# Inertial Iterative Schemes for D-Accretive Mappings in Banach Spaces and Curvature Systems

Authors: Li Wei; Wenwen Yue; Yingzi Shang; Ravi P. Agarwal
Journal: Journal of Mathematics
 (2021)

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## Abstract

We propose and analyze a new iterative scheme with inertial items to approximate a common zero point of two countable d-accretive mappings in the framework of a real uniformly smooth and uniformly convex Banach space. We prove some strong convergence theorems by employing some new techniques compared to the previous corresponding studies. We give some numerical examples to illustrate the effectiveness of the main iterative scheme and present an example of curvature systems to emphasize the importance of the study of d-accretive mappings.


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## Body

## 1. Introduction and Preliminaries

In this paper, we assume thatE is a real Banach space and E∗ is the dual space of E. Suppose that C is a nonempty closed and convex subset of E. The symbols “x,f”, “→” and “⇀” denote the value of f∈E∗ at x∈E, the strong convergence, and the weak convergence either in E or E∗, respectively.The normalized duality mappingJE:E→2E∗ is defined as follows:(1)JEx=x∗∈E∗:x,x∗=x2=x∗2,∀x∈E.Lemma 1.
(see [1]). Assume that E is real uniformly convex and uniformly smooth Banach space. Then (1) JE is single-valued and surjective and, for x∈E and k∈0,+∞, JEkx=kJEx; (2) JE−1=JE∗ is the normalized duality mapping from E∗ to E; (3) both JE and JE−1 are uniformly continuous on each bounded subset of E or E∗, respectively.Definition 1.
(see [2]). The Lyapunov functional ϕ:E×E→R+ is defined as follows: (2)ϕx,y=x2−2x,jEy+y2,∀x,y∈E,jEy∈JEy.
Similarly, the Lyapunov functional defined onE∗×E∗ can be defined and denoted by ϕ¯.Lemma 2 (see [3]).
LetE be a uniformly smooth and uniformly convex Banach space, and let xn and yn be two sequences in E. If either xn or yn is bounded and ϕxn,yn→0, as n→∞, then xn−yn→0, as n⟶∞.Definition 2.
(see [4]). Let Cn be a sequence of nonempty closed and convex subsets of E; then(1)
s−liminfCn, which is called strong lower limit of Cn, is defined as the set of all x∈E such that there exists xn∈Cn for almost all n and it tends to x as n→∞ in the norm(2)
w−limsupCn, which is called weak upper limit of Cn, is defined as the set of all x∈E such that there exists a subsequence Cnm of Cn and xnm∈Cnm for every nm and it tends to x as nm→∞ in the weak topology(3)
Ifs−liminfCn=w−limsupCn, then the common value is denoted by limCnLemma 3.
(see [4]). Let Cn be a decreasing sequence of closed and convex subsets of E, that is, Cn⊂Cm, if n≥m. Then, Cn converges in E and limCn=∩n=1∞Cn.Definition 3.
(see [1, 2]). Suppose that E is a real uniformly smooth and uniformly convex Banach space and C is a nonempty closed and convex subset of E; then, for each x∈E, there exists a unique element v∈C such that x−v=infx−y:y∈C. Such an element v is denoted by PCx and PC is called the metric projection of E onto C.Lemma 4.
(see [5]). Suppose that E is a real uniformly smooth and uniformly convex Banach space and Cn is a sequence of nonempty closed and convex subsets of E. If limCn exists and is not empty, then limn⟶∞PCnx=PlimCnx, for ∀x∈E.Definition 4.
(1)
A mappingT:E⟶E is said to be accretive [6] if Tu1−Tu2,JEu1−u2≥0, ∀ui∈E, i=1,2(2)
A mappingT:E⟶E is said to be d-accretive [7] if Tu1−Tu2,JEu1−JEu2≥0, ∀ui∈E, i=1,2It is easy to see that accretive mappings and d-accretive mappings are identical in a Hilbert space, while they are different in a non-Hilbert space.For a nonlinear mappingA:DA⊂E⟶E, we use FixA=x∈DA:Ax=x and A−10=x∈DA:Ax=0 to denote the fixed point set and zero point set of A, respectively.Lemma 5.
(see [8, 9]). Suppose that E is a real uniformly smooth and uniformly convex Banach space. Let A:E⟶E be d-accretive mapping such that RI+A=E. Under the assumption that A−10≠∅, one has the following: (1)
∀x∈E∗, ∀z∈A−10, and ∀r>0,(3)ϕ¯JEz,JE∗+rAJE∗−1JE∗x+ϕ¯JE∗+rAJE∗−1JE∗x,x≤ϕ¯JEz,x.(2)
Ifxn∈E∗, x∈E∗, xn⟶x, and JE∗+rAJE∗−1JE∗xn⟶x, as n⟶∞, then x=JE∗+rAJE∗−1JE∗x.Definition 5.
(see [10]). Let C be a nonempty closed subset of E and let Q be a mapping of E onto C. Then Q is said to be sunny if QQx+tx−Qx=Qx, for all x∈E and t≥0. A mapping Q:E⟶C is said to be a retraction if Qz=z for every z∈C. If E is a smooth and strictly convex Banach space, then a sunny generalized nonexpansive retraction of E onto C is uniquely decided, which is denoted by RC.Definition 6.
(see [3]). If E is a real uniformly smooth and uniformly convex Banach space and C is a nonempty closed and convex subset of E, then, for each x∈E, there exists a unique element x0∈C satisfying ϕx0,x=infϕz,x:z∈C. In this case, ∀x∈E, define πC:E⟶C by πCx=x0, and πC is called the generalized projection from E onto C.Lemma 6 (see [11]).
Suppose thatE is a real uniformly convex Banach space and r∈0,+∞. Then there exists a continuous and strictly increasing function g:0,2r⟶0,+∞ with g0=0 satisfying(4)αx+1−αy2≤αx2+1−αy2−α1−αgx−y,for ∀α∈0,1, ∀x,y∈E with x≤r and y≤r.Accretive mappings have been extensively studied until now and some works can be seen in [12–16] and the references therein. However, until 2000, some valuable research work has been done on d-accretive mappings. As we know, in 2000, Alber and Reich [17] presented the following iterative schemes for d-accretive mapping T in a real uniformly smooth and uniformly convex Banach space: (5)xn+1=xn−αnTxn,(6)xn+1=xn−αnTxnTxn,n≥0.They proved that the iterative sequencesxn generated by (5) and (6) converge weakly to the zero point of T under the assumption that T is uniformly bounded and demicontinuous.In 2006, Guan [18] presented the following projective method for the d−accretive mapping T in a real uniformly smooth and uniformly convex Banach space E: (7)x1∈DT,yn=I+rnT−1xn,Cn=v∈DT:ϕv,yn≤ϕv,xn,Qn=v∈DT:xn−v,JEx1−JExn≥0,xn+1=πCn∩Qnx1,n≥1.It was shown that the iterative sequencesxn generated by (7) converge strongly to the zero point of T under the assumptions that (1) RI+T=E, (2) T is demicontinuous, and (3) JE is weakly sequentially continuous and satisfies(8)ϕp,I+rnT−1x≤ϕp,x,for ∀x∈E and p∈T−10. The restrictions are extremely strong, and it is hard for us to find such a d−accretive mapping that is demicontinuous and satisfies (8).In 2014, Wei et al. [7] presented the following block iterative schemes for approximating common zero points of d−accretive mappings Tii=1m in a Banach space E: (9)x1∈E,yn=∑i=1mωn,iαn,ixn+1−αn,iI+rn,iTi−1xn,xn+1=∑i=1mηn,iβn,ixn+1−βn,iI+sn,iTi−1yn,n∈N,(10)x1∈E,un=∑i=1mωn,iαn,ixn+1−αn,iI+rn,iTi−1xn,vn+1=∑i=1mηn,iβn,ixn+1−βn,iI+sn,iTi−1un,H1=E,Hn+1=z∈Hn:ϕun,z≤ϕxn,z,xn+1=RHn+1x1,n∈N.Under mild assumptions,xn generated by (9) is proved to be weakly convergent to an element in ∩i=1mTi−10, while (10) is proved to be strongly convergent to an element in ∩i=1mTi−10.In [19], the study on finite d-accretive mappings is extended to that for infinite d-accretive mappings Tii=1∞⊂E×E:(11)u1=v∈E∗,wn,i=I+sn,iJETiJE∗−1un,U1=E∗,Un+1,i=z∈E∗:JE∗un−wn,i,wn,i−z≥0,Un+1=∩i=1∞Un+1,i∩Un,Vn+1=z∈Un+1:v−z2≤PUn+1v−v2+τn+1,un+1∈Vn+1,un¯=JE∗un,n∈N.Then, sequenceun¯ generated by (11) is proved to be strongly convergent to an element in ⋂i=1mTi−10.A new idea can be seen in (11), where the iterative element un∈Vn can be chosen arbitrarily, which is different from the traditional one, for example, (7) in [18]. However, it is found that, for each iterative step n in (11), countable sets Un+1,i should be evaluated for i∈N. To simplify it theoretically, the following iterative scheme is designed in [9]: (12)u1,e1∈E∗,wn=JEαnJE∗un+1−αn∑i=1∞bn,iJE∗JE∗+sn,iTiJE∗−1JE∗un+en,U1=E∗=V1,Un+1=z∈Un:JE∗wn−αnJE∗un−1−αnJE∗un+en,z≥wn2−αnun2−1−αnun+en22,Vn+1=z∈Un+1:u1−z2≤PUn+1u1−u12+τn+1,un+1∈Vn+1,un¯=JE∗un,n∈N.Then,un¯ generated by (12) is proved to be strongly convergent to an element in ∩i=1mTi−10.In 2020, Wei et al. [8] extended the discussion on countable d-accretive mappings Tii=1∞ to that for two groups of countable d-accretive mappings Tii=1∞ and Sii=1∞ and construct two key groups of sets Vn and Yn, where the iterative elements yn and un can be chosen arbitrarily in Vn and Yn, respectively.(13)u1∈E∗,e1∈E∗,ε1∈E∗,vn=JEαnJE∗un+1−αn∑i=1∞an,iJE∗JE∗+rn,iTiJE∗−1JE∗un+en,U1=E∗=V1,Un+1=p∈Xn:JE∗vn−αnJE∗un−1−αnJE∗un+en,p≥vn2−αnun2−1−αnun+en22,Vn+1=p∈Un+1:u1−p2≤PUn+1u1−u12+λn+1,yn∈Vn+1,zn=JEβnJE∗un+1−βn∑i=1∞bn,iJE∗JE∗+sn,iSiJE∗−1JE∗yn+εn,Xn+1=p∈Un+1:JE∗zn−βnJE∗un−1−βnJE∗yn+εn,p≥zn2−βnvn2−1−βnyn+εn22,Yn+1=p∈Xn+1:u1−p2≤PXn+1u1−u12+δn+1,un+1∈Yn+1,un¯=JE∗un,n∈N.Thenun¯ generated by (13) is proved to be strongly convergent to an element in ∩i=1∞Ti−10∩∩i=1∞Si−10.Recall that the inertial-type algorithm was first proposed by Polyak [20] as an acceleration process in solving a smooth convex minimization problem. The inertial-type algorithm involves a two-step iterative method where the next iterate is defined by making use of the previous two iterates. For example, in 2015, Lorenz and Pock [21] proposed the following inertial forward-backward algorithm for approximating zero points of T+S, where T and S are accretive-type mappings in Hilbert space H: (14)u0,u1∈Hchosen arbitrarily,vn=un+θnun−un−1,un+1=I+rnT−1vn−rnSvn,n∈N.In (14), the term θnun−un−1 is called the inertial term.In this paper, motivated by the previous work, some new work is done in the construction of new iterative schemes: (i) the inertial term is inserted for the purpose of possible acceleration; (ii) the combination expressions ofTi or Si are different from those in (11)–(13). Numerical experiments are conducted, and it is very interesting that the rate of convergence is so quick that only eight steps are enough for some special cases and for different choices of iterative elements. To emphasize the importance of the topic, a kind of curvature systems is studied and is taken as an example of d-accretive mappings.


## 2. Iterative Schemes and Strong Convergence Theorems

### 2.1. Iterative Schemes

In this section, we suppose that the following conditions are satisfied:(A1) E is a real uniformly convex and uniformly smooth Banach space; JE:E⟶E∗ and JE∗:E∗⟶E are the normalized duality mappings.(A2) Ti,Si:E⟶E are d-accretive mappings such that RI+Ti=E=RI+Si, for each i∈N.(A3) rn,i, tn,i, ξn and τn are real number sequences in 0,+∞, for i,n∈N. λn and ϑn are real number sequences in −∞,+∞. αnn=1∞ and βnn=1∞ are real number sequences in 0,1.(A4) a0,b0∈0,1, an,ii=0∞ and bn,ii=0∞ are real number sequences in 0,1 such that a0+∑i=1∞an,i=b0+∑i=1∞bn,i=1.(A5) εn1 and εn2 are the error sequences in E∗.(A6) Un˜=JE∗−1a0JE∗+∑i=1∞an,iJE∗JE∗+rn,iTiJE∗−1JE∗; Un˜˜=JE∗−1b0JE∗+∑i=1∞bn,iJE∗JE∗+tn,iSiJE∗−1JE∗JE∗+tn,i−1Si−1JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗.We construct the following iterative scheme:(15)x0,x1,ε11,ε12∈E∗,un=xn+λnxn−xn−1,zn=xn+ϑnxn−xn−1,vn=JE∗−11−αnJE∗un+αnJE∗Un˜zn+εn1,V1=W1=E∗,Vn+1=p∈Vn:ϕ¯p,vn≤1−αnϕ¯p,un+αnϕ¯p,zn+εn1,Wn+1=p∈Vn+1:x1−p2≤PVn+1x1−x12+τn+1,yn∈Wn+1chosen arbitrarily,wn=JE∗−11−βnJE∗un+βnJE∗Un˜˜yn+εn2,U1=X1=E∗,Un+1=p∈Vn+1:ϕ¯p,wn≤1−βnϕ¯p,un+βnϕ¯p,yn+εn2,Xn+1=p∈Un+1:x1−p2≤PUn+1x1−x12+ξn+1,xn+1∈Xn+1chosen arbitrarily,n∈N.


### 2.2. Strong Convergence Theorems

Theorem 1.
Consider∩i=1∞Ti−10∩∩i=1∞Si−10≠∅, infnrn,i>0, infntn,i>0 for i∈N, αn↛0, βn↛0, τn⟶0, λn↛±∞, ϑn↛±∞, ξn⟶0, τn⟶0, εn1⟶0, and εn2⟶0, as n⟶∞. Then, the iterative sequence xn¯=JE∗xn⟶JE∗P∩n=1∞Unx1=JE∗P∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10∈∩i=1∞Ti−10∩∩i=1∞Si−10, as n⟶∞.Proof.
The proof is split into eight steps.Step 1. ∩n=1∞Un≠∅.Since∩i=1∞Ti−10∩∩i=1∞Si−10≠∅, there exists δ0∈E such that Tiδ0=Siδ0=0, ∀i∈N. It follows from Lemma 1 that there exists η0∈E∗ such that JE∗η0=δ0. Therefore, ∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10≠∅.Next, we shall use inductive method to prove that∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10⊂Un, n∈N.∀p∈∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10. For n=1, it is obvious that p∈U1. Suppose that the result is true for n=k. Then, if n=k+1, it follows from the definition of the Lyapunov functional, the convexity of ⋅2, and Lemma 5 that (16)ϕ¯p,vk=p2−21−αkp,JE∗uk−2αkp,JE∗Uk˜zk+εk1+1−αkJE∗uk+αkJE∗Uk˜zk+εk12≤p2−21−αkp,JE∗uk−2αka0p,JE∗zk+εk1−2αkp,∑i=1∞ak,iJE∗JE∗+rk,iTiJE∗−1JE∗zk+εk1+1−αkuk2+αka0zk+εk12+αk∑i=1∞ak,iJE∗+rk,iTiJE∗−1JE∗zk+εk12=1−αkϕ¯p,uk+αka0ϕ¯p,zk+εk1+αk∑i=1∞ak,iϕ¯p,JE∗+rk,iTiJE∗−1JE∗zk+εk1≤1−αkϕ¯p,uk+αkϕ¯p,zk+εk1.Thus,p∈Vk+1. Using Lemma 5 repeatedly, similar to the above discussion, one has (17)ϕ¯p,wk≤1−βkϕ¯p,uk+βkϕ¯p,Uk˜˜yk+εk2≤1−βkϕ¯p,uk+βkb0ϕ¯p,yk+εk2+βk∑i=1∞bk,iϕ¯p,yk+εk2=1−βkϕ¯p,uk+βkϕ¯p,yk+εk2.Thenp∈Uk+1, which implies that ∩n=1∞Un≠∅.Step 2. Both Vn and Un are closed and convex subsets of E∗, for n∈N.Ifn=1, the result is obvious. If n≥2, since(18)ϕ¯p,vn≤1−αnϕ¯p,un+αnϕ¯p,zn+εn1⇔p,1−αnJE∗un+αnJE∗zn+εn1−JE∗vn≤αnzn+εn12+1−αnun2−vn22.Vn is a closed and convex subset of E∗, for n∈N.Since(19)ϕ¯p,wn≤1−βnϕ¯p,un+βnϕ¯p,yn+εn2⇔p,1−βnJE∗un+βnJE∗yn+εn2−JE∗wn≤βnyn+εn22+1−βnun2−wn22.Un is a closed and convex subset of E∗, for n∈N.Step 3.PUnx1⟶P∩n=1∞Unx1, as n⟶∞.The result follows from the results of Steps 1 and 2 and Lemmas3 and 4.Step 4.Wn≠∅ and Xn≠∅.SincePVn+1x1−x1=infq∈Vn+1q−x1; then, for τn+1, there exists θn+1∈Vn+1 such that(20)x1−θn+12≤infq∈Vn+1q−x12+τn+1=PVn+1x1−x12+τn+1.ThenWn≠∅. Similarly, Xn≠∅. This ensures that xn is well defined.Step 5. xn⟶P∩n=1∞Unx1, as n⟶∞.Sincexn+1∈Xn+1, x1−xn+12≤PUn+1x1−x12+ξn+1. It follows from Step 3 and ξn⟶0 that xn is bounded.Sincexn+1∈Xn+1⊂Un+1 and Un is convex, for ∀t∈0,1, tPUn+1x1+1−txn+1∈Un+1. Using Lemma 6, one has (21)PUn+1x1−x12≤tPUn+1x1+1−txn+1−x12≤tPUn+1x1−x12+1−txn+1−x12−t1−tgPUn+1x1−xn+1.Therefore,tgPUn+1x1−xn+1≤xn+1−x12−PUn+1x1−x12≤ξn+1⟶0, as n⟶∞. Therefore, xn+1−PUn+1x1⟶∞, as n⟶∞. Combining with Step 3, xn⟶P∩n=1∞Unx1, n⟶∞.Step 6.un⟶P∩n=1∞Unx1, zn⟶P∩n=1∞Unx1, vn⟶P∩n=1∞Unx1, and yn⟶P∩n=1∞Unx1, as n⟶∞.In fact, sinceun=xn+λnxn−xn−1 with λn↛±∞, un⟶P∩n=1∞Unx1, as n⟶∞. Similarly, zn⟶P∩n=1∞Unx1, as n⟶∞.Sincexn+1∈Xn+1⊂Un+1⊂Vn+1,(22)ϕ¯xn+1,vn≤1−αnϕ¯xn+1,un+αnϕ¯xn+1,zn+εn1=xn+12+1−αnun2−21−αnxn+1,JE∗un+αnzn+εn12−2αnxn+1,JE∗zn+εn1=xn+12−1−αnun2−αnzn+εn12+21−αnun−xn+1,JE∗un+2αnzn+εn1−xn+1,JE∗zn+εn1≤xn+12−zn+εn12+1−αnzn+εn12−un2+21−αnunxn+1−un+2αnzn+εn1zn+εn1−xn+1.Sinceεn1⟶0, it follows from Step 5 and Lemma 2 that xn+1−vn⟶0, as n⟶∞. Therefore, vn⟶P∩n=1∞Unx1, as n⟶∞.Sincexn+1∈Xn+1⊂Un+1,(23)βnϕ¯xn+1,un−ϕ¯xn+1,yn+εn2≤ϕ¯xn+1,un−ϕ¯xn+1,wn⟶0.Sinceβn↛0, there exists a subsequence of n, which is still denoted by n such that ϕ¯xn+1,un−ϕ¯xn+1,yn+εn2⟶0, and then xn+1−yn+εn2⟶0, which ensures that yn⟶P∩n=1∞Unx1, as n⟶∞.Step 7.P∩n=1∞Unx1∈∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10.∀q∈∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10, and using Lemma 5, we have(24)ϕ¯q,Un˜zn+εn1≤a0ϕ¯q,zn+εn1+∑i=1∞an,iϕ¯q,JE∗+rn,iTiJE∗−1JE∗zn+εn1≤a0ϕ¯q,zn+εn1+∑i=1∞an,iϕ¯q,zn+εn1−ϕ¯JE∗+rn,iTiJE∗−1JE∗zn+εn1,un+εn1=ϕ¯q,zn+εn1−∑i=1∞an,iϕ¯JE∗+rn,iTiJE∗−1JE∗zn+εn1,zn+εn1.From iterative scheme (15), we know that(25)ϕ¯q,vn≤1−αnϕ¯q,un+αnϕ¯q,Un˜zn+εn1≤1−αnϕ¯q,un+αnϕ¯q,zn+εn1−αn∑i=1∞an,iϕ¯JE∗+rn,iTiJE∗−1JE∗un+εn1,zn+εn1,which implies that(26)αn∑i=1∞an,iϕ¯JE∗+rn,iTiJE∗−1JE∗zn+εn1,zn+εn1≤1−αnϕ¯q,un+αnϕ¯q,zn+εn1−ϕ¯q,vn≤1−αnun2+αnzn+εn12−vn2−21−αnq,JE∗un−2αnq,JE∗zn+εn1+2q,JE∗vn≤un2−vn2+αnzn+εn12−un2+2qJE∗zn+εn1−JE∗un+2qJE∗un−JE∗vn⟶0.Sinceαn↛0, there exists a subsequence of n, which is still denoted by n such that ϕ¯JE∗+rn,iTiJE∗−1JE∗zn+εn1,zn+εn1⟶0. Then JE∗+rn,iTiJE∗−1JE∗un+εn1⟶P∩n=1∞Vnx1, as n⟶∞.Using Lemma5 again, we have P∩n=1∞Unx1=JE∗+rn,iTiJE∗−1JE∗P∩n=1∞Unx1, ∀i∈N. Therefore, P∩n=1∞Unx1∈∩i=1∞TiJE∗−10.Similarly,(27)ϕ¯q,Un˜˜yn+εn2≤b0ϕ¯q,yn+εn2+∑i=1∞bn,iϕ¯q,JE∗+tn,iSiJE∗−1JE∗JE∗+tn,i−1Si−1JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2≤b0ϕ¯q,yn+εn2+∑i=1∞bn,iϕ¯q,JE∗+tn,i−1Si−1JE∗−1JE∗…JE∗+tn,1S1JE∗−1JE∗yn+εn2−∑i=1∞bn,iϕ¯JE∗+tn,iSiJE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2,JE∗+tn,i−1Si−1JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2.From iterative scheme (15), we have (28)ϕ¯q,wn≤1−βnϕ¯q,un+βnϕ¯q,Un˜˜yn+εn2≤1−βnϕ¯q,un+βnb0ϕ¯q,yn+εn2+βn∑i=1∞bn,iϕq,JE∗+tn,i−1Si−1JE∗−1JE∗yn+εn2−βn∑i=1∞bn,iϕ¯JE∗+tn,iSiJE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2,JE∗+tn,i−1Si−1JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2.Therefore,(29)βn∑i=1∞bn,iϕ¯JE∗+tn,iSiJE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2,JE∗+tn,i−1Si−1JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2≤1−βnϕ¯q,un+βnb0ϕ¯q,yn+εn2+βn∑i=1∞bn,iϕ¯q,yn+εn2−ϕ¯q,wn≤1−βnun2+βnyn+εn22−wn2+2qyn+εn2−un+2qun−wn⟶0.Sinceβn↛0, there exists a subsequence of n, which is still denoted by n such that (30)JE∗+tn,iSiJE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2−JE∗+tn,i−1Si−1JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2⟶0.Repeating the above process,(31)JE∗+tn,i−1Si−1JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2JE∗+tn,i−1Si−2JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2⟶0,JE∗+tn,i−1Si−2JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2−JE∗+tn,i−1Si−3JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2⟶0,⋮JE∗+tn,1S1JE∗−1JE∗yn+εn2−yn+εn2⟶0,n⟶∞.Repeating (30), (31), and Lemma 5, one has P∩n=1∞Unx1=JE∗+tn,iSiJE∗−1JE∗P∩n=1∞Unx1,∀i∈N. Therefore, JE∗P∩n=1∞Unx1∈∩i=1∞Ti−10∩∩i=1∞Si−10.Step 8.xn¯=JE∗xn⟶JE∗P∩n=1∞Unx1∈∩i=1∞Ti−10∩∩i=1∞Si−10, as n⟶∞.Using Steps 1 and 7,P∩n=1∞Unx1−x1=P∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10x1−x1. Since the metric projection is unique, P∩n=1∞Unx1=P∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10x1.
This completes the proof.Remark 1.
Ifλn=ϑn, then un=zn. Two-step inertial iterative scheme (15) reduces to the traditional inertial iterative scheme.Remark 2.
Ifλn=ϑn=0, then un=zn=xn; and two-step iterative scheme (15) extends the corresponding work of (13) in [8].Remark 3.
Ifyn or xn+1 is chosen as PWn+1x1 (or πWn+1x1) or PXn+1x1 (or πXn+1x1), (15) becomes a projection iterative scheme with inertial items.Corollary 1.
In Hilbert spaceH, iterative scheme (15) becomes as follows: (32)x0,x1,ε11,ε12∈H,un=xn+λnxn−xn−1,zn=xn+ϑnxn−xn−1,vn=1−αnun+αnUn¯zn+εn1,V1=W1=H,Vn+1=p∈Vn:21−αnp,un+2αnp,zn+εn1−2p,vn≤1−αnun2+αnzn+εn12−vn2,Wn+1=p∈Vn+1:x1−p2≤PVn+1x1−x12+τn+1,yn∈Wn+1chosen arbitrarily,wn=1−βnun+βnUn¯¯yn+εn2,U1=X1=H,Un+1=p∈Vn+1:21−βnp,un+2βnp,yn+εn2−2p,wn≤1−βnun2+βnyn+εn22−wn2,Xn+1=p∈Un+1:x1−p2≤PUn+1x1−x12+ξn+1,xn+1∈Xn+1chosen arbitrarily,n∈N,where Un¯=a0I+∑i=1∞an,iI+rn,iTi−1 and Un¯¯=b0I+∑i=1∞bn,iI+tn,iSi−1I+tn,i−1Si−1−1⋯I+tn,1S1−1, ∀i,n∈N. Under the assumptions of Theorem 1, the result of Theorem 1 is still true.


## 2.1. Iterative Schemes

In this section, we suppose that the following conditions are satisfied:(A1) E is a real uniformly convex and uniformly smooth Banach space; JE:E⟶E∗ and JE∗:E∗⟶E are the normalized duality mappings.(A2) Ti,Si:E⟶E are d-accretive mappings such that RI+Ti=E=RI+Si, for each i∈N.(A3) rn,i, tn,i, ξn and τn are real number sequences in 0,+∞, for i,n∈N. λn and ϑn are real number sequences in −∞,+∞. αnn=1∞ and βnn=1∞ are real number sequences in 0,1.(A4) a0,b0∈0,1, an,ii=0∞ and bn,ii=0∞ are real number sequences in 0,1 such that a0+∑i=1∞an,i=b0+∑i=1∞bn,i=1.(A5) εn1 and εn2 are the error sequences in E∗.(A6) Un˜=JE∗−1a0JE∗+∑i=1∞an,iJE∗JE∗+rn,iTiJE∗−1JE∗; Un˜˜=JE∗−1b0JE∗+∑i=1∞bn,iJE∗JE∗+tn,iSiJE∗−1JE∗JE∗+tn,i−1Si−1JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗.We construct the following iterative scheme:(15)x0,x1,ε11,ε12∈E∗,un=xn+λnxn−xn−1,zn=xn+ϑnxn−xn−1,vn=JE∗−11−αnJE∗un+αnJE∗Un˜zn+εn1,V1=W1=E∗,Vn+1=p∈Vn:ϕ¯p,vn≤1−αnϕ¯p,un+αnϕ¯p,zn+εn1,Wn+1=p∈Vn+1:x1−p2≤PVn+1x1−x12+τn+1,yn∈Wn+1chosen arbitrarily,wn=JE∗−11−βnJE∗un+βnJE∗Un˜˜yn+εn2,U1=X1=E∗,Un+1=p∈Vn+1:ϕ¯p,wn≤1−βnϕ¯p,un+βnϕ¯p,yn+εn2,Xn+1=p∈Un+1:x1−p2≤PUn+1x1−x12+ξn+1,xn+1∈Xn+1chosen arbitrarily,n∈N.


## 2.2. Strong Convergence Theorems

Theorem 1.
Consider∩i=1∞Ti−10∩∩i=1∞Si−10≠∅, infnrn,i>0, infntn,i>0 for i∈N, αn↛0, βn↛0, τn⟶0, λn↛±∞, ϑn↛±∞, ξn⟶0, τn⟶0, εn1⟶0, and εn2⟶0, as n⟶∞. Then, the iterative sequence xn¯=JE∗xn⟶JE∗P∩n=1∞Unx1=JE∗P∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10∈∩i=1∞Ti−10∩∩i=1∞Si−10, as n⟶∞.Proof.
The proof is split into eight steps.Step 1. ∩n=1∞Un≠∅.Since∩i=1∞Ti−10∩∩i=1∞Si−10≠∅, there exists δ0∈E such that Tiδ0=Siδ0=0, ∀i∈N. It follows from Lemma 1 that there exists η0∈E∗ such that JE∗η0=δ0. Therefore, ∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10≠∅.Next, we shall use inductive method to prove that∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10⊂Un, n∈N.∀p∈∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10. For n=1, it is obvious that p∈U1. Suppose that the result is true for n=k. Then, if n=k+1, it follows from the definition of the Lyapunov functional, the convexity of ⋅2, and Lemma 5 that (16)ϕ¯p,vk=p2−21−αkp,JE∗uk−2αkp,JE∗Uk˜zk+εk1+1−αkJE∗uk+αkJE∗Uk˜zk+εk12≤p2−21−αkp,JE∗uk−2αka0p,JE∗zk+εk1−2αkp,∑i=1∞ak,iJE∗JE∗+rk,iTiJE∗−1JE∗zk+εk1+1−αkuk2+αka0zk+εk12+αk∑i=1∞ak,iJE∗+rk,iTiJE∗−1JE∗zk+εk12=1−αkϕ¯p,uk+αka0ϕ¯p,zk+εk1+αk∑i=1∞ak,iϕ¯p,JE∗+rk,iTiJE∗−1JE∗zk+εk1≤1−αkϕ¯p,uk+αkϕ¯p,zk+εk1.Thus,p∈Vk+1. Using Lemma 5 repeatedly, similar to the above discussion, one has (17)ϕ¯p,wk≤1−βkϕ¯p,uk+βkϕ¯p,Uk˜˜yk+εk2≤1−βkϕ¯p,uk+βkb0ϕ¯p,yk+εk2+βk∑i=1∞bk,iϕ¯p,yk+εk2=1−βkϕ¯p,uk+βkϕ¯p,yk+εk2.Thenp∈Uk+1, which implies that ∩n=1∞Un≠∅.Step 2. Both Vn and Un are closed and convex subsets of E∗, for n∈N.Ifn=1, the result is obvious. If n≥2, since(18)ϕ¯p,vn≤1−αnϕ¯p,un+αnϕ¯p,zn+εn1⇔p,1−αnJE∗un+αnJE∗zn+εn1−JE∗vn≤αnzn+εn12+1−αnun2−vn22.Vn is a closed and convex subset of E∗, for n∈N.Since(19)ϕ¯p,wn≤1−βnϕ¯p,un+βnϕ¯p,yn+εn2⇔p,1−βnJE∗un+βnJE∗yn+εn2−JE∗wn≤βnyn+εn22+1−βnun2−wn22.Un is a closed and convex subset of E∗, for n∈N.Step 3.PUnx1⟶P∩n=1∞Unx1, as n⟶∞.The result follows from the results of Steps 1 and 2 and Lemmas3 and 4.Step 4.Wn≠∅ and Xn≠∅.SincePVn+1x1−x1=infq∈Vn+1q−x1; then, for τn+1, there exists θn+1∈Vn+1 such that(20)x1−θn+12≤infq∈Vn+1q−x12+τn+1=PVn+1x1−x12+τn+1.ThenWn≠∅. Similarly, Xn≠∅. This ensures that xn is well defined.Step 5. xn⟶P∩n=1∞Unx1, as n⟶∞.Sincexn+1∈Xn+1, x1−xn+12≤PUn+1x1−x12+ξn+1. It follows from Step 3 and ξn⟶0 that xn is bounded.Sincexn+1∈Xn+1⊂Un+1 and Un is convex, for ∀t∈0,1, tPUn+1x1+1−txn+1∈Un+1. Using Lemma 6, one has (21)PUn+1x1−x12≤tPUn+1x1+1−txn+1−x12≤tPUn+1x1−x12+1−txn+1−x12−t1−tgPUn+1x1−xn+1.Therefore,tgPUn+1x1−xn+1≤xn+1−x12−PUn+1x1−x12≤ξn+1⟶0, as n⟶∞. Therefore, xn+1−PUn+1x1⟶∞, as n⟶∞. Combining with Step 3, xn⟶P∩n=1∞Unx1, n⟶∞.Step 6.un⟶P∩n=1∞Unx1, zn⟶P∩n=1∞Unx1, vn⟶P∩n=1∞Unx1, and yn⟶P∩n=1∞Unx1, as n⟶∞.In fact, sinceun=xn+λnxn−xn−1 with λn↛±∞, un⟶P∩n=1∞Unx1, as n⟶∞. Similarly, zn⟶P∩n=1∞Unx1, as n⟶∞.Sincexn+1∈Xn+1⊂Un+1⊂Vn+1,(22)ϕ¯xn+1,vn≤1−αnϕ¯xn+1,un+αnϕ¯xn+1,zn+εn1=xn+12+1−αnun2−21−αnxn+1,JE∗un+αnzn+εn12−2αnxn+1,JE∗zn+εn1=xn+12−1−αnun2−αnzn+εn12+21−αnun−xn+1,JE∗un+2αnzn+εn1−xn+1,JE∗zn+εn1≤xn+12−zn+εn12+1−αnzn+εn12−un2+21−αnunxn+1−un+2αnzn+εn1zn+εn1−xn+1.Sinceεn1⟶0, it follows from Step 5 and Lemma 2 that xn+1−vn⟶0, as n⟶∞. Therefore, vn⟶P∩n=1∞Unx1, as n⟶∞.Sincexn+1∈Xn+1⊂Un+1,(23)βnϕ¯xn+1,un−ϕ¯xn+1,yn+εn2≤ϕ¯xn+1,un−ϕ¯xn+1,wn⟶0.Sinceβn↛0, there exists a subsequence of n, which is still denoted by n such that ϕ¯xn+1,un−ϕ¯xn+1,yn+εn2⟶0, and then xn+1−yn+εn2⟶0, which ensures that yn⟶P∩n=1∞Unx1, as n⟶∞.Step 7.P∩n=1∞Unx1∈∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10.∀q∈∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10, and using Lemma 5, we have(24)ϕ¯q,Un˜zn+εn1≤a0ϕ¯q,zn+εn1+∑i=1∞an,iϕ¯q,JE∗+rn,iTiJE∗−1JE∗zn+εn1≤a0ϕ¯q,zn+εn1+∑i=1∞an,iϕ¯q,zn+εn1−ϕ¯JE∗+rn,iTiJE∗−1JE∗zn+εn1,un+εn1=ϕ¯q,zn+εn1−∑i=1∞an,iϕ¯JE∗+rn,iTiJE∗−1JE∗zn+εn1,zn+εn1.From iterative scheme (15), we know that(25)ϕ¯q,vn≤1−αnϕ¯q,un+αnϕ¯q,Un˜zn+εn1≤1−αnϕ¯q,un+αnϕ¯q,zn+εn1−αn∑i=1∞an,iϕ¯JE∗+rn,iTiJE∗−1JE∗un+εn1,zn+εn1,which implies that(26)αn∑i=1∞an,iϕ¯JE∗+rn,iTiJE∗−1JE∗zn+εn1,zn+εn1≤1−αnϕ¯q,un+αnϕ¯q,zn+εn1−ϕ¯q,vn≤1−αnun2+αnzn+εn12−vn2−21−αnq,JE∗un−2αnq,JE∗zn+εn1+2q,JE∗vn≤un2−vn2+αnzn+εn12−un2+2qJE∗zn+εn1−JE∗un+2qJE∗un−JE∗vn⟶0.Sinceαn↛0, there exists a subsequence of n, which is still denoted by n such that ϕ¯JE∗+rn,iTiJE∗−1JE∗zn+εn1,zn+εn1⟶0. Then JE∗+rn,iTiJE∗−1JE∗un+εn1⟶P∩n=1∞Vnx1, as n⟶∞.Using Lemma5 again, we have P∩n=1∞Unx1=JE∗+rn,iTiJE∗−1JE∗P∩n=1∞Unx1, ∀i∈N. Therefore, P∩n=1∞Unx1∈∩i=1∞TiJE∗−10.Similarly,(27)ϕ¯q,Un˜˜yn+εn2≤b0ϕ¯q,yn+εn2+∑i=1∞bn,iϕ¯q,JE∗+tn,iSiJE∗−1JE∗JE∗+tn,i−1Si−1JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2≤b0ϕ¯q,yn+εn2+∑i=1∞bn,iϕ¯q,JE∗+tn,i−1Si−1JE∗−1JE∗…JE∗+tn,1S1JE∗−1JE∗yn+εn2−∑i=1∞bn,iϕ¯JE∗+tn,iSiJE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2,JE∗+tn,i−1Si−1JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2.From iterative scheme (15), we have (28)ϕ¯q,wn≤1−βnϕ¯q,un+βnϕ¯q,Un˜˜yn+εn2≤1−βnϕ¯q,un+βnb0ϕ¯q,yn+εn2+βn∑i=1∞bn,iϕq,JE∗+tn,i−1Si−1JE∗−1JE∗yn+εn2−βn∑i=1∞bn,iϕ¯JE∗+tn,iSiJE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2,JE∗+tn,i−1Si−1JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2.Therefore,(29)βn∑i=1∞bn,iϕ¯JE∗+tn,iSiJE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2,JE∗+tn,i−1Si−1JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2≤1−βnϕ¯q,un+βnb0ϕ¯q,yn+εn2+βn∑i=1∞bn,iϕ¯q,yn+εn2−ϕ¯q,wn≤1−βnun2+βnyn+εn22−wn2+2qyn+εn2−un+2qun−wn⟶0.Sinceβn↛0, there exists a subsequence of n, which is still denoted by n such that (30)JE∗+tn,iSiJE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2−JE∗+tn,i−1Si−1JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2⟶0.Repeating the above process,(31)JE∗+tn,i−1Si−1JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2JE∗+tn,i−1Si−2JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2⟶0,JE∗+tn,i−1Si−2JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2−JE∗+tn,i−1Si−3JE∗−1JE∗⋯JE∗+tn,1S1JE∗−1JE∗yn+εn2⟶0,⋮JE∗+tn,1S1JE∗−1JE∗yn+εn2−yn+εn2⟶0,n⟶∞.Repeating (30), (31), and Lemma 5, one has P∩n=1∞Unx1=JE∗+tn,iSiJE∗−1JE∗P∩n=1∞Unx1,∀i∈N. Therefore, JE∗P∩n=1∞Unx1∈∩i=1∞Ti−10∩∩i=1∞Si−10.Step 8.xn¯=JE∗xn⟶JE∗P∩n=1∞Unx1∈∩i=1∞Ti−10∩∩i=1∞Si−10, as n⟶∞.Using Steps 1 and 7,P∩n=1∞Unx1−x1=P∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10x1−x1. Since the metric projection is unique, P∩n=1∞Unx1=P∩i=1∞TiJE∗−10∩∩i=1∞SiJE∗−10x1.
This completes the proof.Remark 1.
Ifλn=ϑn, then un=zn. Two-step inertial iterative scheme (15) reduces to the traditional inertial iterative scheme.Remark 2.
Ifλn=ϑn=0, then un=zn=xn; and two-step iterative scheme (15) extends the corresponding work of (13) in [8].Remark 3.
Ifyn or xn+1 is chosen as PWn+1x1 (or πWn+1x1) or PXn+1x1 (or πXn+1x1), (15) becomes a projection iterative scheme with inertial items.Corollary 1.
In Hilbert spaceH, iterative scheme (15) becomes as follows: (32)x0,x1,ε11,ε12∈H,un=xn+λnxn−xn−1,zn=xn+ϑnxn−xn−1,vn=1−αnun+αnUn¯zn+εn1,V1=W1=H,Vn+1=p∈Vn:21−αnp,un+2αnp,zn+εn1−2p,vn≤1−αnun2+αnzn+εn12−vn2,Wn+1=p∈Vn+1:x1−p2≤PVn+1x1−x12+τn+1,yn∈Wn+1chosen arbitrarily,wn=1−βnun+βnUn¯¯yn+εn2,U1=X1=H,Un+1=p∈Vn+1:21−βnp,un+2βnp,yn+εn2−2p,wn≤1−βnun2+βnyn+εn22−wn2,Xn+1=p∈Un+1:x1−p2≤PUn+1x1−x12+ξn+1,xn+1∈Xn+1chosen arbitrarily,n∈N,where Un¯=a0I+∑i=1∞an,iI+rn,iTi−1 and Un¯¯=b0I+∑i=1∞bn,iI+tn,iSi−1I+tn,i−1Si−1−1⋯I+tn,1S1−1, ∀i,n∈N. Under the assumptions of Theorem 1, the result of Theorem 1 is still true.


## 3. Numerical Experiments

Theorem 2.
LetE=−∞,+∞, Tix=x/2i, and Six=x/i,∀x∈−∞,+∞, ∀i∈N. Let a0=1/2=b0, an,i=n+2/2n+i+1n+i+2, bn,i=n/2n+1i, and tn,i=rn,i=ni, ∀i,n∈N. Let α1=β1=1; αn=βn=n−1/n, n≥2. λn=ϑn=τn=ξn=1/n; εn1=εn2=0, ∀n∈N. For initial value x0=1, x1=1/3, the iterative sequence xn generated by (32) converges strongly to 0∈∩i=1∞Ti−10∩∩i=1∞Si−10 by the eighth step for two different choices of yn and xn in the corresponding sets Wn+1 and Xn+1, respectively.Proof.
For the special example, we can easily see that all of the assumptions of Corollary1 are satisfied; and the iterative scheme (32) can be simplified as follows: (33)x0=1,x1=13,V2=V1=−∞,+∞=W1,U1=−∞,+∞=X1,un=n+1nxn−1nxn−1,vn=n+52n+4un,Vn+1=Vn∩p:2un−vnp≤un−vnun+vn,Wn+1=Vn+1∩x1−PVn+1x1−x12+1n+1,x1+PVn+1x1−x12+1n+1,yn∈Wn+1,wn=1nun+n−1n+32nn+2yn,setqn=1/nun2+n−1/nyn2−wn221/nun+n−1/nyn−wn,Un+1=Vn+1∩p:2unn+n−1nyn−wnp≤un2n+n−1nyn2−wn2,Xn+1=Un+1∩x1−PUn+1x1−x12+1n+1,x1+PUn+1x1−x12+1n+1,xn+1∈Xn+1,n∈N.
Now, compute step by step and choose two different groups of values ofyn and xn in Wn+1 and Xn+1, respectively; we can get the two following tables.Remark 4.
From Tables1 and 2 derived from the numerical experiments done in Theorem 2, we may find that (1) Wn+1 is an interval that permits us to choose intermediate iterative element yn flexibly; (2) two extreme values of yn in Wn+1, the largest and the smallest, are chosen, from which we can see that the convergence of the iterative sequence xn is not affected.Table 1
yn is chosen as the largest value in Wn+1 and xn+1 is chosen as the smallest value in Xn+1.
Iterative stepValues ofun, vn, Vn+1, and Wn+1Choice ofynValues ofwn, qn, Un+1, and Xn+1Choice ofxn+1n=1u1=v1=−1/3y1=1/3+2/2w1=−1/3x2=1/3−2/2V2=V1=−∞,+∞U2=−∞,+∞W2=13−22,13+22X2=13−22,13+22n=2u2=−0.72732684y2=1/3+3/3w2=−0.0790748x3=1/3−3/3v2=−0.63641098q2 = 1.9704514V3=u2+v2/2,+∞U3=u2+v2/3,q2=−0.68186891,+∞=−0.68186891,1.9704514W3=1/3−3/3,1/3+3/3X3=1/3−3/3,1/3+3/3n=3u3=−0.2007648x4=1/3+4/4w3 = 0.26641173x4=1/3−4/4v3=−0.1606118q3=0.91220228V4=u3+v3/2,+∞U4=u3+v3/2,q3=−0.1806883,+∞=−0.1806901,0.91220228W4=1/3−4/4,1/3+4/4X4=1/3−4/4,1/3+4/4n=4u4=−0.1473291y4=1/3+5/5w4 = 0.30465688x5=1/3−5/5v4=−0.11049682q4=0.75752491V5=u4+v4/2,+∞U5=u4+v4/2,q4=−0.12891296,+∞=−0.12891296,0.75752491W5=1/3−5/5,1/3+5/5X5=1/3−5/5,q4n=5u5=−0.10332302y5=1/3+6/6w5 = 0.31834414x6=1/3−6/6v5=−0.07380214q5=0.67008454V6=u5+v5/2,+∞U6=u5+v5/2,q5=−0.08856258,+∞=−0.08856258,0.67008454W6=1/3−6/6,1/3+6/6X6=1/3−6/6,q5n=6u6=−0.06842073x7=1/3+7/7w6 = 0.32201738x7=1/3−7/7v6=−0.04703925q6=0.61448521V7=u6+v6/2,+∞U7=u6+v6/2,q6=−0.05772999,+∞=−0.05772999,0.61448521W7=1/3−7/7,1/3+7/7X7=1/3−7/7,q6n=7u7=−0.04030488y7=1/3+8/8w7 = 0.32133083y7=1/3−8/8v7=−0.02686992q7=0.5758943V8=u7+v7/2,+∞U8=u7+v7/2,q7=−0.0335874,+∞=−0.0335874,0.5758943W8=1/3−8/8,1/3+8/8X8=1/3−8/8,q7n=8u8=−0.01716868y9=1/3+9/9=2/3w8 = 0.3186872x9=1/3−9/9=0v8=−0.01115964q8=0.5473603V9=u8+v8/2,+∞U9=u8+v8/2,q8=−0.01416416,+∞=−0.01416416,q8W9=1/3−9/9,1/3+9/9X9=1/3−9/9,q8Table 2
yn is chosen as the smallest value in Wn+1 and xn+1 is chosen as the smallest value in Xn+1.
Iterative stepValues ofun, vn, Vn+1, and Wn+1Choice ofynValues ofwn, qn, Un+1, and Xn+1Choice ofxn+1n=1u1=v1=−1/3y1=1/3−2/2w1=−1/3x2=1/3−2/2V2=V1=−∞,+∞U2=−∞,+∞W2=1/3−2/2,1/3+2/2X2=1/3−2/2,1/3+2/2n=2u2=−0.727326847y2=1/3−3/3w2=−0.43991871x3=1/3−3/3v2=−0.636410987q2=−1.100971V3=u2+v2/2,+∞=−0.68186892,+∞U3=u2+v2/2,+∞W3=1/3−3/3,1/3+3/3X3=1/3−3/3,1/3+3/3n=3u3=−0.2007648y3=1/3−4/4w3=−0.13358829x4=q3=−0.1587173v3=−0.1606118q3=−0.1587173V4=u3+v3/2,+∞=−0.1806883,+∞U4=q3,+∞W4=1/3−4/4,1/3+4/4X4=q3,1/3+4/4n=4u4=−0.13739232y4=1/3−5/5w4=−0.0841707x5=q4=−0.10342091v4=−0.10304424q4=−0.10342091V5=u4+v4/2,+∞=−0.12021834,+∞U5=q4,+∞W5=1/3−5/5,1/3+5/5X5=q4,1/3+5/5n=5u5=−0.092361632y5=1/3−6/6w5=−0.05271917x6=q5=−0.06650978v5=−0.06597259q5=−0.06650978V6=u5+v5/2,+∞=−0.079167111,+∞U6=q5,+∞W6=1/3−6/6,1/6+6/6X6=q5,1/3+6/6n=6u6=−0.06035792y6=1/3−7/7w6=−0.0309805x7=q6=−0.04017194v6=−0.04149607q6=−0.04017194V7=u6+v6/2,+∞=−0.050927,+∞U7=q6,+∞W7=1/3−7/7,1/3+7/7X7=q6,1/3+7/7n=7u7=−0.03640939y7=1/3−8/8w7=−0.01482994x8=y7=−0.02022006v7=−0.02427293q7=−0.02076458V8=u7+v7/2,+∞=−0.03034116,+∞U8=q7,+∞W8=1/3−8/8,1/3+8/8X8=1/3−8/8,1/3+8/8n=8u8=−0.01772608y8=1/3−9/9=0w8=−0.00221576x9=1/3−9/9=0v8=−0.01152195U9=V9V9=u8+v8/2,+∞=−0.01462402,+∞X9=1/3−9/9,1/3+9/9W9=1/3−9/9,1/3+9/9


## 4. Curvature Systems

To emphasize the importance of d-accretive mappings, the connection among d-accretive mappings, iterative schemes, and nonlinear boundary value problems is set up.Definition 7.
(see [22]). A single-valued mapping A:DA=E⟶E∗ is hemicontinuous if Ax+ty⇀Ax, as t⟶0, ∀x,y∈E.Definition 8.
(see [22, 23]). A multivalued mapping A:DA⊂E⟶2E∗ is monotone if x−y,u−v≥0, ∀x,y∈DA,u∈Ax, and v∈Ay. The monotone operator A is called maximally monotone if RJE+λA=E∗, ∀λ>0.Lemma 7.
(see [22]). If A:DA=E⟶E∗ is everywhere defined, monotone, and hemicontinuous, then it is maximally monotone.Example 1.
We shall investigate the following curvature systems:(34)−div1+∇ui2si/2∇uimi−1∇ui+εuiri−2ui+uix=hx,x∈Ω,−<ν,1+∇ui2si/2∇uimi−1∇ui>=0,x∈Γ,i∈N,where ⋅ and ⋅,⋅ denote the norm and inner-product in Rn, respectively. Ω is the bounded conical domain of Rnn≥1 with its boundary Γ∈C1, ν is the normal derivative of Γ, ∇ui=∂ui/∂x1,…,∂ui/∂xn, ϵ is a nonnegative constant, and hx is a given function. For i∈N, mi+si+1=qi, mi≥0, and 2n/n+1<qi≤2. If qi≥n, then suppose that 1≤ri<+∞; if qi<n, then suppose that 1≤ri≤nqi/n−qi. We use ⋅qi′ and ⋅1,qi,Ω to denote the norm in Lqi′Ω and W1,qiΩ, respectively, ∀i∈N.Lemma 8.
Fori∈N, define Bi:W1,qi′Ω⟶W1,qi′Ω∗ as follows: ∀u,v∈W1,qi′Ω,(35)v,Biu=∫Ω1+∇uqi′−1sgnuuqi′2−qi′2si/2∇uqi′−1sgnuuqi′2−qi′mi−1∇uqi′−1sgnuuqi′2−qi′∇vqi′−1sgnvvqi′2−qi′dx.
Then,Bi is everywhere defined, hemicontinuous, and monotone, ∀i∈N.Proof.
The proof is split into three steps.Step 1.Bi is everywhere defined.Ifsi≥0, then(36)v,Biu≤∫Ω2max1,∇uqi′−1sgnuuqi′2−qi′2si/2∇uqi′−1sgnuuqi′2−qi′mi∇vqi′−1sgnvvqi′2−qi′dx≤2si/2∫Ω|∇uqi′−1sgnuuqi′2−qi′miqi′dx1/qi′∇vqi′−1sgnvvqi′2−qi′qi1/qi′+2si/2∫Ω∇uqi′−1sgnuuqi′2−qi′qi′dx1/qi′∇vqi′−1sgnvvqi′2−qi′qi1/qi′≤2si/2∇uqi′−1sgnuuqi′2−qi′qi′mi∇vqi′−1sgnvvqi′2−qi′1,qi′,Ω+2si/2∇uqi′−1sgnuuqi′2−qi′1,qi′,Ω∇vqi′−1sgnvvqi′2−qi′1,qi′,Ω.Ifsi<0, then(37)v,Biu≤∫Ω∇uqi′−1sgnuuqi′2−qi′mi+si∇vqi′−1sgnvvqi′2−qi′dx≤∇uqi′−1sgnuuqi′2−qi′qi′qi/qi′∇vqi′−1sgnvvqi′2−qi′1,qi′,Ω.ThusBi is everywhere defined.Step 2.Bi is monotone.For∀u,v∈W1,qi′Ω,(38)u−v,Biu−Biv=∫Ω1+∇uqi′−1sgnuuqi′2−qi′2si/2∇uqi′−1sgnuuqi′2−qi′mi−1∇uqi′−1sgnuuqi′2−qi′−1+∇vqi′−1sgnvvqi′2−qi′2si/2∇vqi′−1sgnvvqi′2−qi′mi−1∇vqi′−1sgnvvqi′2−qi′,∇uqi′−1sgnuuqi′2−qi′−∇vqi′−1sgnvvqi′2−qi′dx≥∫Ω1+∇uqi′−1sgnuuqi′2−qi′2si/2∇uqi′−1sgnuuqi′2−qi′mi−1+∇vqi′−1sgnvvqi′2−qi′2∇vqi′−1sgnvvqi′2−qi′mi×∇uqi′−1sgnuuqi′2−qi′−∇vqi′−1sgnvvqi′2−qi′dx.From the fact thatht:=1+t2si/2tmi, ∀t≥0 is monotone; we know that Bi is monotone.Step 3.Bi is hemicontinuous.∀u,v,w∈W1,qi′Ω and t∈0,1; using Lebesgue’s dominated convergence theorem, one has(39)0≤limt⟶0w,Biu+tv−Biu≤∫Ωlimt⟶0<1+∇u+tvqi′−1sgnuu+tvqi′2−qi′2si2∇u+tvqi′−1sgnuu+tvqi′2−qi′mi−1×∇u+tvqi′−1sgnu+tvu+tvqi′2−qi′−1+∇uqi′−1sgnuuqi′2−qi′2si2∇uqi′−1sgnvuqi′2−qi′mi−1∇uqi′−1sgnvuqi′2−qi′,∇wqi′−1sgnwwqi′2−qi′>dx=0.
Therefore,Bi is hemicontinuous.
This completes the proof.Lemma 9.
(see [8]). For i∈N, define Ci:W1,qi′Ω⟶W1,qi′Ω∗ as follows: ∀u,v∈W1,qi′Ω,(40)v,Ciu=∫Ωuvdx.
ThenCi is maximally monotone, for ∀i∈N.Lemma 10.
(see [22]). For each i∈N, there exist the maximal monotone extension of Bi and the maximal monotone extension of Ci, which are denoted by Bi¯:Lqi′Ω⟶LqiΩ and Ci¯:Lqi′Ω⟶LqiΩ, respectively.Lemma 11.
(see [24]). For i∈N, if qi′≥2, then the normalized duality mapping Ji:Lqi′Ω⟶LqiΩ is defined by Jiu=uqi′−1sgnuuqi′2−qi′, ∀u∈Lqi′Ω. Then, Ji−1:LqiΩ⟶Lqi′Ω is defined by Ji−1u=uqi′−1sgnu, ∀u∈LqiΩ.Based on the above lemmas and imitating Theorems 3.10, 3.11, and 3.12 in [8], one has the following results.Theorem 3.
Fori∈N, define Ti:LqiΩ⟶LqiΩ as follows: ∀u∈LqiΩ, Tiu=Bi¯Ji−1ux. Then Ti is d-accretive and RI+λTi=LqiΩ, ∀λ>0,i∈N.Theorem 4.
Define the mappingSi¯:LqiΩ⟶LqiΩ by Si¯u=Ci¯Ji−1ux, ∀ux∈LqiΩ. Then Si¯ is d-accretive and RI+λSi¯=LqiΩ, ∀λ>0,i∈N.
Define the mappingSi:LqiΩ⟶LqiΩ by Siu=Si¯ux−kqi−1sgnk, ∀ux∈LqiΩ, where k is a constant. Then Si is d-accretive and RI+λSi=LqiΩ, ∀λ>0,i∈N.Theorem 5.
If, in (34), fix≡εkri−1sgnk+k, where k is a constant, then uix≡k is the solution of (34). Moreover, uix≡k⊂∩i=1∞Ti−10∩∩i=1∞Si−10.Proof.
It is obvious thatuix≡k is the solution of (34). If uix≡k, then Tiuix=Bi¯Ji−1k=BiJi−1k=0 and Siuix=Ci¯Ji−1k−kqi−1sgnk=CiJi−1k−kqi−1sgnk. Since v,CiJi−1k−kqi−1sgnk=∫ΩJi−1k−kqi−1sgnkvdx=0, uix≡k⊂∩i=1∞Ti−10∩∩i=1∞Si−10.
This completes the proof.Remark 5.
From Theorem5, we can see the relationship between the solution of curvature systems (34) and common zero points of two groups of d-accretive mappings. This will help us to approximate the solution of curvature systems by using iterative schemes introduced in Section 2.



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