Infinite Product and Its Convergence in CAT(1) Spaces

Abstract

In this paper, we study the convergence of infinite product of strongly quasi-nonexpansive mappings on geodesic spaces with curvature bounded above by one. Our main applications behind this study are to solve convex feasibility by alternating projections, and to solve minimizers of convex functions and common minimizers of several objective functions. To prove our main results, we introduce a new concept of orbital $\Delta$-demiclosed mappings which covers finite products of strongly quasi-nonexpansive, $\Delta$-demiclosed mappings, and hence is applicable to the convergence of infinite products.

1. Introduction

The class of strongly quasi-nonexpansive mappings plays a crucial role in convex optimizations on $\mathrm{CAT}\left(\kappa \right)$ spaces for $\kappa \in \mathbb{R}$. In 1977, Bruck and Reich [1] first introduced the concept of a strongly nonexpansive mapping for the first time, which generalizes the class of firmly nonexpansive mappings. Later, this mapping was generalized to a strongly quasi-nonexpansive mapping by Bruck [2]. An intuitive example of strongly quasi-nonexpansive mappings is the metric projection $P_C$ onto closed convex sets C (see [3,4]), which is crucial for dealing with a convex feasibility problem. In 2012, Ba$\check{\mathrm{c}}$ák, Searston and Sims [5] proved that the sequence $\left\{{\left(P_{C_2}{P}_{C_1}\right)}^{n}x\right\}$$\Delta$-converges to a point in $C_1\cap C_2$ whenever $C_1$ and $C_2$ are closed convex subsets of $\mathrm{CAT}\left(0\right)$ spaces. Following that, Choi, Ji, and Lim [6] proved the same result for any $\mathrm{CAT}\left(\kappa \right)$ spaces with $\kappa >0$. Another important example of a strongly quasi-nonexpansive map is the resolvent mapping of a proper lower semicontinuous convex function f in $\mathrm{CAT}\left(0\right)$ spaces (see Proposition 3.1 of [3]), which was proposed by Jost [7,8] and Mayer [9]. The resolvent mapping of f with respect to $\lambda >0$ is defined by

$$R_{\lambda ,f}\left(x\right)=\underset{y\in X}{arg\; min}\{f\left(y\right)+\frac{1}{\lambda }d{(x,y)}^2\}$$

(1)

for $x\in X$. It was not until 2013 that this resolvent was applied to solve convex optimizations in complete $\mathrm{CAT}\left(0\right)$ spaces by Ba$\check{\mathrm{c}}$ák [10]. This method is called a proximal algorithm and it has been one of the earliest and most successful approximation schemes for convex optimizations. Ba$\check{\mathrm{c}}$ák [11] improved the proximal algorithm to solve the convex optimization when the objective function is usually expressed as the sum of several loss functions, i.e., $f:={\sum }_{i=1}^n{f}_i$ when, for all $i=1,\dots ,n$, $f_i$ is a proper lower semicontinuous convex function. Such a method is called a splitting proximal algorithm. This is a useful technique that can be used to find means and medians phylogenetics on tree spaces (see [12]). Following that, the convex optimization studied on $\mathrm{CAT}\left(\kappa \right)$ spaces grew in popularity.

Among the vast developments, Kimura and Kohsaka [13] introduced a new resolvent on $\mathrm{CAT}\left(1\right)$ spaces by replacing $d{(x,·)}^2$ in (1) with a function corresponding to the metric on the model spaces. The new resolvents still carry the most important properties of being strongly quasi-nonexpansive on $\mathrm{CAT}\left(1\right)$ spaces (see [13] (Theorem 4.6)). In the same year, Espínola and Nicolae [14] developed proximal and splitting proximal algorithms based on these novel resolvents on $\mathrm{CAT}\left(1\right)$ spaces. With a similar idea, Kajimura and Kimura [15] constructed a new resolvent on $\mathrm{CAT}(-1)$ spaces. In [16], the authors subsequently used this resolvent to study splitting methods on $\mathrm{CAT}(-1)$ spaces and applied their results to solve convex feasibility problems, centroid problems, and, particularly, the Karcher means. Note that all preceding studies were investigated and verified using specific technical approaches to handle their results, while the main goal of this research is focused on a broader perspective—viewing the problem through the lens of fixed point theory.

In 2014, Ariza, López and Nicolae [17] achieved new and more broader convergence results to approximate a common fixed point of a finite family of firmly nonexpansive mappings in $\mathrm{CAT}\left(\kappa \right)$ spaces. They studied the asymptotic behavior of the sequence constructed by the composition of firmly nonexpansive mappings on p-uniformly convex spaces, and then the concept of asymptotic regularity was used to investigate the common fixed points of the finite number of firmly nonexpansive mappings. Later, Reich and Salinas [18] generalized the Ariza–López–Nicolae results for $\mathrm{CAT}\left(0\right)$ spaces. They proved a $\Delta$-convergence theorem of infinite products of strongly nonexpansive mappings and they also used their results to solve the convex feasibility problems on complete $\mathrm{CAT}\left(0\right)$ spaces. These results were obtained as consequences of the $\Delta$-convergence of iterations for a sequence formed by strongly quasi-nonexpansive mappings on $\mathrm{CAT}\left(0\right)$ spaces (see [19]). Motivated by the above studies, we show, in this paper, the $\Delta$-convergence of the infinite products of strongly quasi-nonexpansive mappings in the setting of a complete admissible $\mathrm{CAT}\left(1\right)$ space and also the applications to convex feasibility and common minimizer problems. The main results and applications in this paper extend the studies of the Choi–Ji–Lim [6], Ariza–López–Nicolae [17] for $\mathrm{CAT}\left(1\right)$ spaces, and Espínola–Nicolae [14] if the intersection of ${arg\; min}_X{f}_i$ is nonempty.

Our paper is organized as follows. In Section 2, we collect background materials which are required for our main results in subsequent sections and present a new definition of orbital $\Delta$-demiclosedness. In Section 3, we first prove several useful properties for our main results and then conclude with the main results concerning $\Delta$-convergence of the infinite products for strongly quasi-nonexpansive orbital $\Delta$-demiclosed mappings in complete admissible $\mathrm{CAT}\left(1\right)$ spaces. In Section 4, we discuss some possible applications of our main results to solve the convex feasibility and common minimizer problems in $\mathrm{CAT}\left(1\right)$ spaces. In the final section, Section 5, we demonstrate a numerical implementation applied to solve convex feasibility problems.

2. Preliminaries

In this section, some basic notions and useful lemmas necessary for the subsequent results are given. Throughout this paper, the set of all positive integers is denoted by $\mathbb{N}$ and the set of all real numbers is denoted by $\mathbb{R}$.

Let $(X,d)$ be a metric space and $x,y\in X$. A geodesic path joining x and y is a mapping $\gamma :[0,d(x,y\left)\right]\to X$ such that $\gamma \left(0\right)=x$, $\gamma \left(d\right(x,y\left)\right)=y$ and $d\left(\gamma \right(t),\gamma (s\left)\right)=|t-s|$ for any $t,s\in [0,d(x,y\left)\right]$. Let D be a positive real number. A metric space X is said to be (uniquely) $D-$geodesic if for each $x,y\in X$ with $d(x,y)<D$, there exists a (unique) geodesic joining x and y. We say that X is a (uniquely) geodesic metric space if any two elements $x,y\in X$ are connected by a (unique) geodesic. If X is uniquely geodesic, $x,y\in X$, and $\gamma$ is the geodesic path joining x to y, then we write $[x,y]:=\gamma \left(\right[0,d(x,y)\left]\right)$ to denote the geodesic segment of $\gamma$. In this case, we also use the notation $(1-t)x\oplus ty:=\gamma \left(td\right(x,y\left)\right)$. In addition, recall that the geodesic triangle with vertices $x,y,z\in X$, denoted by $\Delta (x,y,z)$, is defined by $[x,y]\cup [y,z]\cup [z,x]$. Recall that the sphere ${\mathbb{S}}^2$ is defined by

$${\mathbb{S}}^2=\{x=(x_1,x_2,x_3)\in {\mathbb{R}}^3|\langle x,x\rangle =1\},$$

where $\langle x,y\rangle ={\sum }_{i=1}^3{x}_i{y}_i$ for all $x,y\in {\mathbb{R}}^3$. The spherical metric $d_{{\mathbb{S}}^2}$ on ${\mathbb{S}}^2$ is defined by

$$d_{{\mathbb{S}}^2}(x,y)=arccos\langle x,y\rangle ,$$

for all $x,y\in X$. From [20], if $x,y\in {\mathbb{S}}^2$, $x\ne y$ and $d_{{\mathbb{S}}^2}(x,y)<\pi$, then the unique geodesic $\gamma$ from x to y is given by

$${\gamma }_{x,y}\left(t\right)=(cos\left(t\right))x+(sin\left(t\right))·\frac{y-\langle x,y\rangle x}{\left|\right|y-\langle x,y\rangle x\left|\right|},$$

(2)

for all $t\in [0,d_{{\mathbb{S}}^2}(x,y)]$. For a triangle $\Delta (x_1,x_2,x_3)$ in a uniquely $\pi -$geodesic space X satisfying $d(x_1,x_2)+d(x_1,x_3)+d(x_2,x_3)<2\pi$, we can find the comparison triangle $\Delta (\overline{x_1},\overline{x_2},\overline{x_3})$ in ${\mathbb{S}}^2$ such that $d(x_i,x_j)=d_{{\mathbb{S}}^2}({\overline{x}}_i,{\overline{x}}_j)$ for all $i,j\in \{1,2,3\}$. Let $\gamma$ be the geodesic joining $x_i$ to $x_j$ and $u:=\gamma \left(t\right)$ for some $t\in [0,d(x_i,x_j)]$. Then, a point $\overline{u}:=\overline{\gamma }\left(t\right)$ is called the comparison point of u, where $\overline{\gamma }$ is the geodesic joining ${\overline{x}}_i$ to ${\overline{x}}_j$. If X is a uniquely $\pi -$geodesic metric space, $x_1,x_2,x_3\in X$, $d(x_1,x_2)+d(x_1,x_3)+d(x_2,x_3)<2\pi$ and every two points $p,q$ in $\Delta (x_1,x_2,x_3)$ and their comparison points $\overline{p},\overline{q}$ in $\Delta (\overline{x_1},\overline{x_2},\overline{x_3})$ satisfy that,

$$d(p,q)\le d_{{\mathbb{S}}^2}(\overline{p},\overline{q}),$$

then X is called a $\mathrm{CAT}\left(1\right)$ space. A $\mathrm{CAT}\left(1\right)$ space X is called admissible if $d(x,y)<\pi /2$ for all $x,y\in X$.

Remark 1

([20]). In general, if $\kappa >0$, then $(X,d)$ is a $\mathrm{CAT}\left(\kappa \right)$ space if and only if $(X,\sqrt{\kappa }d)$ is a $\mathrm{CAT}\left(1\right)$ space.

Let $(X,d)$ be a uniquely geodesic metric space. A subset $C\subset X$ is called convex if $[x,y]\subset C$ for all $x,y\in C$. Let $f:X\to (-\infty ,+\infty ]$ be given. Recall that the effective domain of f is defined by $domf:=\{x\in X|f\left(x\right)<+\infty \}$. If $domf\ne ⌀$, we say that f is proper. In addition, f is said to be convex if

$$f\left(\right(1-t)x\oplus ty)\le (1-t)f\left(x\right)+tf\left(y\right)$$

holds for all $x,y\in X$ and $t\in (0,1)$.

Let $(X,d)$ be a $\mathrm{CAT}\left(1\right)$ space and $\left\{x_n\right\}$ be a sequence of X. The asymptotic center $\mathcal{A}\left(\left\{x_n\right\}\right)$ of a sequence $\left\{x_n\right\}$ is defined by

$$\mathcal{A}\left(\left\{x_n\right\}\right)=\{z\in X|\underset{n\to \infty }{lim\; sup}d(z,x_n)=\underset{y\in X}{inf}\underset{n\to \infty }{lim\; sup}d(y,x_n)\}.$$

The sequence $\left\{x_n\right\}$ is said to be $\Delta$-convergent to an element z of X if

$$\mathcal{A}\left(\left\{x_{n_i}\right\}\right)=\left\{z\right\},$$

for each subsequence $\left\{x_{n_i}\right\}$ of $\left\{x_n\right\}$. In this case, we say that z is the $\Delta$-limit of $\left\{x_n\right\}$. If $\left\{x_n\right\}$ is convergent to z, then it is $\Delta$-convergent to z. If $\left\{x_n\right\}$ is $\Delta$-convergent to z, then it is bounded and every subsequence $\left\{x_{n_i}\right\}$ of $\left\{x_n\right\}$ is $\Delta$-convergent to z. Furthermore, ${\omega }_{\Delta }\left(\left\{x_n\right\}\right)$ denotes the set of all $\overline{z}\in X$ such that there exists a subsequence $\left\{x_{n_i}\right\}$ of $\left\{x_n\right\}$ which is $\Delta$-convergent to $\overline{z}$. A subset $C\subset X$ is called $\Delta$-closed if $y\in C$ whenever $y\in X$ is a $\Delta$-limit of some sequence $\left\{y_n\right\}$ of C. See [21,22,23,24] for more details about the $\Delta$-convergence. A sequence $\left\{x_n\right\}$ is said to be spherically bounded if

$$\underset{y\in X}{inf}\underset{n\to \infty }{lim\; sup}d(x_n,y)<\frac{\pi }{2}.$$

In particular, if $diam\left(X\right)<\pi /2$, then the space X is admissible and every sequence in X is spherically bounded.

Let $(X,d)$ be a $\mathrm{CAT}\left(1\right)$ space and T be a mapping from X into itself. $Fix\left(T\right)=\{x\in X|Tx=x\}$ denotes the set of all fixed points of mapping T. Suppose that the set $Fix\left(T\right)\ne ⌀$. If $d\left(T\right(x),p)\le d(x,p)$ for all $x\in X$ and $p\in Fix\left(T\right)$, then we say that T is a quasi-nonexpansive mapping. If T is quasi-nonexpansive mapping, and for every sequence $\left\{x_n\right\}$ in X and $p\in Fix\left(T\right)$ such that ${sup}_{n\in \mathbb{N}}d(x_n,p)<\pi /2$ and

$$\underset{n\to \infty }{lim}\frac{cosd(x_n,p)}{cosd(T\left(x_n\right),p)}=1,$$

it follows that $\underset{n\to \infty }{lim}d(x_n,T\left(x_n\right))=0$, then T is called a strongly quasi-nonexpan-sive mapping. Moreover, T is said to be $\Delta$-demiclosed in the sense of [4] if for any $\Delta$-convergent sequence $\left\{x_n\right\}$ in X, its $\Delta$-limit belongs to $Fix\left(T\right)$ whenever $\underset{n\to \infty }{lim}d(x_n,T\left(x_n\right))=0$.

Next, we will present two examples of strongly quasi-nonexpansive and $\Delta$-demiclosed mapping in complete admissible $\mathrm{CAT}\left(1\right)$ spaces.

Let $(X,d)$ be a complete admissible $\mathrm{CAT}\left(1\right)$ space and C be a nonempty closed convex subset of X. The metric projection $P_C:X\to C$ is defined by

$$P_C\left(x\right)=\underset{y\in C}{arg\; min}d(x,y),$$

for every $x\in X$. Then for all $x\in X$, there exists a unique $\overline{x}\in C$ such that $P_C\left(x\right)=\overline{x}$. It is known that $Fix\left(P_C\right)=C$ and

$$d(y,P_{C}x)\le d(y,x),$$

for every $y\in C$ and $x\in X$ (see [21] (Proposition 3.5)). In particular, the metric projection onto a nonempty closed and convex subset of a complete admissible $\mathrm{CAT}\left(1\right)$ space is strongly quasi-nonexpansive $\Delta$-demiclosed (see [4]).

On the other hand, Kimura and Kohsaka [13] introduced a new concept of the resolvent $R_{\lambda ,f}$ of a proper lower semicontinuous convex function f of a complete admissible $\mathrm{CAT}\left(1\right)$ space X into $(-\infty ,\infty ]$ with respect to $\lambda >0$ defined by

$$R_{\lambda ,f}\left(x\right)=\underset{y\in X}{arg\; min}\{f\left(y\right)+\frac{1}{\lambda }tand(x,y)sind(x,y)\}$$

(3)

for all $x\in X$. This mapping is well defined as a single-valued mapping (see [13] (Theorem 4.2)), and it is also a strongly quasi-nonexpansive mapping under the admissibility condition of the ambient space X such that $Fix\left(R_{\lambda ,f}\right)={arg\; min}_{X}f$ (see [13] (Theorem 4.6)). In addition, we know that the resolvent $R_{\lambda ,f}$ is a $\Delta$-demiclosed mapping (see [25]).

Now, we propose a new generalization of $\Delta$-demiclosedness.

Definition 1.

Let $(X,d)$ be a complete admissible $\mathrm{CAT}\left(1\right)$ space and let T be a mapping from X to X. The mapping T is called orbital Δ-demiclosed if $x^{*}\in Fix\left(T\right)$ whenever $\left\{x_k\right\}$ is a sequence in X satisfying the following conditions:

1.

$x_k\stackrel{\Delta }{\to }{x}^{*}$;

2.

$d(T\left(x_k\right),x_k)\to 0$;

3.

$\left(x_k\right)$ is a subsequence of $\left\{T^n\left(\widehat{x}\right)\right\}$ for some $\widehat{x}\in X$.

The following lemmas are important for our main results.

Lemma 1

([21]). Let $(X,d)$ be a complete admissible $\mathrm{CAT}\left(1\right)$ space. Suppose that the sequence $\left\{x_n\right\}$ is a spherically bounded sequence in X. Then $\mathcal{A}\left(\left\{x_n\right\}\right)$ is a singleton and $\left\{x_n\right\}$ has a Δ-convergent subseqeunce.

Lemma 2

([26]). Let $(X,d)$ be a complete admissible $\mathrm{CAT}\left(1\right)$ space. Suppose that the sequence $\left\{x_n\right\}$ is a spherically bounded sequence in X and $\left\{d(x_n,z)\right\}$ is convergent for every z in ${\omega }_{\Delta }\left(\left\{x_n\right\}\right)$. Then $\left\{x_n\right\}$ is Δ-convergent to an element $\overline{x}\in X$.

3. Main Results

As the main theorems of this paper, we show that the product of strongly quasi-nonexpansive $\Delta$-demiclosed mappings is a strongly quasi-nonexpansive and orbital $\Delta$-demiclosed mapping. Next, we prove the $\Delta$-convergence of the sequence, which is constructed by a strongly quasi-nonexpansive orbital $\Delta$-demiclosed mapping in the setting of complete admissible $\mathrm{CAT}\left(1\right)$ spaces. Motivated by Reich and Salinas [18], we start with the following lemmas, which are some of the most important tools in determining our results (see also [27]).

Lemma 3.

Let $(X,d)$ be a complete admissible $\mathrm{CAT}\left(1\right)$ space. If $T_i:X\to X$ is a strongly quasi-nonexpansive mapping for all $i\in \{1,2,\dots ,m\}$ and ${\cap }_{i=1}^{m}Fix\left(T_i\right)\ne ⌀$. Then, $Fix(T_m{T}_{m-1}\cdots T_1)={\cap }_{i=1}^{m}Fix\left(T_i\right)$.

Proof. 

Firstly, we will show that ${\cap }_{i=1}^{m}Fix\left(T_i\right)\subset Fix(T_m{T}_{m-1}\cdots T_1)$.

Let $z\in {\cap }_{i=1}^{m}Fix\left(T_i\right)$. Then, we obtain

$$T_m{T}_{m-1}\cdots T_1\left(z\right)=T_m{T}_{m-1}\cdots T_2\left(z\right)=\cdots =T_m\left(z\right)=z.$$

From the above equation, we have $z\in Fix(T_m{T}_{m-1}\cdots T_1)$.

Therefore, ${\cap }_{i=1}^{m}Fix\left(T_i\right)\subset Fix(T_m{T}_{m-1}\cdots T_1)$.

Next, we will show that

$$Fix(T_m{T}_{m-1}\cdots T_1)\subset {\cap }_{i=1}^{m}Fix\left(T_i\right).$$

(4)

Suppose first that $m=2$. Let $z\in Fix\left(T_2{T}_1\right)$ and $T_1\left(z\right)=x\in X$, then $z=T_2{T}_1\left(z\right)=T_2\left(x\right)$. Since $T_1$ and $T_2$ are quasi-nonexpansive, for any $x^{*}\in Fix\left(T_1\right)\cap Fix\left(T_2\right)$, we will obtain

$$d(z,x^{*})=d(T_2\left(x\right),x^{*})\le d(x,x^{*})=d(T_1\left(z\right),x^{*})\le d(z,x^{*}).$$

Hence, $d(T_2\left(x\right),x^{*})=d(x,x^{*})$, and we have

$$\underset{n\to \infty }{lim}\frac{cosd(T_2\left(x\right),x^{*})}{cosd(x,x^{*})}=1.$$

Since ${sup}_{n}d(x,y)=d(x,y)<\frac{\pi }{2}$ for all $y\in Fix\left(T_2\right)$ and $T_2$ is strongly quasi-nonexpansive, we have

$$\underset{n\to \infty }{lim}d(T_2\left(x\right),x)=d(T_2\left(x\right),x)=0,T_2\left(x\right)=x.$$

Then, we see that $z=T_2{T}_1\left(z\right)=T_2\left(x\right)=x=T_1\left(z\right)$, which means that z belongs to $Fix\left(T_1\right)$ and $Fix\left(T_2\right)$. Therefore, $Fix\left(T_2{T}_1\right)\subset Fix\left(T_1\right)\cap Fix\left(T_2\right)$.

Finally, we suppose that (4) holds for $m-1$. Let $z\in Fix(T_m\cdots T_1)$ and $T_{m-1}\cdots T_1\left(z\right)=x\in X$. Then, $T_m\left(x\right)=z$. Since $T_m,T_{m-1},\dots ,T_1$ are quasi-nonexpansive, for any $x^{*}\in {\cap }_{i=1}^{m}Fix\left(T_i\right)$, we have

$$d(z,x^{*})=d(T_m\left(x\right),x^{*})\le d(x,x^{*})=d(T_{m-1}\cdots T_1\left(z\right),x^{*})$$

$\le d(T_{m-2}\cdots T_1\left(z\right),x^{*})\le \cdots \le d(T_1\left(z\right),x^{*})\le d(z,x^{*}).$

Hence, $d(T_m\left(x\right),x^{*})=d(x,x^{*})$. This implies that

$$\underset{n\to \infty }{lim}\frac{cosd(T_m\left(x\right),x^{*})}{cosd(x,x^{*})}=1.$$

Since ${sup}_{n}d(x,y)=d(x,y)<\frac{\pi }{2}$ for all $y\in Fix\left(T_m\right)$, and $T_m$ is strongly quasi-nonexpansive, we have

$$\underset{n\to \infty }{lim}d(T_m\left(x\right),x)=d(T_m\left(x\right),x)=0,T_m\left(x\right)=x,$$

and

$$z=T_m\cdots T_1\left(z\right)=T_m\left(x\right)=x=T_{m-1}\cdots T_1\left(z\right),$$

which means that z belongs to $Fix\left(T_m\right)$ and $z\in Fix(T_{m-1}\cdots T_1)$. By the inductive hypothesis holds for $m-1$, we have ${\cap }_{i=1}^{m-1}Fix\left(T_i\right)$. Therefore, $Fix(T_m\cdots T_1)\subset {\cap }_{i=1}^{m}Fix\left(T_i\right)$. □

Lemma 4.

Let $(X,d)$ be a complete admissible $\mathrm{CAT}\left(1\right)$ space. If $T_i:X\to X$ is a strongly quasi-nonexpansive mapping for all $i\in \{1,2,\dots ,m\}$ and ${\cap }_{i=1}^m$Fix$\left(T_i\right)\ne ⌀$, then $T=T_m{T}_{m-1}\cdots T_1$ is also a strongly quasi-nonexpansive mapping.

Proof. 

We prove this lemma by using mathematical induction.

Suppose first that $m=2$. Assume that $T_1$ and $T_2$ are strongly quasi-nonexpansive mappings such that $Fix\left(T_1\right)\cap Fix\left(T_2\right)\ne ⌀$. Let $T:=T_2{T}_1$ and $\left\{x_n\right\}\subset X$ such that ${sup}_{n}d(x_n,z)<\frac{\pi }{2}$ and

$$\underset{n\to \infty }{lim}\frac{cos\left(d(T\left(x_n\right),z)\right)}{cos\left(d(x_n,z)\right)}=1,$$

(5)

for all $z\in Fix\left(T\right)$.

(i) We will show that T is quasi-nonexpansive.

Take $x\in X$ and $z\in Fix\left(T\right)$. By Lemma 3, $z\in Fix\left(T_2\right)\cap Fix\left(T_1\right)$, we have

$$d(T\left(x\right),z)=d(T_2{T}_1\left(x\right),z)\le d(T_1\left(x\right),z)\le d(x,z).$$

Then T is quasi-nonexpansive.

(ii) We will show that T is strongly quasi-nonexpansive ($\underset{n\to \infty }{lim}d(T\left(x_n\right),x_n)=0$).

Since $T_2$ and $T_1$ are quasi-nonexpansive, we obtain

$$0\le d(T\left(x_n\right),z)=d(T_2{T}_1\left(x_n\right),z)\le d(T_1\left(x_n\right),z)\le d(x_n,z)<\frac{\pi }{2}$$

(6)

this implies that

$$0<cos\left(d(x_n,z)\right)\le cos(d(T_1\left(x_n\right),z)\le cos\left(d(T\left(x_n\right),z)\right).$$

Then we obtain

$$1\le \frac{cos(d(T_1\left(x_n\right),z)}{cos\left(d(x_n,z)\right)}\le \frac{cos\left(d(T\left(x_n\right),z)\right)}{cos\left(d(x_n,z)\right)}$$

and from (5), we have

$$\underset{n\to \infty }{lim}\frac{cos(d(T_1\left(x_n\right),z)}{cos\left(d(x_n,z)\right)}=1.$$

(7)

Since $T_1$ is strongly quasi-nonexpansive and due to (7), we obtain

$$\underset{n\to \infty }{lim}d(T_1\left(x_n\right),x_n)=0.$$

From (6), we have

$$\underset{n}{sup}d(T_1\left(x_n\right),z)<\frac{\pi }{2}$$

(8)

and

$$\underset{n\to \infty }{lim}\frac{cos(d(T_2\left(T_1\left(x_n\right)\right),z)}{cos\left(d(T_1\left(x_n\right),z)\right)}=1.$$

(9)

Combining (8) and (9) with $T_2$ is strongly quasi-nonexpansive, and we obtain

$$\underset{n\to \infty }{lim}d(T_2{T}_1\left(x_n\right),T_1\left(x_n\right))=0.$$

By the triangle inequality, we have

$$0\le d(T\left(x_n\right),x_n)=d(T_2{T}_1\left(x_n\right),x_n)\le d(T_2{T}_1\left(x_n\right),T_1\left(x_n\right))+d(T_1\left(x_n\right),x_n)$$

Thus, it follows that $\underset{n\to \infty }{lim}d(T\left(x_n\right),x_n)=0$, which proves that $T=T_2{T}_1$ is strongly quasi-nonexpansive.

Now, suppose that $T_i\cdots T_1$ is strongly quasi-nonexpansive for $i=m-1$.

Assume that $T_1,\dots ,T_m$ are strongly quasi-nonexpansive such that ${\cap }_{i=1}^{m}Fix\left(T_i\right)\ne ⌀$ and $T_{m-1}\cdots T_1$ is strongly quasi-nonexpansive. Let $T:=T_m\cdots T_1$ and $\left\{x_n\right\}\subset X$ such that ${sup}_{n}d(x_n,z)<\frac{\pi }{2}$ and

$$\underset{n\to \infty }{lim}\frac{cos\left(d(T\left(x_n\right),z)\right)}{cos\left(d(x_n,z)\right)}=1$$

(10)

for all $z\in Fix\left(T\right)$.

(i) We will show that T is quasi-nonexpansive.

Take $x\in X$ and $z\in Fix\left(T\right)$. By Lemma 3, $z\in {\cap }_{i=1}^{m}Fix\left(T_i\right)$, we have

$$d(T\left(x\right),z)=d(T_m\cdots T_1\left(x\right),z)\le \cdots \le d(T_1\left(x\right),z)\le d(x,z).$$

Then, T is quasi-nonexpansive.

(ii) We will show that T is strongly quasi-nonexpansive as $\underset{n\to \infty }{lim}d(T\left(x_n\right),x_n)=0$.

Since $T_i$ is a quasi-nonexpansive for all $i=1,\dots ,m$, we obtain

$$0\le d(T\left(x_n\right),z)=d(T_m\cdots T_1\left(x_n\right),z)\le \cdots \le d(T_1\left(x_n\right),z)\le d(x_n,z)<\frac{\pi }{2}.$$

(11)

This implies that

$$0<cos\left(d(x_n,z)\right)\le cos\left(d(T_1\left(x_n\right),z)\right)\le \cdots \le cos\left(d(T\left(x_n\right),z)\right)\le 1.$$

(12)

By dividing both sides by $cos\left(d(x_n,z)\right)$ in the above inequality, we have

$$1\le \frac{cos(d(T_{m-1}\cdots T_1\left(x_n\right),z)}{cos\left(d(x_n,z)\right)}\le \frac{cos\left(d(T\left(x_n\right),z)\right)}{cos\left(d(x_n,z)\right)}$$

and from (10), we obtain

$$\underset{n\to \infty }{lim}\frac{cos(d(T_{m-1}\cdots T_1\left(x_n\right),z)}{cos\left(d(x_n,z)\right)}=1.$$

(13)

Since $T_{m-1}\cdots T_1$ is strongly quasi-nonexpansive and (13), we have

$$\underset{n\to \infty }{lim}d(T_{m-1}\cdots T_1\left(x_n\right),x_n)=0$$

and from (11), we know that

$$\underset{n}{sup}d(T_{m-1}\cdots T_1\left(x_n\right),z)<\pi /2.$$

(14)

Combining this information with (10) and (12), we obtain

$$\underset{n\to \infty }{lim}\frac{cos(d(T_m(T_{m-1}\cdots T_1\left(x_n\right)),z)}{cos\left(d(T_{m-1}\cdots T_1\left(x_n\right),z)\right)}=1.$$

(15)

From (14) and (15) and as $T_m$ is strongly quasi-nonexpansive, we have

$$\underset{n\to \infty }{lim}d(T_m(T_{m-1}\cdots T_1\left(x_n\right)),T_{m-1}\cdots T_1\left(x_n\right))=0.$$

By the triangle inequality, we obtain

$$\begin{array}{cc}0\hfill & \le d(T_n\left(x_n\right),x_n)=d(T_m\cdots T_1\left(x_n\right),x_n)\hfill \\ & \le d(T_m\cdots T_1\left(x_n\right),T_{m-1}\cdots T_1\left(x_n\right))+d(T_{m-1}\cdots T_1\left(x_n\right),x_n)\hfill \end{array}$$

Thus, it follows that $\underset{n\to \infty }{lim}d(T\left(x_n\right),x_n)=0$, which proves that $T=T_m\cdots T_1$ is strongly quasi-nonexpansive. □

Now, we show the main theorems.

Theorem 1.

Let $(X,d)$ be a complete admissible $\mathrm{CAT}\left(1\right)$ space. If $T_i:X\to X$ is a strongly quasi-nonexpansive Δ-demiclosed mapping for all $i\in \{1,2,\dots ,m\}$ and ${\cap }_{i=1}^m$Fix$\left(T_i\right)\ne ⌀$, then $T=T_m{T}_{m-1}\cdots T_1$ is a strongly quasi-nonexpansive orbital Δ-demiclosed mapping.

Proof. 

Suppose that $T_i:X\to X$ are strongly quasi-nonexpansive $\Delta$-demiclosed mapping for all $1\le i\le m$ and ${\cap }_{i=1}^{n}Fix\left(T_i\right)\ne ⌀$. By Lemma 4, we have that T is strongly quasi-nonexpansive. Then we will prove that only T is orbital $\Delta$-demiclosed.

Let $\left\{x_k\right\}$ be a sequence in X and $x^{*}$ an element of X such that

(i)

$x_k\stackrel{\Delta }{\to }{x}^{*}$;

(ii)

$d(T\left(x_k\right),x_k)\to 0$;

(iii)

$\left\{x_k\right\}$ is a subsequence $\left\{T^{n_k}\left(\widehat{x}\right)\right\}$ for some $\widehat{x}\in X$.

We will show that $x^{*}\in Fix\left(T\right)$ to conclude T is orbital $\Delta$-demiclosed. Since $T_i$ are strongly quasi-nonexpansive for all $1\le i\le m$, by Lemma 3 we have

$$Fix\left(T\right)={\cap }_{i=1}^{m}Fix\left(T_i\right).$$

(16)

Take $z\in Fix\left(T\right)$.

Since T is strongly quasi-nonexpansive, we obtain

$$d(T\left(x_k\right),z)\le d(x_k,z).$$

This implies that

$$0\le d(x_k,z)-d(T\left(x_k\right),z)\le d(T\left(x_k\right),x_k).$$

By condition (ii), we have

$$\underset{k\to \infty }{lim}[d(x_k,z)-d(T\left(x_k\right),z)]=0.$$

(17)

Using the condition (iii), there exists $\widehat{x}\in X$ such that $x_k=T^{n_k}\left(\widehat{x}\right)$, which further gives

$$d(x_k,z)=d(T^{n_k}\left(\widehat{x}\right),z)\le d(T^{n_k-1}\left(\widehat{x}\right),z)\le \cdots \le d(\widehat{x},z)<\frac{\pi }{2}.$$

(18)

Combining (17) and (18), there exists a subsequence $\left\{x_{k_r}\right\}$ of $\left\{x_k\right\}$ satisfying

$$\underset{r\to \infty }{lim}d(x_{k_r},z)=\underset{r\to \infty }{lim}d(T\left(x_{k_r}\right),z).$$

Since T is strongly quasi-nonexpansive, and from (16), we have

$$\underset{r\to \infty }{lim}d(T\left(x_{k_r}\right),z)\le \underset{r\to \infty }{lim}d(T_{m-1}\cdots T_1\left(x_{k_r}\right),z)\le \cdots \le \underset{r\to \infty }{lim}d(\left(x_{k_r}\right),z).$$

This implies that

$$\underset{r\to \infty }{lim}d(T\left(x_{k_r}\right),z)=\underset{r\to \infty }{lim}d(T_{m-1}\cdots T_1\left(x_{k_r}\right),z)=\cdots =\underset{r\to \infty }{lim}d(\left(x_{k_r}\right),z)<\frac{\pi }{2}.$$

(19)

According to (16), we can show that $x^{*}\in {\cap }_{i=1}^{m}Fix\left(T_i\right)$ to conclude that $x^{*}\in Fix\left(T\right)$.

Step 1 First, we will show that $x^{*}\in Fix\left(T_1\right)$.

By (19), we obtain

$$\underset{r\to \infty }{lim}d(T_1\left(x_{k_r}\right),z)=\underset{r\to \infty }{lim}d(\left(x_{k_r}\right),z)<\frac{\pi }{2}.$$

Then

$$0<cos\left(\underset{r\to \infty }{lim}d(\left(x_{k_r}\right),z)\right)=cos\left(\underset{r\to \infty }{lim}d(T_1\left(x_{k_r}\right),z)\right)$$

this implies that

$$\underset{r\to \infty }{lim}\frac{cosd(T_1\left(x_{k_r}\right),z)}{cosd(x_{k_r},z)}=1.$$

Since $T_1$ is strongly quasi-nonexpansive, we obtain

$$\underset{r\to \infty }{lim}d(T_1\left(x_{k_r}\right),x_{k_r})=0$$

(20)

and since $T_1$ is $\Delta$-demiclosed, the $\Delta$-limit of $\left\{x_{k_r}\right\}$ belongs to $Fix\left(T_1\right)$. Then $x^{*}\in Fix\left(T_1\right)$.

Step 2 We next show that $x^{*}\in Fix\left(T_2\right)$.

By (19), we obtain

$$\underset{r\to \infty }{lim}d(T_2{T}_1\left(x_{k_r}\right),z)=\underset{r\to \infty }{lim}d(T_1\left(x_{k_r}\right),z)<\frac{\pi }{2}.$$

Then

$$0<cos\left(\underset{r\to \infty }{lim}d(T_2{T}_1\left(x_{k_r}\right),z)\right)=cos\left(\underset{r\to \infty }{lim}d(T_1\left(x_{k_r}\right),z)\right)$$

this implies that

$$\underset{r\to \infty }{lim}\frac{cosd(T_2\left(T_1\left(x_{k_r}\right)\right),z)}{cosd(T_1\left(x_{k_r}\right),z)}=1.$$

Since $T_2$ is strongly quasi-nonexpansive, we obtain

$$\underset{r\to \infty }{lim}d(T_2\left(T_1\left(x_{k_r}\right)\right),T_1\left(x_{k_r}\right))=0$$

and as $T_2$ is $\Delta$-demiclosed, the $\Delta$-limit of $\left\{T_1\left(x_{k_r}\right)\right\}$ belongs to $Fix\left(T_2\right)$.

Next, we will show that $\left\{T_1\left(x_{k_r}\right)\right\}$$\Delta$-convergent to $x^{*}$.

Let $\left\{T_1\left(x_{k_{r_j}}\right)\right\}$ be an arbitrary subsequence of $\left\{T_1\left(x_{k_r}\right)\right\}$.

$$\begin{array}{ccc}\underset{j\to \infty }{lim\; sup}d(T_1\left(x_{k_{r_j}}\right),x^{*})\hfill & \le \hfill & \underset{j\to \infty }{lim\; sup}d(\left(x_{k_{r_j}}\right),x^{*})\le \underset{j\to \infty }{lim\; sup}d(\left(x_{k_{r_j}}\right),y),\left(\forall y\in X\right)\hfill \\ & \le \hfill & \underset{j\to \infty }{lim\; sup}(d(x_{k_{r_j}},T_1\left(x_{k_{r_j}}\right))+d(T_1\left(x_{k_{r_j}}\right),y)),\left(\forall y\in X\right)\hfill \\ & \le \hfill & \underset{j\to \infty }{lim\; sup}d(x_{k_{r_j}},T_1\left(x_{k_{r_j}}\right))\hfill \\ & & +\underset{j\to \infty }{lim\; sup}d\left(T_1\right(x_{k_{r_j}},)y,),\left(\forall y\in X\right)\hfill \end{array}$$

By (20), we have

$$\underset{j\to \infty }{lim\; sup}d(T_1\left(x_{k_{r_j}}\right),x^{*})\le \underset{j\to \infty }{lim\; sup}d(T_1\left(x_{k_{r_j}}\right),y),\left(\forall y\in X\right)$$

which means that

$$\mathcal{A}\left(\left\{T_1\left(x_{k_{r_j}}\right)\right\}\right)=\left\{x^{*}\right\}$$

for each subsequence $\left\{T_1\left(x_{k_{r_j}}\right)\right\}$ of $\left\{T_1\left(x_{k_r}\right)\right\}$. This implies that $\left\{T_1\left(x_{k_r}\right)\right\}$$\Delta$-convergent to $x^{*}$. Thus, $x^{*}$ belongs to $Fix\left(T_2\right)$.

We now show that $x^{*}\in Fix\left(T_l\right)$ for the remaining $l=3,\dots ,m$. We suppose that step i holds for $i=1,\dots ,l-1$ and we will show that $x^{*}\in Fix\left(T_l\right)$. By (19), we obtain

$$\underset{r\to \infty }{lim}d(T_l\cdots T_1\left(x_{k_r}\right),z)=\underset{r\to \infty }{lim}d(T_{l-1}\cdots T_1\left(x_{k_r}\right),z)<\frac{\pi }{2}.$$

Then

$$0<cos\left(\underset{r\to \infty }{lim}d(T_l\cdots T_1\left(x_{k_r}\right),z)\right)=cos\left(\underset{r\to \infty }{lim}d(T_{l-1}\cdots T_1\left(x_{k_r}\right),z)\right).$$

This implies that

$$\underset{r\to \infty }{lim}\frac{cosd(T_l(T_{l-1}\cdots T_1\left(x_{k_r}\right)),z)}{cosd(T_{l-1}\cdots T_1\left(x_{k_r}\right),z)}=1.$$

Since $T_l$ is strongly quasi-nonexpansive, we obtain

$$\underset{r\to \infty }{lim}d(T_l(T_{l-1}\cdots T_1\left(x_{k_r}\right)),T_{l-1}\cdots T_1\left(x_{k_r}\right))=0$$

and as $T_l$ is $\Delta$-demiclosed, the $\Delta$-limit of $\{T_{l-1}\cdots T_1\left(x_{k_r}\right)\}$ belongs to $Fix\left(T_l\right)$.

Next, we will show that $\{T_{l-1}\cdots T_1\left(x_{k_r}\right)\}$$\Delta$-convergent to $x^{*}$.

Let $\{T_{l-1}\cdots T_1\left(x_{k_{r_j}}\right)\}$ be an arbitrary subsequence of $\{T_{l-1}\cdots T_1\left(x_{k_r}\right)\}$.

$$\begin{array}{ccc}\underset{j\to \infty }{lim\; sup}d(T_{l-1}\cdots T_1\left(x_{k_{r_j}}\right),x^{*})\hfill & \le \hfill & \underset{j\to \infty }{lim\; sup}d(T_{l-2}\cdots T_1\left(x_{k_{r_j}}\right),x^{*})\hfill \\ & \le \hfill & \underset{j\to \infty }{lim\; sup}d(T_{l-2}\cdots T_1\left(x_{k_{r_j}}\right),y),\left(\forall y\in X\right)\hfill \\ & \le \hfill & \underset{j\to \infty }{lim\; sup}(d(T_{l-2}\cdots T_1\left(x_{k_{r_j}}\right),T_{l-1}\cdots T_1\left(x_{k_{r_j}}\right))\hfill \\ & & +d(T_{l-1}\cdots T_1\left(x_{k_{r_j}}\right),y)),\left(\forall y\in X\right)\hfill \\ & \le \hfill & \underset{j\to \infty }{lim\; sup}d(T_{l-2}\cdots T_1\left(x_{k_{r_j}}\right),T_{l-1}\cdots T_1\left(x_{k_{r_j}}\right))\hfill \\ & & +\underset{j\to \infty }{lim\; sup}d(T_{l-1}\cdots T_1\left(x_{k_{r_j}}\right),y),\left(\forall y\in X\right)\hfill \end{array}$$

By the assumption holds for $l-1$, we have

$$\underset{r\to \infty }{lim}d(T_{l-1}\cdots T_1\left(x_{k_r}\right),T_{l-2}\cdots T_1\left(x_{k_r}\right))=0.$$

Then, we obtain

$$\underset{j\to \infty }{lim\; sup}d(T_{l-1}\cdots T_1\left(x_{k_{r_j}}\right),x^{*})\le \underset{j\to \infty }{lim\; sup}d(T_{l-1}\cdots T_1\left(x_{k_{r_j}}\right),y),\left(\forall y\in X\right)$$

which means that

$$\mathcal{A}\left(\{T_{l-1}\cdots T_1\left(x_{k_{r_j}}\right)\}\right)=\left\{x^{*}\right\}$$

for each subsequence $\{T_{l-1}\cdots T_1\left(x_{k_{r_j}}\right)\}$ of $\{T_{l-1}\cdots T_1\left(x_{k_r}\right)\}$. This implies that the sequence $\{T_{l-1}\cdots T_1\left(x_{k_r}\right)\}$$\Delta$-convergent to $x^{*}$. Thus, $x^{*}$ belongs to $Fix\left(T_l\right)$. From step 1 to step m and (16), then $x^{*}\in {\cap }_{i=1}^{m}Fix\left(T_i\right)$; this means that $x^{*}\in Fix\left(T\right)$. Therefore, T is orbital $\Delta$-demiclosed. □

Theorem 2.

Let $(X,d)$ be a complete admissible $\mathrm{CAT}\left(1\right)$ space, $T:X\to X$ is a strongly quasi-nonexpansive orbital Δ-demiclosed mapping, and Fix$\left(T\right)\ne ⌀$. Then, for any initial point $x_0\in X$, the sequence defined for each $n\in \mathbb{N}$ by

$$x_n:=T^n\left(x_0\right)$$

is Δ-convergent to an element $\overline{x}$ in $Fix\left(T\right)$.

Proof. 

Given $x\in X$, consider the sequence $\left\{x_n\right\}$ that is defined by

$$x_1:=x \mathrm{and} x_{n+1}:=T^n\left(x\right):=T\left(x_n\right) \mathrm{for} \mathrm{all} n\in \mathbb{N}.$$

Let $z\in Fix\left(T\right)$. Since T is quasi-nonexpansive, we obtain

$$0\le d(x_{n+1},z)\le d(x_n,z)\le \cdots \le d(x_1,z)=d(x,z)<\frac{\pi }{2}.$$

Observe that the sequence $\left\{d(x_n,z)\right\}$ is bounded and nonincreasing for each $z\in Fix\left(T\right)$. Then the sequence $\left\{d(x_n,z)\right\}$ is convergent to an element of $[0,\frac{\pi }{2})$, for each $z\in Fix\left(T\right)$. We will have

$$0\le \underset{n\to \infty }{lim}d(x_n,z)=\underset{n\to \infty }{lim}d(T\left(x_n\right),z)\le d(x,z)<\frac{\pi }{2}.$$

This implies that

$$0<cos\left(\underset{n\to \infty }{lim}d(x_n,z)\right)=cos\left(\underset{n\to \infty }{lim}d(T\left(x_n\right),z)\right)\le 1,$$

then we obtain

$$\underset{n\to \infty }{lim}\frac{cos\left(d(T\left(x_n\right),z)\right)}{cos\left(d(x_n,z)\right)}=1.$$

Combining this information with the strong quasi-nonexpansivity of T, we have

$$\underset{n\to \infty }{lim}d(x_n,T\left(x_n\right))=0.$$

(21)

Next, we will show that $\left\{x_n\right\}$ is spherically bounded. Since

$$\underset{y\in X}{inf}lim\; supd(T^n\left(x\right),y)\le \underset{y\in Fix\left(T\right)}{inf}lim\; supd(T^n\left(x\right),y)\le \underset{y\in Fix\left(T\right)}{inf}d(x,y)<\frac{\pi }{2},$$

then the sequence $\left\{x_n\right\}$ is spherically bounded. By Lemma 1, the sequence $\left\{x_n\right\}$ has a $\Delta$-convergent subsquence. Take $\overline{x}\in {\omega }_{\Delta }\left(\left\{x_n\right\}\right)$ such that $\left\{x_{n_k}\right\}$ is $\Delta$-convergent to $\overline{x}$, for some subsequence $\left\{x_{n_k}\right\}$ of $\left\{x_n\right\}$. We know that $x_{n_k}=T^{n_k-1}\left(x\right)$ from definition of $\left\{x_n\right\}$. This means that $\left\{x_{n_k}\right\}$ is a subsequence for some orbit. By (21), we have

$$\underset{k\to \infty }{lim}d(x_{n_k},T\left(x_{n_k}\right))=0.$$

Combining this information with T as a orbital $\Delta$-demiclosed mapping, we obtain that $\overline{x}$ belongs to $Fix\left(T\right)$. Thus, ${\omega }_{\Delta }\left(\left\{x_n\right\}\right)\subset Fix\left(T\right)$. This implies that the sequence $\left\{d(x_n,\overline{x})\right\}$ is convergent to an element of $[0,\frac{\pi }{2})$, for all $\overline{x}\in {\omega }_{\Delta }\left(\left\{x_n\right\}\right)$. By Lemma 2, the sequence $\left\{x_n\right\}$ is $\Delta$-convergent to an element of X, since we know that ${\omega }_{\Delta }\left(\left\{x_n\right\}\right)\subset Fix\left(T\right)$. Therefore, the sequence $\left\{x_n\right\}$ is $\Delta$-convergent to an element of $Fix\left(T\right)$. □

4. Applications

In this section, we consider some viable applications of our main results. Particularly, we shall apply our results from the previous section to solve the convex optimizations on $\mathrm{CAT}\left(1\right)$ spaces.

Given a complete admissible $\mathrm{CAT}\left(1\right)$ space X, from Section 2, we know that the resolvent mapping $R_{\lambda ,f}$ of a proper lsc convex function f of X with respect to $\lambda >0$, is a strongly quasi-nonexpansive $\Delta$-demiclosed mapping and $Fix\left(R_{\lambda ,f}\right)={arg\; min}_{X}f$. Then, we can apply our results from the previous section to optimize the function f. Moreover, the convex feasibility problem seeks a common point $\overline{x}$ in the intersection ${\cap }_{i=1}^m{C}_i$, where $C_i$ is a nonempty closed convex subset of X for all $i\in \{1,2,\dots ,m\}$. The metric projection is crucial for dealing with a convex feasibility problem. According to Section 2, we know that the metric projection $P_{C_i}$ from X onto $C_i$ is a strongly quasi-nonexpansive $\Delta$-demiclosed mapping and $Fix\left(P_{C_i}\right)=C_i$ for all $i\in \{1,2,\dots ,m\}$. If ${\cap }_{i=1}^m{C}_i$ is nonempty, we can use the alternating projection mapping for finding the solution of this problem.

4.1. Convex Minimization Problems

Firstly, we will present the application for solving the convex minimization problem of proper lsc convex functions in a complete admissible $\mathrm{CAT}\left(1\right)$ space X. Let $f:X\to (-\infty ,\infty ]$ be a proper lsc convex function and ${\lambda }_i>0$ for all $i\in \{1,2,\dots ,m\}$. According to [13] (Theorem 4.6), we know that the resolvent mapping $R_{{\lambda }_i,f}$ is a strongly quasi-nonexpansive $\Delta$-demiclosed mapping such that $Fix\left(R_{{\lambda }_i,f}\right)$ is equal to $\underset{X}{arg\; min}f$ for all $i\in \{1,2,\dots ,m\}$. Then, we obtain the following corollary.

Corollary 1.

Let $(X,d)$ be a complete admissible $\mathrm{CAT}\left(1\right)$ space and $f:X\to (-\infty ,\infty ]$ be a proper lower semicontinuous convex function. Let $R_{{\lambda }_i,f}:X\to X$ be a resolvent mapping of f with respect to ${\lambda }_i>0$ for all $i\in \{1,2,\dots ,m\}$. If ${arg\; min}_{X}f$ is nonempty, then for any initial point $x_0\in X$, the sequence defined for each $n\in \mathbb{N}$ by

$$x_n:={(R_{{\lambda }_m,f}{R}_{{\lambda }_{m-1},f}\cdots R_{{\lambda }_1,f})}^n\left(x_0\right)$$

is Δ-convergent to an element $\overline{x}$ in $\underset{X}{arg\; min}f$.

Proof. 

Since $R_{{\lambda }_i,f}$ is a strongly quasi-nonexpansive $\Delta$-demiclosed mapping for all $i\in \{1,2,\dots ,m\}$ such that ${\cap }_{i=1}^{m}Fix\left(R_{{\lambda }_i,f}\right)={arg\; min}_{X}f\ne ⌀$, then we have that $R_{{\lambda }_m,f}{R}_{{\lambda }_{m-1},f}\cdots R_{{\lambda }_1,f}$ is a strongly quasi-nonexpansive orbital $\Delta$-demiclosed mapping by using Lemma 1. Let $x_0\in X$. Define a sequence $\left\{x_n\right\}\in X$ by $x_n={(R_{{\lambda }_m,f}{R}_{{\lambda }_{m-1},f}\cdots R_{{\lambda }_1,f})}^n\left(x_0\right)$. By using Theorem 2, the sequence $\left\{x_n\right\}$ is $\Delta$-convergent to some point in $\overline{x}\in Fix(R_{{\lambda }_m,f}{R}_{{\lambda }_{m-1},f}\cdots R_{{\lambda }_1,f})$$={\cap }_{i=1}^{m}Fix\left(R_{{\lambda }_i,f}\right)$. Then, the point $\overline{x}\in \underset{X}{arg\; min}f$. □

In the next application, we consider the sum of several loss functions, given that the objective function f is defined by $f:={\sum }_{i=1}^m{f}_i$ where $f_i$ is a proper lsc convex function for all $i\in \{1,2,\dots ,m\}$. According to [13] (Theorem 4.6), we know that the resolvent mapping $R_{{\lambda }_i,f_i}$ is a strongly quasi-nonexpansive $\Delta$-demiclosed mapping such that $Fix\left(R_{{\lambda }_i,f_i}\right)$ is equal to $\underset{X}{arg\; min}{f}_i$ for all $i\in \{1,2,\dots ,m\}$. By Lemma 3, we have that ${\cap }_{i=1}^{m}Fix\left(R_{{\lambda }_i,f_i}\right)=Fix(R_{{\lambda }_m,f_m}{R}_{{\lambda }_{m-1},f_{m-1}}\cdots R_{{\lambda }_1,f_1})$ and we know that ${\cap }_{i=1}^m\underset{X}{arg\; min}{f}_i=\underset{X}{arg\; min}f$ whenever ${\cap }_{i=1}^m\underset{X}{arg\; min}{f}_i$ is nonempty. Then, we can apply the next corollary for finding the solutions of the objective function.

Corollary 2.

Let $(X,d)$ be a complete admissible $\mathrm{CAT}\left(1\right)$ space and $f:X\to (-\infty ,\infty ]$ defined by $f:={\sum }_{i=1}^m{f}_i$ where $f_i$ is a proper lower semicontinuous convex function for all $i\in \{1,2,\dots ,m\}$. Let $R_{{\lambda }_i,f_i}:X\to X$ be a resolvent mapping of $f_i$ with respect to ${\lambda }_i>0$ for all $i\in \{1,2,\dots ,m\}$. If ${\cap }_{i=1}^m{arg\; min}_X{f}_i$ is nonempty, then for any initial point $x_0\in X$, the sequence defined for each $n\in \mathbb{N}$ by the fact that

$$x_n:={(R_{{\lambda }_m,f_m}{R}_{{\lambda }_{m-1},f_{m-1}}\cdots R_{{\lambda }_1,f_1})}^n\left(x_0\right)$$

is Δ-convergent to an element $\overline{x}$ in $\underset{X}{arg\; min}f$.

Proof. 

Since $R_{{\lambda }_i,f_i}$ is a strongly quasi-nonexpansive $\Delta$-demiclosed mapping for all $i\in \{1,2,\dots ,m\}$ such that ${\cap }_{i=1}^{m}Fix\left(R_{{\lambda }_i,f_i}\right)={\cap }_{i=1}^m{arg\; min}_X{f}_i\ne ⌀$, then $R_{{\lambda }_m,f_m}{R}_{{\lambda }_{m-1},f_{m-1}}\cdots$$R_{{\lambda }_1,f_1}$ is a strongly quasi-nonexpansive orbital $\Delta$-demiclosed mapping by using Lemma 1. Let $x_0\in X$. Define a sequence $\left\{x_n\right\}\in X$ by $x_n={(R_{{\lambda }_m,f_m}{R}_{{\lambda }_{m-1},f_{m-1}}\cdots R_{{\lambda }_1,f_1})}^n\left(x_0\right)$. By using Theorem 2, we obtain that the sequence $\left\{x_n\right\}$ is $\Delta$-convergent to some point in $\overline{x}\in Fix(R_{{\lambda }_m,f_m}{R}_{{\lambda }_{m-1},f_{m-1}}\cdots R_{{\lambda }_1,f_1})$$={\cap }_{i=1}^{m}Fix\left(R_{{\lambda }_i,f_i}\right)$. Then, the point $\overline{x}\in {\cap }_{i=1}^m\underset{X}{arg\; min}{f}_i=\underset{X}{arg\; min}f$. □

4.2. Convex Feasibility Problem

Finally, we consider the feasibility problem with each $C_i$ being closed and convex. We call this particular case the convex feasibility problem. Let $(X,d)$ be a complete admissible CAT$\left(1\right)$ space and let $C_1,C_2,\dots ,C_m$ be nonempty closed convex subsets of X. The convex feasibility problem is to find some point

$$x\in {\cap }_{i=1}^m{C}_i,$$

when this intersection is nonempty.

Now, we will prove the $\Delta$-convergence of alternating projection on complete admissible CAT$\left(1\right)$ space.

Corollary 3.

Let $(X,d)$ be a complete admissible $\mathrm{CAT}\left(1\right)$ space. If $P_{C_i}$ is the metric projection of X onto closed and convex subset $C_i\subset X$ for all $i\in \{1,2,\dots ,m\}$ with ${\cap }_{i=1}^m{C}_i\ne ⌀$, then for any initial point $x_0\in X$, the sequence defined for each $n\in \mathbb{N}$ by

$$x_n:={(P_{C_m}\cdots P_{C_1})}^n\left(x_0\right)$$

is Δ-convergent to an element $\overline{x}\in {\cap }_{i=1}^m{C}_i$.

Proof. 

Since each $P_{C_i}$ is a strongly quasi-nonexpansive and $\Delta$-demiclosed mapping for all $i\in \{1,2,\dots ,m\}$, respectively, according to Lemma 1, the products $P_{C_m}\cdots P_{C_1}$ are strongly quasi-nonexpansive and orbital $\Delta$-demiclosed mapping. By Theorem 2, for each $x\in X$, there exists a point $\overline{x}=\overline{x}\left(x\right)\in Fix(P_{C_m}\cdots P_{C_1})$ such that the sequence $\left\{{(P_{C_m}\cdots P_{C_1})}^n\left(x\right)\right\}$ is $\Delta$-convergent to $\overline{x}$. The fact that $\overline{x}\in {\cap }_{i=1}^{m}Fix\left(P_{C_i}\right)={\cap }_{i=1}^m{C}_i$ follows from Lemma 4. □

Next, we prove the strong convergence alternating projection from the previous corollary by letting $C_j$ be a compact set for some $j\in \{1,2,\dots ,m\}$.

Corollary 4.

Let $(X,d)$ be a complete admissible $\mathrm{CAT}\left(1\right)$ space. If $P_{C_i}$ is the metric projection of X onto a nonempty closed and convex subset $C_i\subset X$ for all $i\in \{1,2,\dots ,m\}$ with ${\cap }_{i=1}^m{C}_i\ne ⌀$ and $C_j$ is a compact set for some $j\in \{1,2,\dots ,m\}$, then for any initial point $x_0\in X$, the sequence defined for each $n\in \mathbb{N}$ by

$$x_n:={(P_{C_m}\cdots P_{C_1})}^n\left(x_0\right)$$

is convergent to an element $\overline{x}\in {\cap }_{i=1}^m{C}_i$.

Proof. 

We will prove this by letting $C_i$ be a compact set for some $i\in \{1,\dots ,m\}$.

Case I Suppose that $C_m$ is a compact set.

Let the starting point $x_0=x$ for some element $x\in X$ and $x_n={(P_{C_m}\cdots P_{C_1})}^n\left(x\right)$ for all $n\in \mathbb{N}$. By Corollary 3, we have that the sequence $\left\{x_n\right\}$ is $\Delta$-convergent to $\overline{x}$ for some $\overline{x}\left(x\right)\in {\cap }_{i=1}^m{C}_i$. Then, every subsequence of $\left\{x_n\right\}$ is $\Delta$-convergent to $\overline{x}$. Since $C_m$ is a compact set and $\left\{x_n\right\}\subset C_m$, then for every subsequence $\left\{x_{n_k}\right\}$ of $\left\{x_n\right\}$ there exists $\left\{x_{n_{k_l}}\right\}\subset \left\{x_{n_k}\right\}$ such that $\underset{n\to \infty }{lim}{x}_{n_{k_l}}=\overline{x}$; this implies that $\underset{n\to \infty }{lim}{x}_n=\overline{x}$.

Case II Suppose that $C_i$ is a compact set for some $i\in \{1,\dots ,m-1\}$.

Define $\widehat{P}:=P_{C_i}{P}_{C_{i-1}}\cdots P_{C_1}{P}_{C_m}{P}_{C_{m-1}}\cdots P_{C_{i+1}}$ and $P:=P_{C_m}{P}_{C_{m-1}}\cdots P_{C_1}$. By case I, we have

$$\underset{n\to \infty }{lim}{\left(\widehat{P}\right)}^n(P_{C_i}{P}_{C_{i-1}}\cdots P_{C_1})\left(x\right)=\overline{x}\in {\cap }_{i=1}^m{C}_i$$

for any $x\in X$. Observe that

$$P^n\left(x\right)={(P_{C_m}\cdots P_{C_1})}^n\left(x\right)=(P_{C_m}{P}_{C_{m-1}}\cdots P_{C_{i+1}}){\left(\widehat{P}\right)}^{n-1}(P_{C_i}\cdots P_{C_1})\left(x\right)$$

and then $d(P^n\left(x\right),\overline{x})\le d({\left(\widehat{P}\right)}^n(P_{C_i}{P}_{C_{i-1}}\cdots P_{C_1})\left(x\right),\overline{x})\to 0$ as $n\to \infty$. Therefore, we can conclude that $\underset{n\to \infty }{lim}{(P_{C_m}\cdots P_{C_1})}^n\left(x\right)=\overline{x}$. □

5. Numerical Implementations

In this section, we implement the proposed alternating projection algorithm to approximate the convex feasibility problem of given closed convex sets which are fitted to $X\subset {\mathbb{S}}^2$, X defined by

$$X=\left\{x\in {\mathbb{S}}^2|d_{{\mathbb{S}}^2}(x,p)\le \delta \right\}$$

(22)

where $p\in {\mathbb{S}}^2$, $0<\delta <\pi /4$, and $d_{{\mathbb{S}}^2}$ is the spherical metric. Recall that X is a complete admissible $\mathrm{CAT}\left(1\right)$ space when equipped with the spherical metric $d_{{\mathbb{S}}^2}$.

Given that $C_1,C_2,\dots ,C_m$ are closed balls on X,

$$C_i=\left\{x\in X|d_{{\mathbb{S}}^2}(x,p_i)\le r_i\right\}$$

(23)

where $p_i\in X$ and $r_i\ge \delta$ for all $i\in \{1,2,\dots ,m\}$. Then $C_1,C_2,\dots ,C_m$ are closed convex subsets of X and $p\in {\cap }_{i=1}^m{C}_i\ne ⌀$. We can define the metric projection $P_{C_i}$ from X onto $C_i$ for all $i=1,2,\dots ,m$ by

$$P_{C_i}\left(x\right)=\left\{\begin{array}{cc}x\hfill & ,x\in C_i;\hfill \\ {\gamma }_{p_i,x}\left(r_i\right)\hfill & ,x\in X\setminus C_i,\hfill \end{array}\right.$$

where ${\gamma }_{p_i,x}\left(r_i\right)$ is defined in (2).

In the numerical implementations, we set $X=\left\{x\in {\mathbb{S}}^2|d_{{\mathbb{S}}^2}(x,p)\le \pi /6\right\}$ where $p=(0,0,1)$, and we use three points, $p_i$, which are elements in X to generate closed convex subsets $C_i$ with $r_i=\pi /6$ for all $i\in \{1,2,3\}$ (see Figure 1).

In the following numerical implementations of Corollary 4, we will show the convergence of alternating sequences by using two distinct starting points $x_0$ in X. Since the dataset is quite large, we only report the implementation results as presented in Figure 2. Therein, the LHS figures represent the center points of $C_1,C_2,C_3$ (green ‘•’ marks), the elements of intersection of $C_i$ (blue ‘•’ marks), and the initial point $x_0$ (black ‘x’ marks), the alternating sequence $x_N$ (black ‘*’ marks), together with approximated solution of CFP (red ‘x’ marks) obtained from the alternating method. The RHS figures show plots of errors after the iteration of N. Note that the errors presented thereby are computed by ${\sum }_{i=1}^{3}d{(x_N,C_i)}^2$. The accepted tolerance in these illustrations is set at $\mathrm{tol}={10}^{-5}$. The calculation was performed using Python in Google Colab.

The features of the alternating sequence’s convergence will vary depending on the chosen initial point, as demonstrated in the example above. Furthermore, we can see that errors fall quickly in the early phases of the computation, but then slow down and eventually halt when errors are fewer than the tol=${10}^{-5}$ value we set.

6. Conclusions

In this paper, we introduced a new concept of orbital $\Delta$-demiclosed mappings, which is a generalization of $\Delta$-demiclosed mappings, and we showed that the product of strongly quasi-nonexpansive $\Delta$-demiclosed mappings is a strongly quasi-nonexpansive orbital $\Delta$-demiclosed mapping in $\mathrm{CAT}\left(1\right)$ spaces. We obtained more general convergence results to approximate a common fixed point of infinite products of strongly quasi-nonexpansive orbital $\Delta$-demiclosed mappings on $\mathrm{CAT}\left(1\right)$ spaces. Our results can be applied to the minimization of convex functions and the sum of finitely many convex functions or to an abstract minimization problem which allows the study of the alternating projection method for the convex feasibility problem for finitely many sets in $\mathrm{CAT}\left(1\right)$ spaces.

However, it is unclear whether the composition of strongly quasi-nonexpansive $\Delta$-demiclosed mappings is still a strongly quasi-nonexpansive $\Delta$-demiclosed mapping in $\mathrm{CAT}\left(1\right)$ spaces.