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 fixed point of contraction mappings with the help of the new iteration method.


Introduction
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.
Very recently the following iterative methods are introduced in [1] and [2], respectively: where is a nonempty convex subset of a Banach space , is a self map of , and { } ∞ =0 , { } ∞ =0 are real sequences in [0, 1].
While the iterative method (1) fails to be named in [1], the iterative method (2) is called Picard-S iteration method in [2]. Since iterative method (1) is a special case of SP iterative method of Phuengrattana and Suantai [3], we will call it here Modified SP iterative method.
It was shown in [1] that Modified SP iterative method (1) is faster than all Picard [4], Mann [5], Ishikawa [6], and [7] 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 [2] showed that Picard-S iteration method (2) is also faster than all Picard [4], Mann [5], Ishikawa [6], [7], 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. The Scientific World Journal Definition 1 (see [8] Definition 2 (see [8]). Assume that for two fixed point iteration processes { } ∞ =0 and {V } ∞ =0 both converging to the same fixed point , the following error estimates, to .

Main Results
Theorem 6. Let be a nonempty closed convex subset of a Banach space and : → a contraction map satisfying condition (3).
converges to a unique fixed point of , say * .
Theorem 7. Let , , and with fixed point * be as in be two iterative sequences defined by (1) and (2), respectively, with real sequences = ∞. Then the following are equivalent: Proof. We will prove (i)⇒(ii). Now by using (1) Since hence the assumption ∑ ∞ =0 = ∞ leads to Thus all conditions of Lemma 4 are fulfilled by (12), and so lim → ∞ ‖ − ‖ = 0. Since Using the same argument as above one can easily show the implication (ii)⇒(i); thus it is omitted here. Proof. The following inequality comes from inequality (10) of Theorem 6:

Theorem 8. Let , , and with fixed point * be as in
The following inequality is due to ([2], inequality (2.5) of Theorem 1): Define Since . (20) Therefore, taking into account assumption (i), we obtain It thus follows from well-known ratio test that ∑ ∞ =0 < ∞. Hence, we have lim → ∞ = 0 which implies that { } ∞ =0 is faster than { } ∞ =0 . 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 [12] for the sake of consistent comparison.
We are now able to establish the following data dependence result.
The Scientific World Journal where > 0 is a fixed number and ∈ (0, 1).