Generalized Bi Quasivariational Inequalities for Quasi Pseudomonotone Type II Operators on Noncompact Sets

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Volume 2010, Article ID 237191,17pages doi:10.1155/2010/237191

Research Article

Generalized Bi-Quasivariational Inequalities

for Quasi-Pseudomonotone Type II Operators

on Noncompact Sets

Mohammad S. R. Chowdhury

1

_{and Yeol Je Cho}

2

1_{Department of Mathematics, Lahore University of Management Sciences (LUMS), Phase II,} Opposite Sector U, D.H.A., Lahore Cantt., Lahore 54792, Pakistan

2_{Department of Mathematics Education and the RINS, Gyeongsang National University,} Chinju 660-701, South Korea

Correspondence should be addressed to Yeol Je Cho,yjcho@gnu.ac.kr

Received 3 November 2009; Accepted 18 January 2010

Academic Editor: Jong Kyu Kim

Copyrightq2010 M. S. R. Chowdhury and Y. J. Cho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove some existence results of solutions for a new class of generalized bi-quasivariational inequalities GBQVI for quasi-pseudomonotone type II and strongly quasi-pseudomonotone type II operators defined on noncompact sets in locally convex Hausdorﬀ topological vector spaces. To obtain these results on GBQVI for pseudomonotone type II and strongly quasi-pseudomonotone type II operators, we use Chowdhury and Tan’s generalized version1996of Ky Fan’s minimax inequality1972as the main tool.

1. Introduction and Preliminaries

In this paper, we obtain some results on generalized bi-quasi-variational inequalities for quasi-pseudo-monotone type II and strongly quasi-pseudo-monotone type II operators defined on noncompact sets in locally convex Hausdorﬀtopological vector spaces. Thus we begin this section by defining the generalized bi-quasi-variational inequalities. For this, we need to introduce some notations which will be used throughout this paper.

LetXbe a nonempty set and let 2X be the family of all nonempty subsets ofX. IfX andY are topological spaces andT :X → 2Y, then the graph ofT is the setGT:{x, y∈

X×Y :y ∈ Tx}. Throughout this paper,Φdenotes either the real fieldRor the complex fieldC.

Let E be a topological vector space overΦ, let F be a vector space over Φand let

·,·:F×E → Φbe a bilinear functional.

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onFgenerated by the family{Wx;:x∈Eand >0}as a subbase for the neighbourhood system at 0 and letδF, Ebe thestrongtopology onFgenerated by the family{UA;:

Ais a nonempty bounded subset ofEand > 0}as a base for the neighbourhood system at 0. We note then thatF, when equipped with theweaktopologyσF, Eor thestrong topologyδF, E, becomes a locally convex topological vector space which is not necessarily Hausdorﬀ. But, if the bilinear functional·,·:F×E → Φseparates points inF, that is, for anyy∈Fwithy /0, there existsx∈Esuch thaty, x/0, thenFalso becomes Hausdorﬀ. Furthermore, for any net{yα}α∈ΓinFandy∈F,

1yα → yinσF, Eif and only ifyα, x → y, xfor anyx∈E,

2yα → yinδF, Eif and only ifyα, x → y, xuniformly for anyx∈A, where a

nonempty bounded subset ofE.

The generalized bi-quasi-variational inequality problem was first introduced by Shih and Tan 1 in 1989. Since Shih and Tan, some authors have obtained many results on generalized quasivariational inequalities, generalized quasivariational-like inequalities and generalized bi-quasi-variational inequalitiessee2–15 .

The following is the definition due to Shih and Tan1 .

Definition 1.1. Let Eand F be a vector spaces over Φ, let·,· : F×E → Φ be a bilinear functional, and letX be a nonempty subset ofE. IfS : X → 2X andM, T : X → 2F, the generalized bi-quasi variational inequality problemGBQVI for the triple S, M,Tis to find

y∈Xsatisfying the following properties:

1y∈Sy,

2infw∈TyRef−w,y−x ≤0 for anyx∈Syandf∈My.

The following definition of the generalized bi-quasi-variational inequality problem is a slight modification ofDefinition 1.1.

Definition 1.2. Let E and F be vector spaces over Φ, let ·,· : F ×E → Φbe a bilinear functional, and letXbe a nonempty subset ofE. IfS:X → 2XandM, T :X → 2F, then the generalized bi-quasivariational inequalityGBQVIproblem for the tripleS,M,Tis:

1to find a pointy∈Xand a pointw ∈Tysuch that

y∈Sy, Ref−w, y−x ≤0, ∀x∈Sy, f ∈My, 1.1

2to find a pointy∈X, a pointw ∈Ty, and a pointf∈Mysuch that

y∈Sy, Ref−w, y−x≤0, ∀x∈Sy. 1.2

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LetXandY be topological spaces and letT :X → 2Y be a set-valued mapping. Then

Tis said to be:

1upper resp., lower semicontinuousat x0 ∈ X if, for each open set G in Y with

Tx0⊂Gresp.,Tx0∩G /∅, there exists an open neighbourhoodUofx0inX

such thatTx⊂Gresp.,Tx∩G /∅for allx∈U,

2upperresp., lower semicontinuousonXifTis upperresp., lowersemicontinuous at each point ofX,

3continuousonXifTis both lower and upper semi-continuous onX.

LetXbe a convex set in a topological vector spaceE. Thenf :X → Ris said to be lower semi-continuousif, for allλ∈R,{x∈X:fx≤λ}is closed inX.

IfXis a convex set in a vector spaceE, thenf :X → Ris said to be concaveif, for all

x, y∈Xand 0≤λ≤1,

fλx 1−λy≥λfx 1−λfy. 1.3

Our main results in this paper are to obtain some existence results of solutions of the generalized bi-quasi-variational inequalities using Chowdhury and Tan’s following definition of quasi-pseudo-monotone type II and strongly quasi-pseudo-monotone type II operators given in3 .

Definition 1.3. LetEbe a topological vector space, letXbe a nonempty subset ofE, and letF be a topological vector space overΦ. Let·,·:F×E → Φbe a bilinear functional. Consider a mappingh:X → Rand two set-valued mappingsM:X → 2F andT:X → 2F.

1Tis called anh-quasi-pseudo-monotoneresp., stronglyh-quasi-pseudo-monotone type II operator if, for anyy ∈ X and every net{yα}α∈Γ inX converging toy resp., weakly toywith

lim sup

α f∈Minfyα inf

u∈Tyα

Ref−u, yα−yhyα−hy≤0,

lim sup

α f∈Minfyαu∈infTyαRe

f−u, yα−xhyα−hx

≥ inf

f∈Mywinf∈TyRe

f−w, y−xhy−hx, ∀x∈X.

1.4

2T is said to be a quasi-pseudo-monotoneresp., strongly quasi-pseudo-monotone type II operator if T is an h-quasi-pseudo-monotone resp., strongly h -quasi-pseudo-monotonetype II operator withh≡0.

The following is an example on quasi-pseudo-monotone type II operators given in3 .

Example 1.4. ConsiderX −1,1 andER. ThenE∗ R. LetM:X → 2Rbe a set-valued mapping defined by

Mx

⎧ ⎨ ⎩

0,2x ifx≥0,

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Again, letT :X → 2Rbe a set-valued mapping defined by

Tx

{1,3} ifx <1,

{1,2,3} ifx1. 1.6

ThenMis lower semi-continuous andT is upper semi-continuous. It can be shown thatT becomes a quasi-pseudo-monotone type II operator onX −1,1 .

iTo show thatMis lower semi-continuous, considerx0 ≥0. ThenMx0 0,2x0 .

Let >0 be given. Then, ifG 2x0−,2x0, thenMx0∩G 0,2x0 ∩2x0−,2x0/∅.

Let >0 be so chosen that 0< x0−/2< x0< x0/2<1. Now, if we takeU x0−/2, x0

/2, then, for allx ∈ U, we have 2x0 − < 2x < 2x0. Thus 2x ∈ Mx∩G. Hence

Mx∩G /∅.

Ifx00,M0 0. Then, for 0∈G , , we can takeU −/2, /2. Thus for all

x∈U,Mx 0,2x ∩, /∅because− < x < /2 implies 2x∈G −, .

Finally, ifx0 <0, thenMx0 2x0,0 . We takeG 2x0−,2x0for some >0

so thatMx0∩G /∅andx0/2 < 0. Thus, for allx ∈U x0−/2, x0/2, we have

2x0− <2x <2x0. Hence 2x∈G∩Mx, whereMx 2x,0 forx <0. Consequently,

Mis lower semi-continuous onX −1,1 .

iiTo show thatT is upper semi-continuous, letx0 ∈ −1,1 be such thatx0 < 1.

ThenTx0 {1,3}. LetG be an open set inR such thatTx0 {1,3} ⊂ G. Let > 0 be

such that−1 < x0− < x0 < x0 < 1. ConsiderU x0−, x0. Then, for allx ∈U,

Tx {1,3} ⊂Gsincex <1. Again, ifx1, thenT1 {1,2,3}. LetGbe an open set inR such thatT1 {1,2,3} ⊂G. Let >0 be such that−1<1− <1 <1. LetU 1−,1 which is an open neighbourhood of 1 inX −1,1 . Then for allx∈U 1−,1 , we have

Tx {1,3}if 1− < x < 1 andTx {1,2,3}ifx 1. Now,T1 {1,2,3} ⊂ G. Also, for allx ∈ Uwith 1− < x < 1, we haveTx {1,3} ⊂ {1,2,3} ⊂ G. HenceT is upper semi-continuous onX −1,1 .

iiiFinally, we will show thatT is also a quasi-pseudo-monotone type II operator. To show this, let us assume first thatyαis a net inX −1,1 such thatyα → yinX −1,1 .

We now show that

lim sup

f−u, yα−y≤0. 1.7

We have

lim sup

f−u, yα−y

lim sup

α ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

inf

f∈0,2yα inf

u∈{1,3}

f−uyα−y

0−3yα−y3y−yα, if 0≤yα<1,

inf

f∈2yα,0 u∈{inf1,3}

f−uyα−y

2yα−3yα−y, if yα<0 and soyα<1,

inf

f∈0,2yαu∈{inf1,2,3}

f−uyα−y

0−3yα−y3y−yα, if yα1, that is, yα≥0

0

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consideringyα−y≥0, the value will be also 0 if we consideryα−y≤0. So, it follows that,

for allx∈−1,1 ,

lim sup

α f∈Minfyα inf

u∈Tyα

f−uyα−x

lim sup α ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ inf

f∈0,2yα inf

u∈{1,3}

f−uyα−x

0−3yα−x3x−yα, if 0≤yα<1,

inf

f∈2yα,0 u∈{inf1,3}

f−uyα−x

2yα−3yα−x, if yα<0 and soyα<1,

inf

f∈0,2yα inf

u∈{1,2,3}

f−uyα−x

0−3yα−x3x−yα, if yα1, that is, yα≥0,

consider yα−x≥0

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

3x−y, if 0≤yα<1,

2y−3y−x, ifyα<0 and soyα<1,

3x−y, ifyα1,

considery−x≥0.

1.9

The values can be obtained similarly for the cases whereyα−x ≤ 0 andy−x ≤ 0. Also, it

follows that, for allx∈X −1,1 ,

inf

f∈Myu∈infTy

f−uy−x

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ inf

f∈0,2y u∈{inf1,3}

f−uy−x

0−3y−x3x−y, if 0≤y <1,

inf

f∈2y,0 u∈{inf1,3}

f−uy−x

2y−3y−x, ify <0 and soy <1,

inf

f∈0,2y u∈{inf1,2,3}

f−uy−x

0−3y−x3x−y, ify1, that is, y≥0,

considery−x≥0

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

3x−y, if 0≤y <1,

2y−3y−x, ify <0 and soy <1,

3x−y, ify1,

considery−x≥0.

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The values can be obtained similarly for the cases wheny−x≤0. Therefore, in all the cases, we have shown that

lim sup

α f∈infMyαu∈infTyα

f−uyα−x≥ inf f∈Myu∈infTy

f−uy−x. 1.11

HenceTis a quasi-pseudo-monotone type II operator.

The above example is a particular case of a more general result on quasi-pseudo-monotone type II operators. We will establish this result in the following proposition.

Proposition 1.5. LetX be a nonempty compact subset of a topological vector spaceE. Suppose that

M:X → 2E∗andT :X → 2E∗are two set-valued mappings such thatMis lower semi-continuous andT is upper semi-continuous. Suppose further that, for anyx ∈ X,Mxand Txare weak∗ -compact sets in E∗. ThenT is both a monotone type II and a strongly quasi-pseudo-monotone type II operator.

Proof. Suppose that{yα}α∈Γis a net inXandy∈Xwithyα → yresp.,yα → yweaklyand

lim sup_αinff∈Myαinfu∈TyαRef−u, yα−y ≤0. Then it follows that, for anyx∈X,

lim sup

α∈Γ f∈Minfyα inf

u∈Tyα

Ref−u, yα−x

≥lim inf

u∈Tyα

Ref−u, yα−x

≥lim inf

u∈Tyα

Ref−u, yα−y

lim inf

α∈Γ f∈infMyα inf

u∈Tyα

Ref−u, y−x

≥0lim inf

α∈Γ f∈infMyα inf

u∈Tyα

Ref−u, y−x

inf

f∈Myu∈infTyRe

f−u, y−x.

1.12

To obtain the above inequalities, we use the following facts. For anyα∈ Γ,uα ∈Tyαand

fα∈Myα. SinceXis compact andTxandMxare weak∗-compact valued for anyx∈X,

using the lower semicontinuity ofMand the upper semicontinuity ofTit can be shown that

details can be verified by the reader easilyuα → u∈Tyandfα → f ∈My. Thus we

obtain

lim inf

α∈Γ f∈infMyαu∈infTyαRe

f−u, y−x inf

f∈Myu∈infTyRe

f−u, y−x 1.13

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In Section 3 of this paper, we obtain some general theorems on solutions for a new class of generalized bi-quasi-variational inequalities for quasi-pseudo-monotone type II and strongly quasi-pseudo-monotone type II operators defined on noncompact sets in topological vector spaces. To obtain these results, we mainly use the following generalized version of Ky Fan’s minimax inequality16 due to Chowdhury and Tan17 .

Theorem 1.6. LetEbe a topological vector space, letX be a nonempty convex subset ofE, and let

f:X×X → R∪ {−∞,∞}be such that

afor anyA∈FXand fixedx∈coA,y→fx, yis lower semi-continuous oncoA,

bfor anyA∈FXandy∈coA,minx∈Afx, y≤0,

cfor anyA∈FXandx, y∈coA, every net{yα}α∈ΓinXconverging toywithftx

1−ty, yα≤0for allα∈Γandt∈0,1 , one hasfx, y≤0,

dthere exist a nonempty closed and compact subsetKofXandx0∈Ksuch thatfx0, y>0

for ally∈X\K.

Then there existsy∈Ksuch thatfx,y≤0for allx∈X.

Now, we use the following lemmas for our main results in this paper.

Lemma 1.7see18 . LetX be a nonempty subset of a Hausdorﬀtopological vector spaceEand letS : X → 2E be an upper semi-continuous mapping such that Sxis a bounded subset of E for any x ∈ X. Then, for any continuous linear functional p on E, the mappingfp : X → R defined by fpy sup_x_∈_S_yRep, x is upper semi-continuous; that is, for anyλ ∈ R, the set

{y∈X:fpy sup_x_∈_S_yRep, x< λ}is open inX.

Lemma 1.8see1,19 . LetX andY be topological spaces, letf : X → Rbe nonnegative and continuous and letg:Y → Rbe lower semi-continuous. Then the mappingF :X×Y → Rdefined byFx, y fxgyfor allx, y∈X×Y is lower semi-continuous.

Theorem 1.9see20,21 . LetX be a nonempty convex subset of a vector space and letY be a nonempty compact convex subset of a Hausdorﬀtopological vector space. Suppose thatf is a real-valued function onX×Y such that, for each fixedx∈X, the mappingy→fx, y, that is,fx,·is lower semi-continuous and convex onY and, for each fixedy∈Y, the mappingx→fx, y, that is,

f·, yis concave onX. Then

min

y∈Y sup_x_∈_Xf

x, ysup

x∈Xminy∈Y f

x, y. _1.14

2. Existence Results

In this section, we will obtain and prove some existence theorems for the solutions to the generalized bi-quasi-variational inequalities of quasi-pseudo-monotone type II and strongly quasi-pseudo-monotone type II operators with noncompact domain in locally convex Hausdorﬀtopological vector spaces. Our results extend and/or generalize the corresponding results in1 .

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Lemma 2.1. LetEbe a Hausdorﬀtopological vector space overΦ, letF be a vector space overΦ, and letX be a nonempty compact subset ofE. Let·,· :F×E → Φbe a bilinear functional such that·,·separates points inF. Suppose that theFequips with theσF, E-topology; for anyw∈F,

x→Rew, xis continuous onEandT,M:X → 2F are upper semi-continuous maps such that

TxandMxare compact for anyx ∈ X. Letx0 ∈X andh :X → R be continuous. Define a

mappingg:X → Rby

gy inf

f∈Mywinf∈TyRe

f−w, y−x0

hy−hx0, ∀y∈X. 2.1

Suppose that·,·is continuous over thecompactsubset_y_∈_XMy−_y_∈_XTy ×XofF×E. Thengis lower semi-continuous onX.

Now, we establish our first main result as follows.

Theorem 2.2. Let E be a locally convex Hausdorﬀ topological vector space over Φ, let X be a nonempty paracompact convex and bounded subset ofE, and letFbe a Hausdorﬀtopological vector space overΦ. Let·,·:F×E → Φbe a bilinear functional which is continuous over compact subsets ofF×X. Suppose that

aS:X → 2Xis upper semi-continuous such that eachSxis compact and convex,

bh:E → Ris convex andhXis bounded,

cT : X → 2F is an h-quasi-pseudo-monotone type II resp., strongly h -quasi-pseudo-monotone type IIoperator and is upper semi-continuous such that eachTxis compact

resp., weakly compactand convex andTXis strongly bounded,

dM : X → 2F is an upper semi-continuous mapping such that each Mx is weakly compact and convex,

ethe setΣ {y∈X: sup_x_∈_S_yinff∈Myinfu∈TyRef−u, y−xhy−hx>0}

is open inX.

Suppose further that there exist a nonempty closed and compact resp., weakly closed and weakly compactsubsetKofXand a pointx0∈Xsuch thatx0∈K∩Syand

inf

w∈Tyf∈infMyRe

f−w, y−x0

hy−hx0>0, ∀y∈X\K. _2.2

Then there exists a pointy∈Xsuch that

1y∈Sy,

2there exist a pointf∈Myand a pointw ∈Tysuch that

Ref−w, y−x≤hx−hy, ∀x∈Sy. 2.3

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Proof. We need to show that there exists a pointy∈Xsuch thaty∈Syand

sup

x∈Sy f∈infMyu∈infTyRe

f−u,y−xhy−hx

≤0. _2.4

Suppose the contrary. Then, for anyy∈X, eithery /∈Syor there existsx∈Sysuch that

inf

f∈Myu∈infTyRe

f−u, y−xhy−hx>0, _2.5

that is, for anyy ∈X, eithery /∈Syory∈Σ. Ify /∈Sy, then, by a separation theorem for convex sets in locally convex Hausdorﬀtopological vector spaces, there exists p ∈ E∗ such that

Rep, y− sup

x∈Sy

Rep, x>0. _2.6

Let

γy sup

x∈Sy inf

f∈Myu∈infTyRe

f−u, y−xhy−hx,

V0 :

y∈X:γy>0 Σ,

2.7

and, for anyp∈E∗, set

Vp:

y∈X: Rep, y− sup

x∈SyRe

p, x>0

. 2.8

ThenX V0 ∪

p∈E∗Vp.Since eachVpis open inX byLemma 1.7and V0 is open inX by

hypothesis,{V0, Vp :p∈E∗}is an open covering forX. SinceXis paracompact, there exists

a continuous partition of unity{β0, βp :p∈E∗}forXsubordinated to the covering{V0, Vp :

p ∈ E∗_}_{see Dugundji}₂₂_{, Theorem VIII, 4.2} _{; that is, for any}_p _∈_E∗_,_β_p _:_X _→ ₀_,₁ _and

β0:X → 0,1 are continuous functions such that, for anyp∈E∗,βpy 0 for ally∈X\Vp

andβ0y 0 for ally ∈ X\V0 and{supportβ0, supportβp : p ∈ E∗}is locally finite and

β0y

p∈E∗βpy 1 for anyy∈X. Note that, for anyA∈FX,his continuous on coA

see23, Corollary 10.1.1 . Define a mappingφ:X×X → Rby

φx, yβ0

y inf

f∈Myu∈infTyRe

f−u, y−xhy−hx

p∈E∗

βpyRep, y−x, ∀x, y∈X.

2.9

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iSinceEis Hausdorﬀ, for anyA∈FXand fixedx∈coA, the mapping

y−→ inf

f∈Myu∈infTyRe

f−u, y−xhy−hx _2.10

is lower semi-continuousresp., weakly lower semi-continuouson coAbyLemma 2.1and so the mapping

y−→β0

y inf

f∈Myu∈infTyRe

2.11

is lower semi-continuous resp., weakly lower semi-continuous on coA byLemma 1.8. Also, for any fixedx∈X,

y−→

p∈E∗

βpyRep, y−x 2.12

is continuous onX. Hence, for anyA∈FXand fixedx∈coA, the mappingy→φx, y is lower semi-continuousresp., weakly lower semi-continuouson coA.

iiFor anyA∈FXandy∈coA, minx∈Aφx, y≤ 0. Indeed, if this is false, then,

for someA{x1, x2, . . . , xn} ∈FXandy∈coA sayyni1λixiwhereλ1, λ2, . . . , λn≥0

withn_i₁λi1, we have min1≤i≤nφxi, y>0. Then, for anyi1,2, . . . , n,

β0

y inf

f∈Myu∈infTyRe

f−u, y−xihy−hxi

p∈E∗

βjyRep, y−xi>0, 2.13

which implies that

0φy, y

β0

y

inf

f∈Myu∈infTyRe

f−u, y−n

i1

λixi

hy−h

_n

i1

λixi

p∈E∗

βjyRe

p, y−n

i1

λixi

≥n

i1

λi ⎧ ⎨ ⎩

⎛ ⎝_β₀_y

⎡ ⎣ inf

f∈Myu∈infTyRe

f−u, y−xihy−hxi ⎤ ⎦

p∈E∗

βpyRep, y−xi ⎞ ⎠

⎫ ⎬ ⎭

>0,

2.14

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iiiSuppose thatA ∈AX,x, y ∈ coA, and{yα}α∈Γis a net inX converging toy

resp., weakly toywithφtx 1−ty, yα≤0 for allα∈Γandt∈0,1 .

Case 1β0y 0. Note thatβ0yα≥0 for anyα∈Γandβ0yα → 0. SinceTXis strongly

bounded and{yα}α∈Γis a bounded net, it follows that

lim sup

α β0

yα' min f∈Myα

min

u∈Tyα

Ref−u, yα−xhyα−hx (

0. 2.15

Also, we have

β0

y min

f∈Myumin∈TyRe

0. 2.16

Thus, from2.15, it follows that

lim sup

α β0

yα' min

f∈Myαumin∈TyαRe

f−u, yα−xhyα−hx (

p∈E∗

βpyRep, y−x

p∈E∗

βpyRep, y−x

β0

y min

f∈Myumin∈TyRef−u, y−xh

y−hx

p∈E∗

βpyRep, y−x.

2.17

Whent1, we haveφx, yα≤0 for allα∈Γ, that is,

β0

yα min

p∈E∗

βpyαRep, yα−x ≤0.

2.18

Therefore, by2.18, we have

lim sup

α β0

yα min

lim inf

α ⎡

⎣

p∈E∗

βpyαRep, yα−x ⎤ ⎦

≤lim sup

α ⎡ ⎣β0

yα_fmin ∈Myα

min

u∈Tyα

Ref−u, yα−xhyα−hx

p∈E∗

≤0,

(12)

which implies that

lim sup

α β0

yα min f∈Myα

min

u∈Tyα

p∈E∗

βpyRep, y−x ≤0.

2.20

Hence, by2.17and2.20, we haveφx, y≤0.

Case 2β0y>0. Sinceβ0yα → β0y, there existsλ∈Γsuch thatβ0yα>0 for anyα≥λ.

Whent0, we haveφy, yα≤0 for allα∈Γ, that is,

β0

yα inf f∈Myα

inf

u∈Tyα

Ref−u, yα−yhyα−hy

p∈E∗

βpyαRep, yα−y ≤0.

2.21

Thus it follows that

lim sup

α ⎡ ⎣β0

yα' inf f∈Myα

inf

u∈TyαR

f−u, yα−yhyα−hy(

p∈E∗

βpyαRep, yα−y ⎤ ⎦_≤0.

2.22

Hence, by2.22, we have

lim sup

α β0

yα' inf f∈Myα

inf

u∈Tyα

Ref−u, yα−yhyα−hy(

lim inf

α ⎡

⎣

p∈E∗

βpyαRep, yα−y ⎤ ⎦

≤lim sup

α ⎡ ⎣β0

yα ⎛ ⎝ _inf

f∈Myα inf

u∈Tyα

Ref−u, yα−yhyα−hy ⎞ ⎠

p∈E∗

βpyαRep, yα−y ⎤ ⎦

≤0.

2.23

Since lim infαp∈E∗βpyαRep, yα−y 0, we have

lim sup

α β0

yα' min

(13)

Sinceβ0yα>0 for allα≥λ, it follows that

β0

ylim sup

α f∈minMyαumin∈TyαRe

f−u, yα−yhyα−hy

lim sup

α β0

yα' min

f−u, yα−yhyα−hy(.

2.25

Sinceβ0y>0, by2.24and2.25, we have

lim sup

α f∈minMyαumin∈TyαRe

f−u, yα−yhyα−hy≤0. 2.26

Since T is an h-quasi-pseudo-monotone type II resp., strongly h-quasi-pseudo-monotone typeIIoperator, we have

lim sup

α f∈minMyα min

u∈Tyα

≥ min

f∈Mywmin∈TyRe

f−w, y−xhy−hx, ∀x∈X.

2.27

Sinceβ0y>0, we have

β0

y lim sup

α '

min

f∈Myα min

u∈Tyα

≥β0

y min

f∈Mywmin∈TyRe

f−w, y−xhy−hx

2.28

and so

β0

y lim sup

α '

min

f∈Myα min

u∈Tyα

p∈E∗

βpyRep, y−x

≥β0

y min

f∈Mywmin∈TyRe

p∈E∗

βpyRep, y−x.

2.29

Whent1, we haveφx, yα≤0 for allα∈Γ, that is,

β0

yα min f∈Myα

min

u∈Tyα

p∈E∗

(14)

and so, by2.29,

0≥lim sup

α ⎡ ⎣β0

yα min f∈Myα

min

u∈Tyα

p∈E∗

≥lim sup

α β0

yα min

lim inf

α ⎡

⎣

p∈E∗

β0

y lim sup

α )

min

f∈Myα min

u∈Tyα

Ref−u, yα−xhyα−hx *

p∈E∗

βpyRep, y−x

≥β0

y min

f∈Mywmin∈TyRe

p∈E∗

βpyRep, y−x.

2.31

Hence we haveφx, y≤0.

ivBy the hypothesis, there exists a nonempty compact and so a closedresp., weakly closed and weakly compactsubsetKofXand a pointx0∈Xsuch thatx0∈K∩Syand

inf

f∈Mywinf∈TyRe

f−w, y−x0

hy−hx0>0, ∀y∈X\K. _2.32

Thus it follows that

β0

y inf

w∈Tyf∈infMyRe

f−w, y−x0

hy−hx0

>0, ∀y∈X\K, 2.33

wheneverβ0y> 0 and Rep, y−x0 >0 wheneverβpy>0 for allp∈E∗. Consequently,

we have

φx0, y

β0

y inf

f∈Mywinf∈TyRe

f−w, y−x0

hy−hx0

p∈E∗

βpyRep, y−x0

>0, ∀y∈X\K.

2.34

IfTis a stronglyh-quasi-pseudo-monotone type II operator, then we equipEwith the weak topology.Thusφsatisfies all the hypotheses ofTheorem 1.6and so, byTheorem 1.6, there exists a pointy∈Ksuch thatφx,y≤0 for allx∈X, that is,

β0

y inf

f∈Myu∈infTyRe

p∈E∗

βpyRep,y−x≤0, ∀x∈X.

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Now, the rest of the proof is similar to the proof in Step 1 of Theorem 1 in24 . Hence we have shown that

sup

x∈Sy f∈infMyu∈infTyRe

≤0. 2.36

Then, by applyingTheorem 1.9as we proved in Step 3 of Theorem 1 in24 , we can show that there exist a pointf∈Myand a pointw ∈Tysuch that

Ref−w, y−x≤hx−hy, ∀x∈Sy. 2.37

We observe from the above proof that the requirement that E is locally convex is needed if and only if the separation theorem is applied to the casey /∈Sy. Thus, ifS:X → 2Xis the constant mappingSx Xfor allx∈X, theEis not required to be locally convex.

Finally, ifT ≡ 0, in order to show that, for any x ∈ X,y → φx, yis lower semi-continuousresp., weakly lower semi-continuous,Lemma 2.1is no longer needed and the weaker continuity assumption on ·,· that, for any f ∈ F, the mapping x → f, x is continuousresp., weakly continuousonXis suﬃcient. This completes the proof.

We will now establish our last result of this section.

Theorem 2.3. Let E be a locally convex Hausdorﬀ topological vector space over Φ, let X be a nonempty paracompact convex and bounded subset ofE, and let F be a vector space over Φ. Let

·,·:F×E → Φbe a bilinear functional such that·,·separates points inF,·,·is continuous over compact subsets ofF ×X, and, for anyf ∈ F, the mappingx → f, xis continuous onX. Suppose thatFequips with the strong topologyδF, Eand

aS:X → 2Xis a continuous mapping such that eachSxis compact and convex,

bh:X → Ris convex andhXis bounded,

cT : X → 2F is an h-quasi-pseudo-monotone type II resp., strongly h -quasi-pseudo-monotone type IIoperator and is an upper semi-continuous mapping such that eachTx is strongly, that is,δF, E-compact and convexresp., weakly, i.e.,σF, E-compact and convex,

dM : X → 2F is an upper semi-continuous mapping such that eachMx isδF, E -compact convex and, for anyy∈Σ,Mis upper semi-continuous at some pointxinSy withinff∈Myinfu∈TyRef−u, y−xhy−hx>0, where

Σ

y∈X: sup

x∈Sy inf

f∈Myu∈infTyRe

>0

. 2.38

Suppose further that there exist a nonempty closed and compact resp., weakly closed and weakly compactsubsetKofXand a pointx0∈Xsuch thatx0∈K∩Syand

inf

f∈Mywinf∈TyRe

f−w, y−x0

(16)

Then there exists a pointy∈Xsuch that

1y∈Sy,

2there exist a pointf∈Myand a pointw ∈Tywith

Ref−w, y−x≤hx−hy, ∀x∈Sy. 2.40

Moreover, ifSx Xfor allx∈X, thenEis not required to be locally convex.

Proof. The proof is similar to the proof of Theorem 2 in24 and so the proof is omitted here.

Remark 2.4. 1Theorems2.2and2.3of this paper are generalizations of Theorems 3.2 and 3.3 in 3 , respectively, on noncompact sets. In Theorems 2.2and 2.3,X is considered to be a paracompact convex and bounded subset of locally convex Hausdorﬀ topological vector space E whereas, in 3 , X is just a compact and convex subset of E. Hence our results generalize the corresponding results in3 .

2The first paper on generalized bi-quasi-variational inequalities was written by Shih and Tan in 1989 in1 and the results were obtained on compact sets where the set-valued mappings were either lower semi-continuous or upper semi-continuous. Our present paper is another extension of the original work in1 using quasi-pseudo-monotone type II operators on noncompact sets.

3 The results in 4 were obtained on compact sets where one of the set-valued mappings is a quasi-pseudo-monotone type I operators which were defined first in4 and extends the results in1 . The quasi-pseudo-monotone type I operators are generalizations of pseudo-monotone type I operators introduced first in17 . In all our results on generalized bi-quasi-variational inequalities, if the operatorsM≡0 and the operatorsT are replaced by

−T, then we obtain results on generalized quasi-variational inequalities which generalize the corresponding results in the literaturesee18 .

4 The results on generalized bi-quasi-variational inequalities given in 5 were obtained for set-valued quasi-semi-monotone and bi-quasi-semi-monotone operators and the corresponding results in2 were obtained for set-valued upper-hemi-continuous operators introduced in6 . Our results in this paper are also further extensions of the corresponding results in2,5 using set-valued quasi-pseudo-monotone type II operators on noncompact sets.

Acknowledgment

This work was supported by the Korea Research Foundation Grant funded by the Korean GovernmentKRF-2008-313-C00050.

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