Vol 4, No 11 (2013)

Available online throug

ISSN 2229 – 5046

ON A MAP OF XIA ET.AL AND SOME FIXED POINT THEOREMS FOR A CLASS OF

CONTRACTIVE MAPPINGS IN G-DISLOCATED METRIC SPACES

CLEMENT AMPADU*

31 Carrolton Road, Boston, Massachusetts, 02132, U.S.A.

(Received on: 14-10-13; Revised & Accepted on: 14-11-13)

ABSTRACT

I

n Xia et.al [Common fixed points for two self-mappings on symmetric sets and complete metric spaces, Advances in Mathematics, vol. 36, no. 4, pp. 415–420, 2007.], the following map is introduced: Let

R

₊

=

[

0

,

∞

)

, let

+ +

→

R

R

T

₁

:

2 , and

T

₂

:

R

₊3

→

R

₊, satisfy the following: (i) if

w

≤

T

₁

(

u

,

v

)

, then there exist

c

∈

(

0

,

1

)

, such that

{ }

u

v

c

w

≤

max

,

; (ii) if

w

≤

T

2

(

u

,

v

,

r

)

, then there exists

c

∈

(

0

,

1

)

, such that

w

≤

c

max

{

u

,

v

,

r

}

. The authors use these maps to prove the following fixed point result: Let

(

X

,

d

)

be a complete metric space, and let

X

X

g

f

,

:

→

be continuous mappings, and for all

x

,

y

∈

X

such that

(

(

,

(

)),

(

,

(

))

)

))

(

),

(

(

f

x

g

y

T

1

d

x

f

x

d

y

g

y

d

≤

, or

d

(

f

(

x

),

g

(

y

))

≤

T

2

(

d

(

x

,

y

),

d

(

x

,

f

(

x

)),

d

(

y

,

g

(

y

))

)

,

then,

f

,

g

have a unique common fixed point. In the present paper we define an analogous map in the setting of G-dislocated metric spaces and use it to obtain fixed point theorems.

AMS Subject Classification: 47H10, 54H25.

I. INTRODUCTION

Fixed point theory, a pivotal branch of analysis, has several applications. One of the most celebrated fixed point theorems is due to Banach [1], and several generalizations of it have appeared in the literature, see [2-9] for examples. In this paper we extend the map of Xia et. al [Common fixed points for two self-mappings on symmetric sets and complete metric spaces, Advances in Mathematics, vol. 36, no. 4, pp. 415–420, 2007], and use it to obtain fixed point theorems in the setting of G-dislocated metric spaces.

II. BASIC NOTIONS AND NOTATIONS

In analogy to Zeyada et.al [A generalization of a fixed point theorem due to Hitzler and Seda in dislocated quasi-metric spaces, The Arabian Journal for Science and Engineering Section A, vol. 31, no. 1, pp. 111–114, 2006], we introduce the following .

Definition 1: Let

X

be a non-empty set. We will say

G

:

X

×

X

×

X

→

R

₊ is a distance function if for all

X

w

z

y

x

,

,

,

∈

(a)

G

(

x

,

y

,

z

)

≥

0

(b)

G

(

x

,

y

,

z

)

=

0

⇔

x

=

y

=

z

(c)

G

(

x

,

y

,

z

)

≤

G

(

w

,

y

,

z

)

+

G

(

x

,

w

,

z

)

+

G

(

x

,

y

,

w

)

Here

(

X

,

G

)

_qwill denote a G-quasi-metric space.

Corresponding author: CLEMENT AMPADU*

Definition: 2 Let

X

be a non-empty set, and

G

:

X

×

X

×

X

→

R

₊ be a distance function . If for all

x

,

y

,

z

,

w

∈

X

(a)

G

(

x

,

y

,

z

)

≥

0

(b)

G

(

x

,

y

,

z

)

=

G

(

y

,

x

,

z

)

=



=

0

⇒

x

=

y

=

z

(c)

G

(

x

,

y

,

z

)

≤

G

(

w

,

y

,

z

)

+

G

(

x

,

w

,

z

)

+

G

(

x

,

y

,

w

)

Here

(

X

,

G

)

_dq will denote a G-dislocated quasi-metric space.

Definition: 3 Let

X

be a non-empty set, and let

G

:

X

×

X

×

X

→

R

₊ be a distance function. If for all

x

,

y

,

z

,

w

∈

X

(a)

G

(

x

,

y

,

z

)

≥

0

(b)

G

(

x

,

y

,

z

)

=

G

(

y

,

x

,

z

)

=



=

0

⇒

x

=

y

=

z

(c)

G

(

x

,

y

,

z

)

=

G

(

y

,

x

,

z

)

=



(d)

G

(

x

,

y

,

z

)

≤

G

(

w

,

y

,

z

)

+

G

(

x

,

w

,

z

)

+

G

(

x

,

y

,

w

)

Here

(

X

,

G

)

_dwill denote G-dislocated metric space

Definition: 4 A sequence

{ }

x

_n in

(

X

,

G

)

_dqwill be called a G-dq-Cauchy sequence if for a given

ε

>

0

, there exist

N

n

0

∈

such that

G

(

x

m

,

x

l

,

x

n

)

<

ε

, or

G

(

x

l

,

x

m

,

x

n

)

<

ε

, or



, that is,

{

(

,

,

),

(

,

,

),

_

}

<

ε

min

G

x

m

x

n

x

l

G

x

n

x

m

x

l for all

m

,

n

,

l

≥

n

0.

Definition: 5 A sequence

{ }

x

_n in the G-metric space

(

X

,

G

)

will be called G-convergent to some

x

∈

X

provided that

lim

G

(

x

_n

,

x

_m

,

x

)

=

lim

G

(

x

_m

,

x

_n

,

x

)

=



=

0

. We will call

x

, the G-limit of

{ }

x

_n .

Definition: 6 We will say the G-metric space

(

X

,

G

)

is G-complete if every G-Cauchy sequence in it converges with respect to

x

∈

X

.

Lemma 7: Every convergent sequence in

(

X

,

G

)

is a Cauchy sequence

Proof: Let

{ }

x

_n be a sequence which converges to some

x

∈

X

. Suppose

ε

>

0

is arbitrary, then there exists

N

n

₀

∈

with

3

)

,

,

(

x

x

x

<

ε

G

_{m n} for all

n

,

m

≥

n

₀. So for

n

,

m

,

l

≥

0

,

We obtain

+

+

<

ε

+

ε

+

ε

=

ε

3

3

3

)

,

,

(

)

,

,

(

)

,

,

(

x

m

x

n

x

G

x

m

x

x

l

G

x

x

n

x

l

G

. Hence

{ }

x

_n is Cauchy.

Lemma: 8 Limit in

(

X

,

G

)

_d are unique.

Proof: Suppose

x

,

y

,

z

are limits of the sequence

{ }

x

_n . Then

x

_n

→

x

,

x

_n

→

y

,

x

_n

→

z

as

n

→

∞

. By triangle inequality,

)

,

,

(

)

,

,

(

)

,

,

(

)

,

,

(

x

y

z

G

x

x

x

G

x

y

x

G

x

x

z

G

≤

_{n n}

+

_{n n}

+

_{n n} . If we take the limit as

n

→

∞

, this implies that

)

,

,

(

)

,

,

(

)

,

,

(

)

,

,

(

x

y

z

G

x

x

x

G

y

y

y

G

z

z

z

G

≤

+

+

. Since

G

is symmetric in the variables we see that

)

,

,

(

)

,

,

(

)

,

,

(

)

,

,

(

)

,

,

(

x

y

z

G

x

y

z

G

x

x

x

G

y

y

y

G

z

z

z

G

=

≤

+

+

=



. Obviously,

,

,

0

)

,

,

(

)

,

,

(

,

0

)

,

,

(

)

,

,

(

y

x

z

−

G

x

y

z

≤

G

z

y

x

−

G

x

y

z

≤



G

etc. So

)

,

,

(

)

,

,

(

)

,

,

(

)

,

,

(

)

,

,

(

)

,

,

(

z

y

x

G

y

x

z

G

x

y

z

G

x

x

x

G

y

y

y

G

z

z

z

G

=

=

≤

+

+

=

)

,

,

(

)

,

,

(

)

,

,

(

)

,

,

(

x

y

z

G

x

x

x

G

x

y

x

G

x

x

z

G

≤

_{n n}

+

_{n n}

+

_{n n} , we see that

G

(

x

,

y

,

z

)

=

0

. So obviously,

0

)

,

,

(

)

,

,

(

)

,

,

(

=

=

=

=

G

z

y

x

G

y

x

z

G

x

y

z



. In particular

x

=

y

=

z

.

Example: 9 Let

X

=

R

₊. Define

G

(

x

,

y

,

z

)

=

x

−

y

+

x

−

z

+

y

−

z

=

d

(

x

,

y

)

+

d

(

x

,

z

)

+

d

(

y

,

z

)

, then

d

G

X

,

)

(

is a metric space. If

{ }

x

_n is an arbitrary sequence in

X

; suppose there exists a positive integer

N

, such

that

k

>

N

gives

6

ε

<

−

a

x

_k , then for any

m

,

n

,

l

>

N

, we see that

ε

ε

ε

ε

₊

₊

₌

≤

+

+

=

3

3

3

)

,

(

)

,

(

)

,

(

)

,

,

(

x

n

x

m

x

l

d

x

n

x

m

d

x

n

x

l

d

x

m

x

l

G

. Since

3

6

6

)

,

(

x

x

=

x

−

x

≤

x

−

a

+

x

−

a

<

ε

+

ε

=

ε

d

_{n m n m n m} . So

{ }

x

_n is a Cauchy sequence in

X

. Also as

∞

→

n

,

x

_n

→

a

∈

X

. Hence every Cauchy sequence in

X

is convergent with respect to

G

. It follows that

d

G

X

,

)

(

is a complete metric space.

III. MAIN RESULTS

Before stating the main result, we introduce some definitions and a lemma that will be useful in the sequel.

Definition: 10 [R. Chen, Fixed Point Theory and Applications, National Defense Industry press, 2012]: There exist

)

(t

φ

that satisfy condition

φ′

, if one lets

φ

:

[

0

,

∞

)

→

[

0

,

∞

)

be a nondecreasing and nonnegative, then

0

)

(

lim

φ

n

t

=

for a given

t

>

0

Lemma: 11[R. Chen, Fixed Point Theory and Applications, National Defense Industry press, 2012]: If

φ

satisfy

φ′

, then

φ

(

t

)

<

t

, for a given

t

>

0

In analogy to Xia et.al [Common fixed points for two self-mappings on symmetric sets and complete metric spaces, Advances in Mathematics, vol. 36, no. 4, pp. 415–420, 2007] we have the following lemma

Lemma: 12 Let

F

:

R

₊4

→

R

₊ be a mapping, and suppose it satisfies the condition

φ

′

, for all

u

,

v

≥

0

, such that,

)

,

,

,

(

v

v

v

u

F

u

≤

, or

u

≤

F

(

v

,

v

,

u

,

v

)

, or

u

≤

F

(

v

,

u

,

v

,

v

)

, or

u

≤

F

(

u

,

v

,

v

,

v

)

, then

u

≤

φ

(v

)

Our main result is as follows.

Theorem: 13 Let

(

X

,

G

)

_dbe a complete metric space, and let

f

,

g

,

h

:

X

×

X

→

R

₊ be mappings such that (i) either

f

,

g

,

h

is continuous, and (ii) there exist

F

satisfying the condition

φ

′

, for all

x

,

y

,

z

,

u

,

v

,

w

∈

X

, such that

)))

,

(

,

,

(

)),

,

(

,

,

(

)),

,

(

,

,

(

),

,

,

(

(

))

,

(

),

,

(

),

,

(

(

f

x

u

g

y

v

h

z

w

F

G

x

y

z

G

x

u

f

x

u

G

y

v

g

y

v

G

z

w

h

z

w

G

≤

, then

h

g

f

,

,

have a unique fixed point.

Proof:

Put

x

_n

=

u

_n

=

( )

fgh

n

(

x

₀

,

u

₀

)

=

fgh

(

x

_n−2

,

u

_n−2

)

,

y

_n

=

v

_n

=

g

( )

fgh

n−1

(

x

₀

,

u

₀

)

,

z

_n

=

w

_n

=

h

( )

fgh

n−2

(

x

₀

,

u

₀

)

, for

n

=

2

,

3

,

4

,

5

,



Obviously,

)

,

(

−1 −1

=

n n

n

g

x

u

y

,

z

n

=

h

(

x

n−2

,

u

n−2

)

,

fg

(

z

n

,

w

n

)

=

y

n,

fg

(

y

n

,

v

n

)

=

x

n,

)

,

(

)

,

(

2 n n n n

n

h

x

u

hfg

y

v

Let

x

=

u

=

gh

(

x

_n

,

u

_n

)

,

y

=

z

=

x

_n,

v

=

w

=

u

_n, then by the condition, we have

2 1 2

(

(

,

),

,

), (

(

,

),

(

,

),

(

,

),

(

,

,

)

(

,

, (

,

)), (

,

, (

,

))

n n n n n n n n n n

n n n

n n n n n n n n

G gh x u

x x

G gh x u

gh x u

fgh x u

G x

y

z

F

G x u g x u

G x u h x u

+ + +





≤ 







By Lemma 12,

(

(

,

,

(

,

)

)

)

,

,

(

x

_{n 2}

y

_{n 1}

z

_{n 2}

G

x

_n

x

_n

gh

x

_n

u

_n

G

_{+ + +}

≤

φ

Also we notice that

(

(

,

,

)

)

))

,

(

,

,

(

x

_n

x

_n

gh

x

_n

u

_n

G

x

_n

x

_n

y

_n

G

≤

φ

By Lemma 11

(

(

,

,

)

)

)

,

,

(

x

_{n 2}

y

_{n 1}

z

_{n 2} 2

G

x

_n

x

_n

y

_n

G

_{+ + +}

≤

φ

By induction, we notice that

(

)

2

2 1 2 2 0 0 0 0

(

_n

,

_n

,

_n

)

n

( , ( ,

), ( ,

)

G x

₊

y

₊

z

₊

≤

ϕ

G x g x u

h x u

Similarly, we obtain

(

)

2 1

1 1 2 2 0 0 0 0

(

n

,

n

,

n

)

n

( , ( ,

), ( ,

)

G y

₊

x

₊

z

₊

≤

ϕ

−

G x g x u

h x u

(

)

2 2

1 2 2 0 0 0 0

(

_n

,

_n

,

_n

)

n

( , ( ,

), ( ,

)

G y

₊

z

₊

x

≤

ϕ

−

G x g x u

h x u

If

n

≥

2

, we obtain

2 1 2 1 2 1 1 2 1 2

(

_n

,

_n

,

_n

)

(

_n

,

_n

,

_n

)

(

_n

,

_n

,

_n

)

(

_n

,

_n

,

_n

)

G x

₊

x

₊

x

≤

G x

₊

y

₊

z

₊

+

G y

₊

x

₊

z

₊

+

G y

₊

z

₊

x

(

)

(

)

(

)

(

)

2 2 1

2 0 0 0 0 2 0 0 0 0

2 2

2 0 0 0 0

2 2

2 0 0 0 0

( , ( ,

), ( ,

)

( , ( ,

), ( ,

)

( , ( ,

), ( ,

)

3

( , ( ,

), ( ,

)

n n

n

n

G x g x u

h x u

G x g x u

h x u

G x g x u

h x u

G x g x u

h x u

ϕ

ϕ

ϕ

ϕ

−

−

−

≤

+

+

≤

Now we observe by the condition

φ

′

for

n

,

m

,

l

∈

N

such that

l

>

m

>

n

, we have

2 1 3 2 1 2 1

( n, m, n m l) ( n , n , n) ( n , n , n ) ( n m l , n m l , n m l)

G x x x+ + ≤G x+ x+ x +G x+ x+ x+ + + G x+ + − x+ + − x+ +

(

)

(

)

(

)

(

)

(

)

2 2 2( 3) 2

2 0 0 0 0 2 0 0 0 0

2( 2 _ 2

2 0 0 0 0

2( 2) 2

2 0 0 0 0

2 2

2 0 0 0 0

2 2

3

( , ( ,

), ( ,

)

3

( , ( ,

), ( ,

)

3

( , ( ,

), ( ,

)

3

( , ( ,

), ( ,

)

3

( , ( ,

), ( ,

)

0

n n

n m l

n m l i

i n

i

i n

G x g x u

h x u

G x g x u

h x u

G x g x u

h x u

G x g x u

h x u

G x g x u

h x u

ϕ

ϕ

ϕ

ϕ

ϕ

− + −

+ + − −

+ + − −

= − ∞

= −

≤

+

+

+

≤

≤

→

∑

∑



It follows that

G

(

x

_n

,

x

_m

,

x

_l

)

→

0

as

m

,

n

,

l

→

∞

. Hence

{ }

x

_n is a Cauchy sequence in

X

.

Since

(

X

,

G

)

_d is complete,

{ }

x

_n converges to some

x

*

∈

X

. In a similar way,

{ }

u

n converges to

u

*

=

x

*

∈

X

, say. If

h,

g

are both continuous, the by their continuity, we see that as.

n

→

∞

z

n+2

=

z

*

=

h

(

x

*

,

u

*

)

, and

)

,

(

* * *

1

y

g

x

u

y

n+

=

=

. So as

n

→

∞

,

G

(

x

n+2

,

y

n+1

,

z

n+2

)

≤

G

(

x

*

,

y

*

,

z

*

)

≤

0

.

So,

(

x

*

,

u

*

)

is a fixed point of

h,

g

.

By the given condition (note :

u

_*

= ∈

x

_*

X

,

say), then,

( ( ,

), ( ,

), ( ,

))

(0, ( ,

, ( ,

), 0, 0 )

It follows that

G

(

x

_*

,

u

_*

,

f

(

x

_*

,

u

_*

))

≤

φ

(

0

)

=

0

, thus,

f

(

x

_*

,

u

_*

)

=

x

_*

=

u

_*. So

(

x

_*

,

u

_*

)

is a common fixed point of

h

g

f

,

,

.

Regarding uniqueness, if

y

_* is another common fixed point of

f

,

g

,

h

, then by the given condition,

* * * * * * * * *

( ,

,

)

( ( ,

), ( ,

), ( ,

))

G x x y

=

G f x x

g x x

h y y

* * * * * * * * * * * * * * *

* * *

( ( ,

,

), ( ,

, ( ,

)), ( ,

, ( ,

)), ( ,

, ( ,

)))

( ( ,

,

), 0, 0, 0 )

F G x x y

G x x f x x

G x x g x x

G y y f y y

F G x x y

≤

=

Since

G

(

x

*

,

x

*

,

y

*

)

≤

φ

(

0

)

=

0

, it follows that

x

*

=

y

*, and uniqueness follows, completing the proof.

IV. CONCLUDING REMARKS

Matthews [Metric domains for completeness. Technical Report 76 [Ph.D. thesis], Department of Computer Science, University of Warwick, Coventry, UK, 1986] generalized Banach contraction mapping theorem in dislocated metric space that is a wider space than metric space. In this paper, we established common fixed point theorems for a class of contractive mappings in the setting of G-dislocated metric spaces.

Let

{ }

x

_n be a sequence of points, we will say that

x

_* is the condensation point of

{ }

x

_n , if there exists subsequence

{ }

x

nk of

{ }

x

n such that

x

nk

→

x

*.

By way of this definition, we have the following theorem

Theorem 14: Let

(

X

,

G

)

_dbe a complete metric space, and let

f

,

g

,

h

:

X

×

X

→

R

₊ be continuous mappings, if

(a) There exists

F

satisfying the condition

φ

′

, for all

x

,

y

,

z

,

u

,

v

,

w

∈

X

such that

)))

,

(

,

,

(

)),

,

(

,

,

(

)),

,

(

,

,

(

),

,

,

(

(

))

,

(

),

,

(

),

,

(

(

f

x

u

g

y

v

h

z

w

F

G

x

y

z

G

x

u

f

x

u

G

y

v

g

y

v

G

z

w

h

z

w

G

≤

(b) There exists

(

x

₀

,

y

₀

)

∈

X

×

X

, such that

{

( )

fgh

n

(

x

₀

,

y

₀

)

}

have a condensation point.

Then

f

,

g

,

h

have a unique common fixed point

REFERENCES

[1] S. Banach, Sur le operations les ensembles abstraits et leur application aux equations integrals, Fund. Math 3 (1922), 131-181

[2] B. K. Dass and S. Gupta, “An extension of Banach contraction principle through rational expression,” Indian Journal of Pure and Applied Mathematics, vol. 6, no. 12, pp. 1455–1458, 1975.

[3] B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the American MathematicalSociety, vol. 226, pp. 257–290, 1977.

[4] S. G. Matthews, Metric domains for completeness. Technical Report 76 [Ph.D. thesis], Department of Computer Science, University of Warwick, Coventry, UK, 1986.

[5] P. Hitzler, Generalized metrices and topology in logic programming semantics [Ph.D. thesis], University College Cork, National University of Ireland, 2001.

[6] P. Hitzler and A. K. Seda, “Dislocated topologies,” Journal of Electrical Engineering, vol. 51, no. 12, pp. 3–7, 2000.

[8] C. T. Aage and J. N. Salunke, “The results on fixed points in dislocated and dislocated quasi-metric space,” Applied Mathematical Sciences, vol. 2, no. 57–60, pp. 2941–2948, 2008.

[9] A. Isufati, “Fixed point theorems in dislocated quasi-metric space,” Applied Mathematical Sciences, vol. 4, no. 5–8, pp. 217–223, 2010.