A Leontief-Type Input-Output Inclusion

Abstract

A Leontief-type input-output inclusion problem based on a set-valued consuming map is studied. By applying nonlinear analysis approach, in particular using the surjection and continuity technique with respect to set-valued maps, solvability and stability results with and without continuity assumption concerning this inclusion are obtained.

1. Introduction

In this paper, we study the solvability and stability of the following input-output inclusion:𝑥∈𝑋⊂𝑅^𝑛₊,𝑐∈(𝐼−𝐴)𝑥=𝑥−𝐴𝑥, (1.1) where 𝑅^𝑛₊ is the set of nonnegative vectors of the 𝑛-dimensional Euclidean space 𝑅^𝑛, 𝑋⊂𝑅^𝑛₊isanonemptyconvexcompactsubset,𝑐∈𝑅^𝑛₊isanetoutputvector,𝐼istheidentitymapfrom𝑅^𝑛toitself,and𝐴∶𝑋→2^𝑅𝑛+isaset-valuedmapfrom𝑋to𝑅^𝑛₊withnonemptyconvexcompactvalues(i.e.,foreach𝑥∈𝑋,𝐴𝑥isanonemptyandconvexcompactsubsetof𝑅^𝑛₊). (1.2)

The present study is essentially a continuation of the investigation initiated in [1–5] where the classical Leontief input-output model was briefly reviewed, several generalized Leontief input-output models were introduced, numerous key references were cited, and some arguments about the assumptions on 𝑋 and 𝐴 were made. For the necessary background material and preliminaries, the reader is referred to [1–5]. Here we will make use of the Rogalski-Cornet Theorem in [6] and the Rogalski-Cornet-type Theorem proved in [5] to prove several solvability and stability theorems with and without continuous conditions concerning 𝐴 for model (1.1). Obviously, for some 𝑐∈𝑅^𝑛₊, inclusion (1.1) may not have solutions. Specifically, if 𝑐 makes (1.1) solvable, then 𝑐∈𝐶(𝐴)=𝑅^𝑛₊⋂(𝐼−𝐴)𝑋,where(𝐼−𝐴)𝑋=⋃_𝑥∈𝑋(𝑥−𝐴𝑥). (1.3) While 𝐶(𝐴) gives us an expression for all possible 𝑐 for which (1.1) has solutions, it is required that all the information regarding 𝐴 is available. It is our intention in this paper to discover some conditions under which (1.1) has solutions for the situation that the information of 𝐴 is only available near the boundary of 𝑋. We also provide a stability analysis for the solution set in terms of closeness, upper semicontinuity and, upper hemicontinuity of certain related set-valued maps.

The paper is organized as follows. In the rest of this section, we review some necessary concepts and several useful results, which are used throughout this paper. In Section 2, we study (1.1) under the assumption that 𝐴 is upper semicontinuous. In Section 3, we recall a Rogalski-Cornet-type theorem appearing in [5], and use it to obtain three solvability and stability results. We give our concluding remarks in Section 4.

In the sequel, we use several classes of maps, including upper and lower semicontinuous (in short, u.s.c. and l.s.c.), upper hemicontinuous (in short, u.h.c.), continuous, and closed set-valued maps between Hausdorff topological (or Hausdorff locally convex) spaces, whose definitions and some other related concepts are given below and can also be found in [6–9].

Definition 1.1. Let 𝑈 and 𝑉 be two Hausdorff topological spaces and 𝐹∶𝑈→2^𝑉 a set-valued map from 𝑈 to 𝑉. The domain of 𝐹 is the set {𝑥∈𝑈∶𝐹𝑥≠∅} denoted by dom𝐹, and the graph of 𝐹 is the set {(𝑥,𝑦)∈𝑈×𝑉∶𝑥∈𝑈,𝑦∈𝐹𝑥} denoted by graph𝐹.(1)We say that 𝐹 is strict if dom𝐹=𝑈, and 𝐹 is closed if graph𝐹 is closed in 𝑈×𝑉.(2)We say that 𝐹 is u.s.c. at 𝑥⁰∈𝑈 if for any neighborhood 𝑁(𝐹𝑥⁰) of 𝐹𝑥⁰, there exists a neighborhood 𝑁(𝑥⁰) of 𝑥⁰ such that 𝐹(𝑁(𝑥⁰))⊆𝑁(𝐹𝑥⁰). 𝐹 is said to be u.s.c. if 𝐹 is u.s.c. at every point 𝑥∈𝑈.(3)We say that 𝐹 is l.s.c. at 𝑥⁰∈𝑈 if for any 𝑦⁰∈𝐹𝑥⁰ and any neighborhood 𝑁(𝑦⁰) of 𝑦⁰, there exists a neighborhood 𝑁(𝑥⁰) of 𝑥⁰ such that 𝐹𝑥⋂𝑁(𝑦₀)≠∅ for any 𝑥∈𝑁(𝑥⁰). 𝐹 is said to be l.s.c. if 𝐹 is l.s.c. at every point 𝑥∈𝑈.(4)If 𝑉 is a Hausdorff locally convex vector space, 𝑉^∗ is its dual and ⟨⋅,⋅⟩ is the duality paring on 𝑉^∗×𝑉. We say that 𝐹 is u.h.c. at 𝑥⁰∈dom𝐹 if for any 𝑝∈𝑉^∗, the function 𝑥↦𝜎^#(𝐹𝑥,𝑝)=sup_{𝑦∈𝐹𝑥}⟨𝑝,𝑦⟩ is upper semicontinuous (in short, u.s.c.) at 𝑥⁰. 𝐹 is said to be u.h.c. if it is u.h.c. at every point of dom𝐹.

We now recall a number of auxiliary results that will be needed in proving our main theorems. They are stated below as lemmas.

Lemma 1.2 (see [8]). Let 𝐹 be an u.s.c. set-valued map from a Hausdorff topological space 𝑈 to a Hausdorff topological space 𝑉 with closed values. Then 𝐹 is closed.

Lemma 1.3 (see [8]). Let 𝐹 be a closed set-valued map from a Hausdorff topological space 𝑈 to a compact Hausdorff topological space 𝑉. Then 𝐹 is u.s.c..

Lemma 1.4 (see [8]). Suppose that 𝑈 is a Hausdorff topological space, 𝑉 a Hausdorff locally convex vector space equipped with the weak topology 𝜎(𝑉,𝑉^∗), and 𝐹∶𝑈→2^{(𝑉,𝜎(𝑉,𝑉∗))} a set-valued map that is u.s.c. at 𝑥₀∈𝑈. Then 𝐹 is u.h.c. at 𝑥₀∈𝑈.

Remark 1.5. If 𝑉=𝑅^𝑛, then the weak topology 𝜎(𝑅^𝑛,𝑅^𝑛∗) on 𝑅^𝑛 coincides with the norm topology.

Lemma 1.6 (see [8]). Suppose that 𝑈 is a complete metric space, and 𝑉 a compact metric space. If 𝐹∶𝑈→2^𝑉 is a closed and strict set-valued map, then the subset of points at which 𝐹 is continuous is residual, that is, the interior of the discontinuous point set of 𝐹 is empty.

In order to use the Rogalski-Cornet Theorem and Rogalski-Cornet type Theorem to discuss the solvability and stability of (1.1), we need some further concepts.

Let 𝑈 be a Hausdorff locally convex vector space (𝑈^∗ its dual, ⟨⋅,⋅⟩ the duality paring on ⟨𝑈^∗,𝑈⟩), 𝑋 a subset of 𝑈, int𝑋 the interior of 𝑋, 𝜕𝑋=𝑋⧵int𝑋 the boundary of 𝑋, and 𝑆∶𝑋→2^𝑈 a set-valued map from 𝑋 to 𝑈. Let 𝑝∈𝑈^∗. The normal cone 𝑁_𝑋(𝑥) to 𝑋 at 𝑥∈𝑋, the supporting set 𝜕(𝑋,𝑝) of 𝑋, and the upper and lower supporting functions 𝑥↦𝜎^#(𝑆𝑥,𝑝) and 𝑥↦𝜎^♭(𝑆𝑥,𝑝) on 𝑋 are defined by 𝑁_𝑋(𝑥)={𝑝∈𝑈^∗∶⟨𝑝,𝑥⟩=𝜎^#(𝑋,𝑝)=sup_𝑦∈𝑋⟨𝑝,𝑦⟩},𝜕(𝑋,𝑝)={𝑥∈𝑋∶⟨𝑝,𝑥⟩=𝜎^♭(𝑋,𝑝)=inf_𝑦∈𝑋𝜎⟨𝑝,𝑦⟩},_#(𝑆𝑥,𝑝)=sup_{𝑦∈𝑆𝑥}⟨𝑝,𝑦⟩,𝜎^♭(𝑆𝑥,𝑝)=inf_{𝑦∈𝑆𝑥}⟨𝑝,𝑦⟩for𝑥∈𝑋. (1.4) We say that𝑆isoutwardif∀𝑝∈𝑈^∗,∀𝑥∈𝜕(𝑋,𝑝),⟨𝑝,𝑥⟩≥𝜎^𝑏(𝑆𝑥,𝑝),𝑆isinwardif∀𝑝∈𝑈^∗,∀𝑥∈𝜕(𝑋,𝑝),⟨𝑝,𝑥⟩≤𝜎^#(𝑆𝑥,𝑝),𝑆isstronglyinwardif∀𝑝∈𝑈^∗,∀𝑥∈𝜕(𝑋,𝑝),𝜎^#(𝑆𝑥,𝑝)≥𝜎^#(𝑋,𝑝). (1.5)

With the help of these concepts, the Rogalski-Cornet theorem can be stated as follows.

Theorem 1.7 (see [6] Rogalski-Cornet). Suppose that 𝑋 is a convex compact subset of 𝑈 supplied with the weak topology, and 𝑆 is an u.h.c. set-valued map from 𝑋 to 𝑈 with nonempty closed convex values. If 𝑆 is either outward or strongly inward, then for any 𝑦∈𝑋, there exists a solution 𝑥∈𝑋 to 𝑦∈𝑆𝑥, that is, 𝑆𝑋=⋃_𝑥∈𝑋𝑆𝑥⊇𝑋.

We use the following notations throughout this paper:|𝐴𝑥|_∞=sup_{𝑧∈𝐴𝑥}‖𝑧‖,𝜇_∞=sup_{𝑥∈𝜕𝑋}|𝐴𝑥|_∞,𝑑(𝑥,𝐹)=inf_𝑧∈𝐹‖𝑥−𝑧‖foreach𝑥∈𝑅^𝑛,𝐹⊆𝑅^𝑛with𝐹≠∅. (1.6)

2. Theorems With u.h.c. Condition

In this section, we assume that 𝐴 of (1.1) is u.s.c. on 𝑋.

Associating this assumption with (1.2), we can show that 𝐴𝑋=⋃_𝑥∈𝑋𝐴𝑥 and 𝐴(𝜕𝑋)=⋃_{𝑥∈𝜕𝑋}𝐴𝑥 are compact. Therefore, 𝐴 is a strict and u.s.c. set-valued map from 𝑋 (a convex compact subset of 𝑅^𝑛₊) to 𝐴𝑋 (a compact subset of 𝑅^𝑛₊) with convex compact values, and 𝜇_∞ defined by (1.6) is finite. Moreover, by Lemmas 1.2 and 1.4 and Remark 1.5, 𝐴 is also closed and u.h.c. on 𝑋. Suppose that 𝐺^𝐴 is a set-valued map from 𝐶(𝐴) to 𝑋 defined by 𝐶(𝐴)=𝑅^𝑛₊𝐺(𝐼−𝐴)𝑋,_𝐴𝑐={𝑥∈𝑋∶𝑐∈𝑥−𝐴𝑥}for𝑐∈𝐶(𝐴). (2.1) Then we have the following results.

Theorem 2.1. If 𝐶(𝐴)≠∅, then 𝐶(𝐴) is compact and 𝐺^𝐴 is closed, u.s.c., and u.h.c., and the subset of points at which 𝐺^𝐴 is continuous is residual.

Proof. Since 𝑋 and 𝐴𝑋 are compact, we know that (𝐼−𝐴)𝑋 is bounded, so is 𝐶(𝐴). Suppose that {𝑐_𝑘∶𝑘≥1}⊆𝐶(𝐴) and 𝑐_𝑘→𝑐₀∈𝑅^𝑛₊ as 𝑘→∞. Then there exist {𝑥_𝑘∶𝑘≥1}⊂𝑋 such that 𝑐_𝑘∈𝑥_𝑘−𝐴𝑥_𝑘. Since {𝑥_𝑘∶𝑘≥1} has a convergent subsequence, we may assume that 𝑥_𝑘→𝑥₀∈𝑋 as 𝑘→∞. As 𝐴 is closed, we then obtain 𝑐₀∈𝑥₀−𝐴𝑥₀, which shows that 𝐶(𝐴) is closed, and hence, also compact.
For the continuity results of 𝐺^𝐴, according to Lemmas 1.3–1.6 and Remark 1.5, we only need to prove that 𝐺^𝐴 is closed because 𝐶(𝐴) and 𝑋 are compact. Suppose that {(𝑐_𝑘,𝑥_𝑘)∶𝑘≥1}⊆graph𝐺^𝐴 such that (𝑐_𝑘,𝑥_𝑘)→(𝑐,𝑥) as 𝑘→∞. Then 𝑐_𝑘∈𝑥_𝑘−𝐴𝑥_𝑘(𝑘≥1). Since 𝐴 is closed, 𝑐_𝑘→𝑐∈𝐶(𝐴) and 𝑥_𝑘→𝑥∈𝑋, we get 𝑐∈𝑥−𝐴𝑥. Hence, 𝐺^𝐴 is closed and has all the continuity results stated in the theorem.

Next, we use Theorem 1.7 to obtain two solvability and stability results by means of the interior and exterior approximation methods used in [2] (three approximation methods have been used to study the single-valued input-output equation in [2]). Besides the assumptions that 𝑋 is convex compact and 𝐴 is strict and u.s.c. with convex compact values, we further assume that int𝑋≠∅ in this section. We have the following lemma. The proof is straightforward and hence omitted.

Lemma 2.2. Let 𝑌⊆𝑅^𝑛 be a nonempty convex set, 𝑦∈𝑌, and 𝜕𝑌=𝑌⧵int𝑌 the boundary of 𝑌. If 𝑝≠0 and 𝑦∈𝜕(𝑌,𝑝) (or 𝑝∈𝑁_𝑌(𝑦)), then 𝑦∈𝜕𝑌.

2.1. Interior Approximation Method

Define a subset 𝑋_∞ of 𝑋 and a set-valued map 𝐹₁ from 𝑋_∞ to 𝑋 by𝑋_∞=𝑋[𝜇_∞,+∞)^𝑛,where[𝜇_∞,+∞)^𝑛={(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑅^𝑛₊∶𝑥_𝑖≥𝜇_∞𝐹},₁𝑐={𝑥∈𝑋∶∃Δ𝑐∈𝑅^𝑛s.t.𝑐+Δ𝑐∈𝑥−𝐴𝑥,‖Δ𝑐‖≤𝜇_∞},for𝑐∈𝑋_∞. (2.2) Then we have the first solvability and stability result for (1.1) as follows.

Theorem 2.3. Suppose 𝑡ℎ𝑎𝑡𝑋_∞≠∅ and 𝑐∈𝑋_∞. Then there exist 𝑥∈𝑋, Δ𝑐∈𝑅^𝑛 such that 𝑐+Δ𝑐∈𝑥−𝐴𝑥 and ‖Δ𝑐‖≤𝜇_∞. Moreover, 𝐹₁ defined by (2.2) is closed, u.s.c., and u.h.c., and the subset of points at which 𝐹₁ is continuous is residual.

Proof. Since 𝐴𝑥 is compact for 𝑥∈𝑋 and 𝐴 is u.s.c., it is easy to see that for 𝜀>0, 𝐵(𝐴𝑥,𝜀)={𝑦∈𝑅^𝑛∶𝑑(𝑦,𝐴𝑥)<𝜀} is an open neighborhood of 𝐴𝑥, and that for each 𝑘≥1 and 𝑥∈𝜕𝑋, there exists a neighborhood 𝑈(𝑥) of 𝑥 with 𝐴(𝑈(𝑥)∩𝑋)⊆𝐵(𝐴𝑥,1/𝑘). As 𝜕𝑋 is compact and 𝜕𝑋⊆⋃_{𝑥∈𝜕𝑋}𝑈(𝑥), there exist {𝑧^𝑖∶1≤𝑖≤𝑚}⊆𝜕𝑋 such that 𝜕𝑋⊆⋃^𝑚_𝑖=1𝑈_𝑖,𝐴(𝑈_𝑖∩𝑋)⊆𝐵(𝐴𝑧^𝑖,1𝑘),for𝑖=1,2,…,𝑚, (2.3) where 𝑈_𝑖=𝑈(𝑧^𝑖). Let 𝑌=𝑋⧵(⋃^𝑚_𝑖=1𝑈_𝑖). Then it is a closed subset of 𝑋 with 𝜕𝑋∩𝑌=∅. Let 𝑑=𝑑(𝜕𝑋,𝑌)=inf_{𝑥∈𝜕𝑋,𝑦∈𝑌}𝑑(𝑥,𝑦). Then we have 𝑑>0. Otherwise, there exist sequences {𝑥^𝑗∶𝑗≥1}⊆𝜕𝑋 and {𝑦^𝑗∶𝑗≥1}⊆𝑌, such that 𝑑(𝑥^𝑗,𝑦^𝑗)→0 as 𝑗→∞. Since 𝑋 is compact, so are 𝜕𝑋 and 𝑌, which imply that there exist convergent subsequences of {𝑥^𝑗∶𝑗≥1} and {𝑦^𝑗∶𝑗≥1}. Without loss of generality, we may assume that {𝑥^𝑗∶𝑗≥1} and {𝑦^𝑗∶𝑗≥1} are convergent to the same point 𝑥 as 𝑗→∞. Since 𝜕𝑋 and 𝑌 are closed, we obtain 𝑥∈𝜕𝑋∩𝑌, a contradiction. Therefore, 𝑑>0. Let 𝑈₀=⋃_𝑦∈𝑌𝐵(𝑦,𝑑/2). Then it is easy to see that 𝑈₀ is an open subset of 𝑋, 𝑋⧵𝑈₀⊆⋃^𝑚_𝑖=1𝑈_𝑖 and 𝐴(𝑋⧵𝑈₀)⊆⋃^𝑚_𝑖=1𝐵(𝐴𝑧^𝑖,1𝑘). (2.4)
For each 𝑘≥1, by the truncation technique of generalized functions in partial differential equations [10], there is a continuous function 𝑔_𝑘 from 𝑋 to 𝑅₊ such that 0≤𝑔_𝑘(𝑥)≤1(𝑥∈𝑋),𝑔_𝑘|_𝑈0=1, supp𝑔_𝑘≡{𝑥∈𝑋∶𝑔_𝑘(𝑥)≠0}⊆ int𝑋,andhence𝑔_𝑘|_𝜕𝑋=0. Here 𝑀 denotes the closure of 𝑀. Define 𝐴_𝑘∶𝑋→2^𝑅𝑛 by 𝐴_𝑘𝑥=𝑔_𝑘(𝑥)𝐴𝑥,for𝑥∈𝑋. (2.5)
We claim that for each 𝑘≥1, 𝐴_𝑘 is a closed, u.s.c., and u.h.c. set-valued map with nonempty convex compact values. Indeed, 𝐴_𝑘𝑥 is clearly a convex compact subset of 𝑅^𝑛 for each 𝑥∈𝑋. Assume that (𝑥^𝑗,𝑦^𝑗)∈graph𝐴_𝑘 (i.e., 𝑦^𝑗∈𝑔_𝑘(𝑥^𝑗)𝐴𝑥^𝑗) such that (𝑥^𝑗,𝑦^𝑗)→(𝑥⁰,𝑦⁰) as 𝑗→∞. Then 𝑥^𝑗→𝑥⁰,𝑦^𝑗→𝑦⁰(𝑗→∞), and for each 𝑗≥1 there exists 𝑤^𝑗∈𝐴𝑥^𝑗 with 𝑦^𝑗=𝑔_𝑘(𝑥^𝑗)w^𝑗. Since 𝐴𝑋 is compact, we may suppose that 𝑤^𝑗→𝑤⁰ as 𝑗→∞. By letting 𝑗→∞ and using the fact that 𝐴 is closed, we get 𝑤⁰∈𝐴𝑥⁰ and 𝑦⁰=𝑔_𝑘(𝑥⁰)𝑤⁰∈𝑔_𝑘(𝑥⁰)𝐴𝑥⁰, that is, (𝑥⁰,𝑦⁰)∈graph𝐴_𝑘. Hence, 𝐴_𝑘 is closed. On the other hand, by (2.5), it is easy to see that 𝐴_𝑘(𝑋) is compact. Hence by Lemmas 1.3 and 1.4 and Remark 1.5, 𝐴_𝑘 is also u.s.c. and u.h.c..
Since 𝜎^#((𝐼−𝐴_𝑘)𝑥,𝑝)=⟨𝑝,𝑥⟩+𝜎^#(𝐴_𝑘𝑥,−𝑝) for 𝑝∈𝑅^𝑛 and 𝑥∈𝑋, it follows that for each 𝑝∈𝑅^𝑛, the function 𝑥→𝜎^#((𝐼−𝐴_𝑘)𝑥,𝑝) is u.s.c. on 𝑋. And so, by Definition 1.1, 𝑆=𝐼−𝐴_𝑘 is u.h.c.
Assume that 𝑝∈𝑅^𝑛 and 𝑥∈𝜕(𝑋,𝑝). If 𝑝=0, then ⟨𝑝,𝑥⟩=0=𝜎^𝑏(𝑥−𝐴_𝑘𝑥,𝑝). If 𝑝≠0, then by Lemma 2.2, 𝑥∈𝜕𝑋, which further implies by 𝑔_𝑘|_𝜕𝑋=0 that 𝐴_𝑘𝑥={0} and hence ⟨𝑝,𝑥⟩=𝜎^𝑏(𝑥−𝐴_𝑘𝑥,𝑝). Therefore, 𝑆=𝐼−𝐴_𝑘 satisfies the outward condition stated in (1.5). In view of Theorem 1.7, we have (𝐼−𝐴_𝑘)𝑋⊇𝑋. Hence for each 𝑐∈𝑋, there exists 𝑥^𝑘∈𝑋 such that 𝑐∈𝑥^𝑘−𝐴_𝑘𝑥^𝑘. (2.6) If 𝑥^𝑘∈𝑈₀, by (2.5) and 𝑔_𝑘|_𝑈0=1, we have 𝑐∈𝑥^𝑘−𝐴𝑥^𝑘. Set Δ𝑐^𝑘=0, then we get 𝑐+Δ𝑐^𝑘∈𝑥^𝑘−𝐴𝑥^𝑘,‖Δ𝑐^𝑘‖≤𝜇_∞+1𝑘. (2.7) If 𝑥^𝑘∈𝑋⧵𝑈₀, by (2.5) and (2.6) there exists 𝑤^𝑘∈𝐴𝑥^𝑘 such that 𝑐=𝑥^𝑘−𝑔_𝑘(𝑥^𝑘)𝑤^𝑘. Let Δ𝑐^𝑘=(𝑔_𝑘(𝑥^𝑘)−1)𝑤^𝑘. Then 𝑐+Δ𝑐^𝑘=𝑥^𝑘−𝑤^𝑘∈𝑥^𝑘−𝐴𝑥^𝑘 and Δ𝑐^𝑘∈(𝑔_𝑘(𝑥^𝑘)−1)𝐴𝑥^𝑘. By virtue of (2.4), (2.5), and (1.6), we see that (2.7) is also true.
Now, we obtain two sequences {𝑥^𝑘∶𝑘≥1}⊆𝑋 and {Δ𝑐^𝑘∶𝑘≥1}⊆𝑅^𝑛 with ‖Δ𝑐^𝑘‖≤𝜇_∞+(1/𝑘). Since 𝑋 is compact and {Δ𝑐^𝑘∶𝑘≥1} is bounded, we may assume that 𝑥^𝑘→𝑥∈𝑋 and Δ𝑐^𝑘→Δ𝑐 as 𝑘→∞. Combining this with (2.7) and also using the fact that 𝐴 is closed, we obtain that 𝑐+Δ𝑐∈𝑥−𝐴𝑥and‖Δ𝑐‖≤𝜇_∞.
Next, we prove the second part of the theorem. Since 𝑋_∞ and 𝑋 are compact, also by Lemmas 1.3–1.6 and Remark 1.5, we only need to prove that 𝐹₁ is closed. Assume that {(𝑐^𝑘,𝑥^𝑘)∶𝑘≥1}⊆graph𝐹₁ with (𝑐^𝑘,𝑥^𝑘)→(𝑐,𝑥)(𝑘→∞). Then for each 𝑘≥1 there exists Δ𝑐^𝑘∈𝑅^𝑛 such that 𝑐^𝑘+Δ𝑐^𝑘∈𝑥^𝑘−𝐴𝑥^𝑘,‖Δ𝑐^𝑘‖≤𝜇_∞. (2.8) We may suppose that Δ𝑐^𝑘→Δ𝑐 as 𝑘→∞. This implies by (2.8) and the closeness of 𝐴 that 𝑐+Δ𝑐∈𝑥−𝐴𝑥 and ‖Δ𝑐‖≤𝜇_∞. Hence, 𝐹₁ is closed and has all the continuity properties stated in the theorem. This completes the proof.

Remark 2.4. In the proof of the theorem, the condition 𝑐∈𝑋_∞ is not used. We impose this requirement in order to make sure that 𝑐+Δ𝑐∈𝑅^𝑛₊.

2.2. Exterior Approximation Method

Define a function 𝛿_∞(𝑥) on 𝑋 and a set-valued map 𝐹₂ from 𝑋_∞ to 𝑋 by𝛿_∞(𝑥)=0if𝑥∈int𝑋,|𝐴𝑥|_∞𝐹if𝑥∈𝜕𝑋,₂𝑐={𝑥∈𝑋∶∃Δ𝑐∈𝑅^𝑛s.t.𝑐+Δ𝑐∈𝑥−𝐴𝑥,‖Δ𝑐‖≤𝛿_∞(𝑥)}for𝑐∈𝑋_∞, (2.9) where 𝑋_∞ is defined as in (2.2). Then we have the following result.

Theorem 2.5. Suppose that 𝑋_∞≠∅ and 𝑐∈𝑋_∞. Then there exist 𝑥∈𝑋, Δ𝑐∈𝑅^𝑛 such that 𝑐+Δ𝑐∈𝑥−𝐴𝑥, and ‖Δ𝑐‖≤𝛿_∞(𝑥). Furthermore, 𝐹₂ defined by (2.9) is closed, u.s.c. and u.h.c., and the subset of points at which 𝐹₂ is continuous is residual.

Proof. Let 𝐵(𝑋,𝜀)={𝑥∈𝑅^𝑛∶𝑑(𝑥,𝑋)≤𝜀} for 𝜀>0. For each 𝑘≥1, let 𝐵_𝑘(𝑋)=𝐵(𝑋,1/𝑘) and let 𝑃_𝑘 be the projection from 𝐵_𝑘(𝑋) to 𝑋, that is, 𝑃_𝑘𝑥∈𝑋 such that ‖𝑃_𝑘𝑥−𝑥‖=inf_𝑦∈𝑋‖𝑦−𝑥‖ for 𝑥∈𝐵_𝑘(𝑋). It is easy to see that 𝐵_𝑘(𝑋) is a convex compact set with nonempty interior, 𝑃_𝑘𝑥=𝑥 if 𝑥∈𝑋 and 𝑃_𝑘𝑥∈𝜕𝑋 if 𝑥∈𝐵_𝑘(𝑋)⧵𝑋, and ‖𝑃_𝑘𝑥−x‖≤1/𝑘 for all 𝑥∈𝐵_𝑘(𝑋).
As in the proof of Theorem 2.3, we assume that 𝑔_𝑘 is a continuous function from 𝐵_𝑘(𝑋) to 𝑅₊ with compact support set such that 0≤𝑔_𝑘(𝑥)≤1for𝑥∈𝐵_𝑘(𝑋),𝑔_𝑘|_𝑋=1,𝑔_𝑘|_{𝜕(𝐵𝑘(𝑋))}=0. (2.10) Let 𝐴_𝑘 be a set-valued map from 𝐵_𝑘(𝑋) to 𝑅^𝑛 defined by 𝐴_𝑘𝑥=𝑔_𝑘(𝑥)𝐴(𝑃_𝑘𝑥)for𝑥∈𝐵_𝑘(𝑋). (2.11)
Utilizing the similar method as in the proof of Theorem 2.3, we can show that 𝐴_𝑘 is a closed, u.s.c., and u.h.c. set-valued map with nonempty convex compact values and satisfies 𝐴_𝑘|_𝑋=𝐴,𝐴_𝑘|_{𝜕(𝐵𝑘(𝑋))}={0}. (2.12) In fact, if (𝑥^𝑚,𝑦^𝑚)∈graph𝐴_𝑘 with (𝑥^𝑚,𝑦^𝑚)→(𝑥⁰,𝑦⁰)(𝑚→∞), then 𝑥⁰∈𝐵_𝑘𝑋, 𝑃_𝑘𝑥^𝑚→𝑃_𝑘𝑥⁰∈𝑋(𝑚→∞), and for each 𝑚≥1, there exists 𝑤^𝑚∈𝐴(𝑃_𝑘𝑥^𝑚) such that 𝑦^𝑚=𝑔_𝑘(𝑥^𝑚)𝑤^𝑚. Since 𝐴(𝑃_𝑘(𝐵_𝑘(𝑋)))⊆𝐴𝑋 and 𝐴𝑋 is compact, we may suppose that 𝑤^𝑚→𝑤⁰(𝑚→∞). By letting 𝑚→∞, from the closeness of 𝐴, Lemmas 1.3 and 1.4, and Remark 1.5, we conclude that 𝑤⁰∈𝐴(𝑃_𝑘𝑥⁰), 𝑦⁰=𝑔_𝑘(𝑥⁰)𝑤⁰∈𝐴_𝑘𝑥⁰, that is, (𝑥⁰,𝑦⁰)∈graph𝐴_𝑘, and thus 𝐴_𝑘 is closed, u.s.c. and u.h.c..
Also as in the proof of Theorem 2.3, we can prove that 𝐼−𝐴_𝑘 satisfies the outward condition stated in (1.5). (In fact, assume that 𝑝∈𝑅^𝑛 and 𝑥∈𝜕(𝐵_𝑘(𝑋),𝑝). If 𝑝=0, then ⟨𝑝,𝑥⟩=0=𝜎^𝑏(𝑥−𝐴_𝑘𝑥,𝑝). If 𝑝≠0, then by Lemma 2.2, 𝑥∈𝜕(𝐵_𝑘(𝑋)), which implies by (2.12) that 𝐴_𝑘𝑥={0}, and hence ⟨𝑝,𝑥⟩=𝜎^𝑏(𝑥−𝐴_𝑘𝑥,𝑝).)By Theorem 1.7,we obtain (𝐼−𝐴_𝑘)𝐵_𝑘(𝑋)⊇𝐵_𝑘(𝑋)⊇𝑋(⊇X_∞). (2.13) Combining this with (2.11), we know that for each 𝑐∈𝑋, there exists 𝑥^𝑘∈𝐵_𝑘(𝑋) such that 𝑐∈𝑥^𝑘−𝑔_𝑘(𝑥^𝑘)𝐴(𝑃_𝑘𝑥^𝑘). (2.14)
Set 𝑥^𝑘=𝑃_𝑘𝑥^𝑘. Then 𝑥^𝑘∈𝑋. If there is 𝑘≥1 such that 𝑥^𝑘∈int𝑋, then from (2.10), (2.12), and (2.14) we have 𝑥^𝑘=𝑥^𝑘 and 𝑐∈𝑥^𝑘−𝐴𝑥^𝑘. Set 𝑥=𝑥^𝑘 and Δ𝑐=0. Then 𝑐+Δ𝑐∈𝑥−𝐴𝑥,‖Δ𝑐‖=0≤𝛿_∞(𝑥). (2.15) If for all 𝑘≥1, 𝑥^𝑘∈𝐵_𝑘(𝑋)⧵int𝑋, then 𝑥^𝑘=𝑃_𝑘𝑥^𝑘∈𝜕𝑋 and ‖𝑥^𝑘−𝑥^𝑘‖≤1/𝑘. By (2.14), there exists 𝑧^𝑘∈𝐴𝑥^𝑘 such that 𝑐=𝑥^𝑘−𝑔(𝑥^𝑘)𝑧^𝑘. Let Δ𝑐^𝑘=𝑥^𝑘−𝑥^𝑘+[𝑔(𝑥^𝑘)−1]𝑧^𝑘. Then we obtain (a)𝑐+Δ𝑐^𝑘=𝑥^𝑘−𝑧^𝑘∈𝑥^𝑘−𝐴𝑥^𝑘,(b)‖Δ𝑐^𝑘‖≤‖𝑥^𝑘−𝑥^𝑘‖+|𝑔(𝑥^𝑘)−1|‖𝑧^𝑘1‖≤𝑘+|𝐴𝑥^𝑘|_∞. (2.16)
Since X is compact and {Δ𝑐^𝑘∶𝑘≥1} is bounded, we may assume that 𝑥^𝑘→𝑥 and Δ𝑐^𝑘→Δ𝑐 as 𝑘→∞. This implies by 𝑥^𝑘∈𝜕𝑋(𝑘≥1) that 𝑥∈𝜕𝑋. On the other hand, if 𝜀>0 and 𝐵(𝐴𝑥,𝜀) is a 𝜀-neighborhood of 𝐴𝑥, then there exists a neighborhood 𝑈(𝑥) of 𝑥 such that 𝐴(𝑈(𝑥)⋂𝑋)⊆𝐵(𝐴𝑥,𝜀) because 𝐴 is u.s.c., and thus for any 𝑥^′∈𝑈(𝑥)∩𝑋, |𝐴𝑥^′|_∞≤|𝐴𝑥|_∞+𝜀. Hence the function 𝑥↦|𝐴𝑥|_∞isuppersemicontinuous(i.e.,u.s.c.)on𝑋. (2.17) By letting 𝑘→∞ and using the closeness of 𝐴, from (2.16) and (2.17), we get 𝑐+Δ𝑐∈𝑥−𝐴𝑥and‖Δ𝑐‖≤|𝐴𝑥|_∞=𝛿_∞(𝑥). Combining this with (2.15), we conclude that the first part of the theorem is true.
As in the proof of Theorem 2.3, we also need to verify that graph𝐹₂ is closed. Suppose that {(𝑐^𝑘,𝑥^𝑘)∶𝑘≥1}⊆graph𝐹₂ and (𝑐^𝑘,𝑥^𝑘)→(𝑐,𝑥) as 𝑘→∞. Then for each 𝑘≥1 there exists Δ𝑐^𝑘∈𝑅^𝑛 such that (a)𝑐^𝑘+Δ𝑐^𝑘∈𝑥^𝑘−𝐴𝑥^𝑘,(b)‖Δ𝑐^𝑘‖≤𝛿_∞(𝑥^𝑘). (2.18) We may suppose Δ𝑐^𝑘→Δ𝑐 as 𝑘→∞. By (2.18)(a) we have 𝑐+Δ𝑐∈𝑥−𝐴𝑥. On the other hand, by (2.9) and (2.17), it is easy to see that the function 𝑥↦𝛿_∞(𝑥) is also u.s.c. on 𝑋. Combining this with (2.18)(b), we obtain ‖Δ𝑐‖≤𝛿_∞(𝑥). Therefore, 𝐹₂ is closed and has the continuity results stated in the theorem.

Remark 2.6. As the discussion in Remark 2.4, the assumption that 𝑐∈𝑋_∞ is used to make sure that 𝑐+Δ𝑐 is a nonnegative net output vector.

Remark 2.7. Another approach to obtain Theorem 2.5 is to use the so-called neighborhood approximation method discussed in [2]. Indeed, if we define the set-valued map 𝑇 from 𝑋 to 𝑅^𝑛 by 𝑇𝑥=𝐴𝑥+𝐵(𝑥,𝛿_∞(𝑥)), then we can prove that 𝑇 is also an upper hemicontinuous set-valued map with convex compact values and satisfies the outward condition stated in (1.5). Hence, by Theorem 1.7, we obtain that 𝑇(𝑋)⊇𝑋⊇𝑋_∞, which can be used to prove Theorem 2.5.

3. Theorems Without Continuity Assumption

In Theorem 1.7 the associated set-valued map is assumed to be u.h.c.. Recently, a Rogalski-Cornet-type theorem without any continuity conditions was proved in [5]. As a simple application, it is briefly applied to (1.1) in [5]. In this section, we develop more solvability and stability results for (1.1). We first review this theorem in the framework of 𝑛-dimensional Euclidean space.

Let 𝑋 be a nonempty convex compact subset of 𝑅^𝑛, 𝑆∶𝑋→2^𝑅𝑛 a set-valued map from 𝑋 to 𝑅^𝑛 such that 𝑆𝑥 is a nonempty closed convex subset of 𝑅^𝑛 for each 𝑥∈𝑋, and 𝑆𝑋=⋃_𝑥∈𝑋𝑆𝑥 its range. For each 𝑝∈𝑅^𝑛, 𝑐∈𝑋, and 𝜀≥0, we set𝑌(𝑆,𝑝,𝑐,𝜀)={𝑥∈𝑋∶𝜎^#(𝑆𝑥−𝑐,𝑝)+𝜀≥0},𝑌(𝑆,𝑝,𝑐)=𝑌(𝑆,𝑝,𝑐,0). (3.1) Let 𝑌(𝑆) be the set of all 𝑐∈𝑋 such that(a)𝑌(𝑆,𝑝,𝑐)isclosed,∀𝑝∈𝑅^𝑛,(b)𝑥∈𝑌(𝑆,𝑝,𝑐)if𝑥∈𝑋,𝑝∈𝑁_𝑋(𝑥). (3.2) Let 𝑌(𝑆) be a subset of 𝑌(𝑆) and 𝐹^𝑆 a set-valued map from 𝑌(𝑆) to 𝑋 defined by(a)𝑌(𝑆)={𝑐∈𝑌(𝑆)∶∀𝑝∈𝑅^𝑛,∀𝜀≥0,𝑌(𝑆,𝑝,𝑐,𝜀)isclosed},(b)𝐹^𝑆𝑐={𝑥∈𝑋∶𝑐∈𝑆𝑥}=𝑋∩𝑆⁻¹𝑐,for𝑐∈𝑌(𝑆). (3.3) Then the following Rogalski-Cornet-type theorem was proved in [5].

In the sequel, we assume that 𝑋 in Lemma 3.1 is precisely the same subset 𝑋 of 𝑅^𝑛₊ as stated in (1.2) and use Lemma 3.1 to obtain three results for (1.1).

Lemma 3.1. (i) (see [5, Theorem 2.8]) If 𝑌(𝑆)≠∅, then 𝑌(𝑆)⊆𝑆𝑋.
(ii) (see [5, Theorem 2.12]) If 𝑌(𝑆)≠∅, then 𝑌(𝑆) is compact. Moreover, 𝐹^𝑆 defined by (3.3) is closed, u.s.c. and u.h.c., and the subset of points at which 𝐹^𝑆 is continuous is residual.

Remark 3.2. (i) Two counter-examples were presented in [5] to show that the set-valued map 𝑆 in Lemma 3.1 does not need the u.s.c., l.s.c., and u.h.c. conditions.
(ii) In case 𝑆 is u.h.c., then for each 𝑝∈𝑅^𝑛, 𝑐∈𝑅^𝑛, and 𝜀≥0, the function 𝑓(𝑥)=𝜎^#(𝑆𝑥−𝑐,𝑝)+𝜀=𝜎^#(𝑆𝑥,𝑝)−⟨𝑝,𝑐⟩+𝜀 is u.s.c. on 𝑋, and thus the upper section 𝑌(𝑆,𝑝,𝑐,𝜀)={𝑥∈𝑋∣𝑓(𝑥)≥0} is closed. This proves that 𝑌(𝑆,𝑝,𝑐,𝜀) is closed for all 𝑝∈𝑅^𝑛, 𝑐∈𝑅^𝑛, and 𝜀≥0 provided 𝑆 is u.h.c..

3.1. The First Result

The first result is similar to Theorem 2.5. However, the u.s.c. assumption concerning 𝐴 introduced in Section 2 has been removed. Let 𝜇_∞,𝑋_∞,𝛿_∞(𝑥), and 𝐹₂ be defined as in (1.6), (2.2), and (2.9), respectively. Define𝐴_∞∶𝑋→2^𝑅𝑛by𝐴_∞𝑥=𝐵(0,𝛿_∞(𝑥))for𝑥∈𝑋. (3.4) Applying Lemma 3.1 to 𝑆=𝐼−(𝐴+𝐴_∞), we have the following results.

Theorem 3.3. Assume that 𝜇_∞ is finite, 𝑋_∞≠∅ and 𝑆=𝐼−(𝐴+𝐴_∞).(i)Let 𝑐⁰∈𝑋_∞. If (3.2)(a) holds for 𝑐⁰, then 𝐹₂𝑐⁰≠∅, that is, ∃𝑥⁰∈𝑋,∃Δ𝑐⁰∈𝑅^𝑛s.t.𝑐⁰+Δ𝑐⁰∈𝑥⁰−𝐴𝑥⁰,‖Δ𝑐‖≤𝜎_∞(𝑥⁰). (3.5)(ii)If 𝑌(𝑆,𝑝,𝑐,𝜀) defined by (3.1) is closed for all 𝑝∈𝑅^𝑛, 𝑐∈𝑋_∞, and 𝜀≥0, then 𝐹₂ defined by (2.9) is closed, u.s.c., and u.h.c., and the subset of points at which 𝐹₂ is continuous is residual.(iii)If 𝐴 is u.s.c., then 𝐹₂ has the same properties as stated in (ii).

Proof. (i) Since 𝐴 has convex compact values, 𝑋 is compact and 𝜇_∞ is finite; by (3.4) it is easy to see that 𝑆=𝐼−(𝐴+𝐴_∞) is a strict set-valued map from 𝑋 to 𝑅^𝑛 with convex compact values. We shall prove 𝑐⁰∈𝑌(𝑆) by showing 𝑐⁰ satisfying (3.2). By the assumption of (i), it is sufficient to prove that (3.2)(b) holds for 𝑐⁰. Indeed, we can prove that (3.2)(b) holds for any 𝑐∈𝑋. Suppose that 𝑥∈𝑋,𝑝∈𝑁_𝑋(𝑥), and 𝑐∈𝑋. If 𝑝=0, then 𝜎^#(𝑆𝑥−𝑐,0)=0. If 𝑝≠0, then 𝑥∈𝜕𝑋 by Lemma 2.2. Hence by (2.9), 𝛿_∞(𝑥)=|𝐴𝑥|_∞, 0∈𝐴𝑥+𝐵(0,|𝐴𝑥|_∞), 𝜎^𝑏(𝐴𝑥+𝐴_∞𝑥,𝑝)≤0, and thus ⟨𝑝,𝑥⟩=sup_𝑦∈𝑋⟨𝑝,𝑦⟩≥⟨𝑝,𝑐⟩≥𝜎^𝑏(𝐴𝑥+𝐴_∞𝑥,𝑝)+⟨𝑝,𝑐⟩, that is, 𝜎^#(𝑆𝑥−𝑐,𝑝)=𝜎^#(𝑥−𝐴𝑥−𝐴_∞𝑥−𝑐,𝑝)≥0. So (3.2)(b) holds for all 𝑐∈𝑋. By Lemma 3.1(i) and 𝑋_∞⊆𝑋, we have 𝑐⁰∈𝑌(𝑆)⊆𝑆𝑋=⋃_𝑥∈𝑋(𝑥−𝐴𝑥−𝐴_∞𝑥). This implies that (3.5) is true, and thus (i) follows.
(ii) By the proof of (i), we have known that (3.2)(b) holds for all 𝑐∈𝑋. Combining this with the assumption of (ii) and using (3.3)(a), we obtain that 𝑋_∞⊆𝑌(𝑆). So 𝑌(𝑆) is nonempty and also compact by Lemma 3.1(ii). In view of Lemmas 1.3–1.6 and Remark 1.5, to complete the proof of (ii), it is enough to show that 𝐹₂ is closed since 𝑋_∞ and 𝑋 are compact. Suppose that {(𝑐^𝑚,𝑥^𝑚)∶𝑚≥1}⊆graph𝐹₂ with (𝑐^𝑚,𝑥^𝑚)→(𝑐⁰,𝑥⁰)(𝑚→∞). Then 𝑐⁰∈𝑋_∞⊂𝑌(𝑆), and for each 𝑚≥1 there exists Δ𝑐^𝑚∈𝐵(0,𝛿_∞(𝑥^𝑚)) such that 𝑐^𝑚+Δ𝑐^𝑚∈𝑥^𝑚−𝐴𝑥^𝑚. By graph𝐹₂⊆𝑋_∞×𝑋⊆𝑌(𝑆)×𝑋 and (3.4), we can see that 𝑐^𝑚∈𝑆𝑥^𝑚, that is, (𝑐^𝑚,𝑥^𝑚)∈graph𝐹^𝑆, where 𝐹^𝑆 is defined by (3.3) for 𝑆=𝐼−(𝐴+𝐴_∞). By Lemma 3.1(ii), 𝐹^𝑆 is closed, and thus (𝑐⁰,𝑥⁰)∈graph𝐹^𝑆, which implies that 𝑥⁰∈𝑋 and 𝑐⁰∈𝑆𝑥⁰=(𝐼−𝐴−𝐴_∞)𝑥⁰. So we can select Δ𝑐⁰∈𝐵(0,𝛿_∞(𝑥⁰)) such that 𝑐⁰+Δ𝑐⁰∈𝑥⁰−𝐴𝑥⁰, that is, (𝑐⁰,𝑥⁰)∈graph𝐹₂. Hence 𝐹₂ is closed.
(iii) Since 𝐴 is u.s.c, by the proof of Theorem 2.5, we have known that the function 𝑥↦𝛿_∞(𝑥) is u.s.c. on 𝑋. Combining this with (3.4), it is easy to see that the set-valued map 𝐴_∞ is u.s.c., and, by Lemma 1.4, also u.h.c.. Thus for each 𝑝∈𝑅^𝑛, the function 𝑥↦𝜎^#(𝑆𝑥,𝑝)=⟨𝑝,𝑥⟩+𝜎^#(𝐴𝑥,−𝑝)+𝜎^#(𝐴_∞𝑥,−𝑝) is u.s.c. on 𝑋. This implies that 𝑆=𝐼−(𝐴+𝐴_∞) is u.h.c. on 𝑋. Hence by Remark 3.2 and 𝑋_∞⊆𝑋, the condition of (ii) is true. This completes the proof.

3.2. The Second Result

Next we use a new number 𝜇₀, a new set 𝑋₀, and a new set-valued map 𝐹₀ from 𝑋₀ to 𝑋 to obtain further solvability and stability result for (1.1). Set|𝐴𝑥|₀=inf_{𝑧∈𝐴𝑥}‖𝑧‖,(𝑥∈𝑋),𝜇₀=sup_{𝑥∈𝜕𝑋}|𝐴𝑥|₀,𝛿₀(𝑥)=0if𝑥∈int𝑋,|𝐴𝑥|₀𝑋if𝑥∈𝜕𝑋,⁰=𝑋[𝜇₀,+∞)^𝑛, (3.6) and define 𝐴₀∶𝑋→2^𝑅𝑛 and 𝐹₀∶𝑋₀→2^𝑋 by𝐴₀𝑥=𝐵(0,𝛿₀𝐹(𝑥))for𝑥∈𝑋,₀𝑐={𝑥∈𝑋∶∃Δ𝑐∈𝑅^𝑛s.t.𝑐+Δ𝑐∈𝑥−𝐴𝑥,‖Δ𝑐‖≤𝛿₀(𝑥)}for𝑐∈𝑋₀. (3.7) Applying Lemma 3.1 to 𝑆=𝐼−(𝐴+𝐴₀), we have the following results.

Theorem 3.4. Assume that 𝜇₀ is finite, 𝑋₀≠∅, and 𝑆=𝐼−(𝐴+𝐴₀).(i)Let 𝑐⁰∈𝑋₀. If (3.2)(a) holds for 𝑐⁰, then 𝐹₀𝑐⁰≠∅, that is, ∃𝑥⁰∈𝑋,∃Δ𝑐⁰∈𝑅^𝑛𝑠.𝑡.𝑐⁰+Δ𝑐⁰∈𝑥⁰−𝐴𝑥⁰,‖Δ𝑐⁰‖≤𝛿₀(𝑥). (3.8)(ii)If 𝑌(𝑆,𝑝,𝑐,𝜀) defined by (3.1) is closed for all 𝑝∈𝑅^𝑛, 𝑐∈𝑋₀ and 𝜀≥0, then 𝐹₀ defined by (3.7) is closed, u.s.c., and u.h.c., and the subset of points at which 𝐹₀ is continuous is residual.(iii)If 𝐴 is u.s.c. on 𝑋 and also l.s.c. on 𝜕𝑋, then the result of (ii) for 𝐹₀ is also true.

Proof. (i) It is easy to see that 𝑆 is a strict set-valued map with convex compact values. As in the proof of Theorem 3.3, we can prove that (3.2)(b) holds for all 𝑐∈𝑋. Thus we can follow the proof of Theorem 3.3(i) to obtain the conclusion of (i). In fact, let 𝑥∈𝑋, 𝑝∈𝑁_𝑋(𝑥), and 𝑐∈𝑋. If 𝑝=0, then 𝜎^#(𝑆𝑥−𝑐,𝑝)=0. If 𝑝≠0, then 𝑥∈𝜕𝑋. So 𝛿₀(𝑥)=|𝐴𝑥|₀, 0∈A𝑥+𝐵(0,𝛿₀(𝑥)) and 𝜎^𝑏(𝐴𝑥+𝐴₀𝑥,𝑝)≤0. This implies ⟨𝑝,𝑥⟩=sup_𝑦∈𝑋⟨𝑝,𝑦⟩≥⟨𝑝,𝑐⟩≥𝜎^𝑏(𝐴𝑥+𝐴₀𝑥,𝑝)+⟨𝑝,𝑐⟩, that is, 𝜎^#(𝑆𝑥−𝑐,𝑝)=𝜎^#(𝑥−𝐴𝑥−𝐴₀𝑥−𝑐,𝑝)≥0. Hence, (3.2)(b) holds for all 𝑐∈𝑋, and by Lemma 3.1(i), 𝑐⁰∈𝑌(𝑆)⊆𝑆𝑋=⋃_𝑥∈𝑋(𝑥−𝐴𝑥−𝐴₀𝑥). Thus (3.8) is true.
(ii) The proof of (ii) is similar to that of Theorem 3.3(ii). Indeed, by the proof of (i) and the assumption of (ii), we can see that 𝑋₀⊆𝑌(𝑆), and thus 𝑌(𝑆) is compact by Lemma 3.1(ii). If {(𝑐^𝑚,𝑥^𝑚)∶𝑚≥1}⊆graph𝐹₀ such that (𝑐^𝑚,𝑥^𝑚)→(𝑐⁰,𝑥⁰) as 𝑚→∞, then 𝑐⁰∈𝑋₀⊂𝑌(𝑆),𝑥⁰∈𝑋, and there exists Δ𝑐^𝑚∈𝐵(0,𝛿₀(𝑥^𝑚)) with 𝑐^𝑚+Δ𝑐^𝑚∈𝑥^𝑚−𝐴𝑥^𝑚. Hence by (3.3) and (3.7), (𝑐^𝑚,𝑥^𝑚)∈graph𝐹^𝑆, where 𝐹^𝑆 is defined by (3.3) for 𝑆=𝐼−(𝐴+𝐴₀). Also by Lemma 3.1(ii), we have (𝑐⁰,𝑥⁰)∈graph𝐹^𝑆, that is, 𝑐⁰∈(𝐼−𝐴−𝐴₀)𝑥⁰. So we can select Δ𝑐⁰∈𝐵(0,𝛿₀(𝑥⁰)) such that 𝑐⁰+Δ𝑐⁰∈𝑥⁰−𝐴𝑥⁰. Therefore, (𝑐⁰,𝑥⁰)∈graph𝐹₀ and 𝐹₀ is closed.
(iii) Similar to the proof of Theorem 3.3(iii), it suffices to show that 𝑆=𝐼−(𝐴+𝐴₀) is u.h.c.. For 𝑥∈int𝑋 and 𝑝∈𝑅^𝑛, we have 𝐴₀𝑥={0}. In this case, since 𝜎^#(𝑆𝑥,𝑝)=⟨𝑥,𝑝⟩+𝜎^#(𝐴𝑥,−𝑝) holds for any 𝑥∈int𝑋 and 𝐴 is u.h.c., we obtain that 𝑆 is u.h.c. in int𝑋. Now, we prove that 𝑆 is also u.h.c. at any point 𝑥⁰∈𝜕𝑋. Because 𝜎^#(𝑆𝑥,𝑝)=⟨𝑥,𝑝⟩+𝜎^#(𝐴𝑥,−𝑝)+𝜎^#(𝐴₀𝑥,−𝑝) holds for any 𝑥∈𝑋 and 𝑝∈𝑅^𝑛, it is sufficient to show that 𝐴₀ is u.h.c. at 𝑥⁰∈𝜕𝑋. Since 𝐴𝑥⁰ is compact, there is 𝑧⁰∈𝐴𝑥⁰ such that ‖𝑧⁰‖=|𝐴𝑥⁰|₀=𝛿₀(𝑥⁰). As 𝐴 is l.s.c. at 𝑥⁰ by the assumption, for 𝜀>0, there exists an open neighborhood 𝑈(𝑥⁰) of 𝑥⁰ such that 𝐴𝑥∩𝐵(𝑧⁰,𝜀)≠∅ for all 𝑥∈𝑋∩𝑈(𝑥⁰). This implies that for each 𝑥∈𝑋∩𝑈(𝑥⁰),𝛿₀(𝑥)≤|𝐴𝑥|₀≤inf_{𝑦∈𝐴𝑥∩𝐵(𝑧0,𝜀)}‖𝑦‖≤sup_{𝑦∈𝐵(𝑧0,𝜀)}‖𝑦‖≤𝛿₀(𝑥⁰)+𝜀, and hence 𝐵(0,𝛿₀(𝑥))⊆𝐵(𝐵(0,𝛿₀(𝑥⁰)),𝜀) for all 𝑥∈𝑋∩𝑈(𝑥⁰). So we conclude that 𝐴₀ is u.s.c. at 𝑥⁰ and, by Lemma 1.4, also u.h.c.. Therefore, 𝑆=𝐼−(𝐴+𝐴₀) is u.h.c. on 𝑋, and the conclusion of (iii) follows. This completes the proof.

Remark 3.5. As in the previous theorems, the assumptions 𝑐⁰∈𝑋_∞ in Theorem 3.3 and 𝑐⁰∈𝑋₀ in Theorem 3.4 are used to make sure that the net output vectors are nonnegative.

3.3. The Third Result

Finally, we use the values of 𝐴 at the boundary 𝜕𝑋 of 𝑋 to consider the solvability and stability of (1.1). Define 𝐹∶𝑋→2^𝑋 by𝑋={𝑐∈𝑋∶∀𝑥∈𝜕𝑋,(𝐴𝑥+𝑐)∩𝑋≠∅},𝐹𝑐={𝑥∈𝑋∶𝑐∈𝑥−𝐴𝑥}for𝑐∈𝑋. (3.9) Applying Lemma 3.1 to 𝑆=𝐼−𝐴, we have the following results.

Theorem 3.6. Assume that 𝑋≠∅ and 𝑆=𝐼−𝐴.(i)If (3.2)(a) holds for all 𝑐∈𝑋, then 𝑋⊆𝑌(𝑆)⊆(𝐼−𝐴)𝑋, and 𝑋 is compact. In particular, if 𝑋∩𝑅^𝑛₊≠∅, then for any 𝑐∈𝑋∩𝑅^𝑛₊, (1.1) is solvable.(ii)If 𝑌(𝑆,𝑝,𝑧,𝜀) defined by (3.1) is closed for all 𝑝∈𝑅^𝑛, 𝑐∈𝑋, and 𝜀≥0, then 𝐹 defined by (3.9) is closed, u.s.c., and u.h.c., and the subset of points at which 𝐹 is continuous is residual.(iii)If 𝐴 is u.s.c., then both results of (i) and (ii) are also true.

Proof. (i) Let 𝑐∈𝑋, 𝑥∈𝑋, and 𝑝∈𝑁_𝑋(𝑥). If 𝑝=0, then 𝜎^#(𝑥−𝐴𝑥−𝑐,0)=⟨0,𝑥⟩. If 𝑝≠0, then 𝑥∈𝜕𝑋. From (3.9), we know that (𝐴𝑥+𝑐)∩𝑋≠∅. This implies that 𝜎^𝑏(𝐴𝑥+𝑐,𝑝)≤𝜎^𝑏((𝐴𝑥+𝑐)∩𝑋,𝑝)≤𝜎^#(𝑋,𝑝)=⟨𝑝,𝑥⟩.Hence (3.2)(b) holds for all 𝑐∈𝑋, and then 𝑐∈𝑌(𝑆). By Lemma 3.1(i), it follows that 𝑋⊆𝑌(𝑆)⊆𝑆𝑋=(𝐼−𝐴)𝑋.
To prove that 𝑋 is compact, we only need to prove that 𝑋 is closed. Suppose that {𝑐^𝑚∶𝑚≥1}⊆𝑋 such that 𝑐^𝑚→𝑐⁰∈𝑋 as 𝑚→∞. Let 𝑥∈𝜕𝑋. Then by (3.9), (𝐴𝑥+𝑐^𝑚)∩𝑋≠∅ for 𝑚≥1. Hence there exists a sequence {𝑦^𝑚∶𝑚≥1}⊆𝑋 such that 𝑦^𝑚∈𝐴𝑥+𝑐^𝑚. Therefore, 𝑦^𝑚−𝑐^𝑚∈𝐴𝑥. Because 𝐴𝑥 is compact, there exists a convergent subsequence of {𝑦^𝑚−𝑐^𝑚∶𝑚≥1}. Since 𝑐^𝑚→𝑐⁰ as 𝑚→∞, we may assume that 𝑦^𝑚→𝑦⁰∈𝑋, which implies 𝑦⁰∈(𝐴𝑥+𝑐⁰)∩𝑋. Hence 𝑐⁰∈𝑋 and 𝑋 is closed. The rest of (i) is clear because 𝑋∩𝑅^𝑛₊⊂(𝐼−𝐴)X.
(ii) By assumption of (ii), the proof of (i), and Lemma 3.1, we can see that 𝑋⊂𝑌(𝑆), 𝑌(𝑆) is compact, and 𝐹^𝑆 defined by (3.3) for 𝑆=𝐼−𝐴 is closed. With the same method as in proving Theorem 3.3(ii), we can show that 𝐹 is closed and has all the continuous properties stated in this theorem (ii).
(iii) By assumption of (iii) and 𝜎^#((𝐼−𝐴)𝑥,𝑝)=⟨𝑝,𝑥⟩+𝜎^#(𝐴𝑥,−𝑝), we can see that 𝑆=𝐼−𝐴 is u.h.c., hence 𝑌(𝑆,𝑝,𝑐,𝜀) is closed for all 𝑝∈𝑅^𝑛, 𝑐∈𝑋 and 𝜀≥0. Hence both conditions of (i) and (ii) are satisfied, and therefore the result of (iii) follows. This completes the proof.

4. Conclusions

In this paper, the Leontief-type input-output inclusion has been studied. First applying Rogalski-Cornet theorem [6], we have proved two solvability and stability theorems (Theorems 2.3 and 2.5) under the assumption that the set-valued map in this inclusion is upper semicontinuous. Then utilizing a Rogalski-Cornet-type theorem proved in [5], we have also proved three solvability and stability theorems (Theorems 3.3–3.6) in which the continuity assumption regarding the set-valued map in this inclusion is no longer needed.