American Journal of Computational Mathematics, 2012, 2, 345-357
http://dx.doi.org/10.4236/ajcm.2012.24048 Published Online December 2012 (http://www.SciRP.org/journal/ajcm)
Strong Convergence of a New Three Step Iterative
Scheme in Banach Spaces
Renu Chugh, Vivek Kumar*, Sanjay Kumar
Department of Mathematics, Maharshi Dayanand University, Rohtak, India
Received August 11, 2012; revised October 12, 2012; accepted November 3, 2012
ABSTRACT
In this paper, we suggest a new type of three step iterative scheme called the CR iterative scheme and study the strong
convergence of this iterative scheme for a certain class of quasi-contractive operators in Banach spaces. We show that
for the aforementioned class of operators, the CR iterative scheme is equivalent to and faster than Picard, Mann,
Ishikawa, Agarwal et al., Noor and SP iterative schemes. Moreover, we also present various numerical examples using
computer programming in C++ for the CR iterative scheme to compare it with the other above mentioned iterative
schemes. Our results show that as far as the rate of convergence is concerned 1) for increasing functions the CR itera-
tive scheme is best, while for decreasing functions the SP iterative scheme is best; 2) CR iterative scheme is best for a
certain class of quasi-contractive operators.
Keywords: Fixed Point; Various Iterative Schemes; Rate of Convergence
1. Introduction
There is a close relationship between the problem of
solving a nonlinear equation and that of approximating
fixed points of a corresponding contractive type operator.
Consequently, there is theoretical and practical interest in
approximating fixed points of various contractive type
operators. Let

,

X
d
,
be a complete metric space and
a self map for X. Suppose that :TX X
 
F
TpXTpp is the set of fixed points of T.
There are several iterative processes in the literature for
which the fixed points of operators have been approxi-
mated over the years by various authors.
In a complete metric space, the Picard iterative scheme
defined by 
0,
nn
x

1,0,1, nn
xTxn
 (1.1)
has been employed to approximate the fixed points of
mappings satisfying the inequality




,dTxTy dxy  , (1.2)
for all ,
x
yX and


0,1  .
Condition (1.2) is called the Banach’s contraction con-
dition. Any operator satisfying (1.2) is called a strict con-
traction.
In 1953, W. R. Mann defined the Mann iterative sche-
me [1] as


11, nnnn
uu 
 
where


n

is a sequence of positive numbers in [0,1].
In 1974, S. Ishikawa defined the Ishikawa iterative
scheme [2] as


11 nnnnn
s
sT 
 t


1
nnnn
ts   ,
n
Ts (1.4)
where


n

and


n

are sequences of positive num-
bers in [0,1].
In 2007, Agarwal et al. defined the Agarwal et al. ite-
rative scheme [3] as


11 nnnnn
s
Ts Tt 



1
nnnn
ts   ,
n
Ts (1.5)
where


n

and


n

are sequences of positive num-
bers in [0,1].
In 2000, M. A. Noor defined the Noor iterative scheme
[4] as


11 nnnn
pp 
 n
Tq


1
nnnn
qp   n Tr


1
nnnn
rp   ,
n
Tp (1.6)
where


n

,


n

and


n

are sequences of posi-
tive numbers in [0,1].
n
Tu (1.3) Recently, Phuengrattana and Suantai defined the SP
iteration scheme [5] as
*Corresponding author.
C
opyright © 2012 SciRes. AJCM

R. CHUGH ET AL. 346


11 nnnnn
x
yT 
 y


1
nnnn
yz  n Tz


1
nnnn
zx   ,
n
Tx (1.7)
where


n

,


n

and

n

are sequences of posi-
tive numbers in [0,1].
Remarks:
1) If , then the Noor iterative scheme (1.6) re-
duces to the Ishikawa iterative scheme (1.4).
0
n
 
2) If , then the Noor iterative scheme (1.6)
reduces to the Mann iterative scheme (1.3).
0
nn
 
3) In addition, when , then the SP iterative
scheme (1.7) reduces to the Mann iterative scheme (1.3).
0
nn
 
In 1972, Zamfirescu [6] obtained the following inter-
esting fixed point theorem.
Theorem 1.1. Let (X, d) be a complete metric space
and a mapping for which there exists real :TX X
numbers a, b and c satisfying 
1
0,1,,0, 2 abc




such that for each pair ,
x
yX, at least one of the fol-
lowing conditions hold

1) ,,dTxTy adxy







2) ,,,dTxTybdxTx dyTy
 






3),,, .dTxTycdxTy dyTx
 
(1.8)
Then T has a fixed point p and the Picard iteration
defined by

0
nn
x

1, 0,1, nn
xTxn

converges to p for any arbitrary but fixed 0
x
X.
The operators satisfying the condition (1.8) are called
Zamfirescu operators.
Berinde [7] introduced a new class of operators on an
arbitrary Banach space satisfying






,2,,dTxTydxTx dxy  , (1.9)
,
x
yX and some .


0, 1  
He proved that this class is wider than the class of
Zamfirescu operators and used the Ishikawa iteration
process to approximate fixed points of this class of ope-
rators in an arbitrary Banach space given in the form of
following:
Theorem 1.2 [7]. Let K be a nonempty closed convex
subset of an arbitrary Banach space X and
be a mapping satisfying (1.9). Let {sn} be defined
through the Ishikawa iteration (1.4) and 0
:TK K
s
K where


n

,


n

are sequences of positive numbers in [0,1]
with


n

satisfying
0
n
n





. Then {sn} converges
strongly to the fixed point of T.
Several authors [5,8-17] have studied the equivalence
between various iterative schemes. S. M. Solutz [15,16]
proved that for quasi-contractive operators the itérative
processes Picard, Mann, Ishikawa and Noor are équi-
valent. Recently, Renu Chugh and Vivek Kumar [17]
proved that for quasi-contractive operators satisfying (1.9)
Picard, Mann, Ishikawa, Noor and SP iterative schemes
are equivalent.
Fixed point iterative procedures are designed to be ap-
plied in solving equations arising in physical formulation
but there is no systematic study of numerical aspects of
these iterative procedures. In computational mathematics,
it is of vital interest to know which of the given iterative
procedures converge faster to a desired solution, com-
monly known as rate of convergence. B. E. Rhoades [18]
compared the Mann and Ishikawa iterative procedures
concerning their rate of convergence. He illustrated the
difference in the rate of convergence for increasing and
decreasing functions. Indeed he used computer programs,
perhaps for the first time to compare the Mann and Ishi-
kawa iterations through examples. S. L. Singh [19] ex-
tended the work of Rhoades. Very recently, Phuengrat-
tana and Suantai [5] proved that SP iterative scheme is
equivalent to and faster than Mann, Ishikawa and Noor
iterative schemes for increasing functions.
Now, we introduce the following CR iterative process:
Let X be a Banach space, a self map of X :TX X
and 0
x
X

. Define the sequence by

0
nn
x





11
1
1,
nnnn
nnnn
nnnnn
n
n
x
yT
yTx
zxT



 
 
 
y
Tz
x
(1.10)
where


n

,


n

and


n

are sequences of posi-
tive numbers in [0,1] with


n

satisfying
0
n
n





.
We shall need the following lemma and definitions.
Lemma 1 [20]. If

is a real number such that
01


 and

0
nn


 is a sequence of positive num-
bers such that lim 0
n
n

, then for any sequence of posi-
tive numbers

0
nn


 satisfying
1,0,1,2, nnn
uu n 
 ,
we have lim 0
n
nu


.
Definition 1.1. Suppose that {an} and {bn} are two
real convergent sequences with limits a and b respec-
tively. Then {an} is said to converge faster than {bn} if
lim 0
n
nn
aa
bb



.

R. CHUGH ET AL. 347
Definition 1.2 [21]. Suppose that {un} and {vn} are
two fixed pointeration procedures both converging to
the sam
t i
e fixed point p with the error estimates
,0,1,
nn
upan 
,0,1,
nn
vpbn 
where {an} and {bn} are two real convergent sequences
converging to 0. If {an} converges faster than {bn}, then
we say {} convefaster than v to p.
rwal et al., SP
an
x subset
a map-
unrges n
The purpose of this paper is to show the convergence
of the CR iterative scheme and to prove equivalence be-
tween Picard, Mann, Ishikawa, Noor, Aga
d CR iterative schemes for quasi-contractive operators
satisfying (1.9). We provide an example for which the
CR iterative scheme is faster than the other above men-
tioned iterative schemes for the aforementioned class of
operators. Also, by using computer programs in C++, we
compare the above mentioned iterative schemes through
examples of increasing and decreasing functions.
2. Result on Strong Convergence
Theorem 2.1. Let K be a nonempty closed conve
of an arbitrary Banach space X and :TK

K
ping satisfying (1.9). Let

0
nn
xbe defined through the
CR iteration (1.10) and 0
x
X

, where


n

,


n

are sequences of positivers in [0,1] with numbe


n

Satisfying n


. Then convetro
0n
to the fixed point of T .
rges sngly
heorem 1hat T
K p

0
nn
x

Proof. T.1 shows t has a unique fixed
point in , say .
From (1.10), we have



1
11
1
,
n
nn
nn
nnn
nn n
nn
xp
yp
Ty p
Tx p
Tz p
Ty p








 
 

(2.1)
Using (1.9), (2.1) yields
1



1
1
1
n
nn
nnn
nn
xp
1
n




x
p
zp
yp




 
 

(2.2)
Using (1.9) and (1.10), (2.2) yields
 








 
1
2
2
2
3
11
11
1
11
11
1
1
1
11 11
1
n
nnn
nnnn
nnn n
nn n
nnn
nnnn
nnnn
nnn
nnn n
nnnn
nnnn
nnn n
xp
xp
xp
Tx p
Tz p
xp
xp
xp
xp
xp
xp
xp
 
 

 
 
 

 
 

1
nn n
Tx p  

 



 

 

 

 



 
 
(2.3)
 


0
0
0
1
0
11
ek
k
n
n
nk
k
n
x
pxp
xp






 





(2.4)
Since 0   < 1, 
k  [0,1] and
0
n
n






t follow
, so
as Hence, i

0
1
e0
k
k
n




 
(2.4) that
n. s from
. Therefore

0
nn
x

1
lim 0
n
nxp



 con-
verges strongly to p.
3.ann,
Ishikawa, Noor, Agarwal et al., SP and CR
Iterative Schemes
. Let K be a nononvex
X and a mapping
itial poe,
Equivalence between Picard, M
Theorem 3.1empty closed csubset
of a Banach space
satisfying (1.9). If the in
:TK K
int is the sam
0,
n
A
nN  , then the followings are equiv
1) The Mann iteration (1.3) converges to p.
2) The Agarwal et al. iteration (1.5) converges to p.
3) The CR iteration (1.10
alent:
) converges to p.
shall show
Using (1.3) and (
Proof. First we prove that 1)  3). Let the Mann it-
eration (1.3) converge to p. We that the CR
iteration (1.10) also converges to p.
1.10), we have


 
11nn
1nnn nnn yuTy Tu    
12
11 2 .
nnnnnnnnn
yu yu uTu
yu uTu


 
  
 
(3.1)
xu
nn
nnnn


R. CHUGH ET AL. 348
Using as and Lemma 1.1, (3.7)
yields
From (1.10), we have

 



 
1
11
121
2
121
,
nn nnnn
nn nnnn
nn nnn
nnnnnn
n nnn
nnn nnn
nnnn n
nn n
Tx uTz u
Tx TuTu u
Tz TuTu u
1nn
nn
yu
x
uTuu
z u
Tu uTu u
xu uTu
zu





 


  



 

 

(3.2)
Again, from (1.10), we have
Tu u   




  
1
1
11 21
nn
nnn nnn
nnn nnnnn
nnn nn
zu
xu Txu
.
n
x
uTxTuTuu
x
uTu


 

 
 

u
(3.3)
Substituting (3.3) in (3.2), we have

 
 
1yu x   
11
12 1 .
nn nnn
nn nn
nnn n
u
xu
Tx u
  
 
  

(3.4)
Substituting (3.4) in (3.1), we obtain
     
   

11
11 1
nn
11 121
2.
nn nnn
nnn
nn n
xu
x
u
uTu

 




 


(3.5)
Also,


 

uTuu pp
1.
nnn n
n
Tu
up 

  (3.6)
Substituting (3.6) in (3.5), we have




 



11
11 1
11 1212
nn
nnnnn
nnn n
n
xu
xu
up
hx uup

1
nn n




 

 



(3.7)
where 1(using

   
11 1
nnn
h   
0,
n
A
nN   ) and






11212l

   

.
n
up n
0
nn
xu

 as
In addition
n.
nnnn
x
pxu up


and this implies that n
x
p
ove that
as
Conversely, we pr
n.
n
x
p impn up. lies
.3), (1.9) and (1.1Using (10 ), we have



11
1
2
112 .
nn
nnn nnn
n n
nn n
nnnnn
xu
y uTy Tu
yu
yTy
y yTy



 



 
 

, we have
1nn nn
yu 

(3.8)
n
u
 
From (1.10)

 









2
11 1
11
11
1
11
nn
n
nnn nn
nn nn n
nn nn nnn
n nn
nnn n
nnn
nn n
nn nn
yu
Tzxxu
Txpxpx
Tzpxpxu
xuxpzp
xu xp
xp
xp
xu x

 

 

 

 


  
 
 
 
 

 
1nn
nnn
Tx u Tz   
11
nn n nnn
Tx xx u    
nn
u
nn n
u 
.
np
(3.9)
Also,







  
1
11 1
11 1
11 1,
nn
nn
n
nn nn
nn nnnn
nn n
yTy
ypTyp
yp
x
pzp
x
p
xp

 

 

 
 
 


 
(3.10)
Substituting (3.9) and (3.10) in (3.8) and rearranging
the terms, we have

R. CHUGH ET AL. 349


   

  




11
2
2
11
11 1(1
2111
11
121 .
nn
nnn
nnn
nnn
nnn
n
xu
xu
n
x
p
xu
xp


 

 


 
 

 



(3.11)
Since 0,
n
A
nN   

1 1,.nN    , so Also 01n n
x
p as n.
Hence, using Lemma 1.1, (3.11) yields 0
nn
xu
as
n.
In addition nnnn
up xuxp  and this im-
plies that as Hence the result.
Next, that 1.
Let as . Using (1.3) and (1.5), we
have
n
up
we show
n.
) 2)
n
up n


 



11
1
12
1
.
nn
nnnnnn
nnnnnnnn n
nnn
nnnn nnn
su
Ts uTt Tu
Ts ut uu Tu
Ts T
Tu






 
  
 
 

win
1n u
11
12
nn nn
Tu
su u
 

u

(3.12)
imates: Now, we have the follog est

1
nn nn
u pTu
n
uTu p
up


  (3.13)

2
21.
nn nnnn
nnn
Ts Tus uu Tu
s
uu



  p
(3.14)
It follows using (3.12), (3.13) and (3.14) that


  




 


11
(1 11
21 11
21
11 .
141
nn
nnnnnn
nn n
nn
nnn n
n
su
11 1
nn n
s
u
up
up
su
up

 
 





 

 
 
(3.15)
Since and as , hence us-
in
up     


0, 1  n
g Lemma 1.1, (3.15) yields
up n
0
nn
su as n.
.
nnnn
s
psu up (3.16)
and this implies that n
s
p
ove that
as
Conversely, we pr
n.
n
s
p implies
Using (1.3), (1.5) and (1.9
n
up.
), we have



11
1
1
2.
nn
nnnnnn
nnnnn
nn nnnn
su
Ts uTt Tu
Tsss u
tu tTt


 


 
  
 
(3.17)
e have twing etes: Now, whe follostima
In addition


1
nn n
s
Ts s  p

 )  (3.18


  
1
111
nn n
nn
tTtt p
s
p
 

 

(3.19)





1
nn nnnnn
tusTsu
1nn n n

1.
n
n n
nn n
nn nn
u
s
u    Tsu
su
su sp





 
(3.20)
It follows from (3.17), (3.18), (3.19) and (3.20
  
) that
 

 


   
11
11 21
11 113
nn
nnn
n
nnn n
su
sp
11
nn n
su
1nn
.
s
us
 
 


 

 
(3.21)
If
p
  
   
0,
n
A
nN   , then


01 11
n
ad .
Also, n
s
p as n
Hence, using Lemma 1.1,
.
(3.21) yields 0us
nn


as n.
In addition
nnnn
upussp


and this implies that as Hence the
result.
Keeping in mind Soltuz’s results [15,16] as well as
Chugh and Kumar’s result [17], Theorem 3.1 leads to the
following corollary:
Corollary 3.2. Let K be a nonempty closed con
subset of a Banach space X and T:
n
up n.
vex
K
K
oint is
a ma
satisfying (1.9). If the initial p the
pping
same,
0,
n
A
nN  
he Picae
he followi
1) Trd it (
, then tuivalent:
ration1.1) coo the fixed
po
) The Mann iteration (1.3) converges to p,
awa iteration (1.4) converges to p,
ngs are eq
nverges t
int p of T,
2
3) The Ishik

R. CHUGH ET AL. 350
4) The Noor iteration (1.6) converges to ,
) conve
.7) conv to p,
ai-Contractive Operators
eration
(1.9). In [22], Qin
sh
ertain quasi-contractive operator. Ciric, Lee
anowed that
Nd Ishi-
kaactive ope-
ra an
exor which the
ite faster than
Mann and Ishikawa iterative schemes.
ter than
Pi -
p
5) The Agarwal et al. iteration (1.5rges to p,
6) The SP iteration (1erges
7) The CR iteration (1.10) converges to p.
4. Results on Fastness of CR Iterative
Scheme for Qus
In [21] Berinde showed that Picard itis faster than
Mann iteration for quasi-contractive operators satisfying
g and Rhoades by taking example
owed that Ishikawa iteration is faster than Mann ite-
ration for a c
d Rafiq [23], by providing an example, sh
oor iterative scheme can be faster than Mann an
wa iterative schemes for some quasi-contr
tor. Recently, Nawab Hussian et al. [24] provide
ample of a quasi-contractive operator f
rative scheme (1.5) due to Agarwal et al. is
Now, we show that the CR iteration is fas
card, Agarwal et al., Noor and SP iterations for quasi
contractive operators satisfying (1.9) as follows:
1) By providing an example 4.1 of a quasi-contractive
operator satisfying (1.9), we show that CR iterative
scheme is faster than Agarwal et al., Noor and SP ite-
rative schemes.
2) By using definition (1.1), we show that CR iterative
scheme is faster than Picard iteration.
1) Example 4.1. Let T:




:0,1 0,1T be defined
by

2
x
Tx, 0
nnn
 , 1, 2,,15n,
4,16
nnn n
n
  
. It is clear that T is a quasi-
contractive operator satisfying (1.9) with a unique fixed
point 0. Also, it is easy to see that T, ,and
nn n


satisfies all the conditions of Theorem 2.1. We show that
C
e scheme.
R iterative scheme is faster than Agarwal et al., Noor
and SP iterative schemes.
Proof. First of all we show that CR iterative scheme is
faster than Noor iterativ
Let 16n and p0 = x0. Then, from [23], for Noor it-
eration (1.6), we have
1
16
248
1.
n
ni0
p
p
iii





 (4.1)
Also, for CR iteration (1
i
.10), we have
10
16
114 8.
2
n
ni
x
x
i
iii





 (4.2)
So,
0
16
114 8
2
n
i
x
x





1
124 8
1
n
n
n
p




 
0
3
2
3
16 2
23 2
1.
28 416
n
i
i
ii i
p
ii
iii



16ii
ii i










 

















It is easy to see that
3
2
3
16 2
16
232
0lim1
28 416
115
lim 1lim0.
n
ni
n
nn
i
ii
iii
in
 
 






 


 




















1
1
lim 0
n
nn
x
p

 
. Hence, we have
Therefore, by definition 1.2, CR iterative scheme con-
verges faster than Noor iterative scheme to the fixed
point 0 of T.
that CR iterative scheme is faster
than SP iterative scheme:
For SP iteration (1.7) we have
Secondly, we show
10
16
612 8
1.
n
ni
x
x
i
iii






So,


16 1
1SP
n
x
i

16
3
2
3
16 2
114 8
CR 2
612 8
1
2
10 3232
1.
2241216
n
i n
n
i
n
i
xi
iii
iii
i
iii
i
iii
ii i
iii
 










16
161
2 8
1
i
151616
n





















 

 


 













It is easy to see that


R. CHUGH ET AL. 351
3
2
3
16 2
16
10 3232
0lim1
2241216
1 1015
lim 1lim0.
1000
n
ni
n
nn
i
ii i
iii
in
 
 



 
 





















Hence, we have


1
1
CR
lim 0
SP
n
nn
x
x

 
. Therefore, by
definition 1.2, CR iterative scheme converges faster than
SP iterative scheme to the fixed point 0 of T.
Next, we show that CR iterative scheme is faster than
Agarwal et al. iterative scheme.
For Agarwal et al. iteration (1.5), we have
1?0n
16
14 .
2
n
i
s
s
i






So,

0
16
1
1
0
16
16
114 8
2
14
2
18
1
14 .
n
i
nn
n
i
n
i
2
x
xi
iii
ss
i
iii

























i




It is easy to see that
16 16
18
nn



  1
0lim1lim1
14
2
15
lim 0.
nn
ii
n
iii
i
i
n
 





 








Hence, we have 1
1
lim 0
n
nn
x
s

 
. Therefore, again by
definition 1.2, CR iterative scheme convergeer than
Agarwal et al. iterative scheme to the fixed point 0 of T.
2) Here we show that CR iteration is faster than Picard
iteration.
Using Picard iteration (1.1) and condition (1.9) we
have
s fast
1
10
.
n
n
x

 

 
1
11
0
11 11
1
,
nn n
nnnn
nn
n
xp
xp
qx pqx p
nn

 





 

 
(4.4)
where

  

 
11 11
11
nn
nnn
qnn



 
 .

(if 0,anN).
n
In order to compare CR and Picard iterations, we must
compare the coefficients of the inequalities (4.4) and
(4.3).
Obviously
11nn
q 

10 n

 as n and hence us-
ing definition (1.1), we can say that CR iteration is faster
than Picard iteration.
Keeping in mind results of example 4.1
Ce conclude that CR iterative
scerative schemes for a certain
class of quasi-contractive operators.
5. Applications
In this section, with the help of computer programs in
compare the rate of convergence of Picard,
Mann, Ishikawa, Noor, Agarwal et al., SP and CR ite-
ration procedures, through examples. The ouome is
listed in the form of Tables 1-4, by taking initial appro-
ation x0 = 0.8d
as well as
iric et al.’s results [23], w
heme is faster than other it
C++, we
tc
xim an

1
2
1n
iterative schemes.
1
nn n
ab g , for all
5.1. Example of Decreasin
Let f: [0,1]→[0,1] be defined by
g Function
m
 
1
f
xx ,
7, 8,m

. Then f is a decreasing function. By taking
m = 7, the comparison of convergence of above men-
tio
tion
e defined as
ned iterative schemes to the exact fixed point p =
0.188348 is listed in the Table 1.
5.2. Example of Increasing Func
pxp 

 (4.3)
Also, from (2.3) we have
Let f : [0,8]→[0,8] b

29x
fx

10
. Then f
equation x3 + x2 − 1 =
to find fixed point of the function
is an increasing function. The comparison of conver-
gence of above mentioned iterative schemes to the exact
fixed point p = 1 is listed in the Table 2.
5.3. Example of Cubic Equation
To find solution of cubic 0 means

12
3
1x as x3 + x2

R. CHUGH ET AL. 352
Table 1. Decreasing function.

R. CHUGH ET AL. 353
Table 2. Increasing function.
CR iteration SP iteration Noor iteration Picard iteration Mann iteration Ishikawa iteration Agarwal et al.
iteration
n fxn x
n+1 fxn x n+1 fxn x n+1 n fxn x n+1 fxn x n+1 n xn+1 fxn
0 0.964 0.998591 0.964 0.998591 0.964 0.9985910.9 0.9 0.964 0.964 0.964 0.99293 0.964 0.99293
- - - - - - - - - - - - - - -
4 1 1 0.999999 0.999999 0.999974 0.9999220.999850.999850.9947140.9828280.999968 0.999937 0.9999960.999997
5 1 1 1 1 0.999984 0.999950.999970.999970.9965950.9884490.999987 0.999973 0.9999990.999999
6 1 1 1 1 0.99999 0.9999660.9999940.9999940.9977030.9919470.999968 0.999937 1 1
7 1 1 1 1 0.999993 0.9999760.9999990.9999990.9983960.9942270.999995 0.999988 1 1
8 1 1 1 1 0.999995 0.9999831 1 0.9988490.9957670.999998 0.999994 1 1
- - - - - - - - - - - - - - -
23 1 1 1 1 1 1 1 1 0.9999640.999850.999999 0.999997 1 1
24 1 1 1 1 1 1 1 1 0.999970.9998740.999999 0.999998 1 1
- - - - - - - - - - - - - - -
33 1 1 1 1 1 1 1 1 0.9999940.9999741 1 1 1
- - - - - - - - - - - - - - -
68 1 1 1 1 1 1 1 1 1 1 1 1 1 1
69 1 1 1 1 1 1 1 1 1 1 1 1 1 1
− 1 = 0 can be rewritten as

12
3
1
x
x. The com-
parison of convergence of above mentioned iterative
schemes to the exact fixed point p = 0.754878 of

12
3
1x is listed in the Table 3.
5.4. Example of Goat Problem
A farmer has a fenced circular pasture of radius a and
wants to tie a goat to the fence with a rope of length b so
as to allow the goat to graze half the pasture. How long
should the rope be to accomplish this?
The length of the rope “b” must be longer than “a” and
shorter than 2a, i.e. 2ab a.
If we let b xa , we get the simplified equation

21 22
42 sin4ππ
2
x
xxx



 x
 and we are
looking for the solution x, with 12x
produce a sequence that
. If we rear-
range the equation, we can will
converge to the solution:

222
ππ 442sin
2
1
x
xxx x

 

or

221
π442sin
2
π
x
xx x
x



.
11x

, Let

221
1
π442sin
2
π
n
nnn
n
x
xx x
x



 




221
π442sin
2.
π
n
nnn x
xx x
fx

 

 and
The comparison of convergence of above mentioned
iterative schemes to the exact fixed point 1.15863 of the
function f(x) is listed in the Table 4. So the rope length
“b” should be approximately 1.15863 a.
For detailed study, these programs are again executed
after changing the parameters and the readings are re-
corded (discussed in the next section).
6. Observations
6.1. Decreasing Function (1 − x)m
1) For m = 8 and xo = 0.8, the Picard scheme never
converges (oscillates between 0 and 1), the Mann scheme
converges in 9 iterations, the Ishikawa scheme converges
in 35 iterations, the Noor scheme converges in 10 itera-
tions, Agarwal et al. iteration does not converges, the CR
scheme converges in 9 iterations and the SP scheme
converges in 7 iterations.
2) For m = 30 and xo = 0.8, the Picard scheme never

R. CHUGH ET AL. 354
n n
Table 3. Cubic equation.
CR iteration SP iteratioNoor iteratioPicard iterationMann iteration Ishikawa iteration Agarwal et al.
iteration
n fxn x
n+1 fxn x 1 fxn x 1 fxn x 1 n+n+n+ fxn x n+1fxn x n+1fxn x n+1
0 0.69857 0.68185 0.698555857577 7 0.68184 0.6987 0.6818450.690.6980.69850.698570.698570.8118480.698570.811848
- - - - - - - - -
0.754709 04839
0.754872 04878 04878 0487055
0.754878 04878 04878 04870.755475 0.755263
- - -
0.754878 04878 04878 04870.37240.7240.7545920.75466
0.754878 04878 04878 0.4878 0.4877 0.48780.92170.92170.78 0.7548780.754097 0.755268 0.755124 0.755071
- - - - - -
0.754878 04878 04878 04878 0.4878 0.48780.22410.2410.754691 0.754722
13 0.754878 0.4878 04878 0.4878 0.4878 0.48780.22590.22590.78 0.7548780.754789 0.754925 0.755054 0.755027
- - - - - - -
0.754878 0.78 0.78 0.78 0.78 0.781 1 0.754878 0.7548780.754877 0.754878 0.759426 0.759083
0.754878 04878 04878 04878 488 0.48780.01130.1130.8780.7548780.754878 0.754878 0.750074 0.750431
0.754878 0.754878 0.754878 0.754878 0.754878 0.7548781 1 0.754878 0.7548780.754878 0.754878 0.759867 0.759516
-- - - - -
3 .7583 0.754992 0.754878 0.754385 0.7549630.82640.8264390.761920.7552480.723666 0.769363 0.7566140.75569
4 .75.75.758 0.754782 0.7549 20.65990.6599550.7544580.7548950.737969 0.762659 0.7539570.75435
5 .75.75.758 0.75485 0.7548860.8441340.8441340.7548580.754880.745922 0.759043
- - - - - - - - - - - -
8 .75.75.758 0.754876 0.754878595930.754878 0.7548780.753476 0.755566
9 .75.7575757588887548
- - -- - - - -
12 .75.75.757575484820.754878 0.7548780.754728 0.754956
75 .7575757594947548
- - - - - - -
27 75487548754875487548
28 .75.75.750.757500000754
29
converges (oscillates between 0 and 1), the Mann scheme
converges in 11 iterations, the Ishikawa scheme con-
ver 37 iterations, the Noor scheme converges in 12
, Agarwal et al. scheme never c
ges in
iterations onverges, the
s and the SP scheme
tions, Noor sche-
m
ergeR scheme converges in 10 itera-
tions a SP sch
4) Taking
CR scheme converges in 13 iteration
converges in 9 iterations.
3) Taking initial guess xo = 0.2 (nearer to the fixed
point), Picard scheme never converges (oscillates be-
tween 0 and 1), Mann scheme converges in 10 iterations,
Ishikawa scheme converges in 40 itera
e converges in 10 iterations, Agarwal et al. scheme
does not conv, the C
nd theeme converges in 8 iterations.

1
4
1
1
nnnn
 

and xo = 0.8, we
obtain that the Mann scheme converges in 23 iterations,
56 iterat
, the CR scheme converges in
11 iterations and the SP scheme converges in
tions.
6.2. Increasing Function
the Ishikawa scheme converges inion, the Noor
scheme converges in 21 iterations, Agarwal et al. scheme
converges in 12 iterations
15 itera-


2910x
1rg
poi
) For xo = 0.8, the Picard scheme convees to a fixed
nt in 8 iterations, the Mann scheme converges in 69
iterations, the Ishikawa scheme converges in 34 itera-
tions, the Noor scheme converges in 24 iterations, Agar-
wal et al. scheme converges in 7 iterations, the SP
scheme converges in 6 iterations and the CR scheme con-
verges in 5 iterations.
2) Taking initial guess xo = 0.6 (away from the fixed
), the Picard scheme converges to a fixed point in 8
ite
point
rations, the Mann scheme converges in 75 iterations,
the Ishikawa scheme converges in 38 iterations, the Noor
scheme converges in 27 iterations, Agarwal et al. scheme
converges in 6 iterations, the SP scheme converges in 7
iterations and the CR scheme converges in 5 iterations.
3) Taking,

1
4
1
1
nnn n
 

, xo = 0.8, we ob-
tain that the Mann sc
Ishikawa scheme con
heme converges in 23 iterations, the
verges in 12 iterations, Noor sche-
al. scheme con-
verges in 6 iterations, t
verges in 9 iterations, the Ishi-
ka
me converges in 9 iterations, Agarwal et
he SP scheme converges in 5 it-
erations and the CR scheme converges in 4 iterations.
6.3. Cubic Equation x3 + x2 –1 = 0
1) For xo = 0.8, the Picard scheme never converges to
the solution of cubic equation (oscillates between 0 and
1), the Mann scheme con
wa scheme converges in 29 iterations, the Noor sche-

R. CHUGH ET AL. 355
Table 4. Goat problem.
CR iteration SP iteration Noor iteration Picard iteration Mann iteration Ishikawa iteration Agarwal et al.
iteration
n fxn x
n+1 fxn x n+1 fxn x n+1 fxn x n+1 fxn x n+1 fxn x n+1 fxn x n+1
0 1.103589 1.150601 1.103589 1.150601 1.103589 1.1506011.1035891.1035891.1035891.1035891.103589 1.137884 1.1035891.137884
1 1.155492 1.1584 1.103589 1.150601 1.155492 1.1552751.1378841.1378841.1378841.1378841.150601 1.149315 1.1506011.15304
1.15863 1.15863 1.158626 1.158625 1.158512 1.1583991.1586011.1586011.1568521.1550111.158273 1.157929 1.1586271.158627
1.15863 1.15863 1.15863 1.15863 1.158574 1.1585151.1586291.1586291.1576931.1566611.158458 1.158278 1.158631.15863
1.15863 1.15863 1.15863 1.15863 1.1586 1.1585671.158631.158631.1580921.1574741.158538 1.158436 1.158631.15863
1.158631.3
1.158631.15863
15863 1.15863 1.15863 1.15863 1.158616 1.1585991.158631.158631.1583491.1580131.158585 1.158533 1.158631.15863
3
3
-
63
63
3
73
- - - - - - - - - - - - - - -
6 1.158629 1.158629 1.15861 1.158608 1.158414 1.1582291.158441.158441.1556931.1528471.157988 1.157426 1.1586031.158608
7 1.15863 1.15863 1.158622 1.15862 1.158472 1.1583281.1585561.1585561.1563661.1540911.158157 1.15772 1.158621.158621
8
9 1.15863 1.15863 1.158628 1.158628 1.158539 1.1584491.1586191.1586191.1572111.1557071.158355 1.15808 1.1586291.158629
10 1.15863 1.15863 1.15863 1.15863 1.158559 1.1584871.1586261.1586261.1574841.1562431.158414 1.158193 1.158631.15863
11
12 1.15863 1.15863 1.15863 1.15863 1.158585 1.1585371.158631.158631.1578571.1569931.158492 1.158344 1.158631.15863
13 1.15863 1.15863 1.15863 1.15863 1.158594 1.1585541.158631.158631.1579871.1572591.158518 1.158395 1.158631.15863
14
15 1.15863 1.15863 1.15863 1.15863 1.158606 1.15857815861.1581761.1576491.158554 1.158468 1.158631.15863
16 1.15863 1.15863 1.15863 1.15863 1.15861 1.1585861.158631.158631.1582451.1577941.158567 1.158494 1.158631.15863
17 1.15863 1.15863 1.15863 1.15863 1.158613 1.1585931.1583021.1579141.158577 1.158516 1.158631.15863
18 1.
- - - - - - - -
47 1.15863 1.15863 1.15863 1.15863 1.15863 1.158631.1586
48 1.15863 1.15863 1.15863 1.15863 1.15863 1.158631.1586
- - - - - - - -
6
- - - - - - -
1.158631.158621.1586071.158629 1.158627 1.158631.15863
1.158631.1586211.1586091.158629 1.158628 1.158631.15863
- -- - - - -
1.158631.1586271.1586231.15863 1.15863 1.158631.15863
- - - - - - -
1.158631.158631.158631.15863 1.15863 1.158631.15863
1.158631.158631.158631.15863 1.15863 1.158631.15863
0 1.15863 1.15863 1.15863 1.15863 1.15863 1.158631.158
- - - - - - - -
89 1.15863 1.15863 1.15863 1.15863 1.15863 1.158631.158
90 1.15863 1.15863 1.15863 1.15863 1.15863 1.158631.1586
me converges in 13 iterations, Agarwal et al. scheme
nees, the CR scheme co
tions and the SP scheme converges in 5 iterations.
ges in 10 iterations and the SP scheme converges
ver convergnverges in 6 itera-
2) Taking initial guess xo = 0.1 (away from the solu-
tion of cubic equation), Picard scheme never converges
(oscillates between 0 and 1), the Mann scheme converges
in 10 iterations, the Ishikawa scheme converges in 30
iterations, Noor scheme converges in 21 iterations,
Agarwal et al. scheme never converges, the CR scheme
conver
in 5 iterations.
3) Taking

1
4
1
nnn n
 

and xo = 0.8, we
1
ob Mann scheme converg
the Ishikawa scheme converges in 27 iterations, the Noor
s to a fixed
converges in 90
iterations, t
tain that thees in 14 iterations,
scheme converges in 9 iterations, Agarwal et al. scheme
converges in 10 iterations, the CR scheme converges in 8
iterations and the SP scheme converges in 8 iterations.
6.4. Goat Problem
1) For xo = 0.8, the Picard scheme converge
point in 13 iterations, the Mann scheme
he Ishikawa scheme converges in 61 itera-
tions, the Noor scheme converges in 48 iterations, Agar-
wal et al. scheme converges in 11 iterations, the SP
scheme converges in 11 iterations and the CR scheme

R. CHUGH ET AL. 356
converges in 8 iterations.
2) Taking initial guess x = 0.6 (away from the fixed
poinhemes to
iterations, the nn sce converges i 104 iterations
the Ishi se conerge71 iterations, tNoor
s
con
12 iterations and the CR scheme converges in 9 itera-
tions.
o
e convergt), the Picard sc a fixed point in 14
Ma
chem
hem
v
n,
kawa s in he
cheme converges in 57 iterations, Agarwal et al. scheme
verges in 12 iterations, the SP scheme converges in
3) Taking

1
4
1
1n
nnn
   o w
o
Ishi
me converges in 14 iterations, Agarwal et al. scheme
c
i
7. Conclusions
For d
m n
f
and x = 0.8,e
btain that Mann scheme converges in 29 iterations, the
kawa scheme converges in 18 iterations, Noor sche-
onverges in 10 iterations ,the SP scheme converges in 7
terations and the CR scheme converges in 7 iterations.
ecreasing functions, we conclude the followings:
1) Picard and Agarwal et al. schees do ot converge
or

1
2
n
1, ra
1
ne
i
M
faster t iteration
) Oincreing the valuef m, Mann, Iikawa
Noo r
iterations t c
e
schemeoor e shows anncreaswhile wa
schemwin iterations
co she.
T speedf iterative schees depds on
 te of convergence of the SP schme
s better than other iterative schemes, while CR and
ann schem
han Is
es show
hikawa
s equivalence. A
.
lso, Noor scheme is
2n as osh,
r, SP and CR schemes require more numbe of
o
For in
onverg
itial gu
e.
ess near3)er to th fixed point, Mann
, Nschem ie, Ishika
e sho
erge. Of
s an dec
course, th
rease
e CR
the num
scheme
ber of
ows no
to
nv chang
4)he omen n

and
n

. If we increase the value of n

and n

, the f
fo
i
nnumr a
sa
xed
ll point is obtained i more ber of iterations
chemes. Agarwal et al. scheme converges for incre sed
value of n

i.e for

1
2
1
nn
 

.
In this case, increasing order of rate of conve
1
rgence
fo
tions increases in each iterative scheme.
Hence, clhe initial guess to the fi
the result is achieved.
3) If we increase the value of
r iterative schemes is Ishikawa, Mann, Noor, Agarwal
et al., SP and CR scheme.
For increasing functions, we conclude the followings:
1) Increasing order of rate of convergence for iterative
schemes is Mann, Ishikawa, Noor, Picard, Agarwal et al.,
SP and CR scheme.
2) For initial guess away from the fixed point, the
number of itera
oser txed point, quicker
, the fixed
btained in less number of iterations for all
schemes. Except CR iterative scheme which remn-
d.
olof cubic euation, we code the
followings:
n

point is o
ains u
affecte
For sution qnclu
1) Picard and Agarwal et al. iterative schemes do not
converge for

1
2
1

1
nn
f
s.
 and rate of convergence o
SP iterative scheme is better than Ishikawa, Noor, Mann
and CR sc
2) Fo
heme
r initial guess away from the solution, the number
of iterations increases in each iterative scheme.
3) If we increase the value of n

and n

, the solu-
tion is obtained in less number of iterations for Noor and
Ishikawa schemes while solution is obtained in more
s
valuef
number of iterations for Mann, SP and CR chemes.
Agarwal et al. iteration converges for increased o
n

In this case, increasing order of rate of conv
equivalence.
s is Mann
o-
e fix pointhe
number of iterations increases in each iterative scheme.
c,r
3) If we icrease the value of
ergence
for iterative schemes is Ishikawa, Mann, Agarwal et al.,
Noor and CR scheme while CR and SP schemes show
For the goat problem we conclude the followings:
1) Increasing order of rate of convergence for iterative
scheme
scheme
, Ishika
and Ag
wa, No
arwal eor, Picar t al. sche
d, SP an
mes sh
d CR
w equi while SP
valence.
2) For initial guess away from thedt,
Hence, loser the initial guess to the fixed point quicke
the result is achieved.
nn and

n

, the xed
poi is oined less mb iterations foall
-
verence fo iterative schems is Mnn, Ishikawa, Nor,
Agwal et al. and CR schee whilR anP sches
c
REFERENCES
fi
r ntbtain nuer fo
schemes. In this case increasing order of rate of con
g
ar
r e
m
a
e C
o
emd S
show equivalene.
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