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Volume 2008, Article ID 363257, 17 pagesdoi:10.1155/2008/363257 Research Article Approximating Common Fixed Points of Lipschitzian Semigroup in Smooth Banach Spaces Shahram Saeidi Depart

Volume 2008, Article ID 363257, 17 pages

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,sh.saeidi@uok.ac.ir

Received 16 August 2008; Accepted 10 December 2008

Recommended by Mohamed Khamsi

Let S be a left amenable semigroup, let S  {Ts : s ∈ S} be a representation of S as Lipschitzian mappings from a nonempty compact convex subset C of a smooth Banach space E into C with a

uniform Lipschitzian condition, let{μ n} be a strongly left regular sequence of means defined on an

S-stable subspace of l∞S, let f be a contraction on C, and let {α n }, {β n }, and {γ n} be sequences

in0, 1 such that α n  β n  γ n  1, for all n Let x n1  α n fx n   β n x n  γ n Tμ n x n , for all n ≥ 1.

Then, under suitable hypotheses on the constants, we show that{x n} converges strongly to some

z in FS, the set of common fixed points of S, which is the unique solution of the variational

inequalityf − Iz, Jy − z ≤ 0, for all y ∈ FS.

Copyrightq 2008 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

Let E be a real Banach space and let C be a nonempty closed convex subset of E A mapping

T : C → C is said to be

i Lipschitzian with Lipschitz constant l > 0 if

ii nonexpansive if

satisfying the property limn → ∞ k n 1 and

Halpern1 introduced the following iterative scheme for approximating a fixed point

of a nonexpansive mapping T on C:

x n1  α n x  1 − α n Tx n , n  1, 2, , 1.4

space with a uniformly Gateaux differentiable norm

to a fixed point of T which is a solution of a variational inequality Recently, many papers have

It is an interesting problem to extend the above results to the nonexpansive semigroup

x n1  α n x  1 − α n Tμ n x n , n  1, 2, , 1.6

C of a smooth and strictly convex Banach space with respect to a left regular sequence {μ n}

refer the readers to20,21

The iterative methods for approximation of fixed points of asymptotically

For a semigroup S, we can define a partial preordering ≺ on S by a ≺ b if and only if

aS ⊃ bS If S is a left reversible semigroup i.e., aS ∩ bS /  ∅ for a, b ∈ S, then it is a directed

set.Indeed, for every a, b ∈ S, applying aS ∩ bS / ∅, there exist a, b ∈ S with aa bb; by

If a semigroup S is left amenable, then S is left reversible 33

Definition 1.1 Let S  {Ts : s ∈ S} be a representation of a left reversible semigroup S as Lipschitzian mappings on C with Lipschitz constants {ks : s ∈ S} We will say that S is an

asymptotically nonexpansive semigroup on C, if there holds the uniform Lipschitzian condition

lims ks ≤ 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

the following viscosity iterative scheme

for an asymptotically nonexpansive semigroup S  {Ts : s ∈ S} on a compact convex

{α n }, {β n } and {γ n } are sequences in 0, 1 such that α n  β n  γ n  1, for all n Then, under

some z in FS, the set of common fixed points of S, which is the unique solution of the

variational inequality

It is remarked that we have not assumed E to be strictly convex and our results are new

11,19

2 Preliminaries

will be denoted byx, x∗ or x∗x With each x ∈ E, we associate the set

Jx 

x∗∈ E∗:x, x∗  x∗ 2 x 2

Banach space E is said to be smooth if the duality mapping J of E is single valued We know

l s ft  fst and r s ft  fts for each t ∈ S and f ∈ l∞S Let X be a subspace of

be left invariantresp., right invariant, that is, l s X ⊂ X resp., r s X ⊂ X for each s ∈ S.

μr s f  μf for each s ∈ S and f ∈ X X is said to be left resp., right amenable if X has

{μ α } of means on X is said to be strongly left regular if

lim

α l∗

subset of E Throughout this paper, S will always denote a semigroup with an identity e S is called left reversible if any two right ideals in S have nonvoid intersection, that is, aS ∩ bS / ∅

for a, b ∈ S In this case, we can define a partial ordering ≺ on S by a ≺ b if and only if aS ⊃ bS.

It is easy too see t ≺ ts, ∀t, s ∈ S Further, if t ≺ s then pt ≺ ps for all p ∈ S If a semigroup S

is left amenable, then S is left reversible But the converse is false.

S  {Ts : s ∈ S} is called a representation of S as Lipschitzian mappings on C if for each s ∈ S, the mapping Ts is Lipschitzian mapping on C with Lipschitz constant ks, and

Tst  TsTt for s, t ∈ S We denote by FS the set of common fixed points of S, and

by C a the set of almost periodic elements in C, that is, all x ∈ C such that {Tsx : s ∈ S} is

P x  tx − P x ∈ B,

A subset D of B is said to be a sunny nonexpansive retract of B if there exists a sunny nonexpansive retraction P of B onto D We know that if E is smooth and P is a retraction

of B onto D, then P is sunny and nonexpansive if and only if for each x ∈ B and z ∈ D,

For more details see20,21

Lemma 2.1 Let S be a left reversible semigroup and S  {Ts : s ∈ S} be a representation of S as

Lipschitzian mappings from a nonempty weakly compact convex subset C of a Banach space E into C, with the uniform Lipschitzian condition lim s ks ≤ 1 on the Lipschitz constants of the mappings Let

X be a left invariant S-stable subspace of l∞S containing 1, and μ be a left invariant mean on X.

Then FS  FTμ ∩ C a

Corollary 2.2 Let {μ n } be an asymptotically left invariant sequence of means on X If z ∈ C a and

lim infn → ∞ Tμ n z − z  0, then z is a common fixed point for S.

Proof From lim inf n → ∞ Tμ n z − z  0, there exists a subsequence {Tμ n k z} of {Tμ n z} that converges strongly to z Since the set of means on X is compact in the weak-star topology,

there exists a subnet{μ n kα : α ∈ Λ} of {μ n k } such that {μ n kα } converges to μ in the weak-star topology Then, it is easy to show that μ is a left invariant mean on X On the other hand, for each x∗∈ E∗, we have



T

μ n kα



z, x∗

Tμz It follows fromLemma 2.1that z is a common fixed point of S.

Lemma 2.3 Let S be a left reversible semigroup and S  {Ts : s ∈ S} be a representation of S as

Lipschitzian mappings from a nonempty weakly compact convex subset C of a Banach space E into C,

with the uniform Lipschitzian condition lim s ks ≤ 1 on the Lipschitz constants of the mappings Let

X be a left invariant subspace of l∞S containing 1 such that the mappings s → Tsx, x∗ be in X

for all x ∈ X and x∗∈ E∗, and {μ n } be a strongly left regular sequence of means on X Then

lim sup

n → ∞

sup

x,y∈C



Proof Consider an arbitrary ε > 0 and take d  diamC Since lim s ks ≤ 1, there exists

s0∈ S such that

sup

s≥s0

ks < 1  ε

From limn → ∞ l∗

l∗

s0μ n − μ n< ε

Tμ n x − Tμ n y2 Tμ n x − Tμ n y, x∗

 μ nsTsx − Tsy, x∗

−l∗s0μ n



s



Tsx − Tsy, x∗

l∗s0μ n



s



Tsx − Tsy, x∗

≤μ n − l∗

s0μ nd x∗  μ nsTs0sx − Ts0sy, x∗

2d dTμ n x − Tμ n y  sup

s∈S

Ts0sx − Ts0syTμ n x − Tμ n y

2Tμ n x − Tμ n y  sup

s∈S

ks0s x − y Tμ n x − Tμ n y.

2.10 Therefore,

Tμ n x − Tμ n y ≤ ε

s∈S

ks0s x − y

s≥s0

ks x − y ≤ ε

2.11

that is,

sup

x,y∈C

Since ε > 0 is arbitrary, the desired result follows.

Remark 2.4 Taking inLemma 2.3

x,y∈C

Corollary 2.5 Let S be a left reversible semigroup and S  {Ts : s ∈ S} be a representation of

S as Lipschitzian mappings from a nonempty compact convex subset C of a Banach space E into C, with the uniform Lipschitzian condition lim s ks ≤ 1 Let X be a left invariant S-stable subspace of

Moreover, if E is smooth, then FS is a sunny nonexpansive retract of C and the sunny nonexpansive retraction of C onto FS is unique.

Proof From 2.14, by taking μ n  μ ∀n, it follows that T μ is nonexpansive So, from

Lemma 2.1, we get FS  FT μ  / ∅ 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 of C and the sunny nonexpansive retraction of C onto FS

is unique19,20 This concludes the result

We will need the following lemmas in what follows

Lemma 2.6 see 20,21 Let X be a real Banach space and let J be the duality mapping Then, for

any given x, y ∈ X and jx  y ∈ Jx  y, there holds the inequality

Lemma 2.7 see 40 Assume {a n } is a sequence of nonnegative real numbers such that

where {γ n } is a sequence in 0, 1 and {δ n } is a sequence in R such that

i∞n1 γ n  ∞;

ii lim supn → ∞ δ n /γ n ≤ 0 or∞

n1 |δ n | < ∞.

Then lim n → ∞ a n  0.

Lemma 2.8 see 41 Let {x n } and {z n } be bounded sequences in a Banach space X and let {β n}

be a sequence in 0, 1 with 0 < lim inf n → ∞ β n and lim sup n → ∞ β n < 1 Suppose

for all integers n ≥ 0 and

lim sup

n → ∞



Then lim n → ∞ x n − z n  0.

3 The main theorem

We are now ready to establish our main theorem

Theorem 3.1 Let S be a left reversible semigroup and S  {Ts : s ∈ S} be a representation of S

as Lipschitzian mappings from a nonempty compact convex subset C of a smooth Banach space E into

C, with the uniform Lipschitzian condition lim s ks ≤ 1 and f be an α-contraction on C for some

0 < α < 1 Let X be a left invariant S-stable subspace of l∞S containing 1, {μ n } be a strongly left

regular sequence of means on X such that lim n → ∞ μ n1 − μ n  0 and {c n } be the sequence defined

by2.13 Let {α n }, {β n } and {γ n } be sequences in 0, 1 such that

i α n  β n  γ n  1, ∀n,

ii limn → ∞ α n  0;

iii∞

n1 α n ∞;

iv lim supn → ∞ c n /α n ≤ 0; (note that, by Remark 2.4 , lim sup n → ∞ c n ≤ 0)

v 0 < lim inf n → ∞ β n≤ lim supn → ∞ β n < 1.

Let {x n } be the following sequence generated by x1 ∈ C and ∀n ≥ 1,

Then {x n } converges strongly to z ∈ FS which is the unique solution of the variational inequality

Equivalently, one has z  P fz, where P is the unique sunny nonexpansive retraction of C onto FS Remark 3.2 For example, we may choose

α n:

⎧

⎪

⎨

⎪

⎩

1

n  √c n if c n ≥ 0,

1

n if c n < 0.

3.3

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

Step 1 Claim Let {ω n } be a sequence in C Then

lim

Put D  sup{ z : z ∈ C} Then

Tμ n1 ω n − Tμ n ω n  sup

z 1

Tμ n1 ω n − Tμ n ω n , z

 sup

z 1 μ n1sTsω n , z

− μ nsTsω n , z

≤ μ n1 − μ n sup

s∈S

3.5

Step 2 Claim lim n → ∞ x n1 − x n  0

Define a sequence{z n } by z n  x n1 − β n x n /1 − β n  so that x n1  β n x n  1 − β n z n

We now compute

z n1 − z n  1

1− β n1



x n2 − β n1 x n1



1− β n



x n1 − β n x n



1− β1n1α n1 f

x n1



 γ n1 T

μ n1



x n1



1− β n



α n fx n   γ n Tμ n x n



1− β1n1α n1 f

x n1



1− α n1 − β n1



T

μ n1



x n1



1− β n



α n fx n 1− α n1 − β n1



Tμ n x n

≤T

μ n1



x n1 − Tμ n x n



 α n1

1− β n1



f

x n1



− Tμ n1



x n1



1− β n1



f

x n1



− Tμ n1



x n1.

3.6

z n1 − z n  ≤ Tμ n1



x n1 − Tμ n x n1   Tμ n x n1 − Tμ n x n   Kα n1  α n



≤T

μ n1



x n1 − Tμ n x n1   x n1 − x n   c n  Kα n1  α n



lim sup

n

z n1 − z n  − x n1 − x n

≤ lim sup

n

T

μ n1



x n1 − Tμ n x n1   c n  Kα n1  α n



ApplyingLemma 2.8, we get limn x n1 − x n  limn 1 − β n  x n − z n  0.

Step 3 Claim The ω-limit set of {x n }, ω{x n }, is a subset of FS.

that

x n1 − x n  α n fx n   1 − β n Tμ n x n − x n  − α n Tμ n x n 3.9 So

x n − Tμ n x n ≤ 1

1− β n

x n1 − x n   α nfx n  − Tμ n x n. 3.10

lim

lim sup

k → ∞

y − T

μ n k



k → ∞

y − x n

k   x n k − Tμ n k



x n k   Tμ n k



x n k − Tμ n k



y

≤ lim sup

k → ∞



2y − x n k  x n k − Tμ n k



x n k  c n k



≤ 0.

3.12

Step 4 Claim The sequence {x n } converges strongly to z  Pfz.

sunny nonexpansive retraction P of C onto FS The Banach Contraction Mapping Principal

We first show

lim sup

n → ∞

Let{x n k } be a subsequence of {x n} such that

lim

k → ∞



f − Iz, Jx n k − z lim sup

n → ∞ f − Iz, Jx n − z. 3.15

y ∈ FS Smoothness of E and a combination of 3.13 and 3.15 give

lim sup

n → ∞



as required Now, taking

x n1 − z2γ n u n − z  β n x n − z

 α n



γfx n  − z2

≤γ n u n − z  β n x n − z2 2α n



fx n  − z, Jx n1 − z

≤ 1 − β n

 γ n

1− β n u n − z

2 β nx n − z2

 2α n



fx n  − fz, Jx n1 − z 2α n



fz − z, J

x n1 − z

≤ γ n2

1− β n

u n − z2 β nx n − z2

 2α n αx n − zx n1 − z  2α n



fz − z, J

x n1 − z

≤ γ n2

1− β n

x n − z2 c n γ2

n

1− β n  β nx n − z2

 α n αx n − z2x n1 − z2

 2α n



fz − z, J

x n1 − z



γ2

n

1− β n  β n  α n α x n − z2

 α n αx n1 − z2 2α n



fz − z, J

x n1 − z c n γ n2

1− β n



1 − α n α − 2α n  2α n α  α

2

n

1− β n x n − z2

 α n αx n1 − z2 2α n



fz − z, J

x n1 − z c n γ2

n

1− β n

3.18

It follows that

x n1 − z2≤

1−21 − ααn

1− α n α x n − z2

1− α n α

2

γfz − z, J

x n1 − z α n

1− β n

x n − z2 c n

α n × γ n2

1− β n

3.19

Corollary 3.3 Let S be a left reversible semigroup and S  {Ts : s ∈ S} be a representation of

S as nonexpansive mappings from a nonempty compact convex subset C of a smooth Banach space

E into C and f be an α-contraction on C for some 0 < α < 1 Let X be a left invariant S-stable subspace of l∞S containing 1 and {μ n } be a strongly left regular sequence of means on X such that

limn → ∞ μ n1 − μ n  0 Let {α n }, {β n } and {γ n } be sequences in 0, 1 such that

i α n  β n  γ n  1, ∀n,

ii limn → ∞ α n  0;

iii∞n1 α n ∞;

iv 0 < lim inf n → ∞ β n≤ lim supn → ∞ β n < 1.

Let {x n } be the sequence generated by x1∈ C and ∀n ≥ 1,

Then {x n } converges strongly to z ∈ FS which is the unique solution of the variational inequality

Equivalently, one has z  P fz, where P is the unique sunny nonexpansive retraction of C onto FS Remark 3.4 If S is a countable left amenable semigroup, then there is a strong left regular

i1 λ i δ x i , λ i≥ 0,n

i1 λ i  1 See 42, Corollary 3.7

Remark 3.5 It is known that if S is a left reversible semigroup, then WAP S, the space of

weakly almost periodic functions on S, has a left invariant mean But the converse is not true

see 43

Problem Can the hypothesis on S ofTheorem 3.1be replaced by WAP S has 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 with ST  TS and f be an α-contraction

on C for some 0 < α < 1 Let {c n } be defined by

c n d

n2

n−1



i0

n−1

j0



1− k i l j



where, d  diamC and k i and l j are defined as

S i x − S i y ≤ k i x − y , T j x − T j y ≤ l j x − y , 4.2