R ENDICONTI
del
S EMINARIO M ATEMATICO
della
U NIVERSITÀ DI P ADOVA
S. P. S INGH
On a fixed point theorem in metric space
Rendiconti del Seminario Matematico della Università di Padova, tome 43 (1970), p. 229-232
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S. P. SINGH
*)
Let
(X, d)
be a metric space. Amapping
T : X --~ X is called acontraction
mapping
if there existsk, o _ k ~ I ,
such thatfor all x, y in X.
The well-known Banach Contraction
Principle
states that a contrac-tion
mapping
on acomplete
metric space X has aunique
fixedpoint.
Recently
Maia[4] proved
thefollowing
theorem:Let X have two metrics d and 8 such that
(1) d(x, y)
for all x, y in X.(2)
X becomplete
with respect to d.(3)
T : X --~ X be amapping
continuous withrespect
to d.and
(4)
T : X -+ X be contraction withrespect
to S.Then there exists a
unique
fixedpoint
of T in X.The aim of this note is to get the same conclusion under much less restricted condition.
Also,
we do not need thecontinuity
of T withrespect to d on
X, just
thecontinuity
at apoint
will serve the purpose.THEOREM. Let X have two metrics d and
8,
and thefollowing
*) Indirizzo dell’A.: Dept. of Mathematics - Memorial University - St John’s, New Foundland, Canada.
This research was partially supported by NCR grant A-3097 .
230
conditions are fulfilled.
(1) d(x, y)
for all x, y inX,
(2)
T : X -+ X is a contractionmapping
withrespect
to6,
i.e.S(Tx, y)
for all x, y in X.(3)
T is continuous atpeX
withrespect
to d.and
(4)
There exists apoint
such that the sequence of iterates has asubsequence f Tnixo convering
to p in metric d.Then T has a
unique
fixedpoint.
PROOF. Given x0EX. We define
It follows from
[4]
that( xn) }
is aCauchy
sequence withrespect
to S. Sinced(x, y) S(x, y)
for all x, y inX,
it follows that is aCauchy
sequence withrespect
to d. Since thesubsequence }
of theCauchy
sequence converges to p, therefore( xn)
converges to p under d i.e. lim xn = p.n-oo
Since T is continuous at p therefore
Thus T has a fixed
point.
We need to prove that T has a
unique
fixedpoint
.Let x and y betwo different fixed
points.
ThenSince
0 _ k 1,
we get a contradiction and therefore T has aunique
fixed
point .
It can be
easily
seen that condition(4)
does notimply
thecomplete-
ness of
X,
with respect to d. Forexample,
let X =[0, 1)
and T : X - X begiven by
4 d is the usual metric. In this case the sequence ofx
iterates has a convergent
subsequence
but the space is notcomplete.
REMARK. The theorem will remain true if condition
(2)
isreplaced by
thefollowing
condition:(2’)
There exists apoint
x0EX such that the sequence of iterates is aCauchy
sequence with respect to S.This condition is so
general
that resultsgiven by
several mathematic- ians can beeasily
unified.It can be
easily
seen that the sequence of iterates of themappings given
below isalways
aCauchy
sequence .EXAM P LE S .
(2) S(Tx, Ty) }
for all x,yeX
and0 oc 2 .
Thismapping
has beengiven by
Kannan[3].
(3) Given E>0,
there exists8>0
such thaty)E+8 implies d(Tx,
This
mapping
has been studiedby
Meir and Keeler[5].
The
mappings
consideredby
Edelstein[2],
Rakotch[6]
andBoyd
and
Wong [1] satisfy
above conditiongiven
inexample (3),
and there-fore the sequence of iterates will be
Cauchy
sequence.REFERENCES
[1] BODY D. W. and WOUNG J. S. W.: On Non-Linear Contractions, Proc. Amer.
Math. Soc. 20 (1969, 458-464.
[2] EDELSTEIN M.: On Fixed and Periodic Points Under Contractive Mappings, Journal London Math. Soc. 37 (1962), 74-79.
232
[3] KANNAN R.: Some Results on Fixed Points II, Amer. Math. Monthly, (1969), 405-408.
[4] MAIA M. G.: Un’osservazione sulle contrazioni metriche, Rend. Semi. Mat.
Univ. Padova XL (1968), 139-143.
[5] MEIR A. and KEELER E.: A Theorem on Contraction Mappings, (to appear).
[6] RAKOTCH R.: A Note on Contractive Mappings, Proc. Amer. Math. Soc. 13
(1962), 459-465.
Manoscritto pervenuto in redazione il 10 ottobre 1969.