@ Appl. Gen. Topol. 16, no. 2(2015), 225-231doi:10.4995/agt.2015.3830
c© AGT, UPV, 2015

Two general fixed point theorems for a

sequence of mappings satisfying implicit

relations in Gp - metric spaces

Valeriu Popa
a
and Alina-Mihaela Patriciu

b

a
“Vasile Alecsandri” University of Bacău, 600115 Bacău, Romania (vpopa@ub.ro)

b
Department of Mathematics and Computer Sciences, Faculty of Sciences and Environment,

“Dunărea de Jos” University of Galaţi, 800201 Galaţi, Romania (Alina.Patriciu@ugal.ro)

Abstract

In this paper, two general fixed point theorem for a sequence of map-

pings satisfying implicit relations in Gp - complete metric spaces are

proved.

2010 MSC: 47H10; 54H25.

Keywords: Gp - complete metric space; sequence of mappings; fixed point;

implicit relation.

1. Introduction and Preliminaries

In this paper we shall investigate the existence and uniqueness of common
fixed point of mappings via implicit relations in the setting of Gp - metric
spaces, inspired from the notion of Gp -metric spaces [25],[4],[6],[7] and other
papers. We remind that Gp - metric is inspired from the notions of G - metric
([15],[16],[1],[3],[14] and other papers) and partial metric ([13], [1], [2], [8], [9],
[10], [11], [12] and other papers).

Several classical fixed point theorems and common fixed point theorems have
been unified considering a general condition by an implicit relation in [17], [18].
Some fixed point theorems for mappings satisfying a implicit relation
in G - metric spaces are established in [19] - [22]. Recently, fixed point for
mappings satisfying implicit relation in partial metric spaces are obtained in

Received 29 April 2015 – Accepted 04 July 2015

http://dx.doi.org/10.4995/agt.2015.3830


V. Popa and A.-M. Patriciu

[5], [9], [10], [24]. Quite recently, a fixed point result for mappings satisfying
an implicit relation in Gp - metric spaces is obtained in [23]. We first recall the
notion of Gp - metric.

Definition 1.1 ([25]). Let X be a nonempty set. A function Gp : X3 → R+
is called a Gp - metric on X if the following conditions are satisfied:

(Gp1) : x = y = z if Gp(x, y, z) = Gp(x, x, x) = Gp(y, y, y) = Gp(z, z, z),
(Gp2) : 0 ≤ Gp(x, x, x) ≤ Gp(x, x, y) ≤ Gp(x, y, z) for all x, y, z ∈ X, with

y 6= z,
(Gp3) : Gp(x, y, z) = Gp(y, z, x) = ... (symmetry in all three variables),
(Gp4) : Gp(x, y, z) ≤ Gp(x, a, a)+Gp(a, y, z)−Gp(a, a, a) for all x, y, z, a ∈ X.
The pair (X, Gp) is called a Gp - metric space.

Definition 1.2 ([25]). Let (X, Gp) be a Gp - metric space and {xn} a sequence
in X. A point x ∈ X is said to be the limit of the sequence {xn} or xn → x
({xn} is Gp - convergent to x) if limn,m→∞ Gp(x, xn, xm) = Gp(x, x, x).
Theorem 1.3 ([4]). Let (X, Gp) be a Gp - metric space. Then, for any {xn} ∈
X and x ∈ X, the following conditions are equivalent:

a) {xn} is Gp - convergent to x,
b) Gp(xn, xn, x) → Gp(x, x, x) as n → ∞,
c) Gp(xn, x, x) → Gp(x, x, x) as n → ∞.

Definition 1.4 ([25]). Let (X, Gp) be a Gp - metric space.
1) A sequence {xn} of X is called a Gp - Cauchy sequence in X if

limn,m→∞ GP(xn, xm, xm) exists and is finite.
2) A Gp - metric space is said to be Gp - complete if every Gp - Cauchy

sequence in X, Gp - converges to x ∈ X such that limn,m→∞ Gp (xn, xm, xm) =
Gp(x, x, x).

Lemma 1.5 ([4]). Let (X, Gp) be a Gp - metric space. Then:
1) If Gp(x, y, z) = 0 then x = y = z,
2) If x 6= y then Gp(x, x, y) > 0.

Lemma 1.6. Let (X, Gp) be a Gp - metric space and {xn} is a sequence in
X which is Gp - convergent to a point x ∈ X with Gp (x, x, x) = 0. Then,
limn→∞ G (xn, y, z) = G (x, y, z) for all y, z ∈ X.
Proof. By (Gp4)

(1.1)
Gp (x, y, z) ≤ Gp (x, xn, xn) + Gp (xn, y, z) − Gp (xn, xn, xn)

≤ Gp (x, xn, xn) + Gp (xn, y, z) ,
which implies

Gp (x, y, z) − Gp (x, xn, xn) ≤ Gp (xn, y, z)
≤ Gp (xn, x, x) + Gp (x, y, z) .

By Theorem 1.3,

Gp (xn, x, x) → Gp (x, x, x) = 0

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 226



Two general fixed point theorems in Gp - metric spaces

and
Gp (x, xn, xn) → Gp (x, x, x) = 0.

Letting n tends to infinity in (1.1) we obtain

lim
n→∞

Gp (xn, y, z) = Gp (x, y, z) .

�

Quite recently, Meena and Nema [14] initiated the study of fixed points for
sequences of mappings in G - metric spaces.

2. Implicit relations

Definition 2.1. Let FGp be the set of all continuous functions F(t1, ..., t5) :
R

5
+ → R satisfying the following conditions:
(F1) : F is non - increasing in variables t2, t3, t4, t5,
(F2) : There exists h ∈ [0, 1) such that for all u, v ≥ 0, F(u, v, u, v, u + v) ≤ 0

implies u ≤ hv.
In the following examples, the proofs of property (F1) are obviously.

Example 2.2. F(t1, ..., t5) = t1 − at2 − bt3 − ct4 − dt5, where a, b, c, d ≥ 0 and
a + b + c + 2d < 1.

(F2) : Let u, v ≥ 0 and F(u, v, u, v, u + v) = u − av − bu − cv − d (u + v) ≤ 0,
which implies u ≤ hv, where 0 ≤ h = a+c+d

1−(b+d)
< 1.

Example 2.3. F(t1, ..., t5) = t1 − k max{t2, t3, t4, t5}, where k ∈
[

0, 1
2

)

.
(F2) : Let u, v ≥ 0 and F(u, v, u, v, u + v) = u − k (u + v) ≤ 0 which implies

u ≤ hv, where 0 ≤ h = k
1−k

< 1.

Example 2.4. F(t1, ..., t5) = t1 − k max
{

t2, t3,
t4+t5

2

}

, where k ∈ [0, 1).
(F2) : Let u, v ≥ 0 and F(u, v, u, v, u + v) = u − k max

{

u, v, u+2v
3

}

≤ 0. If
u > v, then u (1 − k) ≤ 0, a contradiction. Hence u ≤ v, which implies u ≤ hv,
where 0 ≤ h = k < 1.
Example 2.5. F(t1, ..., t5) = t

2
1 − at2t3 − bt3t4 − ct4t5, where a, b, c ≥ 0 and

a + b + 2c < 1.
(F2) : Let u, v ≥ 0 and F(u, v, u, v, u + v) = u2 − auv − buv − cv (u + v) ≤ 0.

If u > v, then u[1 − (a + b + 2c)] ≤ 0, a contradiction. Hence u ≤ v, which
implies u ≤ hv, where 0 ≤ h =

√
a + b + 2c < 1.

Example 2.6. F(t1, ..., t5) = t1 − at2 − b max{2t3, t4 + t5}, where a, b ≥ 0 and
a + 3b < 1.

(F2) : Let u, v ≥ 0 and F(u, v, u, v, u + v) = u − av − b max{2v, u + 2v} ≤ 0.
If u > v, then u[1 − (a + 3b)] ≤ 0, a contradiction. Hence u ≤ v, which implies
u ≤ hv, where 0 ≤ h = a + 3b < 1.
Example 2.7. F(t1, ..., t5) = t1 −at2 −b max {t3 + t4, 2t5}, where a, b ≥ 0 and
a + 4b < 1.

(F2) : Let u, v ≥ 0 and F(u, v, u, v, u+v) = u−av−b max{u+v, 2 (u + v)} =
u − av − 2b (u + v) ≤ 0. Hence u ≤ hv, where 0 ≤ h = a+2b

1−2b
< 1.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 227



V. Popa and A.-M. Patriciu

Example 2.8. F(t1, ..., t5) = t
2
1 − at22 − bt23 − ct4t5, where a, b, c ≥ 0 and

a + b + 2c < 1.
(F2) : Let u, v ≥ 0 be and F(u, v, u, v, u+v) = u2−av2−bu2−cv (u + v) ≤ 0.

If u > v, then u2[1 − (a + b + 2c)] ≤ 0, a contradiction. Hence u ≤ v, which
implies u ≤ hv, where 0 ≤ h =

√
a + b + 2c < 1.

Example 2.9. F(t1, ..., t5) = t1 − a max{t2, t3} − b max{t4, t5}, where a, b ≥ 0
and a + 2b < 1.

(F2) : Let u, v ≥ 0 be and F(u, v, u, v, u+v) = u−a max{u, v}−b (u + v) ≤ 0.
If u > v, then u[1 − (a + 2b)] ≤ 0, a contradiction. Hence u ≤ v, which implies
u ≤ hv, where 0 ≤ h = a + 2b < 1.

3. Main results

Theorem 3.1. Let (X, Gp) be a Gp - complete metric space and {Tn}n∈N :
(X, Gp) → (X, Gp) be a sequence of mappings such that for all x, y, z ∈ X and
i, j, k ∈ N:

(3.1)
F(Gp(Tix, Tjy, Tkz), Gp(x, y, z), Gp(Tix, y, Tkz),

Gp(Tix, z, Tjy), Gp(Tjy, Tkz, x)) ≤ 0
where F ∈ FGp. Then, {Tn}n∈N has a unique common fixed point.

Proof. Let x0 be any arbitrary point of X. We define a sequence {xn} in S
such that xn+1 = Tn+1xn, n = 0, 1, 2, ... .

By (3.1) we have successively

F(Gp(Tnxn−1, Tn+1xn, Tn+2xn+1), Gp(xn−1, xn, xn+1), Gp(Tnxn−1, xn, Tn+2xn+1),
Gp(Tnxn−1, xn+1, Tn+1xn), Gp(Tn+1xn, Tn+2xn+1, xn−1)) ≤ 0

(3.2)
F(Gp(xn, xn+1, xn+2), Gp(xn−1, xn, xn+1), Gp(xn, xn, xn+2),

Gp(xn, xn+1, xn+1), Gp(xn+1, xn+2, xn−1)) ≤ 0.
By (Gp2),

Gp(xn, xn, xn+2) ≤ Gp(xn, xn+1, xn+2)
and

Gp(xn−1, xn, xn) ≤ Gp(xn−1, xn, xn+1).
By (Gp4) and (Gp2)

Gp(xn−1, xn+1, xn+2) ≤ Gp(xn−1, xn, xn) + Gp(xn, xn+1, xn+2)
≤ Gp(xn−1, xn, xn+1) + Gp(xn, xn+1, xn+2).

By (3.2) and (F1) we obtain

F(Gp(xn, xn+1, xn+2), Gp(xn−1, xn, xn+1), Gp(xn, xn+1, xn+2),

Gp(xn−1, xn, xn+1), Gp(xn−1, xn, xn+1) + Gp(xn, xn+1, xn+2)) ≤ 0.
By (F2) we obtain

Gp(xn, xn+1, xn+2) ≤ hGp(xn−1, xn, xn+1)

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 228



Two general fixed point theorems in Gp - metric spaces

which implies

(3.3) Gp(xn, xn+1, xn+2) ≤ hnGp(x0, x1, x2).
Now for any integers k ≥ m ≥ n ≥ 1 we obtain by (Gp4) that
Gp (xn, xm, xk) ≤ Gp (xn, xn+1, xn+2) + Gp (xn+1, xn+2, xn+3) + ... +

+ Gp (xk−2, xk−1, xk)

≤ hn
(

1 + h + ... + hk−n
)

Gp (x0, x1, x2)

≤ h
n

1 − h
G (x0, x1, x2) → 0 as n → ∞.

Since by (Gp2), Gp (xn, xm, xm) ≤ Gp (xn, xm, xk) it follows that
Gp (xn, xm, xm) → 0 as n, m → ∞ and thus, {xn} is a Gp - Cauchy
sequence. Since (X, Gp) is a Gp - complete metric space, by Theorem 1.5, (3.3)
and Definition 1.4, there exists u ∈ X such that limn,m→∞ Gp (xn, xm, xm) =
limn→∞ Gp (u, xn, xn) = Gp (u, u, u) = 0.

Now we prove that u is a common fixed point of {Tn}n∈N.
By (3.1) we have successively

F(Gp(Tnxn−1, Tju, Tku), Gp(xn−1, u, u), Gp(Tnxn−1, u, Tku),
Gp(Tn−1xn−1, u, Tju), Gp(Tju, Tku, xn−1)) ≤ 0,

(3.4)
F(Gp(xn, Tju, Tku), Gp(xn−1, u, u), Gp(xn, u, Tku),

Gp(xn, u, Tju), Gp(Tju, Tku, xn−1)) ≤ 0.
Letting n tends to infinity we obtain

F(Gp(xn, Tju, Tku), 0, Gp(u, u, Tku),
Gp(u, u, Tju), Gp(u, Tju, Tku)) ≤ 0.

By (Gp2) and (F1) we obtain

F(Gp(u, Tju, Tku), Gp(u, Tju, Tku), Gp(u, Tju, Tku),
Gp(u, Tju, Tku), Gp(u, Tju, Tku) + Gp (u, Tju, Tku)) ≤ 0.

By (F2) it follows that

Gp(u, Tju, Tku) ≤ hGp(u, Tju, Tku)
which implies

Gp(u, Tju, Tku) = 0.

By Lemma 1.5 (1), u = Tju = Tku. Thus, u is a common fixed point of
{Tn}n∈N.

Suppose that {Tn}n∈N has another common fixed point v.
Then by (3.1) we have successively

F(Gp(Tiu, Tju, Tkv), Gp(u, u, v), Gp(Tiu, u, Tkv),
Gp(Tiu, v, Tju), Gp(Tju, Tkv, u)) ≤ 0,

F(Gp(u, u, v), Gp(u, u, v), Gp(u, u, v),
Gp(u, v, v), Gp(u, v, v)) ≤ 0.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 229



V. Popa and A.-M. Patriciu

By (F1) we have

F(Gp(u, u, v), Gp(u, u, v), Gp(u, u, v),
Gp(u, u, v), Gp(u, v, v) + Gp (u, u, v)) ≤ 0.

By (F2) we have
Gp(u, u, v) ≤ kGp(u, v, v),

which implies
G(u, v, v) = 0.

By Lemma 1.5 (1), u = v.
Hence, u is the unique common fixed point. �

Theorem 3.2. Let (X, Gp) be a Gp - complete metric space and {Tn}n∈N :
(X, Gp) → (X, Gp) be a sequence of mappings such that for all x, y, z ∈ X and
i, j, k ∈ N:

(3.5)
F(Gp(Tix, Tjy, Tkz), Gp(x, y, z), Gp(Tix, y, z),

Gp(x, Tjy, z), Gp(x, y, Tkz)) ≤ 0
where F ∈ FGp. Then, {Tn}n∈N has a unique common fixed point.
Proof. The proof is similar to the proof of Theorem 3.1. �

Acknowledgements. The authors thank the anonimous reviewers for their

valuable comments, which improved the initial version of the paper.

References

[1] T. Abdeljawad, E. Karapinar and K. Tas, Existence and uniqueness of common fixed
points on partial metric spaces, Applied Math. Lett. 24 (11) (2011), 1894–1899.

[2] I. Altun, F. Sola and H. Simsek, Generalized contractive principle on partial metric
spaces, Topology Appl. 157, no. 18 (2010), 2778–2785.

[3] M. Asadi, E. Karapinar and P. Salimi, A new approach to G - metric spaces and related
fixed point theorems, J. Ineq. Appl. (2013), 2013:454.

[4] H. Aydi, E. Karapinar and P. Salimi, Some fixed point results inGp-metric spaces, J.
Appl. Math. (2012), Article ID 891713.

[5] H. Aydi, M. Jellali and E. Karapinar, Common fixed points for α - implicit contractions
in partial metric spaces. Consequences and Applications, Rev. R. Acad. Cienc. Exactas
Fis. Nat. Ser. A Mat. (DOI: 10.1017/s13398-014-0187-1).

[6] M. A. Barakat and A. M. Zidan, A common fixed point theorem for
weak contractive maps in Gp-metric spaces, J. Egyptean Math. Soc. (DOI:
10.1016/j.joems.2014.06.008).

[7] N. Bilgili, E. Karapinar and P. Salimi, Fixed point theorems for generalized contractions
on Gp-metric spaces, J. Ineq. Appl. (2013), 2013:39.

[8] R. Chi, E. Karapinar and T. D. Than, A generalized contraction principle in partial
metric spaces, Math. Comput. Modelling 55, no. 5-6 (2012), 1673–1681.

[9] S. Guliaz and E. Karapinar, Coupled fixed point results in partially ordered partial metric
spaces through implicit function, Hacet. J. Math. Stat. 429, no. 4 (2013), 347–357.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 230



Two general fixed point theorems in Gp - metric spaces

[10] S. Guliaz, E. Karapinar and I. S. Yuce, CA coupled coincidence point theorem in par-
tially ordered metric spaces with an implicit relation, Fixed Point Theory Appl. (2013),
2013:38.

[11] Z. Kadelburg, H. K. Nashine and S. Radanović, Fixed point results under various con-
tractive conditions in partial metric spaces, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser.
A Mat. 10 (2013), 241–256.

[12] E. Karapinar and I. M. Erhan, Fixed point theorems for operators on partial metric
spaces, Appl. Math. Lett. 24 (11) (2011), 1894–1899.

[13] S. Matthews, Partial metric topology and applications, Proc. 8th Summer Conf. General
Topology and Applications, Ann. New York Acad. Sci. 728 (1994), 183–197.

[14] G. Meena and D. Nema, Common fixed point theorem for a sequence of mappings in
G-metric spaces, Intern. J. Math. Computer Research 2, no. 5 (2014), 403–407.

[15] Z. Mustafa and B. Sims, Some remarks concerning D-metric spaces, Proc. Conf. Fixed
Point Theory Appl., Valencia (Spain) (2003), 189–198.

[16] Z. Mustafa and B. Sims, A new approach to generalized metric spaces, J. Nonlinear
Convex Anal. 7, no. 2 (2006), 289–297.

[17] V. Popa, Fixed point theorems for implicit contractive mappings, St. Cerc. Ştiinţ.. Ser.
Mat. 7 (1997), 129–133.

[18] V. Popa, Some fixed point theorems for compatible mappings satisfying an implicit re-

lation, Demonstr. Math. 32, no. 1 (1999), 157–163.
[19] V. Popa and A.-M. Patriciu, Two general fixed point theorems for pairs of weakly com-

patible mappings in G - metric spaces, Novi Sad J. Math. 42, no. 2 (2013), 49–60.
[20] V. Popa and A.-M. Patriciu, A general fixed point theorem for mappings satisfying an

φ - implicit relation in complete G-metric spaces, Gazi Univ. J. Sci. 25, no. 2 (2012),
403–408.

[21] V. Popa and A.-M. Patriciu, A general fixed point theorem for pair of weakly compatible
mappings in G - metric spaces, J. Nonlinear Sci. Appl. 5, no. 2 (2012), 151–160.

[22] V. Popa and A.-M. Patriciu, Fixed point theorems for mappings satisfying an implicit
relation in complete G – metric spaces, Bul. Instit. Politehn. Iaşi 50 (63), Ser. Mat.
Mec. Teor. Fiz., 2 (2013), 97–123.

[23] V. Popa and A.-M. Patriciu, Well posedness of fixed point problem for mappings satis-
fying an implicit relation in Gp-metric spaces, Math. Sci. Appl. E-Notes 3, no. 1 (2015),
108–117.

[24] C. Vetro and F. Vetro, Common fixed points of mappings satisfying implicit relations
in partial metric spaces, J. Nonlinear Sci. Appl. 6 (2013), 152–161.

[25] M. R. A. Zand and A. N. Nezhad, A generalization of partial metric spaces, J. Contem-
porary Appl. Math. 1, no. 1 (2011), 86–93.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 231