The common solutions of the split feasibility problems and fixed point problems

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R E S E A R C H

Open Access

The common solutions of the split

feasibility problems and ﬁxed point problems

Abdelouahed Hamdi

1

_{, Yeong-Cheng Liou}

2,3

_{, Yonghong Yao}

4*

_{and Chongyang Luo}

4

*_{Correspondence:}

[email protected]

4_{Department of Mathematics,}

Tianjin Polytechnic University, Tianjin, 300387, China Full list of author information is available at the end of the article

Abstract

An extraordinary split problem, which can be regarded as a superimposition of the split feasibility problem and the split ﬁxed point problem, is considered. A

superimposed algorithm is presented. The analysis technique of the suggested algorithm and the corresponding convergence results are demonstrated.

MSC: 47J25; 47H09; 65J15; 90C25

Keywords: split feasibility problem; split ﬁxed point problem; iterative algorithm; quasi-pseudo-contraction; strong convergence

1 Introduction

1.1 Background

The split feasibility problem (SFP) is formulated as ﬁndingu‡_{such that}

u‡∈C and Au‡∈Q oru‡∈C∩A–QwhenA–exists, (.)

whereC (=∅) andQ(=∅) are closed convex subsets of real Hilbert spacesH andH, respectively, andAis a bounded linear operator fromHtoH. The mathematical model of the SFP was refined from phase retrievals and the medical image reconstruction by Censor and Elfving [] in . One effective approach to solve the SFP is algorithmic iteration. There are several effective iterations which are listed as follows.

Existing iterations for the SFP . Simultaneous multiprojections (Censor and Elfving []):

xk+=A–proj_Q

proj_A₍_C₎(Axk), k∈N, (.)

whereC⊂Rn_and_Q_⊂_Rn_{are closed convex sets, and}_A_{is an}_n_×_n_matrix. . Gradient projections (CQ iteration) [–]:

xk+=projC

xk–

AA

T₍_I_–_proj

Q)Axk

, k∈N, (.)

where is a constant andAT_{denotes the transposition of}_A_.

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. Averaged CQ iteration [, ]:

xk+= ( –αk)xk+αkproj_C

xk–

AA

∗₍_I_–_proj Q)Axk

, k∈N, (.)

whereαk∈], [, is a constant andA∗is the adjoint ofA.

. Relaxed CQ iteration [, , ]: Letf:H→Randg:H→Rbe two convex functions. Deﬁne two level sets and the related subdiﬀerentials

C:=x∈H|f(x)≤

and Q:=y∈H|g(y)≤

,

∂f(x) =z∈H|f(u)≥f(x) +u–x,z,u∈H

, ∀x∈C

and

∂g(x) =w∈H|g(v)≥g(y) +v–y,w,v∈H

, ∀y∈Q.

Deﬁne the relaxed CQ iteration as follows:

xk+=proj_C_k

xk–

AA

T₍_I_–_proj

Qk)Axk

, k∈N, (.)

where

Ck=

x∈H|f(xk) +ξk,x–xk ≤

,

whereξk∈∂f(xk), and

Qk=

y∈H|g(Axk) +ηk,y–Axk ≤

,

whereηk∈∂g(Axk).

. Regularized iteration [, ]:

xk+=proj_C

( –αkk)xk–kA∗(I–proj_Q)Axk

, k∈N, (.)

where{αk} ⊂], [ and{k} ∈],_Aαk₊_α k[. . Self-adaptive iteration [–]:

xk+=proj_C

xk–kA∗(I–proj_Q)Axk

, k∈N, (.)

where the step-sizek=

τ_k(I–proj_Q)Axk

A∗₍_I_–proj_Q)Axk in whichτk∈], [. . Halpern-type iteration []:

xk+=αku+ ( –αk)proj_C

xk–kA∗(I–proj_Q)Axk

, k∈N, (.)

whereu∈Cis a ﬁxed point,{αk} ⊂], [ andk=

τk(I–projQ)Axk

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The (two-set) split common ﬁxed point problem (SCFP) can be formulated as ﬁnding u†such that

u†∈Fix(T) and Au†∈Fix(S), (.) whereFix(T) andFix(S) stand for the ﬁxed point sets of the operatorsT :H→Hand

S:H→H.

The SCFP is a natural extension of the SFP and of the convex feasibility problem. The SCFP was ﬁrstly considered by Censor and Segal in [] whereSandT are directed op-erators which include the orthogonal projections and the sub-gradient projectors.

Existing iterations for the SCFP . Censor and Segal’s iteration []:

xk+=T

xk–

AA

∗₍_I_–_S₎_A_xk_, _k_∈_N_. _(.)

. Averaged iteration [, ]: ⎧

⎨ ⎩

yk=xk–_AA∗(I–S)Axk,

xk+= ( –αk)yk+αkTyk, k∈N.

(.)

. Halpern-type iteration []:

xk+=αku+ ( –αk)T

xk–

AA

∗₍_I_–_S₎_A_xk_, _k_∈_N_. _(.)

. Self-adaptive iteration []: ⎧

⎨ ⎩

yk=xk–kA∗(I–S)Axk, xk+= ( –λ)yk+λkTyk, k∈N,

(.)

where the step-sizek=(–τ)(I–S)Axk  A∗(I–S)Axk. . Composite iteration []:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

vk=xk+_AδA∗[( –ζk)I+ζkS(( –ηk)I+ηkS) –I]Axk, uk=αnh(xk) + (I–αkB)vk,

xk+= ( –βk)uk+βkT(( –γk)un+γkTuk), k∈N,

(.)

where {αk}k∈N, {βk}k∈N,{γk}k∈N, {ζk}k∈N and{ηk}k∈N are ﬁve real number sequences in

], [,δ∈], [ is a constant,h:H→His a contraction andB:H→H is a strong positive linear bounded operator.

1.2 Problem statement

The purpose of this paper is to study the following split feasibility problem and ﬁxed point problem:

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Motivated by iterations (.), (.) and (.), we will construct a new iteration to ap-proach the solution of (.). Strong convergence results are given in the third section.

2 Several notions and lemmas

Assume thatHis a real Hilbert space.·,·and · stand for its inner product and norm, respectively. Let (∅ =)C⊂Hbe a closed convex set.

Deﬁnition . An operatorP:C→Cis said to beL-Lipschitzianif

_Pu–Pu†≤Lu–u†, ∀u,u†∈C

for some constantL> .

IfL∈[, [, thenPis calledL-contraction. IfL= , thenPis called nonexpansive.

Deﬁnition . An operatorP:C→Cis said to be ﬁrmly nonexpansive if

_Pu–Pu†≤u–u†–(I–P)u– (I–P)u† (.)

for allu,u†∈C.

Deﬁnition . An operatorP:C→Cis said to be pseudo-contractive if

Pu–Pu†,u–u†≤u–u†

for allu,u†_∈_C_.

Deﬁnition . An operatorP:C→Cis said to be quasi-pseudo-contractive if

_Pu–u†≤u–u†+Pu–u (.)

for allu∈Candu†∈Fix(P).

Deﬁnition . An operatorPis said to be demiclosed if∀un→u‡weakly andP(xn)→u strongly imply thatP(u‡_{) =}_u.

Lemma . ([]) Let {n} ⊂[, +∞[, {ϑn} ⊂], [ and{ηn} be three real number se-quences.Suppose that{n},{ϑn}and{ηn}satisfy the following three conditions:

(i) n+≤( –ϑn)n+ηnϑn,

(ii) ∞_n_=ϑn=∞,

(iii) lim sup_n_→∞ηn≤or

∞

n=|ηnϑn|<∞. Thenlimn→∞n= .

Lemma .([]) Let{ρn}be a sequence of real numbers.Assume that there exists a sub-sequence{ρnk}of{ρn}such thatρnk≤ρnk+for all k≥.For every n≥N,deﬁne an integer sequence{τ(n)}as

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Thenτ(n)→ ∞as n→ ∞and,for all n≥N,

max{ρτ(n),ρn} ≤ρτ(n)+. 3 Algorithms and convergence

In this section, we ﬁrst construct an iterative algorithm for solving problem (.) and subsequently to prove its convergence. Now we give the assumptions on the underlying spaces, involved operators and additional parameters, throughout.

I. Conditions on the underlying spaces:

(UC): HandHare two real Hilbert spaces,

(UC): C⊂HandQ⊂Hare two nonempty closed convex sets.

II. Conditions on the involved operators:

(IO): A:H→His a bounded linear operator with its adjointA∗,

(IO): Bis a strongly positive bounded linear operator onHwith coeﬃcient

σ(> ),

(IO): f :C→His aρ-contraction,

(IO): S:Q→Qis anL-Lipschitzian quasi-pseudo-contractive operator with

L(> ) andT :C→Cis anL-Lipschitzian quasi-pseudo-contractive

operator withL(> ).

III. Conditions on the parameters:

(AP): δandγ are two positive constants,

(AP): {αn}n∈N,{βn}n∈N,{γn}n∈N,{ζn}n∈Nand{ηn}n∈Nare real number sequences in ], [.

We useto denote the set of solutions of problem (.), that is,

=z†|z†∈C∩Fix(T),Az†∈Q∩Fix(S). In the sequel, we assume=∅.

Next, we construct the following iterative algorithm to solve problem (.).

Algorithm . For givenx∈Harbitrarily, deﬁne a sequence{xn}iteratively by ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

zn=proj_QAxn,

vn= ( –ζn)zn+ζnS(( –ηn)zn+ηnSzn), yn=αnγf(xn) + (I–αnB)(xn–δA∗(Axn–vn)), un=proj_Cyn,

xn+= ( –βn)un+βnT(( –γn)un+γnTun)

(.)

for alln∈N.

Theorem . Suppose thatT –IandS–Iare demiclosed at.Assume that the following conditions are satisﬁed:

(C): limn→∞αn= and

∞

n=αn=∞,

(C):  <a<ζn<b<ηn<c<√  +L

+

,

(C):  <a<βn<b<γn<c<√  +L_+,

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Then the sequence{xn}generated by algorithm(.)converges strongly to the unique ﬁxed point of the contractive mappingproj(γf+I–B).

Remark . In the sequel, we denote the unique ﬁxed point of the mappingproj(γf +

I–B) byz†_,_i.e.,_z†₌_proj

(γf+I–B)z†. It is clear thatz†solves the variational inequality

(γf –B)z†_,_z_–_z†_≤_,_∀_z_∈_.

In order to prove Theorem ., we need several helpful propositions.

Proposition .([]) LetHbe a real Hilbert space.LetU:H→Hbe anL-Lipschitzian operator withL> .Then

Fix( –ζ)I+ζUU=FixU( –ζ)I+ζU=Fix(U) for allζ∈(,_L).

Proposition .([]) LetHbe a real Hilbert space.LetU:H→Hbe anL-Lipschitzian quasi-pseudo-contractive operator.Then we have

_U

( –η)x+ηUx–u†≤x–u†+ ( –η)x–U( –η)x+ηUx,

and the operator( –ξ)I+ξU(( –η)I +ηU)is quasi-nonexpansive when <ξ <η< 

√

+L_+,that is,

( –ξ)x+ξU( –η)x+ηUx–u†≤x–u†

for all x∈Hand u†_∈_Fix₍_U_).

Proposition . In any real Hilbert spaceH,the following two equalities hold:

ζu+ ( –ζ)u†=ζu+ ( –ζ)u†–ζ( –ζ)u–u†, ζ ∈[, ] (.)

and

u+u†=u+ u,u†+u† (.)

for all u,u†_∈_H_.

Proposition .([]) LetHbe a real Hilbert space.LetU:H→Hbe anL-Lipschitzian operator withL> .IfI–Uis demiclosed at,thenI–U(( –ζ)I+ζU)is also demiclosed atwhenζ∈(,_L).

Next, we prove Theorem ..

Proof Letz†₌_proj

(γf +I–B)z†. Subsequently, we obtainz†∈C∩Fix(T) andAz†∈

Q∩Fix(S). Note thatproj_Qis ﬁrmly nonexpansive. From (.), we deduce

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Applying Proposition . and noting conditions (C) and (C), we have

FixS( –ηn)I+ηnS

=Fix(S)

and

FixT( –γn)I+γnT

=Fix(T)

for alln∈N.

By condition (C) and Proposition ., we derive

vn–Az†=( –ζn)I+ζnS

( –ηn)I+ηnS

zn–Az†

=( –ζn)I+ζnS

( –ηn)I+ηnS

zn

–( –ζn)I+ζnS

( –ηn)I+ηnS

Az†

≤zn–Az†.

This together with (.) implies that

vn–Az†≤Axn–Az†–Axn–zn. (.)

By condition (C) and Proposition ., we derive

xn+–z†=( –βn)I+βnT

( –γn)I+γnT

un–z†

=( –βn)I+βnT

( –γn)I+γnT

un

–( –βn)I+βnT

( –γn)I+γnT

z†

≤un–z†. (.)

Noting thatproj_Cis nonexpansive, we obtain

un–z†=proj_Cyn–proj_Cz†≤yn–z†. (.)

From (.), we get

_yn_–_z†

=αnγf(xn) + (I–αnB)

xn–δA∗(Axn–vn)–z†

=αnγ

f(xn) –fz†+αn

γfz†–Bz†

+ (I–αnB)

xn–z†–δA∗(Axn–vn)

≤αnγf(xn) –f

z†+αnγf

z†–Bz†

+I–αnBxn–z†–δA∗(Axn–vn)

≤αnγρxn–z†+αnγf

z†–Bz†

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Observe that

xn–z†,A∗(vn–Axn)=Axn–z†,vn–Axn

=Axn–Az†+vn–Axn– (vn–Axn),vn–Axn

=vn–Az†,vn–Axn–vn–Axn. (.)

Using (.), we obtain

vn–Az†,vn–Axn=

vn–Az †

+vn–Axn–Axn–Az†. (.)

From (.), (.) and (.), we get

xn–z†,A∗(vn–Axn)= 

vn–Az †

+vn–Axn–Axn–Az†

–vn–Axn

≤ 

Axn–Az †

–zn–Axn+vn–Axn

–Axn–Az†–vn–Axn

= –

zn–Axn _–

vn–Axn

_. _(.)

According to equality (.), we get

xn–z†+δA∗(vn–Axn) 

=xn–z† 

+δA∗(vn–Axn) 

+ δxn–z†,A∗(vn–Axn)

.

Combining the above equality and (.), we deduce

≤xn–z† 

+δAvn–Axn

–δzn–Axn–δvn–Axn

=xn–z†+δA–δvn–Axn

–δzn–Axn. (.)

In view of condition (C), we know thatδA–δ< . From (.), we have

≤xn–z† 

.

Therefore,

xn–z†+δA∗(vn–Axn)≤xn–z†. (.)

Substituting (.) into (.) we deduce yn–z†≤αnγρxn–z†+αnγf

z†–Bz†+ ( –αnσ)xn–z†

= – (σ–γρ)αnxn–z†+αnγf

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From (.), (.) and (.), we get

xn+–z†≤

 – (σ–γρ)αnxn–z†+αnγf

z†–Bz†

= – (σ–γρ)αnxn–z†+ (σ–γρ)αn

γf(z†_{) –}_B_z†

σ–γρ .

By induction, we get

xn–z†≤max

x–z†,

γf(z†_{) –}_B_z†

σ–γρ

.

Hence, the sequence{xn}is bounded.

Using the ﬁrm nonexpansiveness ofproj_C, we have un–z† =proj_Cyn–z†

≤yn–z†–proj_Cyn–yn =yn–z†



–un–yn. (.)

From (.), (.) and (.), we deduce

xn+–z† 

≤un–z† 

≤yn–z† 

–un–yn

≤ – (σ–γρ)αnxn–z† 

+ αn

σ–γργf

z†–Bz†–un–yn.

It follows that

un–yn≤xn–z†–xn+–z†+

αn

σ–γργf

z†–Bz†. (.)

Next, we consider two possible cases: the sequence {xn–z†}is either monotone de-creasing at inﬁnity (Case ) or not (Case ).

Case . There existsnsuch that the sequence{xn–z†}n≥nis decreasing.

Case . For anyn, there exists an integerm≥nsuch thatxm–z† ≤ xm+–z†. In Case , we assume that there exists some integerm>  such that{xn–z†_}_is de-creasing for alln≥m. Thenlimn→∞xn–z†exists. From (.), we deduce

lim

n→∞un–yn= . (.)

From (.), we have

yn–z†≤αnγρxn–z†+αnγf

z†–Bz†

+ ( –αnσ)xn–z†+δA∗(vn–Axn)

=αnσ

γρxn–z†+γf(z†) –Bz†

σ

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Since{xn}is bounded, there exists a constantM> such that

sup

n

γρxn–z†₊_γ_f_(z†_{) –}_B_z†

σ

<M.

By (.), we deduce

yn–z†≤αnσM+ ( –αnσ)xn–z†+δA∗(vn–Axn). (.)

Combining (.) and (.), we obtain

xn+–z†≤yn–z†

≤( –σ αn)xn–z†+ ( –σ αn)δA–δvn–Axn

– ( –σ αn)δzn–Axn+αnσM.

Hence,

≤( –σ αn)

δ–δAvn–Axn+ ( –σ αn)δzn–Axn

≤( –σ αn)xn–z†–xn+–z† 

+αnσM,

which implies that

lim

n→∞vn–Axn=nlim→∞zn–Axn= . (.) Therefore,

lim

n→∞vn–zn= . (.)

Note thatvn–zn=ζn[S(( –ηn)I+ηnS)zn–zn]. Thus,

lim

n→∞zn–S

( –ηn)I+ηnS

zn= lim

n→∞Axn–S

( –ηn)I+ηnS

Axn= . (.)

Since

Axn–SAxn ≤Axn–S( –ηn)I+ηnS

Axn

+S( –ηn)I+ηnS

Axn–SAxn

≤Axn–S

( –ηn)I+ηnS

Axn+LηnAxn–SAxn,

it follows that

Axn–SAxn ≤ 

 –Lηn

_A_xn–S( –ηn)I+ηnS

Axn.

This together with (.) implies that

lim

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According to (.), we have

yn–xn=αnγf(xn) –δA∗(Axn–vn) –αnB

xn–δA∗(Axn–vn)

≤δAvn–Axn+αnγf(xn) –B

xn–δA∗(Axn–vn).

It follows from (.) and (C) that

lim

n→∞xn–yn= . (.)

From (.) and (.), we have

xn+–z† 

=( –βn)un–z†+βn

T( –γn)un+γnTun

–z†

= ( –βn)un–z†+βnT

( –γn)un+γnTun

–z†

–βn( –βn)T( –γn)un+γnTun

–un. (.)

Applying Proposition ., we get

_T

( –γn)un+γnTun

–z†

≤un–z†+ ( –γn)un–T( –γn)un+γnTun. (.)

From (.), (.) and (.), we deduce

xn+–z† 

≤un–z†–βn(γn–βn)un–T

( –γn)un+γnTun 

≤αnσM+ ( –αnσ)xn–z†+δA∗(vn–Axn) 

–βn(γn–βn)un–T

( –γn)un+γnTun 

≤αnσM+xn–z† 

–βn(γn–βn)un–T

( –γn)un+γnTun 

.

It follows that

βn(γn–βn)un–T

( –γn)un+γnTun

≤xn–z†–xn+–z†+αnσM.

Therefore,

lim

n→∞un–T

( –γn)un+γnTun= . (.)

Observe that

un–Tun ≤un–T

( –γn)un+γnTun+T

( –γn)un+γnTun

–Tun

≤un–T( –γn)un+γnTun+Lγnun–Tun.

Thus,

un–Tun ≤   –Lγn

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This together with (.) implies that

lim

n→∞un–Tun= . (.)

Next, we show that

lim sup

n→∞

γfz†–Bz†,yn–z†≤.

Choose a subsequence{yni}of{yn}such that

lim sup

n→∞

γfz†–Bz†,yn–z†= lim

i→∞

γfz†–Bz†,yni–z †

. (.)

Since the sequence{yni}is bounded, we can choose a subsequence{yn_ij}of{yni}such that yn_ij z. For the sake of convenience, we assume (without loss of generality) thatyniz. Subsequently, we derive from the above conclusions that

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

xniz, yniz,

uniz

(.)

and ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

AxniAz,

AyniAz,

AuniAz.

(.)

Note thatun_i=proj_Cyn_i∈Candzn_i=proj_QAxn_i∈Q. From (.), we deducez∈Cand

Az∈Qby (.). By the demiclosedness ofT –I andS–I, we deducez∈Fix(T) (by (.)) andAz∈Fix(S) (by (.)). To this end, we deducez∈C∩Fix(T) andAz∈Q∩

Fix(S). That is to say,z∈. Therefore,

lim sup

n→∞

γfz†–Bz†,yn–z†= lim

i→∞

γfz†–Bz†,yni–z †

= lim

i→∞

γfz†–Bz†,z–z†

≤. (.)

From (.), we have

yn–z†=αnγ

f(xn) –fz†+αn

γfz†–Bz†

+ (I–αnB)

xn–z†–δA∗(Axn–vn) 

≤ I–αnBxn–z†–δA∗(Axn–vn) 

+ αnγ

f(xn) –fz†,yn–z†+ αn

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≤( –αnσ)xn–z† 

+ αnγf(xn) –f

z†yn–z†

+ αn

γfz†–Bz†,yn–z†

≤( –αnσ)xn–z† 

+αnγρxn–z† 

+αnγρyn–z† 

+ αn

γfz†–Bz†,yn–z†.

It follows that

yn–z†≤

 –(σ–γρ)αn  –γραn

xn–z†+ σ _α

n  –γραn

xn–z†

+ αn  –γραn

γfz†–Bz†,yn–z†.

Therefore,

xn+–z†≤yn–z† 

≤

 –(σ–γρ)αn  –γραn

xn–z†+ σ _α

n  –γραn

xn–z†

+ αn  –γραn

γfz†–Bz†,yn–z†. (.)

Applying Lemma . and (.) to (.), we deducexn→z†_. Case . Assume that there exists an integernsuch that

xn–z†≤xn+–z†.

Setωn={xn–z†}. Then we have

ωn≤ωn+.

Deﬁne an integer sequence{τn}for alln≥nas follows:

τ(n) =max{l∈N|n≤l≤n,ωl≤ωl+}.

It is clear thatτ(n) is a nondecreasing sequence satisfying

lim

n→∞τ(n) =∞

and

ωτ(n)≤ωτ(n)+

for alln≥n.

By a similar argument as that of Case , we can obtain

lim

n→∞uτ(n)–yτ(n)=nlim→∞xτ(n)–yτ(n)= ,

lim

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and

lim

n→∞uτ(n)–Tuτ(n)= .

This implies that

ωw(yτ(n))⊂.

Thus, we obtain

lim sup

n→∞

γfz†–Bz†,yτ(n)–z†

≤. (.)

Sinceωτ(n)≤ωτ(n)+, we have from (.) that

ω_τ₍_n₎≤ω_τ₍_n_)+

≤

 –(σ–γρ)ατ(n)  –γρατ(n)

ω_τ₍_n₎+ σ _α

τ(n)  –γρατ(n)

ω_τ₍_n₎

+ ατ(n)  –γρατ(n)

. (.)

It follows that

ω_τ₍_n₎≤ 

(σ–γρ) –σ_α

τ(n)

. (.)

Combining (.) and (.), we have

lim sup

n→∞ ωτ(n)≤,

and hence

lim

n→∞ωτ(n)= . (.)

By (.), we obtain

lim sup

n→∞ ω



τ(n)+≤lim sup n→∞ ω



τ(n).

This together with (.) implies that

lim

n→∞ωτ(n)+= .

Applying Lemma . we get

≤ωn≤max{ωτ(n),ωτ(n)+}.

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4 Applications

The following results can be deduced directly from Algorithm . and Theorem ..

Algorithm . For givenx∈Harbitrarily, deﬁne a sequence{xn}iteratively by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

zn=proj_QAxn,

vn= ( –ζn)zn+ζnS(( –ηn)zn+ηnSzn), yn= ( –αn)(xn–δA∗(Axn–vn)), un=proj_Cyn,

xn+= ( –βn)un+βnT(( –γn)un+γnTun)

(.)

for alln∈N.

Corollary . Suppose thatT–IandS–Iare demiclosed at.Assume that the following conditions are satisﬁed:

(C): limn→∞αn= and ∞

n=αn=∞,

(C):  <a<ζn<b<ηn<c<√  +L_+,

(C):  <a<βn<b<γn<c<√  +L_+,

(C):  <δ< 

A.

Then the sequence{xn} generated by algorithm(.)converges strongly to the minimum norm solution u♣∈.

Algorithm . For givenx∈Harbitrarily, deﬁne a sequence{xn}iteratively by

xn+=proj_C

αnγf(xn) + (I–αnB)

xn–δA∗(Axn–proj_QAxn) (.) for alln∈N.

Corollary . Assume that the following conditions are satisﬁed:

(C): lim_n_→∞αn= and

∞

n=αn=∞,

(C):  <δ<_A andσ>γρ.

Then the sequence{xn}generated by algorithm(.)converges strongly to u∈(the set of the solutions of (.))provided=∅.

Algorithm . For givenx∈Harbitrarily, deﬁne a sequence{xn}iteratively by

xn+=proj_C

( –αn)xn–δA∗(Axn–proj_QAxn) (.) for alln∈N.

Corollary . Assume that the following conditions are satisﬁed:

n=αn=∞,

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Then the sequence{xn}generated by algorithm(.)converges strongly to the minimum norm solution u♣∈provided=∅.

Algorithm . For givenx∈Harbitrarily, deﬁne a sequence{xn}iteratively by ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

vn= ( –ζn)Axn+ζnS(( –ηn)Axn+ηnSAxn), yn=αnγf(xn) + (I–αnB)(xn–δA∗(Axn–vn)), xn+= ( –βn)yn+βnT(( –γn)yn+γnTyn)

(.)

for alln∈N.

Corollary . Suppose thatT–IandS–Iare demiclosed at.Assume that the following conditions are satisﬁed:

(C): limn→∞αn= and

∞

n=αn=∞,

(C):  <a<ζn<b<ηn<c<√  +L

+

,

(C):  <a<βn<b<γn<c<√  +L_+,

(C):  <δ<_A andσ>γρ.

Then the sequence{xn}generated by algorithm(.)converges strongly to u∈(the set of the solutions of (.))provided=∅.

Algorithm . For givenx∈Harbitrarily, deﬁne a sequence{xn}iteratively by ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

vn= ( –ζn)Axn+ζnS(( –ηn)Axn+ηnSAxn), yn= ( –αn)(xn–δA∗(Axn–vn)),

xn+= ( –βn)yn+βnT(( –γn)yn+γnTyn)

(.)

for alln∈N.

Corollary . Suppose thatT –IandS–I are demiclosed at.Assume that the fol-lowing conditions are satisﬁed:

n=αn=∞,

(C):  <a<ζn<b<ηn<c<√  +L

+

,

(C):  <a<βn<b<γn<c<√  +L_+,

(C):  <δ<_A.

Then the sequence{xn}generated by algorithm(.)converges strongly to the minimum norm solution u♣∈provided=∅.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Statistics and Physics, College of Arts and Science, Qatar University, P.O. Box 2713, Doha,}

Qatar.2_{Department of Information Management, Cheng Shiu University, Kaohsiung, 833, Taiwan.}3_{Center for General}

Education, Kaohsiung Medical University, Kaohsiung, 807, Taiwan.4_{Department of Mathematics, Tianjin Polytechnic}

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Acknowledgements

Yeong-Cheng Liou was supported in part by MOST 101-2628-E-230-001-MY3 and MOST 101-2622-E-230-005-CC3. Abdelouahed Hamdi would like to thank Qatar University for providing excellent research facilities under Grant: QUUG-CAS-DMSP-14/15-4.

Received: 4 August 2015 Accepted: 15 October 2015 References

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