# Iterative Arrangements of the MSCFP for Strictly Pseudocontractive Mappings

Authors: Lin Xue; Huanhuan Cui
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we consider the multiple-set split common fixed point problem in Hilbert spaces. We first study a couple of critical properties of strictly pseudocontractive mappings and particularly the property under mix activity. By utilizing these properties, we propose new iterative strategies for settling this problem as well as several connected issues. Under delicate conditions, we state weak convergence of the proposed strategies that expands the current works from the case of two subsets to the instance of multiple subsets. As an application, we give an exhibit of the theoretical results to the multiple-set split equality problem and the elastic net regularization.


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## Body

## 1. Introduction

Lett and s be the two positive integers, and H1 and H2 stand for two Hilbert spaces. The well-known split feasibility problem (SFP) [1] is formulated as follows: find a point x∈H1 satisfying the property
(1)x∈C,Ax∈Q,where C and Q are nonempty closed convex subset of H1 and H2, respectively, and A is a bounded linear mapping from H1 into H2. There are many generalizations of the SFP, one of which is from two groups to multiple groups, that is, multiple-set split feasibility problem (MSFP) [2]. Actually, it can be formulated as the problem of finding x∈H1 such that
(2)x∈⋂i=1tCi,Ax∈⋂j=1sQj,where A:H1⟶H2 is as above and Cii=1t⊂H1 and Qjj=1s⊂H2 are two classes of nonempty convex closed subsets.The split common fixed point problem (SCFP) [3] is another generalization of the SFP, which requires to find 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 finding x∈H1 such that
(3)x∈FU,Ax∈FT,where A:H1⟶H2 is as above 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 (3) is reduced to the SFP. As a further extension of the SFP, we recall the multiple-set split common fixed point problem (MSCFP). Indeed, the MSCFP extends the SCFP from two groups to the case of multiple groups. Formally, it consists in finding x∈H1 such that
(4)x∈⋂i=1tFUi,Ax∈⋂j=1sFTj,where A:H1⟶H2 is as above 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. Recently, we [4] considered problem (4) whenever the involved mappings are demicontractive. These issues have been concentrated on broadly in different regions like image reconstruction and signal processing [5–9].There are many algorithms in the literature that can solve the SCFP problem (see, e.g., [10–16]). However, in most of these algorithms, the choice of the stepsize is related to A. Thus, to implement these algorithms, 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 [11, 12, 17]. In this connection, Wang [18] recently proposed the following method:
(5)xn+1=xn−τnI−Uxn+A∗I−TAxn,where A∗ is the conjugate of A, I stands for the identity mapping, and τn⊂0,∞ is chosen such that
(6)∑n=0∞τn=∞,∑n=0∞τn2<∞.It is shown that if mappingsU and T are firmly nonexpansive, then the sequence xn generated by (5) converges weakly to a solution of problem (3). It is clear that such a choice of the stepsize does not rely on the norm A. Kraikaew and Saejung [16] weakened condition (6) as follows:
(7)∑n=0∞τn=∞,limn⟶∞τn=0.Furthermore, we [19] extended the above results from the class of firmly nonexpansive mappings to the class of strictly pseudocontractive mappings.Inspired by the above work, we will continue to present and investigate strategies for addressing the MSCFP in Hilbert spaces. We initially explore a few properties of strictly pseudocontractive mappings and track down its soundness under arched combinatorial operation. Exploiting these properties, we propose another iterative algorithm to address the MSCFP, as well as the MSFP. Under gentle conditions, we acquire weak convergence of the proposed algorithm. Our outcomes broaden related work from the instance of two groups to the case of multiple groups.


## 2. Preliminary

Throughout the paper, assume thatH, H1, H2, and H3 are real Hilbert spaces, and FT denotes its fixed point set of a mapping T. For any α,β∈ℝ and x,y∈H, it is well known that [20]
(8)βx+αy2=ββ+αx2+αβ+αy2−βαx−y2.Recall that the mappingT:H⟶H is called nonexpansive if
(9)x−y≤x−y,∀x,y∈H.It is called firmly nonexpansive if(10)Tx−Ty2≤x−y2−I−Tx−I−Ty2,∀x,y∈H.It is calledk-strictly pseudocontractive k<1 if
(11)Tx−Ty2≤x−y2+kI−Tx−I−Ty2,∀x,y∈H.It is clear that the class of strictly pseudocontractive mappings includes the class of nonexpansive mappings, while the latter includes the class of firmly nonexpansive mappings. Indeed, a firmly nonexpansive mapping is−1-strictly pseudocontractive, while a nonexpansive mapping is 0-strictly pseudocontractive. In general, these inclusion are proper (cf. [20, 21]). The following properties of strictly pseudocontractive mappings play an import role in the subsequent analysis. It was shown [21] that if T:H⟶H is k-strictly pseudocontractive, then it follows that
(12)Tx−z,I−Tx≥0,∀z∈FT,x∈H,x−z,I−Tx≥I−Tx2,∀z∈FT,x∈H.Moreover, the fixed point set ofT is convex and closed. We now collect further properties of strictly pseudocontractive mappings.Lemma 1.
A mappingT:H⟶H is k-strictly pseudocontractive with k<1 if and only if there is a nonexpansive mapping R such that
(13)T=11−kR−k1−kI.Proof.
“⇒” Assume T is k-strictly pseudocontractive. Let R=kI+1−kT. It is easy to verify that R fulfils (13). It remains to show that R is nonexpansive. To this end, fix any x,z∈H. It then follows from (8) and the property of strictly pseudocontractive mappings that
(14)Rx−Rz2=kx+1−kTx−kz+1−kTz2=kx−z+1−kTx−Tz2=kx−z2+1−kTx−Tz2−k1−kI−Tx−I−Tz2≤kx−z2+1−kx−z2+kI−Tx−I−Tz2−k1−kI−Tx−I−Tz2=x−z2.
Hence, we haveRx−Rz≤x−z; that is, R is nonexpansive.
“⇐” Assume that there is a nonexpansive mapping R such that (13) follows. Choose any x,z∈H. It then follows from (8) and the property of nonexpansive mappings that
(15)Tx−Tz2=Rx1−k−kx1−k−Rz1−k−kz1−k2=11−kRx−Rz−k1−kx−z2=11−kRx−Rz2−k1−kx−z2+k1−k2I−Rx−I−Rz2≤11−kx−z2−k1−kx−z2+k1−k2I−Rx−I−Rz2x−z2+kI−Tx−I−Tz2.
Hence,T is strictly pseudocontractive, and thus, the proof is complete.Remark 2.
Note that a firmly nonexpansive mapping is−1-strictly pseudocontractive. It is well known that a mapping T is firmly nonexpansive if and only if there is a nonexpansive mapping R such that T=I+R/2. The following lemma can be regarded as an extension of this assertion.Lemma 3.
Assume thatTi:H⟶H is strictly pseudocontractive for each i=1,2⋯t. Let T=∑i=1twiTi, where 0<wi<1,∑i=1twi=1. If ⋂i=1tFTi is nonempty, then
(16)FT=⋂i=1tFTi.Proof.
It suffices to show thatFT⊆⋂i=1tFTi. Fix z∈⋂i=1tFTi and choose any x∈FT. By our hypothesis, there exists ki<1 such that
(17)1−ki2x−Tix2≤x−Tix,x−z,for every i=1,2⋯t. Adding up these inqualities, we have
(18)∑i=1twi1−kix−Tix2≤2∑i=1twix−Tix,x−z=2x−∑i=1twiTix,x−z=2x−Tx,x−z=0.
Thus,∑i=1twi1−kix−Tix2=0. Since wi1−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 4.
For eachi=1,2⋯t, let 0<wi<1 and ∑i=1twi=1, and Ti:H⟶H is strictly pseudocontractive with ki<1. Then, T=∑i=1twiTi is strictly pseudocontractive with
(19)k=1−1∑i=1twi1−ki−1.Proof.
By our hypothesis, for eachi=1,2⋯,t, there exists a nonexpansive mapping Ri such that Ti=1−ki−1Ri−ki1−ki−1I. Now, let us define a mapping R as
(20)R=∑i=1t1−kwi1−kiRi,where k is defined as in (19). It is readily seen that
(21)∑i=1twiTi=∑i=1twi1−kiRi−∑i=1twiki1−kiI=11−k∑i=1t1−kwi1−kiRi−k1−kI=11−kR−k1−kI.
From Lemma1, it remains to show that R is nonexpansive. To this end, choose any x,z∈H. By 1−k=∑i=1twi1−ki−1−1, we have
(22)Rx−Rz=1−k∑i=1twi1−kiRix−∑i=1twi1−kiRiz=∑i=1twi1−ki−1∑i=1twi1−ki−1Rix−Riz≤∑i=1twi1−ki−1∑i=1twi1−ki−1≤∑i=1twi1−ki−1∑i=1twi1−ki−1x−z=x−z.
Hence,R is nonexpansive, and thus, the proof is complete.


## 3. The Case for Strictly Pseudocontractive Mappings

First, let us recall a weak convergence theorem of iterative method (5) for approximating a solution of the two-set split common fixed point problem.Theorem 5 ([19], Theorem 3.1).
Letk,l∈−∞,1. Assume that U and T are, respectively, k- and l-strictly pseudocontractive mappings, and ∑n=1∞τnk¯−τn=∞,0<τn<k¯, where
(23)k¯=1−k1−l1−l+A21−k.
Then, the sequencexn, generated by (5), converges weakly to a solution of problem (3).
We next consider the MSCFP under the following basic assumption.(i)
MSCFP is consistent; that is, it admits at least one solution(ii)
Ui:H1⟶H1,i=1,2,⋯,t is ki-strictly pseudocontractive with ki<1(iii)
Tj:H2⟶H2,j=1,2,⋯,s is lj-strictly pseudocontractive with lj<1Algorithm 1.
Letx0 be arbitrary. Given xn, update the next iteration via
(24)xn+1=xn−τn∑i=1tαiI−Uixn+∑j=1sβjA∗I−TjAxn,where αii=1t⊂0,1 with ∑i=1tαi=1, βjj=1s⊂0,1 with ∑j=1sβj=1, and τn⊂0,∞ are properly chosen stepsizes.Theorem 6.
Assume that conditions (A1)-(A3) hold andτn is chosen so that
(25)∑n=1∞τnk¯−τn=∞,0<τn<k¯,where
(26)k¯=1∑i=1tαi1−ki−1+A2∑j=1sβj1−lj−1.
Then, the sequencexn, generated by Algorithm 1, converges weakly to a solution of MSCFP.Proof.
LetU=∑i=1tαiUi and T=∑j=1sβjTj. By Lemma 4, we conclude that U is k-strictly pseudocontractive with k=1−∑i=1tαi1−ki−1−1, and T is l-strictly pseudocontractive with l=1−∑j=1sβj1−lj−1−1. Hence, by formula (23), we have
(27)1−k1−l1−l+A21−k=∑i=1tαi1−ki−1−1∑i=1sβj1−lj−1−1∑j=1sβj1−lj−1−1+A2∑i=1tαi1−ki−1−1=1∑i=1tαi1−ki−1+A2∑j=1sβj1−lj−1.
Moreover, by Lemma3, FU=⋂i=1tFUi and FT=⋂j=1sFTj. Therefore, by applying Theorem 5, we at once get the assertion as desired.
It seems that the choice of the stepsize above requires the prior information ofki,lj and the norm A. However, as shown below, there is a special case in which the selection of stepsizes ultimately has no relation with ki,lj and the norm A.Corollary 7.
Assume that conditions (A1)-(A3) hold, and the stepsize is chosen so that(28)limn⟶∞τn=0,∑n=1∞τn=∞.
Then, the sequencexn generated by Algorithm 1 converges weakly to a solution of MSFP.
Significantly, if the nonlinear mappings in (4) are all metric projections, then the MSCFP is reduced to the MSFP. Consequently, we can apply our outcome to solve the MSFP. As an application of Algorithm 1, we get the following algorithm for solving problem (2).Algorithm 2.
Letx0 be arbitrary. Given xn, update the next iteration via
(29)xn+1=xn−τn∑i=1tαiI−PCixn+∑j=1sβjA∗I−PQjAxn,where αii=1t⊂0,1 with ∑i=1tαi=1, βjj=1s⊂0,1 with ∑j=1sβj=1, and τn⊂0,∞ are properly chosen stepsize.Corollary 8.
Assume that MSFP is consistent. If the stepsize is chosen so that(30)∑n=1∞τn21+A2−τn=∞,τn<21+A2,then the sequence xn, generated by Algorithm 2, converges weakly to a solution of MSFP.Proof.
LetU=∑i=1tαiPCi and T=∑j=1sβjPQj. By Lemma 4, we conclude that U and T are both −1-strictly pseudocontractive, that is, firmly nonexpansive. In this situation, we have k¯=2/1+A2. By applying Theorem 6, we at once get the assertion as desired.Corollary 9.
Assume MSFP is consistent. If the stepsize is chosen so that(31)∑n=1∞τn=∞,limn⟶∞τn=0,then the sequence xn, generated by Algorithm 2, converges weakly to a solution of MSFP.


## 4. Applications

In this part, we first give an application of our theoretical results to the multiple-set split equality problem (MSEP), which is more general than the original split equality problem [22].Example 1.
The multiple-set split equality problem (MSEP) expects to findx1,x2∈H1×H2 such that
(32)x1,x2∈⋂i=1tFUi×⋂j=1sFTj,A1x1=A2x2,where t and s are two positive integers, A1:H1⟶H3 and A2:H2⟶H3 are two bounded linear mappings, and Ui:H1⟶H1,i=1,2,⋯,t and Tj:H2⟶H2,j=1,2,⋯,s are two classes of nonlinear mappings.
We next consider the MSFP under the following basic assumption.(i)
MSEP is consistent; that is, it admits at least one solution(ii)
Ui:H1⟶H1,i=1,2,⋯,t is ki-strictly pseudocontractive with ki<1(iii)
Tj:H2⟶H2,j=1,2,⋯,s is lj-strictly pseudocontractive with lj<1
Under this situation, we propose a new method for solving problem (32).Algorithm 3.
For an arbitrary initial guessx0,y0, define xn,yn recursively by
(33)xn+1=xn−τnI−∑i=1tαiUixn+A1∗A1xn−A2yn,yn+1=yn−τnI−∑j=1sβjTjyn−A2∗A1xn−A2yn,
whereτn⊂0,∞ is a sequence of positive numbers.
To proceed the convergence analysis, we consider the product spaceH≔H1×H2, in which the inner product and the norm are, respectively, defined by
(34)x,y=x1,y1+x2,y2,x2=x12+x221/2,
wherex=x1,x2,y=y1,y2 with x1,y1∈H1,x2,y2∈H2. Define a linear mapping A:H⟶H3 by
(35)Ax=A1x1−A2x2,∀x=x1,x2.
LetT be the the metric projection onto the set 0⊆H, and define a nonlinear mapping U:H⟶H as
(36)Ux=∑i=1tαiUix1,∑j=1sβjTjx2,∀x=x1,x2,
whereαi and βj are as above.Lemma 10 ([23], Lemma 12).
Let the mappingA be defined as in (35). Then A is linear bounded. Moreover, for x=x1,x2, it follows
(37)A∗Ax=A1∗A1x1−A2x2,−A2∗A1x1−A2x2.Lemma 11.
Let the mappingU be defined as in (36). Then, FU=⋂iFUi×⋂jFTj. Moreover, if conditions (B1)-(B3) are met, then U is k-strictly pseudocontractive with
(38)κ=1−1max∑i=1tαi1−ki−1,∑j=1sβj1−lj−1.Proof.
By Lemma3, it is easy to verify the first assertion. To show the second assertion, fix any x,y∈H. By our hypothesis, ∑i=1tαiUi is k-strictly pseudocontractive with
(39)k=1−1∑i=1tαi1−ki−1,
∑j=1sβjTj is l-strictly pseudocontractive with
(40)l=1−1∑j=1sβj1−lj−1.
It then follows that(41)Ux−Uy2=∑i=1tαiUix1−∑i=1tαiUiy12+∑j=1sβjTjx2−∑j=1sβjTjy22≤x1−y12+kI−∑i=1tαiUix1−I−∑i=1tαiUiy12+x2−y22+lI−∑j=1sβjTjx1−I−∑j=1sβjTjy12≤x−y2+maxk,lI−Ux−I−Uy2.
From (38), we obtain the result as desired.Theorem 12.
Assume that conditions (B1)-(B3) hold. Ifτn is chosen so that ∑n=1∞τnκ¯−τn=∞,0<τn<κ¯, where
(42)κ¯=21−κ2+1−κA12+A22,with κ defined as in (38), then the sequence xn,yn generated by Algorithm 3 converges weakly to a solution of problem (32).Proof.
Letzn=xn,yn and let A,U,T be defined as above. Thus, problem (32) is equivalently changed into finding z∈H such that
(43)z∈FU,Az∈FT.
Moreover, Algorithm3 can be rewritten as
(44)zn+1=zn−τnI−Uzn+A∗I−TAzn.
Note that by Lemma10, U is κ-strictly pseudocontractive and T is −1-strictly pseudocontractive. Hence, by Theorem 5, we conclude that zn converges weakly to some z=x,y such that
(45)z∈FU,Az∈0.
By Lemma11, it is readily seen that x∈⋂iFUi,y∈⋂jFTj and A1x=A2y.We next give an application of our theoretical results to a problem derived from the real world. In statistics and machine learning, least absolute shrinkage and selection operator (LASSO for short) is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the statistical model it produces. It was originally introduced by Tibshirani in [24] who coined out the term and provided further insights into the observed performance.Subsequently, a number of LASSO variants have been created in order to remedy certain limitations of the original technique and to make the method more useful for particular problems. Among them, elastic net regularization adds an additional ridge regression-like penalty which improves performance when the number of predictors is larger than the sample size, allows the method to select strongly correlated variables together, and improves overall prediction accuracy. More specifically, the LASSO is a regularized regression method with theL1 penalty, while the elastic net is a regularized regression method that linearly combines the L1 and L2 penalties of the LASSO and ridge methods. Here, the L1 penalty is defined as x1=∑i=1nxi, and the L2 penalty is defined as x2=∑i=1nxi21/2.Example 2 (see [25]).
The elastic net requires to solve the problem(46)minx∈ℝn12Ax−y122+Ax−y222s.t.x1≤t1,x22≤t2,where A∈ℝm×n, y1,y2∈ℝm, and t1,t2>0 are given parameters. This problem is a specific SCFP with T1x=y1,T2x=y2,∀x∈ℝm and
(47)U1y=y,y1≤t1,y−y1−t1ηy2,y1>t1,where ηy∈∂y1 and
(48)U2y=y,y22≤t2,y−y22−t24y2,y22>t2.Algorithm 4.
Letx0 be arbitrary. Given xn, update the next iteration via
(49)xn+1=xn−τ∑i=12αiI−Uixn+A∗Axn−yi,where αii=12⊂0,1 with ∑i=1tαi=1 and τ is a properly chosen stepsize.It is clear that the above mappings are, respectively, firmly nonexpansive and firmly quasi-nonexpansive, which implies that they are, respectively,−1-strictly pseudocontractive and −1-demicontractive mappings. As an application of Theorem 6, we can deduce that the sequence xn generated by Algorithm 4 converges to a solution to problem (46) provided that the stepsize is chosen so that
(50)0<τ<21+A2.



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