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

We show a Wolff-Denjoy type theorem in the case of a one-parameter continuous semigroups 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.


Introduction
Let us suppose that f : ∆ → ∆ is a holomorphic map of the unit disc ∆ ⊂ C without a fixed point.The classical Wolff-Denjoy theorem asserts that then there is a point ξ ∈ ∂∆ such that the iterates f n converge locally uniformly to ξ on ∆.There is a wide literature concerning various generalizations of this theorem (see [1,5,9,15,13,16] 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 ω f (x) as the set of accumulation points of the sequence f n (x) and the notion of the attractor Ω f = x∈D ω 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 Ω f of a fixed-point free nonexpansive map f : D → D is a star-shaped subset of ∂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 → D is a fixed point free nonexpansive mapping acting on the Hilbert metric space (D, d H ), then there exists a convex set Ω ⊆ ∂D such that for each x ∈ D, all accumulation points ω f (x) of the orbit O(x, f ) lie in Ω.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 Section 3 that if (Y, d) is a (1, κ)-quasi-geodesic space satisfying Axiom 1 ′ and Axiom 4 ′ (introduced in Section 2) and S = {f t : Y → Y } t≥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 → Y is a compact mapping, then there exists ξ ∈ ∂Y such that the semigroup S converges uniformly on bounded sets of Y to ξ.We apply our consideration in the case of bounded strictly convex domains in a Banach space with the metric satisfying condition (C) d(sx + (1 − s)y, z) ≤ max{d(x, z), d(y, z)}.
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 Section 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 ξ on the boundary such that Ω f ⊂ ch(ch(ξ)).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 for any sequences {x n } and {y n } converging to distinct points x and y on the boundary such that the segment [x, y] ∂D, and for any z ∈ D (see Section 4 for more details).
In Section 5 we formulate counterparts of the above results for a one-parameter continuous semigroup of nonexpansive mappings.

Preliminaries
Let (Y, d) be a metric space.Recall that a curve γ : A metric space Y is called (λ, κ)-quasi-geodesic if every pair of points in Y can be connected by a (λ, κ)-quasi-geodesic.A convex bounded domain in a Banach space equipped with the Kobayashi distance is an example of a (1, κ)-quasi geodesic metric space for any κ > 0.
We are interested in considerations related to nonexpansive and contractive mappings in (λ, κ)-quasi-geodesic metric spaces.We call a map f : for every distinct x, y ∈ Y .
Beardon [4] and Karlsson [9] considered some four properties of metric spaces called axioms.Proceeding similarly, we slightly modify these axioms to make them better suited for non-proper metric spaces.
Notice that Axiom 1' implies that if A ⊂ Y is bounded, then the d-closure of A does not intersect the boundary ∂Y and hence coincides with the d-closure of A.
It is not hard to show that Axioms 1 ′ and 2 ′ imply Axiom 3 ′ and Axioms 1 ′ and 3 ′ imply Axiom 4 ′ .We say that a mapping f : 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 ) → 0, as n → ∞, then 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 ξ ∈ ∂Y , pole z 0 ∈ Y and radius r ∈ R by Karlsson in [9] and Lemmens and Nussbaum in [14] note that in a metric space satisfying Axioms 1 and 2, each horoball intersects the boundary of the space at exactly one point.The next lamma shows an important property of big horoballs i.e.Axiom 3 ′ can be regarded as a characteristic that the intersection of horoball's closure consists of a single point.The proof proceeds similarly to the finite-dimensional case that 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 ∈ K dominates x ∈ V if there exists α, β ∈ R such that αy ≤ x ≤ βy.This notion yields on K an equivalence relation ∼ K by x ∼ k y if x dominates y and y dominates x.For all x, y ∈ K such that x ∼ K y and y = 0, define The Hilbert (pseudo-)metric is defined by Moreover, we put d H (0, 0) = 0 and d H (x, y) = ∞ if x ≁ K y.It can be shown that d H is a metric iff x = λy for some λ > 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 Proof.Let {x n }, {y n } ⊂ D be sequences in D and for any n ∈ 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 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 ∈ D is given by where k ∆ denotes the Poincaré metric on the unit disc ∆.
It is well known that if D is a bounded and convex domain in a complex Banach space, then (D, k D ) is a complete metric space satisfying Axiom 1 ′ (see [6], [12]).
The second example of a space that satisfies the Axiom 5 ′ (with respect to the closure in the norm (D, || • ||)) 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 for every x, y ∈ 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: for all x, y, z ∈ D and s ∈ [0, 1] (see, e.g., [7]).

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]).In the further considerations we will use the following lemma, the proof of which can be found in [8].
One of the classical arguments in this line of research is the Ca lka 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 Ca lka theorem for semigroups ( [8], Theorem 3.3).
Now, we can prove the main results of this section.
continuous semigroup of nonexpansive mappings on Y without bounded orbits, and there exists t 0 > 0 such that f t 0 : Y → Y is a compact mapping, then there exists ξ ∈ ∂Y such that the semigroup S converges uniformly on bounded sets of Y to ξ.
Proof.Fix t 0 > 0. We choose a bounded set D ⊂ Y , and define the set We note that K is bounded.Indeed, since D is bounded, there exist x 0 ∈ Y and r > 0 such that D ⊂ B(x 0 , r).By nonexpansivity of f s we get f s (D) ⊂ B(f s (x 0 ), r) for any s > 0. It follows from boundedness of D that It follows from Lemma 3.2 that the set {f s (x 0 ) : 0 ≤ s ≤ t 0 } is compact and hence bounded.Then there exist y 0 ∈ Y and d for t ≥ 0, and by Ca lka's theorem and nonexpansivity of f t 0 we get It follows from Theorem 3.1 that as n → ∞.We note that for every x ∈ C, t > 0 such that t = nt 0 + s and n ∈ N, s ∈ [0, t 0 ), we have Therefore by (3.1),  Proof.In the first step, suppose that the semigroup S = {f t : Y → Y } t≥0 has unbounded orbits.Then the conclusion follows directly from Theorem 3.4.Therefore, we assume that {f t (y)} t≥0 is bounded for every y ∈ Y .By assumption, there exists t 0 > 0 such that 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 ∈ Y .Since f t 0 is nonexpansive, the sequence is nonincreasing and hence it converges to some η, as n → ∞.Therefore, Since f t 0 is contractive, if z 0 and f t 0 (z 0 ) were distinct points, we would have 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 ) → z 0 ∈ Y , we have also f nt 0 (y 0 ) → z 0 , as n → ∞.Now, notice that if we choose any y ∈ Y , then the previous reasoning shows that f nt 0 (y) converges to a fixed point of f t 0 for every y ∈ Y .However, a contractive mapping has at most one fixed point.Therefore f nt 0 (y) → z 0 , as n → ∞, for all y ∈ Y .
Now we show uniform convergence on compact sets.We choose a compact set C ⊂ Y and define the set K as in the proof of Theorem 3.4: Fix ε > 0. Since by Lemma 3.2 the set K is compact, note that for some y 1 , . . ., From the first part of the proof, there exists n 0 such that for any n ≥ n 0 , sup i=1,...,n We choose y ∈ K, then there exists i such that d(y, y i ) < ε 2 .By nonexpansivity of f nt 0 we get Since y ∈ K was chosen arbitrarily, as n → ∞.Note that for every x ∈ C and t > 0 such that t = nt 0 + s, n ∈ N, s ∈ [0, t 0 ), we have Therefore by (3.3), It follows that sup x∈C d(f t (x), z 0 ) → 0, as t → ∞, and the proof is complete.
By Lemma 2.1, we immediately obtain the following conclusion.The following lemma was shown in [7].
Corollary 3.7 and Lemma 3.8 now yields the following result.Theorem 3.9.Suppose that D is a bounded strictly convex domain of a Banach space and (D, d) is (1, κ)-quasi-geodesic space satisfying Axiom 1 ′ and condition (C).If S = {f t : D → D} t≥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 → Y is compact with Fix(f t 0 ) = ∅, then there exists ξ ∈ ∂D such that the semigroup S converges uniformly on bounded sets of D to ξ.
Proof.In the first case, if {f t : D → D} t≥0 has unbounded orbits, then the conclusion follows directly from Theorem 3.4.Thus we can assume that the orbit {f t (y)} t≥0 is bounded for some (hence for any) y ∈ D. Let r({f t (y)}) = inf z∈D lim sup t→∞ d(z, f t (y)), and note that the asymptotic center Since the mapping f t 0 is nonexpansive we note that f t 0 (A ε ) ⊂ A ε .What is more, A ǫ is bounded and closed with respect to d and with respect to the norm.Since f t 0 is compact, Furthermore, which means that f t 0 (A) ⊂ A. The set A is also bounded and closed in ||•|| and f t 0 (A) 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.
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 → D} t≥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 Fix(f t 0 ) = ∅, and the mapping f t 0 is compact, then there exists ξ ∈ ∂D such that the semigroup S converges uniformly on bounded sets of D to ξ.
As discussed in Section 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 → D} t≥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 Fix(f t 0 ) = ∅, and the mapping f t 0 is compact, then there exists ξ ∈ ∂D such that the semigroup S converges uniformly on bounded sets of D to ξ.

Attractor of a nonexpansive mapping
Let V be Banach space and D ⊂ V , f : D → D and y ∈ 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 ω f (y).In other words, The attractor of f is defined as We will need the following lemma (see Lemma 5.2, [7]).Lemma 4.1.Suppose that Y is a (1, κ)-quasi-geodesic space satisfying Axiom 1' and f : Y → Y is a compact nonexpansive mapping without a bounded orbit.Then there exists ξ ∈ ∂Y such that for every z 0 ∈ Y , r ∈ R and a sequence of natural numbers {a n }, there exists z ∈ Y and a subsequence {a n k } of {a n } such that f an k (z) ∈ F z 0 (ξ, r) for every k ∈ N.Moreover, if Y satisfies Axiom 4', then ξ ∈ r∈R F z 0 (ξ, 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, κ)-quasi geodesic space satisfying Axiom 1 ′ and condition (C), whose topology coincides with the norm topology.If f : D → D is a compact and nonexpansive mapping without bounded orbits, then there exists ξ ∈ ∂D such that for some z 0 ∈ D.
It follows from Lemma 3.8 that [η, ζ r ] ⊂ ∂D i.e., for any r ∈ R there exists ζ r ∈ ∂D ∩ as n → ∞.Hence the sequence {ζ rn } ⊂ ∂D and the segments of ends in η and ζ r lie on the boundary.Note that ζ rn ∈ f (D) for any n ∈ N. Hence and by compactness of ∂D ∩ f (D), there is a subsequence Since k 0 was chosen arbitrarily, Fix s ∈ [0, 1] and note that 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, κ)-quasi geodesic space satisfying condition (C), whose topology coincides with the norm topology.If z 0 ∈ D, r > 0 and ξ ∈ ∂D, then for every η ∈ F z 0 (ξ, r) ||•|| we have Proof.Fix η ∈ F z 0 (ξ, r) and choose a sequence {x n } ⊂ D converging to ξ ∈ ∂D and such that the limit lim Since topology of (D, d) and (D, || • ||) coincides on D we get as n → ∞.From (4.1) and (4.2) we have lim inf Hence, for every s ∈ (0, 1), the point sη To complete the proof, consider the case η ∈ ∂F z 0 (ξ, r).So there is such a sequence {y n } ⊂ F z 0 (ξ, r) such that y n → η, as n → ∞.From the previous considerations we have Therefore, Ω f ⊂ ch(ch(ξ)).
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 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) Ω f of a fixed point free nonexpansive mapping f : D → D is a star-shaped subset of ∂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.{f n i (y)} converges to some ξ ∈ ∂D.Fix x ∈ D and choose a subsequence {f a k (x)} of {f n (x)} which converges to some η ∈ ∂D.Fix k ∈ N. Then for sufficiently large n i , we get Hence there exists a subsequence {n i k } of {n i } such that for any k, Then by (4.5) we get It follows from Axiom 2 * that [η, ξ] ⊂ ∂D and hence η ∈ ch(ξ).

Attractor of a semigroup of nonexpansive mappings
The objective of this section is to extend the results of Section 4 to the case of continuous one-parameter semigroups of nonexpansive mappings.
and for every y ∈ Y there is a limit lim t→0 + f t (y) = f 0 (y) = y.Let's mark it with a symbol ω S (x) the set of accumulation points (in norm topology) of a semigroup S defined as ω S (x) = {y ∈ D : ||f tn (x) − y|| → 0 for some increasing sequence {t n } → ∞}.
The attractor of the semigroup S is the set Ω S defined as 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, κ)-quasi geodesic space satisfying Axiom 5 ′ whose topology coincide with the norm topology.Suppose that S = {f t : D → D} t≥0 is a one-parameter continuous semigroup of nonexpansive mappings without bounded orbits, and there exists t 0 such that f t 0 : D → D is a compact mapping.Then for every t 0 > 0, Ω S = Ω ft 0 .
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, κ)-quasi geodesic space satisfying Axiom 1 ′ , Axiom 5 ′ and condition (C) whose topology coincide with the norm topology.If S = {f t : D → D} t≥0 is a one-parameter continuous semigroup of nonexpansive mappings without bounded orbits, and there exists t 0 such that f t 0 : D → D is a compact mapping.Then there exists ξ ∈ ∂D such that for some z 0 ∈ 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, κ)-quasi geodesic space satisfying Axiom 1 ′ , Axiom 5 ′ and condition (C) whose topology coincide with the norm topology.If S = {f t : D → D} t≥0 is a one-parameter continuous semigroup of nonexpansive mappings without bounded orbits, and there exists t 0 such that f t 0 : D → D is a compact mapping, then there exists ξ ∈ ∂D such that Ω S ⊂ ch(ch(ξ)).
Theorem 5.4.Let D be a bounded convex domain in a Banach space V , and let (D, d) be a (1, κ)-quasi geodesic space satisfying Axiom 1 ′ , Axiom 2 * and Axiom 5 ′ whose topology coincide with the norm topology.If S = {f t : D → D} t≥0 is a one-parameter continuous semigroup of nonexpansive mappings without bounded orbits, then there exists ξ ∈ ∂D such that Ω S ⊂ ch(ξ).

Axiom 1 '
. The metric space (Y, d) is an open dense subset of a metric space (Y , d), whose relative topology coincides with the metric topology.For any w and the big horoball byF z 0 (ξ, r) = {y ∈ Y : lim inf w→ξ d(y, w) − d(w, z 0 ) ≤ r}.

Theorem 3 . 1 .
Let (Y, d) be a (1, κ)-quasi-geodesic space satisfying Axiom 1 ′ and Axiom 4 ′ , and suppose that for every ζ ∈ ∂Y and z 0 ∈ Y , the intersection of horoballs' closures r∈R F z 0 (ζ, r) d consists of a single point.If f : Y → Y is a compact nonexpansive mapping without bounded orbits, then there exists ξ ∈ ∂Y such that the iterates f n of f converge uniformly on bounded sets of Y to ξ.

Theorem 3 . 6 .
Let (Y, d) be a (1, κ)-quasi-geodesic space satisfying Axiom 1 ′ and Axiom 4 ′ , and suppose that for every ζ ∈ ∂Y and z 0 ∈ Y , the intersection of horoballs' closures r∈R F z 0 (ζ, r) d consists of a single point.If S = {f t : Y → Y } t≥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 → Y is compact and contractive, then there exists ξ ∈ Y such that the semigroup S converges uniformly on compact sets of Y to ξ.

Corollary 3 . 7 .
Let (Y, d) be a (1, κ)-quasi-geodesic space satisfying Axiom 1 ′ and Axiom 3 ′ .If S = {f t : Y → Y } t≥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 → Y is compact and contractive, then there exists ξ ∈ Y such that the semigroup S converges uniformly on compact sets of Y to ξ.Let V be Banach space and D a bounded convex domain of V .Consider ∂D = D \ D, where D denotes the closure of D in the norm topology.We will always assume that (D, d) is a (1, κ)-quasi geodesic metric space, whose topology coincides with the norm topology.Recall that D ⊂ V is strictly convex if for any z, w ∈ D the open segment (z, w) = {sz + (1 − s)w : s ∈ (0, 1)} lies in D.

Proof.
Fix y ∈ D and a sequence of natural numbers {a n }.Consider a d-closed and bounded set B ⊂ D. It follows from Axiom 1 ′ that O(y) d ∩ B = O(y) d ∩ B is compact in D and hence also in D. That is, (O(y) d , d) is proper and by the Ca lka theorem [7, Theorem 2.1] and nonexpansivity of f we get d(f n (y), y) → ∞, n → ∞.Suppose that f an (y) → η ∈ ∂D.It follows from Lemma 4.1 that we can choose ξ ∈ ∂D and fix z 0 ∈ D and r ∈ R such that there exist z ∈ D and a subsequence {a n k } of {a n } for which f an k (z) ∈ F z 0 (ξ, r), k ∈ N. Without loss of generality, we can assume that

Axiom 2 *
presented a property which we will call Axiom 2 * .If {x n } and {y n } are convergent sequences in D with limits x and y in ∂D, respectively, and the segment [x, y] ∂D, then for each z ∈ D we havelim n→∞ [d(x n , y n ) − max{d(x n , z), d(y n , z)}] = ∞.The next lemma shows that the Hilbert metric satisfies Axiom 2 * (see[15, Theorem  4.13],[14, Proposition 8.3.3]).

Proposition 4 . 5 .
Let D ⊂ 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 ∂D, respectively, and the segment [x, y] ∂D, then for each z ∈ D we havelim n→∞ [d H (x n , y n ) − max{d H (x n , z), d H (y n , z)}] = ∞,that is, (D, d H ) satisfies Axiom 2 * .Proof.Consider a sequence {u n } defined as u n = xn+yn 2 for any n ∈ N. Since the segment [x, y] does not lie on the boundary, we get that u = x+y 2 ∈ D. Note that

Theorem 4 . 6 .
Let D ⊂ V be a bounded convex domain in a Banach space, and let (D, d) be (1, κ)-quasi geodesic space satisfying Axiom 1 ′ and Axiom 2 * , whose topology coincides with the norm topology.Suppose that f : D → D is a compact nonexpansive mapping without bounded orbits, then there exists ξ ∈ ∂D such thatΩ f ⊂ ch(ξ).Proof.Fix y ∈ D and define a sequence {d n } as d n = d(f n (y), y).Consider a d-closed and bounded set B ⊂ D. It follows from Axiom 1 ′ that O(y) d ∩ B = O(y) d ∩ B is compact in D and hence compact in D. That is, (O(y) d , d) is proper and by the Ca lka theorem and nonexpansivity of f we get d(f n (y), y) → ∞, n → ∞.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, . ... By Axiom 1 ′ and since f (D)||•|| is compact (going to another subsequence if necessary) we can assume that We note that ||x n − a n || ≤ d and ||y n − b n || ≤ d for every n ∈ N. Consider subsequences {x n k } and {y n k } of sequences {x n } and {y n }, respectively.If there existed δ > 0 and k 0 Let D be a bounded convex domain in a Banach space V , and suppose that (D, d) is (1, κ)-quasi geodesic space satisfying Axiom 1 ′ and condition (C), whose topology coincides with the norm topology.If f : D → D is a compact nonexpansive mapping without bounded orbits, then there exists ξ ∈ ∂D such thatΩ f ⊂ ch(ch(ξ)).Proof.Fix z 0 ∈ D. It follows from Theorem 4.2 that there exists ξ ∈ ∂D such thatΩ f ⊂ ch( Consider ζ ∈ r∈R F z 0 (ξ, r)||•|| .By Lemma 4.3 we have [ζ, ξ] ⊂ F z 0 (ξ, r) ||•|| for any r ∈ R. Hence [ζ, ξ] ⊂ r∈R F z 0 (ξ, r) ||•|| .Thus ζ ∈ ch(ξ) and hence Since topology of (D, d H ) and (D,|| • ||) coincides on D we have d H (u n , u) → 0, as n → ∞.H (u n , w) = d H (u, w) < ∞.for any w ∈ D. We note thatd H (x n , y n ) = d H (x n , u n ) + d H (u n , y n ) ≥ d H (x n , w) − d H (u n , w) + d H (y n , w) − d H (u n , w).