New Modified Proximal Point Algorithm for Solving Minimization and
Common Fixed Point Problem over CAT(κ) Spaces
CHATUPHOL KHAOFONG 1, PHACHARA SAIPARA2,∗,
SUPHOT SRATHONGLANG1, ANANTACHAI PADCHAROEN3
1Division of Mathematics, Faculty of Science and Technology,
Rajamangala University of Technology Krungthep, Bangkok, THAILAND
2Division of Mathematics, Department of Science, Faculty of
Science and Agricultural Technology, Rajamangala University of
Technology Lanna Nan, Nan, THAILAND
3Department of Mathematics, Faculty of Science and Technology,
Rambhai Barni Rajabhat University, Chanthaburi, THAILAND
∗Corresponding author: splernn@gmail.com
Abstract: -In this paper, we present a newly proximal point algorithm for solving minimization and common fixed
point problems in CAT(1) spaces. Under some mild conditions, we prove strong and ∆-convergence theorems.
Additionally, a convex minimization application and a common fixed point problems in CAT(κ) spaces with the
bounded κ∈(0,∞)are provided. Our findings complement and advance the pertinent recent findings in the
literature.
Key-Words: Geodesic metric space, Convex function, Iteration process; Fixed point problem;
Proximal point algorithm; Minimization problem
Received: March 26, 2023. Revised: October 25, 2023. Accepted: November 26, 2023. Published: February 12, 2024.
1 Introduction
Let (X, d)be a geodesic metric space, Kbe a subset
of X,K=∅and the self mapping Ton Kbe a non-
linear. The set F(T) := {x:T x =x}is called the
set of all fixed points of T. Among the significant an-
alytical issues are ones that relate to fixed points for
certain nonlinear mappings. Now, our attention is on
nonlinear problems such convex minimization prob-
lems and common fixed problems in CAT(1) spaces
under some mild conditions.
In 1976, the concept of ∆-convergence in general
metric spaces was first discussed by the result in, [1].
Let κ∈R. Then, a geodesic space that has a geodesic
triangle that is sufficiently thinner than the compara-
ble comparison triangle in a model space with curva-
ture κis said to be a CAT(κ) space.
The result in, [2], originally investigated the fixed
point theory in CAT(κ)spaces in 2003. Later, many
researchers expanded on the concept of CAT(κ) pro-
vided in, [3], by mainly concentrating on CAT(0)
spaces. Since each CAT(κ) space is a CAT(κ′) space
for any κ′≥κ, the results of a CAT(0) space can
be applied to any CAT(κ) space with κ≤0(see
in, [4]). However, many researchers have studied
CAT(κ) spaces for κ > 0(e.g., [5],[6],[7],[8],[9]).
Now, we introduce some iterative algorithms for
approximating common fixed point as follows. In
2021, the result in, [10], suggested the new iteration
approach for approximating the common fixed point
of three nonexpansive mappings. Let the self map-
pings on J,G1, G2, G3be three nonexpansive, then
the sequence {cn}is generated by c1∈Jand
an= (1 −κn)cn+κnG1cn,
bn= (1 −δn)an+δnG2an,
cn+1 = (1 −µn)G2an,
+µnG3bn+µnG3bn
(1)
where {µn},{δn}and {κn}are real sequences in
(0,1).
On the other hand, let f:X→(−∞,∞]be a
proper and convex function and (X, d)be a geodesic
metric space. The main optimization problem objec-
tive is to find x∈Xsuch that
f(x) = min
y∈Xf(y).
Let argminy∈Xf(y)be the set of minimizers of f.
In 1970, the proximal point algorithm(PPA) was first
developed by the result in, [11]. It is an efficient tech-
nique for tackling this problem. Later on in 1976, the
result in, [12], showed that the PPA converges to the
convex problem’s solution in Hilbert spaces. Let f
be a proper, convex and lower semicontinuous func-
tion on a Hilbert space H. The PPA is generated by
x1∈Hand
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xn+1 =argminy∈Hf(y) + 1
2λn∥y−xn∥2
where for all n∈Nand λn>0. It was proved that
{xn}converges weakly to a minimizer of fprovided
Σ∞
n=1λn=∞. However, the PPA does not always
strongly converge, as demonstrated by the result in,
[13]. The PPA and Halpern’s algorithm, [14], were
merged in 2000 by the result in, [15], who proved the
guarantee of strong convergence.
The asymptotic behavior of the sequences gener-
ated by the PPA for a convex function in geodesic
spaces with curvature constrained above was first
suggested by the result in, [16], in 2017. Addition-
ally, they introduced the PPA in the following way in
aCAT (1)space:
x1∈X,
xn+1 =argminy∈X[g(y)+
1
λn
tan(d(y, xn))sin(d(y, xn))]
(2)
where for all n∈Nand λn>0. By the Fej´er mono-
tonicity, it was proved that, if fhas a minimizer and
Σ∞
n=1λn=∞, then {xn}∆-converges to its mini-
mizer, [17]. A version of split for the PPA was em-
ployed in 2014 by the result in, [18], to minimize the
sum of convex functions in for CAT(0) spaces. Addi-
tional intriguing outcomes can also be studied in the
result in, [19].
Several PPA convergence results have recently
been extended to the context of manifolds from the
usual linear spaces, including the Euclidean, Hilbert
and Banach spaces(see in, [19], [20], [21], [22],
[23]). In analysis and geometry branch, the minimiz-
ers of the objective convex functional in the nonlinear
spaces are extremely important.
The result in, [23], introduced the result of PPA in
CAT (1) spaces Xas follows:
x1∈X,
wn=argminy∈X[g(y)+
1
λn
tan(d(y, xn)) sin(d(y, xn))],
xn+1 =αnxn⊕(1 −αn)T wn
(3)
where {αn}is a real sequences in the interval [0,1],
∀n≥1.
We present a newly modified PPA that is moti-
vated by (1), (2) and (3). Let gbe a proper lower
semi-continuous function from the set Xto (−∞,∞)
and (X, d)be an admissible complete CAT(1) space.
Consider three nonexpansive mappings T1, T2, T3:
K→Ksuch that Ω = F(T1)∩F(T2)∩F(T3)=∅.
Assume that for each a1, a2∈(0,1),{αn}and {βn}
are in [a1, a2]and λnis a sequence where λn≥λ≥
0, for each n≥1and for some λ, then the sequence
{xn}is generated by
wn=argminy∈X[g(y)+
1
λn
tan(d(y, xn))sin(d(y, xn))],
zn= (1 −κn)xn⊕κnT1wn,
yn= (1 −δn)zn⊕δnT2zn,
xn+1 = (1 −µn)T2zn⊕µnT3yn
(4)
where the sequences {µn},{δn}and {κn}are in
(0,1) for all n∈N.
For the purpose to solve minimization problems
and common fixed point problems in CAT(1) spaces,
we introduce a newly PPA in this study and prove
strong and ∆-convergence theorems for this algo-
rithm in CAT(1) spaces. Additionally, a convex mini-
mization application and a common fixed point prob-
lems on CAT(κ) spaces with the bounded positive real
number κare provided.
2 Preliminaries
Let (X, d)be a metric space. A geodesic path joining
xto yis a map γfrom a interval [0, l]⊂Rto the set
Xsuch that γ(0) = x, γ(l) = y, and ρ(γ(t), γ(t′)) =
|t−t′|for all t, t′∈[0, l]and x, y ∈X. Specifi-
cally, γis an isometry and d(x, y) = l. A geodesic
segment joining xand yis a term given to the image
of γ([0, l]) of γ. This geodesic segment is represented
by the symbol [x, y]when it is unique. Accordingly,
z∈[x, y]if and only if there exists α∈[0,1] such
that
d(x, z) = (1 −α)d(x, y)and d(y, z) = αd(x, y).
For this particular case, we can write z=αx ⊕
(1 −α)y. If every two points of Xare joined by a
geodesic which every two points of distance smaller
than D, then the space (X, ρ)is called a geodesic
space or D−geodesic space. If there is exactly one
geodesic joining xand yfor each x, y ∈X, then Xis
called uniquely geodesic or D−uniquely geodesic.
If K⊂Xincludes every geodesic segment joining
any two of its points, then the set Kis called convex.
The set Kis called bounded if
diam(K) := sup{d(x, y) : x, y ∈K}<∞.
The model spaces Mn
κare now introduced; the
reader is referred to, [4], for more information on
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these spaces. Let n∈N. The metric space Rnwith
the usual Euclidean distance is denoted by the sym-
bol En. We symbolize the Euclidean scalar product
in Rn by the symbol (·|·), that is,
(x|y) = x1y1+... +xnynwhere
x= (x1, ..., xn), y = (y1, ..., yn).
Let Snbe the n−dimensional sphere denoted
by
Sn={x=x1, ..., xn+1 ∈Rn+1 : (·|·) = 1},
with metric dSn=arccos(x|y), x, y ∈Sn.
Let En,1be the vector space Rn+1 endowed with
the symmetric bilinear form which associates to vec-
tors u= (u1, ..., un+1)and v= (v1, ..., vn+1)the
real number ⟨u|v⟩denoted by
⟨u|v⟩=−un+1vn+1 +n
i=1 uivi.
Let Hnbe the hyperbolic n −space denoted by
Hn={u= (u1, u2, ..., un+1)∈En,1:⟨u|u⟩=
−1, un+1 >1}
with metric dHnsuch that
cosh(dHn(x, y)) = −⟨x|y⟩, x, y ∈Hn.
Definition 2.1. Let κ∈R, the following metric
spaces are defined by Mn
κ.
(1) if κ= 0 then Mn
0is the Euclidean space En;
(2) if κ > 0then Mn
κis obtained from the spherical
space Snby multiplying the distance function by the
constant 1/√κ;
(3) if κ < 0then Mn
κis obtained from the hyperbolic
space Hnby multiplying the distance function by the
constant 1/√−κ.
A geodesic triangle is made up of three points
in the geodesic space (X, d)(x, y, and z) and three
geodesic segments between each pair of vertices. A
comparison triangle for ∆(x, y, z)in (X, d)is a tri-
angle ∆(x, y, z)in M2
κsuch that
d(x, y) = dM2
κ(x, y), d(x, z) = dM2
κ(x, z)and
ρ(z, x) = dM2
κ(z, x).
If κ≤0then such a comparison triangle always
exists in M2
κ. If κ > 0then such a triangle exists
whenever d(x, y) + d(y, z) + d(z, x)<2Dκ, where
Dκ=π/√κ. A point p∈[x, y]is called a compari-
son point for p∈[x, y]if d(x, p) = dM2
κ(x, p).
A geodesic triangle ∆(x, y, z)in Xis said to sat-
isfy the CAT(κ) inequality if for any p, q ∈∆(x, y, z)
and for their comparison points p, q ∈∆(x, y, z), one
has
d(p, q)≤dM2
κ(p, q).
Definition 2.2. If κ≤0, then Xis called a
CAT(κ)space if and only if Xis a geodesic space
such that all of its geodesic triangles satisfy the
CAT(κ) inequality. If κ > 0, then Xis called a
CAT(κ)space if and only if Xis Dκ-geodesic
and any geodesic triangle ∆(x, y, z)in Xwith
d(x, y) + d(y, z) + d(z, x)<2Dκsatisfies the
CAT(κ)inequality.
Definition 2.3. A self mapping Ton the set Xis
called:
(1) nonexpansive if d(T x, T y)≤d(x, y)for any
x, y ∈X.
(2) demi-compact if, for all {xn} ∈ Csuch that
limn→∞ d(xn, T xn)=0,{xn}has a convergent
subsequence.
Let CAT(1) space be (X, d)such that x, y, z ∈X
satisfy d(x, y) + d(y, z) + d(z, x)<2D1. Then
cos d(αx ⊕(1 −α)y, z)(5)
≥αcos d(x, z) + (1 −α)cos d(y, z)
for all α∈[0,1].
Definition 2.4. ,[24], Let (X, d)be a geodesic met-
ric space.
(1) An open set Uin (X, d)is said to be a CR−
domain for any R∈[0,2] if x, y, z ∈U, any mini-
mal geodesic γ: [0,1] →Xbetween yand zfor all
α∈[0,1],
d2(x, (1 −α)y⊕αz)(6)
≤(1 −α)d2(x, y) + αd2(x, z)
−R
2(1 −α)αd2(y, z).
(2) (X, d)is said to be R−convex for any R∈[0,2]
if Xitself a CR−domain.
(3) (X, d)is said to be locally R −convex for R∈
[0,2] if every point in Xincluded in a CR−domain.
Definition 2.5. Let CAT(1) space be (X, d). A se-
quence {xn}in Xis called △-convergent to x∈X
if xis the unique asymptotic center of every subse-
quence {un}of {xn}.We write △−limn→∞ xn=x
and define W△(xn) := ∪{A({un})}.
The domain of the function g:X→(−∞,∞]is
Dom(g) = {x∈X:g(x)∈R}.
If Dom(g)is nonempty, then gis called proper. If
K={x∈X:g(x)≤β}is closed in Xfor all
β∈R., then gis called lower semi-continuous.
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A CAT(1) space Xis called admissible if
d(v, v′)<π
2for all v, v′∈X. Apart from that, the
{xn}in a CAT(1) space is called spherically bounded
if
inf
y∈Xlim sup
n→∞
d(y, xn)<π
2.
Let gbe a proper lower semi-continuous convex
function. For all λ > 0,the following formulation of
the resolvent of gin the admissible CAT(1) spaces:
Rλ(x) = argmin
y∈Xg(y)
+1
λtan d(y, x)sin d(y, x)
for all x∈X.Rλis well define for all λ >
0. More specifically, F(Rλ)of fixed points of the
resolvent associated with gcoincides with the set
argminy∈Xg(y)of minimizers of g.
Lemma 2.6. Let g:X→(−∞,∞]be a proper
lower semi-continuous convex function and (X, d)be
a admissible complete CAT(1) space. If λ > 0,∈X
and u∈argminXg, then the following inequalities
hold:
π
2A(B−C)≥λ(g(Rλx)−g(u)) (7)
and
B≥C(8)
where
A=1
cos2d(Rλx,x)+ 1,
B=cos d(Rλx, x)cos d(u, Rλx)
and C=cos d(u, x).
Lemma 2.7. Let (X, d)be the admissible complete
CAT(1) space. If g:X→(−∞,∞]is a proper
semi-continuous convex function, then gis ∆−lower
semi-continuous.
Lemma 2.8. Let (X, d)be a complete CAT(1) space
and {xn}be a spherical bounded sequence in X. If
d(dn, ρ)is convergent for all ρ∈W∆({xn}), then
{xn}is ∆−convergent.
Corollary 2.9. Let Cbe a nonempty closed convex
subset of complete CAT(1) space (X, d). Let the self
mapping Ton Cbe a nonepansive. If {xn}is a
bounded sequence such that limn→∞ d(xn, T xn) = 0
and ∆−limn→∞ xn=ω, then ω∈Cand ω=T ω.
3 Main results
The main results can be presented in the following.
Lemma 3.1. Assume that g:X→(−∞,∞]is
a proper lower semi-continuous convex function, let
(X, d)be an admissible complete CAT(1) space. As-
sume that T, S and Rare three nonexpansive map-
pings, such that Ω = F(T1)∩F(T2)∩F(T3)∩
argminx∈Xg(x). Assume that {µn},{δn}and {κn}
are in [a1, a2]for a1, a2∈(0,1) and {λn}is a se-
quence such thatλn≥λ > 0, for each and for some
λ. Assume that for each n≥1, the sequence xnis
generated by (4). Then we have the following:
(1) for all q∈Ω,limn→∞ d(xn, q)exists;
(2) limn→∞ d(xn, zn) = 0;
(3) limn→∞ d(xn, T1xn)
=limn→∞ d(xn, T2xn)
=limn→∞ d(xn, T3xn).
Proof. First, we prove that {xn},{wn}are spherical
bounded. Assume that wn=Rλnxnfor each n≥1.
Let q∈Ω. Then, by (7), we have
min{cos d(wn, xn),cos d(q, wn)}(9)
≥cos d(wn, xn)cos d(q, wn)
≥cos d(q, xn)
it shows that
max{d(wn, xn), d(q, wn)}(10)
≤d(q, xn).
Since T1, T2and T3are three nonexpansive map-
pings and Xis admissible, by (4), we obtain
cos d(q, zn)(11)
=cos d(q, (1 −κn)xn⊕κnT1wn)
≥(1 −κn)cos d(q, xn) + κncos d(q, T1wn)
≥(1 −κn)cos d(q, xn) + κncos d(q, wn)
≥(1 −κn)cos d(q, xn) + κncos d(q, xn)
=cos d(q, xn),
and
cos d(q, yn)(12)
=cos d(q, (1 −δn)zn⊕δnT2zn)
≥(1 −δn)cos d(q, zn) + δncos d(q, T2zn)
≥(1 −δn)cos d(q, zn) + δncos d(q, zn)
≥(1 −δn)cos d(q, xn) + δncos d(q, xn)
=cos d(q, xn),
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and
cos d(q, xn+1)(13)
=cos d(q, (1 −µn)T2zn⊕µnT3yn)
≥(1 −µn)cos d(q, T2zn)
+µncos d(q, T3yn)
≥(1 −µn)cos d(q, zn) + µncos d(q, yn)
≥(1 −µn)cos d(q, xn) + µncos d(q, xn)
=cos d(q, xn),
it shows that
d(q, xn+1)(14)
≤d(q, xn)≤d(q, x1)<π
2.
Thus, the sequence {xn}and {wn}are spherically
bounded. Hence, assertion (1) is true. Now, we prove
that
sup
n≥1
d(xn, wn)<π
2
and limn→∞ d(q, xn)<π
2exists for all q∈Ω. So,
we get
lim
n→∞ d(q, xn) = r≥0.(15)
So, this claim that limn→∞ d(xn, q)exists, for all q∈
Ω. We now claim that limn→∞ d(xn, wn)=0. By
(13), it follows that
cos d(q, xn+1)
=cos d(q, (1 −µn)T2zn⊕µnT3yn)
≥(1 −µn)cos d(q, T2zn)
+µncos d(q, T3yn)
≥(1 −µn)cos d(q, zn) + µncos d(q, yn)
≥(1 −µn)cos d(q, xn) + µncos d(q, yn)
so,
cos d(q, xn+1)
≥cos d(q, xn)−µncos d(q, xn)
+µncos d(q, yn);
µncos d(q, xn)
≥cos d(q, xn)−cos d(q, xn+1)
+µncos d(q, yn);
cos d(q, xn)
≥1
µn
[cos d(q, xn)−cos d(q, xn+1)]
+cos d(q, yn).
Since µn≥a1>0for each n≥1, we get
cos d(q, xn)(16)
≥1
a1
[cos d(q, xn)−cos d(q, xn+1)]
+cos d(q, yn).
So, by (15), (16), we get
r=lim inf
n→∞ cos d(q, xn)(17)
≥lim inf
n→∞ cos d(q, yn).
In contrast, we see from (12) that
lim sup
n→∞
cos d(q, yn)(18)
≥lim sup
n→∞
cos d(q, xn) = r.
So, by (17) and (18), we get
lim
n→∞ cos d(q, yn) = r. (19)
On the same way, by (13), it follows that
cos d(q, xn+1)
=cos d(q, (1 −µn)T2zn⊕µnT3yn)
≥(1 −µn)cos d(q, T2zn)
+µncos d(q, T3yn)
≥(1 −µn)cos d(q, zn) + µncos d(q, yn)
≥(1 −µn)cos d(q, zn) + µncos d(q, xn)
so,
cos d(q, xn+1)
≥(1 −µn)cos d(q, zn)
+µncos d(q, xn);
cos d(q, xn+1)
≥(1 −µn)cos d(q, zn)
+(1 −(1 −µn)) cos d(q, xn);
(1 −µn)cos d(q, xn)
≥(1 −µn)cos d(q, zn)
+cos d(q, xn)−cos d(q, xn+1);
cos d(q, xn)
≥1
1−µn
[cos d(q, xn)−cos d(q, xn+1)]
+cos d(q, zn).
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Since 1−µn≥a1>0for each n≥1, we get
cos d(q, xn)(20)
≥1
a1
[cos d(q, xn)−cos d(q, xn+1)]
+cos d(q, zn).
So, by (15) and (20), we get
r=lim inf
n→∞ cos d(q, xn)(21)
≥lim inf
n→∞ cos d(q, zn).
In contrast, we see from (11) that
lim sup
n→∞
cos d(q, zn)(22)
≥lim sup
n→∞
cos d(q, xn) = r.
So, by (21) and (22), we get
lim
n→∞ cos d(q, zn) = r. (23)
By (9), (10), we get
cos d(q, zn)
= (1 −κn)cos d(q, xn) + κncos d(q, T1wn)
≥(1 −κn)cos d(q, xn) + κncos d(q, wn)
≥cos d(q, xn)−κncos d(q, xn)
+κn
cos d(q, xn)
cos d(wn, xn)
=cos d(q, xn)
+κncos d(q, xn)[ 1
cos d(wn, xn)−1],
that is,
cos d(q, zn)
cos d(q, xn)−1
≥κn[1
cos d(wn, xn)−1].
Since κn≥a1>0for each n≥1, by (15), (19)
and (23), it follows that
1≤1
cos d(wn, xn)
that is,
lim
n→∞ d(wn, xn) = 0.
Thus, we obtain
lim
n→∞ d(Rλnxn, xn) = 0.
Since λn≥λ > 0for each n≥1, we have
lim
n→∞ d(Rλxn, xn) = 0.
Thus, this claim that limn→∞ d(wn, xn) = 0. Hence,
assertion (2) is true. Finally, we prove that
lim
n→∞ d(xn, T1xn) = lim
n→∞ d(xn, T2xn)
=lim
n→∞ d(xn, T3xn)
= 0.
By (5), we obtain
d2(q, zn)
=d2(q, (1 −κn)xn⊕κnT1wn)
≤(1 −κn)d2(q, xn) + κnd2(q, T1wn)
−R
2(1 −κn)κnd2(xn, T1wn)
≤(1 −κn)d2(q, xn) + κnd2(q, wn)
−R
2a1a2d2(xn, T1wn)
≤(1 −κn)d2(q, xn) + κnd2(q, xn)
−R
2a1a2d2(xn, T1wn)
=d2(q, xn)−R
2a1a2d2(xn, T1wn),
it shows that
d2(q, zn)
≤d2(q, xn)−R
2a1a2d2(xn, T1wn);
R
2a1a2d2(xn, T1wn)
≤d2(q, xn)−d2(q, zn);
d2(xn, T1wn)
≤2
Ra1a2
[d2(q, xn)−d2(q, zn)].
This yields
lim
n→∞ d(xn, T1wn) = 0.
So, by the triangle inequality, we have
d(xn, T1xn)≤d(xn, T1wn) + d(T1wn, T1xn)
≤d(xn, T1wn) + d(wn, xn)
→0, as n → ∞
which implies that
lim
n→∞ d(xn, T1xn) = 0.
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Next, we have
d2(q, yn)
=d2(q, (1 −δn)zn⊕δnT2zn)
≤(1 −δn)d2(q, zn) + δnd2(q, T2zn)
−R
2(1 −δn)δnd2(zn, T2zn)
≤(1 −δn)d2(q, zn) + δnd2(q, T2zn)
−R
2a1a2d2(zn, T2zn)
≤(1 −δn)d2(q, zn) + δnd2(q, zn)
−R
2a1a2d2(zn, T2zn)
=d2(q, zn)−R
2a1a2d2(zn, T2zn),
which implies that
d2(q, yn)≤d2(q, zn)
−R
2a1a2d2(zn, T2zn);
R
2a1a2d2(zn, T2zn)≤d2(q, xn)
−d2(q, yn);
d2(zn, T2zn)≤2
Ra1a2
[d2(q, xn)
−d2(q, yn)].
This gives
lim
n→∞ d(zn, T2zn) = 0.
By the triangle inequality, we get
d(xn, T2xn)
≤d(xn, zn) + d(zn, T2xn)
≤d(xn, T2zn) + d(T2zn, zn)
+d(zn, T2zn) + d(T2zn, T2xn)
≤d(xn, zn) + d(T2zn, zn)
+d(zn, T2zn) + d(zn, xn)
→0, as n → ∞.
Lastly, we have
d2(q, xn+1)
=d2(q, (1 −µn)T2zn⊕µnT3yn)
≤(1 −µn)d2(q, T2zn) + µnd2(q, T3yn)
−R
2(1 −µn)µnd2(T2zn, T3yn)
≤(1 −µn)d2(q, zn) + µnd2(q, yn)
=−R
2a1a2d2(zn, T3yn)
≤(1 −µn)d2(q, xn) + µnd2(q, xn)
=−R
2a1a2d2(zn, T3yn)
=d2(q, xn)−R
2a1a2d2(zn, T3yn),
which implies that
d2(q, xn+1)≤d2(q, xn)
−R
2a1a2d2(zn, T3yn);
R
2a1a2d2(zn, T3yn)≤d2(q, xn)
−d2(q, xn+1);
d2(zn, T3yn)≤2
Ra1a2
[d2(q, xn)
−d2(q, xn+1)].
Thus, we get
lim
n→∞ d(zn, T3yn) = 0.
It follows that
d(zn, xn)
≤d((1 −κn)xn⊕κnT1wn, xn)
≤(1 −κn)d(xn, xn) + κnd(T1wn, xn)
→0, as n → ∞,
and
d(yn, xn)
≤d((1 −δn)zn⊕δnT2zn, xn)
≤(1 −δn)d(zn, xn) + δnd(T2zn, xn)
≤d(zn, xn)
→0, as n → ∞.
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By the triangle inequality, we get
d(xn, T3xn)
≤d(xn, zn) + d(zn, T3xn)
≤d(xn, zn) + d(zn, T3yn)
+d(T3yn, T3xn)
≤d(xn, zn) + d(zn, T3yn)
+d(yn, xn)
→0, as n → ∞.
Hence, the assertion 3) is true. The proof is now com-
plete.
Next, suppose that Lemma 3.1’s conclusion is
true. Following are some ∆−convergence results
that we prove.
Theorem 3.2. Assume that g:X→(−∞,∞]is
a proper lower semi-continuous convex function, let
(X, d)be an admissible complete CAT(1) space. Then
{xn}generated by (4) ∆−converges to an element of
Ω = F(T1)∩F(T2)∩F(T3)∩argminx∈Xg(x).
Proof. Let ω∈Ωand assume that wn=Rλnxnfor
each n≥1. Then, for each n > 1, we have g(ω)≤
g(wn). From Lemma 2.6, we have
D≥λn(g(wn)−g(ω)) ≥0(24)
where
D=π
2(1
cos2d(wn, xn)
+1)(cosd(wn, xn)cosd(ω, wn)
−cosd(ω, xn))
Due to the fact that λn>λ>0for each n≥1and
by Lemma 3.1, we can prove that
d(wn, xn)→0as n → ∞,(25)
lim
n→∞ d(ω, xn)and
lim
n→∞ d(ω, wn)exist.
By (24), we get
lim
n→∞ g(wn) = inf g(X).(26)
Next, we prove that W∆({xn})⊂Ω. Let
w∗∈W∆({xn}). Then there exists a subsequence
{xni}of {xn}which ∆−converges to w∗. Since
limn→∞ d(wn, xn), we can observe that the subse-
quence wniof wnalso ∆-converges to the point w∗
according to the definition of the ∆-convergence.
Lemma 2.7 and (26) provide
g(w∗)≤lim inf
i→∞ g(wni)
≤lim
n→∞ g(wn)
=inf g(X).
Hence, w∗∈argminx∈Xg(x)and so W∆({xn})⊂
argminx∈Xg(x). Moreover, since
lim
n→∞ d(xn, T1xn) = lim
n→∞ d(xn, T2xn)
=lim
n→∞ d(xn, T3xn)
= 0,
and {xn}∆−converges to w∗, it follows from Corol-
lary 2.9 that w∗∈F(T1). So, we conclude that
W∆({xn})⊂Ω, we can see that for any w∗∈
W∆({xn}),d(w∗, xn)is convergent. By Lemma 2.8,
{xn}is ∆−convergent to element in Ω. Lemma 2.8
shows that {xn}is ∆−convergent to element in Ω.
The proof is now complete.
Theorem 3.3. Assume that g:X→(−∞,∞]is
a proper lower semi-continuous convex function, let
(X, d)be an admissible complete CAT(1) space. Con-
sequently, these are equivalent.
(A) Strong convergence arises to an element of Ωfor
the sequence xngenerated by (4).
(B) If d(x, Ω) = inf{d(x, x∗) : q∈Ω}, then
lim infn→∞ d(xn,Ω) = 0.
Proof. We start by proving that (A)⇒(B). It is ob-
vious.
Furthermore, we prove that (B)⇒(A). Assume
that lim infn→∞ d(xn,Ω) = 0. Since d(xn+1, q)≤
d(xn, q)for all q∈Ω, we get
d(xn+1,Ω) ≤d(xn,Ω).
Thus, limn→∞ d(xn,Ω) = 0. Then, using the meth-
ods in, [25], we get that {xn}is a Chauchy sequence
in X. This implies that {xn}converges to point
c∈Xand thus d(c, Ω) = 0. Since Ωis closed, c∈Ω.
The proof is now complete.
The mappings T1, T2, T3are called to satisfy the
condition Qif there exists a nondecreasing function
h: [0,∞)→[0,∞)with h(k)≥0for all k∈(0,∞)
such that
d(x, T1x)≥h(d(x, H)),
or
d(x, T2x)≥h(d(x, H)),
or
d(x, T3x)≥h(d(x, H)),
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for all x∈X, where H=H(T1)∩H(T2)∩H(T3).
Applying the condition Qyields the following
result.
Theorem 3.4. Assume that g:X→(−∞,∞]is
a proper lower semi-continuous convex function, let
(X, d)be an admissible complete CAT(1) space. If
Rλ,T1and T2satisfy the condition Q, then {xn}gen-
erated by (4) strongly converges to an element of Ω.
Proof. We prove that limn→∞ d(xn, q)exists for all
q∈Ωby using Lemma 3.1. Additionally, it follows
that limnd(xn,Ω) exists. Applying the condition Q,
we obtain
lim
n→∞ h(d(xn,Ω)) ≤lim
n→∞ d(xn, Rλxn) = 0,
or
lim
n→∞ h(d(xn,Ω)) ≤lim
n→∞ d(xn, T1xn) = 0,
or
lim
n→∞ h(d(xn,Ω)) ≤lim
n→∞ d(xn, T2xn) = 0,
or
lim
n→∞ h(d(xn,Ω)) ≤lim
n→∞ d(xn, T3xn) = 0.
Thus, we obtain
lim
n→∞ h(d(xn,Ω)) = 0
which by using the property of h, results in
limn→∞ d(xn,Ω) = 0. Also, by the remained proof
can be followed by the proof in Theorem 3.3 and
hence, the desired result follows. The proof is now
complete.
Theorem 3.5. Assume that g:X→(−∞,∞]is a
proper lower semi-continuous convex function, let
(X, d)be an admissible complete CAT(1) space. If
Rλor T1or T2is demi-compact, then {xn}generated
by (4) strongly converges to an element of Ω.
Proof. By Lemma 3.1, we obtain
lim
n→∞ d(xn, Rλxn)(27)
=lim
n→∞ d(xn, T1xn)
=lim
n→∞ d(xn, T2xn)
=lim
n→∞ d(xn, T3xn)
= 0
as n→ ∞. Without loss of generality, we assume that
T1, T2, T3or Rλis demi-compact. Therefore, there
exists a subsequence {xni}of {xn}such that {xni}
converges strongly to ρ∗∈X. Hence, from (27) and
the nonexpansiveness of mappings T1, T2, T3, Rλ, it
followed that
d(ρ∗, Rλρ∗) = d(ρ∗, T1ρ∗)
=d(ρ∗, T2ρ∗)
=d(ρ∗, T3ρ∗)
= 0,
which denote that ρ∗is in Ω. Later, we can prove the
strong convergence of {xn}to an element of Ω. The
proof is now complete.
4 Some Applications
Applications for the common fixed point in CAT(κ)
with the bounded positive real number κand some
convex optimization problems, are demonstrated in
this section.
The following assumptions are made throughout
this section:
(A1)Xis a complete CAT(κ) space such that
d(v, v′)<Dκ
2;
(A2)κis a positive real number and Dx=π
√κ;
(A3)g:X→(−∞,∞]be a proper lower semi-
continuous convex function;
(A4)
Rλis the resolvent mapping on Xdefined by
Rλ(x) = argminy∈X[g(y) +
1
λtan(√κd(y, x)) sin(√κd(y, x))]
for all λ > 0and x∈X.
The mapping
Rλis well-defined since (X, √κd)
is the admissible complete CAT(1) space, according
to the result in, [26]. From Theorem 3.2, 3.3, 3.4 and
3.5 and assume that assumptions A1,A2,A3and A4
hold, we get some Corollaries as follows.
Corollary 4.1. Assume that assumptions A1,A2,A3
and A4are hold. Let the mappings T1, T2, T3:C→
Care nonexpansive such that Ω=∅. Suppose that
the sequence {δn},{κn},{µn} ⊆ [a1, a2]for some
a1, a2∈(0,1). Let {λn}be the sequence such that
for each n≥1,λn≥λ > 0for some λ. For any
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x1∈X, generate the sequence {xn} ∈ Cby
wn=argminy∈X[g(y)+
+1
λn
tan(√κd(y, xn)) sin(√κd(y, xn))],
zn= (1 −κn)xn⊕κnT1wn,
yn= (1 −δn)zn⊕δnT2zn,
xn+1 = (1 −µn)T2zn⊕µnT3yn,
(28)
for each n≥1. Then {xn}∆−converges to an
element of Ω.
Corollary 4.2. Assume that assumptions A1,A2,A3
and A4are hold. Let the mappings T1, T2, T3:C→
Care nonexpansive such that Ω=∅. Suppose that
the sequence {δn},{κn},{µn} ⊆ [a1, a2]for some
a1, a2∈(0,1). Let {λn}be the sequence such that
for each n≥1,λn≥λ > 0for some λ. Conse-
quently, these are equivalent.
1) The {xn}generated by (28) converges strongly
to an element of Ω.
2) lim infn→∞ d(xn,Ω) = 0 where d(x, Ω) =
inf{d(x,∗) : q∈Ω}.
Corollary 4.3. Assume that assumptions A1,
A2,A3and A4are hold. Let the mappings
T1, T2, T3:C→Care nonexpansive such
that Ω=∅. Suppose that the sequence
{δn},{κn},{µn} ⊆ [a1, a2]for some a1, a2∈(0,1).
Let {λn}be the sequence such that for each
n≥1,λn≥λ > 0for some λ. If the mappings
Rλ, T1, T2, T3satisfy the condition(Q) then {xn}
generated by (28) converges strongly to an element
of Ω.
Corollary 4.4. Suppose that A1,A2,A3and A4are
hold. Let the mappings T1, T2, T3:C→Care non-
expansive such that Ω=∅. Suppose that the sequence
{δn},{κn},{µn} ⊆ [a1, a2]for some a1, a2∈(0,1).
Let {λn}be the sequence such that for each n≥1,
λn≥λ > 0for some λ. Let {λn}be a sequence such
that, for each n≥1, λn≥λ > 0for some λ. If Rλor
T1or T2or T3is demi-compact, then {xn}generated
by (28) converges strongly to an element of Ω.
5 Conclusion
In this paper, for the minimization problem and the
common fixed point problem in CAT(1) spaces, we
prove a strong and ∆-convergence theorems. Our
main results are a generalization of the results of
various researchers in the literature review (see in,
[17], [18], [19], [20], [21], [22], [23]). Addition-
ally, we discussed about various applications to the
common fixed point problem and the convex mini-
mization problem in CAT(κ) spaces with the bounded
positive real number κ. We further expanded on the
results from the work of Kimura et al., [16], regard-
ing the asymptotic behavior of sequences produced by
the proximal point algorithm for a convex function in
geodesic spaces with curvature bounded above.
Acknowledgment:
The first and third author were supported by
Rajamangala University of Technology Krungthep
(RMUTK). The second author was supported by
Rajamangala University of Technology Lanna
(RMUTL). The last author was supported by
Rambhai Barni Rajabhat University.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Chatuphol Khaofong carried out conceptu-
alization, methodology, supervision, writ-
ing and editing manuscript preparation.
Phachara Saipara carried out conceptualiza-
tion, methodology, supervision, writing orig-
inal draft, editing manuscript preparation
and submitting the manuscript to Journal.
Suphot Srathonglang carried out writing original
draft, writing and editing the manuscript preparation.
Anantachai Padcharoen carried out method-
ology, investigation and validation.
All authors have read and agreed to the
published version of the manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This work was supported by Rajamangala Univer-
sity of Technology Lanna(RMUTL), Rajamangala
University of Technology Krungthep(RMUTK) and
Rambhai Barni Rajabhat University(RBRU)
Conflicts of Interest
The authors have no conflicts of interest to
declare that are relevant to the content of this
article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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