A FIXED POINT THEOREM FOR MAPPINGS
WITH A CONTRACTIVE ITERATE
V. M. SEHGAL
1. Let (X, d) he a metric
satisfies the condition
(1)
space
and /: X—>X a mapping.
If /
d(f(x),f(y))^kd(x,y)
for all x, yEX and for some k<l, then/is
called a contraction.
A well-known theorem of Banach [l] states that for X complete,
each contraction
mapping/
has a unique fixed point u, and the successive approximations
{/"(xo)} converge to u for each Xo£A. In
this paper, we investigate
mappings which are not necessarily contractions and prove the following result.
Theorem.
Let (X, d) be a complete metric space, and f: X—*X a
continuous mapping satisfying the condition: there exists a k < 1 such
that for each x£A, there is a positive integer n(x) such that for all yEX
(2)
d(fnM(y),f^(x))
^ kd(y, x).
Then f has a unique fixed point u and /"(xo)—»w for each XoEX.
Lemma. If f: X—>X be any mapping satisfying the condition of the
above theorem then for each x£A, r(x) =sup„ d(f"(x), x) is finite.
Proof.
Let x£A
and let
l(x) = ma.x{d(fk(x), x): k = 1, 2, ■ • ■ , n(x)}.
If n is a positive integer,
<n^(s + l)-n(x), and
there exists an integer 5^0
such that s-n(x)
d(f"(x), x) ^ <*(/»<*>
■/»-»<*>(*),
fnM(x)) + d(fn^(x),
x)
^ kd(f"-"M(x), x) + l(x)
g /(*) + */(*) + k2l(x) +-h
g l(x)/(l - k)
Hence r(x) =sup„
Proof
xi=/m°(xo)
k'l(x)
for all n ^ 0.
d(fn(x), x) is finite.
of the theorem.
and inductively
Let x0£A
be arbitrary.
Let m0 = n(x0),
w,- = w(x,), xt+i =/•"•'(x,). We show
Received by the editors February 12, 1969.
631
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that
632
V. M. SEHGAL
the sequence
{x„} is a convergent
[December
sequence.
By routine
calculation
we have
d(xn+1, Xn) = d(fm^-f^(Xn-x),f^-1(Xn-x))
^ kd(f^(xn-x),
*»-l) ^
• ■■
g knd(fm»(Xo), Xo).
Therefore,
m>n,
it follows
by lemma
that
m—X
d(xn+1, xn)^knr(x0).
Thus,
for
Un
d(xm, xn) g 2^1 d(xi+i,
Xi) g-r(x0)
i=n
—>0
t
as ra —> oo.
&
The sequence {x„} is therefore Cauchy. Let x„—-*uEX.
lif(u) 9^u, then, there exists a pair of disjoint closed neighborhoods
U and V such that uEU, f(u) £ V and
(3)
P = Ini{d(x,y): xEU,yEV]
Since/ is continuous,
However,
xnEU
and/(x„)£
d(f(Xn), Xn) = d(fm^-f(xn_x),fmn'1(Xn-x))
>0.
V for all n sufficiently
^ kd(f(xn-l),
large.
Xn-l) g • ■ •
< knd(f(xo), x0)->0
as ra—>co, contradicting
(3). Thus/(w)=M.
The uniqueness
of the fixed point follows
To show that/'l(xo)—>u,
set
p* = max{d(fm(xo),u):m
If ra is a sufficiently
immediately
from
(2).
= 0, 1, 2 • • • , (n(u) — 1)}.
large integer,
then
n=r-n(u)+q,
0^q<n(u),
r>0, and
d(f(x0), u) = d(f'-"<-u)+"(xo),fnM(u)) g kd(f^-^-"^+"(xo),
u) g • • •
g kTd(fq(xo), u) g £rp*.
Since ra—>=oimplies r—»°°, we have d(f"(xo),
establishes the theorem.
u)—>0 as ra—»°°. This
2. In this section we give an example of a continuous function /
which satisfies condition (2), but is not a contraction.
The author is
thankful to the referee for showing that no iterate of/is a contraction.
Bryant [2 ] considers a function / (not necessarily continuous) on a
complete metric space X into itself for which there exists a^,0^^<l,
and a positive integer w, such that d(fn(x), f"(y)) ^kd(x, y) for all
x, yEX. That Bryant's
condition is stronger than condition
(2) is
shown below. The author is grateful to Professor David R. Anderson
for his help in the following example.
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1969]
fixed point theorem
for mappings
Example. Let X be the closed unit interval
metric. Write
633
[0, 1 ] with the usual
x = u |~—,—~|u{o},
and let/: X—>A be defined as follows.
For each » = 1, 2 • • • , let
/:r±,_L"urj_,ri
|_2» 2»-!j L2n+1 2»J
be defined by
»+ 2 /
n + 3\
/(x) =-(x-)+
=-
1\1
2"-1/
—'
ifx£-'
2"
1
r 3re + 5
L2"+1(w+2)
in
2"~l J
-
r 1
3» + 5 1
if x E —>-
2B+1
|_2"
2"+1(« + 2)J
and let/(0)=0.
It is obvious that/ is a nondecreasing, continuous function on [0, l]
with 0 as the only fixed point, and that / is not a contraction.
If
xE [l/2n, l/2n_1] and yEX, then by a routine examination of cases
y£[l/2m,
l/2m_1] for m^n
and m^n,
it is easy to verify that/
satisfies
|/(x) -f(y) I ^ n—— \x-y\
for all y £ X.
n+ 4
Therefore, if we choose £ = 1/2 in (2), then for each x£ [1/2", l/2n_1],
n(x) may be taken asn + 3, whereas n(0) may be taken as any integer
greater than one.
To show that the condition in [2] is stronger than (2), let 0^&<1,
and N (a natural number) be given; it will be shown that there exists
x and
y such
that
\fN(x)—fN(y)\>k\x
n> (Nk/1— k))— 2. Since/*
i=l,
is uniformly
—y\.
continuous
Choose and
on
fix
[0, l] for
2, • • ■ , N, there is some 5>0 such that
.
.
\x - y\1 < 8 => \Uf(x) - f(y)
JWn
.
<- (n
n+ N + 3
+ N + 2)2»+"+!
for i = l, 2, • ■ ■, N. Setting x = l/2n_1 and y any member of
[l/2n, 1/2"-1] such that 0< |x-y| <5, it can be shown that/*(x) and
f(y)
are both members of
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634
V. M. SEHGAL
r 3(ra + i) + 5
\_2n+i+1(n+ i+2)'
1 -I
2"+'-1J
for i = 1,2, • • • , N. Thus
i
\f(x)-f(y)\
i
ra + 2 ,
=—— |x-y|,
w+ 3
\f(x) -f2(y) | = ^—4
| x-y| , • • • , |/*(x) -f»(y) |
w+ 4
=-
ra + 2
.
.
.
x — y\ > k\ x — y\ .
n + 2+ N '
'
'
'
The author acknowledges the help of Professor
Reid in the preparation of this paper.
A. T. Bharucha-
References
1. S. Banach, Sur les operations dan les ensembles abstraits et leurs applications
aux
integrates, Fund. Math. 3 (1922), 133-181.
2. V. W. Bryant,
A remark on a fixed point theorem for iterated mappings,
Math. Monthly 75 (1968), 399^00.
University
of Wyoming
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Amer.