A COMMON FIXED POINT THEOREMS FOR A PAIR OF SELF MAPS SATISFYING A GENERAL CONTRACTIVE CONDITION OF INTEGRAL TYPE

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IMPACT FACTOR – 5.088

A COMMON FIXED POINT THEOREMS FOR A PAIR OF SELF

MAPS

SATISFYING A GENERAL CONTRACTIVE CONDITION OF

INTEGRAL TYPE

R. A. Rashwan* H. A. Hammad**

ABSTRACT

In this paper, we prove two common fixed point theorems for a pair of self maps satisfying a general contractive condition of integral type, which extend and improve the results of M. R. Singh, L. Sharmeswar Singh[1].

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1. INTRODUCTION

The first well known result on fixed points for contractive map was Banach's fixed point theorems, published in 1922 (see [3], [6]). In general setting of complete metric space, Smart([13]) presented the following result.

Theorem 1.1. [1] Let (X,d) be a complete metric space,c [0,1), and let T : X X be a map such that for each x, y X ,

d(Tx,Ty) cd(x,y)

Then,T has a unique fixed point z X such that for each x X , Tn x z

n ( )

lim .

After this classical result, many theorems dealing with maps satisfying various types of contractive inequalities have been established (see for details [4], [7]-[12], [14]). In 2002, Branciari ([5]) obtained the following theorem.

Theorem 1.2. [1] Let (X,d) be a complete metric space,c [0,1), and let T : X X be a map such that for each x, y X ,

) , (

0 )

, (

0

) ( )

(

y x d Ty

Tx d

dt t c dt

t ,

where :R R is a Lebesgue - integrable map which is summable, non negative and such that

( ) 0 0

dt

t for each 0 .

Keywords: Lebesgue-integrable map, Complete metric space, weakly compatible mappings, Common fixed point.

2010 Mathematics subject classification:47H10

Then, T has a unique fixed point z X and for each x X, Tn x z

n ( )

lim .

In 2007, Boikanyo [3] proved some fixed point theorems for a self map satisfying

a general contractive condition of integral type as an extension of Branciari’s theorem. In [5], it was mentioned that one can generalize other results related to contractive conditions of some kind, such as in [10].

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IMPACT FACTOR – 5.088

and improve the results of M. R. Singh, L. Sharmeswar Singh[1].

Definition 1.3. [2]Let f and g be two mappings from a metric space (X,d) into itself, f and g is called weakly compatible if they commute at there coincidence point .

i.e fx gx fgx gfx. for somex X .

Definition 1.4. [2] Two self maps f and g of metric space (X,d) is called compatible if 0

) , (

lim _n _n

n d fgx gfx , whenever {xn} is a sequence such that

X. in t some for , lim

limfx gx_n t

n n n

Definition 1. . [2] Maps f and gare said to be commuting if fgx gfx x X .

Definition 1.6. [2] Let f and g are two mappings on a set X, if fx gx for somexin X, thenxis called coincidence point of f and g .

Throughout this paper, N denotes the set of natural numbers.

2.MAIN RESULTS

Theorem 2.1. Let (X,d) be a complete metric space. Let a i_i( 1, 2,3, 4,5) be nonnegative real numbers satisfying

5

1 1 i i

a , T T f₁, , and g₂ are four self maps

ofX satisfying the following conditions: 1-T X₁( ) f X( ) and ( )T X₂ g X( ),

2-the pair ( , ) and ( , g)T f₂ T₁ are weakly compatible,

3-1 2 1

( , ) ( , ) ( , )

1 2

0 0 0

( ) ( ) ( )

d T x T y d fx gy d fx T x

t dt a t dt a t dt

2 1

2

( , ) ( , ) ( , )

5

3 4

0 0 0

( ) ( ) ( )

d gyTy d fxTy d gy T x

a t dt a t dt a t dt, (2.1)

where :R R is a Lebesgue-integrable map which is summable, non-negative and such

that for each 0 , 0

( )t dt 0. Then T₁,T₂,f andg have a unique common fixed

pointz X.

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y

_n

T x

₂ _n

gx

_n ₁ and

y

_n ₁

T x

₁ _n ₁

fx

_n ₂ (2.2)

By interchanging x with y , T₁ with T₂ and f with g, we obtain

2 1 2 1

( , x ) ( , ) ( , ) ( , )

1 2 3

0 0 0 0

( )

d T y T d g y f x d g y T y d f x T x

t d t

a

t d t

a

t d t

a

t d t

1 2

( , ) ( , )

5 4

0 0

( )

d gy T x d fx T y

a

t dt a

t dt

.

(2.3) Now from (2.1) and (2.3) and using symmetric property, we have

1 2 1 2

( , ) ( , ) ( x , T ) ( , )

2 3 2 3

1

0 0 0 0

( )

(

)

( )

(

)

( )

2

d T x T y d fx gy d f x d gy T y

a

t dt

a

t dt

+

2 1

( , ) ( , )

5 5

4 4

0 0

(

)

( )

(

)

( )

2

d fx T y d gy T x

a

t dt

(2.4)

Using (2.4), for even n, we obtain

1 2 1 1 1 1

2 3

1

( , ) ( , ) ( , ) ( , )

( ) ( ) ( ) ( ) ( )

2

0 0 0 0

n n n n n n n n

d y y d T x T x d fx gx _a _a d fx T x

t dt t dt a t dt t dt

+

1 2 1 2 1 1 1

( , ) ( , ) ( , )

2 3 4 5 4 5

0 0 0

(

)

( )

(

)

( )

(

)

( )

2

n n

n n n n

d gx T x d fx T x d gx T x

a

t dt

From (2.2) we have

1 1 1 1

( , ) ( , ) ( , ) ( , )

2 3 2 3

1

0 0 0 0

( )

(

)

( )

(

)

( )

2

n n n n n n n n

d y y d y y d y y d y y

a

t dt

a

t dt

+

1 1

( , ) ( , )

5 5

4 4

0 0

(

)

( )

(

)

( )

2

n n

d y y d y y

a

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IMPACT FACTOR – 5.088

Again using (2.4), for odd n, we obtain

1 1 1 2 1 1 1 1

( , ) ( , ) ( , ) ( , )

2 3

1

0 0 0 0

( )

(

)

( )

2

n n n

n n n n n

d y y d T x T x d fx gx d fx T x

a

t dt

a

t dt

+

1 2 1 1

2 ( , ) ( , )

( , )

5 5

2 3 4 4

0 0 0

(

)

( )

(

)

( )

(

)

( )

2

n n

n n d fxn T x d gx T xn

d gx T x

a

t dt

From (2.2) we get

1 1 1 1

( , ) ( , ) ( , ) ( , )

2 3 2 3

1

0 0 0 0

( )

(

)

( )

(

)

( )

2

n n n n n n n n

d y y d y y d y y d y y

a

t dt

a

t dt

+ 1 1 ( , ) ( , ) 5 5 4 4 0 0

(

)

( )

(

)

( )

2

n n n n

d y y d y y

a

t dt

(2.6)

From (2.5) and (2.6) we observe that

1 1 1 1

( , ) ( , ) ( , ) ( , )

2 3 2 3

1

0 0 0 0

( )

(

)

( )

(

)

( )

2

n n n n n n n n

d y y d y y d y y d y y

a

t dt

a

t dt

+ 1 1 ( , ) ( , ) 5 5 4 4 0 0

(

)

( )

(

)

( )

2

n n n n

d y y d y y

a

t dt

1 1 1

( , ) ( , ) ( , )

2 3 2 3

1

0 0 0

( )

(

)

( )

(

)

( )

2

n n n

d y y d y y d y y

a

t dt

+ ) 1 1 ( , ) ( , 5 5 4 4 0 0

(

)

( )

)

( )

2

(

2

n n

n

d y yn d y y

a

t dt

.

It follows that

1 1

( , ) ( , )

5

1 2 3 4

5

2 3 4

0 0

2 ( )

(

)

( )

2

n n n n

d y y d y y

a

t dt

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1

( , )

0

( )

...

n

d y y

r

t dt

0 1

( , )

0

( )

as

d y y n

r

t dt

n

Since r 1 , owing to the assumption

5

1

i i

a

.Therefore, lim

(

_n

,

_n ₁

)

0

n

d y y

(2.7)

Now we show that { }y_n is a Cauchy sequence in X . Let m n where m n, N. Without loss of generality , we consider two cases arise:

(I) m is even when n is odd (II)m is odd when n is even

Case (I): We choose m and n to be odd and even respectively, by using (2.1) we have

2 1 2

( , ) ( , ) ( , )

( , )

1 2

0 0 0 0

( )

n m m n m m

n m d T x T x d x fx d gx T x

d y y g

t dt

a

t dt a

t dt

1 1 2

( , ) ( , ) ( , )

5

3 4

0 0

( )

n n m n n m

d fx T x d gx T x d fx T x

o

a

t dt a

t dt

By using (2.2) we have

1 1 1 1

( , ) ( , ) ( , )

( , )

1 2 3

0 0 0 0

( )

m n

n m d ym yn d ym y d yn y

d y y

t dt

a

t dt a

t dt

+

1 1

( , ) ( , )

5 4

0 0

( )

n m

m n

d y y d y y

a

t dt a

t dt

.

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IMPACT FACTOR – 5.088

1 1 ( ) ( ) ( , ) ) 1 1

0 0 0 0

( , , ,

1

( )

m m n n

m n

n m d d d y y

d y y y y y y

a

t dt

a

t dt

t dt a

t dt

1 1 1

( , ) ( , ) ( , ) ( , )

2 3 4 4

0 0 0 0

( )

m n m m n

m n m

d y y d y y d y y d y y

a

t dt a

t dt

+ 1 ( , ) ( , ) 5 5 0 0

( )

n n m

n

d y y d y y

a

t dt a

t dt

1 1

( , ) ( , )

( , )

5

1 3

1 2 4

5 5

1 4 1 4

0 0 0

( )

(

)

( )

(

)

( )

1

m n

n m d ym y d yn y

d y y

a

t dt

a

0 1 0 1

( , ) ( )

1 1 3 5 1

1 2 4

5 5

1 4 0 1 4 0

,

(

)

( )

(

)

( )

1

d y y d y

m n

y

a

t dt

a

1

0 1 ( 0 )

( , )

1 1

0 0

,

0 as n,m

( )

d y d y y

m n

y

t dt

r

since r 1.

Case(II): we choose mand n to be even and odd respectively. From (2.1) and repeating the steps of case (I) also we have

( , )

0

.

( )

0 a s ,

n m

d y y

t dt

n m

Then { }y_n is a Cauchy sequence in the complete metric space X , a point z X

Such that lim _n

n y z

2 1 1 1 2

lim

n

lim

n

=z and lim

n

lim

n

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i .e

lim

₂ _n

lim

_n ₁

= lim

₁ _n ₁

lim

_n ₂

n

T x

n

gx

n

T x

n

fx

z

(2.8)

.Since T X₁( ) f X( ), a point

u

X such that z fu. From (2.1) we get

1 2 1 1 1 1

2 ( , ) ( , ) ( , )

( , )

1 2

0 0 0 0

( )

n n n n

d Tx T u d fx gu d fx T x

d z T u

t dt

a

t dt a

t dt

+

1 2 1 1

2 ( , ) ( , )

( , )

5

3 4

0 0 0

( )

n n

d fx T u d gu T x d gu T u

a

t dt a

t dt

By taking the limit as n and by (2.8) we have

2

( , ) ( , ) ( )

1 2

0 0 0

z z,z

( )

d z T u d gu d

t dt

a

t dt a

t dt

+

2 2

( , ) ( , ) ( , )

5

3 4

0 0 0

z z

( )

d gu T u d T u d gu

a

t dt a

t dt

( , )

1 0

( )

d z gu

a

t dt

+

2 2

( , ) ( , )

5

3 3 4

0 0 0 0

( )

d z T u d T u

d guz z d guz

a

t dt

a

t dt a

t dt

2

( , u) ( , )

5

1 3

3 4

0 0

( )

)

( )

1

(

d z T d z gu

dt

a

t dt

t

a

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IMPACT FACTOR – 5.088

( , ) ( , )

0 0

since (r < 1).

( )

d z gu d z gu

r

t dt

(2.9)

If z T u₂ , so we have a contradiction in (2.9) (T₂ g).

Then z T u₂ , so fu T u₂ z. Hence is coincidence point of f and T₂. Since the pair of maps f andT₂are weakly compatible,then

2 2 , i.e T2

T fu fT u z fz. (2.10)

Again since T ( )₂ X g X( ), there exists a point X such that z g . Then by (2.1) and applied the same above steps, we can find that T₁ z. Therefore T₁ g z, so is a coincidence point of T₁ and g.

Also the pair of maps T₁ and g are weakly compatible,

1 1 i.e gz=T1

gT T g z (2.11)

Now we show thatzis a fixed point ofT₂, by using (2.1) we have

1 1 2 1 1 1 1

2 ( , ) ( , ) ( , )

( , )

1 2

0 0 0 0

( )

n n n n

d T x T z d fx gz d fx T x

d z T z

t dt

a

t dt a

t dt

+

1 2 1 1

2 ( , ) ( , )

( , )

5

3 4

0 0 0

( )

n n

d fx T z d gz T x d gz T z

a

t dt a

t dt

Taking the limit as n we get

2 2

1 2

( , ) ( ) ( , ) ( , )

3

0 0 0 0

,

( )

d z T z d zgz d zz d gzT z

t dt

a

t dt a

t dt

+

2

( , ) ( , )

5 4

0 0

( )

d z T z d gz z

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2 2

1

( , ) ( , )

( ) ( ,g ) ( , )

5

3 3 4

0 0 0 0 0

,g

( )

d T z d z T z

d z z d z z z d z gz

a

t dt a

t dt

( , ) ( , ) ( , )

5

1 3

3 4 0 0 0

(

)

( )

1

d z gz d z gz d z gz

a

t dt

r

t dt

a

( since r 1 )

If z T z₂ we have a contradiction, hence z T z₂

i.e from (2.10) we get

z T z

₂

fz

(2.12)

Also by the same way we can show that z is a fixed point of T₁, hence z T z₁

i.e from (2.11) we get z T z₁ gz (2.13)

From (2.12) and (2.13) we obtain that

z

T z

₂

f z

₁z

T

g z

Therefore

z

is a common fixed points of T T f₁, , and g₂ .

For uniqueness of

z

let if possible that

z

and w are common fixed points of

1, , and g2

T T f

Such that(w z),from (2.1) we have

1 2 1

( z, ) ( , )

( , ) ( , )

1 2

0 0 0 0

( )

d T T w d fz T z

d z w d fz gw

t dt

a

t dt a

t dt

+

2 2 1

( , ) ( , ) ( , )

5

3 4

0 0 0

( )

d gw T w d fz T w d gw T z

a

t dt a

t dt

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IMPACT FACTOR – 5.088

( , ) ( , ) ( , ) ( , ) ( , )

5

1 2 3 4

0 0 0 0 0

( )

d z w d z z d w w d z w d w z

t dt a

t dt

a

( , ) ( , ) ( , )

5

1 4

0 0 0

( )

(

)

( )

d z w d z w d z w

t dt

a

t dt

r

t dt

( since r 1 ).

i.e

z

is a unique common fixed point of T T f₁, , and g₂ .

If we put f g in the above theorems we get the following corollary.

Corollary 2.2. Let (X,d) be a complete metric space suppose that the mappings T T₁, ₂ and f

are self maps satisfying the following conditions: 1- T X1( ) f X( ) and ( )T X2 f X( )

2-the pair ( , ) and ( , )T f₂ T g₁ are weakly compatible,

3-1 2 1

( , ) ( , ) ( , )

1 2

0 0 0

( ) ( ) ( )

d T x T y d fx fy d fx T x

t dt a t dt a t dt

+

2 2 1

( , ) ( , ) ( , )

5

3 4

0 0 0

( ) ( ) ( )

d fy T y d fx T y d fy T x

a t dt a t dt a t dt

where :R R is a Lebesgue-integrable map which is summable, non-negative and such

that 0

( )t d t 0 for each 0 . Then T T₁, ₂ and f have a unique common fixed point

z X.

REMARK

(i) Theorem 2.1 (cf.[1]) is a special case of Theorem 2.1 by taking f g I(I is the identity mapping). (ii)By taking ( ) 1t in Theorem 2.1, we obtain the contractive condition of the Theorems 2.1 not involving the integral.

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REFERENCES

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a general contractive condition of integral type”, Kathmandu University Journal of Science

Engineering and Technology, Vol. 6, pp. 20-27, 2010.

2- Vishal Gupta, Anil saini, Ravinder Kumar "common fixed points for weakly compatible

Maps in 2-metric space"International Journal of Mathematical Archive-3(10),

2012,3670-3675.

3- S. Banach, 1922. Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math. 3: 133 – 181 ( in French ) 4- O. A. Boikanyo, 2007. Some fixed point theorems for mappings satisfying a general contractive condition of integral types, Far East J. Math. Sci., (FJMS) 26 (1)219 – 230. 5- A. Branciari, 2002. A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Sci. 29: 531 – 536.

6- R. Caccioppoli, 1930. Un teorema generale sull’ esistenza di elementi uniti in una

transformazione funzionale, Rend. Acad. Dei Lincei 11: 794 – 799 ( in Italian). 7- S.K. Chatterjea, 1972. Fixed – point theorems, C.R. Acad. Balgare Sci. 25: 727 – 730. 8. G. E. Hardy and T. D. Rogers, 1973. A generalization of a fixed point theorem of Reich, Bull.Cand. Math. Soc. 16:201 –206.

9- R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71 – 76. 10- B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977), 257 – 290.

11- B. E. Rhoades, Two fixed point theorems for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 63 (2003), 4007 – 4013.

12- M. Sen Gupta ( Mrs. Das Gupta ), On common fixed points of operators, Bull. Cal. Math. Soc. 66 (1974) 149 –153.

13- D. R. Smart, Fixed point theorems, Cambridge University Press, London, 1974.