Volume 2009, Article ID 936121,16pages doi:10.1155/2009/936121
Research Article
Viscosity Approximation of Common Fixed Points for L-Lipschitzian Semigroup of Pseudocontractive Mappings in Banach Spaces
Xue-song Li,
1Jong Kyu Kim,
2and Nan-jing Huang
11Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2Department of Mathematics, Kyungnam University, Masan, Kyungnam 631-701, South Korea
Correspondence should be addressed to Nan-jing Huang,[email protected] Received 14 January 2009; Accepted 5 March 2009
Recommended by Charles E. Chidume
We study the strong convergence of two kinds of viscosity iteration processes for approximating common fixed points of the pseudocontractive semigroup in uniformly convex Banach spaces with uniformly Gˆateaux differential norms. As special cases, we get the strong convergence of the implicit viscosity iteration process for approximating common fixed points of the nonexpansive semigroup in Banach spaces satisfying some conditions. The results presented in this paper extend and generalize some results concerned with the nonexpansive semigroup inChen and He, 2007 and the pseudocontractive mapping inZegeye et al., 2007to the pseudocontractive semigroup in Banach spaces under different conditions.
Copyrightq2009 Xue-song Li et al. 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.
1. Introduction
LetEbe a real Banach space with the dual spaceE∗andJ:E → 2E∗be a normalized duality mapping defined by
Jx
x∗∈E∗: x, x∗
x2x∗2
, 1.1
where·,·denotes the generalized duality pairing. It is well known thatsee, e.g.,1, pages 107–113
iJis single-valued ifE∗is strictly convex;
iiEis uniformly smooth if and only ifJ is single-valued and uniformly continuous on any bounded subset ofE.
LetKbe a nonempty closed convex subset ofE. A mappingT :K → Eis said to be inonexpansive if
Tx−Ty ≤ x−y, ∀x, y∈K, 1.2
iiL-Lipschitzian if there exists a constantL >0 such that
Tx−Ty ≤Lx−y, ∀x, y∈K, 1.3
iiik-strongly pseudocontractive if there exist a constant k ∈ 0,1 and jx−y ∈ Jx−ysuch that
Tx−Ty, jx−y
≤kx−y2, ∀x, y∈K, 1.4
ivpseudocontractive if there existsjx−y∈Jx−ysuch that Tx−Ty, jx−y
≤ x−y2, ∀x, y∈K. 1.5
It is easy to see that the pseudocontractive mapping is more general than the nonexpansive mapping.
A pseudocontractive semigroup is a family, Γ:
Tt:t≥0
, 1.6
of self-mappings onKsuch that 1T0xxfor allx∈K;
2Ts txTsTtxfor allx∈Kands, t ≥0;
3Ttis pseudocontractive for eacht≥0;
4for eachx∈K, the mappingT·xfromR intoKis continuous.
If the mappingTtin condition3is replaced by 3Ttis nonexpansive for eacht≥0;
thenΓ:{Tt:t≥0}is said to be a nonexpansive semigroup onK.
We denote byFΓthe common fixed points set of pseudocontractive semigroup Γ, that is,
FΓ
t∈R
F Tt
x∈K:Ttxxfor eacht≥0
. 1.7
In the sequel, we always assume thatFΓ/∅.
In recent decades, many authors studied the convergence of iterative algorithms for nonexpansive mappings, nonexpansive semigroup, and pseudocontractive mapping in Banach spacessee, e.g.,2–14. LetΓ:{Tt:t≥0}be a nonexpansive semigroup fromK into itself andf :K → Kbe a contractive mapping. It follows from Banach’s fixed theorem that the following implicit viscosity iteration process is well defined:
xnαnf xn
1−αn T
tn
xn, ∀n≥1, 1.8
whereαn ∈0,1andTtn∈ Γ. Some authors studied the convergence of iteration process 1.8for nonexpansive mappings in certain Banach spaces see5,10. Recently, Xu 11 studied the following implicit iteration process: for anyu∈K,
xnαnu 1−αn
T tn
xn, ∀n≥1, 1.9
whereαn∈0,1,Ttn∈Γ, and obtained the convergence theorem as follows.
Theorem X see 11. Let E be a uniformly convex Banach space having a weakly continuous duality mapJϕwith gaugeϕ,Ka nonempty closed convex subset ofEand
Γ:
Tt:t≥0
1.10
a nonexpansive semigroup onKsuch thatF FixΓ/∅. If
nlim→ ∞tn lim
n→ ∞
αn
tn 0, 1.11
then{xn}generated by1.9converges strongly to a member ofF.
Xu11also proposed the following problem.
Problem X see 11. We do not know if Theorem X holds in a uniformly convex and uniformly smooth Banache.g.,Lp for 1< p <∞.
This problem has been solved by Li and Huang15and Suzuki8, respectively.
Moudafi’s viscosity approximation method has been recently studied by many authors see, e.g.,2, 3,5,10,13,15–17and the references therein. Chen and He 3studied the convergence of1.8constructed from a nonexpansive semigroup and a contractive mapping in a reflective Banach space with a weakly sequentially continuous duality mapping. Zegeye et al.13studied the convergence of 1.8constructed from a pseudocontractive mapping and a contractive mapping.
On the other hand, many authors see 2, 3, 5, 13 studied the following explicit viscosity iteration process: for any giveny0∈K,
yn 1 1−λn
1 θn
yn λnT tn
yn λnθnf yn
, ∀n≥0, 1.12
where Ttn ∈ Γ, λn, θn ∈ 0,1 and λn1 θn ∈ 0,1. Chen and He 3 studied the convergence of 1.12 constructed from a nonexpansive semigroup and obtained some convergence results.
An interesting work is to extend some results involving nonexpansive mapping, nonexpansive semigroup, and pseudocontractive mapping to the semigroup of pseu- docontractive mappings. Li and Huang 15 generalized some corresponding results to pseudocontractive semigroup in Banach spaces. Some further study concerned with approximating common fixed points of the semigroup of pseudocontractive mappings in Banach spaces, we refer to Li and Huang16.
Motivated by the works mentioned above, in this paper, we study the convergence of implicit viscosity iteration process1.8constructed from the pseudocontractive semigroup Γ : {Tt : t ∈ R } and k-strongly pseudocontractive mapping in uniformly convex Banach spaces with uniformly Gˆateaux differential norms. As special cases, we obtain the convergence of the implicit iteration process for approximating the common fixed point of the nonexpansive semigroup in certain Banach spaces. We also study the convergence of the explicit viscosity iteration process 1.12 constructed from the pseudocontractive semigroupΓandk-strongly pseudocontractive mapping in uniformly convex Banach spaces with uniformly Gˆateaux differential norms. The results presented in this paper extend and generalize some results concerned with the nonexpansive semigroup in 3 and the pseudocontractive mapping in 13 to the pseudocontractive semigroup in Banach spaces under different conditions.
2. Preliminaries
A real Banach spaceEis said to have a weakly continuous duality mapping if J is single- valued and weak-to-weak∗ sequentially continuous i.e., if each {xn} is a sequence in E weakly convergent to x, then {Jxn} converges weakly∗ to Jx. Obviously, if E has a weakly continuous duality mapping, thenJ is norm-to-weak∗ sequentially continuous. It is well known thatlp1 < p < ∞posses duality mapping which is weakly continuoussee, e.g.,11.
Letl∞be the Banach space of all bounded real-valued sequences. A Banach limit LIM see1is a linear continuous functional onl∞such that
LIMLIM1 1, LIM
t1, t2, . . .
LIM
t2, t3, . . .
2.1
for eacht t1, t2, . . .∈l∞. If LIM is a Banach limit, then it follows from1, Theorem 1.4.4 that
lim inf
n→ ∞ tn≤LIMt≤lim sup
n→ ∞ tn 2.2
for eacht t1, t2, . . .∈l∞.
A mappingT with domainDTand rangeRTin Eis said to be demiclosed at a pointp∈Eif whenever{xn}is a sequence inDTwhich converges weakly tox∈DTand {Txn}converges strongly top, thenTxp.
For the sake of convenience, we restate the following lemmas that will be used.
Lemma 2.1see18. LetEbe a Banach space,K be a nonempty closed convex subset ofE, and T : K → Kbe a strongly pseudocontractive and continuous mapping. ThenT has a unique fixed point inK.
Lemma 2.2see19. LetEbe a Banach space andJbe the normalized duality mapping. Then for anyx, y∈Eandjx y∈Jx y,
x y2 ≤ x2 2
y, jx y
. 2.3
Lemma 2.3see12. Letr >0. Then a real Banach spaceEis uniformly convex if and only if there exists a continuous and strictly increasing convex functiong :0,∞ → 0,∞withg0 0 such that
λx 1−λy2≤λx2 1−λy2−λ1−λg
x−y
2.4
for allx, y∈Br,λ∈0,1, whereBr {x∈E:x ≤r}.
Lemma 2.4see9. Let{an}be a sequence of nonnegative real numbers such that
an 1≤ 1−τn
an ηn, 2.5
whereτn∈0,1,∀n≥n0,n0∈Nis fixed,∞
n1τn∞, andηnoτn. Then limn→ ∞an 0.
3. Main Results
We first discuss the convergence of implicit viscosity iteration process1.8constructed from a pseudocontractive semigroupΓ:{Tt:t≥0}.
Theorem 3.1. LetKbe a nonempty closed convex subset of a real Banach spaceE. LetΓ :{Tt: t ∈ R } be an L-Lipschitzian semigroup of pseudocontractive mappings and f : K → K be an Lf-Lipschitziank-strongly pseudocontractive mapping. Suppose that for any bounded subsetC⊂K,
lims→0sup
x∈C
Tsx−x0. 3.1
Then the sequence{xn}generated by1.8is well defined. Moreover, if
nlim→ ∞tn lim
n→ ∞
αn
tn 0, 3.2
then limn→ ∞xn−Ttxn0 for anyt∈R . Proof. Let
Tnx:αnfx 1−αn
T tn
x, ∀n≥1. 3.3
Since
Tnx−Tny, jx−y
1−αn T
tn x−T
tn
y, jx−y αn
fx−fy, jx−y
≤ 1−αn1−kx−y2,
3.4
we know that Tn is strongly pseudocontractive and strongly continuous. It follows from Lemma 2.1thatTn has a unique fixed pointsay xn ∈ K, that is,{xn}generated by1.8 is well defined.
Takingp∈FΓ, we have xn−p2 αn
f xn
−p, j
xn−p
1−αn T
tn xn−T
tn p, j
xn−p
≤αnkxn−p2 αn f
p
−p, j
xn−p
1−αnxn−p2
≤ 1−αn1−kxn−p2 αnfp−pxn−p,
3.5
and soxn−p ≤ 1/1−kfp−p. This means{xn}is bounded. By the Lipschitzian conditions ofΓandf, it follows that{Ttnxn}and{fxn}are bounded. Therefore,
xn−T tn
xnαnf xn
−T tn
xn−→0. 3.6
For any givent >0, xn−Ttxnt/tn−1
k0
T
k 1tn xn−T
ktn
xn Ttxn−T t/tn tn
xn
≤t/tnLxn−T tn
xn LT
t− t/tn tn
xn−xn
≤tLαn tnf
xn
−T tn
xn LmaxTsxn−xn: 0≤s≤tn ,
3.7
wheret/tnis the integral part oft/tn. Since limn→ ∞αn/tn 0 andT·x : R → K is continuous for anyx∈K, it follows from3.1that
nlim→ ∞xn−Ttxn0. 3.8
This completes the proof.
Theorem 3.2. LetE be a uniformly convex Banach space with the uniformly Gˆateaux differential norm andKbe a nonempty closed convex subset ofE. LetΓ:{Tt:t∈R }be anL-Lipschitzian semigroup of pseudocontractive mappings satisfying3.1and letf :K → Kbe anLf-Lipschitzian k-strongly pseudocontractive mapping. Suppose that{xn}is a sequence generated by1.8and
1limn→ ∞αn/tn limn→ ∞tn0;
2LIMTtxn−Ttx∗ ≤LIMxn−x∗, ∀x∗∈C, t∈R , whereC:{x∗∈K:Φx∗ minx∈KΦx}withΦx LIMxn−x2for allx∈K.
Then{xn}converges strongly to a common fixed pointx∗ofΓthat is the unique solution inFΓto the following variational inequality:
f x∗
−x∗, j
x∗−p
≥0, ∀p∈FΓ. 3.9
Proof. FromTheorem 3.1, we know that{xn}is bounded and limn→ ∞xn−Ttxn0. It is easy to see thatCis a nonempty bounded closed convex subset ofKsee, e.g.,10.
Now, we show that there exists a common fixed point ofΓinC. For anyt ∈ R and x∗∈C, it follows from limn→ ∞xn−Ttxn0 that
Φ Ttx∗
LIMxn−Ttx∗2 LIMTtxn−Ttx∗2
≤LIMxn−x∗2 Φ
x∗ ,
3.10
and so
TtC⊂C. 3.11
Next, we prove thatCis a singleton. In fact, sinceEis uniformly convex, byLemma 2.3that there exists a continuous and strictly increasing convex functiong : 0,∞ → 0,∞with g0 0 such that, for anyx∗1andx∗2∈C,
xn−x∗1 x∗2 2
2≤ 1
2xn−x∗12 1
2xn−x∗22−1
4gx∗1−x∗2. 3.12 Taking Banach limit LIM on the above inequality, it follows that
1
4gx∗1−x∗2≤ 1
2LIMxn−x∗12 1
2LIMxn−x∗22−LIM
xn− x1∗ x∗2 2
2
≤0.
3.13
This impliesx∗1x∗2and soCis a singleton. Therefore,3.11implies that there existsx∗ ∈C such thatx∗∈FΓ.
For anyp∈FΓ, from1.8, we have xn−f
xn , j
xn−p
1−αn αn
T tn
xn−xn, j
xn−p 1−αn
αn T tn
xn−T tn
p, j
xn−p
−
xn−p, j
xn−p
≤0.
3.14
Sincex∗∈FΓ, it follows from3.14that LIM
xn−f xn
, j
xn−x∗
≤0. 3.15
Furthermore, for anyt∈0,1, byLemma 2.2, we have xn−x∗−t
f xn
−x∗2≤xn−x∗2−2t f
xn
−x∗, j
xn−x∗−t f
xn
−x∗
≤xn−x∗2−2t f
xn
−x∗, j
xn−x∗
−2t f
xn
−x∗, j
xn−x∗−t f
xn
−x∗
−j
xn−x∗ , f
xn
−x∗, j
xn−x∗
≤ 1
2t xn−x∗2−xn−x∗−t f
xn
−x∗2
− f
xn
−x∗, j
xn−x∗−t f
xn
−x∗
−j
xn−x∗ . 3.16
For any >0, sinceEhas a uniformly Gˆateaux differential norm, we know thatJis norm-to- weak∗uniformly continuous on any bounded subset ofEsee, e.g.,1, pages 107–113and so there exists sufficient smallδ>0 such that
f xn
−x∗, j
xn−x∗
≤ 1
2t xn−x∗2−xn−x∗−t f
xn
−x∗2
, ∀t∈0, δ.
3.17
This implies that LIM
f xn
−x∗, j
xn−x∗
≤ 1
2t LIMxn−x∗2−LIMxn−x∗−t f
xn
−x∗2
≤. 3.18
By the arbitrariness of, it follows that LIM
f xn
−x∗, j
xn−x∗
≤0. 3.19
Adding inequalities3.15and3.19, we have LIM
xn−x∗, j
xn−x∗
LIMxn−x∗2≤0. 3.20 This implies that there exists subsequence{xnj} ⊂ {xn}which converges strongly tox∗. From the proof of3.20, we know that LIMxnl −x∗2 ≤0 for any subsequence{xnl} ⊂ {xn}and so there exists subsequence of{xnl}which converges strongly tox∗. If there exists another subsequence{xnk} ⊂ {xn}which converges strongly toy∗, then it follows fromTheorem 3.1 thaty∗∈FΓ. From3.14, we have
x∗−f x∗
, j
x∗−y∗
≤0, y∗−f
y∗ , j
y∗−x∗
≤0. 3.21
Thus
x∗−y∗2≤ f
x∗
−f y∗
, j
x∗−y∗
≤kx∗−y∗2. 3.22 This implies thatx∗−y∗2≤0 and sox∗y∗. Therefore,{xn}converges strongly tox∗∈FΓ.
From 3.14 and the deduction above, we know that x∗ is also the unique solution to the variational inequlity
f x∗
−x∗, j
x∗−p
≥0, ∀p∈FΓ. 3.23
This completes the proof.
Remark 3.3. 1Theorem 3.2extends and generalizes Theorem 3.1 of3from nonexpansive semigroup to Lipschitzian pseudocontractive semigroup in Banach spaces with different conditions;2IfΓis a pseudocontractive mapping, then condition3.1is trivial.
IfΓ : {Tt :t ∈R }is a nonexpansive semigroup, thenΓ :{Tt:t∈R }is anL- Lipschitzian semigroup of pseudocontractive mappings, condition2ofTheorem 3.2holds trivially. FromTheorem 3.2, we have the following result.
Corollary 3.4. LetEbe a uniformly convex Banach space with the uniformly Gˆateaux differential norm andK be a nonempty closed convex subset ofE. LetΓ : {Tt :t ∈R }be a nonexpansive semigroup satisfying3.1and letf :K → Kbe anLf-Lipschitziank-strongly pseudocontractive mapping. Suppose that{xn}is a sequence generated by1.8. If
nlim→ ∞
αn
tn lim
n→ ∞tn0, 3.24
then{xn}converges strongly to a common fixed pointx∗ofΓthat is the unique solution inFΓto VI3.9.
Theorem 3.5. LetEbe a uniformly smooth Banach space andKbe a nonempty closed convex subset ofE. LetΓ:{Tt:t∈R }be a nonexpansive semigroup satisfying3.1and letf:K → Kbe an Lf-Lipschitziank-strongly pseudocontractive mapping. Suppose that{xn}is a sequence generated by 1.8. If
nlim→ ∞
αn tn lim
n→ ∞tn0, 3.25
then{xn}converges strongly to a common fixed pointx∗ofΓthat is the unique solution inFΓto VI3.9.
Proof. For the nonexpansive semigroup Γ, condition 2 of Theorem 3.2 is trivial and so formula3.11holds. Since uniformly smooth Banach spaceEhas the fixed point property for nonexpansive mappingTt see, e.g.,10,Tthas a fixed pointx∗∈C∩FΓ. The rest proof is similar to the proof ofTheorem 3.2and so we omit it. This completes the proof.
Theorem 3.6. Let Ebe a real Hilbert space andK be a nonempty closed convex subset ofE. Let Γ:{Tt:t∈R }be anL-Lipschitzian semigroup of pseudocontractive mappings satisfying3.1 and letf:K → Kbe anLf-Lipschitziank-strongly pseudocontractive mapping. Suppose that{xn} is a sequence generated by1.8. If
nlim→ ∞
αn
tn lim
n→ ∞tn0, 3.26
then{xn}converges strongly to a common fixed pointx∗ofΓthat is the unique solution inFΓto the following variational inequality:
f x∗
−x∗, x∗−p
≥0, ∀p∈FΓ. 3.27
Proof. From the proof of Theorem 3.1, we know that {xn} is bounded and so there exists subsequence{xnj} ⊂ {xn}which converges weakly to some pointx∗ ∈ K. ByTheorem 3.1, we have
nlim→ ∞xn−Ttxn0. 3.28
It follows from20, Theorem 3.18bthatI−Ttis demiclosed at zero for eacht∈R , where Iis an identity mapping. This implies thatx∗∈FΓ.
In addition, from1.8, we have xn−x∗2αn
f xn
−x∗, xn−x∗
1−αn T
tn
xn−x∗, xn−x∗
≤αn f
xn
−f x∗
fx∗
−x∗
, xn−x∗ 1−αn
T tn
xn−T tn
x∗, xn−x∗
≤ 1−αn1−kxn−x∗2 αn f
x∗
−x∗, xn−x∗ ,
3.29
and so
xn−x∗2≤ 1 1−k
f x∗
−x∗, xn−x∗
. 3.30
This implies that{xnj}converges strongly tox∗ ∈FΓ. Similar to the proof ofTheorem 3.2, it is easy to show that{xn}converges strongly tox∗∈FΓthat is also the unique solution to VI3.27. This completes the proof.
Now we turn to discuss the convergence of explicit viscosity iteration process1.12 for approximating the common fixed point of the pseudocontractive semigroupΓ : {Tt: t≥0}.
Theorem 3.7. LetKbe a nonempty closed convex subset of a real Banach spaceE. LetΓ :{Tt: t ∈R }be anL-Lipschitzian semigroup of pseudocontractive mappings withL ≥ 1 such that3.1 holds. Letf : K → Kbe anLf-Lipschitziank-strongly pseudocontractive mapping. Suppose that
the sequence{yn}is generated by1.12and the following conditions hold:
i∞
n1λnθn∞,λn1 θn≤1, for all n≥0;
iiθn/tn → 0,θn−1/θn−1/λnθn → 0,tn → 0 n → ∞;
iiithere exists some constantα >0 such that λn
θn ≤ 1−k
4L2 L Lf1 α, ∀n≥0; 3.31
ivThe following equation holds:
nlim→ ∞
T
tn−tn−1 x−x
λnθ2n 0, ∀x∈K. 3.32
Then limn→ ∞yn−Ttyn0 for anyt∈R .
Proof. Let{xn}denote the sequence defined as in1.8withαn θn/1 θn. By virtue of conditioniiandTheorem 3.1, we know that{xn}is well defined and limn→ ∞xn−Ttxn 0 for anyt∈R . From1.8, we have
1 θn
xnθnf xn
T tn
xn, ∀n≥0, 3.33
λn 1 θn
xn λnθnf xn
λnT tn
xn, ∀n≥0. 3.34
To obtain the assertion ofTheorem 3.7, we first give a serial of estimations: using3.33, we get
xn−xn−12
xn−xn−1, j
xn−xn−1
T tn
xn−T tn−1
xn−1 θn f
xn
−xn
−θn−1 f
xn−1
−xn−1 , j
xn−xn−1
≤ T
tn xn−T
tn
xn−1 T tn
xn−1−T tn−1
xn−1 θn
f xn
−f xn−1
−
xn−xn−1 f
xn−1
−xn−1
−θn−1 fxn−1
−xn−1 , j
xn−xn−1
≤ 1−θn1−kxn−xn−12
θn−θn−1f xn−1
−xn−1xn−xn−1 T
tn−tn−1 T
tn−1
xn−1−T tn−1
xn−1xn−xn−1,
3.35
which implies that
xn−xn−1≤ θn−θn−1 1−kθn
f xn−1
−xn−1 T
tn−tn−1
zn−zn
1−kθn , 3.36
where zn Ttn−1xn−1. From the proof of Theorem 3.1, we know that {xn} is bounded.
Therefore, there exists a constantM >0 such that xn−xn−1≤ M
1−k
θn−θn−1 θn
T
tn−tn−1
zn−zn
1−kθn . 3.37
By using1.12and3.33, we have yn 1−ynλnT
tn yn−
1 θn
yn θnf yn
≤λnT tn
yn−T tn
xn−
1 θn
yn−xn θn
f yn
−f xn
≤λn Lyn−xn
1 θnyn−xn θnLfyn−xn
≤
2 L Lf
λnyn−xn.
3.38
It follows from1.12and3.34that yn 1−xn 1−λn
1 θn
yn−xn λn
T tn
yn−T tn
xn
λnθn f
yn
−f xn
≤ 1−λn 1 θn
λn
L Lfyn−xn
≤ 1 λn
L Lfyn−xn.
3.39
By virtue of1.12,3.34, andLemma 2.2, we have yn 1−xn2 1−λn
1 θn
yn−xn λn
T tn
yn−T tn
xn
λnθn f
yn
−f xn2
≤ 1−λn
1 θn2yn−xn2 2λn T
tn yn−T
tn xn, j
yn 1−xn 2λnθn
f yn
−f xn
, j
yn 1−xn
≤ 1−λn
1 θn2yn−xn2 2λnyn 1−xn2 2λnLyn 1−ynyn 1−xn 2λnθnkyn 1−xn2 2λnθnLfyn 1−ynyn 1−xn.
3.40
Sinceθn → 0, thenλn → 0 by conditioniii. Thus for sufficient largen, we know yn 1−xn2≤ 1−λn
1 θn2yn−xn2 2λ2nL
2 L Lf
1 αyn−xn2 2λn
1 kθnyn 1−xn2. 3.41
Consequently, by conditionivwe can have yn 1−xn2≤ 1−2λn
1 θn λ2n
1 θn2 1−2λn
1 kθn yn−xn2 2λ2nL
2 L Lf 1 α 1−2λn
1 kθn yn−xn2
≤
1−2λnθn1−k−λn/2θn 1 θn2
1−2λn
1 kθn
yn−xn2
2λnθnL
2 L Lf
1 αλn/θn 1−2λn
1 kθn yn−xn2
≤
1−2λnθn1−k−1/41−k 1−2λn
1 kθn
yn−xn2
2λnθn 1/41−k 1−2λn
1 kθnyn−xn2
≤ 1−2λnθn
1−kyn−xn2.
3.42
Squaring on both sides of3.42and using3.37, we get yn 1−xn≤ 1−λnθn1−kyn−xn
≤ 1−λnθn1−kyn−xn−1 xn−xn−1
≤ 1−λnθn1−kyn−xn−1 M 1−k
θn−θn−1 θn
T
tn−tn−1
zn−zn 1−kθn .
3.43
Settingan:yn−xn−1andτn:λnθn1−k, then it follows from conditionsi–ivthat an 1
1−τn an o
τn
. 3.44
ByLemma 2.4, we know thatan → 0, which implies that
nlim→ ∞yn 1−xn0. 3.45
Consequently, sincexn−xn−1 → 0 by3.37, we have
nlim→ ∞yn−xn0. 3.46
Now we prove that limn→ ∞yn−Ttyn0 for anyt∈R . Since yn−Ttyn≤yn−xn xn−Ttxn Ttxn−Ttyn
≤1 Lyn−xn xn−Ttxn, 3.47
byTheorem 3.1and3.46we know that for anyt∈R ,
nlim→ ∞yn−Ttyn0. 3.48
This completes the proof.
Remark 3.8. An example for the conditionsi–iiiofTheorem 3.7is given by tn 1
√4
n 1, θn 1
√3
n 1, λn 1−k
4L
2 L Lf
1 αθn 3.49
for alln≥0, whereαis an any given positive real number. It is easy to see that the conditions with regard toλn andθninTheorem 3.7hold. If the mappingT·x : R → K is Lipschitz continuous for anyx∈K, then conditionivinTheorem 3.7also holds.
Theorem 3.9. LetE be a uniformly convex Banach space with the uniformly Gˆateaux differential norm andKbe a nonempty closed convex subset ofE. LetΓ:{Tt:t∈R }be anL-Lipschitzian semigroup of pseudocontractive mappings withL≥1 such that3.1holds. Letf :K → Kbe anLf- Lipschitziank-strongly pseudocontractive mapping. Suppose that the sequence{yn}is generated by 1.12and conditions (i)–(iv) ofTheorem 3.7hold. Assume further that condition (2) ofTheorem 3.2 holds, where{xn}is generated by1.8withαn θn/1 θn. Then{yn}converges strongly to a common fixed pointx∗ofΓthat is the unique solution inFΓto VI3.9.
Proof. By Theorem3.2, we know that{xn}converges strongly to a fixed pointx∗ofΓthat is the unique solution inFΓto VI3.9, where{xn}is generated by1.8withαnθn/1 θn. It follows from3.46thatyn → x∗∈FΓ. This completes the proof.
Remark 3.10. 1Theorem 3.9extends Theorem 4.1 of13from Lipschitzian pseudocontrac- tive mapping to Lipschitzian pseudocontractive semigroup in Banach spaces under different conditions;2Theorem 3.9also extends Theorem 3.2 of3from nonexpansive semigroup to Lipschitzian pseudocontractive semigroup in Banach spaces under different conditions.
IfΓ : {Tt :t ∈R }is a nonexpansive semigroup, thenΓ :{Tt:t∈R }is anL- Lipschitzian semigroup of pseudocontractive mappings, condition2ofTheorem 3.2holds trivially. Therefore,Theorem 3.9gives the following result.
Corollary 3.11. LetEbe a uniformly convex Banach space with the uniformly Gˆateaux differential norm andK be a nonempty closed convex subset ofE. LetΓ : {Tt :t ∈ R }be a nonexpansive semigroup satisfying 3.1and f : K → K be an Lf-Lipschitzian k-strongly pseudocontractive mapping. Suppose that the sequence{yn}is generated by1.12and conditions (i)–(iv) ofTheorem 3.7 hold. Then{yn}converges strongly to a common fixed pointx∗ofΓthat is the unique solution in F(T) to VI3.9.
Theorem 3.12. LetEbe a uniformly smooth Banach space andKbe a nonempty closed convex subset ofE. LetΓ:{Tt:t∈R }be a nonexpansive semigroup satisfying3.1and letf:K → Kbe an Lf-Lipschitziank-strongly pseudocontractive mapping. Suppose that the sequence{yn}is generated by1.12and conditions (i)–(iv) ofTheorem 3.7hold. Then{yn}converges strongly to a common fixed pointx∗ofΓthat is the unique solution inFΓto VI3.9.
Proof. Let{xn}denote the sequence defined as in1.8withαn θn/1 θn. ByTheorem 3.5, we know that{xn}converges strongly to a fixed pointx∗ofΓthat is the unique solution in FΓto VI3.9. It follows from3.46thatyn → x∗. This completes the proof.
Theorem 3.13. LetEbe a real Hilbert space andK be a nonempty closed convex subset ofE. Let Γ : {Tt : t ∈ R } be anL-Lipschitzian semigroup of pseudocontractive mappings withL ≥ 1 such that3.1holds. Letf :K → Kbe anLf-Lipschitziank-strongly pseudocontractive mapping.
Suppose that the sequence{yn} is generated by1.12and conditions (i)–(iv) of Theorem 3.7hold.
Then{yn}converges strongly to a common fixed pointx∗ofΓthat is the unique solution inFΓto VI3.27.
Proof. Let{xn}denote the sequence defined as in1.8withαn θn/1 θn. ByTheorem 3.6, we know that{xn}converges strongly to a fixed pointx∗ofΓthat is the unique solution in FΓto VI3.27. It follows from3.46thatyn → x∗. This completes the proof.
Acknowledgments
This work was supported by the National Science Foundation of China10671135, 70831005, the Specialized Research Fund for the Doctoral Program of Higher Education20060610005 and the Open FundPLN0703of State Key Laboratory of Oil and Gas Reservoir Geology and ExploitationSouthwest Petroleum University.
References
1 W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.
2 S.-S. Chang, “Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1402–1416, 2006.
3 R. Chen and H. He, “Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space,” Applied Mathematics Letters, vol. 20, no. 7, pp. 751–757, 2007.
4 N.-j. Huang and M.-R. Bai, “A perturbed iterative procedure for multivalued pseudo-contractive mappings and multivalued accretive mappings in Banach spaces,” Computers & Mathematics with Applications, vol. 37, no. 6, pp. 7–15, 1999.
5 J. S. Jung, “Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications. Series A, vol. 64, no. 11, pp. 2536–2552, 2006.
6 S. Reich, “Strong convergence theorems for resolvents of accretive operators in Banach spaces,”
Journal of Mathematical Analysis and Applications, vol. 75, no. 1, pp. 287–292, 1980.
7 T. Suzuki, “On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2133–2136, 2003.
8 T. Suzuki, “Browder’s type convergence theorems for one-parameter semigroups of nonexpansive mappings in Banach spaces,” Israel Journal of Mathematics, vol. 157, no. 1, pp. 239–257, 2007.
9 X. Weng, “Fixed point iteration for local strictly pseudo-contractive mapping,” Proceedings of the American Mathematical Society, vol. 113, no. 3, pp. 727–731, 1991.
10 H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
11 H.-K. Xu, “A strong convergence theorem for contraction semigroups in Banach spaces,” Bulletin of the Australian Mathematical Society, vol. 72, no. 3, pp. 371–379, 2005.
12 H.-K. Xu and R. G. Ori, “An implicit iteration process for nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 22, no. 5-6, pp. 767–773, 2001.
13 H. Zegeye, N. Shahzad, and T. Mekonen, “Viscosity approximation methods for pseudocontractive mappings in Banach spaces,” Applied Mathematics and Computation, vol. 185, no. 1, pp. 538–546, 2007.
14 W.-Y. Zeng, N.-j. Huang, and C.-W. Zhao, “Viscosity approximation methods for generalized mixed equilibrium problems and fixed points of a sequence of nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 714939, 15 pages, 2008.
15 X.-s. Li and N.-j. Huang, “Strong convergence theorems for fixed points of pseudo-contractive semigroup,” Bulletin of the Australian Mathematical Society, vol. 76, no. 3, pp. 441–452, 2007.
16 X.-s. Li and N.-j. Huang, “Viscosity approximation methods for pseudo-contractive semigroup in Banach spaces,” Research Report, Sichuan University, Chengdu, China, 2007.
17 A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
18 K. Deimling, “Zeros of accretive operators,” Manuscripta Mathematica, vol. 13, no. 4, pp. 365–374, 1974.
19 W. V. Petryshyn, “A characterization of strict convexity of Banach spaces and other uses of duality mappings,” Journal of Functional Analysis, vol. 6, pp. 282–291, 1970.
20 F. E. Browder, “Nonlinear mappings of nonexpansive and accretive type in Banach spaces,” Bulletin of the American Mathematical Society, vol. 73, pp. 875–882, 1967.
