Volume 2010, Article ID 907275,16pages doi:10.1155/2010/907275
Research Article
Strong Convergence of a Generalized Iterative
Method for Semigroups of Nonexpansive Mappings in Hilbert Spaces
Husain Piri and Hamid Vaezi
Faculty of Mathematical Sciences, University of Tabriz, Tabriz 51664, Iran
Correspondence should be addressed to Husain Piri,[email protected] Received 20 April 2010; Accepted 18 June 2010
Academic Editor: A. T. M. Lau
Copyrightq2010 H. Piri and H. Vaezi. 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.
Using δ-strongly accretive and λ-strictly pseudocontractive mapping, we introduce a general iterative method for finding a common fixed point of a semigroup of non-expansive mappings in a Hilbert space, with respect to a sequence of left regular means defined on an appropriate space of bounded real-valued functions of the semigroup. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality.
1. Introduction
LetHbe a real Hilbert space. A mappingT ofHinto itself is called non-expansive ifTx− Ty ≤ x−y, for allx, y∈H. By FixT, we denote the set of fixed points ofTi.e., FixT {x∈H:Txx}.
Mann 1 introduced an iteration procedure for approximation of fixed points of a non-expansive mappingT on a Hilbert space as follows. Letx0∈Hand
xn1 1−αnTxnαnxn, n≥0, 1.1
where{αn}is a sequence in0,1. See also2.
On the other hand, Moudafi3introduced the viscosity approximation method for fixed point of non-expansive mappingssee4for further developments in both Hilbert and Banach spaces. Letf be a contraction on a Hilbert space H i.e., fx−fy ≤ αx−y,
for allx, y ∈ Hand 0 ≤α < 1. Starting with an arbitrary initialx0 ∈ H, define a sequence {xn}recursively by
xn1 1−αnTxnαnfxn, n≥0, 1.2 whereαnis sequence in0,1. It is proved in3,4that, under appropriate condition imposed on{αn}, the sequence{xn}generated by1.2converges strongly to the unique solutionx∗in FixTof the variational inequality:
I−f
x∗, x−x∗
≥0, x∈FixT. 1.3
Assume thatAis strongly positive, that is, there is a constantγ >0 with the property
Ax, x ≥γx2, ∀x∈H. 1.4 In4 see also5, it is proved that the sequence{xn}defined by the iterative method below, with the initial guessx0∈Hchosen arbitrarily,
xn1 I−αnATxnαnu, n≥0, 1.5 converges strongly to the unique solution of the minimization problem
x∈FixTmin 1
2Ax, x − x, u, 1.6
provided that the sequence{αn}satisfies certain conditions. Marino and Xu6combined the iterative1.5with the viscosity approximation method1.2and considered the following general iterative methods:
xn1 I−αnATxnαnγfxn, n≥0, 1.7 where 0 < γ < γ/α. They proved that if{αn}is a sequence in0,1satisfying the following conditions:
C1αn → 0, C2∞
n0αn∞, C3either∞
n0|αn1−αn|<∞or limn→ ∞αn1/αn 1,
then, the sequence{xn} generated by 1.7 converges strongly, as n → ∞, to the unique solution of the variational inequality:
A−γf
x∗, x−x∗
≥0, ∀x∈FixT, 1.8
which is the optimality condition for minimization problem
x∈FixTmin 1
2Ax, x −hx, 1.9
wherehis a potential function forγfi.e.,hx γfx, for allx∈H.
LetE∗be the topological dual of a Banach spaceE. The value ofj∈E∗atx∈Ewill be denoted byx, jorjx. With eachx∈E, we associate the set
Jx
j∈E∗: x, j
x2j2 . 1.10 Using the Hahn-Banach theorem, it is immediately clear thatJx/φ for eachx ∈ E. The multivalued mappingJ fromE intoE∗ is said to be thenormalizedduality mapping. A Banach spaceEis said to be smooth if the duality mappingJis single valued. As it is well known, the duality mapping is the identity whenEis a Hilbert space; see7.
Letδ andλbe two positive real numbers such thatδ, λ < 1. Recall that a mappingF with domainDFand rangeRFinEis calledδ-strongly accretive if, for eachx, y∈DF, there existsjx−y∈Jx−ysuch that
Fx−Fy, j x−y
≥δx−y2. 1.11 Recall also that a mappingF is calledλ-strictly pseudo-contractive if, for eachx, y ∈DF, there existsjx−y∈Jx−ysuch that
Fx−Fy, j x−y
≤x−y2−λx−y−Fx−Fy2. 1.12
It is easy to see that1.12can be rewritten as I−Fx−I−Fy, j
x−y
≥λI−Fx−I−Fy2, 1.13 see8.
In this paper, motivated and inspired by Atsushiba and Takahashi9, Lau et al.10, Marino and Xu 6and Xu 4,11, we introduce the iterative below, with the initial guess x0∈Hchosen arbitrarily,
xn1αnγfxn I−αnFTμnxn, n≥0, 1.14 where F is δ-strongly accretive and λ-strictly pseudo-contractive with δ λ > 1, f is a contraction on a Hilbert space H with coefficient 0 < α < 1, γ is a positive real number such thatγ < 1−
1−δ/λ/α, andϕ {Tt : t ∈ S}is a non-expansive semigroup onH such that the set Fixϕ of common fixed point ofϕ is nonempty,X is a subspace ofBS such that 1 ∈ X and the mapping t → Ttx, y is an element ofX for each x, y ∈ H, and{μn}is a sequence of means onX. Our purpose in this paper is to introduce this general iterative algorithm for approximating a common fixed points of semigroups of non-expansive
mappings which solves some variational inequality. We will prove that if{μn}is left regular and{αn}is a sequence in0,1satisfying the conditionsC1andC2, then{xn}converges strongly tox∗∈Fixϕ, which solves the variational inequality:
F−γf
x∗, x−x∗
≥0, ∀x∈Fix ϕ
. 1.15
Various applications to the additive semigroup of nonnegative real numbers and commuting pairs of non-expansive mappings are also presented. It is worth mentioning that we obtain our result without assuming conditionC3.
2. Preliminaries
LetSbe a semigroup and letBSbe the space of all bounded real-valued functions defined onSwith supremum norm. Fors∈Sandf ∈BS, we define elementslsfandrsfinBS by
lsf
t fst, rsf
t fts, ∀t∈S. 2.1
LetXbe a subspace ofBScontaining 1, and letX∗ be its dual. An elementμinX∗is said to be a mean onX ifμ μ1 1. We often writeμtftinstead ofμfforμ∈X∗and f ∈X. LetXbe left invariantresp., right invariant, that is,lsX⊂Xresp.,rsX⊂Xfor eachs∈S. A meanμonXis said to be left invariantright invariantifμlsf μf resp.
μrsf μffor eachs∈Sandf ∈X.Xis said to be leftresp., rightamenable ifXhas a leftresp., rightinvariant mean.X is amenable ifXis both left and right amenable. As it is well known,BSis amenable whenSis a commutative semigroup; see12. A net{μα}of means onXis said to be left regular if
limα ls∗μα−μα0, 2.2
for eachs∈S, wherel∗sis the adjoint operator ofls.
LetCbe a nonempty closed and convex subset of a reflexive Banach spaceE. A family ϕ{Tt:t∈S}of mapping fromCinto itself is said to be a non-expansive semigroup onC ifTtis non-expansive andTts TtTsfor eacht, s∈S. We denote by Fixϕthe set of common fixed points ofϕ, that is,
Fix ϕ
t∈S
{x∈C:Ttxx}. 2.3
The open ball of radiusrcentered at 0 is denoted byBr. For subsetDofE, by coD, we denote the closed convex hull ofD. Weak convergence is denoted by, and strong convergence is denoted by →.
Lemma 2.1see12,13. Let f be a function of semigroupSinto a reflexive Banach spaceEsuch that the weak closure of{ft:t∈S}is weakly compact, and letXbe a subspace ofBScontaining all functionst → ft, x∗withx∗∈E∗. Then, for anyμ∈X∗, there exists a unique elementfμin Esuch that
fμ, x∗ μt
ft, x∗
, 2.4
for allx∗∈E∗. Moreover, ifμis a mean onXthen
ftdμt∈co
ft:t∈S
. 2.5
One can writefμby
ftdμt.
Lemma 2.2see13. LetCbe a closed convex subset of a Hilbert spaceH,ϕ {Tt : t ∈ S}a semigroup fromCintoCsuch that Fixϕ/∅, the mappingt → Ttx, yan element ofX for each x∈Candy∈H, andμa mean onX. If one writesTμxinstead of
Ttxdμt, then the following holds.
iTμis non-expansive mapping fromCintoC.
iiTμx xfor eachx∈Fixϕ.
iiiTμx∈co{Ttx:t∈S}for eachx∈C.
ivIfμis left invariant, thenTμis a non-expansive retraction fromConto Fixϕ.
LetCbe a nonempty subset of a normed spaceE, and letx∈E. An elementy0 ∈Cis said to be the best approximation toxif
x−y0dx, C, 2.6
wheredx, C infy∈Cx−y. The numberdx, Cis called the distance fromxtoCor the error in approximatingxbyC. Thepossibly emptyset of all best approximation fromxto Cis denoted by
PCx
y∈C:x−ydx, C
. 2.7
This defines a mappingPC fromXinto 2Cand is called metricthe nearest pointprojection ontoC.
Lemma 2.3see7. LetCbe a nonempty convex subset of a smooth Banach spaceEand letx∈X andy∈C. Then, the following is equivalent.
iyis the best approximation tox.
iiyis a solution of the variational inequality
y−z, J x−y
≥0, ∀z∈C. 2.8
LetCbe a nonempty subset of a Banach spaceEandT:C → Ea mapping. ThenTis said to be demiclosed atv∈Eif, for any sequence{xn}inC, the following implication holds:
xn u∈C, Txn−→v, implyTuv. 2.9
Lemma 2.4see14. LetCbe a nonempty closed convex subset of a Hilbert spaceHand suppose thatT :C → His non-expansive. Then, the mappingI−T is demiclosed at zero.
The following lemma is well known.
Lemma 2.5. LetHbe a real Hilbert space. Then, for allx, y∈H ix−y2≤ x22y, xy,
iix−y2≥ x22y, x.
Lemma 2.6see11. Let{an}be a sequence of nonnegative real numbers such that
an1 ≤1−bnanbncn, n≥0, 2.10 where{bn}and{cn}are sequences of real numbers satisfying the following conditions:
i{bn} ⊂0,1,∞
n0bn∞, iieither lim supn→ ∞cn≤0 or∞
n0|bncn|<∞.
Then, limn→ ∞an0.
The following lemma will be frequently used throughout the paper. For the sake of completeness, we include its proof.
Lemma 2.7. LetEbe a real smooth Banach space andF :E → Ea mapping.
iIfFisδ-strongly accretive andλ-strictly pseudo-contractive withδλ >1, then,I−Fis contractive with constant
1−δ/λ.
iiIfFisδ-strongly accretive andλ-strictly pseudo-contractive withδλ >1, then, for any fixed numberτ ∈0,1,I−τFis contractive with constant 1−τ1−
1−δ/λ. Proof. iFrom1.11and1.13, we obtain
λI−Fx−I−Fy2≤x−y2−
Fx−Fy, J x−y
≤1−δx−y2. 2.11 Becauseδλ >1⇔
1−δ/λ∈0,1, we have I−Fx−I−Fy≤
1−δ
λ x−y, 2.12
and, therefore,I−Fis contractive with constant
1−δ/λ.
iiBecauseI−Fis contractive with constant
1−δ/λ, for each fixed numberτ ∈ 0,1, we have
x−y−τ
Fx−F
y1−τ x−y
τ
I−Fx−I−Fy
≤1−τx−yτI−Fx−I−Fy
≤1−τx−yτ
1−δ
λ x−y
⎛
⎝1−τ
⎛
⎝1−
1−δ λ
⎞
⎠
⎞
⎠x−y.
2.13
This shows thatI−τFis contractive with constant 1−τ1−
1−δ/λ.
Throughout this paper, F will denote a δ-strongly accretive and λ-strictly pseudo- contractive mapping with δλ > 1, andf is a contraction with coefficient 0 < α < 1 on a Hilbert spaceH. We will also always useγto mean a number in0,1−
1−δ/λ/α.
3. Strong Convergence Theorem
The following is our main result.
Theorem 3.1. Letϕ{Tt:t∈S}be a non-expansive semigroup on a real Hilbert spaceHsuch that Fixϕ/∅. LetXbe a left invariant subspace ofBSsuch that 1∈X, and the functiont → Ttx, y is an element ofXfor eachx, y∈H. Let{μn}be a left regular sequence of means onX, and let{αn} be a sequence in0,1such thatαn → oand∞
n0αn ∞. Letx0 ∈ Hand{xn}be generated by the iteration algorithm1.14. Then,{xn}converges strongly, asn → ∞, tox∗∈Fixϕ, which is a unique solution of the variational inequality1.15. Equivalently, one has
PFixϕ
I−Fγf
x∗x∗. 3.1
Proof. First, we claim that{xn}is bounded. Letp∈Fixϕ; by Lemmas2.2and2.7we have xn1−pαnγfxn I−αnFTμnxn−p
αnγfxn I−αnFTμnxn−I−αnFp−αnF p
≤αnγfxn−F
pI−αnFTμnxn−I−αnFp
≤αnγfxn−γf p αnγf
p
−F p
⎛
⎝1−αn
⎛
⎝1−
1−δ λ
⎞
⎠
⎞
⎠Tμnxn−p
≤
⎛
⎝1−αn
⎛
⎝1−
1−δ λ −γα
⎞
⎠
⎞
⎠xn−pαnγf p
−F p
⎛
⎝1−αn
⎛
⎝1−
1−δ λ −γα
⎞
⎠
⎞
⎠xn−p
αn
1−
1−δ/λ−γα
1−γα−
1−δ/λ γf p
−F p
≤max
⎧⎪
⎨
⎪⎩
⎛
⎝1−
1−δ λ −γα
⎞
⎠
−1γf p
−F
p,xn−p
⎫⎪
⎬
⎪⎭.
3.2
By induction,
xn−p≤max
⎧⎪
⎨
⎪⎩
⎛
⎝1−
1−δ λ −γα
⎞
⎠
−1γf p
−F
p,x0−p
⎫⎪
⎬
⎪⎭M0. 3.3
Therefore,{xn}is bounded and so is{fxn}.
SetD {y ∈ H : y−p ≤ M0}. We remark thatD isϕ-invariant bounded closed convex set and{xn} ⊂D. Now we claim that
lim sup
n→ ∞ sup
y∈D
Tμn
y
−Tt
Tμn
y0, ∀t∈S. 3.4
Let >0. By15, Theorem 1.2, there existsδ >0 such that
coFδTt;D Bδ⊂FTt;D, ∀t∈S. 3.5
Also by15, Corollary 1.1, there exists a natural numberNsuch that
1 N1
N i0
Ttis
y
−Tt
! 1 N1
N i0
Ttis
y"
≤δ, 3.6
for allt, s∈Sandy∈D. Lett∈S. Since{μn}is strongly left regular, there existsn0 ∈Nsuch thatμn−l∗tiμn ≤δ/M0pforn≥n0andi1,2, . . . , N. Then we have
sup
y∈D
Tμn
y
− 1
N1
N i0
Ttis
y
dμns sup
y∈Dsup
z1
####
# Tμn
y , z
−
$ 1 N1
N i0
Ttis
y
dμns, z%##
### sup
y∈Dsup
z1
####
# 1 N1
N i0
μn
s
Ts
y , z
− 1 N1
N i0
μn
s
Ttis
y , z##
###
≤ 1 N1
N i0
sup
y∈Dsup
z1
### μn
s
Ts
y , z
− l∗tiμn
s
Ts
y , z###
≤ max
i0,1,2,...,Nμn−lt∗iμnM0p≤δ, ∀n≥n0.
3.7
ByLemma 2.2we have 1
N1
N i0
Ttis
y
dμns∈co
&
1 N1
N i0
Tti
Ts
y :s∈S
'
. 3.8
It follows from3.5,3.6,3.7, and3.8that
Tμn
y
∈co
&
1 N1
N i0
Ttis
y :s∈S
'
Bδ⊂coFδTt;D Bδ⊂FTt;D, 3.9
for ally∈Dandn≥n0. Therefore, lim sup
n→ ∞ sup
y∈D
Tt
Tμn
y
−Tμn
y≤. 3.10
Since >0 is arbitrary, we get3.4. In this stage, we will show that
n→ ∞limxn−Ttxn0, ∀t∈S. 3.11
Lett∈Sand >0. Then, there existsδ >0, which satisfies3.5. Take
L0
⎡
⎣
⎛
⎝1γα
1−δ λ
⎞
⎠M0γf p
−F p⎤
⎦. 3.12
From limn→ ∞αn 0 and3.4there existsn0 ∈Nsuch thatαn ≤δ/L0andTμnxn ∈FδTt, for alln≥n0. ByLemma 2.7, we have
αnγfxn−FTμnxn
≤αnγfxn−γf
pγf p
−F
Tμnxn
≤αn
γαxn−pγf p
−F
p
αnI−Fp−I−FTμnxnp−Tμnxn
≤αn
⎛
⎝1
1−δ λ γα
⎞
⎠xn−pαnγf p
−F p
≤αn
⎡
⎣
⎛
⎝1
1−δ λ γα
⎞
⎠M0γf p
−F p⎤
⎦
≤αnL0≤δ,
3.13
for alln≥n0. Therefore, we have xn1Tμnxn αn
γfxn F
Tμnxn
∈FδT Bδ⊂FTt, 3.14
for alln≥n0. This shows that
xn−Ttxn ≤, ∀n≥n0. 3.15 Since >0 is arbitrary, we get3.11.
LetQPFixϕ. ThenQI−F−γfis a contraction ofHinto itself. In fact, we see that Q
I−Fγf
x−Q
I−Fγf y
≤I−Fγf x−
I−Fγf y
≤I−Fx−I−F
yγfx−f y
≤
⎛
⎝
1−δ λ γα
⎞
⎠x−y,
3.16
and henceQI−F−γfis a contraction due to
1−δ/λγα∈0,1.
Therefore, by Banach contraction principal,PFixϕγfI−Fhas a unique fixed point x∗. Then usingLemma 2.3,x∗is the unique solution of the variational inequality
F−γf
x∗, x−x∗ ≥0, ∀x∈Fix ϕ
. 3.17
We show that
lim sup
n→ ∞
γfx∗−Fx∗, xn−x∗
≤0. 3.18
Indeed, we can choose a subsequence{xnk}of{xn}such that lim sup
n→ ∞
γfx∗−Fx∗, xn−x∗ lim
k→ ∞
γfx∗−Fx∗, xnk −x∗
. 3.19
Because{xn}is bounded, we may assume thatxn z. In terms ofLemma 2.4and3.11, we conclude thatz∈Fixϕ. Therefore,
lim sup
n→ ∞
γfx∗−Fx∗, xn−x∗
γfx∗−Fx∗, z−x∗
≤0. 3.20
Finally, we prove thatxn → x∗asn → ∞. By Lemmas2.5and2.7we have xn1−x∗2
αnγfxn I−αnFTμnxn−x∗2
αnγfxn−αnFx∗ I−αnFTμnxn−I−αnFx∗2 I−αnFTμnxn−I−αnFx∗22αn
γfxn−Fx∗, xn1−x∗
≤
⎛
⎝1−αn
⎛
⎝1−
1−δ λ
⎞
⎠
⎞
⎠
2
xn−x∗22αn
γfxn−Fx∗, xn1−x∗
≤
⎛
⎝1−αn
⎛
⎝1−
1−δ λ
⎞
⎠
⎞
⎠
2
xn−x∗22αn
γfxn−γfx∗, xn1−x∗ 2αn
γfx∗−Fx∗, xn1−x∗ .
3.21
On the other hand
γfxn−γfx∗, xn1−x∗
≤γαxn−x∗xn1−x∗
≤γα
⎛
⎝1−αn
⎛
⎝1−
1−δ λ
⎞
⎠
⎞
⎠xn−x∗2 γαxn−x∗,
2##γfxn−Fx∗, xn1−x∗##√ αn.
3.22
Since{xn}and{fxn}are bounded, we can take a constantG0>0 such that γαxn−x∗,
2##γfxn−Fx∗, xn1−x∗##< G0, ∀n∈N. 3.23 So from the above, we reach the following:
γfxn−γfx∗, xn1−x∗
≤γα
⎛
⎝1−αn
⎛
⎝1−
1−δ λ
⎞
⎠
⎞
⎠xn−x∗2G0√
αn. 3.24
Substituting3.24in3.21, we obtain xn1−x∗2
≤
⎛
⎝1−αn
⎛
⎝1−
1−δ λ
⎞
⎠
⎞
⎠
2
xn−x∗22αnγα
⎛
⎝1−αn
⎛
⎝1−
1−δ λ
⎞
⎠
⎞
⎠xn−x∗2 2αnG0√
αn2αn
γfxn−Fx∗, xn1−x∗
⎛
⎝1−2αn
⎡
⎣
⎛
⎝1−
1−δ λ
⎞
⎠−αγαnγα
⎛
⎝1−
1−δ λ
⎞
⎠
⎤
⎦
⎞
⎠xn−x∗2
αn
⎡
⎢⎣αn
⎛
⎝1−
1−δ λ
⎞
⎠
2
xn−x∗22G0√ αn2
γfx∗−Fx∗, xn−x∗
⎤
⎥⎦.
3.25
It follows that xn1−x∗2≤
⎛
⎝1−αn
⎡
⎣2
⎛
⎝1−
1−δ λ −αγ
⎞
⎠2αnγα
⎛
⎝1−
1−δ λ
⎞
⎠
⎤
⎦
⎞
⎠xn−x∗2αnβn, 3.26 where
βn
⎡
⎢⎣αn
⎛
⎝1−
1−δ λ
⎞
⎠
2
xn−x∗22G0√ αn2
γfx∗−Fx∗, xn−x∗
⎤
⎥⎦. 3.27
Since{xn}is bounded and limn→ ∞αn0, by3.18, we get lim sup
n→ ∞ βn≤0. 3.28
Consequently, applyingLemma 2.6, to3.26, we conclude thatxn → x∗.
Corollary 3.2. LetX,ϕ,{μn}, and{αn}be as inTheorem 3.1. Suppose thatAa strongly positive bounded linear operator onHwith coefficientγ > 1/2 and 0< ζ < 1−,
2−2γ/α. Let{xn}be defined by the iterative algorithm
xn1αnζfxn I−αnATμnxn, n≥0. 3.29 Then, {xn} converges strongly, as n → ∞, to x∗ ∈ Fixϕ, which is a unique solution of the variational inequality
A−ζf
x∗, x−x∗
≥0, ∀x∈Fix ϕ
. 3.30
Proof. BecauseA is strongly positive bounded linear operator onH with coefficientγ, we have
Ax−Ay, x−y ≥γx−y2. 3.31 Therefore,Aisγ-strongly accretive. On the other hand,
I−Ax−I−Ay2
x−y
−
Ax−Ay ,
x−y
−
Ax−Ay
x−y, x−y
−2
Ax−Ay, x−y
Ax−Ay, Ax−Ay
≤x−y2−2
Ax−Ay, x−y
Ax−y2.
3.32
SinceAis strongly positive if and only if1/AAis strongly positive, we may assume, with no loss of generality, thatA1, so that
Ax−Ay, x−y
≤x−y2−1
2I−Ax−I−Ay2. 3.33 This shows thatAis 1/2-strictly pseudo-contractive. Now applyTheorem 3.1to conclude the result.
Corollary 3.3. Let X,ϕ, {μn} and {αn} be as in Theorem 3.1. Suppose u, x0 ∈ H and define a sequence{xn}by the iterative algorithm
xn1αnu I−αnFTμnxn, n≥0. 3.34 Then, {xn} converges strongly, as n → ∞, to a x∗ ∈ Fixϕ, which is a unique solution of the variational inequality
Fx∗−u, x−x∗ ≥0, ∀x∈Fix ϕ
. 3.35
Proof. It is sufficient to takefuandγ1 inTheorem 3.1.
4. Some Application
Corollary 4.1. LetSandTbe non-expansive mappings on a Hilbert spaceHwithSTTSsuch that FixS∩FixT/∅. Let{αn}be a sequence in0,1satisfying conditionsαn → 0 and∞
n0αn∞.
Letx0∈H,γ∈0,1−
1−δ/λ/αand define a sequence{xn}by the iterative algorithm:
xn1 αnγfxn I−αnF 1 n2
n−1 i0
n−1 j0
SiTixn, n≥0. 4.1
Then, {xn} converges strongly, asn → ∞, to x∗ ∈ FixS∩FixTwhich solves the variational inequality:
F−γf
x∗, x−x∗
≥0, ∀x∈FixS∩FixT. 4.2
Proof. LetTi, j SiTjfor eachi, j ∈ N∪ {0}. Then{Ti, j :i, j ∈ N∪ {0}}is a semigroup of non-expansive mappings onH. Now, for eachn ∈ Nandi, j ∈ BN∪ {0}2, we define μnf 1/n2n−1
i0 n−1
j0 fi, j.Then,{μn}is regular sequence of means16. Next, for each x∈Handn∈N, we have
Tμnx 1 n2
n−1 i0
n−1 j0
SiTjx. 4.3
Therefore, applyingTheorem 3.1, the result follows.
Corollary 4.2. Letϕ{Tt:t∈R}be a strongly continuous semigroup of non-expansive mappings on a Hilbert spaceHsuch that Fixϕ/∅. Letαnbe a sequence in0,1satisfying conditionsαn → 0 and∞
n0αn∞. Letx0∈Handγ ∈0,1−
1−δ/λ/α. Let{xn}be a sequence defined by the iterative algorithm:
xn1αnγfxn I−αnF1 tn
tn
0
Tsxnds, n≥0, 4.4
where{tn}is an increasing sequence in0,∞such that limn→ ∞tn ∞and limn→ ∞tn/tn1 1.
Then,{xn}converges strongly, asn → ∞, tox∗∈Fixϕ, which solves the variational inequality F−γf
x∗, x−x∗
≥0, ∀x∈Fix ϕ
. 4.5
Proof. Forn∈N, we defineμnf 1/tntn
0 ftdtfor eachf∈CR, whereCRdenotes the space of all real-valued bounded continuous functions onRwith supremum norm. Then, {μn} is regular sequence of means 16. Furthermore, for each x ∈ H, we haveTμnx 1/tntn
0 Tsxds. Now, applyTheorem 3.1to conclude the result.
Corollary 4.3. Letϕ{Tt:t∈R}be a strongly continuous semigroup of non-expansive mappings on a Hilbert spaceHsuch that Fixϕ/∅. Letαnbe a sequence in0,1satisfying conditionsαn → 0 and∞
n0αn∞. Letx0∈Handγ ∈0,1−
1−δ/λ/α. Let{xn}be a sequence defined by the iterative algorithm
xn1αnγfxn I−αnFrn
∞
0
exp−rnsTsxnds, n≥0, 4.6
where {rn} is an decreasing sequence in 0,∞ such that limn→ ∞rn 0. Then {xn} converges strongly, asn → ∞, tox∗∈Fixϕ, which solves the variational inequality
F−γf
x∗, x−x∗
≥0, ∀x∈Fix ϕ
. 4.7
Proof. For n ∈ N, we define μnf rn
∞
0 exp−rntftdt for each f ∈ CR. Then {μn} is regular sequence of means 16. Furthermore, for each x ∈ H, we haveTμnx rn
∞
0 exp−rntTtxdt. Now, applyTheorem 3.1to conclude the result.
Corollary 4.4. LetTbe a non-expansive mapping on a Hilbert spaceHsuch that FixT/∅. Letαn
be a sequence in0,1satisfying conditionsαn → 0 and∞
n0αn ∞and letQ {qn,m}be a strongly regular matrix. Letx0∈Handγ∈0,1−
1−δ/λ/α. Let{xn}be a sequence defined by the iterative algorithm
xn1αnγfxn I−αnF∞
m0
qn,mTmxn, n≥0. 4.8
Then,{xn}converges strongly, asn → ∞, tox∗∈FixTwhich solves the variational inequality
F−γf
x∗, x−x∗ ≥0, ∀x∈FixT. 4.9 Proof. For eachn∈N, we define
μn
f ∞
m0
qn,mfm, 4.10
for eachf∈BN∪{0}. SinceQis a strongly regular matrix, for eachm, we haveqn,m → 0, as n → ∞; see17. Then, it is easy to see that{μn}is regular sequence of means. Furthermore, for eachx∈H, we haveTμnx ∞
m0qn,mTmx.Now, applyTheorem 3.1to conclude the result.
Acknowledgments
The authors thank the refereesfor the helpful comments, which improved the presentation of this paper. This paper is dedicated to Professor Anthony To Ming Lau. This paper is based on final report of the research project of the Ph.D. thesis which is done with financial support of research office of the University of Tabriz.
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