@
Appl. Gen. Topol. 22, no. 2 (2021), 483-496

doi:10.4995/agt.2021.16562

© AGT, UPV, 2021

Revisiting Ćirić type nonunique fixed point

theorems via interpolation

Erdal Karapınar *

Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Ho Chi Minh, Vietnam

Department of Mathematics, Cankaya University 06836, Incek, Ankara-Turkey. (erdalkarapinar@yahoo.com)

Communicated by S. Romaguera

Abstract

In this paper, we aim to revisit some non-unique fixed point theorems

that were initiated by Ćirić, first. We consider also some natural con-
sequences of the obtained results. In addition, we provide a simple
example to illustrate the validity of the main result.

2020 MSC: 46T99; 47H10; 54H25.

Keywords: abstract metric space; non-unique fixed point; self-mappings.

1. Introduction and Preliminaries

The notion of ”nonunique fixed point” was suggested and used efficiently by
Ćirić [16] in 1974. Regarding the fact that Banach’s fixed point theorem was
abstracted from the papers of Liouville (1837) and Picard (1890), we underline
the connection of the fixed point theorem and the solution of the differential
equations. As well as the existence, the uniqueness of the solutions of differen-
tial equations is desired in most occasions. On the other hand, there are certain
types of differential equations that have no unique solution. In connection with
this fact, it is necessary to determine that non-unique fixed points are at least
as significant as the unique ones. After the initial work of Ćirić [16], several
authors have published nonunique fixed point results in various conditions in
different abstract spaces, see e.g. [16, 37, 1, 21, 35, 36, 23, 24, 25].

*Dedicated to Professor Ljumbor Ćirić

Received 31 October 2021 – Accepted 25 November 2021

http://dx.doi.org/10.4995/agt.2021.16562


E. Karapınar

On the other hand, recently, the notion of interpolative contraction was
defined in [27] to revisit the well-known results of Kannan [22]. Following this
pioneering result, several papers on the interpolative contraction have appeared
in the literature, see e.g. [28, 19, 9, 4, 26, 8, 34].

One of the most interesting generalization of the metric space is the b-metric
space, defined as follow:

Definition 1.1 ([17, 14]). Let X be a nonempty set and let d : X × X −→
[0,∞) satisfy the following conditions for all x,y,u ∈ X,

(1.1)
(b1) d(x,y) = 0 if and only if x = y(indistancy)
(b2) d(x,y) = d(y,x) (symmetry)
(b3) d(x,y) ≤ s[d(x,u) + d(u,y)] (modified triangle inequality).

Then, the map d is called a b-metric and the space (X,d) a b-metric space.

It is worthy to note that the notion of ”b-metric” was announced also as
”quasi-metric”, see e.g. [13, 14]. It is also interesting to note that the notion of
b-metric has a topology different from that of the standard metric. For example,
closed ball is not a closed set. In the same way, the open ball does not form
an open set. Besides, the b-metric needs not to be continuous. Considering
the above-mentioned features of the b-metric, we can easily understand why so
much research has been done on the b-metric, see e.g. [31, 20, 10, 11, 3, 30, 7,
2, 32, 33, 15, 6, 18, 5].

The following examples are not only standard, but also basic and interesting.

Example 1.2 ([10, 11]). Let X = R. Define
(1.2) d(x,y) = |x−y|p

for p > 1. Then d is a b-metric on R. Clearly, the first two conditions hold.
Since

|x−y|p ≤ 2p−1[|x−z|p + |z −y|p],
the third condition holds with s = 2p−1. Thus, (R,d) is a b-metric space with
a constant s = 2p−1.

Example 1.3 ([10, 11]). For p ∈ (0, 1), take

X = lp(R) =

{
x = {xn}⊂ R :

∞∑
n=1

|xn|p < ∞

}
.

Define

d(x,y) =

(
∞∑
n=1

|xn −yn|p
)1/p

.

Then (X,d) is a b-metric space with s = 21/p.

Example 1.4 ([10, 11]). Let E be a Banach space and 0E be the zero vector
of E. Let P be a cone in E with int(P) 6= ∅ and � be a partial ordering with
respect to P . Let X be a non-empty set. Suppose the mapping d : X×X → E
satisfies:

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 484



Revisiting Ćirić type nonunique fixed point theorems via interpolation

(M1) 0 � d(x,y) for all x,y ∈ X;
(M2) d(x,y) = 0 if and only if x = y;
(M3) d(x,y) � d(x,z) + d(z,y), for all x,y ∈ X;
(M4) d(x,y) = d(y,x) for all x,y ∈ X.

Then d is called a cone metric on X, and the pair (X,d) is called a cone metric
space (CMS).

Recall that a cone P in a Banach space (E,‖ ·‖) is called normal if it there
exist a real number K ≥ 1 satisfies the following condition:

x � y ⇒‖x‖≤ K‖y‖ for all x,y ∈ P.
Let E be a Banach space and P be a normal cone in E with the coefficient of

normality denoted by K. Let X be a non-empty set and D : X ×X → [0,∞)
be defined by D(x,y) = ||d(x,y)||, where d : X × X → E is a cone metric
space. Then (X,D) is a b-metric space with a constant s := K ≥ 1.

In generalization of the contraction condition, several auxiliary functions
were considered in the literature. Among them, we count the notion of com-
parison function which was defined by Rus [38].

Definition 1.5 ([12, 38]). A function φ : [0,∞) → [0,∞) is called a comparison
function if it is increasing and φn(t) → 0 as n →∞ for every t ∈ [0,∞), where
φn is the n-th iterate of φ.

We refer [12, 38] for the basic features and interesting example for compar-
ison functions. Among all, we recollect the following lemma that indicates the
importance of the comparison functions.

Lemma 1.6 ([12, 38]). If φ : [0,∞) → [0,∞) is a comparison function, then

(1) each iterate φk of φ, k ≥ 1 is also a comparison function;
(2) φ is continuous at 0;
(3) φ(t) < t for all t > 0.

Definition 1.7 ([14]). Let s ≥ 1 be a real number. A function φ : [0,∞) →
[0,∞) is called a (b)-comparison function if

(1) φ is increasing;

(2) there exist k0 ∈ N, a ∈ [0, 1) and a convergent nonnegative series
∞∑
k=1

vk

such that
sk+1φk+1(t) ≤ askφk(t) + vk,

for k ≥ k0 and any t ≥ 0.
The collection of all (b)-comparison functions will be denoted by Ψ. Berinde

[14] also proved the following important property of (b)-comparison functions.

Lemma 1.8 ([14]). Let φ : [0,∞) → [0,∞) be a (b)-comparison function.
Then

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 485



E. Karapınar

(1) the series

∞∑
k=0

skφk(t) converges for any t ∈ [0,∞);

(2) the function bs : [0,∞) → [0,∞) defined as bs =
∞∑
k=0

skφk(t) is increas-

ing and is continuous at t = 0.

Remark 1.9. Any (b)-comparison function φ satisfies φ(t) < t and limn→∞φ
n(t) =

0 for each t > 0.

In this paper, we shall reconsider some of well-known nonunique fixed point
theorem via interpolation in the context of b-metric spaces.

2. Non-unique fixed points on b-metric space

We start this section by considering the analog of the notions, ”orbitally
continuous” and ”orbitally complete”, in the framework of b-metric space.

Definition 2.1 (see [16]). Let (X,d) be a b-metric space and T be a self-map
on X.

(1) T is called orbitally continuous if

(2.1) lim
i→∞

Tnix = z

implies

(2.2) lim
i→∞

TTnix = Tz

for each x ∈ X.
(2) (X,d) is called orbitally complete if every Cauchy sequence of type
{Tnix}i∈N converges with respect to τd.

A point z is said to be a periodic point of a function T of period m if
Tmz = z, where T 0x = x and Tmx is defined recursively by Tmx = TTm−1x.

2.1. Ćirić type non-unique fixed point results.

Theorem 2.2. For a nonempty set X, we suppose that the function d : X ×
X → [0,∞) is a b-metric. We presume that a self-mapping T is orbitally
continuous and (X,d,s) forms a T -orbitally complete b-metric space with s ≥ 1.
If there is ψ ∈ Ψ and α ∈ (0, 1) such that

(2.3)
min{

(
dα(Tx,Ty)d1−α(x,Tx)

)
,
(
dα(Tx,Ty)d1−α(y,Ty)

)
}

−min{dα(x,Ty),d1−α(Tx,y)}≤ ψ(d(x,y)),

for all x,y ∈ X, then, for each x0 ∈ X the sequence {Tnx0}n∈N converges to a
fixed point of T .

Proof. Starting from an arbitrary x := x0 ∈ X, we shall built a recursive
sequence {xn} in the following way:

(2.4) x0 := x and xn = Txn−1 for all n ∈ N.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 486



Revisiting Ćirić type nonunique fixed point theorems via interpolation

We presume that

(2.5) xn 6= xn−1 for all n ∈ N.

Indeed, if for some n ∈ N we observe the inequality xn = Txn−1 = xn−1, then,
the proof is completed.

By replacing x = xn−1 and y = xn in the inequality (2.3), we derive that
(2.6)

min{
(
dα(Txn−1,Txn)d

1−α(xn−1,Txn−1)
)
,
(
dα(Txn−1,Txn)d

1−α(xn,Txn)
)
}

−min{dα(xn−1,Txn),d1−α(Txn−1,xn)}
≤ ψ(d(xn−1,xn)).

It yields that

(2.7) min{d1−α(xn,xn+1)dα(xn,xn−1),d(xn,xn+1)}≤ ψ(d(xn−1,xn)).

We shall prove that the sequence (
.
xn−1,xn)} is non-increasing. Suppose, on

the contrary, that there is n0 such that d(xn0,xn0+1) > d(xn0−1,xn0 ). Since
ψ(t) < t for all t > 0, for this case we get

d1−α(xn0,xn0+1)d
α(xn0,xn0−1) ≤ ψ(d(xn0−1,xn0 )) < d(xn0−1,xn0 ),

which implies

d(xn0,xn0−1) ≤ d
1−α(xn0,xn0+1)d

α(xn0,xn0−1) ≤ ψ(d(xn0−1,xn0 )) < d(xn0−1,xn0 ),

that is, a contradiction. Thus, we find that for all n ∈ N,

(2.8) d(xn,xn+1) ≤ ψ(d(xn−1,xn)) < d(xn−1,xn).

Recursively, we derive that
(2.9)
d(xn,xn+1) ≤ ψ(d(xn−1,xn)) ≤ ψ2(d(xn−2,xn−1)) ≤ ···≤ ψn(d(x0,x1)).

Taking (2.8) into account, we note that the sequence {d(xn,xn+1)} is non-
increasing.

In what follows, we shall prove that the sequence {xn} is Cauchy. By using
the triangle inequality (b3), we get
(2.10)

d(xn,xn+k) ≤ s[d(xn,xn+1) + d(xn+1,xn+k)]
≤ sd(xn,xn+1) + s{s[d(xn+1,xn+2) + d(xn+2,xn+k)]}
= sd(xn,xn+1) + s

2d(xn+1,xn+2) + s
2d(xn+2,xn+k)

...
≤ sd(xn,xn+1) + s2d(xn+1,xn+2) + . . .
+ sk−1d(xn+k−2,xn+k−1) + s

k−1d(xn+k−1,xn+k)
≤ sd(xn,xn+1) + s2d(xn+1,xn+2) + . . .
+ sk−1d(xn+k−2,xn+k−1) + s

kd(xn+k−1,xn+k),

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 487



E. Karapınar

since s ≥ 1. Combining (2.9) and (2.10), we derive that
(2.11)

d(xn,xn+k) ≤ sψn(d(x0,x1)) + s2ψn+1d(x0,x1) + . . .
+ sk−1ψn+k−2(d(x0,x1)) + s

kψn+k−1(d(x0,x1))

=
1

sn−1
[snψn(d(x0,x1)) + s

n+1ψn+1d(x0,x1) + . . .

+ sn+k−2ψn+k−2(d(x0,x1)) + s
n+k−1ψn+k−1(d(x0,x1))].

Inevitably, we derive

(2.12) d(xn,xn+k) ≤
1

sn−1
[Pn+k−1 −Pn−1] , n ≥ 1,k ≥ 1,

where Pn =

n∑
j=0

sjψj(d(x0,x1)), n ≥ 1. From Lemma 1.8, the series
∞∑
j=0

sjψj(d(x0,x1))

is convergent and since s ≥ 1, upon taking limit n →∞ in (2.39), we observe

(2.13) lim
n→∞

d(xn,xn+k) ≤ lim
n→∞

1

sn−1
[Pn+k−1 −Pn−1] = 0.

We deduce that the sequence {xn} is Cauchy in (X,d).
Taking into account the T-orbitally completeness, we note that there is

z ∈ X such that xn → z. Owing to the orbital continuity of T, we conclude
that xn → Tz. Consequently, we find z = Tz which terminates the proof. �

Example 2.3. Let the set X = {a,b,c,g,e} and d : X × X → [0,∞) be a
b-metric (with s = 2) defined as follows

d(x,y) a b c g e
a 0 1 9 25 16
b 1 0 4 16 9
c 9 4 0 4 1
g 25 16 4 0 1
e 16 9 1 1 0

Let also the mapping T : X → X be given as

x a b c g e
Tx b b g e e

Thus, we have

d(Tx,Ty) Ta Tb Tc Tg Te
Ta = b 0 0 16 9 9
Tb = b 0 0 16 9 9
Tc = g 16 16 0 1 1
Tg = e 9 9 1 0 0
Te = e 9 9 1 0 0

and

d(x,Ty) a b c g e
Ta = b 1 0 4 16 9
Tb = b 1 0 4 16 9
Tc = g 25 16 4 0 1
Tg = e 16 9 1 1 0
Te = e 16 9 1 1 0

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 488



Revisiting Ćirić type nonunique fixed point theorems via interpolation

We choose α = 1
2

and φ : [0,∞) → [0,∞) as φ(t) = t
2
. We shall denote,

m1(x,y) = min{
(
d1/2(Tx,Ty)d1/2(x,Tx)

)
,
(
d1/2(Tx,Ty)d1/2(y,Ty)

)
}

m2(x,y) = min{d1/2(x,Ty),d1/2(Tx,y)}.

Then we have to consider the following cases:

(1) For x = a,y = b and x = g,y = e, we have d(Tx,Ty) = 0 and
obviously, (2.3) holds.

(2) For x = a,y = c, we have

m1(a,c) = min
{
d1/2(Ta,Tc)d1/2(a,Ta)

)
,
(
d1/2(Ta,Tc)d1/2(c,Tc)

}
= min{4 · 1, 4 · 2} = 4

m2(a,c) = min{5, 2} .

Thus, m1(a,c) −m2(a,c) = 2 < 92 = φ(d(a,c).
(3) For x = a,y = g, we have

m1(a,g) = min
{
d1/2(Ta,Tg)d1/2(a,Ta)

)
,
(
d1/2(Ta,Tg)d1/2(g,Tg)

}
= min{3 · 1, 3 · 1} = 1

m2(a,g) = min{4, 4} ,

and obviously (2.3) holds.
(4) For x = a,y = e, we have

m1(a,e) = min
{
d1/2(Ta,Te)d1/2(a,Ta)

)
,
(
d1/2(Ta,Te)d1/2(e,Te

}
= min{3 · 1, 3 · 0} = 0

m2(a,e) = min{4, 3} = 3,

and (2.3)holds.
(5) For x ∈ {b,e} and y ∈ X, since d(x,Tx) = 0, we have m1(x,y) = 0

and then (2.3) holds.
(6) For x = c and y = g

m1(c,g) = min
{
d1/2(Tc,Tg)d1/2(c,Tc)

)
,
(
d1/2(Tc,Tg)d1/2(g,Tg

}
= min{1 · 2, 1 · 1} = 1

m2(c,g) = min{1, 0} = 0.

Therefore,

m1(c,g) −m2(c,g) = 1 < 2 = φ(d(c,g)).

(7) For x = c and y = e

m1(c,e) = min
{
d1/2(Tc,Te)d1/2(c,Tc)

)
,
(
d1/2(Tc,Te)d1/2(e,Te

}
= min{1 · 2, 1 · 0} = 0

m2(c,e) = min{1, 1} = 1.

Therefore,

m1(c,e) −m2(c,e) = −1 <
1

2
= φ(d(c,e)).

Then the conditions of Theorem 2.2 hold and clearly, T has two fixed points,
x = b and x = e.

In the next corollaries we give some consequences of the Theorem 2.2.

Corollary 2.4. For a nonempty set X, we suppose that the function d : X ×
X → [0,∞) is a b-metric. We presume that a self-mapping T on X is orbitally

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 489



E. Karapınar

continuous and (X,d,s) forms a T -orbitally complete b-metric space with s ≥ 1.
If there is q ∈ [0, 1

s
) and α ∈ (0, 1) such that

(2.14)
min{

(
dα(Tx,Ty)d1−α(x,Tx)

)
,
(
dα(Tx,Ty)d1−α(y,Ty)

)
}

−min{dα(x,Ty),d1−α(Tx,y)}≤ qd(x,y),
for all x,y ∈ X, then for each x0 ∈ X the sequence {Tnx0}n∈N converges to a
fixed point of T .

Proof. It is sufficient to take ψ(t) = qt, where q ∈ [0, 1
s
), in Theorem 2.2. �

Corollary 2.5. Let T be an orbitally continuous self-map on the T -orbitally
complete metric space (X,d). If there is a comparison function ψ and α ∈ (0, 1)
such that

(2.15)
min{

(
dα(Tx,Ty)d1−α(x,Tx)

)
,
(
dα(Tx,Ty)d1−α(y,Ty)

)
}

−min{dα(x,Ty),d1−α(Tx,y)}≤ ψ(d(x,y)),
for all x,y ∈ X, then for each x0 ∈ X the sequence {Tnx0}n∈N converges to a
fixed point of T .

Proof. It is sufficient to take s = 1 in Theorem 2.2. �

Corollary 2.6. Let T be an orbitally continuous self-map on the T -orbitally
complete metric space (X,d). If there is q ∈ [0, 1) and α ∈ (0, 1) such that

(2.16)
min{

(
dα(Tx,Ty)d1−α(x,Tx)

)
,
(
dα(Tx,Ty)d1−α(y,Ty)

)
}

−min{dα(x,Ty),d1−α(Tx,y)}≤ qd(x,y),
for all x,y ∈ X, then for each x0 ∈ X the sequence {Tnx0}n∈N converges to a
fixed point of T .

Proof. It is sufficient to take ψ(t) = qt, where q ∈ [0, 1), in Corollary 2.6. �

2.2. Pachpatte type non-unique fixed point results [37].

Theorem 2.7. For a nonempty set X, we suppose that the function d : X ×
X → [0,∞) is a b-metric. We presume that a self-mapping T is orbitally
continuous and (X,d,s) forms a T -orbitally complete b-metric space with s ≥ 1.
If there exists ψ ∈ Ψ and α ∈ (0, 1) such that
(2.17) m(x,y) −n(x,y) ≤ ψ(dα(x,Tx)d1−α(y,Ty)),
for all x,y ∈ X, where

m(x,y) = min{d(Tx,Ty),dα(x,y)d1−α(Tx,Ty),d(y,Ty)},
n(x,y) = min{dα(x,Tx)d1−α(y,Ty),dα(x,Ty)d1−α(y,Tx)},

then, for each x0 ∈ X the sequence {Tnx0}n∈N converges to a fixed point of T .

Proof. By verbatim, following the initial lines of the proof of the Theorem 2.2,
we shall set-up a recursive sequence {xn = Txn−1}n∈N, by starting from an
arbitrary initial value x0 := x ∈ X.

Replacing in the inequality (2.17) x = xn−1 and y = xn, we obtain that

(2.18) m(xn−1,xn) −n(xn−1,xn) ≤ ψ(dα(xn−1,Txn−1)d1−α(xn,Txn)),

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 490



Revisiting Ćirić type nonunique fixed point theorems via interpolation

where

m(xn−1,xn) = min{d(Txn−1,Txn),dα(xn−1,xn)d1−α(Txn−1,Txn),d(xn,Txn)},
n(xn−1,xn) = min{dα(xn−1,Txn−1)d1−α(xn,Txn),dα(xn−1,Txn)d1−α(xn,Txn−1)}.
By simplifying the above inequality, we get

(2.19) m(xn−1,xn) ≤ ψ(dα(xn−1,xn)d1−α(xn,xn+1)),
where

m(xn−1,xn) = min{d(xn,xn+1),dα(xn−1,xn)d1−α(xn,xn+1)}.
It is clear that the case

m(xn−1,xn) = d
α(xn−1,xn)d

1−α(xn,xn+1)

is not possible for any n ∈ N. If it would be the case, the inequality (2.19)
turns into

(2.20)
dα(xn−1,xn)d

1−α(xn,xn+1) ≤ ψ(dα(xn−1,xn)d1−α(xn,xn+1))
< dα(xn−1,xn)d

1−α(xn,xn+1),

which is a contradiction since ψ(t) < t for all t > 0. Consequently, we derive

(2.21)
d(xn,xn+1) ≤ ψ(dα(xn−1,xn)d1−α(xn,xn+1))

< dα(xn−1,xn)d
1−α(xn,xn+1),

which yields

(2.22) d(xn,xn+1) < d(xn−1,xn).

On account of the fact that the comparison function ψ is nondecreasing, to-
gether with the inequalities (2.21) and (2.22), we find that

(2.23) d(xn,xn+1) ≤ ψ(dα(xn−1,xn)d1−α(xn,xn+1)) < ψ(d(xn−1,xn)),
Recursively, we obtain that

d(xn,xn+1) ≤ ψ(d(xn−1,xn))
≤ ψ2(d(xn−2,xn−1)) ≤ ···≤ ψn(d(x0,x1)).

Hence, we conclude that

lim
n→∞

d(xn+1,xn) = 0.

The remaining part of the proof is verbatim repetition of the related lines
in the proof of Theorem 2.2, so we omit it.

�

Below, we deduce some come consequences of the Theorem 2.7 for particular
choice of the comparison function ans the constant s. In case of ψ(t) = qt in
Theorem 2.7 we deduce the following result.

Corollary 2.8. For a nonempty set X, we suppose that the function d : X ×
X → [0,∞) is a b-metric. We presume that a self-mapping T is orbitally
continuous and (X,d,s) forms a T -orbitally complete b-metric space with s ≥ 1.
Assume that there exists q ∈ [0, 1

s
) and α ∈ (0, 1), such that

(2.24) m(x,y) −n(x,y) ≤ qdα(x,Tx)d1−α(y,Ty),

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 491



E. Karapınar

for all x,y ∈ X, where m(x,y) and n(x,y) are defined as in Theorem 2.7.
Then, for each x0 ∈ X the sequence {Tnx0}n∈N converges to a fixed point of
T .

If the statements of Theorem 2.7 are considered in the context of standard
metric space instead of a b-metric space, we shall obtain the following conse-
quence.

Corollary 2.9. Let T be an orbitally continuous self-map on the T -orbitally
complete metric space (X,d). Suppose that there exist a comparison function
ψ and α ∈ (0, 1) such that

(2.25) m(x,y) −n(x,y) ≤ ψ(dα(x,Tx)d1−α(y,Ty)),

for all x,y ∈ X, where m(x,y) and n(x,y) are defined as in Theorem 2.7. Then
for each x0 ∈ X the sequence {Tnx0}n∈N converges to a fixed point of T .

For ψ(t) = qt in Corollary 2.9, the following results is derived.

Corollary 2.10. Let T be an orbitally continuous self-map on the T -orbitally
complete standard metric space (X,d). Suppose that there exists q ∈ [0, 1) and
α ∈ (0, 1) such that

(2.26) m(x,y) −n(x,y) ≤ qdα(x,Tx)d1−α(y,Ty),

for all x,y ∈ X, where m(x,y) and n(x,y) are defined as in Theorem 2.7.
Then, for each x0 ∈ X the sequence {Tnx0}n∈N converges to a fixed point of
T .

2.3. K-type non-unique fixed point results [23]. The following theorem
is inspired by the main theorem of [23]

Theorem 2.11. For a nonempty set X, we suppose that the function d :
X ×X → [0,∞) is a b-metric. We presume that a self-mapping T is orbitally
continuous and (X,d,s) forms a T -orbitally complete b-metric space with s ≥ 1.
Assume that there exists real numbers α,β,γ ∈ (0, 1) with α + β + γ < 1 and
ψ ∈ Ψ. If the following inequality
(2.27)

dα(Tx,Ty)dβ(x,Tx)dγ(y,Ty)

[
d(y,Tx) + d(x,Ty)

2s

]1−α−β−γ
≤ ψ(d(x,y))

holds for all x,y ∈ X, then, T has at least one fixed point.

Proof. Starting from an arbitrary point x = x0 ∈ X, we shall construct a
sequence {xn} as follows:

(2.28) xn+1 := Txn n = 0, 1, 2, ...

Letting x = xn and y = xn+1, in the inequality (2.27) yields
(2.29)

dα(Txn,Txn+1)d
β(xn,Txn)d

γ(xn+1,Txn+1)[
d(xn+1,Txn)+d(xn,Txn+1)

2s
]1−α−β−γ

≤ ψ(d(xn,xn+1)).

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 492



Revisiting Ćirić type nonunique fixed point theorems via interpolation

On account of (2.28), the statement (2.29) turns into
(2.30)

dα(xn+1,xn+2)d
β(xn,xn+1)d

γ(xn+1,xn+2)[
d(xn+1,xn+1)+d(xn,xn+2)

2s
]1−α−β−γ

≤ ψ(d(xn,xn+1)).

By elementary calculation and simplification, we derive
(2.31)

dα(xn+1,xn+2)d
β(xn,xn+1)d

γ(xn+1,xn+2)[
d(xn,xn+1)+d(xn+1,xn+2)

2
]1−α−β−γ

≤ ψ(d(xn,xn+1)).

Suppose that d(xn,xn+1) < d(xn+1,xn+2). Then the inequality (2.31) turns
into
(2.32)
dα(xn+1,xn+2)d

β(xn,xn+1)d
γ(xn+1,xn+2)[d(xn,xn+1)]

1−α−β−γ

≤ dα(xn+1,xn+2)dβ(xn,xn+1)dγ(xn+1,xn+2)[
d(xn,xn+1)+d(xn+1,xn+2)

2s
]1−α−β−γ

≤ ψ(d(xn,xn+1)).

Then we get

(2.33) d1−α−γ(xn,xn+1)d
1−β(xn+1,xn+2) ≤ ψ(d(xn,xn+1)) < d(xn,xn+1).

The above inequality can be expressed as

(2.34) d1−α−γ(xn,xn+1) < d
1−α−γ(xn,xn+1),

which is a contradiction. Hence, we conclude that

d(xn,xn+1) ≥ d(xn+1,xn+2).

So, the inequality above together with (2.32) yields that

(2.35) d(xn+1,xn+2) ≤ ψ(d(xn,xn+1)) < d(xn,xn+1)

Thus, the sequence {d(xn,xn+1)} is non-increasing.
Recursively, we find that

(2.36)
d(xn,xn+1) ≤ ψ(d(xn−1,xn)) ≤ ψ2(d(xn−2,xn−1)) ≤ ···≤ ψn(d(x0,x1)).

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 493



E. Karapınar

As a next step, we shall show that the sequence {xn} is Cauchy. By em-
ploying the triangle inequality (b3), we get
(2.37)
d(xn,xn+k) ≤ s[d(xn,xn+1) + d(xn+1,xn+k)]

≤ sd(xn,xn+1) + s{s[d(xn+1,xn+2) + d(xn+2,xn+k)]}

= sd(xn,xn+1) + s
2d(xn+1,xn+2) + s

2d(xn+2,xn+k)
...

≤ sd(xn,xn+1) + s2d(xn+1,xn+2) + . . .
+sk−1d(xn+k−2,xn+k−1) + s

k−1d(xn+k−1,xn+k)

≤ sd(xn,xn+1) + s2d(xn+1,xn+2) + . . .
+sk−1d(xn+k−2,xn+k−1) + s

kd(xn+k−1,xn+k),

since s ≥ 1. On account of (2.37) and (2.36), we deduce that
(2.38)
d(xn,xn+k) ≤ sψn(d(x0,x1)) + s2ψn+1d(x0,x1) + . . .

+ sk−1ψn+k−2(d(x0,x1)) + s
kψn+k−1(d(x0,x1))

=
1

sn−1
[snψn(d(x0,x1)) + s

n+1ψn+1d(x0,x1)

+ · · · + sn+k−2ψn+k−2(d(x0,x1)) + sn+k−1ψn+k−1(d(x0,x1))].

Consequently, we have

(2.39) d(xn,xn+k) ≤
1

sn−1
[Pn+k−1 −Pn−1] , n ≥ 1,k ≥ 1,

where Pn =

n∑
j=0

sjψj(d(x0,x1)), n ≥ 1. From Lemma 1.8, the series
∞∑
j=0

sjψj(d(x0,x1))

is convergent and since s ≥ 1, upon taking limit n →∞ in (2.39), we obtain

(2.40) lim
n→∞

d(xn,xn+k) ≤ lim
n→∞

1

sn−1
[Pn+k−1 −Pn−1] = 0.

We deduce that the sequence {xn} is Cauchy in (X,d).
The remaining part of the proof is verbatim repetition of the related lines

in the proof of Theorem 2.2.
�

Finally, we state the following consequence of Theorem 2.11.

Corollary 2.12. Let T be an orbitally continuous self-map on the T -orbitally
complete b-metric space (X,d,s) with s ≥ 1. Suppose there exist real numbers

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 494



Revisiting Ćirić type nonunique fixed point theorems via interpolation

q ∈ [0, 1
s
) and α,β,γ ∈ (0, 1) with α + β + γ < 1. If

(2.41)

dα(Tx,Ty)dβ(x,Tx)dγ(y,Ty)

[
d(y,Tx) + d(x,Ty)

2s

]1−α−β−γ
≤ qd(x,y)

holds for all x,y ∈ X, then, T has at least one fixed point.

Proof. Take ψ(t) = qt in the proof of Theorem 2.11, where q ∈ [0, 1). �

Notice also that the above theorem and corollary of this section are valid in
the setting of standard metric space.

Acknowledgements. The author thanks to his institutes.

References

[1] J. Achari, On Ćirić’s non-unique fixed points, Mat. Vesnik 13 (28), no. 3 (1976), 255–257.

[2] H. Afshari, H. Aydi and E. Karapinar, On generalized α-ψ-Geraghty contractions on

b-metric spaces, Georgian Math. J. 27 (2020), 9–21.
[3] H. Afshari, H. Aydi and E. Karapinar, Existence of fixed points of set-valued mappings

in b-metric spaces, East Asian Mathematical Journal 32, no. 3 (2016), 319–332.

[4] R. P. Agarwal and E. Karapinar, Interpolative Rus-Reich-Ciric type contractions via
simulation functions, An. St. Univ. Ovidius Constanta, Ser. Mat. 27, no. 3 (2019), 137–

152.

[5] U. Aksoy, E. Karapinar and I. M. Erhan, Fixed points of generalized alpha-admissible
contractions on b-metric spaces with an application to boundary value problems, Journal

of Nonlinear and Convex Analysis 17, no. 6 (2016), 1095–1108.

[6] H. Alsulami, S. Gulyaz, E. Karapinar and I. Erhan, An Ulam stability result on quasi-
b-metric-like spaces, Open Mathematics 14, no. 1 (2016), 1087–1103.

[7] M. A. Alghamdi, S. Gulyaz-Ozyurt and E. Karapinar, A note on extended Z-contraction,

Mathematics 8, no. 2 (2020), 195.
[8] H. Aydi, C.-M. Chen and E. Karapinar, Interpolative Ciric-Reich-Rus type contractions

via the Branciari distance, Mathematics 7, no. 1 (2019), 84.

[9] H. Aydi, E. Karapinar and A. F. Roldán López de Hierro, ω-Interpolative Ćirić-Reich-
Rus-type contractions, Mathematics 7 (2019), 57.

[10] H. Aydi, M. F. Bota, E. Karapinar and S. Moradi, A common fixed point for weak

phi-contractions on b-metric spaces, Fixed Point Theory 13, no. 2 (2012), 337–346.

[11] H. Aydi, E. Karapinar, M. F. Bota and S. Mitrovic, A fixed point theorem for set-valued
quasi-contractions in b-metric spaces, Fixed Point Theory Appl. 2012, 2012:88.

[12] V. Berinde, Contracţii Generalizate şi Aplicaţii , Vol. 2, Editura Cub Press, Baie Mare,

Romania, 1997.
[13] V. Berinde, Sequences of operators and fixed points in quasimetric spaces, Mathematica

41, no. 4 (1996), 23–27.
[14] V. Berinde, Generalized contractions in quasi-metric spaces, Seminar on Fixed Point

Theory, Babeş-Bolyai University, Research Sem., (1993), 3–9.

[15] C. Chifu, E. Karapinar and G. Petrusel, Fixed point results in ε-chainable complete
b-metric spaces, Fixed Point Theory 21, no. 2 (2020), 453–464.

[16] L. B. Ćirić, On some maps with a nonunique fixed point, Publ. Inst. Math. 17 (1974),
52–58.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 495



E. Karapınar

[17] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. et Inf. Uni. Ostravien-

sis 1 (1993), 5–11.
[18] A. Fulga, E. Karapinar and G. Petrusel, On hybrid contractions in the context of quasi-

metric spaces, Mathematics 8 (2020), 675.

[19] Y. U. Gaba and E. Karapinar, A new approach to the interpolative contractions, Axioms
2019, 8, 110.

[20] S. Gulyaz-Ozyurt, On some alpha-admissible contraction mappings on Branciari b-

metric spaces, Advances in the Theory of Nonlinear Analysis and its Applications 1
(2017), 1–13.

[21] S. Gupta and B. Ram, Non-unique fixed point theorems of Ćirić type, (Hindi) Vijnana
Parishad Anusandhan Patrika 41, no. 4 (1998), 217–231.

[22] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71–76.

[23] E. Karapinar, A new non-unique fixed point theorem, J. Appl. Funct. Anal. 7, no. 1-2
(2012), 92–97.

[24] E. Karapinar, Some nonunique fixed point theorems of Ćirić type on cone metric spaces,

Abstr. Appl. Anal. 2010 (2010), Article ID 123094.

[25] E. Karapinar, Ciric type nonunique fixed points results: a review, Applied and Compu-
tational Mathematics an International Journal 1 (2019), 3–21.

[26] E. Karapinar, O. Alqahtani and H. Aydi, On interpolative Hardy-Rogers type contrac-

tions, Symmetry 11, no. 1 (2019), 8.
[27] E. Karapinar, Revisiting the Kannan type contractions via interpolation, Adv. Theory

Nonlinear Anal. Appl. 2, no. 2 (2018), 85–87.

[28] E. Karapinar, H. Aydi and Z. D. Mitrovic, On interpolative Boyd-Wong and Matkowski
type contractions, TWMS J. Pure Appl. Math. 11, no. 2 (2020), 204–212.

[29] E. Karapinar, R. Agarwal and H. Aydi, Interpolative Reich-Rus-Ćirić type contractions

on partial metric spaces, Mathematics 6, no. 11 (2018), 256.

[30] E. Karapinar, A. Fulga and A. Petrusel, On Istratescu type contractions in b-metric
spaces, Mathematics 8, no. 3 (2020), 388.

[31] E. Karapinar, A short survey on the recent fixed point results on b-metric spaces, Con-

structive Mathematical Analysis 1, no. 1 (2018), 15–44.
[32] E. Karapinar and C. Chifu, Results in wt-distance over b-metric spaces, Mathematics 8,

no. 2 (2020), 220.

[33] E. Karapinar and A. Fulga, Fixed point on convex b-metric space via admissible map-
pings, TWMS JPAM 12, no. 2 (2021).

[34] E. Karapinar, Interpolative Kannan-Meir-Keeler type contraction, Adv. Theory Nonlin-

ear Anal. 5, no. 4 (2021), 611-614.

[35] Z. Liu, Z. Guo, S. M. Kang and S. K. Lee, On Ćirić type mappings with nonunique fixed
and periodic points, Int. J. Pure Appl. Math. 26, no. 3 (2006), 399–408.

[36] Z. Q. Liu, On Ćirić type mappings with a nonunique coincidence points, Mathematica
(Cluj) 35(58), no. 2 (1993), 221–225.

[37] B. G. Pachpatte, On Ćirić type maps with a nonunique fixed point, Indian J. Pure Appl.
Math. 10, no. 8 (1979), 1039–1043.

[38] I. A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-

Napoca, Romania, 2001.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 496