On Faster Implicit Hybrid Kirk-Multistep Schemes for Contractive-Type Operators

Abstract

The purpose of this paper is to prove strong convergence and T-stability results of some modified hybrid Kirk-Multistep iterations for contractive-type operator in normed linear spaces. Our results show through analytical and numerical approach that the modified hybrid schemes are better in terms of convergence rate than other hybrid Kirk-Multistep iterative schemes in the literature.

1. Introduction

In the recent years, numerous papers have been published on the strong convergence and T-stability of various iterative approximations of fixed points for contractive-type operators. See [1–9]. The Picard iterative scheme defined for was the first iteration to be proved by Banach [10] for a self-map in a complete metric space satisfying (called strict contraction), for all and . Picard iteration (1) which obeys (2) is said to have a fixed point in , where is the set of all fixed points. The Picard iteration will no longer converge to a fixed point of the operator if contractive condition (2) is weaker. Hence, there is a need to consider other iterative procedures.

Mann [11] defined a more general iteration in a Banach space satisfying quasi-nonexpansive operators. For , the Mann iteration is given by where is a sequence of positive numbers in . Putting in (3) yields Picard iteration (1).

A double Mann iteration, called Ishikawa iteration, was introduced by Ishikawa [12]. It is defined for as where and are sequences of positive numbers in .

The three-step iteration, which is more general than Mann and Ishikawa iterations, was defined by Noor [13]. For , the Noor iteration is given as where , , and are sequences in with .

Rhoades and Soltuz [14] defined a multistep iteration in a normed linear space as follows. For , where and , , are sequences of positive numbers in with . Iteration (6) generalized (3), (4), and (5); for example, if in (6), we recover the form (5); if , we have (4); on putting and for each , we have (3).

Another two-step scheme, which is independent of (4), was introduced by Thianwan [15]. Let ; the sequence is defined as where and are sequences of positive numbers in with .

The three-step iteration of (7) called SP iteration was introduced by Phuengrattana and Suantai [16] and it was defined as follows. For , where with . The two-step iteration of (8) can be easily obtained when .

In [17], Gürsoy et al. defined a generalized scheme of forms (7) and (8) in a Banach space. For , where , , with .

Several results have been proved for the strong convergence of the explicit iteration as well as the SP-iterative scheme of fixed points for different types of contractive-like operators in various spaces. See [3, 6, 18, 19]. The stability results of explicit and SP-iterative schemes have been discussed in [2, 4, 9, 20, 21].

Kirk’s iterative procedure was defined by Kirk [22]. For , where is a Banach space and is a self-map of , where is a fixed integer with , for each and .

In order to reduce the cost of computations, Olatinwo [8] introduced two hybrid schemes, namely, Kirk-Mann and Kirk-Ishikawa iterative schemes in a normed linear space. For ,respectively, where and are fixed integers with and and are sequences in satisfying , , , and .

The Kirk-Noor iteration was introduced by Chugh and Kumar [23] as follows: for , where , , and are fixed integers with , and , , and are sequences in satisfying , , , , , and .

In an attempt to generalize (11) and (12), Gürsoy et al. [17] introduced the Kirk-Multistep iteration in an arbitrary Banach space . For , where are fixed integers with ; and are sequences in satisfying , , , and for each .

The Kirk-SP iteration was defined by Hussain et al. [24] as follows: for , where , , and are fixed integers with and , , and are sequences in satisfying , , , , , and . The Kirk-Thianwan iteration can be obtained if in (14).

The Kirk-Multistep-SP, which generalized both Kirk-SP and Kirk-Thianwan schemes, was introduced by Akewe et al. [1] and it was defined as follows: for , where are fixed integers with ; and are sequences in satisfying , , , and for each .

Another modified form of explicit iteration is the implicit iteration. The implicit Mann iteration and implicit Ishikawa iteration were discussed by Ćirić et al. [7] and Xue and Zhang [19], respectively. For , being a closed subset of normed linear space, the implicit Mann and implicit Ishikawa iterations are, respectively, where and are sequences in .

The implicit Noor iteration was defined by Chugh et al. [5] as follows: for where , , and are sequences in with . The implicit Noor iteration (18) is more general than (16) and (17).

The most generalized Banach operator used by several authors is the one proved by Zamfirescu [25].

Let be a complete metric space and let be a self-map of . The operator is Zamfirescu operator if for each pair of points , at least one of the following is true: where , , and are nonnegative constants satisfying , .

The equivalence form of (19) is for and .

Berinde [2] observed that condition (20) implies where .

In [26], Rhoades used a more general contractive condition than (21): for , there exists such that Osilike [20] extended and generalized the contractive condition (22): for , there exists and such that Imoru and Olatinwo [21] employed a more general class of operators than (23) satisfying the following contractive conditions: where and is a monotone increasing function with .

The equivalence form of (24) in a normed linear space is We will need the following definitions and lemmas to prove our main results.

Definition 1 (see [9]). Let be a metric space and a self-mapping. Suppose that is the set of fixed points of . Let be the sequence generated by an iterative procedure involving which is defined by where is the initial approximation and is a function such that . Suppose that converges to a fixed point of . Let and set . Then, iterative procedure (26) is said to be -stable or stable with respect to if and only if implies .

Definition 2 (see [2]). Let and be two nonnegative real sequences which converge to and , respectively. Let (1)if , then converges to faster than to ;(2)if , then both and have the same convergence rate;(3)if , then converges to faster than to .

Lemma 3 (see [2]). Let be a real number such that and is a sequence of nonnegative numbers such that ; then, for any sequence of positive numbers satisfying we have .

Lemma 4 (see [8]). Let be a normed linear space and a map satisfying (25). Let be a subadditive, monotone increasing function such that , , for , . Then, for all , and for all

Note that in (29).

2. Main Results

We present our main results as follows.

Let be an arbitrary Banach space and a self-map. Let ; we define the following iteration, namely, implicit hybrid Kirk-Multistep iterative scheme, as follows: where are fixed integers with ; and are sequences in satisfying , , , and for each with .

If we let in (30), we obtain the implicit Kirk-Noor iteration defined by By setting in (30), we have the implicit Kirk-Ishikawa iteration and we can also obtain the implicit Kirk-Mann iteration when and in (30).

If in (30), then we obtain the implicit multistep iteration (19) with , .

If , , and in (30), we have the implicit Noor scheme (18) with , , and .

If , , and in (30), we have the implicit Ishikawa scheme (17) with , .

If , , and in (30), we have the implicit Mann scheme (16) with .

Throughout, the operator will be assumed as fixed and a fixed point with the condition (29) is unique.

Theorem 5. Let be a normed linear space. Assume is self-map of satisfying the contractive condition (29) with . Then, for , the sequence defined by (30) with converges strongly to the fixed point .

Proof. Let and ; then using (29) and (30) we have This implies that Also, from (30), we have This becomes From (30) again, we have which implies Continuing this way up to in (30), we have implying that Substituting (35)–(39) into (33) it becomes Let ; then Therefore, Similarly, we can easily obtain the following from (40): Applying (42) and (43) in (40) and letting for each , we have As , . Hence, .
Therefore, implicit Kirk-Multistep scheme (30) converges strongly to .

Corollary 6. Let be a normed linear space. Assume is self-map of satisfying the contractive condition (29) with . Then, for , the implicit Kirk-Noor, the implicit Kirk-Ishikawa, and the implicit Kirk-Mann schemes with converge strongly to the fixed point .

Remark 7. The strong convergence results for implicit Noor, implicit Ishikawa, and implicit Mann schemes are obvious from Theorem 5.

Theorem 8. Let be a normed linear space and is a self-map of satisfying contractive condition (29) with . Then, for and , the sequence defined by (30) is T-stable.

Proof. Let be an arbitrary sequence and let , where Suppose and ; by (29) we have This implies that From inequalities (42) and (43), one can easily obtain the following: Then, inequality (47) becomes Letting and by Lemma 3, we have Conversely, suppose for ; then Since , then .
Therefore, iterative scheme (30) is T-stable.

Corollary 9. Let be a normed linear space and is a self-map of satisfying the contractive condition (29) with . Then, for and , the sequence defined by implicit Kirk-Mann, implicit Kirk-Ishikawa, and implicit Kirk-Noor schemes are T-stable.

Remark 10. The stability results for implicit Mann, implicit Ishikawa, and implicit Noor schemes with contractive condition (29) are special cases of Corollary 9.

2.1. Comparison of Several Iterative Schemes

We compare our iterative schemes with others by using the following example.

Example 11. Let and with and fixed point using , , for each , , and .

For the implicit Kirk-Mann iteration (IKM), we have This implies that Also, for implicit Kirk-Ishikawa iteration (IKI), we have with Hence, Similarly, implicit Kirk-Noor iteration (IKN) implies while the implicit multistep Kirk iteration (IMK) gives Now, using Definition 2, we compare the implicit Kirk type iterations as follows: for , we have with

Remark 12. The implicit multistep Kirk iteration (IMK) converges faster than the implicit Kirk-Noor iteration (IKN) for .

Also, with

Remark 13. The implicit Kirk-Noor iteration converges to faster than the implicit Kirk-Ishikawa iteration to .

Similarly, using Definition 2, we have that which implies that the implicit Kirk-Ishikawa iteration converges faster than the implicit Kirk-Mann iteration .

For the Kirk-Mann iteration (KM), we have the following estimate: This implies that The estimates for Kirk-Thianwan (KT), Kirk-SP (KSP), and Kirk-Multistep-SP (KMSP) iterations are, respectively, We compare Kirk-Mann (KM), Kirk-Thianwan (KT), Kirk-SP (KSP), and Kirk-Multistep-SP (KMSP) iterations with our iterative schemes as follows.

Again, using Definition 2, we have with

Remark 14. The implicit Kirk-Mann iteration converges to faster than the Kirk-Mann iteration to .

For the comparison of implicit Kirk-Ishikawa iteration and Kirk-Thianwan iteration , we have with

Remark 15. The implicit Kirk-Ishikawa iteration converges faster than the Kirk-Thianwan iteration .

For the comparison of implicit Kirk-Noor iteration and Kirk-SP iteration , we have with

Remark 16. The implicit Kirk-Noor iteration has better convergence rate than the Kirk-SP iteration .

For the comparison of implicit Kirk-Multistep iteration and Kirk-Multistep-SP iteration , we have with

Remark 17. The implicit Kirk-Multistep iteration has better convergence rate than the Kirk-Multistep-SP iteration for .

3. Conclusion

We have established and proved strong convergence and T-stability results for implicit Kirk-Multistep, implicit Kirk-Noor, implicit Kirk-Ishikawa, and implicit Kirk-Mann iterative schemes of fixed points with contractive-type operators in normed linear spaces. These iterative schemes have better convergence rate when compared with other iterative schemes, namely, multistep Kirk-SP iteration, Kirk-Multistep scheme, Kirk-SP iteration, Kirk-Thianwan scheme, Kirk-Noor scheme, Kirk-Ishikawa scheme, Kirk-Mann scheme, implicit Noor iteration, implicit Ishikawa iteration, implicit Mann iteration, and many more iterative schemes of fixed point in the literature.

Disclosure

Authors agreed to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

Competing Interests

The authors hereby declare that there are no competing interests.