CC BY
5URAL MATHEMATICAL JOURNAL, Vol. 7, No. 1, 2021, pp. 109-119
DOI: 10.15826/umj.2021.1.010
MODIFIED PROXIMAL POINT ALGORITHM FOR MINIMIZATION AND FIXED POINT PROBLEM
IN CAT(0) SPACES
Department of Mathematics, University of Eswatini, Private Bag 4, Kwaluseni, Eswatini
Department of Mathematics and Applied Mathematics, Sefako Makgato Health Science University, P.O. Box 94, Pretoria 0204, South Africa
ugwunnadi4u@yahoo.com
Abstract: In this paper, we study modified-type proximal point algorithm for approximating a common solution of a lower semi-continuous mapping and fixed point of total asymptotically nonexpansive mapping in complete CAT(0) spaces. Under suitable conditions, some strong convergence theorems of the proposed algorithms to such a common solution are proved.
Keywords: Proximal point algorithm, Total asymptotically nonexpansive mapping, Fixed point, A convergence, Strong convergence, CAT(0) space.
In recent years, much attention has been given to develop several iterative methods including the proximal point algorithms (PPA) which was suggested by Martinet [26] for solving convex optimization problems which was extensively developed by Rockafellar [28] in the context of monotone variational inequalities. The main idea of this method consists of replacing the initial problem with a sequence of regularized problems, so that each particular auxiliary problem can be solved by one of the well-known algorithms. Quiet number of different method of proximal point algorithm have been proposed and studied from the classical linear spaces such as Euclidean spaces, Hilbert spaces, and Banach spaces to the setting of manifolds (see [5, 6, 13, 18, 20, 26, 28]).
Recently, the classical proximal point algorithms have been extended from linear spaces such as Hilbert spaces or Banach spaces to the setting of nonlinear version.
In 2013, Bacak [6] introduced the PPA in a CAT(0) space (X, d) as follows: x\ € X and
Godwin Chidi Ugwunnadi
1. Introduction
where Xn > 0, V n > 1. It was shown that if f has a minimizer and
n=1
then the sequence {xn} A—converges to its minimizer [5].
It is a known fact that iterative methods for finding fixed points of nonexpansive mappings have received vast investigations due to its extensive applications in a variety of applied areas of inverse
problem, partial differential equations, image recovery, and signal processing; see [2, 5, 8, 15, 21] and the references therein.
Fixed-point theory in CAT(0) spaces was first studied by Kirk [22, 23]. He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then, the fixed-point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed.
Recently, Suparatulatorn et al. [29] presented a new modified proximal point algorithm for solving the minimization of a convex function and the fixed points of nonexpansive mappings in CAT(0) spaces. Chang et al. [12] proved some strong convergence theorems of the PPA to a common fixed point of asymptotically nonexpansive mappings and to minimizers of a convex function in CAT(0) spaces.
Let C be a nonempty subset of a complete CAT(0) space X and T a mapping from C into itself. Then, a point x € C is called a fixed point of T if Tx = x. We denote by F(T) the set of all the fixed points of T. A mapping T from C into itself is said to be:
(N) nonexpansive if d(Tx,Ty) < d(x,y) for all x,y € C;
(AN) asymptotically nonexpansive, if there is a sequence {un} C [0, to) with un — 0 as n — to such that
d(Tnx,Tny) < (1+ un)d(x,y), Vn > 1, x,y € C; (UL) uniformly L-Lipschitzian, if there exists a constant L > 0 such that
d(Tnx,Tny) < Ld(x,y), Vn > 1, x,y € C.
The concept of total asymptotically nonexpansive mappings was first introduced by Alber et al. [1]. A mapping T : C - C is said to be total asymptotically nonexpansive mapping if there exists nonnegative sequences {^n}, {vn} with — 0, vn — 0 as n — to and a strictly increasing continuous function Z : [0, to) — [0, to) with Z(0) = 0 such that
d(Tnx, Tny) < d(x, y) + VnZ(d(x, y)) + ^n, Vn > 1, x, y € C.
Remark 1. From the definitions, it is known that each nonexpansive mapping is asymptotically nonexpansive mapping with sequence {un = 0}, and each asymptotically nonexpansive mapping is ({^n}, {vn},Z)-total asymptotically nonexpansive mapping with = 0, vn = un, Vn > 1 and Z(t) = t, t > 0. But the opposite may not be true for each of them in a general sense. Furthermore, every asymptotically nonexpansive mapping is a uniformly L-Lipschitzian mapping with
L = sup(1 + un).
n>1
Motivated and inspired by the above works, in this paper, we study a modified algorithm for proximal point and fixed point of total asymptotically nonexpansive mapping in CAT(0) space. Strong convergence of this algorithm is proved. Our method of proof is different from the method in Chang et al. [12].
2. Preliminaries
Let (X, d) be a metric space and x,y € X with d(x,y) = l. A geodesic path from x to y is an isometry c : [0, l] — X such that c(0) = x and c(l) = y. The image of a geodesic path is
called a geodesic segment. A metric space X is a (uniquely) geodesic space, if every two points of X are joined by only one geodesic segment. A geodesic triangle A(x^x2,x3) in a geodesic space X consists of three points xi,x2,x3 of X and three geodesic segments joining each pair of vertices. A comparison triangle of a geodesic triangle A(x1,x2,x3) is the triangle A(x1,x2,x3) := A(x1,x2,x3) in the Euclidean space R2 such that
d(xi,xj) = dR2 (xi,yj), Vi,j = 1,2,3.
A geodesic space X is a CAT(0) space, if for each geodesic triangle A(x1,x2,x3) in X and its comparison triangle A := A(x1,x2,x3) in R2, the CAT(0) inequality d(x,y) < dR2(x,y) is satisfied for all x,y € A and x,y € A.
A thorough discussion of these spaces and their important role in various branches of mathematics are given in [9, 10]. Let x,y € X and A € [0,1], we write \x ® (1 — A)y for the unique point z in the geodesic segment joining from x to y such that
d(z,x) = (1 — A)d(x,y) and d(z,y) = Ad(x,y).
We also denote by [x, y] the geodesic segment joining from x to y, that is,
[x,y] = {Ax ® (1 — A)y : A € [0,1]}.
A subset C of a CAT(0) space is convex if [x, y] C C for all x,y € C.
Berg and Nikolaev [7] introduced the concept of an inner product-like notion (quasi-linearization) in complete CAT(0) spaces to resolve these difficulties as follows:
Let denote a pair (a,b) € X x X by — and call it a vector. The quasilinearization is a map (.,.) : (X x X) x (X x X) ^ R defined by
(al, al) = \ (d2(a, d) + d2(b, c) - d2(a, c) - d2(b, d)), Va, b,c,d£ X. (2.1)
It is easily seen that (—b, —) = (— —), (~ab, —) = — (—a, —) and (a^, —I) + (-—>, —d) = (—b, —d) for all a, b,c,d € X. We say that X satisfies the Cauchy-Schwarz inequality if
(aab, ~cd) < d(a, b)d(c, d)
for all a, b,c,d € X. It is known that a geodesically connected metric space is a CAT(0) space if and only if it satisfies the Cauchy-Schwarz inequality (see [7]).
Lemma 1 [16]. Let X be a CAT(0) .space, x,y,z € X and A € [0,1]. Then d(Ax ® (1 — A)y, z) < Ad(x, z) + (1 — A)d(y, z).
Lemma 2 [16]. Let X be a CAT(0) space, x,y,z € X and A € [0,1]. Then
d2(Ax ® (1 — A)y, z) < Ad2(x, z) + (1 — A)d2(y, z) — A(1 — A)d2(x, y).
Lemma 3 [14]. Let X be a CAT(0) space, x,y,z € X and A € [0,1]. Then
d2(Ax ® (1 — A)y, z) < A2d2(x, z) + (1 — A)2d2(y, z) + 2A(1 — A)(—z, —£).
Let {xn} be a bounded sequence in a complete CAT(0) space X. For x € X, we set
r(x, {xn}) = lim sup d(x, xn).
The asymptotic radius r({xn}) of {xn} is given by
r({x„}) = inf {r(x, {x„}) : x € X},
and the asymptotic center A({xn}) of {xn} is the set
A({xra}) = {x € X : r(x, {xra}) = r({xn}^.
It is well known that in a CAT(0) space A({xn}) consists of exactly one point (see [15, Proposition 7]).
Lemma 4 [24]. Every bounded sequence in a complete CAT(0) space always has a A—convergent subsequence.
Lemma 5 [19]. Let X be a complete CAT(0) space, {xn} be a sequence in X and x € X. Then {xn} A—converges to x if and only if lim supra^^(——)) < 0 for all y € C.
A function f : C c (—to, to] defined on a convex subset C of a CAT(0) space is convex if, for any geodesic
[x, y] := {Yx,y(A) : 0 < A < 1} := {Ax ® (1 — A)y : 0 < A < 1} joining x, y € C, the function f o y is convex, i.e.
f (Yx,y(A)) := f (Ax ® (1 — A)y) < Af (x) + (1 — A)f (y).
For examples of convex functions in CAT(0), see [12]. For any A > 0, define the Moreau-Yosida resolvent of f in CAT(0) space X as
J\(x) = arg min f(y) + -\-d2(y,x)
yex
2A
V x e X.
Let f : X — (-to, to] be a proper convex and lower semi-continuous function. It is shown in [3] that the set F(J^) of fixed points of the resolvent associated with f coincides with the set argminyeX f (y) of minimizers of f. Also for any A > 0, the resolvent Ja of f is nonoexpansive [17].
Lemma 6 (Sub-differential inequality [4]). Let (X, d) be a complete CAT(0) space and f : X — (-to, to] be proper convex and lower semi-continuous. Then, for all x, y € X and A > 0, the following inequality holds:
-^d2(Jxx,y) - ^d2(x,y) + ^d2(x, Jxx) + /(J\x) < f(y).
Lemma 7 [17, 27] (The resolvent identity). Let (X,d) be a complete CAT(0) space and f : X — (-to, to] be proper convex and lower semi-continuous. Then the following identity holds:
/ A - ¿t ¿i \
J\X — ^ ^ J\X © J
all x € X and A > ^ > 0.
Lemma 8 [11]. If C is a closed convex subset of a complete CAT(0) space X and T : C c X be a uniformly L-Lipschitzian and total asymptotically nonexpansive mappings. Let {xn} be a bounded, sequence in C such that xn ^ p and
lim d(xn, Txn) = 0.
n^-x
Then Tp = p.
Lemma 9 [25]. Let {an} be a sequence of real numbers such that there exists a subsequence {n^} of {n} such that ani < a„i+i for all i € N. Then there exists a nondecreasing sequence {mk} C N such that mk c to and the following properties are satisfied by all (sufficiently large) numbers k € N
amk < amk + 1 and afc < amk+1.
In fact,
mk = max{j < k : aj < aj+1}.
Lemma 10. (Xu, [30]) Let {an} be a sequence of nonnegative real numbers satisfying the following relation:
ara+i < (1 — an)a« + a„CT„ + y«, n > 0,
where, (i) {an} C [0,1], = to; (ii) lim sup < 0; (iii) y« > 0; (n > 0), < to.
Then, an c 0 as n c to.
3. Main Result
Theorem 1. Let X be a complete CAT(0) space and C be a nonempty closed convex subset of X. Let f : C — (-to, to] be a proper convex and lower semi-continuous function and T : C — C be L-Lipschitzian and total asymptotically nonexpansive mappings with {un}, {vn} and mappings Z : [0, to) — [0, to) satisfying un < to vn < to such that
Q := F(T) Pi arg minf (y) = 0.
yec
Let {xn}'=1 be a sequence generated by x1 = w € C chosen arbitrarily,
f(y) +
2An
zn = arg min
yec
yn = anw © (1 - an)z„, Xn+1 = (1 - ßn)x„ © ß„Tny„,
(3.1)
where {an}^=1 C (0,1), {ßn}^=1 C [c, d] C (0,1) satisfying
<x
lim an = 0, V^ an = oo, lim — = 0, lim — = 0. n^x ^^ n^x an n^x an
n=1
Assume there exists constant M > 0, such that Z(r) < Mr, Vr > 0. Then {xn}'=1 converges strongly to Q.
Proof. Let p € Q and f (p) < f (y), V y € C. Therefore we obtain
/(P) + < /(y) + ^~d2(y,p), Vj/eC,
2 An 2 An
hence p = J\np, V n > 1. Indeed zn = J\nxn and J\n is nonexpansive [17]. Thus
d(zn,p) = d(JXnXn,J\np) < d(xn,p). Let 5n := anpn(1 + unM). Since there exists N0 > 0 such that
un e(l + UnM) vn
— <-T-r-, — < 1 + UnM),
an M an
for all n > N0 and for some e > 0 satisfying 0 < (l — e)ön < l. For any point p € Ü and n > N0, then we have from (3.1) and from Lemma l that
d(xn+1,p) = d((1 — ßn)xn ® ßnTnyn,p)
< (1 — ßn)d(xn, p) + ßnd(Tn yn,p)
< (1 — ßn)d(xn,p) + ßn(1 + MUn)d(yn,p) + ßn Vn
= (1 — ßn)d(xn,p) + ßn(1 + Mun)[d(anw ® (1 — an)zn,p)] + ßnVn
< (1 — ßn)d(xn,p) + anßn (1 + Mun)d(w,p) + ßn(1 — an)(1 + Mu,n)d(z,n ,p) + ß,nVn
< [1 — ßn + ßn (1 — an)(1 + Mun)]d(xn,p) +anßn(1 + Mun)d(w,p) + ßn Vn
< [1 - (1 - e)6n]d(xn,p) + 5n(l - e){d{W,P) + 1}
< max ^d(xn,[j),
Thus, by induction
d(xn, p) < max | d(xNo, p), + 1) j ^ y n> N0.
It implies that {xn} is bounded, it follows that {yn} and {zn} are also bounded. Furthermore, from (3.1) and Lemma 2 and letting un := 2Mun + un, we obtain
d2(xn+1,p) = d2((1 — ßn)xn ® ßnTnyn,p)
< (1 — ßn )d2(xn,p)+ ßnd2(Tnyn,p) — ßn(1 — ßn)d2(xn,Tnyn)
< (1 — ßn )d2(x n 1 p) + ßn((1 + Mun)d(yn,p) + Vn)2 — ßn(1 — ßn)d2 (x,n,Tnyn) = (1 — ßn )d2(xn,p) + ßn(1 + Mu,n)d2(y,n,'p) + ßnVn[2(1 + Mun)d(yn,p) + Vn]
—ßn(1 — ßn)d2(xn,Tnyn), (3.2)
(1 — e)
(d(w,p) + 1)
also from Lemma 3, we have
d (yn,p) = d (anw ® (1 — an)zn,p)
< and2 (w,p) + (1 — an) 2d2 (zn ,p) + 2an (1 — an)(-^,-5')
< a2nd2(w,p)) + (1 - an)d2(zn,p) + 2an(1 - an)-)-)) (3.3)
< and2(w,p) + (1 - an)d2(xn,p) + 2an(1 - an){w$,—p). (3.4)
From (3.2) and (3.4) and the fact that {yn} is bounded, we have that there exists D > 0 such that
for any n > No, d(yn,p) < D and letting 0n := anwe obtain
d2(xn+1,p) < (1 - ^n)d2(xn,p) + ^n[«nd2(w,p) + (1 - an)d2(xn,p) + 2an(1 - an)(-P,¿nP)] und2(yn,p)^nvn[2(1 + Mun)d(yn,p) + Vn] - £n(1 - ^n)d2(xn,Tnyn)
< (1 - 0n)d2(xn,p) + 0n[and2(w,p) + (1 - an)(wp, inp)]
[unD2 + 2Dvn(1 + Mun) + v2] - ^n(1 - &)d2(xn, Tnyn) (3.5)
< (1 - 0n)d2(xn,p) + 0n[and2(w,p) + (1 - an)(wp, ^p)]
3n[unD2 + 2Dvn (1 + Mun) + vn To complete the proof, we have to consider the following two cases.
+en[UnD2 + 2Dv„ (1 + Mu„) + vn ]. (3.6)
Case 1. Suppose {d(xn,p)} is non-increasing, then {d(xn,p)} is convergent, from (3.5) and boundedness of {zn}, then there exists D1 > 0 such that for any n > N0, d(zn,p) < D1, thus
A*(1 - ^n)d2(xn,Tnyn) < d2(xn,p) - d2(xn+1,p)
+0„[a„d2(w,p) +2(1 - a„)(-p, 5Пp) - d2(x„,p)j +e«[UnD2 + 2Dv„ (1 + Mura) + ]
which implies that hence
< d (xn,p) - d (xn+i,p)
+0„[a„d2(w,p) + 2(1 - a„)Did(w,p) - d2(x„,p)j +en[UnD 2 + 2Dv„ (1 + Mura) + v£j, (3.7)
вп(1 - e«)d2(xra, Tny„) — 0 as n — то,
lim d(x„,Tny„) = 0 (3.8)
and from (3.1), we have
d(yn, ) < a„d(w, ) + (1 — ara)d(zra, zra) —^ 0 as n —у то, (3.9)
also from (3.1) and (3.8), we obtain
d(x„+i,x„) < (1 — e«)d(xn,xra) + end(Tray„,x„) — 0 as n — то. (3.10)
Furthermore from Lemma 6, we see that
-j-d2(z„,p) - -j-d2(xn,p) + -j-d2(xn,zn) < f(p) - f(zn),
2An 2An 2An
since f (p) < f (zn) for all n > 1, it follows that
d2(x„,z„) < d2(x„,p) — d2(z„,p). (3.11)
But from (3.2) and (3.3), we obtain
d2(x„+i,p) < (1 — вп)d2(xn,p) + en[a;;d2(w,p) + (1 — a„)d2(z„,p) + 2a„(1 — a„)(—p,-J)] +era[2v„(1 + Mu„)d(y„ ,p) + vn + u„d2(y„,p)],
therefore, from (2.1) and boundedness of {zn}, we obtain
d2(xn,p) < d2(xn,p) - d2(xn+i,p)) + a?d2(w,p) + (1 - an)d2(zn,p)
ßn (3.12)
+2an(1 — an)D1d(w,p) + 2Dv,n(1 + Mum) + unD2,
from (3.11) and (3.12), we obtain
d2(xn,zn) < -j-(d2(xn,p) - d2(xn+i,p)) + an(and2(w,p) -d2(zn,p))
Pn
+2an(1 - an)Did(w,p) + 2Dvn(1 + Mum) + UnD2, since {xn} and {zn} are bounded and {d(xn,p)} is non-increasing sequence, it follows from that
lim d(xn,zn) = 0, (3.13)
n—^^o
from (3.9) and (3.13), we obtain
lim d(xn, yn) = 0 (3.14)
and
d(yn, Tnyn) < d(yn, xn) + d(xn, Tnyn) P 0 as n p to, (3.15)
also from (3.14) and (3.15), we obtain
d(xn, Tnxn) < d(xn, yn) + d(yn, Tnyn) + d(Tnyn, Tnxn)
< (2 + Mun)d(xn, yn) + d(yn, Tnyn) + vn P 0 (3.16)
as n p to. Observe also that since T is uniformly L-Lipschitzian, we have
d(xn,Txn) < d(xn,Tnxn) + d(Tnxn,Txn)
< d(xn, Tnxn) + Ld(Tn-1xn, xn)
< d(xn,Tnxn) + L[d(Tn-1xn,Tn-1xn-1) + d(Tn-1xn-1,xn-1) + d(xn-1,xra,
< d(xn,Tnxn) + Ld(Tn-1xn-1,xn-1) + L(1 + L)d(xn,xn-1)
it follows from (3.10) and (3.16) that
lim d(xn,Txn) = 0, (3.17)
n—^^o
from (3.14) and (3.17), we obtain
d(yn, Tyn) < d(yn, xn) + d(xn, Txn) + d(Tx,n, Ty,n) < (1 + L)d(yn,xn) + d(xn,Txn) P 0
as n P to. Also since Xn > X> 0, from Lemma 7, we obtain
d(J\xn, J\nxn) = d^J\xn, J\ ^ — J\nXn © -j— ^
< d(xn > (1 - Y" )'hn xn © xn)
< (1 - -£-)d{xn,Zn) p 0
An
as n — то, hence
d(xn, JAxn) < d(xn, zn) + d(zn, JAxn) — 0 as n — то. (3.18)
Moreover, since {xn} is bounded and X is a complete CAT(0) space, by Lemma 4 we choose a subsequence {xni} of {xn} such that A — limxni = v, where v := Pq(w). Then, from (3.15), (3.18), Lemma 8 and the fact that Ja is nonexpansive [17], we have v € F(T), also from Lemma 5, we have
limsup(—v,xnv) < 0. (3.19)
Furthermore, since
(wv, z——) = (wv, zn—) + (w-v ,x—v)
< d(w,v)d(zn , xn) + (w-v, X——),
it follows from (3.13) and (3.19) that
limsup(—v, —< 0. Thus, now putting v := p in inequality (3.6), we get that, for n > No
d2(x„+i,v) < (1 - 6n)d2(x„,v) + 0„[a„d2(w,v) + (1 - a„)(-V,-;V)j
ь ( w, v) + (1 - an
n[und2 + 2Dvn(1 + ^ "<n; "г
(3.20)
Hence where
d2(xn+i, v) < (1 - 0n)d2(xn, v) + + Yn,
CTn := and2(w, v) + (1 — an)(—v, ^P), Yn := вп[ип^2 + 2Dvn(1 + M«„) + vn], it follows from Lemma 10 that d(xn,v) — 0 as n — то. Consequently, xn — v.
Case 2. Suppose that {d(xn,p)}n>1 is non-decreasing sequence. Then, there exists a subsequence {nj of {n} such that
d(xni,p) < d(xni+i,p)
for all i € N. Then, by Lemma 9, there exists an increasing sequence {mj }j>1 such that mj — то, d(xmj,p) < d(xmj+1,p) and d(xj,p) < d(xmj+1,p) for all j > 1. Then from (3.7), we obtain
-"miX*- Hmj)d (xm,',T j ymi) < d (xm,', p) d ^m
emj ( 1 - emj ) d2 (Xmj., Tm ymj ) < d2 (xmj , p) - d2 (xmj + 1, p)
+ömj [amjd2(w,p) + 2(1 - amj)(-p, z-p) - d2(xmj,p)j
+ßmj [Um, D2 + 2Dvm, (1 + Mum, ) + ^, j
< d (Xm, ,p) - d (Xm, + 1,p)
T >
+^mj [am,d2(w,p) +2(1 - am, )Did(w,p) - d2(Xm, ,p)j
+emj [Umj D2 + 2Dvm,. (1 + Mumj ) + V^. ].
This implies d(xmj ,Tmjymj) — 0 as j — то. Thus, as in Case 1, we obtain that d(xmj ,Txmj) — 0 and d(xm., JAxmj ) — 0 as j — то and also following the same argument in Case 1, we get limsup(wV, zmjV} < 0, where v := Pq(w). Also from (3.20), we obtain that,
j2/m „,\ ^ /1 л ^ л Г„, л2
d (Xm,+i,v) < (1 - 0n)d (Xm,, v) + 0m, [am,d (w,v)+(1 - am,)(-v, -
(3.21)
+em, [Um,D2 + 2Dvm, (1 + MUm,) + vm,j.
Since d2(xmj,v) < d2(xmj+1,v), it follows that
0mj d2(xmj ,v) < d2(x mj i v) d (xmj +1,v) + 9mj [amjd2(w,v) + (1 - amj){wtj,zmjV)] +Pmj [Umj D2 + 2Dvmj (1 + Mum, ) + v^ ] < Omj[amjd2(w,v) + (1 - amj){zp,zmjv)]
+Pmj [Umj D2 + 2Dvm3 (1 + MUm3 ) + v2rUj ]. In particular, since dmj > 0, we get
d2(xm.,v) < [am.d2(w,v) + (1 - amj){w$+ -^D2 + 2D^(1 + Mum ) + vmj —
J J J J A/ A/
amj amj amj
Then, since lim sup{wti, zmj v) < 0 and the fact that amj p 0 as j p to and
Umj vmj
lim —- = 0, lim —- = 0
amj j^^ amj
we obtain that d(xmj,v) p 0 as j p to. This together with (3.21) give d(xmj+1, v) p 0 as j p to. But d(xj,v) < d(xmj+1,v), for all j > 1, thus we obtain that xj p v. Therefore, from the above two cases, we can conclude that {xnconverges strongly to an element of Q and the proof is complete. □
4. Conclusion
In this work, we study a modified Halpern-type proximal point algorithm for finding the mini-mizer of a convex lower semi-continuous function which is also a fixed point of total asymptotically nonexpansive mapping. Under some appropriate assumption, we have obtained a strong convergence theorem for the proposed algorithm in the framework of a complete CAT(0) space.
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