On some results related to the Karlsson–Nussbaum conjecture in geodesic spaces

Abstract

We show a Wolff–Denjoy type theorem in the case of a one-parameter continuous semigroup of nonexpansive mappings in which there is a compact mapping. Using the notion of attractor, we are also able to prove some specific properties directly related to the Karlsson–Nussbaum conjecture.

1 Introduction

Let us suppose that $f:\Delta \rightarrow \Delta$ is a holomorphic map of the unit disc $\Delta \subset \mathbb {C}$ without a fixed point. The classical Wolff–Denjoy theorem asserts that then there is a point $\xi \in \partial \Delta$ such that the iterates $f^{n}$ converge locally uniformly to $\xi$ on $\Delta$. There is a wide literature concerning various generalizations of this theorem (see [1, 5, 9, 13, 15, 16] and references therein). One of the important generalizations was given by Beardon who initially considered the Wolff–Denjoy theorem in a purely geometric way depending only on the hyperbolic properties of a metric [3]. In the next step, Beardon developed his results for strictly convex bounded domains with the Hilbert metric [4]. Considering the notion of the omega limit set $\omega _{f}(x)$ as the set of accumulation points of the sequence $f^{n}(x)$ and the notion of the attractor $\Omega _{f}=\bigcup _{x\in D}\omega _{f}(x)$, Karlsson and Noskov showed in [11] that if a bounded convex domain D is endowed with the Hilbert metric $d_{H}$, then the attractor $\Omega _{f}$ of a fixed-point free nonexpansive map $f:D\rightarrow D$ is a star-shaped subset of $\partial D$. This has led to a conjecture formulated by Karlsson and Nussbaum (see [10, 15]) asserting that if D is a bounded convex domain in a finite-dimensional real vector space and $f:D\rightarrow D$ is a fixed-point free nonexpansive mapping acting on the Hilbert metric space $(D,d_{H})$, then there exists a convex set $\Omega \subseteq \partial D$ such that for each $x\in D$, all accumulation points $\omega _{f}(x)$ of the orbit O(x, f) lie in $\Omega$. It remains one of the major problems in the field.

The first objective of this paper is to extend the Wolff–Denjoy type theorem to the case of one-parameter continuous semigroups of nonexpansive (1-Lipschitz) mappings in some quasi-geodesic spaces. We show in Sect. 3 that if (Y, d) is a $(1,\kappa )$-quasi-geodesic space satisfying Axiom $1'$ and Axiom $4'$ (introduced in Sect. 2) and $S=\{f_{t}:Y\rightarrow Y\}_{t\ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings on Y without bounded orbits and there exists $t_0>0$ such that $f_{t_0}:Y \rightarrow Y$ is a compact mapping, then there exists $\xi \in \partial Y$ such that the semigroup S converges uniformly on bounded sets of Y to $\xi$. We apply our consideration in the case of bounded strictly convex domains in a Banach space with the metric satisfying condition

$$\begin{aligned} d(sx+(1-s)y,z)\le \max \{d(x,z),d(y,z)\}. \end{aligned}$$

(C)

In particular, we obtain the Wolff–Denjoy type theorem for Hilbert’s and Kobayashi’s metrics but its applicability is much wider.

The second aim of this note is to show some results related to the Karlsson–Nussbaum conjecture. We start by generalizing the Abate and Raissy result asserting that a big horoball considered in a bounded and convex domain with the Kobayashi distance is a star-shaped set with respect to the center of the horoball. Using this fact we show in Sect. 4 that for a compact nonexpansive mapping f acting on a bounded convex domain in a Banach space equipped with the metric satisfying condition (C), there exists $\xi$ on the boundary such that $\Omega _{f} \subset {{\,\textrm{ch}\,}}({{\,\textrm{ch}\,}}(\xi ))$. Karlsson proved in [10] a special case of the Karlsson–Nussbaum conjecture by showing that the attractor of a fixed-point free nonexpansive mapping is a star-shaped subset of the boundary of the space. In the proof he used the Gromov product. We present a shorter proof of a little more general result, without using the Gromov product, for every metric space (D, d) satisfying condition

$$\begin{aligned} \lim _{n\rightarrow \infty } [d(x_{n},y_{n})-\max \{d(x_{n},z),d(y_{n},z)\}]=\infty \end{aligned}$$

for any sequences $\{x_n\}$ and $\{y_n\}$ converging to distinct points x and y on the boundary such that the segment $[x,y] \subseteq \partial D$, and for any $z \in D$ (see Sect. 4 for more details).

In Sect. 5, we formulate counterparts of the above results for a one-parameter continuous semigroup of nonexpansive mappings.

2 Preliminaries

Let (Y, d) be a metric space. Recall that a curve $\gamma :[a,b]\rightarrow Y$ is called $(\lambda ,\kappa )$-quasi-geodesic if there exists $\lambda \ge 1,\kappa \ge 0$ such that for all $t_{1},t_{2}\in [a,b],$

$$\begin{aligned} \frac{1}{\lambda }\left| t_{1}-t_{2}\right| -\kappa \le d(\gamma (t_{1}),\gamma (t_{2}))\le \lambda \left| t_{1}-t_{2}\right| +\kappa . \end{aligned}$$

A metric space Y is called $(\lambda ,\kappa )$-quasi-geodesic if every pair of points in Y can be connected by a $(\lambda ,\kappa )$-quasi-geodesic. A convex bounded domain in a Banach space equipped with the Kobayashi distance is an example of a $(1, \kappa )$-quasi-geodesic metric space for any $\kappa >0$.

We are interested in considerations related to nonexpansive and contractive mappings in $(\lambda ,\kappa )$-quasi-geodesic metric spaces. We call a map $f:Y\rightarrow Y$ nonexpansive if

$$\begin{aligned} d(f(x),f(y))\le d(x,y) \end{aligned}$$

for every $x,y\in Y$. A map $f:Y\rightarrow Y$ is called contractive if

$$\begin{aligned} d(f(x),f(y))<d(x,y) \end{aligned}$$

for every distinct $x,y\in Y$.

Beardon [4] and Karlsson [9] considered some four properties of metric spaces called axioms.

Axiom 1. The metric space (Y, d) is an open dense subset of a compact metric space $(\overline{Y},\overline{d})$, whose relative topology coincides with the topology of Y. For any $w\in Y$, if $\{x_{n}\}$ is a sequence in Y converging to $\xi \in \partial Y=\overline{Y}{\setminus } Y$, then

$$\begin{aligned} d(x_{n},w)\rightarrow \infty . \end{aligned}$$

Axiom 2. If $\{x_{n}\}$ and $\{y_{n}\}$ are sequences in Y converging to distinct points in $\partial Y$, then for every $w\in Y,$

$$\begin{aligned} d(x_{n},y_{n})-\max \{d(x_{n},w),d(y_{n},w)\}\rightarrow \infty . \end{aligned}$$

If D is a bounded domain of $\mathbb {R}^n$ or $\mathbb {C}^n$ and (D, d) is a complete geodesic space, then the metric space (D, d) is an example of the space which satisfies Axiom 1. The bounded strictly convex domain in $\mathbb {R}^n$ equipped with Hilbert’s metric satisfy Axiom 2.

The Axioms 3 and 4 can be found in [9]. The objective of this paper is to present the results in non-proper case. Since in general, there are sequences without convergent subsequences, we slightly modify three of the axioms to make them better suited for non-proper metric spaces.

Axiom 1’. The metric space (Y, d) is an open dense subset of a metric space $(\overline{Y},\overline{d})$, whose relative topology coincides with the metric topology. For any $w\in Y$, if $\{x_{n}\}$ is a sequence in Y converging to $\xi \in \partial Y=\overline{Y}{\setminus } Y$, then

$$\begin{aligned} d(x_{n},w)\rightarrow \infty . \end{aligned}$$

Notice that Axiom 1’ implies that if $A\subset Y$ is bounded, then the $\overline{d}$-closure of A does not intersect the boundary $\partial Y$ and hence coincides with the d-closure of A.

Axiom 3’. If $\{x_{n}\}$ and $\{y_{n}\}$ are sequences in Y, $x_{n}\rightarrow \xi \in \partial Y$, $y_{n}\rightarrow \eta \in \partial Y$, and if for some $w\in Y,$

$$\begin{aligned} d(x_{n},y_{n})-d(y_{n},w)\rightarrow -\infty , \end{aligned}$$

then $\xi =\eta .$

Axiom 4’. If $\{x_{n}\}$ and $\{y_{n}\}$ are sequences in Y, $x_{n}\rightarrow \xi \in \partial Y$, $y_{n}\rightarrow \eta \in \partial Y$, and if for all n, 

$$\begin{aligned} d(x_{n},y_{n})\le c \end{aligned}$$

for some constant c, then $\xi =\eta .$

The content of the Axiom 2’ coincides with the Axiom 2, the difference is whether we have a proper or a non-proper space under consideration. It is not hard to show that Axioms $1'$ and $2'$ imply Axiom $3'$ and Axioms $1'$ and $3'$ imply Axiom $4'$.

An example of a metric space satisfying Axiom $1'$ is the bounded convex domain D of a Banach space equipped with the Hilbert metric (see Theorem 4.13 [15]). Additionally, the bounded convex domain of a complex Banach space with the Kobayashi distance, is a complete metric space satisfying Axiom $1'$ (see [6, 12]). In general, the Hilbert metric do not satisfy the Axiom $2'$, but provided that it is defined in a strictly convex domain, the Axiom $2'$ is already true [14]. A strictly convex domain equipped with the Kobayashi distance satisfies Axiom $3'$. Moreover, Lemma 3.8, in the third section, shows that Axiom $3'$ is satisfied for a very wide class of metrics. The Gromov hyperbolic metric spaces satisfy the Axiom $2'$, $3'$ and $4'$.

Let us introduce one more property of metric spaces satisfying Axiom 1’ that we will call Axiom $5'$.

Axiom 5’ If $\{x_n\}$ and $\{y_n\}$ are sequences in Y and $d(x_n,y_n) \rightarrow 0$, as $n \rightarrow \infty$, then

$$\begin{aligned} \overline{d}(x_n,y_n) \rightarrow 0, \quad n \rightarrow \infty . \end{aligned}$$

We say that a mapping $f:Y\rightarrow Y$ is compact if $\overline{f(Y)}^{\overline{d}}$, the $\overline{d}$-closure of f(Y), is compact in $(\overline{Y},\overline{d}).$ The following notion of a horoball will be needed throughout the paper. We recall the general definitions introduced by Abate in [1]. Define the small horoball of center $\xi \in \partial Y$, pole $z_{0}\in Y$ and radius $r\in \mathbb {R}$ by

$$\begin{aligned} E_{z_{0}}(\xi ,r)=\{y\in Y:\limsup _{w\rightarrow \xi }d(y,w)-d(w,z_{0})\le r\} \end{aligned}$$

and the big horoball by

$$\begin{aligned} F_{z_{0}}(\xi ,r)=\{y\in Y:\liminf _{w\rightarrow \xi }d(y,w)-d(w,z_{0})\le r\}. \end{aligned}$$

Karlsson [9] and Lemmens and Nussbaum [14] note that in a metric space satisfying Axioms 1 and 2 (see [9] for more details), each horoball intersects the boundary of the space at exactly one point. The next lemma shows an important property of big horoballs, i.e., Axiom $3'$ implies that the intersection of big horoballs’ closures consists of a single point. Moreover, it is the center of all horoballs.

Lemma 2.1

Let (Y, d) satisfy Axioms $1'$ and $3'$, $z_{0}\in Y$ and $\zeta \in \partial Y$. Then,

$$\begin{aligned} \bigcap _{r\in \mathbb {R}}\overline{ F_{z_{0}}(\zeta ,r)}=\{\zeta \}. \end{aligned}$$

The proof proceeds similarly to the case of proper metric which can be found in [7].

We conclude this section by recalling the definitions of Hilbert’s and Kobayashi’s metrics. Let K be a closed normal cone with a non-empty interior in a real Banach space V. We say that $y\in K$ dominates $x\in V$ if there exists $\alpha ,\beta \in \mathbb {R}$ such that $\alpha y\le x\le \beta y$. This notion yields on K an equivalence relation $\sim _{K}$ by $x\sim _{k}y$ if x dominates y and y dominates x. For all $x,y\in K$ such that $x\sim _{K}y$ and $y\ne 0,$ define

$$\begin{aligned} M(x/y)=\inf \{\beta >0:x\le \beta y\} \end{aligned}$$

and

$$\begin{aligned} m(x/y)=\sup \{\alpha >0:\alpha y\le x\}. \end{aligned}$$

The Hilbert (pseudo-)metric is defined by

$$\begin{aligned} d_{H}(x,y)=\log \bigg (\frac{M(x/y)}{m(x/y)}\bigg ). \end{aligned}$$

Moreover, we put $d_{H}(0,0)=0$ and $d_{H}(x,y)=\infty$ if $x\sim _{K}y$. It can be shown that $d_{H}$ is a metric iff $x=\lambda y$ for some $\lambda >0.$

It will be important to us that a bounded convex domain in a Banach space equipped with the Hilbert metric is a complete geodesic space that satisfies Axiom $1'$ [see [15], Theorem 4.13]). Moreover, the following theorem is true.

Theorem 2.2

Let D be a bounded convex domain in a real Banach space and let $\{x_n\},\{y_n\} \subset D$. If $d_H(x_n,y_n) \rightarrow 0$, as $n \rightarrow \infty$, then

$$\begin{aligned} ||x_n-y_n|| \rightarrow 0, \quad n \rightarrow 0. \end{aligned}$$

Proof

Let $\{x_n\},\{y_n\} \subset D$ be sequences in D and for any $n \in \mathbb {N}$ consider a straight line passing through $x_n$ and $y_n$ that intersects the boundary of D in precisely two points $a_n$ and $b_n$ such that $x_n$ is between $a_n$ and $y_n$, and $y_n$ is between $x_n$ and $b_n$. Then,

$$\begin{aligned} d_H(x_n,y_n)= & {} \log \bigg ( \frac{||y_n-a_n ||\cdot ||x_n-b_n||}{||x_n-a_n||\cdot ||y_n-b_n||} \bigg ) \\= & {} \log \bigg ( \frac{(||x_n-a_n||+||x_n+y_n||)(||y_n-b_n||+||x_n-y_n||)}{||x_n-a_n||\cdot ||y_n-b_n||} \bigg ) \\= & {} \log \bigg (\bigg (1+\frac{||x_n-y_n||}{||x_n-a_n||}\bigg )\bigg (1 + \frac{||x_n-y_n||}{||y_n-b_n||}\bigg ) \bigg ). \end{aligned}$$

Let $d = {{\,\textrm{diam}\,}}D$. We note that $||x_n-a_n||\le d$ and $||y_n-b_n||\le d$ for every $n \in \mathbb {N}$. Hence,

$$\begin{aligned} \bigg (1+\frac{||x_n-y_n||}{||x_n-a_n||}\bigg )\bigg (1 + \frac{||x_n-y_n||}{||y_n-b_n||}\bigg ) \ge \bigg (1+\frac{||x_n-y_n||}{d}\bigg )\bigg (1 + \frac{||x_n-y_n||}{d}\bigg ). \end{aligned}$$

Consider subsequences $\{x_{n_k}\}$ and $\{y_{n_k}\}$ of sequences $\{x_{n}\}$ and $\{y_{n}\}$, respectively. If there existed $\delta >0$ and $k_0 \in \mathbb {N}$ such that $||x_{n_k}-y_{n_k}|| \ge \delta$ for $k \ge k_0$, then

$$\begin{aligned} \bigg (1+\frac{||x_{n_k}-y_{n_k}||}{d}\bigg )\bigg (1 + \frac{||x_{n_k}-y_{n_k}||}{d}\bigg ) \ge \bigg (1+\frac{\delta }{d}\bigg )\bigg (1 + \frac{\delta }{d}\bigg ) = \delta _1 >1 \nonumber \\ \end{aligned}$$

(2.1)

that would contradict our assumption $d_H(x_n,y_n) \rightarrow 0$. Therefore, $||x_n-y_n|| \rightarrow 0$, as $n \rightarrow \infty$. $\square$

It means that a convex bounded domain in a Banach space equipped with the Hilbert metric satisfies Axiom $5'$.

Recall that if D is a bounded convex domain of a complex Banach space V, then the Kobayashi distance between $z,w\in D$ is given by

$$\begin{aligned} k_{D}(z,w)=\inf \{k_{\Delta }(0,\gamma )\mid \exists \varphi \in \text{ Hol } (\Delta ,D):\varphi (0)=z,\,\varphi (\gamma )=w\}, \end{aligned}$$

where $k_{\Delta }$ denotes the Poincaré metric on the unit disc $\Delta$.

The second example of a space that satisfies the Axiom $5'$ (with respect to the closure in the norm $(\overline{D},||\cdot ||)$) is a convex bounded domain in a Banach space equipped with the Kobayashi distance (see Theorem 3.4, [12]). It follows from the following theorem.

Theorem 2.3

Suppose D is a bounded convex domain of a Banach space. Then,

$$\begin{aligned} \arg \tanh \bigg (\frac{||x-y||}{{{\,\textrm{diam}\,}}_{||\cdot ||}D}\bigg ) \le k_D(x,y) \end{aligned}$$

for every $x,y \in D$.

It turns out that both Hilbert’s and Kobayashi’s metric d satisfies the following condition (C) that is equivalent to the convexity of balls in D:

$$\begin{aligned} d(sx+(1-s)y,z)\le \max \{d(x,z),d(y,z)\} \end{aligned}$$

(C)

for all $x,y,z\in D$ and $s\in [0,1]$ (see, e.g., [7]).

3 Wolff–Denjoy theorems for semigroups

Although, the subject of this section is the Wolff–Denjoy theorem in the continuous case (i.e., for semigroups of nonexpansive mappings), we will need the Wolff–Denjoy type theorem for iterates of a nonexpansive mapping (see [7]).

Theorem 3.1

Let (Y, d) be a $(1,\kappa )$-quasi-geodesic space satisfying Axiom $1'$ and Axiom $4'$, and suppose that for every $\zeta \in \partial Y$ and $z_{0}\in Y$, the intersection of horoballs’ closures $\bigcap _{r\in \mathbb {R}}\overline{F_{z_{0}}(\zeta ,r)}^{\overline{d}}$ consists of a single point. If $f:Y \rightarrow Y$ is a compact nonexpansive mapping without bounded orbits, then there exists $\xi \in \partial Y$ such that the iterates $f^n$ of f converge uniformly on bounded sets of Y to $\xi$.

In the further considerations, we will use the following lemma, the proof of which can be found in [8].

Lemma 3.2

Let $S=\{f_{t}:Y\rightarrow Y\}_{t\ge 0}$ be a one-parameter continuous semigroup of nonexpansive mappings on Y, C a compact subset of Y and $t_{0}>0.$ Then,

$$\begin{aligned} K=\{f_{s}(x):0\le s\le t_{0},\;x\in C\}=\bigcup _{0\le s\le t_{0}}f_{s}(C) \end{aligned}$$

is compact.

One of the classical arguments in this line of research is the Całka theorem which in the original version concerns a metric space with the property that each bounded subset is totally bounded. We will need the following counterpart of the Całka theorem for semigroups ([8], Theorem 3.3).

Theorem 3.3

Suppose that (Y, d) is a proper metric space. Let $x_{0}\in Y$ and $S=\{f_{t}:Y\rightarrow Y\}_{t\ge 0}$ be a one-parameter continuous semigroup of nonexpansive mappings. If there exists a sequence of real numbers $\{s_{k}\}$ such that a sequence $\{f_{s_{k}}(x_{0})\}$ is bounded, then the orbit $O(x_{0})=\{f_{t}(x_{0}):t\ge 0\}$ of S is bounded.

Now, we can prove the main results of this section.

Theorem 3.4

Let (Y, d) be a $(1,\kappa )$-quasi-geodesic space satisfying Axiom $1'$ and Axiom $4'$, and suppose that for every $\zeta \in \partial Y$ and $z_{0}\in Y$ the intersection of horoballs’ closures $\bigcap _{r\in \mathbb {R}}\overline{F_{z_{0}}(\zeta ,r)}^{\overline{d}}$ consists of a single point. If $S=\{f_{t}:Y\rightarrow Y\}_{t\ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings on Y without bounded orbits, and there exists $t_0>0$ such that $f_{t_0}:Y \rightarrow Y$ is a compact mapping, then there exists $\xi \in \partial Y$ such that the semigroup S converges uniformly on bounded sets of Y to $\xi$.

Proof

Fix $t_0>0$. We choose a bounded set $D \subset Y$, and define the set

$$\begin{aligned} K=\{f_{s}(x):0\le s\le t_{0},\;x\in D\}=\bigcup _{0\le s\le t_{0}}f_{s}(D)\subset Y. \end{aligned}$$

We note that K is bounded. Indeed, since D is bounded, there exist $x_0 \in Y$ and $r>0$ such that $D \subset B(x_0,r)$. By nonexpansivity of $f_s$ we get $f_s(D) \subset B(f_s(x_0),r)$ for any $s>0$. It follows from boundedness of D that

$$\begin{aligned} K = \bigcup _{0\le s\le t_{0}}f_{s}(D) \subset \bigcup _{0\le s\le t_{0}} B(f_s(x_0),r). \end{aligned}$$

It follows from Lemma 3.2 that the set $\{f_s(x_0): \, 0 \le s \le t_0 \}$ is compact and, hence, bounded. Then, there exist $y_0 \in Y$ and $r_1>0$ such that $\{f_s(x_0): \, 0 \le s \le t_0 \} \subset B(y_0,r_1)$. Hence $K = \bigcup _{0\le s\le t_{0}} B(f_s(x_0),r) \subset B(y_0,r+r_1)$ is bounded. Fix $y \in Y$ and note that $f_{nt_0}(y) = f^n_{t_0}(y)$ for any $n \in \mathbb {N}$. Furthermore, $f_t(y) = f_{t_0}(f_{t - t_0}(y)) \in \{\overline{f_{t_0}(Y)}\}^{\overline{d}}$ for any $t \ge t_0$. Since the map $f_{t_0}$ is compact, the $\overline{d}$-closure of the orbit $O(y)= \{f_{t}(y): t \ge t_0\}$ is a compact subset of $\overline{Y}$. Next, consider a d-closed and bounded set $B \subset Y$. It follows from Axiom $1'$ that B is $\overline{d}$-closed. Then, the set $\overline{O(y)}^{\overline{d}} \cap B = \overline{O(y)}^{d} \cap B$ is compact in $\overline{Y}$ and hence compact in Y. That is, $(\overline{O(y)}^{d},d)$ is proper, $f_t: \overline{O(y)}^{d} \rightarrow \overline{O(y)}^{d}$ for $t \ge 0$, and by Całka’s theorem and nonexpansivity of $f_{t_0}$ we get

$$\begin{aligned} d(f_{nt_0}(y),y)\rightarrow \infty , \quad n\rightarrow \infty . \end{aligned}$$

It follows from Theorem 3.1 that

$$\begin{aligned} \sup _{y\in K}\,\overline{d}(f_{nt_{0}}(y),\xi )\rightarrow 0, \end{aligned}$$

(3.1)

as $n\rightarrow \infty$. We note that for every $x\in C$, $t>0$ such that $t = nt_0+s$ and $n\in \mathbb {N}$, $s\in [0,t_{0})$, we have

$$\begin{aligned} f_{t}(x)=f_{nt_{0}+s}(x)=f_{nt_{0}}(f_{s}(x))\in f_{nt_{0}}(K). \end{aligned}$$

Therefore by (3.1),

$$\begin{aligned} \sup _{x\in C}\overline{d}(f_{t}(x),\xi )=\sup _{x\in C}\overline{d} (f_{nt_{0}}(f_{s}(x)),\xi )\le \sup _{y\in K}\overline{d}(f_{nt_{0}}(y),\xi )\rightarrow 0. \end{aligned}$$

It follows that $\sup _{x\in C}\overline{d}(f_{t}(x),\xi )\rightarrow 0,$ as $t\rightarrow \infty$.

$\square$

Lemma 2.1, combined with Theorem 3.4, immediately leads to the following corollary.

Corollary 3.5

Let (Y, d) be a $(1,\kappa )$-quasi-geodesic space satisfying Axiom $1'$ and Axiom $3'$. If $S=\{f_{t}:Y\rightarrow Y\}_{t\ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings on Y without bounded orbits, and there exists $t_0>0$ such that the mapping $f_{t_0}:Y \rightarrow Y$ is compact, then there exists $\xi \in \partial Y$ such that the semigroup S converges uniformly on bounded sets of Y to $\xi$.

The next theorem shows that if there exists in the semigroup $S=\{f_{t}:Y\rightarrow Y\}_{t\ge 0}$ a compact and contractive mapping $f_{t_{0}}:Y\rightarrow Y$, then we get a slightly stronger result.

Theorem 3.6

Let (Y, d) be a $(1,\kappa )$-quasi-geodesic space satisfying Axiom $1'$ and Axiom $4'$, and suppose that for every $\zeta \in \partial Y$ and $z_{0}\in Y$, the intersection of horoballs’ closures $\bigcap _{r\in \mathbb {R}}\overline{F_{z_{0}}(\zeta ,r)}^{\overline{d}}$ consists of a single point. If $S=\{f_{t}:Y\rightarrow Y\}_{t\ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings on Y, and there exists $t_{0}>0$ such that a mapping $f_{t_{0}}:Y\rightarrow Y$ is compact and contractive, then there exists $\xi \in \overline{Y}$ such that the semigroup S converges uniformly on compact sets of Y to $\xi$.

Proof

In the first step, suppose that the semigroup $S=\{f_{t}:Y \rightarrow Y\}_{t\ge 0}$ has unbounded orbits. Then, the conclusion follows directly from Theorem 3.4. Therefore, we assume that $\{f_{t}(y)\}_{t\ge 0}$ is bounded for every $y\in Y$. By assumption, there exists $t_{0} >0$ such that $f_{t_{0}}:Y\rightarrow Y$ is a compact and contractive mapping. Fix $y_0 \in Y$. Note that$f_{nt_0}(y_0) = f^n_{t_0}(y_0)$, $n \in \mathbb {N}$. Since the mapping $f_{t_0}$ is compact and $\{f_{nt_0}(y_0)\}_{n\in \mathbb {N}}$ is bounded we have that $\overline{\{f_{nt_0}(y_0)\}}^{\overline{d}} _{n\in \mathbb {N}} =\overline{\{f_{nt_0}(y_0)\}}^{{d}}_{n\in \mathbb {N}}$ is compact in $\overline{Y}$ and hence also in Y. It follows that there exists a subsequence $\{f_{n_{k}t_{0}}(y_0)\}$ of $\{f_{nt_{0}}(y_0)\}$ converging to some $z_0 \in Y$. Since $f_{t_0}$ is nonexpansive, the sequence

$$\begin{aligned} d_{n}=d(f_{nt_{0}}(y_0),f_{nt_{0}+t_0}(y_0)), \; \; n=1,2,\ldots . \end{aligned}$$

is nonincreasing and, hence, it converges to some $\eta$, as $n\rightarrow \infty$. Therefore,

$$\begin{aligned} \eta \leftarrow d_{n_{k}}=d(f_{n_{k}t_{0}}(y_0),f_{n_{k}t_{0}+t_0}(y_0))\rightarrow d(z_0,f_{t_0}(z_0)) \end{aligned}$$

and

$$\begin{aligned} \eta \leftarrow d_{n_{k}+1}=d(f_{n_{k}t_{0}+t_{0}}(y_0),f_{n_{k}t_{0}+t_{0}+t_0}(y_0))\rightarrow d(f_{t_{0}}(z_0),f_{2t_{0}}(z_0)). \end{aligned}$$

Hence,

$$\begin{aligned} \eta =d(z_0,f_{t_0}(z_0))=d(f_{t_{0}}(z_0),f_{2t_{0}}(z_0)). \end{aligned}$$

(3.2)

Since $f_{t_{0}}$ is contractive, if $z_0$ and $f_{t_0}(z_0)$ were distinct points, we would have

$$\begin{aligned} d(f_{t_{0}}(z_0),f_{2t_{0}}(z_0))<d(z_0,f_{t_0}(z_0)), \end{aligned}$$

a contradiction with (3.2). Thus, $f_{t_0}(z_0)=z_0.$ Moreover, since the sequence $\{d(f_{nt_{0}}(y_0),z_0)\}$ is decreasing and $f_{n_{k}t_{0}}(y_0)\rightarrow z_0\in Y$, we have also $f_{nt_{0}}(y_0)\rightarrow z_0,$ as $n\rightarrow \infty$. Now, notice that if we choose any $y \in Y$, then the previous reasoning shows that $f_{nt_0}(y)$ converges to a fixed point of $f_{t_0}$ for every $y \in Y$. However, a contractive mapping has at most one fixed point. Therefore, $f_{nt_{0}}(y)\rightarrow z_0,$ as $n \rightarrow \infty$, for all $y \in Y$.

Now, we show uniform convergence on compact sets. We choose a compact set $C \subset Y$ and define the set K as in the proof of Theorem 3.4:

$$\begin{aligned} K = \{f_s(x): 0 \le s \le t_0, \, x \in C \}. \end{aligned}$$

Fix $\varepsilon >0$. Since by Lemma 3.2 the set K is compact, note that for some $y_{1},\ldots ,y_{n}\in K$,

$$\begin{aligned} K\subset \bigcup _{i=1}^{n}B \Big (y_{i},\frac{\varepsilon }{2}\Big ). \end{aligned}$$

From the first part of the proof, there exists $n_{0}$ such that for any $n\ge n_{0},$

$$\begin{aligned} \sup _{i=1,\ldots ,n}d(f_{nt_{0}}(y_{i}),z_0)< \frac{\varepsilon }{2}. \end{aligned}$$

We choose $y\in K$, then there exists i such that $d(y,y_{i})<\frac{\varepsilon }{2}$. By nonexpansivity of $f_{nt_0}$ we get

$$\begin{aligned} d(f_{nt_{0}}(y),z_0)&\le d(f_{nt_{0}}(y),f_{nt_{0}}(y_{i}))+d(f_{nt_{0}}(y_{i}),z_0) \\&\le d(y,y_{i})+d(f_{nt_{0}}(y_{i}),z_0) \\&<\frac{\varepsilon }{2}+\frac{\varepsilon }{2}=\varepsilon . \end{aligned}$$

Since $y\in K$ was chosen arbitrarily,

$$\begin{aligned} \sup _{y\in K}d(f_{nt_{0}}(y),z_0)\rightarrow 0, \end{aligned}$$

(3.3)

as $n\rightarrow \infty$. Note that for every $x\in C$ and $t>0$ such that $t = nt_0+s$, $n\in \mathbb {N}$, $s\in [0,t_{0})$, we have

$$\begin{aligned} f_{t}(x)=f_{nt_{0}+s}(x)=f_{nt_{0}}(f_{s}(x))\in f_{nt_{0}}(K). \end{aligned}$$

Therefore by (3.3),

$$\begin{aligned} \sup _{x\in C}d(f_{t}(x),z_0)=\sup _{x\in C}d(f_{nt_{0}}(f_{s}(x)),z_0)\le \sup _{y\in K}d(f_{nt_{0}}(y),z_0). \end{aligned}$$

It follows that $\sup _{x\in C}d(f_{t}(x),z_0)\rightarrow 0,$ as $t\rightarrow \infty$, and the proof is complete.

$\square$

By Lemma 2.1, we immediately obtain the following conclusion.

Corollary 3.7

Let (Y, d) be a $(1,\kappa )$-quasi-geodesic space satisfying Axiom $1'$ and Axiom $3'$. If $S=\{f_{t}:Y\rightarrow Y\}_{t\ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings on Y and there exists $t_0>0$ such that the mapping $f_{t_0}:Y \rightarrow Y$ is compact and contractive, then there exists $\xi \in \overline{Y}$ such that the semigroup S converges uniformly on compact sets of Y to $\xi$.

Let V be Banach space and D a bounded convex domain of V. Consider $\partial D = \overline{D} \setminus D$, where $\overline{D}$ denotes the closure of D in the norm topology. We will always assume that (D, d) is a $(1,\kappa )$-quasi-geodesic metric space, whose topology coincides with the norm topology. Recall that $D\subset V$ is strictly convex if for any $z,w\in \overline{D}$ the open segment

$$\begin{aligned} (z,w)=\{sz+(1-s)w:s\in (0,1)\} \end{aligned}$$

lies in D.

The following lemma was shown in [7].

Lemma 3.8

Suppose that D is a convex domain of V and (D, d) satisfies condition (C). If $\{x_{n}\}$ and $\{y_{n}\}$ are sequences in $D\,$converging to $\xi$ and $\eta$, respectively, in $\partial D$, and if for some $w\in D,$

$$\begin{aligned} d(x_{n},y_{n})-d(y_{n},w)\rightarrow -\infty , \end{aligned}$$

then $[\xi ,\eta ]\subset \partial D.$

Recall that $x_{0}$ is called a fixed point of a mapping $f:D\rightarrow D$ if $f(x_{0})=x_{0}.$ Define

$$\begin{aligned} \textrm{Fix}(f) =\{x\in D:f(x)=x\}. \end{aligned}$$

Corollary 3.7 and Lemma 3.8 now yield the following result.

Theorem 3.9

Suppose that D is a bounded strictly convex domain of a Banach space and (D, d) is $(1,\kappa )$-quasi-geodesic space satisfying Axiom $1'$ and condition (C). If $S=\{f_{t}:D\rightarrow D\}_{t\ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings on Y, and there exists $t_{0}>0$ such that the mapping $f_{t_{0}}:Y\rightarrow Y$ is compact with $\textrm{Fix}(f_{t_0}) = \emptyset ,$ then there exists $\xi \in \partial D$ such that the semigroup S converges uniformly on bounded sets of D to $\xi$.

Proof

In the first case, if $\{f_t:D \rightarrow D\}_{t \ge 0}$ has unbounded orbits, then the conclusion follows directly from Theorem 3.4. Thus, we can assume that the orbit $\{f_{t}(y)\}_{t \ge 0}$ is bounded for some (hence for any) $y \in D$. Let $r(\{f_t(y)\}) = \inf _{z \in D} \limsup _{t \rightarrow \infty } d(z,f_{t}(y))$, and note that the asymptotic center

$$\begin{aligned} A = \{x \in D: \limsup _{t \rightarrow \infty } d(x,f_t(y))=r(\{f_t(y)\}) \} \end{aligned}$$

is non-empty. Indeed, $A = \bigcap _{\varepsilon >0} A_{\varepsilon }$, where

$$\begin{aligned} A_{\epsilon } = \{x \in D: \limsup _{t \rightarrow \infty } d(x,f_t(y))\le r(\{f_t(y)\})+ \epsilon \}. \end{aligned}$$

Since the mapping $f_{t_0}$ is nonexpansive, we note that $f_{t_0}(A_{\varepsilon }) \subset A_{\varepsilon }$. What is more, $A_{\epsilon }$ is bounded and closed with respect to d and with respect to the norm. Since $f_{t_0}$ is compact,

$$\begin{aligned} \emptyset \ne \bigcap _{\varepsilon>0}\overline{f_{t_0}(A_{\varepsilon })}^{||\cdot ||}\subset \bigcap _{\varepsilon >0}A_{\varepsilon }=A. \end{aligned}$$

Furthermore,

$$\begin{aligned} f_{t_0}(A) = f_{t_0}\left( \bigcap _{\epsilon>0}A_{\epsilon }\right) \subset \bigcap _{\epsilon>0} f_{t_0}(A_{\epsilon }) \subset \bigcap _{\epsilon >0} A_{\epsilon } = A, \end{aligned}$$

which means that $f_{t_0}(A) \subset A$. The set A is also bounded and closed in $||\cdot ||$ and $\overline{f_{t_0}(A)}^{||\cdot ||}$ is compact. Since by assumption D is convex, and the metric space (D, d) satisfies condition C, A is convex, too. Therefore, it follows from the Schauder fixed-point theorem that $f_{t_0}$ has a fixed point, which is a contradiction.

$\square$

We said before that any bounded and convex domain in a Banach space can be equipped with the Hilbert metric and become a complete geodesic space satisfying Axiom $1'$ and condition (C). Hence, and from Theorem 3.9, we have the following corollary.

Corollary 3.10

Assume that D is a bounded strictly convex domain in a Banach space. If $S=\{f_{t}:D\rightarrow D\}_{t\ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings with respect to the Hilbert metric $d_{H}$, and there exists $t_0 > 0$ such that $\textrm{Fix}(f_{t_0}) = \emptyset$, and the mapping $f_{t_0}$ is compact, then there exists $\xi \in \partial D$ such that the semigroup S converges uniformly on bounded sets of D to $\xi$.

As discussed in Sect. 2, the Kobayashi distance satisfies all the conditions to formulate the next corollary.

Corollary 3.11

Assume that D is a bounded strictly convex domain in a complex Banach space. If $S=\{f_{t}:D\rightarrow D\}_{t\ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings with respect to the Kobayashi distance $k_{D}$, and there exists $t_0 > 0$ such that $\textrm{Fix}(f_{t_0}) = \emptyset$, and the mapping $f_{t_0}$ is compact, then there exists $\xi \in \partial D$ such that the semigroup S converges uniformly on bounded sets of D to $\xi$.

4 Attractor of a nonexpansive mapping

Let V be Banach space and $D \subset V$, $f:D \rightarrow D$ and $y\in D$. Then, the set of accumulation points (in the norm topology) of the sequence $\{f^{n}(y)\}$ is called the omega limit set of y and is denoted by $\omega _{f}(y)$. In other words,

$$\begin{aligned} \omega _{f}(y)=\{x\in \overline{D}: \exists _{\{n_k\}} \text { an increasing sequence such that} \lim \nolimits _{k \rightarrow \infty } f^{n_k}(y)=x \}. \end{aligned}$$

The attractor of f is defined as

$$\begin{aligned} \Omega _{f}=\bigcup _{y\in D}\omega _{f}(y). \end{aligned}$$

Let $D\subset V$ be a convex domain. Given $\xi \in \partial D,F\subset \partial D$, set

$$\begin{aligned} {{\,\textrm{ch}\,}}(\xi )= & {} \{x\in \partial D:[x,\xi ]\subset \partial D\}, \\ {{\,\textrm{ch}\,}}(F)= & {} \bigcup _{\xi \in F} {{\,\textrm{ch}\,}}(\xi ). \end{aligned}$$

We will need the following lemma (see Lemma 5.2, [7]).

Lemma 4.1

Suppose that Y is a $(1,\kappa )$-quasi-geodesic space satisfying Axiom 1’ and $f:Y\rightarrow Y$ is a compact nonexpansive mapping without a bounded orbit. Then there exists $\xi \in \partial Y$ such that for every $z_{0}\in Y$, $r\in \mathbb {R}$ and a sequence of natural numbers $\{a_{n}\}$, there exists $z\in Y$ and a subsequence $\{a_{n_{k}}\}$ of $\{a_{n}\}$ such that $f^{a_{n_{k}}}(z)\in F_{z_{0}}(\xi ,r)$ for every $k\in \mathbb {N}$. Moreover, if Y satisfies Axiom 4’, then $\xi \in \bigcap _{r\in \mathbb {R}}\overline{F_{z_{0}}(\xi ,r)}.$

The next theorem is a generalization of Theorem 4.10 in [7] which in turn is a generalization of the Abate and Raissy result [2, Theorem 6], who proved it for bounded convex domains with the Kobayashi distance.

Theorem 4.2

Let D be a bounded convex domain in a Banach space V and let (D, d) be a complete $(1,\kappa )$-quasi-geodesic space satisfying Axiom $1'$ and condition (C), whose topology coincides with the norm topology. If $f:D \rightarrow D$ is a compact and nonexpansive mapping without bounded orbits, then there exists $\xi \in \partial D$ such that

$$\begin{aligned} \Omega _f \subset {{\,\textrm{ch}\,}}\left( \bigcap _{r\in \mathbb {R}} \, \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}\right) \end{aligned}$$

for some $z_0 \in D$.

Proof

Fix $y \in D$ and a sequence of natural numbers $\{a_n\}$. Consider a d-closed and bounded set $B \subset D$. It follows from Axiom $1'$ that $\overline{O(y)}^{\overline{d}} \cap B = \overline{O(y)}^{d} \cap B$ is compact in $\overline{D}$ and hence also in D. That is, $(\overline{O(y)}^{d},d)$ is proper and by the Całka theorem [7, Theorem 2.1] and nonexpansivity of f we get $d(f^n(y),y)\rightarrow \infty$, $n\rightarrow \infty .$ Suppose that $f^{a_n}(y) \rightarrow \eta \in \partial D$. It follows from Lemma 4.1 that we can choose $\xi \in \partial D$ and fix $z_0 \in D$ and $r \in \mathbb {R}$ such that there exist $z \in D$ and a subsequence $\{a_{n_k}\}$ of $\{a_n\}$ for which $f^{a_{n_k}}(z) \in F_{z_0}(\xi , r)$, $k \in \mathbb {N}$. Without loss of generality, we can assume that

$$\begin{aligned} f^{a_{n_k}}(z) \rightarrow \zeta _r \in \partial D \cap \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}, \qquad k \rightarrow \infty . \end{aligned}$$

It follows from Lemma 3.8 that $[\eta , \zeta _r] \subset \partial D$, i.e., for any $r \in \mathbb {R}$, there exists $\zeta _r \in \partial D \cap \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}$ such that $\eta \in {{\,\textrm{ch}\,}}(\zeta _r)$. Consider a descending sequence $\{r_n\}$ such that $r_n \rightarrow - \infty$, as $n \rightarrow \infty$. Hence, the sequence $\{\zeta _{r_n}\} \subset \partial D$ and the segments of ends in $\eta$ and $\zeta _r$ lie on the boundary. Note that $\zeta _{r_n} \in \overline{f(D)}$ for any $n \in N$. Hence and by compactness of $\partial D \cap \overline{f(D)}$, there is a subsequence $\{r_{n_k}\}$ of $\{r_n\}$ such that $\zeta _{r_{n_k}} \rightarrow \zeta \in \partial D$, as $k \rightarrow \infty$. Fix $k_0$. Note that for $k \ge k_0$,

$$\begin{aligned} \zeta _{r_{n_k}} \in \overline{F_{z_{0}}(\xi ,r_{n_{k}})}^{||\cdot ||} \subset \overline{F_{z_{0}}(\xi ,r_{n_{k_0}})}^{||\cdot ||} \end{aligned}$$

and, hence,

$$\begin{aligned} \zeta \in \overline{F_{z_{0}}(\xi ,r_{n_{k_0}})}^{||\cdot ||}. \end{aligned}$$

Since $k_0$ was chosen arbitrarily,

$$\begin{aligned} \zeta \in \bigcap _{k\in \mathbb {N}} \overline{F_{z_{0}}(\xi ,r_{n_k})}^{||\cdot ||} = \bigcap _{r\in \mathbb {R}} \overline{F_{z_{0}}(\xi ,r)}^{||\cdot ||}. \end{aligned}$$

Fix $s \in [0,1]$ and note that

$$\begin{aligned} ||s \zeta _{r_{n_k}} + (1-s)\eta - (s \zeta +(1-s)\eta ) || = s ||\zeta _{r_{n_k}} - \zeta || \rightarrow 0, \end{aligned}$$

as $k \rightarrow \infty$, which means that $[\zeta , \eta ] \in \partial D$. Therefore,

$$\begin{aligned} \eta \in {{\,\textrm{ch}\,}}\left( \bigcap _{r\in \mathbb {R}} \, \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}\right) . \end{aligned}$$

$\square$

A big horoball is not always a convex set. However, Abate and Raissy [2] proved that a big horoball considered in a bounded and convex domain with the Kobayashi distance is a star-shaped set with respect to the center of the horoball. We now present a generalization of this fact for all metric spaces satisfying condition (C).

Lemma 4.3

Let D be a bounded convex domain in a Banach space V and suppose that (D, d) is $(1,\kappa )$-quasi-geodesic space satisfying condition (C), whose topology coincides with the norm topology. If $z_0 \in D$, $r > 0$ and $\xi \in \partial D$, then for every $\eta \in \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}$, we have

$$\begin{aligned}{}[\eta ,\xi ] \subset \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}. \end{aligned}$$

Proof

Fix $\eta \in F_{z_0}(\xi ,r)$ and choose a sequence $\{x_n\} \subset D$ converging to $\xi \in \partial D$ and such that the limit

$$\begin{aligned} \lim _{n \rightarrow \infty } d(\eta ,x_n)-d(x_n,z_0) \le r \end{aligned}$$

exists. Fix $s \in (0,1)$. It follows from condition (C) that

$$\begin{aligned} \limsup _{n \rightarrow \infty } d(s\eta +(1-s)x_n,x_n) - d(x_n,z_0) \le \lim _{n \rightarrow \infty } d(\eta ,x_n) - d(x_n,z_0) \le r.\nonumber \\ \end{aligned}$$

(4.1)

Note that

$$\begin{aligned} ||s \eta +(1-s)x_n - (s\eta +(1-s)\xi ) || = (1-s)||x_n - \xi || \rightarrow 0, \quad n \rightarrow \infty . \end{aligned}$$

Since topology of (D, d) and $(\overline{D},||\cdot ||)$ coincides on D we get

$$\begin{aligned} d(s \eta +(1-s)x_n, s\eta +(1-s)\xi ) \rightarrow 0, \end{aligned}$$

as $n \rightarrow \infty$. Hence,

$$\begin{aligned}{} & {} |d(s\eta +(1-s)x_n,x_n) - d(x_n,s\eta +(1-s)\xi )|\nonumber \\{} & {} \quad \le d(s\eta +(1 - s)x_n, s\eta +(1-s)\xi ) \rightarrow 0, \end{aligned}$$

(4.2)

as $n \rightarrow \infty$. From (4.1) and (4.2) we have

$$\begin{aligned}{} & {} \liminf _{w \rightarrow \xi } d(s\eta +(1-s)\xi ,w) - d(w,z_0) \\{} & {} \le \limsup _{n \rightarrow \infty } d(s\eta +(1-s)\xi ,x_n) - d(x_n,z_0) \\{} & {} \quad \le \lim _{n \rightarrow \infty } d(s\eta +(1-s)\xi ,x_n) - d(x_n,s\eta +(1-s)x_n) \\{} & {} \qquad + \limsup _{n \rightarrow \infty } d(s\eta +(1-s)x_n,x_n) - d(x_n,z_0) \le r. \end{aligned}$$

Hence, for every $s \in (0,1)$, the point $s\eta +(1-s)\xi \in F_{z_0}(\xi ,r)$, thus $(\eta ,\xi )\subset F_{z_0}(\xi ,r)$. Therefore, $[\eta ,\xi ]\subset \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}$.

To complete the proof, consider the case $\eta \in \partial F_{z_0}(\xi ,r)$. So there is such a sequence $\{y_n\} \subset F_{z_0}(\xi ,r)$ such that $y_n \rightarrow \eta$, as $n \rightarrow \infty$. From the previous considerations we have $[y_n,\xi ]\subset \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}$ for any $n \in \mathbb {N}$. Fix $s \in [0,1]$. Then,

$$\begin{aligned} ||s\eta +(1-s)\xi -(sy_n+(1-s)\xi )|| = s ||\eta - y_n|| \rightarrow 0, \end{aligned}$$

as $n \rightarrow \infty$ which means that $s\eta +(1-s)\xi \in \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}$. Therefore, $[\eta ,\xi ] \subset \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}$, and the proof is complete. $\square$

Theorem 4.2 combined with Lemma 4.3 gives an immediate result.

Theorem 4.4

Let D be a bounded convex domain in a Banach space V, and suppose that (D, d) is $(1,\kappa )$-quasi-geodesic space satisfying Axiom $1'$ and condition (C), whose topology coincides with the norm topology. If $f:D \rightarrow D$ is a compact nonexpansive mapping without bounded orbits, then there exists $\xi \in \partial D$ such that

$$\begin{aligned} \Omega _f \subset {{\,\textrm{ch}\,}}({{\,\textrm{ch}\,}}(\xi )). \end{aligned}$$

Proof

Fix $z_0 \in D$. It follows from Theorem 4.2 that there exists $\xi \in \partial D$ such that

$$\begin{aligned} \Omega _f \subset {{\,\textrm{ch}\,}}\left( \bigcap _{r\in \mathbb {R}} \, \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}\right) . \end{aligned}$$

Consider $\zeta \in \bigcap _{r\in \mathbb {R}} \, \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}$. By Lemma 4.3 we have $[\zeta ,\xi ] \subset \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}$ for any $r \in \mathbb {R}$. Hence $[\zeta ,\xi ] \subset \bigcap _{r\in \mathbb {R}} \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}$. Thus $\zeta \in {{\,\textrm{ch}\,}}(\xi )$ and hence

$$\begin{aligned} \bigcap _{r\in \mathbb {R}} \, \overline{F_{z_0}(\xi ,r)}^{||\cdot ||} \subset {{\,\textrm{ch}\,}}(\xi ). \end{aligned}$$

Therefore, $\Omega _f \subset {{\,\textrm{ch}\,}}({{\,\textrm{ch}\,}}(\xi )).$ $\square$

Let D be a bounded convex domain in a Banach space V and let (D, d) be a metric space. In the context of Hilbert’s metric, Karlsson presented a property which we will call Axiom $2^{*}$.

Axiom $2^{*}$ If $\{x_{n}\}$ and $\{y_{n}\}$ are convergent sequences in D with limits x and y in $\partial D$, respectively, and the segment $[x,y]\subseteq \partial D$, then for each $z\in D$ we have

$$\begin{aligned} \lim _{n\rightarrow \infty } [d(x_{n},y_{n})-\max \{d(x_{n},z),d(y_{n},z)\}]=\infty . \end{aligned}$$

The next lemma shows that the Hilbert metric satisfies Axiom $2^{*}$ (see [15, Theorem 4.13] for the finite-dimensional case and [14, Proposition 8.3.3]).

Proposition 4.5

Let $D \subset V$ be a bounded convex domain in a Banach space. If $\{x_{n}\}$ and $\{y_{n}\}$ are convergent sequences in D with limits x and y in $\partial D$, respectively, and the segment $[x,y]\subseteq \partial D$, then for each $z\in D$ we have

$$\begin{aligned} \lim _{n\rightarrow \infty } [d_H(x_{n},y_{n})-\max \{d_H(x_{n},z),d_H(y_{n},z)\}]=\infty , \end{aligned}$$

that is, $(D,d_H)$ satisfies Axiom $2^{*}$.

Proof

Consider a sequence $\{u_n\}$ defined as $u_n = \frac{x_n+y_n}{2}$ for any $n \in \mathbb {N}$. Since the segment [x, y] does not lie on the boundary, we get that $u = \frac{x+y}{2} \in D$. Note that

$$\begin{aligned} ||u_n-u || = \Big |\Big |\frac{x_n+y_n}{2} -\frac{x+y}{2}\Big |\Big | \le \frac{1}{2} (||x_n-x|| + ||y_n-y||) \rightarrow 0, \quad n \rightarrow \infty . \end{aligned}$$

Since topology of $(D,d_H)$ and $(\overline{D},||\cdot ||)$ coincides on D we have $d_H (u_n,u) \rightarrow 0$, as $n \rightarrow \infty$. Hence,

$$\begin{aligned} \lim _{n \rightarrow \infty } d_H(u_n, w) = d_H(u,w) < \infty . \end{aligned}$$

(4.3)

for any $w \in D$. We note that

$$\begin{aligned} \begin{aligned} d_H(x_n, y_n)= & {} {}&d_H(x_n, u_n) + d_H(u_n, y_n) \\ {}\ge & {} {}&d_H(x_n, w) - d_H(u_n, w) + d_H(y_n, w) - d_H(u_n, w). \end{aligned} \end{aligned}$$

(4.4)

It follows from Axiom $1'$, (4.3) and (4.4) that

$$\begin{aligned}{} & {} d_H(x_{n},y_{n})-\max \{d_H(x_{n},z),d_H(y_{n},z)\} \ge \min \{d_H(x_{n},z),d_H(y_{n},z)\} \\ {}{} & {} - 2d_H(u_n,w) \rightarrow \infty , \end{aligned}$$

as $n \rightarrow \infty$. This means that $(D,d_H)$ satisfies Axiom $2^{*}$. $\square$

In the case of a convex bounded domain D equipped with the Hilbert metric it was proved in [11] that the attractor (in the norm topology) $\Omega _f$ of a fixed-point free nonexpansive mapping $f: D \rightarrow D$ is a star-shaped subset of $\partial D$. Karlsson [10] proved this theorem using Gromov’s product and it was important that the geodesics are linear segments. At the end of this section, we present a shorter proof of this theorem for all spaces satisfying Axiom $2^{*}$ and not using Gromov’s product.

Theorem 4.6

Let $D \subset V$ be a bounded convex domain in a Banach space, and let (D, d) be $(1,\kappa )$-quasi-geodesic space satisfying Axiom $1'$ and Axiom $2^{*}$, whose topology coincides with the norm topology. Suppose that $f:D \rightarrow D$ is a compact nonexpansive mapping without bounded orbits, then there exists $\xi \in \partial D$ such that

$$\begin{aligned} \Omega _f \subset {{\,\textrm{ch}\,}}(\xi ). \end{aligned}$$

Proof

Fix $y\in D$ and define a sequence $\{d_n\}$ as $d_{n}=d(f^{n}(y),y)$. Consider a d-closed and bounded set $B \subset D$. It follows from Axiom $1'$ that

$\overline{O(y)}^{\overline{d}} \cap B = \overline{O(y)}^{d} \cap B$ is compact in $\overline{D}$ and hence compact in D. That is, $(\overline{O(y)}^{d},d)$ is proper and by the Całka theorem and nonexpansivity of f we get $d(f^n(y),y)\rightarrow \infty$, $n\rightarrow \infty .$ By [9, Observation 3.1], there is a sequence $\{n_i\}$ such that $d_{m}<d_{n_i}$ for $m<n_i, \ i=1,2,\ldots$. By Axiom $1'$ and since $\overline{f(D)}^{||\cdot ||}$ is compact (going to another subsequence if necessary) we can assume that $\{f^{n_i}(y)\}$ converges to some $\xi \in \partial D$. Fix $x \in D$ and choose a subsequence $\{f^{a_{k}}(x)\}$ of $\{f^n(x)\}$ which converges to some $\eta \in \partial D$. Fix $k \in \mathbb {N}$. Then for sufficiently large $n_i$, we get

$$\begin{aligned}{} & {} d(f^{a_k}(x),f^{n_i}(y)) - d(f^{n_i}(y),y) \\{} & {} \quad \le d(f^{a_k}(x),f^{a_k}(y))+ d(f^{a_k}(y),f^{n_i}(y)) - d(f^{n_i}(y),y) \\{} & {} \quad \le d(x,y) + d(f^{n_i -a_k}(y),y) - d(f^{n_i}(y),y) \\{} & {} \quad \le d(x,y) + d(f^{n_i}(y),y) - d(f^{n_i}(y),y) \\{} & {} \quad \le d(x,y). \end{aligned}$$

Hence, there exists a subsequence $\{n_{i_k}\}$ of $\{n_i\}$ such that for any k,

$$\begin{aligned} d(f^{a_k}(x),f^{n_{i_k}}(y)) - d(f^{n_{i_k}}(y),y) \le d(x,y). \end{aligned}$$

(4.5)

Then by (4.5) we get

$$\begin{aligned}{} & {} \liminf _{k \rightarrow \infty } \, d(f^{a_k}(x),f^{n_{i_k}}(y)) - \max \{d(f^{a_k}(x),y),d(f^{n_i}(y),y) \} \\ {}\le & {} \liminf _{k \rightarrow \infty } \, d(f^{a_k}(x),f^{n_{i_k}}(y)) - d(f^{n_{i_k}}(y),y) \\ {}\le & {} d(x,y). \end{aligned}$$

It follows from Axiom $2^{*}$ that $[\eta , \xi ] \subset \partial D$ and hence $\eta \in {{\,\textrm{ch}\,}}(\xi )$. $\square$

5 Attractor of a semigroup of nonexpansive mappings

The objective of this section is to extend the results of Sect. 4 to the case of continuous one-parameter semigroups of nonexpansive mappings.

A family $S=\{f_{t}:Y\rightarrow Y\mid t\in [0,\infty )\}$ is called a one-parameter continuous semigroup if for all $s,t\in [0,\infty )$,

$$\begin{aligned} f_{s+t}=f_{t}\circ f_{s}, \end{aligned}$$

and for every $y\in Y$, there is a limit

$$\begin{aligned} \lim _{t\rightarrow 0^{+}}f_{t}(y)=f_{0}(y)=y. \end{aligned}$$

Let $\omega _{S}(x)$ denote the set of accumulation points (in the norm topology) of a semigroup S defined as

$$\begin{aligned} \omega _{S} (x) = \{y \in \overline{D}: ||f_{t_n}(x)-y|| \rightarrow 0 \; \; \textrm{for some increasing sequence } \{t_n\} \rightarrow \infty \}. \end{aligned}$$

The attractor of the semigroup S is the set $\Omega _{S}$ defined as

$$\begin{aligned} \Omega _{S} = \bigcup _{x \in D} \omega _{S} (x). \end{aligned}$$

The next lemma says that the attractor of the semigroup S is the same set as the attractor of the mapping $f_{t_0}$ for some $t_0>0$.

Lemma 5.1

Let D be a bounded convex domain in a Banach space V, and let (D, d) be a $(1,\kappa )$-quasi-geodesic space satisfying Axiom $5'$ whose topology coincides with the norm topology. Suppose that $S = \{f_t:D \rightarrow D\}_{t \ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings without bounded orbits, and there exists $t_0$ such that $f_{t_0}:D \rightarrow D$ is a compact mapping. Then for every $t_0>0$,

$$\begin{aligned} \Omega _{S} = \Omega _{f_{t_0}}. \end{aligned}$$

Proof

Fix $t_0>0$.

“$\supset$” We consider $\eta \in \Omega _{f_{t_0}}$. It follows from the definition of the attractor that there exist $x_0$ and a subsequence $\{f^{a_n}_{t_0}(x_0)\}$ of $\{f^{n}_{t_0}(x_0)\}$ such that $f^{a_n}_{t_0}(x_0) \rightarrow \eta$, as $n \rightarrow \infty$. Note that $f^n_{t_0}(x) = f_{nt_0}(x)$. Then, $f_{nt_0}(x) \rightarrow \eta$, as $n \rightarrow \infty$. Hence $\eta \in \Omega _{S}$.

“$\subset$” To show the opposite inclusion, choose $\eta \in \Omega _{S}$. Again from the definition of the attractor, there is a monotone sequence $t_1<t_2< \ldots , \;$ $t_n \rightarrow \infty$ and $x_0 \in D$ such that

$$\begin{aligned} ||f_{t_n}(x_0) - \eta || \rightarrow 0, \end{aligned}$$

(5.1)

as $n \rightarrow \infty$. For any $n \in \mathbb {N}$, there exist ${a_n} \in \mathbb {N}$ and $s_n \in [0,t_0)$ such that $t_n = a_nt_0+s_n$. Furthermore, $a_1 \le a_2 \le \ldots$. We can assume (considering a subsequence of this sequence) that $a_1<a_2<\ldots$. Hence,

$$\begin{aligned} f_{t_n}(x_0) = f_{a_nt_0+s_n}(x_0) = f_{t_0}^{a_n}(f_{s_n}(x_0)). \end{aligned}$$

We can suppose that $s_n \rightarrow s_0 \in [0,t_0]$. By continuity of S, we get

$$\begin{aligned} d(f_{s_n}(x_0),f_{s_0}(x_0)) \rightarrow 0, \end{aligned}$$

(5.2)

as $n \rightarrow \infty$. Hence,

$$\begin{aligned} d(f_{t_0}^{a_n}(f_{s_n}(x_0)),f_{t_0}^{a_n}(f_{s_0}(x_0))) \le d(f_{s_n}(x_0),f_{s_0}(x_0)) \rightarrow 0, \end{aligned}$$

as $n \rightarrow \infty$. Since by assumption, Axiom $5'$ is satisfied, we have

$$\begin{aligned} ||f_{t_0}^{a_n}(f_{s_n}(x_0))-f_{t_0}^{a_n}(f_{s_0}(x_0)) || \rightarrow 0, \end{aligned}$$

(5.3)

as $n \rightarrow \infty$. Then, it follows from (5.1) and (5.3) that

$$\begin{aligned} ||f_{t_0}^{a_n}(f_{s_0}(x_0)) - \eta ||\le & {} ||f_{t_0}^{a_n}(f_{s_0}(x_0)) - f_{t_0}^{a_n}(f_{s_n}(x_0))|| + ||f_{t_0}^{a_n}(f_{s_n}(x_0)) - \eta || \\\le & {} ||f_{t_0}^{a_n}(f_{s_0}(x_0)) - f_{t_0}^{a_n}(f_{s_n}(x_0))||+||f_{t_n}(x_0)- \eta || \rightarrow 0, \end{aligned}$$

as $n \rightarrow \infty$. Therefore,

$$\begin{aligned} \eta \in \omega _{f_{t_0}}(f_{s_0}(x_0)) \subset \Omega _{f_{t_0}}. \end{aligned}$$

$\square$

The above lemma in combination with Theorem 4.2 gives the following result, which is the counterpart of Theorem 4.2 for one-parameter continuous semigroups.

Theorem 5.2

Let D be a bounded convex domain in a Banach space V, and let (D, d) be a $(1,\kappa )$-quasi-geodesic space satisfying Axiom $1'$, Axiom $5'$ and condition (C) whose topology coincides with the norm topology. If $S=\{ f_t:D \rightarrow D\}_{t \ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings without bounded orbits, and there exists $t_0$ such that $f_{t_0}:D \rightarrow D$ is a compact mapping. Then, there exists $\xi \in \partial D$ such that

$$\begin{aligned} \Omega _S \subset {{\,\textrm{ch}\,}}\left( \bigcap _{r\in \mathbb {R}} \, \overline{F_{z_0}(\xi ,r)}^{||\cdot ||}\right) \end{aligned}$$

for some $z_0 \in D$.

Using again Lemma 5.1, combined with Theorem 4.4 or Theorem 4.6, respectively, this allows us to obtain the following two results.

Theorem 5.3

Let D be a bounded convex domain in a Banach space V, and let (D, d) be a $(1,\kappa )$-quasi-geodesic space satisfying Axiom $1'$, Axiom $5'$ and condition (C) whose topology coincides with the norm topology. If $S=\{ f_t:D \rightarrow D\}_{t \ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings without bounded orbits, and there exists $t_0$ such that $f_{t_0}:D \rightarrow D$ is a compact mapping, then there exists $\xi \in \partial D$ such that

$$\begin{aligned} \Omega _S \subset {{\,\textrm{ch}\,}}({{\,\textrm{ch}\,}}(\xi )). \end{aligned}$$

Theorem 5.4

Let D be a bounded convex domain in a Banach space V, and let (D, d) be a $(1,\kappa )$-quasi-geodesic space satisfying Axiom $1'$, Axiom $2^{*}$ and Axiom $5'$ whose topology coincides with the norm topology. If $S=\{ f_t:D \rightarrow D\}_{t \ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings without bounded orbits, then there exists $\xi \in \partial D$ such that

$$\begin{aligned} \Omega _S \subset {{\,\textrm{ch}\,}}(\xi ). \end{aligned}$$