a new approach to g-metric and related fixed point theorems

Asadi et al. Journal of Inequalities and Applications 2013, 2013:454http://www.journalofinequalitiesandapplications.com/content/2013/1/454

R E S E A R C H Open Access

A new approach to G-metric and related fixedpoint theoremsMehdi Asadi1, Erdal Karapınar2* and Peyman Salimi3

*Correspondence:[email protected];[email protected] University, Incek, Ankara06836, TurkeyFull list of author information isavailable at the end of the article

AbstractVery recently, Samet et al. and Jleli and Samet reported that most of fixed point resultsin the context of G-metric space, defined by Sims and Zead, can be derived from theusual fixed point theorems on the usual metric space. In this paper, we state andprove some fixed point theorems in the framework of G-metric space that cannot beobtained from the existence results in the context of associated metric space.

1 Introduction and preliminariesIn , Mustafa and Sims introduced the notion of G-metric and investigated the topol-ogy of such spaces. The authors also characterized some celebrated fixed point results inthe context of G-metric space. Following this initial paper, a number of authors have pub-lished many fixed point results on the setting of G-metric space (see, e.g., [–] and thereferences therein). Samet et al. [] and Jleli and Samet [] reported that some pub-lished results can be considered as a straight consequence of the existence theorem in thesetting of the usual metric space. More precisely, the authors of these two papers noticedthat p(x, y) = pG(x, y) = G(x, y, y) is a quasi-metric whenever G : X × X × X → [,∞) is aG-metric. It is evident that each quasi-metric induces a metric. In particular, if the pair(X, p) is a quasi-metric space, then the function defined by

d(x, y) = dG(x, y) = max{

p(x, y), p(y, x)}

, for all x, y ∈ X,

forms a metric on X.The object of this paper is to get some fixed point results in the context of G-metric

space that cannot be concluded from the existence results. This paper can be consideredas a continuation of [], which was inspired by [].

First, we recollect some necessary definitions and results in this direction. The notionof G-metric spaces is defined as follows.

Definition . (See []) Let X be a non-empty set, G : X × X × X → R+ be a function

satisfying the following properties:(G) G(x, y, z) = if x = y = z,(G) < G(x, x, y) for all x, y ∈ X with x �= y,(G) G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X with y �= z,(G) G(x, y, z) = G(x, z, y) = G(y, z, x) = · · · (symmetry in all three variables),(G) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) (rectangle inequality) for all x, y, z, a ∈ X .

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Then 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.

Note that every G-metric on X induces a metric dG on X defined by

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

For a better understanding of the subject, we give the following examples of G-metrics.

Example . Let (X, d) be a metric space. Function G : X × X × X → [, +∞), defined by

G(x, y, z) = max{

d(x, y), d(y, z), d(z, x)}

,

for all x, y, z ∈ X, is a G-metric on X.

Example . (See, e.g., []) Let X = [,∞). Function G : X × X × X → [, +∞), defined by

G(x, y, z) = |x – y| + |y – z| + |z – x|,

for all x, y, z ∈ X, is a G-metric on X.

In their initial paper, Mustafa and Sims [] also defined the basic topological conceptsin G-metric spaces as follows.

Definition . (See []) Let (X, G) be a G-metric space, and let {xn} be a sequence of pointsof X. We say that {xn} is G-convergent to x ∈ X if

limn,m→+∞ G(x, xn, xm) = ,

that is, for any ε > , there exists N ∈ N such that G(x, xn, xm) < ε for all n, m ≥ N . We callx the limit of the sequence and write xn → x or limn→+∞ xn = x.

Proposition . (See []) Let (X, G) be a G-metric space. The following are equivalent:() {xn} is G-convergent to x,() G(xn, xn, x) → as n → +∞,() G(xn, x, x) → as n → +∞,() G(xn, xm, x) → as n, m → +∞.

Definition . (See []) Let (X, G) be a G-metric space. Sequence {xn} is called aG-Cauchy sequence if, for any ε > , there exists N ∈ N such that G(xn, xm, xl) < ε forall m, n, l ≥ N , that is, G(xn, xm, xl) → as n, m, l → +∞.

Proposition . (See []) Let (X, G) be a G-metric space. Then the following are equiva-lent:

() sequence {xn} is G-Cauchy,() for any ε > , there exists N ∈ N such that G(xn, xm, xm) < ε for all m, n ≥ N .

Definition . (See []) A G-metric space (X, G) is called G-complete if every G-Cauchysequence is G-convergent in (X, G).



Definition . Let (X, G) be a G-metric space. Mapping F : X × X × X → X is said to becontinuous if for any three G-convergent sequences {xn}, {yn} and {zn} converging to x, yand z, respectively, {F(xn, yn, zn)} is G-convergent to F(x, y, z).

Mustafa [] extended the well-known Banach [] contraction principle mapping in theframework of G-metric spaces as follows.

Theorem . (See []) Let (X, G) be a complete G-metric space and T : X → X be a map-ping satisfying the following condition for all x, y, z ∈ X:

G(Tx, Ty, Tz) ≤ kG(x, y, z), ()

where k ∈ [, ). Then T has a unique fixed point.

Theorem . (See []) Let (X, G) be a complete G-metric space and T : X → X be a map-ping satisfying the following condition for all x, y ∈ X:

G(Tx, Ty, Ty) ≤ kG(x, y, y), ()


Remark . We notice that condition () implies condition (). The converse is true onlyif k ∈ [,

). For details see [].

Lemma . [] By the rectangle inequality (G) together with the symmetry (G), wehave

G(x, y, y) = G(y, y, x) ≤ G(y, x, x) + G(x, y, x) = G(y, x, x). ()

2 Main resultsTheorem . Let (X, G) be a complete G-metric space and T : X → X be a mapping sat-isfying the following condition for all x, y ∈ X:

G(Tx, Ty, Ty) ≤ kG(x, Tx, y), ()


Proof Let x ∈ X be an arbitrary point, and define the sequence xn by xn = Tn(x). By (),we have

G(xn, xn+, xn+) ≤ kG(xn–, xn, xn). ()

Continuing in the same argument, we will get

G(xn, xn+, xn+) ≤ knG(x, x, x). ()



Moreover, for all n, m ∈ N; n < m, we have by rectangle inequality that

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

+ G(xn+, xn+, xn+) + · · · + G(xm–, xm, xm)

≤ (kn + kn+ + kn+ + · · · + km–)G(x, x, x)

≤ kn

– kG(x, x, x), ()

and so, lim G(xn, xm, xm) = , as n, m → ∞. Thus, {xn} is G-Cauchy sequence. Due to thecompleteness of (X, G), there exists u ∈ X such that {xn} is G-convergent to u.

Suppose that Tu �= u, then

G(xn, Tu, Tu) ≤ kG(xn–, xn, u), ()

taking the limit as n → ∞, and using the fact that function G is continuous, then

G(u, Tu, Tu) ≤ kG(u, u, u). ()

This contradiction implies that u = Tu.To prove uniqueness, suppose that u �= v such that Tv = v, and use Lemma ., then

G(u, u, v) = G(Tu, Tu, Tv) ≤ kG(u, Tu, v) = kG(u, u, v), ()

which implies that u = v. �

Example . Let X = [,∞) and

G(x, y, z) =

⎧⎨⎩

, if x = y = z,

max{x, y, z}, otherwise

be a G-metric on X. Define T : X → X by Tx = x. Then the condition of Theorem .

holds. In fact,

G(Tx, Ty, Ty) =

max{x, y}

and

G(x, Tx, y) = max{x, y},

and so,

G(Tx, Ty, Ty) ≤

G(x, Tx, y).

That is, conditions of Theorem . hold for this example.



Corollary . Let (X, G) be a complete G-metric space and T : X → X be a mapping sat-isfying the following condition for all x, y, z ∈ X:

G(Tx, Ty, Tz) ≤ aG(x, Tx, z) + bG(x, Tx, y),

where ≤ a + b < . Then T has a unique fixed point.

Theorem . Let (X, G) be a complete G-metric space and T : X → X be a mapping sat-isfying the following condition for all x, y ∈ X, where a + b + c + d <

G(Tx, Ty, Ty

) ≤ aG(x, Tx, Tx

)+ bG

(y, Ty, Ty

)+ cG(x, Tx, Ty) + cG

(y, Ty, Tx

). ()

Then T has a unique fixed point.

Proof Take x ∈ X. We construct sequence {xn}∞n= of points in X in the following way:

xn+ = Txn for all n = , , , . . . .

Notice that if xn′ = xn′+ for some n′ ∈ N, then obviously T has a fixed point. Thus, wesuppose that

xn �= xn+

for all n ∈N.That is, we have

G(xn, xn+, xn+) > .

From (), with x = xn– and y = xn, we have

G(Txn–, Txn, Txn

) ≤ aG(xn–, Txn–, Txn–

)+ bG

(xn, Txn, Txn

)+ cG(xn–, Txn–, Txn) + dG

(xn, Txn, Txn–

),

which implies that

G(xn, xn+, xn+) ≤ aG(xn–, xn, xn+) + bG(xn, xn+, xn+)

+ cG(xn–, xn, xn+) + dG(xn, xn+, xn+),

and so,

G(xn, xn+, xn+) ≤ kG(xn–, xn, xn+),

where k = a+c–b–d < . Then

G(xn, xn+, xn+) ≤ knG(x, x, x) ()



for all n ∈N. Note that from (G), we know that

G(xn, xn, xn+) ≤ G(xn, xn+, xn+)

with xn �= xn+, and by Lemma ., we know that

G(xn+, xn+, xn) ≤ G(xn, xn, xn+).

Then by (), we have

G(xn+, xn+, xn) ≤ knG(x, x, x).

Moreover, for all n, m ∈ N; n < m, we have by rectangle inequality that

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

+ G(xn+, xn+, xn+) + · · · + G(xm–, xm, xm)

≤ (kn + kn+ + kn+ + · · · + km–)G(x, x, x)

≤ kn

– kG(x, x, x), ()

and so, lim G(xn, xm, xm) = , as n, m → ∞. Thus, {xn} is G-Cauchy sequence. Due to thecompleteness of (X, G), there exists u ∈ X such that {xn} is G-convergent to z. From (),with x = xn and y = z, we have

G(Txn, Tz, Tz

) ≤ aG(xn, Txn, Txn

)+ bG

(z, Tz, Tz

)+ cG(xn, Txn, Tz) + dG

(z, Tz, Txn

).

Then

G(xn+, Tz, Tz

) ≤ aG(xn, xn+, xn+)+bG(z, Tz, Tz

)+cG(xn, xn+, Tz)+dG(z, Tz, xn+).

Taking limit as n → ∞ in the inequality above, we have

G(z, Tz, Tz

) ≤ (c + d) – b

G(z, z, Tz).

Now, if Tz = Tz, then T has a fixed point. Hence, we assume that Tz �= Tz. Therefore, by(G), we get

G(z, Tz, Tz

) ≤ (c + d) – b

G(z, z, Tz) ≤ (c + d) – b

G(z, Tz, Tz

),

which implies that G(z, Tz, Tz) = , i.e., z = Tz = Tz. �

At first, we assume that

� ={ψ : [,∞) → [,∞) such that ψ is non-decreasing and continuous

}



and

� ={ϕ : [,∞) → [,∞) such that ϕ is lower semicontinuous

},

where ψ(t) = φ(t) = if and only if t = .

Theorem . Let (X, G) be a complete G-metric space and T : X → X be a mapping sat-isfying the following condition for all x, y ∈ X, where ψ ∈ � and φ ∈ � holds

ψ(G

(Tx, Tx, Ty

)) ≤ ψ(G(x, Tx, y)

)– φ

(G(x, Tx, y)

). ()


Proof Take x ∈ X. We construct sequence {xn}∞n= of points in X in the following way:

xn+ = Txn for all n = , , , . . . .

Notice that if xn′ = xn′+ for some n′ ∈ N, then obviously T has a fixed point. Thus, wesuppose that

xn �= xn+

for all n ∈N.By (G), we have

G(xn, xn+, xn+) > .

From (), with x = xn– and y = xn, we have

ψ(G

(Txn–, Txn–, Txn

)) ≤ ψ(G(xn–, Txn–, xn)

)– φ

(G(xn–, Txn–, xn)

),

which implies that

ψ(G(xn, xn+, xn+)

) ≤ ψ(G(xn–, xn, xn)

)– φ

(G(xn–, xn, xn)

)()

≤ ψ(G(xn–, xn, xn)

), ()

then G(xn, xn+, xn+) ≤ G(xn–, xn, xn). So sequence {G(xn, xn+, xn+)} is a decreasing se-quence in R

+, and thus, it is convergent, say t ∈ R+. We claim that t = . Suppose, to the

contrary, that t > . Taking limit as n → ∞ in (), we get

ψ(t) ≤ ψ(t) – φ(t),

which implies φ(t) = . That is, t = , which is a contrary. Hence, t = , i.e.,

limn→∞ G(xn, xn+, xn+) = . ()



We shall show that {xn}∞n= is a G-Cauchy sequence. Suppose, to the contrary, that thereexists ε > , and sequence xn(k) of xn such that

G(xm(k), Txm(k), xn(k)) ≥ ε ()

with n(k) ≥ m(k) > k. Further, corresponding to m(k), we can choose n(k) in such a waythat it is the smallest integer with n(k) > m(k) satisfying (). Hence,

G(xm(k), Txm(k), xn(k)–) < ε. ()

By Lemma . and (G), we have

ε ≤ G(xm(k), Txm(k), xn(k)) = G(xn(k), xm(k), Txm(k))

≤ G(xn(k), xn(k)–, xn(k)–) + G(xn(k)–, Txm(k), xm(k))

≤ G(xm(k), Txm(k), xn(k)–) + sn(k)–

≤ ε + sn(k)–, ()

where sn(k)– = G(xn(k)–, xn(k), xn(k)). Letting k → ∞ in (), we derive that

limk→∞

G(xm(k), Txm(k), xn(k)) = ε. ()

Also, by Lemma . and (G), we obtain the following inequalities:

G(xm(k), Txm(k), xn(k)) ≤ G(xm(k), xm(k)–, xm(k)–) + G(xm(k)–, Txm(k), xn(k))

= G(xm(k), xm(k)–, xm(k)–) + G(xn(k), xm(k)–, Txm(k))

≤ G(xm(k), xm(k)–, xm(k)–) + G(xn(k), xn(k)–, xn(k)–)

+ G(xn(k)–, xm(k)–, Txm(k))

≤ sm(k)– + sn(k)– + G(xn(k)–, xm(k)–, Txm(k)) ()

and

G(xn(k)–, xm(k)–, Txm(k)) ≤ G(xn(k)–, xn(k), xn(k)) + G(xn(k), xm(k)–, Txm(k))

= G(xn(k)–, xn(k), xn(k)) + G(xm(k)–, Txm(k), xn(k))

≤ G(xn(k)–, xn(k), xn(k)) + G(xm(k)–, xm(k), xm(k))

+ G(xm(k), Txm(k), xn(k))

= sn(k)– + sm(k)– + G(xm(k), Txm(k), xn(k)). ()

Letting k → ∞ in () and () and applying (), we find that

limk→∞

G(xn(k)–, xm(k)–, Txm(k)) = ε. ()



Again, by Lemma . and (G), we have

G(xn(k)–, xm(k)–, Txm(k)) = G(Txm(k), xm(k)–, xn(k)–)

= G(xm(k)+, xm(k)–, xn(k)–)

≤ G(xm(k)+, xm(k), xm(k)) + G(xm(k), xm(k)–, xn(k)–)

= G(xm(k)+, xm(k), xm(k)) + G(xm(k)–, xm(k), xn(k)–)

≤ sm(k) + G(xm(k)–, Txm(k)–, xn(k)–), ()

and

G(xm(k)–, Txm(k)–, xn(k)–) = G(xm(k)–, xm(k), xn(k)–)

≤ G(xm(k)–, xm(k)+, xm(k)+) + G(xm(k)+, xm(k), xn(k)–)

≤ G(xm(k)–, xm(k), xm(k)) + G(xm(k), xm(k)+, xm(k)+)

+ G(xm(k)+, xm(k), xn(k)–)

= sm(k)– + sm(k) + G(xm(k)+, xm(k), xn(k)–)

= sm(k)– + sm(k) + G(xm(k), Txm(k), xn(k)–)

< sm(k)– + sm(k) + ε. ()

Taking limit as n → ∞ in () and () and applying (), we have

limk→∞

G(xm(k)–, Txm(k)–, xn(k)–) = ε. ()

By (), with x = xm(k)– and y = xn(k)–, we have

ψ(G(xm(k), Txm(k), xn(k))

)= ψ

(G

(Txm(k)–, Txm(k)–, Txn(k)–

))≤ ψ

(G(xm(k)–, Txm(k)–, xn(k)–)

)– φ

(G(xm(k)–, Txm(k)–, xn(k)–)

).

Taking limit as k → ∞ in the inequality above and applying, we have

ψ(ε) ≤ ψ(ε) – φ(ε),

which implies ε = , which is a contradiction. Then

limm,n→∞ G(xm, Txm, xn) = lim

m,n→∞ G(xm, xm+, xn) = .

That is, {xn}∞ is a Cauchy sequence. Since (X, G) is a G-complete, then there exist z ∈ Xsuch that xn → z as n → ∞. From (), with x = xn and y = z, we have

ψ(G(xn+, xn+, Tz)

)= ψ

(G

(Txn, Txn, Tz

))≤ ψ

(G(xn, Txn, z)

)– φ

(G(xn, Txn, z)

)= ψ

(G(xn, xn+, z)

)– φ

(G(xn, xn+, z)

).



Taking limit as n → ∞, we get

ψ(G(z, z, Tz)

) ≤ ψ() – φ() = .

Then G(z, z, Tz) = , i.e., z = Tz. To prove uniqueness, suppose that z �= u, such that Tu = u.Now, by (), we get

ψ(G

(Tz, Tz, Tu

)) ≤ ψ(G(z, Tz, u)

)– φ

(G(z, Tz, u)

), ()

which implies that φ(G(z, Tz, u)) = , i.e., z = u. �

If we take ψ(t) = t and φ(t) = ( – r)t in Theorem ., where ≤ r < , then we deducethe following corollary.

Corollary . Let (X, G) be a complete G-metric space and T : X → X be a mapping sat-isfying the following condition for all x, y ∈ X, where ≤ r < holds

G(Tx, Tx, Ty

) ≤ rG(x, Tx, y).


Example . Let X = [,∞) and

G(x, y, z) =

⎧⎨⎩

, if x = y = z,

max{x, y} + max{y, z} + max{x, z}, otherwise

be a G-metric on X. Define T : X → X by Tx = x. Then all the conditions of Corollary .

(Theorem .) hold. Indeed,

G(Tx, Tx, Ty

)=

x +

max

{

x, y}

+

max{x, y}

and

G(x, Tx, y) = x + max

{

x, y}

+ max{x, y},

and so,

G(Tx, Tx, Ty

) ≤

G(x, Tx, y)

That is, the conditions of Corollary . (Theorem .) hold for this example.

Corollary . Let (X, G) be a complete G-metric space and T : X → X be a mapping sat-isfying the following condition for all x, y, z ∈ X, where ≤ a + b < holds

G(Tx, Tx, Ty

)+ G

(Tx, Tx, Tz

) ≤ aG(x, Tx, y) + bG(x, Tx, z).




Proof By taking y = z, we get

G(Tx, Tx, Ty

) ≤ (a + b)

G(x, Tx, y),

where ≤ (a+b) < . That is, conditions of Theorem . hold, and T has a unique fixed

point. �

3 Fixed point results for expansive mappingsIn this section, we establish some fixed point results for expansive mappings.

Theorem . Let (X, G) be a complete G-metric space and T : X → X be an onto mappingsatisfying the following condition for all x, y ∈ X, where α > holds

G(Tx, Tx, Ty

) ≥ αG(x, Tx, y). ()


Proof Let x ∈ X, since T is onto, then there exists x ∈ X such that x = Tx. By continuingthis process, we get xn = Txn+ for all n ∈ N ∪ . In case xn = xn+, for some n ∈ N ∪ ,then it is clear that xn is a fixed point of T . Now, assume that xn �= xn+ for all n. From (),with x = xn+ and y = xn, we have

G(xn, xn–, xn–) = G(Txn+, Txn+, Txn

)≥ αG(xn+, Txn+, xn) = αG(xn+, xn, xn),

which implies that

G(xn+, xn, xn) ≤ hG(xn, xn–, xn–), ()

where h = α

< . Then we have

G(xn+, xn, xn) ≤ hnG(x, x, x). ()

By Lemma ., we get

G(xn, xn+, xn+) ≤ G(xn+, xn, xn) ≤ hnG(x, x, x). ()

Following the lines of the proof of Theorem ., we derive that {xn} is a Cauchy sequence.Since (X, G) is complete, then there exists z ∈ X such that xn → z as n → ∞. Consequently,since T is onto, then there exists w ∈ X such that z = Tw. From (), with x = xn+ and y = w,we have

G(xn, xn–, z) = G(Txn+, Txn+, Tw

) ≥ αG(xn+, Txn+, w) = αG(xn+, xn, w).

Taking limit as n → ∞ in the inequality above, we get

G(z, z, w) = limn→∞ G(xn, xn–, z) = .



That is, z = w. Then z = Tw = Tz. To prove uniqueness, suppose that u �= v such that Tv = vand Tu = u. Now by (), we get

G(u, u, v) = G(Tu, Tu, Tv

) ≥ αG(u, Tu, v) ≥ αG(u, u, v) > G(u, u, v),

which is a contradiction. Hence, u = v. �

Theorem . Let (X, G) be a complete G-metric space and T : X → X be a mapping sat-isfying the following condition for all x, y ∈ X, where a >

G(Tx, Ty, Ty

) ≥ αG(x, Tx, Tx

). ()


Proof Let x ∈ X, since T is onto, then there exists x ∈ X such that x = Tx. By continuingthis process, we get xn = Txn+ for all n ∈ N ∪ . In case xn = xn+, for some n ∈ N ∪ ,then it is clear that xn is a fixed point of T . Now, assume that xn �= xn+ for all n. From (),with x = xn+ and y = xn, we have

G(Txn+, Txn, Txn

) ≥ αG(xn+, Txn+, Txn+

),

which implies that

G(xn, xn–, xn–) ≥ αG(xn+, xn, xn–),

and so,

G(xn+, xn, xn–) ≤ hG(xn, xn–, xn–),

where h = α

< . By the mimic of the proof of Theorem ., we can show that {xn} is aCauchy sequence. Since (X, G) is a complete G-metric space, then there exists z ∈ X suchthat xn → z as n → ∞. Consequently, since T is onto, then there exists w ∈ X such thatz = Tw. From (), with x = w and y = xn+, we have

G(z, xn, xn–) = G(Tw, Txn+, Txn+

) ≥ αG(w, Tw, Tw

).

Taking limit as n → ∞ in the inequality above, we have G(w, Tw, Tw) = . That is, w =Tw = Tw. To prove the uniqueness, suppose that u �= v such that Tv = v and Tu = u. �

Competing interestsThe authors declare that there is no conflict of interests regarding the publication of this article.

Authors’ contributionsAll authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Author details1Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, 45156 58145, Iran. 2Atilim University, Incek,Ankara 06836, Turkey. 3Young Researchers and Elite Club, Rasht Branch, Islamic Azad University, Rasht, Iran.



AcknowledgementsThe authors thank to anonymous referees for their remarkable comments, suggestions and ideas that helped to improvethis paper.

Received: 17 July 2013 Accepted: 16 September 2013 Published: 07 Nov 2013

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10.1186/1029-242X-2013-454Cite this article as: Asadi et al.: A new approach to G-metric and related fixed point theorems. Journal of Inequalitiesand Applications 2013, 2013:454


a new approach to g-metric and related fixed point theorems

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