Approximating Common Fixed Points of

Volume 2008, Article ID 363257,17pages doi:10.1155/2008/363257

Research Article

Approximating Common Fixed Points of

Lipschitzian Semigroup in Smooth Banach Spaces

Shahram Saeidi

Department of Mathematics, University of Kurdistan, Sanandaj 416, Kurdistan 66196-64583, Iran

Correspondence should be addressed to Shahram Saeidi,[email protected] Received 16 August 2008; Accepted 10 December 2008

Recommended by Mohamed Khamsi

LetSbe a left amenable semigroup, letS{Ts:s∈S}be a representation ofSas Lipschitzian mappings from a nonempty compact convex subsetCof a smooth Banach spaceEintoCwith a uniform Lipschitzian condition, let{μn}be a strongly left regular sequence of means defined on an S-stable subspace ofl^∞S, letfbe a contraction onC, and let{αn},{βn}, and{γn}be sequences in0, 1such thatαnβnγn 1, for alln. Letxn1αnfxn βnxnγnTμnxn, for alln≥1.

Then, under suitable hypotheses on the constants, we show that{xn}converges strongly to some zin FS, the set of common fixed points ofS, which is the unique solution of the variational inequalityf−Iz, Jy−z ≤0, for ally∈FS.

Copyrightq2008 Shahram Saeidi. 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 and letCbe a nonempty closed convex subset ofE. A mapping T :C → Cis said to be

iLipschitzian with Lipschitz constantl >0 if

Tx−Ty ≤l x−y , ∀x, y∈C; 1.1

iinonexpansive if

Tx−Ty ≤ x−y , ∀x, y∈C; 1.2

iiiasymptotically nonexpansive if there exists a sequence {kn} of positive numbers satisfying the property limn→ ∞kn1 and

Tⁿx−Tⁿy ≤kn x−y , ∀x, y∈C. 1.3

Halpern1introduced the following iterative scheme for approximating a fixed point of a nonexpansive mappingT onC:

x_n1αnx 1−αnTxn, n1,2, . . . , 1.4 wherex1 xis an arbitrary point inCand{αn}is a sequence in0,1. Strong convergence of Halpern type iterative sequence has been widely studied: Wittmann 2discussed such a sequence in a Hilbert space. Shioji and Takahashi3 see also4extended Wittmann’s result and proved strong convergence of{xn}defined by1.4in a uniformly convex Banach space with a uniformly Gateaux differentiable norm.

In particular, Xu5proposed the following viscosity iterative processoriginally due to Moudafi6in a uniformly smooth Banach space:

xn1αnfxn 1−αnTxn, n1,2, . . . , 1.5 where,f:C → Cis a contraction, and proved, under appropriate conditions,{xn}converges to a fixed point ofTwhich is a solution of a variational inequality. Recently, many papers have been devoted to algorithms for finding such solutions, see, for example,7–9.

It is an interesting problem to extend the above results to the nonexpansive semigroup case10–18. Lau, Miyake and Takahashi19considered the following iteration process;

xn1αnx 1−αnTμnxn, n1,2, . . . , 1.6 for a semigroupS {Ts:s ∈S}of nonexpansive mappings on a compact convex subset Cof a smooth and strictly convex Banach space with respect to a left regular sequence{μn} of means defined on an appropriate invariant subspace ofl^∞S; for some related results we refer the readers to20,21.

The iterative methods for approximation of fixed points of asymptotically nonexpan- sive mappings have been studied by authorssee, e.g.,22–32and references therein.

For a semigroupS, we can define a partial preordering≺onSbya≺bif and only if aS ⊃ bS. IfSis a left reversible semigroupi.e.,aS∩bS /∅fora, b ∈S, then it is a directed set.Indeed, for everya, b ∈S, applyingaS∩bS /∅, there exista, b ∈Swithaabb; by takingcaabb, we havecS⊆aS∩bS, and thena≺candb≺c.

If a semigroupSis left amenable, thenSis left reversible33.

Definition 1.1. LetS {Ts: s∈S}be a representation of a left reversible semigroupSas Lipschitzian mappings onCwith Lipschitz constants{ks:s∈S}.We will say thatSis an asymptotically nonexpansive semigroup onC, if there holds the uniform Lipschitzian condition limsks ≤ 1 on the Lipschitz constants.Note that a left reversible semigroup is a directed set.

It is worth mentioning that there is a notion of asymptotically nonexpansive defined dependent on left ideals in a semigroup in34,35.

In this paper, motivated by1.5,1.6and the above-mentioned results, we introduce the following viscosity iterative scheme

xn1αnfxn βnxnγnTμnxn, ∀n≥1, 1.7

for an asymptotically nonexpansive semigroup S {Ts : s ∈ S} on a compact convex subsetCof a smooth Banach spaceEwith respect to a left regular sequence{μn}of means defined on an appropriate invariant subspace ofl^∞S, wheref is a contraction on C, and {αn},{βn}and{γn}are sequences in0,1such thatαnβnγn 1,for alln.Then, under appropriate conditions on constants, we prove that the sequence{xn}converges strongly to somez inFS, the set of common fixed points ofS, which is the unique solution of the variational inequality

f−Iz, Jy−z ≤0, ∀y∈FS. 1.8 It is remarked that we have not assumedE to be strictly convex and our results are new even for nonexpansive mappings. Moreover, our results extend many previous resultse.g., 11,19.

2. Preliminaries

LetEbe a Banach space and letE^∗be the topological dual ofE. The value ofx^∗∈E^∗atx∈E will be denoted byx, x^∗orx^∗x. With eachx∈E, we associate the set

Jx

x^∗∈E^∗:x, x^∗ x^{∗ 2} x ²

. 2.1

Using the Hahn-Banach theorem, it immediately follows that Jx/∅ for each x ∈ E. A Banach spaceEis said to be smooth if the duality mappingJofEis single valued. We know that ifEis smooth, thenJis norm to weak-star continuous; see20,21.

LetSbe a semigroup. We denote byl^∞Sthe Banach space of all bounded real valued functions on S with supremum norm. For each s ∈ S, we define ls and rs on l^∞S by lsft fstand rsft ftsfor eacht ∈ Sandf ∈ l^∞S. LetX be a subspace of l^∞Scontaining 1 and letX^∗be its topological dual. An elementμofX^∗is said to be a mean onX if μ μ1 1. We often writeμtftinstead ofμfforμ ∈X^∗andf ∈X. LetX be left invariantresp., right invariant, that is,lsX⊂ Xresp.,rsX ⊂Xfor eachs∈ S.

A meanμ onX is said to be left invariant resp., right invariant if μlsf 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. A net {μα}of means onXis said to be strongly left regular if

limα l^∗_sμα−μα0, 2.2

for eachs∈S, wherel^∗_sis the adjoint operator ofls. LetCbe a nonempty closed and convex subset ofE. Throughout this paper,Swill always denote a semigroup with an identitye.Sis called left reversible if any two right ideals inShave nonvoid intersection, that is,aS∩bS /∅ fora, b∈S. In this case, we can define a partial ordering≺onSbya≺bif and only ifaS⊃bS.

It is easy too seet≺ts,∀t, s∈S. Further, ift≺sthenpt≺psfor allp∈S. If a semigroupS is left amenable, thenSis left reversible. But the converse is false.

S{Ts:s∈S}is called a representation ofSas Lipschitzian mappings onCif for eachs∈S, the mappingTsis Lipschitzian mapping onCwith Lipschitz constantks, and Tst TsTtfors, t ∈ S. We denote byFSthe set of common fixed points ofS, and

byCathe set of almost periodic elements inC, that is, allx∈Csuch that{Tsx:s∈S}is relatively compact in the norm topology ofE. We will call a subspaceXofl^∞S,S-stable if the functionss→ Tsx, x^∗ands→ Tsx−y onSare inXfor allx, y∈Candx^∗∈E^∗. We know that ifμis a mean onX and if for eachx^∗ ∈ E^∗ the functions → Tsx, x^∗is contained inXandCis weakly compact, then there exists a unique pointx0ofEsuch that

μsTsx, x^∗x0, x^∗, 2.3 for eachx^∗∈E^∗. We denote such a pointx0byTμx. Note thatTμzz, for eachz∈FS;

see36–38. Let D be a subset of B whereB is a subset of a Banach spaceE and letP be a retraction ofBontoD. ThenP is said to be sunny39if for eachx ∈ B andt ≥ 0 with P xtx−P x∈B,

PP xtx−P x P x. 2.4

A subset D of B is said to be a sunny nonexpansive retract of B if there exists a sunny nonexpansive retractionP ofBontoD. We know that ifEis smooth andP is a retraction ofBontoD, thenPis sunny and nonexpansive if and only if for eachx∈Bandz∈D,

x−P x, Jz−P x ≤0. 2.5

For more details see20,21.

We will need the following lemma, which will appear in32.

Lemma 2.1. LetSbe a left reversible semigroup andS {Ts:s∈S}be a representation ofSas Lipschitzian mappings from a nonempty weakly compact convex subsetCof a Banach spaceEintoC, with the uniform Lipschitzian condition limsks≤1 on the Lipschitz constants of the mappings. Let X be a left invariantS-stable subspace ofl^∞Scontaining 1, andμbe a left invariant mean onX.

ThenFS FTμ∩Ca.

Corollary 2.2. Let{μn}be an asymptotically left invariant sequence of means onX. Ifz∈Caand lim inf_{n→ ∞} Tμnz−z 0, thenzis a common fixed point forS.

Proof. From lim inf_{n→ ∞} Tμnz−z 0, there exists a subsequence{Tμnkz} of{Tμnz}

that converges strongly toz. Since the set of means onXis compact in the weak-star topology, there exists a subnet{μn_kα : α∈Λ}of{μnk}such that{μn_kα}converges toμin the weak-star topology. Then, it is easy to show thatμis a left invariant mean onX. On the other hand, for eachx^∗∈E^∗, we have

T μn_kα

z, x^∗

μn_kαT·z, x^∗ −→μT·z, x^∗Tμz, x^∗. 2.6 Now, since{Tμnkz}converges strongly toz, we havez, x^∗ Tμz, x^∗and hencez Tμz. It follows fromLemma 2.1thatzis a common fixed point ofS.

Lemma 2.3. LetSbe a left reversible semigroup andS {Ts:s∈S}be a representation ofSas Lipschitzian mappings from a nonempty weakly compact convex subsetCof a Banach spaceEintoC,

with the uniform Lipschitzian condition limsks≤1 on the Lipschitz constants of the mappings. Let Xbe a left invariant subspace ofl^∞Scontaining 1 such that the mappingss→ Tsx, x^∗be inX for allx∈Xandx^∗∈E^∗, and{μn}be a strongly left regular sequence of means onX. Then

lim sup

n→ ∞ sup

x,y∈C

Tμnx−Tμny − x−y

≤0. 2.7

Proof. Consider an arbitrary ε > 0 and take d diamC. Since limsks ≤ 1, there exists s0∈Ssuch that

sups≥s0

ks<1 ε

2d. 2.8

From lim_{n→ ∞} l_s^∗₀μn−μn 0, we may choose a natural numberNsuch that l^∗_s0μn−μn< ε

2d, ∀n≥N. 2.9

Then, for eachx, y∈C,n≥Nandx^∗∈JTμnx−Tμnywe have Tμnx−Tμny²

Tμnx−Tμny, x^∗ μn_s

Tsx−Tsy, x^∗

− l^∗_s0μn

s

Tsx−Tsy, x^∗

l^∗_s0μn

s

Tsx−Tsy, x^∗

≤μn−l_s^∗₀μnd x^∗ μn_s

Ts0sx−Ts0sy, x^∗

≤ ε

2ddTμnx−Tμnysup

s∈S

Ts0sx−Ts0syTμnx−Tμny

≤ ε

2Tμnx−Tμnysup

s∈Sks0s x−y Tμnx−Tμny.

2.10 Therefore,

Tμnx−Tμny≤ ε 2sup

s∈Sks0s x−y

≤ ε 2sup

s≥s0

ks x−y ≤ ε

2 1 ε 2d

x−y ≤ε x−y ,

2.11

that is,

sup

x,y∈C

Tμnx−Tμny− x−y

≤ε, ∀n≥N. 2.12

Sinceε >0 is arbitrary, the desired result follows.

Remark 2.4. Taking inLemma 2.3 cn sup

x,y∈C

Tμnx−Tμny− x−y

, ∀n, 2.13

we obtain lim sup_{n→ ∞}cn≤0. Moreover,

Tμnx−Tμny≤ x−y cn, ∀x, y∈C. 2.14 Corollary 2.5. LetSbe a left reversible semigroup andS {Ts : s ∈ S}be a representation of Sas Lipschitzian mappings from a nonempty compact convex subsetCof a Banach spaceEintoC, with the uniform Lipschitzian condition lim_sks≤1. LetXbe a left invariantS-stable subspace of l^∞Scontaining 1, andμbe a left invariant mean onX. ThenTμis nonexpansive andFS/∅.

Moreover, ifEis smooth, thenFSis a sunny nonexpansive retract ofCand the sunny nonexpansive retraction ofContoFSis unique.

Proof. From 2.14, by taking μn μ∀n, it follows that Tμ is nonexpansive. So, from Lemma 2.1, we get FS FTμ/∅. On the other hand, it is well-known that the fixed point set of a nonexpansive mapping on a compact convex subset of a smooth Banach space is a sunny nonexpansive retract ofCand the sunny nonexpansive retraction ofContoFS is unique19,20. This concludes the result.

We will need the following lemmas in what follows.

Lemma 2.6see20,21. LetXbe a real Banach space and letJbe the duality mapping. Then, for any givenx, y∈Xandjxy∈Jxy, there holds the inequality

xy ²≤ x ²2y, jxy. 2.15

Lemma 2.7see40. Assume{an}is a sequence of nonnegative real numbers such that

a_n1≤1−γnanδn, n≥0, 2.16 where{γn}is a sequence in0,1and{δn}is a sequence inRsuch that

i_∞

n1γn∞;

iilim sup_{n→ ∞}δn/γn≤0 or_∞

n1|δn|<∞.

Then lim_{n→ ∞}an0.

Lemma 2.8see41. Let{xn}and{zn}be bounded sequences in a Banach spaceXand let{βn} be a sequence in0,1with 0<lim inf_{n→ ∞}βnand lim sup_{n→ ∞}βn<1. Suppose

xn1βnxn 1−βnzn 2.17

for all integersn≥0 and

lim sup

n→ ∞

zn1−zn − xn1−xn

≤0. 2.18

Then lim_{n→ ∞} xn−zn 0.

3. The main theorem

We are now ready to establish our main theorem.

Theorem 3.1. LetSbe a left reversible semigroup andS {Ts :s∈S}be a representation ofS as Lipschitzian mappings from a nonempty compact convex subsetCof a smooth Banach spaceEinto C, with the uniform Lipschitzian condition limsks ≤ 1 andf be anα-contraction onCfor some 0 < α <1. LetXbe a left invariantS-stable subspace ofl^∞Scontaining 1,{μn}be a strongly left regular sequence of means onXsuch that lim_{n→ ∞} μn1−μn 0 and{cn}be the sequence defined by2.13. Let{αn},{βn}and{γn}be sequences in0,1such that

iαnβnγn1, ∀n, iilim_{n→ ∞}αn0;

iii_∞

n1αn∞;

ivlim sup_{n→ ∞}cn/αn≤0; (note that, byRemark 2.4, lim sup_{n→ ∞}cn≤0) v0<lim infn→ ∞βn≤lim sup_{n→ ∞}βn<1.

Let{xn}be the following sequence generated byx1 ∈Cand∀n≥1,

xn1 αnfxn βnxnγnTμnxn. 3.1

Then{xn}converges strongly toz∈FSwhich is the unique solution of the variational inequality f−Iz, Jy−z ≤0, ∀y∈FS. 3.2 Equivalently, one haszP fz, wherePis the unique sunny nonexpansive retraction ofContoFS.

Remark 3.2. For example, we may choose

αn:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 1 n√

cn ifcn≥0, 1

n ifcn<0.

3.3

Proof. We divide the proof into several steps and prove the claim in each step.

Step 1. Claim. Let{ωn}be a sequence inC. Then

n→ ∞limTμn1ωn−Tμnωn0. 3.4

PutDsup{ z :z∈C}. Then Tμn1ωn−Tμnωn sup

z 1

Tμn1ωn−Tμnωn, z sup

z 1

μ_n1s

Tsωn, z

−μn_s

Tsωn, z

≤ μ_n1−μn sup

s∈S Tsωn ≤ μ_n1−μn D−→0, asn−→ ∞.

3.5 Step 2. Claim. lim_{n→ ∞} x_n1−xn 0.

Define a sequence{zn}byzn xn1−βnxn/1−βnso thatxn1βnxn 1−βnzn. We now compute

zn1−zn 1

1−βn1

xn2−βn1xn1

− 1 1−βn

xn1−βnxn

1 1−βn1

α_n1f x_n1

γ_n1T μ_n1

x_n1

− 1 1−βn

αnfxn γnTμnxn

1 1−βn1

αn1f xn1

1−αn1−βn1 T

μn1 xn1

− 1 1−βn

αnfxn

1−α_n1−β_n1

Tμnxn

≤T μ_n1

x_n1−Tμnxn

αn1

1−β_n1 f

xn1

−T μn1

xn1

− αn1

1−β_n1 f

xn1

−T μn1

xn1. 3.6 SinceCis bounded and lim sup_{n→ ∞}βn<1, we have for some big enough constantK >0,

zn1−zn≤T μn1

xn1−Tμnxn1Tμnxn1−TμnxnK

αn1αn

≤T μ_n1

x_n1−Tμnx_n1x_n1−xncnK

α_n1αn

. 3.7

Now, sinceαn → 0 and byStep 1andLemma 2.3, we immediately conclude that lim sup

n

z_n1−zn−x_n1−xn

≤lim sup

n

T μ_n1

x_n1−Tμnx_n1cnK

α_n1αn

≤0. 3.8

ApplyingLemma 2.8, we get lim_n x_n1−xn lim_n1−βn xn−zn 0.

Step 3. Claim. Theω-limit set of{xn},ω{xn}, is a subset ofFS.

Lety ∈ ω{xn}and{xnk}be a subsequence of{xn}converging strongly toy. Note that

x_n1−xnαnfxn 1−βnTμnxn−xn−αnTμnxn. 3.9

So

xn−Tμnxn≤ 1 1−βn

x_n1−xnαnfxn−Tμnxn. 3.10

Hence, byii,vandStep 2, we have

nlim→ ∞xn−Tμnxn0. 3.11

From this andLemma 2.3, we obtain lim sup

k→ ∞

y−T μnk

y≤lim sup

k→ ∞

y−xnkxnk−T μnk

xnkT μnk

xnk−T μnk

y

≤lim sup

k→ ∞

2y−xnkxnk−T μnk

xnkcnk

≤0.

3.12

Therefore, applyingCorollary 2.2, we gety∈FS.

Step 4. Claim. The sequence{xn}converges strongly tozP fz.

We know, fromCorollary 2.5and the proof ofCorollary 2.2, that there exists a unique sunny nonexpansive retractionPofContoFS. The Banach Contraction Mapping Principal guarantees thatP fhas a unique fixed pointzwhich by2.5is the unique solution of

f−Iz, Jy−z ≤0, ∀y∈FS. 3.13

We first show

lim sup

n→ ∞ f−Iz, Jxn−z ≤0. 3.14

Let{xnk}be a subsequence of{xn}such that

klim→ ∞

f−Iz, J

xnk−z

lim sup

n→ ∞ f−Iz, J

xn−z

. 3.15

Without loss of generality, we can assume that{xnk}converges to somey ∈ C. By Step 3, y∈FS. Smoothness ofEand a combination of3.13and3.15give

lim sup

n→ ∞

f−Iz, Jxn−z

f−Iz, Jy−z

≤0, 3.16

as required. Now, taking

unTμnxn, ∀n≥1, 3.17

we have un−z ≤ xn−z cn. By usingLemma 2.6, we have xn1−z²γnun−z βnxn−z

αn

γfxn−z²

≤γnun−z βnxn−z²2αn

fxn−z, J

x_n1−z

≤1−βn γn

1−βnun−z

²βnxn−z² 2αn

fxn−fz, J

x_n1−z 2αn

fz−z, J

x_n1−z

≤ γ_n² 1−βn

un−z²βnxn−z² 2αnαxn−zxn1−z2αn

fz−z, J

xn1−z

≤ γ_n² 1−βn

xn−z² cnγ_n²

1−βn βnxn−z² αnαxn−z²xn1−z²

2αn

fz−z, J

xn1−z γ_n²

1−βn βnαnαxn−z² αnαx_n1−z²2αn

fz−z, J

x_n1−z cnγ_n²

1−βn

1−αnα−2αn2αnα α²_n 1−βn

xn−z² αnαxn1−z²2αn

fz−z, J

xn1−z cnγ_n²

1−βn.

3.18

It follows that

x_n1−z²≤ 1−21−ααn

1−αnα

xn−z²

αn

1−αnα 2

γfz−z, J

xn1−z αn

1−βn

xn−z² cn

αn × γ_n² 1−βn

. 3.19

Now, from conditionsii–v,3.14andLemma 2.7, we get xn−z → 0.

Corollary 3.3. LetSbe a left reversible semigroup andS {Ts : s ∈ S}be a representation of Sas nonexpansive mappings from a nonempty compact convex subsetCof a smooth Banach space

Einto Cand f be anα-contraction onC for some 0 < α < 1. LetX be a left invariant S-stable subspace ofl^∞Scontaining 1 and{μn}be a strongly left regular sequence of means onXsuch that lim_{n→ ∞} μn1−μn 0. Let{αn},{βn}and{γn}be sequences in0,1such that

iαnβnγn1, ∀n, iilimn→ ∞αn0;

iii_∞

n1αn∞;

iv0<lim inf_{n→ ∞}βn≤lim sup_{n→ ∞}βn<1.

Let{xn}be the sequence generated byx1∈Cand∀n≥1,

x_n1 αnfxn βnxnγnTμnxn. 3.20

Then{xn}converges strongly toz∈FSwhich is the unique solution of the variational inequality f−Iz, Jy−z ≤0, ∀y∈FS. 3.21

Equivalently, one haszP fz, wherePis the unique sunny nonexpansive retraction ofContoFS.

Remark 3.4. IfS is a countable left amenable semigroup, then there is a strong left regular sequence onl^∞Sconsisting finite meansμ, that is,μ_n

i1λiδxi,λi≥0,_n

i1λi 1. See42, Corollary 3.7.

Remark 3.5. It is known that ifSis a left reversible semigroup, thenWAPS, the space of weakly almost periodic functions onS, has a left invariant mean. But the converse is not true see43.

Problem. Can the hypothesis onSof Theorem 3.1be replaced byWAPShas a left invariant mean?

4. Applications

Corollary 4.1. Let C be a compact convex subset of a smooth Banach space E and let S, T be asymptotically nonexpansive mappings of C into itself withST TSand f be an α-contraction onCfor some 0< α <1. Let{cn}be defined by

cn d n²

n−1

i0 n−1

j0

1−kilj

, 4.1

where,ddiamCandkiandljare defined as

Sⁱx−Sⁱy ≤ki x−y , T^jx−T^jy ≤lj x−y , 4.2

for allx, y∈C, and limi→ ∞ki limj→ ∞lj1. Let{αn},{βn}and{γn}be sequences in0,1such that

iαnβnγn1, ∀n, iilim_{n→ ∞}αn0;

iii_∞

n1αn∞;

ivlim sup_{n→ ∞}cn/αn≤0; (note that lim_{n→ ∞}cn0) v0<lim infn→ ∞βn≤lim sup_{n→ ∞}βn<1.

Letx1x∈Cand{xn}be a sequence defined by

xn αnfxn βnxnγn

1 n²

n−1 i0

n−1 j0

SⁱT^jxn

4.3

for eachn∈N. Then{xn}converges strongly toz∈FS∩FTwhich is the unique solution of the variational inequality

f−Iz, Jy−z ≤0, ∀y∈FS∩FT. 4.4 Equivalently, one has z P fz, where P is the unique sunny nonexpansive retraction ofC onto FS∩FT.

Proof. LetTi, j SⁱT^jfor eachi, j∈N∪ {0}. Then{Ti, j:i, j∈N∪ {0}}is a semigroup of Lipschitzian mappings onCsuch that for allx, y∈C,

Ti, jx−Ti, jy ≤ki, j x−y 4.5 where ki, j kilj. Hence lim_{i,j→ ∞}ki, j 1. On the other hand, for eachn ∈ N, define μnf 1/n²n−1

i0_n−1

j0fi, jfor eachf ∈ l^∞N∪ {0}². Then, {μn}is a strongly regular sequence of means and limn→ ∞ μ_n1−μn 09,44. Next, for eachx, y∈Candn∈N, we have

Tμnx−Tμny 1 n²

n−1 i0

n−1

j0

SⁱT^jx− 1 n²

n−1

i0 n−1

j0

SⁱT^jy

≤ x−y cn. 4.6

Now, applyTheorem 3.1to conclude the result.

Corollary 4.2. LetCbe a compact convex subset of a smooth Banach spaceEand letS{Tt:t∈ R}be a strongly continuous semigroup of Lipschitzian mappings onCwith the uniform Lipschitzian condition lim_{t→ ∞}kt ≤ 1 and{tn}be an increasing sequence in0,∞such that lim_{n→ ∞}tn ∞ and limn→ ∞tn/tn1 1. Let{αn},{βn}and{γn}be sequences in0,1such that

iαnβnγn1, ∀n, iilim_{n→ ∞}αn0;

iii_∞

n1αn∞;

ivlim sup_{n→ ∞}cn/αn≤0,where

cn sup

x,y∈C

1 tn

_tn

0

Tsxds− 1 tn

_tn

0

Tsyds

− x−y

; 4.7

v0<lim infn→ ∞βn≤lim sup_{n→ ∞}βn<1.

Letx1x∈Cand{xn}be a sequence defined by

x_n1 αnfxn βnxnγn

1 tn

_tn

0

Tsxnds

4.8

for each n ∈ N. Then {xn} converges strongly toz ∈ FS which is the unique solution of the variational inequality

f−Iz, Jy−z ≤0, ∀y∈FS. 4.9 Equivalently, one haszP fz, wherePis the unique sunny nonexpansive retraction ofContoFS.

Proof. Forn∈N, defineμnf 1/tn

_tn

0ftdtfor eachf ∈CR, wheref ∈CRdenotes the space of all real valued bounded continuous functions onRwith supremum norm. Then, {μn}is a strongly regular sequence of means and limn→ ∞ μ_n1−μn 09,44. Further, for eachx∈C, we haveTμnx1/tn

_tn

0Tsxds.Therefore, it suffices to applyTheorem 3.1to conclude the desired result.

Corollary 4.3. LetCbe a compact convex subset of a smooth Banach spaceEand letS{Tt:t∈ R}be a strongly continuous semigroup of Lipschitzian mappings onCwith the uniform Lipschitzian condition lim_{t→ ∞}kt≤1 and{rn}be a decreasing sequence in0,∞such that lim_{n→ ∞}rn0. Let {αn},{βn}and{γn}be sequences in0,1such that

iαnβnγn1, ∀n, iilimn→ ∞αn0;

iii_∞

n1αn∞;

ivlim sup_{n→ ∞}cn/αn≤0,where

cn sup

x,y∈C

rn

_∞

0

exp

−rknt

Ttxdt−rn

_∞

0

exp

−rknt

Ttydt

− x−y

; 4.10

v0<lim infn→ ∞βn≤lim sup_{n→ ∞}βn<1.

Letx1x∈Cand{xn}be a sequence defined by

x_n1αnfxn βnxnγnrn

_∞

0

exp−rnsTsxnds 4.11

for each n ∈ N. Then {xn} converges strongly toz ∈ FS which is the unique solution of the variational inequality

f−Iz, Jy−z ≤0, ∀y∈FS. 4.12

Equivalently, one haszP fz, wherePis the unique sunny nonexpansive retraction ofContoFS.

Proof. Forn ∈ N, defineμnf rn

_∞

0 exp−rkntftdtfor eachf ∈ CR. Then,{μn} is a strongly regular sequence of means and lim_{n→ ∞} μ_n1−μn 09,44. Further, for eachx∈C, we haveTμnxrn

_∞

0 exp−rntTtxdt.Therefore, the result follows fromTheorem 3.1.

Corollary 4.4. Let C be a compact convex subset of a smooth Banach space E and let S be an asymptotically nonexpansive mapping of C into itself and f be an α-contraction on C for some 0< α <1. Let{cn}be defined by

cn d n

n−1

i0

1−ki, 4.13

where,ddiamCandkiis defined as Sⁱx−Sⁱy ≤ki x−y , for allx, y∈C, and limi→ ∞ki1.

Let{αn},{βn}and{γn}be sequences in0,1such that iαnβnγn1, ∀n,

iilim_{n→ ∞}αn0;

iii_∞

n1αn∞;

ivlim sup_{n→ ∞}cn/αn≤0;

v0<lim infn→ ∞βn≤lim sup_{n→ ∞}βn<1.

Letx1x∈Cand{xn}be a sequence defined by

xnαnfxn βnxnγn

∞ m0

qn,mT^mxn 4.14

for eachn ∈ N where Q {qn,m}is a strongly regular matrix. Then {xn} converges strongly to z∈FSwhich is the unique solution of the variational inequality

f−Iz, Jy−z ≤0, ∀y∈FS. 4.15

Equivalently, one haszP fz, wherePis the unique sunny nonexpansive retraction ofContoFS.

Proof. For eachn∈N, define

μnf ^∞

m0

qn,mfm 4.16

for eachf∈l^∞N∪ {0}. SinceQis a strongly regular matrix, for eachm, we haveqn,m → 0, asn → ∞; see 37. Then, it is easy to see that{μn} is a regular sequence of means, and μn1−μn → 044. Further, for eachx∈C, we haveTμnx_∞

m0qn,mT^mx.Now, apply Theorem 3.1to conclude the result.

For deducing some more applications, we refer to, for example,44.

Acknowledgment

The author is very grateful to the referees for their valuable suggestions.

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