Internat. J. Math. & Math. Sci.
VOL. 16 NO. 4 (1993) 669-674
669
A COMMON FIXED POINT THEOREM OF MEIR AND KEELER TYPE
Y.J. CHO
Department of Mathematics
Gyeongsang National University
Jinju 660-701, KOREA
P.P. MURTHY
G. JUNGCK
Balajee Guest House
Street-37, Sector-5
Bhilai (M.P.) 490 006, INDIA
Department of Mathematics
Bradley University
Peoria, Illinois 61625, U.S.A.
(Received June 10, 1992 and in revised form September 22, 1992))
ABSTRACT. In this paper, we introduce the concept of compatible mappings of type (A) on a
metric space, which is equivalent to the concept of compatible mappings under some conditions,
and give a common fixed point theorem of Meir and Keeler type. Our result extends, generalized
and improves some results of Meir-Keeler, Park-Bae, Park-Rhoades, Pant and Rao-Rao, etc.
1991 AMS SUBJECT CLASSIFICATION CODE. 54H25.
KEY WORDS AND PHRASES. Common fixed points, compatible mappings of type (A),
generalized (e, 6) {S, T}-contractions, and {S, T}-iterations.
INTRODUCTION.
In [6], Jungck proved a common fixed point theorem of commuting mappings on a metric
space. Since then, he and many authors extended, generalized and unified this theorem in many
ways ([2], [4], [5], [7]-[10], [15]-[20], [22], [23]). For example, Sessa ([22])introduced the concept
1.
of weakly commuting mappings, which is a generalization of the concept of commuting mappings,
and he and others proved some fixed point theorems for weakly commuting mappings ([20]-[23]).
Recently, Jungck ([8]) proposed a generalization of the concept of weakly commuting
mappings, which is called compatible mappings, and he generalized some fixed point theorems of
Meir-Keeler type, especially, a theorem of Park-Bae ([16]), and in [11], Jungck, Murthy and Cho
introduced the concept of compatible mappings of type (A) on metric spaces and obtained some
fixed point theorems for these mappings.
On the other hand, in [14], Meir and Keeler established a fixed point theorem for a selfmapping f of a metric space (X,d) satisfying the following condition:
For every > 0, there exists a 6 > 0 such that
< d(z,V) < +6 implies d(fz, fv) <
(1.1)
.
In [13], Maiti and Pal also proved a fixed point theorem for a self-mapping
spae (X,d) satisfying the following condition, which is a generalization of (1.1):
For every e > 0, there exists a 6 > 0 such that
e <_ rnaz{d(z,v),d(z, fz),d(y, fv)} < e +6 implies d(fz, fv) <
.
In [17] and [18], Park-Rhoades and RaRao proved
I of
a metric
(1.2)
some fixed point theorems for selfmappings f and g of a metric space (X,d) satisfying the following condition, respectively, which is
a generalization of (1.2):
For every e > 0, there exists a 6 > 0 such that
<_ maz{d(fz, fy),d(fz, gz),d(fy, gy),d(fz, gy)+ d(fy, gx))} < + 6 implies d(gz, gy) < e
(1.3)
Y.J. CHO, P.P. MURTHY AND G. JUNGCK
670
Many other fixed point theorems of Meir-Keeler type
[19]
and
are given in
[1], [3], IS], [12], [15], [16],
[21].
In this paper,
we introduce the concept of compatible mappings of type (A), which is
equivalent to the concept of compatible mappings under some conditions, and give a common
fixed point theorem for compatible mappings of type (A). which extends, generalizes and
improves some common fixed point theorems of Meir-Keeler type.
2.
COMPATIBLE MAPPINGS OF TYPE (A).
In this section, we show that two pairs of compatible mappings and compatible mappings of
type (A) axe equivalent under some conditions and give several properties of compatible mappings
of type (A) for our main results. Throughout this paper, (X,d) denotes a metric spce.
DEFINITION 2.1. Let S,T:(X,d)-.(X,d) be mappings. $ and T axe said to be compatible if
imood(ST(n),TS(zn)
0
for some in X.
whenever {z,} is a sequence in X such that ImooS(z,) T(z,,)
DEFINITION 2.2. Let S,T:(X,d)---,(X,d) be mappings. S and T axe said to be compatible of
type (A) if
lnimood(TS(zn),SS(zn) 0 and ldrnood(ST(zn), rT(zn) 0
for some in X.
whenever {z,} is a sequence in X such that id._mooS(z,, IooT(z,,
’In [11], the following propositions show that Definitions 2.1 and 2.2 are equivalent under
some conditions:
PROPOSITION 2.1. Let S,T X, d)-.( X, d) be continuous mappings. If S and T axe
compatible, then they axe compatible of type (A).
PROPOSITION 2.2. Let $,T (X,d)-,(X,d) be compatible mappings of type (A). If one of S
and T is continuous, then $ and T axe compatible.
The following is a direct consequence of Propositions 2.1 and 2.2:
PROPOSITION 2.3. Let $, T: X, d)-,( X, d) be continuous mappings. Then S and T axe
compatible if and only if they axe compatible of type (A)
The following examples show that Proposition 2.3 is not true if S and T axe discontinuous in
some point of X.
EXAMPLE 2.1. Let X R, the set of real numbers, with the usual metric d(z,y)=
Define S, T: (X, d)-(X, d) as follows:
$(z)
[r if
r
# 0,
T(z)={z--2
and
if z=0
Then S and T axe not continuous at
n2,n 1,2,--.. Then we have, as n-oo,
=0.
so:.)
and
T(tn
if z’#0,
if t=0.
Consider a sequence {r,,} in
0,
1
O,
n4
Id.rnd(ST(xn), TS(xn)
i.rnood(n4, n4)
but
md(ST(zn)’TT(zn)) mood(ns’n)
moo
0
ns- n41
oo
and
Irnd(SS(zn),TS(zn) =/n/mcd(n:, n4) -/n/moo In
n4l
co.
x
defined by
COMMON FIXED POINT THEOREM OF MEIR AND KEELER TYPE
Therefore, S and T
671
compatible of type (A).
X =[0,1] with the usual metric d(x,u)= Ix-yl-
are compatible but are not
EXAMPLE 2.2. Let
Define $, T: [0,1 ]--.[0,1 by
S(z)
{
if
if z
[0,1/2),
[!2’ 1]
and
T(z)
{:-r
if[0,!)
[1/2,1].
if z
Then S and T axe not continuous at t= 1/2. Now, we assert that S and T axe not compatible but
axe compatible of type (A). To see this, suppose that {r,} _c [0,1] and that T(z,,),S(r,,)--.t. By
definition of S and T,
{-,2 1}. Since $ and T agree on [1/2,1], we need only consider 1/2. So we
and that r, < 1/2 for all n. Then T(r,)= l-z,,-- from the right and
can suppose that
from the left. Thus, since r, > for all n
s(r,)
,
r,,--
r,--
ST(xn)
and, since z, < 1/2,
TS(zn) T(zn)
Consequently,
but
S(1 xn)
r,t--"
d( ST( zn), TS( zn) )-’
d(ST(z,,),TT(r.n)= IST(r.)-TT(:,.,)I
I1-T(1-r.)l
I1- I--,0
and
d(TS(rn),SS(zn)
ITS(z,,)- SS(:,,)
2:,, I-’0
I(1 :,,)- r,,
r,,-.
Therefore, S and T are compatible mappings of type (A) but are not compatible.
Next, we give several properties of compatible of type (A) for our main theorems ([11]):
PROPOSITION 2.4. Let S, T: X, d)( X, d) be mappings. If S and T are compatible of type
and
X, then ST(t) TT(t) TS(t) SS(t).
S(t) T(t) for some
(A)
PROPOSITION 2.5. Let S, T: X, d)--,( X, d) be mappings. Let $ and T be compatible of type
(A) and let S(r,,), T(r,,)-,t for some e X. Then we have the following:
as
(1) /,,imooTS(r,, S(t) if S is continuous at t.
(2) ST(t)= TS(t) and S(t)= T(t) if S and T axe continuous at
t.
PROOF. Immediate, from Proposition 2.2 and Proposition 2.2 (2) of [81
3. A COMMON FIXED POINT THEOREM.
Before stating and proving our main theorem, we give some definitions and lemmas:
DEFINITION 3.1 ([8]). Let A,B,S and T be mappings of a metric space (X,d) into itself
such that A(X) T(X) and B(X) C $(X). For o X, any sequence {v,,} defined by
Y2n
Tz2n
Az2n 2’
Y2n SZ2n BZ2n-
1
(3.1)
{S,T}-iteration of z under A and B.
Note that Definition 3.1 assures us that {S,T}-iterations will exist since A(X)CT(X) and
B(X) C S(X), although the sequence {Yn} certainly need not be unique.
DEFINITION 3.2. Let A,B,S and T be mappings of a metric space (X,d) into itself. The
pair {A,B} is called a generalized (,$)- {S,T}-contraction if
(3.2)
A(X) C T(X) and B(X) C S(X),
for
n
1,2,..-, is called an
there exists
,
&(0,oo)-(0,oo) such that, for any e > 0 and (e) <
<_ maz{d(Sr, Ty),d(Sz, Az),d(Ty, By),(d(Sr, By) + d(Ty, Az))} < ()
a function
(3.3)
672
Y.J. CHO, P.P. MURTHY AND G. JUNGCK
implies d(Az, By) < for all z,y X.
For our main theorem, first we give the following:
LEMMA 3.1. Let S and T be mappings of a metric space (X,d) into itself and the pair {A,B}
be a generalized (,6)-{S,T}-contraction. If Zo X and {y.} is an {$,T}-iteration of Zo under A and
B, then we have the following:
(1) for every > 0, < d(y,,yq) < 6(e) implies d(y,+ l,Yq+ i) < where and q are of opposite
parity.
,
,
(2) _m(R)d(n,y. + ) O.
(3) {.} is a Cauchy sequence in X.
PROOF. (1) Since the pair {A,B} is a generalized (,)-{S,T}-contraction, for every > 0,
<_ maz{d(Sz, Ty),d(Sz, Az),d(Ty, By),(d(Sz, By)+d(Ty, Az))} < 6(f)
.
implies d(Az, By) < for all z,y X.
Suppose that < d(yp, y) < 6(e). Putting p 2n and q
have
2m-
in the above inequality, we
d(yp+ 1,Yq+ 1) d(Y2. + 1,Y2m) d(A::tn, Bz2m- 1)
and
<_ d(yp, yq) d(Yan, Y2m_ l) d(Sz2n’Tzam- 1)
<- maz(d{Sz2a, Tz2m- 1)’d(Sz2n’Az2n)’d(Tz2m 1’ Bz2m l)’(d(Sz2n’Bz2m- 1) + d(Tz:m- 1’ Az2-))},
< 6(),
which implies that
d(y,+ 1,Yq+ l) d(Az2n, Bz:tm-l) < "
(2) For Zo
d(u.,
.
+
X, by
(3.3), we have
d( Az., Bz2,,
< maz{d(Sz2n, Tza.- )’d(Sz2n’Az2n)’d(Tz2.- l’Bz2.
l)’d(Sz2.’Bzan- 1) + d(Tzan- , Aza.))}
{d(n, yah l),d(y2n, yn + 1), d(y. 1,yn),d(y2n, y2.) + d(y2._ I’Y2. + 1))}
d(2.
1,
Yah)"
Sillily, we have d(u. + l, Ua. + ) < d(ua., a. + ).
Thus the uen {d(y.,y.+ 1)} is non-increing d nverg to the eatt lower d =0
of its rge rE0.
In ft, other, (1) impH that d(y+,+)<t whenev
Sd(y,u+)<6(t). But since {d(y,y+)} converg to t, there ests a t such that
d(t, ut+ )< 6(t) d
d(yt+ l,Yk+2) < t, which contracts the desiation of t. Therefore, we
have d(y., y. + ) 0.
The prf of (3) follows om the linm of the prf of mma 3.1 (c) ([8]). Ts complet
the prof.
Now we e ready to prove o mn threm:
THEOM 3.2. t A,B,S d T mappings of a complete metc spe (X,d) into it
satisng the conditions (3.2),
(3.4) one of A,B,S, d T is continuous,
(3.5) the prs A,S d B,T e compatible of ty (A) on X,
COMMON FIXED POINT THEOREM OF MEIR AND KEELER TYPE
(3.6) the pair {A,B}
673
is a generalized (e,6)-{S,T}-contraction such that
$
is lower semi-
continuous.
Then A, B,S and T have a unique common fixed point in X.
PROOF. By Lemma 3.1 (3), the {S,T}-iteration of
x under A and B, {y,}, is a Cauchy
is complete, {y,} converges to a point in X. Since
{Sr,,} and {Tr2,_ t} are subsequences of {y,}, they also converge to z.
Suppose that S is continuous. Then we have SS:2,,SAz,Sz as noo. Since A and S are
compatible of type (A), by Proposition 2.5 (1), ASz,-,Sz as n-.oo. Now, we claim that Sz z.
sequence in
x. Since (X,d)
If
Suppose Sz :/: z and let M(x,V) ma{d(Sz, Tv),d(Sz, Ar),d(Tv, Bv),(d(Sx, Bv)+d(Tv, A:))}.
M,= M(Sz,,:2,_,), then we have M,,--..d(Sz, z)# 0 as n---,oo. Let e d(Sz, z) and remember that
() > by definition. Since :(0,o)-.(0,o0) is lower semi-continuous, there exists an a (0,) such
that (t)>e for t(-a,+a). Choose to(-a, ). Then we have O<to<<,(to). But since
M,--, as n--,o, there exists an integer n such that M, e. (to,,(to)) for n >_ no. Therefore, by (3.3),
we
have
d(ASz2,,Bz2,,_ ) <
But d(ASz2n, Bz2n_ )d(Sz, z)
< e for n > no.
as n-o and so we have d(Sz, z)
<
<
, which is
a contradiction.
Tus we have Sz z.
We also claim Az z. Suppose not and let d(Az, z)= and M,[= M(z, z2,_ ). Then we have
Mn’’as n---,o. Now duplicate the argument using the lower semi-continuity of 6 to produce the
contradiction d(Az, z) < ’. Thus we obtain Sz Az z. Since A(X) C T(X), there exists a point
"
w
X such that z
d( z, Bw)
Sz
Az
Tw.
Further,
we claim
that Bw
z.
If Bw
z,
then we have
d( Az, Bw)
< maz{d(Sz, Tw),d(Sz, Az),d(Tw, Bw),1/2(d(Sz, Bw) + d(Tw, Az))}
d(z, Bw),
which is a contradiction. Hence Bw z Tw. Since B and T are compatible of type (A), by
Proposition 2.4, Bz BTw TTw Tz, that is, Bz Tz.
Finally, we shall prove that Bz z. If Bz z, then we have
d(z, Bz) d(Az, Bz)
< maz{d(Sz, Tz),d(Sz, Az),d(Tz, Bz),d(Sz, Bz) +d(Tz, Az))}
=d(z, Bz),
which is a contradiction and so Bz z. Thus, z is a common fixed
uniqueness of the common fixed point z follows easily from (3.6).
Similarly, we can also complete the proof when A or B or T is
the proof.
REMARK. Theorem 3.2 extends, generalizes and improves
Jungck [8], Maiti-Pal [13], Meir-Keeler [14], Pant [15], Park-Bae [16],
[lS], Rhoades [19], etc.
point of A,B,S and T. The
continuous. This completes
Chung [3],
[17], aao-Rao
some results of
Park-Rhoades
REFERENCES
1.
2.
ASSAD, N.A., A fixed point theorem for weakly uniformly strict contractions, Canad. Math.
Bull. 16 (1) (1973), 15-18.
CHANG, S.S., A common fixed point for commuting mappings, Proc. Amer. Math. Soc. 83
(1981), 645-652.
Y.J. CHO, P.P. MURTHY AND G. JUNGCK
674
3.
CHUNG, K.J., On fixed point theorems of Meir and KePler, Math. Japon. 24 (4) (1978),
381-383.
4.
5.
6.
DAS, K.M. & NAIK, K.V., Common fixed point theorems for commuting maps on metric
spaces, Proc. Amer. Math. Soc. 77 (1979), 369-373.
FISHER, B., Common fixed points of commuting mappings, Bull. Inst. Math. Acad. Sinica
9 1981 ), 399-406.
JUNGCK, G., Commuting mappings and fixed points, Amer. Math. Monthly 83 (1976),
261-263.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
JUNGCK, G., Periodic and fixed points, and commuting mappings, Proc. Amer. Math. Soc.
76 (1979), 333-338.
JUNGCK, G., Compatible mappings and common fixed points, Internat. J. Math. & Math.
Sci. 9 (4) (1986), 771-779.
JUNGCK, G., Compatible mappings and common fixed points (2), Internat. J. Math. &
Math. Sci. 11 (1988), 285-288.
JUNGCK, G., Common fixed points of commuting and compatible maps on compacta, Pro(:.
Amer. Math. Soc. 103 (1988), 977-983.
JUNGCK, G.; MURTHY, P.P. & CHO, Y.J., Compatible mappings of type (A), to appear
in Math. Japon.
KHAN, M.A.; KHAN, M.S. & SESSA, S., On a fixed point theorem of Rhoades in
metrically convex spaces, Math. Japon. 31 (1) (1986), 51-55.
MAITI, M. & PAL, T.K., Generalizations of two fixed point theorems, Bull. Calcutta Math.
Soc. 70 (1978), 57-61.
MEIR, A. & KEELER, E., A theorem on contraction mappings, J. Math. Anal. Appl. 28
PANT, R.P., Common fixed points of two pairs of commuting mappings, Indian J. Pure
Appl. Math. 17 (2) (1986), 187-192.
PARK, S. & BAE, J.S., Extensions of a fixed point theorem of Meir and KePler,
Ark. Mat. 19 (1981), 223-228.
PARK, S. & RHOADES, B.E., Meir-Keeler type contractive conditions, Math. Japon. 26 (1)
(1981), 13-20.
RAO, I.H.N. & RAO, K.P.R., Generalizations of fixed point theorems of Meir and KePler
type, Indian J. Pure hppl. Math. 16 (1) (1985), 1249-1262.
RHOADES, B.E., A Meir and Kepler type fixed point theorem for three mappings,
Jnanabha 14 (1984), 17-21.
RHOADES, B.E. & SESSA, S., Common fixed point theorems for three mappings under a
weak commutativity condition, Indian J. Pure Appl. Math. 17 (1986), 47-57.
RHOADES, B.E.; PARK, S. & MOON, K.B., On generalizations of the Meir-Keeler type
contraction maps, J. Math. Anal. Appl. 146 (1990), 482-494.
SESSA, S., On a weak commutativity condition of mappings in fixed point considerations,
Publ. Inst. Math. 32 (46) (1982), 149-153.
SINGH, S.L.; HA, K.S. & CHO, Y.J., Coincidence and fixed points of nonlinear Hybrid
contractions, Internat. J. Math. & Math. Sci. 12 (1989), 147-156.