Research Article  Open Access
On the Rate of Convergence of KirkType Iterative Schemes
Abstract
The purpose of this paper is to introduce Kirktype new iterative schemes called KirkSP and KirkCR schemes and to study the convergence of these iterative schemes by employing certain quasicontractive operators. By taking an example, we will compare KirkSP, KirkCR, KirkMann, KirkIshikawa, and KirkNoor iterative schemes for aforementioned class of operators. Also, using computer programs in C++, we compare the abovementioned 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 contractivetype operator. Consequently, there is a theoretical and practical interest in approximating fixed points of various contractivetype 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 Banach’s 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 KirkMann and KirkIshikawa iterative schemes as follows: where , , , , , and , are fixed integers, called as KirkIshikawa iteration scheme: where , , , and is a fixed integer and is called as KirkMann iteration scheme.
However, from [6], the KirkNoor 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 quasicontractive operators.
Also several authors [11–16] have studied the equivalence between different iterative schemes: Şolutz [13, 14] proved that Picard, Mann, Ishikawa, and Noor iteration schemes are equilvalent for quasicontractive operators. Recently, Chugh and Kumar [17] proved that, for quasicontractive operators satisfying (1.13), Picard, Mann, Ishikawa, Noor, and SP iterative schemes are equivalent.
Fixedpoint 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 KirkMann and KirkIshikawa 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 KirkCR and KirkSP iterative schemes and study the strong convergence of these iterative schemes for quasicontractive operators satisfying (1.14). Moreover, by using C++ programming, comparison for rate of convergences between the abovementioned 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 KirkSP and KirkCR iterative schemes as follows. Let be a Banach space, a selfmap of and . Then, the sequence defined by is called KirkSP iterative scheme and the sequence defined by is called KirkCR iterative scheme, where , , and are fixed integers with and , , are sequences in satisfying , .
Remarks 2. (5) Putting in (2.4), we obtain the KirkMann 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 Kirk’s 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 KirkSP iterative scheme defined by (2.4). Suppose that has a fixed point . Then, the KirkSP iterative scheme converges strongly to .
Proof. Using KirkSP 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, KirkSP 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 KirkCR iterative scheme defined by (2.5). Suppose that has a fixed point . Then, the KirkCR iterative scheme converges strongly to .
Proof. Using KirkCR 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, KirkCR 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 KirkNoor iterative scheme defined by (1.9). Suppose that has a fixed point . Then, the KirkNoor iterative scheme converges strongly to .
Proof. Using KirkNoor 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, KirkNoor iterative scheme converges strongly to .
3. Results on Fastness of KirkType Iterative Schemes for QuasiContractive Operators
In [20], Berinde showed that Picard iteration is faster than Mann iteration for quasicontractive operators satisfying (1.14). In [21], Qing and Rhoades by taking example showed that Ishikawa iteration is faster than Mann iteration for a certain quasicontractive 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 quasicontractive operator. Recently, Hussian et al. [23], provided an example of a quasicontractive 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 Kirktype iterative schemes is as follows: KirkSP, KirkCR, KirkNoor, KirkIshikawa, and KirkMann iterative scheme.
However, after interchanging the parameters the decreasing order of Kirktype iterative schemes is as follows: KirkCR, KirkSP, KirkNoor, KirkIshikawa, and KirkMann.
Example 3.1. Let for some and .
It is clear that is a quasicontractive 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 KirkMann and KirkIshikawa iterative schemes, we have
Now, consider
It is easy to see that
Hence, .
Therefore, by Definition 2.3, KirkIshikawa iterative scheme converges faster than KirkMann iterative scheme to the fixed point 0 of .
Similarly,
with
implies
Therefore, by Definition 2.3, KirkNoor iterative scheme converges faster than KirkIshikawa iterative scheme to the fixed point 0 of .
Again, similarly
with
implies
It shows KirkCR iterative scheme converges faster than KirkNoor iterative scheme to the fixed point 0 of .
Again, let . Then, for KirkCR iterative scheme, we have
So,
It is easy to see that
Hence, we have . It shows KirkSP iterative scheme converges faster than KirkCR iterative scheme to the fixed point 0 of .
The following example shows comparison of simple iterative schemes with their corresponding Kirktype iterative schemes.
Example 3.2. Let , for some and . It is clear that is a quasicontractive 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)KirkMann iterative scheme is faster than Mann iterative scheme, (2)KirkIshikawa iterative scheme is faster than Ishikawa iterative scheme, (3)KirkNoor iterative scheme is faster than Noor iterative scheme, (4)KirkSP iterative scheme is faster than SP iterative scheme,(5)KirkCR iterative scheme is faster than CR iterative scheme.
Proof. Let and with . Then, for KirkMann and Mann iterative schemes, we have
Now, consider
It is easy to see that
Hence, we have .
It shows that KirkMann iterative scheme converges faster than Mann iterative scheme to the fixed point 0 of . Similarly,
with
implies
It shows that KirkIshikawa iterative scheme converges faster than Ishikawa iterative scheme to the fixed point 0 of .
Again, similarly,
with
implies
It shows that KirkNoor iterative scheme converges faster than Noor iterative scheme to the fixed point 0 of .
Again,
with
implies
It shows that KirkSP iterative scheme converges faster than SP iterative scheme to the fixed point 0 of .
Again,
with
implies
It shows that KirkCR 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 Kirktype 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 abovementioned 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 abovementioned 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 abovementioned 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 abovementioned 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 abovementioned 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), KirkMann iterative scheme converges in 9 iterations, KirkIshikawa scheme converges in 9 iterations, KirkNoor iterative scheme converges in 6 iterations, KirkCR and the KirkSP iterative schemes converge in 5 iterations.
(2) Taking and , we observe that KirkMann iterative scheme converges in 10 iterations, KirkIshikawa iterative scheme converges in 12 iteration, KirkNoor scheme converges in 12 iterations, KirkCR iterative scheme converges in 8 iterations, and KirkSP iterative scheme converges in 6 iterations.
5.2. Increasing Functions
(1) Taking initial guess (away from the fixed point), KirkMann iterative scheme converges in 19 iterations, KirkIshikawa iterative scheme converges in 16 iterations, KirkNoor iterative scheme converges in 15 iterations, KirkCR iterative scheme converges in 5 iterations, and KirkSP iterative scheme converges in 3 iterations.
(2) Taking and , we observe that KirkMann iterative scheme converges in 6 iterations, KirkIshikawa iterative scheme converges in 11 iterations, KirkNoor iterative scheme converges in 13 iterations, KirkCR iterative scheme converges in 5 iterations, and KirkSP iterative scheme converges in 4 iterations.
5.3. Functions with Multiple Zeros
(1) Taking initial guess (near the fixed point), KirkMann iterative scheme converges in 12 iterations, KirkIshikawa iterative scheme converges in 10 iterations, KirkNoor iterative scheme converges in 8 iterations, KirkCR iterative scheme converges in 5 iterations and the KirkSP iterative scheme converges in 4 iterations.
(2) Taking and , we observe that KirkMann iterative scheme converges in 9 iterations, KirkIshikawa iterative scheme converges in 13 iterations, KirkNoor iterative scheme converges in 13 iterations, KirkCR iterative scheme converges in 8 iterations, and KirkSP iterative scheme converges in 6 iterations.
5.4. Superlinear Functions with Multiple Roots
(1) Taking initial guess (away from the fixed point), KirkMann, KirkIshikawa and KirkNoor iterative schemes converge in 5 iterations while KirkCR and the KirkSP schemes converge in 4 iterations.
(2) Taking and , we observe that KirkMann, KirkIshikawa, and KirkNoor schemes converge in 11 iterations while KirkCR iterative scheme converges in 4 iterations, and KirkSP iterative scheme converges in 3 iterations.
5.5. Oscillatory Functions
(1) Taking initial guess (near the fixed point), KirkMann iterative scheme converges in 9 iterations, KirkIshikawa, iterative scheme converges in 6 iterations while KirkNoor, KirkCR, and KirkSP iterative schemes converge in 5 iterations.
(2) Taking and , we observe that KirkMann iterative scheme converges in 12 iterations, KirkIshikawa iterative scheme converges in 15 iterations, KirkNoor iterative scheme converges in 13 iterations, KirkCR iterative scheme converges in 9 iterations, and KirkSP iterative scheme converges in 8 iterations.
6. Conclusions
The speed of iterative schemes depends on , , and . From Tables 1–5 and obsevations made in Section 4, we conclude the following.
6.1. Decreasing Cum Sublinear Functions
(1) Decreasing order of rate of convergence of Kirktype iterative schemes is as follows: KirkSP, KirkCR, KirkNoor, KirkIshikawa, and KirkMann.
(2) For initial guess away from the fixed point, KirkSP and KirkIshikawa iterative schemes show an increase while KirkCR, KirkNoor, and KirkMann iterative schemes show no change in the number of iterations to converge.
6.2. Increasing Functions
(1) Decreasing order of rate of convergence of Kirktype iterative schemes is as follows: KirkSP, KirkCR, KirkMann, KirkNoor, and KirkIshikawa.
(2) For initial guess away from the fixed point, the number of iterations increases in case of KirkMann, KirkNoor, and KirkIshikawa iterative schemes. However, KirkSP and KirkCR schemes show no change in the number of iterations.
6.3. Functions with Multiple Zeros
(1) Decreasing order of rate of convergence of Kirktype iterative schemes is as follows: KirkSP, KirkCR, KirkNoor, KirkIshikwa, and KirkMann.
(2) For initial guess near the fixed point, KirkCR, KirkIshikawa, and KirkMann iterative schemes show a decrease while KirkNoor and KirkSP iterative schemes show no change in the number of iterations to converge.
6.4. Superlinear Functions
(1) Decreasing order of rate of convergence of Kirktype iterative schemes is as follows: KirkCR, KirkSP, KirkNoor, and KirkMann, while KirkNoor and KirkIshikawa iterative schemes show equivalence.
(2) For initial guess near the fixed point, KirkCR iterative scheme show an increase, while KirkSP, KirkIshikawa, KirkMann, and KirkNoor 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: KirkCR, KirkIshikawa, and KirkMann, while KirkCR, KirkSP, and KirkNoor iterative schemes show equivalence.
(2) For initial guess near the fixed point, KirkMann and KirkIshikawa iterative schemes show a decrease, while KirkCR, KirkSP, and KirkNoor 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 abovementioned functions. Convergence speed is the highest for .
(10) In Section 4, we have shown comparison between Kirktype iterative schemes for decreasing functions. However, for decreasing functions of the form , Kirktype iterative schemes may not converge.
(11) Hence, KirkSP and KirkCR 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.
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Copyright
Copyright © 2012 Nawab Hussain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.