research article faster multistep iterations for the...
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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 464593, 4 pageshttp://dx.doi.org/10.1155/2013/464593
Research ArticleFaster Multistep Iterations for the Approximation ofFixed Points Applied to Zamfirescu Operators
Shin Min Kang,1 Ljubomir B. SiriT,2 Arif Rafiq,3 Faisal Ali,4 and Young Chel Kwun5
1 Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea2 Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, Belgrade 11000, Serbia3 Department of Mathematics, Lahore Leads University, Lahore 54810, Pakistan4Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 54000, Pakistan5 Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea
Correspondence should be addressed to Young Chel Kwun; [email protected]
Received 13 July 2013; Accepted 7 September 2013
Academic Editor: Salvador Romaguera
Copyright 2013 Shin Min Kang et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By taking a counterexample, we prove that the multistep iteration process is faster than the Mann and Ishikawa iteration processesfor Zamfirescu operators.
1. Introduction
Let be a nonempty convex subset of a normed space , andlet : be a mapping.
(a) For arbitrary 0 , the sequence {
} defined by
+1
= (1 ) + , 0, (
)
where {} is a sequence in [0, 1], is known as the Mann
iteration process [1].(b) For arbitrary
0 , the sequence {
} defined by
+1
= (1 ) + ,
= (1
) +
, 0,
()
where {} and {
} are sequences in [0, 1], is known as the
Ishikawa iteration process [2].(c) For arbitrary
0 , the sequence {
} defined by
+1
= (1 ) + 1
,
= (1
) +
+1
,
1
= (1
1
) + 1
, 0,
()
where {} and {
}, = 1, 2, . . . , 2 ( 2), are sequences
in [0, 1] and denoted by (), is known as the multistep
iteration process [3].
Definition 1 (see [4]). Suppose that {} and {
} are two real
convergent sequences with limits and , respectively. Then,{} is said to converge faster than {
} if
lim
= 0. (1)
Definition 2 (see [4]). Let {} and {V
} be two fixed-point
iteration procedures which, both, converge to the same fixedpoint , say, with error estimates,
,
V , 0, (2)
where lim = 0 = lim
. If {} converges faster than {
},
then {} is said to converge faster than {V
}.
Theorem 3 (see [5]). Let (, ) be a complete metric space,and let : be a mapping for which there exist realnumbers , , and satisfying (0, 1) and , (0, 1/2) suchthat, for each pair , , at least one of the following is true:
(1) (, ) (, ),
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2 Abstract and Applied Analysis
(2) (, ) [(, ) + (, )],(3) (, ) [(, ) + (, )].
Then, has a unique fixed point, and the Picard iteration{} defined by
+1
= , 0, (3)
converges to for any 0 .
Remark 4. An operator , which satisfies the contractionconditions (1)(3) of Theorem 3, will be called a Zam-firescu operator [4, 6, 7].
In [6, 7], Berinde introduced a new class of operators ona normed space satisfying
+ , (4)
for any , , 0 < 1, and 0.He proved that this class is wider than the class of
Zamfirescu operators.The following results are proved in [6, 7].
Theorem5 (see [7]). Let be a nonempty closed convex subsetof a normed space . Let : be an operator satisfying(4). Let {
} be defined through the iterative process (
) and
0 . If () = 0 and
=0= , then {
} converges
strongly to the unique fixed point of .
Theorem6 (see [6]). Let be a nonempty closed convex subsetof an arbitrary Banach space , and let : be anoperator satisfying (4). Let {
} be defined through the iterative
process () and
0 , where {
} and {
} are sequences of
positive numbers in [0, 1] with {} satisfying
=0= .
Then, {} converges strongly to the fixed point of .
The following result can be found in [8].
Theorem 7. Let be a closed convex subset of an arbitraryBanach space . Let theMann and Ishikawa iteration processeswith real sequences {
} and {
} satisfy 0
, 1, and
=0= . Then, (
) and (
) converge strongly to the
unique fixed point of . Let : be a Zamfirescuoperator, and, moreover, the Mann iteration process convergesfaster than the Ishikawa iteration process to the fixed point of.
In [4], Berinde proved the following result.
Theorem 8. Let be a closed convex subset of an arbitraryBanach space , and let : be a Zamfirescu operator.Let {
} be defined by (
) and
0 with a sequence {
}
in [0, 1] satisfying =0
= . Then, {
} converges strongly
to the fixed point of , and, moreover, the Picard iteration {}
converges faster than the Mann iteration.
Remark 9. In [9], Qing and Rhoades by taking a counterex-ample showed that the Mann iteration process convergesmore slowly than the Ishikawa iteration process for Zam-firescu operators.
In this paper, we establish a general theorem to approxi-mate fixed points of quasi-contractive operators in a Banachspace through themultistep iteration process. Our result gen-eralizes and improves upon, amongothers, the correspondingresults of Babu and Vara Prasad [8] and Berinde [4, 6, 7].
We also prove that the Mann iteration process and theIshikawa iteration process converge more slowly than themultistep iteration process for Zamfirescu operators.
2. Main Results
We now prove our main results.
Theorem 10. Let be a nonempty closed convex subset of anarbitrary Banach space , and let : be an operatorsatisfying (4). Let {
} be defined through the iterative process
() and
0 , where {
} and {
}, = 1, 2, . . . , 2 (
2), are sequences in [0, 1] with =0
= . If () = 0, then
() is a singleton, and the sequence {} converges strongly to
the fixed point of .
Proof. Assume that () = 0 and (). Then, using(), we have
+1 =
(1
) + 1
=(1
) ( ) +
(1
)
(1 )
+
1
.
(5)
Now, for = and = 1, (4) gives
1
1
. (6)
By substituting (6) in (5), we obtain
+1 (1 )
+
1
. (7)
In a similar fashion, again by using (), we can get
(1
)
+
+1
, (8)
where = 1, 2, . . . , 2 ( 2) and
1
(1 (1 )
1
)
. (9)
It can be easily seen that, for = 1, 2, . . . , 2 ( 2), wehave
1
(1
1
)
+ 1
2
,
...3
(1
3
)
+ 3
2
,
2
(1
2
)
+ 2
1
.
(10)
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Abstract and Applied Analysis 3
Substituting (9) in (10) gives us
2
(1 (1 )
2
(1 +
1
))
. (11)
It may be noted that, for [0, 1) and {} [0, 1], the
following inequality is always true:
1 1 + 1 + . (12)
From (11) and (12), we get
2
(1 (1 )
2
)
. (13)
By repeating the same procedure, finally from (7) and (10), weyield
+1 [1 (1 ) ]
. (14)
By (14), we inductively obtain
+1
=0
[1 (1 ) ]0
, 0. (15)
Using the fact that 0 < 1, 0 1, and
=0= , it
results that
lim
=0
[1 (1 ) ] = 0, (16)
which, by (15), implies that
lim
+1 = 0. (17)
Consequently, , and this completes the proof.
Now, by a counterexample, we prove that the multistepiteration process is faster than the Mann and Ishikawaiteration processes for Zamfirescu operators.
Example 11. Suppose that : [0, 1] [0, 1] is defined by = (1/2);
= 0 =
1
=
, = 1, 2, . . . , 1 (
2), and = 1, 2, . . . , 15; = 4/ =
1
=
, =
1, 2, . . . , 1 ( 2), and 16. It is clear that is aZamfirescu operator with a unique fixed point 0 and that allof the conditions ofTheorem 10 are satisfied. Also,
= 0=
= , = 1, 2, . . . , 15. Suppose that
0= 0. For the Mann
and Ishikawa iteration processes, we have
= (1
) +
= (1 4
)+
4
1
2
= (1 2
)
= =
=16
(1 2
) 0,
= (1
) + ((1
) +
)
= (1 4
)+
4
1
2(1
2
)
= (1 2
4
)
= =
=16
(1 2
4
) 0,
= (1
) + 1
,
= (1
) +
+1
,
1
= (1
1
) + 1
, 0,
(18)where = 1, 2, . . . , 2 ( 2) implies that
= (1
=1
(2
)
)
= =
=16
(1
=1
(2
)
)0.
(19)
Now, consider
0
0
=
=16(1
=1(2/)
) 0
=16(1 (2/))
0
=
=16(1
=1(2/)
)
=16(1 (2/))
=
=16
(1
=2(2/)
1 (2/)
)
=
=16
(1
=2(2/)
2)
.
(20)
It is easy to see that
0 lim
=16
(1
=2(2/)
2)
lim
=16
(1 1
)
= lim
15
= 0.
(21)
Hence,
lim
0
0
= 0. (22)
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4 Abstract and Applied Analysis
Thus, theMann iteration process converges more slowly thanthe multistep iteration process to the fixed point 0 of .
Similarly,
0
0
=
=16(1
=1(2/)
) 0
=16(1 (2/) (4/))
0
=
=16(1
=1(2/)
)
=16(1 (2/) (4/))
=
=16
(1
=3(2/)
1 (2/) (4/)
)
=
=16
(1
=3(2/)
2 4)
,
(23)
with
0 lim
=16
(1
=3(2/)
2 4)
lim
=16
(1 1
)
= lim
15
= 0,
(24)
implies that
lim
0
0
= 0. (25)
Thus, the Ishikawa iteration process converges more slowlythan the multistep iteration process to the fixed point 0 of .
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
The authors would like to thank the editor and the refereesfor their useful comments and suggestions. This study wassupported by research funds from Dong-A University.
References
[1] W. R. Mann, Mean value methods in iteration, Proceedings ofthe American Mathematical Society, vol. 4, pp. 506510, 1953.
[2] S. Ishikawa, Fixed points by a new iteration method, Proceed-ings of the American Mathematical Society, vol. 44, pp. 147150,1974.
[3] B. E. Rhoades and S. M. Soltuz, The equivalence betweenMann-Ishikawa iterations and multistep iteration, NonlinearAnalysis. Theory, Methods & Applications, vol. 58, no. 1-2, pp.219228, 2004.
[4] V. Berinde, Picard iteration converges faster than Mann iter-ation for a class of quasi-contractive operators, Fixed PointTheory and Applications, no. 2, pp. 97105, 2004.
[5] T. Zamfirescu, Fix point theorems inmetric spaces,Archiv derMathematik, vol. 23, pp. 292298, 1972.
[6] V. Berinde, On the convergence of the Ishikawa iteration inthe class of quasi contractive operators,ActaMathematica Uni-versitatis Comenianae, vol. 73, no. 1, pp. 119126, 2004.
[7] V. Berinde, A convergence theorem for some mean value fixedpoint iteration procedures,Demonstratio Mathematica, vol. 38,no. 1, pp. 177184, 2005.
[8] G. V. R. Babu and K. N. V. V. Vara Prasad, Mann iteration con-verges faster than Ishikawa iteration for the class of Zamfirescuoperators, Fixed Point Theory and Applications, vol. 2006,Article ID 49615, 6 pages, 2006.
[9] Y. Qing and B. E. Rhoades, Comments on the rate of con-vergence between Mann and Ishikawa iterations applied toZamfirescu operators, Fixed PointTheory and Applications, vol.2008, Article ID 387504, 3 pages, 2008.
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