EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 1, 2018, 189-201
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Convergence Rate of Implicit Iteration Process and a
Data Dependence Result

Isa Yildirim1,∗, Mujahid Abbas2

1 Department of Mathematics, Faculty of Science, Ataturk University, Erzurum 25240,
Turkey
2 Department of Mathematics, GC University, Katchery Road, Lahore 54000, Pakistan

Abstract. The aim of this paper is to introduce an implicit S-iteration process and study its
convergence in the framework of W-hyperbolic spaces. We show that the implicit S-iteration
process has higher rate of convergence than implicit Mann type iteration and implicit Ishikawa-
type iteration processes. We present a numerical example to support the analytic result proved
herein. Finally, we prove a data dependence result for a contractive type mapping using implicit
S-iteration process.

2010 Mathematics Subject Classifications: AMS 47H10, 54H25

Key Words and Phrases: Implicit iterations, convergence rate, data dependence

1. Introduction and preliminaries

Throughout this paper, the letter N will denote the set of natural numbers.
The theory of fixed points deals with the conditions which guarantee that a mapping T

of a set X into itself admits one or more fixed points, that is, points x of X which solve an
operator equation x = Tx, called a fixed point equation. Fixed point theory serves as an
essential tool for solving problems arising in various branches of mathematical analysis.
Over the past two decades the development of fixed point theory in metric spaces has
attracted considerable attention due to numerous applications in areas such as variational
and linear inequalities, optimization, and approximation theory.

The set {p ∈ X : p = Tp} of all fixed points of T is denoted by F (T ).
One of the basic and the most widely applied fixed point theorem in all of analysis

is ”Banach ( or Banach- Cassioppoli ) Contraction principle” due to Banach [2]. This
principle lies at the heart of metric fixed point theory. It states that if (X, d) is a complete
metric space and T : X → X satisfies

d(Tx, Ty) ≤ kd(x, y),

∗Corresponding author.

Email addresses: isayildirim@atauni.edu.tr (I. Yildirim), abbas.mujahid@gmail.com (M. Abbas)

http://www.ejpam.com 189 c© 2018 EJPAM All rights reserved.



I. Yildirim, M. Abbas / Eur. J. Pure Appl. Math, 11 (1) (2018), 189-201 190

for all x, y ∈ X, with k ∈ (0, 1), then T has a unique fixed point. Due to its applica-
tions in mathematics and other related disciplines, Banach contraction principle has been
generalized in many directions.

Zamfirescu [21] obtained an important generalization of Banach fixed point theorem
using Zamfirescu mapping.

A self mapping T on a metric space X is called Zamfirescu mapping if there exist real
numbers a, b, c satisfying 0 < a < 1, 0 < b, c < 1/2 such that at least one of the following
is true: 

(z1) d (Tx, Ty) ≤ ad (x, y) ,
(z2) d (Tx, Ty) ≤ b (d (x, Tx) + d (y, Ty)) ,
(z3) d (Tx, Ty) ≤ c (d (x, Ty) + d (y, Tx)) .

(1)

for any x, y ∈ X.
The contractive condition (1) can be reformulated as follows:{

(b1) d (Tx, Ty) ≤ δd (x, y) + 2δd (x, Tx) if one uses (z2) , and
(b2) d (Tx, Ty) ≤ δd (x, y) + 2δd (x, Ty) if one uses (z3) ,

(2)

for all x, y ∈ X, where δ = max
{
a, b

1−b ,
c

1−c

}
([4]). Clearly, δ ∈ [0, 1) .

A mapping satisfying condition (b1) or (b2) is called a quasi-contractive mapping. This
class of mappings is general than the class of Zamfirescu mappings.

Osilike and Udomene [17] extended the above class of mappings and introduced a
mapping T satisfying the following contractive condition:

d (Tx, Ty) ≤ δd (x, y) + Ld (x, Tx) , (3)

for all x, y ∈ X, where L ≥ 0 and δ ∈ [0, 1) .
For more results and discussion in this direction, we refer to [1], [13] and references

mentioned therein.
Imoru and Olantiwo [10] gave the following definition:

Definition 1. A self mapping T on X is called a contractive-like mapping if there exists
a constant δ ∈ [0, 1) and a strictly increasing and continuous function ϕ : [0,∞)→ [0,∞)
with ϕ (0) = 0 such that for any x, y ∈ X, we have

d (Tx, Ty) ≤ δd (x, y) + ϕ (d (x, Tx)) . (4)

Definition 2. [18] Let T, S be two self mappings on X. We say that S is an approximate
operator of T if for all ε > 0, we have d (Tx, Sx) ≤ ε holds for any x ∈ X .

An ambient space equipped with certain convexity structure play a significant role in
solving a fixed point equation. Since Banach space is a vector space, the concept of a line
segment joining any two points of a nonempty subset of a Banach space give rise to the
convexity structure. However, metric spaces do not naturally have this convex structure.
The notion of convex metric spaces was introduced by Takahashi [19] who studied the
fixed points of nonexpansive mappings in the setting of such spaces. All normed spaces



I. Yildirim, M. Abbas / Eur. J. Pure Appl. Math, 11 (1) (2018), 189-201 191

and their convex subsets are convex metric spaces. But there are many examples of convex
metric spaces which are not embedded in any normed space ([19]). Over time, different
convex structures have been introduced on metric spaces.

Kohlenbach [15] introduced W−hyperbolic spaces as follows:

Definition 3. A W−hyperbolic space (X, d,W ) is a metric space (X, d) together with a
convexity mopping W : X2 × [0, 1]→ X satisfying the following properties:

(i) d(u,W (x, y, α)) ≤ (1− α)d(u, x) + αd(u, y)
(ii) d(W (x, y, α),W (x, y, β)) = |α− β| d(x, y)
(iii) W (x, y, α) = W (y, x, 1− α)
(iv) d(W (x, z, α),W (y, w, α)) ≤ (1− α)d(x, y) + αd(z, w)

for all x, y, z, w ∈ X and α, β ∈ [0, 1].

If the triplet (X, d,W ) satisfies the condition (i) only, then it coincides with the convex
metric space introduced by Takahashi [19]. Every hyperbolic space is a convex metric space
but converse does not hold in general ( [6]). A subset E of a W−hyperbolic space X is
convex if W (x, y, α) ∈ E for all x, y ∈ E and α ∈ [0, 1].
Note that every W−hyperbolic space is a geodesic space. CAT(0) spaces, normed linear
space, The Hilbert ball and Busseman spaces are important examples of W− hyperbolic
spaces. A W− hyperbolic space represents a unified approach for both linear and nonlinear
structures simultaneously. There are hyperbolic spaces which are not imbedded in any
Banach space .

On the other hand, different iterative algorithms have been used to approximate the
solution of a fixed point equation ([5, 11, 14, 16, 19]). Implicit iterative schemes are of
great importance from numerical stand point as they provide accurate approximation (
see, [6], [12], [7, 8, 20]).

The motivation of this paper is to define an implicit S-iteration process with higher
rate of convergence when compared with Mann type(7) and Ishikawa type (6) implicit
iterative processes.

Let E be a nonempty convex subset of a W−hyperbolic space X and T : E → E.
Choose x0 ∈ E and define the sequence {xn} as follows:

xn = W (Txn−1, T yn, αn) (5)

yn = W (xn, Txn, βn) , n ∈ N,

where {an} and {βn} are certain real sequences in [0, 1].
We translate implicit Ishikawa and implicit Mann iteration processes introduced by

Ćirić et al. ([8, 9]) in the setup of W−hyperbolic space as follows:

xn = W (xn−1, Tyn, αn) (6)

yn = W (xn, Txn, βn) n ∈ N,

and
xn = W (xn−1, T yn, αn) , n ∈ N. (7)



I. Yildirim, M. Abbas / Eur. J. Pure Appl. Math, 11 (1) (2018), 189-201 192

Remark 1. Note that, the process (5) is independent of (6) and (7) in the sense that
neither of them reduce to the other.

The following definition is due to Berinde [3].

Definition 4. Let {xn} and {un} be two fixed point iteration processes in a metric space
X such that lim

n→∞
xn = lim

n→∞
un = p, where p is a fixed point of a self mapping T on X.

Suppose that
d(xn, p) ≤ an and d(un, p) ≤ bn, n ∈ N.

where {an} and {bn} are two null sequences of positive numbers. If {an} converges faster
than {bn}, then we say {xn} converges faster than {un} to p.

We also need the following lemma in order to prove our main results.

Lemma 1. [18] Let {an} be a nonnegative sequence. If there exists an n0 ∈ N such that
for all n ≥ n0, we have

an+1 ≤ (1− µn)an + µnηn,

where µn ∈ (0, 1),
∑∞

n=0 µn =∞ and ηn ≥ 0 for all n ∈ N. Then the following holds:

0 ≤ lim
n→∞

sup an ≤ lim
n→∞

sup ηn.

2. Main Results

We start with the following result.

Theorem 1. Let E be a nonempty closed convex subset of W−hyperbolic space X and
T : E → E a contractive type mapping with F (T ) 6= ∅. Then, for the sequence {xn}
defined in (5) with

∑
(1− αn) =∞, we have lim

n→∞
xn = p, where p ∈ F (T ).

Proof. Suppose that p ∈ F (T ). Using (5) and (4), we have

d(xn, p) = d(W (Txn−1, T yn, αn) , p) (8)

≤ αnd(Txn−1, p) + (1− αn)d(Tyn, p)

≤ αn [δd(xn−1, p) + ϕ (d (p, Tp))]

+(1− αn) [δd(yn, p) + ϕ (d (p, Tp))]

= αnδd(xn−1, p) + (1− αn)δd(yn, p)

and

d(yn, p) = d(W (xn, Txn, βn) , p) (9)

≤ βnd(xn, p) + (1− βn)d(Txn, p)

≤ βnd(xn, p) + (1− βn) [δd(xn, p) + ϕ (d (p, Tp))]

= βnd(xn, p) + (1− βn)δd(xn, p)

= [βn + (1− βn)δ] d(xn, p).



I. Yildirim, M. Abbas / Eur. J. Pure Appl. Math, 11 (1) (2018), 189-201 193

Therefore,

d(xn, p) ≤ αnδd(xn−1, p) + (1− αn)δ [βn + (1− βn)δ] d(xn, p).

That is,
[1− (1− αn)δ [βn + (1− βn)δ]] d(xn, p) ≤ αnδd(xn−1, p), (10)

which further implies that

d(xn, p) ≤
αnδ

1− (1− αn)δ [βn + (1− βn)δ]
d(xn−1, p). (11)

We set

∆n =
αnδ

1− (1− αn)δ [βn + (1− βn)δ]
.

Then

1−∆n = 1− αnδ

1− (1− αn)δ [βn + (1− βn)δ]

=
1− (1− αn)δ [βn + (1− βn)δ]− αnδ

1− (1− αn)δ [βn + (1− βn)δ]

≥ 1− (1− αn)δ [βn + (1− βn)δ]− αnδ

implies that

∆n ≤ (1− αn)δ [βn + (1− βn)δ] + αnδ (12)

= (1− αn)δ [1− (1− δ) (1− βn)] + αnδ

≤ (1− αn)δ + αn

= 1− (1− αn) (1− δ) .

By (11) and (12), we have

d(xn, p) ≤ [1− (1− αn) (1− δ)] d(xn−1, p) (13)

≤
n∏
i=1

[1− (1− αi) (1− δ)] d(x0, p).

Since a > 0, 1 + a ≤ ea, (13) gives that

d(xn, p) ≤ exp

{
n∑
i=1

(1− αi) (1− δ)

}
d(x0, p) (14)

≤ exp

{ ∞∑
n=1

(1− αn) (1− δ)

}
d(x0, p).

Using the fact that 0 ≤ δ < 1 and
∑

(1−αn) =∞, we conclude that limn→∞ d(xn, p) = 0.



I. Yildirim, M. Abbas / Eur. J. Pure Appl. Math, 11 (1) (2018), 189-201 194

The following result deals with the rate of convergence of implicit S-iteration process.

Theorem 2. Let E be a nonempty closed convex subset of W−hyperbolic space X and
T : E → E a contractive type mapping with F (T ) 6= ∅. Then, the sequence {xn} defined
in (5) with

∑
(1− αn) =∞ converges to the fixed point of T faster than implicit Ishikawa

type (6) and implicit Mann type (7) iterations.
Proof. Let p be a fixed point of T . Using an implicit Ishikawa type (6) iteration process

we have,

d(xn, p) = d(W (xn−1, T yn, αn) , p) (15)

≤ αnd(xn−1, p) + (1− αn)d(Tyn, p)

≤ αnd(xn−1, p) + (1− αn) [δd(yn, p) + ϕ (d (p, Tp))]

= αnd(xn−1, p) + (1− αn)δd(yn, p)

and

d(yn, p) = d(W (xn, Txn, βn) , p) (16)

≤ βnd(xn, p) + (1− βn)d(Txn, p)

≤ βnd(xn, p) + (1− βn) [δd(xn, p) + ϕ (d (p, Tp))]

= βnd(xn, p) + (1− βn)δd(xn, p)

= [βn + δ(1− βn)] d(xn, p).

Therefore,

d(xn, p) ≤ αnd(xn−1, p) + (1− αn)δ [βn + δ(1− βn)] d(xn, p). (17)

That is,

d(xn, p) ≤
αn

1− (1− αn)δ [βn + δ(1− βn)]
d(xn−1, p) (18)

≤ ... ≤ cn

where

cn =

(
αn

1− (1− αn)δ [βn + δ(1− βn)]

)n
d(x0, p). (19)

Using implicit Mann iteration (6), we obtain that

d(xn, p) = d(W (xn−1, Txn, αn) , p) (20)

≤ αnd(xn−1, p) + (1− αn)d(Txn, p)

≤ αnd(xn−1, p) + (1− αn) [δd(xn, p) + ϕ (d (p, Tp))]

= αnd(xn−1, p) + (1− αn)δd(xn, p).

Thus, we have

d(xn, p) ≤
αn

1− (1− αn)δ
d(xn−1, p) (21)



I. Yildirim, M. Abbas / Eur. J. Pure Appl. Math, 11 (1) (2018), 189-201 195

≤ bn

where

bn =

(
αn

1− (1− αn)δ

)n
d(x0, p). (22)

Using implicit S-iteration process and following the arguments of the proof of Theorem 1,
we have

d(xn, p) ≤
αnδ

1− (1− αn)δ [βn + (1− βn)δ] [γn + (1− γn)δ]
d(xn−1, p)

≤ ... ≤ an

where

an =

(
αnδ

1− (1− αn)δ [βn + (1− βn)δ] [γn + (1− γn)δ]

)n
d(x0, p). (23)

Note that limn→∞
an
cn

= 0 and limn→∞
an
bn

= 0. Indeed,

[γn + (1− γn)δ] < 1⇒ [βn + (1− βn)δ] < 1

⇒ (1− αn)δ [βn + (1− βn)δ] < 1

⇒ 1− (1− αn)δ [βn + (1− βn)δ] > 0,

give

αnδ < αn ⇒
αnδ

1− (1− αn)δ [βn + δ(1− βn)]

<
αn

1− (1− αn)δ

⇒
(

αnδ

1− (1− αn)δ [βn + δ(1− βn)]

)n
<

(
αn

1− (1− αn)δ

)n
and

αnδ < αn ⇒
αnδ

1− (1− αn)δ [βn + δ(1− βn)]

<
αn

1− (1− αn)δ [βn + δ(1− βn)]

⇒
(

αnδ

1− (1− αn)δ [βn + δ(1− βn)]

)n
<

(
αn

1− (1− αn)δ [βn + δ(1− βn)]

)n
.



I. Yildirim, M. Abbas / Eur. J. Pure Appl. Math, 11 (1) (2018), 189-201 196

Hence, we have

lim
n→∞

an
cn

= lim
n→∞

(
αnδ

1−(1−αn)δ[βn+δ(1−βn)]

)n
d(x0, p)(

αn
1−(1−αn)δ

)n
d(x0, p)

= 0,

and

lim
n→∞

an
bn

= lim
n→∞

(
αnδ

1−(1−αn)δ[βn+δ(1−βn)]

)n
d(x0, p)(

αn
1−(1−αn)δ[βn+δ(1−βn)]

)n
d(x0, p)

= 0.

We now support our above analytical proof by a numerical example using MATLAB.

Example 1. Let E = [0, 1] and T : E → E a mapping defined by Tx = x
2 . Note that T

is a contractive type mapping.Choose αn = 1 − 1
n and βn = 1 − 1

n , n ≥ 2 and for n = 1,
αn = βn = 0. The comparison of the convergences of the implicit S-iteration (5), implicit
Ishikawa type (6) and implicit Mann type iterations (7) to the fixed point p = 0 are given
in table with the initial value x1 = 1.

The following table presents a comparison of rate of convergence of the implicit S-
iteration process with implicit Ishikawa type and implicit Mann type iteration processes for
the mapping given in Example 1.

n IMI III ISI

2 0.666666666666667 0.615384615384615 0.307692307692308

5 0.406349206349206 0.352704628530670 0.022044039283167

7 0.340992340992341 0.292145335107371 0.004564770861053

10 0.283773192751521 0.240691952056443 0.000470101468860

13 0.248169351176485 0.209336831746067 0.000051107624938

16 0.223294138742407 0.187699995568689 0.000005728149279

20 0.199408653447441 0.167113839554526 0.000000318744353

25 0.178133771931084 0.148920204678483 0.000000008876336

30 0.162477710197415 0.135609685643003 0.000000000252593

35 0.150335628473559 0.125328510781087 0.000000000007295

40 0.140563343828096 0.117078595772533 0.000000000000213

43 0.135541774913220 0.112847389889567 0.000000000000026

46 0.131022580805197 0.109043978938918 0.000000000000003

50 0.125645129018549 0.104523598655989 0.000000000000000

Remark 2. From the example above, we see that our iteration ISI (implicit S-iteration) is
faster than the III (implicit Ishikawa iteration) and IMI (implicit Mann iteration) under
the same control conditions.



I. Yildirim, M. Abbas / Eur. J. Pure Appl. Math, 11 (1) (2018), 189-201 197

The following figure is a graphical presentation of the above results.

This figure confirm that the iteration (5) is converges faster than the iteration methods
mentioned above.

Finally, we present a data dependence result for contractive type mapping T using
implicit S-iteration process.

Theorem 3. Let E be a nonempty closed convex subset of W−hyperbolic space X, T :
E → E a contractive type mapping, S an approximate operator of T and {xn} a sequence
defined in (5). For contractive type mapping T , define an iterative sequence {un} as follows:

un = W (Sun−1, T vn, αn) (24)

vn = W (un, Sun, βn) , n ∈ N,

where {an} and {βn} are real sequences in [0, 1] satisfying an < 1, n ∈ N, and
∑

(1− αn) =

∞. If Tp = p and Sq = q such that un → q as n→∞, then we have, d (p, q) ≤ 2ε

(1− δ)2
,

ε > 0 is some appropriate number.
Proof. By (4), (5) and (24), we have

d(yn, vn) = d (W (xn, Txn, βn) ,W (un, Sun, βn)) (25)

≤ βnd (xn, un) + (1− βn)d (Txn, Sun)

≤ βnd (xn, un) + (1− βn) [d (Txn, Sun) + d (Tun, Sun)]

≤ βnd (xn, un) + (1− βn) [δd (xn, un) + ϕ (d (xn, Txn)) + ε]



I. Yildirim, M. Abbas / Eur. J. Pure Appl. Math, 11 (1) (2018), 189-201 198

= [βn + (1− βn)δ] d (xn, un) + (1− βn) [ϕ (d (xn, Txn)) + ε]

and

d(xn, un) = d (W (Txn−1, Tyn, αn) ,W (Sun−1, Svn, αn)) (26)

≤ αnd (Txn−1, Sun−1) + (1− αn)d (Tyn, Svn)

≤ αn [d (Txn−1, Tun−1) + d (Tun−1, Sun−1)]

+(1− αn) [d (Tyn, T vn) + d (Tvn, Svn)]

≤ αn [δd (xn−1, un−1) + ϕ (d (xn−1, Txn−1)) + ε]

+(1− αn) [δd (yn, vn) + ϕ (d (yn, Tyn)) + ε]

= αnδd (xn−1, un−1) + αnϕ (d (xn−1, Txn−1)) + ε

+(1− αn)δd (yn, vn) + (1− αn)ϕ (d (yn, Tyn)) .

Substituting (25) in (26), we obtain that

d(xn, un) ≤ αnδd (xn−1, un−1) + αnϕ (d (xn−1, Txn−1)) (27)

+(1− αn)ϕ (d (yn, T yn)) + ε+ (1− αn)δ

[[βn + (1− βn)δ] d (xn, un) + (1− βn) [ϕ (d (xn, Txn)) + ε]]

= αnδd (xn−1, un−1) + αnϕ (d (xn−1, Txn−1))

+(1− αn)ϕ (d (yn, T yn)) + ε+ (1− αn)δ

[βn + (1− βn)δ] d (xn, un) + (1− αn)δ

(1− βn)ϕ (d (xn, Txn)) + (1− αn)δ(1− βn)ε.

It follows from ( 27) that

[1− (1− αn)δ [βn + (1− βn)δ]] d(xn, un) (28)

≤ αnδd (xn−1, un−1) + αnϕ (d (xn−1, Txn−1))

+(1− αn)ϕ (d (yn, T yn)) + ε+ (1− αn)δ

(1− βn)ϕ (d (xn, Txn)) + (1− αn)δ(1− βn)ε,

which further implies that

d(xn, un) ≤ αnδ

1− (1− αn)δ [βn + (1− βn)δ]
d (xn−1, un−1) (29)

+

{
αnϕ (d (xn−1, Txn−1)) + (1− αn)ϕ (d (yn, T yn))

+(1− αn)δ(1− βn)ϕ (d (xn, Txn))

}
1− (1− αn)δ [βn + (1− βn)δ]

+ε+ (1− αn)δ(1− βn)ε

1− (1− αn)δ [βn + (1− βn)δ]
.

We set

∆n =
αnδ

1− (1− αn)δ [βn + (1− βn)δ]
.



REFERENCES 199

Following arguments similar to those in the proof of Theorem 1, we have

∆n ≤ 1− (1− αn) (1− δ) . (30)

Therefore,

d(xn, un) ≤ [1− (1− αn) (1− δ)] d (xn−1, un−1)

+

(1− αn) (1− δ)
{ αn

1−αn
ϕ (d (xn−1, Txn−1)) + ϕ (d (yn, T yn))

+δ(1− βn)ϕ (d (xn, Txn)) + 2ε

}
(1− δ) [1− (1− αn)δ [βn + (1− βn)δ]]

.

Note that

1− (1− αn)δ [βn + (1− βn)δ] = 1− (1− αn)δ [1− (1− βn) (1− δ)]
≥ 1− δ.

Hence

d(xn, un) ≤ [1− (1− αn) (1− δ)] d (xn−1, un−1) (31)

+

(1− αn) (1− δ)
{ αn

1−αn
ϕ (d (xn−1, Txn−1)) + ϕ (d (yn, T yn))

+δ(1− βn)ϕ (d (xn, Txn)) + 2ε

}
(1− δ)2

.

That is,
an+1 ≤ (1− µn)an + µnηn

where an+1 = d(xn, un), µn = (1− αn) (1− δ) and

ηn =

αn
1−αn

ϕ (d (xn−1, Txn−1)) + ϕ (d (yn, T yn))

+δ(1− βn)ϕ (d (xn, Txn)) + 2ε

(1− δ)2
.

From Theorem 1, we have limn→∞ d(xn, p) = limn→∞ d(xn−1, p) = 0, and limn→∞ d(un, p) =
0. As ϕ is continuous, limn→∞ ϕ (d (xn−1, Txn−1)) = 0, limn→∞ ϕ (d (yn, T yn)) = 0, and
limn→∞ ϕ (d (xn, Txn)) = 0. Thus all the conditions of Lemma 1 are satisfied, therefore
(31) becomes

d (p, q) ≤ 2ε

(1− δ)2
.

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