Internat. J. Math. & Math. Sci.
VOL. 15 NO. 3 (1992) 435-440
435
FIXED POINT THEOREMS FOR d-COMPLETE TOPOLOGICAL SPACES
TROY L. HICKS
Department of Mathematics
University of Missouri at Rolla
Rolla, Missouri 65401
(Received May 7, 1991)
Generalizations of Banach’s fixed point theorem are proved for a large
ABSTRACT.
These include d-complete symmetric (semi-metric)
class of non-metric spaces.
The distance function used need not be
spaces and complete quasi-metric spaces.
symmetric and need not satisfy the triangular inequality.
KEY WORDS AND PHRASES. Fixed point, d-complete topological spaces.
i80 AMS SUBJECT CLASSIFICATION CODE. 47HI0" 54H25
X x X
Let (X,t) be a topological space and d
y.
and only if x
[0,) such that d(x,y)
E
X is said to be d-complete if
n=l
the sequence {x
n}
Cauchy sequence.
G
X
X, 0(x,)
X
{x,Tx,T2x, ...}
.
[0,) is T-orbitally lower semi-continuous at x
0(x ) and lim x
x if x
n
x* implies G
x implies Tx
n
(X*
<
implies that
In a metric space, such a sequence is a
is convergent in (X,t).
If T
d(Xn,Xn+l)
0 if
< lim inf
G(Xn).
is called the orbit of x.
if
{Xn}
T" X
is a sequence in
X is w-continuous at
Tx.
The basic idea of a d-complete topological space goes back to Kasahara [I] and
[2], Iseki [3], and their L-spaces.
LEMMA i. X is a set, T
X
X and d
X
[0,) such that
(a) d(x,Tx) <_ (x)
(Tx) for all x
d(Tnx,Tn+ix)--
(b)
E
n=O
(c)
If (a) holds for all y
X x X
[0,).
Then there exists
X, if and only if
converges for all x.
0(x,), then E
d(Tnx,Tn+ix)
converges.
n-0
n
PROOF.
4(Ti+ix)]
Suppose (a) holds.
4(x)
4(Tn+ix)
therefore convergent.
For x
_< 4(x).
X, S n
i-0
d(Tix,Ti+Ix)
n
_<
[(Tix)
i-O
{S n) is non-decreasing and bounded above and
T.L. HICKS
436
d(Tix,Ti+ix).
X
Suppose (b) holds and let (x)
4(x)
4(Tx)
d(x,Tx)
i-0
n
n
d(Tix,Ti+ix)
Z
since S -T
n n
d(Ti+Ix,Ti+2x)
Z
d(x,Tx)
d(Tn+ix,Tn+2x)
i-O
i-0
=.
d(x,Tx) as n
The proof of (b) gives (c).
LEMMA 2. X is a topological space, T
X
X and d
X x X
[0,) such that
0 if and only if x
y. Suppose there exists an x
d(x,y)
X such
that lim
Tnx
(a)
Tx
x
x
.
Tx
-imply
PROOF.
x
Assume that Tx
d(x,Tx) is T-orbitally lower semi-continuous at
Tnx
n
d(x ,Tx
x
G(x
. .
x
and (x
d(x*,Tx
Then G(x
n
If x
0
implies G(x)
x
G is T-orbitally lower semi-continuous at x
(b)
lim x
Then"
exists.
n)
and lim inf
d(Tnx,Tn+Ix)
is a sequence in 0(x,) with
0 <_ lim inf
d(Xn,TXn)
lim inf
and G is T-orbitally lower semi-continuous at x
lim inf
G(Xn)
0
lim inf
d(Tnx,Tn+ix)
0.
G(Xn).
then
Thus, Tx
x
The next theorem is a version of Carlsti’s theorem [4 or 5] in this more
general setting.
Caristi’s theorem for metric spaces is a generalization of
Banach’s fixed point theorem.
THEOREM I. Let X be a d-complete topological space. Suppose T
X X and
X
there
exists
such
an x
that
[0,-). Suppose
4
d(y,Ty) S 4(Y)
4(Ty) for all y
O(x,-).
(I)
Then we have:
(a)
(b)
lim
,
Tx
Tnx
x
exists.
if and only if G(x)
x
d(x,Tx) is T-orbitally lower semi-
continuous at x.
From Lemma i,
PROOF.
Tnx
,
Z
n-O
d(Tnx,Tn+Ix)--
is convergent.
X is d-complete so lim
d(T’,Tn+Ix)
exists. Note that lim
0. Now apply Lemma 2.
Even in this general setting one obtains a version of Banach’s theorem
as a corollary of the above theorem.
COROLLARY I. Let X be a d-complete topological space and 0 < k < I.
X X and there exists an x such that
Suppose T
x
d(Ty,T2y)
S k d(y,Ty) for all y
0(x,-).
Then
. Tnx-
(a)
llm
(b)
Tx
x
x
exists.
if and only if G(x)
continuous at x.
d(x,Tx) is T-orbitally lower semi-
(2)
FIXED POINT THEOREMS FOR TOPOLOGICAL SPACES
PROOF.
Set (y)
d(Tn+ix,Tn+2x)
d(y,Ty) for y0(x,).
Tnx
Let y
437
in (2).
Then
d(Tnx,Tn+Ix) and d(Tnx,Tn+ix) k d(Tnx,Tn+Ix) d(Tnx,Tn+ix)
d(Tn+Ix,Tn+2x). Thus, d(Tnx,Tn+ix) l.-Ik [d(Tnx,Tn+Ix) d(Tn+Ix,Tn+2x)] or
k
(y)
d(y,Ty)
(Ty). Apply Theorem i.
In [6], it was shown that many generalizations of Banachs theorem hold
for quasi-metric (d(x,y)
d(y,x)) spaces. However, we do not have the triangular
inequality in the present setting, so standard proofs and theorems do not
necessarily hold. Before proving more theorems, we give the definitions of some of
the special d-complete topological spaces we had in mind when we formulated the
results of this paper.
DEFINITION. A symmetric on a set X is a real-valued function d on
x
X
X such that"
0 and d(x,y)
(a) d(x,y)
0 if and only if x
y- and
(b) d(x,y)
d(y,x).
Let d be a symmetric on a set X and for any
> 0 and any x X, let S(x,)
X" d(x,y) < }. We define a topology t(d) on X by U
{y
t(d) if and only if
for each x
U, some S(x,) <_ U. A symmetric d is a semi-metric if for each x
X
and each
> O, S(x,) is a neighborhood of x in the topology t(d). A topological
space X is said to be symmetrizable (semi-metrizable) if its topology is induced by
X.
Suppose X is a d-complete Hausdorff topological space, T
a symmetric (semi-metric) on
THEOREM 2.
d(Tx,T2x)
is w-continuous and satisfies
[0,), k(O)
kn(d(x,Tx))
Z
<
nl
[k is not assumed to be continuous and
PROOF.
If Tp
p, d(p,Tp)
d(T2x,T3x)
Now
n-i
d(Xn,Xn+ I)
Tnx
In this case, x
n
p
TP.
k(k(a))].
kn(0)
0 for every n.
Tnx.
Assume the condition holds and let x
n
and
.
k2(a)
0, and
X
k(d(x,Tx)) for all x
X, where k’[O,)
Then T has a fixed point if and only if
0, and k is non-decreasing.
there exists an x in X with
X
T2x)
d(Tx
< k(d(x Tx))
k(d(Tx,T2x))
k2(d(x,Tx)). By induction,
d(Xn,Xn+ I) d(Tnx,Tn+Ix) kn(d(x,Tx)).
<
w-contlnuous implies
and X is d-complete.
Xn+ I
T xn
Tp.
Thus
Since
lim x
Xn+ I
p
n
Tp
p exists
T
p
In many special cases of Theorem 2, one has d(Tx,Ty)
k(d(x,y)) for all X, y
X and the special form of k forces w-contlnuity of T. It may also force the
uniqueness of p and enable us to obtain error bounds. In other cases, T need not
be w-continuous so the following theorem is needed.
THEOREM 3. Suppose X is a d-complete Hausdorff topological space, T
X X
and there exists a y such that
k
[0,)
[0,), k(0)
d(Tx,T2x)
k(d(x,Tx)) for every x
0, and k is non-decreasing.
Suppose
0(y,), where
Z
n-i
kn(d(y,Ty))
<
438
T.L. HICKS
E
and {x n) in 0(y,) with
n="I
<
d(Xn,Xn+l)
Tny.
Let x n
imply lim xn exists
Then:
(a)
lim x
(b)
Tp
p exists.
n
p if and only if G(x)
d(x,Tx) is T-orbltally lower seml-contlnuous
at p.
PROOF.
0
The proof of Theorem 2 gives (a).
d(Xn,Xn+ I)
lim
n
d(Tny,Tn+ly)
lim
n
Lemma 2 gives (b) since
0.
Let 0 < A < i. Suppose X is a d-complete Hausdorff topological
X is w-continuous and satisfies d(Tx,Ty) _< A d(x,y) for all x, y
COROLLARY 2.
space, T
X
Then T has a unique fixed point p.
X.
PROOF.
Let k(t)
At.
Then
For any x
X, p
And(x,Tx),
kn(d(x,Tx))
lim
Tnx.
so
Z
kn(d(x,Tx)) <-
n-i
T is w-contlnuous so Theorem 2 gives xn
Tnx p Tp for any x
X. Clearly, p is unique since 0 < A < I.
In Corollary 2, if one replaces topological by symmetrizable, then d(Tx,Ty) <
A d(x,y) forces T to be w-continuous. We now give several examples of a specific
function k where Theorem 2 or Theorem 3 applies to yield another corollary similar
to Corollary 2. To apply the theorems, one needs a non-decreaslng function k and
for any x in X.
an x in
X with
kn(d(x,Tx))
Z
<
n-I
conditions.
.
The following examples satisfy these
The reader can consult [7] for the details that are not obvious and
not provided here.
EXAMPLE I.
k(d(x Tx))
Suppose 0 < A < I.
4: [0,-)
[0,i) and
decreasing, and k
Since
Then
d(Tx,T2x)
_<
And(x,Tx).
Suppose T satisfies d(Tx,Ty) _< /(d(x,y)) d(x,y) for all x,y in X,
EXAMPLE 2.
where
At for t > 0.
Let k(t)
n
k d(x Tx) gives k (d(x,Tx))
4(t) < i,
Z
[0,-)
kn(t)
4
[0,-).
<
Then k(t)
is non-decreaslng.
t
4(t), k is non-
It follows by induction that
kn,(t)
<_ t[(t)]
n.
-.
n-i
EXAMPLE 3.
< i.
Consider k(t)
If t < I, it follows that
theorems apply.
EXAMPLE 4.
(0,I].
k(t)
If 4(t)
t
t
4(t) where 4
kn(t) _<
/(t) where /
t[4(t)]
[0,-)
< I, kn(t) _< (kt)(4(t))
n
[0,-)
n.
[0,-) and 4(t) _<
If k is non-decreaslng, the
[0,-) and /(t) <
for all n _> 2.
4(t) for
If k is non-
decreasing, the theorems apply.
EXAMPLE 5. Assume k is non-decreasing, k is convex on [0,i], and k(t)
all 0 < t < I. Fix t < i. Now k(t) < t gives k(t) -t for some
0
<
(t) < I.
THEOREM 4.
By induction,
kn(t) _< nt
t for t
for all n and thus
Z kn(t) <
n-I
<
-.
Suppose (X,d) is a Hausdorff d-complete symmetrizable space,
t for
FIXED POINT THEOREMS FOR TOPOLOGICAL SPACES
T
[d(x,y)]P
X, and d(Tx,Ty) <
X
d(x,Tx) < i, then x
n
PROOF.
Tnx
Let k(t)
I, and k is convex.
t
p
p for t
Since 0
where p
> i.
439
If there exists x such that
Tp.
> 0.
< i,
k(0) -0, k is increasing, k(t) < t if t <
kn()
E
<
n-1
.
T is w-continuous so Theorem 2
applies.
REMARKS.
Given a topological space (X,t), when does there exist a distance
function d such that X is d-complete?
discrete first countable
X such that t
t
Let X be an infinite set and t any T 1 nonThen there exists a complete metric d for
Now (X,t) is d-
topology for X.
and the metric topology t d is non-dlscrete.
d
complete since Z d(xn Xn+ I) <
implies that ixn} is a d-Cauchy sequence. Thus x
t
x in t and therefore in the topology t. In [8], the construction gives t
E
d
where t is a complete uniform space and the uniformity has a countable base.
E
Hence, the uniformity is metrizable and the compatible metric d must be complete.
It should also be noted that any complete quasl-metrlc space (X,d) (d(x,y)
’d(y,x)) is a d-complete topological space. There are several competing definitions
for a Cauchy sequence, but Ed(Xn,Xn+l) <
will imply that {xn} s a Cauchy
sequence for any reasonable definition. One reasonable defnltion is obtained by
requiring that the filter generated by {xn} be a Cauchy filter in the quasi-
uniformity generated by d.
This gives {x
n}
there exists a positive integer n
o
{y
if
.
X" d(x,y) < }.
Ed(Xn,Xn+l)
<
is a
n() and x
> 0
Cauchy sequence if for each
x()
n
X such that
{xn"
n
_> no
The metric space definition of a Cauchy sequence also holds
REFERENCES
1.
2.
S. Kasahara, On some generalizations of the Banach contraction theorem, Math.
Seminar Notes, 3(1975), 161-169.
S. Kasahara, Some fixed point and coincidence theorems in L-spaces, Math.
Notes, 3(1975), 181-187.
K. Iseki, An approach to fixed point theorems, Math. Seminar Notes,
3(1975), 193-202.
Seminar
3.
4.
Alberta Bollenbacher and T. L. Hicks, A fixed point theorem revisited,
Prof. Amer. Math. Soc. 102, No. 4, (1988), 898-900.
5.
J. Caristi, Fixed point theorems for mappings satisfying inwardness
conditions, Trans. Amer. Math. Soc. 215(1976), 241-251.
6.
T. L. Hicks, Fixed point theorems for quasi-metric spaces, Math. Japonica
33, No. 2(1988), 231-236.
7.
T. L. Hicks, Another view of fixed point theory, Math. Japonica 35, No. 2
(1990), 231-234.
Wilson R. Crisler and Troy L. Hicks, A Note on T 1 topologies, Proc. Amer.
Math. Soc. 46, No. I, (1974), 94-96.
8.