# Strong Convergence of a General Iterative Method for a Countable Family of Nonexpansive Mappings in Banach Spaces

Authors: Chin-Tzong Pang; Eskandar Naraghirad
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We introduce a general algorithm to approximate common fixed points for a countable family of nonexpansive mappings in a real Banach space. We prove strong convergence theorems for the sequences produced by the methods and approximate a common fixed point of a countable family of nonexpansive mappings which solves uniquely the corresponding variational inequality. Furthermore, we apply our results for finding a zero of an accretive operator. It is important to state clearly that the contribution of this paper in relation with the previous works (Marino and Xu, 2006) is a technical method to prove strong convergence theorems of a general iterative algorithm for an infinite family of nonexpansive mappings in Banach spaces. Our results improve and generalize many known results in the current literature.


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## Body

## 1. Introduction

Viscosity approximation method for finding the fixed points of nonexpansive mappings was first proposed by Moudafi [1]. He proved the convergence of the sequence generated by the proposed method. In 2004, Xu [2] proved the strong convergence of the sequence generated by the viscosity approximation method to a unique solution of a certain variational inequality problem defined on the set of fixed points of a nonexpansive map (see also [3]). Marino and Xu [4] considered a general iterative method and proved that the sequence generated by the method converges strongly to a unique solution of a certain variational inequality problem which is the optimality condition for a particular minimization problem. Liu [5] and Qin et al. [6] also studied some applications of the iterative method considered in [4]. Yamada [7] introduced the so-called hybrid steepest-descent method for solving the variational inequality problem and also studied the convergence of the sequence generated by the proposed method. Recently, Tian [8] combined the iterative methods of [4, 7] in order to propose implicit and explicit schemes for constructing a fixed point of a nonexpansive mapping T defined on a real Hilbert space. He also proved the strong convergence of these two schemes to a fixed point of T under appropriate conditions. Related iterative methods for solving fixed point problems, variational inequalities, and optimization problems can be found in [9–14] and the references therein. By virtue of the projection, the authors in [13, 15] extended the implicit and explicit iterative schemes proposed in [8]. The approximation methods for common fixed points of a countable family of nonexpansive mappings have been recently studied by several authors; see, for example, [16, 17].The purpose of this paper is to introduce a general algorithm to approximate common fixed points for a countable family of nonexpansive mappings in a Banach space. We prove strong convergence theorems for the sequences produced by the methods for a common fixed point of a countable family of nonexpansive mappings which solves uniquely the corresponding variational inequality. Furthermore, we apply our results for finding a zero of an accretive operator. Our results improve and generalize many known results in the current literature; see, for example, [4, 7, 8, 13–15, 18–20].


## 2. Preliminaries

Throughout this paper, we denote the set of real numbers and the set of positive integers byℝ and ℕ, respectively. Let E be a Banach space with the norm ∥·∥ and the dual space E*. When {xn} is a sequence in E, we denote the strong convergence of {xn} to x∈E by xn→x and the weak convergence by xn⇀x. For any sequence {xn*} in E*, we denote the strong convergence of {xn*} to x*∈E* by xn*→x*, the weak convergence by xn*⇀x*, and the weak-star convergence by xn*⇀*x*. The normalized duality mapping J:E→2E* is defined by
(1)J(x)={f∈E*:〈x,f〉=∥x∥2,∥x∥=∥f∥},∀x∈E.
The modulus δ of convexity of E is denoted by
(2)δ(ϵ)=inf{1-∥x+y∥2:∥x∥≤1,∥y∥≤1,∥x-y∥≥ϵ}
for every ϵ with 0≤ϵ≤2. A Banach space E is said to be uniformly convex if δ(ϵ)>0 for every ϵ>0. Let S={x∈E:∥x∥=1}. The norm of E is said to be Gâteaux differentiable if for each x,y∈S, the limit
(3)limt→0∥x+ty∥-∥x∥t
exists. In this case, E is called smooth. If the limit (3) is attained uniformly in x,y∈S, then E is called uniformly smooth. The Banach space E is said to be strictly convex if ∥(x+y)/2∥<1 whenever x,y∈S and x≠y. It is well known that E is uniformly convex if and only if E* is uniformly smooth. It is also known that if E is reflexive, then E is strictly convex if and only if E* is smooth; for more details, see [21]. Now, we define a mapping ρ:[0,∞)→[0,∞), the modulus of smoothness of E, as follows:
(4)ρ(t)=sup{12(∥x+y∥+∥x-y∥)-1:ρ(t)=spx,y∈E,∥x∥=1,∥y∥=t12}.
It is well known that E is uniformly smooth if and only if limt→0(ρ(t)/t)=0. Let q∈ℝ be such that 1<q≤2. Then a Banach space E is said to be q-uniformly smooth if there exists a constant cq>0 such that ρ(t)≤cqtq for all t>0. If a Banach space E admits a sequentially continuous duality mapping J from weak topology to weak star topology, then J is single valued and also E is smooth; for more details, see [22]. In this case, the normalized duality mapping J is said to be weakly sequentially continuous; that is, if {xn}⊂E is a sequence with xn⇀x∈E, then J(xn)⇀*J(x) [22]. A Banach space E is said to satisfy the Opial property [23] if for any weakly convergent sequence {xn} in E with weak limit x,
(5)limsupn→∞∥xn-x∥<limsupn→∞∥xn-y∥
for all y∈E with y≠x. It is well known that all Hilbert spaces, all finite dimensional Banach spaces, and the Banach spaces lp (1≤p<∞) satisfy the Opial property; for example, see [22, 23]. It is also known that if E admits a weakly sequentially continuous duality mapping, then E is smooth and enjoys the Opial property; see for more details [22].LetE be a real Banach space and C a nonempty subset of E. Let T:C→E be a mapping. We denote by F(T) the set of fixed points of T; that is, F(T)={x∈C:Tx=x}.Definition 1.
LetC be a nonempty, closed, and convex subset of a real Banach space E. An operator A:C→E is said to be(i)
accretive if there exists j(x-y)∈J(x-y) such that
(6)〈Ax-Ay,j(x-y)〉≥0,∀x,y∈C;(ii)
η-strongly accretive if, for some η>0, there exists j(x-y)∈J(x-y) such that
(7)〈Ax-Ay,j(x-y)〉≥η∥x-y∥2,∀x,y∈C;(iii)
l-Lipschitzian if, for some l>0,
(8)∥Ax-Ay∥≤l∥x-y∥,∀x,y∈C;
in particular, if l∈[0,1), then A is called a contraction;(iv)
nonexpansive if(9)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.A linear bounded operator A:E→E* is said to be strongly positive if there exists γ->0 such that
(10)〈x,Ax〉≥γ-∥x∥2,∀x∈E.Remark 2.
LetC be a nonempty, closed, and convex subset of a real Banach space E and let T:C→C be a nonexpansive mapping. Then I-T is an accretive operator, where I is the identity mapping. Indeed, for any x,y∈C we have
(11)〈(I-T)x-(I-T)y,j(x-y)〉=〈x-y,j(x-y)〉-〈Tx-Ty,j(x-y)〉≥∥x-y∥2-∥Tx-Ty∥∥x-y∥≥∥x-y∥2-∥x-y∥2=0,
which means that I-T is accretive.The following result has been proved in [24].Lemma 3.
LetE be a real 2-uniformly smooth Banach space. Then there exists a best uniformly smooth constant ρ>0 such that
(12)∥x+y∥2≤∥x∥2+2〈y,j(x)〉+2ρ2∥y∥2,
for all x,y∈E.LetC and D be nonempty subsets of real Banach space E with D⊂C. A mapping QD:C→D is said to be sunny if
(13)QD(QDx+t(x-QDx))=QDx
for each x∈E and t≥0. A mapping QD:C→D is said to be a retraction if QDx=x for each x∈C.The following result has been proved in [25].Lemma 4.
LetC and D be nonempty subsets of a real Banach space E with D⊂C and QD:C→D a retraction from C into D. Then QD is sunny and nonexpansive if and only if
(14)〈z-QD(z),j(y-QD(z))〉≤0
for all z∈C and y∈D.Lemma 5 (demiclosedness principle [26]).
LetC be a closed and convex subset of a real 2-uniformly smooth Banach space E and let the normalized duality mapping J:E→E* be weakly sequentially continuous at zero. Suppose that T:C→E is a nonexpansive mapping with F(T)≠⌀. If {xn} is a sequence in C that converges weakly to x and if {(I-T)xn} converges strongly to y, then (I-T)x=y; in particular, if y=0, then x∈F(T).Lemma 6 (see [27]).
Let{sn} be a sequence of nonnegative real numbers satisfying the inequality
(15)sn+1≤(1-γn)sn+γnδn,∀n≥0,
where {γn} and {δn} satisfy the conditions(i)
{γn}⊂[0,1] and ∑n=0∞‍γn=∞, or equivalently, Πn=0∞(1-γn)=0;(ii)
limsupn→∞δn≤0, or(ii)′
∑n=0∞‍γnδn<∞.
Then,limn→∞sn=0.Lemma 7 (see [28]).
Let{xn} and {zn} be two sequences in a Banach space E such that
(16)xn+1=(1-βn)xn+βnzn,n≥1,
where {βn} satisfies the following conditions: 0<liminfn→∞βn≤limsupn→∞βn<1. If limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0, then limn→∞∥xn-zn∥=0.LetC be a subset of a real Banach space E and {Tn}n=1∞ a family of mappings of C such that ∩n=1∞F(Tn)≠⌀. Then {Tn}n=1∞ is said to satisfy the AKTT-condition [29] if for each bounded subset K of C,
(17)∑n=1∞‍sup{∥Tn+1z-Tnz∥:z∈K}<∞.Lemma 8 (see [29]).
LetC be a subset of a real Banach space E and {Tn}n=1∞ a family of mappings of C into itself which satisfies the AKTT-condition. Then, for each x∈C, {Tnx}n=1∞converges strongly to a point in C. Moreover, let the mapping T be defined by
(18)Tx=limn→∞Tnx,∀x∈C.
Then for each bounded subset K of C,
(19)limsupn→∞{∥Tnz-Tz∥:z∈K}=0.
In the sequel, one will write that ({Tn}n=1∞,T) satisfies the AKKT-condition if {Tn}n=1∞ satisfies the AKKT-condition and T is defined by Lemma 8 with F(T)=∩n=1∞F(Tn).We end this section with the following simple examples of mappings satisfying theAKTT-condition (see also Lemma 19).Example 9.
(i) LetE be a Banach space. For any n∈ℕ, let a mapping Tn:E→E be defined by
(20)Tn(x)=xn,∀x∈E.
Then, Tn is a nonexpansive mapping for each n∈ℕ. It could easily be seen that ({Tn}n=1∞,T) satisfies the AKKT-condition, where T(x)=0 for all x∈E.
(ii) LetE be a smooth Banach space and let x0≠0 be any element of E. For any j∈ℕ, we define a mapping Tj:E→E by
(21)Tj(x)={(12+12n+1)x0,ifx=(12+12n)x0;-xj,ifx≠(12+12n)x0,
for all n≥0. We define also a mapping T:E→E by
(22)T(x)={(12+12n+1)x0,ifx=(12+12n)x0;0,ifx≠(12+12n)x0,
for all n≥0. It is easy to verify that ({Tj}j=1∞,T) satisfies the AKKT-condition.
(iii) LetE=l2, where
(23)hvil2={σ=(σ1,σ2,…,σn,…):∑n=1∞‍∥σn∥2<∞},∥σ∥=(∑n=1∞‍∥σn∥2)1/2,∀σ∈l2,〈σ,η〉=∑n=1∞‍σnηn,∀δ=(σ1,σ2,…,σn,…),η=(η1,η2,…,ηn,…)∈l2.
Let {xn}n∈ℕ∪{0}⊂E be a sequence defined by
(24)x0=(1,0,0,0,…)x1=(1,1,0,0,0,…)x2=(1,0,1,0,0,0,…)x3=(1,0,0,1,0,0,0,…)x3⋮xn=(σn,1,σn,2,…,σn,k,…)x3⋮,
where
(25)σn,k={1ifk=1,n+1,0ifk≠1,k≠n+1,
for all n∈ℕ. It is clear that the sequence {xn}n∈ℕ converges weakly to x0. Indeed, for any Λ=(λ1,λ2,…,λn,…)∈l2=(l2)*, we have
(26)Λ(xn-x0)=〈xn-x0,Λ〉=∑k=2∞‍λkσn,k⟶0
as n→∞. It is also obvious that ∥xn-xm∥=2 for any n≠m with n,m sufficiently large. Thus, {xn}n∈ℕ is not a Cauchy sequence. We define a countable family of mappings Tj:E→E by
(27)Tj(x)={nn+1x,ifx=xn;-jj+1x,ifx≠xn,
for all j≥1 and n≥0. It is clear that F(Tj)={0} for all j≥1. It is obvious that Tj is a quasi-nonexpansive mapping for each j∈ℕ. Thus {Tj}j∈ℕ is a countable family of quasi-nonexpansive mappings.
LetTx=limj→∞Tjx for all x∈E. It is easy to see that
(28)T(x)={nn+1x,ifx=xn;-x,ifx≠xn.
Then, we obtain that T is a quasi-nonexpansive mapping with F(T)={0}=F~(T). Let D be a bounded subset of E. Then there exists r>0 such that D⊂Br={z∈E:∥z∥<r}. On the other hand, for any j∈ℕ, we have
(29)∑j=1∞‍sup{∥Tj+1z-Tjz∥:z∈D}=∑j=1∞‍sup{∥-j-1j+2z--jj+1z∥:z∈D}=∑j=1∞‍1(j+2)(j+1)sup{∥z∥:z∈D}<∞.
Furthermore, we have
(30)limsupj→∞{∥Tjz-Tz∥:z∈D}=0.
Therefore, ({Tj}j=1∞,T) satisfies the AKKT-condition.


## 3. Fixed Point and Convergence Theorems

LetE be a 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and let C be a closed and convex subset of E. Let A:C→E be a k-Lipschitzian and η-strongly accretive operator with constants k,η>0, let B:C→E be an l-Lipschitzian mapping with constant l≥0, and let T:C→C be a nonexpansive mapping with F(T)≠⌀. Suppose that 0<η<2kρ, 0<μ<η/k2ρ2. Define a mapping f:[0,1]→ℝ by
(31)f(t)={1-1-2tμ(η-tμk2ρ2)tift∈(0,1],μηift=0.
From the definition of f we deduce that
(32)f(t)≤μη,∀t∈[0,1].
Indeed, for any t∈(0,1], in view of (31) we obtain
(33)f(t)<μη⟺1-1-2tμ(η-tμk2ρ2)t<μη⟺1-1-2tμ(η-tμk2ρ2)<μηt⟺1-μηt<1-2tμ(η-tμk2ρ2)⟺1+μ2η2t2-2μηt<1-2tμ(η-tμk2ρ2)⟺0<η<2kρ.
On the other hand, it is easy to see that f is continuous on compact interval [0,1]. In fact, employing L’Hôpital’s Rule, we conclude that limt→0f(t)=μη. Thus,
(34)∃t0∈[0,1]suchthatf(t0)=min{f(t):t∈[0,1]}.
Set τ0:=τt0=f(t0) and τt:=f(t) if t∈[0,1]. Then we have
(35)0<τ0≤τt≤μη.
Assume now that γ satisfies 0≤γl<τ0. Then we get
(36)0<1μη-γl≤1τt-γl≤1τ0-γl<∞∀t∈[0,1].
In this section, we introduce the following implicit scheme that generates a net {xt}t∈(0,1) in an implicit way:
(37)xt=QC[tγBxt+(I-tμA)Txt].
We prove the strong convergence of {xt} to a fixed point x~ of T which solves the variational inequality
(38)〈(μA-γB)x~,j(x~-z)〉≤0,∀z∈F(T).
We first prove the following extension of Lemma 3.1 in [7] in a 2-uniformly smooth Banach space.Lemma 10.
LetE be a 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and let C be a closed and convex subset of E. Let A:C→E be a k-Lipschitzian and η-strongly accretive operator with 0<η<2kρ, 0<μ<η/k2ρ2, and t∈(0,1). In association with a nonexpansive mapping T:C→C, define the mapping St:C→E by
(39)Stx:=Tx-tμA(Tx),∀x∈C.
Then, St is a contraction with contraction constant τt=1-ct, where ct=1-2tμ(η-tμk2ρ2).Proof.
In view of Lemma3, we conclude that
(40)∥Stx-Sty∥2=∥(T-tμAT)x-(T-tμAT)y∥2=∥(Tx-Ty)-tμ(ATx-ATy)∥2≤∥Tx-Ty∥2-2tμ〈ATx-ATy,j(Tx-Ty)〉+2t2μ2ρ2∥ATx-ATy∥2≤∥Tx-Ty∥2-2tμη∥Tx-Ty∥2+2t2μ2k2ρ2∥Tx-Ty∥2=(1-2tμ(η-tμk2ρ2))∥Tx-Ty∥2≤(1-2tμ(η-tμk2ρ2))∥x-y∥2,
for all x,y∈C. Put ct=1-2tμ(η-tμk2ρ2)∈(0,1). Then by the assumptions t∈(0,1) and 0<η<2kρ, we infer that
(41)∥Stx-Sty∥≤ct∥x-y∥.
Let τt=(1-ct)∈(0,1). Then we have
(42)∥Stx-Sty∥≤(1-τt)∥x-y∥.
Therefore, St is a contraction with contraction constant 1-τt, which completes the proof.Remark 11.
LetE be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and C a closed convex subset of E. Let A:C→E be a k-Lipschitzian and η-strongly accretive operator with constants κ,η>0 and let B:C→H be an l-Lipschitzian mapping with constant l≥0. Assume T:C→C is a nonexpansive mapping with F(T)≠⌀. Let 0<η<2kρ, 0<μ<η/k2ρ2, and 0≤γl<τ0, where τ0=(1-1-2t0μ(η-t0μk2ρ2))/t0 satisfies (34). For any t∈(0,1), let the mapping Rt:C→E be defined by
(43)Rtx:=QC[tγBx+(I-tμA)Tx],∀x∈C.
Using Remark 11, it could easily be seen that
(44)∥Rtx-Rty∥≤(1-t(τ0-γl))∥x-y∥,∀x,y∈C.
Thus in view of Banach contraction principle, the contraction mapping Rt:C→E has a unique fixed point xt in C, which uniquely solves the fixed point equation (37).Remark 12.
LetE be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and C a closed convex subset of E. Let A:C→E be a k-Lipschitzian and η-strongly accretive operator with constants κ,η>0 and let B:C→E be an l-Lipschitzian mapping with constant l≥0. Assume T:C→C is a nonexpansive mapping with F(T)≠⌀. Let 0<η<2kρ, 0<μ<η/k2ρ2, and 0≤γl<τ0, where τ0=(1-1-2t0μ(η-t0μk2ρ2))/t0 satisfies (34). Then
(45)〈(μA-γB)x-(μA-γB)y,j(x-y)〉≥(μη-γl)∥x-y∥2,∀x,y∈C.
That is, μA-γB is strongly accretive with coefficient μη-γl.In the following result, we drive some important properties of the net{xt}t∈(0,1) which will be used in the sequel.Proposition 13.
LetE be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and let C be a closed and convex subset of E. Let A:C→E be a k-Lipschitzian and η-strongly accretive operator with constants κ,η>0 and let B:C→H be an l-Lipschitzian mapping with constant l≥0. Assume T:C→C is a nonexpansive mapping with F(T)≠⌀. Let 0<η<2kρ, 0<μ<η/k2ρ2, and 0≤γl<τ0, where τ0=(1-1-2t0μ(η-t0μk2ρ2))/t0 satisfies (34). For each t∈(0,1), let xt denote a unique solution of the fixed point equation (37). Then, the following properties hold for the net {xt}t∈(0,1):(1)
{xt}t∈(0,1)is bounded;(2)
limt→0∥xt-Txt∥=0;(3)
xtdefines a continuous curve from(0,1)intoC.Proof.
(1) Letp∈F(T) be taken arbitrarily. Then, in view of Lemma 10 we obtain
(46)∥xt-p∥=∥QC[tγBxt+(I-tμA)Txt]-QCp∥≤∥tγBxt+(I-tμA)Txt-p∥=∥(I-tμA)Txt+(I-tμA)ph+t(γBxt-μA(p))∥≤(1-tτt)∥xt-p∥h+t(γl∥xt-p∥+∥γBp-μAp∥)=(1-t(τt-γl))∥xt-p∥+t∥(γB-μA)p∥≤(1-t(τ0-γl))∥xt-p∥+t∥(γB-μA)p∥.
This implies that
(47)∥xt-p∥≤∥(γB-μA)p∥τ0-γl.
This shows that {xt} is bounded.
(2) Since{xt} is bounded, we have that {Bxt} and {ATxt} are bounded too. In view of the definition of {xt} we conclude that
(48)∥xt-Txt∥=∥QC[tγBxt+(I-tμA)Txt]-QC[Txt]∥≤∥tγBxt+(I-tμA)Txt-Txt∥=t∥γBxt-μATxt∥⟶0,
as t→0.
(3) Taket1,t2∈(0,1) arbitrarily. Then, we have
(49)∥xt1-xt2∥=∥QC[t1γBxt1+(I-t1μA)Txt1]h-QC[t2γBxt2+(I-t2μA)Txt2]∥≤∥t1γBxt1+(I-t1μA)Txt1h-[t2γBxt2+(I-t2μA)Txt2]∥=∥(t2-t1)γBxt2+t1γ(Bxt2-Bxt1)h+(t1-t2)μATxt2h+(I-t1μA)Txt2-(I-t1μA)Txt1∥≤(γ∥Bxt2∥+μ∥ATxt2∥)|t1-t2|h+(1-t1(τt1-γl))∥xt1-xt2∥≤(γ∥Bxt2∥+μ∥ATxt2∥)|t1-t2|h+(1-t1(τ0-γl))∥xt1-xt2∥.
This implies that
(50)∥xt2-xt1∥≤γ∥Bxt2∥+μ∥ATxt2∥t1(τ0-γl)|t2-t1|.
The boundedness of {xt} implies that xt defines a continuous curve from (0,1) into C.Theorem 14.
LetE be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and let C be a closed and convex subset of E. Let A:C→E be a k-Lipschitzian and η-strongly accretive operator with constants κ,η>0 and let B:C→H be an l-Lipschitzian mapping with constant l≥0. Assume T:C→C is a nonexpansive mapping with F(T)≠⌀. Let 0<η<2kρ, 0<μ<η/k2ρ2, and 0≤γl<τ0, where τ0=(1-1-2t0μ(η-t0μk2ρ2))/t0 satisfies (34). For each t∈(0,1), let {xt} denote a unique solution of the fixed point equation (37). Then the net {xt} converges strongly, as t→0, to a fixed point x~ of T which solves the variational inequality (38), or equivalently, QF(T)(I-μA+γB)x~=x~.Proof.
In view of Remark11 the variational inequality (38) has a unique solution, say x~∈C. We show that xt→x~ as t→0. To this end, let z∈F(T) be given arbitrary. Set
(51)yt=tγBxt+(I-tμA)Txt,∀t∈(0,1).
Then we have xt=QCyt and hence
(52)xt-z=QCyt-yt+yt-z=QCyt-yt+t(γBxt-μAz)+(I-tμA)Txt-(I-tμA)Tz.
Since QC is a nonexpansive mapping from E onto C, in view of Lemma 4, we conclude that
(53)〈QCyt-yt,j(QCyt-z)〉≤0.
Exploiting Lemma 10, (37), and (52), we obtain
(54)∥xt-z∥2=〈xt-z,j(xt-z)〉=〈QCyt-yt,j(QCyt-z)〉+〈(I-tμA)Txt-(I-tμA)z〉+〈t(γBxt-μAz),j(xt-z)〉≤1τ0[γl∥xt-z∥2+〈γBz-μAz,j(xt-z)〉].
This implies that
(55)∥xt-z∥2≤1τ0-γl〈γBz-μAz,j(xt-z)〉.
Let {tn}⊂(0,1) be such that tn→0+ as n→∞. Letting xn*:=xtn, it follows from Proposition 13(2) that limn→∞∥xn*-Txn*∥=0. The boundedness of {xt} implies that there exists x*∈C such that xn*⇀x* as n→∞. In view of Lemma 5, we deduce that x*∈F. Since xn*⇀x* as n→∞, it follows from (55) that limn→∞∥xn*-x*∥=0. Thus we have limt→0+xt=x* well defined. Next, we show that x* solves the variational inequality (38). We first notice that
(56)xt=QCyt=QCyt-yt+tγBxt+(I-tμA)Txt.
This, together with (52), implies that
(57)(μA-γB)xt=1t(QCyt-yt)(μA-γB)xt-1t(I-T)xt+μ(Axt-ATxt).
Since T is nonexpansive, in view of Remark 2, we conclude that I-T is accretive. This implies that
(58)〈(γB-μA)xt,j(xt-z)〉=1t〈QCyt-yt,j(xt-z)〉-1t〈(I-T)xt-(I-T)z,j(xt-z)〉+μ〈Axt-ATxt,j(xt-z)〉≤μ〈Axt-ATxt,j(xt-z)〉≤μl∥xt-Txt∥∥xt-z∥.
Replacing t by tn in (58), taking the limit n→∞, and noticing that {xt-z}t∈(0,1) is bounded for z∈F(T), we obtain
(59)〈(μA-γB)x*,j(x*-z)〉≤0.
Thus, we have x*∈F(T) a solution of the variational inequality (38). Consequently, x*=x~ by uniqueness. Therefore, xt→x~ as t→0. The variational inequality (38) can be written as
(60)〈(I-μA+γB)x~-x~,j(x~-z)〉≥0,∀z∈F(T).
Thus, in view of Lemma 4, it is equivalent to the following fixed point equation:
(61)QF(T)(I-μA+γB)x~=x~.
This completes the proof.Theorem 15.
LetE be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and let C be a nonempty, closed and convex subset of E. Suppose that the normalized duality mapping J:E→E* is weakly sequentially continuous at zero. Let A:C→E be a k-Lipschitzian and η-strongly accretive operator with constants κ,η>0 and let B:C→H be an l-Lipschitzian mapping with constant l≥0. Let 0<η<2kρ, 0<μ<η/k2ρ2, and 0≤γl<τ0, where τ0=(1-1-2t0μ(η-t0μk2ρ2))/t0 satisfies (34). Assume {Tn}n=1∞ is a sequence of nonexpansive mappings from C into itself such that ∩n=1∞F(Tn)≠⌀. Suppose in addition that T:C→C is a nonexpansive mapping such that ({Tn}n=1∞,T) satisfies the AKTT-condition. For given x1∈C arbitrarily, let the sequence {xn} be generated iteratively by
(62)yn=QC[αnγBxn+(I-αnμA)Tnxn],xn+1=(1-βn)yn+βnTnyn,n∈ℕ,
where QC is the sunny nonexpansive retraction from E onto C and {αn} and {βn} are two real sequences in (0,1) satisfying the following control conditions:
(63)(a):limn→∞αn=0,∑n=1∞‍αn=∞;(b):0<liminfn→∞βn≤limsupn→∞βn<1.
Then, the sequence {xn} converges strongly to x*∈∩n=1∞F(Tn) which solves the variational inequality
(64)〈(μA-γB)x*,j(x*-z)〉≤0,z∈⋂n=1∞F(Tn).Proof.
We divide the proof into several steps.
Step 1. We claim that the sequence {xn} is bounded. Let p∈F be fixed. In view of (62)–(64) and Lemma 10, we obtain
(65)∥yn-p∥=∥QC[αnγBxn+(I-αnμA)Tnxn]-QCp∥≤∥αnγBxn+(I-αnμA)xn-p∥=∥αn(γBxn-μAp)f+(I-αnμA)xn-(I-αnμA)p∥=∥αn(γBxn-γBp)+αn(γBp-μAp)f+(I-αnμA)xn-(I-αnμA)p∥≤αnγl∥xn-p∥+αn∥(γB-μA)p∥f+(1-αnτ0)∥xn-p∥=(1-αn(τ0-γl))∥xn-p∥f+αn∥(γB-μA)p∥≤max{∥xn-p∥,∥(γB-μA)p∥τ0-γl}.
Since Tn is nonexpansive, for all n∈ℕ, it follows from (62) and (65) that
(66)∥xn+1-p∥=∥(1-βn)(xn-p)+βn(Tnyn-p)∥≤(1-βn)∥xn-p∥+βn∥Tnyn-p∥≤(1-βn)∥xn-p∥+βn∥yn-p∥≤(1-βn)∥xn-p∥+βnmax{∥xn-p∥,∥(γB-μA)p∥τ0-γl}≤max{∥xn-p∥,∥(γB-μA)p∥τ0-γl}.
By induction, we conclude that {xn} is bounded. This implies that the sequences {Axn}, {Bxn}, {yn}, and {Tnyn} are bounded too. Let M1=sup{∥xn∥,∥Axn∥,∥Bxn∥,∥yn∥,∥Tnyn∥:n∈ℕ}<∞ and set K={z∈E:∥z∥≤M}. Then we have K a bounded subset of E and {xn,Axn,Bxn,yn,Tnyn}⊂K.
Step 2. We claim that limn→∞∥yn-Tyn∥=0. For this purpose, we denote a sequence {zn} by zn=Tnyn. Then we have
(67)∥zn+1-zn∥=∥Tn+1yn+1-Tnyn∥≤∥Tn+1yn+1-Tn+1yn∥+∥Tn+1yn-Tnyn∥≤∥yn+1-yn∥+sup{∥Tn+1z-Tnz∥:z∈K}.
This implies that
(68)∥zn+1-zn∥-∥yn+1-yn∥≤sup{∥Tn+1z-Tnz∥:z∈K}.
In view of Lemma 8 and (63)(a) we conclude that
(69)limsupn→∞(∥zn+1-zn∥-∥yn+1-yn∥)≤0.
Utilizing Lemma 7, we deduce that
(70)limn→∞∥zn-yn∥=0.
It follows from (63)(b) and (70) that
(71)limn→∞∥yn+1-yn∥=limn→∞(1-βn)∥zn-yn∥=0.
Observe now that
(72)∥yn+1-yn∥=∥QC[αn+1γBxn+1+(I-αn+1μA)Tn+1xn+1]g-QC[αnγBxn+(I-αnμA)Tnxn]∥≤∥αn+1γBxn+1+(I-αn+1μA)Tn+1xn+1g-[αnγBxn+(I-αnμA)Tnxn]∥=∥αn+1γBxn+1+Tn+1xn+1-αn+1μATn+1xn+1g-αnγBxn-Tnxn+αnμATnxn∥≤αn+1γ∥Bxn+1∥+αnγ∥Bxn∥+αn+1μ∥ATn+1xn+1∥+αnμ∥ATnxn∥+∥Tn+1xn+1-Tnxn∥≤(αn+1+αn)(γ+μ)M1+∥Tn+1xn+1-Tn+1xn∥+∥Tn+1xn-Tnxn∥≤(αn+1+αn)(γ+μ)M1+∥xn+1-xn∥+sup{∥Tn+1z-Tnz∥:z∈K}.
This implies that
(73)∥yn+1-yn∥-∥xn+1-xn∥≤(αn+1+αn)(γ+μ)M1+sup{∥Tn+1z-Tnz∥:z∈K}.
Utilizing Lemma 7 and taking into account αn→0, we deduce that
(74)limn→∞∥xn-yn∥=0.
On the other hand, we have
(75)∥yn-Tyn∥≤∥yn-Tnyn∥+∥Tnyn-Tyn∥≤∥yn-zn∥+sup{∥Tnz-Tz∥:z∈K}.
Employing Lemma 8, we obtain
(76)limn→∞∥yn-Tyn∥=0.
Step 3. We prove that there exists x*∈F such that
(77)limsupn→∞〈(μA-γB)x*,j(x*-yn)〉≤0,
where x* is as in Theorem 14. We first note that there exists a subsequence {yni} of {yn} such that
(78)limsupn→∞〈μAx*-γBx*,j(x*-yn)〉=limi→∞〈μAx*-γBx*,j(x*-yni)〉.
Since {yn} is bounded, without loss of generality, we may assume that yni⇀u∈C as i→∞. In view of Lemma 6 and Step 2, we conclude that u∈F. This, together with (78), implies that
(79)limsupn→∞〈μAx*-γBx*,j(x*-yn)〉=limi→∞〈μAx*-γBx*,j(x*-yni)〉=〈μAx*-γBx*,j(x*-u〉)≤0.
Step 4. We claim that limn→∞∥xn-x*∥=0.
For eachn∈ℕ∪{0}, by Lemma 10 and (36) we obtain
(80)∥yn-x*∥2=〈yn-x*,j(yn-x*)〉=〈QC[αnγBxn+(I-αnμA)Tnxn]x-x*,j(yn-x*)〉=〈QC[αnγBxn+(I-αnμA)Tnxn]x-[αnγBxn+(I-αnμA)xxxxx×Tnxn-x*],j(yn-x*)〉+〈αnγBxn+(I-αnμA)Tnxnxxxxx-x*,j(yn-x*)〉≤〈αnγBxn+(I-αnμA)Tnxn222-x*,j(yn-x*)〉=αn〈γBxn-μA(x*),j(yn-x*)〉+〈(I-αnμA)Txnf-(I-αnμA)Tx*,j(yn-x*)〉=αnγ〈Bxn-Bx*,j(yn-x*)〉+αn〈(γB-μA)x*,j(yn-x*)〉+〈(I-αnμA)Txnx-(I-αnμA)Tx*,j(yn-x*)〉≤αnγl∥xn-x*∥∥yn-x*∥+αn〈(γB-μA)x*,j(yn-x*)〉+(1-αnτ0)∥xn-x*∥∥yn-x*∥=(1-αn(τ0-γl))∥xn-x*∥∥yn-x*∥+αn〈(γB-μA)x*,j(yn-x*)〉≤(1-αn(τ0-γl))12(∥xn-x*∥2+∥yn-x*∥2)+αn〈(γB-μA)x*,j(yn-x*)〉.
This implies that
(81)∥yn-x*∥2≤(1-αn(τ0-γl))(1+αn(τ0-γl))∥xn-x*∥2+2αn1+αn(τ0-γl)〈(γB-μA)x*,j(yn-x*)〉≤(1-αn(τ0-γl))∥xn-x*∥2+2αn1+αn(τ0-γl)〈(γB-μA)x*,j(yn-x*)〉=(1-αn(τ0-γl))∥xn-x*∥2+θnξn,
where
(82)θn=αn(τ0-γl),ξn=2(1+αn(τ0-γl))(τ0-γl)×〈(γB-μA)x*,j(yn-x*)〉
In view of (81), we conclude that
(83)∥xn+1-x*∥2=∥(1-βn)xn+βnTnyn-x*∥2≤(1-βn)∥xn-x*∥2+βn∥Tnyn-x*∥2≤(1-βn)∥xn-x*∥2+βn∥yn-x*∥2≤(1-βn)∥xn-x*∥2+βn[(1-αn(τ0-γl))∥xn-x*∥2+θnξn]≤(1-βnαn(τ0-γl))∥xn-x*∥2+βnθnξn=(1-βnαn(τ0-γl))∥xn-x*∥2+βnαn(τ0-γl)ξn=(1-γn)∥xn-x*∥2+γnξn,
where γn=βnαn(τ0-γl). It is easy to show that limn→∞γn=0, ∑n=0∞‍γn=∞, and limsupn→∞ξn≤0. Hence, in view of Lemma 6 and (83), we conclude that the sequence {xn} converges strongly to x*∈F(T). This completes the proof.Remark 16.
Theorem15 improves and extends [19, Theorems 3.1 and 3.2] in the following aspects.(i)
The self-contractive mappingf:C→C in [19, Theorems 3.1 and 3.2] is extended to the case of a Lipschitzian (possibly nonself-) mapping B:C→E on a nonempty closed convex subset C of a Banach space E.(ii)
The identity mappingI is extended to the case of I-A:C→E, where A:C→E is a k-Lipschitzian and η-strongly accretive (possibly nonself-) mapping.(iii)
The contractive coefficientα∈(0,1) in [19, Theorems 3.1 and 3.2] is extended to the case where the Lipschitzian constant l lies in [0,∞).(iv)
In order to find a common fixed point of a countable family of nonexpansive self-mappingsTn:C→C, the Mann type iterations in [19, Theorems 3.1 and 3.2] are extended to develop the new Mann type iteration (62).(v)
The new technique of argument is applied in deriving Theorem14. For instance the characteristic properties (Lemma 4) of sunny nonexpansive retraction play an important role in proving the strong convergence of the net {xt}t∈(0,1) in Theorem 14.(vi)
Whenever we haveC=E,B=f a contraction mapping with coefficient α∈(0,1), A=I the identity mapping on C, and l=α with 0<γα<τ0=(1-1-2t0μ(η-t0μk2ρ2))/t0, Theorem 14 reduces to [19, Theorems 3.1 and 3.2]. Thus, Theorem 14 covers [19, Theorems 3.1 and 3.2] as special cases.Remark 17.
Proposition13 and Theorems 14 and 15 improve and generalize the corresponding results of [4] from Hilbert spaces to Banach spaces.


## 4. Applications

In this section, we apply Theorem15 for finding a zero of an accretive operator. Let E be a real Banach space and let S:E→2E be a mapping. The effective domain of S is denoted by dom(S); that is, dom(S)={x∈E:Sx≠⌀}. The range of S is denoted by R(S). A multivalued mapping S is said to be accretive if for all x,y∈E there exists j∈J(x-y) such that 〈x-y,j〉≥0, where J:E→2E* is the duality mapping. An accretive operator S is m-accretive if R(I+rS)=E for each r≥0. Throughout this section, we assume that S:E→2E is m-accretive and has a zero. For an accretive operator S on E and r>0, we may define a single-valued operator Jr=(I+rS)-1:E→dom(S), which is called the resolvent of S for r>0. Assume S-10={x∈E:0∈Sx}. It is known that S-10=F(Jr) for all r>0.The following lemma has been proved in [21].Lemma 18.
LetE be a real Banach space and let S be an m-accretive operator on E. For r>0, let Jr be the resolvent operator associated with S and r. Then
(84)∥Jrx-Jsx∥≤|r-s|r∥x-Jrx∥,
for all r,s>0 and x∈E.We also know the following lemma from [29].Lemma 19.
LetC be a nonempty, closed, and convex subset of a real Banach space E and let S be an accretive operator on E such that S-10≠⌀ and dom(S)¯⊂C⊂∩r>0R(I+rS). Suppose that {rn} is a sequence of (0,∞) such that inf{rn:n∈ℕ}>0 and ∑n=1∞‍|rn+1-rn|<∞. Then(i)
∑n=1∞‍sup{∥Jrn+1z-Jrnz∥:z∈B}<∞ for any bounded subset B of C;(ii)
limn→∞Jrnz=Jrz for all z∈C and F(Jr)=∩n=1∞F(Jrn), where rn→r as n→∞.As an application of our main result, we include a concrete example in support of Theorem15. Using Theorem 15, we obtain the following strong convergence theorem for m-accretive operators.Theorem 20.
LetE be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant ρ and C a nonempty, closed, and convex subset of E. Suppose that the normalized duality mapping J:E→E* is weakly sequentially continuous at zero. Let A:C→E be a k-Lipschitzian and η-strongly accretive operator with constants κ,η>0 and let B:C→H be an l-Lipschitzian mapping with constant l≥0. Let 0<η<2kρ, 0<μ<η/k2ρ2, and 0≤γl<τ0, where τ0=(1-1-2t0μ(η-t0μk2ρ2))/t0 satisfies (34). Let S be an m-accretive operator from E to E* such that S-1(0)≠⌀. Let rn>0 such that liminfn→∞rn>0, ∑n=1∞‍|rn+1-rn|<∞ and let Jrn=(I+rnS)-1 be the resolvent of S. Let {αn}n=1∞ and {βn}n=1∞ be sequences in [0,1] satisfying the following control conditions:(a)
limn→∞αn=0;(b)
∑n=1∞‍αn=∞;(c)
0<liminfn→∞βn≤limsupn→∞βn<1.
Let{xn}n=1∞ be a sequence generated by
(85)yn=QC[αnγBxn+(I-αnμA)Jrnxn],xn+1=(1-βn)yn+βnJrnyn,n∈ℕ,
where QC is the sunny nonexpansive retraction from E onto C. Then, the sequence {xn} defined in (85) converges strongly to x*∈S-1(0).Proof.
LettingTn=Jrn,∀n∈ℕ, in Theorem 15, from (62), we obtain (85). It is easy to see that Tn satisfies all the conditions in Theorem 15 for all n∈ℕ. Therefore, in view of Theorem 15 we have the conclusions of Theorem 20. This completes the proof.Remark 21.
Theorem20 improves and extends Theorems 4.2, 4.3, 4.4, and 4.5 in [19].



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