INEQUALITY PROBLEMS
ABDUL KHALIQ AND MOHAMMAD RASHID
Received 6 September 2004 and in revised form 3 July 2005
We introduce a class of generalized vector quasivariational-like inequality problems in Banach spaces. We derive some new existence results by using KKM-Fan theorem and an equivalent fixed point theorem. As an application of our results, we have obtained as special cases the existence results for vector quasi-equilibrium problems, generalized vector quasivariational inequality and vector quasi-optimization problems. The results of this paper generalize and unify the corresponding results of several authors and can be considered as a significant extension of the previously known results.
1. Introduction
LetK be a nonempty subset of a spaceXand f :K×K→Rbe a bifunction. The equi-librium problem introduced and studied by Blum and Oettli [4] in 1994 is defined to be the problem of finding a pointx∈K such that f(x,y)≥0 for eachy∈K. If we take f(x,y)= T(x),y−x, whereT:K→X∗(dual ofX) and·,·is the pairing betweenX andX∗then the equilibrium problem reduces to standard variational inequality, intro-duced and studied by Stampacchia [20] in 1964. In recent years this theory has become very powerful and effective tool for studying a wide class of linear and nonlinear prob-lems arising in mathematical programming, optimization theory, elasticity theory, game theory, economics, mechanics, and engineering sciences. This field is dynamic and has emerged as an interesting and fascinating branch of applicable mathematics with wide range of applications in industry, physical, regional, social, pure, and applied sciences. The papers by Harker and Pang [9] and M. A. Noor, K. I. Noor, and T. M. Rassias [18,19] provide some excellent survey on the developments and applications of variational in-equalities whereas for comprehensive bibliography for equilibrium problems we refer to Giannessi [8], Daniele, Giannessi, and Maugeri [5], Ansari and Yao [3] and references therein.
In the present paper, we consider a general type of variational inequality problem which contains equilibrium problems as a special case. So it is interesting to compare these two ways of the problem setting. We establish some existence results for solution to this type of variational inequality problem by using KKM-Fan theorem and an equivalent
fixed point theorem. From special cases, we obtain various known and new results for solving various classes of equilibrium problems, variational inequalities and related prob-lems. Our results generalizes and improves the corresponding results in the literature.
2. Preliminaries
LetX andY be real Banach Spaces. A nonempty subsetP ofX is calledconvex coneif λP⊆P for allλ≥0 andP+P=P. A coneP is calledpointed coneifP is a cone and P(−P)= {0}, where 0 denotes the zero vector. Also, a coneP is calledproper if it is properly contained inX. LetK be a non-empty subset ofX. We will denote by 2K the set of all nonempty subsets of K, clX(K) the closure of K in X,L(X,Y) the space of
all continuous linear operators fromX toY and u,x the evaluation of u∈L(X,Y) at x∈X. Let T:X→2Y be a multifunction, thegraph of T denoted byᏳ(T), is the set{(x,y)∈X×Y :x∈X, y∈T(x)}. TheinverseofT denoted byT−1 is a multifunc-tion fromR(T), range ofT, toX defined byx∈T−1(y) if and only ify∈T(x). AlsoT is said to be upper semicontinuous on X if for each x∈X and each open setU in Y containingT(x), there exists an open neighbourhoodV ofxinX such thatT(y)⊆U, for each y∈V.T is said to beupper hemicontinuousat xif for each y∈X,λ∈[0, 1], the multifunctionλ→T(λy+ (1−λ)x) is upper semicontinuous at 0+. A multifunction T:K→2L(X,Y) is calledgeneralized upper hemicontinuous at x∈K if for each y∈K, λ→ T(λy+ (1−λ)x),η(y,x)is upper semicontinuous at 0+, whereη:K×K→X is a bifunction. LetC:K→2Ybe a multifunction such that for eachx∈K,C(x) is a closed, convex moving cone with intC(x)= ∅, where intC(x) denotes the interior ofC(x). The partial order Cx on Y induced by C(x) is defined by declaring yCxz if and only ifz−y∈C(x) for allx,y,z∈K. We will write y≺Cxz ifz−y∈intC(x) in the case intC(x)= ∅. Let f :K×K →Y, η:K×K→X be bifunctions and T :K →2L(X,Y), S:K→2X be multifunctions. The purpose of this paper is to consider thegeneralized vector quasi-variatonal-like inequality problemof findingx∗∈K∩clXS(x∗) such that, for eachx∈S(x∗) there existst∗∈T(x∗) such that
t∗,ηx,x∗+fx∗,x∈ −/ intYCx∗. (2.1)
If we takeTas single valued mapping then as corollary, we consider the problem of find-ingx∗∈K∩clXS(x∗) such that, for eachx∈S(x∗),
Tx∗,ηx,x∗+fx∗,x∈ −/ intYCx∗. (2.2)
Ifη(x,y)=x−g(y), for allx,y∈K, whereg:K→Kis a mapping, then as corollary, we consider the problem of findingx∗∈K∩clXS(x∗) such that, for eachx∈S(x∗) there
existst∗∈T(x∗) such that
t∗,x−gx∗+fx∗,x∈ −/ intYCx∗. (2.3)
If f ≡0 andS:K→2K be a multifunction with closed values, then (2.1) reduces to the problem of findingx∗∈S(x∗) such that, for eachx∈S(x∗) there existst∗∈T(x∗) such that
t∗,ηx,x∗∈ −/ intYCx∗. (2.4)
It is calledgeneralized vector quasi-variational-like inequality problemconsidered and stud-ied by Ding [6].
If f ≡0 and clXS(x)=K for each x∈K, (2.1) becomes the generalized vector
variational-like inequality problemof findingx∗∈Ksuch that for eachx∈Kthere exists t∗∈T(x∗) such that
t∗,ηx,x∗∈ −/ intYCx∗. (2.5)
This problem was introduced and studied by Ansari [1,2] and B.-S. Lee and G.-M. Lee [16], and ifη(x,y)=x−yfor eachx,y∈K, then (2.5) was considered by Lin, Yang, and Yao [17] and Konnov and Yao [15].
WhenT≡0 andS:K→2K, problem (2.1) reduces to thevector quasi-equilibrium problemof findingx∗∈Ksuch that
x∗∈clXSx∗, fx∗,x∈ −/ intYCx∗ ∀y∈Sx∗. (2.6)
This problem was considered and studied by Khaliq and Krishan [11]. Ifη(x,y)=x−y for eachx,y∈KandS:K→2K, problem (2.1) reduces to the problem of findingx∗∈K such that for eachx∈S(x∗) there existst∗∈T(x∗) such that
x∗∈clXSx∗ and t∗,x−x∗+fx∗,x∈ −/ intYCx∗, (2.7)
which is known asvector quasi-variational inequality problemstudied by Khaliq, Siddiqi, and Krishan [13].
From the above special cases, it is clear that our generalized vector quasi-variational-like inequality problem (2.1) is a more general format of several classes of variational inequalities and equilibrium problems. It includes as special cases the generalized vector quasi-variational and variational-like inequality problems in [1,2,6,8,12,13,14,15,16,
17] as well as the vector quasi-equilibrium problems in [3,4,5,8,11]. Now, we mention some more definitions which will be used in the sequel.
Definition 2.2. LetC:K→2Y be a multifunction such thatC(x) is a proper closed and convex moving cone with intYC(x)= ∅, then a mappingg:K→Y is calledCx-convex
if for eachx,y∈K andλ∈[0, 1], (1−λ)g(x) +λg(y)−g((1−λ)x+λy)∈C(x) and is calledaffineif for eachx,y∈Kandλ∈R,
gλx+ (1−λ)y=λg(x) + (1−λ)g(y). (2.8)
Remark 2.3. If g :K →Y is a Cx-convex vector-valued function then, ni=1λig(yi) −g(ni=1λiyi)∈C(x), for allyi∈K,ti∈[0, 1],i=1,...,nwith
n
i=1λi=1.
Definition 2.4. Let f :K×K→Y, η:K×K→X be bifunctions and T :K →2L(X,Y) be a multifunction, then the pair (T,f) is calledη−Cx-pseudomonotonein K if for all
x,y∈K,
∃u∈T(x), u,η(y,x)+f(x,y)∈ −/ intYC(x)
=⇒ ∀v∈T(y), v,η(y,x)+ f(x,y)∈ −/ intYC(x),
(2.9)
and the pair (T,f) is calledweaklyη−Cx-pseudomonotoneinKif for allx,y∈K,
∃u∈T(x), u,η(y,x)+f(x,y)∈ −/ intYC(x)
=⇒ ∃v∈T(y), v,η(y,x)+f(x,y)∈ −/ intYC(x).
(2.10)
We also need the following KKM-Fan theorem [7] and a fixed point theorem which is a weaker version of Tarafdar’s theorem in [21].
Theorem2.5. LetKbe a nonempty subset of a topological vector spaceXandF:K→2K be aKKM-mapping with closed values. If there is a subsetDcontained in a compact convex subset ofKsuch that∩x∈DF(x)is compact then∩x∈DF(x)= ∅.
Theorem2.6. LetK be a nonempty subset of a Hausdorfftopological vector spaceXand F:K→2K be a multifunction with nonempty convex values such thatF−1(y)is open inK for eachy∈K. If there exists a nonempty subsetDcontained in a compact convex subset of K such thatK\ ∪y∈DF(y)is compact or empty. Then there existsx∗∈K such thatx∗∈
F(x∗).
Remark 2.7. Theorem 2.5has many equivalent formulations in terms of fixed points and is also equivalent toTheorem 2.6.
3. Existence results
and for allα∈(0, 1], (1−α)x+αy∈S(x) and setE= {x∈K:x∈clS(x)}. Assume that the mappingx→Y\(−intYC(x)) for eachx∈K, is a weakly closed mapping, that is, its
graph is closed inX×Y with weak topologies ofXandY.
Theorem3.1. Let f :K×K→Y andη:K×K→Xbe bifunctions andT:K→2L(X,Y)be a multifunction. Suppose the following assumptions holds:
(i)for eachx∈K,η(x,x)=0andf(x,x)∈C(x)∩ −C(x),
(ii)Tis generalized upper hemicontinuous inKwith nonempty compact values, (iii)η(·,·)is affine in the first argument and is continuous in the second argument, f is
Cx-convex in second argument and the pair(T,f)is weaklyη−Cx-pseudomonotone
for eachx∈K,
(iv)for eachx,y∈Kandxλ∈Ksuch thatxλ−−→w x(weak), there exists a subnetxµofxλ
ands∈ f(x,y)−C(x)such that f(xµ,y)−−→w s,
(v)there is a nonempty weakly compact subset D ofK and a subsetDo of a weakly
compact convex subset ofK such that for allx∈K\D, there existsz∈Do∩S(x), T(x),η(z,x)+ f(x,z)⊂ −intYC(x).
Then there existsx∗∈K∩clXS(x∗)such that for eachx∈S(x∗)there existst∗∈T(x∗)
such that
t∗
,ηx,x∗+fx∗,x∈ −/ intYCx∗. (3.1)
Proof. To prove the theorem, we first define the multifunctionsP1andP2for eachx,y∈ Kby
P1(x)=z∈K:T(x),η(z,x)+f(x,z)⊂ −intYC(x) ,
P2(x)=z∈K:T(z),η(z,x)+f(x,z)⊂ −intYC(x) .
(3.2)
Now fori=1, 2 set
Φi(x)=
S(x)∩Pi(x) ifx∈E
K∩S(x) ifx∈K\E (3.3)
andQi(y)=K\Φ−i1(y). Then
Qi(y)=K\x∈K:y∈Φi(x)
=K\x∈E:y∈S(x)∩Pi(x) ∪x∈K\E:y∈S(x)
=K\E∩S−1(y)P−1
i (y) ∪(K\E)∩S−1(y)
=K\E∩P−1
i (y)∪(K\E) ∩S−1(y)
=K\(K\E)∪P−1
i (y) ∩S−1(y)
=K\(K\E)∪P−1
i (y) ∪K\S−1(y)
=E∩K\P−1
i (y) ∪K\S−1(y).
We divide the proof into six steps.
Step 1. Eis nonempty and weakly closed: SinceK∩S(x)= ∅for allx∈K,∪y∈KS−1(y)
=K. By the given assumption and condition (v),S−1(y) is open inK for each y∈K andK\D⊂ ∪y∈DoS−1(y)⊂K. HenceK\ ∪y∈DoS−1(y) is contained inDand is weakly compact. ThusTheorem 2.6implies thatShas a fixed point inKand henceE= ∅. Also weakly closedness of clS(·) implies thatEis weakly closed.
Step 2. Q1is KKM mapping inK: Suppose that there exists a finite subset{y1,...,yn}of
Kandλi≥0,i=1,...,n, withni=1λi=1, such that
xo= n
i=1 λiyi∈/
n
i=1
Q1yi, (3.5)
then we havexo∈Φ−11(yi), which implies that yi∈Φ1(xo) for alli=1,...,n. Ifxo∈E,
thenΦ1(xo)=S(xo)∩P1(xo). Henceyi∈P1(xo), which implies that
Txo,ηyi,xo+fxo,yi⊂ −intYCxo. (3.6)
This implies that for allu∈T(xo),
u,ηyi,xo+fxo,yi∈ −intYCxo i=1,...,n. (3.7)
Which implies
n
i=1
λiu,ηyi,xo+ n
i=1
λifxo,yi∈ −intYCxo. (3.8)
Using (3.8),Cx-convexity off and assumption (i), we have for allu∈T(x0)
0=u,ηxo,xo
=
u,η n
i=1 λiyi,xo
=
n
i=1
λiu,ηyi,xo+ n
i=1
λifxo,yi+f
xo,
n
i=1 λiyi
−
n
i=1
λifxo,yi−fxo,xo∈ −intCxo−Cxo−Cxo
= −intCxo.
(3.9)
Step 3. Q2is KKM mapping inK: Using the definition ofPi(i=1, 2) and weaklyη−Cx -pseudomonotonicity of the pair (T,f) we haveK\P−1
1 (y)⊂K\P2−1(y). ThusQ1(y)⊂ Q2(y) for ally∈Kand henceQ2is also KKM-mapping.
Step 4. Q2(y) for eachy∈K is weakly closed: Weakly closedness ofQ2(y) follows from (3.4), if we prove that for eachy∈K
K\P−1 2 (y)=
x∈K:y /∈P2(x)
=x∈K:T(y),η(y,x)+ f(x,y)−intYC(x)
(3.10)
is weakly closed. Assume that xλ−−→w x and xλ∈K\P2−1(y). Which implies that there existstλ∈T(y) such that
tλ,ηy,xλ+fxλ,y∈ −/ intYCxλ. (3.11)
SinceT(y) is compact, without loss of generality, we can assume that there existst∈T(y) such thattλ→t. Also
tλ,ηy,xλ=tλ−t,ηy,xλ+t,ηy,xλ,
tλ−t,η
y,xλ≤tλ−tηy,xλ−→0. (3.12)
Sincetis also continuous whenXandY are equipped by the weak topologies andηis continuous in the second argument,
t,ηy,xλ−−→w t,η(y,x). (3.13)
Thus (3.11)–(3.13), yields
tλ,ηy,xλ−→t,η(y,x). (3.14)
By assumption (iv) there exists a subnetxµ ofxλands∈ f(x,y)−C(x) such that f(xµ,
y)−−→w s. Therefore, using (3.11), (3.14), assumption (iv), and weak closedness of x→ Y\(−intC(x)) inK, we have
t
,η(x,y)+s∈Y\−intYC(x). (3.15)
Thus
t,η(y,x)+f(x,y)=t,η(y,x)+s+f(x,y)−s∈Y\−intYC(x)+C(x)
=Y\−intYC(x).
(3.16)
Which implies thatx∈K\P−1
Step 5. There existsx∗∈K\ ∪y∈KΦ2−1(y). By (v) for eachx∈K\D, there exists z∈ Do∩S(x) such thatz∈Φ2(x). Which implies thatK\D⊂ ∪z∈DoΦ−21(z). Hence
D⊃
z∈Do K\Φ−1
2 (z)=
z∈Do
Q2(z). (3.17)
Thus all the assumptions ofTheorem 2.5are satisfied and hence there exists
x∗∈
y∈K
K\Φ−1
2 (y)=K\
y∈KΦ
−1
2 (y). (3.18)
Step 6. x∗is a solution of (2.1). Ifx∗∈K\E, (3.18) implies thatΦ2(x∗)= ∅. But given assumption impliesΦ2(x∗)=K∩S(x∗)= ∅, which is a contradiction. Ifx∗∈E, then
Φ2(x∗)=P2(x∗)∩S(x∗)= ∅. Which implies that for eachy∈S(x∗),y /∈P2(x∗). That is for eachy∈S(x∗),
T(y),ηy,x∗+ fx∗,y⊂ −intYCx∗. (3.19)
Suppose thatx∗is not solution of (2.1). Which implies that there existsy∗∈S(x∗),
Tx∗,ηy∗,x∗+fx∗,y∗⊂ −intYCx∗. (3.20)
SinceTis generalized upper hemicontinuous forα >0, small enough
Tαy∗+ (1−α)x∗,ηy∗,x∗+fx∗,y∗⊂ −intYCx∗. (3.21)
On the other hand using assumption (ii), (3.19),η(x,x)=0 andCx-convexity of f, we have
Tαy∗+ (1−α)x∗,ηy∗,x∗+fx∗,y∗
=1αTαy∗+ (1−α)x∗,ηαy∗+ (1−α)x∗,x∗+ fx∗,αy∗+ (1−α)x∗
+1αα fx∗,y∗+ (1−α)fx∗,x∗−fx∗,αy∗+ (1−α)x∗
−1−ααfx∗,x∗
⊂Y\−Cx∗ +Cx∗+Cx∗∩−Cx∗
=Y\−Cx∗ .
(3.22)
Corollary3.2. If inTheorem 3.1we takeT as single valued mapping then there exists x∗∈K∩clXS(x∗)such that, for eachx∈S(x∗),
Tx∗,ηx,x∗+fx∗,x∈ −/ intYCx∗. (3.23)
Corollary3.3. If inTheorem 3.1we takeη(x,y)=x−g(y), for allx,y∈K, whereg: K→K is a mapping, then there existsx∗∈K∩clXS(x∗)such that, for eachx∈S(x∗)
there existst∗∈T(x∗)such that
t∗,x−gx∗+fx∗,x∈ −/ intYCx∗. (3.24)
Theorem 3.4. If we avoid compactness of T(x)for each x∈K and replace the weakly η−Cx-pseudomonotonicity of the pair(T,f)byη−Cx-pseudomonotonicity and the
as-sumption (v) by
(v)othere is a nonempty weakly compact subset D ofK and a subsetDo of a weakly
compact convex subset ofK such that for allx∈K\D, there existsz∈Do∩S(x),
T(x),η(z,x)+ f(x,z)∩ −intYC(x)= ∅
inTheorem 3.1, then there existsx∗∈K∩clXS(x∗)such that for eachx∈S(x∗)there exists
t∗∈T(x∗)such that
t∗,ηx,x∗+fx∗,x∈ −/ intYCx∗. (3.25)
Proof. We first define a multifunctionP3for eachx∈Kby
P3(x)=z∈K:∃t∈T(z) :t,η(z,x)+f(x,z)∈ −intYC(x) . (3.26)
Using P1, P3 with the corresponding Φi and Qi, i=1, 3 analogously to the proof of
Theorem 3.1, we can show thatQ1is a KKM-mapping. By theη−Cx-pseudomonotonicity
of the pair (T,f),K\P−1
1 (y)⊂K\P−31(y) and henceQ1(y)⊂Q3(y) for allx∈K. Thus Q3is also a KKM mapping inK. Now weakly closedness ofQ3(y) follows from (3.4), if we prove that for eachy∈K
K\P−1 3 (y)=
x∈K:y /∈P3(x)
=x∈K:∃t∈T(y) :t,η(y,x)+f(x,y)∈ −/ intYC(x) (3.27)
is weakly closed. Assume thatxλ−−→w xandxλ∈K\P3−1(y). Which implies that for all t∈T(y) we have
t
,ηy,xλ+fxλ,y∈ −/ intYCxλ. (3.28)
Thus assumption (iv) implies that there is a subnetxµ ands∈f(x,y)−C(x) such that f(xµ,y)
w
topolo-gies and weak closedness ofx→Y\ −intC(x) inK, we havet,η(y,x)+s /∈ −intYC(x). Thus
t,η(y,x)+f(x,y)=t,η(y,x)+s+f(x,y)−s∈Y\ −intC(x) +C(x)
=Y\ −intC(x), (3.29)
which shows thatK\P−1
3 (y) is weakly closed and so isQ3(y). Similarly as forQ2, using (v)o,∩z∈DoQ3(z) is weakly compact. Thus all the assumptions ofTheorem 2.5are satisfied and hence there exists
x∗∈
y∈K
K\Φ−1
3 (y)=K\
y∈KΦ
−1
3 (y). (3.30)
Now it remains to show thatx∗is a solution of (2.1), which follows directly fromStep 4
ofTheorem 3.1with minor modifications.
Theorem3.5. Suppose that all the assumptions ofTheorem 3.4are satisfied except weak η−Cx-pseudomonotonicity of the pair(T,f) in (iii) and the condition that generalized upper hemicontinuity ofT is strengthened to the upper semicontinuity of T in the weak topology ofXand norm topology ofL(X,Y). Then there existsx∗∈K∩clXS(x∗)such that,
for eachx∈S(x∗)there existst∗∈T(x∗)such that
t∗,ηx,x∗+fx∗,x∈ −/ intYCx∗. (3.31)
Proof. To prove this theorem it is sufficient to prove that there existsx∗∈ ∩y∈KQ1(y). To
applyTheorem 2.5forQ1, it remains to check only the weak closedness ofQ1(y) for each y∈K, which follows from (3.4), if we prove that for eachy∈K
K\P−1 1 (y)=
x
∈K:y /∈P1(x)
=x∈K:T(x),η(y,x)+f(x,y)−intYC(x) (3.32)
is weakly closed. Assume that xλ−−→w x and xλ∈K\P1−1(y). Which implies that there existstλ∈T(xλ) such that
tλ,ηy,xλ+fxλ,y∈ −/ intYCxλ. (3.33)
Upper semi-continuity ofTimplies that for each>0, there exists a weak neighborhood N(x) such that T(N(x))⊂B(T(x),). We can takexλ∈N(x) and hence there istλ∈
T(x) such thattλ−tλ<. SinceT(x) is compact, without loss of generality, we can
assume that there existst∈T(x) such thattλ→t. Consequently,tλ−t →0. Thus using
arguments similar to those used inTheorem 3.1,Q1(y) is closed and hence the proof is
Remark 3.6. Theorem 3.1 improves and generalizes [11, Theorem 1] and [13, Theo-rem 3.1]. Our proof ofTheorem 3.1depends on the KKM-Fan theorem and fixed point theorem whereas the proof of main results in [11,13] depends on one person game the-orems. IfKis weakly compact, f(x,y)=0,η(x,y)=x−yandA(x)=K, for allx,y∈K,
Theorem 3.1collapses to [17, Theorem 3.1]. Of course, in this case the coercivity assump-tion (v) is omitted. The coercivity assumpassump-tion is unavoidable ifKis only closed and con-vex. We note that whenT≡0 andS:K→2KinCorollary 3.2, we obtain existence results for the vector quasi-equilibrium problem (2.6) and if we take f ≡0 and clXS(x)=Kfor
eachx∈K inTheorem 3.1we obtain existence results for the generalized vector quasi-variational-like inequality problem (2.5).
4. Applications
In this section we establish some existence results for generalized vector quasi variational inequalities and vector quasi optimization problems.
We need the following special case ofDefinition 2.4
Definition 4.1. LetT:K→2L(X,Y)be a multifunction andg:K→Kbe a mapping then Tis calledweakly generalizedCx-pseudomonotoneinKif for allx,y∈K,
∃u∈T(x), u,y−g(x)∈ −/ intYC(x)=⇒ ∃v∈T(y), v,y−g(x)∈ −/ intYC(x).
(4.1)
Theorem4.2. LetT:K→2L(X,Y) be generalized upper hemicontinuous and weakly gen-eralized Cx-pseudomonotone multifunction in K with nonempty compact values. Let η:
K×K→K be affine in the first argument and continuous in the second argument such that for eachx∈K,η(x,x)=0. Suppose that there is a nonempty weakly compact subsetD ofK and a subsetDo of a weakly compact convex subset ofK such that for allx∈K\D,
there existsz∈Do∩S(x),
T(x),η(z,x)⊂ −intYC(x). (4.2)
Then the generalized vector quasi-variational-like inequality problem of findingx∗∈K∩ clXS(x∗)such that for eachx∈S(x∗)there existst∗∈T(x∗)such that
t∗,ηx,x∗∈ −/ intYCx∗, (4.3)
has a solution.
Proof. If we take f =0, then we see that all the assumptions ofTheorem 3.1holds and
hence the proof follows.
Theorem4.3. Let f :K×K→Y be bifunction such that f(x,x)∈C(x)∩ −C(x)and is Cx-convex in second argument. Suppose that for eachx,y∈Kandxλ∈Ksuch thatxλ−−→w x
compact convex subset ofKsuch that for allx∈K\D, there existsz∈Do∩S(x),
f(x,z)∈ −intYC(x). (4.4)
Then the vector quasi-equilibrium problem of findingx∗∈K∩clXS(x∗)such that for each
x∈S(x∗),
fx∗,x∈ −/ intYCx∗, (4.5)
has a solution.
Proof. If we takeT=0 then proof directly follows fromTheorem 3.1. Corollary4.4. If inTheorem 4.2we takeη(x,y)=x−g(y)for all x,y∈K, whereg: K→K be a mapping. Then the generalized vector quasi-variational inequality problem of findingx∗∈K∩clXS(x∗)such that for eachx∈S(x∗)there existst∗∈T(x∗)such that
t∗,x−gx∗∈ −/ intYCx∗, (4.6)
has a solution.
Corollary4.5. If inTheorem 4.3we take f(x,y)=φ(y)−φ(x)for allx,y∈K, whereφ: K→Y be a vector-valued function. Then the vector quasi-optimization problem of finding x∗∈K∩clXS(x∗)such that for eachx∈S(x∗),
φx∗−φ(x)∈ −/ intYCx∗, (4.7)
has a solution.
Remark 4.6. Ding [6] and Kim and Tan [14] respectively employed the scalarization technique and one person game theorems to prove the existence results for the gener-alized vector variational-like inequality problem and the genergener-alized vector quasi-variational inequality problem whereas we have used KKM-Fan theorem and fixed point theorem in our results. The results of this section extends, generalizes and improves the corresponding results in [1,2,6,8,10,12,14,16].
Acknowledgments
The authors wish to express their thanks to Professor A. H. Siddiqi and the anonymous referee for valuable comments and suggestions for improving an earlier draft of the man-uscript.
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Abdul Khaliq: Post Graduate Department of Mathematics, University of Jammu, Jammu and Kashmir-180006, India
Current address: Post Graduate Department of Applied Mathematics, BGSB University, Rajouri, Jammu and Kashmir-185131, India
E-mail address:[email protected]
Mohammad Rashid: Department of Mathematics, Government Degree College, Rajouri, Jammu and Kashmir-185131, India
Current address: Post Graduate Department of Mathematics, University of Jammu, Jammu and Kashmir-180006, India
