# An Inertial Iterative Algorithm with Strong Convergence for Solving Modified Split Feasibility Problem in Banach Spaces

Authors: Huijuan Jia; Shufen Liu; Yazheng Dang
Journal: Journal of Mathematics
 (2021)

---

## Abstract

In this paper, we propose an iterative scheme for a special split feasibility problem with the maximal monotone operator and fixed-point problem in Banach spaces. The algorithm implements Halpern’s iteration with an inertial technique for the problem. Under some mild assumption of the monotonicity of the related mapping, we establish the strong convergence of the sequence generated by the algorithm which does not require the spectral radius ofATA. Finally, the numerical example is presented to demonstrate the efficiency of the algorithm.


---

## Body

## 1. Introduction

The split feasibility problem (shortlySFP) introduced by Censor and Elfving [1] in 1994 can be defined as follows: find a point x∗ satisfying(1)x∗∈C,Ax∗∈Q,where C and Q are nonempty closed-convex subsets of real Hilbert spaces H1 and H2, respectively, and A:H1⟶H2 is a bounded linear operator. The SFP has broad application in modelling real-world problems such as the inverse problem in signal processing, radiotherapy, and data compression ( for example, see [2–4]). Various algorithms have been invented by several authors for solving the SFP and related optimization problems (for example, see [5–10]). The CQ algorithm is one of the most popular solvers for SFP which was first proposed by Byrne [11], taking an initial point arbitrarily and defining the iterative step as(2)xk+1=PCxk−μATI−PQAxk,∀k≥1,where μ∈0,2/ρATA, ρATA is the spectral radius of ATA, and PC and PQ denote the metric projections of H1 onto C and H2 onto Q, respectively, that is, PCx=argminy∈Cx−y over all x∈C. It was proved that the sequence xn generated by (2) converges weakly to a solution of the SFP provided the step size μ∈0,2/ρATA. As an extension of this CQ algorithm, several iterative algorithms have been invented for solving SFP in Hilbert spaces and Banach spaces ( for example, see [12–17]).LetH be a real Hilbert space, F be a strictly convex, reflexive smooth Banach space, JF denotes the duality mapping on F, and C and Q be nonempty closed-convex subsets of H and F, respectively. The following Halpern’s type iteration algorithm was proposed by Alsulami and Takahashi [13] in 2015. Let tk be a sequence in H such that tk⟶t∈H and x1,t1∈H:(3)vk=λktk+1−λkPCxk−rA∗JFI−PQAxk,xk+1=αkxk+1−αkvk,k≥1,where 0<r<∞ and αk⊂0,1. It was proved that the sequence xk defined by (3) converges strongly to a point ω0∈C∩A−1Q; for some ω0=PC∩A−1Qt1,∀a,b∈ℝ if 0<rA2<2, limk⟶∞λk=0, ∑k=1∞λk=∞, and 0<a≤αk≤b<1.However, the previous algorithms only use the current point to get the next iteration which can lead slow convergence. Inertial technique, as an accelerated method, was first proposed by Polyak [18] to speed up the convergence rate of smooth convex minimization. Subsequently, F. Alvarez in [19] combined with a proximal method to solve the problem of finding the zero of a maximal monotone operator. The main idea of this method is to make use of two previous iterates in order to update the next iterate. Due to the fact that the presence of the inertial term in an algorithm speeds up the convergence rate, inertial type algorithms have been widely studied by authors [5, 20, 21].In this paper, we study the following modified SFP in the real Banach space:Findx∈FT∩C such that(4)Ax∈B−10,where C is a nonempty, closed-convex subset of E1, T:C⟶C is a Bregman weak relatively nonexpansive mapping, and B:E2⟶2E2∗ is a maximal monotone operator. E1 and E2 are p-uniformly convex and uniformly smooth real Banach spaces, E1∗ and E2∗ be the duals of E1 and E2, respectively, A:E1⟶E2 be a bounded linear operator, and A∗:E2∗⟶E1∗ be the adjoint of A. We shall denote the value of the functional x∗∈E∗ at x∈E by x∗,x. Obviously, the modified SFP (4) is more general than (1).Motivated by the above results, in this paper, we present an inertial algorithm for solving (4) in p-uniformly convex and uniformly smooth Banach spaces which have strong convergence. Our algorithm is designed to employ previous iterations xk and xk−1 to obtain the next iterative point; all the implementation process does not compute the spectral radius of ATA, which improves the feasibility of the algorithm.The paper is organized as follows. Section2 reviews some preliminaries. Section 3 gives the inertial iterative algorithm and its convergence analysis. Section 4 gives a numerical experiment. Some conclusions are drawn in Section 5.


## 2. Preliminaries

In this section, we recall some basic definitions and preliminaries’ results which will be useful for our convergence analysis in this paper. We denote the strong and weak convergence of the sequencexn to a point x by xk⟶x and xk⇀x, respectively.LetE be a real Banach space and 1<q≤2≤p<∞ and 1/p+1/q=1. Define the modulus of smoothness of E as(5)ρEτ=supx−y+x+y2−1:x=1,y=τ,=supx−τy+x+τy2−1:x=1=y,where E is uniformly smooth if and only if limτ⟶0+ρEτ/τ=0 and E is said to be q-uniformly smooth if there exists a constant Dq>0 such that ρEτ≤Dqτq.Define the modulus of convexity ofE as(6)δEɛ=inf1−x+y2:x=y=1;ɛ=x−y,where E is uniformly convex if and only if δEɛ>0 for all ɛ∈0,2 and E is p-uniformly convex if there is a constant Cp>0 such that δEɛ≥Cpɛp for all ɛ∈0,2. Every uniformly convex Banach space is strictly convex and reﬂexive. It is known that if E is p-uniformly convex and uniformly smooth, then its dual E∗ is q-uniformly smooth and uniformly convex.Definition 1 (see [22]).
Letp>1. Define the generalized duality mapping JEp:E⟶2E∗ as(7)JEp=x∗∈E∗:x∗,x=xp,x∗=xp−1.
It is known that whenE is uniformly smooth, then JEp is norm-to-norm uniformly continuous on bounded subsets of E, and E is smooth if and only if JEp is single valued. JEp is said to be weak-to-weak continuous if(8)xk⇀x⇒JEPxk,y⟶JEpx,y,for anyy∈E.Lemma 1 (see [23]).
Letxandy∈E. If E is a q-uniformly smooth Banach space, then there exists a Dq>0 such that(9)x−yq≤xq−qJEqx,y+Dqyq.Definition 2 (see [24]).
A functionf:E⟶ℝ∪+∞ is said to be(i)
proper if its effective domainf=x∈E:fx<+∞ is nonempty(ii)
convex iffλx+1−λy≤λfx+1−λfy,∀λ∈0,1andxandy∈Df(iii)
lower semicontinuous atx0∈Dfiffx0≤limx⟶x0inffxDefinition 3.
Letf:E⟶R be a differentiable and convex function. The Bregman distance denoted as Δf:domf×domf⟶0,+∞ is defined as(10)Δfx,y=fy−fx−f′x,y−x,x,y∈E,where f′x is the value of the gradient f at x.
It is worthy to note that the duality mappingJEP is actually the gradient of the function fpx=1/pxp,for2≤p<∞. Hence, if f = fp in (12), the Bregman distance with respect to fp now becomes(11)Δpx,y=1pyp−1pxp−JEpx,y−x,=1pyp−xp+JEpx,x−y,=1qxp−yp−JEpx−JEpy,x.
It is generally known that the Bregman distance is not a metric as a result of absence of symmetry, but it possesses some distance-like properties which are stated as follows:(12)Δpx,y=Δpx,z+Δpz,y+JEpx−JEpz,z−y,Δpx,y+Δpy,x=JEpx−JEpy,x−y.
The relationship between the metric and Bregman distance in thep-uniformly convex space is as follows:(13)τx−yp≤Δpx,y≤JEpx−JEpy,x−y,where τ >0 is a fixed number.
LetC be a nonempty closed-convex subset of E. The Bregman projection is defined as(14)ΠCx=argminy∈CΔfpy,x,x∈E.
And, the metric projection can be defined similarly as(15)PCx=argminy∈Cy−x,x∈E.
The Bregman projection is the unique minimizer of the Bregman distance and can be characterized by a variational inequality [25]:(16)JEpx−JEpΠCx,z−ΠCx≤0,∀z∈C,from which we have(17)JEpx−JEpΠCx,z−ΠCx≤0,∀z∈C.
The metric projection which is also the unique minimizer of the norm distance can be characterized by the following variational inequality:(18)JEpx−PCx,z−PCx≤0,∀z∈C.
We define the functionalVp:E×E⟶0,∞associated withfpx=1/pxp by(19)Vpx,x¯=1pxp−x¯,x+1qx¯q,x∈E,x¯∈E∗,where Vpx,x¯≥0. It then follows that(20)Vpx,x¯=Δpx,JE∗qx¯,∀x∈E,x¯∈E∗.
Chuasuk et al. [26] proved the following inequality:(21)Vpx,x¯+y¯,JE∗qx¯−x≤Vpx,x¯+y¯,∀x∈E,x¯,y¯∈E∗.
Furthermore,Vp is convex in the second variable, and thus, for all z∈E,xii=1N,tii=1N⊂0,1, and ∑i=1Nti=1, we have (see [23])(22)Δpz,JE∗q∑i=1NtiJEpxi=Vpz,∑i=1NtiJEpxi≤∑i=1NtiΔpz,xi.
Let C be a nonempty, closed, and convex subset of a smooth Banach spaceE, and let T:C⟶C. A point x∗∈C is called an asymptotic fixed point of T if a sequence xnn∈N exists in C and converges weakly to x∗ such that limn⟶∞xk−Txk=0. We denote the set of all asymptotic fixed points of T by F^T. Moreover, a point x∗∈C is said to be a strong asymptotic fixed point of T if there exists a sequence xkk∈N in C which converges strongly to x∗ such that limk⟶∞xk−Txk=0. We denote the set of all strong asymptotic fixed points of T by F˜T. It follows from the definitions that FT⊂F˜T⊂F^T.Definition 4 (see [27]).
LetT be a mapping such that T:C⟶E. T is said to be(i)
nonexpansive ifTx−Ty≤x−y for each xandy∈C(ii)
quasi-nonexpansive ifTx−y∗≤x−y∗ such that FT≠∅,∀x∈C, and y∗∈FTDefinition 5 (see [28]).
LetT:C⟶E be a mapping. T is said to be(1)
Bregman nonexpansive if(23)ΔpTx,Ty≤Δpx,y,∀xandy∈C.(2)
Bregman quasi-nonexpansive ifFT≠∅ and(24)Δpy∗,Tx≤Δpy∗,x,∀x∈Candy∗∈FT.(3)
Bregman weak relatively nonexpansive ifF˜T≠∅,F˜T=FT, and(25)Δpy∗,Tx≤Δpy∗,x,∀x∈Candy∗∈FT.(4)
Bregman relatively nonexpansive ifFT≠∅,F^T=FT, and(26)Δpy∗,Tx≤Δpy∗,x,∀x∈Candy∗∈FT.
From the definitions, it is evident that the class of Bregman quasi-nonexpansive maps contains the class of Bregman weak relatively nonexpansive maps. The class of Bregman weak relatively nonexpansive maps contains the class of Bregman relatively nonexpansive maps.
LetE be a smooth, strictly convex, and reﬂexive Banach space and A:E⟶2E∗ be a maximal monotone operator. We define a mapping QrA:E⟶DA by (see [25])(27)QrAx=I+rJEp−1A−1x,for allx∈Eandr>0.
This mapping is known as metric resolvent ofA. Obviously, for all r>0, we have(28)0∈JEpQrAx−x+rAQrAx,and FQrA=A−10. Furthermore, for all xandy∈E and by the monotonicity of A, we can show that(29)JEpx−QrAx−JEpy−QrAy,QrAx−QrAy≥0.
From (29), we have for all xandy∈E,(30)JEpx−QrAxr∈AQrAx,(31)JEpy−QrAyr∈AQrAy.
SinceA is monotone, we can obtain (30) from (31) and (32). This implies that for all x∈E,t∈A−10, and whenever A−10≠∅, we have(32)JEPx−QrAx,QrAx−t≥0.Lemma 2 (see [29]).
Let C be a nonempty, closed, and convex subset of a reﬂexive, strictly convex, and smooth Banach spaceE, x0∈C and x∈E. Then, the following assertions are equivalent:(1)
x0=ΠCx(2)
JEpx0−JEpx,z−x0≥0,∀z∈C
Furthermore, for∀y∈C, we have(33)ΔpΠCx,y+Δpx,ΠCx≤Δpx,y.Lemma 3 (see [30]).
LetE be a smooth and uniformly convex real Banach space. Let xn and yn be two bounded sequences in E. Then, limn⟶∞Δpxn,yn=0 if and only if(34)limn⟶∞xn−yn=0.Lemma 4 (see [30]).
Letq≥1 andr>0 be two fixed real numbers, then a Banach space E is uniformly convex if and only if there exists a continuous, strictly, increasing, and convex function g:ℝ+⟶ℝ+,g0=0 such that for all xandy∈Br and0≤α≤1,(35)αx+1−αyq≤αxq+1−αyq−Wqαgx−y,where Wq≔αq1−α+α1−αqandBr≔x∈E:x≤r.


## 3. Inertial Iteration Algorithm and Its Strong Convergence

In this section, we present our inertial iterative algorithm for solving the modified SFP (4) in Banach spaces. We also prove its strong convergence under some suitable conditions.


### 3.1. Inertial Iteration Algorithm

Now, we give our inertial iterative algorithm.Algorithm 3.1.
SupposeΓ=FT∩C∩A−1B−10=∅. Let αk⊂0,1, 0<liminfk⟶∞αk≤limsupk⟶∞αk<1, x0andx1∈C=H1, and θk⊂0,1 be a real sequence, and rk>0. Assuming xk−1andxk have been constructed, we calculate the next iterate xk+1 via the following formulas:(36a)wk=JE1∗qJE1pxk+θkJE1pxk−JE1pxk−1,(36b)vk=ΠCJE1∗JE1pwk−γkμkA∗JE2pI−QrkBAwk,(36c)uk=JE1∗qαkJE1pvk+1−αkJE1pTvk,(36d)Ck=u∈E1:JE1pwk−JE1puk,u≤1qwkp−ukp,(36e)Hk=u∈E1:xk−u,JE1px1−JE1pxk≥0,(36f)xk+1=ΠCk∩Hkx1,∀k∈N,where(37)μk=qI−QrkBAwkpDqA∗JE2pI−QrkBAwkq1/q−1,γkq−1∈0,1.
We can see that during the iteration, it does not require to compute the spectral radius of ATA.


### 3.2. Convergence

SupposeΓ=FT∩C∩A−1B−10≠∅. Now, we prove the following lemmas which will be used to establish the strong convergence.Lemma 5.
xk generated by (36a)–(36f) is well-defined.Proof.
Obviously,Ck∩Hk is a nonempty closed and convex set for ∀k≥1. Now, we show that Γ⊂Ck∩Hk. Let x∗∈Γ. By (22) and Bregman weak relatively nonexpansive of T, we get(38)Δpx∗,uk=Δpx∗,JE1∗qαkJE1pvk+1−αkJE1pTvk,≤αkΔpx∗,vk+1−αkΔpx∗,Tvk,≤αkΔpx∗,vk+1−αnΔpx∗,vk,=Δpx∗,vk.
From (36b), Lemma 1, and the definition of Bregman projection, we have(39)Δpx∗,vk=Δpx∗,ΠCJE1∗qJE1pwk−γkμkA∗JE2pI−QrkBAwk,≤Δpx∗,JE1∗qJE1pwk−γkμkA∗JE2pI−QrkBAwk,=x∗pp−JE1pwk,x∗+γkμkA∗JE2pI−QrkBAwk,x∗+1qJE1pwk−γkμnA∗JE2pI−QrkBAwkq,≤x∗pp−JE1pwk,x∗+γkμkJE2pI−QrkBAwk,Ax∗+1qJE1pwkq−γkμkJE2pI−QrkBAwk,Awk+DqμkqaA∗JE2pI−QrkBAwkq,=x∗pp−JE1pwk,x∗+1qJE1pwkq+γkμkJE2pI−QrkBAwk,Ax∗−Awk+DqγkqμkqqA∗JE2pI−QrkBAwkq,=Δpx∗,wk+μkγkJE2pI−QrkBAwk,Ax∗−QrkBAwk+QrkBAwk−Awk+DqγkqμkqqA∗JE2pI−QrkBAwkq,=Δpx∗,wk+γkμkJE2pI−QrkBAwk,Ax∗−QrkBAwk−γkμkJE2pI−QrkBAwk,Awk−QrkBAwk+Dqγkqμkq−αA∗JE2pI−QrkBAwkq,where the second inequality is from Lemma 2.
Furthermore, from (32), we have(40)Δpx∗,vk≤Δpx∗,wk−γkμkJE2pI−QrkBAwk,I−QrkBAwk+DqγkqμkqqA∗JE2pI−QrkBAwkq.
By the definitions ofμk, we have(41)Δpx∗,vk≤Δpx∗,wk−γkμkI−QrkBAwkp−Dqγkq−1μkq−1qA∗JE2pI−QrkBAwkq,=Δpx∗,wk−γkμk1−γkq−1I−QrkBAwkp.
Sinceγkq−1∈0,1, we have(42)Δpx∗,vk≤Δpx∗,wk.
Then,(43)Δpx∗,uk≤Δpx∗,wk,which implies JE1pwk−JE1puk,u≤1/qwkp−ukp. So Γ⊂Ck, for all k∈ℕ. Since xk+1=ΠCk∩Hkx1, then JE1px1−JE1pxk+1,v−xk+1≤0,∀v∈Ck∩Hk⊂C. Ofcourse, for x∗∈Γ, we also have JE1px1−JE1pnxk+1,x∗−xk+1≤0. It implies that Γ⊂Hk for all k∈ℕ. So, we obtain that Γ⊂Ck∩Hk for all k∈ℕ. Therefore, Ck∩Hk is nonempty, and thus, xk+1=ΠCk∩Hkx1 is well-defined.Lemma 6.
Letxk be a sequence generated by Algorithm 3.1. Then,(1)
limk⟶∞xk+1−xk=0(2)
limk⟶∞xk−wk=0(3)
limk⟶∞Tvk−vk=0(4)
limk⟶∞xk−vk=0(5)
limk⟶∞I−QrkBAwk=0Proof.
(1)
Letz∈Γ. Since Γ⊂Ck∩Hk, ∀k≥1, and xk+1=ΠCk∩Hkx1, we have(44)Δpxk+1,x1≤Δpz,x1,∀k≥1.Thus,Δpxk+1,x1 is bounded.We observe thatxk+1∈Hk, and by (17), we have(45)JE1pxk−JE1px1,xk−xk+1≤0.Also, by (18), we have(46)Δpxk+1,xk≤Δpxk+1,x1−Δpxk,x1,∀k≥1,that is,Δpxk,x1≤Δpxk+1,x1−Δpxk+1,xk.Thus,Δpxk,x1≤Δpxk+1,x1; therefore, Δpxk,x1 is a bounded monotone nondecreasing sequence. Hence, limk⟶∞Δpxk,x1 exists. From (46), we have limn⟶∞Δpxk+1,xk=0. Thus, using Lemma 2.8, we get(47)limk⟶∞xk+1−xk=0.(2)
By the uniform continuity ofJE1p on bounded subsets of E1, from (47) we have(48)limk⟶∞JE1pxk+1−JE1pxk=limk⟶∞JE1pxk−JE1pxk−1=0.From (36a), we obtain(49)wk=JE1∗qJE1pxk+θkJE1pxk−JE1pxk−1.Then,(50)JE1pwk=JE1pxk+θkJE1pxk−JE1pxk−1,which gives(51)JE1pwk−JE1pxk=θkJE1pxk−JE1pxk−1.Therefore,(52)limk⟶∞JE1pwk−JE1pxk=0.SinceJE1∗p is also uniformly continuous on bounded subsets of E1∗, we have(53)limk⟶∞xk−wk=0.(3)
From (47) and (53), we obtain(54)xk+1−wk=xk+1−xk+xk−wk,≤xk+1−xk+xk−wk⟶0ask⟶∞.Note that, from the construction ofCk, we have that(55)Δpxk+1,uk≤Δpxk+1,wk⟶0ask⟶∞.Thus,Δpxk+1,uk⟶0ask⟶∞; also from Lemma 3, we have(56)limk⟶∞xk+1−uk=0.Similarly,xk−uk≤xk−xk+1+xk+1−uk; it then follows that(57)limn⟶∞xk−uk=0.It follows from (53) and (57) that(58)limk⟶∞wk−uk=0.Using Lemma4, from (36c) we have(59)Δpx∗,uk=Δpx∗,JE1∗qαkJE1pvk+1−αkJE1pTvk,=Vpx∗,αkJE1pvk+1−αkJE1pTvk,=1px∗p−αkJE1pvk,x∗−1−αkJE1pTvk,x∗+1qαkJE1pvk+1−αkJE1pTvkq,≤1px∗p−αkJE1pvk,x∗−1−αkJE1pTvk,x∗+1qαkvkp+1−αkqTvkp−WqαkqgJE1pvk−JE1pTvk,=αk1px∗p+1−αk1px∗p−αkJE1pvk,x∗−1−αkJE1pTvk,x∗+1qαkvkp+1−αkqTvkp−WqαkqgJE1pvk−JE1pTvk,=αk1px∗p−JE1pvk,x∗+1qvkp+1−αk1px∗p−JE1pTvk,x∗+1qTvkp−WqαkqgJE1pvk−JE1pTvk,=αkΔpx∗,vk+1−αkΔpx∗,Tvk−WqαkqgJE1pvk−JE1pTvk.SinceT is Bregman weak relatively nonexpansive, we have(60)Δpx∗,uk≤αkΔpx∗,vk+1−αkΔpx∗,vk−WqαkqgJE1pvk−JE1pTvk,=Δpx∗,vk−WqαkqgJE1pvk−JE1pTvk,≤Δpx∗,wk−WqαkqgJE1pvk−JE1pTvk.Hence, from (12) and (13), we get(61)WqαkqgJE1pvk−JE1pTvk≤Δpx∗,wk−Δpx∗,uk,=Δpuk,wk+JE1px∗−JE1puk,uk−wk,≤JE1puk−JE1pwk,uk−wk+JE1px∗−JE1puk,uk−wk,=JE1px∗−JE1pwk,uk−wk.From (58), we have(62)limk⟶∞WqαkqgJE1pvk−JE1pTvk=0,which implies(63)limk⟶∞gJE1pvk−JE1pTvk=0.By the property of mappingg, we obtain(64)limk⟶∞JE1pvk−JE1pTvk=0.SinceJE1∗p is uniformly continuous on bounded subsets of E1∗, we have(65)limk⟶∞Tvk−vk=0.(4)
From Algorithm3.1, we have that(66)JE1Puk−JE1Pvk=1−αkJE1PTvk−JE1Pvk.Since0<liminfk⟶∞αk≤limsupk⟶∞αk<1 and from (65), we have(67)limk⟶∞JE1Puk−JE1Pvk=0.Hence,(68)limk⟶∞uk−vk=0.It is easy to obtain(69)limk⟶∞wk−vk=0,limk⟶∞vk−xk=0.(5)
From (41), we have(70)γkμk1−γkq−1I−QrkBAwkp≤Δpx∗,wk−Δpx∗,vk.
From (13) and (16), we get(71)γkμk1−γkq−1I−QrkBAwkp≤Δpvk,wk+JE1px∗−JE1pvk,vk−wk,≤JE1pvk−JE1pwk,vk−wk+JE1px∗−JE1pvk,vk−wn,=JE1px∗−JE1pwk,vk−wk.
It follows from (69) that(72)limk⟶∞I−QrkBAwkp=0.
Then,(73)limk⟶∞I−QrkBAwk=0.
Now, we present the following strong convergence theorem for Algorithm3.1.Theorem 1.
SupposeΓ=FT∩C∩A−1B−10≠∅. Then, the sequence xn generated by Algorithm 3.1 converges strongly to u∈Γ, where u=ΠΓx1.Proof.
We have known in Lemma6 (1) that limk⟶∞Δpxk,x1 exists. Now, we show that xk⟶x¯∈Γ. Let mandk∈ℕ, then from Lemma 2, we have(74)Δpxm,xk=Δpxm,ΠCk−1∩Hk−1x1≤Δpxm,x1−Δpxk,x1⟶0.
Therefore, by Lemma3, we get that xm−xk⟶0 as mandk⟶∞. Thus, xk is a Cauchy sequence in C. Since C is closed and convex, it implies that there exists x¯∈C such that xk⟶x¯ask⟶∞. Since xk−vk⟶0, Tvk−vk⟶0, and T is a Bregman weak relatively nonexpansive mapping, then x¯∈FT. More so, since xk−wk⟶0, then wk⟶x¯, and by the linearity of A, we have Awk⟶Ax¯. Also from (73), QrkBAwk⟶Ax¯. Since QrkB is a resolvent metric of B for rk>0, then for all k∈ℕ, we have(75)JE2PAwk−QrkBAwkrk∈BQrkBAwk.
So for alls,s∗∈B, we have(76)0≤s−QrkBAwk,s∗−JE2PAwk−QrkBAwkrk.
It follows from (73) that for all s,s∗∈B, we have(77)0≤s∗−0,s−Ax¯.
SinceB is maximal monotone, then it implies that Ax¯∈B−10; hence, x¯∈A−1B−10. Therefore, x¯∈Γ.
Finally, we show thatx¯=ΠΓx1. Suppose there exists y¯∈Γ such that y¯=ΠΓx1. Then,(78)Δpy¯,x1≤Δpx¯,x1.
We have shown in Lemma5 that Γ⊂Ckand∀k≥1, then Δpxk,x1≤Δpx¯,x1. By the lower semicontinuity of the norm, we have(79)Δpx¯,x1=x¯pq−JE1px¯,x1+x1pp,≤liminfk⟶∞x¯pq−JE1pxk,x1+x1pp,=liminfk⟶∞Δpx¯,x1,≤limsupk⟶∞Δpx¯,x1≤Δpy¯,x1.
Combining (78) and (79), we have y¯=x¯. This completes the proof.


## 3.1. Inertial Iteration Algorithm

Now, we give our inertial iterative algorithm.Algorithm 3.1.
SupposeΓ=FT∩C∩A−1B−10=∅. Let αk⊂0,1, 0<liminfk⟶∞αk≤limsupk⟶∞αk<1, x0andx1∈C=H1, and θk⊂0,1 be a real sequence, and rk>0. Assuming xk−1andxk have been constructed, we calculate the next iterate xk+1 via the following formulas:(36a)wk=JE1∗qJE1pxk+θkJE1pxk−JE1pxk−1,(36b)vk=ΠCJE1∗JE1pwk−γkμkA∗JE2pI−QrkBAwk,(36c)uk=JE1∗qαkJE1pvk+1−αkJE1pTvk,(36d)Ck=u∈E1:JE1pwk−JE1puk,u≤1qwkp−ukp,(36e)Hk=u∈E1:xk−u,JE1px1−JE1pxk≥0,(36f)xk+1=ΠCk∩Hkx1,∀k∈N,where(37)μk=qI−QrkBAwkpDqA∗JE2pI−QrkBAwkq1/q−1,γkq−1∈0,1.
We can see that during the iteration, it does not require to compute the spectral radius of ATA.


## 3.2. Convergence

SupposeΓ=FT∩C∩A−1B−10≠∅. Now, we prove the following lemmas which will be used to establish the strong convergence.Lemma 5.
xk generated by (36a)–(36f) is well-defined.Proof.
Obviously,Ck∩Hk is a nonempty closed and convex set for ∀k≥1. Now, we show that Γ⊂Ck∩Hk. Let x∗∈Γ. By (22) and Bregman weak relatively nonexpansive of T, we get(38)Δpx∗,uk=Δpx∗,JE1∗qαkJE1pvk+1−αkJE1pTvk,≤αkΔpx∗,vk+1−αkΔpx∗,Tvk,≤αkΔpx∗,vk+1−αnΔpx∗,vk,=Δpx∗,vk.
From (36b), Lemma 1, and the definition of Bregman projection, we have(39)Δpx∗,vk=Δpx∗,ΠCJE1∗qJE1pwk−γkμkA∗JE2pI−QrkBAwk,≤Δpx∗,JE1∗qJE1pwk−γkμkA∗JE2pI−QrkBAwk,=x∗pp−JE1pwk,x∗+γkμkA∗JE2pI−QrkBAwk,x∗+1qJE1pwk−γkμnA∗JE2pI−QrkBAwkq,≤x∗pp−JE1pwk,x∗+γkμkJE2pI−QrkBAwk,Ax∗+1qJE1pwkq−γkμkJE2pI−QrkBAwk,Awk+DqμkqaA∗JE2pI−QrkBAwkq,=x∗pp−JE1pwk,x∗+1qJE1pwkq+γkμkJE2pI−QrkBAwk,Ax∗−Awk+DqγkqμkqqA∗JE2pI−QrkBAwkq,=Δpx∗,wk+μkγkJE2pI−QrkBAwk,Ax∗−QrkBAwk+QrkBAwk−Awk+DqγkqμkqqA∗JE2pI−QrkBAwkq,=Δpx∗,wk+γkμkJE2pI−QrkBAwk,Ax∗−QrkBAwk−γkμkJE2pI−QrkBAwk,Awk−QrkBAwk+Dqγkqμkq−αA∗JE2pI−QrkBAwkq,where the second inequality is from Lemma 2.
Furthermore, from (32), we have(40)Δpx∗,vk≤Δpx∗,wk−γkμkJE2pI−QrkBAwk,I−QrkBAwk+DqγkqμkqqA∗JE2pI−QrkBAwkq.
By the definitions ofμk, we have(41)Δpx∗,vk≤Δpx∗,wk−γkμkI−QrkBAwkp−Dqγkq−1μkq−1qA∗JE2pI−QrkBAwkq,=Δpx∗,wk−γkμk1−γkq−1I−QrkBAwkp.
Sinceγkq−1∈0,1, we have(42)Δpx∗,vk≤Δpx∗,wk.
Then,(43)Δpx∗,uk≤Δpx∗,wk,which implies JE1pwk−JE1puk,u≤1/qwkp−ukp. So Γ⊂Ck, for all k∈ℕ. Since xk+1=ΠCk∩Hkx1, then JE1px1−JE1pxk+1,v−xk+1≤0,∀v∈Ck∩Hk⊂C. Ofcourse, for x∗∈Γ, we also have JE1px1−JE1pnxk+1,x∗−xk+1≤0. It implies that Γ⊂Hk for all k∈ℕ. So, we obtain that Γ⊂Ck∩Hk for all k∈ℕ. Therefore, Ck∩Hk is nonempty, and thus, xk+1=ΠCk∩Hkx1 is well-defined.Lemma 6.
Letxk be a sequence generated by Algorithm 3.1. Then,(1)
limk⟶∞xk+1−xk=0(2)
limk⟶∞xk−wk=0(3)
limk⟶∞Tvk−vk=0(4)
limk⟶∞xk−vk=0(5)
limk⟶∞I−QrkBAwk=0Proof.
(1)
Letz∈Γ. Since Γ⊂Ck∩Hk, ∀k≥1, and xk+1=ΠCk∩Hkx1, we have(44)Δpxk+1,x1≤Δpz,x1,∀k≥1.Thus,Δpxk+1,x1 is bounded.We observe thatxk+1∈Hk, and by (17), we have(45)JE1pxk−JE1px1,xk−xk+1≤0.Also, by (18), we have(46)Δpxk+1,xk≤Δpxk+1,x1−Δpxk,x1,∀k≥1,that is,Δpxk,x1≤Δpxk+1,x1−Δpxk+1,xk.Thus,Δpxk,x1≤Δpxk+1,x1; therefore, Δpxk,x1 is a bounded monotone nondecreasing sequence. Hence, limk⟶∞Δpxk,x1 exists. From (46), we have limn⟶∞Δpxk+1,xk=0. Thus, using Lemma 2.8, we get(47)limk⟶∞xk+1−xk=0.(2)
By the uniform continuity ofJE1p on bounded subsets of E1, from (47) we have(48)limk⟶∞JE1pxk+1−JE1pxk=limk⟶∞JE1pxk−JE1pxk−1=0.From (36a), we obtain(49)wk=JE1∗qJE1pxk+θkJE1pxk−JE1pxk−1.Then,(50)JE1pwk=JE1pxk+θkJE1pxk−JE1pxk−1,which gives(51)JE1pwk−JE1pxk=θkJE1pxk−JE1pxk−1.Therefore,(52)limk⟶∞JE1pwk−JE1pxk=0.SinceJE1∗p is also uniformly continuous on bounded subsets of E1∗, we have(53)limk⟶∞xk−wk=0.(3)
From (47) and (53), we obtain(54)xk+1−wk=xk+1−xk+xk−wk,≤xk+1−xk+xk−wk⟶0ask⟶∞.Note that, from the construction ofCk, we have that(55)Δpxk+1,uk≤Δpxk+1,wk⟶0ask⟶∞.Thus,Δpxk+1,uk⟶0ask⟶∞; also from Lemma 3, we have(56)limk⟶∞xk+1−uk=0.Similarly,xk−uk≤xk−xk+1+xk+1−uk; it then follows that(57)limn⟶∞xk−uk=0.It follows from (53) and (57) that(58)limk⟶∞wk−uk=0.Using Lemma4, from (36c) we have(59)Δpx∗,uk=Δpx∗,JE1∗qαkJE1pvk+1−αkJE1pTvk,=Vpx∗,αkJE1pvk+1−αkJE1pTvk,=1px∗p−αkJE1pvk,x∗−1−αkJE1pTvk,x∗+1qαkJE1pvk+1−αkJE1pTvkq,≤1px∗p−αkJE1pvk,x∗−1−αkJE1pTvk,x∗+1qαkvkp+1−αkqTvkp−WqαkqgJE1pvk−JE1pTvk,=αk1px∗p+1−αk1px∗p−αkJE1pvk,x∗−1−αkJE1pTvk,x∗+1qαkvkp+1−αkqTvkp−WqαkqgJE1pvk−JE1pTvk,=αk1px∗p−JE1pvk,x∗+1qvkp+1−αk1px∗p−JE1pTvk,x∗+1qTvkp−WqαkqgJE1pvk−JE1pTvk,=αkΔpx∗,vk+1−αkΔpx∗,Tvk−WqαkqgJE1pvk−JE1pTvk.SinceT is Bregman weak relatively nonexpansive, we have(60)Δpx∗,uk≤αkΔpx∗,vk+1−αkΔpx∗,vk−WqαkqgJE1pvk−JE1pTvk,=Δpx∗,vk−WqαkqgJE1pvk−JE1pTvk,≤Δpx∗,wk−WqαkqgJE1pvk−JE1pTvk.Hence, from (12) and (13), we get(61)WqαkqgJE1pvk−JE1pTvk≤Δpx∗,wk−Δpx∗,uk,=Δpuk,wk+JE1px∗−JE1puk,uk−wk,≤JE1puk−JE1pwk,uk−wk+JE1px∗−JE1puk,uk−wk,=JE1px∗−JE1pwk,uk−wk.From (58), we have(62)limk⟶∞WqαkqgJE1pvk−JE1pTvk=0,which implies(63)limk⟶∞gJE1pvk−JE1pTvk=0.By the property of mappingg, we obtain(64)limk⟶∞JE1pvk−JE1pTvk=0.SinceJE1∗p is uniformly continuous on bounded subsets of E1∗, we have(65)limk⟶∞Tvk−vk=0.(4)
From Algorithm3.1, we have that(66)JE1Puk−JE1Pvk=1−αkJE1PTvk−JE1Pvk.Since0<liminfk⟶∞αk≤limsupk⟶∞αk<1 and from (65), we have(67)limk⟶∞JE1Puk−JE1Pvk=0.Hence,(68)limk⟶∞uk−vk=0.It is easy to obtain(69)limk⟶∞wk−vk=0,limk⟶∞vk−xk=0.(5)
From (41), we have(70)γkμk1−γkq−1I−QrkBAwkp≤Δpx∗,wk−Δpx∗,vk.
From (13) and (16), we get(71)γkμk1−γkq−1I−QrkBAwkp≤Δpvk,wk+JE1px∗−JE1pvk,vk−wk,≤JE1pvk−JE1pwk,vk−wk+JE1px∗−JE1pvk,vk−wn,=JE1px∗−JE1pwk,vk−wk.
It follows from (69) that(72)limk⟶∞I−QrkBAwkp=0.
Then,(73)limk⟶∞I−QrkBAwk=0.
Now, we present the following strong convergence theorem for Algorithm3.1.Theorem 1.
SupposeΓ=FT∩C∩A−1B−10≠∅. Then, the sequence xn generated by Algorithm 3.1 converges strongly to u∈Γ, where u=ΠΓx1.Proof.
We have known in Lemma6 (1) that limk⟶∞Δpxk,x1 exists. Now, we show that xk⟶x¯∈Γ. Let mandk∈ℕ, then from Lemma 2, we have(74)Δpxm,xk=Δpxm,ΠCk−1∩Hk−1x1≤Δpxm,x1−Δpxk,x1⟶0.
Therefore, by Lemma3, we get that xm−xk⟶0 as mandk⟶∞. Thus, xk is a Cauchy sequence in C. Since C is closed and convex, it implies that there exists x¯∈C such that xk⟶x¯ask⟶∞. Since xk−vk⟶0, Tvk−vk⟶0, and T is a Bregman weak relatively nonexpansive mapping, then x¯∈FT. More so, since xk−wk⟶0, then wk⟶x¯, and by the linearity of A, we have Awk⟶Ax¯. Also from (73), QrkBAwk⟶Ax¯. Since QrkB is a resolvent metric of B for rk>0, then for all k∈ℕ, we have(75)JE2PAwk−QrkBAwkrk∈BQrkBAwk.
So for alls,s∗∈B, we have(76)0≤s−QrkBAwk,s∗−JE2PAwk−QrkBAwkrk.
It follows from (73) that for all s,s∗∈B, we have(77)0≤s∗−0,s−Ax¯.
SinceB is maximal monotone, then it implies that Ax¯∈B−10; hence, x¯∈A−1B−10. Therefore, x¯∈Γ.
Finally, we show thatx¯=ΠΓx1. Suppose there exists y¯∈Γ such that y¯=ΠΓx1. Then,(78)Δpy¯,x1≤Δpx¯,x1.
We have shown in Lemma5 that Γ⊂Ckand∀k≥1, then Δpxk,x1≤Δpx¯,x1. By the lower semicontinuity of the norm, we have(79)Δpx¯,x1=x¯pq−JE1px¯,x1+x1pp,≤liminfk⟶∞x¯pq−JE1pxk,x1+x1pp,=liminfk⟶∞Δpx¯,x1,≤limsupk⟶∞Δpx¯,x1≤Δpy¯,x1.
Combining (78) and (79), we have y¯=x¯. This completes the proof.


## 4. Numerical Example

In this section, we present one numerical example to compare the performance of Algorithm3.1 with Algorithm (3).Example 1.
LetE1=E2=Rn and A be a n × n randomly generated matrix. Let C=x∈Rn:a,x≥1, where a=−5,−5,0,0,…,0∈Rn. Then,(80)ΠCx=PCx=1−a,xa22a+x.
LetB:Rn⟶2Rn be defined by Bx=2x,andT=PC.We takeθk=3/7k,rk=1/2k,andαk=k/3k+1. We select x0=−1,−1,−1,−1,…,−1∈Rnandx1=−2,−2,−2,−2,…,−2∈Rn as initial two points and select r=1=γk in the two algorithms. The iterative process of Algorithm 3.1 for the example is as follows:(81)wk=xk+37kxk−xk−1,vk=PCwk−μkA∗I−QrkBAwk,uk=k3k+1vk+2k+13k+1PCvk,Ck=u∈E1:wk−uk,u≤wk2−uk2,Hk=u∈E1:xk−u,x1−xk≥0,xk+1=PCk∩Hkx1,∀k∈N,where μk is chosen as defined by (37) and QrkBAwk=k/k+1AwkQ for all k ≥1. We choose various values of n as follows:
CaseI:n=10,CaseII:n=20,CaseIII:n=40,and CaseIV:n=50, and use xk+1−xk22/x2−x122<10−6 as the stopping criterion. We, thus, plot the graph of xk+1−xk22 against the number of iteration in each case and compare the computation results of our algorithm with Algorithm (3) of Alsulami and Takahashi [13]. The computation results can be seen in Figure 1 and Table 1.
From Table1, we can find that Algorithm 3.1 performs better in terms of number of iterations and CPU time-taken for computation than Algorithm (3). From Figure 1, we can see that the error generated by Algorithm 3.1 in the previous iterative steps is ascending; then, it goes down quickly in the latter iterative steps and converges to zero, which is just the effect of the inertial technique, while the error generated by Algorithm (3) always decreases and converges slowly to zero. The results manifest that inertial technique is an effective method for improving the convergence.Figure 1
Example cases. (a) Top-Left Case In = 10; (b) Top-Right Case II = 20; (c) Bottom-Left Case III n = 40; (d) Bottom-Right Case IV n = 50.
(a)(b)(c)(d)Table 1
Computation result for the example.
CaseCPU time (sec.)Iter.Algo. (1.3)Algo. 3.1Algo. (1.3)Algo. 3.1I:n = 100.19600.15724756II:n = 200.20300.163031594III:n = 400.28400.2280422156IV:n = 500.32600.2490472183


## 5. Conclusion

In this paper, we introduced an inertial iterative algorithm for approximating a common solution of the split feasibility problem, monotone inclusion problem, and fixed-point problem for the class of Bregman weak relative nonexpansive mapping inp-uniformly convex and uniformly smooth Banach spaces. Our algorithm is designed in such a way that its implementation does not require to compute the spectral radius of ATA. We also proved a strong convergence theorem under some suitable conditions. Finally, a numerical example is given to test the accuracy and efficiency of our algorithm. The results in this paper improve and extend many related results in the literature.



---

*Source: 9974351-2021-05-11.xml*