S ongly Con e gen App oxima ions o Fixed
Poin s o To al Asymp o ically Nonexpansi e
Mappings
Yako Albe
Depa men o Ma hema ics
The Technion-Is ael Ins i u e o Technology
32000 Hai a, Is ael. Email: albe ya@ echunix. echnion.ac.il
Ra a Esp´ınola∗and Pepa Lo enzo
Depa amen o de An´alisis Ma em´a ico
Uni e sidad de Se illa, P.O. Box 1160
41080-Se ille, Spain. Emails: [email p o ec ed], [email p o ec ed]
∗Co esponding au ho
Abs ac
In his wo k we p o e a new s ong con e gence esul o he egula ized successi e
app oxima ion me hod gi en by
yn+1 =qnz0+ (1 −qn)Tnyn, n = 1,2, ...,
whe e
lim
n→∞
qn= 0 and
∞
X
n=1
qn=∞,
o Ta o al asymp o ically nonexpansi e mapping, i.e., Tis such ha
kTnx−Tnyk ≤ kx−yk+k(1)
nφ(kx−yk) + k(2)
n,
whe e k1
nand k2
na e eal null con e gen sequences and φ:R+→R+is con inuous
and such ha φ(0) = 0 and lim →∞
φ( )
≤C o a ce ain cons an C > 0.
Among o he ea u es, ou esul s essen ially gene alize exis ing esul s on s ong
con e gence o Tnonexpansi e and asymp o ically nonexpansi e. The con e gence
and s abili y analysis is gi en o bo h sel - and nonsel -mappings.
Running i le: S ong con e gence o ixed poin s
Key wo ds: asymp o ically nonexpansi e mappings, bes app oxima ion, ixed
poin , duali y map, i e a ion schemes.
Ma hema ics Subjec Classi ica ion 2000: 47A58, 47H10.
1
2
1 In oduc ion
I e a i e p ocedu es o nonlinea ope a o s ha e been la gely s udied by many au ho s in
he las decades. One o he i s esul s o his na u e was ob ained by B owde [5] o
nonexpansi e sel -mappings de ined on Hilbe spaces. He e B owde s udied he i e a i e
me hod:
xω=ωz0+ (1 −ω)T xω.(1.1)
o Ω a closed and con ex subse o H,z0∈Ω an a bi a y poin and T: Ω →Ω a
nonexpansi e mapping wi h nonemp y ixed poin se N(T) := {x∈Ω : T x =x}.
In [5], B owde p o ed ha lim
ω→0xωexis s and is a ixed poin o T. This esul was ex-
ended by Reich [17] o he case when Xis a uni o mly smoo h Banach space. Fu he mo e,
he showed ha he ixed poin se o Tis a sunny nonexpansi e e ac o Ω.
The ecu si e o mula (explici scheme)
y1∈Ω, yn+1 =qnz0+ (1 −qn)Tyn, n = 1,2, ..., (1.2)
was in oduced by Halpe n [12] who discussed i s con e gence in he amewo k o Hilbe
spaces. La e i has been in es iga ed in [12, 18, 19, 20] wi h di e en addi ional p ope ies
on he sequence {qn}, he ope a o Tand he space X.
B owde ’s and Halpe n’s i e a i e p ocedu es ha e mo i a ed di e en schemes o ind
ixed poin s o asymp o ically nonexpansi e mappings (see Rema k 1.2 o de ini ion). In
his way, T.C. Lim and H.K. Xu [15] s udied he algo i hm o Tasymp o ically nonexpan-
si e which gene a es he sequence (implici scheme)
xn=qnz0+ (1 −qn)Tnxn.(1.3)
They showed ha he sequence {xn}con e ges s ongly o a ixed poin o Tin he ame-
wo k o a uni o mly smoo h Banach space, unde sui able condi ions on he coe icien s.
Ve y ecen ly, in [6], he s ong con e gence o he explici scheme gi en by
y1∈Ωyn+1 =qnz0+ (1 −qn)Tnyn, n = 1,2, ..., (1.4)
whe e z0∈Ω,and
lim
n→∞ qn= 0 and
∞
X
n=1
qn=∞,(1.5)
has been s udied in uni o mly smoo h spaces. I is wo hwhile o poin ou ha he con-
e gence o he implici scheme gi en by (1.3) is an impo an ool in o de o p o e he
s ong con e gence o explici schemes as (1.2).
In his pape we will conside he class o he o al asymp o ically nonexpansi e map-
pings which ha e been in oduced e y ecen ly in [2].
De ini ion 1.1 (c . [2]) A mapping T: Ω →Ωis called o al asymp o ically nonexpansi e
i he e exis nonnega i e eal sequences {k(1)
n}and {k(2)
n}wi h k(1)
n, k(2)
n→0as n→ ∞,
and a con inuous unc ion φ:R+→R+wi h φ(0) = 0 such ha
kTnx−Tnyk ≤ kx−yk+k(1)
nφ(kx−yk) + k(2)
n.(1.6)
3
Rema k 1.2 I φ(λ)≡0 hen (1.6) akes he o m
kTnx−Tnyk ≤ kx−yk+k(2)
n.
Hence, i Ωis a bounded se and TNis con inuous o some in ege N≥1 he mapping T
is o asymp o ically nonexpansi e ype. I φ(λ) = λ hen we can w i e
kTnx−Tnyk ≤ (1 + k(1)
n)kx−yk+k(2)
n.
In addi ion, i k(2)
n= 0 o all n≥1 hen we ob ain he de ini ion o asymp o ically nonex-
pansi e mapping:
kTnx−Tnyk ≤ knkx−yk, kn→1.
I k(1)
n= 0 and k(2)
n= 0 o all n≥1 hen we ob ain he class o nonexpansi e mappings:
kTx −Tyk ≤ kx−yk.
I k(2)
1= 0 hen i ollows om (1.6) ha Tis uni o mly con inuous, howe e , i can be
uni o mly con inuous e en i k(2)
16= 0.
To cons uc he s ong con e gen app oxima ions o solu ions o he equa ion
Tx =x(1.7)
wi h a o al asymp o ically nonexpansi e mapping T, we apply he i e a i e scheme gi en
by (1.4).
Fo nonexpansi e ope a o s T, he algo i hm (1.4) is w i en down in he ollowing o m:
y1∈Ω, yn+1 =qnz0+ (1 −qn)Tyn, n = 1,2, ..., (1.8)
whe e lim
n→∞ qn= 0.
We show nex ha (1.8) is he egula ized successi e app oxima ion me hod o (1.7).
As i is known, equa ion (1.7) is equi alen o
Ax = 0 (1.9)
wi h he acc e i e ope a o A=I−T: Ω →Ω.Tha is, in his case
hAx −Ay, J(x−y)i ≥ 0,(1.10)
whe e Js ands o he no malized duali y map. I x∗is a solu ion o (1.7) hen Ax∗= 0.
In he sequel, we assume ha he ixed poin se N(T) o Tis no emp y. We emphasize
ha he p oblem (1.7) belongs o he class o ill-posed p oblems ( o mo e on ill-pos ed
p oblems see [4]). S ongly con e gen app oxima ions o x∗can be ob ained only by using
some egula iza ion p ocedu e.
Le ωbe a pa ame e such ha 0 < ω < 1 and ω→1.Ob iously, i x∗is a solu ion o
(1.9) hen i is solu ion o he equa ion
ωAx = 0 (1.11)
4
o any ixed ω > 0.Using he gene al heo y (see o example [4], Sec ion 2.7), cons uc
o (1.11) he ope a o egula iza ion me hod wi h egula iza ion pa ame e α= 1−ω→0,
namely,
ωAx + (1 −ω)(x−z0) = 0,(1.12)
whe e z0∈Ω.I is easy o see ha (1.12) is equi alen o
x= (1 −ω)z0+ωT x. (1.13)
Deno e
Tωx= (1 −ω)z0+ωT x.
Since Ω is con ex and closed, we ha e ha Tω: Ω →Ω,and (1.13) can be ew i en as
x=Tωx. (1.14)
Consequen ly, by Banach Con ac ion P inciple, equa ion (1.12) has a unique solu ion
xωand he successi e app oxima ion me hod
x1∈Ωxn+1 = (1 −ω)z0+ωT xn
con e ges s ongly o xω.Le Xbe uni o mly smoo h and ωk→1 as k→ ∞.Conside now
he egula ized equa ion
ωkAx + (1 −ωk)(x−z0) = 0 (1.15)
wi h k ixed and deno e by xki s unique solu ion. Then he e exis s ¯x∗∈ N(T) such ha
xk→¯x∗as k→ ∞.Mo eo e (see [4]), ¯x∗sa is ies he inequali y
h¯x∗−z0, J(¯x∗−x∗)i ≥ 0∀x∗∈ N(T).
I can be shown in he same way ha (1.8) wi h Tnin place o Tis he egula ized
successi e app oxima ion me hod o he equa ion (1.7) wi h o al asymp o ically nonex-
pansi e mapping. Indeed, i o al asymp o ically nonexpansi e mappings a e conside ed in
place o nonexpansi e mappings, hen
hAnx−Any, J(x−y)i ≥ −k(1)
nφ(kx−yk)kx−yk − k(2)
nkx−yk,(1.16)
whe e An=I−Tn.I is clea ha he analysis o s ong con e gence is mo e di icul in
his si ua ion, mo eo e , e y li le is known abou he s uc u e o he solu ion se . In
pa icula he same holds o asymp o ically nonexpansi e mappings o which (1.16) is
hAnx−Any, J(x−y)i ≥ −k(1)
nkx−yk2.(1.17)
The main esul o his pape , Theo em 3.1 in Sec ion 3, s a es a s ong con e gence
esul o he i e a i e scheme (1.4) in e lexi e Banach spaces wi h a weakly con inuous
duali y map on unbounded domains. No ice ha i is an open ques ion we he a e lexi e
Banach space admi ing a weakly sequen ially con inuous duali y mapping is uni o mly
smoo h. An implici scheme con e gence esul is also p o ed. This esul is used o
gua an ee he exis ence o sunny nonexpansi e e ac ions.
In Sec ion 4 we s udy he same i e a i e scheme o o al asymp o ically nonexpansi e
nonsel -mapping. Finally, in Sec ion 5, ou las sec ion we in es iga e he s abili y p ob-
lem o i e a i e schemes wi h espec o pe u ba ions o cons ain se s o nonexpansi e
nonsel -mappings.
5
2 P elimina ies
Le Xbe a eal Banach space wi h no m k·k,le X∗be i s dual space wi h he no m k · k∗
and, as usual, deno e he duali y pai ing o Xand X∗by hϕ, xi,whe e x∈Xand ϕ∈X∗
(in o he wo ds, hϕ, xiis he alue o ϕa x).
I is said ha Xis uni o mly smoo h i o any gi en ε > 0, he e exis s δ > 0 such ha
o all x, y ∈Xwi h kxk= 1 and kyk ≤ δ, he inequali y
2−1(kx+yk+kx−yk)−1≤εkyk
holds. The unc ion
ρX(τ) = sup{2−1(kx+yk+kx−yk)−1 : kxk= 1,kyk=τ}
is called he modulus o smoo hness o he space X.
This unc ion is inc easing and app oaches o ze o as τ→0.Deno e hX(τ) = ρX(τ)
τ.
Obse e ha he space Xis uni o mly smoo h i and only i lim
τ→0hX(τ) = 0.
Le ψ: [0,∞)→[0,∞) be a con inuous s ic ly inc easing unc ion such ha ψ( )→ ∞
as → ∞ and ψ(0) = 0. The gene alized duali y mapping Jψ:X→2X∗associa ed o a
gauge unc ion ψis de ined as
Jψ(x) = {x∗∈X∗:hx, x∗i=ψ(kxk)kxk,kx∗k=ψ(kxk)}, x ∈X.
In he case ha ψ( ) = hen Jψ=Jwhich is he no malized duali y map.
We say ha a Banach space Xhas a weakly con inuous duali y map ([5]) i he e exis s
a gauge unc ion ψ o which he gene alized duali y map Jψis single- alued and weak- o-
weak* sequen ially con inuous.
I is well-known ha Jψis he subdi e en ial, in he sense o con ex analysis, o he
con ex unc ion
Φ( ) = Z
0
ψ(τ)dτ, o τ≥0,
and ha Jψis single- alued i and only i Xis smoo h. We will need he ollowing subdi -
e en ial inequali y which is known o hold in smoo h spaces:
Φ(kx+yk)≤Φ(kxk) + hy, Jψ(x+y)i
o any x, y ∈X.
Nex we in oduce some de ini ions and auxilia y esul s ha will be needed in he
sequel.
De ini ion 2.1 Le Xbe a Banach space and Ca nonemp y closed con ex subse o X.
An ope a o T:C→Xis demiclosed (a y) i T(x) = ywhene e {xn} ⊆ Cis a sequence
weakly con e gen o xand T(xn)→yas n→ ∞.

6
De ini ion 2.2 A Banach space Xsa is ies he Opial’s condi ion i o each sequence {xn}
in X, he ela ion xn* x implies ha
lim sup
n→∞
kxn−xk<lim sup
n→∞
kxn−yk
o all y∈Xwi h x6=y.
De ini ion 2.3 A Banach space Xsa is ies he Gene alized Gossez-Lami Dozo p ope y
(GGLD-p ope y) i
lim in
n→∞ kxnk<lim sup
m→∞
lim sup
n→∞
kxm−xnk
whene e {xn}is a weak null sequence which is no no m con e gen .
The ollowing demiclosedness p inciple can be ound in [11].
Theo em 2.4 Le Xbe a Banach space wi h GGLD-p ope y and Opial’s condi ion. Le
Cbe a weakly compac con ex subse o Xand T:C→Ca uni o mly con inuous mapping
o asymp o ically nonexpansi e ype. Then I−Tis demiclosed a ze o.
The ollowing esul is well-known (see [13] and [14]).
P oposi ion 2.5 I in a e lexi e Banach space X he duali y mapping Jis weakly con in-
uous hen Xsa is ies GGLD-p ope y and Opial’s condi ion.
The nex co olla y ollows as a consequence o his p oposi ion and a ca e ul eading o
he o iginal p oo o Theo em 2.4.
Co olla y 2.6 Le Xbe a e lexi e Banach space wi h a weakly con inuous duali y mapping
J. Le Cbe a closed con ex subse o Xand T:C→Ca uni o mly con inuous mapping
and o al asymp o ically nonexpansi e wi h bounded o bi s. Then I−Tis demiclosed a
ze o.
We will also use he concep o a sunny nonexpansi e e ac ion [10] and, in pa icula ,
i s cha ac e iza ion by means o he duali y map in a smoo h Banach space.
De ini ion 2.7 Le Cbe a non-emp y subse o a Banach space Xand Da subse o C.
A mapping Q:C→Dis said o be
(i) a e ac ion on o Di Q2=Q;
(ii) a nonexpansi e e ac ion i i also sa is ies he inequali y
kQx −Qyk ≤ kx−yk, o all x, y ∈C;
(iii) a sunny e ac ion i o all x∈Cand o all 0≤ < ∞,
Q(Qx + (x−Qx)) = Qx, whene e Qx + (x−Qx)∈C.
7
P oposi ion 2.8 Assume ha Cis a non-emp y closed con ex subse o a smoo h Banach
space Xand Dis a subse o C. Then a nonexpansi e mapping Q:C→Dis a sunny
e ac ion i and only i o all x∈Cand o all ξ∈D,
hx−Qx, J(ξ−Qx)i ≤ 0.
In pa icula , he e is a mos one sunny nonexpansi e e ac ion on D.
Rema k 2.9 P oposi ion 2.5 and Co olla y 2.6 emain s ill alid i he no malized duali y
mapping Jis eplaced by he duali y mapping Jψwi h he gauge unc ion ψ( ).Mo eo e ,
we can use Jψ o cha ac e ize sunny nonexpansi e e ac ions in a smoo h Banach space
gi en by P oposi ion 2.8.
Le G1and G2be nonemp y closed subse s o X. The Hausdo dis ance be ween G1
and G2is de ined by he ollowing o mula:
H(G1, G2) = max{sup
z1∈G1
in
z2∈G2
kz1−z2k,sup
z1∈G2
in
z2∈G1
kz1−z2k}.
We need he ollowing lemma [3] in o de o p o e he main esul o Sec ion 5.
Lemma 2.10 I Xis a uni o mly smoo h Banach space, Ω1and Ω2a e closed con ex
subse s o Xsuch ha he Hausdo dis ance H(Ω1,Ω2)≤σand QΩ1and QΩ2a e he
(unique) sunny nonexpansi e e ac ions on o he subse s Ω1and Ω2, espec i ely, hen
kQΩ1x−QΩ2xk2≤16R(2 +d)hX(16LR−1σ),(2.1)
whe e =kxk, d = max{d1, d2}, R = 2(2 +d) + σand 1<L<1.7is he Figiel cons an
[1, 2, 9]. He e di=dis (θ, Ωi), i = 1,2,and θis he o igin o he space X.
We will o en apply he ollowing lemma on nume ical ecu en inequali ies.
Lemma 2.11 Le {λn}and {γn}be nonnega i e, {αn}be posi i e eal numbe s such ha
λn+1 ≤λn−αnλn+γn,∀n≥1.
Le o all n > 1γn
αn
≤c1and αn≤α. (2.2)
Then λn≤max{λ1, K∗},whe e K∗= (1 + α)c1. In addi ion, i
∞
X
1
αn=∞and γn
αn
→0
hen λn→0as n→ ∞.
8
3 Con e gence Analysis o Successi e App oxima ion Me hod
The goal o his sec ion is o p o e s ong con e gence o he egula ized successi e app ox-
ima ion me hod (1.4). Le us conside he explici scheme (1.4) gi en by
y0∈Ω, yn+1 =qnz0+ (1 −qn)Tnyn, n = 1,2, ...,
wi h
lim
n→∞ qn= 0 and
∞
X
n=1
qn=∞.(3.1)
Theo em 3.1 Le Ωbe a nonemp y closed and con ex subse o a smoo h e lexi e Banach
space Xwi h a weakly sequen ially con inuous duali y map Jψ,T: Ω →Ωa uni o mly
con inuous mapping which is o al asymp o ically nonexpansi e wi h nonemp y ixed poin se
N(T).Le N(T)be such ha he e exis s a sunny nonexpansi e e ac ion Q: Ω → N (T).
Le z0∈Ωand {qn} ⊂ (0,1] a sequence sa is ying (3.1). Le he sequence {yn}be
gene a ed by (1.4). Assume ha
lim
n→∞
k(1)
n+k(2)
n
qn
= 0,(3.2)
and ha he e exis posi i e cons an s M0and M1such ha φ(λ)≤M0λ o λ≥M1.
Suppose ha lim
nkyn−Tynk= 0, hen {yn}con e ges s ongly o he ixed poin ¯x∗=Qz0
o T.
P oo . Fi s ly we obse e ha he sequence {yn} ⊂ Ω because Ω is con ex. Take
x∗∈ N(T).I ollows om (1.4) ha
kyn+1 −x∗k ≤ qnkz0−x∗k+ (1 −qn)kTnyn−Tnx∗k
≤qnkz0−x∗k+ (1 −qn)kyn−x∗k+k(1)
nφ(kyn−x∗k) + k(2)
n.
Deno ing λn=kyn−x∗kwe ha e
λn+1 ≤(1 −qn)λn+qnkz0−x∗k+ (1 −qn)k(1)
nφ(λn) + k(2)
n.(3.3)
Since φis con inuous i a ains i s maximum Mon [0, M1]. Then i is easy o e i y ha
o all λ∈[0,∞)
φ(λ)≤M+M0λ.
The inequali y (3.3) is ew i en as
λn+1 ≤λn−qn−(1 −qn)k(1)
nM0λn+γn,
whe e
γn= (1 −qn)k(1)
nM+k(2)
n+qnkz0−x∗k.
9
Wi hou loss o gene ali y, in iew o (3.2), we assume ha he e exis cons an s α∈(0,1)
and M2>0 such ha o all n≥1
k(1)
n
qn
≤M0(1 −α)
1−qn
,(3.4)
and γn
qn
≤αM2.
Then
λn+1 ≤λn−αqnλn+γn.
By Lemma 2.11, we conclude ha
λn≤max{λ1,(1 + α)M2}.
Thus, he sequence {yn−x∗}is bounded, which, clea ly, implies ha {yn}is a bounded
sequence.
Applying he subdi e en ial inequali y o
yn+1 −Qz0= (1 −qn)(Tnyn−Qz0) + qn(z0−Qz0)
we deduce ha
Φ(kyn+1 −Qz0k)≤Φ((1 −qn)kTnyn−Qz0k) + qnhz0−Qz0, Jψ(yn+1 −Qz0)i.
Since Tis o al asymp o ically nonexpansi e and Qz0∈ N(T), we ha e
kTnyn−Qz0k ≤ kyn−Qz0k+νn,
whe e νn=Mk(1)
n+M0k(1)
nkyn−Qz0k+k(2)
nis bounded and anishes as n→ ∞. Now,
since Φ is a con ex and nondec easing, o nla ge enough we ha e
Φ(kTnyn−Qz0k)≤(1 −νn)Φ(kyn−Qz0k) + νnΦ(kyn−Qz0k+ 1)
≤Φ(kyn−Qz0k) + νnM3
o M3a sui able cons an . Consequen ly
Φ(kyn+1 −Qz0k)≤(1 −qn)Φ(kyn−Qz0k) + (1 −qn)νnM3+ (3.5)
+qnhz0−Qz0, Jψ(yn+1 −Qz0)i.
We claim ha
lim sup
n→∞
hz0−Qz0, Jψ(yn−Qz0)i ≤ 0.
Indeed, since he sequence {yn}is bounded and he space X e lexi e he e exis s a sub-
sequence {ynk}which is weakly con e gen in Ω. Le ¯ybe i s weak limi . We can ix his
subsequence so ha
lim sup
n→∞
hz0−Qz0, Jψ(yn−Qz0)i= lim
k→∞hz0−Qz0, Jψ(ynk−Qz0)i.
16
P oo . Fi s o all, we no e ha N(QΩT) is non-emp y because N(T)⊆ N (QΩT).
We show nex ha {yn} ⊆ Ω is bounded. Take x∗∈ N(T). In iew o he inequali ies
kyn+1 −x∗k ≤ qnkz0−x∗k+ (1 −qn)k(QΩT)nyn−(QΩT)nx∗k
≤qnkz0−x∗k+ (1 −qn)kQΩT(QΩT)n−1yn−QΩT(QΩT)n−1x∗k
≤qnkz0−x∗k+ (1 −qn)kT(QΩT)n−1yn−T(QΩT)n−1x∗k
≤qnkz0−x∗k+ (1 −qn)kyn−x∗k+k(1)
nφ(kyn−x∗k) + k(2)
n,
we conclude, by analogy wi h Theo em 3.1, ha he e exis posi i e cons an s C, C1and
¯
Msuch ha kynk ≤ C, kyn−x∗k ≤ C1and φ(kyn−x∗k)≤¯
M. Since QΩis a nonex-
pansi e mapping, i is no di icul o e i y ha he e exis s a cons an ¯
C > 0 such ha
k(QΩT)nynk ≤ ¯
C.
I ollows om (4.1) ha
lim
n→∞ yn+1 −(QΩT)nyn= 0.(4.3)
On he o he hand, we can show ha
lim
n→∞ yn+1 −(QΩT)nyn+1= 0.(4.4)
Indeed,
kyn+1 −(QΩT)nyn+1k≤kyn+1 −(QΩT)nynk+k(QΩT)nyn−(QΩT)nyn+1k.
Due o he o al asymp o ical nonexpansi eness o T, we ob ain
k(QΩT)nyn−(QΩT)nyn+1k ≤ kT(QΩT)n−1yn−T(QΩT)n−1yn+1k
≤ kyn−yn+1k+k(1)
nφ(kyn−yn+1k) + k(2)
n.
Now he boundedness o {(QΩT)nyn}, he ac ha kyn−yn+1k → 0 and (4.3) p o e
(4.4).
Fu he , we ha e
kyn−QΩTynk ≤ kyn−yn+1k+kyn+1 −(QΩT)nynk+k(QΩT)nyn−QΩT ynk.
F om he uni o m con inui y o T, we es ima e he las e m in he o m:
kQΩT(QΩT)n−1yn−QΩTynk ≤ kT(QΩT)n−1yn−Tynk ≤ ω(k(QΩT)n−1yn−ynk).
By (4.3) and (4.4), one ge s
lim(yn−QΩTyn) = 0.

17
No ice ha QΩTis a mapping as equi ed in Theo em 3.1 and hen we can apply
Theo em 3.1 o he sequence {yn}. Indeed, we w i e
yn+1 −Qz0= (1 −qn)((QΩT)nyn−Qz0) + qn(z0−Qz0).
we use he subdi e en ial inequali y as in Theo em 3.1, and he es o he p oo ollows he
pa e n wi h he only di e ence ha we ob ain ha {yn}s ongly con e ges o Qz0= ¯x∗
which is a ixed poin o QΩT.
5 S abili y Analysis o Nonexpansi e Nonsel -mappings
Nex we s udy he s abili y p oblem o i e a i e p ocesses wi h espec o pe u ba ions
o cons ain se s. Speci ically, we conside a p ocess which in ol es nonexpansi e nonsel -
mappings in he ollowing o m:
y1∈Ω1, yn+1 =qnz0+ (1 −qn)QΩn+1 Tyn, n = 1,2, ..., (5.1)
whe e QΩn:X→Ωnis a sunny nonexpansi e e ac ion, and he p oximi y be ween he
o iginal se Ω and Ωnwi h n= 1,2, ... is gi en by he Hausdo dis ance:
H(Ωn,Ω) ≤σn.(5.2)
Le G=∩nΩnand ¯
G= Ω ∩G6=∅.
Theo em 5.1 Le Xbe a uni o mly smoo h Banach space which has a weakly sequen ially
con inuous duali y map Jψ. Assume ha D⊂Xis a closed con ex se , Ω⊂Dand
Ωn⊂D, n = 1,2, ... a e closed con ex subse s o Xwi h p ope y (5.2). Le T:D→Xbe
a nonexpansi e mapping wi h ixed poin se N(T)such ha N(T)∩Ω6=∅.Take z0some
poin in ¯
G, x∗∈ N(T)∩Ω,{qn}a sequence in (0,1) sa is ying (3.1) and a non-inc easing
sequence σn≤σ, such ha σn→0as n→ ∞.Le he sequence {yn}be gene a ed by (5.1).
We suppose ha he sequence {T yn}is bounded,
lim
n→∞
|qn−qn−1|
qn
= 0 (5.3)
and
lim
n→∞ phX(σn)
qn
= 0.(5.4)
Then {yn}con e ges s ongly o he ixed poin ¯x∗=Qz0o QΩT, whe e Q: Ω → N (QΩT)
is he unique sunny nonexpansi e e ac ion on o N(QΩT).
P oo . We show i s ha {yn}is bounded. I is no di icul o see ha
kyn+1 −x∗k=kqnz0+ (1 −qn)QΩTyn−QΩTx∗+ (1 −qn)(QΩn+1 Tyn−QΩTyn)k
≤qnkz0−x∗k+ (1 −qn)kQΩTyn−QΩTx∗k+ (1 −qn)kQΩn+1 Tyn−QΩTynk.
18
Since {Tyn}is bounded and due o Lemma 2.10, he e exis posi i e cons an s M5and M6
such ha
kQΩn+1 Tyn−QΩTynk ≤ M5qhX(M6σn).
This implies
kyn+1 −x∗k ≤ qnkz0−x∗k+ (1 −qn)kyn−x∗k+M5qhX(M6σn).
Deno ing λn=kyn−x∗kwe ob ain
λn+1 ≤(1 −qn)λn+qnkz0−x∗k+M5qhX(M6σn).
F om Lemma 2.11 i ollows ha {yn−x∗}is bounded. Le kyn−x∗k ≤ C1 o all n.
Nex we e alua e he ollowing di e ence:
yn+1 −yn= (1 −qn)(QΩn+1 T yn−QΩnT yn−1)+(qn−qn−1)(z0−x∗)
+(qn−1−qn)(QΩnTyn−1−QΩTx∗)
= (1 −qn)(QΩn+1 T yn−QΩn+1 T yn−1) + (1 −qn)(QΩn+1 T yn−1−QΩnT yn−1)
+(qn−qn−1)(z0−x∗)+(qn−1−qn)(QΩnTyn−1−QΩnTx∗)
+(qn−1−qn)(QΩnTx∗−QΩT x∗).
Using he ollowing es ima es
kQΩn+1 Tyn−QΩn+1 T yn−1k ≤ kyn−yn−1k,
kQΩnTyn−1−QΩnT x∗k ≤ kyn−1−x∗k,
kQΩnTx∗−QΩT x∗k ≤ M7qhX(M8σn),
and
kQΩn+1 Tyn−1−QΩnT yn−1k ≤ M9qhX(M10σn),
o sui able cons an s M7, ..., M10 (see Lemma 2.10), we ob ain
kyn+1 −ynk ≤ (1 −qn)kyn−yn−1k+ (1 −qn)M9qhX(M10σn)
+|qn−qn−1|kz0−x∗k+C1) + M7qhX(M8σn).
Deno ing λn=kyn−yn−1kone has
λn+1 ≤(1 −qn)λn+γn,
whe e
γn=|qn−qn−1|(kz0−x∗k+C1) + |qn−qn−1|M7qhX(M8σn)
+ (1 −qn)M9qhX(M10σn).
19
Now, by (5.3), (5.4) and Lemma 2.11, we conclude ha kyn−yn−1k → 0.
Nex we show ha
lim
n→∞ kyn−QΩTynk= 0.(5.5)
Indeed,
kyn−QΩTynk≤kyn−QΩnT yn−1k+kQΩnTyn−1−QΩT yn−1k
+kQΩTyn−1−QΩT ynk
≤ kyn−QΩnTyn−1k+M7qhX(M8σn) + kyn−1−ynk,
which s a es ou claim since kyn−yn−1k → 0 and phX(M8σn)→0 as n→ ∞, and
kyn−QΩnTyn−1k=qn(z0−QΩnT yn−1)
which, om he boundedness o {QΩnT yn−1}, also ends o 0.
Now we w i e
yn+1 −Qz0= (1 −qn)(QΩn+1 T yn−Qz0) + qn(z0−Qz0),
and apply he subdi e en ial inequali y o Jψas in Theo em 3.1 o deduce ha
φ(kyn+1 −Qz0k)≤φ((1 −qn)kQΩn+1 Tyn−Qz0k) + qnhz0−Qz0, Jψ(yn+1 −Qz0)i.
Since Qz0∈ N(QΩT)⊆Ω,
kQΩn+1 Tyn−Qz0k ≤ kQΩn+1 T yn−QΩn+1 T Qz0k+kQΩn+1 T Qz0−QΩT Qz0k
≤ kyn−Qz0k+νn,
whe e νn=M7phX(M8σn+1) is bounded and anishes as n→ ∞. F om (5.4) we can
ollow he same easoning as in Theo em 3.1 o ob ain ( o nla ge enough)
φ(kyn+1 −Qz0k)≤(1 −qn)φ(kyn−Qz0k) + (1 −qn)νnM11+ (5.6)
+qnhz0−Qz0, Jψ(yn+1 −Qz0)i.
o M11 a sui able cons an . We claim ha
lim sup
n→∞
hz0−Qz0, Jψ(yn−Qz0)i ≤ 0.(5.7)
Since {yn}is bounded he e exis s a subsequence {ynk}, ynk∈Ωnk o each k, which
weakly con e ges o some poin ¯yand
lim sup
n→∞
hz0−Qz0, Jψ(yn−Qz0)i= lim
k→∞hz0−Qz0, Jψ(ynk−Qz0)i.
Now we apply he ollowing esul :
20
Lemma 5.2 [16] I he se Ωis con ex and closed se in e lexi e Banach space Xand he
sequence o se s Ωn⊆Xsa is y he limi ela ion H(Ωn,Ω) →0as n→ ∞ hen e e y weak
limi poin uo any sequence {un}, un∈Ωn,belongs o he subse Ω.
To deduce ha ¯y∈Ω.The es o he p oo ollows he pa e n o Theo em 3.1 once we
p o e ha ¯yis a ixed poin o QΩT. To p o e his we make use o Opial’s condi ion used
in he ollowing o m:
lim in
l→∞ kynl−¯yk<lim in
l→∞ kynl−QΩT¯yk
≤lim in
l→∞ (kynl−QΩTynlk+kQΩT ynl−QΩT¯yk)
≤lim in
l→∞ kQΩTynl−QΩT¯yk ≤ lim in
l→∞ kynl−¯yk.
Rema k 5.3 No ice ha in his heo em we do no ob ain con e gence o a ixed poin
o T. Con e gence o a ixed poin o Tcan be ob ained i we impose ce ain bounda y
condi ions on Tas, o ins ance, ha T(∂Ω) ⊆Ω. I is no ha d o see ha in his case
N(QΩT) = N(T).
Rema k 5.4 The sunny nonexpansi e e ac ion on o N(QΩT)always exis s in his case
since QΩTis a nonexpansi e sel -mapping [8].
Rema k 5.5 I he condi ion ¯
G= Ω∩G6=∅does no hold we p o e he p e ious heo em in
he ollowing way. Ins ead o Ωnwe conside he collec ion o se s Ω0
n=co(Ω ∪Ωn), whe e
co(A)s ands o he closed con ex closu e o a se A. I is easy o see ha H(Ω,Ω0
n)→0
as n→ ∞. Now i su ices o ollow he same p oo .
Rema k 5.6 The sequence {T yn}is bounded i , o ins ance, {yn}is bounded. This ob i-
ously holds i , o ins ance, Ωis bounded.
Re e ences
[1] Albe , Ya.I.: Gene alized p ojec ion ope a o s in Banach spaces: p ope ies and appli-
ca ions. Func ional Di e en ial Equa ions 1, 1-21 (1994).
[2] Albe , Ya.I., Chidume C.E., Zegeye H.: App oxima ing ixed poin s o o al asymp o -
ically nonexpansi e mapings , Fixed Poin Theo y and Applica ions 2006, a icle ID
10673, 1-20.
[3] Albe , Ya.I., Reich, S., Yao Jen-Chih: I e a i e me hods o sol ing ixed-poin p ob-
lems wi h nonsel -mappings in Banach spaces. Abs ac and Applied Analysis 4, 193-216
(2003).
[4] Albe , Ya.I. and Ryazan se a, I.P., Nonlinea Ill-posed P oblems o Mono one Type,
Sp inge , 2006.
21
[5] B owde , F.E.: Con e gence heo ems o sequences o nonlinea ope a o s in Banach
spaces. Ma h. Z. 100, 201-225 (1967).
[6] Chidume, C.E., Li, J., Udomene, A.: Con e gence o pa hs and app oxima ion o ixed
poin s o asymp o ically nonexpansi e mappings. P oc. Ame . Ma h. Soc. 133, 473-480
(2005).
[7] Chidume, C.E., O oedu, E.U., Zegeye, H.: S ong and weak con e gence heo ems o
asymp o ically nonexpansi e mappings. J. Ma h. Anal. Appl. 280, 364-374 (2003).
[8] Dom´ınguez Bena ides, T., Lo enzo Ram´ı ez, P.: S uc u e o he ixed poin se and
common ixed poin s o asymp o ically nonexpansi e mappings, P oc. Ame . Ma h.
Soc. 129 (12), 3549–3557 (2001).
[9] Figiel, T.: On he moduli o con exi y and smoo hness, S udia Ma hema ica 56, 121-
155 (1976).
[10] K. Goebel, K. and Reich, S., Uni o m Con exi y, Hype bolic Geome y, and Nonex-
pansi e Mappings, Ma cel Dekke , New Yo k and Basel, 1984.
[11] Ga c´ıa-False , J., Sims, B., Smy h, M.: The demiclosedness p inciple o mappings o
asymp o ically nonexpansi e ype, Hous on J. Ma h. 22, 101-108(1996).
[12] Halpe n, B.: Fixed poin s o non-expanding maps. Bull. Ame . Ma h. Soc. 73, 957-961
(1967).
[13] Khamsi, M.A., On uni o m Opial condi ion and uni o m Kadec-Klee p ope y in Ba-
nach and me ic spaces. Nonl. Anal. 26, 1733-1748 (1996).
[14] Lin, P.K., Tan, K.K., Xu, H.K.: Demiclosedness p inciple and asymp o ic beha iou
o asymp o ically nonexpansi e mappings. Nonl. Anal. 24, 929-946 (1995).
[15] Lim, T.C., Xu, H.X.: Fixed poin heo ems o asymp o ically nonexpansi e mappings.
Nonl. Anal. 22, 1345-1355 (1994).
[16] Lisko e s, O.A.: Regula iza ion o ill-posed a ia ional inequali ies on app oxima ely
gi en se s (Russian). Dokl. Akad. Nauk SSSR 317 (2), 300–304 (1991). (T ansla ion
in So ie Ma h. Dokl. 43 (2), 384–387 (1991).)
[17] Reich, S.: S ong con e gence heo ems o esol en s o acc e i e ope a o s in Banach
spaces. J. Ma h. Anal. Appl. 75, 287-292 (1980).
[18] Reich, S.: App oxima ing Fixed Poin s o Nonexpansi e Mapings. Paname ican Ma h-
ema ical Jou nal 423-28 (1994).
[19] Takahashi, W., Kim, G.E.: S ong con e gence o app oximan s o ixed poin s o
nonexpansi e nonsel -mappings in Banach spaces. Nonlinea Anal. 32, 447-454 (1998).
[20] Wi mann R.: App oxima ion o ixed poin s o nonexpansi e mappings. A ch. Ma h.
58, 486-491 (1992).

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[21] Xu, H.K.: S ong con e gence o an i e a i e me hod o nonexpansi e and acc e i e
ope a o s. J. Ma h. Anal. Appl. 314 (2), 631–643 (2006).