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].
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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 \(x_1 \in X\),
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
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
for all \(a,b,c,d\in X\). It can be easily verified that
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
Let X be a CAT(0) space. We say that X satisfies the Cauchy-Schwarz inequality if
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
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
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
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
The set
is a metric space which is called the dual space of (X; d) with metric
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:
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:
It is clear that \(Fix(T) = \{k\}\). Hence for \(x \in [0, k]\) we have
Also for \(x \in (k, 1]\) we have
These show that for \(x \in [0, 1]\) we have
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
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
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
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
If \(x^* \in Fix(TG)\), then
If \(x^* \in Fix(K)\), then
Simplifying we have
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
Simplifying we have
This implies that
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
Since T is quasi-pseudo-contractive, we have
From (2.1), we have
and
Substituting (2.13) and (2.14) into (2.12), after simplifying we have
Substituting (2.15) into (2.11), and after simplifying we have
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:
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:
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
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
It follows from (3.2), (2.1) and Lemma 2.1(ii) that
Also from (3.2), (3.3) and (3.4), we have
This implies that
By induction, we can prove that
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
where
After simplifying and putting \(M = \sup _{n \ge 1}|\xi _n|\), then we have
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
Also from (3.2) and Lemma 2.1(i), we obtain
On the other hand, from (2.8) we have
Also, it follows from (3.5) that
This implies that
Substituting (3.12) into (3.11), after simplifying we have
Since \(\{z_n\}\) is bounded, \(\{d(x_n, p)\}\) is convergent and \(\alpha _n \rightarrow 0\), these show that
Hence from (3.9)–(3.13) and Lemma 2.7 (3) we have
As \(\lambda _n \ge \lambda > 0,\) so by Lemma 2.5 (iii) and (3.13), we have
This together with (3.13) shows that
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
Since \(\limsup _{i \rightarrow \infty }\langle \overrightarrow{ux^*}, \overrightarrow{x_{n_i}x^*}\rangle \le 0\) by Remark 2.1, which shows
By virtue of (3.13), (3.16) and Cauchy-Schwarz inequality we obtain
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
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
Hence by Lemma 2.8, there exists an increasing sequence \(\{m_j\}, \ j\ge 1\) \(m_j \rightarrow \infty \), such that
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
Also from the inequality (3.6) we obtain that
After simplifying we have
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
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
Let \(u \in X\) be a given point. For any given point \(x_1\in X\), let \(\{x_n\}\) be the sequence generated by
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
(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
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:
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
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
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
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).
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References
Ugwunnadi, G.C., Khan, A.R., Abbas, M.: A hybrid proximal point algorithm for finding minimizers and fixed points in CAT(0) spaces. J. Fixed Point Theory Appl. 20, 82 (2018). https://doi.org/10.1007/s11784-018-0555-0
Bridson, M.R., Haeiger, A.: Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999)
Kirk, W.A.: Fixed point theory in CAT(0) spaces and R-trees. Fixed Point Theory Appl. 2004, 309–316 (2004)
Berg, I.D., Nikolaev, I.G.: Quasilinearization and curvature of Alexandrov spaces. Geom. Dedicata 133, 195–218 (2008)
Reich, S., Salinas, Z.: Weak convergence of infinite products of operators in Hadamard spaces. Rend. Circ. Mat. Palermo 65, 55–71 (2016)
Kakavandi, B.A.: Weak topologies in complete CAT(0) metric spaces. Proc. Am. Math. Soc. 141, 1029–1039 (2013)
Ahmadi Kakavandi, B., Amini, M.: Duality and subdifferential for convex functions on complete CAT(0) metric spaces. Nonlinear Anal. 73, 3450–3455 (2010)
Liu, X.D.: Convergence theorems on total asymptotically demicontractive and hemicontractive mappings in CAT(0) spaces. J. Inequal. Appl. 2014, 436 (2014)
Agarwal, R.P., O’Regan, D., Sahu, D.R.: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8, 61–79 (2007)
Jost, J.: Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math. Helv. 70, 659–673 (1995)
Bačák, M., Reich, S.: The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces. J. Fixed Point Theory Appl. 16, 189–202 (2014)
Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16, 899–912 (2008)
Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)
Bačák, M.: The proximal point algorithm in metric spaces. Isr. J. Math. 194, 689–701 (2013)
Cholamjiak, P., Abdou, A.A., Cho, Y.J.: Proximal point algorithms involving fixed points of nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl. 2015, 27 (2015)
Marino, G., Scardamaglia, B., Karapinar, E.: Strong convergence theorem for strict pseudo-contractions in Hilbert spaces. J. Inequal. Appl. 2016, 134 (2016)
Chang, S., Yao, J.C., Wang, L., Qin, L.J.: Some convergence theorems involving proximal point and common fixed points for asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl. 2016, 68 (2016)
Izuchukwu, C., Aremu, K.O., Oyewole, O.K., Mewomo, O.T., Khan, S.H.: On mixed equilibrium problems in Hadamard spaces. J. Mat. 2019, 3210649 (2019). https://doi.org/10.1155/2019/3210649
Sakai, T.: Riemannian Geometry, Translations of Mathematical Monographs. American Mathematical Society, Providence (1996)
Khatibzadeh, H., Ranjbar, S.: Monotone operators and the proximal point algorithm in complete CAT(0) metric spaces. J. Aust. Math. Soc. 103, 70–90 (2017)
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The authors are very thankful to the referees for careful reading the manuscript by the reviewers and for an excellent review and suggested corrections that have helped to improve quality and the presentation of the manuscript.
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This study was supported by the Natural Science Foundation of China (No. 11361070) and by the Natural Science Foundation of China Medical University, Taiwan. This work was also supported by Scientific Research Fund of SiChuan Provincial Education Department (No.14ZA0272),
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Chang, Ss., Wang, L., Zhao, L.C. et al. A proximal point algorithm for finding minimizers and fixed points of quasi-pseudo-contractive mappings in CAT(0) spaces. J. Fixed Point Theory Appl. 23, 5 (2021). https://doi.org/10.1007/s11784-020-00839-9
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DOI: https://doi.org/10.1007/s11784-020-00839-9