 Open Access
 Total Downloads : 208
 Authors : Swatmaram
 Paper ID : IJERTV2IS70323
 Volume & Issue : Volume 02, Issue 07 (July 2013)
 Published (First Online): 16072013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
An Extension of Nesic’s Result for Weakly Compatible Maps
Swatmaram
Department of Mathematics, Chaitanya Bharati Institute of Technology,
Hyderabad500075, Andhra Pradesh State, India
Abstract. A result of Nesic is extended to two pairs of selfmaps through the notions of weak compatibility and
orbital completeness of the metric space.
2000 AMS mathematics subject classification: 54 H 25
Key Words: Weakly compatible selfmaps, orbit and common fixed point

Introduction
In 2003, Nesic [2] proved the following Theorem:
Theorem 1: Let f and g be selfmaps on a metric space satisfying the general inequality
1 adx, y d fx, gy adx, fxdy, gy dx, gy, dy, fx
b maxdx, y, dx, fx, dy, gy, 1 dx, gy dy, fx
2
2
where a 0 and
0 b 1.
for all
x, y X , (1)

If there is a subsequence of the associated sequence <xn> at x0 converging to some z x , where
x2n1
fx2n2
(2)
x2n gx2n1
then f and g have a unique common fixed point.
In this paper we extend theorem1 to two pairs of weakly compatible maps [1] using the notion orbital completeness of the metric space.


Preliminaries:
In this paper (X, d) denotes a metric space and f and g selfmaps on it.
Given a pair of selfmaps S and T on X, an (f, g) orbit at x0 relative to (S, T) is defined by
y2n1 fx2n2 Sx2n1
(3)
y2n gx2n1 Tx2n, n 1,2,3,….
n
n
provided the sequence y
n1
exists [3].
Remark 1: If
S T I X
the identify map on X we get (2) from (3) as a particular case.
Remark 2: Let
f x gx and
gxTx
(4)
and
x0 X. Then by induction on n the (f,g) orbit at xo w.r.t. (S, T) with choice (3) can be defined.
Definition 1: The space X is (f,g) orbitally complete w.r.t. the pair (S,T) at xo if every Cauchy sequence in the orbit (3) converges in x.
Remark 3: If
S T IX
then condition (i) of Theorem 1 follows from orbital completeness.
Definition 2: A point z X is a coincidence point of selfmaps f and T if
fz Tz , while z is a common
coincidence point for pairs (f,T) and (S,g) if
fz gz Sz Tz. .
Definition 3: Selfmaps f and T are said to be weakly compatible [1] if they commute at their coincidence point.
Our Main Result is
Theorem 2: Let f, g, S and T be selfmaps on X satisfying the inclusions (4) and the inequality
1 adTx, Sy d fx, gy adTx, fxdSy, gy dTx, gy, dSy, fx

b maxdTx, Sy, dTx, fx, dSy, gy, 1 dTx, gy dSy, fx

for all
2
x, y X ,
(5)
where the constants a and b have the same choice as in Theorem 1.

Given
x0 X , suppose that X is (f,g) orbitally complete w.r.t. (S,T) at x0.

S and T are onto and

(g,S) and (f,T) are weakly compatible.
Then the four selfmaps will have a common coincidence point, which will also be a unique common fixed point for them.
Proof: Let x0 X . By Remark 2, the (f,g) orbit can be described as in (3).
Write tn dyn, yn1 for n 1. Taking
x x2n2, y x2n1 in the inequality (5) and using (3),
1 adTx2n2, Sx2n1 d fx2n2, gx2n1 a[dTx2n2, fx2n2 dSx2n1, gx2n1
dTx2n2, gx2n1, dSx2n1, fx2n2 ]
bmax{dTx2n2, Sx2n1,dTx2n2, fx2n2 ,dSx2n1, gx2n1,
1 dTx
, gx
dSx , fx
},
2 2n2
2n1
2n1
2n2
1 ady2n2, y2n1 dy2n2, y2n a[dy2n2, y2n1dy2n1, y2n
dy2n2, y2n ,dy2n1, y2n1]
b max{dy2n2, y2n1, dy2n2, y2n1, dy2n1, y2n ,
1 dy , y dy , y
}
2 2n2 2n
2n1
2n1
t2n1 b maxt2n2,t2n1. (6)
Similarly taking x x2n2, y x2n3 in (5) and using (3) and preceding as above we get
t2n2 b maxt2n3, t2n2. (7)
From (6) and (7), we see that
tn b maxtn1, tn for all n 2.
(8)
If maxtn1,tn tn , from (8), tn btn tn a contradiction, and maxtn1tn1
tn 0tn1 0
Vol. 2 Issue 7, July – 2013
Therefore, yn1 yn yn1 and the inequality (8) holds good.
We take maxtn1, tn tn1
So that from (8),
tn btn1
for all n.
for all n 2.
(9)
Repeated application of (9) gives
tn bn1 t1
for all n 2.
(10)
Now for m > n, by triangle inequality and (10),
dym, yn dym, ym1 dym, ym2 dyn1, yn (mn terms)
= tm1 tm2 …. tn bm1 bm2 …. bn1t1
= bn1t 1 b b2 …. bmn bn1t 1 b b2 ….= bn1t1
for all n 1
1 1 1 b
Applying the limit as m, n this gives d(y , y ) , since lim bn1 0 as 0 b 1.
n m n
n
n
Hence yn 1 is Cauchy sequence in the orbit (3). By orbital completeness of X,
n
n
lim y z
n
for some
z X . That is
lim y2n1 lim fx2n2 lim Sx2n1 z
(11)
n
n
n
and
lim y2n lim gx2n1 limTx2n z . (12)
n
n
n
Since S and T are onto,
z Su
and
z Tv
for some u, vX
we prove that
Su gu and Tu
fv .
Put
x x2n2, y u in the inequality (5)
1 adTx2n2, Su d fx2n2, gu a[dTx2n2, fx2n2 dSu, gu dtx2n2, gu, dSu, fx2n2 ]

b max{dTx2n2, Su, dTx2n2, fx2n2 , dSu, gu,
1 dTx , g ds , fx
}
As n , this implies
2 2n2 u
u 2n2
[1 adz, zdz, gu a[d(z, z) d d(z, gu) d(z, gu) d(su, z)] b max{d(z, z), d(z, z), d(z, gu)1 [d(z, gu) d(z, z)]} 2
so that
dz, gub dz, gu or z gu .Thus Su gu z . This and weak compatibility of g and S implies
that
Sgu gsu or Sz gz .
On the other hand, taking x v and y x2n1 in (5)
1 adTv, Sx2n1 d fv, gx2n1 a[dTv, fv)d(Sx2n1, gx2n1 dTv, gx2n1dSx2n1, fv]

b max{dTv, Sx2n1, dTv, fv, dSx2n1, gx2n1,
1 dTv, gx
dSx , fv }
Applying lim as n
2 2n1
2n1
1 adTv, z d fv, z a[dTv, fv)d(z, z) d(Tv, z)d(z, fv]
So that d fv, zbdz, fv or
fTv Tfv fz Tz .

b max{dTv, z, dTv, fv, dz, z,
1 d Tv, z d z, fv } 2
fv z Tv . By weak compatibility of (f, T) we get
Again taking x y z in (5)
1 adTv, Sz d fz, gz a[dTz, fz)d(Sz, gz) d(Tz, gz)d(Sz,Tz)]

b max{dTz, Sz, dTz, fz, dSz, gz,

2
2
1 dTz, gz dSz, fz }
So that 1 ad fz, gzd fz, gz a0 d fz, gzdgz, fz
b max fz, gz ,0,0, 1 dfz, gz dfz, gz
d
2
Or d fz, gzbd fz, gz) fz gz .
Thus fz gz Tz Sz , that is z is a common coincidence point of f, g, T and S.
Finally writing x x2n, y z in (5),
.
1 adTx2n, Sz d fx2n, gz a[dTx2n, fx2n dSz, gz dTx2n, gz, dSz, fx2n ]
b max{dTx2n, Sz, dTx2n, fx2 , dSz, gz,
1 dTx , gz dSz, fx
}.
Appling limit as
n , this gives
2 2n 2n
1 adz, gz dz, gz a[dz, zdgz, gz dz, gz, dgz, z
2
2

b max{dz, gz, dz, z, dgz, gz, 1 dz, gz dgz, z
Or d(z, gz) bdz, gz gz z . Hence fz gz Tz Sz z .
Thus z is a common fixed point of f, g, T and S.
Uniqueness: Let z, z be two common fixed points taking x z, y z' in (5),
1 adTz, Sz' d fz, gz' a[dTz, fzdSz', gz' dTz, gz', dSz', fz]
2
2

b max{dTz, Sz', dTz, fz, dSz', gz', 1 dTz, gz' dSz', fz }
So that d(z, z') bz, z'or z z' . Hence the common fixed point is unique.
Remark 4: It is well known that identity map commutes with every self map and hence (f, T) = (f, I) and (g, S) = (g, I) are weakly compatible pairs. Also I is onto.
In view of Remarks 1, 2 and 3, a common fixed point of f and g is ensured by Theorem 2.
Thus Theorem 2 extends Theorem1 significantly.
References:

Gerald Jungck and Rhoades, B.E., Fixed point for setvalued functions with out continuity, Indian J. pure appl. Math. 29 (3) (1998), 227238.

S.C Nesic , Common fixed point theorems in metric spaces. Bull. math. Soc. Sci. Math. Roumanie 46(94)
(2003), 149155.

T. Phaneendra, Coincidence points of two weakly compatible self maps and common fixe point Theorem through orbits, Ind. J .Math. 46 (23) (2004), 173180.