A modified Mann iteration for zero points of accretive operators

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R E S E A R C H

Open Access

A modiﬁed Mann iteration for zero points of

accretive operators

Jianmin Song

*

_{and Minjiang Chen}

*_{Correspondence:}

[email protected]

Department of Mathematics and Sciences, Shijiazhuang University of Economics, Shijiazhuang, Hebei 050031, China

Abstract

A modiﬁed Mann iteration with computational errors is investigated. A strong convergence theorem for zero points of anm-accretive operator is established in a Banach space.

MSC: 47H06; 47H09; 47J25; 65J15

Keywords: accretive operator; ﬁxed point; nonexpansive mapping; zero point

1 Introduction

In this paper, we are concerned with the problem of finding zero points of accretive oper-ators. Interest in accretive operators stems mainly from their firm connection with equa-tions of evolution, and this is an important class of nonlinear operators. It is known that many physically significant problems can be modelled by initial value problems of the form

x(t) +Ax(t) = , x() =x, (.)

whereAis an accretive operator in an appropriate Banach space. Typical examples where such evolution equations occur can be found in the heat, wave or Schrödinger equations. Ifx(t) is dependent oft, then (.) is reduced to

Au= , (.)

whose solutions correspond to the equilibrium points of (.). An early fundamental result in the theory of accretive operators, due to Browder [], states that the initial value prob-lem (.) is solvable ifAis locally Lipschitz and accretive onE. One of the most popular techniques for solving zero points of accretive operators goes back to the work of Browder []. One of the basic ideas in the case of a Hilbert spaceHis reducing the above equation (.) to a ﬁxed point problem of the operatorRA:H→Hdeﬁned byRA= (I+A)–, which

is called the classical resolvent ofA.

The paper is organized in the following way. In Section , we present the preliminaries that are needed in our work. In Section , a modiﬁed Mann iteration with computational errors is presented. A strong convergence theorem for zero points of anm-accretive op-erator is established in a Banach space. In Section , applications of the main results are discussed.

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2 Preliminaries

LetEbe a real Banach spaceEand letE∗be the dual space ofE. Let·,·denote the pairing betweenEandE∗. The normalized duality mappingJ:E→E∗_{is deﬁned by}

J(x) =f ∈E∗:x,f=x=f

for allx∈E. LetUE={x∈E:x= }.Eis said to be smooth or is said to have a Gâteaux diﬀerentiable norm if the limitlimt→x+ty_t–x exists for eachx,y∈UE.Eis said to have

a uniformly Gâteaux diﬀerentiable norm if for eachy∈UE, the limit is attained uniformly for allx∈UE.E is said to be uniformly smooth or is said to have a uniformly Fréchet diﬀerentiable norm if the limit is attained uniformly forx,y∈UE. In the sequel, we use

jto denote the single-valued normalized duality mapping. It is known that if the norm ofEis uniformly Gâteaux diﬀerentiable, then the duality mappingJis single-valued and uniformly norm to weak∗continuous on each bounded subset ofE.

Recall that a closed convex subsetC of a Banach spaceE is said to have the normal structure if for each bounded closed convex subsetK ofCwhich contains at least two points, there exists an elementxofKwhich is not a diametral point ofK,i.e.,sup{x–y:

y∈K}<d(K), whered(K) is the diameter ofK. It is well known that a closed convex subset of a uniformly convex Banach space has the normal structure and a compact convex subset of a Banach space has the normal structure; see [] for more details.

LetT:C→Cbe a mapping. Recall thatT is said to be contractive if there exits a con-stantα∈(, ) such that

Tx–Ty ≤αx–y, ∀x,y∈C.

For such a case, we also callTanα-contraction.T is said to benonexpansiveif

Tx–Ty ≤ x–y, ∀x,y∈C.

LetDbe a nonempty subset ofC. LetQ:C→D.Qis said to becontractionifQ=Q; sunny if for eachx∈Candt∈(, ), we haveQ(tx+ ( –t)Qx) =Qx; sunny nonexpansive retraction ifQis sunny, nonexpansive, and contraction.K is said to be a nonexpansive retract ofCif there exists a nonexpansive retraction fromContoD.

The following result, which was established in [], describes a characterization of sunny nonexpansive retractions on a smooth Banach space.

LetEbe a smooth Banach space andCbe a nonempty subset ofE. LetQ:E→Cbe a retraction andjbe the normalized duality mapping onE. Then the following are equiva-lent:

() Qis sunny and nonexpansive;

() Qx–Qy≤ x–y,j(Qx–Qy),∀x,y∈E; () x–Qx,j(y–Qx) ≤,∀x∈E,y∈C.

Krasnoselski-Mann iteration generates a sequence{xn}in the following manner:

x∈C, xn+=αnTxn+ ( –αn)xn, ∀n≥. (.)

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and the references therein. In many disciplines, including economics, image recovery, quantum physics, and control theory, problems arise in inﬁnite dimension spaces. In such problems, strong convergence (norm convergence) is often much more desirable than weak convergence, for it translates the physically tangible property that the energyxn–x

of the error between the iteratexnand the solutionxeventually becomes arbitrarily small.

To improve the weak convergence of Krasnoselski-Mann iterative process, diﬀerent mod-iﬁed Mann iterations have been considered; see [–] and the references therein.

LetIdenote the identity operator onE. An operatorA⊂E×Ewith domainD(A) ={z∈ E:Az=∅}and rangeR(A) ={Az:z∈D(A)}is said to beaccretiveif for eachxi∈D(A) andyi∈Axi,i= , , there existsj(x–x)∈J(x–x) such thaty–y,j(x–x) ≥. An

accretive operatorAis said to bem-accretiveifR(I+rA) =Efor allr> . In a real Hilbert space, an operatorAism-accretive if and only ifAis maximal monotone. In this paper, we useA–() to denote the set of zeros ofA.

For an accretive operatorA, we can deﬁne a nonexpansive single-valued mappingJr:

R(I+rA)→D(A) byJr= (I+rA)–_{for each}_r_{> , which is called the}_resolvent_of_A_.

One of classical methods of studying the problem ∈Ax, whereA⊂E×Eis an accretive operator, is the following:

x∈E, xn+=Jrnxn, ∀n≥, (.)

whereJrn= (I+rnA)–and{rn}is a sequence of positive real numbers.

The following iteration also has been extensively investigated:

xn+=αnu+ ( –αn)Jrnxn, ∀n≥, (.)

where{αn}is a real number sequence in (, ),{rn}is a positive real number sequence,

andJrn= (I+rnA)–. It is known that the sequence{xn}generated in the above iteration

converges strongly to a zero point ofAin a Banach space under some restrictions imposed on{αn}and{rn}.

Chenet al.[] investigated the following iteration:

⎧ ⎨ ⎩

yn=βnxn+ ( –βn)Jrnxn,

xn+=αnf(xn) + ( –αn)yn, ∀n≥,

(.)

where {αn}and{βn}are real number sequences in (, ),{rn}is a positive real number

sequence, andJrn= (I+rnA)–. They proved that the sequence{xn}generated in the above

iteration converges strongly to a zero point ofAin a Banach space; for more details, see [] and the references therein.

We also remark that the viscosity approximation method was first introduced by Moudafi [] in the framework of Hilbert spaces. Moudafi proved that the desired so-lution is not only a fixed point of nonlinear mappings but a soso-lution to some variational inequality; for more details, see [] and the references therein.

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points of accretive operators. In this paper, a modiﬁed Mann iteration with computational errors is investigated. A strong convergence theorem for zero points of anm-accretive operator is established in a Banach space. The results mainly improve the corresponding results in Qin and Su [], Hao [], Qinet al.[] and Chenet al.[].

In order to state our main results, we also need the following lemmas.

Lemma .[] Let E be a real reﬂexive Banach space with the uniformly Gâteaux diﬀer-entiable norm and the normal structure,and C be a nonempty closed convex subset of E.

Let S:C→C be a nonexpansive mapping with a fixed point,and f :C→C be a fixed contraction with the coefficientα∈(, ).Let{xt}be a sequence generated by the following xt=tf(xt) + ( –t)Sxt,where t∈(, ).Then{xt}converges strongly as t→to a fixed point x∗of T,which is the unique solution in F(T)to the following variational inequality

f x∗–x∗,j x∗–p≥, ∀p∈F(S).

Lemma .[] Let{xn}and{yn}be bounded sequences in a Banach space E,and{βn} be a sequence in (, )with <lim infn→∞βn≤lim supn→∞βn< .Suppose that xn+=

( –βn)yn+βnxn,∀n≥and lim sup

n→∞

yn+–yn–xn+–xn

≤.

Thenlimn→∞yn–xn= .

Lemma .[] Let{an},{bn},and{cn}be three nonnegative real sequences satisfying an+≤( –tn)an+bn+cn, ∀n≥,

where{tn}is a sequence in(, ).Assume that the following conditions are satisﬁed: (a) ∞_n_=tn=∞andbn=o(tn);

(b) ∞_n_=cn<∞.

Thenlimn→∞an= .

Lemma .[] Let E be a Banach space and let A be an m-accretive operator.Forλ> , μ> ,and x∈E,we have Jλx=Jμ(μ_λx+ ( –μ_λ)Jλx),where Jλ= (I+λA)–and Jμ= (I+μA)–.

3 Main results

Theorem . Let E be a real reﬂexive Banach space with the uniformly Gâteaux diﬀeren-tiable norm.Let A be an m-accretive operator in E such that C:=D(A)is convex and has the normal structure.Let f:C→C be anα-contraction.Let{xn}be a sequence generated in the following manner:x∈C and

⎧ ⎨ ⎩

yn=βnxn+ ( –βn)Jrn(xn+en+),

xn+=αnf(xn) + ( –αn)yn, ∀n≥,

where{αn}and{βn}are real number sequences in(, ),{en}is a sequence in E,{rn}is a positive real number sequence,and Jrn= (I+rnA)–.Assume that A–()is not empty and

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(a) limn→∞αn= and

∞

n=αn=∞;

(b)  <lim infn→∞βn≤lim supn→∞βn< ;

(c) ∞_n_=en<∞;

(d) rn≥rfor eachn≥andlimn→∞|rn–rn+|= .

Then the sequence{xn}converges strongly tox¯,which is the unique solution to the following variational inequalityf(x¯) –x¯,j(p–x¯) ≤,∀p∈A–().

Proof First, we prove that{xn}is bounded. Fixingp∈A–(), we see that

yn–p ≤βnxn–p+ ( –βn)Jrn(xn+en+) –p

≤ xn–p+en+.

It follows that

xn+–p ≤αnf(xn) –p+ ( –αn)yn–p

≤  –αn( –α)

xn–p+αnf(p) –p+en+

≤max

xn–p,f(p) –p  –α

+en+

≤max

xn––p,

f(p) –p

 –α

+en+en+

.. .

≤max

x–p,

f(p) –p

 –α

+ ∞

i=

ei<∞.

This proves that the sequence{xn}is bounded. Ifrn≥rn+, we see from Lemma . that Jrn(xn+en+) –Jrn+(xn++en+)

≤ rn

rn+

(xn+en+) +

 – rn

rn+

Jrn+(xn+en+) – (xn++en+)

= rn

rn+

(xn+en+) – (xn++en+)

+rn+–rn

rn+

Jrn+(xn+en+) – (xn++en+)

≤ xn–xn++en++en++ M

r (rn–rn+), (.)

where M is an appropriate constant such that M ≥supn≥{Jrn+(xn+en+) – (xn++ en+)}. Putzn=xn+_––ββnnxn; that is,

xn+= ( –βn)zn+βnxn, n≥. (.)

Note that

zn+–zn=

αn+f(xn+) + ( –αn+)yn+–βn+xn+

 –βn+

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= αn+  –βn+

f(xn+) –yn+

+Jrn+(xn++en+) –

αn

 –βn

f(xn) –yn

–Jrn(xn+en+).

It follows that

zn+–zn ≤

αn+

 –βn+

_f(xn+) –yn++

αn

 –βn

_f(xn) –yn

+Jrn+(xn++en+) –Jrn(xn+en+). (.)

Substituting (.) into (.), we arrive at

zn+–zn–xn–xn+ ≤

αn+

 –βn+

f(xn+) –yn++

αn

 –βn

f(xn) –yn

+en++en++ M

r (rn–rn+).

In view of restrictions (a), (b) and (c), we ﬁnd that

lim sup

n→∞

zn+–zn–xn–xn+

≤.

By virtue of Lemma ., one obtains that

lim

n→∞zn–xn= . (.)

It follows from (.) thatxn+–xn= ( –βn)(zn–xn). In view of restriction (c), we ﬁnd from

(.) that

lim

n→∞xn+–xn= . (.)

Ifrn≤rn+, we can prove that (.) still holds. On the other hand, we have xn–Jrn(xn+en+)

≤ xn–xn++xn+–yn+yn–Jrn(xn+en+)

≤ xn–xn++αnf(xn) –yn+βnxn–Jrn(xn+en+).

It follows that

( –βn)xn–Jrn(xn+en+)≤ xn–xn++αnf(xn) –yn.

In view of restriction (b), one ﬁnds that

lim

n→∞xn–Jrn(xn+en+)= . (.)

Notice that

xn–Jrnxn ≤xn–Jrn(xn+en+)+Jrn(xn+en+) –Jrnxn

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Since∞_n_=en<∞, from (.) we obtain thatlimn→∞xn–Jrnxn= . Taking a ﬁxed

numberksuch thatr>k> , we arrive at

Jrnxn–Jkxn=

Jk

k rnxn+

 – k

rn

Jrnxn

–Jkxn

≤ – k

rn

(Jrnxn–xn)

≤ Jrnxn–xn. (.)

Since

xn–Jkxn ≤ xn–Jrnxn+Jrnxn–Jkxn ≤xn–Jrnxn,

we therefore ﬁnd that

lim

n→∞xn–Jkxn= . (.)

Now, we are in a position to prove thatlim sup_n_→∞¯x–f(x¯),j(xn–x¯) ≤, wherex¯=

lim_t_→_zt, andzt solves the ﬁxed point equation

zt=tf(zt) + ( –t)Jkzt, ∀t∈(, ). Therefore, we see that

zt–xn= ( –t)

Jkzt–xn,j(zt–xn)

+tf(zt) –xn,j(zt–xn)

= ( –t) Jkzt–Jkxn,j(zt–xn)+Jkxn–xn,j(zt–xn)

+tf(zt) –zt,j(zt–xn)+tzt–xn,j(zt–xn)

≤( –t)zt–xn+Jkxn–xnzt–xn

+tf(zt) –zt,j(zt–xn)+tzt–xn

≤ zt–xn+Jkxn–xnzt–xn+tf(zt) –zt,j(zt–xn), ∀t∈(, ).

This implies that

zt–f(zt),j(zt–xn)≤

tJkxn–xnzt–xn, ∀t∈(, ).

It follows from (.) that

lim sup n→∞

zt–f(zt),j(zt–xn)≤. (.)

Sincezt→ ¯xast→ and from the fact thatjis strong to weak∗uniformly continuous on bounded subsets ofE, we ﬁnd that

zt–f(zt),j(zt–xn)–f(x¯) –x¯,j(xn–x¯)

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+f(x¯) –x¯,j(xn–zt)–zt–f(zt),j(zt–xn)

≤f(x¯) –x¯,j(xn–x¯) –j(xn–zt)+f(x¯) –x¯+zt–f(zt),J(xn–zt)

≤f(x¯) –x¯j(xn–x¯) –j(xn–zt)+f(x¯) –x¯+zt–f(zt)xn–zt.

It follows that

lim t→

zt–f(zt),j(zt–xn)

–f(x¯) –x¯,j(xn–x¯)= .

For any> , there existsκ>  such that∀t∈(,κ) the following inequality holds

f(x¯) –x¯,j(xn–x¯)

≤zt–f(zt),j(zt–xn)

+.

This implies that

lim sup n→∞

f(x¯) –x¯,j(xn–x¯)≤lim sup n→∞

zt–f(zt),j(zt–xn)+.

Note that is arbitrary. In view of (.), we see thatlim sup_n_→∞f(x¯) –x¯,j(xn–x¯) ≤. This implies that

lim sup n→∞

f(x¯) –x¯,j(xn+–x¯)

≤. (.)

Finally, we show thatxn→ ¯xasn→ ∞. Notice that

xn+–x¯≤αn

f(xn) –x¯,j(xn+–x¯)

+ ( –αn)yn–x¯xn+–x¯

≤αn

+ –αn

 yn–x¯

₊_x n+–x¯

.

It follows that

xn+–x¯≤( –αn)xn–x¯+ αn

+dn,

where dn=en+(xn–x¯+en+). In view of restrictions (a) and (c), we know that

_∞

n=dn<∞. Putan=xn–x¯,tn=αn, andcn=dn. In view of Lemma ., we ﬁnd the

desired conclusion.

Iff(x) =u, whereuis a fixed element inC, for anyx∈C, we find the following result. Corollary . Let E be a real reflexive Banach space with the uniformly Gâteaux differen-tiable norm.Let A be an m-accretive operator in E such that C:=D(A)is convex and has the normal structure.Let{xn}be a sequence generated in the following manner:x∈C and

⎧ ⎨ ⎩

yn=βnxn+ ( –βn)Jrnxn,

xn+=αnu+ ( –αn)yn, ∀n≥,

where{αn} and{βn} are real number sequences in(, ),{rn} is a positive real number sequence,and Jrn= (I+rnA)–. Assume that A–()is not empty and the above control

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(a) limn→∞αn= and

∞

n=αn=∞;

(b)  <lim infn→∞βn≤lim supn→∞βn< ;

(c) rn≥rfor eachn≥andlimn→∞|rn–rn+|= . Then the sequence{xn}converges strongly tox¯=QA–_()u.

4 Applications

In this section, we consider solutions of variational inequalities. LetC be a nonempty, closed, and convex subset of a Banach spaceE. LetA:C→E∗be a single-valued mono-tone operator which is hemicontinuous; that is, continuous along each line segment inC

with respect to the weak∗ topology ofE∗. Consider the following variational inequality problem of ﬁnding a pointx∈Csuch that

y–x,Ax ≥, ∀y∈C.

In this section, we use VI(C,A) to denote the solution set of the variational inequality involvingA. The symbolNC(x) stands for the normal cone forCat a pointx∈C; that is,

NC(x) =x∗∈E∗:y–x,x∗≤,∀y∈C.

Theorem . Let E be a real reﬂexive Banach space with the uniformly Gâteaux diﬀer-entiable norm.Let C be a nonempty,closed,and convex subset of E.Let A:C→E∗be a single-valued,monotone,and hemicontinuous operator.Assume that VI(C,A)is not empty and C has the normal structure.Let f :C→C be anα-contraction.Let{xn}be a sequence generated in the following manner:x∈C and

⎧ ⎨ ⎩

yn=βnxn+ ( –βn)VI(C,A+_r_n(I–xn)), xn+=αnf(xn) + ( –αn)yn, ∀n≥,

the above control sequences satisfy the following restrictions: (a) limn→∞αn= and

∞

n=αn=∞;

(b)  <lim inf_n_→∞βn≤lim supn→∞βn< ;

(c) rn≥rfor eachn≥andlimn→∞|rn–rn+|= .

Then the sequence{xn}converges strongly tox¯,which is the unique solution to the following variational inequalityf(x¯) –x¯,j(p–x¯) ≤,∀p∈A–_().

Proof Deﬁne a mappingT⊂E×E∗by

Tx=

⎧ ⎨ ⎩

Ax+NCx, x∈C,

∅, x∈/C.

By Rockafellar [], we know thatTis maximal monotone, andT–() =VI(C,A). For each

rn>  andxn∈E, we see that there exists a uniquexrn∈D(T) such thatxn∈xr+rnT(xrn),

wherexrn= (I+rnT)–xn. Notice that

yn=VI

C,A+ 

rn(I–xn)

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which is equivalent to

y–yn,Ayn+ 

rn(yn–xn)

≥, ∀y∈C,

that is, –Ayn+_r_n(xn–yn)∈NC(yn). This implies thatyn= (I+rnT)–xn. Following the proof

of Theorem ., we can immediately conclude the desired conclusion.

Iff(x) =u, whereuis a fixed element inC, for anyx∈C, we find the following result. Corollary . Let E be a real reflexive Banach space with the uniformly Gâteaux differ-entiable norm.Let C be a nonempty,closed,and convex subset of E.Let A:C→E∗be a single-valued,monotone,and hemicontinuous operator.Assume that VI(C,A)is not empty and C has the normal structure.Let{xn}be a sequence generated in the following manner:

x∈C and ⎧ ⎨ ⎩

yn=βnxn+ ( –βn)VI(C,A+_r_n(I–xn)), xn+=αnu+ ( –αn)yn, ∀n≥,

the above control sequences satisfy the following restrictions: (a) limn→∞αn= and∞n=αn=∞;

(b)  <lim inf_n_→∞βn≤lim supn→∞βn< ;

(c) rn≥rfor eachn≥andlimn→∞|rn–rn+|= . Then the sequence{xn}converges strongly tox¯=QA–_()u.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this manuscript. All authors read and approved the ﬁnal manuscript.

Acknowledgements

This work is supported by the Department of Hebei Education (Z2013110). The authors are grateful to Professor Jin Wang and the anonymous reviewers for useful suggestions which improved the contents of the article.

Received: 1 October 2013 Accepted: 2 December 2013 Published:19 Dec 2013

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