EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 5, 2020, 1110-1130
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Special Issue Dedicated to
Professor Hari M. Srivastava

On the Occasion of his 80th Birthday

Solution of Delay Differential equation via N v
1 iteration

algorithm

Nisha Sharma1, Lakshmi Narayan Mishra2,∗, Vishnu Narayan Mishra3,
Shikha Pandey4

1 Department of Mathematics, Assistant Professor, Pt. J.L.N. Govt. College, Faridabad,
Haryana 121 002, India
2 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology
(VIT) University, Vellore 632 014, Tamil Nadu, India
3 Department of Mathematics, Indira Gandhi National Tribal University, Lalpur,
Amarkantak, Anuppur, Madhya Pradesh 484 887, India
4 Department of Mathematics, School of Sciences and Languages, VIT-AP University,
Amaravati 522 237, Andhra Pradesh, India

Abstract. The aim of this paper is to define a new iteration scheme Nv
1 which converges to a fixed

point faster than some previously existing methods such as Picard, Mann, Ishikawa, Noor, SP, CR,
S, Picard-S, Garodia, K and K∗ methods etc. The effectiveness and efficiency of our algorithm
is confirmed by numerical example and some strong convergence, weak convergence, T -stability
and data dependence results for contraction mapping are also proven. Moreover, it is shown that
differential equation with retarted argument is solved using Nv

1 iteration process.

2020 Mathematics Subject Classifications: 47H09, 47H10

Key Words and Phrases: Iteration schemes, Nv
1 iteration scheme, Convergence analysis, T −

stability, Data Dependency

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i5.3756

Email addresses: nnishaa.bhardwaj@gmail.com (N. Sharma), lakshminarayanmishra04@gmail.com
(L.N. Mishra), vishnunarayanmishra@gmail.com (V.N. Mishra), sp1486@gmail.com (S. Pandey)

https://www.ejpam.com 1110 c© 2020 EJPAM All rights reserved.



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1111

1. Introduction

Throughout in this paper, we will denote set of natural numbers by N and set of real
numbers by R. A mapping T on a subset C of a Banach space E is said to be nonexpansive
if

||Tx− Ty|| ≤ ||x− y||, for all x, y ∈ C.

An element q ∈ C is said to be a fixed point of T if q = T (q). From now on, we will
denote set of all fixed points of T by Tf . A mapping T : C → C is said to be quasi-
nonexpansive mappings if Tf 6= ∅ and ||Tx − Tq|| ≤ ||x − q|| for all x ∈ C and q ∈ Tf .
The existence of fixed points for nonexpansive mappings in the setting of Banach spaces
was studied independently by Browder [3], Gohde [6] and Kirk [8]. They proved that, if
C is nonempty closed bounded and convex subset of a uniformly convex Banach space,
then every nonexpansive mapping T : C → C has at-least one fixed point. A numbers of
generalization of nonexpansive mappings have been considered by some authors in recent
years.
It is natural to study the computation of fixed points for the known existence results,
which is not an easy task. The Banach contraction mapping principle uses Picard iteration
process xn+1 = Txn for approximation of the unique fixed point. Some other well-known
iteration schemes are Mann [9], Ishikawa [7], S [13], Noor [10], Abbas [1], Thakur et. al.
[4] and so on. Speed of convergence plays an important role for an iteration process to be
preferred on another iteration process. Rhoades [12] mentioned that the Mann iteration
process for decreasing function converge faster than the Ishikawa iteration process and
for increasing function the Ishikawa iteration process is better than the Mann iteration
process. More details can be found in [16], [18], [15], [19], [5].
The most popular and simplest iteration method is formulated by{

p0 ∈ C
pn+1 = Tpn, n ∈ N

(1.3)

and is known as Picard iteration method, which is communally used to approximate fixed
point of contraction mappings satisfying

‖Tx− Ty‖ ≤ µ‖x− y‖, µ ∈ (0, 1), (1.4)

for all x, y ∈ C. The subsequent iteration methods are mention to as Mann, Ishikawa,
Noor, SP, S, CR, Picard-S, Garodia’s, K and K∗ iteration methods, respectively:{

v0 ∈ C,
ζn+1 = (1− σ0n)ζn + σ0nTζn, n ∈ N,

(1.5)


v0 ∈ C,
vn+1 = (1− σ0n)vn + σ0nTwn,

wn = (1− σ1n)vn + σ1nTvn, n ∈ N,
(1.6)



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1112


w0 ∈ C,
wn+1 = (1− σ0n)wn + σ0nTwn,

vn = (1− σ1n)wn + σ1nTun,

un = (1− σ2n)wn + σ2nTwn, n ∈ N,

(1.7)


q0 ∈ C,
qn+1 = (1− σ0n)rn + σ0nTrn,

rn = (1− σ1n)sn + σ1nTsn,

sn = (1− σ2n)qn + σ2nTqn, n ∈ N,

(1.8)


t0 ∈ C,
tn+1 = (1− σ0n)Ttn + σ0nTun,

un = (1− σ1n)tn + σ1nTtn, n ∈ N,
(1.9)


u0 ∈ C,
un+1 = (1− σ0n)vn + σ0nTvn,

vn = (1− σ1n)Tun + σ1nTwn,

un = (1− σ2n)un + σ2nTun, n ∈ N,

(1.10)


0 ∈ C,
n+1 = Tkn,

kn = (1− σ0n)Tn + σ0nT`n,

`n = (1− σ1n)n + σ1nTn, n ∈ N,

(1.11)


x′′0 ∈ C,
x′′n+1 = Ty′′n,

y′′n = (1− σ0n)z′′n + σ0nTz
′′
n,

z′′n = Tx′′n, n ∈ N,

(1.12)


ζ0 ∈ C,
ζn+1 = Tηn,

ηn = T ((1− σ0n)Tζn + σ0nTθn),

θn = (1− σ1n)ζn + σ1nTζn, n ∈ N

(1.13)



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1113


x′0 ∈ C,
x′n+1 = Ty′n,

y′n = T ((1− σ0n)z′n + σ0nTz
′
n),

z′n = (1− σ1n)x′n + σ1nTx
′
n, n ∈ N

(1.14)

where αn, βn ∈ (0, 1).

2. Preliminaries

The following definitions about the rate of convergence are due to Berinde [2].

Definition 1. Let {an}∞n=0 and {bn}∞n=0 be two sequences of real numbers with limits a
and C, respectively. Assume that there exists

lim
n→∞

|an − a|
|bn − b|

= `, (1.15)

(i) If ` = 0, then we say that {an}∞n=0 converges faster to a than {bn}∞n=0 to C.
(ii) If 0 < ` <∞, then we say that {an}∞n=0 and {bn}∞n=0 have the same rate of convergence.

Definition 2. Suppose that for two fixed point iteration processes {un}∞n=0 and {vn}∞n=0

both converging to the same fixed point p, the following error estimates

‖un − p‖ ≤ an (1.16)

for all n ∈ N
‖vn − p‖ ≤ bn (1.17)

for all n ∈ N, are available where {an}∞n=0 and {bn}∞n=0 are two sequences of positive
numbers (converging to zero). If {an}∞n=0 converges faster than {bn}∞n=0, then {un}∞n=0

converges faster than {vn}∞n=0 to p.

Recent study of Ullah and Muhammad (1.14), Hussin et. al. (1.13) proved that their
iterative methods converges faster than all the above mentioned iterative methods for a
different class of mappings which include the aforementioned class of contraction opera-
tors. Now, the question arises whether it is possible to find scheme which is faster than
K∗.
Inspired by the works mentioned above, we introduce the following iteration method



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1114

namely Nv
1 iteration: 

ζ0 ∈ C,
θn = T ((1− σ0n)ζn + σ0nTζn),

ηn = Tθn,

ζn+1 = Tηn, n ∈ N,

(1.18)

Let E be a Banach space and C be a nonempty closed convex subset of E. Let {xn} be a
bounded sequence in C. For x ∈ E, set

r(x, {xn}) = lim sup
n→∞

||x− xn||.

The asymptotic radius of {xn} relative to C is given by

r(C, {xn}) = inf{r(x, {xn}) : x ∈ C}.

The asymptotic centre of {xn} relative to C is the set

A(C, {xn}) = {x ∈ C : r(x, {xn}) = r(C, {xn})}.

It is well-known that in a uniformly convex Banach spaces, A(C, xn) consists of exactly
one point. Also, A(C, xn) is nonempty and convex in the case when C is weakly compact
and convex, see e.g., [14, 17]. Following are some basic definitions and results.

Definition 3. A Banach space E is said to be uniformly convex if for each ε ∈ (0, 2],
there is a λ > 0 such that for every x, y ∈ E,

||x|| ≤ 1
||y|| ≤ 1
||x− y|| > ε

 =⇒ 1
2 ||x+ y|| ≤ (1− λ).

Definition 4. [11] A Banach space E is said to have Opial’s property if for each sequence
{xn} in E which weakly converges to x ∈ E and for every y ∈ E, it follows the following

lim sup
n→∞

||xn − x|| < lim sup
n→∞

||xn − y||.

Definition 5. Let E and E
′

be two Banach spaces and let T : E −→ E
′
. Then the mapping

T is said to be demiclosed if x ⇀ x ∈ E and Txn ⇀ y in E
′

imply Tx = y.

Lemma 1. Let C be a non-empty closed convex subset of a uniformly convex Banach
space E and T be a non-expansive on C. Then I − T is demiclosed at 0.



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1115

3. Convergence Analysis

Theorem 1. Suppose that there is a Banach space E, having subset C, which is nonempty
closed and convex. Also, let there be a contraction mapping T : C → C. Let {ζn}∞n=0

be an iterative sequence generated by Nv
1 and with real sequences {σ0n}∞n=0 and {σ1n}∞n=0

∈ [0, 1] such that
∑∞

n=0 σ
0
nσ

1
n = ∞. Then, {ζn}∞n=0 converges strongly to a fixed point of

T .

Proof. It is obvious from Banach contraction theorem that existence and uniqueness
of fixed point xδ is guaranteed. Now, it is to show that ζn → xδ for n → ∞. From Nv

1

iteration scheme it follows that,

‖θn − xδ‖ = ‖T ((1− σ0n)ζn + σ0nTζn)− xδ‖
= ‖T ((1− σ0n)ζn + σ0nTζn)− Txδ‖
≤ ξ‖(1− σ0n)ζn + σ0nTζn − xδ‖
≤ ξ(1− σ0n)‖ζn − xδ‖+ ξσ0n‖Tζn − Txδ‖
≤ ξ(1− σ0n)‖ζn − xδ‖+ σ0nξ

2‖ζn − xδ‖
= ξ(1− (1− ξ)σ0n)‖ζn − xδ‖

also,

‖ηn − xδ‖ ≤ ‖Tθn − xδ‖
= ‖Tθn − Txδ‖
≤ ξ‖θn − xδ‖

using the value of ‖θn − xδ‖, we have

‖ηn − xδ‖ ≤ ξ2(1− (1− ξ)σ0n)‖ζn − xδ‖

similarly, we have

‖ζn+1 − xδ‖ = ‖Tηn − xδ‖
= ‖Tηn − Txδ‖
= ξ‖ηn − xδ‖

using the value of ‖ηn − xδ‖ and ‖ζn − xδ‖, we have

‖ζn+1 − xδ‖ ≤ ξ3(1− (1− ξ)σ0n)‖ζn − xδ‖

Now, inductively using the behaviour of sequence, we have

‖ζn − xδ‖ ≤ ξ3(1− (1− ξ)σ0n−1)‖xn−1 − xδ‖



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1116

‖xn−1 − xδ‖ ≤ ξ3(1− (1− ξ)σ0n−2)‖xn−2 − xδ‖

the repetition results

‖x1 − xδ‖ ≤ ξ3(1− (1− ξ)σ00)‖ζ0 − xδ‖

Proceeding in the same manner, we have

‖ζn+1 − xδ‖ ≤ ‖ζ0 − xδ‖ξ3(n+1)
n∏
k=0

(1− (1− ξ)σ0k)

where (1−σ0n(1− ξ)) ∈ (0, 1) because ξ ∈ (0, 1) and σ0n ∈ [0, 1], for all n ∈ N, Since we
know that 1− x ≤ e−x for all x ∈ [0, 1], so from the above inequality

‖ζn+1 − xδ‖ ≤
‖ζ0 − xδ‖ξ3(n+1)

e(1−ξ)
∑n

k=0 σ
0
k

.

Taking the limit both sides of this inequality, it yields

lim
n→∞

‖ζn+1 − xδ‖ = 0,

which implies that ζn → xδ for n→∞, as required.

Theorem 2. Suppose that there is a Banach space E, having subset C, which is nonempty
closed and convex and also that there is a contraction mapping T on C with a fixed
point xδ. For given x′0 = ζ0 ∈ C, let {ζn}∞n=0 and {x′n}∞n=0 be the iterative sequences
generated by Nv

1 and K∗ respectively, with real sequences {σ0n}∞n=0, {σ1n}∞n=0 ∈ (0, 1) such
that

∑∞
k=0 σ

0
n = ∞ and for all n ∈ N. Then Nv

1 converges to xδ faster than K∗ iteration
scheme.

Proof. Using The result of Theorem 1 it is clear that

‖ζn+1 − xδ‖ ≤ ‖ζ0 − xδ‖ξ3(n+1)
n∏
k=0

(1− (1− ξ)σ0k)

Now, for the K∗ iteration scheme,

‖z′n − xδ‖ = ‖(1− σ1n)x′n + σ0nTx
′
n − xδ‖

≤ (1− σ1n)‖x′n − xδ‖+ σ0n‖Tx′n − Txδ‖
≤ (1− σ1n)‖x′n − xδ‖+ ξσ0n‖x′n − xδ‖
≤ (1− σ1n(1− ξ))‖x′n − xδ‖

Similarly

‖y′n − xδ‖ ≤ ‖T ((σ0nz
′
n + (1− σ0n)Tz′n)− xδ‖



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1117

≤ ξ‖σ0nz′n + (1− σ0n)Tz′n − xδ‖
≤ ξσ0n‖z′n − xδ‖+ (1− σ0n)‖Tz′n − xδ‖
≤ ξσ0n‖z′n − xδ‖+ ξ(1− σ0n)ξ‖z′n − xδ‖
≤ ξ(1− (1− ξ)σ0n)‖z′n − xδ‖
≤ ξ(1− (1− ξ)σ0n)(1− (1− ξ)σ1n)‖x′n − xδ‖

similarly,

‖x′n+1 − xδ‖ = ‖Ty′n − xδ‖
≤ ξ‖y′n − xδ‖

using the value of ‖y′n − xδ‖ and by using the fact that (1− (1− ξ)σ1n) < 0 and finally we
have

‖x′n+1 − xδ‖ ≤ ξ2(1− (1− ξ)σ0n)

‖x′n − xδ‖ ≤ ξ2(ξ − (1− ξ)σ0n−1)‖x′n−1 − xδ‖

also,

‖x′n−1 − xδ‖ ≤ ξ2(1− (1− ξ)σ0n−2)‖x′n−2 − xδ‖

continually, we have

‖x′1 − xδ‖ ≤ ξ2(1− (1− ξ)σ00)‖x′0 − xδ‖

So, it is quite obvious that the following deduction

‖x′n+1 − xδ‖ ≤ ‖x′0 − xδ‖ξ2(n+1)
n∏
k=0

(1− (1− ξ)σ0k)

is correct. Now, let

rn = ‖ζ0 − xδ‖ξ3(n+1)
n∏
k=0

(1− (1− ξ)σ0k)

and

pn = ‖x′0 − xδ‖ξ2(n+1)
n∏
k=1

(1− (1− ξ)σ0k)

Then

pn
rn

=
‖x′0 − xδ‖ξ2(n+1)

∏n
k=0(1− (1− ξ)σ0k)

‖ζ0 − xδ‖ξ3(n+1)
∏n
k=0(1− (1− ξ)σ0k)

approaches to 0 as n approaches to ∞. Thus, {ζn} is a sequence defined in Nv
1 iteration

defined by (1.14), then {ζn} converges faster than the iteration scheme of Ullah and
Muhhamad known as K∗.



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1118

Theorem 3. Suppose that there is a Banach space E, having subset C, which is nonempty
closed and convex and also that there is a contraction mapping T with contraction factor
ξ ∈ (0, 1) such that TF 6= ∅. if {ζn} is a sequence defined in Nv

1 iteration defined by (1.14),
then {x′′n} converges faster than the iteration scheme of Garodia and Uddin defined by
(1.12).

Proof. Using the result of Theorem 1 it is clear that

‖ζn+1 − xδ‖ ≤ ‖ζ0 − xδ‖ξ3(n+1)
n∏
k=0

(1− σ0k(1− ξ))

Now, for scheme (1.12),

‖z′′n − xδ‖ = ‖Tx′′n − xδ‖
≤ ξ‖x′′n − xδ‖

‖y′′n − xδ‖ = ‖(1− σ0n)z′′n − σ0nTz′′n − xδ‖
≤ ‖(1− σ0n)z′′n − σ0nTz′′n − xδ‖
≤ (1− σ0n)‖z′′n − xδ‖+ ξσ0n‖Tz′′n − xδ‖]
≤ (1− σ0n)‖z′′n − xδ‖+ ξσ0n‖z′′n − xδ‖
= (1− (1− ξ)σ0n)‖z′′n − xδ‖

Thus,

‖x′′n+1 − xδ‖ = ‖Ty′′n − xδ‖
≤ ξ‖y′′n − xδ‖

using the value of ‖y′′n − xδ‖ and using the inductive behaviour, we have

‖x′′n+1 − xδ‖ = ξ2(1− (1− ξ)σ0n)‖x′′n − xδ‖
‖x′′n − xδ‖ = ξ2(1− (1− ξ)σ0n−1)‖x′′n−1 − xδ‖
‖x′′n−1 − xδ‖ = ξ2(1− (1− ξ)σ0n−2)‖x′′n−2 − xδ‖

on combining all the inequalities, we have

‖x′′n+1 − xδ‖ ≤ ‖x′′0 − xδ‖ξ2(n+1)
n∏
k=0

(1− σ0k(1− ξ))

Let

rn = ‖ζ0 − xδ‖ξ3(n+1)
n∏
k=0

(1− (1− ξ)σ0k)



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1119

and

pn = ‖x′′0 − xδ‖ξ2(n+1)
n∏
k=0

(1− (1− ξ)σ0k)

Then

pn
rn

=
‖x′′0 − xδ‖ξ2(n+1)

∏n
k=0(1− (1− ξ)σ0k)

‖ζ0 − xδ‖ξ3(n+1)
∏n
k=0(1− (1− ξ)σ0k)

approaches to 0 as n approaches to ∞. Thus {ζn} is a sequence defined in Nv
1 iteration

defined by (1.14), then {ζn} converges faster than the iteration scheme of Garodia and
Uddin defined by (1.12).

Lemma 2. Let C be a nonempty closed convex subset of a uniformly convex Banach
space E and a nonexpansive self mapping T on C with Tf 6= ∅. Let {ζn} be an iterative
sequence defined as Nv

1 . Then limn→∞ ‖ζn − xδ‖ exists for all xδ ∈ Tf .

Proof. From Nv
1 iteration scheme it follows that,

‖θn − xδ‖ = ‖T ((1− σ0n)ζn + σ0nTζn)− xδ‖
= ‖T ((1− σ0n)ζn + σ0nTζn)− Txδ‖
≤ ‖(1− σ0n)ζn + σ0nTζn − xδ‖
≤ (1− σ0n)‖ζn − xδ‖+ σ0n‖Tζn − Txδ‖
≤ (1− σ0n)‖ζn − xδ‖+ σ0n‖ζn − xδ‖
≤ ‖ζn − xδ‖

‖ηn − xδ‖ ≤ ‖Tθn − xδ‖
= ‖Tθn − Txδ‖
≤ ‖θn − xδ‖

using the value of ‖θn − xδ‖, we have

‖ηn − xδ‖ ≤ ‖ζn − xδ‖

similarly, we have

‖ζn+1 − xδ‖ = ‖Tηn − xδ‖
= ‖Tηn − Txδ‖
≤ ‖ηn − xδ‖

using the value of ‖ηn − xδ‖, we have

‖ζn+1 − xδ‖ ≤ ‖ζn − xδ‖



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1120

which confirms the existence of limn→∞ ‖ζn − xδ‖ for all xδ ∈ Tf . Since, {‖ζn − xδ‖} is
bounded and non-increasing for all xδ ∈ Tf .

Now, we prove the weak convergence of Nv
1 iteration process.

Theorem 4. Suppose that there is a uniformly Banach space E, having a nonempty
subset C, which is nonempty closed and convex satisfying Opial’s condition and also that
there is a nonexpansive mapping T : C → C with Tf 6= ∅. If {ζn} is an iterative sequence
defined by Nv

1 , then {ζn} converges weakly to a fixed point of T .

Proof. Let xδ ∈ Tf . Then from Lemma 2 , it is obvious that limn→∞ ‖ζn − xδ‖ exists.
To prove weak convergence of Nv

1 iterative process, it is to be shown that {ζn} has a weak
subsequential limit in Tf . Let {xnu} and {xnv} are the subsequences of {ζn}, converges
to u and v respectively. Using Lemma 2 limn→∞ ‖Tn − ζn‖ = 0, I − T is demiclosed at 0.
So u, v ∈ Tf .
Next, to show the uniqueness, we assume that limn→∞ ‖ζn − u‖ and limn→∞ ‖ζn − v‖
exists. Assuming u 6= v. Then using Opial’s condition, we have

lim
n→∞

‖ζn − u‖ = lim
n→∞

‖xnu − u‖

< lim
n→∞

‖xnu − v‖

= lim
n→∞

‖ζn − v‖

= lim
n→∞

‖xnv − v‖

< lim
n→∞

‖xnv − u‖

= lim
n→∞

‖ζn − u‖

which is a contradiction, so u = v. So, {ζn} converges weakly to a fixed point of T .

Now, we prove the strong convergence of Nv
1 iteration process.

Theorem 5. Suppose that there is a uniformly Banach space E, having a nonempty
subset C, which is nonempty closed and convex. Also, there be a nonexpansive mapping
T : C → C with Tf 6= φ. If {ζn} is an iterative sequence defined by Nv

1 , then {ζn}
converges strongly to a point of Tf iff lim infn→∞ d(ζn, Tf ) = 0.

Proof. If a sequence {ζn} converges to a fixed point q ∈ Tf , then it is obvious that
lim infn→∞ d(ζn, Tf ) = 0.

For converse, lim infn→∞ d(ζn, Tf ) = 0. From Lemma 2, we have the existence of lim infn→∞ ‖ζn−
q‖ for all q ∈ Tf , we have

‖ζn+1 − q‖ ≤ ‖ζn − q‖ for any q ∈ Tf



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1121

which yields
d(ζn+1, Tf ) ≤ d(ζn, Tf )

which implies that {d(ζn, Tf )} is a decreasing sequence which is bounded below by zero.
Results,

limn→∞ d(ζn, Tf ) = 0

Now, to prove that {ζn} is a Cauchy sequence in C. Let ε > 0 be arbitrarily chosen. Since,
lim infn→∞ d(ζn, Tf ) = 0, there is an existence of n0 in such a manner that ∀ n ≥ n0, we
have

d(ζn, Tf ) < ε
4

Particularly,

inf{‖xn0 − q‖ : q ∈ Tf} < ε

so there must be an existence of α ∈ Tf in such a manner that ‖xn0 − α‖ < ε. Thus, for
m,n ≥ n0, we have

‖xn+m − ζn ≤ ‖xn+m − α‖+ ‖ζn − α‖ < 2‖xn0 − α‖ < 2
ε

2
= ε

which proves the Cauchy behaviour of {ζn}. Since it is given that C is a closed subset of a
Banach space E, therefore the convergence of {ζn} in C is confirmed. Let limn→∞ ζn = α
for some α ∈ B.

Now using, limn→∞ ‖Tζn − ζn‖ = 0, we get

‖α− Tα‖ ≤ ‖α− ζn‖+ ‖ζn − Tζn‖+ ‖Tζn − Tα‖

≤ ‖α− ζn‖+ ‖ζn − Tζn‖+ ‖ζn − α‖

which proves that ‖α − Tα‖ approaches to 0 as n approaches to ∞. This shows that
α = Tα. This proves our result.

4. Numerical Example

In this section, an example is to be given which confirms the behaviour of Nv
1 . In order

to support the proof of Theorems 2 and 3, we will use a numerical example as follow

Example. Assuming E = (−∞,∞) and C = [1, 50]. Let T : C → C be mapping
defined as T (x) =

√
x2 − 9x+ 54 for all x ∈ C. Clearly, x = 5 is the fixed point of T . Set

σ0n = σ1n = σ2n = 0.75 for all n ∈ N. choose initial value as 40. Then, we get the following
table and graph. Also, in table Nv

1 is represented by np of iteration values:



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1122



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1123

Figure 1: Comparison Graph based on Numerical example to prove the efficiency of Nv
1 .

Thus, it is evident from the above table and graph that the newly defined iteration
scheme Nv

1 converges much faster and is more efficient than many iteration schemes in
exiting literature.

5. T-stability of N v
1 iteration algorithm

Now, we prove the stability of Nv
1 .

Theorem 6. Suppose that there is a Banach space E, having subset C, which is nonempty
closed and convex. Also, let there be a contraction mapping T : C → C. Let {ζn}∞n=0 be
an iterative sequence generated by Nv

1 and with real sequence {σ0n}
∞
n=0 ∈ [0, 1] such that∑∞

n=0 σ
0
n =∞. Then the iteration algorithm defined as Nv

1 is T − stable.

Proof. Let an arbitrary sequence {tn}∞n=0 ⊂ E in C, generated by Nv
1 and now defined

as ζn+1 = f(T, ζn) converges to a fixed point xδ (by Theorem 1) and εn = ‖tn+1−f(T, tn)‖.
We will prove that limn→∞ tn = p.

Let limn→∞ εn = 0, as from Theorem 1 using the inequality

‖ζn+1 − xδ‖ ≤ ξ3(ξ − (1− ξ)σ0n)‖ζn − xδ‖ (-14)

we have,

‖tn+1 − xδ‖ ≤ ‖tn+1 − f(T, tn)‖+ |f(T, tn)− xδ‖

= εn +

∥∥∥∥∥T
(
T (T ((1− σ0n)ζn)) + σ0nTζn

)
− xδ

∥∥∥∥∥



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1124

≤ ξ3(1− (1− ξ)σ0n)‖tn − xδ‖+ εn.

Define ψn = ‖tn − xδ‖, φn = (1− ξ)σ0n ∈ (0, 1) and ϕ = εn, which implies that ϕn

φn
→ 0 as

n→∞. Thus all the conditions of Lemma 2 are satisfied by above inequality. Hence, we
get limn→∞ tn = p, we have

εn = ‖tn+1 − f(T, tn)‖
≤ ‖tn+1 − xδ‖ − ‖f(T, tn)− xδ‖
≤ ξ3(ξ − (1− ξ)σ0n)‖tn − xδ‖+ εn

This implies that limn→∞ tn = 0. This also implies that Nv
1 is T − stable with respect to

T .

6. Data Dependence Result

In this section we establish some data dependence result.

Theorem 7. Let T̃ be an approximate operator of a contraction mapping T . Let {ζn}∞n=0

be an iterative sequence defines as Nv
1 for T and defined an iterative sequence {ζ̃n}

∞
n=0 for

T̃ , as follows constructed as, for arbitrary ζ̃0 ∈ X by
θ̃n = T̃ ((1− σ0n)ζ̃n + σ0nT̃ ζ̃n)

η̃n = T̃ θ̃n

ζ̃n+1 = T̃ η̃n n ∈ N

where real sequence {σ0n}
∞
n=0 in [0,1] satisfying 1

2 ≤ σ0n, for all n ∈ N and
∑
σ0n = ∞.

Also, if Tp = p and T̃ p̃ = p̃ such that limn→∞ ζ̃ = p̃, then we have

‖p− p̃‖ ≤ 11ε

1− ξ
.

Proof. Using {ζn}∞n=0 and {ζ̃n}
∞
n=0, we have

‖θn − θ̃n‖ =

∥∥∥∥∥T ((1− σ0n)ζn + σ0nTζn)− T̃ ((1− σ0n)ζ̃n + σ0nT̃ ζ̃n)

∥∥∥∥∥
≤

∥∥∥∥∥T
(

(1− σ0n)ζn + σ0nTζn

)
− T

(
(1− σ0n)ζ̃n + σ0nT̃ ζ̃n

)

+ T

(
(1− σ0n)ζ̃n + σ0nT̃ ζ̃n

)
− T̃

(
(1− σ0n)ζ̃n + σ0nT̃ ζ̃n

)∥∥∥∥∥
≤ ξ

(
(1− σ0n)‖ζn − ζ̃n‖+ σ0n‖Tζn − T̃ ζ̃n)‖

)
+ ε



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1125

≤ ξ

(
(1− σ0n)‖ζn − ζ̃n‖+ σ0n

(
‖Tζn − T ζ̃n‖+ ‖T ζ̃n − T̃ ζ̃n)‖

))
+ ε

≤ ξ

(
(1− σ0n)‖ζn − ζ̃n‖+ σ0n

(
ξ(‖ζn − ζ̃n‖) + ε+ ε

))
+ ε

≤ ξ(1− (1− ξ)σ0n)‖ζn − ζ̃n‖+ ξσ0nε+ ε

In similar manner, we have

‖ηn − η̃n‖ = ‖Tθn − T̃ θ̃n‖
‖ηn − η̃n‖ = ‖Tθn − T θ̃n + T θ̃n − T̃ θ̃n‖

≤ ‖Tθn − T θ̃n‖+ ‖T θ̃n − T̃ θ̃n‖
≤ ξ‖θn − θ̃‖+ ε

on substituting the value of ‖θn − θ̃n‖, we have

‖ηn − η̃n‖ ≤ ξ
(
ξ(1− (1− ξ)σ0n)‖ζn − ζ̃n‖+ ξσ0nε+ ε

)
+ ε

In a similar manner, we have

‖ζn+ − ζ̃n+1‖ = ‖Tηn − T̃ η̃n‖
‖ζn − ζ̃n‖ = ‖Tηn − T η̃n + T η̃n − T̃ η̃n‖

≤ ‖Tηn − T η̃n‖+ ‖T η̃n − T̃ η̃n‖
≤ ξ‖ηn − η̃‖+ ε

on substituting the value of ‖ηn − η̃n‖ , we have

‖ζn − ζ̃n‖ ≤ ξ
(
ξ(1− (1− ξ)σ0n)‖ζn − ζ̃n‖+ ξσ0nε+ ε

)
+ ε

using {σ0n}
∞
n=0 in [0,1] and ξ ∈ (0, 1) and combining the above inequalities of same theo-

rem,we have

‖ζn − ζ̃n‖ ≤ (1− (1− ξ)σ0n)‖ζn − ζ̃n‖+ σ0nε+ 5ε

≤ (1− (1− ξ)σ0n)‖ζn − ζ̃n‖+ σ0nε

+ 5(1− σ0n + σ0n)ε

‖ζn − ζ̃n‖ ≤ (1− (1− ξ)σ0n)‖ζn − ζ̃n‖+ σ0n(1− ξ) 11ε

1− ξ



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1126

Let ψn = ‖ζn − ζ̃n‖, φn = (1− σ0n)(1− ξ), ϕn = 11ε
1−ξ , then from the Lemma 2, we have

0 ≤ lim sup
n→∞

‖ζn − ζ̃n‖ ≤ lim sup
n→∞

11ε

1− ξ

considering the result of Theorem 1 we have lim supn→∞ ζn = p and by the assumption
we have that lim supn→∞ ζ̃n = p̃. Using the results together with

‖ζn − ζ̃n‖ ≤ (1− (1− ξ)σ0n)‖ζn − ζ̃n‖+ σ0n(1− ξ) 11ε

1− ξ

we have, ‖p− p̃n‖ ≤ 11ε
1−ξ as required.

7. An Application

Let a Banach space (E([a, b]), ||.||∞) which is space of all continuous real valued func-
tions on a closed interval [a, b] a with endowed chebyshev norm ‖x− y‖∞ = max

t∈[a,b]
|x(t)−

y(t)|. In this section solution of a particular delay differential equation has a solution
generated by Nv

1 iteration scheme.

x′(t) = f(t, x(t), x(t− τ)), t ∈ [t0, b] (7.1)

with initial condition

x(t) = ψ(t), t ∈ [t0 − τ, t0]. (7.2)

We opine that the following conditions are performed

(i) t0, b ∈ R, τ > 0;

(ii) f ∈ E([t0, b]× R2,R);

(iii) ψ ∈ E([t0 − τ, b],R);

(iv) if 2Lf (b− t0) < 1, there exist Lf > 0 such that

|f(t, u1, u2)− f(t, v1, v2)| ≤ Lf
2∑

n=0

|ui − vi|, (7.3)

∀ui, vi ∈ R, i = 1, 2, t ∈ [t0, b],



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1127

By a solution of the problem (7.1)-(7.2) we understand function x ∈ E([t0−τ, b],R)
⋂
E1([t0, b],R).

The problem (7.1)-(7.2) can be reformulated in the following form of integral

x(t) =

{
ψ(t), t ∈ [t0 − τ, t0]
ψ(t0) +

∫ t
t0
f(s, x(s), x(s− τ))ds, t ∈ [t0, b].

(7.4)

Theorem 8. Suppose that conditions (1)-(4) are satisfied. Then the problem (7.1)-(7.4)
has a unique solution in E([t0 − τ, b],R)

⋂
E1([t0, b],R).

Proof. Let {ζn}∞n=0 be an iterative sequence generative by Nv
K iteration method (1.18)

for the operator

Tx(t) =

{
ψ(t), t ∈ [t0 − τ, t0]
ψ(t0) +

∫ t
t0
f(s, x(s), x(s− τ))ds, t ∈ [t0, b].

(7.5)

Let xδ denote the fixed point of T . We will show that ζn → xδ as n→∞.
For t ∈ [t0 − τ, t0], it is easy to see that ζn → xδ as n→∞. For t ∈ [t0, b] we obtain

‖θn − xδ‖∞ = ‖T ((1− σ0n)ζn + σ0nTζn)− xδ‖∞
= ‖T ((1− σ0n)ζn + σ0nTζn)− Txδ‖
≤ ‖(1− σ0n)ζn + σ0nTζn − xδ‖
≤ (1− σ0n)‖ζn − xδ‖∞ + σ0n max

t∈[t0−τ,b]
|Tζn − Txδ|

= (1− σ0n)‖ζn − xδ‖∞ + σ0n max
t∈[t0−τ,b]

∣∣∣∣ψ(t0) +

∫ t

t0

f(s, x(s), x(s− τ))ds

− ψ(t0)−
∫ t

t0

f(s, xδ(s), xδ(s− τ))ds

∣∣∣∣
= (1− σ0n)‖ζn − xδ‖∞ + σ0n max

t∈[t0−τ,b]

∣∣∣∣ψ(t0) +

∫ t

t0

f(s, x(s), x(s− τ))ds

− ψ(t0)−
∫ t

t0

f(s, xδ(s), xδ(s− τ))ds

∣∣∣∣
= (1− σ0n)‖ζn − xδ‖∞ + σ0n max

t∈[t0−τ,b]

∣∣∣∣ ∫ t

t0

f(s, x(s), x(s− τ))ds

−
∫ t

t0

f(s, xδ(s), xδ(s− τ))ds

∣∣∣∣
= (1− σ0n)‖ζn − xδ‖∞ + σ0n max

t∈[t0−τ,b]

∫ t

t0

∣∣∣∣f(s, x(s), x(s− τ))

− f(s, xδ(s), xδ(s− τ))ds

∣∣∣∣



L.N Mishra et al. / Eur. J. Pure Appl. Math, 13 (5) (2020), 1110-1130 1128

= (1− σ0n)‖ζn − xδ‖∞ + σ0n max
t∈[t0−τ,b]

∫ t

t0

Lf

(
|ζn(s)− xδ(s)|

+ |ζn(s− τ)− xδ(s− τ)|
)
ds

= (1− σ0n)‖ζn − xδ‖∞ + σ0n max
t∈[t0−τ,b]

∫ t

t0

Lf

(
|ζn(s)− xδ(s)|+ |ζn(s− τ)

− xδ(s− τ)|
)
ds

= (1− σ0n)‖ζn − xδ‖∞ + σ0nLf

(
max

t∈[t0−τ,b]
|ζn(s)− xδ(s)|+ max

t∈[t0−τ,b]
|ζn(s− τ)

− xδ(s− τ)|
)∫ t

t0

ds

= (1− σ0n)‖ζn − xδ‖∞ + 2σ0nLf (b− t0)‖ζn − xδ‖

‖θn − xδ‖∞ =

[
1− σ0n(1− 2Lf (b− t0))

]
‖ζn − xδ‖ (7.6)

similarly, we have

‖ηn − xδ‖∞ = ‖Tθn − Txδ‖∞

= max
t∈[t0−τ,b]

∣∣∣∣ ∫ t

t0

f(s, θn(s), θn(s− τ))− f(s, xδ(s), xδ(s− τ))]ds

∣∣∣∣
≤ max

t∈[t0−τ,b]

∫ t

t0

∣∣∣∣f(s, θn(s), θn(s− τ))− f(s, xδ(s), xδ(s− τ))

∣∣∣∣ds
= max

t∈[t0−τ,b]

∫ t

t0

Lf

(
|θn(s)− xδ(s)|+ |θn(s− τ)− xδ(s− τ)|

)
ds

‖ηn − xδ‖∞ ≤ 2Lf (b− t0)‖θn − xδ‖∞ (7.7)

and hence, we have

‖ζn+1 − xδ‖∞ = ‖Tηn − Txδ‖∞

= max
t∈[t0−τ,b]

∣∣∣∣ ∫ t

t0

f(s, ηn(s), ηn(s− τ))− f(s, xδ(s), xδ(s− τ))]ds

∣∣∣∣
≤ max

t∈[t0−τ,b]

∫ t

t0

∣∣∣∣f(s, ηn(s), ηn(s− τ))− f(s, xδ(s), xδ(s− τ))

∣∣∣∣ds
= max

t∈[t0−τ,b]

∫ t

t0

Lf

(
|ηn(s)− xδ(s)|+ |ηn(s− τ)− xδ(s− τ)|

)
ds

‖ζn − xδ‖∞ ≤ 2Lf (b− t0)‖ηn − xδ‖∞ (7.7)

using the equations (7.6), (7.7) and (7.8)

‖ζn+1 − xδ‖∞ ≤ 4L2
f (b− t0)2‖ηn − xδ‖∞



REFERENCES 1129

‖ζn+1 − xδ‖∞ ≤ 4L2
f (b− t0)2

[
1− σ0n(1− 2Lf (b− t0))

]
‖ζn − xδ‖

Proceeding in the same manner, we have

‖ζn − xδ‖∞ ≤
[
1− σ0n−1(1− 2Lf (b− t0))

]
‖ζn−1 − xδ‖∞

and

‖ζn−1 − xδ‖∞ ≤
[
1− σ0n−2(1− 2Lf (b− t0))

]
‖ζn−2 − xδ‖∞

and hence we have

‖ζn+1 − xδ‖∞ ≤
n∏
k=0

[
1− σ0k(1− 2Lf (b− t0))

]
‖ζ0 − xδ‖∞ (7.9)

where [1−σ0k(1−2Lf (b− t0)) ∈ (0, 1) because σ0k ∈ (0, 1), for all natural numbers n. Also,
since (1− x) ≤ e−x for all x ∈ [0, 1], from (7.9) we can easily conclude that

‖ζn+1 − xδ‖∞ ≤
‖ζ0 − xδ‖

e(1−(2Lf (b−t0)))
∑∞

k=0 σ
0
k

(7.10)

which led us to limn→∞ ‖ζn+1 − xδ‖∞ = 0 when taking limits of both sides of equation
(7.10).

8. Conclusion

A whole new iteration scheme namely Nv
1 having rate of convergence, faster than

almost all pre-existing iteration schemes to find the solution with minimum possible steps
is established.

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