Iterative methods for split variational inclusion and fixed point problem of nonexpansive semigroup in Hilbert spaces

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R E S E A R C H

Open Access

Iterative methods for split variational

inclusion and ﬁxed point problem of

nonexpansive semigroup in Hilbert spaces

Dao-Jun Wen

*

_{and Yi-An Chen}

*_{Correspondence:}

[email protected] College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China

Abstract

In this paper, we introduce a general iterative method for a split variational inclusion and nonexpansive semigroups in Hilbert spaces. We also prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common ﬁxed points of one-parameter nonexpansive semigroups, which also solves a class of variational inequalities as an optimality condition for a minimization problem. Moreover, a numerical example is given, to illustrate our methods and results, which may be viewed as a reﬁnement and improvement of the previously known results announced by many other authors.

MSC: 47H09; 47H10; 47J22

Keywords: split variational inclusion; nonexpansive semigroup; ﬁxed point; averaged mapping; general iterative method

1 Introduction

LetHandHbe real Hilbert spaces with inner product·,·and norm · , respectively. Recall that a mappingT:H→His called nonexpansive if

Tx–Ty ≤ x–y, ∀x,y∈H.

A one-parameter familyT :={T(s) : ≤s<∞}is said to be a nonexpansive semigroup onHif the following conditions are satisﬁed:

() T()x=xfor allx∈H;

() T(s+t) =T(s)T(t)for alls,t≥;

() T(s)x–T(s)y ≤ x–y, for allx,y∈Hands> ;

() for eachx∈H, the mappings→T(s)xis continuous.

Denote byFix(T) the common ﬁxed point set of the semigroupT,i.e.,Fix(T) :={x∈ H:T(s)x=x,∀s> }. It is well known thatFix(T) is closed and convex (see Lemma  in Browder []).

Recently, the ﬁxed point problem of nonexpansive mappings and its iterative methods have become an attractive subject, and various algorithms have been developed for solving variational inequalities and equilibrium problems; see [–] and the references therein. In

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, Marino and Xu [] introduced the following general iterative methods to approxi-mate a ﬁxed point of a nonexpansive mapping:

xn+=αnγf(xn) + (I–αnB)Txn, (.)

whereαn∈[, ] satisﬁes certain conditions,f is a contraction ofHinto itself, andBis a strongly positive bounded linear operator onH. Moreover, they prove that{xn}converges

strongly tox∗∈Fix(T), the unique solution of the following variational inequality:

(B–γf)x∗,x∗–w≤, ∀w∈Fix(T),

which is also the optimality condition of the minimization problem. Thereafter, Liet al.[] and Cianciarusoet al.[] modiﬁed the general iterative method (.) to the case of non-expansive semigroups and equilibrium problems. To obtain a mean ergodic theorem of nonexpansive mappings, Shehu [] proposed an iterative method for nonexpansive semi-groups, variational inclusions, and generalized equilibrium problems.

Recall also that a multi-valued mappingM:H→His called monotone if, for allx,y∈

H,u∈Mxandv∈Mysuch that

x–y,u–v ≥.

A monotone mappingMis maximal if theGraph(M) is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mappingMis maximal if and only if for (x,u)∈H×H,x–y,u–v ≥ for every (y,v)∈Graph(M) implies that

u∈Mx.

LetM:H→H be a multi-valued maximal monotone mapping. Then the resolvent mappingJM

λ :H→Hassociated withMis deﬁned by

JλM(x) := (I+λM)–(x), ∀x∈H,

for someλ> , whereIstands for the identity operator onH. Note that for allλ>  the resolvent operatorJλMis single-valued, nonexpansive, and ﬁrmly nonexpansive.

In , Moudaﬁ [] introduced the following split monotone variational inclusion prob-lem: Findx∗∈Hsuch that

∈f(x∗) +B(x∗),

y∗=Ax∗∈H: ∈f(y∗) +B(y∗), (.)

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Iff≡ andf≡, then problem (.) reduces to the following split variational inclusion problem: Findx∗∈Hsuch that

∈B(x∗),

y∗=Ax∗∈H: ∈B(y∗), (.)

which constitutes a pair of variational inclusion problems connected with a bounded linear operatorAin two diﬀerent Hilbert spacesHandH. The solution set of problem (.) is denoted byZ ={x∗∈H: ∈B(x∗),y∗=Ax∗∈H: ∈B(y∗)}.

Very recently, Byrneet al.[] studied the weak and strong convergence of the following iterative method for problem (.): For givenx∈Handλ> , compute iterative sequence {xn}generated by the following scheme:

xn+=JλB

xn+A∗

JλB–I

Axn

. (.)

In , Kazmi and Rizvi [] modiﬁed scheme (.) to the case of a split variational inclusion and the ﬁxed point problem of a nonexpansive mapping. To be more precise, they proved the following strong convergence theorem.

Theorem KR Let H and Hbe two real Hilbert spaces and A:H→Hbe a bounded

linear operator.Let f :H→H be a contraction mapping with constant ρ∈(, )and

T:H→Hbe a nonexpansive mapping such that=Fix(T)∩Z =∅.For a given x∈H

arbitrarily,let the iterative sequences{un}and{xn}be generated by

un=JλB[xn+A∗(JλB–I)Axn], xn+=αnf(xn) + ( –αn)Tun,

(.)

whereλ> and∈(, /L),L is the spectral radius of the operator A∗A,and A∗ is the adjoint of A; {αn} is a sequence in (, ) such that limn→∞αn = , ∞n=αn=∞, and

∞

n=|αn–αn–|<∞.Then the sequences{un}and{xn}both converge strongly to z∈, where z=Pf(z).

Inspired and motivated by research going on in this area, we introduce a modiﬁed gen-eral iterative method for a split variational inclusion and nonexpansive semigroups, which is deﬁned in the following way:

xn+=αnγf(xn) + (I–αnB)



sn



T(s)J_λBxn+A∗

J_λB–IAxn

ds, (.)

whereγ ∈[, ] andαn,βn∈[, ],Bis a strongly positive bounded linear operator onH.

Note that, ifγ = ,B=I, andT(s) =T, a nonexpansive mapping, scheme (.) reduces to the approximate method (.), which is mainly due to Byrneet al.[] and has been applied by Kazmi and Rizvi [].

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Moreover, a numerical example is given to illustrate our algorithm and our results, which improve and extend the corresponding results of [, , , , ] and many others.

2 Preliminaries

LetCbe a nonempty, closed, and convex subset of a real Hilbert spaceH. For every point

x∈H, there exists a unique nearest point inC, denoted byPC, such that

x–PCx ≤ x–y, ∀y∈C.

ThenPCis called the metric projection ofHontoC. It is well known thatPCis a

nonex-pansive mapping and the following inequality holds:

x–u,y–u ≤, ∀y∈C,

if and only ifu=PCxfor givenx∈Handu∈C.

Recall that a mappingf :H→H is a contraction, if there exists a constantρ∈(, ) such that

f(x) –f(y)≤ρx–y, ∀x,y∈H.

Throughout the rest of this paper, we always assume thatBis strongly positive; that is, there is a constantγ >  such that

Bx,x ≥γx, ∀x∈H.

A mappingS:H→His said to be averaged if and only if it can be written as the av-erage of the identity mapping and a nonexpansive mapping,i.e.,S:= ( –α)I+αT where α∈(, ) andT:H→His nonexpansive andIis the identity operator onH. We note that averaged mappings are nonexpansive. Further, ﬁrmly nonexpansive mappings (in par-ticular, projections on nonempty, closed, and convex subsets and resolvent operators of maximal monotone operators) are averaged.

In order to prove our main results, we need the following lemmas and propositions.

Lemma . Let Hbe a real Hilbert space.The following well-known results hold:

(i) x+y_≤_x_{+ }_y_{, (}_x₊_y₎_,_∀_x_,_y_∈_H ;

(ii) tx+ ( –t)y₌_t_x_{+ ( –}_t₎_y_–_t_{( –}_t₎_x_–_y_,_t_∈_{[, ]}_,_∀_x_,_y_∈_H .

Lemma .[, ] Let D be a nonempty,bounded, closed, and convex subset of a real Hilbert space H and letT :={T(s) : ≤s<∞}a nonexpansive semigroup on D,then for any u≥,

lim

t→∞sup_x_∈_D

_t t



T(s)x ds–T(u)

t



T(s)x ds= .

Lemma .[, ] Let S:H→H be averaged and T:H→H be nonexpansive;we

have:

(i) W= ( –α)S+αTis averaged,whereα∈(, ).

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Lemma .[] The split variational inclusion problem(.)is equivalent to ﬁnding x∗∈ Hsuch that y∗=Ax∗∈H:x∗=JλB(x∗)and y∗=J

B

λ (y∗)for someλ> .

Lemma .[] Let B be a strongly positive linear bounded operator on a Hilbert space H

with a coeﬃcientγ> and < <B–_._Then_I_– _B_≤_{ –}_γ_.

Lemma .[] Let C be a nonempty,closed,and convex subset of a Hilbert space H.

Assume that f :C→C is a contraction with a coeﬃcientρ∈(, ) and B is a strongly positive linear bounded operator with a coeﬃcientγ> .Then,for <γ <γ/ρ,

x–y, (B–γf)x– (B–γf)y≥(γ –γρ)x–y, ∀x,y∈H.

That is,B–γf is strongly monotone with coeﬃcientγ –γρ.

Lemma .[] Let{an}∞n=be a sequence of nonnegative real numbers such that

an+≤( –γn)an+γnbn+σn,

where{γn}∞n=⊂(, )and{bn}∞n=,{σn}∞n=are sequences inRsuch that

(i) limn→∞γn= and ∞n=γn=∞; (ii) lim sup_n_→∞bn≤;

(iii) σn≥and ∞n=σn<∞. Thenlimn→∞an= .

3 Main results

Theorem . Let H and H be two real Hilbert spaces,let A:H→H be a bounded

linear operator and B be a strongly positive bounded linear operator on Hwith constant γ > .Let B:H→H,B:H→Hbe maximal monotone mappings andT :={T(s) : ≤s<∞}be a one-parameter nonexpansive semigroup on Hsuch thatFix(T)∩Z =∅.

Assume that f :H→H is a contraction mapping with constantρ∈(, ).For anyα∈ (, ),deﬁne the mappingon Hby

(x) =αγf(x) + (I–αB)

t



T(s)J_λBx+A∗J_λB–IAxds,

where t> ,γ ∈(,γ_ρ),and∈(,_L),L is the spectral radius of the operator A∗A,and A∗ is the adjoint of A.Then the mappingis a contraction and has a unique ﬁxed point.

Proof SinceJ_λB andJ_λB are ﬁrmly nonexpansive, they are averaged. For∈(, /L), the mappingI+A∗(JλB –I)Ais averaged; seee.g.[]. It follows from Lemma .(ii) that the mappingJλB(I+A∗(J

B

λ –I)A) is averaged and hence nonexpansive. By Lemma ., for anyx,y∈H, we have

(x) –(y)=αγf(x) + (I–αB)

t



T(s)JλB

x+A∗JλB–I

Axds

–αγf(y) – (I–αB)

t



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≤αγf(x) –f(y)+ ( –αγ)JλB

x+A∗JλB–I

Ax

–J_λBy+A∗J_λB–IAy

≤αγρx–y+ ( –αγ)x–y

= –α(γ–γρ)x–y.

It follows fromγ ∈(,γ_ρ) thatis a contraction mapping. Therefore, by the Banach con-traction principle,(x) has a unique ﬁxed pointxα, that is,

xα=αγf(xα) + (I–αB) 

t



T(s)J_λBxα+A∗

J_λB–IAxα

ds.

Theorem . Let H and Hbe two real Hilbert spaces,let A:H→H be a bounded

linear operator and B be a strongly positive bounded linear operator on H with

con-stant γ > .Let B : H →H, B: H →H be maximal monotone mappings and

T :={T(s) : ≤s<∞} be a one-parameter nonexpansive semigroup on H such that =Fix(T)∩Z =∅.Assume that f :H →H is a contraction mapping with constant ρ ∈(, ),L is the spectral radius of the operator A∗A,and A∗ is the adjoint of A. For given x∈H,λ> ,γ ∈(,γρ),and∈(,



L),suppose that the sequences{αn} ⊂(, ) and{tn} ⊂(,∞)satisfy:

(i) lim_n_→∞αn= , ∞n=αn=∞,and ∞n=|αn–αn–|<∞; (ii) limn→∞sn= +∞andlimn→∞|sn–_ss_nn–|αn= .

Then the sequence{xn}generated by(.)converges strongly to q∈,which is the unique solution of the following variational inequality:

(B–γf)q,q–w≤, ∀w∈.

Proof Takingp∈=Fix(T)∩Z, we havep=J_λBp,Ap=J_λB(Ap), andT(s)p=p. From (.),un=JλB[xn+A∗(JλB–I)Axn], and Lemma ., we estimate

un–p=JλB

xn+A∗

J_λB–IAxn

–J_λBp

≤xn+A∗

J_λB–IAxn–p

≤ xn–p+ 

xn–p,A∗

JλB–I

Axn

+A∗JλB–I

Axn. (.)

By the deﬁnition ofAandA∗, we obtain

A∗J_λB–IAxn≤

J_λB–IAxn,AA∗

J_λB–IAxn

≤LJ_λB–IAxn,

J_λB–IAxn

=LJ_λB–IAxn



. (.)

Using a similar method to [, Theorem .] and [, Theorem .], we have

= xn–p,A∗

JλB–I

Axn

= A(xn–p),

JλB–I

Axn

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= A(xn–p) +

JλB–I

Axn–

JλB–I

Axn,

JλB–I

Axn

= JλBAxn–Ap,

JλB–I

Axn

–JλB–I

Axn  ≤  J

B

λ –I

Axn



–JλB–I

Axn



≤–J_λB–IAxn

 .

Combining (.) and (.), we obtain

un–p≤ xn–p+(L– )JλB–I

Axn. (.)

Settingwn=_s_n

sn

 T(s)undsforn≥, it follows from∈(, 

L) and (.) that

wn–p=

_s_n sn



T(s)un–T(s)p

ds≤ un–p ≤ xn–p. (.)

It follows from (.), (.), and Lemma . that

xn+–p=

αn

γf(xn) –Bp

+ (I–αnB)



sn



T(s)un–T(s)p

ds

≤αnγf(xn) –Bp+ ( –αnγ)

_s_n_snT(s)un–T(s)p

ds

≤αnγf(xn) –f(p)+αnγf(p) –Bp+ ( –αnγ)un–p

≤ –αn(γ –γρ)

xn–p+αnγf(p) –Bp.

By a simple induction, we have

xn–p ≤max

x–p, 

γ –γργf(p) –Bp

. (.)

Therefore,{xn}is bounded, and so are{un}and{wn}.

Now, we show thatlimn→∞xn+–xn= . From (.), we have

xn+–xn=αnγ

f(xn) –f(xn–)

+ (αn–αn–)γf(xn–)

+ (I–αnB)(wn–wn–) – (αn–αn–)Bwn–

≤αnγρxn–xn–+ ( –αnγ)wn–wn–

+|αn–αn–|

Bwn–+γf(xn–). (.)

On the other hand, forp∈, we have

wn–wn–=

_s_n_snT(s)un–T(s)un–

ds +  sn – 

sn–



T(s)un––T(s)p

ds

+ 

sn

sn–

T(s)un––T(s)p

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Given that  v–  w

w= –v–w

v , v,w= .

It follows from (.) that

wn–wn– ≤ un–un–+

_|

sn–sn–| sn

un––p. (.)

Moreover, for∈(,_L), mappingJ_λB[I+A∗(J_λB–I)A] is averaged and hence nonexpan-sive, then we have

un–un–=JλB

xn+A∗

JλB–I

Axn

–JλB

xn–+A∗

JλB–I

Axn–

≤JλB

I+A∗JλB–I

Axn–JλB

I+A∗JλB–I

Axn–

≤ xn–xn–. (.)

Combining (.), (.), and (.), we obtain

xn+–xn ≤αnγρxn–xn–+ ( –αnγ)

xn–xn–+

_|

sn–sn–| sn

un––p

+|αn–αn–|

Bwn–+γf(xn–)

≤ –αn(γ–γρ)

xn–xn–+

|αn–αn–|+|

sn–sn–| sn

M, (.)

whereM=max{supn∈N[Bwn–+γf(xn–)],supn∈Nun––p}. It follows from condi-tions (i)-(ii) and Lemma . that

lim

n→∞xn+–xn= . (.)

Next, we will show thatlimn→∞xn–un= . Note thatwn=_s_n

sn

 T(s)undsand

xn–wn ≤ xn–xn++xn+–wn

=xn–xn++αnγf(xn) + (I–αnB)wn–wn

≤ xn–xn++αnγf(xn) –Bwn.

Together with condition (i) and (.), we obtain

lim

n→∞xn–wn=nlim→∞

xn–



sn



T(s)unds

= . (.)

Observe that

xn–T(u)xn≤

xn–



sn



T(s)unds

+ sn sn 

T(s)unds–T(u)



sn



T(s)unds

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+T(u)

sn



T(s)unds–T(u)xn

≤xn–



sn



T(s)unds

+ sn sn 

T(s)unds–T(u)



sn



T(s)unds

.

It follows from (.) and Lemma . that

lim

n→∞xn–T(u)xn= . (.)

By (.), (.), and Lemma ., we have

xn+–p=wn–p+αn

γf(xn) –Bwn



≤ wn–p+ αn

γf(xn) –Bwn,xn+–p

≤ un–p+ αn

≤xn–p+(L– )JλB–I

Axn

 + αn

≤ xn–p–( –L)JλB–I

Axn



+ αnM, (.)

whereM=max{supn∈Nγf(xn) –Bwn,supn∈Nxn+–p}and∈(,_L), which implies that

( –L)JλB–I

Axn



≤ xn–p–xn+–p+ αnM ≤ xn+–xn

xn–p+xn+–p

+ αnM. (.)

It follows from condition (i) and (.) that

lim

n→∞J

B

λ –I

Axn= . (.)

Furthermore, using (.), (.), and∈(,_L), we obtain

un–p=JλB

xn+A∗

JλB–I

Axn

–JλBp 

≤un–p,xn+A∗

JλB–I

Axn–p

=  

un–p+xn+A∗

–un–p–

xn+A∗

≤  

un–p+xn–p+(L– )JλB–I

Axn



–un–xn–A∗

J_λB–IAxn



≤  

un–p+xn–p–

un–xn+A∗

JλB–I

Axn



– un–xn,A∗

JλB–I

Axn ≤  

un–p+xn–p–un–xn+ A(un–xn)JλB–I

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which implies that

un–p≤ xn–p–un–xn+ A(un–xn)JλB–I

Axn. (.)

It follows from (.) and (.) that

xn+–p≤ un–p+ αnM

≤ xn–p–un–xn+ A(un–xn)JλB–I

Axn+ αnM,

that is,

un–xn≤ xn–p–xn+–p+ A(un–xn)JλB–I

Axn+ αnM

≤ xn–xn+

xn–p+xn+–p

+ A(un–xn)JλB–I

Axn+ αnM.

Combining condition (i), (.), and (.), we have

lim

n→∞un–xn= . (.)

Since{xn}and{un}are bounded, we consider a weak cluster pointwof{xn}. Without

loss of generality, we may assume that subsequence{xnj}of{xn}converges weakly tow,

i.e.,xnjwasj→ ∞. From (.), we have{unj}of{un}, which converges weakly tow. Moreover,unj=J

B

λ [xnj+A∗(J

B

λ –I)Axnj] can be rewritten as

(xnj–unj) +A∗(J

B

λ –I)Axnj

λ ∈Bunj. (.)

By passing to the limitj→ ∞in (.) and by taking into account (.), (.), and the fact that the graph of a maximal monotone operator is weakly-strongly closed, we obtain ∈B(w). Furthermore, since{xn}and{un}have the same asymptotical behavior,{Axnj} weakly converges toAw. From (.) and the fact that the resolventJ_λB is nonexpansive, we obtainAw∈B(Aw). It follows from Lemma . thatw∈Z.

We now show thatlim sup_n_→∞γf(q) –Bq,xn–q ≤, whereq=P(I–B+γf)q. Note

that the subsequence{xnj}of{xn}converges weakly towand

lim sup

n→∞

γf(q) –Bq,xn–q

=lim

j→∞

γf(q) –Bq,xnj–q

. (.)

Assume thatw=T(u)w. By (.) and Opial’s property, we obtain

lim inf

j→∞ xnj–w<lim infj→∞

xnj–T(u)w

≤lim inf

j→∞

xnj–T(u)xnj+T(u)xnj–T(u)w

≤lim inf

j→∞ xnj–T(u)xnj+xnj–w

≤lim inf

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This is a contradiction. Thenw∈Fix(T). Consequently,w∈=Fix(T)∩Z. It follows from (.) that

lim sup

n→∞

γf(q) –Bq,xn–q

=γf(q) –Bq,w–q≤. (.)

On the other hand, we shall show that the uniqueness of a solution of the variational in-equality

(B–γf)x,x–w≤, w∈. (.)

Supposeq∈andqˆ∈both are solutions to (.), then

(B–γf)q,q–qˆ≤ (.)

and

(B–γf)qˆ,qˆ–q≤. (.)

Adding up (.) and (.) one gets

(B–γf)q– (B–γf)ˆq,q–qˆ≤. (.)

By Lemma ., the strong monotonicity ofB–γf, we obtainq=qˆand the uniqueness is proved.

Finally, we show that{xn}converges strongly toqasn→ ∞. From (.), (.), and

Lem-ma ., we have (note thatwn=_s_n

sn

 T(s)unds)

xn+–q=

αnγf(xn) + (I–αnB)wn–q,xn+–q

=αn

γf(xn) –Bq,xn+–q

+(I–αnB)(wn–q),xn+–q

≤αnγ

f(xn) –f(q),xn+–q

+αn

γf(q) –Bq,xn+–q

+ ( –αnγ)wn–qxn+–q

≤αnγρxn–qxn+–q+αn

+ ( –αnγ)xn–qxn+–q

= –αn(γ–γρ)

xn–qxn+–q+αn

≤  –αn(γ–γρ)



xn–q+xn+–q

+αn

≤  –αn(γ–γρ)

 xn–q ₊

xn+–q ₊_α

n

.

It follows that

xn+–q≤

 – (γ –γρ)αn

xn–q+ αn

. (.)

From  < γ < γ_ρ, condition (i), and (.), we can arrive at the desired conclusion

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Theorem . Let Hand H be two real Hilbert spaces and A:H→H be a bounded

linear operator.Let B:H→H,B:H→H be maximal monotone mappings and

T :={T(s) : ≤s<∞}be a one-parameter nonexpansive semigroup on Hsuch that=

Fix(T)∩Z =∅.Assume that f :H→H is a contraction mapping with constantρ∈ (, ),L is the spectral radius of the operator A∗A,and A∗ is the adjoint of A.For given x∈H,λ> ,and∈(,_L),deﬁne{xn}in the following manner:

xn+=αnf(xn) + ( –αn)



sn



T(s)JλB

xn+A∗

JλB–I

Axn

ds, (.)

where the sequence{αn} ⊂(, )satisﬁes the following conditions: (i) lim_n_→∞αn= , ∞n=αn=∞,and ∞n=|αn–αn–|<∞; (ii) limn→∞sn= +∞andlimn→∞|sn–ssnn–|

 αn= .

Then the sequence{xn}converges strongly to q=Pf(q),which is equivalent to the unique

solution of the following variational inequality:

(I–f)q,q–w≤, ∀w∈.

Proof Puttingγ =  andB=I, iterative scheme (.) reduces to viscosity iteration (.). The desired conclusion follows immediately from Theorem .. This completes the

proof.

Theorem . Let Hand H be two real Hilbert spaces,let A:H→Hbe a bounded

linear operator and B be a strongly positive bounded linear operator on Hwith constant γ > .Let B:H→H,B:H→Hbe maximal monotone mappings and T:H→H

be a nonexpansive mapping such that=Fix(T)∩Z = ∅.Assume that f :H→His a

contraction mapping with constantρ∈(, ),L is the spectral radius of the operator A∗A,

and A∗is the adjoint of A.For given x∈H,λ> ,γ ∈(,γ_ρ),and∈(,_L),deﬁne{xn}in the following manner:

un=JλB[xn+A∗(JλB–I)Axn], xn+=αnγf(xn) + (I–αnB)Tun,

(.)

where the sequence {αn} ⊂(, ) satisﬁeslimn→∞αn= , n∞=αn=∞, and ∞n=|αn– αn–|<∞. Then the sequence{xn} converges strongly to q,which is the unique solution of the following variational inequality:

(B–γf)q,q–w≤, ∀w∈.

Proof Clearly, Theorem . is valid for a nonexpansive mapping. Therefore, the desired conclusion follows immediately from Theorem .. This completes the proof.

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[image:13.595.198.396.96.212.2]

Table 1 Numerical results for some initial pointsx1= –1, 0, 1, 2, 15

Iter.(n) x(1)n x(2)n x(3)n xn(4) xn(5)

0 –1.0000 0.0000 1.0000 2.0000 15.000

1 –0.5000 0.0000 0.5000 1.0000 7.5000

2 –0.1891 0.0000 0.1891 0.3781 2.8358

3 –0.0600 0.0000 0.6000 0.1199 0.8996

4 –0.0167 0.0000 0.0167 0.0334 0.2506

5 –0.0042 0.0000 0.0042 0.0084 0.0630

. . . .

8 0.0000 0.0000 0.0000 0.0001 0.0006

9 0.0000 0.0000 0.0000 0.0000 0.0001

10 0.0000 0.0000 0.0000 0.0000 0.0000

Remark . Theorems . and . improve and extend the main results of Marino and Xu [] for nonexpansive mappings, and [–] for nonexpansive semigroups in diﬀerent directions.

4 Numerical results

In this section, we give an example and numerical results to illustrate our algorithm and the main result of this paper.

Example . LetH=H=R. LetBx= xandBx= x. LetT :={T(s) : ≤s<∞}, whereT(s)x=_+_sx,∀x∈R. ThenB,B, andT satisfy all conditions of Theorem . and =Fix(T)∩L ={}. Let{xn}be the sequence generated byxand

xn+=αnγf(xn) + (I–αnB)



sn

   + sJ

B

λ

xn+A∗

J_λB–IAxn

ds. (.)

LetA=B=I, the identity operator, andf(x) = _x,∀x∈R. Puttingγ =λ= ,=_, and αn=√_n,sn=n, then scheme (.) reduces to

xn+=  √n

xn+

 n(

√

n– )ln( + n)xn

. (.)

Settingxn–x∗ ≤–as stop criterion, then we obtain the numerical results of scheme

(.) with diﬀerent initial pointsxin Table .

The computations are performed by Matlab Ra running on a PC Desktop Intel(R) Core(TM)i-M, CPU @. GHz,  MHz, . GB,  GB RAM.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

D-JW carried out the primary studies of a split variational inclusion and the ﬁxed point problem of nonexpansive semigroups and drafted the manuscript. Y-AC participated in algorithm design and convergence analysis. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The authors are grateful to the anonymous editor and referees for valuable remarks suggestions, which helped them very much in improving this manuscript. This work was supported by the National Science Foundation of China (11471059), Basic and Advanced Research Project of Chongqing (cstc2014jcyjA00037), Science and Technology Research Project of Chongqing Municipal Education Commission (KJ1400614, KJ1400618).

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