# An Implicit Iteration Process for Common Fixed Points of Two Infinite Families of Asymptotically Nonexpansive Mappings in Banach Spaces

Authors: Wei-Qi Deng; Peng Bai
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

LetK be a nonempty, closed, and convex subset of a real uniformly convex Banach space E. Let {Tλ}λ∈Λ and {Sλ}λ∈Λ be two infinite families of asymptotically nonexpansive mappings from K to itself with F:={x∈K:Tλx=x=Sλx,λ∈Λ}≠∅. For an arbitrary initial point x0∈K, {xn} is defined as follows: xn=αnxn-1+βn(Tn-1*)mn-1xn-1+γn(Tn*)mnyn, yn=αn′xn+βn′(Sn-1*)mn-1xn-1+γn′(Sn*)mnxn, n=1,2,3,…, where Tn*=Tλin and Sn*=Sλin with in and mn satisfying the positive integer equation: n=i+(m-1)m/2, m≥i; {Tλi}i=1∞ and {Sλi}i=1∞ are two countable subsets of {Tλ}λ∈Λ and {Sλ}λ∈Λ, respectively; {αn}, {βn}, {γn}, {αn′}, {βn′},  and {γn′} are sequences in [δ,1-δ] for some δ∈(0,1), satisfying αn+βn+γn=1 and αn′+βn′+γn′=1. Under some suitable conditions, a strong convergence theorem for common fixed points of the mappings {Tλ}λ∈Λ and {Sλ}λ∈Λ is obtained. The results extend those of the authors whose related researches are restricted to the situation of finite families of asymptotically nonexpansive mappings.


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## Body

## 1. Introduction

LetK be a nonempty, closed, and convex subset of a real uniformly convex Banach space E. A mapping T:K→K is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈K. T is said to be asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞) with kn→1 (n→∞) such that
(1)∥Tnx-Tny∥≤kn∥x-y∥,∀x,y∈K,n=1,2,3,….
It is obvious that a nonexpansive mapping is an asymptotically nonexpansive one, but the converse is not true. Denote by F(T) the set of fixed points of T, that is, F(T)={x∈K:Tx=x}. Throughout this paper, we always assume that F(T)≠∅. As an important generalization of nonexpansive mappings, the class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972, who proved that if K is a nonempty, closed, and convex subset of a real uniformly convex Banach space and T:K→K is an asymptotically nonexpansive mapping, then T has a fixed point.Since then, iterative techniques for approximating fixed points of asymptotically nonexpansive mappings have been studied by various authors (see, e.g., [2–9]). However, these researches are all restricted to the situation of at most finite families of asymptotically nonexpansive mappings. For the extension of finite families to infinite ones, we develop an original method, namely, a specific way of choosing the indexes, for the iterative approximation of common fixed points of the involved mappings.We now cite an announced result as the object of our extension. In 2010, Wang et al. [10] constructed the following iteration process for two asymptotically nonexpansive mappings and obtained some strong convergence theorems for common fixed points of the given mappings in Banach spaces. For an arbitrary initial point x0∈K, {xn},{yn} are defined as follows:
(2)xn=αnxn-1+βnTn-1xn-1+γnTnyn,yn=αn′xn+βn′Sn-1xn-1+γn′Snxn,n=1,2,3,…,
where T,S:K→K are two asymptotically nonexpansive mappings; {αn}, {βn}, {γn}, {αn′}, {βn′}, and {γn′} are real sequences in [0,1) satisfying αn+βn+γn=1 and αn′+βn′+γn′=1.In this paper, a modified iteration scheme of (2) is used for approximating common fixed points of two infinite families of asymptotically nonexpansive mappings; a strong convergence theorem is established in the framework of uniformly convex Banach spaces. The results show the feasibility of the newly developed technique and extend those of the authors whose related researches are restricted to the situation of finite families of such mappings.


## 2. Preliminaries

Throughout this paper, we useF to denote the set of common fixed points of two infinite families of asymptotically nonexpansive mappings {Tλ}λ∈Λ and {Sλ}λ∈Λ, that is, F:={x∈K:Tλx=x=Sλ,λ∈Λ}.LetK be a nonempty, closed, and convex subset of a real Banach space E. Let {Tλi}i=1∞ and {Sλi}i=1∞ be two countable subsets of {Tλ}λ∈Λ and {Sλ}λ∈Λ, respectively. In order to approximate some member of F, we define, from an arbitrary x0∈K, the following implicit iteration scheme:
(3)xn=αnxn-1+βn(Tn-1*)mn-1xn-1+γn(Tn*)mnyn,yn=αn′xn+βn′(Sn-1*)mn-1xn-1+γn′(Sn*)mnxn,n=1,2,3,…,
where {αn}, {βn},  {γn}, {αn′}, {βn′}, and {γn′} are sequences in [δ,1-δ] for some δ∈(0,1), satisfying αn+βn+γn=1 and αn′+βn′+γn′=1; Tn*=Tλin and Sn*=Sλin with in and mn being the solutions to the positive integer equation: n=i+(m-1)m/2(m≥i,n=1,2,3,…), that is, for each n≥1, there exist unique in and mn such that
(4)i1=1,i2=1,i3=2,i4=1,i5=2,i6=3,i7=1,i8=2,…,m1=1,m2=2,m3=2,m4=3,m5=3,m6=3,m7=4,m8=4,….
For convenience, we restate the following concepts and results.A Banach spaceE is said to satisfy Opial’s condition if, for any sequence {xn} in E, xn⇀x implies that
(5)limsupn→∞∥xn-x∥<limsupn→∞∥xn-y∥,
for all y∈E with y≠x, where xn⇀x denotes that {xn} converges weakly to x.A mappingT with domain D(T) and range R(T) in E is said to be demiclosed at p if whenever {xn} is a sequence in D(T) such that {xn} converges weakly to x*∈D(T) and {Txn} converges strongly to p, then Tx*=p.We now need the following lemmas for our main results.Lemma 1 (see [11]).
Let{an},{δn}, and {bn} be sequences of nonnegative real numbers satisfying
(6)an+1≤(1+δn)an+bn,n=1,2,3,….
If ∑n=1∞δn<∞ and ∑n=1∞bn<∞, then limn→∞an exists.Lemma 2 (see [6]).
LetE be a real uniformly convex Banach space, and let a,andb be two constants with 0<a<b<1. Suppose that {tn}⊂[a,b] is a real sequence and {xn}and{yn} are two sequences in E. Then, the conditions
(7)limn→∞∥tnxn+(1-tn)yn∥=d,limsupn→∞∥xn∥≤d,limsupn→∞∥yn∥≤d
imply that limn→∞∥xn-yn∥=0, where d≥0 is a constant.Lemma 3 (see [2]).
LetE be a real uniformly convex Banach space, K a nonempty, closed, convex subset of E, and let T:K→E be an asymptotically nonexpansive mapping with a sequence {kn}⊂[1,∞) and kn→1 as n→∞. Then, I-T is demiclosed at zero.Lemma 4.
The unique solutions to the positive integer equation(8)n=i+(m-1)m2,m≥i,n=1,2,3,…
are
(9)i=n-(m-1)m2,m=-[12-2n+14],n=1,2,3,…,
where [x] denotes the maximal integer that is not larger than x.Proof.
It follows from (8) that
(10)i=n-(m-1)m2,i≤m,n=1,2,3,…,
and hence
(11)1≤i=n-(m-1)m2≤m,n=1,2,3,…,
that is,
(12)(m-1)m2+1≤n≤(m+1)m2,n=1,2,3,…,
which implies that
(13)(m-12)2≤2n-74,(m+12)2≥2n+14,n=1,2,3,….
Thus,
(14)2n+14-12≤m≤12+2n-74,n=1,2,3…,
that is,
(15)-2n-74-12≤-m≤12-2n+14,n=1,2,3,…,
while the difference of the two sides of the inequality above is
(16)1-(2n+14-2n-74)=1-22n+1/4+2n-7/4∈[0,1),n=1,2,3,….
Then, it follows from (15) that (9) holds obviously.


## 3. Main Results

Lemma 5.
LetK be a nonempty, closed, and convex subset of a real uniformly convex Banach space E, and let {Tλi}i=1∞ and {Sλi}i=1∞ be two countable subsets of the asymptotically nonexpansive mappings {Tλ}λ∈Λ and {Sλ}λ∈Λ from K to itself, respectively, with corresponding sequences {kn(i)}⊂[1,∞) and {rn(i)}⊂[1,∞) such that ∑i=1∞∑n=1∞(kn(i)-1)<∞ and ∑i=1∞∑n=1∞(rn(i)-1)<∞. Suppose that {xn} is generated by (3), where γnkn(i)(αn′+γn′rn(i))<1 for all i≥1. If F≠∅, then limn→∞∥xn-q∥ exists for each q∈F.Proof.
Setumn(in)=kmn(in)-1 and vmn(in)=rmn(in)-1 for each positive integer n≥1, where in and mn satisfy the positive integer equation: n=i+(m-1)m/2(m≥i,m∈ℕ). For any q∈F, it follows from (3) that
(17)∥yn-q∥≤αn′∥xn-q∥+βn′rmn-1(in-1)∥xn-1-q∥+γn′rmn(in)∥xn-1-q∥=(αn′+γn′rmn(in))∥xn-q∥+βn′rmn-1(in-1)∥xn-1-q∥,(18)∥xn-q∥≤αn∥xn-1-q∥+βnkmn-1(in-1)∥xn-1-q∥+γnkmn(in)∥yn-q∥=(αn+βnkmn-1(in-1))∥xn-1-q∥+γnkmn(in)∥yn-q∥.
Substituting (17) into (18) yields that
(19)[1-γnkmn(in)(αn′+γn′rmn(in))]∥xn-q∥≤(αn+βnkmn-1(in-1)+γnkmn(in)βn′rmn-1(in-1))∥xn-1-q∥.
Note that an:=1-γnkmn(in)(αn′+γn′rmn(in))>0. We have
(20)∥xn-q∥≤αn+βnkmn-1(in-1)+γnkmn(in)βn′rmn-1(in-1)an∥xn-1-q∥,
which implies that ∥xn-q∥≤(1+bn)∥xn-1-q∥, where
(21)bn=(γnγn′vmn(in)+γnumn(in)+γnγn′umn(in)vmn(in)+βnumn-1(in-1)+γnβn′vmn-1(in-1)+γnβn′umn(in)vmn-1(in-1))×(1-γn(1-βn′)-γnγn′vmn(in)-γnumn(in)(1-βn′)-γnγn′umn(in)vmn(in))-1.
Note that ∑n=1∞‍umn(in)=∑i=1∞‍∑n=i∞‍(kn(i)-1)≤∑i=1∞‍∑n=1∞‍(kn(i)-1)<∞ and ∑n=1∞‍vmn(in)=∑i=1∞‍∑n=i∞‍(rn(i)-1)≤∑i=1∞‍∑n=1∞‍(rn(i)-1)<∞, which implies that limn→∞umn(in)=limn→∞vmn(in)=0. Then, for a given ϵ0∈(0,δ), there exists a positive n0 such that
(22)γn(1-βn′)+γnγn′vmn(in)+γnumn(in)(1-βn′)+γnγn′umn(in)vmn(in)<1-ϵ0,
as n≥n0. Then, it follows from (20) and (22) that
(23)∥xn-q∥≤(1+cn)∥xn-1-q∥,
where cn=(1/ϵ0)[vmn(in)+umn(in)+umn(in)vmn(in)+umn-1(in-1)+vmn-1(in-1)+umn(in)vmn-1(in-1)], and so ∑n=1∞‍cn<∞. Hence, it follows from (23) and Lemma 1 that limn→∞∥xn-q∥ exists for each q∈F. The proof is completed.Remark 6.
Because of the importance of the condition thatF is nonempty, we now give an example satisfying the lemma with the set of common fixed points of {Tλ}λ∈Λ and {Sλ}λ∈Λ being a non single point set. Let E:=ℝ1, and let K:=[-1,1]. Define an infinite family of mappings {Tλ}λ≥1:K→K by
(24)Tλ(x)={1λxλ,x∈[0,1],x,x∈[-1,0),
and an infinite family of mappings {Sλ}λ≥1:K→K by
(25)Sλ(x)={1λsinxλ,x∈[0,1],x,x∈[-1,0).
Then, clearly, {Tλ}λ≥1 and {Sλ}λ≥1 are two infinite families of asymptotically nonexpansive mappings with F=[-1,0].Lemma 7.
LetK, E, {Tλi}i=1∞, {Sλi}i=1∞, and {xn} be the same as those in Lemma 5. If F≠∅, then for each i≥1, there exists a subsequence {xn(i)} of {xn} such that limn→∞∥xn(i)-Tλixn(i)∥=limn→∞∥xn(i)-Sλixn(i)∥=0.Proof.
By Lemma5, we may assume that limn→∞∥xn-q∥=d for a given q∈F, that is,
(26)limn→∞dn=d,
where dn=∥(1-γn)[αn(xn-1-q)/(1-γn)+βn((Tn-1*)mn-1xn-1-q)/(1-γn)]+γn((Tn*)mnyn-q)∥. It follows from (17) that
(27)∥(Tn*)mnyn-q∥≤kmn(in)∥yn-q∥≤kmn(in)[(αn′+γn′rmn(in))∥xn-q∥+βn′rmn-1(in-1)∥xn-1-q∥]=kmn(in)[∥xn-q∥+βn′(∥xn-q∥-∥xn-1-q∥)+γn′vmn(in)∥xn-q∥+βn′vmn-1(in-1)∥xn-1-q∥].
Taking limsup on both sides in (27) yields that
(28)limsupn→∞∥(Tn*)mnyn-q∥≤limsupn→∞∥yn-q∥≤d.
Next, it follows from (26) that
(29)limsupn→∞en≤limsupn→∞(αn1-γn∥xn-1-q∥+βn1-γnkmn-1(in-1)∥xn-1-q∥)=limsupn→∞(1+βn1-γnumn-1(in-1))∥xn-1-q∥=d,
where en=∥αn(xn-1-q)+βn((Tn-1*)mn-1xn-1-q)∥/(1-γn). It then follows from (26), (28), (29), and Lemma 2 that
(30)limn→∞∥αn1-γnxn-1+βn1-γn(Tn-1*)mn-1xn-1-(Tn*)mnyn∥=0,
which, in addition to (3), implies that
(31)limn→∞∥xn-(Tn*)mnyn∥=0.
Now, we show that ∥yn-q∥→d as n→∞. It follows from (18) that
(32)αn+βnγn(∥xn-q∥-∥xn-1-q∥)+∥xn-q∥-βnγnumn-1(in-1)∥xn-1-q∥≤kmn(in)∥yn-q∥.
Taking liminf on both sides in the inequality above yields that
(33)d=liminfn→∞∥xn-q∥≤liminfn→∞∥yn-q∥.
Combining (28) with (33), we have limn→∞∥yn-q∥=d. Then, by ways similar to the preceding ones, it is easily shown that
(34)limn→∞∥βn′1-αn′(Sn-1*)mn-1xn-1+γn′1-αn′(Sn*)mnxn-xn∥=0,
which means that
(35)limn→∞∥yn-xn∥=0.
Set dn′=∥(1-βn)[αn(xn-1-q)/(1-βn)+γn((Tn*)mnyn-q)/(1-βn)]+βn((Tn-1*)mn-1xn-1-q)∥. Since
(36)limn→∞∥xn-q∥=limn→∞dn′=d,
then
(37)limsupn→∞∥(Tn-1*)mn-1xn-1-q∥≤limsupn→∞kmn-1(in-1)∥xn-1-q∥≤d,(38)limsupn→∞en′≤limsupn→∞(αn1-βn∥xn-1-q∥+γn1-βnkmn(in)∥yn-q∥)=d,
where en′=∥αn(xn-1-q)+γn((Tn*)mnyn-q)∥/(1-βn). It then follows from (36)–(38) and Lemma 2 that
(39)limn→∞∥αn1-βnxn-1+γn1-βn(Tn*)mnyn-(Tn-1*)mn-1xn-1∥=0,
which implies that
(40)limn→∞∥xn-(Tn-1*)mn-1xn-1∥=0.
Then, it follows from (31), (35), and (40) that
(41)limn→∞∥xn-xn-1∥=0.
Similarly, we have
(42)limn→∞∥yn-(Sn*)mnxn∥=0,limn→∞∥yn-(Sn-1*)mn-1xn-1∥=0.
Next, for any i≥1, we consider the corresponding subsequence {xl(i)}l∈Γi of {xn}, where l∈Γi:={l:l=i+(j-1)j/2,j≥i,j∈ℕ}. For example, by the definition of Γ1, we have Γ1={1,2,4,7,11,16,…} and i1=i2=i4=i7=i11=i16=⋯=1. For simplicity, {xl(i)}l∈Γi,{yl(i)}l∈Γi, {Tl*(i)}l∈Γi, and {jl(i)}l∈Γi are written as {xn′},{yn′}, {Tn′} and {mn}, respectively. Note that {mn}n∈Γi={i,i+1,i+2,…}, that is, k1(il)=k1(i),mn-1=mn-1, and Tn′=Tλi=Tn-1′ whenever l∈Γi. Then, we have
(43)∥xn′-Tn′xn∥≤∥xn′-(Tn′)mnyn′∥+∥(Tn′)mnyn′-(Tn′)mnxn′∥+∥(Tn′)mnxn′-Tn′xn′∥≤∥xn′-(Tn′)mnyn′∥+kmn(i)∥yn′-xn′∥+k1(i)∥(Tn′)mn-1xn′-xn′∥≤∥xn′-(Tn′)mnyn′∥+kmn(i)∥yn′-xn′∥+k1(i)×[∥(Tn′)mn-1xn′-(Tn-1′)mn-1xn-1′∥+∥(Tn-1′)mn-1xn-1′-xn′∥]≤∥xn′-(Tn′)mnyn′∥+kmn(i)∥yn′-xn′∥+k1(i)[kmn-1(i)∥xn′-xn-1′∥+∥(Tn-1′)mn-1xn-1′-xn′∥].
It hence follows from (31), (35), (40), and (41) that limn→∞∥xn′-Tn′xn′∥=0. That is, for each i≥1, there exists a subsequence {xn(i)} of {xn} such that limn→∞∥xn(i)-Tn*(i)xn(i)∥=0. Since Tn*(i)=Tλi, we have, for each i≥1,
(44)limn→∞∥xn(i)-Tλixn(i)∥=0.
Similarly, it can be shown that, for each i≥1,
(45)limn→∞∥yn(i)-Sλiyn(i)∥=0.
This completes the proof.Remark 8.
The key point of the proof of Lemma7 lies in the use of a specific way of choosing the indexes of the involved mappings, which makes the generalization of finite families of nonlinear mappings to infinite ones possible.Theorem 9.
LetK, E, {Tλi}i=1∞, {Sλi}i=1∞, and {xn} be the same as those in Lemma 5. If F≠∅ and there exist a Tλi0∈{Tλi}i=1∞ or an Sλi0∈{Sλi}i=1∞ and a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(r)>0 for all r∈(0,∞) such that f(d(xn,F))≤∥xn-Tλi0xn∥ or f(d(xn,F))≤∥xn-Sλi0xn∥ for all n≥1, then {xn} converges strongly to some point of F.Proof.
By Lemma7, there exists a subsequence {xn(i0)} of {xn} such that limn→∞∥xn(i0)-Tλi0xn(i0)∥=limn→∞∥xn(i0)-Sλi0xn(i0)∥=0. Since
(46)f(d(xn(i0),F))≤∥xn(i0)-Tλi0xn(i0)∥,
or
(47)f(d(xn(i0),F))≤∥xn(i0)-Sλi0xn(i0)∥,
by taking limsup as n→∞ on both sides in the inequality above, we have
(48)limn→∞f(d(xn(i0),F))=0,
which implies limn→∞d(xn(i0),F)=0 by the definition of the function f.
Now, we will show that{xn(i0)} is a Cauchy sequence. By Lemma 5, there exists a constant M>0 such that ∥xn-q∥≤M∥xm-q∥ as n>m. And for any ϵ>0, there exists a positive integer N such that d(xn(i0),F)<ϵ/2M for all n≥N. Then, for any q∈F and n,m≥N, we have
(49)∥xn(i0)-xm(i0)∥≤∥xn(i0)-q∥+∥xm(i0)-q∥≤2M∥xN(i0)-q∥.
Taking the infimum in the above inequalities for all q∈F yields that
(50)∥xn(i0)-xm(i0)∥≤2Md(xN(i0),F)<ϵ,
which implies that {xn(i0)} is a Cauchy sequence. Therefore, there exists a p∈K such that xn(i0)→p as n→∞ since E is complete. Furthermore, limn→∞d(xn(i0),F)=0 shows that d(p,F)=0, which implies that p∈F since F is closed. It follows from the existence of limn→∞∥xn-p∥ that xn→p as n→∞. This completes the proof.Remark 10.
The result of Theorem9 extends that of Wang et al. [10] whose related research is restricted to the situation of two asymptotically nonexpansive mappings.



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