Data Dependence Results for Multistep and CR Iterative Schemes in the Class of Contractive-Like Operators

Abstract

We intend to establish some results on the data dependence of fixed points of certain contractive-like operators for the multistep and CR iterative processes in a Banach space setting. One of our results generalizes the corresponding results of Soltuz and Grosan (2008) and Chugh and Kumar (2011).

1. Introduction

Throughout this paper, ℕ denotes the set of all nonnegative integers. Let 𝑋 be a Banach space, 𝐸⊂𝑋 a nonempty closed, convex subset of 𝑋, and 𝑇 a self-map on 𝐸. Suppose that 𝐹_𝑇∶={𝑝∈𝑋∶𝑝=𝑇𝑝} is the set of all fixed points of 𝑇. Iterative schemes abound in the literature of fixed point theory for which the fixed points of operators have been approximated over the years by many authors.

It is well known that the Picard iteration procedure [1] is defined by 𝑥₀𝑥∈𝐸,_𝑛+1=𝑇𝑥_𝑛,𝑛∈ℕ. (1) Let {𝛼_𝑛}^∞_𝑛=0, {𝛽_𝑛}^∞_𝑛=0, {𝛾_𝑛}^∞_𝑛=0 and {𝛽^𝑖_𝑛}^∞_𝑛=0, 𝑖=1,𝑘−2, 𝑘≥2 be the real sequences in [0,1) satisfying certain conditions.

The Mann iterative scheme [2] is defined by 𝑥₀𝑥∈𝐸,_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛,𝑛∈ℕ. (2) If 𝛼_𝑛=𝜆 (constant) in (2), then the resulting iteration will be called Krasnoselkij iteration procedure [3].

A sequence {𝑥_𝑛}^∞_𝑛=0, defined by 𝑥₀𝑥∈𝐸,_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛,𝑛∈ℕ, (3) is commonly known as the Ishikawa iterative method [4].

The Noor iterative procedure [5] is defined by 𝑥₀𝑥∈𝐸,_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑧_𝑛,𝑧_𝑛=1−𝛾_𝑛𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛,𝑛∈ℕ. (4) In 2004, Rhoades and Soltuz [6] introduced a multistep iterative process as follows: 𝑥₀𝑥∈𝐸,_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑦¹_𝑛,𝑦^𝑖_𝑛=1−𝛽^𝑖_𝑛𝑥_𝑛+𝛽^𝑖_𝑛𝑇𝑦_𝑛^𝑖+1,𝑦_𝑛^𝑘−1=1−𝛽_𝑛^𝑘−1𝑥_𝑛+𝛽_𝑛^𝑘−1𝑇𝑥_𝑛,𝑛∈ℕ. (5) The iteration processes (2), (3), and (4) can be viewed as the special cases of the iteration procedure (5).

Recently, Chugh et al. introduced a CR iterative scheme in [7] as follows: 𝑥₀𝑥∈𝐸,_𝑛+1=1−𝛼_𝑛𝑦_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑇𝑥_𝑛+𝛽_𝑛𝑇𝑧_𝑛,𝑧_𝑛=1−𝛾_𝑛𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛,𝑛∈ℕ. (6)

Now we mention some important contractive type operators.

Any mapping 𝑇 is called a Kannan mapping, see [8], if there exists 𝑏∈(0,1/2) such that, for all 𝑥,𝑦∈𝑋, 𝑇𝑥−𝑇𝑦≤𝑏‖𝑥−𝑇𝑥‖+𝑦−𝑇𝑦. (7)

Similar mapping is called a Chatterjea mapping, see [9], if there exists 𝑐∈(0,1/2) such that, for all 𝑥,𝑦∈𝑋, 𝑇𝑥−𝑇𝑦≤𝑐𝑥−𝑇𝑦+𝑦−𝑇𝑥. (8)

In [10] Zamfirescu collected these classes of operators and proved an important result which states that an operator 𝑇∶𝑋→𝑋 satisfies condition Z (Zamfirescu condition) if and only if there exist the real numbers 𝑎,𝑏,and𝑐 satisfying 0<𝑎<1, 0<𝑏,and𝑐<1/2 such that, for each pair 𝑥,𝑦∈𝑋, at least one of the following conditions is true: (z₁)‖𝑇𝑥−𝑇𝑦‖≤𝑎‖𝑥−𝑦‖, (z₂)‖𝑇𝑥−𝑇𝑦‖≤𝑏(‖𝑥−𝑇𝑥‖+‖𝑦−𝑇𝑦‖), (z₃)‖𝑇𝑥−𝑇𝑦‖≤𝑐(‖𝑥−𝑇𝑦‖+‖𝑦−𝑇𝑥‖). Then 𝑇 has a unique fixed point 𝑝 and the Picard iteration {𝑥_𝑛}^∞_𝑛=0 defined by (1) converges to 𝑝, for any 𝑥₀∈𝑋.

It is well known, see [11], that the conditions (z₁), (z₂), and (z₃) are independent.

Let 𝑥,𝑦∈𝑋. Since 𝑇 satisfies condition Z, at least one of the conditions from (z₁), (z₂), and (z₃) is satisfied. Then 𝑇 satisfies the inequalities ‖‖,,𝑇𝑥−𝑇𝑦≤𝛿𝑥−𝑦+2𝛿𝑥−𝑇𝑥𝑇𝑥−𝑇𝑦≤𝛿𝑥−𝑦+2𝛿𝑥−𝑇𝑦 (9)(10) for all 𝑥,𝑦∈𝑋 where 𝛿∶=max{𝑎,𝑏/(1−𝑏),𝑐/(1−𝑐)}, 𝛿∈[0,1), and it was shown that this class of operators is wider than the class of Zamfirescu operators; see [12]. Any mapping satisfying condition (9) or (10) is called a quasi-contractive operator.

Osilike and Udomene [13] extended the previous definition by considering an operator satisfying the condition that there exist 𝐿≥0 and 𝛿∈[0,1) such that, for all 𝑥,𝑦∈𝑋, 𝑇𝑥−𝑇𝑦≤𝛿𝑥−𝑦+𝐿‖𝑥−𝑇𝑥‖. (11) Thereafter, Imoru and Olatinwo [14] further generalized and extended the previous definition as follows: an operator 𝑇 is called contractive-like operator if there exist a constant 𝛿∈[0,1) and a strictly increasing and continuous function 𝜑∶[0,∞)→[0,∞) with 𝜑(0)=0, such that, for each 𝑥,𝑦∈𝑋, 𝑇𝑥−𝑇𝑦≤𝛿𝑥−𝑦+𝜑(‖𝑥−𝑇𝑥‖). (12)

Remark 1 (see [15]). A map satisfying (12) need not have a fixed point. However, using (12) it is obvious that if 𝑇 has a fixed point, then it is unique.

It is important to know whether an iterative scheme converges to fixed points of its associated map. In this context, there are numerous works dealing with the convergence of various iterative schemes in the literature, such as [6, 10, 12, 16–27].

As shown by Soltuz and Grosan [26, Theorem 3.1], in a real Banach space 𝑋, the Ishikawa iteration {𝑥_𝑛}^∞_𝑛=0 given by (3) converges to the fixed point of 𝑇, where 𝑇∶𝐸→𝐸 is a mapping satisfying condition (12).

It is known from [28, Corollary 2] that there is equivalence between convergence of iterative procedures (3), (5) and that of some other well-known iterative procedures for the class of operators satisfying (12).

Hussain et al. [29] introduced a Kirk-CR iterative scheme and proved the convergence of this iteration for the class of operators satisfying (12).

Remark 2. Putting 𝑠=𝑡=𝑖=1 in [29, Theorem 2.5], convergence of the CR iteration to a fixed point of contractive-like operators satisfying (12) can be obtained easily.

2. Preliminaries

Definition 3 (see [30]). Let 𝑋 be a Banach space and 𝑇,𝑇∶𝑋→𝑋 two operators. We say that 𝑇 is an approximate operator of 𝑇 if for all 𝑥∈𝑋 and for a fixed 𝜀>0 we have 𝑇𝑥−𝑇𝑥≤𝜀. (13)

Suppose that there exists a certain fixed point iteration that converges to some fixed point 𝑝∈𝐹_𝑇 and 𝑇 has a fixed point 𝑇𝑞∈𝐹 which can be computed by certain method. If it cannot compute fixed point 𝑝 of  𝑇 due to various results, then approximate operator 𝑇 can be used. One can find some of works done under this title in the following list [15, 24–26, 31].

In this paper, we prove the data dependence results for the multistep and CR iterative procedures utilizing the contractive-like operators satisfying (12).

The following lemma will be useful to prove the main results of this work.

Lemma 4 (see [26]). Let {𝑎_𝑛}^∞_𝑛=0 be a nonnegative sequence for which one assumes there exists 𝑛₀(𝜖)∈ℕ, such that for all 𝑛≥𝑛₀ one has satisfied the inequality 𝑎_𝑛+1≤1−𝜇_𝑛𝑎_𝑛+𝜇_𝑛𝜂_𝑛, (14) where 𝜇_𝑛∈(0,1), for all 𝑛∈ℕ, ∑^∞_𝑛=0𝜇_𝑛=∞ and 𝜂_𝑛≥0, for all 𝑛∈ℕ. Then the following holds: 0≤lim_𝑛→∞sup𝑎_𝑛≤lim_𝑛→∞sup𝜂_𝑛. (15)

3. Main Results

For simplicity we use the following notation through this section.

For any iterative process, {𝑥_𝑛}^∞_𝑛=0 and {𝑢_𝑛}^∞_𝑛=0 denote iterative sequences associated to 𝑇 and 𝑇, respectively.

Theorem 5. Let 𝑇∶𝐸→𝐸 be a map satisfying (12) with 𝐹_𝑇≠∅, and let 𝑇 be an approximate operator of 𝑇 as in Definition 3. Let {𝑥_𝑛}^∞_𝑛=0, {𝑢_𝑛}^∞_𝑛=0 be two iterative sequences defined by the multistep iteration (5) and with real sequences {𝛽^𝑖_𝑛}^∞_𝑛=0⊂[0,1), 𝑖=1,𝑘−1, and {𝛼_𝑛}^∞_𝑛=0⊂[0,1) satisfying ∑𝛼_𝑛=∞. If 𝑝=𝑇𝑝 and 𝑞=𝑇𝑞, then one has 𝑝−𝑞≤𝑘𝜀.1−𝛿 (16)

Proof. For a given 𝑥₀∈𝐸 and 𝑢₀∈𝐸 we consider the following multistep iteration for 𝑇 and 𝑇: 𝑥₀𝑥∈𝐸,_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑦¹_𝑛𝑦,𝑛∈ℕ,_𝑖𝑛=1−𝛽^𝑖_𝑛𝑥_𝑛+𝛽^𝑖_𝑛𝑇𝑦_𝑛^𝑖+1,𝑖=𝑦1,𝑘−2,_𝑛𝑘−1=1−𝛽_𝑛^𝑘−1𝑥_𝑛+𝛽_𝑛^𝑘−1𝑇𝑥_𝑛𝑢,𝑘≥2,₀𝑢∈𝐸,_𝑛+1=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑇𝑣¹_𝑛𝑣,𝑛∈ℕ,_𝑖𝑛=1−𝛽^𝑖_𝑛𝑢_𝑛+𝛽^𝑖_𝑛𝑇𝑣_𝑛^𝑖+1,𝑖=𝑣1,𝑘−2,_𝑛𝑘−1=1−𝛽_𝑛^𝑘−1𝑢_𝑛+𝛽_𝑛^𝑘−1𝑇𝑢_𝑛,𝑘≥2. (17)
Then from (17), we get 𝑥_𝑛+1−𝑢_𝑛+1=1−𝛼_𝑛𝑥_𝑛−𝑢_𝑛+𝛼_𝑛𝑇𝑦¹_𝑛−𝑇𝑣¹_𝑛. (18) Thus, we have the following estimates by using (18) and (12): 𝑥_𝑛+1−𝑢_𝑛+1=1−𝛼_𝑛𝑥_𝑛−𝑢_𝑛+𝛼_𝑛𝑇𝑦¹_𝑛−𝑇𝑣¹_𝑛≤1−𝛼_𝑛𝑥_𝑛−𝑢_𝑛+𝛼_𝑛𝑇𝑦¹_𝑛−𝑇𝑣¹_𝑛=1−𝛼_𝑛𝑥_𝑛−𝑢_𝑛+𝛼_𝑛𝑇𝑦¹_𝑛−𝑇𝑣¹_𝑛+𝑇𝑣¹_𝑛−𝑇𝑣¹_𝑛≤1−𝛼_𝑛𝑥_𝑛−𝑢_𝑛+𝛼_𝑛𝑇𝑦¹_𝑛−𝑇𝑣¹_𝑛+𝛼_𝑛𝑇𝑣¹_𝑛−𝑇𝑣¹_𝑛≤1−𝛼_𝑛𝑥_𝑛−𝑢_𝑛+𝛼_𝑛𝛿𝑦¹_𝑛−𝑣¹_𝑛+𝛼_𝑛𝜑𝑦¹_𝑛−𝑇𝑦¹_𝑛+𝛼_𝑛𝑦𝜀,_1𝑛−𝑣¹_𝑛=1−𝛽¹_𝑛𝑥_𝑛−𝑢_𝑛+𝛽¹_𝑛𝑇𝑦²_𝑛−𝑇𝑣²_𝑛≤1−𝛽¹_𝑛𝑥_𝑛−𝑢_𝑛+𝛽¹_𝑛𝑇𝑦²_𝑛−𝑇𝑣²_𝑛≤1−𝛽¹_𝑛𝑥_𝑛−𝑢_𝑛+𝛽¹_𝑛𝑇𝑦²_𝑛−𝑇𝑣²_𝑛+𝛽¹_𝑛𝑇𝑣²_𝑛−𝑇𝑣²_𝑛≤1−𝛽¹_𝑛𝑥_𝑛−𝑢_𝑛+𝛽¹_𝑛𝛿𝑦²_𝑛−𝑣²_𝑛+𝛽¹_𝑛𝜑𝑦²_𝑛−𝑇𝑦²_𝑛+𝛽¹_𝑛𝑦𝜀,_2𝑛−𝑣²_𝑛=1−𝛽²_𝑛𝑥_𝑛−𝑢_𝑛+𝛽²_𝑛𝑇𝑦³_𝑛−𝑇𝑣³_𝑛≤1−𝛽²_𝑛𝑥_𝑛−𝑢_𝑛+𝛽²_𝑛𝑇𝑦³_𝑛−𝑇𝑣³_𝑛≤1−𝛽²_𝑛𝑥_𝑛−𝑢_𝑛+𝛽²_𝑛𝑇𝑦³_𝑛−𝑇𝑣³_𝑛+𝛽²_𝑛𝑇𝑣³_𝑛−𝑇𝑣³_𝑛≤1−𝛽²_𝑛𝑥_𝑛−𝑢_𝑛+𝛽²_𝑛𝛿𝑦³_𝑛−𝑣³_𝑛+𝛽²_𝑛𝜑𝑦³_𝑛−𝑇𝑦³_𝑛+𝛽²_𝑛𝑦𝜀,_3𝑛−𝑣³_𝑛=1−𝛽³_𝑛𝑥_𝑛−𝑢_𝑛+𝛽³_𝑛𝑇𝑦⁴_𝑛−𝑇𝑣⁴_𝑛≤1−𝛽³_𝑛𝑥_𝑛−𝑢_𝑛+𝛽³_𝑛𝑇𝑦⁴_𝑛−𝑇𝑣⁴_𝑛≤1−𝛽³_𝑛𝑥_𝑛−𝑢_𝑛+𝛽³_𝑛𝑇𝑦⁴_𝑛−𝑇𝑣⁴_𝑛+𝛽³_𝑛𝑇𝑣⁴_𝑛−𝑇𝑣⁴_𝑛≤1−𝛽³_𝑛𝑥_𝑛−𝑢_𝑛+𝛽³_𝑛𝛿𝑦⁴_𝑛−𝑣⁴_𝑛+𝛽³_𝑛𝜑𝑦⁴_𝑛−𝑇𝑦⁴_𝑛+𝛽³_𝑛𝜀. (19)(20)(21)(22) Combining (19), (20), (21), and (22) we obtain 𝑥_𝑛+1−𝑢_𝑛+1≤1−𝛼_𝑛+𝛼_𝑛𝛿1−𝛽¹_𝑛+𝛼_𝑛𝛿²𝛽¹_𝑛1−𝛽²_𝑛+𝛼_𝑛𝛿³𝛽¹_𝑛𝛽²_𝑛1−𝛽³_𝑛𝑥_𝑛−𝑢_𝑛+𝛼_𝑛𝛿⁴𝛽¹_𝑛𝛽²_𝑛𝛽³_𝑛𝑦⁴_𝑛−𝑣⁴_𝑛+𝛼_𝑛𝛿³𝛽¹_𝑛𝛽²_𝑛𝛽³_𝑛𝜑𝑦⁴_𝑛−𝑇𝑦⁴_𝑛+𝛼_𝑛𝛿³𝛽¹_𝑛𝛽²_𝑛𝛽³_𝑛𝜀+𝛼_𝑛𝛿²𝛽¹_𝑛𝛽²_𝑛𝜑𝑦³_𝑛−𝑇𝑦³_𝑛+𝛼_𝑛𝛿²𝛽¹_𝑛𝛽²_𝑛𝜀+𝛼_𝑛𝛿𝛽¹_𝑛𝜑𝑦²_𝑛−𝑇𝑦²_𝑛+𝛼_𝑛𝛿𝛽¹_𝑛𝜀+𝛼_𝑛𝜑𝑦¹_𝑛−𝑇𝑦¹_𝑛+𝛼_𝑛𝜀. (23) Thus, inductively, we get 𝑥_𝑛+1−𝑢_𝑛+1≤1−𝛼_𝑛+𝛿𝛼_𝑛1−𝛽¹_𝑛+𝛿²𝛼_𝑛𝛽¹_𝑛1−𝛽²_𝑛+𝛿³𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛1−𝛽³_𝑛+𝛿⁴𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛𝛽³_𝑛1−𝛽⁴_𝑛+⋯+𝛿^𝑘−2𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛⋯𝛽_𝑛^𝑘−31−𝛽_𝑛^𝑘−2𝑥_𝑛−𝑢_𝑛+𝛿^𝑘−1𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛⋯𝛽_𝑛^𝑘−2𝑦_𝑛^𝑘−1−𝑣_𝑛^𝑘−1+𝛿^𝑘−2𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛⋯𝛽_𝑛^𝑘−2𝜑𝑦_𝑛^𝑘−1−𝑇𝑦_𝑛^𝑘−1+⋯+𝛿³𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛𝛽³_𝑛𝜑𝑦⁴_𝑛−𝑇𝑦⁴_𝑛+𝛿²𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛𝜑𝑦³_𝑛−𝑇𝑦³_𝑛+𝛿𝛼_𝑛𝛽¹_𝑛𝜑𝑦²_𝑛−𝑇𝑦²_𝑛+𝛼_𝑛𝜑𝑦¹_𝑛−𝑇𝑦¹_𝑛+𝛿^𝑘−2𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛⋯𝛽_𝑛^𝑘−2𝜀+⋯+𝛿²𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛𝜀+𝛿𝛼_𝑛𝛽¹_𝑛𝜀+𝛼_𝑛𝜀. (24) Using now (17) and (12), we get 𝑦_𝑛^𝑘−1−𝑣_𝑛^𝑘−1=1−𝛽_𝑛^𝑘−1𝑥_𝑛−𝑢_𝑛+𝛽_𝑛^𝑘−1𝑇𝑥_𝑛−𝑇𝑢_𝑛≤1−𝛽_𝑛^𝑘−1𝑥_𝑛−𝑢_𝑛+𝛽_𝑛^𝑘−1𝑇𝑥_𝑛−𝑇𝑢_𝑛≤1−𝛽_𝑛^𝑘−1𝑥_𝑛−𝑢_𝑛+𝛽_𝑛^𝑘−1𝑇𝑥_𝑛−𝑇𝑢_𝑛+𝛽_𝑛^𝑘−1𝑇𝑢_𝑛−𝑇𝑢_𝑛≤1−𝛽_𝑛^𝑘−1𝑥_𝑛−𝑢_𝑛+𝛽_𝑛^𝑘−1𝛿𝑥_𝑛−𝑢_𝑛+𝛽_𝑛^𝑘−1𝜑𝑥_𝑛−𝑇𝑥_𝑛+𝛽_𝑛^𝑘−1𝜀. (25) Substituting (25) in (24) we have 𝑥_𝑛+1−𝑢_𝑛+1≤1−𝛼_𝑛+𝛿𝛼_𝑛1−𝛽¹_𝑛+𝛿²𝛼_𝑛𝛽¹_𝑛1−𝛽²_𝑛+𝛿³𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛1−𝛽³_𝑛+𝛿⁴𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛𝛽³_𝑛1−𝛽⁴_𝑛+⋯+𝛿^𝑘−2𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛⋯𝛽_𝑛^𝑘−31−𝛽_𝑛^𝑘−2+𝛿^𝑘−1𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛⋯𝛽_𝑛^𝑘−21−𝛽_𝑛^𝑘−1+𝛿^𝑘𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛⋯𝛽_𝑛^𝑘−2𝛽_𝑛^𝑘−1𝑥_𝑛−𝑢_𝑛+𝛿^𝑘−1𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛⋯𝛽_𝑛^𝑘−2𝛽_𝑛^𝑘−1𝜑𝑥_𝑛−𝑇𝑥_𝑛+𝛿^𝑘−2𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛⋯𝛽_𝑛^𝑘−2𝜑𝑦_𝑛^𝑘−1−𝑇𝑦_𝑛^𝑘−1+⋯+𝛿³𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛𝛽³_𝑛𝜑𝑦⁴_𝑛−𝑇𝑦⁴_𝑛+𝛿²𝛼_𝑛𝛽¹_n𝛽²_𝑛𝜑𝑦³_𝑛−𝑇𝑦³_𝑛+𝛿𝛼_𝑛𝛽¹_𝑛𝜑𝑦²_𝑛−𝑇𝑦²_𝑛+𝛼_𝑛𝜑𝑦¹_𝑛−𝑇𝑦¹_𝑛+𝛿^𝑘−1𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛⋯𝛽_𝑛^𝑘−2𝛽_𝑛^𝑘−1𝜀+𝛿^𝑘−2𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛⋯𝛽_𝑛^𝑘−2𝜀+⋯+𝛿²𝛼_𝑛𝛽¹_𝑛𝛽²_𝑛𝜀+𝛿𝛼_𝑛𝛽¹_𝑛𝜀+𝛼_𝑛𝜀. (26) If this inequality is rearranged using {𝛽^𝑖_𝑛}^∞_𝑛=0⊂[0,1), 𝛿^𝑖+1<𝛿^𝑖 for each 𝑖=1,𝑘−1, then we get the following inequality as follows: 𝑥_𝑛+1−𝑢_𝑛+1≤1−𝛼_𝑛(1−𝛿)𝑥_𝑛−𝑢_𝑛+𝛼_𝑛()𝜑𝑥1−𝛿^𝑛−𝑇𝑥_𝑛1−𝛿+𝛼_𝑛()×𝜑𝑦1−𝛿_𝑛𝑘−1−𝑇𝑦_𝑛^𝑘−1+⋯+𝜑𝑦¹_𝑛−𝑇𝑦¹_𝑛+𝑘𝜀.1−𝛿 (27) Denote 𝑎_𝑛∶=𝑥_𝑛−𝑢_𝑛,𝜇_𝑛∶=𝛼_𝑛()∈(),𝜂1−𝛿0.1_𝑛𝜑𝑥∶=^𝑛−𝑇𝑥_𝑛+𝜑𝑦1−𝛿_𝑛𝑘−1−𝑇𝑦_𝑛^𝑘−1+⋯+𝜑𝑦¹_𝑛−𝑇𝑦¹_𝑛+𝑘𝜀.1−𝛿 (28) From [26, Theorem 3.1] and [28, Corollary 2] we have lim_𝑛→∞‖𝑥_𝑛−𝑝‖=0. Since 𝑇 satisfies condition (12), and 𝑝∈𝐹_𝑇, that is, 𝑇𝑝=𝑝, it follows from (12) that 0≤𝑥_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛−𝑝+𝑇𝑝−𝑇𝑥_𝑛≤𝑥_𝑛−𝑝+𝛿𝑝−𝑥_𝑛=(𝑥+𝜑𝑝−𝑇𝑝1+𝛿)_𝑛−𝑝⟶0as𝑛⟶∞. (29) Considering 𝛽^𝑖_𝑛∈[0,1), for all 𝑛∈ℕ, 𝑖=1,𝑘−1, and using (12) and (5) we have 0≤𝑦¹_𝑛−𝑇𝑦¹_𝑛=𝑦¹_𝑛−𝑝+𝑝−𝑇𝑦¹_𝑛≤𝑦¹_𝑛−𝑝+𝑇𝑝−𝑇𝑦¹_𝑛≤𝑦¹_𝑛−𝑝+𝛿𝑝−𝑦¹_𝑛=(𝑦+𝜑𝑝−𝑇𝑝1+𝛿)¹_𝑛=(−𝑝1+𝛿)1−𝛽¹_𝑛𝑦²_𝑛+𝛽¹_𝑛𝑇𝑦²_𝑛−𝑝1−𝛽¹_𝑛+𝛽¹_𝑛≤(1+𝛿)1−𝛽¹_𝑛𝑦²_𝑛−𝑝+𝛽¹_𝑛𝑇𝑦²_𝑛≤(−𝑇𝑝1+𝛿)1−𝛽¹_𝑛𝑦²_𝑛−𝑝+𝛽¹_𝑛𝛿𝑦²_𝑛=(−𝑝1+𝛿)1−𝛽¹_𝑛(1−𝛿)𝑦²_𝑛=(−𝑝1+𝛿)1−𝛽¹_𝑛(×1−𝛿)1−𝛽²_𝑛𝑦³_𝑛+𝛽²_𝑛𝑇𝑦³_𝑛−𝑝1−𝛽²_𝑛+𝛽²_𝑛≤(1+𝛿)1−𝛽¹_𝑛(×1−𝛿)1−𝛽²_𝑛𝑦³_𝑛−𝑝+𝛽²_𝑛𝑇𝑦³_𝑛≤(−𝑇𝑝1+𝛿)1−𝛽¹_𝑛(×1−𝛿)1−𝛽²_𝑛(1−𝛿)𝑦³_𝑛⋮≤(−𝑝1+𝛿)1−𝛽¹_𝑛(⋯1−𝛿)1−𝛽_𝑛^𝑘−2(1−𝛿)𝑦_𝑛^𝑘−1≤(−𝑝1+𝛿)1−𝛽¹_𝑛(⋯1−𝛿)1−𝛽_𝑛^𝑘−1(1−𝛿)𝑥_𝑛≤(𝑥−𝑝1+𝛿)_𝑛−𝑝⟶0as𝑛⟶∞. (30) It is easy to see from (30) that this result is also valid for ‖𝑇𝑦²_𝑛−𝑦²_𝑛‖,…,‖𝑇𝑦_𝑛^𝑘−1−𝑦_𝑛^𝑘−1‖.
Making use of the fact that 𝜑 is a continuous map we have lim_𝑛→∞𝜑𝑥_𝑛−𝑇𝑥_𝑛=lim_𝑛→∞𝜑𝑦¹_𝑛−𝑇𝑦¹_𝑛=⋯=lim_𝑛→∞𝜑𝑦_𝑛^𝑘−1−𝑇𝑦_𝑛^𝑘−1=0. (31) Hence an application of Lemma 4 to (27) leads to 𝑝−𝑞≤𝑘𝜀.1−𝛿 (32)

Now we prove result on data dependence for the CR iterative procedure.

Theorem 6. Let 𝑇∶𝐸→𝐸 be a map satisfying (12) with 𝐹_𝑇≠∅, and let 𝑇 be an approximate operator of 𝑇 as in Definition 3. Let {𝑥_𝑛}^∞_𝑛=0, {𝑢_𝑛}^∞_𝑛=0 be two iterative sequences defined by the CR iteration (6) and with real sequences {𝛽_𝑛}^∞_𝑛=0, {𝜇_𝑛}^∞_𝑛=0,{𝛼_𝑛}^∞_𝑛=0⊂[0,1] satisfying (i)1/2≤𝛼_𝑛, for all 𝑛∈ℕ, and (ii)∑^∞_𝑛=0𝛼_𝑛=∞. If 𝑝=𝑇𝑝 and 𝑞=𝑇𝑞, then one has 𝑝−𝑞≤5𝜀.1−𝛿 (33)

Proof. For a given 𝑥₀∈𝐸 and 𝑢₀∈𝐸 we consider the following iteration for 𝑇 and 𝑇: 𝑥_𝑛+1=1−𝛼_𝑛𝑦_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑇𝑥_𝑛+𝛽_𝑛𝑇𝑧_𝑛,𝑧_𝑛=1−𝜇_𝑛𝑥_𝑛+𝜇_𝑛𝑇𝑥_𝑛,𝑢_𝑛+1=1−𝛼_𝑛𝑣_𝑛+𝛼_𝑛𝑇𝑣_𝑛,𝑣_𝑛=1−𝛽_𝑛𝑇𝑢_𝑛+𝛽_𝑛𝑇𝑤_𝑛,𝑤_𝑛=1−𝜇_𝑛𝑢_𝑛+𝜇_𝑛𝑇𝑢_𝑛. (34) Then from (34) we have 𝑥_𝑛+1−𝑢_𝑛+1=1−𝛼_𝑛𝑦_𝑛−𝑣_𝑛+𝛼_𝑛𝑇𝑦_𝑛−𝑇𝑣_𝑛,𝑦_𝑛−𝑣_𝑛=1−𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑢_𝑛+𝛽_𝑛𝑇𝑧_𝑛−𝑇𝑤_𝑛,𝑧_𝑛−𝑤_𝑛=1−𝜇_𝑛𝑥_𝑛−𝑢_𝑛+𝜇_𝑛𝑇𝑥_𝑛−𝑇𝑢_𝑛. (35) Thus, by considering (35), it follows from (6) and (12) that 𝑥_𝑛+1−𝑢_𝑛+1=1−𝛼_𝑛𝑦_𝑛−𝑣_𝑛+𝛼_𝑛𝑇𝑦_𝑛−𝑇𝑣_𝑛≤1−𝛼_𝑛𝑦_𝑛−𝑣_𝑛+𝛼_𝑛𝑇𝑦_𝑛−𝑇𝑣_𝑛=1−𝛼_𝑛𝑦_𝑛−𝑣_𝑛+𝛼_𝑛𝑇𝑦_𝑛−𝑇𝑣_𝑛+𝑇𝑣_𝑛−𝑇𝑣_𝑛≤1−𝛼_𝑛𝑦_𝑛−𝑣_𝑛+𝛼_𝑛𝑇𝑦_𝑛−𝑇𝑣_𝑛+𝛼_𝑛𝑇𝑣_𝑛−𝑇𝑣_𝑛≤1−𝛼_𝑛𝑦_𝑛−𝑣_𝑛+𝛼_𝑛𝛿𝑦_𝑛−𝑣_𝑛+𝛼_𝑛𝜑𝑦_𝑛−𝑇𝑦_𝑛+𝛼_𝑛𝜀=1−𝛼_𝑛(1−𝛿)𝑦_𝑛−𝑣_𝑛+𝛼_𝑛𝜑𝑦_𝑛−𝑇𝑦_𝑛+𝛼_𝑛𝑦𝜀,_𝑛−𝑣_𝑛=1−𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑢_𝑛+𝛽_𝑛𝑇𝑧_𝑛−𝑇𝑤_𝑛≤1−𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑢_𝑛+𝛽_𝑛𝑇𝑧_𝑛−𝑇𝑤_𝑛=1−𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑢_𝑛+𝑇𝑢_𝑛−𝑇𝑢_𝑛+𝛽_𝑛𝑇𝑧_𝑛−𝑇𝑤_𝑛+𝑇𝑤_𝑛−𝑇𝑤_𝑛≤1−𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑢_𝑛+𝑇𝑢_𝑛−𝑇𝑢_𝑛+𝛽_𝑛𝑇𝑧_𝑛−𝑇𝑤_𝑛+𝑇𝑤_𝑛−𝑇𝑤_𝑛≤1−𝛽_𝑛𝛿𝑥_𝑛−𝑢_𝑛+𝛽_𝑛𝛿𝑧_𝑛−𝑤_𝑛+1−𝛽_𝑛𝜑𝑥_𝑛−𝑇𝑥_𝑛+1−𝛽_𝑛𝜀+𝛽_𝑛𝜑𝑇𝑧_𝑛−𝑇𝑤_𝑛+𝛽_𝑛𝑧𝜀,_𝑛−𝑤_𝑛=1−𝜇_𝑛𝑥_𝑛−𝑢_𝑛+𝜇_𝑛𝑇𝑥_𝑛−𝑇𝑢_𝑛≤1−𝜇_𝑛𝑥_𝑛−𝑢_𝑛+𝜇_𝑛𝑇𝑥_𝑛−𝑇𝑢_𝑛=1−𝜇_𝑛𝑥_𝑛−𝑢_𝑛+𝜇_𝑛𝑇𝑥_𝑛−𝑇𝑢_𝑛+𝑇𝑢_𝑛−𝑇𝑢_𝑛≤1−𝜇_𝑛𝑥_𝑛−𝑢_𝑛+𝜇_𝑛𝑇𝑥_𝑛−𝑇𝑢_𝑛+𝑇𝑢_𝑛−𝑇𝑢_𝑛≤1−𝜇_𝑛𝑥_𝑛−𝑢_𝑛+𝜇_𝑛𝛿𝑥_𝑛−𝑢_𝑛+𝜑𝑥_𝑛−𝑇𝑥_𝑛=+𝜀1−𝜇_𝑛(1−𝛿)𝑥_𝑛−𝑢_𝑛+𝜇_𝑛𝜑𝑥_𝑛−𝑇𝑥_𝑛+𝜇_𝑛𝜀. (36)(37)(38) Combining (36), (37), and (38) we obtain 𝑥_𝑛+1−𝑢_𝑛+1≤1−𝛼_𝑛(×1−𝛿)1−𝛽_𝑛𝛿+𝛽_𝑛𝛿1−𝜇_𝑛(1−𝛿)𝑥_𝑛−𝑢_𝑛+1−𝛼_𝑛(×1−𝛿)1−𝛽_𝑛1−𝛿𝜇_𝑛𝜑𝑥_𝑛−𝑇𝑥_𝑛+𝛽_𝑛𝜇_𝑛+𝛿𝜀1−𝛽_𝑛𝜀+𝛽_𝑛𝜑𝑧_𝑛−𝑇𝑧_𝑛+𝛽_𝑛𝜀+𝛼_𝑛𝜑𝑦_𝑛−𝑇𝑦_𝑛+𝛼_𝑛𝜀. (39) It may be noted that for {𝛽_𝑛}^∞_𝑛=0, {𝜇_𝑛}^∞_𝑛=0⊂[0,1), and 𝛿∈[0,1) the following inequalities are always true: 1−𝜇_𝑛(1−𝛿)<1,𝛽_𝑛𝜇_𝑛𝛿<1,𝛿𝜇_𝑛<𝛿. (40) Using now the inequality 𝛿𝜇_𝑛<𝛿 we get 1−𝛽_𝑛1−𝛿𝜇_𝑛<1−𝛽_𝑛(1−𝛿)<1. (41) Therefore an application of (40) and (41) to (39) gives us 𝑥_𝑛+1−𝑢_𝑛+1≤1−𝛼_𝑛(1−𝛿)𝑥_𝑛−𝑢_𝑛+1−𝛼_𝑛(×𝜑𝑥1−𝛿)_𝑛−𝑇𝑥_𝑛+𝜀+𝜀+𝜑𝑧_𝑛−𝑇𝑧_𝑛+𝛼_𝑛𝜑𝑦_𝑛−𝑇𝑦_𝑛+𝛼_𝑛𝜀. (42) Now, by the condition (i) 1/2≤𝛼_𝑛, for all 𝑛∈ℕ we have 1−𝛼_𝑛≤𝛼_𝑛. (43) Utilizing (43) in (42) we obtain 𝑥_𝑛+1−𝑢_𝑛+1≤1−𝛼_𝑛(1−𝛿)𝑥_𝑛−𝑢_𝑛+𝛼_𝑛()𝑥1−𝛿2𝜑_𝑛−𝑇𝑥_𝑛+𝜑𝑦_𝑛−𝑇𝑦_𝑛1−𝛿+𝛼_𝑛()𝑧1−𝛿2𝜑_𝑛−𝑇𝑧_𝑛+5𝜀.1−𝛿 (44) Denote 𝑎_𝑛∶=𝑥_𝑛−𝑢_𝑛,𝜇_𝑛∶=𝛼_𝑛()∈(),𝜂1−𝛿0,1_𝑛𝑥∶=2𝜑_𝑛−𝑇𝑥_𝑛+𝜑𝑦_𝑛−𝑇𝑦_𝑛+2𝜑𝑧_𝑛−𝑇𝑧_𝑛+5𝜀.1−𝛿 (45)
From Remark 2, we have lim_𝑛→∞‖𝑥_𝑛−𝑝‖=0. Since 𝑇 satisfies condition (12), and 𝑝∈𝐹_𝑇, that is, 𝑇𝑝=𝑝, using similar arguments as in the proof of Theorem 5, we get lim_𝑛→∞𝑥_𝑛−𝑇𝑥_𝑛=lim_𝑛→∞𝑦_𝑛−𝑇𝑦_𝑛=lim_𝑛→∞𝑧_𝑛−𝑇𝑧_𝑛=0. (46) Making use of the fact that 𝜑 is a continuous map we have lim_𝑛→∞𝜑𝑥_𝑛−𝑇𝑥_𝑛=lim_𝑛→∞𝜑𝑦_𝑛−𝑇𝑦_𝑛=lim_𝑛→∞𝜑𝑧_𝑛−𝑇𝑧_𝑛=0. (47) Hence an application of Lemma 4 to (44) leads to 𝑝−𝑞≤5𝜀.1−𝛿 (48)

Corollary 7. Since the Mann (2), Ishikawa (3), and Noor (4) iterative processes are special cases of the multistep iterative scheme (5), the data dependence results of these iterative processes can be obtained similarly.