I nternat. J. ath. & Math. Sci.
Vol. 2 #3 (1979) 473-480
473
A POINTWISE CONTRACTION CRITERIA FOR THE
EXISTENCE OF FIXED POINTS
V. M. SEHGAL
Department of Mathematics
University of Wyoming
Laramie, Wyoming 82071
U.S.A.
(Received May 7, 19 79)
ABSTRACT.
mapping.
QT(x,y]
Let
S
be a subset of a metric space
(X,d)
and
T: S + X
be a
In this paper, we define the notion of lower directional increment
of
T
conditions for
at
T
x
S
in the direction of
to have a fixed point when
y
X
and give sufficient
QT(x,Tx] < I
for each
x
S.
The results herein generalize the recent theorems of Clarke (Caned. Math. Bull.
Vol. 21(1), 1978, 7-11) and also improve considerably some earlier results.
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
Primary 47HI0, secondary 54H25.
INTRODUCTION.
In a recent paper [2], Clarke introduced the notion of lower derivative
D_T(x,Tx)
for a mapping
T: X + X
on a metric space
cient conditions for a continuous mapping
when
DT(x,Tx) < i
for each
x
X.
T
X
and obtained suffi-
to have a fixed point in
However, in order that
DT(x,y)
X
be
474
V.M. SEHGAL
(x,y) (to be defined later) contain points
finite, it is necessary that
arbitrary close to
whenever
x
x
# y.
The purpose of this paper is
(a)
ove restriction by introducing the notion of lower directional
to remove the
to consider mappings that are not necessarily con-
increment (see below), (b)
S
tinuous and are def,ned on a subset
X
of
As a
X.
with values in
consequence of our main result, we obtain the results contained in [2] and also
some other results (see [3] and [5]).
i.
PRELIMINARIES.
A function
X.
a nonempty subset of
(l.s.c.) on
semicontinuous
open for each real
S
iff
:
/
R+
x
for each
induces a partial order
the function
S
(nonnegatlve reals) is lower
S
o
S: (x) > r}
{x
It is easy to verify that given a function
r.
x < y
S
in
iff
S
denote a complete metric space and
(X,d)
Throughout this paper, let
< in
S
:
is
+,
S + R
given by
d(x,y) < (y)
(1.1)
(x).
The following Lemma is well-known (see Brondsted [i] or Kasahara [4]).
LEMMA i.
function on
Let
S
Then there is an element
S.
X
be a closed subset of
to partial order (i.i) in
u
and
S
G
:
S
/
+
R
be a
l.s.c.
which is minimal with respect
S.
As a consequence of Lemma i, we have
LEMMA 2.
function on
Let
S.
S
be a closed subset of
Then for each
e
with
X
and
:
S
/
+
R
be a l.s.c.
0 < e < i, there exists a
u
u(e)
S
such that
@(u) < @(x) + ed(x,u),
for each
x
S.
(1.2)
CONTRACTION CRITERIA FOR FIXED POINTS
PROOF.
d e, d
The proof is immediate by Lemma I (if we replace the metric
S.
S
Let
{x }
If for a sequence
for each
n
O(Xo)
xS, then
with a cluster point
S
c
+
S + R
:
and
by
< O(x)
f(y) < lim
n-=
x
be a l.s.c.
o
d(X,Xn),
for each
S.
x
f,
The proof is immediate since for any l.s.c,
PROOF.
2.
X
be a closed subset of
i
O(xn) < O(x) +
that
d
e.d).
e
LEMMA 3.
on
475
d(Yn,y)
+
0
implies
f(yn ).
MAIN RESULTS.
Let
For an
X.
be a subset of
S
S and
x
X
y
x
with
# y,
define
note that
Let
y
x
and
d(x,z) + d(z,y)
be a mapping.
For
x
QT(x,y]
T
{z
(x,y]
directional increment
0,
oo,
of
at
S
y
and
X, define the lower
in the direction
x
y
as
y,
if x
(Tx Tz)
inf{d d(x,z)
z
if (x,y] N S
.
(x,y]n S},
For the convenience of the notation
if
d(x,y)}
(x,y].
T: S /X
QT(x,y]
X: z
if (x,y] n S
we shall denote
# #,
d(Tx,Ty)
d(x,y)
0(x,y)
x-#y.
REMARK.
It may be noted that if
there is a sequence
{z }
n
c
(x,y]
S
QT(x,y]
such that
is finite and
O(x,z n
+
x
# y, then
QT(x y).
The following is the main result of this paper and is related to the lines
of argument in [2].
476
V.M. SEHGAL
THEOREM i.
Let
S
X
be a closed subset of
T: S
and
/
X be a mapping
satisfying the following conditions:
:S
The mapping
For each
sup{QT(x,Tx]: x
{x }
then any sequence
{x }
sequence
T
u
S
c
n
if
a
< I
then either (a)
QT(xn
for which
TXn]
or (b) if
i
i implies that the
/
(2.3)
S.
It follows by Lemmc 2, that for each positive integer
n, there
such that
i
(un) <_ (x) +
for each
(2.1)
is l.s.c, on S,
has a cluster point.
n
S
n
d(x,Tx)
(x)
(2.2)
S}
has a fixed point in
PROOF.
is a
defined by
S, QT(x,Tx] < i,
x
If
Then
+
R
/
< i
We assert that if
S.
x
QT(un,Tun
i, then
remark, for each fixed
Suppose
i.
/
n, there exists
(2.4)
d(X,Un),
then
Tu
u
n
n
TUn
Un
for any
{Zm }
sequence
for some
n.
c_ (un,Tun]
n
Then
0
S
and
by the
such
that
as
m +
.
0(Un,Zm)
/
It now follows by (2.4) that for each
(un) <_ (z m) + i d(un,zm)
Since for each
d(Un,Zm)
m,
+
<_
m +
,
Consequently, if
(2.3a) holds, then
u
TZm)
+
i
d(un
We have for each
$(Un).
(2.6)
Zm).
m,
i
) <_ 0(Un,Zm).
it follows by
(I
m,
d(zm, TUn) + d(Tun
d(Zm,TUn)
(I
Therefore, as
(2.5)
QT(Un,TUn]
i
)
(2.5) and (2.2) that for each fixed
< QT(un, Tu
n
n,
< i
n)
n
u
n
this case, otherwise by
+ i.
Therefore, if
for any n, then QT(Un,TU
Tu
n
for some n and the theorem is established in
--Tu
n
(2.3b), the sequence
{u }
n
has a cluster point
u
S.
CONTRACTION CRITERIA FOR FIXED POINTS
477
It follows by Lemma 3, that
(u) < (x),
for each
S.
x
We assert that
remark, there is a sequence
Tu
=
z
u.
(2.7)
Suppose
(u,Tu] n S
0(u z n
/
Then again by the
u.
such that as
n
/
,
(2.8)
QT(u,ru]
d(U,Zn
However, by (2.7) and the relation
Tu
+ d(z n,Tu)
(u), we have for each
n,
d(u,z n) +
(u) _<
d(Zn,TU)
(Zn) _< d(Zn,TU)
+
d(Tu,Tzn )"
0(u z n > i for each n and hence by (2.8)
Tu.
This contradicts (2.2). Thus u
This implies that
3.
SOME APPLICATIONS.
T: X
For a mapping
T
QT(u Tu] > i.
at
x
/
X, Clarke [2] defined lower derivative D_T(x,y)
y
in the direction of
DT(x,y)
0,
if
y,
x
O(x,z),
lira
as
(x,y)
if
(x,y]\{y}
,
Z-+X
where
z
(x,y)
,
if
O(x,z)
lira
O(x,z)].
(x,y)
d(z,x) < g
inf
lira
e+O
z+x
z
z
(x,y)
x,y
Since for any
,
(x,y)
X, QT(x,y]
<_ D_T(x,y),
the following results in [2] are
special cases of Theorem i.
COROLLARY I.
sup{D_T(x,Tx):
x
COROLLARY 2.
Let
T: X
X} < i.
Let
/
T
Then
T: X
/
be a continuous mapping such that
X
X
has a fixed point.
be a continuous mapping such that
of
V.M. SEHGAL
478
DT(x,Tx) < i
DT(xn Txn)
for each
I
/
X.
x
{xn
If for any sequence
implies that the sequence
{x }
X
in
with
then
has a cluster point
n
T
has a fixed point.
The following simple examples show that both Corollaries i and 2 are
indeed special cases of Theorem i.
EXAMPLE I.
{0,i}
X
Let
Tx
constant mapping defined by
,
DT(I,TI)
T
since
T
for each
0
QT(x,Tx]
is continuous and
EXAMPLE 2.
T: X
/
.
Clearly, T
(x,Tx)
X
X.
x
X
Let
X, T
0
for each
T
has a fixed point.
be the closed interval
x
i
[,3]
/
X
(I,TI)
Since
,
be a
However,
does not satisfy the conditions of Corollary i.
conditions of Theorem i and it follows
Let
T: X
with the discrete metric and
satisfies
with the usual metric.
be the mapping defined by
Tx=l+
X
1.
is continuous, strictly decreasing and for each
x # z
Further, it is easy to verify that for any
and therefore, for any
x # Tx,
X with
x
D_T(x,Tx)
x
=-.1
Tx # x,
with
0(x z)
1
XZ
Consequently, if
X
x < i, DT(x,Tx) > 1
and hence
However, since for any
X
Since
x
1+/5
2
boundary.
constant
(x,Tx],
i
(x,Tx]} < x’Tx
z
S
c
A mapping
i
x+l
< I
satisfies conditions of Theorem i.
only fixed point of
is the
k < i
{i__:
xz
inf
is compact, T
For a set
# Tx, Tx
x
QT(x Tx]
does not satisfy conditions of Corollary 2.
T
X, let
T: S
in
X.
denote the interior of
S
/
T
X
In this case
S
and
S
its
is a contraction mapping if there exists a
such that for all
x,y
S, d(Tx,Ty) < kd(x,y).
As another
CONTRACTION CRITERIA FOR FIXED POINTS
479
consequence of Theorem I, we have
COROLLARY 3.
T: S
X
/
Let
S
be a contraction mapping.
PROOF.
T
Since
0 < % < i
such that
S(x,e) n (x,Tx]
Tx
S
(1-%)
l-Txll
c
Tx
has a fixed point.
.
S
S.
.
S, 0(x,z) < k < i,
x,z
{y: lly-xll < e} c__ S.
Then
(x,Tx]
T
S, then
and
x # Tx, (x,Tx] n S
with
< e.
(x,Tx]
and hence
and consequently
S
x
e > 0, S(x,e)
then for some
S
x
T(6S)
If
is continuous and for any
it suffices to show that for any
if
X
be a closed subset of a Banach space
%,
Choose a
(%x + (l-%)Tx)
z
If
Now,
S,
x
then by hypothesis
S} < i.
sup{QT(x,Tx]: x
Thus
The result below was obtained by Su and the author [5] (see also
Edelstein [3]) and is again a consequence of Theorem i.
COROLLARY 4.
T: S
/
X
lx-ryll
Let
S
be a compact subset of a Banach space
be a mapping satisfying the condition:
<
I-YlI.
PROOF.
(x,Tx] s S #
QT(x,Tx] < i.
.
If
x,y
for all
X
and
S, x
y,
T(S) c__ S, then r has a fixed point.
with
x
# Tx,
Therefore, it follows by hypothesis that for any
x
S,
As in the proof of Corollary 3, for any
Since compactness implies
of Theorem I and has a fixed point in
S.
(2.3b), T
x
S
satisfies the conditions
480
V.M. SEHGAL
REFERENCES
i.
Brondsted, Arne. Fixed Points and Partial Orders, Proc. Amer. Math.
Soc. 60 (1976) 365-366.
2.
Clarke, Frank H.
3.
Edelsteln, M. On Fixed and Periodic Points under Contractive Mappings,
J. London Math. Soc. 37 (1962) 74-79.
4.
Kasahara, S. On Fixed Points in Partially Ordered Sets and KirkCaristl Theorem, Math. Seminar Notes, Kobe Unlversltv, (3),2(1975).
5.
Some Fixed Point Theorems for NonSu, C. H. and Sehgal, V.M.
expansive Mappings in Locally Convex Spaces, Boll. Un. Mat. Ital. (4),
10(1974) 598-601.
Pointwlse Contraction Criteria for the Existence of
Fixed Points, Canad. Math. Bull. 21 (i) (1978) 7-11.
KEV WORDS AND PHRASES.
Fxed’ point ’theor
Low dtivtive, Lower directional increment,