# A General Iterative Method for a Nonexpansive Semigroup in Banach Spaces with Gauge Functions

Authors: Kamonrat Nammanee; Suthep Suantai; Prasit Cholamjiak
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

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 mappingJφ, 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.


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## Body

## 1. Introduction

LetE 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(i)
T(0)x=x for all x∈K,(ii)
T(t+s)x=T(t)T(s)x for all t,s≥0 and x∈K,(iii)
for eachx∈K, the mapping T(·)x is continuous,(iv)
for eacht≥0, T(t) is nonexpansive, that is,(1.1)‖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:(1.2)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:C1:limn→∞αn=0; C2: ∑n=0∞αn=∞; C3: limn→∞(αn-αn-1)/αn2=0.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:C1:limn→∞αn=0; C2: ∑n=0∞αn=∞; C3: ∑n=0∞|αn+1-αn|<∞.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).Letf:K→K be a contraction. In 2000, Moudafi [13] introduced the explicit viscosity approximation method for a nonexpansive mapping T as follows:(1.3)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.LetA be a strongly positive bounded linear operator on a real Hilbert space H, that is, there is a constant γ¯>0 such that(1.4)〈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 spaceH:(1.5)min⁡x∈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:(1.6)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(1.7)〈(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:(1.8)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:(1.9)xn=αnf(xn)+(1-αn)T(tn)xn,n≥1,(1.10)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]).LetE 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(1.11)〈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:(1.12)xn=αnγf(xn)+(I-αnA)T(tn)xn,n≥1,(1.13)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 spaceE 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 by(2.1)δE(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 if(2.2)lim⁡t→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.LetρE:[0,∞)→[0,∞) be the modulus of smoothness of E defined by(2.3)ρE(t)=sup⁡{12(‖x+y‖+‖x-y‖)-1:x∈S(E),‖y‖≤t}.A Banach spaceE 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 functionφ:[0,∞)→[0,∞) is said to be gauge function if φ(0)=0 and limt→∞φ(t)=∞.Definition 2.2.
LetE be a normed space and φ a gauge function. Then the mapping Jφ:E→2E* defined by
(2.4)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 caseφ(t)=t, the duality mapping Jφ=J is called the normalized duality mapping.In the caseφ(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 functionφ, the function Φ:[0,∞)→[0,∞) defined by
(2.5)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 if0≤k≤1, then φ(kx)≤φ(x). Further(2.6)Φ(kt)=∫0ktφ(s)ds=k∫0tφ(kx)dx≤k∫0tφ(x)dx=kΦ(t).Remark 2.4.
For eachx in a Banach space E, Jφ(x)=∂Φ(∥x∥), where ∂ denotes the sub-differential.We also know the following facts:(i)
Jφ is a nonempty, closed, and convex set in E* for each x∈E,(ii)
Jφ is a function when E* is strictly convex,(iii)
IfJφ is single-valued, then(2.7)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 [44]).
Assume that a Banach spaceE has a weakly continuous duality mapping Jφ with gauge φ.(i)
For allx,y∈E, the following inequality holds:
(2.8)Φ(‖x+y‖)≤Φ(‖x‖)+〈y,Jφ(x+y)〉.
In particular, for all x,y∈E,
(2.9)‖x+y‖2≤‖x‖2+2〈y,J(x+y)〉.(ii)
Assume that a sequence{xn} in E converges weakly to a point x∈E. Then the following holds:
(2.10)limsup⁡n→∞Φ(‖xn-y‖)=limsup⁡n→∞Φ(‖xn-x‖)+Φ(‖x-y‖)
for all x,y∈E.Lemma 2.6 (See [23]).
Assume that a Banach spaceE 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 [12]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.11)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|<∞.
Thenlimn→∞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.
LetE 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:
(3.1)〈(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
(3.2)〈(A-γf)(p),Jφ(p-q)〉≤0,〈(A-γf)(q),Jφ(q-p)〉≤0.
Adding the above inequalities, we get
(3.3)〈A(p)-A(q)-γ(f(p)-f(q)),Jφ(p-q)〉≤0.
This shows that
(3.4)〈A(p-q),Jφ(p-q)〉≤γ〈f(p)-f(q),Jφ(p-q)〉,
which implies by the strong positivity of A(3.5)γ¯‖p-q‖φ(‖p-q‖)≤〈A(p-q),Jφ(p-q)〉≤γα‖p-q‖φ(‖p-q‖).
Since φ is invariant on [0,1],
(3.6)φ(1)γ¯‖p-q‖φ(‖p-q‖)≤γα‖p-q‖φ(‖p-q‖).
It follows that
(3.7)(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(3.8)‖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
(3.9)‖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(3.10)‖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
(3.11)limsup⁡j→∞Φ(‖xj-T(t)p‖)≤limsup⁡j→∞Φ(‖xj-p‖).
On the other hand, by Lemma 2.5 (ii), we have
(3.12)limsup⁡j→∞Φ(‖xj-T(t)p‖)=limsup⁡j→∞Φ(‖xj-p‖)+Φ(‖T(t)p-p‖).
Combining (3.11) and (3.12), we have
(3.13)Φ(‖T(t)p-p‖)≤0.
This implies that p∈F. Further, we see that
(3.14)∥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
(3.15)‖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
(3.16)Φ(‖xj-p‖)≤‖xj-p‖φ(‖xj-p‖).
Hence
(3.17)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 thatp is a solution in F to the variational inequality (3.1). For any w∈F, we see that(3.18)〈(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
(3.19)(A-γf)(xn)=-1αn(I-αnA)(I-T(tn))xn,
which implies
(3.20)〈(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
(3.21)‖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
(3.22)〈(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 Theorem3.1, we consider the explicit one in a reflexive Banach space which admits the duality mapping Jφ.Theorem 4.1.
LetE 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.
Sinceαn→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
(4.1)‖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
(4.2)‖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:
(4.3)〈(A-γf)q,Jφ(q-w)〉≤0,∀w∈F.
Next we prove that
(4.4)limsup⁡n→∞〈(A-γf)q,Jφ(q-xn+1)〉≤0.
Indeed, we can choose a subsequence {xnj} of {xn} such that
(4.5)limsup⁡n→∞〈(A-γf)q,Jφ(q-xn)〉=limsup⁡j→∞〈(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
(4.6)‖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
(4.7)limsup⁡n→∞Φ(‖xj-T(t)p‖)=limsup⁡n→∞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,
(4.8)limsup⁡n→∞〈(A-γf)q,Jφ(q-xn+1)〉=limsup⁡j→∞〈(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
(4.9)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.
Theorems3.1 and 4.1 improve and extend the main results proved in [15] in the following senses:(i)
from a nonexpansive mapping to a nonexpansive semigroup,(ii)
from a real Hilbert space to a reflexive Banach space which admits a weakly continuous duality mapping with gauge functions.



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