An Extension of Nesic’s Result for Weakly Compatible Maps

An Extension of Nesic’s Result for Weakly Compatible Maps

Swatmaram

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

   2

   2

   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

      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.

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

   (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 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

2

x, y X ,

(5)

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

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 (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

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:

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.