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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; yckwun@dau.ac.kr

    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) (, ) (, ),

  • 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)

  • 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

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