# On Mann’s Type Method for Nonexpansive and Strongly Quasinonexpansive Mappings in Hilbert Spaces

Authors: Nawab Hussain; Giuseppe Marino; Badriah A. S. Alamri
Journal: Journal of Function Spaces
 (2015)

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## Abstract

In the setting of Hilbert spaces, we study Mann’s type method to approximate strong solutions of variational inequalities. We show that these solutions are fixed points of a nonexpansive mapping and/or a strongly quasinonexpansive mapping, depending on the coefficients involved in the algorithm.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and induce a norm ·. Let T:H→H be a nonlinear mapping. Fix(T)≔{x∈H:x=Tx} denotes the fixed point set of T.The problem of approximating fixed points of nonlinear mapping has been widely investigated by many authors.The starting point of the present paper is the following scheme introduced by Iemoto and Takahashi in [1]: (1)xn+1=1-αnxn+αnβnTxn+1-βnSxn,where T is a nonexpansive mapping and S is a nonspreading mapping.Depending on the coefficientsαn and βn, in [1] was obtained weak convergence of the method either to a fixed point of T or to a fixed point of S or to a common fixed point of T and S.Motivated by this result, with the aim of obtaining strong convergence, in [2] we introduced a viscosity in a previous scheme and we have been able to prove the following result, in which the most important role was played by the average type mappings Tδ=(1-δ)I+δT and Sδ=(1-δ)I+δS, δ∈(0,1).Theorem 1 (see [2]).
LetT:H→H be a nonexpansive mapping and let S:H→H be a nonspreading mapping. Let (2)x1∈C,xn+1=αnfxn+1-αnβnTωxn+1-βnSωxn,n≥1,with (αn)n⊂(0,1), αn→0, and ∑n‍αn=∞. Then(i)
if∑n‍(1-βn)<∞, ∑n‍|αn-αn+1|<∞, and Fix(T)≠∅, then (xn) strongly converges to p¯=PFix(T)f(p¯) which is the unique point in Fix(T) that satisfies the variational inequality(3)I-fp¯,x-p¯≥0,∀x∈FixT;(ii)
if∑n‍βn<∞ and Fix(S)≠∅, then (xn)n converges strongly to p~=PFix(S)f(p~) which is the unique point in Fix(S) that satisfies the variational inequality(4)I-fp~,x-p~≥0,∀x∈FixS;(iii)
ifliminfnβn(1-βn)>0 and Fix(T)∩Fix(S)≠∅, 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(5)I-fp0,x-p0≥0,∀x∈FixT∩FixS.At this point, three questions can be posed:(1)
Is it possible to replace the coefficient(1-αn) with the term (I-αnA) where A:H→H is a linear, bounded, strongly positive operator? In doing so, we would be able to solve some minimization problem.(2)
Is it possible to replace the type average mappingsTδ and Sδ with the original mappings T and S?(3)
Is it possible to replace the nonspreading mappingS with a more general type of mappings?We will see here that the answers are all positive.A natural class of mappings containing the nonspreading mappings is given by the class of hybrid mappings. Following Aoyama et al. [3] we say that S:H→H is an L-hybrid mapping, with L≥0, if (6)Sx-Sy2≤x-y2+Lx-Sx,y-Sy∀x,y∈Hor equivalently (7)2Sx-Sy2≤x-Sy2+y-Sx2-21-L2x-Sx,y-Sy.We denote byHL the family of L-hybrid mappings defined on the space H.Note that particular choices ofL give important families of nonlinear mappings:(i)
H0 is the family of nonexpansive mappings (i.e., Sx-Sy≤x,y for all x,y∈H).(ii)
H2 is the family of nonspreading mappings (i.e., Sx-Sy2≤x-y2+2〈x-Sx,y-Sy〉, ∀x,y∈H).Relevant properties ofL-hybrid mappings are the following:(HL-1)
IfS∈HL and Fix(S)≠∅, then S is quasinonexpansive (i.e., Tx-p≤x-p for all p∈Fix(T), for all x∈H) (the proof is immediate by the definition).(HL-2)
IfS∈HL⇒I-S is demiclosed in 0∈H (for the proof see [1]).(HL-3)
IfS∈HL⇒Fix(S) is a closed and convex set (for the proof, see [4]).(HL-4)
IfS∈HL⇒Sδ∈HL/δ, where Sδ is the type average mapping (8)Sδ=1-δI+δS(for the proof, see [5]).A first improvement of Theorem1 [2] that one can obtain is to replace S with an L-hybrid mapping. (This will be a corollary of our main result, thanks to (HL-1), (HL-3), and next Proposition 2.)But we can do better. Inspired by the paper of Wongchan and Saejung [6], we can consider a mapping S that has only the properties of being strongly quasinonexpansive and demiclosed.We recall that the concept of strongly nonexpansive mapping was introduced by Bruck and Reich in 1977 [7] as follows: a mapping T is said to be strongly nonexpansive if T is nonexpansive and whenever (xn-yn) is bounded and xn-yn-Txn-Ty→0, it follows that (xn-yn)-(Txn-Tyn)→0.To our knowledge, Saejung [8] in 2010 introduced the concept of strong quasi-nonexpansivity: a mapping S is said strongly quasi-nonexpansive if Fix(S)≠∅, S is quasinonexpansive and xn-Sxn→0 whenever (xn) is a bounded sequence such that xn-p-Sxn-p→0 for some p∈Fix(S).In [7] was proved that an average mapping defined on a uniformly convex Banach space is strongly nonexpansive.Following the same line on the proof, one can show that a type average mappingSδ=(1-δ)I+δS of a quasinonexpansive mapping is strongly quasinonexpansive. For the sake of completeness, we propose here the following proof, lightly different, valid in Hilbert spaces.Proposition 2.
LetS:C→H be a quasinonexpansive mapping, where C is a closed convex subset of H. Then the type average mapping Sδ=(1-δ)I+δS is a strongly quasinonexpansive mapping.Proof.
Suppose(xn)⊂C is a bounded sequence such that xn-p-Sδxn-p→0 for some p∈Fix(S). Now (9)Sδxn-p=1-δxn+δSxn-p≤1-δxn-p+δSxn-p⟹xn-p-Sδxn-p≥δxn-p-Sxn-p.So(10)by  quasinonexpansivity  ofS0≤xn-p-Sxn-p≤1δxn-p-Sδxn-p⟶0⟹xn-p-Sxn-p⟶0too.Now, let (xnk-p) be a subsequence of (xn-p) that admits limit, xnk-p→d, as k→∞. Then, from the assumption that xn-p-Sδxn-p→0, it follows that Sδxnk-p→d as k→∞; that is, (1-δ)(xn-p)+δ(Sxn-p)→d. But then(11)
d
2
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.By the boundness of the sequence (xn), this is sufficient, by the standard argument of compactness, to ensure xn-Sδxn→0 as n→∞.So, by the quasinonexpansivity ofL-hybrid mappings, it follows that in Theorem 1 [2] the assumption that S is a strongly quasinonexpansive mapping instead of type average nonspreading (or also L-hybrid) mapping is a better assumption.Of course, one can ask if some importantL-hybrid mapping, as a nonspreading mapping, or a nonexpansive mapping, is already strongly quasinonexpansive. This is not always true, as shown in the following proposition.Proposition 3.
There exist nonexpansive mappings that are not strongly quasinonexpansive. Moreover, there exist nonspreading mappings that are not strongly quasinonexpansive.Proof.
LetT:H→H be such that Tx=-x. Then T is nonexpansive but not strongly quasinonexpansive.
Moreover letX=A∪B∪C⊂H, where (12)A=x∈H:x≤1,B=x∈H:1<x<2,C=x∈H:2≤x≤3.Define S:X→X by(13)Sx≔x,ifx∈A,xx,ifx∈B,0,ifx∈C.One can see that S is a nonspreading mapping, distinguishing three cases: (x∈A, y∈B), (x∈A, y∈C), and (x∈B, y∈C).
To see thatS is not strongly quasinonexpansive, take x0 with x0=1.
Thenx0∈Fix(S). Moreover, define zn=2+(1/n)x0. Then Szn=0 and zn-x0-Szn-x0=1+(1/n)-1→0 but Szn-zn=(2+(1/n))x0→2x0.Conversely, we have the following.Proposition 4.
There exist strongly quasinonexpansive mappings that are notL-hybrid mappings for any L (and so, thanks to (HL-4), that are not ever type average Sδ with SL-hybrid).Proof.
LetH=R. Define (14)T0=0,Tn=-n+1n+1,∀n∈N,n>0,T-n=n-1n+1,∀n∈N,n>0.Then define in linear way T on each interval [n,n+1], n∈Z.
One can see easily thatFix(T)=0 and T is strongly quasinonexpansive.
Moreover, from the fact that, for largen∈Z, T is definedalmost as -I, one can prove that can not be L-hybrid for any L≥0.Finally, we show that the hypothesis on demiclosedness ofI-T is necessary in Theorem  2.3 (the main Theorem) of [6].Proposition 5.
There exist strongly quasinonexpansive mappings such that(I-S) are not demiclosed in 0.Proof.
LetH=R. Define S:[0,1]→[0,1] by(15)Sx≔x2,if0≤x<1,12,ifx=1.Then Fix(S)={0}, S is strongly quasinonexpansive, but I-S is not demiclosed on 0 (xn=1-(1/n) is such that xn→1 but xn-Sxn→0).Reassuming, the main result of [6] has allowed us to improve Theorem 1 [2] with an emphasis on the concept of strong quasinonexpansivity instead of nonspreading.Our reasoning is inspired by the ideas contained in [2, 6, 9].The solid bases on which our proof rests are given by the following lemmas.Lemma 6 (Xu’s Lemma [10]).
Assume(an)n∈N is a sequence of nonnegative numbers such that (16)an+1≤1-snan+snσn+γn,n≥0,where (sn)n is a sequence in [0,1] and (δn)n is a sequence in R such that (1)
∑n=1∞‍sn=∞;(2)
limsupn→∞σn≤0;(3)
γn≥0, ∑n=1∞‍γn<∞.Then limn→∞an=0.Lemma 7 (Maingé’s Lemma [11]).
Let(γn)n be real sequence that has a subsequence (γnj) which satisfies γnj<γnj+1 for all j. Then there exists an increasing sequence of integers (τ(n))n≥n0 satisfying(1)
limnτn=+∞;(2)
γτ(n)<γτ(n)+1, for all n≥n0;(3)
γn<γτ(n)+1, for all n≥n0.


## 2. Main Results

In all this section we denote byO(1) any bounded real sequence (so, e.g., O(1)+O(1)=O(1)).Theorem 8.
LetH be a Hilbert space. Suppose (i)
T:H→H is a nonexpansive mapping and S:H→H is a strongly quasinonexpansive mapping such that I-S is demiclosed in 0, for which Fix(T)∩Fix(S)≠∅;(ii)
A:H→H is a linear bounded self-adjoint γ¯- strongly positive operator, γ¯<1;(iii)
f:H→H is a contraction, with ρ<γ¯ (of course this is not a real restriction, but this is useful for convenience of notations);(iv)
(αn)n,(βn)n are two real sequences in [0,1].Consider the iteration scheme(17)x1∈C,xn+1=αnfxn+I-αnAβnTxn+1-βnSxn,n≥1.Then(1)
if∑n‍(1-βn)<∞, αn→0, ∑n‍αn=∞, and ∑n‍|αn-αn+1|<∞, then (xn) strongly converges to p¯∈Fix(T) which is the unique point in Fix(T) that solves the variational inequality(18)A-fp¯,x-p¯≥0,∀x∈FixT,which is the optimality condition for the minimization problem(19)minx∈FixT⁡12Ax,x-hx,whereh is a potential function for f (i.e., h′(x)=f(x), for all x∈H);(2)
if∑n‍βn<∞, αn→0, ∑n‍αn=+∞, and βn/αn→0, then (xn)n converges strongly to p~∈Fix(S) which is the unique solution in Fix(S) of the variational inequality(20)A-fp~,x-p~≥0,∀x∈FixS;(3)
ifliminfnβn(1-βn)>0, αn→0, and ∑n‍αn=+∞, then (xn) strongly converges to p0∈Fix(T)∩Fix(S) which is the unique solution in Fix(T)∩Fix(S) of the variational inequality (21)A-fp0,x-p0≥0,∀x∈FixT∩FixS.Proof.
First of all, we see that(xn) is a bounded sequence.
Indeed, letUn=βnT+(1-βn)S and z∈Fix(T)∩Fix(S). Then xn+1-z=αnf(xn)+(I-αnA)Unxn-p (by linearity of A) =(I-αnA)(Unxn-z)+αn(f(z)-Az)-αn(f(z)-f(xn))≤ (since I-αnA≤1-αnγ¯, [9]) ≤(1-αnγ¯+αnρ)xn-z+αnfz-Az=[1-αn(γ¯-ρ)]xn-z+αn(γ¯-ρ)(f(z)-Az/γ¯ρ) (from convexity) ≤maxxn-z,f(z)-Az/γ¯ρ (from induction) ≤maxx1-z,f(z)-A(z)/γ¯ρ.
Then(xn) is bounded.
Moreover(22)xn+1-Unxn=αnfxn-AUnxn⟶0,n⟶∞,since αn→0.
Proof of (1). The the key is to prove that (23)xn+1-xn⟶0.To see this, we calculate (24)
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by  nonexpansivity  of
T
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,where sn=1-βn+αn(γ¯βn-ρ)≥αnO(1); definitively, γn=|αn-1-αn|O(1)+[(1-αnγ¯)(1-βn)+|βn-βn-1|]O(1).
Thanks to hypotheses onαn,βn, we see that sn→0, ∑n‍sn=+∞, and ∑n‍γn<∞. This is sufficient, from Xu’s Lemma, to conclude xn+1-xn→0.
From this and (22), it follows immediately that(25)xn-Unxn⟶0.Indeed, xn-Unxn=(xn-xn+1)-(xn+1-Unxn).
Moreover(26)xn-Unxn=xn-βnTxn-1-βnSxn≥xn-βnTxn-1-βnSxn⟹xn-βnTxn≤xn-Unxn+1-βnO1,and so from (25) and hypotheses ∑n‍(1-βn)<∞, we have also (27)xn-βnTxn⟶0.From this we deduce also that(28)xn-Txn⟶0,and this gives that any weak limit of (xn) is in Fix(T), since T is nonexpansive, and thus the Principle of Demiclosedness is satisfied.
Now we can show thatxn→p¯, where p¯ is the unique solution in Fix(T) of variational inequality (18). We show first that(29)limsupn⁡Unxn-p¯,fp¯-Ap¯≤0.Indeed, let (xnk) be a subsequence of (xn) such that(30)limsupn⁡xn-p¯,fp¯-Ap¯=limk⁡xnk-p¯,fp¯-Ap¯,and xnk⇀z. Then z∈Fix(T) and so, from (30), (31)limsupn⁡xn-p¯,fp¯-Ap¯=z-p¯,fp¯-Ap¯,and this is nonpositive by definition of p¯.
So,(32)
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Finally(33)xn+1-p¯=I-αnAUnxn-p¯+αnfxn-Ap¯2≤1-αnγ¯2Unxn-p¯2+αn2fxn-Ap¯2+2αnI-αnAUnxn-p¯,fxn-Ap¯≤1-αnγ¯2βnTxn-p¯+1-βnSxn-p¯2+αn2fxn-Ap¯2+2αnUnxn-p¯,fxn-Ap¯-2αn2AUnxn-p¯,fxn-Ap¯≤1-αnγ¯2βnxn-p¯+1-βnO12+αn2O1+2αnUnxn-p¯,fxn-Ap¯+2αn2O1≤1-αnγ¯2xn-p¯2+1-βnO1+αn2O1+2αnUnxn-p¯,fxn-fp¯+2αnUnxn-p¯,fp¯-Ap¯≤1-αnγ¯2xn-p¯2+1-βnO1+αn2O1+2αnUnxn-p¯+fp¯-Ap¯+2αnβnxn-p¯+1-βnO1ρxn-p¯≤1-αnγ¯2+2αnρxn-p¯2+αnαnO1+2Unxn-p¯,fp¯-Ap¯+1-βnO1=1-snxn-p¯2+αnσn+γn,where (34)sn=1-1-αnγ¯2-2αnρ=1-1-αn2γ¯2+2αnγ¯-2αnρ=αnαnγ¯2+2γ¯-ρ,σn=αnO1+2Unxn-p¯,fp¯-Ap¯,γn=1-βnO1.Thanks to the hypotheses on αn,βn and (29), from Xu’s Lemma again, we obtain xn→p¯.
Proof of (2). Let p~ be the unique solution of variational inequality (20).
We want to show thatxn→p~. Now,(35)xn+1-p~2=αnfxn+I-αnAUnxn-p~+αnp~-αnp~+αnAz-αnAz2=αnfxn-Az+I-αnAUnxn-p~2by  the  subdifferential  inequalityx+y2≤x2+2y,x+y≤1-αnγ¯2Unxn-p~2+2αnfxn-Ap~,xn+1-p~=1-αnγ¯2βnTxn-p~+1-βnSxn-p~2+2αnfxn-fp~,xn+1-p~+2αnfp~-Ap~,xn+1-p~by  the  quasinonexpansivity  ofS≤1-αnγ¯2xn-p~2+βnO1+2αnρxn-p~xn+1-p~+2αnxn+1-p~,fp~-Ap~.At this point we distinguish two cases: or the sequence xn-p~ is definitively not increasing or not.
Alternative 1. (xn-p~) is definitively not increasing, so xn+1-p~≤xn-p~, ∀n≥N.
Then (35) furnish(36)xn+1-p~2≤1-αnγ¯2+2αnρxn-p~+βnO1+2αnxn+1-p~-fp~-Ap~.Putting sn=1-(1-αnγ¯)2-2αnρ=αnγ¯2-2αnγ¯+2αnρ=αn[αnγ¯2+2γ¯-ρ]≥0, σn=〈xn+1-p~,f(p~)-Ap~〉, and γn=βnO(1), we can rewrite (36) as(37)xn+1-p~2≤1-snxn-p~2+αnσn+γn,so the thesis xn→p~ will follow again by Xu’s Lemma if we are able to show that(38)limsupn⁡xn+1-p~,fp~-Ap~≤0.(Note that until now we have not used the hypothesis of strong quasinonexpansivity of S.) Now, since (xn-p~) is definitively not increasing, there exists the limnx-n-p~. Then (39)
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by  hypothesis
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=
0
.Thus(40)limn⁡Sxn-p~-xn-p~=0.From the strong quasinonexpansivity of S, we deduce(41)Sxn-xn⟶0.At this point, by using the demiclosedness of (I-S) in 0, we can proceed as in the Proof of (1) to show (38).
The statement is proved when Alternative 1 holds.
Alternative 2. (xn-p~) is not definitively not increasing; that is, there exists a subsequence (xnj-p~) such that(42)xnj-p~<xnj+1-p~,∀j∈N.From Maingé’s Lemma, it follows that there exists an increasing sequence of integers (τ(n))n∈⋉ satisfying (43)limnτn=+∞,xτn-p~<xτn+1-p~,(44)xn-p~<xτn-p~,∀n≥n0.Then (45)0≤liminfn⁡xτn+1-p~-xτn-p~.Retracing the same inequalities used to have (41) with τ(n) instead of n, we obtain (46)limnSxτn-p~-xτn-p~=0.Again the strong quasinonexpansivity yields(47)Sxτn-xτn⟶0,And, from the demiclosedness of I-S in 0, we deduce as above that(48)limsupn⁡xτn+1-p~,fp~-Ap~≤0.Incidentally, we observe that (49)xτn+1-xτn=ατnfxτn-Axτn+1-ατnγ¯βτnO1+1-ατnγ¯Sxτn-xτn,and so, from (47), it follows also that xτ(n)+1-xτ(n)→0.
Now rewrite (35) with τ(n) instead of n: (50)xτn+1-p~≤1-ατnγ¯2xτn-p~2+2ατnρxτn-p~xτn+1-p~+2ατnxτn+1-p~,fp~-Ap~+βτnO1by  property43≤1-ατnγ¯2+2ατnρxτn+1-p~2+2ατnxτn+1-p~,fp~-Ap~+βτnO1⟹2ατnγ¯-ρxτn+1-p~2≤ατnγ¯2xτn+1-p~2+2ατnxτn+1-p~,fp~-Ap~+βτnO1⟹dividing by ατ(n)(51)0≤2γ¯-ρxτn+1-p~2≤ατnγ¯2xτn+1-p~2+2xτn+1-p~,fp~-Ap~+βτnατnO1.Passing to limsup and recalling the hypothesis βn/αn→0 and (48), we obtain limnxτ(n)-p~≤limnxτ(n)+1-p~=0. Equation (44) ensures that also xn→p~.
Proof of (3). Let p0 be the unique point in Fix(S)∩F that satisfies variational inequality (21). Then (52)Unxn-p02=βnTxn-p0+1-βnSxn-p02=βnTxn-p02+1-βnSxn-p02-βn1-βnTxn-Sxn2≤xn-p02-βn1-βnTxn-Sxn2⟹xn+1-p02=αnfxn-Ap0+I-αnAUnxn-p02≤1-αnγ¯2Unxn-p02+αn2O1+αnO1≤xn-p02-βn1-βnTxn-Sxn2+αn2O1+αnO1.Also now we distinguish two cases.
Alternative 1. (xn-p0) is definitively not increasing, xn+1-p0≤xn-p0, ∀n≥N.
Then there existslimnxn-p~, so (52) furnish (53)βn1-βnTxn-Sxn≤xn-p02-xn+1-p02+αn2O1+αnO1,and so, by hypothesis liminfnβn(1-βn)>0, we deduce(54)Txn-Sxn⟶0.Moreover, (55)
0
=
lim
n
⁡
x
n
+
1
-
p
0
-
x
n
-
p
0
≤
liminf
⁡
α
n
f
x
n
-
A
p
0
+
1
-
α
n
γ
¯
U
n
x
n
-
p
0
-
x
n
-
p
0
since
α
n
⟶
0
≤
liminf
n
⁡
U
n
x
n
-
p
0
-
x
n
-
p
0
=
liminf
n
⁡
β
n
T
x
n
-
p
0
+
1
-
β
n
S
x
n
-
p
0
-
x
n
-
p
0
≤
liminf
n
⁡
β
n
T
x
n
-
p
0
-
x
n
-
p
0
+
1
-
β
n
S
x
n
-
p
0
-
x
n
-
p
0
≤
limsup
n
⁡
β
n
T
x
n
-
p
0
-
x
n
-
p
0
+
1
-
β
n
·
S
x
n
-
p
0
-
x
n
-
p
0
by  the  quasinonexpansivity  of
T
and
S
≤
0
.But then(56)
lim
n
⁡
β
n
T
x
n
-
p
0
-
x
n
-
p
0
+
1
-
β
n
S
x
n
-
p
0
-
x
n
-
p
0
=
0
.Since both of the addends are nonpositive and the limit of the sum is zero, it follows that(57)limnβn⁡Txn-p0-xn-p0=limn⁡1-βnSxn-p0-xn-p0=0.But then the hypothesis liminfnβn(1-βn)>0 implies that(58)limn⁡Txn-p0-xn-p0=limn⁡Sxn-p0-xn-p0=0.From strong quasinonexpansivity of S, it follows that (59)Sxn-xn⟶0.Again xn-Txn=xn-Sxn+Sxn-Txn, so, by (54) and (57),(60)xn-Txn⟶0.We show now that(61)limsupn⁡xn-p0,fp0-Ap0≤0.Indeed, select a subsequence xnk⇀z such that (62)limsupn⁡xn-p0,fp0-Ap0=limk⁡xnk-p0,fp0-Ap0=z-p0,fp0-Ap0.But by the demiclosedness of both T and S and by (59) and (60), one deduces that (63)z∈FixS∩FixT,and  so,  by  definition  ofp0,(61) is obtained.
Moreover, from (59) and (60) at once, it follows that(64)Unxn-xn⟶0.Finally we are able to show xn→p0. Indeed, (65)
x
n
+
1
-
p
0
2
=
I
-
α
n
A
U
n
x
n
-
p
0
+
α
n
f
x
n
-
A
p
0
2
=
1
-
α
n
γ
¯
2
U
n
x
n
-
p
0
2
+
α
n
2
f
x
n
-
A
p
0
2
+
2
α
n
I
-
α
n
A
U
n
x
n
-
p
0
,
f
x
n
-
A
p
0
≤
1
-
α
n
γ
¯
2
x
n
-
p
0
2
+
α
n
2
f
x
n
-
A
p
0
2
-
2
A
U
n
x
n
-
p
0
,
f
x
n
-
A
p
0
+
2
α
n
U
n
x
n
-
x
n
,
f
x
n
-
A
p
0
+
2
α
n
x
n
-
p
0
,
f
x
n
-
f
p
0
+
2
α
n
x
n
-
p
0
,
f
p
0
-
A
p
0
≤
1
-
α
n
γ
¯
2
+
2
α
n
ρ
x
n
-
p
0
2
+
α
n
2
O
1
+
2
α
n
U
n
x
n
-
x
n
,
f
x
n
-
A
p
0
+
2
α
n
x
n
-
p
0
,
f
p
0
-
p
0
.Put(66)sn=1-1-αγ¯2-2αnρ,σn=2Unxn-xn,fxn-Ap0+2xn-p0,fp0-p0+αnO1,and the thesis follows again by Xu’s Lemma, taking account of (61) and (64).
Alternative 2. (xn-p0) is not definitively not increasing; that is, there exists a subsequence (xnj-p0) such that xnj-p0<xnj+1-p0, ∀j∈N.
From Maingé’s Lemma, it follows that there exists an increasing sequence of integers(τ(n))n∈N satisfying (43) and (44).
Then(67)
0
≤
liminf
n
⁡
x
τ
n
+
1
-
p
0
-
x
τ
n
-
p
0
≤
limsup
n
⁡
x
τ
n
+
1
-
p
0
-
x
τ
n
-
p
0
≤
limsup
n
⁡
x
n
+
1
-
p
0
-
x
n
-
p
0
≤
limsup
n
⁡
I
-
α
n
A
U
n
x
n
-
p
0
+
α
n
f
x
n
-
A
p
0
-
x
n
-
p
0
≤
1
-
α
n
γ
¯
x
n
-
p
0
+
α
n
O
1
-
x
n
-
p
0
⟶
0
,
as
n
⟶
∞
.Hence(68)limn⁡xτn+1-p0-xτn-p0=0.Now, retracing the same inequalities used to obtain (59) with τ(n) instead of n, we have(69)Sxτn-xτn⟶0.Moreover, we can rewrite (52) as (70)0≤βτn1-βτnTxτn-Sxτn2≤xτn-p02-xτn+1-p02+αnO1,and so, from (68) and the hypothesis liminfnβn(1-βn)>0,(71)Txτn-Sxτn⟶0.Again, (72)xτn-Txτn=xτn-Sxτn+Sxτn-Txτn,so, by (69) and (71),(73)xτn-Txτn⟶0,so also(74)xτn-Uτnxτn⟶0.The same reasoning used to have (61) can be now repeated with τ(n) instead of n obtaining(75)limsupn⁡xτn-p0,fp0-Ap0≤0.Analogously, we can repeat all the reasoning used to deduce (68) with τ(n) instead of n and obtain (76)xτn+1-p02≤1-sτnxτn-p02+sτnστn,where (77)
s
τ
n
=
1
-
1
-
α
τ
n
γ
¯
-
2
α
τ
n
ρ
,
σ
τ
n
=
α
τ
n
s
τ
n
2
U
τ
n
x
τ
n
-
x
τ
n
,
f
x
τ
n
-
A
p
0
+
2
x
τ
n
-
p
0
,
f
p
0
-
A
p
0
+
α
n
O
1
,So, from Xu’s Lemma, taking in account (74) and (75), we have (78)xτn-p02⟶0.Once again by (44) we deduce xn→p0.Remark 9.
Theorem  1.1 in [2] is a corollary of our main result, but the technique of proof is completely different. Here we used the strong quasinonexpansivity of the mapping S. In Theorem 1 we used the following properties of a type average Sδ=(1-δ)I+δS of a nonspreading mapping S: (i)
Sδx-Sδy2≤x-y2+(2/δ)〈x-Sδx,y-Sδy〉-1-δx-Sδx-Sδy2.(ii)
δ/2x-Sx2≤〈x-Sδx,x-q〉, ∀q∈Fix(S).From Proposition 3 we know that there exist nonspreading mappings that are not strongly quasinonexpansive mappings.
So we do not know if the main result remains true if we take a nonspreading mappingS instead of a strongly quasinonexpansive mapping, or of type average Sδ of a nonspreading mapping.
The question remains open. However we would like to note that, in all the examples that we know, the algorithm works also with nonspreading mappings.



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*Source: 671983-2015-07-02.xml*