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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 526503, 22 pages
http://dx.doi.org/10.1155/2012/526503
Research Article

On the Rate of Convergence of Kirk-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
3Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore, Pakistan

Received 29 January 2012; Revised 21 April 2012; Accepted 14 May 2012

Academic Editor: B. V. Rathish Kumar

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.

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 𝑇𝑋𝑋 a selfmap of 𝑋. Suppose that 𝐹(𝑇)={𝑝𝑋,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 𝑥𝑛+1=𝑇𝑥𝑛,𝑛=0,1,,(1.1) which is used to approximate the fixed points of mappings satisfying the inequality: 𝑑(𝑇𝑥,𝑇𝑦)𝛼𝑑(𝑥,𝑦)(1.2) for all 𝑥,𝑦𝑋 and 𝛼[0,1). Condition (1.2) is called Banach’s contraction condition.

The following iteration schemes are now well known: 𝑢𝑛+1=1𝛼𝑛𝑢𝑛+𝛼𝑛𝑇𝑢𝑛,(1.3) where {𝛼𝑛} is a sequences of positive numbers in [0,1], due to Mann [1]. 𝑥𝑛+1=𝑘𝑖=0𝛼𝑖𝑇𝑖𝑥𝑛,𝑘𝑖=0𝛼𝑖=1,(1.4) where {𝛼𝑛} is a sequences of positive numbers in [0,1], due to Kirk [2]. 𝑥𝑛+1=1𝛼𝑛𝑥𝑛+𝛼𝑛𝑇𝑦𝑛,𝑦𝑛=1𝛽𝑛𝑥𝑛+𝛽𝑛𝑇𝑥𝑛,(1.5) where {𝛼𝑛} and {𝛽𝑛} are sequences of positive numbers in [0,1], due to Ishikawa [3]. 𝑥𝑛+1=1𝛼𝑛𝑥𝑛+𝛼𝑛𝑇𝑦𝑛,𝑦𝑛=1𝛽𝑛𝑥𝑛+𝛽𝑛𝑇𝑧𝑛,𝑧𝑛=1𝛾𝑛𝑥𝑛+𝛾𝑛𝑇𝑥𝑛,(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: 𝑥𝑛+1=𝛼𝑛,0𝑥𝑛+𝑘𝑖=1𝛼𝑛,𝑖𝑇𝑖𝑦𝑛,𝑘𝑖=0𝛼𝑛,𝑖𝑦=1,𝑛=𝑠𝑗=0𝛽𝑛,𝑗𝑇𝑗𝑥𝑛,𝑠𝑗=0𝛽𝑛,𝑗=1,𝑛=0,1,2,,(1.7) where 𝑘𝑠, 𝛼𝑛,𝑖0,𝛼𝑛,00, 𝛽𝑛,𝑗0, 𝛽𝑛,00, 𝛼𝑛,𝑖,𝛽𝑛,𝑗[0,1], and 𝑘, 𝑠 are fixed integers, called as Kirk-Ishikawa iteration scheme: 𝑢𝑛+1=𝑘𝑖=0𝛼𝑛,𝑖𝑇𝑖𝑢𝑛,𝑘𝑖=0𝛼𝑛,𝑖=1,𝑛=0,1,2,,(1.8) where 𝛼𝑛,𝑖0, 𝛼𝑛,00, 𝛼𝑛,𝑖[0,1], and 𝑘 is a fixed integer and is called as Kirk-Mann iteration scheme.

However, from [6], the Kirk-Noor iterative scheme is given by 𝑥𝑛+1=𝛼𝑛,0𝑥𝑛+𝑘𝑖=1𝛼𝑛,𝑖𝑇𝑖𝑦𝑛,𝑘𝑖=0𝛼𝑛,𝑖𝑦=1,𝑛=𝛽𝑛,0𝑥𝑛+𝑠𝑟=1𝛽𝑛,𝑟𝑇𝑟𝑧𝑛,𝑠𝑗=0𝛽𝑛,𝑗𝑧=1,𝑛=𝑡𝑙=0𝛾𝑛,𝑙𝑇𝑙𝑥𝑛,𝑡𝑙=0𝛾𝑛,𝑙=1,𝑛=0,1,2,.(1.9) In [7], Phuengrattana and Suantai defined the SP iteration scheme as follows: 𝑥𝑛+1=1𝛼𝑛𝑦𝑛+𝛼𝑛𝑇𝑦𝑛,𝑦𝑛=1𝛽𝑛𝑧𝑛+𝛽𝑛𝑇𝑧𝑛,𝑧𝑛=1𝛾𝑛𝑥𝑛+𝛾𝑛𝑇𝑥𝑛,(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]: 𝑥𝑛+1=1𝛼𝑛𝑦𝑛+𝛼𝑛𝑇𝑦𝑛,𝑦𝑛=1𝛽𝑛𝑇𝑥𝑛+𝛽𝑛𝑇𝑧𝑛,𝑧𝑛=1𝛾𝑛𝑥𝑛+𝛾𝑛𝑇𝑥𝑛,(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 𝑇𝑋𝑋 a mapping for which there exists real numbers 𝑎, 𝑏, and c satisfying 𝑎(0,1),𝑏,𝑐(0,1/2) such that for each pair 𝑥,𝑦𝑋 at least one of the following conditions holds: [𝑑],[].(i)𝑑(𝑇𝑥,𝑇𝑦)𝑎𝑑(𝑥,𝑦),(ii)𝑑(𝑇𝑥,𝑇𝑦)𝑏(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦)(iii)𝑑(𝑇𝑥,𝑇𝑦)𝑐𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥)(1.12) Then, 𝑇 has a unique fixed point 𝑝 and the Picard iteration scheme {𝑥𝑛} defined by (1.1) converges to 𝑝 for any arbitrary but fixed 𝑥0𝑋.

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(𝑇𝑥,𝑇𝑦)2𝛿𝑑(𝑥,𝑇𝑥)+𝛿𝑑(𝑥,𝑦)(1.13) for all 𝑥,𝑦𝑋 and 𝛿[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 𝑇𝐾𝐾 a mapping satisfying (1.13). Let {𝑥𝑛}𝑛=0 be defined through the Ishikawa iteration scheme (1.5) and 𝑥0𝐾, where {𝛼𝑛}, {𝛽𝑛} are sequences of positive real numbers in [0,1] with {𝛼𝑛} satisfying 𝑛=0𝛼𝑛=. Then, {𝑥𝑛}𝑛=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 [1116] 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 𝑎[0,1) and a monotone increasing function 𝜑𝑅+𝑅+ with 𝜑(0)=0, such that 𝑇𝑥𝑇𝑦𝜑(𝑥𝑇𝑥)+𝑎𝑥𝑦𝑥,𝑦𝑋.(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 0𝛿<1 and {𝑛}𝑛=0 is a sequence of nonnegative numbers such that lim𝑛𝑛=0, then, for any sequence of positive numbers {𝑢𝑛}𝑛=0 satisfying 𝑢𝑛+1𝛿𝑢𝑛+𝑛,𝑛=0,1,2,,(2.1) one has lim𝑛𝑢𝑛=0.

Lemma 2.2 (see [5]). Let (𝑋,) be a normed linear space and let 𝑇𝑋𝑋 be a selfmap of 𝑋 satisfying (1.13). Let 𝜑𝑅+𝑅+ be a subadditive, monotone increasing function such that 𝜑(0)=0, 𝜑(𝐿𝑢)𝐿𝜑(𝑢), 𝐿0, 𝑢𝑅+. Then, for all 𝑖𝑁, 𝐿0 and for all 𝑥,𝑦𝑋, 𝑇𝑖𝑥𝑇𝑖𝑦𝑖𝑗=1𝑖𝑗𝑎𝑖𝑗𝜑𝑗(𝑥𝑇𝑥)+𝑎𝑖𝑥𝑦.(2.2)

Definition 2.3 (see [20]). Suppose {𝑎𝑛} and {𝑏𝑛} are two real convergent sequences with limits 𝑎 and 𝑏, respectively. Then, {𝑎𝑛} is said to converge faster than {𝑏𝑛} if lim𝑛||||𝑎𝑛𝑎𝑏𝑛||||𝑏=0.(2.3)

Now, we define Kirk-SP and Kirk-CR iterative schemes as follows. Let 𝑋 be a Banach space, 𝑇𝑋𝑋 a selfmap of 𝑋 and 𝑥0𝑋. Then, the sequence {𝑥𝑛}𝑛=0 defined by 𝑥𝑛+1=𝛼𝑛,0𝑦𝑛+𝑘𝑖=1𝛼𝑛,𝑖𝑇𝑖𝑦𝑛,𝑘𝑖=0𝛼𝑛,𝑖𝑦=1,𝑛=𝛽𝑛,0𝑧𝑛+𝑠𝑟=1𝛽𝑛,𝑟𝑇𝑟𝑧𝑛,𝑠𝑗=0𝛽𝑛,𝑗𝑧=1,𝑛=𝑡𝑙=0𝛾𝑛,𝑙𝑇𝑙𝑥𝑛,𝑡𝑙=0𝛾𝑛,𝑙=1,𝑛=0,1,2,,(2.4) is called Kirk-SP iterative scheme and the sequence {𝑥𝑛}𝑛=0 defined by 𝑥𝑛+1=𝛼𝑛,0𝑦𝑛+𝑘𝑖=1𝛼𝑛,𝑖𝑇𝑖𝑦𝑛,𝑘𝑖=0𝛼𝑛,𝑖𝑦=1,𝑛=𝛽𝑛,0𝑇𝑥𝑛+𝑠𝑟=1𝛽𝑛,𝑟𝑇𝑟𝑧𝑛,𝑠𝑟=0𝛽𝑛,𝑟𝑧=1,𝑛=𝑡𝑙=0𝛾𝑛,𝑙𝑇𝑙𝑥𝑛,𝑡𝑙=0𝛾𝑛,𝑙=1,𝑛=0,1,2,(2.5) is called Kirk-CR iterative scheme, where 𝑘, 𝑠, and 𝑡 are fixed integers with 𝑘𝑠𝑡 and 𝛼𝑛,𝑖, 𝛽𝑛,𝑟, 𝛾𝑛,𝑙 are sequences in [0,1] satisfying 𝛼𝑛,𝑖0,𝛼𝑛,00, 𝛽𝑛,𝑟0,𝛽𝑛,00,𝛾𝑛,𝑙0,𝛾𝑛,00.

Remarks 2. (5) Putting 𝑡=𝑠=0 in (2.4), we obtain the Kirk-Mann iterative scheme (1.9).
(6) Putting 𝑠=0,𝑘=1,𝑡=0 in (2.4), we get usual Mann iterative scheme (1.3). With 1𝑖=0𝛼𝑛,𝑖=1,𝛼𝑛,1=𝛼𝑛.
(7) Putting 𝑠=0,𝑡=0, and 𝛼𝑛,𝑖=𝛼𝑖 in (2.4), we obtain the usual Kirk’s iterative scheme (1.4).
(8) Putting 𝑠=𝑡=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,) be a normed linear space and 𝑇𝑋𝑋 a selfmap of 𝑋 satisfying the contractive condition (1.14) and 𝜑𝑅+𝑅+ a subadditive monotone increasing function such that 𝜑(0)=0 and 𝜑(𝐿𝑢)𝐿𝜑(𝑢),𝐿0,𝑢𝑅+. Let 𝑥0𝑋 and {𝑥𝑛}𝑛=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 𝑥𝑛+1𝑝𝛼𝑛,0𝑦𝑛+𝑝𝑘𝑖=1𝛼𝑛,𝑖𝑇𝑖𝑦𝑛𝑝𝛼𝑛,0𝑦𝑛+𝑝𝑘𝑖=1𝛼𝑛,𝑖𝑖𝑗=1𝑖𝑗𝑎𝑖𝑗𝜑𝑗(𝑝𝑇𝑝)+𝑎𝑖𝑦𝑛𝑝=𝛼𝑛,0𝑦𝑛+𝑝𝑘𝑖=1𝛼𝑛,𝑖𝑎𝑖𝑦𝑛=𝑝𝑘𝑖=0𝛼𝑛,𝑖𝑎𝑖𝑦𝑛.𝑝(2.6) Now, we have the following estimates: 𝑦𝑛𝑝𝛽𝑛,0𝑧𝑛+𝑝𝑠𝑟=1𝛽𝑛,𝑟𝑇𝑟𝑧𝑛𝑝𝛽𝑛,0𝑧𝑛+𝑝𝑠𝑟=1𝛽𝑛,𝑟𝑟𝑗=1𝑟𝑗𝑎𝑟𝑗𝜑𝑗(𝑝𝑇𝑝)+𝑎𝑟𝑧𝑛=𝑝𝑘𝑟=0𝛽𝑛,𝑟𝑎𝑟𝑧𝑛,𝑧𝑝𝑛𝑝𝛾𝑛,𝑙𝑥𝑛+𝑝𝑡𝑙=1𝛾𝑛,𝑙𝑇𝑙𝑥𝑛𝑝𝛾𝑛,𝑙𝑥𝑛+𝑝𝑡𝑙=1𝛾𝑛,𝑙𝑙𝑗=1𝑙𝑗𝑎𝑙𝑗𝜑𝑗(𝑝𝑇𝑝)+𝑎𝑙𝑥𝑛=𝑝𝑡𝑙=0𝛾𝑛,𝑙𝑎𝑙𝑥𝑛.𝑝(2.7) It follows from (2.6), (2.7) that 𝑥𝑛+1𝑝𝑘𝑖=0𝛼𝑛,𝑖𝑎𝑖𝑠𝑟=0𝛽𝑛,𝑖𝑎𝑖𝑡𝑙=0𝛾𝑛,𝑙𝑎𝑙𝑥𝑛.𝑝(2.8) Since 𝑎𝑖[0,1) and 𝑘𝑖=0𝛼𝑛,𝑖=𝑠𝑟=0𝛽𝑛,𝑟=𝑡𝑙=0𝛾𝑛,𝑙=1, hence 𝑘𝑖=0𝛼𝑛,𝑖𝑎𝑖𝑠𝑟=0𝛽𝑛,𝑟𝑎𝑟𝑡𝑙=0𝛾𝑛,𝑙𝑎𝑙<𝑘𝑖=0𝛼𝑛,𝑖𝑠𝑟=0𝛽𝑛,𝑟𝑡𝑙=0𝛾𝑛,𝑙=1.(2.9) Using (2.9) and Lemma 2.1, (2.8) yields lim𝑛𝑥𝑛=𝑝. Thus, Kirk-SP iterative scheme converges strongly to 𝑝.

Theorem 2.5. Let (𝑋,) be a normed linear space and 𝑇𝑋𝑋 a selfmap of 𝑋 satisfying the contractive condition (1.14) and 𝜑𝑅+𝑅+ a subadditive monotone increasing function such that 𝜑(0)=0 and 𝜑(𝐿𝑢)𝐿𝜑(𝑢),𝐿0,𝑢𝑅+. Let 𝑥0𝑋 and {𝑥𝑛}𝑛=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 𝑥𝑛+1𝑝𝑘𝑖=0𝛼𝑛,𝑖𝑎𝑖𝑦𝑛.𝑝(2.10) Now, we have the following estimates: 𝑦𝑛+1𝑝𝛽𝑛,𝑜𝑇𝑥𝑛+𝑝𝑠𝑟=1𝛽𝑛,𝑟𝑇𝑟𝑧𝑛𝑝𝑎𝛽𝑛,0𝑥𝑛+𝑝𝑠𝑟=1𝛽𝑛,𝑟𝑟𝑗=1𝑟𝑗𝑎𝑟𝑗𝜑𝑗(𝑝𝑇𝑝)+𝑎𝑟𝑧𝑛𝑝=𝑎𝛽𝑛,0𝑥𝑛+𝑝𝑠𝑟=1𝛽𝑛,𝑟𝑎𝑟𝑧𝑛,𝑧𝑝𝑛𝑝𝑡𝑙=0𝛾𝑛,𝑙𝑎𝑙𝑥𝑛.𝑝(2.11) It follows from (2.10), (2.11) that 𝑥𝑛+1𝑝𝑎𝑘𝑖=0𝛼𝑛,𝑖𝑎𝑖𝛽𝑛,0𝑥𝑛+𝑝𝑘𝑖=0𝛼𝑛,𝑖𝑎𝑖𝑠𝑟=1𝛽𝑛,𝑟𝑎𝑟𝑡𝑙=0𝛾𝑛,𝑙𝑎𝑙𝑥𝑛=𝑝𝑘𝑖=0𝛼𝑛,𝑖𝑎𝑖𝑎𝛽𝑛,0+𝑠𝑟=1𝛽𝑛,𝑟𝑎𝑟𝑡𝑙=0𝛾𝑛,𝑙𝑎𝑙𝑥𝑛.𝑝(2.12) Since 𝑎𝑖[0,1) and 𝑘𝑖=0𝛼𝑛,𝑖=𝑠𝑟=0𝛽𝑛,𝑟=𝑡𝑙=0𝛾𝑛,𝑙=1 with 𝛼𝑛,00,𝛽𝑛,00,𝛾𝑛,00, hence 𝑘𝑖=0𝛼𝑛,𝑖𝑎𝑖𝑎𝛽𝑛,0+𝑠𝑟=1𝛽𝑛,𝑟𝑎𝑟𝑡𝑙=0𝛾𝑛,𝑙𝑎𝑙<𝑘𝑖=0𝛼𝑛,𝑖𝛽𝑛,0+𝑠𝑟=1𝛽𝑛,𝑟𝑡𝑙=0𝛾𝑛,𝑙=𝛽𝑛,0+𝑠𝑟=1𝛽𝑛,𝑟=1.(2.13) Using (2.13) and Lemma 2.1, (2.12) yields lim𝑛𝑥𝑛=𝑝. Thus, Kirk-CR iterative scheme converges strongly to 𝑝.

Theorem 2.6. Let (𝑋,) be a normed linear space and 𝑇𝑋𝑋 a selfmap of 𝑋 satisfying the contractive condition (1.14) and 𝜑𝑅+𝑅+ a subadditive monotone increasing function such that 𝜑(0)=0 and 𝜑(𝐿𝑢)𝐿𝜑(𝑢), 𝐿0,𝑢𝑅+. Let 𝑥0𝑋 and {𝑥𝑛}𝑛=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 𝑥𝑛+1𝑝𝛼𝑛,0𝑥𝑛+𝑝𝑘𝑖=1𝛼𝑛,𝑖𝑇𝑖𝑦𝑛𝑝𝛼𝑛,0𝑥𝑛+𝑝𝑘𝑖=1𝛼𝑛,𝑖𝑖𝑗=1𝑖𝑗𝑎𝑖𝑗𝜑𝑗(𝑝𝑇𝑝)+𝑎𝑖𝑦𝑛𝑝=𝛼𝑛,0𝑥𝑛+𝑝𝑘𝑖=1𝛼𝑛,𝑖𝑎𝑖𝑦𝑛.𝑝(2.14) Now, we have the following estimates: 𝑦𝑛𝑝𝛽𝑛,0𝑥𝑛+𝑝𝑠𝑟=1𝛽𝑛,𝑟𝑇𝑟𝑧𝑛𝑝𝛽𝑛,0𝑥𝑛+𝑝𝑠𝑟=1𝛽𝑛,𝑟𝑟𝑗=1𝑟𝑗𝑎𝑟𝑗𝜑𝑗(𝑝𝑇𝑝)+𝑎𝑟𝑧𝑛𝑝=𝛽𝑛,0𝑥𝑛+𝑝𝑘𝑟=1𝛽𝑛,𝑟𝑎𝑟𝑧𝑛,𝑧𝑝𝑛𝑝𝛾𝑛,𝑙𝑥𝑛+𝑝𝑡𝑙=1𝛾𝑛,𝑙𝑇𝑙𝑥𝑛𝑝𝛾𝑛,𝑙𝑥𝑛+𝑝𝑡𝑙=1𝛾𝑛,𝑙𝑙𝑗=1𝑙𝑗𝑎𝑙𝑗𝜑𝑗(𝑝𝑇𝑝)+𝑎𝑙𝑥𝑛=𝑝𝑡𝑙=0𝛾𝑛,𝑙𝑎𝑙𝑥𝑛.𝑝(2.15) It follows from (2.14), (2.15) that 𝑥𝑛+1𝑝𝛼𝑛,0𝑥𝑛+𝑝𝑘𝑖=1𝛼𝑛,𝑖𝑎𝑖𝛽𝑛,0𝑥𝑛+𝑝𝑘𝑖=1𝛼𝑛,𝑖𝑎𝑖𝑠𝑟=1𝛽𝑛,𝑖𝑎𝑖𝑡𝑙=0𝛾𝑛,𝑙𝑎𝑙𝑥𝑛<𝛼𝑝𝑛,0+1𝛼𝑛,0𝛽𝑛,0+1𝛼𝑛,01𝛽𝑛,0𝑥𝑛=𝑥𝑝𝑛.𝑝(2.16) Using Lemma (2.2), (2.16) yields lim𝑛𝑥𝑛=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 𝑇[0,1][0,1]=𝑥/2,𝛼𝑛,1=𝛽𝑛,1=𝛾𝑛,1=𝛼𝑛,0=𝛽𝑛,0=𝛾𝑛,0=4/𝑛,𝑛=1,2,,𝑛0 for some 𝑛0𝑁 and 𝛼𝑛,2=𝛽𝑛,2=𝛾𝑛,2=18/𝑛,𝑛𝑛0.

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 𝑛64 and 𝑢0=𝑥0 with 𝑥00. Then, for Kirk-Mann and Kirk-Ishikawa iterative schemes, we have 𝑢𝑛+1=𝑛𝑖=6414+4𝑖𝑢0,𝑥𝑛+1=𝑛𝑖=641+516𝑖𝑥0.(3.1) Now, consider ||||𝑥𝑛+1𝑢𝑛+1||||=||||||𝑛𝑖=641/16+5/𝑖𝑥0𝑛𝑖=641/4+4/𝑖𝑢0||||||=||||||𝑛𝑖=6413/161/𝑖1/4+4/𝑖||||||=||||||𝑛𝑖=6431𝑖164||||||𝑖+64.(3.2) It is easy to see that 0lim𝑛𝑛𝑖=6431𝑖164𝑖+64lim𝑛𝑛𝑖=6411𝑖=lim𝑛63𝑛=0.(3.3) Hence, lim𝑛|𝑥𝑛+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, ||||𝑥𝑛+1(Kirk-Noor)𝑥𝑛+1||||=||||||(Kirk-Ishikawa)𝑛𝑖=641/64+21/4𝑖𝑥0𝑛𝑖=641/16+5/𝑖𝑥0||||||=||||||𝑛𝑖=6413/641/4𝑖1/16+5/𝑖||||||=||||||𝑛𝑖=6431𝑖164||||||,𝑖+320(3.4) with 0lim𝑛𝑛𝑖=6431𝑖164𝑖+320lim𝑛𝑛𝑖=6411𝑖=lim𝑛63𝑛=0,(3.5) implies lim𝑛||||𝑥𝑛+1(Kirk-Noor)𝑥𝑛+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 ||||𝑥𝑛+1(Kirk-CR)𝑥𝑛+1||||=||||||(Kirk-Noor)𝑛𝑖=641/64+1/𝑥𝑖+12/𝑖0𝑛𝑖=641/64+21/4𝑖𝑥0||||||=||||||𝑛𝑖=64117/4𝑖12/𝑖1/64+21/4𝑖||||||=||||||𝑛𝑖=6416417𝑖484𝑖+1344𝑖||||||,(3.7) with 0lim𝑛𝑛𝑖=6416417𝑖484𝑖+1344𝑖lim𝑛𝑛𝑖=6411𝑖=lim𝑛63𝑛=0,(3.8) implies lim𝑛||||𝑥𝑛+1(Kirk-CR)𝑥𝑛+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 𝑛300. Then, for Kirk-CR iterative scheme, we have 𝑥𝑛+1=𝑛𝑖=3001+3644𝑖+12𝑛+64𝑖3/2𝑥0.(3.10) So, ||||𝑥𝑛+1(Kirk-SP)𝑥𝑛+1||||=||||||(Kirk-CR)𝑛𝑖=3001/64+3/4𝑖+12/𝑖+64/𝑖3/2𝑥0𝑛𝑖=3001/64+1/𝑥𝑖+12/𝑖0||||||=||||||𝑛𝑖=30011/4𝑖64/𝑖3/21/64+1/||||||=|||||𝑖+12/𝑖𝑛𝑖=300164𝑖163844𝑖3/2+256𝑖+3072𝑖|||||.(3.11) It is easy to see that 0lim𝑛𝑛𝑖=300164𝑖163844𝑖3/2+256𝑖+3072𝑖lim𝑛𝑛𝑖=30011𝑖=lim𝑛299𝑛=0.(3.12) Hence, we have lim𝑛|𝑥𝑛+1(Kirk-SP)/𝑥𝑛+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 𝑇[0,1][0,1]=𝑥/2,𝛼𝑛,0=𝛽𝑛,0=𝛾𝑛,0=𝛼𝑛,1=𝛽𝑛,1=𝛾𝑛,1=4/𝑛,𝑛=1,2,,𝑛0, for some 𝑛0𝑁 and 𝛼𝑛,3=𝛽𝑛,3=𝛾𝑛,3=18/𝑛,𝑛𝑛0. 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 𝑛70 and 𝑢0=𝑥0 with 𝑥00. Then, for Kirk-Mann and Mann iterative schemes, we have 𝑢𝑛+1=𝑛𝑖=6414+4𝑛𝑢0,𝑥𝑛+1=𝑛𝑖=7021𝑖𝑥0.(3.13) Now, consider ||||𝑥𝑛+1𝑢𝑛+1||||=||||||𝑛𝑖=701/4+4/𝑖𝑥0𝑛𝑖=7012/𝑖𝑢0||||||=||||||𝑛𝑖=7013/46/𝑖12/𝑖||||||=||||||𝑛𝑖=7031𝑖244||||||𝑖8.(3.14) It is easy to see that 0lim𝑛𝑛𝑖=7031𝑖244𝑖8lim𝑛𝑛𝑖=7011𝑖=lim𝑛69𝑛=0.(3.15) Hence, we have lim𝑛|𝑥𝑛+1/𝑢𝑛+1|=0.
It shows that Kirk-Mann iterative scheme converges faster than Mann iterative scheme to the fixed point 0 of 𝑇. Similarly, ||||𝑥𝑛+1(Kirk-Ishikawa)𝑥𝑛+1||||=||||||(Ishikawa)𝑛𝑖=701/16+5/𝑖𝑥0𝑛𝑖=7012/𝑥𝑖4/𝑖0||||||=||||||𝑛𝑖=70115/167/𝑖4/𝑖12/||||||=||||||𝑖4/𝑖𝑛𝑖=70115𝑖112𝑖6416𝑖32||||||,𝑖64(3.16) with 0lim𝑛𝑛𝑖=70115𝑖112𝑖6416𝑖32𝑖64lim𝑛𝑛𝑖=7011𝑖=lim𝑛69𝑛=0,(3.17) implies lim𝑛||||𝑥𝑛+1(Kirk-Ishikawa)𝑥𝑛+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, ||||𝑥𝑛+1(Kirk-Noor)𝑥𝑛+1||||=(Noor)𝑛𝑖=701/64+21/4𝑖𝑥0𝑛𝑖=7012/𝑖4/𝑖8/𝑖3/2𝑥0=𝑛𝑖=70163/6429/4𝑖4/𝑖8/𝑖3/212/𝑖4/𝑖8/𝑖3/2=𝑛𝑖=70163𝑖3/2464𝑖256𝑖51264𝑖3/2128𝑖256,𝑖512(3.19) with 0𝑛𝑖=70163𝑖3/2464𝑖256𝑖51264𝑖3/2128𝑖256𝑖512lim𝑛𝑛𝑖=7011𝑖=lim𝑛69𝑛=0,(3.20) implies lim𝑛||||𝑥𝑛+1(Kirk-Noor)𝑥𝑛+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, ||||𝑥𝑛+1(Kirk-SP)𝑥𝑛+1||||=||||||(SP)𝑛𝑖=701/64+3/4𝑖+12/𝑖+64/𝑖3/2𝑥0𝑛𝑖=7016/𝑖+12/𝑖8/𝑖3/2𝑥0||||||=||||||𝑛𝑖=70163/6427/4𝑖72/𝑖3/216/𝑖+12/𝑖8/𝑖3/2||||||=|||||𝑛𝑖=70163𝑖3/2432𝑖460864𝑖3/2384𝑖768|||||,𝑖512(3.22) with 0lim𝑛𝑛𝑖=70163𝑖3/2432𝑖460864𝑖3/2384𝑖768𝑖512lim𝑛𝑛𝑖=7011𝑖=lim𝑛69𝑛=0,(3.23) implies lim𝑛||||𝑥𝑛+1(Kirk-SP)𝑥𝑛+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, ||||𝑥𝑛+1(Kirk-CR)𝑥𝑛+1||||=(CR)𝑛𝑖=701/64+1/𝑥𝑖+12/𝑖0𝑛𝑖=701/21/𝑖4/𝑖+8/𝑖3/2𝑥=𝑛𝑖=70131/642/𝑖16/𝑖+8/𝑖3/21/21/𝑖4/𝑖+8/𝑖3/2=𝑛𝑖=70131𝑖3/2128𝑖1024𝑖+51232𝑖3/264𝑖256,𝑖+512(3.25) with 0lim𝑛𝑛𝑖=70131𝑖3/2128𝑖1024𝑖+51232𝑖3/264𝑖256𝑖+512lim𝑛𝑛𝑖=7011𝑖=lim𝑛69𝑛=0,(3.26) implies lim𝑛||||𝑥𝑛+1(Kirk-CR)𝑥𝑛+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=1𝛼𝑛,1𝛼𝑛,2, and 𝛽𝑛,0=1𝛽𝑛,1𝛽𝑛,2, 𝛾𝑛,0=1𝛾𝑛,1𝛾𝑛,2 for all iterative schemes.

tab1
Table 1: Decreasing cum sublinear functions.
tab2
Table 2: Increasing functions.
tab3
Table 3: Functions with multiple zeros.
tab4
Table 4: Superlinear functions.
tab5
Table 5: Oscillatory functions.
4.1. Decreasing Cum Sublinear Functions

The function 𝑓[0,1][0,1] defined by 𝑓(𝑥)=(1𝑥3)1/2 is a decreasing and sublinear function. By taking initial approximation 𝑥0=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 𝑓[0,2][0,2] be defined by 𝑓(𝑥)=(𝜋+𝑥𝑛4𝑥𝑛2(42𝑥𝑛2)sin1(𝑥𝑛/2))/𝜋. Then, 𝑓 is an increasing function. By taking initial approximation 𝑥0=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(𝑥)=(1𝑥)2 is a function with multiple zeros. By taking initial approximation 𝑥0=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(𝑥)=2𝑥37𝑥2+8𝑥2 is a superlinear function with multiple real roots. By taking initial approximation 𝑥0=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(𝑥)=1/𝑥 is an oscillatory function. By taking initial approximation 𝑥0=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 𝑥𝑜=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 𝑥𝑜=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 𝑥𝑜=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 𝑥𝑜=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 𝑥𝑜=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 𝑥𝑜=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 𝑥𝑜=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 𝑥𝑜=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 𝑥𝑜=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 𝑥𝑜=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 15 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 𝑘=𝑠=𝑡>2, the convergence speed of each iterative scheme decreases for all type of the above-mentioned functions. Convergence speed is the highest for 𝑘=𝑠=𝑡=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(𝑥)=(1𝑥)𝑚,𝑚=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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