Data dependence results of new multi-step and S-iterative schemes for contractive-like operators
© Gürsoy et al.; licensee Springer. 2013
Received: 5 December 2012
Accepted: 10 March 2013
Published: 28 March 2013
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.
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.
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.
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.
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.
Definition 1 
for all .
Lemma 1 
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 1 Let 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.
It is easy to see from (2.13) that this result is also valid for .
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 3 Let 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 of T.
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.
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 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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