# Strong Convergence Theorem for Finding a Common Solution of Convex Minimization and Fixed Point Problems in CAT(0) Spaces

Authors: P. V. Ndlovu; L. O. Jolaoso; Maggie Aphane
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

In this paper, we introduce a proximal point algorithm for approximating a common solution of finite family of convex minimization problems and fixed point problems fork-demicontractive mappings in complete CAT(0) spaces. We prove a strong convergence result and obtain other consequence results which generalize and extend some recent results in the literature. We further provide a numerical example to illustrate the convergence behaviour of the sequence generated by our algorithm.


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## Body

## 1. Introduction

LetX,d be a geodesic metric space and f:X⟶−∞,∞ be a proper, convex, and lower semicontinuous function. One of the problems arising in optimization theory is to find a minimizer of f, that is, find z∈X such that
(1)fz=mink∈Xfk.We denote the set of minimizers off by argminf. A strong tool used for solving such problems is known as the proximal point algorithm (PPA) which was first initiated by Martinet [1]. Rockafellar [2] generalized the study to Hilbert spaces H as follows: for s1∈H,(2)sn+1=argminfp+12σn∥sn−p∥2,p∈H,where σn>0 for all n∈ℤ+. In 2013, Bačák [3] extended the PPA to CAT(0) spaces as follows: s1∈X(3)sn+1=argminfp+12σnd2p,sn,p∈X,where σn>0 for all n∈ℤ+. It was shown that if f has a minimizer and liminfn⟶∞σn>0, then the sequence snΔ-converges to its minimizer. Bačák [4] further employed a split version of the PPA for sum of convex operators in CAT(0) spaces. Furthermore, Cholamjiak [5] modified the PPA in CAT(0) space X using the Halpern method as follows: s1∈X,(4)sn=argp∈Zminfp+12σd2p,sn,sn+1=bnu⊕1−bnsn,n∈ℤ+,where p∈X,σ>0, limn⟶∞bn=0, and ∑n=1∞bn=∞. He showed that the sequence sn strongly converges to its minimizer.On the other hand, one of the most fundamental findings in fixed point theory is the Banach contraction principle [6]. It has been used to prove the existence of the solutions of different nonlinear functional equations. Recently, efforts have been done to obtain a fixed point in partially ordered sets. Ran and Reurings (see [7]) generalized the Banach contraction to ordered metric space. Nieto and Rodriguez-Lopez [8] extended the results of fixed point theory in partially ordered metric space and used them to study the existence of the solution of differential equations. Fixed point theory in CAT(0) spaces was introduced by Kirk [9]; since then, many others have studied fixed point theory for different types of mappings, such as nonexpansive, Lipschitz, quasinonexpansive, k-strictly pseudocontractive, and k-demicontractive together with nonlinear optimization problems (see [10–15]).Definition 1 (see [16]).
LetC be a nonempty subset of a metric space X,d and T:C⟶C be a nonlinear mapping. A point x∈C is called a fixed point of T if Tx=x. The set of fixed points of T is denoted by FT, i.e., FT=x∈C:Tx=x.Definition 2 (see [17]).
LetX be a Hadamard space and C be a nonempty closed and convex subset of X. A mapping T:C⟶C is said to be
(i)
a contraction, if there existsα∈0,1, such that(5)dTx,Ty≤αdx,y,∀x,y∈C,when α=1, then T is said to be nonexpansive.
(ii)
quasinonexpansive, ifFT≠∅ and(6)dp,Tx≤dp,x,∀p∈FT,x∈C.(iii)
k-strictly pseudocontractive, if there exists k∈0,1 such that(7)d2Tx,Ty≤d2x,y+kdx,Tx+dx,Ty2,∀x,y∈C.(iv)
k-demicontractive, if FT≠∅ and there exists k∈0,1 such that(8)d2Tx,p≤d2x,p+kd2Tx,x,∀x∈C,p∈FT.We note that the class ofk-demicontractive mappings contains the class of nonexpansive, quasinonexpansive, and k-strictly pseudocontractive mappings (see [18, 19]).In 2014, Cholamjiak and Abdou [20] modified (4) in CAT(0) spaces for approximating the minimizers of f and common fixed point of nonexpansive mappings T1 and T2 as follows: given s1∈X,(9)qn=argp∈Xminfp+12σndp,sn,rn=1−ansn⊕anT1qn,sn+1=1−bnT1sm⊕bnT2rn∀n∈ℤ+.Also, they established theΔ-convergence for sequence sn generated by (9). Motivated by the results of Liu and Chang [17], the authors in [21] introduced a modified PPA for finding a common solution of convex minimization problems and fixed point of nonexpansive mappings in CAT(0) spaces. In particular, the authors proved the Δ-convergence for the following results.Theorem 3.
LetD be a nonempty closed convex subset of a complete CAT(0) space. Let Kii=1N be a finite family of proper, convex, and lower semicontinuous functions of D into −∞,∞ and let Tii=1N be a finite family of nonexpansive mappings of D into itself. Suppose that F=⋂i=1Narga∈Dminki⋂⋂i=1NFTi is nonempty. For xi∈D, let xn be a sequence in D defined by
(10)yni=arga∈Dminkia+12σnid2a,xn,zn=βn0⊕βn1yn1⊕βn2yn2⊕⊕βnNynN,wn=rn0zn⊕rn1T1zn⊕rn2T2zn⊕⊕rnNTNzn,xn+1=αnxn⊕1−αnwn,∀n≥1,where αn, βni, and rni are sequences in 0,1 such that 0<a≤αn,βni,rni≤b<1, ∑i=1Nβni=1, and ∑i=1Nrni=1 for all n≤1 and λni⊂0,∞ for all n≥1,i=1,2,3,N. Then, the sequence xn generated by (1.3) Δ-converges to an element w∈F.Recently, Chidume et al. [22] proved the following theorem for approximating the common fixed points of finite family of k-demicontractive mappings in CAT(0) spaces.Theorem 4 (see [22]).
LetX be a nonempty closed convex subset of a complete CAT(0) space. Let Ti:X⟶CBX,i=1,2,⋯,m be a family of demicontractive mappings with constants kj∈0,1, i=1,2,⋯,m, such that ∩i=1mFTi≠∅. Suppose that Ti is Δ-demiclosed at 0 for all i=1,2,⋯,m and Tip=p for all p∈∩i=1nFTi for arbitrary x1∈H, define a sequence xn by
(11)xn+1=α0xn⊕α1yn1⊕α2yn2⊕⋯⊕αnym,n≥1,where yni∈Tixn,i=1,2,3⋯,m,α0∈k,1,αi∈0,1,i=1,2,3⋯,m, such that ∑i=0mαi=1 and k=maxki. Then, xnΔ-converges to a point p∈∩i=1mFTi.Motivated by the above results, in this paper, we introduce a modified Halpern method together with the PPA method for approximating a common solution of finite family of convex minimization problems and common fixed point ofk-demicontractive mappings in complete CAT(0) spaces. We prove a strong convergence result and provide some consequence results which generalize some other related results in the literature. We also give a numerical example to show the convergence behaviour of the sequence generated by our algorithm.


## 2. Preliminaries

Definition 5 (see [23]).
LetX,d be a metric space and x,y∈X. A geodesic path from x to y is a map c:0,1⟶X such that c0=x, cl=y, and dct,ct′=∣t−t′∣ for all t,t′∈0,1. Its image is called a geodesic segment. If every pair of points x,y∈X are joined by a geodesic, we call the space X,d a geodesic space, and a space X,d is said to be a uniquely geodesic if every pair of points of X are joined by exactly one geodesic segment. A geodesic triangle Δx1,x2,x3 in a geodesic metric space X,d contains three points x1,x2,x3∈X (vertices of Δ), and the geodesic segment between each pair of vertices (edge of Δ) of a comparison triangle for the geodesic triangle Δx1,x2,x3∈X,d is a triangle Δ¯≔Δx¯1,x¯2,x¯3 in the Euclidean plane ℝ2 such that dℝ2x¯i,x¯j=dx¯i,x¯j for all i,j∈1,2,3. A geodesic space is said to be a CAT(0) space if for each geodesic triangle Δx1,x2,x3 in X and its comparison Δ¯=Δx¯1,x¯2,x¯3∈ℝ2, the CAT(0) space inequality
(12)dx,y≤dℝ2x¯,y¯,is satisfied for all x,y∈Δ and comparison points x¯,y¯∈Δ¯. If y0 is the midpoint and x,y1,y2 are points in a CAT(0) space, then the CAT(0) inequality implies
(13)d2x,y0≤12d2x,y1+12d2x,y2−14dy1,y2.Equation (13) is called the CN-inequality of Bruhat and Tits [23]. Moreover, a geodesic space is a CAT(0) space if and only if CN-inequality is satisfied. Examples of CAT(0) spaces are complete, simply connected Riemannian manifold having nonpositive sectional curvature, ℝ-Trees, Euclidean buildings, pre-Hilbert spaces, the complex Hilbert ball, with hyperbolic spaces. A complete CAT(0) space is called the Hadamard space. The normed linear spaces satisfy the CN-inequality if and only if it satisfies the parallelogram identity (see [24]). It is not so unexpected to have an inner product-like notion in Hadamard spaces.Definition 6 (see [25]).
The concept of quasilinearization was introduced by Berg and Nikolaev [25] as follows: let us first denote a pair a,b∈X×X by ab⟵ and call it a vector. Then, quasilinearization is defined as a map .,.:X×X×X×X⟶ℝ defined by
(14)ab⟵,cd⟵=12d2a,d+d2b,c−d2a,c−d2b,d,∀a,b,c,d∈X.
It can be seen thatab⟵,cd⟵=cd⟵,ab⟵, ab⟵,cd⟵=−ba⟵,cd⟵, and ax⟵,cd⟵+xb⟵,cd⟵=ab⟵,cd⟵ for all a,b,c,d,x∈X. Also the Cauchy-Schwartz inequality is satisfied if and only if
(15)ab⟵,cd⟵≤da,bdc,d,∀a,b,c,d∈X.Lemma 7.
LetX be CAT(0) space, w,x,y,z∈X, and t,s∈0,1. Then,
(i)
d2tx⊕1−ty,z≤t2d2x,z+1−t2d2y,z+2t1−txz⟵,yz⟵ [26](ii)
d2tx⊕1−ty,z≤td2x,z+1−td2y,z−t1−td2x,y [10](iii)
d1−tx⊕ty,z≤1−tdx,z+tdy,z [23]Definition 8 (see [10]; demiclosedness principle).
LetC be a nonempty closed and convex subset of a Hadarmard space X and T:C⟶C be a mapping. T is said to be Δ-demiclosed at zero if there exists a bounded sequence xn⊂X such that Δ-converges to p and limn⟶∞=0, then p∈FT.
The notion of asymptotic properties was introduced in a general framework of Hadamard space as follows.Definition 9 (see [10]).
Letxn be a bounded sequence in Hadamard space X,d. For any x∈X, we put rx,xn=limn⟶∞supdx,xn.(1)
The asymptotic radiusrxn of xn is given by rxn=infrx,xn:x∈X.(2)
The asymptotic centreAxn of xn is the set(16)Axn=x∈X:rx,xn=rxn.In Hadamard space,Axn consist of exactly one point.Lemma 10 (see [10]).
Ifxn is a bounded sequence in Hadamard space with Axn=x,un is the subsequence of xn with Aun=u and the sequence dxn,u converges, then x=u.
Letf:X⟶−∞,∞ be the proper, convex, and lower semicontinuous function. For all λ>0, we define the Moreau-Yosida resolvent of f in Hadamard space X as follows:
(17)Jλx=argminx∈Xfy+12λd2y,x∀y∈X.
It was proven in [24] that the set FJλ of a fixed point of the resolvent Jλ associated with f coincides with the set argminf, i.e., the set of minimizers of f. Also, for λ>0, the resolvent Jλ of f is nonexpansive (see [27]).Lemma 11 (see [28]).
LetX,d be a Hadamard space and f:X⟶−∞,∞ be a proper, convex, and lower semicontinuous function. Then, for all x,y∈X and λ>0, we have
(18)12λd2Jλx,y−12λd2x,y+12λd2x,Jλx+fJλx≤fy.Lemma 12 (see [29]).
LetX,d be a CAT(0) space and x∈X. Let x0,x1,x2,⋯,xN∈X and γ0,γ1,⋯,γN, be the real numbers in 0,1 such that ∑i=0Nγi=1. Then, the following inequality holds:
(19)dx,⊕i=0Nγixi2≤∑i=0Nγidx,xi2−∑i,j=0,i≠jNγiγjdxi,xj2.Lemma 13 (see [30]).
Letsn be a sequence of nonnegative real numbers, αn be a sequence of real numbers in 0,1 with ∑n=1∞αn=∞,un a sequence of nonnegative real numbers with ∑n=1∞un<∞,un and tn be a sequence of real numbers with limsupn⟶∞tn≤0. Suppose that
(20)sn+1≤1−αnsn+αntn+un,for all ∈ℕ. Then, limn⟶∞sn=0.Lemma 14 (see [31]).
LetX be a complete CAT(0) space, xn a sequence in X, and x∈X, then xnΔ-convergence to x if and only if limsupn⟶∞xny⟵,yny⟵≤0 for all y∈X.


## 3. Iterative Algorithm

In this section, we present our main result which deals with approximation of the common solution of the finite family of convex minimization problems and common fixed point ofk-demicontractive mappings in complete CAT(0) space. First, we give a coincise statement for our algorithm as follows.LetX be a complete CAT(0) space and C be a nonempty closed convex subset of X. For i=1,2,⋯,N, let fi:X⟶ℝ⋃+∞ be a finite family of proper, convex, and lower semicontinuous functions, and for j=1,2,⋯,M, let Tj:X⟶X be a finite family of kj-demicontractive mappings such that Tj is demiclosed at zero. Suppose the solution set
(21)Ω=⋂i=1Nargminfi∩⋂j=1MFTj≠∅.Letαn and γn,jj=0M be sequences in 0,1 such that(C1)limn⟶∞αn=0 and ∑n=0∞αn=+∞(C2)∑j=0Mγn,j=1 and liminfn⟶∞γn,0−k>0, where k=maxkj,1≤j≤M(C3)liminfn⟶∞λn,i>0Choosex0∈X arbitrarily and let xn be a sequence generated by the following algorithm.Algorithm 15.
(22)yn,1=argminy∈Xf1y+12λn,1d2y,xn,yn,2=argminy∈Xf2y+12λn,2d2y,yn,1,⋮yn,N−1=argminy∈XfN−1y+12λn,N−1d2y,yn,N−2,yn=yn,N=argminy∈XfNy+12λn,Nd2y,yn,N−1,zn=γn,0yn⊕⊕j=1Mγn,jTjyn,xn+1=αnx0⊕1−αnzn,∀n≥0.We begin the proof of the convergence of Algorithm15 with the following lemmas.Lemma 16.
The sequencexn generated by Algorithm 15 is bounded.Proof.
Letp∈Ω, then fip≤fiy for all y∈X,i=1,2,⋯,N and Tjp=p for all j=1,2,⋯,M. Thus, we have
(23)fNp+12λn,Ndp,p2≤fNy+12λn,Ndy,p2,∀y∈X.
Hence,p=Jλn,Np for n∈ℕ. Since yn,N=Jλn,iyn,N−1 and Jλn,i is nonexpansive, then we have
(24)dyn,N,p=dJλn,NyN−1,Jλn,Np≤dyn,N−1,p.
Similarly,p=Jλn,N−1p for all n∈ℕ0 and yn,N−1=Jλn,iyn,N−2. Thus, we obtain
(25)dyn,N−1,p=dJλn,N−1yN−2,Jλn,N−1p≤dyn,N−2,p⋮≤dyn,1,p.
Moreover,yn,1=Jλn,1xn, then we obtain
(26)dyn,1,p≤dJλn,1xn,Jλn,1p≤dxn,p.
Therefore, from (24), (25), and (26), we have
(27)dyn,p=dyn,N,p≤dxn,p.
Furthermore, it follows from Lemma7 (ii) and Lemma 12 that
(28)dzn,p2=dγn,0yn⊕⊕j=1Mγn,jTjyn,p2≤γn,0dyn,p2+∑j=1Mγn,jdTjyn,p2≤γn,0dyn,p2+∑j=1Mγn,jdTjyn,p2−∑j=1Mγn,0γn,jdyn,Tjyn2≤γn,0dyn,p2+∑j=1Mγn,jdyn,p+kjdyn,Tjyn−∑j=1Mγn,0γn,jdyn,Tjyn2=dyn,p2+∑j=1Mγn,jkjdyn,Tjyn2−∑j=1Mγn,0γn,jdyn,Tjyn2≤dyn,p2−∑j=1Mγn,jγn,0−kdyn,Tjyn2.
Thus, from (C2), we obtain(29)dzn,p≤dyn,p≤dxn,p.
Therefore, from Lemma7 (v), we get
(30)dxn+1,p=dαnx0⊕1−αnzn,p≤αndx0,p+1−αndzn,p≤αndx0,p+1−αndxn,p≤maxdx0,p,dxn,p⋮≤maxdx0,p,dx0,p=dx0,p.
This implies thatdxn,p is bounded. Therefore, xn,yn,zn are bounded.Lemma 17.
Letxnk be a subsequence of the sequence xn generated by Algorithm 15 such that liminfk⟶∞dxnk+1,p−dxnk,p≥0 for all p∈Ω. Thus, the following conclusions hold:
(i)
limk⟶∞dxnk,p exists for all p∈Ω(ii)
limk⟶∞dynk,i,ynk,i−1=0 for i=1,2,⋯,N(iii)
limk⟶∞dynk,Tjynk=0 for j=1,2,⋯,MProof.
From (30), we have
(31)0≤liminfk⟶∞dxnk+1,p−dxnk,p≤liminfk⟶∞αnkdx0,p+1−αnkdxnk,p−dxnk,p≤liminfk⟶∞αnkdx0,p−dxnk,p=0.
Hence,limk⟶∞dxnk+1,xnk=0. Therefore, limk⟶∞dxnk,p exists.
(ii) Also, from Lemma7, we have
(32)d2xnk+1,p=d2αnkx0⊕1−αnkznk,p≤αnk2d2x0,p+1−αnk2d2znk,p+2αnk1−αnkx0p⟵,znkp⟵≤αnkd2x0,p+1−αnkd2ynk,p+2αnk1−αnkx0p⟵,znkp⟵.
Also from Lemma11, we obtain
(33)12λnkdynk,1,p2−12λnkdxnk,p2+12λnkdxnk,ynk,12≤f1p−f1ynk,1.
Thus, from (32), we get
(34)d2xnk,ynk,1≤d2xnk,p−d2ynk,1,p=d2xnk,p−d2xnk+1,p+d2xnk+1,p−d2ynk,1,p≤d2xnk,p−d2xnk+1,p+αnkd2x0,p+1−αnkd2ynk,p+2αnk1−αnkx0p⟵,znkp⟵−d2ynk,1,p=d2xnk,p−d2xnk+1,p+αnkd2x0,p−dynk,1,p+21−αnkx0p⟵,znkp⟵.
Therefore,(35)limk⟶∞dxnk,ynk,1=0.
Now, we show thatdynk,1,ynk,2=0. From the fact that Jλnk,1xnk=ynk,1 and Jλnk,1ynk,1=ynk,2, we get
(36)dynk,1,ynk,2=dJλnk,1xnk,Jλnk,1ynk,1≤dxnk,ynk,1.
Hence,(37)limk⟶∞dynk,1,ynk,2=0.
Continuing in this manner, we get(38)dynk,N−1,ynk,N=dJλnk,N−1ynk−2,Jλnk,N−1ynk,N−1≤dynk,N−2,ynk,N−1.
Hence, by induction,(39)limk⟶∞dynk,i−1,ynk,i=0,fori=1,2,⋯,N.
In addition, using the triangle inequality with (35), (37), and (39), we get the following:
(40)dxnk,ynk=dxnk,ynk,N≤dxnk,ynk,1+dynk,1,ynk,2+⋯+dynk,N−1,ynk,N.
Therefore,(41)limk⟶∞dxnk,ynk=0.
(iii) Now, we show that(42)limn⟶∞dynk,Tjynk,i−1=0.
From (28), we have
d2znk,p≤d2yn,p−∑j=1Mγn,jγnk,0−kjdynk,Tjynk.(30)
Thus,(43)∑j=1Mγjγnk,0−kjdynk,Tjyn≤d2ynk,p−d2znk,p≤d2xnk,p−d2znk,p=d2xnk,p−d2xnk+1,p+d2xnk+1,p−d2znk,p.
From (32), we have
(44)∑j=1Mγjγnk,0−kjdynk,Tjynk≤d2xnk,p−d2xnk+1,p+αnkd2x0,p+1−αnkd2znk,p+2αnk1−αnkx0p⟵,znkp⟵−dznk,p2=d2xnk,p−d2xnk+1,p+αnkd2x0,p−dznk,p+21−αnkx0p⟵,znkp⟵.
Hence,(45)limn⟶∞∑j=1Mγjγnk,0−kjdyn,Tjynk=0.
This implies that(46)limn⟶∞dynk,Tjynk,i−1=0.Now, we prove the strong convergence of the iterative scheme.Theorem 18.
The sequencexn generated by Algorithm 15 converges strongly to an element p∈Ω, where Ω=⋂i=1Nargminfi∩⋂j=1MFTj.Proof.
Letp∈Ω. From Lemma 7 (i), we obtain
(47)d2xn+1,p=d2αnx0⊕1−αnzn,p≤αn2d2x0,p+1−αn2d2zn,p+2αn1−αnx0p⟵,znp⟵.
Since(48)dzn,p≤dxn,p,so from (47), we obtain
(49)d2xn+1,p≤αn2d2x0,p+1−αn2d2xn,p+2αn1−αnx0p⟵,znp⟵.
Moreover, sinceαn∈0,1, we have
(50)d2xn+1,p≤1−αnd2xn,p+αnαnd2x0,p+21−αnx0p⟵,znp⟵.
Hence, from (50), we have
(51)Sn+1≤1−αnSn+αnbn+cn,where
(52)Sn=d2xn+1,p,bn=αnd2x0,p+21−αnx0p⟵,znp⟵cn=0.
Obviously,d2xn,p≥0, and from condition (C1), we have ∑n=0∞αn=+∞. Also, since αn⟶0, we obtain αnd2xn,p⟶0, and from Lemma 14, we have limsupn⟶∞x0p⟵,znp⟵≤0. Thus, limsupn⟶∞bn≤0. Clearly, ∑n=0∞cn<∞. Therefore, from Lemma 13, we obtain that limn⟶∞Sn=0. This implies that limn⟶∞dxn,p=0. Therefore, xn converges strongly to p∈Ω.The following can be obtained as consequence results from our main theorem.(i) IfN=M=1, then we have the following result for approximating a common solution of minimization and fixed point problem for a demicontractive mapping T.Corollary 19.
LetX be a complete CAT(0) space and C be a nonempty closed convex subset of X. Let f:X⟶ℝ∪+∞ be a proper, convex, and lower semicontinuous function, and let T:C⟶C be k-demicontractive mappings such that I−T is demiclosed at zero. Suppose Γ=argminf∩FT≠∅. Let αn and γnj=0M be sequences in 0,1 such that
(C1)limn⟶∞αn=0 and ∑n=0∞αn=+∞
(C2)liminfn⟶∞γn−k>0
(C3)liminfn⟶∞λn>0Choosex0∈X arbitrarily and let xn be a sequence generated by the following algorithm.Algorithm 20.
(53)yn=argminy∈Xfy+12λnd2y,xn,zn=γnyn⊕1−γnTyn,xn+1=αnx0⊕1−αnzn,∀n≥1.Then, the sequencexn converges strongly to a solution p∈Γ.Also, changingTj from kj demicontractive into quasinonexpansive mapping, we have the following result.Corollary 21.
LetX be a complete CAT(0) space and C be a nonempty closed convex subset of X. For i=1,2,⋯,N, let fi:X⟶ℝ⋃+∞ be a finite family of proper, convex, and lower semicontinuous functions, and for j=1,2,⋯,M, let Tj:C⟶C be a finite family of quasinonexpansive mappings such that I−Tj is demiclosed at zero. Suppose
(54)Ω=⋂i=1Nargminfi∩⋂j=1MFTj≠∅.
Letαn and γn,jj=0M be sequences in 0,1 such that
(C1)limn⟶∞αn=0 and ∑n=0∞αn=+∞
(C2)∑j=0Mγn,j=1 and liminfn⟶∞γn,i>0
(C3)liminfn⟶∞λn,i>0Choosex0∈X arbitrarily and let xn be a sequence generated by the following algorithm.Algorithm 22.
(55)yn,1=argminy∈Xf1y+12λn,1d2y,xn,yn,2=argminy∈Xf2y+12λn,2d2y,yn,1,⋮yn,N−1=argminy∈XfN−1y+12λn,N−1d2y,yn,N−2,yn=yn,N=argminy∈XfNy+12λn,Nd2y,yn,N−1,zn=γn,0yn⊕⊕j=1Mγn,jTjyn,xn+1=αnx0⊕1−αnzn,∀n≥1.Then, the sequencexn converges strongly to a solution p∈Ω.


## 4. Numerical Example

In this section, we present a computational example to illustrate the convergence behaviour of the proposed iterative scheme. We perform the numerical experiment using MATLAB programming.Example 23.
LetX=ℝm (Euclidean m-dimensional space) with the Euclidean norm defined by
(56)∥x∥=∑i=1mxi21/2,and inner product
(57)x,y=∑i=1mxiyi∀x,y∈ℝm.It is clear thatX is a complete CAT(0) space. Let
(58)C=x=x1,x2,⋯,xm∈ℝm:−5≤xi≤5,i=1,2,⋯,m.Also, fori=1,2,3,⋯,N, let fi:X⟶ℝ∪+∞ be defined by
(59)fix=12∥Aix−bi∥2,where Ai is a m×m matrix and bi is a m×1 vector for all i=1,2,⋯,N. It is easy to see that fi is proper, convex, and lower semicontinuous.Also, forj=1,2,⋯,M, let Tj:ℝm⟶ℝm be defined by
(60)Tjx=−j+12x,∀x∈ℝm,j=1,2,⋯,M.We need to show thatTj is a kj-demicontractive mapping, for all j=1,2,3,⋯,M.Recall thatTj is kj-demicontactive if
(61)FTj≠∅,d2Tjx,p≤d2x,p+kjd2x,Tjx∀x∈ℝ,p∈FTj,j=1,2,3,⋯,M.Clearly,(62)0=0,0,⋯,0m∈FTjforallj=1,2,3,⋯,M.Also,(63)d2Tjx,p=−j+1x2−02=−j+1x22=j+124x2.Then,(64)d2x,p=∥x−0∥2=∥x∥2,d2x,Tjx=x−−j+1x22=x+j+1x22=2x+j+1x22=3x+jx22=3+jx22=3+j24∥x∥2.Now, since(65)d2Tjx,p=j+124∥x∥2=∥x∥2+j2+2j−34∥x∥2=∥x∥2+j2+2j−39+6j+j2×3+j24∥x∥2=∥x∥2+j−1j+3∥x−Tjx∥2.Therefore,(66)d2Tjx,p=d2x,p+kjd2x,Tjx,where kj=j−1/j+3. Hence, Tj is kj-demicontractive, where kj=j−1/j+3.Sincefj0≤fjx;forallx∈ℝm, and j=1,2,⋯,m, then 0∈argminfj. Hence,
(67)0∈Ω=⋂i=0Nargminfj∩⋂j=0MFTj≠∅.We chooseαn=1/n,γn,j=1/M+1; clearly,
(68)limn⟶∞αn=0,∑i=0∞αn=+∞,∑j=0Mγn,j=1.Sincek=maxkj=M−1/M+3, then
(69)liminfn⟶∞γn,0−k=liminfn⟶∞1M+1−M−1M+3>0,for M>2. Also, λn,i=n+i/5n+9, limn⟶∞αn,i>0. Hence, conditions (1)–(3) are satisfied. In this case, our algorithm becomes
(70)yn,1=argminy∈Xf1y+5n+92n+1d2y,xn,yn,2=argminy∈Xf2y+5n+92n+2d2y,yn,1,⋮yn,N−1=argminy∈XfN−1y+5n+92n+N−1d2y,yn,N−2,yn=yn,N=argminy∈XfNy+5n+92n+Nd2y,yn,N−1,zn=1M+1yn+∑j=1M1M+1−j+12yn,xn+1=1nx0+n−1nzn,∀n≥1.In our computation, the initial values are generated randomly inℝm where the values of m,N, and M are as follows:
(i)
Case I:m=10,N=10,M=30(ii)
Case II:m=15,N=10,M=20(iii)
Case III:m=20,N=5,M=10(iv)
Case IV:m=50,N=25,M=10We compare the performance of Algorithm15 with Algorithm (3) (which refers to Iterative scheme 10 in Theorem 3), choosing Tix=x/2i,αn=1/n+1,βni=1/N+1,rni=1/2i, and λn,i=n+i/5n+9. We also use ∥xn+1−xn∥<10−4 as the tolerance level for our computation. The numerical results are shown in Table 1 and Figure 1.Table 1
Computation result for Example23.
Algorithm15Algorithm (3)Case IIter.712Time (s)0.12810.1708Case IIIter.1524Time (s)2.25545.8941Case IIIIter.1422Time (s)1.59144.8935Case IVIter.1626Time (s)1.61344.0589Figure 1
Example23: (a) Case I; (b) Case II; (c) Case III; (d) Case IV.
(a)(b)(c)(d)


## 5. Conclusions

In this paper, we introduced a modified Halpern-type algorithm for approximating the common solution of the finite family of convex minimization problems and fixed point problems fork-demicontractive mappings in CAT(0) spaces. A strong result is proven under a mild condition, and a numerical example is provided to illustrate the convergence behaviour of the sequence generated by our iterative scheme. This result improves and generalizes many other recent results such as [32] from Hilbert to CAT(0) spaces. We are hopeful that this result can be extend to the split minimization, split inclusion, and the split fixed problems (see, e.g., [32–34]) in CAT(0) spaces. We wish to consider this in our future research.



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