Internat. J. Math. & Math. Sci.
VOL. 13 NO. 4 (1990) 779-782
779
REMARKS ON A FIXED-POINT THEOREM OF GERALD JUNGCK
MAIBAM RANJIT SINGH
Department of Mathemat
Man[put Unlvers[ty
Imphal- 795003, India
(Received August 18, 1988 and in revised form May 29, 1989)
Jungck [I] obtained a flxed-polnt theorem for a pair of continuous selfRecently, Barada K. Ray [2] extended the theorem
ABSTRACT.
mappings on a complete metric space.
of Jungck [I]
for three self-mapplngs on a complete metric space.
In the present
paper we omit the continuity of the mapping used by Ray [2] and replace his four
by a single condition.
conditions
Our results so obtained generalize and/or unify
fixed-polnt theorems of Jungck [I], Ray [2], Rhoades [3], Cirl
[4], Pal and
Malti
[5], and Sharma and Yuel [6].
KEYWORDS AND PHRASES.
Fixed Point Theorem,
Continuous Self-Mapplngs,
and Complete
Metric Space.
54H25, 47HI0.
1980 AMS SUBJECT CLASSIFICATION CODE.
INTRODUCTION.
We quote two theorems:
Theorem I. (Jungck [I]).
If S and T are continuous mappings of a complete metric
space (X,d) into itself such that
i) s(x) cT(X),
ll) ST
TS, and
ill)
d(Sx,Sy) < a d(Tx, Ty) for every pair of points
X and for a
x,y
[O,l), then
F
S
where
F
and
FS, T
S
F
FS, T
T
{x e X: x
{x
E
X: x
Theorem 2 (Ray [2]).
{u} for some u in X,
Sx}, FT
Sx
{x
TT
il)
U
Ill)
2
1
TiT
i
X:,
x
Tx}
Tx}.
Let T be a continuous mapping and T
two mappings of a complete metric space
i)
E
and T
2
be any other
(X,d) into itself such that
1,2,
T i(X) c_ T(X), and
at least one of the following is satisfied for every pair of points x,y in X:
MAIBAM RANJIT SINGH
780
ad(Ty, T2Y)d(Tx,
+ d(Tx, Ty)
d(Tlx’ T2Y)
a, 13, a + 13
where 0
<
d(TlX T2Y)
TlX)
+ 13 d(Tx,Ty),
I,
X max (d(Tx,Ty),
I/2[d(Tx,TlX)+d(Ty,T2Y)]
I/2 [d(Tx,T2Y)+d(Ty,TIX)
<
where 0
(1.2)
1,
d(Tlx,T2Y)
max {d(Tx,Ty),
d(Tx,TlY) d(Ty,T2Y),
d(Tx,T2Y), d(Ty,TIX)}
<
where 0
1/2,
d(TlX,T2Y)
max
{IKId(Tx,Ty) K2d(Tx,TIX) I,
[KId(Tx,Ty
where -I
Then
<
K
<
K
2
FT, TI,T 2
2.
2
+
<
<
2, K
I.
is non-empty, where
FT,TI,T 2
Furthermore,
K
K2d(Ty,T2Y )]}
{x
FTI FT2
X: x
Tx
FT,T IT2
TlX T2x}
{u}, for some u in X.
MAIN RESULTS.
Now we give our result.
Let (X,d) be a complete metric space. Let T,TI,T2: X X satisfy
(I), (li) of Theorem 2 and (i) let the following conditions hold for every pair of
points x,y in X:
THEOREM 2.1.
d(TITX,T2TY)
{d(x,TiTx) d(y,T2TY), d(y,TITx),d(x,T2TY),
a[ l+d(y,T2TY) ]d(x,TITX)
[d(x’TITx)+d(y’T2TY)]’
-I:-d(x,y-)-
I max
+
13[d(x,TiTx)+d(y,T2TY)]+ v[d(y,TiTx)+d(x,T2TY)
+
, d(x,y)],
IZld(X,y)-Z2d(x,T1Tx)l,
K d (x, y)-K2d (y ,T
2TY
REMARKS ON A FIXED-POINT THEOREM OF GERALD JUNGCK
I,
0
where
(S + v + )
l-u(a + S + ,)
0
Then
6
)
I, -I
<
K
<
<
K
2
<
+ 6
+
+
0,
+ uK
2
<
<
I, 2 +
2, K
<
I,
I.
is non-empty, where
FT,TIT2
FT,TI,T
Furthermore, F
PROOF
,,,
781
F
FT2
T
Let x
E
O
T2X2n_l,
X2n
TlX T2x}
{u}, for some u in X.
T’TI ’T2
X, define
TlX2n
X2n+l
Tx
{x e X: x
2
n
O, 1,2...
n
1,2,3...
Then, using Theorem 2. l, (i), we have
d(X2n+l X2n)
where K
max
{,
-I-’
X
d(X2n X2n_l)
(S + v + )
r}
l-(a + B "+ )’
K
K2}’ KI > 0,
-K
{K
Z2,
I_---K--’--}, K < 0.
max [K
whe re
max
I- K2,
{xn} is a Cauchy sequence.
u as
x
n/
+ t
X such that
Since X is complete there exists u
(R).
Now,
d(TiTu, X2n)
d(TITU,T2Tx2n_l).
Then using Theorem 2.1 (i) and allowing n
we have u
TITU.
d(X2n+l,T2Tu)
such that
Hence u
TITU TTIU using Theorem 2
d(TiTX2n,T2Tu).
u etc,
u, X2n+1
X2n
-such that
which yields Tu
T2u.
Hence, u
we have u
T2Tu. Hence
TIT and T2T.
T1u
T2u
u
T2Tu TT2u.
From Theorem 2.1 (1), it is
Tu we have
d(TITTu,T2Tu
u using Theorem 2.1 (i).
Tu
u etc,
Again using Theorem 2 (i) and allowing n
Now, let v denote any common fixed point of
v since 2 + 6 < I. For proving u
easy to see that u
d(Tu,u)ffid(TT1Tu,T2Tu
(i).
u, X2n_l
Further,
X2n
Hence u
which shows that
TITU TlU.
FT,FT ,FT 2 are
Similarly, u
non-empty.
T2Tu
Then we
MAIBAM RANJIT SINGH
782
can see that F
Let X
EXAMPLE.
x
Here
l,
,
x
TIX
T,TI,T2,
O. Take x
F
T
R,
FT2 [0,I]T’TIT2
{u} for some u in X.
with Euclidean metric d. Let Tx
0
.
x
<
l,
TIX .--,
1,
x
are all dtscount[uuous at x
y-
This completes the proof.
T2x
x
-,
0
x, 0
x
<
x
1,
<
I, Tx
T2x =-t-"
=I/2,
x
l.
and have a unique common fixed point x
Obviously all the conditions (t), (it) of Theorem 2 and (i) of
Hence the result.
(1) Contractive Definition 20 of Rhoades [3] is a special case of
o Ctrct [4], Theorem of Pal and Haiti
(2) Theorem
condition (i) of Theorem 2.1.
[5], and Theorem 4 of Sharma and Yuel [6] are special cases of Theorem 2.1.
Theorem 2.
hold true.
REHARKS.
My sincere thanks are due to Professor A.K. ChatterJee for his
guidance in the presentation of this paper and also to the anonymous referee for his
valuable comments tn the mprovement o the same.
ACKNOWLEDGEMENT.
REFERENCES
I.
JUNGCK, G., Commuting Mappings and Fixed Points, Amer. Math.
Monthl
83 (1976),
26t-263.
2.
RAY, B.K., Remarks on a Flxed-polnt Theorem of Gerald Jungck, Jour.
(Science) 12(2) (1985), 169-171.
3.
RHOADES,
4.
Cirlc, L.B., A Generalization of
Math. Soc. 45 (1974), 267-273.
5.
PAL, T.K. and MAITI, M., Extension of Fixed-polnt Theorems of Rhoades and Cirlc,
Proc. Amer. Math. Soc. 64 (1977), 283-286.
6.
SHARMA, P.L. and YUEL, A.K., A Unique Fixed Point Theorem in Metric Space, Bull.
Cal. Math. Soc. 76 (1984), 153-156.
B.E., A Comparison of Various Definitions
Trans. Amer. Math. Soc. 226 (1977), 257-290.
of
Univ. Kuwait
Contractive
Banach’s Contraction Principle,
Mappings
Proc.
Amer.