AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 169410 10.1155/2012/169410 169410 Research Article The Equivalence of Convergence Results between Mann and Multistep Iterations with Errors for Uniformly Continuous Generalized Weak Φ-Pseudocontractive Mappings in Banach Spaces Lv Guiwen 1 Zhou Haiyun 2 Yao Jen-Chih 1 Department of Mathematics and Physics Shijiazhuang Tiedao University Shijiazhuang 050043 China stdu.edu.cn 2 Department of Mathematics Shijiazhuang Mechanical Engineering College Shijiazhuang 050003 China 2012 26 12 2012 2012 24 11 2012 07 12 2012 2012 Copyright © 2012 Guiwen Lv and Haiyun Zhou. 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.

We prove the equivalence of the convergence of the Mann and multistep iterations with errors for uniformly continuous generalized weak Φ-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.

1. Introduction

Let E be a real Banach space, E* be its dual space, and J:E2E* be the normalized duality mapping defined by (1.1)J(x)={fE*:x,f=x·f=f2}, where ·,· denotes the generalized duality pairing. The single-valued normalized duality mapping is denoted by j.

Definition 1.1.

A mapping T:EE is said to be

strongly accretive if for all x,yE, there exist a constant k(0,1) and j(x-y)J(x-y) such that (1.2)Tx-Ty,j(x-y)kx-y2;

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,yE;

generalized Φ-accretive if, for any x,yE, 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.

Let N(T)={xE:Tx=0}. If x,yE in the formulas of Definition 1.1 is replaced by xE, qN(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 mapping T with domain D(T) and range R(T) is said to be

strongly pseudocontractive if there exist a constant k(0,1) and j(x-y)J(x-y) such that for each x,yD(T), (1.5)Tx-Ty,j(x-y)kx-y2;

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-y2-ϕ(x-y)x-y,x,yD(T);

generalized Φ-pseudocontractive if, for any x,yD(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-y2-Φ(x-y).

Definition 1.4.

Let F(T)={xE:Tx=x}. The mapping T is called Φ-strongly pseudocontractive, generalized Φ-pseudocontractive, if, for all xD(T), qF(T), the formula (2), (3) in the above Definition 1.3 hold.

Definition 1.5.

A mapping T is said to be

generalized weak Φ-accretive if, for all x,yE, 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-y2+Φ(x-y);

generalized weak Φ-quasi-accretive if, for all xE, qN(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-q2+Φ(x-q);

generalized weak Φ-pseudocontractive if, for any x,yD(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-y2-Φ(x-y)1+x-y2+Φ(x-y);

generalized weak Φ-hemicontractive if, for any xK, qF(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-q2-Φ(x-q)1+x-q2+Φ(x-q).

It is very well known that a mapping T 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  that the class of strongly pseudocontractive mappings is a proper subclass of ϕ-strongly pseudocontractive mappings. Furthermore, an example in  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.

Let E=(-,+) be real number space with usual norm and K=[0,+). T:KE defined by (1.12)Tx=x+x3+x2x-x1+xx+x2,xK. Then T has a fixed point 0F(T). Φ:[0,+)[0,+) defined by Φ(t)=t3/2 is a strictly increasing function with Φ(0)=0. For all xK and 0F(T), we have (1.13)Tx-T0,j(x-0)=x+x3+x2x-x1+xx+x2-0,x-0Tx-T0,j(x-0)=x2+x4+x3x-xx1+xx+x2=x2-x3/21+x3/2+x2Tx-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 mapping T:EE is called Lipschitz, if there exists a constant L>0 such that (1.14)Tx-TyLx-y,x,yE. 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. Let K be a nonempty convex subset of E, and let {Ti}i=1M be a finite family of self-maps of K. For x0K, 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,p2, 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.

If p=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,n0.

If βn=ηn=0, for all n0, 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,n0, 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 . 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 .

Lemma 1.8 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

Let E be a real normed space. Then, for all x,yE, the following inequality holds: (1.18)x+y2x2+2y,j(x+y),j(x+y)J(x+y).

Lemma 1.9 (see [<xref ref-type="bibr" rid="B14">14</xref>]).

Let {ρn} be nonnegative sequence which satisfies the following inequality: (1.19)ρn+1(1-λn)ρn+σn,nN, where λn(0,1),limnλn=0 and n=0λn=,σn=o(λn). Then ρn0 as n.

Lemma 1.10.

Let {θn},{cn},{en} and {tn} be four nonnegative real sequences satisfying the following conditions: (i) limntn=0; (ii) n=0tn=; (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,n0. If {θn} is bounded, then θn0 as n.

Proof.

Since limntn=0,{θn} is bounded, we set R=max{supn0tn,supn0θn}, 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+121+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 θn0 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 θnj0 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+1ε. 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 θn0 as n.

2. Main Results Theorem 2.1.

Let K be a nonempty closed convex subset of a Banach space E. Suppose that Tn=Tn(modM), and Ti:KK,iI={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 u0K 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+δn1, (ii) n=0αn=, (iii) limnαn=0, (iv) δn=o(αn). Then the iteration sequence {un} converges strongly to the unique fixed point of T.

Proof.

Since F=i=1MF(Ti), set qF. 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-y2-Φl(x-y)1+x-y2+Φl(x-y),x,yK,lI.

Firstly, we claim that there exists u0K with u0Tu0 such that t0=u0-Tu0·u0-q·[1+u0-q2+Φ1(u0-q)]R(Φ1). In fact, if u0=Tu0, then we have done. Otherwise, there exists the smallest positive integer n0N such that un0Tun0. We denote un0=u0, then we will obtain that t0R(Φ1). Indeed, if R(Φ1)=[0,+), then t0R(Φ). If R(Φ1)=[0,A] with 0<A<+, then for qK, there exists a sequence {wn}K such that wnq as n with wnq, and we also obtain that the sequence {wn-Twn} is bounded. So there exists n0N such that wn-Twn·wn-q·[1+wn-q2+Φ1(wn-q)]R(Φ1) for nn0, then we redefine u0=wn0, let ω0=Φ1-1(t0)>0.

Next we shall prove un-qω0 for n0. Clearly, u0-qω0 holds. Suppose that un-qω0, for some n, then we want to prove un+1-qω0. 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-qω0}+supnωn. Since limnαn=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},n0.

From (1.17), we have (2.3)un+1-q=(1-αn-δn)(un-q)+αn(Tnun-q)+δn(ωn-q)un+1-qun-q+αnTnun-q+δωn-qun+1-qω0+αnTnun-q+δnωn-qun+1-qω0+M(αn+δn)ω0+2Mαn32ω0,(2.4)un+1-un=αnTnun+δnωn-(αn+δn)unun+1-unαnTnun-q+δnωn-q+(αn+δn)un-qun+1-un(αn+δn)(M+ω0)<δ.

Since Tn are uniformly continuous mappings, so Tnun+1-Tnun<ε0.

Applying Lemma 1.8, the recursion (1.17), and the above inequalities, we obtain (2.5)un+1-q2=(1-αn-δn)(un-q)+αn(Tnun-q)+δn(ωn-q)2(1-αn-δn)2un-q2+2αnTnun-q,j(un+1-q)+2δnωn-q·un+1-q(1-αn)2un-q2+2αnTnun+1-Tq,j(un+1-q)+2αnTnun-Tnun+1·un+1-q+2δnωn-q·un+1-q(1-αn)2un-q2+2αn[un+1-q2-Φl(un+1-q)1+Φl(un+1-q)+un+1-q2]+2αnTnun-Tnun+1·un+1-q+2δnωn-q·un+1-q. Inequality (2.5) implies (2.6)un+1-q2un-q2-2αnΦl(un+1-q)1+Φl(un+1-q)+un+1-q2+αn21-2αnun-q2+2αn1-2αnTnun-Tnun+1·un+1-q+2δn1-2αnωn-q·un+1-qω02-2αnΦl(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-qω0; that is, the sequence {un} is bounded. Let N=supnun-q. From (2.4), we have (2.7)un+1-un(αn+δn)(M+ω0)0,n, that is, limnun+1-un=0. Since T is on uniformly continuous, so (2.8)limnTnun+1-Tnun=0. Again using (2.5), we have (2.9)un+1-q2un-q2-2αnΦl(un+1-q)1+Φl(un+1-q)+un+1-q2]+An, where (2.10)An=αn2N2+2αnNTnun-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 limnun-q=0.

Theorem 2.2.

Let E be a Banach space and K be a nonempty closed convex subset of E, Tn are as in Theorem 2.1. For x0,u0K, 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:

0αn+δn1,0βni+ηni1,1ip-1;

n=0αn=;

limnαn=0;

limnβni=limnηni=0,i=1,,p-1;

δn=o(αn).

Then the following two assertions are equivalent:

the iteration sequence {xn} strongly converges to the common point of F(Ti),iI;

the sequence iteration {un} strongly converges to the common point of F(Ti),iI.

Proof.

Since F=i=1MF(Ti), set qF. 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 Theorem 2.1, there exists x0K with x0T1x0 such that r0=x0-T1x0·x0-q·[1+x0-q2+Φ1(x0-q)]R(Φ). Setting a0=Φ1-1(r0), we have x0-qa0. Set B1={x-qa0:xK}, B2={x-q2a0:xK}. Since Tl are bounded mappings and {ωni}(i=1,,p-1),{νn} are some bounded sequences in K, we can set M=max{supxB2Tnx-q;supnNωni-q;supnNνn-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,yB2. 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,n0.

Now we claim that if xnB1, then yniB2,1ip-1.

From (1.15), we obtain that (2.11)ynp-1-q(1-βnp-1-ηnp-1)xn-q+βnp-1Tnxn-q+ηnp-1ωnp-1-qynp-2-qxn-q+(βnp-1+ηnp-1)Mynp-2-qxn-q+2τ0M2a0,ynp-2-q(1-βnp-2-ηnp-2)xn-q+βnp-2Tnynp-1-q+ηnp-2ωnp-2-qynp-2-qxn-q+(βnp-2+ηnp-2)Mynp-2-qxn-q+2τ0M2a0,we also obtain that(2.12)yn1-q2a0.

Now we suppose that xn-qa0 holds. We will prove that xn+1-qa0. 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-qxn-q+αnTnyn1-q+δnνn-qxn+1-qxn-q+(αn+δn)Mxn+1-qxn-q+2τ0Mxn-q+14a054a0. 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+αnTnyn1-q+βn1Tnyn2-q+δnνn-q+ηn1ωn1-qxn+1-yn14τ0(μ0+M)δ. Since Tn are uniformly continuous mappings, we get (2.15)Tnxn+1-Tnyn1ϵ0. Using (2.1), Lemma 1.8, and the recursion formula (1.15), we have (2.16)xn+1-q2=(1-αn-δn)(xn-q)+αn(Tnyn1-q)+δn(νn-q)2(1-αn)2xn-q2+2αnTnyn1-q,j(xn+1-q)+2δnνn-q·xn+1-q(1-αn)2xn-q2+2αnTnxn+1-q,j(xn+1-q)+2αnTnyn1-Tnxn+1,j(xn+1-q)+2δnνn-q·xn+1-q(1-αn)2xn-q2+2αn[xn+1-q2-Φl(xn+1-q)1+Φl(xn+1-q)+xn+1-q2]+2αnTnyn1-Tnxn+1·xn+1-q+2δnM·xn+1-q. Which implies (2.17)xn+1-q2xn-q2-2αn1-2αnΦl(xn+1-q)1+Φl(xn+1-q)+xn+1-q2+αn21-2αnxn-q2+2αn1-2αnTnyn1-Tnxn+1·xn+1-q+2δn1-2αnM·xn+1-qa02-2αn1-2αnΦl(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-qμ0; that is, the sequence {xn-q} is bounded. Since unq, as n, so the sequence {xn-un} is bounded.

Step 2. We prove limnxn-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{supn0xn-un,supn0Tnxn-un,supn0Tnyni-un,supn0μn-un,supn0νn-un,supn0ωn1-un}, (i=1,2,,p-1). From (1.15) and (1.17), we obtain (2.18)xn+1-un+12=(1-αn-δn)(xn-un)+αn(Tnyn1-Tnun)+δn(νn-μn)2(1-αn)2xn-un2+2αnTnyn1-Tnun,j(xn+1-un+1)+2δnνn-μn·xn+1-un+1(1-αn)2xn-un2+2αnTnxn+1-Tnun+1,j(xn+1-un+1)+2αnTnyn1-Tnxn+1+Tnun+1-Tnun,j(xn+1-un+1)+2δnνn-μn·xn+1-un+1(1+αn2)xn-q2-2αnΦl(xn+1-un+1)1+Φl(xn+1-un+1)+xn+1-un+12+2αnTnyn1-Tnxn+1·xn+1-un+1+2αnTnun+1-Tnun·xn+1-un+1+2δnM·xn+1-un+1(2.19)xn+1-yn1(βn1+αn+ηn1+δn)xn-un+αnTnnyn1-unxn+1-yn1+βn1Tnnyn2-un+δnνn-un+ηn1ωn1-unxn+1-yn12(βn1+αn+ηn1+δn)L.

By the conditions (iii)–(v), we have (2.20)limnxn+1-yn1=0.

Since limnun-q=0, so (2.21)un+1-unun-q+un+1-q. That is: (2.22)limnun+1-un=0.

By the uniform continuity of T, we obtain (2.23)limnTnxn+1-Tnyn1=0,limnTnun+1-Tnun=0.

From (2.23) and the conditions (iii) and (v), (2.18) becomes (2.24)xn+1-un+12(1+αn2)xn-un2-2αnΦl(xn+1-un+1)1+Φl(xn+1-un+1)+xn+1-un+12+o(αn).

By Lemma 1.10, we get limnxn-un=0. Since limnun-q=0, and the inequality 0xn-qxn-un+un-q, so limnxn-q=0.

From Theorems 2.1 and 2.2, we can obtain the following corollary.

Corollary 2.3.

Let E be a Banach space and K be a nonempty closed convex subset of E, Tn are as in Theorem 2.1. For x0K, 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:

0αn+δn1,0βni+ηni1,1ip-1;

n=0αn=;

limnαn=0;

limnβni=limnηni=0,i=1,,p-1;

δn=o(αn).

Then the iteration sequence {xn} strongly converges to the common point of F(Ti), iI.

Corollary 2.4.

Let Tn=Sn(modM), Tl:EE,lI={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:

0αn+δn1,0βni+ηni1,1ip-1;

n=0αn=;

limnαn=0;

limnβni=limnηni=0,i=1,,p-1;

δn=o(αn).

Let the sequence {xn} in E be generated iteratively from some x0E 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,p2, where Slx:=x-Tlx for all xE and {νn},{ωni} are any bounded sequences in K.

Then {xn} defined by (2.25) converges strongly to x*.

Proof.

We simply observe that Sl:=I-Tl,lI are M uniformly continuous generalized weak Φ-hemicontractive mappings. The result follows from Corollary 2.3.

Acknowledgments

This project was supported by Hebei Province Natural Science Foundation (A2011210033); Shijiazhuang Tiedao University Foundation (Q64).

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