A Rogalski-Cornet Type Inclusion Theorem Based on Two Hausdorff Locally Convex Vector Spaces

Abstract

A Rogalski-Cornet type inclusion theorem based on two Hausdorff locally convex vector spaces is proved and composed of two parts. An example is presented to show that the associated set-valued map in the first part does not need any conventional continuity conditions including upper hemicontinuous. As an application, solvability results regarding an abstract von Neumann inclusion system are obtained.

1. Introduction

The Rogalski-Cornet inclusion theorem, based on a Hausdorff locally convex vector space and as an useful tool to deal with some inclusion problems described by an upper hemicontinuous set-valued map, can be stated as follows.

Theorem 1.1 (see [1]). Let 𝑋 be a convex compact subset of a Hausdorff locally convex vector space 𝑈, and 𝑆an upper hemicontinuous set-valued map from 𝑋 to 𝑈 with nonempty closed convex values. If 𝑆 is outward, that is, ∀𝑝∈𝑈^∗,∀𝑥∈𝑋 with ⟨𝑝,𝑥⟩=inf_𝑦∈𝑋⟨𝑝,𝑦⟩,⟨𝑝,𝑥⟩≥inf_{𝑦∈𝑆𝑥}⟨𝑝,𝑦⟩, then 𝑆𝑋=∪_𝑥∈𝑋𝑆𝑥⊇𝑋.

Two comments are in order.

(1) Theorem 1.1 can be used to deal with those inclusion problems described by a set-valued map whose domain and range are contained in a same space. For example, if 𝑅^𝑛₊ denotes the set of all nonnegative vectors of the 𝑛 dimensional Euclidean space 𝑅^𝑛, 𝑐∈𝑅^𝑛₊ is an expected demand of the market, 𝑋⊂𝑅^𝑛₊ some enterprise’s admission output bundle set, and 𝐴 or 𝐹 is the enterprise’s consuming map or correspondence (namely, set-valued map) from 𝑋 to 𝑅^𝑛₊. Then a class of Leontief type input-output inclusion system, as a nonlinear extension to the classical input-output equation [2–4], is composed of(a)(b)𝑥∈𝑋s.t.𝑥−𝐴𝑥=𝑐,𝑥∈𝑋s.t.𝑥−𝐹𝑥∋𝑐, (1.1) and has been studied by Sandberg [5] and Fujimoto [6] with the nonlinear analysis methods. Moreover, some primary extensions to Theorem 1.1, also based on a Hausdorff locally convex vector space, have been made by Liu and Zhang [7] such that the associated correspondence 𝑆 no longer needs the upper hemicontinuous condition, and the obtained results (see [7]) also have been used to deal with the solvability of (1.1)(b) (see [8]).

(2) However, if a problem is concerned with two different spaces, then Theorem 1.1 is generally useless even if this problem can be changed into an inclusion system. For example, assume that 𝑐∈𝑅^𝑚₊ is an expected demand of the market, 𝑋⊂𝑅^𝑛₊ is some enterprise’s raw material bundle set, and 𝐵,𝐴 (or 𝐺,𝐹) are the enterprise’s output and input maps (or correspondences) from 𝑋 to 𝑅^𝑚₊, respectively. If the semiordering in 𝑅^𝑚 is defined by 𝑦¹≥𝑦² iff 𝑦¹−𝑦²∈𝑅^𝑚₊, then the following von Neumann type input-output inequality system, composed of a single-valued inequality and a set-valued inequality,(a)(b)𝑥∈𝑋s.t.𝐵𝑥−𝐴𝑥≥𝑐,𝑥∈𝑋s.t.∃𝑦∈𝐺𝑥,∃𝑧∈𝐹𝑥with𝑦−𝑧≥𝑐, (1.2) has been studied by Liu and Zhang [9], and Liu [10, 11] with the nonlinear analysis methods including the minimax and saddle point techniques attributed to [1, 12, 13]. We claim that (1.2) also includes some economic growth problems. For example, if we restrict 𝜆≥𝜆₀>0 (where 𝜆₀ may be viewed as the minimal growth fact of the output regarding input accepted by the enterprise) and replace 𝐴 by 𝜆𝐴, respectively, then (1.2)(a) reduces to a single-valued von Neumann economic growth model 𝜆≥𝜆₀,s.t.∃𝑥∈𝑋with𝐵𝑥≥𝜆𝐴𝑥+𝑐, which has been studied by Medvegyev [14], and Bidard and Hosoda [15]. Some other research regarding economic growth has also been made by Jones [16, 17], and Jones, Williams [18]. It is easy to see that (1.2)(b) yields a set-valued economic growth problem. However, up to now, no any corresponding references can be seen. This shows that to study (1.2)(b) is also useful.

Returning to (1.2), if we set 𝑆₁𝑥=𝐴𝑥+𝑅^𝑚₊ and 𝑆₂𝑥=𝐹𝑥+𝑅^𝑚₊ for 𝑥∈𝑋, then (1.2) equals to the following inclusion system:(a)𝑥∈𝑋s.t.𝐵𝑥−𝑆₁(b)𝑥∋𝑐,𝑥∈𝑋s.t.𝐺𝑥−𝑆₂𝑥∋𝑐, (1.3) which is difficult to be handled by Theorem 1.1 except for 𝑛=𝑚. Moreover, the following(a)(b)𝑥∈𝑋s.t.𝐵𝑥−𝐴𝑥=𝑐,𝑥∈𝑋s.t.𝐺𝑥−𝐹𝑥∋𝑐 (1.4) (as an inclusion system described by the preceding 𝐵,𝐴 or 𝐺,𝐹, which is clearly more practical than (1.1) and more fine than (1.2)) can also hardly be tackled by Theorem 1.1 even along with the minimax method.

Now let 𝑣∈𝑅^𝑚,𝑌⊂𝑅^𝑚, and 𝑆∶𝑋⊂𝑅^𝑛→𝑅^𝑚 be a set-valued map, then (1.2)–(1.4) (equivalently, (1.3)-(1.4)) can be viewed as the special examples of the first inclusion in the following von Neumann type inclusion system(a)(b)(c)𝑥∈𝑋s.t.𝑆𝑥∋𝑣,thatis,𝑣∈𝑆𝑋,𝑥∈𝑋s.t.𝑆𝑥∩𝑌≠∅,thatis,𝑆𝑋∩𝑌≠∅,∀𝑦∈𝑌,∃𝑥∈𝑋s.t.𝑆𝑥∋𝑦,thatis,𝑌⊆𝑆𝑋. (1.5) We claim (1.5) an abstract von Neumann inclusion system if 𝑅^𝑛 and 𝑅^𝑚 are replaced by two Hausdorff locally convex vector spaces 𝑈 and 𝑉, respectively.

In the sequel, we attempt to extend Theorem 1.1 to a new situation, that is, to present a Rogalski-Cornet type theorem composed of existence and continuity parts (as the main result proved in Section 3), such that the abstract von Neumann inclusion system (1.5) could be tackled, which means that the domain and range of 𝑆 may probably be contained in two different Hausdorff locally convex vector spaces 𝑈 and 𝑉, respectively. Since Theorem  15.1.9 in [1] and Theorem  6.4.10 in [12] are such type of results while their correspondences need the upper hemicontinuous assumption, and a new paper completed by Lignola [19] provides some Ky Fan inequalities and Nash equilibrium points without semicontinuity and compactness, we concentrate our attention to the instance such that the associated correspondence no longer needs any conventional continuity conditions including upper hemicontinuous.

The paper is arranged as follows. We introduce some necessary concepts in the next section. In Section 3, we first prove a Rogalski-Cornet type inclusion theorem based on two different Hausdorff locally convex vector spaces, followed by an example to show that the first part of this theorem does not need any conventional continuity conditions such as upper semicontinuous, lower semicontinuous, and upper hemicontinuous. Then, as an application of this theorem, we provide some solvability results for (1.5). Finally we present the conclusion in Section 4.

2. Preliminary

In the sections below, without any special explanations, we always assume 𝑈,𝑉 are two Hausdorff locally convex vector spaces, 𝑈^∗,𝑉^∗ their duals and ⟨⋅,⋅⟩ the duality paring on ⟨𝑈^∗,𝑈⟩ or ⟨𝑉^∗,𝑉⟩. We need some concepts with respect to a function 𝑓 defined on 𝑈 or 𝑉 such as convex or concave, upper semicontinuous or lower semicontinuous (in short, u.s.c. or l.s.c.) and continuous, and concepts to a set-valued map 𝑆 from a subset 𝑋⊂𝑈 to 𝑉 including closed, upper, and lower semicontinuous (in short, u.s.c. and l.s.c.) and continuous, whose definitions can be consulted in [1, 12, 13], so the details are omitted here.

Let 𝑍⊂𝑉 and 𝑞∈𝑉^∗. Denote𝜎^♭𝑍,𝑞=inf_𝑦∈𝑍𝑞,𝑦,𝜎^♯𝑍,𝑞=sup_𝑦∈𝑍,𝜕=𝑞,𝑦𝑍,𝑞𝑦∈𝑍∶𝑞,𝑦=𝜎^♭𝑍,𝑞thesupportsetof𝑍at𝑞. (2.1)

We also need the following.

Definition 2.1. (see [1, 12, 13]). Let 𝑈 be a Hausdorff topological space and 𝑆 a set-valued map from 𝑈 to 𝑉. Then 𝑆 is said to be upper hemicontinuous (in short, u.h.c.) if for any 𝑞∈𝑉^∗,𝑥↦𝜎^♯(𝑆𝑥,𝑞)=sup_{𝑣∈𝑆𝑥}⟨𝑞,𝑣⟩ is u.s.c. on 𝑈.

3. Main Theorem and Application to (1.5)

In this section, we always assume that(a)(b)(c)()𝑋⊂𝑈,𝑌⊂𝑉aretwononemptyconvexcompactsubsets,𝑆∶𝑋→𝑉isasetvaluedmapwithnonemptyclosedconvexvalues,𝐿∈ℒ𝑈,𝑉isacontinuouslinearmapfrom𝑈to𝑉. (3.1)

Under some additional assumptions, we first prove a Rogalski-Cornet type inclusion theorem for a set-valued map 𝑆 from 𝑋⊂𝑈 to 𝑉 with and without u.h.c. condition, then present an application of this theorem to the abstract von Neumann inclusion system (1.5).

3.1. Main Theorem and a Counterexample

The Rogalski-Cornet type inclusion theorem we will prove is the following.

Theorem 3.1. (A) Results without u.h.c. Condition.
(i)Existence. (1) Assume that (a)∀𝑞∈𝑉^∗,𝑥∈𝑋∶𝜎^♭𝑆𝑥,𝑞≤𝜎^♯(b)𝑌,𝑞𝑖𝑠𝑐𝑙𝑜𝑠𝑒𝑑,∀𝑞∈𝑉^∗,∀𝑥∈𝑋𝑤𝑖𝑡ℎ𝐿𝑥∈𝜕𝐿𝑋,𝑞,𝜎^♭𝑆𝑥,𝑞≤𝜎^♯𝑌,𝑞, (3.2) then 𝑌∩𝑆𝑋≠∅, that is, there exist 𝑦∈𝑌 and 𝑥∈𝑋 such that 𝑦∈𝑆𝑥. (2) If 𝑌₀ is the set of all 𝑦∈𝑌such that (a)∀𝑞∈𝑉^∗,𝑥∈𝑋∶𝜎^♭≤(b)𝑆𝑥,𝑞𝑞,𝑦𝑖𝑠𝑐𝑙𝑜𝑠𝑒𝑑,∀𝑞∈𝑉^∗,∀𝑥∈𝑋𝑤𝑖𝑡ℎ𝐿𝑥∈𝜕𝐿𝑋,𝑞,𝜎^♭𝑆𝑥,𝑞≤𝑞,𝑦, (3.3) then 𝑌₀⊂𝑆𝑋, that is, for each 𝑦∈𝑌₀, there exists 𝑥∈𝑋 such that 𝑦∈𝑆𝑥.(ii)Continuity. Assume that 𝑌₁, the set of all 𝑦∈𝑌  such that (a)∀𝑞∈𝑉^∗,∀𝑐≥0,𝑥∈𝑋∶𝜎^♭≤(b𝑏)𝑆𝑥,𝑞𝑞,𝑦+𝑐𝑖𝑠𝑐𝑙𝑜𝑠𝑒𝑑,)(3.3)(ℎ𝑜𝑙𝑑𝑠𝑎𝑙𝑠𝑜, (3.4) is nonempty, then 𝑌₁⊆𝑆𝑋. Moreover, 𝑌₁ is compact, and the inverse of 𝑆 restricting to 𝑌₁ defined by 𝑆_𝑌⁻¹₁∶𝑌₁⟶𝑋∶𝑦⟼𝑆_𝑌⁻¹₁𝑦=𝑥∈𝑋∶𝑦∈𝑆𝑥 (3.5) is a u.s.c. and u.h.c. set-valued map.(B) Results with u.h.c. Condition.
Assume that 𝑆is u.h.c., then the following are true. (i)If (3.2)(b) holds, then 𝑌∩𝑆𝑋 is nonempty and compact, and the inverse of 𝑆 restricting to 𝑌∩𝑆𝑋 defined by 𝑆⁻¹_{𝑌∩𝑆𝑋}∶𝑌∩𝑆𝑋⟶𝑋∶𝑦⟼𝑆⁻¹_{𝑌∩𝑆𝑋}𝑦=𝑥∈𝑋∶𝑦∈𝑆𝑥 (3.6) is a u.s.c. and u.h.c. set-valued map.(ii)If 𝑌₁, the set of all 𝑦∈𝑌  satisfying (3.3)(b), is nonempty, then 𝑌₁ is compact with 𝑌₁⊂𝑆𝑋, and the inverse of 𝑆 restricting to 𝑌₁ defined by (3.5) is also a u.s.c and u.h.c. set-valued map.

To prove this theorem, we need some known results and state them in lemmas as follows.

Lemma 3.2 (see [1]). Let 𝐾 be a convex compact subset of 𝑈 and let 𝜙∶𝐾×𝐾→𝑅 satisfy that ∀𝑦∈𝐾,𝑥↦𝜙(𝑥,𝑦) is lower semicontinuous, ∀𝑥∈𝐾,𝑦↦𝜙(𝑥,𝑦) is quasi-concave, and sup_𝑦∈𝐾𝜙(𝑦,𝑦)≤0. Then there exists 𝑥∈𝐾  such that sup_𝑦∈𝐾𝜙(𝑥,𝑦)≤0.

Lemma 3.3 (see [12]). Let 𝑈 be a Hausdorff topological space, 𝑉 a compact Hausdorff topological space, and 𝑆 a closed set-valued map from 𝑈 to 𝑉. Then 𝑆 is u.s.c.

Lemma 3.4 (see [12]). Let 𝑈 be a Hausdorff topological space, 𝑉 supplied with the weak topology 𝜎(𝑉,𝑉^∗), and 𝑆 a u.s.c. set-valued map from 𝑈 to 𝑉. Then 𝑆 is u.h.c.

Remark 3.5. Since the original vector topology 𝜏 on 𝑉 is stronger than the weak topology 𝜎(𝑉,𝑉^∗), this Lemma is also true if 𝑉 is supplied with the original topology 𝜏.

Lemma 3.6 (see [12]). Let 𝑈 be a Hausdorff topological space, 𝑉 supplied with the weak topology, and let 𝑆 be a u.h.c. set-valued map from 𝑈 to 𝑉 with nonempty closed convex values. Then the graph of 𝑆 denoted by graph 𝑆 is closed, that is, 𝑆 is closed.

With these lemmas, we proceed to prove Theorem 3.1.

Proof. Proof of Part A. (i) First we prove (i).
(1) Under the assumptions (3.1) and (3.2), we will prove 𝑌∩𝑆𝑋≠∅ by contradiction. If 𝑌∩𝑆𝑥=∅ for all 𝑥∈𝑋, then by (3.1), we see that for each 𝑥∈𝑋, 𝑆𝑥−𝑌 is closed convex with 0∈𝑆𝑥−𝑌. So the Hahn-Banach separation theorem implies that ∀𝑥∈𝑋,∃𝑞=𝑞_𝑥∈𝑉^∗s.t.𝜎^♭𝑆𝑥−𝑌,𝑞=inf_{𝑣∈𝑆𝑥−𝑌}𝑞,𝑣>𝑞,0=0. (3.7)
Setting 𝑋^♭(𝑞)={𝑥∈𝑋∶𝜎^♭(𝑆𝑥−𝑌,𝑞)>0} for 𝑞∈𝑉^∗, from (3.7) we have 𝑋=⋃_{𝑞∈𝑉∗}𝑋^♭(𝑞). Since 𝑋 is compact and 𝑋^♭(𝑞)={𝑥∈𝑋∶𝜎^♭(𝑆𝑥,𝑞)−𝜎^♯(𝑌,𝑞)>0}=𝑋⧵{𝑥∈𝑋∶𝜎^♭(𝑆𝑥,𝑞)≤𝜎^♯(𝑌,𝑞)}, by (3.2)(a) we know that {𝑋^♭(𝑞)∶𝑞∈𝑉^∗} forms an open covering of 𝑋. Therefore, 𝑋=⋃^𝑛_𝑖=1𝑋^♭(𝑞_𝑖) holds for some finite subset {𝑞₁,𝑞₂,…,𝑞_𝑛}⊂𝑉^∗, and there exists a continuous partition of unit {𝛼₁,𝛼₂,…,𝛼_𝑛} associated to this finite covering of 𝑋 such that (a)𝛼_𝑖(𝑥)(1≤𝑖≤𝑛)arenonnegativecontinuouson𝑋with^𝑛_𝑖=1𝛼_𝑖(𝑥)(b)=1,supp𝛼_𝑖⊂𝑋^♭𝑞_𝑖,hence,𝜎^♭𝑆𝑥−𝑌,𝑞_𝑖>0if𝛼_𝑖(𝑥)>0(1≤𝑖≤𝑛), (3.8) where  supp𝛼_𝑖={𝑥∈𝑋∶𝛼_𝑖(𝑥)>0} is the closure of the set {𝑥∈𝑋∶𝛼_𝑖(𝑥)>0}.
Define 𝜑(⋅,⋅) on 𝑋×𝑋 by 𝜑𝑥,𝑦=^𝑛_𝑖=1𝛼_𝑖(𝑥)𝑞_𝑖,𝐿𝑥−𝑦,𝑥,𝑦∈𝑋×𝑋. (3.9) Clearly, 𝜑 satisfies the assumptions of Lemma 3.2. Indeed, it is easy to see that for any 𝑦∈𝑋, 𝑥↦𝜑(𝑥,𝑦) is continuous, for any 𝑥∈𝑋, 𝑦↦𝜑(𝑥,𝑦) is affine, and sup_𝑦∈𝑋𝜑(𝑦,𝑦)=0. So there exists 𝑥∈𝑋 such that sup_𝑦∈𝑋𝜑(𝑥,𝑦)≤0. Take 𝑞=∑^𝑛_𝑖=1𝛼_𝑖(𝑥)𝑞_𝑖 and 𝐼={𝑖∈{1,2,…,𝑛}∶𝛼_𝑖(𝑥)>0}, then from (3.8) and (3.9) we know that 𝐼≠∅ and 0≤⟨𝑞,𝐿𝑥⟩−inf_𝑦∈𝑋⟨𝑞,𝐿𝑦⟩=sup_𝑦∈𝑋⟨𝑞,𝐿(𝑥−𝑦)⟩=sup_𝑦∈𝑌𝜑(𝑥,𝑦)≤0. This further implies (a)𝑞,𝐿𝑥=inf_𝑦∈𝑋𝑞,𝐿𝑦=𝜎^♭𝐿𝑋,𝑞,thatis,𝐿𝑥∈𝜕𝐿𝑋,𝑞,(b)∀𝑖∈𝐼,𝛼_𝑖(𝑥)>0,𝜎^♭𝑆𝑥−𝑌,𝑞_𝑖>0,𝑞=_𝑖∈𝐼𝛼_𝑖(𝑥)𝑞_𝑖. (3.10) Associating this with (3.2)(b), we obtain that 0≥𝜎^♭𝑆𝑥,𝑞−𝜎^♯𝑌,𝑞=𝜎^♭𝑆𝑥−𝑌,_𝑖∈𝐼𝛼_𝑖(𝑥)𝑞_𝑖≥_𝑖∈𝐼𝛼_𝑖(𝑥)𝜎^♭𝑆𝑥−𝑌,𝑞_𝑖>0. (3.11) This is a contradiction. Therefore, 𝑌∩𝑆𝑋≠∅.
(2) Statement (2) immediately follows from (1) because for each 𝑦∈𝑌₀, {𝑦} is compact, and (3.2) holds for 𝑌={𝑦} because of the assumption (3.3).(ii) Then we prove (ii).
By (3.4) and statement (i)(2), it is easy to see that 𝑌₁⊆𝑆𝑋, so the left is to show that 𝑌₁ is compact and 𝑆_𝑌⁻¹₁ defined by (3.5) is u.s.c. and u.h.c.
(1) Since 𝑌 is compact and 𝑌₁⊆𝑌, it is sufficient to verify that 𝑌₁ is closed, that is, to verify that if {𝑦_𝜆∶𝜆∈Λ}⊂𝑌₁ is a generalized sequence such that 𝑦_𝜆→𝑦₀∈𝑌, then 𝑦₀∈𝑌₁. To this end, for each 𝑞∈𝑉^∗,𝑦∈𝑌 and each 𝑐≥0, we define (a)𝑋^♭𝑆,𝑞,𝑦,𝑐=𝑥∈𝑋∶𝜎^♭≤,(b)𝑋𝑆𝑥,𝑞𝑞,𝑦+𝑐_♭𝑆,𝑞,𝑦=𝑋^♭𝑆,𝑞,𝑦,0, (3.12) and will prove that (a)∀𝑞∈𝑉^∗,∀𝑐≥0,𝑋^♭𝑆,𝑞,𝑦₀(b),𝑐isclosed,∀𝑞∈𝑉^∗,∀𝑥∈𝑋with𝐿𝑥∈𝜕𝐿𝑋,𝑞,𝑥∈𝑋^♭𝑆,𝑞,𝑦₀, (3.13) which implies by (3.4) and (3.12) that 𝑦₀∈𝑌₁, and thus 𝑌₁ is closed.
(a) First we prove (3.13)(a). Suppose that 𝑞∈𝑉^∗,𝑐≥0 are fixed, and {𝑥_𝛼∶𝛼∈𝐼}⊂𝑋^♭(𝑆,𝑞,𝑦₀,𝑐) is a generalized sequence such that 𝑥_𝛼→𝑥₀. By (3.12)(a) we have 𝜎^♭𝑆𝑥_𝛼,𝑞≤𝑞,𝑦₀+𝑐,∀𝛼∈𝐼. (3.14) Since 𝑦_𝜆→𝑦₀, for each 𝜀>0 there exists 𝜆₀∈Λ such that 𝑞,𝑦_𝜆0−𝑦₀<𝜀2. (3.15) Associating this with (3.14), we obtain 𝜎^♭𝑆𝑥_𝛼,𝑞−𝑞,𝑦_𝜆0𝜀−𝑐−2≤𝜎^♭𝑆𝑥_𝛼,𝑞−𝑞,𝑦₀𝜀−𝑐−2−𝑞,𝑦₀−𝑦_𝜆0≤0forany𝛼∈𝐼. (3.16) As 𝑦_𝜆0∈𝑌₁, from (3.4)(a), (3.12)(a), and (3.16), we conclude that {𝑥_𝛼}⊆𝑋^♭(𝑆,𝑞,𝑦_𝜆0,𝑐+𝜀/2) and 𝑥₀∈𝑋^♭(𝑆,𝑞,𝑦_𝜆0,𝑐+𝜀/2) because 𝑋^♭(𝑆,𝑞,𝑦_𝜆0,𝑐+𝜀/2) is closed. Hence, 𝜎^♭(𝑆𝑥₀,𝑞)≤⟨𝑞,𝑦_𝜆0⟩+𝑐+𝜀/2, which together with (3.15) yields 𝜎^♭𝑆𝑥₀,𝑞−𝑞,𝑦₀−𝑐−𝜀≤𝜎^♭𝑆𝑥₀,𝑞−𝑞,𝑦_𝜆0𝜀−𝑐−2−𝜀2−𝑞,𝑦_𝜆0−𝑦₀≤0. (3.17)
By taking 𝜀→0, from (3.17) we obtain that 𝜎^♭(𝑆𝑥₀,𝑞)≤⟨𝑞,𝑦₀⟩+𝑐. Hence, 𝑥₀∈𝑋^♭(𝑆,𝑞,𝑧₀,𝑐) and (3.13)(a) follows.
(b) Then we prove (3.13)(b). Suppose that 𝑞∈𝑉^∗,𝑥∈𝑋 satisfy 𝐿𝑥∈𝜕(𝐿𝑋,𝑞), and (3.15) holds for this 𝑞 and some 𝜆₀∈Λ. Since 𝑦_𝜆0∈𝑌₁, by (3.4)(b), we have 𝜎^♭(𝑆𝑥,𝑞)≤⟨𝑞,𝑦_𝜆0⟩. In view of (3.15), we get 𝜎^♭𝑆𝑥,𝑞−𝑞,𝑦₀≤𝜎^♭𝑆𝑥,𝑞−𝑞,𝑦_𝜆0+𝑞,𝑦_𝜆0−𝑦₀≤𝜀2. (3.18) Thus, 𝜎^♭(𝑆𝑥,𝑞)≤⟨𝑞,𝑦₀⟩ because 𝜀 is arbitrary. This shows that 𝑥∈𝑋^♭(𝑆,𝑞,𝑦₀) and (3.13)(b) is also true. Therefore, 𝑦₀∈𝑌₁.
(2) To prove the continuity of 𝑆_𝑌⁻¹₁, by Lemmas 3.3 and 3.4 and Remark 3.5, it is sufficient to verify that 𝑆_𝑌⁻¹₁∶𝑌₁→𝑋(⊂𝑈) is closed because 𝑋 is compact. Assume that {(𝑦_𝛼,𝑥_𝛼)∶𝛼∈𝐼}⊂graph𝑆_𝑌⁻¹₁ satisfy (𝑦_𝛼,𝑥_𝛼)→(𝑦₀,𝑥₀)∈𝑌₁×𝑋, then 𝑦_𝛼∈𝑆𝑥_𝛼(𝛼∈𝐼). This implies that ∀𝑞∈𝑉^∗,∀𝛼∈Λ,𝜎^♭𝑆𝑥_𝛼,𝑞≤𝑞,𝑦_𝛼. (3.19) Let 𝑞∈𝑉^∗ be fixed. Since 𝑦_𝛼→𝑦₀∈𝑌₁, for any 𝜀>0, there exists 𝛼₀∈𝐼 such that |⟨𝑞,𝑦_𝛼−𝑦₀⟩|<𝜀for𝛼≻𝛼₀, which together with (3.19) yields 𝜎^♭𝑆𝑥_𝛼,𝑞−𝑞,𝑦₀−𝜀≤𝜎^♭𝑆𝑥_𝛼,𝑞−𝑞,𝑦_𝛼−𝜀−𝑞,𝑦_𝛼−𝑦₀≤0𝛼≻𝛼₀. (3.20) Therefore, {𝑥_𝛼∶𝛼≻𝛼₀}⊂𝑋^♭(𝑆,𝑞,𝑦₀,𝜀). This implies 𝑥₀∈𝑋^♭(𝑆,𝑞,𝑦₀,𝜀) because 𝑦₀∈𝑌₁ and thus 𝑋^♭(𝑆,𝑞,𝑦₀,𝜀) is closed. Hence, 𝜎^♭(𝑆𝑥₀,𝑞)≤⟨𝑞,𝑦₀⟩+𝜀. By letting 𝜀→0, it follows that 𝜎^♭(𝑆𝑥₀,𝑞)≤⟨𝑞,𝑦₀⟩ holds for any fixed 𝑞∈𝑉^∗. Since 𝑆𝑥₀ is closed convex by assumption (3.1), we have 𝑦₀∈𝑆𝑥₀ (also thanks to the Hahn-Banach separation theorem). Combing this with the fact 𝑦₀∈𝑌₁, we conclude that (𝑦₀,𝑥₀)∈ graph 𝑆_𝑌⁻¹₁. Therefore, 𝑆_𝑌⁻¹₁ defined by (3.5) is closed, and statement (ii) follows.
Proof of Part B. Since 𝑆 is u.h.c., by Definition 2.1 we know that for each 𝑞∈𝑉^∗, 𝑥↦𝜎^♭(𝑆𝑥,𝑞)=−𝜎^♯(𝑆𝑥,−𝑞) is lower semicontinuous. Hence the lower sections {𝑥∈𝑋∶𝜎^♭(𝑆𝑥,𝑞)≤𝜎^♯(𝑌,𝑞)} and {𝑥∈𝑋∶𝜎^♭(𝑆𝑥,𝑞)≤⟨𝑞,𝑦⟩+𝑐} of the function 𝑥↦𝜎^♭(𝑆𝑥,𝑞) are closed in 𝑋 for all 𝑞∈𝑉^∗,𝑦∈𝑌 and all 𝑐∈𝑅¹. This implies that all the conditions (a) from (3.2) to (3.4) are satisfied, and thus all the statements of part (A) are true. So the remaining is to show that both 𝑌∩𝑆𝑋 and graph 𝑆⁻¹_{𝑌∩𝑆𝑋} (where 𝑆⁻¹_{𝑌∩𝑆𝑋} is defined by (3.6)) are closed also thanks to Lemmas 3.3 and 3.4 and Remark 3.5.
(1) Assume that {𝑦_𝛼∶𝛼∈𝐼} is a generalized sequence of 𝑌∩𝑆𝑋 with 𝑦_𝛼→𝑦₀, then 𝑦₀∈𝑌 (because 𝑌 is compact), and for each 𝛼∈𝐼 there exists 𝑥_𝛼∈𝑋 such that 𝑦_𝛼∈𝑆𝑥_𝛼. As 𝑋 is compact, choosing a generalized subsequence if necessary, we may assume 𝑥_𝛼→𝑥₀∈𝑋. On the other hand, it is easy to see that in Definition 2.1, if 𝜏 denotes the original vector topology on 𝑉, then 𝑉 can be supplied with any compatible topologies 𝕋 of 𝜏 including the weak topology 𝜎(𝑉,𝑉^∗) because (𝑉,𝕋)^∗=(𝑉,𝜎(𝑉,𝑉^∗))^∗=𝑉^∗. Associating this with (3.1)(b) and using Hahn-Banach’s separation theorem, we conclude that 𝑆∶𝑋⊂𝑈→𝑉 is also a u.h.c. correspondence with nonempty closed convex values when 𝑉 is supplied with the weak topology 𝜎(𝑉,𝑉^∗), which implies by Lemma 3.6 that 𝑆 is closed. Therefore, 𝑦₀∈𝑆𝑥₀ and 𝑌∩𝑆𝑋 is closed.
(2) Assume that {(𝑦_𝛼,𝑥_𝛼)∶𝛼∈Λ}⊂graph𝑆⁻¹_{𝑌∩𝑆𝑋} is a generalized sequence such that (𝑦_𝛼,𝑥_𝛼)→(𝑦₀,𝑥₀), then 𝑦₀∈𝑌∩𝑆𝑋 (because 𝑌∩𝑆𝑋 is closed), 𝑥₀∈𝑋 and 𝑦_𝛼∈𝑆𝑥_𝛼(𝛼∈Λ). Thus, 𝑦₀∈𝑆𝑥₀ by the closeness of 𝑆, and 𝑆⁻¹_{𝑌∩𝑆𝑋} is closed. This completes the proof.

Remark 3.7. We claim that the part (A) of Theorem 3.1 does not need any conventional continuity conditions such as u.s.c., l.s.c. and u.h.c. See the following counterexample.

Example 3.8. Let 𝑈=𝑅,𝑉=𝑅^𝑛(𝑛≥2), 𝑒_𝑖=(0,…,0^{𝑖0𝑥0𝑓𝑒37},1,0,…,0)(𝑖=1,2,…,𝑛), 𝑋=[0,1],𝑌=[0,1]^𝑛, and let 𝑄₀ be the set of all rational numbers of [0,1], 𝑃₀=[0,1]⧵𝑄₀. Assume that 𝑘∈{1,2,…,𝑛} is a fixed number, 𝐿 a single-valued map from 𝑅¹ to 𝑅^𝑛, and 𝑆 a set-valued map from 𝑋 to 𝑅^𝑛 defined by 𝐿𝑥=𝑥𝑒_𝑘,for𝑥∈𝑅¹[],𝑆𝑥=𝑥−3,𝑥+4^𝑛if𝑥∈𝑄₀,[𝑥−4,𝑥+3]^𝑛if𝑥∈𝑃₀. (3.21) Then we have the following (i) It is easy to see that 𝐿∈ℒ(𝑅¹,𝑅^𝑛) (i.e., 𝐿 is a continuous linear map from 𝑅¹ to 𝑅^𝑛), 𝑆𝑥 is a nonempty convex compact subset of 𝑅^𝑛 for each 𝑥∈𝑋, and 𝑆 is not u.s.c. or l.s.c. at any point of 𝑋. It can also be shown that 𝑆 is not u.h.c. on 𝑋. Indeed, if 𝑞=(𝑞₁,𝑞₂,…,𝑞_𝑛)∈𝑉^∗=𝑅^𝑛 with 𝑞_𝑖>0(𝑖=1,2,…,𝑛), then 𝑥↦𝜎^#(𝑆𝑥,𝑞) is not u.s.c. at any points of 𝑃₀ because 𝜎^♯𝑆𝑥,𝑞=sup_{𝑦1,𝑦2,…,𝑦𝑛}^𝑛_{∈𝑆𝑥𝑖=1}𝑞_𝑖,𝑦_𝑖=^𝑛𝑖=1𝑞_𝑖(𝑥+4)if𝑥∈𝑄₀,_𝑛𝑖=1𝑞_𝑖(𝑥+3)if𝑥∈𝑃₀. (3.22)
(ii) Now we verify that (3.2)–(3.4) hold for this example.
(1) It is easy to see that ∀𝑝∈𝑅¹,∀𝑥∈𝑋=[0,1], and ∀𝑦∈[0,1], we have 𝜎^♭]−[]=𝑝()𝑝()𝜎[𝑥−3,𝑥+40,1,𝑝𝑥−4<0,if𝑝>0,𝑥+4<0,if𝑝<0,_♭[]=𝑝𝑝𝜎𝑥−3,𝑥+4−𝑦,𝑝𝑥−3−𝑦<0,if𝑝>0,𝑥+4−𝑦<0,if𝑝<0,_♭[]−[]=𝑝()𝑝()𝜎𝑥−4,𝑥+30,1,𝑝𝑥−5<0,if𝑝>0,𝑥+3<0,if𝑝<0,_♭[]=𝑝𝑝𝑥−4,𝑥+3−𝑦,𝑝𝑥−4−𝑦<0,if𝑝>0,𝑥+3−𝑦<0,if𝑝<0. (3.23) Since ∏^𝑛_𝑖=1𝐴_𝑖−∏^𝑛_𝑖=1𝐵_𝑖=∏^𝑛_𝑖=1(𝐴_𝑖−𝐵_𝑖) and 𝜎^♭(∏^𝑛_𝑖=1𝐹_𝑖,𝑞)=∑^𝑛_𝑖=1𝜎^♭(𝐹_𝑖,𝑞_𝑖) for 𝐴_𝑖,𝐵_𝑖,𝐹_𝑖⊂𝑅¹(𝑖=1,2,…,𝑛) and 𝑞=(𝑞₁,𝑞₂,…,𝑞_𝑛)∈𝑅^𝑛, from (3.23) and using the fact that 𝜎^♭(𝐹₀,0)=0 for any 𝐹₀⊆𝑅¹, we obtain that ∀𝑥∈𝑋=[0,1], ∀𝑞=(𝑞₁,𝑞₂,…,𝑞_𝑛)∈𝑅^𝑛 and ∀𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)∈𝑌=[0,1]^𝑛, 𝜎^♭𝑆𝑥−𝑌,𝑞=^𝑛_𝑖=1𝜎^♭[𝑥−3,𝑥+4]−[0,1],𝑞_𝑖𝜎≤0,_♭𝑆𝑥−𝑦,𝑞=^𝑛_𝑖=1𝜎^♭[𝑥−3,𝑥+4]−𝑦_𝑖,𝑞_𝑖≤0,if𝑥∈𝑄₀𝜎_♭,𝑆𝑥−𝑌,𝑞=^𝑛_𝑖=1𝜎^♭[𝑥−4,𝑥+3]−[0,1],𝑞_𝑖𝜎≤0,_♭𝑆𝑥−𝑦,𝑞=^𝑛_𝑖=1𝜎^♭[𝑥−4,𝑥+3]−𝑦_𝑖,𝑞_𝑖≤0,if𝑥∈𝑃₀. (3.24)(3.25) Both (3.24) and (3.25) imply that ∀𝑞∈𝑅^𝑛,∀𝑦∈𝑌,∀𝑐≥0,𝑥∈𝑋∶𝜎^♭𝑆𝑥,𝑞≤𝜎^♯𝑌,𝑞=[0,1],𝑥∈𝑋∶𝜎^♭𝑆𝑥,𝑞≤𝑞,𝑦=[0,1],areclosed,𝑥∈𝑋∶𝜎^♭𝑆𝑥,𝑞≤𝑞,𝑦+𝑐=[0,1]. (3.26) Therefore, all (a) of (3.2)–(3.4) are satisfied.
(2) As 𝐿𝑥=(0,…,0^{𝑘0𝑥0𝑓𝑒37},𝑥,0,…,0)(𝑥∈𝑋=[0,1]), 𝐿𝑋=∏^𝑛_𝑖=1𝑋_𝑖, where 𝑋_𝑘=𝑋 and 𝑋_𝑖={0} for 𝑖≠𝑘, then for any 𝑞=(𝑞₁,𝑞₂,…,𝑞_𝑛)∈𝑅^𝑛 and any 𝑥∈𝑋, we see that 𝑞,𝐿𝑥=𝜎^♭𝐿𝑋,𝑞⟺𝑞_𝑘𝑥=𝜎^♭𝑋,𝑞_𝑘=0,𝑞_𝑘𝑞≥0_𝑘,𝑞_𝑘⟺<0𝑥=0,if𝑞_𝑘>0,𝑥=1,if𝑞_𝑘<0,𝑥∈𝑋,if𝑞_𝑘=0. (3.27) From (3.26) and (3.27), it follows that ∀𝑞=(𝑞₁,𝑞₂,…,𝑞_𝑛)∈𝑅^𝑛 and ∀𝑥∈𝑋=[0,1] with 𝐿𝑥∈𝜕(𝐿𝑋,𝑞), 𝜎^♭=𝜎𝑆𝑥−𝑌,𝑞^♭𝑆0−𝑌,𝑞≤0,if𝑞_𝑘𝜎>0,_♭𝑆1−𝑌,𝑞≤0,if𝑞_𝑘𝜎<0,_♭𝑆𝑥−𝑌,𝑞≤0,if𝑞_𝑘=0,(𝑥∈𝑋), (3.28)∀𝑦∈𝑌,𝜎^♭=𝜎𝑆𝑥−𝑦,𝑞^♭𝑆0−𝑦,𝑞≤0,if𝑞_𝑘𝜎>0,_♭𝑆1−𝑦,𝑞≤0,if𝑞_𝑘𝜎<0,_♭𝑆𝑥−𝑦,𝑞≤0,if𝑞_𝑘=0,(𝑥∈𝑋). (3.29) Both (3.28) and (3.29) show that ∀𝑞∈𝑅^𝑛,∀𝑥∈𝑋with𝐿𝑥∈𝜕(𝐿𝑋,𝑞), we have 𝜎^♭𝑆𝑥,𝑞≤𝜎^♯𝑌,𝑞,𝜎^♭𝑆,𝑞≤𝑞,𝑦𝑦∈𝑌. (3.30) Hence, all (b) of (3.2)-(3.4) are satisfied. Therefore, Theorem 3.1(A) holds for this example. Indeed, we have 𝑌₀=𝑌₁=𝑌⊂𝑆𝑋.

Remark 3.9. In case 𝑈=𝑉 and 𝐿=𝐼 (the identity map from 𝑈 to itself), then part (A) of Theorem 3.1 is just the main results in [7] and part (B) returns to the Theorem 1.1.

Remark 3.10. From the proof of Theorem 3.1(A), we see that the Ky-Fan inequality is very important to the existence part. Regarding this inequality, some extensions have been made by Lignola [19], Lin and Simons [20], Alzer [21], as well as S. J. Li and X. B. Li [22], we think that probably in the future, these new Ky-Fan inequalities could also be used to obtain some new useful Rogalski-Cornet type theorems.

3.2. Application to (1.5)

From Theorem 3.1, we can obtain a solvability theorem to (1.5) as follows.

Theorem 3.11. (i)  If 𝑣∈𝑉, and (3.2) holds for 𝑌={𝑣}, then 𝑣∈𝑆𝑋, that is, (1.5)(a) is solvable.
(ii) Assume that ∀𝑞∈𝑉^∗,∀𝑥∈𝑋𝑤𝑖𝑡ℎ𝐿𝑥∈𝜕𝐿𝑋,𝑞,𝜎^♭𝑆𝑥,𝑞≤𝑞,𝐿𝑥. (3.31) Then (1) 𝐿𝑋⊂𝑆𝑋 if ∀𝑞∈𝑉^∗,∀𝑦∈𝐿𝑋,𝑥∈𝑋∶𝜎^♭𝑆𝑥,𝑞≤𝑞,𝑦𝑖𝑠𝑐𝑙𝑜𝑠𝑒𝑑. (3.32) In particular, (a) Equation (1.5)(a) is solvable (i.e., 𝑣∈𝑆𝑋) when 𝑣∈𝐿𝑋,(b) Equation (1.5)(b) has a solution (i.e., 𝑆𝑋∩𝑌≠∅) when 𝑌∩𝐿𝑋≠∅,(c) Equation (1.5)(c) is solvable (i.e., 𝑌⊂𝑆𝑋) when 𝑌⊂𝐿𝑋. (2) The inverse of 𝑆 restricting to LX defined by 𝑆⁻¹_𝐿𝑋∶𝐿𝑋→𝑋∶𝑦↦𝑆⁻¹_𝐿𝑋(𝑦)={𝑥∈𝑋∶𝑦∈𝑆𝑥} is u.s.c. and u.h.c. if ∀𝑦∈𝐿𝑋,∀𝑞∈𝑉^∗,∀𝑐≥0,𝑥∈𝑋∶𝜎^♭𝑆𝑥,𝑞≤𝑞,𝑦+𝑐𝑖𝑠𝑐𝑙𝑜𝑠𝑒𝑑. (3.33)
(iii) If 𝑆 is u.h.c., then the assumptions (3.2)(a), (3.32), and (3.33) in statements (i) and (ii) can be removed.

Proof. (i) Applying Theorem 3.1(A)(i)(1) to 𝑌={𝑣}, we see that the statement (i) is true.
(ii) Set 𝑌=𝐿𝑋. Since 𝑋 is a convex compact subset of 𝑈, and 𝐿 a continuous linear map from 𝑈 to 𝑉, we see that 𝑌 is a convex compact subset of 𝑉. Moreover, for each 𝑦∈𝐿𝑋, as 𝐿𝑥∈𝜕(𝐿𝑋,𝑞) equals to ⟨𝑞,𝐿𝑥⟩=𝜎^♭(𝐿𝑋,𝑞), from (3.31), we obtain that ∀𝑞∈𝑉^∗,∀𝑥∈𝑋with𝐿𝑥∈𝜕𝐿𝑋,𝑞,𝜎^♭𝑆𝑥,𝑞≤𝑞,𝐿𝑥=𝜎^♭𝐿𝑋,𝑞≤𝑞,𝑦. (3.34) This means that the assumption (3.3)(b) holds for any 𝑦∈𝐿𝑋.
(1)Associating (3.32) with (3.34) we know that (3.3) holds for all 𝑦∈𝑌, which implies by Theorem 3.1(A)(i)(2), for 𝑌=𝐿𝑋, that 𝐿𝑋=𝑌₀⊂𝑆𝑋, and thus all the statements (a), (b), and (c) of (ii)(1) are true.(2)From (3.33) and (3.34), we see that 𝑌₁ defined by (3.4) is precisely equal to 𝐿𝑋. Hence by Theorem 3.1(A)(ii), the statement (ii)(2) follows.
(iii) The conclusion of (iii) is clear. This completes the proof.

4. Conclusion

In this paper, we have proved a Rogalski-Cornet type theorem (namely, Theorem 3.1) based on two Hausdorff locally convex vector spaces, and presented a counterexample (i.e., Example 3.8) to show that the first part of this theorem does not need any conventional continuity conditions such as upper semicontinuous, lower semicontinuous and upper hemicontinuous conditions. Applying this theorem, by Theorem 3.11 we have also provided the solvability results for a class of abstract von Neumann input-output inclusion system (namely, (1.5)).

Acknowledgment

This work is supported by the National Science Foundation of China (no. 61179027).