# An Inertial Method for Split Common Fixed Point Problems in Hilbert Spaces

Authors: Haixia Zhang; Huanhuan Cui
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we consider the split common fixed point problem in Hilbert spaces. By using the inertial technique, we propose a new algorithm for solving the problem. Under some mild conditions, we establish two weak convergence theorems of the proposed algorithm. Moreover, the stepsize in our algorithm is independent of the norm of the given linear mapping, which can further improve the performance of the algorithm.


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## Body

## 1. Introduction

In recent years, there has been growing interest in the study of the split common fixed point problem because of its various applications in signal processing and image reconstruction [1–3]. More specifically, the problem consists in finding x¯∈H1 satisfying(1)x¯∈FU,Ax¯∈FT,where FU and FT stand for the fixed point sets of mappings U:H1⟶H1 and T:H2⟶H2, respectively, and A:H1⟶H2 is a bounded linear mapping. Here, H1 and H2 are two Hilbert spaces. In particular, if we let the mappings in (1) be the projections, then it is reduced to the well-known split feasibility problem (SFP): find x¯∈H1 such that(2)x¯∈C,Ax¯∈Q,where C⊆H1 and Q⊆H2 are two nonempty closed convex subsets and A:H1⟶H2 is a bounded linear mapping; see, e.g., [1, 4–7].There are several algorithms for solving the split common fixed point problem. Among them, Censor and Segal [8] introduced an algorithm as (3)xk+1=Uxk−τA∗I−TAxk,where I stands for the identity mapping, A∗ is the adjoint mapping of A, and the stepsize τ is a constant in 0,2A−2. In particular, when U=PC and T=PQ, then the above algorithm is reduced to the well-known CQ algorithm for solving the split feasibility problem [4]. Note that this choice of the stepsize requires the exact value or estimation of the norm A. To avoid the calculation of A, Cui and Wang [9] proposed a variable stepsize as(4)τk=I−TAxk2A∗I−TAxk2.It is readily seen that the above choice of the stepsize does not need any prior knowledge of the linear operator. Recently, Wang [10] introduced a new method for solving (1) as(5)xk+1=xk−τkI−Uxk+A∗I−TAxk,where the stepsize is set as(6)τk=I−Uxk2+I−TAxk2I−Uxk+A∗I−TAxk2.Recently, the above algorithms were further extended to the general case; see, e.g., [2, 10–17].The inertial method was first introduced in [18], and now, it has been successfully applied to solving various optimization problems arising from some applied sciences [19, 20]. In particular, this method was also applied for solving the split feasibility problem [21, 22]. By applying the inertial technique, Dang et al. [21] recently proposed the inertial relaxed CQ algorithm, which is defined as(7)wk=xk+θkxk−xk−1,xk+1=PCwk−τA∗I−PQAwk,where 0≤θk<θ<1 and 0<τ<2/A2. It is clear that the constant stepsize requires the estimation of the norm A. To avoid the estimation of the norm, Gibali et al. [23] modified the above stepsize as(8)τk=ρkI−PQAwk2ηk2,ηk=max1,A∗I−PQAwk,with 0<ρk<4. It is shown that the inertial relaxed CQ algorithm converges weakly toward a solution of the SFP provided that ∑k=1∞θkxk−xk−12<∞. The main advantage of the inertial method is that it can indeed speed up the convergence of the original algorithm. It is thus natural to extend it to the split common fixed point problem. Recently, Cui et al. [24] proposed a modified algorithm of (3) as(9)wk=xk+θkxk−xk−1,xk+1=Uwk−τkA∗I−TAwk,where 0≤θk<θ<1 and τk is defined as in (6). It was shown that algorithm (9) converges weakly to a solution of the problem provided that ∑k=1∞θkxk−xk−12<∞.In this paper, we aim to continue the study of the split common fixed point problem in Hilbert spaces. Motivated by the inertial method, we propose a new algorithm for solving the split common fixed point problem that greatly improves the performance of the original algorithm. Moreover, the stepsize in our algorithm is independent of the normA. Under some mild conditions, we establish two weak convergence theorems of the proposed algorithm.


## 2. Preliminary

In the following, we shall assume that problem (1) is consistent, that is, its solution set denoted by f is nonempty. The notation “⟶” stands for strong convergence, “⇀” weak convergence, and ωwxn the set of weak cluster points of a sequence xn. Let C be a nonempty closed convex subset. For a mapping T defined on C, we let FT=x∈C:Tx=x be its fixed point set and T′=I−T be its complement.Definition 1.
A mappingT:C⟶H is said to be nonexpansive if(10)Tx−Ty≤x−y,∀x,y∈C.
T is called quasi-nonexpansive if FT≠∅, and(11)Tx−y≤x−y,∀x∈C,y∈FT.Definition 2.
LetT:C⟶H be a mapping with FT≠∅. Then, T′ is said to be demiclosed at 0 if, for any xk in C, there holds the following implication:(12)xk⇀xT′xk⟶0⇒x∈FT.
It is well known that ifT is a nonexpansive mapping, then T′ is demiclosed at 0; see [25].Lemma 1 (see [25]).
IfT:C⟶H is quasi-nonexpansive, then(13)2x−z,T′x≥T′x2,∀z∈FT,x∈C.Lemma 2 (see [25]).
Assume thatxk is a sequence in H such that(i)
For eachz∈C, the limit of xk−z exists(ii)
Any weak cluster point ofxk belongs to C
Then,xk is weakly convergent to an element in C.Lemma 3 (see [18]).
Letϕk and δk be two nonnegative real sequences such that ∑k=0∞δk<∞ and(14)ϕk+1−ϕk≤θkϕk−ϕk−1+δk,where 0≤θk≤θ<1. Then, the sequence ϕk is convergent.Lemma 4 (see [25]).
Lets,t∈ℝ and x,y∈H. It then follows that(15)tx+sy2=tt+sx2+st+sy2−tsx−y2.


## 3. The Proposed Algorithm

Algorithm 1.
Letx0,x1 be arbitrary. Given xk,xk−1, choose θk∈0,1, and set(16)wk=xk+θkxk−xk−1.
IfU′wk+A∗T′Awk=0, then stop; otherwise, update the next iteration via(17)xk+1=wk−τkU′wk+A∗T′Awk,where(18)τk=U′wk2+T′Awk22U′wk+A∗T′Awk2.Remark 1.
In comparison, our stepsize (18) is independent of the norm A so that the calculation or estimation of A is avoided.Remark 2.
IfU′wk+A∗T′Awk=0 for some k∈ℕ, then wk is a solution of the problem. To see this, let z∈f. It then follows from Lemma 1 that U′wk2≤2U′wk,wk−z, and(19)T′Awk2≤2T′Awk,Awk−Az=A∗T′Awk,wk−z.
Combining these inequalities yields(20)U′wk2+T′Awk2≤2U′wk+A∗T′Awk,wk−z≤2U′wk+A∗T′Awkwk−z.
This yieldsU′wk=T′Awk=0, which implies wk∈f.
If we letθk≡0 in (16), then we get a new algorithm for problem (1).Algorithm 2.
Letx0 be arbitrary. Given xk, if U′xk+A∗T′Axk=0, then stop; otherwise, update the next iteration via(21)xk+1=xk−τkU′xk+A∗T′Axk,where(22)τk=U′xk2+T′Axk22U′xk+A∗T′Axk2.


## 4. Convergence Analysis

In this section, we shall establish the convergence of the proposed algorithm. By Remark2, we may assume that Algorithm 1 generates an infinite iterative sequence. To proceed, we first prove the following lemma.Lemma 5.
Letxk and wk be the sequences generated by Algorithm 1. Let δk=1/41+A2U′wk2+T′Awk2. Then, for any z∈S, it follows that(23)xk+1−z2≤wk−z2−δk.Proof.
SinceU is quasi-nonexpansive, we have(24)xk+1−z2=wk−τkU′wk+A∗T′Awk−z2=wk−z2+τk2U′wk+A∗T′Awk2−2τkU′wk,wk−z−2τkT′Awk,Awk−Az≤wk−z2+τk2U′wk+A∗T′Awk2−τkU′wk2−τkT′Awk2.
In view of (18), we have(25)xk+1−z2≤wk−z2−U′wk2+T′Awk224U′wk+A∗T′Awk2.
To finish the proof, it suffices to note that(26)U′wk2+T′Awk22U′wk+A∗T′Awk2≥U′wk2+T′Awk22U′wk+AT′Awk2≥U′wk2+T′Awk221+A2U′wk2+T′Awk2=11+A2U′wk2+T′Awk2.
This completes the proof.Theorem 1.
Assume thatU is quasi-nonexpansive such that U′ is demiclosed at 0, and T is quasi-nonexpansive such that T′ is demiclosed at 0. If, for each k∈ℕ, θk≤θ<1 such that(c1)
∑k=1∞θkxk−xk−12<∞,then the sequencexk generated by Algorithm 1 converges weakly to an element in f.Proof.
We first show that the sequencexk−z is convergent for any z∈f. From Lemma 4, we deduce(27)wk−z2=1+θkxk−z−θkxk−1−z2=1+θkxk−z2−θkxk−1−z2+θk1+θkxk−xk−12.
By Lemma5, this yields(28)xk+1−z2≤1+θkxk−z2−θkxk−1−z2+2θkxk−xk−12−δk.
Letϕk:=xk−z2. Then, the above inequality can be rewritten as(29)ϕk+1−ϕk≤θkϕk−ϕk−1+2θkxk−xk−12−δk.
By condition (c1), we then apply Lemma3 to deduce that ϕk is convergent, and so is the sequence xk−z.
We next show that each weak cluster point ofxk belongs to f. Since ϕk is convergent, this implies that ϕk−ϕk+1 converges to 0 as n⟶∞. It then follows from (29) that(30)δk≤ϕk−ϕk+1+θkϕk−ϕk−1+2θkxk−xk−12≤ϕk−ϕk+1+ϕk−ϕk−1+2θkxk−xk−12.
Note thatlimkθkxk−xk−12=0 by condition (c1). By passing to the limit in the above inequality, we have δk converging to 0 so that(31)limk⟶∞U′wk=limk⟶∞T′Awk=0.
Moreover, it is clear thatxk is bounded; thus, the set ωwxn is nonempty. Now, take any x∈ωwxk, and take a subsequence xkl such that it weakly converges to x. On the contrary, we deduce from (c1) that(32)wk−xk2=θk2xk−xk−1≤θkxk−xk−1⟶0so that wkl also weakly converges to x and Awkl weakly converges to Ax. Since U′ and T′ are both demiclosed at 0, this together with (31) indicates x∈FU and Ax∈FT; that is, x is an element in f.
Finally, by Lemma2, the sequence xk converges weakly to a solution of problem (1).Remark 3.
We now construct a sequence satisfying condition (c1). For eachk∈ℕ, let(33)θk=min0.5,1k+12xk−xk−12,xk≠xk−1,0.5,xk=xk−1.
We next study the convergence of Algorithm1 under another condition. To proceed, we need the following lemma.Lemma 6.
Letxk and wk be the sequences generated by Algorithm 1. For any z∈f, let ϕk=xk−z2−θkxk−1−z2+θk/23+θkxk−xk−12. If θk is nondecreasing, then(34)2ϕk−ϕk+1≥1−4θk+1−θk+12xk−xk+12+δk,where δk is defined as in Lemma 5.Proof.
In view of (17) and (18), we get(35)xk+1−wk2=U′wk2+T′Awk224U′wk+A∗T′Awk2.
It then follows from inequality (25) that(36)xk+1−z2≤wk−z2−12xk+1−wk2−12δk.
Moreover, it follows from (27) that(37)xk+1−z2≤1+θkxk−z2−θkxk−1−z2+2θkxk−xk−12−12xk+1−wk2−12δk.
On the contrary, we have(38)wk−xk+12=xk−xk+1+θkxk−xk−12=xk−xk+12+θk2xk−xk−12+2θkxk−xk+1,xk−xk−1≥xk−xk+12+θk2xk−xk−12−2θkxk−xk+1xk−xk−1≥xk−xk+12+θk2xk−xk−1‖2−θkxk−xk+12+xk−xk−12=1−θkxk−xk+12−θk1−θkxk−xk−12.
Substituting this into (21), we have(39)xk+1−z2≤1+θkxk−z2−θkxk−1−z2+θk1+θkxk−xk−12−121−θkxk−xk+12+θk21−θkxk−xk−12−12δk.
Sinceθk is nondecreasing, this implies(40)xk+1−z2≤1+θkxk−z2−θkxk−1−z2+θk23+θkxk−xk−12−121−θkxk−xk+12−12δk≤1+θk+1xk−z2−θkxk−1−z2+θk23+θkxk−xk−12−121−θk+1xk−xk+12−12δk.
From the definition ofϕk, we get the desired inequality.Theorem 2.
Assume thatU is quasi-nonexpansive such that U′ is demiclosed at 0, and T is quasi-nonexpansive such that T′ is demiclosed at 0. If(c2)
θk is nondecreasing and converges to θ∈0,5−2,then the sequencexk generated by Algorithm 1 converges weakly to an element in f.Proof.
We first show thatxk−z is convergent for each z∈f. It then follows from Lemma 6 and the range of θk that(41)2ϕk−ϕk+1≥1−4θ−θ2xk−xk+12+δk≥0so that ϕk is nonincreasing. From the definition of ϕk, we get(42)xk−z2≤θkxk−1−z2+ϕk≤θxk−1−z2+ϕ1.
By induction, we have(43)xk−z2≤x0−z2+ϕ11−θ.
Thus,xk is bounded. Moreover, from the definition of ϕk,(44)ϕk+1≥−θk+1xk−z2≥−xk−z2≥−x0−z2−ϕ11−θ,which implies that ϕk is bounded from below, and thus, it is convergent. Passing to the limit in (41) yields(45)limk⟶∞δk=limk⟶∞xk+1−xk=0.
On the contrary,(46)θkxk−1−z2−xk−z2=θkxk−1−z−xk−zxk−1−z+xk−z≤xk−1−xkxk−1−z+xk−z⟶0,from which it follows that(47)limk⟶∞xk−z2=11−θlimk⟶∞ϕk.
Here, we used the fact (by the definition ofϕk) that(48)xk−z2=11−θkϕk+θkxk−1−z2−xk−z2−θk3+θk2xk−xk−12.
Thus,xk−z is convergent.
We next show that the sequencexk converges weakly to a solution of problem (1). By Lemma 2, it suffices to show that each weak cluster point of xk belongs to f. Moreover, it is clear that xk is bounded; thus, the set ωwxn is nonempty. Now, take any x∈ωwxk. On the contrary, we deduce from (16) and (45) that(49)wk−xk=θkxk−xk−1≤xk−xk−1⟶0.
In a similar way, we deduce thatx∈FU and Ax∈FT; that is, x is an element in f. Hence, the proof is complete.
If we letθk≡0, then it satisfies (c1) and (c2). As a result, we get the following conclusion.Corollary 1.
Assume thatU is quasi-nonexpansive such that U′ is demiclosed at 0, and T is quasi-nonexpansive such that T′ is demiclosed at 0. Then, the sequence xk generated by Algorithm 2 converges weakly to an element in f.


## 5. Concluding Remarks

The main contribution of this paper is to propose a new algorithm for solving the split common fixed point problem in Hilbert spaces. There are two advantages of the proposed algorithm. Compared with the original algorithm for solving the problem, our proposed algorithm is faster in convergence rate. Furthermore, the stepsize in the proposed algorithm is independent of the norm of the given linear mapping, which can further improve its performance.



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*Source: 6661682-2021-01-04.xml*