We study strong convergence of the sequence generated by implicit and
explicit general iterative methods for a one-parameter nonexpansive semigroup
in a reflexive Banach space which admits the duality mapping Jφ, where φ is a gauge function on [0,∞). Our results improve and extend those announced by G. Marino and H.-K. Xu (2006) and many authors.

1. Introduction

Let E be a real Banach space and E* the dual space of E. Let K be a nonempty, closed, and convex subset of E. A (one-parameter) nonexpansive semigroup is a family 𝔉={T(t):t≥0} of self-mappings of K such that

T(0)x=x for all x∈K,

T(t+s)x=T(t)T(s)x for all t,s≥0 and x∈K,

for each x∈K, the mapping T(·)x is continuous,

for each t≥0, T(t) is nonexpansive, that is,

‖T(t)x-T(t)y‖≤‖x-y‖,∀x,y∈K.
We denote F by the common fixed points set of 𝔉, that is, F:=⋂t≥0F(T(t)).

In 1967, Halpern [1] introduced the following classical iteration for a nonexpansive mapping T:K→K in a real Hilbert space:xn+1=αnu+(1-αn)Txn,n≥0,
where {αn}⊂(0,1) and u∈K.

In 1977, Lions [2] obtained a strong convergence provide the real sequence {αn} satisfies the following conditions:

Reich [3] also extended the result of Halpern from Hilbert spaces to uniformly smooth Banach spaces. However, both Halpern’s and Lion’s conditions imposed on the real sequence {αn} excluded the canonical choice αn=1/(n+1).

In 1992, Wittmann [4] proved that the sequence {xn} converges strongly to a fixed point of T if {αn} satisfies the following conditions:

Shioji and Takahashi [5] extended Wittmann’s result to real Banach spaces with uniformly Gâteaux differentiable norms and in which each nonempty closed convex and bounded subset has the fixed point property for nonexpansive mappings. The concept of the Halpern iterative scheme has been widely used to approximate the fixed points for nonexpansive mappings (see, e.g., [6–12] and the reference cited therein).

Let f:K→K be a contraction. In 2000, Moudafi [13] introduced the explicit viscosity approximation method for a nonexpansive mapping T as follows:xn+1=αnf(xn)+(1-αn)Txn,n≥0,
where αn∈(0,1). Xu [14] also studied the iteration process (1.3) in uniformly smooth Banach spaces.

Let A be a strongly positive bounded linear operator on a real Hilbert space H, that is, there is a constant γ¯>0 such that〈Ax,x〉≥γ¯‖x‖2,∀x∈H.

A typical problem is to minimize a quadratic function over the fixed points set of a nonexpansive mapping on a Hilbert space H:minx∈C12〈Ax,x〉-〈x,b〉,
where C is the fixed points set of a nonexpansive mapping T on H and b is a given point in H.

In 2006, Marino and Xu [15] introduced the following general iterative method for a nonexpansive mapping T in a Hilbert space H:xn+1=αnγf(xn)+(I-αnA)Txn,n≥1,
where {αn}⊂(0,1), f is a contraction on H, and A is a strongly positive bounded linear operator on H. They proved that the sequence {xn} generated by (1.6) converges strongly to a fixed point x*∈F(T) which also solves the variational inequality〈(A-γf)x*,x-x*〉≥0,∀x∈F(T),
which is the optimality condition for the minimization problem: minx∈C(1/2)〈Ax,x〉-h(x), where h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H).

Suzuki [16] first introduced the following implicit viscosity method for a nonexpansive semigroup {T(t):t≥0} in a Hilbert space:xn=αnu+(1-αn)T(tn)xn,n≥1,
where {αn}⊂(0,1) and u∈K. He proved strong convergence of iteration (1.8) under suitable conditions. Subsequently, Xu [17] extended Suzuki’s [16] result from a Hilbert space to a uniformly convex Banach space which admits a weakly sequentially continuous normalized duality mapping.

Motivated by Chen and Song [18], in 2007, Chen and He [19] investigated the implicit and explicit viscosity methods for a nonexpansive semigroup without integral in a reflexive Banach space which admits a weakly sequentially continuous normalized duality mapping:xn=αnf(xn)+(1-αn)T(tn)xn,n≥1,xn+1=αnf(xn)+(1-αn)T(tn)xn,n≥1,
where {αn}⊂(0,1).

In 2008, Song and Xu [20] also studied the iterations (1.9) and (1.10) in a reflexive and strictly convex Banach space with a Gâteaux differentiable norm. Subsequently, Cholamjiak and Suantai [21] extended Song and Xu’s results to a Banach space which admits duality mapping with a gauge function. Wangkeeree and Kamraksa [22] and Wangkeeree et al. [23] obtained the convergence results concerning the duality mapping with a gauge function in Banach spaces. The convergence of iterations for a nonexpansive semigroup and nonlinear mappings has been studied by many authors (see, e.g., [24–38]).

Let E be a real reflexive Banach space which admits the duality mapping Jφ with a gauge φ. Let {T(t):t≥0} be a nonexpansive semigroup on E. Recall that an operator A is said to be strongly positive if there exists a constant γ¯>0 such that〈Ax,Jφ(x)〉≥γ¯‖x‖φ(‖x‖),‖αI-βA‖=sup‖x‖≤1|〈(αI-βA)x,Jφ(x)〉|,
where α∈[0,1] and β∈[-1,1].

Motivated by Chen and Song [18], Chen and He [19], Marino and Xu [15], Colao et al. [39], and Wangkeeree et al. [23], we study strong convergence of the following general iterative methods:xn=αnγf(xn)+(I-αnA)T(tn)xn,n≥1,xn+1=αnγf(xn)+(I-αnA)T(tn)xn,n≥1,
where {αn}⊂(0,1), f is a contraction on E and A is a positive bounded linear operator on E.

2. Preliminaries

A Banach space E is called strictly convex if ∥x+y∥/2<1 for all x,y∈E with ∥x∥=∥y∥=1 and x≠y. A Banach space E is called uniformly convex if for each ϵ>0 there is a δ>0 such that for x,y∈E with ∥x∥,∥y∥≤1 and ∥x-y∥≥ϵ,∥x+y∥≤2(1-δ) holds. The modulus of convexity of E is defined byδE(ϵ)=inf{1-‖12(x+y)‖:‖x‖,‖y‖≤1,‖x-y‖≥ϵ},
for all ϵ∈[0,2]. E is uniformly convex if δE(0)=0, and δE(ϵ)>0 for all 0<ϵ≤2. It is known that every uniformly convex Banach space is strictly convex and reflexive. Let S(E)={x∈E:∥x∥=1}. Then the norm of E is said to be Gâteaux differentiable iflimt→0‖x+ty‖-‖x‖t
exists for each x,y∈S(E). In this case E is called smooth. The norm of E is said to be Fréchet differentiable if for each x∈S(E), the limit is attained uniformly for y∈S(E). The norm of E is called uniformly Fréchet differentiable, if the limit is attained uniformly for x,y∈S(E). It is well known that (uniformly) Fréchet differentiability of the norm of E implies (uniformly) Gâteaux differentiability of the norm of E.

Let ρE:[0,∞)→[0,∞) be the modulus of smoothness of E defined byρE(t)=sup{12(‖x+y‖+‖x-y‖)-1:x∈S(E),‖y‖≤t}.

A Banach space E is called uniformly smooth if ρE(t)/t→0 as t→0. See [40–42] for more details.

We need the following definitions and results which can be found in [40, 41, 43].

Definition 2.1.

A continuous strictly increasing function φ:[0,∞)→[0,∞) is said to be gauge function if φ(0)=0 and limt→∞φ(t)=∞.

Definition 2.2.

Let E be a normed space and φ a gauge function. Then the mapping Jφ:E→2E* defined by
Jφ(x)={f*∈E*:〈x,f*〉=‖x‖φ(‖x‖),‖f*‖=φ(‖x‖)},x∈E,
is called the duality mapping with gauge function φ.

In the particular case φ(t)=t, the duality mapping Jφ=J is called the normalized duality mapping.

In the case φ(t)=tq-1,q>1, the duality mapping Jφ=Jq is called the generalized duality mapping. It follows from the definition that Jφ(x)=φ(∥x∥)/∥x∥J(x) and Jq(x)=∥x∥q-2J(x),q>1.

Remark 2.3.

For the gauge function φ, the function Φ:[0,∞)→[0,∞) defined by
Φ(t)=∫0tφ(s)ds
is a continuous convex and strictly increasing function on [0,∞). Therefore, Φ has a continuous inverse function Φ-1.

It is noted that if 0≤k≤1, then φ(kx)≤φ(x). FurtherΦ(kt)=∫0ktφ(s)ds=k∫0tφ(kx)dx≤k∫0tφ(x)dx=kΦ(t).

Remark 2.4.

For each x in a Banach space E, Jφ(x)=∂Φ(∥x∥), where ∂ denotes the sub-differential.

We also know the following facts:

Jφ is a nonempty, closed, and convex set in E* for each x∈E,

Jφ is a function when E* is strictly convex,

If Jφ is single-valued, then

Jφ(λx)=sign(λ)φ(‖λx‖)φ(‖x‖)Jφ(x),∀x∈E,λ∈R,〈x-y,Jφ(x)-Jφ(y)〉≥(φ(‖x‖)-φ(‖y‖))(‖x‖-‖y‖),∀x,y∈E.
Following Browder [43], we say that a Banach space E has a weakly continuous duality mapping if there exists a gauge φ for which the duality mapping Jφ is single-valued and continuous from the weak topology to the weak* topology, that is, for any {xn} with xn⇀x, the sequence {Jφ(xn)} converges weakly* to Jφ(x). It is known that the space ℓp has a weakly continuous duality mapping with a gauge function φ(t)=tp-1 for all 1<p<∞. Moreover, φ is invariant on [0,1].Lemma 2.5 (See [<xref ref-type="bibr" rid="B24">44</xref>]).

Assume that a Banach space E has a weakly continuous duality mapping Jφ with gauge φ.

For all x,y∈E, the following inequality holds:
Φ(‖x+y‖)≤Φ(‖x‖)+〈y,Jφ(x+y)〉.
In particular, for all x,y∈E,
‖x+y‖2≤‖x‖2+2〈y,J(x+y)〉.

Assume that a sequence {xn} in E converges weakly to a point x∈E. Then the following holds:
limsupn→∞Φ(‖xn-y‖)=limsupn→∞Φ(‖xn-x‖)+Φ(‖x-y‖)
for all x,y∈E.

Lemma 2.6 (See [<xref ref-type="bibr" rid="B39">23</xref>]).

Assume that a Banach space E has a weakly continuous duality mapping Jφ with gauge φ. Let A be a strongly positive bounded linear operator on E with coefficient γ¯>0 and 0<ρ≤φ(1)∥A∥-1. Then ∥I-ρA∥≤φ(1)(1-ργ¯).

Lemma 2.7 (See [<xref ref-type="bibr" rid="B42">12</xref>]).

Assume that {an} is a sequence of nonnegative real numbers such that
an+1≤(1-γn)an+γnδn,n≥1,
where {γn} is a sequence in (0,1) and {δn} is a sequence in ℝ such that

(a) ∑n=1∞γn=∞; (b) limsupn→∞δn≤0 or ∑n=1∞|γnδn|<∞.

Then limn→∞an=0.

3. Implicit Iteration Scheme

In this section, we prove a strong convergence theorem of an implicit iterative method (1.12).

Theorem 3.1.

Let E be a reflexive which admits a weakly continuous duality mapping Jφ with gauge φ such that φ is invariant on [0,1]. Let 𝔉={T(t):t≥0} be a nonexpansive semigroup on E such that F≠∅. Let f be a contraction on E with the coefficient α∈(0,1) and A a strongly positive bounded linear operator with coefficient γ¯>0 and 0<γ<γ̅φ(1)/α. Let {αn} and {tn} be real sequences satisfying 0<αn<1, tn>0 and limn→∞tn=limn→∞αn/tn=0. Then {xn} defined by (1.12) converges strongly to q∈F which solves the following variational inequality:
〈(A-γf)(q),Jφ(q-w)〉≤0,∀w∈F.

Proof.

First, we prove the uniqueness of the solution to the variational inequality (3.1) in F. Suppose that p,q∈F satisfy (3.1), so we have
〈(A-γf)(p),Jφ(p-q)〉≤0,〈(A-γf)(q),Jφ(q-p)〉≤0.
Adding the above inequalities, we get
〈A(p)-A(q)-γ(f(p)-f(q)),Jφ(p-q)〉≤0.
This shows that
〈A(p-q),Jφ(p-q)〉≤γ〈f(p)-f(q),Jφ(p-q)〉,
which implies by the strong positivity of Aγ¯‖p-q‖φ(‖p-q‖)≤〈A(p-q),Jφ(p-q)〉≤γα‖p-q‖φ(‖p-q‖).
Since φ is invariant on [0,1],
φ(1)γ¯‖p-q‖φ(‖p-q‖)≤γα‖p-q‖φ(‖p-q‖).
It follows that
(φ(1)γ¯-γα)‖p-q‖φ(‖p-q‖)≤0.
Therefore p=q since 0<γ<(γ̅φ(1))/α.

We next prove that {xn} is bounded. For each w∈F, by Lemma 2.6, we have‖xn-w‖=‖αnγf(xn)+(I-αnA)T(tn)xn-w‖=‖(I-αnA)T(tn)xn-(I-αnA)w+αn(γf(xn)-A(w))‖≤φ(1)(1-αnγ¯)‖xn-w‖+αn(γα‖xn-w‖+‖γf(w)-A(w)‖)≤‖xn-w‖-αnφ(1)γ¯‖xn-w‖+αnγα‖xn-w‖+αn‖γf(w)-A(w)‖,
which yields
‖xn-w‖≤1φ(1)γ¯-γα‖γf(w)-A(w)‖.
Hence {xn} is bounded. So are {f(xn)} and {AT(tn)xn}.

We next prove that {xn} is relatively sequentially compact. By the reflexivity of E and the boundedness of {xn}, there exists a subsequence {xnj} of {xn} and a point p in E such that xnj⇀p as j→∞. Now we show that p∈F. Put xj=xnj, βj=αnj and sj=tnj for j∈ℕ, fix t>0. We see that‖xj-T(t)p‖≤∑k=0[t/sj]-1‖T((k+1)sj)xj-T(ksj)xj+1‖+‖T([tsj]sj)xj-T([tsj]sj)p‖+‖T([tsj]sj)p-T(t)p‖≤[tsj]‖T(sj)xj-xj‖+‖xj-p‖+‖T(t-[tsj]sj)p-p‖=[tsj]βj‖AT(sj)xj-γf(xj)‖+‖xj-p‖+‖T(t-[tsj]sj)p-p‖≤tβjsj‖AT(sj)xj-γf(xj)‖+‖xj-p‖+max{‖T(s)p-p‖:0≤s≤sj}.
So we have
limsupj→∞Φ(‖xj-T(t)p‖)≤limsupj→∞Φ(‖xj-p‖).
On the other hand, by Lemma 2.5 (ii), we have
limsupj→∞Φ(‖xj-T(t)p‖)=limsupj→∞Φ(‖xj-p‖)+Φ(‖T(t)p-p‖).
Combining (3.11) and (3.12), we have
Φ(‖T(t)p-p‖)≤0.
This implies that p∈F. Further, we see that
∥xj-p∥φ(∥xj-p∥)=〈xj-p,Jφ(xj-p)〉=〈(I-βjA)T(sj)xj-(I-βjA)p,Jφ(xj-p)〉+βj〈γf(xj)-γf(p),Jφ(xj-p)〉+βj〈γf(p)-A(p),Jφ(xj-p)〉≤φ(1)(1-βjγ¯)‖xj-p‖φ(‖xj-p‖)+βjγα‖xj-p‖φ(‖xj-p‖)+βj〈γf(p)-A(p),Jφ(xj-p)〉.
So we have
‖xj-p‖φ(‖xj-p‖)≤1φ(1)γ¯-γα〈γf(p)-A(p),Jφ(xj-p)〉.
By the definition of Φ, it is easily seen that
Φ(‖xj-p‖)≤‖xj-p‖φ(‖xj-p‖).
Hence
Φ(‖xj-p‖)≤1φ(1)γ¯-γα〈γf(p)-A(p),Jφ(xj-p)〉.
Therefore Φ(∥xj-p∥)→0 as j→∞ since Jφ is weakly continuous; consequently, xj→p as j→∞ by the continuity of Φ. Hence {xn} is relatively sequentially compact.

Finally, we prove that p is a solution in F to the variational inequality (3.1). For any w∈F, we see that〈(I-T(tn))xn-(I-T(tn))w,Jφ(xn-w)〉=〈xn-w,Jφ(xn-w)〉-〈T(tn)xn-T(tn)w,Jφ(xn-w)〉≥‖xn-w‖φ‖xn-w‖-‖T(tn)xn-T(tn)w‖‖Jφ(xn-w)‖≥‖xn-w‖φ‖xn-w‖-‖xn-w‖‖Jφ(xn-w)‖=0.
On the other hand, we have
(A-γf)(xn)=-1αn(I-αnA)(I-T(tn))xn,
which implies
〈(A-γf)(xn),Jφ(xn-w)〉=-1αn〈(I-T(tn))xn-(I-T(tn))w,Jφ(xn-w)〉+〈A(I-T(tn))xn,Jφ(xn-w)〉≤〈A(I-T(tn))xn,Jφ(xn-w)〉.
Observe
‖xj-T(sj)xj‖=βj‖γf(xj)-AT(sj)xj‖→0,
as j→∞. Replacing n by nj and letting j→∞ in (3.20), we obtain
〈(A-γf)(p),Jφ(p-w)〉≤0,∀w∈F.
So p∈F is a solution of variational inequality (3.1); and hence p=q by the uniqueness. In a summary, we have proved that {xn} is relatively sequentially compact and each cluster point of {xn} (as n→∞) equals q. Therefore xn→q as n→∞. This completes the proof.

4. Explicit Iteration Scheme

In this section, utilizing the implicit version in Theorem 3.1, we consider the explicit one in a reflexive Banach space which admits the duality mapping Jφ.

Theorem 4.1.

Let E be a reflexive Banach space which admits a weakly continuous duality mapping Jφ with gauge φ such that φ is invariant on [0,1]. Let {T(t):t≥0} be a nonexpansive semigroup on E such that F≠∅. Let f be a contraction on E with the coefficient α∈(0,1) and A a strongly positive bounded linear operator with coefficient γ¯>0 and 0<γ<γ̅φ(1)/α. Let {αn} and {tn} be real sequences satisfying 0<αn<1, ∑n=1∞αn=∞, tn>0 and limn→∞tn=limn→∞αn/tn=0. Then {xn} defined by (1.13) converges strongly to q∈F which also solves the variational inequality (3.1).

Proof.

Since αn→0, we may assume that αn<φ(1)∥A∥-1 and 1-αn(φ(1)γ¯-γα)>0 for all n. First we prove that {xn} is bounded. For each w∈F, by Lemma 2.6, we have
‖xn+1-w‖=‖αnγf(xn)+(I-αnA)T(tn)xn-w‖=‖(I-αnA)T(tn)xn-(I-αnA)w+αn(γf(xn)-A(w))‖≤φ(1)(1-αnγ¯)‖xn-w‖+αnγα‖xn-w‖+αn‖γf(w)-A(w)‖=(φ(1)-αn(φ(1)γ¯-γα))‖xn-w‖+αn‖γf(w)-A(w)‖≤(1-αn(φ(1)γ¯-γα))‖xn-w‖+αn(φ(1)γ¯-γα))‖γf(w)-A(w)‖φ(1)γ¯-γα.
It follows from induction that
‖xn+1-w‖≤max{‖x1-w‖,‖γf(w)-A(w)‖φ(1)γ¯-γα},n≥1.
Thus {xn} is bounded, and hence so are {f(xn)} and {AT(tn)xn}. From Theorem 3.1, there is a unique solution q∈F to the following variational inequality:
〈(A-γf)q,Jφ(q-w)〉≤0,∀w∈F.
Next we prove that
limsupn→∞〈(A-γf)q,Jφ(q-xn+1)〉≤0.
Indeed, we can choose a subsequence {xnj} of {xn} such that
limsupn→∞〈(A-γf)q,Jφ(q-xn)〉=limsupj→∞〈(A-γf)q,Jφ(q-xnj)〉.
Further, we can assume that xnj⇀p∈E by the reflexivity of E and the boundedness of {xn}. Now we show that p∈F. Put xj=xnj,βj=αnj and sj=tnj for j∈ℕ, fix t>0. We obtain
‖xj+1-T(t)p‖≤∑k=0[t/sj]-1‖T((k+1)sj)xj-T(ksj)xj+1‖+‖T([tsj]sj)xj-T([tsj]sj)p‖+‖T([tsj]sj)p-T(t)p‖≤[tsj]‖T(sj)xj-xj+1‖+‖xj-p‖+‖T(t-[tsj]sj)p-p‖=[tsj]βj‖AT(sj)xj-γf(xj)‖+‖xj-p‖+‖T(t-[tsj]sj)p-p‖≤tβjsj‖AT(sj)xj-γf(xj)‖+‖xj-p‖+max{‖T(s)p-p‖:0≤s≤sj}.
It follows that limsupn→∞Φ(∥xj-T(t)p∥)≤limsupn→∞Φ(∥xj-p∥). From Lemma 2.5 (ii) we have
limsupn→∞Φ(‖xj-T(t)p‖)=limsupn→∞Φ(‖xj-p‖)+Φ(‖T(t)p-p‖).
So we have Φ(∥T(t)p-p∥)≤0 and hence p∈F. Since the duality mapping Jφ is weakly sequentially continuous,
limsupn→∞〈(A-γf)q,Jφ(q-xn+1)〉=limsupj→∞〈(A-γf)q,Jφ(q-xnj+1)〉=〈(A-γf)q,Jφ(q-p)〉≤0.
Finally, we show that xn→q. From Lemma 2.5 (i), we have
Φ(‖xn+1-q‖)=Φ(‖(I-αnA)T(tn)xn-(I-αnA)q+αn(γf(xn)-γf(q))+αn(γf(q)-A(q))‖)≤Φ(‖(I-αnA)(T(tn)xn-q)+αn(γf(xn)-γf(q))‖)+αn〈γf(q)-A(q),Jφ(xn+1-q)〉≤Φ(φ(1)(1-αnγ¯)‖xn-q‖+αnγα‖xn-q‖)+αn〈γf(q)-A(q),Jφ(xn+1-q)〉=Φ((φ(1)-αn(φ(1)γ¯-γα))‖xn-q‖)+αn〈γf(q)-A(q),Jφ(xn+1-q)〉≤(1-αn(φ(1)γ¯-γα))Φ(‖xn-q‖)+αn〈γf(q)-A(q),Jφ(xn+1-q)〉.
Note that ∑n=1∞αn=∞ and limsupn→∞〈γf(q)-A(q),Jφ(xn+1-q)〉≤0. Using Lemma 2.7, we have xn→q as n→∞ by the continuity of Φ. This completes the proof.

Remark 4.2.

Theorems 3.1 and 4.1 improve and extend the main results proved in [15] in the following senses:

from a nonexpansive mapping to a nonexpansive semigroup,

from a real Hilbert space to a reflexive Banach space which admits a weakly continuous duality mapping with gauge functions.

Acknowledgments

The authors wish to thank the editor and the referee for valuable suggestions. K. Nammanee was supported by the Thailand Research Fund, the Commission on Higher Education, and the University of Phayao under Grant MRG5380202. S. Suantai and P. Cholamjiak wish to thank the Thailand Research Fund and the Centre of Excellence in Mathematics, Thailand.

HalpernB.Fixed points of nonexpanding mapsLionsP.-L.Approximation de points fixes de contractionsReichS.Approximating fixed points of nonexpansive mappingsWittmannR.Approximation of fixed points of nonexpansive mappingsShiojiN.TakahashiW.Strong convergence of approximated sequences for nonexpansive mappings in Banach spacesAoyamaK.KimuraY.TakahashiW.ToyodaM.Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach spaceChidumeC. E.ChidumeC. O.Iterative approximation of fixed points of nonexpansive mappingsChoY. J.KangS. M.ZhouH.Some control conditions on iterative methodsKimT.-H.XuH.-K.Strong convergence of modified Mann iterationsReichS.Strong convergence theorems for resolvents of accretive operators in Banach spacesXuH.-K.Another control condition in an iterative method for nonexpansive mappingsXuH.-K.Iterative algorithms for nonlinear operatorsMoudafiA.Viscosity approximation methods for fixed-points problemsXuH.-K.Viscosity approximation methods for nonexpansive mappingsMarinoG.XuH.-K.A general iterative method for nonexpansive mappings in Hilbert spacesSuzukiT.On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spacesXuH.-K.A strong convergence theorem for contraction semigroups in Banach spacesChenR.SongY.Convergence to common fixed point of nonexpansive semigroupsChenR.HeH.Viscosity approximation of common fixed points of nonexpansive semigroups in Banach spaceSongY.XuS.Strong convergence theorems for nonexpansive semigroup in Banach spacesCholamjiakP.SuantaiS.Viscosity approximation methods for a nonexpansive semigroup in Banach spaces with gauge functionsJournal of Global Optimization. In press10.1007/s10898-011-9756-4WangkeereeR.KamraksaU.Strong convergence theorems of viscosity iterative methods for a countable family of strict pseudo-contractions in Banach spacesWangkeereeR.rattanapornw@nu.ac.thPetrotN.narinp@nu.ac.thWangkeereeR.rabianw@nu.ac.thThe general iterative methods for nonexpansive mappings in Banach spacesArgyrosI. K.ChoY. J.QinX.On the implicit iterative process for strictly pseudo-contractive mappings in Banach spacesChangS.-S.ChoY. J.LeeH. W. J.ChanC. K.Strong convergence theorems for Lipschitzian demicontraction semigroups in Banach spacesChoY. J.ĆirićL.WangS.-H.Convergence theorems for nonexpansive semigroups in CAT(0) spacesChoY. J.KangS. M.QinX.Some results on k-strictly pseudo-contractive mappings in Hilbert spacesChoY. J.KangS. M.QinX.Strong convergence of an implicit iterative process for an infinite family of strict pseudocontractionsChoY. J.QinX.KangS. M.Strong convergence of the modified Halpern-type iterative algorithms in Banach spacesGuoW.ChoY. J.On the strong convergence of the implicit iterative processes with errors for a finite family of asymptotically nonexpansive mappingsHeH.LiuS.ChoY. J.An explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappingsKangJ.SuY.ZhangX.General iterative algorithm for nonexpansive semigroups and variational inequalities in Hilbert spacesLiX. N.GuJ. S.Strong convergence of modified Ishikawa iteration for a nonexpansive semigroup in Banach spacesLiS.LiL.SuY.General iterative methods for a one-parameter nonexpansive semigroup in Hilbert spaceLinQ.Viscosity approximation to common fixed points of a nonexpansive semigroup with a generalized contraction mappingPlubtiengS.PunpaengR.Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spacesQinX.ChoY. J.KangS. M.ZhouH.Convergence of a modified Halpern-type iteration algorithm for quasi-ϕ-nonexpansive mappingsSuzukiT.Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integralsColaoV.MarinoG.XuH.-K.An iterative method for finding common solutions of equilibrium and fixed point problemsAgarwalR. P.O'ReganD.SahuD. R.ChidumeC.TakahashiW.BrowderF. E.Convergence theorems for sequences of nonlinear operators in Banach spacesLimT.-C.XuH. K.Fixed point theorems for asymptotically nonexpansive mappings