@ Appl. Gen. Topol. 22, no. 1 (2021), 109-120doi:10.4995/agt.2021.14045
© AGT, UPV, 2021

From interpolative contractive mappings to

generalized Ćirić-quasi contraction mappings

Kushal Roy and Sayantan Panja

Department of Mathematics, The University of Burdwan, Purba Bardhaman-713104, West Ben-

gal, India. (kushal.roy93@gmail.com, spanja1729@gmail.com)

Communicated by E. Karapinar

Abstract

In this article we consider a restricted version of Ćirić-quasi contrac-

tion mapping for showing that this mapping generalizes several known

interpolative type contractive mappings. Also here we introduce the

concept of interpolative strictly contractive type mapping T and prove

a fixed point theorem for such mapping over a T -orbitally compact

metric space. Some examples are given in support of our established

results. Finally we give an observation regarding (λ, α, β)-interpolative
Kannan contractions introduced by Gaba et al.

2010 MSC: 47H10; 54H25.

Keywords: fixed point; restricted Ćirić-quasi contraction mapping; inter-

polative strictly contractive type mapping; T -orbitally compact

metric space.

1. Introduction and preliminaries

In the year 1922, S. Banach had established a remarkable fixed point theo-
rem, known as ’Banach Contraction Principle’ which is given as follows:

Theorem 1.1 ([2]). If a mapping T from a complete metric space (X, d) to
itself satisfies the following condition

(1.1) d(T x, T y) ≤ αd(x, y) for all x, y ∈ X,

for some α ∈ [0, 1) then T possesses a unique fixed point in X.

Received 18 July 2020 – Accepted 30 January 2021

http://dx.doi.org/10.4995/agt.2021.14045


K. Roy and S. Panja

Several generalizations of this theorem have been made by researchers, work-
ing in the area of fixed point theory, by means of different new type contractive
mappings.

Recently E. Karapinar [7] proposed a new Kannan-type contractive mapping
via the notion of interpolation and proved a fixed point theorem over metric
space. In his paper, Karapinar assumed that the interpolative Kannan-type
contractive mapping T over a metric space X satisfies the contractive condition
for all x, y ∈ X with x 6= T x. But in this situation it is to be noted that if this
mapping T has a fixed in X then it will be a constant mapping and therefore T
has a unique fixed point trivially. To remove such triviality the authors in [8]
assumed that interpolative type mappings satisfy the contractive condition for
all x, y ∈ X \ Fix(T ), where Fix(T ) is the set of all fixed points of T. Though
in this case an interpolative contractive type mapping may possesses more than
one fixed point.

Definition 1.2 ([7]). In a metric space (X, d), a mapping T : X → X is said
to be interpolative Kannan-type contractive mapping if it satisfies

(1.2) d(T x, T y) ≤ λ[d(x, T x)]α[d(y, T y)]1−α for all x, y ∈ X \ Fix(T ),

for some λ ∈ [0, 1) and for some α ∈ (0, 1).

Theorem 1.3. [7] Let (X, d) be a complete metric space and T : X → X be an
interpolative Kannan-type contractive mapping. Then T has at least one fixed
point in X.

As an extension of interpolative Kannan-type contractive mappings, Karap-
inar et al. introduced interpolative Reich-Rus-Ćirić type contractions (See [8]).
The definition is given below.

Definition 1.4 ([8]). In a metric space (X, d), a mapping T : X → X is called

interpolative Reich-Rus-Ćirić type contraction mapping if it satisfies
(1.3)

d(T x, T y) ≤ λ[d(x, y)]β[d(x, T x)]α[d(y, T y)]1−α−β for all x, y ∈ X \ Fix(T ),

for some λ ∈ [0, 1) and for α, β ∈ (0, 1).

Theorem 1.5 ([8]). Let (X, d) be a complete metric space and T : X → X be

an interpolative Reich-Rus-Ćirić type contraction mapping. Then T has a fixed
point in X.

Further extension of interpolative Kannan-type contractive mappings has
been given by Karapinar et al. [9], which is known as interpolative Hardy-
Rogers type contraction. The definition is given as follows.

Definition 1.6 ([9]). In a metric space (X, d), a mapping T : X → X is said
to be interpolative Hardy-Rogers type contraction mapping if it satisfies
(1.4)

d(T x, T y) ≤ λ[d(x, y)]β[d(x, T x)]α[d(y, T y)]γ
[

1

2
(d(x, T y) + d(y, T x))

]1−α−β−γ

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 110



From interpolative contractive mappings...

for all x, y ∈ X \ Fix(T ), for some λ ∈ [0, 1) and for α, β, γ ∈ (0, 1) with
α + β + γ < 1.

Theorem 1.7 ([9]). Let (X, d) be a complete metric space and T : X → X be
an interpolative Hardy-Rogers type contraction mapping. Then T has at least
one fixed point in X.

Recently C. B. Ampadu [1] has defined interpolative Berinde weak operator
in his paper. The definition is given as follows:

Definition 1.8 ([1]). Let (X,d) be a metric space. We say T : X → X is
(i) an interpolative Berinde weak operator if it satisfies

(1.5) d(T x, T y) ≤ λ[d(x, y)]α[d(x, T x)]1−α for all x, y ∈ X \ Fix(T ),

for some λ ∈ [0, 1) and for some α ∈ (0, 1).
(ii) an alternate interpolative Berinde Weak operator if it satisfies

(1.6) d(T x, T y) ≤ λ
√

d(x, y)d(x, T x) for all x, y ∈ X \ Fix(T ),

where λ ∈ [0, 1).
Any interpolative Berinde weak operator is an alternate interpolative Berinde
Weak operator.

Theorem 1.9 ([1]). In a complete metric space (X, d) an interpolative Berinde
weak operator T always possesses a fixed point.

As a generalization of ’Banach Contraction Principle’, Ćirić [3] had intro-

duced a new contractive mapping known as Ćirić-quasi contraction mapping
and proved a fixed point theorem for such mappings.

Theorem 1.10 ([3]). Let (X, d) be a complete metric space and T : X → X
be a self mapping. If T satisfies the contractive condition

d(T x, T y) ≤ k max{d(x, y), d(x, T x), d(y, T y),
d(x, T y) + d(y, T x)

2
} for all x, y ∈ X,

then T has a unique fixed point in X.

In the next section we find some new forms of interpolative contractive
mappings and show that these interpolative contractive mappings are nothing

but Ćirić-quasi contraction mappings.

2. Main results

Let (X, d) be a metric space, ∆IK be the set of all interpolative Kannan type

contractions on X and ∆SK = {T : X → X : d(T x, T y) ≤ λ
√

d(x, T x)d(y, T y)
for all x, y ∈ X \ Fix(T ), where λ ∈ [0, 1)}.

Theorem 2.1. In a metric space (X, d), ∆IK = ∆SK.

Proof. Clearly ∆SK ⊂ ∆IK. Now let T ∈ ∆IK be chosen as arbitrary. Then
there exists λ ∈ [0, 1) and α ∈ (0, 1) such that

d(T x, T y) ≤ λd(x, T x)αd(y, T y)1−α for all x, y ∈ X \ Fix(T ).

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 111



K. Roy and S. Panja

Now for any x, y ∈ X \ Fix(T ) we have

d(T x, T y) ≤ λd(x, T x)αd(y, T y)1−α(2.1)

and also due to symmetry

d(T x, T y) = d(T y, T x) ≤ λd(y, T y)αd(x, T x)1−α.(2.2)

Multiplying the inequalities (2.1) and (2.2) it follows that

d(T x, T y) ≤ λ
√

d(x, T x)d(y, T y),

which proves that T ∈ ∆SK and hence ∆IK = ∆SK. �

In a metric space (X, d), let ∆IR be the set of all interpolative Reich-

Rus-Ćirić type contractions on X and ∆SR = {T : X → X : d(T x, T y) ≤

λd(x, y)α{d(x, T x)d(y, T y)}
1−α
2 for all x, y ∈ X \ Fix(T ), where λ ∈ [0, 1), α ∈

(0, 1)}.

Theorem 2.2. In a metric space (X, d), ∆IR = ∆SR.

Proof. It is clearly seen that ∆SR ⊂ ∆IR. Now let T ∈ ∆IR be chosen arbi-
trarily. Then there exists λ ∈ [0, 1) and α, β ∈ (0, 1) such that

d(T x, T y) ≤ λd(x, y)αd(x, T x)βd(y, T y)1−α−β for all x, y ∈ X \ Fix(T ).

Now for any x, y ∈ X \ Fix(T ) we have

d(T x, T y) ≤ λd(x, y)αd(x, T x)βd(y, T y)1−α−β(2.3)

and also due to symmetry we get

d(T x, T y) = d(T y, T x) ≤ λd(y, x)αd(y, T y)βd(x, T x)1−α−β.(2.4)

Multiplying the inequalities (2.3) and (2.4) it follows that

d(T x, T y) ≤ λd(x, y)α{d(x, T x)d(y, T y)}
1−α
2 ,

which proves that T ∈ ∆SR and hence ∆IR = ∆SR. �

Remark 2.3. From the Theorem 2.2 we observe that, β has no importance to

define interpolative Reich-Rus-Ćirić type contraction mappings.

Let us take ∆IH as the set of all interpolative Hardy-Rogers type contrac-
tions and ∆SH = {T : X → X : d(T x, T y) ≤ λd(x, y)α{d(x, T x)d(y, T y)}ξ

(

1
2
[d(x, T y) + d(y, T x)]

)

1−α−2ξ

for all x, y ∈ X \ Fix(T ), where λ ∈ [0, 1), α, ξ ∈ (0, 1) such that α + 2ξ < 1}.

Theorem 2.4. In a metric space (X, d), ∆IH = ∆SH.

Proof. ∆SH ⊂ ∆IH trivially. Now let T ∈ ∆IH be taken as arbitrary. Then
there exists λ ∈ [0, 1) and α, β, γ ∈ (0, 1) with α + β + γ < 1 such that

d(T x, T y) ≤ λd(x, y)αd(x, T x)βd(y, T y)γ
(

1

2
[d(x, T y) + d(y, T x)]

)1−α−β−γ

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 112



From interpolative contractive mappings...

for all x, y ∈ X \ Fix(T ). Now for any x, y ∈ X \ Fix(T ) we have

d(T x, T y) ≤ λd(x, y)αd(x, T x)βd(y, T y)γ
(

1

2
[d(x, T y) + d(y, T x)]

)1−α−β−γ

(2.5)

and also due to the symmetry of d we get

d(T x, T y) = d(T y, T x) ≤ λd(y, x)αd(y, T y)βd(x, T x)γ
(

1

2
[d(y, T x) + d(x, T y)]

)1−α−β−γ

.

(2.6)

Multiplying the inequalities (2.5) and (2.6) it follows that

d(T x, T y) ≤ λd(x, y)α{d(x, T x)d(y, T y)}
β+γ
2

(

1

2
[d(y, T x) + d(x, T y)]

)1−α−β−γ

,

which proves that T ∈ ∆SH and hence ∆IH = ∆SH. �

Remark 2.5. From Theorem 2.1, 2.2 and 2.4 it is clear that in each of the
Definitions, T can be expressed by fewer constants used as powers in the R.H.S.

Now we consider a version of Ćirić-quasi contraction mapping and show that
interpolative contractive mappings are special cases of such type of mappings.

Definition 2.6. Let (X, d) be a metric space. A non-identity mapping T :

X → X is said to be restricted Ćirić-quasi contraction mapping if there exists
λ ∈ [0, 1) such that

d(T x, T y) ≤ λM(x, y) for all x, y ∈ X \ Fix(T ),(2.7)

where M(x, y) = max{d(x, y), d(x, T x), d(y, T y), 1
2
[d(y, T x) + d(x, T y)]}.

Theorem 2.7. In a complete metric space (X, d), a restricted Ćirić-quasi con-
traction mapping possesses at least one fixed point in X.

Proof. The proof is straight forward so we omit the proof. �

Clearly any Ćirić-quasi contraction mapping is also a restricted Ćirić-quasi
contraction mapping but the converse is not true in general. The following
examples proves our assertion.

Example 2.8. (i) Let X = [0, 1] be the metric space endowed with the usual
metric and T : X → X be defined by

T (x) =











0 if x = 0
1−x
2

if 0 < x < 1

1 if x = 1.

Then it can be easily checked that T is a restricted Ćirić-quasi contraction
mapping but it is not a Ćirić-quasi contraction mapping, because T has three
fixed points 0, 1

3
and 1.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 113



K. Roy and S. Panja

(ii) Let X = [1, 2] together with the usual metric and T : X → X be defined
by

T (x) =

{

x+1
2

if 1 ≤ x < 2

2 if x = 2.

Then it can be easily checked that T is a restricted Ćirić-quasi contraction
mapping but it is not a Ćirić-quasi contraction mapping, since T has two fixed
points 1 and 2.

(iii) Let X = [−1, 1] be the metric space endowed with the usual metric and
T : X → X be defined by

T (x) =











1
2

if x = −1

x if − 1 < x < 1

−1
2

if x = 1.

Then it can be easily checked that T is a restricted Ćirić-quasi contraction map-
ping but it is not a Ćirić-quasi contraction mapping, because T has infinitely
many fixed points in X.

Let ∆IW and ∆IC be the collections of all alternate interpolative Berinde
weak mappings and restricted Ćirić-quasi contraction mappings respectively.
Now we prove the following theorem.

Theorem 2.9. In a metric space (X, d) if T ∈ ∆IK ∪ ∆IR ∪ ∆IH ∪ ∆IW then
T ∈ ∆IC.

Proof. Let T ∈ ∆IK. Then there exists λ ∈ [0, 1) such that d(T x, T y) ≤

λ
√

d(x, T x)d(y, T y) for all x, y ∈ X \ Fix(T ). Thus for any x, y ∈ X \ Fix(T )
we have

d(T x, T y) ≤ λ
√

d(x, T x)d(y, T y)

≤ λ
√

M(x, y)2 = λM(x, y).(2.8)

If T ∈ ∆IR then there exist λ ∈ [0, 1) and α ∈ (0, 1) such that d(T x, T y) ≤

λd(x, y)α{d(x, T x)d(y, T y)}
1−α
2 for all x, y ∈ X \ Fix(T ). Thus for any x, y ∈

X \ Fix(T ) we have

d(T x, T y) ≤ λd(x, y)α{d(x, T x)d(y, T y)}
1−α
2

≤ λM(x, y)α{M(x, y)2}
1−α
2 = λM(x, y).(2.9)

Choose T ∈ ∆IH. Then there exist λ ∈ [0, 1) and α, ξ ∈ (0, 1) with α+2ξ < 1

such that d(T x, T y) ≤ λd(x, y)α{d(x, T x)d(y, T y)}ξ
(

1
2
[d(x, T y) + d(y, T x)]

)1−α−2ξ

for all x, y ∈ X \ Fix(T ). Thus for any x, y ∈ X \ Fix(T ) we get

d(T x, T y) ≤ λd(x, y)α{d(x, T x)d(y, T y)}ξ
(

1

2
[d(x, T y) + d(y, T x)]

)1−α−2ξ

≤ λM(x, y)α{M(x, y)2}ξ (M(x, y))
1−α−2ξ

= λM(x, y).(2.10)

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 114



From interpolative contractive mappings...

Consider T ∈ ∆IW . Then there exists λ ∈ [0, 1) such that d(T x, T y) ≤

λ
√

d(x, y)d(x, T x) for all x, y ∈ X \ Fix(T ). Thus for any x, y ∈ X \ Fix(T )
we have

d(T x, T y) ≤ λ
√

d(x, y)d(x, T x)

≤ λ
√

M(x, y)2 = λM(x, y).(2.11)

Hence from (2.8), (2.9), (2.10) and (2.11) we have in any case T ∈ ∆IC. �

Theorem 2.2 [7], Corollary 1 [8], Theorem 4 [9] and Theorem 1.2 [1] follow
from our next corollary.

Corollary 2.10. In a complete metric space (X, d) if T ∈ ∆IK ∪∆IR ∪∆IH ∪
∆IW then T has a fixed point in X.

Proof. From Theorem 2.9 we see that if T ∈ ∆IK ∪ ∆IR ∪ ∆IH ∪ ∆IW then
T ∈ ∆IC. Also Theorem 2.7 says that a mapping T ∈ ∆IC always possesses
fixed point in X. Hence the corollary. �

Any mapping T ∈ ∆IC may not be a member of ∆IK ∪ ∆IR ∪ ∆IH ∪ ∆IW .
The next example supports our contention.

Example 2.11. Let us consider X = [0, 1] equipped with the usual metric.
Also let T : X → X be defined by

T (x) =











0 if x = 0
x
2

if 0 < x < 1

1 if x = 1.

Then clearly T is a restricted Ćirić-quasi contraction mapping for 1
2
≤ λ < 1

but not an usual Ćirić-quasi contraction mapping. Also by taking x = ǫ and
y = 1 − δ with 0 < ǫ, δ < 1 and letting ǫ, δ → 0 we see that T /∈ ∆IK ∪ ∆IR ∪
∆IH ∪ ∆IW .

In metric fixed point theory, our main objective is to check whether a map-
ping T over a complete metric space X into itself possesses a fixed point in X.
In order to satisfy the interpolative Kannan type contractive condition (1.2)
for a mapping T : X → X, we have to know the set Fix(T ) and whenever
we know the whole set Fix(T ) why we bother about, whether the mapping T
satisfies the contractive condition (1.2) ?

In one word, to check the existence of fixed points for an interpolative con-
tractive mapping T in X, we have to know the set Fix(T ) in advance, which
is quite absurd.

Moreover, Theorem 2.7 shows that, if any one of the contractive condition
(like Banach, Kannan, Chatterjea) holds ”for all x, y ∈ X \ Fix(T )” instead
of ”for all x, y ∈ X” then we can easily remove the part uniqueness from the
like Theorems (Banach, Kannan, Chaterjea), but in each case we have to know
first the set Fix(T ).

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 115



K. Roy and S. Panja

From this point of view we can conclude that the Theorem 1.3 has no real
significance.

To avoid such a situation we can redefine the contractive condition (1.2) in
the way that is given below.

Remark 2.12. In a metric space (X, d) if we define an interpolative mapping
T : X → X satisfying

(2.12) d(T x, T y) ≤ λ
√

max{d(x, T x), d(x, y)}. max{d(y, T y), d(x, y)}

for all x, y ∈ X and for some λ ∈ [0, 1), then it is seen that T can be a non-
constant function even if T has a fixed point in X.
Clearly the contractive condition (2.12) can also be taken as

d(T x, T y) ≤ λ[max{d(x, T x), d(x, y)}]α[max{d(y, T y), d(x, y)}]1−α

for all x, y ∈ X, for α ∈ (0, 1) and for some λ ∈ [0, 1).

Remark 2.13. It is to be noted that if Ba(X), Mi(X) and Ci(X) are the
set of all Banach contractions, interpolative contractive mappings satisfying
condition (2.12) and Ćirić quasi contractions on X respectively then Ba(X) ⊂
Mi(X) ⊂ Ci(X). Therefore it is clear that in a complete metric space (X, d)
an interpolative contractive mapping T satisfying condition (2.12) has a unique
fixed point.

3. Interpolative strictly contractive mappings over a compact
metric space

In this section we prove some fixed point theorems for interpolative strictly
contractive type mappings in the framework of a metric space which is weaker
than compact metric space. First we recall the definitions of T -orbitally com-
pact metric space with respect to a self mapping T and orbital continuity of a
self mapping over a metric space.

Definition 3.1 ([4]). A metric space (X, d) is said to be T -orbitally compact
with respect to a mapping T : X → X if for all x ∈ X, every sequence in
the orbit of T at x ∈ X given by O(x, T ) = {x, T x, T 2x, ...} has a convergent
subsequence in X.

Definition 3.2 ([5]). Let (X, d) be a metric space. A mapping T : (X, d) →
(X, d) is said to be orbitally continuous if u ∈ X and such that u = limi→∞ T

nix
for some x ∈ X, then T u = limi→∞ T T

nix.

Theorem 3.3. Let (X, d) be a metric space and T : X → X be a mapping
which satisfies

d(T x, T y) < Θ(x, y) for all x, y /∈ Fix(T ),(3.1)

where Θ(x, y) = max{
√

d(x, T x)d(y, T y), d(x, y)µ{d(x, T x)d(y, T y)}
1−µ
2 ,

d(x, y)ν{d(x, T x)d(y, T y)}τ
(

1
2
[d(x, T y) + d(y, T x)]

)1−ν−2τ
, d(x, y)ξ

[

(d(x,T x)+1)d(y,T y)
1+d(x,y)

]1−ξ

}

with µ, ν, τ, ξ ∈ (0, 1) and ν +2τ < 1. If X is compact (or, T -orbitally compact)

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 116



From interpolative contractive mappings...

then T has atleast one fixed point in X, provided that T is orbitally continuous
in X.

Proof. Let x0 ∈ X be chosen as arbitrary. Let us construct an iterative se-
quence {xn}, where xn = T

nx0 for all n ≥ 1. If xn = xn+1 for some n ∈ N∪{0}
then xn will be a fixed point of T. So without loss of generality we assume that
xn 6= xn+1 for all n ≥ 0. Now from the contractive condition (3.1) we have

d(xn, xn+1) = d(T xn−1, T xn) < Θ(xn−1, xn) for all n ≥ 1.(3.2)

Now we have to consider four cases.
Case-I: If Θ(xn−1, xn) =

√

d(xn−1, xn)d(xn, xn+1) then we get

d(xn, xn+1) <
√

d(xn−1, xn)d(xn, xn+1)

⇒ d(xn, xn+1) < d(xn−1, xn).(3.3)

Case-II: If Θ(xn−1, xn) = d(xn−1, xn)
µ{d(xn−1, xn)d(xn, xn+1)}

1−µ
2 then

we have

d(xn, xn+1) < d(xn−1, xn)
µ{d(xn−1, xn)d(xn, xn+1)}

1−µ
2

⇒ d(xn, xn+1)
1+µ
2 < d(xn−1, xn)

1+µ
2

⇒ d(xn, xn+1) < d(xn−1, xn).(3.4)

Case-III: If Θ(xn−1, xn) = d(xn−1, xn)
ν{d(xn−1, xn)d(xn, xn+1)}

τ ×
(

1
2
[d(xn−1, xn+1) + d(xn, xn)]

)1−ν−2τ
then we obtain that

d(xn, xn+1) < d(xn−1, xn)
ν{d(xn−1, xn)d(xn, xn+1)}

τ

(

1

2
[d(xn−1, xn+1) + d(xn, xn)]

)1−ν−2τ

≤ d(xn−1, xn)
ν{d(xn−1, xn)d(xn, xn+1)}

τ

(

1

2
[d(xn−1, xn) + d(xn, xn+1)]

)1−ν−2τ

.

(3.5)

If d(xn−1, xn) ≤ d(xn, xn+1) then from (3.5) it follows that

d(xn, xn+1) < d(xn−1, xn)
ν{d(xn−1, xn)d(xn, xn+1)}

τ

(

1

2
[d(xn−1, xn) + d(xn, xn+1)]

)1−ν−2τ

≤ d(xn, xn+1), a contradiction.

(3.6)

Which implies that d(xn, xn+1) < d(xn−1, xn).

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 117



K. Roy and S. Panja

Case-IV: If Θ(xn−1, xn) = d(xn−1, xn)
ξ
[

(d(xn−1,xn)+1)d(xn,xn+1)
1+d(xn−1,xn)

]1−ξ

then

we have

d(xn, xn+1) < d(xn−1, xn)
ξ

[

(d(xn−1, xn) + 1)d(xn, xn+1)

1 + d(xn−1, xn)

]1−ξ

= d(xn−1, xn)
ξd(xn, xn+1)

1−ξ

⇒ d(xn, xn+1)
ξ < d(xn−1, xn)

ξ

⇒ d(xn, xn+1) < d(xn−1, xn).(3.7)

Thus from equations (3.3), (3.4), (3.5) and (3.7) we see that d(xn, xn+1) <
d(xn−1, xn) for all n ∈ N. So {d(xn−1, xn)} is a monotonically decreasing
sequence which is bounded below. Therefore there exists some l ≥ 0 such that
d(xn−1, xn) → l as n → ∞.

Now since X is compact (or, T -orbitally compact), {xn} has a convergent
subsequence {xnk}, which converges to some u ∈ X. Due to the orbital con-
tinuity of T it follows that {xnk+1} converges to T u and {xnk+2} converges
to T 2u respectively. Therefore the continuity of the metric d implies that
limk→∞ d(xnk , xnk+1) = d(u, T u) and limk→∞ d(xnk+1, xnk+2) = d(T u, T

2u).
So d(u, T u) = l = d(T u, T 2u). If l > 0 then u, T u /∈ Fix(T ) and therefore

d(T u, T 2u) < Θ(u, T u) implies that d(T u, T 2u) < d(u, T u), a contradiction.

(3.8)

Hence l = 0 and T u = u that is u is a fixed point of T. �

From the above theorem we get the following immediate corollaries.

Corollary 3.4. Let (X, d) be a metric space and T : X → X be a mapping
which satisfies

d(T x, T y) < d(x, T x)γd(y, T y)1−γ for all x, y /∈ Fix(T ), γ ∈ (0, 1).(3.9)

If X is compact (or, T -orbitally compact) then T has a fixed point in X, pro-
vided that T is orbitally continuous in X.

Example 3.5. Let X = [0, ∞) with the usual metric, M = {n +
(

n + 1
n

)2
:

n ≥ 2} and T : X → X be defined by

T (x) =

{

n if x = n +
(

n + 1
n

)2
, n ≥ 2

x if x ∈ X \ M.

Then T satisfies the contractive condition (3.1) in particular the contractive
condition (3.9). Also X is T -orbitally compact and T is orbitally continuous
on X. Here we see that T has infinitely many fixed points in X.

Corollary 3.6. Let (X, d) be a metric space and T : X → X be a mapping
which satisfies

d(T x, T y) < d(x, y)γd(x, T x)δd(y, T y)1−γ−δ for all x, y /∈ Fix(T ), γ, δ ∈ (0, 1).

(3.10)

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 118



From interpolative contractive mappings...

If X is compact (or, T -orbitally compact) then T has atleast one fixed point in
X, provided that T is orbitally continuous in X.

Corollary 3.7. Let (X, d) be a metric space and T : X → X be a mapping
which satisfies

d(T x, T y) < d(x, y)γd(x, T x)δd(y, T y)ζ
(

1

2
[d(x, T y) + d(y, T x)]

)1−γ−δ−ζ

(3.11)

for all x, y /∈ Fix(T ), where γ, δ, ζ ∈ (0, 1) with γ + δ + ζ < 1. If X is compact
(or, T -orbitally compact) then T has a fixed point in X, provided that T is
orbitally continuous in X.

Corollary 3.8. Let (X, d) be a metric space and T : X → X be a mapping
which satisfies

d(T x, T y) < d(x, y)ξ
[

(d(x, T x) + 1)d(y, T y)

1 + d(x, y)

]1−ξ

for all x, y /∈ Fix(T ), ξ ∈ (0, 1).

(3.12)

If X is compact (or, T -orbitally compact) then T has atleast one fixed point in
X, provided that T is orbitally continuous in X.

4. A remark on interpolative Kannan contractivity conditions

In [6] the authors have defined (λ, α, β)-interpolative Kannan contraction
and prove a fixed point theorem for such mappings. The definition of the
mapping is given as follows:

Definition 4.1 ([6]). Let (X, d) a metric space and T : X → X be a self map.
T is called a (λ, α, β)-interpolative Kannan contraction, if there exist λ ∈ [0, 1)
and α, β ∈ (0, 1) with α + β < 1 such that

d(T x, T y) ≤ λd(x, T x)αd(y, T y)β for all x, y ∈ X \ Fix(T ).(4.1)

Theorem 4.2 ([6]). Let (X, d) a complete metric space and T : X → X be
a (λ, α, β)-interpolative Kannan contraction with λ ∈ [0, 1) and α, β ∈ (0, 1),
α + β < 1. Then T has a fixed point in X.

Theorem 4.2 is not true in general. The next example proves our assertion.

Example 4.3. Let X = {1
3
, 1
2
} with usual metric and T : X → X be given by

T (x) =

{

1
2

if x = 1
3

1
3

if x = 1
2
.

Then T is a (λ, α, β)-interpolative Kannan contraction with λ = 3
5
and α =

β = 1
3
. Here X is complete but T has no fixed point in X.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 119



K. Roy and S. Panja

Comment/s: (The reason/s why the proof of Theorem 2 in [6] fails)
In the proof of Theorem 2 (See the line number 5 of Theorem 2 in Page 2

of [6]) the authors used the fact that

d(xn, xn+1)
1−β ≤ λd(xn−1, xn)

α ≤ λd(xn−1, xn)
1−β whenever α < 1 − β,

(4.2)

which is actually not true in case 0 < d(xn−1, xn) < 1.

Therefore the contractive condition

d(T x, T y) ≤ λd(x, T x)αd(y, T y)1−α

can not be replaced by the contractive condition (4.1).

E. Karapinar have pointed out a similar idea in Example 2 of [10], where he
forewarned about the mappings T : {x0, y0} → {x0, y0} defined by T x0 = y0
and T y0 = x0. These particular type of mappings defined on two point sets
satisfy the contractive condition (3) (See [10]) but are fixed-points free.

Acknowledgements. First and second authors acknowledge financial sup-

port awarded by the Council of Scientific and Industrial Research, New Delhi,

India, through research fellowship for carrying out research work leading to the

preparation of this manuscript.

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