Aclass of extended ishikawa iterative processes in Banachspaces for nonexpansive mappings

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(School of Mathematics and Systems Science, Shenyang Normal University, Shenyang, 110034, China)

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Aclass of extended ishikawa iterative processes in Banachspaces for nonexpansive mappings

CHENGCongdian,GUANHongyan

(School of Mathematics and Systems Science, Shenyang Normal University, Shenyang, 110034, China)

The problem whether a iterative process of nonexpansive mappingTin real Banach Spaces converges to its fixed point(IPNMCFP)has be extensively studied. Particularly, in 2004, Xu H K constructed a kind of extended Mann iterative process for nonexpansive mapping T, and by the so called viscosity approximation methods, he proved that the iterative process converges strongly to a fixed point of Tunder the uniformly smooth Banach Spaces. In 2007, Zhang S S developed the work of Xu H K. This paper further studies the problem IPNMCFP. By replacing constants with operators, the Ishikawa iterative process is generalized as a class of extended Ishikowa iterative process. Under some conditions, the strong convergence of the iterative process is proved in the viscosity approximation methods, using the theory of duality mapping and variational inequality. And in a special case the iterative process converges strongly to a fixed point ofTis also proved. For the introduced iterative process involves many kinds of Mann and Ishikawa iterative processes, the main conclusion of the present work extends and generalizes some recent results of this research area.

nonexpansive mapping; fixed point; sequence; Mann iterative process; Ishikawa iterative process

0 Introduction and preliminaries

Throughout the present work, we always assume thatEis a real Banach space,E*is the duality space of E, 〈·,·〉 is the dual pair betweenEandE*, andJ:E→2E*is the normalized duality mapping defined by

(1)

We also assume thatDis a nonempty closed convex subset ofE,T:D→Dis a mapping, andF(T) denotes the set of all the fixed points ofT. In addition, we useΠDrepresenting all the contractions onD, i.e.ΠD={f|f:D→D, and there existsα∈(0,1) such that ‖f(x)-f(y)‖≤α‖x-y‖ for allx,y∈D}.

The following result is well known (see Goebel and Reich[1]).

Proposition Let E be uniformly smooth. Then the duality mappingJdefined by (1) is single valued, and it is uniformly continuous on the bounded subset ofEwith the norm topologies ofEandE*.

Recall that the sequences

(2)

(3)

(4)

(5)

are respectively called Mann iterative process, Ishikawa iterative process, modified Mann iterative process with error and modified Ishikawa iterative process with error ofT, wherex0,u∈Dandn≥0. The problem for these iterative sequences converging to the fixed point ofTwas studied by lots of authors, e.g., Halpern[2], Reich[3], Zhang and Tian[4], Chidume[5], Liu[6], Liu Q H and Liu Y[7], Zhao and Zhang[8]. In particular, Xu[9]generalized (2) to the iterative process

(6)

Under a certain conditions, he proved that {xn} converges strongly to a fixed point ofTand other related results. In 2007, Zhang[10]extended and improved the work of Xu.

Motivated and inspired by the contributions above, the present work addresses the following iterative process.

(7)

Lemma 1[9]LetXbe a uniformly smooth Banach space,Cbe a closed convex subset ofX,T:C→Cbe a nonexpansive withF(T)≠φ, andf∈ΠC. Then {xt} defined byxt=tf(xt)+(1-t)Txtconverges strongly to a point inF(T). If we defineQ:ΠC→F(T) by

(8)

thenQ(f) solves the variational inequality

In particular, iff=u∈Cis a constant, then (8) is reduced to the sunny nonexpansive retraction of Reich fromContoF(T),

Lemma 2[11]LetXbe a real Banach space andJp:X→2X*,1

Lemma 3[6]Let {an},{bn} and {cn} be three nonnegative real sequences satisfying

1 Main results

In this section, we address the strong convergence of the iterative sequence (7).

Lemma 4 Letf,fn∈ΠD,tn∈(0,1), letTbe a nonexpansive mapping, and letznbe the unique solution of the equationz=tnfn(z)+(1-tn)Tzfor alln≥0. Thenzn→Q(f)(defined by (8) ) astn→0 (strongly) if {fn(x)} converges uniformly tof(x) onD.

This leads to

Proof Since ‖Txn-xn‖→0, we can choose {tn} such that ‖Txn-xn‖=o(tn). Letznbe the unique solution of the fixed point of equationz=tnfn(z)+(1-tn)Tz. Then {zn} converges strongly toQ(f) by Lemma 4. Letz=Q(f). Then we have

(9)

(10)

On the other hand, we have

(11)

(12)

(Note:zn→zand {xn} is bounded.) and

Substitute in (11) the (12), (13) and (14), we obtain

(15)

(Note: In terms of Proposition 1,jis uniformly continuous on bounded subset.) Combining (10), (11) and (15), we also obtain

This further leads to

By Lemma 2,xn+1→z. This completes the proof.

2 Special cases

Whenβn=1,gn(x)=xandfn(x)=f(x), (7) reduces to (6), and that {fn(x)} converges uniformly tofholds obviously. By Theorem 1, we can immediately obtain the following conclusion, which is the major conclusion of [10, Theorem 1].

In addition,Yao[12]also studied the sequence

(16)

which can be transformed as

Thus, we can easily know the following conclusion holds from Theorem 1, which can be taken as a complementary result of [12, Theorem 3.1].

3 Conclusion

A class of extended Ishikowa iterative process for a nonexpansive mappingTin real Banach Spaces, which involves many kinds of Mann and Ishikawa iterative processes, is introduced and studied. Under some conditions, the strong convergence of the iterative process is proved by the viscosity approximation methods. And in a special case, the iterative process converges strongly to a fixed point ofTis also proved. The main conclusion of the present work extends and generalizes some recent results of this research area.

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1673-5862(2016)02-0201-05

關(guān)于巴拿赫空間中非膨脹映射的一類擴(kuò)展的石川迭代序列

程叢電, 關(guān)洪巖

(沈陽師范大學(xué) 數(shù)學(xué)與系統(tǒng)科學(xué)學(xué)院, 沈陽 110034)

關(guān)于巴拿赫空間中非膨脹映射的迭代序列是否收斂到該映射的不動(dòng)點(diǎn)問題已有許多研究工作;2004年,徐洪坤建立了一種擴(kuò)展的曼恩迭代序列,并用黏性逼近方法在一致光滑巴拿赫空間的框架下證明了其收斂到該映射的不動(dòng)點(diǎn);2007年,張石生推廣與改進(jìn)了徐洪坤的工作?；谝酝嘘P(guān)工作,進(jìn)一步探討巴拿赫空間中非膨脹映射的迭代序列的收斂性與非膨脹映射的不動(dòng)點(diǎn)問題。利用算子替換常數(shù)值與向量給出了一類擴(kuò)展的石川迭代序列;基于對(duì)偶映射與變分不等式理論,采用黏性逼近方法,證明了該迭代序列的某種強(qiáng)收斂性及一個(gè)有關(guān)不動(dòng)點(diǎn)定理。由于所建立的迭代序列概括了多種類型的曼恩和石川迭代序列,此項(xiàng)工作發(fā)展與推廣了該領(lǐng)域的許多近期研究成果。

非膨脹映射; 序列; 曼恩(Mann)迭代; 石川(Ishikawa)迭代; 不動(dòng)點(diǎn)

O177 Document code: A

10.3969/ j.issn.1673-5862.2016.02.016

理論與應(yīng)用研究