# On Strong Convergence of Halpern’s Method Using Averaged Type Mappings

Authors: F. Cianciaruso; G. Marino; A. Rugiano; B. Scardamaglia
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

Under suitable hypotheses on control coefficients, we study Halpern’s method to approximate strongly common fixed points of a nonexpansive mapping and of a nonspreading mapping or a fixed point of one of them. A crucial tool in our results is the regularization with the averaged type mappings.


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## Body

## 1. Introduction

LetH be a real Hilbert space with the inner product 〈·,·〉, which induces the norm ∥·∥.LetC be a nonempty, closed, and convex subset of H. Let T be a nonlinear mapping of C into itself; we denote by Fix⁡(T) the set of fixed points of T, that is, Fix⁡(T)={z∈C:Tz=z}.We recall that a mappingT is said to be nonexpansive if
(1)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.The problem of finding fixed points of nonexpansive mappings has been widely investigated by many authors.Halpern [1] was the first to consider the following explicit method:
(2)x1∈C,xn+1=αnu+(1-αn)Txn,∀n≥1,
where (αn)n∈N⊂[0,1] and u∈C is fixed.Moreover, Halpern proved in [1] the following theorem on the convergence of (2) for a particular choice of (αn)n∈N.Theorem 1.
LetC be a bounded, closed, and convex subset of Hilbert space H and let T:C→C be a nonexpansive mapping. For any initialization x1∈C and anchor u∈C, define a sequence (xn)n∈N in C by
(3)xn+1=n-θu+(1-n-θ)Txn,∀n≥1,
where θ∈(0,1). Then (xn)n∈N converges strongly to the element of Fix⁡(T) nearest to u.He also showed that the control conditions,(C1)lim⁡n→∞αn=0,(C2)∑n=1∞‍αn=∞,are necessary for the convergence of (2) to a fixed point of T.Subsequently, several authors carefully studied the following problem: are the control conditions(C1) and (C2) sufficient for the convergence of (2)?In this direction, C. E. Chidume and C. O. Chidume [2] and Suzuki [3], independently, proved that the conditions (C1) and (C2) are sufficient to assure the strong convergence to a fixed point of T of the following iterative sequence:
(4)x1,u∈C;xn+1=αnu+(1-αn)(λxn+(1-λ)Txn),∀n≥1.
Recently, in the setting of Banach spaces, Song and Chai [4], under the same conditions (C1) and (C2) but under stronger hypotheses on the mapping, obtained strong convergence of Halpern iterations (2). In particular, they assumed that E is a real reflexive Banach space with a uniformly Gateâux differentiable norm and with the fixed point property for nonexpansive self-mappings, and considered an important subclass of nonexpansive mappings which is the firmly type nonexpansive mappings.LetT be a mapping with domain D(T). T is said to be firmly type nonexpansive [4] if for all x,y∈D(T), there exists k∈(0,+∞) such that
(5)∥Tx-Ty∥2≤∥x-y∥2-k∥(x-Tx)-(y-Ty)∥2.
A more general class of firmly type nonexpansive mappings is the class of the strongly nonexpansive mappings. Recall that a mapping T:C→C is said to be strongly nonexpansive if (1)
T is nonexpansive;(2)
xn-yn-(Txn-Tyn)→0, whenever (xn)n∈N and (yn)n∈N are sequences in C such that (xn-yn)n∈N is bounded and ∥xn-yn∥-∥Txn-Tyn∥→0.Saejung [5] proved the strong convergence of Halpern’s iterations (2) for strongly nonexpansive mappings in a Banach space E such that one of the following conditions is satisfied:(i)
E is uniformly smooth;(ii)
E is reflexive, strictly convex with a uniformly Gateâux differentiable norm.In the setting of Hilbert spaces, Kohsaka and Takahashi [6] defined T:C→C a nonspreading mapping if
(6)2∥Tx-Ty∥2≤∥Tx-y∥2+∥x-Ty∥2,∀x,y∈C.
The following Lemma is a useful characterization of nonspreading mapping.Lemma 2 (see [7]).
LetC be a nonempty closed subset of Hilbert space H. Then, a mapping T:C→C is nonspreading if and only if
(7)∥Tx-Ty∥2≤∥x-y∥2+2〈x-Tx,y-Ty〉,∀x,y∈C.Observe that ifT is a nonspreading mapping from C into itself and Fix⁡(T)≠∅, then T is quasi-nonexpansive; that is,
(8)∥Tx-p∥≤∥x-p∥,∀x∈C,∀p∈Fix⁡(T).Further, the set of fixed points of a quasi-nonexpansive mapping is closed and convex [8].Osilike and Isiogugu [9] studied Halpern’s type for k-strictly pseudononspreading mappings T, which are a more general class of the nonspreading mappings.To obtain the strong convergence of (2), they replaced the mapping T with the averaged type mapping Tδ, that is, with the mapping
(9)Tδ=(1-δ)I+δT,δ∈(0,1).
Iemoto and Takahashi [7] approximated common fixed points of a nonexpansive mapping T and of a nonspreading mapping S in a Hilbert space using Moudafi’s iterative scheme [10]. They obtained the following Theorem that states the weak convergence of their iterative method.Theorem 3.
LetH be a Hilbert space and let C be a nonempty, closed, and convex subset of H. Assume that Fix⁡(S)∩Fix⁡(T)≠∅. Define a sequence (xn)n∈N as follows:
(10)x1∈Cxn+1=(1-αn)xn+αn[βnSxn+(1-βn)Txn],
for all n∈N, where (αn)n∈N, (βn)n∈N⊂[0,1]. Then, the following hold. (i)
Iflim⁡inf⁡n→∞⁡αn(1-αn)>0 and ∑n=1∞‍(1-βn)<∞, then (xn)n∈N converges weakly to p∈Fix⁡(S).(ii)
If∑n=1∞‍αn(1-αn)=∞ and ∑n=1∞‍βn<∞, then (xn)n∈N converges weakly to p∈Fix⁡(T).(iii)
Iflim⁡inf⁡n→∞⁡αn(1-αn)>0 and lim⁡inf⁡n→∞⁡βn(1-βn)>0, then (xn)n∈N converges weakly to p∈Fix⁡(S)∩Fix⁡(T).In this paper, inspired by Iemoto and Takahashi [7], we introduce an iterative method of Halpern’s type involving the averaged type mappings Tδ and Sδ, where T is a nonexpansive mapping and S is a nonspreading mapping. The averaged type mappings Tδ and Sδ have a regularizing role in order to prove the strong convergence of our iterative scheme. In particular, we prove that the method strongly converges to the unique solution z of the variational inequality
(11)〈u-z,y-z〉≤0,∀y∈D,
where u∈C is an anchor and, depending on the hypotheses on control coefficients, D is the set of fixed points of T, the set of fixed points of S, or the set of common fixed points of T and S.Suitable tools in our proofs are Maingé’s Lemma [11] and some techniques used by Maingé in [12] to study the strong convergence of the viscosity approximation method. However, Wongchan and Saejung [13] found a small mistake in Maingé’s proof.


## 2. Preliminaries

To begin, we collected some lemmas which we will use in our proofs in the next section.In the sequel, we denote byH a real Hilbert space and by C a nonempty closed convex subset of H.Lemma 4.
The following known results hold:(1)
∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2,for allx,y∈H and for all t∈[0,1],(2)
∥x+y∥2≤∥x∥2+2〈y,x+y〉,for allx,y∈H.We recall that for every pointx∈H, there exists a unique nearest point in C, denoted by PCx, such that
(12)∥x-PCx∥≤∥x-y∥,∀y∈C.
Such PC is called the metric projection of H onto C.Lemma [14] characterizes the projection PC.Lemma 5.
LetC be a closed and convex subset of a real Hilbert space and let PC be the metric projection from H onto C. Given that x∈H and z∈C, then z=PCx if and only if there holds the inequality
(13)〈x-z,y-z〉≤0,∀y∈C.By Lemma5, if u∈C is fixed, z=PCu is the unique solution of the variational inequality (13).To prove our main theorem, we need some fundamental properties of the involved mappings in the variational inequality.The following result summarizes some significant properties ofI-T if T is a nonexpansive mapping ([15, 16]).Lemma 6.
LetC be a nonempty closed convex subset of H and let T:C→C be nonexpansive. Then, (1)
I-T:C→H is (1/2)-inverse strongly monotone, that is,
(14)12∥(I-T)x-(I-T)y∥2≤〈x-y,(I-T)x-(I-T)y〉,for allx,y∈C;(2)
moreover, ifFix⁡(T)≠∅, I-T is demiclosed at 0; that is, for every sequence (xn)n∈N weakly convergent to p such that xn-Txn→0 as n→∞, it follows p∈Fix⁡(T).Iemoto and Takahashi showed the demiclosedness ofI-S at 0 and a suitable property of I-S. These results are summarized in the following two Lemmas.Lemma 7 (see [7]).
LetC be a nonempty, closed, and convex subset of H. Let S:C→C be a nonspreading mapping such that Fix⁡(S)≠∅. Then, I-S is demiclosed at 0.Lemma 8 (see [7]).
LetC be a nonempty, closed, and convex subset of H. Let S:C→C be a nonspreading mapping. Then,
(15)∥(I-S)x-(I-S)y∥2≤〈x-y,(I-S)x-(I-S)y〉+12(∥x-Sx∥2+∥y-Sy∥2),
for all x,y∈C.IfT is a nonexpansive mapping of C into itself, Byrne [15] defined the averaged mapping as follows:
(16)Tδ=(1-δ)I+δT=I-δ(I-T),
where δ∈(0,1).Moreover, Byrne [15] and successively Moudafi [17] proved some properties of the averaged mappings; in particular, they showed that Tδ is a nonexpansive mapping. In this paper, inspired by [15, 17], we definethe averaged type mappingVδ as in (16) for a nonlinear mapping V:C→C; we notice that Fix⁡(V)=Fix⁡(Vδ). It is easy to verify that if S is a nonspreading mapping of C into itself and Fix⁡(S)≠∅, the averaged type mapping Sδ is quasi-nonexpansive and consequently the set of fixed points of Sδ is closed and convex.Actually, it follows from [9] that Sδ is quasi-firmly type nonexpansive mapping; that is, it is a firmly type nonexpansive mapping (5) on fixed points of S. For completeness we include the easy proof.Proposition 9.
LetC be a nonempty closed and convex subset of H and let S:C→C be a nonspreading mapping such that Fix⁡(S) is nonempty. Then the averaged type mapping
(17)Sδ=(1-δ)I+δS
is quasi-firmly type nonexpansive mapping with coefficient k=(1-δ)∈(0,1).Proof.
We obtain(18)∥Sδx-Sδy∥2=∥(1-δ)(x-y)+δ(Sx-Sy)∥2(byLemma4)=(1-δ)∥x-y∥2+δ∥Sx-Sy∥2-δ(1-δ)∥(x-Sx)-(y-Sy)∥2(by(7))≤(1-δ)∥x-y∥2+δ[∥x-y∥2+2〈x-Sx,y-Sy〉]-δ(1-δ)∥(x-Sx)-(y-Sy)∥2=∥x-y∥2+2δ〈δ(x-Sx),δ(y-Sy)〉-1-δδ∥δ(x-Sx)-δ(y-Sy)∥2(by(17))=∥x-y∥2+2δ〈x-Sδx,y-Sδy〉-1-δδ∥(x-Sδx)-(y-Sδy)∥2≤∥x-y∥2+2δ〈x-Sδx,y-Sδy〉-(1-δ)∥(x-Sδx)-(y-Sδy)∥2.
Hence, we have
(19)∥Sδx-Sδy∥2≤∥x-y∥2+2δ〈x-Sδx,y-Sδy〉-(1-δ)∥(x-Sδx)-(y-Sδy)∥2.
In particular, choosing in (19) y=p, where p∈Fix⁡(S)=Fix⁡(Sδ) we obtain
(20)∥Sδx-p∥2≤∥x-p∥2-(1-δ)∥x-Sδx∥2.The following lemma is useful in the proof of our main result.Lemma 10.
LetC be a nonempty closed and convex subspace of H, u∈C fixed, T a nonexpansive mapping from C into itself, and S a nonspreading mapping from C into itself such that Fix⁡(T)∩Fix⁡(S)≠∅. Consider a bounded sequence (yn)n∈N⊂C. Then, (1)
if∥yn-Tyn∥→0, as n→∞, then
(21)limsup⁡n→∞〈u-p¯,yn-p¯〉≤0,wherep¯=PFix⁡(T)u is the unique point in Fix⁡(T) that satisfies the variational inequality
(22)〈u-p¯,x-p¯〉≤0,∀x∈Fix⁡(T).(2)
If∥yn-Syn∥→0, as n→∞, then
(23)limsup⁡n→∞〈u-p^,yn-p^〉≤0,wherep^=PFix⁡(S)u is the unique point in Fix⁡(S) that satisfies the variational inequality
(24)〈u-p^,x-p^〉≤0,∀x∈Fix⁡(S).(3)
If∥yn-Syn∥→0 and ∥yn-Tyn∥→0, as n→∞, then
(25)limsup⁡n→∞〈u-p0,yn-p0〉≤0,wherep0=PFix⁡(T)∩Fix⁡(S)u is the unique point in Fix⁡(T)∩Fix⁡(S) that satisfies the variational inequality
(26)〈u-p0,x-p0〉≤0,∀x∈Fix⁡(T)∩Fix⁡(S).Proof.
(
1
) Let p¯ satisfy (22). Let (ynk)k∈N be a subsequence of (yn)n∈N for which
(27)limsup⁡n→∞〈p¯-u,yn-p¯〉=lim⁡k→∞〈p¯-u,ynk-p¯〉.
Select a subsequence (ynkj)j∈N of (ynk)k∈N such that ynkj⇀v (this is possible by boundedness of (yn)n∈N). By the hypothesis ∥yn-Tyn∥→0, as n→∞, and by demiclosedness of I-T at 0 we have v∈Fix⁡(T) and
(28)limsup⁡n→∞〈p¯-u,yn-p¯〉=lim⁡j→∞〈p¯-u,ynkj-p¯〉=〈p¯-u,v-p¯〉,
so the claim follows by (22).
(
2
) The proof is the same of (1) since also S is demiclosed in 0.
(
3
) Select a subsequence (ynk)k∈N of (yn)n∈N such that
(29)limsup⁡n→∞〈p0-u,yn-p0〉=lim⁡k→∞〈p0-u,ynk-p0〉,
where p0 satisfies (26). Now select a subsequence (ynkj)j∈N of (ynk)k∈N such that ynkj⇀w. Then by demiclosedness of I-T and I-S at 0 and by the hypotheses ∥yn-Tyn∥→0 and ∥yn-Syn∥→0, as n→∞, we obtain that w=Tw=Sw, that is, w∈Fix⁡(T)∩Fix⁡(S). So,
(30)limsup⁡n→∞〈p0-u,yn-p0〉=lim⁡j→∞〈p0-u,ynkj-p0〉=〈p0-u,w-p0〉,
so the claim follows by (26).A pertinent tool for us is the well-known lemma of Xu [18].Lemma 11.
Let(an)n∈N be a sequence of nonnegative real numbers satisfying the following relation:
(31)an+1≤(1-αn)an+αnσn+θn,n≥0,
where,(i)
(αn)n∈N⊂[0,1], ∑n=1∞‍αn=∞;(ii)
limsup⁡n→∞⁡σn≤0;(iii)
θn≥0, ∑n=1∞‍θn<∞.Then,
(32)lim⁡n→∞an=0.Finally, a crucial tool for our results is the following lemma proved by Maingé.Lemma 12 (see [11]).
Let(γn)n∈N be a sequence of real numbers such that there exists a subsequence (γnj)j∈N of (γn)n∈N such that γnj<γnj+1, for all j∈N. Consider the sequence of integers (τ(n))n∈N defined by
(33)τ(n)∶=max⁡{k≤n:γk<γk+1}.
Then, (τ(n))n∈N is a nondecreasing sequence for all n≥n0, satisfying (i)
lim⁡n→∞τ(n)=∞;(ii)
γτ(n)<γτ(n)+1, ∀n≥n0;(iii)
γn<γτ(n)+1, ∀n≥n0.


## 3. Main Result

In all sections we denote byT:C→C a nonexpansive mapping, S:C→C a nonspreading mapping, and Tδ,Sδ:C→C the respectively averaged type mappings. Moreover, (βn)n∈N⊂[0,1] denotes a real sequence and Un:C→C denotes the convex combination of Tδ and Sδ, that is,
(34)Un=βnTδxn+(1-βn)Sδxn,n∈N.
Further we assume that (i)
Fix⁡(S)∩Fix⁡(T)≠∅;(ii)
(αn)n∈N⊂(0,1) a real sequence such that lim⁡n→∞αn=0 and ∑n=1∞‍αn=0;(iii)
O(1) is any bounded real sequence.We start with the following lemma:Lemma 13.
Letu∈C be an anchor and let (xn)n∈N be the sequence defined by
(35)xn+1=αnu+(1-αn)Unxn,
where
(36)Un=βnTδ+(1-βn)Sδ.
Then(1)
Un is quasi-nonexpansive for all n∈N;(2)
(xn)n∈N, (Sxn)n∈N, (Txn)n∈N, (Sδxn)n∈N, (Tδxn)n∈N, (Unxn)n∈N are bounded sequences.Proof.
(
1
) Any convex combination of quasi-nonexpansive mappings is quasi-nonexpansive too. So is every Un, since Tδ and Sδ are quasi-nonexpansive.
(
2
) The boundedness of (xn)n∈N follows the fact that Un is quasi-nonexpansive. In fact, let q∈Fix⁡(T)∩Fix⁡(S). Then
(37)∥xn+1-q∥=∥αn(u-q)+(1-αn)(Unxn-q)∥≤αn∥u-q∥+(1-αn)∥Unxn-q∥≤αn∥u-q∥+(1-αn)∥xn-q∥.
Since
(38)∥x1-q∥≤max⁡{∥u-q∥,∥x1-q∥},
and by induction we assume that
(39)∥xn-q∥≤max⁡{∥u-q∥,∥x1-q∥},
and then
(40)∥xn+1-q∥≤αn∥u-q∥+(1-αn)max⁡⁡{∥u-q∥,∥x1-q∥}≤αnmax⁡⁡{∥u-q∥,∥x1-q∥}+(1-αn)max⁡⁡{∥u-q∥,∥x1-q∥}=max⁡{∥u-q∥,∥x1-q∥}.
Thus (xn)n∈N is bounded. The boundedness of the other sequences follows by boundedness of (xn)n∈N and by the quasi-nonexpansivity of involved mappings.Now, we prove our strong convergence theorem.Theorem 14.
LetH be a Hilbert space and let C be a nonempty closed and convex subset of H. Let T:C→C be a nonexpansive mapping and let S:C→C be a nonspreading mapping such that Fix⁡(S)∩Fix⁡(T)≠∅. Let Tδ and Sδ be the averaged type mappings. Suppose that (αn)n∈N is a real sequence in (0,1) satisfying the conditions (1)
lim⁡n→∞αn=0,(2)
∑n=1∞‍αn=∞.If (βn)n∈N is a sequence in [0,1], we define a sequence (xn)n∈N as follows:
(41)x1∈Cxn+1=αnu+(1-αn)[βnTδxn+(1-βn)Sδxn],n∈N.
Then, the following hold. (i)
If∑n=1∞‍(1-βn)<∞, then (xn)n∈N strongly converges to p¯=PFix⁡(T)u which is the unique solution in Fix⁡(T) of the variational inequality 〈u-p¯,x-p¯〉≤0, for all x∈Fix⁡(T).(ii)
If∑n=1∞‍βn<∞, then (xn)n∈N strongly converges to p^=PFix⁡(S)u which is the unique solution in Fix⁡(S) of the variational inequality 〈u-p^,x-p^〉≤0, for all x∈Fix⁡(S).(iii)
Iflim⁡inf⁡n→∞⁡βn(1-βn)>0, then (xn)n∈N strongly converges to p0=PFix⁡(T)∩Fix⁡(S)u which is the unique solution in Fix⁡(T)∩Fix⁡(S) of the variational inequality 〈u-p0,x-p0〉≤0, for all x∈Fix⁡(T)∩Fix⁡(S).Proof.
(i) We rewrite the sequence(xn+1)n∈N as
(42)xn+1=αnu+(1-αn)Tδxn+(1-βn)En,
where En=(1-αn)(Sδxn-Tδxn) is bounded, that is, ∥En∥≤O(1).
We begin to prove thatlim⁡n→∞∥xn-Tδxn∥=0.
Letp¯∈Fix⁡(T)=Fix⁡(Tδ) the unique solution in Fix⁡(T) of the variational inequality
(43)〈u-p¯,x-p¯〉≤0,∀x∈Fix⁡(T).
We have
(44)∥xn+1-p¯∥2=∥αnu+(1-αn)(1-δ)xn+(1-αn)δTxn+(1-βn)En-p¯∥2=∥[(1-αn)δ(Txn-xn)+xn-p¯]+[αn(u-xn)+(1-βn)En]∥2(byLemma4)≤∥(1-αn)δ(Txn-xn)+xn-p¯∥2+2〈αn(u-xn)+(1-βn)En,xn+1-p¯〉≤∥(1-αn)δ(Txn-xn)+xn-p¯∥2+2αn〈u-xn,xn+1-p¯〉+2(1-βn)〈En,xn+1-p¯〉≤(1-αn)2δ2∥Txn-xn∥2+∥xn-p¯∥2-2(1-αn)δ〈xn-p¯,xn-Txn〉+2αn∥u-xn∥∥xn+1-p¯∥+2(1-βn)∥En∥∥xn+1-p¯∥=(1-αn)2δ2∥xn-Txn∥2+∥xn-p¯∥2((I-T)p¯=0)-2(1-αn)δ〈xn-p¯,(I-T)xn-(I-T)p¯〉+αnO(1)+(1-βn)O(1)(byLemma6)≤∥xn-p¯∥2+(1-αn)2δ2∥xn-Txn∥2-(1-αn)δ∥(I-T)xn-(I-T)p¯∥2+αnO(1)+(1-βn)O(1)=∥xn-p¯∥2-(1-αn)δ[1-δ(1-αn)]∥xn-Txn∥2+αnO(1)+(1-βn)O(1)
and hence
(45)0≤(1-αn)δ[1-δ(1-αn)]∥xn-Txn∥2≤∥xn-p¯∥2-∥xn+1-p¯∥2+αnO(1)+(1-βn)O(1).
We turn our attention to the monotony of the sequence (∥xn-p¯∥)n∈N.
We consider the following two cases.
Case  A. ∥xn+1-p¯∥ is definitively nonincreasing.
Case  B. There exists a subsequence (xnk)k∈N such that
(46)∥xnk-p-∥<∥xnk+1-p-∥∀k∈N.
Case  A. Since (∥xn-p¯∥)n∈N is definitively nonincreasing, lim⁡n→∞∥xn-p¯∥2 exists. From (45), lim⁡n→∞αn=0, and ∑n=1∞‍(1-βn)<∞, we have
(47)0≤limsup⁡n→∞((1-αn)δ[1-δ(1-αn)]∥xn-Txn∥2)≤limsup⁡n→∞(∥xn-p¯∥2-∥xn+1-p¯∥2≤limsup⁡n→∞h+αnO(1)+(1-βn)O(1)∥xn-p¯∥2-∥xn+1-p¯∥2)=0,
so, we can conclude that
(48)lim⁡n→∞∥xn-Txn∥=0,(49)lim⁡n→∞∥xn-Tδxn∥=lim⁡n→∞δ∥xn-Txn∥=0.
By Lemma10, it follows that
(50)limsup⁡n→∞〈u-p¯,xn-p¯〉≤0.
Finally, we prove that(xn)n∈N converges strongly to p¯.
We compute that(51)∥xn+1-p¯∥2=∥αn(u-p¯)+(1-αn)(Tδxn-p¯)+(1-βn)En∥2(byLemma4)≤∥αn(u-p¯)+(1-αn)(Tδxn-p¯)∥2+2(1-βn)〈En,xn+1-p¯〉≤αn2∥u-p¯∥2+(1-αn)2∥Tδxn-p¯∥2+2αn(1-αn)〈u-p¯,Tδxn-p¯〉+(1-βn)O(1)(Tδnonexpansive)≤(1-αn)∥xn-p¯∥2+αn2O(1)+2αn(1-αn)〈u-p¯,Tδxn-xn〉+2αn(1-αn)〈u-p¯,xn-p¯〉+(1-βn)O(1)≤(1-αn)∥xn-p¯∥2+αn2O(1)+αnO(1)∥Tδxn-xn∥+2αn(1-αn)〈u-p¯,xn-p¯〉+(1-βn)O(1).
If we putσn=αnO(1)+O(1)∥Tδxn-xn∥+2(1-αn)〈u-p¯,xn-p¯〉 and θn=(1-βn)O(1), we have
(52)∥xn+1-p¯∥2≤(1-αn)∥xn-p¯∥2+αnσn+θn.
Hence, from assumption∑n=1∞‍αn=∞ and ∑n=1∞‍(1-βn)<∞, from (49) and lim⁡sup⁡n→∞⁡〈u-p¯,xn-p¯〉≤0 we can apply Xu’s Lemma 11.
Case  B. There exists a subsequence (xnk)k∈N such that
(53)∥xnk-p¯∥<∥xnk+1-p¯∥∀k∈N.
Then by Maingé Lemma12 there exists a sequence of integers (τ(n))n∈N that satisfies (a)
(τ(n))n∈N is nondecreasing;(b)
lim⁡n→∞τ(n)=∞;(c)
∥xτ(n)-p¯∥<∥xτ(n)+1-p¯∥;(d)
∥xn-p¯∥<∥xτ(n)+1-p¯∥.
Consequently,(54)0≤liminf⁡n→∞(∥xτ(n)+1-p¯∥-∥xτ(n)-p¯∥)≤limsup⁡n→∞(∥xτ(n)+1-p¯∥-∥xτ(n)-p¯∥)≤limsup⁡n→∞(∥xn+1-p¯∥-∥xn-p¯∥)(by(42))=limsup⁡n→∞(∥αn(u-Tδxn)+Tδxn-p¯+(1-βn)En∥=limsup⁡n→∞-∥xn-p¯∥)(Tδnonexpansive)≤limsup⁡n→∞(αnO(1)+∥xn-p¯∥+(1-βn)O(1)-∥xn-p¯∥)=0,
so
(55)lim⁡n→∞(∥xτ(n)+1-p¯∥-∥xτ(n)-p¯∥)=0.
By (45), we have
(56)0≤(1-ατ(n))δ[1-δ(1-ατ(n))]∥xτ(n)-Txτ(n)∥2≤∥xτ(n)-p¯∥2-∥xτ(n)+1-p¯∥2+αnO(1)+(1-βτ(n))O(1),
and from (55), lim⁡n→∞αn=0 and ∑n=1∞‍(1-βn)<∞ we get
(57)lim⁡n→∞∥xτ(n)-Txτ(n)∥=0.
By Lemma 10 and (57) we have
(58)limsup⁡n→∞〈u-p¯,xτ(n)-p¯〉≤0.
Finally, we show that (xn)n∈N converges strongly to p¯.
As in CaseA, we can obtain
(59)lim⁡n→∞∥xτ(n)-p¯∥=0;
then, from property (d) of Maingé Lemma 12 and (55) we can conclude
(60)lim⁡n→∞∥xn-p¯∥=0.Proof.
(ii) Now, we rewrite the sequence(xn+1)n∈N as
(61)xn+1=αnu+(1-αn)Sδxn+βnEn,
where En=(1-αn)(Tδxn-Sδxn) is bounded, that is, ∥En∥≤O(1).
We begin to prove thatlim⁡n→∞∥xn-Sδxn∥=0.
Letp^∈Fix⁡(S)=Fix⁡(Sδ) the unique solution in Fix⁡(S) of the variational inequality 〈u-p^,x-p^〉≤0, for all x∈Fix⁡(S). We have
(62)∥xn+1-p^∥2=∥αnu+(1-αn)(1-δ)xn+(1-αn)δSxn+βnEn-p^∥2=∥[(1-αn)δ(Sxn-xn)+xn-p^]+[αn(u-xn)+βnEn]∥2(byLemma4)≤∥(1-αn)δ(Sxn-xn)+xn-p^∥2+2〈αn(u-xn)+βnEn,xn+1-p^〉≤∥(1-αn)δ(Sxn-xn)+xn-p^∥2+2αn〈u-xn,xn+1-p^〉+2βn〈En,xn-p^〉≤(1-αn)2δ2∥Sxn-xn∥2+∥xn-p^∥2-2(1-αn)δ〈xn-p^,xn-Sxn〉+αnO(1)+βnO(1)=(1-αn)2δ2∥xn-Sxn∥2+∥xn-p^∥2((I-S)p^=0)-2(1-αn)δ〈xn-p^,(I-S)xn-(I-S)p^〉+αnO(1)+βnO(1)(byLemma8)≤∥xn-p^∥2+(1-αn)2δ2∥xn-Sxn∥2-2(1-αn)δ[12(∥xn-Sxn∥2+∥p-Sp∥2)∥(I-S)xn-(I-S)p^∥2-2(1-αn)δ-12(∥xn-Sxn∥2+∥p-Sp∥2)]+αnO(1)+βnO(1)≤∥xn-p^∥2+(1-αn)2δ2∥xn-Sxn∥2-(1-αn)δ∥xn-Sxn∥2+αnO(1)+βnO(1)=∥xn-p^∥2-(1-αn)δ[1-δ(1-αn)]∥xn-Sxn∥2+αnO(1)+βnO(1)
and hence
(63)0≤(1-αn)δ[1-δ(1-αn)]∥xn-Sxn∥2≤∥xn-p^∥2-∥xn+1-p^∥2+αnO(1)+βnO(1).
Again, we turn our attention to the monotony of the sequence (∥xn-p^∥)n∈N. We consider the following two cases.
Case  A. ∥xn+1-p^∥ is definitively nonincreasing.
Case  B. There exists a subsequence (xnk)k∈N such that
(64)∥xnk-p^∥<∥xnk+1-p^∥∀k∈N.
Case  A. Since (∥xn-p^∥)n∈N is definitively nonincreasing, lim⁡n→∞∥xn-p^∥2 exists. From (45), lim⁡n→∞αn=0, and ∑n=1∞‍βn<∞, we have
(65)0≤limsup⁡n→∞((1-αn)δ[1-δ(1-αn)]∥xn-Sxn∥2)≤limsup⁡n→∞(∥xn-p^∥2-∥xn+1-p^∥2≤limsup⁡n→∞+αnO(1)+βnO(1)∥xn-p^∥2-∥xn+1-p^∥2)=0,
and hence
(66)lim⁡n→∞∥xn-Sxn∥=0,(67)lim⁡n→∞∥xn-Sδxn∥=lim⁡n→∞δ∥xn-Sxn∥=0.
By Lemma 10, it follows that
(68)limsup⁡n→∞〈u-p^,xn-p^〉≤0.
Finally, we prove that (xn)n∈N converges strongly to p^.
We compute that(69)∥xn+1-p^∥2=∥αn(u-p^)+(1-αn)(Sδxn-p^)+βnEn∥2(byLemma4)≤∥αn(u-p^)+(1-αn)(Sδxn-p^)∥2+2βn〈En,xn+1-p^〉≤αn2∥u-p^∥2+(1-αn)2∥Sδxn-p^∥2+2αn(1-αn)〈u-p^,Sδxn-p^〉+βnO(1)(Sδquasi-nonexpansive)≤(1-αn)∥xn-p^∥2+αn2O(1)+2αn(1-αn)〈u-p^,Sδxn-xn〉+2αn(1-αn)〈u-p^,xn-p^〉+βnO(1)≤(1-αn)∥xn-p^∥2+αn2O(1)+αnO(1)∥Sδxn-xn∥+2αn(1-αn)〈u-p^,xn-p^〉+βnO(1).
If we putσn=αnO(1)+O(1)∥Sδxn-xn∥+2(1-αn)〈u-p^,xn-p^〉 and θn=βnO(1), we have
(70)∥xn+1-p^∥2≤(1-αn)∥xn-p^∥2+αnσn+θn.
So, from assumption ∑n=1∞‍αn=∞ and ∑n=1∞‍βn<∞, from (67) and lim⁡sup⁡n→∞⁡〈p^-u,xn-p^〉≥0 we can apply Xu’s Lemma 11.
Case  B. There exists a subsequence (xnk)k∈N such that
(71)∥xnk-p^∥<∥xnk+1-p^∥∀k∈N.
Then by Maingé Lemma there exists a sequence of integers(τ(n))n∈N that satisfies (a)
(τ(n))n∈N is nondecreasing;(b)
lim⁡n→∞τ(n)=∞;(c)
∥xτ(n)-p^∥<∥xτ(n)+1-p^∥;(d)
∥xn-p^∥<∥xτ(n)+1-p^∥.
Consequently,(72)0≤liminf⁡n→∞(∥xτ(n)+1-p^∥-∥xτ(n)-p^∥)≤limsup⁡n→∞(∥xτ(n)+1-p^∥-∥xτ(n)-p^∥)≤limsup⁡n→∞(∥xn+1-p^∥-∥xn-p^∥)(by(61))=limsup⁡n→∞(∥αn(u-Sδxn)+Sδxn-p^+βnEn∥-∥xn-p^∥)(Sδquasi-nonexpansive)≤limsup⁡n→∞(αnO(1)+∥xn-p^∥+βnO(1)-∥xn-p^∥)=0,
so
(73)lim⁡n→∞(∥xτ(n)+1-p^∥-∥xτ(n)-p^∥)=0.
By (63), we obtain
(74)0≤(1-ατ(n))δ[1-δ(1-ατ(n))]∥xτ(n)-Sxτ(n)∥2≤∥xτ(n)-p^∥2-∥xτ(n)+1-p^∥2+αnO(1)+βτ(n)O(1),
and from (73), lim⁡n→∞αn=0 and ∑n=1∞‍βn<∞ we get
(75)lim⁡n→∞∥xτ(n)-Sxτ(n)∥=0.
By Lemma 10 and (75) we get
(76)limsup⁡n→∞〈u-p^,xτ(n)-p^〉≤0.
Finally, we show that (xn)n∈N converges strongly to p¯.
As in CaseA, we obtain
(77)lim⁡n→∞∥xτ(n)-p^∥=0,
and then from property (d) of Maingé Lemma and (73) we can conclude that
(78)lim⁡n→∞∥xn-p^∥=0.Proof.
(iii) We recall that the sequence(xn+1)n∈N is defined as
(79)xn+1=αnu+(1-αn)Unxn,
where Un=βnTδxn+(1-βn)Sδxn.
We first show thatlim⁡n→∞∥xn-Unxn∥=0.
Letp0∈Fix⁡(T)∩Fix⁡(S) be the unique solution of the variational inequality 〈u-p0,x-p0〉≤0, for all x∈Fix⁡(T)∩Fix⁡(S). We compute that
(80)∥Unxn-p0∥2=∥βn(Tδxn-p0)+(1-βn)(Sδxn-p0)∥2(byLemma4)=βn∥Tδxn-p0∥2+(1-βn)∥Sδxn-p0∥2-βn(1-βn)∥Tδxn-Sδxn∥2(Tδnonexpansiveandby(20))≤βn∥xn-p0∥2+(1-βn)∥xn-p0∥2-(1-βn)(1-δ)∥xn-Sδxn∥2-βn(1-βn)∥Tδxn-Sδxn∥2=∥xn-p0∥2-(1-βn)(1-δ)∥xn-Sδxn∥2-βn(1-βn)∥Tδxn-Sδxn∥2.
So, we get
(81)∥Unxn-p0∥2≤∥xn-p0∥2-(1-βn)(1-δ)∥xn-Sδxn∥2-βn(1-βn)∥Tδxn-S-δxn∥2.
We have
(82)∥xn+1-p0∥2=∥Unxn-p0+αn(u-Unxn)∥2≤∥Unxn-p0∥2+αn(αn∥u-Unxn∥2+2∥Unxn-p0∥∥u-Unxn∥)≤∥Unxn-p0∥2+αnO(1)(by(81))≤∥xn-p0∥2-(1-βn)(1-δ)∥xn-Sδxn∥2-βn(1-βn)∥Tδxn-Sδxn∥2+αnO(1).
From (82), we derive
(83)(1-βn)(1-δ)∥xn-Sδxn∥2≤∥xn-p0∥2-∥xn+1-p0∥2+αnO(1),(84)βn(1-βn)∥Tδxn-Sδxn∥2≤∥xn-p0∥2-∥xn+1-p0∥2+αnO(1).
Now, also we consider two cases.
Case  A. ∥xn+1-p0^∥ is definitively nonincreasing.
Case  B. There exists a subsequence (xnk)k∈N such that
(85)∥xnk-p0^∥<∥xnk+1-p0^∥∀k∈N.
Case  A. Since (∥xn-p0∥)n∈N is definitively nonincreasing, lim⁡n→∞∥xn-p0∥2 exists. From (83) and lim⁡n→∞αn=0 and since lim⁡inf⁡n→∞⁡βn(1-βn)>0 we conclude that
(86)lim⁡n→∞∥xn-Sδxn∥=lim⁡n→∞δ∥xn-Sxn∥=0.
Furthermore, from (84) we have
(87)lim⁡n→∞∥Sδxn-Tδxn∥=lim⁡n→∞δ∥Sxn-Txn∥=0;
since
(88)∥xn-Txn∥≤∥xn-Sxn∥+∥Sxn-Txn∥,
by (86) and (87) we obtain
(89)lim⁡n→∞∥xn-Tδxn∥=lim⁡n→∞δ∥xn-Txn∥=0.
Then, since(90)∥Unxn-xn∥≤βn∥Tδxn-xn∥+(1-βn)∥Sδxn-xn∥,
by (86) and (89) we get
(91)lim⁡n→∞∥xn-Unxn∥=0.
By Lemma 10, we have
(92)limsup⁡n→∞〈u-p0,xn-p0〉≤0.
Finally, (xn)n∈N converges strongly to p0.
We compute that(93)∥xn+1-p0∥2≤∥(1-αn)(Unxn-p0)+αn(u-p0)∥=(1-αn)2∥Unxn-p0∥2+αn2∥u-p0∥2+2αn(1-αn)〈Unxn-p0,u-p0〉(Unquasi-nonexpansive)≤(1-αn)2∥xn-p0∥2+αn2O(1)+2αn(1-αn)〈Unxn-xn,u-p0〉+2αn(1-αn)〈xn-p0,u-p0〉≤(1-αn)∥xn-p0∥2+αn2O(1)+αnO(1)∥Unxn-xn∥+2αn(1-αn)〈xn-p0,u-p0〉.
If we set σn=αnO(1)+O(1)∥Unxn-xn∥+2(1-αn)〈xn-p0,u-p0〉, we have
(94)∥xn+1-p0∥2≤(1-αn)∥xn-p0∥2+αnσn.
From ∑n=1∞‍αn=∞, (91), and lim⁡sup⁡n→∞⁡〈u-p0,xn-p0〉≤0 we conclude that (xn)n∈N strongly converges to p0.
Case  B. (∥xn-p0∥)n∈N is not definitively nonincreasing. This means that there exists a subsequence (xnk)k∈N such that
(95)∥xnk-p0∥<∥xnk+1-p0∥∀k∈N.
Then by Maingé Lemma 12 there exists a sequence of integers (τ(n))n∈N that satisfies some properties defined previously.
Consequently,(96)0≤liminf⁡n→∞(∥xτ(n)+1-p0∥-∥xτ(n)-p0∥)≤limsup⁡n→∞(∥xτ(n)+1-p0∥-∥xτ(n)-p0∥)≤limsup⁡n→∞(∥xn+1-p0∥-∥xn-p0∥)(by(79))=limsup⁡n→∞(∥αn(u-p0)+(1-αn)(Unxn-p0)∥=limsup⁡n→∞-∥xn-p0∥)(Unquasi-nonexpansive)≤limsup⁡n→∞(αnO(1)+∥xn-p0∥-∥xn-p0∥)=0,
and hence
(97)lim⁡n→∞(∥xτ(n)+1-p0∥-∥xτ(n)-p0∥)=0.
By (83) we get
(98)(1-βτ(n))(1-δ)∥xτ(n)-Sδxτ(n)∥2≤∥xτ(n)-p0∥2-∥xn+1-p0∥2+ατ(n)O(1),
and by(84) we have
(99)βτ(n)(1-βτ(n))∥Tδxτ(n)-Sδxτ(n)∥2≤∥xτ(n)-p0∥2-∥xn+1-p0∥2+ατ(n)O(1).
As in Case  A, we get(a)
lim⁡n→∞∥xτ(n)-Sxτ(n)∥=0,(b)
lim⁡n→∞∥xτ(n)-Txτ(n)∥=0,(c)
lim⁡n→∞∥xτ(n)-Uτ(n)xτ(n)∥=0.
By Lemma10, (a) and (b) we have
(100)limsup⁡n→∞〈u-p0,xτ(n)-p0〉≤0.
Finally, we prove that (xn)n∈N converges strongly to p0.
As in CaseA, using (c), the assumption ∑n=1∞‍αn=∞, and (100) we can apply Xu’s Lemma 11 and conclude that
(101)lim⁡n→∞∥xτ(n)-p^∥=0,
and then from property (d) of Maingé Lemma and (97) we can derive that
(102)lim⁡n→∞∥xn-p0∥=0.Remark 15.
We remark that in order to prove that(103)∥xn-Txn∥⟶0,∥xn-Sxn∥⟶0,∥xn-Unxn∥⟶0,
we do not follow the line of the proof of Song and Chai in [4] because their techniques seem questionable.Remark 16.
The main result of this paper gives a positive answer to the question of Kurokawa and Takahashi; see Remark page1567 in [19].



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