Associated Sequence And A Generalized Common Fixed Point Theorem Under A New Condition

Associated Sequence And A Generalized Common Fixed Point Theorem Under A New Condition

Neena B. Gupta

Department of Mathematics, Career College, Bhopal, M.P., India

ABSTRACT :

The purpose of this paper is to prove a generalized common fixed point theorem by changing the condition, used by

V. Srinivas and B.V.B. Reddy [9]. To prove this theorem we use the definition of weakly compatible mapping & associated sequence.

Keywords :

Fixed Point, Self maps, compatible mappings, weakly compatible mappings, Cauchy sequence, associated sequence.

Introduction :

G. Jungck [3] introduced the concept of compatible maps which is weaker than weakly commuting mappings. After words Jungck and Phoades [5] defined weaker class of maps known as weakly compatible maps. Further Srinivas and B.V.B. Reddy [9] used the concept of associated sequence.

1. Definition and Preliminaries :

   1. Compatible Mappings :

      If (X,d) be a metric space. Then two self maps A and B of (X,d) are said to be compatible

      mappings if lim

      n

      d ( ABxn , BAxn

      ) =0, whenever {x }

      is a sequence in X such that

      n n1

      n n1

      lim

      n

      Ax lim Bx u, for some u X .

      n n

      n n

      n

   2. Weakly Compatible

      Let A and B be mappings from a metric space (X,d) into itself. Then A and B are said to be weakly compatible if they commute at their coincident point

      i.e. Ax = Bx , for some x X

      ABx = BAx

      It is clear that every compatible pair is weakly compatible but its converse need not be true.

   3. Cauchy sequence

      A sequence {xn }n1 in a metric space (X,d) is called Cauchy sequence if for given 0 ,

      there corresponds

      d (xm, xn )

      n0 N

      such that for all m, n n0 , we have

   4. Associated sequence

      Suppose A, B, S and T are self maps of a metric space (X,d) satisfying the following condition

      A (X) T(X) and B (X) S(X) (1.1.4.1)

      Then for an arbitrary x X

      such that A x Tx1 and for this point

      x1 , there exists a

      point

      x2 in X such that

      Bx1 Sx2 and so on. Proceeding in similar manner, we can define

      n n1

      n n1

      a sequence {y }

      in X such that

      y2n1 Ax2n Tx2n1 and y2n Bx2n1 Sx2n

      for n 0

      —————- (1.1.4.2)

      Then this sequence is called Associated sequence of x relative to the four self maps A,B,S and T.

   5. In (1998) Brijendra Singh and M.S. Chauhan [1] proved that common fixed point theorem for self maps A,B,S and T in metric space (X,d) by using the condition (1.1.4.1) and

      [d ( Ax, By)]2 k [d ( Ax, Sx) d (By,Ty) d (By, Sx) d ( Ax,Ty)]

      1

      • k2[d ( Ax, Sx) d ( Ax,Ty) (By,Ty) d (By, Sx)]

        (1.1.5.1)

        Where 0 K1 2K2 1, K1, K2 0

   6. In (2012) V. Srinivas and B.V. B Reddy[9] established a generalize common fixed point theorem by using weakly compatible mapping and Associated sequence under the condition (1.1.4.1) and (1.1.4.2).

      Now we generalize the theorem using new condition under weakly compatible mapping and associated sequence.

      Now we prove a lemma which plays an important role in our main theorem.

   7. Lemma: Let A, B, S and T be a self mapping from a complete metric space (X,d) into itself satisfying the following conditions

A(X ) T (X ) and B(X ) S(X ) ——— (1.1.8)

One of A, B, S or T is continuous such that

d ( Ax, Sx). d (By,Ty),

[d ( Ax, By]2 .max d (By, Sx).d ( Ax,Ty), ……………………….(1.1.9)

d ( Ax, Sx).d ( Ax,Ty)

n n1

n n1

Then the associated sequence {y }

relative to four self maps is a Cauchy

sequence in X.

Proof: From conditions (1.1.8) & (1.1.9) and from the definition of associated sequence, we have

[d( y , y )]2 [d( Ax , Bx

)]2 ,

By (1.1.4.2)

2n1 2n 2n 2n1

d ( Ax2n , Sx2n ) d (Bx2n1,Tx2n1 ),

.max d (Bx , Sx ) d ( Ax ,Tx ), ………………………………….. By (1.1.9)

2n1 2n 2n 2n1

d ( Ax , Sx ) d ( Ax ,Tx ),

2n 2n 2n 2n1

d ( y2n1, y2n ) d ( y2n , y2n1 ),

2n

2n

.max d ( y

, y2n

) d ( y

)

)

2n1, y2n1 ),

d ( y

2n1, y2n

) d ( y

2n1, y2n1

d ( y2n1, y2n ) d ( y2n , y2n1 ),

.max 0,

)

)

d ( y

2n1, y2n

) d ( y

2n1, y2n1

[d ( y

, y )]2

d ( y2n1, y2n ).d ( y2n , y2n1 ),

.max 0,

2n1 2n

)

)

d ( y

2n1, y2n

) d ( y

2n1, y2n1

d ( y2n1, y2n1 ) ,

[d ( y

, y )] .max 0,

2n1 2n

( y , y )

2n1 2n1

d ( y

, y )

d ( y2n , y2n1 ) ,

.max 0,

2n1 2n

d ( y , y ) d ( y , y )

2n1 2n

2n 2n1

By triangular Inequality

d( y2n1, y2n1 ) d( y2n1, y2n ) d( y2n , y2n1)

d( y2n1, y2n ) [d( y2n1, y2n ) d( y2n , y2n1)]

d( y2n1, y2n ) d( y2n1, y2n ) d ( y2n , y2n1)

(1) d ( y2n1, y2n ) d( y2n , y2n1)

d ( y

, y )

d ( y , y )

2n1 2n

1

2n 2n1

d ( y2n1, y2n ) d( y2n , y2n1)

Where

1 1

1 0 (1 ) 1

Now d ( y2n , y2n1)

( y2n1, y2n )

Then d ( yn , yn1)

( yn1, yn )

n2 n1

n2 n1

2 d( y , y )

n3 n2

n3 n2

3 d ( y , y )

..

..

0 1

0 1

n d ( y , y )

Now d( yn1, yn2 )

d( yn , yn1)

n1 n

n1 n

2 d( y , y )

n2 n3

n2 n3

2 d( y , y )

..

..

n1 d ( y , y )

0, 1

Similarly we can show

d ( y

, y ) n2 d ( , y )

n2, n3 0 1

d ( y

, y ) n3 d ( y , y )

n3, n4 0 1

..

..

d( yn p1,

, yn p )

n p1 d( y , y ) ,

for everyinteger p 0

0 1

0 1

Now d( yn, , yn p ) d( yn , yn1) d( yn1, , yn2 ) d ( yn2 , yn3 )

…………………………… d (yn p1 , yn1 )

0 1 0 1 0 1 0 1

0 1 0 1 0 1 0 1

n d( y , y ) n1 d( y , y ) n2 d( y , y ) ………….. n p1 d( y , y )

0 1

0 1

[ n n1 n2 …………………. n p1 ]d ( y , y )

0 1

0 1

n[1 2 …………………. n p ]d ( y , y )

1 n 0 ,

as n

d( yn , yn p ) 0,

asn

, for everyinteger p 0

n n1

n n1

This showsthat{ y }

is a Cauchy sequencein X .

X is acompletemetricspace.

n n1

n n1

=>{y }

converges to z X

Theorem :

Let A, B, S and T are self maps of a metric space (X,d) atisfying the condition (1.1.8)&(1.1.9)

And the pairs (A,S) and (B,T) are weakly compatible. Further,

The associated sequence relative to self maps A, B, S and T such that the sequence

Ax0 , Bx1, Ax2 , Bx3 , …………………………….., Ax2n , Bx2n1,………………….Convergsto z X asn .

Then A, B, S and T have a unique common fixed point z in X.

Proof

B (X ) S(X )

X such that z=S ..(1.1.10) We have to prove A = S

Now consider

d ( A , Sv).d (Bx2n1,Tx2n1 ) ,

[d ( A , Bx

)]2 .max d (Bx , Sv).d ( Av,Tx ),

2n1

2n1 2n1

d ( Av, Sv) d ( Av,Tx )

as n

, Bx2n1,

Tx2n1 z

2n1

d ( A , z) d (z, z),

[d ( A , z)]2 .max d (z, z) d ( Av, z),

d ( Av, z) d ( Av, z)

.max{0, 0,[d ( A , z)]2}

[d ( Av, z)]2

(d (Av, z)]2 [d(Av, z)]2 0

(d (Av, z)]2 (1) 0

0 1

0 1 1

d(A , z) 0

A z

A Sv z,

Byusing

(1.1.10)

………………………………..(1.1.11)

The pair (A, S)isweakly Compatible.

ASv SAv

Az Sz By (1.1.11)

…………………………………(1.1.12)

Now

A(X ) T (X )

X suchthat z Tw………………………………..(1.1.13)

Now we prove B T

Now consider

d ( A , Sv) d (B,T),

[d ( A , B)]2 .max d (B, Sv) d ( Av,T), ,

By (1.1.9)

d ( Av, Sv) d ( Av,T)

d (z, z) d (B, z),

[d (z, B)]2 .max d (B, z) d (z, z), , By

(1.1.11) & (1.1.13)

d (z, z) d (z, z)

[d(z,b)]2 0

But square of distance may not be less than zero.

[d(z, B]2 mustbeequalto Zero. we get (d (z, B]2 0

d (z, B] 0

z B………………………………….(1.1.14)

By (1.1.13) & (1.1.14)

B T z…………………………….(1.1.15)

Again since the pair (B,T) is weakly compatible.

Bz

Tz,

By (1.1.15)

…………………………….(1.1.16)

Now Consider

[d(Az, z)]2 (d(Az, B)]2

, By

(1.1.15)

d ( Az, Sz) d (B,T),

.max d (B, Sz) d ( Az,T), ,

By (1.1.9)

d ( Az, Sz) d ( Az,T)

d ( Az, Az) d (Z , Z ) ,

.max d (Z , Az) d ( Az, Z ), ,

d ( Az, Az) d ( Az, z)

. d( Az, z)]2

[d(Az, Az)]2 [d( Az, z)]2

By (1.1.12) & (1.1.15)

[d(Az, z)]2 (1 )

[d(Az, z)]2 0

[d(Az, z) 0

0

1

0 (1)

Az

z ……………………..1.1.17

Az

Sz z ,

By (1.1.12) &

1.1.17

…………………………………………….. (1.1.18)

Again consider

[d(z, Bz)]2 (d(Av, Bz)]2

d ( Av, Sv) d (Bz,Tz),

.max d (Bz, Sv) d ( Av,Tz), ,

By (1.1.19)

d ( Av, Sv) d ( Av,Tz)

d (z, z) d (Bz, Bz),

.max d (Bz, z) d (z, Bz),

d (z, z) d (z, Bz)

[d(z, Bz)]2 .max{0,[d(Bz, z)]2 , 0}

d(z, Bz)]2 (d(z, Bz)]2

(d(z, Bz)]2 (1) 0

Av Sv BwTw z and Bz Tz

(d(z, Bz)]2 0

d(z, Bz) 0

1 0 (1)

Bz z

Bz Tz z ,

By (1.1.16)

……………………………………………….(1.1.19)

By (1.1.18) & (1.1.19)

Az Sz Bz Tz z……………………………………..(1.1.20)

We get z is a common fixed point of A, B, S and T.

Uniqueness of Common fixed point :

Let z1, and z2 are two common fixed point of A, B, S and T. Then by using (1.1.20)

Az1 = Sz1 = Bz1 = Tz1 = z1 . (1.1.21) And also

Az2 = Sz2 = Bz2 = Tz2 = z2 . (1.1.22)

Consider

1 2 1 2

1 2 1 2

(d(z , z )]2 [d(Az , Az )]2

, By (1.1.21) & (1.1.22)

d ( Az1, Sz1 ) d ( Az2 ,Tz2 ) ,

.max d ( Az , Sz ) d ( Az ,Tz ) , ,

By (1.1.9)

2 1 1 2

d ( Az , Sz ) d ( Az ,Tz )

1 1 1 2

d (z1, z1 ) d (z2 , z2 ) ,

.max d (z , z ) d (z , z ) , ,

By (1.1.21) & (1.1.22)

2 1 1 2

d (z , z ) d (z , z )

1 1 1 2

1 2 1 2

1 2 1 2

[d(z , z )]2 .max{0,[d(z , z )]2 , 0}

1 2 1 2

1 2 1 2

d(z , z )]2 (d(z , z )]2

1 2 1 2

1 2 1 2

(d(z , z )]2 (d(z , z )]2 0

1 2

1 2

[d(z , z )]2 (1) 0

1 2

1 2

[d(z , z )]2 0

d (z1, z2 ) 0

z1 z2

1

Hence there exists an unique common fixed point of A, B, S and T.

References:

1. Bijendra Sing and S. Chauhan, (1998) on common fixed point of four mappings, Bull. Cal.Maths.Soc., 88,301-308.

2. B.Fisher, (1983) Common fixed points of four mappings, Bull. Inst. Math. Acad.Sinica, 11,103.

3. Jungck.G,(1986) Compatible mappings and common fixed points, Internat.J.Math & Math. Sci.9,771-778.

4. Jungck,G.(1988) compatible mappings and common fixed points(2), Internat.J.Math.&Math.Sci.11,285-288

5. Jungck.G. and Rhoades.B.E.,(1998) Fixed point for set valued functions without continuity, Indian J. Pure.Appl.Math., 29 (3),227-238.

6. R.P.Pant,(1999) A Common fixed point theorem under a new condition, Indian J. of Pure and App. Math., 30(2),147-152.

7. Srinivas.Vand Umamaheshwar Rao.R(2008), A fixed point theorem for four self maps under weakly compatible, Proceeding of world congress on engineering, vol.II,WCE 2008,London, U.K.

8. Srinivas. V&Umamaheshwar Rao.R, Common Fixed Point of Four Self Maps Vol.3,No.2, 2011,113-118.

9. V.Srinivas, B.V.B. Reddy.R.Umameheshwar Rao (2012), A Common fixed point theorem using weakly compatible mapping, Vol.2, No.3, 2012