# Multiple-Sets Split Common Fixed-Point Problems for Demicontractive Mappings

Authors: Huanhuan Cui
Journal: Journal of Mathematics
 (2021)

---

## Abstract

In this paper, we are concerned with the multiple-sets split common fixed-point problems whenever the involved mappings are demicontractive. We first study several properties of demicontractive mappings and particularly their connection with directed mappings. By making use of these properties, we propose some new iterative methods for solving multiple-sets split common fixed-point problems, as well as multiple-sets spit feasibility problems. Under mild conditions, we establish their weak convergence of the proposed methods.


---

## Body

## 1. Introduction

The split common fixed-point problem (SCFP) requires finding an element in a fixed-point set such that its image under a linear transformation belongs to another fixed-point set. Formally, it consists in findingx∈H1 such that(1)x∈FU,Ax∈FT,where A:H1⟶H2 is a bounded linear mapping from a Hilbert space H1 into another Hilbert space H2, and FU and FT are respectively the fixed-point sets of nonlinear mappings U:H1⟶H1 and T:H2⟶H2. Specially, if U and T are both metric projections, then problem (1) is reduced to the well-known split feasibility problem (SFP) [1]. Actually, the SFP can be formulated as finding x∈H1 such that(2)x∈C,Ax∈Q,where C⊆H1 and Q⊆H2 are nonempty closed convex sets, and mapping A is as above. These two problems recently have been extensively investigated since they play an important role in various areas including signal processing and image reconstruction [2–6].We assume throughout the paper that problem (1) is consistent, which means that its solution set is nonempty. Censor and Segal [7] studied problem (1) when U and T are directed mappings. In this situation, they proposed the following method:(3)xn+1=Uxn−τnA∗I−TAxn,where A∗ is the conjugate of A, I stands for the identity mapping, and τn is a properly chosen stepsize. It is shown that if τn is chosen in 0,2/A2, then (7) converges weakly to a solution of (1). Subsequently, this result was extended to more general cases (see, e.g., [8–17]). Since the choice of the stepsize is related to A, thus to implement (7), one has to compute (or at least estimate) the norm A, which is generally not easy in practice. A way avoiding this is to adopt variable stepsize which ultimately has no relation with A [9, 10, 18]. In this connection, Wang and Cui [10] proposed the following stepsize:(4)τn=I−TAxn2A∗I−TAxn2,I−TAxn≠0;0,I−TAxn=0.On the other hand, Wang [19] proposed a new method:(5)xn+1=xn−τnI−Uxn+A∗I−TAxn,where τn⊂0,∞ is chosen such that(6)τn=I−Uxn2+I−TAxn2I−Uxn+A∗I−TAxn2.It is clear that the selection of stepsizes (8) and (6) does not rely on the norm A, which in turn improves the performance of the original algorithm. Assume that U and T are both directed such that I−T and I−U are demiclosed at 0. It is shown that the sequence xn generated by (7) and (8) or (5) and (6) converges weakly to a solution of problem (1).Now, let us consider the multiple-sets split common fixed-point problem (MSCFP) that is more general than the SCFP. Formally, it consists in findingx∈H1 such that(7)x∈∩i=1tFUi,Ax∈∩j=1sFTj,where t and s are two positive integers, A:H1⟶H2 is a bounded linear mapping from a Hilbert space H1 into another Hilbert space H2, and FUi and FTj are respectively the fixed-point sets of nonlinear mappings Ui:H1⟶H1,i=1,2,…,t and Tj:H2⟶H2,j=1,2,…,s. Specially, if these nonlinear mappings are all metric projections, problem (7) is reduced to the well-known MSFP [20]. Actually, it can be formulated as the problem of finding x∈H1 such that(8)x∈∩i=1tCi,Ax∈∩j=1sQj,where t and s are two positive integers, A:H1⟶H2 is as above, and Cii=1t⊂H1 and Qjj=1s⊂H2 are two classes of nonempty convex closed subsets.Inspired by the works mentioned above, we are aimed to introduce and analyze iterative methods for solving the MSCFP in Hilbert spaces. We first study several properties of demicontractive mappings and especially find its connection with the directed mapping. By making use of these properties, we propose a new iterative algorithm for solving the MSCFP, as well as MSFP. Under mild conditions, we obtain the weak convergence of the proposed algorithm. Our results extend the related works from the case of two-sets to the case of multiple-sets.


## 2. Preliminary

Throughout the paper, assume thatH,H1,H2 are real Hilbert spaces, and FT denotes its fixed-point set of a mapping T. The following formula plays an important role in the subsequent analysis.Lemma 1 (see [21]).
Lets,t∈ℝ and x,y∈H. It then follows that(9)tx+sy2=tt+sx2+st+sy2−tsx−y2.We next recall the definition of several important classes of nonlinear mappings.Definition 1 (see [21]).
LetT be a mapping from H into H.(i)
T is nonexpansive if(10)Tx−Ty≤x−y,∀x,y∈H.(ii)
T is firmly nonexpansive if(11)Tx−Ty2≤x−y2−I−Tx−I−Ty2,∀x,y∈H.(iii)
T is k-strictly pseudocontractive k<1 if(12)Tx−Ty2≤x−y2+kI−Tx−I−Ty2,∀x,y∈H.Definition 2 (see [21]).
LetT:H⟶H be a mapping with FT≠∅.(i)
T is quasinonexpansive if(13)Tx−y≤x−y,∀x,y∈H×FT.(ii)
T is directed if(14)Tx−y2≤x−y2−I−Tx2,∀x,y∈H×FT.(iii)
T is k-demicontractive k<1 if(15)Tx−y2≤x−y2+kI−Tx2,∀x,y∈H×FT.It is clear that a directed mapping is−1-demicontractive, while a quasinonexpansive mapping is 0-demicontractive. It is also clear that a firmly nonexpansive mapping is −1-strictly pseudocontractive, while a nonexpansive mapping is 0-strictly pseudocontractive.It is well known that a mappingT is firmly nonexpansive if and only if 2T−I is nonexpansive (cf. [21]). Analogously, we can easily get the following lemma, which presents a characteristic of directed mappings by using quasinonexpansive mappings.Lemma 2.
A mappingT is directed if and only if 2T−I is quasinonexpansive.We now study properties of demicontractive mappings.Lemma 3 (see [22]).
LetT:H⟶H be k-demicontractive k<1 with FT≠∅. Then, the following hold.(i)
Tx−z,I−Tx≥0,∀z∈FT,x∈H;(ii)
x−z,I−Tx≥I−Tx2,∀z∈FT,x∈H.Lemma 4.
For eachi=1,2,…,t, assume that Ti:H⟶H is ki-demicontractive with ki<1. Let T=1/2∑i=1tωi1+kiI+1−kiTi, where 0<ωi<1,∑i=1tωi=1. If ∩i=1tFTi is nonempty, then(16)FT=∩i=1tFTi.Proof.
We first show∩i=1tFTi⊆FT. Pick x∈∩i=1tFTi. It then follows that(17)Tx=12∑i=1tωi1+kix+1−kiTix=12∑i=1tωi1+kix+1−kix=12∑i=1tωi2x=x.Since x is chosen arbitrarily, we have ∩i=1tFTi⊆FT.
It suffices to show thatFT⊆∩i=1tFTi. Fix z∈∩i=1tFTi and choose any x∈FT. Since Tx=x and Ti is ki-demicontractive, we have(18)0=4Tx−x,x−z=2∑i=1tωi1−kiTix−x,x−z≥∑i=1tωi1−ki2Tix−x2.Thus, ∑i=1tωi1−ki2x−Tix2=0. Since ωi1−ki>0, we have x−Tix=0 for all i=1,2…t. Moreover, since x is chosen arbitrarily, we get FT⊆∩i=1tFTi. Hence, the proof is complete.Lemma 5.
For eachi=1,2…t, assume that Ti:H⟶H is ki-demicontractive with ki<1. Let T=1/2∑i=1tωi1+kiI+1−kiTi, where 0<ωi<1,∑i=1tωi=1. If ∩i=1tFTi is nonempty, then T is directed. Moreover, if for each i=1,2…t, I−Ti is demiclosed at 0, then I−T is also demiclosed at 0.Proof.
By Lemma4, we have FT=∩i=1tFTi≠∅. By Lemma 2, it suffices to show that 2T−I=∑i=1tωikiI+1−kiTi is quasinonexpansive. To this end, fix any x,z∈H×FT. By Lemma 1 and the property of demicontractions that(19)kix+1−kiTix−z2=kix−z+1−kiTix−z2=kix−z2+1−kiTix−z2−ki1−kiI−Tix2≤kix−z2+1−kix−zi2+kiI−Tix2−ki1−kiI−Tix2=x−z2,hence kix+1−kiTix−z≤x−z for all i=1,2…t. It then follows that(20)2T−Ix−z=∑i=1tωikix+1−kiTix−z≤∑i=1tωikix+1−kiTix−z≤∑i=1tωix−z=x−z.
Thus,2T−I is quasinonexpansive, which implies T is directed.
Let us now prove the second assertion. By Lemma4, we have FT=∩i=1tFTi≠∅. Let xn⊂H be such that xn⇀x and xn−Txn⟶0 as n⟶∞. Fix z∈FT. Since Ti is ki-demicontractive, we have(21)4Txn−xn,xn−z=2∑i=1tωi1−kiTixn−xn,xn−z≥∑i=1tωi1−ki2Tixn−xn2.
Sinceωi1−ki>0, we have limnxn−Tixn=0, which, by our hypothesis, implies limnx−Tix=0 for all i=1,2…t, that is, x∈∩i=1tFTi. By Lemma 4, the proof is complete.Finally, we end this section by recalling two weak convergence theorems of iterative methods for approximating a solution of the two-sets SCFP (1).Theorem 1 (see [10], Theorem 3.1).
(Assume thatU and T are both directed such that I−U and I−T are both demiclosed at 0. Then, the sequence xn, generated by (7) and (8), converges weakly to a solution of problem (1).Theorem 2 (see [19], Theorem 3.4).
Assume thatU and T are both directed such that I−U and I−T are both demiclosed at 0. Then, the sequence xn, generated by (5) and (6), converges weakly to a solution of problem (1).


## 3. The Case for Demicontractive Mappings

In this section, we are concerned with the multiple-sets split common feasibility problem and we assume that (7) is consistent, which means that its solution set is nonempty. First, motivated by (7) and (8), we propose the first algorithm for solving problem (7).Algorithm 1.
Letx0 be arbitrary and choose αii=1t⊂0,1 with ∑i=1tαi=1, βjj=1s⊂0,1 with ∑j=1sβj=1. Given xn, update the next iteration via(22)yn=xn−τn∑j=1sβj1−ljA∗I−TjAxnxn+1=12∑i=1tαi1+kiyn+1−kiUiyn,where τn=0 if ∑j=1sβj1−ljI−TjAxn=0; otherwise,(23)τn=∑j=1sβj1−ljI−TjAxn2∑j=1sβj1−ljA∗I−TjAxn2.Theorem 3.
Assume thatUi and Tj are respectively ki and lj-demicontractive such that I−Ui and I−Tj are demiclosed at 0 for i=1,2,…,t and j=1,2,…,s. Then, the sequence xn, generated by Algorithm 1, converges weakly to a solution of (7).Proof.
LetU=1/2∑i=1tαi1+kiI+1−kiUi and T=1/2∑j=1sβj1+ljI+1−ljTj. Thus, we can rewrite Algorithm 1 as(24)xn+1=Uxn−τnA∗I−TAxn,where τn=0 if I−TAxn=0; otherwise,(25)τn=I−TAxn2A∗I−TAxn2.By Lemma 5, U and T are both directed such as I−T and I−U are demiclosed at 0. It then follows from Theorem 1 that xn weakly converges to a point x that satisfies x∈FU and Ax∈FT. Moreover, by Lemma 4, we conclude that x∈∩iFUi and Ax∈∩jFTj, that is, x is a solution of problem (7).Motivated by (5) and (6), we propose the second algorithm for solving problem (7).Algorithm 2.
Letx0 be arbitrary and choose αii=1t⊂0,1 with ∑i=1tαi=1, βjj=1s⊂0,1 with ∑j=1sβj=1. Given xn, if(26)∑i=1tαi1−kiI−Uixn+∑j=1sβj1−ljA∗I−TjAxn=0,then stop; otherwise, update the next iteration via(27)xn+1=xn−τn∑i=1t1−kiI−Uixn+∑j=1sβj1−ljA∗I−TjAxn,where(28)τn=∑i=1tαiI−Ui1−kixn2+∑j=1sβj1−ljA∗I−TjAxn22∑i=1t1−kiI−Uixn+∑j=1sβj1−ljAxn2.Theorem 4.
Assume thatUi and Tj are respectively ki and lj-demicontractive such that I−Ui and I−Tj are demiclosed at 0 for i=1,2,…,t and j=1,2,…,s. Then, the sequence xn, generated by Algorithm 2, converges weakly to a solution of (7).Proof.
LetU=1/2∑i=1tαi1+kiI+1−kiUi and T=1/2∑j=1sβj1+ljI+1−ljTj. Thus, we can rewrite Algorithm 2 as xn+1=xn−τnI−Uxn+A∗I−TAxn, where(29)τn=I−Uxn2+I−TAxn2I−Uxn+A∗I−TAxn2.
By Lemma5, U and T are both directed such as I−T and I−U are demiclosed at 0. It then follows from Theorem 2 that xn weakly converges to a point x that satisfies x∈FU and Ax∈FT. Moreover, by Lemma 4, we conclude that x∈∩iFUi and Ax∈∩jFTj, that is, x is a solution of problem (7).


## 4. Multiple-Sets Split Feasibility Problem

In this section, we apply the previous result to approximate a solution of the multiple-sets split feasibility problem (MSFP). Also, we assume that problem (8) is consistent, which means that its solution set is nonempty. By applying Algorithm 1, we obtain the first algorithm for solving (8).Algorithm 3.
Letx0 be arbitrary and choose αii=1t⊂0,1 with ∑i=1tαi=1, βjj=1s⊂0,1 with ∑j=1sβj=1. Given xn, update the next iteration via(30)xn+1=∑i=1tαiPCixn−τnA∗∑j=1sβjI−PQjAxn,where τn=0 if ∑j=1sβj1−ljI−TjAxn=0; otherwise,(31)τn=∑j=1sβjI−PQjAxn2∑j=1sβjA∗I−PQjAxn2.Theorem 5.
The sequencexn, generated by Algorithm 3, converges weakly to a solution of (2).Proof.
It suffices to notice that bothPCi and PQj are −1-demicontractive, which implies ki=lj=−1 for all i=1,…,t,j=1,…,s. Applying Theorem 3 yields the desired assertion.Next, we propose the second algorithm for solving (8) by applying Algorithm 2.Algorithm 4.
Letx0 be arbitrary and choose αii=1t⊂0,1 with ∑i=1tαi=1, βjj=1s⊂0,1 with ∑j=1sβj=1. Given xn, if(32)∑i=1tαiI−PCixn+∑j=1sβjA∗I−PQjAxn=0,then stop; otherwise, update the next iteration via(33)xn+1=xn−τn∑i=1tαiI−PCixn+∑j=1sβjA∗I−PQjAxn,where(34)τn=∑i=1tαiI−PCixn2+∑j=1sβjI−PQjAxn2∑i=1tαiI−PCixn+∑j=1sβjA∗I−PQjAxn2.Theorem 6.
The sequencexn, generated by Algorithm 4, converges weakly to a solution of (8).Proof.
It suffices to notice that bothPCi and PQj are −1-demicontractive, which implies ki=lj=−1 for all i=1,…,t,j=1,…,s. Applying Theorem 4 yields the desired assertion.


## 5. Conclusion

In this paper, we consider the MSCFP whenever the involved mappings are demicontractive. We obtained several properties of demicontractive mappings and particularly their connection with directed mappings. These properties enable us to propose some new iterative methods for solving MSCFP, as well as MSFP. Under mild conditions, we establish their weak convergence of the proposed methods. Our results extend the existing works from the case of two-sets to the case of multiple-sets.



---

*Source: 3962348-2021-12-17.xml*