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
,
a
Department of Mathematics, Jaypee Institute of
Information Technology University, Noida
Komal Goyalb
b
Department 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 andY X S id  , the identity map on X in (1.4) becomes Picard iterative procedure.
Similarly, on putting andY 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 n
n
d y f T y

 , we have lim n
n
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
1
0 1,0 and
2
      ,
with max{ , , }, 0 <1
1 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 anySz 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 n
n 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 ) , 0n 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 n
n i
n n i j i i i i i
i j i j
n n
n 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 n
n
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 n
n
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 0n
n


 . 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 i
n
j i j
a r d Sx Tx d Sy Ty 

  
   
0 1
and lim (1 ) 0.
n n
n i
j i j
n
j i j
r   

  
  
Finally, condition (iii) of iterative scheme implies
1
lim (1 ) 0
n
i i
n
i j
a 

 
  
Hence it follows from inequality that lim .n
n
Sy p


This completes the proof.
Example 3.1. Consider the non-linear equation 2 cos( 2) 0x x
e e    . Let us take cos( 2)x
Tx e  and
2x
Sx 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 n
n i
n n i j i i i i i
i j i j
n n
n 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 n
n
y p

n



On putting andY 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 n
n
Sy p

n



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.
5. References
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metric spaces," Fixed Point Theory and Applications, vol. 2012, pp. 1-8, 2012.
[2] V. Berinde, “Picard Iteration Converges Faster than Mann Iteration for a Class of Quasi-Contractive Operators,"
Fixed Point Theory and Applications, vol. 2, pp. 97-105, 2004.
[3] M. Boriceanu, "Strict fixed point theorems for multivalued operators in b-metric spaces," Int. J. Mod. Math, vol. 4,
pp. 285-301, 2009.
[4] S. Czerwik, "Nonlinear set-valued contraction mappings in b-metric spaces," Atti Del Seminario Matematico E
Fisico Universita Di Modena, vol. 46, pp. 263-276, 1998.
[5] A. M. Harder and T. L. Hicks, "A stable iteration procedure for nonexpansive mappings," Math. Japonica, vol. 33,
pp. 687-692, 1988.
[6] A. M. Harder and T. L. Hicks, "Stability results for fixed point iteration procedures," Math. Japonica, vol. 33, pp.
693-706, 1988.
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vol. 44, pp. 147-150, 1974.
[8] G. Jungck, "Commuting mappings and fixed points," The American Mathematical Monthly, vol. 83, pp. 261-263,
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[9] W. R. Mann, "Mean value methods in iteration," Proceedings of the American Mathematical Society, vol. 4, pp.
506-510, 1953.
[10] M. Olatinwo and C. O. Imoru, "A generalization of some results on multi-valued weakly Picard mappings in b-
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[11]A. Ostrowski, "The Round‐off Stability of Iterations," ZAMM‐Journal of Applied Mathematics and
Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, vol. 47, pp. 77-81, 1967.
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(ICECT), 2011 3rd International Conference on, vol. 5, pp. 231-234, 2011.
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[19]S. L. Singh, C. Bhatnagar and S. N. Mishra, "Stability of Jungck-type iterative procedures," International Journal of
Mathematics and Mathematical Sciences, vol. 2005, pp. 3035-3043, 2005.
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479-489, 1956.

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