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

Authors: A. P. Farajzadeh; A. Amini-Harandi; D. O'Regan
Journal: Abstract and Applied Analysis
 (2008)

---

## 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).


---

## Body

## 1. Introduction and Preliminaries

Throughout this paper, unless  otherwise specified, we
always letX and Y be real Hausdorff topological  vector spaces, K⊆X a nonempty convex set, C:K→2Y 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 2Y 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→2Y be a set-valued mapping. We need the following
definitions and results in the sequel.Definition 1.1.
LetF:K×K→2Y be a set-valued mapping. One says that F is
(i)
stronglyC-pseudomonotone if, for any given x and y∈K,(1.1)⊈−intC(x)⇒F(y,x)⊆−C(y).(ii)
C-pseudomonotone if, for any given x and y∈K,(1.2)F(x,y)⊈−C(x)∖{0}⇒F(y,x)⊆−C(y).Remark 1.2.
(1) StronglyC-pseudomonotonicity implies C-pseudomonotonicity.
(2) LetT:K→2L(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.
LetX and Y be two topological spaces. A set-valued
mapping G:X→2Y 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 {xα}⊆K,xα converges to x, and G(xα)⊆C,
for all α imply that G(x)⊆C.Lemma 1.5 (see [2]).
LetX and Y be two topological spaces. Suppose that G:X→2Y is a set-valued mapping. Then the following
statements are true.
(a)
IfG is closed and compact, then G is u.s.c.(b)
Let, for anyx∈X,G(x) be compact. If G is u.s.c. on X then for any net {xα}⊂X such that xα→x and for every yα∈G(xα), there exist y∈G(x) and a subnet {yβ} of {yα} such that yβ→y.For the converse of (b) in Lemma1.5, we refer the
reader to [3].Definition 1.6.
LetX be a topological vector space and Y a topological space. A set-valued mapping G:X→2Y is called upper hemicontinuous if the
restriction of G on straight lines is upper semicontinuous.Definition 1.7.
One says that the mappingG:K×K→2Y is C-upper sign continuous if, for all x,y∈K, the following implication holds: (1.3)G((1−t)x+ty,y)∩C((1−t)x+ty)≠∅,∀t∈]0,1[⇒G(x,y)∩C(x)≠∅.Remark 1.8.
Letf: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)
findx∈K such that F(x,y)⊈−intC(x),∀y∈K, and(GVEP2)
findx∈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 fixedx∈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 zt=y+t(z−y),
where t∈]0,1[ and y∈K as above. By (II), we have (2.1)F(zt,y)⊆−C(zt),∀t∈]0,1[.  We claim that F(zt,z)∩C(zt)≠∅. Suppose F(zt,z)∩C(zt)=∅, for some t∈]0,1[. Then (2.2)F(zt,z)⊆Y∖C(zt),forthist∈]0,1[,  and so F(zt,zt)⊆(1−t)F(zt,y)+tF(zt,z)−C(zt)⊆−C(zt)+(Y∖C(zt))−C(zt)=(Y∖C(zt))−C(zt)⊆Y∖C(zt), 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(zt)−C(zt)⊆Y∖C(zt)). Therefore, for all t∈]0,1[, the set F(zt,z)∩C(zt) 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 mappingC:K→2Y 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 ut∈F(zt,y)∩C(zt)≠∅, where zt=(1−t)x+ty,t∈]0,1[. By Lemma 1.5(b), there exists a subnet of (ut) (without loss of generality (ut)) and u∈F(x,y) such that ut→u, where t→0. Now, since C:K→2Y has closed graph (note ut→u and zt→x as t→0) and ut∈C(zt), 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 Lemma2.1 and using Remark 2.2, we can deduce the following result.Lemma 2.3.
Suppose that
(i)
for each fixedz∈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 mappingx→W(x)=Y∖(−intC(x)), for each x∈K, has closed graph;(v)
for each fixedx∈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.
LetT:K→2L(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 Lemma2.1,
the solution set of (GVEP2) is convex.Proof.
Letx1 and x2 be solutions of (GVEP2). By Lemma 2.1, we have (2.3)F(z,xi)⊆−C(z),∀z∈K,i=1,2.  From this and
condition (iv) of Lemma 2.1, for all t∈]0,1[, we deduce that (2.4)F(z,(1−t)x1+tx2)⊆(1−t)F(z,x1)+tF(z,x2)−C(z)⊆−C(z), for all z∈K. Hence, from  Lemma 2.1, we get (2.5)F((1−t)x1+tx2,z)⊈−C((1−t)x1+tx2))∖{0},∀z∈K. 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 Lemma2.3, the solution set of (GVEP1) is convex.Remark 2.7.
Lemma2.5 extends Theorem 3 of
Yin and Xu [6] and Lemma 2.5 of
Fang and Huang [1].Definition 2.8.
LetK0 be a nonempty subset of K.
A set-valued mapping Γ:K0→2K 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]).
LetK be a nonempty subset of a topological vector
space X and Γ:K→2X 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(2.6)⋂x∈KΓ(x)≠∅.Lemma 2.10 (see [8]).
LetK be a convex subset of a metrizable topological
vector space X and F:K→2K 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 Lemma2.1 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.
DefineΓ,Γ^:K→2K by (2.7)Γ(y)={x∈K:F(x,y)⊈−C(x)∖{0}},Γ^(y)={x∈K:F(y,x)⊆−C(y)}.  We claim that Γ is a KKM mapping. If not, there exist y1,y2,…,yn∈K and ti>0,∑i=1nti=1 such that z=∑i=1ntiyi∉Γ(yi), that is, (2.8)F(z,yi)⊆−C(z)∖{0},i=1,2,3,…,n,  and so, since −C(z) is a closed convex pointed cone, (2.9)∑i=1ntiF(z,yi)⊆−C(z)∖{0}.   It follows from
condition (iv) of Lemma 2.1 that (2.10)F(z,z)⊆∑i=1ntiF(z,yi)−C(z). Now, by
combining (2.9) and (2.10), we get (2.11)F(z,z)⊆∑i=1ntiF(z,yi)−C(z)⊆−C(z)∖{0}−C(z)⊆−C(z)∖{0},  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 (2.12)F(y,z)⊆−C(z),∀y∈K. Now, it follows
from Lemma 2.1 that (2.13)F(z,y)⊈−C(y)∖{0},∀y∈K, 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  Theorem2.11, we derive the
existence result for a solution of the following problem which consists of
finding a u∈K such that (2.14)〈A(u,u),v−g(u)〉⊈−C(u)∖{0},∀v∈K,  where A:K×K→2L(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.
LetX be metriziable topological vector space, K nonempty convex subset of X, A:K×K→2L(X,Y), and g:K→K be two mappings. Assume that
(i)
for each fixedw∈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 fixedv∈K, the mapping (w,u)→〈A(w,u),u−g(v)〉 is lower semicontinuous;(iv)
for each finite dimensional subspaceM of X with KM=K∩M≠∅,
there exist compact subset BM and compact convex subset DM of KM such that ∀(w,z)∈KM×(KM∖BM), ∃u∈DM such that 〈A(w,u),z−g(u)〉⊈−C(u).  Then there
exists u∈K such that (2.15)〈A(u,u),v−g(u)〉⊈−C(u)∖{0},∀v∈K.Proof.
LetM⊂X be a finite dimensional subspace with KM=K∩M≠∅.
For each fixed w∈K,
consider the problem of finding a u∈KM such that(2.16)〈A(w,u),v−g(u)〉⊈−C(u)∖{0},∀v∈KM.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∈KM,
we define a set-valued mapping T:KM→2KM by(2.17)T(w)={u∈KM:〈A(w,u),v−g(u)〉⊈−C(u)∖{0},∀v∈KM}.Then T(w) is a nonempty closed subset of BM,
in fact, T(w) is the solution set of (2.16) corresponding to w.
By Lemma 2.1, we have(2.18)T(w)={u∈KM:〈A(w,v),u−g(v)〉⊆−C(v),∀v∈KM},which is a convex set. By
condition (iii) via Remark 1.4, T is closed. By (iv), we have T(KM)=⋃w∈KMT(w)⊆BM.
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∈KM,
that is,(2.19)〈A(w0,v),w0−g(v)〉⊆−C(v),∀v∈KM.Set ℳ={M⊂X:M is a finite dimensional subspace with KM≠∅} and for M∈ℳ,(2.20)WM={u∈KM:〈A(u,v),u−g(v)〉⊆−C(v),∀v∈KM}.By (2.19) and conditions (iii)
and (iv), WM is a nonempty and closed subset of the compact
set BM and hence WM is compact in K.
Let {Mi}i=1n be a finite subset of ℳ.
From the definition of WM,
we have W∪iMi⊂⋂i=1nWMi and so {WM:M∈ℳ} has the finite intersection property, so,
there is u∈⋂M∈ℳWM (note, if ⋂M∈ℳWM=∅, then WM0⊆⋃M≠M0K∖WM, where M0 is an arbitrary element of ℳ, so the family {K∖WM}M≠M0 is an open covering for the compact set WM0 and so there exist Mi1,…,Min such that WM0⊆⋃j=1nK∖Wij, which implies that ⋂j=1nWij⋂WM0=∅, which is a contradiction).
We claim that(2.21)〈A(u,u),v−g(u)〉⊈−C(u)∖{0},∀v∈K.
Indeed, for eachv∈K, there is Mv∈ℳ such that v∈KM.
Hence, by u∈WMv (note, u∈⋂M∈ℳWM) and the definition of WMv,
we have (2.22)〈A(u,v),u−g(v)〉⊆−C(v),  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(2.23)〈A(u,u),v−g(u)〉⊈−C(u)∖{0},∀v∈K, and so the
proof of the theorem is complete.



---

*Source: 968478-2009-01-14.xml*