Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 560671, 18 pages
doi:10.1155/2011/560671
Research Article
A New Composite General Iterative Scheme for
Nonexpansive Semigroups in Banach Spaces
Pongsakorn Sunthrayuth and Poom Kumam
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi
(KMUTT), Bangmod, Bangkok 10140, Thailand
Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th
Received 1 February 2011; Accepted 19 March 2011
Academic Editor: Yonghong Yao
Copyright q 2011 P. Sunthrayuth and P. Kumam. 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.
We introduce a new general composite iterative scheme for finding a common fixed point of
nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous
duality mapping. A strong convergence theorem of the purposed iterative approximation method
is established under some certain control conditions. Our results improve and extend announced
by many others.
1. Introduction
Throughout this paper we denoted by Æ and Ê the set of all positive integers and all positive
real numbers, respectively. Let X be a real Banach space, and let C be a nonempty closed
convex subset of X. A mapping T of C into itself is said to be nonexpansive if Tx−Ty ≤ x−y
for each x, y ∈ C. We denote by FT the set of fixed points of T. We know that FT is
nonempty if C is bounded; for more detail see 1. A one-parameter family S {Tt : t ∈ Ê }
from C of X into itself is said to be a nonexpansive semigroup on C if it satisfies the following
conditions:
i T0x x for all x ∈ C;
ii Ts t Ts ◦ Tt for all s, t ∈ Ê ;
iii for each x ∈ C the mapping t → Ttx is continuous;
iv Ttx − Tty ≤ x − y for all x, y ∈ C and t ∈ Ê .
We denote by FS the set of all common fixed points of S, that is, FS : ∩t∈Ê FTt {x ∈ C : Ttx x}. We know that FS is nonempty if C is bounded; see 2. Recall that
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a self-mapping f : C → C is a contraction if there exists a constant α ∈ 0, 1 such that
fx − fy ≤ αx − y for each x, y ∈ C. As in 3, we use the notation C to denote the
collection of all contractions on C, that is, C {f : C → C a contraction}. Note that each
f ∈ C has a unique fixed point in C.
In the last ten years, the iterative methods for nonexpansive mappings have recently
been applied to solve convex minimization problems; see, for example, 3–5. Let H be a real
Hilbert space, whose inner product and norm are denoted by ·, ·
and · , respectively. Let
A be a strongly positive bounded linear operator on H: that is, there is a constant γ > 0 with
property
Ax, x
≥ γx2
∀x ∈ H.
1.1
A typical problem is to minimize a quadratic function over the set of the fixed points of a
nonexpansive mapping on a real Hilbert space H:
1
min Ax, x
− x, b
,
x∈F 2
1.2
where C is the fixed point set of a nonexpansive mapping T on H and b is a given point in H.
In 2003, Xu 3 proved that the sequence {xn } generated by
x0 ∈ C chosen arbitrarily,
xn1 I − αn ATxn αn u,
∀n ≥ 0,
1.3
converges strongly to the unique solution of the minimization problem 1.2 provided that
the sequence {αn } satisfies certain conditions. Using the viscosity approximation method,
Moudafi 6 introduced the iterative process for nonexpansive mappings see 3, 7 for
further developments in both Hilbert and Banach spaces and proved that if H is a real
Hilbert space, the sequence {xn } generated by the following algorithm:
x0 ∈ C chosen arbitrarily,
xn1 αn fxn 1 − αn Txn ,
∀n ≥ 0,
1.4
where f : C → C is a contraction mapping with constant α ∈ 0, 1 and {αn } ⊂ 0, 1 satisfies
certain conditions, converges strongly to a fixed point of T in C which is unique solution x∗
of the variational inequality:
f − I x∗ , y − x∗ ≤ 0,
∀y ∈ FT.
1.5
In 2006, Marino and Xu 8 combined the iterative method 1.3 with the viscosity
approximation method 1.4 considering the following general iterative process:
x0 ∈ C chosen arbitrarily,
xn1 αn γ fxn I − αn ATxn ,
∀n ≥ 0,
1.6
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where 0 < γ < γ/α. They proved that the sequence {xn } generated by 1.6 converges strongly
to a unique solution x∗ of the variational inequality:
γ f − A x∗ , y − x∗ ≤ 0,
∀y ∈ FT,
1.7
which is the optimality condition for the minimization problem:
1
min Ax, x
− hx,
x∈C 2
1.8
where C is the fixed point set of a nonexpansive mapping T and h is a potential function for
γ f i.e., h x γ fx for x ∈ H. Kim and Xu 9 studied the sequence generated by the
following algorithm:
x1 ∈ C chosen arbitrarily,
yn αn xn 1 − αn Txn ,
xn1 βn u 1 − βn yn , ∀n ≥ 0,
1.9
and proved strong convergence of scheme 1.9 in the framework of uniformly smooth
Banach spaces. Later, yao, et al. 10 introduced a new iteration process by combining the
modified Mann iteration 9 and the viscosity approximation method introduced by Moudafi
6. Let C be a closed convex subset of a Banach space, and let T : C → C be a nonexpansive
mapping such that FT /
∅ and f ∈ C . Define {xn } in the following way:
x1 ∈ C chosen arbitrarily;
xn1
yn αn xn 1 − αn Txn ;
βn fxn 1 − βn yn , ∀n ≥ 0,
1.10
where {αn } and {βn } are two sequences in 0, 1. They proved under certain different control
conditions on the sequences {αn } and {βn } that {xn } converges strongly to a fixed point of T.
Recently, Chen and Song 11 studied the sequence generated by the algorithm in a uniformly
convex Banach space, as follows:
xn1
x1 ∈ C chosen arbitrarily;
1 tn
αn fxn 1 − αn Tsxn ds,
tn 0
∀n ∈ Æ ,
1.11
and they proved that the sequence {xn } defined by 1.11 converges strongly to the unique
solution of the variational inequality:
f − I x∗ , J y − x∗ ≤ 0,
∀y ∈ FT.
1.12
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In 2010, Sunthrayuth and Kumam 12 introduced the a general iterative scheme generated
by
xn1
x0 ∈ C chosen arbitrarily;
1 tn
αn γ fxn βn xn 1 − βn I − αn A
Tsxn ds,
tn 0
∀n ≥ 0,
1.13
for the approximation of common fixed point of a one-parameter nonexpansive semigroup in
a Banach space under some appropriate control conditions. They proved strong convergence
theorems of the iterative scheme which solve some variational inequality. Very recently,
Kumam and Wattanawitoon 13 studied and introduced a new composite explicit viscosity
iteration method of fixed point solutions of variational inequalities for nonexpansive
semigroups in Hilbert spaces. They proved strong convergence theorems of the composite
iterative schemes which solve some variational inequalities under some appropriate conditions. In the same year, Sunthrayuth et al. 14 introduced a general composite iterative
scheme for nonexpansive semigroups in Banach spaces. They established some strong convergence theorems of the general iteration scheme under different control conditions.
In this paper, motivated by Yao et al. 10, Sunthrayuth, and Kumam 12 and Kumam
and Wattanawitoon 13 we introduce a new general iterative algorithm 3.23 for finding
a common point of the set of solution of some variational inequality for nonexpansive
semigroups in Banach spaces which admit a weakly continuous duality mapping and then
proved the strong convergence theorem generated by the proposed iterative scheme. The
results presented in this paper improve and extend some others from Hilbert spaces to Banach
spaces and some others as special cases.
2. Preliminaries
Throughout this paper, we write xn x resp., xn ∗ x to indicate that the sequence {xn }
weakly resp., weak∗ converges to x; as usual xn → x will symbolize strong convergence;
also, a mapping I denote the identity mapping. Let X be a real Banach space, and let X ∗ be
its dual space. Let U {x ∈ X : x 1}. A Banach space X is said to be uniformly convex
if, for each ∈ 0, 2, there exists a δ > 0 such that for each x, y ∈ U, x − y ≥ implies
x y/2 ≤ 1 − δ. It is known that a uniformly convex Banach space is reflexive and strictly
convex see also 15. A Banach space is said to be smooth if the limit limt → 0 x ty − x/t
exists for each x, y ∈ U. It is also said to be uniformly smooth if the limit is attained uniformly
for x, y ∈ U.
Let ϕ : 0, ∞ : Ê → Ê be a continuous strictly increasing function such that
ϕ0 0 and ϕt → ∞ as t → ∞. This function ϕ is called a gauge function . The duality
∗
mapping Jϕ : X → 2X associated with a gauge function ϕ is defined by
Jϕ x f ∗ ∈ X ∗ : x, f ∗ xϕx, f ∗ ϕx, ∀x ∈ X ,
2.1
where ·, ·
denotes the generalized duality paring. In particular, the duality mapping with
the gauge function ϕt t, denoted by J, is referred to as the normalized duality mapping.
0 see 16.
Clearly, there holds the relation Jϕ x ϕx/xJx for each x /
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Browder 16 initiated the study of certain classes of nonlinear operators by means of
the duality mapping Jϕ . Following Browder 16, we say that Banach space X has a weakly
continuous duality mapping if there exists a gauge function ϕ for which the duality mapping
Jϕ x is single-valued and continuous from the weak topology to the weak∗ topology; that
is, for each {xn } with xn x, the sequence {Jxn } converges weakly∗ to Jϕ x. It is known
that lp has a weakly continuous duality mapping with a gauge function ϕt tp−1 for all
t
1 < p < ∞. Set Φt 0 ϕτdτ, for all t ≥ 0; then Jϕ x ∂Φx, where ∂ denotes the
subdifferential in the sense of convex analysis recall that the subdifferential of the convex
function φ : X → Ê at x ∈ X is the set ∂φx {x∗ ∈ X; φy ≥ φx x∗ , y − x
, for all y ∈
X}.
In a Banach space having a weakly continuous duality mapping Jϕ with a gauge
function ϕ, we defined an operator A is to be strongly positive see 17 if there exists a
constant γ > 0 with the property
Ax, Jϕ x ≥ γxϕx,
aI − bA sup aI − bAx, Jϕ x ,
x≤1
2.2
a ∈ 0, 1, b ∈ −1, 1.
2.3
If X : H is a real Hilbert space, then the inequality 2.2 reduces to 1.1.
The first part of the next lemma is an immediate consequence of the subdifferential
inequality and the proof of the second part can be found in 18.
Lemma 2.1 see 18. Assume that a Banach space X has a weakly continuous duality mapping Jϕ
with gauge ϕ.
i For all x, y ∈ X, the following inequality holds:
Φ x y ≤ Φx y, Jϕ x y .
2.4
In particular, for all x, y ∈ X,
x y2 ≤ x2 2 y, J x y .
2.5
ii Assume that a sequence {xn } in X converges weakly to a point x ∈ X. Then the following
identity holds:
lim sup Φ xn − y lim sup Φxn − x Φ y − x ,
n→∞
n→∞
∀x, y ∈ X.
2.6
Lemma 2.2 see 17. Assume that a Banach space X has a weakly continuous duality mapping Jϕ
with gauge ϕ. Let A be a strongly positive linear bounded operator on X with a coefficient γ > 0 and
0 < ρ ≤ ϕ1A−1 . Then I − ρA ≤ ϕ11 − ργ.
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Lemma 2.3 see 11. Let C be a closed convex subset of a uniformly convex Banach space X and
∅. Then, for each r > 0
let S {Tt : t ∈ Ê } be a nonexpansive semigroup on C such that FS /
and h ≥ 0,
t
1 t
1
Tsx ds − Th
Tsx ds 0.
lim sup t → ∞x∈C∩B t 0
t
0
r
2.7
Lemma 2.4 see 19. Assume that {an } is a sequence of nonnegative real numbers such that
an1 ≤ 1 − μn an δn ,
2.8
where {μn } is a sequence in 0, 1 and {δn } is a sequence in Ê such that
i ∞
n0 μn ∞;
ii lim supn → ∞ δn /μn ≤ 0 or ∞
n0 |δn | < ∞.
Then, limn → ∞ an 0.
3. Main Results
Let X be a Banach space which admits a weakly continuous duality mapping Jϕ with gauge
ϕ such that ϕ is invariant on 0, 1, and let C be a nonempty closed convex subset of X such
that C ± C ⊂ C. Let S {Tt : t ∈ Ê } be a nonexpansive semigroup from C into itself, let
f be a contraction mapping with a coefficient α ∈ 0, 1, let A be a strongly positive linear
bounded operator with a coefficient γ > 0 such that 0 < γ < γ ϕ1/α, and let t ∈ 0, 1 such
f
that t ≤ ϕ1A−1 which satisfies t → 0. Define the mapping Tt : C → C by
f
Tt : tγ f I − tA
1
λt
λt
Tsds
3.1
0
to be a contraction mapping. Indeed, for each x, y ∈ C,
λ
t
1
f f
Tsx − Tsy ds Tt x − Tt y tγ fx − f y I − tA
λt 0
λ
t
1
Tsx − Tsy ds
≤ tγ fx − f y I − tA
λt 0
≤ tγ αx − y ϕ1 1 − tγ x − y
≤ 1 − t ϕ1γ − γ α x − y.
3.2
Thus, by Banach contraction mapping principle, there exists a unique fixed point xt ∈ C, that
is,
1
xt tγ fxt I − tA
λt
λt
Tsxt ds.
0
3.3
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Remark 3.1. We note that space lp has a weakly continuous duality mapping with a gauge
function ϕt tp−1 for all 1 < p < ∞. This shows that ϕ is invariant on 0, 1.
Lemma 3.2. Let X be a uniformly convex Banach space which admits a weakly continuous duality
mapping Jϕ with gauge ϕ such that ϕ is invariant on 0, 1, and let C be a nonempty closed convex
subset of X such that C ± C ⊂ C. Let S {Tt : t ∈ Ê } be a nonexpansive semigroup from C
into itself such that FS /
∅, let f be a contraction mapping with a coefficient α ∈ 0, 1, let A be a
strongly positive linear bounded operator with a coefficient γ > 0 such that 0 < γ < γ ϕ1/α, and let
t ∈ 0, 1 such that t ≤ ϕ1A−1 which satisfies t → 0. Then the net {xt } defined by 3.3 with
{λt }0<t<1 is a positive real divergent sequence, converges strongly as t → 0 to a common fixed point
x∗ , in which x∗ ∈ FS, and is the unique solution of the variational inequality:
γ fx∗ − Ax∗ , Jϕ x − x∗ ≤ 0,
∀x ∈ FS.
3.4
Proof. Firstly, we show the uniqueness of a solution of the variational inequality 3.4.
Suppose that x,
x∗ ∈ FS are solutions of 3.4; then
γ fx∗ − Ax∗ , Jϕ x − x∗ ≤ 0,
γ fx
− Ax,
Jϕ x∗ − x
≤ 0.
3.5
Adding up 3.5, we obtain
γ fx∗ − Ax∗ − γ fx
− Ax , Jϕ x − x∗ − fx∗ , Jϕ x − x∗ Ax − x∗ , Jϕ x − x∗ − γ fx
− fx∗ Jϕ x − x∗ ≥ γ x − x∗ ϕx − x∗ − γ fx
0≥
≥ γ Φx − x∗ − γ αΦx − x∗ γ − γ α Φx − x∗ ≥ ϕ1γ − γ α Φx − x∗ ,
3.6
which is a contradiction, we must have x x∗ , and the uniqueness is proved. Here in after,
we use x to denote the unique solution of the variational inequality 3.4.
Next, we show that {xt } is bounded. Indeed, for each p ∈ FS, we have
λ
t
1
xt − p Tsxt − p ds t γ fxt − Ap I − tA
λt 0
1 λt Tsxt − pds
≤ tγ fxt − Ap I − tA
λt 0
≤ tγ fxt − f p tγ f p − Ap ϕ1 1 − tγ xt − p
≤ 1 − t ϕ1γ − γ α xt − p tγ f p − Ap.
3.7
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It follows that
xt − p ≤
1
γ f p − Ap.
ϕ1γ − γ α
3.8
λ
Hence, {xt } is bounded, so are {fxt } and {A1/λt 0 t Tsxt ds}.
Next, we show that xt − Thxt → 0 as t → 0. We note that
1 λt
1 λt
Tsxt ds tγ fxt − A
Tsxt ds .
xt −
λt 0
λt 0
3.9
Moreover, we note that
λ
1 λt
t
1
1 λt
Tsxt ds Tsxt ds − Th
Tsxt ds xt − Thxt ≤ xt −
λt 0
λt 0
λt 0
λ
t
1
Tsxt ds − Thxt Th
λt 0
λ
1 λt
t
1
1 λt
≤ 2xt −
Tsxt ds Tsxt ds − Th
Tsxt ds ,
λt 0
λt 0
λt 0
3.10
for all h ≥ 0. Define the set K {z ∈ C : z − p ≤ γ fp − Ap/ϕ1γ − γ α}; then K is a
nonempty bounded closed convex subset of C which is Ts-invariant for each h ≥ 0. Since
{xt } ⊂ K and K is bounded, there exists r > 0 such that K ⊂ Br , and it follows by Lemma 2.3
that
λ
1 λt
t
1
Tsxt ds − Th
Tsxt ds 0,
lim 3.11
λt → ∞ λt 0
λt 0
for each h ≥ 0. From 3.9-3.10, letting t → 0 and noting 3.11 then, for each h ≥ 0, we
obtain
xt − Thxt −→ 0.
3.12
Assume that {tn }∞
n1 ⊂ 0, 1 is such that tn → 0 as n → ∞. Put xn : xtn and λn : λtn . We
will show that {xn } contains a subsequence converging strongly to x ∈ FS. Since {xn } is
bounded sequence and Banach space X is a uniformly convex, hence it is reflexive, and there
exists a subsequence {xnj } of {xn } which converges weakly to some x ∈ C as j → ∞. Again,
since Jϕ is weakly sequentially continuous, we have by Lemma 2.1 that
lim sup Φ xnj − z lim sup Φ xnj − x Φz − x.
j→∞
j →∞
Let Hz lim supj → ∞ Φxnj − z, for all z ∈ C.
3.13
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It follows that Hz Hx
Φz − x,
for all z ∈ C. From 3.12, we have
HThx
lim sup Φ xnj − Thx
j →∞
lim sup Φ Thxnj − Thx
j →∞
3.14
≤ lim sup Φ xnj − x Hx.
j →∞
On the other hand, we note that
HThx
lim sup Φ xnj − x ΦThx − x
j →∞
3.15
Hx
ΦThx − x.
Combining 3.14 with 3.15, we obtain ΦThx − x
≤ 0. This implies that Thx x,
t
that is, x ∈ FS. In fact, since Φt 0 ϕτdτ, for all t ≥ 0 and ϕ : Ê → Ê is the gauge
function, then for 1 ≥ k ≥ 0, ϕky ≤ ϕy and
Φkt kt
0
ϕτdτ k
t
0
ϕ ky dy ≤ k
t
ϕ y dy kΦt.
3.16
0
By Lemma 2.1, we have
λ
t
1
Φxn − x
Φ tn γ fxn − Ax I − tA
Tsxn ds − x λn 0
1 λn
Jϕ xn − x
≤ Φ I − tn A
Tsxn − xds
tn γ fxn − Ax,
λn 0
≤ Φ ϕ1 1 − tn γ xn − x
tn γ fxn − γ fx,
Jϕ xn − x
tn γ fx
− Ax,
Jϕ xn − x
≤ ϕ1 1 − tn γ Φxn − x
tn γ αxn − x
Jϕ xn − x
tn γ fx
− Ax,
Jϕ xn − x
ϕ1 1 − tn γ Φxn − x
− Ax,
Jϕ xn − x
tn γ αΦxn − x
tn γ fx
≤ 1 − tn ϕ1γ − γ α Φxn − x
tn γ fx
.
− Ax,
Jϕ xn − x
3.17
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This implies that
Φxn − x
≤
1
.
γ fx
− Ax,
Jϕ xn − x
ϕ1γ − γ α
3.18
1
γ fx
− Ax,
Jϕ xnj − x .
ϕ1γ − γ α
3.19
In particular, we have
Φ xnj − x ≤
Since the mapping Jϕ is single-valued and weakly continuous, it follows from 3.19 that
→ 0 as j → ∞. This implies that xnj → x as j → ∞.
Φxnj − x
Next, we show that x solves the variational inequality 3.4, for each x ∈ FS. From
3.3, we derive that
λ
t
1
1
γ f − A xt − I − tA
Tsxt ds − xt .
t
λt 0
3.20
Now, we observe that
1
λt
λt
0
1
I − Tsx ds −
λt
x − xt , Jϕ x − xt −
λt
I − Tsxt ds, Jϕ x − xt 0
1
λt
1
≥ x − xt Jϕ x − xt −
λt
λt
Tsx − Tsxt ds, Jϕ x − xt 0
λt
Ts x − Tsxt dsJϕ x − xt 0
≥ Φx − xt − x − xt Jϕ x − xt Φx − xt − Φx − xt 0.
3.21
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It follows from 3.20 that
1
γ f − A xt , Jϕ x − xt −
t
1
−
t
I − tA
I − tA
1
λt
1
λt
λt
Tsxt ds − xt , Jϕ x − xt 0
λt
0
1
Tsxt ds −
λt
λt
xt ds , Jϕ x − xt 0
1 1 λt
1 λt
−
I − Tsx ds −
I − Tsxt ds, Jϕ x − xt t λt 0
λt 0
λ
t
1
A
Ts − Ixt ds , Jϕ x − xt λt 0
λ
t
1
≤ A
Ts − Ixt ds , Jϕ x − xt .
λt 0
3.22
Now, replacing t and λt with tnj and λnj , respectively, in 3.22, and letting j → ∞, and we
Jϕ x− x
≤
notice that Ts−Ixnj → Ts−Ix 0 for x ∈ FS, we obtain that γ f −Ax,
0. That is, x is a solution of the variational inequality 3.4. By uniqueness, as x x∗ , we have
shown that each cluster point of the net {xt } is equal to x∗ . Then, we conclude that xt → x∗
as t → 0. This proof is complete.
Theorem 3.3. Let X be a uniformly convex Banach space which admits a weakly continuous duality
mapping Jϕ with the gauge function ϕ such that ϕ is invariant in 0, 1, and let C be a nonempty closed
convex subset of X such that C ± C ⊂ C. Let S {Tt : t ∈ Ê } be a nonexpansive semigroup from
C into itself such that FS /
∅, let f be a contraction mapping with a coefficient α ∈ 0, 1, and let
A be a strongly positive linear bounded operator with a coefficient γ > 0 such that 0 < γ < γ ϕ1/α.
∞
∞
∞
Let {αn }∞
n0 , {βn }n0 , {γn }n0 be the sequences in 0, 1 and let {tn }n0 be a positive real divergent
sequence. Assume that the following conditions hold:
C1 limn → ∞ α n 0 and ∞
n0 αn ∞,
C2 limn → ∞ γn 0,
C3 βn oαn ,
Then the sequence {xn } defined by
x0 ∈ C chosen arbitrarily;
1 tn
Tsxn ds;
zn γn xn 1 − γn
tn 0
yn αn γ fzn I − αn Azn ;
xn1 βn xn 1 − βn yn , ∀n ≥ 0,
converges strongly to the common fixed point x∗ that is obtained in Lemma 3.2.
3.23
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Proof. From the condition C1, we may assume, with no loss of generality, that αn ≤
ϕ1A−1 for each n ≥ 0. From Lemma 2.2, we have I − αn A ≤ ϕ11 − αn γ.
Firstly, we show that {xn } is bounded. Let p ∈ FS; we get
1 tn zn − p γn xn − p 1 − γn
Tsxn − p ds
tn 0
1 tn Tsxn − pds
≤ γn xn − p 1 − γn
tn 0
≤ γn xn − p 1 − γn xn − p
xn − p,
yn − p αn γ fzn − Ap I − αn A zn − p ≤ αn γ fzn − f p αn γ f p − Ap I − αn Azn − p
≤ 1 − αn ϕ1γ − γ α xn − p αn γ f p − Ap.
3.24
It follows that
xn1 − p βn xn − p 1 − βn yn − p ≤ βn xn − p 1 − βn yn − p
≤ βn xn − p 1 − βn
1 − αn ϕ1γ − γ α xn − p αn γ f p − Ap
γ f p − Ap
1 − αn ϕ1γ − γ α 1 − βn xn − p αn ϕ1γ − γ α 1 − βn
ϕ1γ − γ α
γ f p − Ap
.
≤ max xn − p,
ϕ1γ − γ α
3.25
By induction on n, we have
γ f p − Ap
xn − p ≤ max x0 − p,
,
ϕ1γ − γ α
Thus, {xn } is bounded. Since {xn } is bounded, then 1/tn t
{A1/tn 0n Tsxn ds} and {fzn } are also bounded.
tn
0
∀n ≥ 0.
3.26
Tsxn ds − p ≤ xn − p and
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Next, we show that limn → ∞ xn − Thxn 0, for all h ≥ 0. From 3.23, we note that
1 tn
1 tn
Tsxn ds ≤ xn1 − yn yn − zn zn −
Tsxn ds
xn1 −
tn 0
tn 0
tn
1
≤ βn xn − yn αn γ fzn − Azn γn xn −
Tsxn ds.
tn 0
3.27
By the conditions C1–C3, then 3.27, we obtain
1 tn
Tsxn ds 0.
lim xn1 −
n → ∞
tn 0
3.28
Moreover, we note that
1 tn
1 tn
1 tn
Tsxn ds Tsxn ds − Th
Tsxn ds xn1 − Thxn1 ≤ xn1 −
tn 0
tn 0
tn 0
t
1 n
Th
Tsxn ds − Thxn1 tn 0
t
1 tn
1 n
1 tn
≤ 2xn1 −
Tsxn ds Tsxn ds−Th
Tsxn ds .
tn 0
tn 0
tn 0
3.29
Define the set K {z ∈ C : z − p ≤ x0 − p γ fp − Ap/ϕ1γ − γ α}. Then K is a
nonempty bounded closed convex subset of C, which is Ts—invariant for each s ≥ 0 and
contains {xn }; it follows from Lemma 2.3 that
t
1 tn
1 n
lim Tsxn ds − Th
Tsxn ds 0,
n → ∞ tn 0
tn 0
∀h ≥ 0.
3.30
Then, for all h ≥ 0, from 3.28 and 3.30, into 3.29, we obtain limn → ∞ xn1 − Thxn1 0,
and hence
lim xn − Thxn 0,
n→∞
∀h ≥ 0.
3.31
Next, we show that lim supn → ∞ γ fx∗ − Ax∗ , Jϕ xn − x∗ ≤ 0. We can take subsequence
{xnj } ⊂ {xn } such that
lim γ fx∗ − Ax∗ , Jϕ xnj − x∗
lim sup γ fx∗ − Ax∗ , Jϕ xn − x∗ .
j→∞
n→∞
3.32
14
International Journal of Mathematics and Mathematical Sciences
By the assumption that X is uniformly convex, hence it is reflexive and {xn } is bounded; then
there exists a subsequence {xnj } which converges weakly to some x ∈ C as j → ∞. Since Jϕ
is weakly continuous, from Lemma 2.1, we have
lim sup Φ xnj − z lim sup Φ xnj − x Φz − x,
j →∞
j →∞
∀z ∈ C.
3.33
Let Hz lim supj → ∞ Φxnj − z, for all z ∈ C.
It follows that Hz Hx Φz − x, for all z ∈ C.
From 3.31, we have
HThx lim sup Φ xnj − Thx
j →∞
lim sup Φ Thxnj − Thx
3.34
j →∞
≤ lim sup Φ xnj − x Hx.
j →∞
On the other hand, we note that
HThx lim sup Φ xnj − x ΦThx − x
j →∞
3.35
Hx ΦThx − x.
Combining 3.34 with 3.35, we obtain ΦThx − x ≤ 0.
This implies that Thx x; that is, x ∈ FS.
Since the duality map Jϕ is single-valued and weakly continuous, we get that
lim sup γ fx∗ − Ax∗ , Jϕ xn − x∗ lim γ fx∗ − Ax∗ , Jϕ xnj − x∗
j →∞
n→∞
∗
∗
∗
3.36
γ fx − Ax , Jϕ x − x ≤ 0,
as required. Hence,
lim sup γ fx∗ − Ax∗ , Jϕ xn1 − x∗ ≤ 0.
n→∞
3.37
Since xn1 − yn βn xn − yn , by condition C3, we obtain that limn → ∞ xn1 − yn 0. It
follows from 3.37, that
lim sup γ fx∗ − Ax∗ , Jϕ yn − x∗ ≤ 0.
n→∞
3.38
International Journal of Mathematics and Mathematical Sciences
15
Finally, we show that xn → x∗ as n → ∞. Now, from Lemma 2.1, we have
Φ yn − x∗ Φ αn γ fzn − Ax∗ I − αn Azn − x∗ Φ αn γ fzn − γ fx∗ αn γ fx∗ − Ax∗ I − αn Azn − x∗ ≤ Φ αn γ fzn − γ fx∗ I − αn Azn − x∗ αn γ fx∗ − Ax∗ , Jϕ yn − x∗
≤ 1 − αn ϕ1γ − γ α Φxn − x∗ αn γ fx∗ − Ax∗ , Jϕ yn − x∗ .
3.39
On the other hand, we note that
Φxn1 − x∗ Φ βn xn − x∗ 1 − βn yn − x∗ ≤ 1 − βn Φ yn − x∗ βn xn − x∗ , Jϕ xn1 − x∗ .
3.40
It follows from 3.40 that
Φxn1 − x∗ ≤ 1 − αn ϕ1γ − γ α Φxn − x∗ αn γ fx∗ − Ax∗ , Jϕ yn − x∗
βn xn − x∗ , Jϕ xn1 − x∗ ∗
∗
∗
∗ βn
≤ 1−αn ϕ1γ −γ α Φxn −x αn γ fx −Ax , Jϕ yn −x M ,
αn
3.41
where M supn≥0 {xn − x∗ ϕxn1 − x∗ }.
Put μn : αn ϕ1γ − γ α and δn : αn γ fx∗ − Ax∗ , Jϕ yn − x∗ βn /αn M.
Then 3.41 reduces to formula Φxn1 − x∗ ≤ 1 − μn Φxn − x∗ δn . By conditions
C1 and C3 and noting 3.38, it is easy to see that ∞
n0 μn ∞ and lim supn → ∞ δ n /μn ∗
∗
∗
lim supn → ∞ 1/ϕ1γ −γ αγ fx −Ax , Jϕ yn −x βn /αn M ≤ 0. Applying Lemma 2.4,
we obtain Φxn − x∗ → 0 as n → ∞ this implies that xn → x∗ as n → ∞. This completes
the proof.
Taking γn 0 in 3.23, we can get the following corollary easily.
Corollary 3.4. Let X be a uniformly convex Banach space which admits a weakly continuous duality
mapping Jϕ with the gauge function ϕ such that ϕ invariant in 0, 1, C be a nonempty closed convex
subset of X such that C ± C ⊂ C. Let S {Tt : t ∈ Ê } be a nonexpansive semigroup from C
into itself such that FS /
∅, f be a contraction mapping with a coefficient α ∈ 0, 1 and A be a
strongly positive linear bounded operator with a coefficient γ > 0 such that 0 < γ < γ ϕ1/α. Let
∞
∞
{αn }∞
n0 , {βn }n0 be the sequences in 0, 1 and {tn }n0 be a positive real divergent sequence. Assume
the following conditions are hold:
C1 limn → ∞ αn 0 and ∞
n0 αn ∞;
C2 βn oαn .
16
International Journal of Mathematics and Mathematical Sciences
Then the sequence {xn } defined by
x0 ∈ C chosen arbitrarily,
t
1 tn
yn αn γ f
Tsxn ds I − αn A
Tsxn ds,
tn 0
0
xn1 βn xn 1 − βn yn , ∀n ≥ 0,
1
tn
n
3.42
converges strongly to the common fixed point x∗ , in which x∗ ∈ FS is the unique solution of the
variational inequality:
γ fx∗ − Ax∗ , Jϕ x − x∗ ≤ 0,
∀x ∈ FS.
3.43
A strong mean convergence theorem for nonexpansive mapping was first established
by Baillon 20 and it was generalized to that for nonlinear semigroups by Reich et al. 21–
23. It is clear that Theorem 3.3 are valid for nonexpansive mappings. Thus, we have the
following mean ergodic theorem of viscosity iteration process for nonexpansive mappings in
Hilbert spaces.
Corollary 3.5. Let H be a real Hilbert space, and letC be a nonempty closed convex subset of H
such that C ± C ⊂ C. Let T be a nonexpansive mapping from C into itself such that FT /
∅, f be
a contraction mapping with a coefficient α ∈ 0, 1, and let A be a strongly positive linear bounded
∞
∞
operator with a coefficient γ > 0 such that 0 < γ < γ ϕ1/α. Let {αn }∞
n0 , {βn }n0 , and{γn }n0 be
∞
the sequences in 0, 1 and let {tn }n0 be a positive real divergent sequence. Assume that the following
conditions are hold:
C1 limn → ∞ αn 0 and ∞
n0 αn ∞;
C2 limn → ∞ γn 0;
C3 βn oαn .
Then the sequence {xn } defined by
x0 ∈ C chosen arbitrarily,
n
1 T j xn ,
zn γn xn 1 − γn
n 1 j0
3.44
yn αn γ fzn I − αn Azn ,
xn1 βn xn 1 − βn yn , ∀n ≥ 0,
converges strongly to the common fixed point x∗ , in which x∗ ∈ FT is the unique solution of the
variational inequality:
γ fx∗ − Ax∗ , x − x∗ ≤ 0,
∀x ∈ FT.
3.45
International Journal of Mathematics and Mathematical Sciences
17
Acknowledgments
The authors are grateful for the reviewers for the careful reading of the paper and for the
suggestions which improved the quality of this work. They would like to thank the National
Research University Project of Thailand’s Office of the Higher Education Commission for
financial support under NRU-CSEC project no. 54000267.
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