# A New Iteration Process for Approximation of Common Fixed Points  for Finite Families of Total Asymptotically Nonexpansive Mappings

Authors: C. E. Chidume; E. U. Ofoedu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

LetE be a real Banach space, and K a closed convex nonempty subset of E.
Let T1,T2,…,Tm:K→K be m total asymptotically nonexpansive mappings. A
simple iterative sequence {xn}n≥1 is constructed in E and necessary and sufficient
conditions for this sequence to converge to a common fixed point of {Ti}i=1m are given. Furthermore, in the case that E is a uniformly convex real Banach space,
strong convergence of the sequence {xn}n=1∞ to a common fixed point of the family
{Ti}i=1m is proved. Our recursion formula is much simpler and much more applicable than those recently announced by several authors for the same problem.


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## Body

## 1. Introduction

LetK be  a nonempty subset of a normed real linear space E. A mapping T:K→K is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈K.The mappingT is called asymptotically nonexpansive if there exists a sequence {μn}n≥1⊂[0,∞) with limn→∞μn=0 such that for all x,y∈K,(1.1)∥Tnx-Tny∥≤(1+μn)∥x-y∥∀n≥1.
The mapping T is called uniformlyL-Lipschitzian if there exists a constant L≥0 such that for all x,y∈K,(1.2)∥Tnx-Tny∥≤L∥x-y∥∀n≥1.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalization of the class of nonexpansive mappings. They proved that if K is a nonempty closed convex bounded subset of a uniformly convex real Banach space and T is an asymptotically nonexpansive self-mapping of K, then T has a fixed point.A mappingT is said to be asymptotically nonexpansive in the intermediate sense (see, e.g., [2]) if it is continuous and the following inequality holds:(1.3)lim supn→∞supx,y∈K(∥Tnx-Tny∥-∥x-y∥)≤0.
Observe that if we define(1.4)an:=supx,y∈K(∥Tnx-Tny∥-∥x-y∥),σn=max{0,an},
then σn→0 as n→∞ and (1.3) reduces to(1.5)∥Tnx-Tny∥≤∥x-y∥+σn,∀x,y∈K,n≥1.
The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [2]. It is known [3] that if K is a nonempty closed convex bounded subset of a uniformly convex real Banach space E and T is a self-mapping of K which is asymptotically nonexpansive in the intermediate sense, then T has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings (see, e.g., [4]).Sahu [5], introduced the class of nearly Lipschitzian mappings. Let K be a nonempty subset of a normed space E and let {an}n≥1 be a sequence in [0,+∞) such that limn→∞an=0. A mapping T:K→K is called nearly Lipschitzian with respect to {an}n≥1 if for each n∈ℕ, there exists kn≥0 such that(1.6)∥Tnx-Tny∥≤kn(∥x-y∥+an)∀x,y∈K.
Define(1.7)η(Tn):=sup{∥Tnx-Tny∥∥x-y∥+an:x,y∈K,x≠y}.
Observe that for any sequence {kn}n≥1 satisfying (1.6), η(Tn)≤knforalln∈ℕ and that(1.8)∥Tnx-Tny∥≤η(Tn)(∥x-y∥+an)∀x,y∈K,n∈ℕ.η(Tn) is called the nearly Lipschitz constant. A nearly Lipschitzian mapping T is said to be(i)
nearly contraction if η(Tn)<1 for all n∈ℕ;(ii)
nearly nonexpansive if η(Tn)=1 for all n∈ℕ;(iii)
nearly asymptotically nonexpansive if η(Tn)≥1 for all n∈ℕ and  limn→∞η(Tn)=1;(iv)
nearly uniformL-Lipschitzian if η(Tn)≤L for all n∈ℕ;(v)
nearly uniformk-contraction if η(Tn)≤k<1 for all n∈ℕ.Example 1.1.
LetE=ℝ,K=[0,1]. Define T:K→K by
(1.9)Tx={12,ifx∈[0,12],0,ifx∈(12,1].
It is obvious that T is not continuous, and thus, not Lipschitz. However, T is nearly nonexpansive. In fact, for a real sequence {an}n≥1 with a1=1/2 and an→0 as n→∞, we have
(1.10)∥Tx-Ty∥≤∥x-y∥+a1∀x,y∈K,∥Tnx-Tny∥≤∥x-y∥+an∀x,y∈K,n≥2.
This is because Tnx=1/2forallx∈[0,1],n≥2.Remark 1.2.
IfK is a bounded domain of an asymptotically nonexpansive mapping T, then T is nearly nonexpansive. In fact, for all x,y∈K and n∈ℕ, we have
(1.11)∥Tnx-Tny∥≤(1+μn)∥x-y∥≤∥x-y∥+diam(K)μn.
Furthermore, we easily observe that every nearly nonexpansive mapping is nearly asymptotically nonexpansive with η(Tn)≡1foralln∈ℕ.Remark 1.3.
IfK is a bounded domain of a nearly asymptotically nonexpansive mapping T, then T is asymptotically nonexpansive in the intermediate sense. To see this, let T be a nearly asymptotically nonexpansive mapping. Then,
(1.12)∥Tnx-Tny∥≤η(Tn)(∥x-y∥+an)∀x,y∈K,n≥1,
which implies that
(1.13)supx,y∈K(∥Tnx-Tny∥-∥x-y∥)≤(η(Tn)-1)diam(K)+η(Tn)an,n≥1.
Hence,
(1.14)lim supn→∞supx,y∈K(∥Tnx-Tny∥-∥x-y∥)≤0.We observe from Remarks1.2 and 1.3 that the classes of nearly nonexpansive mappings and nearly asymptotically nonexpansive mappings are intermediate classes between the class of asymptotically nonexpansive mappings and that of asymptotically nonexpansive in the intermediate sense mappings.The main tool for approximation of fixed points of generalizations of nonexpansive mappings remainsiterative technique. Several authors have studied approximation of fixed points of generalizations of nonexpansive mappings using Mann and Ishikawa iterative methods (see, e.g., [6–19]).Bose [20] proved that if K is a nonempty closed convex bounded subset of a uniformly convex real Banach space E satisfying Opial's condition [21] (i.e., for all sequences {xn} in E such that {xn} converges weakly to some x∈E, the inequality lim infn→∞∥xn-x∥<lim infn→∞∥xn-y∥ holds for all y≠x in E) and T:K→K is an asymptotically nonexpansive mapping, then the sequence {Tnx} converges weakly to a fixed point of T provided that  T is asymptotically regular atx∈K; that is, the limit(1.15)limn→∞∥Tnx-Tn+1x∥=0
holds. Passty [13] and also Xu and Noor [22] showed that the requirement of Opial's condition can be replaced by the Fréchet differentiability of the space norm. Furthermore, Tan and Xu [23, 24] established that the asymptotic regularity of T at a point x can be weakened to the so-called weakly asymptotic regularity ofTatx, defined as follows: T is weakly asymptotic regular at x∈K if(1.16)ω-limn→∞(Tnx-Tn+1x)=0
holds, where ω-lim denotes the weak limit.In [17, 18], Schu introduced a modified Mann iteration scheme for approximation of fixed points of asymptotically nonexpansive self-mappings defined on a nonempty closed convex and bounded subset K of a uniformly convex real Banach space E. He proved that the iterative sequence {xn} generated by(1.17)x1∈K,xn+1=(1-αn)xn+αnTnxn,n≥1
converges weakly to some fixed point of T if Opial's condition holds, {μn}⊂[0,∞) for all n≥1,∑n=1∞μn<∞,{αn} is a real sequence such that 0<a≤αn≤b<1,n≥1, for some positive constants a and b. Neither condition (1.15) nor condition (1.16) is required with Schu's scheme. Schu's result, however, does not apply, for instance, to Lp spaces with p>1,p≠2 because none of these spaces satisfies Opial's condition.Rhoades [15] obtained a strong convergence theorem for asymptotically nonexpansive mappings in uniformly convex real Banach spaces using the modified Ishikawa-type iteration method. Osilike and Aniagbosor proved in [12] that the results of [15, 17, 18] remain true without the boundedness requirement imposed on K, provided that F(T):={x∈K:Tx=x}≠∅. Tan and Xu [25] extended the theorem of Schu [18] to uniformly convex Banach space with a Fréchet differentiable norm without assuming that the space satisfies Opial's condition. Thus, their result applies to Lp spaces with 1<p<∞.Chang et al. [26] established weak convergence theorems for asymptotically nonexpansive mappings in Banach spaces without assuming any of the following conditions: (i) E satisfies the Opial's condition; (ii) T is asymptotically regular or weakly asymptotically regular; (iii) K is bounded. Their results improve and generalize the corresponding results of Bose [20], Górnicki [27], Passty [13], Schu [18],Tan and Xu [23–25], Xu and Noor [22], and many others.G. E. Kim and T. H. Kim [4] studied the strong convergence of the Mann and Ishikawa-type iteration methods with errors for mappings which are asymptotically nonexpansive in the intermediate sense in real Banach spaces.In all the above papers, the mappingT remains a self-mapping of nonempty closed convex subset K of a uniformly convex real Banach space E. If, however, the domain D(T) of T is a proper subset of E, then the Mann and Ishikawa-type iterative processes and Schu's modifications of type (1.17) may fail to be well defined.Chidume et al. [28] proved convergence theorems for asymptotically nonexpansive nonself-mappings in Banach spaces and extended the corresponding results of [12, 15, 17, 18, 26].Alber et al. [29] introduced a more general class of asymptotically nonexpansive mappings called total asymptotically nonexpansive mappings and studied methods of approximation of fixed points of mappings belonging to this class.Definition 1.4.
A mappingT:K→K is said to be total asymptotically nonexpansive if there exist nonnegative real sequences {μn} and {ln},n≥1 with μn,ln→0 as n→∞ and strictly increasing continuous function ϕ:ℝ+→ℝ+ with ϕ(0)=0 such that for all x,y∈K,(1.18)∥Tnx-Tny∥≤∥x-y∥+μnϕ(∥x-y∥)+ln,n≥1.Remark 1.5.
Ifϕ(λ)=λ, then (1.18) reduces to
(1.19)∥Tnx-Tny∥≤(1+μn)∥x-y∥+ln,n≥1.
In addition, if ln=0 for all n≥1, then total asymptotically nonexpansive mappings coincide with asymptotically nonexpansive mappings. If μn=0 and ln=0 for all n≥1, we obtain from (1.18) the class of mappings that includes the class of nonexpansive mappings. If μn=0 and ln=σn=max{0,an}, where an:=supx,y∈K(∥Tnx-Tny∥-∥x-y∥) for all n≥1, then (1.18) reduces to (1.5) which has been studied as mappings asymptotically nonexpansive in the intermediate sense.The idea of Definition1.4 is to unify various definitions of classes of mappings associated with the class of asymptotically nonexpansive mappings and to prove a general convergence theorems applicable to all these classes of nonlinear mappings.Another class of nonlinear mappings introduced as a further generalization of nonexpansive mappings with nonempty fixed point sets is the class of asymptotically quasi-nonexpansive mappings which properly contains the class of asymptotically nonexpansive operators with nonempty fixed point sets (see, e.g., [8, 16, 30–33]).A mappingT is said to be quasi-nonexpansive if F(T)≠∅ and
(1.20)∥Tx-x*∥≤∥x-x*∥,∀x∈D(T),x*∈F(T).T is called asymptotically quasi-nonexpansive if F(T)≠∅ and there exists a sequence {μn}∈[0,∞) with limn→∞μn=0 such that for all x∈D(T) and x*∈F(T),(1.21)∥Tnx-x*∥≤(1+μn)∥x-x*∥,n≥1.T is said to be asymptotically quasi-nonexpansive in intermediate sense if it is continuous and(1.22)lim supn→∞{supx∈D(T),x*∈F(T)[∥Tnx-x*∥-∥x-x*∥]}≤0.Remark 1.6.
Observe that if we define(1.23)an*:=supx∈D(T,x*∈F(T)[∥Tnx-x*∥-∥x-x*∥],σn*=max{0,an*},
then σn*→0 as n→∞ and (1.22) reduces to
(1.24)∥Tnx-x*∥≤∥x-x*∥+σn*,∀x,y∈K,n≥1.Existence theorems for common fixed points of certain families of nonlinear mappings have been established by various authors (see, e.g., [2, 34–37]).Within the past 30 years or so, research on iterative approximation of common fixed points of generalizations of nonlinear nonexpansive mappings surged. Considerable research efforts have been devoted to developing iterative methods for approximating common fixed points (when they exist) of finite families of this class of mappings (see, e.g., [33, 38–46]).In [16], Shahzad and Udomene established necessary and sufficient conditions for convergence of Ishikawa-type iteration sequences involving two asymptotically quasi-nonexpansive mappings to a common fixed point of the mappings in arbitrary real Banach spaces. They also established a sufficient condition for the convergence of the Ishikawa-type iteration sequences involving two uniformly continuous asymptotically quasi-nonexpansive mappings to a common fixed point of the mappings in real uniformly convex Banach spaces.Recently, Chidume and Ofoedu [47] introduced an iterative scheme for approximation of a common fixed point of a finite family of total asymptotically nonexpansive mappings in Banach spaces. More precisely, they proved the following theorems.Theorem CO1
LetE be a real Banach space, let K be a nonempty closed convex subset of E, and Ti:K→K,i=1,2,…,m be m total asymptotically nonexpansive mappings with sequences {μin},{lin},n≥1,i=1,2,…,m such that F:=⋂i=1mF(Ti)≠∅. Let {xn} be given by
(1.25)x1∈E,xn+1=(1-αn)xn+αnT1nxn,ifm=1,n≥1,x1∈E,xn+1=(1-αn)xn+αnT1ny1n,y1n=(1-αn)xn+αnT2ny2n,⋮y(m-2)n=(1-αn)xn+αnTm-1ny(m-1)n,y(m-1)n=(1-αn)xn+αnTmnxn,ifm≥2,n≥1.
Suppose ∑n=1∞μin<∞,∑n=1∞lin<∞i=1,2,…,m and suppose that there exist Mi,Mi*>0 such that ϕi(λi)≤Mi*λi for all λi≥Mi,i=1,2,…,m. Then the sequence {xn} is bounded and limn→∞∥xn-p∥ exists, p∈F. Moreover, the sequence {xn} converges strongly to a common fixed point of Ti,i=1,2,…,m if and only if lim infn→∞d(xn,F)=0, where d(xn,F)=infy∈F∥xn-y∥,n≥1.Theorem CO2
LetE be a uniformly convex real Banach space, let  K be a nonempty closed convex subset of E, and Ti:K→K,i=1,2,…,m be m uniformly continuous total asymptotically nonexpansive mappings with sequences {μin},{lin}⊂[0,∞) such that ∑n=1∞μin<∞,∑n=1∞lin<∞,i=1,2,…,m and F:=⋂i=1mF(Ti)≠∅. Let {αin}⊂[ε,1-ε] for some ε∈(0,1). From arbitrary x1∈E, define the sequence {xn} by (1.25). Suppose that there exist Mi,Mi*>0 such that ϕi(λi)≤Mi*λi whenever λi≥Mi,i=1,2,…,m; and that one of T1,T2,…,Tm is compact, then {xn} converges strongly to some p∈F.It is our purpose in this paper to constructa new iterative sequence much simpler than (1.25) for approximation of common fixed points of finite families of total asymptotically nonexpansive mappings and give necessary and sufficient conditions for the convergence of the scheme to common fixed points of the mappings in arbitrary real Banach spaces. A sufficient condition for convergence of the iteration process to a common fixed point of mappings under our setting is also established in uniformly convex real Banach spaces. Our theorems unify, extend and generalize the corresponding results of Alber et al. [29], Sahu [5], Shahzad and Udomene [16], and a host of other results recently announced for the approxima tion of common fixed points of finite families of several classes of nonlinear mappings. Our iteration process is also of independent interest.


## 2. Preliminary

In the sequel, we shall need the following lemmas.Lemma 2.1.
Let{an}, {αn}, and {bn} be sequences of nonnegative real numbers such that
(2.1)an+1≤(1+αn)an+bn.
Suppose that ∑n=1∞αn<∞ and ∑n=1∞bn<∞. Then {an} is bounded and limn→∞an exists. Moreover, if in addition, lim infn→∞an=0, then limn→∞an=0.Lemma 2.2 (Zeidler [48, pages 484-485]).
LetE be a uniformly convex real Banach space and t∈(0,1). Suppose that {xn} and {yn} are two sequences of E such that
(2.2)lim supn→∞∥xn∥≤r,lim supn→∞∥yn∥≤r,limn→∞∥(1-t)xn+tyn∥=r
hold for some r≥0, then limn→∞∥xn-yn∥=0.


## 3. Main Results

LetK be a nonempty closed convex subset of a real normed space E. Let T1,T2,…,Tm:K→K be m total asymptotically nonexpansive mappings. We define the iterative sequence {xn} by(3.1)x1∈K,xn+1=(1-α0n)xn+∑i=1mαinTinxn,n≥1,
where {αin}n=1∞,i=0,1,2,…m are sequences in (0,1) such that ∑i=0mαin=1.We now state and prove our main theorems.Theorem 3.1.
LetE be a real Banach space, let  K be a nonempty closed convex subset of E, and let  Ti:K→K,i=1,2,…,m, be m total asymptotically nonexpansive mappings with sequences {μin},{lin},n≥1,i=1,2,…,m such that F:=⋂i=1mF(Ti)≠∅. Let {xn} be given by (3.1). Suppose ∑n=1∞μin<∞,∑n=1∞lin<∞i=1,2,…,m and suppose that there exist Mi,Mi*>0 such that ϕi(λi)≤Mi*λi for all λi≥Mi,i=1,2,…,m. Then the sequence {xn} is bounded and limn→∞∥xn-p∥ exists, p∈F.Proof.
Letp∈F. Then we have from (3.1) that
(3.2)∥xn+1-p∥=∥(1-∑i=1mαin)(xn-p)+∑i=1mαin(Tinxn-p)∥≤(1-∑i=1mαin)∥xn-p∥+∑i=1mαin[∥xn-p∥+μinϕi(∥xn-p∥)+lin].
Since ϕi is an increasing function, it follows that ϕi(λi)≤ϕi(Mi) whenever λi≤Mi and (by hypothesis) ϕi(λi)≤Mi*λi if λi≥Mi. In either case, we have
(3.3)ϕi(∥xn-p∥)≤ϕi(Mi)+Mi*∥xn-p∥
for some Mi>0,Mi*>0. Thus,
(3.4)∥xn+1-p∥≤(1+k0∑i=1mμin)∥xn-p∥+k0∑i=1m(μin+lin)
for some constant k0>0. Hence,
(3.5)∥xn+1-p∥≤(1+δn)∥xn-p∥+γn,n≥1,
where δn=k0∑i=1mμin and γn=k0∑i=1m(μin+lin). Observe that ∑n=1∞δn<∞ and ∑n=1∞γn<∞. So, from (3.5) and by Lemma 2.1 we obtain that the sequence {xn} is bounded and that limn→∞∥xn-p∥ exists. This completes the proof.


### 3.1. Necessary and Sufficient Conditions for Convergence in Real Banach Spaces

Theorem 3.2.
LetE be a real Banach space, let  K be a nonempty closed convex subset of E, and let  Ti:K→K,i=1,2,…,m be m continuous total asymptotically nonexpansive mappings with sequences {μin},{lin}n≥1,i=1,2,…,m such that F:=⋂i=1mF(Ti)≠∅. Let {xn} be given by (3.1). Suppose ∑n=1∞μin<∞,∑n=1∞lin<∞,i=1,2,…,m and suppose that there exist Mi,Mi*>0 such that ϕi(λi)≤Mi*λi for all λi≥Mi,i=1,2,…,m. Then the sequence {xn} converges strongly to a common fixed point of Ti,i=1,2,…,m if and only if  lim infn→∞d(xn,F)=0, where d(xn,F)=infy∈F∥xn-y∥,n≥1.Proof.
It suffices to show thatlim infn→∞d(xn,F)=0 implies that {xn} converges to a common fixed point of Ti,i=1,2,…,m.Necessity
Since (3.5) holds for all p∈F, we obtain from it that
(3.6)d(xn+1,F)≤(1+δn)d(xn,F)+γn,n≥1.
Lemma 2.1 then implies that limn→∞d(xn,F) exists. But, lim infn→∞d(xn,F)=0. Hence,limn→∞d(xn,F)=0.Sufficiency
Next, we first show that{xn} is a Cauchy sequence in E. For all integer m≥1, we obtain from inequality (3.5) that
(3.7)∥xn+m-p∥≤∏i=nn+m-1(1+δi)∥xn-p∥+(∑i=nn+m-1γi)∏i=nn+m-1(1+δi)≤exp(∑i=nn+m-1δi)∥xn-p∥+(∑i=nn+m-1γi)exp(∑i=nn+m-1δi),
so that for all integers m≥1 and all p∈F,(3.8)∥xn+m-xn∥≤∥xn+m-p∥+∥xn-p∥≤[1+exp(∑i=nn+m-1δi)]∥xn-p∥+(∑i=nn+m-1γi)exp(∑i=nn+m-1δi).
We therefore have that
(3.9)∥xn+m-xn∥≤D∥xn-p∥+D(∑i=n∞γi)
for some constant D>0. Taking infimum over p∈F in (3.9) gives
(3.10)∥xn+m-xn∥≤Dd(xn,F)+D(∑i=n∞γi).
Now, since limn→∞d(xn,F)=0 and ∑i=1∞γi<∞, given ϵ>0, there exists an integer N1>0 such that for all n≥N1,d(xn,F)<ϵ/2(D+1) and ∑i=n∞γi<ϵ/2(D+1). So for all integers n≥N1,m≥1, we obtain from (3.10) that
(3.11)∥xn+m-xn∥<ϵ.Hence, {xn} is a Cauchy sequence in E, and since E is complete there exists l*∈E such that xn→l* as n→∞. We now show that l* is a common fixed point of Ti,i=1,2,…,m, that is, we show that l*∈F. Suppose for contradiction that l*∈Fc (where Fc denotes the complement of F). Since F is a closed subset of E (recall each Ti,i=1,2,…,m is continuous), we have that d(l*,F)>0. But, for all p∈F, we have
(3.12)∥l*-p∥≤∥l*-xn∥+∥xn-p∥.
This implies
(3.13)d(l*,F)≤∥xn-l*∥+d(xn,F),
so that as n→∞ we obtain d(l*,F)=0 which contradicts d(l*,F)>0. Thus, l* is a common fixed point of Ti,i=1,2,…,m. This completes the proof.Remark 3.3.
IfT1,T2,…,Tm are asymptotically nonexpansive mappings, then lin=0 for all n≥1,i=1,2,…,m and ϕi(λi)=λi so that the assumption that there exist Mi,Mi*>0 such that ϕi(λi)≤Mi*λi for all λi≥Mi,i=1,2,…,m in the above theorems is no longer needed.Thus, we have the following corollary.Corollary 3.4.
LetE be a real Banach space, let  K be a nonempty closed convex subset of E, and let  Ti:K→K,i=1,2,…,m be m continuous asymptotically nonexpansive mappings with sequences {μin},n≥1,i=1,2,…,m such that F=⋂i=1mF(Ti)≠∅. Let {xn} be given by (3.1). Suppose ∑n=1∞μin<∞,i=1,2,…,m. Then the sequence {xn} is bounded and limn→∞∥xn-p∥ exists, p∈F. Moreover, {xn} converges strongly to a common fixed point of Ti,i=1,2,…,m if and only if lim infn→∞d(xn,F)=0.


### 3.2. Convergence Theorem in Real Uniformly Convex Banach Spaces

Theorem 3.5.
LetE be a uniformly convex real Banach space, K be a nonempty closed convex subset of E and Ti:K→K,i=1,2,…,m be m uniformly continuous total asymptotically nonexpansive mappings with sequences {μin},{lin}⊂[0,∞) such that ∑n=1∞μin<∞,∑n=1∞lin<∞,i=1,2,…,m and F:=⋂i=1mF(Ti)≠∅. From arbitrary x1∈E, define the sequence {xn} by (3.1). Suppose that there exist Mi,Mi*>0 such that ϕi(λi)≤Mi*λi whenever λi≥Mi,i=1,2,…,m. Then limn→∞∥xn-Tinxn∥=0,i=1,2,…,m.Proof.
Letp∈F. Then, by Theorem 3.1, limn→∞∥xn-p∥ exists. Let limn→∞∥xn-p∥=r. If r=0, then by continuity of Ti,i=1,2,…,m, we are done. Now suppose r>0. We show that limn→∞∥xn-Tinxn∥=0,i=1,2,…,m. We observe that
(3.14)∥Tinxn-p∥≤∥xn-p∥+μinϕi(∥xn-p∥)+lin≤∥xn-p∥+μinϕi(Mi)+μinMi*∥xn-p∥+lin(i=1,2,…,m).
so that taking lim sup on both sides of this inequality, we obtain
(3.15)lim supn→∞∥Tinxn-p∥≤r(i=1,2,…,m).
Let {tn}n≥1∈(0,1) be such that tn→0 as n→∞ and define
(3.16)yin:=(1-tn)xn+tnTinxn(i=1,2,…,m).
Then,
(3.17)∥yin-xn∥≤tn∥Tinxn-xn∥≤tnM(i=1,2,…,m)
for some M>0. Thus,
(3.18)limn→∞∥yin-xn∥=0(i=1,2,…,m).
Furthermore,
(3.19)|∥yin-p∥-∥xn-p∥|≤∥yin-xn∥(i=1,2,…,m).
This implies that
(3.20)limn→∞∥yin-p∥=limn→∞∥xn-p∥=r(i=1,2,…,m).
But,
(3.21)∥yin-p∥=∥(1-tn)(xn-p)+tn(Tinxn-p)∥(i=1,2,…,m).
So,
(3.22)limn→∞∥(1-tn)(xn-p)+tn(Tinxn-p)∥=r,(i=1,2,…,m).
Hence, by Lemma 2.2, we obtain
(3.23)limn→∞∥xn-Tinxn∥=0,(i=1,2,…,m).
This completes the proof.Theorem 3.6.
LetE be a uniformly convex real Banach space, let  K be a nonempty closed convex subset of E, and let  Ti:K→K,i=1,2,…,m be m uniformly continuous total asymptotically nonexpansive mappings with sequences {μin},{lin}⊂[0,∞) such that ∑n=1∞μin<∞,∑n=1∞lin<∞,i=1,2,…,m and F:=⋂i=1mF(Ti)≠∅. From arbitrary x1∈E, define the sequence {xn} by (1.24). Suppose that there exist Mi,Mi*>0 such that ϕi(λi)≤Mi*λi whenever λi≥Mi,i=1,2,…,m; and that one of T1,T2,…,Tm is compact, then {xn} converges strongly some p∈F.Proof.
We obtain from Theorem3.5 that
(3.24)limn→∞∥Tinxn-xn∥=0,i=1,2,…,m.
Using the recursion formula (3.1), we observe that
(3.25)∥xn+1-xn∥=∥α0nxn+∑i=1mαinTinxn-xn∥=∥∑i=1mαin(Tinxn-xn)∥≤∑i=1mαin∥Tinxn-xn∥.
It then follows from (3.24) and (3.25) that
(3.26)limn→∞∥xn+1-xn∥=0.
Without loss of generality, let T1 be compact. Since T1 is continuous and compact, it is completely continuous. Thus, there exists a subsequence {T1nkxnk} of {T1nxn} such that T1nkxnk→x* as k→∞ for some x*∈E. Thus T1nk+1xnk→T1x* as k→∞ and from (3.24), we have that limk→∞xnk=x*. Also from (3.24) T2nkxnk→x*,T3nkxnk→x*,…,Tmnkxnk→x* as k→∞. Thus, T2nk+1xnk→T2x*,T3nk+1xnk→T3x*,…,Tmnk+1xnk→Tmx* as k→∞. Now, since from (3.26), ∥xnk+1-xnk∥→0 as k→∞, it follows that xnk+1→x* as k→∞. Next, we show that x*∈F. Observe that
(3.27)∥x*-Tix*∥≤∥x*-xnk+1∥+∥xnk+1-Tink+1xnk+1∥+∥Tink+1xnk+1-Tink+1xnk∥+∥Tink+1xnk-Tix*∥.
Taking limit as k→∞ and using the fact that Ti(i=1,2,…m) are uniformly continuous we have that x*=Tix*(i=1,2,…m) and so x*∈F(Ti)(i=1,2,…m). But by Theorem 3.1, limn→∞∥xn-p∥ exists, p∈F. Hence, {xn} converges strongly to x*∈F. This completes the proof.In view of Remark12, the following corollary is now obvious.Corollary 3.7.
LetE be a uniformly convex real Banach space, let  K be a nonempty closed convex subset of E, and let Ti:K→K,i=1,2,…,m be m asymptotically nonexpansive mappings with sequences {μin}⊂[0,∞) such that ∑n=1∞μin<∞,i=1,2,…,m; F=⋂i=1mF(Ti)≠∅ and that one of T1,T2,…,Tm is compact. From arbitrary x1∈E, define the sequence {xn} by (3.1). Then {xn} converges strongly to to some p∈F.Remark 3.8.
Observe that the theorems of this paper remain true for mappingsT1,T2,…,Tm satisfying (1.5) provided that  ∑n=1∞σn<∞. In this case, the requirement that there exist Mi,Mi*>0 such that ϕi(λi)≤Mi*λi for all λi≥Mi,i=1,2,…,m is not needed.Remark 3.9.
A prototype forϕ:[0,∞)→[0,∞) satisfying the conditions of our theorems is ϕ(λ)=λs,0<s≤1. Prototypes for the sequences {αin}n=1∞foralli=0,1,2,…,m in this paper are the following:
(3.28)α0n:=n2n+1;αin:=n+1m(2n+1)∀n∈ℕ,i=1,2,…,m.Remark 3.10.
Addition ofbounded (or the so called mean) error terms to the iteration process studied in this paper leads to no further generalization.Definition 3.11.
A mappingT:K→K is said to be total asymptotically quasi-nonexpansive if F(T)≠∅ and there exist nonnegative real sequences {μn} and {ln},n≥1 with μn,ln→0 as n→∞ and strictly increasing continuous function ϕ:ℝ+→ℝ+ with ϕ(0)=0 such that for all x∈E,x*∈F(T),(3.29)∥Tnx-x*∥≤∥x-x*∥+μnϕ(∥x-x*∥)+ln,n≥1.Remark 3.12.
Ifϕ(λ)=λ, then (3.29) reduces to
(3.30)∥Tnx-x*∥≤(1+μn)∥x-x*∥+ln,n≥1.
In addition, if ln=0 for all n≥1, then total asymptotically quasi-nonexpansive mappings coincide with asymptotically quasi-nonexpansive mappings studied by various authors. If μn=0 and ln=0 for all n≥1, we obtain from (3.30) the class of quasi-nonexpansive mappings. Observe that the class of total asymptotically nonexpansive mappings with nonempty fixed point sets belongs to the class of total asymptotically quasi-nonexpansive mappings. Moreover, if μn=0 and ln=σn*, then (3.29) reduces to (1.24).It is trivial to observe that all the theorems of this paper carry over to the class of total asymptotically quasi-nonexpansive mappings with little or no modifications.A subsetK of a real normed linear space E is said to be a retract of E if there exists a continuous map P:E→K such that Px=x for all x∈K. It is well known (see, e.g., [28]) that every closed convex nonempty subset of a uniformly convex Banach space is a retract. A map P:E→E is said to be a retraction if P2=P. It follows that if a map P ia a retraction, then Py=y for all y in the range of P. The mapping p is called a sunny nonexpansive retraction if for all x∈E and t∈(0,1),P((1-t)x+tP(x))=P(x).Definition 3.13.
LetK be a nonempty closed and convex subset of E. Let P:E→K be the nonexpansive retraction of E onto K. A nonself map T:K→E is said to be total asymptotically nonexpansive if there exist sequences {μn}n≥1,{ln}n≥1 in [0,+∞) with μn,ln→0 as n→∞ and a strictly increasing continuous function ϕ:[0,+∞)→[0,+∞) with ϕ(0)=0 such that for all x,y∈K,(3.31)∥T(PT)n-1x-T(PT)n-1y∥≤∥x-y∥+μnϕ(∥x-y∥)+ln,n≥1.
Let T1,T2,…,Tm:K→E be m total asymptotically nonexpansive nonself maps; assuming existence of common fixed points of these operators, our theorems and method of proof easily carry over to this class of mappings using the iterative sequence {xn} defined by
(3.32)x1∈K,xn+1=P(α0nxn+∑i=1rαinTi(PTi)n-1xn)
instead of (3.1) provided that the well definedness of P as a sunny nonexpansive retraction is guaranteed.Remark 3.14.
It is clear that the recursion formula (3.1) introduced and studied in this paper is much simpler than the recursion formulas (1.25) studied earlier for this problem.Remark 3.15.
Our theorems unify, extend, and generalize the corresponding results of Alber et al. [29], Sahu [5], Shahzad and Udomene [16], and a host of other results recently announced (see, e.g., [8, 16, 22, 28, 30, 39, 40, 44, 47, 49–58]) for the approximation of common fixed points of finite families of several classes of nonlinear mappings.


## 3.1. Necessary and Sufficient Conditions for Convergence in Real Banach Spaces

Theorem 3.2.
LetE be a real Banach space, let  K be a nonempty closed convex subset of E, and let  Ti:K→K,i=1,2,…,m be m continuous total asymptotically nonexpansive mappings with sequences {μin},{lin}n≥1,i=1,2,…,m such that F:=⋂i=1mF(Ti)≠∅. Let {xn} be given by (3.1). Suppose ∑n=1∞μin<∞,∑n=1∞lin<∞,i=1,2,…,m and suppose that there exist Mi,Mi*>0 such that ϕi(λi)≤Mi*λi for all λi≥Mi,i=1,2,…,m. Then the sequence {xn} converges strongly to a common fixed point of Ti,i=1,2,…,m if and only if  lim infn→∞d(xn,F)=0, where d(xn,F)=infy∈F∥xn-y∥,n≥1.Proof.
It suffices to show thatlim infn→∞d(xn,F)=0 implies that {xn} converges to a common fixed point of Ti,i=1,2,…,m.Necessity
Since (3.5) holds for all p∈F, we obtain from it that
(3.6)d(xn+1,F)≤(1+δn)d(xn,F)+γn,n≥1.
Lemma 2.1 then implies that limn→∞d(xn,F) exists. But, lim infn→∞d(xn,F)=0. Hence,limn→∞d(xn,F)=0.Sufficiency
Next, we first show that{xn} is a Cauchy sequence in E. For all integer m≥1, we obtain from inequality (3.5) that
(3.7)∥xn+m-p∥≤∏i=nn+m-1(1+δi)∥xn-p∥+(∑i=nn+m-1γi)∏i=nn+m-1(1+δi)≤exp(∑i=nn+m-1δi)∥xn-p∥+(∑i=nn+m-1γi)exp(∑i=nn+m-1δi),
so that for all integers m≥1 and all p∈F,(3.8)∥xn+m-xn∥≤∥xn+m-p∥+∥xn-p∥≤[1+exp(∑i=nn+m-1δi)]∥xn-p∥+(∑i=nn+m-1γi)exp(∑i=nn+m-1δi).
We therefore have that
(3.9)∥xn+m-xn∥≤D∥xn-p∥+D(∑i=n∞γi)
for some constant D>0. Taking infimum over p∈F in (3.9) gives
(3.10)∥xn+m-xn∥≤Dd(xn,F)+D(∑i=n∞γi).
Now, since limn→∞d(xn,F)=0 and ∑i=1∞γi<∞, given ϵ>0, there exists an integer N1>0 such that for all n≥N1,d(xn,F)<ϵ/2(D+1) and ∑i=n∞γi<ϵ/2(D+1). So for all integers n≥N1,m≥1, we obtain from (3.10) that
(3.11)∥xn+m-xn∥<ϵ.Hence, {xn} is a Cauchy sequence in E, and since E is complete there exists l*∈E such that xn→l* as n→∞. We now show that l* is a common fixed point of Ti,i=1,2,…,m, that is, we show that l*∈F. Suppose for contradiction that l*∈Fc (where Fc denotes the complement of F). Since F is a closed subset of E (recall each Ti,i=1,2,…,m is continuous), we have that d(l*,F)>0. But, for all p∈F, we have
(3.12)∥l*-p∥≤∥l*-xn∥+∥xn-p∥.
This implies
(3.13)d(l*,F)≤∥xn-l*∥+d(xn,F),
so that as n→∞ we obtain d(l*,F)=0 which contradicts d(l*,F)>0. Thus, l* is a common fixed point of Ti,i=1,2,…,m. This completes the proof.Remark 3.3.
IfT1,T2,…,Tm are asymptotically nonexpansive mappings, then lin=0 for all n≥1,i=1,2,…,m and ϕi(λi)=λi so that the assumption that there exist Mi,Mi*>0 such that ϕi(λi)≤Mi*λi for all λi≥Mi,i=1,2,…,m in the above theorems is no longer needed.Thus, we have the following corollary.Corollary 3.4.
LetE be a real Banach space, let  K be a nonempty closed convex subset of E, and let  Ti:K→K,i=1,2,…,m be m continuous asymptotically nonexpansive mappings with sequences {μin},n≥1,i=1,2,…,m such that F=⋂i=1mF(Ti)≠∅. Let {xn} be given by (3.1). Suppose ∑n=1∞μin<∞,i=1,2,…,m. Then the sequence {xn} is bounded and limn→∞∥xn-p∥ exists, p∈F. Moreover, {xn} converges strongly to a common fixed point of Ti,i=1,2,…,m if and only if lim infn→∞d(xn,F)=0.


## 3.2. Convergence Theorem in Real Uniformly Convex Banach Spaces

Theorem 3.5.
LetE be a uniformly convex real Banach space, K be a nonempty closed convex subset of E and Ti:K→K,i=1,2,…,m be m uniformly continuous total asymptotically nonexpansive mappings with sequences {μin},{lin}⊂[0,∞) such that ∑n=1∞μin<∞,∑n=1∞lin<∞,i=1,2,…,m and F:=⋂i=1mF(Ti)≠∅. From arbitrary x1∈E, define the sequence {xn} by (3.1). Suppose that there exist Mi,Mi*>0 such that ϕi(λi)≤Mi*λi whenever λi≥Mi,i=1,2,…,m. Then limn→∞∥xn-Tinxn∥=0,i=1,2,…,m.Proof.
Letp∈F. Then, by Theorem 3.1, limn→∞∥xn-p∥ exists. Let limn→∞∥xn-p∥=r. If r=0, then by continuity of Ti,i=1,2,…,m, we are done. Now suppose r>0. We show that limn→∞∥xn-Tinxn∥=0,i=1,2,…,m. We observe that
(3.14)∥Tinxn-p∥≤∥xn-p∥+μinϕi(∥xn-p∥)+lin≤∥xn-p∥+μinϕi(Mi)+μinMi*∥xn-p∥+lin(i=1,2,…,m).
so that taking lim sup on both sides of this inequality, we obtain
(3.15)lim supn→∞∥Tinxn-p∥≤r(i=1,2,…,m).
Let {tn}n≥1∈(0,1) be such that tn→0 as n→∞ and define
(3.16)yin:=(1-tn)xn+tnTinxn(i=1,2,…,m).
Then,
(3.17)∥yin-xn∥≤tn∥Tinxn-xn∥≤tnM(i=1,2,…,m)
for some M>0. Thus,
(3.18)limn→∞∥yin-xn∥=0(i=1,2,…,m).
Furthermore,
(3.19)|∥yin-p∥-∥xn-p∥|≤∥yin-xn∥(i=1,2,…,m).
This implies that
(3.20)limn→∞∥yin-p∥=limn→∞∥xn-p∥=r(i=1,2,…,m).
But,
(3.21)∥yin-p∥=∥(1-tn)(xn-p)+tn(Tinxn-p)∥(i=1,2,…,m).
So,
(3.22)limn→∞∥(1-tn)(xn-p)+tn(Tinxn-p)∥=r,(i=1,2,…,m).
Hence, by Lemma 2.2, we obtain
(3.23)limn→∞∥xn-Tinxn∥=0,(i=1,2,…,m).
This completes the proof.Theorem 3.6.
LetE be a uniformly convex real Banach space, let  K be a nonempty closed convex subset of E, and let  Ti:K→K,i=1,2,…,m be m uniformly continuous total asymptotically nonexpansive mappings with sequences {μin},{lin}⊂[0,∞) such that ∑n=1∞μin<∞,∑n=1∞lin<∞,i=1,2,…,m and F:=⋂i=1mF(Ti)≠∅. From arbitrary x1∈E, define the sequence {xn} by (1.24). Suppose that there exist Mi,Mi*>0 such that ϕi(λi)≤Mi*λi whenever λi≥Mi,i=1,2,…,m; and that one of T1,T2,…,Tm is compact, then {xn} converges strongly some p∈F.Proof.
We obtain from Theorem3.5 that
(3.24)limn→∞∥Tinxn-xn∥=0,i=1,2,…,m.
Using the recursion formula (3.1), we observe that
(3.25)∥xn+1-xn∥=∥α0nxn+∑i=1mαinTinxn-xn∥=∥∑i=1mαin(Tinxn-xn)∥≤∑i=1mαin∥Tinxn-xn∥.
It then follows from (3.24) and (3.25) that
(3.26)limn→∞∥xn+1-xn∥=0.
Without loss of generality, let T1 be compact. Since T1 is continuous and compact, it is completely continuous. Thus, there exists a subsequence {T1nkxnk} of {T1nxn} such that T1nkxnk→x* as k→∞ for some x*∈E. Thus T1nk+1xnk→T1x* as k→∞ and from (3.24), we have that limk→∞xnk=x*. Also from (3.24) T2nkxnk→x*,T3nkxnk→x*,…,Tmnkxnk→x* as k→∞. Thus, T2nk+1xnk→T2x*,T3nk+1xnk→T3x*,…,Tmnk+1xnk→Tmx* as k→∞. Now, since from (3.26), ∥xnk+1-xnk∥→0 as k→∞, it follows that xnk+1→x* as k→∞. Next, we show that x*∈F. Observe that
(3.27)∥x*-Tix*∥≤∥x*-xnk+1∥+∥xnk+1-Tink+1xnk+1∥+∥Tink+1xnk+1-Tink+1xnk∥+∥Tink+1xnk-Tix*∥.
Taking limit as k→∞ and using the fact that Ti(i=1,2,…m) are uniformly continuous we have that x*=Tix*(i=1,2,…m) and so x*∈F(Ti)(i=1,2,…m). But by Theorem 3.1, limn→∞∥xn-p∥ exists, p∈F. Hence, {xn} converges strongly to x*∈F. This completes the proof.In view of Remark12, the following corollary is now obvious.Corollary 3.7.
LetE be a uniformly convex real Banach space, let  K be a nonempty closed convex subset of E, and let Ti:K→K,i=1,2,…,m be m asymptotically nonexpansive mappings with sequences {μin}⊂[0,∞) such that ∑n=1∞μin<∞,i=1,2,…,m; F=⋂i=1mF(Ti)≠∅ and that one of T1,T2,…,Tm is compact. From arbitrary x1∈E, define the sequence {xn} by (3.1). Then {xn} converges strongly to to some p∈F.Remark 3.8.
Observe that the theorems of this paper remain true for mappingsT1,T2,…,Tm satisfying (1.5) provided that  ∑n=1∞σn<∞. In this case, the requirement that there exist Mi,Mi*>0 such that ϕi(λi)≤Mi*λi for all λi≥Mi,i=1,2,…,m is not needed.Remark 3.9.
A prototype forϕ:[0,∞)→[0,∞) satisfying the conditions of our theorems is ϕ(λ)=λs,0<s≤1. Prototypes for the sequences {αin}n=1∞foralli=0,1,2,…,m in this paper are the following:
(3.28)α0n:=n2n+1;αin:=n+1m(2n+1)∀n∈ℕ,i=1,2,…,m.Remark 3.10.
Addition ofbounded (or the so called mean) error terms to the iteration process studied in this paper leads to no further generalization.Definition 3.11.
A mappingT:K→K is said to be total asymptotically quasi-nonexpansive if F(T)≠∅ and there exist nonnegative real sequences {μn} and {ln},n≥1 with μn,ln→0 as n→∞ and strictly increasing continuous function ϕ:ℝ+→ℝ+ with ϕ(0)=0 such that for all x∈E,x*∈F(T),(3.29)∥Tnx-x*∥≤∥x-x*∥+μnϕ(∥x-x*∥)+ln,n≥1.Remark 3.12.
Ifϕ(λ)=λ, then (3.29) reduces to
(3.30)∥Tnx-x*∥≤(1+μn)∥x-x*∥+ln,n≥1.
In addition, if ln=0 for all n≥1, then total asymptotically quasi-nonexpansive mappings coincide with asymptotically quasi-nonexpansive mappings studied by various authors. If μn=0 and ln=0 for all n≥1, we obtain from (3.30) the class of quasi-nonexpansive mappings. Observe that the class of total asymptotically nonexpansive mappings with nonempty fixed point sets belongs to the class of total asymptotically quasi-nonexpansive mappings. Moreover, if μn=0 and ln=σn*, then (3.29) reduces to (1.24).It is trivial to observe that all the theorems of this paper carry over to the class of total asymptotically quasi-nonexpansive mappings with little or no modifications.A subsetK of a real normed linear space E is said to be a retract of E if there exists a continuous map P:E→K such that Px=x for all x∈K. It is well known (see, e.g., [28]) that every closed convex nonempty subset of a uniformly convex Banach space is a retract. A map P:E→E is said to be a retraction if P2=P. It follows that if a map P ia a retraction, then Py=y for all y in the range of P. The mapping p is called a sunny nonexpansive retraction if for all x∈E and t∈(0,1),P((1-t)x+tP(x))=P(x).Definition 3.13.
LetK be a nonempty closed and convex subset of E. Let P:E→K be the nonexpansive retraction of E onto K. A nonself map T:K→E is said to be total asymptotically nonexpansive if there exist sequences {μn}n≥1,{ln}n≥1 in [0,+∞) with μn,ln→0 as n→∞ and a strictly increasing continuous function ϕ:[0,+∞)→[0,+∞) with ϕ(0)=0 such that for all x,y∈K,(3.31)∥T(PT)n-1x-T(PT)n-1y∥≤∥x-y∥+μnϕ(∥x-y∥)+ln,n≥1.
Let T1,T2,…,Tm:K→E be m total asymptotically nonexpansive nonself maps; assuming existence of common fixed points of these operators, our theorems and method of proof easily carry over to this class of mappings using the iterative sequence {xn} defined by
(3.32)x1∈K,xn+1=P(α0nxn+∑i=1rαinTi(PTi)n-1xn)
instead of (3.1) provided that the well definedness of P as a sunny nonexpansive retraction is guaranteed.Remark 3.14.
It is clear that the recursion formula (3.1) introduced and studied in this paper is much simpler than the recursion formulas (1.25) studied earlier for this problem.Remark 3.15.
Our theorems unify, extend, and generalize the corresponding results of Alber et al. [29], Sahu [5], Shahzad and Udomene [16], and a host of other results recently announced (see, e.g., [8, 16, 22, 28, 30, 39, 40, 44, 47, 49–58]) for the approximation of common fixed points of finite families of several classes of nonlinear mappings.



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*Source: 615107-2009-12-15.xml*