# Strong Convergence Theorems for a Countable Family of Lipschitzian Mappings

Authors: Weerayuth Nilsrakoo; Satit Saejung
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We modify the iterative method introduced by Kim and Xu (2006) for a countable family of Lipschitzian mappings by the hybrid method of Takahashi et al. (2008). Our results include recent ones concerning asymptotically nonexpansive mappings due to Plubtieng and Ungchittrakool (2007) and Zegeye and Shahzad (2008, 2010) as special cases.


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## Body

## 1. Introduction

LetC be a  nonempty closed convex subset of a real Hilbert space H. A mapping T:C→C is said to be Lipschitzian if there exists a positive constant L such that (1.1)∥Tx-Ty∥≤L∥x-y∥∀x,y∈C.
In this case, T is also said to be L-Lipschitzian. Clearly, if T is L1-Lipschitzian and L1<L2, then T is L2-Lipschitzian. Throughout the paper, we assume that every Lipschitzian mapping is L-Lipschitzian with L≥1. If L=1, then T is known as a nonexpansive mapping. We denote by F(T) the set of fixed points of T. If C is nonempty bounded closed convex and T is a nonexpansive of C into itself, then F(T)≠Ø  (see [1]). There are many methods for approximating fixed points of a nonexpansive mapping. In 1953, Mann [2] introduced the iteration as follows: a sequence {xn} defined by(1.2)xn+1=αnxn+(1-αn)Txn,
where the initial guess element x1∈C is arbitrary and {αn} is a real sequence in [0,1]. Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results is proved by Reich [3]. In an infinite-dimensional Hilbert space, Mann iteration can conclude only weak convergence [4]. Attempts to modify the Mann iteration method (1.2) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [5] proposed the following modification of Mann iteration method (1.2):(1.3)x1=x∈Cisarbitrary,yn=αnxn+(1-αn)Txn,Cn={z∈C:∥yn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x-xn〉≥0},xn+1=PCn∩Qnx,n=1,2,3,…,
where PK denotes the metric projection from H onto a closed convex subset K of H. They prove that if the sequence {αn} bounded above from one, then {xn} defined by (1.3) converges strongly to PF(T)x. Takahashi et al. [6] modified (1.3) so-called the shrinking projection method for a countable family of nonexpansive mappings {Tn}n=1∞ as follows:(1.4)x1=x∈H,C1=C,yn=αnxn+(1-αn)Tnxn,Cn+1={z∈Cn:∥yn-z∥≤∥xn-z∥},xn+1=PCn+1x,n=1,2,3,…,
and prove that if the sequence {αn} bounded above from one, then {xn} defined by (1.4) converges strongly to P∩n=1∞F(Tn)x.Recently, the present authors [7] extended (1.3) to obtain a strong convergence theorem for common fixed points of a countable family of Ln-Lipschitzian mappings {Tn}n=1∞ by (1.5)x1=x∈Cisarbitrary,yn=αnxn+(1-αn)Tnxn,Cn={z∈C:∥yn-z∥2≤∥xn-z∥2+θn},Qn={z∈C:〈xn-z,x-xn〉≥0},xn+1=PCn∩Qnx,n=1,2,3,…,
where θn=(1-αn)(Ln2-1)(diamC)2→0 as n→∞ and prove that {xn} defined by (1.5) converges strongly to P∩n=1∞F(Tn)x.In this paper, we establish strong convergence theorems for finding common fixed points of a countable family of Lipschitzian mappings in a real Hilbert space. Moreover, we also apply our results for asymptotically nonexpansive mappings.


## 2. Preliminaries

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Then,(2.1)∥x-y∥2=∥x∥2-∥y∥2-2〈x-y,y〉,(2.2)∥λx+(1-λ)y∥2=λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)∥x-y∥2,
for all x,y∈H and λ∈[0,1]. For any n points x1,x2,…,xn in H, the following generalized identity holds:(2.3)‖∑i=1nλixi‖2=∑i=1nλi∥xi∥2-∑i<jλiλj∥xi-xj∥2,
where λi∈[0,1] and ∑i=1nλi=1.We writexn→x (xn⇀x, resp.) if {xn} converges strongly (weakly, resp.) to x. It is also known that H satisfies:  (1)
theOpial's condition [8] that is, for any sequence {xn} with xn⇀x,
(2.4)liminf⁡n→∞∥xn-x∥<liminf⁡n→∞∥xn-y∥
holds for every y∈H with y≠x(2)
theKadec-Klee property [9, 10]; that is, for any sequence {xn} with xn⇀x and ∥xn∥→∥x∥ together imply xn→x.LetC be a nonempty closed convex subset of H. Then, for any x∈H, there exists a unique nearest point in C, denoted by PCx, such that (2.5)∥x-PCx∥≤∥x-y∥∀y∈C.
Such a mapping PC is called the metric projection of H onto C. We know that PC is nonexpansive. Furthermore, for x∈H and z∈C, (2.6)z=PCxiff〈x-z,z-y〉≥0,∀y∈C.
To deal with a family of mappings, the following conditions are introduced: let C be a subset of a Banach space, let {Tn} and 𝒯 be families of mappings of C with ⋂n=1∞F(Tn)=F(𝒯)≠Ø, where F(𝒯) is the set of all common fixed points of all mappings in 𝒯. {Tn} is said to satisfy  (a)
theAKTT-condition [11] if for each bounded subset B of C,
(2.7)∑n=1∞sup⁡{∥Tn+1z-Tnz∥:z∈B}<∞,(b)
theNST-condition (I) with𝒯 [12] if for each bounded sequence {zn} in C,
(2.8)lim⁡n→∞∥zn-Tnzn∥=0implieslim⁡n→∞∥zn-Tzn∥=0∀T∈T,(c)
theNST-condition (II) [12] if for each bounded sequence {zn} in C,
(2.9)lim⁡n→∞∥zn+1-Tnzn∥=0implieslim⁡n→∞∥zn-Tmzn∥=0∀m∈N,(d)
NST*-condition with𝒯 [13] if for each bounded sequence {zn} in C,
(2.10)lim⁡n→∞∥zn-Tnzn∥=0,lim⁡n→∞∥zn-zn+1∥=0,
imply limn→∞∥zn-Tzn∥=0 for all T∈𝒯.In particular, if𝒯={T}, then we simply say that {Tn} satisfies the NST-condition (I) with T (NST*-condition with T, resp.) rather than NST-condition (I) with {T} (NST*-condition with {T}, resp.).Remark 2.1.
It follows directly from the definitions above that(i)
if{Tn} satisfies the NST-condition (I) with 𝒯, then {Tn} satisfies the NST*-condition with 𝒯(ii)
if{Tn} satisfies the NST-condition (II), then {Tn} satisfies the NST*-condition with {Tn}.Lemma 2.2 . (see [11, Lemma 3.2]).
LetC be a nonempty closed subset of a Banach space, and let {Tn} be a family of mappings of C into itself which satisfies the AKTT-condition, then the mapping T:C→C defined by
(2.11)Tx=lim⁡n→∞Tnx∀x∈C
satisfies
(2.12)lim⁡n→∞sup⁡{∥Tz-Tnz∥:z∈B}=0,
for each bounded subset B of C.From now on, we will write({Tn},T) satisfies AKTT-condition if {Tn} satisfies AKTT-condition and T is defined by (2.11).Lemma 2.3 . (see [13, Lemma 2.6]).
LetC be a nonempty closed subset of a Banach space. Suppose that ({Tn},T) satisfies AKTT-condition and F(T)=⋂n=1∞F(Tn)≠Ø. Then, {Tn} satisfies the NST-condition (I) with T. Consequently, {Tn} satisfies the NST*-condition with T.Remark 2.4.
There are families of mappings{Tn} and 𝒯 such that  (1)
{Tn} satisfies the NST*-condition with 𝒯, and(2)
{Tn} fails the NST-condition (I) with 𝒯 and the NST-condition (II).The following example shows that the NST*-condition with𝒯 is strictly weaker than NST-condition (I) with 𝒯 and the NST-condition (II).Example 2.5 (see [13, Example 2.9]).
LetH:=ℝ2 and C:=[0,1]×[0,1]. Define T1,T2:C→C as follows:
(2.13)T1(x,y)=(x,1-y),T2(x,y)=(1-x,y),
for all (x,y)∈C. Hence, T1 and T2 are nonexpansive mappings with
(2.14)F(T1)∩F(T2)=([0,1]×{12})∩({12}×[0,1])={(12,12)}≠Ø.
Let Tn=Tn-1(mod2)+1. Then, {Tn} satisfies NST*-condition but it fails NST-condition (I) with 𝒯 and the NST-condition (II).Lemma 2.6.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Tn} and {Sn} be two families of tn-Lipschitzian and sn-Lipschitzian mappings of C into itself, respectively. Let {Un} be a family of mappings of C into itself defined by
(2.15)Un=Tn(βnI+(1-βn)Sn)∀n∈N,
where {βn} is a sequence in [a,b] for some a,b∈(0,1) and I is an identity mapping. Assume that {tn} and {sn} are two sequences such that tn→1 and sn→1. Then, the following statements hold.  (i)
{Un} is a family of Ln-Lipschitzian mappings of C into itself, where Ln=(βntn2+(1-βn)tn2sn2)1/2 and Ln→1.(ii)
Suppose that𝒯1 and 𝒯2 are families of mappings of C into itself such that F(𝒯1)=⋂n=1∞F(Tn), F(𝒯2)=⋂n=1∞F(Sn) and F(𝒯1)∩F(𝒯2)≠Ø. If {Tn} and {Sn} satisfy the NST*-condition with 𝒯1 and 𝒯2, respectively, then {Un} satisfies the NST*-condition with 𝒯1∪𝒯2 and(2.16)⋂n=1∞F(Un)=F(T1)∩F(T2).Proof.
(i) We first observe that(2.17)∥Unx-Uny∥2≤tn2∥βn(x-y)+(1-βn)(Snx-Sny)∥2≤tn2(βn∥x-y∥2+(1-βn)∥Snx-Sny∥2)≤βntn2∥x-y∥2+(1-βn)tn2sn2∥x-y∥2=Ln2∥x-y∥2,
for all x,y∈C. That is, Un is Ln-Lipschitzian. Since tn→1 and sn→1, it follows that Ln→1.
(ii) Let{zn} be a bounded sequence in C such that limn→∞∥zn-Unzn∥=limn→∞∥zn+1-zn∥=0. Let p∈F(𝒯1)∩F(𝒯2), and let M=sup{∥zn-Unzn∥,∥zn-p∥:n∈ℕ}. Then
(2.18)∥zn-p∥2≤(∥zn-Unzn∥+∥Unzn-p∥)2=∥zn-Unzn∥2+2‖zn-Unzn‖‖Unzn-p‖+∥Unzn-p∥2≤3M∥zn-Unzn∥+∥Tn(βnzn+(1-βn)Snzn)-Tnp∥2≤3M∥zn-Unzn∥+tn2∥βn(zn-p)+(1-βn)(Snzn-p)∥2=3M∥zn-Unzn∥+βntn2∥zn-p∥2+(1-βn)tn2∥Snzn-p∥2-βn(1-βn)tn2∥zn-Snzn∥2≤3M∥zn-Unzn∥+βntn2∥zn-p∥2+(1-βn)tn2sn2∥zn-p∥2-a(1-b)∥zn-Snzn∥2=3M∥zn-Unzn∥+Ln2∥zn-p∥2-a(1-b)∥zn-Snzn∥2,
for all n∈ℕ. In particular,
(2.19)a(1-b)∥zn-Snzn∥2≤3M∥zn-Unzn∥+(Ln2-1)∥zn-p∥2.
So, we get
(2.20)lim⁡n→∞∥zn-Snzn∥=0.
Since {Sn} satisfies the NST*-condition with 𝒯2, we have
(2.21)lim⁡n→∞∥zn-Szn∥=0∀S∈T2.
Since
(2.22)∥zn-Tnzn∥≤∥zn-Unzn∥+∥Unzn-Tnzn∥≤∥zn-Unzn∥+(1-βn)tn∥zn-Snzn∥,
it follows that
(2.23)lim⁡n→∞∥zn-Tnzn∥=0.
Since {Tn} satisfies the NST*-condition with 𝒯1, we have
(2.24)lim⁡n→∞∥zn-Tzn∥=0∀T∈T1.
It is easy to see that F(𝒯1)∩F(𝒯2)⊂⋂n=1∞F(Un). To see the reverse inclusion, let z∈⋂n=1∞F(Un) follow the first part of the proof above but now let zn≡z. Then, z∈F(𝒯1)∩F(𝒯2)=F(𝒯1∪𝒯2). Hence, {Un} satisfies the NST*-condition with 𝒯1∪𝒯2.Lemma 2.7.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Tni}n=1∞ be families of Lni-Lipschitzian mappings of C into itself for i=1,2,…,r, respectively. Let {Tn} be a family of mappings of C into itself defined by
(2.25)Tn=∑i=1rβniTni∀n∈N,
where {βni}n=1∞ are sequences in [a,b] for some a,b∈(0,1) satisfying ∑i=1rβni=1 for all n∈ℕ. Assume that {Lni}n=1∞ are sequences such that Lni→1 as n→∞ for all i=1,2,…,r. Then, the following statements hold.  (i)
{Tn} is a family of Ln-Lipschitzian mappings of C into itself, where Ln=(∑i=1rβniLni2)1/2 and Ln→1.(ii)
Suppose that𝒯i are families of mappings of C into itself such that F(𝒯i)=⋂n=1∞F(Tni) for i=1,2,…,r and ⋂i=1rF(𝒯i)≠Ø. If {Tni} satisfies the NST*-condition with 𝒯i for all i=1,2,…,r, then {Tn} satisfies the NST*-condition with ⋃i=1r𝒯i and(2.26)⋂n=1∞F(Tn)=⋂i=1rF(Ti).Proof.
(i) From (2.3), we have
(2.27)∥Tnx-Tny∥2=‖∑i=1rβni(Tnix-Tniy)‖2≤∑i=1rβni∥Tnix-Tniy∥2≤(∑i=1rβniLni2)∥x-y∥2=Ln2∥x-y∥2,
for all x,y∈C. That is, Tn is Ln-Lipschitzian. Since Lni→1 for i=1,2,…,r and ∑i=1rβni=1, it follows that Ln→1.
(ii) Let{zn} be a bounded sequence in C such that limn→∞∥zn-Tnzn∥=limn→∞∥zn+1-zn∥=0. Let p∈⋂i=1rF(𝒯i), and let M=sup{∥zn-Tnzn∥,∥zn-p∥:n∈ℕ}; it follows from (2.3) that
(2.28)∥zn-p∥2≤(∥zn-Tnzn∥+∥Tnzn-p∥)2=∥zn-Tnzn∥2+2‖zn-Tnzn‖‖Tnzn-p‖+∥Tnzn-p∥2≤3M∥zn-Tnzn∥+‖∑i=1rβni(Tnizn-p)‖2=3M∥zn-Tnzn∥+∑i=1rβni∥Tnizn-p∥2-∑i<jβniβnj∥Tnizn-Tnjzn∥2≤3M∥zn-Tnzn∥+∑i=1rβniLni2∥zn-p∥2-a2∑i<j∥Tnizn-Tnjzn∥2=3M∥zn-Tnzn∥+Ln2∥zn-p∥2-a2∑i<j∥Tnizn-Tnjzn∥2.
So, by (i), we get
(2.29)lim⁡n→∞∥Tnizn-Tnjzn∥=0∀i,j∈{1,2,…,r}.
For each k=1,2,…,r, we have
(2.30)∥zn-Tnkzn∥≤∥zn-Tnzn∥+∥Tnzn-Tnkzn∥=∥zn-Tnzn∥+‖∑i=1rβni(Tnizn-Tnkzn)‖≤∥zn-Tnzn∥+∑i=1rβni‖Tnizn-Tnkzn‖⟶0.
Since each family {Tnk}n=1∞ satisfies the NST*-condition with 𝒯k,
(2.31)lim⁡n→∞∥zn-Tzn∥=0∀T∈⋃i=1rTi.
It is easy to see that ⋂i=1rF(𝒯i)⊂⋂n=1∞F(Tn). To see the reverse inclusion, let z∈⋂n=1∞F(Tn). Follow the first part of the proof above but now let zn≡z. Then, z∈⋂i=1rF(𝒯i)=F(∪i=1r𝒯i). Hence, {Tn} satisfies the NST*-condition with ⋃i=1r𝒯i.Lemma 2.8.
LetC be a nonempty closed convex subset of a real Hilbert space H, and let {Tn} be a family of Ln-Lipschitzian mappings of C into itself with Ln→1 and ⋂n=1∞F(Tn)≠Ø. If {Tn} satisfies the NST*-condition with 𝒯, where 𝒯 is a family of mappings of C into itself such that F(𝒯)=⋂n=1∞F(Tn), then F(𝒯) is closed and convex.Proof.
It follows from the continuity ofTn that F(Tn) is closed and so is F(𝒯)=⋂n=1∞F(Tn). Now, we prove that F(𝒯) is convex. To this end, let x,y∈F(𝒯). Put z=tx+(1-t)y, where t∈(0,1). From (2.2), we have
(2.32)∥z-Tnz∥2=t∥x-Tnz∥2+(1-t)∥y-Tnz∥2-t(1-t)∥x-y∥2≤tLn2∥x-z∥2+(1-t)Ln2∥y-z∥2-t(1-t)∥x-y∥2=t(1-t)(Ln2-1)∥x-y∥2.
So, we get
(2.33)lim⁡n→∞∥z-Tnz∥=0.
Since {Tn} satisfies the NST*-condition with 𝒯, we have ∥z-Tz∥=0 for all T∈𝒯. Then, z∈F(𝒯) and so F(𝒯) is convex.Remark 2.9.
The conclusions of Lemmas2.6, 2.7, and 2.8 remain true if we replace a Hilbert space with a uniformly convex Banach space. Recall a Banach space X is uniformly convex if for any ɛ>0, there exists δ>0 such that ∥x∥=∥y∥=1 and ∥x-y∥≥ɛ imply ∥(x+y)/2∥≤1-δ.


## 3. Main Results

In this section, using the method introduced by Takahashi et al.  [6], we obtain a strong convergence theorem for a countable family of Lipschitzian mappings.Recall that a mappingT:C→C is closed (demiclosed, resp.) at y if whenever {xn} is a sequence in C satisfying xn→x (xn⇀x, resp.) and Txn→y, then x∈C and Tx=y.Theorem 3.1.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H. Let {Tn} be a family of Ln-Lipschitzian mappings of C into itself with a common fixed point. Assume that {αn} is a sequence in [0,b] for some b∈(0,1). For x1=x∈H and C1=C, one defines a sequence {xn} of C as follows:
(3.1)yn=αnxn+(1-αn)Tnxn,Cn+1={z∈Cn:∥yn-z∥2≤∥xn-z∥2+θn},xn+1=PCn+1x,n=1,2,3,…,
where
(3.2)θn=(1-αn)(Ln2-1)(diam⁡C)2⟶0asn⟶∞.
Suppose that 𝒯 is a family of mappings of C into itself such that F(𝒯)=⋂n=1∞F(Tn) and I-T is closed at 0 for all T∈𝒯. If {Tn} satisfies the NST*-condition with 𝒯, then {xn} converges strongly to PF(𝒯)x.Proof.
By Lemma2.8, we have F(𝒯) is closed and convex. We now prove that Cn is closed and convex for each n∈ℕ by induction. It is obvious that C1=C is closed and convex. Assume that Ck is closed and convex for some k∈ℕ. For z∈Ck, we know that
(3.3)∥yk-z∥2≤∥xk-z∥2+θk
is equivalent to
(3.4)2〈xk-yk,z〉≤∥xk∥2-∥yk∥2+θk.
It follows that Ck+1 is closed and convex. Next, we show that
(3.5)F(T)⊂Cn∀n∈N.
It is clear that F(𝒯)⊂C1=C. Suppose that F(𝒯)⊂Ck for some k∈ℕ. Then, for p∈F(𝒯),
(3.6)∥yk-p∥2=∥αkxk+(1-αn)Tkxk-p∥2≤αk∥xk-p∥2+(1-αk)∥Tkxk-p∥2≤αk∥xk-p∥2+(1-αk)Lk2∥xk-p∥2=∥xk-p∥2+(1-αk)(Lk2-1)∥xk-p∥2≤∥xk-p∥2+θk,
we have p∈Ck+1. Therefore, we obtain (3.5). Now, the sequence {xn} is well defined. As xn=PCnx,
(3.7)∥xn-x∥≤∥z-x∥∀z∈Cn,∀n∈N.
In particular, since F(𝒯)⊂Cn,
(3.8)∥xn-x∥≤∥p-x∥∀p∈F(T),∀n∈N.
On the other hand, from xn=PCnx and xn+1∈Cn+1⊂Cn, we have
(3.9)∥xn-x∥≤∥xn+1-x∥∀n∈N.
Therefore, {∥xn-x∥} is nondecreasing and bounded. So,
(3.10)lim⁡n→∞∥xn-x∥exists.
Noticing again that xn=PCnx and for any positive integer k, xn+k∈Cn+k-1⊂Cn, we have
(3.11)〈xn-xn+k,x-xn〉≥0.
It follows from (2.1) that
(3.12)∥xn+k-xn∥2=∥(xn+k-x)-(xn-x)∥2=∥xn+k-x∥2-∥xn-x∥2-2〈xn+k-xn,xn-x〉≤∥xn+k-x∥2-∥xn-x∥2.
It then follows from the existence of limn→∞∥xn-x∥2 that {xn} is a Cauchy sequence. Moreover,
(3.13)lim⁡n→∞∥xn+1-xn∥=0.
We now assume that xn→w for some w∈C. Now, since xn+1∈Cn+1 and Cn+1⊂Cn, ∥yn-xn+1∥2≤∥xn-xn+1∥2+θn which implies that
(3.14)∥yn-xn+1∥≤∥xn-xn+1∥+θn⟶0.
From αn≤b<1, we get
(3.15)∥xn-Tnxn∥=11-αn∥yn-xn∥≤11-b(∥yn-xn+1∥+∥xn-xn+1∥)⟶0.
Since {Tn} satisfies the NST*-condition with 𝒯, we have
(3.16)lim⁡n→∞∥xn-Txn∥=0∀T∈T.
Since I-T is closed at 0 for all T∈𝒯, we have (I-T)w=0. This implies that w∈F(𝒯). Furthermore, by (3.8),
(3.17)∥w-x∥=lim⁡n→∞∥xn-x∥≤∥p-x∥∀p∈F(T).
Hence, w=PF(𝒯)x. This completes the proof.Lemma 3.2 . (see [9, Theorem 10.4]).
LetC be a nonempty closed convex subset of a real Hilbert space, and let T:C→C be a nonexpansive mapping. Then, I-T is demiclosed at 0.It is not difficult to see from the proof of Theorem3.1 that the boundedness of C can be discarded if {Tn} is a family of nonexpansive mappings. So, we immediately obtain the following theorem.Theorem 3.3.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Tn} and 𝒯 be two families of nonexpansive mappings of C into itself such that ⋂n=1∞F(Tn)=F(𝒯)≠Ø and suppose that {Tn} satisfies the NST*-condition with 𝒯. Assume that {αn} is a sequence in [0,b] for some b∈(0,1). Then, the sequence {xn} in C defined by (1.4) converges strongly to PF(𝒯)x.Remark 3.4.
Theorem3.3 includes [6, Theorem 3.3] as a special case since the NST-condition (I) with 𝒯 implies the NST*-condition with 𝒯.Theorem 3.5.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H. Let {Tn} be a family of Ln-Lipschitzian mappings of C into itself with a common fixed point. Suppose that 𝒯 is a family of mappings from C into itself such that F(𝒯)=⋂n=1∞F(Tn) and I-T is demiclosed at 0 for all T∈𝒯. Assume that {αn} is a sequence in [0,b] for some b∈(0,1). If {Tn} satisfies the NST*-condition with 𝒯, then the sequence {xn} in C defined by (1.5) converges strongly to PF(𝒯)x.Proof.
The proof is analogous to the proof of [7, Theorem 10]  and Theorem 3.1, so it is omitted.


## 4. Deduced Results

LetC be a subset of a real Hilbert space H. A mapping T:C→C is said to be an asymptotically nonexpansive if there exists a sequence {kn} of real numbers such that kn∈[1,∞), kn→1, and (4.1)∥Tnx-Tny∥≤kn∥x-y∥∀x,y∈C,n∈N.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [14] as an important generalization of the class of nonexpansive mappings. They proved that if C is nonempty bounded closed convex and T is an asymptotically nonexpansive self-mapping of C, then T has a fixed point.In this section, we use the NST*-condition to obtain recent results proved by Kim and Xu [15], Plubtieng and Ungchittrakool [16], and Zegeye and Shahzad [17, 18]. We start with the following auxiliary result.Lemma 4.1.
LetC be a nonempty closed convex subset of a Hilbert space H, and let T be an asymptotically nonexpansive mappings of C into itself with a sequence {kn} in [1,∞) satisfying kn→1 and F(T)≠Ø. Then, {Tn} is a family of kn-Lipschitzian mappings of C into itself and satisfies the NST*-condition with T.Proof.
We note that{Tn} is a family of kn-Lipschitzian mappings of C into itself. Let {zn} be a bounded sequence in C such that
(4.2)lim⁡n→∞∥zn-Tnzn∥=0,lim⁡n→∞∥zn+1-zn∥=0.
Since
(4.3)∥zn+1-Tzn+1∥≤∥zn+1-Tn+1zn+1∥+∥Tn+1zn+1-Tzn+1∥≤∥zn+1-Tn+1zn+1∥+k1∥Tnzn+1-zn+1∥≤∥zn+1-Tn+1zn+1∥+k1(∥Tnzn+1-Tnzn∥+∥Tnzn-zn∥+∥zn+1-zn∥)≤∥zn+1-Tn+1zn+1∥+k1(kn+1)∥zn+1-zn∥+k1∥Tnzn-zn∥,
it follows that
(4.4)lim⁡n→∞∥zn-Tzn∥=0.
It is easy to see that F(T)⊂⋂n=1∞F(Tn). To see the reverse inclusion, let z∈⋂n=1∞F(Tn) following from the first part of the proof above, but now let zn≡z. Then, z∈⋂n=1∞F(Tn), and hence ⋂n=1∞F(Tn)⊂F(T). This implies that {Tn} satisfies the NST*-condition with T.Lemma 4.2 . (see [19]).
LetC be a nonempty bounded closed convex subset of a Hilbert space H, and let T be an asymptotically nonexpansive mappings of C into itself. Then, I-T is demiclosed at 0.Using Theorem3.1 and Lemmas 2.6 and 4.1, we have the following result.Theorem 4.3.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H, and let S, T be two asymptotically nonexpansive mappings of C into itself with sequences {sn} and {tn}, respectively, and F(S)∩F(T)≠Ø. Assume that {αn} is a sequence in [0,b] and {βn} is a sequence in [a,b] for some a,b∈(0,1). For x1=x∈H and C1=C, one defines a sequence {xn} of C as follows:
(4.5)zn=βnxn+(1-βn)Snxn,yn=αnxn+(1-αn)Tnzn,Cn+1={z∈Cn:∥yn-z∥2≤∥xn-z∥2+θn},xn+1=PCn+1x,n=1,2,3,…,
where
(4.6)θn=(1-αn)((tn2-1)+(1-βn)tn2(sn2-1))(diam⁡C)2⟶0asn⟶∞.
Then, {xn} converges strongly to PF(S)∩F(T)x.Using Theorem3.5 and Lemmas 2.6 and 4.1, we have the following result.Theorem 4.4 . (see [16, Theorem 3.1]).
LetC be a nonempty bounded closed convex subset of a real Hilbert space H, and let S, T be two asymptotically nonexpansive mappings of C into itself with sequences {sn} and {tn}, respectively, and F(S)∩F(T)≠Ø. Assume that {αn} is a sequence in [0,b] and {βn} is a sequence in [a,b] for some a,b∈(0,1). For x1=x∈C, one defines a sequence {xn} of C as follows:
(4.7)zn=βnxn+(1-βn)Snxn,yn=αnxn+(1-αn)Tnzn,Cn={z∈C:∥yn-z∥2≤∥xn-z∥2+θn},Qn={z∈C:〈xn-z,x-xn〉≥0},xn+1=PCn∩Qnx,n=1,2,3,…,
where
(4.8)θn=(1-αn)((tn2-1)+(1-βn)tn2(sn2-1))(diam⁡C)2⟶0asn⟶∞.
Then, {xn} converges strongly to PF(S)∩F(T)x.Using Theorem3.1 and Lemmas 2.7 and 4.1, we have the following result.Theorem 4.5.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H. Let {Ti}i=1r be a finite family of asymptotically nonexpansive mappings of C into itself with sequences {kni} for i=1,2,…,r, respectively, and suppose that ⋂i=1rF(Ti)≠Ø. Assume that {αni}n=1∞ are sequences in [0,1) such that αn0≤b<1, αni≥a>0 for some a,b∈(0,1) and ∑i=0rαni=1 for all n∈ℕ. For x1=x∈H and C1=C, one defines a sequence {xn} of C as follows:
(4.9)yn=αn0xn+αn1T1nxn+αn2T2nxn+⋯+αnrTrnxn,Cn+1={z∈Cn:∥yn-z∥2≤∥xn-z∥2+θn},xn+1=PCn+1x,n=1,2,3,…,
where
(4.10)θn=(αn1(kn12-1)+αn2(kn22-1)+⋯+αnr(knr2-1))(diam⁡C)2⟶0asn⟶∞.
Then, {xn} converges strongly to P∩i=1rF(Ti)x.Using Theorem3.5 and Lemmas 2.7 and 4.1, we have the following two results which were proved by Zegeye and Shahzad [17, 18].Theorem 4.6.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H. Let {Ti}i=1r be a finite family of asymptotically nonexpansive mappings of C with sequences {kni} for i=1,2,…,r, respectively, and suppose that ⋂i=1rF(Ti)≠Ø. Assume that {αni}n=1∞ are sequences in [0,1) such that αn0≤b<1, αni≥a>0 for some a,b∈(0,1) and ∑i=0rαni=1 for all n∈ℕ. For x1=x∈C, one defines a sequence {xn} of C as follows:
(4.11)yn=αn0xn+αn1T1nxn+αn2T2nxn+⋯+αnrTrnxn,Cn={z∈C:∥yn-z∥2≤∥xn-z∥2+θn},Qn={z∈C:〈xn-z,x-xn〉≥0},xn+1=PCn∩Qnx,n=1,2,3,…,
where
(4.12)θn=(αn1(kn12-1)+αn2(kn22-1)+⋯+αnr(knr2-1))(diam⁡C)2⟶0asn⟶∞.
Then, {xn} converges strongly to P∩i=1rF(Ti)x.Theorem 4.7.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H. Let 𝔗i={Ti(t):t∈ℝ+,i=1,2,…,r}  be a finite family of asymptotically nonexpansive semigroups such that F=⋂i=1rF(𝔗i)≠Ø. Assume that {αni}n=1∞ are sequences in [0,1) such that
(4.13)αn0≤b<1,αni≥a>0,
for some a,b∈(0,1) and ∑i=0rαni=1 for all n∈ℕ. Let {tni,i=1,2,…,r} be finite positive and divergent real sequences. For x1=x∈C, one defines a sequence {xn} of C as follows:
(4.14)yn=αn0xn+αn11tn1∫0tn1T1(u)xndu+αn21tn2∫0tn2T2(u)xndu+⋯+αnr1tnr∫0tnrTr(u)xndu,Cn={z∈C:∥yn-z∥2≤∥xn-z∥2+θn},Qn={z∈C:〈xn-z,x-xn〉≥0},xn+1=PCn∩Qnx,n=1,2,3,…,
where
(4.15)θ¯n=(αn1(Lu12-1)+αn2(Lu22-1)+⋯+αnr(Lur2-1))(diam⁡C)2⟶0asn⟶∞,
with Lui=(1/tni)∫0tniLiTidu. Then {xn} converges strongly to PFx.



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*Source: 739561-2010-12-01.xml*