Let C 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.

1. Introduction

A mapping T 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‖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 andxk+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 byxn+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 byxk+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 space X 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.

A geodesic triangle ▵(x1,x2,x3) in a geodesic space (X,d) consists of three points x1,x2,x3 in X (the vertices 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¯∈▵¯,
d(x,y)≤dE2(x¯,y¯).

Let x,y∈X, by Lemma 2.1(iv) of [14] for each t∈[0,1], there exists a unique point z∈[x,y] such thatd(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.

Let X be a complete CAT(0) space.

[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.

[14, Lemma 2.4] For x,y,z∈X and t∈[0,1], we have
d((1-t)x⊕ty,z)≤(1-t)d(x,z)+td(y,z).

[14, Lemma 2.5] For x,y,z∈X and t∈[0,1], we have
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 of Δ-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
r(x,{xn})=limsupn→∞d(x,xn).
The asymptotic radius r({xn}) of {xn} is given by
r({xn})=inf{r(x,{xn}):x∈X},
and the asymptotic center A({xn}) of {xn} is the set
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 [<xref ref-type="bibr" rid="B27">28</xref>, <xref ref-type="bibr" rid="B31">29</xref>]).

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.

Let X be a complete CAT(0) space.

[28, page 3690] Every bounded sequence in X has a Δ-convergent subsequence.

[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.

[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 mapping T:X→X is said to be nonexpansive [31] if
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 thatd(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 thatd(Tnx,Tny)≤αn(x)d(x,y),∀n≥1,x,y∈X,
where limsupn→∞αn(x)≤1. The following implications hold. 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.

Let C 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

Let M 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 [<xref ref-type="bibr" rid="B17">2</xref>]).

Let ℱ 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.

Let X 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.

Let C 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.

Let 𝒯 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
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,
⋂j=1kF(Tj)=F(Tk)⋂A≠∅.
By induction, ⋂j=1nF(Tj)≠∅. Hence, 𝒯 has the finite intersection property. Since 𝒞(X) is compact,
F(S)=⋂T∈TT≠∅.
Obviously, the set is closed and convex.

As a consequence of Lemma 2.1(i) and Theorem 3.2, we obtain an analog of Bruck's theorem [35].

Corollary 3.3.

Let C 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 haved(Tinx,Tiny)≤αn(x)d(x,y),limsupn→∞αn(x)≤1.
Let an(x)=max{αn(x),1}. Again, without loss of generality, we can assume thatd(Tinx,Tiny)≤an(x)d(x,y),limn→∞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.

Define 𝒯r(C) as a class of all T∈𝒯(C) such that
∑n=1∞supx∈Cbn(x)<∞,anisaboundedfunctionforeveryn∈ℕ.

Let T1,…,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

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 [<xref ref-type="bibr" rid="B40">36</xref>, Lemma 2.2]).

Let {an} and {un} be sequences of nonnegative real numbers satisfying
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 [<xref ref-type="bibr" rid="B18">37</xref>, <xref ref-type="bibr" rid="B32">38</xref>]).

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,
limn→∞d(un,vn)=0.

Lemma 4.4.

Let C 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,

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,

there exists a constant M>0 such that d(xk+l,p)≤Md(xk,p), for all p∈F and k,l∈ℕ.

Proof.

(a) Let p∈F and vk=supx∈Cbnk(x) for all k∈ℕ. Since ∑k=1∞supx∈Cbnk(x)<∞, we have ∑k=1∞vk<∞. Now,
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,
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
d(yik,p)≤(1+vk)id(xk,p),∀i=1,2,…,m-1.
This implies
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
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.

Let C 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 Lemma 4.4(a), we have
d(xk+1,p)≤(1+vk)md(xk,p),∀p∈F,∀k∈N.
This implies
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
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
d(xk,F)<ɛ2M,∀k≥k1.
Hence, there exists z1∈F such that
d(xk1,z1)<ɛ2M.
By (4.14) and (4.16), for k≥k1, we have
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
d(xk,q)<ϵ2(1+L),∀k≥k2.
Since limk→∞d(xk,F)=0, there exists k3≥k2 such that
d(xk,F)<ϵ2(1+L),∀k≥k3.
Thus, there exists z2∈F such that
d(xk3,z2)<ϵ2(1+L).
By (4.18) and (4.20), for each i=1,2,…,m, we have
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 Theorem 4.5, we obtain the following.

Corollary 4.6.

Let C 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.

Let C 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,

limk→∞d(xk,p) exists for all p∈F,

limk→∞d(xk,Tjnky(j-1)k)=0, for all j=1,2,…,m,

if the set 𝒥={k∈ℕ:nk+1=1+nk} is quasiperiodic, then limk→∞d(xk,Tjxk)=0, for all j=1,2,…,m.

Proof.

(i) Follows from Lemmas 4.2(i) and 4.4(a).

(ii) Let p∈F, then, by (i), we have limk→∞d(xk,p) exists. Let
limk→∞d(xk,p)=c.
By (4.9) and (4.22), we get that
limsupk→∞d(yjk,p)≤c,for1≤j≤m-1.
Note that
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,
d(xk,p)≤d(xk,p)δm-j-d(xk+1,p)δm-j(1+vk)m-j+d(yjk,p),
so that
c≤liminfk→∞d(yjk,p),for1≤j≤m-1.
From (4.23) and (4.26), we have
limk→∞d(yjk,p)=c,foreachj=1,2,…,m-1.
That is
limk→∞d((1-tjk)xk⊕tjkTjnky(j-1)k,p)=c,
for each j=1,2,…,m-1.

We also obtain from (4.23) that
limsupk→∞d(Tjnky(j-1)k,p)≤c,foreachj=1,2,…,m-1.
By Lemma 4.3, we get that
limk→∞d(Tjnky(j-1)k,xk)=0,foreachj=1,2,…,m-1.
For the case j=m, by (4.1), we have
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
limsupk→∞d(Tmnky(m-1)k,p)≤c.
Moreover,
limk→∞d((1-tmk)xk⊕tmkTmnky(m-1)k,p)=limk→∞d(xk+1,p)=c.
Again, by Lemma 4.3, we get that
limk→∞d(Tmnky(m-1)k,xk)=0.
Thus, (4.30) and (4.34) imply that
limk→∞d(Tjnky(j-1)k,xk)=0,for eachj=1,2,…,m.(iii) For j=1, from (ii), we havelimk→∞d(T1nkxk,xk)=0.
If j=2,3,…,m, then we have
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
limsupk→∞d(Tjnkxk,xk)=0,forj=2,3,…,m.
By (4.36) and (4.38), we have
limk→∞d(Tjnkxk,xk)=0,∀j=1,2,…,m.
By the construction of the sequence {xk}, we have from (4.35) that
limk→∞d(xk+1,xk)=0.
Next, we show that
limk→∞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
limk→∞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
d(xl+1,xl)<ɛ3pM,∀l≥N2.
This implies that
d(xil,xl)≤d(xil,xil-1)+⋯+d(xl+1,xl)≤p(ɛ3pM)=ɛ3M.
By the definition of T, we have
d(Tjxil,Tjxl)≤Md(xil,xl)≤M(ɛ3M)=ɛ3.

Let N=max{N1,N2}. Then, for l≥N, we have from (4.42), (4.44), and (4.45) that
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
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.

Let C 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.

Let C 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 Lemmas 2.3 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.

Let C 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 our Δ-convergence theorem.

Theorem 4.12.

Let C 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.

Let p∈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 4.11ωw(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 mapping T: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.

Let C 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 Lemma 4.8, we have
limk→∞d(xk,Tixk)=0,fori=1,…,m.
Let i∈{1,…,m} be such that Til is semicompact. Thus, by Lemma 4.9,
limk→∞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
d(q,Tiq)=limj→∞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. Let C be a nonempty closed convex subset of a Banach space X and fix x1∈C. Define a sequence {xk} in C asxk+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.

Let X 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.

Let C 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}.

Acknowledgments

This research is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The first author also thanks the Graduate School of Chiang Mai University, Thailand.

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