Vol 9, No 3 (2018)

(1)

COMMON FIXED POINTS

FOR FOUR MAPS USING

α

−

ADMISSIBLE FUNCTIONS IN METRIC SPACES

K.P.R. RAO*

1

_{AND P. RANGASWAMY}

2

1

_{Department of Mathematics,}

Acharya Nagarjuna University, Nagarjuna Nagar – 522 510, (A.P.), India.

2

_{Department of Mathematics, IIIT Ongole, RGUKT-(A.P.), India.}

(Received On: 05-01-18; Revised & Accepted On: 06-02-18)

ABSTRACT

I

n this paper, we introduce

α

−

admissible function associated with four maps and obtain a unique common fixed point theorem. We also give an example to illustrate our main theorem.

Keywords: Complete metric spaces,

α

−

admissible functions, Compatible mappings

Mathematics Subject Classification: 54H25, 47H10.

1. INTRODUCTION AND PRELIMINARIES

In 1973, Geraghty [3] introduced an interesting class of auxiliary functions to refine the Banach contraction mapping principle. Letℱ denote a set of all functions

β

:

[

0 ,

∞

)

→

[

0 ,

1 )

satisfying the condition

( )

1 lim

=

∞

→ n

n

β

t

implies

lim

n→∞

t

n

=

0 .

By using the function

β

∈

ℱ, Geraghty [3] proved the following remarkable theorem.

Theorem 1.1 [3]: Let

(

X d

,

)

be a complete metric space and

T

:

X

→

X

be an operation. If T satisfies

(

Tx

Ty

)

(

d

( )

x

y

)

d

x

y

for

all

x

y

X

where

d

,

≤

β

,

(

,

)

,

∈

,

β

∈

ℱ, then T has a unique fixed point in X.

Definition 1.2: Let Ψ denote the class of all functions

ψ

:

[

0 ,

∞

)

→

[

0 ,

∞

)

which satisfy the following conditions (a)

ψ

is non-decreasing and continuous,

(b)

ψ

(

t

)

=

0 ⇔

t

=

0 .

Definition 1.3 [4]: Let

f

and

g

be self mappings on a metric space

(

X d

,

)

. The pair

(

f g

,

)

is said to be compatible if

d fgx gfx

(

_n

,

_n

)

→

0

whenever there exists a sequence

{ }

x

_n in X such that

fx

_n

→

z

and

.

n

gx

→

z for some z

∈

X

Samet et.al [6] introduced the notion of

α

−

admissible mappings as follows

Definition 1.4 [6]: Let X be a non empty set,

T

:

X

→

X

and

α

:

X

×

X

→

[

0 ,

∞

)

be mappings. Then T is called

α

−

admissible if for all

x

,

y

∈

X

, we have

α

( )

x

,

y

≥

1 implies

α

(

Tx

,

Ty

)

≥

1 .

Corresponding Author: K.P.R. Rao*

1

_,

1

_{Department of Mathematics,}

(2)

Some interesting examples of such mappings are given in [6]. Actually, they proved the following

Theorem 1.5 [6]: Let

(

X

,

d

)

be a complete metric space. Suppose that

α

:

X

×

X

→

[

0 ,

∞

)

and ϕ: 0,

[ ) [ )

∞ → 0,∞ , where

φ

is non-decreasing and

∑

φ

n

(

t

)

<

∞

for each

t

>

0 .

_{Suppose that}

T

:

X

→

X

satisfies

( ) (

x

,

y

d

Tx

,

Ty

)

φ

(

d

( )

x

,

y

)

α

≤

for all

x

,

y

∈

X

.

Assume the following (i) T is

α

−

admissible,

(ii) there exists

x

₀

∈

X

such that

α

(

x

₀

,

Tx

₀

)

≥

1 ,

(iii) either T is continuous or if

{ }

x

_n is a sequence in X with

α

(

x

_n

,

x

_n₊₁

)

≥

1

for all

n

∈

𝒩(the set of all natural numbers)

and

x

_n

→

x

as

n

→

∞

,

then

α

(

x

_n

,

x

)

≥

1

for all

n

∈

𝒩.

Then T has a fixed point in X.

Further, if for any

x

,

y

∈

X

,

there exists

z

∈

X

such that

α

( )

x

,

z

≥

1

and

α

( )

y

,

z

≥

1

then T has a unique fixed point in X.

Recently. Karapinar et.al [5] defined the notion of triangular

α

−

admissible mappings as follows

T X

:

→

X

and

α

:

X

×

X

→

[

0,

∞

)

. Then T is called triangular

α

−

admissible if

(i)

x

,

y

∈

X

,

α

( )

x

,

y

≥

1 ⇒

α

(

Tx

,

Ty

)

≥

1

(ii)

x

,

y

,

z

∈

X

,

α

( )

x

,

z

≥

1 and

α

( )

z

,

y

≥

1 ⇒

α

( )

x

,

y

≥

1 .

Later Shahi et.al [7] and Abdeljawad [1] defined the following

f g X

, :

→

X

and

α

:

X

×

X

→

[

0,

∞

)

. Then

f

is said to be

α

−

admissible with respect to

g

if

α

(

gx

,

gy

)

≥

1 implies

α

(

fx

,

fy

)

≥

1 for

all

x

,

y

∈

X

.

f g X

, :

→

X

and

α

:

X

×

X

→

[

0,

∞

)

. Then the pair

(

f

,

g

)

is said to be

α

−

admissible if

α

( )

x y

,

≥

1 implies

α

(

fx gy

,

)

≥

1 and

α

(

gx fy

,

)

≥

1 for all x y

,

∈

X

.

Using these definitions, we introduce the following

Definition 1.9: Let X be a non empty set and

f

,

g

,

S

,

T

:

X

→

X

and

α

:

X

×

X

→

[

0 ,

∞

)

. The pair

(

f

,

g

)

is said to be

α

−

admissible w.r.to the pair

(

S

,

T

)

(

,

)

1 (

,

)

1 (

,

)

1 (

,

)

1. if

α

Sx Ty

≥

implies

α

fx gy

≥

and

α

Tx Sy

≥

implies

α

gx fy

≥

Definition 1.10:

(

f

,

g

)

is called triangular

α

−

admissible w.r.to (S,T) if (i)

(

f

,

g

)

is

α

−

admissible w.r.to (S,T) and

(ii)

α

( )

x

,

y

≥

1 and

α

( )

y

,

z

≥

1 ⇒

α

( )

x

,

z

≥

1 for

all

x

,

y

,

z

∈

X

.

Recently Shahi et.al [7] and Cho et.al [2] proved the following

Theorem 1.11 (Theorem 3.1, [7]): Let

(

X d

,

)

f g X

, :

→

X

be such that

(

)

(

).

f X

⊆

g X

Assume the following

(1.12.1)

f

is

α

−

admissible with respect to

g

,

(1.12.2)

α

(

gx

,

gy

) (

d

fx

,

fy

)

≤

ψ

(

M

(

gx

,

gy

)

,

where

(

gx

gy

)

d

(

gx

gy

) (

d

gx

fx

) (

d

gy

fy

) (

d

gx

fy

) (

d

gy

fx

)

and

M













+

=

2 ,

,

2 ,

,

(3)

ψ

:

[

0 ,

∞

)

→

[

0 ,

∞

)

is continuous, nondecreasing and

∑

∞

=

>

∞

<

1

,

0 )

(

n n

t

all

for

t

ψ

(1.12.3) there exists

x

₀

∈

X

such that

α

(

gx

₀

,

fx

₀

)

≥

1 ,

(1.12.4) if

{ }

gx

_n is a sequence in X such that

α

(

gx

_n

,

gx

_n₊₁

)

≥

1 ,

for

all

n

and

gx

n

→

gz

∈

g

(

X

),

then there exists a subsequence

{ }

gx

nk

of

{ }

gx

n such that

α

(

gx

nk

,

gz

)

≥

1,

for all k

.

Also suppose that

g

(

X

)

is closed.

Then

f

and

g

have a coincidence point.

Theorem 1.12 (Theorem 2.1, [2]): Let

(

X

,

d

)

be a complete metric space

α

:

X

×

X

→

[

0 ,

∞

)

be a function and

.

:

X

T

→

suppose that the following conditions are satisfied

(1.13.1)

α

( ) (

x

,

y

d

Tx

,

Ty

)

≤

β

(

M

( )

x

,

y

) ( )

M

x

,

y

for

all

x

,

y

∈

X

,

where

β

∈

ℱ and

M

( )

x

,

y

=

max

{

d

( ) (

x

,

y

,

d

x

,

Tx

) (

,

d

y

,

Ty

)

}

,

(1.13.2) T is triangular

α

−

admissible,

(1.13.3) there exists

x

₁

∈

X

such that

α

(

x

₁

,

Tx

₁

)

≥

1 ,

(1.13.4) T is continuous. Then T has a fixed point.

Now we prove our main result to generalize Theorems 1.11 and 1.12.

2. MAIN RESULT

Theorem 2.1: Let

(

X

,

d

)

α

:

X

×

X

→

[

0,

∞

)

be a function. Let , ,f g S and T be self

mappings on X

satisfying

(2.1.1)

f X

(

)

⊆

T X

(

),

g X

(

)

⊆

S X

(

),

(2.1.2)

α

(

Sx Ty

,

)

ψ

(

d fx gy

(

,

)

≤

β ψ

(

M x y

( )

,

)

ψ

(

M x y

( )

,

)

for all x y

,

∈

X

where

β

∈

ℱ,

ψ

∈

Ψ and

( )

(

) (

)

[

(

,

) (

,

)

]

,

2

1 ,

,

max

,













₊

=

d

Sx

Ty

d

Sx

fx

d

Ty

gy

d

Sx

gy

d

Ty

fx

y

x

M

(2.1.3) the pairs

(

f S and

,

)

( )

g, T

are compatible and S and T are continuous on X,

(2.1.4)

(

f

, g

)

is triangular

α

−

admissible w.r.to

(

S, T ,

)

(2.1.5)

α

(

Sx

1

,

fx

1

)

≥

1

and

α

(

fx

1

,

Sx

1

)

≥

1

for some

x

1

∈

X

,

(2.1.6) Assume that

α

(

Sy

₂_n

, y

₂_n₋₁

)

≥

1,

α

(

y

₂_n

, Ty

₂_n₊₁

)

≥

1,

α

(

z, y

₂_n₋₁

)

≥

1 and

α

( )

z z

,

≥

1

whenever there exists a sequence

{ }

y

n

in X such that

α

(

y

n

,

y

n+₁

)

≥

1 for n

=

1, 2, 3,...

and y

n

→

z for some z

∈

X

.

Then

f g S and T

, ,

have a common fixed point.

(2.1.7) Further if

α

( )

u

, v

≥

1

whenever

u

and

v

are common fixed points of

f g S and T

, ,

then

f g S and T

, ,

have unique common fixed point in X.

Proof: From (2.1.5), we have

α

(

Sx fx

₁

,

₁

)

≥

1 for some x

₁

∈

X

.

From (2.1.1), define the sequences

{ }

x

_n

and

{ }

y

_n

as follows

:

1 1 2

,

2 2 3

,

3 3 4

,

4 4 5

,....

y

=

fx

=

Tx

y

=

gx

=

Sx

y

=

fx

=

Tx

y

=

gx

=

Sx

2n 1 2n 1 2n 2

,

2n 2 2n 2 2n 3

,

0,1, 2,....

(4)

Now

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

1 1 1 2

1 2 1 2

2 3

2 3 2 3

3 4

3 4 3 4

,

1 ,

1 , g

1,

(2.1.4), .

, y

1 ,

1 g

,

1,

(2.1.4), .

, y

1 ,

1 , g

1,

(2.1.4), .

, y

1 Sx fx

Sx Tx

fx

x

from

i e

y

Tx Sx

x

fx

from

i e

y

Sx Tx

fx

x

from

i e

y

α

≥

⇒

≥

⇒

≥

⇒

≥

⇒

≥

⇒

≥

⇒

≥

Continuing in this way, we have

(

y

n

, y

n 1

)

1 for n

1, 2, 3,...

α

₊

≥

=

(1)

Similarly using

α

(

fx

₁

,

Sx

₁

)

≥

1

, we have

α

(

y

_n₊₁

,

y

_n

)

≥

1

for n=1,2,……. (1)1

By (2.1.4), using triangular property, we have

(

y

_m

, y

_n

)

1 for m

n

.

α

≥

<

(2)

Case-(a): Suppose

y

₂_m

=

y

₂_m₊₁

.

Then

α

(

Sx

₂_m₊₁

, T

x

₂_m₊₂

)

=

α

(

y

₂_m

, y

₂_m₊₁

)

≥

1 from

(1).

Now

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

2 1 2 2 2 1 2 2

, y

,

, T

,

m m m m

d y

d fx

gx

Sx

x

d fx

gx

M x

x

M x

x

ψ

α

ψ

β ψ

ψ

+ + + +

=

≤

where

(

)

(

) (

)

(

)

(

)

(

)

2 2 1 2 2 1 2 1 2 2

2 1 2 2

2 2 2 2 1 2 1

2 1 2 2

,

max

₁

,

2 ,

m m m m m m

m m

m m m m

m m

d y

y

d y

y

d y

y

M x

x

d y

y

d y

y

d y

y

+ + + +

+ +

+ + +

+ +







=

_

_



+















=

Thus

ψ

(

d y

(

2m+1

, y

2m+2

)

≤

β ψ

(

d y

(

2m+1

,

y

2m+2

)

ψ

(

d y

(

2m+1

,

y

2m+2

)

Heance

(

)

(

)

(

)

[

1 −

β

ψ

d

y

₂_m₊₁

,

y

₂_m₊₂

]

ψ

(

d

(

y

₂_m₊₁

,

y

₂_m₊₂

)

≤

0

which in turn yields that

(

)

(

d

y

₂_m₊₁

,

y

₂_m₊₂

)

ψ

=0 so that

y

₂_m₊₁

=

y

₂_m₊₂

.

Continuing in this way we get

y

₂_m

=

y

₂_m₊₁

=

y

₂_m₊₂

=

....

Hence

{ }

y

_n is Cauchy.

Case-(b): Suppose

y

_n

≠

y

_n₊₁

for all n

.

As in Case (a), we have

(

)

(

d y

2n 1

, y

2n 2

)

(

M x

(

2n 1

,

x

2n 2

)

(

M x

(

2n 1

,

x

2n 2

)

(5)

where

(

)

{

(

) (

)

}

(

x

) (

d

y

)

then

we

get

M

If

y

d

y

d

x

M

n n n n n n n n n n 2 2 1 2 2 2 1 2 2 2 1 2 1 2 2 2 2 1 2

,

max

,

+ + + + + + + + +

=

(

)

(

)

(

)

(

)

(

)

(

)

2 1 2 2 2 1 2 2 2 1 2 2

2 1 2 2

, y

,

n n n n n n

n n

d y

y

d y

y

d y

y

ψ

β ψ

ψ

+ + + + + +

+ +

≤

<

It is a contradiction. Hence

(

)

(

)

(

)

(

)

(

)

(

)

2 1 2 2 2 2 1 2 2 1

2 2 1

, y

,

n n n n n n

n n

d y

y

d y

y

d y

y

ψ

β ψ

ψ

+ + + +

+

≤

<

d y

(

₂_n₊₁

, y

₂_n₊₂

)

<

d y

(

₂_n

, y

₂_n₊₁

)

.

Similarly using (2.1.2) and (1)1, we can show that

d y

(

₂_n

, y

₂_n₊₁

)

<

d y

(

₂_n₋₁

, y

₂_n

)

.

Thus

{

d y

(

_n

,

y

_n₊₁

)

}

is a decreasing sequence of non-negative real numbers.

Hence it converges to some real number

r

≥

0

such that

(

1

)

lim

_n

,

_n

.

n→∞

d y y

+

=

r

Suppose

r

>

0.

From (3),

(

)

(

)

(

2 1 2 2

)

(

2 1 2 2

)

2 1 2 2

, y

, x

1. , x

n n n n n n

d y

M x

ψ

β ψ

ψ

+ + + + + +

≤

<

Letting

n

→ ∞

and using the continuity of

ψ

, we get

(

)

(

)

(

2 1 2 2

)

1 lim

_n

, x

_n

1

n→∞

β ψ

M x

+ +

≤

so that

lim

(

₂ ₊₁

,

₂ ₊₂

)

=

0

∞

→ n n

n

ψ

M

x

ψ

( )

r

=

0 and hence r

=

0.

It is a contradiction. Thus

(

1

)

lim

_n

,

_n

0

n→∞

d y y

+

=

(4)

Now we prove that

{ }

y

_n is a Cauchy sequence. In view of (4), it is sufficient to show that

{ }

y

₂_n is Cauchy.

Assume on the contrary that

{ }

y

₂_n is not a Cauchy sequence. Then there exists

∈>

0

for which we can find two

subsequences

{ } { }

₂ ₂

{ }

₂

k k

m n n

y

and y

of

y

so that

n

_kis the smallest positive integer such that

2 n

_k

>

2 m

_k

>

k

,

(

2m_k

,

2n_k

)

d y

y

≥∈

₍₅₎

(

2m_k

,

2n_k 2

)

d y

y

₋

<∈

(6)

Now from (5) and (6), we have

(

) (

)

(

k k

) (

k k

)

(6)

Letting

k

→ ∞

and using (4), we get

lim

(

₂

,

₂

)

k k

m n

k→∞

d y

y

so that

∈≤

≤∈

(

2 2

)

lim

,

k k

m n

k→∞

d y

y

=∈

₍₇₎

We have

(

₂ ₁

,

₂

) (

₂

,

₂

) (

₂ ₁

,

₂

)

k k k k k k

m n m n m m

d y

₊

y

−

d y

y

≤

d y

₊

y

Letting

k

→ ∞

and using (4) and (7), we get

(

2 1 2

)

lim

,

k k

m n

k→∞

d y

+

y

=∈

(8)

We have

(

₂

,

₂ ₁

) (

₂

,

₂

) (

₂ ₁

,

₂

)

k k k k k k

m n m n n n

d y

y

₋

−

d y

y

≤

d y

₋

y

Letting

k

→ ∞

(

2 2 1

)

lim

,

k k

m n

k→∞

d y

y

−

=∈

(9)

We have

(

₂ ₁

,

₂ ₁

) (

₂

,

₂

) (

₂ ₁

,

₂

) (

₂

,

₂ ₁

)

k k k k k k k k

n m m n n n m m

d y

₋

y

₊

−

d y

y

≤

d y

₋

y

+

d y

y

₊

Letting

k

→ ∞

(

2 1 2 1

)

lim

,

k k

n m

k→∞

d y

−

y

+

=∈

(10)

(

Sx

2m_k 1

,

Tx

2n_k

) (

y

2m_k

,

y

2n_k 1

)

1 α

₊

=

α

₋

≥

from (2)

(

)

(

d y

2m_k 1

,

y

2n_k

)

(

d fx

(

2m_k 1

,

gx

2n_k

)

ψ

₊

=

ψ

₊

(

Sx

2m_k 1

,

Tx

2n_k

)

₍

d fx

(

2m_k 1

,

gx

2n_k

)

₎

α

₊

ψ

₊

≤

(

)

(

)

(

M x

2mk 1

,

x

2nk

)

(

M x

(

2mk 1

,

x

2nk

)

β ψ

₊

ψ

₊

≤

where

(

)

(

) (

)

(

) (

)

2 2 1 2 2 1 2 1 2

2 1 2

2 2 2 1 2 1

,

max

₁

,

2

k k k k k k

k k

k k k k

m n m m n n

m n

m n n m

d y

y

d y

y

d y

y

M x

x

d y

y

d y

y

− + −

+

− +







=

_

_



₊





_

_







,

(9), (4), (7), (10).

as k

from

→∈

→ ∞

From (11), we get

(

)

(

)

(

)

(

)

(

,

)

1 .

,

2 1 2 2

1 2

2 1 2

<

≤

₊

+ +

k k k

n m n

m n m

x

M

x

M

y

d

ψ

β

ψ

Letting

k

→

∞

and using (8) and the continuity of

ψ

,

we get

(

)

(

)

(

,

)

1 lim

1 ≤

_→_∞ 2m_k+1 2n_k

≤

k

β

ψ

M

x

so that

lim

_→_∞

(

2m_k+1

,

2n_k

)

=

0

k

ψ

M

x

and hence

ψ

(

∈

)

=

0 .

Thus

∈=

0 .

It is a contradiction.

(7)

Since X is complete, there exists

z

∈

X

such

that

y

_n

→

z

and hence

.

lim

fx

₂ ₁

Tx

₂ ₂

gx

₂ ₂

Sx

₂_n ₁

z

n

n n

n n n

n→∞ +

=

→∞ +

=

→∞ +

=

→∞ +

=

Since S is Continuous, we have

2

2 1 2 1

lim

_n

lim

_n

.

n→∞

S x

+

=

Sz and

n→∞

Sfx

+

=

Sz

(

)

(

2 1 2 1

)

Sin

,

lim

_n

,

_n

0.

n

ce the pair

f S is compatible we have

d fSx

₊

Sfx

₊

→∞

=

2 1

lim

_n

.

n

Hence

fSx

₊

Sz

→∞

=

Now

(

SSx

2n 1

,

Tx

2n

)

(

Sy

2n

,

y

2n 1

)

1 α

₊

=

α

₋

≥

from (2.1.6)

(

)

(

d fSx

2n 1

,

gx

2n

)

(

SSx

2n 1

,

Tx

2n

)

(

d fSx

(

2n 1

,

gx

2n

)

ψ

₊

≤

α

₊

ψ

₊

≤

β ψ

(

M Sx

(

2n+1

,

x

2n

)

ψ

(

M Sx

(

2n+1

,

x

2n

)

,

where

(

)

(

) (

)

(

) (

)

2 1 2 2 1 2 1 2 2

2 1 2

2 1 2 2 2 1

,

max

₁

,

2

n n n n n n

n n

n n n n

d SSx

Tx

d SSx

fSx

d Tx

gx

M Sx

x

d SSx

gx

d Tx

fSx

+ + +

+

+ +







=

_

_

+

















(

,

)

.

d Sz z as n

→

→ ∞

From (12), we get

(

)

(

)

(

)

(

,

)

(

,

)

1 .

,

2 1 2 2

1 2

2 1

2

≤

<

+ +

+

n n n

n n

n

_M

_Sx

_x

x

Sx

M

gx

fSx

d

_β

_ψ

ψ

Letting

n

→

∞

ψ

,

we get

(

)

(

)

(

,

)

1 lim

1 ≤

_→_∞ 2n+1 2n

≤

n

β

ψ

M

Sx

x

lim

_→_∞

(

2n+1

,

2n

)

=

0

n

ψ

M

Sx

x

so that

ψ

(

d

(

Sz

,

z

)

=

0 .

Hence

Sz

=

z

.

Since T is continuous, we have

2

2 2 2 2

lim

_n

lim

_n

.

n→∞

T x

+

=

Tz and

n→∞

Tgx

+

=

Tz

(

)

(

2 2 2 2

)

Sin

,

lim

_n

,

_n

0.

n

ce the pair g T is compatible we have

d Tgx

₊

gTx

₊

→∞

=

2 2

lim

_n

.

n

Hence

gTx

₊

Tz

→∞

=

Now

(

Sx

2n 1

,

TTx

2n 2

)

(

y

2n

,

Ty

2n 1

)

1 α

₊ ₊

=

α

₊

≥

from (2.1.6)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

2 1 2 2 2 1 2 2 2 1 2 2

2 1 2 2 2 1 2 2

,

n n n n n n

n n n n

d fx

gTx

Sx

TTx

d fx

gTx

M x

Tx

M x

Tx

ψ

α

ψ

β ψ

ψ

+ + + + + +

+ + + +

≤

(13)

where

(

)

(

) (

)

(

) (

)

(

)

2 1 2 2 2 1 2 1 2 2 2 2

2 1 2 2

2 1 2 2 2 2 2 1

,

max

₁

,

2 ,

.

n n n n n n

n n

n n n n

d Sx

TTx

d Sx

fx

d TTx

gTx

M x

Tx

d Sx

gTx

d TTx

fx

d z Tz a sn

+ + + + + +

+ +

+ + + +







=

_

_

+

















(8)

From (13), we get

(

)

(

)

(

)

(

,

)

(

,

)

1 .

,

2 2 1 2 2 2 1 2 2 2 1

2

≤

<

+ + + + + + n n n n n n

Tx

x

M

Tx

x

M

gTx

fx

d

_β

_ψ

ψ

Letting

n

→

∞

ψ

,

we get

(

)

(

)

(

,

)

1 lim

1 ≤

₂ ₊₁ ₂ ₊₂

≤

∞

→ n n

n

β

ψ

M

x

Tx

lim

(

₂ ₊₁

,

₂ ₊₂

)

=

0

∞

→ n n

n

ψ

M

x

Tx

so that

ψ

(

d

(

z

,

Tz

)

=

0 .

Hence

Tz

=

z

.

Now

(

Sz Tx

,

2n

)

(

z y

,

2n 1

)

1 α

=

α

₋

≥

from (2.1.6)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

2 2 2

2 2

,

n n n

n n

d fz gx

Sz Tx

d fz gx

M z x

ψ

α

ψ

β ψ

ψ

≤

(14) where

(

)

(

) (

)

(

) (

)

(

)

2 2 2

2

2 2

,

max

₁

,

2 ,

.

n n n

n

n n

d Sz Tx

d Sz fz d Tx

gx

M z x

d Sz gx

d Tx

fz

d z fz as n







=

_

_

+

















→

→ ∞

(

)

(

)

(

)

(

,

)

(

,

)

1 .

,

2 2

2

≤

<

n n n

x

z

M

x

z

M

gx

fz

d

_β

_ψ

ψ

Letting

n

→

∞

ψ

,

we get

(

)

(

)

(

,

)

1 lim

1 ≤

₂

≤

∞

→ n

n

β

ψ

M

z

x

lim

(

,

₂

)

=

0

∞

→ n

n

ψ

M

z

x

so that

ψ

(

d

(

z

,

fz

)

=

0 .

Hence

fz

=

z

.

Now

(

Sz Tz

,

)

( )

z z

,

1 α

=

α

≥

from (2.1.6)

(

)

(

)

(

)

(

)

(

)

( )

(

)

(

)

(

( )

)

,

d z gz

d fz gz

Sz Tz

d fz gz

M z z

ψ

α

ψ

β ψ

ψ

=

≤

(15)

where

( )

(

) (

)

(

) (

)

(

)

,

max

₁

,

2 ,

.

d Sz Tz d Sz fz d Tz gz

M z z

d Sz gz

d Tz fz

d z gz

as n







=

_

_

+

















→

→ ∞

From (15), we have

(

)

(

)

(

)

[

1 −

β

ψ

d

z

,

gz

]

ψ

(

d

(

z

,

gz

)

≤

0 ψ

(

d

(

z

,

gz

)

=

0

. Thus gz = z.

(9)

Suppose u and v are two common fixed points of

f g S

, ,

and

T

Then

α

(

Su Tv

,

)

=

α

( )

u v

,

≥

1

from (2.1.7)

( )

(

)

(

)

(

)

(

)

( )

(

)

(

)

(

( )

)

,

d u v

d fu gv

Su Tv

d fu gv

M u v

ψ

α

ψ

β ψ

ψ

=

≤

(16)

where

( )

( ) ( ) ( )

( ) ( )

( )

1 ,

max

,

2 ,

.

M u v

d u v d u u

d v v

d u v

d v u

d u v





=

_



_

+



_

_





=

From (16), we have

( )

(

)

(

)

[

1 −

β

ψ

d

u

,

v

]

ψ

(

d

( )

u

,

v

)

≤

0

which in turn yields that so that

ψ

(

d

( )

u

,

v

)

=

0

. Hence

u

=

v

.

Thus

f g S and T

, ,

have a unique common fixed point.

Now, we give an example to illustrate Theorem 2.1.

Example 2.2: Let

X

=

[

0 ,

∞

)

be endowed with the metric

d

( )

x

,

y

=

x

−

y

for

all

x

,

y

∈

X

.

Define

,

.

3

27 ,

2 ,

18 :

,

2 2

X

y

x

all

for

x

Tx

and

x

gx

x

Sx

x

fx

by

X

T

S

g

f

→

=

∈

Define

[

)

( )

[ ]







∈

=

∞

→

×

otherwise

y

x

if

y

x

by

X

,

0

1 ,

0 ,

,

1 ,

,

0 :

α

Define

[

)

[

)

by

( )

t

for

all

t

[

0 ,

)

and

:

[

0 ,

)

[

0 ,

1 )

by

2 ,

0 ,

0 :

∞

→

∞

ψ

=

∈

∞

β

∞

→

ψ

[

0 ,

)

.

9

1 )

(

t

=

for

all

t

∈

∞

β

Clearly the conditions

(

2 .

1 .

1 ) (

,

2 .

1 .

3 ) (

,

2 .

1 .

4 )

and condition

(

2 .

1 .

5 )

with

x

₁

=

0

are satisfied.

If

[ ]

0 ,

1 (

,

)

1 .

3

1 ,

0

2

=

∈

=

∈

=

x

and

Ty

y

then

Sx

Ty

Sx

α

(

)

(

)

2

1 ,

,

2 18

27 x

y

Sx Ty

d fx gy

α

ψ

=

−

(

)

(

)

( )

(

)

( )

(

)

(

)

(

( )

)

2

1 9 4

6

1 ,

9

1 ,

9 ,

,

x

y

d Sx Ty

M x y

ψ

β ψ

ψ

=

−

=

≤

=

If

[ ]

0 ,

1 (

,

)

0 .

3

1 ,

0

2

=

∉

=

∉

=

x

or

Ty

y

then

Sx

Ty

Sx

α

(10)

Clearly

(

2 .

1 .

6 )

is satisfied if we take

y

_n

1 for all n

.

n

=

Clearly ‘0’ is a common fixed point of

f g S and T

, ,

.

The condition

(

2 .

1 .

7 )

is clear and ‘0’ is unique common fixed point of

f g S and T

, ,

.

REFERENCES

1. Abdeljawad,T.(2013). Meier-Keeler α-contractive fixed and common fixed point theorems, Fixed point theory Appl., 2013:19,10 pages, doi:10.1186/1687-1812-2013-19.

2. Cho,S.H., Bae,J.S. and Karapinar,E.(2013).Fixed point theorems for α-Geraghaty contraction type maps in metric spaces, Fixed point theory. Appl., 2013:329, 11Pages, doi: 10.1186/1687-1812-2013-329.

3. Geraghty,M.(1973).On Contractive mappings,Proc.Amer.Math.Soc.,40, 604-608.

4. Jungck,G.(1986).Compatible mappings and common fixed points, Int.J.Math.Math.Sci., 9(4),771-779.

5. Karapinar,E.Kumam,P. and Salimi,P.(2013).On α − ψ -Meier-Keeler contractive mappings, Fixed point theory.Appl.,2013:94,12Pages, doi:10-1186/1687-1812-2013-94.

6. Samet,B..Vetro,C. and Vetro,P.(2012). Fixed point theorems for α −ψ-contractive type mappings, Nonlinear Analysis, 75, 2154-2165.

7. Shahi,P. Kumar,J. and Bhatia,S.S.(2015). Coincidence and common fixed point results for generalized α − ψ -contractive type mappings with applications, Bull.Belg.Math.Soc.Simon Stevin, Vol.22 (2), 299-318.

Source of support: Nil, Conflict of interest: None Declared.