Existence Results for Generalized Vector Equilibrium Problems with Multivalued Mappings via KKM Theory

Abstract

We first define upper sign continuity for a set-valued mapping and then we consider two types of generalized vector equilibrium problems in topological vector spaces and provide sufficient conditions under which the solution sets are nonempty and compact. Finally, we give an application of our main results. The paper generalizes and improves results obtained by Fang and Huang in (2005).

1. Introduction and Preliminaries

Throughout this paper, unless otherwise specified, we always let 𝑋 and 𝑌 be real Hausdorff topological vector spaces, 𝐾⊆𝑋 a nonempty convex set, 𝐶∶𝐾→2^𝑌 with pointed closed cone convex values (we recall that a subset 𝐴 of 𝑌 is convex cone and pointed whenever 𝐴+𝐴⊆𝑌,𝑡𝐴⊆𝐴, for 𝑡≥0, and 𝐴∩−𝐴={0}, resp.), where 2^𝑌 denotes all the subsets of 𝑌. Denote by 𝐿(𝑋,𝑌) the set of all continuous linear mappings from 𝑋 into 𝑌. For any given 𝑙∈𝐿(𝑋,𝑌),𝑥∈𝑋, let ⟨𝑙,𝑥⟩ denote the value of 𝑙 at 𝑥. Let 𝑇∶𝐾→𝐿(𝑋,𝑌) and 𝑔∶𝐾→𝐾 be two mappings. Finally, let 𝐹∶𝐾×𝐾→2^𝑌 be a set-valued mapping. We need the following definitions and results in the sequel.Definition 1.1. Let 𝐹∶𝐾×𝐾→2^𝑌 be a set-valued mapping. One says that 𝐹 is
(i)strongly 𝐶-pseudomonotone if, for any given 𝑥 and 𝑦∈𝐾,⊈−int𝐶(𝑥)⇒𝐹(𝑦,𝑥)⊆−𝐶(𝑦).(1.1)(ii)𝐶-pseudomonotone if, for any given 𝑥 and 𝑦∈𝐾,𝐹(𝑥,𝑦)⊈−𝐶(𝑥)⧵{0}⇒𝐹(𝑦,𝑥)⊆−𝐶(𝑦).(1.2)Remark 1.2. (1) Strongly 𝐶-pseudomonotonicity implies 𝐶-pseudomonotonicity.
(2) Let 𝑇∶𝐾→2^{𝐿(𝑋,𝑌)} be a set-valued mapping and let 𝑔∶𝐾→𝐾 be a mapping. If we define 𝐹(𝑥,𝑦)=⟨𝑇𝑥,𝑦−𝑔(𝑥)⟩, for each (𝑥,𝑦)∈𝐾×𝐾, then strongly 𝐶-pseudomonotonicity and 𝐶-pseudomonotonicity reduce to the strongly 𝐶-pseudomonotonicity and 𝐶-pseudomonotonicity, of 𝑇 with respect to 𝑔, respectively, introduced in [1].

Definition 1.3. Let 𝑋 and 𝑌 be two topological spaces. A set-valued mapping 𝐺∶𝑋→2^𝑌 is called
(i)upper semicontinuous (u.s.c.) at 𝑥∈𝑋 if for each open set 𝑉 containing 𝐺(𝑥), there is an open set 𝑈 containing 𝑥 such that for each 𝑡∈𝑈,𝐺(𝑡)⊆𝑉;𝐺 is said to be u.s.c. on 𝑋 if it is u.s.c. at all 𝑥∈𝑋;(ii)lower semicontinuous (l.s.c.) at 𝑥∈𝑋 if for each open set 𝑉 with 𝐺(𝑥)∩𝑉≠∅, there is an open set 𝑈 containing 𝑥 such that for each 𝑡∈𝑈,𝐺(𝑡)∩𝑉≠∅;𝐺 is said to be l.s.c. on 𝑋 if it is l.s.c. at all 𝑥∈𝑋;(iii)closed if the graph of 𝐺, that is, the set {(𝑥,𝑦)∶𝑥∈𝑋,𝑦∈𝐺(𝑥)}, is a closed set in 𝑋×𝑌;(iv)compact if the closure of range 𝐺, that is, cl𝐺(𝑋), is compact, where 𝐺(𝑋)=⋃_𝑥∈𝑋𝐺(𝑥).Remark 1.4. One can see that (ii) is equivalent to the following statement:
𝐺 is l.s.c. at 𝑥∈𝑋 if for each closed set 𝐶⊆𝑌, any net {𝑥_𝛼}⊆𝐾,𝑥_𝛼 converges to 𝑥, and 𝐺(𝑥_𝛼)⊆𝐶, for all 𝛼 imply that 𝐺(𝑥)⊆𝐶.

Lemma 1.5 (see [2]). Let 𝑋 and 𝑌 be two topological spaces. Suppose that 𝐺∶𝑋→2^𝑌 is a set-valued mapping. Then the following statements are true.
(a)If 𝐺 is closed and compact, then 𝐺 is u.s.c.(b)Let, for any 𝑥∈𝑋,𝐺(𝑥) be compact. If 𝐺 is u.s.c. on 𝑋 then for any net {𝑥_𝛼}⊂𝑋 such that 𝑥_𝛼→𝑥 and for every 𝑦_𝛼∈𝐺(𝑥_𝛼), there exist 𝑦∈𝐺(𝑥) and a subnet {𝑦_𝛽} of {𝑦_𝛼} such that 𝑦_𝛽→𝑦.

For the converse of (b) in Lemma 1.5, we refer the reader to [3].

Definition 1.6. Let 𝑋 be a topological vector space and 𝑌 a topological space. A set-valued mapping 𝐺∶𝑋→2^𝑌 is called upper hemicontinuous if the restriction of 𝐺 on straight lines is upper semicontinuous.Definition 1.7. One says that the mapping 𝐺∶𝐾×𝐾→2^𝑌 is 𝐶-upper sign continuous if, for all 𝑥,𝑦∈𝐾, the following implication holds: 𝐺((1−𝑡)𝑥+𝑡𝑦,𝑦)∩𝐶((1−𝑡)𝑥+𝑡𝑦)≠∅,∀𝑡∈]0,1[⇒𝐺(𝑥,𝑦)∩𝐶(𝑥)≠∅.(1.3)Remark 1.8. Let 𝑓∶𝐾×𝐾→ℝ be a mapping. If we define 𝐺(𝑥,𝑦)={𝑓(𝑥,𝑦)}, for all 𝑥,𝑦∈𝐾 and 𝐶(𝑥)=[0,∞), then Definition 1.7 reduces to the upper sign continuous introduced by Bianchi and Pini in [4]. The upper sign continuity notion was first introduced by Hadjisavvas [5] for a single-valued mapping in the framework of variational inequality problems.

2. Main Results

In this section, we consider the following generalized vector equilibrium problems (for short, GVEPs) in the topological vector space setting:

(GVEP₁) find 𝑥∈𝐾 such that 𝐹(𝑥,𝑦)⊈−int𝐶(𝑥),∀𝑦∈𝐾, and(GVEP₂) find 𝑥∈𝐾 such that 𝐹(𝑥,𝑦)⊈−𝐶(𝑥)⧵{0},∀𝑦∈𝐾.

Clearly, a solution of GVEP₂ is also a solution of problem GVEP₁. We need the following lemma in the sequel.Lemma 2.1. Suppose that
(i)𝐹 is 𝐶-pseudomonotone;(ii)𝐹(𝑥,𝑥)∩𝐶(𝑥)≠∅ for each 𝑥∈𝐾;(iii)𝐹 is 𝐶-upper sign continuous;(iv) for each fixed 𝑥∈𝐾, the mapping 𝑧→𝐹(𝑥,𝑧) is convex, that is, 𝐹(𝑥,(1−𝑡)𝑦+𝑡𝑧)⊆(1−𝑡)𝐹(𝑥,𝑦)+𝑡𝐹(𝑥,𝑧)−𝐶(𝑥),∀𝑦∈𝐾,∀𝑡∈]0,1[. Then for any given 𝑦∈𝐾, the following are equivalent:
(I)𝐹(𝑦,𝑧)⊈−𝐶(𝑦)⧵{0},∀𝑧∈𝐾;(II)𝐹(𝑧,𝑦)⊆−𝐶(𝑧),∀𝑧∈𝐾.
Proof. (I)⇒(II) is obvious from the definition of 𝐶-pseudomonotonicity of 𝐹. Suppose that (II) holds. For each 𝑧∈𝐾, put 𝑧_𝑡=𝑦+𝑡(𝑧−𝑦), where 𝑡∈]0,1[ and 𝑦∈𝐾 as above. By (II), we have 𝐹(𝑧_𝑡,𝑦)⊆−𝐶(𝑧_𝑡),∀𝑡∈]0,1[.(2.1) We claim that 𝐹(𝑧_𝑡,𝑧)∩𝐶(𝑧_𝑡)≠∅. Suppose 𝐹(𝑧_𝑡,𝑧)∩𝐶(𝑧_𝑡)=∅, for some 𝑡∈]0,1[. Then 𝐹(𝑧_𝑡,𝑧)⊆𝑌⧵𝐶(𝑧_𝑡),forthis𝑡∈]0,1[,(2.2) and so 𝐹(𝑧_𝑡,𝑧_𝑡)⊆(1−𝑡)𝐹(𝑧_𝑡,𝑦)+𝑡𝐹(𝑧_𝑡,𝑧)−𝐶(𝑧_𝑡)⊆−𝐶(𝑧_𝑡)+(𝑌⧵𝐶(𝑧_𝑡))−𝐶(𝑧_𝑡)=(𝑌⧵𝐶(𝑧_𝑡))−𝐶(𝑧_𝑡)⊆𝑌⧵𝐶(𝑧_𝑡), which contradicts (ii) (note the first inclusion follows from (iv), the second inclusion follows from (2.1) and (2.2), and the third follows from the relation 𝑌⧵𝐶(𝑧_𝑡)−𝐶(𝑧_𝑡)⊆𝑌⧵𝐶(𝑧_𝑡)). Therefore, for all 𝑡∈]0,1[, the set 𝐹(𝑧_𝑡,𝑧)∩𝐶(𝑧_𝑡) is nonempty. Thus, by (iii) there is a 𝑢∈𝐹(𝑦,𝑧)∩𝐶(𝑦). Hence, since 𝐶(𝑦)∩−𝐶(𝑦)⧵{0}=∅, we get 𝑢∉(−𝐶(𝑦)⧵{0}). Consequently, 𝐹(𝑦,𝑧)⊈−𝐶(𝑦)⧵{0}. This completes the proof.Remark 2.2. If the set-valued mapping 𝐶∶𝐾→2^𝑌 has closed graph and for each fixed 𝑧∈𝐾 the mapping 𝑥→𝐹(𝑥,𝑧) is upper hemicontinuous with nonempty compact values, then condition (iii) in Lemma 2.1 holds. To see this, let 𝑥 and 𝑦 be arbitrary elements of 𝐾 and 𝑢_𝑡∈𝐹(𝑧_𝑡,𝑦)∩𝐶(𝑧_𝑡)≠∅, where 𝑧_𝑡=(1−𝑡)𝑥+𝑡𝑦,𝑡∈]0,1[. By Lemma 1.5(b), there exists a subnet of (𝑢_𝑡) (without loss of generality (𝑢_𝑡)) and 𝑢∈𝐹(𝑥,𝑦) such that 𝑢_𝑡→𝑢, where 𝑡→0. Now, since 𝐶∶𝐾→2^𝑌 has closed graph (note 𝑢_𝑡→𝑢 and 𝑧_𝑡→𝑥 as 𝑡→0) and 𝑢_𝑡∈𝐶(𝑧_𝑡), we have 𝑢∈𝐶(𝑥). Hence, 𝑢∈𝐹(𝑥,𝑦)∩𝐶(𝑥) and so 𝐹(𝑥,𝑦)∩𝐶(𝑥)≠∅. This shows that 𝐹 is 𝐶-upper sign continuous. Therefore, Lemma 2.1 improves Lemma 2.3 in [1].

By a similar argument as in Lemma 2.1 and using Remark 2.2, we can deduce the following result.Lemma 2.3. Suppose that
(i)for each fixed 𝑧∈𝐾, the mapping 𝑥→𝐹(𝑥,𝑧) is upper semicontinuous with compact values;(ii)𝐹 is strongly 𝐶-pseudomonotone;(iii)𝐹(𝑥,𝑥)⊈−int𝐶(𝑥), for each 𝑥∈𝐾;(iv) the mapping 𝑥→𝑊(𝑥)=𝑌⧵(−int𝐶(𝑥)), for each 𝑥∈𝐾, has closed graph;(v) for each fixed 𝑥∈𝐾, the mapping 𝑧→𝐹(𝑥,𝑧) is convex. Then for any given 𝑦∈𝐾, the following are equivalent:
(I) 𝐹(𝑦,𝑧)⊈−int𝐶(𝑦),∀𝑧∈𝐾;(II) 𝐹(𝑧,𝑦)⊆−𝐶(𝑧),∀𝑧∈𝐾.Remark 2.4. Let 𝑇∶𝐾→2^{𝐿(𝑋,𝑌)} be a set-valued mapping. If we define 𝐹(𝑥,𝑦)=⟨𝑇𝑥,𝑦−𝑥⟩, where 𝑥,𝑦∈𝐾, then Lemma 2.3 reduces to Lemma 3 of Yin and Xu [6].Lemma 2.5. Under the assumptions of Lemma 2.1, the solution set of (GVEP₂) is convex.Proof. Let 𝑥₁ and 𝑥₂ be solutions of (GVEP₂). By Lemma 2.1, we have 𝐹(𝑧,𝑥_𝑖)⊆−𝐶(𝑧),∀𝑧∈𝐾,𝑖=1,2.(2.3) From this and condition (iv) of Lemma 2.1, for all 𝑡∈]0,1[, we deduce that 𝐹(𝑧,(1−𝑡)𝑥₁+𝑡𝑥₂)⊆(1−𝑡)𝐹(𝑧,𝑥₁)+𝑡𝐹(𝑧,𝑥₂)−𝐶(𝑧)⊆−𝐶(𝑧),(2.4) for all 𝑧∈𝐾. Hence, from Lemma 2.1, we get 𝐹((1−𝑡)𝑥₁+𝑡𝑥₂,𝑧)⊈−𝐶((1−𝑡)𝑥₁+𝑡𝑥₂))⧵{0},∀𝑧∈𝐾.(2.5) This means that (1−𝑡)𝑥₁+𝑡𝑥₂ is a solution of (GVEP₂). The proof is complete.

Similarly, we can prove the following lemma.Lemma 2.6. Under the assumptions of Lemma 2.3, the solution set of (GVEP₁) is convex.Remark 2.7. Lemma 2.5 extends Theorem 3 of Yin and Xu [6] and Lemma 2.5 of Fang and Huang [1].Definition 2.8. Let 𝐾₀ be a nonempty subset of 𝐾. A set-valued mapping Γ∶𝐾₀→2^𝐾 is said to be a KKM map if co𝐴⊆⋃_𝑥∈𝐴Γ(𝑥), for every finite subset 𝐴 of 𝐾₀, where co denotes the convex hull.Lemma 2.9 ((Fan-KKM lemma) [7]). Let 𝐾 be a nonempty subset of a topological vector space 𝑋 and Γ∶𝐾→2^𝑋 be a KKM mapping with closed values. Assume that there exists a nonempty compact convex subset 𝐵 of 𝐾 such that ⋂_𝑥∈𝐵Γ(𝑥) is compact. Then_𝑥∈𝐾Γ(𝑥)≠∅.(2.6)Lemma 2.10 (see [8]). Let 𝐾 be a convex subset of a metrizable topological vector space 𝑋 and 𝐹∶𝐾→2^𝐾 be a compact upper semicontinuous set-valued mapping with nonempty closed convex values. Then 𝐹 has a fixed point in 𝐾.Theorem 2.11. Let all the assumptions of Lemma 2.1 hold and for each fixed 𝑥∈𝐾, the mapping 𝑦→𝐹(𝑥,𝑦) is lower semicontinuous, where 𝑦∈𝐾. If there exist a nonempty compact subset 𝐵 of 𝐾 and a nonempty convex compact subset 𝐷 of 𝐾 such that, for each 𝑥∈𝐾⧵𝐵 there exists 𝑦∈𝐷 such that 𝐹(𝑦,𝑥)⊈−𝐶(𝑦), then the solution set of problem (GVEP₂) is nonempty and compact in 𝐾.Proof. Define Γ,Γ∶𝐾→2^𝐾 by Γ(𝑦)={𝑥∈𝐾∶𝐹(𝑥,𝑦)⊈−𝐶(𝑥)⧵{0}},Γ(𝑦)={𝑥∈𝐾∶𝐹(𝑦,𝑥)⊆−𝐶(𝑦)}.(2.7) We claim that Γ is a KKM mapping. If not, there exist 𝑦₁,𝑦₂,…,𝑦_𝑛∈𝐾 and 𝑡_𝑖>0,∑^𝑛_𝑖=1𝑡_𝑖=1 such that 𝑧=∑^𝑛_𝑖=1𝑡_𝑖𝑦_𝑖∉Γ(𝑦_𝑖), that is, 𝐹(𝑧,𝑦_𝑖)⊆−𝐶(𝑧)⧵{0},𝑖=1,2,3,…,𝑛,(2.8) and so, since −𝐶(𝑧) is a closed convex pointed cone, ^𝑛_𝑖=1𝑡_𝑖𝐹(𝑧,𝑦_𝑖)⊆−𝐶(𝑧)⧵{0}.(2.9) It follows from condition (iv) of Lemma 2.1 that 𝐹(𝑧,𝑧)⊆^𝑛_𝑖=1𝑡_𝑖𝐹(𝑧,𝑦_𝑖)−𝐶(𝑧).(2.10) Now, by combining (2.9) and (2.10), we get 𝐹(𝑧,𝑧)⊆^𝑛_𝑖=1𝑡_𝑖𝐹(𝑧,𝑦_𝑖)−𝐶(𝑧)⊆−𝐶(𝑧)⧵{0}−𝐶(𝑧)⊆−𝐶(𝑧)⧵{0},(2.11) which is a contradiction to condition (ii) of Lemma 2.1. Therefore, Γ is a KKM mapping and so Γ is also a KKM mapping (note, Γ(𝑦)⊆Γ(𝑦), for all 𝑦∈𝐾). By Remark 1.4, the values of Γ are closed in 𝐾 (note, for each fixed 𝑥∈𝐾, the mapping 𝑦→𝐹(𝑥,𝑦) is lower semicontinuous) and by our assumption, we obtain that ⋂_𝑦∈𝐷Γ(𝑦) is a closed subset of the compact set 𝐵 and hence ⋂_𝑦∈𝐷Γ(𝑦) is compact in 𝐾. Therefore, Γ fulfils all the assumptions of Lemma 2.9 and so ⋂_𝑦∈𝐾Γ(𝑦)≠∅. This means that there exists 𝑧∈𝐾 such that 𝐹(𝑦,𝑧)⊆−𝐶(𝑧),∀𝑦∈𝐾.(2.12) Now, it follows from Lemma 2.1 that 𝐹(𝑧,𝑦)⊈−𝐶(𝑦)⧵{0},∀𝑦∈𝐾,(2.13) and hence 𝑧 is a solution of the problem (GVEP₂). This proves that the solution set of (GVEP₂) is nonempty. By Lemma 2.1, the solution set of (GVEP₂) equals ⋂_𝑦∈𝐾Γ(𝑦) and so it is a compact set in 𝐾 (note, in the above that the set ⋂_𝑦∈𝐾Γ(𝑦) is a closed subset of the compact set 𝐵). The proof is complete.

As an application of Theorem 2.11, we derive the existence result for a solution of the following problem which consists of finding a 𝑢∈𝐾 such that ⟨𝐴(𝑢,𝑢),𝑣−𝑔(𝑢)⟩⊈−𝐶(𝑢)⧵{0},∀𝑣∈𝐾,(2.14) where 𝐴∶𝐾×𝐾→2^{𝐿(𝑋,𝑌)} and 𝑔∶𝐾→𝐾.

This problem was considered by Fang and Huang [1] in reflexive Banach spaces setting for a set-valued mapping which is demi-𝐶-pseudomonotone.Theorem 2.12. Let 𝑋 be metriziable topological vector space, 𝐾 nonempty convex subset of 𝑋, 𝐴∶𝐾×𝐾→2^{𝐿(𝑋,𝑌)}, and 𝑔∶𝐾→𝐾 be two mappings. Assume that
(i)for each fixed 𝑤∈𝐾, the mapping (𝑢,𝑣)→⟨𝐴(𝑤,𝑢),𝑣−𝑔(𝑢)⟩ is 𝐶-pseudomonotone and 𝐶-upper sign continuous;(ii)⟨𝐴(𝑤,𝑢),𝑢−𝑔(𝑢)⟩∩𝐶(𝑢)≠∅, for each (𝑤,𝑢);(iii) for each fixed 𝑣∈𝐾, the mapping (𝑤,𝑢)→⟨𝐴(𝑤,𝑢),𝑢−𝑔(𝑣)⟩ is lower semicontinuous;(iv) for each finite dimensional subspace 𝑀 of 𝑋 with 𝐾_𝑀=𝐾∩𝑀≠∅, there exist compact subset 𝐵_𝑀 and compact convex subset 𝐷_𝑀 of 𝐾_𝑀 such that ∀(𝑤,𝑧)∈𝐾_𝑀×(𝐾_𝑀⧵𝐵_𝑀), ∃𝑢∈𝐷_𝑀 such that ⟨𝐴(𝑤,𝑢),𝑧−𝑔(𝑢)⟩⊈−𝐶(𝑢). Then there exists 𝑢∈𝐾 such that ⟨𝐴(𝑢,𝑢),𝑣−𝑔(𝑢)⟩⊈−𝐶(𝑢)⧵{0},∀𝑣∈𝐾.(2.15)Proof. Let 𝑀⊂𝑋 be a finite dimensional subspace with 𝐾_𝑀=𝐾∩𝑀≠∅. For each fixed 𝑤∈𝐾, consider the problem of finding a 𝑢∈𝐾_𝑀 such that⟨𝐴(𝑤,𝑢),𝑣−𝑔(𝑢)⟩⊈−𝐶(𝑢)⧵{0},∀𝑣∈𝐾_𝑀.(2.16)By Theorem 2.11, the problem (2.16) has a nonempty compact solution set in 𝐾 (note, in Theorem 2.11 take 𝐹(𝑢,𝑣)=⟨𝐴(𝑤,𝑢),𝑣−𝑔(𝑢)⟩,(𝑢,𝑣)∈𝐾×𝐾). For 𝑤∈𝐾_𝑀, we define a set-valued mapping 𝑇∶𝐾_𝑀→2^𝐾𝑀 by𝑇(𝑤)={𝑢∈𝐾_𝑀∶⟨𝐴(𝑤,𝑢),𝑣−𝑔(𝑢)⟩⊈−𝐶(𝑢)⧵{0},∀𝑣∈𝐾_𝑀}.(2.17)Then 𝑇(𝑤) is a nonempty closed subset of 𝐵_𝑀, in fact, 𝑇(𝑤) is the solution set of (2.16) corresponding to 𝑤. By Lemma 2.1, we have𝑇(𝑤)={𝑢∈𝐾_𝑀∶⟨𝐴(𝑤,𝑣),𝑢−𝑔(𝑣)⟩⊆−𝐶(𝑣),∀𝑣∈𝐾_𝑀},(2.18)which is a convex set. By condition (iii) via Remark 1.4, 𝑇 is closed. By (iv), we have 𝑇(𝐾_𝑀)=⋃_{𝑤∈𝐾𝑀}𝑇(𝑤)⊆𝐵_𝑀. Hence, Lemma 1.5(a) implies that 𝑇 is upper semicontinuous. Hence, 𝑇 satisfies all the assumptions of Lemma 2.1 and so 𝑇 has a fixed point 𝑤₀∈𝐾_𝑀, that is,⟨𝐴(𝑤₀,𝑣),𝑤₀−𝑔(𝑣)⟩⊆−𝐶(𝑣),∀𝑣∈𝐾_𝑀.(2.19)Set ℳ={𝑀⊂𝑋∶𝑀 is a finite dimensional subspace with 𝐾_𝑀≠∅} and for 𝑀∈ℳ,𝑊_𝑀={𝑢∈𝐾_𝑀∶⟨𝐴(𝑢,𝑣),𝑢−𝑔(𝑣)⟩⊆−𝐶(𝑣),∀𝑣∈𝐾_𝑀}.(2.20)By (2.19) and conditions (iii) and (iv), 𝑊_𝑀 is a nonempty and closed subset of the compact set 𝐵_𝑀 and hence 𝑊_𝑀 is compact in 𝐾. Let {𝑀_𝑖}^𝑛_𝑖=1 be a finite subset of ℳ. From the definition of 𝑊_𝑀, we have 𝑊_{∪𝑖𝑀𝑖}⊂⋂^𝑛_𝑖=1𝑊_𝑀𝑖 and so {𝑊_𝑀∶𝑀∈ℳ} has the finite intersection property, so, there is 𝑢∈⋂_𝑀∈ℳ𝑊_𝑀 (note, if ⋂_𝑀∈ℳ𝑊_𝑀=∅, then 𝑊_𝑀0⊆⋃_𝑀≠𝑀0𝐾⧵𝑊_𝑀, where 𝑀₀ is an arbitrary element of ℳ, so the family {𝐾⧵𝑊_𝑀}_𝑀≠𝑀0 is an open covering for the compact set 𝑊_𝑀0 and so there exist 𝑀_𝑖1,…,𝑀_𝑖𝑛 such that 𝑊_𝑀0⊆⋃^𝑛_𝑗=1𝐾⧵𝑊_𝑖𝑗, which implies that ⋂^𝑛_𝑗=1𝑊_𝑖𝑗⋂𝑊_𝑀0=∅, which is a contradiction).
We claim that ⟨𝐴(𝑢,𝑢),𝑣−𝑔(𝑢)⟩⊈−𝐶(𝑢)⧵{0},∀𝑣∈𝐾.(2.21)
Indeed, for each 𝑣∈𝐾, there is 𝑀_𝑣∈ℳ such that 𝑣∈𝐾_𝑀. Hence, by 𝑢∈𝑊_𝑀𝑣 (note, 𝑢∈⋂_𝑀∈ℳ𝑊_𝑀) and the definition of 𝑊_𝑀𝑣, we have ⟨𝐴(𝑢,𝑣),𝑢−𝑔(𝑣)⟩⊆−𝐶(𝑣),(2.22) and so since 𝑣 was an arbitrary element of 𝐾, then (2.21) is true, for all 𝑣∈𝐾. This completes the proof of claim. From (2.21) and Lemma 2.1, we have⟨𝐴(𝑢,𝑢),𝑣−𝑔(𝑢)⟩⊈−𝐶(𝑢)⧵{0},∀𝑣∈𝐾,(2.23) and so the proof of the theorem is complete.

Acknowledgments

The authors wish to thank an anonymous referee for useful comments that improved the presentation of the paper. (The first author was in part supported by a Grant from IPM (no. 97490015).)