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We introduce Kirk-multistep-SP and Kirk-S iterative algorithms and we prove some convergence and stability results for these iterative algorithms. Since these iterative algorithms are more general than some other iterative algorithms in the existing literature, our results generalize and unify some other results in the literature.

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, [

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, [

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.

The following multistep-SP iteration was employed in [

By taking

A sequence

Continuing the above trend, we will introduce and employ the following iterative schemes which are called Kirk-multistep-SP and Kirk-S iterations, respectively:

By taking

We end this section with some definitions and lemmas which will be useful in proving our main results.

Let

By taking

A map satisfying (

Let

If

Let

For simplicity we assume in the following four theorems that

Let

The uniqueness of

Using Kirk-multistep-SP iterative process (

Let

Let

It follows from (

Again using (

Now suppose that

Using Lemma

Hence (

Let

The uniqueness of

Using Kirk-S iterative process (

Let

Let

It follows from (

Now suppose that

Using Lemma

Hence (

Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research has been supported by Yıldız Technical University Scientific Research Projects Coordination Department, Project no. BAPK 2012-07-03-DOP02.