Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 526503 | 22 pages | https://doi.org/10.1155/2012/526503

On the Rate of Convergence of Kirk-Type Iterative Schemes

Academic Editor: B. V. Rathish Kumar
Received29 Jan 2012
Revised21 Apr 2012
Accepted14 May 2012
Published31 Jul 2012

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 (𝑋,𝑑) be a complete metric space and π‘‡βˆΆπ‘‹β†’π‘‹ a selfmap of 𝑋. Suppose that 𝐹(𝑇)={π‘βˆˆπ‘‹,𝑇𝑝=𝑝} is the set of fixed points of 𝑇. 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 π‘₯𝑛+1=𝑇π‘₯𝑛,𝑛=0,1,…,(1.1) which is used to approximate the fixed points of mappings satisfying the inequality: 𝑑(𝑇π‘₯,𝑇𝑦)≀𝛼𝑑(π‘₯,𝑦)(1.2) for all π‘₯,π‘¦βˆˆπ‘‹ and π›Όβˆˆ[0,1). Condition (1.2) is called Banach’s contraction condition.

The following iteration schemes are now well known: 𝑒𝑛+1=ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘’π‘›+𝛼𝑛𝑇𝑒𝑛,(1.3) where {𝛼𝑛} is a sequences of positive numbers in [0,1], due to Mann [1]. π‘₯𝑛+1=π‘˜ξ“π‘–=0𝛼𝑖𝑇𝑖π‘₯𝑛,π‘˜ξ“π‘–=0𝛼𝑖=1,(1.4) where {𝛼𝑛} is a sequences of positive numbers in [0,1], due to Kirk [2]. π‘₯𝑛+1=ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘₯𝑛+𝛼𝑛𝑇𝑦𝑛,𝑦𝑛=ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛+𝛽𝑛𝑇π‘₯𝑛,(1.5) where {𝛼𝑛} and {𝛽𝑛} are sequences of positive numbers in [0,1], due to Ishikawa [3]. π‘₯𝑛+1=ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘₯𝑛+𝛼𝑛𝑇𝑦𝑛,𝑦𝑛=ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛+𝛽𝑛𝑇𝑧𝑛,𝑧𝑛=ξ€·1βˆ’π›Ύπ‘›ξ€Έπ‘₯𝑛+𝛾𝑛𝑇π‘₯𝑛,(1.6) where {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are sequences of positive numbers in [0,1], due to Noor [4].

In [5], Olatinwo introduced the Kirk-Mann and Kirk-Ishikawa iterative schemes as follows: π‘₯𝑛+1=𝛼𝑛,0π‘₯𝑛+π‘˜ξ“π‘–=1𝛼𝑛,𝑖𝑇𝑖𝑦𝑛,π‘˜ξ“π‘–=0𝛼𝑛,𝑖𝑦=1,𝑛=𝑠𝑗=0𝛽𝑛,𝑗𝑇𝑗π‘₯𝑛,𝑠𝑗=0𝛽𝑛,𝑗=1,𝑛=0,1,2,…,(1.7) where π‘˜β‰₯𝑠, 𝛼𝑛,𝑖β‰₯0,𝛼𝑛,0β‰ 0, 𝛽𝑛,𝑗β‰₯0, 𝛽𝑛,0β‰ 0, 𝛼𝑛,𝑖,𝛽𝑛,π‘—βˆˆ[0,1], and π‘˜, 𝑠 are fixed integers, called as Kirk-Ishikawa iteration scheme: 𝑒𝑛+1=π‘˜ξ“π‘–=0𝛼𝑛,𝑖𝑇𝑖𝑒𝑛,π‘˜ξ“π‘–=0𝛼𝑛,𝑖=1,𝑛=0,1,2,…,(1.8) where 𝛼𝑛,𝑖β‰₯0, 𝛼𝑛,0β‰ 0, 𝛼𝑛,π‘–βˆˆ[0,1], and π‘˜ is a fixed integer and is called as Kirk-Mann iteration scheme.

However, from [6], the Kirk-Noor iterative scheme is given by π‘₯𝑛+1=𝛼𝑛,0π‘₯𝑛+π‘˜ξ“π‘–=1𝛼𝑛,𝑖𝑇𝑖𝑦𝑛,π‘˜ξ“π‘–=0𝛼𝑛,𝑖𝑦=1,𝑛=𝛽𝑛,0π‘₯𝑛+π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘Ÿπ‘‡π‘Ÿπ‘§π‘›,𝑠𝑗=0𝛽𝑛,𝑗𝑧=1,𝑛=𝑑𝑙=0𝛾𝑛,𝑙𝑇𝑙π‘₯𝑛,𝑑𝑙=0𝛾𝑛,𝑙=1,𝑛=0,1,2,….(1.9) In [7], Phuengrattana and Suantai defined the SP iteration scheme as follows: π‘₯𝑛+1=ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘¦π‘›+𝛼𝑛𝑇𝑦𝑛,𝑦𝑛=ξ€·1βˆ’π›½π‘›ξ€Έπ‘§π‘›+𝛽𝑛𝑇𝑧𝑛,𝑧𝑛=ξ€·1βˆ’π›Ύπ‘›ξ€Έπ‘₯𝑛+𝛾𝑛𝑇π‘₯𝑛,(1.10) where {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are sequences of positive numbers in [0,1].

Recently, Chugh and Kumar introduced the following iteration scheme [8]: π‘₯𝑛+1=ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘¦π‘›+𝛼𝑛𝑇𝑦𝑛,𝑦𝑛=ξ€·1βˆ’π›½π‘›ξ€Έπ‘‡π‘₯𝑛+𝛽𝑛𝑇𝑧𝑛,𝑧𝑛=ξ€·1βˆ’π›Ύπ‘›ξ€Έπ‘₯𝑛+𝛾𝑛𝑇π‘₯𝑛,(1.11) where {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are sequences of positive numbers in [0,1].

Remarks 1. (1) If 𝛾𝑛=0, then (1.6) reduces to the Ishikawa iteration scheme (1.5).
(2) If 𝛽𝑛=𝛾𝑛=0, then (1.6) reduces to the Mann iteration scheme (1.3).
(3) If 𝛽𝑛=0, then (1.5) reduces to the Mann iteration scheme (1.3).
(4) If 𝛽𝑛=𝛾𝑛=0, 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 (𝑋,𝑑) be a complete metric space and π‘‡βˆΆπ‘‹β†’π‘‹ a mapping for which there exists real numbers π‘Ž, 𝑏, and c satisfying π‘Žβˆˆ(0,1),𝑏,π‘βˆˆ(0,1/2) such that for each pair π‘₯,π‘¦βˆˆπ‘‹ at least one of the following conditions holds: [𝑑],[].(i)𝑑(𝑇π‘₯,𝑇𝑦)β‰€π‘Žπ‘‘(π‘₯,𝑦),(ii)𝑑(𝑇π‘₯,𝑇𝑦)≀𝑏(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑇𝑦)(iii)𝑑(𝑇π‘₯,𝑇𝑦)≀𝑐𝑑(π‘₯,𝑇𝑦)+𝑑(𝑦,𝑇π‘₯)(1.12) Then, 𝑇 has a unique fixed point 𝑝 and the Picard iteration scheme {π‘₯𝑛} defined by (1.1) converges to 𝑝 for any arbitrary but fixed π‘₯0βˆˆπ‘‹.

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 𝑑(𝑇π‘₯,𝑇𝑦)≀2𝛿𝑑(π‘₯,𝑇π‘₯)+𝛿𝑑(π‘₯,𝑦)(1.13) for all π‘₯,π‘¦βˆˆπ‘‹ and π›Ώβˆˆ[0,1). 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 𝐾 be a nonempty closed convex subset of an arbitrary Banach space 𝑋 and π‘‡βˆΆπΎβ†’πΎ a mapping satisfying (1.13). Let {π‘₯𝑛}βˆžπ‘›=0 be defined through the Ishikawa iteration scheme (1.5) and π‘₯0∈𝐾, where {𝛼𝑛}, {𝛽𝑛} are sequences of positive real numbers in [0,1] with {𝛼𝑛} satisfying βˆ‘βˆžπ‘›=0𝛼𝑛=∞. Then, {π‘₯𝑛}βˆžπ‘›=0 converges strongly to the fixed point of 𝑇.

However, in [11], Rafiq studied the convergence of the Noor iteration scheme [4] involving quasi-contractive 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 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 π‘Žβˆˆ[0,1) and a monotone increasing function πœ‘βˆΆπ‘…+→𝑅+ with πœ‘(0)=0, such that ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€πœ‘(β€–π‘₯βˆ’π‘‡π‘₯β€–)+π‘Žβ€–π‘₯βˆ’π‘¦β€–βˆ€π‘₯,π‘¦βˆˆπ‘‹.(1.14) 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 𝛿 is a real number such that 0≀𝛿<1 and {βˆˆπ‘›}βˆžπ‘›=0 is a sequence of nonnegative numbers such that limπ‘›β†’βˆžβˆˆπ‘›=0, then, for any sequence of positive numbers {𝑒𝑛}βˆžπ‘›=0 satisfying 𝑒𝑛+1≀𝛿𝑒𝑛+βˆˆπ‘›,𝑛=0,1,2,…,(2.1) one has limπ‘›β†’βˆžπ‘’π‘›=0.

Lemma 2.2 (see [5]). Let (𝑋,β€–β‹…β€–) be a normed linear space and let π‘‡βˆΆπ‘‹β†’π‘‹ be a selfmap of 𝑋 satisfying (1.13). Let πœ‘βˆΆπ‘…+→𝑅+ be a subadditive, monotone increasing function such that πœ‘(0)=0, πœ‘(𝐿𝑒)β‰€πΏπœ‘(𝑒), 𝐿β‰₯0, π‘’βˆˆπ‘…+. Then, for all π‘–βˆˆπ‘, 𝐿β‰₯0 and for all π‘₯,π‘¦βˆˆπ‘‹, ‖‖𝑇𝑖π‘₯βˆ’π‘‡π‘–π‘¦β€–β€–β‰€π‘–ξ“π‘—=1βŽ›βŽœβŽœβŽπ‘–π‘—βŽžβŽŸβŽŸβŽ π‘Žπ‘–βˆ’π‘—πœ‘π‘—(β€–π‘₯βˆ’π‘‡π‘₯β€–)+π‘Žπ‘–β€–π‘₯βˆ’π‘¦β€–.(2.2)

Definition 2.3 (see [20]). Suppose {π‘Žπ‘›} and {𝑏𝑛} are two real convergent sequences with limits π‘Ž and 𝑏, respectively. Then, {π‘Žπ‘›} is said to converge faster than {𝑏𝑛} if limπ‘›β†’βˆž||||π‘Žπ‘›βˆ’π‘Žπ‘π‘›||||βˆ’π‘=0.(2.3)

Now, we define Kirk-SP and Kirk-CR iterative schemes as follows. Let 𝑋 be a Banach space, π‘‡βˆΆπ‘‹β†’π‘‹ a selfmap of 𝑋 and π‘₯0βˆˆπ‘‹. Then, the sequence {π‘₯𝑛}βˆžπ‘›=0 defined by π‘₯𝑛+1=𝛼𝑛,0𝑦𝑛+π‘˜ξ“π‘–=1𝛼𝑛,𝑖𝑇𝑖𝑦𝑛,π‘˜ξ“π‘–=0𝛼𝑛,𝑖𝑦=1,𝑛=𝛽𝑛,0𝑧𝑛+π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘Ÿπ‘‡π‘Ÿπ‘§π‘›,𝑠𝑗=0𝛽𝑛,𝑗𝑧=1,𝑛=𝑑𝑙=0𝛾𝑛,𝑙𝑇𝑙π‘₯𝑛,𝑑𝑙=0𝛾𝑛,𝑙=1,𝑛=0,1,2,…,(2.4) is called Kirk-SP iterative scheme and the sequence {π‘₯𝑛}βˆžπ‘›=0 defined by π‘₯𝑛+1=𝛼𝑛,0𝑦𝑛+π‘˜ξ“π‘–=1𝛼𝑛,𝑖𝑇𝑖𝑦𝑛,π‘˜ξ“π‘–=0𝛼𝑛,𝑖𝑦=1,𝑛=𝛽𝑛,0𝑇π‘₯𝑛+π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘Ÿπ‘‡π‘Ÿπ‘§π‘›,π‘ ξ“π‘Ÿ=0𝛽𝑛,π‘Ÿπ‘§=1,𝑛=𝑑𝑙=0𝛾𝑛,𝑙𝑇𝑙π‘₯𝑛,𝑑𝑙=0𝛾𝑛,𝑙=1,𝑛=0,1,2…,(2.5) is called Kirk-CR iterative scheme, where π‘˜, 𝑠, and 𝑑 are fixed integers with π‘˜β‰₯𝑠β‰₯𝑑 and 𝛼𝑛,𝑖, 𝛽𝑛,π‘Ÿ, 𝛾𝑛,𝑙 are sequences in [0,1] satisfying 𝛼𝑛,𝑖β‰₯0,𝛼𝑛,0β‰ 0, 𝛽𝑛,π‘Ÿβ‰₯0,𝛽𝑛,0β‰ 0,𝛾𝑛,𝑙β‰₯0,𝛾𝑛,0β‰ 0.

Remarks 2. (5) Putting 𝑑=𝑠=0 in (2.4), we obtain the Kirk-Mann iterative scheme (1.9).
(6) Putting 𝑠=0,π‘˜=1,𝑑=0 in (2.4), we get usual Mann iterative scheme (1.3). With βˆ‘1𝑖=0𝛼𝑛,𝑖=1,𝛼𝑛,1=𝛼𝑛.
(7) Putting 𝑠=0,𝑑=0, and 𝛼𝑛,𝑖=𝛼𝑖 in (2.4), we obtain the usual Kirk’s iterative scheme (1.4).
(8) Putting 𝑠=𝑑=1 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 (𝑋,β€–β‹…β€–) be a normed linear space and π‘‡βˆΆπ‘‹β†’π‘‹ a selfmap of 𝑋 satisfying the contractive condition (1.14) and πœ‘βˆΆπ‘…+→𝑅+ a subadditive monotone increasing function such that πœ‘(0)=0 and πœ‘(𝐿𝑒)β‰€πΏπœ‘(𝑒),𝐿β‰₯0,π‘’βˆˆπ‘…+. Let π‘₯0βˆˆπ‘‹ and {π‘₯𝑛}βˆžπ‘›=0 be the Kirk-SP iterative scheme defined by (2.4). Suppose that 𝑇 has a fixed point 𝑝. Then, the Kirk-SP iterative scheme converges strongly to 𝑝.

Proof. Using Kirk-SP iterative scheme (2.4) and Lemma 2.1, we have β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘β‰€π›Όπ‘›,0‖‖𝑦𝑛‖‖+βˆ’π‘π‘˜ξ“π‘–=1𝛼𝑛,π‘–β€–β€–π‘‡π‘–π‘¦π‘›β€–β€–βˆ’π‘β‰€π›Όπ‘›,0‖‖𝑦𝑛‖‖+βˆ’π‘π‘˜ξ“π‘–=1𝛼𝑛,π‘–βŽ§βŽͺ⎨βŽͺβŽ©π‘–ξ“π‘—=1βŽ›βŽœβŽœβŽπ‘–π‘—βŽžβŽŸβŽŸβŽ π‘Žπ‘–βˆ’π‘—πœ‘π‘—(β€–π‘βˆ’π‘‡π‘β€–)+π‘Žπ‘–β€–β€–π‘¦π‘›β€–β€–βŽ«βŽͺ⎬βŽͺβŽ­βˆ’π‘=𝛼𝑛,0‖‖𝑦𝑛‖‖+ξƒ©βˆ’π‘π‘˜ξ“π‘–=1𝛼𝑛,π‘–π‘Žπ‘–ξƒͺ‖‖𝑦𝑛‖‖=ξƒ©βˆ’π‘π‘˜ξ“π‘–=0𝛼𝑛,π‘–π‘Žπ‘–ξƒͺ‖‖𝑦𝑛‖‖.βˆ’π‘(2.6) Now, we have the following estimates: β€–β€–π‘¦π‘›β€–β€–βˆ’π‘β‰€π›½π‘›,0‖‖𝑧𝑛‖‖+βˆ’π‘π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘Ÿβ€–β€–π‘‡π‘Ÿπ‘§π‘›β€–β€–βˆ’π‘β‰€π›½π‘›,0‖‖𝑧𝑛‖‖+βˆ’π‘π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘ŸβŽ§βŽͺ⎨βŽͺβŽ©π‘Ÿξ“π‘—=1βŽ›βŽœβŽœβŽπ‘Ÿπ‘—βŽžβŽŸβŽŸβŽ π‘Žπ‘Ÿβˆ’π‘—πœ‘π‘—(β€–π‘βˆ’π‘‡π‘β€–)+π‘Žπ‘Ÿβ€–β€–π‘§π‘›β€–β€–βŽ«βŽͺ⎬βŽͺ⎭=ξƒ©βˆ’π‘π‘˜ξ“π‘Ÿ=0𝛽𝑛,π‘Ÿπ‘Žπ‘Ÿξƒͺ‖‖𝑧𝑛‖‖,β€–β€–π‘§βˆ’π‘π‘›β€–β€–βˆ’π‘β‰€π›Ύπ‘›,𝑙‖‖π‘₯𝑛‖‖+βˆ’π‘π‘‘ξ“π‘™=1𝛾𝑛,𝑙‖‖𝑇𝑙π‘₯π‘›β€–β€–βˆ’π‘β‰€π›Ύπ‘›,𝑙‖‖π‘₯𝑛‖‖+βˆ’π‘π‘‘ξ“π‘™=1𝛾𝑛,π‘™βŽ§βŽͺ⎨βŽͺβŽ©π‘™ξ“π‘—=1βŽ›βŽœβŽœβŽπ‘™π‘—βŽžβŽŸβŽŸβŽ π‘Žπ‘™βˆ’π‘—πœ‘π‘—(β€–π‘βˆ’π‘‡π‘β€–)+π‘Žπ‘™β€–β€–π‘₯π‘›β€–β€–βŽ«βŽͺ⎬βŽͺ⎭=ξƒ©βˆ’π‘π‘‘ξ“π‘™=0𝛾𝑛,π‘™π‘Žπ‘™ξƒͺβ€–β€–π‘₯𝑛‖‖.βˆ’π‘(2.7) It follows from (2.6), (2.7) that β€–β€–π‘₯𝑛+1β€–β€–β‰€ξƒ©βˆ’π‘π‘˜ξ“π‘–=0𝛼𝑛,π‘–π‘Žπ‘–ξƒͺξƒ©π‘ ξ“π‘Ÿ=0𝛽𝑛,π‘–π‘Žπ‘–ξƒͺ𝑑𝑙=0𝛾𝑛,π‘™π‘Žπ‘™ξƒͺβ€–β€–π‘₯𝑛‖‖.βˆ’π‘(2.8) Since π‘Žπ‘–βˆˆ[0,1) and βˆ‘π‘˜π‘–=0𝛼𝑛,𝑖=βˆ‘π‘ π‘Ÿ=0𝛽𝑛,π‘Ÿ=βˆ‘π‘‘π‘™=0𝛾𝑛,𝑙=1, hence ξƒ©π‘˜ξ“π‘–=0𝛼𝑛,π‘–π‘Žπ‘–ξƒͺξƒ©π‘ ξ“π‘Ÿ=0𝛽𝑛,π‘Ÿπ‘Žπ‘Ÿξƒͺ𝑑𝑙=0𝛾𝑛,π‘™π‘Žπ‘™ξƒͺ<ξƒ©π‘˜ξ“π‘–=0𝛼𝑛,𝑖ξƒͺξƒ©π‘ ξ“π‘Ÿ=0𝛽𝑛,π‘Ÿξƒͺ𝑑𝑙=0𝛾𝑛,𝑙ξƒͺ=1.(2.9) Using (2.9) and Lemma 2.1, (2.8) yields limπ‘›β†’βˆžπ‘₯𝑛=𝑝. Thus, Kirk-SP iterative scheme converges strongly to 𝑝.

Theorem 2.5. Let (𝑋,β€–β‹…β€–) be a normed linear space and π‘‡βˆΆπ‘‹β†’π‘‹ a selfmap of 𝑋 satisfying the contractive condition (1.14) and πœ‘βˆΆπ‘…+→𝑅+ a subadditive monotone increasing function such that πœ‘(0)=0 and πœ‘(𝐿𝑒)β‰€πΏπœ‘(𝑒),𝐿β‰₯0,π‘’βˆˆπ‘…+. Let π‘₯0βˆˆπ‘‹ and {π‘₯𝑛}βˆžπ‘›=0 be the Kirk-CR iterative scheme defined by (2.5). Suppose that 𝑇 has a fixed point 𝑝. Then, the Kirk-CR iterative scheme converges strongly to 𝑝.

Proof. Using Kirk-CR iterative scheme (2.5) and Lemma 2.2, we have β€–β€–π‘₯𝑛+1β€–β€–β‰€ξƒ©βˆ’π‘π‘˜ξ“π‘–=0𝛼𝑛,π‘–π‘Žπ‘–ξƒͺ‖‖𝑦𝑛‖‖.βˆ’π‘(2.10) Now, we have the following estimates: ‖‖𝑦𝑛+1β€–β€–βˆ’π‘β‰€π›½π‘›,π‘œβ€–β€–π‘‡π‘₯𝑛‖‖+βˆ’π‘π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘Ÿβ€–β€–π‘‡π‘Ÿπ‘§π‘›β€–β€–βˆ’π‘β‰€π‘Žπ›½π‘›,0β€–β€–π‘₯𝑛‖‖+βˆ’π‘π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘ŸβŽ§βŽͺ⎨βŽͺβŽ©π‘Ÿξ“π‘—=1βŽ›βŽœβŽœβŽπ‘Ÿπ‘—βŽžβŽŸβŽŸβŽ π‘Žπ‘Ÿβˆ’π‘—πœ‘π‘—(β€–π‘βˆ’π‘‡π‘β€–)+π‘Žπ‘Ÿβ€–β€–π‘§π‘›β€–β€–βŽ«βŽͺ⎬βŽͺβŽ­βˆ’π‘=π‘Žπ›½π‘›,0β€–β€–π‘₯𝑛‖‖+ξƒ©βˆ’π‘π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘Ÿπ‘Žπ‘Ÿξƒͺ‖‖𝑧𝑛‖‖,β€–β€–π‘§βˆ’π‘π‘›β€–β€–β‰€ξƒ©βˆ’π‘π‘‘ξ“π‘™=0𝛾𝑛,π‘™π‘Žπ‘™ξƒͺβ€–β€–π‘₯𝑛‖‖.βˆ’π‘(2.11) It follows from (2.10), (2.11) that β€–β€–π‘₯𝑛+1β€–β€–ξƒ©βˆ’π‘β‰€π‘Žπ‘˜ξ“π‘–=0𝛼𝑛,π‘–π‘Žπ‘–ξƒͺ𝛽𝑛,0β€–β€–π‘₯𝑛‖‖+ξƒ©βˆ’π‘π‘˜ξ“π‘–=0𝛼𝑛,π‘–π‘Žπ‘–ξƒͺξƒ©π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘Ÿπ‘Žπ‘Ÿξƒͺ𝑑𝑙=0𝛾𝑛,π‘™π‘Žπ‘™ξƒͺβ€–β€–π‘₯𝑛‖‖=ξƒ©βˆ’π‘π‘˜ξ“π‘–=0𝛼𝑛,π‘–π‘Žπ‘–ξƒͺξƒ¬π‘Žπ›½π‘›,0+ξƒ©π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘Ÿπ‘Žπ‘Ÿξƒͺ𝑑𝑙=0𝛾𝑛,π‘™π‘Žπ‘™β€–β€–π‘₯ξƒͺ𝑛‖‖.βˆ’π‘(2.12) Since π‘Žπ‘–βˆˆ[0,1) and βˆ‘π‘˜π‘–=0𝛼𝑛,𝑖=βˆ‘π‘ π‘Ÿ=0𝛽𝑛,π‘Ÿ=βˆ‘π‘‘π‘™=0𝛾𝑛,𝑙=1 with 𝛼𝑛,0β‰ 0,𝛽𝑛,0β‰ 0,𝛾𝑛,0β‰ 0, hence ξƒ©π‘˜ξ“π‘–=0𝛼𝑛,π‘–π‘Žπ‘–ξƒͺξƒ¬π‘Žπ›½π‘›,0+ξƒ©π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘Ÿπ‘Žπ‘Ÿξƒͺ𝑑𝑙=0𝛾𝑛,π‘™π‘Žπ‘™<ξƒͺξƒ­π‘˜ξ“π‘–=0𝛼𝑛,𝑖𝛽ξƒͺ𝑛,0+ξƒ©π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘Ÿξƒͺ𝑑𝑙=0𝛾𝑛,𝑙ξƒͺξƒ­=𝛽𝑛,0+π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘Ÿ=1.(2.13) Using (2.13) and Lemma 2.1, (2.12) yields limπ‘›β†’βˆžπ‘₯𝑛=𝑝. Thus, Kirk-CR iterative scheme converges strongly to 𝑝.

Theorem 2.6. Let (𝑋,β€–β‹…β€–) be a normed linear space and π‘‡βˆΆπ‘‹β†’π‘‹ a selfmap of 𝑋 satisfying the contractive condition (1.14) and πœ‘βˆΆπ‘…+→𝑅+ a subadditive monotone increasing function such that πœ‘(0)=0 and πœ‘(𝐿𝑒)β‰€πΏπœ‘(𝑒), 𝐿β‰₯0,π‘’βˆˆπ‘…+. Let π‘₯0βˆˆπ‘‹ and {π‘₯𝑛}βˆžπ‘›=0 be the Kirk-Noor iterative scheme defined by (1.9). Suppose that 𝑇 has a fixed point 𝑝. Then, the Kirk-Noor iterative scheme converges strongly to 𝑝.

Proof. Using Kirk-Noor iterative scheme (1.9) and Lemma 2.2, we have β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘β‰€π›Όπ‘›,0β€–β€–π‘₯𝑛‖‖+βˆ’π‘π‘˜ξ“π‘–=1𝛼𝑛,π‘–β€–β€–π‘‡π‘–π‘¦π‘›β€–β€–βˆ’π‘β‰€π›Όπ‘›,0β€–β€–π‘₯𝑛‖‖+βˆ’π‘π‘˜ξ“π‘–=1𝛼𝑛,π‘–βŽ§βŽͺ⎨βŽͺβŽ©π‘–ξ“π‘—=1βŽ›βŽœβŽœβŽπ‘–π‘—βŽžβŽŸβŽŸβŽ π‘Žπ‘–βˆ’π‘—πœ‘π‘—(β€–π‘βˆ’π‘‡π‘β€–)+π‘Žπ‘–β€–β€–π‘¦π‘›β€–β€–βŽ«βŽͺ⎬βŽͺβŽ­βˆ’π‘=𝛼𝑛,0β€–β€–π‘₯𝑛‖‖+ξƒ©βˆ’π‘π‘˜ξ“π‘–=1𝛼𝑛,π‘–π‘Žπ‘–ξƒͺ‖‖𝑦𝑛‖‖.βˆ’π‘(2.14) Now, we have the following estimates: β€–β€–π‘¦π‘›β€–β€–βˆ’π‘β‰€π›½π‘›,0β€–β€–π‘₯𝑛‖‖+βˆ’π‘π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘Ÿβ€–β€–π‘‡π‘Ÿπ‘§π‘›β€–β€–βˆ’π‘β‰€π›½π‘›,0β€–β€–π‘₯𝑛‖‖+βˆ’π‘π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘ŸβŽ§βŽͺ⎨βŽͺβŽ©π‘Ÿξ“π‘—=1βŽ›βŽœβŽœβŽπ‘Ÿπ‘—βŽžβŽŸβŽŸβŽ π‘Žπ‘Ÿβˆ’π‘—πœ‘π‘—(β€–π‘βˆ’π‘‡π‘β€–)+π‘Žπ‘Ÿβ€–β€–π‘§π‘›β€–β€–βŽ«βŽͺ⎬βŽͺβŽ­βˆ’π‘=𝛽𝑛,0β€–β€–π‘₯𝑛‖‖+ξƒ©βˆ’π‘π‘˜ξ“π‘Ÿ=1𝛽𝑛,π‘Ÿπ‘Žπ‘Ÿξƒͺ‖‖𝑧𝑛‖‖,β€–β€–π‘§βˆ’π‘π‘›β€–β€–βˆ’π‘β‰€π›Ύπ‘›,𝑙‖‖π‘₯𝑛‖‖+βˆ’π‘π‘‘ξ“π‘™=1𝛾𝑛,𝑙‖‖𝑇𝑙π‘₯π‘›β€–β€–βˆ’π‘β‰€π›Ύπ‘›,𝑙‖‖π‘₯𝑛‖‖+βˆ’π‘π‘‘ξ“π‘™=1𝛾𝑛,π‘™βŽ§βŽͺ⎨βŽͺβŽ©π‘™ξ“π‘—=1βŽ›βŽœβŽœβŽπ‘™π‘—βŽžβŽŸβŽŸβŽ π‘Žπ‘™βˆ’π‘—πœ‘π‘—(β€–π‘βˆ’π‘‡π‘β€–)+π‘Žπ‘™β€–β€–π‘₯π‘›β€–β€–βŽ«βŽͺ⎬βŽͺ⎭=ξƒ©βˆ’π‘π‘‘ξ“π‘™=0𝛾𝑛,π‘™π‘Žπ‘™ξƒͺβ€–β€–π‘₯𝑛‖‖.βˆ’π‘(2.15) It follows from (2.14), (2.15) that β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘β‰€π›Όπ‘›,0β€–β€–π‘₯𝑛‖‖+ξƒ©βˆ’π‘π‘˜ξ“π‘–=1𝛼𝑛,π‘–π‘Žπ‘–ξƒͺ𝛽𝑛,0β€–β€–π‘₯𝑛‖‖+ξƒ©βˆ’π‘π‘˜ξ“π‘–=1𝛼𝑛,π‘–π‘Žπ‘–ξƒͺξƒ©π‘ ξ“π‘Ÿ=1𝛽𝑛,π‘–π‘Žπ‘–ξƒͺ𝑑𝑙=0𝛾𝑛,π‘™π‘Žπ‘™ξƒͺβ€–β€–π‘₯𝑛‖‖<ξ€Ίπ›Όβˆ’π‘π‘›,0+ξ€·1βˆ’π›Όπ‘›,0𝛽𝑛,0+ξ€·1βˆ’π›Όπ‘›,0ξ€Έξ€·1βˆ’π›½π‘›,0β€–β€–π‘₯𝑛‖‖=β€–β€–π‘₯βˆ’π‘π‘›β€–β€–.βˆ’π‘(2.16) Using Lemma (2.2), (2.16) yields limπ‘›β†’βˆžπ‘₯𝑛=𝑝. Thus, Kirk-Noor iterative scheme converges strongly to 𝑝.

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 π‘‡βˆΆ[0,1]β†’[0,1]∢=π‘₯/2,𝛼𝑛,1=𝛽𝑛,1=𝛾𝑛,1=𝛼𝑛,0=𝛽𝑛,0=𝛾𝑛,0√=4/𝑛,𝑛=1,2,…,𝑛0 for some 𝑛0βˆˆπ‘ and 𝛼𝑛,2=𝛽𝑛,2=𝛾𝑛,2√=1βˆ’8/𝑛,𝑛β‰₯𝑛0.

It is clear that 𝑇 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 𝑛β‰₯64 and 𝑒0=π‘₯0 with π‘₯0β‰ 0. Then, for Kirk-Mann and Kirk-Ishikawa iterative schemes, we have 𝑒𝑛+1=𝑛𝑖=6414+4βˆšπ‘–ξƒͺ𝑒0,π‘₯𝑛+1=𝑛𝑖=641+516βˆšπ‘–ξƒͺπ‘₯0.(3.1) Now, consider ||||π‘₯𝑛+1𝑒𝑛+1||||=||||||βˆπ‘›π‘–=64ξ‚€βˆš1/16+5/𝑖π‘₯0βˆπ‘›π‘–=64ξ‚€βˆš1/4+4/𝑖𝑒0||||||=||||||𝑛𝑖=64βŽ‘βŽ’βŽ’βŽ£ξ‚€βˆš1βˆ’3/16βˆ’1/π‘–ξ‚ξ‚€βˆš1/4+4/π‘–ξ‚βŽ€βŽ₯βŽ₯⎦||||||=||||||𝑛𝑖=64βŽ‘βŽ’βŽ’βŽ£ξ‚€3√1βˆ’ξ‚π‘–βˆ’16ξ‚€4βˆšξ‚βŽ€βŽ₯βŽ₯⎦||||||𝑖+64.(3.2) It is easy to see that 0≀limπ‘›π‘›β†’βˆžξ‘π‘–=64βŽ‘βŽ’βŽ’βŽ£ξ‚€3√1βˆ’ξ‚π‘–βˆ’16ξ‚€4βˆšξ‚βŽ€βŽ₯βŽ₯βŽ¦π‘–+64≀limπ‘›π‘›β†’βˆžξ‘π‘–=64ξ‚€11βˆ’π‘–ξ‚=limπ‘›β†’βˆž63𝑛=0.(3.3) Hence, limπ‘›β†’βˆž|π‘₯𝑛+1/𝑝𝑛+1|=0.
Therefore, by Definition 2.3, Kirk-Ishikawa iterative scheme converges faster than Kirk-Mann iterative scheme to the fixed point 0 of 𝑇.
Similarly, ||||π‘₯𝑛+1(Kirk-Noor)π‘₯𝑛+1||||=||||||∏(Kirk-Ishikawa)𝑛𝑖=64ξ‚€βˆš1/64+21/4𝑖π‘₯0βˆπ‘›π‘–=64ξ‚€βˆš1/16+5/𝑖π‘₯0||||||=||||||𝑛𝑖=64βŽ‘βŽ’βŽ’βŽ£ξ‚€βˆš1βˆ’3/64βˆ’1/4π‘–ξ‚ξ‚€βˆš1/16+5/π‘–ξ‚βŽ€βŽ₯βŽ₯⎦||||||=||||||𝑛𝑖=64βŽ‘βŽ’βŽ’βŽ£ξ‚€3√1βˆ’ξ‚π‘–βˆ’16ξ‚€4βˆšξ‚βŽ€βŽ₯βŽ₯⎦||||||,𝑖+320(3.4) with 0≀limπ‘›π‘›β†’βˆžξ‘π‘–=64βŽ‘βŽ’βŽ’βŽ£ξ‚€3√1βˆ’ξ‚π‘–βˆ’16ξ‚€4βˆšξ‚βŽ€βŽ₯βŽ₯βŽ¦π‘–+320≀limπ‘›π‘›β†’βˆžξ‘π‘–=64ξ‚€11βˆ’π‘–ξ‚=limπ‘›β†’βˆž63𝑛=0,(3.5) implies limπ‘›β†’βˆž||||π‘₯𝑛+1(Kirk-Noor)π‘₯𝑛+1||||(Kirk-Ishikawa)=0.(3.6) Therefore, by Definition 2.3, Kirk-Noor iterative scheme converges faster than Kirk-Ishikawa iterative scheme to the fixed point 0 of 𝑇.
Again, similarly ||||π‘₯𝑛+1(Kirk-CR)π‘₯𝑛+1||||=||||||∏(Kirk-Noor)𝑛𝑖=64ξ‚€βˆš1/64+1/π‘₯𝑖+12/𝑖0βˆπ‘›π‘–=64ξ‚€βˆš1/64+21/4𝑖π‘₯0||||||=||||||𝑛𝑖=64βŽ‘βŽ’βŽ’βŽ£ξ‚€βˆš1βˆ’17/4ξ‚π‘–βˆ’12/π‘–ξ‚€βˆš1/64+21/4π‘–ξ‚βŽ€βŽ₯βŽ₯⎦||||||=||||||𝑛𝑖=64βŽ‘βŽ’βŽ’βŽ£ξ‚€βˆš1βˆ’6417ξ‚π‘–βˆ’48ξ‚€βˆš4𝑖+1344π‘–ξ‚βŽ€βŽ₯βŽ₯⎦||||||,(3.7) with 0≀limπ‘›π‘›β†’βˆžξ‘π‘–=64βŽ‘βŽ’βŽ’βŽ£ξ‚€βˆš1βˆ’6417ξ‚π‘–βˆ’48ξ‚€βˆš4𝑖+1344π‘–ξ‚βŽ€βŽ₯βŽ₯βŽ¦β‰€limπ‘›π‘›β†’βˆžξ‘π‘–=64ξ‚€11βˆ’π‘–ξ‚=limπ‘›β†’βˆž63𝑛=0,(3.8) implies limπ‘›β†’βˆž||||π‘₯𝑛+1(Kirk-CR)π‘₯𝑛+1||||(Kirk-Noor)=0.(3.9) It shows Kirk-CR iterative scheme converges faster than Kirk-Noor iterative scheme to the fixed point 0 of 𝑇.
Again, let 𝑛β‰₯300. Then, for Kirk-CR iterative scheme, we have π‘₯𝑛+1=𝑛𝑖=3001+3644βˆšπ‘–+12𝑛+64𝑖3/2ξƒͺπ‘₯0.(3.10) So, ||||π‘₯𝑛+1(Kirk-SP)π‘₯𝑛+1||||=||||||∏(Kirk-CR)𝑛𝑖=300ξ‚€βˆš1/64+3/4𝑖+12/𝑖+64/𝑖3/2π‘₯0βˆπ‘›π‘–=300ξ‚€βˆš1/64+1/π‘₯𝑖+12/𝑖0||||||=||||||𝑛𝑖=300βŽ‘βŽ’βŽ’βŽ£ξ‚€βˆš1βˆ’1/4π‘–βˆ’64/𝑖3/2ξ‚βˆš1/64+1/⎀βŽ₯βŽ₯⎦||||||=|||||𝑖+12/𝑖𝑛𝑖=3001βˆ’64π‘–βˆ’163844𝑖3/2√+256𝑖+3072𝑖|||||.(3.11) It is easy to see that 0≀limπ‘›π‘›β†’βˆžξ‘π‘–=3001βˆ’64π‘–βˆ’163844𝑖3/2√+256𝑖+3072𝑖≀limπ‘›π‘›β†’βˆžξ‘π‘–=30011βˆ’π‘–ξ‚„=limπ‘›β†’βˆž299𝑛=0.(3.12) Hence, we have limπ‘›β†’βˆž|π‘₯𝑛+1(Kirk-SP)/π‘₯𝑛+1(Kirk-CR)|=0. It shows Kirk-SP iterative scheme converges faster than Kirk-CR iterative scheme to the fixed point 0 of 𝑇.

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

Example 3.2. Let π‘‡βˆΆ[0,1]β†’[0,1]∢=π‘₯/2,𝛼𝑛,0=𝛽𝑛,0=𝛾𝑛,0=𝛼𝑛,1=𝛽𝑛,1=𝛾𝑛,1√=4/𝑛,𝑛=1,2,…,𝑛0, for some 𝑛0βˆˆπ‘ and 𝛼𝑛,3=𝛽𝑛,3=𝛾𝑛,3√=1βˆ’8/𝑛,𝑛β‰₯𝑛0. It is clear that 𝑇 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 𝑛β‰₯70 and 𝑒0=π‘₯0 with π‘₯0β‰ 0. Then, for Kirk-Mann and Mann iterative schemes, we have 𝑒𝑛+1=𝑛𝑖=6414+4βˆšπ‘›ξƒͺ𝑒0,π‘₯𝑛+1=𝑛𝑖=7021βˆ’βˆšπ‘–ξƒͺπ‘₯0.(3.13) Now, consider ||||π‘₯𝑛+1𝑒𝑛+1||||=||||||βˆπ‘›π‘–=70ξ‚€βˆš1/4+4/𝑖π‘₯0βˆπ‘›π‘–=70ξ‚€βˆš1βˆ’2/𝑖𝑒0||||||=||||||𝑛𝑖=70βŽ‘βŽ’βŽ’βŽ£ξ‚€βˆš1βˆ’3/4βˆ’6/π‘–ξ‚ξ‚€βˆš1βˆ’2/π‘–ξ‚βŽ€βŽ₯βŽ₯⎦||||||=||||||𝑛𝑖=70βŽ‘βŽ’βŽ’βŽ£ξ‚€3√1βˆ’ξ‚π‘–βˆ’24ξ‚€4βˆšξ‚βŽ€βŽ₯βŽ₯⎦||||||π‘–βˆ’8.(3.14) It is easy to see that 0≀limπ‘›π‘›β†’βˆžξ‘π‘–=70βŽ‘βŽ’βŽ’βŽ£ξ‚€3√1βˆ’ξ‚π‘–βˆ’24ξ‚€4βˆšξ‚βŽ€βŽ₯βŽ₯βŽ¦π‘–βˆ’8≀limπ‘›π‘›β†’βˆžξ‘π‘–=70ξ‚€11βˆ’π‘–ξ‚=limπ‘›β†’βˆž69𝑛=0.(3.15) Hence, we have limπ‘›β†’βˆž|π‘₯𝑛+1/𝑒𝑛+1|=0.
It shows that Kirk-Mann iterative scheme converges faster than Mann iterative scheme to the fixed point 0 of 𝑇. Similarly, ||||π‘₯𝑛+1(Kirk-Ishikawa)π‘₯𝑛+1||||=||||||∏(Ishikawa)𝑛𝑖=70ξ‚€βˆš1/16+5/𝑖π‘₯0βˆπ‘›π‘–=70ξ‚€βˆš1βˆ’2/π‘₯π‘–βˆ’4/𝑖0||||||=||||||𝑛𝑖=70βŽ‘βŽ’βŽ’βŽ£ξ‚€βˆš1βˆ’15/16βˆ’7/ξ‚π‘–βˆ’4/π‘–ξ‚€βˆš1βˆ’2/ξ‚βŽ€βŽ₯βŽ₯⎦||||||=||||||π‘–βˆ’4/𝑖𝑛𝑖=70βŽ‘βŽ’βŽ’βŽ£ξ‚€βˆš1βˆ’15π‘–βˆ’112ξ‚π‘–βˆ’64ξ‚€βˆš16π‘–βˆ’32ξ‚βŽ€βŽ₯βŽ₯⎦||||||,π‘–βˆ’64(3.16) with 0≀limπ‘›π‘›β†’βˆžξ‘π‘–=70βŽ‘βŽ’βŽ’βŽ£ξ‚€βˆš1βˆ’15π‘–βˆ’112ξ‚π‘–βˆ’64ξ‚€βˆš16π‘–βˆ’32ξ‚βŽ€βŽ₯βŽ₯βŽ¦π‘–βˆ’64≀limπ‘›π‘›β†’βˆžξ‘π‘–=70ξ‚€11βˆ’π‘–ξ‚=limπ‘›β†’βˆž69𝑛=0,(3.17) implies limπ‘›β†’βˆž||||π‘₯𝑛+1(Kirk-Ishikawa)π‘₯𝑛+1||||(Ishikawa)=0.(3.18) It shows that Kirk-Ishikawa iterative scheme converges faster than Ishikawa iterative scheme to the fixed point 0 of 𝑇.
Again, similarly, ||||π‘₯𝑛+1(Kirk-Noor)π‘₯𝑛+1||||=∏(Noor)𝑛𝑖=70ξ‚€βˆš1/64+21/4𝑖π‘₯0βˆπ‘›π‘–=70ξ‚€βˆš1βˆ’2/π‘–βˆ’4/π‘–βˆ’8/𝑖3/2π‘₯0=𝑛𝑖=70βŽ‘βŽ’βŽ’βŽ£ξ‚€βˆš1βˆ’63/64βˆ’29/4π‘–βˆ’4/π‘–βˆ’8/𝑖3/2ξ‚βˆš1βˆ’2/π‘–βˆ’4/π‘–βˆ’8/𝑖3/2⎀βŽ₯βŽ₯⎦=𝑛𝑖=70βŽ‘βŽ’βŽ’βŽ£ξ‚€1βˆ’63𝑖3/2βˆšβˆ’464π‘–βˆ’256ξ‚π‘–βˆ’51264𝑖3/2βˆšβˆ’128π‘–βˆ’256⎀βŽ₯βŽ₯⎦,π‘–βˆ’512(3.19) with 0≀𝑛𝑖=70βŽ‘βŽ’βŽ’βŽ£ξ‚€1βˆ’63𝑖3/2βˆšβˆ’464π‘–βˆ’256ξ‚π‘–βˆ’51264𝑖3/2βˆšβˆ’128π‘–βˆ’256⎀βŽ₯βŽ₯βŽ¦π‘–βˆ’512≀limπ‘›π‘›β†’βˆžξ‘π‘–=7011βˆ’π‘–ξ‚„=limπ‘›β†’βˆž69𝑛=0,(3.20) implies limπ‘›β†’βˆž||||π‘₯𝑛+1(Kirk-Noor)π‘₯𝑛+1||||(Noor)=0.(3.21) It shows that Kirk-Noor iterative scheme converges faster than Noor iterative scheme to the fixed point 0 of 𝑇.
Again, ||||π‘₯𝑛+1(Kirk-SP)π‘₯𝑛+1||||=||||||∏(SP)𝑛𝑖=70ξ‚€βˆš1/64+3/4𝑖+12/𝑖+64/𝑖3/2π‘₯0βˆπ‘›π‘–=70ξ‚€βˆš1βˆ’6/𝑖+12/π‘–βˆ’8/𝑖3/2π‘₯0||||||=||||||𝑛𝑖=70βŽ‘βŽ’βŽ’βŽ£ξ‚€βˆš1βˆ’63/64βˆ’27/4π‘–βˆ’72/𝑖3/2ξ‚ξ‚€βˆš1βˆ’6/𝑖+12/π‘–βˆ’8/𝑖3/2ξ‚βŽ€βŽ₯βŽ₯⎦||||||=|||||𝑛𝑖=701βˆ’63𝑖3/2ξ€Έβˆ’432π‘–βˆ’460864𝑖3/2βˆšβˆ’384π‘–βˆ’768ξƒ­|||||,π‘–βˆ’512(3.22) with 0≀limπ‘›π‘›β†’βˆžξ‘π‘–=701βˆ’63𝑖3/2ξ€Έβˆ’432π‘–βˆ’460864𝑖3/2βˆšβˆ’384π‘–βˆ’768ξƒ­π‘–βˆ’512≀limπ‘›π‘›β†’βˆžξ‘π‘–=70ξ‚€11βˆ’π‘–ξ‚=limπ‘›β†’βˆž69𝑛=0,(3.23) implies limπ‘›β†’βˆž||||π‘₯𝑛+1(Kirk-SP)π‘₯𝑛+1||||(SP)=0.(3.24) It shows that Kirk-SP iterative scheme converges faster than SP iterative scheme to the fixed point 0 of 𝑇.
Again, ||||π‘₯𝑛+1(Kirk-CR)π‘₯𝑛+1||||=∏(CR)𝑛𝑖=70ξ‚€βˆš1/64+1/π‘₯𝑖+12/𝑖0βˆπ‘›π‘–=70ξ‚€βˆš1/2βˆ’1/π‘–βˆ’4/𝑖+8/𝑖3/2π‘₯=𝑛𝑖=70βŽ‘βŽ’βŽ’βŽ£ξ‚€βˆš1βˆ’31/64βˆ’2/π‘–βˆ’16/𝑖+8/𝑖3/2ξ‚βˆš1/2βˆ’1/π‘–βˆ’4/𝑖+8/𝑖3/2⎀βŽ₯βŽ₯⎦=𝑛𝑖=70βŽ‘βŽ’βŽ’βŽ£ξ‚€1βˆ’31𝑖3/2βˆšβˆ’128π‘–βˆ’1024𝑖+51232𝑖3/2βˆšβˆ’64π‘–βˆ’256⎀βŽ₯βŽ₯⎦,𝑖+512(3.25) with 0≀limπ‘›π‘›β†’βˆžξ‘π‘–=70βŽ‘βŽ’βŽ’βŽ£ξ‚€1βˆ’31𝑖3/2βˆšβˆ’128π‘–βˆ’1024𝑖+51232𝑖3/2βˆšβˆ’64π‘–βˆ’256⎀βŽ₯βŽ₯βŽ¦π‘–+512≀limπ‘›π‘›β†’βˆžξ‘π‘–=7011βˆ’π‘–ξ‚„=limπ‘›β†’βˆž69𝑛=0,(3.26) implies limπ‘›β†’βˆž||||π‘₯𝑛+1(Kirk-CR)π‘₯𝑛+1||||(CR)=0.(3.27) It shows that Kirk-CR iterative scheme converges faster than CR iterative scheme to the fixed point 0 of 𝑇.

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 𝛼𝑛,1=𝛽𝑛,1=𝛾𝑛,1=𝛼𝑛,2=𝛽𝑛,2=𝛾𝑛,2=1/(1+𝑛)1/2,𝛼𝑛,0=1βˆ’π›Όπ‘›,1βˆ’π›Όπ‘›,2, and 𝛽𝑛,0=1βˆ’π›½π‘›,1βˆ’π›½π‘›,2, 𝛾𝑛,0=1βˆ’π›Ύπ‘›,1βˆ’π›Ύπ‘›,2 for all iterative schemes.


Kirk-CR iterationKirk-SP iterationKirk-Noor iterationKirk-Mann iterationKirk-Ishikawa iteration
𝑛 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1

00.698570.7108070.698570.7237260.698570.7103840.698570.7104180.698570.698227
10.8005420.759840.787990.755650.8009410.7708590.8009090.7651420.8121570.779333
20.74920.7548190.7540020.7548770.7361640.7525950.7430040.754010.7257170.750854
30.7549440.7548780.7548780.7548780.757450.7548790.7558590.7548120.7593960.754853
40.7548780.7548780.7548780.7548780.7548770.7548780.7549520.7548660.7549050.754875
50.7548780.7548780.7548780.7548780.7548780.7548780.7548910.7548750.7548810.754877
60.7548780.7548780.7548780.7548780.7548780.7548780.7548810.7548770.7548780.754878
70.7548780.7548780.7548780.7548780.7548780.7548780.7548790.7548770.7548780.754878
80.7548780.7548780.7548780.7548780.7548780.7548780.7548780.7548780.7548780.754878
90.7548780.7548780.7548780.7548780.7548780.7548780.7548780.7548780.7548780.754878
100.7548780.7548780.7548780.7548780.7548780.7548780.7548780.7548780.7548780.754878


Kirk-CR iterationKirk-SP iterationKirk-Noor iterationKirk-Mann iterationKirk-Ishikawa iteration
𝑛 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1

01.1035891.1168581.1035891.174641.1035891.4517921.1035891.2414731.1035891.365883
11.1427231.1584531.1650021.158631.3004290.9976181.1932721.1586971.2532541.076407
21.1585611.1586311.158631.158631.1028711.1896021.1586571.1586411.1282931.167741
31.1586311.158631.158631.158631.1710961.159291.1586351.1586331.1622371.159321
41.158631.158631.158631.158631.158891.1587311.1586321.1586311.1589021.158763
51.158631.158631.158631.158631.158671.1586551.1586311.1586311.1586821.158667
61.158631.158631.158631.158631.158641.1586381.1586311.1586311.1586451.158643
71.158631.158631.158631.158631.1586331.1586331.1586311.1586311.1586351.158635
81.158631.158631.158631.158631.1586321.1586321.158631.158631.1586321.158632
91.158631.158631.158631.158631.1586311.1586311.158631.158631.1586311.158631
101.158631.158631.158631.158631.1586311.1586311.158631.158631.1586311.158631
111.158631.158631.158631.158631.1586311.1586311.158631.158631.1586311.158631
121.158631.158631.158631.158631.158631.158631.158631.158631.1586311.158631
131.158631.158631.158631.158631.158631.158631.158631.158631.158631.158631
141.158631.158631.158631.158631.158631.158631.158631.158631.158631.15863
151.158631.158631.158631.158631.158631.158631.158631.158631.158631.15863


Kirk-CR iterationKirk-SP iterationKirk-Noor iterationKirk-Mann iterationKirk-Ishikawa iteration
𝑛 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1

00.010.1297630.010.1814070.010.1942680.010.09010.01βˆ’0.04247
10.7573120.4196040.6700950.384790.6492050.4879410.8279180.5690451.0867430.695256
20.3368590.3821040.3784840.3819660.2622040.3632230.1857220.4020060.0928690.341163
30.3817950.3819660.3819660.3819660.4054850.3819130.3575960.3851390.4340660.381282
40.3819650.3819660.3819660.3819660.3820310.381960.3780540.3827210.3828120.381873
50.3819660.3819660.3819660.3819660.3819740.3819650.3810330.3821950.3820810.381946
60.3819660.3819660.3819660.3819660.3819680.3819660.3816830.3820470.3819910.38196
70.3819660.3819660.3819660.3819660.3819660.3819660.3818660.3819980.3819730.381964
80.3819660.3819660.3819660.3819660.3819660.3819660.3819260.381980.3819680.381965
90.3819660.3819660.3819660.3819660.3819660.3819660.3819490.3819720.3819670.381966
100.3819660.3819660.3819660.3819660.3819660.3819660.3819580.3819690.3819660.381966
110.3819660.3819660.3819660.3819660.3819660.3819660.3819620.3819680.3819660.381966
120.3819660.3819660.3819660.3819660.3819660.3819660.3819640.3819670.3819660.381966
130.3819660.3819660.3819660.3819660.3819660.3819660.3819650.3819660.3819660.381966
140.3819660.3819660.3819660.3819660.3819660.3819660.3819650.3819660.3819660.381966
150.3819660.3819660.3819660.3819660.3819660.3819660.3819660.3819660.3819660.381966


Kirk-CR iterationKirk-SP iterationKirk-Noor iterationKirk-Mann iterationKirk-Ishikawa iteration
𝑛 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1

00.9880.9943720.9881.0835830.9881.0928280.9881.0878530.9881.093597
10.9999680.9999960.9941820.9936720.9929830.9603310.9936380.9590830.9928790.95975
2110.9999591.0000230.9983021.0061150.9981891.0052820.9982491.006211
311110.99996310.9999720.9999860.9999621
411111110.9999991 𝟏
51111111111
61111111111
71111111111
81111111111
91111111111
101111111111


Kirk-CR iterationKirk-SP iterationKirk-Noor iterationKirk-Mann iterationKirk-Ishikawa iteration
𝑛 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1 𝑓 π‘₯ 𝑛 π‘₯ 𝑛 + 1

00.250.250.250.250.250.250.250.250.250.25
141.29429840.96690541.76896242.9016542.191915
20.7726190.9952451.0342281.0001360.5653030.8898310.3446311.4253550.4562220.833093
31.00477810.99986411.12380810.701581.0634671.2003461.000137
41111110.940321.0083940.9998631.000015
51111110.9916761.0015690.9999851.000003
61111110.9984341.0003840.9999971.000001
71111110.9996161.0001120.9999991
81111110.9998881.00003711
91111110.9999631.00001411
101111110.9999861.00000511
111111110.9999951.00000211
121111110.9999981.00000111
131111110.999999111
141111111111

4.1. Decreasing Cum Sublinear Functions

The function π‘“βˆΆ[0,1]β†’[0,1] defined by 𝑓(π‘₯)=(1βˆ’π‘₯3)1/2 is a decreasing and sublinear function. By taking initial approximation π‘₯0=0.8, the comparison of convergence of the above-mentioned iterative schemes to the exact fixed point 𝑝=0.754878 is listed in Table 1.

4.2. Increasing Functions

Let π‘“βˆΆ[0,2]β†’[0,2] be defined by 𝑓(π‘₯)=(πœ‹+π‘₯π‘›βˆš4βˆ’π‘₯𝑛2βˆ’(4βˆ’2π‘₯𝑛2)sinβˆ’1(π‘₯𝑛/2))/πœ‹. Then, 𝑓 is an increasing function. By taking initial approximation π‘₯0=1, the comparison of convergence of the above-mentioned iterative schemes to the exact fixed point 𝑝=1.15863 of 𝑓 is listed in Table 2.

4.3. Functions with Multiple Zeros

The function defined by 𝑓(π‘₯)=(1βˆ’π‘₯)2 is a function with multiple zeros. By taking initial approximation π‘₯0=0.9, the comparison of convergence of the above-mentioned iterative schemes to the exact fixed point 𝑝=0.381966 is listed in Table 3.

4.4. Superlinear Functions with Multiple Roots

The function defined by 𝑓(π‘₯)=2π‘₯3βˆ’7π‘₯2+8π‘₯βˆ’2 is a superlinear function with multiple real roots. By taking initial approximation π‘₯0=0.9, the comparison of convergence of the above-mentioned iterative schemes to the exact fixed point 𝑝=1 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 𝑓(π‘₯)=1/π‘₯ is an oscillatory function. By taking initial approximation π‘₯0=4, the comparison of convergence of the above-mentioned iterative schemes to the exact fixed point 𝑝=1 is listed in Table 5.

5. Observations

5.1. Decreasing Functions

(1) Taking initial guess π‘₯π‘œ=0.5 (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 𝛼𝑛=𝛽𝑛=𝛾𝑛=1/(1+𝑛)1/4 and π‘₯π‘œ=0.8, 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 π‘₯π‘œ=0.8 (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 𝛼𝑛=𝛽𝑛=𝛾𝑛=1/(1+𝑛)1/4 and π‘₯π‘œ=1, 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 π‘₯π‘œ=0.6 (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 𝛼𝑛=𝛽𝑛=𝛾𝑛=1/(1+𝑛)1/4 and π‘₯π‘œ=0.9, 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 π‘₯π‘œ=0.6 (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 𝛼𝑛=𝛽𝑛=𝛾𝑛=1/(1+𝑛)1/4 and π‘₯π‘œ=0.9, 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 π‘₯π‘œ=0.6 (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 𝛼𝑛=𝛽𝑛=𝛾𝑛=1/(1+𝑛)1/4 and π‘₯π‘œ=4, 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 𝛼𝑛, 𝛽𝑛, and 𝛾𝑛. 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 Kirk-type iterative schemes is as follows:  Kirk-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: Kirk-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:  Kirk-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: Kirk-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:  Kirk-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 π‘˜=𝑠=𝑑>2, the convergence speed of each iterative scheme decreases for all type of the above-mentioned functions. Convergence speed is the highest for π‘˜=𝑠=𝑑=2.
(10) In Section 4, we have shown comparison between Kirk-type iterative schemes for decreasing functions. However, for decreasing functions of the form 𝑓(π‘₯)=(1βˆ’π‘₯)π‘š,π‘š=7,8,9…, 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.

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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.

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