CUBO A Mathematical Journal
Vol.14, No¯ 03, (85–101). October 2012

A Common Fixed Point Theorem in G-Metric Spaces

S.K.Mohanta and Srikanta Mohanta

Department of Mathematics,

West Bengal State University,

Barasat, 24 Pargans (North), West Bengal, Kolkata 700126,

email: smwbes@yahoo.in

ABSTRACT

We prove a common fixed point theorem for a pair of self mappings in complete G-

metric spaces. Our result will improve and supplement some recent results in the setting

of G-metric spaces.

RESUMEN

Probamos un teorema de punto fijo genérico para un par de auto-aplicaciones en es-

pacios G-métricos completos. Nuestro resultado mejorará y complementará algunos de

los resultados recientes en el marco de los espacios G-métricos.

Keywords and Phrases: G-metric space, G-Cauchy sequence, G-continuity, common fixed point.

2010 AMS Mathematics Subject Classification: 54H25, 47H10.



86 S.K.Mohanta and Srikanta Mohanta CUBO
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1 Introduction

The study of metric 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. Different generalizations of the usual notion of a metric space have been proposed

by Gähler [4, 5] and by Dhage [2, 3]. Unfortunately, it was found that most of the results claimed

by Dhage are invalid. These errors were pointed out by Mustafa and Sims in [13]. They also

introduced a more appropriate concept of generalized metric space called G-metric space [9] and

developed a new fixed point theory for various mappings in this new structure. Our aim in this

study is to prove a common fixed point theorem in a complete G-metric space. This theorem

generalizes the fixed point results of [1], [10] and [11].

2 Preliminaries

In this section, we present some basic definitions and results for G-metric spaces which will be

needed in the sequel. Throughout this paper we denote by N the set of positive integers.

Definition 2.1. (see[9]) Let X be a nonempty set, and let G : X × X × X → R+ be a function

satisfying the following axioms:

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

(G2) 0 < G(x, x, y), for all x, y ∈ X, with x 6= y,

(G3) G(x, x, y) ≤ G(x, y, z), for all x, y, z ∈ X, with z 6= y,

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

(G5) G(x, y, z) ≤ G(x, a, a) + G(a, y, z), for all x, y, z, a ∈ X, (rectangle inequality).

Then the function G is called a generalized metric , or, more specifically a G-metric on X, and

the pair (X, G) is called a G-metric space.

Proposition 2.1. (see[9]) Let (X, G) be a G-metric space. Then for any x, y, z, and a ∈ X, it follows

that

(1) if G(x, y, z) = 0 then x = y = z,



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A Common Fixed Point Theorem in G-Metric Spaces ... 87

(2) G(x, y, z) ≤ G(x, x, y) + G(x, x, z),

(3) G(x, y, y) ≤ 2G(y, x, x),

(4) G(x, y, z) ≤ G(x, a, z) + G(a, y, z),

(5) G(x, y, z) ≤ 2
3
(G(x, y, a) + G(x, a, z) + G(a, y, z)),

(6) G(x, y, z) ≤ G(x, a, a) + G(y, a, a) + G(z, a, a).

Definition 2.2. (see[9]) Let (X, G) be a G-metric space, let (xn) be a sequence of points of X, we

say that (xn) is G-convergent to x if lim
n,m→∞

G(x, xn, xm) = 0; that is, for any ǫ > 0, there exists

n0 ∈ N such that G(x, xn, xm) < ǫ, for all n, m ≥ n0. We call x as the limit of the sequence

(xn) and write xn −→ x.

Proposition 2.2. (see[9]) Let (X, G) be a G-metric space, then the following are equivalent.

(1) (xn) is G − convergent to x.

(2) G(xn, xn, x) → 0, as n → ∞.

(3) G(xn, x, x) → 0, as n → ∞.

(4) G(xn, xm, x) → 0, as n, m → ∞.

Definition 2.3. (see[9]) Let (X, G) be a G-metric space, a sequence (xn) is called G-Cauchy if

given ǫ > 0, there is n0 ∈ N such that G(xn, xm, xl) < ǫ, for all n, m, l ≥ n0; that is, if

G(xn, xm, xl) → 0 as n, m, l → ∞.

Definition 2.4. (see[9]) Let (X, G) and (X
′

, G
′

) be G-metric spaces and let f : (X, G) → (X
′

, G
′

)

be a function, then f is said to be G-continuous at a point a ∈ X if given ǫ > 0, there exists δ > 0

such that x, y ∈ X; G(a, x, y) < δ implies G
′

(f(a), f(x), f(y)) < ǫ. A function f is G-continuous on

X if and only if it is G-continuous at all a ∈ X.

Proposition 2.3. (see[9]) Let (X, G) and (X
′

, G
′

) be G-metric spaces, then a function f : X → X
′

is

G-continuous at a point x ∈ X if and only if it is G-sequentially continuous at x; that is, whenever

(xn) is G-convergent to x, (f(xn)) is G-convergent to f(x).

Proposition 2.4. (see[9]) Let (X, G) be a G-metric space, then the function G(x, y, z) is jointly

continuous in all three of its variables.

Proposition 2.5. (see[9]) Every G-metric space (X, G) will define a metric space (X, dG) by

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



88 S.K.Mohanta and Srikanta Mohanta CUBO
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Definition 2.5. (see[9]) A G-metric space (X, G) is said to be G-complete (or a complete G-metric

space) if every G-Cauchy sequence in (X, G) is G-convergent in (X, G).

Proposition 2.6. (see[9]) A G-metric space (X, G) is G-complete if and only if (X, dG) is a complete

metric space.

3 Main Results

Theorem 3.1. Let (X, G) be a complete G-metric space, and let T1, T2 be mappings from X into

itself satisfying

max






G(T1(x), T2(T1(x)), T2(T1(x))),

G(T2(x), T1(T2(x)), T1(T2(x)))





≤ r min






G(x, T1(x), T1(x)),

G(x, T2(x), T2(x))





(3.1)

for every x ∈ X, where 0 ≤ r < 1 and that

inf [G(x, y, y) + min {G(x, T1(x), T1(x)), G(x, T2(x), T2(x))} : x ∈ X] > 0

for every y ∈ X with y is not a common fixed point of T1 and T2.

Then T1 and T2 have a common fixed point in X.

Proof. Let x0 ∈ X be arbitrary and define a sequence (xn) by

xn = T1(xn−1), if n is odd

= T2(xn−1), if n is even.

Then for any odd positive integer n ∈ N, we have

G(xn, xn+1, xn+1) = G(T1(xn−1), T2(xn), T2(xn))

= G(T1(xn−1), T2(T1(xn−1)), T2(T1(xn−1)))

≤ max






G(T1(xn−1), T2(T1(xn−1)), T2(T1(xn−1))),

G(T2(xn−1), T1(T2(xn−1)), T1(T2(xn−1)))






≤ r min






G(xn−1, T1(xn−1), T1(xn−1)),

G(xn−1, T2(xn−1), T2(xn−1))





, by (3.1)

≤ r G(xn−1, T1(xn−1), T1(xn−1))

= r G(xn−1, xn, xn).



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A Common Fixed Point Theorem in G-Metric Spaces ... 89

If n is even, then by (3.1), we have

G(xn, xn+1, xn+1) = G(T2(xn−1), T1(xn), T1(xn))

= G(T2(xn−1), T1(T2(xn−1)), T1(T2(xn−1)))

≤ max






G(T1(xn−1), T2(T1(xn−1)), T2(T1(xn−1))),

G(T2(xn−1), T1(T2(xn−1)), T1(T2(xn−1)))






≤ r min






G(xn−1, T1(xn−1), T1(xn−1)),

G(xn−1, T2(xn−1), T2(xn−1))






≤ r G(xn−1, T2(xn−1), T2(xn−1))

= r G(xn−1, xn, xn).

Thus for any positive integer n, it must be the case that

G(xn, xn+1, xn+1) ≤ r G(xn−1, xn, xn). (3.2)

By repeated application of (3.2), we obtain

G(xn, xn+1, xn+1) ≤ r
n G(x0, x1, x1). (3.3)

Then, for all n, m ∈ N, n < m, we have by repeated use of the rectangle inequality and (3.3)

that

G(xn, xm, xm) ≤ G(xn, xn+1, xn+1) + G(xn+1, xn+2, xn+2)

+G(xn+2, xn+3, xn+3) + · · · + G(xm−1, xm, xm)

≤
(

rn + rn+1 + · · · + rm−1
)

G(x0, x1, x1)

≤
rn

1 − r
G(x0, x1, x1).

Then, lim G(xn, xm, xm) = 0, as n, m → ∞, since lim
r
n

1−r
G(x0, x1, x1) = 0, as n, m → ∞. For

n, m, l ∈ N, (G5) implies that

G(xn, xm, xl) ≤ G(xn, xm, xm) + G(xl, xm, xm),

taking limit as n, m, l → ∞, we get G(xn, xm, xl) → 0. So (xn) is a G-Cauchy sequence. By

completeness of (X, G), there exists u ∈ X such that (xn) is G-convergent to u.

Let n ∈ N be fixed. Then, since (xm) G-converges to u and G is continuous on its variables,

we have

G(xn, u, u) = lim
m→∞

G(xn, xm, xm) ≤
rn

1 − r
G(x0, x1, x1).



90 S.K.Mohanta and Srikanta Mohanta CUBO
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Assume that u is not a common fixed point of T1 and T2. Then, by hypothesis, we have

0 < inf [ G(x, u, u) + min {G(x, T1(x), T1(x)), G(x, T2(x), T2(x))} : x ∈ X]

≤ inf [ G(xn, u, u) + min { G(xn, T1(xn), T1(xn)), G(xn, T2(xn), T2(xn))} : n ∈ N]

≤ inf

[

rn

1 − r
G(x0, x1, x1) + G(xn, xn+1, xn+1) : n ∈ N

]

≤ inf

[

rn

1 − r
G(x0, x1, x1) + r

n G(x0, x1, x1) : n ∈ N

]

= 0,

which is a contradiction. Therefore, u is a common fixed point of T1 and T2.

Theorem 3.2. Let (X, G) be a complete G-metric space, and let T1, T2 be mappings from X into

itself satisfying

max






G(T1(x), T1(x), T2(T1(x))),

G(T2(x), T2(x), T1(T2(x)))






≤ r min






G(x, x, T1(x)),

G(x, x, T2(x))






(3.4)

for every x ∈ X, where 0 ≤ r < 1 and that

inf [G(x, x, y) + min {G(x, x, T1(x)), G(x, x, T2(x))} : x ∈ X] > 0

for every y ∈ X with y is not a common fixed point of T1 and T2.

Then T1 and T2 have a common fixed point in X.

Proof. Let x0 ∈ X be arbitrary and define a sequence (xn) by

xn = T1(xn−1), if n is odd

= T2(xn−1), if n is even.

Then by the argument similar to that used in Theorem 3.1, we have for any positive integer n,

G(xn, xn, xn+1) ≤ r
n G(x0, x0, x1). (3.5)

Then, for all n, m ∈ N, n < m, we have by repeated use of the rectangle inequality and (3.5) that

G(xm, xn, xn) ≤ G(xm, xm−1, xm−1) + G(xm−1, xm−2, xm−2)

+G(xm−2, xm−3, xm−3) + · · · + G(xn+1, xn, xn)

≤
(

rn + rn+1 + · · · + rm−1
)

G(x0, x0, x1)

≤
rn

1 − r
G(x0, x0, x1).



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A Common Fixed Point Theorem in G-Metric Spaces ... 91

Thus (xn) becomes a G-Cauchy sequence. By completeness of (X, G), there exists u ∈ X such that

(xn) is G-convergent to u.

Let n ∈ N be fixed. Then since (xm) G-converges to u and G is continuous on its variables,

we have

G(xn, xn, u) = lim
m→∞

G(xn, xn, xm) ≤
rn

1 − r
G(x0, x0, x1).

The argument similar to that used in the proof of Theorem 3.1 establishes that u is a common

fixed point of T1 and T2.

Combining Theorem 3.1 and Theorem 3.2, we state the following Theorem:

Theorem 3.3. Let (X, G) be a complete G-metric space, and let T1, T2 be mappings from X into

itself satisfying one of the following conditions:

max






G(T1(x), T2(T1(x)), T2(T1(x))),

G(T2(x), T1(T2(x)), T1(T2(x)))






≤ r min






G(x, T1(x), T1(x)),

G(x, T2(x), T2(x))






for every x ∈ X, where 0 ≤ r < 1 and that

inf [G(x, y, y) + min {G(x, T1(x), T1(x)), G(x, T2(x), T2(x))} : x ∈ X] > 0

for every y ∈ X with y is not a common fixed point of T1 and T2.

or

max






G(T1(x), T1(x), T2(T1(x))),

G(T2(x), T2(x), T1(T2(x)))






≤ r min






G(x, x, T1(x)),

G(x, x, T2(x))






for every x ∈ X, where 0 ≤ r < 1 and that

inf [G(x, x, y) + min {G(x, x, T1(x)), G(x, x, T2(x))} : x ∈ X] > 0

for every y ∈ X with y is not a common fixed point of T1 and T2.

Then T1 and T2 have a common fixed point in X.

As an application of Theorem 3.3, we have the following Corollary.



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Corollary 1. Let (X, G) be a complete G-metric space, and let T : X → X be a mapping satisfying

one of the following conditions:

G(T(x), T2(x), T2(x)) ≤ r G(x, T(x), T(x)) (3.6)

for every x ∈ X, where 0 ≤ r < 1 and that

inf [G(x, y, y) + G(x, T(x), T(x)) : x ∈ X] > 0 (3.7)

for every y ∈ X with y 6= T(y).

or

G(T(x), T(x), T2(x)) ≤ r G(x, x, T(x)) (3.8)

for every x ∈ X, where 0 ≤ r < 1 and that

inf [G(x, x, y) + G(x, x, T(x)) : x ∈ X] > 0 (3.9)

for every y ∈ X with y 6= T(y).

Then T has a fixed point in X.

Proof. Take T1 = T2 = T in Theorem 3.3.

We now supplement Corollary 1 by examination of condition (3.6)(or, (3.8)) and condition

(3.7)(or, (3.9)) in respect of their independence. In fact, we furnish Example 3.1 and Example 3.2

below to show that these two conditions are independent in the sense that Corollary 1 shall fall

through by dropping one in favour of the other.

Example 3.1. Let X = {0} ∪
{

1
2n

: n ≥ 1
}
. Define G : X × X × X → R+ by

G(x, y, z) =
1

4
| x − y | +

1

4
| y − z | +

1

4
| z − x |, for all x, y, z ∈ X.

Then (X, G) is a complete G-metric space.

Define T : X → X by T(0) = 1
2
and T

(

1
2n

)

= 1
2n+1

for n ≥ 1.Clearly, T has got no fixed point in X.

Also, it is easy to check that

G(T(x), T2(x), T2(x)) =
1

2
G(x, T(x), T(x))

for every x ∈ X.

Thus, condition (3.6) in Corollary 1 is satisfied. However, T(y) 6= y for all y ∈ X and so

inf {G(x, y, y) + G(x, T(x), T(x)) : x, y ∈ X with y 6= T(y)}



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A Common Fixed Point Theorem in G-Metric Spaces ... 93

= inf {G(x, y, y) + G(x, T(x), T(x)) : x, y ∈ X}

= inf

{
1

2
| x − y | +

1

2
| x − T(x) |: x, y ∈ X

}

= 0.

Thus condition (3.7) in Corollary 1 does not hold.

Similarly, we can verify that condition (3.8) in Corollary 1 is also satisfied but condition (3.9) fails.

Clearly, the conclusion of Corollary 1 is not valid.

Example 3.2. Take X = {0, 1} ∪ [2, ∞). Define G : X × X × X → R+ by

G(x, y, z) =
1

4
| x − y | +

1

4
| y − z | +

1

4
| z − x |, for all x, y, z ∈ X.

Then (X, G) is a complete G-metric space.

Define T : X → X by

T(x) = 0, for x 6= 0

= 1, for x = 0.

Clearly, T possesses no fixed point in X.

Since T(y) 6= y for all y ∈ X, we have

inf {G(x, y, y) + G(x, T(x), T(x)) : x, y ∈ X with y 6= T(y)}

= inf {G(x, y, y) + G(x, T(x), T(x)) : x, y ∈ X}

= inf

{
1

2
| x − y | +

1

2
| x − T(x) |: x, y ∈ X

}

> 0.

Thus condition (3.7) in Corollary 1 is satisfied.

However, for x = 0, we have

G(T(x), T2(x), T2(x)) =
1

2
| T(x) − T2(x) |=

1

2
> r G(x, T(x), T(x))

for any r ∈ [0, 1).

This shows that condition (3.6) in Corollary 1 does not hold.

Similarly, we can check that condition (3.9) in Corollary 1 is also satisfied but condition (3.8) fails.

Obviously, Corollary 1 is invalid in this case.

As an application of Corollary 1, we have the following results.



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Corollary 2. Let (X, G) be a complete G-metric space, and let T : X → X be a G-continuous

mapping satisfying one of the following conditions:

G(T(x), T2(x), T2(x)) ≤ r G(x, T(x), T(x)) (3.10)

or

G(T(x), T(x), T2(x)) ≤ r G(x, x, T(x)) (3.11)

for every x ∈ X, where 0 ≤ r < 1. Then T has a fixed point in X.

Proof. Suppose that T satisfies condition (3.10) for every x ∈ X. Assume that there exists y ∈ X

with y 6= T(y) and

inf [G(x, y, y) + G(x, T(x), T(x)) : x ∈ X] = 0.

Then there exists a sequence (xn) in X such that

lim
n→∞

{G(xn, y, y) + G(xn, T(xn), T(xn))} = 0,

which implies that,

G(xn, y, y) → 0 and G(xn, T(xn), T(xn)) → 0 as n → ∞.

So, by Proposition 2.2, the sequence (xn) is G-convergent to y.

But by (G5), we have

G(T(xn), y, y) ≤ G(T(xn), xn, xn) + G(xn, y, y)

≤ 2 G(xn, T(xn), T(xn)) + G(xn, y, y)

→ 0 as n → ∞.

Again, by Proposition 2.2, the sequence (T(xn)) is G-convergent to y. So G-continuity of T

implies that, (T2(xn)) G-converges to T(y).

Then by use of the rectangle inequality and (3.10) that

G(xn, T
2(xn), T

2(xn)) ≤ G(xn, T(xn), T(xn)) + G(T(xn), T
2(xn), T

2(xn))

≤ G(xn, T(xn), T(xn)) + r G(xn, T(xn), T(xn)).

Taking the limit as n → ∞, and using the fact that the function G is continuous on its variables,

we have

G(y, T(y), T(y)) ≤ G(y, y, y) + r G(y, y, y)

= 0,



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A Common Fixed Point Theorem in G-Metric Spaces ... 95

which implies that, y = T(y). This is a contradiction.

Hence, if y 6= T(y), then

inf [G(x, y, y) + G(x, T(x), T(x)) : x ∈ X] > 0.

If T satisfies condition (3.11), then using the same methods as above one can prove that

inf [G(x, x, y) + G(x, x, T(x)) : x ∈ X] > 0.

So, using Corollary 1, we have the desired result.

The following Corollary is a generalization of the result [[1], Theorem2.1].

Corollary 3. Let (X, G) be a complete G-metric space, and let T : X → X be a mapping satisfying

one of the following conditions:

G(T(x), T(y), T(z)) ≤ k max






G(x, y, z), G(x, T(x), T(x)),

G(y, T(y), T(y)), G(z, T(z), T(z)),

G(x,T(y),T(y))+G(z,T(x),T(x))

2
,

G(x,T(y),T(y))+G(y,T(x),T(x))

2
,

G(y,T(z),T(z))+G(z,T(y),T(y))

2
,

G(x,T(z),T(z))+G(z,T(x),T(x))

2






(3.12)

or

G(T(x), T(y), T(z)) ≤ k max






G(x, y, z), G(x, x, T(x)),

G(y, y, T(y)), G(z, z, T(z)),

G(x,x,T(y))+G(z,z,T(x))

2
,

G(x,x,T(y))+G(y,y,T(x))

2
,

G(y,y,T(z))+G(z,z,T(y))

2
,

G(x,x,T(z))+G(z,z,T(x))

2






(3.13)



96 S.K.Mohanta and Srikanta Mohanta CUBO
14, 3 (2012)

for all x, y, z ∈ X, where 0 ≤ k < 1. Then T has a unique fixed point (say u) in X and T is

G-continuous at u.

Proof. Suppose that T satisfies condition (3.12) for all x, y, z ∈ X. Then replacing y and z by T(x),

we obtain from (3.12) and using (G5) that

G(T(x), T2(x), T2(x)) ≤ k max






G(x, T(x), T(x)), G(x, T(x), T(x)),

G(T(x), T2(x), T2(x)), G(T(x), T2(x), T2(x)),

G(x,T
2
(x),T

2
(x))+G(T(x),T(x),T(x))

2
,

G(x,T
2
(x),T

2
(x))+G(T(x),T(x),T(x))

2
,

G(T(x),T
2
(x),T

2
(x))+G(T(x),T

2
(x),T

2
(x))

2
,

G(x,T
2
(x),T

2
(x))+G(T(x),T(x),T(x))

2






≤ k max






G(x, T(x), T(x)), G(T(x), T2(x), T2(x)),

G(x,T(x),T(x))+G(T(x),T
2
(x),T

2
(x))

2






(3.14)

Without loss of generality we may assume that T(x) 6= T2(x). For, otherwise, T has a fixed point.

So, (3.14) leads to the following cases,

(1) G(T(x), T2(x), T2(x)) ≤ k
G(x,T(x),T(x))+G(T(x),T

2
(x),T

2
(x))

2
,

(2) G(T(x), T2(x), T2(x)) ≤ k G(x, T(x), T(x)).

In the first case, we have

G(T(x), T2(x), T2(x)) ≤
k

2 − k
G(x, T(x), T(x)).

Put r = k
2−k

. Then 0 ≤ r < 1.

Thus, in each case we must have

G(T(x), T2(x), T2(x)) ≤ r G(x, T(x), T(x))

for every x ∈ X, where 0 ≤ r < 1.

Assume that there exists y ∈ X with y 6= T(y) and

inf [G(x, y, y) + G(x, T(x), T(x)) : x ∈ X] = 0.



CUBO
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A Common Fixed Point Theorem in G-Metric Spaces ... 97

Proceeding exactly the same way as in the proof of Corollary 2, there exists a sequence (xn) in X

such that (xn) is G-convergent to y and (T(xn)) is G-convergent to y.

Now applying (3.12), we have

G(T(xn), T(y), T(y)) ≤ k max






G(xn, y, y), G(xn, T(xn), T(xn)),

G(y, T(y), T(y)), G(y, T(y), T(y)),

G(xn,T(y),T(y))+G(y,T(xn),T(xn))

2
,

G(xn,T(y),T(y))+G(y,T(xn),T(xn))

2
,

G(y,T(y),T(y))+G(y,T(y),T(y))

2
,

G(xn,T(y),T(y))+G(y,T(xn),T(xn))

2






.

Taking the limit as n → ∞, and using the fact that the function G is continuous on its

variables, we obtain

G(y, T(y), T(y)) ≤ k G(y, T(y), T(y)),

which is a contradiction.

Hence, if y 6= T(y), then

inf [G(x, y, y) + G(x, T(x), T(x)) : x ∈ X] > 0.

Now Corollary 1 applies to obtain a fixed point (say u) of T.

The proof using (3.13) is similar. Uniqueness of u and G-continuity of T at u may be verified in

the usual way by using any one of condition (3.12) and condition (3.13) that T satisfies.

Remark 1. We see that special cases of Corollary 3 are Theorem 2.1 of [1], Theorem 2.1 of [11]

and Theorems 2.1, and 2.4 of [10].

The following Corollary is the result [[1], Theorem 2.2].

Corollary 4. Let (X, G) be a complete G-metric space, and let T : X → X be a mapping satisfying

one of the following conditions:

G(T(x), T(y), T(z)) ≤ k max






G(x, y, z), G(x, T(x), T(x)),

G(y, T(y), T(y)), G(x, T(y), T(y)),

G(y, T(x), T(x)), G(z, T(z), T(z))






(3.15)



98 S.K.Mohanta and Srikanta Mohanta CUBO
14, 3 (2012)

or

G(T(x), T(y), T(z)) ≤ k max






G(x, y, z), G(x, x, T(x)),

G(y, y, T(y)), G(x, x, T(y)),

G(y, y, T(x)), G(z, z, T(z))






(3.16)

for all x, y, z ∈ X, where 0 ≤ k < 1. Then T has a unique fixed point (say u) in X and T is

G-continuous at u.

Proof. Suppose that T satisfies condition (3.15) for all x, y, z ∈ X. Then replacing z by x ; y and

x by T(x) in (3.15), we have

G(T2(x), T2(x), T(x)) ≤ k max






G(T(x), T(x), x), G(T(x), T2(x), T2(x)),

G(T(x), T2(x), T2(x)), G(T(x), T2(x), T2(x)),

G(T(x), T2(x), T2(x)), G(x, T(x), T(x))






≤ k max
{

G(x, T(x), T(x)), G(T(x), T2(x), T2(x))
}
.

Without loss of generality we may assume that T(x) 6= T2(x). For, otherwise, T has a fixed

point.

So, it must be the case that,

G(T(x), T2(x), T2(x)) ≤ k G(x, T(x), T(x))

for every x ∈ X, where 0 ≤ k < 1.

By the same argument used in the proof of Corollary 3, we see that if y 6= T(y), then

inf [G(x, y, y) + G(x, T(x), T(x)) : x ∈ X] > 0.

Now Corollary 1 applies to obtain a fixed point (say u) of T.

The proof using (3.16) is similar. Uniqueness of u and G-continuity of T at u are obtained by the

same argument used in Corollary 3.

The following Corollary is the result [[11], Theorem2.9].

Corollary 5. Let (X, G) be a complete G-metric space, and let T : X → X be a mapping satisfying

one of the following conditions:

G(T(x), T(y), T(y)) ≤ a {G(x, T(y), T(y)) + G(y, T(x), T(x))} (3.17)



CUBO
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A Common Fixed Point Theorem in G-Metric Spaces ... 99

or

G(T(x), T(y), T(y)) ≤ a {G(x, x, T(y)) + G(y, y, T(x))} (3.18)

for all x, y ∈ X, where 0 ≤ a < 1
2
. Then T has a unique fixed point (say u) in X and T is

G-continuous at u.

Proof. Suppose that T satisfies condition (3.17) for all x, y ∈ X. Then replacing y by T(x) in (3.17),

we have

G(T(x), T2(x), T2(x)) ≤ a
{
G(x, T2(x), T2(x)) + G(T(x), T(x), T(x))

}

≤ a
{
G(x, T(x), T(x)) + G(T(x), T2(x), T2(x))

}
, by (G5).

So, it must be the case that,

G(T(x), T2(x), T2(x)) ≤
a

1 − a
G(x, T(x), T(x)).

Put r = a
1−a

. Then 0 ≤ r < 1 since 0 ≤ a < 1
2
.

Thus,

G(T(x), T2(x), T2(x)) ≤ r G(x, T(x), T(x)).

for every x ∈ X, where 0 ≤ r < 1.

Assume that there exists y ∈ X with y 6= T(y) and

inf [G(x, y, y) + G(x, T(x), T(x)) : x ∈ X] = 0.

As in the proof of Corollary 2, there exists a sequence (xn) in X such that (xn) is G-convergent to

y and (T(xn)) is G-convergent to y.

Now using (3.17), we have

G(T(xn), T(y), T(y)) ≤ a {G(xn, T(y), T(y)) + G(y, T(xn), T(xn))} .

Taking the limit as n → ∞, and using the fact that the function G is continuous on its variables,

we have

G(y, T(y), T(y)) ≤ a {G(y, T(y), T(y)) + G(y, y, y)}

= a G(y, T(y), T(y)),

which is a contradiction.

Hence, if y 6= T(y), then

inf [G(x, y, y) + G(x, T(x), T(x)) : x ∈ X] > 0.

Now applying Corollary 1, we obtain a fixed point (say u) of T.

The proof using (3.18) is similar. Uniqueness of u and G-continuity of T at u are obtained by the

same argument used above.

Received: May 2011. Revised: June 2012.



100 S.K.Mohanta and Srikanta Mohanta CUBO
14, 3 (2012)

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