@
Appl. Gen. Topol. 23, no. 1 (2022), 213-223

doi:10.4995/agt.2022.11902

© AGT, UPV, 2022

Numerical reckoning fixed points via

new faster iteration process

Kifayat Ullah a,∗ , Junaid Ahmad b and Fida Muhammad Khan c

a Department of Mathematical Sciences, University of Lakki Marwat, Lakki Marwat - 28420,

Pakistan (kifayatmath@yahoo.com)
b Department of Mathematics, International Islamic University, Islamabad - 44000, Pakistan

(ahmadjunaid436@gmail.com)
c Department of Mathematics, University of Science and Technology, Bannu - 28100, Pak-

istan (fidamuhammad809@gmail.com)

Communicated by M. Abbas

Abstract

In this paper, we propose a new iteration process which is faster than the
leading; S [J. Nonlinear Convex Anal. 8, no. 1 (2007), 61–79], Thakur
et al. [App. Math. Comp. 275 (2016), 147–155] and M [Filomat 32,
no. 1 (2018), 187–196] iterations for numerical reckoning fixed points.
Using this new iteration process, some fixed point convergence results
for generalized α-nonexpansive mappings in the setting of uniformly
convex Banach spaces are proved. At the end of paper, we offer a
numerical example to compare the rate of convergence of the proposed
iteration process with the leading iteration processes.

2020 MSC: 47H09; 47H10.

Keywords: generalized α-nonexpansive mappings; uniformly convex Banach
space; iteration process; weak convergence; strong convergence.

1. Introduction

Throughout this paper, we will denote the set of natural numbers by N.
Let X be a Banach space and M be a nonempty subset of X. A mapping

Received 27 May 2019 – Accepted 24 November 2021

http://dx.doi.org/10.4995/agt.2022.11902
https://orcid.org/0000-0002-6991-4287


K. Ullah, J. Ahmad and F. M. Khan

T : M → M is said to nonexpansive if
||Tx−Ty|| ≤ ||x−y||, for all x,y ∈ M.

An element p ∈ M is said to be a fixed point of T if p = T(p). From now
on, we will denote the set of all fixed points of T by F(T). A mapping
T : M → M is said to be a quasi-nonexpansive mapping if F(T) 6= ∅ and
||T(x) −T(p)|| ≤ ||x−p|| for all x ∈ M and p ∈ F(T). It is well-known that
F(T) is nonempty in the case when X is uniformly convex, T is nonexpansive
and M is closed, bounded and convex; see [6, 7, 10]. A number of generaliza-
tions of nonexpansive mappings have been considered by some researchers in
recent years. Suzuki [17] introduced a new class of mappings known as Suzuki
generalized nonexpansive mappings which is a condition on mappings called
condition (C) and obtained some convergence and existence results for such
mappings. Note that, a mapping T : M → M is said to satisfy condition (C) if

1

2
||x−Tx|| ≤ ||x−y|| implies ||Tx−Ty|| ≤ ||x−y||,

for each x,y ∈ M.
Aoyama and Kohsaka [4] introduced the class of α-nonexpansive mappings

in the framework of Banach spaces and obtained some fixed point results for
such mappings. A mapping T : M → M is said to be α-nonexpansive if there
exists a real number α ∈ [0, 1) such that for all x,y ∈ M,

||Tx−Ty||2 ≤ α||Tx−y||2 + α||x−Ty||2 + (1 − 2α)||x−y||2.
Ariza-Puiz et al. [5] proved that the concept of α-nonexpansive is trivial for

α < 0. It is obvious that every nonexpansive mapping is 0-nonexpansive and
also every α-nonexpansive mapping with F(T) 6= ∅ is a quasi-nonexpansive.
Note that, in general condition (C) and α-nonexpansive mappings are not con-
tinuous (see [17] and [14] ).

Recently, Pant and Shukla [14] introduced an interesting class of generalized
nonexpansive mappings in Banach spaces known as generalized α-nonexpansive
mappings which contains the class of Suzuki generalized nonexpansive map-
pings. A mapping T : M → M is said to generalized α-nonexpansive if there
exists a real number α ∈ [0, 1) such that for each x,y ∈ M,
1

2
||x−Tx|| ≤ ||x−y|| ⇒ ||Tx−Ty|| ≤ α||Tx−y||+α||Ty−x||+(1−2α)||x−y||.

Once the existence result of a fixed point for a mapping is established, an
algorithm to find the value of the fixed point is desirable. The famous Banach
contraction mapping principle uses Picard iteration xn+1 = Txn for approx-
imation of fixed point. Some other well-known iterations are the Mann [11],
Ishikawa [9], S [3], Picard-S [8], Noor [12], Abbas [1], Thakur et al. [19] and so
on. Speed of convergence plays an important role for an iteration process to be
preferred on another iteration process. Rhoades [15] mentioned that the Mann
iteration process for a decreasing function converges faster than the Ishikawa
iteration process and for an increasing function the Ishikawa iteration process
is better than the Mann iteration process.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 214



Numerical reckoning fixed points via new faster iteration process

The well-known Mann [11] and Ishikawa [9] iteration schemes are respec-
tively defined as:

(1.1)

{
x1 ∈ M,
xn+1 = (1 −αn)xn + αnTxn,n ∈ N,

where αn ∈ (0, 1).

(1.2)




x1 ∈ M,
yn = (1 −βn)xn + βnTxn,
xn+1 = (1 −αn)xn + αnTyn,n ∈ N,

where αn,βn ∈ (0, 1).
In 2007, Agarwal et al. [3] introduced the following iteration process known

as S iteration:

(1.3)




x1 ∈ M,
yn = (1 −βn)xn + βnTxn,
xn+1 = (1 −αn)Txn + αnTyn,n ∈ N,

where αn,βn ∈ (0, 1). They proved that the rate of convergence of iteration
process (1.3) is same to the Picard iteration xn+1 = Txn and faster than the
Mann [11] iteration process in the class of contraction mappings.

In 2016, Thakur et al. [19] introduced the following iteration scheme:

(1.4)




x1 ∈ M,
zn = (1 −βn)xn + βnTxn,
yn = T ((1 −αn)xn + αnzn) ,
xn+1 = Tyn,n ∈ N,

where αn,βn ∈ (0, 1). With the help of a numerical example, they proved that
(1.4) is faster than the Picard, Mann [11], Ishikawa [9], S [3], Noor [12] and
Abbas [1] iteration processes in the class of Suzuki generalized nonexpansive
mappings.

Recently in 2018, Ullah and Arshad [20] used a new iteration process known
as M iteration:

(1.5)




x1 ∈ M,
zn = (1 −αn)xn + αnTxn,
yn = Tzn,
xn+1 = Tyn,n ∈ N,

where αn ∈ (0, 1). With the help of a numerical example, they proved that
(1.5) is faster than S [3], Picard-S [8] and Thakur et al. [19] iteration processes
for Suzuki generalized nonexpansive mappings.

Problem 1.1. Is it possible to develop an iteration process whose rate of con-
vergence is even faster than the iteration process (1.5) ?

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 215



K. Ullah, J. Ahmad and F. M. Khan

As an answer, we introduce the following new iteration called KF iteration
scheme:

(1.6)




x1 ∈ M,
zn = T((1 −βn)xn + βnTxn),
yn = Tzn,
xn+1 = T((1 −αn)Txn + αnTyn),n ∈ N,

where αn,βn ∈ (0, 1).
With the help of numerical example, we compare the rate of convergence

of iteration (1.6) with the leading S (1.3), Thakur et al. (1.4) and M (1.5)
iteration.

2. Preliminaries

In this section, we give some preliminaries.
Let X be a Banach space and M be a nonempty closed convex subset of X.

Let {xn} be a bounded sequence in M. For x ∈ X, set
r(x,{xn}) = lim sup

n→∞
||x−xn||.

The asymptotic radius of {xn} relative to M is given by
r(M,{xn}) = inf{r(x,{xn}) : x ∈ M}.

The asymptotic center of {xn} relative to M is the set
A(M,{xn}) = {x ∈ M : r(x,{xn}) = r(M,{xn})}.

It is well-known that in a uniformly convex Banach space setting, A(M,xn)
consists of exactly one point. Also, A(M,xn) is nonempty and convex when M
is weakly compact and convex (see, [18] and [2]). A Banach space X is said to
uniformly convex if for all ε > 0, there is a λ > 0 such that, for x,y ∈ X with
||x|| ≤ 1, ||y|| ≤ 1 and ||x − y|| ≤ ε, ||x + y|| ≤ 2(1 − λ) holds. Note that, a
Banach space X is said to have Opial’s property [13] if for each sequence {xn}
in X which weakly converges to x ∈ X and for every y ∈ X, it follows the
following

lim sup
n→∞

||xn −x|| < lim sup
n→∞

||xn −y||.

Examples of Banach spaces satisfying this condition are Hilbert spaces and all
lp spaces (1 < p < ∞).

We now list some basic facts about generalized α-nonexpansive mappings,
which can be found in [14].

Proposition 2.1. Let X be a Banach space, M be a nonempty subset of X
and T : M → M be a mapping.

(i) If T is a Suzuki generalized nonexpansive mapping, then T is a gener-
alized α-nonexpansive mapping.

(ii) If T is a generalized α-nonexpansive mapping and has a fixed point,
then T is a quasi-nonexpansive mapping.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 216



Numerical reckoning fixed points via new faster iteration process

(iii) If T is a generalized α-nonexpansive mapping. Then F(T) is closed.
Moreover, if X is strictly convex and M is convex, then F(T) is also
convex.

(iv) If T is a generalized α-nonexpansive mapping. Then for each x,y ∈ M,

||x−Ty|| ≤
(

3 + α

1 −α

)
||x−Tx|| + ||x−y||.

(v) If X has Opial property, T is generalized α-nonexpansive, {xn} con-
verges weakly to a point v and limn→∞ ||Txn−xn|| = 0, then v ∈ F(T).

Lemma 2.2 ([16]). Let X be a uniformly convex Banach space and 0 < p ≤
αn ≤ q < 1 for every n ∈ N. If {xn} and {yn} are two sequences in X such
that lim supn→∞ ||xn|| ≤ t, lim supn→∞ ||yn|| ≤ t and limn→∞ ||αnxn + (1 −
αn)yn|| = t for some t ≥ 0 then, limn→∞ ||xn −yn|| = 0.

3. Main Results

We open this section with the following important lemma.

Lemma 3.1. Let M be a nonempty closed convex subset of a Banach space X
and T : M → M be a generalized α-nonexpansive mapping with F(T) 6= ∅. Let
{xn} be a sequence generated by (1.6), then limn→∞ ||xn − p|| exists for each
p ∈ F(T).

Proof. Let p ∈ F(T). By Proposition 2.1 part (ii), we have
||zn −p|| = ||T((1 −βn)xn + βnTxn) −p||

≤ ||(1 −βn)xn + βnTxn −p||
≤ (1 −βn)||xn −p|| + βn||Txn −p||
≤ (1 −βn)||xn −p|| + βn||xn −p||
≤ ||xn −p||,

and

||yn −p|| = ||Tzn −p||
≤ ||zn −p||.

They imply that,

||xn+1 −p|| = ||T((1 −αn)Txn + αnTyn) −p||
≤ ||(1 −αn)Txn + αnTyn −p||
≤ (1 −αn)||Txn −p|| + αn||Tyn −p||
≤ (1 −αn)||xn −p|| + αn||yn −p||
≤ (1 −αn)||xn −p|| + αn||zn −p||
≤ (1 −αn)||xn −p|| + αn||xn −p||
≤ ||xn −p||.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 217



K. Ullah, J. Ahmad and F. M. Khan

Thus {||xn−p||} is bounded and nonincreasing, which implies that limn→∞ ||xn−
p|| exists for all p ∈ F(T). �

The following theorem is necessary for the next results.

Theorem 3.2. Let M be a nonempty closed convex subset of a uniformly con-
vex Banach space X and T : M → M a generalized α-nonexpansive mapping.
Let {xn} be a sequence generated by (1.6). Then, F(T) 6= ∅ if and only if {xn}
is bounded and limn→∞ ||Txn −xn|| = 0.

Proof. Suppose that F(T) 6= ∅ and p ∈ F(T). Then, by Lemma 3.1, limn→∞ ||xn−
p|| exists and {xn} is bounded. Put

(3.1) lim
n→∞

||xn −p|| = t.

In view of the proof of Lemma 3.1 together with (3.1), we have

(3.2) lim sup
n→∞

||zn −p|| ≤ lim sup
n→∞

||xn −p|| = t.

By Proposition 2.1 part (ii), we have

(3.3) lim sup
n→∞

||Txn −p|| ≤ lim sup
n→∞

||xn −p|| = t.

Again by the proof of Lemma 3.1, we have

||xn+1 −p|| ≤ (1 −αn)||xn −p|| + αn||zn −p||.

It follows that,

||xn+1 −p||− ||xn −p|| ≤
||xn+1 −p||− ||xn −p||

αn
≤ ||zn −p||− ||xn −p||.

So, we can get ||xn+1 −p|| ≤ ||zn −p|| and from (3.1), we have

(3.4) t ≤ lim inf
n→∞

||zn −p||.

From (3.2) and (3.4), we obtain

(3.5) t = lim
n→∞

||zn −p||.

From (3.1) and (3.5), we have

t = lim
n→∞

||zn −p||

= lim
n→∞

||T((1 −βn)xn + βnTxn) −p||

≤ lim
n→∞

||(1 −βn)xn + βnTxn −p||

= lim
n→∞

||(1 −βn)(xn −p) + βn(Txn −p)||

≤ lim
n→∞

(1 −βn)||xn −p|| + lim
n→∞

βn||Txn −p||

≤ lim
n→∞

(1 −βn)||xn −p|| + lim
n→∞

βn||xn −p||

≤ t.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 218



Numerical reckoning fixed points via new faster iteration process

Hence,

(3.6) t = lim
n→∞

||(1 −βn)(xn −p) + βn(Txn −p)||.

Now from (3.1), (3.3) and (3.6) together with Lemma 2.2, we obtain

lim
n→∞

||Txn −xn|| = 0.

Conversely, we assume that {xn} is bounded and limn→∞ ||Txn−xn|| = 0. Let
p ∈ A(M,{xn}). By proposition 2.1 part (iv), we have

r(Tp,{xn}) = lim sup
n→∞

||xn −Tp||

≤
(

3 + α

1 −α

)
lim sup
n→∞

||Txn −xn|| + lim sup
n→∞

||xn −p||

= lim sup
n→∞

||xn −p||

= r(p,{xn}).

Hence, we conclude that Tp ∈ A(M,{xn}. Since X is uniformly convex,
A(M,{xn}) consist of a unique element. Thus, we have p = T(p). �

First we prove our weak convergence result.

Theorem 3.3. Let X be a uniformly Banach space with Opial property, M
a nonempty closed convex subset of X and T : M → M be generalized α-
nonexpansive mapping with F(T) 6= ∅. Then, {xn} generated by (1.6) con-
verges weakly to an element of F(T).

Proof. By Theorem 3.2, {xn} is bounded and limn→∞ ||Txn −xn|| = 0. Since
X is uniformly convex, X is reflexive. So, a subsequence {xni} of {xn} exists
such that {xni} converges weakly to some v1 ∈ M. By Proposition 2.1 part
(v), we have v1 ∈ F(T). It is sufficient to show that {xn} converges weakly
to v1. In fact, if {xn} does not converges weakly to v1. Then, there exists a
subsequence {xnj} of {xn} and v2 ∈ M such that {xnj} converges weakly to
v2 and v2 6= v1. Again by Proposition 2.1 part (v), v2 ∈ F(T). By Lemma 3.1
together with Opial property, we have

lim
n→∞

||xn −v1|| = lim
i→∞

||xni −v1||

< lim
i→∞

||xni −v2||

= lim
n→∞

||xn −v2||

= lim
j→∞

||xnj −v2||

< lim
j→∞

||xnj −v1||

= lim
n→∞

||xn −v1||.

This is a contradiction, so, v1 = v2. Thus, {xn} converges weakly to v1 ∈
F(T). �

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 219



K. Ullah, J. Ahmad and F. M. Khan

We now prove our strong convergence result.

Theorem 3.4. Let M be a nonempty closed convex subset of a uniformly con-
vex Banach space X and T : M → M be a generalized α-nonexpansive map-
ping. If F(T) 6= ∅ and lim infn→∞dist(xn,F(T)) = 0 (where dist(x,F(T)) =
inf{||x−p|| : p ∈ F(T)}). Then, {xn} generated by (1.6) converges strongly to
an element of F(T).

Proof. By Lemma 3.1, limn→∞ ||xn − p|| exists, for each p ∈ F(T). So,
limn→∞dist(xn,F(T)) exists, thus

lim
n→∞

dist(xn,F(T)) = 0.

Therefore, there exists a subsequence {xnk} of {xn} and {vk} in FT such that
||xnk − vk|| ≤

1
2k

for each k ∈ N. By the proof of Lemma 3.1, {xn} is nonin-
creasing, so

||xnk+1 −vk|| ≤ ||xnk −vk|| ≤
1

2k
.

Therefore,

||vk+1 −vk|| ≤ ||vk+1 −xnk+1|| + ||xnk+1 −vk||

≤
1

2k+1
+

1

2k

≤
1

2k−1
→ 0, as k →∞.

Hence, {vk} is a Cauchy sequence in F(T) and so it converges to some p. Since,
by Proposition 2.1 part (iii), F(T) is closed, we have p ∈ F(T). By Lemma
3.1, limn→∞ ||xn −p|| exists, hence {xn} converges strongly to p ∈ F(T). �

4. example

We compare rate of convergence of our new KF iteration (1.6) with leading
S (1.3), M (1.5) Thakur et al. (1.4) in slightly general setting using Exam-
ple 4.1, in which T is generalized α-nonexpansive but not Suzuki generalized
nonexpansive.

Example 4.1. Let M = [0,∞) with absolute valued norm. Define a mapping
T : M → M by

Tx =

{
0 if x ∈

[
0, 1

5000

)
x
2

if x ∈
[

1
5000

,∞
)
.

Choose x = 1
8000

and y = 1
5000

. We see that, 1
2
|x − Tx| < |x − y| but

|Tx − Ty| > |x − y|. Thus, T does not satisfy condition (C) and so T is
not Suzuki generalized nonexpansive. On the other hand, T is a generalized
α-nonexpansive mapping. In fact, for α = 1

3
, we have:

Case I: When x,y ∈
[
0, 1

5000

)
, then clearly

1

3
|Tx−y| +

1

3
|x−Ty| +

1

3
|x−y| ≥ 0 = |Tx−Ty|.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 220



Numerical reckoning fixed points via new faster iteration process

Case II: When x ∈
[

1
5000

,∞
)

and y ∈
[
0, 1

5000

)
, we have

1

3
|Tx−y| +

1

3
|x−Ty| +

1

3
|x−y| =

1

3

∣∣∣x
2
−y
∣∣∣ + 1

3
|x− 0| +

1

3
|x−y|

≥
1

3

∣∣∣(x
2
−y
)
− (x−y)

∣∣∣ + 1
3
|x|

=
1

3

∣∣∣x
2

∣∣∣ + 1
3
|x|

≥
1

3

∣∣∣x
2

+ x
∣∣∣

=
1

2
|x|

= |Tx−Ty|.

Case III: When x,y ∈ [ 1
5000

,∞), we have

1

3
|Tx−y| +

1

3
|x−Ty| +

1

3
|x−y| =

1

3

∣∣∣x
2
−y
∣∣∣ + 1

3

∣∣∣x− y
2

∣∣∣ + 1
3
|x−y|

≥
1

3

∣∣∣(x
2
−y
)

+
(
x−

y

2

)∣∣∣ + 1
3
|x−y|

=
1

2
|x−y| +

1

3
|x−y|

≥
1

2
|x−y|

= |Tx−Ty|.

Hence, T is a generalized α-nonexpansive mapping with F(T) = {0}. Take
αn = 0.70 and βn = 0.65. The iterative values for x1 = 10 are given in Table
1. Figure 1 shows the convergence behaviors of different iterative schemes.
Clearly the new KF iteration process is moving fast to the fixed point of T as
compared to other iteration processes.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 221



K. Ullah, J. Ahmad and F. M. Khan

Table 1. Sequences generated by KF (1.6), M (1.5), Thakur
et al. (1.4) and S (1.3) iteration schemes for mapping T of
Example 4.1.

KF (1.6) M (1.5) Thakur et al. (1.4) S (1.3)

x1 10 10 10 10
x2 1.0453120000 1.62500000000 1.9312500000 3.8625000000
x3 0.1092678222 0.26406250000 0.3729726562 1.4918906250
x4 0.0114219020 0.04291015625 0.0720303442 0.5762427539
x5 0.0011939456 0.00697290039 0.0139108602 0.2225737636
x6 0.0001248046 0.00113309631 0.0026865348 0.0859691162
x7 0 0.00018412815 0.0005188370 0.0332055711
x8 0 0 0.0001002004 0.0128256518
x9 0 0 0 0.0049539080
x10 0 0 0 0.0019134469
x11 0 0 0 0.0007390688
x12 0 0 0 0.0002854653

O

O
O O O O

O

O

O

O O O O O

O

O

O

O O O O O O

O

O

O

O
O O O O O O O O O

O

0 2 4 6 8 10 12 14

0

1

2

3

4

5

Number of iteration

V
a
lu
e
o
f
x
n

KF

M

Thakur et al.

S

Figure 1. Convergence behaviors of KF, M, Thakur et al.
and S iteration processes to the fixed point of the mapping
defined in Example 4.1 where x1 = 10.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 222



Numerical reckoning fixed points via new faster iteration process

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