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.
Keywords:new multi-step iteration; S-iteration; data dependence; contractive-like operator
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., [1-12] 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 [13-22].
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 , Rus et al.[24,25], Berinde , Espínola and Petruşel , Markin , Chifu and Petruşel , Olantiwo [30,31], Şoltuz [32,33], Şoltuz and Grosan , Chugh and Kumar  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 , 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 , and Chugh and Kumar , 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  Zamfirescu established an important generalization of the Banach fixed point theorem using the following contractive condition. For a mapping , there exist real numbers a, b, c satisfying , such that, for each pair , at least one of the following is true:
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 , the contractive condition (1.1) leads to
for all , where , , 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  considered operators T for which there exist real numbers and such that for all ,
Imoru and Olantiwo  gave a more general definition: An operator T is called a contractive-like operator if there exists a constant and a strictly increasing and continuous function , with , such that for each ,
A map satisfying (1.4) need not have a fixed point, even if E is complete. For example, let and define T by
WLOG, assume that . Then, for or , , and (1.4) is automatically satisfied.
If , then .
Define φ by for any . Then φ is increasing, continuous, and . Also, so that .
for any , and (1.4) is satisfied for . 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 be a nonempty closed, convex subset of X, and let T be a self-map on E. Define to be the set of fixed points of T. Let , , and , , be real sequences in satisfying certain conditions.
In  Rhoades and Şoltuz introduced a multi-step iterative procedure given by
The sequence defined by
Thianwan  defined a two-step iteration by
Recently Phuengrattana and Suantai  introduced an SP iteration method defined by
We shall employ the following iterative process. For an arbitrary fixed order ,
or, in short,
Remark 1 If each , then SP iteration (1.8) reduces to two-step iteration (1.7). By taking and in (1.10), we obtain iterations (1.8) and (1.7), respectively.
We shall need the following definition and lemma in the sequel.
Let be two operators. We say that is an approximate operator for T if, for some , we have
for all .
Let be a nonnegative sequence for which one assumes that there exists an such that for all ,
is satisfied, where for all , and , . Then the following holds:
2 Main results
For simplicity we use the following notation throughout this section.
For any iterative process, and denote iterative sequences associated to T and , respectively.
Theorem 1Let be a map satisfying (1.4) with , and let be a sequence defined by (1.10), then the sequence converges to the unique fixed point of T.
Proof The proof can be easily obtained by using the argument in the proof of (, Theorem 3.1). □
This result allows us to give the next theorem.
Theorem 2Let be a map satisfying (1.4) with , and let be an approximate operator ofTas in Definition 1. Let , be two iterative sequences defined by (1.10) with real sequences satisfying (i) , , (ii) . If and , then we have
Proof For given and , we consider the following multi-step iteration for T and :
Thus, from (1.4), (2.1) and (2.2), we have the following inequalities.
Combining (2.3), (2.4) and (2.5), we obtain
Thus, by induction, we get
Again, using (1.4), (2.1) and (2.2), we get
Substituting (2.8) in (2.7), we have
Since and , for , we have
From inequality (2.10) and assumption (i) in (2.9), it follows
From Theorem 1 it follows that . Since T satisfies condition (1.4) and ,
Since , , , using (1.4) and (1.10), we have
It is easy to see from (2.13) that this result is also valid for .
Since φ is continuous, we have
Hence an application of Lemma 1 to (2.11) leads to
As shown by Hussain et al. (, Theorem 8), in an arbitrary Banach space X, the S-iteration given by (1.6) converges to the fixed point of T, where is a mapping satisfying condition (1.3).
Theorem 3Let be a map satisfying (1.4) with , and let be defined by (1.6) with real sequences satisfying . Then the sequence converges to the unique fixed point ofT.
Proof The argument is similar to the proof of Theorem 8 of , and is thus omitted. □
We now prove the result on data dependence for the S-iterative procedure by utilizing Theorem 3.
Theorem 4LetT, be two operators as in Theorem 2. Let , be S-iterations defined by (1.6) with real sequences satisfying (i) , , and (ii) . If and , then we have
Proof For a given and , we consider the following iteration for T and :
Using (1.4), (2.16) and (2.17), we obtain the following estimates:
Combining (2.18) and (2.19), we get
For and ,
It follows from assumption (i) that
Therefore, combining (2.22) and (2.21) to (2.20) gives
From Theorem 3, we have . Since T satisfies condition (1.4), and , using an argument similar to that in the proof of Theorem 2,
Using the fact that φ is continuous, we have
An application of Lemma 1 to (2.24) leads to
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  and Theorem 2.1 of . By taking and in Theorem 2, data dependence results for the iterative schemes (1.8) and (1.7) can be easily obtained. For , Theorem 2 reduces to Theorem 3.2 of . Since condition (1.4) is more general than condition (1.3), Theorem 3 generalizes Theorem 8 of .
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
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.
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