Fixed Point Results in α, β Partial b-Metric Spaces Using C-Contraction Mappings

(1)

Copyright @2022, JNSMR, ISSN: 2460-4453

Available online at http://journal.walisongo.ac.id/index.php/jnsmr

Fixed point results in α, β partial b-metric spaces using C-contraction type mapping and its generalization

Ahmad Ansar¹, Syamsuddin Mas’ud²

1 Mathematics Department, Universitas Sulawesi Barat, Indonesia

2 Mathematics Department, Universitas Negeri Makassar, Indonesia

Abstract

Corresponding author:

[email protected] Received: 26 Aug 2022 Revised : 20 Dec 2022 Accepted: 30 Dec 2022

Banach contraction mapping has main role in nonlinear analysis courses and has been modified to get new kind of generalizations in some abstract spaces to produce many fixed point theory. Fixed point theory has been proved in partial metric spaces and b-metric spaces as generalizations of metric spaces to obtain new theorems. In addition, using modified of contraction mapping we get some fixed point that have been used to solve differential equations or integral equations, and have many applications. Therefore, this area is actively studied by many researchers. The goal of this article is present and prove some fixed point theorems for extension of contraction mapping in α, β partial b-metric spaces. In this research, we learn about notions of b-metric spaces and partial metric that are combined to generated partial b-metric spaces from many literatures. Afterwards, generalizations are made to get α, β partial b-metric spaces. Using the properties of convergence, Cauchy sequences, and notions of completeness in α, β partial b- metric spaces, we prove some fixed point theorem. Fixed point theory that we generated used C-contraction mapping and its generalizations with some conditions. Existence and uniqueness of fixed point raised for some restrictions of α, β conditions. Some corollaries of main results are also proved. Our main theorems extend and increase some existence in the previous results.

Keywords: b-metric; partial metric; fixed point

1. Introduction

Partial metric spaces are notions of distance in which two similar points is not

always equal to zero. This metric was studied by Matthews [1] in 1992 and come as a generalization of metric spaces. The other type of metric spaces is b-metric spaces introduced by Bakhtin [2] and extensively used by Czerwik

(2)

[3] in prove some fixed point theorem.

Afterwards, Suzana Aleksic, et. al. [4] establish some new theorems for multivalued mapping in b-metric spaces. Also, Jain and Kaur [5] devoted some new theorems related to b-metric-like spaces and Nazam [6] construct common fixed point theorems in Partial b-metric spaces. Other researchers [7], [8], [9], [10] [11] have studied fixed point theorems involving different model of contraction mapping in b-metric space. Fixed point results in b-metric space is very active topic studied by researchers because it has many application in other fields such as in fractional differential equations, boundary value problems, or integral equations [6], [12], [13], [14] [15].

Shukla [16], in 2014, proposed partial b- metric spaces that generalize both of b-metric space and partial metric space. Shukla also prove Banach and Kannan type contraction principle.

Research about partial b-metric spaces has been attracted many researchers to modified or make generalized to find new kind of fixed point theory [17], [18], [19], [20], [21]. Recently, Pravin and Virath [22] devised of α, β partial b- metric space and prove Banach contraction principal in . In this paper, we shall observe Chatterjea [23] type fixed point and some other generalization in α, β partial b- metric space.

2. Methods

The method of this research is literature study. Research begins with study about theory in b-metric spaces, partial metric spaces and related fixed point. Afterwards, we learn fixed point of partial b-metric spaces as generalization of b-metric spaces. From article Pravin and Virath [22] , concept partial b-metric space is defined and prove fixed point theorems. Using concept partial b-metric space, in this article we formulate and prove main fixed point results to extends previous results.

In this part, we write some definitions, theorems and symbols that underlies main results.

Definition 2.1 [16]

Given a non-empty set and function . Function is called partial - metric on , if there exist a real number for all hold

(i) iff

(ii) (iii) (iv)

The pair is called partial b-metric spaces with coefficient .

Given a non-empty set and function . Function is partial - metric on if there exists two real numbers

for all satisfied

(i) if and only if

(ii) (iii) (iv)

Furthermore, is called an partial -metric space. Clearly, for in ,then we get as partial b-metric space.

Example 2.1 [22]

Let and let be a

function with

(1)

for every . Then is an partial -metric space.

Every partial -metric on have a open balls

and

with and .

(

^{X p}^,

) (

^{X p}^,

)

a b, a b,

X :

b X X´ ®R b b

X a³1

, , x y z XÎ

( ) ( ) ( )

^, ^, ^,

b x x =b y y =b x y x y=

( ) ( )

^, ^,

b x x £b y x

( ) ( )

^, ^,

b x y =b y x

( )

^,

( ) ( )

^, ^,

( )

^,

b x y £ éaëb x z +b z y ù -û b z z

(

^{X b}^,

)

a

X

p:

b X X´ ®R bp a b, b X

, 1

a b³ x y z X, , Î

( )

^,

( )

^,

(

^,

)

p p p

b x x =b x y =b y y x y=

( )

^,

( )

^,

p p

b x x £b x y

( )

^,

( )

^,

p p

b x y =b y x

( )

^,

( )

^,

( )

^,

( )

^,

p p p p

b x y £ab x z +bb z y -b z z

(

X b^, p

)

a b, b

a b=

(

^{X b}^, ^p

) (

X b^, p

)

( )

^1,3

A= b A A_p: ´ ®R

( )

, ^{x y} ¹2

b x yp =e ^- +

, ,

x y z AÎ

(

^{A b}^, ^p

)

^{a b}^,

b

a b, b X

( )

^,

{

^:

( )

^,

( )

^,

}

bp p p

N xe = y X b x yÎ < +e b x x

( )

^,

{

^:

( )

^,

( )

^,

}

bp p p

N xe = y X b x yÎ £e+b x x x XÎ e >0

(3)

Let be an partial -metric space, and sequence and , we get

(i) converges to , if

.

(ii) is a Cauchy in , provide , exists and finite.

The partial -metric space is said complete if any Cauchy sequence in satisfied

for some . Definition 2.5 [23]

A mapping is said C-contraction in is a metric space if there is exist

such that for every satisfied (2)

Theorem 2.1 [22]

Given be a complete partial b- metric space and a mapping such that for all satisfied the following condition

with . Then has unique fixed point.

3. Result and Discussion

Now, we are ready to prove fixed point results in partial -metric space. The first

result related to mapping that satisfied condition in Definition 2.5.

Theorem 3.1

Given be partial -metric space with complete properties and

be a mapping such that

(3) for all in X with . Then has a unique fixed point.

Proof.

Given and we defined sequence in that satisfied

For any , we obtain

Repeating the above process for all , we obtain

Therefore, because

.

Supposed . For ,

(

X b^, p

)

a b, ^b

{ }

xn ÍX x XÎ

{ }

xn ^x

( ) ( )

lim _p _n, _p ,

n b x x b x x

®¥ =

{ }

xn

(

^{X b}^, ^p

)

( )

,lim _p _n, _m

n m b x x

®¥

a b, ^b

(

X b^, p

)

{ }

xn X

( ) ( ) ( )

,lim _p _n, _m lim _p _n, _p ,

n m b x x n b x x b x x

®¥ = ®¥ =

x XÎ

: F X ®X

(

^{X d}^,

) ( )

0,¹2

lÎ x y X, Î

(

^,

) ( (

^,

) (

^,

) )

d Fx Fy £l d x Fy +d y Fx

( X p , )

^{a b}^,

: F X ®X ,

x y XÎ

(

^,

) ( )

^,

p p

b Fx Fy £lb x y

[

^0,1

)

lÎ F

a b, ^b

(

X b^, p

)

a b, b

: F X ®X

(

^,

) (

^,

) (

^,

)

p p p

b Fx Fy £léëb x Fy +b y Fx ùû ,

x y ^lÎ ëé^0,^{a b}¹+

)

F

x0ÎX

{ }

xn

X

1 0

n

n n

x =Fx_- =F x nÎN

( ) ( )

1 2 1

2 1 1 2

2 1 1

1 1 1 1

2 1 1

, ,

p n n p n n

b x x b Fx Fx

b x Fx b x Fx b x x b x x

b x x b x x

l l

l a b

- - -

- - - -

- - -

- - - -

- - -

=

é ù

£ ë + û

é ù

= ë + û

é

£ ë + +

- + ùû

é ù

= ë + û

nÎN

( ) ( )

( )

1 2 1

1

0 1

, ,

1 1 ,

p n n p n n

n p

b x x b x x

b x x al

bl al

bl

- - -

-

£ -

æ ö

£ çè - ÷ø

(

1

)

lim _n , _n 0

n p x_- x

®¥ =

1 1 al

bl^<

-

1 g al

= bl

- m n, ÎN

(4)

Since , then as

and for . Then implies

that . Therefore, the

sequence is Cauchy sequence in , i.e.

Since is complete partial -metric space then is convergence sequence, and say converge to , so

For any , we obtain

Therefore,

Taking limit for both sides, we obtained . From Definition 2.2,

and .

Therefore,

. So, is a fixed point of .

We shall present the prove of uniqueness of fixed point . Suppose that are distinct fixed point of . We obtain,

Therefore that implies that . Hence, just has one fixed point.

The following result extended fixed point theorem by Misrah et.al [24] in b-metric space.

Theorem 3.2

Given be partial b-metric spaces with complete properties and

be a mapping such that

for every with are positive real numbers that satisfied . Then has a unique fixed point.

Proof.

Firstly, we make a sequence such that for any holds

( ) ( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

1 1

1 1 2

2

2 3

1

1 1

0 1 0 1

2 2

0 1

1 11

0 1

0

, , ,

,

, ,

, , ,

p n n m p n n p n n m

p n n

p n n p n n m

p n n p n n

p n n

m

p n m n m

n n

p p

n p

m n m

p n

p

b x x b x x b x x b x x

b x x b x x b x x b x x

b x x b x x

b x x b x x

b x x b x x b x x

a b

a ba

b a b

ag bag

b ag b g g

+ + + +

+ +

+ + +

+ +

- + - +

+ +

- + -

£ + +

-

£ +

£ + +

+ +

£ + +

+ +

£

!

( ) [

( ) ( ( ) ) ( )

2 2 1

1 1

1

1 0 1

, 1

1

m m

m n m

b x xp

a abg ab g b g

a bg

g bg

bg

- -

-

+ + +

+ ùû

é - ù

ê ú

= +

ê - ú

ê ú

ë û

!

1 1 g al

= bl<

- 0

1 al n

bl

æ ö ®

ç - ÷

è ø

n® ¥

( ) bg

ⁿ^®⁰ ⁿ^{® ¥}

( )

,lim _p _n, _{n m} 0

n m b x x₊

®¥ =

{ }

xn X

( ) ( ) ( )

,lim _p _n, _m lim _p _n, _p , 0

n m b x x n b x x b x x

®¥ = ®¥ = =

X a b, b

{ }

xn

{ }

xn x XÎ

( ) ( )

lim _p _n, _p , 0

n b x x b x x

®¥ = =

nÎN

( ) ( ) ( ) ( )

( ) ( )

( ) ( ( ) ( ) )

( ) ( ) ( )

( ) ( )

1 1 1 1

1

1 1

2

, , , ,

, ,

, , ,

, ,

p p n p n p n n

p n p n

p n p n p n

p p

b x Fx b x x b x Fx b x x b x x b Fx Fx

b x x b x Tx b x x

b x x b x x b x x

b x Tx b x x

a b

a bl

a bl abl

b l bl

+ + + +

+

+ +

£ + -

£ +

£ + +

£ + + +

-

( ) ( ) ( )

( )

1 1

2 2

2

, , ,

1 1

1 ,

p p n p n

p n

b x Tx b x x b x x

b x x

a bl

b l b l

abl b l

+ +

£ + +

- -

-

(

^,

)

⁰

b x Fxp = b x xp

( )

^, =⁰

(

^,

)

⁰

b Tx Txp =

( )

^,

(

^,

) (

^,

)

⁰

p p p

b x x =b Fx Fx =b x Fx = x F

F x y X, Î

( )

F

( )

( ) ( )

( )

( ) ( )

( )

, ,

2 ,

p p

p

b x y b Fx Fy

b x Fy b y Fx b x y b y x b x y

l l l

=

£ +

= +

£

( )

^, ⁰

b x yp = x y=

F

a b,

(

X b^, p

)

a b,

: F X ®X

(

^,

) (

^,

) (

^,

) ( )

^,

p Tx Ty £ap x Tx +bp y Ty +cp x y ,

x y XÎ a b c, ,

(

^{a b c}

)

¹

b + + < F

{ }

xn ÍX x0ÎX

1 0

n

n n

x =Tx_- =T x

(5)

For all , we have

Therefore, . If we

continued the process, we get

For , we get

Since , then , so

. Taking limit for both side with ,

we have . This gives us

for . Hence, the sequence is Cauchy sequence in i.e.

Since is complete partial b-metric space, then is convergent sequence.

Let the sequence convergent to in X.

In this step, as a fixed point of will be presented. From definition of partial b- metric space, we get

Therefore,

or

Since and , we

have . We get and

. Thus, that give us as a fixed point of . Furthermore, prove of a unique fixed point is given. Suppose

are distinct fixed point of . We have

Hence, or . So, just has one fixed point in .

Corollary 3.3

If be a complete partial b- metric space and be a mapping satisfying

nÎN

( ) ( )

( )

( ) ( ) ( )

1 1 2

1 1 2 2

2 1

1 2 1 2 1

1 2 1

, ,

,

, , ,

, ,

p n n p n n

p n n

p n n p n n p n n

p n n p n n

b x x b Fx Fx

ab x Fx bb x Fx cb x x

ab x x bb x x cb x x ab x x b c b x x

- - -

- - - -

- -

- - - - -

- - -

=

£ + +

= + +

(

, 1

) (

2, 1

)

p n n 1 p n n

b x x b cb x x

- a - -

£ + -

(

, 1

)

¹

(

0, 1

)

1

n

p n n p

b x x b c b x x a

- -

æ + ö

£ çè - ÷ø ,

n mÎN

( ) ( ) ( )

( )

1 1

2

0 1

1

, , ,

,

1 , 1

1

p n n m p n n p n n m

p n n

n p

m m

b x x b x x b x x b x x

b c b c

b x x

a a

b c a

a b

a b b

+ + + +

+ +

-

£ + +

-

é

+ +

æ ö æ ö

£çè - ÷ø êë çè - ÷ø+ + ù

æ ö

+ çè - ÷ø úûú

!

(

^{a b c}

)

¹

b + + < a b c+ + <1 1 1

b c a + <

- n® ¥

1 0 b c

a + ®

(

^,

)

⁰-

p n n m

b x x ₊ ® n m, ® ¥

{ }

xn X

( ) ( ) ( )

lim, _p _n, _m lim _p _n, _p , 0

n m b x x n b x x b x x

®¥ = ®¥ = =

X a b,

{ }

xn

{ }

xn ^x

x T

a b,

( ) ( ) ( ) ( )

( ) ( )

1

1 1

1

, , , ,

, ,

p p n p n p n n

p n p n

p n p n n

p p n

b x Fx b x x b x Fx b x x b x x b Fx Fx

b x x ab x Fx bb x Fx cb x x

a b

-

- -

-

£ + -

£ +

£ + éë

+ ùû

( ) ( ) ( ) ( )

( )

1 1

1

1 , , ,

,

p p n p n n

p n

b b x Fx b x x a b x Fx c b x x

b a b

b

- -

-

- £ + +

( ) ( ) ( )

( )

( ) ( )

( )

1 1

1

, , ,

1 1

1 ,

, ,

1 1

1 ,

p p n p n n

p n

p n p n n

p n

b x Fx b x x a b x Fx

b b

c b x x b

b x x a b x x

b b

c b x x b

a b

b b

a b

b b

- -

-

£ + +

- -

-

£ + +

- -

-

b¹ 1_b ^limn b x xp

(

n^,

)

b x xp

( )

^, ⁰

®¥ = =

(

^,

)

⁰

b x Fxp = ^{p x x}

( )

^, ⁼⁰

(

^,

)

⁰

b Fx Fxp = Fx x= x XÎ F

, u x XÎ

( ) ( )

F

( ) ( ) ( )

( ) ( )

, ,

, , ,

,

p p

p p p

p

b u x b Fu Fx

ab u Fu bb y Fx cb u x ab u u bb x x cb u x ab u x bb u x cb u x

a b c b u x

=

£ + +

( )

^, ⁰

b u xp = x y= F X

(

X b^, p

)

a b,

: F X ®X

(6)

for all with positive real numbers that satisfied and for some fixed , then has one fixed point.

Proof.

Write . From Theorem 3.2, then has just one fixed point i.e. there is such

that . Hence, . For some ,

we have

Therefore,

Hence, , this

implies Thus, or is

fixed point of .

Next, prove of the uniqueness is start with assume is another a fixed point .

It is implied that or . So, has a single fixed point.

The next theorem is a generalization of result from Pravin and Virath using Ciric [25] type contraction.

Theorem 3.4

If be a complete partial b- metric space and be a mapping such that for every satisfied

for , then has a unique fixed point.

Proof.

Define a sequence in such that

with .

For , we obtain

If

then . We

got contradiction, so we have Now, we divide into two cases.

Case 1:

( ) ( ) ( )

( )

, , ,

,

n n n n

p p p

p

b F x F y ab x F x bb y F y cb x y

£ + +

,

x y XÎ a b c, ,

(

^{a b c}

)

¹

b + + <

b a a

b < F

S T= n S

u XÎ

( )

G u =u F u uⁿ = nÎN

( ) ( ( ) )

( ( ) )

( ( ) ) ( ) ( )

( ) ( ) ( )

, ,

,

, , ,

n

p p

n p

n n

p p p

n

p p p

b Fu u b F F u u b F Fu u

ab Fu F Fu bb u F u cb Fu u ab Fu F F u bb u u cb Fu u ab Fu Fu bb u u cb Tu u

=

£ + +

= + +

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

1 , , ,

, ,

, 1 ,

,

p p p

p p

p

c b Fu u ab Fu Fu bb u u a b Fu u a b u Fu

b u u bb u u a b Fu u b b u u a b Fu u

a b

a b a b

- £ +

£ + +

- +

= + - -

£ +

(

,

) ( ) ( )

,

p 1 p

b Fu u a b Fu u c

a b+

£ -

(

^,

)

⁰

b Fu up = Fu u= u XÎ F

w F

( ) ( )

( )

( ) ( ) ( )

( ) ( )

, ,

,

, , ,

,

p p

n n

p

n n

p p p

p

b u w b Tu Tw b T u T w

ab u T u bb w T w cb u w ab u u bb w w cb u w ab u w bb u w cb u w

a b c b u w

=

£ + +

= + +

£ + +

= + +

(

^,

)

⁰

b u wp = u w= F

(

X b^, p

)

a b,

: F X ®X ,

x y XÎ

( ) { ( ) ( ) ( )

( ) ( ) }

, max , , , , , ,

, , ,

p p p p

p p

b Fx Fy b x y b x Fx b y Fy b x Fy b y Fx

l

£

)

0,_b1

l_Îë^é F

X

1 0

n

n n

x =Tx_- =T x x0ÎX

nÎN

( ) ( )

( )

{ ( )

( ) ( )

( ) }

( )

{ ( )

( ) ( ) ( ) }

( )

{ ( )

{

( ) }

1 1

1

1 1 1

1

1 1

1 1 1

1 1

, ,

max , , , ,

, , , ,

,

max , , , ,

, , , , ,

max , , , ,

,

p n n p n n

p n n

p n n p n n

p n n p n n p n n

p n n p n n

p n n

b x x b Fx Fx

b x x b x Fx b x Fx b x Fx b x Fx

b x x b x x

b x x b x x b x x b x x b x x b x x

l

l l

+ -

-

- - -

-

- +

- - +

- +

=

£

=

£

( ) ( ) ( )

{ }

( )

1 1 1 1

1

max , , , , ,

,

p n n p n n p n n

p n n

b x x b x x b x x b x x

- + - +

+

=

(

1,

) (

, 1

) (

, 1

)

p n n p n n p n n

b x ₊ x £lb x x₊ <b x x ₊

(

1,

)

max

{ (

, 1

) (

, 1, 1

) }

p n n p n n p n n

b x₊ x £l b x x_- b x _- x₊

(7)

. We get

Continuing the process, we have

Following the step in Theorem 3.1, we can establish that for

Since and , then

. Thus, is Cauchy sequence in . Because is complete partial b-metric space, then converge to

i.e.

We shall prove that is a fixed point of mapping . For , we have

Therefore, or is a fixed

point of . Case 2:

We get

We have

Iterating the process, we obtain

Hence, . So, the sequence

is Cauchy in . Therefore, there is

such that . Therefore, we

have

Since , then .

Thus, is a fixed point of . From two case, we know that has a fixed point. Next, suppose that has two fixed point. We have

We have contradiction. Thus, or .

Corollary 3.5

If be a complete partial b- metric space and be a mapping such that

for all with , then has single fixed point.

4. Conclusion

Through this article, we give some notions of partial b-metric spaces and related theorems about partial b-metric spaces. Our main results are some fixed point theorems in partial b-metric spaces using C-contraction mapping and its modification. Fixed point is formulated with some conditions and proved to get extended from previous results. Existence

( ) ( )

{

1 1 1

} (

1

)

max b x x_p _n, _n_- ,b x_p _n_- ,x_n₊ =b x x_p _n, _n_-

(

1,

) (

, 1

)

p n n p n n

b x₊ x £lb x x_-

(

1,

)

ⁿ

(

1, 0

)

p n n p

b x₊ x £l b x x

, m nÎN

(

,

) (

0, 1

)

1

n

p n n m p

b x x l a b x x

+ £ bl

)

- 0,_b1

l_Îë^é b³1

( )

lim _p _n, _{n m} 0

n b x x₊

®¥ =

{ }

xn

X X a b,

{ }

xn

x XÎ

( ) ( )

lim _p _n, _p ,

n b x x b x x

®¥ =

x F nÎN

( ) ( ) ( ) ( )

( ) ( )

1 1

, , , ,

, ,

p p n p n p n n

p n p n

b Fx x b x x b x Fx b x x b x x b Fx Fx

b x x b x x

a b

a bl

- -

£ + -

£ +

(

^,

)

⁰

b x Fxp = x XÎ F

( ) ( )

{

1 1 1

} (

1 1

)

max b x x_p _n, _n_- ,b x_p _n_- ,x_n₊ =b x_p _n_-,x_n₊

( ) ( )

1 1 1

1 1

, ,

p n n p n n

b x x b x x

b x x b x x l

la lb

+ - +

- +

£

£ +

(

1,

) (

1,

)

p n n 1 p n n

b x x la b x x

+ £ lb -

-

(

1,

) (

0, 1

)

1

n

p n n p

b x x la b x x

+ lb

æ ö

£ çè - ÷ø

( )

lim _p _n, _{n m} 0

n b x x₊

®¥ =

{ }

xn X x XÎ

( ) ( )

lim _p _n, _p ,

n b x x b x x

®¥ =

( ) ( ) ( ) ( )

( ) ( )

1 1 1 1

1

, , , ,

, ,

,

p p n p n p n n

p n p n

p n

b x Tx b x x b x Tx b x x b x x b x Tx

b x x

a b

a bl

+ + + +

+

£ + -

£ +

( ) ( )

lim _p _n, _p ,

n b x x b x x

®¥ = ^{p x Tx}

(

^,

)

⁼⁰

x F

F

( )

F

( )

( ) ( ) ( )

{

( ) ( ) }

( ) ( ) ( )

{

( ) ( ) }

( ) ( ) ( )

{ }

( ) ( )

, ,

max , , , , , ,

, , ,

max , , , , , ,

, , ,

max , , , , ,

, ,

p p

p p p

p p

p p p

p p

p p p

p p

b x y b Fx Fy

b x y b x Fx b y Fy b x Fy b y Fx

b x y b x x b y y b x y b y x

b x y b x x b y y b x y

b x y l l

l l

=

£

=

£

<

( )

^, ⁰

b x yp = x y=

(

X b^, p

)

a b,

: F X ®X

(

^,

)

^max

{ ( ) (

^, ^, ^,

) (

^, ^,

) }

p p p p

b Fx Fy £l b x y b x Fy b y Fx ,

x y XÎ l_Îë^é^0,b¹

)

T

a b,

(8)

and uniqueness of fixed point are proved using convergence, Cauchy sequence, and completeness in partial b-metric space. In addition, some corollaries from the main results are given and proved.

Acknowledgments

Authors thank those who contributed to write this article and give some valuable Comments.

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