# The Equivalence of Convergence Results between Ishikawa and Mann Iterations with Errors for Uniformly Continuous Generalized Φ-Pseudocontractive Mappings in Normed Linear Spaces

Authors: Guiwen Lv; Zhiqun Xue
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We prove the equivalence of the convergence of the Mann and Ishikawa iterations with errors for uniformly continuous generalized Φ-pseudocontractive mappings in normed
linear spaces. Our results extend and improve the corresponding results of Xu, 1998, Kim et al., 2009, Ofoedu, 2006, Chidume and Zegeye, 2004, Chidume, 2001, Chang et al. 2002, Liu, 1995, Hirano and huang, 2003, C. E. Chidume and C. O. Chidume, 2005, and huang, 2007.


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## Body

## 1.  Introduction

LetE be a real normed linear space, E* its dual space, and J:E→2E* the normalized duality mapping defined by(1.1)J(x)={f∈E*:〈x,f〉=‖x‖⋅‖f‖=‖f‖2},
where 〈·,·〉 denotes the generalized duality pairing. The single-valued normalized duality mapping is denoted by j.Definition 1.1.
A mappingT:E→E is said to be(1)
strongly accretive if, for allx,y∈E, there exist a constant k∈(0,1) and j(x-y)∈J(x-y) such that
(1.2)〈Tx-Ty,j(x-y)〉≥k‖x-y‖2,(2)
ϕ-strongly accretive if there exist j(x-y)∈J(x-y) and a strictly increasing function ϕ:[0,+∞)→[0,+∞) with ϕ(0)=0 such that
(1.3)〈Tx-Ty,j(x-y)〉≥ϕ(‖x-y‖)‖x-y‖,∀x,y∈E,(3)
generalizedΦ-accretive if, for all x,y∈E, there exist j(x-y)∈J(x-y) and a strictly increasing function Φ:[0,+∞)→[0,+∞) with Φ(0)=0 such that
(1.4)〈Tx-Ty,j(x-y)〉≥Φ(‖x-y‖).Definition 1.2.
LetN(T)={x∈E:Tx=0}≠∅. The mapping T is called strongly quasi-accretive if, for all x∈E,q∈N(T), there exist a constant k∈(0,1) and j(x-y)∈J(x-y) such that 〈Tx-Tq,j(x-q)〉≥k∥x-q∥2;T is called ϕ-strongly quasi-accretive if, for all x∈E,q∈N(T), there exists a function ϕ such that 〈Tx-Tq,j(x-q)〉≥ϕ(∥x-q∥)∥x-q∥, where ϕ is as in Definition 1.1. Finally, T is called generalized Φ-quasi-accretive if, for each x∈E,q∈N(T), there exist j(x-q)∈J(x-q) and a strictly increasing function Φ:[0,+∞)→[0,+∞) with Φ(0)=0 such that 〈Tx-Tq,j(x-q)〉≥Φ(∥x-q∥).Closely related to the class of accretive-type mappings are those of pseudocontractive types.Definition 1.3.
A mappingT with domain D(T) and range R(T) is said to be(1)
strongly pseudocontractive if there exist a constantk∈(0,1) and j(x-y)∈J(x-y) such that, for each x,y∈D(T),
(1.5)〈Tx-Ty,j(x-y)〉≤k‖x-y‖2,(2)
ϕ-strongly pseudocontractive if there exist j(x-y)∈J(x-y) and a strictly increasing function ϕ:[0,+∞)→[0,+∞) with ϕ(0)=0 such that
(1.6)〈Tx-Ty,j(x-y)〉≤‖x-y‖2-ϕ(‖x-y‖)‖x-y‖,∀x,y∈D(T),(3)
generalizedΦ-pseudocontractive if, for all x,y∈D(T), there exist j(x-y)∈J(x-y) and a strictly increasing function Φ:[0,+∞)→[0,+∞) with Φ(0)=0 such that
(1.7)〈Tx-Ty,j(x-y)〉≤‖x-y‖2-Φ(‖x-y‖).Definition 1.4.
LetF(T)={x∈E:Tx=x}≠∅. The mapping T is called generalized Φ-hemi-pseudocontractive if, for all x∈D(T),q∈F(T), there exist j(x-q)∈J(x-q) and a strictly increasing function Φ:[0,+∞)→[0,+∞) with Φ(0)=0 such that
(1.8)〈Tx-Tq,j(x-q)〉≤‖x-q‖2-Φ(‖x-q‖).Obviously, a mappingT is strongly pseudocontractive, ϕ-strongly pseudocontractive, generalized Φ-pseudocontractive, and generalized Φ-hemicontractive if and only if (I-T) is strongly accretive, ϕ-strongly accretive, generalized Φ-accretive, and generalized Φ-quasi-accretive, respectively.It is shown in [1] that the class of strongly pseudocontractive mappings is a proper subclass of ϕ-strongly pseudocontractive mappings. Furthermore, an example in [2] shows that the class of ϕ-strongly pseudocontractive mappings with the nonempty fixed point set is a proper subclass of Φ-hemicontractive mappings. Hence, the class of generalized Φ-hemicontractive mappings is the most general among those defined above.Definition 1.5.
The mappingT:E→E is called Lipschitz if exists a constant L>0 such that
(1.9)‖Tx-Ty‖≤L‖x-y‖,∀x,y∈E.
It is clear that if T is Lipschitz then it must be uniformly continuous. Otherwise, it is not true.The following iteration schemes were introduced by Xu [3] in 1998. Let K be a nonempty convex subset of E. For any given x0∈K, the sequence {xn}n=0∞ is defined by(1.10)xn+1=αnxn+βnTyn+γnδn,yn=αn′xn+βn′Txn+γn′ηn,∀n≥0,
is called the Ishikawa iteration sequence with errors, where {δn}n=0∞,{ηn}n=0∞ are arbitrary bounded sequences in K and {αn}n=0∞,{βn}n=0∞,{γn}n=0∞,{αn′}n=0∞,{βn′}n=0∞, and {γn′}n=0∞ are six real sequences in [0,1] such that αn+βn+γn=αn′+βn′+γn′=1 for all n≥0 and satisfy certain conditions.Ifβn′=γn′=0, for all n≥0, then, from (1.10), we get the Mann iteration sequence with errors {un}n=0∞ defined by(1.11)∀u0∈K,un+1=αnun+βnTun+γnεn,∀n≥0,
where {εn}n=0∞ is an arbitrary bounded sequence in K.Numerous convergence results have been proved through iterative methods of approximating fixed points of Lipschitz pseudocontractive- (accretive-) type nonlinear mappings [3–10]. Most of these results have been extended to uniformly continuous mappings by some authors. Recently, C. E. Chidume and C. O. Chidume in [11] gave the most general result for uniformly continuous generalized Φ-hemicontractive mappings in normed linear. Their results are as follows.Theorem 1.6 (see [11, Theorem 2.3]).
LetE be a real normed linear space, K nonempty subset of E, and T:E→K a uniformly continuous generalized Φ-hemicontractive mapping, that is, there exist x*∈F(T) and a strictly increasing function Φ:[0,+∞)→[0,+∞),Φ(0)=0, such that, for all x∈K, there exists j(x-x*)∈J(x-x*) such that
(1.12)〈Tx-x*,j(x-x*)〉≤‖x-x*‖2-Φ(‖x-x*‖).(a)
Ify*∈K is a fixed point of T, then y*=x* and so T has at most one fixed point in K.(b)
Suppose there existsx0∈K such that the sequence {xn} defined by
(1.13)xn+1=anxn+bnTxn+cnun,∀n≥0,
is contained in K, where {an},{bn}, and {cn} are real sequences satisfying the following conditions:
(i)
an+bn+cn=1,(ii)
∑n=0∞(bn+cn)=∞,(iii)
∑n=0∞(bn+cn)2<∞,(iv)
∑n=0∞cn<∞.Then, {xn} converges strongly to x*. In particular, if y* is a fixed point of T in K, then {xn} converges strongly to y*.Unfortunately, the control conditions (iii)∑n=0∞(bn+cn)2<∞ and (iv) ∑n=0∞cn<∞ cannot assure that the result in [11] holds. In the proof course of p552, let ϵ>0 be any given, we can choose (also in view of conditions (iii) and (iv)) an integer N1>0 such that for all n>N1 the following inequality M1cn<(Φ(ϵ)/4)αn holds, where αn=bn+cn. The inequality above M1cn<(Φ(ϵ)/4)αn implies that cn=o(αn). But conditions (iii) and (iv) of [11] can not assure that  cn=o(αn). On the one hand, let cn=1/n2,n=1,2,3,…;α1=0,α2=1/2,α3=0,α4=1/4,α5=0,α6=1/6,…; then ∑n=0∞cn<∞, but cn≠o(αn). On the other hand, set cn=1/n,αn=2/n; then cn=o(αn), but ∑n=0∞cn=∞.The purpose of this paper is that we obtain the convergence result of the Mann iteration with errors, and we also prove the equivalence of convergence between the Ishikawa iteration with errors defined by (1.10) and the Mann iteration with errors defined by (1.11). We also show that the Ishikawa iteration with errors defined by (1.10) converges to the unique fixed point of T. Our results extend and improve the corresponding results of [3–12]. For this, in the sequel, we will need the following lemmas.Lemma 1.7 (see [13]).
LetE be a real normed space. Then, for all x,y∈E, the following inequality holds:
(1.14)‖x+y‖2≤‖x‖2+2〈y,j(x+y)〉,∀j(x+y)∈J(x+y).Lemma 1.8 (see [14]).
LetΦ:[0,+∞)→[0,+∞) be a strictly increasing continuous function with Φ(0)=0 and  {θn},{σn}, and {λn} nonnegative three sequences that satisfy the following inequality:
(1.15)θn+12≤θn2-2λnΦ(θn+1)+σn,n≥N,
where λn∈(0,1),limn→∞λn=0, and ∑n=0∞λn=∞,σn=o(λn). Then θn→0 as n→∞.


## 2.  Main Results

Theorem 2.1.
LetK be a nonempty closed convex subset of a real normed linear space E. Suppose that  T:K→K is a uniformly continuous generalized Φ-hemicontractive mapping with F(T)≠∅. Let {un} be a sequence in K defined iteratively from some u0∈K by (1.11), where {εn} is an arbitrary bounded sequence in K and {αn},{βn}, and {γn} are three sequences in [0,1] satisfying the following conditions:(i)
αn+βn+γn=1,(ii)
∑n=0∞βn=∞,(iii)
limn→∞βn=0,(iv)
γn=o(βn).Then, the iteration sequence {un} converges strongly to the unique fixed point of T.Proof.
Letq∈F(T). The uniqueness of the fixed point of T comes from Definition 1.4.
First, we prove that there existsu0∈K with u0≠Tu0 such that t0=∥u0-Tu0∥·∥u0-q∥∈R(Φ). In fact, if u0=Tu0, then we are done. Otherwise, there exists the smallest positive integer n0∈N such that un0≠Tun0. We denote un0=u0, and then we obtain that t0=∥u0-Tu0∥·∥u0-q∥∈R(Φ). Indeed, if R(Φ)=[0,+∞), then t0∈R(Φ). If R(Φ)=[0,A) with 0<A<+∞, then, for q∈K, there exists a sequence {wn}⊂K such that wn→q as n→∞ with wn≠q, and we also obtain that the sequence {wn-Twn} is bounded. So there exists n0∈N such that ∥wn-Twn∥·∥wn-q∥∈R(Φ) for n≥n0, and then we redefine u0=wn0. Let ω0=Φ-1(∥(u0-q)-(Tu0-q)∥·∥u0-q∥)>0.
Next forn≥0 we will prove ∥un-q∥≤ω0 by induction. Clearly, ∥u0-q∥≤ω0 holds. Suppose that ∥un-q∥≤ω0, for some n; then we want to prove ∥un+1-q∥≤ω0. If it is not the case, then ∥un+1-q∥>ω0. Since T is a uniformly continuous mapping, setting ϵ0=Φ(ω0)/12ω0, there exists δ>0 such that ∥Tx-Ty∥<Φ(ω0)/12ω0 whenever ∥x-y∥<δ and T is a bounded operator. Set M=sup{∥Tx∥:∥x-q∥≤ω0}+supn∥εn∥. Since limn→∞βn=0,γn=o(βn), without loss of generality, let
(2.1)βn,γnβn<min⁡{ω04(M+‖q‖),δ2M+2ω0+2‖q‖,Φ(ω0)6ω0,Φ(ω0)6(ω0+M+‖q‖)2},n≥0.
From (1.10), we have
(2.2)‖un+1-q‖=‖αn(un-q)+βnTun+γnε-(βn+γn)q‖≤‖un-q‖+(βn‖Tun‖+γn‖εn‖+(βn+γn)‖q‖)≤ω0+βn(‖Tun‖+‖εn‖+2‖q‖)≤ω0+βn(2M+2‖q‖)≤32ω0,∥un+1-un∥=‖βnTun+γnεn-(βn+γn)un‖≤βn‖Tun‖+γn‖εn‖+(βn+γn)‖un-q‖+(βn+γn)‖q‖≤βn(‖Tun‖+‖εn‖+2‖un-q‖+2‖q‖)≤βn(2M+2ω0+2‖q‖).
Applying Lemma1.7, the recursion (1.11), and the inequalities above, we obtain
(2.3)‖un+1-q‖2=‖(un-q)+βnTun+γnεn-(βn+γn)un‖2≤‖un-q‖2+2〈βnTun+γnεn-(βn+γn)un,j(un+1-q)〉≤‖un-q‖2+2βn〈Tun+1-Tq,j(un+1-q)〉+2βn〈(Tun-Tun+1)+(un+1-un)-(un+1-q),j(un+1-q)〉+2γn(‖un-q‖+‖q‖+‖εn‖)‖un+1-q‖≤‖un-q‖2-2βnΦ(‖un+1-q‖)+2βn‖Tun-Tun+1‖⋅‖un+1-q‖+2βn‖un-un+1‖⋅‖un+1-q‖+2γn(ω0+‖q‖+M)‖un+1-q‖≤ω02-2βnΦ(ω0)+2βnΦ(ω0)12ω03ω02+2βnΦ(ω0)6ω03ω02+2βnΦ(ω0)6ω03ω02<ω02,
which is a contraction with the assumption ∥un+1-q∥>ω0. Then, ∥un+1-q∥≤ω0, that is, the sequence {un} is bounded. It leads to
(2.4)lim⁡n→∞‖un+1-un‖=0,lim⁡n→∞‖Tun+1-Tun‖=0.
Again using Lemma1.7, we have
(2.5)‖un+1-q‖2=‖(un-q)+βnTun+γnεn-(βn+γn)un‖2≤‖un-q‖2+2〈βnTun+γnεn-(βn+γn)un,j(un+1-q)〉≤‖un-q‖2+2βn〈Tun+1-Tq,j(xu+1-q)〉+2βn〈(Tun-Tun+1)+(un+1-un)-(un+1-q),j(un+1-q)〉+2γn(‖un-q‖+‖q‖+‖εn‖)‖un+1-q‖≤‖un-q‖2-2βnΦ(‖un+1-q‖)+2βn‖Tun-Tun+1‖⋅‖un+1-q‖+2βn‖un-un+1‖⋅‖un+1-q‖+2γn(ω0+∥q∥+M)‖un+1-q‖≤‖un-q‖2-2βnΦ(‖un+1-q‖)+An,
where
(2.6)An=2βn‖Tun-Tun+1‖⋅‖un+1-q‖+2βn‖un-un+1‖⋅‖un+1-q‖+2γn(ω0+‖q‖+M)‖un+1-q‖=o(βn).
Therefore, (2.5) becomes
(2.7)‖un+1-q‖2≤‖un-q‖2-2βnΦ(‖un+1-q‖)+o(βn).
From Lemma 1.8, we obtain limn→∞∥un-q∥=0.Theorem 2.2.
LetK be a nonempty closed convex subset of a real normed linear space E and  T:K→K a uniformly continuous generalized Φ-pseudocontractive mapping with F(T)≠∅. Let {un},{xn} be two sequences in K defined iteratively from some u0≠x0∈K by (1.11) and (1.10), where {δn},{ηn},{εn} are three arbitrary bounded sequences in K and {αn},{βn},{γn},{αn′},{βn′}, and {γn′} are six sequences in [0,1] satisfying the following conditions:(i)
αn+βn+γn=αn′+βn′+γn′=1,(ii)
∑n=0∞βn=∞,(iii)
limn→∞βn=limn→∞βn′=limn→∞γn′=0,(iv)
γn=o(βn).Then, the following two assertions are equivalent:(1)
Iteration (1.11) converges strongly to the unique fixed point of T,(2)
Iteration (1.10) converges strongly to the unique fixed point of T.Proof.
Letq∈F(T). The uniqueness of q comes from Definition 1.4. If the Ishikawa iteration with errors converges to q, then setting βn′=γn′=0, for all n≥0 in (1.10), we can get the convergence of the Mann iteration with errors. Conversely, we only prove that (1)⇒(2), that is, if the Mann iteration with errors converges to q, we want to prove the convergence of the Ishikawa iteration with errors.
First, we will prove that there existsx0∈K with x0≠Tx0 such that t0=∥x0-Tx0∥·∥x0-q∥∈R(Φ).
In fact, ifx0=Tx0, then we are done. Otherwise, there exists the smallest positive integer N0∈N such that xN0≠TxN0. We denote xN0=x0, and then we obtain that t0=∥x0-Tx0∥·∥x0-q∥∈R(Φ). Indeed if R(Φ)=[0,+∞), then t0∈R(Φ). If R(Φ)=[0,A) with 0<A<+∞, then for q∈K there exists a sequence {wn}⊂K such that wn→q as n→∞ with wn≠q, and we also obtain that the sequence {wn-Twn} is bounded. So there exists n0∈N such that ∥wn-Twn∥·∥wn-q∥∈R(Φ),n≥n0, and then we redefine x0=wn0. Let μ0=Φ-1(∥(x0-q)-(Tx0-q)∥·∥x0-q∥)>0.
Second, we will prove that the sequence{xn-q} is a bounded sequence.
Set(2.8)B1={‖x-q‖≤μ0:x∈K},B2={‖x-q‖≤2μ0:x∈K},M1=max⁡{sup⁡x∈B2‖Tx‖;sup⁡n∈N‖εn‖;sup⁡n∈N‖δn‖;sup⁡n∈N‖ηn‖;sup⁡n∈N‖un‖;sup⁡n∈N‖Tun‖}.
Since T is uniformly continuous, for ϵ=Φ(μ0)/5μ0, there exists δ>0 such that ∥Tx-Ty∥<ϵ whenever ∥x-y∥<δ. Now let k=min{1,μ0/4(M1+μ0+∥q∥),δ/2(4M1+μ0+∥q∥),Φ(μ0)/20μ0(M1+μ0+∥q∥)}. By the control conditions (iii) and (iv), without loss of generality, set |αn-αn′|,βn,γn,βn′,γn′<k, for alln≥0.
Observe that ifxn∈B1, we obtain yn∈B2. Indeed
(2.9)‖yn-q‖=‖(1-βn′-γn′)xn+βn′Txn+γn′ηn-q‖≤‖xn-q‖+βn′‖Txn‖+(βn′+γn′)‖xn-q‖+γn′‖ηn‖+(βn′+γn′)‖q‖≤μ0+k(2M1+2μ0+2‖q‖)≤2μ0.
Next, by induction, we provexn∈B1. Clearly, from (2.1), we obtain ∥x0-q∥≤μ0, that is, x0∈B1. Set ∥xn-q∥≤μ0, for some n; then we will prove that ∥xn+1-q∥≤μ0. If it is not the case, we assume that ∥xn+1-q∥>μ0. From (1.10), we obtain the following inequalities:
(2.10)‖xn+1-q‖=‖αnxn+βnTyn+γnδn-q‖≤‖(1-βn-γn)(xn-q)+βnTyn+γnδn+(βn+γn)q‖≤‖xn-q‖+(βn+γn)‖xn-q‖+βn‖Tyn‖+γn‖δn‖+(βn+γn)‖q‖≤μ0+k(2μ0+2M1+2‖q‖)≤54μ0,‖xn+1-xn‖=‖βnTyn+γnδn-(βn+γn)xn‖≤(βn+γn)‖xn-q‖+βn‖Tyn‖+γn‖δn‖+(βn+γn)‖q‖≤k(2μ0+2M1+2‖q‖)≤Φ(μ0)10μ0,‖xn+1-yn‖=‖(αn-αn′)xn+βnTyn-βn′Txn+γnδn-γn′ηn‖≤|αn-αn′|⋅‖xn-q‖+βn‖Tyn‖+βn′‖Txn‖+γn‖δn‖+γn′‖ηn‖+|αn-αn′|⋅‖q‖≤k(μ0+4M1+‖q‖)≤δ.
SinceT is a uniformly continuous mapping, then
(2.11)‖Txn+1-Tyn‖≤ϵ=Φ(μ0)5μ0.
Sinceγn=o(βn), we let
(2.12)γn<βn⋅18⋅Φ(μ0)⋅(45μ0)2.
From (2.9)–(2.12), we have
(2.13)‖xn+1-q‖2=‖(1-βn-γn)xn+βnTyn+γnδn-q‖2=‖(xn-q)+βnTyn+γnδn-(βn+γn)xn‖2≤‖xn-q‖2+2〈βnTyn+γnδn-(βn+γn)xn,j(xn+1-q)〉≤‖xn-q‖2+2βn〈Txn+1-Tq,j(xn+1-q)〉+2βn〈(Tyn-Txn+1),j(xn+1-q)〉+2(βn+γn)〈(xn+1-xn),j(xn+1-q)〉+2γn‖δn‖⋅‖xn+1-q‖-2(βn+γn)〈(xn+1-q),j(xn+1-q)〉≤‖xn-q‖2-2βnΦ(‖xn+1-q‖)+2βn‖Txn+1-Tyn‖⋅‖xn+1-q‖+2(βn+γn)‖xn+1-xn‖⋅‖xn+1-q‖+2γn‖δn‖‖xn+1-q‖+2γn‖xn+1-q‖2<μ02-2βnΦ(μ0)+2βnΦ(μ0)5μ0⋅5μ04+4βnΦ(μ0)10μ0⋅5μ04+2βnΦ(μ0)5μ0⋅5μ04+2βn⋅18⋅Φ(μ0)⋅(45μ0)2⋅(5μ04)2<μ02,
which is a contradiction with the assumption ∥xn+1-q∥>μ0. Hence, xn∈B1, that is, the sequence {xn} is bounded. From (2.9), the sequence {yn} is also bounded. So we obtain
(2.14)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞‖un+1-un‖=lim⁡n→∞‖xn+1-yn‖=0.
Since T is a uniformly continuous mapping, then
(2.15)lim⁡n→∞‖Tun+1-Tun‖=lim⁡n→∞‖Txn+1-Tyn‖=0.
Using the recursion formula (1.10), we compute as follows:
(2.16)‖xn+1-un+1‖2≤‖xn-un‖2-2βnΦ(‖xn+1-un+1‖)+2βn‖Txn+1-Tyn‖⋅‖xn+1-un+1‖+2βn‖Tun-Tun+1‖⋅‖xn+1-un+1‖+2(βn+γn)‖un-un+1‖⋅‖xn+1-un+1‖+2(βn+γn)‖xn+1-xn‖⋅‖xn+1-un+1‖+2γn(‖δn‖+‖εn‖)‖xn+1-un+1‖=‖xn-un‖2-2βnΦ(‖xn+1-un+1‖)+An,
where
(2.17)An=2βn‖Txn+1-Tyn‖⋅‖xn+1-un+1‖+2βn‖Tun-Tun+1‖⋅‖xn+1-un+1‖+2(βn+γn)‖un-un+1‖⋅‖xn+1-un+1‖+2(βn+γn)‖xn+1-xn‖⋅‖xn+1-un+1‖+2γn(‖δn‖+‖εn‖)‖xn+1-un+1‖.
Since the sequence{xn-un} is bounded, from (2.15), (2.16), and the control condition (iv), we get An=o(βn). Then, in (2.18), set θn=∥xn-un∥,λn=βn, and σn=An. From Lemma 1.8, we obtain limn→∞∥xn+1-un+1∥=0. Since limn→∞∥un-q∥=0, using inequality 0≤∥xn-q∥≤∥xn-un∥+∥un-q∥, then limn→∞∥xn-q∥=0, and we complete the proof of Theorem 2.1.From Theorems2.1 and 2.2, we obtain the following corollary.Corollary 2.3.
LetK be a nonempty convex subset of a real normed linear space E. T:K→K is a uniformly continuous generalized Φ-hemicontractive mapping with F(T)≠∅. Let {xn} be a sequence in K defined iteratively from some x0∈K by (1.10), where {δn},{ηn} are arbitrary bounded sequences in K and {αn},{βn},{γn},{αn′},{βn′}, and {γn′} are sequences in [0,1] satisfying the following conditions:(i)
αn+βn+γn=αn′+βn′+γn′=1,(ii)
∑n=0∞βn=∞,(iii)
limn→∞βn=limn→∞βn′=limn→∞γn′=0,(iv)
γn=o(βn).Then, the iteration sequence {xn} defined by (1.10) converges strongly to the unique fixed point of T.Corollary 2.4.
LetT:E→E be a uniformly continuous generalized Φ-accretive mapping with N(T)≠∅. For some u0,x0∈E, the iteration sequences {un},{xn} in E are  defined as follows:
(2.18)un+1=αnun+βnSun+γnεn,∀n≥0,(2.19)xn+1=αnxn+βnSyn+γnδn,yn=αn′xn+βn′Sxn+γn′ηn,∀n≥0,
where Sx:=x-Tx for all x∈E. {δn},{ηn}, and {εn} are arbitrary bounded sequences in K and {αn},{βn},{γn},{αn′},{βn′}, and {γn′} are sequences in [0,1] satisfying the following conditions:(i)
αn+βn+γn=αn′+βn′+γn′=1,(ii)
∑n=0∞βn=∞,(iii)
limn→∞βn=limn→∞βn′=limn→∞γn′=0,(iv)
γn=o(βn).Then, the following two assertions are equivalent:(1)
{xn} converges strongly to the unique fixed point of S,(2)
{un} converges strongly to the unique fixed point of S.Proof.
LetS=I-T, and observe that T is a uniformly continuous generalized Φ-accretive mapping if and only if S is a uniformly continuous generalized Φ-pseudocontractive mapping. The result follows from Theorem 2.2.Remark 2.5.
Our results improve and extend the corresponding results of [11] in the following sense.(i)
We point out the gaps of C. E. Chidume and C. O. Chidume [11] in their proof and provide a counterexample.(ii)
In the proof method, Theorem2.1 differs from Theorem  2.3 in [11].(iii)
The control conditions∑n=0∞(bn+cn)2<∞ and ∑n=0∞cn<∞ in Theorem  2.3 in [11] are replaced by the condition bn=o(cn). Under the new condition, we obtain the convergence theorem of the Mann iteration sequence with errors.(iv)
We also obtain the equivalence of convergence results between the  Ishikawa and Mann iterations with errors.(v)
The Mann iteration sequence with errors is extended to the Ishikawa iteration sequence with errors.



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*Source: 395098-2012-05-31.xml*