Convergence theorems for modified generalized f projections and generalized nonexpansive mappings

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R E S E A R C H

Open Access

Convergence theorems for modiﬁed

generalized

f

-projections and generalized

nonexpansive mappings

Qingqing Cheng, Yongfu Su

*

_{and Jingling Zhang}

*_{Correspondence:}

[email protected]

Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300160, China

Abstract

The purpose of this paper is to study a sequence of modiﬁed generalized

f-projections in a reﬂexive, smooth, and strictly convex Banach space and show that Mosco convergence of their ranges implies their pointwise convergence to the generalizedf-projection onto the limit set. Furthermore, we prove a strong convergence theorem for a countable family of

α-nonexpansive mappings in a

uniformly convex and smooth Banach space using the properties of a modiﬁed generalizedf-projection operator. Our main results generalize the results of Ziming Wang, Yongfu Su, and Jinlong Kang and enrich the research contents of

α-nonexpansive mappings.

MSC: 47H05; 47H09; 47H10

Keywords:

α-nonexpansive mappings; monotone hybrid algorithm; modiﬁed

generalizedf-projection operator; Mosco convergence; common ﬁxed point

1 Introduction

LetEbe a real Banach space andCbe a nonempty closed convex subset ofE. A mapping

T:C→Cis said to be nonexpansive if

Tx–Ty ≤ x–y, ∀x,y∈C. (.)

Lots of iterative schemes for nonexpansive mappings have been introduced (see [–]); furthermore, many strong convergence theorems for nonexpansive mappings have been proved. On the other hand, there are many nonlinear mappings which are more general than the nonexpansive mapping. Compared to the existing problem of a fixed point of those mappings, the iterative methods for finding a fixed point are also very useful in studying the fixed point theory and the theory of equations in other fields.

In , Gobel and Pineda [] introduced and studied a new mapping, calledα -nonex-pansive mapping. The mapping is more general than the nonex-nonex-pansive mapping.

Deﬁnition . For a given multi-index,α= (α,α, . . . ,αn) satisﬁesαi≥,i= , , . . . ,n

andn_i_=αi= . A mappingT:C→Cis said to beα-nonexpansive if n

i=

αiTix–Tiy≤ x–y, ∀x,y∈C. (.)

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In order to show that the class ofα-nonexpansive mappings is more general than the one of nonexpansive mappings, we give an example [].

Example . LetE=R_{, and}

T(x) = ⎧ ⎨ ⎩

 ifx= ;



x ifx∈(, +∞).

ThenT is not nonexpansive butα-nonexpansive.

Proof Obviously,T is not nonexpansive. Takingx=_,y= , by the deﬁnition ofTx, we have

Tx–Ty=| – |> – 

=x–y.

On the other hand, for everyx,y∈[, +∞), we have Tx–Ty=x–y.

Therefore, we can aﬃrm that

Tx–Ty+Tx–Ty=x–y,

whereα= (α,α) = (, ). ThenT is anα-nonexpansive mapping but not a nonexpansive

one.

IfT is a nonexpansive self-mapping, we can imply thatT must be anα-nonexpansive one, whereα= (α,α, . . . ,αn) = (_n, . . . ,_n).

For technical reasons, we always assume that the first coefficientαis nonzero, that is, α> . In this case the mappingTsatisfies the Lipschitz condition

Tx–Ty ≤  α

x–y, ∀x,y∈C.

For theα-nonexpansive mappingT,α= (α,α,α, . . .αn), it is obvious that the mapping

Tαx= n

i=

αiTix, ∀x∈C (.)

is nonexpansive. However, the nonexpansiveness ofTαis much weaker than (.), for

in-stance, it does not entail the continuity ofT(see []).

In , Klin-eam and Suantai [] introduced the relation of ﬁxed point sets between an

α-nonexpansive operator and aTαoperator. They gave the following theorem.

Theorem .(see Theorem . of Klin-eam and Suantai []) Let C be a closed convex subset of a Banach space E and for all n∈N,letα= (α,α, . . . ,αn)such thatαi≥,i=

, , . . . ,n,α> ,and

n

i=αi= .Let T be anα-nonexpansive mapping from C into itself. Ifα> n–√

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At the same time, they have succeeded in proving the demiclosedness principle for the

α-nonexpansive mappings.

Theorem .(see Theorem . of Klin-eam and Suantai []) Let C be a closed convex subset of a Banach space E and for all n∈N,letα= (α,α, . . . ,αn)such thatαi≥,i=

, , . . . ,n,α> ,and

n

i=αi= .Let T be anα-nonexpansive mapping from C into itself. Ifα> n–√

,if{xn} ⊂C converges weakly to x and{xn–Txn} converges strongly toas n→ ∞,then x∈F(T).

Recently, Wanget al.[] proposed the following hybrid algorithm for anα-nonexpansive mapping in a Banach space:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

yn= ( –βn)xn+βnTxn,

Cn+={z∈Cn:yn–z ≤ xn–z}, xn+=Cn+x, n∈N.

(.)

As we know that ifCis a nonempty closed convex subset of a Hilbert spaceHand recall that the (nearest point) projectionPCfromHontoCassigns to eachx∈H, and the unique pointPCx∈Csatisﬁes the propertyx–PCx=min_y∈Cx–y, it is well known thatPCis nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. We consider the functional deﬁned by

φ(y,x) =y– y,Jx+x, ∀x,y∈E,

whereJis the normalized duality mapping and the Banach space is smooth. In this con-nection, Alber [] introduced a generalized projectionCfromEtoCas follows:

C(x) =arg min

y∈Cφ(y,x), ∀x∈E.

It is obvious from the deﬁnition of functionalφthat

y–x≤φ(y,x)≤y+x, ∀x,y∈E.

IfEis a Hilbert space, thenφ(y,x) =y–xandCbecomes the metric projection ofE

ontoC. The generalized projectionC:E→Cis a map that assigns to an arbitrary point x∈Ethe minimum point of the functionalφ(y,x), that is,Cx=x¯, wherex¯is the solution to the minimization problem

φ(x¯,x) =inf

y∈Cφ(y,x).

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convex and uniformly smooth Banach spaces. Moreover, Alber [] presented some appli-cations of the generalized projections to approximately solve variational inequalities and von Neumann intersection problem in Banach spaces. In , Li [] extended the gener-alized projection operator from uniformly convex and uniformly smooth Banach spaces to reﬂexive Banach spaces and studied some properties of the generalized projection op-erator with applications to solve the variational inequality in Banach spaces. Later, Wu and Huang [] introduced a new generalized f-projection operator in Banach spaces. They extended the deﬁnition of generalized projection operators introduced by Abler [] and proved some properties of the generalizedf-projection operator. In , Fanet al.

[] presented some basic results for the generalizedf-projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces.

The purpose of this paper is to study a sequence of modified generalizedf-projections in a reflexive, smooth, and strictly convex Banach space and show that Mosco convergence of their ranges implies their pointwise convergence to the generalizedf-projection onto the limit set. Furthermore, we prove strong convergence theorem for a countable family ofα -nonexpansive mappings in a uniformly convex and smooth Banach space using the prop-erties of a modified generalizedf-projection operator. Our main results generalize the results of Wanget al.[] and enrich the research contents ofα-nonexpansive mappings.

2 Preliminaries

A Banach spaceEis said to be strictly convex ifx+_y<  forx,y∈Ewithx=y=  and

x=y. It is said to be uniformly convex if for each>  there isδ>  such that forx,y∈E

withx,y ≤ andx–y ≥,x+y ≤( –δ) holds. The spaceEis said to be smooth if the limit

lim

t→

x+ty–x

t (.)

exists for allx,y∈S(E) ={x∈E:x= }. AndEis said to be uniformly smooth if the limit (.) exists uniformly for allx,y∈S(E).

Remark . The following basic properties of a Banach space E can be found in Cio-ranescu []:

(i) ifEis uniformly convex, thenEis reﬂexive and strictly convex;

(ii) a Banach spaceEis uniformly smooth if and only ifE∗is uniformly convex; (iii) each uniformly convex Banach spaceEhas the Kadec-Klee property,i.e., for any

sequence{xn} ⊂E, ifxnx∈Eandxn → x, thenxn→x.

LetEbe a real Banach space with the dualE∗. We denote byJthe normalized duality mapping fromEto E∗_{deﬁned by}

Jx=f ∈E∗:x,f=x=f, x∈E.

Many properties of the normalized duality mappingJcan be found in Takahashi [] or Vainberg []. We list some properties below for easy reference:

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(iii) Jis a continuous operator in smooth Banach spaces;

(iv) Jis a uniformly continuous operator on each bounded set in uniformly smooth Banach spaces;

(v) Jis a bijection in smooth, reﬂexive, and strictly convex Banach spaces; (vi) Jis the identity operator in Hilbert spaces.

Next, we recall the concept of generalizedf-projector operator, together with its prop-erties. LetG:C×E∗→R∪ {+∞}be a functional deﬁned as follows:

G(ξ,ϕ) =ξ– ξ,ϕ+ϕ+ ρf(ξ), (.) whereξ∈C,ϕ∈E∗,ρis a positive number andf :C→R∪ {+∞}is proper, convex, and lower semi-continuous. From the deﬁnitions ofGandf, it is easy to see the following properties:

(i) G(ξ,ϕ)is convex and continuous with respect toϕwhenξ is ﬁxed;

(ii) G(ξ,ϕ)is convex and lower semi-continuous with respect toξwhenϕis ﬁxed.

Deﬁnition .([]) LetEbe a real Banach space with its dualE∗. LetCbe a nonempty, closed, and convex subset ofE. We say thatf_C:E∗→C is a generalizedf-projection operator if

f_Cϕ=

u∈C:G(u,ϕ) =inf

ξ∈CG(ξ,ϕ)

, ∀ϕ∈E∗. (.)

For the generalizedf-projection operator, Wu and Huang [] proved the following basic properties.

Lemma .([]) Let E be a real reﬂexive Banach space with its dual E∗,and let C be a nonempty,closed,and convex subset of E.Then the following statements hold:

(i) f_Cϕis a nonempty closed convex subset ofCfor allϕ∈E∗. (ii) IfEis smooth,then for allϕ∈E∗,x∈f_Cϕif and only if

x–y,ϕ–Jx+ρf(y) –ρf(x)≥, ∀y∈C.

(iii) IfEis strictly convex andf :C→R∪ {+∞}is positive homogeneous(i.e.,

f(tx) =tf(x)for allt> such thattx∈C,wherex∈C),thenf_Cis a single-valued

mapping.

Fanet al.[] showed that the conditionf is positive homogeneous, which appeared in Lemma ., can be removed.

Lemma .([]) Let E be a real reﬂexive Banach space with its dual E∗,and let C be a nonempty,closed,and convex subset of E.Then if E is strictly convex,thenf_Cis a single-valued mapping.

Recall thatJis a single-valued mapping whenEis a smooth Banach space. There exists a unique elementϕ∈E∗such thatϕ=Jxfor eachx∈E. This substitution in (.) gives

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Deﬁnition . LetEbe a real Banach space andCbe a nonempty, closed, and convex subset ofE. We say thatf_C:E→C_{is a generalized}_f_{-projection operator if}

f_C(x) =

u∈C:G(u,Jx) =inf

ξ∈CG(ξ,Jx)

, ∀x∈E.

We know that the following lemmas hold for the operatorf_C.

Lemma .([]) Let C be a nonempty,closed,and convex subset of a smooth and reﬂexive Banach space E.Then the following statements hold:

(i) f_Cxis a nonempty closed and convex subset ofCfor allx∈E. (ii) For allx∈E,xˆ∈f_Cxif and only if

ˆx–y,Jx–Jxˆ+ρf(y) –ρf(x)≥, ∀y∈C.

(iii) IfEis strictly convex,thenf_Cis a single-valued mapping.

Now, we introduce a modiﬁed generalizedf-projection operator. LetG:C×E∗→Rbe a functional deﬁned as follows:

G(ξ,ϕ) =ξ– ξ,ϕ+ϕ+ ρf(ξ), (.)

whereξ∈C,ϕ∈E∗,ρis a positive number andf:C→Ris convex and weakly continu-ous. From the deﬁnitions ofGandf, it is easy to see the following properties:

(i) G(ξ,ϕ)is convex and continuous with respect toϕwhenξ is ﬁxed;

(ii) G(ξ,ϕ)is convex and weakly lower semi-continuous with respect toξ whenϕis ﬁxed.

Obviously, the other deﬁnitions and lemmas hold respectively.

Next, we give the following example [] which shows that metric projection, general-ized projection and generalgeneral-izedf-projection are diﬀerent.

Example . LetX=R_{be provided with the norm}

(x,x,x)=

x +x

+x

+x

.

This is a smooth strictly convex Banach space andC={x∈R_|_x

= ,x= }is a closed

and convex subset ofX. It is a simple computation; we getPC(, , ) = (, , ),C(, , ) = (, , ).

We setρ=  is a positive number and deﬁnef :C→Rby

f(x) = – – √.

Thenf is convex and weakly continuous. Simple computations show that

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LetEbe a Banach space, and letC,C,C, . . . be a sequence of weakly closed subsets

ofE. We denote bys–LinCnthe set of limit points of{Cn}, that is,x∈s–LinCnif and only if there exists{xn} ⊂Esuch that{xn}converges strongly toxand thatxn∈Cnfor alln∈N.

Similarly, we denote byw–LsnCnthe set of cluster points of{Cn},y∈w–LsnCnif and only if there exists{yni}such that{yni}converges weakly toyand that{yni} ∈Cni for all

i∈N. Using these definitions, we define the Mosco convergence [] ofCn_i. IfCsatisfies

s–LinCn=C=w–LsnCn, (.)

we say thatCnis a Mosco convergent sequence toCand write

C=M–_n→∞lim Cn. (.)

Notice that the inclusions–LinCn⊂w–LsnCnis always true. Therefore, in order to show the existence ofM–lim_n→∞Cn, it is suﬃcient to provew–LsnCn⊂s–LinCn. For more

details, see [].

3 Main results

3.1 Generalized Mosco convergence theorems

Theorem . Let E be a smooth,reﬂexive,and strictly convex Banach space and C be a nonempty closed convex subset of E.Let C,C,C, . . .be nonempty closed convex subsets of C,f :E→R be a convex and weakly continuous mapping with C⊂int(D(f)).If C= M–limn→∞Cnexists and is nonempty,then Cis a closed convex subset of C and,for each x∈C,{f_C

nx}converges weakly to

f Cx.

Proof It is easy to prove thatCis closed and convex ifCnis a closed convex subset ofC

for eachn∈N. Fixx∈C. For the sake of simplicity, we writexninstead off_C_nxforn∈N. SinceC=M–limn→∞Cn, we have that for eachy∈C, there exists{yn} ⊂Esuch that yn→yasn→ ∞and thatyn∈Cnfor eachn∈N. From Lemma ., we have

xn–yn,Jx–Jxn+ρf(yn) –ρf(x)≥.

Hence, we obtain

≤ xn–x,Jx–Jxn+x–yn,Jx–Jxn+ρf(yn) –ρf(x)

≤–x–xn+x+xnx–yn+ρf(yn) –ρf(x),

thus,

x–xn≤x+xnx–yn+ρf(yn) –ρf(x).

Suppose that{xn}is not bounded. Then there exists a subsequence{xni}of{xn}such that

xni → ∞. It follows that

x

xni

– x+xni ≤

 + x

xni

x–yni+

ρf(yni) –ρf(x)

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for a suﬃciently large numberi∈N. Asi→ ∞, we obtain +∞ ≤ x–yni< +∞. This is

a contradiction. Hence we have that{xn}is bounded.

Since{xn}is bounded, there exists a subsequence, again denoted by{xn}, such that it

converges weakly tox∈C. From the deﬁnition ofC, we getx∈C.

Now, we prove thatf_C

x=x. From weak lower semi-continuity of the norm and weak continuity off, we have

lim inf

n→∞ G(xn,Jx) =lim infn→∞ xn

_{– }_xn_,_Jx₊_x₊_ρf₍_xn₎

≥ x– x,Jx+x+ρf(x)

=G(x,Jx).

On the other hand, we get

lim inf

n→∞ G(xn,Jx)≤lim infn→∞ G(yn,Jx)

=lim inf

n→∞ yn

_{– }_yn_,_Jx₊_x₊_ρf₍_yn₎

=G(y,Jx).

So,

G(x,Jx)≤G(y,Jx), ∀y∈C,

that is,

G(x,Jx) = inf y∈C

G(y,Jx).

Hence we getf_C_x=x.

According to our consideration above, each sequence{xn}has, in turn, a subsequence

which converges weakly to the unique pointf_C

x. Therefore, the sequence{xn}converges weakly tof_C

x. This completes the proof.

A Banach spaceEis said to have the Kadec-Klee property if a sequence{xn}ofE satis-fying thatxnxandxn → xconverges strongly tox. It is known thatE∗has a

Fréchet diﬀerentiable norm if and only ifEis reﬂexive, strictly convex, and has the Kadec-Klee property; see, for example, [].

Theorem . Let E be a smooth Banach space such that E∗ has a Fréchet diﬀerentiable norm. Let C be a nonempty closed convex subset of E. Let C,C,C, . . . be nonempty closed convex subsets of C,f :E→R be a convex and weakly continuous mapping with C⊂int(D(f)).If C=M–limn→∞Cnexists and is nonempty,then Cis a closed convex subset of C and,for each x∈C,{f_C

nx}converges strongly to

f Cx.

Proof Fixx∈Carbitrarily. We writexn=fCnxandx=

f

Cx. By Theorem ., we ob-tainxnx. SinceE∗has a Fréchet diﬀerentiable norm,Ehas the Kadec-Klee property.

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a sequence{yn} ⊂Csuch thatyn→xasn→ ∞andyn∈Cnfor eachn∈N. It follows

that

G(x,Jx)≤lim inf_n→∞ G(xn,Jx)

≤lim sup

n→∞ G(xn,Jx)

≤lim sup

n→∞ G(yn,Jx)

=G(x,Jx).

Hence we obtainG(x,Jx) =limn→∞G(xn,Jx). Sincexn,J(x)converges tox,J(x)andf

is weakly continuous, we get

lim

n→∞xn=x.

Using the Kadec-Klee property ofE, we obtain that{xn}converges strongly tox. This

completes the proof.

Deﬁnition .([]) LetCbe a closed convex subset of a Banach spaceE, let{Tn}∞n=be

a countable family of mappings ofCinto itself with the nonempty common ﬁxed point setF. The{Tn}∞_n_=is said to be uniformly closed ifxn→xandxn–Tnxn → asn→ ∞

impliesx∈F.

3.2 Strong convergence theorems

Lemma .(see Lemma . of Klin-eam and Suantai []) Let C be a closed convex subset of a Banach space E and for all n∈N,letα= (α,α, . . . ,αn)such thatαi≥,i= , , . . . ,n, α> ,and

n

i=αi= .Let T be anα-nonexpansive mapping from C into itself.Ifα> n–√

, let{xm}be a bounded sequence in C,thenxm–Txm →if and only ifxm–Tαxm → as m→ ∞.

Lemma .([]) Let C be a closed convex subset of a Banach space E,and for all n∈N,

letα= (α,α, . . . ,αn)such thatαi≥,i= , , . . . ,n,α> ,and

n

i=αi= .Let T be an α-nonexpansive mapping from C into itself.Ifα> n–√

,let{xm} ⊂C converge strongly to x andxm–Txm →converge strongly toas m→ ∞,then x∈F(T).

Lemma .([]) Let C be a closed convex subset of a uniformly convex and smooth Banach space E,and for all n∈N,letα= (α,α, . . . ,αn)such thatαi≥,i= , , . . . ,n,α> ,and

n

i=αi= .Let T be anα-nonexpansive mapping from C into itself.Ifα> n–√

,then F(T) is closed and convex.

Theorem . Let C be a closed convex subset of a uniformly convex and smooth Banach

space E,let{Tn}∞_n_=be a uniformly closed countable family ofαn-nonexpansive mappings of C into itself such that F:=∞_n_=F(Tn)=∅,letαn= (αn,αn, . . . ,αnN)such thatαni≥,

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sequence{xn}in C by the following algorithm: ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

yn= ( –βn)xn+βnTnxn,

Cn+={z∈Cn:yn–z ≤ xn–z}, xn+=fCn+x, n∈N,

(.)

where <a≤βn≤for all n∈N.Ifαn> N–√_,then{xn}converges strongly to x

∗₌f Fx.

Proof Step . We show thatCnis closed and convex for eachn≥.

From the deﬁnitions ofCn, it is obvious thatCnis closed for eachn≥. Moreover, since

yn–z ≤ xn–zis equivalent to

yn–xn+ yn–xn,Jxn–Jz ≤,

soCnis convex for eachn≥.

Step . We show thatF⊂Cnfor alln≥. For allp∈F, we have that

yn–p=( –βn)xn+βnTnxn–p

≤( –βn)xn–p+βnTnxn–p

= ( –βn)xn–p+βnαn(Tnxn–Tnp)

+αn

Tnxn–T_np+· · ·+αnN

Tnxn–TN

n p

≤( –βn)xn–p+βn

 –αN–

n

αN–

n

xn–Tnp

≤( –βn)xn–p+βnxn–p

=xn–p.

It implies thatp∈Cnfor alln≥. So, we haveF⊂Cnfor alln≥.

Step . We show thatlim_n→∞xn=x∗=fC¯xandx

∗_∈_F_{, where}_C_¯ ₌∞

n=Cn. Indeed,

since{Cn}is a decreasing sequence of closed convex subsets ofEsuch thatC¯=∞_n_=Cnis nonempty, it follows that

M– lim

n→∞Cn=C¯= ∞

n= Cn=∅.

By Theorem ., we get

xn→x∗ asn→ ∞. (.)

Noticing thatxn+=fCn+x∈Cn+, we obtain that

yn–xn+ ≤ xn–xn+.

In view of (.), we have that

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and

yn–xn ≤ yn–xn++xn+–xn → asn→ ∞.

Fromyn= ( –βn)xn+βnTnxn, we have

xn–Tnxn= 

βn

yn–xn.

Because of the assumption that  <a≤βn≤, we have

lim

n→∞xn–Tnxn= .

Since{xn}is uniformly closed, thenx∗∈F.

Step . We show thatx∗=f_Fx. Sincex∗=f_C_¯x∈FandFis a nonempty closed convex

subset ofC¯ =∞_n_=Cn, we conclude thatx∗=fFx. This completes the proof.

Corollary .([]) Let C be a closed convex subset of a uniformly convex and smooth Banach space E,let T be anα-nonexpansive mapping of C into itself such that F(T)=∅,

letα= (α,α, . . . ,αN)such thatαi≥,i= , , . . . ,N,α> ,and N

i=αi= .For any given Gauss x∈E,C=C,and x=Cx,deﬁne a sequence{xn}in C by the following

algorithm:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

yn= ( –βn)xn+βnTxn,

Cn+={z∈Cn:yn–z ≤ xn–z}, xn+=Cn+x, n∈N,

(.)

where <a≤βn≤for all n∈N.Ifα> N_–√

,then{xn}converges strongly to x

∗₌_F_x

.

Proof SubstitutingTtoTnin the proof of Theorem . and puttingf(x)≡, we can draw

from Theorem . the desired conclusion immediately.

Remark . Theorem . extends the main results of [] from a single mapping to a countable family of mappings and from the generalized projection operator to the modi-ﬁed generalizedf-projection operator by a new method.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal manuscript.

Acknowledgements

This project is supported by the National Natural Science Foundation of China under grant (11071279).

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