Advances in Pure Mathematics, 2011, 1, 163-169
doi:10.4236/apm.2011.14030 Published Online July 2011 (http://www.SciRP.org/journal/apm)
On Some Fixed Point Theorems for 1-Set Weakly
Contractive Multi-Valued Mappings with Weakly
Sequentially Closed Graph
Afif Ben Amar1, Aneta Sikorska-Nowak2
1Département de Mathéma tiques Université de Gafsa Faculté des sciences d e Gafsa Cité
Universitaire Zarrou k, Gafsa Tunisie
2Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznan, Poland
Received February 20, 2011; revised March 11, 20 1 1; acce pted March 21, 2011
Abstract
In this paper we prove Leray-Schauder and Furi-Pera types fixed point theorems for a class of multi-valued
mappings with weakly sequentially closed graph. Our results improve and extend previous results for weakly
sequentially closed maps and are very important in applications, mainly for the investigating of boundary
value problems on noncompact intervals.
Keywords: Measures of Weak Noncompactness, Weakly Condensing, Weakly Nonexpansive
1. Introduction




 


bd
cv
cl,cv
:,
=: ,
=: ,,
PXAXA is nonemptyandbounded
PXAXA isnonemptyandconvex
PX
XA isnonemptyclosedandconvex
 


Fixed point theory for weakly completely continuous
multi-valued mappings takes an important role for the
existence of solutions for operator inclusions, positive
solutions of elliptic equation with discontinuous non-
linearities and periodic and boundary value problems for
second order differential inclusions (see [1-3]) and others.
In [4] O’Regan has proved a number of fixed point
theorems for multi-valued maps defined on bounded
domains with weakly compact convex values and which
are weakly contractive and have weakly sequentially
closed graph. The aim of the present paper is to extend
and improve these theorems to the case of weakly
condensing and 1-set weakly contractive multi-valued
maps with weakly sequentially closed graph. Further-
more, we do not assume that they are from a point into
weakly compact convex set. We note that the domains of
all of the multi-valued maps discussed here are not
assumed to be bounded. Our results generalize and
extend relevant and recent ones (see [4-10]). The main
condition in our results is formulated in terms of
axioma t ic measures of w eak noncom- pactness.
A
Now we shall introduce the notation and give preli-
minary results which will be need ed in the paper.
Let
X
be a Hausdorff linear topological space, then
we define
Let
Z
a non-empty subset of a Banach space Y
and be a multi-valued mapping. We denote
:2
X
FZ





 
==,:
yZ
RFFyand GrFzxZXxFz

 

the range and the graph of
F
respectively. Moreover,
for every subset
A
of
X
, we put




 
1=:FAzZFzA 

 and




 
=:AzZFzA
.
F

F
is called upper semicontinuous on Z if


1
F
A
is closed, for every closed subset

A
of
X
(or,
equivalently, if


F
A
 is open, for every open
subset
A
of
X
).

F
is called weakly upper semicontinuous if
F
is
upper semicontinuous with respect to the weak
topologies of
Z
and
X
.
Now we suppose that
X
is a Banach space and
Z
is weakly closed in
Y
.

F
is said to be weakly compact if the set

RF is
relatively weakly compact in
X
. Moreover,
F
is said

164 A. B. AMAR ET AL.
to have weakly sequentially closed graph if for every
sequence nn

x
Z with w
n
x
x

 in
Z
and for
every sequence with

nn
y
,,
n
n y

nn
x w yF y in X implies


yFx,
where w

 denotes weak convergence.
F
is called
weakly completely continuous if
F
has a weakly
sequentially closed graph and, if
A
is bounded subset
of
Z
, then

F
A is a weakly relative compact subset
of
X
.
Definition 1.1 Let be a Banach space and a
lattice with a least elemen t, which is denoted by . By a
measure of weak non-compactness ()
E C
0
M
NWC E on ,
we mean a function defined on the set of all
bounded subsets of with values in , such that for
any :

E

bd
PE C
12
,
(1)


1
=



1
conv , where conv denotes the
closed convex hull of .


(2)


1 2

aE

 1 
12

 
=a 
=0 E
C
 

,
(3) for all .

11

(4) 1 if and only if is weakly rela-
tively compact in .
If the lattice is a cone of vector space, then the
M
NWC


is said positive homogenous provided
for all

= 

 >0

and


PE
bd
The notion above is a generalization of the important
well known DeBlasi measure of weak non-compactness
 .

(see [11]) defined on each bounded set

of
by E

weakly

= iheret sa
s hat

nf>0:t
compact set
exis
uch t
E
DDB   
 
where
E
B is the closed unit ball of . E
It is well known that

enjoys these properties: for
any ,

bd
PE

12
,
(5).

 
1 2
,  
1>0
12
=max
 
1
=  
  .
(6) for all

 

.
(7) .

12
  



1 2
  
:2


Now let us introduce the following definitions.
Definition 1.2 Let be a nonempty subset of
Banach space E and a MNWC on E. If
E
F ,
we say that
(a)
F
is -condensing if 
 


<
F
DD

 for
all bounded sets with .
D

D0
(b) F is -nonexpansive map if






F
DD
E

for all bounded sets . D
In the sequel, we shall need also the following multi-
valued fixed point theorem.
Theorem 1.1 (see [12], p. 206). Let be a nonvoid,
convex and closed subset of a locally convex space .
Let



:FP
cl,cv



F
be an upper semicontinuous
multi-valued mapping such that is relatively
compact. Then
F
has a fixed point.
2. Sadovskii Type Fixed Point Theorems
We begin with the following interesting property of
multi-valued maps with weakly sequentially closed
graph which is a basic tool for achieving our aim.
Theorem 2.1 Let

be a non-empty, weakly compact
subset of a Banach space E. Suppose :2
E
F has
weakly sequentially closed graph and is rela-
tively weakly compact. Then


F
F
has a weakly closed
graph.


=EEEE
ww
w
Proof. Since

E

(w E the space
endowed its weak topology), it follows that w
F 
EE

is a weakly compact subset of . Also,
w
GrF F GrF
 
. So, is relatively weakly com-
pact. Let ,w xy F

 
GrF
 
,
nn
n
be weakly adherent to
, then by the Eberlein-Šmulian theorem [13,
theorem
x
8.12.4, p. 549] we can find yGrF such
that


,w
nnn
yFxx xw
n
and
y
y


 E in .
Because F has weakly sequentially closed graph,




yFx and so
x
,yGrFF
GrF
. Therefore, Gr is
weakly closed.
Q.E.D.
Remark 2.1 With the conditions of Theorem 2.1, we
prove that is weakly compact.
Theorem 2.2 Let

be a non-empty, closed, convex
subset of a Banach space . Suppose


Ecv :FP
has weakly sequentially closed graph and


F

is
weakly relatively compact. Then
F
ha s a fixed poi nt.

 
=KconvF
K
Proof. Set . It follows, using the
Krein-Šmulian theorem (see [14, p. 434]) that is a
weakly compact convex set. We have


FK .
Notice also that


cv
KPK. By Theorem 2.1
F
:
F
has weakly closed graph, and so

F
x is weakly closed
for every
x
K

. Thus, by [15, Proposition 14.5, p. 69]
F
is weakly upper semicontinuous. Because
endowed with its weak topology is a Hausdorff locally
convex space, we apply Theorem 1.1 to get that
E
F
has
a fixed point xK


()a
()b
.
Q.E.D.
Remark 2.2 Theorem 2.2 improves Theorem 2.2
in [4] and Theorem 2.1 in [1] for the case of Banach
spaces.
Because every single-valued and weakly sequen-
tially continuous map has weakly sequentially closed

A. B. AMAR ET AL.
165


:


F graph, then Theorem 2.2 extends and improves a fixed
point theorem of Arino, Gautier and Penot for the case of
Banach spaces [5], Theorem 2.5 in [9] and Corollary 2.3
in [10].
F
is

-nonexpansive and
Now, we are ready to prove some fixed point theorems
for a broader class of multi-valued mappings with
weakly sequentially closed graph, in which the operators
have the property that the image of any set is in a certain
sense more weakly compact than the original set itself.
Theorem 2.3 Let be a non-empty, closed, convex
subset of a Banach space . Assume a MNWC on
and cv has weakly sequentially closed
graph. In addition, suppose that
E

E

PF
F
is -condensing
and is bounded. Then



F
F
has a fixed point.
Proof. Let 0. We consider the family x
F
of all
closed bounded convex subsets of such that D
0
x
D and

F
xD for all
x
D. Obviously
F
is
non-empty, since
 


0
con . We denote v FxF 
=DF
K
D

0 . We have that K is closed convex and
x
K

.
If
x
K, then


F
xD for all and hence DF


F
xK. Consequently,
K
F

. We will prove that
K
is weakly compact. Denoting by





*0
=
K
convFKx* , we have
K
K, which
implies that
 
*
F
xFKK for all *
x
K. There-
fore *
KF
 and *
K
K=
. Consequently, *
K
K. Since
K
is weakly closed, it suffices to show that
K
is
relatively weakly compact. If , we obtain

K>0
 




0
=
  
<
K
conv FKxF K K
K which is a contradiction. Hence, and so
is relatively weakly compact. Now,

=0K


cv
:
F
KPK has
weakly sequentially closed graph. Fr om Theorem 2. 2 ,
F
has a fixed point in . K
Q.E.D.
Remark 2.3 Theorem 2.3 improves and extends
Theorem 2.3 in [4].
Using Theorem 2.3, we obtain a sharpening of
Theorem 2.2.
Corolla ry 2 .1 Let be a non-empty, closed, convex
subset of a Banach space . Assume
E


cv
FP :
a weakly completely continuous map with


F

is
bounded. Then
F
has a fixed point.
Proof. This is an immediate consequence of Theorem
2.3, since F is clearly -condensing where 

is any
MNWC on . E
Q.E.D.
Theorem 2.4 Let be a Banach space,
E

be a
non-empty, closed, convex subset of and  a posi-
tive homogenous MNWC on . Assume
has weakly sequentially closed graph,
also suppose

is
bounded. In addition, suppose that the implication


,
nnn
nn
f
xwithyFxforalln and
xy asn 
 
 

,then thereexistsxwithxFx
(1)
holds. Then
F
has a fixed point.
Remark 2.4

means the zero of the space . E
=for=1,2,
nn
FtF n

Proof. Let , where

nn
t is a
sequence of


0,1 1
n such that t. Since  and




is convex, it follows that

:FP   
n
cvn. Also
F
has a weakly sequentially closed graph. Let


bd
DP. Then, we have









=
nnn
F
DtFDtD


0D we have So, if



  
<
n
F
DD
n
F
is Therefore,

-condensing on

. From
Theorem 2.3, n
F
has a fixed point, in . For all ,
let n


nn
y
Fx=nn with n
x
ty. Clearly the sequence


nn
y

=1
nn
xy ty is bounded and n n


n 1
n
tn
x
as
, since as . Thus (1) implies that
there exists 

with
x
Fx.

Q.E.D.
As a consequence we obtain
Corolla ry 2 .2 Let

be a non-empty, closed, convex
subset of a Banach space and
E

. In addition
assume


:FP
cv

 has weakly sequentially closed
graph,

-nonexpansive and is bounded. Also,
suppose (1) holds. Then

F
F
has a fixed point.
Proof. This is an immediate consequence of Theorem
2.4, since the DeBlasi measure

is positive homo-
genous.
Q.E.D.
3. Leray-Schauder and Furi-Pera Types
Fixed Point Theorems
In applications, the construction of a set such that 


F

 :2 is very difficult and sometimes impossible.
In that line, we investigate maps
E
F with
weakly sequentially closed graph.
Lemma 3.1 Let

be a weakly closed subset of a
Banach space with
E
E

FP


cv
:
E

 :2. Assume
E
F
has weakly sequentially closed graph with


F

is
bounded. Let


x

nn and n

be a real sequence.
If w
n
 , then the condition
x
x

 n and



nn n
Fx  

for all n implies
x
x
Fx  
n

n
.
Proof. For all , there exists n
y
Fx
= such that
nnn


w
n
x
x
yF

. If =0

, then


 (

is
bounded) and


x   0. If

, then without loss


166
0
A. B. AMAR ET AL.
of generality, we can suppose that n

 for all . So,
nnn
n
1=
x
y 
1
for all implies n nw
y
x


. Since
F
has weakly sequentially closed graph, we have


y
Fx,
which means that


x
Fx  
E
.
Q.E.D.
Our first result is of Leray-Schauder type.
Theorem 3.1 Let be a Banach space,

be a
non-empty, closed, convex subset of . and be a
weakly open subset of with E U U

. Assume

a
MNWC on and E


cv has a weakly
sequentially closed graph. In addition, suppose
:w FU P
F
is
-condensing and

w
F
U is bounded. Then, either
(A1)
F
has a fixed point, or
(A2) there is a point
x
U


(0,1) (the weak boundary of in ) and U

 with

x
Fx  .
Remark 3.1 In view of the convexity, the set is
weakly closed and therefore the weak closure of U and
the weak closure of U in coincide for any U.
()a

(b) For , we have U =\
ww
U
2
()
UU
.
Proof. Suppose
A
does not hold and
F
does not
have a fixed point in (otherwise, we are finished,
i.e. U


1
A
occurs). Let t he set defi ne d b y D


: ,[0,1]
w
UxFxfome   
D
=Dx
D
rso
is non-empty and bounded, because


and

w
F
U

conv  
is bo u nded. We have
  
DFD0D. So,


 implies
  


nv
= : ,[0,1]
w
DF
xFxfor me



 
=0D
D
co
xU
D
so
D
which is a contradiction. Hence, and is
weakly relatively compact. Now, we prove that is

weakly sequentially closed. For that, let


nn
x
a
sequence of such that Dw
n
x
x

 , w
x
U. For all
, there exists a n[0,1]
n

 such that


=
nnn
x
Fx  .
[0,1]
n

, we can extract a subsequence
 nj
j

[0,
such
that 1]
nj

  . We put =
nnn
j
jj
x
y

, where
 nn
j
j
yFx. Applying Lemma 3.1, we deduce that
x
D. Let w
x
UD, be weakly adherent to . Since
w
D is weakly compact, by the Eberlein-Šmulian
theorem [13, Theorem 8.12.4, p. 549], there exists a
sequence

nn
x
D such that w
n
x
x

 , so
x
D

.
Hence =
w
DD D and is a weakly closed subset of
the weakly compact set . Therefore is weakly
compact. Because
Dw
U
EE

\=DU 
: [0,1]
endowed with its weak topology
is a Hausdorff locally convex space, we have that is
completely regular [17, p. 16]. Since ,
then by [16, p. 146], there is a weakly continuous
function 

=1x  , such that for
x
D



and


=0x  \ for
x
U. Since  is convex,

, and


F
with nonempty convex values, we can
define the multi-valued map by:

*cv
: FP 
 

*,i,
=,i \
w
w
xFxfx U
Fx fx U









*
F

is bounded. Because Clearly, =w UU


[0,1] 
,
is compact,

is weakly continuous and
F
has
a weakly sequentially closed graph, by Lemma 3.1, we
have *
F
has weakly sequentially closed graph. Let
X

 
, bounded. Then, since

 
*conv w
FX FXU  


we have






*w
F
XXUFX






*<


and
F
XX if 0X * . So,
F
is

-condensing. Therefore, all of the assumptions of
Theorem 2.3 are satisfied for *
F
. Consequently there
exists 

*
00
with


x
Fx0 . If
0
x
x
U
,


0=0x
0=x
and U

, which contradicts the hypothesis


.
Then 0
x
U

and 000

x
xFx , which implies
that 0
x
D

, and so and the proof is
complete.

0=1x

Q.E.D.
Remark 3.2 (a) Theorem 3.1 extends Theorem 2.5 in
[4] and shows that the condition
F
has weakly closed
graph can be replaced by
F
has weakly sequentially
closed graph.
(b) Theorem 3.1 extends and improves Theorem 2.4 in
[4] and shows that the condition w U
E
is weakly compact
in the statement of this theorem is redundant.
(c) Theorem 3.1 extends Theorem 3.3 in [8] in the
context of single-valued and weakly sequentially
continuous maps to the case of multi-valued maps with
weakly sequentially closed graph.
Corollary 3.1 Let be a Banach space,

be a
non-empty, closed, convex subset of . and U be a
weakly open subset of E

with U

. Assume

a
MNWC on and

cv
:w FU P has a weakly E
sequentially closed graph,

-condensing with

w

F
U
is bounded. In addition, suppose F satisfying the Leray-
Schauder boundary condition


(0,1)xFxfor everyxUand 



A. B. AMAR ET AL.
167
F
has a fixed point in w
U
E
then .
Corollary 3.2 Let be a Banach space,

be a
non-empty, closed, convex subset of . and be a
weakly open subset of with E U U 

. Assume that

cv
wP:FU a weakly completely continuous map
with

w
F
U
(0,1) and  
is bounded. In addition, suppose F
satisfying the Lera y-Schauder boundary condition

xFxfor everyxU


then
F
has a fixed point in w U
 
U
.
Proof. Since F is weakly completely continuous, it
follows that F is -condensing on for any MNWC
on . Now, it suffices to apply the Theorem 3.1. Q.E.D.
We now use Theorem 2.4 to obtain a nonlinear
alternative of Leray-Schauder type for multi-valued 1-set
weakly contractive maps.
E
Theorem 3.2 Let be a non-empty, closed, convex
subset of a Banach space , let be a weakly open
subset of with
E
U

. Assume a positive
homogenous MNWC on ,

E

cv
wP

:FU has a
weakly sequentially closed graph, -nonexpansive,

w
F
U is bounded and (1) holds on w U. In addition,
suppose
F
satisfies the following Leray-Schauder con-
dition

xFxfor everyxU 

(0,1)and   (2)
then
F
has a fixed point in w U.
Proof. Let =
nn
F
tF, for , wh ere =1n,2,


nn
t is
a sequence of such that . Since (0,1) 1
n
t



and is convex, it follows that 

wP
cv .
Also n
:FU
n
F
is -condensing and has a weakly sequen-
tially closed graph. Suppose that


nnnn
y
Fy 
y

 1)

(0, for
some and for some
nUn

. Then we
have nn

nnn
y
tF y
(0,1)

t
 which contradicts the hypothesis
(2) since nn

. Now, applying Corollary 3.1, the
remainder of the proof is similar to that of Theorem 2.4.
Q.E.D.
Remark 3.3 Theorem 3.2 extends and improves
Theorem 3.4 in [8].
In applications, it is extremely difficult to construct a
weakly open set U as in Theorem 3.1, so we are
motivated to construct a Furi-Pera type fixed point
theorems ([18]) for a multi-valued mapping :2
E
FM
with weakly sequentially closed graph. Here
M
is a
closed convex subset of with (possible) an empty
weak interior. The last mentioned case is very important
in applications, mainly for the investigation of boundary
value problems on noncompact intervals (see [18]).
E
E
Our next result extends and improves Theorem 2.4 in
[6].
Theorem 3.3 Let be a separable Banach space,

a closed convex subset of , and
E
M
a closed
convex subset of

with
M

. Also, assume


:FM P
:rE M
>0
cv has weakly sequentially closed graph
and weakly compact map. In addition, assume the
following conditions are satisfied:
There exists a weakly sequentially continuous retra-
ction (3)
There exists

and a weakly compact set
M

with





K
=:, ,=xEdxMMheredxyx y



 
(4)


 
=:,,0<KxEdxM

for any





, if


=1
,
jjj
x 
[0,1]M
,
w
jKj
xxM

is a seque nce in with


0<1xFx  



 and
(5)


jj
then,
F
xM  K for j sufficiently large here
M


denotes the weak boundary of M in
K

.
F
has a fixed point in
M
. Then
Proof. Consider



=,Nx xFrx 
N

We first show . Notice


cv
:Fr P 
r is
weakly compact. Since is weakly sequentially conti-
nuous,
M
is weakly closed, and
F
has weakly se-
quentially closed graph, it follows that
F
r has weakly
sequentially closed graph. Theorem 2.2 implies that
F
r
N has a fixed point, so

N
 
NFr FM N

n
Next we show that is
weakly compact. Indeed, , so
is relatively weakly compact. Now, let

n
x
a se-
quence of such that Nw
n
x
x
x

 , E
n. For all

 we have


nn
x
Fr x

and



wrxinM
n
rx .
Because
F
has weakly sequentially closed graph,


x
Fr x. Hence,
x
N

and is weakly sequen-
tially closed. Applying again the Eberlein-Šmulian
theorem [13, theorem 8.12.4, p. 549], we obtain that
is weakly compact. We now show . To do
this, we argue by contradiction and we use some ideas in
[18]. Suppose
N
N
=MN
= MN

N. Then since is compact
and
M
is closed we have by [19, p. 65] that




,=inf:,>0dNMx y xNyM
,0< . Thus, there exists



, with

, here =KN




=,,KxEdxM

K

. We have


is closed,
convex, so weakly closed and
K
M


. Using (4), we
obtain
K

is weakly compact. Because is separable,
the weak topology on E
K

is metrizable (see [20]), and

168 A. B. AMAR ET AL.
let

denote this metric. For , let

1,2,i 

 
,<xM i=,UxK
 i


1,


2,
We fix i. Now i
U is open in
K

with
respect to the topology generated by

, and so is
weakly open in i
U
K

. Also we have


,,
wK xMi==
ii
UU x



 
and



,xMi=,
Ki
UxK



 

dN


,>M
Since

, then =
w
i. Applying
Corollary 3.2, we get that there exists
NU


0,1  
iK
y



i
and
i
U such that ii
y
Fr y
y
 . In particular, since
, then
iK


i
U


1,2,i

rsome[0,1]  
R
forM


=:ExFr
each
ii
Fr y  (6)
Now we investigate
,foRx x
R is non-empty, because


. Also,
 
M

RconvF

, so by the Krein-Šmulian
theorem (see [14, p. 434]),
R
is relatively weakly com-
pact. Since F
r
has weakly sequentially closed graph
and is compact, we deduce by Lemma 3.1 that [0,1]
R
is weakly sequentially closed. This together with


for 1,2,j,=, [0,1]
jj
yM j


implies that we may assume without loss of generality
that
00
and \
ww
jj
yyMK

 =
w
K
M M


Also since

j
ji
y
Fr  y we have


000
y
Fr y  .
If 0=1

, then

0

0
y
Fr y
[0,1)
, which contradicts
=MN 0 . Thus,



=,=
ji
. But (5) with

00
=
K
x
ryMxyryM

 implies

ji
F
ry M  for sufficiently large. This contra- j
=
dicts (6). Thus . Accordingly, there exists MN
x
M such that
 
=
x
FrxF x
()a* E
Q.E.D.
Remark 3.4 If is separable and
M
is
weakly compact, then there exists a weakly continuous
and so weakly sequentially continuous retraction onto
M
(see [21]).
()bE If is reflexive then it suffices to take


F
M
is bounded, since a subset of a reflexive Banach space is
weakly compact iff it is closed in the weak topology and
bounded in the norm topology.
Remark 3.5 Theorem 3.3 extends Theorem 2.5
in [7]. ()a
()b As a corollary of Theorem 3.3 we find Theorem
2.6 in [8].
Remark 3.6 If in the statements of Theorem 3.3, the
convex set

is weakly compact, then a special case of
(5) which is useful in applications is


If

=1
,
jjj
x 
[0,1]M
,
w
jj
xx
is a sequence in with




and

x
Fx  0<1 with

, then


jj
F
xM   for j sufficiently large.. (7)
Remark 3.7 The set
M
can be with void weak inte-
rior.
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