On Multistep Iterative Scheme for Approximating the Common Fixed Points of Contractive-Like Operators

Abstract

We introduce the Jungck-multistep iteration and show that it converges strongly to the unique common fixed point of a pair of weakly compatible generalized contractive-like operators defined on a Banach space. As corollaries, the results show that the Jungck-Mann, Jungck-Ishikawa, and Jungck-Noor iterations can also be used to approximate the common fixed points of such maps. The results are improvements, generalizations, and extensions of the work of Olatinwo and Imoru (2008), Olatinwo (2008). Consequently, several results in literature are generalized.

1. Introduction

The convergence of Picard, Mann, Ishikawa, Noor and multistep iterations have been commonly used to approximate the fixed points of several classes of single quasicontractive operators, for example, see [1–6].

Let be a Banach space, , a nonempty convex subset of and a self-map of .

Definition 1.1. Let . The Picard iteration scheme is defined by

Definition 1.2. For any given , the Mann iteration scheme [7] is defined by where are real sequences in [0,1) such that .

Definition 1.3. Let . The Ishikawa iteration scheme [8] is defined by where are real sequences in [0,1) such that .

Observe that if for each , then the Ishikawa iteration process (1.3) reduces to the Mann iteration scheme (1.2).

Definition 1.4. Let . The Noor iteration (or three-step) scheme [9] is defined by where are real sequences in such that.

For motivation and the advantage of using Noor’s iteration, see [5, 9, 10].

Observe that if for each , then the Noor iteration process (1.4) reduces to the Ishikawa iteration scheme (1.3).

Definition 1.5. Let . The multistep iteration scheme [11] is defined by where , are real sequences in such that .

Observe that the multistep iteration is a generalization of the Noor, Ishikawa, and the Mann iterations. In fact, if in (1.5), we have the Mann iteration (1.2), if in (1.5), we have the Ishikawa iteration (1.3), and if , we have the Noor iterations (1.4).

We note that while many authors have worked on the existence of fixed points for a pair of quasicontractive maps, for example, see [1, 12–15], little is known about the approximations of those common fixed points using the convergence of iteration techniques. Jungck was the first to introduce an iteration scheme, which is now called Jungck iteration scheme [13] to approximate the common fixed points of what is now called Jungck contraction maps. Singh et al. [15] of recent introduced the Jungck-Mann iteration procedure and discussed its stability for a pair of contractive maps. Olatinwo and Imoru [16], Olatinwo [17, 18] built on that work to introduce the Jungck-Ishikawa and Jungck-Noor iteration schemes and used their convergences 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. However, a coincidence point for a pair of quasicontractive maps needs not to be a common fixed point. We introduce the Jungck-multistep iteration and show that its convergence can be used to approximate the common fixed points of those pairs of quasicontractive maps without assuming the injectivity of any of the operators. Hence the iterative sequence used is a generalization of that used in [16–18]. The fact that the injectivity of any of the maps is not assumed in our results and the common fixed points of those maps are approximated and not just the coincidence points make the corollary of our results an improvement of the results of Olaleru [19], Olatinwo and Imoru [16]. Consequently, a lot of results dealing with convergence of Picard, Mann, Ishikawa, and multistep iterations for single quasicontractive operators on Banach spaces are generalized.

2. Preliminaries

Let be a Banach space, an arbitrary set, and such that .

Then we have the following definitions.

Definition 2.1 (see [13]). For any , there exists a sequence such that . The Jungck iteration is defined as the sequence such that This procedure becomes Picard iteration when and , where is the identity map on .

Similarly, the Jungck contraction maps are the maps satisfying

If and , then maps satisfying (2.2) become the well-known contraction maps.

Definition 2.2 (see [15]). For any given , the Jungck-Mann iteration scheme is defined by where are real sequences in [0,1) such that .

Definition 2.3 (see [18]). Let . The Jungck-Ishikawa iteration scheme is defined by where are real sequences in such that .

Definition 2.4 (see [18]). Let . The Jungck-Noor iteration (or three-step) scheme is defined by where , and are real sequences in such that .

Definition 2.5. Let . The Jungck-multistep iteration scheme is defined by where , are real sequences in [0,1) such that .

Observe that the Jungck-multistep iteration is a generalization of the Jungck-Noor, Jungck-Ishikawa and the Jungck-Mann iterations. In fact, if in (2.6), we have the Jungck-Mann iteration (2.3), if in (2.6), we have the Jungck-Ishikawa iteration (2.4) and if , we have the Jungck-Noor iterations (2.5).

Observe that if and , then the Jungck-multistep (2.6), Jungck-Noor (2.5), Jungck-Ishikawa (2.4), and the Jungck-Mann (2.3) iterations, respectively, become the multistep (1.5), Noor (1.4), Ishikawa (1.3), and the Mann (1.2) iterative procedures.

One of the most general contractive-like operators which has been studied by several authors is the Zamfirescu operators.

Suppose that is a Banach space. The map is called a Zamfirescu operator if

where see [6].

It is known that the operators satisfying (2.7) are generalizations of Kannan maps [4] and Chatterjea maps [3]. Zamfirescu [6] proved that the Zamfirescu operator has a unique fixed point which can be approximated by Picard iteration (1.1). Berinde [2] showed that Ishikawa iteration can be used to approximate the fixed point of a Zamfirescu operator when is a Banach space while it was shown by the first author [20] that if is generalised to a complete metrizable locally convex space (which includes Banach spaces), the Mann iteration can be used to approximate the fixed point of a Zamfirescu operator. Several researchers have studied the convergence rate of these iterations with respect to the Zamfirescu operators. For example, it has been shown that the Picard iteration (1.1) converges faster than the Mann iteration (1.2) when dealing with the Zamfirescu operators. For example, see [21]. It is still a subject of research as to conditions under which the Mann iteration will converge faster than the Ishikawa or vice-versa when dealing with the Zamfirescu operators.

We now consider the following conditions. is a Banach space and a nonempty set such that and . For and :

where is a monotone increasing sequence with .

Remark 2.6. Observe that if and , (2.8) is the same as the Zamfirescu operator (2.7) already studied by several authors; (2.9) becomes the operator studied by Rhoades [22]; while (2.10) becomes the operator introduced by Osilike [23]. Operators satisfying (2.11) and (2.12) were introduced by Olatinwo [16].

A comparison of the four maps show the following.

Proposition 2.7. (2.8)(2.9)(2.10)(2.11)(2.12) but the converses are not true.

Proof. (2.8)(2.9): This follows immediately since (2.9)(2.10): We consider each of the possibilities.Case 1. Suppose and consequently, . Setting completes the proof.Case 2. Suppose After computing we have. Setting and completes the proof.Case 3.
(2.10)(2.11): Suppose and in (2.11), we have (2.10).
(2.11)(2.12): This follows from the fact that

We need the following definition.

Definition 2.8 (see [1]). A point is called a coincident point of a pair of self-maps if there exists a point (called a point of coincidence) in such that . Self-maps and are said to be weakly compatible if they commute at their coincidence points, that is, if for some , then .

Olatinwo and Imoru [16] proved that the Jungck-Mann and Jungck-Ishikawa converge to the of defined by (2.8) when is an injective operator. It was shown in [19] that the Jungck-Ishikawa iteration converges to the coincidence point of defined by (2.12) when is an injective operator while the same convergence result was proved for Jungck-Noor when are defined by (2.11) [18]. (We note that the maps satisfying (2.9) and of course (2.10)–(2.12) need not have a coincidence point [15].) We rather prove the convergence of multistep iteration to the unique of defined by (2.12), without assuming that is injective, provided the coincident point exist for .

3. Main Results

The following lemma is well known.

Lemma 3.1. Let be a sequence of nonnegative numbers such that for any , where and . Then converges to zero.

Theorem 3.2. Let be a Banach space and for an arbitrary set such that (2.12) holds and . Assume that and have a coincidence point such that . For any , the Jungck-multistep iteration (2.6) converges to .

Further, if and commute at (i.e., and are weakly compatible), then is the unique common fixed point of .

Proof. In view of (2.6) and (2.12) coupled with the fact that , we have An application of (2.6) and (2.12) gives Substituting (3.2) in (3.1), we have Similarly, an application of (2.6) and (2.12) give Substituting (3.4) in (3.3) we have Similarly, an application of (2.6) and (2.12) gives Substituting (3.6) in (3.5) we have Continuing the above process we have Hence by Lemma 3.1.
Next we show that is unique. Suppose there exists another point of coincidence . Then there is an such that . Hence, from (2.12) we have
Since , then and so is unique.
Since are weakly compatible, then and so . Hence is a coincidence point of and since the coincidence point is unique, then and hence and therefore is the unique common fixed point of and the proof is complete.

Remark 3.3. Weaker versions of Theorem 3.2 are the results in [16, 18] where is assumed injective and the convergence is not to the common fixed point but to the coincidence point of . Furthermore, the Jungck-multistep iteration used in Theorem 3.2 is more general than the Jungck-Ishikawa and the Jungck-Noor iteration used in [17, 18].

It is already shown in [1, 20] that if or is a complete subspace of , then maps satisfying the generalized Zamfirescu operators (2.8) have a unique coincidence point. Hence we have the following results.

Theorem 3.4. Let be a Banach space and such that and . Assume that and are weakly compatible. For any , the Jungck-multistep iteration (2.6) converges to the unique common fixed point of .

Since the Jungck-Noor, Jungck-Ishikawa and Jungck-Mann iterations are special cases of Jungck-multistep iteration, then we have the following consequences.

Corollary 3.5. Let be a Banach space and such that and . Assume and are weakly compatible. For any , the Jungck-Noor iteration (2.5) converges to the unique common fixed point of .

Corollary 3.6. Let be a Banach space and such that and . Assume that and are weakly compatible. For any , the Jungck-Ishikawa iteration (2.4) converges to the unique common fixed point of .

Remark 3.7. (i) A weaker version of Corollary 3.6 is the main result of [16] where the convergence is to the coincidence point of and is assumed injective.
(ii) If in Corollary 3.5, then we have the main result of [2].

Corollary 3.8. Let be a Banach space and such that and. Assume that and are weakly compatible. For any , the Jungck-Mann iteration (2.3) converges to the unique common fixed point of .

Remark 3.9. If , Corollary 3.8 gives the result of [20].

It is already shown in [1, 2] that if or is a complete subspace of , then maps satisfying the operators (2.9) has a unique coincidence point. Hence we have the following results.

Theorem 3.10. Let be a Banach space space and such that and . Assume that and are weakly compatible. For any , the Jungck-multistep iteration (2.6) converges to the unique common fixed point of .

Since the Jungck-Noor, Jungck-Ishikawa, and Jungck-Mann iterations are special cases of Jungck-multistep iteration, then we have the following consequences.

Corollary 3.11. Let be a Banach space and such that and . Assume that and are weakly compatible. For any , the Jungck-Noor iteration (2.5) converges to the unique common fixed point of .

Corollary 3.12. Let be a Banach space and such that and . Assume that and are weakly compatible. For any , the Jungck-Ishikawa iteration (2.4) converges to the unique common fixed point of .

Corollary 3.13. Let be a Banach space and such that and . Assume that and are weakly compatible. For any , the Jungck-Mann iteration (2.3) converges to the unique common fixed point of .

Acknowledgments

The research is supported by the African Mathematics Millennium Science Initiative (AMMSI). The first author is grateful to the Hampton University, Virginia, USA, for hospitality.

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Copyright

Copyright © 2010 J. O. Olaleru and H. Akewe. 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.