Contents lists available atScienceDirectNonlinear Analysis journal homepage:www.elsevier.com/locate/na An existence theorem for generalized variational inequalities with B.T.. Leeb,∗ aDe
Trang 1Contents lists available atScienceDirect
Nonlinear Analysis
journal homepage:www.elsevier.com/locate/na
An existence theorem for generalized variational inequalities with
B.T Kiena, G.M Leeb,∗
aDepartment of Information and Technology, Hanoi National University of Civil Engineering, 55 Giai Phong, Hanoi, Viet Nam
bDepartment of Applied Mathematics, Pukyong National University, Busan, 608-737, Republic of Korea
a r t i c l e i n f o
Article history:
Received 10 June 2010
Accepted 11 October 2010
Keywords:
B-pseudomonotone operator
K-pseudomonotone operator
C-pseudomonotone operator
Generalized variational inequality
Solution existence
a b s t r a c t
This paper gives a solution existence theorem for a generalized variational inequality problem with an operator which is defined on an infinite dimensional space, which
is C-pseudomonotone in the sense of Inoan and Kolumbán [D Inoan, J Kolumbán, On pseudomonotone set-valued mappings, Nonlinear Analysis 68 (2008) 47–53], but which may not be upper semicontinuous on finite dimensional subspaces The proof of the theorem provides a new technique which reduces infinite variational inequality problems
to finite ones Two examples are given and analyzed to illustrate the theorem Moreover,
an example is presented to show that the C-pseudomonotonicity of the operator cannot be omitted in the theorem
© 2010 Elsevier Ltd All rights reserved
1 Introduction
The theory of pseudomonotone operators plays an important role in nonlinear analysis, optimization and variational inequalities (see [1–10]) There are some kinds of pseudomonotone operators which were studied One type of pseudomonotone operators was introduced by Karamardian in 1976 [6] which has been frequently used in optimization problems (see [2,4,11]) Another type of pseudo-monotone operators was introduced by Brezis in 1968 [12] which has been used in the study of solution existence of partial differential equations and integral equations (see [13,14])
It is known that the two mentioned classes of pseudo-monotone operators are different However, recently, it has been shown by [3,5] that both notions have a common generalization which is useful for studying generalized variational
inequalities Namely, under certain conditions, both classes satisfy a property of the so-called C -pseudomonotone operators Let us recall some notations and concepts which are related to our problem Throughout the paper we assume that X is
a topological vector space which satisfies the Hausdoff separation axiom and X∗is its dual Suppose K ⊂X is a nonempty
closed convex set andΦ : K → 2X∗ is a multifunction from K to X∗(which is equipped with the weak∗topology) We denote by Gr(Φ)the graph ofΦ, which is defined by
Gr(Φ) = {(x,x∗) ∈K×X∗:x∗∈Φ(x)}.
The generalized variational inequality defined byΦand K , denoted by GVI(Φ,K), is the problem of finding a point x∈K and
x∗∈Φ(x)such that
✩ This work was supported by the Korea Science and Engineering Foundation (KOSEF) NRL Program grant funded by the Korea government (MEST) (No ROA-2008-000-20010-0).
∗Corresponding author.
E-mail addresses:kienbt@nuce.edu.vn (B.T Kien), gmlee@pknu.ac.kr (G.M Lee).
0362-546X/$ – see front matter © 2010 Elsevier Ltd All rights reserved.
Trang 2Recall thatΦis said to be monotone if for all(x,x∗), (y,y∗) ∈Gr(Φ)one has
⟨x∗−y∗,x−y⟩ ≥0.
Φis said to be pseudomonotone in the sense of Karamardian (K -pseudomonotone, for short) if for any(x,x∗), (y,y∗) ∈
Gr(Φ)the following implication holds:
⟨y∗,x−y⟩ ≥0H⇒ ⟨x∗,x−y⟩ ≥0.
It is clear that monotonicity implies K -pseudomonotonicity.
Φ : K → 2X∗ is called pseudomonotone in the sense of Brezis (B-pseudomonotone, for short) if for any u ∈ K and
every net{u i}with u i ⇀ u,u∗i ∈ Φ(u i), and lim sup⟨u∗i,u i−u⟩ ≤ 0 then for eachv ∈ K , there exists u∗v ∈ Φ(u)such that⟨u∗
v,u− v⟩ ≤lim inf⟨u∗
i,u i− v⟩ It is known that ifΦis monotone and hemicontinuous thenΦis B-pseudomonotone
(see [15, Proposition 27.6]) Moreover, when X is a finite dimensional space, any continuous single-valued mapφ :K→X∗
is B-pseudomonotone But it is not necessary to be K -pseudomonotone.
According to [5 Φ:K →2X∗is said to be C -pseudomonotone if for any x,y∈K and net{x i}in K with x i ⇀x,
sup
x∗∈ Φ (xi)
⟨x∗, (1−t)x+ty−x i⟩ ≥0, ∀t ∈ [0,1] , ∀i∈I
implies
sup
x∗∈ Φ (x)
⟨x∗,y−x⟩ ≥0.
The C -pseudomonotone maps appeared in the literature also under the name of 0-segmentary closed maps, see
for instance [16] The implications ‘‘B-pseudomonotone implies C -pseudomonotone’’ and ‘‘K -pseudomonotone and hemicontinuous implies C -pseudomonotone’’ are proved in [5] in the presence of some supplementary conditions Based on
a characteristic property of C -pseudomonotone operators and using a generalization of the Fan intersection lemma, Inoan
and Kolumbán [5] established an important existence theorem for generalized variational inequalities
Observe that, in almost all previous results and the result of [5] on the solution existence of GVIs, the continuity of operatorΦis often required Namely, solution existence of GVIs is guaranteed under assumptions thatΦis monotone or pseudomonotone (in some sense) andΦis upper semicontinuous on finite dimensional subspaces of X (see for instance
[7, Chapter III Theorem 1.4] and [5, Theorem 15])
One may ask whether the solution existence of GVIs for C -pseudomonotone operators is still valid if operators are not contin-uous on finite dimensional subspaces.
The aim of this paper is to answer the above question We will show that ifΦis C -pseudomonotone and satisfies some
additional conditions, the solution existence of GVI(Φ,K)is still guaranteed even thoughΦis discontinuous
Notice that, there are some papers dealing with solution existence of variational inequalities with discontinuous data
in the literature (see for instance [17,18]) But GVIs were considered in the finite dimensional setting, and the obtained results were proved and followed directly from the so-called KKM lemma In this paper the problem is considered in infinite dimensional spaces Besides, it seems that our main result cannot follow directly from the Fan intersection lemma [19] The obtained theorem is proved by a technique which reduce infinite problems to finite problems Although the obtained result
is modest, the contribution here is to give a new method of proof
We are now ready for stating our result
Theorem 1.1 Let X be a topological vector space and K⊂X be convex, closed and nonempty Suppose thatΦ:K→2X∗is an operator which satisfies the following conditions:
(i) Φis C -pseudomonotone,
(ii) there exist weakly compact subsets B0,B of K , where B0⊆B and B0lies in a finite dimensional subspace, such that for every
x∈K\B there exists z∈B0satisfying
sup
f∈ Φ (x)
⟨f,z−x⟩ <0,
(iii) for each z∈K−K , the set{x∈K:supx∗∈Φ (x)⟨x∗,z⟩ ≥0}is closed,
(iv) Φhas convex and weakly∗compact values.
Then GVI(Φ,K)has a solution, that is, there exist x0∈B and x∗
0∈Φ(x0)such that
⟨x∗0,x−x0⟩ ≥0, ∀x∈K.
Trang 32 Proof of Theorem 1.1
Let us denote byLthe family of finite dimensional subspaces L⊂X such that B0⊂L For each L∈L, we set B L=B∩L
and K L=K∩L It is clear that B L and K L are closed Fixing any L∈L, we define a mappingΦL:L∩K→L∗by the formula
whereαL:X∗→L∗is given by
⟨ αL x∗,y⟩ = ⟨x∗,y⟩ ∀y∈L. (3)
To prove the theorem we use the following lemma
Lemma 2.1 ([ 18 , Theorem 1.2]) Suppose K ⊂R n is a nonempty closed convex set and T :K →2R n is a multifunction which satisfies the following two conditions:
(a) for each z∈K−K , the set{x∈K:supf∈T(x)⟨f,z⟩ ≥0}is closed,
(b) there exists a compact subset B⊂K with the property that for every x∈K\B, there exists z∈B such that
sup
f∈T(x)
⟨f,z−x⟩ <0,
(c) T has convex and compact values.
Then there exist x0∈B and f0∈T(x0)such that
⟨f0,x−x0⟩ ≥0, ∀x∈K.
Consider the problem GVI(ΦL,K L) We shall show that GVI(ΦL,K L)satisfies all conditions ofLemma 2.1
(a) For each z∈K L−K L,z∈L and so by(2)we have
{x∈K L: sup
f∈ ΦL(x)
⟨f,z⟩ ≥0} = {x∈K L: sup
x∗∈ Φ (x)
⟨ αL x∗,z⟩ ≥0}
= {x∈K L: sup
x∗∈ Φ (x)
⟨x∗,z⟩ ≥0} = {x∈K: sup
x∗∈ Φ (x)
⟨x∗,z⟩ ≥0} ∩L,
which is a closed set by condition (iii) ofTheorem 1.1
(b) Since K L\B L⊂K\B, it follows from condition (ii) ofTheorem 1.1that for each x∈K L\B L there exists z∈B0⊂B Lsuch that
sup
f∈ ΦL(x)
⟨f,z−x⟩ = sup
x∗∈ Φ (x)
⟨x∗,z−x⟩ <0.
(c) SinceΦhas convex and weakly∗compact values,ΦL has convex and compact values on K L
Thus all conditions ofLemma 2.1are fulfilled According toLemma 2.1, for each L ∈L, there exists x L ∈B Lwhich is a solution of GVI(ΦL,K L), that is,
sup
f∈ ΦL(xL)
⟨f,y−x L⟩ ≥0 ∀y∈K L.
This is equivalent to
sup
x∗∈ Φ (xL)
For each Y ∈Lwe denote by S Ythe set of allˆx∈B such that there exists a subspace L⊇Y with the property that xˆ ∈B L
and
sup
x∗∈ Φ (ˆx)
⟨x∗,y− ˆx⟩ ≥0 ∀y∈K L.
We claim that the family{S Y}has the finite intersection property, where S Y is the weak closure of S Y In fact, for each Y ∈L,
by putting L = Y , we have from(4)that x Y ∈ S Y Hence S Y is nonempty Take subspaces L1,L2, ,L n ∈ Land put
M=span{L1,L2, ,L n} Then we have M∈Land
S M⊂
n
i=1
S Li.
This implies that
∅ ̸=S M⊆S M⊆
n
i=1
S Li⊆
n
i=1
S Li.
The claim is proved
Since S Y ⊂B and B is weakly compact, the finite intersection property of{S Y}implies
S Y ̸= ∅
Trang 4This means that there exists a point x0∈B such that x0∈S Y for all Y ∈L Fix any y∈K and choose Y ∈Lsuch that Y contains y and x0 Since x0∈S Y , there exists a net x i∈S Y such that x i ⇀x0 By definition of S Ywe have
sup
x∗∈ Φ (xi)
⟨x∗, v −x i⟩ ≥0 ∀ v ∈K Y.
In particular, forv =ty+ (1−t)x0, we get
sup
x∗∈ Φ (xi)
⟨x∗,ty+ (1−t)x0−x i⟩ ≥0 ∀t ∈ [0,1] , ∀i∈I. (5) SinceΦis C -pseudomonotone, from(5)we get
sup
x∗∈ Φ (x0 )
To obtain the conclusion we need the following Sion’s minimax theorem (see also [20, Theorem 1])
Lemma 2.2 ([ 21 , Theorem 3.4]) Let P be a compact convex set in a topological vector space X and Q be a convex subset of a
topological vector space Y Let h be a real-valued function on P×Q such that
(i) h(x, ·)is upper semicontinuous and quasi-concave on Q for each x∈P,
(ii) h(·,y)is lower semicontinuous and quasi-convex on P for each y∈Q Then
min
x∈P sup
y∈Q
h(x,y) =sup
y∈Q
min
x∈P h(x,y).
Since y is arbitrary, from(6)we have
sup
x∗∈ Φ (x0 )
⟨x∗,x−x0⟩ ≥0, ∀x∈K.
This is equivalent to
min
x∗∈ Φ (x0 )⟨x
Hence
sup
x∈K
min
x∗∈ Φ (x0 )⟨x
∗,x0−x⟩ ≤0.
ByLemma 2.2, we get
min
x∗∈ Φ (x0 )supx∈K
⟨x∗,x0−x⟩ =sup
x∈K
min
x∗∈ Φ (x0 )⟨x
Since the functionφ(x∗) := supx∈K⟨x∗,x0−x⟩is lower semicontinuous in the weakly star tolology of X∗, there exist
x∗
0∈Φ(x0)such thatφ(x∗
0) =minx∗∈ Φ (x0 )φ(x∗) Hence from(7)we obtain
⟨x∗0,x0−x⟩ ≤0, ∀x∈K.
The proof of the theorem is complete
3 A special case and examples
Let us give a special case ofTheorem 1.1whenΦis upper semicontinuous
Recall that a multifunction G:K⊂X→2Y , where Y is a topological space, is said to be upper semicontinuous on K if for any closed set V ⊂Y , the set{x:G(x) ∩V̸= ∅}is closed
Corollary 3.1 (Cf [ 5 , Theorem 15]) Let X be a topological vector space and K ⊂ X be convex, closed, nonempty Suppose that
Φ:K→2X∗is an operator which satisfies the following conditions.
(a) Φis C -pseudomonotone,
(b) there exist a weakly compact subset B⊂X and z0∈B such that
sup
f∈ Φ (x)
⟨f,z0−x⟩ <0
for all x∈K\B,
(c) for every finite dimensional subspace Z of X,Φis upper semicontinuous on K∩Z , with the weak∗topology in X∗,
(d)Φ(x)is convex and weakly∗compact for every x∈K
Then GVI(Φ,K)has a solution.
Trang 5Proof For the proof we put B0 = {z0}and show that for each L ∈ L,GVI(ΦL,K L)has a solution x L ∈ B L = B∩L To do
this we verify all conditions ofLemma 2.1 It is clear that conditions (b) and (c) ofLemma 2.1are automatically fulfilled for GVI(ΦL,K L) It remains to check condition (a)
For each z∈K L−K L, we have
{x∈K: sup
x∗∈ Φ (x)
⟨x∗,z⟩ ≥0} ∩L= {x∈K L: sup
f∈ ΦL(x)
⟨f,z⟩ ≥0}
= {x∈K L:ΦL(x) ∩W ̸= ∅} ,
where W = {f ∈L∗ : ⟨f,z⟩ ≥0}, which is a closed set in L∗ By (c),ΦL is upper semicontinuous on K L, it follows that the set{x∈K L :ΦL(x) ∩W ̸= ∅}is closed Hence (a) ofLemma 2.1is valid The conclusion follows from the proof method of
Theorem 1.1
Remark 3.2 The conditions of [5, Theorem 15] are the same asCorollary 3.1except for the following:
(b)′
there exist a weakly compact subset B⊂X and z0∈K such that
sup
f∈ Φ (x)
⟨f,z0−x⟩ <0 ∀x∈K\B;
(d)′ Φ(x)is weakly∗compact for every x∈K
The following are some illustrative examples forTheorem 1.1, where solution existence of GVI(Φ,K)is guaranteed even thoughΦis discontinuous
Example 3.3 Suppose X = R,K = [0,1]andφ : K → R is defined byφ(0) = 0, φ(x) = 1 for all x ̸= 0 Thenφis
C -pseudomonotone and discontinuous at 0 For each z∈K−K we have
{x∈K: ⟨ φ(x),z⟩ ≥0} = {0} if z<0
[0,1] if z≥0.
Hence all conditions ofTheorem 1.1are fulfilled ButCorollary 3.1or Theorem 15 in [5] cannot apply to this example By a simple computation, we see that VI(φ,K)has a solution x0=0
Let us give an example for the case of infinite dimensional spaces
Example 3.4 Let
X=l2=
x= (x1,x2, ,x n, ) :
∞
−
i=1
|x i|2< +∞
and
K= {x= (x1,x2, ,x k, ) : ‖x‖l2 ≤1}
Then K is a convex and weakly compact set in l2 LetΦ:K→l2be an operator which is defined by
Φ(x) = { { θ}y0= (1,0,0, ,0, )} if x if x= ̸= θ θ,
whereθ = (0,0, ,0, ) We would like to check the conditions ofTheorem 1.1 It is obvious that conditions (ii) and (iv) are automatically fulfilled
Taking any z∈K−K , we assume that z= (z1,z2, ,z n, ) Then we have
{x∈K: sup
x∗∈ Φ (x)
⟨x∗,z⟩ ≥0} = { θ} if z1<0
K if z1≥0.
Hence condition (iii) ofTheorem 1.1is valid We now claim thatΦis C -pseudomonotone on K Let x,y∈ K,x j ∈ K and
x j⇀x such that
sup
f∈ Φ (x j)
⟨f,z−x j⟩ ≥0, ∀z∈ [x,y] , ∀j∈N.
Let x= (x1,x2, ,x k, )and x j= (x j1,x j2, ,x j k, ) If x= θthen
sup
Φ (θ)
⟨f,y− θ⟩ ≥0.
Trang 6If x̸= θthen there exists j0such that x j̸= θfor all j≥j0, and so we have
sup
f∈ Φ (xj)
⟨f,z−x j⟩ = ⟨y0,z−x j⟩ ≥0, ∀z∈ [x,y] , ∀j≥j0. (8)
Note that since x j⇀x, we have x j k→x k for all k=1,2, Putting z=y and letting j→ ∞, we get from(8)that sup
f∈ Φ (x)
⟨f,y−x⟩ = ⟨y0,y−x⟩ ≥0.
HenceΦis C -pseudomonotone and so condition (i) ofTheorem 1.1is valid Thus all conditions ofTheorem 1.1are fulfilled
Observe that for any z∈K we have
sup
f∈ Φ (θ)
⟨f,z− θ⟩ =0.
Henceθis a solution of VI(Φ,K) Besides,Φis discontinuous atθ
FromExample 3.4one may ask whether GVI(Φ,K)still has a solution if the assumption on C-pseudomonotonicity ofΦ
is omitted The following example shows that condition (i) ofTheorem 1.1plays an essential role for the solution existence
of GVI(Φ,K)
Example 3.5 Suppose X =l2and K is the unit ball in X LetΦ:K →l2defined byΦ(x) = (1− ‖x‖2,x1,x2, ,x k, )
−x with x= (x1,x2, ,x k, ) It is clear thatΦis continuous Hence condition (iii) is valid Besides, conditions (ii) and (iv) are automatically fulfilled However,Φis not C -pseudomonotone on K In fact, taking x= θ = (0,0, ,0, ),y= (−1,0,0, ,0, )and a sequence x j= (0,0, ,0,1,0,0, ), where 1 is at the j-th position, we see that x jconverges
weakly to x and for all t ∈ [0,1]one has
⟨Φ(x j),ty+ (1−t)x−x j⟩ =
t+1 if j=1,
1 if j>1.
But we have⟨Φ(x),y−x⟩ = −1<0 ThusΦis not C -pseudomonotone on K
We now show that GVI(Φ,K)has no solution Conversely, suppose x is a solution of the problem If‖x‖ < 1 then we getΦ(x) =0 This implies that(1− ‖x‖2,x1,x2, ,x k, ) =x Hence‖x‖ =1, which is absurd If‖x‖ =1 then the condition
⟨Φ(x),y−x⟩ ≥0 ∀y∈K,
yieldsΦ(x) = −λx for someλ ≥0 This implies 1= |1− λ|and soλ =0 orλ =2 Whenλ =0 then we get(1− ‖x‖2,
x1,x2, ,x k, ) = x From this we obtain x = θ Also, whenλ = 2 we get(1− ‖x‖2,x1,x2, ,x k, ) = −x It
follows that x= θ, which contradicts‖x‖ =1 Therefore GVI(Φ,K)has no solution
Acknowledgements
The authors wish to thank the anonymous referees for their suggestions and comments This work was also supported
by the Vietnamese Natural Foundation of Science and Technology Development (NAFOSTED)
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