International Journal of Analysis and Applications
ISSN 2291-8639
Volume 2, Number 2 (2013), 173-177
http://www.etamaths.com

FIXED POINT OF ORDER 2 ON G-METRIC SPACE

ANIMESH GUPTA

Abstract. In this article we introduce a new concept of fixed point that is

fixed point of order 2 on G-metric space and some results are achieved.

1. Introduction and preliminaries

In 2003, Mustafa and Sims [4] introduced a more appropriate and robust notion
of a generalized metric space as follows.

Definition 1.1. [4] Let X be a nonempty set, and let G : X ×X ×X → [0,∞) be
a function satisfying the following axioms:

(1) G(x,y,z) = 0 if and only if x = y = z;
(2) G(x,x,y) > 0, for all x 6= y;
(3) G(x,y,z) ≥ G(x,x,y), for all x,y,z ∈ X;
(4) G(x,y,z) = G(x,z,y) = G(z,y,x) = · · · (symmetric in all three variables);
(5) G(x,y,z) ≤ G(x,w,w) + G(w,y,z), for all x,y,z,w ∈ X.

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.

Definition 1.2. Suppose that (X,G) is a G-metric space, T : X → X is a function
and x0 ∈ X is fixed point of T. We call x0 is a fixed pointof order 2 if it is not
alone point and the following satisfies:

lim
x→x0

G(Tx,Tx,x0)

G(x,x,x0)
= 1(1.1)

We remember the following definitions. We will show that for the case (a) there
is not fixed point of order 2 but in two other cases there is fixed point of order 2.

Definition 1.3. Suppose that (X,G) is a G-metric space, T : X → X is a function.
(a) T is a contraction, if there exist k ∈ [0, 1) such that G(Tx,Ty,Tz) ≤

kG(x,y,z) for all x,y,z ∈ X.
(b) T is a contractive mapping, if G(Tx,Ty,Tz) < G(x,y,z) for all x,y,z ∈ X

which x 6= y 6= z.
(c) T is non-expansive mapping, if G(Tx,Ty,Tz) ≤ G(x,y,z) for all x,y,z ∈

X.

In the following we consider first some properties for fixed point of order 2.

2010 Mathematics Subject Classification. Primary 47H10; Secondary 54H25, 55M20.
Key words and phrases. Fixed point, fixed point of order 2, contraction mapping, non expansive

mapping.

c©2013 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

173



174 GUPTA

2. Main Results

Proposition 2.1. If x0 ∈ X is a fixed point of order 2 for T on X. Then T is
continuous at x0.

Proof. limn→∞G(Tx,Tx,x0) = limx→x0
G(T x,T x,x0)
G(x,x,x0)

G(x,x,x0)

limx→x0
G(T x,T x,x0)
G(x,x,x0)

limx→x0 G(x,x,x0) = 0. �

Proposition 2.2. Let (X,G) be a metric space and T : X → X be a function such
that x0 ∈ X is a fixed point for T , not alone point for X and alone point for T(X).
Then x0 is not fixed point of order 2 for T .

Proof. According to assumption x0 is alone point for T(X). There is a neighbor-
hood of x0, like N(x0) such that N(x0) ∩T(X) and each x ∈ N(x0) implies that
G(Tx,Tx,x0) = 0. Therefore, limx→x0

G(T x,T x,x0)
G(x,x,x0)

= 0, i.e; x0 is not a fixed point

of order 2 for T . �

Proposition 2.3. Suppose that x0 ∈ X be a fixed point for Ti : X → X which
i = 1, 2, ...,n where (n ∈ N) and also limx→x0

G(Tix,Tix,x0)
G(x,x,x0)

= λi. Then x0 is a

fixed point of order 2 for T1T2...Tn if and only if λ1λ2...λn = 1.

Proof. Ti is continuous at x0 for all i = 1, 2, ...,n by a simple change of variable
that

lim
x→x0

G(Tk(Tk+1...Tnx),Tk(Tk+1...Tnx),x0)

G(Tk+1...Tnx,Tk+1...Tnx,x0)
= lim

t→x0

G(Tkt,Tkt,x0)

t,t,x0

and the last limit is equal with λk for k = 1, 2, ...,n. Hence,

lim
x→x0

G(T1T2...Tnx,T1T2...Tnx,x0)

G(x,x,x0)
=

lim
x→x0

G(T1(T2...Tn)x,T1(T2...Tn)x,x0)

G(T2...Tn,T2...Tn,x0)

G(T2(T3...Tn)x,T2(T3...Tn)x,x0)

G(T3...Tn,T3...Tn,x0)
...
G(Tnx,Tnx,x0)

G(x,x,x0)

λ1λ2...λn

�

Proposition 2.4. Letx0 ∈ X be a fixed point for Ti : X → X for i = 1, 2, ...,n and
n ∈ N.

(a) If x0 is fixed point of order 2 for all Ti, then x0 is fixed point for T1T2...Tn.
(b) If x0 is fixed point order 2 for T1T2 and T2 , then x0 is fixed point of order

2 for T1.

Proof. (a) By proposition 2.1.

(b) x0 is fixed point of order 2 for T1T2 and T2. Thus, limx→x0
G(T1T2x,T1T2x,x0)

G(x,x,x0)
=

1, limx→x0
G(T2x,T2x,x0)

G(x,x,x0)
= 1. Since T is continuous at x0 for t = T2x.

1 =
lim

x→x0
G(T1T2x,T1T2x,x0)

G(x,x,x0)

limx→x0
G(T2x,T2x,x0)

G(x,x,x0)

= lim
x→x0

G(T1T2x,T1T2x,x0)

G(T2x,T2x,x0)
= lim

t→x0

G(T1t,T1t,x0)

G(t,t,x0)

�



FIXED POINT OF ORDER 2 ON G-METRIC SPACE 175

Proposition 2.5. Suppose that x0 is not alone point and is a fixed point for Ti :
X → X for i = 1, 2, ...,n and n ∈ N.

(a) If Ti be a contractive mapping or non expansive mapping for i = 1, 2, ...,n

and n ∈ N and limx→x0
G(Tix,Tix,x0)

G(x,x,x0)
= λi. Then x0 ∈ X is a fixed point of

order 2 for T1T2...Tn if and only if x0 is a fixed point of order 2 for all Ti.

(b) If limx→x0
G(T1x,T1x,x0)

G(x,x,x0)
= λ then x0 is a fixed point of order 2 for T1 if and

only if x0 be a fixed point of order 2 for T
n
1 where n is arbitrary positive

integer.
(c) If T1 be a contractive mapping or non-expansive mapping, then x0 is a fixed

point of order 2 for T1 if and only if there exist n ∈ N such that x0 be a
fixed point of order 2 for Tn1 .

Proof. (a) Let Ti be a contractive mapping for all i = 1, 2, ...,n. If x0 is a fixed
point of order 2 for all Ti then by proposition 2.3, x0 is a fixed point of order 2
for T1T2...Tn. Now assume that x0 is a fixed point of order 2 for T1T2...Tn then

by proposition 2.2, 1 = limx→x0
G(T1T2...Tnx,T1T2...Tnx,x0)

G(x,x,x0)
= λ1λ2...λn. But all Ti

are contractive mappings so
G(T1x,T1x,x0)

G(x,x,x0)
< 1 which implies that λi ≤ 1 for all

i = 1, 2, ...n. Hence, λ1 = λ2 = ... = λn = 1. Proof for non expansive is similar.

(b) By proposition 2.2, limx→x0
G(T n1 x,,x0)
G(x,x,x0)

= λn. Then λn = 1 if and only if

λ = 1 because λ ≥ 0.
(c) Let T1 be a contractive mapping and there exists n ∈ N such that x0 is a

fixed point of order 2 for Tn1 .T1 is a contractive mapping. So

G(Tn1 x,T
n
1 x,x0) < ... < G(T1x,T1x,x0) < G(x,x,x0)

1 = lim
x→x0

G(Tn1 x,T
n
1 x,x0)

G(x,x,x0
≤
G(T1x,T1x,x0)

G(x,x,x0
≤ 1.

Therefore, limx→x0
G(T1x,T1x,x0)

G(x,x,x0
= 1. �

Proposition 2.6. Suppose that (X,G) is a metric space, T : X → X is a function
and x0 is a fixed point of T . If T is contraction then x0 is not a fixed point of order
2 for T .

Proof. Since T is a contractive mapping so there exists α ∈ [0, 1) such that G(Tx,Ty,Tz) ≤
αG(x,y,z) for all x,y,z ∈ X. Therefore G(T x,T x,x0)

G(x,x,x0)
≤ α < 1 and x0 can not be a

fixed point of order 2 for T. �

Proposition 2.7. Suppose that x0 ∈ X be a fixed point of order 2 for T : X → X
where T is one to one and g is left inverse of T . Then x0 is also a fixed point of
order 2 for g.

Proof. It is clear that x0 is a fixed point for g. On the other hand, since T is
continuous at x0 for t = Tx so



176 GUPTA

1 = lim
x→x0

G(Tx,Tx,x0)

G(x,x,x0)
= lim

x→x0

G(g(T(Tx)),g(T(Tx)),x0)

G(gTx,gTx,x0)

= lim
t→x0

G(g(Tt),g(Tt),x0)

G(gt,gt,x0)

= lim
t→x0

G(t,t,x0)

G(gt,gt,x0)
= lim

t→x0

1
G(gt,gt,x0)
G(t,t,x0)

Therefore, limt→x0
G(gt,gt,x0)
G(t,t,x0)

= 1. �

In the following we give another condition for the fixed point of order 2.

Proposition 2.8. Suppose that x0 is not alone point and is a fixed point for T :
x → X.

(a) If limx→x0
G(T x,T x,x)
G(x,x,x0)

= 0 then x0 is a fixed point of order 2 for T .

(b) If limx→x0
G(T x,T x,x)
G(T x,T x,x0)

= 0 then x0 is a fixed point of order 2 for T .

Proof. (a) From the definition of G-metric space we have

| G(x,x,x0) −G(Tx,Tx,x0) | ≤ G(Tx,Tx,x)

1 −
G(Tx,Tx,x0)

G(x,x,x0)
≤

G(Tx,Tx,x)

G(x,x,x0)

≤ 1 +
G(Tx,Tx,x0)

G(x,x,x0)

limx→x0
G(T x,T x,x0)
G(x,x,x0)

= 1.

(b) Prove of this part is similarly as prove of (a). �

Proposition 2.9. Suppose that x0 is a fixed point for T : X → X and ψ : X → R+
is a real valued function.

(a) If x0 be a fixed point of order 2 for T then limx→x0
G(T x,T x,x)
G(x,x,x0)

≤ 2.
(b) If G(Tx,Tx,x) ≤ 2ψ(x) −ψ(Tx) ≤ G(x,x,x0) for all x ∈ X then x0 is a

fixed point of order 2 for T if and only if limx→x0
G(T x,T x,x)
G(x,x,x0)

= 0.

Proof. (a) From the inequality

G(Tx,Tx,x) ≤ G(Tx,x0,x0) + G(x0,Tx,x)
≤ G(Tx,Tx,x0) + G(x,x,x0)

G(Tx,Tx,x)

G(x,x,x0)
≤

G(Tx,Tx,x0)

G(x,x,x0)
+ 1.

Therefore, limx→x0
G(T x,T x,x)
G(x,x,x0)

≤ 2.
(b) From inequality G(Tx,Tx,x) ≤ 2ψ(x) −ψ(Tx) ≤ G(x,x,x0),

G(x,x,Tx) + G(Tx,Tx,T2x) + ... + G(Tn−1x,Tn−1x,Tnx) ≤ Σni=12ψ(T
i−1x) −ψ(Tix)

= 2ψ(x) −ψ(Tnx)



FIXED POINT OF ORDER 2 ON G-METRIC SPACE 177

and

G(Tn−1x,Tn−1x,Tnx

G(x,x,x0)
=

G(Tn−1x,Tn−1x,Tnx)

G(Tn−1x,Tn−1x,Tn−2x)

G(Tn−1x,Tn−1x,Tn−2x)

G(Tn−2x,Tn−2x,Tn−3x)
...

= ...
G(T2x,T2x,x0)

G(Tx,Tx,x0)

G(Tx,Tx,x0)

G(x,x,x0
,

since limx→x0
G(T n−1x,T n−1x,T nx)

G(x,x,x0)
= limx→x0

G(T x,T x,x)
G(x,x,x0)

and limx→x0
G(T n−kx,T n−kx,T nx)

G(x,x,x0)
=

1 which k = 1, 2, ...n − 1, so limx→x0
G(T n−1x,T n−1x,T nx)

G(x,x,x0)
= limx→x0

G(T x,T x,x)
G(x,x,x0)

.

From inequality G(Tx,Tx,x) ≤ 2ψ(x) − ψ(Tx) ≤ G(x,x,x0). It is clear that
ψ(Tnx) is strict decreasing.

n
G(Tx,Tx,x)

G(x,x,x0)
≤ lim

x→x0

2ψ(x) −ψ(Tnx)
G(x,x,x0)

≤ lim
x→x0

2ψ(x) −ψ(Tnx)
2ψ(x) −ψ(Tx)

≤ lim
x→x0

2ψ(x) −ψ(Tnx)
2ψ(x) −ψ(Tnx)

= 1.

Hence, limx→x0
G(T x,T x,x)
G(x,x,x0)

= 1
n

. Since n is arbitrary positive integer, limx→x0
G(T x,T x,x)
G(x,x,x0)

=

0. �

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Department of Applied Mathematics, Sagar Institute of Science, Technology & Re-

search, Ratibad, Bhopal - 462043