An Extension of Nesic’s Result for Weakly Compatible Maps

DOI : 10.17577/IJERTV2IS70323

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An Extension of Nesic’s Result for Weakly Compatible Maps


Department of Mathematics, Chaitanya Bharati Institute of Technology,

Hyderabad-500075, Andhra Pradesh State, India

Abstract. A result of Nesic is extended to two pairs of self-maps 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 self-maps, orbit and common fixed point

  1. Introduction

    In 2003, Nesic [2] proved the following Theorem:

    Theorem 1: Let f and g be self-maps 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



    where a 0 and

    0 b 1.

    for all

    x, y X , (1)

    1. If there is a subsequence of the associated sequence <xn> at x0 converging to some z x , where




      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.

  2. Preliminaries:

    In this paper (X, d) denotes a metric space and f and g self-maps on it.

    Given a pair of self-maps S and T on X, an (f, g) orbit at x0 relative to (S, T) is defined by

    y2n1 fx2n2 Sx2n1


    y2n gx2n1 Tx2n, n 1,2,3,….



    provided the sequence y


    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




    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 self-maps 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: Self-maps 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 self-maps 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


x, y X ,


where the constants a and b have the same choice as in Theorem 1.

  1. Given

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

  2. S and T are onto and

  3. (g,S) and (f,T) are weakly compatible.

Then the four self-maps 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




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



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.


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.


Repeated application of (9) gives

tn bn1 t1

for all n 2.


Now for m > n, by triangle inequality and (10),

dym, yn dym, ym1 dym, ym2 dyn1, yn (m-n 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



Hence yn 1 is Cauchy sequence in the orbit (3). By orbital completeness of X,



lim y z


for some

z X . That is

lim y2n1 lim fx2n2 lim Sx2n1 z






lim y2n lim gx2n1 limTx2n z . (12)




Since S and T are onto,

z Su


z Tv

for some u, vX

we prove that

Su gu and Tu

fv .


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


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


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,



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



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



  • 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]



  • 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.


  1. Gerald Jungck and Rhoades, B.E., Fixed point for set-valued functions with out continuity, Indian J. pure appl. Math. 29 (3) (1998), 227-238.

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

    (2003), 149-155.

  3. T. Phaneendra, Coincidence points of two weakly compatible self maps and common fixe point Theorem through orbits, Ind. J .Math. 46 (2-3) (2004), 173-180.

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