On split common solution problems: new nonlinear feasible algorithms, strong convergence results and their applications

(1)

R E S E A R C H

Open Access

On split common solution problems: new

nonlinear feasible algorithms, strong

convergence results and their applications

Zhenhua He

1

_{and Wei-Shih Du}

2*

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

National Kaohsiung Normal University, Kaohsiung, 824, Taiwan Full list of author information is available at the end of the article

Abstract

In this paper, we study and give examples for classes of generalized contractive mappings. We establish some new strong convergence theorems of feasible iterative algorithms for the split common solution problem (SCSP) and give some applications of these new results.

MSC: 47H06; 47J25; 47H09; 65K10

Keywords: Lipschitzian; demicontractive mapping; pseudocontractive mapping; quasi-nonexpansive mapping; nonexpansive mapping; split common solution problem (SCSP); iterative algorithm; strong convergence theorem

1 Introduction and preliminaries

LetKbe a closed convex subset of a real Hilbert spaceHwith the inner product·,·and the norm · . The following inequalities are known and useful.

• x+y≤ y+ x,x+y;

• x–y₌_x₊_y_{– }_x_,_y_{for all}_x_,_y_∈_H_;

• αx+ ( –α)y₌_α_x_{+ ( –}_α₎_y_–_α_{( –}_α₎_x_–_y_{for all}_x_,_y_∈_H_and_α_∈_{[, ].} For each pointx∈H, there exists a unique nearest point inK, denoted byPKx, such that

x–PKx ≤ x–y for ally∈K.

The mappingPK is called themetric projectionfromHontoK. It is well known thatPK

has the following properties:

(i) x–y,PKx–PKy ≥ PKx–PKyfor everyx,y∈H.

(ii) Forx∈Handz∈K,z=PKx⇔ x–z,z–y ≥for ally∈K.

(iii) Forx∈Handy∈K,

y–PKx+x–PKx≤ x–y. (.)

LetHandHbe two Hilbert spaces. LetA:H→HandA∗:H→Hbe two bounded linear operators.A∗is called the adjoint operator (or adjoint) ofAif

Az,w=z,A∗w for allz∈Handw∈H.

(2)

It is known that the adjoint operator of a bounded linear operator on a Hilbert space always exists and is bounded linear and unique. Moreover, it is not hard to show that ifA∗is an adjoint operator ofA, thenA=A∗. The symbolsNandRare used to denote the sets of positive integers and real numbers, respectively.

LetHandHbe two real Hilbert spaces. LetCbe a closed convex subset ofHandKbe a closed convex subset ofH. LetT:C→CwithF(T)=∅andS:K→KwithF(S)=∅ be two mappings. LetA:H→Hbe a bounded linear operator. The mathematical model ofthe split common solution problem(SCSP in short) is deﬁned as follows:

(SCSP) Findp∈Csuch thatTp=pandu:=Ap∈KsatisfyingSu=u.

In fact, SCSP contains several important problems as special cases and many authors have studied and introduced some new iterative algorithms for SCSP and presented some strong and weak convergence theorems for SCSP; see, for instance, [–] and the refer-ences therein. Motivated and inspired by their works, in this paper, we study and establish new strong convergence results by using new iterative algorithms of SCSP for pseudocon-tractive mappings andk-demicontractive mappings in Hilbert spaces.

The paper is divided into four sections. In Section , we study and give examples for classes of generalized contractive mappings. Some new strong convergence theorems of feasible iterative algorithms for SCSP are established in Section . Finally, some applica-tions and further remarks for our new results are given in Section . Consequently, in this paper, some of our results are original and completely diﬀerent from these known related results in the literature.

2 Classes of generalized contractive mappings and their examples

LetT be a mapping with domainD(T) and rangeR(T) in a Hilbert spaceH. Recall that T is said to be

(i) pseudocontractiveif

Tx–Ty,x–y ≤ x–y, ∀x,y∈D(T),

or, equivalently,

Tx–Ty≤ x–y+(I–T)x– (I–T)y, ∀x,y∈D(T);

(ii) demicontractiveif, for allx∈D(T)andp∈F(T),

Tx–p,x–p ≤ x–p

or, equivalently,

Tx–p≤ x–p+(I–T)x;

(iii) k-demicontractiveif there exists a constantk∈[, )such that

(3)

(iv) quasi-nonexpansiveif it is-demicontractive, that is,

Tx–p ≤ x–p for allx∈D(T)andp∈F(T);

(v) Lipschitzianif there existsL> such that

Tx–Ty ≤Lx–y, ∀x,y∈D(T);

(vi) nonexpansiveif it is Lipschitzian withL= ; (vii) contractiveif it is Lipschitzian withL< .

A Banach space (X, · ) is said to satisfyOpial’s conditionif, for each sequence{xn}in

Xwhich converges weakly to a pointx∈X, we have

lim inf

n→∞ xn–x<lim infn→∞ xn–y, ∀y∈X,y=x.

It is well known that any Hilbert space satisﬁes Opial’s condition.

Deﬁnition .(see []) LetKbe a nonempty closed convex subset of a real Hilbert space HandT be a mapping fromKintoK. The mappingT is said to bedemiclosedif, for any sequence{xn}which weakly converges toy, and if the sequence{Txn}strongly converges

toz, thenTy=z.

Remark . In Deﬁnition ., the particular case of demiclosedness at zero is frequently used in some iterative convergence algorithms, which is the particular case whenz=θ, the zero vector ofH; for more details, one can refer to [].

The following concept of zero-demiclosedness was introduced as follows.

Deﬁnition .(see [, Deﬁnition .]) LetK be a nonempty closed convex subset of a real Hilbert space andT be a mapping fromKintoK. The mappingT is calledzero -demiclosedif{xn}inKsatisfyingxn–Txn → andxnz∈KimpliesTz=z.

The following result was essentially proved in [], but we give the proof for the sake of completeness and the reader’s convenience.

Theorem .(see [, Proposition .]) Let K be a nonempty closed convex subset of a real Hilbert space with zero vectorθ.Then the following statements hold.

(a) LetTbe a mapping fromKintoK.ThenT is zero-demiclosed if and only ifI–Tis demiclosed atθ.

(b) LetTbe a nonexpansive mapping fromHinto itself.If there is a bounded sequence

{xn} ⊂Hsuch thatxn–Txn →asn→,thenT is zero-demiclosed.

Proof Obviously, the conclusion (a) holds. To see (b), since{xn} is bounded, there is a

subsequence{xnk} ⊂ {xn}andz∈H such thatxnk z. One can claimTz=z. Indeed, if

Tz=z, it follows from Opial’s condition that

lim inf

k→∞ xnk–z< lim infk→∞ xnk–Tz

≤lim inf

k→∞

xnk–Txnk+Txnk–Tz

(4)

=lim inf

k→∞ Txnk–Tz

≤lim inf

k→∞ xnk–z,

which is a contradiction. SoTz=zand henceTis zero-demiclosed.

Now, we give some examples to show the existence of these generalized contractive map-pings (i)-(vi) which also expound the relation between them.

Example A LetH=Rwith the absolute-value norm| · |andC= [–, ]. LetT:C→C

be deﬁned by

Tx=

x_{– } _if_x_∈_{[–, ],} – ifx∈[–, –].

ThenF(T) ={–}. Since

Tx– (–)≤x– (–)+ |Tx–x|

 _{for all}_x_∈_C_,

we know thatTis a_-demicontractive mapping. However, due to

T – 

– (–)>– – (–)

,

Tis not quasi-nonexpansive.

Example B LetH=Rwith the absolute-value norm| · |andC= [_, ]. LetT:C→Cbe deﬁned by

Tx=

x, ∀x∈C.

ThenF(T) ={}. Since

|Tx– |_{≤ |}_x_{– }_|₊ |Tx–x|

 _{for all}_x_∈_C_,

Tis a_-demicontractive mapping. Moreover,Tis also a pseudocontractive mapping.

Example C LetH=Rwith the absolute-value norm| · |. LetT:H→Hbe deﬁned by

Tx=

–√–( +x) ifx≤–, x+  ifx≥–.

It is easy to see that

|Tx–Ty| ≤ |x–y| for allx,y∈H.

(5)

The following example shows that there exists a continuous quasi-nonexpansive map-ping which is not nonexpansive.

Example D(see []) LetH=Rwith the absolute-value norm| · |andC= [, +∞). Deﬁne T:C→Cby

Tx=x _{+ }

 +x, ∀x∈C.

Obviously,F(T) ={}. It is easy to see that

|Tx– |= x

 +x|x– | ≤ |x– | for allx∈C

and

T() –T  

= 

>  –

 .

HenceTis a continuous quasi-nonexpansive mapping but not nonexpansive.

The following example shows that there exists a demicontractive mapping which is nei-ther pseudocontractive nork-demicontractive for allk∈[, ).

Example E LetH=Rwith the absolute-value norm| · |. LetT:H→Hbe deﬁned by

Tx=

x–x+  ifx∈(–∞, ],

x_+

+x ifx∈[, +∞).

|Tx– |≤ |x– |+|Tx–x| for allx∈H,

Tis a demicontractive mapping. However,T is not a pseudocontractive mapping due to the fact that whenx= – andy= –., we have

|Tx–Ty|>|x–y|+(x–Tx) – (y–Ty).

It is easy to see thatT is not ak-demicontractive mapping for allk∈[, ).

The following example shows that there exists a discontinuous pseudocontractive map-ping which is not a demicontractive mapmap-ping.

Example F LetH=Rwith the absolute-value norm| · |. LetT:H→Hbe deﬁned by

Tx=

x_{+ } _if_x_∈_(–_∞_{, ],} – –x ifx∈(, +∞).

ThenF(T) =∅. Due to

(6)

we know thatTis a discontinuous pseudocontractive mapping but not a demicontractive mapping.

The following example shows that there exists a pseudocontractive mapping which is notk-demicontractive for allk∈[, ).

Example G LetH=Rwith the absolute-value norm| · |. LetT:H→Hbe deﬁned by

Tx= ⎧ ⎪ ⎨ ⎪ ⎩

 –x _if_x_∈_{[, ],}  –x ifx∈[, ],  ifx∈[, +∞).

|Tx–Ty|≤ |x–y|+(I–T)x– (I–T)y for allx∈H,

Tis a pseudocontractive mapping. It is easy to see thatTis not ak-demicontractive map-ping for allk∈[, ).

The following example shows that there exists a discontinuousk-demicontractive map-ping for somek∈[, ) as well as being demiclosed atθwhich is neither pseudocontractive nor quasi-nonexpansive.

Example H LetH=Rwith the absolute-value norm| · |andC= [–, ]. LetT:C→C be deﬁned by

Tx= ⎧ ⎪ ⎨ ⎪ ⎩

x_{– } _if_x_∈_{[–, ],} –

 ifx= –  ,

– ifx∈[–, –_)∪(–_, –].

Then the following statements hold. (a) Tis discontinuous _-demicontractive. (b) Tis demiclosed atθ.

(c) Tis not pseudocontractive. (d) Tis not quasi-nonexpansive.

Proof Clearly,F(T) ={–}. Since

Tx– (–)≤x– (–)+

(I–T)x 

for allx∈C,

Tis a discontinuous_-demicontractive mapping and (a) is proved. Now, we verify (b). In fact, let{xn} ⊂[–, ] withxn→zandxn–Txn→ asn→ ∞. If allxn∈[–, ], we can

proveTz=zandz= –∈F(T) easily. If there exists a subsequence{xnk} ⊂[–, –], then,

fromxn–Txn→ asn→ ∞, we can ﬁnd a subsequence{xn_ki}of{xnk}such thatxn_ki= –

  for alli. Hence we have

(7)

which impliesz= –∈F(T). To see (c) and (d), note that

T –



–T – 

>–

– –  

+(I–T) – 

– (I–T) – 



and T –



– (–)> – 

– (–),

soT is neither pseudocontractive nor quasi-nonexpansive. The proof is completed.

3 New feasible iterative algorithms for SCSP and strong convergence theorems

In this section, we establish some new strong convergence theorems by using feasible it-erative algorithms for SCSP.

Theorem . Let Hand Hbe two real Hilbert spaces andθibe the zero vector of Hifor

i= , .Let C be a nonempty closed convex subset of Hand A:H→Hbe a bounded lin-ear operator with its adjoint B.Let T:C→C be a Lipschitzian pseudocontractive mapping with Lipschitz constant L> andF(T)=∅,and let S:H→Hbe a k-demicontractive mapping withF(S)=∅which is demiclosed atθ.Let C=C and{xn}be a sequence

gen-erated by the following algorithm:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C chosen arbitrarily, yn= ( –α)xn+αTxn,

zn=βxn+ ( –β)Tyn,

wn=PC(zn+ξB(S–I)Azn),

Cn+={v∈Cn:wn–v ≤ zn–v ≤ xn–v},

xn+=PCn+(x), ∀n∈N,

(.)

where <  –β<α< 

√+L,ξ∈(,

–k

B)and PCnis the projection operator from Hinto Cnfor n∈N.Suppose that

=p∈F(T) :Ap∈F(S)=∅.

Then there exists q∈such that (a) xn→qasn→ ∞,

(b) Axn→Aqasn→ ∞.

Proof We will show the conclusion by proceeding with the following steps. Step . For anyp∈, we prove

wn–p≤ zn–p–ξ

 –k–ξB(S–I)Azn



. (.)

Indeed, since

wn–p

≤zn+ξB(S–I)Azn–p

(8)

=zn–p+ξB(S–I)Azn



+ ξzn–p,B(S–I)Azn



+ ξAzn–Ap, (S–I)Azn



+ ξAzn–Ap+ (S–I)Azn– (S–I)Azn, (S–I)Azn



+ ξSAzn–Ap, (S–I)Azn

– ξ(S–I)Azn



≤ zn–p+ξB(S–I)Azn



+ ξSAzn–Ap, (S–I)Azn

– ξ(S–I)Azn



and

ξSAzn–Ap, (S–I)Azn

=ξSAzn–Ap+(S–I)Azn



–Azn–Ap

≤ξAzn–Ap+k(S–I)Azn+(S–I)Azn–Azn–Ap

≤ξ–Azn–Azn+k(S–I)Azn+(S–I)Azn



=ξk(S–I)Azn+(S–I)Azn

 ,

we get

wn–p≤ zn–p–ξ

 –k–ξB(S–I)Azn,

and our desired result is proved. Step . We prove

zn–p ≤ xn–p for alln∈N. (.)

For anyn∈N, by (.), we have

zn–p

=βxn–p+ ( –β)Tyn–p– ( –β)βTyn–xn

≤βxn–p+ ( –β)yn–p+ ( –β)Tyn–yn– ( –β)βTyn–xn

≤βxn–p+ ( –β)

( –α)xn–p+αTxn–xn– ( –α)αTxn–xn

+ ( –β)Tyn–yn– ( –β)βTyn–xn

≤ xn–p+ ( –β)

αTxn–xn– ( –α)αTxn–xn

– ( –β)βTyn–xn

+ ( –β)( –α)(xn–Tyn) +α(Txn–Tyn)



≤ xn–p+ ( –β)

– ( –β)βTyn–xn

+ ( –β)( –α)xn–Tyn+αTxn–Tyn– ( –α)αTxn–xn

≤ xn–p+ ( –β)

– ( –β)βTyn–xn

+ ( –β)( –α)xn–Tyn+αLxn–yn– ( –α)αTxn–xn

≤ xn–p+ ( –β)

(9)

+ ( –β)( –α)xn–Tyn+αLxn–Txn– ( –α)αTxn–xn

=xn–p– ( –β)(α+β– )Tyn–xn

– ( –β)α – α–αLTxn–xn. (.)

Sinceα+β>  andα< 

√+L, from (.), we havezn–p

_≤_x

n–p, or, equivalently,

zn–p ≤ xn–p. (.)

Step . We show thatCnis a nonempty closed convex set for anyn∈N.

For anyp∈, by taking into account (.) and (.), we obtain

wn–p ≤ zn–p ≤ xn–p for alln∈N.

So we know⊂Cnand henceCn=∅for alln∈N. It is easy to verify thatCnis closed and

convex for alln∈N.

Step . We prove that{xn}is a Cauchy sequence inCandxn→qasn→ ∞for some

q∈C.

Since⊂Cn+⊂Cnandxn+=PCn+(x)⊂Cn, we get

xn+–x ≤ p–x for allp∈

and

xn–x ≤ xn+–x for alln∈N,

which show that{xn}is bounded and{xn–x}is nondecreasing in [,∞). So

lim

n→∞xn–x ≥

exists. For anym,n∈Nwithm>n, fromxm=PCm(x)⊂Cnand (.), we have

xm–xn+x–xn=xm–PCn(x) 

+x–PCn(x) 

≤ xm–x. (.)

Inequality (.) implies

lim

m,n→∞xn–xm= .

So{xn}is a Cauchy sequence. Clearly,

lim

n→∞xn+–xn= . (.)

By the completeness ofC, there existsq∈Csuch thatxn→qasn→ ∞.

Step . Finally, we show that the following hold: (i) q∈,

(10)

For anyn∈N, sincexn+=PCn+(x)∈Cn+⊂Cn, from (.), we have

zn–xn ≤ zn–xn++xn+–xn ≤xn+–xn (.)

and

wn–xn ≤ wn–xn++xn+–xn ≤xn+–xn. (.)

From inequalities (.), (.) and (.), we deduce

lim

n→∞zn–xn= ,

lim

n→∞wn–xn= 

(.)

and hence

lim

n→∞wn–zn= . (.)

By taking into account (.) and (.), we get

α – α–αLTxn–xn+ (α+β– )Tyn–xn

≤   –β

xn–p–zn–p

≤ 

 –βxn–znxn–p → asn→ ∞.

So, we obtain

lim

n→∞Txn–xn=nlim→∞Tyn–xn= . (.)

Sincexn→qasn→ ∞, from (.) and the continuity of the norm · and the

Lips-chitzian pseudocontractive mappingT, we can deduce thatTq=q, namelyq∈F(T). On the other hand, from (.) and (.), we have

ξ –k–ξB(S–I)Azn

≤ zn–p–wn–p

≤ zn–wn

zn–p–wn–p

→ asn→ ∞,

which yields that

lim

n→∞(S–I)Azn= . (.)

Since thek-demicontractive mappingSis demiclosed atθ, taking into accountxn→q,

Axn→Aq,zn–xn → and (.), we have

(11)

and

Aq∈F(S).

Hence we conﬁrmq∈. The proof is completed.

By virtue of Theorem ., we can establish the following:

(i) Strong convergence algorithms for the split common solution problem for Lipschitzian pseudocontractive mappings and nonexpansive mappings (see Corollary . below).

(ii) Strong convergence algorithms for the split common solution problem for Lipschitzian pseudocontractive mappings and quasi-nonexpansive mappings (see Corollary . below).

Corollary . Let Hand Hbe two real Hilbert spaces andθibe the zero vector of Hifor

i= , .Let C be a nonempty closed convex subset of H and A:H→Hbe a bounded linear operator with its adjoint B.Let T:C→C be a Lipschitzian pseudocontractive map-ping with Lipschitz constant L> andF(T)=∅,and let S:H→Hbe a nonexpansive mapping withF(S)=∅.Let C=C and{xn}be a sequence generated by the following

algo-rithm:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

xn+=PCn+(x), ∀n∈N,

where <  –β<α< 

√+L,ξ∈(,



B)and PCnis the projection operator from Hinto Cnfor n∈N.Suppose that

=p∈F(T) :Ap∈F(S)=∅.

Proof Since the mappingS is nonexpansive, it is -demicontractive. Hence the desired conclusion follows from Theorem . immediately by takingk= .

Corollary . Let Hand Hbe two real Hilbert spaces andθibe the zero vector of Hifor

(12)

gen-erated by the following algorithm:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

xn+=PCn+(x), ∀n∈N,

where <  –β<α< 

√+L,ξ∈(,



B)and PCnis the projection operator from Hinto

Cnfor n∈N.Suppose that

=p∈F(T) :Ap∈F(S)=∅.

Example . LetH=Rwith the absolute-value norm| · |. LetH= [√_, √

]with the normα= (a +a)



 for_α= (_a_,_a_)∈_H_ and the inner product_α,_β=

i=aibifor

α= (a,a) andβ= (b,b)∈H. LetA:H→Hbe deﬁned byAx= (x,x) forx∈R. Then Ais a bounded linear operator with its adjoint operatorBz=z+zforz= (z,z)∈H. Clearly,A=B=√. LetC= [√

, √

]. LetT:C→CandS:H→Hbe deﬁned by

Tx=

x forx∈C

and

Sz=  z

,  z

forz= (z,z)∈H,

respectively. It is easy to see that • F(T) ={};

• F(S) ={(, )};

• ={p∈F(T) :Ap∈F(S)}={} =∅;

• Tis a Lipschitzian pseudocontractive mapping with Lipschitz constantL=√; • TandSboth are 

-demicontractive mappings. By using algorithm (.) with  <  –β<α< 

√andξ∈(, 

), we can verifyxn→ and Axn→A() = (, )∈F(S) asn→ ∞.

4 Some applications and further remarks for Theorem 3.1

LetCbe a nonempty subset of a Hilbert spaceH. Recall that a mappingU:C→Cis said to beaccretiveif

(13)

Obviously,U:C→Cis accretive if and only ifI–U:C→Cis pseudocontractive. More-over,

F(I–U) =U–(θ) :={x∈C:Ux=θ},

whereθ is the zero vector ofH.

At the end of this paper, by applying Theorem ., we obtain the following: (i) Strong convergence algorithms for the split common solution problem for

Lipschitzian accretive mappings and demicontractive nonexpansive mappings (see Theorem . below).

(ii) Strong convergence algorithms for the split common solution problem for Lipschitzian accretive mappings and nonexpansive mappings (see Corollary . below).

(iii) Strong convergence algorithms for the split common solution problem for Lipschitzian accretive mappings and quasi-nonexpansive mappings (see Corollary . below).

Theorem . Let Hand Hbe two real Hilbert spaces andθibe the zero vector of Hifor

i= , .Let A:H→Hbe a bounded linear operator with its adjoint B and U:H→H be a Lipschitzian accretive mapping with Lipschitz constant L>  and U–(θ)=∅. Let S:H→Hbe a k-demicontractive mapping withF(S)=∅which is demiclosed atθ.Let {xn}be a sequence generated by the following algorithm:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈H chosen arbitrarily, yn=xn–αUxn,

zn=βxn+ ( –β)(I–U)yn,

wn=zn+ξB(S–I)Azn,

xn+=PCn+(x), ∀n∈N,

(.)

where <  –β<α< 

√+L,ξ∈(,

–k

=p∈U–(θ) :Ap∈F(S)=∅.

Proof LetC=H. Then the iterative process (.) can be rewritten as follows: ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C chosen arbitrarily, yn= ( –α)xn+α(I–U)xn,

(14)

SetT:=I–U, thenF(T) =U–₍_θ_{) and}_T _{is a Lipschitzian pseudocontractive mapping} with Lipschitz constant  +L. Therefore the desired conclusion follows from Theorem .

immediately.

The following interesting results are immediate from Theorem ..

Corollary . Let Hand Hbe two real Hilbert spaces andθibe the zero vector of Hifor

i= , .Let A:H→Hbe a bounded linear operator with its adjoint B and U:H→H be a Lipschitzian accretive mapping with Lipschitz constant L>  and U–₍_θ_)₌_∅. _Let S:H→Hbe a quasi-nonexpansive mapping withF(S)=∅which is demiclosed atθ. Let{xn}be a sequence generated by the following algorithm:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

xn+=PCn+(x), ∀n∈N,

where <  –β<α< 

√+L,ξ∈(,



=p∈U–(θ) :Ap∈F(S)=∅.

Corollary . Let Hand Hbe two real Hilbert spaces andθibe the zero vector of Hifor

i= , .Let A:H→Hbe a bounded linear operator with its adjoint B.Let U:H→H be a Lipschitzian accretive mapping with Lipschitz constant L>  and U–₍_θ_)₌_∅_. _Let S:H→Hbe a nonexpansive mapping withF(S)=∅.Let{xn}be a sequence generated

by the following algorithm:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

xn+=PCn+(x), ∀n∈N,

where <  –β<α< 

√+L,ξ∈(,



=p∈U–(θ) :Ap∈F(S)=∅.

(15)

(a) xn→qasn→ ∞,

Remark . In Theorems . and ., the control coeﬃcientsαandβcan be respectively replaced with the sequences{αn}and{βn}satisfying  <ε<  –βn<αn<_√_+_L for some

positive real numberε.

Remark . Obviously, all results in this paper are true ifH=H. They generalize and improve many results in the literature; see, for instance, [, , –].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally and signiﬁcantly in writing this paper. Both authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China.}2_{Department of Mathematics, National}

Kaohsiung Normal University, Kaohsiung, 824, Taiwan.

Acknowledgments

The ﬁrst author was supported by the Candidate Foundation of Youth Academic Experts at Honghe University (2014HB0206); the second author was supported by Grant No. MOST 103-2115-M-017-001 of the Ministry of Science and Technology of the Republic of China.

Received: 12 July 2014 Accepted: 26 September 2014 Published:22 Oct 2014 References

1. Censor, Y, Segal, A: The split common fixed point problem for directed operators. J. Convex Anal.16, 587-600 (2009) 2. Moudafi, A: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal.74,

4083-4087 (2011)

3. Moudaﬁ, A: Split monotone variational inclusions. J. Optim. Theory Appl.150, 275-283 (2011)

4. Yao, Y, Postolache, M, Liou, Y-C: Strong convergence of a self-adaptive method for the split feasibility problem. Fixed Point Theory Appl.2013, 201 (2013)

5. Censor, Y, Gibali, A, Reich, S: Algorithms for the split variational inequality problem. Numer. Algorithms59(2), 301-323 (2012)

6. Zhao, J, He, S: Strong convergence of the viscosity approximation process for the split common ﬁxed-point problem of quasi-nonexpansive mapping. J. Appl. Math.2012, Article ID 438023 (2012). doi:10.1155/2012/438023

7. He, Z: The split equilibrium problems and its convergence algorithms. J. Inequal. Appl.2012, 162 (2012) 8. He, Z, Du, W-S: Nonlinear algorithms approach to split common solution problems. Fixed Point Theory Appl.2012,

130 (2012)

9. He, Z, Du, W-S: On hybrid split problem and its nonlinear algorithms. Fixed Point Theory Appl.2013, 47 (2013) 10. Li, C-l, Liou, Y-C, Yao, Y: A damped algorithm for the split feasibility and ﬁxed point problems. J. Inequal. Appl.2013,

379 (2013)

11. Moudafi, A: The split common fixed-point problem for demicontractive mappings. Inverse Probl.26, 055007 (2010) 12. Ceng, L-C, Petrusel, A, Yao, J-C: Strong convergence of modified implicit iterative algorithms with perturbed

mappings for continuous pseudocontractive mappings. Appl. Math. Comput.209, 162-176 (2009)

13. Chen, R, Song, Y, Zhou, H: Convergence theorems for implicit iteration process for a ﬁnite family of continuous pseudocontractive mappings. J. Math. Anal. Appl.314, 701-709 (2006)

14. Chidume, CO, Souza, GD: Convergence of a Halpern-type iteration algorithm for a class of pseudo-contractive mappings. Nonlinear Anal.69, 2286-2292 (2008)

15. Morales, CH, Jung, JS: Convergence of paths for pseudocontractive mappings in Banach spaces. Proc. Am. Math. Soc.

128, 3411-3419 (2000)

16. Yao, Y, Liou, Y-C, Chen, R: Strong convergence of an iterative algorithm for pseudocontractive mappings in Banach spaces. Nonlinear Anal.67, 3311-3317 (2007)

17. Zhou, H: Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces. Nonlinear Anal.70, 4039-4046 (2009)

18. Schu, J: Approximating ﬁxed points of Lipschitzian pseudocontractive mappings. Houst. J. Math.19, 107-115 (1993) 19. Udomene, A: Path convergence, approximation of ﬁxed points and variational solutions of Lipschitz

pseudocontractions in Banach spaces. Nonlinear Anal.67, 2403-2414 (2007)

20. Song, Y: Strong convergence of viscosity approximation methods with strong pseudocontraction for Lipschitz pseudocontractive mappings. Positivity13, 643-655 (2009)

(16)

22. Ishikawa, S: Fixed point by a new iteration method. Proc. Am. Math. Soc.4(1), 147-150 (1974)

23. He, Z: A new iterative scheme for equilibrium problems and ﬁxed point problems of strict pseudo-contractive mappings and its application. Math. Commun.17, 411-422 (2012)

24. Yao, Y, Postolache, M, Liou, Y-C: Coupling Ishikawa algorithms with hybrid techniques for pseudocontractive mappings. Fixed Point Theory Appl.2013, 211 (2013)

25. Du, W-S, He, Z: Feasible iterative algorithms for split common solution problems. J. Nonlinear Convex Anal. (in press) 26. Zhou, H: Convergence theorems of ﬁxed points for Lipschitz pseudo-contractions in Hilbert spaces. J. Math. Anal.

Appl.343, 546-556 (2008)

27. Tang, YC, Peng, JG, Liu, LW: Strong convergence theorem for pseudocontractive mappings in Hilbert spaces. Nonlinear Anal.74, 380-385 (2011)

28. Chidume, CE, Zegeye, H: Approximate ﬁxed point sequences and convergence theorems for Lipschitz pseudo-contractive maps. Proc. Am. Math. Soc.132, 831-840 (2003)

29. Shahzad, N, Zegeye, H: Approximating a common point of ﬁxed points of a pseudocontractive mapping and zeros of sum of monotone mappings. Fixed Point Theory Appl.2014, 85 (2014)

10.1186/1687-1812-2014-219