Sufficient conditions for perturbations to define the resolvent of the equilibrium problem on complete geodesic spaces

Abstract

In this paper, we consider sufficient conditions for perturbations to define a resolvent of the equilibrium problem on complete geodesic spaces with curvature bounded above.

1 Introduction

Equilibrium problem on a metric space X is defined by Blum and Oettli as to find $z \in K$ such that $f(z, y) \ge 0$ for all $y \in K$, where K is a closed subset of X and f is a real function on $K^2$. They studied the equilibrium problem on Banach spaces in [1]. Since then, equilibrium problems have been studied as it relates to several problems such as fixed point problems, convex minimization problems, Nash equilibria, and so on. For instance, to find $z \in K$ such that $g(y) - g(z) \ge 0$ for all $y \in K$, is a convex minimization problem for a convex real function g.

Let X be a Hilbert space and K a nonempty closed convex subset of X. In 2005, Combettes and Hirstoaga [4] showed a set-valued mapping ${R_{f}} :X \rightarrow 2^K$ named a resolvent defined by

$$\begin{aligned} {R_{f}} x = \left\{ z \in K \mathrel {}\bigg |\mathrel {} \inf _{y \in K} \left( f(z, y) + \langle z - x, y - z \rangle \right) \ge 0 \right\} \end{aligned}$$

for $x \in X$ can be defined as a single-valued mapping for each real function $f :K^2 \rightarrow \mathbb {R}$ satisfying conditions (E1)–(E4) defined later. The set ${R_{f}} x$ is identical to ${{\,\textrm{Equil}\,}}h_x$, which is the set of all solution $z \in K$ of the equilibrium problem for $h_x$ defined by $h_x(z, y) = f(z, y) + \langle z - x, y - z \rangle$ for $z, y \in K$.

In 2018, Kimura and Kishi [9] proposed a resolvent operator of equilibrium problems defined by a solution of an equilibrium problem on CAT(0) spaces. A CAT(0) space (X, d) is one of the geodesic spaces having a generally nonlinear structure, and the class of CAT(0) spaces includes Hilbert spaces. Its resolvent is used a square of the metric $d^2$ as a perturbation function. In fact, they define a resolvent operator ${R_{f}} :K^2 \rightarrow \mathbb {R}$ by

$$\begin{aligned} {R_{f}} x = {{\,\textrm{Equil}\,}}h_x = \left\{ z \in K \mathrel {}\bigg |\mathrel {} \inf _{y \in K} h_x(z, y) \ge 0 \right\} \end{aligned}$$

for $x \in X$, a nonempty closed convex subset K of X, a function f satisfying (E1)–(E4), and a function $h_x :K^2 \rightarrow \mathbb {R}$ defined by $h_x(z, y) = f(z, y) + \frac{1}{2} d(x, y)^2 - \frac{1}{2} d(x, z)^2$ for $z, y \in K$.

In 2021, Kimura [8] showed that a resolvent of an equilibrium problem can be defined by using $-\log \cos d$ as the perturbation function on CAT(1) spaces. That is, they defined a resolvent ${R_{f}}$ by ${R_{f}} x = {{\,\textrm{Equil}\,}}h_x$, where $h_x$ is defined by a formula $h_x(z, y) = f(z, y) - \log \cos d(x, y) + \log \cos d(x, z)$.

Recently, in 2022, Kimura and Ogihara [12] show that a new type of perturbation function $\cosh d$ can be used in defining the resolvent on a CAT($-1$) space. This resolvent is superior to other resolvents in that its properties are given by simpler inequalities than the resolvent using the perturbation $-\log \cos d$.

As shown above, we can define different types of resolvent operators for an equilibrium problem. Each operator has independent properties, which may cause some differences when we consider approximation schemes of the solutions, such as assumptions for convergence, the convergence rate of the procedure, and others.

This paper finds sufficient conditions for perturbation functions to define the resolvent of an equilibrium problem on CAT($\kappa$) spaces. Our results describe the properties of resolvent operators with different perturbation functions in a unified way. These results will make it possible to apply to resolvent operators with perturbation functions which will be newly proposed in the future.

2 Preliminaries

Let X be a metric space and $x, y \in X$. A mapping $\gamma :[0, 1] \rightarrow X$ is called a geodesic joining x and y if $\gamma (0) = y$, $\gamma (1) = x$, and $d(\gamma (s), \gamma (t)) = |s - t| d(x, y)$ hold for any $s, t \in [0, 1]$. X is called a uniquely D-geodesic space if a geodesic joining x and y always exists uniquely for two points $x, y \in X$, such that $d(x, y) < D$.

Let X be a uniquely D-geodesic space and let $x, y \in X$. For a unique geodesic $\gamma$ joining x and y, put $t x \oplus (1 - t) y = \gamma (t)$ for each $t \in [0, 1]$. The point $t x \oplus (1 - t) y$ is called a convex combination of x and y. In what follows, put $[x, y] = [y, x] = \{ t x \oplus (1 - t) y \mid {x, y \in X, t \in [0, 1]}{} \}$. A subset $C \subset X$ is said to be convex if $t x \oplus (1 - t) y \in C$ holds for any $x, y \in C$ and $t \in \mathopen {]}0, 1 \mathclose {[}$.

Let $M_\kappa$ be a 2-dimensional model space with a with a metric $d'$ and a constant curvature $\kappa \in \mathbb {R}$ defined by

$$\begin{aligned} M_\kappa = {\left\{ \begin{array}{ll} \frac{1}{\sqrt{\kappa }} \, \mathbb {S}^2 &{} (\text {if }\kappa > 0), \\ \mathbb {R}^2 &{} (\text {if }\kappa = 0), \\ \frac{1}{\sqrt{-\kappa }} \, \mathbb {H}^2 &{} (\text {if }\kappa < 0). \end{array}\right. } \end{aligned}$$

Let $D_\kappa$ be a diameter of $M_\kappa$, which is equal to

$$\begin{aligned} D_\kappa = {\left\{ \begin{array}{ll} \infty &{} (\text{ if } \kappa \le 0), \\ \frac{\pi }{\sqrt{\kappa }} &{} (\text{ if } \kappa > 0). \end{array}\right. } \end{aligned}$$

For $\kappa \in \mathbb {R}$, let X be a uniquely $D_\kappa$-geodesic space. We define a geodesic triangle $\triangle (x, y, z)$ by $[x, y] \cup [y, z] \cup [z, x]$ for any $x, y, z \in X$ such that $d(x, y) + d(y, z) + d(z, x) < 2 D_\kappa$. We call three points x, y, z vertices of $\triangle (x, y, z)$. For each $\triangle (x, y, z)$, there exists three points $\overline{x}, \overline{y}, \overline{z} \in M_\kappa$ such that $d(x, y) = d'(\overline{x}, \overline{y})$, $d(y, z) = d'(\overline{y}, \overline{z})$, $d(z, x) = d'(\overline{z}, \overline{x})$. For these points $\overline{x}, \overline{y}, \overline{z}$, the triangle $\overline{\triangle }(\overline{x}, \overline{y}, \overline{z})$ defined by $[\overline{x}, \overline{y}] \cup [\overline{y}, \overline{z}] \cup [\overline{z}, \overline{x}]$ is called the comparison triangle. For a point $p \in \triangle (x, y, z)$, there exists a point $\overline{p} \in \overline{\triangle }(\overline{x}, \overline{y}, \overline{z})$ corresponding to p, such that the distances from two adjacent vertices are identical. That point $\overline{p}$ is called a comparison point of p.

Let $\kappa \in \mathbb {R}$. A uniquely $D_\kappa$-geodesic space X is called a CAT($\kappa$) space if, for any $\triangle \mathrel {{:}{=}} \triangle (x, y, z)$ and its comparison triangle $\overline{\triangle } \mathrel {{:}{=}} \overline{\triangle }(\overline{x}, \overline{y}, \overline{z})$, and for any two points $p, q \in \triangle$ and these comparison points $\overline{p}, \overline{q} \in \overline{\triangle }$, the inequality $d(p, q) \le d'(\overline{p}, \overline{q})$ holds. A CAT($\kappa$) space X is said to be admissible if $d(x, y) < D_\kappa / 2$ holds for every $x, y \in X$. X is always admissible if $\kappa \le 0$, since $D_\kappa = \infty$ for any $\kappa \le 0$. Notice that every CAT($\kappa$) space is also a CAT($\kappa '$) space if $\kappa < \kappa '$. Furthermore, the class of the CAT(0) space encompasses Hilbert spaces, and thus, the CAT(0) space is a generalization of Hilbert spaces in a different way from Banach spaces. For more details, see [2] and [3].

Let X be a CAT($\kappa$) space. For a subset C of X, we define a convex hull ${{\,\textrm{co}\,}}C$ by ${{\,\textrm{co}\,}}C = \bigcup _{n = 1}^\infty C_n$, where $C_1 = C$ and $C_{n + 1} = \{ t u \oplus (1 - t) v \mid {u, v \in C_n, t \in [0, 1]}{} \}$ for $n \in \mathbb {N}$.

Let X be a metric space. A sequence $\{ x_n \} \subset X$ is said to $\Delta$-converge to $x_0$ if $\limsup _{i \rightarrow \infty } d(x_{n_i}, x_0) < \limsup _{i \rightarrow \infty } d(x_{n_i}, u)$ for any subsequence $\{ x_{n_i} \}$ of $\{ x_n \}$ and for any $u \in X {\setminus } \{ x_0 \}$. We write the $\Delta$-convergence of the sequence $\{ x_n \}$ to $x_0$ by $x_n {\mathop {\rightharpoonup }\limits ^{\Delta }} x_0$. We know that every convergent sequence to $x_0$ $\Delta$-converges to $x_0$. A subset $C \subset X$ is said to be $\Delta$-compact if every sequence $\{ x_n \} \subset C$ has a $\Delta$-convergent subsequence to a point in C. A subset $C \subset X$ is said to be $\Delta$-closed if a $\Delta$-limit of every $\Delta$-convergent sequence always belongs to C.

Lemma 2.1

(Kirk and Panyanak [13]). Let X be a complete CAT(0) space and M a closed convex subset of X. Then, M is $\Delta$-compact.

Lemma 2.2

(Kirk and Panyanak [13]). Let X be a complete CAT(0) space and M a bounded closed convex subset of X. Then, M is $\Delta$-closed.

Let X be a metric space and T a mapping from X into itself. In what follows, we name the following condition ($\Delta *$):

• ($\Delta *$) if a sequence $\{ x_n \} \subset X$ and $x_0 \in X$ satisfies

  $$\begin{aligned} \limsup _{n \rightarrow \infty } d(x_n, x_0) < \limsup _{n \rightarrow \infty } d(x_n, u) \end{aligned}$$

  for any $u \in X {\setminus } \{ x_0 \}$ and $\lim _{n \rightarrow \infty } d(T x_n, x_n) = 0$, then $x_0 \in F(T)$.

It is a weaker condition than $\Delta$-demiclosedness of T.

Let X be a CAT(0) space and f a function from X into $\mathbb {R}$. We say f is coercive if $f(y) \rightarrow \infty$ whenever $d(x, y) \rightarrow \infty$ for some $x \in X$. A function $f :X \rightarrow \mathbb {R}$ is said to be convex if for any $x, y \in X$ and $t \in \mathopen {]}0, 1 \mathclose {[}$, $f(t x \oplus (1 - t) y) \le t f(x) + (1 - t) f(y)$ holds. We call f a upper hemicontinuous function if $\limsup _{t \downarrow 0} f(t x \oplus (1 - t) y) \le f(y)$ holds for any $x, y \in X$.

Let X be a metric space and T be a mapping from X into itself. The point $p \in X$ is called a fixed point of T if $p = T p$ holds, and F(T) denotes a set of all fixed points of T. T is said to be quasinonexpansive if $F(T) \ne \emptyset$ and $d(T x, p) \le d(x, p)$ holds for any $x \in X$ and $p \in F(T)$. T is said to be asymptotically regular if $\lim _{n \rightarrow \infty } d(T^{n + 1} x, T^n x) = 0$ for every $x \in X$. Furthermore, T is said to be $\Delta$-demiclosed if a $\Delta$-limit of any $\Delta$-convergent sequences $\{ x_n \} \subset X$ with $\lim _{n \rightarrow \infty } d(x_n, Tx_n) = 0$ belongs to F(T).

Lemma 2.3

(Mayer [15]). Let X be a complete CAT(0) space and f a lower semicontinuous convex function from X into $\mathbb {R}$. Then, there exists $L \in \mathopen {]}-\infty , 0 ]$ such that

$$\begin{aligned} \liminf _{\begin{array}{c} d(u, v) \rightarrow \infty \\ v :\text {fixed} \end{array}} \frac{f(u)}{d(u, v)} \ge L \end{aligned}$$

for any $v \in X$.

Lemma 2.4

(Kimura and Kohsaka [15]). For $\kappa > 0$, let X be an admissible complete CAT($\kappa$) space and f a lower semicontinuous convex function from X into $\mathbb {R}$. Then, f is bounded below.

For $\kappa \in \mathbb {R}$, define a function $c_\kappa :\mathopen {]}-\infty , \infty ]\rightarrow \mathbb {R}$ by

$$\begin{aligned} c_\kappa (d) = {\left\{ \begin{array}{ll} \frac{1}{-\kappa } (\cosh (\sqrt{-\kappa }\, d) - 1) &{} \text {(if }\kappa < 0\text {)}, \\ \frac{1}{2} d^2 &{} \text {(if }\kappa = 0\text {)}, \\ \frac{1}{\kappa } (1 - \cos (\sqrt{\kappa }\, d)) &{} \text {(if }\kappa > 0\text {)} \end{array}\right. } \end{aligned}$$

for $d \in \mathbb {R}$ and $c_\kappa (\infty ) = \infty$. Then, the function $c_\kappa$ is infinitely differentiable on $\mathbb {R}$, and we get

$$\begin{aligned} c'_\kappa (d) = {\left\{ \begin{array}{ll} \frac{1}{\sqrt{-\kappa }} \sinh (\sqrt{-\kappa }\, d) &{} \text {(if }\kappa < 0\text {)}, \\ d &{} \text {(if }\kappa = 0\text {)}, \\ \frac{1}{\sqrt{\kappa }} \sin (\sqrt{\kappa }\, d) &{} \text {(if }\kappa > 0\text {)} \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} c''_\kappa (d) = 1 - \kappa c_\kappa (d) = {\left\{ \begin{array}{ll} \displaystyle \cosh (\sqrt{-\kappa }\, d) &{} \text {(if }\kappa < 0\text {)}, \\ \displaystyle 1 &{} \text {(if }\kappa = 0\text {)}, \\ \displaystyle \cos (\sqrt{\kappa }\, d) &{} \text {(if }\kappa > 0\text {)} \end{array}\right. } \end{aligned}$$

for $d \in \mathbb {R}$. From these equations, we can get the following for $\kappa \in \mathbb {R}$:

• $c_\kappa (0) = 0$, and $c_\kappa (d) > 0$ for any $d \in \mathopen {]}0, D_\kappa / 2 \mathclose {[}$,

• $c'_\kappa (0) = 0$, and $c'_\kappa (d) > 0$ for any $d \in \mathopen {]}0, D_\kappa \mathclose {[}$,

• $c''_\kappa (0) = 1$, and $c''_\kappa (d) > 0$ for any $d \in \mathopen {]}0, D_\kappa / 2 \mathclose {[}$,

• $c'_\kappa$ is an odd function, and $c''_\kappa$ is an even function,

• $c_\kappa$ is convex on $[0, D_\kappa / 2]$,

• $c''_\kappa (d) + \kappa c_\kappa (d) = 1$ for any $d \in \mathbb {R}$,

• $c''_\kappa (d)^2 + \kappa c'_\kappa (d)^2 = 1$ for any $d \in \mathbb {R}$.

The value of $c_\kappa (d)$ can be expressed by $c_\kappa (d) = \sum _{n = 1}^\infty (-1)^{n - 1} \kappa ^{n - 1} d^{2 n} / (2 n)!$ when $0^0 {:}{=}1$.

Lemma 2.5

For any $\kappa \in \mathbb {R}$ and $d_1, d_2 \in \mathbb {R}$

$$\begin{aligned} c_\kappa (d_1) + c''_\kappa (d_1) c_\kappa (d_2) = c_\kappa (d_2) + c''_\kappa (d_2) c_\kappa (d_1) \end{aligned}$$

holds. In addition, these are equal to $\dfrac{1 - c''_\kappa (d_1) c''_\kappa (d_2)}{\kappa }$ if $\kappa \ne 0$.

Proof

For any $\kappa \in \mathbb {R}$ and $d_1, d_2 \in \mathbb {R}$, we get

$$\begin{aligned} c_\kappa (d_1) + c''_\kappa (d_1) c_\kappa (d_2)&= c_\kappa (d_1) + (1 - \kappa c_\kappa (d_1)) c_\kappa (d_2) \\&= c_\kappa (d_2) + (1 - \kappa c_\kappa (d_2)) c_\kappa (d_1) \\&= c_\kappa (d_2) + c''_\kappa (d_2) c_\kappa (d_1). \end{aligned}$$

Moreover, if $\kappa \ne 0$, then

$$\begin{aligned} c_\kappa (d_1) + c''_\kappa (d_1) c_\kappa (d_2) = \frac{1 - c''_\kappa (d_1)}{\kappa } + \frac{c''_\kappa (d_1) \left( 1 - c''_\kappa (d_2) \right) }{\kappa } = \frac{1 - c''_\kappa (d_1) c''_\kappa (d_2)}{\kappa } \end{aligned}$$

holds; therefore, we get the conclusion. $\square$

For each $t \in [0, 1]$, $d \in \mathopen {]}0, D_\kappa / 2 \mathclose {[}$ and $\kappa \in \mathbb {R}$, put

$$\begin{aligned} (t)_{d}^{\kappa } = \frac{c'_\kappa (t d)}{c'_\kappa (d)}. \end{aligned}$$

This notation was first proposed in [11].

Let X be a CAT(0) space. Then, the following inequality holds for any $x, y, z \in X$ with $d(x, y) + d(y, z) + d(z, x) < 2 D_\kappa$ and $t \in [0, 1]$:

$$\begin{aligned} d(t x \oplus (1 - t) y, z) \le t d(x, z)^2 + (1 - t)^2 d(y, z)^2 - t (1 - t) d(x, y)^2. \end{aligned}$$

This inequality is called a parallelogram law on the CAT(0) space. If X is a Hilbert space, the above inequality holds as an equation.

In general CAT($\kappa$) space X, Kimura and Sudo [11] showed that parallelogram law on the CAT($\kappa$) space is expressed by the following form:

$$\begin{aligned}{} & {} c_\kappa (d(t x \oplus (1 - t) y, z)) \le (t)_{d(x, y)}^{\kappa } (c_\kappa (d(x, z)) - c_\kappa ((1 - t) d(x, y))) \\{} & {} \quad + (1 - t)_{d(x, y)}^{\kappa } (c_\kappa (d(y, z)) - c_\kappa (t d(x, y))), \end{aligned}$$

where $x, y, z \in X$, $d(x, y) + d(y, z) + d(z, x) < 2 D_\kappa$, $x \ne y$, and $t \in [0, 1]$.

Let X be a CAT(0) space. In 2019, Kohsaka [14] proposed the concept of metrically nonspreading mappings and firmly metrically nonspreading mappings. In [14], a metrically nonspreading mapping is simply said to nonspreading mapping, but the word “metrically” is often added to distinguish it from other metrically nonspreading-like concepts. A mapping $T :X \rightarrow X$ is said to be metrically nonspreading if

$$\begin{aligned} 2 d(T x, T y)^2 \le d(x, T y)^2 + d(y, T x)^2 \end{aligned}$$

holds for any $x, y \in X$. Moreover, a mapping $T :X \rightarrow X$ is said to be firmly metrically nonspreading if

$$\begin{aligned} 2 d(T x, T y)^2 \le d(x, T y)^2 - d(x, T x)^2 + d(y, T x)^2 - d(y, T y)^2 \end{aligned}$$

holds for any $x, y \in X$. It is clear that every firmly metrically nonspreading mapping is metrically nonspreading.

Recently, the notion of nonspreading mapping has been proposed for CAT($\kappa$) spaces for $\kappa \ne 0$.

Let X be an admissible CAT(1) space. A mapping $T :X \rightarrow X$ is said to be spherically nonspreading of product type [10] if

$$\begin{aligned} \cos ^2 d(T x, T y) \ge \cos d(x, T y) \cos d(y, T x) \end{aligned}$$

holds for any $x, y \in X$. In addition, a mapping $T :X \rightarrow X$ is said to be spherically nonspreading of sum type [7] if

$$\begin{aligned} 2 \cos d(T x, T y) \ge \cos d(x, T y) + \cos d(y, T x) \end{aligned}$$

holds for any $x, y \in X$. Furthermore, a mapping $T :X \rightarrow X$ is said to be firmly spherically nonspreading [10] if

$$\begin{aligned} \cos ^2 d(T x, T y) \ge \frac{2}{\cos d(x, T x) + \cos d(y, T y)} \cos d(x, T y) \cos d(y, T x) \end{aligned}$$

holds for any $x, y \in X$. From these definitions, we evidently obtain the following relationship:

• If T is spherically nonspreading of sum type, then T is spherically nonspreading of product type;

• if T is firmly spherically nonspreading, then T is spherically nonspreading of product type.

Let X be a CAT($-1$) space. A mapping $T :X \rightarrow X$ is said to be hyperbolically nonspreading [5] if

$$\begin{aligned} 2 \cosh d(T x, T y) \le \cosh d(x, T y) + \cosh d(y, T x) \end{aligned}$$

holds for any $x, y \in X$.

In 2019, Kajimura and Kimura [6] proposed the notion of vicinal mappings with $\psi$.

Let X be an admissible CAT($\kappa$) space and take $\psi :[0, D_\kappa / 2 \mathclose {[}\rightarrow [0, \infty \mathclose {[}$ such that it is right continuous at 0. A mapping $T :X \rightarrow X$ is said to be vicinal with $\psi$ [6] if

$$\begin{aligned}{} & {} \bigl ( \psi (d(x, T x)) + \psi (d(y, T y)) \bigr ) c_\kappa (d(T x, T y)) \\{} & {} \quad \le \psi (d(x, T x)) c_\kappa (d(x, T y)) + \psi (d(y, T y)) c_\kappa (d(y, T x)) \end{aligned}$$

holds for any $x, y \in X$. A mapping $T :X \rightarrow X$ is said to be firmly vicinal with $\psi$ [6] if

$$\begin{aligned}{} & {} \bigl ( \psi (d(x, T x)) c_\kappa (d(x, T x)) + \psi (d(y, T y)) c_\kappa (d(y, T y)) \bigr ) c''_\kappa (d(T x, T y)) \\{} & {} \qquad + \bigl ( \psi (d(x, T x)) + \psi (d(y, T y)) \bigr ) c_\kappa (d(T x, T y)) \\{} & {} \quad \le \psi (d(x, T x)) c_\kappa (d(x, T y)) + \psi (d(y, T y)) c_\kappa (d(y, T x)) \end{aligned}$$

holds for any $x, y \in X$. We can obtain easily that every firmly vicinal mapping with $\psi$ is vicinal with $\psi$.

The notion of vicinality with $\psi$ plays a role in unifying expression of several nonspreading mappings. Note that there are the following relations between nonspreadingness and vicinality with $\psi$: for a CAT$(\kappa$) space X and a mapping $T :X \rightarrow X$:

• T is metrically nonspreading if and only if T is vicinal with any positive constant function for $\kappa = 0$;

• T is spherically nonspreading of sum type if and only if T is vicinal with any positive constant function for $\kappa = 1$;

• T is hyperbolically nonspreading if and only if T is vicinal with any positive constant function for $\kappa = -1$;

• T is firmly metrically nonspreading if and only if T is firmly vicinal with any positive constant function for $\kappa = 0$;

• T is firmly hyperbolically nonspreading if and only if T is firmly vicinal with any positive constant function for $\kappa = -1$.

Lemma 2.6

(Kajimura and Kimura [6]). Let X be an admissible CAT($\kappa$) space. Suppose that $T :X \rightarrow X$ is vicinal with some $\psi$. Then, T satisfies condition $(\Delta *)$, and hence, T is $\Delta$-demiclosed. Furthermore, if F(T) is nonempty, then T is quasinonexpansive.

Lemma 2.7

(Kajimura and Kimura [6]). Let X be an admissible CAT($\kappa$) space. Suppose that $T :X \rightarrow X$ is firmly vicinal with some $\psi$. If F(T) is nonempty, then T is asymptotically regular.

Lemma 2.8

Let X be an admissible CAT($\kappa$) space and let $\psi :[0, \infty \mathclose {[}\rightarrow [0, \infty \mathclose {[}$, such that $\psi$ is right continuous at 0. Then, for a mapping $T :X \rightarrow X$, the following are equivalent:

• (i) T is firmly vicinal with $\psi$;

• (ii) for any $x, y \in X$

  $$\begin{aligned}&\left( \psi (d(x, T x)) c''_\kappa (d(x, T x)) + \psi (d(y, T y)) c''_\kappa (d(y, T y)) \right) c_\kappa (d(T x, T y)) \\ \qquad&\le \psi (d(x, T x)) \left( c_\kappa (d(x, T y)) - c_\kappa (d(x, T x)) \right) \\ \qquad \quad&{} + \psi (d(y, T y)) \left( c_\kappa (d(y, T x)) - c_\kappa (d(y, T y)) \right) ; \end{aligned}$$

• (iii) for any $x, y \in X$

  $$\begin{aligned}&\frac{1}{\kappa } \left( \psi (d(x, T x)) c''_\kappa (d(x, T x)) + \psi (d(y, T y)) c''_\kappa (d(y, T y)) \right) c''_\kappa (d(T x, T y)) \\ \qquad&\ge \frac{1}{\kappa } \left( \psi (d(x, T x)) c''_\kappa (d(x, T y)) + \psi (d(y, T y)) c''_\kappa (d(y, T x)) \right) , \end{aligned}$$

where (iii) is considered only when $\kappa \ne 0$.

Proof

Using Lemma 2.5, we get

$$\begin{aligned}&T\text { is firmly vicinal with }\psi \\&\quad \iff \psi (d(x, T x)) \left( c_\kappa (d(x, T x)) c''_\kappa (d(T x, T y)) + c_\kappa (d(T x, T y)) \right) \\&\quad \qquad {} + \psi (d(y, T y)) \left( c_\kappa (d(y, T y)) c''_\kappa (d(T x, T y)) + c_\kappa (d(T x, T y)) \right) \\&\qquad {} \le \psi (d(x, T x)) c_\kappa (d(x, T y)) + \psi (d(y, T y)) c_\kappa (d(y, T x)) \\&\quad \iff \psi (d(x, T x)) \left( c_\kappa (d(T x, T y)) c''_\kappa (d(x, T x)) + c_\kappa (d(x, T x)) \right) \\&\quad \qquad {} + \psi (d(y, T y)) \left( c_\kappa (d(T x, T y)) c''_\kappa (d(y, T y)) + c_\kappa (d(y, T y)) \right) \\&\qquad {} \le \psi (d(x, T x)) c_\kappa (d(x, T y)) + \psi (d(y, T y)) c_\kappa (d(y, T x)) \\&\quad \iff \left( \psi (d(x, T x)) c''_\kappa (d(x, T x)) + \psi (d(y, T y)) c''_\kappa (d(y, T y)) \right) c_\kappa (d(T x, T y)) \\&\qquad {} \le \psi (d(x, T x)) \left( c_\kappa (d(x, T y)) - c_\kappa (d(x, T x)) \right) \\&\quad \qquad {} + \psi (d(y, T y)) \left( c_\kappa (d(y, T x)) - c_\kappa (d(y, T y)) \right) \end{aligned}$$

for $x, y \in X$ and thus (i) and (ii) are equivalent. In addition, if $\kappa \ne 0$, then (ii) is equivalent to

$$\begin{aligned}{} & {} \left( \psi (d(x, T x)) c''_\kappa (d(x, T x)) + \psi (d(y, T y)) c''_\kappa (d(y, T y)) \right) \cdot \frac{1 - c''_\kappa (d(T x, T y))}{\kappa } \\{} & {} \quad \le \psi (d(x, T x)) \left( \frac{1 - c''_\kappa (d(x, T y))}{\kappa } - \frac{1 - c''_\kappa (d(x, T x))}{\kappa } \right) \\{} & {} \qquad + \psi (d(y, T y)) \left( \frac{1 - c''_\kappa (d(y, T x))}{\kappa } - \frac{1 - c''_\kappa (d(y, T y))}{\kappa } \right) , \end{aligned}$$

and so is

$$\begin{aligned}{} & {} \frac{1}{\kappa } \left( \psi (d(x, T x)) c''_\kappa (d(x, T x)) + \psi (d(y, T y)) c''_\kappa (d(y, T y)) \right) c''_\kappa (d(T x, T y)) \\{} & {} \qquad \ge \frac{1}{\kappa } \left( \psi (d(x, T x)) c''_\kappa (d(x, T y)) + \psi (d(y, T y)) c''_\kappa (d(y, T x)) \right) \end{aligned}$$

for $x, y \in X$. Hence, (ii) and (iii) are equivalent if $\kappa \ne 0$. $\square$

Let X be a CAT($\kappa$) space and K a nonempty closed convex subset of X. Then, for any $x \in X$, there exists a unique point $p_x$, such that $p_x \in K$ and $d(x, p_x) = \inf _{y \in K} d(x, y)$. Thus, we can define a mapping $P_K$ from X onto K by $x \longmapsto p_x$, and this is called a metric projection onto K.

Lemma 2.9

Let X be an admissible complete CAT($\kappa$) space and K a nonempty closed convex subset of X. Then

$$\begin{aligned} c_\kappa (d(x, P_K x)) c''_\kappa (d(P_K x, z)) \le c_\kappa (d(x, z)) - c_\kappa (d(P_K x, z)) \end{aligned}$$

(1)

and

$$\begin{aligned} c_\kappa (d(P_K x, z)) c''_\kappa (d(x, P_K x)) \le c_\kappa (d(x, z)) - c_\kappa (d(x, P_K x)) \end{aligned}$$

(2)

holds for any $x \in X$ and $z \in K$.

Proof

Put $u = P_K x$. Then, we get

$$\begin{aligned} c_\kappa (d(x, u))&\le c_\kappa (d(x, t z \oplus (1 - t) u)) \\&\le (t)_{d(u, z)}^{\kappa } (c_\kappa (d(x, z)) - c_\kappa ((1 - t) d(u, z))) \\&\quad {} + (1 - t)_{d(u, z)}^{\kappa } (c_\kappa (d(x, u)) - c_\kappa (t d(u, z))) \end{aligned}$$

for any $t \in \mathopen {]}0, 1 \mathclose {[}$, and hence

$$\begin{aligned}{} & {} \frac{1 - (1 - t)_{d(u, z)}^{\kappa }}{(t)_{d(u, z)}^{\kappa }} c_\kappa (d(x, u)) \\{} & {} \quad \le c_\kappa (d(x, z)) - c_\kappa ((1 - t) d(u, z)) - \frac{c_\kappa (t d(u, z))}{(t)_{d(u, z)}^{\kappa }} (1 - t)_{d(u, z)}^{\kappa } \end{aligned}$$

holds for any $t \in \mathopen {]}0, 1 \mathclose {[}$. Letting $t \rightarrow 0$, we obtain the inequality (1). Using the inequality (1) and Lemma 2.5, we also get (2). $\square$

Corollary 2.10

Let X be an admissible complete CAT($\kappa$) space and K a nonempty closed convex subset of X. Then, $d(P_K x, z) \le d(x, z)$ holds for any $x \in X$ and $z \in K$. Therefore, the metric projection $P_K$ is quasinonexpansive.

Proof

From Lemma 2.9 (1), we get $0 \le c_\kappa (d(x, z)) - c_\kappa (d(P_K x, z))$, and hence, we get the conclusion. $\square$

Let X be an admissible CAT($\kappa$) space and K a nonempty closed convex subset of X. For a bifunction $f :K^2 \rightarrow \mathbb {R}$, we name the following conditions (E1)–(E4):

• (E1) $f(z, z) = 0$ for all $z \in K$,

• (E2) $f(z, y) + f(y, z) \le 0$ for all $z, y \in K$,

• (E3) $f(z, \cdot ) :K \rightarrow \mathbb {R}$ is lower semicontinuous and convex for every $z\,{\in }\,K$,

• (E4) $f(\cdot , y) :K \rightarrow \mathbb {R}$ is upper hemicontinuous for every $y \in K$.

A point $z \in K$ is called a solution of the equilibrium problem for f if $f(z, y) \ge 0$ holds for any $y \in K$, or equivalently, $\inf _{y \in K} f(z, y) \ge 0$. For a bifunction f, ${{\,\textrm{Equil}\,}}f$ denotes the set of all solutions to the equilibrium problem. Namely

$$\begin{aligned} {{\,\textrm{Equil}\,}}f = \left\{ z \in K \mathrel {}\bigg |\mathrel {} \inf _{y \in K} f(z, y) \ge 0 \right\} . \end{aligned}$$

Henceforth, we consider only cases that f satisfies (E1)–(E4) for the equilibrium problem for f.

3 Main result

In 2018, Kimura and Kishi [9] proposed the resolvent $Q_f :X \rightarrow 2^K$ of the equilibrium problem for $f :K^2 \rightarrow \mathbb {R}$ defined by

$$\begin{aligned} Q_f x = \left\{ z \in K \mathrel {}\bigg |\mathrel {} \inf _{y \in K} \left( f(z, y) + \frac{1}{2} d(x, y)^2 - \frac{1}{2} d(x, z)^2 \right) \ge 0 \right\} \end{aligned}$$

on a complete CAT(0) space X and $K \subset X$. $Q_f$ can be defined as a single-valued mapping under the appropriate conditions of X, K, and f. Actually, they assume that X has the convex hull finite property (explained later), K is a nonempty closed convex subset of X, and f satisfies (E1)–(E4). The above definition can be expressed by

$$\begin{aligned} Q_f x = \left\{ z \in K \mathrel {}\bigg |\mathrel {} \inf _{y \in K} \left( f(z, y) + \varphi (c_0 (d(x, y))) - \varphi (c_0 (d(x, z))) \right) \ge 0 \right\} , \end{aligned}$$

where $\varphi (t) \,{:}{=}\, t$ for $t \in [0, \infty \mathclose {[}$.

In 2021, Kimura [8] shows that the resolvent $R_f$ defined by the following form becomes a single-valued mapping under the appropriate conditions on an admissible complete CAT(1) space:

$$\begin{aligned} {R_{f}} x = \left\{ z \in K \mathrel {}\bigg |\mathrel {} \inf _{y \in K} \left( f(z, y) - \log \cos d(x, y) + \log \cos d(x, z) \right) \ge 0 \right\} . \end{aligned}$$

The above equation can be rewritten as

$$\begin{aligned} {R_{f}} x = \left\{ z \in K \mathrel {}\bigg |\mathrel {} \inf _{y \in K} \left( f(z, y) + \varphi (c_1 (d(x, y))) - \varphi (c_1 (d(x, z))) \right) \ge 0 \right\} , \end{aligned}$$

where $\varphi (t) \mathrel {{:}{=}} -\log \left( 1 - t \right)$ for $t \in [0, 1 \mathclose {[}$.

Similarly, the resolvent $S_f$ on a complete CAT($-1$) space produced by Kimura and Ogihara [12] defined by the formula

$$\begin{aligned} {S_{f}} x = \left\{ z \in K \mathrel {}\bigg |\mathrel {} \inf _{y \in K} \left( f(z, y) + \cosh d(x, y) - \cosh d(x, z) \right) \ge 0 \right\} \end{aligned}$$

can be expressed by

$$\begin{aligned} {S_{f}} x = \left\{ z \in K \mathrel {}\bigg |\mathrel {} \inf _{y \in K} \left( f(z, y) + \varphi (c_{-1} (d(x, y))) - \varphi (c_{-1} (d(x, z))) \right) \ge 0 \right\} , \end{aligned}$$

where $\varphi (t) \mathrel {{:}{=}} t$ for $t \in [0, \infty \mathclose {[}$.

Therefore, in general CAT($\kappa$) spaces, we expect a resolvent defined using a perturbation $\varphi (c_\kappa (d))$ to be a single-valued mapping with the appropriate conditions. In this section, we consider the sufficient conditions for the function $\varphi$ to define the resolvent as a single-valued mapping on an admissible CAT($\kappa$) space.

The convex hull finite property is the property defined by [17] for CAT(0) spaces originally. In this paper, we define the convex hull finite property by the condition: for any finite subset E of X and every continuous mapping T from ${{\,\textrm{cl}\,}}{{\,\textrm{co}\,}}E$ into itself, T has a fixed point.

In what follows, put $[n] = \{ 1, 2, \dotsc , n \}$ for each $n \in \mathbb {N}$.

Lemma 3.1

Let X be a uniquely geodesic space and $E = \{ y_1, y_2, \dotsc , y_n \}$ a subset of X. For a nonempty set A, let h be a bifunction from $A \times X$ into $\mathbb {R}$. Suppose that the function $h(z, \cdot ) :X \rightarrow \mathbb {R}$ is convex for any $z \in A$. Then, for any $v \in {{\,\textrm{co}\,}}E$, there exists $\{ \mu _1, \mu _2, \dotsc , \mu _n \} \subset [0, 1]$ such that $\sum _{i = 1}^n \mu _i = 1$ and $h(z, v) \le \sum _{i = 1}^n \mu _i h(z, y_i)$ for any $z \in A$.

Proof

Put $F_1 = E$ and $F_{j + 1} = \{ t u \oplus (1 - t) u' \mid u, u' \in X,\ t \in [0, 1] \}$ for $j \in \mathbb {N}$. Then, ${{\,\textrm{co}\,}}E$ coincides with $\bigcup _{j \in \mathbb {N}} F_j$. Thus, we need to show that the existence of such $\{ \mu _1, \mu _2, \dotsc , \mu _n \} \subset [0, 1]$ for any $j \in \mathbb {N}$ and $v \in F_j$. We show it by induction for $j \in \mathbb {N}$.

Suppose $j = 1$ and let $v \in F_j = E$. Then, there exists $i_n \in [n]$ such that $v = y_{i_0}$. Thus, putting $\mu _{i_0} = 1$ and $\mu _i = 0$ for $i \in [n] {\setminus } \{ i_0 \}$, we get $h(z, v) = \sum _{i = 1}^n h(z, y_i)$ for any $z \in A$.

Next, assume that is true for some $j \in \mathbb {N}$. Let $v \in F_{j + 1}$. Then, there exists $t \in [0, 1]$ and $u, u' \in F_j$ such that $v = t u \oplus (1 - t) u'$. Hence, from the assumption, there exists $\{ \mu _1, \mu _2, \dotsc , \mu _n \} \subset [0, 1]$ such that $\sum _{i = 1}^n \mu _i = 1$ and $h(z, u) \le \sum _{i = 1}^n \mu _i h(z, y_i)$ for any $z \in A$. Similarly, we have the existence of $\{ \mu '_1, \mu '_2, \dotsc , \mu '_n \} \subset [0, 1]$ such that $\sum _{i = 1}^n \mu '_i = 1$ and $h(z, u') \le \sum _{i = 1}^n \mu '_i h(z, y_i)$ for any $z \in A$. Let $z \in A$ arbitrarily. Then

$$\begin{aligned} h(z, v) \le t h(z, u) + (1 - t) h(z, u') \le \sum _{i = 1}^n \left( t \mu _i + (1 - t) \mu '_i \right) h(z, y_i) \end{aligned}$$

and $\sum _{i = 1}^n \left( t \mu _i + (1 - t) \mu '_i \right) = 1$ hold, and thus, we get the conclusion. $\square$

Lemma 3.2

(Kimura [8]). For $\kappa > 0$, let X be an admissible complete CAT($\kappa$) space having the convex hull finite property, and C a nonempty subset of X. Suppose that a mapping $M :C \rightarrow 2^X$ satisfies that M(y) is closed for any $y \in X$. If ${{\,\textrm{cl}\,}}{{\,\textrm{co}\,}}E \subset \bigcup _{y \in E} M(y)$ holds for any finite subset E of X, then $\{ M(y) \mid y \in C \}$ has the finite intersection property.

Lemma 3.3

(Niculescu and Rovenţa [16]). Let X be a complete CAT(0) space having the convex hull finite property and C a nonempty subset of X. Suppose that a mapping $M :C \rightarrow 2^X$ satisfies that M(y) is nonempty closed convex for any $y \in X$. If ${{\,\textrm{cl}\,}}{{\,\textrm{co}\,}}E \subset \bigcup _{y \in E} M(y)$ holds for any finite subset E of X, then $\{ M(y) \mid y \in C \}$ has the finite intersection property.

Lemma 3.4

(Kimura [8]). For $\kappa > 0$, Let X be an admissible complete CAT($\kappa$) space and C a nonempty closed convex subset of X satisfying that $\inf _{y \in C} \sup _{x \in C} d(x, y) < D_\kappa / 2$. Let $\mathcal {M}$ be a family of closed convex subsets of X and suppose that $\mathcal {M}$ has the finite intersection property. Then, $\bigcap \mathcal {M} \ne \emptyset$ holds.

Lemma 3.5

(Kimura and Kishi [9]). Let X be a complete CAT(0) space and C a $\Delta$-compact subset of X. Let $\mathcal {M}$ be a family of $\Delta$-closed subsets of X and suppose that $\mathcal {M}$ has the finite intersection property. Then, $\bigcap \mathcal {M} \ne \emptyset$ holds.

Lemma 3.6

Let X be an admissible complete CAT($\kappa$) space. Suppose that X has the convex hull finite property. Let K be a nonempty closed convex subset of X and suppose that the function $h :K^2 \rightarrow \mathbb {R}$ satisfies (E1), (E2) and (E3). Let C be a nonempty closed convex subset of K and define a set M(y) by $M(y) = \{ z \in C \mid h(y, z) \le 0 \}$ for each $y \in C$. Then, the following properties hold:

• (i) M(y) is nonempty closed convex subset of C for any $y \in C$;

• (ii) the set $\{ M(y) \mid y \in C \}$ has the finite intersection property;

• (iii) if $\inf _{y \in C} \sup _{x \in C} d(x, y) < D_\kappa / 2$, then $\bigcap _{y \in C} M(y) \ne \emptyset$.

Proof

(i) Let $y \in C$. Since h satisfies (E1), we get $y \in M(y)$ and hence M(y) is nonempty. Therefore, since h satisfies (E3), we obtain M(y) is closed and convex.

(ii) Let $E = \{ y_1, y_2, \dotsc , y_n \} \subset C$. We show ${{\,\textrm{co}\,}}E \subset \bigcup _{i = 1}^n M(y_i)$. Assume that it is false, and let $v \in {{\,\textrm{co}\,}}E \setminus \bigcup _{i = 1}^n M(y_i)$. Then, we get $h(y_i, v) > 0$ for any $i \in [n]$. Since $v \in {{\,\textrm{co}\,}}E$, by Lemma 3.1, there exists $\{ \mu _1, \mu _2, \dotsc , \mu _n \} \subset [0, 1]$, such that $\sum _{i = 1}^n \mu _i = 1$ and $h(y_k, v) \le \sum _{i = 1}^n \mu _i h(y_k, y_i)$ for any $k \in [n]$. Thus, we obtain

$$\begin{aligned}{} & {} 0 < \sum _{k = 1}^n \mu _k h(y_k, v) \le \sum _{k = 1}^n \sum _{i = 1}^n \mu _k \mu _i h(y_k, y_i) \\{} & {} \quad = \frac{1}{2} \sum _{k = 1}^n \sum _{i = 1}^n \mu _k \mu _i (h(y_k, y_i) + h(y_i, y_k)) \le 0. \end{aligned}$$

It is a contradiction. Hence, ${{\,\textrm{co}\,}}E \subset \bigcup _{i = 1}^n M(y_i)$ is true. It implies that

$$\begin{aligned} {{\,\textrm{cl}\,}}{{\,\textrm{co}\,}}E \subset {{\,\textrm{cl}\,}}\bigcup _{i = 1}^n M(y_i) = \bigcup _{i = 1}^n M(y_i), \end{aligned}$$

and thus, we get $\{ M(y) \mid y \in C \}$ has the finite intersection property by Lemma 3.2 or Lemma 3.3.

(iii) First, we consider the case of $\kappa > 0$. By the result of (ii), $\{ M(y) \mid y \in C \}$ has the finite intersection property. Hence, we get $\bigcap _{y \in C} M(y) \ne \emptyset$ by using Lemma 3.4. We consider the case of $\kappa \le 0$. Suppose $\inf _{y \in C} \sup _{x \in C} d(x, y) < D_\kappa / 2 = \infty$. It means that C is bounded, and hence, C is $\Delta$-compact by Lemma 2.1. Furthermore, M(y) is $\Delta$-closed for any $y \in C$, since $M(y) \subset C$ from Lemma 2.2. Thus, from Lemma 3.5 and (ii), we have $\bigcap _{y \in C} M(y) \ne \emptyset$. $\square$

Lemma 3.7

Let X be an admissible complete CAT($\kappa$) space. Suppose that X has the convex hull finite property. Let K be a nonempty closed convex subset of X and h a real function on $K^2$ with conditions (E1)–(E4). Take $u \in K$, $R \in \mathopen {]}0, D_\kappa / 2 \mathclose {[}$ and put $C = \{ z \in K \mid d(u, z) \le R \}$. Suppose that $h(z, u) \le 0$ for any $z \in C$ with $d(u, z) = R$, and assume that $\inf _{y \in C} \sup _{x \in C} d(x, y) < D_\kappa / 2$. Then, there exists $z_0 \in C$ such that $\inf _{y \in K} h(z_0, y) \ge 0$.

Proof

Put $M(y) = \{ z \in C \mid h(y, z) \le 0 \}$ for any $y \in C$. C is a nonempty closed convex subset of K. Therefore, from Lemma 3.6, we obtain $\bigcap _{y \in C} M(y) \ne \emptyset$.

Let $z_0 \in \bigcap _{y \in C} M(y)$. Then, we get $h(y, z_0) \le 0$ for any $y \in C$, and $d(u, z_0) \le R$. Let $w \in C$ and $t \in \mathopen {]}0, 1 \mathclose {[}$ arbitrarily. Then, $t w \oplus (1 - t) z_0 \in C$, and hence

$$\begin{aligned} 0 = h(t w \oplus (1 - t) z_0, t w \oplus (1 - t) z_0) \le t h(t w \oplus (1 - t) z_0, w) \end{aligned}$$

holds using the condition (E3). It implies that $h(t w \oplus (1 - t) z_0, w) \ge 0$. Since h satisfies the condition (E4), we obtain

$$\begin{aligned} h(z_0, w) \ge \limsup _{t \rightarrow 0} h(t w \oplus (1 - t) z_0, w) \ge 0. \end{aligned}$$

Therefore, $h(z_0, w) \ge 0$ holds for any $w \in C$.

We show that $h(z_0, y) \ge 0$ holds for each $y \in K$. Let $y \in K$ and put

$$\begin{aligned} u_0 = {\left\{ \begin{array}{ll} u &{} (\text {if }d(u, z_0) = R), \\ z_0 &{} (\text {if }d(u, z_0) < R). \end{array}\right. } \end{aligned}$$

Then, we have $d(u, u_0) < R$. In fact, if $d(u, z_0) = R$, then we get $d(u, u_0) = d(u, u) = 0$. On the other hand, if $d(u, z_0) < R$, then we have $d(u, u_0) = d(u, z_0) < R$.

Since $d(u, u_0) < R$, we can take a sufficiently small $t_0 \in \mathopen {]}0, 1 \mathclose {[}$ satisfying $t_0 c_\kappa (d(u, y)) + (1 - t_0) c_\kappa (d(u, u_0)) < c_\kappa (R)$. Then, we get

$$\begin{aligned} c_\kappa (d(u, t_0 y \oplus (1 - t_0) u_0)) \le t_0 c_\kappa (d(u, y)) + (1 - t_0) c_\kappa (d(u, u_0)) < c_\kappa (R), \end{aligned}$$

and thus, $d(u, t_0 y \oplus (1 - t_0) u_0) < R$. Since K is convex, we get $t_0 y \oplus (1 - t_0) u_0 \in K$. Hence, by the definition of C, we obtain $t_0 y \oplus (1 - t_0) u_0 \in C$. Therefore

$$\begin{aligned} 0 \le h(z_0, t_0 y \oplus (1 - t_0) u_0) \le t_0 h(z_0, y) + (1 - t_0) h(z_0, u_0). \end{aligned}$$

Incidentally, we also have $h(z_0, u_0) \le 0$, and hence, $h(z_0, y) \ge 0$. Indeed, if $d(u, z_0) = R$, then $h(z_0, u_0) = h(z_0, u) \le 0$ holds by the assumption, and if $d(u, z_0) < R$, then $h(z_0, u_0) = h(z_0, z_0) = 0$. Thus, we get the conclusion. $\square$

Remark 3.8

In the assumptions of Lemma 3.7, there need not exist $z \in C$, such that $d(u, z) = R$.

Theorem 3.9

Let X be an admissible complete CAT($\kappa$) space. Suppose that X has the convex hull finite property. Let K be a nonempty closed convex subset of X and let $f :K^2 \rightarrow \mathbb {R}$ with conditions (E1)–(E4). Let $\Phi :[0, D_\kappa / 2 \mathclose {[}\rightarrow [0, \infty \mathclose {[}$ be a continuous convex function satisfying $\lim _{d \rightarrow D_\kappa / 2} \Phi d / d = \infty$, where $\Phi d \mathrel {{:}{=}} \Phi (d)$. Assume that $\Phi$ is strictly increasing only when $\kappa > 0$. Define a function $h :K^2 \rightarrow \mathbb {R}$ by

$$\begin{aligned} h(z, y) = f(z, y) + \Phi d(x, y) - \Phi d(x, z) \end{aligned}$$

for any $(z, y) \in K^2$. Then, for any $x \in X$, there exists $z_0 \in K$ such that $\inf _{y \in K} h(z_0, y) \ge 0$.

Proof

Since we assume that $\Phi$ is continuous and convex, we have h satisfies conditions (E1)–(E4).

First, we consider the case where $\kappa \le 0$. Note that we only need to take into account the case where $\kappa = 0$. Then, $\lim _{d \rightarrow \infty } \Phi d / d = \infty$ holds.

Suppose that K is unbounded, and let $u \in K$. Then, we have

$$\begin{aligned} h(z, u)&= f(z, u) + \Phi d(x, u) - \Phi d(x, z) \\&\le -f(u, z) + \Phi d(x, u) - \Phi d(x, z) \\&= \Phi d(x, u) - (f(u, z) + \Phi d(x, z)) \end{aligned}$$

for any $z \in K$. By Lemma 2.3, there exists $L \in \mathopen {]}-\infty , 0 ]$, such that

$$\begin{aligned} \liminf _{\begin{array}{c} d(u, z) \rightarrow \infty \\ z \in K \end{array}} \frac{f(u, z)}{d(u, z)} \ge L. \end{aligned}$$

It yields that

and hence, $f(u, z) + \Phi d(x, z) \rightarrow \infty$ when $d(u, z) \rightarrow \infty$. It means that $h(z, u) \rightarrow -\infty$ if $d(u, z) \rightarrow \infty$, and thus, we can take $R > 0$ such that $h(z, u) \le 0$ for any $z \in K$ with $d(u, z) = R$.

On the other hand, if K is bounded, then we can take $R > 0$ such that $d(u, z) < R$ for any $z \in K$.

Put $C = \{ z \in K \mid d(u, z) \le R \}$. Then, since $u \in C$, we have

Therefore, from Lemma 3.7, there exists $z_0 \in C$ such that $\inf _{y \in K} h(z_0, y) \ge 0$. Thus, we get the conclusion if $\kappa \le 0$.

In the following, we consider the case where $\kappa > 0$. By the assumptions for $\Phi$, we can assume that $\Phi$ is bijective onto $[k, \infty \mathclose {[}$ for $k \mathrel {{:}{=}} \Phi 0$. Thus, there exists the inverse $\Phi ^{-1} :[k, \infty \mathclose {[}\rightarrow [0, D_\kappa / 2 \mathclose {[}$ of $\Phi$.

Let $u = P_K x$ and put $L = \inf _{y \in K} f(u, y) - \Phi d(x, u)$. Since f satisfies the condition (E3), we obtain that $f(u, \cdot )$ is bounded below by Lemma 2.4. Hence, we have

$$\begin{aligned} -\infty < L \le f(u, u) - \Phi d(x, u) = -\Phi d(x, u) \le -k. \end{aligned}$$

If $L = -k$, then we obtain

$$\begin{aligned} h(u, y)&= f(u, y) + \Phi d(x, y) - \Phi d(x, u) \ge -k + \Phi d(x, y) \ge 0 \end{aligned}$$

for all $y \in K$, and thus, we get the conclusion.

Suppose $L < -k$. Using Corollary 2.10, we get $d(u, z) \le d(x, z)$ and it implies $\Phi d(u, z) \le \Phi d(x, z)$. Thus, we have

$$\begin{aligned} h(z, u)&= f(z, u) + \Phi d(x, u) - \Phi d(x, z) \\&\le -f(u, z) + \Phi d(x, u) - \Phi d(u, z) \\&\le -L - \Phi d(u, z) \end{aligned}$$

for any $z \in K$. Put $R = \Phi ^{-1} (-L)$. Then, we obtain

$$\begin{aligned} -L - \Phi d(u, z) = 0 \iff d(u, z) = R \end{aligned}$$

for any $z \in K$. It implies that $h(z, u) \le 0$ holds for any $z \in K$ with $d(u, z) = R$.

Put $C = \{ z \in K \mid d(u, z) \le R \}$ and $M(y) = \{ z \in C \mid h(y, z) \le 0 \}$ for any $y \in C$. Then

Therefore, from Lemma 3.7, there exists $z_0 \in C$ such that $\inf _{y \in K} h(z_0, y) \ge 0$. Thus, we get the conclusion. $\square$

Remark 3.10

In the case where $\kappa \le 0$ in Theorem 3.9, the assumption $\lim _{d \rightarrow D_\kappa / 2} \Phi d / d = \infty$ is unnecessarily strong. The conclusion of Theorem 3.9 can be obtained even if we change the assumption $\lim _{d \rightarrow D_\kappa / 2} \Phi d / d = \lim _{d \rightarrow \infty } \Phi d / d = \infty$ to

$$\begin{aligned} \liminf _{\begin{array}{c} d(v, z) \rightarrow \infty \\ z \in K \end{array}} \frac{f(v, z)}{d(v, z)} + \liminf _{d \rightarrow \infty } \frac{\Phi d}{d} > 0 \quad (\forall v \in K). \end{aligned}$$

In fact, the above inequality implies that

$$\begin{aligned} \liminf _{\begin{array}{c} d(u, z) \rightarrow \infty \\ z \in K \end{array}} \frac{f(u, z) + \Phi d(x, z)}{d(u, z)}&\ge \liminf _{\begin{array}{c} d(u, z) \rightarrow \infty \\ z \in K \end{array}} \frac{f(u, z)}{d(u, z)} + \liminf _{\begin{array}{c} d(u, z) \rightarrow \infty \\ z \in K \end{array}}\frac{\Phi d(x, z)}{d(u, z)} \\&= \liminf _{\begin{array}{c} d(u, z) \rightarrow \infty \\ z \in K \end{array}} \frac{f(u, z)}{d(u, z)} + \liminf _{d \rightarrow \infty } \frac{\Phi d}{d} > 0 \end{aligned}$$

for $u \in K$. This inequality serves as an alternative to the formula ($\mathfrak {F}$).

Lemma 3.11

Let X be an admissible complete CAT($\kappa$) space. Suppose that X has the convex hull finite property. Let K be a nonempty closed convex subset of X and f a real function on $K^2$ with conditions (E1)–(E4). Let $\varphi :[0, c_\kappa (D_\kappa / 2) \mathclose {[}\rightarrow [0, \infty \mathclose {[}$ be a strictly increasing and differentiable function, such that $\varphi '$ is continuous. Define $\Phi :[0, D_\kappa / 2 \mathclose {[}\rightarrow [0, \infty \mathclose {[}$ by $\Phi d = \varphi (c_\kappa (d))$ for $d \in [0, D_\kappa / 2 \mathclose {[}$ and suppose $\lim _{d \rightarrow D_\kappa / 2} \Phi d / d = \infty$. Suppose that $\Phi$ is convex. Define a subset $R_f x$ of K by

$$\begin{aligned} R_f x = \left\{ z \in K \mathrel {}\bigg |\mathrel {} \inf _{y \in K} \left( f(z, y) + \Phi d(x, y) - \Phi d(x, z) \right) \ge 0 \right\} \end{aligned}$$

for each $x \in X$. Then, $R_f x$ is nonempty for any $x \in X$. Moreover

$$\begin{aligned} 0 \le f(z, w) + \varphi '(c_\kappa (d(x, z))) \cdot \frac{D}{c'_\kappa (D)} \left( c_\kappa (d(x, w)) - c_\kappa (D) - c''_\kappa (D) c_\kappa (d(x, z)) \right) \end{aligned}$$

holds for any $z \in R_f x$ and $w \in K {\setminus } \{ z \}$, where $D = d(z, w)$.

Proof

By the definition of $\Phi$, it is continuous and convex. In addition, since $c_\kappa$ is strictly increasing on $[0, D_\kappa / 2]$, so is $\Phi$ on $[0, D_\kappa / 2 \mathclose {[}$. Thus, by Theorem 3.9, $R_f x$ is nonempty for every $x \in X$.

Let $z \in R_f x$ and $w \in K$ and suppose $w \ne z$. Since $t w \oplus (1 - t) z \in K$, we have

$$\begin{aligned} 0&\le f(z, t w \oplus (1 - t) z) + \Phi d(x, t w \oplus (1 - t) z) - \Phi d(x, z) \\&\le t f(z, w) + (1 - t) f(z, z) + \Phi d(x, t w \oplus (1 - t) z) - \Phi d(x, z) \\&= t f(z, w) + \Phi d(x, t w \oplus (1 - t) z) - \Phi d(x, z) \end{aligned}$$

for any $t \in \mathopen {]}0, 1 \mathclose {[}$, and thus

$$\begin{aligned} 0 \le f(z, w) + \frac{\Phi d(x, t w \oplus (1 - t) z) - \Phi d(x, z)}{t} \end{aligned}$$

holds for any $t \in \mathopen {]}0, 1 \mathclose {[}$. Put $D = d(z, w) > 0$ and

$$\begin{aligned} \Upsilon (t) = (t)_{D}^{\kappa } \left( c_\kappa (d(x, w)) - c_\kappa ((1 - t) D) \right) + (1 - t)_{D}^{\kappa } \left( c_\kappa (d(x, z)) - c_\kappa (t D) \right) \end{aligned}$$

for every $t \in [0, 1 \mathclose {[}$. Using the parallelogram law on CAT($\kappa$) spaces, we obtain

$$\begin{aligned} \Phi d(x, t w \oplus (1 - t) z) - \Phi d(x, z)&= \varphi (c_\kappa (d(x, t w \oplus (1 - t) z))) - \Phi d(x, z) \\&\le \varphi (\Upsilon (t)) - \Phi d(x, z). \end{aligned}$$

for any $t \in \mathopen {]}0, 1 \mathclose {[}$. Put $L(t) = \varphi (\Upsilon (t)) - \Phi d(x, z)$ for every $t \in \mathopen {]}0, 1 \mathclose {[}$. Then, L is differentiable on $\mathopen {]}0, 1 \mathclose {[}$, and we get $0 \le f(z, w) + L(t) / t$ for any $t \in \mathopen {]}0, 1 \mathclose {[}$. Now, we have

$$\begin{aligned} \frac{d}{d t} (t)_{D}^{\kappa } = \frac{D c''_\kappa (t D)}{c'_\kappa (D)} \end{aligned}$$

and

$$\begin{aligned} \frac{d}{d t} (1 - t)_{D}^{\kappa } = -\frac{D c''_\kappa ((1 - t) D)}{c'_\kappa (D)} \end{aligned}$$

for any $t \in \mathopen {]}0, 1 \mathclose {[}$, and thus, we get

$$\begin{aligned} \Upsilon '(t)&= \frac{D c''_\kappa (t D)}{c'_\kappa (D)} \left( c_\kappa (d(x, w)) - c_\kappa ((1 - t) d(w, z)) \right) + D (t)_{D}^{\kappa } c_\kappa ((1 - t) D) \\&\quad {} - \frac{D c''_\kappa ((1 - t) D)}{c'_\kappa (D)} \left( c_\kappa (d(x, z)) - c_\kappa (t d(w, z)) \right) - D (1 - t)_{D}^{\kappa } c_\kappa (t D). \end{aligned}$$

Since $(t)_{\kappa }^{D}$ is continuous at 0 and 1, we obtain

$$\begin{aligned} \lim _{t \rightarrow 0} \Upsilon '(t) = \Upsilon '(0) = \frac{D}{c'_\kappa (D)} \left( c_\kappa (d(x, w)) - c_\kappa (D) - c''_\kappa (D) c_\kappa (d(x, z)) \right) . \end{aligned}$$

Since $\varphi '$ is continuous, we also have

$$\begin{aligned} \lim _{t \rightarrow 0} \varphi '(\Upsilon (t)) = \varphi '(\Upsilon (0)) = \varphi '(c_\kappa (d(x, z))), \end{aligned}$$

and hence

$$\begin{aligned}&\lim _{t \rightarrow 0} \frac{dL}{dt}(t) \\&\quad = \lim _{t \rightarrow 0} \varphi '(\Upsilon (t)) \Upsilon '(t) \\&\quad = \varphi '(c_\kappa (d(x, z))) \cdot \frac{D}{c'_\kappa (D)} \left( c_\kappa (d(x, w)) - c_\kappa (D) - c''_\kappa (D) c_\kappa (d(x, z)) \right) . \end{aligned}$$

Since $\lim _{t \rightarrow 0} L(t) / t = \lim _{t \rightarrow 0} dL(t) / dt$, we obtain

$$\begin{aligned} 0 \le f(z, w) + \varphi '(c_\kappa (d(x, z))) \cdot \frac{D}{c'_\kappa (D)} \left( c_\kappa (d(x, w)) - c_\kappa (D) - c''_\kappa (D) c_\kappa (d(x, z)) \right) \end{aligned}$$

for any $z \in {R_{f}} x$ and $w \in K {\setminus } \{ z \}$, where $D = d(z, w)$. $\square$

Remark 3.12

Similar to Lemma 3.7, if $\kappa \le 0$, then we can change the assumption $\lim _{d \rightarrow D_\kappa / 2} \Phi d / d = \infty$ of Lemma 3.11 to

$$\begin{aligned} \liminf _{\begin{array}{c} d(v, z) \rightarrow \infty \\ z \in K \end{array}} \frac{f(v, z)}{d(v, z)} + \liminf _{d \rightarrow \infty } \frac{\Phi d}{d} > 0 \quad (\forall v \in K). \end{aligned}$$

In Theorem 3.13, we can relax the assumption in the same way.

Now, we show the main result of the well-definedness of the resolvent for equilibrium problems using a perturbation function $\Phi = \varphi \circ c_\kappa$.

Theorem 3.13

Let X be an admissible complete CAT($\kappa$) space. Suppose that X has the convex hull finite property. Let K be a nonempty closed convex subset of X and f a real function on $K^2$ with conditions (E1)–(E4). Let $\varphi :[0, c_\kappa (D_\kappa / 2) \mathclose {[}\rightarrow [0, \infty \mathclose {[}$ be a strictly increasing and differentiable function, such that $\varphi '$ is continuous and nondecreasing. Define $\Phi :[0, D_\kappa / 2 \mathclose {[}\rightarrow [0, \infty \mathclose {[}$ by $\Phi d = \varphi (c_\kappa (d))$ for $d \in [0, D_\kappa / 2 \mathclose {[}$ and suppose $\lim _{d \rightarrow D_\kappa / 2} \Phi d / d = \infty$. Define a subset $R_f x$ of K by

$$\begin{aligned} R_f x = \left\{ z \in K \mathrel {}\bigg |\mathrel {} \inf _{y \in K} \left( f(z, y) + \Phi d(x, y) - \Phi d(x, z) \right) \ge 0 \right\} \end{aligned}$$

for each $x \in X$. Then, the following hold:

• (i) $R_f x$ consists of one point for every $x \in X$, and thus, $R_f :X \rightarrow K$ is defined as a single-valued mapping;

• (ii) $R_f$ satisfies the following inequality for any $x_1, x_2 \in X$:

  $$\begin{aligned}{} & {} \left( \psi (D_1) c''_\kappa (D_1) + \psi (D_2) c''_\kappa (D_2) \right) c_\kappa (d({R_{f}} x_1, {R_{f}} x_2)) \\{} & {} \quad \le \psi (D_1) \left( c_\kappa (d(x_1, {R_{f}} x_2)) - c_\kappa (D_1) \right) \\{} & {} \qquad + \psi (D_2) \left( c_\kappa (d(x_2, {R_{f}} x_1)) - c_\kappa (D_2) \right) , \end{aligned}$$

  where $\psi \mathrel {{:}{=}}\varphi ' \circ c_\kappa$, $D_1 = d(x_1, {R_{f}} x_1)$, $D_2 = d(x_2, {R_{f}} x_2)$;

• (iii) ${R_{f}}$ is firmly vicinal with $\psi$;

• (iv) ${R_{f}}$ is $\Delta$-demiclosed; and if $F({R_{f}}) \ne \emptyset$, then ${R_{f}}$ is quasinonexpansive and asymptotically regular;

• (v) $F({R_{f}}) = {{\,\textrm{Equil}\,}}f$ holds, and thus, ${{\,\textrm{Equil}\,}}f$ is closed and convex.

Proof

By Lemma 3.11, ${R_{f}} x$ is always nonempty for $x \in X$. Furthermore, the nondecreasingness of $\varphi '$ yields the convexity of $\varphi$. Thus, $\Phi$ is also convex, since $c_\kappa$ is convex on $[0, D_\kappa / 2 \mathclose {[}$.

First of all, we show

for any $x_1, x_2 \in X$, $z_1 \in R_f x_1$ and $z_2 \in R_f x_2$.

Let $x_1, x_2 \in X$, $z_1 \in {R_{f}} x_1$ and $z_2 \in {R_{f}} x_2$, and put $D = d(z_1, z_2)$. If $z_1 = z_2$, then ($*$) holds obviously. Considering the case of $z_1 \ne z_2$, we have

$$\begin{aligned}{} & {} 0 \le f(z_1, z_2) + \varphi '(c_\kappa (d(x_1, z_1))) \\{} & {} \quad \cdot \frac{D}{c'_\kappa (D)} \left( c_\kappa (d(x_1, z_2)) - c_\kappa (D) - c''_\kappa (D) c_\kappa (d(x_1, z_1)) \right) \end{aligned}$$

and

$$\begin{aligned}{} & {} 0 \le f(z_2, z_1) + \varphi '(c_\kappa (d(x_2, z_2))) \\{} & {} \quad \cdot \frac{D}{c'_\kappa (D)} \left( c_\kappa (d(x_2, z_1)) - c_\kappa (D) - c''_\kappa (D) c_\kappa (d(x_2, z_2)) \right) \end{aligned}$$

from Lemma 3.11. Summing up these inequalities and dividing by $D / c'_\kappa (D)$, we obtain

$$\begin{aligned} 0&\le \varphi '(c_\kappa (d(x_1, z_1))) \left( c_\kappa (d(x_1, z_2)) - c_\kappa (D) - c''_\kappa (D) c_\kappa (d(x_1, z_1)) \right) \\&\quad {} + \varphi '(c_\kappa (d(x_2, z_2))) \left( c_\kappa (d(x_2, z_1)) - c_\kappa (D) - c''_\kappa (D) c_\kappa (d(x_2, z_2)) \right) . \end{aligned}$$

Let H be the right-hand side of the above inequality. Since $c''_\kappa (d) = 1 - \kappa c_\kappa (d)$ holds for any $d \in \mathbb {R}$, we have

$$\begin{aligned} H&= \varphi '(c_\kappa (d(x_1, z_1))) \cdot \big ( c_\kappa (d(x_1, z_2)) - c_\kappa (D) - c_\kappa (d(x_1, z_1)) \\&\quad {} + \kappa c_\kappa (D) c_\kappa (d(x_1, z_1))\big ) + \varphi '(c_\kappa (d(x_2, z_2))) \\&\quad {} \cdot \left( c_\kappa (d(x_2, z_1)) - c_\kappa (D) - c_\kappa (d(x_2, z_2)) + \kappa c_\kappa (D) c_\kappa (d(x_2, z_2)) \right) \\&= \varphi '(c_\kappa (d(x_1, z_1))) \left( c_\kappa (d(x_1, z_2)) - c_\kappa (d(x_1, z_1)) \right) \\&\quad {} + \varphi '(c_\kappa (d(x_2, z_2))) \left( c_\kappa (d(x_2, z_1)) - c_\kappa (d(x_2, z_2)) \right) \\&\quad {} - \Bigl ( \varphi '(c_\kappa (d(x_1, z_1))) \left( 1 - \kappa c_\kappa (d(x_1, z_1)) \right) \\&\qquad {} + \varphi '(c_\kappa (d(x_2, z_2))) \left( 1 - \kappa c_\kappa (d(x_2, z_2)) \right) \Bigr ) c_\kappa (D) \\&= \varphi '(c_\kappa (d(x_1, z_1))) \left( c_\kappa (d(x_1, z_2)) - c_\kappa (d(x_1, z_1)) \right) \\&\quad {} + \varphi '(c_\kappa (d(x_2, z_2))) \left( c_\kappa (d(x_2, z_1)) - c_\kappa (d(x_2, z_2)) \right) \\&\quad {} - \Bigl ( \varphi '(c_\kappa (d(x_1, z_1)))\, c''_\kappa (d(x_1, z_1)) + \varphi '(c_\kappa (d(x_2, z_2)))\, c''_\kappa (d(x_2, z_2)) \Bigr ) c_\kappa (D), \end{aligned}$$

and thus, we obtain ($*$).

(i) Let $x \in X$ and $z_1, z_2 \in {R_{f}} x$. Since $\varphi$ is strictly increasing, we have $\varphi '(t) > 0$ for any $t \in \mathopen {]}0, c_\kappa (D_\kappa / 2) \mathclose {[}$, and thus, we get

$$\begin{aligned}{} & {} c_\kappa (d(z_1, z_2)) \\{} & {} \qquad \le \frac{\left( \varphi '(c_\kappa (d(x, z_1))) - \varphi '(c_\kappa (d(x, z_2))) \right) \left( c_\kappa (d(x, z_2)) - c_\kappa (d(x, z_1)) \right) }{\varphi '(c_\kappa (d(x, z_1))) c''_\kappa (d(x, z_1)) + \varphi '(c_\kappa (d(x, z_2))) c''_\kappa (d(x, z_2))}. \end{aligned}$$

Since we are now assuming that $\varphi '$ is nondecreasing, we obtain $c_\kappa (d(z_1, z_2)) \le 0$, and thus, $z_1 = z_2$. Therefore, we can consider $R_f$ to be a single-valued mapping from X into K.

(ii) Let $x_1, x_2 \in X$ and let $z_1 = {R_{f}} x_1$, $z_2 = {R_{f}} x_2$. Then, by the formula ($*$), we obtain the conclusion.

(iii) The inequality (ii) implies that ${R_{f}}$ is firmly vicinal with $\varphi ' \circ c_\kappa$ from Lemma 2.8.

(iv) If $F({R_{f}}) \ne \emptyset$ holds, then we get (iv) by Lemmas 2.6 and 2.7.

(v) We show $F({R_{f}}) = {{\,\textrm{Equil}\,}}f$. Suppose $z \in F({R_{f}})$. Then, $z = {R_{f}} z$. Let $y \in K$ and put $D = d(z, y)$. Then

$$\begin{aligned} 0&\le f(z, y) + \varphi '(c_\kappa (d(x, z))) \\&\quad {} \cdot \frac{D}{c'_\kappa (D)} \left( c_\kappa (d({R_{f}} z, y)) - c_\kappa (D) - c''_\kappa (D) c_\kappa (d({R_{f}} z, z)) \right) \\&= f(z, y) + \varphi '(c_\kappa (d(x, z))) \\&\quad {} \cdot \frac{D}{c'_\kappa (D)} \left( c_\kappa (D) - c_\kappa (D) - c''_\kappa (D) \cdot 0 \right) \\&= f(z, y) \end{aligned}$$

holds if $z \ne y$; and $f(z, y) \ge 0$ is obviously holds even if $z = y$. Thus, we get $f(z, y) \ge 0$ for any $y \in K$. It implies $z \in {{\,\textrm{Equil}\,}}f$, and thus, $F({R_{f}}) \subset {{\,\textrm{Equil}\,}}f$.

On the other hand, we suppose $z \in {{\,\textrm{Equil}\,}}f$, or equivalently, $\inf _{y \in K} f(z, y) \ge 0$. Thus, we obtain

$$\begin{aligned} \inf _{y \in K} \left( f(z, y) + \Phi d(z, y) - \Phi d(z, z) \right) \ge \inf _{y \in K} f(z, y) \ge 0, \end{aligned}$$

and hence, $z = {R_{f}} z$, that is, $z \in F({R_{f}})$. It concludes that $F({R_{f}}) = {{\,\textrm{Equil}\,}}f$.

If $F({R_{f}})$ is nonempty, then ${R_{f}}$ is quasinonexpansive by (iv). Thus the set $F({R_{f}})$ is closed and convex, and so is ${{\,\textrm{Equil}\,}}f$. $\square$

As a result, we obtain the perturbation function $\Phi = \varphi \circ c_\kappa :[0, D_\kappa / 2 \mathclose {[}\rightarrow [0, \infty \mathclose {[}$ makes $R_f$ a single-valued mapping if $\varphi :[0, c_\kappa (D_\kappa / 2) \mathclose {[}\rightarrow [0, \infty \mathclose {[}$ has the following conditions (a)–(d):

(a):

$\varphi$ is strictly increasing and differentiable;

(b):

$\varphi '$ is continuous;

(c):

$\varphi '$ is nondecreasing;

(d-1):

if $\kappa \le 0$, then $\Phi$ satisfies

$$\begin{aligned} \liminf _{\begin{array}{c} d(v, z) \rightarrow \infty \\ z \in K \end{array}} \frac{f(v, z)}{d(v, z)} + \liminf _{d \rightarrow \infty } \frac{\Phi d}{d} > 0 \end{aligned}$$

for all $v \in K$;

(d-2):

if $\kappa > 0$, then $\Phi$ satisfies $\lim _{d \rightarrow D_\kappa / 2} \Phi d = \infty$.

The condition (d-2) is equivalent to $\lim _{t \uparrow 1} \varphi (t / \kappa ) = \infty$ for $\kappa > 0$. Note that the conditions (d-1) and (d-2) always hold if $\lim _{d \rightarrow D_\kappa / 2} \Phi d / d = \infty$ holds. Moreover, if $\kappa \le 0$ and $\lim _{t \rightarrow \infty } \varphi '(t) = \infty$, then (d-1) is true. Indeed, if $\lim _{t \rightarrow \infty } \varphi '(t) = \infty$, then we have $\lim _{t \rightarrow \infty } \varphi (t)/t = \infty$ and hence $\lim _{d \rightarrow \infty } \Phi d / d = \infty$.

We now introduce some specific cases of perturbations for Theorem 3.13. In the following four cases, $R_f$ defined in Theorem 3.13 becomes a single-valued mapping.

Corollary 3.14

If $\kappa = 1$ and $\varphi (t) = -\log \left( 1 - t \right)$ for $t \in [0, 1 \mathclose {[}$, that is, $\Phi d = - \log \cos d$ for $d \in [0, \pi / 2 \mathclose {[}$, then a single-valued mapping $R_f$ defined in Theorem 3.13 satisfies

$$\begin{aligned} 2 \cos d({R_{f}} x_1, {R_{f}} x_2) \ge \frac{\cos d(x_1, {R_{f}} x_2)}{\cos d(x_1, {R_{f}} x_1)} + \frac{\cos d(x_2, {R_{f}} x_1)}{\cos d(x_2, {R_{f}} x_2)} \end{aligned}$$

(A)

for any $x_1, x_2 \in X$. Moreover, ${R_{f}}$ is spherically nonspreading of sum type.

Proof

Put $\psi = \varphi ' \circ c_\kappa$, then $\psi (d) = 1 / \cos d$ for $d \in [0, \pi / 2 \mathclose {[}$. By Theorem 3.13 (iii), ${R_{f}}$ is firmly vicinal with $\psi$. Therefore, we get (A). From this inequality, we can deduce the spherical nonspreadingness of sum type of ${R_{f}}$, see [8]. $\square$

The above corollary is a result that is the same as the result of Kimura [8]. The following Corollary 3.15 is the same result as in Kimura and Kishi [9].

Corollary 3.15

If $\kappa = 0$ and $\varphi (t) = t$ for $t \in [0, \infty \mathclose {[}$, that is, $\Phi d = \frac{1}{2} d^2$ for $d \in [0, \infty \mathclose {[}$, then a single-valued mapping ${R_{f}}$ defined in Theorem 3.13 is firmly metrically nonspreading.

Proof

Put $\psi = \varphi ' \circ c_\kappa$, i.e., $\psi (d) = 1$ for any $d \in [0, \infty \mathclose {[}$. Then, ${R_{f}}$ is firmly vicinal with a constant function $\psi$, and hence, we get the conclusion. $\square$

The result of Corollary 3.16 is the same as the result of Kimura and Ogihara [12].

Corollary 3.16

We consider the case of $\kappa = -1$ and $\varphi (t) = t + 1$ for $t \in [0, \infty \mathclose {[}$, that is, $\Phi d = \cosh d$ for $d \in [0, \infty \mathclose {[}$. Then, a single-valued mapping ${R_{f}}$ defined in Theorem 3.13 satisfies

$$\begin{aligned}{} & {} \cosh d({R_{f}} x_1, {R_{f}} x_2) \\{} & {} \quad \le \frac{1}{\cosh d(x_1, {R_{f}} x_1) + \cosh d(x_2, {R_{f}} x_2)} \left( \cosh d(x_1, {R_{f}} x_2) + \cos d(x_2, {R_{f}} x_1) \right) \end{aligned}$$

for any $x_1, x_2 \in X$. It implies that ${R_{f}}$ is hyperbolically nonspreading.

Proof

Put $\psi = \varphi ' \circ c_\kappa$, that is, $\psi (d) = 1$ for any $d \in [0, \infty \mathclose {[}$. Then, ${R_{f}}$ is firmly vicinal with $\psi$, and thus, we get the desired inequality. Since $\cosh d \ge 1$ for any $d \in [0, \infty \mathclose {[}$, we also have the hyperbolical nonspreadingness of ${R_{f}}$. $\square$

Example 3.17

We consider the case of $\kappa = 1$ and $\varphi (t) = t$ for $t \in [0, 1 \mathclose {[}$, that is, $\Phi d = 1 - \cos d$ for $d \in [0, \pi / 2 \mathclose {[}$. Then, we do not know from Theorem 3.13 whether the mapping ${R_{f}}$ defined by Theorem 3.13 is a single-valued mapping or not, since $\lim _{d \rightarrow D_\kappa / 2} \Phi d / d = 2 / \pi \ne \infty$.

The following two corollaries are our new result.

Corollary 3.18

If $\kappa = 1$ and $\varphi (t) = 1 / (1 - t) - (1 - t)$ for $t \in [0, 1 \mathclose {[}$, that is, $\Phi d = \tan d \sin d$ for $d \in [0, \pi / 2 \mathclose {[}$, then a single-valued mapping $R_f$ defined in Theorem 3.13 satisfies

$$\begin{aligned}&\cos d({R_{f}} x_1, {R_{f}} x_2) \\&\quad \ge \frac{ \left( \frac{1}{\cos ^2 d(x_1, {R_{f}} x_1)} + 1 \right) \cos d(x_1, {R_{f}} x_2) + \left( \frac{1}{\cos ^2 d(x_2, {R_{f}} x_2)} + 1 \right) \cos d(x_2, {R_{f}} x_1) }{ \left( \frac{1}{\cos ^2 d(x_1, {R_{f}} x_1)} + 1 \right) \cos d(x_1, {R_{f}} x_1) + \left( \frac{1}{\cos ^2 d(x_2, {R_{f}} x_2)} + 1 \right) \cos d(x_2, {R_{f}} x_2) } \end{aligned}$$

(B)

for any $x_1, x_2 \in X$. Moreover, ${R_{f}}$ is firmly spherically nonspreading.

Proof

Put $\psi = \varphi ' \circ c_\kappa$, that is, $\psi (d) = \frac{1}{\cos ^2 d} + 1$ for $d \in [0, \pi / 2 \mathclose {[}$, then $R_f$ is firmly vicinal with $\psi$. Therefore, we get the inequality (B).

Let $x_1, x_2 \in X$ and put $\varphi _1 = \cos d(x_1, {R_{f}} x_2)$, $\varphi _2 = \cos d(x_2, {R_{f}} x_1)$ and put $C_1 = \cos d(x_1, {R_{f}} x_1)$ and $C_2 = \cos d(x_2, {R_{f}} x_2)$. Then, we get

$$\begin{aligned} \cos d({R_{f}} x_1, {R_{f}} x_2)&\ge \frac{\left( \frac{1}{C_1^{\, 2}} + 1 \right) \varphi _1 + \left( \frac{1}{C_2^{\, 2}} + 1 \right) \varphi _2}{\left( \frac{1}{C_1^{\, 2}} + 1 \right) C_1 + \left( \frac{1}{C_2^{\, 2}} +1 \right) C_2} \\&= \frac{\frac{C_2}{C_1} \varphi _1 + \frac{C_1}{C_2} \varphi _2 + C_1 C_2 (\varphi _1 + \varphi _2)}{(C_1 + C_2) (1 + C_1 C_2)} \\&\ge \frac{2 \sqrt{\varphi _1 \varphi _2} + 2 C_1 C_2 \sqrt{\varphi _1 \varphi _2}}{(C_1 + C_2) (1 + C_1 C_2)} \\&= \frac{2 \sqrt{\varphi _1 \varphi _2}}{C_1 + C_2}, \end{aligned}$$

and thus

$$\begin{aligned} \cos ^2 d({R_{f}} x_1, {R_{f}} x_2)&\ge \left( \frac{2}{C_1 + C_2} \right) ^2 \varphi _1 \varphi _2 \\&\ge \frac{2}{C_1 + C_2} \cos d(x_1, {R_{f}} x_2) \cos d(x_2, {R_{f}} x_1) \end{aligned}$$

holds for any $x_1, x_2 \in X$. It implies that ${R_{f}}$ is firmly spherically nonspreading. $\square$

In 2016, the well-definedness of the resolvent of the convex function defined by using the perturbation $\tan d \sin d$ was proved by Kimura and Kohsaka [10]. From the results of Corollary 3.18, we obtain that we can use the same perturbation $\tan d \sin d$ to define the resolvent of the equilibrium problem as the single-valued mapping.

Corollary 3.19

We consider the case of $\kappa = -1$ and $\varphi (t) = \log \left( t + 1 \right)$ for $t \in [0, \infty \mathclose {[}$, that is, $\Phi d = \log \cosh d$ for $d \in [0, \infty \mathclose {[}$. Suppose that $f :K^2 \rightarrow K$ satisfies (E1)–(E4) and

$$\begin{aligned} \liminf _{\begin{array}{c} d(v, z) \rightarrow \infty \\ z \in K \end{array}} \frac{f(v, z)}{d(v, z)} + 1 > 0 \end{aligned}$$

for all $v \in K$. Then, a mapping ${R_{f}}$ in Theorem 3.13 is well defined as a single-valued mapping, and it satisfies

$$\begin{aligned} 2 \cosh d({R_{f}} x_1, {R_{f}} x_2) \le \frac{\cosh d(x_1, {R_{f}} x_2)}{\cosh d(x_1, {R_{f}} x_1)} + \frac{\cosh d(x_2, {R_{f}} x_1)}{\cosh d(x_2, {R_{f}} x_2)} \end{aligned}$$

for any $x_1, x_2 \in X$.

Proof

Since $\lim _{d \rightarrow \infty } \Phi d / d = 1$, we obtain the well-definedness of the resolvent ${R_{f}}$ from Theorem 3.13. Put $\psi = \varphi ' \circ c_\kappa$, that is, $\psi (d) = 1 / \cosh d$ for $d \in [0, \infty \mathclose {[}$, then ${R_{f}}$ is firmly vicinal with $\psi$, and hence, we get the conclusion. $\square$

Appendix

For the sake of information, let us describe some properties that hold for resolvent ${R_{f}}$ as follows.

Theorem 4.1

Let X be an admissible complete CAT($\kappa$) space. Let K be a nonempty closed convex subset of X and f a real function on $K^2$ with conditions (E2). Let $\varphi :[0, c_\kappa (D_\kappa / 2) \mathclose {[}\rightarrow [0, \infty \mathclose {[}$ be a differentiable function. Suppose that for each $\nu > 0$, there exists a mapping $R_{\nu f} :X \rightarrow K$, such that

$$\begin{aligned}{} & {} 0 \le \nu f(R_{\nu f} x, w) + \varphi '(c_\kappa (d(x, R_{\nu f} x))) \\{} & {} \quad \cdot \frac{E}{c'_\kappa (E)} \left( c_\kappa (d(x, w)) - c_\kappa (E) - c''_\kappa (E) c_\kappa (d(x, R_{\nu f} x)) \right) \end{aligned}$$

holds for any $x \in X$ and $w \in K {\setminus } \{ R_{\nu f} x \}$, where $E = d(R_{\nu f} x, w)$. Then, for any $\lambda , \mu > 0$ and $x, y \in X$, the following inequalities hold:

$$\begin{aligned}{} & {} 0 \le \lambda \varphi '(c_\kappa (D_{yy})) \left( c_\kappa (D_{yx}) - c_\kappa (D) - c''_\kappa (D) c_\kappa (D_{yy}) \right) \\{} & {} \qquad + \mu \varphi '(c_\kappa (D_{xx})) \left( c_\kappa (D_{xy}) - c_\kappa (D) - c''_\kappa (D) c_\kappa (D_{xx}) \right) , \end{aligned}$$

(i)

$$\begin{aligned}{} & {} \bigl ( \lambda \varphi '(c_\kappa (D_{yy})) c''_\kappa (D_{yy}) + \mu \varphi '(c_\kappa (D_{xx})) c''_\kappa (D_{xx}) \bigr ) c_\kappa (D) \\{} & {} \quad \le \lambda \varphi '(c_\kappa (D_{yy})) \bigl ( c_\kappa (D_{yx}) - c_\kappa (D_{yy}) \bigr ) \\{} & {} \qquad + \mu \varphi '(c_\kappa (D_{xx})) \bigl ( c_\kappa (D_{xy}) - c_\kappa (D_{xx}) \bigr ), \end{aligned}$$

(ii)

$$\begin{aligned}{} & {} \frac{1}{\kappa } \bigl ( \lambda \varphi '(c_\kappa (D_{yy})) c''_\kappa (D_{yy}) + \mu \varphi '(c_\kappa (D_{xx})) c''_\kappa (D_{xx}) \bigr ) c''_\kappa (D) \\{} & {} \quad \ge \frac{1}{\kappa } \bigl ( \lambda \varphi '(c_\kappa (D_{yy})) c''_\kappa (D_{yx}) + \mu \varphi '(c_\kappa (D_{xx})) c''_\kappa (D_{xy}) \bigr ), \end{aligned}$$

(iii)

where $D_{xx} = d(x, R_{\lambda f} x)$, $D_{xy} = d(x, R_{\mu f} y)$, $D_{yx} = d(y, R_{\lambda f} x)$, $D_{yy} = d(y, R_{\mu f} y)$, and $D = d(R_{\lambda f} x, R_{\mu f} y)$. The inequality (iii) is considered only when $\kappa \ne 0$.

Proof

Let $\lambda , \mu > 0$ and $x, y \in X$. If $R_{\lambda f} x = R_{\mu f} y$, then we get the conclusion obviously. Suppose that $R_{\lambda f} x \ne R_{\mu f} y$. Then, for a function $\mu f :K^2 \rightarrow \mathbb {R}$, we have

$$\begin{aligned} 0 \le \mu f (R_{\mu f} y, R_{\lambda f} x) {+} \varphi '(c_\kappa (D_{yy})) \cdot \frac{D}{c'_\kappa (D)} \left( c_\kappa (D_{yx}) {-} c_\kappa (D) {-} c''_\kappa (D) c_\kappa (D_{yy}) \right) , \end{aligned}$$

and hence

$$\begin{aligned}{} & {} 0 \le \frac{c'_\kappa (D)}{D} \lambda \mu f(R_{\mu f} y, R_{\lambda f} x) \\{} & {} \quad + \lambda \varphi '(c_\kappa (D_{yy})) \left( c_\kappa (D_{yx}) - c_\kappa (D) - c''_\kappa (D) c_\kappa (D_{yy}) \right) . \end{aligned}$$

Similarly, for a function $\lambda f$, we get

$$\begin{aligned} 0{} & {} \le \frac{c'_\kappa (D)}{D} \mu \lambda f(R_{\lambda f} x, R_{\mu f} y) \\{} & {} \quad + \mu \varphi '(c_\kappa (D_{xx})) \left( c_\kappa (D_{xy}) - c_\kappa (D) - c''_\kappa (D) c_\kappa (D_{xx}) \right) . \end{aligned}$$

Summing up these two inequalities, we obtain (i), since f satisfies (E2).

Using Lemma 2.5, we get inequality (i) is equivalent to

$$\begin{aligned} 0{} & {} \le \lambda \varphi '(c_\kappa (D_{yy})) \left( c_\kappa (D_{yx}) - c_\kappa (D_{yy}) - c''_\kappa (D_{yy}) c_\kappa (D) \right) \\{} & {} \quad + \mu \varphi '(c_\kappa (D_{xx})) \left( c_\kappa (D_{xy}) - c_\kappa (D_{xx}) - c''_\kappa (D_{xx}) c_\kappa (D) \right) , \end{aligned}$$

and so is (ii).

Assume that $\kappa \ne 0$. Then, using Lemma 2.5 again, we obtain that (i) is equivalent to

$$\begin{aligned} 0{} & {} \le \lambda \varphi '(c_\kappa (D_{yy})) \left( \frac{1 - c''_\kappa (D_{yx})}{\kappa } - \frac{1 - c''_\kappa (D_{yy}) c''_\kappa (D)}{\kappa } \right) \\{} & {} \quad + \mu \varphi '(c_\kappa (D_{xx})) \left( \frac{1 - c''_\kappa (D_{xy})}{\kappa } - \frac{1 - c''_\kappa (D_{xx}) c''_\kappa (D)}{\kappa } \right) , \end{aligned}$$

and so is (iii). $\square$

Corollary 4.2

Resolvents $R_{\lambda f}$, $R_{\mu f}$ of the equilibrium problem for $\lambda f$ and $\mu f$ defined by Theorem 3.13 satisfy inequalities (i), (ii) and (iii) for any $\lambda , \mu > 0$ and $x, y \in X$.