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 [1] [2] introduced 2-metric spaces and claimed them as generalizations of metric spaces. But many researchers proved that there was no relation between these two spaces. These considerations led Dhage [3] to initiate a study of general metric spaces called D-metric spaces. As a probable modification to D-metric spaces, Shaban Sedghi, Nabi Shobe and Haiyun Zhou [4] have introduced D*-metric spaces. In 2006, Zead Mustafa and Brailey Sims [5] initiated G -metric spaces. Several researchers proved many common fixed point theorems on G -metric spaces.

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: ( [5] , Definition 3) Let X be a non-empty set and G : X 3 → [ 0 ,   ∞ ) be a function satisfying:

(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: ( [5] , Definition 4) A G-metric Space ( X , G ) is said to be symmetric if

(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: ( [5] , Example 1): Let X = { a , b } Define G : X 3 → [ 0 ,   ∞ ) by

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): ( [5] , p. 292) If ( X , G ) is a G-metric space then G ( x , y , y ) ≤ 2 G ( y , x , x ) for all x , y ∈ X .

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: ( [5] , Proposition 6) Let ( X , G ) be a G-metric Space, then for a sequence { x n } ⊆ X and point x ∈ X the following are equivalent.

(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: ( [5] , Definition 8) Let ( X , G ) be a G-metric space, then a sequence { x n } ⊆ X is said to be G-Cauchy if for each ε > 0 , there exists a natural number N such that G ( x n , x m , x l ) < ε for all n , m , l ≥ N .

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

Definition 2.9: ( [5] , Definition 9) A G-metric space ( X , G ) is said to be G- complete if every G -Cauchy sequence in ( X , G ) is G-convergent in ( X , G ) .

Gerald Jungck [6] initiated the notion of weakly compatible mappings, as a generalization of commuting maps. We now give the definition of weakly compatibility in a G-metric space.

Definition 2.10: [7] Suppose f and g are selfmaps of a G-metric space ( X , G ) . The pair ( f , g ) is said to be weakly compatible if G ( f g x ,   g f x ,   g f x ) = 0 whenever G ( f x ,   g x ,   g x ) = 0.

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.

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