Vol 6, No 10 (2015)

(1)

International Journal of Mathematical Archive- 6(10), Oct. – 2015 254

SUFFICIENT CONDITIONS FOR THE EXISTENCE OF FIXED POINTS

OF WEAK GENERALIZED GERAGHTY CONTRACTIONS IN COMPLETE METRIC SPACES

K. P. R. SASTRY*

1

, K. K. M. SARMA

2

, CH.SRINIVASA RAO

3

**_{, VEDULA PERRAJU*}**

1

D.No.45-42-25/A, Opp. Indian Overseas Bank,

Akkayyapalem, Visakhapatnam, Pin: 530016, Andhra Pradesh, India.

India (Received On: 09-10-15; Revised & Accepted On: 30-10-15)

ABSTRACT

W

e introduce the notion of

(

α

,

ϕ

,

β

)

-weak generalized Geraghty contractions via triangular

α

-admissible mappings to prove some sufficient conditions for the existence of fixed points of such maps in complete metric space, where

ϕ

is an altering distance function and

β

∈

S

where

S

=

{

β

:

(0,

∞

)

→

[0,1)

satisfying

β

(

t

_n

)

→

1 ⇒

t

_n

→

0}

. Examples are provided to illustrate our results.

Key words: Fixed points, Weak contraction, Altering distance function, Geraghty type contraction. triangular

α

- admissible, complete metric space,

(

α

,

ϕ

,

β

)

- weak generalized Geraghty contraction.

Mathematics Subject Classification (2010): 54H25,47H10.

1. INTRODUCTION AND PRELIMINARIES

Most of the fixed point theorems in nonlinear analysis usually start with Banach [9] contraction principle. A huge amount of literature is witnessed on applications, generalizations and extensions of this principle carried out by several authors in different directions like weakening the hypothesis and considering different mappings. Fixed point theory is an essential tool in the study of various varieties of problems in control theory, economic theory, nonlinear analysis and global analysis.

But all the generalizations may not be from this principle. In 1989, Bakktin [8] introduced the concept of a b-metric space as a generalization of a metric space. In 1993, Czerwik[11] extended many results related to the b - metric space. In 1994, Matthews [20] introduced the concept of partial metric space in which the self distance of any point of space may not be zero. In 1996, O’Neill [28] generalized the concept of partial metric space by admitting negative distances. In 2013, Shukla [34] generalized both the concepts of b-metric and partial metric space by introducing the notation of partial b-metric spaces. Many authors recently studied the existence of fixed points of self maps in different types of metric spaces [16,35,23,32,37]. Some authors [4,20,24,30,31] obtained some fixed point theorems in b-metric spaces. Some authors proved

α

−

ψ

versions of certain fixed point theorems in different types of metric spaces [3,16]. Recently Samet et.al [29] and Jalal Hassanzadeasl [14] obtained fixed point theorems for

α

−

ψ

contractive mappings. Mustafa [24] gave a generalization of Banach contraction principle in complete ordered partial b-metric space by introducing the notion of a generalized

α

−

ψ

weakly contractive mapping. G.V.R. Babu et.al [5] proved coupled fixed point theorems by using

(

α

,

ϕ

,

β

)

-weak generalized Geraghty contraction. In 2012, Mohammad Mursaleen et.al [22] proved coupled fixed point theorems for

α

-

ψ

contractive type mappings in partially ordered metric spaces.

In this paper we modify the concepts of G. V. R. Babu et.al [6] to study sufficient conditions for the existence of fixed points of weak generalized Geraghty contractions in a complete metric space. In fact, we obtain sufficient conditions for the existence of fixed points of weak generalized Geraghty contractions in a complete metric space. A supporting example is also given. Further an open problem is also given at the end of this paper.

Corresponding Author: Vedula Perraju*

1

D.No.45-42-25/A, Opp. Indian Overseas Bank,

(2)

Notation: G. V. R. Babu. et.al [6] used the following notation in their paper.

(

X

,

d

)

denotes a metric space. Let

X

T

:

→

be a self map of

X

and Fix

(

T

)

denotes the set of all fixed points of

T

. We denote

= { : (0, )

[0,1) /

( )

_n

1

_n

0},

S

β

∞ →

β

t

→ ⇒ →

t

and

Φ

=

{

ϕ

:

[0,

∞

)

→

[0,

∞

)/

ϕ

is non-decreasing, continuous and

ϕ

(

t

)

=

0 ⇔

t

=

0}

. We call the elements of

Φ

as altering distance functions. Further, we use the following notation: for sub sequences

{

}

n h

x

and

{

}

n k

x

in X with

n k n

h

x

≠

, we write

( (

), (

))

= (

,

), =

hn kn

n h_n k_n n

n

d T x

T x

d

d x

x

d

∆

and

( ( (

), (

))

=

,

(

)

h_n k_n

n

d T x

T x

n

d

φ

∆

∀

.

We denote the set of all real numbers by

R

, the set of all non-negative reals by

R

+and the set of all natural numbers by

N

.

Definition 1.1: (B.Samet.et.al.[29]) Let

T

:

X

→

X

be a self map and

α

:

X

×

X

→

R

be a function. Then T is said to be

α

- admissible if

α

(

x

,

y

)

≥

1 ⇒

α

(

Tx

,

Ty

)

≥

1

.

Definition 1.2: (E. Karapinar. et.al. [18]) An

α

-admissible map

T

is said to be triangular

α

- admissible if

1 )

,

(

x

z

≥

α

and

α

(

z

,

y

)

≥

1 ⇒

α

(

x

,

y

)

≥

1

.

For more details and examples on

α

-admissible and triangular

α

-admissible maps, one can refer [17], [18] and [29].

Lemma 1.3: (E. Karapinar. et.al. [18])

Let

T

:

X

→

X

be triangular

α

-admissible map. Assume that there exists

x

₁

∈

X

such that

α

(

x

₁

,

Tx

₁

)

≥

1

. Define the sequence

x

_n₊₁ by

x

_n₊₁

=

Tx

_n,

n

=

1,2,...

. Then we have

α

(

x

_n

,

x

_m

)

≥

1

for all

m

,

n

∈

N

with

n

<

m

.

Definition 1.4: (S. H. Cho. et.al.[13]) Let

(

X

,

d

)

be a metric space, and let

α

:

X

×

X

→

R

be a function. A map

X

T

:

→

is called an

α

- Geraghty type contraction if there exists

β

∈

S

such that

( , ) (

x y d Tx Ty

,

) ( ( , )) ( , )

d x y

α

≤

β

for all

x

,

y

∈

X

. (1.4.1)

Definition 1.5: (G. V. R. Babu. et.al.[6]) Let

(

X

,

d

)

be a metric space and

T

:

X

→

X

be a self map. If there exist

Φ

∈

→

×

,

:

ϕ

α

X

R

and

L

≥

0

such that

( , ) (( (

x y

d Tx Ty

,

))) < ((

M x y

( , ))) . ( , )

L N x y

α

φ

+

(1.5.1) for all

x

,

y

∈

X

,

x

≠

y

where

1 ( , ) = { ( , ), ( ,

), ( ,

), [ ( ,

)

( ,

)]},

2 M x y

max d x y d x Tx d y Ty

d x Ty

+

d y Tx

( , ) = { ( ,

), ( ,

)}

N x y

min d x Tx d x Ty d y Tx

then

T

is said to be an almost generalized

α

- contractive map w.r.to the altering distance function

ϕ

.

Definition 1.6: (G. V. R. Babu. et.al.[6]) Let

(

X

,

d

)

be a metric space let

T

:

X

→

X

be a self map. If there exist

:

X

R

,

S

,

α

×

→

β

∈

φ

∈Φ

and

L

≥

0

such that

( , ) (( (

x y

d Tx Ty

,

)))

( (

M x y

( , ))) (

M x y

( , ))

L N x y

. ( , )

α

φ

≤

β φ

φ

+

(1.6.1) for all

x

,

y

∈

X

,

where

1 ( , ) =

{ ( , ), ( ,

), ( ,

), [ ( ,

)

( ,

)]},

2 M x y

max d x y d x Tx d y Ty

d x Ty

+

d y Tx

( , ) =

{ ( ,

), ( ,

)}

(3)

Theorem 1.7: (G. V. R. Babu. et.al.[6]) Let

T

be a self map on a complete metric space

X

. Let

α

:

X

×

X

→

R

be a function. Assume that there exist

ϕ

∈

Φ

, and

L

≥

0

such that

)

,

(

.

))

,

(

<

))

,

(

)

,

(

x

y

ϕ

d

Tx

Ty

ϕ

M

x

y

+

L

N

x

y

α

(1.7.1) for all

x

,

y

∈

X

,

x

≠

y

where

1 ( , ) =

{ ( , ), ( ,

), ( ,

), [ ( ,

)

( ,

)]},

2 M x y

max d x y d x Tx d y Ty

d x Ty

+

d y Tx

( , ) =

{ ( ,

), ( ,

)}.

N x y

min d x Tx d x Ty d y Tx

Further, assume that

(i)

T

is

α

- admissible, and

(ii) there exists

x

₀

∈

X

such that

α

(

x

₀

,

Tx

₀

)

≥

1

and set

x

_n

=

Tx

_n₋₁ for

n

=

1,2,3,...

. Then the sequence

}

{

x

_n converges to

z

and

z

is a unique fixed point of

T

in

X

if and only if for any two sub-sequences

}

{

n h

x

and

{

}

n k

x

of

{

x

_n

}

with

n k n

h

x

≠

, we have that

∆

ϕn

→

1 ⇒

d

n

→

0

, provided that

T

is

continuous at

z

.

Theorem 1.8: (G. V. R. Babu. et.al.[6]) Let

(

X

,

d

)

be a complete metric space,

α

:

X

×

X

→

R

be a function and let

T

:

X

→

X

be a self map. Suppose the following conditions hold:

(i)

T

is

(

α

,

ϕ

,

β

)

- weak generalized Geraghty contraction;

(ii)

T

is triangular

α

- admissible,

(iii)there exists

x

₀

∈

X

such that

α

( ,

x Tx

₀ ₀

) 1

≥

and set

x

_n

=

Tx

_n₋₁ for

n

=

1,2,3,...

(iv)either (a)

T

is continuous (or) (b) if

{

x

_n

}

is a sequence in

X

such that

α

(

x

_n

,

x

_n₊₁

)

≥

1

for all n and

x

_n

→

as

n

→

∞

, then there exists a sub-sequence

{

}

k n

x

of

{

x

_n

}

such that

(

x

,

x

)

≥

1

k n

α

for all

k

. Then

T

has a fixed point

x

in

X

, provided that

β

is continuous on

(0,

∞

)

2. MAIN RESULT

In this section we obtain sufficient conditions for the existence of fixed points of weak generalized Geraghty contractions in a complete space. We begin this section with the following lemma, which can be easily established.

Lemma 2.1: Let

(

X

,

d

)

be a metric space. Let

{

x

_n

}

be a sequence in X such that

d

(

x

_n₊₁

,

x

_n

)

→

0

as

n

→

∞

. If

}

{

x

_n is not a Cauchy sequence then there exist an

ε

>

0

and sequences of positive integers

{

m

_k

}

and

{

n

_k

}

with

k

n

m

_k

>

_k

>

and

(i)

lim

(

−1

,

+1

)

=

ε

∞

→ mk nk

k

x

d

(ii)

_lim

(

,

)

=

ε

k n k m k

x

d

∞ →

(iii)

_lim

(

₁

,

)

=

ε

k n k m k

x

d

₋

∞ →

(iv)

lim

(

,

)

=

ε

1

+ ∞

→ mk nk

k

x

d

Notation: We adopt the following notation throughout this paper.

(

X

,

d

)

denotes a metric space. Let

T

:

X

→

X

be a self map of

X

and Fix

(

T

)

denotes the set of all fixed points of

T

.

We denote

= { : (0, )

[0,1) /

( )

_n

1

_n

0},

S

β

∞ →

β

t

→ ⇒ →

t

and

Φ

=

{

ϕ

:

[0,

∞

)

→

[0,

∞

)/

ϕ

is non-decreasing, continuous and

ϕ

(

t

)

=

0 ⇔

t

=

0}

. We call the elements of

Φ

as altering distance functions. Further, we use the following notation: for any sequences

{

a

_n

}

and

{

b

_n

}

in X with

a

_n

≠

b

_n

,

we write

n n n n

n n n

d

b

T

a

T

d

b

a

d

=

(

,

),

∆

=

(

),

(

))

and

=

( ( (

), ( ))

,

(

)

n n

n

d T a

T b

n

d

φ

∆

∀

We denote the set of all real

(4)

Theorem 2.2: Let

T

X

. Let

α

:

X

×

X

→

R

be a continuous function and

α

(

x

,

x

)

>

1 ∀

x

∈

X

. Assume that there exist

ϕ

∈

Φ

, and

L

≥

0

such that

)

,

(

.

))

,

(

<

))

,

(

)

,

(

x

y

ϕ

d

Tx

Ty

ϕ

M

x

y

+

L

N

x

y

α

(2.2.1) for all

x

,

y

∈

X

,

x

≠

y

where

)]},

,

(

)

,

(

[

2

1 ),

,

(

),

,

(

),

,

(

{

=

)

,

(

x

y

max

d

x

y

d

x

Tx

d

y

Ty

d

x

Ty

d

y

Tx

M

+

and

)}

,

(

),

,

(

),

,

(

{

=

)

,

(

x

y

min

d

x

Tx

d

x

Ty

d

y

Tx

N

(2.2.2)

(i)

T

is

α

- admissible, and

(ii) there exists

x

₀

∈

X

such that

α

(

x

₀

,

Tx

₀

)

≥

1

. Set

x

_n

=

Tx

_n₋₁ for

n

=

1,2,3,...

. (iii)forany two sequences

{

a

_n

}

and

{

b

_n

}

of

X

with

a

_n

≠

b

_n, we have that

∆

ϕ_n

→

1

0 →

⇒

d

_n (2.2.3)

Then the sequence

{

x

_n

}

converges to a point

z

∈

X

and

z

is a fixed point of

T

in

X

. Further if

y

,

z

are fixed point of

T

in

X

, then

α

(

y

,

z

)

<

1

or

y

=

z

Proof: Let

x

₀

∈

X

be such that

α

(

x

₀

,

Tx

₀

)

≥

1

. We define

{

x

_n

}

in

X

by

x

_n₊₁

=

Tx

_n for each

n

=

0,1,2,3,..

.

If

x

_n

=

x

_n₊₁ for some

n

∈

N

, then

x

_n

=

Tx

_n and hence

x

_n is a fixed point of

T

. Hence, without loss of generality, we assume that

x

_n

≠

x

_n₊₁ for all

n

∈

N

. By using the

α

- admissibility of

T

, we have

1 )

,

(

=

)

,

(

x

₀

x

₁

α

x

₀

Tx

₀

≥

α

⇒

α

(

x

₁

,

x

₂

)

=

α

(

Tx

₀

,

Tx

₁

)

=

≥

1

. Now, by mathematical induction, it is easy to see that

α

(

x

_n

,

x

_n₊₁

)

≥

1

for all

n

∈

N

. By taking

x

=

x

_n₋₁ and

y

=

x

_n in the inequality (2.2.1), we get

)

,

(

))

,

(

))

,

(

)

,

(

))

,

(

=

))

,

(

d

x

_n

x

_n₊₁

ϕ

d

Tx

_n₋₁

Tx

_n

≤

α

x

_n₋₁

x

_n

ϕ

d

Tx

_n₋₁

Tx

_n

≤

ϕ

M

x

_n₋₁

x

_n

+

LN

x

_n₋₁

x

_n

ϕ

(2.2.4)

where

)]}

,

(

)

(

[

2

1 ),

,

(

),

,

(

),

,

(

{

=

)

,

(

₋₁ ₋₁ ₋₁ ₋₁ ₋_1,

+

_n _n₋₁

n Tx n n

n n

n n n n

n

x

max

d

x

d

x

Tx

d

x

Tx

d

x

d

x

Tx

x

M

)]}

,

(

)

,

(

[

2

1 ),

,

(

),

,

(

),

,

(

{

=

max

d

x

_n₋₁

x

_n

d

x

_n₋₁

x

_n

d

x

_n

x

_n₊₁

d

x

_n₋₁

x

_n₊₁

+

d

x

_n

x

_n

=

max

{

d

(

x

n−1

,

x

n

),

d

(

x

n

,

x

n+1

)}

(2.2.5)

)}

,

(

),

,

(

),

,

(

{

=

)

,

(

x

_n ₁

x

_n

min

d

x

_n ₁

x

_n

d

x

_n ₁

Tx

_n

d

Tx

_n ₁

x

_n

N

₋ ₋ ₋ ₋

=

min

{

d

(

x

n−1

,

x

n

),

d

(

x

n−1

,

Tx

n

),

d

(

x

n

,

x

n

)}

=

0

(2.2.6)

If

max

{

d

(

x

_n₋₁

,

x

_n

),

d

(

x

_n

,

x

_n₊₁

)}

=

d

(

x

_n

,

x

_n₊₁

)

for some

n

∈

N

_{then from (2.2.4),(2.2.5) and (2.2.6), we have}

))

,

(

=

))

,

(

<

))

,

(

d

x

_n

x

_n₊₁

ϕ

M

x

_n₋₁

x

_n

ϕ

d

x

_n

x

_n₊₁

ϕ

, a contradiction.

Thus, we have

max

{

d

(

x

_n₋₁

,

x

_n

),

d

(

x

_n

,

x

_n₊₁

)}

=

d

(

x

_n₋₁

,

x

_n

)

for all

n

∈

N

and hence

)

,

(

<

)

,

(

x

_n

x

_n ₁

d

x

_n ₁

x

_n

d

₊ ₋ for all

n

∈

N

. (2.2.7)

Thus it follows that

{

d

(

x

_n

,

x

_n₊₁

)}

is a non-negative, decreasing sequence of real numbers. Suppose that

_lim

(

,

₊₁

)

=

,

≥

0

∞

→

d

x

n

x

n

r

n

Now we prove that

r

=

0.

(5)

By choosing

a

_n

=

x

_n

,

b

_n

=

x

_n₊₁,

n

d

x

T

x

T

d

n n n n

)

(

))

(

),

(

=

1

∀

∆

+

ϕ

,

On taking limit as

n

→

∞

,

then we have

_lim

∞ → n 1 2 1 2 1 1

(

_lim

(

,

))

( (

,

))

( )

=

_lim

=

= 1

( (

,

))

(

_lim

(

,

))

( )

n n

n n n

n n

n n n n

n

d x

x

d x

x

r

d x x

r

φ

+ +

+ + →∞

→∞ ₊ ₊

→∞

∆

.

Hence by our assumption

ϕ

(

d

_n

)

→

0

as

n

→

∞

i.e.,

_lim

∞ → n

0 =

))

,

(

d

x

_n

x

_n₊₁

ϕ

0 =

)

(

r

ϕ

⇒

0 =

r

⇒

, contradicts our assumption that

r

>

0

0 =

)

,

(

lim

=

₊₁

∞ →

∴

_n _n

n

x

d

r

Now, we show that

{

x

_n

}

is a Cauchy sequence in

X

.

Suppose that

{

x

_n

}

is not a Cauchy sequence.

Then there exist

ε

>

0

, and sub-sequences

{

}

k m

x

and

{

}

k n

x

of

{

x

_n

}

with

m

_k

>

n

_k

>

k

such that

ε

≥

)

,

(

k n k m

x

d

and

(

₁

,

)

<

ε

k n k

m

x

d

₋ and we have the following identities.

(1)

lim

(

+1

,

+1

)

=

ε

∞

→ mk nk

k

x

d

(2)

_lim

(

,

)

=

ε

k n k m k

x

d

∞ → Hence

))

,

(

))

,

(

lim

k n k m k n k m

k

d

x

Tx

d

ϕ

∞ → = 1 1

( (

,

))

lim

( (

,

))

m_k n_k

k

m_k n_k

d x

x

d x

x

φ

+ +

→∞ =

₍

₎

=

1. )

(

ε

ϕ

ε

ϕ

Now, by our assumption

ϕ

(

d

_n

)

→

0

as

n

→

∞

i.e.,

lim

n→∞

0 =

))

,

(

k n k m

x

d

ϕ

⇒

lim

∞ → n

0 =

)

,

(

k n k m

x

d

, a contradiction to (2).

Therefore

{

x

_n

}

is a Cauchy sequence in

X

.

Since

X

is complete, there exists

z

∈

X

such that

x

_n

z

n

=

lim

∞

→ .

Now, we show that z is a fixed point of T .

Then the following are the cases.

Case- (1): Suppose there is a sequence

n

x

z

k n

k

}

=

{

∋

, we suppose

x

z

k n ₋₁

≠

z

x

k

n ₋₁

→



,

α

(

z

,

z

)

>

1

and

α

is continuous,

1 >

)

,

(

1

z

x

k n ₋

(6)

Now,

))

,

(

d

z

Tz

ϕ

=

(

d

(

x

,

Tz

))

k n

ϕ

=

ϕ

(

d

(

Tx

nk−1

,

Tz

))

≤

α

(

x

nk−1

,

z

)

ϕ

(

d

(

Tx

nk−1

,

Tz

))

<

ϕ

(

M

(

x

nk−1

,

z

))

+

L

.

N

(

x

n

,

z

)

(2.2.8)

But

1 1 1

(

_n

, ) =

{ (

_n

, ), (

_n

,

_n

),

k k k k

M x

z

ma

d x

x

z d x

x

− − − 1

1 ( ,

), [ (

,

)

(

, )]}

2

nk nk

d z Tz

d x

Tz

d x

z

−

+

1

=

{ (

_n

,

_n

),

k k

max d x

x

−

d x

(

nk

,

x

nk+1

)}

=

d

(

x

nk−1

,

x

nk

)

0 =

))

,

(

<

))

,

(

1 nk

k

n

x

d

Tz

z

d

−

∴

ϕ

0 =

))

,

(

d

z

Tz

ϕ

∴

( ,

) = 0

d z Tz

⇒

Tz

z

=

∴

Case-(2): Suppose

x

_n

≠

z

and

z

≠

Tz

))

,

(

=

))

,

(

d

x

_n ₁

Tz

ϕ

d

Tx

_n

Tz

ϕ

₊

∴

≤

α

(

x

n

,

z

)

ϕ

(

d

(

Tx

n

,

Tz

))

(since

α

is continuous and

α

(

z

,

z

)

>

1

)

<

ϕ

(

M

(

x

n

,

z

))

+

L

.

N

(

x

n

,

z

)

(2.2.9)

But

)

,

(

=

)]}

,

(

)

,

(

[

2

1 ),

,

(

),

,

(

),

,

(

{

=

)

,

(

₁

d

x

₁

z

d

x

Tz

d

z

Tz

s

Tz

z

d

x

d

z

x

d

max

z

x

M

_n _n _n _n₊ _n₊

+

_n for large

n

1

( , ) =

_n

{ ( , ), ( ,

_n _n _n

), ( ,

)} = 0

N x z

min d x z d x x

₊

d z Tz

for large

n

∴

ϕ

(

d

(

x

_n₊₁

,

Tz

))

≤

α

(

x

_n

,

z

)

ϕ

(

d

(

Tx

_n

,

Tz

))

<

ϕ

(

d

(

z

,

Tz

))

+

L

.0

On letting

n

→

∞

, we get

))

,

(

))

,

(

)

,

(

))

,

(

d

z

Tz

α

z

ϕ

d

z

Tz

ϕ

d

z

Tz

ϕ

≤

1 =

)

,

(

z

α

⇒

, a contradiction.

Hence

z

=

Tz

.

Consequently

z

is a fixed point of

T

in

X

.

Suppose

y

,

z

are two fixed points of

T

in

X

z

Tz

y

Ty

=

,

=

⇒

(2.2.10)

If

α

(

y

,

z

)

<

1

, then there is nothing to prove.

(7)

Then

))

,

(

d

Ty

Tz

ϕ

≤

α

(

y

,

z

)

ϕ

(

d

(

Ty

,

Tz

))

<

ϕ

(

M

(

y

,

z

))

+

L

.

N

(

y

,

z

)

(2.2.11)

But

[

(

,

)

(

,

)]}

=

(

,

)

2

1 ),

,

(

),

,

(

),

,

(

{

=

)

,

(

y

z

max

d

y

z

d

y

Ty

d

z

Tz

d

y

Tz

d

Ty

z

d

y

z

M

+

0 =

)}

,

(

),

,

(

),

,

(

{

=

)

,

(

y

z

min

d

y

z

d

y

Ty

d

z

Tz

N

∴

ϕ

(

d

(

Ty

,

Tz

))

≤

α

(

y

,

z

)

ϕ

(

d

(

Ty

,

Tz

))

<

ϕ

(

d

(

y

,

z

))

⇒

ϕ

(

d

(

y

,

z

))

≤

α

(

y

,

z

)

ϕ

(

d

(

y

,

z

))

<

ϕ

(

d

(

y

,

z

))

, a contradiction.

Hence

y

=

z

.

The following corollary can be easily established.

Corollary 2.3: Let

T

X

. Let

α

:

X

×

X

→

R

be a continuous function. Assume that there exist

ϕ

∈

Φ

, such that

)

,

(

.

))

,

(

<

))

,

(

)

,

(

x

y

ϕ

d

Tx

Ty

ϕ

M

x

y

+

L

N

x

y

α

(2.3.1) for all

x

,

y

∈

X

,

x

≠

y

where

1 ( , ) =

{ ( , ), ( ,

), ( ,

), [ ( ,

)

(

, )]},

2 ( , ) =

{ ( ,

), ( ,

)}.

M x y

max d x y d x Tx d y Ty

d x Ty

d Tx y

N x y

min d x Tx d x Ty d y Tx

+

(i)

T

is

α

- admissible, and

(ii) there exists

x

₀

∈

X

such that

α

(

x

₀

,

Tx

₀

)

≥

1

and set

x

_n

=

Tx

_n₋₁ for

n

=

1,2,3,...

. (iii) for any two sequences

{

a

_n

}

and

{

b

_n

}

of

X

with

a

_n

≠

b

_n,

We have that

∆

ϕn

→

1 ⇒

d

n

→

0

(2.3.2)

Then the sequence

{

x

_n

}

is a Cauchy sequence. Suppose

{

x

_n

}

converges to

z

and

α

(

z

,

z

)

>

1

.

Then

z

is a fixed point of

T

in

X

.

Theorem 2.4: Let

(

X

,

d

)

be a complete metric space,

α

:

X

×

X

→

R

be a continuous function and let

X

T

:

→

be a self map. Suppose the following conditions hold: (i)

T

is

(

α

,

ϕ

,

β

)

- weak generalized Geraghty contraction i.e.,

)

,

(

.

))

,

(

)))

,

(

))

,

(

)

,

(

x

y

ϕ

d

Tx

Ty

≤

β

ϕ

M

x

y

ϕ

M

x

y

+

L

N

x

y

α

_(2.4.1) where

L

≥

0

for all

x

,

y

∈

X

, where

)}.

,

(

),

,

(

),

,

(

{

=

)

,

(

)]},

,

(

)

,

(

[

2

1 ),

,

(

),

,

(

),

,

(

{

=

)

,

(

Tx

y

d

Ty

x

d

Tx

x

d

min

y

x

N

and

y

Tx

d

Ty

x

d

Ty

y

d

Tx

x

d

y

x

d

max

y

x

M

+

(ii)

T

is triangular

α

- admissible,

(iii)there exists

x

₀

∈

X

such that

α

( ,

x Tx

₀ ₀

) 1

≥

and set

x

_n

=

Tx

_n₋₁for

n

=

1,2,3,...

.Then

{

x

n

}

_{is a Cauchy}

sequence.

(8)

Proof: As in the theorem 2.2, Let

x

₀

∈

X

be such that

α

(

x

₀

,

Tx

₀

)

≥

1

. We define

{

x

_n

}

in

X

by

x

_n

=

Tx

_n₋₁ for each

n

=

1,2,3,...

If

x

_n

=

x

_n₊₁ for some

n

∈

N

, then

x

_n

=

Tx

_n and hence

x

_n is a fixed point of

T

. Hence, without loss of generality, we assume that

x

_n

≠

x

_n₊₁ for all

n

∈

N

. By using the

α

- admissibility of

T

, we have

1 )

,

(

=

)

,

(

x

₀

x

₁

α

x

₀

Tx

₀

≥

α

1 )

,

(

=

)

,

(

₁ ₂ ₀ ₁

≥

⇒

α

x

α

Tx

.

Now, by mathematical induction, it is easy to see that

α

(

x

_n

,

x

_n₊₁

)

≥

1

for all

n

∈

N

.

By taking

x

=

x

_n₋₁ and

y

=

x

_n in the inequality (2.4.1),

We get

1 1

( ( ,

d x x

_n _n

)) = ( (

d Tx

_n

,

Tx

_n

))

φ

₊

φ

₋

≤

α

(

x

_n₋₁

,

x

_n

) ( (

φ

d Tx

_n₋₁

,

Tx

_n

))

≤

α

(

x

_n₋₁

,

x

_n

) ( (

φ

d Tx

_n₋₁

,

Tx

_n

))

(2.4.2) where

)]}

,

(

)

(

[

2

1 ),

,

(

),

,

(

),

,

(

{

=

)

,

(

x

n−1

x

n

max

d

x

n−1

x

n

d

x

n−1

Tx

n−1

d

x

n

Tx

n

d

x

n−1,Tx_n

+

d

x

n

Tx

n−1

M

)]}

,

(

)

,

(

[

2

1 ),

,

(

),

,

(

),

,

(

{

=

max

d

x

_n₋₁

x

_n

d

x

_n₋₁

x

_n

d

x

_n

x

_n₊₁

d

x

_n₋₁

x

_n₊₁

+

d

x

_n

x

_n

=

max

{

d

(

x

n−1

,

x

n

),

d

(

x

n

,

x

n+1

)}

(2.4.3)

)}

,

(

),

,

(

),

,

(

{

=

)

,

(

x

_n ₁

x

_n

min

d

x

_n ₁

x

_n

d

x

_n ₁

Tx

_n

d

Tx

_n ₁

x

_n

N

₋ ₋ ₋ ₋

=

min

{

d

(

x

n−1

,

x

n

),

d

(

x

n−1

,

Tx

n

),

d

(

x

n

,

x

n

)}

=

0

(2.4.4)

If

max

{

d

(

x

_n₋₁

,

x

_n

),

d

(

x

_n

,

x

_n₊₁

)}

=

d

(

x

_n

,

x

_n₊₁

)

for some

n

∈

N

_{then from (2.4.2),(2.4.3) and (2.4.4), we have}

))

,

(

=

))

,

(

<

))

,

(

d

x

_n

x

_n₊₁

ϕ

M

x

_n₋₁

x

_n

ϕ

d

x

_n

x

_n₊₁

ϕ

, a contradiction.

Thus, we have

max

{

d

(

x

_n₋₁

,

x

_n

),

d

(

x

_n

,

x

_n₊₁

)}

=

d

(

x

_n₋₁

,

x

_n

)

for all

n

∈

N

and hence

)

,

(

<

)

,

(

x

_n

x

_n ₁

d

x

_n ₁

x

_n

d

₊ ₋ for all

n

∈

N

. (2.4.5)

Thus it follows that

{

d

(

x

_n

,

x

_n₊₁

)}

is a non-negative, decreasing sequence of real numbers. Suppose that

0 ,

=

)

,

(

lim

+1

≥

∞

→

d

x

n

x

n

r

n

Now we prove that

r

=

0.

Assume that

r

>

0

.

∴

α

( ,

x x

n n+1

) ( (

φ

d Tx Tx

n

,

n+1

)) ( (

≤

β φ

M x x

( ,

n n+1

))) (

φ

M x x

( ,

n n+1

))

+

L N x x

. ( ,

n n+1

)

⇒

ϕ

(

d

(

Tx

_n

,

Tx

_n₊₁

))

≤

β

(

ϕ

(

M

(

x

_n

,

x

_n₊₁

)))

ϕ

(

d

(

x

_n

,

x

_n₊₁

)

+

L

.

N

(

x

_n

,

x

_n₊₁

)

(9)

On letting

n

→

∞

1 2 1 1

( (

,

))

( ( ( ,

))) ( ( ,

))

lim

n n

liminf

n n n n

n n

d x

x

d x x

φ

₊ ₊

β φ

₊

φ

₊

→∞

≤

→∞

limsup

( ( (

n

,

n 1

))) ( (

n

,

n 1

))

n

d x x

β φ

₊

φ

₊

→∞

≤

lim

( ( ,

n n 1

))

n

d x x

φ

₊ →∞

≤

)

(

)

(

))

,

(

limsup

)

(

)))

,

(

liminf

)

(

r

d

x

1

r

d

x

n

x

n 1

r

n n n n

ϕ

β

ϕ

β

ϕ

≤

∴

₊ ∞ → + ∞ →

0 =

)

(

r

ϕ

⇒

or

lim

(

,

+1

)))

=

1

∞

→ n n

n

x

d

ϕ

β

1 =

)))

,

(

lim

+1

∞ →

∴

_n _n

n

x

d

ϕ

β

0 =

)

,

(

lim

+1

∞ →

⇒

_n _n

n

x

d

0 =

r

⇒

, a contradiction for our assumption of

r

>

0

.

0 =

r

∴

Hence

_lim

(

,

₊₁

)

=

0

∞

→ n n

n

x

d

Let us suppose

{

x

_n

}

is not a Cauchy sequence.

Then there exist

ε

>

0

, and sub-sequences

{

}

k m

x

and

{

}

k n

x

of

{

x

_n

}

with

m

_k

>

n

_k

>

k

such that

ε

≥

)

,

(

k n k m

x

d

and

(

₁

,

)

<

ε

k n k

m

x

d

₋ and we have the following identities.

(i)

lim

(

+1

,

+1

)

=

ε

∞

→ mk nk

k

x

d

(ii)

lim

(

,

)

=

ε

k n k m k

x

d

∞ →

Also since

T

is triangular

α

- admissible and

α

(

x

_n

,

x

_n₊₁

)

≥

1 ∀

n

∈

N

,

1 )

,

(

≥

∴

k n k m

x

α

Hence 1 1

( (

_m

,

_n

)) = ( (

_m

,

_n

))

k k k k

d x

x

d Tx

Tx

φ

₊ ₊

φ

(

_m

,

_n

) ( (

_m

,

_n

))

k k k k

x

d Tx

Tx

α

φ

≤

( (

(

_m

,

_n

)) (

(

_m

,

_n

))

(

_m

,

_n

)

k k k k k k

M x

x

M x

x

LN x

x

β φ

φ

≤

+

(2.4.6) where

1

(

_m

,

_n

) =

{ (

_m

,

_n

), (

_m

,

_m

),

k k k k k k

M x

x

ma

d x

x

d x

x

₊ 1 1 1

1 (

,

), [ (

,

)

(

,

)]}

2

n_k n_k m_k n_k m_k n_k

d x

x

₊

d x

x

₊

+

d x

₊

x

=

d

(

x

mk

,

x

nk

)

(2.4.7)

1 1

(

_m

,

_n

) =

{ (

_m

,

_n

), (

_m

,

_m

), (

_n

,

_n

)}

k k k k k k k k

N x

x

min d x

x

d x

x

₊

d x

x

₊

=

0

for large

k

(2.4.8)

Therefore from (2.4.6),(2.4.7) and (2.4.8), we have

)

,

(

))

,

(

))

,

(

))

,

(

₁ ₁

k n k m k n k m k n k m k n k

m

x

d

x

d

x

LN

x

d

₊ ₊

≤

β

ϕ

+