Resolvents for Convex Functions on Geodesic Spaces and Their Nonspreadingness

Abstract

The convex optimization problems have been considered by many researchers on geodesic spaces. In these problems, the resolvent operators play an important role. In this paper, we propose new resolvents on geodesic spaces, and we show that they have better properties than other known resolvent operators.

1. Introduction

Resolvents for a convex function play an important role to consider convex optimization problems. On a complete CAT(0) space X, Jost [1] and Mayer [2] proposed the resolvent for a proper lower semicontinuous convex function f of X into $\left]-\infty ,\infty \right]$ by

$$J_f\left(x\right)=\underset{y\in X}{argmin}\left\{f\left(y\right)+{\displaystyle \frac{1}{2}}d{(y,x)}^2\right\}$$

for each $x\in X$. It is known that $J_f$ is metrically nonspreading, that is,

$$2d{(J_{f}x,J_{f}y)}^2\leqq d{(J_{f}x,y)}^2+d{(x,J_{f}y)}^2$$

for all $x,y\in X$. For more details regarding metrical nonspreadingness, see [3].

On the other hand, on an admissible complete CAT(1) space X, two kinds of resolvents for a proper lower semicontinuous convex function were defined. The first one was proposed by Kimura and Kohsaka [4] as follows:

$$T_1\left(x\right)=\underset{y\in X}{argmin}\{f\left(y\right)+tand(y,x)sind(y,x)\}$$

for each $x\in X$. They also defined a spherically nonspreading mapping of product type and showed that the resolvent $T_1$ satisfies it, that is,

$${cos}^{2}d(T_{1}x,T_{1}y)\geqq cosd(T_{1}x,y)cosd(x,T_{1}y)$$

for all $x,y\in X$. The second one was proposed by the Kajimura and Kimura [5] as follows:

$$T_2\left(x\right)=\underset{y\in X}{argmin}\{f\left(y\right)-logcosd(y,x)\}$$

for each $x\in X$. Motivated by the spherical nonspreadingness of product type, they proposed the concept of spherical nonspreadingness of sum type and showed that $T_2$ satisfies this property, that is,

$$2cosd(T_{2}x,T_{2}y)\geqq cosd(T_{2}x,y)+cosd(x,T_{2}y)$$

for all $x,y\in X$. The spherical nonspreadingness of sum type implies the spherical nonspreadingness of product type by using the arithmetic–geometric mean inequality. Therefore, we know that $T_2$ is also the spherically nonspreading of product type. However, we do not know whether $T_1$ is a spherically nonspreading of sum type or not.

On a complete CAT($-1$) space X, motivated by the definition of the resolvent $T_1$, Kajimura and Kimura [6] proposed the resolvent for a proper lower semicontinuous convex function f by

$$T_3\left(x\right)=\underset{y\in X}{argmin}\{f\left(y\right)+tanhd(y,x)sinhd(y,x)\}$$

for each $x\in X$. They also showed its hyperbolical nonspreadingness, that is,

$$2coshd(T_{3}x,T_{3}y)\leqq coshd(T_{3}x,y)+coshd(x,T_{3}y)$$

for all $x,y\in X$.

Resolvent operators are applied to generate approximate sequences which converge to a minimizer of given functions; for instance, see [7,8,9,10,11,12] for Banach spaces, [13,14,15,16] for the Riemannian manifold with nonpositive sectional curvature, [17,18] for CAT($\kappa$) spaces, and the references therein.

In this paper, we propose new resolvents for a convex function on a CAT(1) space and a CAT($-1$) space, respectively. We also show that those resolvents satisfy the $\kappa$-nonspreadingness, which contains the class of metrically nonspreading mapping, spherically nonspreading mapping of sum type, and hyperbolically nonspreading mapping.

2. Preliminaries

Let X be a metric space. X is called a uniquely geodesic space if for each $x,y\in X$, there exists a unique mapping ${\gamma }_{xy}:\left[0,l\right]\to X$ such that ${\gamma }_{xy}\left(0\right)=x$, ${\gamma }_{xy}\left(l\right)=y$ and $d({\gamma }_{xy}\left(s\right),{\gamma }_{xy}\left(t\right))=\left|s-t\right|$ for all $s,t\in \left[0,l\right]$. The mapping ${\gamma }_{xy}$ is called a geodesic joining x and y. For $D>0$, a metric space X is called a D-uniquely geodesic space if for each $x,y\in X$ with $d(x,y)<D$, there exists the unique geodesic joining x and y. Let X be a D-uniquely geodesic space such that $d(u,v)<D$ for all $u,v\in X$. For each $x,y\in X$, the convex combination between x and y is defined by

$$tx\oplus (1-t)y={\gamma }_{xy}\left((1-t)d(x,y)\right)$$

for all $t\in \left[0,1\right]$. A function f of X into $\left]-\infty ,\infty \right]$ is said to be the following:

• Proper if $domf\ne \varnothing$, where $domf=\{x\in X\mid f(x)\in \mathbb{R}\}$;
• Lower semicontinuous if the set $\{x\in X\mid f(x)\leqq a\}$ is closed for all $a\in \mathbb{R}$;
• Convex if $f(tx\oplus (1-t\left)y\right)\leqq tf\left(x\right)+(1-t)f\left(y\right)$ for all $x,y\in X$ and $t\in \left]0,1\right[$.

For $\kappa \in \mathbb{R}$, the two-dimensional model space $M_{\kappa }^2$ is defined by

$$\begin{array}{c}\hfill M_{\kappa }^2=\left\{\begin{array}{cc}(1/\sqrt{\kappa }){\mathbb{S}}^2\hfill & (\kappa >0);\hfill \\ {\mathbb{R}}^2\hfill & (\kappa =0);\hfill \\ (1/\sqrt{-\kappa }){\mathbb{H}}^2\hfill & (\kappa <0),\hfill \end{array}\right.\end{array}$$

where ${\mathbb{S}}^2$ is the two-dimensional unit sphere on ${\mathbb{R}}^3$, and ${\mathbb{H}}^2$ is the two-dimensional hyperbolic space. The diameter of $M_{\kappa }^2$ is denoted by $D_{\kappa }$, that is, $D_{\kappa }=\pi /\sqrt{\kappa }$ for $\kappa >0$ and $D_{\kappa }=\infty$ for $\kappa \leqq 0$.

For $\kappa \in \mathbb{R}$, let X be a $D_{\kappa }$-uniquely geodesic space and $x,y,z\in X$ such that $d(x,y)+d(y,z)+d(z,x)<2D_{\kappa }$. The set $▵(x,y.z)=\mathrm{Im}{\gamma }_{xy}\cup \mathrm{Im}{\gamma }_{yz}\cup \mathrm{Im}{\gamma }_{zx}$ is called a geodesic triangle. Its comparison triangle on $M_{\kappa }^2$ is defined by the set $\overline{▵}(\overline{x},\overline{y},\overline{z})\subset M_{\kappa }^2$ satisfying $d(x,y)=d(\overline{x},\overline{y})$, $d(y,z)=d(\overline{y},\overline{z})$, and $d(z,x)=d(\overline{z},\overline{x})$. A point $\overline{p}\in \overline{▵}(\overline{x},\overline{y},\overline{z})$ is called a comparison point of $p\in ▵(x,y.z)$ if one of the following conditions holds:

• If $p\in \mathrm{Im}{\gamma }_{xy}$, then $\overline{p}\in \mathrm{Im}{\gamma }_{\overline{x}\overline{y}}$ and $d(x,p)=d(\overline{x},\overline{p})$;
• If $p\in \mathrm{Im}{\gamma }_{yz}$, then $\overline{p}\in \mathrm{Im}{\gamma }_{\overline{y}\overline{z}}$ and $d(y,p)=d(\overline{y},\overline{p})$;
• If $p\in \mathrm{Im}{\gamma }_{zx}$, then $\overline{p}\in \mathrm{Im}{\gamma }_{\overline{z}\overline{x}}$ and $d(z,p)=d(\overline{z},\overline{p})$.

A $D_{\kappa }$-uniquely geodesic space X is called a CAT($\kappa$) space if $d_X(p,q)\leqq d_{M_{\kappa }^2}(\overline{p},\overline{q})$ for all $p,q\in ▵(x,y.z)$. If ${\kappa }^{\prime }<\kappa$, then every CAT(${\kappa }^{\prime }$) space is a CAT($\kappa$) space. For more details regarding CAT($\kappa$) space, see [19].

A CAT($\kappa$) space X is said to be admissible if $d(x,y)<D_{\kappa }/2$ for all $x,y\in X$. Notice that every CAT($\kappa$) space is admissible if $\kappa \leqq 0$, since $D_{\kappa }=\infty$ for such a case. In a CAT($\kappa$) space, the following inequality, which is called a CN-inequality, is well known.

Lemma 1.

For $\kappa \in \{1,0,-1\}$, let X be a CAT$\left(\kappa \right)$ space and $x,y,z\in X$, satisfying $d(x,y)+d(y,z)+d(z,x)<2D_{\kappa }$. Then, the following inequalities hold:

• If $\kappa =1$, then

  $$\begin{array}{l}cosd(tx\oplus (1-t)y,z)sind(x,y)\\ \geqq cosd(x,z)sin\left(td\right(x,y\left)\right)+cosd(y,z)sin\left(\right(1-t\left)d\right(x,y\left)\right);\end{array}$$
• If $\kappa =0$, then

  $$d{(tx\oplus (1-t)y,z)}^2\leqq td{(x,z)}^2+(1-t)d{(y,z)}^2-t(1-t)d{(x,y)}^2;$$
• If $\kappa =-1$, then

  $$\begin{array}{c}coshd(tx\oplus (1-t)y,z)sinhd(x,y)\hfill \\ \leqq coshd(x,z)sinh\left(td\right(x,y\left)\right)+coshd(y,z)sinh\left(\right(1-t\left)d\right(x,y\left)\right).\hfill \end{array}$$

By using the properties of the functions cosine and hyperbolic cosine, we obtain the following corollaries.

Corollary 1.

For $\kappa \in \{1,0,-1\}$, let X be a CAT$\left(\kappa \right)$ space and $x,y,z\in X$, satisfying $d(x,y)+d(y,z)+d(z,x)<2D_{\kappa }$. Then, the following inequalities hold:

• If $\kappa =1$, then

  $$cos\left({\displaystyle \frac{1}{2}}x\oplus {\displaystyle \frac{1}{2}}y,z\right)cos(d(x,y)/2)\geqq {\displaystyle \frac{1}{2}}cosd(x,z)+{\displaystyle \frac{1}{2}}cosd(y,z);$$
• If $\kappa =0$, then

  $$d{\left({\displaystyle \frac{1}{2}}x\oplus {\displaystyle \frac{1}{2}}y,z\right)}^2\leqq {\displaystyle \frac{1}{2}}d{(x,z)}^2+{\displaystyle \frac{1}{2}}d{(y,z)}^2-{\displaystyle \frac{1}{4}}d{(x,y)}^2;$$
• If $\kappa =-1$, then

  $$coshd\left({\displaystyle \frac{1}{2}}x\oplus {\displaystyle \frac{1}{2}}y,z\right)cosh(d(x,y)/2)\leqq {\displaystyle \frac{1}{2}}coshd(x,z)+{\displaystyle \frac{1}{2}}coshd(y,z).$$

Corollary 2.

For $\kappa \in \{1,0,-1\}$, let X be an admissible CAT$\left(\kappa \right)$ space and $x,y,z\in X$. Then, the following inequalities hold:

• If $\kappa =1$, then

  $$cos\left(tx\oplus (1-t)y,z\right)\geqq tcosd(x,z)+(1-t)cosd(y,z);$$
• If $\kappa =0$, then

  $$d{\left(tx\oplus (1-t)y,z\right)}^2\leqq td{(x,z)}^2+(1-t)d{(y,z)}^2;$$
• If $\kappa =-1$, then

  $$coshd\left(tx\oplus (1-t)y,z\right)\leqq tcoshd(x,z)+(1-t)coshd(y,z).$$

Corollary 2 shows the concavity or the convexity of each function.

The following theorem is directly obtained by [20] [Theorem 4.1].

Theorem 1.

For $\kappa \in \{1,0,-1\}$, let X be an admissible CAT$\left(\kappa \right)$ space and f a convex function of X into $\left]-\infty ,\infty \right]$. Then, the following properties hold:

• On a CAT(1) space, if $Tp={argmin}_{u\in X}\{f\left(u\right)+\psi (1-cosd(u,p))\}$ is a singleton for each $p\in X$, then

  $$\begin{array}{cc}\hfill & ({\psi }^{\prime }\left(K_{1}x\right)cosd(Tx,x)+{\psi }^{\prime }\left(K_{1}y\right)cosd(Ty,y))cosd(Tx,Ty)\hfill \\ \hfill & \geqq {\psi }^{\prime }\left(K_{1}y\right)cosd(Tx,y)+{\psi }^{\prime }\left(K_{1}x\right)cosd(x,Ty)\hfill \end{array}$$

  for all $x,y\in X$, where $\psi :\left[0,1\right[\to \left[0,\infty \right[$ is an increasing function of class $C^1$ and $K_{1}z=1-cosd(Tz,z)$ for all $z\in X$;
• On a CAT(0) space, if $Tp={argmin}_{u\in X}\{f\left(u\right)+\psi (1/2d{(u,p)}^2)\}$ is a singleton for each $p\in X$, then

  $$\begin{array}{cc}\hfill & {\psi }^{\prime }\left(K_{0}x\right)d{(Tx,x)}^2+{\psi }^{\prime }\left(K_{0}y\right)d{(Ty,y)}^2+({\psi }^{\prime }\left(K_{0}x\right)+{\psi }^{\prime }\left(K_{0}y\right))d{(Tx,Ty)}^2\hfill \\ \hfill & \leqq {\psi }^{\prime }\left(K_{0}y\right)d{(Tx,y)}^2+{\psi }^{\prime }\left(K_{0}x\right)d{(x,Ty)}^2\hfill \end{array}$$

  for all $x,y\in X$, where $\psi :\left[0,\infty \right[\to \left[0,\infty \right[$ is an increasing function of class $C^1$ and $K_{0}z=1/2d{(Tz,z)}^2$ for all $z\in X$;
• On a CAT($-1$) space, if $Tp={argmin}_{u\in X}\{f\left(u\right)+\psi (coshd(u,p)-1)\}$ is a singleton for each $p\in X$, then

  $$\begin{array}{cc}\hfill & ({\psi }^{\prime }\left(K_{-1}x\right)coshd(Tx,x)+{\psi }^{\prime }\left(K_{-1}y\right)coshd(Ty,y))coshd(Tx,Ty)\hfill \\ \hfill & \leqq {\psi }^{\prime }\left(K_{-1}y\right)coshd(Tx,y)+{\psi }^{\prime }\left(K_{-1}x\right)coshd(x,Ty)\hfill \end{array}$$

  for all $x,y\in X$, where $\psi :\left[0,\infty \right[\to \left[0,\infty \right[$ is an increasing function of class $C^1$ and $K_{-1}z=coshd(Tz,z)-1$ for all $z\in X$.

3. A Resolvent on a CAT(1) Space

In this section, we show that the resolvent for a convex function on admissible CAT(1) spaces is well defined as a single valued mapping. Furthermore, we show its spherical nonspreadingness of sum type.

Lemma 2

([4]). Let X be an admissible complete CAT(1) space, f be a proper lower semicontinuous convex function of X into $\left]-\infty ,\infty \right]$, and p be an element of X. If $f\left(x_n\right)\to \infty$ whenever $\left\{x_n\right\}$ is a sequence of X with $d(p,x_n)\to \pi /2$, then f has a minimizer. Furthermore, if

$$f\left({\displaystyle \frac{1}{2}}x\oplus {\displaystyle \frac{1}{2}}y\right)<{\displaystyle \frac{1}{2}}f\left(x\right)+{\displaystyle \frac{1}{2}}f\left(y\right)$$

for $x,y\in domf$ with $x\ne y$, then the minimizer of f is unique.

Lemma 3

([5]). Let X be an admissible complete CAT(1) space, f be a proper lower semicontinuous convex function of X into $\left]-\infty ,\infty \right]$, and p be an element of X. Then, the function

$$g(·)=f(·)-log(cosd(·,p\left)\right)$$

satisfies the following properties:

• g is a proper lower semicontinuous convex function;
• $g\left(x_n\right)\to \infty$ whenever $\left\{x_n\right\}$ is a sequence of X with $d(p,x_n)\to \pi /2$;
• For $x,y\in domg$ with $x\ne y$,

  $$g\left({\displaystyle \frac{1}{2}}x\oplus {\displaystyle \frac{1}{2}}y\right)<{\displaystyle \frac{1}{2}}g\left(x\right)+{\displaystyle \frac{1}{2}}g\left(y\right).$$

Theorem 2.

Let X be an admissible complete CAT(1) space, f be a proper lower semicontinuous convex function of X into $\left]-\infty ,\infty \right]$, and p be an element of X. Then,

$$\underset{y\in X}{argmin}\{f\left(y\right)+1-cosd(y,p)-logcosd(y,p)\}$$

consists of one point.

Proof. 

By Corollary 2 and 3, we know that $f(·)+1-cosd(·,p)-logcosd(·,p)$ is a convex function. Moreover, it is also a proper lower semicontinuous function of $\left[0,\pi /2\right[\to \left]-\infty ,\infty \right]$. Since $t\to cost$ is bounded for $t\in \left[0,\pi /2\right[$, Lemma 3 implies that $(f\left(x_n\right)+1-cosd(x_n,p)-logcosd(x_n,p))\to \infty$ whenever $\left\{x_n\right\}$ is a sequence of X with $d(x_n,p)\to \infty$. Therefore, by Lemma 2, it has a minimizer.

We next show its uniqueness. Let $x,y\in domf$ with $x\ne y$. It follows from Corollary 1 and Lemma 3 that

$$\begin{array}{cc}\hfill & f\left({\displaystyle \frac{1}{2}}x\oplus {\displaystyle \frac{1}{2}}y\right)+1-cosd\left({\displaystyle \frac{1}{2}}x\oplus {\displaystyle \frac{1}{2}}y,p\right)-logcosd\left({\displaystyle \frac{1}{2}}x\oplus {\displaystyle \frac{1}{2}}y,p\right)\hfill \\ \hfill & <{\displaystyle \frac{1}{2}}(f\left(x\right)+1-cosd(x,p)-logcosd(x,p))\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}(f\left(y\right)+1-cosd(y,p)-logcosd(y,p)).\hfill \end{array}$$

By Lemma 2, we obtain the conclusion.  □

From the theorem above,

$$Q_{f}x=\underset{y\in X}{argmin}\{f\left(y\right)+1-cosd(y,x)-logcosd(y,x))\}$$

is well defined as a single-valued mapping of X into itself, where X is an admissible complete CAT(1) space, and f is a proper lower semicontinuous convex function of X into $\left]-\infty ,\infty \right]$. We define a resolvent of f on X by $Q_f:X\to X$.

Theorem 3.

Let X be an admissible complete CAT(1) space, f be a proper lower semicontinuous convex function of X into $\left]-\infty ,\infty \right]$, and $Q_f$ be a resolvent of f. Then, $Q_f$ is a spherically nonspreading of sum type, that is,

$$2cosd(Q_{f}x,Q_{f}y)\geqq cosd(Q_{f}x,y)+cosd(x,Q_{f}y)$$

for all $x,y\in X$.

Proof. 

Put $C_z=cosd(Q_{f}z,z)$ for all $z\in X$. By the definition of $Q_f$, we can express

$$Q_f\left(x\right)=\underset{y\in X}{argmin}\{f\left(y\right)+\psi (1-cosd(y,x)),\}$$

for $x\in X$, where $\psi \left(t\right)=t-log(1-t)$ for $t\in \left[0,1\right[$. Since $\psi :\left[0,1\right[\to \left[0,\infty \right[$ is an increasing function of class $C^1$, it follows from Theorem 1 that

$$\begin{array}{cc}\hfill & \left(\left(1+{\displaystyle \frac{1}{C_x}}\right)C_x+\left(1+{\displaystyle \frac{1}{C_y}}\right)C_y\right)cosd(Q_{f}x,Q_{f}y)\hfill \\ \hfill & \geqq \left(1+{\displaystyle \frac{1}{C_y}}\right)cosd(Q_{f}x,y)+\left(1+{\displaystyle \frac{1}{C_x}}\right)cosd(x,Q_{f}y)\hfill \end{array}$$

and hence,

$$\begin{array}{cc}\hfill & (C_x+C_y+2)cosd(Q_{f}x,Q_{f}y)\hfill \\ \hfill & \geqq \left(1+{\displaystyle \frac{1}{C_y}}\right)cosd(Q_{f}x,y)+\left(1+{\displaystyle \frac{1}{C_x}}\right)cosd(x,Q_{f}y)\hfill \end{array}$$

for all $x,y\in X$. Since $0<cost\leqq 1$ for $0\leqq t<\pi /2$, we obtain

$$2cosd(Q_{f}x,y)\geqq cosd(Q_{f}x,y)+cosd(x,Q_{f}y).$$

Thus we obtain the conclusion.  □

4. A Resolvent on a CAT($-1$) Space

In this section, we define a new resolvent for a convex function on CAT($-1$) spaces. We also show its hyperbolical nonspreadingness.

Lemma 4

([2]). Let X be a complete CAT(0) space and f be a proper lower semicontinuous convex function. Then, there exists a non-negative real number C which depends on only f such that

$$\underset{d(u,v)\to \infty }{lim\; inf}{\displaystyle \frac{f\left(u\right)}{d(u,v)}}\geqq -C$$

for all $v\in X$.

Lemma 5

([6]). Let X be a complete CAT(0) space and f be a proper lower semicontinuous convex function of X into $\left]-\infty ,\infty \right]$. Suppose that $f\left(x\right)\to \infty$ whenever $d(x,p)\to \infty$ for some $p\in X$. Then, ${argmin}_{X}f$ is nonempty. Furthermore, if

$$f\left({\displaystyle \frac{1}{2}}x\oplus {\displaystyle \frac{1}{2}}y\right)<{\displaystyle \frac{1}{2}}f\left(x\right)+{\displaystyle \frac{1}{2}}f\left(y\right)$$

holds for all $x,y\in X$ with $x\ne y$, then ${argmin}_{X}f$ consists of one point.

Theorem 4.

Let X be a complete CAT($-1$) space, f be a proper lower semicontinuous convex function of X into $\left]-\infty ,\infty \right]$, and p be an element of X. Then, the function

$$g(·)=f(·)+coshd(·,p)-1+logcoshd(·,p)$$

has a unique minimizer.

Proof. 

Let $x,y\in X$ and $t\in \left]0,1\right[$. By the convexity of $d(·,p)$ and $log(cosht)$, we have

$$\begin{array}{cc}\hfill log(coshd(tx\oplus (1-t)y,p\left)\right)& \leqq log(cosh(td(x,p)+(1-t)d(y,p)\left)\right)\hfill \\ \hfill & \leqq tlog(coshd(x,p\left)\right)+(1-t)log(coshd(y,p\left)\right).\hfill \end{array}$$

Therefore, by Corollary 2, g is convex. It is obvious that g is a proper lower semicontinuous function of X into $\left]-\infty ,\infty \right]$. On the other hand, by Lemma 4, we obtain

$$\begin{array}{cc}\hfill & \underset{d(x,p)\to \infty }{lim\; inf}{\displaystyle \frac{f\left(x\right)+coshd(x,p)-1+log(coshd(x,p\left)\right)}{d(x,p)}}\hfill \\ \hfill & \geqq \underset{d(x,p)\to \infty }{lim\; inf}{\displaystyle \frac{f\left(x\right)}{d(x,p)}}+\underset{d(x,p)\to \infty }{lim\; inf}{\displaystyle \frac{coshd(x,p)-1}{d(x,p)}}+\underset{d(x,p)\to \infty }{lim\; inf}{\displaystyle \frac{log(coshd(x,p\left)\right)}{d(x,p)}}\hfill \\ \hfill & \geqq -C+\underset{d(x,p)\to \infty }{lim\; inf}{\displaystyle \frac{coshd(x,p)-1}{d(x,p)}}\hfill \end{array}$$

and hence, $g\to \infty$ as $d(x,p)\to \infty$. From Lemma 5, we know that g has a minimizer. We next show its uniqueness. Let $x,y\in X$ with $x\ne y$. By the convexity of g and Corollary 1, we have

$$g\left({\displaystyle \frac{1}{2}}x\oplus {\displaystyle \frac{1}{2}}y\right)<{\displaystyle \frac{1}{2}}g\left(x\right)+{\displaystyle \frac{1}{2}}g\left(y\right).$$

Therefore, by Lemma 5, we obtain the conclusion.  □

From the theorem above,

$$R_{f}x=\underset{y\in X}{argmin}\{f\left(y\right)+coshd(y,x)-1+logcoshd(y,x)\}$$

is a single-valued mapping of a CAT($-1$) space X to X. We next show the hyperbolical nonspreadingness of $R_f$.

Theorem 5.

Let X be a complete CAT($-1$) space, f be a proper lower semicontinuous convex function of X into $\left]-\infty ,\infty \right]$, and $R_f$ be a resolvent of f. Then, $R_f$ is hyperbolically nonspreading, that is,

$$2coshd(R_{f}x,R_{f}y)\leqq coshd(R_{f}x,y)+coshd(x,R_{f}y)$$

for all $x,y\in X$.

Proof. 

Put $C_z=coshd(Tz,z)$ for all $z\in X$. Let $x,y\in X$ and $\psi \left(t\right)=t+log(1+t)$ for $t\in \left[0,\infty \right[$. Then, we can express $R_f$ by

$$R_f\left(u\right)=\underset{v\in X}{argmin}\{f\left(v\right)+\psi (coshd(v,u)-1)\}$$

for all $u\in X$. We also know that $\psi :\left[0,\infty \right[\to \left[0,\infty \right[$ is an increasing function of class $C^1$. By Theorem 1, we have

$$\begin{array}{cc}\hfill & \left(\left(1+{\displaystyle \frac{1}{C_x}}\right)C_x+\left(1+{\displaystyle \frac{1}{C_y}}\right)C_y\right)coshd(R_{f}x,R_{f}y)\hfill \\ \hfill & \leqq \left(1+{\displaystyle \frac{1}{C_y}}\right)coshd(R_{f}x,y)+\left(1+{\displaystyle \frac{1}{C_x}}\right)coshd(x,R_{f}y)\hfill \end{array}$$

and hence,

$$\begin{array}{cc}\hfill & (C_x+C_y+2)coshd(R_{f}x,R_{f}y)\hfill \\ \hfill & \leqq \left(1+{\displaystyle \frac{1}{C_y}}\right)coshd(R_{f}x,y)+\left(1+{\displaystyle \frac{1}{C_x}}\right)coshd(x,R_{f}y).\hfill \end{array}$$

Since $cosh\left(t\right)\geqq 1$ for $t\in \mathbb{R}$, we obtain

$$2coshd(Tx,Ty)\leqq coshd(Tx,y)+coshd(x,Ty).$$

This is the desired result.  □

5. A Resolvent on a CAT($\kappa$) Space

For $\kappa >0$, it is known that $(X,d)$ is an admissible CAT($\kappa$) space if and only if $(X,\sqrt{\kappa }d)$ is an admissible CAT(1) space. Similarly, for $\kappa <0$, it is also known that $(X,d)$ is a CAT($\kappa$) space if and only if $(X,\sqrt{-\kappa }d)$ is a CAT($-1$) space.

In this section, we unify the three kinds of resolvents $J_f$, $Q_f$, and $R_f$ by using the function ${\phi }_{\kappa }$ of $\left[0,D_{\kappa }/2\right[$ into $\left[0,\infty \right[$, which is defined as follows:

$$\begin{array}{cc}\hfill {\phi }_{\kappa }\left(t\right)& =t^2+{\displaystyle \frac{\kappa }{24}}{t}^4+{\displaystyle \frac{17{\kappa }^2}{720}}{t}^6+{\displaystyle \frac{271{\kappa }^3}{40320}}{t}^8+{\displaystyle \frac{7937{\kappa }^4}{3628800}}{t}^{10}+\cdots \hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{1}{\kappa }}(1-cos\left(\sqrt{\kappa }t\right)-log(cos\left(\sqrt{\kappa }t\right)))\hfill & (\kappa >0);\hfill \\ t^2\hfill & (\kappa =0);\hfill \\ {\displaystyle \frac{1}{-\kappa }}(cosh\left(\sqrt{-\kappa }t\right)-1+log(cosh\left(\sqrt{-\kappa }t\right)))\hfill & (\kappa <0).\hfill \end{array}\right.\hfill \end{array}$$

We know that the function ${\phi }_{\kappa }$ is increasing, strictly convex, continuous, ${\phi }_{\kappa }\left(0\right)=0$, and $\phi \to \infty$ as $t\uparrow D_{\kappa }/2$ for each $\kappa \in \mathbb{R}$. These conditions play an important role to define a resolvent for a convex function.

Using the results in this paper, the following resolvent

$$S_{f}x=\underset{y\in X}{argmin}\{f\left(y\right)+{\phi }_{\kappa }\left(d(y,x)\right)\}$$

is well defined as a single-valued mapping for each $x\in X$, where X is an admissible CAT($\kappa$) space for $\kappa \in \mathbb{R}$. It is easy to show $S_{i_C}=P_C$, where $i_C$ is an indicator function for some nonempty closed convex subset C of X. In fact, we have

$$\begin{array}{cc}\hfill S_{i_C}x& =\underset{y\in C}{argmin}\{i_C\left(y\right)+{\phi }_{\kappa }\left(d(y,x)\right)\}\hfill \\ \hfill & =\underset{y\in C}{argmin}\left\{{\phi }_{\kappa }\left(d(y,x)\right)\right\}\hfill \\ \hfill & =\underset{y\in C}{argmin}\left\{d(y,x)\right\}=P_{C}x\hfill \end{array}$$

for all $x\in X$. We also know that $\mathcal{F}\left(S_f\right)={argmin}_{X}f$; for more details, see [4].

We also use the function $c_{\kappa }:\left[0,D_{\kappa }/2\right[\to \left[0,\infty \right[$ defined by

$$\begin{array}{cc}\hfill c_{\kappa }\left(t\right)& ={\displaystyle \frac{t^2}{2}}-{\displaystyle \frac{\kappa t^4}{24}}+{\displaystyle \frac{{\kappa }^2{t}^6}{720}}+\cdots \hfill \\ \hfill & =\sum _{n=1}^{\infty }{\displaystyle \frac{{(-1)}^{n-1}{\kappa }^{n-1}{t}^{2n}}{\left(2n\right)!}}\hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{1}{\kappa }}\left(1-cos\left(\sqrt{\kappa }t\right)\right)\hfill & (\kappa >0);\hfill \\ {\displaystyle \frac{1}{2}}{t}^2\hfill & (\kappa =0);\hfill \\ {\displaystyle \frac{1}{-\kappa }}\left(cosh\left(\sqrt{-\kappa }t\right)-1\right)\hfill & (\kappa <0),\hfill \end{array}\right.\hfill \end{array}$$

for all $t\in \left[0,D_{\kappa }/2\right[$. It is a convex and increasing function, with $c_{\kappa }\left(0\right)=0$. For more details, see [20]. Using this function, we define $\kappa$-nonspreadingness of a mapping of a metric space X into itself.

Let X be a metric space with $d(x,y)<D_{\kappa }/2$ for $x,y\in X$. A mapping T of X into itself is said to be $\kappa$-nonspreading if

$$2c_{\kappa }\left(d(Tx,Ty)\right)\leqq c_{\kappa }\left(d(Tx,y)\right)+c_{\kappa }\left(d(x,Ty)\right)$$

for all $x,y\in X$. Then 0-nonspreadingness, 1-nonspreadingness, and ($-1$)-nonspreadingness naturally coincide with metrical nonspreadingness [3], spherical nonspreading of sum type [5], and hyperbolical nonspreadingness [6], respectively. We directly obtain the following theorem.

Theorem 6.

Let κ be a real number, $(X,d)$ be an admissible CAT($\kappa$) space with the metric d, f be a proper lower semicontinuous convex function of X into $\left]-\infty ,\infty \right]$, and $S_f$ be the resolvent of f. Then, $S_f$ is κ-nonspreading, that is,

$$2c_{\kappa }\left(d(S_{f}x,S_{f}y)\right)\leqq c_{\kappa }\left(d(S_{f}x,y)\right)+c_{\kappa }\left(d(x,S_{f}y)\right)$$

for all $x,y\in X$.

Proof. 

If $\kappa =0$, it obviously holds. We first show the case of $\kappa >0$. Let $x,y\in X$, and put $d^{\prime }(x,y)=\sqrt{\kappa }d(x,y)$. Then, we know that $(X,d^{\prime })$ is an admissible CAT(1) space. Thus, we obtain

$$2cos{d}^{\prime }(S_{f}x,S_{f}y)\geqq cos{d}^{\prime }(S_{f}x,y)+cos{d}^{\prime }(x,S_{f}y)$$

and hence,

$${\displaystyle \frac{2}{\kappa }}(1-cos\sqrt{\kappa }d(S_{f}x,S_{f}y))\leqq {\displaystyle \frac{1}{\kappa }}(1-cos\sqrt{\kappa }d(S_{f}x,y))+{\displaystyle \frac{1}{\kappa }}(1-cos\sqrt{\kappa }d(x,S_{f}y)).$$

This is the desired result.

For $\kappa <0$, let $x,y\in X$, and put $d^{\prime }(x,y)=\sqrt{-\kappa }d(x,y)$. Since $(X,d^{\prime })$ is a CAT($-1$) space, we have

$$2cosh{d}^{\prime }(S_{f}x,S_{f}y)\leqq cosh{d}^{\prime }(S_{f}x,y)+cosh{d}^{\prime }(x,S_{f}y)$$

and hence,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{2}{-\kappa }}(cosh\sqrt{-\kappa }d(S_{f}x,S_{f}y)-1)\hfill \\ \hfill & \leqq {\displaystyle \frac{1}{-\kappa }}(cosh\sqrt{-\kappa }d(S_{f}x,y)-1)+{\displaystyle \frac{1}{-\kappa }}(cosh\sqrt{-\kappa }d(x,S_{f}y)-1).\hfill \end{array}$$

Thus, we obtain the conclusion.  □

6. Comparison with Known Resolvents

In the previous section, we unified two definitions of the resolvent operators by using the function ${\phi }_{\kappa }$ defined by the power series including $\kappa$. Namely, we proposed the resolvent $S_0$ defined by

$$S_0\left(x\right)=S_f\left(x\right)=\underset{y\in X}{argmin}\{f\left(y\right)+{\phi }_{\kappa }\left(d(y,x)\right)\}.$$

Using the same technique, we can unify several known resolvents as follows:

$$\begin{array}{cc}\hfill S_1\left(x\right)& =\left\{\begin{array}{cc}{\displaystyle \underset{y\in X}{argmin}\left\{f\left(y\right)+{\displaystyle \frac{1}{\kappa }}tan\left(\sqrt{\kappa }d(y,x)\right)sin\left(\sqrt{\kappa }d(y,x)\right)\right\}}\hfill & (\kappa >0);\hfill \\ {\displaystyle \underset{y\in X}{argmin}\left\{f\left(y\right)+{\displaystyle \frac{1}{2}}d{(y,x)}^2\right\}}\hfill & (\kappa =0);\hfill \\ {\displaystyle \underset{y\in X}{argmin}\left\{f\left(y\right)+{\displaystyle \frac{1}{-\kappa }}tanh\left(\sqrt{-\kappa }d(y,x)\right)sinh\left(\sqrt{-\kappa }d(y,x)\right)\right\}}\hfill & (\kappa <0);\hfill \end{array}\right.\hfill \\ \hfill S_2\left(x\right)& =\left\{\begin{array}{cc}{\displaystyle \underset{y\in X}{argmin}\left\{f\left(y\right)-{\displaystyle \frac{1}{\kappa }}log(cos\sqrt{\kappa }d(y,x))\right\}}\hfill & (\kappa >0);\hfill \\ {\displaystyle \underset{y\in X}{argmin}\left\{f\left(y\right)+{\displaystyle \frac{1}{2}}d{(y,x)}^2\right\}}\hfill & (\kappa =0);\hfill \\ {\displaystyle \underset{y\in X}{argmin}\left\{f\left(y\right)+{\displaystyle \frac{1}{-\kappa }}log(cosh\sqrt{-\kappa }d(y,x))\right\}}\hfill & (\kappa <0);\hfill \end{array}\right.\hfill \\ \hfill S_3\left(x\right)& =\left\{\begin{array}{cc}{\displaystyle \underset{y\in X}{argmin}\left\{f\left(y\right)+{\displaystyle \frac{1}{\kappa }}(1-cos\sqrt{\kappa }d(y,x))\right\}}\hfill & (\kappa >0);\hfill \\ {\displaystyle \underset{y\in X}{argmin}\left\{f\left(y\right)+{\displaystyle \frac{1}{2}}d{(y,x)}^2\right\}}\hfill & (\kappa =0);\hfill \\ {\displaystyle \underset{y\in X}{argmin}\left\{f\left(y\right)+{\displaystyle \frac{1}{-\kappa }}(cosh\sqrt{-\kappa }d(y,x)-1)\right\}}\hfill & (\kappa <0).\hfill \end{array}\right.\hfill \end{array}$$

We first discuss the well definedness of these resolvents. As mentioned in the previous section, for any $\kappa \in \mathbb{R}$, the newly proposed resolvent $S_0$ is well defined as a single-valued mapping for every proper lower semicontinuous convex function f. The operator $S_1$ is also well defined for any $\kappa \in \mathbb{R}$ and f. On the other hand, $S_2$ is not well defined for some f if $\kappa$ is negative. Indeed, we have the following example.

Example 1.

Let $X=\mathbb{R}$ with the metric defined by $d(x,y)=\left|x-y\right|$ for $x,y\in \mathbb{R}$. Then, we know that X is an admissible complete CAT$\left(\kappa \right)$ space for any $\kappa \le 0$. Fix $\kappa <0$ and define $f:X\to \left]-\infty ,\infty \right]$ by

$$f\left(x\right)=-{\displaystyle \frac{2}{\sqrt{-\kappa }}}x$$

for $x\in X$. Then, since

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{\displaystyle \frac{log(cosh\sqrt{-\kappa }t)}{t}}=\sqrt{-\kappa },\end{array}$$

we have

$$\begin{array}{cc}\hfill & \underset{y\to \infty }{lim}\left\{f\left(y\right)+{\displaystyle \frac{1}{-\kappa }}log(cosh\sqrt{-\kappa }d(y,0))\right\}\hfill \\ \hfill & =\underset{y\to \infty }{lim}\left\{-{\displaystyle \frac{2}{\sqrt{-\kappa }}}y+{\displaystyle \frac{\sqrt{-\kappa }}{-\kappa }}y\right\}\hfill \\ \hfill & =\left\{-{\displaystyle \frac{2}{\sqrt{-\kappa }}}+{\displaystyle \frac{1}{\sqrt{-\kappa }}}\right\}\underset{y\to \infty }{lim}y=-{\displaystyle \frac{1}{\sqrt{-\kappa }}}\underset{y\to \infty }{lim}y=-\infty .\hfill \end{array}$$

This fact implies that $S_2\left(0\right)$ is empty, and therefore, $S_2$ is not well defined as a single-valued mapping.

We do not know whether the operator $S_3$ is well defined or not in general. Indeed, if $\kappa =1$, the perturbation function $g=1-cosd(·,x)$ is not coercive, that is, the assumption $d(x,p)\to \pi /2$ for some $p\in X$ does not imply $g\left(x\right)\to \infty$. Therefore, we cannot apply Lemma 2 for this function, and thus, the well definedness of $S_3$ is unknown.

Next, we discuss the properties of the resolvent operators. Theorem 6 shows the $\kappa$-nonspreadingness of $S_0$ for any $\kappa \in \mathbb{R}$. Moreover, we can see that $S_2$ and $S_3$ also have this property from the facts shown in the Introduction. However, we only know that the resolvent $S_1$ is $\kappa$-nonspreading for $\kappa \le 0$. If $\kappa >0$, then $S_1$ is a spherically nonspreading of product type in the sense that

$${cos}^2\left(\sqrt{\kappa }d(S_{1}x,S_{1}y)\right)\ge cos\left(\sqrt{\kappa }d(S_{1}x,y)\right)cos\left(\sqrt{\kappa }d(x,S_{1}y)\right)$$

for any $x,y\in X$, and we do not know whether this property implies the $\kappa$-nonspreadingness or not.

The facts described above are summarized in

Table 1. Properties of resolvents.

Operator	Well Definedness			$\mathit{\kappa }$-Nonspreading
	$\mathit{\kappa }>\mathbf{0}$	$\mathit{\kappa }=\mathbf{0}$	$\mathit{\kappa }<\mathbf{0}$	$\mathit{\kappa }>\mathbf{0}$	$\mathit{\kappa }=\mathbf{0}$	$\mathit{\kappa }<\mathbf{0}$
$S_0$	✓	✓	✓	✓	✓	✓
$S_1$	✓	✓	✓		✓	✓
$S_2$	✓	✓		✓	✓	✓
$S_3$		✓	✓	✓	✓	✓

. From these observations, we conclude that the proposed resolvent operator $S_0$ has sufficiently good properties compared to the others.