# A New Iterative Construction for Approximating Solutions of a Split Common Fixed Point Problem

Authors: Huimin He; Qinwei Fan; Rudong Chen
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we aim to construct a new strong convergence algorithm for a split common fixed point problem involving the demicontractive operators. It is proved that the vector sequence generated via the Halpern-like algorithm converges to a solution of the split common fixed point problem in norm. The main convergence results presented in this paper extend and improve some corresponding results announced recently. The highlights of this paper shed on the novel algorithm and the new analysis techniques.


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## Body

## 1. Introduction

LetH1 and H2 be the Hilbert spaces and C and Q be nonempty closed and convex subsets of H1 and H2, respectively.The split feasibility problem (SFP) is known to find(1)x∈C,such thatAx∈Q,where A:H1⟶H2 is a linear bounded operator.In [1], the split feasibility problem (SFP) in the finite-dimensional Hilbert spaces was introduced by Censor and Elfving. This problem is equivalent to a number of nonlinear optimization problems and finds numerous real applications, such as signal processing and medical imaging (see, e.g., [2–7]).For this split problem, simultaneous multiprojections algorithm was employed by Censor and Elfving in the finite-dimensional spaceRn to obtain the algorithm as follows: (2)xn+1=A−1PQPACAxn,where both C and Q are convex and closed subsets of Rn, the linear bounded operator A of Rn is an n×n matrix, and PQ is the orthogonal projection operator onto the sets Q.The above algorithm (2) involves the matrix A−1 (one always assumes the existence of A−1) at every iterative step. Calculating A−1 is very much time-consuming, if the dimensions are large scale, in particular, and thus it does not become popular.In order to overcome the fault, Byrne [2, 8] proposed the following novel algorithm CQ, which is under the spotlight of recent research(3)xn+1=PCxn−γA∗I−PQAxn,n≥0,where PC and PQ are the orthogonal projection operators onto the sets C and Q, respectively, and 0<γ<2/ρ with ρ being the spectral radius of the composite mapping A∗A. But, the CQ algorithm’s step-size is fixed, and it is related to spectral radius of A∗A. On the other hand, the orthogonal projection onto the subsets C and Q in Hilbert space H1 is not easily calculated generally except the special cases, such as balls and polyhedrals. With the real applications (intensity-modulated radiation therapy and medical imaging) of the SFP in signal processing, the SFP has obtained much attention. Now, the approximate solutions of the SFP have been studied extensively by scholars and engineers (see, e.g., [9–13]).In (1), if CandQ are the intersections of fixed point sets of finite many nonlinear operators, the SFP becomes the split common fixed point problem (SCFPP). The SCFPP was studied first by Censor and Segal [14] in 2009, which consists of finding an element x∈H1 with(4)x∈∩i=1mFixTi,s.t.Ax∈∩j=1nFixSj,where FixTi denotes the fixed point set of Ti:H1⟶H1 and FixSj denotes the fixed point sets of Si:H2⟶H2, respectively.In particular, ifm=n=1, then(5)x∈FixT,s.t.Ax∈FixS,and T:H1⟶H1, S:H2⟶H2, and FixT denotes the fixed point set of T, and FixS denotes the fixed point set of S.The SCFPP becomes a specific case of SFP and closely related to SFP. To solve this problem, the original algorithm for the directed operator was introduced by Censor and Segal [14] in 2009 as follows: (6)xn+1=Txn−ρA∗I−SAxn,n≥0,where ρ satisfies the constraint condition 0<ρ<2/A2, and the authors got the weak convergence of the sequence xn for solving the SCFPP (5) if the SCFPP consists, that is, its solution set is nonempty.Recently, Cui and Wang [15] studied the following algorithm, and they got the weak convergence of the sequence xn for solving the SCFPP (5):(7)xn+1=Uλxn−ρnA∗I−TAxn,where Uλ=1−λI+λU and ρn is given in the following pattern:(8)ρn=1−τI−TAxn22A∗I−TAxn2,Axn≠TAxn,0,otherwise.The step-size of this algorithmρn does not depend on the norm of the operator A and searches automatically.In 2015, Boikanyo [16] extended the main results of Cui and Wang [15] and constructed the Halpern-type algorithm for demicontractive operators that converge to a solution of the SCFPP (5) strongly:(9)xn+1=αnu+1−αnUλxn−ρnA∗I−TAxn,where ρn is given as (8). In this result, the resolvent I−ρnA∗I−TA plays an important role. Indeed, the techniques of resolvents is quite popular, and it acts as a bridge between fixed point problems and a number of optimization problems (see, e.g., [17–21] and the references therein).Motivated by the above results, we propose a novel algorithm on demicontractive operators for approximating a solution of the SCFPP (5):(10)un=xn−ρnA∗I−TAxn,xn+1=1−αn1−ξnI+ξnU1−ηnI+ηnUun+αnu,where ρn is also obtained by (8). Our algorithm is also based on the Halpern iteration. Indeed, it is a core for many algorithms in split problems (see, e.g., [22–26]). We get the strong convergence of the iterative sequence xn generated by (10) for solving the SCFPP (5). Our main results are in two folds. First, we construct a novel iterative algorithm to solve the split common fixed point problem for the demicontractive operators. Second, we permit step-size to be selected self-adaptively by the self-adaptive method, which avoids to depend on the norm of the nonlinear operator A. Our results extend and improve some results of Boikanyo [16], Cui and Wang [15], Yao et al. [27], and many others.


## 2. Preliminaries

In this section, we will present some lemmas, which are useful to prove our main results as follows.LetH be a Hilbert space, which is endowed with the inner product ·,·, norm ·. Then, the following inequalities hold:(11)u+v2≤u2+2v,u+v,∀u,v∈H,(12)tu+1−tv2=tu2+1−tv2−t1−tu−v2,∀t∈Rand∀u,v∈H.Definition 1.
LetT:H⟶H be an operator, then I−T called demiclosed at zero, if the following implication holds for any xn in H:(13)xn⇀xI−Txn⟶0⇒x=Tx.Note that the nonexpansive operator is demiclosed at zero [28].Lemma 1.
(see [29]). Let an be a sequence of real nonnegative numbers with(14)an+1≤1−γnan+δn,where γn is a sequence in 0,1 and δn is a real sequence such that(i)
∑n=1∞γn=∞(ii)
limsupn⟶∞δn/γn≤0 or ∑n=1∞δn<∞
Then,limn⟶∞an=0.Lemma 2.
(see [15]). Let A:H1⟶H2 be a linear bounded operator and T:H2⟶H2 a τ−demicontractive mapping with τ<1. If A−1FixT≠∅, then it is as follows:(a)
I−TAx^=0⇔A∗I−TAx^=0, ∀x^∈H1.(b)
In addition, forz∈A−1FixT,(15)x−z−ρA∗I−TAx^2+1−τ2I−TAx^44A∗I−TAx^2≤x^−z2,where x∈H1, Ax≠TAx and(16)ρ≔1−τI−TAx^22A∗I−TAx^2.Lemma 3.
(see [30]). Let H be a Hilbert space and let T be an L-Lipschitzian mapping defined on H with the module L≥1. Set(17)K≔ξTηT+1−ηI+1−ξI.
If0<ξ<η<1/1+1+L2, then the following conclusions hold:(1)
K is demiclosed at zero point 0, if T is demiclosed at 0(2)
FixT=FixTηT+1−ηI=FixK(3)
IfT:H⟶H is a quasi-pseudo-contractive operator, then the operator K is quasi-non-expansiveLemma 4.
(see [31]). Let sk be a real numbers sequence that does not decrease at infinity in the sense that there exists a subsequence skj of sk such that skj<skj+1 for all j≥0. Define an integer sequence mkk≥k0 by(18)mk=maxk0≤l≤k:sl<sl+1.
Then,mk⟶∞ as k⟶∞ and(19)smk+1≥maxsmk,sk,for all k≥k0.


## 3. Some Nonlinear Operators

Definition 2.
An operatorT:H⟶H is said to be L−Lipschitzian if and only if there exists L>0 such that(20)Tx−Ty≤Lx−y,for all x,y∈C.Definition 3.
An operatorT:H⟶H is said to be nonexpansive if and only if(21)Tx−Ty≤x−z,∀x∈H.Definition 4.
An operatorT:H⟶H is said to be quasi-non-expansive if and only if FixT≠∅ and(22)Tx−z≤x−z,∀x∈H,∀z∈FixT.Definition 5.
An operatorT:H⟶H is said to be firmly nonexpansive if and only if(23)Tx−Ty2≤x−y2−I−Tx−I−Ty2,∀x,y∈H.Definition 6.
An operatorT:H⟶H is said to be firmly quasi-non-expansive if and only if FixT≠∅ and(24)Tx−z2≤x−z2−I−Tx2,∀x∈H,∀z∈FixT.Definition 7.
An operatorT:H⟶H is said to be pseudocontractive if and only if(25)Tx−Ty,x−y≤x−y2,∀x,y∈H.Note thatT is pseudocontractive if and only if the operator I−T is monotone. There is also an alternative definition for pseudocontractive operators, that is, T is said to be pseudocontractive if and only if(26)Tx−Ty2≤x−y2+I−Tx−I−Ty2,∀x,y∈H.Definition 8.
An operatorT:H⟶H is said to be quasi-pseudo-contractive if and only if FixT≠∅ and(27)Tx−x∗2≤x−x∗2+Tx−x2,∀x∈H,∀x∗∈FixT.Definition 9.
An operatorT:H⟶H is said to be strictly pseudocontractive if and only if there exists k∈0,1 such that(28)Tx−Ty2≤x−y2+kI−Tx−I−Ty2,∀x,y∈H.Definition 10.
A operatorT:H⟶H is said to be directed if and only if(29)z−Tx,x−Tx≤0,∀x∈H,∀z∈FixT.Definition 11.
An operatorT:H⟶H is said to be τ−demicontractive with τ<1 if and only if(30)Tx−z2≤x−z2+τx−Tx2,∀x∈H,∀z∈FixT.It is easy to obtain that (29) is equivalent to(31)z−Tx2+x−Tx2−x−z2≤0,∀x∈H,∀z∈FixT.Remark 1.
The classes ofk-demicontrative mappings, directed mappings, quasi-non-expansive mappings, and nonexpansive mappings are closely related. By the above definitions, we obtain the following conclusion relations easily (see Figures 1–7).(1)
The nonexpansive mapping withFixT≠∅ is quasi-non-expansive mapping(2)
The quasi-non-expansive mapping is0−demicontrative mapping(3)
The firmly nonexpansive mapping is nonexpansive mapping(4)
The firmly quasi-non-expansive mapping is quasi-non-expansive mapping(5)
The firmly nonexpansive mapping is firmly quasi-non-expansive mapping(6)
The directed mapping is demicontractive mapping(7)
The demicontractive mapping is quasi-pseudo-contractive mapping(8)
The strictly pseudocontractive mapping is pseudocontractive mapping(9)
The pseudocontractive mapping is quasi-pseudo-contractive mappingFigure 1
The relations of some nonlinear operators.Figure 2
The relations of some nonlinear operators.Figure 3
The relations of some nonlinear operators.Figure 4
The relations of some nonlinear operators.Figure 5
The relations of some nonlinear operators.Figure 6
The relations of some nonlinear operators.Figure 7
The relations of some nonlinear operators.


## 4. Main Results

In this section, some assumptions are as follows:(1)
H1 and H2 are two Hilbert spaces, A:H1⟶H2 is a linear bounded operator, and A∗ is the adjoint of A(2)
U:H1⟶H1 and T:H2⟶H2 are two L−Lipschitzian operators with L≥1, FixU≠∅, andFixT≠∅(3)
U:H1⟶H1 is a κ-demicontractive operator (κ<1), and T:H2⟶H2 is a τ-demicontractive operator (τ<1)(4)
I−U and I−T are two demiclosed operators at O(5)
The set of solutions of SCFPP (5), denoted by S, is nonemptyThe strong convergence of a sequencexn to a point x∈H is denoted by xn⟶x.Now, we give the new algorithm to findx∗∈S.where A is a bounded and linear mapping, A∗ is the adjoint of operator A, and ρn is obtained as follows:(33)ρn=1−τI−TAxn22A∗I−TAxn2,Axn≠TAxn,0,otherwise.Algorithm 1.
H1 is a real Hilbert space, and FixU≠∅. Take an initial point x0∈H1 arbitrarily, and fix u∈H1 and θn⊂0,1. If the n−th iteration xn is available, then the n+1−th iteration is constructed via the following formula:(32)un=xn−ρnA∗I−TAxn,xn+1=θnu+1−θn1−μnI+μnU1−νnI+νnUun,Lemma 5.
Assume thatH1 is a Hilbert space, U:H1⟶H1 is a κ-demicontractive operator with κ≤1, L−Lipschitzian mappings (L≥1), and FixU≠∅. Denote Uμ,ν≔1−μI+μU1−νI+νU with 0<μ<ν<2−κ/1+1+L22−κ. Then, for all x∈H1,(34)z−Uμ,ν2≤x−z2−μν2−2ν−κ−ν2L2Ux−x2,where z∈FixU. Moreover,(35)z−Uμ,ν≤z−x.
That is,Uμ,ν is quasi-non-expansive.Proof.
Sincez∈FixU, we get from (30) that(36)U1−νI+νUx−z2≤1−νI+νUx−z2+κ1−νI+νUx−U1−νI+νUx2≤1−νx−z+νUx−z2+κ1−νI+νUx−U1−νI+νUx2.
Based on the fact thatU is L−Lipschitzian, we get(37)Ux−U1−νI+νUx≤νLx−Ux.
Also, from (30) and (12), we can get(38)1−νx−z+νUx−z2=1−νx−z2+νUx−z2−ν1−νx−Ux2≤1−νx−z2+νx−z2+κUx−x2−ν1−νx−Ux2=x−z2+νν+κ−1Ux−x2.
By (12) and (37), we get(39)1−νI+νUx−U1−νI+νUx2=1−νx−U1−νI+νUx+νUx−U1−νI+νUx2=1−νx−U1−νI+νUx2+νUx−U1−νI+νUx2−ν1−νx−Ux2≤1−νx−U1−νI+νUx2+νν2L2Ux−x2−ν1−νx−Ux2=1−νx−U1−νI+νUx2−ν1−ν−ν2L2x−Ux2.
Substituting (38) and (39) into (36), we have(40)U1−νI+νUx−z2≤x−z2+νν+κ−1Ux−x2+1−νx−U1−νI+νUx2−ν1−ν−ν2L2x−Ux2=x−z2+1−νx−U1−νI+νUx2−ν2−2ν−κ−ν2L2x−Ux2.
Sinceμ<ν, combining (12) and (40), we get(41)1−μx+μU1−νI+νUx−z2=1−μx−z+μU1−νI+νUx−z2=1−μx−z2+μU1−νI+νUx−z2−μ1−μU1−νI+νUx−x2=1−μx−z2−μ1−μU1−νI+νUx−x2+μx−z2+1−νx−U1−νI+νUx2−ν2−2ν−κ−ν2L2x−Ux2=x−z2+μμ−νx−U1−νI+νUx2−ν2−2ν−κ−ν2L2x−Ux2≤x−z2−ν2−2ν−κ−ν2L2x−Ux2.
Sinceν<2−κ/1+1+L22−κ, we deduce(42)2−2ν−κ−ν2L2>0.
Hence,(43)1−μx+μU1−νI+νUx−z2≤x−z2.
That is,Uμ,ν is quasi-non-expansive.Theorem 1.
Assume that problem (5) is consistent S≠∅. Let H1,H2,A,U,T,xn be the same as above. If θn⊂0,1 satisfies limn⟶∞θn=0 and ∑n=0∞θn=∞, where a and b are constants and μn and νn satisfies 0<a<μn<νn<b<2−κ/1+1+L22−κ,∀n≥1, then the sequence xn converges to a point x¯∈S in norm and x¯ is the nearest point S to ux¯=PSu.Proof.
This proof is split into three parts as follows.Step 1.
Prove thatxn is a bounded sequence.
Takep∈S. From Theorem 1, we know that Uμn,νn is quasi-non-expansive. From (32), we have(44)xn+1−p=θnu+1−θnUμn,νnun−p=θnu−p+1−θnUμn,νnun−p≤θnu−p+1−θnUμn,νnun−p≤θnu−p+1−θnun−p≤θnu−p+1−θnxn−p.
By induction, we get(45)xn−p≤maxu−p,x0−p.
Thus,xn is bounded.Step 2.
(46)xn+1−x¯2≤1−θnxn−x¯2+2θnu−x¯,xn+1−x¯,where x¯=PSu.
Consider the caseρn≠0. From (32), (35), and (11), we get(47)xn+1−x¯2=θnu+1−θnUμn,νnun−x¯=θnu−x¯+1−θnUμn,νnun−x¯≤1−θn2Uμn,νnun−x¯2+2θnu−x¯,xn+1−x¯≤1−θnUμn,νnun−x¯2+2θnu−x¯,xn+1−x¯≤1−θnun−x¯2+2θnu−x¯,xn+1−x¯≤1−θnxn−x¯2−1−τ24I−TAxn4A∗I−TAxn2+2θnu−x¯,xn+1−x¯≤1−θnxn−x¯2+2θnu−x¯,xn+1−x¯.
Hence,(48)xn+1−x¯2≤1−θnxn−x¯2+2θnu−x¯,xn+1−x¯.
Consider the caseρn=0. From (32) and (11), we get(49)xn+1−x¯2=θnu+1−θnUμn,νnun−x¯=θnu−x¯+1−θnUμn,νnun−x¯≤1−θn2Uμn,νnun−x¯2+2θnu−x¯,xn+1−x¯≤1−θnUμn,νnun−x¯2+2θnu−x¯,xn+1−x¯≤1−θnun−x¯2+2θnu−x¯,xn+1−x¯≤1−θnxn−x¯2+2θnu−x¯,xn+1−x¯.
Hence,(50)xn+1−x¯2≤1−θnxn−x¯2+2θnu−x¯,xn+1−x¯.Step 3.
Prove thatxn⟶x¯ as n⟶∞.
This step is divided into two cases. Denotesn≔xn−x¯2.Case 1.
Assume there exists a positive integern0 and the sequence sn is decreasing for any n≥n0. Then, sn converges to some point strongly by the monotonic bounded principle.
First, we show that(51)limsupn⟶∞u−x¯,xn−x¯≤0.
Using the choice (33) of the step-size ρn, (32), (34), (35), and (11), we get(52)xn+1−x¯2=θnu+1−θnUμn,νnun−x¯=θnu−x¯+1−θnUμn,νnun−x¯≤1−θn2Uμn,νnun−x¯2+2θnu−x¯,xn+1−x¯≤Uμn,νnun−x¯2+2θnu−x¯,xn+1−x¯≤un−x¯2−μnνn2−2νn−κ−νn2L2Uun−un2+2θnu−x¯,xn+1−x≤xn−x¯2−I−TAxn4A∗I−TAxn21−τ24−μnνn2−2νn−κ−νn2L2Uun−un2+2θnu−x¯,xn+1−x¯.
So,(53)μnνn2−2νn−κ−νn2L2Uun−un2≤sn−sn+1+θnL,0≤1−τ2I−TAxn44A∗I−TAxn2≤sn−sn+1+θnL,where L is a nonnegative real constant such that supn∈N2fxn−x¯,xn+1−x¯≤L. Based on the fact that sn is convergent, we have(54)un−Uun⟶0,asn⟶∞,(55)I−TAxn2A∗I−TAxn⟶0,asn⟶∞.
Moreover,(56)I−TAxn2A∗I−TAxn≥I−TAxn2I−TAxn·A≥I−TAxnA.
Hence,(57)Axn−TAxn⟶0.
Since(58)xn−un=ρnA∗I−TAxn=1−τI−TAxn22A∗I−TAxn⟶0,asn⟶∞.
Sincexn⇀q, we have un⇀q due to (58). From (54) and as I−U is demiclosed at zero, we have(59)q∈FixU.
From (55) and I−T is demiclosed at zero, we have(60)Aq∈FixT.
Thus,q∈S by (59) and (60). Hence, it follows from x¯=PSu that(61)limsup\limits_n⟶∞u−x¯,xn−x¯=u−x¯,q−x¯≤0.
Secondly, we show that(62)xn+1−xn⟶0,asn⟶∞.
From (32), we have(63)Uμn,νnun−un=μnun−U1−νnI+νnUun=μnun−Uun+Uun−U1−νnI+νnUun≤μnun−Uun+μnUun−U1−νnI+νnUun≤μnun−Uun+μnLun−1−νnI+νnUun=μnun−Uun+μnνnLun−Uun=μn1+νnLun−Uun.
From the above equation and (32), (54), and (58), we have(64)xn+1−xn≤θnu−xn+1−θnxn−Uμn,νnun≤θnu−xn+xn−un+un−Uμn,νnun≤θnu−xn+xn−un+μn1+νnLun−Uun≤θnu−xn+xn−un+b1+bLun−Uun.
Combining (54) and 58, we get(65)xn+1−xn⟶0,asn⟶∞.
Thirdly, we show thatxn⟶x¯ as n⟶∞.
Together with (51) and (62), we get(66)limsupn⟶∞u−x¯,xn+1−x¯≤0.
Applying Lemma2 to (46), which together with the assumption of θn and (66), we get xn⟶x¯ as n⟶∞ easily.Case 2.
Assume that there is no positive integern0 and a decreasing sequence sn for any n≥n0. That is, there is a subsequence ski of sk such that ski<ski+1 for any i∈N.
From Lemma4, we can define a nondecreasing sequence mk⊂N such that mk⟶∞ as k⟶∞ and(67)smk≤smk+1.
Firstly, we show(68)limsupn⟶∞u−x¯,xmk−x¯≤0.
It follows from (52) and (67) and the boundedness of xmk that(69)μmkνmk2−2νmk−κ−νmk2L2Uumk−umk2≤smk−smk+1+αmkL≤αmkL,0≤1−τ24I−TAxmk4A∗I−TAxmk2≤smk−smk+1+αmkL≤αmkL.
Thus,(70)umk−Uumk⟶0,asn⟶∞,I−TAxmk2A∗I−TAxmk⟶0,asn⟶∞.
Moreover,(71)1AI−TAxmk≤I−TAxmk2A·I−TAxmk≤I−TAxmk2A∗I−TAxmk.
Hence,(72)Axmk−TAxmk⟶0,due to(73)xmk−umk=ρmkA∗I−TAxmk=1−τI−TAxmk22A∗I−TAxmk⟶0,asn⟶∞.
Sincexmk⇀q, then umk⇀q. So, we have q∈S by the similar proofs in Case 1. Hence, it follows from x¯=PSu that(74)limsupn⟶∞u−x¯,xmk−x¯=u−x¯,q−x¯≤0.
Secondly, we show(75)xmk+1−xmk⟶0,ask⟶∞.
From (32), we have(76)Uμmk,νmkumk−umk=μmkumk−U1−νmkI+νmkUumk=μmkumk−Uumk+Uumk−U1−νmkI+νmkUumk≤μmkumk−Uumk+μmkUumk−U1−νmkI+νmkUumk≤μmkumk−Uumk+μmkLumk−1−νmkI+νmkUumk=μmkumk−Uumk+μmkνmkLumk−Uumk=μmk1+νmkLumk−Uumk.
By the above equation and (32), we have(77)xmk+1−xmk≤αmku−xmk+1−αmkxmk−Uμmk,νmkumk≤αmku−xmk+xmk−umk+umk−Uμmk,νmkumk≤αmku−xmk+xmk−umk+μmk1+νmkLumk−Uumk≤αmku−xmk+xmk−umk+b1+bLumk−Uumk.
Combining (54) and the (58), we get(78)xmk+1−xmk⟶0,asn⟶∞.
Thirdly, we show thatxmk⟶x¯ as n⟶∞.
Using (68) and (75), we get(79)limsupn⟶∞u−x¯,xmk+1−x¯≤0.
Based onsmk≤smk+1,∀k∈N and (46), we get(80)αmksmk+1+1−αmksmk+1−smk≤2αmku−x¯,xmk+1−x¯.
So,(81)αmksmk+1≤2αmku−x¯,xmk+1−x¯,that is,(82)smk+1≤2u−x¯,xmk+1−x¯.
Taking the limitk⟶∞, using (79), we obtain(83)smk+1⟶0,ask⟶∞.
Thus,(84)sk⟶0,ask⟶∞,due to sk≤smk+1. The proof is completed.


## 5. Numerical Example

In the section, we present a numerical experiment to demonstrate the convergence of this algorithm.AssumeH1=H2=R3,·2 and T,U:R3⟶R3 is defined by(85)Txyz=13xyz,Uabc=0ab.Let the bounded linear operatorA be defined by(86)A=5−5−7−42−4−7−45.Clearly, bothU and T are 0−demicontractive mappings.Choose the parameters as follows:(87)θn=1n,μn=1n,νn=1n,∀n≥1.ρn is chosen in the following way:(88)ρn=1−τI−TAxn22A∗I−TAxn2,Axn≠TAxn,0,otherwise,where A is a bounded and linear mapping and A∗ is its adjoint. Then, the iterative algorithm (10) becomes as follows:(89)un=xn−ρnA∗I−TAxn,xn+1=1nu+1−1n1−1nI+1nU1−1nI+1nUun,where u=1−12 is a fixed point in R3, and the initial point x1=a1b1c1=1−25 and xn=anbncn is generated by the algorithm (10). We plot the numbers of iterations and xn+1−xn2 in the following graphs (Figures 8 and 9), the numbers of iterations and xn=an,bn,cn.Figure 8
The iterative curves of algorithm (21) under different n.Figure 9
The iterative curves of algorithm (21) under different n.


## 6. Conclusion

In this paper, we proposed a new iteration algorithm (10) and we obtained the strong convergence of the sequence xn for split common fixed point problems (5). The main result is an extension of the related results announced in [15, 16, 27]. The research highlights of this paper are novel algorithms and their analysis techniques. The improvement on the extension of the operator, such as the demicontractive mappings, the directed operators, the quasi-non-expansive operators, and quasi-pseudo-contractive operators will be of interest for further research in the future.



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*Source: 6659079-2021-01-30.xml*