Third International Conference on Applications of Mathematics to Nonlinear Sciences,

Electronic Journal of Differential Equations, Conference 27 (2024), pp. 13–26.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.conf.27.k1

CONVERGENCE THEOREMS OF IMPLICIT TYPE ITERATIONS

IN GEODESIC SPACES WITH NEGATIVE CURVATURE

YASUNORI KIMURA, KAZUYA SASAKI, KAKERU TORII

Abstract. In this article, we prove convergence theorems of the implicit iter-
ative methods in the sense of Browder type and Xu-Ori type with (−1)-convex

combination in CAT(−1) spaces.

1. Introduction

In recent years, fixed point theory has been investigated by many mathemati-
cians. In particular, approximating fixed points of a nonlinear mapping is one of
the main topics in this theory. Researchers have investigated some types of ap-
proximating iteration to find a fixed point of a mapping in several spaces, such as
Banach spaces and geodesic spaces.

This paper considers two types of iterative schemes: explicit type schemes and
implicit type schemes. This research field utilizes explicit iteration types, particu-
larly Halpern and Mann types. However, implicit type methods, like Browder [10]
and Xu-Ori [11] types, also have their significance.

Explicit type schemes generate a sequence {xn} by explicitly expressing xn+1 in
terms of xn. Halpern and Mann types iteration are explicit type schemes to find a
fixed point of a mapping T : X → X. These define a sequence {xn} as follows:

• Halpern type: xn+1 := αnu⊕ (1− αn)Txn;
• Mann type: xn+1 := αnxn ⊕ (1− αn)Txn

for n ∈ N. On the other hand, there are some implicit type schemes such as
Browder type and Xu-Ori type. These generate a sequence {xn} by finding the
unique element xn satisfying the following equations:

• Browder type: xn = αnu⊕ (1− αn)Txn;
• Xu-Ori type: xn = αnxn−1 ⊕ (1− αn)Txn

for n ∈ N. In this article, we consider implicit type schemes in geodesic spaces,
particularly complete CAT(−1) spaces.

Recently, Kimura [6] proved the following convergence theorem with multiple
anchor points {uk} in a complete CAT(0) space (which is also known as a Hadamard
space). It uses the Browder type iterative scheme for multiple anchor points.

2020 Mathematics Subject Classification. 47H09.
Key words and phrases. Fixed point; nonexpansive mapping; implicit type iteration;

Geodesic space; convergence theorem; ∆-convergence; CAT(−1) space.
©2024 This work is licensed under a CC BY 4.0 license.

Published August 20, 2024.

13



14 Y. KIMURA, K. SASAKI, K. TORII EJDE-2024/CONF/27

Theorem 1.1 (Kimura [6, Theorem 3.3]). Let X be a Hadamard space and let
T : X → X be a nonexpansive mapping such that F (T ) ̸= ∅, where F (T ) is a set
of all fixed points of T . Suppose that {αn} ⊂ ]0, 1[ such that αn → 0 as n → ∞.
For k = 1, 2, . . . , r, let {βk

n} ⊂ [0, 1] such that
∑r

k=1 β
k
n = 1 for all n ∈ N and

βk
n → βk ∈ [0, 1] as n → ∞. Let u1, u2, . . . , ur ∈ X and define {xn} ⊂ X by

xn = argminy∈X

(
αn

r∑
k=1

βk
nd(y, uk)

2 + (1− αn)d(y, Txn)
2
)

for n ∈ N. Then, {xn} converges to the unique minimizer of a function g : F (T ) →
R defined by

g(y) =

r∑
k=1

βkd(y, uk)
2

for y ∈ F (T ).

Furthermore, Kimura also proved the following ∆-convergence theorem with an
implicit iterative scheme for a finite family of nonexpansive mappings by using the
Xu-Ori type iterative scheme.

Theorem 1.2 (Kimura [7, Theorem 3.2]). Let X be a Hadamard space. For k =

1, 2, . . . , N , let Tk : X → X be a nonexpansive mapping such that
⋂N

k=1 F (Tk) ̸= ∅.
For k = 0, 1, . . . , N , suppose {αk

n} ⊂ [a, b] ⊂ ]0, 1[ such that
∑N

k=0 α
k
n = 1. For

given x1 ∈ X, generate a sequence {xn} ⊂ X satisfying

xn+1 = argminy∈X

(
α0
nd(xn, y)

2 +

N∑
k=1

αk
nd(Tkxn+1, y)

2
)

for n ∈ N. Then, {xn} is well-defined and ∆-convergent to some x0 ∈
⋂N

k=1 F (Tk).

In this article, we prove convergence theorems for implicit iterative methods in
the sense of Browder and Xu-Ori types with (−1)-convex combination in complete
CAT(−1) spaces.

2. Preliminaries

Let (X, d) be a metric space. For x, y ∈ X and l ≥ 0, a mapping c : [0, l] → X
is called a geodesic with endpoints x, y ∈ X if it satisfies c(0) = x, c(l) = y, and
d(c(t), c(s)) = |t− s| for every t, s ∈ [0, l]. Then l = d(c(0), c(l)) = d(x, y). We say
X is a geodesic space if a geodesic with endpoints x and y exists for all x, y ∈ X.
In this paper, we assume X has the unique geodesic for every x, y ∈ X. Then, we
denote the image of the geodesic with endpoints x, y ∈ X by [x, y], which is well
defined. We call [x, y] a geodesic segment with endpoints x and y.

Let E2 be the 2-dimensional Euclidean space, and let H2 be the 2-dimensional
hyperbolic space, which are both geodesic spaces. For κ ≤ 0, let M2

κ be a 2-
dimensional space with constant curvature κ defined by

M2
κ =

{
E2 if κ = 0;

1√
−κ

H2 if κ < 0,

where 1√
−κ

H2 is a geodesic space defined from H2 by multiplying the metric on H2

by 1/
√
−κ.



EJDE-2024/CONF/27 IMPLICIT TYPE ITERATIONS IN GEODESIC SPACES 15

Let (X, d) be a geodesic space. For x, y, z ∈ X, a geodesic triangle △(x, y, z) is
defined as the union of three segments [x, y], [y, z], and [z, x]. Fix κ ≤ 0 and let
M2

κ be a model space with a metric ρκ. For each geodesic triangle △(x, y, z) on X,
its comparison triangle △̄(x̄, ȳ, z̄) is defined as the triangle in M2

κ whose length of
each corresponding edge is identical with that of the original triangle:

d(x, y) = ρκ(x̄, ȳ), d(y, z) = ρκ(ȳ, z̄), d(z, x) = ρκ(z̄, x̄).

A point p̄ ∈ △̄(x̄, ȳ, z̄) is called a comparison point for p ∈ △(x, y, z) if d(u, p) =
ρκ(ū, p̄) and d(v, p) = ρκ(v̄, p̄), where u, v are adjacent endpoints of p. A ge-
odesic space X is called a CAT(κ) space if for all triangles △(x, y, z), points
p, q ∈ △(x, y, z), and their comparison points p̄, q̄ ∈ △̄(x̄, ȳ, z̄), the inequality

d(p, q) ≤ ρκ(p̄, q̄) (2.1)

holds. The inequality (2.1) is called the CAT(κ) inequality.
It is clear that the n-dimensional Euclidean space (En, dE) is an example of

the complete CAT(0) spaces, since it always satisfies dE(p, q) = ρ0(p̄, q̄) in (2.1).
More generally, the class of complete CAT(0) spaces consists of the class of Hilbert
spaces. A complete CAT(0) space is often called a Hadamard space. We know that
a Banach space is not a CAT(0) space in general. Furthermore, the n-dimensional
hyperbolic space Hn is a complete CAT(−1) space, but the n-dimensional Euclidean
space En is not a CAT(−1) space.

Let (X, d) be a geodesic space. Then, for x, y ∈ X and t ∈ [0, 1], there exists the
unique point z ∈ [x, y] such that d(x, z) = (1−t)d(x, y) and d(z, y) = td(x, y). Such
a point z is called a convex combination of x and y. We denote it by tx⊕ (1− t)y.

Let (X, d) be a CAT(0) space and let (E2, ρ) be the 2-dimensional Euclidean
space. Let △(x, y, z) be a geodesic triangle on X and take its comparison triangle
△̄(x̄, ȳ, z̄) on E2. Then we know that the following equation, known as Stewart’s
theorem, holds for all t ∈ [0, 1]:

ρ(z̄, tx̄⊕ (1− t)ȳ)2 = tρ(z̄, x̄)2 + (1− t)ρ(z̄, ȳ)2 − t(1− t)ρ(x̄, ȳ)2.

This can be obtained by the following calculation in R2:

∥z̄ − (tx̄+ (1− t)ȳ)∥2 = ⟨z̄ − (tx̄+ (1− t)ȳ), z̄ − (tx̄+ (1− t)ȳ)⟩
= t∥z̄ − x̄∥2 + (1− t)∥z̄ − ȳ∥2 − t(1− t)∥x̄− ȳ∥2.

Note that R2 is one of the models of 2-dimensional Euclidean space. Moreover, since
X is a CAT(0) space, we have d(z, tx ⊕ (1 − t)y) ≤ ρ(z̄, tx̄ ⊕ (1 − t)ȳ). Therefore,
since d(z, x) = ρκ(z̄, x̄), d(z, y) = ρκ(z̄, ȳ), and d(x, y) = ρκ(x̄, ȳ), we obtain an
inequality

d(z, tx⊕ (1− t)y)2 ≤ td(z, x)2 + (1− t)d(z, y)2 − t(1− t)d(x, y)2 (2.2)

for all t ∈ [0, 1]. We introduce the following characterization of CAT(0) spaces.

Theorem 2.1 ([1, Theorem 1.3.3]). For a geodesic space (X, d), the following two
conditions are equivalent:

(a) (X, d) is a CAT(0) space;
(b) the inequality (2.2) holds for all x, y, z ∈ X and t ∈ [0, 1].

Similarly, the following inequality holds for every CAT(−1) space X:

cosh d(z, tx⊕ (1− t)y) sinh d(x, y)

≤ cosh d(z, x) sinh(td(x, y)) + cosh d(z, y) sinh((1− t)d(x, y))



16 Y. KIMURA, K. SASAKI, K. TORII EJDE-2024/CONF/27

for every x, y, z ∈ X and t ∈ [0, 1]. This is obtained by the following equation on
the 2-dimensional hyperbolic space (H2, ρ):

cosh ρ(z̄, tx̄⊕ (1− t)ȳ) sinh ρ(x̄, ȳ)

= cosh ρ(z̄, x̄) sinh(tρ(x̄, ȳ)) + cosh ρ(z̄, ȳ) sinh((1− t)ρ(x̄, ȳ))

for every x̄, ȳ, z̄ ∈ H2 and t ∈ [0, 1].
We know that any CAT(κ) is a CAT(κ′) for κ < κ′. Therefore, every results for

CAT(0) spaces can apply to any CAT(κ) spaces with κ ≤ 0. For more details, see
[2].

LetX be a CAT(0) space. A subset C ofX is said to be convex if tx⊕(1−t)y ∈ C
for all x, y ∈ C and t ∈ ]0, 1[.

Let X be a Hadamard space, and let C be a nonempty closed convex subset of
X. Then there exists the unique point px ∈ C such that d(x, px) = infy∈C d(x, y)
for each x ∈ X. We define the metric projection PC from X onto C by PCx = px
for all x ∈ X.

Let X be a CAT(0) space. For a bounded sequence {xn} in X, let r(x, {xn}) =
lim supn→∞ d(x, xn) for x ∈ X, and define the asymptotic radius r({xn}) of {xn}
by

r({xn}) = inf
x∈X

r(x, {xn}).

The asymptotic center AC({xn}) of {xn} is a set of all points p ∈ X such that

r(p, {xn}) = r({xn}).

If a CAT(0) space X is complete, then an asymptotic center of a bounded sequence
{xn} on X is unique, see [3, Proposition 7].

Let X be a CAT(0) space. We say a sequence {xn} on X is ∆-convergent to
x0 ∈ X if x0 is the unique element of the asymptotic center of any subsequence of
{xn}. Then x0 is called a ∆-limit of {xn}.

Theorem 2.2 (Kirk and Panyanak [8, Proposition 3.5]). Let X be a Hadamard
space and let {xn} be a bounded sequence on X. Then there exists a ∆-convergent
subsequence of {xn}.

Let X be a CAT(0) space. A mapping T : X → X is said to be nonexpansive if

d(Tx, Ty) ≤ d(x, y)

for every x, y ∈ X. We know the set F (T ) = {z ∈ X : z = Tz} of all fixed points of
a nonexpansive mapping T is closed and convex. A mapping U : X → X is called
a contraction if there exists α ∈ [0, 1[ such that for all x, y ∈ X,

d(Ux,Uy) ≤ αd(x, y).

If X is complete, then the Banach contraction principle guarantees the existence
and uniqueness of a fixed point of U . Let f be a real function on X and let C be
a nonempty subset of X. Then argminx∈C f(x) stands for the set of all minimizers
of f on C. Furthermore, if argminx∈C f(x) consists of exactly one point, then
argminx∈C f(x) directly denotes such a point.

In this article, we use the notion of (−1)-convex combination introduced by
Kimura and Sasaki defined as follows:



EJDE-2024/CONF/27 IMPLICIT TYPE ITERATIONS IN GEODESIC SPACES 17

Definition 2.3 (Kimura and Sasaki [9, Definition 3.6]). Let X be a geodesic space.
Then for all u, v ∈ X, and α ∈ [0, 1], the set

argminx∈X (α cosh d(u, x) + (1− α) cosh d(v, x))

is a singleton. Thus define a (−1)-convex combination of u and v by

αu
−1
⊕ (1− α)v := argminx∈X (α cosh d(u, x) + (1− α) cosh d(v, x)).

We know that αu
−1
⊕ (1− α)v ∈ [u, v] for all u, v ∈ X and α ∈ [0, 1]. Namely,

αu
−1
⊕ (1− α)v = argminx∈[u,v] (α cosh d(u, x) + (1− α) cosh d(v, x))

holds, see [9, Lemma 3.5].

Lemma 2.4 ([9]). Let X be a geodesic space. For x, y ∈ X with x ̸= y and
α ∈ [0, 1], an equation

αx
−1
⊕ (1− α)y = σx⊕ (1− σ)y

holds, where

σ =
1

d(x, y)
tanh−1 α sinh d(x, y)

1− α+ α cosh d(x, y)
.

It is obvious that αx
−1
⊕ (1− α)y = αx⊕ (1− α)y if x = y.

Lemma 2.5 ([9, Corollary 3.9]). Let X be a CAT(−1) space and x, y, z ∈ X. Then
for all α ∈ [0, 1],

cosh d(αx
−1
⊕ (1− α)y, z) ≤ α cosh d(x, z) + (1− α) cosh d(y, z).

Lemma 2.6 ([9, Lemma 3.7]). For any d > 0 and α ∈ [0, 1],

1

d
tanh−1 α sinh d

1− α+ α cosh d
+

1

d
tanh−1 (1− α) sinh d

α+ (1− α) cosh d
= 1.

Lemma 2.7 ([9, Lemma 3.4]). For fixed d > 0 and α ∈ [0, 1], let

σ =
1

d
tanh−1 α sinh d

1− α+ α cosh d
.

Define a function g : [0, 1] → R by

g(t) = α cosh((1− t)d) + (1− α) cosh td

for t ∈ [0, 1]. Then g is strictly convex and infinitely differentiable. Moreover,
g′(σ) = 0 holds and hence σ is the unique minimizer of g.

The following results play important roles in the main results.

Theorem 2.8 (He, Fang, Lopez and Li [4, Proposition 2.3]). Let X be a Hadamard

space and {xn} a bounded sequence on X such that xn
∆
⇀ x ∈ X. Then, for all

u ∈ X, the following holds:

d(u, x) ≤ lim inf
n→∞

d(u, xn).

Lemma 2.9 (Kimura [5, Lemma 3.1]). Let {xn} be a ∆-convergent sequence in
a Hadamard space X with its ∆-limit x ∈ X. If {d(xn, u)} converges for some
u ∈ X, then {xn} converges to x.



18 Y. KIMURA, K. SASAKI, K. TORII EJDE-2024/CONF/27

3. Main results

In this section, we prove a convergence theorem with Browder and Xu-Ori type
iteration in complete CAT(−1) spaces, respectively. To prove our main result, we
first show the following lemmas.

Lemma 3.1. Let α ∈ [0, 1[ and define a function f : [0,∞[ → R by

f(x) = x tanh−1 (1− α) sinhx

α+ (1− α) coshx

for x ∈ R. Then, f is strictly increasing.

Proof. Fix α ∈ [0, 1[ and define f1 : R → ]−1, 1[ by

f1(x) =
(1− α) sinhx

α+ (1− α) coshx

for x ∈ R. Then
f ′
1(x) =

(1− a)(1 + a(coshx− 1))

(1 + (1− a)(coshx− 1))2
> 0

for all x ∈ R and hence f1 is strictly increasing. Thus a function f2 : [0,∞[ → [0,∞[
defined by f2(x) = tanh−1(f1(x)) for x ∈ [0,∞[ is also strictly increasing. This
follows the strict increasingness of f . □

Lemma 3.2. Let d > 0. Define f : ]0,∞[ → R, by

f(t) =
sinh td

t

for t ∈ ]0,∞[. Then, f is strictly increasing.

Proof. We have

f ′(t) =
td cosh td− sinh td

t2
=

1

t2

∫ td

0

x sinhx dx > 0.

Thus we obtain the desired result. □

Lemma 3.3. For fixed d > 0 and α ∈ ]0, 1/2[, let

σ =
1

d
tanh−1 α sinh d

1− α+ α cosh d
.

Then α < σ < 1/2.

Proof. Define g : [0, 1] → R by

g(t) = α cosh((1− t)d) + (1− α) cosh td

for t ∈ [0, 1]. Then σ is the unique minimizer of g from Lemma 2.7. Moreover,
from the strict convexity of g, we have g′(x) < 0 for all x ∈ ]0, σ[, and g′(x) > 0
for all x ∈ ]σ, 1[. Since g′(1/2) = d(1 − 2α) sinh(d/2) > 0, we have σ < 1/2. By
α < 1/2 < 1− α and Lemma 3.2, we have

g′(α) = −αd sinh((1− α)d) + (1− α)d sinhαd

= dα(1− α)
(
− sinh((1− α)d)

1− α
+

sinhαd

α

)
< 0.

Therefore, α < σ. This is the desired result. □



EJDE-2024/CONF/27 IMPLICIT TYPE ITERATIONS IN GEODESIC SPACES 19

Lemma 3.4. For fixed d > 0, assume that α, σ ∈ [0, 1] satisfy the equation

σ =
1

d
tanh−1 α sinh d

1− α+ α cosh d
.

Then, α = 1/2 if and only if σ = 1/2.

Proof. The given equation is equivalent to

α =
sinhσd

sinhσd+ sinh((1− σ)d)
.

From this we derives the conclusion using basic calculations. □

Lemma 3.5. For fixed d1, d2 ≥ 0 and α ∈ ]0, 1/2[, let

σ1 =


1

d1
tanh−1 α sinh d1

1− α+ α cosh d1
if d1 ̸= 0;

α if d1 = 0

and

σ2 =


1

d2
tanh−1 α sinh d2

1− α+ α cosh d2
if d2 ̸= 0;

α if d2 = 0.

Then, σ1 > σ2 if and only if d1 > d2. Moreover, σ1 = σ2 if and only if d1 = d2.

Proof. We consider the following cases:

(i) d1 = 0 or d2 = 0:
(a) d1 = 0 and d2 = 0;
(b) d1 ̸= 0 and d2 = 0;
(c) d1 = 0 and d2 ̸= 0,

(ii) d1 ̸= 0 and d2 ̸= 0:
(d) d1 = d2;
(e) d1 ̸= d2.

First, we consider case (i).
(a) If d1 = d2 = 0, then it is obvious that σ1 = α = σ2.
(b) Suppose that d1 ̸= 0 and d2 = 0. Then d1 > d2. Furthermore, from Lemma
3.3, we have σ1 > α = σ2.
(c) Similar to (b), if d1 = 0 and d2 ̸= 0, then d1 < d2 and σ1 = α < σ2 from Lemma
3.3.

Next, consider the case (ii). We hereinafter suppose that d1 ̸= 0 and d2 ̸= 0.
Define a function g : [0, 1] → R by

g(t) = α cosh((1− t)d1) + (1− α) cosh td1

for t ∈ [0, 1]. Then from Lemma 2.7, σ1 is the unique minimizer of g. This follows
that g′(σ2) > 0 if and only if σ1 < σ2, and g′(σ2) < 0 if and only if σ1 > σ2. We
also get α < σ1 < 1/2 and α < σ2 < 1/2 by Lemma 3.3. By the definition of σ2,
we obtain

α =
sinhσ2d2

sinhσ2d2 + sinh((1− σ2)d2)
.

Therefore,

g(t) =
sinhσ2d2 cosh((1− t)d1) + sinh((1− σ2)d2) cosh td1

sinhσ2d2 + sinh((1− σ2)d2)



20 Y. KIMURA, K. SASAKI, K. TORII EJDE-2024/CONF/27

for all t ∈ [0, 1]. It follows that

g′(t) =
d1(− sinhσ2d2 sinh((1− t)d1) + sinh((1− σ2)d2) sinh td1)

sinhσ2d2 + sinh((1− σ2)d2)

for all t ∈ ]0, 1[. Put C = d1/(sinhσ2d2 + sinh((1− σ2)d2)) > 0. Then

g′(σ2) = C · (− sinhσ2d2 sinh((1− σ2)d1) + sinh((1− σ2)d2) sinhσ2d1).

Put p = (d1 + d2)/2, q = (d2 − d1)/2, and k = 1 − 2σ2. Then p > 0, |q| < p,
0 < k < 1, and

g′(σ2) = C ·
(
− sinh

(
(p+ q)

(1
2
− 1

2
k
))

sinh
(
(p− q)

(1
2
+

1

2
k
))

+ sinh
(
(p+ q)

(1
2
+

1

2
k
))

sinh
(
(p− q)

(1
2
− 1

2
k
)))

=
1

2
C · (− cosh(p− kq) + cosh(−kp+ q) + cosh(p+ kq)− cosh(kp+ q))

= C · (− sinh kp sinh q + sinh kq sinh p)

= C sinh p sinh q(−f(p) + f(q)),

where we define f : R → ]0, k] by

f(x) =


sinh kx

sinhx
if x ̸= 0;

k if x = 0

for x ∈ R. Then f is a differentiable even function and it satisfies f ′(x) > 0 for all
x < 0, and f ′(x) < 0 for all x > 0.

(d): Suppose that d1 = d2. Then we have q = 0 and hence g′(σ2) = 0. It implies
that σ1 = σ2.

(e): Suppose that d1 ̸= d2. Then since |q| < p, we obtain −f(p) + f(q) > 0.
Therefore, g′(σ2) > 0 if and only if q > 0, that is, d2 − d1 > 0. In other words, if
d1 < d2, then σ1 < σ2; if d1 > d2, then σ1 > σ2.

From (i) and (ii), conditions σ1 > σ2 and d1 > d2 are equivalent, and so are
conditions σ1 = σ2 and d1 = d2. □

Let X be a CAT(0) space. Then as noted in the preliminaries, the following
inequality holds for every x, y, z ∈ X and t ∈ ]0, 1[:

d(tx⊕ (1− t)y, z)2 ≤ td(x, z)2 + (1− t)d(y, z)2 − t(1− t)d(x, y)2.

Since every CAT(−1) space is a CAT(0) space, the above inequality also holds in
CAT(−1) spaces.

Theorem 3.6. Let X be a CAT(−1) space and let T : X → X be a nonexpansive
mapping. Let u ∈ X and α ∈ ]0, 1

2 ]. Define U : X → X by

Ux = αu
−1
⊕ (1− α)Tx

for x ∈ X. Then, U is a contraction.

Proof. Let x, y ∈ X. If d(Ux,Uy) = 0, then obviously there exists β ∈ [0, 1[ such
that d(Ux,Uy) ≤ βd(x, y). Thus, we consider the case where d(Ux,Uy) ̸= 0. Then
from Lemma 2.4, we have

d(Ux,Uy)2



EJDE-2024/CONF/27 IMPLICIT TYPE ITERATIONS IN GEODESIC SPACES 21

= d(αu
−1
⊕ (1− α)Tx, αu

−1
⊕ (1− α)Ty)2

= d(σ1u⊕ (1− σ1)Tx, σ2u⊕ (1− σ2)Ty)
2

≤ σ1d(u, σ2u⊕ (1− σ2)Ty)
2 + (1− σ1)d(Tx, σ2u⊕ (1− σ2)Ty)

2

− σ1(1− σ1)d(u, Tx)
2

≤ σ1(1− σ2)
2d(u, Ty)2 + (1− σ1)(σ2d(u, Tx)

2 + (1− σ2)d(Tx, Ty)
2

− σ2(1− σ2)d(u, Ty)
2)− σ1(1− σ1)d(u, Tx)

2

= (σ1 − σ2)((1− σ2)d(u, Ty)
2 − (1− σ1)d(u, Tx)

2) + (1− σ1)(1− σ2)d(Tx, Ty)
2,

where

σ1 =


1

d(u, Tx)
tanh−1 α sinh d(u, Tx)

1− α+ α cosh d(u, Tx)
if u ̸= Tx;

α if u = Tx;

σ2 =


1

d(u, Ty)
tanh−1 α sinh d(u, Ty)

1− α+ α cosh d(u, Ty)
if u ̸= Ty;

α if u = Ty.

We consider the following two cases: (i) σ1 ≥ σ2, and (ii) σ2 ≥ σ1.
First, we consider the case (i). From Lemma 2.6, we have

1− σ1 =


1

d(u, Tx)
tanh−1 (1− α) sinh d(u, Tx)

α+ (1− α) cosh d(u, Tx)
if u ̸= Tx;

1− α if u = Tx;

1− σ2 =


1

d(u, Ty)
tanh−1 (1− α) sinh d(u, Ty)

α+ (1− α) cosh d(u, Ty)
if u ̸= Ty;

1− α if u = Ty.

Therefore,

(1− σ1)d(u, Tx)
2 = d(u, Tx) tanh−1 (1− α) sinh d(u, Tx)

α+ (1− α) cosh d(u, Tx)

and

(1− σ2)d(u, Ty)
2 = d(u, Ty) tanh−1 (1− α) sinh d(u, Ty)

α+ (1− α) cosh d(u, Ty)
.

Using Lemmas 3.5 and 3.1, we obtain

(1− σ2)d(u, Ty)
2 ≤ (1− σ1)d(u, Tx)

2.

Similarly, we consider the case (ii) and then we obtain

(1− σ1)d(u, Tx)
2 ≤ (1− σ2)d(u, Ty)

2.

Therefore, in both cases (i) and (ii), we have

d(Ux,Uy)2 ≤ (1− σ1)(1− σ2)d(Tx, Ty)
2.

By Lemmas 3.3 and 3.4, we have σ1 ≥ α and σ2 ≥ α. Thus

(1− σ1)(1− σ2) ≤ (1− α)2,

and it follows that

d(Ux,Uy)2 ≤ (1− α)2d(Tx, Ty)2 ≤ (1− α)2d(x, y)2.



22 Y. KIMURA, K. SASAKI, K. TORII EJDE-2024/CONF/27

Therefore,

d(Ux,Uy) ≤ (1− α)d(x, y),

and hence U is a contraction. □

Henceforth, we consider implicit-type iterative schemes. Now we prove a con-
vergence theorem using Browder type iteration in complete CAT(−1) spaces.

Theorem 3.7. Let X be a complete CAT(−1) space, and let T : X → X be a
nonexpansive mapping with F (T ) ̸= ∅. Let u ∈ X and {αn} ⊂ ]0, 1

2 ] such that
αn → 0 as n → ∞. Define {xn} ⊂ X by

xn = αnu
−1
⊕ (1− αn)Txn.

Then, {xn} is well-defined and convergent to PF (T )u.

Proof. We know that Theorem 3.6 implies the well-definedness of xn for every
n ∈ N. Let p = PF (T )u. Then

d(p, u) = inf
y∈F (T )

d(y, u).

By Lemma 2.5, we have

cosh d(xn, p) = cosh d(αnu
−1
⊕ (1− αn)Txn, p)

≤ αn cosh d(u, p) + (1− αn) cosh d(Txn, p)

≤ αn cosh d(u, p) + (1− αn) cosh d(xn, p).

for all n ∈ N. Thus,
cosh d(xn, p) ≤ cosh d(u, p)

for all n ∈ N and hence we obtain

d(Txn, p) ≤ d(xn, p) ≤ d(u, p)

for all n ∈ N. It implies that {xn} and {Txn} are bounded. Since

d(xn, Txn) ≤ d(xn, p) + d(p, Txn),

we have {d(xn, Txn)} is also bounded.
Fix n ∈ N and put D = d(xn, p). From the definition of xn, we have

(αn cosh d(xn, u) + (1− αn) cosh d(xn, Txn)) sinhD

≤ (αn cosh d(txn ⊕ (1− t)p, u) + (1− αn) cosh d(txn ⊕ (1− t)p, Txn)) sinhD

≤ αn(cosh d(xn, u) sinh tD + cosh d(p, u) sinh(1− t)D)

+ (1− αn)(cosh d(xn, Txn) sinh tD + cosh d(p, Txn) sinh(1− t)D)

= (αn cosh d(xn, u) + (1− αn) cosh d(xn, Txn)) sinh tD

+ (αn cosh d(p, u) + (1− αn) cosh d(p, Txn)) sinh(1− t)D

for all t ∈ ]0, 1[. Thus

(αn cosh d(xn, u) + (1− αn) cosh d(xn, Txn))
sinhD − sinh tD

sinh(1− t)D

≤ αn cosh d(p, u) + (1− αn) cosh d(p, Txn).

Letting t → 1, we obtain

(αn cosh d(xn, u) + (1− αn) cosh d(xn, Txn)) cosh d(xn, p)



EJDE-2024/CONF/27 IMPLICIT TYPE ITERATIONS IN GEODESIC SPACES 23

≤ αn cosh d(p, u) + (1− αn) cosh d(Txn, p)

≤ αn cosh d(p, u) + (1− αn) cosh d(xn, p).

Therefore,

αn cosh d(xn, u) + (1− αn) cosh d(xn, Txn) ≤ αn
cosh d(p, u)

cosh d(xn, p)
+ (1− αn). (3.1)

Thus, since αn → 0 and {xn} is bounded, we obtain lim supn→∞ cosh d(xn, Txn) ≤
1 from (3.1), and hence we have

lim
n→∞

d(xn, Txn) = 0.

From (3.1), we obtain

αn cosh d(xn, u) ≤ αn cosh d(xn, u) + (1− αn)(cosh d(xn, Txn)− 1)

≤ αn
cosh d(p, u)

cosh d(xn, p)

≤ αn cosh d(p, u).

Thus, we obtain cosh d(xn, u) ≤ cosh d(p, u) and it follows that

d(xn, u) ≤ d(p, u) (3.2)

for all n ∈ N.
To show that {xn} converges to p, we prove that {xn} is ∆-convergent to p.

Thus, we take a subsequence {xni
} ⊂ {xn} arbitrarily, and let v be an element of

the asymptotic center of {xni}. Then, taking subsequence repeatedly, we can find
{x′

j} ⊂ {xni} such that

lim
j→∞

d(x′
j , p) = lim sup

i→∞
d(xni

, p) (3.3)

and there exists q ∈ X such that x′
j

∆
⇀ q from Theorem 2.2. Then q ∈ AC({x′

j}).
We show q = p. Since T is nonexpansive, we have

lim sup
j→∞

d(x′
j , T q) ≤ lim sup

j→∞
(d(x′

j , Tx
′
j) + d(Tx′

j , T q))

≤ lim sup
j→∞

d(x′
j , Tx

′
j) + lim sup

j→∞
d(Tx′

j , T q)

≤ lim sup
j→∞

d(x′
j , q).

From the uniqueness of the element of AC({x′
j}), we obtain q ∈ F (T ). By Theo-

rem 2.8 and (3.2), we have

d(q, u) ≤ lim inf
j→∞

d(x′
j , u) ≤ d(p, u).

Since p is the unique nearest point of u on F (T ), the above inequality implies that
q = p and p ∈ AC({x′

j}). From (3.3), we have

lim sup
i→∞

d(xni
, p) = lim

j→∞
d(x′

j , p) ≤ lim sup
j→∞

d(x′
j , v) ≤ lim sup

i→∞
d(xni

, v).

Hence p ∈ AC({xni}) and it implies that v = p. Since v is an asymptotic center
of {xni

} ⊂ {xn}, which is arbitrarily chosen, and it coincides with p, {xn} is ∆-
convergent to p.



24 Y. KIMURA, K. SASAKI, K. TORII EJDE-2024/CONF/27

We finally show the convergence of {xn} to p. Since {xn} is ∆-convergent to p
and from (3.2), we have

d(p, u) ≤ lim inf
n→∞

d(xn, u) ≤ lim sup
n→∞

d(xn, u) ≤ d(p, u),

and hence we obtain

lim
n→∞

d(xn, u) = d(p, u).

Therefore, xn → p by Lemma 2.9, which is the desired result. □

We obtain the convergence theorem in the sense of Browder type with (−1)-
convex combination in a complete CAT(−1) space. Next, we consider the conver-
gence theorem in the sense of Xu-Ori type iteration in the same space.

Theorem 3.8. Let X be a complete CAT(−1) space and let T : X → X be a
nonexpansive mapping with F (T ) ̸= ∅. Suppose that {αn} ⊂ R and a ∈ R satisfies
0 < a ≤ αn ≤ 1

2 for all n ∈ N. Let x1 ∈ X and generate {xn} as follows: For
n ∈ N and given xn ∈ X, let xn+1 be the unique point in X satisfying that

xn+1 = αnxn

−1
⊕ (1− αn)Txn+1.

Then, {xn} is well-defined and ∆-convergent to some x0 ∈ F (T ).

Proof. Fix n ∈ N and define a mapping Vn : X → X by

Vnx = argminy∈X (αn cosh d(y, xn) + (1− αn) cosh d(y, Tx))

for x ∈ X. In the same way as Theorem 3.6, we obtain Vn is a contraction and
thus it has the unique fixed point xn+1 ∈ X. That is, it satisfies that

xn+1 = Vnxn+1 = argminy∈X (αn cosh d(y, xn) + (1− αn) cosh d(y, Txn+1)) ,

and hence {xn} is well-defined.
Next, we show {xn} is ∆-convergent to some element in F (T ). Let p ∈ F (T )

and t ∈ ]0, 1[. Fix n ∈ N and put D = d(xn+1, p). Then,

(αn cosh d(xn, xn+1) + (1− αn) cosh d(Txn+1, xn+1)) sinhD

= (αn cosh d(xn, Vnxn+1) + (1− αn) cosh d(Txn+1, Vnxn+1)) sinhD

≤ αn cosh d(xn, txn+1 ⊕ (1− t)p) sinhD

+ (1− αn) cosh d(Txn+1, txn+1 ⊕ (1− t)p) sinhD

≤ αn(cosh d(xn, xn+1) sinh tD + cosh d(xn, p) sinh(1− t)D)

+ (1− αn)(cosh d(Txn+1, xn+1) sinh tD

+ cosh d(Txn+1, p) sinh(1− t)D)

= (αn cosh d(xn, xn+1) + (1− αn) cosh d(Txn+1, xn+1)) sinh tD

+ (αn cosh d(xn, p) + (1− αn) cosh d(Txn+1, p)) sinh(1− t)D.

Thus

(αn cosh d(xn, xn+1) + (1− αn) cosh d(Txn+1, xn+1))
sinhD − sinh tD

sinh(1− t)D

≤ αn cosh d(xn, p) + (1− αn) cosh d(Txn+1, p).

Letting t → 1, we obtain

(αn cosh d(xn, xn+1) + (1− αn) cosh d(Txn+1, xn+1)) cosh d(xn+1, p)



EJDE-2024/CONF/27 IMPLICIT TYPE ITERATIONS IN GEODESIC SPACES 25

≤ αn cosh d(xn, p) + (1− αn) cosh d(Txn+1, p)

≤ αn cosh d(xn, p) + (1− αn) cosh d(xn+1, p).

Hence we have

cosh d(xn+1, p) ≤ αn cosh d(xn, p) + (1− αn) cosh d(xn+1, p).

Therefore, since {αn} ⊂ ]0, 1
2 ], we obtain

cosh d(xn+1, p) ≤ cosh d(xn, p).

This implies that the real sequence {d(xn, p)} is nonincreasing and bounded below.
Thus there exists a limit

lim
n→∞

d(xn, p) = cp ∈ R

and hence

1 ≤ αn cosh d(xn, xn+1) + (1− αn) cosh d(Txn+1, xn+1)

≤ (αn cosh d(xn, p) + (1− αn) cosh d(xn+1, p))
1

cosh d(xn+1, p)

≤ αn(cosh d(xn, p)− cosh d(xn+1, p))

cosh d(xn+1, p)
+ 1

→ 1

as n → ∞. This implies

lim
n→∞

(αn cosh d(xn, xn+1) + (1− αn) cosh d(Txn+1, xn+1)) = 1.

Then
lim
n→∞

cosh d(xn, xn+1) = lim
n→∞

cosh d(Txn+1, xn+1) = 1.

Indeed, we assume {cosh d(xn, xn+1)} does not converge to 1. Then there ex-
ist ε > 0 and a subsequence {cosh d(xni

, xni+1)} of {cosh d(xn, xn+1)} such that
cosh d(xni

, xni+1) ≥ 1 + ε for i ∈ N. Furthermore, since {αni
} ⊂ [a, 1

2 ], we may

assume that αni
→ α0 ∈ [a, 1

2 ] without loss of generality. Then we have

1 = lim
i→∞

(αni
cosh d(xni

, xni+1) + (1− αni
) cosh d(Txni

, xni+1))

≥ α0 lim inf
i→∞

cosh d(xni , xni+1) + (1− α0) lim inf
i→∞

cosh d(Txni+1, xni+1)

≥ α0(1 + ε) + (1− α0) = 1 + α0ε > 1.

This is a contradiction. Thus we have limn→∞ cosh d(xn, xn+1) = 1, and similarly
we obtain limn→∞ cosh d(Txn+1, xn+1) = 1. Hence we obtain

lim
n→∞

d(xn, xn+1) = lim
n→∞

d(Txn+1, xn+1) = 0.

Let x0 ∈ X be the unique asymptotic center of a sequence {xn} and let u ∈ X be
an asymptotic center of any subsequence {xni} of {xn}. We will show that u = x0.
From the definition of the asymptotic center, we have

r({xni
}) = lim sup

i→∞
d(xni

, u)

≤ lim sup
i→∞

d(xni , Tu)

≤ lim sup
i→∞

(d(xni
, Txni

) + d(Txni
, Tu))

= lim sup
i→∞

d(Txni , Tu)



26 Y. KIMURA, K. SASAKI, K. TORII EJDE-2024/CONF/27

≤ lim sup
i→∞

d(xni
, u) = r({xni

}).

This implies Tu ∈ AC({xni
}). From the uniqueness of an asymptotic center, we

obtain u ∈ F (T ). It follows that {d(xn, u)} is convergent to cu ∈ R. Therefore,
r({xn}) = lim sup

n→∞
d(xn, x0)

≤ lim sup
n→∞

d(xn, u) = cu = lim
i→∞

d(xni
, u)

≤ lim sup
i→∞

d(xni
, x0)

≤ lim sup
n→∞

d(xn, x0) = r({xn}).

Thus u ∈ AC({xn}). From the uniqueness of an asymptotic center, we obtain
u = x0. Hence, {xn} is ∆-convergent to x0 ∈ F (T ). This is the desired result. □

Acknowledgments. This work was partially supported by JSPS KAKENHI Grant
Number JP21K03316.

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Yasunori Kimura

Department of Information Science, Toho University, Miyama, Funabashi, Chiba 274-
8510, Japan

Email address: yasunori@is.sci.toho-u.ac.jp

Kazuya Sasaki

Faculty of Science, Toho University, Miyama, Funabashi, Chiba 274-8510, Japan

Email address: kazuya.sasaki@sci.toho-u.ac.jp

Kakeru Torii
Japan

Email address: torikake.1123.8@gmail.com