WITH SET-VALUED MAPPINGS
JIAN WEN PENG AND DAO LI ZHUReceived 26 October 2005; Revised 5 March 2006; Accepted 12 April 2006
A new mathematical model of generalized vector quasiequilibrium problem with set-valued mappings is introduced, and several existence results of a solution for the gen-eralized vector quasiequilibrium problem with and withoutΦ-condensing mapping are shown. The results in this paper extend and unify those results in the literature.
Copyright © 2006 J. W. Peng and D. L. Zhu. This is an open access article distributed un-der the Creative Commons Attribution License, which permits unrestricted use, distri-bution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Throughout this paper, letZ,E, andFbe topological vector spaces, letX⊆EandY⊆F be nonempty, closed, and convex subsets. LetD:X→2X,T:X→2Y andΨ:X×Y× X→2Z be set-valued mappings, and letC:X→2Z be a set-valued mapping such that C(x) is a closed pointed and convex cone with intC(x)= ∅for eachx∈X, where intC(x) denotes the interior of the setC(x). Then the generalized vector quasi-equilibrium prob-lem with set-valued mappings (GVQEP) is to find (x,y) inX×Y such that
x∈D(x), y∈T(x), Ψ(x,y,z)−intC(x), ∀z∈D(x). (1.1)
The GVQEP is a new, interesting, meaningful, and general mathematical model, which contains many mathematical models as special cases, for some examples, we have the following.
(i) IfΨis replaced by a single-valued function f :X×Y×X→ZandC(x)=Cfor all x∈X, then the GVQEP reduces to finding (x,y) inX×Y such that
x∈D(x), y∈T(x), f(x,y,z)∈ −/ intC, ∀z∈D(x). (1.2)
It was investigated by Chiang et al. in [7].
IfΨis replaced by a scalar function f :X×Y×X→RandC(x)= {r∈R:r≥0}for allx∈X, then the GVQEP reduces to finding (x,y) inX×Y such that
x∈D(x), y∈T(x), f(x,y,z)≥0, ∀z∈D(x). (1.3)
This was investigated in [5,6,12,13,19] and contains the generalized quasi-variational inequality in [4,8,20,21] as a special case.
(ii) IfD(x)=Xfor allx∈Xand f = −Ψ, then the GVQEP reduces to finding (x,y) inX×Ysuch that
x∈X, y∈T(x), f(x,y,z)⊆intC(x), ∀z∈X. (1.4)
It is the model of GVEP3 by Fu and Wan in [10]. Fu and Wan also introduce another kind of general vector equilibrium problem (i.e., GVEP1 in [10]) which is to findxinXsuch that for allz∈X,∃y∈T(x), f(x,y,z)⊆intC(x).
(iii) IfY = {y} andT(x)= {y}for all x∈X, define a function ϕ:X×X→2Z as ϕ(x,z)=F(x,y,z), then the GVQEP reduces to findingxinXsuch that
x∈D(x), ϕ(x,z)⊆ −intC(x), ∀z∈D(x). (1.5)
This was studied in [1,16].
In this paper, by some maximal element theorems, we prove the existence results of a solution for the GVQEP with and withoutΦ-condensing mappings, and we also present some existence results of a solution for the special cases of the GVQEP. The results in this paper extend and unify those results in [1,5–7,10,12,13,15,19] and the references therein.
2. Preliminaries
In this section, we recall some definitions and some well-known results we need. Definition 2.1(see [2]). LetC:X→2Z be a set-valued mapping with intC(x)= ∅for all x∈X. Let ϕ:X×X→2Z be a set-valued mapping. Thenϕ(x,z) is said to be Cx -quasiconvex-like if for allx∈X,y1,y2∈X, andα∈[0, 1], either
ϕx,αy1+ (1−α)y2
⊆ϕx,y1
−C(x) (2.1)
or
ϕx,αy1+ (1−α
y2)⊆ϕ
x,y2
−C(x). (2.2)
{z1,z2,...,zn}inX, and for allx∈X withx∈Co{z1,z2,...,zn}, there exists some j∈
{1, 2,...,n}such thatϕ(x,zj)⊆ −intC(x).
Remark 2.3. (i) It is clear that ifZ=RandC(x)= {r∈R:r≥0}for allx∈X, andϕ is a single-valued function, then theC(x)-0-diagonal quasiconvexity ofϕreduces to the 0-diagonal quasiconvexity in [22,23], hereγ=0.
(ii) The following example shows that theCx-0-diagonal quasiconvexity of a set-valued
mapFis a true generalization ofCx-convex-likeness of the sameF.
LetEbe a real normed space with dual spaceE∗,X⊂E,Z=R. Let • denote the norm onE. LetC:X→2Zbe defined asC(x)=[x, +∞), for allx∈X, and let [e1,e2]
denote the line segment joininge1 ande2. Choose p∈E∗, we define a set-valued map
F:X×X→2Zas
F(x,z)=u,z−x:u∈−2xzp,−xzp, ∀x∈X. (2.3)
Then,F isCx-0-diagonal quasiconvex in the second argument. Otherwise, there exists finite set{z1,z2,...,zn} ⊆X, and there isx∈Xwithx=
n
j=1αjzj(αj≥0,
n
j=1αj=1)
such that for all j=1, 2,...,n,F(x,zj)⊆ −intC(x). Then for eachj, for allλj∈[0, 1], we
have
λj−2xzjp+1−λj− xzjp,zj−x<−x ≤0. (2.4)
It follows that
p,zj−x>0, j=1, 2,...,n. (2.5)
Then we have
0<
n
j=1
αj p,zj−x= p,x−x =0, (2.6)
a contradiction.
Definition 2.4(see [3,14]). LetEandFbe two topological spaces and letT:E→2Fbe a set-valued mapping.
(1) A subsetX⊆Eis said to be compactly open (resp., compactly closed) inEif for any nonempty compact subsetKofE,X∩Kis open (resp., closed) inK.
(2)Tis said to be upper semicontinuous if the set{x∈E:T(x)⊆V}is open inEfor every open subsetVofF.
(3)T is said to have open (resp., compactly open) lower sections if the setT−1(y)=
{x∈E:y∈T(x)}is open (resp., compactly open) inEfor eachy∈F.
Remark 2.5. Clearly each open (resp., closed) subset ofEis compactly open (resp., com-pactly closed), and the converse is not true in general.
(i)Φ(M)=0 if and only ifMis precompact (i.e., it is relatively compact); (ii)Φ(CoM)=Φ(M), wherecoMdenotes the convex closure ofM; (iii)Φ(M∪N)=max{Φ(M),Φ(N)}.
Definition 2.7(see [9]). LetΦ: 2E→Lbe a measure of noncompactness onEandD⊆E. A set-valued mappingT:D→2E is calledΦ-condensing provided that ifM⊆Dwith Φ(T(M))≥Φ(M), thenMis relatively compact.
Remark 2.8. Note that every set-valued mapping defined on a compact set is Φ-condensing for any measure of noncompactnessΦ. IfEis locally convex andT:D→2E is a compact set-valued mapping (i.e.,T(X) is precompact), thenTisΦ-condensing for any measure of noncompactnessΦ. It is clear that ifT:D→2E isΦ-condensing and T∗:D→2EsatisfiesT∗(x)⊆T(x) for allx∈D, thenT∗is alsoΦ-condensing.
Let CoA denote the convex hull of the setA.
Lemma2.9 (see [11]). LetXbe a nonempty convex subset of a Hausdorfftopological vector spaceEand letS:X→2Xbe a set-valued mapping such that for eachx∈X,x /∈Co(S(x)) and for eachy∈X,S−1(y)is open inX. Suppose further that there exist a nonempty compact
subsetNofX and a nonempty compact convex subsetBofXsuch thatCo(S(x))∩B= ∅ for allx∈X\N. Then there exists a pointx∈Xsuch thatS(x)= ∅.
Lemma2.10 (see [14]). LetXbe a nonempty closed and convex subset of a locally convex topological vector spaceEand letΦ: 2E→Lbe a measure of noncompactness onE. Suppose thatS:X→2Xis a set-valued mapping such that the following conditions are satisfied:
(i)for eachx∈X,x /∈S(x);
(ii)for eachy∈X,S−1(y)is compactly open inX;
(iii)the set-valued mappingS:X→2X isΦ-condensing. Then there existsx∈X such thatS(x)= ∅.
3. Existence results
Some existence results of a solution for the GVQEP withoutΦ-condensing mappings are first shown.
Theorem3.1. LetZbe a topological vector space, letEandFbe two Hausdorfftopological vector spaces, letX⊆EandY ⊆Fbe nonempty and convex subsets, letC:X→2Zbe a set-valued mapping such thatC(x)is a closed pointed and convex cone withintC(x)= ∅for eachx∈X, letD:X→2X andT:X→2Ybe set-valued mappings with nonempty convex values and open lower sections, and the setW= {(x,y)∈X×Y:x∈D(x)andy∈T(x)} is closed inX×Y. LetΨ:X×Y×X→2Zbe a set-valued mapping. Assume that
(i)M=Z\(−intC) :X→2Zis upper semicontinuous; (ii)for eachy∈Y,Ψ(x,y,z)isCx-0-diagonally quasiconvex;
(iii)for allz∈X,(x,y)→Ψ(x,y,z)is upper semicontinuous onX×Y with compact values;
Then, there exists(x,y)inX×Ysuch that
x∈D(x), y∈T(x), Ψ(x,y,z)⊆ −intC(x), ∀z∈D(x). (3.1)
That is, the solution set of the GVQEP is nonempty. Proof. Define a set-valued mappingP:X×Y→2Xby
P(x,y)=z∈X:Ψ(x,y,z)⊆ −intC(x), ∀(x,y)∈X×Y. (3.2)
It is needed to prove thatx /∈Co(P(x,y)) for all (x,y)∈X×Y. To see this, suppose, by way of contradiction, that there exist some points (x,y)∈X×Ysuch thatx∈Co(P(x,y)). Then there exist finite pointsz1,z2,...,zn inX, andαj≥0 with
n
j=1αj=1 such that
x=nj=1αjzjandzj∈P(x,y) for all j=1, 2,...,n. That is,Ψ(x,y,zj)⊆ −intC(x), j=
1, 2,...,n, which contradicts the fact thatΨ(x,y,z) isCx-0-diagonal quasiconvex.
There-fore,x /∈Co(P(x,y)), for all (x,y)∈X×Y. Now, it is needed to prove that the setP−1(z)=
{(x,y)∈X×Y:Ψ(x,y,z)⊆ −intC(x)}is open for eachz∈X. That is,Phas open lower sections onX×Y. It is only needed to prove thatQ(z)= {(x,y)∈X×Y :Ψ(x,y,z)⊆ −intC(x)}is closed for allz∈X. In fact, consider a net (xt,yt)∈Q(z) such that (xt,yt)→ (x,y)∈X×Y. Since (xt,yt)∈Q(z), there existsut∈Ψ(xt,yt,z) such thatut∈ −int/ C(xt).
From the upper semicontinuity and compact values ofΨ onX×Y and [17, Proposi-tion 1], it suffices to find a subset{utj}which converges to someu∈Ψ(x,y,z), where utj ∈Ψ(xtj,ytj,z). Since (xtj,ytj)→(x,y), by [3, Proposition 7, page 110] and the up-per semicontinuity ofM, it follows thatu /∈ −intC(x), and hence (x,y)∈Q(z),Q(z) is closed.
Hence,Phas open lower sections, and by [18, Lemma 2], we know that CoP:X×Y→ 2X also has open lower sections. Also define another set-valued mappingS:X×Y → 2X×Yby
S(x,y)=
⎧ ⎪ ⎨ ⎪ ⎩
D(x)∩CoP(x,y)×T(x) if (x,y)∈W,
D(x)×T(x) if (x,y)∈/ W. (3.3)
Then, it is clear that for all (x,y)∈X×Y,S(x,y) is convex, and (x,y)∈/ Co(S(x,y))= S(x,y).
Since for all (u,v)∈X×Y,
S−1(u,v)=CoP−1(u)∩D−1(u)×Y∩T−1(v)×Y
∪(X×Y\W)∩D−1(u)×Y∩T−1(v)×Y, (3.4)
and D−1(u)×Y,T−1(v)×Y, CoP−1(u), and X×Y\W are open in X×Y, we have
S−1(u,v) open inX×Y.
nonempty, we have (x,y)∈WandD(x)∩CoP(x,y)= ∅. This implies that (x,y)∈W andD(x)∩P(x,y)= ∅. Therefore,
x∈D(x), y∈T(x), Ψ(x,y,z)⊆ −intC(x), ∀z∈D(x). (3.5)
That is, the solution set of the GVQEP is nonempty. The proof is completed.
Theorem3.2. LetZbe a topological vector space, letEandFbe two Hausdorfftopological vector spaces, letX⊆EandY ⊆Fbe nonempty and convex subsets, letC:X→2Zbe a set-valued mapping such thatC(x)is a closed pointed and convex cone withintC(x)= ∅for eachx∈X, letD:X→2X andT:X→2Ybe set-valued mappings with nonempty convex values and open lower sections, and the setW= {(x,y)∈X×Y:x∈D(x)andy∈T(x)} is closed inX×Y. LetΨ:X×Y×X→2Zbe a set-valued mapping. Assume that
(i)M=Z\(−intC) :X→2Zis upper semicontinuous; (ii)for allx∈X, for ally∈Y,Ψ(x,y,x)⊆ −intC(x);
(iii)for each(x,y)∈X×Y, the setP(x,y)= {z∈X:Ψ(x,y,z)⊆ −intC(x)}is a con-vex set;
(iv)for allz∈X,(x,y)→Ψ(x,y,z)is upper semicontinuous onX×Y with compact values;
(v)there exist nonempty and compact subsetsN⊆X andK⊆Y and nonempty, com-pact, and convex subsetsB⊆X,A⊆Ysuch that for all(x,y)∈X×Y\N×K∃u∈ B,v∈Asatisfyingu∈D(x),v∈T(x), andΨ(x,y,u)⊆ −intC(x).
Then, there exists(x,y)inX×Ysuch that
x∈D(x), y∈T(x), Ψ(x,y,z)⊆ −intC(x), ∀z∈D(x). (3.6)
That is, the solution set of the GVQEP is nonempty.
Proof. ByTheorem 3.1, it is only needed to prove thatΨ(x,y,z) isCx-0-diagonally
qua-siconvex for all y∈Y. If not, then there exist y∈Y and some finite set{z1,z2,...,zn}
inX, and some pointx∈X withx∈Co{z1,z2,...,zn}, such that for each j=1, 2,...,n,
Ψ(x,y,zj)⊆ −intC(x). SinceP(x,y)= {z∈X:Ψ(x,y,z)⊆ −intC(x)}is a convex set,
x∈P(x,y), that is,Ψ(x,y,x)⊆ −intC(x), which contradicts the condition (ii). The proof
is completed.
Then, some existence results of a solution for the GVQEP withΦ-condensing map-pings are shown as follows.
Theorem3.3. LetZ be a topological vector space, letEandFbe two locally convex topo-logical vector spaces, let X⊆Eand Y ⊆F be nonempty, closed, and convex subsets, let C:X→2Z be a set-valued mapping such that C(x)is a closed pointed and convex cone withintC(x)= ∅for each x∈X,D:X→2X, andT:X→2Y be set-valued mappings with nonempty convex values and compactly open lower sections, and the setW= {(x,y)∈ X×Y:x∈D(x)andy∈T(x)}is compactly closed inX×Y. LetΨ:X×Y×X→2Zbe a set-valued mapping andΦ: 2E→Lbe a measure of noncompactness onE. Assume that
(iii)for allz∈X,(x,y)→Ψ(x,y,z)is upper semicontinuous on each compact subset of X×Y with compact values;
(iv)the set-valued mapD×T:X×X→2X×Y defined as(D×T)(x,y)=D(x)×T(y), for all(x,y)∈X×X, isΦ-condensing.
Then, there exists(x,y)inX×Ysuch that
x∈D(x), y∈T(x), Ψ(x,y,z)⊆ −intC(x), ∀z∈D(x). (3.7)
That is, the solution set of the GVQEP is nonempty.
Proof. LetP:X×Y →2X and S:X×Y →2X×Y be the same as defined in the proof ofTheorem 3.1. Following similar argument in the proof ofTheorem 3.1, we have for all (x,y)∈X×Y,S(x,y) is convex, (x,y)∈/ S(x,y), andShas compactly open lower sections inX×Y.
By the definition ofS,S(x,y)⊆D(x)×T(x) for all (x,y)∈X×Y. SinceD×TisΦ -condensing, so isS. Hence, byLemma 2.10,∃(x,y)∈X×Y such thatS(x,y)= ∅. Since for all (x,y)∈X×X,D(x), and T(y) are nonempty, we have (x,y)∈W and D(x)∩ CoP(x,y)= ∅. This implies that (x,y)∈WandD(x)∩P(x,y)= ∅. Therefore,
x∈D(x), y∈T(x), Ψ(x,y,z)⊆ −intC(x), ∀z∈D(x). (3.8)
That is, the solution set of the GVQEP is nonempty. The proof is completed. ByTheorem 3.3, and by similar argument to those in the proof ofTheorem 3.2, it is easy to obtain the following result.
Theorem3.4. LetZ be a topological vector space, letEandFbe two locally convex topo-logical vector spaces, let X⊆Eand Y ⊆F be nonempty, closed, and convex subsets, let C:X→2Z be a set-valued mapping such that C(x)is a closed pointed and convex cone withintC(x)= ∅for eachx∈X, letD:X→2X andT:X→2Ybe set-valued mappings with nonempty convex values and compactly open lower sections, and the setW= {(x,y)∈ X×Y:x∈D(x)andy∈T(x)}is compactly closed inX×Y. LetΨ:X×Y×X→2Zbe a set-valued mapping and letΦ: 2E→Lbe a measure of noncompactness onE. Assume that
(i)M=Z\(−intC) :X→2Zis upper semicontinuous on each compact subset ofX; (ii)for allx∈X, for ally∈Y,Ψ(x,y,x)⊆ −intC(x);
(iii)for each(x,y)∈X×Y, the setP(x,y)= {z∈X:Ψ(x,y,z)⊆ −intC(x)}is a con-vex set;
(iv)for allz∈X,(x,y)→Ψ(x,y,z)is upper semicontinuous on each compact subset of X×Y with compact values;
(v)the set-valued mapD×T:X×X→2X×Y defined as(D×T)(x,y)=D(x)×T(y), for all(x,y)∈X×X, isΦ-condensing.
Then, there exists(x,y)inX×Ysuch that
x∈D(x), y∈T(x), Ψ(x,y,z)⊆ −intC(x), ∀z∈D(x). (3.9)
Remark 3.5. If for eachy∈Y,Ψ(x,y,z) isCx-convex-like, then the condition (iii) in both
Theorems3.2and3.4holds. In fact, for anyz1,z2∈P(x,y), that is,z1,z2∈X,Ψ(x,y,z1)⊆
−intC(x) andΨ(x,y,z2)⊆ −intC(x). Then, for allλ∈[0, 1],λz1+ (1−λ)z2∈XsinceX
is convex. And sinceΨ(x,y,z) isCx-quasiconvex-like for ally∈Y, we have either
Ψx,y,λz1+ (1−λ)z2
⊆Ψx,y,z1
−C(x)⊆ −C(x)−intC(x)⊆ −intC(x), (3.10)
or
Ψx,y,λz1+ (1−λ)z2
⊆Ψx,y,z2
−C(x)⊆ −C(x)−intC(x)⊆ −intC(x). (3.11)
In both cases, we getΨ(x,y,λz1+ (1−λ)z2)⊆ −intC(x). Hence,λz1+ (1−λ)z2∈P(x,y)
for all (x,y)∈X×Y, and thereforeP(x,y) is convex.
Remark 3.6. Theorems3.1,3.2,3.3, and3.4, respectively, generalize those results in [5– 7,12,13,19] from scalar or vector-valued case to set-valued case with noncompact and nonmonotone conditions.
By [10, Lemma 2], we know that ifxis a solution of GVEP3, then it is also is a solution of GVEP1. Fu and Wan [10] got some existence results of a solution for GVEP1. Let f = −ΨandD(x)=Xfor allx∈X, by Theorems3.1and3.3, respectively, we can obtain the existence results of a solution for GVEP3 as follows.
Corollary3.7. LetZbe a topological vector space, letEandFbe two Hausdorfftopological vector spaces, letX⊆EandY⊆F be nonempty and convex subsets, letC:X→2Z be a set-valued mapping such thatC(x)is a closed pointed and convex cone withintC(x)= ∅ for eachx∈X, letT:X→2Y be a set-valued mapping with nonempty convex values and open lower sections, and the setW= {(x,y)∈X×Y:y∈T(x)}is closed inX×Y. Let
f :X×Y×X→2Zbe a set-valued mapping. Assume that (i)M=Z\(−intC) :X→2Zis upper semicontinuous; (ii)for eachy∈Y,−f(x,y,z)isCx-0-diagonally quasiconvex;
(iii)for allz∈X,(x,y)→ −f(x,y,z)is upper semicontinuous onX×Y with compact values;
(iv)there exist nonempty and compact subsetsN⊆X,K⊆Y and nonempty, compact, and convex subsetsB⊆X,A⊆Y such that for all(x,y)∈X×Y\N×K ∃u∈B, v∈Asatisfyingv∈T(x)and f(x,y,u)⊆intC(x).
Then, there existsxinXandy∈T(x)such thatf(x,y,z)⊆intC(x), for allz∈X. That is, the solution set of the GVEP3 is nonempty.
Corollary 3.8. Let Z be a topological vector space, let E andF be two locally convex topological vector spaces, letX⊆EandY⊆F be nonempty, closed and convex subsets, let C:X→2Zbe a set-valued mapping such thatC(x)is a closed pointed and convex cone with intC(x)= ∅for eachx∈X, letT:X→2Ybe set-valued mapping with nonempty convex values and compactly open lower sections, and the set W= {(x,y)∈X×Y :y∈T(x)} is compactly closed inX×Y. Let f :X×Y×X→2Z be a set-valued mapping and let Φ: 2E→Lbe a measure of noncompactness onE. Assume that
(iii)for allz∈X,(x,y)→ −f(x,y,z)is upper semicontinuous on each compact subset of X×Y with compact values;
(iv)the set-valued mapT:X→2YisΦ-condensing.
Then, there existsxinXandy∈T(x)such thatf(x,y,z)⊆intC(x), for allz∈X. That is, the solution set of the GVEP3 is nonempty.
Remark 3.9. The condition (iii) of both Corollaries3.7and3.8can be replaced by the following conditions:
(a) for allx∈X, for ally∈Y, f(x,y,x)⊆intC(x);
(b) for each (x,y)∈X×Y, the setP(x,y)= {z∈X:f(x,y,z)⊆intC(x)}is a convex set.
IfC(x)=Cfor allx∈X and Ψis replaced by a single-valued mapping f, then by Theorems3.1and3.3, respectively, we have the following two results which are general-izations of [7, Theorems 3.2 and 3.5] and [19, Theorems 6 and 7].
Corollary3.10. LetZbe a topological vector space, letEandF be two Hausdorff topo-logical vector spaces, let X⊆E and Y ⊆F be nonempty and convex subsets, let C⊆Z be a closed pointed and convex cone withintC= ∅, letD:X→2X and T:X→2Y be set-valued mappings with nonempty convex values and open lower sections, and the set W= {(x,y)∈X×Y:x∈D(x)andy∈T(x)}is closed inX×Y. Let f :X×Y×X→Z be a single-valued mapping. Assume that
(i)for eachy∈Y, f(x,y,z)isC-0-diagonally quasiconvex; (ii)for allz∈X,(x,y)→f(x,y,z)is continuous onX×Y;
(iii)there exist nonempty and compact subsetN⊆XandK⊆Yand nonempty, compact, and convex subsetB⊆X,A⊆Y such that for all(x,y)∈X×Y\N×K ∃u∈B, v∈Asatisfyingu∈D(x),v∈T(x)andf(x,y,u)∈ −intC.
Then, there exists(x,y)inX×Ysuch that
x∈D(x), y∈T(x) f(x,y,z)∈ −int/ C, ∀z∈D(x). (3.12)
Corollary 3.11. Let Z be a topological vector space, letEandF be two locally convex topological vector spaces, letX⊆EandY⊆F be nonempty, closed, and convex subsets, let C⊆Zbe a closed pointed and convex cone withintC= ∅, letD:X→2XandT:X→2Y be set-valued mappings with nonempty convex values and compactly open lower sections, and the setW= {(x,y)∈X×Y:x∈D(x)andy∈T(x)}is compactly closed inX×Y. Let f :X×Y×X→Zbe a single-valued mapping and letΦ: 2E→Lbe a measure of non-compactness onE. Assume that
(i)for eachy∈Y, f(x,y,z)isC-0-diagonally quasiconvex;
(ii)for allz∈X,(x,y)→f(x,y,z)is continuous on each compact subset ofX×Ywith compact values;
(iii)the set-valued mapD×T:X×X→2X×Y defined as(D×T)(x,y)=D(x)×T(y), for all(x,y)∈X×X, isΦ-condensing.
Then, there exists(x,y)inX×Ysuch that
Remark 3.12. The condition (ii) in both Corollaries3.10and3.11can be replaced by the following conditions:
(a) for allx∈X, for ally∈Y, f(x,y,x)∈ −int/ C;
(b) for each (x,y)∈X×Y, the setP(x,y)= {z∈X:f(x,y,z)∈ −intC}is a convex set.
LetY= {y}. Define a set-valued mappingT:X→2YasT(x)= {y}for allx∈Xand define another set-valued mappingΨ:X×Y×XasΨ(x,y,z)=ϕ(x,z), for all (x,y,z)∈ X×Y×X. Then by Theorems3.1and3.3, respectively, we have following results which are generalizations of [1, Corollary 3.1].
Corollary3.13. LetZbe a topological vector space, letEbe a Hausdorfftopological vector space, letX⊆Ebe a nonempty and convex subset, letC:X→2Zbe a set-valued mapping such thatC(x)is a closed pointed and convex cone withintC(x)= ∅for eachx∈X, let D:X→2Xbe a set-valued mapping with nonempty convex values and open lower sections, and the setW= {x∈X:x∈D(x)} is closed inX. Letϕ:X×X→2Z be a set-valued mapping. Assume that
(i)M=Z\(−intC) :X→2Zis upper semicontinuous; (ii)ϕ(x,z)isCx-0-diagonally quasiconvex;
(iii)for allz∈X,x→ϕ(x,z)is upper semicontinuous onXwith compact values; (iv)there exist nonempty and compact subsetN⊆Xand nonempty, compact, and convex
subsetB⊆X such that for allx∈X\N ∃u∈Bsatisfyingu∈D(x)andϕ(x,u)⊆ −intC(x).
Then, there existsxinXsuch that
x∈D(x), ϕ(x,z)⊆ −intC(x), ∀z∈D(x). (3.14)
Corollary3.14. LetZ be a topological vector space, letEbe a locally convex topological vector space, letX⊆Ebe nonempty, closed, and convex subset, letC:X→2Zbe a set-valued mapping such that C(x)is a closed pointed and convex cone withintC(x)= ∅for each x∈X, letD:X→2X be set-valued mapping with nonempty convex values and compactly open lower sections, and the setW= {x∈X:x∈D(x)}is compactly closed inX. Letϕ: X×X→2Z be a set-valued mapping, and letΦ: 2E→Lbe a measure of noncompactness onE. Assume that
(i)M=Z\(−intC) :X→2Zis upper semicontinuous on each compact subset ofX; (ii)ϕ(x,z)isCx-0-diagonally quasiconvex;
(iii)for allz∈X,x→ϕ(x,z)is upper semicontinuous on each compact subset ofXwith compact values;
(iv)the set-valued mapD:X→2XisΦ-condensing. Then, there existsxinXsuch that
x∈D(x), ϕ(x,z)⊆ −intC(x), ∀z∈D(x). (3.15)
Remark 3.15. The condition (ii) in both Corollaries3.13and3.14can be replaced by the following conditions:
(a) for allx∈X,ϕ(x,x)⊆ −intC(x);
Acknowledgments
This paper was supported by the National Natural Science Foundation of China (Grant no. 10171118), Education Committee Project, Research Foundation of Chongqing (Grant no. 030801), and Natural Science Foundation of Chongqing (no. 8409).
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Jian Wen Peng: Department of Management Science, School of Management, Fudan University, Shanghai 200433, China
Current address: College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China
E-mail address:jwpeng6@yahoo.com.cn
Dao Li Zhu: Department of Management Science, School of Management, Fudan University, Shanghai 200433, China
