# Implicit and Explicit Iterations with Meir-Keeler-Type Contraction  for a Finite Family of Nonexpansive Semigroups  in Banach Spaces

Authors: Jiancai Huang; Shenghua Wang; Yeol Je Cho
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce an implicit and explicit iterative schemes for a finite family of nonexpansive semigroups with the Meir-Keeler-type contraction in a Banach space. Then we prove the strong convergence for the implicit and explicit iterative schemes. Our results extend and improve some recent ones in literatures.


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## Body

## 1. Introduction

LetC be a nonempty subset of a Banach space E and T:C→C be a mapping. We call T nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈E. The set of all fixed points of T is denoted by Fix(T), that is, Fix(T)={x∈C:x=Tx}.One parameter family𝒯={T(t):t≥0} is said to a semigroup of nonexpansive mappings or nonexpansive semigroup on C if the following conditions are satisfied: (1)
T(0)x=x for all x∈C;(2)
T(s+t)=T(s)T(t) for all s,t≥0;(3)
for eacht≥0, ∥T(t)x-T(t)y∥≤∥x-y∥ for all x,y∈C;(4)
for eachx∈C, the mapping T(·)x from ℝ+, where ℝ+ denotes the set of all nonnegative reals, into C is continuous.We denote byFix(𝒯) the set of all common fixed points of semigroup 𝒯, that is, Fix(𝒯)={x∈C:T(t)x=x,0≤t<∞} and ℕ by the set of natural numbers.Now, we recall some recent work on nonexpansive semigroup in literatures. In [1], Shioji and Takahashi introduced the following implicit iteration for a nonexpansive semigroup in a Hilbert space:(1.1)xn=αnx+(1-αn)1tn∫0tnT(s)xnds,∀n∈N,
where {αn}⊂(0,1) and {tn}⊂(0,∞). Under the certain conditions on {αn} and {tn}, they proved that the sequence {xn} defined by (1.1) converges strongly to an element in Fix(𝒯).In [2], Suzuki introduced the following implicit iteration for a nonexpansive semigroup in a Hilbert space:(1.2)xn=αnu+(1-αn)T(tn)xn,∀n∈N,
where {αn}⊂(0,1) and {tn}⊂(0,∞). Under the conditions that limn→∞tn=limn→∞αn/tn=0, he proved that {xn} defined by (1.2) converges strongly to an element of Fix(𝒯). Later on, Xu [3] extended the iteration (1.2) to a uniformly convex Banach space that admits a weakly sequentially continuous duality mapping. Song and Xu [4] also extended the iteration (1.2) to a reflexive and strictly convex Banach space.In 2007, Chen and He [5] studied the following implicit and explicit viscosity approximation processes for a nonexpansive semigroup in a reflexive Banach space admitting a weakly sequentially continuous duality mapping:(1.3)xn=αnf(xn)+(1-αn)T(tn)xn,yn+1=βnf(yn)+(1-βn)T(tn)yn,∀n∈N,
where f is a contraction, {αn}⊂(0,1) and {tn}⊂(0,∞). They proved the strong convergence for the above iterations under some certain conditions on the control sequences.Recently, Chen et al. [6] introduced the following implicit and explicit iterations for nonexpansive semigroups in a reflexive Banach space admitting a weakly sequentially continuous duality mapping:(1.4)yn=αnxn+(1-αn)T(tn)xn,xn=βnf(xn)+(1-βn)yn,∀n∈N,(1.5)yn=αnxn+(1-αn)T(tn)xn,xn+1=βnf(xn)+(1-βn)yn,∀n∈N,
where f is a contraction, {αn}⊂(0,1) and {tn}⊂(0,∞). They proved that {xn} defined by (1.4) and (1.5) converges strongly to an element q of Fix(𝒯), which is the unique solution of the following variation inequality problem:(1.6)〈(f-I),j(x-q)〉≤0,∀x∈Fix⁡(T).For more convergence theorems on implicit and explicit iterations for nonexpansive semigroups, refer to [7–13].In this paper, we introduce an implicit and explicit iterative process by a generalized contraction for a finite family of nonexpansive semigroups in a Banach space. Then we prove the strong convergence for the iterations and our results extend the corresponding ones of Suzuki [2], Xu [3], Chen and He [5], and Chen et al. [6].


## 2. Preliminaries

LetE be a Banach space and E* the duality space of E. We denote the normalized mapping from E to 2E* by J defined by(2.1)J(x)={j∈E*:〈x,jx〉=‖x‖2=‖j‖},∀x∈E,
where 〈·,·〉 denotes the generalized duality pairing. For any x,y∈E with j(x)∈J(x) and j(x+y)∈J(x+y), it is well known that the following inequality holds:
(2.2)‖x‖2+2〈y,j(x)〉≤‖x+y‖2≤‖x‖2+2〈y,j(x+y)〉.The dual mappingJ is called weakly sequentially continuous if J is single valued, and {xn}⇀x∈E, where ⇀ denotes the weak convergence, then J(xn) weakly star converges to J(x) [14–16]. A Banach space E is called to satisfy Opial’s condition [17] if for any sequence {xn} in E, xn⇀x, (2.3)limsup⁡n→∞‖xn-x‖<limsup⁡n→∞‖xn-y‖,∀y∈Ewithx≠y.
It is known that if E admits a weakly sequentially continuous duality mapping J, then E is smooth and satisfies Opial’s condition [14].A functionψ:ℝ+→ℝ+ is said to be an L-function if ψ(0)=0, ψ(t)>0 for any t>0, and for every t>0 and s>0, there exists u>s such that ψ(t)≤s, for all t∈[s,u]. This implies that ψ(t)<t for all t>0.Letf:C→C be a mapping. f is said to be a (ψ,L)-contraction if there exists a L-function ψ:ℝ+→ℝ+ such that ∥f(x)-f(y)∥<ψ(∥x-y∥) for all x,y∈C with x≠y. Obviously, if ψ(t)=kt for all t>0, where k∈(0,1), then f is a contraction. f is called a Meir-Keeler-type mapping if for each ϵ>0, there exists δ(ϵ)>0 such that for all x,y∈C, if ϵ<∥x-y∥<ϵ+δ, then ∥f(x)-f(y)∥<ϵ.In this paper, we always assume thatψ(t) is continuous, strictly increasing and limt→∞η(t)=∞, where η(t)=t-ψ(t), is strictly increasing and onto.The following lemmas will be used in next section.Lemma 2.1 (see [18]).
Let(X,d) be a metric space and  f:X→X be a mapping. The following assertions are equivalent: (i)
f is a Meir-Keeler-type mapping;(ii)
there exists anL-function  ψ:ℝ+→ℝ+ such that  f is a  (ψ,L)-contraction.Lemma 2.2 (see [19]).
LetE be a Banach space and  C be a convex subset of  E. Let  T:C→C be a nonexpansive mapping and  f be a  (ψ,L)-contraction. Then the following assertions hold: (i)
T∘f is a  (ψ,L)-contraction on  C and has a unique fixed point in  C;(ii)
for eachα∈(0,1), the mapping  x↦αf(x)+(1-α)Tx is of Meir-Keeler-type and it has a unique fixed point in  C.Lemma 2.3 (see [20]).
LetE be a Banach space and  C be a convex subset of  E. Let  f:C→C be a Meir-Keeler-type contraction. Then for each  ϵ>0 there exists  r∈(0,1) such that, for each  x,y∈C with  ∥x-y∥≥ϵ,∥f(x)-f(y)∥≤r∥x-y∥.Lemma 2.4 (see [21]).
Let C be a closed convex subset of a strictly convex Banach space E. LetTm:C→C be a nonexpansive mapping for each  1≤m≤r, where  r is some integer. Suppose that  ∩m=1rFix(Tm) is nonempty. Let  {λn} be a sequence of positive numbers with  ∑n=1rλn=1. Then the mapping  S:C→C defined by
(2.4)Sx=∑m=1rλmTmx,∀x∈C,
is well defined, nonexpansive and Fix(S)=∩m=1rFix(Tm) holds.Lemma 2.5 (see [22]).
Assume that{αn} is a sequence of nonnegative real numbers such that
(2.5)αn+1≤(1-γn)αn+δn,n∈N,
where {γn} is a sequence in (0,1) and {δn} is a sequence in ℝ such that(i)
limn→∞γn=0;(ii)
∑n=1∞γn=∞;(iii)
limsupn→∞δn/γn≤0 or ∑n=1∞|δn|<∞.Then limn→∞αn=0.


## 3. Main Results

In this section, by a generalized contraction mapping we mean a Meir-Keeler-type mapping or(ψ,L)- contraction. In the rest of the paper we suppose that ψ from the definition of the (ψ,L)-contraction is continuous, strictly increasing and η(t) is strictly increasing and onto, where η(t)=t-ψ(t), for all t∈ℝ+. As a consequence, we have the η(t) is a bijection on ℝ+.Theorem 3.1.
LetC be a nonempty closed convex subset of a reflexive Banach space  E which admits a weakly sequentially continuous duality mapping  J from  E into  E*. For every  i=1,…,N(N≥1), let  𝒯i={Ti(t):t≥0} be a semigroup of nonexpansive mappings on  C such that  ℱ=∩i=1NFix(𝒯i)≠∅ and  f:C→C be a generalized contraction on  C. Let  {αn},{βn}⊂[0,1) and  {tn}⊂(0,∞) be the sequences satisfying  limn→∞tn=limn→∞(αn/tn)=0 and  limsupn→∞βn<1. Let  {xn} be a sequence generated by
(3.1)xn=αnf(xn)+1-αnN∑i=1Nyin,yin=βnxn+(1-βn)Ti(tn)xn,i=1,…,N.
Then {xn} converges strongly to a point x*∈ℱ, which is the unique solution to the following variational inequality:
(3.2)〈(f-I)x*,j(x-x*)〉≤0,∀x∈F.Proof.
First, we show that the sequence{xn} generated by (3.1) is well defined. For every n∈ℕ and i=1,…,N, let Uin=βnI+(1-βn)Ti(tn) and define Wn:C→C by
(3.3)Wnx=αnf(x)+(1-αn)Gnx,∀x∈C,
where Gnx=(1/N)∑i=1NUinx. Since Uin is nonexpansive, Gn is nonexpansive. By Lemma 2.2 we see that Wn is a Meir-Keeler-type contraction for each n∈ℕ. Hence, each Wn has a unique fixed point, denoted as xn, which uniquely solves the fixed point equation (3.3). Hence {xn} generated by (3.1) is well defined.
Now we prove that{xn} generated by (3.1) is bounded. For any p∈ℱ, we have
(3.4)‖yin-p‖≤βn‖xn-p‖+(1-βn)‖Ti(tn)xn-p‖≤‖xn-p‖.
Using (3.4), we get
(3.5)∥xn-p∥2=〈αnf(xn)+1-αnN∑i=1Nyin-p,j(xn-p)〉=αn〈f(xn)-f(p),j(xn-p)〉+αn〈f(p)-p,j(xn-p)〉+1-αnN∑i=1N〈yin-p,j(xn-p)〉≤αnψ(‖xn-p‖)‖xn-p‖+αn‖f(p)-p‖‖xn-p‖+1-αnN∑i=1N‖yin-p‖‖xn-p‖=αnψ(‖xn-p‖)‖xn-p‖+αn‖f(p)-p‖‖xn-p‖+(1-αn)‖xn-p‖2
and hence
(3.6)‖xn-p‖≤ψ(‖xn-p‖)+‖f(p)-p‖,
which implies that
(3.7)η(‖xn-p‖)=‖xn-p‖-ψ(‖xn-p‖)≤‖f(p)-p‖.
Hence
(3.8)‖xn-p‖≤η-1(‖f(p)-p‖).
This shows that {xn} is bounded, and so are {Ti(tn)xn}, {f(xn)} and {yin}.
SinceE is reflexivity and {xn} is bounded, there exists a subsequence {xnj}⊂{xn} such that xnj⇀x* for some x*∈C as j→∞. Now we prove that x*∈ℱ. For any fixed t>0, we have
(3.9)∑i=1N‖xnj-Ti(t)x*‖≤∑i=1N[∑k=0[t/tni]-1‖Ti((k+1)tnj)xnj-Ti(ktnj)xnj‖+‖Ti([ttnj]tnj)xnj-Ti([ttnj]tnj)x*‖+‖Ti([ttnj]tnj)xnj-Ti(t)x*‖]≤∑i=1N[[ttnj]‖Ti(tnj)xnj-xnj‖+‖xnj-x*‖+‖Ti(t-[ttnj]tnj)xnj-x*‖]≤∑i=1N[[ttnj]‖Ti(tnj)xnj-xnj‖+‖xnj-x*‖+max⁡{‖Ti(s)x*-x*‖:0≤s≤tnj}]≤Nαnj[t/tnj](1-αnj)((1-βnj))‖xnj-f(xnj)‖+N‖xnj-x*‖+∑i=1Nmax⁡{‖Ti(s)x*-x*‖:0≤s≤tnj}≤Nt(1-αnj)(1-βnj)αnjtnj‖xnj-f(xnj)‖+N‖xnj-x*‖+∑i=1Nmax⁡{‖Ti(s)x*-x*‖:0≤s≤tnj}.
By hypothesis on {tn},{αn}, {βn}, we have
(3.10)lim⁡j→∞Nt(1-αnj)(1-βnj)αnjtnj=0.
Further, from (3.9) we get
(3.11)limsup⁡j→∞∑i=1N‖xnj-Ti(t)x*‖≤limsup⁡j→∞N‖xnj-x*‖.
Since E admits a weakly sequentially duality mapping, we see that E satisfies Opial’s condition. Thus if x*∉ℱ, we have
(3.12)limsup⁡j→∞N‖xnj-x*‖<limsup⁡j→∞∑i=1N‖xnj-Tix*‖.
This contradicts (3.11). So x*∈ℱ.
In (3.5), replacing p with x* and n with nj, we see that
(3.13)‖xnj-x*‖2=αnj〈f(xnj)-f(x*),j(xnj-x*)〉+αnj〈f(x*)-x*,j(xnj-x*)〉+1-αnjN∑i=1N〈yinj-x*,j(xnj-x*)〉≤αnjψ(‖xnj-x*‖)‖xnj-x*‖+αnj〈f(x*)-x*,j(xnj-x*)〉+1-αnjN∑i=1N‖yinj-x*‖‖xnj-x*‖≤αnjψ(‖xnj-x*‖)‖xnj-x*‖+αnj〈f(x*)-x*,j(xnj-x*)〉+(1-αnj)‖xn-p‖2,
which implies that
(3.14)‖xnj-x*‖(ψ(‖xnj-x*‖)-‖xnj-x*‖)≤〈f(x*)-x*,j(xnj-x*)〉.
Now we prove that {xn} is relatively sequentially compact. Since j is weakly sequentially continuous, we have
(3.15)lim⁡j→∞‖xnj-x*‖(ψ(‖xnj-x*‖)-‖xnj-x*‖)≤0,
which implies that
(3.16)lim⁡j→∞‖xnj-x*‖=0,orlim⁡j→∞(ψ(‖xnj-x*‖)-‖xnj-x*‖)=0.
If limj→∞∥xnj-x*∥=0, then {xn} is relatively sequentially compact. If limj→∞(ψ(∥xnj-x*∥)-∥xnj-x*∥)=0, we have limj→∞∥xnj-x*∥=limj→∞ψ(∥xnj-x*∥). Since ψ is continuous, limj→∞∥xnj-x*∥=ψ(limj→∞∥xnj-x*∥). By the definition of ψ, we conclude that limj→∞∥xnj-x*∥=0, which implies that {xn} is relatively sequentially compact.
Next, we prove thatx* is the solution to (3.2). Indeed, for any x∈ℱ, we have
(3.17)‖xn-x‖2=〈αn(f(xn)-xn+xn-x),j(xn-x)〉+1-αnN∑i=1N〈yin-x,j(xn-x)〉=αn〈f(xn)-xn,j(xn-x)〉+αn〈xn-x,j(xn-x)〉+1-αnN∑i=1N[βn〈xn-x,j(xn-x)〉+(1-βn)〈Ti(tn)xn-x,j(xn-x*)〉]≤αn〈f(xn)-xn,j(xn-x)〉+αn‖xn-x‖2+1-αnN∑i=1N[βn‖xn-x‖2+(1-βn)‖Ti(tn)xn-x‖‖xn-x‖]≤αn〈f(xn)-xn,j(xn-x)〉+αn‖xn-x‖2+1-αnN∑i=1N[βn‖xn-x‖2+(1-βn)‖xn-x‖2]=αn〈f(xn)-xn,j(xn-x)〉+‖xn-x‖2.
Therefore,
(3.18)〈f(xn)-xn,j(x-xn)〉≤0.
Since xnj⇀x* and j is weakly sequentially continuous, we have
(3.19)〈f(x*)-x*,j(x-x*)〉=lim⁡j→∞〈f(xnj)-xnj,j(x-xnj)〉≤0.
This shows that x* is the solution of the variational inequality (3.2).
Finally, we prove thatx* is the unique solution of the variational inequality (3.2). Assume that x̂∈ℱ with x̂≠x* is another solution of (3.2). Then there exists ϵ>0 such that ∥x̂-x*∥>ϵ. By Lemma 2.3 there exists r∈(0,1) such that ∥f(x̂)-f(x*)∥≤r∥x̂-x*∥. Since both x̂ and x* are the solution of (3.2), we have
(3.20)〈f(x*)-x*,j(x̂-x*)〉≤0,〈f(x̂)-x̂,j(x*-x̂)〉≤0.
Adding the above inequalities, we get
(3.21)0<(1-r)ϵ2<(1-r)‖x̂-x*‖2≤〈(I-f)x*-(I-f)x̂〉,j(x*-x̂)≤0,
which is a contradiction. Therefore, we must have x̂=x*, which implies that x* is the unique solution of (3.2).
In a similar way it can be shown that each cluster point of sequence{xn} is equal to x*. Therefore, the entire sequence {xn} converges strongly to x*. This completes the proof.If lettingβn=0 for all n∈ℕ in Theorem 3.1, then we get the following.Corollary 3.2.
LetC be a nonempty closed convex subset of a reflexive Banach space  E which admits a weakly sequentially continuous duality mapping  J from  E into  E*. For every  i=1,…,N ( N≥1), let  𝒯i={Ti(t):t≥0} be a semigroup of nonexpansive mappings on  C such that  ℱ=∩i=1NFix(𝒯i)≠∅ and  f:C→C be a generalized contraction on  C. Let  {αn}⊂[0,1) and  {tn}⊂(0,∞) be sequences satisfying  limn→∞tn=limn→∞(αn/tn)=0. Let  {xn} be a sequence generated by
(3.22)xn=αnf(xn)+1-αnN∑i=1NTi(tn)xn.
Then {xn} converges strongly to a point x*∈ℱ, which is the unique solution to the following variational inequality:
(3.23)〈(f-I)x*,j(x-x*)〉≤0,∀x∈F.Theorem 3.3.
LetC be a nonempty closed convex subset of a reflexive and strictly convex Banach space  E which admits a weakly sequentially continuous duality mapping  J from  E into  E*. For every  i=1,⋯,N(N≥1), let  𝒯i={Ti(t):t≥0} be a semigroup of nonexpansive mappings on  C such that  ℱ=∩i=1NFix(𝒯i)≠∅ and  f:C→C be a generalized contraction on  C. Let  {αn},{βn}⊂[0,1) and  {tn}⊂(0,∞) be the sequences satisfying  limn→∞tn=limn→∞(βn/tn)=0. Let  {xn} be a sequence generated
(3.24)yin=αnxn+(1-αn)Ti(tn)xn,i=1,…,N,xn+1=βnf(xn)+1-βnN∑i=1Nyin,∀n∈N.
Then {xn} converges strongly to a point x*∈ℱ, which is the unique solution of variational inequality (3.2).Proof.
Letp∈ℱ and M=max{∥x1-p∥,η-1(∥f(p)-p∥}. Now we show by induction that
(3.25)‖xn-p‖≤M,∀n∈N.
It is obvious that (3.25) holds for n=1. Suppose that (3.25) holds for some n=k, where k>1. Observe that
(3.26)‖yik-p‖=‖αk(xk-p)+(1-αk)(Ti(tk)xk-p)‖≤αk‖xk-p‖+(1-αk)‖Ti(tk)xk-p‖≤‖xk-p‖.
Now, by using (3.24) and (3.26), we have
(3.27)∥xk+1-p∥=‖βk(f(xk)-p)+1-βkN∑i=1N(yik-p)‖≤βk‖f(xk)-f(p)‖+βk‖f(p)-p‖+1-βkN∑i=1N‖yik-p‖≤βkψ(‖xk-p‖)+βk‖f(p)-p‖+1-βkN∑i=1N‖xk-p‖=βkψ(‖xk-p‖)+βk‖f(p)-p‖+(1-βk)‖xk-p‖=βkψ(‖xk-p‖)+βkη(η-1‖f(p)-p‖)+(1-βk)‖xk-p‖≤βkψ(M)+βkη(M)+(1-βk)M=βkψ(M)+βk(M-ψ(M))+(1-βk)M=M.
By induction we conclude that (3.25) holds for all n∈ℕ. Therefore, {xn} is bounded and so are {f(xn)}, {yin}, {Ti(tn)xn}.
For eachi=1,…,N and n∈ℕ, define the mapping U(tn)=(1/N)∑i=1NSi(tn), where Si(tn)=αnI+(1-αn)Ti(tn). Then we rewrite the sequence (3.24) to
(3.28)xn+1=βnf(xn)+(1-βn)U(tn)xn.
Obviously, each U(tn) is nonexpansive. Since {xn} is bounded and E is reflexive, we may assume that some subsequence {xnj} of {xn} converges weakly to p. Next we show that p∈ℱ. Put xj=xnj, βj=βnj, and tj=tnj for each j∈ℕ. Fix t>0. By (3.28) we have
(3.29)‖xj-U(t)p‖=∑k=0[t/tj]-1‖U((k+1)tj)xj-U(ktj)xj‖+‖U([ttj]tj)xj-U([ttj]tj)p‖+‖U([ttj]tj)p-U(t)p‖≤[ttj]‖U(tj)xj-xj+1‖+‖xj+1-p‖+‖U(t-[ttj]tj)p-p‖=[ttj]βj‖U(tj)xj-f(xj)‖+‖xj+1-p‖+‖U(t-[ttj]tj)p-p‖≤tβjtj‖U(tj)xj-f(xj)‖+‖xj+1-p‖+max⁡{‖U(s)p-p‖:0≤s≤tj}.
So, for all j∈ℕ, we have
(3.30)limsup⁡j→∞‖xj-U(t)p‖≤limsup⁡j→∞‖xj+1-p‖=limsup⁡j→∞‖xj-p‖.
SinceE has a weakly sequentially continuous duality mapping satisfying Opials’ condition, this implies p=U(t)p. By Lemma 2.4, we have Fix(U(t))=∩i=1NFix(Ti(t)) for each t>0. Therefore, p∈ℱ. In view of the variational inequality (3.2) and the assumption that duality mapping J is weakly sequentially continuous, we conclude that
(3.31)limsup⁡n→∞〈(f-I)q,j(xn+1-q)〉=lim⁡j→∞〈(f-I)q,j(xnj+1-q)〉=〈(I-f)q,j(p-q)〉≤0.
Finally, we prove thatxn→q as n→∞. Suppose that ∥xn-q∥↛0. Then there exists ϵ>0 and subsequence {xnj} of {xn} such that ∥xnj-q∥≥ϵ for all j∈ℕ. Put xj=xnj, βj=βnj and tj=tnj. By Lemma 2.3 one has ∥f(xj)-f(q)∥≤r∥xj-q∥ for all j∈ℕ. Now, from (2.2) and (3.28) we have
(3.32)∥xj+1-q∥2=‖(1-βn)(U(tj)xj-q)+βn(f(xj)-q)‖2≤(1-βj)2‖U(tj)xj-q‖2+2βj〈f(xj)-q,j(xj+1-q)〉≤(1-βj)2‖xj-q‖2+2βn〈f(xj)-f(q),j(xj+1-q)〉+2βj〈f(q)-q,j(xj+1-q)〉≤(1-βj)2‖xj-q‖2+2βjr‖xj-q‖‖xj+1-q‖+2βn〈f(q)-q,j(xj+1-q)〉≤(1-βj)2‖xj-q‖2+βjr(‖xj-q‖2+‖xj+1-q‖2)+2βj〈f(q)-q,j(xj+1-q)〉=((1-βj)2+βjr)‖xj-q‖2+βjr‖xj+1-q‖2+2βj〈f(q)-q,j(xj+1-q)〉.
It follows that
(3.33)‖xj+1‖≤1-(2-r)βj+βj21-βjr‖xj-q‖2+2βj1-βjr〈f(q)-q,j(xj+1-q)〉≤1-βjr-2(1-r)βj1-βjr‖xj-q‖2+2βj1-βjr〈f(q)-q,j(xj+1-q)〉+βj2M=(1-2(1-r)βj1-βjr)‖xj-q‖2+2βj1-βjr〈f(q)-q,j(xj+1-q)〉+βj2M≤(1-2(1-r)βj)‖xj-q‖2+βj(21-r〈f(q)-q,j(xj+1-q)〉+βjM),
where M is a constant.
Letγj=2(1-r)βj and δj=βj((2/(1-r))〈f(q)-q,j(xj+1-q)〉+βjM). It follows from (3.33) that
(3.34)‖xj+1-q‖≤(1-γj)‖xj-q‖+δj.
It is easy to see that γj→0, ∑j=1∞γj=∞ and (noting (3.28))
(3.35)limsup⁡j→∞δjγj=limsup⁡1(1-r)2〈f(q)-q,j(xj+1-q)〉+M2(1-r)βj,limsup⁡n→∞1(1-r)2〈f(q)-q,j(xj+1-q)〉≤0.
Using Lemma 2.5, we conclude that ∥xj-q∥→0 as j→∞. It is a contradiction. Therefore, xn→q as n→∞. This completes the proof.If lettingαn=0 for all n∈ℕ in Theorem 3.3, then we get the following.Corollary 3.4.
LetC be a nonempty closed convex subset of a reflexive and strictly convex Banach space  E which admits a weakly sequentially continuous duality mapping  J from  E into  E*. For every  i=1,…,N(N≥1), let  𝒯i={Ti(t):t≥0} be a semigroup of nonexpansive mappings on  C such that  ℱ=∩i=1NFix(𝒯i)≠∅ and  f:C→C be a generalized contraction on  C. Let  {βn}⊂[0,1) and  {tn}⊂(0,∞) be sequences satisfying  limn→∞tn=limn→∞(βn/tn)=0. Let  {xn} be a sequence generated
(3.36)xn+1=βnf(xn)+1-βnN∑i=1NTi(tn)xn,∀n∈N.
Then {xn} converges strongly to a point x*∈ℱ, which is the unique solution of variational inequality (3.2).Remark 3.5.
Theorem3.1 and Corollary 3.2 extend the corresponding ones of Suzuki [2], Xu [3], and Chen and He [5] from one nonexpansive semigroup to a finite family of nonexpansive semigroups. But Theorem 3.3 and Corollary 3.4 are not the extension of Theorem 3.2 of Chen and He [5] since Banach space in Theorem 3.3 and Corollary 3.4 is required to be strictly convex. But if letting N=1 in Theorem 3.3 and Corollary 3.4, we can remove the restriction on strict convexity and hence they extend Theorem 3.2 of Chen and He [5] from a contraction to a generalized contraction.Remark 3.6.
Our Theorem3.1 extends and improves Theorems 3.2 and 4.2 of Song and Xu [4] from a nonexpansive semigroup to a finite family of nonexpansive semigroups and a contraction to a generalized contraction. Our conditions on the control sequences are different with ones of Song and Xu [4].



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*Source: 720192-2012-04-11.xml*