#### Abstract

The purpose of this paper is to introduce Kirk-type new iterative schemes called Kirk-SP and Kirk-CR schemes and to study the convergence of these iterative schemes by employing certain quasi-contractive operators. By taking an example, we will compare Kirk-SP, Kirk-CR, Kirk-Mann, Kirk-Ishikawa, and Kirk-Noor iterative schemes for aforementioned class of operators. Also, using computer programs in C++, we compare the above-mentioned iterative schemes through examples of increasing, decreasing, sublinear, superlinear, and oscillatory functions.

#### 1. Introduction and Preliminaries

There is a close relationship between the problem of solving a nonlinear equation and that of approximating fixed points of a corresponding contractive-type operator. Consequently, there is a theoretical and practical interest in approximating fixed points of various contractive-type operators. Let be a complete metric space and a selfmap of . Suppose that is the set of fixed points of . There are several iterative schemes in the literature for which the fixed points of operators have been approximated over the years by various authors. In a complete metric space, the Picard iterative scheme is defined by which is used to approximate the fixed points of mappings satisfying the inequality: for all and . Condition (1.2) is called Banachβs contraction condition.

The following iteration schemes are now well known: where is a sequences of positive numbers in , due to Mann [1]. where is a sequences of positive numbers in , due to Kirk [2]. where and are sequences of positive numbers in , due to Ishikawa [3]. where , , and are sequences of positive numbers in , due to Noor [4].

In [5], Olatinwo introduced the Kirk-Mann and Kirk-Ishikawa iterative schemes as follows: where , , , , , and , are fixed integers, called as Kirk-Ishikawa iteration scheme: where , , , and is a fixed integer and is called as Kirk-Mann iteration scheme.

However, from [6], the Kirk-Noor iterative scheme is given by In [7], Phuengrattana and Suantai defined the SP iteration scheme as follows: where , , and are sequences of positive numbers in .

Recently, Chugh and Kumar introduced the following iteration scheme [8]: where , , and are sequences of positive numbers in .

*Remarks 1. *(1) If , then (1.6) reduces to the Ishikawa iteration scheme (1.5).

(2) If , then (1.6) reduces to the Mann iteration scheme (1.3).

(3) If , then (1.5) reduces to the Mann iteration scheme (1.3).

(4) If , then (1.10) reduces to the Mann iteration scheme (1.3).

In [9], Zamfirescu obtained the following interesting fixed point theorem.

Theorem 1.1. *Let be a complete metric space and a mapping for which there exists real numbers , , and c satisfying such that for each pair at least one of the following conditions holds:
**
Then, has a unique fixed point and the Picard iteration scheme defined by (1.1) converges to for any arbitrary but fixed .*

The operators satisfying the condition (1.12) are called Zamfirescu operators.

Berinde in [10] introduced a new class of operators on an arbitrary Banach space and satisfying for all and . He proved that this class is wider than the class of Zamfirescu operators and used the Ishikawa iteration scheme to approximate fixed points of this class of operators in an arbitrary Banach space given in the form of following theorem.

Theorem 1.2 (see [10]). * Let be a nonempty closed convex subset of an arbitrary Banach space and a mapping satisfying (1.13). Let be defined through the Ishikawa iteration scheme (1.5) and , where , are sequences of positive real numbers in with satisfying . Then, converges strongly to the fixed point of .*

However, in [11], Rafiq studied the convergence of the Noor iteration scheme [4] involving quasi-contractive operators.

Also several authors [11β16] have studied the equivalence between different iterative schemes: Εolutz [13, 14] proved that Picard, Mann, Ishikawa, and Noor iteration schemes are equilvalent for quasi-contractive operators. Recently, Chugh and Kumar [17] proved that, for quasi-contractive operators satisfying (1.13), Picard, Mann, Ishikawa, Noor, and SP iterative schemes are equivalent.

Fixed-point iterative schemes are designed to be applied in solving equations arising in physical formulation but there is no systematic study of numerical aspects of these iterative schemes. In computational mathematics, it is of vital interest to know which of the given iterative scheme converges faster to a desired solution, commonly known as rate of convergence. Rhoades in [18] compared the Mann and Ishikawa iterative schemes by concerning their rate of convergences. He illustrated the difference in the rate of convergence for increasing and deceasing functions (see also [19]). However, Olatinwo [5] proved the stability of Kirk-Mann and Kirk-Ishikawa iterative schemes for the following operator which is more general than (1.13). Indeed, he employed the following contractive definition: there exist and a monotone increasing function with , such that Motivated by the work of Olatinow [5] and Phuengrattana and Suantai [7], in this paper, we introduce the Kirk-CR and Kirk-SP iterative schemes and study the strong convergence of these iterative schemes for quasi-contractive operators satisfying (1.14). Moreover, by using C++ programming, comparison for rate of convergences between the above-mentioned Kirk type iterative schemes is also shown for increasing, decreasing, sublinear, superlinear, and oscillatory functions, respectively.

#### 2. Main Results

We will need the following lemmas and definition in the sequel.

Lemma 2.1 (see [10]). *If is a real number such that and is a sequence of nonnegative numbers such that , then, for any sequence of positive numbers satisfying
**
one has .*

Lemma 2.2 (see [5]). * Let be a normed linear space and let be a selfmap of satisfying (1.13). Let be a subadditive, monotone increasing function such that , , , . Then, for all , and for all ,
*

*Definition 2.3 (see [20]). *Suppose and are two real convergent sequences with limits and , respectively. Then, is said to converge faster than if

Now, we define Kirk-SP and Kirk-CR iterative schemes as follows. Let be a Banach space, a selfmap of and . Then, the sequence defined by is called Kirk-SP iterative scheme and the sequence defined by is called Kirk-CR iterative scheme, where , , and are fixed integers with and , , are sequences in satisfying , .

*Remarks 2. *(5) Putting in (2.4), we obtain the Kirk-Mann iterative scheme (1.9).

(6) Putting in (2.4), we get usual Mann iterative scheme (1.3). With .

(7) Putting , and in (2.4), we obtain the usual Kirkβs iterative scheme (1.4).

(8) Putting in (2.4) and (2.5), we obtain the SP (1.10) and CR (1.11) iterative schemes, respectively.

We now prove our main results.

Theorem 2.4. * Let be a normed linear space and a selfmap of satisfying the contractive condition (1.14) and a subadditive monotone increasing function such that and . Let and be the Kirk-SP iterative scheme defined by (2.4). Suppose that has a fixed point . Then, the Kirk-SP iterative scheme converges strongly to .*

*Proof. * Using Kirk-SP iterative scheme (2.4) and Lemma 2.1, we have
Now, we have the following estimates:
It follows from (2.6), (2.7) that
Since and , hence
Using (2.9) and Lemma 2.1, (2.8) yields . Thus, Kirk-SP iterative scheme converges strongly to *. *

Theorem 2.5. * Let be a normed linear space and a selfmap of satisfying the contractive condition (1.14) and a subadditive monotone increasing function such that and . Let and be the Kirk-CR iterative scheme defined by (2.5). Suppose that has a fixed point . Then, the Kirk-CR iterative scheme converges strongly to .*

*Proof. * Using Kirk-CR iterative scheme (2.5) and Lemma 2.2, we have
Now, we have the following estimates:
It follows from (2.10), (2.11) that
Since and with , hence
Using (2.13) and Lemma 2.1, (2.12) yields . Thus, Kirk-CR iterative scheme converges strongly to *. *

Theorem 2.6. * Let be a normed linear space and a selfmap of satisfying the contractive condition (1.14) and a subadditive monotone increasing function such that and , . Let and be the Kirk-Noor iterative scheme defined by (1.9). Suppose that has a fixed point . Then, the Kirk-Noor iterative scheme converges strongly to .*

*Proof. * Using Kirk-Noor iterative scheme (1.9) and Lemma 2.2, we have
Now, we have the following estimates:
It follows from (2.14), (2.15) that
Using Lemma (2.2), (2.16) yields . Thus, Kirk-Noor iterative scheme converges strongly to .

#### 3. Results on Fastness of Kirk-Type Iterative Schemes for Quasi-Contractive Operators

In [20], Berinde showed that Picard iteration is faster than Mann iteration for quasi-contractive operators satisfying (1.14). In [21], Qing and Rhoades by taking example showed that Ishikawa iteration is faster than Mann iteration for a certain quasi-contractive operator. Ciric et al. [22], by providing an example, showed that Noor iterative scheme can be faster than Mann and Ishikawa iterative schemes for some quasi-contractive operator. Recently, Hussian et al. [23], provided an example of a quasi-contractive operator for which the iterative scheme due to Agarwal et al. is faster than Mann and Ishikawa iterative schemes.

Now, by providing Example 3.1, we prove that the decreasing order of Kirk-type iterative schemes is as follows: Kirk-SP, Kirk-CR, Kirk-Noor, Kirk-Ishikawa, and Kirk-Mann iterative scheme.

However, after interchanging the parameters the decreasing order of Kirk-type iterative schemes is as follows: Kirk-CR, Kirk-SP, Kirk-Noor, Kirk-Ishikawa, and Kirk-Mann.

*Example 3.1. *Let for some and .

It is clear that is a quasi-contractive operator satisfying (1.14) with a unique fixed point 0. Also, it is easy to see that Example 3.1 satisfies all the conditions of Theorems 2.4, 2.5, and 2.6.

*Proof. * Let and with . Then, for Kirk-Mann and Kirk-Ishikawa iterative schemes, we have
Now, consider
It is easy to see that
Hence, .

Therefore, by Definition 2.3, Kirk-Ishikawa iterative scheme converges faster than Kirk-Mann iterative scheme to the fixed point 0 of .

Similarly,
with
implies
Therefore, by Definition 2.3, Kirk-Noor iterative scheme converges faster than Kirk-Ishikawa iterative scheme to the fixed point 0 of .

Again, similarly
with
implies
It shows Kirk-CR iterative scheme converges faster than Kirk-Noor iterative scheme to the fixed point 0 of .

Again, let . Then, for Kirk-CR iterative scheme, we have
So,
It is easy to see that
Hence, we have . It shows Kirk-SP iterative scheme converges faster than Kirk-CR iterative scheme to the fixed point 0 of .

The following example shows comparison of simple iterative schemes with their corresponding Kirk-type iterative schemes.

*Example 3.2. *Let , for some and . It is clear that is a quasi-contractive operator satisfying (1.14) with a unique fixed point 0. Also, it is easy to see that Example 3.2 satisfies all the conditions of Theorems 2.4, 2.5, and 2.6. We will show the following:(1)Kirk-Mann iterative scheme is faster than Mann iterative scheme, (2)Kirk-Ishikawa iterative scheme is faster than Ishikawa iterative scheme, (3)Kirk-Noor iterative scheme is faster than Noor iterative scheme, (4)Kirk-SP iterative scheme is faster than SP iterative scheme,(5)Kirk-CR iterative scheme is faster than CR iterative scheme.

*Proof. *Let and with . Then, for Kirk-Mann and Mann iterative schemes, we have
Now, consider
It is easy to see that
Hence, we have .

It shows that Kirk-Mann iterative scheme converges faster than Mann iterative scheme to the fixed point 0 of . Similarly,
with
implies
It shows that Kirk-Ishikawa iterative scheme converges faster than Ishikawa iterative scheme to the fixed point 0 of .

Again, similarly,
with
implies
It shows that Kirk-Noor iterative scheme converges faster than Noor iterative scheme to the fixed point 0 of .

Again,
with
implies
It shows that Kirk-SP iterative scheme converges faster than SP iterative scheme to the fixed point 0 of .

Again,
with
implies
It shows that Kirk-CR iterative scheme converges faster than CR iterative scheme to the fixed point 0 of .

#### 4. Applications

In this section, with the help of computer programs in C++, we compare the rate of convergence of Kirk-type iterative schemes, through examples. The outcome is listed in the form of Tables 1, 2, 3, 4, and 5, by taking , and , for all iterative schemes.

##### 4.1. Decreasing Cum Sublinear Functions

The function defined by is a decreasing and sublinear function. By taking initial approximation , the comparison of convergence of the above-mentioned iterative schemes to the exact fixed point is listed in Table 1.

##### 4.2. Increasing Functions

Let be defined by . Then, is an increasing function. By taking initial approximation , the comparison of convergence of the above-mentioned iterative schemes to the exact fixed point of is listed in Table 2.

##### 4.3. Functions with Multiple Zeros

The function defined by is a function with multiple zeros. By taking initial approximation , the comparison of convergence of the above-mentioned iterative schemes to the exact fixed point is listed in Table 3.

##### 4.4. Superlinear Functions with Multiple Roots

The function defined by is a superlinear function with multiple real roots. By taking initial approximation , the comparison of convergence of the above-mentioned iterative schemes to the exact fixed point is listed in Table 4.

For detailed study, these programs are again executed after changing the parameters and some observations are made as given below.

##### 4.5. Oscillatory Functions

The function defined by is an oscillatory function. By taking initial approximation , the comparison of convergence of the above-mentioned iterative schemes to the exact fixed point is listed in Table 5.

#### 5. Observations

##### 5.1. Decreasing Functions

(1) Taking initial guess (away from the fixed point), Kirk-Mann iterative scheme converges in 9 iterations, Kirk-Ishikawa scheme converges in 9 iterations, Kirk-Noor iterative scheme converges in 6 iterations, Kirk-CR and the Kirk-SP iterative schemes converge in 5 iterations.

(2) Taking and , we observe that Kirk-Mann iterative scheme converges in 10 iterations, Kirk-Ishikawa iterative scheme converges in 12 iteration, Kirk-Noor scheme converges in 12 iterations, Kirk-CR iterative scheme converges in 8 iterations, and Kirk-SP iterative scheme converges in 6 iterations.

##### 5.2. Increasing Functions

(1) Taking initial guess (away from the fixed point), Kirk-Mann iterative scheme converges in 19 iterations, Kirk-Ishikawa iterative scheme converges in 16 iterations, Kirk-Noor iterative scheme converges in 15 iterations, Kirk-CR iterative scheme converges in 5 iterations, and Kirk-SP iterative scheme converges in 3 iterations.

(2) Taking and , we observe that Kirk-Mann iterative scheme converges in 6 iterations, Kirk-Ishikawa iterative scheme converges in 11 iterations, Kirk-Noor iterative scheme converges in 13 iterations, Kirk-CR iterative scheme converges in 5 iterations, and Kirk-SP iterative scheme converges in 4 iterations.

##### 5.3. Functions with Multiple Zeros

(1) Taking initial guess (near the fixed point), Kirk-Mann iterative scheme converges in 12 iterations, Kirk-Ishikawa iterative scheme converges in 10 iterations, Kirk-Noor iterative scheme converges in 8 iterations, Kirk-CR iterative scheme converges in 5 iterations and the Kirk-SP iterative scheme converges in 4 iterations.

(2) Taking and , we observe that Kirk-Mann iterative scheme converges in 9 iterations, Kirk-Ishikawa iterative scheme converges in 13 iterations, Kirk-Noor iterative scheme converges in 13 iterations, Kirk-CR iterative scheme converges in 8 iterations, and Kirk-SP iterative scheme converges in 6 iterations.

##### 5.4. Superlinear Functions with Multiple Roots

(1) Taking initial guess (away from the fixed point), Kirk-Mann, Kirk-Ishikawa and Kirk-Noor iterative schemes converge in 5 iterations while Kirk-CR and the Kirk-SP schemes converge in 4 iterations.

(2) Taking and , we observe that Kirk-Mann, Kirk-Ishikawa, and Kirk-Noor schemes converge in 11 iterations while Kirk-CR iterative scheme converges in 4 iterations, and Kirk-SP iterative scheme converges in 3 iterations.

##### 5.5. Oscillatory Functions

(1) Taking initial guess (near the fixed point), Kirk-Mann iterative scheme converges in 9 iterations, Kirk-Ishikawa, iterative scheme converges in 6 iterations while Kirk-Noor, Kirk-CR, and Kirk-SP iterative schemes converge in 5 iterations.

(2) Taking and , we observe that Kirk-Mann iterative scheme converges in 12 iterations, Kirk-Ishikawa iterative scheme converges in 15 iterations, Kirk-Noor iterative scheme converges in 13 iterations, Kirk-CR iterative scheme converges in 9 iterations, and Kirk-SP iterative scheme converges in 8 iterations.

#### 6. Conclusions

The speed of iterative schemes depends on , , and . From Tables 1β5 and obsevations made in Section 4, we conclude the following.

##### 6.1. Decreasing Cum Sublinear Functions

(1) Decreasing order of rate of convergence of Kirk-type iterative schemes is as follows: βKirk-SP, Kirk-CR, Kirk-Noor, Kirk-Ishikawa, and Kirk-Mann.

(2) For initial guess away from the fixed point, Kirk-SP and Kirk-Ishikawa iterative schemes show an increase while Kirk-CR, Kirk-Noor, and Kirk-Mann iterative schemes show no change in the number of iterations to converge.

##### 6.2. Increasing Functions

(1) Decreasing order of rate of convergence of Kirk-type iterative schemes is as follows:βKirk-SP, Kirk-CR, Kirk-Mann, Kirk-Noor, and Kirk-Ishikawa.

(2) For initial guess away from the fixed point, the number of iterations increases in case of Kirk-Mann, Kirk-Noor, and Kirk-Ishikawa iterative schemes. However, Kirk-SP and Kirk-CR schemes show no change in the number of iterations.

##### 6.3. Functions with Multiple Zeros

(1) Decreasing order of rate of convergence of Kirk-type iterative schemes is as follows: βKirk-SP, Kirk-CR, Kirk-Noor, Kirk-Ishikwa, and Kirk-Mann.

(2) For initial guess near the fixed point, Kirk-CR, Kirk-Ishikawa, and Kirk-Mann iterative schemes show a decrease while Kirk-Noor and Kirk-SP iterative schemes show no change in the number of iterations to converge.

##### 6.4. Superlinear Functions

(1) Decreasing order of rate of convergence of Kirk-type iterative schemes is as follows:βKirk-CR, Kirk-SP, Kirk-Noor, and Kirk-Mann, while Kirk-Noor and Kirk-Ishikawa iterative schemes show equivalence.

(2) For initial guess near the fixed point, Kirk-CR iterative scheme show an increase, while Kirk-SP, Kirk-Ishikawa, Kirk-Mann, and Kirk-Noor iterative schemes show no change in the number of iterations to converge.

##### 6.5. Oscillatory Functions

(1) Decreasing order of rate of convergence of Kirk type iterative schemes is as follows: βKirk-CR, Kirk-Ishikawa, and Kirk-Mann, while Kirk-CR, Kirk-SP, and Kirk-Noor iterative schemes show equivalence.

(2) For initial guess near the fixed point, Kirk-Mann and Kirk-Ishikawa iterative schemes show a decrease, while Kirk-CR, Kirk-SP, and Kirk-Noor iterative schemes show no change in the number of iterations to converge.

*Remarks 3. *(9) It is observed from experiments that, on taking , the convergence speed of each iterative scheme decreases for all type of the above-mentioned functions. Convergence speed is the highest for .

(10) In Section 4, we have shown comparison between Kirk-type iterative schemes for decreasing functions. However, for decreasing functions of the form , Kirk-type iterative schemes may not converge.

(11) Hence, Kirk-SP and Kirk-CR iterative schemes have a good potential for further applications.

#### Acknowledgments

The authors would like to thank the referees for valuable suggestions on the paper and N. Hussain gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.