l’iternat. J. Math. & Math. Sci.
Vol. 8 No. 3 (1985) 469-475
469
CONTRACTIVE MAPPINGS ON A PREMETRIC SPACE
C.Y. SHEN and A. SOUND
Mathematics and Statistics Department
Simon Fraser University
Burnaby, B.C.
VSA IS6 CANADA
(Received April 30, 1984)
In this
ABSTRACT.
’;ontractive
p’per, we study the fixed point
property of certain types of
mappings defined on a premetric space.
The applications of these
results to topologic:l vector spaces and to metric spaces are also discussed.
Premetric space, sequentially
<EY WORDS AND PHRASES.
.weometric
t-contractive mappings,
mean contraction, iteratively t-contractive mapping, topological
vector space.
1980 MATHEMATICS SUBJECT CLASSIFICATION. 47H10.
i.
INTRODUCTION.
The classical Banach contraction mapping principle can be simply stated as
follows
(X,d)
Let
be a complete metric space, and let
exists a constant
k, 0
d(Tx,)
for all
x,y e X.
T:X
X.
Suppose there
k < I, such that
T
Then
k
d(x,y)
has a unique fixed point.
It is well known that this
principle is central to many existence proofs in mathematical analysis.
Since its
publication, much work has been done to generalize this result by allowing
X
to
be a more general type of space than the metric space or by relaxing the estimate
d(Tx,Ty)
S k
d(x,y).
In one of his papers [i], Wong introduced the concept of
local iterative contraction.
inequality
d(Tx,Ty)
k
His idea was based on the observation that the
d(x,y)
can be expressed as
d(Tx,)
@:[0,)
[0,)
(d(x,y))
@(t)
kt.
Banach’s result by replacing the function
@(t)
if one defines
by
Of course, one can then generalize
kt
with some more general type
of function.
On the other h’d, Kasahara [2] proved, in
principle for premetric spaces.
1968,
the Banach contraction
Roughly speaking, a premetric space is different
from a metric space by not having the symmetry property and the property that
C. Y. SHEN AND A. SOUND
470
d(x,y) > 0
@ y.
x
if
Kasahara demonstrated in his paper that there exists a
.lose connection between the premetric spaces and a certain class of subsets of a
topological vector space (see next section).
As a consequence, he showed that his
result implied a fixed point theorem by Dubinsky for topological vector space.
@ to
In this paper, we take
f
@k(t)
Z
eries
be a monotone increasing function such that the
t > 0
of its iterations converges for all
This characterization
allows us to prove that certain mappings defined on a premetric space haw
@
We introduce, in Section 2, the notion of locally
:he fixed point property.
teratively
<-contractive mappings,
and prove that they have fixed points.
The
.ppli:ation of this result to mappings defined on a topological space is also
,.ontained in that section.
In Section 3, we study the sequentially @-contractive
This concept has its origin from a paper by Yun [3].
r.appings on a premetric space.
manuscript, Yun introduced the notion of geometric mean contraction for a
]n his
’.;equence of mappings defined on a metric space, and stated:
any sequential
mapping with geometric mean contraction of a complete non-empty metric space
M
nto
M
has a unique fixed point in
This theorem in its present form,
:owever, is incorrect, and we have shown a modified version follows from our
’fneorem 3.1.
2.
LOCALLY ITERATIVELY CONTRACTIVE MAPPINGS.
X
Let
D
be a non-empty set, let
X, and let
X
be a subset of
non-negative real-valued function defined on D.
The triple
(X,D,d)
be a
d
is called a
premetric space if the following two conditions are satisfied:
0.
x a X, (x,x)
D and d(x,x)
(x,y),(y,z)
D, then (x,z) g D and d(x,z)
for all
i.
ii.
if
d(x,y) + d(y,z).
A sequence
For convenience, we denote the set of all positive integers by I.
(Xn
X
in
n c N
and
(right-convergent) to
is r-convergent
d(x,x n
0
as
g,m
s N
(x,x n)
if
g
with
g
(x
A sequence
(xg,xm)
m,
g
n
and
D
X
in
r-convergent to some point in
X
x
to
n
By letting
d(x,y)
Ix-yl,
as
X
is
n
for all
((x,x):
N
and
d(Xn,X)
X} U ((l,x): 0
x
{i-
i}
n
X
in
n
x _<-
is
0
as
i,
and
is r-convergent to
i
but it
Thus, in a premetric space, not every r-convergent sequence is
a r-Cauchy sequence.
d
(Xn,X)
[0,), D
(x
for example, a sequence
D
we see that the sequence
is not r-Cauchy.
we let
X
if
0
Similarly, one can define g-convergent
X
sequences, g-Caehy sequences, etc.
g-convergent
all
is a r-Cauchy
d(xg,xm)
A premetric space is r-complete if every r-Cauchy sequence in
m
fo’r
D
A premetrie space is r-separated if the limit
n
of every r-convergent sequence is unique.
sequence if for all
x X
Next let
X be any non-empty set and let
0, then every sequence in
X
D
X
X.
If
is r-convergent but the limit is not
uni q ue.
In an early paper by Kasahara [2], the author provided an interesting example
of a premetric space which illustrated the close connection between the premetric
spaces and the topological vector spaces. We shall describe this example here.
B
Let
be a non-empty, bounded, star-shaped
(i.e., XB
c B
onvex subset of a iisdorff topological vector pace E.
E
E
suc
(x,y) a D if and only if
(x,y)
D, we define d(x,y)
that
For each pair
x-y s
inf
for all
i),
0
Let D bt a subset of
XB for some real
XB}.
{ > 0: x-y
> 0
Then
CONTRACTIVE MAPPINGS ON A PREMETRIC SPACE
(E,D,d) is an
(E,D,d)
471
It is easy to see
r-separated and g-separated premetric space.
E
To show that
is a premetric space.
that
is r-separated, we
follow Kasahara’s argument by first noting that for every neighborhood
in
E
V+V c U.
such that
y
and
Take a
Let
{x
Suppose
6 > 0
with
n
Since
sufficiently large
c V+V c U. But E
is
n
6B c V
V
Ben
y-x
and
n
o(t)
Ill
l)
6B
belong to
n
x
for
n
y.
(t)
E
(X,D,d)
Let
0,(0(t)
t >
converges for all
(Tm,z,Tnz)
converges for all
e D
z g X
N
g
m,n
for all
Ck(t)
t,
($k-l(t)).
n(z)
and
{Tnz
n E
Let
@
is continuous at 0 with
PROOF.
(x,y) c D
such that for all
(d(x,y)).
Tn(z)x,n z )y
Note that from part (i) of the above definition, we have
LEMMA 2.1.
is
y) e D,
d(Tn(Zlx,Tn(Z)y)
x,y E
X
if
{0,1,2,...},
0
T: X
A mapping
be a premetric space.
C-contractive at
there exists a positive integer
Tn(Z)x ,Tn( z
for all
0
of
(x -x) + (-y+x)
n
x-v
x
0
DEFINITION 2.1.
ii.
x-x
U
which r-converges to
is circle, we have
Hausdorff, it follows that
said to be iteratively
and
E
for all
be a function which maps the non-negative reals into itself such
0,($k(t)
i.
lV c V
is a sequence in
is monotone increasing and
that
t >
V (i.e.,
there exists a circled neighborhood
NO}.
Next,
@k(t)
< t
t > 0
for all
and
0
t > 0
Suppose there exists a
monotone increasing,
@(t)
Then
be given as above.
@(0)
e D
we prove
t > 0
$(t)
= t.
Hence,
lira
k
such that
k E N.
for all
@k(t)
Since
is
$k(t)
t > 0
X
<
0
The second part follows easily from the inequality
which contradicts the fact that
(0)
-< lim (t) _<- lim t
t0 +
t+0 +
Before we state our main theorem for this section, we point out that if
0
(X,D,d),
a subset of a premetric space
(Y
Y)
THEOREM 2.1.
Let
D’
with
D
be iteratively
and
-contractive at
(Y,D’,d’)
is also a premetric space
be a r-separated premetric space and let
z
X
If
X
{x
0
X:
(Tnx,z)
D
T: X
for all
N
is r-complete, then T has a unique fixed point in X
and the
O
0
successive iterations u
are r-convergent to the fixed point.
TUk_l, u0 z
k
n
PROOF.
Let
{vk}
shall show that
on
T
for all
T
S
n(z
m,n
E N
O
and define recursively
Vo
X
forms a r-Cauchy sequence in
v g X
0
k
By condition
it is clear that
for all
(ii) on
k e N
T
0
z, vk
(m,k e N O
Since
d(Vk,Vk_l)+
k-l(d(Vl,V0))
(7(d(Sz,z))
+
+
SVk_ 1
We
From condition (i)
{0,1,2,...
0
we have
d(Vm+k,Vm)" Cm(d(Vk,V0)),
d(Vk,V0)
is
did
d’
(X,D,d)
then
Y
d(Vl,V0)
+ d(Vl,V0)
and
(Vm, v)
n
e D
X
C.Y. SHEN AND A. SOUND
472
is monotone increasing, we obtain
and
d(Vm+k,vm)
for all
m,k
6
N
X
x
0
X
g
k
N
{Vm
@
and the fact that
is r-convergent to
(Tix0,z)
Because
(T ix0,vk) c D
for all
{vk}
is r-complete, the sequence
0
for all
N
k
Sx
g
X
Since
0
D
(Z,Vk)
and
N
i,k
O
r-convergent to
k
i,k
Y0
Tx0;
X0
in
(Sko ’sz
0
Sx
N
g
Tx
whence,
such that
x
0
0
Then
TY0 Y0
((o’Z)
--<
{uk} where
for some
x
z, uk
0
To this end, note that for all
0
m,k e N O
u
Then
is r-convergent to
x
TUk_ I
is
n(z),
Thus, we have the following inequality-
k
that
x
0
we have
0
0
(x0,z) d(smx0,smk) =< m((Xo,kz)).
0,1,...,n(z)}.
d(x0,Tgz)
max.{d(x0,z)
{Tgz
Let
for all
the sequence
x0
r-convergent to the fixed point
n(z) +
@(d(x0,vk))
@(0)
with
for all
Finally, we will show that the sequence
m
0
is r-convergent to some point
d(TSkx0,skz) __< @k(d(Tx0,z))
(o’k)
Y0
m
X
r-separated,
D
as
In particular,
Now suppose there exists another
which implies that
is
0
=0
t
for all
d(Sx0,vk+I)
is continuous at
d(Tx0,vk)
Thus, {vk} is
0.
0
is a r-Cauchy sequence in
It follows from the inequality
0
O
{Vk}
@re(t)
Using the fact that
O
we have shown that
Since
$m(o(d(Sz,z))
_<-
.
$m()
which
snows
0
As an immediate consequence of Theorem 2.1, we have the following fixed point
theorem in the setting of a topological vector space:
COROLLARY 2.1.
Let
E
be a Hausdorff topological vector space, and let
be a bounded star-shaped convex subset of
E
T: E
Let
E
B
be a mapping such
that the following two conditions are satisfied:
there exists a
i.
n(z)
there exists a positive integer
ii.
n(z)x-T n(z)y $(I)B where
{x E E: Tnx-z
nB for some
XB
for
X0
sequentially complete, then T
x,y, x-y
such that for all
XB
is given as before.
T
implies
3.
Tmz-Tnz
m,n e N
such that for all
0.
some
If
z E E
n=
and for all
0
X
has a unique fixed point in
N}
n
is
0.
SEQUENTIAL CONTRACTIVE MAPPINGS.
Let
(X,D,d)
{T.
be a premetric space and let
i
N}
be a sequence of
1
then we say this
N
T.T.z for all i,j
X into X. If T.T.
z j
j
we define
sequence of mappings is commutative. For each positive integer n
n
1
n+l
or equal
than
less
integer
a
is
i
positive
If
T
with
T
T
T
1
1
we mean
to n
by
mappings from
Tn+
Tn(i)’
Tn(i)
T
n
We shall adopt the convention that
DEFINITION 3.1.
if
(x,y) e D implies
A sequence
that
T
Ti+iTi-i
T
n
[T.}
1
(i)
T
n
1
if
if
i > n
is said to be sequentially
-contractive
473
CONTRACTIVE MAPPINGS ON A PREMETRIC SPACE
(Tix,TiY)
N,
i.
for all
i e
ii.
for all
n,i e N,
d(T n
C’.s
where
(i)
x ,T
are congtant and
THEOREI,I
(i) y)
{y
(TiY,3/)
D
g
for all
E
i > n
C in(d(x,y))
if
i _<- n
{Tnz}
First, we’ll show that
n
belong to
{Tnz}
D
propert3/ of
D
Y
N
n
for all
(Tn+iz,Tnz)
and
m-l(t m
m-i
n
Next,
N
g
Moreover,
Y
Now let
which shows
D
for all
3/0
+
+
e D
mn+l
d(y 0 ’(i)z
Y,
y
for all
N.
i
N.
n
m
Y.
From
Using the transitive
n
d(Tn+iz,Tnz)
n( tn+l
{Tnz}
__
i
with
m
N, the sequence
it r-converges to
n
is r-convergent to some
proof).
i.
YO
{Tn(i)z}
is also
(It turns out that
It is easy to see that
{Tn(i)z}
We have
d(Tm(i)z’Tn(i)z) "-" cim-l(tm + + Ci #n(tn+l)
that {Tn(i)z} is a r-Cauch3/ sequence. Since (Tiz,z)
mn+l
(Tn+iz’(i)z)
that
N
m,n
g
is commutative, it follows that
1
+
we claim that for each fixed
Y
{T.}
with
the r-Cauch3/ propert3/ is not needed in our
is in
TiY0
Thus, the sequence
n
a r-Cauch3/ sequence in
(3/,z)
and
for all
+
Y
3/0
z E Y
#k(tk+l)
7
which goes to zero as
and
d(Tmz,Tm-lz)
<_
be a
If the set
is a r-Cauchy sequence in
e D
m,n
we have, for all
d(Tmz,Tnz)
point
{T.
is r-complete and if there exists a
d(TkZ,Z)
(T z,z)
the facts
and
if
-contractive mappings.
N}
-(t
PROOF.
Cid(x,y)
[n(d(x,))
7.
<
where t
k
1
then there exists a unique 3/0 6 Y such that
such that
_<-
is a function as given in Section 2.
@
sequence of commutative, sequentially
Y
d(Tix,TiY)
and
X be a r-separated premetric space, and let
Let
3.1.
n
e D
D, we see
Hence,
)n+l
d(3/0,Tn+iz) + d(Tn+iz,(i)z
=< d(Yo,Tn+iz + ciCn+l(d(T.z,z))
=<
1
which goes to zero as
Finall3/,
n
(YO,Tnz), (TiYo,Y0) e D implies that (TiYo,Tnz)
d(Ti3/O,Tnz) d(TiY0,Tn(i)Tiz)
-<
ci d(0
which goes to zero as
for all
3/1
Y
n
(i)z
Since
YO
X
is r-separated, we have
such that
is unique, we proceed as follows:
Ti3/l
i
(3/1 ’Tnz
nd
and
Ti3/0
3/0
i g N.
To show
g
n
T
D
for all
(Tn
i
Then
Tnz
e D
suppose there exists another
(yl,Z)
g
D
which implies that
C. Y. SHEN AND A. SOUND
474
d(y I ,Tnz
(Tnz}
Hence,
cn( d(yI ,z
__<
is r-convergent to
).
X
and by r-separateness of
Yl
we see that
Y0
Yl
One can obtain an immediate corollary of the above theorem if we first define
{T i} of self-mappings on a premetric space
A sequence
DEFINITION 3.2.
said to be
C-contractive if
d(Tix,TiY)
=< (d(x,y))
(x,y) e D implies
for all
i e I
(Tix,TiY)
that
e D
X
is
and
as given before.
with
{T i} be a sequence of commutative -contractive selfIf the set
mappings defined on a r-separated premetric space X
D for all i e N} is r-complete and if there exists a z e Y
{y: (Tiy,y)
Y
Let
COROLLARY 3.1.
k-l( k)
k,z,z)
7.
<
d(T
where t
t
k
i
e Y such that
then there exists a unique
such that
TiY0 Y0
Y0
{T.
One only needs to observe that
PROOF.
(y,z) e D
and
y e Y
for all
i e 11
for all
is also sequentially
i for all i
with C.
In a manuscript by Yun [3], the author introduced the notion of geometrical
of self-mappings defined on a metric space X.
mean contraction for a sequence {T.
-contractive
i
After a careful study of his work, we believe his definition and also the
statement of Theorem 1 in his paper need to be modified.
We shall give here our
version.
DEFINITION 3.3.
self-mappings on
(X,d)
Let
X
be a metric space and let
We say that
mean contraction if for each
{T.
1
i g N
r.
_<-
(rl... rn )I/n
where
0 -< G < 1
Clearly, if
to
X
@(t)
Gt, t
X
< G
Y
is a metric space, then the set
{T.
is a sequential mapping with
i
is sequentially
=
N
n
is a constant.
Also, if
{T i}
then
(x,y e X)
rid(x,y)
is a positive constant, and, for all
i
-contractive provided
we
in Theorem 3.1 is equal
eometric mean contraction,
l-l-} and
let C.
max. {r
i
COROLLARY 3.2.
X
r.
z
i
0
Let
{T.
X be a complete metric space and let
1
commutative sequential mapping with geometric mean contraction.
x
be a sequence of
we have
d(Tix,TiY)
where
{T i}
is a sequential mapping with geometric
such that the set
{d(Tix,x):
i e
N}
is bounded
then
be a
If there exists an
{T
i
has a unique
common fixed point.
PROOF.
Let
b
be an upper bound of
{d(Tix,x):
i e
N).
Since
is mono-
tone increasing we have
k-l(tk) k-l(b)
Hence,
k-l(tk) <
__<
where
t
d(TkX,X).
k
easily from Theorem 3.1.
7.
and the corollary now follows
We shall remark here the Corollary 3.2 remains valid if we replace the
requirement that
{T.
be commutative by the following less stringent condition:
n
n
for all n e N.
T
T
n+l
I
(Ti} is said to be weakly commutative if T
Tn+
CONTRACTIVE MAPPINGS ON A PREMETRIC SPACE
ACKNOWLEDGEMENT.
475
This research is supported by the Natural Sciences and
Engineering Research Council Grant
A8511.
REFEREN CES
Two extensions of the Banach contraction mapping principle,
J.
i.
W0NG, J.
2.
!,lath_..Analysi_s_ and Appl., Vol 2.2., No. 2, 1968, p. h38-4h3.
KASAHARA, S. A remark on the contraction principle, Proc. Japan Acad.,
44, 1968,
3.
YUN, Tian-quan.
p.
of
21-26.
Fixed point theorems of sequential mapping with geometric
mean contraction and its applications, Preprint.