stability of iteration for some general operators in b-metric

78 B. Prasad, Komal Goyal

International Journal of Computer & Mathematical Sciences

IJCMS

ISSN 2347 – 8527

Volume 5, Issue 4

April 2016

Stability of Iteration for Some General Operators in b-Metric

Spaces

B. Prasada,

aDepartment of Mathematics, Jaypee Institute of

Information Technology University, Noida

Komal Goyalb

bDepartment of Mathematics, Jaypee Institute of

Information Technology University, Noida

Abstract

Various real life problems are modeled in the form of some functional equation. Such an equation can be easily

transformed into a fixed point equation of the type fx x , where f is an operator defined on certain space and x is any

point of it. Such problems are popularly solved by computing a sequence of gradually accurate iterates obtained after

following certain iterative procedure. Thus iterative methods have emerged as important tools for solving many real

problems ranging from various branches of science to engineering. However, while computing the solutions manually or

through software, we usually continue the computation on the basis of an approximate sequence which is quite close to

the sequence generated by the selected iteration procedure. The qualitative properties such as convergence and stability

of the iteration procedure under the defined operator have a great role to play in such cases. Our aim is to discuss the

stability of the Jungck-Mann iteration scheme for the operators satisfying some general contractive condition in the

settings of b-metric spaces.

Keywords: Jungck-Mann iteration procedure, Stability result, Fixed point iteration procedure, b-metric space.

1. Introduction

Let ( , )X d be a complete metric space and :T X X . The point which satisfies Tx x is called fixed

point of metric space. Several iterative processes have been defined in the literature to approximate the fixed

point of a map in the given space.

Let 0{ }n nx X

be the sequence generated by iteration procedure involving the operator T, then the Picard

iterative procedure is defined as,

1 ( , ) , 0,1,... (1.1)n n nx f T x Tx n

If ( , ) (1 )n n n n nf T x x Tx , then for 0x X

and 0{ } [0,1]n n

, Mann iteration process [9] is

defined as,

1 (1 ) , 0,1,... (1.2)n n n n nx x Tx n

On putting { } 1n in (1.2) becomes the Picard iterative process (1.1).

In 1976, Jungck defined a new iterative scheme.

Definition 1.1 [8]. Let Y be an arbitrary non empty set and ( , )X d be a metric space. Let , :S T Y X and

( ) ( )T Y S Y for some 0x Y , consider

1 ( , ), 0,1,2... (1.3)n nSx f T x n

For and ( , )n nY X f T x Tx , the iterative procedure (1.3) yields the Jungck iteration namely

1 , 0,1,2... (1.4)n nSx Tx n

In 2005, Singh et al. [19] defined Junck-Mann iteration process as follows:

1 (1 ) , 0,1,2... (1.5)n n n n nSx Sx Tx n

where 0{ }n n

is a sequence in [0,1] .



IJCMS

ISSN 2347 – 8527

Volume 5, Issue 4

April 2016

On putting and Y X S id , the identity map on X in (1.4) becomes Picard iterative procedure.

Similarly, on putting and Y X S id , the identity map on X, (1.5) becomes Mann iterative procedure.

2. Preliminaries The first stability result of T-stable mapping was given by Ostrowski [11] for Banach contraction principle.

Later, Harder and Hick [5, 6] defined it in a formal way as follows.

Definition 2.1 [5]. An iterative procedure 1 ( , )n nx f T x is said to be T–stable with respect to a mapping T

if { }nx

converges to a fixed point q of T and whenever{ }ny

is a sequence in X with

1lim ( , ( , )) 0n nn

d y f T y

, we have lim nn

y q

.

They proved that the sequence { }nx generated by Picard iterative process in complete metric space converges

strongly to the fixed point of T, satisfying

( , ) 2 ( , ) ( , )d Tx Ty d x Tx d x y for each pair ,x y X , where

10 1,0 and

2 ,

with max{ , , }, 0 <11 1

.

They also established the stability of the iterative procedure with respect to T.

In 2005, Singh et al. [19] defined ( , )S T stability in the following manner.

Definition 2.2 [19]. Let , : , ( ) ( )S T Y X T Y S Y and ‘z’ a coincidence point of T and S that is

0 (say), for any Sz Tz p x Y , let the sequence { }nSx , generated by iterative procedure (1.4),

converges to ‘p’. Let { }nSy X be an arbitrary sequence, and set 1( , ( , )), 0,1,2...n n nd Sy f T y n then

the iterative procedure ( , )nf T x will be called ( , )S T stable if and only if lim 0 limn nn n

Sy p

The concept of b-metric space was introduced by Czerwik [4]. Since then, several papers have been published

on the fixed point theory of various classes of single-valued and multi-valued operators in b-metric spaces

(see, for instances [1],[3],[10]).

Definition 2.3 [4]. Let X be a set and 1r be a given real number. A function :d X X R is said to be

a b-metric if and only if for all , ,x y z X , the following conditions are satisfied:

(i) ( , ) 0 iff ,

(ii) ( , ) ( , ),

(iii) ( , ) [ ( , ) ( , )]

d x y x y

d x y d y x

d x z r d x y d y z

A pair ( , )X d is called a b-metric space.

Our aim is to discuss the convergence of the Jungck-Mann iteration scheme in the settings of b-metric

spaces. In this paper, we prove stability result for Junck-Mann iterative procedure by using contractive

condition

( , ) { ( , ) ( , )}, (2.1)d Tx Ty a d Sx Tx d Sy Ty

3. Main Result

Theorem 3.1. Let ( , )X d be a b-metric space and S and T be the maps on an arbitrary set Y with values in X

such that ( ) ( )T Y S Y and ( ) or ( )S Y T Y is a complete subspace of X. Let z be a coincidence point of T

and S, that is .Sz Tz p Let 0x Y and let the sequence{ }nSx , generated by



IJCMS

ISSN 2347 – 8527

Volume 5, Issue 4

April 2016

1 (1 ) , 0 n n n n nSx Sx Tx n

where { }n satisfies for n=0, 1, 2…

0

0 1

(i) 1 (ii) 0 1, 0

(iii) (iv) (1 )

n

n n

n j i i

j i j

n

a

converges to p. Let { }nSy X and define 1( , (1 ) ), 0n n n n n nd Sy Sy Ty n . If the pair ( , )S T

satisfies (2.1) for all ,x y Y . Then,

2

1 1 0 0

0 0 1

0 1

(I) ( , ) ( , ) (1 ) ( , ) (1 )[ ( , ) ( , )]

(1 )

n n nn i

n n i j i i i i i

i j i j

n nn i

j i j

j i j

d p Sy d p Sx r d Sx Sy a r d Sx Tx d Sy Ty

r

(II) lim nn

Sy p

if and only if lim 0.n

n

Proof: By the triangle inequality and the condition (1.5)

1 1 1 1

1 1

1 1

( , ) [ ( , ) ( , )]

[ ( , ) ((1 ) , )]

( , ) [ ((1 ) ,(1 ) ) ((1 ) , )]

n n n n

n n n n n n

n n n n n n n n n n n n n n

d p Sy r d p Sx d Sx Sy

r d p Sx d Sx Tx Sy

rd p Sx r d Sx Tx Sy Ty d Sy Ty Sy

1

1

( , ) [(1 ) ( , ) ( , )]

( , ) (1 ) ( , ) [ ( , ) ( , )] (3.1)

n n n n n n n n

n n n n n n n n n n

rd p Sx r d Sx Sy d Tx Ty r

rd p Sx r d Sx Sy ra d Sx Tx d Sy Ty r

also

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1

( , ) [ ((1 ) ,(1 ) ) ((1 ) , )]

[(1 ) ( , ) ( , )]

n n n n n n n n n n n n n n n

n n n n n n n

d Sx Sy r d Sx Tx Sy Ty d Sy Ty Sy

r d Sx Sy d Tx Ty r

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1

(1 ) ( , ) [ ( , ) ( , )]

(1 ) ( , ) ( , ) (

n n n n n n n n n

n n n n n n n

r d Sx Sy ra d Sx Tx d Sy Ty r

r d Sx Sy ra d Sx Tx ra d Sy

1 1 1, ) (3.2)n n nTy r

Therefore,

1 1

1 1 1 1 1 1 1 1 1 1

( , ) ( , ) (1 ) ( , ) [ ( , ) ( , )]

( , ) (1 )[ (1 ) ( , ) ( , ) ( , )

n n n n n n n n n n n


n

d p Sy rd p Sx r d Sx Sy ra d Sx Tx d Sy Ty r

rd p Sx r r d Sx Sy ra d Sx Tx ra d Sy Ty

r

1

2 2

1 1 1 1 1 1 1 1 1

2

1

] [ ( , ) ( , )]

( , ) (1 )(1 ) ( , ) (1 ) [ ( , ) ( , )]

(1 ) [ ( , ) ( , )]

n n n n n n


n n n n n n n

ra d Sx Tx d Sy Ty r

rd p Sx r d Sx Sy r a d Sx Tx d Sy Ty

r ra d Sx Tx d Sy Ty r

(3.3)

n

This process when repeated ( 1)n times, yields (I).

To prove (II), suppose that lim nn

Sy p

. Then,



IJCMS

ISSN 2347 – 8527

Volume 5, Issue 4

April 2016

1

1

1

1

1

1

( ,(1 ) )

( , ) ( , (1 ) )

( , ) ((1 ) ,(1 ) )

( , ) (1 ) ( , ) ( , )

( , ) (1 ) ( , ) ( , )

( , )

n n n n n n

n n n n n

n n n n n n n

n n n n n

n n n n n

n

d Sy Sy Ty

d Sy p d p Sy Ty

d Sy p d p Sy Ty

d Sy p d p Sy d p Ty

d Sy p d p Sy d Tp Ty

d Sy p

(1 ) ( , ) [ ( , ) ( , )] 0 as .n n n n nd p Sy a d Sp Tp d Sy Ty n

Now suppose that lim 0nn

. Let A denotes the lower triangular matrix with entries

1

(1 )n

nj j i

i j

Then A is multiplicative, so that

0 1

lim (1 )[ ( , ) ( , )] 0,n n

n i

j i i i i in

j i j

a r d Sx Tx d Sy Ty

0 1

and lim (1 ) 0.n n

n i

j i jn

j i j

r

Finally, condition (iii) of iterative scheme implies 1

lim (1 ) 0n

i in

i j

a

Hence it follows from inequality that lim .nn

Sy p

This completes the proof.

Example 3.1. Consider the non-linear equation 2 cos( 2) 0x xe e . Let us take cos( 2)xTx e and

2xSx e . If we choose initial guess 0 1.5x . It is observed that Jungck-Mann iteration scheme converges

in 7 iterations (with 0.95 ) whereas Jungck – Noor iteration scheme takes 14 iterations (with

0.95, 0.12 and =0.25 ) and Jungck – Ishikawa iteration scheme evaluates the solution in 14

iteration (with 0.95, 0.12 ).

Table 1 Comparison of convergence of iterative procedures

n

Jungck Noor

( 0.95, 0.12 and =0.25)

n

Jungck Ishikawa

( 0.95, 0.12 )

n

Jungck Mann

( 0.95 )

nTx 1nSx 1nx nTx 1nSx 1nx nTx 1nSx 1nx

0 -0.7901 -0.3465 0.5029 0 -0.7901 -0.3465 0.5029 0 -0.7901 -0.6265 0.3174

1 0.9406 0.9152 1.0699 1 0.9406 0.9152 1.0699 1 0.8101 0.7383 1.0073

2 0.6096 0.6521 0.9754 2 0.6096 0.6521 0.9754 2 0.7396 0.7398 1.0078

. . . . . . . . . . . .

. . . . . . . . . . . .

14 0.7391 0.7391 1.0076 14 0.7391 0.7391 1.0076 7 0.7391 0.7391 1.0076

This shows Jungck – Mann iterative procedure converges faster than Jungck – Ishikawa and Jungck – Noor

iterative procedures for this example.



IJCMS

ISSN 2347 – 8527

Volume 5, Issue 4

April 2016

Corollary 3.1. Let ( , )X d be a b-metric space and :T X X . Let p be the fixed point of T and 0x X .

The sequence{ }nTx , generated by

1 (1 ) , 0n n n n nx x Tx n


0

0 1

(i) 1 (ii)0 1, 0

(iii) (iv) (1 )

n

nn

n j i i

j i j

n

a

converges to p. Let { }nTy X and define 1( ,(1 ) ), 0n n n n n nd y y Ty n . If the mapping T satisfies

(2.1) for all , ,x y Y . Then,

2

1 1 0 0

0 0 1

0 1

(I) ( , ) ( , ) (1 ) ( , ) (1 )[ ( , ) ( , )]

(1 )

n n nn i

n n i j i i i i i

i j i j

n nn i

j i j

j i j

d p y d p x r d x y a r d x Tx d y Ty

r

(II) lim nn

y p

if and only if lim 0.nn

On putting and Y X S id , the identity map on X in Theorem 3.1, we get the result for Mann iterative

procedure.

Corollary 3.2. Let ( , )X d be a metric space and let S and T be maps on an arbitrary set Y with values in X

such that ( ) ( )T Y S Y and ( ) or ( )S Y T Y is a complete subspace of X. Let z be a coincidence point of T and

S, that is .Sz Tz p Let 0x Y and let the sequence{ }nSx , generated by

1 (1 ) , 0n n n n nSx Sx Tx n


0

0 1

(i) 1 (ii)0 1, 0

(iii) (iv) (1 )

n

nn

n j i i

j i j

n

a

converges to p. Let { }nSy X and define 1( , (1 ) ), 0n n n n n nd Sy Sy Ty n . If the pair ( , )S T is

for all , ,x y Y satisfies (2.1). Then,

1 1 0 0

0 0 1

0 1

(I) ( , ) ( , ) (1 ) ( , ) (1 )[ ( , ) ( , )]

(1 )

n n n

n n i j i i i i i

i j i j

n n

j i j

j i j

d p Sy d p Sx d Sx Sy a d Sx Tx d Sy Ty

(II) lim nn

Sy p

if and only if lim 0.nn

If we put 1r in Theorem 3.1, we get desired result.



IJCMS

ISSN 2347 – 8527

Volume 5, Issue 4

April 2016

4. Conclusion

A stability result regarding the ( , )S T stability of Jungck-Mann iterative scheme is established.

Comparative study in case of Jungck-Noor, Jungck-Ishikawa and Junck-Mann iterative procedures is also

presented as an example.

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[2] V. Berinde, “Picard Iteration Converges Faster than Mann Iteration for a Class of Quasi-Contractive Operators,"

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[3] M. Boriceanu, "Strict fixed point theorems for multivalued operators in b-metric spaces," Int. J. Mod. Math, vol. 4,

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Fisico Universita Di Modena, vol. 46, pp. 263-276, 1998.

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stability of iteration for some general operators in b-metric

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