On iterative solutions of a split feasibility problem with nonexpansive mappings

R E S E A R C H

Open Access

On iterative solutions of a split feasibility

problem with nonexpansive mappings

M.A. Kutbi

_{, A. Latif}

_{and X. Qin}

*_{Correspondence:[email protected]}

1_{Department of Mathematics, King} Abdulaziz University, Jeddah, Saudi Arabia

Full list of author information is available at the end of the article

Abstract

We analyze iterative solutions of a split feasibility problem with common fixed point constraints of a family of nonexpansive mappings. We present solution theorems of the feasibility problem under some weak assumptions imposed on different mappings and control sequences.

MSC: 47H05; 47H09; 47J20; 65K15

Keywords: Fixed Point; Hilbert space; Monotone operator; Nonexpansive operator; Variational inequality

1 Introduction-preliminaries

LetH1andH2be Hilbert spaces, and letCandQbe nonempty convex closed sets inH1

andH2, respectively. LetA:H1→H2be a bounded linear mapping.

In 1994, Censor and Elfving [10] introduced the well-known split feasibility problem for modeling inverse problems formulated as follows:

Find x∗∈C such thatAx∗∈Q. (1.1)

It can be formulated as the following convex feasibility problem:

Find x∗∈C∩A–1(Q).

Both split feasibility and convex feasibility problems are much related to a number of real-world applications, for example, signal processing, intensity-modulated radiation therapy, and image reconstruction; see [9,11,35] and the references therein. Recently, a number of regularized iterative methods have been introduced and investigated for solutions of the feasibility problems in either Banach or Hilbert spaces by many authors; see [1–5,16,17, 19,28,31] and the references therein.

LetHbe a real Hilbert space endowed with inner product·,·and induced norm · . LetSbe a mapping onH.Fix(S) stands for a fixed point set ofS. Recall thatSis said to be nonexpansive if

Sx–Sy ≤ x–y, ∀x,y∈H.

It is well known that every nonexpansive mapping satisfies the following property:

2Sx–Sy, (y–Sy) – (x–Sx)≤(x–Sx) – (y–Sy)2, ∀x,y∈H.

The mappingSis said to be quasinonexpansive if

x–Sy ≤ x–y, ∀x∈Fix(S)=∅,y∈H.

It is obvious that quasinonexpansive mappings may not be continuous beyond their fixed-point sets. Every quasinonexpansive mappingSsatisfies the following property:

2x–Sy, (y–Sy)≤ y–Sy2, ∀x∈Fix(S)=∅,y∈H. (1.2)

It is said to be firmly nonexpansive if

Sx–Sy2_{≤ Sx–Sy,x–y, ∀x,y∈H.}

It is is said to be firmly quasinonexpansive if

x–Sy2≤ x–Sy,x–y, ∀x∈Fix(S)=∅,y∈H.

It is is said to be contractive if there exists a constantκ∈(0, 1) such that

Sx–Sy2≤κx–y, ∀x,y∈H.

Contractive mappings and their extensions are important classes of nonlinear mappings since they are connected with differential equations and nonsmooth optimization; see [7, 8,14,21] and the references therein. Recently, they have been extensively analyzed via projection-based iterative methods. It deserves mentioning that the methods based on nearest-point projections are not efficient from the viewpoint of numerical computation. LetProjH_C be the nearest-point (metric) projection fromHontoC, that is,

ProjH_Cy:=x∈C:x–y=dist_C(y),

wheredist_C(y) :=inf_x∈Cx–yfory∈H.

To avoid using nearest projections, Yamada [33] recently studied a descent method, which is known as the Yamada descent algorithm. This algorithm is as follows:

un+1= (I–αn+1μF)Tun, ∀n∈N,

Now we recall some useful notions. LetF:C→Hbe a nonself single-valued operator. It is called

(i) monotone if

x∗–x,Fx∗–Fx≥0, ∀x∗,x∈C;

(ii) strongly monotone if there exists a positive constantη> 0such that

ηx∗–x2≤x∗–x,Fx∗–Fx, ∀x∗,x∈C. (iii) L-Lipschitz if there existsL> 0such that

Fx–Fx∗≤Lx–x∗, ∀x∗,x∈C.

Let M:H →2H _{be a set-valued monotone mapping. The zero-point set ofM is} de-noted byM–1_{(0). Recall thatMis said to be monotone if, for allx,y∈H, u∈Mx, and}

v∈My

x–y,u–v ≥0.

It is said to be maximal if its graphGraph(M) is not properly contained in the graph of any other monotone mapping. IfMis maximally monotone, thenGraph(M) is weakly strongly closed; see [24] and the references therein. A well-known fact is that for (x,u)∈H×H, x–y,u–v ≥0 for all (y,v)∈Graph(M) implies thatu∈M(x) iffMis maximal. LetNbe a maximal monotone operator with domainDom(N) and rangeH. Define the mapping ResN

λ :H→Dom(M) associated with indexλby

ResN_λx= (λN+ Id)–1(x), ∀x∈H,

where Id is the identity operator on H. If N is the subdifferential of proper convex lower semicontinuous functions, then the resolvent operator is the known proximity op-erator. The resolve operator plays a significant role in nonsmooth optimization prob-lems. A variety of nonlinear problems, including variational inequalities and equilib-rium problems, can be formulated as finding a zero of a maximal monotone opera-tor. It is known that Fix(ResN

λ) =N–1(0); see [15, 18, 20, 27, 34] and the references therein.

LetN be a set-valued maximal monotone operator onH1, and letMbe a set-valued

maximal monotone operator onH2. We consider the following split inclusion problem:

findx∗∈H1such that

0∈Nx∗,y∗=Ax∗∈H2 solves 0∈M

y∗, (1.3)

whereAis a linear bounded mapping fromH1toH2. We denote bySIP(M,N) the solution

set of problem (1.3).

theo-rems of the feasibility problem under some weak assumptions imposed on different map-pings. For our main result, we also need the following tools.

LetSibe a nonexpansive mapping onC, and letηibe real numbers with 0 <ηi< 1 for eachi≥1. LetWnbe a mapping onCdefined for eachn≥1 by

Un,n+1=I,

Un,n= (1 –ηn)I+ηnSnUn,n+1,

Un,n–1= (1 –ηn–1)I+ηn–1Sn–1Un,n,

.. .

Un,k= (1 –ηk)I+ηkSkUn,k+1,

Un,k–1= (1 –ηk–1)I+ηk–1Sk–1Un,k,

.. .

Un,2= (1 –η2)I+η2S2Un,3,

Un,1= (1 –η1)I+η1S1Un,2,

Wn=Un,1.

(1.4)

It is clear thatWn:C→C, governed byS1,S2, . . . ,Snandη1,η2, . . . ,ηn, is a nonexpansive

mapping; see [29] and the references therein. We further assume that 0 <ηi≤η< 1 for

i≥1, whereηis a constant in (0, 1).

Lemma 1.1([29]) Let C be a convex and closed set in a Hilbert space H, and let Si be

nonexpansive mappings on C with fixed points.If ∞_i=1Fix(Si)=∅,then (1) limn→∞Un,kexists for each positive integerkand eachx∈C;

(2) the mappingW:C→Cdefined by

Wx:= lim

n→∞Wnx=nlim→∞Un,1x, x∈C, (1.5)

is a nonexpansive mapping withFix(W) = ∞_i=1Fix(Si) =Fix(Wn).

Lemma 1.2 ([12]) Let C be a convex and closed set in a Hilbert space H, and let Si

be a nonexpansive mappings on C with fixed points.Assume that ∞_i=1F(Si)=∅.Then

limn→∞supx∈KWnx–Wx= 0for any bounded set K⊂C.

Lemma 1.3([33]) Let H be a Hilbert space.Let F be anL-Lipschitz continuous andη -strongly monotone mapping on the space H.Let Tα_{be a mapping on the space H defined}

by Tα_{x=x–μαFx for x∈H,whereαis a real number in(0, 1).If0 <L}2_{μ∈(0, 2η)and} τ = 1 –1 –μ(2η–μL2_{)∈(0, 1],then}

_Tα

x–Tαy≤(1 –τ α)x–y, ∀x,y∈H.

Lemma 1.4([32]) Let{αn},{βn},and{γn}be sequences of real numbers such thatαn∈ [0, 1],∞_n=1αn=∞,lim supn→∞βn≤0,and

∞

non-negative real numbers such that

λn+1≤(1 –αn)λn+αnβn+γn.

Thenlim_n→∞λn= 0.

Lemma 1.5([25]) Let{xn}be a sequence in a real Hilbert space H.If xn x,then

lim inf

n→∞ xn–x<lim infn→∞ xn–y

for any y∈X with y=x.This is also equivalent to

lim sup n→∞

xn–x<lim sup n→∞

xn–y.

Lemma 1.6([6, resolvent equality]) Let H be a Hilbert space.Let N be a set-valued max-imal operator on H.For parametersλ> 0andμ> 0,we have

ResN_μ

1 –μ

λ

ResN_λx+μ

λx

=ResN_λx, ∀x∈H. (1.6)

Lemma 1.7([30]) Let H be a Hilbert space.Let{xn}and{yn}be bounded sequences in H with xn+1= (1 –βn)yn+βnxnfor all n≥0and

lim sup n→∞

yn+1–yn–xn+1–xn

≤0,

where{βn}is a sequence in(0, 1)such thatlim infn→∞βn> 0andlim supn→∞βn< 1.Then limn→∞yn–xn= 0.

2 Main results

Theorem 2.1 Let H1and H2be Hilbert spaces,and let N and M be set-valued maximal

monotone mappings on H1and H2,respectively.Let Sibe nonexpansive mappings on H1for

all integers i≥1.Let F:H1→H1be anL-Lipschitz continuous andτ-strongly monotone

mapping.Let A be a linear bounded operator from H1 to H2, and let A∗ be its adjoint

operator.Assume that ∞_i=1Fix(Si)∩SIP(M,N)=∅.Let{xn}be a vector sequence in H1

generated by the iterative process

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x1∈H1,

yn=γnResNsn(xn+γA

∗_(ResM

rn–I)Axn) + (1 –γn)xn, xn+1=βn(I–μαnF)Wnyn+ (1 –βn)xn, n≥1,

whereγ andμare two positive real numbers,{sn}and{rn}are two positive real number se-quences,{αn},{βn},and{γn}are real number sequences in(0, 1).Suppose thatγ ∈(0,_A12), μ∈(0,_L2τ2),lim infn→∞sn> 0,limn→∞|sn–sn+1|<∞,lim infn→∞rn> 0,limn→∞|rn–rn+1|<

∞,∞_n=1αn=∞,{βn}is number sequence in[β¯,β¯],whereβ¯andβ¯are two real numbers

such thatlimn→∞|γn+1–γn|= 0.Then the sequence{xn}converges strongly tox∈H1,which

is a unique solution of the variational inequality

x–y,Fx ≤0, ∀y∈

∞

i=1

Fix(Si)∩SIP(M,N).

Proof The proof is split into four steps.

Step 1. We prove that{xn}is a bounded vector sequence inH1.

For any fixedp∈ ∞_i=1Fix(Si)∩SIP(M,N), we concludeAp=ResM_rnAp,p=ResN_snp, and

p=Sipfor eachi≥1. SinceApis a fixed point ofResMrnandRes

M

rnis a (firmly) nonexpansive

mapping, we have

ResM_rn–IAxn,ResM_rnAxn–Ap

≤ResMrnAxn–Axn 2

2 . (2.1)

Putting

zn=ResNsn

xn+γA∗

ResM_rn–IAxn

,

(2.1) sends us to

zn–p2

≤γA∗ResM_r

n–I

Axn+ (xn–p)2

≤γ2A2ResM_r

n–I

Axn2+ 2γ

A∗ResM_r

n–I

Axn,xn–p

+xn–p2

=γγA2– 2ResM_r

n–I

Axn2

+ 2γResM_r

n–I

Axn,ResM_r

nAxn–Ap

+xn–p2

≤γγA2– 1ResM_r

n–I

Axn2+xn–p2, (2.2)

which leads to

yn–p2≤γnzn–p2+ (1 –γn)xn–p2

≤ xn–p2–γnγ

1 –γA2ResM_r

n–I

Axn

2

). (2.3)

The restriction imposed on parameterγ tells us thatyn–p ≤ xn–p. SinceWnis a nonexpansive mapping for eachn, we find from Lemma1.3that

xn+1–p ≤βn(I–μαnF)Wnyn– (I–μαnF)p–μαnFp+ (1 –βn)xn–p ≤βn(I–μαnF)Wnyn– (I–μαnF)p+μβnαnFp+ (1 –βn)xn–p ≤βn(1 –τ αn)Wnyn–Wnp+μβnαnFp+ (1 –βn)xn–p

≤βn(1 –τ αn)yn–p+μβnαnFp+ (1 –βn)xn–p

≤τ αnβn

Fpμ

τ + (1 –τ αnβn)xn–p

≤max

Fpμ

τ ,xn–p

,

Step 2. We prove thatlimn→∞xn+1–xn= 0. From resolvent equality (1.6) in Lemma1.6

we see that

zn–zn+1 ≤ResNsnρn–Res

N

sn+1ρn+Res

N

sn+1ρn–Res

N sn+1ρn+1

≤ResN_snρn–ResNsn+1ρn+ρn–ρn+1

=ResN_s

n+1

sn+1

sn

ρn+

1 –sn+1

sn

ResN_s

nρn

–ResN_s

n+1ρn

+ρn–ρn+1

=

sn+1

sn

ρn+

1 –sn+1

sn

ResN_s

nρn

–ρn

+ρn–ρn+1

≤1 –sn+1

sn

ρn–ResNsnρn+ρn+1–ρn, (2.4)

where

ρn=xn+γA∗

ResM_r

n–I

Axn.

It is easy to see that

(xn+1–xn) –γA∗(Axn+1–Axn)

=

xn+1–xn2– 2γ

xn+1–xn,A∗(Axn+1–Axn)

+γA∗(Axn+1–Axn) 2

=1 –γA2xn+1–xn,

which sends us to

ρn+1–ρn

≤γA∗ResM_r

n+1Axn+1–Res

M

rnAxn+(xn+1–xn) –γA

∗_(Ax

n+1–Axn)

≤ xn+1–xn+γA

1 –rn+1

rn

Axn–ResMrnAxn. (2.5)

Inequalities (2.4) and (2.5) yield

zn–zn+1 ≤

1 –sn+1

sn

ρn–ResNsnρn+xn+1–xn

+γA1 –rn+1

rn

Axn–ResMrnAxn,

which further leads us to

yn–yn+1 ≤γnzn–zn+1+ (1 –γn)xn–xn+1+|γn–γn+1|zn+1–xz+1

≤γn

1 –sn+1

sn

ρn–ResNsnρn+xn+1–xn

+γnγA

1 –rn+1

rn

Axn–ResMrnAxn

From Lemma1.1we arrive at

Wn+1yn+1–Wnyn

≤ Wn+1yn+1–Wnyn+1+Wnyn+1–Wnyn ≤sup

x∈Ψ

Wn+1x–Wx+Wx–Wnx

+yn+1–yn

≤sup x∈Ψ

Wn+1x–Wx+Wx–Wnx

+γn

1 –sn+1

sn

ρn–ResNsnρn

+xn+1–xn+γnγA

1 –rn+1

rn

Axn–ResMrnAxn

+|γn–γn+1|zn+1–xz+1, (2.6)

whereΨ is a bounded set containing{yn}. Inequality (2.6) ensures that

(I–μαn+1F)Wn+1yn+1– (I–μαnF)Wnyn ≤(I–μαn+1F)Wn+1yn+1– (I–μαn+1F)Wnyn

+(I–μαn+1F)Wnyn– (I–μαnF)Wnyn

≤(1 –τ αn+1)Wn+1yn+1–Wnyn+|αn+1–αn|μFWnyn ≤(1 –τ αn+1)Wn+1yn+1–Wnyn+|αn+1–αn|μFWnyn ≤(1 –τ αn+1)sup

x∈Ψ

Wn+1x–Wx+Wx–Wnx

+ (1 –τ αn+1)γn

1 –sn+1

sn

ρn–ResNsnρn

+ (1 –τ αn+1)xn+1–xn+ (1 –τ αn+1)γnγA

1 –rn+1

rn

Axn–ResMrnAxn

+ (1 –τ αn+1)|γn–γn+1|zn+1–xz+1+|αn+1–αn|μFWnyn.

This further leads to

(I–μαn+1F)Wn+1yn+1– (I–μαnF)Wnyn–xn+1–xn

≤sup x∈Ψ

Wn+1x–Wx+Wx–Wnx

+γn

1 –sn+1

sn

ρn–ResNsnρn

+γnγA

1 –rn+1

rn

Axn–ResMrnAxn

+|γn–γn+1|zn+1–xz+1+

|αn+1|+|αn|

μFWnyn.

Using Lemma1.2, the boundedness of operatorA, and the restrictions on the parameter sequences{αn},{γn},{sn}, and{rn}, we obtain that

lim sup n→∞

(I–μαn+1F)Wn+1yn+1– (I–μαnF)Wnyn–xn+1–xn

With the aid of Lemma1.7, we conclude that

lim

n→∞(I–μαnF)Wnyn–xn= 0. (2.7)

Sinceαn→0 asn→ ∞, we also have

lim

n→∞Wnyn–xn= 0. (2.8)

From (2.7) we see that

lim

n→∞xn+1–xn= 0. (2.9)

Since{xn}is a bounded vector sequence inH1, we find that there is a subsequence{xni}of

{xn}that converges weakly tox¯.

Step 3. We prove thatx∈ ∞_i=1Fix(Si)∩SIP(M,N). Put

ϕn= (I–μαnF)Wnyn.

For anyp∈ ∞_i=1Fix(Si)∩SIP(M,N), we conclude from (2.3) that ϕn–p2≤(I–μαnF)Wnyn– (I–μαnF)Wnp

2

– 2μαnϕn–p,Fp

≤(1 –τ αn)2Wnyn–Wnp2– 2μαnϕn–p,Fp ≤(1 –τ αn)2yn–p2– 2μαnϕn–p,Fp ≤(1 –τ αn)2xn–p2–γ

1 –γA2(1 –τ αn)2ResM_rnAxn–Axn

2

+ 2μαnFpϕn–p.

This shows us that

xn+1–p2≤βnϕn–p2+ (1 –βn)xn–p2 ≤ xn–p2–βnγ

1 –γA2(1 –τ αn)2ResMrnAxn–Axn 2

+ 2μαnβnFpϕn–p.

It follows that

γ1 –γA2(1 –τ αn)2βnAxn–ResMrnAxn 2

≤ xn–xn+1

xn–p+xn+1–p

+ 2μαnβnFpϕn–p.

Limit (2.9) and the fact thatαn→0 asn→ ∞lead us to

lim

n→∞Axn–Res M

rnAxn= 0. (2.10)

Next, we have

2zn–p2

≤2γA∗ResM_rn–IAxn+xn–p,zn–p

=γ2A∗ResM_rn–IAxn

2

+ 2γA∗ResM_rn–IAxn,xn–p

+xn–p2

–xn+γA∗

ResM_rn–IAxn–yn

2

+zn–p2

≤γ2A2ResM_rnAxn–Axn

2

+ 2γResM_rnAxn–Ap,ResMrnAxn–Axn

–ResM_rnAxn–Axn

2

+zn–p2+xn–p2–zn–xn2– 2γ

A∗ResM_rn–IAxn,xn–zn

–γA∗ResM_rn–IAxn

2

)

≤ xn–p2+zn–p2+ 2Aγxn–znResMrnAxn–Axn–xn–zn 2_,

that is,

zn–p2≤ xn–p2+ 2Aγzn–xnResNsnAxn–Axn–xn–zn 2_.

This sends us to

ϕn–p2≤(1 –τ αn)2Wnyn–Wnp2– 2μαnϕn–p,Fp

≤(1 –τ αn)2γnzn–p2+ (1 –τ αn)2(1 –γn)xn–p2+ 2μαnϕn–pFp ≤(1 –τ αn)2xn–p2+ 2(1 –τ αn)2γnAγzn–xnResNsnAxn–Axn

– (1 –τ αn)2γnxn–zn2+ 2μαnϕn–pFp.

It follows that

xn+1–p2≤βnϕn–p2+ (1 –βn)xn–p2

≤ xn–p2+ 2βn(1 –τ αn)2γnAγzn–xnResNsnAxn–Axn

–βn(1 –τ αn)2γnxn–zn2+ 2μαnβnϕn–pFp.

Hence

βn(1 –τ αn)2γnxn–zn2

≤ xn–xn+1

xn–p+xn+1–p

+ 2Aγzn–xnResNsnAxn–Axn

+ 2μαnϕn–pFp.

Using (2.9) and (2.10), we have thatxn–zn→0 asn→ ∞, that is,

lim

n→∞xn–Res

N sn

xn+γA∗

ResM_rn–IAxn= 0. (2.11)

Sincexn–zn→0 asn→ ∞, we have that{zn}converges weakly tox¯. Further,{zni}

con-verges weakly tox¯ asi→ ∞. The graphs of maximal monotone mappings are weakly-strongly closed. Observe that

xni–zni sni

+γA∗

ResM

r_niAxni–Axni sni

So 0∈N(¯x). Fixing a positive real numberp, Lemma1.6yields thatAxni–Res

M

p Axni →

asi→ ∞, which implies 0∈M(Ax¯).

We are now in a position to show thatx¯∈ ∞_i=1Fix(Si) =Fix(W). We have yni–Wniyni ≤ yni–Wniyni+Wniyni–Wyni

≤ yni–Wniyni+sup

x∈Ψ

Wnix–Wx.

Relations (2.8) and (2.11) yield thatlimi→∞yni–Wniyni= 0. Ifx¯=Wx¯, then the Opial

condition, Lemma1.5, sends us to

lim sup

i→∞ ¯x–yni <lim supi→∞ Wx¯–yni

≤lim sup i→∞

Wyni–yni+Wx¯–Wyni

≤lim sup

i→∞ ¯x–yni,

a contradiction. Thusx¯∈Fix(W), that is,x¯∈ ∞_i=1Fix(Si). Step 4. We prove that the sequence{xn}is strongly convergent.

SinceFis strongly monotone and Lipschitz continuous, we get that the following vari-ational inequality has a unique solution:

x–y,Fx ≤0, ∀y∈

∞

i=1

Fix(Si)∩SIP(M,N).

Thus

lim sup n→∞

x–ϕn,Fx ≤0. (2.12)

Lemma1.1and Lemma1.3send us to

xn+1–x2

≤βnϕn–x2+ (1 –βn)xn–x2

≤βn(I–μαnF)Wnyn– (I–μαnF)x

2

– 2μαn

ϕn–x,F(x)

+ (1 –βn)xn–x2

≤βn

(1 –τ αn)2yn–x¯2– 2μαn

F(x),ϕn–x

+ (1 –βn)xn–x2

≤(1 – 2τβnαn)xn–x2+ 2τβnαnΠn,

whereΠ=μ_τFx¯,x–ϕn+τ αn

2 xn–x

2_{. In light of Lemma1.4, we find thatx}

n–x →0

asn→ ∞. This completes the proof.

From Theorem2.1we have the following subresult on split inclusion problem (1.3).

Corollary 2.1 Let H1and H2be Hilbert spaces,and let N and M be set-valued maximal

monotone mappings on H1and H2,respectively.Let F:H1→H1 be anL-Lipschitz

H2,and let A∗be its the adjoint operator.Assume thatSIP(M,N)=∅.Let{xn}be the vector

sequence in H1generated by the iterative process

x1∈H1, xn+1=βn(I–μαnF)ResNsn

xn+γA∗

ResM_r

n–I

Axn

+ (1 –βn)xn, n≥1,

whereγ andμare two positive real numbers,{sn}and{rn}are two positive real number sequences,and{αn}and{βn}are real number sequences in(0, 1).Suppose thatγ ∈(0,_A12), μ∈(0,_L2τ2),lim infn→∞sn> 0,limn→∞|sn–sn+1|<∞,lim infn→∞rn> 0,limn→∞|rn–rn+1|<

∞,∞_n=1αn=∞,{βn}is a number sequence in[β¯,β¯],whereβ¯andβ¯are two real numbers

in(0, 1),such thatlim_n→∞|βn+1–βn|= 0.Then the sequence{xn}converges strongly tox∈

H1,which is a unique solution of the variational inequalityx–y,Fx ≤0,∀y∈SIP(M,N).

Remark2.1 In this paper, we investigated the descent iterative methods for split inclusion problem with a common fixed point constraint of an infinite family of nonexpansive map-pings. It deserves mentioning that our method does not involve projections. A solution theorem of the problem was established in the framework of Hilbert spaces under some weak assumptions imposed on different mappings and control sequences.

Acknowledgements

The authors thank the learned referees for their comments and appreciation.

Funding

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. KEP-2-130-39. Therefore the authors acknowledge with thanks DSR for technical and financial support.

Availability of data and materials

All data generated or analyzed during this study are included in this paper.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and approved the final manuscript.

Author details

1_{Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia.}2_{General Education Center, National}

Yunlin University of Science and Technology, Douliou, Taiwan.

Publisher’s Note

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Received: 23 April 2019 Accepted: 7 August 2019

References

1. Al-Mazrooei, A.E., Latif, A., Qin, X., Yao, Y.-C.: Fixed point algorithms for split feasibility problems. Fixed Point Theory20, 245–254 (2019)

2. Ansari, Q.H., Babu, F., Yao, J.-C.: Regularization of proximal point algorithms in Hadamard manifolds. J. Fixed Point Theory Appl.21, Article ID 25 (2019)

3. Ansari, Q.H., Rehan, A.: Iterative methods for generalized split feasibility problems in Banach spaces. Carpath. J. Math. 33, 9–26 (2017)

4. Ansari, Q.H., Rehan, A., Wen, C.-F.: Implicit and explicit algorithms for split common fixed point problems. J. Nonlinear Convex Anal.17, 1381–1397 (2016)

5. Ansari, Q.H., Rehan, A., Yao, J.-C.: Split feasibility and fixed point problems for asymptoticallyk-strict pseudo-contractive mappings in intermediate sense. Fixed Point Theory18, 57–68 (2017)

6. Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Amsterdam (1976) 7. Ceng, L.C., Ansari, Q.H., Yao, J.-C.: Relaxed extragradient methods for finding minimum-norm solutions of the split

feasibility problem. Nonlinear Anal.75, 2116–2125 (2012)

8. Ceng, L.C., Ansari, Q.H., Yao, J.-C.: An extragradient method for solving split feasibility and fixed point problems. Comput. Math. Appl.64, 633–642 (2012)

10. Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)

11. Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications. Inverse Probl. 21, 2071–2084 (2005)

12. Chang, S.S., Lee, H.W.J., Chan, C.K.: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal.70, 3307–3319 (2009)

13. Chang, S.S., Wen, C.F., Yao, J.-C.: Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces. Optimization67, 1183–1196 (2018)

14. Chang, S.S., Wen, C.F., Yao, J.-C.: Zero point problem of accretive operators in Banach spaces. Bull. Malays. Math. Sci. Soc.42, 105–118 (2019)

15. Chang, S.S., Yao, J.-C., Wen, C.F., Wang, L.: A modified viscosity implicit-type proximal point algorithm for monotone inclusions and asymptotically nonexpansive mappings in Hadamard spaces. J. Inequal. Appl.2018, 235 (2018) 16. Cho, S.Y.: Generalized mixed equilibrium and fixed point problems in a Banach space. J. Nonlinear Sci. Appl.9,

1083–1092 (2016)

17. Cho, S.Y.: Strong convergence analysis of a hybrid algorithm for nonlinear operators in a Banach space. J. Appl. Anal. Comput.8, 19–31 (2018)

18. Cho, S.Y.: On the convergence of a modified Mann-type algorithms in Hilbert spaces. J. Nonlinear Convex Anal.20, 461–469 (2019)

19. Cho, S.Y., Kang, S.M.: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci.34, 1607–1618 (2012)

20. Cho, S.Y., Li, W., Kang, S.M.: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013, Article ID 199 (2013)

21. Dehaish, B.A.B., Qin, X., Latif, A., Bakodah, H.O.: Weak and strong convergence of algorithms for the sum of two accretive operators with applications. J. Nonlinear Convex Anal.16, 1321–1336 (2015)

22. Liu, L.: A hybrid steepest descent method for solving split feasibility problems involving nonexpansive mappings. J. Nonlinear Convex Anal.20, 471–488 (2019)

23. Majee, P., Nahak, C.: On inertial proximal algorithm for split variational inclusion problems. Optimization67, 1701–1716 (2018)

24. Moudafi, A.: On the difference of two maximal monotone operators: regularization and algorithmic approaches. Appl. Math. Comput.202, 446–452 (2008)

25. Opial, Z.: Weak convergence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc.73, 591–597 (1967)

26. Peng, C.: A hybrid steepest descent method for a split feasibility problem in Hilbert spaces. J. Nonlinear Convex Anal. 20, 471–488 (2019)

27. Rezapour, S., Zakeri, S.H.: Strong convergence theorems forδ-inverse strongly accretive operators in Banach spaces. Appl. Set-Valued Anal. Optim.1, 39–52 (2019)

28. Shang, M.: A descent-like method for fixed points and split conclusion problems. J. Appl. Numer. Optim.1, 91–101 (2019)

29. Shimoji, K., Takahashi, W.: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwan. J. Math.5, 387–404 (2001)

30. Suzuki, T.: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl.2005, 685918 (2005)

31. Takahashi, W., Wen, C.F., Yao, J.-C.: The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space. Fixed Point Theory19, 407–419 (2018)

32. Xue, Z., Zhou, H., Cho, Y.J.: Iterative solutions of nonlinear equations for m-accretive operators in Banach spaces. J. Nonlinear Convex Anal.1, 313–320 (2000)

33. Yamada, Y.: The hybrid steepest-descent method for variational inequalities problems over the intersection of the fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473–504. North-Holland, Amsterdam (2001)

34. Yang, Y., Yuan, Q.: A hybrid descent iterative algorithm for a split inclusion problem. J. Nonlinear Funct. Anal.2018, Article ID 42 (2018)