Data dependence results of new multi-step and S-iterative schemes for contractive-like operators

  • Faik Gürsoy1,

    Affiliated with

    • Vatan Karakaya2Email author and

      Affiliated with

      • Billy E Rhoades3

        Affiliated with

        Fixed Point Theory and Applications20132013:76

        DOI: 10.1186/1687-1812-2013-76

        Received: 5 December 2012

        Accepted: 10 March 2013

        Published: 28 March 2013

        Abstract

        In this paper, we prove that the convergence of a new iteration and S-iteration can be used to approximate the fixed points of contractive-like operators. We also prove some data dependence results for these new iteration and S-iteration schemes for contractive-like operators. Our results extend and improve some known results in the literature.

        MSC: 47H10.

        Keywords

        new multi-step iteration S-iteration data dependence contractive-like operator

        1 Introduction

        Contractive mappings and iteration procedures are some of the main tools in the study of fixed point theory. There are many contractive mappings and iteration schemes that have been introduced and developed by several authors to serve various purposes in the literature of this highly active research area, viz., [112] among others.

        Whether an iteration method used in any investigation converges to a fixed point of a contractive type mapping corresponding to a particular iteration process is of utmost importance. Therefore it is natural to see many works related to the convergence of iteration methods such as [1322].

        Fixed point theory is concerned with investigating a wide variety of issues such as the existence (and uniqueness) of fixed points, the construction of fixed points, etc. One of these themes is data dependency of fixed points. Data dependency of fixed points has been the subject of research in fixed point theory for some time now, and data dependence research is an important theme in its own right.

        Several authors who have made contributions to the study of data dependence of fixed points are Rus and Muresan [23], Rus et al.[24, 25], Berinde [26], Espínola and Petruşel [27], Markin [28], Chifu and Petruşel [29], Olantiwo [30, 31], Şoltuz [32, 33], Şoltuz and Grosan [34], Chugh and Kumar [35] and the references therein.

        This paper is organized as follows. In Section 1 we present a brief survey of some known contractive mappings and iterative schemes and collect some preliminaries that will be used in the proofs of our main results. In Section 2 we show that the convergence of a new multi-step iteration, which is a special case of the Jungck multistep-SP iterative process defined in [36], and S-iteration (due to Agarwal et al.) can be used to approximate the fixed points of contractive-like operators. Motivated by the works of Şoltuz [32, 33], Şoltuz and Grosan [34], and Chugh and Kumar [35], we prove two data dependence results for the new multi-step iteration and S-iteration schemes by employing contractive-like operators.

        As a background of our exposition, we now mention some contractive mappings and iteration schemes.

        In [37] Zamfirescu established an important generalization of the Banach fixed point theorem using the following contractive condition. For a mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq1_HTML.gif , there exist real numbers a, b, c satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq3_HTML.gif such that, for each pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq4_HTML.gif , at least one of the following is true:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ1_HTML.gif
        (1.1)
        A mapping T satisfying the contractive conditions (z1), (z2) and (z3) in (1.1) is called a Zamfirescu operator. An operator satisfying condition (z2) is called a Kannan operator, while the mapping satisfying condition (z3) is called a Chatterjea operator. As shown in [13], the contractive condition (1.1) leads to
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ2_HTML.gif
        (1.2)

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq5_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq7_HTML.gif , and it was shown that this class of operators is wider than the class of Zamfirescu operators. Any mapping satisfying condition (b1) or (b2) is called a quasi-contractive operator.

        Extending the above definition, Osilike and Udomene [20] considered operators T for which there exist real numbers http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq8_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq7_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq5_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ3_HTML.gif
        (1.3)
        Imoru and Olantiwo [38] gave a more general definition: An operator T is called a contractive-like operator if there exists a constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq7_HTML.gif and a strictly increasing and continuous function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq9_HTML.gif , with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq10_HTML.gif , such that for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq11_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ4_HTML.gif
        (1.4)
        A map satisfying (1.4) need not have a fixed point, even if E is complete. For example, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq12_HTML.gif and define T by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equa_HTML.gif

        WLOG, assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq13_HTML.gif . Then, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq14_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq15_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq16_HTML.gif , and (1.4) is automatically satisfied.

        If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq17_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq18_HTML.gif .

        Define φ by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq19_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq20_HTML.gif . Then φ is increasing, continuous, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq10_HTML.gif . Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq21_HTML.gif so that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq22_HTML.gif .

        Therefore
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equb_HTML.gif

        for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq23_HTML.gif , and (1.4) is satisfied for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq17_HTML.gif . But T has no fixed point.

        However, using (1.4) it is obvious that if T has a fixed point, then it is unique.

        From now on, we demand that ℕ denotes the set of all nonnegative integers. Let X be a Banach space, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq24_HTML.gif be a nonempty closed, convex subset of X, and let T be a self-map on E. Define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq25_HTML.gif to be the set of fixed points of T. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq26_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq27_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq28_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq29_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq31_HTML.gif be real sequences in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq32_HTML.gif satisfying certain conditions.

        In [5] Rhoades and Şoltuz introduced a multi-step iterative procedure given by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ5_HTML.gif
        (1.5)
        The sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq33_HTML.gif defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ6_HTML.gif
        (1.6)

        is known as the S-iteration process (see [12, 17, 39]).

        Thianwan [6] defined a two-step iteration http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq34_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ7_HTML.gif
        (1.7)
        Recently Phuengrattana and Suantai [7] introduced an SP iteration method defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ8_HTML.gif
        (1.8)
        We shall employ the following iterative process. For an arbitrary fixed order http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq31_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ9_HTML.gif
        (1.9)
        or, in short,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ10_HTML.gif
        (1.10)
        where
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ11_HTML.gif
        (1.11)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ12_HTML.gif
        (1.12)

        Remark 1 If each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq35_HTML.gif , then SP iteration (1.8) reduces to two-step iteration (1.7). By taking http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq36_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq37_HTML.gif in (1.10), we obtain iterations (1.8) and (1.7), respectively.

        We shall need the following definition and lemma in the sequel.

        Definition 1[40]

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq38_HTML.gif be two operators. We say that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq39_HTML.gif is an approximate operator for T if, for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq40_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equc_HTML.gif

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq41_HTML.gif .

        Lemma 1[34]

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq42_HTML.gif be a nonnegative sequence for which one assumes that there exists an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq43_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq44_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equd_HTML.gif
        is satisfied, where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq45_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq46_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq47_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq48_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq49_HTML.gif . Then the following holds:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Eque_HTML.gif

        2 Main results

        For simplicity we use the following notation throughout this section.

        For any iterative process, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq50_HTML.gif denote iterative sequences associated to T and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq39_HTML.gif , respectively.

        Theorem 1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq1_HTML.gif be a map satisfying (1.4) with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq51_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq33_HTML.gif be a sequence defined by (1.10), then the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq52_HTML.gif converges to the unique fixed point of T.

        Proof The proof can be easily obtained by using the argument in the proof of ([36], Theorem 3.1). □

        This result allows us to give the next theorem.

        Theorem 2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq1_HTML.gif be a map satisfying (1.4) with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq53_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq39_HTML.gif be an approximate operator of T as in Definition 1. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq54_HTML.gif be two iterative sequences defined by (1.10) with real sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq55_HTML.gif satisfying (i) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq56_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq57_HTML.gif , (ii) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq58_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq60_HTML.gif , then we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equf_HTML.gif
        Proof For given http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq61_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq62_HTML.gif , we consider the following multi-step iteration for T and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq39_HTML.gif :
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ13_HTML.gif
        (2.1)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ14_HTML.gif
        (2.2)
        Thus, from (1.4), (2.1) and (2.2), we have the following inequalities.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ15_HTML.gif
        (2.3)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ16_HTML.gif
        (2.4)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ17_HTML.gif
        (2.5)
        Combining (2.3), (2.4) and (2.5), we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ18_HTML.gif
        (2.6)
        Thus, by induction, we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ19_HTML.gif
        (2.7)
        Again, using (1.4), (2.1) and (2.2), we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ20_HTML.gif
        (2.8)
        Substituting (2.8) in (2.7), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ21_HTML.gif
        (2.9)
        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq7_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq63_HTML.gif , for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq57_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ22_HTML.gif
        (2.10)
        From inequality (2.10) and assumption (i) in (2.9), it follows
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ23_HTML.gif
        (2.11)
        Define
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equg_HTML.gif
        From Theorem 1 it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq64_HTML.gif . Since T satisfies condition (1.4) and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq65_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ24_HTML.gif
        (2.12)
        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq66_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq67_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq57_HTML.gif , using (1.4) and (1.10), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ25_HTML.gif
        (2.13)

        It is easy to see from (2.13) that this result is also valid for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq68_HTML.gif .

        Since φ is continuous, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ26_HTML.gif
        (2.14)
        Hence an application of Lemma 1 to (2.11) leads to
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ27_HTML.gif
        (2.15)

         □

        As shown by Hussain et al. ([22], Theorem 8), in an arbitrary Banach space X, the S-iteration http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq33_HTML.gif given by (1.6) converges to the fixed point of T, where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq1_HTML.gif is a mapping satisfying condition (1.3).

        Theorem 3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq1_HTML.gif be a map satisfying (1.4) with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq53_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq33_HTML.gif be defined by (1.6) with real sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq69_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq70_HTML.gif . Then the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq33_HTML.gif converges to the unique fixed point of T.

        Proof The argument is similar to the proof of Theorem 8 of [22], and is thus omitted. □

        We now prove the result on data dependence for the S-iterative procedure by utilizing Theorem 3.

        Theorem 4 Let T, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq39_HTML.gif be two operators as in Theorem 2. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq34_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq50_HTML.gif be S-iterations defined by (1.6) with real sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq71_HTML.gif satisfying (i) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq72_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq67_HTML.gif , and (ii) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq70_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq60_HTML.gif , then we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equh_HTML.gif
        Proof For a given http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq61_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq62_HTML.gif , we consider the following iteration for T and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq39_HTML.gif :
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ28_HTML.gif
        (2.16)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ29_HTML.gif
        (2.17)
        Using (1.4), (2.16) and (2.17), we obtain the following estimates:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ30_HTML.gif
        (2.18)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ31_HTML.gif
        (2.19)
        Combining (2.18) and (2.19), we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ32_HTML.gif
        (2.20)
        For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq73_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq7_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ33_HTML.gif
        (2.21)
        It follows from assumption (i) that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ34_HTML.gif
        (2.22)
        Therefore, combining (2.22) and (2.21) to (2.20) gives
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ35_HTML.gif
        (2.23)
        or, equivalently,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ36_HTML.gif
        (2.24)
        Now define
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equi_HTML.gif
        From Theorem 3, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq64_HTML.gif . Since T satisfies condition (1.4), and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq65_HTML.gif , using an argument similar to that in the proof of Theorem 2,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ37_HTML.gif
        (2.25)
        Using the fact that φ is continuous, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ38_HTML.gif
        (2.26)
        An application of Lemma 1 to (2.24) leads to
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_Equ39_HTML.gif
        (2.27)

         □

        3 Conclusion

        Since the iterative schemes (1.7) and (1.8) are special cases of the iterative process (1.10), Theorem 1 generalizes Theorem 2.1 of [19] and Theorem 2.1 of [18]. By taking http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq36_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq37_HTML.gif in Theorem 2, data dependence results for the iterative schemes (1.8) and (1.7) can be easily obtained. For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_411_IEq36_HTML.gif , Theorem 2 reduces to Theorem 3.2 of [35]. Since condition (1.4) is more general than condition (1.3), Theorem 3 generalizes Theorem 8 of [22].

        Declarations

        Acknowledgements

        The first two authors would like to thank Yıldız Technical University Scientific Research Projects Coordination Unit under project number BAPK 2012-07-03-DOP02 for financial support during the preparation of this manuscript.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Faculty of Science and Letters, Yildiz Technical University
        (2)
        Department of Mathematical Engineering, Faculty of Chemical and Metallurgical Engineering, Yildiz Technical University
        (3)
        Department of Mathematics, Indiana University

        References

        1. Rhoades BE: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 1977, 226:257–290.MathSciNetMATHView Article
        2. Mann WR: Mean value methods in iterations. Proc. Am. Math. Soc. 1953, 4:506–510.MATHView Article
        3. Ishikawa S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44:147–150.MathSciNetMATHView Article
        4. Noor MA: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 2000, 251:217–229.MathSciNetMATHView Article
        5. Rhoades BE, Şoltuz SM: The equivalence between Mann-Ishikawa iterations and multistep iteration. Nonlinear Anal. 2004, 58:219–228.MathSciNetMATHView Article
        6. Thianwan S: Common fixed points of new iterations for two asymptotically nonexpansive nonself mappings in a Banach space. J. Comput. Appl. Math. 2008. doi:10.1016/j.cam.2008.05.051
        7. Phuengrattana W, Suantai S: On the rate of convergence of Mann, Ishikawa, Noor and SP iterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math. 2011, 235:3006–3014.MathSciNetMATHView Article
        8. Glowinski R, Le Tallec P: Augmented Langrangian and Operator Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia; 1989.View Article
        9. Xu B, Noor MA: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations. J. Math. Anal. Appl. 1998, 224:91–101.MathSciNetMATHView Article
        10. Takahashi W: Iterative methods for approximation of fixed points and their applications. J. Oper. Res. Soc. Jpn. 2000, 43:87–108.MATHView Article
        11. Das G, Debata JP: Fixed points of quasi-nonexpansive mappings. Indian J. Pure Appl. Math. 1986, 17:1263–1269.MathSciNetMATH
        12. Agarwal RP, O’Regan D, Sahu DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 2007, 8:61–79.MathSciNetMATH
        13. Berinde V: On the convergence of the Ishikawa iteration in the class of quasi contractive operators. Acta Math. Univ. Comen. 2004, 73:119–126.MathSciNetMATH
        14. Chidume CE, Chidume CO: Convergence theorem for fixed points of uniformly continuous generalized phihemicontractive mappings. J. Math. Anal. Appl. 2005, 303:545–554.MathSciNetMATHView Article
        15. Chidume CE, Chidume CO: Iterative approximation of fixed points of nonexpansive mappings. J. Math. Anal. Appl. 2006, 318:288–295.MathSciNetMATHView Article
        16. Suantai S: Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings. J. Math. Anal. Appl. 2005, 311:506–517.MathSciNetMATHView Article
        17. Sahu DR: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory Appl. 2011, 12:187–204.MathSciNetMATH
        18. Yıldırım İ, Özdemir M, Kızıltunç H: On the convergence of a new two-step iteration in the class of quasi-contractive operators. Int. J. Math. Anal. 2009, 3:1881–1892.MATH
        19. Chugh R, Kumar V: Strong convergence of SP iterative scheme for quasi-contractive operators in Banach spaces. Int. J. Comput. Appl. 2011, 31:21–27.
        20. Osilike MO, Udomene A: Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings. Indian J. Pure Appl. Math. 1999, 30:1229–1234.MathSciNetMATH
        21. Rafiq A: On the convergence of the three step iteration process in the class of quasi-contractive operators. Acta Math. Acad. Paedagog. Nyházi. 2006, 22:305–309.MathSciNetMATH
        22. Hussain N, Rafiq A, Damjanović B, Lazović R: On rate of convergence of various iterative schemes. Fixed Point Theory Appl. 2011., 2011: Article ID 45. doi:10.1186/1687–1812–2011–45
        23. Rus IA, Muresan S: Data dependence of the fixed points set of weakly Picard operators. Stud. Univ. Babeş-Bolyai, Math. 1998, 43:79–83.MathSciNetMATH
        24. Rus IA, Petruşel A, Sîntamarian A: Data dependence of the fixed points set of multivalued weakly Picard operators. Stud. Univ. Babeş-Bolyai, Math. 2001, 46:111–121.MATH
        25. Rus IA, Petruşel A, Sîntamarian A: Data dependence of the fixed point set of some multivalued weakly Picard operators. Nonlinear Anal., Theory Methods Appl. 2003, 52:1947–1959.MATHView Article
        26. Berinde V: On the approximation of fixed points of weak contractive mappings. Carpath. J. Math. 2003, 19:7–22.MathSciNetMATH
        27. Espínola R, Petruşel A: Existence and data dependence of fixed points for multivalued operators on gauge spaces. J. Math. Anal. Appl. 2005, 309:420–432.MathSciNetMATHView Article
        28. Markin JT: Continuous dependence of fixed point sets. Proc. Am. Math. Soc. 1973, 38:545–547.MathSciNetMATHView Article
        29. Chifu C, Petruşel G: Existence and data dependence of fixed points and strict fixed points for contractive-type multivalued operators. Fixed Point Theory Appl. 2007. doi:10.1155/2007/34248
        30. Olatinwo MO: Some results on the continuous dependence of the fixed points in normed linear space. Fixed Point Theory Appl. 2009, 10:151–157.MathSciNetMATH
        31. Olatinwo MO:On the continuous dependence of the fixed points for -contractive-type operators. Kragujev. J. Math. 2010, 34:91–102.MathSciNetMATH
        32. Şoltuz SM: Data dependence for Mann iteration. Octogon Math. Mag. 2001, 9:825–828.
        33. Şoltuz SM: Data dependence for Ishikawa iteration. Lect. Mat. 2004, 25:149–155.MathSciNet
        34. Şoltuz SM, Grosan T: Data dependence for Ishikawa iteration when dealing with contractive like operators. Fixed Point Theory Appl. 2008., 2008: Article ID 242916. doi:10.1155/2008/242916
        35. Chugh R, Kumar V: Data dependence of Noor and SP iterative schemes when dealing with quasi-contractive operators. Int. J. Comput. Appl. 2011, 40:41–46.
        36. Akewe H: Strong convergence and stability of Jungck-multistep-SP iteration for generalized contractive-like inequality operators. Adv. Nat. Sci. 2012, 5:21–27.
        37. Zamfirescu T: Fix point theorems in metric spaces. Arch. Math. 1972, 23:292–298.MathSciNetMATHView Article
        38. Imoru CO, Olantiwo MO: On the stability of Picard and Mann iteration processes. Carpath. J. Math. 2003, 19:155–160.MATH
        39. Agarwal RP, O’Regan D, Sahu DR: Fixed Point Theory for Lipschitzian Type-Mappings with Applications. Springer, New York; 2009.MATH
        40. Berinde V: Iterative Approximation of Fixed Points. Springer, Berlin; 2007.MATH

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        © Gürsoy et al.; licensee Springer. 2013

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