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The modified general iterative methods for
nonexpansive semigroups in banach spaces
†
Rabian Wangkeeree
*
and Pakkapon Preechasilp
* Correspondence:
th
Department of Mathematics,
Faculty of Science, Naresuan
University, Phitsanulok 65000,
Thailand
Abstract
In this paper, we introduce the modified general iterative approximation methods for
finding a common fixed point of nonexpansive semigroups which is a unique
solution of some variational inequalities. The strong conve rgence theorems are
established in the framework of a reflexive Banach space which admits a weakly
continuous duality mapping. The main result extends various result s existing in the
current literature.
Mathematics Subject Classification (2000) 47H05, 47H09, 47J25, 65J15
Keywords: nonexpansive semigroups, strong convergence theorem, Banach space,
common fixed point
1. Introduction
Let C be a nonempty subset of a normed linear space E. Recall that a mapping T: C ®
C is called nonexpansive if
Tx − Ty≤ x − y, ∀x, y ∈ E.
(1:1)
We use F(T) to denote the set of fixed points of T, that is, F(T)={x Î E: Tx = x}. A
self mapping f: E ® E is a contra ction on E if there exists a constant a Î (0, 1) and x,
y Î E such that

f (x) − f(y)≤αx − y.
(1:2)
We use Π
E
to denote the collection of all contractions on E.Thatis,Π
E
={f: f is a
contraction on E}.
Here, we consider a scheme for a semigroup of nonexpans ive mappings. Let C be a
closed convex subset of a Banach space E. Then, a family
S = {T(s):0≤ s < ∞}
of
mappings of C into itself is called a nonexpansive semigroup on E if it satisf ies the fol-
lowing conditions:
(i) T(0)x = x for all x Î C ;
(ii) T(s + t)=T(s)T (t) for all s, t ≥ 0;
(iii) ||T(s)x - T (s)y|| ≤ ||x - y|| for all x, y Î C and s ≥ 0;
(iv) for all x Î C, the mapping s ↦ T(s)x is continuous.
We denote by
F(S )
the set of all common fixed points of
S
, that is,
Wangkeeree and Preechasilp Fixed Point Theory and Applications 2011, 2011:76
/>© 2011 Wangkeeree and Preechasilp; licensee Springer. This is an Open Acc ess article distribute d under the terms of the Creative
Commons Attribution License (http ://creativecommons. org/licenses/by/2.0), which permits unrestricted use, distribut ion, and
reproduction in any medium, provided the original work is properly cited.
F(S ):={x ∈ E : T(s)x = x,0 ≤ s < ∞} = ∩
s≥0
F( T(s)).

One classical way to study nonexpansive mappings is to use contractions to approxi-
mate a non-expansive mapping ([1-3]). More precisely, take t Î (0, 1) and define a
contraction T
t
:E® E by
T
t
x = tu +(1− t)Tx, ∀x ∈ E,
(1:3)
where u Î E is a fixed point. Banach’s contraction mapping principle guarantees that
T
t
has a uniqu e fixed poi nt x
t
in E. It is un clear, in general, what is the behavi or of x
t
as t ® 0, even if T has a fixed point. However, in the case of T having a fixed point,
Browder [1] proved that if E is a Hilbert space, then x
t
converges strongly to a fixed
point of T. Reich [2] extended Browder’s result to the setting of Banach spaces and
proved that if E is a uniformly smooth Banach space, then {x
t
} converges stron gly to a
fixed point of T and the limit defines the (unique) sunny nonexpansive retraction from
E onto F(T). Xu [3] proved Reich’s results hold in reflexiv e Banach spaces which have
a weakly continuous duality mapping.
In the last ten years or so, the iterative methods for no nexpansive mappings have
recently been applied to solve convex minimization problems; see, e.g., [4-6] and the
references therein.

By a gauge function , we mean a continuous strictly increasing function : [0, ∞) ®
[0, ∞) such that (0) = 0 and (t) ® ∞ as t ® ∞.LetE* be the dual space of E.The
duality mapping
J
ϕ
: E → 2
E
∗
associated with a gauge function  is defined by
J
ϕ
(x)={f
∗
∈ E
∗
: x, f
∗
 = xϕ(x), f
∗
 = ϕ(x)}, ∀x ∈ E.
In particular, the duality mapping with the gauge function (t)=t, denoted by J,is
referred to as the normalized duality mapping. Clearly, there holds the relation
J
ϕ
(x)=
ϕ(x)
x
J(x)
for all x ≠ 0 (see [7]).
Browder [7] initiated the study of certain classes of nonline ar operators by means of

thedualitymappingJ

. Following Browder [7], we say that a Banach space E has a
weakly continuous duality mapping if there exists a gauge  for which the duality map-
ping J

(x) is sing le-valued and continuous from the weak topology to the weak* topol-
ogy, that is, for any {x
n
}withx
n
⇀ x, the sequence {J

(x
n
)} converges weakly* to J

(x).
It is known that l
p
has a weakly continuous duality mapping with a gauge function (t)
= t
p-1
for all 1 <p < ∞. Set
(t)=
t

0
ϕ(τ )dτ , ∀t ≥ 0,
then

J
ϕ
(x)=∂(x), ∀x ∈ E,
where ∂ denotes the sub-differential in the sense of convex analysis.
In a Banach space E having a weakly continuous duality mapping J

with a gauge
function ,anoperatorA is said to be strongly positive [8] if there exists a constant
¯γ>0
with the property
Wangkeeree and Preechasilp Fixed Point Theory and Applications 2011, 2011:76
/>Page 2 of 15
Ax, J
ϕ
(x)≥ ¯γ xϕ(x)
(1:4)
and
αI − βA =sup
x≤1
|(αI − βA)x, J
ϕ
(x)|, α ∈ [0, 1], β ∈ [−1, 1],
(1:5)
where I is the identity mapping. If E:=H is a real Hilbert space, then the inequality
(1.4) reduces to
Ax, x≥ ¯γx
2
for all x ∈ H.
(1:6)
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:
min
x∈C
1
2
Ax, x−x, b,
(1:7)
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 ([5]) proved that the sequence { x
n
} defined by the iterative
method below, with the initial guess x
0
Î H chosen arbitrarily:
x
n+1
=(I − α
n
A)Tx
n
+ α
n
u, n ≥ 0,
(1:8)
converges strongly to the unique solution o f the mini mizatio n problem (1.7) pro-
vided the sequence {a
n
} satisfies certain conditions. Using the viscosity approximation
method, Moudafi [9] introduced the following iterative iterative process for nonexpan-
sive mappings (see [10,11] for further developments in both Hilbert and Banach

spaces). Let f be a contraction on H. Starting with an arbitrary initial x
0
Î H,definea
sequence {x
n
} recursively by
x
n+1
=(1− σ
n
)Tx
n
+ σ
n
f (x
n
), n ≥ 0,
(1:9)
where {s
n
} is a sequence in (0, 1). It is proved [9,11] that under certain appropriate
conditions imposed on {s
n
}, the sequence {x
n
} generated by (1.9) strongly converges to
the unique solution x*inC of the variational inequality
(I − f )x
∗
, x − x

∗
≥0, x ∈ H.
(1:10)
In [12], Marino and Xu mixed the iterative method (1.8) and t he viscosity a pproxi-
mation method (1.9) and considered the following general iterative method:
x
n+1
=(I − α
n
A)Tx
n
+ α
n
γ f (x
n
), n ≥ 0,
(1:11)
where A is a strongly positive bounded linear operator on H. They proved that if the
sequence {a
n
} of parameters satisfies the following conditions
(C1) lim
n®∞
a
n
=0,
(C2)

∞
n=1

α
n
= ∞
, and
(C3)

∞
n
=1
| α
n+1
− α
n
|<
∞
,
then the sequence {x
n
} generated by (1.11) converges strongly to the unique solutio n
x*inH of the variational inequality
(A − γ f )x
∗
, x − x
∗
≥0, x ∈ H
(1:12)
Wangkeeree and Preechasilp Fixed Point Theory and Applications 2011, 2011:76
/>Page 3 of 15
which is the optimality condition for the minimization problem:
min

x∈C
1
2
Ax, x−h(x)
,whereh is a potential function for gf(i.e ., h’(x)=gf(x)forx Î
H).
Very recently, Wangkeeree et al. [8] introduced the following general iterative
approximation method in the fr amework of a reflexive Banach space E which admits a
weakly continuous duality mapping:
⎧
⎨
⎩
x
0
= x ∈ E,
y
n
= β
n
x
n
+(1− β
n
)T
n
x
n
,
x
n+1

= α
n
γ f (x
n
)+(I − α
n
A)y
n
, n ≥ 0,
(1:13)
where A is strongly positive bounded line ar operator on E and proved the strong
convergence theorems for a c ountable family of nonexpansive mappings
{T
n
: E → E}
∞
n=1
. Other investigati ons of approximating c ommon fixed points for a
countable family of nonexpansive mappings can be found in Refs. [1,3,8,10-14] and
many results not cited here.
Inspired and m otivated by the iterative (1.13) given above, we give the following
modified general iterative scheme for a nonexpansive semigroup {T(t): t >0}:forany
{T(t
n
): t
n
>0,n Î N} ⊂ {T(t): t > 0},
⎧
⎨
⎩

x
0
= x ∈ E,
y
n
= β
n
x
n
+(1− β
n
)T(t
n
)x
n
,
x
n+1
= α
n
γ f (x
n
)+(I − α
n
A)y
n
, n ≥ 0,
(1:14)
where {a
n

}, {b
n
} and {t
n
} are real sequence satisfying appropriate control conditions,
A is strongly positive bounded linear operator on E and f is a contraction on E.The
strong convergence theorems are proved in the framework of a reflexive Banach space
which admits a weakly continuous duality mapping. Furthermore, by using these
results, we obtain strong convergence theorems of the following new iterative schemes
{u
n
} and {w
n
} defined by
⎧
⎨
⎩
u
0
= u ∈ E,
v
n
= β
n
u
n
+(1− β
n
)T(t
n

)u
n
,
u
n+1
= α
n
γ f (T(t
n
)u
n
)+(I − α
n
A)v
n
, n ≥ 0,
(1:15)
and
⎧
⎨
⎩
w
0
= c ∈ E,
v
n
= β
n
w
n

+(1− β
n
)T(t
n
)w
n
,
w
n+1
= T(t
n
)

α
n
γ f (w
n
)+(I − α
n
A)v
n

, n ≥ 0.
(1:16)
The results presented in this paper improve and extend the corresponding results
announced by Marino and Xu [12], Wangkeeree et al. [8], and Li et al. [15] many
others.
2. Preliminaries
Throughout this paper, let E be a real Banach space and E* be its dual space. We write
x

n
⇀ x (respectively x
n
⇀* x) to i ndicate that the sequence {x
n
} weakly (respectively
weak*) converges to x;asusualx
n
® x will symbolize strong c onvergence. Let U ={x
Î E:||x|| = 1}. A Banach space E is said to uniformly c onvex if, for any ε Î (0, 2],
Wangkeeree and Preechasilp Fixed Point Theory and Applications 2011, 2011:76
/>Page 4 of 15
there exists δ > 0 such that, for any x, y Î U ,||x - y|| ≥ ε implies

x+y
2
≤1 − δ
.Itis
known that a uniformly convex Bana ch space is reflexi ve and strictly convex (see also
[16]). A Banac h space E is sa id to be smooth if the limit
lim
t→0
x+ty−x
t
exists for al l
x, y Î U. It is also said to be uniformly smooth if the limit is attained uniformly for x,
y Î U.
Now we collect some useful lemmas for proving the convergence result of this paper.
The first part of the next lemma is an immediate consequence of the subd ifferential
inequality and the proof of the second part can be found in [17].

Lemma 2.1. ([17]) Assume that a Banach space E has a weakly continuous duality
mapping J

with gauge .
(i) For all x, y Î E, the following inequality holds:
(x + y) ≤ (x)+y, J
ϕ
(x + y).
In particular, for all x, y Î E,
x + y
2
≤x
2
+2y, J(x + y).
(ii) Assume that a sequence {x
n
} in E converges weakly to a point x Î E.
Then the following identity holds:
lim sup
n→∞
(x
n
− y) = lim sup
n→∞
(x
n
− x)+(y − x), ∀x, y ∈ E.
Now, we present the concept of uniformly asymptotically regular semigroup.
S
is

said to be uniformly asymptotically regular (in short, u.a.r.) on C if for all h ≥ 0and
any bounded subset B of C,
lim
s→∞
sup
x∈B
T(h)(T(s)x) − T(s)x  =0.
The nonexpansi ve semigroup {s
t
: t > 0} defined by the following lemma is an exam-
ple of u.a.r. nonexpansive semigroup. Other examples of u.a.r. operator semigroup can
be found in [[18], Examples 17,18].
Lemma 2.2. (see [[19], Lemma 2.7]). LetCbeanonemptyclosedconvexsubsetofa
uniformly convex Banach space E, B a bounded closed convex subset of C, and
S = {T(s):0≤ s < ∞}
a nonexpansive semigroup on C such that
F(S ) = ∅
.Foreachh
>0,set
σ
t
(x)=
1
t
t

0
T(s)xds
, then
lim

t→∞
sup
x∈B
σ
t
(x) − T(h)σ
t
(x) =0.
(2:1)
Example 2.3.Theset{s
t
: t > 0} defined by Lemma 2.2 is u.a.r. nonexpansive semi-
group. In fact, it is obvious that {s
t
: t > 0} is a nonexpansive semigroup. For each h >
0, we have
Wangkeeree and Preechasilp Fixed Point Theory and Applications 2011, 2011:76
/>Page 5 of 15
σ
t
(x) − σ
h
σ
t
(x) =







σ
t
(x) −
1
h
h

0
T(s)σ
t
(x)ds






=






1
h
h

0

(σ
t
(x) − T(s)σ
t
(x))ds






≤
1
h
h

0
σ
t
(x) − T(s)σ
t
(x)ds.
Applying Lemma 2.2, we have
lim
t→∞
sup
xinB
σ
t
(x) − σ

h
σ
t
(x)≤
1
h
h

0
lim
t→∞
sup
x∈B
σ
t
(x) − σ
h
σ
t
(x)ds =0.
The next valuable lemma is proved for applying our main results.
Lemma 2.4. [[8], Lemma 3.1] Assume that a Banach space E has a weakly c ontinu-
ous duality mapp ing J

with gauge . Let A be a strong positive linear bounded opera-
tor on E with coefficient
¯γ>0
and 0<r ≥ (1)||A||
-1
. Then

I − ρA≤ϕ(1)(1 − ρ ¯γ )
.
Lemma 2.5. ([6]) Assume that {a
n
} is a sequence of nonnegative real numbers such
that
a
n+1
≤ (1 − α
n
)a
n
+ b
n
,
where {a
n
} is a sequence in (0, 1) and {b
n
} is a sequence such that
(a)

∞
n=1
α
n
= ∞
;
(b) lim sup
n®∞

b
n
/a
n
≤ 0 or

∞
n=1
b
n
 < ∞
.
Then lim
n®∞
a
n
=0.
3. Main results
Let E be a Banach space which admits a we akly continuous duality mapping J

with
gauge  such that  is invari ant on [0, 1], i.e., ([0, 1]) ⊂ [0, 1]. Let
S = {T(s):s ≥ 0}
be a nonexpansive semigroups from C into itself. For f Î Π
E
, t Î (0, 1), and A is a
strongly positive bounded linear operator with coefficient
¯γ>0
and
0 <γ <

¯γϕ(1)
α
,
the mapping S
t
:E® E defined by
S
t
(x)=tγ f (x)+(I − tA)T(λ
t
)x, ∀x ∈ E
is a contraction mapping. Indeed, for any x, y Î E,
S
t
(x) − S
t
(y) = tγ (f (x) − f (y)) + (I − tA)(T(λ
t
)x − T(λ
t
)y)
≤ tγ f (x) − f(y) + I − tAT(λ
t
)x − T( λ
t
)y
≤ tγαx − y + ϕ(1)(1 − t ¯γ )x − y
≤

1 − t(ϕ(1) ¯γ − γα)


x − y.
(3:1)
Thus, by Banach contraction mapping principle, there exists a unique fixed point x
t
in E, that is,
Wangkeeree and Preechasilp Fixed Point Theory and Applications 2011, 2011:76
/>Page 6 of 15
x
t
= tγ f(x
t
)+(I − tA)T(λ
t
)x
t
.
(3:2)
Remark 3.1.Wenotethatl
p
space has a weakly continuous duality mapping with a
gauge function (t)=t
p-1
for all 1 <p < ∞. It is clear that  is invariant on [0, 1].
Lemma 3.2. Let E be a reflexive Banach space which admits a weakly continuous
duality mapping J

with gauge  such that  is invariant on [0, 1]. Let
S = {T(s):s ≥ 0}
be a nonexpansive semigroup with

F(S ) = ∅
, and f Î Π
E
, let A be a
strongly positive bounded linear operator with coefficient
¯γ>0
and
0 <γ <
¯γϕ(1)
α
,
and let t Î (0, 1) which satisfying t ® 0. Then the net {x
t
} defined by (3.2) with {l
t
}
0<t
<1
is a positive real divergent sequenc e; converges strongly as t ® 0 to a common fixed
point
˜
x
in
F(S )
which solves the variational inequality:
(A − γ f )
˜
x, J
ϕ
(

˜
x − z)≤0, z ∈ F(S).
(3:3)
Proof. We first show that the uniqueness of a solution of the variational inequality
(3.3). Suppose both
˜
x ∈ F (
S)
and
x
∗
∈ F(S)
are solutions to (3.3), then
(A − γ f )
˜
x, J
ϕ
(
˜
x − x
∗
)≤0
(3:4)
and
(A − γ f )x
∗
, J
ϕ
(x
∗

−
˜
x)≤0.
(3:5)
Adding (3.4) and (3.5), we obtain
(A − γ f )
˜
x − (A − γ f )x
∗
, J
ϕ
(
˜
x − x
∗
)≤0.
(3:6)
Noticing that for any x, y Î E,
(A − γ f)x − (A − γ f )y, J
ϕ
(x − y) = A(x − y), J
ϕ
(x − y)−γ f (x) − f(y), J
ϕ
(x − y)
≥¯γ x − yϕ(x − y) − γ f (x) − f (y)J
ϕ
(x − y)
≥¯γ(x − y) − γα(x − y)
=(¯γ − γα)(x − y)

≥ ( ¯γϕ(1) − γα)(x − y) ≥ 0.
(3:7)
Using (3.6) and
0 < ¯γϕ(1) − γα
in the last inequality, we g et that
(
˜
x − x
∗
)=0
.
Therefore,
˜
x = x
∗
and the uniqueness is proved. Below we use
˜
x
to denote the unique
solution of (3.3). Next, we will rove that {x
t
} is bounded. Take a
p ∈ F(S)
,thenwe
have
x
t
− p = tγ f(x
t
)+(I − tA)T(λ

t
)x
t
− p
= (I − tA)T(λ
t
)x
t
− (I − tA)p + t(γ f (x
t
) − A(p))
≤ϕ(1)(1 − t ¯γ )x
t
− p + t(γαx
t
− p + γ f (p) − A(p)).
It follows that
x
t
− p≤
1
¯γϕ(1) − γα
γ f (p) − A(p).
Hence, {x
t
} is bounded, so are {f(x
t
)} and {AT(x
t
)}. The definition of {x

t
} implies that
x
t
− T(λ
t
)x
t
 = tγ f (x
t
) − A(T(λ
t
)x
t
)→0ast → 0.
(3:8)
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/>Page 7 of 15
Next, we show that ||x
t
- T(h)x
t
|| ® 0 for all h ≥ 0. Since {T(t): t ≥ 0} is u.a.r. nonex-
pansive semigroup and lim
t®0
l
t
= ∞, then, for all h > 0 and for any bounded subset D
of C containing {x
t

},
lim
t→0
T(h)(T(λ
t
)x
t
) − T(λ
t
)x
t
≤lim
t→0
sup
x∈D
T(h)(T(λ
t
)x
t
) − T(λ
t
)x
t
 =0.
Hence, when t ® 0, for all h > 0, we have
x
t
− T(h)x
t
≤x

t
− T(λ
t
)x
t
 + T(λ
t
)x
t
− T(h)(T(λ)x
t
) + T(h)(T(λ
t
)x
t
) − T(h)x
t

≤ 2x
t
− T(λ
t
)x
t
 + T(λ
t
)x
t
− T(h)(T(λ
t

)x
t
)→0.
(3:9)
Assume that
{t
n
}
∞
n=1
⊂ (0, 1)
is such that t
n
® 0asn ® ∞.Put
x
n
:= x
t
n
and
λ
n
:= λ
t
n
.Weshowthat{x
n
} contains a subsequence converging strongly to
˜
x ∈ F (

S)
.
It follo ws from reflexivit y of E and the boundedne ss of sequence {x
n
} that there exists
{x
n
j
}
which is a subsequence of {x
n
} converging weakly to w Î E as n ® ∞. Since J

is
weakly sequentially continuous, we have by Lemma 2.1 that
lim sup
j→∞
(x
n
j
− x) = lim sup
j→∞
(x
n
j
− w)+(x − w), for all x ∈ E.
Let
H(x) = lim sup
j→∞
(x

n
j
− x), for all x ∈ E.
It follows that
H(x)=H(w)+(x − w), for all x ∈ E.
For h ≥ 0, from (3.9) we obtain
H(T(h)w) = lim sup
j→∞
(x
n
j
− T(h)w) = lim sup
j→∞
(T(h)x
n
j
− T(h)w)
≤ lim sup
j→∞
(x
n
j
− w)=H(w).
(3:10)
On the other hand, however,
H(T(h)w)=H(w)+(T(h)w − w).
(3:11)
It follows from (3.10) and (3.11) that
(T(h)w − w)=H(T(h)w) − H(w) ≤ 0.
This implies that T(h)w = w fo r all h ≥ 0, and so

w ∈ F(S )
.Next,weshowthat
x
n
j
→ w
as j ® ∞. In fact, since
(t)=

t
0
ϕ(τ )dτ
, ∀t ≥ 0, and :[0,∞) ® [0, ∞)isa
gauge function, then for 1 ≥ k ≥ 0, (kx) ≤ (x) and
(kt)=
kt

0
ϕ(τ )dτ = k
t

0
ϕ(kx)dx ≤ k
t

0
ϕ(x)dx = k(t).
Wangkeeree and Preechasilp Fixed Point Theory and Applications 2011, 2011:76
/>Page 8 of 15
Following Lemma 2.1, we have

(x
n
− w)=((I − t
n
A)T(t
n
)x
n
− (I − t
n
A)w + t
n
(γ f (x
n
) − A(w)))
= ((I − t
n
A)T(t
n
)x
n
− (I − t
n
A)w)+t
n
γ f (x
n
) − A(w), J
ϕ
(x

n
− w)
≤ (ϕ(1)(1 − t
n
¯γ )x
n
− w)+t
n
γ f (x
t
n
) − f (w), J
ϕ
(x
n
− w)
+ t
n
γ f (w) − A(w), J
ϕ
(x
n
− w)
≤ ϕ(1)(1 − t
n
¯γ )(x
n
− w)+t
n
γ f (x

n
) − f (w)J
ϕ
(x
n
− w)
+ t
n
γ f (w) − A(w), J
ϕ
(x
n
− w)
≤ ϕ(1)(1 − t
n
¯γ )(x
n
− w)+t
n
γαx
n
− wJ
ϕ
(x
n
− w)
+ t
n
γ f (w) − A(w), J
ϕ

(x
n
− w)
= ϕ(1)(1 − t
n
¯γ )(x
n
− w)+t
n
γα(x
n
− w)
+ t
n
γ f (w) − A(w), J
ϕ
(x
n
− w)
=(1− t
n
( ¯γϕ(1) − γα))(x
n
− w)+t
n
γ f (w) − A(w), J
ϕ
(x
n
− w)

.
(3:12)
This implies that
(x
n
j
− w) ≤
1
¯γϕ(1) − γα
γ f (w) − A(w), J
ϕ
(x
n
j
− w).
Now observing that x
n
⇀ w implies J

(x
n
- w) ⇀ 0, we conclude from the last
inequality that
(x
n
j
− w) → 0asj →∞.
Hence,
x
n

j
→ w
as j ® ∞.Next,weprovethatw solves the variational inequality
(3.3). For any
z ∈ F (S )
, we observe that
(I − T(λ
t
))x
t
− (I − T(λ
t
))z, J
ϕ
(x
t
− z) = x
t
− z, J
ϕ
(x
t
− z) + T(λ
t
)x
t
− T(λ
t
)z, J
ϕ

(x
t
− z)
= (x
t
− z) −T(λ
t
)z − T(λ
t
)x
t
, J
ϕ
(x
t
− z)
≥ (x
t
− z) −T(λ
t
)z − T(λ
t
)x
t
J
ϕ
(x
t
− z)
≥ (x

t
− z) −z − x
t
J
ϕ
(x
t
− z)
= (x
t
− z) − (x
t
− z)=0.
(3:13)
Since
x
t
= tγ f(x
t
)+(I − tA)T(λ
t
)x
t
,
we can derive that
(A − γ f )(x
t
)=−
1
t

(I − T(λ
t
))x
t
+(A(I − T ( λ
t
))x
t
).
Thus,
(A − γ f)(x
t
), J
ϕ
(x
t
− z) = −
1
t
(I − T(λ
t
))x
t
− (I − T(λ
t
))z, J
ϕ
(x
t
− z) + A(I − T(λ

t
))x
t
, J
ϕ
(x
t
− z)
≤A(I − T(λ
t
))x
t
, J
ϕ
(x
t
− z) .
(3:14)
Noticing that
x
n
j
− T(λ
t
n
j
)x
n
j
→ 0.

Now replacing t and l
t
with n
j
and
t
n
j
in (3.14) and letting j ® ∞, we have
(A − γ f )w, J
ϕ
(w − z)≤0.
Wangkeeree and Preechasilp Fixed Point Theory and Applications 2011, 2011:76
/>Page 9 of 15
So, w Î F(T) is a solution of the variational inequality (3.3), and hence,
w =
˜
x
by th e
uniqueness. In a summary, we have shown that each cluster point of {x
t
}(at t ® 0)
equals
˜
x
. Therefore,
x
t
→
˜

x
as t ® 0. This completes the proof.
Theorem 3.3. Let E be a reflexive Banach space which admits a w eakly continuous
duality mapping J

with gauge  such that  is invariant on [0, 1]. Let {T(s): s ≥ 0} be
a u.a.r. semigroup of nonexpansive mappings with
F(S ) = ∅
, and f Î Π
E
, let A be a
strongly positive bounded linear operator with coefficient
¯γ>0
and
0 <γ <
¯γϕ(1)
α
. Let
the sequence {x
n
} be generated by the following:
⎧
⎨
⎩
x
0
= x ∈ E,
y
n
= β

n
x
n
+(1− β
n
)T(t
n
)x
n
,
x
n+1
= α
n
γ f (x
n
)+(I − α
n
A)y
n
, n ≥ 0
(3:15)
where {a
n
} ⊂ (0, 1) and {b
n
} ⊂ [0, 1] are real sequences satisfying the following condi-
tions:
(C1) lim
n®∞

a
n
=0and

∞
n=1
α
n
= ∞
(C2) lim
n®∞
b
n
=0,
(C3) lim
n®∞
t
n
= ∞.
Then {x
n
} converges strongly to
˜
x
that is obtained in Lemma 3.2.
Proof. Since lim
n®∞
a
n
=0,wemayassume,withoutlossofgenerality,thata

n
<
(1)||A||
-1
for all n. By Lemma 2.4, we have
I − α
n
A≤ϕ(1)(1 − α
n
¯γ )
.Wefirst
observe that {x
n
} is bounded. Indeed, pick any
p ∈ F(S)
to obtain
y
n
− p = β
n
x
n
+(1− β
n
)T(t
n
)x
n
− p
= β

n
(x
n
− p)+(1− β
n
)(T(t
n
)x
n
− T(t
n
)p)
≤ β
n
x
n
− p +(1− β
n
)x
n
− p
= x
n
− p,
(3:16)
and so
x
n+1
− p = α
n

γ f (x
n
)+(I − α
n
A)y
n
− p
= α
n
(γ f (x
n
) − A(p)) + (I − α
n
A)y
n
− (I − α
n
A)p
≤ α
n
γ f (x
n
) − A(p) + ϕ(1)(1 − α
n
¯γ )y
n
− p
≤ α
n
γ f (x

n
) − f (p) + α
n
γ f (p) − A(p) + ϕ(1)(1 − α
n
¯γ )y
n
− p
≤ α
n
γαx
n
− p + α
n
γ f (p) − A(p) + ϕ(1)(1 − α
n
¯γ )x
n
− p
≤ (1 − α
n
( ¯γϕ(1) − γα))x
n
− p + α
n
γ f (x
n
) − A(p)
=(1− α
n

( ¯γϕ(1) − γα))x
n
− p + α
n
( ¯γϕ(1) − γα)
γ f (x
n
) − A(p)
¯γϕ(1) − γα
.
It follows from induction that
x
n
− p≤max

x
0
− p,
γ f (p) − A(p)
¯γϕ(1) − γα

, n ≥ 0.
(3:17)
The boundedness of {x
n
} implies that {y
n
}, {T(t
n
)x

n
} and {f(x
n
)} are bounded.
Wangkeeree and Preechasilp Fixed Point Theory and Applications 2011, 2011:76
/>Page 10 of 15
Thus by (3.29), (C1) and (C2), we have
y
n
− T(t
n
)x
n
 = β
n
x
n
− T(t
n
)x
n
→0
and there by,
x
n+1
− T(t
n
)x
n
≤y

n
− T(t
n
)x
n
 + α
n
γ f (x
n
) − A(y
n
)→0.
Since {T(t): t ≥ 0} is u.a.r. nonexpansive semigroup and lim
n®∞
t
n
= ∞, then, for all h
> 0 and for any bounded subset D of C containing {x
n
},
lim
n→∞
T(h)(T(t
n
)x
n
) − T(t
n
)x
n

≤ lim
n→∞
sup
x∈D
T(h)(T(t
n
)x
n
) − T(t
n
)x
n
 =0.
Hence, when n ® ∞, for all h > 0, we have
x
n+1
− T( h) x
n+1
≤x
n+1
− T( t
n
)x
n
 + T(t
n
)x
n
− T( h)(T(t
n

)x
n
) + T(h)(T(t
n
)x
n
) − T( h) x
n+1

≤ 2x
n+1
− T( t
n
)x
n
 + T(t
n
)x
n
− T(h)(T(t
n
)x
n
)→0
(3:18)
Next, we prove that
lim sup
n→∞
γ f (
˜

x) − A
˜
x, J
ϕ
(x
n
−
˜
x)≤0,
(3:19)
Let
{x
n
k
}
be a subsequence of {x
n
} such that
lim
k→∞
γ f (
˜
x) − A
˜
x, J
ϕ
(x
n
k
−

˜
x) = lim sup
n→∞
γ f (
˜
x) − A
˜
x, J
ϕ
(x
n
−
˜
x).
(3:20)
If follows from reflexivity of E and the boundedness of sequence
{x
n
k
}
that there
exists
{x
n
k
i
}
which is a subsequ ence of
{x
n

k
}
converging weakly to w Î E as i ® ∞.
Since J

is weakly continuous, we have by Lemma 2.1 that
lim sup
n→∞
(x
n
k
i
− x) = lim sup
n→∞
(x
n
k
i
− w)+(x − w), for all x ∈ E.
Let
H(x) = lim sup
n→∞
(x
n
k
i
− x), for all x ∈ E.
It follows that
H(x)=H(w)+(x − w), for all x ∈ E.
From (3.18), for each h > 0, we obtain

H(T(h)w) = lim sup
i→∞
(x
n
k
i
− T(h)w) = lim sup
i→∞
(T(h)x
n
k
i
− T(h)w)
≤ lim sup
i→∞
(x
n
k
i
− w)=H(w)
(3:21)
On the other hand, however,
H(T(h)w)=H(w)+(T(h)w − w)
(3:22)
It follows from (3.21) and (3.22) that
(T(h)w − w)=H(T(h)w) − H(w) ≤ 0.
This implies that T(h )w = w for all h > 0, and so
w ∈ F(S )
. Since the duality map J


is single-valued and weakly continuous, we get that
Wangkeeree and Preechasilp Fixed Point Theory and Applications 2011, 2011:76
/>Page 11 of 15
lim sup
n→∞
γ f (
˜
x) − A
˜
x, J
ϕ
(x
n
−
˜
x) = lim
k→∞
γ f (
˜
x) − A
˜
x, J
ϕ
(x
n
k
−
˜
x)
= lim

i→∞
γ f (
˜
x) − A
˜
x, J
ϕ
(x
n
k
i
−
˜
x)
= (A − γ f)
˜
x, J
ϕ
(
˜
x − w)≤0
as required.
Finally, we show that
x
n
→
˜
x
as n ® ∞.
(||x

n+1
−
˜
x||)=(||α
n
(γ f(x
n
)) + (I − α
n
A)y
n
−
˜
x||)
= (||α
n
(γ f(x
n
) − A
˜
x)+(I − α
n
A)(y
n
−
˜
x)||)
= (||α
n
(γ f(x

n
) − γ f(
˜
x)) + α
n
(γ f(
˜
x) − A
˜
x)+(I − α
n
A)(y
n
−
˜
x)||)
≤ (||α
n
(γ f(x
n
) − γ f(
˜
x)) + (I − α
n
A)(y
n
−
˜
x)||)+α
n

γ f(
˜
x) − A
˜
x, J
ϕ
(x
n+1
−
˜
x)
≤ (||α
n
(γ f(x
n
) − γ f(
˜
x))|| + ||(I − α
n
A)(y
n
−
˜
x)||)+α
n
γ f(
˜
x) − A
˜
x, J

ϕ
(x
n+1
−
˜
x)

≤ (α
n
γα||x
n
−
˜
x|| + ϕ(1)(1 − α
n
¯γ )||y
n
−
˜
x||)) + α
n
γ f(
˜
x) − A
˜
x, J
ϕ
(x
n+1
−

˜
x)
≤ (α
n
γα||x
n
−
˜
x|| + ϕ(1)(1 − α
n
¯γ )||x
n
−
˜
x||)) + α
n
γ f(
˜
x) − A
˜
x, J
ϕ
(x
n+1
−
˜
x)
= ((ϕ(1) − α
n
(ϕ(1) ¯γ − γα))||x

n
−
˜
x||)+α
n
γ f(
˜
x) − A
˜
x, J
ϕ
(x
n+1
−
˜
x)
≤ (1 − α
n
(ϕ(1) ¯γ − γα))(||x
n
−
˜
x||)) + α
n
γ f(
˜
x) − A
˜
x, J
ϕ

(x
n+1
−
˜
x).
(3:23)
Apply Lemma 2.5 to (3.23) to conclude
(x
n+1
−
˜
x) → 0
as n ® ∞,thatis,
x
n
→
˜
x
as n ® ∞. This completes the proof. □
Corollary 3.4. Let E be a reflexive Banach space which admits a weakly continuous
duality mapping J

with gauge  such that  is invariant on [0, 1]. Let {T(s): s ≥ 0} be
a u.a.r. semigroup of nonexpansive mappings with
F(S ) = ∅
, and f Î Π
E
, let A be a
strongly positive bounded linear operator with coefficient
¯γ>0

and
0 <γ <
¯γϕ(1)
α
. Let
the sequence {x
n
} be generated by the following:
⎧
⎨
⎩
u
0
= u ∈ E,
v
n
= β
n
u
n
+(1− β
n
)T(t
n
)u
n
,
u
n+1
= α

n
γ f (T(t
n
)u
n
)+(I − α
n
A)v
n
, n ≥ 0
(3:24)
where {a
n
} ⊂ (0, 1) and {b
n
} ⊂ [0, 1] are real sequences satisfying the following condi-
tions:
(C1) lim
n®∞
a
n
=0and

∞
n=1
α
n
= ∞
(C2) lim
n®∞

b
n
=0,
(C3) lim
n®∞
t
n
= ∞.
Then {u
n
} converges strongly to
˜
x
that is obtained in Lemma 3.2.
Proof. Let {x
n
} be the sequence in given by x
0
= u
0
and

y
n
= β
n
x
n
+(1− β
n

)T(t
n
)x
n
,
x
n+1
= α
n
γ f (x
n
)+(I − α
n
A)y
n
, n ≥ 0.
(3:25)
From Theorem 3.3,
x
n
→
˜
x
. We claim that
u
n
→
˜
x
. From (3.26) and (3.25), we have

y
n
− v
n
 = β
n
x
n
+(1− β
n
)T(t
n
)x
n
− β
n
u
n
− (1 − β
n
)T(t
n
)u
n

≤ β
n
x
n
− u

n
 +(1− β
n
)T(t
n
)x
n
− T(t
n
)u
n

≤ β
n
x
n
− u
n
 +(1− β
n
)x
n
− u
n

= x
n
− u
n
.

Wangkeeree and Preechasilp Fixed Point Theory and Applications 2011, 2011:76
/>Page 12 of 15
Again, it then follows that
x
n+1
− u
n+1
 = α
n
γ f (x
n
)+(I − α
n
A)y
n
− α
n
γ f (T(t
n
)u
n
) − (I − α
n
A)v
n

≤ α
n
γ f( x
n

) − f (T(t
n
)u
n
) + I − α
n
Ay
n
− v
n

≤ α
n
γαx
n
− T(t
n
)u
n
 + ϕ(1)(1 − α
n
¯γ )x
n
− u
n

≤ α
n
γαx
n

− T(t
n
)
˜
x + α
n
γαT(t
n
)
˜
x − T(t
n
)u
n
 + ϕ(1)(1 − α
n
¯γ )x
n
− u
n

≤ α
n
γαx
n
−
˜
x + α
n
γα

˜
x − u
n
 + ϕ(1)(1 − α
n
¯γ )x
n
− u
n

= α
n
γαx
n
−
˜
x + α
n
γα
˜
x − x
n
 + α
n
γαx
n
− u
n
 + ϕ(1)(1 − α
n

¯γ )x
n
− u
n

=(ϕ(1)(1 − α
n
¯γ )+α
n
γα)x
n
− u
n
 +(α
n
γα+ α
n
γα)x
n
−
˜
x
≤ (1 − α
n
(ϕ(1) ¯γ − γα))x
n
− u
n
 + α
n

(ϕ(1) ¯γ − γα)
2γα
(ϕ(1) ¯γ − γα)
x
n
−
˜
x.
It follows fr om

∞
n=1
α
n
= ∞
,
lim
n→∞
x
n
−
˜
x =0
, and Lemma 2.5 that ||x
n
- u
n
||
® 0. Consequently,
u

n
→
˜
x
as required. □
Corollary 3.5. Let E be a reflexive Banach space which admits a weakly continuous
duality mapping J

with gauge  such that  is invariant on [0, 1]. Let {T(s): s ≥ 0} be
a u.a.r. semigroup of nonexpansive mappings with
F(S ) = ∅
, and f Î Π
E
, let A be a
strongly positive bounded linear operator with coefficient
¯γ>0
and
0 <γ <
¯γϕ(1)
α
. Let
the sequence {x
n
} be generated by the following:
⎧
⎨
⎩
w
0
= w ∈ E,

v
n
= β
n
w
n
+(1− β
n
)T(t
n
)w
n
,
w
n+1
= T(t
n
)

α
n
γ f (w
n
)+(I − α
n
A)v
n

, n ≥ 0
(3:26)

where {a
n
} ⊂ (0, 1) and {b
n
} ⊂ [0, 1] are real sequences satisfying the following condi-
tions:
(C1) lim
n®∞
a
n
=0and

∞
n=1
α
n
= ∞
(C2) lim
n®∞
b
n
=0,
(C3) lim
n®∞
t
n
= ∞.
Then {w
n
} converges strongly to

˜
x
that is obtained in Lemma 3.2.
Proof. Define the sequence {u
n
} and {s
n
}by
u
n
= α
n
γ f (w
n
)+(I − α
n
A)w
n
, σ
n
= α
n+1
∀n ≥ 0.
(3:27)
Taking
p ∈ F(S)
, we have
w
n+1
− p = T(t

n
)u
n
− T(t
n
)p≤u
n
− p
= α
n
γ f (w
n
)+(I − α
n
A)w
n
− (I − α
n
A)p − α
n
Ap
≤ α
n
γ f (w
n
) − Ap + I − α
n
Aw
n
− p

≤ α
n
γ f (w
n
) − Ap + ϕ(1)(1 − α
n
¯γ )w
n
− p
≤ α
n
γ f (w
n
) − γ f(p) + α
n
γ f (p) − Ap + ϕ(1)(1 − α
n
¯γ )w
n
− p
≤ α
n
γαw
n
− p + α
n
γ f (p) − Ap + ϕ(1)(1 − α
n
¯γ )w
n

− p
=(1− α
n
( ¯γϕ(1) − γα))w
n
− p + α
n
( ¯γϕ(1) − γα)
γ f (p) − Ap
( ¯γϕ(1) − γα)
.
Wangkeeree and Preechasilp Fixed Point Theory and Applications 2011, 2011:76
/>Page 13 of 15
It follows from induction that
w
n+1
− p≤max

w
0
− p,
γ f (p) − A(p)
¯γϕ(1) − γα

, n ≥ 0.
Thus, both {u
n
} and {w
n
} are bounded. We observe that

u
n+1
= α
n+1
f (w
n+1
)+(I − α
n+1
A)w
n+1
= σ
n
f (T(t
n
)u
n
)+(I − σ
n
A)T(t
n
)u
n
.
Thus, Corollary 3.4 implies that{u
n
} converges strongly to some point
˜
x
. In this case,
we also have

w
n
−
˜
x≤w
n
− u
n
 + u
n
−
˜
x = α
n
γ f (w
n
) − Aw
n
 + u
n
−
˜
x→0.
Hence, the sequence {w
n
} c onverges strongly to some point
˜
x
. This completes the
proof. □

By Lemma 2.2, we obtain the following corollary.
Corollary 3.6. Let E be a uniformly convex Banach space which admits a weakly con-
tinuous duality mapp ing J

with gauge  such that  is invariant on [0, 1]. Let C be a
nonempty closed convex subset of E and
S = {T(s):s ≥ 0}
a nonexpansive semigroup
from C into itself such that
F(S ) = ∅
.
Let f Î Π
E
, and let A be a strongly positive linear bounded operator with a coefficient
0 <γ <
¯γϕ(1)
α
and
0 <γ <
¯γϕ(1)
α
. Let the sequence {x
n
} be generated by the following:
⎧
⎪
⎪
⎨
⎪
⎪

⎩
x
0
= x ∈ E,
y
n
= β
n
x
n
+(1− β
n
)
1
t
n
t
n

0
T(s)x
n
ds,
x
n+1
= α
n
γ f (x
n
)+(I − α

n
A)y
n
, n ≥ 0
(3:28)
where {a
n
} ⊂ (0, 1) and {b
n
} ⊂ [0, 1] are real sequences satisfying the following condi-
tions:
(C1) lim
n®∞
a
n
=0and

∞
n=1
α
n
= ∞
(C2) lim
n®∞
b
n
=0,
(C3) lim
n®∞
t

n
= ∞.
Then {x
n
} converges strongly to
˜
x
that is obtained in Lemma 3.2.
Setting E ≡ H and b
n
≡ 0 a real Hilbert space in Corollary 3.6, we have the following
result.
Corollary 3.7.[[15],Theorem3.2]LetHbearealHilbertspace.LetCbeanone-
mpty closed convex subset of E and
S = {T(s):s ≥ 0}
a nonexpansive semigroup from C
into itself such that
F(S ) = ∅
. Let f Î Π
E
, and let A b e a strongly positive linear
bounded operator with a coefficient
¯γ>0
and
0 <γ <
¯
γ
α
. Let the sequence {x
n

} be gen-
erated by the following:
⎧
⎨
⎩
x
0
= x ∈ E,
x
n+1
= α
n
γ f (x
n
)+(I − α
n
A)
1
t
n
t
n

0
T(s)x
n
ds, n ≥ 0
(3:29)
Wangkeeree and Preechasilp Fixed Point Theory and Applications 2011, 2011:76
/>Page 14 of 15

where {a
n
} ⊂ (0, 1) is a real sequences satisfying the following conditions:
(C1) lim
n®∞
a
n
=0and

∞
n=1
α
n
= ∞
(C2) lim
n®∞
t
n
= ∞.
Then {x
n
} converges strongly to
˜
x
that is obtained in Lemma 3.2. Then {x
n
} converges
strongly to
˜
x

which solves the variational inequality (1.12).
Acknowledgements
The project was supported by Naresuan university, Thailand.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final
manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 6 July 2011 Accepted: 7 November 2011 Published: 7 November 2011
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doi:10.1186/1687-1812-2011-76
Cite this article as: Wangkeeree and Preechasilp: The modified general iterative methods for nonexpansive
semigroups in banach spaces
†
. Fixed Point Theory and Applications 2011 2011:76.
Wangkeeree and Preechasilp Fixed Point Theory and Applications 2011, 2011:76
/>Page 15 of 15