The purpose of this paper is to propose an algorithm for solving the split common fixed point problems for total asymptotically strictly pseudocontractive mappings in infinite-dimensional Hilbert spaces. The results presented in the paper improve and extend some recent results of Moudafi (2011 and 2010), Xu (2010 and 2006), Censor and Segal (2009), Censor et al. (2005), Masad and Reich (2007), Censor et al. (2007), Yang (2004), and others.

1. Introduction

Throughout this paper, we always assume that H1,H2 are real Hilbert spaces, “→,⇀” denote by strong and weak convergence, respectively, and F(T) is the fixed point set of a mapping T.

The split common fixed point problem (SCFP) is a generalization of the split feasibility problem (SEP) and the convex feasibility problem (CFP). It is worth mentioning that SFP in finite-dimensional spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [3–5].

SEP in an infinite-dimensional Hilbert space can be found in [2, 4, 6–8]. Moreover the convex feasibility formalism is at the core of the modeling of many inverse problems and has been used to model significant real-world problems.

The split common fixed point problems for a class of quasi-nonexpansive mappings and demicontractive mappings in the setting of Hilbert space were first introduced and studied by Moudafi [9, 10].

The purpose of this paper is to introduce and study the following split common fixed point problem for a more general class of total asymptotically strict pseudocontraction (SCFP) in the framework of an infinite-dimensional Hilbert spaces which contains the quasi-nonexpansive mappings and the demicontractive mappings as its special cases:findx*∈CsuchthatAx*∈Q,
where A:H1→H2 is a bounded linear operator, S:H1→H1 and T:H2→H2 are mappings C:=F(S), and Q:=F(T). In the sequel we use Γ to denote the set of solutions of (SCFP), that is,Γ={x∈C,Ax∈Q}.

2. Preliminaries

We first recall some definitions, notations, and conclusions which will be needed in proving our main results.

Let E be a Banach space. A mapping T:E→E is said to be demiclosed at origin, if for any sequence {xn}⊂E with xn⇀x* and ∥(I-T)xn∥→0, x*=Tx*.

A Banach space E is said to have the Opial property, if for any sequence {xn} with xn⇀x*,liminfn→∞‖xn-x*‖<liminfn→∞‖xn-y‖,∀y∈Ewithy≠x*.

Remark 2.1.

It is well known that each Hilbert space possesses the Opial property.

Definition 2.2.

Let H be a real Hilbert space, and let K be nonempty and closed convex subset of H.

A mapping G:K→K is said to be (γ,{μn},{ξn},ϕ)-totally asymptotically strictly pseudocontractive, if there exist a constant γ∈[0,1) and sequences {μn}⊂[0,∞) and {ξn}⊂[0,∞) with μn→0 and ξn→0 such that for all x,y∈K‖Gnx-Gny‖2≤‖x-y‖2+γ‖(I-Gn)x-(I-Gn)y‖2+μnϕ(‖x-y‖)+ξn,∀n≥1,
where ϕ:[0,∞)→[0,∞) is a continuous and strictly increasing function with ϕ(0)=0.

A mapping G:K→K is said to be (γ,{kn})-asymptotically strictly pseudocontractive, if there exist a constant γ∈[0,1) and a sequence {kn}⊂[1,∞) with kn→1 such that

‖Gnx-Gny‖2≤kn‖x-y‖2+γ‖(I-Gn)x-(I-Gn)y‖2,∀x,y∈K.

Especially, if there exists γ∈[0,1) such that
‖Gx-Gy‖2≤‖x-y‖2+γ‖(I-G)x-(I-G)y‖2,∀x,y∈K,
then G:K→K is called a γ-strictly pseudocontractive mapping.

A mapping G:K→K is said to be uniformly L-Lipschitzian, if there exists a constant L>0, such that
‖Gnx-Gny‖≤L‖x-y‖,∀x,y∈K,n≥1.

A mapping G:K→K is said to be semicompact, if for any bounded sequence {xn}⊂K with limn→∞∥xn-Gxn∥=0, there exists a subsequence {xni}⊂{xn} such that {xni} converges strongly to some point x*∈K.

Remark 2.3.

If ϕ(λ)=λ2,λ≥0, and ξn=0, then a (γ,{μn},{ξn},ϕ)-total asymptotically strictly pseudocontractive mapping is an (γ,{kn})-asymptotically strict pseudocontractive mapping, where {kn=1+μn}.

Proposition 2.4.

Let G:K→K be a (γ,{μn},{ξn},ϕ)-total asymptotically strictly pseudocontractive mapping. If F(G)≠∅, then for each q∈F(G) and for each x∈K, the following inequalities hold and they are equivalent:
‖Gnx-q‖2≤‖x-q‖2+γ‖x-Gnx‖2+μnϕ(‖x-q‖)+ξn;〈x-Gnx,x-q〉≥1-γ2‖x-Gnx‖2-μn2ϕ(‖x-q‖)-ξn2;〈x-Gnx,q-Gnx〉≤γ+12‖Gnx-x‖2+μn2ϕ(‖x-q‖)+ξn2.

Proof.

(I) Inequality (2.6) can be obtained from (2.2) immediately.

(II) (2.6) ⇔ (2.7) In fact, since
‖Gnx-q‖2=‖Gnx-x+x-q‖2=‖Gnx-x‖2+‖x-q‖2+2〈Gnx-x,x-q〉,∀x∈K,q∈F(G),
from (2.6) we have that
‖Gnx-x‖2+‖x-q‖2+2〈Gnx-x,x-q〉≤‖x-q‖2+γ‖x-Gnx‖2+μnϕ(‖x-q‖)+ξn.
Simplifying it, the inequality (2.7) is obtained.

Conversely, from (2.7) the inequality (2.6) can be obtained immediately.

(III) (2.7) ⇔ (2.8) In fact, since〈x-Gnx,x-q〉=〈x-Gnx,x-Gnx+Gnx-q〉=‖x-Gnx‖2+〈x-Gnx,Gnx-q〉,
it follows from (2.7) that
‖x-Gnx‖2+〈x-Gnx,Gnx-q〉≥1-γ2‖x-Gnx‖2-μn2ϕ(‖x-q‖)-ξn2.
Simplifying it, the inequality (2.8) is obtained.

Conversely, the inequality (2.7) can be obtained from (2.8) immediately.

This completes the proof of Proposition 2.4.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B11">11</xref>]).

Let {an}, {bn}, and {δn} be sequences of nonnegative real numbers satisfying
an+1≤(1+δn)an+bn,∀n≥1.
If ∑i=1∞δn<∞ and ∑i=1∞bn<∞, then the limit limn→∞an exists.

Lemma 2.6 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Let H be a real Hilbert space. If {xn} is a sequence in H weakly convergent to z, then
limsupn→∞‖xn-y‖2=limsupn→∞‖xn-z‖2+‖z-y‖2,∀y∈H.

Proposition 2.7.

Let H be a real Hilbert space and let T:H→H be a uniformly L-Lipschitzian and γ,{μn},{ξn},ϕ-total asymptotically strictly pseudocontractive mapping. Then the demiclosedness principle holds for T in the sense that if {xn} is a sequence in H such that xn⇀x*, and limsupm→∞limsupn→∞∥xn-Tmxn∥=0, then (I-T)x*=0. In particular, if xn⇀x*, and ∥(I-T)xn∥→0, then (I-T)x*=0, that is, T is demiclosed at origin.

Proof.

Since {xn} is bounded, we can define a function f on H by
f(x)=limsupn→∞‖xn-x‖2,∀x∈H.
Since xn⇀x*, it follows from Lemma 2.6 that
f(x)=f(x*)+‖x-x*‖2,∀x∈H.
In particular, for each m≥1,
f(Tmx*)=f(x*)+‖Tmx*-x*‖2.
On the other hand, since T is a (γ,{μn},{ξn},ϕ)-total asymptotically strictly pseudocontraction mapping, we get
f(Tmx*)=limsupn→∞‖xn-Tmx*‖2=limsupn→∞‖xn-Tmxn+Tmxn-Tmx*‖2=limsupn→∞(‖xn-Tmxn‖2+2〈xn-Tmxn,Tmxn-Tmx*〉+‖Tmxn-Tmx*‖2)≤limsupn→∞‖xn-Tmxn‖(‖xn-Tmxn‖+2L‖xn-x*‖)+limsupn→∞(‖xn-x*‖2+γ‖xn-Tmxn-(x*-Tmx*)‖2+μmϕ(‖xn-x*‖)+ξm).
Taking limsupm→∞ on both sides and observing the facts that limm→∞μm=0, limm→∞ξm=0, and limsupm→∞limsupn→∞∥xn-Tmxn∥=0, we derive that
limsupm→∞f(Tmx*)≤limsupm→∞‖xn-x*‖2+γlimsupm→∞‖x*-Tmx*‖2=f(x*)+γlimsupm→∞‖x*-Tmx*‖2.
On the other hand, it follows from (2.17) that
limsupm→∞f(Tmx*)=f(x*)+limsupm→∞‖x*-Tmx*‖2.
Since κ<1, this together with (2.19) shows that limsupm→∞∥x*-Tmx*∥2=0. That is, limm→∞Tmx*=x*; hence Tx*=x*.

3. Split Common Fixed Point Problem

For solving the split common fixed point problem (1.1), let us assume that the following conditions are satisfied.

(1) H1 and H2 are two real Hilbert spaces, and A:H1→H2 is a bounded linear operator.

(2) S:H1→H1 is a uniformly L-Lipschitzian and (β,{μn(1)},{ξn(1)},ϕ1)-total asymptotically strictly pseudocontractive mapping and T:H2→H2 is a uniformly L̃-Lipschitzian and (κ,{μn(2)},{ξn(2)},ϕ2)-total asymptotically strictly pseudocontractive mapping satisfying the following conditions:

C:=F(S)≠∅, Q:=F(T)≠∅;

μn=max{μn(1),μn(2)},ξn=max{ξn(1),ξn(2)},n≥1, and ∑n=1∞μn<∞, ∑n=1∞ξn<∞;

ϕ=max{ϕ1,ϕ2} and there exist two positive constants M and M* such that ϕ(λ)≤M*λ2 for all λ≥M.

We are now in a position to give the following result.

Theorem 3.1.

Let H1,H2,A,S,T,C,Q,β,κ,L,L̃,{μn},{ξn}, and ϕ be the same as mentiond before. Let {xn} be the sequence generated by
x1∈H1chosenarbitrarily,xn+1=(1-αn)un+αnSn(un),un=xn+γA*(Tn-I)Axn,∀n≥1,
where {αn} is a sequence in [0,1] and γ>0 is a constant satisfying the following conditions:

αn∈(δ,1-β),for all n≥1 and γ∈(0,(1-κ)/∥A∥2), where δ∈(0,1-β) is a positive constant.

If Γ≠∅ (where Γ is the set of solutions to (SCFP)-(1.1)), then {xn} converges weakly to a point x*∈Γ.

In addition, if S is also semicompact, then {xn} and {un} both converge strongly to x*∈Γ.

Proof.

The following is the proof of Theorem 3.1.

The Proof of Conclusion (I)

(1) First we prove that for each p∈Γ, the following limits exist and
limn→∞‖xn-p‖=limn→∞‖un-p‖.

In fact, since ϕ is a continuous and increasing function, it results that ϕ(λ)≤ϕ(M), if λ≤M, and ϕ(λ)≤M*λ2, if λ≥M. In either case, we can obtain that
ϕ(λ)≤ϕ(M)+M*λ2,∀λ≥0.

For any given p∈Γ, hence p∈C:=F(S), and Ap∈Q:=F(T), from (3.1) and (2.7) we have
‖xn+1-p‖2=‖un-p-αn(un-Snun)‖2=‖un-p‖2-2αn〈un-p,un-Snun〉+αn2‖un-Snun‖2≤‖un-p‖2-αn(1-β)‖un-Snun‖2+αnμnϕ(‖un-p‖)+αnξn+αn2‖un-Snun‖2(by(2.7))≤‖un-p‖2-αn(1-β-αn)‖un-Snun‖2+αnμn(ϕ(M)+M*(‖un-p‖)2)+αnξn=(1+αnμnM*)‖un-p‖2-αn(1-β-αn)‖un-Snun‖2+αnμnϕ(M)+αnξn.
On the other hand, since
‖un-p‖2=‖xn-p+γA*(Tn-I)Axn‖2=‖xn-p‖2+γ2‖A*(Tn-I)Axn‖2+2γ〈xn-p,A*(Tn-I)Axn〉,γ2‖A*(Tn-I)Axn‖2=γ2〈A*(Tn-I)Axn,A*(Tn-I)Axn〉=γ2〈AA*(Tn-I)Axn,(Tn-I)Axn〉≤γ2‖A‖2‖TnAxn-Axn‖2,2γ〈xn-p,A*(Tn-I)Axn〉=2γ〈Axn-Ap,(Tn-I)Axn〉=2γ〈Axn-Ap)+(Tn-I)Axn-(Tn-I)Axn,(Tn-I)Axn〉=2γ{〈TnAxn-Ap,TnAxn-Axn〉-‖(Tn-I)Axn‖2}.
In (2.8) taking x=Axn,Gn=Tn, and q=Ap and noting Ap∈F(T), from (2.8) we have
〈TnAxn-Ap,TnAxn-Axn〉-‖(Tn-I)Axn‖2≤1+κ2‖(Tn-I)Axn‖2+μn2ϕ(‖Axn-Ap‖)+ξn2-‖(Tnn-I)Axn‖2≤κ-12‖(Tn-I)Axn‖2+μn2(ϕ(M)+M*‖Axn-Ap‖2)+ξn2≤κ-12‖(Tn-I)Axn‖2+μn2M*‖A‖2‖xn-p‖2+μn2ϕ(M)+ξn2.
Substituting (3.8) into (3.7), after simplifying it and then substituting the resultant result into (3.5), we have
‖un-p‖2≤‖xn-p‖2+γ2‖A‖2‖TnAxn-Axn‖2+γ(κ-1)‖(Tn-I)Axn‖2+γμnM*‖A‖2‖xn-p‖2+γμnϕ(M)+γξn=‖xn-p‖2-γ(1-κ-γ‖A‖2)‖TnAxn-Axn‖2+γμnM*‖A‖2‖xn-p‖2+γμnϕ(M)+γξn≤(1+γμnM*‖A‖2)‖xn-p‖2-γ(1-κ-γ‖A‖2)‖TnAxn-Axn‖2+γμnϕ(M)+γξn.
Substituting (3.9) into (3.4) and simplifying it we have
‖xn+1-p‖2≤(1+αnμnM*){(1+γμnM*‖A‖2)‖xn-p‖2-γ(1-κ-γ‖A‖2)‖TnAxn-Axn‖2+γμnϕ(M)+γξn}-αn(1-β-αn)‖un-Snun‖2+αnμnϕ(M)+αnξn=(1+δn)‖xn-p‖2-γ(1-κ-γ‖A‖2)‖TnAxn-Axn‖2-αn(1-β-αn)‖un-Snun‖2+bn,
where
δn=αnμnM*+γμnM*‖A‖2+γ‖A‖2αnμn2(M*)2,bn=((1+αnμnM*)γ+αn)μnϕ(M)+((1+αnμn⋅M*)γ+αn)ξn.
By condition (iii) we have
‖xn+1-p‖2≤(1+δn)‖xn-p‖2+bn.
By condition (ii), ∑n=1∞δn<∞ and ∑n=1∞bn<∞. Hence it follows from Lemma 2.5 that the following limit exists:
limn→∞‖xn-p‖.
Consequently, from (3.10) and (3.13) we have that
γ(1-κ-γ‖A‖2)‖(Tn-I)Axn‖2+αn(1-β-αn)‖un-Snun‖2≤‖xn-p‖2-‖xn+1-p‖2+δn‖xn-p‖2⟶0(asn⟶∞).
This together with the condition (iii) implies that
limn→∞‖un-Snun‖=0;limn→∞‖(Tn-I)Axn‖=0.
It follows from (3.5), (3.13), and (3.16) that the limit limn→∞∥un-p∥ exists and
limn→∞‖un-p‖=limn→∞‖xn-p‖.
The conclusion (1) is proved.

(2) Next we prove that
limn→∞‖xn+1-xn‖=0,limn→∞‖un+1-un‖=0.

In fact, it follows from (3.1) that
‖xn+1-xn‖=‖(1-αn)un+αnSn(un)-xn‖=‖(1-αn)(xn+γA*(Tn-I)Axn)+αnSn(un)-xn‖=‖(1-αn)γA*(Tn-I)Axn+αn(Sn(un)-xn)‖=‖(1-αn)γA*(Tn-I)Axn+αn(Sn(un)-un)+αn(un-xn)‖=‖(1-αn)γA*(Tn-I)Axn+αn(Sn(un)-un)+αnγA*(Tn-I)Axn‖=‖γA*(Tn-I)Axn+αn(Sn(un)-un)‖.
In view of (3.15) and (3.16) we have that
limn→∞‖xn+1-xn‖=0.
Similarly, it follows from (3.1), (3.16), and (3.20) that
‖un+1-un‖=‖xn+1+γA*(Tn+1-I)Axn+1-(xn+γA*(Tn-I)Axn)‖≤‖xn+1-xn‖+γ‖A*(Tn+1-I)Axn+1‖+γ‖A*(Tn-I)Axn‖⟶0(asn⟶∞).
The conclusion (3.18) is proved.

(3) Next we prove that
‖un-Sun‖⟶0,‖Axn-TAxn‖⟶0(asn⟶∞).

In fact, from (3.15) we have
ζn:=‖un-Snun‖⟶0(asn⟶∞).
Since S is uniformly L-Lipschitzian continuous, it follows from (3.18) and (3.23) that
‖un-Sun‖≤‖un-Snun‖+‖Snun-Sun‖≤ζn+L‖Sn-1un-un‖≤ζn+L{‖Sn-1un-Sn-1un-1‖+‖Sn-1un-1-un‖}≤ζn+L2‖un-un-1‖+L‖Sn-1un-1-un-1+un-1-un‖≤ζn+L(1+L)‖un-un-1‖+Lζn-1⟶0(asn⟶∞).

Similarly, from (3.16) we have
‖Axn-TnAxn‖⟶0(asn⟶∞).
Since T is uniformly L̃-Lipschitzian continuous, by the same way as above, from (3.18) and (3.25), we can also prove that
‖Axn-TAxn‖⟶0(asn⟶∞).

(4) Finally we prove that xn⇀x* and un⇀x* which is a solution of (SCFP)-(1.1).

Since {un} is bounded, there exists a subsequence {uni}⊂{un} such that uni⇀x* (some point in H1). From (3.22) we have
‖uni-Suni‖⟶0(asni⟶∞).
By Proposition 2.7, S is demiclosed at zero; hence we know that x*∈F(S).

Moreover, from (3.1) and (3.16) we have
xni=uni-γA*(Tni-I)Axni⇀x*.
Since A is a linear bounded operator, it gets Axni⇀Ax*. In view of (3.22) we have
‖Axni-TAxni‖⟶0(asni⟶∞).
Again by Proposition 2.7, T is demiclosed at zero, and we have Ax*∈F(T). Summing up the above argument, it shows that x*∈Γ; that is, x* is a solution to the (SCFP)-(1.1).

Now we prove that xn⇀x* and un⇀x*.

Suppose to the contrary, if there exists another subsequence {unj}⊂{un} such that unj⇀y*∈Γ with y*≠x*, then by virtue of (3.2) and the Opial property of Hilbert space, we have
liminfni→∞‖uni-x*‖<liminfni→∞‖uni-y*‖=limn→∞‖un-y*‖=limnj→∞‖unj-y*‖<liminfnj→∞‖unj-x*‖=limn→∞‖un-x*‖=liminfni→∞‖uni-x*‖.
This is a contradiction. Therefore, un⇀x*. By using (3.1) and (3.16), we have
xn=un-γA*(Tnn-I)Axn⇀x*.

The Proof of Conclusion (II)

By the assumption that S is semicompact, it follows from (3.27) that there exists a subsequence of {uni} (without loss of generality, we still denote it by {uni}) such that uni→u*∈H (some point in H). Since uni⇀x*, this implies that x*=u*. And so uni→x*∈Γ. By virtue of (3.2) we know that limn→∞∥un-x*∥=0 and limn→∞∥xn-x*∥=0; that is, {un} and {xn} both converge strongly to x*∈Γ.

This completes the proof of Theorem 3.1.

The following theorem can be obtained from Theorem 3.1 immediately.

Theorem 3.2.

Let H1 and H2 be two real Hilbert spaces, let A:H1→H2 be a bounded linear operator, let S:H1→H1 be a uniformly L-Lipschitzian and (β,{kn(1)})-asymptotically strictly pseudocontractive mapping, and let T:H2→H2 be a uniformly L̃-Lipschitzian and (κ,{kn(2)})- asymptotically strictly pseudocontractive mapping satisfying the following conditions:

C:=F(S)≠∅, Q:=F(T)≠∅;

kn=max{kn(1),kn(2)}, and ∑n=1∞(kn-1)<∞.

Let {xn} be the sequence defined by (3.1), where {αn} is a sequence in [0,1] and γ>0 is a constant satisfying the following condition:

αn∈(δ,1-β), for all n≥1 and γ∈(0,(1-κ)/∥A∥2), where δ∈(0,1-β) is a constant. If Γ≠∅, then the conclusions of Theorem 3.1 still hold.

From Theorems 3.1 and 3.2 we can obtain the following.

Theorem 3.3.

Let H1 and H2 be two real Hilbert spaces, let A:H1→H2 be a bounded linear operator, S:H1→H1 be a uniformly L-Lipschitzian and β-strictly pseudocontractive mapping, and let T:H2→H2 be a uniformly L̃-Lipschitzian and κ-strictly pseudocontractive mapping satisfying the following conditions:

C:=F(S)≠∅, Q:=F(T)≠∅;

T and S both are demiclosed at origin.

Let {xn} be the sequence generated by
x1∈H1chosenarbitrarily,xn+1=(1-αn)un+αnSun,un=xn+γA*(T-I)Axn,∀n≥1,
where {αn} is a sequence in [0,1] and γ>0 is a constant satisfying the following condition:

αn∈(δ,1-β), for all n≥1 and γ∈(0,(1-κ)/∥A∥2), where δ∈(0,1-β) is a constant. If Γ≠∅, then the conclusions of Theorem 3.1 still hold.

Proof.

By the same way as given in the proof of Theorems 3.1 and 3.2 and noting that in the case of strictly pseudocontractive mapping the sequence {kn=1} in Theorem 3.2. Therefore we can prove that for each p∈Γ, the limits limn→∞∥xn-p∥ and limn→∞∥un-p∥ exist and
limn→∞‖xn-p‖=limn→∞‖un-p‖;‖un-Sun‖⟶0;‖Axn-TAxn‖⟶0;‖un-un+1‖⟶0;‖xn-xn+1‖⟶0;xn⇀x*;un⇀x*∈Γ.
In addition, if S is also semicompact, we can also prove that {xn} and {un} both converge strongly to x*.

Remark 3.4.

Theorems 3.1 and 3.2 improve and extend the corresponding results of Censor et al. [4, 5], Yang [7], Moudafi [9, 10], Xu [13], Censor and Segal [14], Masad and Reich [15], and others.

Acknowledgments

The authors would like to express their thanks to the editors and the referees for their helpful comments and suggestions. This work was supported by the Natural Science Foundation of Yunnan Province and Yunnan University of Finance and Economics.

CensorY.ElfvingT.A multiprojection algorithm using Bregman projections in a product spaceByrneC.Iterative oblique projection onto convex sets and the split feasibility problemCensorY.BortfeldT.MartinB.TrofimovA.A unified approach for inversion problem in intensity-modulated radiation therapyCensorY.ElfvingT.KopfN.BortfeldT.The multiple-sets split feasibility problem and its applications for inverse problemsCensorY.MotovaA.SegalA.Perturbed projections and subgradient projections for the multiple-sets split feasibility problemXuH.-K.A variable Krasnosel'skii-Mann algorithm and the multiple-set split feasibility problemYangQ.The relaxed CQ algorithm solving the split feasibility problemZhaoJ.YangQ.Several solution methods for the split feasibility problemMoudafiA.The split common fixed-point problem for demicontractive mappingsMoudafiA.A note on the split common fixed-point problem for quasi-nonexpansive operatorsAoyamaK.KimuraY.TakahashiW.ToyodaM.Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach spaceMartinez-YanesC.XuH.-K.Strong convergence of the CQ method for fixed point iteration processesXuH. K.Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spacesCensorY.SegalA.The split common fixed point problem for directed operatorsMasadE.ReichS.A note on the multiple-set split convex feasibility problem in Hilbert space