Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 237191,17 pages doi:10.1155/2010/237191

Research Article

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

Mohammad S. R. Chowdhury1 and Yeol Je Cho2

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 YeolJe Cho, yjcho@gnu.ac.kr

Received 3 November 2009; Accepted 18 January 2010

Academic Editor: Jong Kyu Kim

Copyright © 2010 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 quasi-pseudomonotone type II and strongly quasi-pseudomonotone type II operators, we use Chowdhury and Tan's generalized version (1996) of Ky Fan's minimax inequality (1972) as 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 Hausdorff topological 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.

Let X be a nonempty set and let 2X be the family of all nonempty subsets of X. If X and Y are topological spaces and T : X ^ 2Y, then the graph of T is the set G(T) := {(x, y) e X x Y : y e T(x)}. Throughout this paper, ® denotes either the real field R or the complex field C.

Let E be a topological vector space over let F be a vector space over ® and let (■, ■) : F x E ^ ® be a bilinear functional.

For any x0 e E, any nonempty subset A of E, and any e > 0, let W(x0; e) := {y e F : |(y,xo)| < e} and U(A; e) := {y e F : supxeA|(y,x)| < e}. Let o(F,E) be the (weak) topology

on F generated by the family {W(x; e) : x e E and e > 0} as a subbase for the neighbourhood system at 0 and let 6(F,E) be the (strong) topology on F generated by the family {U(A; e) : A is a nonempty bounded subset of E and e > 0} as a base for the neighbourhood system at 0. We note then that F, when equipped with the (weak) topology a(F,E) or the (strong) topology 6(F,E), becomes a locally convex topological vector space which is not necessarily Hausdorff. But, if the bilinear functional {■, ■) : F x E ^ ® separates points in F, that is, for any y e F with y ^ 0, there exists x e E such that {y, x) ^ 0, then F also becomes Hausdorff. Furthermore, for any net {ya.}aer in F and y e F,

(1) ya ^ y in a{F,E) if and only if {ya,x) ^ {y,x) for any x e E,

(2) ya ^ y in 6{F,E) if and only if {ya,x) ^ {y,x) uniformly for any x e A, where a nonempty bounded subset of E.

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 (quasi)variational inequalities, generalized (quasi)variational-like inequalities and generalized bi-quasi-variational inequalities (see [2-15]).

The following is the definition due to Shih and Tan [1].

Definition 1.1. Let E and F be a vector spaces over let {■, ■) : F x E ^ ® be a bilinear functional, and let X be a nonempty subset of E. If S : X ^ 2X and M, T : X ^ 2F, the generalized bi-quasi variational inequality problem (GBQVI) for the triple (S, M, T) is to find y e X satisfying the following properties:

(1) y e S(y),

(2) infweTy Re{/ - w,y - x) < 0 for any x e S(y) and f e M(y).

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

Definition 1.2. Let E and F be vector spaces over let {■, ■) : F x E ^ ® be a bilinear functional, and let X be a nonempty subset of E. If S : X ^ 2X and M, T : X ^ 2F, then the generalized bi-quasivariational inequality (GBQVI) problem for the triple (S, M, T) is:

(1) to find a point y e X and a point w e T(y) such that

y e S(y), Re/ - w,y - x) < 0, Vx e S(y), f e M(y), (1.1)

(2) to find a point y e X, a point w e T(y), and a point f e M(y) such that

y e S(y), Re( f - w,y - x) < 0, Vx e S(y). (1.2)

Let X be a nonempty subset of E and let T : X ^ 2E' be a set-valued mapping. Then T is said to be monotone on X if, for any x,y e X, u e T(x), and w e T(y),Re{w - u,y - x) > 0.

Let X and Y be topological spaces and let T : X ^ 2Y be a set-valued mapping. Then T is said to be:

(1) upper (resp., lower) semicontinuous at x0 e X if, for each open set G in Y with T(xo) c G (resp., T(xo) n G / 0), there exists an open neighbourhood U of xo in X such that T(x) c G (resp., T(x) n G / 0) for all x e U,

(2) upper (resp., lower) semicontinuous on X if T is upper (resp., lower) semicontinuous at each point of X,

(3) continuous on X if T is both lower and upper semi-continuous on X.

Let X be a convex set in a topological vector space E. Then f : X ^ R is said to be lower semi-continuous if, for all 1 e R, {x e X : f (x) < 1} is closed in X.

If X is a convex set in a vector space E, then f : X ^ R is said to be concave if, for all x,y e X and 0 < 1 < 1,

f (.1x + (1 - 1)y) > 1f (x) + (1 - 1)f (y). (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 in [3].

Definition 1.3. Let E be a topological vector space, let X be a nonempty subset of E, and let F be a topological vector space over Let (■, •) : F x E ^ ® be a bilinear functional. Consider a mapping h : X ^ R and two set-valued mappings M : X ^ 2F and T : X ^ 2f .

(1) T is called an h-quasi-pseudo-monotone (resp., strongly h-quasi-pseudo-monotone) type II operator if, for any y e X and every net {ya}aer in X converging to y (resp., weakly to y) with

lim sup

lim sup

f inf ) iif ) Re(f - U,ya - y) + hy) - hy

f eM(y.) ueT(y„)

inf inf R^ f - u,ya - ^ + h(ya) - h(x)

f eM(ya) ueTy)

> inf inf R^f - w,y - x + - h(x), Vx e X.

f eM(y) weT(y)

(2) T is said to be a quasi-pseudo-monotone (resp., strongly quasi-pseudo-monotone) type II operator if T is an h-quasi-pseudo-monotone (resp., strongly h-quasi-pseudo-monotone) type II operator with h = 0.

The following is an example on quasi-pseudo-monotone type II operators given in [3] .

Example 1.4. Consider X = [-1,1] and E = R. Then E* = R. Let M : X ^ 2R be a set-valued mapping defined by

f [0,2x] if x > 0, M(x) ^ (1.5)

I [2x, 0] if x < 0.

Again, let T : X ^ 2R be a set-valued mapping defined by

T (x) =

(1,3} (1,2,3}

if x < 1, if x = 1.

Then M is lower semi-continuous and T is upper semi-continuous. It can be shown that T becomes a quasi-pseudo-monotone type II operator on X = [-1,1].

(i) To show that M is lower semi-continuous, consider xo > 0. Then M(xo) = [0,2xo]. Let e > Obe given. Then, if G = (2x0 -e, 2x0+e), then M(x0)nG = [0,2x0] n(2x0 -e, 2x0+e) = 0. Let e > 0 be so chosen that 0 < x0 - e/2 < x0 < x0 + e/2 < 1. Now, if we take U = (x0 - e/2,x0 + e/2), then, for all x e U, we have 2x0 - e < 2x < 2x0 + e. Thus 2x e M(x) n G. Hence M(x) n G / 0.

If x0 = 0, M(0) = 0. Then, for0 e G = (e,e),we can take U = (-e/2, e/2). Thus for all x e U, M(x) = [0,2x] n (e, e) / 0 because -e < x < e/2 implies 2x e G = (-e,e).

Finally, if x0 < 0, then M(x0) = [2x0,0]. We take G = (2x0 - e, 2x0 + e) for some e > 0 so that M(x0) n G = 0 and x0 + e/2 < 0. Thus, for all x e U = (x0 - e/2,x0 + e/2), we have 2x0 - e < 2x < 2x0 + e. Hence 2x e G n M(x), where M(x) = [2x, 0] for x < 0. Consequently, M is lower semi-continuous on X = [-1,1].

(ii) To show that T is upper semi-continuous, let x0 e [-1,1] be such that x0 < 1. Then T(x0) = (1,3}. Let G be an open set in R such that T(x0) = (1,3} c G. Let e > 0 be such that -1 < x0 - e < x0 < x0 + e < 1. Consider U = (x0 - e,x0 + e). Then, for all x e U, T(x) = (1,3} c G since x < 1. Again, if x = 1, then T(1) = (1,2,3}. Let G be an open set in R suchthat T (1) = (1,2,3} c G. Let e> 0 be suchthat -1 < 1 - e < 1 < 1 + e. Let U = (1 - e, 1] which is an open neighbourhood of 1 in X = [-1,1]. Then for all x e U = (1 - e, 1], we have T(x) = (1,3} if 1 - e < x < 1 and T(x) = (1,2,3} if x = 1. Now, T(1) = (1,2,3} c G. Also, for all x e U with 1 - e < x < 1, we have T(x) = (1,3} c (1,2,3} c G. Hence T is upper semi-continuous on X = [-1,1].

(iii) Finally, we will show that T is also a quasi-pseudo-monotone type II operator. To show this, let us assume first that {ya) is a net in X = [-1,1] such that ya ^ y in X = [-1,1]. We now show that

lim sup

inf inf Re( f - u,ya - y)

feM(ya) ueT(ya) J

We have

lim sup

inf inf Re f - u, y - y

f eM(ya ) ueT (ya ) ^

f i0f 1 f f - a - y)

f e[0,2ya] ue(1,3}

= lim sup

= (0 - 3)(y« - y) = 3(y - ya), if0 < ya< L infm f f - u)y - y)

f e[2ya,0] ue(1,3}

= (2ya - 3) (ya - y), if ya< 0 and so ya < 1,

f i0f ] if f - - y)

= (0 - 3) (ya - y) = 3(y - ya), if ya = 1, that is, ya > 0

(considering ya - y > 0, the value will be also 0 if we consider y a - y < 0). So, it follows that, for all x e [-1,1],

lim sup

inf inf (f - u) (ya - x)

f eM(ya) ueT(ya) v

f ftf ] f f - a - X

f e[0,2ya] ue{1,3}

= (0 - 3) (ya - x) = 3(x - ya),

= lim sup

fi2nfm f f - - x)

f e [2ya,0] ue{1,3}

= (2ya - 3) (ya -

inf inf (f - u) (ya - x)

f e[0,2yj ue{ 1,2,3}

if 0 < ya< 1,

if ya< 0 and so ya < 1,

= (0 - 3) (ya - x = 3(x - ya), if ya = 1, that is, ya > 0, (consider ya - x > 0)

3( x - y), if 0 < ya< 1,

(2y - 3) (y - x), if ya< 0 and so ya < 1, 3(x - y), if ya = 1,

consider y - x > 0 .

The values can be obtained similarly for the cases where ya - x < 0 and y - x < 0. Also, it follows that, for all x e X = [-1,1],

inf inf f - u y - x

f eM(y) ueT(y)

inf inf f - u y - x

f e[0,2y] ue{1,3} ^ y

=(0 - 3K y - x) = 3(x -

inf inf f - u y - x

fe[2y,0] ue{1,3}

= 2 y - 3 y - x ,

inf inf f - u y - x

f e [0,2y] ue{ 1,2,3} /w 7

if 0 < y < 1,

if y < 0 and so y < 1,

if y = 1, that is, y > 0,

(1.10)

= (0 - 3)(y - x) = 3(x - y^

consider y - x > 0

3(x - y), if 0 < y< 1,

(2y - 3) (y - x), if y < 0 and so y < 1, 3(x - y^ if y = 1,

(consider y - x > 0).

The values can be obtained similarly for the cases when y - x < 0. Therefore, in all the cases, we have shown that

lim sup

inf inf (f - u) (ya - x)

f eM(ya) ueT(yaV

> f f vfP - u(y - (1.11)

f M(y) u T(y)

Hence T is 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. Let X be a nonempty compact subset of a topological vector space E. Suppose that M : X ^ 2e* and T : X ^ 2E* are two set-valued mappings such that M is lower semi-continuous and T is upper semi-continuous. Suppose further that, for any x e X, M(x) and T(x) are weak*-compact sets in E*. Then T is both a quasi-pseudo-monotone type II and a strongly quasi-pseudo-monotone type II operator.

Proof. Suppose that {ya}a er is a net in X and y e X with ya ^ y (resp., ya ^ y weakly) and limsupa[inffeM(ya)infueT(ya) Re(f - u,ya - y)] < 0. Then it follows that, for any x e X,

lim sup

inf inf Re f - u, ya - x

f e M(ya) u eT (ya)

> lim inf

inf inf Re( f - u,ya - x)

f e M(ya) u eT (ya)

> lim inf

inf inf Re( f - u,ya - y)

f e M(ya) u eT (ya)

lim inf

inf inf Re f - u, y - x

f e M(ya) u eT (ya)

(1.12)

> 0 + lim inf

inf inf Re f - u, y - x

f e M(ya) u eT (ya)

= inf inf Re f - u, y - x .

f M(y) u T(y)

To obtain the above inequalities, we use the following facts. For any a e r, ua e T(ya) and fa e M(ya). Since X is compact and T(x) and M(x) are weak*-compact valued for any x e X, using the lower semicontinuity of M and the upper semicontinuity of T it can be shown that (details can be verified by the reader easily) ua ^ u e T(y) and fa ^ f e M(y). Thus we obtain

lim inf

f e M(ya) u eT (ya)

inf inf Ref - u,y - ^ = inf inf Re(f - u,y - ^ (1.13)

f M(y) u T(y)

in the last inequality above. Consequently, T is both a quasi-pseudo-monotone type II and a strongly quasi-pseudo-monotone type II operator. □

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 inequality [16] due to Chowdhury and Tan [17].

Theorem 1.6. Let E be a topological vector space, let X be a nonempty convex subset of E, and let f : X x X ^ R u {-to, be such that

(a) for any A e F(X) and fixed x e co(A), y ^ f (x, y) is lower semi-continuous on co(A),

(b) for any A e F(X) and y e co(A), minx e Af (x, y) < 0,

(c) for any A e F(X) and x,y e co(A), every net {ya}aer in X converging to y with f (tx + (1 - t)y,ya) < 0 for all ae r and t e [0,1], one has f (x,y) < 0,

(d) there exist a nonempty closed and compact subset K of X and x0 e K such that f (x0, y) > 0 for all y eX \ K.

Then there exists ye K such that f (x, y) < 0 for all x eX.

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

Lemma 1.7 (see [18]). Let X be a nonempty subset of a Hausdorff topological vector space E and let S : X ^ 2e be an upper semi-continuous mapping such that S(x) is a bounded subset of E for any x e X. Then, for any continuous linear functional p on E, the mapping fp : X ^ R defined by fp(y) = supxeSy Re(p,x) is upper semi-continuous; that is, for any X e R, the set {y eX : fp (y) = supxeS(y) Re(p,x) < X} is open in X.

Lemma 1.8 (see [1, 19]). Let X and Y be topological spaces, let f : X ^ R be nonnegative and continuous and let g : Y ^ R be lower semi-continuous. Then the mapping F : X x Y ^ R defined by F(x, y) = f (x)g(y) for all (x,y) eX x Y is lower semi-continuous.

Theorem 1.9 (see [20, 21]). Let X be a nonempty convex subset of a vector space and let Y be a nonempty compact convex subset of a Hausdorff topological vector space. Suppose that f is a real-valued function on X x Y such that, for each fixed x eX, the mapping y ^ f (x, y), that is, f (x, ■) is lower semi-continuous and convex on Y and, for each fixed y eY, the mapping x ^ f (x,y), that is, f (■, y) is concave on X. Then

minsup f(x,y) = supmin f(x,y). (1.14)

yeY xeX xeX yeY v y

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 Hausdorff topological vector spaces. Our results extend and/or generalize the corresponding results in [1].

Before we establish our main results, we state the following result which is Lemma 3.1

in [3].

Lemma 2.1. Let E be a Hausdorff topological vector space over O, let F be a vector space over O, and let X be a nonempty compact subset of E. Let (■, ■) : F x E ^ O be a bilinear functional such that (■, ■) separates points in F. Suppose that the F equips with the a(F,E)-topology;for any w e F, x ^ Re(w,x) is continuous on E and T, M : X ^ 2F are upper semi-continuous maps such that T(x) and M(x) are compact for any x e X. Let x0 e X and h : X ^ R be continuous. Define a mapping g : X ^ R by

Suppose that (■, ■) is continuous over the (compact) subset [UyeX M(y) - UyeX T(y)] x X of F x E. Then g is lower semi-continuous on X.

Now, we establish our first main result as follows.

Theorem 2.2. Let E be a locally convex Hausdorff topological vector space over O, let X be a nonempty paracompact convex and bounded subset of E, and let F be a Hausdorff topological vector space over O.Let (■, ■) : F x E ^ O be a bilinear functional which is continuous over compact subsets ofF x X. Suppose that

(a) S : X ^ 2X is upper semi-continuous such that each S(x) is compact and convex,

(b) h : E ^ R is convex and h(X) is bounded,

(c) T : X ^ 2f is an h-quasi-pseudo-monotone type II (resp., strongly h-quasi-pseudo-

monotone type II) operator and is upper semi-continuous such that each T(x) is compact (resp., weakly compact) and convex and T(X) is strongly bounded,

(e) the set 2 = {y e X : supxeS(y)(inffeM(y)infueT(y) Re(f - u,y - x) + h(y) - h(x)) > 0} is open in X.

Suppose further that there exist a nonempty closed and compact (resp., weakly closed and weakly compact) subset K of X and a point x0 e X such that x0 e K n S(y) and

inf inf Re(f - w,y - x0) + h(y) - h(xo), Vy e X.

f eM(y) weT(y)

(d) M : X ^ 2f is an upper semi-continuous mapping such that each M(x) is weakly compact and convex,

inf inf Re(f - w,y - x^ + h(y) - h(x0) > 0, Vy e X \ K.

weT(y) f eM(y)

Then there exists a point ye X such that

(1) y e S(y),

(2) there exist a point f e M(y) and a point W e T(y) such that

Moreover, if S(x) = X for all x e X, then E is not required to be locally convex, and if T = 0, then the continuity assumption on (■, ■) can be weakened to the assumption that, for any f e F, the mapping x ^ (f,x) is continuous (resp., weakly continuous) on X.

Proof. We need to show that there exists a point ye X such that y e S(y) and

xeS(y)

inf inf Ref - u,y - x) + h(y) - h(x)

f eM(y) ueT(y)

< 0. (2.4)

Suppose the contrary. Then, for any y eX, either y / S(y) or there exists x e S(y) such

, ^f ) ^) Re(f - u,y - x + hy - h(x) > 0, (2.5)

f eM(y) ueT(y) ^ '

that is, for any y eX, either y / S(y) or ye 2. If y / S(y), then, by a separation theorem for convex sets in locally convex Hausdorff topological vector spaces, there exists p e E* such that

Re<P,y - sup Re(P,x) > 0. (2 6)

xeS(y) K ' >

r(y) = sup in() in(f) Re(f - u,y - x) + %) - h(x),

xeS(y) f eM(y) ueT(y) (2 7)

V) := {yeX : Y(y) > 0} =2,

and, for any p e E*, set

Vp := i yeX : Re( p,y) - sup Re( p,x) > 0 I. (2.8)

I xeS(y) J

Then X = Vo U{JpeE, Vp. Since each Vp is open in X by Lemma 1.7 and Vo is open in X by hypothesis, {V0,Vp : p e E*} is an open covering for X. Since X is paracompact, there exists a continuous partition of unity : p e E*} for X subordinated to the covering {V0,Vp :

p e E*} (see Dugundji [22, Theorem VIII, 4.2]); that is, for any p e E*, ftp : X ^ [0,1] and ¡30 : X ^ [0,1] are continuous functions such that, for any p e E*, (y) = 0 for all yeX \ Vp and ft0(y) = 0 for all yeX \ V0 and {support ft0, support : p e E*} is locally finite and (y) + ^peE* ftp(y) = 1 for any yeX. Note that, for any A e F(X), h is continuous on co(A) (see [23, Corollary 10.1.1]). Define a mapping $ : X x X ^ R by

Hx,y) = inf mf) Ref - u,y - x) + %) - h(x)

f eM(y) ueT(yy)

_ (2.9)

+ E My) Re(p, y- x), Vx, yeX.

p e E'

Then we have the following.

(i) Since E is Hausdorff, for any A e F(X) and fixed x e co(A), the mapping

y i—> inf inf Re( f - u,y - x) + h(y) - h(x)

f eM(y) ueT(y)

(2.10)

is lower semi-continuous (resp., weakly lower semi-continuous) on co(A) by Lemma 2.1 and so the mapping

y 1—» p0(y) inf inf Re(f - u,y - x) + h(y) - h(x)

f eM(y) u eT(y)

(2.11)

is lower semi-continuous (resp., weakly lower semi-continuous) on co(A) by Lemma 1.8. Also, for any fixed x e X,

y E My) Re(p>y - x)

p e £*

(2.12)

is continuous on X. Hence, for any A e F(X) and fixed x e co(A), the mapping y ^ $(x,y) is lower semi-continuous (resp., weakly lower semi-continuous) on co(A).

(ii) For any A e F(X) and y e co(A), minx e Aty(x,y) < 0. Indeed, if this is false, then, for some A = {x1,x2,...,xn} e F(X) and y e co(A) (say y = ^ "=1 ^¿xj where 11/12/ ...,Xn > 0 with£ n=1 ^ = 1), we have min1<j<n^(xj/ y) > 0. Then, for any j = 1,2,...,n,

Po(y fiMfy) Jry f - u,y - + h(y - h(xi) + 2^ y P'y - > (2.13)

p e£*

which implies that

0 = Hy'y)

= M y)

, ) in(f ) Re\ f - u,y - ^ Xixi) + hy - h5>

feM(y) ueT(y) \ /

2>(y) rK p,y -Ë ^

p e £*

¿¿ii^o(y)

i=1 L \

inf inf Re(f-u,y-xù + h(y)-h(xi)

f e M(y) u eT (y) v ^ '

+X My) Re( P'y- xi)

p e £*

(2.14)

which is a contradiction.

(iii) Suppose that A e A(X), x,y e co(A), and {ya}aer is a net in X converging to y (resp., weakly to y) with $(tx + (1 - t)y,ya) < 0 for all a e r and t e [0,1].

Case 1 (ft0(y) = 0). Note that ft0(ya) > 0 for any a e r and (ya) ^ 0. Since T(X) is strongly bounded and {ya}aer is a bounded net, it follows that

lim sup

ßo(yM f mfr min Re(f - u,ya - x) + h(ya) - h(x)

\f €M(ya) UeT(ya)

(2.15)

Also, we have

ß0(y) min min Re(f - u,y - x) + h(y) - h(x)

f M(y) U T(y)

(2.16)

Thus, from (2.15), it follows that

lim sup

ßo{ya)( min mp Ref - u,ya - x) + h{ya) - h(x) )

f M(ya) U T(ya)

+ XMy) Re(V'y - x) = Eßpy Re(^y - x)

peE' peE'

= M y)

min minRe(f - u,y - x) + h(y) - h(x)

f M(y) U T(y)

+ Eßpy Re(p>y - x).

When t = 1, we have $(x, ya) < 0 for all a e r, that is,

(2.17)

ßo(ya) min min Re f - u,ya - x) + h(y^) - h(x) + ^ ßp(ya) Re(p,ya - x) <0

f M(ya) U T(ya)

(2.18)

Therefore, by (2.18), we have

lim sup

Mya) t min mn Re(f - u, ya - x) + %«) - h(x)

f e M(ya) U eT (ya)

lim inf

< lim sup

5ußp(ya) Re(p^a - x)

ß0 (ya) f m;(n ) min) Re(f -U' ya-x) +Hya)-h(x) + X frfa) Re (p ya-x

f e M(ya) U eT (ya)

(2.19)

which implies that

lim sup

Pdya) min min Ref - u,ya - x) + h(ya) - h(x) + ^ ftpty) Re(p,y - x) < 0.

f e M(ya) u eT (ya)

(2.20)

Hence, by (2.17) and (2.20), we have $(x,y) < 0.

Case 2 (ft0(y) > 0). Since ft0(ya) ^ ft0(y), there exists X e r such that ft0(ya) > 0 for any a > X. When t = 0, we have $(y, ya) < 0 for all a e r, that is,

P0( ya) mf ) iT(f ) Re( f - u,ya - y) + h(ya) - h{y) + ^ Pp^a) Re(p,ya - y) < 0

f e M(ya) u eT (ya)

(2.21)

Thus it follows that

lim sup

Mya)( mf ) mf )Rf-u,ya-y) + %«) -

\f e M(ya ) u eT (ya)

+ X ftp(ya) Re(p,ya-y)

< 0. (2.22)

Hence, by (2.22), we have

lim sup

Mya)( mf ) jnf ) Re(f - u,ya - y) + %a) -

\f e M(ya) u eT (ya)

lim inf

YuMa) Re(p,ya - y)

< lim sup

Mya) ( inf ) irif ) Re(f - u,ya - y) + %a) - %)

\ f e M(ya) u eT (ya)

(2.23)

+ X ftp(ya) Re(p,ya - y)

Since liminfa[%peE* ftp(ya) Re(p,ya - y)] = 0, we have

lim sup

ftdya){ min min Re(f - u,ya - y) + h(ya) -

\feM(ya)ueT (ya)

(2.24)

Journal of Inequalities and Applications Since ¡0(ya) > 0 for all a > X, it follows that

¡0 ( y) lim sup

, mi(n ) min) f - u,ya - y) + h(ya) - %)

f eMy) u eT(ya)

= lim sup

¡0(yM mi(n ) min)Ref - u,ya - y> + %a) -

\f e M(ya) u eT(ya)

(2.25)

Since ¡¡0(y) > 0, by (2.24) and (2.25), we have

lim sup

^mi(n ) min) Re(f - u,ya - y) + h(ya) - My)

f e M(ya) u eT(ya)

(2.26)

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

lim sup

min min Re(f-u,ya-x) + h(ya)-h(x)

feM(ya) ueT(ya) V ^ ' W y

> min min Re(/ - w, y - x) + h(y) - h(x), Vx e X.

/ eM(y) w eT(y)

(2.27)

Since ¡ 0(y) > 0, we have

lim sup min min Re f - u, y - x + h y - h(x)

f M(y ) u T(y )

min min Re(/ - w,y - x) + h(y) - h(x)

f e M(y) w eT(y)

(2.28)

and so

¡0 (y) lim sup ( min min Re(/ - u, ya - x) + h(ya) - h(x) )

a V e M(ya) u eT (ya) /

+ X My) Re<p,y - x)

>¡^(^ min)Re(f - w,y-x> + %)- h(x)

f M(y) w T(y)

+ X ¡^(^ Re( ^y - x).

(2.29)

When f = 1, we have $(x, ya) < 0 for all a e r, that is,

¡0{y a) min min Ref - u,y a - x) + h{y a)- h(x) + X ¡¡p(ya)Re(p,ya - x) < 0 (2.30)

f M(y ) u T(y )

and so, by (2.29),

0 > lim sup

> lim sup

+ lim inf

во{уа) min min Re(f-u,ya-x)+h(ya)-h(x) + ^ fiP(ya) Re(p,ya-x

f eM(y„) ueT(y„)

p e E'

Poiy«) ,mAn ч min R^f - U,ya - X + h(ya) - h(x)

feM(ya) u eT(y„)

%Pp(ya) Re(P,ya - x)

= Po(y) limsuH min mn Re(f-u,ya-x) +hy)- h(xH +£ fipiy) Re(p,y-x)

i f e M(y«) u eTы j peE'

> My) min) min) Re(f - ^ у - x> + %) - h(x) + X Re(P'y -

fM(y) ro eT (y) peE'

(2.31)

Hence we have $(x, y) < 0.

(iv) By the hypothesis, there exists a nonempty compact and so a closed (resp., weakly closed and weakly compact) subset K of X and a point x0 e X such that x0 e K n S(y) and

, f inf ^Re(f - w,y - x°) + hy - h(xo) > 0, Vy e X \ K. (2.32)

/ e M(y) weT(y) v '

Thus it follows that

inf w ) Re(f - - + h(y) - h(x0)

ГО eT(y) f e M(y)

> 0, Vy e X \ K,

(2.33)

whenever ¡0(y) > 0 and Re(p,y - x0) > 0 whenever ¡p (y) > 0 for all peE*. Consequently, we have

^x0,y = fi0(y) inf inf Ref - ro,y - x0) + h(y) - h(x0)

f e M(y) roeT(y)

+ Xh(y) Re(py - x^ > 0, Vy e X \ K.

(2.34)

(If T is a strongly h-quasi-pseudo-monotone type II operator, then we equip E with the weak topology.) Thus ф satisfies all the hypotheses of Theorem 1.6 and so, by Theorem 1.6, there exists a point y e K such that ф(x, y) < 0 for all x e X, that is,

в0 (y) inf inf Re(f - uy - x) + h(y) - h(x) + ^ fiP(y) Re(p,y - x) < 0, VxeX.

f e M(y) ueT(y)

(2.35)

Now, the rest of the proof is similar to the proof in Step 1 of Theorem 1 in [24]. Hence we have shown that

xeS(y)

inf inf Re( f - u,y - x) + h(y) - h(x)

f eM(y) ueT(y) V y '

< 0. (2.36)

Then, by applying Theorem 1.9 as we proved in Step 3 of Theorem 1 in [24], we can show that there exist a point f e M(y) and a point w e T(y) such that

Re^f - w,y - x} < h(x) - h(y), VxeS(y). (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 case y / S(y). Thus, if S : X ^ 2X is the constant mapping S(x) = X for all x e X, the E is not required to be locally convex.

Finally, if T = 0, in order to show that, for any x e X, y ^ ty(x,y) is lower semi-continuous (resp., weakly lower semi-continuous), Lemma 2.1 is no longer needed and the weaker continuity assumption on (■, ■) that, for any f e F, the mapping x ^ (f,x) is continuous (resp., weakly continuous) on X is 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 let X be a nonempty paracompact convex and bounded subset of E, and let F be a vector space over Let (■, ■) : F x E ^ ® be a bilinear functional such that (■, ■) separates points in F, (■, ■) is continuous over compact subsets of F x X, and, for any f e F, the mapping x ^ (f,x) is continuous on X. Suppose that F equips with the strong topology S(F,E) and

(a) S : X ^ 2X is a continuous mapping such that each S(x) is compact and convex,

(b) h : X ^ R is convex and h(X) is bounded,

(c) T : X ^ 2f is an h-quasi-pseudo-monotone type II (resp., strongly h-quasi-pseudo-monotone type II) operator and is an upper semi-continuous mapping such that each T(x) is strongly, that is, 6(F,E)-compact and convex (resp., weakly, i.e., a(F,E)-compact and convex),

(d) M : X ^ 2f is an upper semi-continuous mapping such that each M(x) is 6(F,E)-compact convex and, for any ye 2, M is upper semi-continuous at some point x in S(y) with inffeM(y)infueT(y) Re(f - u,y - x) + h(y) - h(x) > 0, where

inf inf Re( f - u,y - x) + h(y) - h(x)

f eM(y) ueT(y) V V '

2 = \ y eX : sup

x S(y)

> 0 \. (2.38)

Suppose further that there exist a nonempty closed and compact (resp., weakly closed and weakly compact) subset K of X and a point x0 eX such that x0 e K n S(y) and

t f inf ^ Re(f - w,y - x0) + hy - h(x0) > 0, VyeX \ K. (2.39)

f eM(y) weT(y) ^ '

Then there exists a point ye X such that

(1) y e S(y),

(2) there exist a point f e M(y) and a point w e T(y) with

Re^ f - w, y - x^ < h(x) - h(y), Vx e S(y). (2.40)

Moreover, if S(x) = X for all x eX, then E is not required to be locally convex.

Proof. The proof is similar to the proof of Theorem 2 in [24] and so the proof is omitted here.

Remark 2.4. (1) Theorems 2.2 and 2.3 of this paper are generalizations of Theorems 3.2 and 3.3 in [3], respectively, on noncompact sets. In Theorems 2.2 and 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 in [3].

(2) The first paper on generalized bi-quasi-variational inequalities was written by Shih and Tan in 1989 in [1] 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 in [1] 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 in [4] and extends the results in [1]. The quasi-pseudo-monotone type I operators are generalizations of pseudo-monotone type I operators introduced first in [17]. In all our results on generalized bi-quasi-variational inequalities, if the operators M = 0 and the operators T are replaced by -T, then we obtain results on generalized quasi-variational inequalities which generalize the corresponding results in the literature (see [18]).

(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 in [2] were obtained for set-valued upper-hemi-continuous operators introduced in [6]. Our results in this paper are also further extensions of the corresponding results in [2, 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 Government (KRF-2008-313-C00050).

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