 Research
 Open Access
 Published:
On total asymptotically nonexpansive mappings in CAT(\kappa ) spaces
Journal of Inequalities and Applications volume 2014, Article number: 336 (2014)
Abstract
In this article, we obtain the demiclosed principle, fixed point theorems and convergence theorems for the class of total asymptotically nonexpansive mappings on CAT(\kappa ) spaces with \kappa >0. Our results generalize the results of Chang et al. (Appl. Math. Comput. 219:26112617, 2012), Tang et al. (Abstr. Appl. Anal. 2012:965751, 2012), Karapınar et al. (J. Appl. Math. 2014:738150, 2014) and many others.
1 Introduction
For a real number κ, a CAT(\kappa ) space is a geodesic metric space whose geodesic triangle is thinner than the corresponding comparison triangle in a model space with curvature κ. The precise definition is given below. The letters C, A, and T stand for Cartan, Alexandrov, and Toponogov, who have made important contributions to the understanding of curvature via inequalities for the distance function.
Fixed point theory in CAT(\kappa ) spaces was first studied by Kirk [1, 2]. His works were followed by a series of new works by many authors, mainly focusing on CAT(0) spaces (see, e.g., [3–11]). Since any CAT(\kappa ) space is a CAT({\kappa}^{\prime}) space for {\kappa}^{\prime}\ge \kappa, all results for CAT(0) spaces immediately apply to any CAT(\kappa ) space with \kappa \le 0. However, there are only a few articles that contain fixed point results in the setting of CAT(\kappa ) spaces with \kappa >0.
The concept of total asymptotically nonexpansive mappings was first introduced in Banach spaces by Alber et al. [12]. It generalizes the concept of asymptotically nonexpansive mappings introduced by Goebel and Kirk [13] as well as the concept of nearly asymptotically nonexpansive mappings introduced by Sahu [14]. In 2012, Chang et al. [15] studied the demiclosed principle and Δconvergence theorems for total asymptotically nonexpansive mappings in the setting of CAT(0) spaces. Since then the convergence of several iteration procedures for this type of mappings has been rapidly developed and many of articles have appeared (see, e.g., [16–24]). Among other things, under some suitable assumptions, Karapınar et al. [24] obtained the demiclosed principle, fixed point theorems, and convergence theorems for the following iteration.
Let K be a nonempty closed convex subset of a CAT(0) space X and T:K\to K be a total asymptotically nonexpansive mapping. Given {x}_{1}\in K, and let \{{x}_{n}\}\subseteq K be defined by
where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are sequences in [0,1].
In this article, we extend Karapınar et al.’s results to the general setting of CAT(\kappa ) space with \kappa >0.
2 Preliminaries
Let (X,\rho ) be a metric space. A geodesic path joining x\in X to y\in X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0,l]\subset \mathbb{R} to X such that c(0)=x, c(l)=y, and \rho (c(t),c({t}^{\prime}))=t{t}^{\prime} for all t,{t}^{\prime}\in [0,l]. In particular, c is an isometry and \rho (x,y)=l. The image c([0,l]) of c is called a geodesic segment joining x and y. When it is unique, this geodesic segment is denoted by [x,y]. This means that z\in [x,y] if and only if there exists \alpha \in [0,1] such that
In this case, we write z=\alpha x\oplus (1\alpha )y. The space (X,\rho ) is said to be a geodesic space (Dgeodesic space) if every two points of X (every two points of distance smaller than D) are joined by a geodesic, and X is said to be uniquely geodesic (Duniquely geodesic) if there is exactly one geodesic joining x and y for each x,y\in X (for x,y\in X with \rho (x,y)<D). A subset K of X is said to be convex if K includes every geodesic segment joining any two of its points. The set K is said to be bounded if
Now we introduce the model spaces {M}_{\kappa}^{n}, for more details on these spaces the reader is referred to [25]. Let n\in \mathbb{N}. We denote by {\mathbb{E}}^{n} the metric space {\mathbb{R}}^{n} endowed with the usual Euclidean distance. We denote by (\cdot \cdot ) the Euclidean scalar product in {\mathbb{R}}^{n}, that is,
Let {\mathbb{S}}^{n} denote the ndimensional sphere defined by
with metric {d}_{{\mathbb{S}}^{n}}(x,y)=arccos(xy), x,y\in {\mathbb{S}}^{n}.
Let {\mathbb{E}}^{n,1} denote the vector space {\mathbb{R}}^{n+1} endowed with the symmetric bilinear form which associates to vectors u=({u}_{1},\dots ,{u}_{n+1}) and v=({v}_{1},\dots ,{v}_{n+1}) the real number \u3008uv\u3009 defined by
Let {\mathbb{H}}^{n} denote the hyperbolic nspace defined by
with metric {d}_{{\mathbb{H}}^{n}} such that
Definition 2.1 Given \kappa \in \mathbb{R}, we denote by {M}_{\kappa}^{n} the following metric spaces:

(i)
if \kappa =0, then {M}_{0}^{n} is the Euclidean space {\mathbb{E}}^{n};

(ii)
if \kappa >0, then {M}_{\kappa}^{n} is obtained from the spherical space {\mathbb{S}}^{n} by multiplying the distance function by the constant 1/\sqrt{\kappa};

(iii)
if \kappa <0, then {M}_{\kappa}^{n} is obtained from the hyperbolic space {\mathbb{H}}^{n} by multiplying the distance function by the constant 1/\sqrt{\kappa}.
A geodesic triangle \mathrm{\u25b3}(x,y,z) in a geodesic space (X,\rho ) consists of three points x, y, z in X (the vertices of △) and three geodesic segments between each pair of vertices (the edges of △). A comparison triangle for a geodesic triangle \mathrm{\u25b3}(x,y,z) in (X,\rho ) is a triangle \overline{\mathrm{\u25b3}}(\overline{x},\overline{y},\overline{z}) in {M}_{\kappa}^{2} such that
If \kappa \le 0, then such a comparison triangle always exists in {M}_{\kappa}^{2}. If \kappa >0, then such a triangle exists whenever \rho (x,y)+\rho (y,z)+\rho (z,x)<2{D}_{\kappa}, where {D}_{\kappa}=\pi /\sqrt{\kappa}. A point \overline{p}\in [\overline{x},\overline{y}] is called a comparison point for p\in [x,y] if \rho (x,p)={d}_{{M}_{\kappa}^{2}}(\overline{x},\overline{p}).
A geodesic triangle \mathrm{\u25b3}(x,y,z) in X is said to satisfy the CAT(\kappa ) inequality if for any p,q\in \mathrm{\u25b3}(x,y,z) and for their comparison points \overline{p},\overline{q}\in \overline{\mathrm{\u25b3}}(\overline{x},\overline{y},\overline{z}), one has
Definition 2.2 If \kappa \le 0, then X is called a CAT(\kappa ) space if and only if X is a geodesic space such that all of its geodesic triangles satisfy the CAT(\kappa ) inequality.
If \kappa >0, then X is called a CAT(\kappa ) space if and only if X is {D}_{\kappa}geodesic and any geodesic triangle \mathrm{\u25b3}(x,y,z) in X with \rho (x,y)+\rho (y,z)+\rho (z,x)<2{D}_{\kappa} satisfies the CAT(\kappa ) inequality.
Notice that in a CAT(0) space (X,\rho ) if x,y,z\in X, then the CAT(0) inequality implies
This is the (CN) inequality of Bruhat and Tits [26]. This inequality is extended by Dhompongsa and Panyanak [27] as
for all \alpha \in [0,1] and x,y,z\in X. In fact, if X is a geodesic space, then the following statements are equivalent:

(i)
X is a CAT(0) space;

(ii)
X satisfies (CN);

(iii)
X satisfies (CN^{∗}).
Let R\in (0,2]. Recall that a geodesic space (X,\rho ) is said to be Rconvex for R (see [28]) if for any three points x,y,z\in X, we have
It follows from (CN^{∗}) that a geodesic space (X,\rho ) is a CAT(0) space if and only if (X,\rho ) is Rconvex for R=2. The following lemma is a consequence of Proposition 3.1 in [28].
Lemma 2.3 Let \kappa >0 and (X,\rho ) be a CAT(\kappa ) space with diam(X)\le \frac{\pi /2\epsilon}{\sqrt{\kappa}} for some \epsilon \in (0,\pi /2). Then (X,\rho ) is Rconvex for R=(\pi 2\epsilon )tan(\epsilon ).
The following lemma is also needed.
Lemma 2.4 ([[25], p.176])
Let \kappa >0 and (X,\rho ) be a complete CAT(\kappa ) space with diam(X)\le \frac{\pi /2\epsilon}{\sqrt{\kappa}} for some \epsilon \in (0,\pi /2). Then
for all x,y,z\in X and \alpha \in [0,1].
We now collect some elementary facts about CAT(\kappa ) spaces. Most of them are proved in the setting of CAT(1) spaces. For completeness, we state the results in CAT(\kappa ) with \kappa >0.
Let \{{x}_{n}\} be a bounded sequence in a CAT(\kappa ) space (X,\rho ). For x\in X, we set
The asymptotic radius r(\{{x}_{n}\}) of \{{x}_{n}\} is given by
and the asymptotic center A(\{{x}_{n}\}) of \{{x}_{n}\} is the set
It is known from Proposition 4.1 of [8] that in a CAT(\kappa ) space X with diam(X)<\frac{\pi}{2\sqrt{\kappa}}, A(\{{x}_{n}\}) consists of exactly one point. We now give the concept of Δconvergence and collect some of its basic properties.
A sequence \{{x}_{n}\} in X is said to Δconverge to x\in X if x is the unique asymptotic center of \{{u}_{n}\} for every subsequence \{{u}_{n}\} of \{{x}_{n}\}. In this case we write \mathrm{\Delta}\text{}{lim}_{n}{x}_{n}=x and call x the Δlimit of \{{x}_{n}\}.
Lemma 2.6 Let \kappa >0 and (X,\rho ) be a complete CAT(\kappa ) space with diam(X)\le \frac{\pi /2\epsilon}{\sqrt{\kappa}} for some \epsilon \in (0,\pi /2). Then the following statements hold:

(i)
[[8], Corollary 4.4] Every sequence in X has a Δconvergent subsequence;

(ii)
[[8], Proposition 4.5] If \{{x}_{n}\}\subseteq X and \mathrm{\Delta}\text{}{lim}_{n}{x}_{n}=x, then x\in {\bigcap}_{k=1}^{\mathrm{\infty}}\overline{conv}\{{x}_{k},{x}_{k+1},\dots \}, where \overline{conv}(A)=\bigcap \{B:B\supseteq A\mathit{\text{and}}B\mathit{\text{is closed and convex}}\}.
By the uniqueness of asymptotic centers, we can obtain the following lemma (cf. [[27], Lemma 2.8]).
Lemma 2.7 Let \kappa >0 and (X,\rho ) be a complete CAT(\kappa ) space with diam(X)\le \frac{\pi /2\epsilon}{\sqrt{\kappa}} for some \epsilon \in (0,\pi /2). If \{{x}_{n}\} is a sequence in X with A(\{{x}_{n}\})=\{x\} and \{{u}_{n}\} is a subsequence of \{{x}_{n}\} with A(\{{u}_{n}\})=\{u\} and the sequence \{\rho ({x}_{n},u)\} converges, then x=u.
Definition 2.8 Let K be a nonempty subset of a CAT(\kappa ) space (X,\rho ). A mapping T:K\to K is called total asymptotically nonexpansive if there exist nonnegative real sequences \{{\nu}_{n}\}, \{{\mu}_{n}\} with {\nu}_{n}\to 0, {\mu}_{n}\to 0 as n\to \mathrm{\infty} and a strictly increasing continuous function \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with \psi (0)=0 such that
A point x\in K is called a fixed point of T if x=T(x). We denote with F(T) the set of fixed points of T. A sequence \{{x}_{n}\} in K is called approximate fixed point sequence for T (AFPS in short) if
Algorithm 1 The sequence \{{x}_{n}\} defined by {x}_{1}\in K and
is called an Ishikawa iterative sequence (see [30]).
If {\beta}_{n}=0 for all n\in \mathbb{N}, then Algorithm 1 reduces to the following.
Algorithm 2 The sequence \{{x}_{n}\} defined by {x}_{1}\in K and
is called a Mann iterative sequence (see [31]).
The following lemma is also needed.
Lemma 2.9 ([[32], Lemma 1])
Let \{{s}_{n}\} and \{{t}_{n}\} be sequences of nonnegative real numbers satisfying
If {\sum}_{n=1}^{\mathrm{\infty}}{t}_{n}<\mathrm{\infty}, then {lim}_{n\to \mathrm{\infty}}{s}_{n} exists.
3 Main results
3.1 Existence theorems
Theorem 3.1 Let \kappa >0 and (X,\rho ) be a complete CAT(\kappa ) space with diam(X)\le \frac{\pi /2\epsilon}{\sqrt{\kappa}} for some \epsilon \in (0,\pi /2). Let K be a nonempty closed convex subset of X, and let T:K\to K be a continuous total asymptotically nonexpansive mapping. Then T has a fixed point in K.
Proof Fix x\in K. We can consider the sequence {\{{T}^{n}(x)\}}_{n=1}^{\mathrm{\infty}} as a bounded sequence in K. Let \varphi :K\to [0,\mathrm{\infty}) be a function defined by
Then there exists w\in K such that \varphi (w)=inf\{\varphi (u):u\in K\}. Since T is total asymptotically nonexpansive, for each n,m\in \mathbb{N}, we have
Let M=diam(K). Taking n\to \mathrm{\infty} in (2), we get that
This implies that
In view of (1), we have
Taking n\to \mathrm{\infty}, we get that
yielding
By (3) and (4), we have {lim}_{m,h\to \mathrm{\infty}}\rho {({T}^{m}(w),{T}^{h}(w))}^{2}\le 0. Therefore, {\{{T}^{n}(w)\}}_{n=1}^{\mathrm{\infty}} is a Cauchy sequence in K and hence converges to some point v\in K. Since T is continuous,
□
From Theorem 3.1 we shall now derive a result for CAT(0) spaces which can also be found in [24].
Corollary 3.2 Let (X,\rho ) be a complete CAT(0) space and K be a nonempty bounded closed convex subset of X. If T:K\to K is a continuous total asymptotically nonexpansive mapping, then T has a fixed point.
Proof It is well known that every convex subset of a CAT(0) space, equipped with the induced metric, is a CAT(0) space (cf. [25]). Then (K,\rho ) is a CAT(0) space and hence it is a CAT(\kappa ) space for all \kappa >0. Notice also that K is Rconvex for R=2. Since K is bounded, we can choose \epsilon \in (0,\pi /2) and \kappa >0 so that diam(K)\le \frac{\pi /2\epsilon}{\sqrt{\kappa}}. The conclusion follows from Theorem 3.1. □
3.2 Demiclosed principle
Theorem 3.3 Let \kappa >0 and (X,\rho ) be a complete CAT(\kappa ) space with diam(X)\le \frac{\pi /2\epsilon}{\sqrt{\kappa}} for some \epsilon \in (0,\pi /2). Let K be a nonempty closed convex subset of X, and let T:K\to K be a uniformly continuous total asymptotically nonexpansive mapping. If \{{x}_{n}\} is an AFPS for T such that \mathrm{\Delta}\text{}{lim}_{n}{x}_{n}=w, then w\in K and w=T(w).
Proof By Lemma 2.6, w\in K. As in Theorem 3.1, we define \varphi (u):={lim\hspace{0.17em}sup}_{n}\rho ({x}_{n},u) for each u\in K. Since {lim}_{n}\rho ({x}_{n},T({x}_{n}))=0, by induction we can show that {lim}_{n}\rho ({x}_{n},{T}^{m}({x}_{n}))=0 for all m\in \mathbb{N} (cf. [16]). This implies that
In (5), taking u={T}^{m}(w), we have
Hence
In view of (1), we have
where R=(\pi 2\epsilon )tan(\epsilon ). Since \mathrm{\Delta}\text{}{lim}_{n}{x}_{n}=w, letting n\to \mathrm{\infty}, we get that
yielding
By (6) and (7), we have {lim}_{m\to \mathrm{\infty}}\rho (w,{T}^{m}(w))=0. Since T is continuous,
□
As we have observed in Corollary 3.2, we can derive the following result from Theorem 3.3.
Corollary 3.4 ([[24], Theorem 12])
Let (X,\rho ) be a complete CAT(0) space, K be a nonempty bounded closed convex subset of X, and T:K\to K be a uniformly continuous total asymptotically nonexpansive mapping. If \{{x}_{n}\} is an AFPS for T such that \mathrm{\Delta}\text{}{lim}_{n}{x}_{n}=w, then w\in K and w=T(w).
3.3 Convergence theorems
We begin this section by proving a crucial lemma.
Lemma 3.5 Let \kappa >0 and (X,\rho ) be a complete CAT(\kappa ) space with diam(X)\le \frac{\pi /2\epsilon}{\sqrt{\kappa}} for some \epsilon \in (0,\pi /2). Let K be a nonempty closed convex subset of X, and T:K\to K be a uniformly continuous total asymptotically nonexpansive mapping with {\sum}_{n=1}^{\mathrm{\infty}}{\nu}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}. Let {x}_{1}\in K and \{{x}_{n}\} be a sequence in K defined by
where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are sequences in (0,1) such that {lim\hspace{0.17em}inf}_{n}{\alpha}_{n}{\beta}_{n}(1{\beta}_{n})>0. Then \{{x}_{n}\} is an AFPS for T and {lim}_{n}\rho ({x}_{n},p) exists for all p\in F(T).
Proof It follows from Theorem 3.1 that F(T)\ne \mathrm{\varnothing}. Let p\in F(T) and M=diam(K). Since T is total asymptotically nonexpansive, by Lemma 2.4 we have
This implies that
Since {\sum}_{n=1}^{\mathrm{\infty}}{\nu}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}, by Lemma 2.9 {lim}_{n}\rho ({x}_{n},p) exists. Next, we show that \{{x}_{n}\} is an AFPS for T. In view of (1), we have
This implies that
Again by (1), we have
Substituting this into (8), we get that
yielding
Since {\sum}_{n=1}^{\mathrm{\infty}}{\nu}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}, we have
This implies by {lim\hspace{0.17em}inf}_{n}{\alpha}_{n}{\beta}_{n}(1{\beta}_{n})>0 that
By the uniform continuity of T, we have
It follows from (9) and the definitions of {x}_{n+1} and {y}_{n} that
By (9), (10), and (11), we have
□
Now, we are ready to prove our Δconvergence theorem.
Theorem 3.6 Let \kappa >0 and (X,\rho ) be a complete CAT(\kappa ) space with diam(X)\le \frac{\pi /2\epsilon}{\sqrt{\kappa}} for some \epsilon \in (0,\pi /2). Let K be a nonempty closed convex subset of X, and let T:K\to K be a uniformly continuous total asymptotically nonexpansive mapping with {\sum}_{n=1}^{\mathrm{\infty}}{\nu}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}. Let {x}_{1}\in K and \{{x}_{n}\} be a sequence in K defined by
where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are sequences in (0,1) such that {lim\hspace{0.17em}inf}_{n}{\alpha}_{n}{\beta}_{n}(1{\beta}_{n})>0. Then \{{x}_{n}\} Δconverges to a fixed point of T.
Proof Let {\omega}_{w}({x}_{n}):=\bigcup A(\{{u}_{n}\}) where the union is taken over all subsequences \{{u}_{n}\} of \{{x}_{n}\}. We can complete the proof by showing that {\omega}_{w}({x}_{n}) is contained in F(T) and {\omega}_{w}({x}_{n}) consists of exactly one point. Let u\in {\omega}_{w}({x}_{n}), then there exists a subsequence \{{u}_{n}\} of \{{x}_{n}\} such that A(\{{u}_{n}\})=\{u\}. By Lemma 2.6, there exists a subsequence \{{v}_{n}\} of \{{u}_{n}\} such that \mathrm{\Delta}\text{}{lim}_{n}{v}_{n}=v\in K. Hence v\in F(T) by Lemma 3.5 and Theorem 3.3. Since {lim}_{n}\rho ({x}_{n},v) exists, u=v by Lemma 2.7. This shows that {\omega}_{w}({x}_{n})\subseteq F(T). Next, we show that {\omega}_{w}({x}_{n}) consists of exactly one point. Let \{{u}_{n}\} be a subsequence of \{{x}_{n}\} with A(\{{u}_{n}\})=\{u\}, and let A(\{{x}_{n}\})=\{x\}. Since u\in {\omega}_{w}({x}_{n})\subseteq F(T), by Lemma 3.5 {lim}_{n}\rho ({x}_{n},u) exists. Again, by Lemma 2.7, x=u. This completes the proof. □
As a consequence of Theorem 3.6, we obtain the following.
Corollary 3.7 ([[24], Theorem 17])
Let (X,\rho ) be a complete CAT(0) space, K be a nonempty bounded closed convex subset of X, and T:K\to K be a uniformly continuous total asymptotically nonexpansive mapping with {\sum}_{n=1}^{\mathrm{\infty}}{\nu}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}. Let {x}_{1}\in K and \{{x}_{n}\} be a sequence in K defined by
where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are sequences in (0,1) such that {lim\hspace{0.17em}inf}_{n}{\alpha}_{n}{\beta}_{n}(1{\beta}_{n})>0. Then \{{x}_{n}\} Δconverges to a fixed point of T.
Recall that a mapping T:K\to K is said to be semicompact if K is closed and each bounded AFPS for T in K has a convergent subsequence. Now, we prove a strong convergence theorem for uniformly continuous total asymptotically nonexpansive semicompact mappings.
Theorem 3.8 Let \kappa >0 and (X,\rho ) be a complete CAT(\kappa ) space with diam(X)\le \frac{\pi /2\epsilon}{\sqrt{\kappa}} for some \epsilon \in (0,\pi /2). Let K be a nonempty closed convex subset of X, and let T:K\to K be a uniformly continuous total asymptotically nonexpansive mapping with {\sum}_{n=1}^{\mathrm{\infty}}{\nu}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}. Let {x}_{1}\in K and \{{x}_{n}\} be a sequence in K defined by
where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are sequences in (0,1) such that {lim\hspace{0.17em}inf}_{n}{\alpha}_{n}{\beta}_{n}(1{\beta}_{n})>0. Suppose that {T}^{m} is semicompact for some m\in \mathbb{N}. Then \{{x}_{n}\} converges strongly to a fixed point of T.
Proof By Lemma 3.5, {lim}_{n}\rho ({x}_{n},T({x}_{n}))=0. Since T is uniformly continuous, we have
as n\to \mathrm{\infty}. That is, \{{x}_{n}\} is an AFPS for {T}^{m}. By the semicompactness of {T}^{m}, there exist a subsequence \{{x}_{{n}_{j}}\} of \{{x}_{n}\} and p\in K such that {lim}_{j\to \mathrm{\infty}}{x}_{{n}_{j}}=p. Again, by the uniform continuity of T, we have
That is, p\in F(T). By Lemma 3.5, {lim}_{n}\rho ({x}_{n},p) exists, thus p is the strong limit of the sequence \{{x}_{n}\} itself. □
Corollary 3.9 ([[24], Theorem 22])
Let (X,\rho ) be a complete CAT(0) space, K be a nonempty bounded closed convex subset of X, and T:K\to K be a uniformly continuous total asymptotically nonexpansive mapping with {\sum}_{n=1}^{\mathrm{\infty}}{\nu}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}. Let {x}_{1}\in K and \{{x}_{n}\} be a sequence in K defined by
where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are sequences in (0,1) such that {lim\hspace{0.17em}inf}_{n}{\alpha}_{n}{\beta}_{n}(1{\beta}_{n})>0. Suppose that {T}^{m} is semicompact for some m\in \mathbb{N}. Then \{{x}_{n}\} converges strongly to a fixed point of T.
Remark 3.10 The results in this article also hold for the class of weakly total asymptotically nonexpansive mappings in the following sense. A mapping T:K\to K is called weakly total asymptotically nonexpansive if there exist nonnegative real sequences \{{\nu}_{n}\}, \{{\mu}_{n}\} with {\nu}_{n}\to 0, {\mu}_{n}\to 0 as n\to \mathrm{\infty} and a nondecreasing function \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) such that
Author’s contributions
The author completed the paper himself. The author read and approved the final manuscript.
References
Kirk WA: Geodesic geometry and fixed point theory. Colecc. Abierta 64. In Seminar of Mathematical Analysis. Univ. Sevilla Secr. Publ., Seville; 2003:195–225. (Malaga/Seville, 2002/2003)
Kirk WA: Geodesic geometry and fixed point theory II. In International Conference on Fixed Point Theory and Applications. Yokohama Publishers, Yokohama; 2004:113–142.
Dhompongsa S, Kaewkhao A, Panyanak B:Lim’s theorems for multivalued mappings in CAT(0) spaces. J. Math. Anal. Appl. 2005, 312: 478–487. 10.1016/j.jmaa.2005.03.055
Chaoha P, Phonon A:A note on fixed point sets in CAT(0) spaces. J. Math. Anal. Appl. 2006, 320: 983–987. 10.1016/j.jmaa.2005.08.006
Leustean L:A quadratic rate of asymptotic regularity for CAT(0)spaces. J. Math. Anal. Appl. 2007, 325: 386–399. 10.1016/j.jmaa.2006.01.081
Kirk WA, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal. 2008, 68: 3689–3696. 10.1016/j.na.2007.04.011
Shahzad N, Markin J:Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces. J. Math. Anal. Appl. 2008, 337: 1457–1464. 10.1016/j.jmaa.2007.04.041
Espinola R, FernandezLeon A:CAT(k)Spaces, weak convergence and fixed points. J. Math. Anal. Appl. 2009, 353: 410–427. 10.1016/j.jmaa.2008.12.015
Saejung S:Halpern’s iteration in CAT(0) spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 471781
Cho YJ, Ćirić L, Wang S:Convergence theorems for nonexpansive semigroups in CAT(0) spaces. Nonlinear Anal. 2011, 74: 6050–6059. 10.1016/j.na.2011.05.082
Abkar A, Eslamian M:Common fixed point results in CAT(0) spaces. Nonlinear Anal. 2011, 74: 1835–1840. 10.1016/j.na.2010.10.056
Alber YI, Chidume CE, Zegeye H: Approximating fixed points of total asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2006., 2006: Article ID 10673
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1972, 35: 171–174. 10.1090/S00029939197202985003
Sahu DR: Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces. Comment. Math. Univ. Carol. 2005, 46: 653–666.
Chang SS, Wang L, Joseph Lee HW, Chan CK, Yang L:Demiclosed principle and Δconvergence theorems for total asymptotically nonexpansive mappings in CAT(0) spaces. Appl. Math. Comput. 2012, 219: 2611–2617. 10.1016/j.amc.2012.08.095
Tang JF, Chang SS, Joseph Lee HW, Chan CK:Iterative algorithm and Δconvergence theorems for total asymptotically nonexpansive mappings in CAT(0) spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 965751
Basarir M, Sahin A:On the strong and Δconvergence for total asymptotically nonexpansive mappings on a CAT(0) space. Carpath. Math. Publ. 2013, 5: 170–179.
Chang SS, Wang L, Joseph Lee HW, Chan CK:Strong and Δconvergence for mixed type total asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 122
Wan LL: ΔConvergence for mixedtype total asymptotically nonexpansive mappings in hyperbolic spaces. J. Inequal. Appl. 2013., 2013: Article ID 553
Wang L, Chang SS, Ma Z:Convergence theorems for total asymptotically nonexpansive nonself mappings in CAT(0) spaces. J. Inequal. Appl. 2013., 2013: Article ID 135
Yang L, Zhao FH:Strong and Δconvergence theorems for total asymptotically nonexpansive nonself mappings in CAT(0) spaces. J. Inequal. Appl. 2013., 2013: Article ID 557
Zhao LC, Chang SS, Kim JK: Mixed type iteration for total asymptotically nonexpansive mappings in hyperbolic spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 353
Zhao LC, Chang SS, Wang XR: Convergence theorems for total asymptotically nonexpansive mappings in hyperbolic spaces. J. Appl. Math. 2013., 2013: Article ID 689765
Karapınar E, Salahifard H, Vaezpour SM:Demiclosedness principle for total asymptotically nonexpansive mappings in CAT(0) spaces. J. Appl. Math. 2014., 2014: Article ID 738150
Bridson M, Haefliger A: Metric Spaces of NonPositive Curvature. Springer, Berlin; 1999.
Bruhat F, Tits J: Groupes réductifs sur un corps local. I. Données radicielles valuées. Publ. Math. Inst. Hautes Études Sci. 1972, 41: 5–251. 10.1007/BF02715544
Dhompongsa S, Panyanak B:On Δconvergence theorems in CAT(0) spaces. Comput. Math. Appl. 2008, 56: 2572–2579. 10.1016/j.camwa.2008.05.036
Ohta S: Convexities of metric spaces. Geom. Dedic. 2007, 125: 225–250. 10.1007/s1071100791593
Lim TC: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 1976, 60: 179–182. 10.1090/S0002993919760423139X
Ishikawa S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147–150. 10.1090/S00029939197403364695
Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S00029939195300548463
Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 1993, 178: 301–308. 10.1006/jmaa.1993.1309
Acknowledgements
The author thanks Chiang Mai University for financial support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Panyanak, B. On total asymptotically nonexpansive mappings in CAT(\kappa ) spaces. J Inequal Appl 2014, 336 (2014). https://doi.org/10.1186/1029242X2014336
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X2014336
Keywords
 fixed point
 total asymptotically nonexpansive mapping
 demiclosed principle
 Δconvergence
 CAT(\kappa ) space