@
Appl. Gen. Topol. 23, no. 2 (2022), 391-404

doi:10.4995/agt.2022.16701

© AGT, UPV, 2022

Fixed point results of enriched interpolative

Kannan type operators with applications

Mujahid Abbas a, Rizwan Anjum b and Shakeela Riasat c

a Department of Mathematics, Government College University Katchery Road, Lahore 54000,

Pakistan and Department of Mathematics and Applied Mathematics, University of Pretoria Hat-

field 002, Pretoria, South Africa (mujahid.abbas@up.ac.za)
b Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Riphah

International University, Lahore, Pakistan. (rizwan.anjum@riphah.edu.pk)
c Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600,

Pakistan. (shakeelariasat@sms.edu.pk)

Communicated by E. Karapinar

Abstract

The purpose of this paper is to introduce the class of enriched in-
terpolative Kannan type operators on Banach space that contains the
classes of enriched Kannan operators, interpolative Kannan type con-
traction operators and some other classes of nonlinear operators. Some
examples are presented to support the concepts introduced herein. A
convergence theorem for the Krasnoselskii iteration method to approx-
imate fixed point of the enriched interpolative Kannan type operators
is proved. We study well-posedness, Ulam-Hyers stability and periodic
point property of operators introduced herein. As an application of the
main result, variational inequality problem is solved.

2020 MSC: 47H10; 47H09; 47J25; 49J40.

Keywords: fixed point; enriched Kannan operators; interpolative Kannan
type contraction; Krasnoselskii iteration; well-posedness; peri-
odic point; Ulam-Hyers stability; variational inequality problem.

Received 20 November 2021 – Accepted 14 March 2022

http://dx.doi.org/10.4995/agt.2022.16701


Mujahid Abbas, Rizwan Anjum and Shakeela Riasat

1. Introduction and Preliminaries

Let (X,d) be a metric space and T : X → X. We denote the set {x ∈ X :
Tx = x} of fixed points of T by Fix(T). Solving a fixed point problem of an
operator T, denote by FPP(T) is to show that the set Fix(T) is nonempty.
Define T0 = I (the identity map on X) and Tn = Tn−1 ◦ T, called the nth-
iterate of T for n ≥ 1. The most simplest iteration procedure to approximate
the solution of a fixed point equation Tx = x is the method of successive
approximations (or Picard iteration) given by

(1.1) xn = T
nx0, n = 1, 2, . . . ,

where x0 is an initial guess in domain of an operator T.
A sequence {xn}∞n=0 in X given by (1.1) is called a T-orbital sequence around
x0. The collection of all such sequences is denoted by O(T,x0). If there exists
x∗ in X such that Fix(T) = {x∗} and the Picard iteration associated with T
converges to x∗ for any initial guess x0 in X, then T is called a Picard operator
(see [1, 7, 40]).
If there exists k ∈ [0, 1) such that for any x,y ∈ X, we have

(1.2) d(Tx,Ty) ≤ k d(x,y).

Then T is called a Banach contraction operator which, if defined on a complete
metric space, is a classical example of a continuous Picard operator. Thus,
it was natural to ask the question whether in the framework of an complete
metric space, a discontinuous operator satisfying somewhat similar contractive
conditions is a Picard operator. This was answered in an affirmative by Kannan
[26] in 1968.
An operator T on X is called Kannan contraction operator if there exists
a ∈ [0, 0.5) such that for any x,y ∈ X, we have

(1.3) d(Tx,Ty) ≤ a{d(x,Tx) + d(y,Ty)}.

Kannan contraction operator defined on a complete metric space is an example
of a discontinuous Picard operator ([25], [26]).

Subrahmanyam [41] proved that a metric space X is complete if and only if
every Kannan contraction operator on X has a fixed point. Moreover, Connell
[18] gave an example of an incomplete metric space X on which every Banach
contraction operator has a fixed point. This shows that the Banach contrac-
tion operators do not characterize the completeness of their domain. As long as
contractive conditions are concerned, the classes of Banach and Kannan con-
traction operators are incomparable ([27]), but the class of Kannan contraction
operators attracted the attention of several mathematicians because of its con-
nection with a characterization of its metric completeness.
Kannan’s theorem has been generalized in different ways by many authors to
extend the limits of metric fixed point theory in different directions.
Karapinar introduced a new class of Kannan type operators called interpolative

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 392



Enriched interpolative Kannan type operators

Kannan type operators and proved a fixed point result for such operators in
the setup of complete metric spaces ([28]).
An operator T : X → X is called an interpolative Kannan type if there exists
a ∈ [0, 1) such that for all x,y ∈ X \Fix(T), we have

(1.4) d(Tx,Ty) ≤ a[d(x,Tx)]α[d(y,Ty)]1−α,
where α ∈ (0, 1).
The main result in [28] is stated as follows.

Theorem 1.1 ([28]). Let (X,d) be a complete metric space and T an inter-
polative Kannan type operator. Then T is a Picard operator.

For more results in this direction, we refer to [6, 8, 9, 19, 20, 21, 22, 29, 30,
31, 32, 33, 34, 35, 36, 38]) and references mentioned therein.
Existence, uniqueness, stability, approximation and characterization of fixed
points of certain operators are some of the main concerns of a metric fixed
point theory. Contractive conditions on operators and distance structure of
operator’s domain play a vital role to ensure the convergence of iterative meth-
ods.
If for any x,y ∈ X, we have

d(Tx,Ty) ≤ d(x,y).

Then T is called a nonexpansive operator. An operator T on X is called
asymptotically regular on X if for all x in X,

d(Tn+1x,Tnx) → 0 as n →∞.
The concept of asymptotic regularity plays an important role in approximating
the fixed points of operators. Picard iteration method fails to converge to a
fixed point of certain contractive mappings such as nonexpansive mappings
on metric spaces. This led to the study of a variety of fixed point iteration
procedures in the setup of Banach spaces.
In this paper, we shall approximate the fixed point of some nonlinear mappings
through Krasnoselskii iteration method.
Let λ ∈ [0, 1]. A sequence {xn}∞n=0 given by
(1.5) xn+1 = (1 −λ)xn + λTxn, n = 0, 1, 2, . . .
is called the Krasnoselskii iteration.
Note that, Krasnoselskii iteration {xn}∞n=0 sequence given by (1.5) is exactly
the Picard iteration corresponding to an averaged operator

(1.6) Tλ = (1 −λ)I + λT.
Moreover, for λ = 1 the Krasnoselskii iteration method reduces to Picard iter-
ation method. Also, Fix(T) = Fix(Tλ), for all λ ∈ (0, 1].
On the other hand, Browder and Petryshyn [16] introduced the concept of
asymptotic regularity in connection with the study of fixed points of nonex-
pansive mappings. As a matter of fact, the same property was used in 1955 by

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 393



Mujahid Abbas, Rizwan Anjum and Shakeela Riasat

Krasnoselskii [37] to prove that if K is a compact convex subset of a uniformly
convex Banach space and T : K → K is a nonexpansive mapping, then for any
x0 ∈ K, the sequence

(1.7) xn+1 =
1

2
(xn + Txn), n ≥ 0,

converges to fixed point of T.
In proving the above result, Krasnoselskii used the fact that if T is nonexpan-
sive which, in general, is not asymptotically regular, then the averaged mapping
T1

2
is asymptotically regular.

Therefore, an averaged operator Tλ enriches the class of nonexpansive map-
pings with respect to the asymptotic regularity. This observation suggested
that one could enrich the classes of contractive mappings studied in the frame-
work of metric spaces by imposing certain contractive condition on Tλ instead
of T itself.
Employing this approach, the classes such as enriched contractions and en-
riched φ-contractions [11], enriched Kannan contractions [12], enriched Chat-
terjea mappings [14], enriched nonexpansive mappings in Hilbert spaces [13],

enriched multivalued contractions [3] and enriched Ćirić-Reich-Rus contraction
[15], enriched cyclic contraction [5], enriched modified Kannan pair [2], enriched
quasi contraction [4] were introduced and studied.
Abbas et al. [3] proved fixed point results by imposing the condition that Tλ-
orbital subset is a complete subset of a normed space (see, Theorem 3 of [3]).

Similarly, Górnicki and Bisht[24] considered the enriched Ćirić-Reich-Rus con-
traction operators and proved a fixed point theorem by imposing the condition
that Tλ is asymptotically regular mapping (see, Theorem 3.1 of [24]).

Consistent with [12], let (X,‖·‖) be a normed space. A operator T : X → X
is called an enriched Kannan contraction or (b,a)-enriched Kannan contraction
if there exist b ∈ [0,∞) and a ∈ [0, 0.5) such that for all x,y ∈ X, the following
holds:

(1.8) ‖b(x−y) + Tx−Ty‖≤ a(‖x−Tx‖ + ‖y −Ty‖).
As shown in [12], several well-known contractive conditions in the existing lit-
erature imply the (b,a)-enriched Kannan contraction condition. It was proved
in [12] that any enriched Kannan contraction operator defined on a Banach
space has a unique fixed point which can be approximated by means of the
Krasnoselskii iterative scheme.

Motivated by the work of Berinde and Păcurar [12], Absas et al. [3] and
Górnicki and Bisht [24], we propose a new class of enriched interpolative Kan-
nan type operators. The purpose of this paper is to prove the existence of fixed
point of such operators. Moreover, we study the well-posedness, Ulam-Hyers
stability and periodic point property of the operators introduced herein. Fi-
nally, an application of our result to solve variational inequality problems is
also given.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 394



Enriched interpolative Kannan type operators

2. Fixed point approximation of enriched interpolative Kannan
type operators

In the sequel, the notations N and R will denote the set of all natural numbers
and the set of all real numbers, respectively.
In this section, we present a new class of enriched interpolative Kannan type
operators, which is first of its kind in the existing literature on metric fixed-
point theory. Existence and convergence results of such class of operators are
also obtained.
First, we introduce the following concept

Definition 2.1. Let (X,‖·‖) be a normed space. A mapping T : X → X is
called enriched interpolative Kannan type operator if there exist b ∈ [0,∞), a ∈
[0, 1) and α ∈ (0, 1) such that for all x,y ∈ X, we have

(2.1) ‖b(x−y) + (Tx−Ty)‖≤ a
(
‖x−Tx‖

)α(‖y −Ty‖)1−α.
To highlight an involvement of parameters a,b and α in (2.1), we call T a
(a,b,α)-enriched interpolative Kannan type operator.

Example 2.2. Any interpolative Kannan type operator T satisfying (1.4) is a
(a, 0,α)-enriched interpolative Kannan contraction operator, that is, T satisfies
(2.1) with b = 0.

We now give an example of an enriched interpolative Kannan type operator
which is not a interpolative Kannan type contraction operator.

Example 2.3. Let (Y,µ) be a finite measure space. The classical Lebesgue
space X = L2(Y,µ) is defined as the collection of all Borel measurable functions
f : Y → R such that

∫
Y
|f(y)|2dµ(y) < ∞. We know that the space X equipped

with the norm ‖f‖X =
(∫

Y
|f|2dµ

)1
2

is a Banach space.

Define the operator T : L2(Y,µ) → L2(Y,µ) by

Tf = g − 3f,

where g(y) = 1, ∀ y ∈ Y. Clearly, g ∈ L2(Y,µ) as µ(Y ) < ∞.
Note that T is an (0.5, 3, 0.5)-enriched interpolative Kannan type operator but
not an interpolative Kannan type contraction operator.
Indeed, if T would be a interpolative Kannan type operator then, by (1.4),
there would exist a ∈ [0, 1) and α ∈ (0, 1) such that

‖−3f + 3h‖X ≤ a
(
‖4f −g‖X

)α(‖4h−g‖X )1−α ∀ f,h ∈ X,
which on taking f(y) = 0 and h(y) = 1, for all y ∈ Y gives 3α ≤ a < 1, a
contradiction.

We now present the following result.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 395



Mujahid Abbas, Rizwan Anjum and Shakeela Riasat

Theorem 2.4. Let (X,‖·‖) be a normed space, T : X → X a (a,b,α)-enriched
interpolative Kannan type mapping.
Then,

(1) Fix(T) = {x∗};
(2) There exist a Tλ-orbital sequence {xn}∞n=0 around x0, given by

(2.2) xn+1 = (1 −λ)xn + λTxn; n ≥ 0,

converges to x∗ provided that, for x0 ∈ X, Tλ-orbital subset O(Tλ,x0)
is a complete subset of X, where λ = 1

b+1
.

Proof. We divide the proof into the following two cases.
Case 1. If b > 0. Then λ = 1

b+1
∈ (0, 1) and the enriched interpolative Kannan

type operator (2.1) satisfies the following contraction condition:∥∥∥∥
(

1

λ
− 1
)

(x−y) + Tx−Ty
∥∥∥∥ ≤ a‖x−Tx‖α‖y −Ty‖1−α

and hence

‖(1 −λ)(x−y) + λ(Tx−Ty)‖≤ aλ‖x−Tx‖α‖y −Ty‖1−α

which can be written in an equivalent form as follows:

(2.3) ‖Tλx−Tλy‖≤ a‖x−Tλx‖
α‖y −Tλy‖

1−α
, ∀ x,y ∈ X.

In view of (1.6), the Krasnoselskii iterative sequence defined by (2.2) is exactly
the Picard’s iteration associated with Tλ, that is,

(2.4) xn+1 = Tλxn, n ≥ 0.

Take x := xn and y := xn−1 in (2.3) to get

‖xn+1 −xn‖ = ‖Tλxn −Tλxn−1‖

≤ a‖xn −Tλxn‖
α‖xn−1 −Tλxn−1‖

1−α

≤ a‖xn −xn+1‖
α‖xn−1 −xn‖

1−α

which implies that

(2.5) ‖xn+1 −xn‖
1−α ≤ a‖xn−1 −xn‖

1−α
.

As α ∈ (0, 1),

(2.6) ‖xn+1 −xn‖≤ a‖xn−1 −xn‖ .

Inductively, we obtain that

(2.7) ‖xn+1 −xn‖≤ an‖x0 −x1‖ .

By (2.7) and triangular inequality, we have

(2.8) ‖xn −xn+r‖≤
an

1 −a
‖x0 −x1‖ , r ∈ N, n ≥ 1,

which, in view of 0 < a < 1 gives that {xn}∞n=0 is a Cauchy sequence in the
complete subset O(Tλ,x0) of X.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 396



Enriched interpolative Kannan type operators

Next, we assume that there exists an element x∗ in O(Tλ,x0) such that limn→∞xn =
x∗. Note that

‖x∗ −Tλx∗‖≤‖x∗ −xn+1‖ + ‖xn+1 −Tλx∗‖
≤‖x∗ −xn+1‖ + ‖Tλxn −Tλx∗‖

≤‖x∗ −xn+1‖ + a‖xn −Tλxn‖
α‖x∗ −Tλx∗‖

1−α

≤‖x∗ −xn+1‖ + a‖xn −xn+1‖
α‖x∗ −Tλx∗‖

1−α
.

On taking the limit as n →∞ on both sides of the above inequality, we get
that x∗ = Tλx

∗.

Assume that x∗ and y∗ are two fixed points of T. Then from (2.3), we have

‖x∗ −y∗‖ = ‖Tλx∗ −Tλy∗‖≤ a‖x∗ −Tλx∗‖
α‖y∗ −Tλy∗‖

1−α

≤ a‖x∗ −x∗‖α‖y∗ −y∗‖1−α ,

which, gives x∗ = y∗.
Case 2. b = 0. In this case, the enriched interpolative Kannan type operator
(2.1) becomes

(2.9) ‖Tx−Ty‖≤ a‖x−Tx‖α‖y −Ty‖1−α ∀ x,y ∈ X,

where a ∈ (0, 1). That is, T is an interpolative Kannan type contraction oper-
ator and hence by Theorem 2.2 of [28], T has a unique fixed point. �

The following example illustrate the above theorem.

Example 2.5. Let X = R \
{

1
5
, 4
5

}
be endowed with the usual norm and

T : X → X be defined by Tx = 1 −x, ∀ x ∈ X. Clearly T is a (1, 1)-enriched
interpolative Kannan type operator. Now λ = 1

1+b
gives that λ = 1

2
. Let

x0 = 1/2 be the fixed in X. Then, we have

T1
2
(x0) =

1

2
x0 +

1

2
Tx0 =

1

2

(
1

2

)
+

1

2

(
1

2

)
=

1

2
.

Pick x1 = T1
2
x0 = 1/2. Continuing this way, we obtained xn+1 = T1

2
xn, where

xn+1 = (
1
2
, 1
2
, 1
2
, . . . ). Note that {xn}∞n=0 is a Cauchy sequence which converges

to 1/2 and 1/2 is the fixed point of T.

We now present the following fixed point theorem for (a,b,α)-enriched in-
terpolative Kannan type operator on a Banach space.

Corollary 2.6. Let (X,‖·‖) be a Banach space and T : X → X a (a,b,α)-
enriched interpolative Kannan type operator. Then T has a unique fixed point.

Proof. Following arguments similar to those in proof Theorem 2.4, the result
follows. �

If we take b = 0 in the Corollary 2.6, we obtain Theorem 2.2 of [28] in the
setting of Banach spaces.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 397



Mujahid Abbas, Rizwan Anjum and Shakeela Riasat

Corollary 2.7 ([28]). Let T be an interpolative Kannan type operator on a
Banach space (X,‖·‖). Then T is a Picard operator.

By Corollary 2.6, we obtain the following result.

Corollary 2.8 ([12]). Let (X,‖·‖) be a Banach space and T : X → X an
(b,a)-enriched Kannan contraction, that is, for all x,y ∈ X, it satisfies the
following inequality;

(2.10) ‖b(x−y) + Tx−Ty‖≤ a
{
‖x−Tx‖ + ‖y −Ty‖

}
with b ∈ [0,∞) and a ∈ [0, 0.5). Then T has a unique fixed point.

Proof. Take λ = 1
b+1

. Obviously, 0 < λ < 1 and the (b,a)-enriched Kannan

contraction condition (2.10) becomes∥∥∥∥
(

1

λ
− 1
)

(x−y) + Tx−Ty
∥∥∥∥

≤ a
{
‖x−Tx‖ + ‖y −Ty‖

}
, ∀ x,y ∈ X,

which can be written in an equivalent form as follows;

(2.11) ‖Tλx−Tλy‖≤ a
{
‖x−Tλx‖ + ‖y −Tλy‖

}
, ∀ x,y ∈ X.

By (2.11), Tλ is a Kannan contraction. It follows from [28] that Tλ satisfies
condition (2.11) and condition (2.3). Since, for λ = 1

b+1
, the inequality (2.3) is

same as the condition (2.1). This suggests that T is an enriched interpolative
Kannan type operator and then the Corollary 2.6 leads to the conclusion. �

3. Well-Posedness, Perodic Point and Ulam-Hyers Stability
Results

We now present the well-Posedness, perodic point and Ulam-Hyers stability
results for (a,b,α)-enriched interpolative Kannan type operators.

3.1. Well-Posedness.
Let us start with the following definition.

Definition 3.1 ([39]). Let (X,d) be a metric space and T : X → X. The fixed
point problem FPP(T) is said to be well-posed if T has unique fixed point
x∗(say) and for any sequence {xn} in X satisfying limn→∞d(Txn,xn) = 0, we
have limn→∞xn = x

∗.

Since Fix(T) = Fix(Tλ), we conclude that the fixed point problem of T is
well-posed if and only if the fixed point problem of Tλ is well-posed.
Well-posedness of certain fixed point problems has been studied by several
mathematicians, see for example, [10], [39] and references mentioned therein.
We now study the well-posedness of a fixed point problem of mappings in
Theorem 2.4 and Corollary 2.6.

Theorem 3.2. Let (X,‖·‖) be a Banach space. Suppose that T is an operator
on X as in the Theorem 2.4. Then, FPP(T) is well-posed.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 398



Enriched interpolative Kannan type operators

Proof. It follows from Theorem 2.4 that x∗ is the unique fixed point of T.
Suppose that limn→∞‖Tλxn −xn‖ = 0. Using (2.3) we have,

‖xn −x∗‖≤‖xn −Tλxn‖ + ‖Tλxn −x∗‖
= ‖xn −Tλxn‖ + ‖Tλxn −Tλx∗‖

≤‖xn −Tλxn‖ + a‖xn −Tλxn‖
α‖x∗ −Tλx∗‖

1−α

that is,

(3.1) ‖xn −x∗‖≤‖xn −Tλxn‖ .
It follows from (3.1) that limn→∞xn = x

∗ provided that limn→∞‖Tλxn −xn‖ =
0. This complete the proof. �

Corollary 3.3. Let (X,‖·‖) be Banach space. Suppose that T is an operator
on X as in the Corollary 2.6. Then the fixed point problem is well-posed.

Proof. Following arguments similar to those in the proof Theorem 3.2, the
result follows. �

3.2. Perodic Point Result. Clearly, a fixed point x∗ of T is also a fixed
point of Tn for every n ∈ N. However, the converse is false. For example, if
we take, X = [0, 1] and define an operator T on X by Tx = 1 − x. Then T
has a unique fixed point 1/2, and for each even integer n, nth-iterate of T is
an identity map and hence every point of [0, 1] is a fixed point of Tn. Also, if
X = [0,π], Tx = cos x, then every iterate of T has the same fixed point as T.
If a map T satisfies Fix(T) = Fix(Tn) for each n ∈ N, then it is said to have
a periodic point property P [23].
Since Fix(T) = Fix(Tλ), we conclude that the mapping T has property P if
and only the mapping Tλ has property P.

Theorem 3.4. Let (X,‖.‖) be a Banach space. Suppose that T is an operator
on X as in the Theorem 2.4. Then T has property P.

Proof. From Theorem 2.4, T has a fixed point. Let y∗ ∈ Fix(Tn). Now from
(2.3), we have

‖y∗ −Tλy∗‖ =
∥∥Tnλ y∗ −Tλ(Tnλ y∗)∥∥

=
∥∥Tλ(Tn−1λ y∗)−Tλ(Tnλ y∗)∥∥

≤ a
∥∥Tn−1λ y∗ −Tnλ y∗∥∥α ∥∥Tnλ y∗ −Tn+1λ y∗∥∥1−α ,

that is,

(3.2)
∥∥Tnλ y∗ −Tn+1λ y∗∥∥α ≤ a∥∥Tn−1λ y∗ −Tnλ y∗∥∥α .

Since α ∈ (0, 1), then (3.2) becomes

‖y∗ −Tλy∗‖ =
∥∥Tnλ y∗ −Tn+1λ y∗∥∥ ≤ a∥∥Tn−1λ y∗ −Tnλ y∗∥∥ ≤

≤ a2
∥∥Tn−2λ y∗ −Tn−1λ y∗∥∥ ≤ . . . ≤ an‖y∗ −Tλy∗‖ .

Now, 0 ≤ a < 1 implies that ‖y∗ −Tλy∗‖ = 0 and hence y∗ = Ty∗. �

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 399



Mujahid Abbas, Rizwan Anjum and Shakeela Riasat

3.3. Ulam-Hyers Stability. Let (X,d) be a metric space, T : X → X and
� > 0 . A point w∗ ∈ X called an �-solution of the fixed point problem FPP(T)
if w∗ satisfies the following inequality

d(w∗,Tw∗) ≤ �.

Let us recall the notion of Ulam-Hyers stability.

Definition 3.5 ([42]). Let (X,d) be a metric space, T : X → X and � > 0.
The fixed point problem FPP(T) is called generalized Ulam-Hyers stable if and
only if there exists an increasing and continuos function φ : [0,∞) → [0,∞)
with φ(0) = 0 such that for each �-solution w∗ ∈ X of the fixed point equation
Tx = x , there exists a solution x∗ of Tx = x in X such that

d(x∗,w∗) ≤ φ(�).

Remark 3.6. If the function φ in the above definition is given by φ(t) = mt
for all t ≥ 0, where m > 0, then the fixed point equation Tx = x is said to be
Ulam-Hyers stable.

Theorem 3.7. Let (X,‖·‖) be a Banach space. Suppose that T is an operator
on X as in the Theorem 2.4. Then the fixed point problem is Ulam-Hyers
stable.

Proof. Since Fix(T) = Fix(Tλ), it follows that the fixed point problem Tx = x
is equivalent to the fixed point problem

(3.3) x = Tλx.

Let w∗ be �-solution of the fixed point equation (3.3), that is,

(3.4) d(w∗,Tλw
∗) ≤ �.

Using (2.3) and (3.4), we get

‖x∗ −w∗‖ = ‖Tλx∗ −w∗‖≤‖Tλx∗ −Tλw∗‖ + ‖Tλw∗ −w∗‖

≤ a‖x∗ −Tλx∗‖
α‖w∗ −Tλw∗‖

1−α
+ �

= �.

�

4. Application to Variational inequality problem

Variational inequality theory provides some important tools to handle the
problems arising in economic, engineering, mechanics, mathematical program-
ming, transportation and others. Many numerical methods have been con-
structed for solving variational inequalities and optimization problems. The
aim of this section is to present generic convergence theorems for Krasnoselskii
type algorithms that solve variational inequality problems.

Let H be a real Hilbert space with inner product
〈
·, ·
〉
, C ⊂ H a closed and

convex set and S : H → H.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 400



Enriched interpolative Kannan type operators

The variational inequality problem with respect to S and C, denoted by V IP(S,C),
is to find x∗ ∈ C such that〈

Sx∗,x−x∗
〉
≥ 0, ∀ x ∈ H.

It is well known [17] that if γ > 0, then x∗ ∈ C is a solution of V IP(S,C) if
and only if x∗ is a solution of fixed point problem of PC ◦ (I −γS), where PC
is the nearest point projection onto C.
Amongst many others results in [17], it was proved that if I − γS and PC ◦
(I − γS) are averaged nonexpansive operators, then, under some additional
assumptions, the iterative algorithm {xn}∞n=0 defined by

xn+1 = PC(I −γS)xn, n ≥ 0,

converges weakly to a solution of the V IP(S,C), if such solutions exist.
In the case of averaged nonexpansive mappings, the problem of replacing the
weak convergence in the above result with strong convergence has received a
much attention of researchers.
Our alternative is to consider V IP(S,C) for enriched interpolative Kannan
type contraction operators, which are in general discontinuous operators in
contrast of nonexpansive operators which are always continuous.
In this case, we shall have V IP(S,C) with a unique solution, as shown in the
next theorem. Moreover, the considered algorithm (4.1) will converge strongly
to the solution of the V IP(S,C).

Theorem 4.1. Assume that for γ > 0, PC(I − γS) is enriched interpolative
Kannan type contraction operator. Then there exists λ ∈ (0, 1] such that the
iterative algorithm {xn}∞n=0 defined by

(4.1) xn+1 = (1 −λ)xn + λPC(I −γG)xn, n ≥ 0,

converges strongly to the unique solution x∗ of the V IP(S,C) for any x0 ∈ C.

Proof. Since C is closed, we take X := C and T := PC(I − γS) and apply
Corollary 2.6. �

Example 4.2. Let X = R2 and for any x = (x1,x2) and y = (y1,y2) in X,
the inner product is defined by〈

x,y
〉

= x1y1 + x2y2.

Then X equipped with the above inner product is a Hilbert space. The above
inner product gives the norm given by

‖x‖ = (< x,y >)1/2.

Define S : X → X by

S(x) =
(1, 0) + x

γ
∀ x ∈ X,

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 401



Mujahid Abbas, Rizwan Anjum and Shakeela Riasat

where γ > 0 be fixed real number.
For a mapping PC : X → C defined by

PC(x) =

{
x
‖x‖ ; x /∈ C
x ; x ∈ C,

where C = {x ∈ X : ‖x‖≤ 1}, it is easy to check that PC(I −γS) is a (a, 0,α)
enriched interpolative Kannan type contraction.
By Corollary 2.6, PC(I −γS) has a unique solution, which in turns a solution
for V IP(S,C).

5. Conclusions

(1) We introduced a large class of contractive operators, called enriched
interpolative Kannan type operators, that includes usual interpolative
Kannan type operators and enriched Kannan operators.

(2) We presented examples to show that the class of enriched interpolative
Kannan type operators strictly includes the interpolative Kannan type
operator in the sense that there exist operators which are not inter-
polative Kannan type and belong to the class of enriched interpolative
Kannan type operators.

(3) We studied the set of fixed point (Theorem 2.4) and constructed an
algorithm of Krasnoselskii type in order to approximate fixed point of
enriched interpolative Kannan type operators and we proved a strong
convergence theorem.

(4) We also obtained Theorems 3.2, 3.4 and 3.7 for a well-posedness, per-
odic point and Ulam-Hyers stability problem of the fixed point problem
for enriched interpolative Kannan type operators, respectively.

(5) As application of our main results (Corollary 2.6), we presented two
Krasnoselskii projection type algorithms for solving variational inequal-
ity problems for the class of enriched interpolative Kannan type oper-
ators, thus improving the existence and weak convergence results for
variational inequality problems in [17] to existence and uniqueness as
well as to strong convergence theorems.

Acknowledgements. Authors are grateful to the editor in chief and the
reviewers for their useful comments and constructive remarks that helped to
improve the presentation of the paper.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 402



Enriched interpolative Kannan type operators

References

[1] R. Anjum and M. Abbas, Fixed point property of a nonempty set relative to the class of
friendly mappings, Rev. R. Acad. Cienc. Exactas F́ıs. Nat. Ser. A Mat. RACSAM 116,

no. 1 (2022), Paper no. 32.

[2] R. Anjum and M. Abbas, Common fixed point theorem for modified Kannan enriched
contraction pair in Banach spaces and its applications, Filomat 35, no. 8 (2021) 2485–

2495.

[3] M. Abbas, R. Anjum and V. Berinde, Enriched multivalued contractions with applica-
tions to differential inclusions and dynamic programming, Symmetry 13, no. 8 (2021),

Paper no. 1350.
[4] M. Abbas, R. Anjum and V. Berinde, Equivalence of certain iteration processes obtained

by two new classes of operators, Mathematics 9, no. 18 (2021), Paper no. 2292.

[5] M. Abbas, R. Anjum and H. Iqbal, Generalized enriched cyclic contractions with appli-
cation to generalized iterated function system, Chaos, Solitons and Fractals 154 (2022),

Paper no. 111591.

[6] R. P. Agarwal and E. Karapinar, Interpolative Rus-Reich-Ćirić type contractions via

simulation functions, An. St. Univ. Ovidius Constanta, Ser. Mat. 27, no. 3 (2019), 137–

152.
[7] R. P. Agarwal, E. Karapinar, D. O’Regan and A. F. Roldán López de Hierro, Fixed

point theory in metric type spaces, Springer International Publishing, 2015.

[8] H. Aydi, E. Karapinar and A. F. Roldán López de Hierro, w-Interpolative Ćirić-Reich-
Rus-type contractions, Mathematics 7, no. 57 (2019), Paper no. 57.

[9] H. Aydi, C. M. Chen and E. Karapinar, Interpolative Ćirić-Reich-Rustypes via the

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

[10] F. S. D. Blasi and J. Myjak, Sur la porosité de l’ensemble des contractions sans point
fixe, C. R. Acad. Sci. Paris. 308 (1989), 51–54.

[11] V. Berinde and M. Păcurar, Approximating fixed points of enriched contractions in

Banach spaces, Journal of Fixed Point Theory and Applications 22, no. 2 (2020), 1–10.
[12] V. Berinde and M. Păcurar, Kannan’s fixed point approximation for solving split feasi-

bility and variational inequality problems, Journal of Computational and Applied Math-

ematics 386 (2021), Paper no. 113217.
[13] V. Berinde, Approximating fixed points of enriched nonexpansive mappings by Kras-

noselskii iteration in Hilbert spaces, Carpathian J. Math. 35, no. 3 (2019), 293–304.

[14] V. Berinde and M. Păcurar, Approximating fixed points of enriched Chatterjea con-
tractions by Krasnoselskii iterative algorithm in Banach spaces, Journal of Fixed Point

Theory and Applications 23, no. 4 (2021), 1–16.

[15] V. Berinde and M. Păcurar, Fixed point theorems for enriched Ćirić-Reich-Rus con-

tractions in Banach spaces and convex metric spaces, Carpathian J. Math. 37 (2021),
173–184.

[16] F. E. Browder and W. V. Petryshyn, The solution by iteration of nonlinear functional

equations in Banach spaces, Bull. Am. Math. Soc. 72 (1966), 571–575.
[17] C. Byrne, A unified treatment of some iterative algorithms in signal processing and

image reconstruction, Inverse Probl. 20, no. 1 (2004), 103–120.

[18] E. Connell, Properties of fixed point spaces, Proc. Amer. Math. Soc. 10 (1959), 974–979.
[19] I. C. Chifu and E. Karapinar, Admissible hybrid Z-contractions in b-metric spaces,

Axioms. 9, no. 1, (2020), Paper no. 2.

[20] P. Debnath and M. de La Sen, Set-valued interpolative Hardy-Rogers and set-valued

Reich-Rus-Ćirić-type contractions in b-metric spaces, Mathematics. 7, no. 9 (2019),
Paper no. 849.

[21] Y. U. Gaba and E. Karapinar, A new approach to the interpolative contractions, Axioms

8, no. 4 (2019), Paper no. 110.
[22] Y. U. Gaba, H. Aydi and N. Mlaik, (ρ,η,µ)-interpolative Kannan contractions I, Axioms.

10, no. 3 (2021), Paper no. 212.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 403



Mujahid Abbas, Rizwan Anjum and Shakeela Riasat

[23] G. S. Jeong and B. E. Rhoades, Maps for which F(T) = F(Tn), Fixed Point Thoery

Appl. 6 (2005), 87–131.

[24] J. Górnicki and R. K. Bisht, Around averaged mappings, Journal of Fixed Point Theory
and Applications 23 (2021), Paper no. 48.

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

[26] R. Kannan, Some results o fixed points, II, Amer. Math. Monthly 76 (1969), 405–408.
[27] M. Kikkawa and T. Suzuki, Some similarity between contractions and Kannan mappings

II, Bull. Kyushu Inst. Technol. Pure Appl. Math. 55 (2008) 1–13.

[28] E. Karapinar, Revisiting the Kannan type contractions via interpolation, Adv. Theory
Nonlinear Anal. Appl. 2, no. 2 (2018), 85–87.

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

tions, Symmetry. 11, no. 1 (2019), Paper no. 8.

[30] E. Karapinar, R. Agarwal and H. Aydi, Interpolative Reich-Rus-Ćirić type contractions
on partial metric spaces, Mathematics 6, no. 11 (2018), Paper no. 256.

[31] E. Karapinar and A. Fulga, New hybrid contractions on b-metric spaces, Mathematics

7, no. 7 (2019), Paper no. 578.
[32] 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.

[33] E. Karapinar, A. Fulga and A. F. Roldán López de Hierro, Fixed point theory in the
setting of (α,β,ψ,φ)-interpolative contractions, Advances in Difference Equations 1,

(2021), 1–16.

[34] E. Karapinar, Interpolative Kannan-Meir-Keeler type contractions, Advances in the
Theory of Nonlinear Analysis and its Applications 5, no. 4 (2021), 611–614.

[35] M. S. Khan, Y. M. Singh and E. Karapinar, On the interpolative (φ,ψ)-type Z-

contractions, U. P. B. Sci. Bull. Series A. 83, no. 2 (2021), 25–38.
[36] E. Karapinar, A. Fulga and S. S. Yesilkaya, New Results on Perov-interpolative contrac-

tions of Suzuki type mappings, Journal of Functional Spaces 2021, Art. ID 9587604.
[37] M. A. Krasnoselskii, Two remarks on the method of successive approximations, Uspekhi

Mat. Nauk. 10 (1955), 123–127.

[38] M. Noorwali, Common fixed point for Kannan type via interpolation, J. Math. Anal. 9,
no. 6 (2018), 92–94.

[39] S. Reich and A. J. Zaslavski, Well-posedness of fixed point problems, Far East Journal

of Mathematical Sciences (FJMS) 2001, Special volume, Part III, 393–401.
[40] I. A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-

Napoca, 2001.

[41] P. V. Subrahmanyam, Remarks on some fixed point theorems related to Banach’s con-
traction principle, J. Math. Phys. Sci. 8 (1974), 445–457.

[42] W. Sintunavarat, Generalized Ulam-Hyres stability, well-posedness and limit shadowing

of fixed point problems for α-β-contraction mapping in metric spaces, The Sci. World
J. 2014, Article ID 569174.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 404