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Convergence Analysis for a Modified SP Iterative Method
We consider a new iterative method due to Kadioglu and Yildirim (2014) for further investigation. We study convergence analysis of this iterative method when applied to class of contraction mappings. Furthermore, we give a data dependence result for fi…xed point of contraction mappings with the help of the new iteration method.
Recent progress in nonlinear science reveals that iterative methods are most powerful tools which are used to approximate solutions of nonlinear problems whose solutions are inaccessible analytically. Therefore, in recent years, an intensive interest has been devoted to developing faster and more effective iterative methods for solving nonlinear problems arising from diverse branches in science and engineering.
While the iterative method (1) fails to be named in , the iterative method (2) is called Picard-S iteration method in . Since iterative method (1) is a special case of SP iterative method of Phuengrattana and Suantai , we will call it here Modified SP iterative method.
It was shown in  that Modified SP iterative method (1) is faster than all Picard , Mann , Ishikawa , and  iterative methods in the sense of Definitions 1 and 2 given below for the class of contraction mappings satisfying Using the same class of contraction mappings (3), Gürsoy and Karakaya  showed that Picard-S iteration method (2) is also faster than all Picard , Mann , Ishikawa , , and some other iterative methods in the existing literature.
In this paper, we show that Modified SP iterative method converges to the fixed point of contraction mappings (3). Also, we establish an equivalence between convergence of iterative methods (1) and (2). For the sake of completness, we give a comparison result between the rate of convergences of iterative methods (1) and (2), and it thus will be shown that Picard-S iteration method is still the fastest method. Finally, a data dependence result for the fixed point of the contraction mappings (3) is proven.
The following definitions and lemmas will be needed in order to obtain the main results of this paper.
Definition 1 (see ). Let and be two sequences of real numbers with limits and , respectively. Suppose that exists.(i)If , then we say that converges faster to than to .(ii)If , then we say that and have the same rate of convergence.
Definition 2 (see ). Assume that for two fixed point iteration processes and both converging to the same fixed point , the following error estimates, are available where and are two sequences of positive numbers (converging to zero). If converges faster than , then converges faster than to .
Definition 3 (see ). Let be two operators. We say that is an approximate operator of if for all and for a fixed we have
Lemma 4 (see ). Let and be nonnegative real sequences and suppose that for all , , , and as holds. Then .
Lemma 5 (see ). Let be a nonnegative sequence such that there exists , for all ; the following inequality holds. Consider where , for all , and , . Then
2. Main Results
Theorem 6. Let be a nonempty closed convex subset of a Banach space and a contraction map satisfying condition (3). Let be an iterative sequence generated by (1) with real sequences , in satisfying . Then converges to a unique fixed point of , say .
Proof. The well-known Picard-Banach theorem guarantees the existence and uniqueness of . We will show that as . From (3) and (1) we have By induction on the inequality (10), we derive Since , taking the limit of both sides of inequality (11) yields ; that is, as .
Theorem 7. Let , , and with fixed point be as in Theorem 6. Let , be two iterative sequences defined by (1) and (2), respectively, with real sequences , in satisfying . Then the following are equivalent:(i) converges to ;(ii) converges to .
Proof. We will prove (i)(ii). Now by using (1), (2), and condition (3), we have
Define Since and , which implies as . Since also , for all hence the assumption leads to Thus all conditions of Lemma 4 are fulfilled by (12), and so . Since
Using the same argument as above one can easily show the implication (ii)(i); thus it is omitted here.
Theorem 8. Let , , and with fixed point be as in Theorem 6. Let , be real sequences in satisfying(i).
For given , consider iterative sequences and defined by (1) and (2), respectively. Then converges to faster than does.
Proof. The following inequality comes from inequality (10) of Theorem 6:
The following inequality is due to (, inequality of Theorem 1):
Since Therefore, taking into account assumption (i), we obtain It thus follows from well-known ratio test that . Hence, we have which implies that is faster than .
In order to support analytical proof of Theorem 8 and to illustrate the efficiency of Picard-S iteration method (2), we will use a numerical example provided by Sahu  for the sake of consistent comparison.
Example 9. Let and . Let be a mapping and for all . is a contraction with contractivity factor and ; see . Take with initial value . Tables 1, 2, and 3 show that Picard-S iteration method (2) converges faster than all SP , Picard , Mann , Ishikawa , , CR , , Noor , and Normal-  iteration methods including a new three-step iteration method due to Abbas and Nazir .
We are now able to establish the following data dependence result.
Theorem 10. Let be an approximate operator of satisfying condition (3). Let be an iterative sequence generated by (1) for and define an iterative sequence as follows: where , are real sequences in satisfying (i) , (ii) for all , and (iii) . If and such that as , then we have where is a fixed number and .
Proof. It follows from (1), (3), (22), and assumption (ii) that
From assumption (i) we have and thus, inequality (24) becomes
Denote that It follows from Lemma 5 that From Theorem 6 we know that . Thus, using this fact together with the assumption we obtain
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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