A Result on Fixed Point in Complete G-Metric Space

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IJSRR, 7(3) July – Sep., 2018 Page 1400

Research article

Available online www.ijsrr.org

ISSN: 2279–0543

International Journal of Scientific Research and Reviews

A Result on Fixed Point in Complete G-Metric Space

Raju Naga V.

*

Department of Mathematics, University College of Engineering Osmania University Hyderabad,Telangana, India

Email:[email protected]

ABSTRACT

In this paper , we define a generalized weakly contraction mapping in G-metric space and

present a fixed point theorem for such mapping in complete G-metric space which generalizes a

result in the recent literature.

KEYWORDS: G-metric space , fixed point, self map, generalized weakly contraction mapping

*Corresponding author

V.Naga Raju

Department of Mathematics,

University College of Engineering Osmania University

Hyderabad,Telangana, India

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INTRODUCTION AND PRILIMINARIES

In 2006, Mustafa and Sims2 introduced a notion of generalized metric space called G-metric

space . Later, several authors established many fixed point results for self mappings in G-metric spaces under certain contractions.

Definition 1.1. [2]

Let be a non empty set and let : × × →[0,∞) be a function

satisfying the following properties:

( , , ) = 0 if = = .

( , , ) > 0 for all , ∈ , with ≠ .

( , , )≤ ( , , ) for all , , ∈ , with ≠ .

( , , ) = ( , , ) = ( , , ) =..., (symmetry in all three variables)

( , , )≤ ( , , ) + ( , , ), for all , , , ∈ (rectangular inequality)

Then the function is called a -metric on and the pair ( , ) is called a -metric space.

Example.[2] Let

( , ) be a usual metric space. Then ( , ) is -metric space, where

( , , ) = ( , ) + ( , ) + ( , ), for all , , ∈ .

Definition 1.2.[3] Let

( , ) and ( ′_, ′₎_{be -metric spaces and} _{: ( , )}_→₍ ′_, ′₎_{be a}

function, then is said to be -continuous at a point ∈ if for given > 0 there exist > 0 such

that for , ∈ , ( , , ) < implies that ( , , ) < .

A function is -continuous on if and only if it is -continuous at all ∈ .

Definition 1.3. [3]

Let ( , ) be a -metric space and let { } be a sequence of points in . We

say that { } is -convergent to if lim , →∞ ( , , ) = 0. That is,for any > 0,there exist

N∈n such that ( , , ) < for all , ≥ n. We call as the limit of the sequence { } and we

write → as →∞ or lim _→_∞ = .

Proposition 1.4. [3]

Let ( , ) and ( ′_, ′₎_{be metric spaces. A function} _: _→ _is

-continuous at a point ∈ if and only if it is -sequentially continuous, that is, { } is

-convergent to ( ) whenever { } is -convergent to .

Proposition 1.5.[4]

Let ( , ) be a -metric space. Then the following statements are equivalent:

{ } is -convergent to .

( , , )→0 as → ∞.

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Definition 1.6.[3] Let

( , ) be a -metric space. A sequence { } is called -Cauchy sequence

if given > 0, there is N∈ n such that ( , , ) < for all , , ≥N, that is,

( , , )→0as , , →∞.

Definition1.7.[3]

A -metric space ( , ) is said to be -complete if every -cauchy sequence

in ( , ) is -convergent in ( , ).

Proposition 1.8. [4] Every -metric space

( , ) defines a metric space ( , ), where

defined by

( , ) = ( , , ) + ( , , ) for all , ∈ .

Proposition 1.9. [2] Let

( , ) be a -metric space. Then for any , , , ∈ , the following

hold.

If ( , , ) = 0 then = = .

( , , )≤ ( , , ) + ( , , ).

( , , )≤2 ( , , ).

( , , )≤ ( , , ) + ( , , ).

( , , )≤ { ( , , ) + ( , , ) + ( , , )}.

Definition 1.10. [5]

A mapping : [0,∞)→[0,∞) is called an altering distance function if is

continuous and non-decreasing

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

Definition 1.11. [1]

A self mapping : → , where ( , ) is a metric space, is said to be a

generalized weakly contractive mapping if

( , ) ≤ ( , ) − ∅(max{ ( , ), ( , )})for all x,y where ( , ) =

max{ ( , ), ( , ), ( , ), ( , ) + ( , ) } , is an altering distance function and

∅: [0,∞)→[0,∞) is continuous and non-decreasing function such that ∅( ) = 0 if and only if = 0.

B.S.Choudhury et al. proved the following theorem in 1.

Theorem 1.12. Let (X,d) be a complete metric space and T be a generalized weakly contractive

self mapping of X. Then T has a unique fixed point.

Motivated by the above result, we prove the same result in G-metric spaces.

Definition 1.13.

A mapping : → , where ( , ) is a -metric space, is said to be a

generalized weakly contractive mapping if

( ( , , )≤ ( ( , , ))− ∅(max{ ( , , ), ( , , ), ( , , )})

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IJSRR, 7(3) July – Sep., 2018 Page 1403 where ( , , ) = max{ ( , , ), ( , , ), ( , , ), ( , , ),

{ ( , , ) + ( , , ) + ( , , )},

is an altering distance function and ∅: [0,∞)→[0,∞) is continuous and non-decreasing function

such that ∅( ) = 0 if and only if = 0.

MAIN RESULT

Theorem 2.1. A generalized weakly contractive self mapping of a complete

-metric space ( , )

has a unique fixed point.

Proof :

Let ∈ be arbitrary . Then we can define a sequence { } in X such that =

for all ≥1.

If = for some n, then T has a fixed point.

So assume that ≠ for all ≥1

Now, from (1.14) we have for all ≥1

( , , ) ≤

( , , ) − ∅ max{ ( , , ), ( , , ),

( , , ), } . (2.2)

where,

( , , ) = max ( , , ), ( , , ), ( , , ),

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

( , , ) =

max

⎩ ⎪ ⎨ ⎪

⎧ ( , , ), ( , , ),

( , , ),

1

3 ( , , ) + ( , , ) + ( , , )

⎭ ⎪ ⎬ ⎪ ⎫

= max ( , , ), ( , , ), 1

3 ( , , )

+ ( , , )

But ( , , ) + ( , , ) + ( , , )

≤ max{ ( , , ), ( , , )}

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IJSRR, 7(3) July – Sep., 2018 Page 1404 Now from (1.14),we have

( , , ) ≤ (max{ ( , , ), ( , , ))−

∅(max{ ( , , ), ( , , ), }). (2.3)

Suppose ( , , ) ≤ ( , , ) for some integer n.

Then from (2.3), we have

( , , ) ≤ ( , , ) − ∅ ( , , )

∅ ( , , ) ≤ 0 implies

∅ ( , , ) = 0 which gives`

( , , ) = 0,that is, ( , , ) = 0

or = ,a contradiction.

Hence ( , , ) < ( , , )∀ n≥1

Thus { ( , , )} is monotonic decreasing sequence and hence is converging to some

real number r ≥ 0.

i.e., lim

n ( , , ) = r

Suppose r > 0.

( , , ) ≤ ( , , ) − ∅ ( , , ) .

Letting n→∞, we get ( )≤ ( )− ∅( ) implies

∅( ) ≤0,a contradiction.

Hence r = 0.

or lim

n ( , , ) = 0.

i.e., lim

n ( , , ) = 0 (2.4)

We now show that the sequence { } is Cauchy.

Suppose { } is not Cauchy. Then there exists an ε > 0 and two sequences { ( )} and { ( )} with

n(k) > m(k) > k for all positive integers k such that _{( )}, _{( )}, _{( )} ≥ ε and

( ), ( ) , ( ) < ε . (2.5)

Now, by rectangular inequality, (2.4) and (2.5), we obtain

ε ≤ ( ), ( ), ( ) ≤ ( ), ( ) , ( ) + ( ) , ( ), ( )

So ε ≤ _{( )}, _{( )}, _{( )} <ε + _{( )} , _{( )}, _{( )}

Letting k→∞ in the above inequalities, we get

ε ≤ lim

k ( ), ( ), ( ) <ε + 0

Therefore lim

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IJSRR, 7(3) July – Sep., 2018 Page 1405 Again by rectangular inequality, we have

( ), ( ), ( ) ≤ ( ), ( ) , ( ) + ( ) , ( ) , ( ) +

( ) , ( ), ( )

and

( ) , ( ) , ( ) ≤ ( ) , ( ), ( ) + ( ), ( ), ( ) +

( ), ( ) , ( )

Letting k→∞, we get from (2.4) and (2.6),

lim

k ( ) , ( ) , ( ) = ε

Again by rectangular inequality,

( ), ( ), ( ) ≤ ( ), ( ) , ( ) + ( ) , ( ), ( ) and

( ), ( ) , ( ) ≤ ( ), ( ), ( ) + ( ), ( ) , ( ) .

Letting k→∞ in the above inequalities, we get

lim

k ( ), ( ) , ( ) = ε.

Similarly, we can show that

lim

k ( ), ( ) , ( ) = ε.

Now, put = ( ), = ( ) , = ( ) in (1.14)

( ), ( ), ( ) ≤ M{ ( ), ( ), ( )

−∅ max ( ), ( ), ( ) , ( ), ( ), ( ) ,

( ), ( ), ( )

(2.7)

where

( ), ( ), ( )

= max _{( )}, _{( )}, _{( )} , _{( )}, _{( )}, _{( )} , _{( )}, _{( )}, _{( )} ,

( ), ( ), ( ) ,

1

3 ( ), ( ), ( ) + ( ), ( ), ( )

+ _{( )}, _{( )}, _{( )}

= max ( ) , ( ) , ( ) , ( ) , ( ), ( ) ,

( ) , ( ), ( ) ,

1

3( ( ) , ( ), ( )

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IJSRR, 7(3) July – Sep., 2018 Page 1406 Letting k→∞, and using (2.4), (2.6), we get

lim

k ( ), ( ), ( ) =max {ε, 0, 0, (ε +0+ ε )} = ε

Now , taking limit as k→∞ in (2.6), we get

(ε)≤ (ε)− ∅(ε) implies∅(ε) ≤0, a contraction.

Thus { } is Cauchy in X. Since X is a complete G- metric space, we can find a t∈X such that lim n

= t.

We shall now show that t is a fixed point of T.

Put x = , y = t, z = t in (1.14), we get

( , , )

≤ (M{ ( , , ))− ∅(max{ ( , , ), ( , , ), ( , , ) })

= max

( , , ), ( , , ), ( , , ), ( , , ), 1

3 ( , , ) + ( , , ) + ( , , )

− ∅ max{ ( , , ), ( , , ) ( , , )}

( , , ) ≤

max 0, ( , , ),2

3 ( , , ) − ∅max{ ( , , ), ( , , )}

Taking limit as n→∞,

( , , ) ≤ ( ( , , )) − ∅ ( ( , , ))

So ( , , )= 0 which gives Tt = t. This shows that t is a fixed point of T.

To prove uniqueness, lett¹ be another fixed point of T.

Put x = t, y =t¹, z =t¹ in (1.14), we get

( , t¹ , t¹) ≤ ( , t¹, t¹) − ∅(max( ( , t , t ), (t , t , t ), (t , t , t )))

where ( , t¹, t¹) = max{ ( , t¹, t¹), ( , , ), (t¹, t¹, t¹), (t¹, t¹, t¹), ( , t¹, t¹) +

(t¹, t¹, t¹) + (t¹, , ) }

= max ( , t , t ), 0,0, 0, 1

3 ( , t , t ) + (t , t , t ) + (t , , )

= ( , t¹, t¹), since (t , , )≤2 ( , t , t )

So ( , t¹ , t¹) ≤ ( , t¹, t¹) − ∅( ( , t , t ))

which implies ( , t¹, t¹) = 0 and therefore t = t¹ .

Corollory 2.8. Suppose

( , ) is a complete -metric space.If the

mapping : → satisfies the following condition ∀ , , ∈

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(max{ ( , , ), ( , , ), ( , , ), ( , , ))−

∅(max{ ( , , ), ( , , ), ( , , )}) , where

is an altering distance function and ∅: [0,∞)→[0,∞) is continuous and non-decreasing function

such that ∅( ) = 0 if and only if = 0.

hen T has a unique fixed point in X.

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