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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 132626, 15 pages
http://dx.doi.org/10.1155/2013/132626
Research Article

On Rate of Convergence of Jungck-Type Iterative Schemes

1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics, M.D. University, Rohtak 124001, India

Received 27 February 2013; Accepted 3 April 2013

Academic Editor: Yisheng Song

Copyright © 2013 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.

Abstract

We introduce a new iterative scheme called Jungck-CR iterative scheme and study the stability and strong convergence of this iterative scheme for a pair of nonself-mappings using a certain contractive condition. Also, convergence speed comparison and applications of Jungck-type iterative schemes will be shown through examples.

1. Introduction and Preliminaries

Let be a Banach space, an arbitrary set, and such that . For , consider the following iterative scheme: This scheme is called Jungck iterative scheme and was essentially introduced by Jungck [1] in 1976 and it becomes the Picard iterative scheme when (identity mapping) and .

For , Singh et al. [2] defined the Jungck-Mann iterative scheme as For , Olatinwo defined the Jungck-Ishikawa [3] (see also [4, 5]) and Jungck-Noor [6] iterative schemes as respectively.

Chugh and Kumar [7] defined the Jungck-SP iterative scheme as where ,  , and are sequences of positive numbers in .

Remark 1. If and (identity mapping), then the Jungck-SP (5), Jungck-Noor (4), Jungck-Ishikawa (3), and the Jungck-Mann (2) iterative schemes, respectively, become the SP [8], Noor [9], Ishikawa [10] and the Mann [11] iterative schemes.

Jungck [1] used the iterative scheme (1) to approximate the common fixed points of the mappings and satisfying the following Jungck contraction:

Olatinwo [3] used the following more general contractive definition than (6) to prove the stability and strong convergence results for the Jungck-Ishikawa iteration process: there exists a real number and a monotone increasing function : such that and for all , we have Olatinwo [6] used the convergences of Jungck-Noor iterative scheme (4) to approximate the coincidence points (not common fixed points) of some pairs of generalized contractive-like operators with the assumption that one of each of the pairs of maps is injective.

Motivated by the above facts, for , , and , we introduce the following iterative scheme: and call it Jungck-CR iterative scheme.

Remark 2. Putting and , in Jungck-CR iterative scheme, we get Jungck versions of Agarwal et al. [12] and Sahu and Petruşel [13] iterative schemes, respectively, as defined below:

We will need the following definitions and lemma.

Definition 3 (see [14]). Let and be two fixed-point iteration procedures that converge to the same fixed point on a normed space such that the error estimates are available, where and are two sequences of positive numbers (converging to zero). If converge faster than , then we say that converges faster to than .

Definition 4 (see [15]). Suppose that and are two real convergent sequences with limits and , respectively. Then, is said to converge faster than if

Definition 5 (see [16, 17]). Let and be two self-maps on . A point in is called (1) a fixed point of if ; (2) coincidence point of a pair if ; (3) common fixed point of a pair if . If for some in , then is called a point of coincidence of and . A pair is said to be weakly compatible if and commute at their coincidence points.

Lemma 6 (see [18]). If is a real number such that and is a sequence of positive numbers such that , then for any sequence of positive numbers satisfying one has .

Definition 7 (see [2]). Let , be non-self-operators for an arbitrary set such that and a point of coincidence of and . Let , be the sequence generated by an iterative procedure where is the initial approximation and is some function. Suppose that converges to . Let be an arbitrary sequence and set . Then, the iterative procedure (11) is said to be -stable or stable if and only if implies .

The purpose of this paper is to study the stability and strong convergence of Jungck-CR iterative scheme for nonself-mappings in an arbitrary Banach space by employing the contractive conditions (7) and then to compare convergence rates of Jungck-type iterative schemes. Moreover, applications of Jungck-type iterative schemes in recurrent neural networks (RNN) analysis will be discussed.

2. Strong Convergence in an Arbitrary Banach Space

Theorem 8. Let be an arbitrary Banach space, and let , be nonself-operators on an arbitrary set satisfying contractive condition (7). Assume that , is a complete subspace of and (say). For , let be the Jungck-CR iterative scheme defined by , where , are sequences of positive numbers in with satisfying .Then, the Jungck-CR iterative scheme converges strongly to . Also, will be the unique common fixed point of , provided that , and and are weakly compatible.

Proof. First, we prove that Jungck-CR iterative scheme converges strongly to .
It follows from and (7) that Now, we have the following estimates: It follows from (13) that Using and , inequality (14) yields It follows from (15) and (12) that Since and , so as .
Hence, it follows from (16) that . Therefore, converges strongly to .
Now, we prove that is unique common fixed point of and .
Let there exist another point of coincidence say . Then, there exists such that . But from (7), we have which implies that as .
Now, as and are weakly compatible and , so and hence . Therefore, is a point of coincidence of , and since the point of coincidence is unique then . Thus, , and therefore is unique common fixed point of and .

Corollary 9. Let be an arbitrary Banach space, and, , be nonself-operators on an arbitrary set satisfying contractive condition (7). Assume that , is a complete subspace of and (say). For , let be the iterative scheme defined by , where are sequences of positive numbers in with satisfying . Then the Jungck-Agarwal iterative scheme converges strongly to . Also, will be the unique common fixed point of , provided that , and and are weakly compatible.

Proof. Putting and , in iterative scheme , convergence of iterative scheme can be proved on the same lines as in Theorem 8.

Corollary 10. Let be an arbitrary Banach space and , and let be nonself-operators on an arbitrary set satisfying contractive condition (7). Assume that , is a complete subspace of and (say). For , let be the Jungck-S iterative scheme defined by , where are sequences of positive numbers in with satisfying . Then the Jungck-S iterative scheme converges strongly to . Also, will be the unique common fixed point of , provided that , and and are weakly compatible.

Proof. Putting and in iterative scheme , convergence of iterative scheme can be proved on the same lines as in the Theorem 8.

The following examples reveal the validity of our results.

Example 11. Let . Define and by It is clear that and are quasicontractive operators satisfying (7) but do not satisfy contractive condition (6), with a unique common fixed point 0.

Using computer programming in C++ with initial approximation , convergence of Jungck-CR iterative scheme to the common fixed point 0 is shown in Table 1.

tab1
Table 1

Example 12. Let . Define and by , , and . It is clear that and are weakly compatible quasicontractive operators satisfying (7) with a unique common fixed point 0.5.

Using computer programming in C++ with initial approximation , convergence of Jungck-CR iterative scheme to the common fixed point 0.5 is shown in Table 2.

tab2
Table 2

Theorem 13. Let be an arbitrary Banach space and , and let be nonself operators on an arbitrary set Y satisfying contractive condition (7). Assume that , S(Y) is a complete subspace of , and (say). For and , let be the Jungck-CR iterative scheme converging to , where are sequences in with satisfying for all . Then, the Jungck-CR iterative scheme is -stable.

Proof. Suppose that be an arbitrary sequence, , , where and let .
Then, for Jungck-CR iterative scheme , we have Now, we have the following estimates: It follows from (19), (20) that Using and , we have .
Hence using Lemma 6, (21) yields .
Conversely, let . Then, using contractive condition (7) and the triangle inequality, we have By using estimates (20), (22), yields Hence, .
Therefore, the JCR iterative scheme is stable.

3. Results on Direct Comparison of Jungck-Type Iterative Schemes

Various authors [7, 1315, 1922] have worked on convergence speed of iterative schemes. In [14], Berinde showed that Picard iteration is faster than Mann iteration for quasicontractive operators. In [15], Qing and Rhoades by taking an example showed that Ishikawa iteration is faster than Mann iteration for a certain quasicontractive operator. In [20], Hussain et al. 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. Recently, Chugh and Kumar [19] showed that SP iterative scheme with error terms converges faster than Ishikawa and Noor iterative schemes for accretive-type mappings. For recent work in this direction, we refer the reader to [2327] and references therein.

Theorem 14. Let be an arbitrary Banach space, and let , be nonself-operators on an arbitrary set satisfying contractive condition (7). Assume that , is a complete subspace of X and (say). For , let Jungck-Mann iterative scheme be defined by (JM) and Jungck-Ishikawa iterative scheme be defined by (JI), with ,  , for some  and satisfying .
Then, the Jungck-Ishikawa iterative scheme converges faster than Jungck-Mann iterative scheme to .

Proof. For Jungck-Mann iterative scheme, we have Also, for Jungck-Ishikawa iterative scheme, we have But Hence, Using (24) and (27), we have But we observe that Using (29) together with , we have As , so (28) yields .
Therefore, by Definition 4, Jungck-Ishikawa iterative scheme converges faster than Jungck-Mann iterative scheme to .

Theorem 15. Let be an arbitrary Banach space, and let , be nonself-operators on an arbitrary set Y satisfying contractive condition (7). Assume that , is a complete subspace of , and (say). For , let Jungck-Noor iterative scheme be defined by (JN) and Jungck-Ishikawa iterative scheme defined by (JI), with , for some and satisfying . Then, the Jungck-Noor iterative scheme converges faster than Jungck-Ishikawa iterative scheme to .

Proof. For Jungck-Ishikawa iterative scheme, we have Also, for Jungck-Noor iterative scheme, we have Using (31) and (32), we have Making the same calculations as in Theorem 14, (33) yields By Definition 4, Jungck-Noor iterative scheme converges faster than Jungck-Ishikawa iterative scheme to .

Theorem 16. Let be an arbitrary Banach space and , be nonself operators on an arbitrary set satisfying contractive condition (7). Assume that , is a complete subspace of and (say). For , let Jungck-Noor iterative scheme be defined by (JN) and Jungck-SP iterative scheme defined by (JSP), with , for some satisfying . Then, the Jungck-SP iterative scheme converges faster than Jungck-Noor iterative scheme to .

Proof. For Jungck-Noor iterative scheme, we have Also, for Jungck-SP iterative scheme, we have Using (35) and (36), we have We observe that Using (38) together with , (37) yields Therefore, by Definition 4, Jungck-SP iterative scheme converges faster than Jungck-Noor iterative scheme .

Theorem 17. Let be an arbitrary Banach space, and let , be nonself operators on an arbitrary set satisfying contractive condition (7). Assume that , is a complete subspace of and (say). For , let Jungck-Agarwal’s et al. iterative scheme be defined by and Jungck-SP iterative scheme be defined by (JSP) with , , and . Then, the Jungck-Agarwal iterative scheme converges faster than Jungck-SP iterative scheme to .

Proof. For Jungck-SP iterative scheme, we have Also, for Jungck-Agarwal iterative scheme, we have Using (40) and (41), we have Since and .
Hence from (42), we have Therefore, by Definition 4, Jungck-SP iterative scheme converges faster than Jungck-Agarwal et al.’s iterative scheme to .

Theorem 18. Let be an arbitrary Banach space, and let , be nonself-operators on an arbitrary set satisfying contractive condition (7). Assume that , is a complete subspace of and (say). For , let Jungck-S iterative scheme be defined by and Jungck-Agarwal iterative scheme defined by . Then, the Jungck-S iterative scheme converges faster than Jungck-Agarwal iterative scheme to .

Proof. For Jungck-S iterative scheme, we have Also, for Jungck-Agarwal iterative scheme, we have It is obvious that Hence by Definition 3, Jungck-S iterative scheme converges faster than Jungck-Agarwal iterative scheme.

Theorem 19. Let be an arbitrary Banach space, and let , be nonself operators on an arbitrary set Y satisfying contractive condition (7). Assume that , is a complete subspace of and (say). For , let Jungck-S iterative scheme be defined by and Jungck-CR iterative scheme be defined by . Then, the Jungck-CR iterative scheme converges faster than Jungck-S iterative scheme to .

Proof. For Jungck-S iterative scheme, we have Also, for Jungck-CR iterative scheme, we have It is obvious that Hence by Definition 3, Jungck-CR iterative scheme converges faster than Jungck-S iterative scheme.

The following example supports the above results.

Example 20. Let , , , , , for some , and , . It is clear that and are quasicontractive operators satisfying (7) with the unique common fixed point 0. Also, it is easy to see that Example 20 satisfies all the conditions of Theorem 8 and Theorems 1419.

Proof . For JM, JI, JN, JA, JS, JSP, and JCR iterative schemes with initial approximation , we have Now, for , consider It is easy to see that Hence, .
Therefore, by Definition 4, Jungck-Ishikawa iterative scheme converges faster than Jungck-Mann iterative scheme to the common fixed point 0 of and .
Similarly, for , with implies Therefore, by Definition 4, JN iterative scheme converges faster than JI iterative scheme to the common fixed point 0 of and .
Again, similarly, for , with implies Therefore, by Definition 4, JSP iterative scheme converges faster than JN iterative scheme to the common fixed point 0 of and .
Again, similarly, for , with implies Therefore, by Definition 4, JA iterative scheme converges faster than JSP iterative scheme to the common fixed point 0 of and .
Again, for , with implies Therefore, by Definition 4, JS iterative scheme converges faster than JA iterative scheme to the common fixed point 0 of and .
Similarly, again, for , with implies Therefore, by Definition 4, JCR iterative scheme converges faster than JS iterative scheme to the common fixed point 0 of and .
From Example 20, we observe that the decreasing order of Jungck-type iterative schemes is as follows:
JCR, JS, JA, JSP, JN, JI, and JM.

4. Applications

4.1. Jungck-Type Iterative Schemes in RNN Analysis

Recurrent neural networks (RNNs) are a class of densely connected single-layer nonlinear networks of perceptrons. RNNs not only operate on an input space but also on an internal statespace. This is equivalent to a with-memory Iterated Function System [28]. The state space enables the representation (and learning) of temporally/sequentially extended dependencies over unspecified (and potentially infinite) intervals according to Because of the network’s nonlinearity, a number of undesirable local energy minima emerge from the learning procedure. This has been shown to significantly affect the network’s performance. The iterative schemes like Mann, Ishikawa and -iteration may be used to estimate the number of iterations required to achieve a stable state in recurrent autoassociative neural networks.

4.1.1. Decreasing Function

In order to solve this function by Jungck-type iterative schemes, we write it in the form , where the functions , are defined as and , respectively. By taking initial approximation and , the obtained results are listed in Table 3 showing convergence of different Jungck-type schemes to .

tab3
Table 3: Decreasing function.
4.1.2. Increasing Function

In order to solve this function by Jungck-type iterative schemes, we write it in the form , where the functions , are defined as and , respectively. By taking initial approximation and , the obtained results are listed in Table 4 showing convergence of different Jungck-type schemes to .

tab4
Table 4: Increasing function.
4.1.3. Oscillating Function

In order to solve this function by Jungck-type iterative schemes, we write it in the form , where the functions , are defined as and , respectively. By taking initial approximation and , the obtained results are listed in Table 5 showing convergence of different Jungck type schemes to .

tab5
Table 5: Oscillatory function.
4.1.4. Biquadratic Equation

In order to solve this equation, we rewrite it in the form , where the functions , are defined as and , respectively. Taking initial approximation and , the obtained results are listed in Table 6 showing convergence of different Jungck-type schemes to .

tab6
Table 6: Biquadratic equation.

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

Decreasing Function(1)Taking initial guess (near common fixed point), Jungck-Noor iterative scheme converges in 14 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 5 iterations, and Jungck-S iterative scheme converges in 25 iterations while Jungck-Mann iterative scheme shows strange constant behavior.(2)Taking and , we observe that Jungck-Noor iterative scheme converges in 13 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 11 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 8 iterations, and Jungck-S iterative scheme converges in 27 iterations while Jungck-Mann iterative scheme shows strange constant behavior.

Increasing Functions(1)Taking initial guess (near coincidence point), Jungck-Noor iterative scheme converges in 7 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 7 iterations while Jungck-Mann iterative scheme converges in 13 iterations.(2)Taking and , we observe that Jungck-Noor iterative scheme converges in 7 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 7 iterations while Jungck-Mann iterative scheme converges in 14 iterations.

Oscillatory Function(1)Taking initial guess (near common fixed point), Jungck-Noor iterative scheme converges in 8 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 6 terations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 5 iterations, Jungck-S iterative scheme converges in 11 iterations while Jungck-Mann iterative scheme converges in 19 iterations.(2)Taking and , we observe that Jungck-Noor iterative scheme converges in 8 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 9 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, Jungck-S iterative scheme converges in 12 iterations while Jungck-Mann iterative scheme converges in 21 iterations.

Biquadratic Equation(1)Taking initial guess (near coincidence point), Jungck-Noor iterative scheme converges in 11 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 7 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 4 iterations, and Jungck-S iterative scheme converges in 18 iterations while Jungck-Mann iterative scheme converges in 35 iterations.(2)Taking and , we observe that Jungck-Noor iterative scheme converges in 12 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 19 iterations while Jungck-Mann iterative scheme converges in 37 iterations.

5. Conclusions

The speed of iterative schemes depends on , , and . From Tables 36 and observations made above, we make the following conjectures.

5.1. Decreasing Function

(1)Decreasing order of rate of convergence of Jungck type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), Jungck-Noor, and Jungck-S iterative scheme.(2)For initial guess near to common fixed point, Jungck-CR (Jungck-SP), Jungck-Noor, and Jungck-S iterative schemes show a decrease while Jungck-Agarwal (Jungck-Ishikawa) iterative scheme shows no change in the number of iterations to converge.

5.2. Increasing Functions

(1)Decreasing order of rate of convergence of Jungck-type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-S (Jungck-Noor), Jungck-Agarwal (Jungck-Ishikawa), and Jungck Mann iterative scheme.(2)For initial guess near to the coincidence point, all Jungck-type iterative schemes show a decrease in the number of iterations to converge.

5.3. Oscillatory Functions

(1)Decreasing order of rate of convergence of Jungck-type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), Jungck-Noor, Jungck-S, and Jungck-Mann iterative scheme.(2)For initial guess near to the common fixed point, Jungck-Mann and Jungck-S iterative schemes show a decr ease while Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), and Jungck-Noor iterative schemes show no change in the number of iterations to converge.

5.4. Biquadratic Equation

(1)Decreasing order of rate of convergence of Jungck type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), Jungck-Noor, Jungck-S, and Jungck-Mann iterative scheme.(2)For initial guess near to the coincidence point, all Jungck-type iterative schemes show a decrease in the number of iterations to converge.

Remark 21. In each case mentioned above, Jungck-CR and Jungck-SP iterative schemes have better convergence rate as compared to other iterative schemes and hence have a good potential for further applications.

Acknowledgments

This research was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The first and third authors acknowledge with thanks DSR, KAU, Saudi Arabia, for financial support.

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