The split common fixed point problem for asymptotically nonexpansive semigroups in Banach spaces

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R E S E A R C H

Open Access

The split common ﬁxed point problem for

asymptotically nonexpansive semigroups in

Banach spaces

Lin Wang

1*

_{and Zhaoli Ma}

2

*_{Correspondence:}

[email protected]

1_{College of Statistics and}

Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, P.R. China

Abstract

In this paper, we propose an iteration method for ﬁnding a split common ﬁxed point of asymptotically nonexpansive semigroups in the setting of two Banach spaces, and we obtain some weak and strong convergence theorems of the iteration scheme proposed. The results presented in the paper are new and improve and extend some recent corresponding results.

MSC: 47H09; 47J25

Keywords: split common ﬁxed point problem; convergence; asymptotically nonexpansive semigroup; Banach space

1 Introduction

LetEbe a real normed linear space andCbe a nonempty closed convex subset ofE. The mappingT:C→Cis said to be nonexpansive if for allx,y∈C

Tx–Ty ≤ x–y. (.)

The mappingT:C→Cis said to be asymptotically nonexpansive if there exists a se-quence{kn} ⊂[,∞) withlimn→∞kn=  such that for allx,y∈Cand eachn≥

Tnx–Tny≤knx–y. (.)

The class of nonexpansive mappings is one of the most important classes of mappings in nonlinear science. The class of asymptotically nonexpansive mappings is an important generalization of the class of nonexpansive mappings, which was introduced by Goebel and Kirk [] in . They proved that ifCis a nonempty closed convex subset of a real uniformly convex Banach space andTis an asymptotically nonexpansive mapping, then

Thas a ﬁxed point.

Example .([]) LetCbe a unit ball in a real Hilbert spacel_{and let}_S_:_C_→_C_{be a} mapping deﬁned by

S: (x,x, . . .)→

,x_,ax,ax, . . .

,

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It is proved in Goebal and Kirk [] that:

(i) Sx–Sy ≤x–y, for allx,y∈C;

(ii) Snx–Sny ≤n_j_=ajx–y, for allx,y∈C,∀n≥.

Takingaj= 

– 

j–,j≥, it is easy to see that ∞

j=aj=_. So we can takek=  and

kn= n_j_=aj,n≥, then

lim

n→∞kn=nlim→∞ n

j= –



j– _{= .} _(.)

ThereforeS is an asymptotically nonexpansive mapping fromC into itself withF(S) = {(, , . . . , , . . .)}.

Deﬁnition .([]) A one-parameter familyF:={T(t) :t≥}ofEinto itself is called a strongly continuous semigroup of Lipschitzian mappings onEif it satisﬁes the following conditions:

(i) T()x=x, for allx∈E;

(ii) T(s+t) =T(s)T(t), for alls,t≥;

(iii) for eachx∈E, the mappingt→T(t)xis continuous;

(iv) for eacht> , there exists a bounded measurable functionL(t) : [,∞)→[,∞)

such that

T(t)x–T(t)y≤L(t)x–y, for allx,y∈E. (.)

If the bounded measurable function L(t) in (.) is such thatL(t)≥ for each t> ,

L(t) is nonincreasing int, andlimt→∞L(t) = , then the strong continuous semigroup of

Lipschitzian mappings is said to be an asymptotically nonexpansive semigroup. We denote byF(F) the set of all common ﬁxed points ofF, that is,

F(F) :=x∈E:T(t)x=x, ≤t<∞=

t≥

FT(t). (.)

IfFsatisﬁes (i)-(iii) and

lim sup t→∞,x∈D

T(t)x–T(s)T(t)x= , for alls>  and boundedD⊆C, (.)

thenFis called uniformly asymptotically regular onC.

Example .([] (Example of asymptotically nonexpansive semigroup)) LetEbe an uni-formly convex Banach space which admits a weakly continuous duality mapping. LetL(E) be the space of all bounded linear operators onE. For∈L(E), deﬁneF:={T(t) :t∈R+}

of bounded linear operators by using the following exponential expression:

T(t) =e–t:=

∞

k= (–)k

k! t

kk_.

Then the family F:={T(t) :t∈R+_} _{satisﬁes the semigroup properties. Moreover, this} family forms a one-parameter semigroup of self-mappings ofEbecause et_{= [}_e–t_]–_:

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In , in ﬁnite dimensional Hilbert spaces, Censor and Elfving [] introduced the split feasibility problem for modeling inverse problems which arise from phase retrievals and in medical imagine reconstruction []. It has been found that split feasibility problems can be used in various disciplines, such as imagine restoration, computer tomograph and radiation therapy treatment planning [–].

LetHandHbe two real Hilbert spaces,CandQbe nonempty closed convex subsets ofHandH, respectively. The split feasibility problem is formulated as ﬁnding a point

q∈Hsuch that

q∈C and Aq∈Q, (.)

whereA:H→His a bounded linear operator.

If (.) has a solution, it is not hard to see thatx∈Csolves (.) if and only if it solves the following ﬁxed point equation:

x=PCI–γA∗(I–PQ)Ax, x∈C, (.)

wherePCandPQare the projections ontoCandQ, respectively,γ is a positive constant,

andA∗denotes the adjoint ofA.

WhenCandQin (.) are the sets of ﬁxed points of two nonlinear mappings, andCand

Qare nonempty closed convex subsets ofHandH, respectively, then the split feasibility problem (.) is also said to be split common fixed point problem []. It is well known that each nonempty closed convex subset of a Hilbert space is the set of fixed points of its projection, therefore, the split common fixed point problem may be considered as a generalization of split feasibility problem.

In the setting of two Hilbert spaces, for demicontractive mappings, Moudafi [] pro-posed an iteration scheme and obtained a weak convergence theorem of the split common fixed point problem. Since then, the split common fixed point problems of other nonlinear mappings in the setting of two Hilbert spaces have been studied by some authors; see, for instance, [, –]. Especially, Cholamjiaket al.[] obtained a strong convergence the-orem of split common fixed point problem involving a uniformly asymptotically regular nonexpansive semigroup and a total asymptotically strict pseudo-contractive mapping in Hilbert spaces.

In , in the setting of one Hilbert space and one Banach space, Takahashi [] inves-tigated the split feasibility problem and split common null point problem, and obtained some strong and weak convergence theorems under some mild control conditions.

Recently, in the setting of two Banach spaces, Tanget al.[] obtained a weak conver-gence theorem and a strong converconver-gence theorem of the split common ﬁxed point problem involving a quasi-strict pseudo-contractive mapping and an asymptotically nonexpansive mapping under the following assumptions:

() Eis a real uniformly convex and-uniformly smooth Banach space having the

Opial property and the best smoothness constantksatisfying <k<√

.

() Eis a real Banach space.

() A:E→Ebe a bounded linear operator andA∗is the adjoint ofA.

() S:E→Eis an{ln}-asymptotically nonexpansive mapping with{ln} ⊂(,∞)and

ln→.T:E→Eis aτ-quasi-strict pseudo-contractive mapping withF(S)=∅

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Theorem .([]) Let E,E,A,S,T,and{ln}be the same as above.For each x∈E,let {xn}be the sequence generated by

zn=xn+γJ_–A∗J(T–I)Axn,

xn+= ( –αn)zn+αnSn(zn), ∀n≥,

(.)

where{αn}is a sequence in(, )withlim infn→∞αn( –αn) > ,γ is a positive constant satisfying  <γ <min{– k

A,–_Aτ}, {ln} is a sequence in[,∞) with L=supn≥{ln} and

∞

n=(ln– ) <∞.

(I) If={p∈F(S) :Ap∈F(T)} =φ,then the sequence{xn}converges weakly to a point

x∗∈.

(II) In addition,if={p∈F(S) :Ap∈F(T)} =φandSis semi-compact,then{xn}

converges strongly to a pointx∗∈.

This naturally brings about the following question:

Question Can we obtain the convergence results of split common ﬁxed point problem for asymptotically nonexpansive semigroups in the setting of two Banach spaces?

In this paper, motivated and inspired by the recent research going on in the direction of split feasibility problems and split common fixed point problems, we construct an it-eration scheme to approximate a split common fixed point of two asymptotically nonex-pansive semigroups in the setting of two Banach spaces. Under some suitable conditions on parameters, the iteration scheme proposed is shown to converge strongly and weakly to a split common fixed point of asymptotically nonexpansive semigroups in two Banach spaces.

2 Preliminaries

We now recall some deﬁnitions and elementary facts which will be used in the proofs of our main results.

LetEbe a real Banach space with the dualE∗. The normalized duality mappingJfrom

Eto E∗is deﬁned by

Jx=x∗∈E∗:x,x∗=x=x∗, ∀x∈E, (.)

where·,·denotes the generalized duality pairing betweenEandE∗.

A Banach spaceEis said to be strictly convex ifx+_y≤ for allx,y∈U={z∈E:z= } withx=y. The modulus of convexity ofEis deﬁned by

δE( ) =inf

 – (x+y)

:x,y ≤,x–y ≥

,

for all ∈[, ].Eis said to be uniformly convex ifδE() = , andδE( ) >  for all  < ≤.

A Hilbert space is -uniformly convex, whileLp _is_max_{_p_{, }-uniformly convex for every} p> .

LetρE: [,∞)→[,∞) be the modulus of smoothness ofEdeﬁned by

ρE(t) =sup

 

x+y+x–y–  :x∈U,y ≤t

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A Banach spaceEis said to be uniformly smooth if ρE(t)

t → ast→. A typical

exam-ple of uniformly smooth Banach space isLp, wherep> . More precisely,Lp ismin{p,

}-uniformly smooth for everyp> . Letqbe a ﬁxed real number withq> , then a Banach spaceEis said to beq-uniformly smooth if there exists a constantc>  such thatρE(t)≤ctq

for allt> . It is well known that everyq-uniformly smooth Banach space is uniformly smooth.

Lemma .([]) Given a number r> .A real Banach space E is uniformly convex if

and only if there exists a continuous strictly increasing function g: [,∞)→[,∞)with g() = such that

tx+ ( –t)y≤tx+ ( –t)y–t( –t)gx–y,

for all x,y∈E,t∈[, ],withx ≤r andy ≤r.

LetEandEbe two real Banach spaces,CandQbe nonempty closed convex subsets ofEandE, respectively.A:E→Eis a bounded linear operator such thatA= . Then the split common ﬁxed point problem is to ﬁnd a pointq∈Ewith the property:

q∈F(S) and Aq∈F(T), i.e.,q∈F(S)∩A–F(T), (.)

whereF(S) andF(T) denote the sets of ﬁxed points ofSandT, respectively.

LetT :C→Cbe a mapping withF(T)=∅. ThenTis said to be demiclosed at zero if for any{xn} ⊂Cwithxnxandxn–Txn →,x=Tx.

A mappingT:C→Cis said to be semi-compact, if for any sequence{xn}inCsuch that

xn–Txn → (n→ ∞), there exists a subsequence{xn_j}of{xn}such that{xx_j}converges strongly tox∗∈C.

A Banach spaceEis said to satisfy Opial property if for any sequence{xn}inE,xnx, for anyy∈Ewithy=x, we have

lim inf

n→∞ xn–x<lim infn→∞ xn–y. (.)

Lemma .([]) Let E be a real uniformly convex Banach space,C be a nonempty closed

subset of E,and let T:C→C be an asymptotically nonexpansive mapping.Then I–T is demiclosed at zero,that is,if{xn} ⊂C converges weakly to a point p∈C andlimn→∞xn– Txn= ,then p=Tp.

Lemma .([]) Let{an}and{αn}be two nonnegative real number sequences and satisfy

an+≤( +αn)an, ∀n≥,

where an≥,αn≥and

∞

n=αn<∞.Then () limn→∞anexists;

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Lemma .([]) Let E be a-uniformly smooth Banach space with the best smoothness constants K> .Then the following inequality holds:

x+y≤ x+ y,Jx+ Ky, ∀x,y∈E. (.)

3 Main results

Theorem . Let E be a real uniformly convex and-uniformly smooth Banach space

satisfying Opial’s condition and with the best smoothness constant k satisfying <k<√

,

Ebe a real Banach space,and A:E→Ebe a bounded linear operator and A∗be the

ad-joint of A.Let{S(t) :t≥}:E→Ebe a uniformly asymptotically regular asymptotically

nonexpansive semigroup with a bounded measurable function L()(t) : [,∞)→[,∞) sat-isfying limt→∞L()(t) = and C:=t≥F(S(t))=∅, {T(t) :t≥}:E→E be a

uni-formly asymptotically regular family of asymptotically nonexpansive semigroup with a bounded measurable function L()(t) : [,∞)→[,∞)satisfyinglimt→∞L()(t) =  and Q:=_t_≥_F(T(t))=∅,respectively.Let{xn}be a sequence generated by x∈E,

zn=xn+γJ_–A∗J(T(tn) –I)Axn, ∀n≥,

xn+= ( –αn)zn+αnS(tn)(zn),

(.)

where{tn}is a sequence of real numbers,{αn}is a sequence in(, ),andγ is a positive constant satisfying

() tn> andlimn→∞tn=∞; () L(t) =max{L()₍_t_),_L()₍_t_)}_and∞

n=(L(tn) – ) <∞;

() M=sup_nL₍_t

n),lim infn→∞αn( –αn) > and <γ <min{–k



A_M,_A}.

(I) If={p∈C:Ap∈Q} =∅,then the sequence{xn}converges weakly to a split common ﬁxed pointx∗∈.

(II) In addition,if={p∈C:Ap∈Q} =∅and there exists at least one

S(t)∈ {S(t) :t≥}that is semi-compact,then{xn}converges strongly to a split common ﬁxed pointx∗∈.

Proof Now we prove the conclusion (I). We shall divide the proof into four steps.

Step. We ﬁrst show that the limitlimn→∞xn–pexists for anyp∈.

For any givenp∈, thenp∈CandAp∈Q. It follows from (.), (.), and Lemma . that

zn–p=(xn–p) +γJ_–A∗J

T(tn) –IAxn

≤γJ_–A∗J

T(tn) –IAxn+ γxn–p,A∗J

T(tn) –IAxn

+ kxn–p

≤γAT(tn) –I

Axn



+ γAxn–Ap,J

T(tn) –I

Axn

+ kxn–p

=γAT(tn) –I

Axn



+ kxn–p

+ γAxn–T(tn)Axn+T(tn)Axn–T(tn)Ap,J

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=γAT(tn) –IAxn– γAxn–T(tn)Axn

+ γT(tn)Axn–T(tn)Ap,J

T(tn) –IAxn+ kxn–p

≤γAT(tn) –IAxn– γT(tn) –IAxn

+γT(tn)Axn–T(tn)Ap+T(tn) –IAxn+ kxn–p

≤γγA– T(tn) –IAxn+γAL(tn)xn–p+ kxn–p

=γAL(tn) + kxn–p–γ –γAT(tn) –IAxn. (.)

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

xn+–p=zn–p+αn

S(tn)zn–zn

≤( –αn)zn–p+αnS(tn)zn–p



–αn( –αn)gzn–S(tn)zn

≤( –αn)zn–p+αnL(tn)zn–p–αn( –αn)gzn–S(tn)zn

= +αn

L(tn) – 

zn–p–αn( –αn)gzn–S

n tn

zn

≤ +αn

L(tn) – γAL(tn) + kxn–p

–γ –γAT(tn) –IAxn–αn( –αn)gzn–S(tn)zn

≤ +αn

L(tn) – γAM+ kxn–p

– +αn

L(tn) – γ –γAT(tn) –IAxn

–αn( –αn)gzn–S(tn)zn. (.)

Since limtn→∞L(tn) = ,

∞

n=(L(tn) – ) < ∞,  < k< √,  < γ < –k

A_M, so  < γA_M_{+ }_k_{< , and from (.) and Lemma . we see that the}_lim

n→∞xn–p

ex-ists. This implies that{xn}is bounded. Further, it follows from (.) that{zn}is bounded,

too.

Step. We prove thatlimn→∞xn+–xn=  andlimn→∞zn+–zn= . It follows from (.) that

xn+–p≤

 +αn

L(tn) – γAM+ kxn–p

–γ –γAT(tn) –IAxn–αn( –αn)gzn–S(tn)zn. (.)

From (.), we have

γ –γAT(tn) –IAxn+αn( –αn)gzn–S(tn)zn

≤ +αn

L(tn) – 

γAL(tn) + k

xn–p–xn+–p

=xn–p+αn

L(tn) – γAL(tn) + kxn–p–xn+–p. (.)

This implies that

lim

n→∞T(tn) –I

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and

lim

n→∞gzn–S(tn)zn= . (.)

By virtue of Lemma . and the property ofg, we may get

lim

n→∞zn–S(tn)zn= . (.)

It follows from (.) that

xn+–xn=( –αn)zn+αnS(tn)zn–xn

=( –αn)

xn+γJ_–A∗J

T(tn) –IAxn+αnS(tn)zn–xn

=( –αn)γJ–A∗J

T(tn) –IAxn+αn

S(tn)zn–xn

=( –αn)γJ–A∗J

T(tn) –IAxn+αn

S(tn)zn–zn+αn(zn–xn)

=( –αn)γJ–A∗J

T(tn) –IAxn+αn

S(tn)zn–zn

+αnγJ–A∗J

T(tn) –I

Axn

=γJ_–A∗J

T(tn) –I

Axn+αn

S(tn)zn–zn

≤γJ_–A∗J

T(tn) –I

Axn+αnS(tn)zn–zn. (.)

It follows from (.), (.), and (.) that

lim

n→∞xn+–xn= . (.)

In addition, since

T(tn) –IAxn+=T(tn) –I

Axn+–

T(tn) –IAxn+T(tn) –IAxn

≤L(tn) + Axn+–xn+T(tn) –I

Axn, it follows from (.) and (.) that

lim

n→∞T(tn) –I

Axn+= . (.)

Similarly,

zn+–zn

=xn++γJ–A∗J

T(tn) –I

Axn+–xn–γJ–A∗J

T(tn) –I

Axn

≤(xn+–xn) +γJ–A∗J

T(tn) –I

Axn+–γJ–A∗J

T(tn) –I

Axn

≤ xn+–xn+γJ–A∗J

T(tn) –I

Axn++γJ–A∗J

T(tn) –I

Axn. (.)

In view of (.), (.), (.), and (.), we have

lim

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In addition,

xn–zn=J(xn–zn)=γA∗J

T(tn) –IAxn≤γAT(tn) –IAxn, (.)

from (.), we have

lim

n→∞xn–zn= . (.)

Step. We prove thatlim_n_→∞zn–S(t)zn=  andlimn→∞(T(t) –I)Azn=  for all t≥.

Since {S(t) : t ≥ } and {T(t) : t ≥ } are uniformly asymptotically regular, and

lim_n_→∞tn=∞, for allt≥,

lim

n→∞S(t)S(tn)zn–S(tn)zn≤_nlim sup_→∞_,_x_∈_CS(t)S(tn)x–S(tn)x=  (.)

and

lim n→∞T(t)

T(tn) –IAzn–T(t) –IAzn

≤ lim sup n→∞,x∈C

T(t)T(tn) –IAx–T(t) –IAx= . (.)

Since{S(t)x}is continuous ontfor allx∈E, and

zn–S(t)zn≤zn–S(tn)zn+S(tn)zn–S(t)S(tn)zn

+S(t)S(tn)zn–S(t)zn, (.)

it follows from (.) and (.) that

lim

n→∞zn–S(t)zn= . (.)

Similarly,

lim

n→∞T(t) –I

Axn= . (.)

Step. We prove that{xn}converges weakly to a pointx∗∈.

By the reﬂexivity of Banach spaceEand boundedness of{xn}, there exists a subsequence

{xni}of{xn}converging weakly tox∗. By using (.) this implies that{zni}of{zn}converges weakly tox∗, too. SinceS(t) is asymptotically nonexpansive for allt≥, it is demiclosed at zero, we know from Lemma . thatx∗∈F(S(t)).

On the other hand, sinceAis a bounded linear operator, we know that{Axni}converges weakly toAx∗. It follows from (.) thatlimni→∞(T(t) –I)Axni= . SinceT(t) is

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In fact, if there exists another subsequence{xnj}of{xn}such that{xnj}converges weakly toy∗∈E, we also know thaty∗∈F(T(t)). By the assumption thatEsatisﬁes Opial’s con-dition, we have

lim inf ni→∞xni–x

∗_<_{lim inf} ni→∞xni–y

∗

=lim inf n→∞xn–y

∗

=lim inf nj→∞

_xnj_–_y∗

<lim inf ni→∞xni–x

∗

=lim inf n→∞xn–x

∗₌_{lim inf} ni→∞xni–x

∗_. _(.)

This is a contradiction. Therefore{xn}converges weakly tox∗∈. The proof of conclusion (I) is completed.

Next, we prove conclusion (II).

Since there exists at least oneS(t)∈ {S(t) :t≥}that is semi-compact andlimn→∞zn– S(t)zn=  for allt≥, there exists subsequence{znj}of{zn}such that{znj}converges strongly to μ∗∈E. By using (.) again, we know that the subsequence {xnj} of {xn} converges strongly toμ∗. Due to{xn}converging weakly tox∗, we obtainμ∗=x∗. Since limn→∞xn–x∗exists andlimn→∞xnj–x∗= , we know that{xn}converges strongly

tox∗∈. This completes the proof of conclusion (II).

Corollary . Let E be a real uniformly convex and-uniformly smooth Banach space

satisfying Opial’s condition and with the best smoothness constant k satisfying <k<√

,

Ebe a real Banach space,and A:E→Ebe a bounded linear operator and A∗ is the

adjoint of A,{S(t) :t≥}:E→Eand{T(t) :t≥}:E→Ebe two-parameter

nonex-pansive semigroups satisfying C:=_t_≥_F(S(t))=∅and Q:=_t_≥_F(T(t))=∅,respectively.

Let{xn}be a sequence generated by x∈E,

zn=xn+γJ–

 A∗J(T(tn) –I)Axn, ∀n≥,

(.)

where{tn} is a sequence of real numbers satisfying tn> andlimn→∞tn=∞,{αn} is a sequence in(, )satisfyinglim infn→∞αn( –αn) > ,γ is a positive constant satisfying

 <γ<–_Ak.

(I) If={p∈C:Ap∈Q} =∅,then the sequence{xn}converges weakly to a split common ﬁxed pointx∗∈.

(II) In addition,if={p∈C:Ap∈Q} =∅and there exists at least one

S(t)∈ {S(t) :t≥}that is semi-compact,then{xn}converges strongly to a split common ﬁxed pointx∗∈.

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Lemma .([]) Let E be a smooth,strictly convex,and reﬂexive Banach space,and K be a nonempty,closed,and convex subset of E.Let x∈E and z∈K.Then the following conditions are equivalent:

(i) z=PKx;

(ii) z–y,J(x–z) ≥,∀y∈K,whereJis the normalized duality mapping onE.

Deﬁnition .([]) LetEbe a smooth, strictly convex, and reﬂexive Banach space, and

Kbe a nonempty, closed and convex subset ofE. LetS:K→Kbe a nonlinear mapping withF(S) being a nonempty, closed, and convex subset ofKandV:K→Kbe a nonlinear mapping. The so-called hierarchical variational inequality problem for a mappingSwith respect to a mappingVin Banach spaces is to ﬁndx∈F(S) such that

x∗–x,JVx∗–x∗≥, ∀x∈F(S). (.)

By Lemma ., the hierarchical variational inequality problem in Banach space (.) is equivalent to the following ﬁxed point equation:

x∗=PF(S)V

x∗. (.)

LettingC=F(S) andQ=F(PF(S)V) (the ﬁxed point set ofPF(S)V) andA=I(the identity mapping onE), then the hierarchical variational inequality problem (.) for a mappingS

with respect to a mappingVin Banach space is equivalent to the following split common ﬁxed point problem in Banach space:

to ﬁndx∈Csuch thatx∈Q. (.)

Therefore the set of solutions of hierarchical variational inequality problem (.) is just the set of solutions of split common ﬁxed point problem (.).

In Theorem ., we takeE=E =E, A=I,T(t) =PF(S(t))V(t), J =J=J (whereJ is the normalized duality mapping on E), the following conclusion can be obtained from Theorem . immediately.

Theorem . Let E be a real uniformly convex and -uniformly smooth Banach space satisfying Opial’s condition and with the best smoothness constant k satisfying <k<_√

,

Let {S(t) :t≥}:E→E be uniformly asymptotically regular family of asymptotically nonexpansive semigroup with a bounded measurable function L()₍_t_{) : [,}_∞₎_→_[,_∞₎

satisfying limt→∞L()(t) =  and C:=t≥F(S(t))=∅. Let{V(t) :t≥}:E→E be a

semigroup and{T(t) :t≥}:={PF(S(t))V(t) :t≥}be uniformly asymptotically regular

family of asymptotically nonexpansive semigroup with a bounded measurable function L()₍_t_{) : [,}_∞)_→_[,_∞)_satisfying_lim

t→∞L()(t) = and Q:=t≥F(T(t))=∅.Let{xn}

be a sequence generated by x∈E,

zn=xn+γ(T(tn) –I)xn, ∀n≥,

(.)

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() tn> andlimn→∞tn=∞;

() L(t) =max{L()(t),L()(t)}and∞_n_=(L(tn) – ) <∞; () M=sup_nL₍_t

n),lim infn→∞αn( –αn) > ,and <γ <min{–k



A_M,_A}.

(I) If=∅(the set of solutions of hierarchical variational inequality problem(.)is nonempty),then the sequence{xn}converges weakly to a split common ﬁxed point

x∗∈.

(II) In addition,if={p∈C:Ap∈Q} =∅and there exists at least one

S(t)∈ {S(t) :t≥}that is semi-compact,then{xn}converges strongly to a split

common ﬁxed pointx∗∈.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Author details

1_{College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, P.R.}

China.2_{School of Information Engineering, The College of Arts and Sciences Yunnan Normal University, Kunming,}

Yunnan 650222, P.R. China.

Acknowledgements

The authors would like to express their thanks to the reviewers and editors for their helpful suggestions and advices. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).

Received: 30 January 2016 Accepted: 22 September 2016

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