Some Convergence and Stability Results for Two New Kirk Type Hybrid Fixed Point Iterative Algorithms

Fixed point theory has an important role in the study of nonlinear phenomena. This theory has been applied in a wide range of disciplines in various areas such as science, technology, and economics; see, for example, [1–5]. The importance of this theory has attracted researchers’ interest, and consequently numerous fixed point theorems have been put forward; see, for example, [6–17] and the references included therein. In this highly dynamic area, one of themost celebrated theorems amongst hundreds is Banach fixed point theorem (also known as the contraction mapping theorem or contraction mapping principle) [7]. An important process which is called iteration method arises naturally during proving of this theorem. A fixed point iteration method is given by a general form as follows:


Introduction and Preliminaries
Fixed point theory has an important role in the study of nonlinear phenomena.This theory has been applied in a wide range of disciplines in various areas such as science, technology, and economics; see, for example, [1][2][3][4][5].The importance of this theory has attracted researchers' interest, and consequently numerous fixed point theorems have been put forward; see, for example, [6][7][8][9][10][11][12][13][14][15][16][17] and the references included therein.In this highly dynamic area, one of the most celebrated theorems amongst hundreds is Banach fixed point theorem (also known as the contraction mapping theorem or contraction mapping principle) [7].An important process which is called iteration method arises naturally during proving of this theorem.A fixed point iteration method is given by a general form as follows: where  is an ambient space,  0 is an arbitrary initial point,  :  →  is an operator, and  is some function.For example, if (,   ) =   in (1), then we obtain well-known Picard iteration [18] as follows: Iterative methods are important instruments commonly used in the study of fixed point theory.These powerful and useful tools enable us to find solutions for a wide variety of problems that arise in many branches of the above mentioned areas.This is a reason, among a number of reasons, why researchers are seeking new iteration methods or trying to improve existing methods over the years.In this respect, it is not surprising to see a number of papers dealing with the study of iterative methods to investigate various important themes; see, for example, [19][20][21][22][23][24][25][26][27].
The purpose of this paper is to introduce two new Kirk type hybrid iteration methods and to show that these iterative methods can be used to approximate fixed points of certain class of contractive operators.Furthermore, we prove that these iterative methods are stable with respect to the same class of contractive operators.
As a background to our exposition, we describe some iteration schemes and contractive type mappings.
Remark 3 (see [20,28]).A map satisfying (7) need not have a fixed point.However, using (7), it is obvious that if  has a fixed point, then it is unique.
We will say that the iterative sequence {  } ∞ =0 is -stable or stable with respect to  if and only if lim Lemma 5 (see [8]).If  is a real number such that  ∈ [0, 1) and {  } ∞ =0 is a sequence of nonnegative numbers such that lim  → ∞   = 0, then, for any sequence of positive numbers one has lim  → ∞   = 0.

Main Results
For simplicity we assume in the following four theorems that  is a normed linear space,  is a self map of  satisfying the contractive condition (7) with some fixed point  ∈   , and  : R + → R + is a subadditive monotone increasing function such that (0) = 0 and () ≤ (),  ≥ 0,  ∈ R + .
Theorem 7. Let {  } ∈N be a sequence generated by the Kirkmultistep-SP iterative scheme (5).Then the iterative sequence {  } ∈N converges strongly to .
Proof.The uniqueness of  follows from (7).We will now prove that   → .
Using Lemma 6 we have , − Combining ( 29), (30), and (31) we obtain Thus, by induction, we get Again using (5) and Lemma 6 we have Substituting ( 34) into (33) we derive Again define Using the same argument as that of the first part of the proof we obtain  ∈ (0, 1).Hence (35) becomes It therefore follows from assumption lim  → ∞   =  that   → 0 as  → ∞.
Theorem 9. Let {  } ∈N be a sequence generated by the Kirk-S iterative scheme (6).Then, the iterative sequence {  } ∈N converges strongly to .
Proof.The uniqueness of  follows from (7).We will now prove that   → .