# On Mann’s Method with Viscosity for Nonexpansive and Nonspreading Mappings in Hilbert Spaces

Authors: Nawab Hussain; Giuseppe Marino; Afrah A. N. Abdou
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

In the setting of Hilbert spaces, inspired by Iemoto and Takahashi (2009), we study a Mann’s method with viscosity to approximate strongly (common) fixed points of a nonexpansive mapping and a nonspreading mapping. A crucial tool in our results is the nonspreading-average type mapping.


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## Body

## Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 that induces the norm ∥·∥=〈·,·〉. Let T:Dom⁡(T)→H be a mapping. We denote by Fix⁡(T) the set of fixed points of T, Fix⁡(T)={x∈Dom⁡(T):Tx=x}.A mappingT is said to be(i)
nonexpansive [1] (1967) if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈Dom⁡(T);(ii)
firmly nonexpansive [1] (1967) if 〈x-y,Tx-Ty〉≥∥Tx-Ty∥2 for all x,y∈Dom⁡(T);(iii)
firmly type nonexpansive [2] (2009) if
(1)∥Tx-Ty∥2≤∥x-y∥2-k∥(I-T)x-(I-T)y∥2
for all x,y∈Dom⁡(T) for all x,y∈Dom⁡(T);(iv)
strongly nonexpansive [3] (1977) if  T  is nonexpansive and (xn)n∈N,(yn)n∈N⊂Dom⁡(T), ∥xn-yn∥ bounded and (∥xn-yn∥-∥Txn-Tyn∥)→0 then ∥(I-T)xn-(I-T)yn∥→0 for all x,y∈Dom⁡(T);(v)
nonspreading [4] (2008) if 2∥Tx-Ty∥2≤∥Tx-y∥2+∥x-Ty∥2 for all x,y∈Dom⁡(T);(vi)
k-strict nonspreading [5] (2011) if
(2)∥Tx-Ty∥2≤∥x-y∥2+2〈x-Tx,y-Ty〉+k∥(I-T)x-(I-T)y∥2;(vii)
quasi-nonexpansive if∥Tx-y∥≤∥x-y∥ for all x∈Dom⁡(T) and for all y∈Fix⁡(T).Of course,firmly nonexpansive⇒ firmly type nonexpansive ⇒ strongly nonexpansive ⇒ nonexpansive ⇒ quasi-nonexpansive ⇐ nonspreading ⇐  k-strict pseudononspreading.IfC is a nonempty, closed, and convex subset of H, we denote by PC:H→C the metric projection on C; that is, for any x∈H, PCx is the unique element in C such that
(3)∥x-PCx∥≤∥x-y∥,∀y∈C.
It is well known (see [1]) that PC is a firmly nonexpansive mapping and that PC is characterized by the variational inequality
(4)〈PCx-x,z-PCx〉≥0,∀z∈C.
The firm nonexpansivity has many equivalent formulations.Theorem 1.
LetT:C→C a mapping. There are equivalents.(1)
T is firmly nonexpansive; that is, 〈x-y,Tx-Ty〉≥∥Tx-Ty∥2 for all x,y∈Dom(T).(2)
For eachx,y∈C, the convex function Φx,y:[0,1]→R defined by
(5)Φx,y(t)=∥(1-t)(x-y)+t(Tx-Ty)∥
is nonincreasing on [0,1].(3)
The mapping(2t-I) is nonexpansive.(4)
T=(1/2)(I+N) with N is nonexpansive.(5)
∥Tx-Ty∥≤Φx,y(t) for all t∈[0,1].(6)
The mapping(I-T) is firmly nonexpansive.(7)
∥Tx-Ty∥2≤∥x-y∥2-∥(I-t)x-(I-T)y∥2 holds.(8)
One has2∥Tx-Ty∥2+∥Tx-x∥2+∥Ty-y∥2≤∥Tx-y∥2+∥Ty-x∥2.Proof.
The equivalences (1) to (5) are proved in [6]. The equivalences (1) and (6) are proved in [7]. Let us prove that (1) is equivalent to (8):
(6)2∥Tx-Ty∥2+∥Tx-x∥2+∥Ty-y∥2≤∥Tx-y∥2+∥Ty-x∥2⟺2∥Tx-Ty∥2-2〈Tx,x〉-2〈Ty,y〉+∥Tx∥2+∥y∥2≤∥Tx∥2+∥y∥2+∥Ty∥2+∥x∥2-2〈Tx,y〉-2〈Ty,x〉⟺2∥Tx-Ty∥2-2〈Tx,x〉-2〈Ty,y〉≤-2〈Tx,y〉-2〈Ty,x〉⟺2∥Tx-Ty∥2≤2〈Tx,x〉+2〈Ty,y〉-2〈Tx,y〉-2〈Ty,x〉=2〈Tx,x-y〉-2〈Ty,x-y〉=2〈Tx-Ty,x-y〉⟺∥Tx-Ty∥2≤〈Tx-Ty,x-y〉.
Finally, we prove that (1) is equivalent to (7). (I-T) is firmly nonexpansive ⇔〈x-y,(I-T)x-(I-T)y〉  ≥  ∥(I-T)x-(I-T)y∥2⇔∥(I-T)x-(I-T)y∥2  ≤  ∥x-y∥2-〈x-y,Tx-Ty〉  ≤  ∥x-y∥2-∥Tx-Ty∥2.Two important classes of mappings containing the firmly nonexpansive mappings are the average mappings and the nonspreading mappings.After [7], we say that T is an nonexpansive-average mappings if
(7)T=(1-α)I+αN
for some α∈(0,1), and N is a nonexpansive mapping.Definition 2.
LetM be a class of mappings. One says that T is a M-average mapping if
(8)T=Nα=(1-α)I+αN
for some α∈(0,1) where N is a mapping belonging to the class M.Of courseFix⁡(Nα)=Fix⁡(N).The nonexpansive-average mappingregularizes a nonexpansive mapping N according to the celebrated Schaefer’s result [8].Theorem 3.
Any orbit(Nαkx)k∈N of a nonexpansive-average mapping Nα=(1-α)I-αN converges weakly to a fixed point of  N whenever such points exist.Here we are interested in nonspreading and nonspreading-average mappings.Theorem 4.
LetS:C→C be a mapping. The following are equivalent.(1)
S is nonspreading; that is, 2∥Tx-Ty∥2≤∥Tx-y∥2+∥x-Ty∥2;(2)
One has∥Sx-Sy∥2≤∥x-y∥2+2〈x-Sx,y-Sy〉;(3)
∥Sx-Sy∥2≤∥x-y∥2+∥x-Sx∥2+∥y-Sy∥2-∥(I-S)x-(I-S)y∥2.
Moreover, letS be a nonspreading mapping. Then(a)
Fix⁡(S) is closed and convex;(b)
(I-S) is demiclosed;(c)
One has∥(I-S)x-(I-S)y∥2≤〈x-y,(I-S)x-(I-S)y〉+(1/2)∥(I-S)x∥2+(1/2)∥(I-S)y∥2.
IfSω=(1-ω)I+ωS is a nonspreading-average mapping, then one has the following.(i)
(9)∥Sωx-Sωy∥2≤∥x-y∥2+2ω〈x-Sωx,y-Sωy〉-(1-ω)∥(x-Sωx)-(y-Sωy)∥2.
In particular Sω is quasi firmly nonexpansive; that is,
(10)∥Sωx-q∥2≤∥x-q∥2-(1-ω)∥x-Sωx∥2.(ii)
(ω/2)∥x-Sx∥2≤〈x-Sωx,x-q〉, for all q∈Fix⁡(S).Proof.
The equivalence of (1) and (2) is proved in Lemma 3.2 of [9].
The equivalence of (1) and (3) follows by the fact that(11)2〈x-Sωx,x-q〉=-∥(x-Sx)-(y-Sy)∥2+∥x-Sx∥2+∥y-Sy∥2.
The item (a) is proved in [4], while (b) and (c) are proved in [9].
The item (i) is proved in Theorem 3.1 of [5].
Now we prove (ii). Since(12)〈x-Sωx,x-q〉=〈x-(1-ω)x-Sx,x-q〉=ω〈x-Sx,x-q〉
thus we need to show that
(13)12∥x-Sx∥2≤〈x-Sx,x-q〉.
This follows by quasi-nonexpansivity of S. Indeed
(14)12∥x-Sx∥2=12∥(x-q)+(q-Sx)∥2=12[∥x-q∥2+∥q-Sx∥2hhhh+2〈x-q,q-Sx〉∥x-q∥2+∥q-Sx∥2](bythequasi-nonexpansivityofS)≤∥x-q∥2+〈x-q,q-Sx〉=〈x-q,x-q〉+〈x-q,q-Sx〉=〈x-q,x-Sx〉.Recently, Song and Chai [2] in the general setting of Banach spaces obtained strong convergence of Halpern’s iteration
(15)xn+1=αnu+(1-αn)Txn
for firmly type nonexpansive mapping T. (Saejung in [10] noted that their proof seems to be questionable, but the result is true as a consequence of a more general result proved in [10]). Indeed, in [10] is proved the strong convergence of Halpern iteration for strongly nonexpansive mappings (and it is easy to see that the class of strongly nonexpansive mappings contains the class of firmly type nonexpansive mappings).Osilike and Isiogugu [5] studied the Halpern iteration for k-strict pseudo-non-spreading mappings. They showed that if one considers the k-strict pseudo-non-spreading-average mapping, then Halpern’s iteration converges strongly to a fixed point of such a mapping.On the other hand, Iemoto and Takahashi [9] approximated weakly fixed points of nonexpansive mappings and/or a nonspreading mapping in a Hilbert space using Moudafi’s iteration scheme [11]. Specifically, they proved the following result.Theorem 5.
LetH be a Hilbert space and let C be a nonempty closed convex subset of H. Let S be a nonspreading mapping on C into itself and let T be a nonexpansive mappings on C into itself such that Fix⁡(T)∩Fix⁡(S)≠∅. Define a sequence (xn)n∈N as follows:
(16)x1∈C,xn+1=(1-αn)xn+αn[βnSxn+(1-βn)Txn]
for all n∈N, where (αn)n∈N, (βn)n∈N, are in [0,1]. Then the following holds.(I)
ifliminf⁡αn(1-αn)>0 and ∑‍(1-βn)<∞, then (xn)n weakly converges to v∈Fix⁡(S).(II)
if∑‍αn(1-αn)=∞ and ∑‍βn<∞, then (xn)n weakly converges to v∈Fix⁡(T).(III)
ifliminf⁡αn(1-αn)>0 and liminf⁡βn(1-βn)>0, then (xn)n weakly converges to v∈Fix⁡(T)∩Fix⁡(S).In [12], the authors obtained strong convergence for the Halpern method by using type average mappings, with assumptions on the coefficients very similar to Theorem 5.So one can ask if this result holds for Moudafi’s viscosity method [13]. We cannot take advantage of using the above positive results on Halpern’s iteration and invoke Suzuki’s result [14] that affirms that Halpern’s approximation convergence implies Moudafi’s viscosity approximation convergence. Indeed, as proved by Suzuki, this is true for nonexpansive mappings not for nonspreading mappings.In spite of this we obtain the affirmative answer in our main result.Our proofs took inspiration by [5, 12, 15, 16]. Related papers in which there are not nonspreading but other types of mappings or semigroups of nonexpansive mappings are [17–23].


## 2. Main Results

In this section, we always will assume the following.(i)
H is a Hilbert space.(ii)
C is a closed and convex subset of H.(iii)
T:C→C is a nonexpansive mapping.(iv)
Tω:C→C is an average mapping of T, Tω=(1-ω)I+ωT.(v)
S:C→C is a nonspreading mapping.(vi)
Sω:C→C is a nonspreading-average mapping of S, Sω=(1-ω)I+ωS.(vii)
Un:C→C is a convex combination of Tω and Sω, Un=(1-βn)Sω+βnTω.(viii)
Fix⁡(T)∩Fix⁡(S)≠∅.(ix)
f:C→C is a ρ-contraction; that is, ∥f(x)-f(y)∥≤ρ∥x-y∥, 0<ρ<1.(x)
(αn)n∈N⊂[0,1] is a real sequence satisfying αn→0 and ∑n=1∞‍αn=∞.(xi)
O(1) denote any bounded real sequence (so O(1)+O(1)=O(1)).The following lemmas are the keys to obtain our main result.Lemma 6 (see [24]).
Assume that(an)n∈N is a sequence of nonnegative numbers such that
(17)an+1≤(1-γn)an+γnδn+σn,n≥0,
where (γn)n is a sequence in (0,1) and (δn)n is a sequence in R and (σn)n⊂R+ such that, (1)
∑n=1∞‍γn=∞;(2)
lim⁡sup⁡n→∞δn≤0 and ∑n=1∞‍|σn|<∞.Then, lim⁡n→∞an=0.Lemma 7.
Let(xn)n be the sequence defined by
(18)xn+1=αnf(xn)+(1-αn)Unxn.
Then, (i) Un is quasi nonexpansive; (ii) (xn)n,(Sxn)n,(Txn)n,(Sωxn)n,(Tωxn)n, and (Unxn)n are bounded sequences.Proof.
(i) Any convex combination of quasi nonexpansive mappings is, in turn, quasi nonexpansive. So isUn, since Tω and Sω are quasi nonexpansive (see Theorem 4, (i)).
(ii) We see that the boundedness of(xn)n follows by the quasi nonexpansivity of Un. For this let q∈Fix⁡(T)∩Fix⁡(S). Then
(19)∥xn+1-q∥≤αn∥f(xn)-q∥+(1-αn)∥Unxn-q∥≤αn∥f(xn)-f(q)∥+αn∥f(q)-q∥+(1-αn)∥Unxn-q∥≤αnρ∥xn-z∥+αn∥f(q)-q∥+(1-αn)∥xn-q∥≤(1-αn(1-ρ))∥xn-q∥+αn∥f(q)-q∥=(1-αn(1-ρ))∥xn-q∥+αn(1-ρ)∥f(q)-q∥1-ρ(byconvexityof∥·∥)≤max⁡{∥xn-q∥,∥f(q)-q∥1-ρ}⟹∥xn+1-q∥≤max⁡{∥x1-q∥,∥f(q)-q∥1-ρ}.
The boundedness of (xn)n is proved. The boundedness of the other sequences in (ii) follows by this last (since Fix⁡(T)∩Fix⁡(S)≠∅).Lemma 8.
Let(yn)n be a bounded sequence in C. Then one has the following. (i)
If∥yn-Tyn∥→0, then
(20)limsup⁡〈(I-f)p-,yn-p-〉≥0,
where p-=PFix⁡(T)f(p-) is the unique point in Fix⁡(T) that satisfies the variational inequality
(21)〈(I-f)p-,x-p-〉≥0,∀x∈Fix⁡(T).(ii)
If∥yn-Syn∥→0, then
(22)limsup⁡〈(I-f)p~,yn-p~〉≥0,
where p~=PFix⁡(S)f(p~) is the unique point in Fix⁡(S) that satisfies the variational inequality
(23)〈(I-f)p~,x-p~〉≥0,∀x∈Fix⁡(T).(iii)
If both∥yn-Syn∥→0 and ∥yn-Tyn∥→0, then
(24)limsup⁡〈(I-f)p0,yn-p0〉≥0,
where p0=PFix⁡(T)∩Fix⁡(S)f(p0) is the unique point in Fix⁡(T)∩Fix⁡(S) that satisfies the variational inequality
(25)〈(I-f)p0,x-p0〉≥0,∀x∈Fix⁡(T)∩Fix⁡(S).Proof.
(i) Letp- satisfy (21). Let (ynk) be a subsequence of (yn) for which
(26)limsup⁡n〈(I-f)p-,yn-p-〉=lim⁡k〈(I-f)p-,ynk-p-〉.
Select a subsequence (ynkj)j  of (ynk)k  such that ynkj⇀v (this, of course, is possible by boundedness of (yn)n). From the assumption ∥yn-Tyn∥→0 and demiclosedness of T (see [1]) we have v∈Fix⁡(T), and
(27)limsup⁡n〈(I-f)p-,yn-p-〉=lim⁡j〈(I-f)p-,ynk-p-〉=〈(I-f)p-,x-p-〉
so the claim follows by (21).
(ii) It follows as in (i) sinceS is demicloded too (see Theorem 4, (b)).
(iii) Select a subsequence(ynk)k of (yn)n such that
(28)limsup⁡n〈(I-f)p0,yn-p0〉=lim⁡k〈(I-f)p0,ynk-p0〉,
where p0 satisfies (25). Now select a subsequence (ynkj)j of (ynk)k such that ynkj⇀w. Then, by demiclosedness of both T and S, and by the hypotheses ∥yn-Tyn∥→0 and ∥yn-Syn∥→0, we obtain that Tw=Sw=w; that is, w∈Fix⁡(T)∩Fix⁡(S). So the claim follows by (25) and
(29)limsup⁡n〈(I-f)p0,yn-p0〉=lim⁡j〈(I-f)p0,ynk-p0〉=〈(I-f)p0,x-p0〉.Lemma 9 (see [6]).
LetC be a nonempty closed convex subset of H and let T:C→C be a nonexpansive mapping. Then (I-T):C→H is 1/2-inverse strongly monotone; that is,
(30)12∥(I-T)x-(I-T)y∥2≤〈x-y,(I-T)x-(I-T)y〉.Lemma 10 (Maingé [25]).
Let(τn)n be real sequence that has a subsequence (τnj) which satisfies τnj<τnj+1 for all j. Then the sequence of integers (δ(n))n defined by δ(n)=max⁡{k≤n:τk<τk+1} has the following properties:(1)
δ(n)≤δ(n+1);(2)
δ(n)→∞ as n→∞;(3)
τδ(n)<τδ(n)+1;(4)
τn<τδ(n)+1.Theorem 11.
Let(31)x1∈C,xn+1=αnf(xn)+(1-αn)[βnTωxn+(1-βn)Sωxn],n≥1
with (αn)n⊂(0,1),αn→0 and ∑n‍αn=∞. Then one has the following.(i)
If∑n‍(1-βn)<∞ and ∑n‍|αn-αn+1|<∞, then (xn) strongly converges to p-=PFix⁡(T)f(p-) that is the unique point in Fix⁡(T) that satisfies the variational inequality
(32)〈(I-f)p-,x-p-〉≥0,∀x∈Fix⁡(T).(ii)
If∑n‍βn<∞, then (xn)n converges strongly to p~=PFix⁡(S)f(p~) that is the unique point in Fix⁡(S) that satisfies the variational inequality
(33)〈(I-f)p~,x-p~〉≥0,∀x∈Fix⁡(S).(iii)
Ifliminf⁡nβn(1-βn)>0, then (xn) strongly converges to p0=PFix⁡(T)∩Fix⁡(S)f(p0) which is the unique point in Fix⁡(T)∩Fix⁡(S) that satisfies the variational inequality
(34)〈(I-f)p0,x-p0〉≥0,∀x∈Fix⁡(T)∩Fix⁡(S).Proof.
By Lemma7, we obtain that (xn)n is bounded.
Proof of (i).Let p- be as in (i) of Lemma 8; that is,(35)〈(I-f)p-,x-p-〉≥0,∀x∈Fix⁡(T).
Step
1.  One has limsup⁡n(∥xn-p-∥-∥xn+1-p-∥)=0.
Proof of Step
1.This immediately follows by the asymptotic regularity of (xn)n. So we prove that (xn) is asymptotically regular; that is, ∥xn-xn+1∥→0:
(36)xn+1-xn=αnf(xn)+(1-αn)Unxn-αn-1f(xn-1)-(1-αn-1)Un-1xn-1±(1-αn)Un-1xn-1=(1-αn)[Unxn-Un-1xn-1]+(αn-1-αn)Un-1xn-1-αn-1f(xn-1)+αnf(xn)-αnf(xn-1)+αnf(xn-1)=(1-αn)[Unxn-Un-1xn-1]+(αn-1-αn)(Un-1xn-1-f(xn-1))+αn(f(xn)-f(xn-1))=(αn-1-αn)(Un-1xn-1-f(xn-1))+αn(f(xn)-f(xn-1))+(1-αn)[βnTωxn+(1-βn)Sωxnhhhhhhhhhhh-βn-1Tωxn-1-(1-βn-1)Sωxn-1hhhhhhhhhhh±βnTωxn-1±(1-βn)Sωxn-1]=(αn-1-αn)(Un-1xn-1-f(xn-1))+αn(f(xn)-f(xn-1))+(1-αn)[βn(Tωxn-Tωxn-1)hhhhhhhhh+(1-βn)(Sωxn-Sωxn-1)hhhhhhhhhhh+(βn-βn-1)(Tωxn-1-Sωxn-1)].
So
(37)∥xn+1-xn∥≤|αn-1-αn|O(1)+αnρ∥xn-xn-1∥+(1-αn)[βn∥xn-xn-1∥+(1-βn)O(1)hhhhhhhhhh+|βn-βn-1|O(1)]=((1-αn)βn+αnρ)∥xn-xn-1∥+γn=1-[1-((1-αn)βn+αnρ)]∥xn-xn-1∥+γn,
where γn=|αn-αn-1|O(1)+(1-αn)(1-βn)O(1)+|βn-βn-1|O(1) is such that ∑n‍γn<∞, thanks to the assumptions ∑n‍|αn-αn-1|<∞ and ∑n‍(1-βn)<∞.
So if we putδn=1-βn+αnβn-αnρ we have
(38)∥xn+1-xn∥≤(1-δn)∥xn-1-xn∥+γn.
From the assumption ∑n‍αn=∞ we deduce immediately ∑n‍δn=∞. This is sufficient for Xu’s Lemma 6, to conclude that (xn)n is asymptotically regular.
Step
2. One has ∥xn-Tωxn∥→0,and∥xn-Txn∥=(1/ω)∥xn-Tωxn∥→0.
Proof of Step
2.We define an auxiliary sequence (zn)n by
(39)zn+1=αnf(xn)+(1-αn)Tωxn.
Observe that
(40)∥zn+1-Tωxn∥=αn∥f(xn)-Tωxn∥⟶0
and so
(41)∥zn+1-xn+1∥=∥αnf(xn)+(1-αn)Tωxnhhh-αnf(xn)-(1-αn)Unxn∥=(1-αn)∥Tωxn-Uxn∥=(1-αn)∥Tωxn-βnTωxn-(1-βn)Sωxn∥=(1-αn)(1-βn)∥Tωxn-Sωxn∥⟶0;
hence we get
(42)∑n‍∥xn-zn∥<∞.
Now
(43)∥zn+1-p-∥2=∥αnf(xn)+(1-αn)(1-ω)xnhhh+(1-αn)ωTxn-p-∥2=∥[(1-αn)ω(Txn-xn)+xn-p-]hhh+αn(f(xn)-xn)∥2hhi(bythewellknowninequalityhhhh∥x+y∥2≤∥x∥2+2〈y,x+y〉)≤∥(1-αn)ω(Txn-xn)+xn-p-∥2+2αn〈f(xn)-xn,zn+1-p-〉≤(1-αn)2ω2∥Txn-xn∥2+∥xn-p-∥2-2(1-αn)ω〈xn-p-,xn-Txn〉+2αn∥f(xn)-xn∥∥zn+1-p-∥≤(1-αn)2ω2∥Txn-xn∥2+∥xn-p-∥2-2(1-αn)ω〈xn-p-,(I-T)xn-(I-T)p-〉+2αn∥f(xn)-xn∥∥zn+1-p-∥(bytheinversestrongmonotonicityhhhof(I-T),Lemma9)≤(1-αn)2ω2∥Txn-xn∥2+∥xn-p-∥2-(1-αn)ω∥(I-T)xn-(I-T)p-∥2+2αn∥f(xn)-xn∥∥zn+1-p-∥=∥xn-p-∥2-(1-αn)ω[1-(1-αn)ω]×∥xn-Txn∥2+2αn∥f(xn)-xn∥∥zn+1-p-∥
and hence
(44)0≤(1-αn)ω[1-(1-αn)ω]∥xn-Txn∥2≤-∥zn+1-p-∥2+∥xn-p-∥2+2αn∥f(xn)-xn∥∥zn+1-p-∥.
Passing to lim⁡supn, the last member goes to zero thanks to Step 1, to boundedness of (xn)n and (41). So we obtain
(45)ω(1-ω)lim⁡n∥xn-Txn∥=0.
From this immediately we have also ∥zn-Tzn∥→0.
From Step 2 and Lemma8(i) we obtain
(46)limsup⁡n〈(I-f)p-,xn-p-〉≥0.
Moreover, from Step 2 and Lemma 8 we also have(47)limsup⁡n〈(I-f)p-,zn-p-〉≥0.Step3. One has zn→p-.
Proof of Step
3. By using the auxiliary sequence (zn)n, we can write xn as
(48)xn+1=zn+1+(1-βn)En,
where En=(1-αn)(Sωxn-Tωxn) is a bounded sequence and so
(49)∥xn-zn∥≤(1-βn)O(1),(50)∥zn+1-p-∥2=∥αn(f(xn)-p-)+(1-αn)(Tωxn-p-)∥2=(1-αn)2∥Tωxn-p-∥2+αn2∥f(xn)-p-∥2+2αn(1-αn)〈f(xn)-p-,Tωxn-p-〉≤(1-2αn+αn2)∥xn-p-∥2+αn2∥f(xn)-p-∥2+2αn(1-αn)〈f(xn)-f(p-),Tωxn-p-〉+2αn(1-αn)〈f(p-)-p-,Tωxn-p-〉≤(1-2αn+αn2)∥xn-p-∥2+αn2∥f(xn)-p-∥2+2αn(1-αn)ρ∥xn-p-∥2+2αn(1-αn)〈f(p-)-p-,Tωxn-xn〉+2αn(1-αn)〈f(p-)-p-,xn-p-〉≤[1-2αn+αn2+2αn(1-αn)ρ]×(∥xn-zn∥+∥zn-p-∥)2+2αn(1-αn)〈f(p-)-p-,Tωxn-xn〉+2αn(1-αn)〈f(p-)-p-,xn-p-〉+αn2∥f(xn)-p-∥2=[1-2αn+αn2+2αn(1-αn)ρ]∥zn-p-∥2+(1-βn)O(1)+2αn(1-αn)〈f(p-)-p-,Tωxn-xn〉×2αn(1-αn)〈f(p-)-p-,xn-p-〉+αn2∥f(xn)-p-∥2.
So putting δn=αn[2-αn-2ρ(1-αn)], γn=(1-βn)O(1), and
(51)σn=2(1-αn)〈f(p-)-p-,Tωxn-xn〉+2(1-αn)〈f(p-)-p-,xn-p-〉+αn∥f(xn)-p-∥2
one has easily that δn∈(0,1), ∑n‍δn=∞, ∑n‍γn<∞, and
(52)limsup⁡nσn=limsup⁡n〈xn-p-,f(p-)-p-〉≤0.
Thus, we can rewrite (50) as
(53)∥zn+1-p-∥2≤(1-δn)∥zn-p-∥2+αnσn+γn.
This is sufficient, for Xu’s Lemma 6, to conclude that zn→p-. Lastly, by (49) immediately follows xn→p-.
Proof of (ii). Rewrite xn+1 as
(54)xn+1=αnf(xn)+(1-αn)Sωxn+βnEn,
where En=(1-αn)(Tωxn+Sωxn) is bounded (i.e., ∥En∥≤O(1)).
Now,(55)∥xn+1-p~∥2=∥αnf(xn)+(1-αn)(1-ω)xnh+(1-αn)ωSxn+βEn-p~∥2=∥[(1-αn)ω(Sxn-xn)+xn-p~]hhh+[αn(f(xn)-xn)+βnEn]∥2(bythewellknowninequalityhhh∥x+y∥2≤∥x∥2+2〈y,x+y〉)≤∥(1-αn)ω(Sxn-xn)+xn-p~∥2+2αn〈f(xn)-xn,xn+1-p~〉+βnO(1)≤(1-αn)2ω2∥Sxn-xn∥2+∥xn-p~∥2-2(1-αn)ω〈xn-Sxn,xn-p~〉+2αn∥f(xn)-xn∥∥xn+1-p~∥+βnO(1)(by(c)ofTheorem4)≤(1-αn)2ω2∥Sxn-xn∥2+∥xn-p~∥2-2(1-αn)ω[∥xn-Sxn∥2-12∥xn-Sxn∥2]+2αn∥f(xn)-xn∥∥xn+1-p~∥+βnO(1)=∥xn-p~∥2+[(1-αn)2ω2-(1-αn)ω]∥xn-Sxn∥2+2αn∥f(xn)-xn∥∥xn+1-p~∥+βnO(1)=∥xn-p~∥2-(1-αn)ω[1-(1-αn)ω]∥xn-Sxn∥2+2αn∥f(xn)-xn∥∥xn+1-p~∥+βnO(1)
and hence
(56)(1-αn)ω[1-(1-αn)ω]∥xn-Sxn∥2-2αn∥f(xn)-xn∥∥xn+1-p~∥-βnO(1)≤∥xn-p~∥2-∥xn+1-p~∥2.
Now we distinguish two alternatives.
Alternative
1.
(
∥
x
n
-
p
~
∥
)
n is definitively nonincreasing.
Then there existslim⁡n∥xn-p~∥2 and so, passing to the lim⁡sup⁡ in (56), we obtain
(57)∥xn-Sxn∥⟶0,∥xn-Sωxn∥=ω∥xn-Sxn∥⟶0.
By Lemma 8(ii) it follows that
(58)〈(I-f)p~,x-p~〉≥0.
So
(59)∥xn+1-p~∥2=∥αn(f(xn)-p~)hhh+(1-αn)(Sωxn-p~)+βnEn∥2≤∥αn(f(xn)-p~)+(1-αn)(Sωxn-p~)∥2+βnO(1)=(1-αn)2∥Sωxn-p~∥2+αn2O(1)+2αn(1-αn)〈f(xn)-p~,Sωxn-p~〉+βnO(1)(bythequasi-nonexpansivityofSω)≤(1-2αn+αn2)∥xn-p~∥2+αn2O(1)+βnO(1)+2αn(1-αn)〈f(xn)-f(p~),Sωxn-p~〉+2αn(1-αn)〈f(p~)-p~,Sωxn-p~〉≤(1-2αn+αn2+2αn(1-αn)ρ)∥xn-p~∥2+αn2O(1)+βnO(1)+2αn(1-αn)〈f(p~)-p~,Sωxn-xn〉+2αn(1-αn)〈f(p~)-p~,xn-p~〉=(1-2αn+αn2+2αn(1-αn)ρ)∥xn-p~∥2+αn2O(1)+αn[∥Sωxn-xn∥O(1)hhhhhhhhhhhh+(1-α)〈f(p~)-p~,xn-p~〉]+βnO(1).
So, as in Step 1, thanks to (57), (58), and Xu’s Lemma, we obtain xn→p~.
Alternative
2.
(
∥
x
n
-
p
~
∥
)
n is not definitively nonincreasing.
This means that there exists a subsequence(xnk)k such that
(60)∥xnk-p~∥2≤∥xnk+1-p~∥2.
Then, thanks to Maingé’s Lemma, we know that there exists a sequence of integers (δ(n))n that satisfies the following.
(i)δ(n) is nondecreasing, (ii) δ(n)→∞, (iii) ∥xδ(n)-p~∥2<∥xδ(n)+1-p~∥2, and (iv) ∥xn-p~∥<∥xδ(n)+1-p~∥, for all n≥n1.
Consequently,(61)0≤liminf⁡n∥xδ(n)+1-p~∥-∥xδ(n)-p~∥≤limsup⁡n∥xδ(n)+1-p~∥-∥xδ(n)-p~∥≤limsup⁡n∥xn+1-p~∥-∥xn-p~∥≤limsup⁡n∥αn(f(xn)-Sωxn)+(Sωxn-p~)+βnEn∥-∥xn-p~∥(bythequasi-nonexpansivityofSω)≤limsup⁡αnO(1)+∥xn-p~∥+βnO(1)-∥xn-p~∥=0.
So
(62)lim⁡n(∥xδ(n)+1-p~∥-∥xn-p~∥)=0.
If we rewrite (56) as
(63)(1-αn)ω[1-(1-αn)ω]∥xn-Sxn∥2≤αnO(1)+βnO(1)+∥xn-p~∥2-∥xn+1-p~∥2≤αnO(1)+βnO(1)+(∥xn-p~∥-∥xn+1-p~∥)O(1),
then (62) implies that
(64)∥xδ(n)-Sxδ(n)∥⟶0
and this, in turn, by using Lemma 8(ii), means that
(65)limsup⁡〈(I-f)p~,yn-p~〉≥0.
At this point it is clear that we can continue as in Alternative 1 and we obtain ∥xδ(n)-p~∥→0.
Then (62) furnishes
(66)∥xδ(n)+1-p~∥⟶0,
and finally by property (iv) of Maingé’s Lemma, that is, ∥xn-p~∥<∥xδ(n)+1-p~∥, for all n≥n1, we point out that xn→p~.
Proof of (iii). Let be p0 as in (iii) of Lemma 8; that is,
(67)〈(I-f)p0,x-p0〉≥0,∀x∈Fix⁡(T)∩Fix⁡(S).
Now,
(68)∥Unxn-p0∥2=∥βn(Tωxn-p0)+(1-βn)(Sωxn-p0)∥2=βn∥Tωxn-p0∥2+(1-βn)∥Sωxn-p0∥2-βn(1-βn)∥Tωxn-Sωxn∥2(bythequasi-nonexpansivityofTω,khhbyquasifirmlynonexpansivityhhhofSωandTheorem4(i))≤βn∥xn-p0∥2+(1-βn)∥xn-p0∥2-(1-βn)(1-ω)∥xn-Sωxn∥2-βn(1-βn)∥Tωxn-Sωxn∥2(69)=∥xn-p0∥2-(1-βn)(1-ω)∥xn-Sωxn∥2-βn(1-βn)∥Tωxn-Sωxn∥2
so
(70)∥xn+1-p0∥2=∥αn(f(xn)-p0)h+(1-αn)(Unxn-p0)∥2≤∥Unxn-p0∥2+αn(αnO(1)+O(1))≤by(68)≤∥xn-p0∥2-(1-βn)(1-ω)∥xn-Sωxn∥2-βn(1-βn)∥Tωxn-Sωxn∥2.
From this we derive the following inequalities:
(71)(1-βn)(1-ω)∥xn-Sωxn∥2≤∥xn-p0∥2-∥xn+1-p0∥+αnO(1),βn(1-βn)∥Tωxn-Sωxn∥2≤∥xn-p0∥2-∥xn+1-p0∥2+αnO(1).
Now, also here consider two alternatives.
Alternative
1.
(
∥
x
n
-
p
0
∥
)
n is definitively nonincreasing.
Then there existlim⁡∥xn-p0∥ and (|∥xn+1-p0∥-∥xn-p0∥|)→0.
So, passing to thelim⁡sup⁡ in (71), the assumption lim⁡infnβn(1-βn)>0 yields:
(72)∥xn-Sωxn∥⟶0,∥Sωxn-Tωxn∥⟶0.
Moreover, xn-Sωxn=ω(xn-Sxn) so,
(73)∥xn-Sxn∥⟶0
and Tωxn-Sωxn=ω(Txn-Sxn) so
(74)∥Txn-Sxn∥⟶0
so that from xn-Txn=xn-Sxn+Sxn-Txn it follows at once that
(75)∥xn-Txn∥⟶0.
From Lemma 8(iii) we obtain
(76)limsup⁡〈(I-f)p0,xn-p0〉≥0.
Further, from Unxn-xn=βn(Tωxn-xn)+(1-βn)(Sωxn-xn), we get
(77)∥Unxn-xn∥⟶0.
Now we are able to show that xn→p0.
Indeed,(78)∥xn+1-p0∥2=∥(1-αn)(Unxn-p0)h+αn(f(xn)-p0)∥2=(1-αn)2∥Unxn-p0∥2+αn2∥f(xn)-p0∥2+2(1-αn)αn〈Unxn-p0,f(xn)-p0〉≤(1-2αn+αn2)∥xn-p0∥2+αn2∥f(xn)-p0∥2+2(1-αn)αn〈Unxn-p0,f(xn)-f(p0)〉+2(1-αn)αn〈Unxn-p0,f(p0)-p0〉≤[1-2αn+αn2+2αn(1-αn)ρ]∥xn-p0∥2+αn2∥f(xn)-p0∥2+2(1-αn)αn〈Unxn-xn,f(p0)-p0〉+2(1-αn)αn〈xn-p0,f(p0)-p0〉.
So, put δn=αn[2-αn-2ρ(1-αn)], σn=αn∥f(xn)-p0∥2  +  2(1-αn)〈Unxn-xn,f(p0)-p0〉  +  2(1-αn)〈xn-p0,f(p0)-p0〉 one easily has δn∈(0,1),   ∑n‍δn=∞, limsup⁡nσn  =  limsup⁡n〈xn-p0  ,  f(p0)-p0〉  ≤  0, and
(79)∥xn+1-p0∥2≤(1-δn)∥xn-p0∥2+αnσn.
This is sufficient, for Xu’s Lemma 6, to ensure that xn→p0.
Alternative
2.
(
∥
x
n
-
p
0
∥
)
n is not definitively nonincreasing.
This means that there exists a subsequence(∥xnk-p0∥)k such that
(80)∥xnk-p0∥≤∥xnk+1-p0∥.
Then, thanks to Maingé’s Lemma, we know that there exists a sequence of integers (δn) that satisfies the following.
(i)δ(n) is nondecreasing, (ii) δ(n)→∞, (iii) ∥xδ(n)-p0∥2<∥xδ(n)+1-p0∥2, and (iv) ∥xn-p0∥  <  ∥xδ(n)+1-p0∥2, for all n≥n1.
Consequently,(81)0≤liminf⁡n∥xδ(n)+1-p0∥-∥xδ(n)-p0∥≤limsup⁡n∥xδ(n)+1-p0∥-∥xδ(n)-p0∥≤limsup⁡n∥xn+1-p~∥-∥xn-p~∥≤limsup⁡n[∥αn(f(xn)-p0)+(1-αn)(Unxn-p0)∥hhhhhhhh-∥xn-p0∥∥αn(f(xn)-p0+(1-αn)(Unxn-p0)∥]≤limsup⁡αn[∥f(xn)-p0∥-∥xn-p0∥]=0.
Hence
(82)lim⁡n(∥xδ(n)+1-p0∥-∥xδ(n)-p0∥)=0.
So, passing to limsup⁡ on δ(n) in (71), one obtains, as in the Alternative 1,
(83)∥xδ(n)-Sωxδ(n)∥⟶0,∥Tωxδ(n)-Sωxδ(n)∥⟶0,∥xδ(n)-Sxδ(n)∥⟶0,∥Txδ(n)-Sxδ(n)∥⟶0,∥xδ(n)-Txδ(n)∥⟶0,∥Uδ(n)xδ(n)-xδ(n)∥⟶0.
Again from Lemma 8(iii) it follows that
(84)limsup⁡〈(I-f)p0,xδ(n)-p0〉≥0.
Following the reasoning of Alternative 1, one obtains that
(85)∥xδ(n)-p0∥⟶0.
Then (82) furnishes ∥xδ(n)+1-p0∥→0 and finally by the property (iv) of Maingé Lemma; that is,
(86)∥xn-p0∥<∥xδ(n)+1-p0∥,
we obtain xn→p0 as required.Remark 12.
The main result of this paper contains as a particular case the positive answer to the question raised by Kurokawa and Takahashi page  1567 in [26].Remark 13.
Our reasoning, different from that of Tian and Jin [27] and Deng et al. [28], has allowed us to prove our results without having ω<1/2.



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*Source: 152530-2014-07-09.xml*