# Strong Convergence Theorems for a Finite Family ofλi-Strict Pseudocontractions in 2-Uniformly Smooth Banach Spaces by Metric Projections

Authors: Xin-dong Liu; Shih-sen Chang; Xiong-rui Wang
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

A new hybrid projection algorithm is considered for a finite family ofλi-strict
pseudocontractions. Using the metric projection, some strong convergence theorems of common
elements are obtained in a uniformly convex and 2-uniformly smooth Banach space. The results
presented in this paper improve and extend the corresponding results of Matsushita and Takahshi, 2008, Kang and Wang, 2011, and many others.


---

## Body

## 1.  Introduction

LetE be a real Banach space and let 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 closed convex subset of a real Banach space E. A mapping T:C→C is said to be nonexpansive if
(1.2)∥Tx-Ty∥≤∥x-y∥,
for all x,y∈C. Also a mapping T:C→C is called a λ-strict pseudocontraction if there exists a constant λ∈(0,1) such that for every x,y∈C and for some j(x-y)∈J(x-y), the following holds:
(1.3)〈Tx-Ty,j(x-y)〉≤∥x-y∥2-λ∥(I-T)x-(I-T)y∥2.From (1.3) we can prove that if T is λ-strict pseudo-contractive, then T is Lipschitz continuous with the Lipschitz constant L=(1+λ)/λ.It is well-known that the classes of nonexpansive mappings and pseudocontractions are two kinds important nonlinear mappings, which have been studied extensively by many authors (see [1–8]).In [9] Reich considered the Mann iterative scheme {xn}(1.4)xn+1=(1-αn)xn+αnTxn,x1∈C
for nonexpansive mappings, where {αn} is a sequence in (0,1). Under suitable conditions, the author proved that {xn} converges weakly to a fixed point of T. In 2005, Kim and Xu [10] proved a strong convergence theorem for nonexpansive mappings by modified Mann iteration. In 2008, Zhou [11] extended and improved the main results of Kim and Xu to the more broad 2-uniformly smooth Banach spaces for λ-strict pseudocontractive mappings.On the other hand, by using metric projection, Nakajo and Takahashi [12] introduced the following iterative algorithms for the nonexpansive mapping T in the framework of Hilbert spaces:
(1.5)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 (1.5) converges strongly to a fixed point of T.In 2006, Xu [13] extended Nakajo and Takahashi's theorem to Banach spaces by using the generalized projection.In 2008, Matsushita and Takahashi [14] presented the following iterative algorithms for the nonexpansive mapping T in the framework of Banach spaces:
(1.6)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 (1.6) converges strongly to a fixed point of nonexpansive mapping T.Recently, Kang and Wang [15] introduced the following hybrid projection algorithm for a pair of nonexpansive mapping T in the framework of Banach spaces:
(1.7)x0=x∈C,yn=αnT1xn+(1-αn)T2xn,Cn=co¯{z∈C:∥z-T1z∥+∥z-T2z∥≤tn∥xn-yn∥},xn+1=PCnx,n=0,1,2,…,
where co¯C denotes the convex closure of the set C, {αn} is a sequence in [0, 1], {tn} is a sequence in (0,1) with tn→0, and PCn is the metric projection from E onto Cn. Then, they proved that {xn} generated by (1.7) converges strongly to a fixed point of two nonexpansive mappings T1 and T2.In this paper, motivated by the research work going on in this direction, we introduce the following iterative for finding fixed points of a finite family ofλi-strict pseudocontractions in a uniformly convex and 2-uniformly smooth Banach space:
(1.8)x0=x∈C,yn=∑i=1N‍αn,iTixn,Cn=co¯{z∈C:∑i=1N‍∥z-Tiz∥≤tn∥xn-yn∥},xn+1=PCnx,n=1,2,…,
where co¯C denotes the convex closure of the set C, {αn,i} is N sequences in [0,1] and ∑i=1N‍αn,i=1 for each n≥0, {tn} is a sequence in (0,1) with tn→0, and PCn is the metric projection from E onto Cn. we prove defined by (1.8) converges strongly to a common fixed point of a finite family of λi-strictly pseudocontractions, the main results of Kang and Wang is extended and improved to strictly pseudocontractions.


## 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. We also assume that E* is the dual space of E, and J:E→2E* is the normalized duality mapping. Some properties of duality mapping have been given in [16].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 [17–20].In the sequel, we make use the following lemmas for our main results.Lemma 2.1 (see [21]).
LetE be a real 2-uniformly smooth Banach space 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 2.2 (see [11]).
LetC be a nonempty subset of a real 2-uniformly smooth Banach space E with the best smooth constant K>0 and let T:C→C be a λ-strict pseudocontraction. For α∈(0,1)∩(0,λ/K2], we define Tαx=(1-α)x+αTx. Then Tα:C→E is nonexpansive such that F(Tα)=F(T).Lemma 2.3 (demiclosed principle, see [22]).
LetE be a real uniformly convex Banach space, let C be a nonempty closed convex subset of E, and let T:C→C be a continuous pseudocontractive mapping. Then, I-T is demiclosed at zero.Lemma 2.4 (see [23]).
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 closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E with the best smooth constant K>0, and T:C→C be a λ-strict pseudocontraction. Then for each r>0, there exists a strictly increasing convex continuous function γ:[0,∞)→[0,∞) such that γ(0)=0 and
(3.1)γ(α∥T(∑j=0m‍μjzj)-∑j=0m‍μjTzj∥)≤αmax0≤j<k≤m(∥zj-Tzj∥+∥zk-Tzk∥),
for all m≥1, {μj}j=0m∈Δm,{zj}j=0m⊂C∩Br, where α∈(0,1)∩(0,λ/K2], Δm={{μ0,μ1,…,μm}:0≤μj(0≤j≤m)  and ∑j=0m‍μj=1}, Br={x∈E:||x||≤r}.Proof.
Define the mappingTα:C→C as Tαx=(1-α)x+αTx, for all x∈C. Then Tα is nonexpansive. From Lemma 2.4, there exists a strictly increasing convex continuous function γ:[0,∞)→[0,∞) with γ(0)=0 and such that
(3.2)γ(∥Tα(∑j=0m‍μjzj)-∑j=0m‍μjTαzj∥)≤max0≤j<k≤m(∥zj-zk∥-∥Tαzj-Tαzk∥).
Hence
(3.3)γ(α∥T(∑j=0m‍μjzj)-∑j=0m‍μjTzj∥)=γ(∥Tα(∑j=0m‍μjzj)-∑j=0m‍μjTαzj∥)≤max0≤j<k≤m(∥zj-zk∥-∥Tαzj-Tαzk∥)≤max0≤j<k≤m(∥zj-Tαzj∥+∥zk-Tαzk∥)=αmax0≤j<k≤m(∥zj-Tzj∥+∥zk-Tzk∥).
This completes the proof.Theorem 3.2.
LetC be a nonempty closed subset of a uniformly convex and 2-uniformly smooth Banach space E with the best smooth constant K>0, assume that for each i(i=1,2,…,N), Ti:C→C is a λi-strict pseudocontraction for some 0<λi<1 such that ℱ=∩i=1Nℱ(Ti)≠∅. Let {αn,i} be N sequences in [0,1] with ∑i=1N‍αn,i=1 for each n≥0 and {tn} be a sequence in (0,1) with tn→0. Let {xn} be a sequence generated by (1.8), where co¯{z∈C:∑i=1N‍||z-Tiz||≤tn||xn-yn||} denotes the convex closure of the set {z∈C:∑i=1N‍||z-Tiz||≤tn||xn-yn||} 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-Tixn||→0 as n→∞ for all i∈{1,2,…,N}.
Sincexn+1∈Cn, there exist some positive integer m∈ℕ (ℕ denotes the set of all positive integers), {μi}∈Δm and {zi}i=0m⊂C such that
(3.5)∥xn+1-∑j=0m‍μjzj∥<tn,(3.6)∑i=1N‍∥zj-Tizj∥≤tn∥xn-yn∥
for all j∈{0,1,…,m}. Put r0=supn≥1||xn-p|| and λ=min1≤i≤N{λi}. Take α∈(0,1)∩(0,λ/K2]. It follows from Lemma 2.2 and (3.5) that
(3.7)∥xn-Tixn∥=1α∥(Tiαxn-p)+(p-xn)∥≤2r0α,(3.8)∥Ti(∑j=0m‍μjzj)-Tixn+1∥≤1α(∥Tiα(∑j=0m‍μjzj)-Tiαxn+1∥+(1-α)∥∑j=0m‍μjzj-xn+1∥)≤(2α-1)∥∑j=0m‍μjzj-xn+1∥≤(2α-1)tn
for all i∈{1,2,…,N}. Moreover, (3.7) implies
(3.9)∥xn-yn∥≤2r0α.
It follows from Lemma 3.1, (3.5)–(3.9) that
(3.10)∑i=1N‍∥xn+1-Tixn+1∥≤∑i=1N‍(∥xn+1-∑j=0m‍μjzj∥+∥∑j=0m‍μj(zj-Tizj)∥∑i=1N‍+∥∑j=0m‍μjTizj-Ti(∑j=0m‍μjzj)∥+∥Ti(∑j=0m‍μjzj)-Tixn+1∥)≤2Nα∥xn+1-∑j=0m‍μjzj∥+∑j=0m‍μj(∑i=1N‍∥zj-Tizj∥)+∑i=1N‍∥∑j=0m‍μjTizj-Ti(∑j=0m‍μjzj)∥≤2Nαtn+tn∥yn-xn∥+∑i=1N‍1αγ-1(αmax0≤k<j≤m(∥zk-Tizk∥+∥zj-Tizj∥))≤2N+2r0αtn+Nαγ-1(4r0tn)→0asn→∞.
This shows that
(3.11)∥xn-Tixn∥→0asn→∞
for all i∈{1,2,…,N}.
(III) Finally, we prove thatxn→p=Pℱx.
It follows from the boundedness of{xn} that there exists {xni}⊂{xn} such that xni⇀v as i→∞. Since for each i∈{0,1,…,N}, Ti is a λi-strict pseudocontraction, then Ti is demiclosed. one has v∈ℱ.
From the weakly lower semicontinuity of the norm and (3.4), we have
(3.12)∥p-x∥≤||v-x||≤liminfi→∞∥xni∥-x≤limsupi→∞∥xni-x∥≤∥p-x∥.
This shows p=v and hence xni⇀p as i→∞. Therefore, we obtain xn⇀p. Further, we have that
(3.13)limn→∞∥xn-x∥=∥p-x∥.
Since E is uniformly convex, we have xn-x→p-x. This shows that xn→p. This completes the proof.Corollary 3.3.
LetC be a nonempty closed subset of a uniformly convex and 2-uniformly smooth Banach space E with the best smooth constant K>0, assume that T:C→C is a λ-strict pseudocontraction for some 0<λ<1 such that ℱ(T)≠∅. Let {xn} be a sequence generated by
(3.14)x0=x∈C,Cn=co¯{z∈C:||z-Tz||≤tn||xn-Txn||},xn+1=PCnx,n=0,1,2,…,
where {tn} is a sequence in (0,1) with tn→0. co¯{z∈C:||z-Tz||≤tn||xn-Txn||} denotes the convex closure of the set {z∈C:||z-Tz||≤tn||xn-Txn||} and PCn is the metric projection from E onto Cn. Then {xn} converges strongly to Pℱ(T)x.Proof.
SetT1=T,Tk=I for all 2≤k≤N, and αn,1=1, αn,k=0 for all 2≤k≤N in Theorem 3.2. The desired result can be obtained directly from Theorem 3.2.Remark 3.4.
At the end of the paper, we would like to point out that concerning the convergence problem of iterative sequences for strictly pseudocontractive mappings has been considered and studied by many authors. It can be consulted the references [24–37].



---

*Source: 759718-2012-10-11.xml*