# A New Composite General Iterative Scheme for Nonexpansive Semigroups in Banach Spaces

Authors: Pongsakorn Sunthrayuth; Poom Kumam
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

---

## Abstract

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.


---

## Body

## 1. Introduction

Throughout this paper we denoted byℕ and ℝ+ 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 nonexpansiveif ∥Tx-Ty∥≤∥x-y∥ for each x,y∈C. We denote by F(T) the set of fixed points of T. We know that F(T) is nonempty if C is bounded; for more detail see [1]. A one-parameter family 𝒮={T(t):t∈ℝ+} from C of X into itself is said to be a nonexpansive semigroup on C if it satisfies the following conditions:  (i)
T(0)x=xforallx∈C;(ii)
T(s+t)=T(s)∘T(t) for all s,t∈ℝ+;(iii)
for eachx∈C the mapping t↦T(t)x is continuous;(iv)
∥T(t)x-T(t)y∥≤∥x-y∥ for all x,y∈C and t∈ℝ+.We denote byF(𝒮) the set of all common fixed points of 𝒮, that is, F(𝒮)∶=∩t∈ℝ+F(T(t))={x∈C:T(t)x=x}. We know that F(𝒮) is nonempty if C is bounded; see [2]. Recall that a self-mapping f:C→C is a contraction if there exists a constant α∈(0,1) such that ∥f(x)-f(y)∥≤α∥x-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→Cacontraction}. 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(1.1)〈Ax,x〉≥γ¯‖x‖2∀x∈H.
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:(1.2)min⁡x∈F12〈Ax,x〉-〈x,b〉,
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(1.3)x0∈Cchosenarbitrarily,xn+1=(I-αnA)Txn+αnu,∀n≥0,
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:(1.4)x0∈Cchosenarbitrarily,xn+1=αnf(xn)+(1-αn)Txn,∀n≥0,
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:(1.5)〈(f-I)x*,y-x*〉≤0,∀y∈F(T).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:(1.6)x0∈Cchosenarbitrarily,xn+1=αnγf(xn)+(I-αnA)Txn,∀n≥0,
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:(1.7)〈(γf-A)x*,y-x*〉≤0,∀y∈F(T),
which is the optimality condition for the minimization problem:(1.8)min⁡x∈C12〈Ax,x〉-h(x),
where C is the fixed point set of a nonexpansive mapping T and h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H). Kim and Xu [9] studied the sequence generated by the following algorithm:(1.9)x1∈Cchosenarbitrarily,yn=αnxn+(1-αn)Txn,xn+1=βnu+(1-βn)yn,∀n≥0,
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 F(T)≠∅ and f∈∏C. Define {xn} in the following way:(1.10)x1∈Cchosenarbitrarily;yn=αnxn+(1-αn)Txn;xn+1=βnf(xn)+(1-βn)yn,∀n≥0,
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:(1.11)x1∈Cchosenarbitrarily;xn+1=αnf(xn)+(1-αn)1tn∫0tnT(s)xnds,∀n∈N,
and they proved that the sequence {xn} defined by (1.11) converges strongly to the unique solution of the variational inequality:(1.12)〈(f-I)x*,J(y-x*)〉≤0,∀y∈F(T).
In 2010, Sunthrayuth and Kumam [12] introduced the a general iterative scheme generated by(1.13)x0∈Cchosenarbitrarily;xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)1tn∫0tnT(s)xnds,∀n≥0,
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 writexn⇀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 ϵ∈(0,2], there exists a δ>0 such that for each x,y∈U, ∥x-y∥≥ϵ 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.Letφ:[0,∞)∶=ℝ+→ℝ+ be a continuous strictly increasing function such that φ(0)=0 and φ(t)→∞ 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 (2.1)Jφ(x)={f*∈X*:〈x,f*〉=‖x‖φ(‖x‖),‖f*‖=φ(‖x‖),∀x∈X},
where 〈·,·〉 denotes the generalized duality paring. In particular, the duality mapping with the gauge function φ(t)=t, denoted by J, is referred to as the normalized duality mapping. Clearly, there holds the relation Jφ(x)=(φ(∥x∥)/∥x∥)J(x) for each x≠0 (see [16]).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 {J(xn)} converges weakly* to Jφ(x). It is known that lp has a weakly continuous duality mapping with a gauge function φ(t)=tp-1 for all 1<p<∞. Set Φ(t)=∫0tφ(τ)dτ,  forallt≥0; then Jφ(x)=∂Φ(∥x∥), where ∂ denotes the subdifferential in the sense of convex analysis (recall that the subdifferential of the convex function ϕ:X→ℝ at x∈X is the set ∂ϕ(x)={x*∈X;ϕ(y)≥ϕ(x)+〈x*,y-x〉,forally∈X}).In a Banach space having a weakly continuous duality mappingJφ 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(2.2)〈Ax,Jφ(x)〉≥γ¯‖x‖φ(‖x‖),(2.3)‖aI-bA‖=sup⁡‖x‖≤1|〈(aI-bA)x,Jφ(x)〉|,a∈[0,1],b∈[-1,1].
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 spaceX has a weakly continuous duality mapping Jφ with gauge φ.  (i)
For allx,y∈X, the following inequality holds:
(2.4)Φ(‖x+y‖)≤Φ(‖x‖)+〈y,Jφ(x+y)〉.
In particular, for all x,y∈X,(2.5)‖x+y‖2≤‖x‖2+2〈y,J(x+y)〉.(ii)
Assume that a sequence{xn} in X converges weakly to a point x∈X. Then the following identity holds:(2.6)limsup⁡n→∞Φ(‖xn-y‖)=limsup⁡n→∞Φ(‖xn-x‖)+Φ(‖y-x‖),∀x,y∈X.Lemma 2.2 (see [17]).
Assume that a Banach spaceX 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≤φ(1)∥A∥-1. Then ∥I-ρA∥≤φ(1)(1-ργ¯).Lemma 2.3 (see [11]).
LetC be a closed convex subset of a uniformly convex Banach space X and let 𝒮={T(t):t∈ℝ+} be a nonexpansive semigroup on C such that F(𝒮)≠∅. Then, for each r>0 and h≥0,
(2.7)lim⁡t→∞⁡sup⁡x∈C∩Br‖1t∫0tT(s)xds-T(h)(1t∫0tT(s)xds)‖=0.Lemma 2.4 (see [19]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.8)an+1≤(1-μn)an+δn,
where {μn} is a sequence in (0,1) and {δn} is a sequence in ℝ such that  (i)
∑n=0∞μn=∞;(ii)
limsupn→∞(δn/μn)≤0 or ∑n=0∞|δn|<∞. Then, limn→∞an=0.


## 3. Main Results

LetX 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={T(t):t∈ℝ+} 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¯φ(1)/α, and let t∈(0,1) such that t≤φ(1)∥A∥-1 which satisfies t→0. Define the mapping Ttf:C→C by (3.1)Ttf∶=tγf+(I-tA)1λt∫0λtT(s)ds
to be a contraction mapping. Indeed, for each x,y∈C,(3.2)‖Ttfx-Ttfy‖=‖tγ(f(x)-f(y))+(I-tA)(1λt∫0λt(T(s)x-T(s)y)ds)‖≤tγ‖f(x)-f(y)‖+‖I-tA‖(1λt∫0λt‖T(s)x-T(s)y‖ds)≤tγα‖x-y‖+φ(1)(1-tγ¯)‖x-y‖≤(1-t(φ(1)γ¯-γα))‖x-y‖.
Thus, by Banach contraction mapping principle, there exists a unique fixed point xt∈C, that is,(3.3)xt=tγf(xt)+(I-tA)1λt∫0λtT(s)xtds.Remark 3.1.
We note that spacelp has a weakly continuous duality mapping with a gauge function φ(t)=tp-1 for all 1<p<∞. This shows that φ is invariant on [0,1].Lemma 3.2.
LetX 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={T(t):t∈ℝ+} be a nonexpansive semigroup from C into itself such that F(𝒮)≠∅, 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¯φ(1)/α, and let t∈(0,1) such that t≤φ(1)∥A∥-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*∈F(𝒮), and is the unique solution of the variational inequality:
(3.4)〈γf(x*)-Ax*,Jφ(x-x*)〉≤0,∀x∈F(S).Proof.
Firstly, we show the uniqueness of a solution of the variational inequality (3.4). Suppose that x̃,x*∈F(𝒮) are solutions of (3.4); then
(3.5)〈γf(x*)-Ax*,Jφ(x̃-x*)〉≤0,〈γf(x̃)-Ax̃,Jφ(x*-x̃)〉≤0.
Adding up (3.5), we obtain
(3.6)0≥〈(γf(x*)-Ax*)-(γf(x̃)-Ax̃),Jφ(x̃-x*)〉=〈A(x̃-x*),Jφ(x̃-x*)〉-γ〈f(x̃)-f(x*),Jφ(x̃-x*)〉≥γ¯‖x̃-x*‖φ‖x̃-x*‖-γ‖f(x̃)-f(x*)‖‖Jφ(x̃-x*)‖≥γ¯Φ(‖x̃-x*‖)-γαΦ(‖x̃-x*‖)=(γ¯-γα)Φ(‖x̃-x*‖)≥(φ(1)γ¯-γα)Φ(‖x̃-x*‖),
which is a contradiction, we must have x̃=x*, and the uniqueness is proved. Here in after, we use x̃ to denote the unique solution of the variational inequality (3.4).
Next, we show that{xt} is bounded. Indeed, for each p∈F(𝒮), we have
(3.7)‖xt-p‖=‖t(γf(xt)-Ap)+(I-tA)(1λt∫0λt(T(s)xt-p)ds)‖≤t‖γf(xt)-Ap‖+‖I-tA‖1λt∫0λt‖T(s)xt-p‖ds≤tγ‖f(xt)-f(p)‖+t‖γf(p)-Ap‖+φ(1)(1-tγ¯)‖xt-p‖≤(1-t(φ(1)γ¯-γα))‖xt-p‖+t‖γf(p)-Ap‖.
It follows that
(3.8)‖xt-p‖≤1φ(1)γ¯-γα‖γf(p)-Ap‖.
Hence, {xt} is bounded, so are {f(xt)} and {A((1/λt)∫0λtT(s)xtds)}.
Next, we show that∥xt-T(h)xt∥→0 as t→0. We note that
(3.9)‖xt-1λt∫0λtT(s)xtds‖=t‖γf(xt)-A(1λt∫0λtT(s)xtds)‖.
Moreover, we note that
(3.10)‖xt-T(h)xt‖≤‖xt-1λt∫0λtT(s)xtds‖+‖1λt∫0λtT(s)xtds-T(h)(1λt∫0λtT(s)xtds)‖+‖T(h)(1λt∫0λtT(s)xtds)-T(h)xt‖≤2‖xt-1λt∫0λtT(s)xtds‖+‖1λt∫0λtT(s)xtds-T(h)(1λt∫0λtT(s)xtds)‖,
for all h≥0. Define the set K={z∈C:∥z-p∥≤∥γf(p)-Ap∥/(φ(1)γ¯-γα)}; then K is a nonempty bounded closed convex subset of C which is T(s)-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
(3.11)lim⁡λt→∞‖1λt∫0λtT(s)xtds-T(h)(1λt∫0λtT(s)xtds)‖=0,
for each h≥0. From (3.9)-(3.10), letting t→0 and noting (3.11) then, for each h≥0, we obtain
(3.12)‖xt-T(h)xt‖⟶0.
Assume that {tn}n=1∞⊂(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 x̃∈F(𝒮). 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 x̃∈C as j→∞. Again, since Jφ is weakly sequentially continuous, we have by Lemma 2.1 that
(3.13)limsup⁡j→∞Φ(‖xnj-z‖)=limsup⁡j→∞Φ(‖xnj-x̃‖)+Φ(‖z-x̃‖).
Let H(z)=limsupj→∞Φ(∥xnj-z∥), for all z∈C.
It follows thatH(z)=H(x̃)+Φ(∥z-x̃∥),  forallz∈C. From (3.12), we have
(3.14)H(T(h)x̃)=limsup⁡j→∞Φ(‖xnj-T(h)x̃‖)=limsup⁡j→∞Φ(‖T(h)xnj-T(h)x̃‖)≤limsup⁡j→∞Φ(‖xnj-x̃‖)=H(x̃).
On the other hand, we note that
(3.15)H(T(h)x̃)=limsup⁡j→∞Φ(‖xnj-x̃‖)+Φ(‖T(h)x̃-x̃‖)=H(x̃)+Φ(‖T(h)x̃-x̃‖).
Combining (3.14) with (3.15), we obtain Φ(∥T(h)x̃-x̃∥)≤0. This implies that T(h)x̃=x̃, that is, x̃∈F(𝒮). In fact, since Φ(t)=∫0tφ(τ)dτ, forallt≥0 and φ:ℝ+→ℝ+ is the gauge function, then for 1≥k≥0, φ(ky)≤φ(y) and
(3.16)Φ(kt)=∫0ktφ(τ)dτ=k∫0tφ(ky)dy≤k∫0tφ(y)dy=kΦ(t).
By Lemma 2.1, we have
(3.17)Φ(‖xn-x̃‖)=Φ(‖tn(γf(xn)-Ax̃)+(I-tA)(1λn∫0λtT(s)xnds-x̃)‖)≤Φ(‖(I-tnA)(1λn∫0λn(T(s)xn-x̃)ds)‖)+tn〈γf(xn)-Ax̃,Jφ(xn-x̃)〉≤Φ(φ(1)(1-tnγ¯)(‖xn-x̃‖))+tn〈γf(xn)-γf(x̃),Jφ(xn-x̃)〉+tn〈γf(x̃)-Ax̃,Jφ(xn-x̃)〉≤φ(1)(1-tnγ¯)Φ(‖xn-x̃‖)+tnγα‖xn-x̃‖‖Jφ(xn-x̃)‖+tn〈γf(x̃)-Ax̃,Jφ(xn-x̃)〉=φ(1)(1-tnγ¯)Φ(‖xn-x̃‖)+tnγαΦ(‖xn-x̃‖)+tn〈γf(x̃)-Ax̃,Jφ(xn-x̃)〉≤(1-tn(φ(1)γ¯-γα))Φ(‖xn-x̃‖)+tn〈γf(x̃)-Ax̃,Jφ(xn-x̃)〉.
This implies that
(3.18)Φ(‖xn-x̃‖)≤1φ(1)γ¯-γα〈γf(x̃)-Ax̃,Jφ(xn-x̃)〉.
In particular, we have
(3.19)Φ(‖xnj-x̃‖)≤1φ(1)γ¯-γα〈γf(x̃)-Ax̃,Jφ(xnj-x̃)〉.
Since the mapping Jφ is single-valued and weakly continuous, it follows from (3.19) that Φ(∥xnj-x̃∥)→0 as j→∞. This implies that xnj→x̃ as j→∞.
Next, we show thatx̃ solves the variational inequality (3.4), for each x∈F(𝒮). From (3.3), we derive that
(3.20)(γf-A)xt=-1t(I-tA)(1λt∫0λtT(s)xtds-xt).
Now, we observe that
(3.21)〈1λt∫0λt(I-T(s))xds-1λt∫0λt(I-T(s))xtds,Jφ(x-xt)〉=〈x-xt,Jφ(x-xt)〉-〈1λt∫0λt(T(s)x-T(s)xt)ds,Jφ(x-xt)〉≥‖x-xt‖‖Jφ(x-xt)‖-1λt∫0λt‖T(s)x-T(s)xt‖ds‖Jφ(x-xt)‖≥Φ(‖x-xt‖)-‖x-xt‖‖Jφ(x-xt)‖=Φ(‖x-xt‖)-Φ(‖x-xt‖)=0.
It follows from (3.20) that
(3.22)〈(γf-A)xt,Jφ(x-xt)〉=-1t〈(I-tA)(1λt∫0λtT(s)xtds-xt),Jφ(x-xt)〉=-1t〈(I-tA)(1λt∫0λtT(s)xtds-1λt∫0λtxtds),Jφ(x-xt)〉=-1t〈1λt∫0λt(I-T(s))xds-1λt∫0λt(I-T(s))xtds,Jφ(x-xt)〉+〈A(1λt∫0λt(T(s)-I)xtds),Jφ(x-xt)〉≤〈A(1λt∫0λt(T(s)-I))xtds),Jφ(x-xt)〉.
Now, replacing t and λt with tnj and λnj, respectively, in (3.22), and letting j→∞, and we notice that (T(s)-I)xnj→(T(s)-I)x̃=0 for x̃∈F(𝒮), we obtain that 〈(γf-A)x̃,Jφ(x-x̃)〉≤0. That is, x̃ is a solution of the variational inequality (3.4). By uniqueness, as x̃=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.
LetX 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={T(t):t∈ℝ+} be a nonexpansive semigroup from C into itself such that F(𝒮)≠∅, 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¯φ(1)/α. Let {αn}n=0∞,{βn}n=0∞,{γn}n=0∞ be the sequences in (0,1) and let {tn}n=0∞ be a positive real divergent sequence. Assume that the following conditions hold:  (C1)
limn→∞α_n=0 and ∑n=0∞αn=∞,(C2)
limn→∞γn=0,(C3)
βn=o(αn), Then the sequence {xn} defined by
(3.23)x0∈Cchosenarbitrarily;zn=γnxn+(1-γn)1tn∫0tnT(s)xnds;yn=αnγf(zn)+(I-αnA)zn;xn+1=βnxn+(1-βn)yn,∀n≥0,
converges strongly to the common fixed point x* that is obtained in Lemma 3.2.Proof.
From the condition (C1), we may assume, with no loss of generality, thatαn≤φ(1)∥A∥-1 for each n≥0. From Lemma 2.2, we have ∥I-αnA∥≤φ(1)(1-αnγ¯).
Firstly, we show that{xn} is bounded. Let p∈F(𝒮); we get
(3.24)‖zn-p‖=‖γn(xn-p)+(1-γn)1tn∫0tn(T(s)xn-p)ds‖≤γn‖xn-p‖+(1-γn)1tn∫0tn‖T(s)xn-p‖ds≤γn‖xn-p‖+(1-γn)‖xn-p‖=‖xn-p‖,‖yn-p‖=‖αn(γf(zn)-Ap)+(I-αnA)(zn-p)‖≤αnγ‖f(zn)-f(p)‖+αn‖γf(p)-Ap‖+‖I-αnA‖‖zn-p‖≤(1-αn(φ(1)γ¯-γα))‖xn-p‖+αn‖γf(p)-Ap‖.
It follows that
(3.25)‖xn+1-p‖=‖βn(xn-p)+(1-βn)(yn-p)‖≤βn‖xn-p‖+(1-βn)‖yn-p‖≤βn‖xn-p‖+(1-βn)((1-αn(φ(1)γ¯-γα))‖xn-p‖+αn‖γf(p)-Ap‖)=(1-αn(φ(1)γ¯-γα)(1-βn))‖xn-p‖+αn(φ(1)γ¯-γα)(1-βn)‖γf(p)-Ap‖φ(1)γ¯-γα≤max⁡{‖xn-p‖,‖γf(p)-Ap‖φ(1)γ¯-γα}.
By induction on n, we have
(3.26)‖xn-p‖≤max⁡{‖x0-p‖,‖γf(p)-Ap‖φ(1)γ¯-γα},∀n≥0.
Thus, {xn} is bounded. Since {xn} is bounded, then ∥(1/tn)∫0tnT(s)xnds-p∥≤∥xn-p∥ and {A((1/tn)∫0tnT(s)xnds)} and {f(zn)} are also bounded.
Next, we show thatlimn→∞∥xn-T(h)xn∥=0, forallh≥0. From (3.23), we note that
(3.27)‖xn+1-1tn∫0tnT(s)xnds‖≤‖xn+1-yn‖+‖yn-zn‖+‖zn-1tn∫0tnT(s)xnds‖≤βn‖xn-yn‖+αn‖γf(zn)-Azn‖+γn‖xn-1tn∫0tnT(s)xnds‖.
By the conditions (C1)–(C3), then (3.27), we obtain
(3.28)lim⁡n→∞‖xn+1-1tn∫0tnT(s)xnds‖=0.
Moreover, we note that
(3.29)‖xn+1-T(h)xn+1‖≤‖xn+1-1tn∫0tnT(s)xnds‖+‖1tn∫0tnT(s)xnds-T(h)(1tn∫0tnT(s)xnds)‖+‖T(h)(1tn∫0tnT(s)xnds)-T(h)xn+1‖≤2‖xn+1-1tn∫0tnT(s)xnds‖+‖1tn∫0tnT(s)xnds-T(h)(1tn∫0tnT(s)xnds)‖.
Define the set K={z∈C:∥z-p∥≤∥x0-p∥+∥γf(p)-Ap∥/(φ(1)γ¯-γα)}. Then K is a nonempty bounded closed convex subset of C, which is T(s)—invariant for each s≥0 and contains {xn}; it follows from Lemma 2.3 that
(3.30)lim⁡n→∞‖1tn∫0tnT(s)xnds-T(h)(1tn∫0tnT(s)xnds)‖=0,∀h≥0.
Then, for all h≥0, from (3.28) and (3.30), into (3.29), we obtain limn→∞∥xn+1-T(h)xn+1∥=0, and hence
(3.31)lim⁡n→∞‖xn-T(h)xn‖=0,∀h≥0.
Next, we show that limsupn→∞〈γf(x*)-Ax*,Jφ(xn-x*)〉≤0. We can take subsequence {xnj}⊂{xn} such that
(3.32)lim⁡j→∞〈γf(x*)-Ax*,Jφ(xnj-x*)〉=limsup⁡n→∞〈γf(x*)-Ax*,Jφ(xn-x*)〉.
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
(3.33)limsup⁡j→∞Φ(‖xnj-z‖)=limsup⁡j→∞Φ(‖xnj-x‖)+Φ(‖z-x‖),∀z∈C.
Let H(z)=limsupj→∞Φ(∥xnj-z∥), forallz∈C.
It follows thatH(z)=H(x)+Φ(∥z-x∥), forallz∈C.
From (3.31), we have
(3.34)H(T(h)x)=limsup⁡j→∞Φ(‖xnj-T(h)x‖)=limsup⁡j→∞Φ(‖T(h)xnj-T(h)x‖)≤limsup⁡j→∞Φ(‖xnj-x‖)=H(x).
On the other hand, we note that
(3.35)H(T(h)x)=limsup⁡j→∞Φ(‖xnj-x‖)+Φ(‖T(h)x-x‖)=H(x)+Φ(‖T(h)x-x‖).
Combining (3.34) with (3.35), we obtain Φ(∥T(h)x-x∥)≤0.
This implies thatT(h)x=x; that is, x∈F(𝒮).
Since the duality mapJφ is single-valued and weakly continuous, we get that
(3.36)limsup⁡n→∞〈γf(x*)-Ax*,Jφ(xn-x*)〉=lim⁡j→∞〈γf(x*)-Ax*,Jφ(xnj-x*)〉=〈γf(x*)-Ax*,Jφ(x-x*)〉≤0,
as required. Hence,
(3.37)limsup⁡n→∞〈γf(x*)-Ax*,Jφ(xn+1-x*)〉≤0.
Since ∥xn+1-yn∥=βn∥xn-yn∥, by condition (C3), we obtain that limn→∞∥xn+1-yn∥=0. It follows from (3.37), that
(3.38)limsup⁡n→∞〈γf(x*)-Ax*,Jφ(yn-x*)〉≤0.
Finally, we show that xn→x* as n→∞. Now, from Lemma 2.1, we have
(3.39)Φ(‖yn-x*‖)=Φ(‖αn(γf(zn)-Ax*)+(I-αnA)(zn-x*)‖)=Φ(‖αn(γf(zn)-γf(x*))+αn(γf(x*)-Ax*)+(I-αnA)(zn-x*)‖)≤Φ(‖αn(γf(zn)-γf(x*))+(I-αnA)(zn-x*)‖)+αn〈γf(x*)-Ax*,Jφ(yn-x*)〉≤(1-αn(φ(1)γ¯-γα))Φ(‖xn-x*‖)+αn〈γf(x*)-Ax*,Jφ(yn-x*)〉.
On the other hand, we note that
(3.40)Φ(‖xn+1-x*‖)=Φ(‖βn(xn-x*)+(1-βn)(yn-x*)‖)≤(1-βn)Φ(‖yn-x*‖)+βn〈xn-x*,Jφ(xn+1-x*)〉.
It follows from (3.40) that
(3.41)Φ(‖xn+1-x*‖)≤(1-αn(φ(1)γ¯-γα))Φ(‖xn-x*‖)+αn〈γf(x*)-Ax*,Jφ(yn-x*)〉+βn〈xn-x*,Jφ(xn+1-x*)〉≤(1-αn(φ(1)γ¯-γα))Φ(‖xn-x*‖)+αn[〈γf(x*)-Ax*,Jφ(yn-x*)〉+βnαnM],
where M=supn≥0{∥xn-x*∥φ(∥xn+1-x*∥)}.
Putμn∶=αn(φ(1)γ¯-γα) and δn∶=αn[〈γf(x*)-Ax*,Jφ(yn-x*)〉+(βn/αn)M]. Then (3.41) reduces to formula Φ(∥xn+1-x*∥)≤(1-μn)Φ(∥xn-x*∥)+δn. By conditions (C1)  and (C3) and noting (3.38), it is easy to see that ∑n=0∞μn=∞ and limsupn→∞(δn/μn)=limsupn→∞(1/φ(1)γ¯-γα)[〈γf(x*)-Ax*,Jφ(yn-x*)〉+(βn/αn)M]≤0. Applying Lemma 2.4, we obtain Φ(∥xn-x*∥)→0 as n→∞ this implies that xn→x* as n→∞. This completes the proof.Takingγn=0 in (3.23), we can get the following corollary easily.Corollary 3.4.
LetX 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={T(t):t∈ℝ+} be a nonexpansive semigroup from C into itself such that F(𝒮)≠∅, 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¯φ(1)/α. Let {αn}n=0∞,{βn}n=0∞ be the sequences in (0,1) and {tn}n=0∞ be a positive real divergent sequence. Assume the following conditions are hold:  (C1)
limn→∞αn=0 and ∑n=0∞αn=∞;(C2)
βn=o(αn). Then the sequence {xn} defined by
(3.42)x0∈Cchosenarbitrarily,yn=αnγf(1tn∫0tnT(s)xnds)+(I-αnA)1tn∫0tnT(s)xnds,xn+1=βnxn+(1-βn)yn,∀n≥0,
converges strongly to the common fixed point x*, in which x*∈F(𝒮) is the unique solution of the variational inequality:
(3.43)〈γf(x*)-Ax*,Jφ(x-x*)〉≤0,∀x∈F(S).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.
LetH 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 F(T)≠∅, 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¯φ(1)/α. Let {αn}n=0∞,{βn}n=0∞,and{γn}n=0∞ be the sequences in (0,1) and let {tn}n=0∞ be a positive real divergent sequence. Assume that the following conditions are hold:  (C1)
limn→∞αn=0 and ∑n=0∞αn=∞;(C2)
limn→∞γn=0;(C3)
βn=o(αn). Then the sequence {xn} defined by
(3.44)x0∈Cchosenarbitrarily,zn=γnxn+(1-γn)1n+1∑j=0nTjxn,yn=αnγf(zn)+(I-αnA)zn,xn+1=βnxn+(1-βn)yn,∀n≥0,
converges strongly to the common fixed point x*, in which x*∈F(T) is the unique solution of the variational inequality:
(3.45)〈γf(x*)-Ax*,x-x*〉≤0,∀x∈F(T).



---

*Source: 560671-2011-05-22.xml*