Convergence theorems for total asymptotically nonexpansive map**s in $\operatorname{CAT} (\kappa )$ spaces

Abstract

The purpose of this paper is to study the convergence theorems in $\operatorname{CAT} (\kappa )$ spaces with $k > 0$ for total asymptotically nonexpansive map**s which are essentially wider than nonexpansive map**s, asymptotically nonexpansive map**, and asymptotically nonexpansive map**s in the intermediate sense. Our results generalize, unify, and improve several comparable results in the existing literature.

1 Introduction and preliminaries

Let C be a nonempty subset of a metric space $(X, d)$.

Recall that a map** $T : C \to X$ is said to be:

(i) nonexpansive if $d(T x, T y) \le d(x, y)$ for all $x, y\in C$;

(ii) asymptotically nonexpansive [1] if there exists a sequence $\{k_{n}\} \subset [1, \infty )$ with $\lim_{n\to \infty} k_{n} = 1$ such that $d(T^{n} x, T^{n} y) \le k_{n} d(x, y)$ for all $x, y \in C$ and $n \in \mathscr{N}$, where $\mathscr{N}$ denotes the set of positive integers. The class of asymptotically nonexpansive map**s includes a class of nonexpansive map**s as a proper subclass.

(iii) In 1993, Bruck, Kuczumow, and Reich [2] introduced the concept of asymptotically nonexpansive map** in the intermediate sense. A map** $T : C \to C$ is said to be asymptotically nonexpansive in the intermediate sense if T is uniformly continuous and the following inequality holds:

$$ \limsup_{n \to \infty} \sup_{x,y\in C}\bigl\{ d \bigl(T^{n} x, T^{n} y\bigr)- d (x, y) \bigr\} \le 0. $$

(1.1)

It is easy to know that the class of asymptotically nonexpansive map**s in the intermediate sense is more general than the class of asymptotically nonexpansive map**s.

Definition 1.1

([3])

A map** $T : C \to C$ is said to be $(\{\mu _{n}\}, \{\nu _{n}\}, \zeta )$-total asymptotically nonexpansive if there exist nonnegative sequences $\{\mu _{n}\}$, $\{\nu _{n}\}$ with $\mu _{n} \to 0$, $\nu _{n} \to 0$ and a strictly increasing continuous function $\zeta : [0, \infty ) \to [0, \infty )$ with $\zeta (0) = 0$ such that

$$ d\bigl(T^{n} x, T^{n} y\bigr) \le d(x,y) + \nu _{n} \zeta \bigl(d(x,y)\bigr) + \mu _{n}, \quad \forall n \ge 1, x, y \in C. $$

(1.2)

The concept of total asymptotically nonexpansive map**s is more general than that of asymptotically nonexpansive map**s in the intermediate sense. In fact, if $T : C \to C$ is an asymptotically nonexpansive map** in the intermediate sense, denote by $\mu _{n} = \max \{\sup_{x,y \in C} (d(T^{n} x, T^{n} y) - d(x, y)), 0\}$. Then $\mu _{n} \ge 0$, $\lim_{n \to \infty}\mu _{n} = 0$, and

$$ d\bigl(T^{n} x, T^{n} y\bigr) \le d(x, y) + \mu _{n},\quad \forall x, y, \in C, n \ge 1. $$

(1.3)

Taking $\{\nu _{n} = 0\}$, $\zeta = t$, $t \ge 0$, then (1.3) can be written as

$$ d\bigl(T^{n} x, T^{n} y\bigr) \le d(x,y) + \nu _{n} \zeta \bigl(d(x,y)\bigr) + \mu _{n}, \quad \forall n \ge 1, x, y \in C, $$

i.e., T is a total asymptotically nonexpansive map**.

Definition 1.2

A map** $T: C \to C$ is said to be uniformly L-Lipschitzian if there exists a constant $L > 0$ such that

$$ d\bigl(T^{n} x, T^{n} y\bigr) \le L d(x, y) \quad \forall x, y, \in C \text{ and } \forall n \ge 1. $$

In recent years, CAT(0) spaces (the precise definition of a CAT(0) space is given below) have attracted the attention of many authors because they have played a very important role in different aspects of geometry [4]. Kirk [5, 6] showed that a nonexpansive map** defined on a bounded closed convex subset of a complete CAT(0) space has a fixed point.

In 2012, Chang et al. [7] studied the demiclosedness principle and Δ-convergence theorems for total asymptotically nonexpansive map**s in the setting of CAT(0) spaces. Since then the convergence of several iteration procedures for this type of map**s has been rapidly developed, and many of articles have appeared (see, e.g., [8–17]). In 2013, under some suitable assumptions, Karapinar et al. [9] obtained the demiclosedness principle, fixed point theorems, and convergence theorems for the following iteration:

Let C be a nonempty closed convex subset of a CAT(0) space X and $T : C \to C$ be a total asymptotically nonexpansive map**. Given $x_{1} \in C$, let $\{x_{n}\} \subset C$ be defined by

$$ x_{n+1} = (1 - \alpha _{n})x_{n} \oplus \alpha _{n} T^{n} \bigl((1 -\beta _{n})x_{ \oplus } \beta _{n} T^{n} (x_{n})\bigr), \quad n \in \mathscr{N}, $$

where $\{\alpha _{n}\}$ and $\{\beta _{n}\}$ are sequences in $[0; 1]$.

It is well known that any $\operatorname{CAT} (\kappa )$ space is a $\operatorname{CAT} (\kappa _{1})$ space for $\kappa _{1} \ge \kappa$. Thus, all results for CAT(0) spaces immediately apply to any $\operatorname{CAT} (\kappa )$ space with $\kappa \le 0$.

Very recently, Panyanak [10] obtained the demiclosedness principle, fixed point theorems, and convergence theorems for total asymptotically nonexpansive map**s on $\operatorname{CAT} (\kappa )$ space with $\kappa > 0$, which generalize the results of Chang et al. [7], Tang et al. [8], Karapinar et al. [9].

Motivated by the work going on in this direction, in this paper we aim to study the strong convergence of a sequence generated by an infinite family of total asymptotically nonexpansive map**s in $\operatorname{CAT} (\kappa )$ spaces with $\kappa > 0$. Our results are new, they extend and improve the corresponding results of Chang et al. [7], Tang et al. [8], Karapinar et al. [9], Panyanak [10], Hea et al. [18], and many others.

2 Preliminaries

In this section, we first recall some definitions, notations, and conclusions that will be needed in our paper.

Let $(X, d)$ be a metric space. A geodesic path joining $x \in X$ to $y \in X$ is a map** c from a closed interval $[0, l] \subset \mathscr{R}$ to X such that $c(0) = x$, $c(l) = y$, and $d(c(t); c(t^{\prime })) = |t - t^{\prime }|$ for all $t, t^{\prime } \in [0; l]$. In particular, c is an isometry and $d(x, y) = l$. 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

$$ d(x, z) = (1 - \alpha ) d(x,y), \quad \text{and} \quad d(y, z) = \alpha d(x; y). $$

In this case, we write $z = \alpha x \oplus (1-\alpha )y$.

A metric space $(X, d)$ is said to be a geodesic space (D-geodesic 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 (D-uniquely geodesic) if there is exactly one geodesic joining x and y for each $x, y \in X$ (for $x, y \in X$ with $d(x, y) < D$). A subset C of X is said to be convex if C includes every geodesic segment joining any two of its points.

Now we introduce the concept of model spaces $M^{n}_{\kappa}$. For more details on these spaces, the reader is referred to [4, 14]. Let $n \in \mathscr{N}$. We denote by $E^{n}$ the metric space $\mathscr{R}^{n}$ endowed with the usual Euclidean distance. We denote by $(\cdot | \cdot )$ the Euclidean scalar product in $\mathscr{R}^{n}$, that is,

$$ (x|y) = x_{1}y_{1} + \cdots + x_{n} y_{n} \quad \text{where } x = (x_{1}, \ldots x_{n}), y = (y_{1},\ldots y_{n}). $$

Let $\mathscr{S}^{n}$ denote the n-dimensional sphere defined by

$$ \mathscr{S}^{n} = \bigl\{ x = (x_{1}, \ldots , x_{n+1})\in \mathscr{R}^{n+1} : (x|x) = 1\bigr\} $$

with metric

$$ d_{\mathscr{S}^{n}}(x, y) = \arccos(x|y), \quad x, y\in \mathscr{S}^{n}. $$

Let $E^{n;1}$ denote the vector space $\mathscr{R}^{n+1}$ endowed with the symmetric bilinear form that associates to vectors $u = (u_{1}, \ldots , u_{n+1})$ and $v = (v_{1}, \ldots , v_{n+1})$, and the real number $\langle u|v \rangle$ is defined by

$$ \langle u|v \rangle = u- u_{n+1} v_{n+1} + \sum _{i=1}^{n} u_{i} v_{i}. $$

Let $\mathscr{H}^{n}$ denote the hyperbolic n-space defined by

$$ \mathscr{H}^{n} = \bigl\{ u = (u_{1}, \ldots , u_{n+1)} \in E^{n;1} : \langle u|u \rangle = -1, u_{n+1} > 0\bigr\} $$

with metric $d_{\mathscr{H}^{n}}$ such that

$$ \cosh d_{\mathscr{H}^{n}}(x,y) = - \langle x| y \rangle , \quad x, y \in \mathscr{H}^{n}. $$

Definition 2.1

Given $\kappa \in \mathscr{R}$, we denote by $M^{n}_{\kappa}$ the following metric spaces:

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

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

(iii) if $\kappa < 0$, then $M^{n}_{\kappa}$ is obtained from the hyperbolic space $\mathscr{H}^{n}$ by multiplying the distance function by the constant $\frac{1}{\sqrt{ - \kappa}}$.

A geodesic triangle $\Delta (x, y, z)$ in a geodesic space $(X, d)$ consists of three points x, y, z in X (the vertices of Δ) and three geodesic segments between each pair of vertices (the edges of Δ). A comparison triangle for a geodesic triangle $\Delta (x; y; z)$ in $(X, d)$ is a triangle $\bar{\Delta}(\bar{x}, \bar{y},\bar{z})$ in $M^{2}_{\kappa}$ such that

$$ d(x; y) = d_{M^{2}_{\kappa}}(\bar{x}, \bar{y}), \qquad d(y,z) = d_{M^{2}_{ \kappa}}( \bar{y}, \bar{z}), \qquad d(z,x) = d_{M^{2}_{\kappa}}(\bar{z}, \bar{x}). $$

If $\kappa < 0$, then such a comparison triangle always exists in $M^{2}_{\kappa}$. If $\kappa > 0$, then such a triangle exists whenever $d(x, y) + d(y, z) + d(z, x) < 2D_{\kappa}$, where $D_{\kappa}= \frac{\pi}{\sqrt{\kappa}}$.

A point $\bar{p} \in [\bar{x}, \bar{y}]$ is called a comparison point for $p \in [x, y]$ if $d(x, p) = d_{M^{2}_{\kappa}}(\bar{x}, \bar{p})$.

A geodesic triangle $\Delta (x, y, z)$ in X is said to satisfy the CAT(κ) inequality if for any $p, q \in \Delta (x, y, z)$ and for their comparison points $\bar{p}, \bar{q} \in \Delta (\bar{x}, \bar{y}, \bar{z})$, one has

$$ d(p, q) \le d_{M^{2}_{\kappa}} (\bar{p},\bar{q}). $$

Definition 2.2

A metric space $(X, d)$ is called a CAT(0) space if X is a geodesic space such that all of its geodesic triangles satisfy the $\operatorname{CAT} (\kappa )$ inequality.

If $\kappa > 0$, then X is called a $\operatorname{CAT} (\kappa )$ space if X is $D_{\kappa}$-geodesic and any geodesic triangle $\Delta (x, y, z)$ in X with $d(x, y) + d(y, z) + d(z, x) < 2D_{\kappa}$ satisfies the $\operatorname{CAT} ( \kappa )$ inequality.

Definition 2.3

A geodesic space $(X, d)$ is said to be R-convex with $R \in (0, 2]$ (see [16]) if for any three points $x, y, z \in X$, we have

$$ d^{2}\bigl(x, (1-\alpha )y \oplus \alpha z\bigr)\le (1-\alpha ) d^{2}(x,y) + \alpha d^{2}(x,z) - \frac{R}{2}\alpha (1- \alpha )d^{2}(y, z). $$

(2.1)

Notice that if $(X,d)$ is a geodesic space, then the following statements are equivalent:

(i) $(X,d)$ is a CAT(0) space;

(ii) $(X, d)$ is R-convex with $R=2$, i.e., it satisfies the following inequality:

$$ d^{2}\bigl(x, (1-\alpha )y \oplus \alpha z\bigr)\le (1-\alpha ) d^{2}(x,y) + \alpha d^{2}(x,z) - \alpha (1-\alpha )d^{2}(y; z) $$

(2.2)

for all $\alpha \in (0, 1]$ and $x, y, z \in X$.

The following lemma is a consequence of Proposition 3.1 in [19].

Lemma 2.4

Let $\kappa > 0$ and $(X, d)$ be a CAT(κ) space with $\operatorname{diam}(X)\le \frac{\frac{\pi}{2} - \epsilon}{\sqrt{\kappa}}$ for some $\epsilon \in (0, \frac{\pi}{2})$. Then $(X,d)$ is R-convex with $R = (\pi - 2\epsilon ) \tan(\epsilon )$.

Lemma 2.5

([20, page 176])

Let $\kappa > 0$ and $(X,d)$ be a complete CAT(κ) space with $\operatorname{diam}(X)\le \frac{\frac{\pi}{2} - \epsilon}{\sqrt{\kappa}}$ for some $\epsilon \in (0, \frac{\pi}{2})$. Then

$$ d\bigl((1 -\alpha )x \oplus \alpha y, z\bigr) \le (1 -\alpha ) d(x, z) + \alpha d(y, z) $$

(2.3)

for all $x, y, z \in X$ and $\alpha \in [0, 1]$.

We now collect some elementary facts about $\operatorname{CAT}(\kappa )$ spaces, $\kappa > 0$.

Let $\{x_{n}\}$ be a bounded sequence in a CAT(κ) space $(X, d)$. For $x \in X$, we set

$$ r\bigl(x, \{x_{n}\}\bigr) = \lim \sup_{n \to \infty} d(x, x_{n}). $$

The asymptotic radius $r(\{x_{n}\})$ of $\{x_{n}\}$ is given by

$$ r\bigl(\{x_{n}\}\bigr) = \inf \bigl\{ r\bigl(x, \{x_{n}\} \bigr) : x \in X\bigr\} . $$

The asymptotic radius $r(\{x_{n}\})$ of $\{x_{n}\}$ is given by

$$ r\bigl(\{x_{n}\}\bigr) = \inf \bigl\{ r\bigl(x, \{x_{n}\} \bigr) : x \in X\bigr\} . $$

(2.4)

The asymptotic center $A(\{x_{n}\})$ of $\{x_{n}\}$ is the set

$$ A\bigl(\{x_{n}\}\bigr) = \bigl\{ x \in X: r\bigl(x, \{x_{n}\} \bigr) = r\bigl(\{x_{n}\}\bigr)\bigr\} . $$

(2.5)

It is known from Proposition 4.1 of [21] that in a CAT(κ) space X with $\operatorname{diam}(X) < \frac{\pi}{2\sqrt{\kappa}}$, $A(\{x_{n}\})$ consists of exactly one point.

We now give the concept of Δ-convergence and collect some of its basic properties.

Definition 2.6

([22, 23])

A sequence $\{x_{n}\}$ in X is said to Δ-converge 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 $\Delta -\lim_{n \to \infty} x_{n} = x$, and x is called the Δ-limit of $\{x_{n}\}$.

Lemma 2.7

Let $(X, d)$ be a complete CAT(κ) space with $\kappa > 0$ and $\operatorname{diam}(X)\le \frac{\frac{\pi}{2} - \epsilon}{\sqrt{\kappa}}$ for some $\epsilon \in (0, \frac{\pi}{2})$. Then the following statements hold:

• (i) [17, Corollary 4.4] Every sequence in X has a Δ-convergence subsequence;

• (ii) [17, Proposition 4.5] If $\{x_{n}\} \subset X$ and $\Delta -\lim x_{n} = x$, then

  $$ x\in \bigcap_{n=1}^{\infty }\overline{ \mathrm{conv}}\{x_{n}, x_{n+1}, \ldots \}, $$

where $\overline{\mathrm{conv}}(A) = \bigcap \{B : B \supseteq A \textit{and} B \textit{is} \textit{closed} \textit{and} \textit{convex}\}$.

By the uniqueness of asymptotic centers, we can obtain the following lemma (cf. [24, Lemma 2.8]).

Lemma 2.8

Let $(X,d)$ be a complete CAT(κ) space with $\kappa > 0$ and $\operatorname{diam}(X)\le \frac{\frac{\pi}{2} - \epsilon}{\sqrt{\kappa}}$ for some $\epsilon \in (0, \frac{\pi}{2})$. If $\{x_{n}\}$ is a sequence in X with $A(\{x_{n}\}) = \{x\}$ and if $\{u_{n}\}$ is a subsequence of $\{x_{n}\}$ with $A(\{u_{n}\}) = \{u\}$ and the sequence $\{d(x_{n}, u)\}$ converges, then $x = u$.

In the sequel, we use $F(T)$ to denote the fixed point set of a map** T.

Definition 2.9

([25])

(1) A triple $(X, d, W)$ is called a hyperbolic space if $(X, d)$ is a metric space and $W : X \times X \times [0, 1] \to X$ is a map** such that $\forall x, y, z, w \in X$, $\alpha , \beta , \in [0, 1]$, the following hold:

(W1) $d(z, W(x, y, \alpha ) \le \alpha d(z, x) + (1-\alpha )d(z, y)$;

(W2) $d(W(x, y, \alpha ), W(x, y, \beta ) = |\alpha - \beta | d(x, y)$;

(W3) $W(x, y, \alpha ) = W(y, x, 1-\alpha )$;

(W4) $d(W(x, z, \alpha ), W(y, w, \alpha )\le \alpha d(x, y) + (1- \alpha )d(z, w)$.

(2) A hyperbolic space $(X, d, W)$ is called uniformly convex if for any $r > 0$ and $\epsilon \in (0, 2]$, there exists $\delta \in (0, 1]$ such that, for all $x, y, z\in X$,

$$ \left. \begin{aligned} &d(x, z) \le r \\ &d(y, z) \le r \\ &d(x, y) \ge \epsilon \cdot r \end{aligned} \right\} \Rightarrow d\biggl( \frac{1}{2} x \oplus \frac{1}{2} y, z\biggr) \le (1 - \delta )r. $$

(2.6)

A map** $\eta : (0, \infty ) \times (0, 2] \to (0, 1]$ providing such $\delta : = \eta (r, \epsilon )$ for given $r > 0$ and $\epsilon \in (0, 2]$ is called a modulus of uniform convexity.

Lemma 2.10

([25])

Let $(X, d, W)$ be a uniformly convex hyperbolic space with modulus of uniform convexity η. For any $r > 0$, $\epsilon \in (0, 2]$, $\lambda \in [0, 1]$, and $x, y, z \in X$,

$$ \left. \begin{aligned} &d(x, z) \le r \\ &d(y, z) \le r \\ &d(x, y) \ge \epsilon \cdot r \end{aligned} \right\} \Rightarrow d\bigl((1 - \lambda ) x \oplus \lambda y, z\bigr) \le \bigl(1 - 2\lambda (1-\lambda )\eta (r, \epsilon )\bigr)r. $$

(2.7)

Proposition 2.11

Let $(X,d)$ be a complete uniformly convex CAT(κ) space $\kappa > 0$ with modulus of uniform convexity η and $\operatorname{diam}(X)\le \frac{\frac{\pi}{2} - \epsilon}{\sqrt{\kappa}}$ for some $\epsilon \in (0, \frac{\pi}{2})$. Let $x\in X$ be a given point and $\{t_{n}\}$ be a sequence in $[b, c]$ with $b, c \in (0, 1)$ and $0 < b(1-c) \le \frac{1}{2}$. Let $\{x_{n}\}$ and $\{y_{n}\}$ be any sequences in X such that

$$\begin{aligned}& \limsup_{n \to \infty} d(x_{n}, x)\le r, \qquad \limsup_{n \to \infty}d(y_{n}, x)\le r\quad\textit{and} \\& \lim_{n \to \infty}d\bigl((1-t_{n})x_{n} \oplus t_{n} y_{n}\bigr), x) = r \quad \textit{for some } r \ge 0. \end{aligned}$$

Then

$$ \lim_{n \to \infty}d(x_{n}, y_{n})=0. $$

(2.8)

Proof

By the assumption that $(X,d)$ is a complete CAT(κ) space $\kappa > 0$ and $\operatorname{diam}(X)\le \frac{\frac{\pi}{2} - \epsilon}{\sqrt{\kappa}}$ for some $\epsilon \in (0, \frac{\pi}{2})$, it follows from Lemma 2.5 that for all $x, y, z \in X$ and $\alpha \in [0, 1]$

$$ d\bigl((1 -\alpha )x \oplus \alpha y, z\bigr) \le (1 -\alpha ) d(x, z) + \alpha d(y, z). $$

Letting $W(x, y, \alpha ): = (1 -\alpha )x \oplus \alpha y$. It is easy to prove that $W(x, y, \alpha )$ satisfies conditions $(W1)-(W4)$. Hence $(X, d, W)$ is a hyperbolic space. Again, since $(X,d)$ is uniformly convex with modulus of uniform convexity η, this implies that $(X, d, W)$ is a uniformly convex hyperbolic space with modulus of uniform convexity η.

Now we consider two cases.

1. If $r = 0$, then the conclusion of Proposition 2.11 is obvious.

2. The case of $r > 0$. If it is not the case that $d(x_{n}, y_{n}) \to 0$ as $n \to \infty$, then there are subsequences (denoted by $\{x_{n}\}$ and $\{y_{n}\}$ again) such that

$$ \inf_{n} d(x_{n}, y_{n}) > 0. $$

(2.9)

Choose $\epsilon \in (0, 1]$ such that

$$ d(x_{n}, y_{n}) \ge \epsilon (r + 1) > 0,\quad \forall n \in \mathscr{N}. $$

(2.10)

Since $0 < b(1-c) < \frac{1}{2}$ and $0 < \eta (r,\epsilon ) \le 1$, $0 < 2b(1-c)\eta (r,\epsilon ) \le 1$. This implies $0 \le 1 - 2b(1-c)\eta (r,\epsilon ) < 1$. Choose $R \in (r, r +1)$ such that

$$ (1 - 2b(1- c)\eta (r,\epsilon )R < r. $$

(2.11)

Since

$$ \lim \sup_{n} d(x_{n}, x) \le r,\qquad \lim \sup _{n} d(y_{n}, x) \le r,\quad r < R, $$

(2.12)

there are further subsequences again denoted by $\{x_{n}\}$ and $\{y_{n}\}$ such that

$$ d(x_{n}, x) \le R,\qquad d(y_{n}, x) \le R,\qquad d(x_{n}, y_{n}) \ge \epsilon R,\quad \forall n \in \mathscr{N}. $$

(2.13)

Then, by Lemma 2.10 and (2.11),

$$ \begin{aligned} d\bigl((1 - t_{n})x_{n}, t_{n}y_{n}, x\bigr) &\le \bigl(1- 2t_{n}(1- t_{n})\eta (R, \epsilon )\bigr)R \\ & \le \bigl(1 - 2b(1 -c)\eta (r,\epsilon )\bigr)R < r \end{aligned} $$

(2.14)

for all $n \in \mathscr{N}$. Taking $n\to \infty$, we obtain

$$ \lim_{n \to \infty} d\bigl((1-t_{n}) x_{n} \oplus t_{n} y_{n}, x\bigr) < r, $$

(2.15)

which contradicts the hypothesis.

The conclusion of Proposition 2.11 is proved. □

Lemma 2.12

Let $\{ a_{n}\}$, $\{ \lambda _{n}\}$, and $\{ c_{n}\}$ be the sequences of nonnegative numbers such that

$$ a_{n+1} \le (1+ \lambda _{n})a_{n} + c_{n}, \quad \forall n \ge 1. $$

If $\sum_{n = 1}^{\infty }\lambda _{n} < \infty$ and $\sum_{n = 1}^{\infty }c_{n} < \infty$, then $\lim_{n \to \infty} a_{n}$ exists. In addition, if there exists a subsequence $\{a_{n_{i}}\} \subset \{a_{n}\}$ such that $a_{n_{i}} \to 0$, then $\lim_{n \to \infty} a_{n} = 0$.

3 Strong convergence theorems for total asymptotically nonexpansive map**s in CAT(κ) spaces

Lemma 3.1

([26])

(1) For each positive integer $n\ge 1$, the unique solutions $i(n)$ and $k(n)$ with $k(n) \ge i(n)$ to the following positive integer equation

$$ n = i(n) + \frac{(k(n) -1)k(n)}{2} $$

(3.1)

are as follows:

$$ \textstyle\begin{cases} i(n) = n - \frac{(k(n)-1)k(n)}{2}, \\ k(n) =[\frac{1}{2} + \sqrt[2]{2n - \frac{7}{4}}], & k(n) \ge i(n), \end{cases} $$

and $k(n) \to \infty$ (as $n \to \infty$), where $[x]$ denotes the maximal integer that is not larger than x.

(2) For each $i \ge 1$, denote by

$$ \textstyle\begin{cases} \Gamma _{i}: = \{n \in \mathscr{N}: n = i + \frac{(k(n)-1)k(n)}{2}, k(n) \ge i\},\quad \textit{and} \\ K_{i}: = \{k(n): n \in \Gamma _{i}, n = i + \frac{(k(n)-1)k(n)}{2}, k(n) \ge i\}, \end{cases} $$

then $k(n)+1 = k(n+1)$, $\forall n \in \Gamma _{i}$.

In this section we prove some strong convergence theorems for the following iterative scheme:

$$ \textstyle\begin{cases} x_{1} \in C, \\ x_{n+1} = (1 - \alpha _{n}) x_{n} \oplus \alpha _{n} T_{i(n)}^{k(n)}y_{n}, \\ y_{n} = (1-\beta _{n})x_{n} \oplus \beta _{n} T_{i(n)}^{k(n)}x_{n}, \end{cases}\displaystyle n \ge 1, $$

(3.2)

where C is a nonempty closed and convex subset of a complete CAT(κ) space X, $\kappa > 0$, for each $i \ge 1$, $T_{i}: C\rightarrow C$ is uniformly $L_{i}$-Lipschitzian and $(\{\nu _{n}^{(i)}\}, \{\mu _{n}^{(i)}\}, \zeta ^{(i)})$-total asymptotically nonexpansive map**s defined by (1.2); and for each positive integer $n\ge 1$, $i(n)$ and $k(n)$ are the unique solutions of the positive integer equation (3.1).

Theorem 3.2

Let $(X,d)$ be a complete uniformly convex CAT(κ) space with $\kappa > 0$ and $\operatorname{diam}(X)\le \frac{\frac{\pi}{2} - \epsilon}{\sqrt{\kappa}}$ for some $\epsilon \in (0, \frac{\pi}{2})$. Let C be a nonempty closed and convex subset of X and, for each $i \ge 1$, let $T_{i}: C\rightarrow C$ be uniformly $L_{i}$-Lipschitzian and $(\{\nu _{n}^{(i)}\}, \{\mu _{n}^{(i)}\}, \zeta ^{(i)})$-total asymptotically nonexpansive map**s defined by (1.2) such that

• (i) $\sum_{i=1}^{\infty }\sum_{n=1}^{\infty }\nu _{n}^{(i)} < \infty$, $\sum_{i=1}^{\infty}\sum_{n=1}^{\infty }\mu _{n}^{(i)} < \infty$,

• (ii) there exists a constant $M_{*} > 0$ such that $\zeta ^{(i)}(r)\leq M_{*} r$, $\forall r\geq 0$, $i= 1, 2, \ldots$ ;

• (iii) there exist constants $a, b \in (0, 1)$ with $0 < b(1-a) \le \frac{1}{2}$ such that $\{\alpha _{n}\}, \{\beta _{n}\} \subset [a, b]$.

If $\mathscr{F} := \bigcap_{i = 1}^{\infty }F(T_{i})\neq \emptyset$ and there exist a map** $T_{n_{0}} \in \{T_{i}\}_{i=1}^{\infty}$ and a nondecreasing function $f: [0, \infty ) \to [0, \infty )$ with f(0)= 0 and $f(r) > 0$ $\forall r > 0$ such that

$$ f\bigl(d(x_{n}, \mathscr{F})\bigr) \le d(x_{n}, T_{n_{0}} x_{n}), \quad \forall n \ge 1, $$

(3.3)

then the sequence $\{x_{n}\}$ defined by (3.2) converges strongly (i.e., in metric topology) to some point $p^{*} \in \mathscr{F}$.

Proof

First we observe that for each $i\ge 1$, $T_{i}: C \to C$ is a $(\{\nu _{n}^{(i)}\}, \{\mu _{n}^{(i)}\}, \zeta ^{(i)})$-total asymptotically nonexpansive map**. By condition (ii), for each $n \ge 1$ and any $x, y \in C$, we have

$$ d\bigl(T_{i}^{n} x, T_{i}^{n} y\bigr) \le d(x,y) + \nu _{n}^{(i)}\zeta ^{i}\bigl( d(x,y) \bigr)+ \mu _{n}^{(i)} \le \bigl(1+ \nu _{n}^{(i)} M_{*}\bigr)d(x,y) + \mu _{n}^{(i)}. $$

(3.4)

(I) First we prove that the following limits exist:

$$ \lim_{n \to \infty}d(x_{n}, \mathscr{F}), \quad \text{and} \quad \lim_{n \to \infty}d(x_{n}, p)\quad\text{for each } p \in \mathscr{F}. $$

(3.5)

In fact, since $p \in \mathscr{F}$ and $T_{i}$, $i \ge 1$ is a total asymptotically nonexpansive map**, it follows from (3.4) and Lemma 2.5 that

$$ \begin{aligned} d(y_{n}, p)& = d\bigl((1-\beta _{n}) x_{n} \oplus \beta _{n} T_{i(n)}^{k(n)}x_{n}, p\bigr) \\ & \le (1-\beta _{n})d( x_{n}, p) + \beta _{n} d \bigl(T_{i(n)}^{k(n)}x_{n}, p\bigr) \\ & = (1-\beta _{n}) d(x_{n}, p) + \beta _{n} \bigl\{ d(x_{n}, p)+ \nu _{k(n)}^{i(n)} \zeta ^{i(n)} \bigl(d(x_{n}, p)\bigr)+ \mu ^{i(n)}_{k(n)}\bigr\} \\ & \le d(x_{n}, p)+ \nu _{k(n)}^{i(n)}\zeta ^{i(n)}\bigl(d(x_{n}, p)\bigr)+ \mu _{k(n)}^{i(n)} \\ & \le \bigl(1+ \nu _{k(n)}^{i(n)} M_{*} \bigr)d(x_{n}, p) + \mu _{k(n)}^{i(n)} \end{aligned} $$

(3.6)

and

$$ \begin{aligned} d(x_{n+1}, p)& = d\bigl((1 - \alpha _{n}) x_{n} \oplus \alpha _{n} T_{i(n)}^{k(n)}y_{n}, p\bigr) \\ & \le (1- \alpha _{n}) d( x_{n}, p) + \alpha _{n} d\bigl(T_{i(n)}^{k(n)}y_{n}, p\bigr) \\ & = (1- \alpha _{n})d(x_{n}, p) + \alpha _{n} \bigl\{ d(y_{n}, p)+ \nu ^{i(n)}_{k(n)} \zeta ^{i(n)}\bigl( d(y_{n}, p)\bigr) + \mu ^{i(n)}_{k(n)} \bigr\} \\ & \le (1- \alpha _{n}) d(x_{n}, p) + \alpha _{n} \bigl\{ \bigl(1+ \nu ^{i(n)}_{k(n)} M_{*} \bigr)d(y_{n}, p) + \mu ^{i(n)}_{k(n)}\bigr\} . \end{aligned} $$

(3.7)

Substituting (3.6) into (3.7) and simplifying it, we have

$$ d(x_{n+1}, p) \le (1 + \sigma _{n})d(x_{n}, p) + \xi _{n}, \quad \forall n \ge 1 \text{ and } p\in \mathscr{F}, $$

(3.8)

and so

$$ d(x_{n+1}, \mathscr{F}) \le (1 + \sigma _{n})d(x_{n}, \mathscr{F}) + \xi _{n}, \quad \forall n \ge 1, $$

(3.9)

where $\sigma _{n} = b\nu ^{i(n)}_{k(n)} M_{*}(2 + \nu ^{i(n)}_{k(n)} M_{*})$, $\xi _{n} = b (2 + \nu ^{i(n)}_{k(n)} M_{*})\mu ^{i(n)}_{k(n)}$. By virtue of condition (i),

$$ \sum_{n=1}^{\infty }\sigma _{n} < \infty \quad \text{and}\quad \sum_{n=1}^{ \infty } \xi _{n} < \infty . $$

(3.10)

By Lemma 2.12, the limits $\lim_{n \to \infty}d(x_{n}, \mathscr{F})$ and $\lim_{n \to \infty}d(x_{n}, p)$ exist for each $p \in \mathscr{F}$.

(II) Next we prove that for each $i \ge 1$ there exists some subsequence $\{x_{m(\in \Gamma _{i})}\} \subset \{x_{n}\}$ such that

$$ \lim_{m(\in \Gamma _{i})\to \infty}d(x_{m}, T_{i} x_{m}) = 0, $$

(3.11)

where $\Gamma _{i}$ is the set of positive integers defined by Lemma 3.1(2).

In fact, it follows from (3.5) that for each given $p \in \mathscr{F}$, the limit $\lim_{n \to \infty}d(x_{n}, p)$ exists. Without loss of generality, we can assume that

$$ \lim_{n \to \infty}d(x_{n}, p) = r \ge 0. $$

(3.12)

From (3.6) we have

$$ \limsup_{n \to \infty}d(y_{n}, p)\le \lim _{n \to \infty} \bigl\{ \bigl(1+ \nu _{k(n)}^{i(n)} M_{*}\bigr)d(x_{n}, p) + \mu _{k(n)}^{i(n)} \bigr\} = r. $$

(3.13)

Since

$$ \begin{aligned} d\bigl(T_{i(n)}^{k(n)} y_{n}, p\bigr) & \le d(y_{n}, p) + \nu ^{i(n)}_{k(n)} \zeta ^{i(n)}\bigl(d(y_{n}, p)\bigr) + \nu ^{i(n)}_{k(n)} \\ & \le \bigl(1 + \nu ^{i(n)}_{k(n)} M_{*}\bigr) d(y_{n}, p) + \mu ^{i(n)}_{k(n)}, \quad \forall n \ge 1, \end{aligned} $$

from (3.13) we have

$$ \limsup_{n \to \infty}d\bigl(T_{i(n)}^{k(n)} y_{n}, p\bigr)\le r. $$

(3.14)

In addition, it follows from (3.8) that

$$ \begin{aligned} d(x_{n+1}, p)& = d\bigl((1 - \alpha _{n}) x_{n} \oplus \alpha _{n} T_{i(n)}^{k(n)}y_{n}, p\bigr) \\ &\le (1 + \sigma _{n})d(x_{n}, p) + \xi _{n}. \end{aligned} $$

This implies that

$$ \lim_{n \to \infty}d\bigl((1 - \alpha _{n}) x_{n} \oplus \alpha _{n} T_{i(n)}^{k(n)}y_{n}, p \bigr)= r. $$

(3.15)

From (3.12), (3.14), (3.15), and Proposition 2.11, we have

$$ \lim_{n \to \infty}d\bigl( x_{n}, T_{i(n)}^{k(n)}y_{n} \bigr) = 0. $$

(3.16)

Since

$$ \begin{aligned} d(x_{n}, p) &\le d\bigl(x_{n}, T_{i(n)}^{k(n)}y_{n}\bigr) + d\bigl(T_{i(n)}^{k(n)}y_{n}, p\bigr) \\ & \le d\bigl(x_{n}, T_{i(n)}^{k(n)}y_{n} \bigr) + \bigl\{ d(y_{n}, p) +\nu ^{i(n)}_{k(n)} \zeta ^{i(n)}\bigl(d(y_{n}, p)\bigr) + \mu ^{i(n)}_{k(n)} \bigr\} \\ & \le d\bigl(x_{n}, T_{i(n)}^{k(n)}y_{n} \bigr) + \bigl(1 + \nu ^{i(n)}_{k(n) }M_{*} \bigr)d(y_{n}, p) + \mu ^{i(n)}_{k(n)}. \end{aligned} $$

Taking lim inf on both sides of the above inequality, from (3.16) we have

$$ \liminf_{n \to \infty}d(y_{n}, p) \ge r. $$

This together with (3.13) shows that

$$ \lim_{n \to \infty}d(y_{n}, p) = r. $$

(3.17)

Using (3.6) we have

$$ \begin{aligned} r &= \lim_{n \to \infty}d(y_{n}, p) = \lim_{n \to \infty}\bigl\{ d\bigl((1- \beta _{n}) x_{n} \oplus \beta _{n} T_{i(n)}^{k(n)}x_{n}, p\bigr)\bigr\} \\ &\le \lim_{n \to \infty} \bigl\{ \bigl(1+ \nu _{k(n)}^{i(n)} M_{*}\bigr)d(x_{n}, p) + \mu _{k(n)}^{i(n)} \bigr\} = r. \end{aligned} $$

(3.18)

This implies that

$$ \lim_{n \to \infty}\bigl\{ d\bigl((1-\beta _{n}) x_{n} \oplus \beta _{n} T_{i(n)}^{k(n)}x_{n}, p\bigr)\bigr\} =r. $$

(3.19)

Similarly, we can also prove that

$$ \limsup_{n \to \infty} d\bigl(T_{i(n)}^{k(n)}x_{n}, p\bigr) \le \limsup_{n \to \infty}\bigl\{ d(x_{n},p) + \nu _{k(n)}^{i(n)}\zeta ^{i(n)}\bigl(d(x_{n},p) \bigr) + \mu _{k(n)}^{i(n)}\bigr\} \le r. $$

This together with (3.12), (3.19), and Lemma 2.11 gives that

$$ \lim_{n \to \infty}d\bigl(x_{n},T_{i(n)}^{k(n)}x_{n} \bigr) = 0. $$

(3.20)

Therefore we have

$$ \begin{aligned} d(x_{n}, y_{n}) & = d \bigl(x_{n}, (1-\beta _{n})x_{n} \oplus \beta _{n} T_{i(n)}^{k(n)}x_{n}\bigr) \\ & \le \beta _{n} d\bigl(x_{n},T_{i(n)}^{k(n)}x_{n} \bigr) \to 0 \quad (\text{as } n \to \infty ). \end{aligned} $$

(3.21)

Furthermore, it follows from (3.16) that

$$ \begin{aligned} d(x_{n +1}, x_{n}) & = d((1 - \alpha _{n}) x_{n} \oplus \alpha _{n} d \bigl(T_{i(n)}^{k(n)}y_{n}, x_{n}\bigr) \\ & \le \alpha _{n} d\bigl(T_{i(n)}^{k(n)}y_{n}, x_{n}\bigr) \to 0 \quad (\text{as } n \to \infty ). \end{aligned} $$

(3.22)

This together with (3.21) shows that

$$ d(x_{n +1}, y_{n}) \le d(x_{n +1}, x_{n}) + d(x_{n}, y_{n}) \to 0 \quad (\text{as } n \to \infty ). $$

(3.23)

From Lemma 3.1, (3.16), (3.20), (3.22), and (3.23), we have that for each given positive integer $i \ge 1$, there exist subsequences $\{x_{m}\}_{m\in \Gamma _{i}}$, $\{y_{m}\}_{m \in \Gamma _{i}}$, and $\{k(m)\}_{m \in \Gamma _{i}} \subset K_{i}: = \{k(m): m \in \Gamma _{i}, m = i + \frac{(k(m)-1)k(m)}{2}, k(m) \ge i\}$ such that

$$ \begin{aligned} d(x_{m}, T_{i} x_{m}) \le{}& d\bigl(x_{m}, T_{i}^{k(m)}x_{m} \bigr) + d\bigl(T_{i}^{k(m)}x_{m}, T_{i}^{k(m)}y_{m-1}\bigr) + d\bigl(T_{i}^{k(m)}y_{m-1}, T_{i} x_{m}\bigr) \\ \le{}& d\bigl(x_{m}, T_{i}^{k(m)}x_{m} \bigr) + \bigl\{ d(x_{m}, y_{m-1}) + \nu _{k(m)}^{(i)} \zeta ^{(i)}\bigl(d(x_{m},y_{m-1})\bigr) + \mu _{k(m)}^{(i)}\bigr\} \\ & {} + L_{i} (d\bigl(T_{i}^{k(m)-1}y_{m-1}, x_{m}\bigr) \\ \le{}& d\bigl(x_{m}, T_{i}^{k(m)}x_{m} \bigr) + \bigl\{ d(x_{m},y_{m-1}) + \nu _{k(m)}^{(i)} \zeta ^{(i)}\bigl(d(x_{m},y_{m-1})\bigr) + \mu _{k(m)}^{(i)}\bigr\} \\ & {} + L_{i} (d\bigl(T_{i}^{k(m-1)}y_{m-1}, x_{m-1}\bigr) + L_{i} (d(x_{m-1}, x_{m}) \to 0 (\text{as} m \to \infty ). \end{aligned} $$

(3.24)

The conclusion (3.11) is proved.

(III) Now we prove that $\{x_{n}\}$ converges strongly (i.e., in the metric topology) to some point $p^{*} \in \mathscr{F}$.

In fact, it follows from (3.11) and (3.24) that for given map** $T_{n_{0}}$ there exists some subsequence $\{x_{m}\}_{m \in \Gamma _{n_{0}}}$ of $\{x_{n}\}$ such that

$$ \lim_{m(\in \Gamma _{n_{0}}) \to \infty}d(x_{m}, T_{n_{0}} x_{m}) = 0. $$

By (3.3) we have

$$ f\bigl(d(x_{m}, \mathscr{F})\bigr) \le d(x_{m}, T_{n_{0}} x_{m})\quad \forall m \ge 1. $$

Let $m \to \infty$, and taking limsup on the above inequality, we have $\lim_{m\to \infty} f(d(x_{m}, \mathscr{F})) = 0$. By the property of f, this implies that

$$ \lim_{m(\in \Gamma _{n_{0}}) \to \infty} d(x_{m}, \mathscr{F}) = 0. $$

(3.25)

Next we prove that $\{x_{m}\}_{m \in \Gamma _{n_{0}}}$ is a Cauchy sequence in C. In fact, it follows from (3.8) that for any $p \in \mathscr{F}$

$$ d(x_{m+1}, p) \le (1 + \sigma _{m})d(x_{m}, p) + \xi _{m}, \quad \forall m ( \in \Gamma _{n_{0}}) \ge 1, $$

where $\sum_{m = 1}^{\infty }\sigma _{m} < \infty$ and $\sum_{m = 1}^{\infty }\xi _{m} < \infty$. Hence, for any positive integers $j, n \in \Gamma _{n_{0}}$, $n > j$, and $n = m + j$ for some positive integer m, we have

$$ \begin{aligned} d(x_{n}, x_{j}) &= d(x_{j+m}, x_{j}) \le d(x_{j+m},p) + d(x_{j}, p) \\ & \le (1 + \sigma _{j+m-1})d(x_{j+m-1}, p) + \xi _{j+m-1} + d(x_{j}, p). \end{aligned} $$

Since for each $x \ge 0$, $1+x \le e^{x}$, we have

$$ \begin{aligned} d(x_{n}, x_{j}) ={}& d(x_{j+m}, x_{j}) \\ \le{}& e^{\sigma _{j+m-1}} d(x_{j+m-1}, p) + \xi _{j+m-1} + d(x_{j}, p) \\ \le{}& e^{\sigma _{j+m-1} + \sigma _{j+m-2}}d(x_{j+m-2}, p) + e^{ \sigma _{j+m-1}}\xi _{j+m-2} + \xi _{j+m-1} + d(x_{j}, p) \\ \le{}& \cdots \\ \le{}& e^{\sum _{i=j}^{j+m-1}\sigma _{i}} d(x_{j}, p) + e^{\sum _{i=j+1}^{j+m-1} \sigma _{i}}\xi _{j} + e^{\sum _{i=j+2}^{j+m-2}\sigma _{i}}\xi _{j+1} + \cdots \\ & {} + e^{\sigma _{j+m-1}}\xi _{j+m-2} + \xi _{j+m-1} + d(x_{j}, p) \\ \le {}& (1+ M)d(x_{j}, p) + M\sum_{i=j}^{j+m - 1} \xi _{i} \\ ={}& (1+ M)d(x_{j}, p) + M\sum_{i=j}^{n - 1} \xi _{i}, \quad \text{for each } p\in \mathscr{F}. \end{aligned} $$

Therefore we have

$$ d(x_{n}, x_{j}) = d(x_{j+m}, x_{j}) \le (1+ M)d(x_{j}, \mathscr{F}) + M \sum_{i=j}^{n-1} \xi _{i}, $$

where $M = e^{\sum _{i=1}^{\infty}}\sigma _{i} < \infty$. By (3.25) we have

$$ d(x_{n}, x_{j}) \le (1+ M)d(x_{j}, \mathscr{F}) + M\sum_{i=j}^{n-1} \xi _{i} \to 0 \quad \bigl(\text{as } n, j (\in \Gamma _{n_{0}}) \to \infty \bigr). $$

This shows that the subsequence $\{x_{m}\}_{m \in \Gamma _{n_{0}}}$ is a Cauchy sequence in C. Since C is a closed subset in a complete CAT(κ) space X, it is complete. Without loss of generality, we can assume that the subsequence $\{x_{m}\}$ converges strongly (i.e., in metric topology in X) to some point $p^{*} \in C$. It is easy to know that $\mathscr{F}$ is a closed subset in C. Since $\lim_{m \to \infty} d(x_{m}, \mathscr{F}) = 0$, $p^{*} \in \mathscr{F}$. By using (3.5), it yields that the whole sequence $\{x_{n}\}$ converges strongly to $p^{*} \in \mathscr{F}$.

This completes the proof of Theorem 3.2. □

Remark 3.3

It should be pointed out that if $(X, d)$ is a CAT(0) space, then X is uniformly convex, its modulus of uniform convexity $\eta (r,\epsilon ) = \frac{\epsilon ^{2}}{8}$ [25, 27] and all of its geodesic triangles satisfy the $\operatorname{CAT} (\kappa )$ inequality. These imply that if $(X, d)$ is a CAT(0) space, then the conditions that appeared in Theorem 3.2 “$(X,d)$ is uniformly convex and $\operatorname{diam}(X)\le \frac{\frac{\pi}{2} - \epsilon}{\sqrt{\kappa}}$ for some $\epsilon \in (0, \frac{\pi}{2})$” are of no use here. Therefore from Theorem 3.2 we can obtain the following.

Theorem 3.4

Let $(X,d)$ be a complete CAT(0) space. Let C be a nonempty closed and convex subset of X, and for each $i \ge 1$, let $T_{i}: C\rightarrow C$ be uniformly $L_{i}$-Lipschitzian and $(\{\nu _{n}^{(i)}\}, \{\mu _{n}^{(i)}\}, \zeta ^{(i)})$-total asymptotically nonexpansive map**s defined by (1.2) such that

• (i) $\sum_{i=1}^{\infty }\sum_{n=1}^{\infty }\nu _{n}^{(i)} < \infty$, $\sum_{i=1}^{\infty}\sum_{n=1}^{\infty }\mu _{n}^{(i)} < \infty$,

• (ii) there exists a constant $M_{*} > 0$ such that $\zeta ^{(i)}(r)\leq M_{*} r$, $\forall r\geq 0$, $i= 1, 2, \ldots$ ;

• (iii) there exist constants $a, b \in (0, 1)$ with $0 < b(1-a) \le \frac{1}{2}$ such that $\{\alpha _{n}\}, \{\beta _{n}\} \subset [a, b]$.

If $\mathscr{F} := \bigcap_{i = 1}^{\infty }F(T_{i})\neq \emptyset$ and there exist a map** $T_{n_{0}} \in \{T_{i}\}_{i=1}^{\infty}$ and a nondecreasing function $f: [0, \infty ) \to [0, \infty )$ with $f(0)= 0$ and $f(r) > 0$ $\forall r > 0$ such that

$$ f\bigl(d(x_{n}, \mathscr{F})\bigr) \le d(x_{n}, T_{n_{0}} x_{n}),\quad \forall n \ge 1, $$

then the sequence $\{x_{n}\}$ defined by (3.2) converges strongly (i.e., in metric topology) to some point $p^{*} \in \mathscr{F}$.