑦¹≥(or>)𝑦² in 𝑅^𝑘 by 𝑦¹−𝑦²∈𝑅^𝑘₊ (or ∈ int𝑅^𝑘₊).

Let 𝜆∈𝑅₊=𝑅¹₊, 𝐹⊆𝑅^𝑚₊, 𝑋⊆𝑅^𝑛₊, and 𝑇=(𝑇₁,…,𝑇_𝑚), 𝑆=(𝑆₁,…,𝑆_𝑚)∶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:(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₊.(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)⎧⎪⎨⎪⎩𝜆>0∶∃𝑥∈𝑋s.t.𝑇𝑥≥𝜆𝑆𝑥+𝑐,(b)⎧⎪⎨⎪⎩0<𝜆⟶max∶∃𝑥∈𝑋s.t.𝑇𝑥≥𝜆𝑆𝑥+𝑐.(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).

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:(a)⎧⎪⎨⎪⎩∃𝑥∈𝑋s.t.𝑥−𝐴𝑥=𝑐,(b)⎧⎪⎨⎪⎩∃𝑥∈𝑋s.t.𝑥−𝑆𝑥∋𝑐,(1.3) 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^𝑅𝑛+ 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 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→ 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:(a)⎧⎪⎨⎪⎩𝑥∈𝑋s.t.𝑇𝑥−𝑆𝑥∈𝐹+𝑅^𝑚₊,(b)⎧⎪⎨⎪⎩𝑥∈𝑋s.t.𝑇𝑥−𝑆𝑥≥𝑐.(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₊→ 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:(a)⎧⎪⎨⎪⎩𝜆∈𝜆₀,+∞∶∃𝑥∈𝑋s.t.𝑇𝑥≥𝜆𝑆𝑥,(b)⎧⎪⎨⎪⎩𝜆₀≤𝜆⟶max∶∃𝑥∈𝑋s.t.𝑇𝑥≥𝜆𝑆𝑥.(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^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:(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.(1.6) Both (a) and (b) are just the static descriptions of the dynamic Leontief model (a)𝜇>1or(b)1<𝜇⟶max∶∃𝑥∈𝑋s.t.𝑥(𝑡)=𝜇^𝑡𝑥with(𝐼−𝐴+𝐵)𝑥(𝑡)≥𝐵𝑥(𝑡+1),𝑡=1,2,….(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.

1.3. Questions and Assumptions

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

Question 1 (Existence). If 𝜆>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 eigenvalue 𝜆=𝜆(𝐹) to (1.1) exist? How to describe it?

Question 3 (Stability). If the answer to the Question 2 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 all 𝑖=1,2,…,𝑚, 𝑇_𝑖∶𝑋→int𝑅₊ is upper semicontinuous and concave, 𝑆_𝑖∶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

Let 𝑓,𝑔_𝛼(𝛼∈Λ)∶𝑋⊂𝑅^𝑘→𝑅 and 𝜑∶𝑃×𝑋⊂𝑅^𝑚×𝑅^𝑛→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.

Definition 2.1. (1) If inf_𝑝∈𝑃sup_𝑥∈𝑋𝜑(𝑝,𝑥)=sup_𝑥∈𝑋inf_𝑝∈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 sup_𝑥∈𝑋𝜑(𝑝,𝑥)=inf_𝑝∈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 inf_𝑝∈𝑃sup_𝑥∈𝑋𝜑(𝑝,𝑥)≤sup_𝑥∈𝑋inf_𝑝∈𝑃𝜑(𝑝,𝑥),(2)(𝑝,𝑥)∈𝑆(𝜑) if and only if (𝑝,𝑥)∈𝑃×𝑋 with sup_𝑥∈𝑋𝜑(𝑝,𝑥)≤inf_𝑝∈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)if 𝑆(𝜑)≠∅, then 𝑣(𝜑) exists and 𝑣(𝜑)=𝜑(𝑝,𝑥)=sup_𝑥∈𝑋𝜑(𝑝,𝑥)=inf_𝑝∈𝑃𝜑(𝑝,𝑥) for all (𝑝,𝑥)∈𝑆(𝜑).

Definition 2.3. Let 𝑓 be a function from 𝑅^𝑘 to 𝑅∪{+∞} with the domain dom(𝑓)={𝑥∈𝑅^𝑘∶𝑓(𝑥)<+∞} and 𝑔 a function from 𝑅^𝑘∗(=𝑅^𝑘) 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 𝑅^𝑘×𝑅 defined by epi(𝑓)={(𝑥,𝑎)∈𝑅^𝑘×𝑅∶𝑓(𝑥)≤𝑎}.(2)The conjugate functions of 𝑓 and 𝑔 are the functions 𝑓^∗∶𝑅^𝑘∗→𝑅∪{+∞} and 𝑔^∗∶𝑅^𝑘→𝑅∪{+∞} defined by 𝑓^∗(𝑝)=sup_{𝑥∈𝑅𝑘}[⟨𝑝,𝑥⟩−𝑓(𝑥)] for 𝑝∈𝑅^𝑘∗ and 𝑔^∗(𝑥)=sup_{𝑝∈𝑅𝑘}[⟨𝑝,𝑥⟩−𝑔(𝑝)] for 𝑥∈𝑅^𝑘, respectively. The biconjugate 𝑓^∗∗ of 𝑓 is therefore defined on 𝑅^𝑘∗∗(=𝑅^𝑘) by 𝑓^∗∗=(𝑓^∗)^∗.(3)If 𝑓 is a proper function from 𝑅^𝑘 to 𝑅∪{+∞} and 𝑥₀∈dom(𝑓), then the subdifferential of 𝑓 at 𝑥₀ is the (possibly empty) subset 𝜕𝑓(𝑥₀) of 𝑅^𝑘∗ defined by 𝜕𝑓(𝑥₀)={𝑝∈𝑅^𝑘∗∶𝑓(𝑥₀)−𝑓(𝑥)≤⟨𝑝,𝑥₀−𝑥⟩ for all 𝑥∈𝑅^𝑘}.

Remark 2.4. If 𝑓 is a proper function from 𝑅^𝑘 to 𝑅∪{−∞}, 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)≠∅.

Definition 2.5. 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∈𝐴𝑑(𝑥,𝑦).(2)Let 𝜌(𝐴,𝐵)=sup_𝑥∈𝐴𝑑(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)}.

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

Lemma 2.6 (see [9]). (1) A proper function 𝑓∶𝑅^𝑘→𝑅∪{+∞} is convex or l.s.c. if and only if its epigraph epi(𝑓) is convex or closed in 𝑅^𝑘×𝑅.
(2) The upper envelope sup_𝛼∈Λ𝑓_𝛼(𝑥) of proper convex (or l.s.c.) functions 𝑓_𝛼(𝑥)∶𝑅^𝑘→𝑅∪{+∞}(𝛼∈Λ) is also proper convex (or l.s.c.) when the dom(sup_𝛼∈Λ𝑓_𝛼)={𝑥∈𝑅^𝑘∶sup_𝛼∈Λ𝑓_𝛼(𝑥)<+∞} is nonempty.
(3) The lower envelope inf_𝛼∈Λ𝑔_𝛼(𝑥) of proper concave (or u.s.c.) functions 𝑔_𝛼(𝑥)∶𝑅^𝑘→𝑅∪{+∞}(𝛼∈Λ) is also proper concave (or u.s.c.) when the dom(inf_𝛼∈Λ𝑔_𝛼)={𝑥∈𝑅^𝑘∶inf_𝛼∈Λ𝑔_𝛼(𝑥)>−∞} is nonempty.

Remark 2.7. Since epi(sup_𝛼∈Λ𝑓_𝛼)=⋂_𝛼∈Λepi(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.

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_𝑦∈𝑌𝑓(𝑥,𝑦) (or, 𝑘∶𝑋→𝑅 defined by 𝑘(𝑥)=sup_𝑦∈𝑌𝑓(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. Then inf_𝑝∈𝑃sup_𝑥∈𝑋𝜑(𝑝,𝑥)=sup_𝑥∈𝑋inf_𝑝∈𝑃𝜑(𝑝,𝑥) and there exists (𝑝,𝑥)∈𝑃×𝑋 such that 𝜑(𝑝,𝑥)=sup_𝑥∈𝑋𝜑(𝑝,𝑥)=inf_𝑝∈𝑃𝜑(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^∗∗.

Lemma 2.11 (see [8]). Let 𝑓 be a proper function defined on 𝑅^𝑘, and 𝑝₀∈𝑅^𝑘∗. 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. If 𝑓 is a finite function from 𝑋⊆𝑅^𝑘 to 𝑅, define 𝑓_𝑋 by 𝑓_𝑋(𝑥)=𝑓(𝑥) if 𝑥∈𝑋, or =+∞ if 𝑥∈𝑅^𝑘⧵𝑋, then we can use the preceding associated concepts and lemmas for 𝑓 by identifying 𝑓 with 𝑓_𝑋.

3. Solvability Results to (1.1)

3.1. Auxiliary Functions

In the sequel, we assume that(1)Assumptions1-3inSection1aresatisfied,and𝜆∈𝑅₊,𝐹∈𝔹^𝑚₊,(2)𝑃⊆𝑅^𝑚₊⧵{0}isaconvexcompactsubsetwith𝑅₊𝑃=𝑅^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 by𝑓_𝜆,𝐹(𝑝,𝑥)=sup_𝑐∈𝐹⟨𝑝,𝑇𝑥−𝜆𝑆𝑥−𝑐⟩=sup_{𝑐1,𝑐2,…,𝑐𝑚∈𝐹}^𝑚_𝑖=1𝑝_𝑖𝑇_𝑖𝑥−𝜆𝑆_𝑖𝑥−𝑐_𝑖,(3.2)𝑔_𝐹(𝑝,𝑥)=sup_𝑐∈𝐹⟨𝑝,𝑇𝑥−𝑐⟩⟨𝑝,𝑆𝑥⟩=sup_{(𝑐1,𝑐2,…,𝑐𝑚)∈𝐹}∑^𝑚_𝑖=1𝑝_𝑖𝑇_𝑖𝑥−𝑐_𝑖∑^𝑚_𝑖=1𝑝_𝑖𝑆_𝑖𝑥.(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,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 that0<𝜀₀=inf_{𝑝∈𝑃,𝑥∈𝑋}⟨𝑝,𝑆𝑥⟩<+∞,0<𝜀₁=sup_{𝑝∈𝑃,𝑥∈𝑋}⟨𝑝,𝑇𝑥⟩<+∞,(3.4) 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↦𝑓̂‌_𝜆,𝐹(𝑝,𝑥) to 𝑥↦−𝑓_𝜆,𝐹(𝑝,𝑥) (for each fixed 𝑝∈𝑃) and 𝑝↦𝑓̃‌_𝜆,𝐹(𝑝,𝑥) to 𝑝↦𝑓_𝜆,𝐹(𝑝,𝑥) (for each fixed 𝑥∈𝑋) by𝑓̂‌_𝜆,𝐹(𝑝,𝑥)=⎧⎪⎨⎪⎩−𝑓_𝜆,𝐹(𝑝,𝑥),𝑥∈𝑋,+∞,𝑥∈𝑅^𝑛⧵𝑋,𝑓̃‌_𝜆,𝐹(𝑝,𝑥)=⎧⎪⎨⎪⎩𝑓_𝜆,𝐹(𝑝,𝑥),𝑝∈𝑃,+∞,𝑝∈𝑅^𝑚⧵𝑃.(3.5) According to Definition 2.3, the conjugate and biconjugate functions of 𝑥↦𝑓̂‌_𝜆,𝐹(𝑝,𝑥) and 𝑝↦𝑓̃‌_𝜆,𝐹(𝑝,𝑥) are then denoted by𝑞⟼𝑓̂‌^∗_𝜆,𝐹(𝑝,𝑞),𝑞∈𝑅^𝑛,𝑥⟼𝑓̂‌^∗∗_𝜆,𝐹(𝑝,𝑥),𝑥∈𝑅^𝑛(foreachfixed𝑝∈𝑃),𝑟⟼𝑓̃‌^∗_𝜆,𝐹(𝑟,𝑥),𝑟∈𝑅^𝑚,𝑝⟼𝑓̃‌^∗∗_𝜆,𝐹(𝑝,𝑥),𝑝∈𝑅^𝑚(foreachfixed𝑥∈𝑋).(3.6) By Definition 2.5, the Hausdorff distance in 𝔹^𝑚₊ (see Assumption 3) is provided by𝑑_𝐻𝐹₁,𝐹₂=maxsup_{𝑐1∈𝐹1}𝑑𝑐₁,𝐹₂,sup_{𝑐2∈𝐹2}𝑑𝑐₂,𝐹₁for𝐹₁,𝐹₂∈𝔹^𝑚₊.(3.7)

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(𝑓_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₊ with 𝑣(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.

Theorem 3.2. (1)  𝜆 is a lower eigenvalue to (1.1) and 𝑥 its eigenvector if and only if inf_𝑝∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥) ≥0 if and only if inf_𝑝∈𝑃𝑔_𝐹(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(𝑓_𝜆,𝐹)≥0,(b)𝑓_𝜆,𝐹(𝑝,̂𝑥̂)≥0 for (𝑝,̂𝑥̂)∈𝑆(𝑓_𝜆,𝐹),(c)𝑣(𝑔_𝐹) exists with 𝑣(𝑔_𝐹)≥𝜆,(d)𝑆(𝑔_𝐹)≠∅ and 𝑔_𝐹(𝑝,̂𝑥̂)≥𝜆 for (𝑝,̂𝑥̂)∈𝑆(𝑔_𝐹).
(3) The following statements are equivalent:
(a)System (1.1) has at least one lower eigenvalue,(b)𝑣(𝑓_0,𝐹)>0,(c)𝑣(𝑔_𝐹) exists with 𝑣(𝑔_𝐹)>0,(d)𝑆(𝑔_𝐹)≠∅ and 𝑔_𝐹(𝑝,̂𝑥̂)>0 for (𝑝,̂𝑥̂)∈𝑆(𝑔_𝐹).

Theorem 3.3. (1) 𝜆 exists if and only if one of the following statements is true:
(a)𝑣(𝑓_0,𝐹)>0,(b)𝑓_0,𝐹(𝑝,̂𝑥̂)>0 for (𝑝,̂𝑥̂)∈𝑆(𝑓_0,𝐹),(c)𝑣(𝑓_𝜆,𝐹)=0,(d)𝑣(𝑔_𝐹) exists with 𝑣(𝑔_𝐹)=𝜆,(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).
(2) If 𝑣(𝑓_0,𝐹)>0, 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∈𝑃𝑔_𝐹(𝑝,𝑥)=𝜆.(b)There exist 𝑥̂∈𝑋,𝑐̂∈𝐹 and 𝑖₀∈{1,2,…,𝑚} such that 𝑇𝑥̂≥𝜆𝑆𝑥̂+𝑐̂ and 𝑇_𝑖0𝑥̂=𝜆𝑆_𝑖0𝑥̂+𝑐̂_𝑖0.(c)𝜆=𝜆(𝐹) 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 𝑥∈𝜕𝑓̂‌^∗_𝜆,𝐹(𝑝,0)and 𝑝∈𝜕𝑓̃‌^∗_𝜆,𝐹(0,𝑥). Where 𝜕𝑓̂‌^∗_𝜆,𝐹(𝑝,0) and 𝜕𝑓̃‌^∗_𝜆,𝐹(0,𝑥) are the subdifferentials of 𝑓̂‌^∗_𝜆,𝐹(𝑝,𝑞) at 𝑞=0 and 𝑓̃‌^∗_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)].
(3) Let ℂ^𝑚₊={𝐹∈𝔹^𝑚₊∶𝑣(𝑓_0,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(𝑔_𝐹),(b)for all 𝐹₁,𝐹₂∈ℂ^𝑚₊, |𝜆(𝐹₁)−𝜆(𝐹₂)|≤(sup_𝑝∈𝑃‖𝑝‖/𝜀₀)𝑑_𝐻(𝐹₁,𝐹₂), where 𝜀₀ is defined by (3.4). Thus, 𝐹↦𝜆(𝐹) is Lipschitz on ℂ^𝑚₊ with the Hausdorff distance 𝑑_𝐻(⋅,⋅).

Remark 3.4. If we take 𝑃=Σ^𝑚−1={𝑝∈𝑅^𝑚₊∶∑^𝑚_𝑖=1𝑝_𝑖=1}, then Σ^𝑚−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 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∈𝑋𝑓_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∈𝑃𝑓_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 (a)∀𝑥∈𝑋,∀𝑐∈𝐹,𝑝⟼⟨𝑝,𝑇𝑥−𝜆𝑆𝑥−𝑐⟩isconvexl.s.c.on𝑃,(b)∀𝑝∈𝑃,(𝑥,𝑐)⟼⟨𝑝,𝑇𝑥−𝜆𝑆𝑥−𝑐⟩isu.s.c.on𝑋×𝐹.(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)isconvexl.s.c.on𝑃andinf_𝑝∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥)isfi,∀𝑝∈𝑃,𝑥⟼𝑓_𝜆,𝐹(𝑝,𝑥)isu.s.c.on𝑋andsup_𝑥∈𝑋𝑓_𝜆,𝐹(𝑝,𝑥)isfi.(3.9)
If 𝑥^𝑖∈𝑋(𝑖=1,2), then by (3.2), there exist 𝑐^𝑖=(𝑐^𝑖₁,𝑐^𝑖₂,…,𝑐^𝑖_𝑚)∈𝐹(𝑖=1,2) such that 𝑓_𝜆,𝐹(𝑝,𝑥^𝑖)=⟨𝑝,𝑇𝑥^𝑖−𝜆𝑆𝑥^𝑖−𝑐^𝑖⟩. Since 𝑇_𝑖,−𝜆𝑆_𝑖(𝑖=1,2,…,𝑚) are concave, 𝑋,𝐹 are convex and 𝑝∈𝑃 is nonnegative, we have for each 𝛼∈[0,1],𝑓_𝜆,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=𝛼𝑓_𝜆,𝐹𝑝,𝑥¹+(1−𝛼)𝑓_𝜆,𝐹𝑝,𝑥²,thatis,𝑥⟼𝑓_𝜆,𝐹(𝑝,𝑥)isconcaveon𝑋.(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∈𝑋𝑓_𝜆,𝐹(𝑝^𝑖,𝑥)=inf_𝑝∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥^𝑖) for 𝑖=1,2. By (1) and (2) (i.e., 𝑝↦sup_𝑥∈𝑋𝑓_𝜆,𝐹(𝑝,𝑥) is convex on 𝑃 and 𝑥↦inf_𝑝∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥) is concave on 𝑋), we have sup_𝑥∈𝑋𝑓_𝜆,𝐹𝛼𝑝¹+(1−𝛼)𝑝²,𝑥≤𝛼sup_𝑥∈𝑋𝑓_𝜆,𝐹𝑝¹,𝑥+(1−𝛼)sup_𝑥∈𝑋𝑓_𝜆,𝐹𝑝²,𝑥=𝛼inf_𝑝∈𝑃𝑓_𝜆,𝐹𝑝,𝑥¹+(1−𝛼)inf_𝑝∈𝑃𝑓_𝜆,𝐹𝑝,𝑥²≤inf_𝑝∈𝑃𝑓_𝜆,𝐹𝑝,𝛼𝑥¹+(1−𝛼)𝑥².(3.11) This implies by Remark 2.2(2) that 𝛼(𝑝¹,𝑥¹)+(1−𝛼)(𝑝²,𝑥²)∈𝑆(𝑓_𝜆,𝐹), and thus 𝑆(𝑓_𝜆,𝐹) is convex.
If (𝑝^𝑘,𝑥^𝑘)∈𝑆(𝑓_𝜆,𝐹) with (𝑝^𝑘,𝑥^𝑘)→(𝑝⁰,𝑥⁰)∈𝑃×𝑋(𝑘→∞), then sup_𝑥∈𝑋𝑓_𝜆,𝐹(𝑝^𝑘,𝑥)=inf_𝑝∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥^𝑘) for all 𝑘=1,2,…. By taking 𝑘→∞, from (1) and (2) (that is, 𝑝↦sup_𝑥∈𝑋𝑓_𝜆,𝐹(𝑝,𝑥) is l.s.c. on 𝑃 and 𝑥↦inf_𝑝∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥) is u.s.c. on 𝑋), we obtain that sup_𝑥∈X𝑓_𝜆,𝐹𝑝⁰,𝑥≤liminf_𝑘→∞sup_𝑥∈𝑋𝑓_𝜆,𝐹𝑝^𝑘,𝑥≤limsup_𝑘→∞inf_𝑝∈𝑃𝑓_𝜆,𝐹𝑝,𝑥^𝑘≤inf_𝑝∈𝑃𝑓_𝜆,𝐹𝑝,𝑥⁰.(3.12) Hence by Remark 2.2(2), (𝑝⁰,𝑥⁰)∈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 𝑅₊ with 𝑣(𝑓_+∞,𝐹)=lim_𝜆→+∞𝑣(𝑓_𝜆,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 𝑅₊×𝑃→𝑅∶(𝜆,𝑝)⟼𝑓_𝜆,𝐹(𝑝,𝑥)=sup_𝑐∈𝐹⟨𝑝,𝑇𝑥−𝜆𝑆𝑥−𝑐⟩isl.s.c.,𝑅₊×𝑋→𝑅∶(𝜆,𝑥)⟼𝑓_𝜆,𝐹(𝑝,𝑥)=sup_𝑐∈𝐹⟨𝑝,𝑇𝑥−𝜆𝑆𝑥−𝑐⟩isu.s.c..(3.13) From Lemma 2.6(2)-(3), it follows that 𝑅₊×𝑃⟼𝑅∶(𝜆,𝑝)⟼sup_𝑥∈𝑋𝑓_𝜆,𝐹(𝑝,𝑥)isl.s.c.,𝑅₊×𝑋⟼𝑅∶(𝜆,𝑥)⟼inf_𝑝∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥)isu.s.c..(3.14) First applying Lemma 2.8 to both functions of (3.14), and then using Lemma 3.5(3), we further obtain that 𝑅₊⟼𝑅∶𝜆⟼inf_𝑝∈𝑃sup_𝑥∈𝑋𝑓_𝜆,𝐹(𝑝,𝑥)isl.s.c.,𝑅₊⟼𝑅∶𝜆⟼sup_𝑥∈𝑋inf_𝑝∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥)isu.s.c.,(3.15) and thus 𝜆↦𝑣(𝑓_𝜆,𝐹) is continuous on 𝑅₊.
Suppose that 𝜆₂>𝜆₁≥0, then by (3.2), 𝑓_𝜆1,𝐹(𝑝,𝑥)=𝑓_𝜆2,𝐹(𝑝,𝑥)+(𝜆₂−𝜆₁)⟨𝑝,𝑆𝑥⟩ for all (𝑝,𝑥)∈𝑃×𝑋. This implies by (3.4) that 𝑣(𝑓_𝜆1,𝐹)≥𝑣(𝑓_𝜆2,𝐹)+(𝜆₂−𝜆₁)𝜀₀, where 𝜀₀=inf_{𝑝∈𝑃,𝑥∈𝑋}⟨𝑝,𝑆𝑥⟩∈(0,+∞). Hence 𝜆↦𝑣(𝑓_𝜆,𝐹) is strictly decreasing.
By Lemma 3.5(3), Remark 2.2(3) and (3.2), it is easily to see that for each 𝜆∈𝑅₊ and (𝑝,𝑥)∈𝑆(𝑓_𝜆,𝐹), 𝑣𝑓_𝜆,𝐹=sup_𝑐∈𝐹𝑝,𝑇𝑥−𝜆𝑆𝑥−𝑐≤sup_{𝑝∈𝑃,𝑥∈𝑋}⟨𝑝,𝑇𝑥⟩−𝜆inf_{𝑝∈𝑃,𝑥∈𝑋}⟨𝑝,𝑆𝑥⟩=𝜀₁−𝜆𝜀₀.(3.16) 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 inf_𝑝∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥)≥0.
(2) 𝜆 is a lower eigenvalue to (1.1) if and only if 𝑣(𝑓_𝜆,𝐹)≥0 if and only if 𝑓_𝜆,𝐹(𝑝,̂𝑥̂)≥0 for (𝑝,̂𝑥̂)∈𝑆(𝑓_𝜆,𝐹).

Proof. (1) If 𝜆>0 and (𝑥,𝑐)∈𝑋×𝐹 satisfy 𝑇𝑥≥𝜆𝑆𝑥+𝑐, then for each 𝑝∈𝑃(⊆𝑅^𝑚₊), 𝑓_𝜆,𝐹(𝑝,𝑥)≥⟨𝑝,𝑇𝑥−𝜆𝑆𝑥−𝑐⟩≥0. Hence, inf_𝑝∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥)≥0.
If 𝜆>0 and 𝑥∈𝑋 satisfy inf_𝑝∈𝑃𝑓_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 𝑝̂∈𝑃 with 𝑝̂=𝑡𝑝^∗. It follows that inf_𝑝∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥)≤𝑓_𝜆,𝐹(𝑝,̂𝑥)=𝑡sup_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∈𝑃𝑓_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,̂𝑥̂)=𝑣(𝑓_𝜆,𝐹)≥0 for all (𝑝,̂𝑥̂)∈𝑆(𝑓_𝜆,𝐹). If (𝑝,̂𝑥̂)∈𝑆(𝑓_𝜆,𝐹) with 𝑓_𝜆,𝐹(𝑝,̂𝑥̂)≥0, then inf_𝑝∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥̂)=𝑓_𝜆,𝐹(𝑝,̂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(𝑓_0,𝐹)>0. (c)𝑓_0,𝐹(𝑝,̂𝑥̂)>0 for (𝑝,̂𝑥̂)∈𝑆(𝑓_0,𝐹).(d)There is a unique 𝜆̂‌>0 with 𝑣(𝑓𝜆̂‌_,𝐹)=0.(e)The maximal lower eigenvalue 𝜆=𝜆(𝐹) to (1.1) exists. In particular, 𝜆̂‌=𝜆 if either 𝑣(𝑓_0,𝐹)>0 or one of the 𝜆̂‌ 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 𝜆₀(>0) is a lower eigenvalue to (1.1), then by Lemmas 3.6 and 3.7(2), 𝑣(𝑓_0,𝐹)>𝑣(𝑓_𝜆0,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,̂𝑥̂)>0 for any (𝑝,̂𝑥̂)∈𝑆(𝑓_0,𝐹). If 𝑣(𝑓_0,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. If 𝐹∈𝔹^𝑚₊, then one has the following. (1)𝑝↦𝑔_𝐹(𝑝,𝑥)(𝑥∈𝑋) and 𝑝↦sup_𝑥∈𝑋𝑔_𝐹(𝑝,𝑥) are continuous on 𝑃.(2)𝑥↦𝑔_𝐹(𝑝,𝑥)(𝑝∈𝑃) and 𝑥↦inf_𝑝∈𝑃𝑔_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,𝑇𝑥−𝑐⟩/⟨𝑝,𝑆𝑥⟩ is continuous on 𝑃, by (3.3), and Lemma 2.6(2), we see that 𝑝↦𝑔_𝐹(𝑝,𝑥)=sup_𝑐∈𝐹(⟨𝑝,𝑇𝑥−𝑐⟩/⟨𝑝,𝑆𝑥⟩)(𝑥∈𝑋) and 𝑝↦sup_𝑥∈𝑋𝑔_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,𝑇𝑥−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,𝑇𝑥−𝑐⟩/⟨𝑝,𝑆𝑥⟩) on 𝑃×𝑋 and 𝑝↦sup_𝑥∈𝑋𝑔_𝐹(𝑝,𝑥) on 𝑃 are u.s.c., so is 𝑝↦𝑔_𝐹(𝑝,𝑥). Hence (1) is true.
(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↦inf_𝑝∈𝑃𝑔_𝐹(𝑝,𝑥) because of Lemma 2.6(3).
(3) By Remark 2.2(3), we only need to prove the necessary part. Assume 𝑣(𝑔_𝐹) exists, that is, inf_𝑝∈𝑃sup_𝑥∈𝑋𝑔_𝐹(𝑝,𝑥)=sup_𝑥∈𝑋inf_𝑝∈𝑃𝑔_𝐹(𝑝,𝑥), then both (1) and (2) imply that there exist 𝑝∈𝑃 and 𝑥∈𝑋 with sup_𝑥∈𝑋𝑔_𝐹(𝑝,𝑥)=inf_𝑝∈𝑃𝑔_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∈𝑃𝑔_𝐹(𝑝,𝑥)≥𝜆.
(2)  𝜆 is a lower eigenvalue to (1.1) if and only if sup_𝑥∈𝑋inf_𝑝∈𝑃𝑔_𝐹(𝑝,𝑥)≥𝜆.

Proof. (1) Suppose 𝜆>0 and 𝑥∈𝑋. Since for each 𝑝∈𝑃, 𝑔_𝐹(𝑝,𝑥)=sup_𝑐∈𝐹(⟨𝑝,𝑇𝑥−𝑐⟩/⟨𝑝,𝑆𝑥⟩)≥𝜆 equals to 𝑓_𝜆,𝐹(𝑝,𝑥)=sup_𝑐∈𝐹⟨𝑝,𝑇𝑥−𝜆𝑆𝑥−𝑐⟩≥0, which implies that inf_𝑝∈𝑃𝑔_𝐹(𝑝,𝑥)≥𝜆 if and only if inf_𝑝∈𝑃𝑓_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∈𝑋inf_𝑝∈𝑃𝑔_𝐹(𝑝,𝑥)≥𝜆(>0), then Lemma 3.9(2) shows that there exists 𝑥_𝜆∈𝑋 with inf_𝑝∈𝑃𝑔_𝐹(𝑝,𝑥_𝜆)=sup_𝑥∈𝑋inf_𝑝∈𝑃𝑔_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,̂𝑥̂)≥𝜆 for (𝑝,̂𝑥̂)∈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,̂𝑥̂)>0 for (𝑝,̂𝑥̂)∈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∈𝑋inf_𝑝∈𝑃𝑔_𝐹(𝑝,𝑥),𝜆^∗=inf_𝑝∈𝑃sup_𝑥∈𝑋𝑔_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∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥)=𝑣(𝑓_𝜆,𝐹)=0. This shows that sup_𝑐∈𝐹⟨𝑝,𝑇𝑥−𝜆𝑆𝑥−𝑐⟩=𝑓_𝜆,𝐹(𝑝,𝑥)≥0 for any 𝑝∈𝑃, that is, 𝜆≤sup_𝑐∈𝐹(⟨𝑝,𝑇𝑥−𝑐⟩/⟨𝑝,𝑆𝑥⟩)=𝑔_𝐹(𝑝,𝑥)(𝑝∈𝑃). Hence, 𝜆≤inf_𝑝∈𝑃𝑔_𝐹(𝑝,𝑥)≤𝜆_∗. On the other hand, since for each 𝑝∈𝑃, 𝜆^∗≤sup_𝑥∈𝑋𝑔_𝐹(𝑝,𝑥), by Lemma 3.9(2), there exists 𝑥_𝑝∈𝑋 such that 𝜆^∗≤sup_𝑥∈𝑋𝑔_𝐹(𝑝,𝑥)=𝑔_𝐹(𝑝,𝑥_𝑝)=sup_𝑐∈𝐹(⟨𝑝,𝑇𝑥_𝑝−𝑐⟩/⟨𝑝,𝑆𝑥_𝑝⟩). It follows that sup_𝑥∈𝑋𝑓_𝜆∗,𝐹(𝑝,𝑥)≥𝑓_𝜆∗,𝐹(𝑝,𝑥_𝑝)=sup_𝑐∈𝐹⟨𝑝,𝑇𝑥_𝑝−𝜆^∗𝑆𝑥_𝑝−𝑐⟩≥0 for any 𝑝∈𝑃. Hence by Lemma 3.5(3), 𝑣(𝑓_𝜆∗,𝐹)=inf_𝑝∈𝑃sup_𝑥∈𝑋𝑓_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∈𝑃𝑔_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.
(2) If 𝜆>0 is a lower eigenvalue to (1.1), then Lemmas 3.8(1), 3.10(2) and statement (1) imply that 𝑣(𝑓_0,𝐹)>0,𝑣(𝑔_𝐹) exists and 𝑣(𝑔_𝐹)=sup_𝑥∈𝑋inf_𝑝∈𝑃𝑔_𝐹(𝑝,𝑥)≥𝜆. If 𝑣(𝑔_𝐹) exists 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,̂𝑥̂)=𝑣(𝑔_𝐹)≥𝜆 for (𝑝,̂𝑥̂)∈𝑆(𝑔_𝐹). If 𝑆(𝑔_𝐹)≠∅ and 𝑔_𝐹(𝑝,̂𝑥̂)≥𝜆 for (𝑝,̂𝑥̂)∈𝑆(𝑔_𝐹), then by Remark 2.2(3) and Lemma 3.10(1), we see that inf_𝑝∈𝑃𝑔_𝐹(𝑝,𝑥̂)=𝑔_𝐹(𝑝,̂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) 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 𝑃×𝑋.
(2) The first three statements of Theorem 3.3(2) are true.
(3) Theorem 3.3(3) is true.

Proof. (1) By Lemma 3.11(1), 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 (𝑝^𝑘,𝑥^𝑘)∈𝑆(𝑔_𝐹) such that (𝑝^𝑘,𝑥^𝑘)→(𝑝⁰,𝑥⁰)∈𝑃×𝑋 as 𝑘→∞, then sup_𝑥∈𝑋𝑔_𝐹𝑝⁰,𝑥=liminf_𝑘→∞sup_𝑥∈𝑋𝑔_𝐹𝑝^𝑘,𝑥≤limsup_𝑘→∞inf_𝑝∈𝑃𝑔_𝐹𝑝,𝑥^𝑘≤inf_𝑝∈𝑃𝑔_𝐹𝑝,𝑥⁰.(3.17) 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∈𝑃𝑔_𝐹(𝑝,𝑥)≥𝜆. On the other hand, by Lemma 3.9(1), there exists 𝑝∈𝑃 such that 𝑣(𝑔_𝐹)=sup_𝑥∈𝑋𝑔_𝐹(𝑝,𝑥). So we obtain that sup_𝑥∈𝑋𝑔_𝐹(𝑝,𝑥)=𝜆≤inf_𝑝∈𝑃𝑔_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∈𝑃𝑔_𝐹(𝑝,𝑥)=𝑣(𝑔_𝐹)=𝜆. If inf_𝑝∈𝑃𝑔_𝐹(𝑝,𝑥)=𝜆, then Lemma 3.10(1) shows that 𝑥 is an optimal eigenvector. Hence, Theorem 3.3(2)(a) follows.
Next we prove statement (b). By Lemmas 3.5(2) and 3.8(1), there exists 𝑥̂∈𝑋 with0=𝑣𝑓_𝜆,𝐹=sup_𝑥∈𝑋inf_𝑝∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥)=inf_𝑝∈𝑃𝑓_𝜆,𝐹(𝑝,𝑥̂)=inf_𝑝∈𝑃sup_𝑐∈𝐹𝑝,𝑇𝑥̂−𝜆𝑆𝑥̂−𝑐.(3.18) By applying Lemma 2.9 to 𝜑(𝑝,𝑐)=⟨𝑝,𝑇𝑥̂−𝜆𝑆𝑥̂−𝑐⟩ on 𝑃×𝐹, this leads to sup_𝑐∈𝐹inf_𝑝∈𝑃𝑝,𝑇𝑥̂−𝜆𝑆𝑥̂−𝑐=inf_𝑝∈𝑃sup_𝑐∈𝐹𝑝,𝑇𝑥̂−𝜆𝑆𝑥̂−𝑐=0.(3.19) Since 𝑐↦inf_𝑝∈𝑃⟨𝑝,𝑇𝑥̂−𝜆𝑆𝑥̂−𝑐⟩ is u.s.c. on 𝐹 and 𝑝↦⟨𝑝,𝑇𝑥̂−𝜆𝑆𝑥̂−𝑐⟩ is continuous on 𝑃, from (3.19), first there exists 𝑐̂=(𝑐₁̂,𝑐₂̂,…,𝑐_𝑚̂)∈𝐹 and then there exists 𝑝̂=(𝑝₁̂,𝑝₂̂,…,𝑝_𝑚̂)∈𝑃 such that 0=sup_𝑐∈𝐹inf_𝑝∈𝑃𝑝,𝑇𝑥̂−𝜆𝑆𝑥̂−𝑐=inf_𝑝∈𝑃𝑝,𝑇𝑥̂−𝜆𝑆𝑥̂−𝑐̂=𝑝,̂𝑇𝑥̂−𝜆𝑆𝑥̂−𝑐̂.(3.20) As 𝑅₊𝑃=𝑅^𝑚₊, for each 𝑖=1,2,…,𝑚, there exists 𝑡_𝑖>0 with 𝑝^𝑖=(0,…,0,𝑡^{𝑖}_𝑖,0,…,0_{𝑚})∈𝑃, which implies by (3.20) that for each 𝑖=1,2,…,𝑚, 𝑡_𝑖𝑇_𝑖𝑥̂−𝜆𝑆_𝑖𝑥̂−𝑐_𝑖̂=𝑝^𝑖,𝑇𝑥̂−𝜆𝑆𝑥̂−𝑐̂≥0,thatis,𝑇𝑥̂≥𝜆𝑆𝑥̂+𝑐.̂(3.21)
On the other hand, we can see that 𝐼𝑝̂={𝑖∶𝑝_𝑖̂>0} is nonempty because 𝑝̂∈𝑃⊆𝑅^𝑚₊⧵{0}. This gives, by (3.20) and (3.21), that for each 𝑖₀∈𝐼𝑝̂,0≤𝑝̂_𝑖0𝑇_𝑖0𝑥̂−𝜆𝑆_𝑖0𝑥̂−𝑐̂_𝑖0≤𝑝,̂𝑇𝑥̂−𝜆𝑆𝑥̂−𝑐̂=0,thatis,𝑇_𝑖0𝑥̂=𝜆𝑆_𝑖0𝑥̂+𝑐̂_𝑖0.(3.22) 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 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 say𝑓_𝜆,𝐹𝑝,𝑥≤𝑓_𝜆,𝐹𝑝,𝑥=0≤𝑓_𝜆,𝐹𝑝,𝑥,(𝑝,𝑥)∈𝑃×𝑋,(3.23) because for each (𝑝,𝑥)∈𝑃×𝑋, 𝑔_𝐹𝑝,𝑥=sup_𝑐∈𝐹𝑝,𝑇𝑥−𝑐𝑝,𝑆𝑥≤𝜆iff𝑓_𝜆,𝐹𝑝,𝑥=sup_𝑐∈𝐹𝑝,𝑇𝑥−𝜆𝑆𝑥−𝑐≤0,𝑔_𝐹𝑝,𝑥=sup_𝑐∈𝐹𝑝,𝑇𝑥−𝑐𝑝,𝑆𝑥=𝜆iff𝑓_𝜆,𝐹𝑝,𝑥=sup_𝑐∈𝐹𝑝,𝑇𝑥−𝜆𝑆𝑥−𝑐=0,𝑔_𝐹𝑝,𝑥=sup_𝑐∈𝐹𝑝,𝑇𝑥−𝑐𝑝,𝑆𝑥≥𝜆iff𝑓_𝜆,𝐹𝑝,𝑥=sup_𝑐∈𝐹𝑝,𝑇𝑥−𝜆𝑆𝑥−𝑐≥0.(3.24) In view of (3.5), we know that (3.23) is also equivalent to min_{𝑥∈𝑅𝑛}𝑓̂‌_𝜆,𝐹𝑝,𝑥=min_𝑥∈𝑋−𝑓_𝜆,𝐹𝑝,𝑥=0=−𝑓_𝜆,𝐹𝑝,𝑥=𝑓̂‌_𝜆,𝐹𝑝,𝑥,min_{𝑝∈𝑅𝑚}𝑓̃‌_𝜆,𝐹𝑝,𝑥=min_𝑝∈𝑃𝑓_𝜆,𝐹𝑝,𝑥=0=𝑓_𝜆,𝐹𝑝,𝑥=𝑓̃‌_𝜆,𝐹𝑝,𝑥.(3.25) Also by (3.5), we have epi𝑓̂‌_𝜆,𝐹(𝑝,⋅)={(𝑥,𝑎)∈𝑋×𝑅∶−𝑓_𝜆,𝐹(𝑝,𝑥)≤𝑎} and epi𝑓̃‌_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 epi𝑓̂‌_𝜆,𝐹(𝑝,⋅) (or epi𝑓̃‌_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↦𝑓̂‌_𝜆,𝐹(𝑝,𝑥)(𝑥∈𝑅^𝑛) and 𝑝↦𝑓̃‌_𝜆,𝐹(𝑝,𝑥)(𝑝∈𝑅^𝑚) are proper convex and l.s.c. with 𝑓̂‌_𝜆,𝐹𝑝,𝑥=𝑓̂‌^∗∗_𝜆,𝐹𝑝,𝑥(𝑥∈𝑅^𝑛),𝑓̃‌_𝜆,𝐹𝑝,𝑥=𝑓̃‌^∗∗_𝜆,𝐹𝑝,𝑥(𝑝∈𝑅^𝑚).(3.26) Applying Lemma 2.11 to the functions 𝑥↦𝑓̂‌_𝜆,𝐹(𝑝,𝑥)−⟨𝑞₀,𝑥⟩ on 𝑅^𝑛 with 𝑞₀=0 and 𝑝↦𝑓̃‌_𝜆,𝐹(𝑝,𝑥)−⟨𝑟₀,𝑝⟩ on 𝑅^𝑚 with 𝑟₀=0, and using (3.25) and (3.26), we conclude that (3.23) holds if and only if 𝑥∈𝜕𝑓̂‌^∗_𝜆,𝐹(𝑝,0) and 𝑝∈𝜕𝑓̃‌^∗_𝜆,𝐹(0,𝑥). Hence Theorem 3.3(2)(c) is also true.
(3) Finally we prove Theorem 3.3(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, 𝑐_0𝑖<𝑇_𝑖𝑥₀ for 𝑖=1,2,…,𝑚). Take 𝐹_𝑥0={𝑐₀}, then sup_𝑥∈𝑋𝑓_0,𝐹𝑥0(𝑝,𝑥)=sup_𝑥∈𝑋⟨𝑝,𝑇𝑥−𝑐₀⟩≥⟨𝑝,𝑇𝑥₀−𝑐₀⟩>0 for any 𝑝∈𝑃. Hence, 𝑣(𝑓_0,𝐹𝑥0)=inf_𝑝∈𝑃sup_𝑥∈𝑋𝑓_0,𝐹𝑥0(𝑝,𝑥)>0 because 𝑝↦sup_𝑥∈𝑋𝑓_0,𝐹𝑥0(𝑝,𝑥) is l.s.c. on 𝑃. This shows that ℂ^𝑚₊={𝐹∈𝔹^𝑚₊∶𝑣(𝑓_0,𝐹)>0} is nonempty. Moreover, Lemma 3.11(1) implies that 𝜆(𝐹) exists with 𝜆(𝐹)=𝑣(𝑔_𝐹) for any 𝐹∈ℂ^𝑚₊. Hence statement (a) follows.
(b) Let 𝐹_𝑖∈ℂ^𝑚₊(𝑖=1,2), then we have 𝜆(𝐹_𝑖)=𝑣(𝑔_𝐹𝑖)(𝑖=1,2). Suppose that (𝑝,𝑥)∈𝑃×𝑋. Since 𝐹_𝑖(𝑖=1,2) are compact, we can select 𝑐^𝑖∈𝐹_𝑖(𝑖=1,2) such that 𝑔_𝐹1(𝑝,𝑥)=sup_𝑐∈𝐹1(⟨𝑝,𝑇𝑥−𝑐⟩/⟨𝑝,𝑆𝑥⟩)=⟨𝑝,𝑇𝑥−𝑐¹⟩/⟨𝑝,𝑆𝑥⟩ and ‖𝑐¹−𝑐²‖=𝑑(𝑐¹,𝐹₂). This deduces that𝑔_𝐹1(𝑝,𝑥)=𝑝,𝑇𝑥−𝑐²+𝑐²−𝑐¹⟨𝑝,𝑆𝑥⟩≤𝑔_𝐹2(𝑝,𝑥)+sup_𝑝∈𝑃‖𝑝‖𝜀₀𝑑_𝐻𝐹₁,𝐹₂,(3.27) because ⟨𝑝,𝑇𝑥−𝑐²⟩/⟨𝑝,𝑆𝑥⟩≤𝑔_𝐹2(𝑝,𝑥), ‖𝑐¹−𝑐²‖=𝑑(𝑐¹,𝐹₂)≤𝑑_𝐻(𝐹₁,𝐹₂) and 𝜀₀=inf_{𝑝∈𝑃,𝑥∈𝑋}⟨𝑝,𝑆𝑥⟩ is positive. By taking minimax values for both sides of (3.27), we have 𝑣(𝑔_𝐹1)≤𝑣(𝑔_𝐹2)+(sup_𝑝∈𝑃‖𝑝‖/𝜀₀)𝑑_𝐻(𝐹₁,𝐹₂). Therefore, |𝜆(𝐹₁)−𝜆(𝐹₂)|=|𝑣(𝑔_𝐹1)−𝑣(𝑔_𝐹2)|≤(sup_𝑝∈𝑃‖𝑝‖/𝜀₀)𝑑_𝐻(𝐹₁,𝐹₂) because 𝑑_𝐻(𝐹₁,𝐹₂)=𝑑_𝐻(𝐹₂,𝐹₁), and the last lemma follows.

Proofs of Theorems 3.1–3.3
Proof. (i) For Theorem 3.1. (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 Theorem 3.2. (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 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.

4. Solvability Results to (1.2)

Let 𝐹={𝑐}(𝑐∈𝑅^𝑚₊), 𝜆∈𝑅₊, 𝑓_𝜆,𝐹=𝑓_𝜆,𝑐, 𝑔_𝐹=𝑔_𝑐, and 𝜆(𝐹)=𝜆(𝑐) (if exists), then∀(𝑝,𝑥)∈𝑃×𝑋,𝑓_𝜆,𝑐(𝑝,𝑥)=⟨𝑝,𝑇𝑥−𝜆𝑆𝑥−𝑐⟩,𝑔_𝑐(𝑝,𝑥)=⟨𝑝,𝑇𝑥−𝑐⟩⟨𝑝,𝑆𝑥⟩,(4.1) and the functions 𝑓̂‌_𝜆,𝑐(𝑝,𝑥),𝑓̃‌_𝜆,𝑐(𝑝,𝑥),𝑓̂‌^∗_𝜆,𝑐(𝑝,𝑞),𝑓̂‌^∗∗_𝜆,𝑐(𝑝,𝑥),𝑓̃‌^∗_𝜆,𝑐(𝑟,𝑥) and 𝑓̃‌^∗∗_𝜆,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(𝑓_𝜆,𝑐) exists and 𝑆(𝑓_𝜆,𝑐) is a nonempty convex compact subset of 𝑃×𝑋. Furthermore, 𝜆↦𝑣(𝑓_𝜆,𝑐) is continuous and strictly decreasing on 𝑅₊ with 𝑣(𝑓_+∞,𝑐)=lim_𝜆→+∞𝑣(𝑓_𝜆,𝑐)=−∞.
(2)𝑣(𝑔_𝑐) exists if and only if 𝑆(𝑔_𝑐)≠∅. Moreover, if 𝑣(𝑓_0,𝑐)>0, then 𝑣(𝑔_𝑐) exists and 𝑆(𝑔_𝑐) 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 inf_𝑝∈𝑃𝑓_𝜆,𝑐(𝑝,𝑥)≥0 if and only if inf_𝑝∈𝑃𝑔_𝑐(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(𝑓_𝜆,𝑐)≥0,(b)𝑓_𝜆,𝑐(𝑝,̂𝑥̂)≥0 for (𝑝,̂𝑥̂)∈𝑆(𝑓_𝜆,𝑐),(c)𝑣(𝑔_𝑐) exists with 𝑣(𝑔_𝑐)≥𝜆,(d)𝑆(𝑔_𝑐)≠∅ and 𝑔_𝑐(𝑝,̂𝑥̂)≥𝜆 for (𝑝,̂𝑥̂)∈𝑆(𝑔_𝑐).
(3) The following statements are equivalent.
(a)System (1.2) has at least one lower eigenvalue,(b)𝑣(𝑓_0,𝑐)>0,(c)𝑣(𝑔_𝑐) exists with 𝑣(𝑔_𝑐)>0,(d)𝑆(𝑔_𝑐)≠∅ and 𝑔_𝑐(𝑝,̂𝑥̂)>0 for (𝑝,̂𝑥̂)∈𝑆(𝑔_𝑐).

Theorem 4.3. (1)𝜆 exists if and only if one of the following statements is true.
(a)𝑣(𝑓_0,𝑐)>0.(b)𝑓_0,𝑐(𝑝,̂𝑥̂)>0 for (𝑝,̂𝑥̂)∈𝑆(𝑓_0,𝑐).(c)𝑣(𝑓_𝜆,𝑐)=0.(d)𝑣(𝑔_𝑐) exists with 𝑣(𝑔_𝑐)=𝜆.(e)𝑆(𝑔_𝑐)≠∅ and 𝑔_𝑐(𝑝,𝑥)=𝜆 for (𝑝,𝑥)∈𝑆(𝑔_𝑐). Where 𝜆=𝜆(𝑐)(>0) is the maximal lower eigenvalue to (1.2).
(2) If 𝑣(𝑓_0,𝑐)>0, or equivalently, if 𝑣(𝑔_𝑐) exists with 𝑣(𝑔_𝑐)>0, then
(a)𝑥 is an optimal eigenvector if and only if there exists 𝑝∈𝑃 with (𝑝,𝑥)∈𝑆(𝑔_𝑐) if and only if inf_𝑝∈𝑃𝑔_𝑐(𝑝,𝑥)=𝜆.(b)There exist 𝑥̂∈𝑋 and 𝑖₀∈{1,2,…,𝑚} such that 𝑇𝑥̂≥𝜆𝑆𝑥̂+𝑐 and 𝑇_𝑖0𝑥̂=𝜆𝑆_𝑖0𝑥̂+𝑐_𝑖0.(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∈𝜕𝑓̂‌^∗_𝜆,𝑐(𝑝,0) and 𝑝∈𝜕𝑓̃‌^∗_𝜆,𝑐(0,𝑥). Where 𝜕𝑓̂‌^∗_𝜆,𝑐(𝑝,0) and 𝜕𝑓̃‌^∗_𝜆,𝑐(0,𝑥) are the subdifferentials of 𝑓̂‌^∗_𝜆,𝑐(𝑝,𝑞) at 𝑞=0 and 𝑓̃‌^∗_𝜆,𝑐(𝑟,𝑥) at 𝑟=0, respectively.(d)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(𝑐₁)−𝜆(𝑐₂)|≤(sup_𝑝∈𝑃‖𝑝‖/𝜀₀)‖𝑐₁−𝑐₂‖(𝑐₁,𝑐₂∈ℂ₊), where 𝜀₀ 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 Theorems 3.1–3.3 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)

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.

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 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.
(a)There exists 𝜆≥1 with 𝑣(𝑓_𝜆,𝑐)=0.(b)𝑣(𝑔_𝑐) exists with 𝑣(𝑔_𝑐)≥1.(c)𝑆(𝑔_𝑐)≠∅ and 𝑔_𝑐(𝑝,𝑥)≥1 for (𝑝,𝑥)∈𝑆(𝑔_𝑐).

5.2. 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,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 inf_𝑝∈𝑃𝑓_𝜆,0(𝑝,𝑥)≥0 if and only if inf_𝑝∈𝑃𝑔₀(𝑝,𝑥)≥𝜆≥𝜆₀.
(2)𝜆 is a growth factor to (1.5) if and only if 𝜆≥𝜆₀ with 𝑣(𝑓_𝜆,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(𝑓_𝜆,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(𝑓_𝜆,0)=0 if and only if 𝑣(𝑔₀)=𝜆≥𝜆₀.
(5) If 𝑣(𝑔₀)=𝜆≥𝜆₀, then there exist 𝑥∈𝑋 and 𝑖₀∈{1,2,…,𝑚} such that 𝑇𝑥≥𝜆𝑆𝑥 and 𝑇_𝑖0𝑥=𝜆𝑆_𝑖0𝑥.
(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∈𝜕𝑓̂‌^∗_𝜆,0(𝑝,0) and 𝑝∈𝜕𝑓̃‌^∗_𝜆,0(0,𝑥).

5.3. Solvability to (1.6)

To present the solvability results to (1.6), we assume that(a)𝑋⊂int𝑅^𝑛₊isconvexcompact,(b)𝐴,𝐵∈𝑅^𝑛2₊with(𝐼−𝐴)𝑋⊂int𝑅^𝑛₊,𝐵𝑋⊂int𝑅^𝑛₊,(5.1) and define 𝑓_𝜆(𝑝,𝑥)=𝑓_𝜆,0(𝑝,𝑥) and 𝑔(𝑝,𝑥)=𝑔₀(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.

Theorem 5.3. If (5.1) holds and 𝑓_𝜆,𝑔 be defined by (5.1). Then one has the following. (1)There exists 𝜆>0 such that 𝑣(𝑓_𝜆)=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.