# Monotone Hybrid Projection Algorithms for an Infinitely Countable Family of Lipschitz Generalized Asymptotically Quasi-Nonexpansive Mappings

Authors: Watcharaporn Cholamjiak; Suthep Suantai
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

We prove a weak convergence theorem of the modified Mann iteration process for a uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mapping in a uniformly convex Banach space. We also introduce two kinds of new monotone hybrid methods and obtain strong convergence theorems for an infinitely countable family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings in a Hilbert space. The results improve and extend the corresponding ones announced by Kim and Xu (2006) and Nakajo and Takahashi (2003).


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## Body

## 1. Introduction

LetC be a  nonempty, closed, and convex subset of a real Banach space E. We denote by F(T) the set of fixed points of T, that is, F(T)={x∈C:x=Tx}. Then  T is said to be(i)
nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C;(ii)
asymptotically nonexpansive if there exists a sequence kn≥1, limn→∞kn=1 and
(1.1)∥Tnx-Tny∥≤kn∥x-y∥
for all x,y∈C and n≥1;(iii)
asymptotically quasi-nonexpansive if there exists a sequence kn≥1, limn→∞kn=1 and
(1.2)∥Tnx-p∥≤kn∥x-p∥
for all x∈C,p∈F(T) and n≥1;(iv)
generalized asymptotically nonexpansive [1] if there exist nonnegative real sequences {kn} and {cn} with kn≥1, limn→∞kn=1 and limn→∞cn=0 such that
(1.3)∥Tnx-Tny∥≤kn∥x-y∥+cn
for all x,y∈C and n≥1;(v)
generalized asymptotically quasi-nonexpansive [1] if there exist nonnegative real sequences {kn} and {cn} with kn≥1, limn→∞kn=1 and limn→∞cn=0 such that
(1.4)∥Tnx-p∥≤kn∥x-p∥+cn
for all x∈C,  p∈F(T) and n≥1;(vi)
asymptotically nonexpansive in the weak sense [2] if
(1.5)lim supn→∞supy∈C(∥Tnx-Tny∥-∥x-y∥)≤0
for all x∈C.(vii)
asymptotically nonexpansive in the intermediate sense [3] if
(1.6)lim supn→∞supx,y∈C(∥Tnx-Tny∥-∥x-y∥)≤0;(viii)
uniformly L-Lipschitzianif there exists a constant L>0 such that
(1.7)∥Tnx-Tny∥≤L∥x-y∥
for all x,y∈C and n≥1.It is clear that a generalized asymptotically quasi-nonexpansive mapping is to unify various classes of mappings associated with the class of generalized asymptotically nonexpansive mapping, asymptotically nonexpansive mappings, and nonexpansive mappings. However, the converse of each of above statement may be not true. The example shows that a generalized asymptotically quasi-nonexpansive mapping is not an asymptotically quasi-nonexpansive mapping; see [1]. Note that if T is asymptotically nonexpansive in the weak sense, we have that(1.8)∥Tnx-Tny∥≤kn∥x-y∥+cn
for all x,y∈C, where cn=max{0,supx,y∈C(∥Tnx-Tny∥-∥x-y∥)} so that limn→∞cn=0. Hence, T is a generalized asymptotically nonexpansive mapping.The mappingT:C→C is said to be demiclosed at 0 if for each sequence {xn} in C converging weakly to x and {Txn} converging strongly to 0, we have Tx=0.A Banach spaceE is said to satisfy Opial's property, see [4], if for each x∈E and each sequence {xn} weakly convergent to x, the following condition holds for all x≠y:(1.9)lim infn→∞∥xn-x∥<lim infn→∞∥xn-y∥.Letτ be a Hausdorff linear topology and let  E satisfy the uniform τ-Opial property. In 1993, Bruck, Kuczumow, and Reich proved that {Tnx} is τ-convergent if and only if {Tnx} is τ-asymptotically regular, that is,(1.10)Tn+1x-Tnx→τ0.
Moreover, they also proved that the τ-limit of {Tnx} is a fixed point of T.In 1953, Mann [5] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping T in a Hilbert space H:(1.11)xn+1=αnxn+(1-αn)Txn,∀n∈ℕ,
where the initial point x0 is taken in C arbitrarily and {αn} is a sequence in [0,1].However, we note that Mann's iteration process (1.11) has only weak convergence, in general; for instance, see [6–8].In 2003, Nakajo and Takahashi [9] proposed the following modification of the Mann iteration for a single nonexpansive mapping T in a Hilbert space. They proved the following theorem.Theorem 1.1.
LetC be a closed and convex subset of a Hilbert space H and let T:C→C be a nonexpansive mapping such that F(T)≠∅. Assume that {αn}n=0∞ is a sequence in [0,1] such that αn≤1-δ for some δ∈(0,1]. Define a sequence {xn}n=0∞ in C by the following algorithm:
(1.12)x0∈Cchosenarbitrarily,yn=αnxn+(1-αn)Txn,Cn={z∈C:∥yn-z∥≤∥xn-z∥},Qn={z∈C:〈x0-xn,xn-z〉≥0},xn+1=PCn∩Qnx0.
Then {xn} defined by (1.12)  converges strongly to PF(T)x0.Recently, Kim and Xu [10] extended the result of Nakajo and Takahashi [9] from nonexpansive mappings to asymptotically nonexpansive mappings. They proved the following theorem.Theorem 1.2.
Let C be a nonempty, bounded, closed, and convex subset of a Hilbert space H and letT:C→C be an asymptotically nonexpansive mapping with a sequence {kn} such that kn→1 as n→∞. Assume that {αn}n=0∞ is a sequence in [0,1] such that lim supn→∞αn<1. Define a sequence {xn} in C by the following   algorithm:
(1.13)x0∈Cchosen  arbitrarily,yn=αnxn+(1-αn)Tnxn,Cn={z∈C:∥yn-z∥2≤∥xn-z∥2+θn},Qn={z∈C:〈x0-xn,xn-z〉≥0},xn+1=PCn∩Qnx0,
where θn=(1-αn)(kn2-1)(diam C)2→0, as n→∞. Then {xn} defined by (1.13) converges strongly to PF(T)x0.Since 2003, the strong convergence problems of the CQ method for fixed point iteration processes in a Hilbert space or a Banach space have been studied by many authors; see [9–20].Let{Ti}i=1∞ be an infinitely family of uniformly Li-Lipschitzian and generalized asymptotically quasi-nonexpansive mappings and let F:=⋂i=1∞F(Ti). In this paper, motivated by Kim and Xu [10] and  Nakajo and Takahashi [9], we introduce two kinds of new algorithms for finding a common fixed point of a countable family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings which are modifications of the normal Mann iterative scheme. Our iterative schemes are defined as follows.Algorithm 1.3.
For an initial pointx0∈C, compute the sequence {xn} by the iterative process:
(1.14)yi,n=αi,nxn+(1-αi,n)Tinxn,Ci,n={z∈C:∥yi,n-z∥2≤∥xn-z∥2-αi,n(1-αi,n)∥Tinxn-xn∥2+(1-αi,n)θi,n},Cn=⋂i=1∞Ci,n,Q0=C,Qn={z∈Qn-1:〈z-xn,x0-xn〉≤0},n≥1,xn+1=PCn∩Qnx0,n≥0,
where θi,n=(ki,n2-1)∇n2+2ki,nci,n∇n+ci,n2, ∇n=supn∈ℕ{∥xn-z∥:z∈F}<∞.Algorithm 1.4.
For an initial pointx0∈C, compute the sequence {xn} by the iterative process:
(1.15)Ci,0=C,yi,n=αi,nxn+(1-αi,n)Tinxn,Ci,n+1={z∈Ci,n:∥yi,n-z∥2≤∥xn-z∥2-αi,n(1-αi,n)∥Tinxn-xn∥2+(1-αi,n)θi,n},Cn+1=⋂i=1∞Ci,n+1,xn+1=PCn+1x0,n≥0,
where θi,n=(ki,n2-1)∇n2+2ki,nci,n∇n+ci,n2, ∇n=supn∈ℕ{∥xn-z∥:z∈F}<∞.


## 2. Preliminaries

In this section, we present some useful lemmas which will be used in our main results.Lemma 2.1 (see [21]).
Let{an}, {bn}, and {cn} be three sequences of nonnegative numbers such that bn≥1 and
(2.1)an+1≤bnan+cn,∀n≥1,
if ∑n=1∞cn<∞ and ∑n=1∞(bn-1)<∞, then limn→∞an exists.Lemma 2.2 (see [22]).
Letp>1,  r>0 be two fixed numbers. Then a Banach space E is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(2.2)∥λx+(1-λ)y∥p≤λ∥x∥p+(1-λ)∥y∥p-ωp(λ)g(∥x-y∥)
for all x,y∈Br(0)={x∈X:∥x∥≤r} and λ∈[0,1] where ωp(λ)=λ(1-λ)p+λp(1-λ).Lemma 2.3.
LetC be a nonempty subset of a Banach space E and let  T:C→C be a uniformly L-Lipschitzian mapping. Let {xn} be a sequence in C such that limn→∞∥xn+1-xn∥=0  and  limn→∞∥xn-Tnxn∥=0. Then limn→∞∥xn-Txn∥=0.Proof.
SinceT is uniformly L-Lipschitzian, we have
(2.3)∥xn-Txn∥≤∥xn-xn+1∥+∥xn+1-Tn+1xn+1∥+∥Tn+1xn+1-Tn+1xn∥+∥Tn+1xn-Txn∥≤∥xn-xn+1∥+∥xn+1-Tn+1xn+1∥+L∥xn-xn+1∥+L∥Tnxn-xn∥.
It follows that  limn→∞∥xn-Txn∥=0.The following lemmas give some characterizations and useful properties of the metric projectionPC in a Hilbert space.LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥ and let C be a closed and convex subset of H. For every point x∈H, there exists a unique nearest point in C, denoted by PCx, such that
(2.4)∥x-PCx∥≤∥x-y∥,∀y∈C.PC is called the metric projection of H onto C. We know that PC is a nonexpansive mapping of H onto C.Lemma 2.4 (see [12]).
LetC be a closed and convex subset of a real Hilbert space H and let PC be the metric projection from H onto C. Given x∈H and z∈C, then z=PCx  if and only if the following holds:
(2.5)〈x-z,y-z〉≤0,∀y∈C.Lemma 2.5 (see [9]).
LetC be a nonempty, closed, and convex subset of a real Hilbert space H and let  PC:H→C be the matric projection from H onto C. Then the following inequality holds:
(2.6)∥y-PCx∥2+∥x-PCx∥2≤∥x-y∥2,∀x∈H,∀y∈C.Lemma 2.6 (see [12]).
LetH be a real Hilbert space. Then the following equations hold:  (i)
∥x-y∥2=∥x∥2-∥y∥2-2〈x-y,y〉, for all  x,y∈H;(ii)
∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2, for all t∈[0,1] and x,y∈H.Lemma 2.7 (see [10]).
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Given x,y,z∈H and also given a∈ℝ, the set
(2.7){v∈C:∥y-v∥2≤∥x-v∥2+〈z,v〉+a}
is convex and closed.Lemma 2.8.
LetC be a closed and convex subset of a real Hilbert space H. Let  T:C→C be a uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive mapping with nonnegative real sequences {kn},  {cn} such that kn≥1, limn→∞kn=1 and limn→∞cn=0. Then F(T) is a closed and convex subset of C.Proof.
SinceT is continuous, F(T) is closed. Next, we show that F(T) is convex. Let p1,p2∈F(T) and t∈(0,1). Put p=tp1+(1-t)p2. By Lemma 2.6, we have
(2.8)∥p-Tnp∥2=∥t(Tnp-p1)+(1-t)(Tnp-p2)∥2=t∥Tnp-p1∥2+(1-t)∥Tnp-p2∥2-t(1-t)∥p1-p2∥2≤t(kn∥p-p1∥+cn)2+(1-t)(kn∥p-p2∥+cn)2-t(1-t)∥p1-p2∥2=t(1-t)2kn2∥p1-p2∥2+(1-t)t2kn2∥p1-p2∥2-t(1-t)∥p1-p2∥2+2tkncn∥p-p1∥+tcn2+2(1-t)kncn∥p-p2∥+(1-t)cn2=((1-t)t(kn2-1))∥p1-p2∥2+2tkncn∥p-p1∥+tcn2+2(1-t)kncn∥p-p2∥+(1-t)cn2.
This implies that limn→∞∥p-Tnp∥=0. Since T is continuous, we have limn→∞∥Tp-Tn+1p∥=0, so that p=Tp. Hence p∈F(T).


## 3. Main Results

First, we prove a weak convergence theorem for a single uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mapping in a uniformly convex Banach space.Theorem 3.1.
LetE be a uniformly convex Banach space E which satisfies Opial's property. Let C be a nonempty, closed, and convex subset of E, and T:C→C a uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive mapping with nonnegative real sequences {kn}, {cn} such that kn≥1, limn→∞kn=1 and limn→∞cn=0. Assume that I-T is demiclosed at 0, where I is the identity mapping of C and  {αn} is a sequence in [0,1] such that 0<lim infn→∞αn≤lim supn→∞αn<1 and ∑n=1∞(kn-1)<∞,∑n=1∞cn<∞. Let {xn} be the sequence in C generated by the modified Mann iteration process:
(3.1)xn+1=αnxn+(1-αn)Tnxn,∀n∈ℕ.
Then {xn} converges weakly to a fixed point of F(T).Proof.
Letp∈F(T), we have
(3.2)∥xn+1-p∥≤αn∥xn-p∥+(1-αn)∥Tnxn-p∥≤αn∥xn-p∥+(1-αn)kn∥xn-p∥+(1-αn)cn≤(1+(kn-1)(1-αn))∥xn-p∥+(1-αn)cn.
Since Σn=1∞(kn-1)<∞,  Σn=1∞cn<∞, then by Lemma 2.1 and (3.2), we obtain that
(3.3)limn→∞∥xn-p∥
exists.
This implies that{Tnxn-p} is bounded. Put r=max{supn∈N∥Tnxn-p∥,supn∈N∥xn-p∥}. By Lemma 2.2, there is a continuous strictly increasing convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(3.4)∥xn+1-p∥2=∥αn(xn-p)+(1-αn)(Tnxn-p)∥2≤αn∥xn-p∥2+(1-αn)∥Tnxn-p∥2-αn(1-αn)g(∥Tnxn-xn∥)≤αn∥xn-p∥2+(1-αn)kn2∥xn-p∥2+2(1-αn)kncn∥xn-p∥+(1-αn)cn2-αn(1-αn)g(∥Tnxn-xn∥).
It follows that
(3.5)αn(1-αn)g(∥Tnxn-xn∥)≤(kn2-αn(kn2-1))∥xn-p∥2-∥xn+1-p∥2+2(1-αn)kncn∥xn-p∥+(1-αn)cn2.
By our assumptions and (3.3), we get limn→∞g(∥Tnxn-xn∥)=0. Since g is continuous strictly increasing with g(0)=0, we can conclude that limn→∞∥Tnxn-xn∥=0. Observe that ∥xn+1-xn∥≤αn∥Tnxn-xn∥→0. It follows from Lemma 2.3 that limn→∞∥xn-Txn∥=0. Since {xn} is bounded, there exists a subsequence {xnk} of {xn} converging weakly to some q1∈C. From limn→∞∥xn-Txn∥=0 and I-T is demiclosed at 0, we obtain that  Tq1=q1. That is, q1∈F(T). Next, we show that {xn} converges weakly to q1 and  take another subsequence {xmk} of {xn} converging weakly to some q2∈C. Again, as above, we can conclude that q2∈F(T). Finally, we show that q1=q2. Assume q1≠q2. Then by   Opial's property of E, we have
(3.6)limn→∞∥xn-q1∥=limk→∞∥xnk-q1∥<limk→∞∥xnk-q2∥=limn→∞∥xn-q2∥=limk→∞∥xmk-q2∥<limk→∞∥xmk-q1∥=limn→∞∥xn-q1∥,
which is a contradiction. Therefore q1=q2. This shows that {xn} converges weakly to q1∈F(T).Remark 3.2.
(1) In [3, Theorem 2], Bruck et al. proved that if T:C→C is asymptotically nonexpansive in the weak sense and T is τ-asymptotically regular, then   Picard’s iterated sequence {Tnx} is τ-convergent to a fixed point of T. Since every asymptotically nonexpansive in the weak sense is a generalized asymptotically nonexpansive mapping and if its fixed point set is nonempty, then it is a generalized quasiasymptotically nonexpansive mapping. So we can apply Theorem 3.1 with a mapping T which is asymptotically nonexpansive in the weak sense when its fixed point set is nonempty and obtain that the sequence {xn} generated by the modified Mann iteration process
(3.7)xn+1=αnxn+(1-αn)Tnxn,∀n∈ℕ
converges weakly to a fixed point of T without asymptotically regularity condition of T.
(2) In [3, Theorem 4], Bruck et al. showed that if C is a bounded convex subset of a uniformly convex Banach space and C is sequentially τ-compact, T:C→C is asymptotically nonexpansive in the intermediate sense and ∑i=1∞cni<∞ where cn=max{0,supx,y∈C(∥Tnx-Tny∥-∥x-y∥)} for some sequence of nonnegative integers {ni}, then the sequence {xi} generated by
(3.8)xi+1=αixi+(1-αi)Tnixi
is τ-convergent to a fixed point of T. Note that every asymptotically nonexpansive mapping in the intermediate sense is a generalized asymptotically nonexpansive mapping. Hence, Theorem 3.1 can be applied to the class of asymptotically nonexpansive mappings in the intermediate sense to obtain weak convergence of the sequence {xn} generated by
(3.9)xn+1=αnxn+(1-αn)Tnxn,∀n∈ℕ
without the boundedness and compactness conditions on C.Note that the modified Mann's iteration in Theorem3.1 has only weak convergence.Question 1.
How can we modify the modified Mann's iteration in order to obtain strong convergence?In the following theorem, we introduce a monotone hybrid method with the modified Mann's iteration to obtain a strong convergence theorem for an infinite family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings.Theorem 3.3.
LetC be a closed and convex subset of a real Hilbert space H. Let {Ti}i=1∞ be an infinitely countable family of uniformly Li-Lipschitzian and generalized asymptotically quasi-nonexpansive mappings of C into itself with nonnegative real sequences {ki,n}, {ci,n} such that ki,n≥1, limn→∞ki,n=1, limn→∞ci,n=0, for all i∈N. Assume that F:=⋂i=1∞F(Ti)≠∅ and the sequence αi,n∈[0,1), for all  i,n∈ℕ. Then the sequence {xn} generated by Algorithm 1.3 converges strongly to PFx0.Proof.
We split the proof into six steps.
Step 1.
Show thatPFx0 is well defined for every x0∈C.
By Lemma2.8, we obtain that F(Ti) is a closed and convex subset of C for every i∈ℕ. Hence, F:=⋂i=1∞F(Ti) is a nonempty, closed, and convex subset of C; consequently, PFx0 is well defined for every x0∈C.Step 2.
Show thatPCn∩Qnx0 is well defined.
From the definition ofCn and Qn, it is obvious that Qn is closed and convex for each n∈ℕ. It follows from Lemma 2.7 that Ci,n is closed and convex for all i,n∈ℕ. This implies that Cn is closed and convex for each n∈ℕ. Next, we will show that F⊂Cn∩Qn, for all n≥0. First, we prove that F⊂Ci,n for all n≥0 and i∈ℕ. Since Ti is a generalized asymptotically quasi-nonexpansive mapping for all i∈ℕ, we have that for any p∈F,
(3.10)∥yi,n-p∥2=∥αi,nxn+(1-αi,n)Tinxn-p∥2=αi,n∥xn-p∥2+(1-αi,n)∥Tinxn-p∥2-αi,n(1-αi,n)∥Tinxn-xn∥2≤αi,n∥xn-p∥2+(1-αi,n)(ki,n∥xn-p∥+ci,n)2-αi,n(1-αi,n)∥Tinxn-xn∥2=∥xn-p∥2-αi,n(1-αi,n)∥Tinxn-xn∥2+(1-αi,n)((ki,n2-1)∥xn-p∥2+2ki,nci,n∥xn-p∥+ci,n2)≤∥xn-p∥2-αi,n(1-αi,n)∥Tinxn-xn∥2+(1-αi,n)θi,n.
Hence, p∈Ci,n for all n≥0 and i∈ℕ. This proves that F⊂Ci,n for all n≥0 and i∈ℕ. Hence, F⊂Cn for all n≥0.
As shown in Marino and Xu [12], by induction, we can show that F⊂Qn for all n≥0. Hence F⊂Cn∩Qn, for all n≥0,  and  so PCn∩Qnx0 is well defined.Step 3.
Show thatlimn→∞∥xn-x0∥ exists.
Fromxn=PQnx0 and xn+1∈Qn, we have
(3.11)∥xn-x0∥≤∥xn+1-x0∥,∀n≥0.
On the other hand, as F⊂Qn, we obtain
(3.12)∥xn-x0∥≤∥z-x0∥,∀n≥0,∀z∈F.
So we have that the sequence {xn} is bounded and nondecreasing. Therefore limn→∞∥xn-x0∥ exists.Step 4.
Show thatxn→q, where q∈C.
Form>n, by the definition of Qn, we see that Qm⊂Qn. Noting that xm=PQmx0 and xn=PQnx0, by Lemma 2.5, we conclude that
(3.13)∥xm-xn∥2≤∥xm-x0∥2-∥xn-x0∥2.
It follows from Step 3 that {xn} is a Cauchy. So, we can assume that xn→q as n→∞ for some q∈C. In particular, we have that
(3.14)limn→∞∥xn+1-xn∥=0.Step 5.
Show thatlimn→∞∥xn-Tixn∥=0, for all  i∈ℕ.
Leti,n∈ℕ. Since xn+1∈Cn, it follows from (3.10) that
(3.15)∥yi,n-xn+1∥2≤∥xn-xn+1∥2-αi,n(1-αi,n)∥Tinxn-xn∥2+(1-αi,n)θi,n.
Moreover, by Lemma 2.6, we have
(3.16)∥yi,n-xn+1∥2=∥αi,nxn+(1-αi,n)Tinxn-xn+1∥2=αi,n∥xn-xn+1∥2+(1-αi,n)∥Tinxn-xn+1∥2-αi,n(1-αi,n)∥Tinxn-xn∥2.
It follows from (3.15) and (3.16) that
(3.17)(1-αi,n)∥Tinxn-xn+1∥2≤(1-αi,n)∥xn-xn+1∥2+(1-αi,n)θi,n.
Since θi,n→0 as n→∞ for all i∈ℕ, we have by (3.14) and (3.17) that
(3.18)limn→∞∥Tinxn-xn+1∥=0.
This implies that
(3.19)∥Tinxn-xn∥≤∥Tinxn-xn+1∥+∥xn+1-xn∥→0
for all i∈ℕ. By Lemma 2.3 and (3.14), we get that limn→∞∥xn-Tixn∥=0 for all i∈ℕ.Step 6.
Show thatq=PFx0.
Sincexn→q and  limn→∞∥xn-Tixn∥=0 for all i∈ℕ, we have q=Tiq for all i∈ℕ. Hence q∈F. By Lemma 2.4, we obtain
(3.20)〈z-xn,x0-xn〉≤0
for all z∈Qn-1∩Cn-1. Since F⊂Qn-1∩Cn-1, we have
(3.21)〈z-q,x0-q〉≤0
for all z∈F. Again by Lemma 2.4, we obtain that q=PFx0. This completes the proof.Theorem 3.4.
LetC be a closed and convex subset of a real Hilbert space H. Let {Ti}i=1∞ be an infinitely countable family of uniformly Li-Lipschitzian and generalized asymptotically quasi-nonexpansive mappings of C into itself with nonnegative real sequences {ki,n}, {ci,n} such that ki,n≥1, limn→∞ki,n=1, limn→∞ci,n=0, for all  i∈ℕ. Assume that F:=⋂i=1∞F(Ti)≠∅ and the sequence {αi,n}⊂[0,1), for all  i,n∈ℕ. Then the sequence {xn} generated by Algorithm 1.4 converges strongly to PFx0.Proof.
We divide our proof into four steps.
Step 1.
Show thatCn is closed and convex and F⊂Cn for all n≥1.
It follows from Lemma2.7 that Ci,n is closed and convex for all i,n∈ℕ. This implies that Cn is closed and convex for each n∈ℕ. Next, we will show that F⊂Ci,n for all n∈ℕ. For n=1, F⊂Ci,1=C. Assume that F⊂Ci,n for n∈ℕ. It follows from (3.10) and the definition of Ci,n+1  that  F⊂Ci,n+1.Step 2.
Show thatlimn→∞∥xn-x0∥ exists.
Fromxn=PCnx0, Cn+1⊂Cn and xn+1∈Cn, for all  n≥0, we have
(3.22)∥xn-x0∥≤∥xn+1-x0∥,∀n≥0.
On the other hand, as F⊂Cn, we obtain
(3.23)∥xn-x0∥≤∥z-x0∥,∀n≥0,∀z∈F.
So we have that the sequence {xn} is bounded and nondecreasing. Therefore limn→∞∥xn-x0∥ exists.Step 3.
Show thatxn→q, where q∈C.
Form>n, by the definition of Cn, we see that xm=PCmx0∈Cm⊂Cn. By Lemma 2.5, we obtain that
(3.24)∥xm-xn∥2≤∥xm-x0∥2-∥xn-x0∥2.
From Step 2, we obtain that {xn} is Cauchy. Hence xn→q as n→∞ for some q∈C and limn→∞∥xn+1-xn∥=0. By using the same proof as in Step 5 of Theorem 3.3, we can show that limn→∞∥xn-Tixn∥=0, for all  i∈ℕ.Step 4.
Show thatq=PFx0.
Sincexn→q and limn→∞∥xn-Tixn∥=0, for all  i∈ℕ, we  have q=Tiq, for all  i∈ℕ. Hence q∈F. Since xn=PCnx0, by Lemma 2.4, we have
(3.25)〈z-xn,x0-xn〉≤0
for all z∈Cn, and hence,
(3.26)〈z-q,x0-q〉≤0
for all z∈F. This shows that q=PFx0, which completes the proof.Since a generalized asymptotically quasi-nonexpansive mapping is to unify various classes of mappings associated with the class of generalized asymptotically nonexpansive mapping, we have the following.Corollary 3.5.
LetC be a closed and convex subset of a real Hilbert space H. Let {Ti}i=1∞ be an infinitely countable family of uniformly Li-Lipschitzian and generalized asymptotically nonexpansive mappings of C into itself with nonnegative real sequences {ki,n}, {ci,n} such that ki,n≥1, limn→∞ki,n=1, limn→∞ci,n=0, for all  i∈N. Assume that F:=⋂i=1∞F(Ti)≠∅ and the sequence {αi,n}⊂[0,1), for all  i,n∈ℕ. Let a sequence {xn} be generated by the following manner:
(3.27)x0∈Cchosen  arbitrarily,yi,n=αi,nxn+(1-αi,n)Tinxn,Ci,n={z∈C:∥yi,n-z∥2≤∥xn-z∥2-αi,n(1-αi,n)∥Tinxn-xn∥2+(1-αi,n)θi,n},Cn=⋂i=1∞Ci,n,Q0=C,Qn={z∈Qn-1:〈z-xn,x0-xn〉≤0},n≥1,xn+1=PCn∩Qnx0,n≥0,
where θi,n=(ki,n2-1)∇n2+2ki,nci,n∇n+ci,n2, ∇n=supn∈ℕ{∥xn-z∥:z∈F}<∞. Then the sequence {xn} converges strongly to PFx0.Corollary 3.6.
LetC be a closed and convex subset of a real Hilbert space H. Let {Ti}i=1∞ be an infinitely countable family of uniformly Li-Lipschitzian and generalized asymptotically nonexpansive mappings of C into itself with nonnegative real sequences {ki,n}, {ci,n} such that ki,n≥1, limn→∞ki,n=1, limn→∞ci,n=0, for all  i∈N. Assume that F:=⋂i=1∞F(Ti)≠∅ and the sequence {αi,n}⊂[0,1), for all  i,n∈ℕ. Let a sequence {xn} be generated by the following manner:
(3.28)x0∈Cchosen arbitrarily,Ci,0=C,yi,n=αi,nxn+(1-αi,n)Tinxn,Ci,n+1={z∈Ci,n:∥yi,n-z∥2≤∥xn-z∥2-αi,n(1-αi,n)∥Tinxn-xn∥2+(1-αi,n)θi,n},Cn+1=⋂i=1∞Ci,n+1,xn+1=PCn+1x0,n≥0,
where θi,n=(ki,n2-1)∇n2+2ki,nci,n∇n+ci,n2, ∇n=supn∈ℕ{∥xn-z∥:z∈F}<∞. Then the sequence {xn} converges strongly to PFx0.Remark 3.7.
(i)  Corollaries3.5 and  3.6 improve and extend the main result in [10].
(ii)  If we takeTi=T, ki,n=1 and ci,n=0 for all i,n∈ℕ where T is a nonexpansive mapping, then Corollary 3.5 reduces to  [9, Theorem 3.4].



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*Source: 297565-2009-12-31.xml*