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Some convergence theorems involving proximal point and common fixed points for asymptotically nonexpansive mappings in \(\operatorname {CAT}(0)\) spaces
Fixed Point Theory and Applications volume 2016, Article number: 68 (2016)
Abstract
In this paper, a new modified proximal point algorithm involving fixed point iterates of asymptotically nonexpansive mappings in \(\operatorname {CAT}(0)\) spaces is proposed and the existence of a sequence generated by our iterative process converging to a minimizer of a convex function and a common fixed point of asymptotically nonexpansive mappings is proved.
Introduction
Recently, many convergence results by the proximal point algorithm (shortly, the PPA) which was initiated by Martinet [1] in 1970 for solving optimization problems have been extended from the classical linear spaces such as Euclidean spaces, Hilbert spaces, and Banach spaces to the setting of manifolds (see [1–9]).
For example, in 2013, Bačák [6] introduced the PPA in a \(\operatorname{CAT}(0)\) space \((X, d)\) as follows: \(x_{1} \in X\) and
where \(\lambda_{n} > 0\), \(\forall n \ge1\). It was shown that if f has a minimizer and \(\Sigma_{n=1}^{\infty}\lambda_{n} = \infty\), then the sequence \(\{x_{n}\}\) △converges to its minimizer (see [7]).
Also in 2015, CholamjiakAbdouCho [10] established the strong convergence of the sequence to minimizers of a convex function and to fixed points of nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces.
Motivated and inspired by the research going on in this direction, it is naturally to put forward the following.
Open question
Can we establish the strong convergence of the sequence to minimizers of a convex function and to a common fixed point of asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces?
The purpose of this paper is to propose the modified proximal point algorithm using the Stype iteration process for four asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces and to prove some △ and strong convergence theorems of the proposed processes under suitable conditions.
Our results not only give an affirmative answer to the above open question but also generalize the corresponding results of Bačák [6], ArizaRuiz et al. [7], CholamjiakAbdouCho [10], Agarwal et al. [11], DhompongsaPanyanak [12], KhanAbbas [13], and many others.
Preliminaries
Recall that a metric space \((X, d)\) is called a \(\operatorname{CAT}(0)\) space, if it is geodesically connected and if every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane. A subset K of a \(\operatorname{CAT}(0)\) space X is convex if, for any \(x, y \in K\), we have \([x, y] \subset K\), where \([x, y]: = \{\lambda x \oplus(1\lambda)y: 0 \le\lambda\le 1 \}\) is the unique geodesic joining x and y.
It is well known that a geodesic space \((X, d)\) is a \(\operatorname{CAT}(0)\) space, if and only if the inequality
is satisfied for all \(x,y,z\in X\) and \(t\in[0,1]\). In particular, if x, y, z are points in a \(\operatorname{CAT}(0)\) space \((X, d)\) and \(t\in[0,1]\), then
In order to save space, we will not repeat the geometric properties, some conclusions, and the △convergence of \(\operatorname {CAT}(0)\) space here. The interested reader may refer to (for example) [12, 14–16].
In the sequel, we denote by \(F(T)\) the fixed point set of a mapping T.
Recall that a mapping \(T : C \to C\) is said to be asymptotically nonexpansive, if there exists a sequence \(\{k_{n}\} \subset[1, \infty)\) with \(k_{n} \to1\) such that
Recall that a function \(f : C \to(\infty, \infty]\) defined on a convex subset C of a \(\operatorname{CAT}(0)\) space is convex if, for any geodesic \([x,y]: = \{\gamma_{x,y}(\lambda): 0 \le \lambda\le1\} : = \{\lambda x \oplus(1\lambda)y: 0 \le\lambda\le 1 \}\) joining \(x, y \in C\), the function \(f\circ\gamma\) is convex, i.e.,
Examples of convex functions in \(\operatorname{CAT}(0)\) space X:
Example 1
The function \(y \mapsto d(x, y): X \to[0, \infty)\) is convex.
Example 2
For a nonempty, closed, and convex subset \(C \subset X\), the indicator function defined by
is a proper, convex, and lower semicontinuous function.
Example 3
The function \(y \mapsto d^{2}(z, y): X \to[0, \infty )\) is convex.
Indeed, for each two points \(x, y\in X\), there is a unique geodesic \(\gamma_{x, y}(\lambda)\) joining x and y such that
This implies that the function \(y \mapsto d^{2}(z, y): X \to[0, \infty )\) is convex.
For any \(\lambda> 0\), define the MoreauYosida resolvent of f in \(\operatorname{CAT}(0)\) space X as
Let \(f : X \to( \infty, \infty]\) be a proper convex and lower semicontinuous function. It was shown in [11] that the set \(F(J_{\lambda})\) of fixed points of the resolvent associated with f coincides with the set \(\operatorname{arg\,min}y_{X} f (y)\) of minimizers of f. Also for any \(\lambda> 0\), the resolvent \(J_{\lambda}\) of f is nonexpansive [17].
Lemma 2.1
(Subdifferential inequality [18])
Let \((X, d)\) be a complete \(\operatorname{CAT}(0)\) space and \(f : X \to(\infty, \infty ]\) be proper convex and lower semicontinuous. Then, for all \(x, y \in X\) and \(\lambda> 0\), the following inequality holds:
Lemma 2.2
(Demiclosed principle [16])
Assume C is a closed convex subset of a complete \(\operatorname{CAT}(0)\) space X and \(T : C \to C\) be an asymptotically nonexpansive mapping. Let \(\{ x_{n}\}\) be a bounded sequence in C such that \(\bigtriangleup\mbox{} \lim x_{n} = p\) and \(\lim_{n\to\infty} d(x_{n}, Tx_{n}) = 0\). Then \(Tp = p\).
Lemma 2.3
[19]
Let \(\{a_{n}\}\) be a sequence of nonnegative real numbers satisfying the following conditions:
where \(b_{n} \ge0\) and \(\sum_{n =1}^{\infty}b_{n} < \infty\), then the limit \(\lim_{n \to\infty}a_{n}\) exists.
Lemma 2.4
Let X be a \(\operatorname{CAT}(0)\) space, C be a nonempty, closed, and convex subset of X. Let \(\{x_{i}\}_{i= 1}^{n} \) be any finite subset of C, and \(\alpha_{i} \in(0, 1)\), \(i = 1, 2, \ldots, n\) such that \(\sum_{n = 1}^{n} \alpha_{i} = 1\). Then the following inequalities hold:
Lemma 2.5
(The resolvent identity [17])
Let \((X,d)\) be a complete \(\operatorname{CAT}(0)\) space and \(f: X \to( \infty, \infty]\) be a proper convex and lower semicontinuous function. Then the following identity holds:
Some △convergence theorems involving proximal point and common fixed points for asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces
We are now in a position to give the main results of the paper.
Theorem 3.1
Assume that

(1)
\((X,d)\) is a complete \(\operatorname{CAT}(0)\) space, and C is a nonempty, closed, and convex subset of X;

(2)
\(f: C \to(\infty, \infty]\) is a proper convex and lower continuous function;

(3)
\(T_{i}:C\rightarrow C\) and \(S_{i}:C\rightarrow C\), \(i=1,2\) all are \(\{k_{n}\}\)asymptotically nonexpansive mappings with \(k_{n} \in[1, \infty )\), \(k_{n} \to1\) and \(\sum_{i = 1}^{\infty}(k_{n}  1) < \infty\) such that
$$ \Omega: = F(T_{1})\cap F(T_{2})\cap F(S_{1})\cap F(S_{2})\cap \mathop{\operatorname{arg\,min}}_{y \in C}f(y) \neq \emptyset; $$(3.1) 
(4)
\(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\), \(\{\delta_{n}\}\), \(\{ \eta_{n}\}\), \(\{\xi_{n}\}\) are sequences in \([0, 1]\) with
$$ \begin{aligned} &\alpha_{n} + \beta_{n} + \gamma_{n} = 1, \\ &\delta_{n} + \eta_{n} + \xi_{n} = 1,\quad 0 < a \le \alpha_{n}, \beta_{n}, \gamma_{n}, \delta_{n}, \eta_{n}, \xi_{n} < 1, \forall n \ge1, \end{aligned} $$(3.2)where a is a positive constant in \((0, 1)\);

(5)
\(\{\lambda_{n}\}\) is a sequence such that \(\lambda_{n} \ge\lambda> 0\) for all \(n \ge1\) and some λ.
Let \(\{x_{n}\}\) be the sequence generated in the following manner:
Then \(\{x_{n}\}\) △converges to a point \(x^{*} \in\Omega\) which is a minimizer of f in C as well as a common fixed point of \(T_{i}\), \(S_{i}\), \(i = 1,2\).
Proof
Let \(q \in\Omega\). Then \(q = T_{1} q = T_{2} q = S_{1} q = S_{2} q\) and \(f(q) \le f(y)\), \(\forall y \in C\). Therefore we have
and hence \(q = J_{\lambda_{n}} q\), \(\forall n \ge1\).
(I) First we prove that the limit \(\lim_{n \to\infty} d(x_{n}, q)\) exists.
Indeed, \(z_{n} = J_{\lambda_{n}}x_{n}\), and \(J_{\lambda_{n}}\) is nonexpansive [17]. Hence we have
Also, by (3.3), (3.4), and (2.8), we have
Similarly, by (3.3) and (3.5), we obtain
where \(L = 1+ \sup_{n \ge1} k_{n}\). By Lemma 2.3, the limit \(\lim_{n \to \infty} d(x_{n}, q)\) exists. Without loss of generality, we can assume that
Therefore \(\{x_{n}\}\) is bounded, and so are \(\{z_{n}\}\), \(\{y_{n}\}\), \(\{ T^{n}_{i}x_{n}\}\), \(i = 1, 2\), \(\{S_{1}^{n} x_{n}\}\), \(\{T^{n}_{2} z_{n}\}\), \(\{S^{n}_{2} y_{n}\}\).
(II) Now we prove that \(\lim_{n \to\infty} d(x_{n}, z_{n}) = 0\).
Indeed, by the subdifferential inequality (2.7) we have
Since \(f(q) \le f(z_{n})\), \(\forall n \ge1\), it follows that
Furthermore, it follows from (3.6) that
Simplifying we have
This together with (3.7) shows that
On the other hand it follows from (3.5) that
This together with (3.9) implies that
Also, by (3.5) we have
which can be rewritten as
This together with (3.10) shows that
From (3.4), it follows that
This shows that \(\lim_{n \to\infty} d(z_{n}, q) = c\). Therefore it follows from (3.8) that
(III) Now we prove that
Indeed, it follows from (2.9) that
By virtue of (3.7) and (3.10) we have
By condition (4) we have
Since
this together with (3.14) and (3.12) shows that
Also from (3.12), (3.14), (3.15), and (2.9) we have
(IV) Now we prove that
In fact, it follows from (3.3) and (2.9) that
which can be rewritten as
This implies that
This together with \(\lim_{n \to\infty} d(x_{n}, T_{i}^{n} x_{n}) = 0\), \(\lim_{n \to\infty}d(x_{n}, z_{n}) =0\), \(\lim_{n \to\infty}d(y_{n}, z_{n}) =0\) shows that
By the way, it follows from (3.18) that
(V) Next we prove that
In fact, it follows from (3.14), (3.15), and (3.19) that, for each \(i = 1, 2\),
Similarly we can also prove that \(\lim_{n \to\infty} d(x_{n}, S_{i} x_{n}) = 0\), \(i = 1, 2\).
(VI) Next we prove that
In fact, it follows from (3.12) and Lemma 2.5 that
(VII) Next we prove that
where \(A(\{u_{n}\})\) is the asymptotic center of \(\{u_{n}\}\) (for the definition of the asymptotic center see, for example, [15, 16]).
Let \(u \in w_{\bigtriangleup }(x_{n})\), then there exists a subsequence \(\{ u_{n}\}\) of \(\{x_{n}\}\) such that \(A(\{u_{n}\}) = \{u\}\). Therefore there exists a subsequence \(\{v_{n}\}\) of \(\{u_{n}\}\) such that \(\bigtriangleup\mbox{} \lim_{n\to\infty} v_{n} = v\) for some \(v \in C\). In view of (3.14), (3.18), and (3.21)
By Lemma 2.2, \(v\in\Omega\). So, by (3.7), the limit \(\lim_{n \to \infty} d(x_{n}, v)\) exists and \(u = v\) [12]. This shows that \(w_{\bigtriangleup }(x_{n}) \subset\Omega\).
Finally, we show that the sequence \(\{x_{n}\}\) △converges to a point in Ω. To this end, it suffices to show that \(w_{\bigtriangleup}(x_{n})\) consists of exactly one point. Let \(\{u_{n}\}\) be a subsequence of \(\{x_{n}\}\) with \(A(\{u_{n}\}) = \{u\}\) and let \(A(\{x_{n}\}) = \{x\}\). Since \(u \in w_{\bigtriangleup}(x_{n}) \subset\Omega\) and \(\{d(x_{n},u)\}\) converges, we have \(x = u\) [12]. Hence \(w_{\bigtriangleup}(x_{n}) = \{x\}\).
This completes the proof of Theorem 3.1. □
Remark 3.2

1.
Theorem 3.1 generalizes the main results in Agarwal et al. [11] and KhanAbbas [13] from one nonexpansive mapping to four asymptotically nonexpansive mappings involving the convex and lower semicontinuous function in \(\operatorname{CAT}(0)\) spaces.

2.
Theorem 3.1 extends the main result in Bačák [6], and the corresponding results in ArizaRuiz et al. [7] and Cholamjiak et al. [10]. In fact, we present a new modified proximal point algorithm for solving the convex minimization problem as well as the fixed point problem of asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces.
Since every real Hilbert space H is a complete \(\operatorname {CAT}(0)\) space, the following result can be obtained from Theorem 3.1 immediately.
Corollary 3.3
Let H be a real Hilbert space and C be a nonempty closed and convex subset of H. Let \(T_{1}\), \(T_{2}\), \(S_{1}\), \(S_{2}\), \(\{k_{n}\}\), f, \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\), \(\{\delta_{n}\}\), \(\{\eta_{n}\}\), \(\{\xi_{n}\}\), \(\{\lambda_{n}\}\), λ, and Ω satisfy the conditions (1)(5) in Theorem 3.1. Let \(\{x_{n}\}\) be the sequence generated in the following manner:
Then the sequence \(\{x_{n}\}\) converges weakly to an element in Ω.
Remark 3.4
Corollary 3.3 is an improvement and generalization of the main result in Agarwal et al. [11], Rockafellar [2], and Güler [3].
Some strong convergence theorems involving proximal point and common fixed points for asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces
Let \((X,d)\) be a \(\operatorname{CAT}(0)\) space, and C be a nonempty, closed, and convex subset of X.
Recall that a mapping \(T: C \to C\) is said to be demicompact, if for any bounded sequence \(\{x_{n}\}\) in C such that \(d(x_{n}, Tx_{n}) \to 0\) (as \(n \to\infty\)), there exists a subsequence \(\{x_{n_{i}}\} \subset \{x_{n}\}\) such that \(\{x_{n_{i}}\}\) converges strongly (i.e., in metric topology) to some point \(p \in C\).
Theorem 4.1
Under the assumptions of Theorem 3.1, if, in addition, one of \(S_{1}\), \(S_{2}\), \(T_{1}\), and \(T_{2}\) is demicompact, then the sequence \(\{ x_{n}\}\) defined by (3.3) converges strongly (i.e., in metric topology) to a point \(x^{*} \in\Omega\).
Proof
In fact, it follows from (3.20) and (3.21) that
and
Again by the assumption that one of \(S_{1}\), \(S_{2}\), \(T_{1}\), and \(T_{2}\) is demicompact, without loss of generality, we can assume \(T_{1}\) is demicompact, and it follows from (4.1) that there exists a subsequence \(\{x_{n_{i}}\} \subset\{x_{n}\}\) such that \(\{x_{n_{i}}\}\) converges strongly to some point \(p \in C\). Since \(J_{\lambda}\) is nonexpansive, it is demiclosed at 0. Again since \(S_{1}\), \(S_{2}\), \(T_{1}\), and \(T_{2}\) are asymptotically nonexpansive, by Lemma 2.2, they are also demiclosed at 0. Hence \(p \in\Omega\). Again by (3.7) the limit \(\lim_{n \to\infty}d(x_{n}, p)\) exists. Hence we have \(\lim_{n \to \infty}d(x_{n}, p) =0\).
This completes the proof of Theorem 4.1. □
Theorem 4.2
Under the assumptions of Theorem 3.1, assume, in addition, there exists a nondecreasing function \(g: [0, \infty) \to[0, \infty)\) with \(g(0) = 0\), \(g(r) > 0\), \(\forall r > 0\), such that
Then the sequence \(\{x_{n}\}\) defined by (3.3) converges strongly (i.e., in metric topology) to a point \(p^{*} \in \Omega\).
Proof
It follows from (3.20) and (3.21) that for each \(i = 1, 2\) and each λ, \(0 < \lambda\le\lambda_{n}\) we have
Therefore we have \(\lim_{n \to\infty}g(d(x_{n}, \Omega)) = 0\). Since g is nondecreasing with \(g(0) = 0\) and \(g(r) > 0\), \(r > 0\), we have
By virtue of the definition of \(\{x_{n}\}\) and (4.5), it is easy to prove that \(\{x_{n}\}\) is a Cauchy sequence in C. Since C is a closed subset in a complete \(\operatorname{CAT}(0)\) space X, it is complete. Without loss of generality, we can assume that \(\{x_{n}\}\) converges strongly to some point \(p^{*}\). It is easy to see that \(F(J_{\lambda})\), \(F(T_{i})\), and \(F(S_{i})\), \(i = 1, 2\), all are closed subsets in C, so is Ω. Since \(\lim_{n \to \infty} d(x_{n}, \Omega) = 0\), \(p^{*} \in\Omega\). This completes the proof of Theorem 4.2. □
References
 1.
Martinet, B: Réularisation d’inéquations variationnelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 4, 154158 (1970)
 2.
Rockafellar, RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877898 (1976)
 3.
Güler, O: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403419 (1991)
 4.
Kamimura, S, Takahashi, W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106, 226240 (2000)
 5.
Halpern, B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957961 (1967)
 6.
Bačák, M: The proximal point algorithm in metric spaces. Isr. J. Math. 194, 689701 (2013)
 7.
ArizaRuiz, D, Leustean, L, López, G: Firmly nonexpansive mappings in classes of geodesic spaces. Trans. Am. Math. Soc. 366, 42994322 (2014)
 8.
Boikanyo, OA, Morosanu, G: A proximal point algorithm converging strongly for general errors. Optim. Lett. 4, 635641 (2010)
 9.
Marino, G, Xu, HK: Convergence of generalized proximal point algorithm. Commun. Pure Appl. Anal. 3, 791808 (2004)
 10.
Cholamjiak, P, Abdou, AA, Cho, YJ: Proximal point algorithms involving fixed points of nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces. Fixed Point Theory Appl. 2015, 227 (2015)
 11.
Agarwal, RP, O’Regan, D, Sahu, DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8, 6179 (2007)
 12.
Dhompongsa, S, Panyanak, B: On △convergence theorems in \(\operatorname {CAT}(0)\) spaces. Comput. Math. Appl. 56, 25722579 (2008)
 13.
Khan, SH, Abbas, M: Strong and △convergence of some iterative schemes in \(\operatorname{CAT}(0)\) spaces. Comput. Math. Appl. 61, 109116 (2011)
 14.
Kirk, WA: Geodesic geometry and fixed point theory. In: Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), pp. 195225. Univ. Sevilla Secr. Publ., Seville (2003)
 15.
Kirk, WA: Fixed point theory in \(\operatorname{CAT}(0)\) spaces and \(\mathbb{R}\)trees. Fixed Point Theory Appl. 2004(4), 309316 (2004)
 16.
Chang, SS, Wang, L, Lee, HWJ, Chan, CK, Yang, L: Demiclosed principle and △convergence theorems for total asymptotically nonexpansive mappings in \(\operatorname {CAT}(0)\) spaces. Appl. Math. Comput. 219, 26112617 (2012)
 17.
Jost, J: Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math. Helv. 70, 659673 (1995)
 18.
Ambrosio, L, Gigli, N, Savare, G: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Lectures in Mathematics ETH Zurich. Birkhäuser, Basel (2008)
 19.
Xu, HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659678 (2003)
 20.
Tang, JF, Chang, SS, Dong, J: Strong convergence theorems of Cesárotype means for nonexpansive mapping in \(\operatorname{CAT}(0)\) space. Fixed Point Theory Appl. 2015, 100 (2015)
 21.
Chidume, CE, Bello, AU, Ndambomve, P: Strong and △convergence theorems for common fixed points of a finite family of multivalued demicontractive mappings in \(\operatorname{CAT}(0)\) spaces. Abstr. Appl. Anal. 2014, Article ID 805168 (2014)
Acknowledgements
The authors would like to express their thanks to the referees for their helpful comments and advices. This study was supported by the National Natural Science Foundation of China (Grant No. 11361070) and the Natural Science Foundation of China Medical University, Taiwan.
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Chang, S., Yao, J., Wang, L. et al. Some convergence theorems involving proximal point and common fixed points for asymptotically nonexpansive mappings in \(\operatorname {CAT}(0)\) spaces. Fixed Point Theory Appl 2016, 68 (2016). https://doi.org/10.1186/s1366301605597
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Keywords
 convex minimization problem
 resolvent identity
 \(\operatorname{CAT}(0)\) space
 proximal point algorithm
 asymptotically nonexpansive mapping