Stability Result of Iterative procedure in normed space

* Department of Mathematics Jaypee Institute of Information Technology, Noida, India, Email: b_prasad10@yahoo.com
** Department of Mathematics Jaypee Institute of Information Technology, Noida, India, Email: Komal.goyal0988@gmail.com
Stability Result of Iterative
Procedure in Normed Space
B. Prasad* and Komal Goyal**
ABSTRACT
The intent of this paper is to study the stability of Jungck-Noor iteration schemes for maps satisfying a general
contractive condition in normed space. Our result contains some of the results of Berinde [2-3], [5], Bosede and
Rhoades [6], Bosede [7], Imoru and Olatinwo [12], Olatinwo et al. [18].
Keywords: Jungck-Mann iteration, Jungck-Noor iteration, Stability of iterations, Fixed point iteration, Stability
results in normed space, (S, T) stability.
1. INTRODUCTION AND PRILIMINIRIES
Let ( , )X d be a complete metric space and :T X X� . Let 0
{ }n n
x X�
�
� be the sequence generated by iteration
procedure involving the operator T, if
1
( , ) , 0, 1,n n n
x f T x Tx n�
� � � ... (1.1)
then it is called Picard iteration process. The Picard iteration can be used to approximate the unique fixed
point for strict type contractive operator. There was a need of some other iterative procedures for slightly
weaker contractive conditions.
If for 0
x X� , the sequence 0{ }n nx �
� is defined by,,
1
(1 ) , 0, 1,...n n n n n
x x Tx n�
� � � �� (1.2)
where 0
{ } [0,1]n n
�
�
�� is called Mann iteration process [16].
And
1
(1 ) ,n n n n n
x x Tz�
� � ��
if
(1 ) , 0, 1,...n n n n n
z x Tx n� � � �� (1.3)
where 0 0{ } and { }n n n n
� �
� �� are the real sequences in [0, 1], then it is called Ishikawa iteration process [13].
The sequence is defined by,
1
(1 ) ,n n n n n
x x Ty�
� � ��
(1 ) ,n n n n n
y x Tz� � ��
I J C T A, 9(20), 2016, pp. 149-158
© International Science Press

150 B. Prasad and Komal Goyal
(1 ) , 0, 1,...n n n n n
z x Tx n� � � �� (1.4)
where 0 0 0{ } , { } and { }n n n n n n
� � �
� � �� are the real sequences in [0, 1], then it is called Jungck-Noor iterative
scheme.
On putting { } 1n
�� in (1.2), it becomes Picard iterative process. Similarly, if 0n
�� for each ‘n’ in
(1.3), then it reduces to (1.2). If we put 0n �� for each ‘n’ in (1.4), then it becomes (1.3).
Definition 1.1 [14]. Let Y be an arbitrary non emptyset and (X, d) be a metric space. Let , :S T Y X� and
( ) ( )T Y S Y� for some 0
,x Y� consider
1
, 0, 1, 2...n n
Sx Tx n�
� � (1.5)
If
1
(1 ) , 0, 1, 2...,n n n n n
Sx Sx Tx n�
� � � �� (1.6)
where 0{ }n n
�
�� is a sequence in [0, 1], then it is called Junck-Mann iteration process [36].
Olatinwo and Imoru [19] defined 0{ }n nSx �
� as
1
(1 ) ,n n n n n
Sx Sx Tz�
� � ��
(1 ) , 0, 1,...n n n n n
Sz Sx Tx n� � � �� (1.7)
where 0 0{ } and { }n n n n
� �
� �� are the real sequences in [0, 1], this scheme is called Jungck-Ishikawa iteration.
Further, Olatinwo [20] defined 0
{ }n n
Sx �
�
for three step iteration procedure as follows.
Definition 1.2 [20]. Let , :S T T X� and ( ) ( )T X S X� . Define
1
(1 ) ,n n n n n
Sx Sx Tz�
� � ��
(1 ) ,n n n n n
Sz Sx Tr� � ��
(1 )n n n n n
Sr Sx Tx� � �� (1.8)
where 0, 1,...n � and { },{ } and { }n n n
� � � satisfy
(i) 0
1��
(ii) 0 , , 1, 0n n n
n� � ��
(iii) ��n
= �
(iv)
10
(1 )
n n
j i i
i jj
a
� ��
� �� converges.
This is called Jungck-Noor iteration scheme [20].
The first result on the stability is due to Ostrowoski [22]. However Harder and Hick [10-11] defined T-
stability as follows:

Stability Result of Iterative Procedure in Normed Space 151
Definition 1.3 [10-11]. The iterative procedure 1 ( , )n nx f T x� � is said to be T-stable with respect to T if
{ }nx converges to a fixed point q of T and whenever { }ny is a sequence in X with 1lim ( , ( , )) 0,n n
n
d y f T y�
��
�
we have lim n
n
y q
��
� .
The (S, T) stability mapping is defined by Singh et al. [36] in the following manner.
Definition 1.4 [36]. Let , : , ( ) ( )S T Y X T Y S Y� � and “z” a coincidence point of T and S that is
0(say), for anySz Tz p x Y� � � , let the sequence { }n
Sx , generatedby iterative procedure (1.4) , converges
to ‘p’. Let { }nSy X� be an arbitrarysequence, and set 1
( , ( , )), 0, 1, 2...n n n
d Sy f T y n�
� �� thenthe iterative
procedure ( , )nf T x will be called ( , )S T stable if and only if lim 0 lim .n n
n n
Sy p
��
� � ��
Harder and Hick [10-11] obtained stability results for Zamfirescu operator (Z-operator) for Picard and
Mann iterative procedures.
Suppose X is a Banach space and Y a nonempty set such that ( ) ( )T Y S Y� . Then , :S T Y X� is called
Zamfirescu operator if for ,x y Y� and (0, 1),h �
max{ , , }.
2 2
Sx Tx Sy Ty Sx Ty Sy Tx
Tx Ty h Sx Sy
� � � � � �
� � � (1.9)
Rhoades [34-35] obtained fixed point results for Mann and Ishikawa iteration procedures in uniformly
Banach space. Berinde [4] used these iterative procedures for approximating the fixed point of Z-operator
in arbitrary Banach space. Several authors used Z-operator for different iterative procedures in the setting
of different spaces. Motivated by rich literature of Z-operator, Osilike [21] established stability results for
Picard, Mann and Ishikawa iterative procedures for a large class of mappings and introduced the following
contractive condition.
, 0, 0 1.Tx Ty Sx Sy L Sx Tx L� � � � � � � �� (1.10)
It cn be seen that (1.9) (1.10)� . It can be understood it better by taking cases one by one.
Case I: On putting and 0h L� �� in (1.10), we get first part.
Case II:
2
and
2 2
h h
L
h h
� �
� �
� gives second part of (1.9).
Case III: .
2
Sx Ty Sy Tx
Tx Ty h Sy Tx h Sy Sx h Sx Tx
� � �
� � � � � � � �
Hence andh L h� �� completes the proof.
Olantinwo [20] generalized the above contractive condition as follows.
( ), 0 1,Tx Ty Sx Sy Sx Tx� � � � � � �� (1.11)
where : R R� �
�� is a monotone decreasing sequence with (0) 0�� . If we take ( )u Lu�� in (1.11), we
get (1.10) which shows (1.10) (1.11)� . We see (1.9) (1.10) (1.11)� � . Thus the theory of stability of fixed
point iteration has been widely studied in the literature and interesting fixed point results are obtained by a
number of authors in various settings, see for instance [1-6], [10-12], [18-32] and several reference thereof.

The following lemma of Berinde [4] is required for the sequel.
Lemma 1.1 [4]. If � is a real number such that 00 1 and { }n n
�
�� is a sequence of positive number
such that lim 0,n
n��
�� then for any sequence of positive numbers 0{ }n nu �
� satisfying
1 , 0, 1, 2...,n n nu u n� � � �� we have lim 0.n
n
u
��
�
2. MAIN RESULT
Theorem 2.1. Let ( , )X � be a normed space and , :S T Y X� be non-self maps on an arbitrary set Y such
that ( ) ( ),T Y S Y� where S(Y) is a complete subspace of X and S an injective operator. Let z be a coincidence
point of S and T i.e; Sz Tz p� � (say). Suppose S and T satisfy,,
) ( ) ( , ), [0,1), (0) 0.Tx Ty Sx Tx d Sx Sy� � � � � �� (2.1)
for 0
.x Y� Let 0{ }n nSx �
� be Jungck-Noor iterative scheme (1.8) converging to p, where { },{ },{ }n n n� � � are
sequences of positive number in [0, 1] with { }n� satisfying 0 .n n� � �� Then the Jungck-Noor iterative
scheme is (S, T) stable.
Proof. Suppose that 0
{ } ,n n
Sy X�
�
� 1
(1 ) , 0, 1, 2, 3..,n n n n n n
Sy Sy Ts n�
� � � � ��
where
(1 ) ,n n n n n
Ss Sy Tq� � ��
(1 ) .n n n n n
Sq Sy Ty� � ��
and let lim 0.n
n��
�� Then it follows from (1.8) and (2.1) that
1 1
(1 ) (1 ) (1 )n n n n n n n n n n n n
Sy p Sy Sy Ts Sy Ts p� �
� � � � � � � � � � ��
(1 )n n n n n
Sy p Ts p� � � � � ��
(1 )n n n n n
Sy p Ts Tz� � � � � ��
(1 ) [ ( )]n n n n n
Sy p Sz Ss Sz Tz� � � � � � � ��
(1 ) + (0)n n n n n n
Sy p p Ss� � � � � ��
(1 ) + .0n n n n n n
Sy p p Ss� � � � � ��
(1 ) .n n n n n
Sy p p Ss� � � � � �� (2.2)
Now we have the following equation
(1 ) (1 )n n n n n n n
p Ss p Sy Tq� � � � � � ��
(1 )n n n n
p Sy p Tq� � � � ��
(1 )n n n n
p Sy Tz Tq� � � � ��
(1 ) [ ( )]n n n n
p Sy Sz Sq Sz Tz� � � � � � ��
(1 ) (0)n n n n n
p Sy Sz Sq� � � � � ��

(1 ) .0n n n n n
p Sy Sz Sq� � � � � ��
(1 ) .n n n n
p Sy p Sq� � � � �� (2.3)
Also we have
(1 ) (1 )n n n n n n n
p Sq p Sy Ty� � � � � � ��
(1 )n n n n
p Sy p Ty� � � � ��
(1 )n n n n
p Sy Tz Ty� � � � ��
(1 ) [ ( )]n n n n
p Sy Sz Sy Sz Tz� � � � � � ��
(1 ) (0)n n n n n
p Sy Sz Sy� � � � � ��
(1 ) .0n n n n n
p Sy Sz Sy� � � � � ��
(1 ) .n n n n
p Sy p Sy� � � � �� (2.4)
It follows from (2.2), (2.3) and (2.4) that
1
[1 {1 (1 )}] .n n n n n n n n n
Sy p p Sy�
� � � � � � � � � �� (2.5)
Using 0 and [0, 1),n
� � �� we have
[1 {1 (1 )}] 1.n n n n n n
� � � � � � ��
Hence, using lemma (2.1), (2.5) yields 1lim .n
n
Sy p�
��
�
Conversely, let 1lim .n
n
Sy p�
��
� Then using contractive condition (2.1) and triangle inequality, we have
1
(1 )n n n n n n
Sy Sy Ts�
� � � ��
1
(1 ) (1 )n n n n n n n
Sy p p Sy Ts�
� � � � � � � ��
1
(1 )n n n n n
Sy p p Sy p Ts�
� � � � � � ��
1
(1 )n n n n n
Sy p p Sy Tz Ts�
� � � � � � ��
1
(1 ) [ ( )]n n n n n
Sy p p Sy Sz Ss Sz Tz�
� � � � � � � � ��
1
(1 ) (0)n n n n n n
Sy p p Sy Sz Ss�
� � � � � � � ��
1
(1 ) .0n n n n n n
Sy p p Sy Sz Ss�
� � � � � � � ��
1
(1 ) .n n n n n
Sy p p Sy p Ss�
� � � � � � �� (2.6)
Again using (2.3) and (2.4), it yields
1
[1 {1 (1 )}] 0 as .n n n n n n n n n
Sy p Sy p n�
� � � � � � � � � � � � ��
Hence, the iterative procedure defined in (1.8) is stable with respect to pair (S, T).
Example 2.1. Let X R�
� . Define , :S T X X� by , and ( ) ,
2 4 3
x x x
Sx Tx x� � �� where : R R� �
��
with (0)=0� and (X, d) has the usual metric. Then T satisfies contractive condition (2.1) and ( ) 0.F T � Also
Jungck-Noor iterative scheme (1.8) is stable.

Proof: Now 0p � is the coincidence point. Taking
1
for each 1.
2
n n n
n� � � ��
Let
2
2
n
y
n
�
�
.
Then, lim 0n
n
y p
��
� �
1
(1 ) , 0,1,2,3..,n n n n n n
Sy Sy Ts n�
� � � � ��
where
(1 ) ,n n n n n
Ss Sy Tq� � ��
(1 ) .n n n n n
Sq Sy Ty� � ��
1 1 3
(1 ) ,
2 2 2 4 8
n n
n n
y y
Sq y� � � � �
3 3
2( ) .
8 4
n n n
q y y� � �
1 1 1 3 11
(1 ) ,
2 2 2 4 4 32
n
n n n
y
Ss y y� � � � � �
11 11
2( ) .
32 16
n n n
s y y� � �
1
1 1 1 11 33
(1 ) ,
2 2 2 4 16 128
n
n n n
y
Sy y y�
� � � � � �
33 33
2( ) .
128 64
n n n
s y y� � �
1
(1 ) , 0,1,2,3..n n n n n n
Sy Sy Ts n�
� � � � ��
1 2 33 2
= ( ) 0 as
2 2 ( 1) 64 2
n
n n
� � � � �
� � �
Hence, lim 0.n
n��
��
Therefore, Jungck-Noor iteration is stable.
On putting Y X E� � and ,S id� the identity map on X, a�� and considering p, a fixed point of T,
that is, p Tx x� � in Theorem 2.1, we get Theorem 3.1of Bosede [7].
Corollary 2.1 [7]. Let ( , )E � be a Banach space, :T E E� be a selfmap of E with a fixed point p,
satisfying the contractive condition, p Ty a p y� � � such that for each and 0 1y E a� � � where p is a
fixed point. For 0
,x E� let � � 0n n
x
�
�
be the Noor iterative process defined as,
1 (1 ) ,
(1 ) ,
(1 ) .
n n n n n
n n n n n
n n n n n
x x Tq
q x Tr
r x Tx
� � � �
� � �
� � �
� �
� �
� �

converging to p, (i.e; Tp p� ), where � � � � � �0 0 0
, andn n nn n n
� � �
� � �
� � � are sequences of real numbers in [0, 1]
such that 0 , 0 and 0n n n� � � � � �� for all n. Then, the Noor iteration process is T--stable.
In the same way, on putting Y X E� � and ,S id� the identity map on X, 0,n
a� �� and considering
p Tx x� � in Theorem 2.1, we get Theorem 3.2 of Bosede [7].
On putting ,Y X E� � S id� and 0,n n
a� � �� in Theorem 2.1, we get Theorem 2.2 of Bosede
and Rhoades [6].
Corollary 2.2 [6]. Let E be a Banach space, T a selfmap of E with a fixed point p and satisfying
for some 0 1 and for each .p Ty a p y a y X� � � � � �
The Mann iteration with 0 for all ,n
n� �� is T-stable.
If we put n
�� in Corollary 2.2, we get T-stability for Kransnoselskij iterative procedure where
0 1.� ��
And if we put 1n
�� in Corollary 2.2, we get T-stability for Picard iterative procedure.
On putting
1
, , , 0 and
2
,n n n
Y X E S id b� � � � � � �� we get Theorem 1 of Olatinwo et al. [18].
Corollary 2.3 [18]. Let 0
{ }n n
y E�
�
� and 1
1
( )
2
n n n ny y Ty�� . Let ( ,E � ) be a normed linear space
and :T E E� a selfmap of E satisfying
( ) , 0 b<1.Tx Ty x Tx b x y� � � � � ��
Suppose T has a fixed point p. For arbitrary 0
,x E� define sequence 0{ }n nx �
� iteratively by;
1
1
( , ) ( ), 0.
2
n n n n
x f T x x Tx n�
� � � �
Let : R R� �
�� be monotonic increasing with (0) 0.�� Then, the Krasnolseskij process is T-stable.
On putting , , , 0 andn n n
Y X E S id b a� � � � � � �� , we get Theorem 2 of Olatinwo [18].
On putting , , , 0,n
Y X E S id b� � � � �� we get Theorem 3 of Olatinwo [18].
On putting , , , 0,n n
Y X E S id b� � � � � �� we get Theorem 3.2 of Imoru and Olatinwo [12].
Corollary 2.4 [12]. Let ( , )E � be a normed linear space and let :T E E� be a selfmap of E satisfying
( ) , 0 b<1.Tx Ty x Tx b x y� � � � � ��
Suppose T has a fixed point *
.p Let 0
x E� and suppose that 1
( , ) (1 ) , 0,n n n n n n
x f T x x Tx n�
� � � � ��
where 0
{ }n n
�
�
� is a real sequence in [0, 1] such that 0 , 0, 1, 2,...n
n� � �� Suppose also that : R R� �
��
be monotonic increasing with (0) 0.�� Then, the Mann iteration is T-stable.

We have already proved that (13) (15)� and on putting , and 0,n n
Y X S id� � � �� we get Theorem
3 of Berinde [2].
Corollary 2.5 [2]. Let ( , )X � be a normed linear space and :T X X� be a Zamfirescu contraction.
Suppose there exists ( )p F T� such that the Mann iteration 0{ }n nx �
� with 0
x X� and 0
{ }n n
�
�
� satisfying
0
,n
n
�
�
� �� converges to p. Then the Mann iteration procedure is T-stable.
On putting , , 0 and ( ) ( ) where ( , ),n
Y X E S id u L u u d x Tx� � � � � �� we get Theorem 1 of Berinde
[5].
Corollary 2.6 [5]. Let E be a normed linear space, K a closed convex subset of E, and :T K K� an
operator with ( ) ,F T � � satisfying (14). Let 0{ }n nx �
� be the Ishikawa iteration and 0
,x K� arbitrary, where
{ }n
� and { }n
� are sequences in [0, 1] with { }n
� satisfying
0
.n
n
�
�
� �� Then { }nx converges strongly to the
unique fixed point of T.
We have already proved that (13) (15)� and on putting , and 0,n
Y X E S id� � � �� we get Corollary 2
of Berinde [5].
Similarly using (13) (15),� , and 0,n n
Y X E S id� � � � �� we get Corollary 2 of Berinde [5].
On putting Y X� and ,S id� the identity map on X, 1, 0n n n
� � �� and
( ) ( ) where ( , )u L u u d x Tx� �� in Theorem 2.1, we get that { }nx is stable with respect to T. From remark
and example 1 in [2], it is clear that any stable iteration procedure is also almost stable and it is obvious that
any almost stable iteration procedure is also summably almost stable, since
0
( , ) lim .n n
n
n
d y p y p
�
��
�
� � � ��
Hence, we get result Theorem 1 of Berinde [3].
Corollary 2.7 [3]. Let ( , )X d be a metric spaceand :T X X� a mapping satisfying contractive condition
( , ) ( , ) ( , ) for [0,1), 0 ,d Tx Ty ad x y Ld x Tx a L x y X� � � � � � .
Suppose T has a fixed point p. Let 0
x X� and 1
, 0,n n
x Tx n�
� � then { }nx converges strongly to to p and
is summable almost stable with respect to T.
Similar to above reason, On putting Y X� and ,S id� the identity map on X, 0n
�� and
( ) ( ) where ( , )u L u u d x Tx� �� in Theorem 2.1, we get Theorem 2 of Berinde [3].
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