Applied Mathematics, 2010, 1, 81-86
doi:10.4236/am.2010.11010 Published Online May 2010 (http://www.SciRP.o rg/journal/am)
Nonzero Solutions of Generalized Variational Inequalities*
Jue Li, Yisheng Lai
Department of Information and Computer Science, Zhejiang Gongshang University, Hangzhou, China
Received March 17, 2010; revised April 22, 2010; accepted April 29, 2010
Abstract
The existence of nonzero solutions for a class of generalized variational inequalities is studied by fixed point
index approach for multivalued mappings in finite dimensional spaces and reflexive Banach spaces. Some
new existence theorems of nonzero solutions for this class of generalized variational inequalities are estab-
lished.
Keywords: Variational Inequality, Fixed Point Index of Multivalued Mappings, Nonzero Solution
1. Introduction
Variational inequality theory with applications are an
important part of nonlinear analysis and have been ap-
plied intensively to mechanics, differential equation,
cybernetics, quantitative economics, optimization theory
and nonlinear programming etc. (see [1-4]).
Variational inequalities, generalized variational in-
equalities and generalized quasivariational inequalities
were studied intensively in the last 30 years with topo-
logical method, variational method, semi-ordering me-
thod, fixed point method, minimax theorem of Ky Fan
and KKM technique ([1-4]). In 1998, motivated by the
paper [5], Zhu [6] studied a system of variational inequa-
lities involving the linear operators in reflexive Banach
spaces by using the coincidence degree theory due to
Mawhin [7]. Some existence results of positive solutions
for this system of variational inequalities in reflexive
Banach spaces were proved.
Let
X
be a real Banach space,
*
X
its dual and (·, ·)
the pair between
*
X
and
X
. Suppose that
K
is a
nonempty closed convex subset of
X
.
Find
uK∈
,
0u=
/, and
()
w gu∈ such that
(,)(,),Auvuwvuv K− ≥−∀∈
(1)
where mapping * :AK X→ is nonlinear and
*
:2
X
gK→
is a multi-valued mapping.
The existence of nonzero solutions for variational in-
equalities is an important topic of variational inequality
theory. [8] discussed the variational inequality (1) when
A
is coercive or monotone and g is set-contractive or
upper semi-continuous. [9] considered the variational
inequa l i t y (1) when
A
is single-valued continuous and
g is set-contractive.
On the other hand, recently, under some different con-
ditions, [10,11] obtained some existence theorems of
nonzero solutions for a class of generalized variational
inequalities by fixed point index approach for mul-
ti-valued mappings in reflexive Banach space.
Based on the importance of studying the existence of
nonzero solutions for variational inequalities, and motivated
and inspired by recent research works in this field, in this
paper, we discuss the existence of nonzero solutions for a
class of generalized variational inequalities as follows:
Find
,0u Ku∈≠
such that
( ,)()()
((), )(,),
Au vuj vju
guvufvuv K
−+ −
≥− +−∀∈
(2)
where
*
,:Ag KX→
are two nonlinear mapping and
*
fX∈
.
A mapping
*
:AX X→
is called hemicontinuous at
0
xX
∈ if for each
yX∈
,
00
()
n
Aty Awxx
∗
+
when
0
n
t→+
. A multivalued mapping
:T
*
() 2
X
DT X⊂→
is said to be locally bounded in v if there exists a neigh-
bourhood
V
of
x
for each
xX∈
such that the set
(( ))TV DT∩
is bounded in
*
X
. Suppose that
K
is a
closed convex subset of
X
with
0K∈
. For such
K
,
the recession cone
rcK
of
K
is defined by
rcK =
{:, }
wXv wKvK∈+ ∈∀∈. It is easily seen that the
recession cone is indeed a cone and we have that
rcK = ∅
/. For a proper lower semicontinuous convex
functional
: {}jX R→ ∪∞
with (0) 0j= and
()jK⊂
[0, )R
+
= +∞
, in the virtue of [12], the limit
1
lim( )
tj tw
t
→+∞ =
*
This work was supported by The Zhejiang Provincial Natural
Science Foundation(No.Y7080068) and the Foundation of Depart-
ment of Education of Zhejiang Province(No.20070628)

J. LI ET AL.
82
()jw
∞
exists in
{}R∪∞
for every
wX∈
and
j
∞
is also a lower semicontinuous convex functional with
(0) 0j∞= and with the property that
()ju v+≤
()( ),,juj vuvX
∞
+ ∀∈
.
Suppose that
K
is a closed convex subset of
X
and
U
is an open subset of
X
with
K
U UK=∩=
/
∅
. The closure and boundary of
K
U
relative to
K
are denoted by
K
U
and
()
K
U∂
respectively. Assume
that
:2
K
K
TU →
is an upper semicontinuous mapping
with nonempty compact convex values and
T
is also
condensing, i.e., (()) ()TS S
αα
< where
α
is the
Kuratowski measure of noncompactness on
X
. If
()x Tx∉
for
()
K
xU∈∂
, then the fixed point index,
(, )
K
i TU
, is well defined (see[13]).
Proposition 1[13] Let
K
be a nonempty closed
convex subset of real Banach space
X
and
U
be an
open subset of
X
. Suppose that
:2
K
K
TU →
is an
upper semicontinuous mapping with nonempty compact
convex values and
()
x Tx∉ for
()
K
xU
∈∂ . Then the
index,
(, )
K
i TU
, has the following properties:
1) If
(, )0
K
iTU =
/
, then
T
has a fixed point;
2) For mapping

0
X
with constant value
0
{}x
, if
0K
xU∈
, then

0
( ,)1
K
i XU=
;
3) Let
12
,
UU
be two open subsets of
X
with
12
UU∩=∅
.
If
()x Tx∉
when
12
(())(())
KK
xU U∈∂ ∪∂
, then
1212
(,)(, )(,)
K KK
i TUUi TUi TU∪= +
;
4) Let
: [0,1]2
K
K
HU×→
be an upper semicontinu-
ous mapping with nonempty compact convex values and
(([0,1]))( )H QQ αα ×<
whenever
( )0,K Q QU α = ⊂
/
.
If
(, )xHtx∉
for every
[0,1],( )
K
t xU∈ ∈∂
, then
( )
( 1,,)((0,),)
KK
iHU iHU⋅= ⋅
.
For every
*
qX∈
, let
()
Uq be the set of solutions
in
K
of the following variational inequality
( ,)()()
(,)(,),
Au vuj vju
qvufvuvK
−+ −
≥− +−∀∈
(3)
Define a mapping
*
:2
K
A
KX→
by
*
(): (),.
A
K qUqqX= ∈
Obvio usl y, ()
A
Kq= ∅ if and only if the variational
inequa l i t y (3) has no solution in
K
.
2. Nonzero Solutions in n
R
Lemma 1 Let
X
be a separable reflexive Banach
space. Suppose that
*
:AX X→
is a bounded mono-
tone hemicontinuous mapping (i.e., for every bounded
subset
D
of
X
,
()AD
is bounded) and
:(, ]jK→−∞ +∞
is a proper convex lower semic o n-
tinuous functional. Assume that there exists
0
vK∈
sa-
tisfying
00
,
inf [(,lim )()()]0
uuKAu uvj uj v →+∞ ∈−+ −> ‖‖
(4)
Then for any given
*
fX∈
there exists
uX∈
such
that
( ,)()()
( ,),.
Au vuj vj u
fv uvX
−+ −
≥− ∀∈
(5)
Proof. Without loss of generality, assume that
0f=
,
otherwise, set
()()( ,)jvjvf v= −

. Let
{:
r
K xX= ∈
}xr≤‖
‖. Because
X
is a separable reflexive Banach
space, for given
r
, there exists a closed convex sets
sequences ,1, 2,,
m
Km=… satisfying the following con-
ditions:
)a
1
,1, 2,;
r
mm
K KKm
+
⊂⊂ =…
)b
,
mm
KX⊂
m
X
is
m
-dimensional subspace of
X
;
)c
1m m
K
∞
=

is dense in
r
K
.
First, we shall verify that for each
m
, there exists
mm
uK
∈ such that
(,)()()0,.
mm m m
Auvujvj uvK−+−≥ ∀∈
(6)
Because
m
X
is a finite dimensional subspace (deno-
ted its inner product by [.,.] ), there exists a linear conti-
nuous mapping
*
:
m
XX π →
such that
(, )g ω =
[ ,]g πω
for all m K ω ∈. Thus inequality (6) can be
written
[() ,)]
( )(),.
m
Auuuvu
jvjuv K
π −+− −
≤ −∀∈
(7)
Define a function
( ):(,]
mm
JvX→−∞ +∞
by
( ),
() , \.
m
m
mm
jvv K
Jv vX K
∈ 
=+∞ ∈

Then inequality (7) can be written
[() ,)]
( )(),
mm m
Auuuvu
Jv JuvK
π −+− −
≤ −∀∈
(8)
which is equivalent to the equality
()
m
J
uPAu u π =−+
(9)
by [2,3], where
m
J
P
is an approximate mapping of m J.

J. LI ET AL.
83
Obviously,
( ):
m
J mm
PAI KK π −+ →
is continuous.
According to Brouwer's fixed point theorem (see [2,3]),
there exists
mm
uK∈
satisfying the equality (9), that is,
m
u
is a solution of the variational inequality (6).
Second, we shall verify that for each
r
, there exists
r
r
uK∈
such that
(,)( )()0,.
r
rr r
Auvujvj uvK−+−≥ ∀∈
(10)
In fact,
r
m
KK⊂
and
A
is a bounded mapping,
whi ch implies that there constant
C
such that
m
Au C≤‖‖
for
1, 2,m=…
. Since
X
is a reflexive and
r
K
is weakly closed, there exists a subsequence
{}{}
m
uu
µ
⊂
such that
r
w
uu
µ
 →
and
r
r
uK∈
. Because
1m m
K
∞
=

is dense in
r
K
, for any given
0 ε >
, there
exists 01m
m
uK
∞
=
∈

such that
0
.
r
uu ε −≤‖‖
It then
follows from (6) that
00
(,)()().Au uujuju
µµ µ
−≤ −
(11)
when
µ
is sufficiently large. Thus we have
00
0
sup( ,
sup( ,)s
l
up( ,)
s
im )
lim lim
up( ()())lim .
r
r
Au uu
Au uuAu uu
ju juC
µµ
µ
µµ µ
µµ
µ
µ
ε
−
≤−+ −
≤− +⋅
Since
j
is a lower semicontinuous function and
ε
is an arbitrary positive number, we have
sup(,) 0.lim r Au uu
µµ
µ
−≤
(12)
This together with
A
being a monotone hemiconti-
nuous mapping impl ies that
inf (,)
( ,),.
lim
r
rr
Au uv
Au uvvK
µµ
µ
−
≥− ∀∈ (13)
If
1m m
vK
∞
=
∈
, it then follows from (6) that
(,)()()Auuvj vj u
µµ µ
−≤ −
(14)
when
µ
is sufficiently large. It thus follows from
(13) that
1
lim
lim
(,)inf (,)
inf (()())
( )(),.
rr
rm
m
Au uvAuuv
jv ju
jv juvK
µµ
µ
µ
µ
∞
=
−≤ −
≤−
≤ −∀∈

(15)
Because
1m m
K
∞
=

is dense in
r
K
, the above in-
equality holds for all
r
vK∈
. therefore r
u
is a solution
of the variational inequality (10).
New we shall verify that the variational inequality (5)
has a solution. Taking
0
vv=
in (10), we have
00
(,)()() 0
rr r
Auuvj uj v−+−≤
(16)
and so it then follows from condition (4) that there ex-
ists constant C > 0 such that
r
uC≤‖‖
. Taking r > C
then r
ur
<
‖
‖ and so
r
u
is an inner point of
r
B
.
Thus for any given
X ω ∈
, we have
(1 )
r
tu−+
r
tB
ω ∈by taking
(0,1)t∈
small enough. Let
(1) r vtu t
ω =−+
in (10), then we obtain
(,)(()())0
rr r
tAuutjj u ωω −+− ≥
by
j
being a convex lower semicontinuous function.
Thus
(,)( )()0,.
rr r
AuujjuX ωω ω −+−≥∀∈
Therefore r u is a solution of the variational inequalit y
(5).
Theorem 1 Let
K
be a nonempty unbounded closed
convex set in
n
XR=
with
0K∈
. Suppose that
*
XX
→ is a bounded monotone hemicontinuous map-
pingwith (, )0()Au uuK≥ ∀∈and
:jK→
(,]
−∞ +∞ is a
bounded proper convex lowersemicontinuous functional
with
(0) 0j=
(i.e., for every bounded subset
D
of
K
,
()jD
is bounded). Give a continuous mapping
*
:gK X→
and
*
fX∈
. Assume
a)
0
( ,)()
lim
u
Au uj u
u
→
+= +∞
‖‖
‖‖
;
b) there exists constant
0 α ≥
such that
1
( ,)()()
inf suplimli ()m
uu
Au uj ug uuK
uu
αα +
→+∞ →+∞
+>∈
‖‖ ‖‖
‖‖
‖‖‖‖
;
c) there exists a point 0\ {0}u rcK∈ such that
0
(,) 0fu =
/
Then (2) has a nonzero solution.
Proof. It is easy to see from condition (b) and Lemma
1 that the variational inequality (3) has a solution in
K
for every
*
qX∈
. Define a mapping
:2
K
A
KgK →
by
()( ):(( )),
AA
K guKguuK=∈
Then A
Kg
is an upper semi-continuous mapping
with nonempty compact convex values by [10, Lemma
1]. Let
{: }
R
KxK x R=∈≤‖‖
. We shall verify that
(, )1
R
KA
iK gK=
for large enough
R
and
( ,)
r
KA
iK gK
0=
for small enough
r
.

J. LI ET AL.
84
Firstly, define a mapping by
: [0,1]2,
RK
HK×→
( ,)(())
A
Ht utKg u=
. It is easily seen that
(, )Htu
is an
upper semicontinuous mapping with nonempty compact
convex values. We claim that there exists large enough
R
such that
(, )u Htu∉
for all
(0,1),t∈
()
R
uK∈∂
.
Otherwise, there exist two sequences
{ },{},[0,1],
n nn
t ut∈
0,
nn
tu=→ +∞
/‖‖
such that
n
u∈
(,)( ())
n nnAn
Ht utKgu=
or ( ())
n
An
n
uK gu
t∈.
Thus
((),) () ()
((), )(, ),
nn n
nn n
nn
n
nn
uu u
Avjvj
tt t
uu
guvf vuK
tt
−+ +
≥− +−∀∈
(17)
Letting
0v=
and denoting n
n
n
u
zu =‖‖ in (17), we
obtain from (17) that
11
( )((),)( )()
()
(,)( )(,)
nnnnn
nnn nn
nn
nn n
n n
tuutu
Aj
utt ut
gu t
t zfz
u u
αα
αα
α
++
+
≤+
‖‖‖‖
‖‖ ‖‖
(18)
Denote n
n
n
u
yK
t
=∈ . Then
n
y→ +∞‖‖
.We can ob-
tain from (18) that
1
(, )()()
() .
nn nn
n
n nn
n
nn
Ayyjyguf
t
y uy
gu f
uy
α
α αα
αα
+
+≤+
≤+
‖‖
‖‖‖‖‖‖
‖‖
‖‖‖‖
‖‖
‖‖
(19)
Hence we have
1
( ,)()()
inf suliml pim
uu
Au uj ugu
uu
αα +
→+∞ →+∞
+≤
‖‖ ‖‖
‖‖
‖‖‖‖
which contradicts to condition (b). Therefore
( ,) ((1,),)
((0, ),)
ˆ
(0,) 1
RR
KAK
R
K
R
K
iKgKiHK
iH K
iK
= ⋅
= ⋅
= =
(20)
by Proposition 1(4) and (2).
Secondly, we shall verify that
(, )0
r
KA
iK gK=
for
small enough
r
(
1r<
). In fact, there exist consta nt s
12
,, 0
CC M> fr om the boundedness of
j
, locally
boundedness of
A
and condition (b) such that for all
1
uK
∈, we have
012
| ()()|(),ju ujuCguC+− ≤≤‖‖,
0 20
00
((),) |,
|(, )|
g uuCuAuM
Au uMu
≤≤
≤
‖‖‖,‖
‖‖ (21)
Since
0
(,) 0fu =
/
, let
0
(,)0fu <
. Take
N
large
enough such that
0 120
(1)( ,)()NfuCCM u−>+ +
(22)
Define a mapping by
[0,1]2(,, )
rK
H KHtu×→ =
(() )
A
Kg utNf−. Then
H
is an upper semi- continuous
mapping with nonempty compact convex values. We
claim that there exists a small enough
r
such that
(, )u Htu∉ for all
( ),[0,1]
r
u Kt∈∂ ∈
. Otherwise, there
exist sequences
{ },{},[0,1],
n nn
t ut∈
( ),0
r
nn
u Ku∈∂→‖‖
such that
(,)
n nn
uHt u∈=
(( ))
A nn
Kg utNf−. Thus
(,)()()
(() ,)(,),
nn n
nn nn
Auvuj vj u
guNtfvufv uvK
−+ −
≥−− +−∀∈
Taking
0, n
n
n
u
vz u = =‖‖
, we have
()
1(,)
((),)(1)(, )
n
nn
nn
nnn n
ju
Au u
uu
guztNf z
+
≤ +−
‖‖‖‖
Since (,) ()
nn n
n
Auuj u
u
+→ +∞
‖‖ and
2
((), )(1)(, )
( )(1)
(1 ),
nnn n
n
guztNf z
guN f
C Nf
+−
≤ ++
≤ ++
‖‖‖‖
‖‖
we obtain a contradiction. Therefore
( ,)
r
KA
iK gK=
((0,),)((1,),)
rr
KK
iHK iHK⋅= ⋅
by Proposition 1 (4). If
((1, ),)0
r
K
iH K⋅=
/
, then the mapping
(1,):2
K
HK⋅→
has a fixed point
u
in
r
K
by Proposition 1(1), i.e.,
(1,)(( ))
A
uHuKg uNf∈= −
.
Thus
( ,)()()
(() ,)(,),
Au vuj vj u
guNfv ufv uvK
−+ −
≥−− +−∀∈
Taking
0
vuu= +
, we have
00
00
(, )()()
((), )(1)(, )
Auujuuju
guuNf u
+ +−
≥+− (23)
Hence
0
00 0
0120 201
(1)( ,)
(, )()()((), )
()
N fu
Auujuujugu u
MuCCu CMuC
−
≤+ +−−
≤++=++‖‖‖‖‖‖

J. LI ET AL.
85
by (21) and (23). That contradicts to (22). Therefore,
((1, ),)0
r
K
iH K⋅=
and then
(, )0
r
KA
K gKi=
.
It follows from Proposition 1(3) that
(,
KA
i Kg
\ )1
Rr
KK=
. Therefore there exists a fixed point
\
Rr
uK K∈
which is a nonzero solution of (2).
3. Nonzero Solutions in Reflexive Banach
Spaces
Theorem 2 Let
X
be a reflexive Banach space and
KX⊂
a nonempty unbounded closed convex set with
0K∈
. Suppose that * :AX X→ is a bounded mono-
tone hemicontinuous mapping with
(,) 0Au u≥
for
uK∈
and
:(, ]jK→−∞ +∞
is a bounded convex
lower semicontinuous functional with (0) 0j=. Assume
that
*
:gK X→
is continuous from the weak topology
on
X
to the strong topology on
*
X
. Give
*
fX∈
.
The following conditions are assumed to
be satisfied
(a)
0
(,) 0fu =
/
for some
0
\ {0}u rcK∈
;
(b) there constant
0 α ≥
such that
1
( ,)()()
inf suplimli ()m
uu
Au uj ug uuK
uu
αα +
→+∞ → +∞
+>∈
‖‖ ‖‖
‖‖
‖‖‖‖
;
(c)
0
infim ()0lw
s
s
u
ju
→
>
.
Then (2) has a nonzero solution.
Proof. It is easily seen that
0
(, )()
lim
u
Auu ju
u
→
+=
‖‖
‖‖
+∞
by the condition (c). Let
FX⊂
be a finite di-
mensional subspace containing 0 u. We shall show that
all conditions in Theorem 1 are satisfied on space
F
.
Denote
F
K KF= ∩
which is a nonempty un-
bounded closed convex set. Let
:
F
jF X→
be an in-
jective mapping and
** *
:
F
jX F→
its dual mapping.
Denote
* **
(|):,( |):
FFFFF F
AjAFFF gjgKK= →=
*
F→. We know that
*
,
F FF
Aj Aj=
*
F FF
gj gj=. Then,
,
FF
Ag
are hemicontinuous and continuous respective-
l y.
For 12 ,F xx K∈, we have
1212
**
1212
1 212
121 2
(( )(),)
(( )(),)
(,())
(, )0
FF
FF
F
AxAx xx
j Axj Axxx
AxAx j xx
AxAx xx
−−
=−−
=−−
= −−≥
by the monotony of
A
. This means that F
A
is mono-
tone. On the other hand,
**
F
jf F∈
and
*
0
( ,)
F
j fu=
00
(,)(,) 0
F
f jufu= =
/
. Similarly, we have
1
(,) ()()
inf sup
(
lim lim
).
FF
uu
F
Auujugu
uu
uK
αα +
→+∞ →+∞
+>
∈
‖‖ ‖‖
‖‖
‖‖‖‖
Therefore all conditions in Theorem 1 are satisfied on
space
F
and so there exists , 0
F FF
u Ku
∈=
/ such
that
*
( (),)()()
((), )(, ),
FF FF
FFF FFF
A uvujvju
guvujfvuvK
−+ −
≥−+−∀∈
It yields that
((),) () ()
((), )(, ),
FF F
FFF F
Auv ujvju
guvufvuvK
−+ −
≥− +−∀∈
Taking
0v=
, we get
(,) ()((),)(,)
FFFF FF
Auuj uguufu
+≤+. Hence
1
(,) ()().
FF FF
F FF
Auuj uguf
u uu
α αα +
+≤+ ‖‖ ‖‖
‖‖‖‖‖‖
This together with condition (b) implies that there ex-
ists a constant M > 0 such that F
uM
≤
‖
‖ for all finite
dimensional subspace
F
containing 0 u. Since
X
is
reflexive and
K
is weakly closed, with a similar argu-
ment to that in the proof of Theorem 2 in [10] (also see
[8]), we shall show that there exists
uK
′∈
such that
for every finite dimensional subspace
F
containing
0
u,
u′
is in the weak closure of the set
1
1
{}
FF
FF
Vu
⊂
=
where 1
F
is a finite dimensional subspace in
X
.
In fact, since
F
V
is bounded, we know that
()
w
F
V
(the weak closure of the set
F
V
) is weakly compact.
On the other hand, let
12
, ,,
m
FF F
be finite dimen-
siona l subspaces containing 0 u. Define ()
:
m
F
=
12
{, ,,}
m
span FFF
. Then
()m
F
containing 0
u
is a
finite dimensional subspace. Hence,
1
i
m
F
i
V
=
=

11
()
11
1
( {}){}
im
m
FF
iFFFF
uu
=⊂⊂
== ∅
/
 
, then
()
w
F
F
V=
/

∅
. T hat is to say, there exists
uK
′∈
such that for
every finite dimensional subspace
F
containing 0 u,
u′
is in the weak closure of the set
11
{}
FFF F
Vu
⊂
= ∪
.
Now let
vK∈
and
F′
a finite dimensional subspace
of
X
which contains 0 u and
v
. Since
u′
belongs to

J. LI ET AL.
86
the weak closure of the set
1
1
{}
FF
FF
Vu
′
′⊂
=
. We may
find a sequence
{}
F
u
α
in
F
V
′
such that
w
F
uu
α
→′
.
However,
F
u
α
satisfies the following inequality
(,)()( )
((),)(,)
FF F
FF F
Auvuj vj u
gu vufvu
αα α
ααα
−+−
≥−+−
(24)
The monotony of A implies that
( ,)()()
((),)(,)
FF
FF F
Av vujvj u
gu vufvu
αα
ααα
−+−
≥−+−
Letting
w
F
uu
α
′
→
yields tha t
(,)()()
((), )(, ),
Av vujvj u
guvufvuv K
′′
−+ −
′′′
≥−+−∀∈
Thus
(,)()()
((), )(, ),
Auvuj vju
guvufvuv K
′′ ′
−+ −
′′ ′
≥− +−∀∈
by Minty’s Theorem [2,3]. We claim that
0u′=
/
.
Otherwise,
0
w
F
u
α
→
. Taking
0v=
in (24) yields that
() (,)((),)(,)
((),)(,)
FFFFFF
FFF
j uAuug uufu
guuf u
αααα αα
αα α
≤− ++
≤+
The right side of the above inequality tends to 0,
which contradicts to the condition (c). Therefore
u′
is
a nonzero solution of (2).
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