Volume 2010, Article ID 820535,21pages doi:10.1155/2010/820535
Research Article
Implicit Iteration Methods with Errors
for Discrete Non-Lipschitzian Families with
Perturbed Mappings
Tae-Hwa Kim and Kui-Yeon Kim
Division of Mathematical Sciences, Pukyong National University, Busan 608-737, South Korea
Correspondence should be addressed to Tae-Hwa Kim,[email protected]
Received 31 October 2009; Accepted 3 March 2010
Academic Editor: Yeol J. E. Cho
Copyrightq2010 T.-H. Kim and K.-Y. Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Motivated and inspired by the recent works on implicit iteration methods, we prove necessary and sufficient conditions for strong convergence of the eventually implicit iteration methods with errors to a common fixed point of a discrete family which is continuous total asymptotically nonexpansivein brief, TANonq-uniformly smooth Banach spaces with a perturbed mapping F, 1< q≤2, under some suitable control conditions of parameters. Some applications to viscosity approximation methods or to the eventually implicit algorithms with errors for a finite family of TAN self-mappings in real Banach spaces are also added.
1. Introduction
LetXbe a real Banach space with norm · and letX∗be the dual ofX. Denote by·,·the duality product. When{xn}is a sequence inX, we denote the strong convergence of{xn}to x∈Xbyxn → xand the weak convergence byxn x. LetKbe a nonempty closed convex
subset ofX and letT : K → K be a mapping. Now let FixTbe the fixed point set ofT; namely,
FixT:{x∈K:Txx}. 1.1
Also, for viscosity approximation methods inSection 4, we useΠKto denote the collection
Recently, Alber et al.1introduced the wider class of total asymptotically nonexpan-sive mappings to unify various definitions of classes of nonlinear mappings associated with the class of asymptotically nonexpansive mappings; see also Definition 1 of2. They say that a mappingT :K → Kis said to betotal asymptotically nonexpansivein brief, TAN 1 or2
if there exist two nonnegative real sequences{cn}and{dn}withcn, dn → 0, andφ∈ΓR
such that
Tnx−Tny≤x−yc
nφx−ydn, 1.2
for allx, y∈Kandn≥1, whereR: 0,∞and
φ∈ΓR⇐⇒φis strictly increasing,continuous onRandφ0 0. 1.3
In this case, T is often said to be TAN on K with respect toin short, w.r.t. {cn}, {dn}, andφ. Sometimes we also writecnT,dnT, andφT instead ofcn,dn, andφwhenever the
distinction of the mappingT is needed.
Remark 1.1. Note firstly that ifT is continuous, the property1.2withcn 0 for all n ≥ 1
is equivalent to the following one which is said to be asymptotically nonexpansive in the intermediate sense3:
lim sup
n→ ∞ x,ysup∈K
Tnx−Tny−x−y≤0.
1.4
Indeed, takingcn ≡0 in1.2firstly, we have
sup
x,y∈K
Tnx−Tny−x−y≤d
n 1.5
for eachn ≥1, and next taking the lim sup on both sides asn → ∞immediately gives the property1.4becausedn → 0 asn → ∞. Conversely, taking
dnmax
0,sup
x,y∈K
Tnx−Tny−x−y
1.6
for eachn ≥ 1,1.4immediately impliesdn → 0 asn → ∞; see also2. Note also that a
mapping satisfying the property1.4is non-Lipschitzian; see4. Also, if we takeφt t for allt≥0 anddn 0 for alln≥ 1 in1.2, it can be reduced to the well-known concept of asymptotically nonexpansivemapping5as
Tnx−Tny≤1c
for allx, y∈Kandn≥1. Furthermore, in addition, takingcn0 for alln≥1 in1.2, it can
be reduced to the concept ofnonexpansivemapping as
Tx−Ty≤x−y 1.8
for allx, y∈K.
Recently, motivated and stimulated by1.2, Kim and Park6introduced a discrete familyI {Tn : K → K}of non-Lipschitzian mappings, called TAN on K, namely, I {Tn :K → K}is said to be TAN onKw.r.t.{cn},{dn}, andφif there exist nonnegative real
sequences{cn}and{dn}, n≥1 withcn, dn → 0, andφ∈ΓRsuch that
Tnx−Tny≤x−ycnφx−ydn, 1.9
for allx, y∈Kandn≥1. Furthermore, we say thatIiscontinuousonKprovided eachTn ∈I
is continuous onK; see6for examples of continuousTAN families. In particular, we say thatI {Tn : K → K}is simply TAN whenK X. Then they established necessary and
sufficient conditions for strong convergence of the sequence{xn}defined recursively by the following explicit algorithm:
xn1Tnxn, n≥1, 1.10
starting from an initial guess x1 ∈ K, to a common fixed point of I in Banach spaces as
follows.
Theorem 1.2see 6. LetX be a real Banach space K a nonempty closed convex subset of X. Let a discrete familyI {Tn : K → K}be continuous TAN onKw.r.t.{cn},{dn}, andφwith
C: ∞n1FixTn/∅. Assume that{cn},{dn}, andφsatisfy the following two conditions:
C1there existα, β >0such thatφt≤αtfor allt≥β;
C2∞n1cn <∞,∞n1dn<∞.
Then the sequence{xn}defined by the explicit iteration method1.10converges strongly to a common fixed point ofIif and only iflim infn→ ∞dxn, C 0, wheredxn, C infz∈Cxn−z.
Now let us briefly investigate the recent history concerning the implicit iterative algorithms with errors for nonexpansive mappings and asymptotically nonexpansive mappings. Let{Ti}Ni1be a finite family of nonexpansive self-mappings of a nonempty closed
convex subsetKof a Hilbert spaceHwith Ni1FixTi/∅. In 2001, Xu and Ori7introduced
the following iterative algorithm defined implicitly inH:
xnαnxn−1 1−αnTnxn, n≥1, 1.11
where {αn} ⊂ 0,1,x0 ∈ K is arbitrarily chosen, and Tn : TnmodN, namely, the mod
function takes values in the set{1,2, . . . , N}asTn TN forr 0;Tn Tr for 0 < r < N
αn → 0, they proved the weak convergence of the sequence{xn} defined by the implicit
iteration process1.11to a common fixed point of{Ti}Ni1.
In 2002, Zhou and Chang8introduced the following implicit iterative process with errors for a finite family{Ti}Ni1ofNasymptotically nonexpansive self-mappings ofK:
xnαnxn−1βnTnnxnγnun, n≥1, 1.12
where x0 ∈ K is arbitrarily chosen, {αn},{βn}, and {γn} are sequences in 0,1such that
αnβnγn 1 for alln ≥1, and{un}is a bounded sequence inK. AsC: Ni1FixTi/∅
and∞n1cn < ∞, assuming, in addition, that there exists a constantL >0 such that for any
i, j∈ {1,2, . . . , N}withi /j,
Tinx−Tjny≤Lx−y, n≥1, 1.13
for allx, y ∈ K, they established the weak and strong convergence of the implicit iterative process1.12with errors in uniformly convex Banach spaces, under some suitable conditions of parameters and the assumption 1.13. Subsequently, Sun 9 introduced the following modified implicit process for a finite family{Ti}Ni1 ofN asymptotically nonexpansive
self-mappings ofK:
xnαnxn−1 1−αnTknnxn, n≥1, 1.14
whereknis given askn kforr 0;kn k1 for 0< r < Nwhenevern kNr for some integers k ≥ 0 and 0 ≤ r < N in this case, note that kn → ∞ asn → ∞, kn−N kn−1 andTn−NTnforn≥N, and he studied the necessary and sufficient
conditions for the strong convergence of the implicit iteration scheme1.14for such a finite family{Ti}Ni1 as the conclusion inTheorem 1.2, under assumption of the existence of{xn}
generated implicitly by1.14. Recently, for removing the condition1.13, Chang et al.10
also introduced the following modified implicit iteration scheme with errors for a finite family
{Ti}Ni1ofNasymptotically nonexpansive self-mappings ofKsatisfyingKK⊂K:
xnαnxn−1 1−αnTknnxnun, n≥1, 1.15
where{un}is a bounded sequence inK. In case the control sequence{αn}is bounded away from 0 and 1, they studied the weak and strong convergence of the sequence{xn}generated implicitly by1.15.
On the other hand, Zeng and Yao11recently consider the following iterative process defined implicitly for a finite family{Ti}Ni1of nonexpansive mappings from a whole Hilbert
spaceHinto itself with a perturbed mappingF:
xnαnxn−1 1−αn
Tnxn−λnμF
Tnxn
wherex0 ∈His arbitrarily chosen,{αn} ⊂0,1,{λn} ⊂0,1,μ∈0, ρfor someρ >0, and
F:H → Hisκ-Lipschitzian andη-strongly monotone, that is,
Fx−Fy≤κx−y,
Fx−Fy, x−y≥ηx−y2 1.17
for allx, y ∈ X and for someκ > 0 and η > 0. They established necessary and sufficient conditions for the strong convergence of the implicit iteration scheme1.16for such a finite family{Ti}Ni1with the perturbed mappingF.
Remark 1.3. Notice that, in the implicit iterative algorithm1.16, taking all theλn ≡0 reduces
to1.11equipped withKH.
Let a discrete familyI{Tn:X → X}be continuous TAN with a perturbed mapping
F, namely,F :X → Xisκ-Lipschitzian andη-strongly accretive onX, that is,
Fx−Fy≤κx−y,
Fx−Fy, Jx−y≥ηx−y2 1.18
for all x, y ∈ X and for someκ > 0 and η > 0, whereJ denotes the normalized duality mapping onX. Then we consider the following eventuallyimplicit iterative algorithm with errors for such a familyI{Tn:X → X}with the perturbed mappingF:
xn αnxn−1
1−αn−βn
Tnxn−λnμnFTnxn
βnwn 1.19
for all sufficiently largen ≥ n0, wherexn0−1 : u ∈X is arbitrarily chosen,αn ∈ 0,1,βn ∈
0,1−αn,λn ∈0,1,μn ∈0, ρfor someρ >0, and{wn}is bounded inX. We next exhibit
that the sequence{xn}defined implicitly and eventually for the familyI {Tn : X → X}
with the perturbed mappingFas in1.19is well defined for all sufficiently largen, using the well-known Meir-Keeler’s theorem due to Meir and Keeler12.
Remark 1.4. Note that taking all theλn ≡ 0 in 1.19 reduces to the following eventually
implicit algorithms:
xnαnxn−1
1−αn−βn
Tnxnβnwn 1.20
for all sufficiently largen, whereαn ∈ 0,1andβn ∈ 0,1−αn. In this case, note also that
the domainXof all the self-mappingsTncan be restricted within a nonempty closed convex
subsetKofX.
Finally, inspired and motivated by recent works of Chidume and Ofoedu2, Zhou and Chang8, Sun 9, Chang et al.10, and Zeng and Yao11, we shall give necessary and sufficient conditions for strong convergence of the eventually implicit iteration processes
1.19 with errors to a common fixed point of such a discrete familyI {Tn : X → X}
with a perturbed mappingF on reflexive, strictly convex andq-uniformly smooth Banach spaces, 1< q ≤ 2, under some suitable conditions of parameters and ∞n1FixTn/∅. Some
applications to viscosity approximation methods or to the eventually implicit algorithm
2. Preliminaries
LetXbe a real Banach space and letX∗be its dual. For 1< p <∞, the mappingJp:X → 2X
∗
defined by
Jpx
x∗∈X∗:x, x∗x∗x,x∗xp−1 2.1
for eachx∈Xis called thegeneralizedduality mappingonX. In particular,J :J2is called
thenormalized duality mappingonX. It is well known thatJpx xp−2Jxforx /0; see13
or14for more properties of duality mappings.
The moduli of convexity and smoothness ofX are functionsδX :0,2 → 0,1and
ρX:0,∞ → 0,∞defined respectively by
δX inf
1−xy 2
:x ≤1, y≤1, x−y≥
,
ρXt sup
xyx−y
2−1 :x ≤1, y≤t
.
2.2
Xis said to be uniformly convexifδX>0 for all >0 anduniformly smoothif limt↓0ρXt/t
0. Letq >1 be a given real number. ThenXis said to beq-uniformly smoothif there is a constant c > 0 such thatρXt ≤ ctq; see also14–16for more details. It is well known17that no
Banach space isq-uniformly smooth forq >2, and also that if 1< r ≤q≤2, thenq-uniformly smooth space isr-uniformly smooth. Hilbert spaces,Lp orpspaces, 1 < p < ∞, and the
Sobolev spaces, Wmp,1 < p < ∞, are q-uniformly smooth. Hilbert spaces are 2-uniformly
smooth while
LporporWp mis
⎧ ⎨ ⎩
p-uniformly smooth if 1< p≤ 2;
2-uniformly smooth if p≥ 2.
2.3
The following result due to Xu18is very useful for our argument.
Lemma 2.1see18. Let1< q≤2be a given number.Xisq-uniformly smooth if and only if there exists a constantcq>0such that
xyq
≤ xqqy, Jqx
cqyq 2.4
for allx, y∈X.
Remark 2.2. Note thatc21 in a Hilbert spaceHbecausexy2x22y, xy2for
LetX be aq-uniformly smooth Banach space with 1< q ≤2 and let the mappingF : X → Xbeκ-Lipschitzian andη-strongly accretive onX. Then using2.4,Jqx xq−2Jx
and1.18, we compute for allx, y∈X,
I−μFx−I−μFyq
x−y−μFx−Fyq
≤x−yq−qμFx−Fy, Jq
x−y cqμ
Fx−Fyq
≤ x−yq−qμηx−yq cqμqκqx−yq
1−qμη cqμqκqx−yq
1−μqη− cqμq−1κq
x−yq,
2.5
whereIdenotes the identity operator onX. Letμ∈0, ρ, where
ρ:min
⎧ ⎨ ⎩
1 qη,
qη cqκq
1/q−1⎫⎬
⎭ 2.6
the choice ofρ <1/qηis just a way to ensure that 1−μqη− cqμq−1κq>0 to take theq-root..
ThenI−μFis a contraction because
I−μF
x−I−μFy≤q 1−
μqη− cqμq−1κq x−y 2.7
for allx, y∈Xand 0<q 1−μqη− c
qμq−1κq<1 forμ∈0, ρ.
Let a discrete familyI{Tn :X → X}be TAN. Givenλ∈0,1,μ >0, andk≥1, let
the mappingΦλ,μ,k:X → Xbe defined by
Φλ,μ,kx:T
kx−λμFTkx 2.8
for allx∈X. Then the eventually implicit iteration algorithm1.19is simply expressed as
xnαnxn−1
1−αn−βn
Φλn,μnnx
nβnwn 2.9
for all sufficiently largen.
The following lemmas will be used frequently throughout this paper.
Lemma 2.3. If0≤λ <1,0< μ < ρ, andk≥1, then there holds forΦλ,μ,k:X → X,
Φλ,μ,kx−Φλ,μ,ky≤1−λτT
kx−Tky, x, y∈X, 2.10
whereτ :1−q 1−μqη−c
Proof. Using2.7, we get for allx, y∈X,
Φλ,μ,kx−Φλ,μ,kyT
kx−λμFTkx−
Tky−λμF
Tky
λI−μFTkx 1−λTkx−
λI−μFTky 1−λTky
≤λI−μFTkx−
I−μFTky 1−λTkx−Tky
≤λq
1−μqη− cqμq−1κq Tkx−Tky 1−λTkx−Tky
1−λτTkx−Tky.
2.11
This completes the proof.
In 1969, Meir and Kleeler12established the following fixed point theorem which is a remarkable generalization of the Banach contraction principle.
Theorem 2.4see12. LetX, dbe a complete metric space and letΦbe a Meir-Keeler contraction (MKC, for short) onX, namely, for each >0, there existsδ >0such that
dx, y< δ impliesdΦx,Φy< 2.12
for allx, y∈X. ThenΦhas a unique fixed pointz∈Xand also{Φnx}converges strongly tozfor allx∈X.
Then the following easy observation is crucial for the construction of the eventually implicit iteration algorithm2.9.
Proposition 2.5. Let a discrete familyI {Tn : X → X}be TAN,t ∈ 0,1, andk ≥ 1. Then
1−tTkis an MKC onXfor all sufficiently largek.
Proof. Given >0, chooseδt/21−t. On setting
An: 1−t
δcnφδ dn
, 2.13
An → 1−tδasn → ∞becausecn, dn → 0, and so we can see
for all sufficiently largek. Letx, y ∈ X with x−y < δ. Then use 1.9, the strictly increasing property ofφ, and2.14, in turn, to derive
1−tTkx−1−tTky 1−tTkx−Tky
≤1−tx−yckφx−ydk
<1−tδckφδ dk
Ak
≤1−tδ t
2 .
2.15
Hence1−tTkis an MKC onXfor all sufficiently largek.
Lemma 2.6see19,20. Let{an},{αn}, and{βn}be sequences of nonnegative real numbers such that
an1≤1αnanβn 2.16
for alln≥1. Suppose that∞n1αn <∞and∞n1βn <∞. Thenlimn→ ∞anexists. Moreover, if in addition,lim infn→ ∞an0, thenlimn→ ∞an0.
3. Necessary and Sufficient Conditions for Convergence
LetXbe aq-uniformly smooth Banach space with 1< q≤2 and let the mappingF :X → X beκ-Lipschitzian andη-strongly accretive onX. Let a discrete familyI {Tn :X → X}be
TAN. Lett∈0,1, s∈0,1−t. Letk≥1,λ∈0,1, andμ∈0, ρ, whereρis the constant in
2.6. Fixu, w∈Xand let the mappingΓt,s,k :X → Xbe defined by
Γt,s,kxtu 1−t−sΦλ,μ,kxsw 3.1
for allx∈X. UsingLemma 2.3, we have
Γt,s,kx−Γt,s,ky1−t−sΦλ,μ,kx−1−t−sΦλ,μ,ky
1−t−sΦλ,μ,kx−Φλ,μ,ky
≤1−t−s1−λτTkx−1−λτTky
3.2
for allx, y∈X. First, in case ofλ /0, byProposition 2.5,1−λτTkin3.2is an MKC onXfor
all sufficiently largekand hence so isΓt,s,k. In the other case ofλ0, note thatΦλ,μ,k T
k
and1−t−sΦλ,μ,k 1−t−sT
k. ApplyingProposition 2.5again,1−t−sTkhence,Γt,s,k
is an MKC onXfor all sufficiently largek. Therefore, in any case,Γt,s,kis an MKC onXfor
all sufficiently largek. ApplyingTheorem 2.4Meir-Keeler, there exists a unique fixed point xt,s,kofΓt,s,kinX, that is,
for all sufficiently largek, where t ∈ 0,1, s ∈ 0,1−t,λ ∈ 0,1, andμ ∈ 0, ρ. This equation3.3exhibits that the eventually implicit iteration schemes1.19with perturbed mappingF are well defined and so we can present necessary and sufficient conditions for strong convergence of the sequence{xn}defined implicitly and eventually by 1.19onq -uniformly smooth Banach spaces.
Theorem 3.1. Let1 < q ≤ 2be a real number andN ≥1. LetXbe aq-uniformly smooth Banach space. LetF :X → X beκ-Lipschitzian andη-strongly accretive for some constantsκ >0,η >0. Let also a discrete family I {Tn : X → X} be continuous TAN withC : ∞n1FixTn/∅. Let{xn}be the sequence defined by the implicit iteration method1.19with bounded errors{wn} in X. Assume that{cn}and{dn}satisfy the condition (C2) inTheorem 1.2, and thatφsatisfies the following property:
C1there existα >0, β≥0such thatφt≤αtfor allt≥β.
Assume also that{αn},{βn},{λn}, and{μn}are sequences of nonnegative real numbers satisfying the following control conditions:
C3{λn} ⊂0,1for all sufficiently largenand∞n1λn <∞;
C40 < μn < ρ for for all sufficiently large n, where ρ and cp are constants in2.6and
Lemma 2.1, respectively;
C50< a:lim infn→ ∞αn≤lim supn→ ∞αn<1;
C6βn∈0,1−αnfor all sufficiently largenand ∞
n1βn<∞.
Then{xn}converges strongly to a common fixed point ofIif and only iflim infn→ ∞dxn, C 0.
Lemma 3.2. Under the same hypotheses asTheorem 3.1, there hold the following properties:
ilimn→ ∞xn−pexists for allp∈C, and hence{xn}and{Φλn,μn,nxn}are bounded;
iilimn→ ∞dxn, Cexists.
Proof. First, note that it follows fromC1and the strictly increasing property ofφthat
φt≤φβαt, t≥0. 3.4
In fact, ift ≤ β, sinceφis nondecreasing, we haveφt ≤ φβ. For anyt ≥ β, byC1, we getφt≤αt. Hence3.4is required. Now to provei, letp ∈Cand letn≥1 be arbitrarily given. Use2.8,2.10,1.9, and3.4, in turn, to derive
Φλn,μn,nx
n−p≤Φλn,μn,nxn−Φλn,μn,npΦλn,μn,np−p ≤1−λnτnTnxn−TnpλnμnFp
≤1−λnτnxn−pcnφxn−pdn
λnμnFp ≤1−λnτn
1αcnxn−pφ
βcndn
λnμnFp,
whereτn:1−q
1−μnqη−cqμnq−1κq∈0,1as inLemma 2.3. This inequality3.5together
withC4yields
xn−p≤αnxn−1−p
1−αn−βnΦλn,μn,nxn−pβnwn−p
≤αnxn−1−p
1−αn−βn
×1−λnτn1αcnxn−pφ
βcndn
λnμnFp
βn
wnp
≤αnxn−1−p 1−αnαnλnτn1αcnxn−p
φβcndnλnρFpβnM1
≤αnxn−1−p 1−αnαnλn1αcnxn−p
φβcndnλnρFpβnM1
≤αnxn−1−p 1−αnαnλn1αcn αcnxn−p
φβcndnλnρFpβnM1
3.6
for all sufficiently largen, whereM1sup{wn:n≥1}p<∞. Sincealim infn→ ∞αn∈
0,1,λn → 0, andcn → 0, we can choose
0≤ηn:λn1αcn αcn
αn <
1 3.7
for all sufficiently largenandηn → 0 asn → ∞. Now using3.7, we get
xn−p≤ 1 1−ηn
xn−1−p 1 αn
1−ηn
φβcndnλnρFpβnM1
≤ 1 ηn 1−ηn
!
xn−1−p
1 a1−ηn
φβcndnλnρFpβnM1
≤1M2ηnxn−1−pM2
a
φβcndnλnρFpβnM1
,
3.8
for all sufficiently largen, whereM2:sup{1/1−ηn:n≥1}<∞and hence there exists a
suitable constantM >0 such that
xn−p≤
1Mηnxn−1−pMσn 3.9
for all sufficiently largen, whereσn :cndnλnβn. Since 0≤ηn< λnαcn11/a, it follows
fromC2,C3, andC6thatηn<∞andσn<∞. Hence the limit limn→ ∞xn−pexists
fromLemma 2.6. Since{xn}is bounded, so is{Tnxn}because
for a fixedp∈Cby1.9. Then it follows fromLemma 2.3that{Φλn,μn,nxn}is also bounded.
Henceiis obtained.
Now to showii, taking the infimum over allp ∈ Con the both sides of inequality
3.9, we obtain
dxn, C≤
1Mηn
dxn−1, C Mσn 3.11
for all sufficiently largen. ApplyingLemma 2.6again,iiis quickly obtained.
Proof ofTheorem 3.1. It suffices to show the sufficiency. Assume that
lim inf
n→ ∞ dxn, C 0. 3.12
Then it follows from ii ofLemma 3.2that limn→ ∞dxn, C 0. Since 0 ≤ ηn < 1 for all
sufficiently largenin3.7andηn<∞in the proving process ofLemma 3.2, we see that
1≤K:"1Mηn
≤eMηn <∞. 3.13
Given >0, since limn→ ∞dxn, C 0 andMσn<∞, we can choose a positive integern0
sufficiently large so that3.9holds for alln≥n0, and
dxn, C<
4K,
∞
#
in
Mσi<
4K, n≥n0. 3.14
Let n, m ≥ n0 and p ∈ C. First, use the inequality3.9repeatedly together with 3.13 to
derive
xn−p≤ "n in01
1Mηixn0−p n−1 #
in01 Mσi
n "
ki1
1Mηk
Mσn
≤K
$
xn0−p #n in01
Mσi %
,
3.15
which implies that
xn−xm ≤xn−pxm−p
≤K
$ xn
0−p n #
in01 Mσi
%
K
$ xn
0−p m #
in01 Mσi
%
≤2K
$ xn
0−p
∞
#
in01 Mσi
%
.
Taking the infimum over allp∈Cfirstly on both sides and next using3.14, we have
xn−xm ≤2K $
dxn0, C
∞
#
in01 Mσi
%
≤2K 4K
4K
, n, m≥n0.
3.17
This shows that{xn}is a Cauchy sequence inX. Sayxn → x∗ ∈ X. Finally, we claim that
x∗∈C. In fact, note first that
x∗−p≤ x∗−xnx
n−p 3.18
for allp∈Candn≥1. Taking the infimum again over allp∈Con both sides ensures that
dx∗, C≤ x∗−xndxn, C−→0 3.19
as n → ∞. Since C is closed by continuity of I, it follows that x∗ ∈ C and the proof is complete.
Corollary 3.3. Under the same hypotheses asTheorem 3.1, the sequence{xn}converges strongly to a common fixed pointp ∈Iif and only if there exists a subsequence{xnk}of{xn}which converges strongly top.
As taking all theλn 0 in1.19, in view ofRemark 1.4, we have the following direct
consequence ofTheorem 3.1in real Banach spaces.
Theorem 3.4. LetN≥1, letKbe a nonempty closed convex subset of a real Banach spaceX, and let a discrete familyI {Tn :K → K}be continuous TAN onK w.r.t.{cn},{dn}, andφwithC :
∞
n1FixTn/∅. Let{xn}be the sequence defined implicitly and eventually by1.20with bounded errors{wn}inK. Assume that{cn}and{dn}satisfy the condition (C2) andφsatisfies the property
C1, and also that{αn}and{βn}are sequences in0,1satisfying (C5) and (C6) inTheorem 3.1. Then{xn}converges strongly to a common fixed point ofIinKif and only iflim infn→ ∞dxn, C
0.
Remark 3.5. Theorem 3.4is just an implicit iterative version with error terms ofTheorem 1.2.
4. Viscosity Approximation Methods
Recall firstly that a mappingψ : R → R is said to be anL-functionifψ0 0,φt > 0 for eacht > 0, and for everys > 0 there existsu > ssuch thatψt ≤ s t ∈ s, u. As a consequence, everyL-functionψ satisfiesψt < tfor each t > 0. Also, for a metric space
X, d, a mappingf :X → Xis said to beψ, L-contractionifψ :R → Ris anL-function anddfx, fy< ψdx, yfor allx, y∈Xwithx /y; see21for more details.
Theorem 4.1see22. LetX, dbe a metric space and letf :X → Xbe a mapping. Thenf is an MKC if and only if there exists anL-functionψ:R → Rsuch thatfis aψ, L-contraction.
LetKbe a nonempty closed convex subset of a real Banach spaceXand letT :K → K be a nonexpansive mapping. Given a real numbert∈ 0,1and anf ∈ ΠK, we defineTtf :
K → Kby
Ttfxtfx 1−tTx 4.1
for allx∈K. For simplicity, we will writeTtforTtf provided no confusion occurs. Obviously,
Tt is a contraction onK and the well-known Banach Contraction Principle guarantees the
unique fixed point ofTt, sayxt. Thenxtis the unique solution of the following fixed point
equation
xttfxt 1−tTxt, 4.2
which is later called thecontinuousimplicit method, while thediscreteimplicit algorithm is specially expressed as
xnαnfxn 1−αnTxn, 4.3
where{αn}is a sequence in0,1tending to zero. On the other hand, the first explicit viscosity algorithm is given in 2000 year by Moudafi23as
xn1 αnfxn 1−αnTxn, n≥1, 4.4
wherex1 ∈ Cis arbitrarily chosen and{αn} is a sequence in0,1. Recently, Petrus¸el and
Yao21introduced the following implicit viscosity approximation scheme for a uniformly asymptotically regular sequence {Tn} of nonexpansive mappings from K into itself in a reflexive Banach spaceX:
xntnfxn 1−tnTnxn, n≥1, 4.5
where{tn}is a sequence in0,1withtn → 0, andf :K → Kis a generalized contraction
recall that it is called a generalized contraction in 21 wheneverf is either an MKC or a
ψ, L-contraction in the sense ofTheorem 4.1. AsC: n∞1FixTn/∅, they studied strong
convergence of the sequence{xn}generated by4.5to a common fixed pointpof{Tn}such thatp∈Cis the unique solution to the variational inequality
fp−p, Jy−p≤0, y∈C; 4.6
contraction and a nonexpansive semigroup in reflexive Banach spaces; see also 25 for approximating fixed points of nonexpansive mappings.
From now on, unless other specified, assume that K is a nonempty closed convex subset of a real Banach spaceX and a discrete family I {Tn : X → X}is TAN. In this
section, as a special case of1.20, we shall consider the following eventually implicit iteration method with no errors and a fixed anchorx0 ∈Kfor such a familyI:
xnαnx0 1−αnTnxn 4.7
for all sufficiently largen, whereαn∈a,1for somea∈0,1.
Here we need the following properties for Meir-Keeler contractions, studied recently by Suzuki26.
Lemma 4.2see26. LetKbe a convex subset of a Banach spaceX. LetΦbe an MKC onK. Then there hold the following properties:
igiven >0, there existsr∈0,1such that, for allx, y∈K,
x−y≥ impliesΦx−Φy≤rx−y; 4.8
iiifT :K → Kis nonexpansive, thenT◦Φis an MKC onK.
Proposition 4.3. Fixa∈0,1,αn∈a,1for alln. LetΦbe an MKC onKand let a discrete family I{Tn:X → X}be TAN onK. Then a mappingx→αnΦx 1−αnTnxis an MKC onKfor all sufficiently largen.
Proof. We employ the ideas of Proposition 3iiin26andProposition 2.5. Define
Γnx:αnΦx 1−αnTnx 4.9
for allx∈K. Given >0, byiofLemma 4.2, there existsr ∈0,1satisfying4.8. Choose
δ: a1−r
21−a1−r. 4.10
Fixx, y∈Kwithx−y< δ. Then we must claim thatΓnx−Γny< for all sufficiently
largen. Indeed, in case ofx−y ≥ , we first use4.8,1.9, and the strictly increasing property ofφto derive
Γnx−Γny≤αnΦx−Φy 1−αnTnx−Tny
≤αnrx−y 1−αnx−ycnφx−ydn
≤1−αn1−rx−ycnφx−ydn ≤1−a1−rδ cnφδ dn
:An,
whereAn → 1−a1−rδasn → ∞becausecn, dn → 0. Then we can see
0≤An≤1−a1−rδ a
1−r 2
1−a1−r a1−r 21−a1−r
!
a1−r
2
4.12
for all sufficiently largen. In the other case of 0<x−y< becausexyis trivial, we can takesuch thatx−y< < . For this>0, it follows from the definition of MKC in
Theorem 2.4that there existsδ>0 such that
x−y< δ impliesΦx−Φy< 4.13
for allx, y∈K. Now repeating the previous techniques, we have
Γnx−Γny≤αnΦx−Φy 1−αnTnx−Tny
< αn 1−αnx−ycnφx−ydn
< αn 1−an
cnφ
dn
< cnφ
dn
:Bn,
4.14
whereBn → asn → ∞. Then we can also see
0≤Bn<
− 4.15
for all sufficiently largen. This completes the proof.
Remark 4.4. LetT :K → Kbe nonexpansive. If we takeTn T,αn a∈0,1for alln≥1,
then a mappingx→aΦx 1−aTxis an MKC onK; seeiiof Proposition 3 in26.
Proposition 4.5. Letφsatisfy the conditionC1inTheorem 3.1. Assume thatlimn→ ∞xnexists for any anchorx0 ∈ K, where the sequence{xn}equipped with the anchorx0is defined implicitly and eventually asxnαnx0 1−αnTnxnfor all sufficiently largen. Define
Px0:nlim→ ∞xn. 4.16
ThenP is nonexpansive onK.
Proof. Letx0, y0 ∈Kbe any different anchors. Let{xn}and{yn}be defined as
xnαnx0 1−αnTnxn,
ynαny0 1−αnTnyn
for all sufficiently largen. By virtue of1.9andC1, we can compute
xn−yn≤αnx0−y0 1−αnTnxn−Tnyn
≤αnx0−y0 1−αnxn−yncnφxn−yndn
≤αnx0−y0 1−αn1αcnxn−yncnφ
βdn
≤αnx0−y0 1−αnαcnxn−yncnφ
βdn
4.18
for all sufficiently largen. A simple calculation yields
xn−yn≤ 1 αcn a−αcn
!
x0−y0
1 a−αcn
cnφb dn
4.19
becauseαn ∈a,1for somea∈0,1. Sincecn, dn → 0 asn → ∞, we haveP x0−P y0 ≤ x0−y0and the proof is complete.
Now, in case all theβn ≡ 0 in1.20, we prove strong convergence of the following
viscosity approximation methods for a discrete familyI{Tn:K → K}which is continuous
TAN on a nonempty closed convex subsetKof a real Banach spaceX.
Theorem 4.6. Under the same hypotheses of K, X, I, C, {cn}, {dn}, φ, {αn}, and {βn} as
Theorem 3.4, assume that P x0 limn→ ∞xn exists for any anchorx0 ∈ K. LetΦbe an MKC on
K, and let{yn}be the sequence defined implicitly and eventually as
ynαnΦ
yn
1−αnTnyn 4.20
for all sufficiently largen. Then{yn}converges strongly to the unique pointz∈CsatisfyingP◦Φz z.
Proof. Our proving method employs the idea used for proving Theorem 7 in26. Note first thatP◦ΦandαnΦ 1−αnTnare MKCs onKfromiiofLemma 4.2andProposition 4.3,
respectively. Hence,Theorem 2.4Meir-Keelerensures the existence and uniqueness of yn
andz. By our assumption,P◦ΦzPΦz limn→ ∞xnz∈Cexists for the anchorΦz∈K,
where{xn}is a sequence defined implicitly and eventually by
xnαnΦz 1−αnTnxn 4.21
for all sufficiently largen. We must claim thatyn → zasn → ∞, too. Indeed, repeat the
proving technique used for arriving at4.19to get
yn−xn≤ 1 αcn a−αcn
!
Φyn−Φz
1 a−αcn
cnφ
βdn
for all sufficiently large n, where a lim infn→ ∞αn ∈ 0,1. Now to prove the claim by
contradiction, assume that {yn} does not converge to z. Then there exist > 0 and a subsequence {ynk}of {yn}such that ynk −z ≥ for allk ≥ 1. Byiof Lemma 4.2, for
this >0, there existsr ∈ 0,1satisfying4.8. Then using4.22combined with4.8, we can compute
yn
k−z≤ynk−xnkxnk−z
≤ 1 αcnk a−αcnk
!
Φynk−Φz
1 a−αcnk
cnkφ
βdnk
xnk−z
≤ 1 αcnk a−αcnk
!
rynk −z
1 a−αcnk
cnkφ
βdnk
xnk−z
4.23
for all sufficiently largek. Sincecnk, dnk → 0, andxnk → zask → ∞, taking the lim sup as k → ∞on both sides yields
lim sup
k→ ∞
ynk−z≤rlim sup
k→ ∞
ynk−z, 4.24
which immediately shows that limk→ ∞ynk z. This contradicts to the construction of{ynk}.
Therefore, it must beyn → zasn → ∞. This completes the proof.
Remark 4.7. Note that if all the Tn : K → K are nonexpansive, our Theorem 4.6 can
be reduced to Theorem 7 of 26. However, if all the Tn : K → K are asymptotically
nonexpansive, it still seems new.
5. Applications to a Finite Family of TAN Self-Mappings
Let X be a smooth Banach space and let N ≥ 1 be fixed. Let {Ti}Ni1 be a finite family
of N continuous TAN mappings defined on X; more precisely, for each 1 ≤ i ≤ N,Ti is
continuous TAN w.r.t.{cnTi},{dnTi}, andφTi. In this section, as special cases, we consider
the following eventually implicit iteration algorithm with errors for such a finite family{Ti}Ni1
with a perturbed mappingF:
xnαnxn−1
1−αn−βn
Anxn−λnμnFAnxn
βnwn 5.1
for all sufficient large n, where An is any one amongTnn,Tknn, and Ni1λ n
i T
n
i , all the
λin∈0,1withni1λin1 andλi:inf{λin :n≥1}>0 for 1≤i≤N; see Proposition 1.2
in6for more details.
Remark 5.1. Note that the discrete familyI: {An :X → X}is obviously continuous TAN
cn: max
1≤i≤NcnTi, dn:1max≤i≤NdnTi, φ:1max≤i≤NφTi, 5.2
and also that Ni1FixTi ⊂ ∞n1FixAn. As in Remark 1.4, taking all the λn ≡ 0 in 5.1
reduces to the following eventually implicit iteration algorithm for a finite family{Ti}Ni1 of
Ncontinuous TAN self-mappings ofK, a nonempty closed convex subset of a real Banach spaceX:
xnαnxn−1
1−αn−βn
Anxnβnwn 5.3
for all sufficiently largen.
As direct consequences of Theorems3.1and3.4, we have the following necessary and sufficient conditions for strong convergence of the eventually implicit iteration methods5.1
and5.3, respectively, for such a finite family{Ti}Ni1ofNcontinuous TAN self-mappings.
Theorem 5.2. Let1 < q ≤ 2be a real number andN ≥1. LetXbe aq-uniformly smooth Banach space. LetF :X → Xbeκ-Lipschitzian andη-strongly accretive for some constantsκ >0,η >0. Let {Ti}Ni1 be a finite family ofN continuous TAN mappings defined onX withC: Ni1FixTi/∅. Let{xn}be the sequence defined by the eventually implicit iteration method5.1with bounded errors {wn}inX. Assume that{cn},{dn}, andφgiven as in5.2satisfy the conditions (C2) andC1, and also that{αn},{βn},{λn}and{μn} are sequences satisfying the control conditions (C2)–(C6) in Theorem 3.1. Then {xn} converges strongly to a common fixed point of {Ti}Ni1 if and only if
lim infn→ ∞dxn, C 0.
As taking all theλn ≡0 in5.1, in view of Remarks5.1and1.4, we have the following
direct consequence ofTheorem 5.2.
Theorem 5.3. LetN ≥ 1, letKbe a nonempty closed convex subset of a real Banach spaceX, and let{Ti}Ni1 be a finite family ofN continuous TAN self-mappings ofK withC : Ni1FixTi/∅. Let{xn}be the sequence defined implicitly by5.3with bounded errors{wn}in K. Assume that {cn},{dn}, andφgiven as in5.2satisfy the conditions (C2) andC1, and also that{αn}and{βn} are sequences in0,1satisfying (C5) and (C6) inTheorem 3.1. Then{xn}converges strongly to a common fixed point of{Ti}Ni1inKif and only iflim infn→ ∞dxn, C 0.
Remark 5.4. i Theorem 5.2improves and extends the corresponding Theorem 2.2 due to Zeng and Yao11for a finite family of nonexpansive self-mappings in Hilbert space settings in case whenμn:μis fixed andβn0 for alln≥1.
ii Our results are still new when all the λn 0 and βn 0; compare with the
corresponding results of Xu and Ori7in Hilbert spaces.
iii Theorem 5.3is just an implicit iterative version of Theorem 8 of Chidume and Ofoedu2for a finite family ofNcontinuous TAN self-mappings in real Banach spaces, and it still seems new.
Acknowledgment
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