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By using weakly compatible conditions of selfmapping pairs, we prove a com-mon fixed point theorem for six mappings in generalized complete metric spaces. An example is provided to support our result.

The study of fixed point theory has been at the centre of vigorous activity and it has a wide range of applications in applied mathematics and sciences. Over the past two decades, a considerable amount of research work for the development of fixed point theory have executed by several authors.

In 1963, Gahler [

The purpose of this paper is to prove a common fixed point theorem for six weakly compatible selfmaps of a complete G -metric space. Now we recall some basic definitions and results on G -metric space.

We begin with

Definition 2.1: ( [

(G1) G ( x , y , z ) = 0 if x = y = z .

(G2) 0 < G ( x , x , y ) for all x , y ∈ X with x ≠ y .

(G3) G ( x , x , y ) < G ( x , y , z ) for all x , y , z ∈ X with y ≠ z .

(G4) G ( x , y , z ) = G ( σ ( x , y , z ) ) for all x , y , z ∈ X , where σ ( x , y , z ) is a permutation of the set { x , y , z } .

And

(G5) G ( x , y , z ) < G ( x , w , w ) + G ( w , y , z ) for all x , y , z , w ∈ X .

Then G is called a G-metric on X and the pair ( X , G ) is called a G-metric Space.

Definition 2.2: ( [^{ }

(G6) G ( x , y , y ) = G ( x , x , y ) for all x , y ∈ X .

The example given below is a non-symmetric G-metric space.

Example 2.3: ( [

G ( a , a , a ) = G ( b , b , b ) = 0 ; G ( a , a , b ) = 1 , G ( a , b , b ) = 2 and extend G to all of X 3 by using (G4).

Then it is easy to verify that ( X , G ) is a G-metric space. Since G ( a , a , b ) ≠ G ( a , b , b ) , the space ( X , G ) is non-symmetric, in view of (G6).

Example 2.4: Let ( X , d ) be a metric space. Define G s d : X 3 → [ 0 , ∞ ) by

G s d ( x , y , z ) = 1 3 [ d ( x , y ) + d ( y , z ) + d ( z , x ) ] for x , y , z ∈ X .Then ( X , G s d ) is a G-metric Space.

Lemma (2.5): ( [

Definition 2.6: Let ( X , G ) be a G-metric Space. A sequence { x n } in X is said to be G-convergent if there is a x 0 ∈ X such that to each ε > 0 there is a natural number N for which G ( x n , x n , x 0 ) < ε for all n ≥ N .

Lemma 2.7: ( [

(1) { x n } is G- convergent to x .

(2) d G ( x n , x ) → 0 as n → ∞ (that is { x n } converges to x relative to the metric d G ).

(3) G ( x n , x n , x ) → 0 as n → ∞ .

(4) G ( x n , x , x ) → 0 as n → ∞ .

(5) G ( x m , x n , x ) → 0 as m , n → ∞ .

Definition 2.8: ( [

Note that every G-convergent sequence in a G-metric space ( X , G ) is G- Cauchy.

Definition 2.9: ( [

Gerald Jungck [

Definition 2.10: [

Theorem 3.1: Suppose f , g , h , p , Q and R are six selfmaps of a complete G -metric space ( X , G ) satisfying the following conditions.

(3.1.1) f g ( X ) ⊆ R ( X ) and h p ( X ) ⊆ Q ( X ) ,

(3.1.2)

G ( h p x , f g y , f g y ) ≤ α G ( R x , Q y , Q y ) + β [ G ( R x , h p x , h p x ) + G ( Q y , f g y , f g y ) ] + γ [ G ( R x , f g y , f g y ) + G ( h p x , Q y , Q y ) ]

for all x , y ∈ X and α , β , γ are non-negative real numbers such that α + 2 β + 2 γ < 1 ,

(3.1.3) one of R ( X ) , Q ( X ) is closed sub subset of X ,

(3.1.4) ( f g , Q ) and ( h p , R ) are weakly compatible pairs,

(3.1.5) The pairs ( h , p ) , ( h , R ) , ( f , g ) , and ( f , Q ) are commuting.

Then f , g , h , p , Q and R have a unique common fixed point in X .

Proof: Let x 0 ∈ X be an arbitrary point. Since f g ( X ) ⊆ R ( X ) and h p ( X ) ⊆ Q ( X ) there exists x 1 , x 2 ∈ X such that h p x 0 = Q x 1 and f g x 1 = R x 2 again there exists x 3 , x 4 ∈ X such that h p x 2 = Q x 3 and f g x 3 = R x 4 , continuing in the same manner for each n ≥ 0 , we obtain a sequence { x n } in X such that

y 2 n = h p x 2 n = Q x 2 n + 1 , y 2 n + 1 = f g x 2 n + 1 = R x 2 n + 2 for n ≥ 0. (3.1.6)

From condition (3.1.2), we have

G ( y 2 n , y 2 n + 1 , y 2 n + 1 ) = G ( h p x 2 n , f g x 2 n + 1 , f g x 2 n + 1 ) ≤ α G ( R x 2 n , Q x 2 n + 1 , Q x 2 n + 1 ) + β [ G ( R x 2 n , h p x 2 n , h p x 2 n ) + G ( Q x 2 n + 1 , f g x 2 n + 1 , f g x 2 n + 1 ) ] + γ [ G ( R x 2 n , f g x 2 n + 1 , f g x 2 n + 1 ) + G ( h p x 2 n , Q x 2 n + 1 , Q x 2 n + 1 ) ] = α G ( y 2 n − 1 , y 2 n , y 2 n ) + β [ G ( y 2 n − 1 , y 2 n , y 2 n ) + G ( y 2 n , y 2 n + 1 , y 2 n + 1 ) ] + γ [ G ( y 2 n − 1 , y 2 n + 1 , y 2 n + 1 ) + G ( y 2 n , y 2 n , y 2 n ) ] ≤ ( α + β + γ ) G ( y 2 n − 1 , y 2 n , y 2 n ) + ( β + γ ) G ( y 2 n , y 2 n + 1 , y 2 n + 1 ) .

Therefore

( 1 − β − γ ) G ( y 2 n , y 2 n + 1 , y 2 n + 1 ) ≤ ( α + β + γ ) G ( y 2 n − 1 , y 2 n , y 2 n ) G ( y 2 n , y 2 n + 1 , y 2 n + 1 ) ≤ ( α + β + γ ) ( 1 − β − γ ) G ( y 2 n − 1 , y 2 n , y 2 n ) G ( y 2 n , y 2 n + 1 , y 2 n + 1 ) ≤ k G ( y 2 n − 1 , y 2 n , y 2 n ) (3.1.7)

where k = ( α + β + γ ) ( 1 − β − γ ) < 1 .

Similarly, we can show that

G ( y 2 n + 1 , y 2 n + 2 , y 2 n + 2 ) ≤ k G ( y 2 n , y 2 n + 1 , y 2 n + 1 ) . (3.18)

From (3.1.7) and (3.1.8) we have

G ( y n , y n + 1 , y n + 1 ) ≤ k G ( y n − 1 , y n , y n ) ≤ ⋯ ≤ k n G ( y 0 , y 1 , y 1 ) .

Now for every n , m ∈ N such that m > n we have

G ( y n , y m , y m ) ≤ G ( y n , y n + 1 , y n + 1 ) + G ( y n + 1 , y n + 2 , y n + 2 ) + ⋯ + G ( y m − 1 , y m , y m ) ≤ k n G ( y 0 , y 1 , y 1 ) + k n + 1 G ( y 0 , y 1 , y 1 ) + ⋯ + k m − 1 G ( y 0 , y 1 , y 1 ) ≤ k n ( 1 + k + k 2 + ⋯ + k m − n + 1 ) G ( y 0 , y 1 , y 1 ) ≤ k n ( 1 − k m − n ) 1 − k G ( h x 0 , h x 1 , h x 1 ) → 0 as n → ∞ .

Since k < 1.

Therefore, { y n } is a Cauchy sequence in X . Since X is a complete G-metric space, then there exists a point z ∈ X such that

lim n → ∞ h p x 2 n = lim n → ∞ Q x 2 n + 1 = lim n → ∞ f g x 2 n + 1 = lim n → ∞ R x 2 n + 2 = z . (3.1.9)

If R ( X ) is a closed subset of X , then there exists a point u ∈ X such that z = R u .

Now from (3.1.2), we have

G ( h p u , f g x 2 n + 1 , f g x 2 n + 1 ) ≤ α G ( R u , Q x 2 n + 1 , Q x 2 n + 1 ) + β [ G ( R u , h p u , h p u ) + G ( Q x 2 n + 1 , f g x 2 n + 1 , f g x 2 n + 1 ) ] + γ [ G ( R u , f g x 2 n + 1 , f g x 2 n + 1 ) + G ( h p u , Q x 2 n + 1 , Q x 2 n + 1 ) ] . (3.1.10)

Letting n → ∞ in (3.1.10) and by the continuity of G we have

G ( h p u , z , z ) ≤ α G ( z , z , z ) + β [ G ( z , h p u , h p u ) + G ( z , z , z ) ] + γ [ G ( z , z , z ) + G ( h p u , z , z ) ] ≤ ( 2 β + γ ) G ( h p u , z , z ) ,

which leads to a contradiction as 2 β + γ < 1 .

Hence G ( h p u , z , z ) = 0 , which implies h p u = z .

Therefore,

h p u = R u = z . (3.1.11)

Now since h p ( X ) ⊆ Q ( X ) then there exists a point v ∈ X such that z = Q v .

Then we have by (3.1.2)

G ( h p u , f g v , f g v ) ≤ α G ( R u , Q v , Q v ) + β [ G ( R u , h p u , h p u ) + G ( Q v , f g v , f g v ) ] + γ [ G ( R u , f g v , f g v ) + G ( h p u , Q v , Q v ) ] (3.1.12)

G ( z , f g v , f g v ) ≤ α G ( z , z , z ) + β [ G ( z , z , z ) + G ( z , f g v , f g v ) ] + γ [ G ( z , f g v , f g v ) + G ( z , z , z ) ] ≤ ( β + γ ) G ( z , f g v , f g v ) ,

which leads to a contradiction, since β + γ < 1 . Hence f g v = z .

Therefore,

f g v = Q v = z . (3.1.13)

From (3.1.11) and (3.1.13) we have R u = h p u = f g v = Q v = z .

Since the pair ( f g , Q ) is weakly compatible then f g Q v = Q f g v which gives f g z = Q z .

Now (3.1.2) we have

G ( z , f g z , f g z ) = G ( h p u , f g z , f g z ) ≤ α G ( R u , Q z , Q z ) + β [ G ( R u , h p u , h p u ) + G ( Q z , f g z , f g z ) ] + γ [ G ( R u , f g z , f g z ) + G ( h p u , Q z , Q z ) ] = α G ( z , f g z , f g z ) + β [ G ( z , z , z ) + G ( f g z , f g z , f g z ) ] + γ [ G ( z , f g z , f g z ) + G ( z , f g z , f g z ) ] = ( α + 2 γ ) G ( z , f g z , f g z )

which is a contradiction, since α + 2 γ < 1. Hence G ( z , f g z , f g z ) = 0 thus f g z = z .

Showing that z is a common fixed point of f g and Q .

Since the pair ( h p , R ) is weakly compatible then h p R u = R h p u which gives h p z = R z .

Then we have by (3.1.2)

G ( h p z , z , z ) = G ( h p z , f g z , f g z ) ≤ α G ( R z , Q z , Q z ) + β [ G ( R z , h p z , h p z ) + G ( Q z , f g z , f g z ) ] + γ [ G ( R z , f g z , f g z ) + G ( h p z , Q z , Q z ) ] = α G ( h p z , z , z ) + β [ G ( h p z , h p z , h p z ) + G ( z , z , z ) ] + γ [ G ( h p z , z , z ) + G ( h p z , z , z ) ] = ( α + 2 γ ) G ( h p z , z , z ) ,

which is a contradiction, since α + 2 γ < 1 . Hence G ( h p z , z , z ) = 0 thus h p z = z .

Showing that z is a common fixed point of h p and R .

Therefore, z is a common fixed point of f g , h p , R and Q .

By commuting conditions of the pairs in (3.1.5), we have

f z = f ( f g z ) = f ( g f z ) = f g ( f z ) , f z = f ( Q z ) = Q ( f z ) .

And

h z = h ( h p z ) = h ( p h z ) = h p ( h z ) , h z = h ( R z ) = R ( h z ) .

From (3.1.2)

G ( z , f z , f z ) = G ( h p z , f g f z , f g f z ) ≤ α G ( R z , Q f z , Q f z ) + β [ G ( R z , h p z , h p z ) + G ( Q f z , f g f z , f g f z ) ] + γ [ G ( R z , f g f z , f g f z ) + G ( h p z , Q f z , Q f z ) ] = α G ( z , f z , f z ) + β [ G ( z , z , z ) + G ( f z , f z , f z ) ] + γ [ G ( z , f z , f z ) + G ( z , f z , f z ) ] = ( α + 2 γ ) G ( z , f z , f z ) .

Since α + 2 γ < 1 , we have G ( z , f z , f z ) = 0 thus f z = z .

Also g z = g f z = f g z = z .

Therefore, we have f z = g z = R z = f g z = z .

Similarly, we have h z = p z = Q z = h p z = z .

Therefore, z is a common fixed point of f , g , h , p , Q and R .

The proof is similar in case if Q ( X ) is a closed subset of X .

We now prove the uniqueness of the common fixed point.

If possible, assume that w is another common fixed point of f , g , h , p , Q and R .

By condition (3.1.2) we have

G ( z , w , w ) = G ( h p z , f g w , f g w ) ≤ α G ( R z , Q w , Q w ) + β [ G ( R z , h p z , h p z ) + G ( Q w , f g w , f g w ) ] + γ [ G ( R z , f g w , f g w ) + G ( h p z , Q w , Q w ) ] = α G ( z , w , w ) + β [ G ( z , z , z ) + G ( w , w , w ) ] + γ [ G ( z , w , w ) ) + G ( z , w , w ) ] = ( α + 2 γ ) G ( z , w , w ) ,

which is a contradiction, since α + 2 γ < 1 .

Hence G ( z , w , w ) = 0 which gives z = w .

Therefore, z is a unique common fixed point of f , g , h , p , Q and R .

As an example, we have the following.

Let X = [ 0 , 1 ] with G ( x , y , z ) = | x − y | + | y − z | + | z − x | for x , y , z ∈ X . Then G is a G-metric on X .

Define

f : X → X , g : X → X , h : X → X , p : X → X , Q : X → X , R : X → X

by

f x = h x = x + 1 3 , ∀ x ∈ X , g x = p x = 3 x + 1 5 , ∀ x ∈ X , Q x = R x = x , ∀ x ∈ X .

f g x = f ( 3 x + 1 5 ) = x + 2 5 , h p x = h ( 3 x + 1 5 ) = x + 2 5 ,

f g X = [ 2 5 , 3 5 ] , h p X = [ 2 5 , 3 5 ] , R X = [ 0 , 1 ] , Q X = [ 0 , 1 ]

f g X ⊆ R X , h p X ⊆ Q X .

Proving the condition (3.1.1) of the Theorem (3.1).

R X and Q X are closed subsets of X . Proving the condition (3.1.3) of the Theorem (3.1).

Since f g ( 1 2 ) = 1 2 and Q ( 1 2 ) = 1 2 then f g Q ( 1 2 ) = Q f g ( 1 2 ) , showing that the pair ( f g , Q ) is weakly compatible.

Also, the pair ( h p , R ) is weakly compatible.

Proving the condition (3.1.4) of the Theorem (3.1).

h p ( x ) = x + 2 5 = p h ( x ) , h R ( x ) = h ( x ) = R h ( x ) , f g ( x ) = x + 2 5 = g f ( x ) , f Q ( x ) = f ( x ) = Q f ( x ) ,

showing that ( h , R ) , ( f , Q ) , ( h , p ) and ( f , g ) are commuting pairs.

Proving the condition (3.1.5) of the Theorem (3.1).

Now we prove the condition (3.1.2) of the Theorem (3.1).

On taking α = 1 10 , β = 1 8 , γ = 1 12 then α + 2 β + 2 γ = 31 60 < 1.

Now G ( h p x , f g y , f g y ) = 2 | h p x − f g y | = 2 5 | x − y |

G ( R x , Q y , Q y ) = 2 | R x − Q y | = 2 | x − y | , G ( R x , h p x , h p x ) = 2 | R x − h p x | = 4 5 | 2 x − 1 | , G ( Q y , f g y , f g y ) = 2 | f g y − Q y | = 4 5 | 1 − 2 y | , G ( R x , f g y , f g y ) = 2 | R x − f g y | = 2 5 | 5 x − y − 2 | , G ( h p x , Q y , Q y ) = 2 | h p x − Q y | = 2 5 | x + 2 − 5 y |

α G ( R x , Q y , Q y ) + β [ G ( R x , h p x , h p x ) + G ( Q y , f g y , f g y ) ] + γ [ G ( R x , f g y , f g y ) + G ( h p x , Q y , Q y ) ] = 2 α | x − y | + 4 5 β ( | 2 x − 1 | + | 1 − 2 y | ) + 2 5 γ ( | 5 x − y − 2 | + | x − 5 y − 2 | ) ≥ 2 α | x − y | + 4 5 β | 2 x − 2 y | + 2 5 γ | 6 x − 6 y | = ( 2 α + 8 β 5 + 12 5 γ ) | x − y | = 3 5 | x − y | ≥ 2 5 | x − y | = G ( f g x , h p y , h p y ) .

Therefore,

G ( h p x , f g y , f g y ) ≤ α G ( R x , Q y , Q y ) + β [ G ( R x , h p x , h p x ) + G ( Q y , f g y , f g y ) ] + γ [ G ( R x , f g y , f g y ) + G ( h p x , Q y , Q y ) ] .

Proving the condition (3.1.2) of the Theorem (3.1).

Hence all the conditions of the Theorem (3.1) are satisfied.

Therefore, 1 2 is a unique common fixed point of f , g , h , p , Q and R .

Suppose f , p , Q and R are four selfmaps of a complete G -metric space ( X , G ) satisfying the following conditions:

(3.1.1) f ( X ) ⊆ R ( X ) and p ( X ) ⊆ Q ( X ) ,

(3.1.2) G ( p x , f y , f y ) ≤ α G ( R x , Q y , Q y ) + β [ G ( R x , p x , p x ) + G ( Q y , f y , f y ) ] + γ [ G ( R x , f y , f y ) + G ( p x , Q y , Q y ) ]

for all x , y ∈ X and α , β , γ are non-negative real numbers such that α + 2 β + 2 γ < 1 ,

(3.1.3) One of R ( X ) , Q ( X ) is closed sub subset of X,

(3.1.4) ( p , R ) and ( f , Q ) are weakly compatible pairs,

Then f , p , Q and R have a unique common fixed point in X .

Proof: Follows from the Theorem (3.1) if g = h = I the identity map.

Suppose f , p and R are three selfmaps of a complete G -metric space ( X , G ) satisfying the following conditions:

(3.1.1) f ( X ) ⊆ R ( X ) and p ( X ) ⊆ R ( X ) ,

(3.1.2) G ( p x , f y , f y ) ≤ α G ( R x , R y , R y ) + β [ G ( R x , p x , p x ) + G ( R y , f y , f y ) ] + γ [ G ( R x , f y , f y ) + G ( p x , R y , R y ) ]

for all x , y ∈ X and α , β , γ are non-negative real numbers such that α + 2 β + 2 γ < 1 ,

(3.1.3) R ( X ) is closed sub subset of X,

(3.1.4) ( p , R ) and ( f , R ) are weakly compatible pairs.

Then f , p and R have a unique common fixed point in X .

Proof: Follows from the Theorem (3.1) if g = h = I the identity map, and Q = R .

Goud, J.N. and Rangamma, M. (2017) Common Fixed Point Theorem for Six Selfmaps of a Complete G-Metric Space. Advances in Pure Mathematics, 7, 290-297. https://doi.org/10.4236/apm.2017.73015