,,  yxyxbyxd yxyxP  0,,  yMx int P  yx x  0

New Contractive Conditions Of Integral Type And Fixed Point Theorems In Cone Metric Space Deepak Kumar

Faculty of Mathematics, Govt. Post Graduate College, Ambala Cantt.-133001, Haryana (India) E-mail: [email protected]

Abstract: The aim of this paper is to extend the concept of F. Khojasteh, Z. Goodarzi and A. Razani to some new contractive conditions of integral type in cone metric space.

[Kumar, D. New Contractive Conditions Of Integral Type And Fixed Point Theorems In Cone Metric Space.

Academ Arena 2019;11(7):8-14]. ISSN 1553-992X (print); ISSN 2158-771X (online).

http://www.sciencepub.net/academia. 2. doi:10.7537/marsaaj110719.02.

Key words: Cone Metric Space, Contractive Conditions, Fixed Point.

1. Introduction:

The concept of cone metric space was introduced by Huang and Zhang [1] in 2007 and some fixed point theorems was proved. Initially Branciari [2]

introduced the contractive condition of integral type and extended Banach fixed point theorem. Later on F.

Khojasteh, Z. Goodarzi and A. Razani [3] gave the concept of cone integrable function and proved Branciari’s theorem in cone metric space. The aim of

this paper is to extend the concept of [3], to some new contractive conditions of integral type in cone metric space.

The following definitions and lemmas are useful for us to prove the main results.

Definition 1.1[1]: Let Ε be a real Banach space and P a subset of Ε

. P is called a cone if the following hold.

(1) P is closed, non-empty and P 0 _.

(2) If a,bR _and a,b 0_{, then} axbyP_, x, yP_. (3) xP _and xP _implies x0_.

Let PΕ be a cone. We define a partial ordering with respect to P as x y

if and only if yxP _and y

x will imply that x y _but xy_{, while} x  y will mean that yxintP_{, where} intP _denotes

the interior of P.

The cone P is called normal if there is a numberM 0 such that 0 x  y _implies x  M y

Ε

 x, y . The least positive number M is called the normal constant.

Example: Suppose ΕR²

, ^P^{x, y}^{Ε x, y0}

. X R_{. Let} ^{d :XX Ε} be defined as

x y bx y x y

d ,   , 

where bR _and b 0_{. Then} X, d

is cone metric space.

Definition 1.2[1]: Let X, d be a cone metric space and let  x_n

be a sequence in X. Then (1)  x_n

is said to converges to some x X if for every cΕ _with 0  c_,  a natural number N such that  n  N _, dx_n, xc

. (2)  x_n

is said to be Cauchy sequence if for every cΕ _with 0 c,  a natural number N such that N

n

m 

 , _, dx_n, x_mc

. x,yP (3) A cone metric space X, d

is complete if every Cauchy sequence is convergent.

Definition 1.3[3]: Let P be a normal cone in Ε

and ,  Ε _where   . Then we define

,x^Ε:s 1s,s0,1_,

, x^Ε:s1s, s0,1_.

Definition 1.4[3]: The set P₁ x₀, x₁,x₂,...,x_n 

is called a partition of , if and only if the sets xj xjⁿ_j

1, 1

 

are pairwise disjoint and ,ⁿ_j1x_j1, x_j 

. Definition 1.5[3]: Let P₁ x₀, x₁, x₂,..., x_n 

be a partition of ,

and  ^, ^{ P}

be an increasing function. We define cone lower sum and cone upper sum as

    1

1

0

, 1 _









 j j j

n

j con

n P x x x

L  

,

   1 1

1

0

, 1 _{ }







 ^{j j j}

n

j con

n P x x x

U  

, respectively.

The function  is called cone integrable function on , if and only if for all partitions P¹ _of ,

 , ₁ lim  , ₁

limL P S U_n^con P

n con con

n n    

, where S^con is unique. We shall write S^{con }  dp



 or ^  t dp t

 _.

Lemma 1.1[3]: If ,  , _then  dp ^  dp





 

 ^ _for ¹X, P

a₁ b₂dp a ₁dp b ^ ₂dp









  

 ^{  } _for ₁,₂¹X, P _and a, bR

where ¹X, P

denotes the set all cone integrable functions.

Definition 1.6[3]: A function :PΕ is said to be subadditive cone integrable function if and only if P



, 

dp dp

dp  

 ^{ }



  

0^{ } 0 ^ 0 .

2. Main Results:

Theorem 2.1: Let X, d

be a complete cone metric space with normal cone P. Let : P P _{be a} nonvanishing and subadditive cone integrable map on each ,  P_{for which} 0

0 

^{ dp} _, ^{  0}_{. Let}

X X

T :  be a mapping such that

   

       

dp c

dp ^{d xT y d yT x}

y T x T

d   

 ^ 0 ^{,  ,}

,

0 for each



 





 2

, 1 0 ,

, y X c

x

. Then T has a unique fixed point in X.

Proof: Let xX_{, choose} x₁X

such that x₁ T x

. Let x₂X

be such that x₂ T x

. Continuing in this way we can define x_n Tx_n1Tⁿ x

for n1,2,3,...

      

dp

dp ^{n n}

n

n x dT x T x

x

d   

 ^{1 } 0 ^{, 1}

, 0

   

dp c^{d xn xnd xn xn} 

 ^{, 1, 1}

0

 

dp c^{d xn xn} 

 ^{1, 1}

0

But dx_n1,x_n1 dx_n1,x_ndx_n,x_n1

, therefore

 

n dp

n x

x

d 

0 ^,

1    

dp c^{d xn xnd xnxn} 

 ^{1, , 1}

0

Since  is cone subadditive, so

     

dp c

dp c

dp ^{n n n n}

n

n x d x x d x x

x

d     

 ^{1  1 } 0 ^{, 1}

, 0 ,

0



     

dp k

c dp

dp c ^{n n n n}

n

n x d x x d x x

x

d     

 ^{1 } _ ^{1 } 0 ^{, 1}

, 0 ,

0 1 _{, where} c

k c

  1



 

dp kⁿ ^{d x x} 

 ^1,0

0

     

dp k

dp ^{n dT x x}

x x

d _{n n}



 

 ^{ } 0 ^,

, 0

1

Since 0k 1_{, and} 0

0 

^{ dp} _{for each}  ^{ 0}_{, so}

 

0

lim ^,

0

1 

^{d xn xn dp}

n 

, which implies, that lim  _n1, _n0

n d x x

. To show  x_n

is Cauchy sequence, we shall show that ^lim    ^,  ^0



 n n

n d T x _ T x

for each positive integer

_.

Let  0 be any integer. By triangular inequality

xn xn dxn xn  dxn xn  dxn xn

d _,  _, _1  _1, _2 ... _1,

     

dp

dp ^{n n n n}

n

n x d x x d x x

x

d ^   ^{ } 

 ^{ } 0 ^{ ,   ... ,} ,

0

1 1

   

       

dp dp

dp ^{n n n n n n}

n

n T x dx x dx x d x x

x T

d ^   ^   ^{ } 

 ^{    } 0 ^{ ,   ... ,}

, 0 ,

0

1 1

1

Since  is cone subadditive

     

dp dp

dp ^{d x x d x x}

x x

d _{n n n n n}





 ^{ }



  

 ^{        }

 ^{1 1 2 1,1}

0 ,

0 ,

0 ...

^{kn  kn   kn}_^{dx x  dp}

 ^{    1, 0}

0 2

1 ...

^{knkn  kn kn} _^{dT x x dp}

 ^,

0 1 2

1 ...

 

 

k dp k^{n dT x x}

 

  ^,

1 0

Letting n_,

   

 

0

lim ^,

0

1 

 ^{ }



 ^{dT xn T xn} dp

n ^ 

. Which implies that ^lim    ^{,  }^0



 n n

n d T x _ T x

for each positive integer _.

Hence  x_n

is a Cauchy sequence. Since X is complete cone metric space so  x_n

is convergent to some X

z _{. i.e.} x_n z

n 

lim

.

 

      

dp

dp ⁿ

n dT z T x

x z T

d   

 ^{ } 0 ^,

, 0

1

    

dp c ^{d zxn d xnT z} 

 ^{, ,}

0

1

    

dp c

dp

c ^{d zxn}  ^ ^{d xnT z} 

 ^{ ,}

0 ,

0

1

As n

 

    

dp c

dp ^{d zT z}

z z T

d   

 ^ 0 ^,

, 0

which implies that dT z ,z0 _i.e. T z z_. Thus z is a fixed point of T.

Uniqueness: Let T has two fixed point z and w i.e. T z z _and T w w_.

            

dp c

dp

dp ^{dT z T w d zT w dwT z}

w z

d     

 ^{ } 0 ^{,  ,}

, 0 ,

0

   

dp c

dp

c ^{d zw}  ^ ^{d wz} 

 ^,

0 ,

0



   

c dp dp c ^{d zw}

w z

d   

 ^ _ 0 ^,

,

0 1

 

dp k^{d zw} 

 ^,

0 where c

k c

  1

Which implies that dz, w0 _i.e. zw_. This shows that T has a unique fixed point in X.

Theorem 2.2: Let X, d

be a complete cone metric space with normal cone P. Let : P P _{be a} nonvanishing and subadditive cone integrable map on each ^{, }^{ P}_{for which} 0^{ dp}  ⁰_, ^{  0}_{. Let}

X X

T :  be a mapping such that

   

      

dp b

dp a

dp ^{d xy d yT x}

y T x T

d     

 ^{ } 0 ^,

, 0 ,

0 . For a,bR _s.t. a 12b _and 2

0b 1

. Then T has unique fixed point.

Proof: Let xX_{, choose} x₁X such that x₁ T x _{. Let} x₂X be such that x₂ T x . Continuing in this way we can define x_n Tx_n1Tⁿ x

for n1,2,3,...

      

dp

dp ^{n n}

n

n x dT x T x

x

d   

 ^{1 } 0 ^{, 1}

, 0

   

dp b

dp

a^{d xnxn}  ^ ^{d xn xn} 

 ^{1 1, 1}

0 ,

0

Using triangle inequality and cone subadditivity,

     

dp b

dp b

dp

a^{d xnxn}  ^ ^{d xn xn}  ^ ^{d xn xn} 

 ^{1 1 , 1}

0 ,

0 ,

0

     

dp k

b dp b

dp a ^{n n n n}

n

n x d x x d x x

x

d     

 ^{1 } _^{ 1 } 0 ^{, 1}

, 0 ,

0 1 _{, where} b

b k a



  1



       

dp k

dp k

dp ^{n d x x n dT x x}

x x

d _{n n}





  

 ^{  } 0 ^,

, 0 ,

0

0 1 1

Since

1 1



  b

b k a

then as n_,

 

0

lim ^,

0

1 

^{d xn xn dp}

n 

Which implies that ^lim  _n1^, _n⁰

n d x x

. It is easy to show that  x_n

is a Cauchy sequence (See previous theorem). Since X is complete cone metric space so there is some zX such that

z x_n

n 

lim

.