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

Authors: Vatan Karakaya; Faik Gürsoy; Kadri Doğan; Müzeyyen Ertürk
Journal: Abstract and Applied Analysis
 (2013)

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## 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).


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## Body

## 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 FT:={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
(1)x0∈E,xn+1=Txn,n∈ℕ.
Let {αn}n=0∞, {βn}n=0∞, {γn}n=0∞ and {βni}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
(2)x0∈E,xn+1=(1-αn)xn+αnTxn,n∈ℕ.
If αn=λ (constant) in (2), then the resulting iteration will be called Krasnoselkij iteration procedure [3].A sequence{xn}n=0∞, defined by
(3)x0∈E,xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,n∈ℕ,
is commonly known as the Ishikawa iterative method [4].The Noor iterative procedure [5] is defined by
(4)x0∈E,xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTzn,zn=(1-γn)xn+γnTxn,n∈ℕ.
In 2004, Rhoades and Soltuz [6] introduced a multistep iterative process as follows:
(5)x0∈E,xn+1=(1-αn)xn+αnTyn1,yni=(1-βni)xn+βniTyni+1,ynk-1=(1-βnk-1)xn+βnk-1Txn,n∈ℕ.
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:
(6)x0∈E,xn+1=(1-αn)yn+αnTyn,yn=(1-βn)Txn+βnTzn,zn=(1-γn)xn+γnTxn,n∈ℕ.Now we mention some important contractive type operators.Any mappingT is called a Kannan mapping, see [8], if there exists b∈(0,1/2) such that, for all x,y∈X,
(7)∥Tx-Ty∥≤b(∥x-Tx∥+∥y-Ty∥).Similar mapping is called a Chatterjea mapping, see [9], if there exists c∈(0,1/2) such that, for all x,y∈X,
(8)∥Tx-Ty∥≤c(∥x-Ty∥+∥y-Tx∥).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 {xn}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.Letx,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
(9)∥Tx-Ty∥≤δ∥x-y∥+2δ∥x-Tx∥,(10)∥Tx-Ty∥≤δ∥x-y∥+2δ∥x-Ty∥,
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,
(11)∥Tx-Ty∥≤δ∥x-y∥+L∥x-Tx∥.
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,
(12)∥Tx-Ty∥≤δ∥x-y∥+φ(∥x-Tx∥).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 {xn}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.
Puttings=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]).
LetX 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 ɛ>0 we have
(13)∥Tx-T~x∥≤ɛ.Suppose that there exists a certain fixed point iteration that converges to some fixed pointp∈FT and T~ has a fixed point q∈FT~ 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{an}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
(14)an+1≤(1-μn)an+μnηn,
where μn∈(0,1), for all n∈ℕ, ∑n=0∞μn=∞ and ηn≥0, for all n∈ℕ. Then the following holds:
(15)0≤limn→∞supan≤limn→∞supηn.


## 3. Main Results

For simplicity we use the following notation through this section.For any iterative process,{xn}n=0∞ and {un}n=0∞ denote iterative sequences associated to T and T~, respectively.Theorem 5.
LetT:E→E be a map satisfying (12) with FT≠∅, and let T~ be an approximate operator of T as in Definition 3. Let {xn}n=0∞, {un}n=0∞ be two iterative sequences defined by the multistep iteration (5) and with real sequences {βni}n=0∞⊂[0,1), i=1,k-1¯, and {αn}n=0∞⊂[0,1) satisfying ∑‍αn=∞. If p=Tp and q=T~q, then one has
(16)∥p-q∥≤kɛ1-δ.Proof.
For a givenx0∈E and u0∈E we consider the following multistep iteration for T and T~:
(17)x0∈E,xn+1=(1-αn)xn+αnTyn1,n∈ℕ,yni=(1-βni)xn+βniTyni+1,i=1,k-2¯,ynk-1=(1-βnk-1)xn+βnk-1Txn,k≥2,u0∈E,un+1=(1-αn)un+αnT~vn1,n∈ℕ,vni=(1-βni)un+βniT~vni+1,i=1,k-2¯,vnk-1=(1-βnk-1)un+βnk-1T~un,k≥2.
Then from (17), we get
(18)xn+1-un+1=(1-αn)(xn-un)+αn(Tyn1-T~vn1).
Thus, we have the following estimates by using (18) and (12):
(19)∥xn+1-un+1∥=∥(1-αn)(xn-un)+αn(Tyn1-T~vn1)∥≤(1-αn)∥xn-un∥+αn∥Tyn1-T~vn1∥=(1-αn)∥xn-un∥+αn∥Tyn1-Tvn1+Tvn1-T~vn1∥≤(1-αn)∥xn-un∥+αn∥Tyn1-Tvn1∥+αn∥Tvn1-T~vn1∥≤(1-αn)∥xn-un∥+αnδ∥yn1-vn1∥+αnφ(∥yn1-Tyn1∥)+αnɛ,(20)∥yn1-vn1∥=∥(1-βn1)(xn-un)+βn1(Tyn2-T~vn2)∥≤(1-βn1)∥xn-un∥+βn1∥Tyn2-T~vn2∥≤(1-βn1)∥xn-un∥+βn1∥Tyn2-Tvn2∥+βn1∥Tvn2-T~vn2∥≤(1-βn1)∥xn-un∥+βn1δ∥yn2-vn2∥+βn1φ(∥yn2-Tyn2∥)+βn1ɛ,(21)∥yn2-vn2∥=∥(1-βn2)(xn-un)+βn2(Tyn3-T~vn3)∥≤(1-βn2)∥xn-un∥+βn2∥Tyn3-T~vn3∥≤(1-βn2)∥xn-un∥+βn2∥Tyn3-Tvn3∥+βn2∥Tvn3-T~vn3∥≤(1-βn2)∥xn-un∥+βn2δ∥yn3-vn3∥+βn2φ(∥yn3-Tyn3∥)+βn2ɛ,(22)∥yn3-vn3∥=∥(1-βn3)(xn-un)+βn3(Tyn4-T~vn4)∥≤(1-βn3)∥xn-un∥+βn3∥Tyn4-T~vn4∥≤(1-βn3)∥xn-un∥+βn3∥Tyn4-Tvn4∥+βn3∥Tvn4-T~vn4∥≤(1-βn3)∥xn-un∥+βn3δ∥yn4-vn4∥+βn3φ(∥yn4-Tyn4∥)+βn3ɛ.
Combining (19), (20), (21), and (22) we obtain
(23)∥xn+1-un+1∥≤[(1-αn)+αnδ(1-βn1)+αnδ2βn1(1-βn2)+αnδ3βn1βn2(1-βn3)]∥xn-un∥+αnδ4βn1βn2βn3∥yn4-vn4∥+αnδ3βn1βn2βn3φ(∥yn4-Tyn4∥)+αnδ3βn1βn2βn3ɛ+αnδ2βn1βn2φ(∥yn3-Tyn3∥)+αnδ2βn1βn2ɛ+αnδβn1φ(∥yn2-Tyn2∥)+αnδβn1ɛ+αnφ(∥yn1-Tyn1∥)+αnɛ.
Thus, inductively, we get
(24)∥xn+1-un+1∥≤[(1-αn)+δαn(1-βn1)+δ2αnβn1(1-βn2)+δ3αnβn1βn2(1-βn3)+δ4αnβn1βn2βn3(1-βn4)+⋯+δk-2αnβn1βn2⋯βnk-3(1-βnk-2)]∥xn-un∥+δk-1αnβn1βn2⋯βnk-2∥ynk-1-vnk-1∥+δk-2αnβn1βn2⋯βnk-2φ(∥ynk-1-Tynk-1∥)+⋯+δ3αnβn1βn2βn3φ(∥yn4-Tyn4∥)+δ2αnβn1βn2φ(∥yn3-Tyn3∥)+δαnβn1φ(∥yn2-Tyn2∥)+αnφ(∥yn1-Tyn1∥)+δk-2αnβn1βn2⋯βnk-2ɛ+⋯+δ2αnβn1βn2ɛ+δαnβn1ɛ+αnɛ.
Using now (17) and (12), we get
(25)∥ynk-1-vnk-1∥=∥(1-βnk-1)(xn-un)+βnk-1(Txn-T~un)∥≤(1-βnk-1)∥xn-un∥+βnk-1∥Txn-T~un∥≤(1-βnk-1)∥xn-un∥+βnk-1∥Txn-Tun∥+βnk-1∥Tun-T~un∥≤(1-βnk-1)∥xn-un∥+βnk-1δ∥xn-un∥+βnk-1φ(∥xn-Txn∥)+βnk-1ɛ.
Substituting (25) in (24) we have
(26)∥xn+1-un+1∥≤[(1-αn)+δαn(1-βn1)+δ2αnβn1(1-βn2)+δ3αnβn1βn2(1-βn3)+δ4αnβn1βn2βn3(1-βn4)+⋯+δk-2αnβn1βn2⋯βnk-3(1-βnk-2)+δk-1αnβn1βn2⋯βnk-2(1-βnk-1)+δkαnβn1βn2⋯βnk-2βnk-1]∥xn-un∥+δk-1αnβn1βn2⋯βnk-2βnk-1φ(∥xn-Txn∥)+δk-2αnβn1βn2⋯βnk-2φ(∥ynk-1-Tynk-1∥)+⋯+δ3αnβn1βn2βn3φ(∥yn4-Tyn4∥)+δ2αnβn1βn2φ(∥yn3-Tyn3∥)+δαnβn1φ(∥yn2-Tyn2∥)+αnφ(∥yn1-Tyn1∥)+δk-1αnβn1βn2⋯βnk-2βnk-1ɛ+δk-2αnβn1βn2⋯βnk-2ɛ+⋯+δ2αnβn1βn2ɛ+δαnβn1ɛ+αnɛ.
If this inequality is rearranged  using {βni}n=0∞⊂[0,1), δi+1<δi for each i=1,k-1¯, then we get the following inequality as follows:
(27)∥xn+1-un+1∥≤[1-αn(1-δ)]∥xn-un∥+αn(1-δ)φ(∥xn-Txn∥)1-δ+αn(1-δ)×{φ(∥ynk-1-Tynk-1∥)+⋯+φ(∥yn1-Tyn1∥)+kɛ}1-δ.
Denote
(28)an∶=∥xn-un∥,μn∶=αn(1-δ)∈(0.1),ηn∶=φ(∥xn-Txn∥)1-δ+φ(∥ynk-1-Tynk-1∥)+⋯+φ(∥yn1-Tyn1∥)+kɛ1-δ.
From [26, Theorem 3.1] and [28, Corollary 2] we have limn→∞∥xn-p∥=0. Since T satisfies condition (12), and p∈FT, that is, Tp=p, it follows from (12) that
(29)0≤∥xn-Txn∥≤∥xn-p∥+∥Tp-Txn∥≤∥xn-p∥+δ∥p-xn∥+φ(∥p-Tp∥)=(1+δ)∥xn-p∥⟶0asn⟶∞.
Considering βni∈[0,1), for all n∈ℕ, i=1,k-1¯, and using (12) and (5) we have
(30)0≤∥yn1-Tyn1∥=∥yn1-p+p-Tyn1∥≤∥yn1-p∥+∥Tp-Tyn1∥≤∥yn1-p∥+δ∥p-yn1∥+φ(∥p-Tp∥)=(1+δ)∥yn1-p∥=(1+δ)∥(1-βn1)yn2+βn1Tyn2-p(1-βn1+βn1)∥≤(1+δ){(1-βn1)∥yn2-p∥+βn1∥Tyn2-Tp∥}≤(1+δ){(1-βn1)∥yn2-p∥+βn1δ∥yn2-p∥}=(1+δ)[1-βn1(1-δ)]∥yn2-p∥=(1+δ)[1-βn1(1-δ)]×∥(1-βn2)yn3+βn2Tyn3-p(1-βn2+βn2)∥≤(1+δ)[1-βn1(1-δ)]×{(1-βn2)∥yn3-p∥+βn2∥Tyn3-Tp∥}≤(1+δ)[1-βn1(1-δ)]×[1-βn2(1-δ)]∥yn3-p∥⋮≤(1+δ)[1-βn1(1-δ)]⋯[1-βnk-2(1-δ)]∥ynk-1-p∥≤(1+δ)[1-βn1(1-δ)]⋯[1-βnk-1(1-δ)]∥xn-p∥≤(1+δ)∥xn-p∥⟶0asn⟶∞.
It is easy to see from (30) that this result is also valid for ∥Tyn2-yn2∥,…,∥Tynk-1-ynk-1∥.
Making use of the fact thatφ is a continuous map we have
(31)limn→∞φ(∥xn-Txn∥)=limn→∞φ(∥yn1-Tyn1∥)=⋯=limn→∞φ(∥ynk-1-Tynk-1∥)=0.
Hence an application of Lemma 4 to (27) leads to
(32)∥p-q∥≤kɛ1-δ.Now we prove result on data dependence for the CR iterative procedure.Theorem 6.
LetT:E→E be a map satisfying (12) with FT≠∅, and let T~ be an approximate operator of T as in Definition 3. Let {xn}n=0∞, {un}n=0∞ be two iterative sequences defined by the CR iteration (6) and with real sequences {βn}n=0∞, {μn}n=0∞,{αn}n=0∞⊂[0,1] satisfying (i)1/2≤αn, for all n∈ℕ, and (ii)∑n=0∞αn=∞. If p=Tp and q=T~q, then one has
(33)∥p-q∥≤5ɛ1-δ.Proof.
For a givenx0∈E and u0∈E we consider the following iteration for T and T~:
(34)xn+1=(1-αn)yn+αnTyn,yn=(1-βn)Txn+βnTzn,zn=(1-μn)xn+μnTxn,un+1=(1-αn)vn+αnT~vn,vn=(1-βn)T~un+βnT~wn,wn=(1-μn)un+μnT~un.
Then from (34) we have
(35)xn+1-un+1=(1-αn)(yn-vn)+αn(Tyn-T~vn),yn-vn=(1-βn)(Txn-T~un)+βn(Tzn-T~wn),zn-wn=(1-μn)(xn-un)+μn(Txn-T~un).
Thus, by considering (35), it follows from (6) and (12) that
(36)∥xn+1-un+1∥=∥(1-αn)(yn-vn)+αn(Tyn-T~vn)∥≤(1-αn)∥yn-vn∥+αn∥Tyn-T~vn∥=(1-αn)∥yn-vn∥+αn∥Tyn-Tvn+Tvn-T~vn∥≤(1-αn)∥yn-vn∥+αn∥Tyn-Tvn∥+αn∥Tvn-T~vn∥≤(1-αn)∥yn-vn∥+αnδ∥yn-vn∥+αnφ(∥yn-Tyn∥)+αnɛ=[1-αn(1-δ)]∥yn-vn∥+αnφ(∥yn-Tyn∥)+αnɛ,(37)∥yn-vn∥=∥(1-βn)(Txn-T~un)+βn(Tzn-T~wn)∥≤(1-βn)∥Txn-T~un∥+βn∥Tzn-T~wn∥=(1-βn)∥Txn-Tun+Tun-T~un∥+βn∥Tzn-Twn+Twn-T~wn∥≤(1-βn){∥Txn-Tun∥+∥Tun-T~un∥}+βn{∥Tzn-Twn∥+∥Twn-T~wn∥}≤(1-βn)δ∥xn-un∥+βnδ∥zn-wn∥+(1-βn)φ(∥xn-Txn∥)+(1-βn)ɛ+βnφ(∥Tzn-Twn∥)+βnɛ,(38)∥zn-wn∥=∥(1-μn)(xn-un)+μn(Txn-T~un)∥≤(1-μn)∥xn-un∥+μn∥Txn-T~un∥=(1-μn)∥xn-un∥+μn∥Txn-Tun+Tun-T~un∥≤(1-μn)∥xn-un∥+μn{∥Txn-Tun∥+∥Tun-T~un∥}≤(1-μn)∥xn-un∥+μn{δ∥xn-un∥+φ(∥xn-Txn∥)+ɛ}=[1-μn(1-δ)]∥xn-un∥+μnφ(∥xn-Txn∥)+μnɛ.
Combining (36), (37), and (38) we obtain
(39)∥xn+1-un+1∥≤[1-αn(1-δ)]×[(1-βn)δ+βnδ[1-μn(1-δ)]]∥xn-un∥+[1-αn(1-δ)]×{[1-βn(1-δμn)]φ(∥xn-Txn∥)+βnμnδɛ+(1-βn)ɛ+βnφ(∥zn-Tzn∥)+βnɛ}+αnφ(∥yn-Tyn∥)+αnɛ.
It may be noted that for {βn}n=0∞, {μn}n=0∞⊂[0,1), and δ∈[0,1) the following inequalities are always true:
(40)1-μn(1-δ)<1,βnμnδ<1,δμn<δ.
Using now the inequality δμn<δ we get
(41)1-βn(1-δμn)<1-βn(1-δ)<1.
Therefore an application of (40) and (41) to (39) gives us
(42)∥xn+1-un+1∥≤[1-αn(1-δ)]∥xn-un∥+[1-αn(1-δ)]×{φ(∥xn-Txn∥)+ɛ+ɛ+φ(∥zn-Tzn∥)}+αnφ(∥yn-Tyn∥)+αnɛ.
Now, by the condition (i) 1/2≤αn, for all n∈ℕ we have
(43)1-αn≤αn.
Utilizing (43) in (42) we obtain
(44)∥xn+1-un+1∥≤[1-αn(1-δ)]∥xn-un∥+αn(1-δ){2φ(∥xn-Txn∥)+φ(∥yn-Tyn∥)}1-δ+αn(1-δ){2φ(∥zn-Tzn∥)+5ɛ}1-δ.
Denote
(45)an∶=∥xn-un∥,μn∶=αn(1-δ)∈(0,1),ηn∶={2φ(∥xn-Txn∥)+φ(∥yn-Tyn∥)+2φ(∥zn-Tzn∥)+5ɛ}1-δ.
From Remark2, we have limn→∞∥xn-p∥=0. Since T satisfies condition (12), and p∈FT, that is, Tp=p, using similar arguments as in the proof of Theorem 5, we get
(46)limn→∞∥xn-Txn∥=limn→∞∥yn-Tyn∥=limn→∞∥zn-Tzn∥=0.
Making use of the fact that φ is a continuous map we have
(47)limn→∞φ(∥xn-Txn∥)=limn→∞φ(∥yn-Tyn∥)=limn→∞φ(∥zn-Tzn∥)=0.
Hence an application of Lemma 4 to (44) leads to
(48)∥p-q∥≤5ɛ1-δ.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.



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