# A New Hybrid Algorithm forλ-Strict Asymptotically Pseudocontractions in 2-Uniformly Smooth Banach Spaces

Authors: Xin-dong Liu; Shih-sen Chang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A new hybrid projection algorithm is considered for aλ-strict asymptotically pseudocontractive mapping. Using the metric projection, a strong convergence theorem is obtained in a uniformly convex and 2-uniformly smooth Banach spaces. The result presented in this paper mainly improves and extends the corresponding results of Matsushita and Takahashi (2008), Dehghan (2011) Kang and Wang (2011), and many others.


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## Body

## 1. Introduction

LetE be a real Banach space and E* be the dual spaces of E. Assume that J is the normalized duality mapping from E into 2E* defined by
(1.1)J(x)={x*∈E*:〈x,x*〉=∥x∥2=∥x*∥2},∀x∈E,
where 〈·,·〉 is the generalized duality pairing between E and E*.LetC be a nonempty closed convex subset of a real Banach space E.Definition 1.1.
LetT:C→C be a mapping:(1)
Tis said to be nonexpansive if for all x,y∈C,
(1.2)∥Tx-Ty∥≤∥x-y∥,(2)
T is said to be asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞) with kn→1 such that for all x,y∈C,
(1.3)∥Tx-Ty∥≤kn∥x-y∥,(3)
T is said to be λ-strictly pseudocontractive in the terminology of Browder-Petryshyn [1] if there exists a constant λ∈(0,1) such that for all x,y∈C,
(1.4)〈Tx-Ty,j(x-y)〉≤∥x-y∥2-λ∥(I-T)x-(I-T)y∥2,∀j(x-y)∈J(x-y),(4)
T is said to be (λ,{kn})-strict asymptotically pseudocontractive if there exist a constant λ∈(0,1) and a sequence {kn}⊂[1,∞) with kn→1 such that for all x,y∈C and for all j(x-y)∈J(x-y),
(1.5)〈Tnx-Tny,j(x-y)〉≤kn∥x-y∥2-λ∥(I-Tn)x-(I-Tn)y∥2,∀n≥1,(5)
T is said to be uniformly L-Lipschitzian if there exists a constant L>0 such that
(1.6)∥Tnx-Tny∥≤L∥x-y∥,∀n≥1.
It is wellknown that the class of (λ,{kn})-strictly asymptotically pseudocontractive mappings was first introduced in Hilbert spaces by Liu [2]. In the case of Hilbert spaces, it is shown by [2] that (1.5) is equivalent to the inequality
(1.7)∥Tnx-Tny∥2≤kn∥x-y∥2+λ∥(I-Tn)x-(I-Tn)y∥2.
Concerning the convergence problem of iterative sequences for strictly pseudocontractive mappings has been studied by several authors (see [1, 3–20]). Concerning the class of strictly asymptotically pseudocontractive mappings, Liu [2] proved the following results.Theorem 1.2.
LetH be a real Hilbert space, let C be a nonempty closed convex and bounded subset of H, and let T:C→C be a completely continuous uniformly L-Lipschitzian (λ,{kn})-strictly asymptotically pseudocontractive mapping such that ∑n=1∞‍(kn2-1)<∞. Let {αn}⊂(0,1) be a sequence satisfying the following condition:
(1.8)0<ϵ≤αn≤1-λ-ϵ,∀n≥1andsomeϵ>0.
Then, the sequence {xn} generated from an arbitrary x1∈C by
(1.9)xn+1=(1-αn)xn+αnTnxn,∀n≥1
converges strongly to a fixed point of T.In 2007,  Osilike  et al. [21] proved the following theorem.Theorem 1.3.
LetE be a real q-uniformly smooth Banach space which is also uniformly convex, let C be a nonempty closed convex subset of E, let T:C→C be a (λ,{kn})-strictly asymptotically pseudocontractive mapping such that ∑n=1∞‍(kn2-1)<∞, and let F(T)≠∅. Let {αn}⊂(0,1) be a real sequence satisfying the following condition:
(1.10)0<a≤αn≤b<q(1-k)2cq(1+L)-(q-2),∀n≥1.
Let {xn} be the sequence defined by (1.9). Then, {xn} converges weakly to a fixed point of T.On the other hand, by using the metric projection, Nakajo and Takahashi [22] introduced the following iterative algorithms for the nonexpansive mapping T in the framework of Hilbert spaces:
(1.11)x0=x∈C,yn=αnxn+(1-αn)Txn,Cn={z∈C:∥z-yn∥}≤∥z-xn∥,Qn={z∈C:〈xn-z,x-xn〉≥0},xn+1=PCn∩Qnx,n=0,1,2,…,
where {αn}⊂[0,α],α∈[0,1) and PCn∩Qn is the metric projection from a Hilbert space H onto Cn∩Qn. They proved that {xn} generated by (1.11) converges strongly to a fixed point of T.In 2006, Xu [23] extended Nakajo and Takahashi's theorem to Banach spaces by using the generalized projection.In 2008, Matsushita and Takahashi [24] presented the following iterative algorithms for the nonexpansive mapping T in the framework of Banach spaces:
(1.12)x0=x∈C,Cn=co¯{z∈C:∥z-Tz∥≤tn∥xn-Txn∥},Dn={z∈C:〈xn-z,J(x-xn)〉≥0},xn+1=PCn∩Dnx,n=0,1,2,…,
where co¯C denotes the convex closure of the set C, J is normalized duality mapping, {tn} is a sequence in (0, 1) with tn→0, and PCn∩Dn is the metric projection from E onto Cn∩Dn. Then, they proved that {xn} generated by (1.12) converges strongly to a fixed point of nonexpansive mapping T.Recently, Dehghan [25] introduced the following hybrid projection algorithm for an asymptotically nonexpansive mapping T in the framework of Banach spaces:
(1.13)x0=x∈C,C0=D0=C,Cn=co¯{z∈Cn-1:∥z-Tnz∥≤tn∥xn-Tnxn∥},Dn={z∈Dn-1:〈xn-z,J(x-xn)〉≥0},xn+1=PCn∩Dnx,n=1,2,…,
where co¯C  denotes the convex closure of the set C, {tn} is a sequence in (0, 1) with tn→0, and PCn∩Dn is the metric projection from E onto Cn∩Dn. Then, he proved that {xn} generated by (1.13) converges strongly to a fixed point of an asymptotically nonexpansive mappings T.Motivated by the research work going on in this direction, the purpose of this paper is to introduce the following iteration for finding a fixed point of(λ,{kn})-strict asymptotically pseudocontraction in a uniformly convex and 2-uniformly smooth Banach spaces:
(1.14)x0=x∈C,C0=C,Cn=co¯{z∈Cn-1:∥z-Tnz∥≤tn∥xn-Tnxn∥},xn+1=PCnx,n=1,2,…,
where co¯C denotes the convex closure of the set C, {tn} is a sequence in (0, 1) with tn→0, and PCn is the metric projection from E onto Cn. Under suitable conditions some strong convergence theorem for the sequence {xn} defined by (1.14) to converge a fixed point of an asymptotically λ-strictly pseudocontraction. The result presented in the paper extends and improves the main results of Matsushita and Takahashi [24], Dehghan [25], Kang and Wang  [26], and others.


## 2. Preliminaries

In this section, we recall the well-known concepts and results which will be needed to prove our main results. Throughout this paper, we assume thatE is a real Banach space and C is a nonempty subset of E. When {xn} is a sequence in E, we denote strong convergence of {xn} to x∈E by xn→x and weak convergence by xn⇀x.A Banach spaceE is said to be strictly convex if ∥x+y∥/2<1 for all x,y∈U={z∈E:∥z∥=1} with x≠y. E is said to be uniformly convex if for each ϵ>0 there is a δ>0 such that for x,y∈E with ||x||,||y||≤1 and ||x-y||≥ϵ,||x+y||≤2(1-δ) holds. The modulus of convexity of E is defined by
(2.1)δE(ϵ)=inf{1-∥x+y2∥:∥x∥,∥y∥≤1,∥x-y∥≥ϵ},E is said to be smooth if the limit
(2.2)limt→0∥x+ty∥-∥x∥t
exists for all x,y∈U. The modulus of smoothness of E is defined by
(2.3)ρE(t)=sup{12(∥x+y∥+∥x-y∥)-1:∥x∥≤1,∥y∥≤t}.
A Banach space E is said to be uniformly smooth if ρE(t)/t→0 as t→0. A Banach space E is said to be q-uniformly smooth, if there exists a fixed constant c>0 such that ρE(t)≤ctq.IfE is a reflexive, strictly convex and smooth Banach space, then for any x∈E, there exists a unique point x0∈C such that
(2.4)∥x0-x∥=miny∈C∥y-x∥.
The mapping PC:E→C defined by PCx=x0 is called the metric projection from E onto C. Let x∈E and u∈C. Then it is known that u=PCx if and only if
(2.5)〈u-y,J(x-u)〉≥0,∀y∈C.
For the details on the metric projection, refer to [27–30].In the sequel, we make use of the following lemmas for our main results.Lemma 2.1 (see [31]).
LetE be a real Banach space, C a nonempty subset of E, and T:C→C a (λ,{kn})-strictly asymptotically pseudocontractive mapping. Then T is uniformly L-Lipschitzian.Lemma 2.2 (see [32]).
LetE be a real 2-uniformly smooth Banach spaces with the best smooth constant K. Then the following inequality holds:
(2.6)∥x+y∥2≤∥x∥2+2〈y,J(x)〉+2∥Ky∥2,
for any x,y∈E.Lemma 2.3 (demiclosed principle [21]).
LetE be a real q-uniformly smooth Banach space which is also uniformly convex. Let C be a nonempty closed convex subset of E and T:C→C a (λ,{kn})-strictly asymptotically pseudocontractive mapping with a nonempty fixed point set. Then I-T is demiclosed at zero, where I is the identical mapping.Lemma 2.4 (see [33]).
LetC be a closed convex subset of a uniformly convex Banach space. Then for each r>0, there exists a strictly increasing convex continuous function γ:[0,∞)→[0,∞) such that γ(0)=0 and
(2.7)γ(∥T(∑j=0mμjzj)-∑j=0m‍μjTzj∥)≤max0≤j<k≤m(∥zj-zk∥-∥Tzj-Tzk∥),
for all m≥1, {μj}j=0m∈Δm,{zj}j=0m⊂C∩Br and T∈Lip(C,1), where Δm={{μ0,μ1,…,μm}:0≤μj(0≤j≤m)  and  ∑j=0m‍μj=1}, Br={x∈E:||x||≤r} and Lip(C,1) is the set of all nonexpansive mappings from C into E.


## 3. Main Results

Now we are ready to give our main results in this paper.Lemma 3.1.
LetC be a nonempty subset of a real 2-uniformly smooth Banach space E with the best smooth constant K, and T:C→C be a (λ,{kn})-strict asymptotically pseudocontraction. For α∈(0,1)∩(0,λ/K2], one defines
(3.1)Sn,αx=12α(kn-1)+1[(1-α)x+αTnx],
for all x∈C and each n≥1. Then Sn,α:C→E is a nonexpansive.Proof.
For anyx,y∈C, put βn,α=2α(kn-1)+1, we compute
(3.2)∥Sn,αx-Sn,αy∥2=∥1βn,α[(1-α)x+αTnx]-1βn,α[(1-α)y+αTny]∥2=1βn,α2∥(x-y)+α(Tnx-Tny-(x-y))∥2≤1βn,α2(∥x-y∥2+2α〈Tnx-Tny-(x-y),j(x-y)〉+2K2α2∥Tnx-Tny-(x-y)∥2)=1βn,α2(∥x-y∥2+2α〈Tnx-Tny,j(x-y)〉-2α∥x-y∥2+2K2α2∥Tnx-Tny-(x-y)∥2)≤1βn,α2(∥x-y∥2+2α(kn∥x-y∥2-λ∥Tnx-Tny-(x-y)∥2)-2α∥x-y∥2+2K2α2∥Tnx-Tny-(x-y)∥2)=1βn,α2{(1+2αkn-2α)∥x-y∥2-2αλ∥Tnx-Tny-(x-y)∥2+2K2α2∥Tnx-Tny-(x-y)∥2}≤∥x-y∥2,
which shows that Sn,α is a nonexpansive mapping. This completes the proof.Theorem 3.2.
LetC be a nonempty bounded and closed convex subset of a uniformly convex and 2-uniformly smooth Banach spaces E with the best smooth constant K>0, assume that T:C→C is a (λ,{kn})-strict asymptotically pseudocontraction such that ℱ=ℱ(T)≠∅. Let {tn} be a sequence in (0, 1) with tn→0. Let {xn} be a sequence generated by (1.14), where
(3.3)co¯{z∈Cn-1:∥z-Tnz∥≤tn∥xn-Tnxn∥}
denotes the convex closure of the set {z∈Cn-1:||z-Tnz||≤tn||xn-Tnxn||} and PCn is the metric projection from E onto Cn. Then {xn} converges strongly to Pℱx.Proof.
(I) First we prove that{xn} is well defined and bounded.
It is easy to check thatCn is closed and convex and ℱ⊂Cn for all n≥0. Therefore, {xn} is well defined.
Putp=Pℱx. Since ℱ⊂Cn and xn+1=PCnx, we have that
(3.4)∥xn+1-x∥≤∥p-x∥,
for all n≥0. Hence, {xn} is bounded.
(II) Now we prove that||xn-Tn-kxn||→0 as n→∞ for any k∈ℕ(ℕ denotes the set of all positive integers).
Fixk∈ℕ and put l=n-k,n>k. Since xn=PCn-1x, we have xn∈Cn-1⊂⋯⊂Cl. Then there exist some positive integer m∈ℕ, {μi}∈Δm and {zi}i=0m{zi}i=1m⊂Cl-1 such that
(3.5)∥xn-∑j=0m‍μjzj∥<tl,(3.6)∥zj-Tlzj∥≤tl∥xl-Tlxl∥,
for all j∈{0,1,…,m}. Take α∈(0,1)∩(0,λ/K2]. Put M=supx∈C||x||,p=Pℱx, r0=supn≥1||xn-p|| and βl,α=2α(kl-1)+1. we define
(3.7)Sn,αx=12α(kn-1)+1[(1-α)x+αTnx],
for all x∈C and each n≥1, then Sl,αp=(1/βl,α)p. It follows from Lemma 3.1 and (3.5) that
(3.8)∥xl-Tlxl∥=βl,αα∥1βl,α(αTlxl+(1-α)xl)-1βl,αxl∥=βl,αα∥(Sl,αxl-Sl,αp)+1βl,α(p-xl)∥≤βl,α+1αr0,∥Tl(∑j=0m‍μjzj)-Tlxn∥≤βl,αα(∥Sl,α(∑j=0m‍μjzj)-Sl,αxn∥+1-αβl,α∥∑j=0m‍μjzj-xn∥)≤(βl,α+1α-1)∥∑j=0m‍μjzj-xn∥≤(βl,α+1α-1)tl.
Moreover, from Lemmas 2.4 and 3.1, we have
(3.9)∥Tl(∑j=0m‍μjzj)-∑j=0m‍μjTlzj∥=βl,αα∥Sl,α(∑j=0m‍μjzj)-∑j=0m‍μjSl,αzj∥≤βl,ααγ-1[max0≤j<k≤m(∥zj-zk∥-∥Sl,αzj-Sl,αzk∥)]≤βl,ααγ-1[max0≤j<k≤m(∥zj-Sl,αzj∥+∥zk-Sl,αzk∥)]≤βl,ααγ-1{max0≤j<k≤m[αβl,α(∥zj-Tlzj∥+∥zk-Tlzk∥)+(1-1βl,α)(∥zj∥+∥zk∥)]}≤βl,ααγ-1{2αtlβl,α∥xl-Tlxl∥+2(1-1βl,α)M}≤βl,ααγ-1{2αtlβl,α{βl,α+1αr0}+2(1-1βl,α)M}=βl,ααγ-1{2(1+1βl,α)r0tl+2(1-1βl,α)M}.
Observe that βl,α→1 as n→∞, it follows from (3.5)–(3.9) that
(3.10)∥xn-Tlxn∥≤∥xn-∑j=0m‍μjzj∥+∥∑j=0m‍μj(zj-Tlzj)∥+∥∑j=0m‍μjTlzj-Tl(∑j=0m‍μjzj)∥+∥Tl(∑j=0m‍μjzj)-Tlxn∥≤tl+tl∥xl-Tlxl∥+βl,ααγ-1{2(1+1βl,α)r0tl+2(1-1βl,α)M}+(βl,α+1α-1)tl≤tl+βl,α+1αr0tl+βl,ααγ-1{2(1+1βl,α)r0tl+2(1-1βl,α)M}+(βl,α+1α-1)tl→0asn→∞.
This shows that
(3.11)∥xn-Tlxn∥→0asn→∞.
(III) we prove that ||xn-Txn||→0 as n→∞.
SinceT is a uniformly L-Lipschitzian, we have
(3.12)∥xn-Txn∥≤∥xn-Tn-1xn∥+∥Tn-1xn-Txn∥≤∥xn-Tn-1xn∥+L∥Tn-2xn-xn∥→0asn→∞.
(IV) Finally, we prove that xn→p=Pℱx.
It follows from the boundedness of{xn} that for each subsequence {xni}⊂{xn} there exists a subsequence (without loss of generality we can still denote it by) {xni} such that xni⇀v as i→∞. Since T:C→C is a uniformly L-Lipschitzian and (λ,{kn})-strict asymptotically pseudocontraction, from Lemma 2.3, we know that T is demiclosed. Hence we have v∈ℱ.
From the weakly lower semicontinuity of the norm and (3.4), it follows that
(3.13)∥p-x∥≤∥v-x∥≤liminfi→∞∥xni-x∥≤limsupi→∞∥xni-x∥≤∥p-x∥.
This shows p=v and hence xni⇀p as i→∞. By the arbitrariness of {xni}⊂{xn}, we obtain xn⇀p. Further, it follows from (3.13) that
(3.14)limn→∞∥xn-x∥=∥p-x∥.
Since E is uniformly convex, it has the Kadec-Klee property. Hence, we have xn-x→p-x, that is, xn→p. This completes the proof.



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