# The Equivalence of Convergence Results between Mann and Multistep Iterations with Errors for Uniformly Continuous Generalized WeakΦ-Pseudocontractive Mappings in Banach Spaces

Authors: Guiwen Lv; Haiyun Zhou
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We prove the equivalence of the convergence of the Mann and multistep iterations with errors for uniformly continuous generalized weakΦ-pseduocontractive mappings in Banach spaces. We also obtain the convergence results of Mann and multistep iterations with errors. Our results extend and improve the corresponding results.


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## Body

## 1. Introduction

LetE be a real Banach space, E* be its dual space, and J:E→2E* be 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 any 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∥).Remark 1.2.
LetN(T)={x∈E:Tx=0}≠∅. If x,y∈E in the formulas of Definition 1.1 is replaced by x∈E, q∈N(T), then T is called strongly quasi-accretive, ϕ-strongly quasi-accretive, generalized Φ-quasi-accretive mapping, respectively.Closely related to the class of accretive-type mappings are those of pseudocontractive type mappings.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 any 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 Φ-strongly pseudocontractive, generalized Φ-pseudocontractive, if, for all x∈D(T), q∈F(T), the formula (2), (3) in the above Definition 1.3 hold.Definition 1.5.
A mappingT is said to be(1)
generalized weakΦ-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
(1.8)〈Tx-Ty,j(x-y)〉≥Φ(∥x-y∥)1+∥x-y∥2+Φ(∥x-y∥);(2)
generalized weakΦ-quasi-accretive if, for all 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
(1.9)〈Tx-q,j(x-q)〉≥Φ(∥x-q∥)1+∥x-q∥2+Φ(∥x-q∥);(3)
generalized weakΦ-pseudocontractive if, for any 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
(1.10)〈Tx-Ty,j(x-y)〉≤∥x-y∥2-Φ(∥x-y∥)1+∥x-y∥2+Φ(∥x-y∥);(4)
generalized weakΦ-hemicontractive if, for any x∈K, 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
(1.11)〈Tx-q,j(x-q)〉≤∥x-q∥2-Φ(∥x-q∥)1+∥x-q∥2+Φ(∥x-q∥).It is very well known that a mappingT is strongly pseudocontractive (hemicontractive), ϕ-strongly pseudocontractive (ϕ-strongly hemicontractive), generalized Φ-pseudocontractive (generalized Φ-hemicontractive), generalized weak Φ-pseudocontractive (generalized weak Φ-hemicontractive) if and only if (I-T) are strongly accretive (quasi-accretive), ϕ-strongly accretive (ϕ-strongly quasi-accretive), (I-T) is generalized Φ-accretive (generalized Φ-quasi-accretive), generalized weak Φ-accretive (weak 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 hemicontractive mappings with the nonempty fixed point set is a proper subclass of generalized Φ-hemicontractive mappings. Obviously, generalized Φ-hemicontractive mapping must be generalized weak Φ-hemicontractive, but, on the contrary, it is not true. We have the following example.Example 1.6.
LetE=(-∞,+∞) be real number space with usual norm and K=[0,+∞). T:K→E defined by
(1.12)Tx=x+x3+x2x-x1+xx+x2,∀x∈K.
Then T has a fixed point 0∈F(T). Φ:[0,+∞)→[0,+∞) defined by Φ(t)=t3/2 is a strictly increasing function with Φ(0)=0. For all x∈K and 0∈F(T), we have
(1.13)〈Tx-T0,j(x-0)〉=〈x+x3+x2x-x1+xx+x2-0,x-0〉〈Tx-T0,j(x-0)〉=x2+x4+x3x-xx1+xx+x2=x2-x3/21+x3/2+x2〈Tx-T0,j(x-0)〉=|x-0|2-Φ(x)1+Φ(x)+x2=|x-0|2-σ(x)〈Tx-T0,j(x-0)〉≥|x-0|2-Φ(x).
Then T is a generalized weak Φ-hemicontractive map, but it is not a generalized Φ-hemicontractive map; that is, the class of generalized weak Φ-hemicontractive maps properly contains the class of generalized Φ-hemicontractive maps. Hence the class of generalized weak Φ-hemicontractive mappings is the most general among those defined above.Definition 1.7.
The mappingT:E→E is called Lipschitz, if there exists a constant L>0 such that
(1.14)∥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. For example, the function f(x)=x,x∈[0,+∞) is uniformly continuous but it is not Lipschitz.Now let us consider the multi-step iteration with errors. LetK be a nonempty convex subset of E, and let {Ti}i=1M be a finite family of self-maps of K. For x0∈K, the sequence {xn} is generated as follows:
(1.15)xn+1=(1-αn-δn)xn+αnTnyn1+δnνn,yni=(1-βni-ηni)xn+βniTnyni+1+ηniωni,i=1,…,p-2,ynp-1=(1-βnp-1-ηnp-1)xn+βnp-1Tnxn+ηnp-1ωnp-1,p≥2,
where {νn},{ωni} are any bounded sequences in K and {αn},{δn},{βni},{ηni},(i=1,2,…,p-1) are sequences in [0,1] satisfying certain conditions.Ifp=2, (1.15) becomes the Ishikawa iteration sequence with errors {xn}n=0∞ defined iteratively by
(1.16)xn+1=(1-αn-δn)xn+αnTnyn1+δnνn,yn1=(1-βn-ηn)xn+βnTnxn+ηnωn,∀n≥0.Ifβn=ηn=0, for all n≥0, then from (1.16), we get the Mann iteration sequence with errors {un}n=0∞ defined by
(1.17)un+1=(1-αn-δn)un+αnTnun+δnμn,∀n≥0,
where {μn}⊂K is bounded.Recently, many authors have researched the iteration approximation of fixed points by Lipschitz pseudocontractive (accretive) type nonlinear mappings and have obtained some excellent results [3–12]. In this paper we prove the equivalence between the Mann and multi-step iterations with errors for uniformly continuous generalized weak Φ-pseduocontractive mappings in Banach spaces. Our results extend and improve the corresponding results [3–12].Lemma 1.8 (see [13]).
LetE be a real normed space. Then, for all x,y∈E, the following inequality holds:
(1.18)∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉,∀j(x+y)∈J(x+y).Lemma 1.9 (see [14]).
Let{ρn} be nonnegative sequence which satisfies the following inequality:
(1.19)ρn+1≤(1-λn)ρn+σn,n≥N,
where λn∈(0,1),limn→∞λn=0 and ∑n=0∞λn=∞,σn=o(λn). Then ρn→0 as n→∞.Lemma 1.10.
Let{θn},{cn},{en} and {tn} be four nonnegative real sequences satisfying the following conditions: (i) limn→∞tn=0; (ii) ∑n=0∞‍tn=∞; (iii) cn=o(tn),en=o(tn). Let Φ:[0,+∞)→[0,+∞) be a strictly increasing and continuous function with Φ(0)=0 such that
(1.20)θn+12≤(1+cn)θn2-tnΦ(θn+1)1+Φ(θn+1)+θn+12+en,n≥0.
If {θn} is bounded, then θn→0 as n→∞.Proof.
Sincelimn→∞tn=0,{θn} is bounded, we set R=max{supn≥0tn,supn≥0θn}, γ=liminfn→∞(Φ(θn+1)/(1+θn+12)[1+Φ(R)+R2]), then γ=0. Otherwise, we assume that γ>0, then there exists a constant δ>0 with δ=min{1,γ} and a natural number N1 such that
(1.21)Φ(θn+1)>(δ+δθn+12)[1+Φ(R)+R2]>δθn+12[1+Φ(R)+R2],
for n>N1.
Then, from (1.20), we get
(1.22)θn+12≤1+cn1+δtnθn2+en.
Since cn=o(tn), there exists a nature number N2>N1, such that cn<(δ/2)tn,n>N2. Hence (1+cn)/(1+δtn)<1-(δ/2)tn and (1.22) becomes
(1.23)θn+12≤(1-δ2tn)θn2+en.
By Lemma 1.9, we obtain that θn→0 as n→∞. Since Φ is strictly increasing and continuous with Φ(0)=0. Hence γ=0, which is contradicting with the assumption γ>0. Then γ=0, there exists a subsequence {θnj} of {θn} such that θnj→0 as j→∞. Let 0<ε<1 be any given. Since cn=o(tn), en=o(tn), then there exists a natural number N3>N2, such that
(1.24)θnj<ε,cnj<Φ(ε)2M2(1+R2+Φ(R))tnj,enj<Φ(ε)2(1+R2+Φ(R))tnj
for all j>N3. Next, we will show that θnj+m<ε for all m=1,2,3,…. First, we want to prove that θnj+1<ε. Suppose that it is not the case, then θnj+1≥ε. Since Φ is strictly increasing,
(1.25)Φ(θnj+1)≥Φ(ε).
From (1.24) and (1.25), we obtain that
(1.26)θnj+12≤(1+cnj)θnj2-2tnjΦ(ε)1+R2+Φ(R)+enjθnj+12<θnj2-Φ(ε)(1+R2+Φ(R))tnj<θnj2<ε2.
That is θnj+1<ε, which is a contradiction. Hence θnj+1<ε. Now we assume that θnj+m<ε holds. Using the similar way, it follows that θnj+m+1<ε. Therefore, this shows that θn→0 as n→∞.


## 2. Main Results

Theorem 2.1.
LetK be a nonempty closed convex subset of a Banach space E. Suppose that Tn=Tn(modM), and Ti:K→K,i∈I={1,2,…,M} are M uniformly continuous generalized weak Φ-hemicontractive mappings with F=⋂i=1MF(Ti)≠∅. Let {un} be a sequence in K defined iteratively from some u0∈K by (1.17), where {μn} is an arbitrary bounded sequence in K and {αn},{δn} are two sequences in [0,1] satisfying the following conditions: (i) αn+δn≤1, (ii) ∑n=0∞αn=∞, (iii) limn→∞αn=0, (iv) δn=o(αn). Then the iteration sequence {un} converges strongly to the unique fixed point of T.Proof.
SinceF=⋂i=1MF(Ti)≠∅, set q∈F. Since the mapping Tn are generalized weak Φ-hemicontractive mappings, there exist strictly increasing functions Φl:[0,+∞)→[0,+∞) with Φl(0)=0 and j(x-y)∈J(x-y) such that
(2.1)〈Tlx-Tly,j(x-y)〉≤∥x-y∥2-Φl(∥x-y∥)1+∥x-y∥2+Φl(∥x-y∥),∀x,y∈K,l∈I.
Firstly, we claim that there existsu0∈K with u0≠Tu0 such that t0=∥u0-Tu0∥·∥u0-q∥·[1+∥u0-q∥2+Φ1(∥u0-q∥)]∈R(Φ1). In fact, if u0=Tu0, then we have done. Otherwise, there exists the smallest positive integer n0∈N such that un0≠Tun0. We denote un0=u0, then we will obtain that t0∈R(Φ1). Indeed, if R(Φ1)=[0,+∞), then t0∈R(Φ). If R(Φ1)=[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∥·[1+∥wn-q∥2+Φ1(∥wn-q∥)]∈R(Φ1) for n≥n0, then we redefine u0=wn0, let ω0=Φ1-1(t0)>0.
Next we shall prove∥un-q∥≤ω0 for n≥0. 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=Φl(ω0)/12ω0[1+Φl((3/2)ω0)+((3/2)ω0)2], there exists δ>0 such that ∥Tnx-Tny∥<ϵ0, whenever ∥x-y∥<δ; and Tn are bounded operators, set M=sup{∥Tnx∥:∥x-q∥≤ω0}+supn∥ωn∥. Since limn→∞αn=0, δn=o(αn), without loss of generality, let
(2.2)αn,δnαn<min{14,ω04M,δ4(M+ω0),Φl(ω0)4ω0[1+Φl((3/2)ω0)+((3/2)ω0)2],Φl(ω0)12[1+Φl((3/2)ω0)+((3/2)ω0)2]Mω0},n≥0.
From (1.17), we have
(2.3)∥un+1-q∥=∥(1-αn-δn)(un-q)+αn(Tnun-q)+δn(ωn-q)∥∥un+1-q∥≤∥un-q∥+αn∥Tnun-q∥+δ∥ωn-q∥∥un+1-q∥≤ω0+αn∥Tnun-q∥+δn∥ωn-q∥∥un+1-q∥≤ω0+M(αn+δn)≤ω0+2Mαn≤32ω0,(2.4)∥un+1-un∥=∥αnTnun+δnωn-(αn+δn)un∥∥un+1-un∥≤αn∥Tnun-q∥+δn∥ωn-q∥+(αn+δn)∥un-q∥∥un+1-un∥≤(αn+δn)(M+ω0)<δ.
SinceTn are uniformly continuous mappings, so ∥Tnun+1-Tnun∥<ε0.
Applying Lemma1.8, the recursion (1.17), and the above inequalities, we obtain
(2.5)∥un+1-q∥2=∥(1-αn-δn)(un-q)+αn(Tnun-q)+δn(ωn-q)∥2≤(1-αn-δn)2∥un-q∥2+2αn〈Tnun-q,j(un+1-q)〉+2δn∥ωn-q∥·∥un+1-q∥≤(1-αn)2∥un-q∥2+2αn〈Tnun+1-Tq,j(un+1-q)〉+2αn∥Tnun-Tnun+1∥·∥un+1-q∥+2δn∥ωn-q∥·∥un+1-q∥≤(1-αn)2∥un-q∥2+2αn[∥un+1-q∥2-Φl(∥un+1-q∥)1+Φl(∥un+1-q∥)+∥un+1-q∥2]+2αn∥Tnun-Tnun+1∥·∥un+1-q∥+2δn∥ωn-q∥·∥un+1-q∥.
Inequality (2.5) implies
(2.6)∥un+1-q∥2≤∥un-q∥2-2αnΦl(∥un+1-q∥)1+Φl(∥un+1-q∥)+∥un+1-q∥2+αn21-2αn∥un-q∥2+2αn1-2αn∥Tnun-Tnun+1∥·∥un+1-q∥+2δn1-2αn∥ωn-q∥·∥un+1-q∥≤ω02-2αnΦl(ω0)1+Φl((3/2)ω0)+((3/2)ω0)2+2αnΦl(ω0)4[1+Φl((3/2)ω0)+((3/2)ω0)2]ω02·ω02+4αnΦl(ω0)12[1+Φl((3/2)ω0)+((3/2)ω0)2]ω0·3ω02+4αnΦl(ω0)12[1+Φl((3/2)ω0)+((3/2)ω0)2]Mω0·3Mω02<ω02,
which is a contradiction with the assumption ∥un+1-q∥>ω0. Then ∥un+1-q∥≤ω0; that is, the sequence {un} is bounded. Let N=supn∥un-q∥. From (2.4), we have
(2.7)∥un+1-un∥≤(αn+δn)(M+ω0)→0,n→∞,
that is, limn→∞∥un+1-un∥=0. Since T is on uniformly continuous, so
(2.8)limn→∞∥Tnun+1-Tnun∥=0.
Again using (2.5), we have
(2.9)∥un+1-q∥2≤∥un-q∥2-2αnΦl(∥un+1-q∥)1+Φl(∥un+1-q∥)+∥un+1-q∥2]+An,
where
(2.10)An=αn2N2+2αnN∥Tnun-Tnun+1∥+2δnMN.
By (2.8), the conditions (iii) and (iv), we get An=o(αn). So applying Lemma 1.10 on (2.9), we obtain limn→∞∥un-q∥=0.Theorem 2.2.
LetE be a Banach space and K be a nonempty closed convex subset of E, Tn are as in Theorem 2.1. For x0,u0∈K, the sequence iterations {xn},{un} are defined by (1.15) and (1.17), respectively. {αn},{δn},{βni},{ηni},i=1,2,…,p-1 are sequences in [0,1] satisfying the following conditions:(i)
0≤αn+δn≤1,0≤βni+ηni≤1,1≤i≤p-1;(ii)
∑n=0∞αn=∞;(iii)
limn→∞αn=0;(iv)
limn→∞βni=limn→∞ηni=0,i=1,…,p-1;(v)
δn=o(αn).
Then the following two assertions are equivalent:(I)
the iteration sequence{xn} strongly converges to the common point of F(Ti),i∈I;(II)
the sequence iteration{un} strongly converges to the common point of F(Ti),i∈I.Proof.
SinceF=⋂i=1MF(Ti)≠∅, set q∈F. If the iteration sequence {xn} strongly converges to q, then setting p=2,βn=δn=0, we obtain the convergence of the iteration sequence {un}. Conversely, we only prove that (II)⇒(I). The proof is divided into two parts.
Step 1. We show that {xn-un} is bounded.
By the proof method of Theorem2.1, there exists x0∈K with x0≠T1x0 such that r0=∥x0-T1x0∥·∥x0-q∥·[1+∥x0-q∥2+Φ1(∥x0-q∥)]∈R(Φ). Setting a0=Φ1-1(r0), we have ∥x0-q∥≤a0. Set B1={∥x-q∥≤a0:x∈K}, B2={∥x-q∥≤2a0:x∈K}. Since Tl are bounded mappings and {ωni}(i=1,…,p-1),{νn} are some bounded sequences in K, we can set M=max{supx∈B2∥Tnx-q∥;supn∈N∥ωni-q∥;supn∈N∥νn-q∥}. Since Ti are uniformly continuous mappings, given ϵ0=Φl(a0)/4a0[1+(5a0/4)2+Φl(5a0/4)],∃δ>0, such that ∥Tx-Ty∥<ϵ0 whenever ∥x-y∥<δ, for all x,y∈B2. Now, we define τ0=min{1/2,a0/8M,a0/8(M+a0),δ/8(M+a0),Φl(a0)/5a02[1+(5a0/4)2+Φl(5a0/4)],Φl(a0)/5a0M[1+(5a0/4)2+Φl(5a0/4)]}. Since the control conditions (iii)-(iv), without loss of generality, we let 0<αn,δn/αn,βni,ηni<τ0,n≥0.
Now we claim that ifxn∈B1, then yni∈B2,1≤i≤p-1.
From (1.15), we obtain that
(2.11)∥ynp-1-q∥≤(1-βnp-1-ηnp-1)∥xn-q∥+βnp-1∥Tnxn-q∥+ηnp-1∥ωnp-1-q∥∥ynp-2-q∥≤∥xn-q∥+(βnp-1+ηnp-1)M∥ynp-2-q∥≤∥xn-q∥+2τ0M≤2a0,∥ynp-2-q∥≤(1-βnp-2-ηnp-2)∥xn-q∥+βnp-2∥Tnynp-1-q∥+ηnp-2∥ωnp-2-q∥∥ynp-2-q∥≤∥xn-q∥+(βnp-2+ηnp-2)M∥ynp-2-q∥≤∥xn-q∥+2τ0M≤2a0,we also obtain that(2.12)∥yn1-q∥≤2a0.
Now we suppose that∥xn-q∥≤a0 holds. We will prove that ∥xn+1-q∥≤a0. If it is not the case, we assume that ∥xn+1-q∥>a0. From (1.15), we obtain that
(2.13)∥xn+1-q∥=∥(1-αn-δn)(xn-q)+αn(Tnyn1-q)+δn(νn-q)∥∥xn+1-q∥≤∥xn-q∥+αn∥Tnyn1-q∥+δn∥νn-q∥∥xn+1-q∥≤∥xn-q∥+(αn+δn)M∥xn+1-q∥≤∥xn-q∥+2τ0M≤∥xn-q∥+14a0≤54a0.
Consequently, by (2.11) and (2.12), we obtain
(2.14)∥xn+1-yn1∥=∥[(βn1-αn)+(ηn1-δn)](xn-q)+αn(Tnyn1-q)-βn1(Tnnyn2-q)+δn(νn-q)-ηn1(ωn1-q)∥∥xn+1-yn1∥≤(βn1+αn+ηn1+δn)∥xn-q∥+αn∥Tnyn1-q∥+βn1∥Tnyn2-q∥+δn∥νn-q∥+ηn1∥ωn1-q∥∥xn+1-yn1∥≤4τ0(μ0+M)≤δ.
Since Tn are uniformly continuous mappings, we get
(2.15)∥Tnxn+1-Tnyn1∥≤ϵ0.
Using (2.1), Lemma 1.8, and the recursion formula (1.15), we have
(2.16)∥xn+1-q∥2=∥(1-αn-δn)(xn-q)+αn(Tnyn1-q)+δn(νn-q)∥2≤(1-αn)2∥xn-q∥2+2αn〈Tnyn1-q,j(xn+1-q)〉+2δn∥νn-q∥·∥xn+1-q∥≤(1-αn)2∥xn-q∥2+2αn〈Tnxn+1-q,j(xn+1-q)〉+2αn〈Tnyn1-Tnxn+1,j(xn+1-q)〉+2δn∥νn-q∥·∥xn+1-q∥≤(1-αn)2∥xn-q∥2+2αn[∥xn+1-q∥2-Φl(∥xn+1-q∥)1+Φl(∥xn+1-q∥)+∥xn+1-q∥2]+2αn∥Tnyn1-Tnxn+1∥·∥xn+1-q∥+2δnM·∥xn+1-q∥.
Which implies
(2.17)∥xn+1-q∥2≤∥xn-q∥2-2αn1-2αnΦl(∥xn+1-q∥)1+Φl(∥xn+1-q∥)+∥xn+1-q∥2+αn21-2αn∥xn-q∥2+2αn1-2αn∥Tnyn1-Tnxn+1∥·∥xn+1-q∥+2δn1-2αnM·∥xn+1-q∥≤a02-2αn1-2αnΦl(a0)1+(5a0/4)2+Φl(5a0/4)+2αn1-2αnΦl(a0)4a02[1+(5a0/4)2+Φl(5a0/4)]a02+2αn1-2αnΦl(a0)5a0[1+(5a0/4)2+Φl(5a0/4)]·5a04+2αn1-2αnΦl(a0)5a0M[1+(5a0/4)2+Φl(5a0/4)]·5a0M4<a02
which is a contradiction with the assumption ∥xn+1-q∥>μ0, then ∥xn+1-q∥≤μ0; that is, the sequence {xn-q} is bounded. Since un→q, as n→∞, so the sequence {xn-un} is bounded.
Step
2. We prove limn→∞∥xn-q∥=0.
Since{xn-un} is bounded, again applying (2.11) and (2.12), we get the boundedness of {yni-un},i=1,2,…,p-1. Since Tn=Tn(modM) are bounded mappings, set L=max{supn≥0∥xn-un∥,supn≥0∥Tnxn-un∥,supn≥0∥Tnyni-un∥,supn≥0∥μn-un∥,supn≥0∥νn-un∥,supn≥0∥ωn1-un∥}, (i=1,2,…,p-1). From (1.15) and (1.17), we obtain
(2.18)∥xn+1-un+1∥2=∥(1-αn-δn)(xn-un)+αn(Tnyn1-Tnun)+δn(νn-μn)∥2≤(1-αn)2∥xn-un∥2+2αn〈Tnyn1-Tnun,j(xn+1-un+1)〉+2δn∥νn-μn∥·∥xn+1-un+1∥≤(1-αn)2∥xn-un∥2+2αn〈Tnxn+1-Tnun+1,j(xn+1-un+1)〉+2αn〈Tnyn1-Tnxn+1+Tnun+1-Tnun,j(xn+1-un+1)〉+2δn∥νn-μn∥·∥xn+1-un+1∥≤(1+αn2)∥xn-q∥2-2αnΦl(∥xn+1-un+1∥)1+Φl(∥xn+1-un+1∥)+∥xn+1-un+1∥2+2αn∥Tnyn1-Tnxn+1∥·∥xn+1-un+1∥+2αn∥Tnun+1-Tnun∥·∥xn+1-un+1∥+2δnM·∥xn+1-un+1∥(2.19)∥xn+1-yn1∥≤(βn1+αn+ηn1+δn)∥xn-un∥+αn∥Tnnyn1-un∥∥xn+1-yn1∥+βn1∥Tnnyn2-un∥+δn∥νn-un∥+ηn1∥ωn1-un∥∥xn+1-yn1∥≤2(βn1+αn+ηn1+δn)L.
By the conditions (iii)–(v), we have(2.20)limn→∞∥xn+1-yn1∥=0.
Sincelimn→∞∥un-q∥=0, so
(2.21)∥un+1-un∥≤∥un-q∥+∥un+1-q∥.
That is:
(2.22)limn→∞∥un+1-un∥=0.
By the uniform continuity ofT, we obtain
(2.23)limn→∞∥Tnxn+1-Tnyn1∥=0,limn→∞∥Tnun+1-Tnun∥=0.
From (2.23) and the conditions (iii) and (v), (2.18) becomes
(2.24)∥xn+1-un+1∥2≤(1+αn2)∥xn-un∥2-2αnΦl(∥xn+1-un+1∥)1+Φl(∥xn+1-un+1∥)+∥xn+1-un+1∥2+o(αn).
By Lemma1.10, we get limn→∞∥xn-un∥=0. Since limn→∞∥un-q∥=0, and the inequality 0≤∥xn-q∥≤∥xn-un∥+∥un-q∥, so limn→∞∥xn-q∥=0.From Theorems2.1 and 2.2, we can obtain the following corollary.Corollary 2.3.
LetE be a Banach space and K be a nonempty closed convex subset of E, Tn are as in Theorem 2.1. For x0∈K, the sequence iterations {xn} is defined by (1.15). {αn},{δn},{βni},{ηni},(i=1,2,…,p-1) are sequences in [0,1] satisfying the following conditions:(i)
0≤αn+δn≤1,0≤βni+ηni≤1,1≤i≤p-1;(ii)
∑n=0∞αn=∞;(iii)
limn→∞αn=0;(iv)
limn→∞βni=limn→∞ηni=0,i=1,…,p-1;(v)
δn=o(αn).
Then the iteration sequence{xn} strongly converges to the common point of F(Ti), i∈I.Corollary 2.4.
LetTn=Sn(modM), Tl:E→E,l∈I={1,2,…,M} are M uniformly continuous generalized weak Φ-quasi-accretive mappings. Suppose N(F)=⋂i=1MN(Fi)≠∅, that is, there exists x*∈N(F). Let {αn},{δn},{βni},{ηni},(i=1,2,…,p-1) be sequences in [0,1] satisfying the following conditions:(i)
0≤αn+δn≤1,0≤βni+ηni≤1,1≤i≤p-1;(ii)
∑n=0∞αn=∞;(iii)
limn→∞αn=0;(iv)
limn→∞βni=limn→∞ηni=0,i=1,…,p-1;(v)
δn=o(αn).
Let the sequence{xn} in E be generated iteratively from some x0∈E by
(2.25)xn+1=(1-αn-δn)xn+αnSnyn1+δnνn,yni=(1-βni-ηni)xn+βniSnyni+1+ηniωni,i=1,…p-2,ynp-1=(1-βnp-1-ηnp-1)xn+βnp-1Snxn+ηnp-1ωnp-1,p≥2,
where Slx:=x-Tlx for all x∈E and {νn},{ωni} are any bounded sequences in K.
Then{xn} defined by (2.25) converges strongly to x*.Proof.
We simply observe thatSl:=I-Tl,l∈I are M uniformly continuous generalized weak Φ-hemicontractive mappings. The result follows from Corollary 2.3.



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*Source: 169410-2012-12-26.xml*