# Strong Convergence on the Split Feasibility Problem by MixingW-Mapping

Authors: Fugen Gao; Xiaoxiao Liu; Xiaochun Li
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we concern with the split feasibility problem (SFP) in real Hilbert space whenever the sets involved are nonempty, closed, and convex. By mixingW-mapping with the viscosity, we introduce a new iterative algorithm for solving the split feasibility problem, and we prove that our proposed algorithm is convergent strongly to a solution of the split feasibility problem.


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## Body

## 1. Introduction

Throughout this article, we assume thatH1 and H2 are two real Hilbert spaces. The split feasibility problem (SFP) was introduced by Censor and Elfving [1], and it is formulated as finding a point x in C such that Ax is in Q, namely,(1)x∈C,Ax∈Q,where C and Q are nonempty, closed, and convex subsets of real Hilbert spaces H1 and H2, respectively, and A is a bounded linear operator from H1 to H2.Many inverse problems arising from various fields of science and technology, such as intensity-modulated radiation therapy [2], signal processing, and image reconstruction, can be summarized as SFP. Due to its applications, many algorithms have been invented to solve SFP (see, for instance, [3–11]).To solve problem (1), in 2002, Byren [3] introduced a popular algorithm which is called the CQ-algorithm as follows:(2)xn+1=PCI−μnA∗I−PQAxn,where I is the identity operator on H, PC and PQ denote the metric projection onto the closed convex subsets C and Q, respectively, and A∗ is the adjoint operator of A and 0<μn<2/A2. In 2018, Wang [10] proposed his algorithm as follows:(3)xn+1=xn−τnI−PCnxn+A∗I−PQnAxn,where Cn, Qn, and τn were given by(4)Cn=x∈H1:cxn≤ξn,xn−x,ξn∈∂cxn,Qn=y∈H2:qAxn≤ζn,Axn−y,ζn∈∂qAxn,in which c:H1⟶R and q:H2⟶R are two given convex functions, and(5)τn=λnxn−PCnxn2+I−PQnAxn22I−PCnxn+A∗I−PQnAxn2,in which λn∈0,4.To obtain strong convergence theorem, Wang [10] modified his algorithm as follows:(6)yn=xn−τnI−PCnxn+A∗I−PQnAxn,xn+1=αnu+1−αnyn,where Cn, Qn, and τn were given as the same to the weak convergence theorem; αn is a sequence in 0,1 which is chosen so that(7)limn⟶∞αn=0,∑n=0∞αn=∞,either∑n=0∞αn+1−αn<∞orlimn⟶∞αn+1αn=1.On the other hand, another problem which is similar to the split feasibility problem is the proximal split feasibility problem (PSFP), and the proximal split feasibility problem is to find a pointx∗ satisfying the property:(8)x∗∈argminf,Ax∗∈argming,where f:H1⟶R∪∞ and g:H2⟶R∪∞ are two proper and lower semicontinuous convex functions, A:H1⟶H2 is a linear bounded operator, and argminf:=x¯∈H1:fx¯≤fx,∀x∈H1, argming:=y¯∈H2:gy¯≤gy,∀y∈H2.To solve problem (8), in 2014, Moudafi and Thakur [12] introduced the split proximal algorithm by the following iterative scheme:(9)xn+1=proxλμnfxn−μnA∗I−proxλgAxn,where the stepsize μn was the suitable positive real number sequence, and they also proved the weak convergence of the sequence generated by the above equation to a solution of problem (8). In 2015, Shehu et al. [13] introduced a viscosity-type algorithm for solving proximal split feasibility problems as follows:(10)yn=xn−μnA∗I−proxλgAxn,xn+1=αnψxn+1−αnproxλμnfyn,where ψ:H1⟶H1 is a contraction mapping. They also proved a strong convergence of the sequence generated by iterative scheme (10) in Hilbert spaces. Recently, Shehu and Iyiola [14] introduced the following algorithm for solving split proximal problems and fixed point problems in Hilbert spaces:(11)un=1−αnxn,yn=proxλγnfun−γnA∗I−proxλgAun,xn+1=1−βnyn+βnTyn,where T is a k-strictly pseudocontractive mapping. They also showed that, under certain assumptions imposed on the parameters, the sequence xn was generated by the algorithm that they introduced converges strongly to x∗∈FT∩argminf∩A−1argming.If we definedf=iC and g=iQ as indicated functions of sets C and Q, where C and Q are nonempty, closed, and convex sets of H1 and H2, respectively, then the proximal split feasibility problem (8) becomes the split feasibility problem (1). In this paper, inspired and motivated by these works that have been done, we focus on the split feasibility problem in Hilbert spaces.The rest of this paper is organized as follows. In Section2, we review some definitions and lemmas that we need. In Section 3, we introduce a new iterative algorithm based on the viscosity method and W-mapping which is defined in Section 2 for finding a solution of the split feasibility problem and prove a strong convergence theorem under some mild conditions.


## 2. Preliminaries

Throughout this paper, letHii=1,2 be a real Hilbert space with the inner product ⋅,⋅ and norm ⋅. We denote by I the identity operator on Hii=1,2 and by wwxn the set of all weak cluster points of xn. The notation  ⟶  stands for strong convergence and  ⇀  stands for weak convergence.Definition 1.
(see [15]). Let T:H⟶H be a nonlinear mapping. Then, T is(1)
Nonexpansive if(12)Tx−Ty≤x−y,for∀x,y∈H1.(2)
Firmly nonexpansive if(13)Tx−Ty2+I−Tx−I−Ty2≤x−y2,for∀x,y∈H1.Definition 2.
LetC be a nonempty closed convex subset of H1. Then, an orthogonal projection PC:H1⟶C is defined by(14)PCx=argminy∈Cx−y2,x∈H1.Lemma 1.
LetC be a nonempty closed convex subset of H1, then(1)
x−PCx,z−PCx≤0,∀x∈H1,z∈C.(2)
PC and I−PC both are (firmly) nonexpansive.(3)
PCx−PCy,x−y≥PCx−PCy2,∀x,y∈H1.(4)
x−z,x−PCx≥x−PCx2,∀x∈H1,z∈C.Definition 3.
LetT:H1⟶H1 be an operator with FixT≠∅. If for any xn in H1, xn⇀x and I−Txn⟶0, we can obtain x∈FixT, then we claim that I−T is demiclosed at zero.Lemma 2.
IfT:H1⟶H1 is a nonexpansive operator, then I−T is demiclosed at zero.Lemma 3.
For allx,y∈H1, and η>0, we have(1)
x+y2≤y2+2x,x+y.(2)
x+y2≤1+ηx2+1+1/ηy2.Definition 4.
If for anyn≥1 and z∈C, let Si1≤i≤n be a nonexpansive mapping on C and ηi be real numbers with 0<ηi≤η<1. We define a mapping Wn on C for each n≥1 by(15)Un,n+1=IUn,n=1−ηnI+ηnSnUn,n+1Un,n−1=1−ηn−1I+ηn−1Sn−1Un,n⋮Un,k=1−ηkI+ηkSkUn,k+1Un,k−1=1−ηk−1I+ηk−1Sk−1Un,k⋮Un,2=1−η2I+η2S2Un,3Un,1=1−η1I+η1S1Un,2Wn=Un,1,then we say that Wn is the W-mapping generated by Sn,Sn−1,…,S1 and ηn,ηn−1,…,η1.Lemma 4.
(see [16]). Let Wn:H1⟶H1 be a W-mapping, then Wn is a nonexpansive mapping.Lemma 5.
(see [17]). Assume that sk is a sequence of nonnegative real number such that(16)sk+1≤1−λksk+λkbk+ck,where λk, bk, and ck satisfy the following conditions:(1)
∑k=0∞λk=∞;(2)
eitherlimsupk⟶∞bk≤0 or ∑k=0∞λkbk<∞;(3)
ck≥0 for all k and ∑k=0∞ck<∞.
Then,limk⟶∞sk=0.


## 3. Main Results

To introduce our iterative algorithm for solving the split feasibility problem in real Hilbert spaces, firstly, we shall assume that problem (1) is consistent, namely, its solution set, denoted by S, is nonempty. Secondly, we need to define a special W-mapping Wnn≥1 as follows:(17)Un,n+1=IUn,n=1−ηnI+ηnPCUn,n+1Un,n−1=1−ηn−1I+ηn−1PCUn,n⋮Un,k=1−ηkI+ηkPCUn,k+1Un,k−1=1−ηk−1I+ηk−1PCUn,k⋮Un,2=1−η2I+η2PCUn,3Un,1=1−η1I+η1PCUn,2Wn=Un,1,where ηi=1/n−i+11≤i≤n. From Lemma 4, we know that Wn is nonexpansive.Now, we will introduce our iterative algorithms for the split feasibility problem.Algorithm 1.
Given an initial pointx0∈H1, let ψ:H1⟶H1 be a contraction mapping with δ∈0,1. Assume that xn has been constructed and compute xn+1 by the following iterative scheme:(18)yn=xn−τnxn−PCxn+A∗I−PQAxn,xn+1=αnψxn+1−αnWnyn,where τn and αn satisfy(1)
0<ɛ≤τn≤2/1+A2−ɛ;(2)
limn⟶∞αn=0,∑0∞αn=∞.Theorem 1.
Letxn be the sequence generated by Algorithm 1, then xn converges strongly to a solution x∗ of the SFP (1), where x∗=PSψx∗.Proof.
Sinceψ:H1⟶H1 is a contraction mapping with δ∈0,1 and the fact that PS is nonexpansive, it is clear that PSψ:H1⟶S is also a contraction mapping. By Banach fixed point theorem, there exists x∗∈S, such that x∗=PSψx∗.
Sincex∗∈S, that is, x∗∈C and Ax∗∈Q. By the definition of Wn, we have x∗=Wnx∗. In what follows, we will divide the proof into four steps.
Firstly, we prove that the sequencexn is bounded.
From (18) and Lemmas 1 and 3, we have(19)yn−x∗2=xn−x∗−τnxn−PCxn+A∗I−PQAxn2=xn−x∗2−2τnxn−x∗,xn−PCxn+Axn−PQAxn,Axn−Ax∗+τn2xn−PCxn+A∗I−PQAxn2≤xn−x∗2−2τnxn−PCxn2+Axn−PQAxn2+τn21+A2xn−PCxn2+1+1A2Axn−PQAxn2≤xn−x∗2−τn2−τn1+A2xn−PCxn2+Axn−PQAxn2.
By condition (1), we have τn2−τn1+A2>0, so yn−x∗≤xn−x∗. Therefore, from (18), we obtain(20)xn+1−x∗=αnψxn−x∗+1−αnWnyn−x∗≤αnψxn−x∗+1−αnWnyn−x∗≤αnψxn−ψx∗+αnψx∗−x∗+1−αnWnyn−x∗≤αnδxn−x∗+αnψx∗−x∗+1−αnyn−x∗≤αnδxn−x∗+αnψx∗−x∗+1−αnxn−x∗=1−αn1−δxn−x∗+αn1−δψx∗−x∗1−δ≤maxxn−x∗,ψx∗−x∗1−δ.
By introduction, we obtain(21)xn+1−x∗≤maxx0−x∗,ψx∗−x∗1−δ,for all n≥0. The above inequality implies that the sequence xn is bounded. Combining with (18), we know that yn, Wnyn, and ψxn are also bounded.
Secondly, we show that the following inequality holds:(22)xn+1−x∗2≤1−αn˜xn−x∗2+αn˜δn,where αn˜=αn1−δ2 and(23)δn=2αnψx∗−x∗,xn+1−x∗−tnxn−PCxn2+Axn−PQAxn2αn1−δ2,with tn=τn1−αn2−τn1+A2.
From equations (18) and (19) and Lemma 3, we have(24)xn+1−x∗2=αnψxn−ψx∗+1−αnWnyn−x∗+αnψx∗−x∗2≤αnψxn−ψx∗+1−αnWnyn−x∗2+2αnψx∗−x∗,xn+1−x∗≤αnψxn−ψx∗2+1−αnWnyn−x∗2+2αnψx∗−x∗,xn+1−x∗≤αnδ2xn−x∗2+1−αnyn−x∗2+2αnψx∗−x∗,xn+1−x∗≤αnδ2xn−x∗2+1−αnxn−x∗2−τn1−αn2−τn1+A2×xn−PCxn2+Axn−PQAxn2+2αnψx∗−x∗,xn+1−x∗=1−αn1−δ2xn−x∗2+αn1−δ22ψx∗−x∗,xn+1−x∗1−δ2−τn1−αn2−τn1+A2xn−PCxn2+Axn−PQAxn2αn1−δ2.
So, inequality (22) holds.
Thirdly, we show thatlimsupn⟶∞δn is finite.
Sincexn is bounded, we have(25)δn≤2ψx∗−x∗,xn+1−x∗1−δ2≤2ψx∗−x∗⋅xn+1−x∗1−δ2<∞.
This implies thatlimsupn⟶∞δn<∞. Next, we will show that limsupn⟶∞δn<−1 by contraction. If we assume that limsupn⟶∞δn<−1, then there exists n0∈N, such that δn≤−1 for all n≥n0. From (22), we obtain(26)xn+1−x∗2≤1−αn˜xn−x∗2+αn˜δn≤1−αn˜xn−x∗2−αn˜≤xn−x∗2−αn˜.
By introduction, we have(27)xn+1−x∗2≤xn0−x∗2−∑i=n0nαi˜.
Since∑i=0∞αi=∞, ∑i=0∞αi˜=1−δ2∑i=0∞αi=∞, then there exists N>n0, such that ∑i=n0Nαi˜>xn0−x∗2. Combining with the last inequality, we have(28)xN+1−x∗2≤xn0−x∗2−∑i=n0Nαi˜<0,which is contradicted with the fact that xN+1−x∗2 is nonnegative. Thus, limsupn⟶∞δn≥−1. So, limsupn⟶∞δn is finite.
Lastly, we show thatlimsupn⟶∞δn≤0.
Sincelimsupn⟶∞δn is finite, there exists a subsequence nk such that(29)limsupn⟶∞δn=limk⟶∞δnk=limk⟶∞2αnkψx∗−x∗,xnk+1−x∗αnk1−δ2−tnkxnk−PCxnk2+Axnk−PQAxnk2αnk1−δ2.
Sinceψx∗−x∗,xnk+1−x∗ is bounded, without loss of generality, we may assume the limit of ψx∗−x∗,xnk+1−x∗ exists. From (29), we may also assume the following limit exists:(30)limk⟶∞tnkxnk−PCxnk2+Axnk−PQAxnk2αnk1−δ2.
These conditionslimn⟶∞αn=0,0<ɛ≤τnk≤2/1+A2−ɛ and tn=τn1−αn2−τn1+A2 imply tnk/αnk1−δ2⟶∞k⟶∞. So, we obtain(31)limk⟶∞xnk−PCxnk2+Axnk−PQAxnk2=0,that is,(32)limk⟶∞xnk−PCxnk=limk⟶∞Axnk−PQAxnk2=0.
Next, we prove that any weak cluster point of the sequencexnk is a solution of the SFP (1).
Sincexnk is bounded, let x¯ be a weak cluster point of the sequence xnk; without loss of generality, we assume that xnk⇀x¯; then, we obtain Axnk⇀Ax¯. From the fact that PC and PQ are nonexpansive, Lemma 2 implies I−PC and I−PQ are demiclosed at zero; from (32), we obtain x¯=PCx¯ and Ax¯=PQAx¯, i.e., x¯∈C,Ax¯∈Q, hence x¯∈S.
Finally, we show thatxnk+1−xnk⟶0k⟶∞.
From (18) and the definition of Wn, we know(33)Wnk−ynk=1−η1ynk+η1PCUnk,2ynk−ynk≤η1ynk−PCUnk,2ynk=ynk−PCUnk,2ynknk⟶0,k⟶∞,and, from (32), we have(34)ynk−xnk=τnkxnk−PCxnk+A∗I−PQAxnk≤τnkxnk−PCxnk+A⋅Axnk−PQAxnk⟶0,k⟶∞.
So,(35)xnk+1−xnk=αnkψxnk+1−αnkWnkynk−xnk≤αnkψxnk−xnk+1−αnkWnkynk−xnk+1−αnkynk−xnk⟶0,k⟶∞.
This implies that any weak cluster point ofxnk+1 also belongs to S. Without loss of generality, we assume that xnk+1 converges weakly to x^∈S. Now, combing (29), Lemma 1, and the fact that x∗=PSψx∗, we can obtain(36)limsupn⟶∞δn≤limk⟶∞2ψx∗−x∗,xnk+1−x∗1−δ2=2ψx∗−x∗,x^−x∗1−δ2≤0.
From Lemma5, we get limn⟶∞xn−x∗=0, which ends the proof.
From Theorem1, we obtain the following subresult on the split feasibility problem (1).Algorithm 2.
Given an initial pointx0∈H1, let u∈H1 be fixed. Assume that xn has been constructed and compute xn+1 by the following iterative scheme:(37)yn=xn−τnxn−PCxn+A∗I−PQAxn,xn+1=αnu+1−αnWnyn,where τn and αn satisfy(1)
0<ɛ≤τn≤2/1+A2−ɛ;(2)
limn⟶∞αn=0,∑0∞αn=∞.Corollary 1.
Letxn be the sequence generated by Algorithm 2, then xn converges strongly to a solution x∗ of the SFP (1), where x∗=PSu.



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*Source: 9924937-2021-05-27.xml*