# Common Fixed Points for Asymptotic Pointwise Nonexpansive Mappings in Metric and Banach Spaces

Authors: P. Pasom; B. Panyanak
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

LetC be a nonempty bounded closed convex subset of a complete CAT(0) space X. We prove that the common fixed point set of any commuting family of asymptotic pointwise nonexpansive mappings on C is nonempty closed and convex. We also show that, under some suitable conditions, the sequence {xk}k=1∞ defined by xk+1=(1-tmk)xk⊕tmkTmnky(m-1)k,y(m-1)k=(1-t(m-1)k)xk⊕t(m-1)kTm-1nky(m-2)k,y(m-2)k=(1-t(m-2)k)xk⊕t(m-2)kTm-2nky(m-3)k,…,y2k=(1-t2k)xk⊕t2kT2nky1k,y1k=(1-t1k)xk⊕t1kT1nky0k,y0k=xk,k∈ℕ, converges to a common fixed point of T1,T2,…,Tm where they are asymptotic pointwise nonexpansive mappings on C, {tik}k=1∞ are sequences in [0,1] for all i=1,2,…,m, and {nk} is an increasing sequence of natural numbers. The related results for uniformly convex Banach spaces are also included.


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## Body

## 1. Introduction

A mappingT on a subset C of a Banach space X is said to be asymptotic pointwise nonexpansive if there exists a sequence of mappings αn:C→[0,∞) such that(1.1)‖Tnx-Tny‖≤αn(x)‖x-y‖,
where limsupn→∞αn(x)≤1, for all x,y∈C. This class of mappings was introduced by Kirk and Xu [1], where it was shown that if C is a bounded closed convex subset of a uniformly convex Banach space X, then every asymptotic pointwise nonexpansive mapping T:C→C always has a fixed point. In 2009, Hussain and Khamsi [2] extended Kirk-Xu's result to the case of metric spaces, specifically to the so-called CAT(0) spaces. Recently, Kozlowski [3] defined an iterative sequence for an asymptotic pointwise nonexpansive mapping T:C→C by x1∈C and(1.2)xk+1=(1-tk)xk+tkTnkyk,yk=(1-sk)xk+skTnkxk,k∈N,
where {tk} and {sk} are sequences in [0,1] and {nk} is an increasing sequence of natural numbers. He proved, under some suitable assumptions, that the sequence {xk} defined by (1.2) converges weakly to a fixed point of T where X is a uniformly convex Banach space which satisfies the Opial condition and {xk} converges strongly to a fixed point of T provided Tr is a compact mapping for some r∈ℕ. On the other hand, Khan et al. [4] studied the iterative process defined by(1.3)xn+1=(1-αmn)xn+αmnTmny(m-1)n,y(m-1)n=(1-α(m-1)n)xn+α(m-1)nTm-1ny(m-2)n,y(m-2)n=(1-α(m-2)n)xn+α(m-2)nTm-2ny(m-3)n,⋮y2n=(1-α2n)xn+α2nT2ny1n,y1n=(1-α1n)xn+α1nT1ny0n,y0n=xn,n∈N,
where T1,…,Tm are asymptotically quasi-nonexpansive mappings on C and {αin}n=1∞ are sequences in [0,1] for all i=1,2,…,m.In this paper, motivated by the results mentioned above, we ensure the existence of common fixed points for a family of asymptotic pointwise nonexpansive mappings in a CAT(0) space. Furthermore, we obtain▵ and strong convergence theorems of a sequence defined by(1.4)xk+1=(1-tmk)xk⊕tmkTmnky(m-1)k,y(m-1)k=(1-t(m-1)k)xk⊕t(m-1)kTm-1nky(m-2)k,y(m-2)k=(1-t(m-2)k)xk⊕t(m-2)kTm-2nky(m-3)k,⋮y2k=(1-t2k)xk⊕t2kT2nky1k,y1k=(1-t1k)xk⊕t1kT1nky0k,y0k=xk,k∈N,
where T1,…,Tm are asymptotic pointwise nonexpansive mappings on a subset C of a complete CAT(0) space and {tik}k=1∞ are sequences in [0,1] for all i=1,2,…,m, and {nk} is an increasing sequence of natural numbers. We also note that our method can be used to prove the analogous results for uniformly convex Banach spaces.


## 2. Preliminaries

A metric spaceX is a CAT(0) space if it is geodesically connected and if every geodesic triangle in X is at least as “thin” as its comparison triangle in the Euclidean plane. It is well-known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include Pre-Hilbert spaces (see [5]), ℝ-trees (see [6]), Euclidean buildings (see [7]), and the complex Hilbert ball with a hyperbolic metric (see [8]). For a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [5].Fixed point theory in CAT(0) spaces was first studied by Kirk (see [9, 10]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed, and many papers have appeared (see, e.g., [2, 11–22] and the references therein). It is worth mentioning that fixed point theorems in CAT(0) spaces (specially in ℝ-trees) can be applied to graph theory, biology, and computer science (see, e.g., [6, 23–26]).Let(X,d) be a metric space. A geodesic path joining x∈X to y∈X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0,l]⊂ℝ to X such that c(0)=x,c(l)=y, and d(c(t),c(t′))=|t-t′| for all t,t′∈[0,l]. In particular, c is an isometry and d(x,y)=l. The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denoted by [x,y]. The space (X,d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x,y∈X. A subset Y⊂X is said to be convex if Y includes every geodesic segment joining any two of its points.Ageodesic triangle▵(x1,x2,x3) in a geodesic space (X,d) consists of three points x1,x2,x3 in X (thevertices of ▵) and a geodesic segment between each pair of vertices (the edges of ▵). A comparison triangle for geodesic triangle ▵(x1,x2,x3) in (X,d) is a triangle ▵̅(x1,x2,x3)∶=▵(x¯1,x¯2,x¯3) in the Euclidean plane 𝔼2 such that d𝔼2(x¯i,x¯j)=d(xi,xj) for i,j∈{1,2,3}.A geodesic space is said to be a CAT(0) space if all geodesic triangles satisfy the following comparison axiom.CAT(0): Let▵ be a geodesic triangle in X, and let ▵̅ be a comparison triangle for ▵. Then, ▵ is said to satisfy the CAT(0) inequality if for all x,y∈▵ and all comparison points x¯,y¯∈▵¯,
(2.1)d(x,y)≤dE2(x¯,y¯).Letx,y∈X, by Lemma  2.1(iv) of [14] for each t∈[0,1], there exists a unique point z∈[x,y] such that(2.2)d(x,z)=td(x,y),d(y,z)=(1-t)d(x,y).
We will use the notation (1-t)x⊕ty for the unique point z satisfying (2.2). We now collect some elementary facts about CAT(0) spaces.Lemma 2.1.
LetX be a complete CAT(0) space.(i)
[5, Proposition  2.4] If C is a nonempty closed convex subset of X, then, for every x∈X, there exists a unique point P(x)∈C such that d(x,P(x))=inf{d(x,y):y∈C}. Moreover, the map x↦P(x) is a nonexpansive retract from X onto C.(ii)
[14, Lemma  2.4] For x,y,z∈X and t∈[0,1], we have
(2.3)d((1-t)x⊕ty,z)≤(1-t)d(x,z)+td(y,z).(iii)
[14, Lemma  2.5] For x,y,z∈X and t∈[0,1], we have
(2.4)d((1-t)x⊕ty,z)2≤(1-t)d(x,z)2+td(y,z)2-t(1-t)d(x,y)2.We now give the concept ofΔ-convergence and collect some of its basic properties. Let {xn} be a bounded sequence in a CAT(0) space X. For x∈X, we set
(2.5)r(x,{xn})=limsup⁡n→∞d(x,xn).
The asymptotic radiusr({xn}) of {xn} is given by
(2.6)r({xn})=inf⁡⁡{r(x,{xn}):x∈X},
and the asymptotic centerA({xn}) of {xn} is the set
(2.7)A({xn})={x∈X:r(x,{xn})=r({xn})}.It is known from Proposition  7 of [27] that, in a CAT(0) space, A({xn}) consists of exactly one point.Definition 2.2 (see [28, 29]).
A sequence{xn} in a CAT(0) space X is said to Δ-converge to x∈X if x is the unique asymptotic center of {un} for every subsequence {un} of {xn}. In this case, we write Δ-limnxn=x and call x the Δ-limit of {xn}.Lemma 2.3.
LetX be a complete CAT(0) space.(i)
[28, page 3690] Every bounded sequence in X has a Δ-convergent subsequence.(ii)
[30, Proposition  2.1] If C is a closed convex subset of a complete CAT(0) space and if {xn} is a bounded sequence in C, then the asymptotic center of {xn} is in C.(iii)
[14, Lemma  2.8] If {xn} is a bounded sequence in a complete CAT(0) space with A({xn})={x} and {un} is a subsequence of {xn} with A({un})={u} and the sequence {d(xn,u)} converges, then x=u.Recall that a mappingT:X→X is said to be nonexpansive [31] if
(2.8)d(Tx,Ty)≤d(x,y),∀x,y∈X,where T is called asymptotically nonexpansive [32] if there is a sequence {kn} of positive numbers with the property limn→∞kn=1 and such that(2.9)d(Tnx,Tny)≤knd(x,y),∀n≥1,x,y∈X,where T is called an asymptotic pointwise nonexpansive mapping [1] if there exists a sequence of functions αn:X→[0,∞) such that(2.10)d(Tnx,Tny)≤αn(x)d(x,y),∀n≥1,x,y∈X,
where limsupn→∞αn(x)≤1. The following implications hold. (2.11)Tisnonexpansive⟹Tisasymptoticallynonexpansive⟹Tisasymptoticpointwisenonexpansive.
A point x∈X is called a fixed point of T if x=Tx. We shall denote by F(T) the set of fixed points of T. The existence of fixed points for asymptotic pointwise nonexpansive mappings in CAT(0) spaces was proved by Hussain and Khamsi [2] as the following result.Theorem 2.4.
LetC be a nonempty bounded closed convex subset of a complete CAT(0) space X. Suppose that T:C→C is an asymptotic pointwise nonexpansive mapping. Then, F(T) is nonempty closed and convex.


## 3. Existence Theorems

LetM be a metric space and ℱ a family of subsets of M. Then, we say that ℱ defines a convexity structure on M if it contains the closed balls and is stable by intersection.Definition 3.1 (see [2]).
Letℱ be a convexity structure on M. We will say that ℱ is compact if any family {Aα}α∈Γ of elements of ℱ has a nonempty intersection provided ⋂α∈FAα≠∅ for any finite subset F⊂Γ.LetX be a complete CAT(0) space. We denote by 𝒞(X) the family of all closed convex subsets of X. Then, 𝒞(X) is a compact convexity structure on X (see, e.g., [2]).The following theorem is an extension of Theorem  4.3 in [33]. For an analog of this result in uniformly convex Banach spaces, see [34].Theorem 3.2.
LetC be a nonempty bounded closed and convex subset of a complete CAT(0) space X. Then, for any commuting family 𝒮 of asymptotic pointwise nonexpansive mappings on C, the set ℱ(𝒮) of common fixed points of 𝒮 is nonempty closed and convex.Proof.
Let𝒯 be the family of all finite intersections of the fixed point sets of mappings in the commutative family 𝒮. We first show that 𝒯 has the finite intersection property. Let T1,T2,…,Tn∈𝒮. By Theorem 2.4, F(T1) is a nonempty closed and convex subset of C. We assume that A∶=⋂j=1k-1F(Tj) is nonempty closed and convex for some k∈ℕ with 1<k≤n. For x∈A and j∈ℕ with 1≤j<k, we have
(3.1)Tk(x)=Tk∘Tj(x)=Tj∘Tk(x).
Thus, Tk(x) is a fixed point of Tj, which implies that Tk(x)∈A; therefore, A is invariant under Tk. Again, by Theorem 2.4, Tk has a fixed point in A, that is,
(3.2)⋂j=1kF(Tj)=F(Tk)⋂A≠∅.
By induction, ⋂j=1nF(Tj)≠∅. Hence, 𝒯 has the finite intersection property. Since 𝒞(X) is compact,
(3.3)F(S)=⋂T∈TT≠∅.
Obviously, the set is closed and convex.As a consequence of Lemma2.1(i) and Theorem 3.2, we obtain an analog of Bruck's theorem [35].Corollary 3.3.
LetC be a nonempty bounded closed and convex subset of a complete CAT(0) space X. Then, for any commuting family 𝒮 of nonexpansive mappings on C, the set ℱ(𝒮) of common fixed points of 𝒮 is a nonempty nonexpansive retract of C.


## 4. Convergence Theorems

Throughout this section,X stands for a complete CAT(0) space. Let C be a closed convex subset of X. We shall denote by 𝒯(C) the class of all asymptotic pointwise nonexpansive mappings from C into C. Let T1,…,Tm∈𝒯(C), without loss of generality, we can assume that there exists a sequence of mappings αn:C→[0,∞) such that for all x,y∈C,  i=1,…,m, and n∈ℕ, we have(4.1)d(Tinx,Tiny)≤αn(x)d(x,y),limsup⁡n→∞⁡αn(x)≤1.
Let an(x)=max{αn(x),1}. Again, without loss of generality, we can assume that(4.2)d(Tinx,Tiny)≤an(x)d(x,y),lim⁡n→∞an(x)=1,an(x)≥1,
for all x,y∈C,i=1,…,m, and n∈ℕ. We define bn(x)=an(x)-1, then, for each x∈C, we have limn→∞bn(x)=0.The following definition is a mild modification of [3, Definition  2.3].Definition 4.1.
Define𝒯r(C) as a class of all T∈𝒯(C) such that
(4.3)∑n=1∞sup⁡x∈C⁡bn(x)<∞,anisaboundedfunctionforeveryn∈ℕ.
LetT1,…,Tm∈𝒯r(C), and let {tik}k=1∞⊂(0,1) be bounded away from 0 and 1 for all i=1,2,…,m, and {nk} an increasing sequence of natural numbers. Let x1∈C, and define a sequence {xk} in C as(4.4)xk+1=(1-tmk)xk⊕tmkTmnky(m-1)k,y(m-1)k=(1-t(m-1)k)xk⊕t(m-1)kTm-1nky(m-2)k,y(m-2)k=(1-t(m-2)k)xk⊕t(m-2)kTm-2nky(m-3)k,⋮y2k=(1-t2k)xk⊕t2kT2nky1k,y1k=(1-t1k)xk⊕t1kT1nky0k,y0k=xk,k∈N.
We say that the sequence {xk} in (4.4) is well defined if limsupk→∞ank(xk)=1. As in [3], we observe that limk→∞ak(x)=1 for every x∈C. Hence, we can always choose a subsequence {ank} which makes {xk} well defined.Lemma 4.2 (see [36, Lemma  2.2]).
Let{an} and {un} be sequences of nonnegative real numbers satisfying
(4.5)an+1≤(1+un)an,∀n∈N,∑n=1∞un<∞.
Then, (i) limnan exists, (ii) if liminfnan=0, then limnan=0.Lemma 4.3 (see [37, 38]).
Suppose{tn} is a sequence in [b,c] for some b,c∈(0,1) and {un}, {vn} are sequences in X such that limsupnd(un,w)≤r,  limsupnd(vn,w)≤r, and limnd((1-tn)un⊕tnvn,w)=r for some r≥0. Then,
(4.6)lim⁡n→∞d(un,vn)=0.Lemma 4.4.
LetC be a nonempty closed convex subset of X and T1,…,Tm∈𝒯r(C). Let {tik}k=1∞⊂[a,b]⊂(0,1) and {nk}⊂ℕ be such that {xk} in (4.4) is well defined. Assume that F∶=⋂i=1mF(Ti)≠∅. Then,(a)
there exists a sequence{vk} in [0,∞) such that ∑k=1∞vk<∞ and d(xk+1,p)≤(1+vk)md(xk,p),   for all p∈F and all k∈ℕ,(b)
there exists a constantM>0 such that d(xk+l,p)≤Md(xk,p),   for all p∈F and k,l∈ℕ.Proof.
(a) Letp∈F and vk=supx∈Cbnk(x)  for all k∈ℕ. Since ∑k=1∞supx∈Cbnk(x)<∞, we have ∑k=1∞vk<∞. Now,
(4.7)d(y1k,p)≤(1-t1k)d(xk,p)+t1kd(T1nkxk,p)≤(1-t1k)d(xk,p)+t1k(1+bnk(p))d(xk,p)=(1+t1kbnk(p))d(xk,p)≤(1+vk)d(xk,p).
Suppose that d(yjk,p)≤(1+vk)jd(xk,p) holds for some 1≤j≤m-2. Then,
(4.8)d(y(j+1)k,p)≤(1-t(j+1)k)d(xk,p)+t(j+1)kd(Tj+1nkyjk,p)≤(1-t(j+1)k)d(xk,p)+t(j+1)k(1+bnk(p))d(yjk,p)≤(1-t(j+1)k)d(xk,p)+t(j+1)k(1+vk)j+1d(xk,p)=[1-t(j+1)k+t(j+1)k(1+∑r=1j+1(j+1)j⋯(j+2-r)r!vkr)]d(xk,p)=[1+t(j+1)k∑r=1j+1(j+1)j⋅⋅⋅(j+2-r)r!vkr]d(xk,p)≤(1+vk)j+1d(xk,p).
By induction, we have
(4.9)d(yik,p)≤(1+vk)id(xk,p),∀i=1,2,…,m-1.
This implies
(4.10)d(xk+1,p)≤(1-tmk)d(xk,p)+tmkd(Tmnky(m-1)k,p)≤(1-tmk)d(xk,p)+tmk(1+bnk(p))d(y(m-1)k,p)≤(1-tmk)d(xk,p)+tmk(1+vk)(1+vk)m-1d(xk,p)≤(1-tmk)d(xk,p)+tmk(1+vk)md(xk,p)=[1-tmk+tmk(1+∑r=1mm(m-1)⋯(m-r+1)r!vkr)]d(xk,p)=[1+tmk∑r=1mm(m-1)⋅⋅⋅(m-r+1)r!vkr]d(xk,p)≤(1+vk)md(xk,p).
This completes the proof of (a).
(b) We observe that(1+α)n≤enα holds for all n∈ℕ and α≥0. Thus, by (a), for k,l∈ℕ, we have
(4.11)d(xk+l,p)≤(1+vk+l-1)md(xk+l-1,p)≤exp⁡{mvk+l-1}d(xk+l-1,p)≤⋯≤exp⁡{m∑i=1k+l-1vi}d(xk,p)≤exp⁡{m∑i=1∞vi}d(xk,p).
The proof is complete by setting M=exp{m∑i=1∞vi}.Theorem 4.5.
LetC be a nonempty closed convex subset of X and T1,…,Tm∈𝒯r(C). Let {tik}k=1∞⊂[a,b]⊂(0,1) and {nk}⊂ℕ be such that {xk} in (4.4) is well defined. Assume that F≠∅. Then, {xk} converges to some point in F if and only if liminfk→∞d(xk,F)=0, where d(x,F)=infp∈Fd(x,p).Proof.
The necessity is obvious. Now, we prove the sufficiency. From Lemma4.4(a), we have
(4.12)d(xk+1,p)≤(1+vk)md(xk,p),∀p∈F,∀k∈N.
This implies
(4.13)d(xk+1,F)≤(1+vk)md(xk,F)=(1+∑r=1mm(m-1)⋅⋅⋅(m-r+1)r!vkr)d(xk,F).
Since ∑k=1∞vk<∞, then ∑k=1∞∑r=1m(m(m-1)···(m-r+1)/r!)vkr<∞. By Lemma 4.2(ii), we get limk→∞d(xk,F)=0. Next, we show that {xk} is a Cauchy sequence. From Lemma 4.4(b), there exists M>0 such that
(4.14)d(xk+l,p)≤Md(xk,p),∀p∈F,k,l∈N.
Since limk→∞d(xk,F)=0, for each ɛ>0, there exists k1∈ℕ such that
(4.15)d(xk,F)<ɛ2M,∀k≥k1.
Hence, there exists z1∈F such that
(4.16)d(xk1,z1)<ɛ2M.
By (4.14) and (4.16), for k≥k1, we have
(4.17)d(xk+l,xk)≤d(xk+l,z1)+d(xk,z1)≤Md(xk1,z1)+Md(xk1,z1)<2M(ɛ2M)=ɛ.
This shows that {xk} is a Cauchy sequence and so converges to some q∈C. We next show that q∈F. Let L=sup{a1(x):x∈C}. Then, for each ϵ>0, there exists k2∈ℕ such that
(4.18)d(xk,q)<ϵ2(1+L),∀k≥k2.
Since limk→∞d(xk,F)=0, there exists k3≥k2 such that
(4.19)d(xk,F)<ϵ2(1+L),∀k≥k3.
Thus, there exists z2∈F such that
(4.20)d(xk3,z2)<ϵ2(1+L).
By (4.18) and (4.20), for each i=1,2,…,m, we have
(4.21)d(Tiq,q)≤d(Tiq,Tixk3)+d(Tixk3,z2)+d(z2,xk3)+d(xk3,q)≤Ld(xk3,q)+Ld(xk3,z2)+d(xk3,z2)+d(xk3,q)≤(1+L)d(xk3,q)+(1+L)d(xk3,z2)<(1+L)ϵ2(1+L)+(1+L)ϵ2(1+L)=ϵ.
Since ϵ is arbitrary, we have Tiq=q for all i=1,2,…,m. Hence, q∈F.As an immediate consequence of Theorem4.5, we obtain the following.Corollary 4.6.
LetC be a nonempty closed convex subset of X and T1,…,Tm∈𝒯r(C). Let {tik}k=1∞⊂[a,b]⊂(0,1) and {nk}⊂ℕ be such that {xk} in (4.4) is well defined. Assume that F≠∅. Then, {xk} converges to a point p∈F if and only if there exists a subsequence {xkj} of {xk} which converges to p.Definition 4.7.
A strictly increasing sequence{nk}⊂ℕ is called quasiperiodic [39] if the sequence {nk+1-nk} is bounded or equivalently if there exists a number p∈ℕ such that any block of p consecutive natural numbers must contain a term of the sequence {nk}. The smallest of such numbers p will be called a quasiperiod of {nk}.Lemma 4.8.
LetC be a nonempty closed convex subset of X and T1,…,Tm∈𝒯r(C). Let {tik}k=1∞⊂[δ,1-δ] for some δ∈(0,1/2) and {nk}⊂ℕ be such that {xk} in (4.4) is well defined. Then,(i)
limk→∞d(xk,p) exists for all p∈F,(ii)
limk→∞d(xk,Tjnky(j-1)k)=0,   for all j=1,2,…,m,(iii)
if the set𝒥={k∈ℕ:nk+1=1+nk} is quasiperiodic, then limk→∞d(xk,Tjxk)=0,   for all j=1,2,…,m.Proof.
(i) Follows from Lemmas4.2(i) and 4.4(a).
(ii) Letp∈F, then, by (i), we have limk→∞d(xk,p) exists. Let
(4.22)lim⁡k→∞d(xk,p)=c.
By (4.9) and (4.22), we get that
(4.23)limsup⁡k→∞d(yjk,p)≤c,for1≤j≤m-1.
Note that
(4.24)d(xk+1,p)≤(1-tmk)d(xk,p)+tmkd(Tmnky(m-1)k,p)≤(1-tmk)d(xk,p)+tmk(1+vk)d(y(m-1)k,p)⋮≤(1-tmkt(m-1)k⋅⋅⋅t(j+1)k)(1+vk)m-jd(xk,p)+tmkt(m-1)k⋅⋅⋅t(j+1)k(1+vk)m-jd(yjk,p).
Thus,
(4.25)d(xk,p)≤d(xk,p)δm-j-d(xk+1,p)δm-j(1+vk)m-j+d(yjk,p),
so that
(4.26)c≤lim⁡inf⁡k→∞d(yjk,p),for1≤j≤m-1.
From (4.23) and (4.26), we have
(4.27)lim⁡k→∞d(yjk,p)=c,foreachj=1,2,…,m-1.
That is
(4.28)lim⁡k→∞d((1-tjk)xk⊕tjkTjnky(j-1)k,p)=c,
for each j=1,2,…,m-1.
We also obtain from (4.23) that
(4.29)limsup⁡k→∞d(Tjnky(j-1)k,p)≤c,foreachj=1,2,…,m-1.
By Lemma 4.3, we get that
(4.30)lim⁡k→∞d(Tjnky(j-1)k,xk)=0,foreachj=1,2,…,m-1.
For the case j=m, by (4.1), we have
(4.31)d(Tmnky(m-1)k,p)≤(1+bnk(p))d(y(m-1)k,p)≤(1+bnk(p))(1+vnk)m-1d(xk,p).
But since limk→∞d(xk,p)=c, then
(4.32)limsup⁡k→∞d(Tmnky(m-1)k,p)≤c.
Moreover,
(4.33)lim⁡k→∞d((1-tmk)xk⊕tmkTmnky(m-1)k,p)=lim⁡k→∞d(xk+1,p)=c.
Again, by Lemma 4.3, we get that
(4.34)lim⁡k→∞d(Tmnky(m-1)k,xk)=0.
Thus, (4.30) and (4.34) imply that
(4.35)lim⁡k→∞d(Tjnky(j-1)k,xk)=0,for eachj=1,2,…,m.(iii) For j=1, from (ii), we have(4.36)lim⁡k→∞d(T1nkxk,xk)=0.
If j=2,3,…,m, then we have
(4.37)d(Tjnkxk,xk)≤d(Tjnkxk,Tjnky(j-1)k)+d(Tjnky(j-1)k,xk)≤ank(xk)d(xk,y(j-1)k)+d(Tjnky(j-1)k,xk)≤ank(xk)t(j-1)kd(xk,Tj-1nky(j-2)k)+d(Tjnky(j-1)k,xk).
By (ii) and limsupk→∞ank(xk)=1, we get
(4.38)limsup⁡k→∞⁡d(Tjnkxk,xk)=0,forj=2,3,…,m.
By (4.36) and (4.38), we have
(4.39)lim⁡k→∞d(Tjnkxk,xk)=0,∀j=1,2,…,m.
By the construction of the sequence {xk}, we have from (4.35) that
(4.40)lim⁡k→∞d(xk+1,xk)=0.
Next, we show that
(4.41)lim⁡k→∞d(Tjxk,xk)=0,∀j=1,2,…,m.
It is enough to prove that d(Tjxk,xk)→0 as k→∞ though 𝒥. Indeed, let p be a quasiperiod of 𝒥, and let ɛ>0 be given. Then, there exists N1∈ℕ such that
(4.42)lim⁡k→∞d(Tjxk,xk)<ɛ3,∀k∈Jsuchthatk≥N1.
By the quasiperiodicity of 𝒥, for each l∈ℕ, there exists il∈𝒥 such that |l-il|≤p. Without loss of generality, we can assume that l≤il≤l+p (the proof for the other case is identical). Let M=sup{a1(x):x∈C}. Then, M≥1. Since liml→∞d(xl+1,xl)=0 by (4.40), there exists N2∈ℕ such that
(4.43)d(xl+1,xl)<ɛ3pM,∀l≥N2.
This implies that
(4.44)d(xil,xl)≤d(xil,xil-1)+⋯+d(xl+1,xl)≤p(ɛ3pM)=ɛ3M.
By the definition of T, we have
(4.45)d(Tjxil,Tjxl)≤Md(xil,xl)≤M(ɛ3M)=ɛ3.
LetN=max{N1,N2}. Then, for l≥N, we have from (4.42), (4.44), and (4.45) that
(4.46)d(xl,Tjxl)≤d(xl,xil)+d(xil,Tjxil)+d(Tjxil,Tjxl)<ɛ3M+ɛ3+ɛ3≤ɛ.
To prove that d(Tjxk,xk)→0 as k→∞ though 𝒥. Since 𝒥={k∈ℕ:nk+1=nk+1} is quasiperiodic, for each k∈𝒥, we have
(4.47)d(xk,Tjxk)≤d(xk,xk+1)+d(xk+1,Tjnk+1xk+1)+d(Tjnk+1xk+1,Tjnk+1xk)+d(Tjnk+1xk,Tjxk)≤d(xk,xk+1)+d(xk+1,Tjnk+1xk+1)+ank+1(xk+1)d(xk+1,xk)+a1(xk)d(Tjnkxk,xk).
From this, together with (4.39) and (4.40), we can obtain that d(Tjxk,xk)→0 as k→∞ through 𝒥.The following lemmas can be found in [3] (see also [2]).Lemma 4.9.
LetC be a nonempty closed convex subset of X, and let T∈𝒯r(C). If limn→∞d(xn,Txn)=0, then limn→∞d(xn,Tlxn)=0 for every l∈ℕ.Lemma 4.10.
LetC be a nonempty closed convex subset of X, and let T∈𝒯r(C). Suppose {xn} is a bounded sequence in C such that limnd(xn,Txn)=0 and Δ-limnxn=w. Then, Tw=w.By using Lemmas2.3 and 4.10, we can obtain the following result. We omit the proof because it is similar to the one given in [38].Lemma 4.11.
LetC be a closed convex subset of X, and let T:C→C be an asymptotic pointwise nonexpansive mapping. Suppose {xn} is a bounded sequence in C such that limnd(xn,T(xn))=0 and d(xn,v) converges for each v∈F(T), then ωw(xn)⊂F(T). Here, ωw(xn)=⋃A({un}) where the union is taken over all subsequences {un} of {xn}. Moreover, ωw(xn) consists of exactly one point.Now, we are ready to prove ourΔ-convergence theorem.Theorem 4.12.
LetC be a nonempty closed convex subset of X and T1,…,Tm∈𝒯r(C). Let {tik}k=1∞⊂[δ,1-δ] for some δ∈(0,1/2) and {nk}⊂ℕ be such that {xk} in (4.4) is well defined. Suppose that F∶=⋂i=1mF(Ti)≠∅ and the set 𝒥={k∈ℕ:nk+1=1+nk} is quasiperiodic. Then, {xk}Δ-converges to a common fixed point of the family {Ti:i=1,2,…,m}.Proof.
Letp∈F, by Lemma 4.8,  limk→∞d(xk,p) existsm and hence {xk} is bounded. Since limk→∞d(xk,Tjxk)=0 for all j=1,2,…,m, then by Lemma 4.11  ωw(xk)⊂F(Tj) for all j=1,2,…,m, and hence ωw(xk)⊂⋂j=1mF(Tj)=F. Since ωw(xn) consists of exactly one point, then {xk}Δ-converges to an element of F.Before proving our strong convergence theorem, we recall that a mappingT:C→C is said to be semicompact if C is closed and, for any bounded sequence {xn} in C with limn→∞d(xn,Txn)=0, there exists a subsequence {xnj} of {xn} and x∈C such that limk→∞xnk=x.Theorem 4.13.
LetC be a nonempty closed convex subset of X and T1,…,Tm∈𝒯r(C) such that Til is semicompact for some i∈{1,…,m} and l∈ℕ. Let {tik}k=1∞⊂[δ,1-δ] for some δ∈(0,1/2) and {nk}⊂ℕ be such that {xk} in (4.4) is well defined. Suppose that F∶=⋂i=1mF(Ti)≠∅ and the set 𝒥={k∈ℕ:nk+1=1+nk} is quasiperiodic. Then, {xk} converges to a common fixed point of the family {Ti:i=1,2,…,m}.Proof.
By Lemma4.8, we have
(4.48)lim⁡k→∞d(xk,Tixk)=0,fori=1,…,m.
Let i∈{1,…,m} be such that Til is semicompact. Thus, by Lemma 4.9,
(4.49)lim⁡k→∞d(xk,Tilxk)=0.
We can also find a subsequence {xnj} of {xk} such that limj→∞xkj=q∈C. Hence, from (4.48), we have
(4.50)d(q,Tiq)=lim⁡j→∞d(xkj,Tixkj)=0,∀i=1,…,m.
Thus, q∈F, and, by Corollary 4.6, {xk} converges to q. This completes the proof.


## 5. Concluding Remarks

One may observe that our method can be used to obtain the analogous results for uniformly convex Banach spaces. LetC be a nonempty closed convex subset of a Banach space X and fix x1∈C. Define a sequence {xk} in C as(5.1)xk+1=(1-tmk)xk+tmkTmnky(m-1)k,y(m-1)k=(1-t(m-1)k)xk+t(m-1)kTm-1nky(m-2)k,y(m-2)k=(1-t(m-2)k)xk+t(m-2)kTm-2nky(m-3)k,⋮y2k=(1-t2k)xk+t2kT2nky1k,y1k=(1-t1k)xk+t1kT1nky0k,y0k=xk,k∈N,
where T1,…,Tm∈𝒯r(C),  {tik}k=1∞ are sequences in [0,1] for all i=1,2,…,m, and {nk} is an increasing sequence of natural numbers.Theorem 5.1.
LetX be a uniformly convex Banach space with the Opial property, and let C be a nonempty closed convex subset of X. Let T1,…,Tm∈𝒯r(C),  {tik}k=1∞⊂[δ,1-δ] for some δ∈(0,1/2), and let {nk}⊂ℕ be such that {xk} in (5.1) is well defined. Suppose that F∶=⋂i=1mF(Ti)≠∅ and the set 𝒥={k∈ℕ:nk+1=1+nk} is quasiperiodic. Then, {xk} converges weakly to a common fixed point of the family {Ti:i=1,2,…,m}.Theorem 5.2.
LetC be a nonempty closed convex subset of a uniformly convex Banach space X and T1,…,Tm∈𝒯r(C) such that Til is semicompact for some i∈{1,…,m} and l∈ℕ. Let {tik}k=1∞⊂[δ,1-δ] for some δ∈(0,1/2), and let {nk}⊂ℕ be such that {xk} in (5.1) is well defined. Suppose that F∶=⋂i=1mF(Ti)≠∅ and the set 𝒥={k∈ℕ:nk+1=1+nk} is quasiperiodic. Then, {xk} converges strongly to a common fixed point of the family {Ti:i=1,2,…,m}.



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