# Common fixed points of R-weakly commuting maps in generalized metric spaces

## Abstract

In this paper, using the setting of a generalized metric space, a unique common fixed point of four R-weakly commuting maps satisfying a generalized contractive condition is obtained. We also present example in support of our result.

2000 MSC: 54H25; 47H10; 54E50.

## 1 Introduction and preliminaries

The study of unique common fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity. Mustafa and Sims [1] generalized the concept of a metric, in which the real number is assigned to every triplet of an arbitrary set. Based on the notion of generalized metric spaces, Mustafa et al. [26] obtained some fixed point theorems for mappings satisfying different contractive conditions. Study of common fixed point theorems in generalized metric spaces was initiated by Abbas and Rhoades [7]. Abbas et al. [8] obtained some periodic point results in generalized metric spaces. While, Chugh et al. [9] obtained some fixed point results for maps satisfying property p in G-metric spaces. Saadati et al. [10] studied some fixed point results for contractive mappings in partially ordered G-metric spaces. Recently, Shatanawi [11] obtained fixed points of Φ-maps in G-metric spaces. Abbas et al. [12] gave some new results of coupled common fixed point results in two generalized metric spaces (see also [13]).

The aim of this paper is to initiate the study of unique common fixed point of four R-weakly commuting maps satisfying a generalized contractive condition in G-metric spaces.

Consistent with Mustafa and Sims [2], the following definitions and results will be needed in the sequel.

Definition 1.1. Let X be a nonempty set. Suppose that a mapping G :

X × X × XR+ satisfies:

G1 : G(x, y, z) = 0 if x = y = z;

G2 : 0 < G(x, y, z) for all x, y, z X, with xy;

G3 : G(x, x, y) ≤ G(x, y, z) for all x, y, z X, with yz;

G4 : G(x, y, z) = G(x, z, y) = G(y, z, x) = ··· (symmetry in all three variables); and

G5 : G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, a X.

Then G is called a G-metric on X and (X, G) is called a G-metric space.

Definition 1.2. A sequence {x n } in a G-metric space X is:

1. (i)

a G-Cauchy sequence if, for any ε > 0, there is an n 0 N (the set of natural numbers) such that for all n, m, l ≥ n 0, G(x n , x m , x l ) < ε,

2. (ii)

a G-convergent sequence if, for any ε > 0, there is an x X and an n 0 N, such that for all n, mn 0, G(x, x n , x m ) < ε.

A G-metric space on X is said to be G-complete if every G-Cauchy sequence in X is G-convergent in X. It is known that → 0 as n, m → ∞.

Proposition 1.3. Let X be a G-metric space. Then the following are equivalent:

1. (1)

{x n } is G-convergent to x.

2. (2)

G(x n , x m , x) → 0 as n, m → ∞.

3. (3)

G(x n , x n , x) → 0 as n → ∞.

4. (4)

G(x n , x, x) → 0 as n → ∞.

Definition 1.4. A G-metric on X is said to be symmetric if G(x, y, y) = G(y, x, x) for all x, y X.

Proposition 1.5. Every G-metric on X will define a metric d G on X by

$d G ( x , y ) = G ( x , y , y ) + G ( y , x , x ) , ∀ x , y ∈ X .$
(1.1)

For a symmetric G-metric,

$d G ( x , y ) = 2 G ( x , y , y ) , ∀ x , y ∈ X .$
(1.2)

However, if G is non-symmetric, then the following inequality holds:

$3 2 G ( x , y , y ) ≤ d G ( x , y ) ≤ 3 G ( x , y , y ) , ∀ x , y ∈ X .$
(1.3)

It is also obvious that

$G ( x , x , y ) ≤ 2 G ( x , y , y ) .$

Now, we give an example of a non-symmetric G-metric.

Example 1.6. Let X = {1, 2} and a mapping G : X × X × XR+ be defined as

$( x , y , z ) G ( x , y , z ) ( 1 , 1 , 1 ) , ( 2 , 2 , 2 ) 0 ( 1 , 1 , 2 ) , ( 1 , 2 , 1 ) , ( 2 , 1 , 1 ) 0 . 5 ( 1 , 2 , 2 ) , ( 2 , 1 , 2 ) , ( 2 , 2 , 1 ) 1 .$

Note that G satisfies all the axioms of a generalized metric but G(x, x, y) ≠ G(x, y, y) for distinct x, y in X. Therefore, G is a non-symmetric G-metric on X.

In 1999, Pant [14] introduced the concept of weakly commuting maps in metric spaces. We shall study R-weakly commuting and compatible mappings in the frame work of G-metric spaces.

Definition 1.7. Let X be a G-metric space and f and g be two self-mappings of X. Then f and g are called R-weakly commuting if there exists a positive real number R such that G(fgx, fgx, gfx) ≤ RG(fx, fx, gx) holds for each x X.

Two maps f and g are said to be compatible if, whenever {x n } in X such that {fx n } and {gx n } are G-convergent to some t X, then limn→∞G(fgx n , fgx n , gfx n ) = 0.

Example 1.8. Let X = [0, 2] with complete G-metric defined by

$G ( x , y , z ) = max { | x - y | , | x - z | , | y - z | } .$

Let f, g, S, T : XX defined by

and

Then note that the pairs {f, S} and {g, T} are R-weakly commuting as they commute at their coincidence points. The pair {f, S} is continuous compatible while the pair {g, T} is non-compatible. To see that g and T are non-compatible, consider a decreasing sequence {x n } in X such that x n → 1. Then $g x n → 1 2$, $T x n → 1 2$. $gT x n = 4 - x n 4 → 3 4$ and $Tg x n = 2 - x n 4 → 1 4$. □

## 2 Common fixed point theorems

In this section, we obtain some unique common fixed point results for four mappings satisfying certain generalized contractive conditions in the framework of a generalized metric space. We start with the following result.

Theorem 2.1. Let X be a complete G-metric space. Suppose that {f, S} and {g, T} be pointwise R-weakly commuting pairs of self-mappings on X satisfying

$G ( f x , f x , g y ) ≤ h max { G ( S x , S x , T y ) , G ( f x , f x , S x ) , G ( g y , g y , T y ) , [ G ( f x , f x , T y ) + G ( g y , g y , S x ) ] / 2 }$
(2.1)

and

$G ( f x , g y , g y ) ≤ h max { G ( S x , T y , T y ) , G ( f x , S x , S x ) , G ( g y , T y , T y ) , [ G ( f x , T y , T y ) + G ( g y , S x , S x ) ] / 2 }$
(2.2)

for all x, y X, where h [0, 1). Suppose that fX TX, gX SX, and one of the pair {f, S} or {g, T} is compatible. If the mappings in the compatible pair are continuous, then f, g, S and T have a unique common fixed point.

Proof. Suppose that f and g satisfy the conditions (2.1) and (2.2). If G is symmetric, then by adding these, we have

$d G ( f x , g y ) ≤ h 2 max { d G ( S x , T y ) , d G ( f x , S x ) , d G ( g y , T y ) , [ d G ( f x , T y ) + d G ( g y , S x ) ] ∕ 2 } + h 2 max { d G ( S x , T y ) , d G ( f x , S x ) , d G ( g y , T y ) , [ d G ( f x , T y ) + d G ( g y , S x ) ] ∕ 2 } = h max { d G ( S x , T y ) , d G ( f x , S x ) , d G ( g y , T y ) , [ d G ( f x , T y ) + d G ( g y , S x ) ] ∕ 2 } ,$

for all x, y X with 0 ≤ h < 1, the existence and uniqueness of a common fixed point follows from [14]. However, if X is non-symmetric G-metric space, then by the definition of metric d G on X and (1.3), we obtain

$d G ( f x , g y ) = G ( f x , f x , g y ) + G ( f x , g y , g y ) ≤ 2 h 3 max { d G ( S x , T y ) , d G ( f x , S x ) , d G ( g y , T y ) , [ d G ( f x , T y ) + d G ( g y , S x ) ] ∕ 2 } + 2 h 3 max { d G ( S x , T y ) , d G ( f x , S x ) , d G ( g y , T y ) , [ d G ( f x , T y ) + d G ( g y , S x ) ] ∕ 2 } = 4 h 3 max { d G ( S x , T y ) , d G ( f x , S x ) , d G ( g y , T y ) , [ d G ( f x , T y ) + d G ( g y , S X ) ] ∕ 2 } ,$

for all x, y X. Here, the contractivity factor $4 h 3$ needs not be less than 1. Therefore, metric d G gives no information. In this case, let x0 be an arbitrary point in X. Choose x1 and x2 in X such that gx0 = Sx1 and fx1 = Tx2. This can be done, since the ranges of S and T contain those of g and f, respectively. Again choose x3 and x4 in X such that gx2 = Sx 3 and fx3 = Tx4. Continuing this process, having chosen x n in X such that gx2n= Sx2n+1and fx2n+1= Tx2n+2, n = 0, 1, 2, .... Let

$y 2 n = S x 2 n + 1 = g x 2 n a n d y 2 n + 1 = T x 2 n + 2 = f x 2 n + 1 f o r a l l n = 0 , 1 , 2 , … .$

For a given n N, if n is even, so n = 2k for some k N. Then from (2.1)

$G ( y n + 1 , y n + 1 , y n ) = G ( y 2 k + 1 , y 2 k + 1 , y 2 k ) = G ( f x 2 k + 1 , f x 2 k + 1 , g x 2 k ) ≤ h max { G ( S x 2 k + 1 , S x 2 k + 1 , T x 2 k ) , G ( f x 2 k + 1 , f x 2 k + 1 , S x 2 k + 1 ) , G ( g x 2 k , g x 2 k , T x 2 k ) , [ G ( f x 2 k + 1 , f x 2 k + 1 , T x 2 k ) + G ( g x 2 k , g x 2 k , S x 2 k + 1 ) ] / 2 } = h max { G ( y 2 k , y 2 k , y 2 k − 1 ) , G ( y 2 k + 1 , y 2 k + 1 , y 2 k ) , G ( y 2 k , y 2 k , y 2 k − 1 ) , [ G ( y 2 k + 1 , y 2 k + 1 , y 2 k − 1 ) + G ( y 2 k , y 2 k , y 2 k ) ] / 2 } ≤ h max { G ( y 2 k , y 2 k , y 2 k − 1 ) , G ( y 2 k + 1 , y 2 k + 1 , y 2 k ) , [ G ( y 2 k + 1 , y 2 k + 1 , y 2 k ) + G ( y 2 k , y 2 k , y 2 k − 1 ) ] / 2 } = h max { G ( y n , y n , y n − 1 ) , G ( y n + 1 , y n + 1 , y n ) } .$

This implies that

$G ( y n + 1 , y n + 1 , y n ) ≤ h G ( y n , y n , y n - 1 ) .$

If n is odd, then n = 2k + 1 for some k N. In this case (2.1) gives

$G ( y n + 1 , y n + 1 , y n ) = G ( y 2 k + 2 , y 2 k + 2 , y 2 k + 1 ) = G ( f x 2 k + 2 , f x 2 k + 2 + g x 2 k + 1 ) ≤ h max { G ( S x 2 k + 2 , S x 2 k + 2 , T x 2 k + 1 ) , G ( f x 2 k + 2 , f x 2 k + 2 , S x 2 k + 2 ) , G ( g x 2 k + 1 , g x 2 k + 1 , T x 2 k + 1 ) , [ G ( f x 2 k + 2 , f x 2 k + 2 , T x 2 k + 1 ) + G ( g x 2 k + 1 , g x 2 k + 1 , S x 2 k + 2 ) ] / 2 } = h max { G ( y 2 k + 1 , y 2 k + 1 , y 2 k ) , G ( y 2 k + 2 , y 2 k + 2 , y 2 k + 1 ) , G ( y 2 k + 1 , y 2 k + 1 , y 2 k ) , [ G ( y 2 k + 2 , y 2 k + 2 , y 2 k ) + G ( y 2 k + 1 , y 2 k + 1 , y 2 k + 1 ) ] / 2 } ≤ h max { G ( y 2 k + 1 , y 2 k + 1 , y 2 k ) , G ( y 2 k + 2 , y 2 k + 2 , y 2 k + 1 ) , [ G ( y 2 k + 2 , y 2 k + 2 , y 2 k + 1 ) + G ( y 2 k + 1 , y 2 k + 1 , y 2 k ) ] / 2 } = h max { G ( y 2 k + 1 , y 2 k + 1 , y 2 k ) , G ( y 2 k + 2 , y 2 k + 2 , y 2 k + 1 ) } = h max { G ( y n , y n , y n − 1 ) , G ( y n + 1 , y n + 1 , y n ) } ,$

that is,

$G ( y n + 1 , y n + 1 , y n ) ≤ h G ( y n , y n , y n - 1 ) .$

Continuing the above process, we have

$G ( y n + 1 , y n + 1 , y n ) ≤ h n G ( y 1 , y 1 , y 0 ) .$

Thus, if y0 = y1, we get G(y n , yn+1, yn+1) = 0 for each n N. Hence, y n = yn+1for each n N. Therefore, {y n } is G-Cauchy. So we may assume that y0y1.

Let n, m N with m > n,

$G ( y n , y m , y m ) ≤ G ( y n , y n + 1 , y n + 1 ) + G ( y n + 1 , y n + 2 , y n + 2 ) + ⋯ + G ( y m - 1 , y m , y m ) ≤ h n G ( y 0 , y 1 , y 1 ) + h n + 1 G ( y 0 , y 1 , y 1 ) + ⋯ + h m - 1 G ( y 0 , y 1 , y 1 ) = h n G ( y 0 , y 1 , y 1 ) ∑ i = 0 m - n - 1 h i ≤ h n 1 - h G ( y 0 , y 1 , y 1 ) ,$

and so G(y n , y m , y m ) → 0 as m, n → ∞. Hence {y n } is a Cauchy sequence in X. Since X is G-complete, there exists a point z X such that limn→∞y n = z.

Consequently

$lim n → ∞ y 2 n = lim n → ∞ S x 2 n + 1 = lim n → ∞ g x 2 n = z$

and

$lim n → ∞ y 2 n + 1 = lim n → ∞ T x 2 n + 2 = lim n → ∞ f x 2 n + 1 = z .$

Let f and S be continuous compatible mappings. Compatibility of f and S implies that limn→∞G(fSx2n+1, fSx2n+1, Sfx2n+1) = 0, that is G(fz, fz, Sz) = 0 which implies that fz = Sz. Since fX TX, there exists some u X such that fz = Tu. Now from (2.1), we have

(2.3)

Also, from (2.2)

(2.4)

Combining above two inequalities, we get

$G ( f z , f z , g u ) ≤ h 2 G ( f z , f z , g u ) .$

Since h < 1, so that fz = gu. Hence, fz = Sz = gu = Tu. As the pair {g, T} is R-weakly commuting, there exists R > 0 such that

$G ( g T u , g T u , T g u ) ≤ R G ( g u , g u , T u ) = 0 ,$

that is, gTu = Tgu. Moreover, ggu = gTu = Tgu = TTu. Similarly, the pair {f, S} is R-weakly commuting, there exists some R > 0 such that

$G ( f S z , f S z , S f z ) ≤ R G ( f z , f z , S z ) = 0 ,$

so that fSz = Sfz and ffz = fSz = Sfz = SSz.

Now by (2.1)

$G ( f f z , f f z , f z ) = G ( f f z , f f z , g u ) ≤ h max { G ( S f z , S f z , T u ) , G ( f f z , f f z , S f z ) , G ( g u , g u , T u ) , [ G ( f f z , f f z , T u ) + G ( g u , g u , S f z ) ] / 2 } = h max { G ( f f z , f f z , g u ) , G ( f f z , f f z , f f z ) , G ( g u , g u , g u ) , [ G ( f f z , f f z , g u ) + G ( g u , g u , f f z ) ] / 2 } = h max { G ( f f z , f f z , f z ) , [ G ( f f z , f f z , f z ) + G ( f z , f z , f f z ) ] / 2 } = h 2 [ G ( f f z , f f z , f z ) + G ( f z , f z , f f z ) ] ,$

so that

$G ( f f z , f f z , f z ) ≤ h G ( f z , f z , f f z ) .$
(2.5)

Again from (2.2), we have

$G ( f f z , f z , f z ) = G ( f f z , g u , g u ) ≤ h max { G ( S f z , T u , T u ) , G ( f f z , S f z , S f z ) , G ( g u , T u , T u ) , [ G ( f f Z , T u , T u ) + G ( g u , S f z , S f z ) ] / 2 } = h max { G ( S f z , g u , g u ) , G ( f f z , f f z , f f z ) , G ( g u , g u , g u ) , [ G ( f f z , g u , g u ) + G ( g u , f f z , f f z ) ] / 2 } = h max { G ( f f z , f z , f z ) , [ G ( f f z , f z , f z ) + G ( f z , f f z , f f z ) ] / 2 } = h 2 [ G ( f f z , f z , f z ) + G ( f f z , f f z , f z ) ] ,$

which implies

$G ( f f z , f z , f z ) ≤ h G ( f f z , f f z , f z ) .$
(2.6)

From (2.5) and (2.6), we obtain

$G ( f f z , f f z , f z ) ≤ h 2 G ( f f z , f f z , f z ) ,$

and since h2 < 1 so that ffz = fz. Hence, ffz = Sfz = fz, and fz is the common fixed point of f and S. Since gu = fz, following arguments similar to those given above we conclude that fz is a common fixed point of g and T as well. Now we show the uniqueness of fixed point. For this, assume that there exists another point w in X which is the common fixed point of f, g, S and T. From (2.1), we obtain

$G ( f z , f z , w ) = G ( f f z , f f z , g w ) ≤ h max { G ( S f z , S f z , T w ) , G ( f f z , f f z , S f z ) , G ( g w , g w , T w ) , [ G ( f f z , f f z , T w ) + G ( g w , g w , S f z ) ] / 2 } = h max { G ( f z , f z , w ) , G ( f z , f z , f z ) , G ( w , w , w ) , [ G ( f z , f z , w ) + G ( w , w , f z ) ] / 2 } = h 2 [ G ( f z , f z , w ) + G ( w , w , f z ) ] ,$

which implies that

$G ( f z , f z , w ) ≤ h G ( w , w , f z ) .$
(2.7)

From (2.2), we get

$G ( f z , w , w ) = G ( f f z , g w , g w ) ≤ h max { G ( S f z , T w , T w ) , G ( f f z , S f z , S f z ) , G ( g w , T w , T w ) , [ G ( f f z , T w , T w ) + G ( g w , S f z , S f z ) ] / 2 } = h max { G ( f z , w , w ) , G ( f z , f z , f z ) , G ( w , w , w ) , [ G ( f z , w , w ) + G ( w , f z , f z ) ] / 2 } = h 2 [ G ( f z , w , w ) + G ( w , f z , f z ) ] ,$

which implies

$G ( f z , w , w ) ≤ h G ( f z , f z , w ) .$
(2.8)

Now (2.7) and (2.8) give

$G ( f z , f z , w ) ≤ h 2 G ( f z , f z , w ) ,$

and fz = w. This completes the proof.

Example 2.2. Let X = {0, 1, 2} with G-metric defined by

is a non-symmetric G-metric on X because G(0, 0, 1) ≠ G(0, 1, 1).

Let f, g, S, T : XX defined by

$x f ( x ) g ( x ) S ( x ) T ( x ) 0 0 0 0 0 1 0 2 2 1 2 0 0 1 1$

Then fX TX and gX SX, with the pairs {f, S} and {g, T} are R-weakly commuting as they commute at their coincidence points.

Now to get (2.1) and (2.2) satisfied, we have the following nine cases: (I) x, y = 0, (II) x = 0, y = 2, (III) x = 1, y = 0, (IV) x = 1, y = 2, (V) x = 2, y = 0, (VI) x = 2, y = 2. For all these cases, f(x) = g(y) = 0 implies G(fx, fx, gy) = 0 and (2.1) and (2.2) hold.

1. (VII)

For x = 0, y = 1, then fx = 0, gy = 2, Sx = 0, Ty = 1.

$G ( f x , f x , g y ) = G ( 0 , 0 , 2 ) = 1 ≤ h max { 1 , 0 , 2 , 1 } = h max { G ( 0 , 0 , 1 ) , G ( 0 , 0 , 0 ) , G ( 2 , 2 , 1 ) , [ G ( 0 , 0 , 1 ) + G ( 2 , 2 , 0 ) ] / 2 } = h max { G ( S x , S x , T y ) , G ( f x , f x , S x ) , G ( g y , g y , T y ) , [ G ( f x , f x , T y ) + G ( g y , g y , S x ) ] / 2 } .$

Thus, (2.1) is satisfied where $h= 4 5$.

Also

$G ( f x , g y , g y ) = G ( 0 , 2 , 2 ) = 1 ≤ h max { 2 , 0 , 2 , 1.5 } = h max { G ( 0 , 1 , 1 ) , G ( 0 , 0 , 0 ) , G ( 2 , 1 , 1 ) , [ G ( 0 , 1 , 1 ) + G ( 2 , 0 , 0 ) ] / 2 } = h max { G ( S x , T y , T y ) , G ( f x , S x , S x ) , G ( g y , T y , T y ) , [ G ( f x , T y , T y ) + G ( g y , S x , S x ) ] / 2 } .$

Thus, (2.2) is satisfied where $h= 4 5$.

1. (VIII)

Now when x = 1, y = 1, then fx = 0, gy = 2, Sx = 2, Ty = 1.

$G ( f x , f x , g y ) = G ( 0 , 0 , 2 ) = 1 ≤ h max { 2 , 1 , 2 , 0.5 } = h max { G ( 2 , 2 , 1 ) , G ( 0 , 0 , 2 ) , G ( 2 , 2 , 1 ) , [ G ( 0 , 0 , 1 ) + G ( 2 , 2 , 2 ) ] / 2 } = h max { G ( S x , S x , T y ) , G ( f x , f x , S x ) , G ( g y , g y , T y ) , [ G ( f x , f x , T y ) + G ( g y , g y , S x ) ] / 2 } .$

Thus, (2.1) is satisfied where $h= 4 5$.

And

$G ( f x , g y , g y ) = G ( 0 , 2 , 2 ) = 1 ≤ h max { 2 , 1 , 2 , 1 } = h max { G ( 2 , 1 , 1 ) , G ( 0 , 2 , 2 ) , G ( 2 , 1 , 1 ) , [ G ( 0 , 1 , 1 ) + G ( 2 , 2 , 2 ) ] / 2 } = h max { G ( S x , T y , T y ) , G ( f x , S x , S x ) , G ( g y , T y , T y ) , [ G ( f x , T y , T y ) + G ( g y , S x , S x ) ] / 2 } .$

Thus, (2.2) is satisfied where $h= 4 5$.

1. (IX)

If x = 2, y = 1, then fx = 0, gy = 2, Sx = 1, Ty = 1 and

$G ( f x , f x , g y ) = G ( 0 , 0 , 2 ) = 1 ≤ h max { 0 , 1 , 2 , 1.5 } = h max { G ( 1 , 1 , 1 ) , G ( 0 , 0 , 1 ) , G ( 2 , 2 , 1 ) , [ G ( 0 , 0 , 1 ) + G ( 2 , 2 , 1 ) ] / 2 } = h max { G ( S x , S x , T y ) , G ( f x , f x , S x ) , G ( g y , g y , T y ) , [ G ( f x , f x , T y ) + G ( g y , g y , S x ) ] / 2 } .$

Thus, (2.1) is satisfied where $h= 4 5$.

Also

$G ( f x , g y , g y ) = G ( 0 , 2 , 2 ) = 1 ≤ h max { 0 , 2 , 2 , 2 } = h max { G ( 1 , 1 , 1 ) , G ( 0 , 1 , 1 ) , G ( 2 , 1 , 1 ) , [ G ( 0 , 1 , 1 ) + G ( 2 , 1 , 1 ) ] / 2 } = h max { G ( S x , T y , T y ) , G ( f x , S x , S x ) , G ( g y , T y , T y ) , [ G ( f x , T y , T y ) + G ( g y , S x , S x ) ] / 2 } .$

Thus, (2.2) is satisfied where $h= 4 5$.

Hence, for all x, y X, (2.1) and (2.2) are satisfied for $h= 4 5 <1$ so that all the conditions of Theorem 2.1 are satisfied. Moreover, 0 is the unique common fixed point for all of the mappings f, g, S and T.

In Theorem 2.1, if we take f = g, then we have the following corollary.

Corollary 2.3. Let X be a complete G-metric space. Suppose that {f, S} and {f, T} be pointwise R-weakly commuting pairs of self-mappings on X satisfying

$G ( f x , f x , f y ) ≤ h max { G ( S x , S x , T y ) , G ( f x , f x , S x ) , G ( f y , f y , T y ) , [ G ( f x , f x , T y ) + G ( f y , f y , S x ) ] / 2 }$
(2.9)

and

$G ( f x , f y , f y ) ≤ h max { G ( S x , T y , T y ) , G ( f x , S x , S x ) , G ( f y , T y , T y ) } [ G ( f x , T y , T y ) + G ( f y , S x , S x ) ] / 2 }$
(2.10)

for all x, y X, where h [0, 1). Suppose that fX SX TX, and one of the pairs {f, S} or {f, T} is compatible. If the mappings in the compatible pair are continuous, then f, S and T have a unique common fixed point.

Also, if we take S = T in Theorem 2.1, then we get the following.

Corollary 2.4. Let X be a complete G-metric space. Suppose that {f, S} and {g, S} are pointwise R-weakly commuting pairs of self-maps on X and

$G ( f x , f x , g y ) ≤ h max { G ( S x , S x , S y ) , G ( f x , f x , S x ) , G ( g y , g y , S y ) , [ G ( f x , f x , S y ) + G ( g y , g y , S x ) ] / 2 }$
(2.11)

and

$G ( f x , g y , g y ) ≤ h max { G ( S x , S y , S y ) , G ( f x , S x , S x ) , G ( g y , S y , S y ) , [ G ( f x , S y , S y ) + G ( g y , S x , S x ) ] / 2 }$
(2.12)

hold for all x, y X, where h [0, 1). Suppose that fX gX SX and one of the pairs {f, S} or {g, S} is compatible. If the mappings in the compatible pair are continuous, then f, g and S have a unique common fixed point.

Corollary 2.5. Let X be a complete G-metric space. Suppose that f and g are two self-mappings on X satisfying

$G ( f x , f x , g y ) ≤ h max { G ( x , x , y ) , G ( f x , f x , x ) , G ( g y , g y , y ) , [ G ( f x , f x , y ) + G ( g y , g y , x ) ] / 2 }$
(2.13)

and

(2.14)

for all x, y X, where h [0, 1). Suppose that one of f or g is continuous, then f and g have a unique common fixed point.

Proof. Taking S and T as identity maps on X, the result follows from Theorem 2.1.

Corollary 2.6. Let X be a complete G-metric space and f be a self-map on X such that

$G ( f x , f x , f y ) ≤ h max { G ( x , x , y ) , G ( f x , f x , x ) , G ( f y , f y , y ) , [ G ( f x , f x , y ) + G ( f y , f y , x ) ] / 2 }$
(2.15)

and

$G ( f x , f y , f y ) ≤ h max { G ( x , y , y ) , G ( f x , x , x ) , G ( f y , y , y ) , [ G ( f x , y , y ) + G ( f y , x , x ) ] / 2 }$
(2.16)

hold for all x, y X, where h [0, 1). Then f has a unique fixed point.

Proof. If we take f = g, and S and T as identity maps on X, then from f has a unique fixed point by Theorem 2.1.

## 3 Application

Let Ω = [0, 1] be bounded open set in , L2(Ω), the set of functions on Ω whose square is integrable on Ω. Consider an integral equation

$p ( t , x ( t ) ) = ∫ Ω q ( t , s , x ( s ) ) d s$
(3.1)

where p : Ω × and q : Ω × Ω × be two mappings. Define G : X × X × X+ by

$G ( x , y , z ) = sup t ∈ Ω | x ( t ) - y ( t ) | + sup t ∈ Ω | y ( t ) - z ( t ) | + sup t ∈ Ω | z ( t ) - x ( t ) | .$

Then X is a G-complete metric space. We assume the following that is there exists a function G : Ω × +:

1. (i)

p(s, v(t)) ≥ ∫Ω q(t, s, u(s)) dsG(s, v(t)) for each s, t Ω..

2. (ii)

p(s, v(t)) - G(s, v(t)) ≤ h |p(s, v(t)) - v(t)|.

Then integral equation (3.1) has a solution in L2(Ω).

Proof. Define (fx)(t) = p(t, x(t)) and (gx)(t) = ∫Ωq(t, s, x(s)) ds. Now

$G ( f x , f x , g y ) = 2 sup t ∈ Ω | ( f x ) ( t ) - ( g y ) ( t ) | (1) = 2 sup t ∈ Ω p ( t , x ( t ) ) - ∫ Ω q ( t , s , y ( t ) ) d t (2) ≤ 2 sup t ∈ Ω | p ( t , x ( t ) ) - G ( t , x ( t ) ) | (3) ≤ 2 h sup t ∈ Ω | p ( t , x ( t ) ) - x ( t ) | (4) = h G ( f x , f x , x ) . (5) (6)$

Thus

$G ( f x , f x , g y ) ≤ h max { G ( x , x , y ) , G ( f x , f x , x ) , G ( g y , g y , y ) , [ G ( f x , f x , y ) + G ( g y , g y , x ) ] / 2 }$

is satisfied. Similarly (2.14) is satisfied. Now we can apply Corollary 2.5 to obtain the solution of integral equation (3.1) in L2(Ω).

Remark 1. Theorems 2.8-2.9 in [3] and Corollaries 2.6-2.8 in [4] are special cases of our results Theorem 2.1 and Corollaries 2.3-2.6.

Remark 2. A G-metric naturally induces a metric d G given by d G (x, y) = G(x, y, y) + G(x, x, y). If the G-metric is not symmetric, the inequalities (2.1) and (2.2) do not reduce to any metric inequality with the metric d G . Hence, our theorems do not reduce to fixed point problems in the corresponding metric space (X, d G ).

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Abbas, M., Khan, S.H. & Nazir, T. Common fixed points of R-weakly commuting maps in generalized metric spaces. Fixed Point Theory Appl 2011, 41 (2011). https://doi.org/10.1186/1687-1812-2011-41