# A Viscosity Approximation Method with a Weakly Contractive Mapping

Authors: Pongsakorn Sunthrayuth; Chanan Sudsukh; Poom Kumam
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

We introduce a new viscosity approximation method with a weakly contractive mapping of
general iterative processes for finding common fixed point of nonexpansive semigroups{T(t):t∈ℝ+} in the
framework of Banach spaces. We proved that under some mild conditions these iterative processes converge
strongly to the common fixed point of {T(t):t∈ℝ+}, which is the unique solution of some variational
inequality. The results obtained in this paper extend and improve on the recent results of Li et al. (2009), Chen and He (2007), and many others as special cases.


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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) as the set of fixed points of T. We know that F(T) is nonempty if C is bounded; for more detail see [1]. Throughout this paper we denote by ℕ and ℝ+ the set of all positive integers and all positive real numbers, respectively. 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.Definition 1.1.
A mappingϕ:C→C is said to be weakly contractive if there exists a continuous and strictly increasing function ψ:ℝ+→ℝ+,ψ(0)=0 such that
(1.1)‖ϕ(x)-ϕ(y)‖≤‖x-y‖-ψ(‖x-y‖),∀x,y∈C.Remark 1.2.
As a special case, if we considerψ(t):=(1-α)t, for each t∈[0,∞) and α∈(0,1), then we get the contraction mapping with the coefficient α.In the last ten years or so, 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 γ̅>0 with property (1.2)〈Ax,x〉≥γ̅‖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: (1.3)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 [5] proved that the sequence {xn} defined by the iterative method below, with the initial guess x0∈H chosen arbitrarily: (1.4)xn+1=(I-αnA)Txn+αnu,∀n≥0,
converges strongly to the unique solution of the minimization problem (1.3) provided the sequence {αn} satisfies certain conditions. Using the viscosity approximation method, Moudafi [6] introduced the following iterative process for nonexpansive mappings (see [7, 8] for further developments in both Hilbert and Banach spaces). Let f be a contraction on H. Starting with an arbitrary initial x0∈H, we defined the sequence {xn} recursively by (1.5)xn+1=σnf(xn)+(1-σn)Txn,∀n≥0,
where {σn} is a sequence in (0,1). It is proved in [6, 8] that under certain appropriate conditions imposed on {σn}, the sequence {xn} generated by (1.5) strongly converges to a unique solution x* of the variational inequality (1.6)〈(f-I)x*,ω-x*〉≤0,∀ω∈F(T).
Recently, Marino and Xu [9] combined the iterative method (1.4) with the viscosity approximation method (1.5) considering the following general iterative process: (1.7)xn+1=αnγf(xn)+(I-αnA)Txn,∀n≥0,
where 0<γ<(γ̅/α). They proved that the sequence {xn} generated by (1.7) converges strongly to a unique solution x* of the variational inequality (1.8)〈(γf-A)x*,ω-x*〉≤0,∀ω∈F(T),
which is the optimality condition for the minimization problem: minx∈C(1/2)〈Ax,x〉-h(x), where h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H).On the other hand, Shioji and Takahashi [10] introduced in a Hilbert space the implicit iteration as follows: (1.9)xn=αnu+(1-αn)1tn∫0tnT(s)xnds,∀n∈N,
where {αn} is a sequence in (0,1)and{tn} is a positive real divergent sequence and for u∈C. They proved, under certain restrictions on the sequence {αn}, that the sequence {xn} defined by (1.9) converges strongly to a member of F(𝒮) (see also [11]).Chen and Song [12] studied the strong convergence of the following sequence (1.10) for a nonexpansive semigroup 𝒮={T(t):t∈ℝ+} with F(𝒮)≠∅ in a uniformly convex Banach space: (1.10)xn=αnf(xn)+(1-αn)1tn∫0tnT(s)xnds,∀n∈N.
Suzuki [13] was the first to introduced again in a Hilbert space the following implicit iteration process: (1.11)xn=αnu+(1-αn)T(tn)xn,∀n∈N,
for a nonexpansive semigroup case. Xu [14] established a Banach space version of the sequence (1.11) of Suzuki [13].In 2007, Chen and He [15] extended the result of Suzuki [13] and Xu [14] and studied the strong convergence theorem of viscosity implicit iteration process for a nonexpansive semigroup 𝒮={T(t):t∈ℝ+} with F(𝒮)≠∅, in Banach spaces as follows: (1.12)xn=αnf(xn)+(1-αn)T(tn)xn,∀n∈N.
Very recently, S. Li et al. [16] considered a general iterative process for a nonexpansive semigroup 𝒮={T(t):t∈ℝ+} in a Hilbert space as follows: (1.13)xn=αnγf(xn)+(I-αnA)1tn∫0tnT(s)xnds,∀n∈N,
where {αn}⊂(0,1] and {tn} are two sequences satisfying certain conditions. They proved that the sequence {xn} defined by (1.13) converges strongly to x*∈F(𝒮), which solves the following variational inequality (1.8).Question 1.
Can Theorem of S. Li et al. [16] be extended from Hilbert space to a general Banach space?Question 2.
Can we extend the iterative method of algorithms (1.10) and (1.12) to general algorithms?Question 3.
We know that the weakly contractive mapping is more general than the contractive mapping. What happens if the contractive mapping is replaced by the weakly contractive mapping?The purpose of this paper is to give affirmative answer to these questions mentioned above. Motivated by the iterative process (1.10), (1.12), and (1.13), we introduce a new viscosity approximation method with a weakly contractive mapping of general iterative processes for nonexpansive semigroups, which is a unique solution of some variational inequality. We prove the strong convergence theorems of these iterative processes in a Banach space which admits a weakly sequentially continuous duality mapping. The results presented in this paper improve and extend the corresponding results announced by Chen and He [15] and S. Li et al. [16] and many others as special cases.


## 2. Preliminaries

Throughout this paper, letX be a real Banach space and  C a closed convex subset of X. Let J:X→2X* be a normalized duality mapping by 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 notations ⇀ 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 (2.1)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 γ̅>0 with the property (2.2)〈Ax,J(x)〉≥γ̅‖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 (2.1) 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 (2.1) 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) if the limit (2.1) 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 and uniformly convex if δX(ϵ)>0 for all ϵ>0, where δX(ϵ) is modulus of convexity of X defined by δX(ϵ)=inf{1-(∥x+y∥/2):∥x∥≤1,∥y∥≤1,∥x+y∥≥ϵ}, for all ϵ∈[0,2]. A uniformly convex Banach space X is reflexive and strictly convex (see Theorems 4.1.6, and 4.1.2 of [18]) and every uniformly smooth Banach space X is a reflexive Banach with uniformly Gâteaux differentiable norm (see Theorems 4.3.7, and 4.1.6 of [18] and also [19]).In the sequel we will use the following lemmas, which will be used in the proofs for the main results in the next section.Lemma 2.1 (see [17]).
Assume thatA is a strongly positive linear bounded operator on a smooth Banach space X with coefficient γ̅>0 and 0<ρ≤∥A∥-1. Then  ∥I-ρA∥≤1-ργ̅.Lemma 2.2 (see [20]).
Let(X,d) be a complete metric space and T:X→X a weakly contractive mapping. Then, T has a unique fixed point in X.If a Banach spaceX admits a sequentially continuous duality mapping J from weak topology to weak star topology, then, by Lemma 1 of [21], 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 [21–23] 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
(2.3)limsup⁡n→∞⁡‖xn-x‖<limsup⁡n→∞‖xn-y‖∀y∈X,withx≠y.By Theorem 1 in [21], 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 2.3 ([22] (Demiclosed Principle)).
LetC be a nonempty closed convex subset of a reflexive Banach space X which satisfies Opial's condition, and that suppose T:C→X is nonexpansive. Then the mapping I-T is demiclosed at zero, that is, xn⇀x and xn-Txn→0 imply that x=Tx.Lemma 2.4 (see [15]).
LetC be a closed convex subset of a uniformly convex Banach space X and 𝒮={T(t):t∈ℝ+} a nonexpansive semigroup on C such that  F(𝒮)≠∅. Then, for each  r>0 and  h≥0,
(2.4)lim⁡t→∞⁡sup⁡x∈C∩Br⁡‖1t∫0tT(s)xds-T(h)(1t∫0tT(s)xds)‖=0.


## 3. Main Results

In this section, we prove our main results.Theorem 3.1.
LetX be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping J from X into X* and C 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 ϕ be a weakly contractive mapping and A a strongly positive linear bounded operator with a coefficient γ̅>0 such that 0<γ<γ̅. Let {xn} be a sequence defined by
(3.1)yn=βnxn+(1-βn)1tn∫0tnT(s)xnds,xn=αnγϕ(xn)+(I-αnA)yn,∀n≥0,
where {αn},{βn} are two sequences in (0,1) and {tn} is a positive real divergent sequence satisfying the following conditions:  (C1)
limn→∞αn=0,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1.Then the sequence {xn} defined by (3.1) converges strongly to the common fixed point x*∈F(𝒮), where x* is the unique solution of the variational inequality
(3.2)〈γϕ(x*)-Ax*,J(ω-x*)〉≤0,∀ω∈F(S).Proof.
Firstly, we show that{xn} defined by (3.1) is well define. Since limn→∞αn=0, we may assume, with no loss of generality, that αn<∥A∥-1 for each n≥0. Define the mapping Tnϕ:C→C by
(3.3)Tnϕ:=αnγϕ+(I-αnA)[βnI+(1-βn)1tn∫0tnT(s)ds].
Indeed, from Lemma 2.1, we have, for all x,y∈C,
(3.4)‖Tnϕx-Tnϕy‖=αnγ(ϕ(x)-ϕ(y))+βn(I-αnA)(x-y)+(1-βn)(I-αnA)1tn∫0tn(T(s)x-T(s)y)ds≤αnγ‖ϕ(x)-ϕ(y)‖+βn‖I-αnA‖‖x-y‖+(1-βn)‖I-αnA‖1tn∫0tn‖T(s)x-T(s)y‖ds≤αnγ[‖x-y‖-ψ(‖x-y‖)]+βn(1-αnγ̅)‖x-y‖+(1-βn)(1-αnγ̅)‖x-y‖=αnγ[‖x-y‖-ψ(‖x-y‖)]+(1-αnγ̅)‖x-y‖=[1-αn(γ̅-γ)]‖x-y‖-αnγψ(‖x-y‖)≤‖x-y‖-αnγψ(‖x-y‖).
This shows that Tnϕ is weakly contractive. It follows from Lemma 2.2 that Tnϕ has a unique fixed point xn∈C, that is, {xn} defined by (3.1) is well defined.
Next, we show that{xn} is bounded. Letting p∈F(𝒮), we get(3.5)‖yn-p‖=‖βn(xn-p)+(1-βn)1tn∫0tn(T(s)xn-p)ds‖≤βn‖xn-p‖+(1-βn)1tn∫0tn‖T(s)xn-p‖ds≤βn‖xn-p‖+(1-βn)‖xn-p‖=‖xn-p‖,‖xn-p‖2=〈αn(γϕ(xn)-Ap)+(I-αnA)(yn-p),J(xn-p)〉=αn〈(γϕ(xn)-γϕ(p))+(γϕ(p)-Ap),J(xn-p)〉+〈(I-αnA)(yn-p),J(xn-p)〉≤αnγ‖ϕ(xn)-ϕ(p)‖‖J(xn-p)‖+αn〈γϕ(p)-Ap,J(xn-p)〉+‖I-αnA‖‖yn-p‖‖J(xn-p)‖≤αn[γ‖xn-p‖-γψ(‖xn-p‖)]‖xn-p‖+αn〈γϕ(p)-Ap,J(xn-p)〉+(1-αnγ̅)‖xn-p‖2=[1-αn(γ̅-γ)]‖xn-p‖2-αnγψ(‖xn-p‖)‖xn-p‖+αn〈γϕ(p)-Ap,J(xn-p)〉,
and so
(3.6)γψ(‖x-p‖)‖xn-p‖+(γ̅-γ)‖xn-p‖≤〈γϕ(p)-Ap,J(xn-p)〉.
Thus,
(3.7)γψ(‖x-p‖)‖xn-p‖≤〈γϕ(p)-Ap,J(xn-p)〉≤‖γϕ(p)-Ap‖‖J(xn-p)‖=‖γϕ(p)-Ap‖‖xn-p‖.
It follows that γψ(∥xn-p∥)≤∥γϕ(p)-Ap∥. Hence,
(3.8)‖xn-p‖≤ψ-1(‖γϕ(p)-Ap‖γ),
which implies that {xn} is bounded. Since ϕ is weakly contractive, we have
(3.9)‖ϕ(xn)-ϕ(p)‖≤‖xn-p‖-ψ(‖xn-p‖)≤‖xn-p‖.
Then, {ϕ(xn)} is bounded. From (3.1), we have
(3.10)(I-αnA)yn=xn-αnγϕ(xn)‖yn‖≤11-αnγ̅‖xn‖+αnγ1-αnγ̅‖ϕ(xn)‖.
Thus, {yn} is also bounded. We denoted zn:=(1/tn)∫0tnT(s)xnds. And since
(3.11)zn=11-βnyn-βn1-βnxn,‖zn‖≤11-βn‖yn‖+βn1-βn‖xn‖,{zn} is also bounded by the boundedness of {xn} and {yn}.
Next, we show that∥xn-T(h)xn∥→0 as n→∞, for all h≥0. We note that(3.12)‖xn-zn‖≤‖xn-yn‖+‖yn-zn‖=αn‖γϕ(xn)-Ayn‖+βn‖xn-zn‖.
It follows that
(3.13)(1-βn)‖xn-zn‖≤αn‖γϕ(xn)-Ayn‖.
By conditions (C1) and (C2), we obtain
(3.14)lim⁡n→∞‖xn-zn‖=0.
Moreover, we note that, for all h≥0,
(3.15)‖xn-T(h)xn‖≤‖xn-zn‖+‖zn-T(h)zn‖+‖T(h)zn-T(h)xn‖≤2‖xn-zn‖+‖zn-T(h)zn‖.
Define the set K={z∈C:∥z-p∥≤ψ-1(∥γϕ(p)-Ap∥/γ)}; then K is a nonempty bounded closed convex subset of C, which isT(h)-invariant for each h≥0 (i.e., T(h)C⊂C). Since {xn}⊂C and C is bounded, there exists r>0 such that K⊂Br, and it follows from Lemma 2.4 that
(3.16)lim⁡n→∞‖zn-T(h)zn‖=0,∀h≥0.
Noting (3.14) and (3.16), then, from (3.15), we obtain
(3.17)lim⁡n→∞‖xn-T(h)xn‖=0,∀h≥0.
Next, we show that{xn} contains a subsequence converging strongly to x̃∈F(𝒮). Since {xn} is bounded and Banach space X is a uniformly convex, it is reflexive and there exists a subsequence {xnj}⊂{xn}, which converges weakly to some x̃∈C as j→∞. Again since Banach space X has a weakly sequentially continuous duality mapping satisfying Opial's condition, noting (3.17) and by Lemma 2.3, we have x̃∈F(𝒮). From (3.7), replace p by x̃ to obtain(3.18)γψ(‖x-x̃‖)‖xnj-x̃‖≤〈γϕ(x̃)-Ax̃,J(xnj-x̃)〉.
Since J is single valued and weakly sequentially continuous from X to X*, we get that
(3.19)lim⁡j→∞γψ(‖xnj-x̃‖)‖xnj-x̃‖≤lim⁡j→∞〈γϕ(x̃)-Ax̃,J(xnj-x̃)〉=0.
Thus, xnj→x̃ as j→∞.
Next, we show thatx̃ is a solution of the variational inequality (3.2). Firstly, since(3.20)‖xn-yn‖=αn‖γϕ(xn)-Ayn‖,
by condition (C1), we obtain limn→∞∥xn-yn∥=0. Since xnj→x̃∈F(𝒮), then ynj→x̃∈F(𝒮).
From (3.1), we derive that γϕ(xn)-Ayn=(1/αn)(xn-yn). Then, for each ω∈F(𝒮),(3.21)〈γϕ(xn)-Ayn,J(ω-xn)〉=1αn〈xn-yn,J(ω-xn)〉=1αn(1-βn)〈xn-1tn∫0tnT(s)xnds,J(ω-xn)〉=1αn(1-βn)[〈ω-1tn∫0tnT(s)xnds,J(ω-xn)〉-‖ω-xn‖2]≤1αn(1-βn)[‖ω-xn‖2-‖ω-xn‖2]=0.
Therefore,
(3.22)〈γϕ(xn)-Ayn,J(ω-xn)≤0.
Since the duality mapping J is single-valued and weakly sequentially continuous duality mapping from X to X*, for each ω∈F(𝒮),xnj→x̃ and ynj→x̃, then, from (3.22), we obtain
(3.23)lim⁡j→∞〈γϕ(xnj)-Aynj,J(ω-xnj)〉=〈γϕ(x̃)-Ax̃,J(ω-x̃)〉≤0.
That is, x̃∈F(𝒮) is a solution of the variational inequality (3.2).
Next, we show the uniqueness of the solution of the variational inequality (3.2). Suppose that x̃,x*∈F(𝒮) satisfy (3.2). Then,(3.24)〈γϕ(x̃)-Ax̃,J(x*-x̃)〉≤0,〈γϕ(x*)-Ax*,J(x̃-x*)〉≤0.
Adding up (3.24), we get
(3.25)0≥〈(γϕ-A)x̃-(γϕ-A)x*,J(x*-x̃)〉=〈A(x*-x̃),J(x*-x̃)〉-γ〈ϕ(x*)-ϕ(x̃),J(x*-x̃)〉≥γ̅‖x*-x̃‖2-γ‖ϕ(x*)-ϕ(x̃)‖‖J(x*-x̃)‖≥γ̅‖x*-x̃‖2-γ‖x*-x̃‖2+γψ(‖x*-x̃‖)‖x*-x̃‖.
Thus γψ(∥x*-x̃∥)≤(γ-γ̅)∥x*-x̃∥. By the property of ψ, we must have x̃=x* and the uniqueness is proved.
Finally, we show that{xn} converges strongly to x̃∈F(𝒮). Suppose that there exists another subsequence xni→x̂ as i→∞. We note that x̂∈F(𝒮) is the solution of the variational inequality (3.2). Hence, x̃=x̂=x* by uniqueness. In summary, we have shown that {xn} is sequentially compact and each cluster point of the sequence {xn} is equal to x*. Therefore, we conclude that xn→x* as n→∞. This proof is complete.Remark 3.2.
(1) Theorem3.1 improves and generalizes Theorem 3.1 of S. Li et al. [16] from a contractive mapping to a weakly contractive mapping and from Hilbert spaces to Banach spaces.
(2) Theorem3.1 also improves and generalizes Theorem 3.2 of Marino and Xu [9] from a nonexpansive mapping to a nonexpansive semigroup, from a contractive mapping to a weakly contractive mapping and from Hilbert spaces to Banach spaces.A strong mean convergence theorem for nonexpansive mapping was first established by Baillon [24], and it was generalized to that for nonlinear semigroups by Reich [25–27]. It is clear that Theorem 3.1 is valid for nonexpansive mappings. Thus, we have the following mean ergodic theorem of viscosity iteration process for nonexpansive mappings in Hilbert spaces.Corollary 3.3.
LetH be a real Hilbert space and C 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)≠∅.  Let ϕ be a weakly contractive mapping and A a strongly positive linear bounded operator with a coefficient γ̅>0 such that 0<γ<γ̅.  Let {xn} be a sequence defined by
(3.26)yn=βnxn+(1-βn)1n+1∑j=0nTjxn,xn=αnγϕ(xn)+(I-αnA)yn,∀n≥0,
where {αn},{βn} are two sequences in (0,1) satisfying the following conditions:  (C1)
limn→∞αn=0,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1.Then the sequence {xn} defined by (3.26) converges strongly to the common fixed point x*∈F(𝒮), where x* is the unique solution of the variational inequality
(3.27)〈γϕ(x*)-Ax*,J(ω-x*)〉≤0,∀ω∈F(S).TakingA=I and γ=1 in Theorem 3.1, we get the following corollary.Corollary 3.4.
LetX be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping J from X into X*, and C 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 ϕ be a weakly contractive mapping. Let {xn} be a sequence defined by
(3.28)yn=βnxn+(1-βn)1tn∫0tnT(s)xnds,xn=αnϕ(xn)+(1-αn)yn,∀n≥0,
where {αn}, {βn} are two sequences in (0,1) and {tn} is a positive real divergent sequence satisfying the following conditions:  (C1)
limn→∞αn=0,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1.Then the sequence {xn} defined by (3.28) converges strongly to the common fixed point x*∈F(𝒮), where x* is the unique solution of the variational inequality
(3.29)〈ϕ(x*)-x*,J(ω-x*)〉≤0,∀ω∈F(S).Next, we prove a strong convergence theorem under different conditions.Theorem 3.5.
LetX be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping J from X into X* and C 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 ϕ be a weakly contractive mapping and A a strongly positive linear bounded operator with a coefficient γ̅>0 such that 0<γ<γ̅. Let {xn} be a sequence defined by
(3.30)yn=βnxn+(1-βn)T(tn)xn,xn=αnγϕ(xn)+(I-αnA)yn,∀n≥0,
where {αn}, {βn} are two sequences in (0,1) and {tn} is a positive real sequence satisfying the following conditions:  (C1)
limn→∞αn=0,(C2)
limn→∞tn=limn→∞(αn/tn)=0.Then, the sequence {xn} defined by (3.30) converges strongly to the common fixed point x*∈F(𝒮), where x* is the unique solution of the variational inequality
(3.31)〈γϕ(x*)-Ax*,J(ω-x*)〉≤0,∀ω∈F(S).Proof.
Firstly, we show that{xn} defined by (3.30) is well defined. Since limn→∞αn=0, we may assume, with no loss of generality, that αn<∥A∥-1 for each n≥0. Define the mapping Tnϕ:C→C by
(3.32)Tnϕ:=αnγϕ+(I-αnA)[βnI+(1-βn)T(tn)].
From Lemma 2.1, we have for all x,y∈C,
(3.33)‖Tnϕx-Tnϕy‖=αnγ(ϕ(x)-ϕ(y))+βn(I-αnA)(x-y)+(1-βn)(I-αnA)(T(tn)x-T(tn)y)≤αnγ‖ϕ(x)-ϕ(y)‖+βn‖I-αnA‖‖x-y‖+(1-βn)‖I-αnA‖‖T(tn)x-T(tn)y‖≤αnγ[‖x-y‖-ψ(‖x-y‖)]+βn(1-αnγ̅)‖x-y‖+(1-βn)(1-αnγ̅)‖x-y‖=αnγ[‖x-y‖-ψ(‖x-y‖)]+(1-αnγ̅)‖x-y‖=[1-αn(γ̅-γ)]‖x-y‖-αnγψ(‖x-y‖)≤‖x-y‖-αnγψ(‖x-y‖).
This show that Tnϕ is weakly contractive. It follows from Lemma 2.2 that Tnϕ has a unique fixed point xn∈C, that is, {xn} defined by (3.30) is well defined.
Next, we show that{xn} is bounded. Letting p∈F(𝒮), we get(3.34)‖yn-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-p‖2=〈αn(γϕ(xn)-Ap)+(I-αnA)(yn-p),J(xn-p)〉=αn〈(γϕ(xn)-γϕ(p))+(γϕ(p)-Ap),J(xn-p)〉+〈(I-αnA)(yn-p),J(xn-p)〉≤αnγ‖ϕ(xn)-ϕ(p)‖‖J(xn-p)‖+αn〈γϕ(p)-Ap,J(xn-p)〉+‖I-αnA‖‖yn-p‖‖J(xn-p)‖≤αn[γ‖xn-p‖-γψ(‖xn-p‖)]‖xn-p‖+αn〈γϕ(p)-Ap,J(xn-p)〉+(1-αnγ̅)‖xn-p‖2=[1-αn(γ̅-γ)]‖xn-p‖2-αnγψ(‖xn-p‖)‖xn-p‖+αn〈γϕ(p)-Ap,J(xn-p)〉,
and so
(3.35)γψ(‖x-p‖)‖xn-p‖+(γ̅-γ)‖xn-p‖≤〈γϕ(p)-Ap,J(xn-p)〉.
Thus,
(3.36)γψ(‖x-p‖)‖xn-p‖≤〈γϕ(p)-Ap,J(xn-p)〉≤‖γϕ(p)-Ap‖‖J(xn-p)‖=‖γϕ(p)-Ap‖‖xn-p‖.
It follows that γψ(∥xn-p∥)≤∥γϕ(p)-Ap∥. Hence,
(3.37)‖xn-p‖≤ψ-1(‖γϕ(p)-Ap‖γ).
This implies that {xn} is bounded, so are {ϕ(xn)}, {Ayn} and {T(tn)xn}.
Next, we show that{xn} contains a subsequence converging strongly to x̃∈F(𝒮). By reflexivity of X and boundedness of the sequence {xn}, there exists subsequence {xnj}⊆{xn} such that xnj⇀x̃∈F(𝒮) as j→∞. Now, we show that x̃∈F(𝒮). Put xj∶=xnj, yj∶=ynj, αj∶=αnj, βj∶=βnj, and tj∶=tnj for j∈ℕ, and fix t>0. We note that(3.38)‖xj-T(t)x̃‖≤∑k=0[t/tj]-1‖T((k+1)tj)xj-T(ktj)xj‖+‖T([ttj]tj)xj-T([ttj]tj)x̃‖+‖T([ttj]tj)x̃-T(t)x̃‖≤[ttj]‖T(tj)xj-xj‖+‖xj-x̃‖+‖T(t-[ttj]tj)x̃-x̃‖=[ttj]αj1-βj‖γϕ(xj)-Ayj‖+‖xj-x̃‖+‖T(t-[ttj]tj)x̃-x̃‖≤t1-βjαjtj‖γϕ(xj)-Ayj‖+‖xj-x̃‖+max⁡{‖T(s)x̃-x̃‖:0≤s≤tj}.
For all j∈ℕ, we have
(3.39)limsup⁡j→∞‖xj-T(t)x̃‖≤limsup⁡j→∞‖xj-x̃‖.
By the assumption that Banach space X has a weakly sequentially continuous duality mapping satisfying Opial’s condition, (3.39) implies that T(t)x̃=x̃, and we get that x̃∈F(𝒮). From (3.36), replace p by x̃ to obtain
(3.40)γψ(‖xj-x̃‖)‖xj-x̃‖≤〈γϕ(x̃)-Ax̃,J(xj-x̃)〉.
Since J is singlevalued and weakly sequentially continuous from X to X*, we get that
(3.41)lim⁡j→∞γψ(∥xj-x̃∥)∥xj-x̃∥≤lim⁡j→∞〈γϕ(x̃)-Ax̃,J(xj-x̃)〉=0.
Thus, xj→x̃ as j→∞, namely, there is a subsequence {xnj}⊆{xn} such that xnj→x̃ as j→∞.
Next, we show thatx̃ is a solution of the variational inequality (3.31). Firstly, since(3.42)‖xn-yn‖=αn‖γϕ(xn)-Ayn‖.
by condition (C1), we obtain limn→∞∥xn-yn∥=0. Since xnj→x̃∈F(𝒮), then ynj→x̃∈F(𝒮).
From (3.30), we derive that γϕ(xn)-Ayn=(1/αn)(xn-yn). Then, for each ω∈F(𝒮),(3.43)〈γϕ(xn)-Ayn,J(ω-xn)〉=1αn〈xn-yn,J(ω-xn)〉=1αn(1-βn)〈xn-T(tn)xn,J(ω-xn)〉=1αn(1-βn)[〈ω-T(tn)xn,J(ω-xn)〉-‖ω-xn‖2]≤1αn(1-βn)[‖ω-xn‖2-‖ω-xn‖2]=0.
Therefore,
(3.44)〈γϕ(xn)-Ayn,J(ω-xn)〉≤0.
By using the same argument and techniques as those of Theorem 3.1, we note that the variational inequality (3.31) has a unique solution. We denoted by x*∈F(𝒮) the unique solution of (3.31). Therefore, xn→x* as n→∞. The proof is completed.Remark 3.6.
(1) Theorem3.5 improves and generalizes Theorem 3.2 of Marino and Xu [9] from a nonexpansive mapping to a nonexpansive semigroup, from a contractive mapping to a weakly contractive mapping, and from Hilbert spaces to Banach spaces.
(2) Theorem3.5 also improves and generalizes Theorem 3.1 of Chen and He [15] from a contractive mapping to a weakly contractive mapping.TakingA=I and γ=1 in Theorem 3.5, we get the following corollary.Corollary 3.7.
LetX be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping J from X into X* and C 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 ϕ be a weakly contractive mapping. Let {xn} be a sequence defined by
(3.45)yn=βnxn+(1-βn)T(tn)xn,xn=αnϕ(xn)+(1-αn)yn,∀n≥0,
where {αn}, {βn} are two sequences in (0,1) and {tn} is a positive real sequence satisfying the following conditions:  (C1)
limn→∞αn=0,(C2)
limn→∞tn=limn→∞(αn/tn)=0.Then the sequence {xn} defined by (3.45) converges strongly to the common fixed point x*∈F(𝒮), where x* is the unique solution of the variational inequality
(3.46)〈ϕ(x*)-x*,J(ω-x*)〉≤0,∀ω∈F(S).



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*Source: 484061-2011-09-14.xml*