Cone C-class Function with Common Fixed Point Theorems for Cone b-metric Space

Vol. 8, 2017, 83-94

ISSN: 2349-0632 (P), 2349-0640 (online) Published 29 August 2017

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/jmi.v8a9

83

Journal of

Cone C-class Function with Common Fixed Point

Theorems for Cone b-metric Space

R. Krishnakumar1 and D.Dhamodharan2

1

Department of Mathematics,Urumu Dhanalakshmi College Tiruchirappalli-620019, India

Email: [email protected]

2

Department of Mathematics, Jamal Mohamed College (Autonomous) Tiruchirapplli-620020, India

Corresponding author. Email: [email protected]

Received 1 August 2017; accepted 17 August 2017

Abstract. In this paper, we discuss the common fixed point theorem obtain sufficient conditions for the existence of common fixed points of a pair of mapping satisfying generalized contraction involving rational expressions in cone b metric spaces via cone

C class functions

Keywords: Common fixed point,cone b-metric space,cone metric space cone C-class function

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

1. Introduction and mathematical preliminaries

In 1089, Bakhtin [2] introduced b-metric space as a generalization of metric space. He proved the contraction principle in b -metric spaces that generalized the famous contraction principle in metric space. In 2007, Huang and Zhang [4] are introduce the concept of cone metric space and also they discussed some properties of the convergence of sequences and proved the fixed point theorems of a contraction mapping for cone metric spaces; Any mapping T of a complete cone metric space X into itself that satisfies, for some 0≤k<1, the inequality d(Tx,Ty)≤kd(x,y),∀x,y∈X has a unique fixed point. In 2011, Hussain and Saha [6] introduced the concept of cone b-metric space as a generalization of b -metric spaces and cone metric spaces. They established some topological properties in such spaces and improved some recent results about KKM mappings in the setting of a cone b-metric space. Note on

ϕ

−

ψ

-contractive type mappings and related fixed point are proved by Ansari [11].

In this paper, we investigate the common fixed point theorem obtain sufficient conditions for the existence of common fixed points of a pair of mapping satisfying generalized contraction involving rational expressions in cone b metric spaces via cone

84

Definition 1.1. Let E be the real Banach space. A subset P of E is called a cone if and only if:

• P is closed, non empty and P≠∅

• ax+by∈P for all x,y∈P and non negative real numbers a,b • P∩( P− )={0}.

Given a cone P⊂E, we define a partial ordering ≤ with respect to P by x≤y if and only if y−x∈P. We will write x < y to indicate that x≤y but x≠ y, while

y

x, will stand for y−x∈intP, where intP denotes the interior of P. The cone P is called normal if there is a number K >0 such that 0≤ x≤ y implies PxP≤KPyP for all x,y∈E. The least positive number satisfying the above is called the normal constant.

Example 1.2. [10]Let K >1. be given. Consider the real vector space with ,1]} 1 1 ; , : { =

k x R b a b ax

E + ∈ ∈ −

with supremum norm and the cone

0} 0, : {

= ax+b a≥ b≤ P

in E. The cone P is regular and so normal.

Definition 1.3. Let X be a nonempty set. A mapping d:X×X →E is said to be cone metric if and only if, for all x,y,z∈X , the following conditions are satisfied:

• d(x,y)=0 if and only if x = y, • d(x,y)=d(y,x),

• d(x,z)≤d(x,y)+d(y,z).

Then (X,d) is called a cone metric space (CMS).

Example 1.4. Let E= R2

0} , : ) , {(

= x y x y≥ P

R

X = and d:X×X →E such that

|) | |, (| = ) ,

(x y x y x y

d −

α

−

where

α

≥0 is a constant. Then (X,d) is a cone metric space.

Definition 1.5. [6] Let X be a nonempty set and s≥1 be a given real number. A mapping d:X×X →E is said to be cone b-metric if and only if, for all x,y,z∈X , the following conditions are satisfied:

• d(x,y)=0 if and only if x = y, • d(x,y)=d(y,x),

• d(x,z)≤s[d(x,y)+d(y,z)].

85

Example 1.6. [5] Let E= R2

0} , : ) , {(

= x y x y≥ P

R

X = and d:X×X →E such that

) | | , | (| = ) ,

(x y x y p x y p

d −

α

−

where

α

≥0 and p>1 are two real constants. Then (X,d) is a cone b-metric space

with the coefficient s=2p >1, but not a cone metric space. In fact, we only need to prove (iii) in Definition 1.5 as follows:

Let x,y,z∈X . Setu=x−z,v=z−y, so x−y=u+v. From the inequality

0. , forall ) (

2 }) , { max (2 )

(a+b p ≤ a b p ≤ p ap +bp a b≥ we have

) | | | (| 2 = ) | | | (| 2 |) | | (| | =| |

|x−y p u+v p≤ u + v p ≤ p u p + v p p x−z p + z−y p

which implies that d(x,y)≤s[d(x,z)+d(z,y)] with s=2p >1. But p p

p

y z z x y

x | | | | |

| − ≤ − + −

is impossible for all x>z> y. Indeed, taking account of the inequality

0, , forall >

)

(a+b p ap+bp a b≥ we arrive at

p p

p p p p

p

y z z x v u v u v u y

x | =| | =( ) > =| | | |

| − + + + − + −

for all x > z > y. Thus, (c₃) in Definition 1.3 is not satisfied, i.e., (X, d) is not a cone metric space.

Example 1.7. [5] Let X =lp with 0< p<1, where ={{ } : | | < }.

1 =

∞

⊂

∑

∞ p

n n n

p

x R x l

Let d :X×X →R₊ define by ( , )=( | | ) ,

1

1 =

p p n n n

y x y

x

d

∑

−

∞

where

. } { = }, {

= x_n y y_n lp

x ∈ Then (X,d) is a b-metric space [3]. Put 1}

forall 0, : } {{ = ,

=_l1 _{P x} ∈_{E x} ≥ _n≥

E _{n n} . Letting the mapping d*:X×X →E

be

defined by } 1,

2 ) , ( { = ) , (

* ≥

n y x d y x

d _n we conclude that (X,d*) is a cone b-metric

space with the coefficient =2 >1,

1

p

s but it is not a cone metric space.

Definition 1.8. [6] Let (X,d) be a cone b-metric space, x∈X and {x_n} be a sequence in X. Then

1. {x_n} converges to x whenever, for every c∈E with 0=c, there is a

natural number N such that d(xn,x)=c for all n≥N. We denote this by

x xn n

= lim

∞

86

2. {xn} is a Cauchy sequence whenever, for every c∈E with 0=c,

there is a natural number N such that d(x_n,x_m)=c for all n,m≥ N

3. (X,d) is a complete cone b-metric space if every Cauchy sequence is convergent.

Definition 1.9. A function

ψ

:P→P is called an altering distance function if the following properties are satisfied:

)

(i

ψ

is non-decreasing and continuous, )

(ii

ψ

(t)=0 if and only if t=0.

Definition 1.10. An ultra altering distance function is a continuous, nondecreasing

mapping

ϕ

:P→P such that

ϕ

(t)>0 , t>0 and

ϕ

(0)≥0.

We denote this set with Φu

Definition 1.11. [12] A mapping F:P2 →P is called cone C-class function if it is continuous and satisfies following axioms:

1. F(s,t)≤s;

2. F(s,t)=s implies that either s=0 or t=0; for all s,t∈P.

We denote cone C-class functions as C.

Example 1.12. [12] The following functions F:P2 →P are elements of C, for all )

0, ,t∈ ∞ s :

1. F(s,t)=s−t,

2. F(s,t)=ks, where 0<k <1,

3. F(s,t)=s

β

(s), where

β

:[0,∞)→[0,1),

4. F(s,t)=Ψ(s), where Ψ:P→P ,Ψ(0)=0 ,Ψ(s)>0 for all P

s∈ with s≠0 and Ψ(S)≤s for all s∈P.,

5. F(s,t)=s−

ϕ

(s), where

ϕ

:[0,∞)→0,∞) is a continuous function such that

ϕ

(t)=0⇔t=0;

6. F(s,t)=s−h(s,t), where h:[0,∞)×0,∞)→0,∞) is a continuous function such that h(s,t)=0⇔t=0 for all t, s>0.

7. F(s,t)=

ϕ

(s),F(s,t)=s⇒s=0_{, here}

ϕ

:[0,∞)→[0,∞) is a upper semi continuous function such that

ϕ

(0)=0 and

ϕ

(t <) t for t>0.

Lemma 1.13. Let

ψ

and

ϕ

are altering distance and ultra altering distance functions respectively , F∈C and {s_n} a decreasing sequence in P such that

)) ( ), ( ( )

(sn 1 F

ψ

sn

ϕ

sn

ψ

+ ≤

for all n≥1. Then lim n =0 n

s

∞

87

2. Main result

Theorem 2.1. Let (X,d) be a complete cone b-metric space (CCbMS) with the co-efficient s≥1 and P be a normal cone with normal constant K . Suppose that the mappings S,T:X →X satisfy:

)])) , ( ) , ( [ )] , ( ) , ( [ ) , ( 1 ) , ( )] , ( [1 ) , ( ( )]), , ( ) , ( [ )] , ( ) , ( [ ) , ( 1 ) , ( )] , ( [1 ) , ( ( ( )) , ( ( 4 3 2 1 4 3 2 1 Sx y d Ty x d a Ty y d Sx x d a y x d Ty y d Sx x d a y x d a Sx y d Ty x d a Ty y d Sx x d a y x d Ty y d Sx x d a y x d a F Ty Sx d + + + + + + + + + + + + + + ≤

ϕ

ψ

ψ

(2.1) for all x,y∈X, where a1,a2,a3,a4 are non negative reals with

1) ( 1)

( ₃

2

1+a + s+ a +s s+

sa a₄ <1.

ψ

and

ϕ

are altering distance and ultra altering distance functions respectively ,F∈C such that

ψ

(t+s)≤

ψ

(t)+

ψ

(s). Then S and T have a unique common fixed point in X.

Proof: Let x₀ be an arbitrary point X and define

⋯ 0,1,2, = , = ,

= 2 2 2 2 1

1

2 Sx x Tx k

x k+ k k+ k+ Then form 2.1 we have

88 ))) , ( ) ( ) , ( ) (( )), , ( ) ( ) , ( ) (( ( = )])) , ( ) , ( [ ) , ( ) ( ) , ( ) (( )]), , ( ) , ( [ ) , ( ) ( ) , ( ) (( ( ))) , ( ) , ( ) ( ) , ( ) (( )), , ( ) , ( ) ( ) , ( ) (( ( = 2 2 1 2 4 3 2 1 2 2 4 3 1 2 2 1 2 4 3 2 1 2 2 4 3 1 2 2 1 2 1 2 2 4 2 2 1 2 3 2 1 2 2 3 1 2 2 1 2 1 2 2 4 2 2 1 2 3 2 1 2 2 3 1 2 2 2 4 2 2 1 2 3 2 1 2 2 3 1 2 2 2 4 2 2 1 2 3 2 1 2 2 3 1 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ≤ + + + + + + + + k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k x x d sa a a x x d sa a a x x d sa a a x x d sa a a F x x d x x d sa x x d a a x x d a a x x d x x d sa x x d a a x x d a a F x x d a x x d a a x x d a a x x d a x x d a a x x d a a F

ϕ

ψ

ϕ

ψ

ϕ

ψ

This implies that ( , ).

1 ) ,

( 2 2 1

4 3 1 4 3 1 2 2 1

2 + + +

− − −

+ +

≤ k k

k

k d x x

sa a a sa a a x x

d Similarly, we have

)) , (

(d x_2k+2 x_2k+3

ψ

)])) , ( ) ( [ )] ( ) , ( [ ) , ( 1 ) , ( )] , ( [1 ) , ( ( )]), , ( ) ( [ )] ( ) , ( [ ) , ( 1 ) , ( )] , ( [1 ) , ( ( ( )) ( ( = 1 2 2 2 2 2 1, 2 4 2 2 2, 2 1 2 1 2 3 2 2 1 2 2 2 2 2 1 2 1 2 2 2 2 1 2 1 1 2 2 2 2 2 1, 2 4 2 2 2, 2 1 2 1 2 3 2 2 1 2 2 2 2 2 1 2 1 2 2 2 2 1 2 1 2 2 1, 2 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ≤ k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k Sx x d Tx x d a Tx x d Sx x d a x x d Tx x d Sx x d a x x d a Sx x d Tx x d a Tx x d Sx x d a x x d Tx x d Sx x d a x x d a F Tx Sx d

ϕ

ψ

ψ

)])) , ( ) ( [ )] ( ) , ( [ ) , ( 1 ) , ( )] , ( [1 ) , ( ( )]), , ( ) ( [ )] ( ) , ( [ ) , ( 1 ) , ( )] , ( [1 ) , ( ( ( = 2 2 2 2 3 2 1, 2 4 3 2 2, 2 2 2 1 2 3 2 2 1 2 3 2 2 2 2 2 1 2 2 2 2 1 2 1 2 2 2 2 3 2 1, 2 4 3 2 2, 2 2 2 1 2 3 2 2 1 2 3 2 2 2 2 2 1 2 2 2 2 1 2 1 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k x x d x x d a x x d x x d a x x d x x d x x d a x x d a x x d x x d a x x d x x d a x x d x x d x x d a x x d a F

ϕ

ψ

)])) , ( ) , ( [ ) , ( ) ( ) , ( ) (( )]), , ( ) , ( [ ) , ( ) ( ) , ( ) (( ( ))) , ( ) , ( ) ( ) , ( ) (( )), , ( ) , ( ) ( ) , ( ) (( ( = 3 2 2 2 2 2 1 2 4 3 2 2 2 3 2 2 2 1 2 3 1 3 2 2 2 2 2 1 2 4 3 2 2 2 3 2 2 2 1 2 3 1 3 2 1 2 4 3 2 2 2 3 2 2 2 1 2 3 1 3 2 1 2 4 3 2 2 2 3 2 2 2 1 2 3 1 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ≤ + + + + + + + + k k k k k k k k k k k k k k k k k k k k k k k k k k k k x x d x x d sa x x d a a x x d a a x x d x x d sa x x d a a x x d a a F x x d a x x d a a x x d a a x x d a x x d a a x x d a a F

ϕ

ψ

ϕ

ψ

))). , ( ) ( ) , ( ) (( )), , ( ) ( ) , ( ) (( ( = 3 2 2 2 4 3 2 2 2 1 2 4 3 1 3 2 2 2 4 3 2 2 2 1 2 4 3 1 + + + + + + + + + + + + + + + + + + k k k k k k k k x x d sa a a x x d sa a a x x d sa a a x x d sa a a F

ϕ

ψ

This implies that

). , ( 1 ) ,

( 2 1 2 2

4 3 1 4 3 1 3 2 2

2 + + + +

− − −

+ +

≤ k k

k

k d x x

89 Putting 4 3 1 4 3 1 1 = sa a a sa a a − − − + +

λ

As sa₁+a₂+(s+1)a₃+s(s+1)a₄ <1, it is clear that <1/s, we have , ). , ( ) , ( ) ,

( 0 1

1 1

2

1 x x x d x x

x

d n n n n n

+ +

+

+ ≤

λ

≤⋯≤

λ

Hence for any m> n, we have

). , ( ] 1 [ ) , ( ] ) ( [1 = ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( = )] , ( ) , ( [ ) , ( ) , ( ) , ( = )] , ( ) , ( [ ) , ( 0 1 0 1 1 3 3 2 2 0 1 1 0 1 2 3 0 1 1 2 0 1 1 1 3 2 3 2 1 2 1 2 2 2 1 2 1 2 2 1 2 1 1 1 1 1 x x d s s x x d s s s s s x x d s x x d s x x d s x x d s x x d s x x d s x x d s x x sd x x d s x x d s x x sd x x d x x d s x x sd x x sd x x sd x x d x x d s x x d n m n m n m n n n m m n m n n n n n n n m n n n n n m n n n n n m n n n m n n n m n

λ

λ

λ

λ

λ

λ

λ

δ

λ

λ

λ

− ≤ + + + + + + + + + ≤ + + + + ≤ + + + + ≤ + + ≤ − − + + + − + − + + + + + + + + + + + + + + + + + + ⋯ ⋯ ⋯

Since P is a normal cone with normal constant K, so we get

P P

P

P ( , )

1 )

,

( d x1 x0

s s K x x d n m n

λ

λ

− ≤ .

This implies Pd(xn,xm)P→0asn,m→∞ since 0<s

λ

<1.

Hence {x_n} is a Cauchy sequence. Since (X,d) is a complete cone b-metric

space, there exists p∈X such that x_n→ p as n→∞. Now, since

)) , ( (d pTp

ψ

≤ψ(s[d(p,x2n+1)+d(x2n+1,Tp)]) )) , ( ) , ( (

=

ψ

sd Sx2n Tp +sd p x2n+1

)) , ( ( )) , ( (

=

ψ

sd Sx_2n Tp +

ψ

sd p x_2n+1

) , ( 1 ) , ( )] , ( [1 ) , ( [ ) , ( ( ( 2 2 2 2 2 1 1 2 p x d Tp p d Sx x d a p x d a s x p sd F n n n n n + + + + ≤

ψ

+ ))]), , ( ) , ( ( )) , ( ) , (

( 2 2 1 4 2 2

3 d x n x n d p Tp a d x n Tp d p Sx n

a + + +

+ + ) , ( 1 ) , ( )] , ( [1 ) , ( [ ) , ( ( 2 2 2 2 2 1 1 2 p x d Tp p d Sx x d a p x d a s x p sd n n n n n ₊ + + + + )) , ( ( ))])) , ( ) , ( ( )) , ( ) , (

( 2 2 1 4 2 2 2 1

3 + + + + + +

+a d x n x n d p Tp a d x n Tp d p Sx n

ψ

sd p x n

_{1 ( , )}

) , ( )] , ( [1 ) , ( [ ) , ( ( ( = 2 1 2 2 2 2 1 1 2 p x d Tp p d x x d a p x d a s x p sd F n n n n n + + + + + +

ψ

))]), , ( ) , ( ( )) , ( ,

( _{2 2 1 4 2 2}

3d x n x n d pTp a d x n Tp d p Sxn

a + + +

90 ) , ( 1 ) , ( )] , ( [1 ) , ( [ ) , ( (( 2 1 2 2 2 2 1 1 2 p x d Tp p d x x d a p x d a s x p sd n n n n n + + + + + +

ϕ

)) , ( ( ))]))) , ( ) , ( ( )) , ( ,

( _{2 2 1 4 2 2 2 1}

3 + + + + + +

+ad x n x n d pTp a d x n Tp d p Sxn

ψ

sd p x n

) , ( 1 ) , ( )] , ( [1 ) , ( [ ) , ( ( ( 2 1 2 2 2 2 1 1 2 p x d Tp p d x x d a p x d a s x p sd F n n n n n ₊ + + + ≤ + +

ψ

))]), , ( ) , ( ( )) , ( ,

( 2 2 1 4 2 2

3d x n x n d pTp a d x n Tp d p Sxn

a + + +

+ + ) , ( 1 ) , ( )] , ( [1 ) , ( [ ) , ( (( 2 1 2 2 2 2 1 1 2 p x d Tp p d x x d a p x d a s x p sd n n n n n + + + + + +

ϕ

))]))) , ( ) , ( ( )) , ( ,

( _{2 2 1 4 2 2}

3d x n x n d pTp a d x n Tp d p Sx n

a + + +

+ +

As x_n → p and x_n+1→ p as

n

→

∞

, we get

0 ] ) , ( ) ( ) , ( [ ) , ( )

(1−sa2−sa3−sa4 Pd p Tp P≤K sa1Pd x2n pP+s a4 Pd p x2n+1 P →

as n→∞.

Hence Pd(Tp,p)P=0. Since (1−sa₂−sa₃−sa₄)>0. Thus we get Tp= p, that is, p is a fixed point of T. Uniqueness:

Let q be another fixed point common to S and T, that is Sq=Tq=q such that p≠q.

Then from 2.1 we have ))

91

which is a contradiction. Hence Pd(p,q)P=0 and so p= q. Thus p is a unique common fixed point of S and T.

This completes the proof. Putting S= T, we have the following results.

Corollary 2.2. Let (X,d) be a complete cone b-metric space (CCbMS) with the co-efficient s≥1 and P be a normal cone with normal constant K. Suppose that the mappings S,T:X →X satisfy:

)])) , ( ) , ( [ )] , ( ) , ( [

) , ( 1

)] , ( ) , ( [1 ) , ( ( )]), , ( ) , ( [

)] , ( ) , ( [ )

, ( 1

)] , ( ) , ( [1 ) , ( ( ( )) , ( (

4 3

2 1

4

3 2

1

Tx y d Ty x d a Ty y d Tx x d a

y x d

Ty y d Tx x d a y x d a Tx

y d Ty x d a

Ty y d Tx x d a y

x d

Ty y d Tx x d a y x d a F Ty

Tx d

+ +

+

+ +

+ + +

+

+ +

+ + + ≤

ϕ

ψ

ψ

(2.2) for all x,y∈X , where a1,a2,a3,a4 are non negative reals with

1 < 1) ( 1)

( 3 4

2

1 a s a s s a

sa + + + + + .

ψ

and

ϕ

are altering distance and ultra altering distance functions respectively ,F∈C such that

ψ

(t+s)≤

ψ

(t)+

ψ

(s). Then T has a unique common fixed point in X.

Proof: The proof of corollary (2.2) immediately follows from Theorem (2.1) by taking

T

S = . This completes the proof.

Corollary 2.3. Let (X,d) be a complete cone b-metric space (CCbMS) with the co-efficient s≥1 and P be a normal cone with normal constant K. Suppose that the mappings S,T:X →X satisfy:

)) , (

(d TnxTny

ψ

)])) ,

( ) , ( [ )] , ( ) , ( [

) , ( 1

)] , ( ) , ( [1 )

, ( ( )]), ,

( ) , ( [

)] , ( ) , ( [ )

, ( 1

)] , ( ) , ( [1 )

, ( ( (

4 3

2 1

4

3 2

1

x T y d y T x d a y T y d x T x d a

y x d

y T y d x T x d a y x d a x T y d y T x d a

y T y d x T x d a y

x d

y T y d x T x d a y x d a F

n n

n n

n n

n n

n n

n n

+ +

+

+ +

+ + +

+

+ +

+ + + ≤

ϕ

ψ

(2.3) for all x,y∈X , where a1,a2,a3,a4 are non negative reals with

1 < 1) ( 1)

( 3 4

2

1 a s a s s a

sa + + + + + .

ψ

and

ϕ

are altering distance and ultra altering distance functions respectively ,F∈C such that

ψ

(t+s)≤

ψ

(t)+

ψ

(s). Then T has a unique common fixed point in X.

92 )) , ( ) 2 (( =

)])) , ( ) , ( [ )] , ( ) , ( [

) , ( 1

) , ( )] , ( [1 ) , ( (

)]), , ( ) , ( [ )] , ( ) , ( [

) , ( 1

) , ( )] , ( [1 ) , ( ( ( =

)])) ,

( ) , ( [ )] , ( ) , ( [

) , ( 1

) , ( )] ,

( [1 ) , ( (

)]), ,

( ) , ( [ )] , ( ) , ( [

) , ( 1

) , ( )] ,

( [1 ) , ( ( (

)])) ,

( ) , ( [ )] , ( ) , ( [

) , ( 1

) , ( )] ,

( [1 ) , ( (

)]), ,

( ) , ( [ )] , ( ) , ( [

) , ( 1

) , ( )] ,

( [1 ) , ( ( (

)) , ( ( =

)) , ( ( = )) , ( (

4 1

4 3

2 1

4 3

2 1

4 3

2 1

4 3

2 1

4 3

2 1

4 3

2 1

u Tu d a a

Tu u d u Tu d a u u d Tu Tu d a

u Tu d

u u d Tu Tu d a u Tu d a

Tu u d u Tu d a u u d Tu Tu d a

u Tu d

u u d Tu Tu d a u Tu d a F

u TT u d u T Tu d a u T u d u TT Tu d a

u Tu d

u T u d u TT Tu d a u Tu d a

u TT u d u T Tu d a u T u d u TT Tu d a

u Tu d

u T u d u TT Tu d a u Tu d a F

Tu T u d u T Tu d a u T u d Tu T Tu d a

u Tu d

u T u d Tu T Tu d a u Tu d a

Tu T u d u T Tu d a u T u d Tu T Tu d a

u Tu d

u T u d Tu T Tu d a u Tu d a F

u T Tu T d

u T u TT d u

Tu d

n n

n n

n n

n n

n n

n n

n n

n n

n n

n n

n n

n n

n n

n n

+

+ +

+

+ +

+ +

+ +

+

+ +

+ +

+ +

+

+ +

+ +

+ +

+

+ +

+ + ≤

+ +

+

+ +

+ +

+ +

+

+ +

+ + ≤

ψ

ϕ

ψ

ϕ

ψ

ϕ

ψ

ψ

ψ

ψ

and so d(Tu,u)=0. Thus, Tu= u. This show that T has a unique fixed point X. This completes the proof.

Putting a1=k,a2 =a3=a4 =0 in Corollary (2.2) then we have the following result.

Corollary 2.4. Let (X,d) be a complete cone b-metric space (CCbMS) with the co-efficient s≥1 and P be a normal cone with normal constant K. Suppose that the mappings S,T:X →X satisfy:

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

(d Tx Ty F

ψ

kd x y

ϕ

kd x y

ψ

≤

For all x,y∈X, where k∈(0,1) is a constant with sk<1.

ψ

and

ϕ

are altering distance and ultra altering distance functions respectively, F∈C such that

). ( ) ( )

(t s

ψ

t

ψ

s

ψ

+ ≤ + Then T has a unique common fixed point in X.

Note 2.5. Corollary 2.4 extends well known Banach contraction principle from complete

metric space to that setting of complete cone b-metric space via cone C-class function consider in this paper.

93

Corollary 2.6. Let (X,d) be a complete cone b-metric space (CCbMS) with the co-efficient s≥1 and P be a normal cone with normal constant K. Suppose that the mappings S,T:X →X satisfy:

)])) , ( ) , ( [ ( )]), , ( ) , ( [ ( ( )) , (

(d Tx Ty ≤F

ψ

k d xTx +d y Ty

ϕ

k d xTx +d y Ty

ψ

For all x,y∈X, where ) 1 1 (0,

+ ∈

s

k is a constant with k(s+1)<1.

ψ

and

ϕ

are

altering distance and ultra altering distance functions respectively ,F∈C such that ).

( ) ( )

(t s

ψ

t

ψ

s

ψ

+ ≤ + Then T has a unique common fixed point in X.

Note 2.7. Corollary 2.6 extends well known Kannan contraction principle from complete

metric space to that setting of complete cone b-metric space via cone C-class function consider in this paper.

Putting a4 =k,a1=a2 =a3 =0 in corollary (2.2), then we have the following result.

Corollary 2.8. Let (X,d) be a complete cone b-metric space (CCbMS) with the co-efficient s≥1 and P be a normal cone with normal constant K. Suppose that the mappings S,T:X →X satisfy:

)])) , ( ) , ( [ ( )]), , ( ) , ( [ ( ( )) , (

(d Tx Ty ≤F

ψ

k d xTy +d yTx

ϕ

k d xTy +d yTx

ψ

For all x,y∈X, where ) 1 1 (0,

+ ∈

s

k is a constant with k(s+1)<1.

ψ

and

ϕ

are

altering distance and ultra altering distance functions respectively ,F∈C such that ).

( ) ( )

(t s

ψ

t

ψ

s

ψ

+ ≤ + Then T has a unique common fixed point in X.

Note 2.9. Corollary 2.8 extends well known Kannan contraction principle from complete

metric space to that setting of complete cone b-metric space via cone C-class function consider in this paper.

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