# Convergence Theorems of a General Composite Iterative Method for Nonexpansive Semigroups in Banach Spaces

Authors: Pitipong Sunthrayuth; Kriengsak Wattanawitoon; Poom Kumam
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

We introduce a general composite iterative scheme for nonexpansive semigroups in
Banach spaces. We establish some strong convergence theorems of the general iteration scheme under
different control conditions. The results presented in this paper improve and extend the corresponding
results of Marino and Xu (2006), and others, from Hilbert spaces to Banach spaces.


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## Body

## 1. Introduction

LetX 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 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 𝒯={T(t):0≤t<∞} 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=x for all x∈C;(ii)
T(s+t)=T(s)∘T(t) for all s,t≥0;(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≥0.We denote byF(𝒯) the set of all common fixed points of 𝒯, that is F(𝒯)=⋂t≥0F(T(t)). We know that F(𝒯) 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 ΠC to denote the collection of all contraction on C, that is, ΠC={f:C→Cacontraction}. Note that each f∈ΠC has a unique fixed point in C. Iterative methods for nonexpansive mappings have recently been applied to solve minimization problems; see, for example, [4–10] and references therein.A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH:(1.1)min⁡x∈F12〈Ax,x〉-〈x,u〉,
where F is the fixed point set of a nonexpansive mapping T on H and u is a given point in H. Assume A is strongly positive; that is there is a constant γ¯ such that 〈Ax,x〉≥γ¯∥x∥2 for all x∈H.In 2003, Xu [7] proved that the sequence {xn} generated by(1.2)x0=x∈Hchosenarbitrarily;xn+1=αnu+(1-αn)Txn,∀n≥0,
converges strongly to the unique solution of the minimization problem (1.1) provided the sequence {αn} satisfies certain conditions. On the other hand, Moudafi [11] introduced the viscosity approximation method and proved that if H is a real Hilbert space, the sequence {xn} generated by the following algorithm:(1.3)x0=x∈Hchosenarbitrarily;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 the unique solution of the following variational inequality:(1.4)〈(f-I)x*,y-x*〉≤0,∀y∈F(T).Recently, Marino and Xu [12] combined the iterative method (1.2) with the viscosity approximation method (1.3) considering the following general iterative process:(1.5)x0=x∈Hchosenarbitrarily;xn+1=αnγf(xn)+(I-αnA)Txn,∀n≥0.
They proved the sequence {xn}, generated by (1.5), converges strongly to the unique solution of the variational inequality(1.6)〈(γf-A)x*,y-x*〉≤0,∀y∈F(T),
which is the optimality condition for the minimization problem(1.7)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 all x∈H). Xu [3] studied further the viscosity approximation method for nonexpansive semigroup in uniformly smooth Banach spaces. This result extends Theorem 2.2 of Moudafi [11] to a Banach space. Kim and Xu [13] studied the sequence generated by the algorithm(1.8)x1=x∈Cchosenarbitrarily;yn=αnxn+(1-αn)Txn;xn+1=βnu+(1-βn)yn,∀n≥1,
and proved strong convergence of scheme (1.8) in the framework of a uniformly smooth Banach space. Later, Yao et al. [14] introduced a new iteration process by combining the modified Mann iteration [13] and the viscosity method introduced by Moudafi [11]. Let C be a closed convex subset of a Banach space, T:C→C a nonexpansive mapping such that F(T)≠∅; and f∈ΠC. Define {xn} in the following way:(1.9)x1=x∈Cchosenarbitrarily;yn=αnxn+(1-αn)Txn;xn+1=βnf(xn)+(1-βn)yn,∀n≥1,
where {αn} and {βn} are two sequences in (0,1). They proved, under different control conditions on the sequences {αn} and {βn}, that {xn} converge strongly to a fixed point of T.In 2008, Sahu and O’Regan [15] studied several strong convergence theorems for a family of nonexpansive or pseudocontractive nonself-mappings in a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm. Recently, Li and Gu [16] studied the sequence generated by the algorithm in Banach spaces, as follows:(1.10)x1=x∈Cchosenarbitrarily;yn=αnxn+(1-αn)T(tn)xn;xn+1=βnf(xn)+(1-βn)yn,∀n≥1,
and they proved the sequence {xn} defined by (1.10) that, converges strongly to the unique solution of the variational inequality:(1.11)〈(f-I)x*,J(y-x*)〉≤0,∀y∈F(T).
Very recently, Kumam and Wattanawitoon [5] introduced the following new composite explicit iterative schemes defined by given x0=x∈C and(1.12)xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)1tn∫0tnT(s)ynds,yn=δnγf(xn)+γnxn+((1-γn)I-δnA)xn,∀n≥0,
for the approximation of common fixed point of a one parameter nonexpansive semigroup in a real Hilbert space under some appropriate control conditions. They proved strong convergence theorems of the composite iterative schemes which solve some variational inequalities under some appropriate conditions.Question 1.
Can the theorem of Marino and Xu [12] be extend from a Hilbert space to a general Banach space, such as  uniformly smooth Banach space?Question 2.
Can we extend the iterative method of algorithm (1.10) to a general iterative process?The purpose of this paper is to give affirmative answer to these questions mentioned above. In this paper, inspired and motivated by the iterative sequence (1.5) given by Marino and Xu [12] and (1.10) given by Li and Gu [16], we introduce a composite iterative algorithm {xn} in a Banach space as follows:(1.13)x1=x∈Cchosenarbitrarily;yn=βnxn+(1-βn)T(tn)xn;xn+1=αnγf(xn)+δnxn+((1-δn)I-αnA)yn,∀n≥1,
where f:C→C is a contraction mapping, T(tn) is a nonexpansive semigroup and A is a strongly positive linear bounded operator, and prove, under certain different control conditions on the sequences {αn}, {βn}, and {δn}, that {xn} defined by (1.13) converges strongly to a common fixed point, which solves some variational inequality in Banach spaces. The results presented in this paper extend the corresponding results announced by Marino and Xu [12] and some others from Hilbert spaces to Banach spaces.


## 2. Preliminaries

Throughout this paper, letX be a real Banach space, C be a closed convex subset of X. Let J:X→2X* be a normalized duality mapping by (2.1)J(x)={f*∈X*:〈x,f*〉=‖x‖2=‖f*‖2},
where X* denotes the dual space of X and 〈·,·〉 denotes the generalized duality paring. In the following, the notation ⇀ and → denote the weak and strong convergence, respectively. Also, a mapping I:C→C denotes the identity mapping.The norm of a Banach spaceX is said to be Gâteaux differentiable if the limit(2.2)lim⁡t→0‖x+ty‖-‖x‖t
exists for each x,y∈C on the unit sphere S(X) of X. In this case X is smooth. Recall that the Banach space X is said to be smoothif duality mapping J is single valued. In a smooth Banach space, we always assume that A is strongly positive (see [17]), that is, a constant γ¯>0 with the property(2.3)〈Ax,J(x)〉≥γ¯‖x‖2,‖aI-bA‖=sup⁡‖x‖≤1‖〈(aI-bA)x,J(x)〉‖,a∈[0,1],b∈[-1,1].Moreover, if for eachy in S(X) the limit (2.2) is uniformly attained for x∈S(X), we say that the norm X is uniformly Gâteaux differentiable. The norm of X is said to be Frêchet differentiable, if for each x∈S(X), the limit (2.2) is attained uniformly for y∈S(X). The norm of X is said to be uniformly Frêchet differentiable (or X is said to be uniformly smooth), the limit (2.2) is attained uniformly for (x,y)∈S(X)×S(X). A Banach space X is said to be strictly convex if ∥x∥=∥y∥=1, x≠y implies ∥x+y∥/2<1;  uniformly convex if δX(ϵ)>0 for all ϵ>0, where δX(ϵ) is modulus of convexity of X defined by (2.4)δX(ϵ)=inf⁡{1-‖x+y‖2:‖x‖≤1,‖y‖≤1,‖x+y‖≥ɛ},∀ϵ∈[0,2].
A uniformly convex Banach space X is reflexive and strictly convex (see [18, Theorems 4.1.6 and 4.1.2]) and every uniformly smooth Banach space X is a reflexive Banach with uniformly Gâteaux differentiable norm (see [18, Theorems 4.3.7 and 4.1.6]) (also see [19]).Now, we present the concept of a uniformly asymptotically regular semigroup (see [20–22]). Let C be a nonempty closed convex subset of a Banach space X, 𝒯={T(t):t>0} is a continuous operator semigroup on C. Then 𝒯 is said to uniformly asymptotically regular (in short, u.a.r.) on C if for all h≥0 and any bounded subset B of C,(2.5)lim⁡t→∞sup⁡x∈B‖T(h)T(t)x-T(t)x‖=0.Lemma 2.1 (Chen and Song [23]).
LetC be a nonempty closed convex subset of a uniformly Banach space X, B a bounded closed convex subset of C. If we denote 𝒯={T(t):t>0} a nonexpansive semigroup on C such that F:=⋂t>0F(T(t))≠∅. For all h≥0, the set σt(x)=(1/t)∫0tT(s)xds, then
(2.6)lim⁡t→∞sup⁡x∈B‖σt(x)-T(h)σt(x)‖=0.It is easy to check that the set{σt:t>0} defined by Lemma 2.1 is a u.a.r. nonexpansive semigroup on C (see [24] for more detail).Lemma 2.2 (Cai and Hu [17]).
Assume thatA is a strongly positive linear bounded operator on a smooth Banach space X with coefficient γ¯>0 and 0<ρ≤∥A∥-1. Then ∥I-ρA∥≤1-ργ¯.Lemma 2.3 (Suzuki [25]).
Let{xn} and {yn} be bounded sequences in a Banach space X and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose xn+1=(1-βn)yn+βnxn for all integers n≥0 and limsupn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then, limn→∞∥yn-xn∥=0.If a Banach spaceX admits a sequentially continuous duality mapping J from weak topology to weak star topology, then by Lemma 1 of [26], we have that duality mapping J is a single value. In this case, the duality mapping J is said to be a weakly sequentially continuous duality mapping, that is, for each {xn}⊂X with xn⇀x, we have J(xn)⇀*J(x) (see [26–28] for more details).A Banach spaceX is said to be satisfying Opial's condition if for any sequence xn⇀x for all x∈X implies (2.7)limsup⁡n→∞‖xn-x‖<limsup⁡n→∞‖xn-y‖∀y∈X,withx≠y.By Theorem 1 in [26], it is well known that if X admits a weakly sequentially continuous duality mapping, then X satisfies Opial's condition and X is smooth.Lemma 2.4 (Demiclosed principle (Jung [27])).
LetC be a nonempty closed convex subset of a reflexive Banach space X which satisfies Opial's condition, and suppose T:C→X is nonexpansive. Then the mapping I-T is demiclosed at zero, that is, xn⇀x and xn-Txn→0 implies x=Tx.Lemma 2.5 (Liu [29]).
LetX be a real Banach space and J:X→2X* be the normalized duality mapping. Then, for any x,y∈X, we have
(2.8)‖x+y‖2≤‖x‖2+2〈y,j(x+y)〉,
for all j(x+y)∈J(x+y) with x≠y.Lemma 2.6 (Aoyama et al. [30]).
Let{sn} be a sequence of nonnegative real numbers, {an} a sequence of [0,1] with ∑n=1∞an=∞, {cn} a sequence of nonnegative real number with ∑n=1∞cn<∞, and {bn} a sequence of real numbers with limsupn→∞bn≤0. Suppose that
(2.9)sn+1=(1-an)sn+anbn+cn,
for all n∈ℕ. Then limn→∞sn=0.


## 3. Main Results

We need following the lemma to prove our theorem.Lemma 3.1.
LetC be a nonempty bounded closed convex subset of a reflexive, smooth Banach space X with admits a weakly sequentially continuous duality mapping J from X to X* with C±C⊂C and let 𝒯={T(t):0≤t<∞} be a nonexpansive semigroup on C such that F(𝒯)≠∅. Let f:C→C be a contraction mapping with a coefficient α∈(0,1), A be a strongly positive linear bounded operator with a coefficient γ¯>0 such that 0<γ<γ¯/α and {λt}0<t<1 be a net of positive real sequence such that limt→0λt=∞. Suppose that {T(λt)} is a u.a.r. nonexpansive semigroup. Then the sequence {xt} define by
(3.1)xt=tγf(xt)+(I-tA)T(λt)xt,
converges strongly to the common fixed point x*, where x* solves the variational inequality
(3.2)〈γf(x*)-Ax*,J(y-x*)〉≤0,∀y∈F(T).Proof.
Observe that fort∈(0,∥A∥-1). By Lemma 2.2, we have ∥I-tA∥≤1-tγ¯.
Firstly, we show that{xt} defined by (3.1) is well define. Define the mapping Sf:C→C provided by Sf:=tγf+(I-tA)T(λt), for each t∈(0,1). Then, for each x,y∈C that (3.3)‖Sfx-Sfy‖=‖tγ(f(x)-f(y))+(I-tA)(T(λt)x-T(λt)y)‖≤tγα‖x-y‖+(1-tγ¯)‖x-y‖=(1-t(γ¯-γα))‖x-y‖<‖x-y‖.
This show that Sf is a contraction mapping. Thus, by Banach's contraction principle guarantees that Sf has a unique fixed point xt∈C, that is, xt defined by (3.1) is well define.
Next, we show the uniqueness of a solution of the variational inequality (3.2). Supposing x*,x̃∈F(𝒯) satisfy the inequality (3.2), we have(3.4)〈(γf-A)x̃,J(x*-x̃)〉≤0,〈(γf-A)x*,J(x̃-x*)〉≤0.
Adding up both equations of (3.4), we get that
(3.5)0≥〈(γf-A)x̃-(γf-A)x*,J(x*-x̃)〉=〈A(x*-x̃),J(x*-x̃)〉-γ〈f(x*)-f(x̃),J(x*-x̃)〉≥γ¯‖x*-x̃‖2-γ‖f(x*)-f(x̃)‖‖J(x*-x̃)‖≥γ¯‖x*-x̃‖2-γα‖x*-x̃‖2=(γ¯-γα)‖x*-x̃‖2.
Since 0<γ<γ¯/α this implies  that γ¯-γα>0, which is a contradiction. Hence x̃=x* and the uniqueness is proved.
Next, we show that{xt} is bounded. Indeed, for any p∈F(𝒯), we have(3.6)‖xt-p‖=‖tγf(xt)+(I-tA)T(λt)xt-p‖=‖t(γf(xt)-Ap)+(I-tA)(T(λt)xt-p)‖≤t‖γf(xt)-Ap‖+‖I-tA‖‖T(λt)xt-p‖≤t‖γf(xt)-Ap‖+(1-tγ¯)‖xt-p‖≤t‖γ(f(xt)-f(p))+γf(p)-Ap‖+(1-tγ¯)‖xt-p‖≤t(γα‖xt-p‖+‖γf(p)-Ap‖)+(1-tγ¯)‖xt-p‖=(1-t(γ¯-γα))‖xt-p‖+t‖γf(p)-Ap‖.
It follows that ∥xt-p∥≤∥γf(p)-Ap∥/(γ¯-γα). Hence {xt} is bounded.
Next, we show that∥xt-T(h)xt∥→0 as t→0. We observe that(3.7)‖xt-T(h)xt‖≤‖xt-T(λt)xt‖+‖T(λt)xt-T(h)T(λt)xt‖+‖T(h)T(λt)xt-T(h)xt‖≤2‖xt-T(λt)xt‖+‖T(λt)xt-T(h)T(λt)xt‖
for all h≥0. On the other hand, we note that
(3.8)xt-T(λt)xt=t(γf(xt)-AT(λt)xt),
for all t>0. By assuming that limt→0λt=∞ and {T(λt)} be a u.a.r. nonexpansive semigroup, then for all h≥0, we get
(3.9)lim⁡λt→∞‖T(λt)xt-T(h)T(λt)xt‖≤lim⁡λt→∞sup⁡z∈C‖T(λt)z-T(h)T(λt)z‖=0.
From (3.7)–(3.9), letting t→0, we get
(3.10)lim⁡t→0‖xt-T(h)xt‖=0
for all h≥0. Assume {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 converges strongly to x*, where x*∈F(𝒮). Since {xn} is bounded sequence and Banach space X is reflexive, there exists a subsequence {xnj} of {xn} which converges weakly to x*∈C as j→∞. Again since Banach space X has a weakly sequentially continuous duality mapping satisfying Opial's condition. It follows by Lemma 2.4 and noting (3.10), we have x*∈F(𝒯). For each n≥1, we note that
(3.11)xn-x*=tnγf(xn)+(I-tnA)T(λn)xn-x*=tn(γf(xn)-Ax*)+(I-tnA)(T(λn)xn-x*).
Thus, we have
(3.12)‖xn-x*‖2=tn〈γf(xn)-Ax*,J(xn-x*)〉+〈(I-tnA)(T(λn)xn-x*),J(xn-x*)〉≤tn〈γf(xn)-Ax*,J(xn-x*)〉+‖I-tnA‖‖T(λn)xn-x*‖‖J(xn-x*)‖≤tn〈γf(xn)-Ax*,J(xn-x*)〉+(1-tnγ¯)‖xn-x*‖2.
It follows that
(3.13)‖xn-x*‖2≤1γ¯〈γf(xn)-Ax*,J(xn-x*)〉=1γ¯[〈γf(xn)-γf(x*),J(xn-x*)〉+〈γf(x*)-Ax*,J(xn-x*)〉]≤1γ¯[γα‖xn-x*‖2+〈γf(x*)-Ax*,J(xn-x*)〉].
Hence,
(3.14)‖xn-x*‖2≤1γ¯-γα〈γf(x*)-Ax*,J(xn-x*)〉.
In particular, we have
(3.15)‖xnj-x*‖2≤1γ¯-γα〈γf(x*)-Ax*,J(xnj-x*)〉.
Since {xn} is bounded and the duality mapping J is single-valued and weakly sequentially continuous from X into X*, it follows (3.15), we have xnj→x* as j→∞.
Next, we show thatx* solves the variational inequality (3.2). Since(3.16)xt=tγf(xt)+(I-tA)T(λt)xt,
we derive that
(3.17)(γf-A)xt=-1t(I-tA)(T(λt)xt-xt).
Notice that
(3.18)〈(T(λt)xt-xt)-(T(λt)y-y),J(y-xt)〉≥‖xt-y‖2-‖T(λt)xt-T(λt)y‖‖xt-y‖≥‖xt-y‖2-‖xt-y‖2=0.
It follows that, for any y∈F(𝒯),
(3.19)〈(γf-A)xt,J(y-xt)〉=-1t〈(I-tA)(T(λt)xt-xt),J(y-xt)〉=-1t〈(T(λt)xt-xt)-(T(λt)y-y),J(y-xt)〉+〈A(T(λt)xt-xt),J(y-xt)〉≤〈A(T(λt)xt-xt),J(y-xt)〉.
Now, replacing t and λt with tnj and λnj, respectively in (3.19), and letting j→∞, noting (3.8), we obtain 〈(γf-A)x*,J(y-x*)〉≤0. That is, x* is a solution of variational inequality (3.2). By uniqueness, as x*=x̃, we have shown that each cluster point of the sequence {xt} is equal to x*. Then, we conclude that xt→x* as t→0. This completes the proof.Now, we prove the following theorem which is the main result of this paper.Theorem 3.2.
LetC be a nonempty bounded closed convex subset of a uniformly smooth Banach space X which admit a weakly sequentially continuous duality mapping from X into X* with C±C⊂C and 𝒯={T(t):0≤t<∞} be a nonexpansive semigroup on C such that F(𝒯)≠∅. Let f:C→C be a contraction mapping with a coefficient α∈(0,1) and A be a strongly positive bounded linear operator with coefficient γ¯>0 such that 0<γ<γ¯/α. Let {αn}n=1∞, {βn}n=1∞, {δn}n=1∞ be the sequences in (0,1) and {tn}n=1∞ be a positive real divergent sequence such that tn≤tn+1 for all n∈ℕ. Assume the following control conditions are hold: (C1)
limn→∞αn=0 and ∑n=1∞αn=∞;(C2)
∑n=1∞|αn+1-αn|<∞, ∑n=1∞|βn+1-βn|<∞ and ∑n=1∞|δn+1-δn|<∞;(C3)
∑n=1∞supz∈C{∥T(h)T(tn)z-T(tn)z∥}<∞ for all h≥0.Then the sequence {xn} defined by (1.13) converges strongly to the common fixed point x*, where x* is the unique solution in F(𝒯) of the variational inequality (3.2).Proof.
First, we show that{xn} is bounded. By the control condition (C1), we may assume, without loss of generality, that αn≤(1-δn)∥A∥-1 for all n≥0. Since A is a linear bounded operator on X, by (2.3), we have ∥A∥=sup{|〈Au,J(u)〉|:u∈X,∥u∥=1}. Observe that
(3.20)〈((1-δn)I-αnA)u,J(u)〉=1-δn-αn〈Au,J(u)〉≥1-δn-αn‖A‖≥0,
that is to say (1-δn)I-αnA is positive. It follows that
(3.21)‖(1-δn)I-αnA‖=sup⁡{〈((1-δn)I-αnA)u,J(u)〉:u∈X,‖u‖=1}=sup⁡{1-δn-αn〈Au,J(u)〉:u∈X,‖u‖=1}≤1-δn-αnγ¯.
For any p∈F(𝒯), we compute
(3.22)‖yn-p‖=‖βnxn+(1-βn)T(tn)xn-p‖=‖βn(xn-p)+(1-βn)(T(tn)xn-p)‖≤βn‖xn-p‖+(1-βn)‖T(tn)xn-p‖≤βn‖xn-p‖+(1-βn)‖xn-p‖=‖xn-p‖,‖xn+1-p‖=‖αnγf(xn)+δnxn+((1-δn)I-αnA)yn-p‖=‖αn(γf(xn)-Ap)+δn(xn-p)+((1-δn)I-αnA)(yn-p)‖≤αn[γ‖f(xn)-f(p)‖+‖γf(p)-Ap‖]+δn‖xn-p‖+‖(1-δn)I-αnA‖‖yn-p‖≤αn[γα‖xn-p‖+‖γf(p)-Ap‖]+δn‖xn-p‖+(1-δn-αnγ¯)‖xn-p‖=[1-(γ̅-γα)αn]‖xn-p‖+(γ̅-γα)αn‖γf(p)-Ap‖γ̅-γα.
By induction, we get
(3.23)‖xn+1-p‖≤max⁡{‖x1-p‖,‖γf(p)-Ap‖γ¯-γα},
for n≥1. Hence {xn} is bounded, so are {yn}, {f(xn)} and {T(tn)xn}.
Next, we show thatlimn→∞∥xn+1-xn∥=0. From definition of {yn}, observing that (3.24)yn=βnxn+(1-βn)T(tn)xn,yn+1=βn+1xn+1+(1-βn+1)T(tn+1)xn+1.
We note that
(3.25)yn+1-yn=βn+1xn+1+(1-βn+1)T(tn+1)xn+1-βnxn-(1-βn)T(tn)xn=(1-βn+1)(T(tn+1)xn+1-T(tn)xn)+βn+1(xn+1-xn)+(βn+1-βn)(xn-T(tn)xn).
It follows that
(3.26)‖yn+1-yn‖≤(1-βn+1)‖T(tn+1)xn+1-T(tn)xn‖+βn+1‖xn+1-xn‖+|βn+1-βn|‖xn-T(tn)xn‖.
Now, we consider the first term on the right side of (3.2), we have
(3.27)‖T(tn+1)xn+1-T(tn)xn‖≤‖T(tn+1)xn+1-T(tn+1)xn‖+‖T(tn+1)xn-T(tn)xn‖≤‖xn+1-xn‖+‖T(tn+1)xn-T(tn)xn‖.
Substituting (3.27) into (3.26), we get(3.28)‖yn+1-yn‖≤(1-βn+1)‖xn+1-xn‖+(1-βn+1)‖T(tn+1)xn-T(tn)xn‖+βn+1‖xn+1-xn‖+|βn+1-βn|‖xn-T(tn)xn‖≤‖xn+1-xn‖+‖T(tn+1)xn-T(tn)xn‖+|βn+1-βn|‖xn-T(tn)xn‖=‖xn+1-xn‖+‖T(tn+1-tn)T(tn)xn-T(tn)xn‖+|βn+1-βn|‖xn-T(tn)xn‖≤‖xn+1-xn‖+sup⁡z∈C‖T(tn+1-tn)T(tn)z-T(tn)z‖+|βn+1-βn|‖xn-T(tn)xn‖.
Similarly, from definition of {xn}, observing that
(3.29)xn+1=αnγf(xn)+δnxn+((1-δn)I-αnA)yn,xn+2=αn+1γf(xn+1)+δn+1xn+1+((1-δn+1)I-αn+1A)yn+1.
We note that
(3.30)xn+2-xn+1=αn+1γf(xn+1)+δn+1xn+1+((1-δn+1)I-αn+1A)yn+1-αnγf(xn)-δnxn-((1-δn)I-αnA)yn=((1-δn+1)I-αn+1A)(yn+1-yn)-(αn+1-αn)Ayn-(δn+1-δn)yn+(δn+1xn+1-δnxn)+γ(αn+1f(xn+1)-αnf(xn))=((1-δn+1)I-αn+1A)(yn+1-yn)+(δn+1xn+1-δn+1xn)+γαn+1(f(xn+1)-f(xn))+(αn+1-αn)(γf(xn)-Ayn)+(δn+1-δn)(xn-yn).
It follows that
(3.31)‖xn+2-xn+1‖≤‖(1-δn+1)I-αn+1A‖‖yn+1-yn‖+δn+1‖xn+1-xn‖+γαn+1‖f(xn+1)-f(xn)‖+|αn+1-αn|‖γf(xn)-Ayn‖+|δn+1-δn|‖xn-yn‖≤(1-δn+1-αn+1γ¯)‖yn+1-yn‖+δn+1‖xn+1-xn‖+γααn+1‖xn+1-xn‖+|αn+1-αn|(γ‖f(xn)‖+‖Ayn‖)+|δn+1-δn|‖xn-yn‖.
Substituting (3.28) into (3.31), we get
(3.32)‖xn+2-xn+1‖≤[1-(γ¯-γα)αn+1]‖xn+1-xn‖+(1-δn+1-αn+1γ¯)sup⁡z∈C‖T(tn+1-tn)T(tn)z-T(tn)z‖+(1-δn+1-αn+1γ¯)|βn+1-βn|‖xn-T(tn)xn‖+|αn+1-αn|(γ‖f(xn)‖+‖Ayn‖)+|δn+1-δn|(1-βn)‖T(tn)xn-xn‖≤[1-(γ¯-γα)αn+1]‖xn+1-xn‖+sup⁡z∈C‖T(tn+1-tn)T(tn)z-T(tn)z‖+|βn+1-βn|‖xn-T(tn)xn‖+|αn+1-αn|(γ‖f(xn)‖+‖Ayn‖)+|δn+1-δn|‖T(tn)xn-xn‖≤[1-(γ¯-γα)αn+1]‖xn+1-xn‖+sup⁡z∈C‖T(tn+1-tn)T(tn)z-T(tn)z‖+M1(|αn+1-αn|+|βn+1-βn|+|δn+1-δn|),
where M1 is an appropriate constant such that
(3.33)M1≥‖xn-T(tn)xn‖+γ‖f(xn)‖+‖Ayn‖∀n∈N.
Putting cn:=M1(|αn+1-αn|+|βn+1-βn|+|δn+1-δn|)+supz∈C∥T(tn+1-tn)T(tn)z-T(tn)z∥, observing control conditions (C2) and (C3), we have
(3.34)∑n=1∞cn=∑n=1∞(M1(|αn+1-αn|+|βn+1-βn|+|δn+1-δn|)+sup⁡z∈C‖T(tn+1-tn)T(tn)z-T(tn)z‖)≤M1∑n=1∞(|αn+1-αn|+|βn+1-βn|+|δn+1-δn|)+∑n=1∞sup⁡z∈C{‖T(tn+1-tn)T(tn)z-T(tn)z‖}<∞.
Hence, by Lemma 2.6 to (3.32), we get that
(3.35)lim⁡n→∞‖xn+1-xn‖=0.
Next, we show that
(3.36)lim⁡n→∞‖T(h)xn-xn‖=0.
Observe that
(3.37)‖yn-xn‖≤‖xn-xn+1‖+‖xn+1-yn‖=‖xn-xn+1‖+‖αnγf(xn)+δn(xn-yn)-αnAyn‖≤‖xn-xn+1‖+δn‖xn-yn‖+αn(γ‖f(xn)‖+‖Ayn‖).
It follows that
(3.38)‖yn-xn‖≤11-δn(‖xn-xn+1‖+αn(γ‖f(xn)‖+‖Ayn‖)).
Observing control condition (C1) and noting (3.10), we have
(3.39)lim⁡n→∞‖yn-xn‖=0.
Moreover, we note that
(3.40)‖xn-T(tn)xn‖≤‖xn-yn‖+‖yn-T(tn)xn‖=‖xn-yn‖+βn‖xn-T(tn)xn‖.
It follows that
(3.41)‖xn-T(tn)xn‖≤11-βn‖xn-yn‖.
Noting (3.19), hence
(3.42)lim⁡n→∞‖xn-T(tn)xn‖=0.
On the other hand, we note that
(3.43)‖T(h)xn-xn‖≤‖T(h)xn-T(h)T(tn)xn‖+‖T(h)T(tn)xn-T(tn)xn‖+‖T(tn)xn-xn‖≤2‖xn-T(tn)xn‖+‖T(h)T(tn)xn-T(tn)xn‖≤2‖xn-T(tn)xn‖+sup⁡z∈C‖T(h)T(tn)z-T(tn)z‖.
Therefor, by (3.26) and control condition (C3) imply that limn→∞supz∈C∥T(h)T(tn)z-T(tn)z∥=0, we have
(3.44)lim⁡n→∞‖T(h)xn-xn‖=0.
Next, we show that(3.45)limsup⁡n→∞〈γf(x*)-Ax*,J(xn-x*)〉≤0.
For eacht∈(0,1), let xt be a unique point of C such that xt=tγf(xt)+(I-tA)T(λt)xt. By Lemma 3.1, we have(3.46)‖xt-xn‖2≤‖(I-tA)(T(λt)xt-xn)‖2+2t〈γf(xt)-Axn,J(xt-xn)〉≤‖I-tA‖2‖T(λt)xt-xn‖2+2t〈γf(xt)-Axn,J(xt-xn)〉≤(1-tγ¯)2(‖T(λt)xt-T(λt)xn‖+‖T(λt)xn-xn‖)2+2t〈γf(xt)-Axn,J(xt-xn)〉≤(1-2γ¯t+γ¯2t2)‖xt-xn‖2+fn(t)+2t〈γf(xt)-Axt,J(xt-xn)〉+2t〈Axt-Axn,J(xt-xn)〉,
where fn(t):=(2∥xt-xn∥+∥xn-T(λt)xn∥)∥xn-T(λt)xn∥.
On the other hand, observe thatA is a strongly positive linear bounded operator, it follows from (2.3), we have(3.47)〈Axt-Axn,J(xt-xn)〉=〈A(xt-xn),J(xt-xn)〉≥γ¯‖xt-xn‖2.
Combining (3.47) with (3.46), we have
(3.48)2t〈Axt-γf(xt),J(xt-xn)〉≤(γ¯2t2-2γ¯t)‖xt-xn‖2+an(t)+2t〈Axt-Axn,J(xt-xn)〉≤(γ¯t2-2t)〈Axt-Axn,J(xt-xn)〉+2t〈Axt-Axn,J(xt-xn)〉+fn(t)=γ¯t2〈Axt-Axn,J(xt-xn)〉+fn(t).
It follows that
(3.49)〈Axt-γf(xt),J(xt-xn)〉≤γ¯t2〈Axt-Axn,J(xt-xn)〉+12tfn(t).
Now, taking limit superior as n→∞ firstly, and then as t→0 in (3.49) (using (3.44)), we have
(3.50)limsup⁡n→∞〈Axt-γf(xt),J(xt-xn)〉≤t2M2,
where M2>0 is a constant such that M2≥γ¯〈Axt-Axn,J(xt-xn)〉 for all t∈(0,1) and n≥1. Now, taking limit superior as t→0 in (3.50). Hence, we get
(3.51)limsup⁡t→0limsup⁡n→∞〈Axt-γf(xt),J(xt-xn)〉≤0.
Moreover, we note that
(3.52)〈γf(x*)-Ax*,J(xn-x*)〉=〈γf(x*)-Ax*,J(xn-x*)〉-〈γf(x*)-Ax*,J(xn-xt)〉+〈γf(x*)-Ax*,J(xn-xt)〉-〈γf(x*)-Axt,J(xn-xt)〉+〈γf(x*)-Axt,J(xn-xt)〉-〈γf(xt)-Axt,J(xn-xt)〉+〈γf(xt)-Axt,J(xn-xt)〉=〈γf(x*)-Ax*,J(xn-x*)-J(xn-xt)〉+〈Axt-Ax*,J(xn-xt)〉+〈γf(x*)-γf(xt),J(xn-xt)〉+〈γf(xt)-Axt,J(xn-xt)〉.
Taking limit superior as n→∞ in (3.52), we have
(3.53)limsup⁡n→∞〈γf(x*)-Ax*,J(xn-x*)〉≤limsup⁡n→∞〈γf(x*)-Ax*,J(xn-x*)-J(xn-xt)〉+‖A‖‖xt-x*‖limsup⁡n→∞‖xn-xt‖+‖γf(x*)-γf(xt)‖limsup⁡n→∞‖xn-xt‖+limsup⁡n→∞〈γf(xt)-Axt,J(xn-xt)〉≤limsup⁡n→∞〈γf(x*)-Ax*,J(xn-x*)-J(xn-xt)〉+(‖A‖+γα)‖xt-x*‖limsup⁡n→∞‖xn-xt‖+limsup⁡n→∞〈γf(xt)-Axt,J(xn-xt)〉.
By Lemma 3.1, xt→x*∈F(𝒯) as t→0. Since X is a uniformly smooth Banach space, imply that J is norm-to-norm uniformly continuous on bounded subset of C (see, e.g., [18, Lemma 1]), we obtain
(3.54)limsup⁡t→0limsup⁡n→∞〈γf(x*)-Ax*,J(xn-x*)-J(xn-xt)〉=0.
Therefore, from (3.53), we have
(3.55)limsup⁡n→∞〈γf(x*)-Ax*,J(xn-x*)〉=limsup⁡t→0limsup⁡n→∞〈γf(x*)-Ax*,J(xn-x*)〉≤limsup⁡t→0limsup⁡n→∞〈γf(xt)-Axt,J(xn-xt)〉≤0.
Finally, we show thatlimn→∞∥xn-x*∥=0. By Lemma 2.5, we have(3.56)‖xn+1-x*‖2=‖αnγf(xn)+δnxn+((1-δn)I-αnA)yn-x*‖2=‖((1-δn)I-αnA)(yn-x*)+δn(xn-x*)+αn(γf(xn)-Ax*)‖2≤‖((1-δn)I-αnA)(yn-x*)+δn(xn-x*)‖2+2αn〈γf(xn)-Ax*,J(xn+1-x*)〉≤[‖((1-δn)I-αnA)(yn-x*)‖+δn‖xn-x*‖]2+2αn〈γf(xn)-Ax*,J(xn+1-x*)〉≤[‖(1-δn)I-αnA‖‖yn-x*‖+δn‖xn-x*‖]2+2αn〈γf(xn)-Ax*,J(xn+1-x*)〉≤[‖1-δn-αnγ¯‖‖xn-x*‖+δn‖xn-x*‖]2+2αn〈γf(xn)-Ax*,J(xn+1-x*)〉≤(1-αnγ¯)2‖xn-x*‖2+2αn〈γf(xn)-γf(x*),J(xn+1-x*)〉+2αn〈γf(x*)-Ax*,J(xn+1-x*)〉≤(1-αnγ¯)2‖xn-x*‖2+2αnγα‖xn-x*‖‖xn+1-x*‖+2αn〈γf(x*)-Ax*,J(xn+1-x*)〉≤(1-αnγ¯)2‖xn-x*‖2+αnγα(‖xn-x*‖2+‖xn+1-x*‖2)+2αn〈γf(x*)-Ax*,J(xn+1-x*)〉.
It follows that
(3.57)‖xn+1-x*‖2≤[(1-αnγ¯)2+αnγα1-αnγα]‖xn-x*‖2+2αn1-αnγα〈γf(x*)-Ax*,J(xn+1-x*)〉=[1-2αnγ¯+αnγα1-αnγα]‖xn-x*‖2+αn2γ¯21-αnγα‖xn-x*‖2+2αn1-αnγα〈γf(x*)-Ax*,J(xn+1-x*)〉=[1-2αn(γ¯-γα)1-αnγα]‖xn-x*‖2+αn2γ¯21-αnγα‖xn-x*‖2+2αn1-αnγα〈γf(x*)-Ax*,J(xn+1-x*)〉.
Put an:=2αn(γ¯-γα)/(1-αnγα) and bn:=(αnγ¯2/2(γ¯-γα))∥xn-x*∥2+(1/(γ¯-γα))〈γf(x*)-Ax*,J(xn+1-x*)〉. The above reduces to formula ∥xn+1-x*∥2≤(1-an)∥xn-x*∥2+anbn. Observing control condition (C1) and noting (3.55), it is easily seen that ∑n=1∞an=∞ and limsupn→∞bn≤0. By Lemma 2.6, we conclude that limn→∞∥xn-x*∥=0. This completes the proof.Corollary 3.3.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H with C±C⊂C and 𝒯={T(t):0≤t<∞} be a nonexpansive semigroup on C such that F(𝒯)≠∅. Let f:C→C be a contraction mapping with a coefficient α∈(0,1) and A be a strongly positive bounded linear operator with coefficient γ¯>0 such that 0<γ<γ¯/α. Let {αn}n=1∞, {βn}n=1∞, {δn}n=1∞ be the sequences in (0,1) and {tn}n=1∞ be a positive real divergent sequence such that tn≤tn+1 for all n∈ℕ. Assume that the following control conditions that  hold: (C1)
limn→∞αn=0 and ∑n=1∞αn=∞;(C2)
∑n=1∞|αn+1-αn|<∞, ∑n=1∞|βn+1-βn|<∞, and ∑n=1∞|δn+1-δn|<∞;(C3)
∑n=1∞supz∈C{∥T(h)T(tn)z-T(tn)z∥}<∞ for all h≥0.Then the sequence {xn} defined by (1.13) converges strongly to the common fixed point x*, where x* is the unique solution in F(𝒯) of the variational inequality (1.6).Proof.
From Theorem3.2, if X:=H is a real Hilbert space, we get the desired conclusion easily.Corollary 3.4.
LetC be a nonempty bounded closed convex subset of a uniformly smooth Banach space X which admit a weakly sequentially continuous duality mapping from X into X* and 𝒯={T(t):0≤t<∞} be a nonexpansive semigroup mapping on C such that F(𝒯)≠∅. Let f:C→C be a contraction mapping with a coefficient α∈(0,1). Let {αn}n=1∞, {βn}n=1∞, {δn}n=1∞ be the sequences in (0,1) and {tn}n=1∞ be a positive real divergent sequence such that tn≤tn+1 for all n∈ℕ. Assume that the following control conditions hold: (C1)
limn→∞αn=0 and ∑n=1∞αn=∞;(C2)
∑n=1∞|αn+1-αn|<∞, ∑n=1∞|βn+1-βn|<∞, and ∑n=1∞|δn+1-δn|<∞;(C3)
∑n=1∞supz∈C{∥T(h)T(tn)z-T(tn)z∥}<∞ for all h≥0.Then the sequence {xn} defined by
(3.58)x1=x∈Cchosenarbitrarily,yn=βnxn+(1-βn)T(tn)xn;xn+1=αnf(xn)+δnxn+(1-δn-αn)yn,∀n≥1,
converges strongly to the fixed point x*, where x* is the unique solution in F(𝒯) of the variational inequality (1.11).Proof.
TakingA=I and γ=1, then (1.13) is reduced to (3.58).Corollary 3.5 (Marino and Xu [12]).
LetH be a real Hilbert space. Let T be a nonexpansive mapping on H such that F(T)≠∅, f be a contraction mapping with a coefficient α∈(0,1) and A be a strongly positive bounded linear operator with a coefficient γ¯>0 such that 0<γ<γ¯/α. Assume that  the following control conditions hold: (C1)
limn→∞αn=0 and ∑n=1∞αn=∞;(C2)
∑n=1∞|αn+1-αn|<∞.Then the sequence {xn} defined by
(3.59)x0=x∈Hchosenarbitrarily;xn+1=αnγf(xn)+(I-αnA)Txn,∀n≥0,
converges strongly to the fixed point x*, where x* is the unique solution in F(T) of the variational inequality (1.6).Proof.
IfX:=H is a real Hilbert space, we get the desired conclusion easily. Taking δn=0, βn=0, and T(tn)=T, then (1.13) is reduced to (3.59).Now, we prove the following theorem, under certain different control conditions on the sequence{αn}, {βn} and {δn}.Theorem 3.6.
LetC be a nonempty bounded closed convex subset of a uniformly smooth Banach space X which admit a weakly sequentially continuous duality mapping from X into X* with C±C⊂C and 𝒯={T(t):0≤t<∞} be a nonexpansive semigroup on C such that F(𝒯)≠∅. Let f:C→C be a contraction mapping with a coefficient α∈(0,1) and A be a strongly positive bounded linear operator with coefficient γ¯>0 such that 0<γ<γ¯/α. Let {αn}n=1∞, {βn}n=1∞, {δn}n=1∞ be the sequences in (0,1) and {tn}n=1∞ be a positive real divergent sequence such that tn+1=h+tn for all n∈ℕ and for all h≥0. Assume that the following control conditions are hold: (C1)
limn→∞αn=0 and ∑n=1∞αn=∞;(C2)
limn→∞|βn+1-βn|=0;(C3)
0<liminfn→∞δn≤limsupn→∞δn<1.Suppose that {T(tn)} is a u.a.r. nonexpansive semigroup on C. Then the sequence {xn} defined by (1.13) converges strongly to the common fixed point x*, where x* is the unique solution in F(𝒯) of the variational inequality (3.2).Proof.
By using the same argument and techniques as those of Theorem3.2, we note that {xn} is bounded, and so are the set {yn}, {f(xn)} and {T(tn)xn}. We will show that limn→∞∥xn+1-xn∥=0.
From Theorem3.2, we note that(3.60)‖yn+1-yn‖≤‖xn+1-xn‖+‖T(tn+1)xn-T(tn)xn‖+|βn+1-βn|‖xn-T(tn)xn‖.
Definexn+1=(1-δn)ln+δnxn. That is ln=(xn+1-δnxn)/(1-δn). Now, we compute ln+1-ln.
Then, we note that(3.61)ln+1-ln=xn+2-δn+1xn+11-δn+1-xn+1-δnxn1-δn=αn+1γf(xn+1)+((1-δn+1)I-αn+1A)yn+11-δn+1-αnγf(xn)+((1-δn)I-αnA)yn1-δn=αn+11-δn+1(γf(xn+1)-Ayn+1)+αn1-δn(Ayn-γf(xn))+yn+1-yn,
one has
(3.62)‖ln+1-ln‖≤αn+11-δn+1‖γf(xn+1)-Ayn+1‖+αn1-δn‖Ayn-γf(xn)‖+‖yn+1-yn‖.
Substituting (3.60) into (3.62), we have
(3.63)‖ln+1-ln‖≤αn+11-δn+1‖γf(xn+1)-Ayn+1‖+αn1-δn‖Ayn-γf(xn)‖+‖xn+1-xn‖+‖T(tn+1)xn-T(tn)xn‖+|βn+1-βn|‖xn-T(tn)xn‖.
By assuming that tn+1=h+tn, it follows that
(3.64)‖ln+1-ln‖-‖xn+1-xn‖≤αn+11-δn+1‖γf(xn+1)-Ayn+1‖+αn1-δn‖Ayn-γf(xn)‖+‖T(h)T(tn)xn-T(tn)xn‖+|βn+1-βn|‖xn-T(tn)xn‖.
Since {T(tn)} is a u.a.r. nonexpansive semigroup and limn→∞tn=∞, then for all h≥0, we have
(3.65)lim⁡n→∞‖T(h)T(tn)xn-T(tn)xn‖≤lim⁡n→∞sup⁡z∈C‖T(h)T(tn)z-T(tn)z‖=0.
Moreover, since {xn}, {T(tn)xn} are bounded, observing control conditions (C1), (C2), (C3), and noting (3.65) then, inequality (3.64) implies that
(3.66)limsup⁡n→∞(‖ln+1-ln‖-‖xn+1-xn‖)≤0.
It follows by Lemma 2.5 that
(3.67)lim⁡n→∞‖ln-xn‖=0.
Observing that
(3.68)xn+1-xn=(1-δn)ln+δnxn-xn=(1-δn)(ln-xn),
and hence,
(3.69)lim⁡n→∞‖xn+1-xn‖=0.
By the same proof in Theorem 3.2, we note that
(3.70)‖yn-xn‖≤11-δn(‖xn-xn+1‖+αn(γ‖f(xn)‖+‖Ayn‖)).
Hence limn→∞∥yn-xn∥=0. Moreover, by Theorem 3.2, we note that
(3.71)lim⁡n→∞‖xn-T(tn)xn‖=0.
Next, we show that limn→∞∥T(h)xn-xn∥=0. We note that
(3.72)‖T(h)xn-xn‖≤‖T(h)xn-T(h)T(tn)xn‖+‖T(h)T(tn)xn-T(tn)xn‖+‖T(tn)xn-xn‖≤2‖xn-T(tn)xn‖+‖T(h)T(tn)xn-T(tn)xn‖.
Since {T(tn)} is a u.a.r. nonexpansive semigroup, and noting (3.65), we have
(3.73)lim⁡n→∞‖T(h)xn-xn‖=0
for all h≥0. By using the same argument and techniques as those of Theorem 3.2, we have {xn} converges strongly to a common fixed point x*∈F(𝒯). This completes the proof.Corollary 3.7.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H with C±C⊂C and let 𝒯={T(t):0≤t<∞} be a nonexpansive semigroup on C such that F(𝒯)≠∅. Let f:C→C be a contraction mapping with a coefficient α∈(0,1) and A be a strongly positive bounded linear operator with coefficient γ¯>0 such that 0<γ<γ¯/α. Let {αn}n=1∞, {βn}n=1∞, {δn}n=1∞ be the sequences in (0,1) and {tn}n=1∞ be a positive real divergent sequence such that tn+1=h+tn for all n∈ℕ and for all h≥0. Assume that the following control conditions are hold: (C1)
limn→∞αn=0 and ∑n=1∞αn=∞;(C2)
limn→∞|βn+1-βn|=0;(C3)
0<liminfn→∞δn≤limsupn→∞δn<1.Suppose that {T(tn)} is a u.a.r. nonexpansive semigroup on C. Then the sequence {xn} defined by (1.13) converges strongly to the fixed point x*, where x* is the unique solution in F(𝒯) of the variational inequality (1.6).Proof.
From Theorem3.6, if X:=H is a real Hilbert space, we get the desired conclusion easily.If we takingA=I and γ=1, then (1.13) is reduced to (3.74). So, we obtain the following corollary.Corollary 3.8.
LetC be a nonempty bounded closed convex subset of a uniformly smooth Banach space X which admit a weakly sequentially continuous duality mapping from X into X* and let 𝒯={T(t):0≤t<∞} be a nonexpansive semigroup on C such that F(𝒯)≠∅. Let f:C→C be a contraction mapping with a coefficient α∈(0,1) and {αn}n=1∞, {βn}n=1∞, {δn}n=1∞ be the sequences in (0,1) and {tn}n=1∞ be a positive real divergent sequence such that tn+1=h+tn for all n∈ℕ and for all h≥0. Assume that the following control conditions hold: (C1)
limn→∞αn=0 and ∑n=1∞αn=∞;(C2)
limn→∞|βn+1-βn|=0;(C3)
0<liminfn→∞δn≤limsupn→∞δn<1.Suppose that {T(tn)} is a u.a.r. nonexpansive semigroup on C. Then the sequence {xn} defined by
(3.74)x1=x∈Cchosenarbitrarily;yn=βnxn+(1-βn)T(tn)xn;xn+1=αnf(xn)+δnxn+(1-δn-αn)yn,∀n≥1,
converges strongly to the fixed point x*, where x* is the unique solution in F(𝒯) of the variational inequality (1.11).



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*Source: 576135-2011-01-26.xml*