Some Convergence And Stability Results For The Kirk Multistep And Kirk-Sp Fixed Point Iterative Algorithms For Contractive-Like Operators In Normed Linear Spaces

The purpose of this paper is to introduce a new Kirk type iterative algorithm called Kirk multistep iteration and to study its convergence. We also prove some theorems related with the stability results for the Kirk-multistep and Kirk-SP iterative processes by employing certain contractive-like operators. Our results generalize and unify some other results in the literature.


Introduction and Preliminaries
This paper is organized as follows. Section 1 outlines some known contractive mappings and iterative schemes and collects some preliminaries that will be used in the proofs of our main results. We then propose a new Kirk type iterative process called Kirk multistep iteration. Section 2 presents a result dealing with the convergence of this new iterative procedure, which unifies and extends some other iterative schemes in the existing literature. Also we prove some theorems related to the stability of the Kirk multistep and Kirk-SP iterative processes by employing certain contractivelike operators.
Fixed point iterations are commonly used to solve nonlinear equations arising in physical systems. Such equations can be transformed into a fixed point equation = which is solved by some iterative processes of form +1 = ( , ), = 0, 1, 2, . . ., that converges to a fixed point of . This is a reason, among a number of reasons, why there is presently a great deal of interest in the introduction and development of various iterative algorithms. Consequently iteration schemes abound in the literature of fixed point theory, for which fixed points of operators have been approximated over the years by various authors, for example, [1][2][3][4][5][6][7][8][9][10].
A particular fixed point iteration generates a theoretical sequence { } ∞ =0 . In applications, various errors (e.g., roundoff or discretization of the function etc.) occur during computation of the sequence { } ∞ =0 . Because of these errors we cannot obtain the theoretical sequence { } ∞ =0 , but an approximate sequence { } ∞ =0 instead. We will say that the iterative process is -stable or stable with respect to if and only if { } ∞ =0 converges to a fixed point of , then { } ∞ =0 converges to = .
The initiator of this kind study is Urabe [18] while a formal definition for the stability of general iterative schemes is given by Harder and Hicks [19,20] as follows.
A pioneering result on the stability of iterative procedures established in metric space for the Picard iteration is due to Ostrowski [21], which states that: Let ( , ) be a complete metric space and : → a Banach contraction mapping, that is, where ∈ [0, 1). Let ∈ be the fixed point of , 0 ∈ , and +1 = , = 0, 1, 2, . . .. Suppose that { } ∞ =0 is a sequence in and = ( +1 , ). Then Moreover, lim → ∞ = ⇔ lim → ∞ = 0. Using Definition 1, Harder and Hicks [19,20] proved some stability theorems for well-known Picard, Mann, and Kirk's iterations by employing several classes of contractive type operators. Rhoades [23,24] Later Osilike [27] further generalized and extended some of the results in [23] by using a large class of contractive type operators satisfying the following condition, which is more general than those of Rhoades [23,24] and Harder and Hicks [20]: for some ∈ [0, 1), ≥ 0, and for all , ∈ . By employing the contractive condition (9), Osilike and Udomene proved some stability results for the Picard, Ishikawa, and Kirk's iteration in [29] where a new and shorter method than those mentioned above was used. Using the same method of proof as in [29], Berinde [26] again established the stability results in Harder and Hicks [20].
Remark 2 (see [2,11]). A map satisfying (10) need not have a fixed point. However, using (10), it is obvious that if has a fixed point, then it is unique.
Recently Chugh and Kumar [15] improved and extended the results of [16] and some of the references cited therein by introducing the Kirk-Noor iterative algorithm.
We end this section with some lemmas which will be useful in proving our main results.

Main Results
For simplicity we assume in the following three theorems that is a normed linear space, is a self-map of satisfying the contractive condition (10) with ̸ = 0, and : R + → R + is a subadditive monotone increasing function such that (0) = 0 and ( ) ≤ ( ), ≥ 0, ∈ R + . Theorem 6. Let { } ∈N be a sequence generated by the Kirk multistep iterative scheme (4). Suppose that has a fixed point . Then the iterative sequence { } ∈N converges strongly to .
Proof. The uniqueness of follows from (12). We will now prove that → . Using (4) and Lemma 4, we get 4 Abstract and Applied Analysis By combining (13), (14), (15), and (16) we obtain Continuing the above process we have Abstract and Applied Analysis 5 Using again (4) and Lemma 4, we get Substituting (19) into (18) we derive Define ) .
Using Lemma 4 we have 8 Abstract and Applied Analysis It follows from the relation (32), (33), and (34) that Thus, by induction, we get Utilizing (4) and Lemma 4, we obtain Abstract and Applied Analysis 9 Substituting (37) into (36) gives Again define ) .