New △-convergence theorems of iterative sequences for asymptotically nonexpansive mappings in CAT(0) spaces are obtained. Consider an asymptotically nonexpansive self-mapping T of a closed convex subset C of a CAT(0) space X. Consider the iteration process {xn}, where x0∈C is arbitrary and xn+1=αnxn⊕(1-αn)Tnyn or xn+1=αnTnxn⊕(1-αn)yn,yn=βnxn⊕(1-βn)Tnxn for n≥1, where {αn},{βn}⊂(0,1). It is shown that under certain appropriate conditions on αn,βn,{xn} △-converges to a fixed point of T.

1. Introduction and Preliminaries

Let C be a nonempty subset of a metric space (X,d). A mapping T:C→C is a contraction if there exists k∈[0,1) such that for all x,y∈C, we have d(Tx,Ty)<kd(x,y). It is said to be nonexpansive if for all x,y∈C, we have d(Tx,Ty)≤d(x,y). T is said to be asymptotically nonexpansive if there exists a sequence {kn}∈[1,∞) with kn→1 such that d(Tnx,Tny)≤knd(x,y) for all integers n≥1 and all x,y∈C. Clearly, every contraction mapping is nonexpansive and every nonexpansive mapping is asymptotically nonexpansive with sequence kn=1, for all n≥1. There are, however, asymptotically nonexpansive mappings which are not nonexpansive (see, e.g., [1]). As a generalization of the class of nonexpansive mappings, the class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [2] in 1972 and has been studied by several authors (see, e.g., [3–5]). Goebel and Kirk proved that if C is a nonempty closed convex and bounded subset of a uniformly convex Banach space (more general than a Hilbert space, i.e., CAT(0) space), then every asymptotically nonexpansive self-mapping of C has a fixed point. The weak and strong convergence problems to fixed points of nonexpansive and asymptotically nonexpansive mappings have been studied by many authors.

We will denote by F(T) the set of fixed points of T. In 1967, Halpern [6] introduced an explicit iterative scheme for a nonexpansive mapping T on a subset C of a Hilbert space by taking any point u,x1∈C and defined the iterative sequence {xn} by
(1)xn+1=αnu+(1-αn)Txn,forn≥1,
where αn∈[0,1]. He pointed out that under certain appropriate conditions on αn,{xn} converges strongly to a fixed point of T. In 1994, Tan and Xu [7] introduced the following iterative scheme for asymptotically nonexpansive mapping on uniformly convex Banach space:
(2)x0∈C,xn+1=αnf(xn)+(1-αn)Tnyn,n≥0,yn=γnxn+(1-γn)Tnxn,n≥0,
where {αn},{γn}⊆(0,1). They proved that under certain appropriate conditions on αn,γn,{xn} converges weakly to a fixed point of T.

In 2012, we [8] studied the viscosity approximation methods for nonexpansive mappings on CAT(0) space. For a contraction f on C, consider the iteration process {xn}, where x0∈C is arbitrary and
(3)xn+1=αnf(xn)⊕(1-αn)Txn,
for n≥1, where {αn}⊂(0,1). We proved that under certain appropriate conditions on αn,{xn} converges strongly to a fixed point of T which solves some variational inequality.

The purpose of this paper is to study the iterative scheme defined as follows: consider an asymptotically nonexpansive self-mapping T of a closed convex subset C of a CAT(0) space X with coefficient kn. consider the iteration process {xn}, where x0∈C is arbitrary and
(4)xn+1=αnxn⊕(1-αn)Tnyn,yn=βnxn⊕(1-βn)Tnxn,
or
(5)xn+1=αnTnxn⊕(1-αn)yn,yn=βnxn⊕(1-βn)Tnxn,
for n≥1, where {αn},{βn}⊂(0,1). We show that {xn}△-converges to a fixed point of T under certain appropriate conditions on αn,βn, and kn.

We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results.

Lemma 1.

Let X be a CAT(0) space. Then, one has the following:

(see [9, Lemma 2.4]) for each x,y,z∈X and t∈[0,1], one has
(6)d((1-t)x⊕ty,z)≤(1-t)d(x,z)+td(y,z),

(see [10]) for each x,y,z∈X and t,s∈[0,1] one has
(7)d((1-t)x⊕ty,(1-s)x⊕sy)≤|t-s|d(x,y),

(see [5, Lemma 3]) for each x,y,z∈X and t∈[0,1], one has
(8)d((1-t)z⊕tx,(1-t)z⊕ty)≤td(x,y),

(see [9]) for each x,y,z∈X and t∈[0,1], one has
(9)d2((1-t)x⊕ty,z)≤td2(x,z)+(1-t)d2(y,z)-t(1-t)d2(x,y).

Let X be a complete CAT(0) space and let {xn} be a bounded sequence in a complete X and for x∈X set
(10)r(x,{xn})=limsupn→∞d(x,xn).

The asymptotic radius r({xn}) of {xn} is given by
(11)r({xn})=inf{r(x,{xn}):x∈X},
and the asymptotic center A({xn}) of {xn} is the set
(12)A({xn})={x∈X:r(x,{xn})=r({xn})}.

It is known (see, e.g., [11, Proposition 7]) that in a CAT(0) space, A({xn}) consists of exactly one point.

A sequence {xn} in X is said to △-converge to x∈X if x is the unique asymptotic center of {un} for every subsequence {un} of {xn}. In this case, we write △-limnxn=x and call x the △-limit of {xn}.

Lemma 2.

Assume that X is a CAT(0) space. Then, one has the following:

(see [12]) every bounded sequence in X has a △-convergent subsequence;

(see [13]) if K is a closed convex subset of X and T:K→X is an asymptotically nonexpansive mapping, then the conditions {xn}△-converge to x and d(xn,T(xn))→0, imply x∈K and x∈F(T).

Lemma 3 (see [<xref ref-type="bibr" rid="B3">14</xref>, <xref ref-type="bibr" rid="B16">15</xref>]).

Let {an},{bn},and{cn} be three nonnegative real sequences satisfying the following condition:
(13)an+1≤(1+bn)an+cn,∀n≥n0,
where n0 is some nonnegative integer, ∑n=1∞bn<∞, ∑n=1∞cn<∞. Then the limit limn→∞an exists.

2. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M125"><mml:mrow><mml:mo>△</mml:mo></mml:mrow></mml:math></inline-formula>-Convergence of the Iteration Sequences

In this section, we will study the △-convergence of the iteration sequence for asymptotically nonexpansive mappings in CAT(0) spaces.

Suppose that X be a CAT(0) space, C a closed convex subset of X, and T:C→C an asymptotically nonexpansive mapping with coefficient kn. Firstly, we consider the iteration process:
(14)x0∈C,xn+1=αnxn⊕(1-αn)Tnyn,n≥0,yn=βnxn⊕(1-βn)Tnxn,n≥0,
where {αn},{βn}⊆(0,1) and kn satisfy the following.

There exist positive integers n0, n1, and δ>0, 0<b<min{1,1/L}, where L=supnkn, such that
(15)0<δ<αn<1-δ,n≥n0,0<1-βn<b,n≥n1,

Consider ∑n=1∞(kn2-1)<∞.

We will prove that {xn}△-converges to a fixed point of T.

Lemma 4.

Let X be a CAT(0) space, C a closed convex subset of X, T:C→C an asymptotically nonexpansive mapping with coefficient kn, and ∑n=1∞(kn2-1)<∞. If F(T)≠∅, {αn},{βn}⊆(0,1). Let x0∈C, {xn} be generated by xn+1=αnxn⊕(1-αn)Tnyn, yn=βnxn⊕(1-βn)Tnxn, n≥0. Then the limit limn→∞d(xn,p) exists for all p∈F(T).

Proof.

Taking p∈F(T), we have
(16)d(xn+1,p)=d(αnxn⊕(1-αn)Tnyn,p)≤αnd(xn,p)+(1-αn)d(Tnyn,p)≤αnd(xn,p)+(1-αn)knd(yn,p)≤αnd(xn,p)+(1-αn)kn{βnd(xn,p)+(1-βn)d(Tnxn,p)}≤αnd(xn,p)+(1-αn)kn{βnd(xn,p)+(1-βn)knd(xn,p)}={1+(1-αn)(kn-1)×[kn(1-βn)+1]}d(xn,p)≤{1+(kn2-1)}d(xn,p).
By Lemma 3, we can get that limn→∞d(xn,p) exists.

Remark 5.

The above lemma implies that {xn} is bounded and so is the sequence {Txn}. Moreover, let L=supnkn, then we have
(17)d(Tnxn,p)≤knd(xn,p)≤Ld(xn,p),d(yn,p)≤βnd(xn,p)+(1-βn)d(Tnxn,p)≤Ld(xn,p)d(Tnyn,p)≤knd(yn,p)≤L2d(xn,p).
It follows that the sequences {Tnxn}, {yn}, {Tnyn} are bounded.

Proposition 6.

Let X be a CAT(0) space, C a closed convex subset of X, and T:C→C an asymptotically nonexpansive mapping with coefficient kn. If F(T)≠∅, {αn},{βn}⊆(0,1). Let x0∈C, {xn} be generated by xn+1=αnxn⊕(1-αn)Tnyn, yn=βnxn⊕(1-βn)Tnxn, n≥0. Then under the hypotheses (i) and (ii), one can get that limn→∞d(xn,Tnyn)=0.

Proof.

By the assumption, F(T) is nonempty. Take p∈F(T), by Lemma 1(iv), we have
(18)d2(xn+1,p)=d2(αnxn⊕(1-αn)Tnyn,p)≤αnd2(xn,p)+(1-αn)d2(Tnyn,p)-αn(1-αn)d2(xn,Tnyn)≤d2(xn,p)+(1-αn){d2(Tnyn,p)-d2(yn,p)}+(1-αn){d2(yn,p)-d2(xn,p)}-αn(1-αn)d2(xn,Tnyn),d2(yn,p)-d2(xn,p)=d2(βnxn⊕(1-βn)Tnxn,p)-d2(xn,p)≤βnd2(xn,p)+(1-βn)d2(Tnxn,p)-βn(1-βn)d2(xn,Tnxn)-d2(xn,p)≤βnd2(xn,p)+(1-βn)d2(Tnxn,p)-d2(xn,p),
which implies that
(19)d2(yn,p)-d2(xn,p)≤(1-βn)[d2(Tnxn,p)-d2(xn,p)]≤(1-βn)(kn2-1)d2(xn,p).

Therefore, we have
(20)d2(xn+1,p)≤d2(xn,p)+(1-αn)(kn2-1)d2(yn,p)+(1-αn)(1-βn)(kn2-1)d2(xn,p)-αn(1-αn)d2(xn,Tnyn).

Since {xn} and {yn} are bounded and 0<δ<αn<1-δ for all n≥n0. we have
(21)δ2d2(xn,Tnyn)≤d2(xn,p)-d2(xn+1,p)+(1-αn)(kn2-1)d2(yn,p)+(1-αn)(1-βn)(kn2-1)d2(xn,p).

By the conditions (i) and (ii), we have
(22)∑n=1∞δ2d2(xn,Tnyn)<∞,
which implies that
(23)limn→∞d2(xn,Tnyn)=0.

Theorem 7.

Let X be a CAT(0) space, C a closed convex subset of X, and T:C→C an asymptotically nonexpansive mapping with coefficient kn. If F(T)≠∅, {αn},{βn}⊆(0,1). Let x0∈C, {xn} be generated by xn+1=αnxn⊕(1-αn)Tnyn,yn=βnxn⊕(1-βn)Tnxn,n≥0. Then under the hypotheses (i) and (ii), one can get that {xn}△-converges to a fix point of T.

Proof.

We first show that limn→∞d(xn,Tnxn)=0. Indeed
(24)d(xn,yn)=d(xn,βnxn⊕(1-βn)Tnxn)≤(1-βn)d(xn,Tnxn)≤(1-βn){d(xn,Tnyn)+d(Tnyn,Tnxn)}≤(1-βn){d(xn,Tnyn)+Ld(yn,xn)};
it follows that
(25)[1-L(1-βn)]d(xn,yn)≤(1-βn)d(xn,Tnyn).
By the conditions (i) and (ii) and Proposition 6, we get limn→∞d(xn,yn)=0.

And then,
(26)d(xn,Tnxn)≤d(xn,Tnyn)+d(Tnyn,Tnxn)≤d(xn,Tnyn)+Ld(yn,xn).
By Proposition 6, we get that limn→∞d(xn,Tnxn)=0.

We claim that limn→∞d(xn,Txn)=0. Indeed we have
(27)d(yn,Tnxn)=d(βnxn⊕(1-βn)Tnxn,Tnxn)≤βnd(xn,Tnxn)→0.d(xn+1,xn)=d(αnxn⊕(1-αn)Tnyn,xn)≤(1-αn)d(xn,Tnyn)→0.d(xn-1,Tn-1xn)≤d(xn-1,Tn-1xn-1)+d(Tn-1xn-1,Tn-1xn)≤d(xn-1,Tn-1xn-1)+Ld(xn-1,xn)→0.d(xn,Tn-1xn)≤d((1-αn-1)Tn-1yn-1,Tn-1xnαn-1xn-1⊕(1-αn-1)Tn-1yn-1,Tn-1xn)≤αn-1d(xn-1,Tn-1xn)+(1-αn-1)d(Tn-1yn-1,Tn-1xn)≤αn-1d(xn-1,Tn-1xn)+(1-αn-1)Ld(yn-1,xn)≤αn-1d(xn-1,Tn-1xn)+(1-αn-1)L[d(yn-1,xn-1)+d(xn-1,xn)]→0.
Thus,
(28)d(xn,Txn)≤d(xn,Tnxn)+d(Tnxn,Txn)≤d(xn,Tnxn)+Ld(Tn-1xn,xn)→0.

Since {xn} is bounded, we may assume that {xn}△-converges to a point x^. By Lemma 2, we have x^∈F(T).

Next we will consider another iteration process:
(29)x0∈C,xn+1=αnTnxn⊕(1-αn)yn,n≥0,yn=βnxn⊕(1-βn)Tnxn,n≥0,
where {αn},{βn}⊆(0,1), and kn satisfy the following

There exist positive integers n0 and δ>0, such that
(30)0<δ<αn<1-δ,n≥n0;1-βn→0;

∑n=1∞(kn-1)<∞.

We will prove that {xn} also △-converges to a fixed point of T.

Lemma 8.

Let X be a CAT(0) space, C a closed convex subset of X, T:C→C an asymptotically nonexpansive mapping with coefficient kn, and ∑n=1∞(kn-1)<∞. If F(T)≠∅, {αn},{βn}⊆(0,1). Let x0∈C, {xn} be generated by xn+1=αnTnxn⊕(1-αn)yn,yn=βnxn⊕(1-βn)Tnxn,n≥0. Then the limit limn→∞d(xn,p) exists for all p∈F(T).

Proof.

Taking p∈F(T), we have
(31)d(xn+1,p)=d(αnTnxn⊕(1-αn)yn,p)≤αnknd(xn,p)+(1-αn)d(yn,p)≤αnknd(xn,p)+(1-αn){βnd(xn,p)+(1-βn)d(Tnxn,p)}≤αnknd(xn,p)+(1-αn){βnd(xn,p)+(1-βn)knd(xn,p)}={1+(kn-1)[1-(1-αn)βn]}d(xn,p).
By Lemma 3, we can get that limn→∞d(xn,p) exists.

Next, we will prove limn→∞d(Tnxn,yn)=0.

Proposition 9.

Let X be a CAT(0) space, C a closed convex subset of X, and T:C→C an asymptotically nonexpansive mapping with coefficient kn. If F(T)≠∅, {αn},{βn}⊆(0,1). Let x0∈C, {xn} be generated by xn+1=αnTnxn⊕(1-αn)yn,yn=βnxn⊕(1-βn)Tnxn,n≥0. Then under the hypotheses (H1) and (H2), one can get that limn→∞d(Tnxn,yn)=0.

Proof.

By the assumption, F(T) is nonempty. Take p∈F(T), let L=supnkn, then we have
(32)d(Tnxn,p)≤knd(xn,p)≤Ld(xn,p),d(yn,p)≤βnd(xn,p)+(1-βn)d(Tnxn,p)≤Ld(xn,p)d(Tnyn,p)≤knd(yn,p)≤L2d(xn,p).
It follows that the sequences {xn},{Tnxn},{yn},{Tnyn} are bounded.

By Lemma 1, we have
(33)d2(xn+1,p)=d2(αnTnxn⊕(1-αn)yn,p)≤αnkn2d2(xn,p)+(1-αn)d2(yn,p)-αn(1-αn)d2(Tnxn,yn)≤d2(xn,p)+(1-αn){d2(yn,p)-d2(xn,p)}+αn(kn2-1)d2(xn,p)-αn(1-αn)d2(Tnxn,yn).

Similar to the proof of Proposition 6, we can get
(34)d2(yn,p)-d2(xn,p)≤(1-βn)(kn2-1)d2(xn,p).

Therefore, we have
(35)d2(xn+1,p)≤d2(xn,p)+(1-αn)(1-βn)×(kn2-1)d2(xn,p)+αn(kn2-1)d2(xn,p)-αn(1-αn)d2(Tnxn,yn).
Since {xn},{yn} are bounded and 0<δ<αn<1-δ for all n≥n0. we have
(36)δ2d2(Tnxn,yn)≤d2(xn,p)-d2(xn+1,p)+(1-αn)(1-βn)(kn2-1)d2(xn,p)+αn(kn2-1)d2(xn,p).

By the conditions (H1) and (H2), we have ∑n=1∞(kn2-1)<∞ and
(37)∑n=1∞δ2d2(Tnxn,yn)<∞,
which implies that
(38)limn→∞d2(Tnxn,yn)=0.

Theorem 10.

Let X be a CAT(0) space, C a closed convex subset of X, and T:C→C an asymptotically nonexpansive mapping with coefficient kn. If F(T)≠∅, {αn},{βn}⊆(0,1). Let x0∈C, {xn} be generated by xn+1=αnTnxn⊕(1-αn)yn,yn=βnxn⊕(1-βn)Tnxn,n≥0. Then under the hypotheses (H1) and (H2), one can get that {xn}△-converges to a fix point of T.

Proof.

We first show that limn→∞d(xn,Tnxn)=0. Indeed, by Lemma 1, and βn→1, we can get
(39)d(xn,yn)=d(xn,βnxn⊕(1-βn)Tnxn)≤(1-βn)d(xn,Tnxn)→0.
And then,
(40)d(xn,Tnxn)≤d(xn,yn)+d(yn,Tnxn).
By Proposition 9, we obtain that limn→∞d(xn,Tnxn)=0.

We claim that limn→∞d(xn,Txn)=0. Indeed we have
(41)d(xn+1,xn)=d(αnTnxn⊕(1-αn)yn,xn)≤αnd(Tnxn,xn)+(1-αn)d(xn,yn)→0.d(xn,Tn-1xn)≤d(αn-1Tn-1xn-1⊕(1-αn-1)yn-1,Tn-1xn)≤αn-1d(Tn-1xn-1,Tn-1xn)+(1-αn-1)d(yn-1,Tn-1xn)≤αn-1kn-1d(xn-1,xn)+(1-αn-1)[d(yn-1,Tn-1xn-1)+d(Tn-1xn-1,Tn-1xn)]≤αn-1kn-1d(xn-1,xn)+(1-αn-1)[d(yn-1,Tn-1xn-1)+kn-1d(xn-1,xn)(yn-1,Tn-1xn-1)]→0.
Thus,
(42)d(xn,Txn)≤d(xn,Tnxn)+d(Tnxn,Txn)≤d(xn,Tnxn)+Ld(Tn-1xn,xn)→0.

Since {xn} is bounded, we may assume that {xn}△-converges to a point x^. By Lemma 2, we have x^∈F(T).

Acknowledgment

This research was supported by NSFC Grants nos. 11071279 and 11226125.

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