A proximal point algorithm for finding minimizers and fixed points of quasi-pseudo-contractive mappings in CAT(0) spaces

Abstract

The purpose of this article is twofold. One is to establish a proximal point algorithm for finding a minimizer of a proper convex and lower semi-continuous function and fixed points of quasi-pseudo-contractive mappings in CAT(0) spaces. The other is to point out and correct a basic and conceptual error in a paper of Ugwunnadi et al. [Theorem 3.1, J. Fixed Point Theory Appl. (2018) 20: 82].

1 Introduction

Recently, Ugwunnadi et al. [1] introduced a hybrid proximal point algorithm and established some strong convergence theorems to a common solution of proximal point for a proper convex and lower semi-continuous function and a fixed point of a k-demicontractive mapping in the framework of a CAT(0) space. Particular, the following main result is given:

TheoremUKA

[1, Theorem 3.1]. Let (X, d) be a complete CAT(0) space, $f : X \rightarrow (-\infty , + \infty ]$ be a proper convex and lower semi-continuous function and $T : X \rightarrow X$ be an L-Lipschitzian k-demicontractive mapping such that T is $\Delta$-demiclosed. If $\{\alpha _n\}$ and $\{\beta _n\}$ are sequences in (0,1) satisfying the following conditions:

• (c1) $lim_{n \rightarrow \infty } \alpha _n = 0$;

• (c2) $\sum _{n =1}^\infty \alpha _n = \infty$;

• (c3) $0< \varepsilon \le \beta _n < 1 - 2k, \ \forall n \ge 1$, where $\varepsilon$ and $k \in [0, 1)$ are some positive constants,

and $\Omega := Fix(T)\bigcap argmin_{y \in X} f(y) \not = \emptyset$, then the sequence $\{x_n\}$ generated by given $x_1 \in X$,

$$\begin{aligned} \left\{ \begin{aligned} z_n&= arg min_{y \in X}\left[ f(y) + \frac{1}{2\lambda _n}d^2(y, x_n)\right] ,\\ y_n&= (1 -\alpha _n)z_n,\\ x_{n+1}&= (1- \beta _n)z_n \oplus \beta _n Ty_n \end{aligned} \right. \end{aligned}$$

(1.1)

converges strongly to some point $p \in \Omega$.

During carefully reading Theorem UKA and its proof, we found that there exist some basic and conceptual errors in it. Since (X, d) is a CAT(0) space, it is not linear. Therefore it does not have a scalar multiplication and element 0. These show that the sequences $\{y_n\}$ and $\{x_n\}$ defined by (1.1) are ill-posed. And the proof of Theorem UKA is also lack of rationality.

The main purpose of this paper is to establish a proximal point algorithm for finding minimizers of a proper convex and lower semi-continuous function and fixed points of quasi-pseudo-contractive mappings in CAT(0) spaces and to point out and correct a basic and conceptual error in Ugwunnadi et al. [1, Theorem 3.1].

2 Preliminaries

Let (X, d) be a metric space and $x, y \in X$. A geodesic path joining x to y is an isometry $c : [0,\ d(x; y)] \rightarrow X$ such that $c(0) = x$ and $c(d(x; y)) = y$. The image of a geodesic path joining x to y is called a geodesic segment between x and y. A metric space X is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic segment joining x and y for each $x, y \in X$.

Let X be a uniquely geodesic space. We write $(1-t)x \oplus ty$ for the unique point z in the geodesic segment joining x to y such that $d(x, z) = td(x, y)$ and $d(y, z) = (1-t)d(x, y)$. We also denote by [x, y] the geodesic segment joining x to y, that is, $[x, y] = \{(1-t) x \oplus ty: 0 \le t \le 1 \}$. A subset C of X is convex if $[x, y] \subset C$ for all $x, y \in C$.

A uniquely geodesic space (X; d) is a CAT(0) space, if and only if

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

(2.1)

for all $x, y,z \in X$ and all $t \in [0, 1]$.

It is well-known that any complete and simply connected Riemannian manifold having non-positive sectional curvature is a CAT(0) space. Other examples of CAT(0) spaces include pre-Hilbert spaces [2], R-trees, Euclidean buildings [3].

Let X be a metric space, $\{x_n\}$ be a bounded sequence in X, and $r(., \{x_n\}) : X \rightarrow [0, \infty )$ be a continuous functional defined by $r(x, \{x_n\}) = \limsup _{n \rightarrow \infty } d(x, x_n)$. The asymptotic radius of $\{x_n\}$ is given by $r(\{x_n\}):= \inf \{r(x, {x_n}) : x \in X\}$ while the asymptotic center of $\{x_n\}$ is the set $A(\{x_n\}) = \{x \in X : r(x, \{x_n\}) = r(\{x_n\})\}$. It is generally known that in a CAT(0) space, $A(\{x_n\})$ consists of exactly one point. A sequence $\{x_n\}$ in X is said to be $\Delta$-convergent to a point $x \in X$ if $A(\{x_{n_k}\}) = \{x\}$ for every subsequence $\{x_{n_k}\}$ of $\{x_n\}$. In this case, we write $\Delta -\lim _{n \rightarrow \infty } x_n = x$.

In 2008 Berg and Nikolaev [4] (see also, Reich and Salinas [5]) introduced the concept of quasilinearization in CAT(0) space X as follows:

Denote a pair $(a,b)\in X\times X$ by $\overrightarrow{ab}$ and call it a vector. Quasi-linearization in CAT(0) space X is defined as a mapping $\langle \cdot ,\cdot \rangle : (X\times X)\times (X\times X)\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} \langle \overrightarrow{ab},\overrightarrow{cd}\rangle =\frac{1}{2}(d^2(a,d)+d^2(b,c)-d^2(a,c)-d^2(b,d)) \end{aligned}$$

(2.2)

for all $a,b,c,d\in X$. It can be easily verified that

$$\begin{aligned}&\langle \overrightarrow{ab}, \overrightarrow{ab} \rangle = d^2(a,b), \ \langle \overrightarrow{ba}, \overrightarrow{cd} \rangle = - \langle \overrightarrow{ab}, \overrightarrow{cd} \rangle , \ and \ \\&\quad \langle \overrightarrow{ab}, \overrightarrow{cd} \rangle = \langle \overrightarrow{ae}, \overrightarrow{cd} \rangle + \langle \overrightarrow{eb}, \overrightarrow{cd} \rangle \ \forall a, b, c, d, e \in X. \end{aligned}$$

Remark 2.1

[6] It is well known that if X is a complete CAT(0) space, then $\{x_n\}$ $\Delta$-converges to $x^* \in X$ if and only if

$$\begin{aligned} \lim sup_{n \rightarrow \infty } \langle \overrightarrow{x^*x_n}, \overrightarrow{x^*y} \rangle \le 0, \ \forall y \in X. \end{aligned}$$

Let X be a CAT(0) space. We say that X satisfies the Cauchy-Schwarz inequality if

$$\begin{aligned} \langle \overrightarrow{ab}, \overrightarrow{cd} \rangle \le d(a, b) d(c, d), \ \forall a, b, c, d \in X. \end{aligned}$$

(2.3)

It is known ( [4], Corollary 3) that a geodesically connected metric space is a CAT(0) space if and only if it satisfies the Cauchy-Schwarz inequality.

Ahmadi Kakavandi and Amini [7] have introduced the concept of dual space of a complete CAT(0) space X, based on a work of Berg and Nikolaev [4], as follows.

Consider the map $\Theta : {\mathbb {R}} \times X \times X \rightarrow C(X, {\mathbb {R}})$ defined by

$$\begin{aligned} \Theta (t, a, b)(x) = t \langle \overrightarrow{ab}, \ \overrightarrow{ax}\rangle , \ (t \in {\mathbb {R}}, a, b, x \in X), \end{aligned}$$

where $C(X, {\mathbb {R}})$ is the space of all continuous real-valued functions on X. Then the Cauchy-Schwartz inequality implies that $\Theta (t, a, b)$ is a Lipschitz function with Lipschitz semi-norm $L(\Theta (t, a, b)) = |t|d(a, b)$, $(t\in {\mathbb {R}}, \ a, b \in X)$, where

$$\begin{aligned} L(\phi ) = \sup \left\{ \frac{\phi (x)- \phi (y)}{d(x;y)} : x, y \in X, \ x \not = y\right\} \end{aligned}$$

is the Lipschitz semi-norm for any function $\phi : X \rightarrow {\mathbb {R}}$. A pseudometric D on ${\mathbb {R}} \times X \times X$ is defined by

$$\begin{aligned} D((t, a, b), (s, c, d)) = L(\Theta (t, a, b) - \Theta (s, c, d)), \ (t, s \in {\mathbb {R}}, \ a, b, c, d \in X). \end{aligned}$$

For a complete CAT(0) space (X, d), the pseudometric space $({\mathbb {R}} \times X \times X, D)$ can be considered as a subspace of the pseudometric space of all real-valued Lipschitz functions $(Lip(X, {\mathbb {R}}), L)$. And $D((t, a, b), (s, c, d)) = 0$ if and only if $t\langle \overrightarrow{ab},\ \overrightarrow{xy} \rangle = s\langle \overrightarrow{cd},\ \overrightarrow{xy} \rangle$, for all $x, y \in X$. Hence D imposes an equivalent relation on ${\mathbb {R}} \times X \times X$, where the equivalence class of (t, a, b) is

$$\begin{aligned}{}[t\overrightarrow{ab}] = \{s\overrightarrow{cd}: D((t, a, b), (s, c, d))=0\}. \end{aligned}$$

The set

$$\begin{aligned} X^* = \{[t\overrightarrow{ab}]: (t,a,b) \in {\mathbb {R}} \times X \times X\} \end{aligned}$$

is a metric space which is called the dual space of (X; d) with metric

$$\begin{aligned} D([t\overrightarrow{ab}], [s\overrightarrow{cd}]): = D((t, a, b), (s, c, d)), \end{aligned}$$

The following inequalities can be proved easily.

Lemma 2.1

Let X be a CAT(0) space. For all $x, y, z, u, w\in X$ and $t \in [0,1]$, the following inequalities hold:

• (i) $d(t x\oplus (1-t)y,\,z)\le t d(x,z) +(1-t) d(y,z)$;

• (ii) $d^2((1-t)x \oplus t y, z)\le (1-t)^2 d^2(x, z) + t^2 d^2(y,z) + 2 t(1-t)\langle \overrightarrow{xz}, \overrightarrow{yz} \rangle$;

• (iii) $d(tx\oplus (1-t)y,\ tu\oplus (1-t)w)\le t d(x,u) +(1-t) d(y,w)$.

In the sequel, we always assume that X is a complete CAT(0) space, C is a nonempty and closed convex subset of X and Fix(T) is the fixed point set of a mapping T.

Definition 2.2

A mapping $T : C \rightarrow C$ is said to be

• (1) contractive if there exists a constant $k \in (0, 1)$ such that

  $$\begin{aligned} d(T x, T y) \le k d(x, y), \ \ \forall x, y \in C; \end{aligned}$$

  if $k =1$, then T is said to be nonexpansive;

• (2) quasinonexpansive if $Fix(T) \ne \emptyset$ and

  $$\begin{aligned} d(Tx, p) \le d(x, p),\forall p\in Fix(T), \ x \in C; \end{aligned}$$

• (3) firmly nonexpansive if

  $$\begin{aligned} d^2(Tx, Ty) \le \langle \overrightarrow{TxTy}, \overrightarrow{xy} \rangle , \ \forall x, y \in C; \end{aligned}$$

  (2.4)

• (4) k-demicontractive [8] if $Fix(T) \ne \emptyset$ and there exists a constant $k \in [0; 1)$ such that

  $$\begin{aligned} d^2(T x, p) \le d^2(x, p) + k d^2(x, Tx), \ \forall x \in C, \ p \in Fix(T); \end{aligned}$$

• (5) quasi-pseudo-contractive if $Fix(T) \ne \emptyset$ and

  $$\begin{aligned} d^2(T x, p) \le d^2(x, p) + d^2(x, Tx), \ \forall x \in C, \ p \in Fix(T); \end{aligned}$$

  (2.5)

Remark 2.3

From the definitions above, it is easy to see that if $Fix(T)\not = \emptyset$, then the following implications hold:

$$\begin{aligned} (3) \Longrightarrow (2) \Longrightarrow (4) \Longrightarrow (5). \end{aligned}$$

But the converse is not true. These show that the class of quasi-pseudo-contractive mappings is more general than the classes of k-demicontractive mappings, quasinonexpansive mappings.

Definition 2.4

Let (X, d) be a complete CAT(0) space. A mapping $T : X \rightarrow X$ is said to be $\Delta$-demiclosed, if for any bounded sequence $\{x_n\}$ in X such that $\Delta -\lim _{n \rightarrow \infty }x_n = p$ and $\lim _{n \rightarrow \infty } d(x_n, Tx_n) = 0$, then $Tp = p$.

Example of quasi-pseudo-contractive mappings Let H be the closed interval [0, 1] with the absolute value as norm. Let $T : H \rightarrow H$ be the mapping defined by:

$$\begin{aligned} Tx = \left\{ \begin{aligned}&k, \ \ \ x \in [0, k], \ k \in (0, 1)\\&0, \ \ \ x \in (k, 1]. \end{aligned} \right. \end{aligned}$$

(2.6)

It is clear that $Fix(T) = \{k\}$. Hence for $x \in [0, k]$ we have

$$\begin{aligned} |Tx - k|^2 = 0 \le |x - k|^2 + |x - Tx|^2. \end{aligned}$$

Also for $x \in (k, 1]$ we have

$$\begin{aligned} |Tx - k|^2 = k^2 \le |x - k|^2 + |Tx - x|^2. \end{aligned}$$

These show that for $x \in [0, 1]$ we have

$$\begin{aligned} |Tx - k|^2 \le |x - k|^2 + |x - Tx|^2, \end{aligned}$$

i.e., T is a quasi-pseudo-contractive mapping. Also it is easy to see that T is demiclosed.

Definition 2.5

A function $f : C \rightarrow (-\infty , \infty ]$ is said to be convex if for all $x,y \in C$ and all $\lambda \in [0,1]$ the following inequality holds

$$\begin{aligned} f(\lambda x \oplus (1-\lambda ) y) \le \lambda f(x) + (1- \lambda ) f(y). \end{aligned}$$

Lemma 2.6

[9, 10]. Let $f : X \rightarrow (- \infty , \infty ]$ be a proper convex and lower semi-continuous function. For any $\lambda > 0$, define the Moreau-Yosida resolvent of f in CAT(0) space X as

$$\begin{aligned} J_\lambda ^f(x) = argmin_{y \in X}\left[ f(y) + \frac{1}{2\lambda }d^2(y,x)\right] ,\ \ \forall x \in X. \end{aligned}$$

(2.7)

Then

• (i) the set $Fix(J_\lambda ^f)$ of fixed points of the resolvent of f coincides with the set $argmin_{y\in X}f (y)$ of minimizers of f, and for any $\lambda > 0$, the resolvent $J^f_\lambda$ of f is a firmly nonexpansive mapping. Hence it is nonexpansive;

• (ii) Since $J^f_\lambda$ is a firmly nonexpansive mapping, if $Fix(J^f_\lambda ) \not = \emptyset$, then from (2.4) we have

  $$\begin{aligned} d^2(J^f_\lambda x, p) \le d^2(x, p) - d^2(J^f_\lambda x, x), \ \forall x \in X, \ p \in Fix(J^f_\lambda ). \end{aligned}$$

  (2.8)

• (iii) For any $x \in X$, and $\lambda> \mu > 0$, the following identity holds:

  $$\begin{aligned} J_{\lambda }^f(x) = J_{\mu }^f \left( \frac{\lambda -\mu }{\lambda } J^f_{\lambda }(x) \oplus \frac{\mu }{\lambda }x\right) . \end{aligned}$$

Lemma 2.7

(see also [11]). Let X be a complete CAT(0) space and $T : X \rightarrow X$ be a L-Lipschitzian mapping with $L \ge 1$. Let $G: X \rightarrow X$ and $K: X \rightarrow X$ be two mappings defined by

$$\begin{aligned} K(x) := (1 - \xi )x \oplus \xi T(Gx); \ \ \ G(x): = (1-\eta )x \oplus \eta Tx, \;\; x \in X. \end{aligned}$$

(2.9)

If $0< \xi< \eta < \frac{1}{1 + \sqrt{1 + L^2}}$, then the following conclusions hold:

• (1) $Fix(T) = Fix(T(G)) = Fix(K)$;

• (2) If T is $\Delta$-demiclosed, then K is also $\Delta$-demiclosed;

• (3) $K: X \rightarrow X$ is $L^2$-Lipschitzian;

• (4) In addition, if $T : X \rightarrow X$ is quasi-pseudo-contractive, then $K: X \rightarrow X$ is a quasi-nonexpansive mapping, i.e., for any $x \in X$ and $p \in Fix(K)( = Fix(T))$

  $$\begin{aligned} d^2(Kx, p) \le d^2(x,p) - \xi \eta (1- 2\eta - L^2 \eta ^2)d^2(x, Tx) \le d^2(x, p). \end{aligned}$$

  (2.10)

Proof

Now we prove the conclusion (1).

If $x^* \in Fix(T)$, then

$$\begin{aligned}&d(x^*, TGx^*)= d(x^*, T((1-\eta )x^* \oplus \eta Tx^*)\\&= d(x^*, Tx^*) =0, \ \ i.e., x^* \in Fix(TG). \end{aligned}$$

If $x^* \in Fix(TG)$, then

$$\begin{aligned} \begin{aligned} d(x^*, Kx^*)&= d(TG(x^*),(1 - \xi )x^* \oplus \xi TG(x^*))\\&= (1 - \xi )d(TG(x^*), x^*)=0,\ \ i.e., x^* \in Fix(K). \end{aligned} \end{aligned}$$

If $x^* \in Fix(K)$, then

$$\begin{aligned} \begin{aligned} d(x^*, Tx^*)&= d((1 - \xi )x^* \oplus \xi TG(x^*), Tx^*)\\&\le (1 - \xi )d(x^*, Tx^*)+ \xi d(TG(x^*), Tx^*)\\&\le (1 - \xi )d(x^*, Tx^*)+ \xi L d(G(x^*), x^*).\\ \end{aligned} \end{aligned}$$

Simplifying we have

$$\begin{aligned} d(x^*, Tx^*) \le Ld(x^*, Gx^*)= L d(x^*,(1-\eta )x^* \oplus \eta Tx^*) \le L \eta d(x^*, Tx^*). \end{aligned}$$

Since $L \eta < 1$, this implies that $x^* \in Fix(T)$. The conclusion (1) is proved.

Now we prove the conclusion (2).

For any sequence $\{x_n\} \subset X$ with $\Delta -\lim _{n \rightarrow \infty }x_n = x$, and $\lim _{n \rightarrow \infty }d(x_n, Kx_n) = 0$, we show that $x \in Fix(K)$. By conclusion (1), it is sufficient to prove that $x \in Fix(T)$. In fact, since T is L-Lipschizian, we have

$$\begin{aligned} \begin{aligned} d(x_n, Tx_n)&\le d(x_n, Kx_n) + d(Kx_n, Tx_n) = d(x_n, Kx_n) \\&\quad + d((1 - \xi )x_n \oplus \xi T(Gx_n), Tx_n)\\&\le d(x_n, Kx_n) + (1 - \xi ) d(x_n,Tx_n) + \xi d(T(Gx_n), Tx_n).\\ \end{aligned} \end{aligned}$$

Simplifying we have

$$\begin{aligned} \begin{aligned} d(x_n, Tx_n)&\le \frac{1}{\xi }d(x_n, Kx_n) + d(T(Gx_n), Tx_n)\\&\le \frac{1}{\xi }d(x_n, Kx_n) + L d((1-\eta )x_n \oplus \eta Tx_n, x_n)\\&\le \frac{1}{\xi }d(x_n, Kx_n) + L \eta d(Tx_n, x_n).\\ \end{aligned} \end{aligned}$$

This implies that

$$\begin{aligned} (1-L\eta )d(x_n, Tx_n) \le \frac{1}{\xi }d(x_n, Kx_n). \end{aligned}$$

Since $(1-L\eta ) > 0$ and $d(x_n, Kx_n) \rightarrow 0$, this implies that $d(x_n, Tx_n) \rightarrow 0$. Since T is $\Delta$-demiclosed, $x \in Fix(T)$. Hence $x \in Fix(K)$, i.e., K is $\Delta$-demiclosed.

The conclusion (2) is proved.

The conclusion (3) is obvious, the proof is omitted.

Now we prove the conclusion (4).

For any $p \in Fix(T)$ and $x \in X$, it follows from (2.1) that

$$\begin{aligned} \begin{aligned} d^2(Kx,p)&= d^2((1 - \xi )x \oplus \xi T((1-\eta )x \oplus \eta Tx), p)\\&\le (1 - \xi )d^2(x, p) + \xi d^2(T((1-\eta )x \oplus \eta Tx), p)\\&\quad -\xi (1 - \xi ) d^2(x,T((1-\eta )x \oplus \eta Tx)).\\ \end{aligned} \end{aligned}$$

(2.11)

Since T is quasi-pseudo-contractive, we have

$$\begin{aligned} \begin{aligned}&d^2(T((1-\eta )x \oplus \eta Tx), p) \le d^2((1-\eta )x \oplus \eta Tx), p)\\&\quad + d^2((1-\eta )x \oplus \eta Tx),T((1-\eta )x \oplus \eta Tx)).\\ \end{aligned} \end{aligned}$$

(2.12)

From (2.1), we have

$$\begin{aligned} \begin{aligned} d^2((1-\eta )x \oplus \eta Tx), p)&\le (1-\eta )d^2(x, p) + \eta d^2(Tx, p) -\eta (1-\eta )d^2(x, Tx)\\&\le (1-\eta )d^2(x, p) \\&\quad + \eta \{d^2(x, p)+ d^2(x, Tx)\} -\eta (1-\eta )d^2(x, Tx)\\&= d^2(x, p) + \eta ^2 d^2(x, Tx), \\ \end{aligned} \end{aligned}$$

(2.13)

and

$$\begin{aligned} \begin{aligned} d^2((1-\eta )x&\oplus \eta Tx, T((1-\eta )x \oplus \eta Tx)) \le (1-\eta )d^2(x, T((1-\eta )x \oplus \eta Tx))\\&\quad + \eta d^2(Tx,T((1-\eta )x \oplus \eta Tx))\\&\quad - \eta (1-\eta )d^2(x, Tx)\\&\le (1-\eta )d^2(x, T((1-\eta )x \oplus \eta Tx)) + \eta L^2 d^2(x, (1-\eta )x \\&\quad \oplus \eta Tx)- \eta (1-\eta )d^2(x, Tx)\\&\le (1-\eta )d^2(x, T((1-\eta )x \oplus \eta Tx)) \\&\quad + \eta ^3 L^2 d^2(x, Tx)- \eta (1-\eta )d^2(x, Tx)\\&\le (1-\xi )d^2(x, T((1-\eta )x \oplus \eta Tx)) \\&\quad - \eta ((1-\eta -L^2\eta ^2)d^2(x, Tx) \ (since \ \xi < \eta ).\\ \end{aligned} \end{aligned}$$

(2.14)

Substituting (2.13) and (2.14) into (2.12), after simplifying we have

$$\begin{aligned} \begin{aligned} d^2(T((1-\eta )x \oplus \eta Tx), p)&\le d^2(x, p) + (1-\xi )d^2(x, T((1-\eta )x \oplus \eta Tx))\\&\quad - \eta (1- 2\eta - L^2 \eta ^2)d^2(x, Tx).\\ \end{aligned} \end{aligned}$$

(2.15)

Substituting (2.15) into (2.11), and after simplifying we have

$$\begin{aligned} d^2(Kx,p) \le d^2(x, p) - \xi \eta (1- 2\eta - L^2 \eta ^2)d^2(x, Tx) \le d^2(x, p). \end{aligned}$$

This completes the proof of Lemma 2.7. $\square$

Lemma 2.8

[12]. If $\{a_n\}$ is a sequence of real numbers and there exists a subsequence $\{n_i\}$ of $\{n\}$ such that $a_{n_i} < a_{n_i+1}$ for all $i \in N$, then there exists a nondecreasing sequence $\{m_k\} \subset N$ such that $m_k \rightarrow \infty$ and the following properties are satisfied:

$$\begin{aligned} a_{m_k} \le a_{m_k+1}\ and \ a_k \le a_{m_k+1}. \end{aligned}$$

for all sufficiently large numbers $k \in N$. In fact, $m_k = max\{j \le k : a_j < a_{j+1}\}$.

Lemma 2.9

([13]). If $\{a_n\}$ is a sequence of nonnegative real numbers satisfying the following conditions:

$$\begin{aligned} a_{n+1} \le (1 - \delta _n)a_n + \delta _n \sigma _n + \gamma _n, n \ge 0. \end{aligned}$$

where $\{\delta _n\} \subset [0, 1]$, $\sum _{n=0}^{\infty } \delta _n = \infty$; $\limsup _{n \rightarrow \infty } \sigma _n \le 0$; $\gamma _n \ge 0$, and $\sum _{n = 0}^\infty \gamma _n < \infty$. Then $a_n \rightarrow 0, \ as \ n \rightarrow \infty$.

3 Main results

Throughout this section we assume that

• (1) (X, d) is a complete CAT(0) space;

• (2) $f:X \rightarrow (-\infty ,+ \infty ]$ is a proper convex and lower semi-continuous function, and $J_{\lambda _n}^f: X \rightarrow X$ is the Moreau-Yosida resolvent of f;

• (3) $T: X \rightarrow X$ is an L-Lipschitzian quasi-pseudo contractive mapping with $L \ge 1$ and T is $\Delta$-demiclosed;

• (4) Define the mappings $G: X \rightarrow X$ and $K: X \rightarrow X$ by

  $$\begin{aligned} K(x) := (1 - \xi )x \oplus \xi T(Gx); \ \ \ G(x): = (1-\eta )x \oplus \eta Tx, \;\; x \in X, \end{aligned}$$

  (3.1)

  where $0< \xi< \eta < \frac{1}{1 + \sqrt{1 + L^2}}$.

Theorem 3.1

Let $(X,d), \ f, \ J_{\lambda _n}^f, \ T,\ K, \ G$ satisfy the conditions (1)-(4) as above. Let $u \in X$ be a given point. For any given point $x_1\in X$, let $\{x_n\}$ be the sequence generated by

$$\begin{aligned} \left\{ \begin{aligned} z_n&= J_{\lambda _n}^f (x_n) : = argmin_{y \in X}\left[ f(y)+ \frac{1}{2\lambda _n} d^2(y, x_n)\right] ,\\ y_n&= \alpha _n u \oplus (1-\alpha _n)z_n,\\ x_{n+1}&= (1-\beta _n)z_n \oplus \beta _n Ky_n, \end{aligned} \right. \ \ \ n \ge 1. \end{aligned}$$

(3.2)

If $\Omega : = Fix(T)\bigcap argmin_{y \in X} f(y) \not = \emptyset$ and the sequences $\{\alpha _n\}$, $\{\beta _n\}$ and $\{\lambda _n\}$ satisfy the following conditions:

• (c1) $\{\alpha _n\} \subset (0, 1)$, $\lim _{n \rightarrow \infty }\alpha _n = 0$ and $\sum _{n=1}^\infty \alpha _n = \infty$,

• (c2) $0< \varepsilon \le \beta _n \le b < 1, \ \lambda _n> \lambda > 0, \ \forall n \ge 1$, where $\varepsilon , \ b$ and $\lambda$ are some positive constants,

then the sequence $\{x_n\}$ converges strongly to some point in $\Omega$.

Proof

First we observe that by the assumptions of Theorem 3.1, Lemma 2.7 and Lemma 2.5 we know that

• (a) the mapping $K: X \rightarrow X$ is quasi-nonexpansive, $\Delta$-demiclosed, $L^2$-Lipschitzian and $Fix(T) = Fix(K)$;

• (b) $J^f_{\lambda _n}$ is nonexpansive, so it is $\Delta$-demiclosed, and $Fix(J_{\lambda _n}^f) = argmin_{y\in X}f (y)$.

(I) Now we prove that the sequence $\{x_n\}$ is bounded.

In fact, if $p\in \Omega$, then $p = J^f_{\lambda _n}(p), \forall n \ge 1$, and $p \in Fix(T)=Fix(K)$. Since $J^f_{\lambda _n}$ is a nonexpansive mapping, we have

$$\begin{aligned} d(z_n, p) = d(J^f_{\lambda _n}(x_n), J^f_{\lambda _n}(p)) \le d(x_n, p). \end{aligned}$$

(3.3)

It follows from (3.2), (2.1) and Lemma 2.1(ii) that

$$\begin{aligned} \begin{aligned} d^2(y_n, p)&= d^2(\alpha _n u \oplus (1-\alpha _n)z_n, p)\\&\le \alpha _n d^2(u, p) +(1-\alpha _n)d^2(z_n, p) - \alpha _n (1-\alpha _n)d^2(u, z_n)\\&= \alpha _n^2 d^2(u, p) +(1-\alpha _n)^2d^2(z_n, p) + 2\alpha _n (1-\alpha _n)\langle \overrightarrow{up}, \overrightarrow{z_np} \rangle .\\ \end{aligned} \end{aligned}$$

(3.4)

Also from (3.2), (3.3) and (3.4), we have

$$\begin{aligned} \begin{aligned} d^2(x_{n+1}, p)&= d^2((1-\beta _n)z_n \oplus \beta _n Ky_n, p)\\&\le (1-\beta _n)d^2(z_n, p) + \beta _n d^2(Ky_n, p) - \beta _n (1-\beta _n)d^2(z_n, Ky_n)\\&\le (1-\beta _n)d^2(z_n, p) + \beta _n d^2(y_n, p) - \beta _n (1-\beta _n)d^2(z_n, Ky_n)\\&\le (1-\beta _n)d^2(z_n, p) + \beta _n \{\alpha _n d^2(u, p) +(1-\alpha _n)d^2(z_n, p)\\&\quad - \alpha _n (1-\alpha _n)d^2(u, z_n)\} - \beta _n (1-\beta _n)d^2(z_n, Ky_n).\\ \end{aligned} \end{aligned}$$

(3.5)

This implies that

$$\begin{aligned} \begin{aligned} d^2(x_{n+1}, p)&\le (1-\alpha _n\beta _n)d^2(x_n, p) + \alpha _n \beta _n d^2(u, p)\\&\le max\{d^2(x_n, p), d^2(u, p)\}. \end{aligned} \end{aligned}$$

By induction, we can prove that

$$\begin{aligned} d^2(x_{n}, p) \le max\{d^2(x_1, p), d^2(u, p)\},\ \ \forall n \ge 1. \end{aligned}$$

This implies that $\{x_n\}$ is a bounded sequence. So are $\{z_n\}$, $\{y_n\}$ and $\{Ky_n\}$. The conclusion (I) is proved. $\square$

(II) Next we prove that $\{x_n\}$ converges strongly to some point in $\Omega$.

In fact, from (3.2) and (3.4) we have

$$\begin{aligned} \begin{aligned} d^2(x_{n+1}, p)&= d^2((1-\beta _n)z_n \oplus \beta _n Ky_n, p)\\&\le (1-\beta _n)d^2(z_n, p) + \beta _n d^2(Ky_n, p) - \beta _n (1-\beta _n)d^2(z_n, Ky_n)\\&\le (1-\beta _n)d^2(z_n, p) + \beta _n d^2(y_n, p) - \beta _n (1-\beta _n)d^2(z_n, Ky_n)\\&\le (1-\beta _n)d^2(z_n, p) - \beta _n (1-\beta _n)d^2(z_n, Ky_n)\\&\quad + \beta _n \{\alpha _n^2 d^2(u, p) +(1-\alpha _n)^2 d^2(z_n, p) + 2\alpha _n (1-\alpha _n)\langle \overrightarrow{up}, \overrightarrow{z_np} \rangle \}\\&\le (1-\alpha _n \beta _n)d^2(z_n, p) + \alpha _n \beta _n \{\alpha _n d^2(u, p) + 2 (1-\alpha _n)\langle \overrightarrow{up}, \overrightarrow{z_np} \rangle \}\\&\quad - \beta _n (1-\beta _n)d^2(z_n, Ky_n)\\&\le (1-\alpha _n \beta _n)d^2(x_n, p) + \alpha _n \beta _n \{\alpha _n d^2(u, p) + 2 (1-\alpha _n)\langle \overrightarrow{up}, \overrightarrow{z_np} \rangle \}\\&\quad - \beta _n (1-\beta _n)d^2(z_n, Ky_n).\\&= d^2(x_n, p) + \alpha _n \beta _n \xi _n - \beta _n (1-\beta _n)d^2(z_n, Ky_n),\\ \end{aligned}\nonumber \\ \end{aligned}$$

(3.6)

where

$$\begin{aligned} \xi _n = - d^2(x_n, p) + \alpha _n d^2(u, p) + 2 (1-\alpha _n)\langle \overrightarrow{up}, \overrightarrow{z_np} \rangle . \end{aligned}$$

After simplifying and putting $M = \sup _{n \ge 1}|\xi _n|$, then we have

$$\begin{aligned} \beta _n (1-\beta _n)d^2(z_n, Ky_n) \le d^2(x_n, p) - d^2(x_{n+1}, p) + \alpha _n \beta _n M, \end{aligned}$$

(3.7)

Now we consider the following two cases:

Case 1: Assume that $\{d(x_n, p)\}$ is eventually nonincreasing. Hence there exists a sufficiently large positive integer $n_0$ such that $d(x_{n+1}, p) \le d(x_n, p), \ \forall n \ge n_0$. Since $\{x_n\}$ is bounded, the limit $\lim _{n\rightarrow \infty } d(x_n, p)$ exists. Since $\alpha _n \rightarrow 0$ and $0< \varepsilon \le \beta _n \le b < 1$ (by conditions $(c_1)$ and $(c_2)$), from (3.7) we have that

$$\begin{aligned} d(z_n, Ky_n) \rightarrow 0,\ as \ n \rightarrow \infty . \end{aligned}$$

(3.8)

Also from (3.2) and Lemma 2.1(i), we obtain

$$\begin{aligned} \begin{aligned} d(y_n, z_n)&= (\alpha _n u \oplus (1-\alpha _n)z_n, z_n)\\&\le \alpha _n d(u,z_n) \rightarrow 0, \ as \ n \rightarrow \infty .\\ \end{aligned} \end{aligned}$$

(3.9)

From (3.8) and (3.9) we have

$$\begin{aligned} d(y_n, Ky_n) \rightarrow 0, \ as \ n \rightarrow \infty . \end{aligned}$$

(3.10)

On the other hand, from (2.8) we have

$$\begin{aligned} d^2(x_n, z_n) \le d^2(x_n, p) - d^2(z_n, p). \end{aligned}$$

(3.11)

Also, it follows from (3.5) that

$$\begin{aligned} d^2(x_{n+1}, p)\le (1-\beta _n)d^2(x_n, p) + \beta _n \{\alpha _n d^2(u, p) +(1-\alpha _n)d^2(z_n, p)\}.\\ \end{aligned}$$

This implies that

$$\begin{aligned} d^2(x_n, p) \le \frac{1}{\beta _n}\{d^2(x_n, p) - d^2(x_{n+1}, p)\} + \alpha _n d^2(u, p) +(1-\alpha _n)d^2(z_n, p).\nonumber \\ \end{aligned}$$

(3.12)

Substituting (3.12) into (3.11), after simplifying we have

$$\begin{aligned} d^2(x_n, z_n) \le \frac{1}{\beta _n} (d^2(x_n, p) - d^2(x_{n+1}, p)) + \alpha _n (d^2(u, p) - d^2(z_n, p)). \end{aligned}$$

Since $\{z_n\}$ is bounded, $\{d(x_n, p)\}$ is convergent and $\alpha _n \rightarrow 0$, these show that

$$\begin{aligned} \lim _{n \rightarrow \infty } d(x_n, z_n) = 0. \end{aligned}$$

(3.13)

Hence from (3.9)–(3.13) and Lemma 2.7 (3) we have

$$\begin{aligned} \begin{aligned} d(x_n, Kx_n)&\le d(x_n, z_n) + d(z_n, y_n) + d(y_n, Ky_n) + d(Ky_n, Kx_n)\\&\le d(x_n, z_n) + d(z_n, y_n)\\&\quad + d(y_n, Ky_n) + L^2 d(y_n, x_n) \rightarrow 0, \ as \ n \rightarrow \infty .\\ \end{aligned} \end{aligned}$$

(3.14)

As $\lambda _n \ge \lambda > 0,$ so by Lemma 2.5 (iii) and (3.13), we have

$$\begin{aligned} \begin{aligned}&d(J_{\lambda }^f(x_n), z_n) = d(J_{\lambda }^f(x_n), J_{\lambda _n}^f(x_n))\\&\quad = d(J_{\lambda }^f(x_n),J_{\lambda }^f\left( \frac{\lambda _n -\lambda }{\lambda _n}J_{\lambda _n}^f (x_n) \oplus \frac{\lambda }{\lambda _n}x_n\right) \\&\quad \le d\left( x_n,\frac{\lambda _n -\lambda }{\lambda _n}J_{\lambda _n}^f (x_n) \oplus \frac{\lambda }{\lambda _n}x_n\right) \\&\quad \le \left( 1 - \frac{\lambda }{\lambda _n}\right) d(x_n, J_{\lambda _n}^f (x_n))\\&\quad = \left( 1 - \frac{\lambda }{\lambda _n}\right) d(x_n, z_n) \rightarrow 0, \ as \ n \rightarrow \infty . \end{aligned} \end{aligned}$$

This together with (3.13) shows that

$$\begin{aligned} d(x_n, J_{\lambda }^f (x_n)) \le d(x_n, z_n)+ d(z_n,J_{\lambda }^f (x_n)) \rightarrow 0, \ as \ n \rightarrow \infty . \end{aligned}$$

(3.15)

Since $\{x_n\}$ is bounded, there exists a subsequence $\{x_{n_i}\} \subset \{x_n\}$ such that $\Delta -\lim _{i \rightarrow \infty } x_{n_i}= x^* \in X$ and

$$\begin{aligned} \limsup _{n \rightarrow \infty }\langle \overrightarrow{ux^*}, \overrightarrow{x_nx^*}\rangle = \limsup _{i \rightarrow \infty }\langle \overrightarrow{ux^*}, \overrightarrow{x_{n_i}x^*}\rangle . \end{aligned}$$

(3.16)

Since $\limsup _{i \rightarrow \infty }\langle \overrightarrow{ux^*}, \overrightarrow{x_{n_i}x^*}\rangle \le 0$ by Remark 2.1, which shows

$$\begin{aligned} \limsup _{n \rightarrow \infty }\langle \overrightarrow{ux^*}, \overrightarrow{x_nx^*}\rangle \le 0. \end{aligned}$$

(3.17)

By virtue of (3.13), (3.16) and Cauchy-Schwarz inequality we obtain

$$\begin{aligned} \begin{aligned} \limsup _{n \rightarrow \infty }&\langle \overrightarrow{ux^*}, \overrightarrow{z_nx^*} \rangle \le \limsup _{n \rightarrow \infty }\langle \overrightarrow{ux^*}, \overrightarrow{z_nx_n} \rangle + \limsup _{n \rightarrow \infty }\langle \overrightarrow{ux^*}, \overrightarrow{x_nx^*} \rangle \\&\le \limsup _{n \rightarrow \infty } d(u, x^*) d(z_n, x_n) \rangle + \limsup _{n \rightarrow \infty }\langle \overrightarrow{ux^*}, \overrightarrow{x_nx^*} \rangle \le 0.\\ \end{aligned} \end{aligned}$$

(3.18)

On the other hand, since K is $\Delta$-demiclosed, from (3.14), $x^* \in Fix(K)$. Also since $J_{\lambda }^f$ is nonexpansive, it is also $\Delta$-demiclosed. From (3.15) $x^* \in Fix(J_{\lambda }^f)$. Hence $x^* \in \Omega$.

Taking $p = x^*$ in (3.6), we obtain

$$\begin{aligned}&d^2(x_{n+1}, x^*) \le (1-\alpha _n \beta _n)d^2(x_n, x^*) + \alpha _n \beta _n \{\alpha _n d^2(u, x^*) \nonumber \\&\quad + 2 (1-\alpha _n)\langle \overrightarrow{ux^*}, \overrightarrow{z_nx^*} \rangle \}. \end{aligned}$$

(3.19)

Putting $a_n = d^2(x_n, x^*), \ \delta _n = \alpha _n \beta _n, \ \sigma _n = \alpha _n d^2(u, x^*) + 2 (1-\alpha _n)\langle \overrightarrow{ux^*}, \overrightarrow{z_nx^*} \rangle$ and $\gamma _n = 0$ in Lemma 2.9, we obtain that $d(x_n, x^*) \rightarrow 0$, i.e., $x_n \rightarrow x^* \in \Omega$.

Case 2: Assume that $\{d(x_n, p)\}$ is not eventually nonincreasing. Hence there exists a subsequence $\{n_i\} \subset \{n\}$ such that

$$\begin{aligned} d(x_{n_i}, p) < d(x_{n_i+1}, p), \ \ \forall i \in N. \end{aligned}$$

Hence by Lemma 2.8, there exists an increasing sequence $\{m_j\}, \ j\ge 1$ $m_j \rightarrow \infty$, such that

$$\begin{aligned} d(x_{m_j}, p) \le d(x_{m_j+1}, p), \ and \ \ d(x_j, p) \le d(x_{m_j +1}, p), \ \forall j \ge 1. \end{aligned}$$

(3.20)

Also from (3.7) and the fact that $\alpha _{m_j} \rightarrow 0, \ as \ m_j \rightarrow \infty$ we obtain $d(z_{m_j}, Ky_{m_j}) \rightarrow 0, \ as \ j \rightarrow \infty .$ Following arguments similar to those in the proof of Case 1, we can get

$$\begin{aligned} \limsup _{j \rightarrow \infty }\langle \overrightarrow{ux^*}, \overrightarrow{x_{m_j} x^*}\rangle \le 0. \end{aligned}$$

(3.21)

Also from the inequality (3.6) we obtain that

$$\begin{aligned} \begin{aligned} d^2(x_{m_j +1}, x^*)&\le (1-\alpha _{m_j} \beta _{m_j})d^2(x_{m_j}, x^*) \\&\quad + \alpha _{m_j} \beta _{m_j} \{\alpha _{m_j} d^2(u, x^*) + 2 (1-\alpha _{m_j})\langle \overrightarrow{ux^*}, \overrightarrow{z_{m_j}x^*} \rangle \}.\\ \end{aligned} \end{aligned}$$

(3.22)

After simplifying we have

$$\begin{aligned} \begin{aligned} \alpha _{m_j} \beta _{m_j} d^2(x_{m_j}, x^*)&\le d^2(x_{m_j}, x^*) - d^2(x_{m_j +1}, x^*) + \alpha _{m_j} \beta _{m_j} \{\alpha _{m_j} d^2(u, x^*)\\&\quad + 2 (1-\alpha _{m_j})\langle \overrightarrow{ux^*}, \overrightarrow{z_{m_j}x^*} \rangle \}\\&\le \alpha _{m_j} \beta _{m_j} \{\alpha _{m_j} d^2(u, x^*) + 2 (1-\alpha _{m_j})\langle \overrightarrow{ux^*}, \overrightarrow{z_{m_j}x^*} \rangle \}.\\ \end{aligned} \end{aligned}$$

This implies that $d^2(x_{m_j}, x^*) \rightarrow 0, \ as \ j \rightarrow \infty$. From (3.22) it follows that $d^2(x_{m_j+1}, x^*) \rightarrow 0, \ as \ j \rightarrow \infty$. Hence from (3.20) we have that

$$\begin{aligned} \lim _{j \rightarrow \infty } d(x_j, x^*) \le \lim _{j \rightarrow \infty } d(x_{m_j +1}, x^*) = 0, \ i.e., \ \lim _{j \rightarrow \infty } x_j = x^* \in \Omega . \end{aligned}$$

This completes the proof of Theorem 3.1. $\square$

In Theorem 3.1, if the mapping $T: X \rightarrow X$ is replaced by a k-demicontractive mapping, then the following result can be obtained from Theorem 3.1 immediately.

Corollary 3.2

Let $(X,d), \ f, \ J_{\lambda _n}^f$ be the same as in Theorem 3.1. Let $T: X \rightarrow X$ be a L-Lipschitzian, k-demicontractive and $\Delta$-demiclosed mapping with $L \ge 1$ and $k \in (0, 1)$. Denote the mapping $S : X \rightarrow X$ by

$$\begin{aligned} Sx : = \delta x \oplus (1-\delta )Tx, \ x \in X, \ 0< k \le \delta < 1. \end{aligned}$$

(3.23)

Let $u \in X$ be a given point. For any given point $x_1\in X$, let $\{x_n\}$ be the sequence generated by

$$\begin{aligned} \left\{ \begin{aligned} z_n&= J_{\lambda _n}^f (x_n) : = argmin_{y \in X}\left[ f(y)+ \frac{1}{2\lambda _n} d^2(y, x_n)\right] ,\\ y_n&= \alpha _n u \oplus (1-\alpha _n)z_n,\\ x_{n+1}&= (1-\beta _n)z_n \oplus \beta _n Sy_n,\\ \end{aligned} \right. \ \ \forall \ n \ge 1. \end{aligned}$$

(3.24)

If $\Omega : = Fix(T)\bigcap argmin_{y \in X} f(y) \not = \emptyset$ and the sequences $\{\alpha _n\}$, $\{\beta _n\}$ and $\{\lambda _n\}$ satisfying the following conditions:

• (c1) $\{\alpha _n\} \subset (0, 1)$, $\lim _{n \rightarrow \infty }\alpha _n = 0$ and $\sum _{n=1}^\infty \alpha _n = \infty$;

• (c2) $0< \varepsilon \le \beta _n \le b < 1, \ \lambda _n> \lambda > 0, \ \forall n \ge 1$, where $\lambda , \ b$ and $\varepsilon$ are some positive constants,

then the sequence $\{x_n\}$ converges strongly to some point in $\Omega$.

Proof

In order to prove Corollary 3.2, it is sufficient to prove that the mapping $S: X \rightarrow X$ defined by (3.23) has the following properties:

(1) Fix(T)=Fix(S); (2) S is demiclosed; (3) S is L-lipschitzian; (4) S is a quasi-nonexpensive mappings.

It is easy to prove that S has the properties (1)-(3). Next we prove that S has the property (4). In fact, since $Fix(T) = Fix(S)$, hence for any $p\in Fix(T) = Fix(S)$ and $x \in X$ it follows from (3.23) that

$$\begin{aligned} \begin{aligned} d^2(Sx, p)&= d^2(\delta x \oplus (1-\delta )Tx, \ p) \\&\le \delta d^2(x, p) + (1-\delta )d^2(Tx,\ p) - \delta (1-\delta )d^2(x, Tx)\\&\le \delta d^2(x, p) + (1-\delta )\{d^2(x,\ p) + k d^2(x, Tx)\} - \delta (1-\delta )d^2(x, Tx)\\&= d^2(x, p) + (1-\delta )(k - \delta )d^2(x, Tx) \le d^2(x, p) \ (since \ k \le \delta ). \end{aligned} \end{aligned}$$

(4) is proved. This completes the proof of Corollary 3.2. $\square$

Remark 3.3

Theorem 3.1 not only corrects some basic errors in Ugwunnadi et al. [1], but also extends the main results in [1] from k-demi-contractive mappings to quasi-pseudo-contractive mappings in CAT(0) space. Theorem 3.1 extends the result of Ba$\breve{c}\acute{a}$k [14] from weak convergence to strong convergence and the result of Cholamjiak et al. [15] from nonexpanvive mapping to Lipschitzian quasi-pseudo mapping. Also Theorem 3.1 extended the result in [16] from strict pseudo-contractive mapping in a real Hilbert space to Lipschitzian quasi-pseudo mapping in a more general space than Hilbert space. We studied a new hybrid proximal point algorithm for solving convex minimization problem as well as fixed point problem of Lipschitzian quasi-pseudo mapping in CAT(0) spaces. Our method of proof is different from that of Cholamjiak et al. [15] and Chang et al. [17].

4 Applications

Throughout this section we assume that (X, d) is a complete CAT(0) space and C is a non-empty closed and convex subset of X.

4.1 Application to convex minimization problem and equilibrium problem in CAT(0) space

The “so called” equilibrium problem for a bifunction $F: C \times C \rightarrow {\mathbb {R}}$ is to find a $x^* \in C$ such that

$$\begin{aligned} F(x^*, y) \ge 0,\ \forall y \in C, \end{aligned}$$

(4.1)

where $F: C \rightarrow {\mathbb {R}}$ satisfies the following conditions:

• (A1) $F (x, x) = 0, \ \forall x \in C$;

• (A2) F is monotone, i.e. , $F (x, y) + F ( y, x) \le 0, \ \forall x, y \in C$;

• (A3) The function $y \mapsto F(x, y)$ is convex for all $x \in C$;

By using F we define a mapping $T_r: X \rightarrow C,\ r > 0$ as follows:

$$\begin{aligned} T_r(x): = \{z \in C: F(z,y) - \frac{1}{r} \langle {\overline{yz}}, \ {\overline{xz}}\rangle \}\ge 0, \ \forall y \in C\}. \end{aligned}$$

(4.2)

We have the following result

Lemma 4.1

[18] Let C be a nonempty closed convex subset of a complete CAT(0) space X. Let $F: C \times C \rightarrow {\mathbb {R}}$ be a bifunction satisfying the conditions (A1)–(A3). If the following condition is satisfied

(A4) For each ${\bar{x}} \in X$ and $r > 0$ , there exists a compact subset $D_{{\bar{x}}} \subset C$ containing a point $y_{{\bar{x}}}\in D_{{\bar{x}}} \subset C$ such that

$$\begin{aligned} F(x,y_{{\bar{x}}})- \frac{1}{r}\langle \overline{xy_{{\bar{x}}}}, \ \overline{x{\bar{x}}}\rangle < 0, \forall x \in C\setminus D_{{\bar{x}}}, \end{aligned}$$

then, the following conclusions hold:

• (a) $T_r$ is well defined in X and $T_r$ is a single-valued mapping;

• (b) $T_r$ is firmly nonexpansive restricted to C, i.e., $\forall x, y \in C$

  $$\begin{aligned} d^2(T_r x,\ T_r y ) \le \langle \overrightarrow{T_r x T_r y}, \ {\overline{xy}} \rangle ; \end{aligned}$$

  Therefore $T_r$ is a a nonexpansive (i.e., 1-Lipschitzian) and demiclosed mapping restricted to C. In addition, if $Fix(T_r) \not = \emptyset$, then $T_r$ is quasi-nonexpansive.

• (c) $Fix (T_r) = \Omega _1$, where $\Omega _1$ is the solution set of problem (4.1);

• (d) If $Fix(T_r)\not = \emptyset$, we have

  $$\begin{aligned} d^2(T_r x, x) \le d^2(x,p) - d^2(T_r x, p), \ \forall x \in C, \ and \ \forall p \in Fix(T_r). \end{aligned}$$

Taking $T = K = T_r$ in Theorem 3.1, then the following theorem can be obtained from Theorem 3.1 and Lemma 4.1 immediately.

Theorem 4.2

Let X be a complete CAT(0) space, C be a nonempty closed and convex subset of X. Let $f:C\rightarrow {\mathbb {R}}$ be a proper convex and lower semi-continuous function, and $J_{\lambda _n}^f: C \rightarrow C$ be the Moreau-Yosida resolvent of f. Let $F: C \times C \rightarrow {\mathbb {R}}$ be a bifunction satisfying the conditions (A1)- (A4) and $T_r, \ r \ >0$ be the mapping defined by (4.2). Let $u \in X$ be a given point. For any given point $x_1\in X$, let $\{x_n\}$ be the sequence generated by

$$\begin{aligned} \left\{ \begin{aligned} z_n&= J_{\lambda _n}^f (x_n) : = argmin_{y \in X}\left[ f(y)+ \frac{1}{2\lambda _n} d^2(y, x_n)\right] ,\\ y_n&= \alpha _n u \oplus (1-\alpha _n)z_n,\\ x_{n+1}&= (1-\beta _n)z_n \oplus \beta _n T_r y_n, \end{aligned} \right. \ \ \ n \ge 1. \end{aligned}$$

(4.3)

If $\Omega _2: = Fix(T_r)\bigcap argmin_{y \in X} f(y) \not = \emptyset$ and the sequences $\{\alpha _n\}$, $\{\beta _n\}$ and $\{\lambda _n\}$ satisfy the following conditions:

• (c1) $\{\alpha _n\} \subset (0, 1)$, $\lim _{n \rightarrow \infty }\alpha _n = 0$ and $\sum _{n=1}^\infty \alpha _n = \infty$,

• (c2) $0< \varepsilon \le \beta _n \le b < 1, \ \lambda _n> \lambda > 0, \ \forall n \ge 1$, where $\varepsilon , \ b$ and $\lambda$ are some positive constants,

  then (1) the sequence $\{x_n\}$ converges strongly to some point in $\Omega _2$. (2) Especially, if $f \equiv 0$, then $J_{\lambda _n}^f =I$ (identity mapping) for all $n \ge 1$. Hence the sequence $\{x_n\}$ defined by

  $$\begin{aligned} \left\{ \begin{aligned} y_n&= \alpha _n u \oplus (1-\alpha _n)x_n,\\ x_{n+1}&= (1-\beta _n)x_n \oplus \beta _n T_r y_n, \end{aligned} \right. \ \ \ n \ge 1. \end{aligned}$$

  (4.4)

  converges strongly to a solution of equilibrium problem (4.1).

4.2 Application to saddle point problem in CAT(0) spaces

Let $X_1$ and $X_2$ be complete CAT(0) spaces. Then the product space $X = X_1 \times X_2$ is also a complete CAT(0) space (see [19, Page 239]. A function $H : X_1 \times X_2 \rightarrow {\mathbb {R}}$ is called a saddle function if

• (i) $y \mapsto H(x, y)$ is convex on $X_2$ for each $x \in X_1$ and

• (ii) $x\mapsto H(x, y)$ is concave, i.e., $x \mapsto - H(x, y)$ is convex on $X_1$ for each $y \in X_2$.

  A point $z^* = (x^*, y^*) \in X_1 \times X_2$ is said to be a saddle point of H if

  $$\begin{aligned} H(x, y^*) \le H(x^*, y^*) \le H(x^*, y), \ \forall z = (x, y) \in X_1 \times X_2. \end{aligned}$$

  (4.5)

  We denote by $\Omega _3$ the set of saddle points of problem (4.5). Let $V_H : X= X_1 \times X_2 \rightarrow 2^{X_1^*} \times 2^{X_2^*}$ be a multivalued mapping associated with saddle function H (where $X^*_i$ is the dual space of $X_i, \ i = 1,2,$ (see, (2.4)) defined by

  $$\begin{aligned} V_H(x, y) = \partial (-H(., y))(x) \times \partial (H(x, .))(y),\ \forall (x,y) \in X_1 \times X_2, \end{aligned}$$

  (4.6)

  Let us define the resolvent $J_{\lambda }^{V_H} : X = X_1 \times X_2 \rightarrow 2^{X_1 \times X_2}$ of $V_H$ of order $\lambda > 0$ by

  $$\begin{aligned} J_{\lambda }^{V_H}(x): = \{z \in X: [\frac{1}{\lambda }\overrightarrow{zx}] \in V_H(z)\},\ x \in X= X_1 \times X_2. \end{aligned}$$

  (4.7)

The following results hold.

Lemma 4.3

[20] Let $X_1$ and $X_2$ be complete CAT(0) spaces, H be a saddle function on $X = X_1 \times X_2$ and $V_H$ be the multivalued mapping defined by (4.6). Then

• (1) $J_{\lambda }^{V_H}: X \rightarrow X, \ \lambda > 0$ is a single-valued and firmly nonexpansive mapping;

• (2) A point $z^* = (x^*, y^*) \in X$ is a saddle point of H if and only if $z^* \in Fix(J_{\lambda }^{V_H})$.

In Theorem 3.1, taking $f \equiv 0, \ T = K = J_{\lambda }^{V_H}$, then the following result can be obtained from Theorem 3.1 immediately.

Theorem 4.4

Let $X_1$, $X_2, \ X$, H, $V_H$ and $J_{\lambda }^{V_H}$ be the same as in Lemma 4.3. Let $u \in X$ be a given point. For any given point $x_1\in X$, let $\{x_n\}$ be the sequence generated by

$$\begin{aligned} \left\{ \begin{aligned} y_n&= \alpha _n u \oplus (1-\alpha _n)x_n,\\ x_{n+1}&= (1-\beta _n)x_n \oplus \beta _n J_{\lambda }^{V_H}(y_n), \end{aligned} \right. \ \ \ n \ge 1. \end{aligned}$$

(4.8)

If $Fix(J_{\lambda }^{V_H}) \not = \emptyset$ and the sequences $\{\alpha _n\}$, and $\{\beta _n\}$ satisfy the following conditions:

• (c1) $\{\alpha _n\} \subset (0, 1)$, $\lim _{n \rightarrow \infty }\alpha _n = 0$ and $\sum _{n=1}^\infty \alpha _n = \infty$,

• (c2) $0< \varepsilon \le \beta _n \le b < 1, \ \forall n \ge 1$, where $\varepsilon , \ b$ are some positive constants,

  then the sequence $\{x_n\}$ converges strongly to a saddle point of problem (4.5).