# Convergence Theorem for a Family of Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces

Authors: Bashir Ali; G. C. Ugwunnadi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

LetE be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let 𝔍={T(t):t≥0} be a family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup of E, with functions u,v:[0,∞)→[0,∞). Let F:=F(𝔍)=∩t≥0F(T(t))≠∅ and f:K→K be a weakly contractive map. For some positive real numbers λ and δ satisfying δ+λ>1, let G:E→E be a δ-strongly accretive and λ-strictly pseudocontractive map. Let {tn} be an increasing sequence in [0,∞) with limn→∞tn=∞, and let {αn} and {βn} be sequences in (0,1] satisfying some conditions. Strong convergence of a viscosity iterative sequence to common fixed points of the family 𝔍 of uniformly asymptotically regular asymptotically nonexpansive semigroup, which also solves the variational inequality 〈(G-γf)p,j(p-x)〉≤0,
for all
x∈F, is proved in a framework of a real Banach space.


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## Body

## 1. Introduction

LetE be a real Banach space. We denote by J the normalized duality map from E to 2E* (E* is the dual space of E), and it is defined by
(1.1)J(x)={f∈E*:〈x,f〉=‖x‖2=‖f‖2}.A mappingT:E→E is said to be contractive if ∥Tx-Ty∥≤α∥x-y∥, for x,y∈E, and some constant α∈[0,1). It is said to be weakly contractive if there exists a nondecreasing function ψ:[0,∞)→[0,∞) satisfying ψ(t)=0 if and only if t=0 and ∥Tx-Ty∥≤∥x-y∥-ψ(∥x-y∥), for all x,y∈E. It is known that the class of weakly contractive maps contain properly the class of contractive ones, see [1, 2]. A mapping T:E→E is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥, for all x,y∈E and asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞) with kn→1 as n→∞ and ∥Tnx-Tny∥≤kn∥x-y∥, for all x,y∈E. We denote by F(T)={x∈K:Tx=x} the set of fixed points of a map T.A mappingT:E→E is said to be total asymptotically nonexpansive (see [3]) if there exist nonnegative real sequences {un} and {vn}, with un→0 and vn→0 as n→∞ and strictly increasing and continuous functions ψ:ℝ+→ℝ+ with ψ(0)=0 such that
(1.2)‖Tnx-Tny‖≤‖x-y‖+unψ(‖x-y‖)+vn,∀x,y∈K.Remark 1.1.
Ifψ(λ)=λ, the total asymptotically nonexpansive mapping coincides with generalized asymptotically nonexpansive mapping. In addition, for all n∈ℕ, if vn=0, then generalized asymptotically nonexpansive mapping coincides with asymptotically nonexpansive mapping; if un=0,vn=max{0,pn} where pn:=supx,y∈K(∥Tnx-Tny∥-∥x-y∥), then generalized asymptotically nonexpansive mapping coincide with asymptotically nonexpansive mapping in the intermediate sense; if un=0, and vn=0 then we obtain from (1.2) the class of nonexpansive mapping.
A one-parameter family of generalized asymptotically nonexpansive semigroup is a family𝔍={T(t):t≥0} of self-mapping of E such that  (i)
T(0)x=x for x∈E,(ii)
T(s+t)x=T(s)T(t)x for all t,s≥0 and x∈E,(iii)
limt→0T(t)x=x for x∈E,(iv)
there exist functionsu,v:[0,∞)→[0,∞) such that u(t)→0,v(t)→0 as t→∞, and
(1.3)‖T(t)x-T(t)y‖≤(1+u(t))‖x-y‖+v(t)∀x,y∈E.
We will denote byF  the common fixed-point set of 𝔍, that is,
(1.4)F:=Fix⁡(J)={x∈E:T(t)x=x,t≥0}=⋂t≥0⁡Fix⁡(T(t)).
The family𝔍={T(t):t≥0} is said to be asymptotically regular if
(1.5)lim⁡s→∞‖T(s+t)x-T(s)x‖=0,
for all t∈[0,∞) and x∈E. It is said to be uniformly asymptotically regular if, for any t≥0 and for any bounded subset C of E,
(1.6)lim⁡s→∞sup⁡x∈C‖T(s+t)x-T(s)x‖=0.
For some positive real numbersδ and λ, a mapping G:E→E is said to be δ-strongly accretive if for any x,y∈E, there exists j(x-y)∈J(x-y) such that
(1.7)〈Gx-Gy,j(x-y)〉≥δ‖x-y‖2,
and it is called λ-strictly pseudocontractive if
(1.8)〈Gx-Gy,j(x-y)〉≤‖x-y‖2-λ‖(I-G)x-(I-G)y‖2.
LetE be a real Banach space, and let δ,λ, and τ be positive real numbers satisfying δ+λ>1 and τ∈(0,1). Let G:E→E be a δ-strongly accretive and λ-strictly pseudocontractive, then the following holds, see [4], for x,y∈E:(1.9)‖(I-G)x-(I-G)y‖≤(1-δλ)‖x-y‖,‖(I-τG)x-(I-τG)y‖≤1-τ(1-1-δλ)‖x-y‖,
that is, (I-G) and (I-τG) are contractive mappings.
LetC be a nonempty closed-convex subset of E and T:E→E a map. Then, a variational inequality problem with respect to C and T is found to be x*∈C such that
(1.10)〈Tx*,j(y-x*)〉≥0,∀y∈C,j(y-x*)∈J(y-x*).
Recently, convergence theorems for fixed points of nonexpansive mappings, common fixed points of family of nonexpansive mappings, nonexpansive semigroup, and their generalisation have been studied by numerous authors (see, e.g., [5–21]).
Acedo and Suzuki [22], recently, proved the strong convergence of the Browder's implicit scheme, x0,u∈C,
(1.11)xn=αnu+(1-αn)T(tn)xn,n≥0,
to a common fixed point of a uniformly asymptotically regular family {T(t):t≥0} of nonexpansive semigroup in the framework of a real Hilbert space.
Li et al. [23] proved strong convergence theorems for implicit viscosity schemes for common fixed points of family of generalized asymptotically nonexpansive semigroups in Banach spaces.
LetS be a semigroup and B(S) the subspace of all bounded real-valued functions defined on S with supremum norm. For each s∈S, the left translator operator l(s) on B(S) is defined by (l(s)f)(t)=f(st) for each t∈S and f∈B(S). Let X be a subspace of B(S) containing 1, and let X* be its topological dual. An element μ of X* is said to be a mean on X if ∥μ∥=μ(1)=1. Let X be ls invariant, that is, ls(X)⊂X for each s∈S. A mean μ on X is said to be left invariant if μ(lsf)=μ(f) for each s∈S and f∈X.
Recently, Saeidi and Naseri [24] studied the problem of approximating common fixed point of a family of nonexpansive semigroup and solution of some variational inequality problem in a real Hilbert space. They proved the following theorem.Theorem 1.2 (Saeidi and Naseri [24]).
Let𝔍={T(t):t∈S} be a nonexpansive semigroup in a real Hilbert space H such that F(𝔍)≠∅. Let X be a left invariant subspace of B(S) such that 1∈X, and the function t→〈T(t)x,y〉 is an element of X for each x,y∈H. Let f:E→E be a contraction with constant α, and let G:H→H be strongly positive map with constant γ¯>0. Let {μn} be a left regular sequence of means on X, and let {αn} be a sequence in (0,1) such that limn→∞αn=0 and  ∑n=1∞αn=∞. Let γ∈(0,γ¯/α), and let {xn} be a sequence generated by x0∈H,
(1.12)xn+1=(I-αnG)T(μn)xn+αnγf(xn),n≥0.
Then,{xn} converges strongly to a common fixed point of the family 𝔍 which is the unique solution of the variational inequality 〈(G-γf)x*,j(x-x*)〉≥0 for all x∈F(𝔍). Equivalently one has PF(𝔍)(I-G+γf)x*=x*.More recently, as commented by Golkarmanesh and Naseri [25], Piri and Vaezi [4] gave a minor variation of Theorem 1.2 as follows.Theorem 1.3 (Piri and Vaezi [4]).
Let𝔍={T(t):t∈S} be a nonexpansive semigroup on a real Hilbert space H such that F(𝔍)≠∅. Let X be a left invariant subspace of B(S) such that 1∈X, and the function t→〈T(t)x,y〉 is an element of X for each x,y∈H. Let f:E→E be a contraction with constant α, and let G:H→H be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1. Let {μn} be a left regular sequence of means on X, and let {αn} be a sequence in (0,1) such that limn→∞αn=0 and ∑n=1∞αn=∞. Let {xn} be a sequence generated by x0∈H,
(1.13)xn+1=(I-αnG)T(μn)xn+αnγf(xn),n≥0,
where 0<γ<(1-(1-δ/λ))/α, then, {xn} converges strongly to a common fixed point of the family F(𝔍) which is the unique solution of the variational inequality 〈(G-γf)x*,j(x-x*)〉≥0 for all x∈F(𝔍). Equivalently one has PF(𝔍)(I-G+γf)x*=x*.Very recently, Ali [26] continued the study of the problem in [4, 24] and proved a strong convergence theorem in a Banach space setting much more general than Hilbert space. He actually proved the following theorem.Theorem 1.4 (Ali [26]).
LetE be a real Banach space with local uniform Opial's property whose duality mapping is sequentially continuous. Let 𝔍={T(t):t≥0} be a uniformly asymptotically regular family of asymptotically nonexpansive semigroup of E with function k:[0,∞)→[0,∞) and F:=F(𝔍)=∩t≥0F(T(t))≠∅. Let f:E→E be weakly contractive, and let G:E→E be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1. Let η:=(1-(1-δ)/λ) and γ∈(0,min{η,δ/2}). Let {βn} and {αn} be sequences in (0,1], and let {tn} be an increasing sequence in [0,∞) satisfying the following conditions:
(1.14)lim⁡n→∞αn=0,lim⁡n→∞knαn=0,∑‍n=1∞αn=∞,0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1.
Define a sequence{xn} by x0∈E,
(1.15)xn+1=βnxn+(1-βn)yn,yn=(I-αnG)T(tn)xn+αnγnf(xn),n≥0.
Then, the sequence{xn} converges strongly to a common fixed point of the family 𝔍 which solves the variational inequality
(1.16)〈(G-γf)q,j(x-q)〉≥0,∀x∈F.Remark 1.5.
It is well known that alllp(1<p<∞) spaces satisfy Opial's condition and possess a weakly sequentially continuous duality mapping. However, Lp  (1<p<∞) spaces and consequently all Sobolev spaces do not satisfy either of the properties.
It is our purpose in this paper to prove a strong convergence theorem for approximating common fixed points of family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup in a real reflexive and strictly convex Banach spaceE with a uniformly Gâteaux differentiable norm. Our theorem is applicable in Lp(ℓp) spaces, 1<p<∞ (and consequently in sobolev spaces). Our theorem extends and improves some recent important results. For instance, our theorem presents a convergence of an explicit scheme that extends Theorem 1.4 to a more general setting of Banach spaces that includes Lp(1<p<∞) spaces on one hand and for more general class of maps on the other hand.


## 2. Preliminaries

LetS:={x∈E:||x||=1} denote the unit sphere of a real Banach space E. E is said to have a Gâteaux differentiable norm if the limit
(2.1)lim⁡t→0‖x+ty‖-‖x‖t
exists for each x,y∈S; E is said to have a uniformly Gâteaux differentiable norm if for each y∈S, the limit is attained uniformly for x∈S. A Banach space E is said to be strictly convex if ∥x+y∥/2<1 for x≠y and ∥x∥=∥y∥=1.LetK be a nonempty, closed, convex, and bounded subset of a real Banach space E, and let the diameter of K be defined by d(K):=sup{∥x-y∥:x,y∈K}.   For each x∈K, let r(x,K):=sup{∥x-y∥:y∈K}   and r(K):=inf{r(x,K):x∈K}  denote the Chebyshev radius of K relative to itself. The normal structure coefficientN(E) of E (introduced in 1980 by Bynum [27], see also Lim [28] and the references contained therein) is defined by N(E):=inf{(d(K)/r(K)): K is a closed convex and bounded subset of E with d(K)>0}. A space E such that N(E)>1 is said to have uniform normal structures. It is known that every space with a uniform normal structure is reflexive, and that all uniformly convex and uniformly smooth Banach spaces have uniform normal structure (see, e.g., [29]).LetE be a real Banach space with uniformly Gâteaux differentiable norm, then the normalized duality mapping J:E→2E*, defined by (1.1), is singled valued and uniformly continuous from the norm topology of E to the weak* topology of E* on each bounded subset of E, see, for example [30].Definition 2.1.
Letμ be a continuous linear functional on l∞, and let (a0,a1,…)∈l∞. We write μn(an) instead of μ(a0,a1,…). The function μ is called a Banach limit when μ satisfies ||μ||=μn(1)=1 and μn(an+1)=μn(an) for each (a0,a1,…)∈l∞.
For a Banach limitμ, it is known that liminfn→∞an≤μn(an)≤limsupn→∞an  for every a=(a0,a1,…)∈l∞. So if a=(a0,a1,…)∈l∞ and an-bn→0 as n→∞, we have μn(an)=μn(bn).
We will make use of the following well-known result.Lemma 2.2.
LetE be a real-normed linear space. Then, the following inequality holds:
(2.2)‖x+y‖2≤‖x‖2+2〈y,j(x+y)〉∀x,y∈E,j(x+y)∈J(x+y).In the sequel, we shall also make use of the following lemmas.Lemma 2.3 (Suzuki [31]).
Let{xn} and {yn} be bounded sequences in a real Banach space E, and let {βn} be a sequence in [0,1] with 0<liminfβn≤limsupβn<1. Suppose that xn+1=βnyn+(1-βn)xn for all integer n≥1 and limsupn→∞(||yn+1-yn||-||xn+1-xn||)≤0. Then, limn→∞||yn-xn||=0.Lemma 2.4 (Shioji and Takahashi [32]).
Let(a0,a1,a2,…)∈l∞ be such that μnan≤0 for all Banach limits μ. If limsupn→∞(an+1-an)≤0, then limsupn→∞an≤0.Lemma 2.5 (Xu [33]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relation:
(2.3)an+1≤(1-αn)an+αnσn+γn,n≥0,
where (i){αn}⊂[0,1],∑n=0∞αn=∞  (ii)limsupn→∞σn≤0   (iii)γn≥0 and (n≥0),∑n=0∞γn<∞. Then, an→0 as n→∞.


## 3. Main Results

Theorem 3.1.
LetE be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm, and let 𝔍={T(t):t≥0} be uniformly asymptotically regular family of generalized asymptotically nonexpansive semigroup of E, with functions u,v:[0,∞)→[0,∞) and F:=F(𝔍)=∩t≥0F(T(t))≠∅. Let f:E→E be weakly contractive, and let G:E→E be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1. Let η:=(1-(1-δ)/λ) and γ∈(0,min{δ,η/2}). Let {βn} and {αn} be sequences in (0,1] and {tn} an increasing sequence in [0,∞) satisfying the following conditions:
(3.1)lim⁡n→∞αn=0,lim⁡n→∞u(tn)αn=0,lim⁡n→∞v(tn)αn=0,∑‍n=1∞αn=∞,0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1,lim⁡n→∞tn=∞.
Define a sequence{xn} by x0∈E,
(3.2)xn+1=βnxn+(1-βn)yn,yn=(I-αnG)T(tn)xn+αnγf(xn),n≥0.
Then, the sequence{xn} converges strongly to a common fixed point of the family 𝔍 which solves the variational inequality
(3.3)〈(G-γf)q,j(x-q)〉≥0,∀x∈F.Proof.
We start by showing that solution of the variational inequality (3.3) in F is at most one. Assume that q,p∈F are solutions of the variational inequality (3.3), then
(3.4)〈(G-γf)p,j(q-p)〉≥0,〈(G-γf)q,j(p-q)〉≥0.
Adding these two inequalities, we get(3.5)〈(G-γf)p-(G-γf)q,j(p-q)〉≤0.
Therefore,(3.6)0≥〈(G-γf)p-(G-γf)q,j(p-q)〉=〈G(p)-G(q),j(p-q)〉-γ〈f(p)-f(q),j(p-q)〉≥δ‖p-q‖2-γ‖f(p)-f(q)‖‖p-q‖≥δ‖p-q‖2+γψ(‖p-q‖)‖p-q‖-γ‖p-q‖2=(δ-γ)‖p-q‖2+γψ(‖p-q‖)‖p-q‖.
Since δ>γ, we obtain that p=q, and so the solution is unique in F.
Now, letp∈F, since (1-αnη)(u(tn)/αn)→0 and (1-αnη)(v(tn)/αn)→0 as n→∞, then there exists n0∈ℕ such that (1-αnη)(u(tn)/αn)<(η-γ)/2 and (1-αnη)(v(tn)/αn)<(η-γ)/2 for all n≥n0. Hence, for n≥n0, we have the following:
(3.7)‖yn-p‖≤‖(I-αnG)(T(tn)xn-p)‖+αn‖γf(xn)-G(p)‖≤(1-αnη)[(1+u(tn))‖xn-p‖+v(tn)]+αnγ‖f(xn)-f(p)‖+αn‖γf(p)-G(p)‖≤[1-αn(η-γ)+(1-αnη)u(tn)]‖xn-p‖+(1-αnη)v(tn)+αn‖γf(p)-G(p)‖,
so that(3.8)‖xn+1-p‖≤βn‖xn-p‖+(1-βn)‖yn-p‖≤[βn+(1-βn)[1-αn(η-γ)+(1-αnη)u(tn)]]‖xn-p‖+(1-αnη)(1-βn)v(tn)+αn(1-βn)‖γf(p)-G(p)‖≤[1-αn(1-βn)((η-γ)-(1-αnη)u(tn)αn)]‖xn-p‖+αn(1-βn)[‖γf(p)-G(p)‖+(1-αnη)v(tn)αn]≤[1-αn(1-βn)((η-γ)-(1-αnη)u(tn)αn)]‖xn-p‖+αn(1-βn)((η-γ)-(1-αnη)u(tn)αn)×2[‖γf(p)-G(p)‖+(1-αnη)(v(tn)/αn)]η-γ≤max⁡{‖xn-p‖,2‖γf(p)-G(p)‖η-γ+1}.
By induction, we have(3.9)‖xn-p‖≤max⁡{‖xn0-p‖,2‖γf(p)-G(p)‖η-γ+1},∀n≥0.
Thus,{xn} is bounded and so are {T(tn)xn},{GT(tn)xn},{yn}, and {f(xn)}.
Observe that(3.10)yn+1-yn=((I-αn+1G)T(tn+1)xn+1-(I-αn+1G)T(tn+1)xn)+((I-αn+1G)T(tn+1)xn-(I-αnG)T(tn+1)xn)+((I-αnG)T(tn+1)xn-(I-αnG)T(tn)xn)+(αn+1γf(xn+1)-αn+1γf(xn))+(αn+1γf(xn)-αnγf(xn)),
so that
(3.11)‖yn+1-yn‖≤(1-αn+1η)(1+u(tn+1))‖xn+1-xn‖+(1-αn+1η)v(tn+1)+|αn-αn+1|‖GT(tn+1)xn‖+(1-αnη)‖T((tn+1-tn)+tn)xn-T(tn)xn‖+αn+1γ‖f(xn+1)-f(xn)‖+|αn+1-αn|γ‖f(xn)‖≤(1-αn+1η)(1+u(tn+1))‖xn+1-xn‖+(1-αn+1η)v(tn+1)+|αn-αn+1|‖GT(tn+1)xn‖+(1-αnη)sup⁡z∈{xn},s∈R+‖T(s+tn)z-T(tn)z‖+αn+1γ‖f(xn+1)-f(xn)‖+|αn+1-αn|γ‖f(xn)‖.
From this, we obtain that(3.12)‖yn+1-yn‖-‖xn+1-xn‖≤[(1-αn+1η)(1+u(tn+1))-1]‖xn+1-xn‖+(1-αn+1η)v(tn+1)+|αn-αn+1|‖GT(tn+1)xn‖+(1-αnη)sup⁡z∈{xn},s∈R+‖T(s+tn)z-T(tn)z‖+αn+1γ‖f(xn+1)-f(xn)‖+|αn+1-αn|γ‖f(xn)‖,
which implies that
(3.13)limsup⁡n→∞(‖yn+1-yn‖-‖xn+1-xn‖)≤0,
and by Lemma 2.3,
(3.14)lim⁡n→∞‖yn-xn‖=0.
Thus,(3.15)‖xn+1-xn‖=(1-βn)‖yn-xn‖⟶0asn⟶∞.
Next, we show thatlimn→∞∥yn-T(t)yn∥=0, for all t≥0.
Since(3.16)‖xn-T(tn)xn‖≤‖xn-xn+1‖+‖xn+1-T(tn)xn‖≤‖xn-xn+1‖+βn‖xn-T(tn)xn‖+(1-βn)‖yn-T(tn)xn‖,
we have
(3.17)(1-βn)‖xn-T(tn)xn‖≤‖xn-xn+1‖+(1-βn)‖yn-T(tn)xn‖=‖xn-xn+1‖+αn(1-βn)‖γf(xn)-GT(tn)xn‖.
Fromαn→0 as n→∞ and (3.15), we obtain
(3.18)lim⁡n→∞‖xn-T(tn)xn‖=0.
Also,(3.19)‖yn-T(tn)yn‖≤‖yn-xn‖+‖xn-T(tn)xn‖+‖T(tn)xn-T(tn)yn‖≤(2+u(tn))‖yn-xn‖+v(tn)+‖xn-T(tn)xn‖⟶0asn⟶∞.
Sincelimn→∞tn=∞ and {T(t):t≥0} is uniformly asymptotically regular,
(3.20)lim⁡n→∞⁡‖T(t)T(tn)xn-T(tn)xn‖≤lim⁡n→∞sup⁡x∈C‖T(t)T(tn)x-T(tn)x‖=0,lim⁡n→∞⁡‖T(t)T(tn)yn-T(tn)yn‖≤lim⁡n→∞sup⁡y∈C‖T(t)T(tn)y-T(tn)y‖=0,
where C is any bounded subset of E containing {xn}. Since {T(t)} is continuous, we get that
(3.21)‖yn-T(t)yn‖≤‖yn-T(tn)yn‖+‖T(tn)yn-T(t)(T(tn)yn)‖+‖T(t)(T(tn)yn)-T(t)yn‖.
This implies that(3.22)lim⁡n→∞‖yn-T(t)yn‖=0,∀t≥0.
Next, we show that(3.23)limsup⁡n→∞〈(γf-G)p,j(yn-p)〉≤0.
Define a mapϕ:E→ℝ by
(3.24)ϕ(y):=μn‖yn-y‖2,∀y∈E.
Then,ϕ(y)→∞ as ∥y∥→∞, ϕ is continuous and convex, so as E is reflexive, there exists q∈E such that ϕ(q)=minu∈Eϕ(u). Hence, the set
(3.25)K*:={y∈E:ϕ(y)=min⁡u∈Eϕ(u)}≠∅.
Sincelimn→∞∥yn-T(t)yn∥=0,limt→∞u(t)=0,limt→∞v(t)=0, and ϕ is continuous for all z∈K*, we have
(3.26)ϕ(lim⁡t→∞T(t)z)=lim⁡t→∞ϕ(T(t)z)=lim⁡t→∞μn‖yn-T(t)z‖2≤lim⁡t→∞μn((1+u(t))‖yn-z‖+(v(t)))2=μn‖yn-z‖2=ϕ(z).
Hence,limt→∞T(t)z∈K*.
Letp∈F. Since K* is a closed-convex set, there exists a unique q∈K* such that
(3.27)‖p-q‖=min⁡x∈K*‖p-x‖.
Sincep=limt→∞T(t)p and limt→∞T(t)q∈K*,
(3.28)‖p-lim⁡t→∞T(t)q‖=‖lim⁡t→∞T(t)p-lim⁡t→∞T(t)q‖=lim⁡t→∞‖T(t)p-T(t)q‖≤lim⁡t→∞((1+u(t))‖p-q‖+v(t))≤‖p-q‖.
Therefore,limt→∞T(t)q=q. Since T(s+h)x=T(s)T(h)x for all x∈E and s≥0, we have
(3.29)q=lim⁡t→∞T(t)q=lim⁡t→∞T(s+t)q=lim⁡t→∞T(s)T(t)q=T(s)lim⁡t→∞T(t)q=T(s)q.
Therefore,q∈F and so K*∩F≠∅.
Letp∈K*∩F(T) and τ∈(0,1). Then, it follows that ϕ(p)≤ϕ(p-τ(G-γf)p), and using Lemma 2.2, we obtain that
(3.30)‖yn-p+τ(G-γf)p‖2≤‖yn-p‖2+2τ〈(G-γf)p,j(yn-p+τ(G-γf)p)〉,
which implies that
(3.31)μn〈(γf-G)p,j(yn-p+τ(G-γf)p)〉≤0.
Moreover,(3.32)μn〈(γf-G)p,j(yn-p)〉=μn〈(γf-G)p,j(yn-p)-j(yn-p+τ(G-γf)p)〉+μn〈(γf-G)p,j(yn-p+τ(G-γf)p)〉≤μn〈(γf-G)p,j(yn-p)-j(yn-p+τ(G-γf)p)〉.
Sincej is norm-to-weak* uniformly continuous on bounded subsets of E, we have that
(3.33)μn〈(γf-G)p,j(yn-p)〉≤0.
Observe that from (3.14) and (3.15), we have
(3.34)lim⁡n→∞‖yn+1-yn‖=0.
This implies that(3.35)limsup⁡n→∞[〈(γf-G)p,j(yn-p)〉-〈(γf-G)p,j(yn+1-p)〉]≤0,
and so we obtain by Lemma 2.4 that
(3.36)limsup⁡n→∞〈(γf-G)p,j(yn-p)〉≤0.
Finally, we show thatxn→p as n→∞. Since limn→∞(u(tn)/αn)=0, if we denote by σ(tn) the value 2u(tn)+u(tn)2, then we clearly have limn→∞(σ(tn)/αn)=0. Let N0∈ℕ be large enough such that (1-αnη)(σ(tn)/αn)<(η-2γ)/2, for all n≥N0, and let M be a positive real number such that ||xn-p||≤M for all n≥0. Then, using the recursion formula (3.2) and for n≥N0, we have
(3.37)‖yn-p‖2=‖αn(γf(xn)-G(p))+(I-αnG)(T(tn)xn-p)‖2≤(1-αnη)‖T(tn)xn-p‖2+2αn〈γf(xn)-G(p),j(yn-p)〉≤(1-αnη)[(1+u(tn))‖xn-p‖+v(tn)]2+2αn〈γf(xn)-γf(p)+γf(p)-G(p),j(yn-p)〉≤(1-αnη)[(1+u(tn))2‖xn-p‖2+2(1+u(tn))v(tn)‖xn-p‖2+v(tn)2]+2αn〈γf(p)-G(p),j(yn-p)〉-2αnγ‖yn-p‖ψ(‖xn-p‖)+2αnγ‖(yn-xn)+(xn-p)‖‖xn-p‖≤[(1-αnη)(1+σ(tn))+2αnγ]‖xn-p‖2+αn[2〈γf(p)-G(p),j(yn-p)〉+2(1-αnη)(1+u(tn))v(tn)αn‖xn-p‖2+(1-αnη)v(tn)2αn+2γ‖yn-xn‖‖xn-p‖]=[1-αn((η-2γ)-(1-αnη)σnαn)]‖xn-p‖2+αn[v(tn)2αn2〈γf(p)-G(p),j(yn-p)〉+2(1-αnη)(1+u(tn))v(tn)αn‖xn-p‖2+(1-αnη)v(tn)2αn+2γ‖yn-xn‖‖xn-p‖],
so that(3.38)‖xn+1-p‖2≤βn‖xn-p‖2+(1-βn)‖yn-p‖2≤(βn+(1-βn)[1-αn((η-2γ)-(1-αnη)σnαn)])‖xn-p‖2+αn(1-βn)[v(tn)2αn2〈γf(p)-G(p),j(yn-p)〉+2(1-αnη)(1+u(tn))×v(tn)αn‖xn-p‖2+(1-αnη)v(tn)2αn+2γ‖yn-xn‖‖xn-p‖]≤[1-αn(1-βn)((η-2γ)-(1-αnη)σnαn)]‖xn-p‖2+αn(1-βn)[2〈γf(p)-G(p),j(yn-p)〉+2(1-αnη)(1+u(tn))v(tn)αnM2+(1-αnη)v(tn)2αn+2γ‖yn-xn‖M]=[1-αn(1-βn)((η-2γ)-(1-αnη)σnαn)]‖xn-p‖2+αn(1-βn)((η-2γ)-(1-αnη)σnαn)×[2〈γf(p)-G(p),j(yn-p)〉+2(1-αnη)(1+u(tn))(v(tn)αn)M2+An]((η-2γ)-(1-αnη)(σnαn)),
where 𝒜n denotes (1-αnη)(v(tn)2/αn)+2γ∥yn-xn∥M.
Observe  that∑n=1∞αn(1-βn)((η-2γ)-(1-αnη)(σn/αn))=∞ and
(3.39)limsup⁡n→∞(2〈γf(p)-G(p),j(yn-p)〉+2(1-αnη)(1+u(tn))(v(tn)/αn)M2+An((η-2γ)-(1-αnη)(σn/αn)))≤0.
Applying Lemma2.5, we obtain ∥xn-p∥→0 as n→∞. This completes the proof.The following corollaries follow from Theorem3.1.Corollary 3.2.
LetE be a real uniformly convex and uniformly smooth Banach space, 𝔍={T(t):t≥0}, and let F,f,G,δ,λ,η,γ,{βn},{αn},{tn} and {xn} be as in Theorem 3.1. Then, the sequence {xn} converges strongly to a common fixed point of the family 𝔍 which solves the variational inequality (3.3).Corollary 3.3.
LetE=H be a real Hilbert space,  and let 𝔍={T(t):t≥0},F,f,G,δ,λ,η,γ,{βn},{αn},{tn} and {xn} be as in Theorem 3.1. Then, the sequence {xn} converges strongly to a common fixed point of the family 𝔍 which solves the variational inequality
(3.40)〈(G-γf)q,x-q〉≥0,∀x∈F.Corollary 3.4.
Let𝔍={T(t):t≥0} be a family of nonexpansive semigroup of a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm E, and let F,f,G,δ,λ,η,γ,{βn},{αn},{tn}, and {xn} be as in Theorem 3.1. Then, the sequence {xn} converges strongly to a common fixed point of the family 𝔍 which solves the variational inequality (3.3).



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*Source: 986426-2012-07-22.xml*