A new iterative scheme for numerical reckoning fixed points of total asymptotically nonexpansive mappings
 Adoon Pansuwan^{1} and
 Wutiphol Sintunavarat^{1}Email author
https://doi.org/10.1186/s1366301605739
© Pansuwan and Sintunavarat 2016
Received: 9 February 2016
Accepted: 25 July 2016
Published: 5 August 2016
Abstract
In this paper, we propose a new iterative algorithm to approximate fixed points of total asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces. We also provide two examples to illustrate the convergence behavior of the proposed algorithm and numerically compare the convergence of the proposed iteration scheme with the existing schemes.
Keywords
MSC
1 Introduction and basic definitions
Let \((X, d)\) be a metric space. A geodesic path joining \(x \in X\) to \(y \in X\) (or, more briefly, a geodesic from x to y) is a mapping \(c:[0, l] \subseteq\mathbb{R} \rightarrow X\) such that \(c(0) = x\), \(c(l) = y\) and \(d(c(a), c(b)) = a  b\) for all \(a,b \in[0, l]\). It is easy to see that c is an isometry and \(d(x, y) = l\). The image \(c([0,l])\) is called a geodesic (or metric) segment joining x and y and is denoted by \([x, y]\) if it is unique.
The metric space \((X, d)\) 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 joining x and y for each \(x, y \in X\). A subset C of X is said to be convex if C includes every geodesic segment joining any two of its point.
A geodesic triangle \(\triangle(x_{1}, x_{2}, x_{3})\) in a geodesic metric space \((X, d)\) consists of three points \(x_{1}, x_{2}, x_{3} \in X\) (the vertices of △) and a geodesic segment between each pair of vertices (the edges of △). A comparison triangle for the geodesic triangle \(\triangle(x_{1}, x_{2}, x_{3})\) in \((X, d)\) is a triangle \(\overline{\triangle}(x_{1}, x_{2}, x_{3}) := \triangle(\overline{x}_{1}, \overline{x}_{2}, \overline{x}_{3})\) in the Euclidean plane \(\mathbb{R}^{2}\) such that \(d_{\mathbb{R}^{2}} (\overline{x}_{i}, \overline{x}_{j}) = d(x_{i}, x_{j})\) for all \(i, j \in\{1, 2, 3\}\). Bridson and Haefliger [1] have shown that such a triangle always exists.

Let △ be a geodesic triangle in X and let \(\overline {\triangle} \subseteq\mathbb{R}^{2}\) be a comparison triangle for △. Then △ is said to satisfy the \(\operatorname{CAT}(0)\) inequality if for all \(x, y \in\triangle\) and all comparison points \(\overline{x}, \overline{y} \in\overline{\triangle}\),$$ d(x,y)\leq d_{\mathbb{R}^{2}}(\overline{x},\overline{y}). $$(1.1)
Definition 1.1
 1.The asymptotic radius \(r(\{x_{n}\})\) of \(\{x_{n}\}\) is given bywhere \(r(x,\{x_{n}\}) :=\limsup_{n\rightarrow\infty} d(x,x_{n})\).$$r\bigl(\{x_{n}\}\bigr) := \inf_{x\in X}\bigl\{ r\bigl(x, \{x_{n}\}\bigr)\bigr\} , $$
 2.The asymptotic center \(A(\{x_{n}\})\) of \(\{x_{n}\}\) is the set$$A\bigl(\{x_{n}\}\bigr) := \bigl\{ x\in X : r\bigl(x,\{x_{n}\} \bigr) = r\bigl(\{x_{n}\}\bigr)\bigr\} . $$
In 2006, Dhompongsa et al. [4] showed that \(A(\{ x_{n}\})\) consists of exactly one point for each bounded sequence \(\{x_{n}\}\) in a \(\operatorname{CAT}(0)\) space (see Proposition 7 in [4])).
Next, we give the concept of Δconvergent sequence in a \(\operatorname{CAT}(0)\) spaces.
Definition 1.2
Let \((X,d)\) be a \(\operatorname{CAT}(0)\) space. A sequence \(\{x_{n}\}\) in X is said to Δconverge to \(x\in X\) if and only if x is the unique asymptotic center of all subsequences of \(\{x_{n}\}\). In this case, we write \(\Delta\mbox{}\!\lim_{n\rightarrow\infty} x_{n} = x\) and x is called the Δlimit of \(\{x_{n}\}\).
Let us recall some basics for nonlinear mappings on \(\operatorname {CAT}(0)\) spaces.
Definition 1.3
Definition 1.4
([5])
Definition 1.5
Definition 1.6
([6])
Remark 1.7
From Definitions 1.3, 1.4, 1.5, and 1.6, we note that each nonexpansive mapping is an asymptotically nonexpansive mapping with a sequence \(\{k_{n}:=1\}\) for all \(n\in\mathbb{N}\) and each asymptotically nonexpansive mapping is a \((\{v_{n}\}, \{u_{n}\}, \zeta )\)total asymptotically nonexpansive mapping with two sequences \(\{v_{n}:=k_{n}1\}\) and \(\{u_{n}:=0\}\) for all \(n\in\mathbb {N}\) and ζ is an identity mapping. Also, we see that each asymptotically nonexpansive mapping is a uniformly LLipschitzian mapping with \(L:=\sup_{n\in\mathbb {N}} \{k_{n}\}\).
Lemma 1.8
([7], Theorem 2.8)
Let C be a closed convex subset of a complete \(\operatorname{CAT}(0)\) space \((X,d)\) and \(T : C \rightarrow C\) be a total asymptotically nonexpansive and uniformly LLipschitzian mapping. If \(\{x_{n}\}\) is a bounded sequence in C such that \(\lim_{n\rightarrow\infty} d(x_{n},Tx_{n}) = 0 \) and \(\Delta\mbox{}\!\lim_{n\rightarrow\infty} x_{n} =p\), then \(Tp = p\).
In 2014, Panyanak [8] gave the following existence result of fixed points for total asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces which is also need in our main results.
Theorem 1.9
([8], Corollary 3.2)
Let C be a nonempty bounded closed convex subset of a complete \(\operatorname{CAT}(0)\) space \((X,d)\) and \(T : C \rightarrow C\) be a continuous total asymptotically nonexpansive mapping. Then T has a fixed point.
Motivated by the above recorded studies, in this work, we introduce a new iterative algorithm called ‘modified PicardIshikawa hybrid’ to approximate fixed points of total asymptotically nonexpansive mappings on \(\operatorname{CAT}(0)\) spaces. Our results are refinements and generalizations of many recent results from the current literature. We also provide two numerical examples to illustrate the convergence behavior of the proposed algorithm.
Before we show our main results in the next section, let us recall some useful lemmas.
Lemma 1.10
([10], Lemma 2)
Lemma 1.11
 (C_{1}):

every bounded sequence in X always has a Δconvergent subsequence [11], p.3690;
 (C_{2}):

if \(\{x_{n}\}\) is a bounded sequence in a closed convex subset C of X, then the asymptotic center of \(\{x_{n}\}\) is in C [12], Proposition 2.1;
 (C_{3}):

if \(\{x_{n}\}\) is a bounded sequence in X with \(A(\{x_{n}\} ) = \{p\}\), \(\{u_{n}\}\) is a subsequence of \(\{x_{n}\}\) with \(A(\{u_{n}\}) = \{u\}\) and the sequence \(\{d(x_{n},u)\}\) converges, then \(p = u\) [13], Lemma 2.8.
Lemma 1.12
([14], Lemma 4.5)
2 Main results
Theorem 2.1
 (S_{1}):

\(\sum_{n=1}^{\infty} v_{n} < \infty\) and \(\sum_{n=1}^{\infty} u_{n} < \infty\);
 (S_{2}):

there exist constants \(a, b\) with \(0 < a\leq\alpha_{n} \leq b < 1\) for all \(n\in\mathbb{N}\) and \(0< a(1b) \leq\frac{1}{2}\);
 (S_{3}):

there exists a constant \(M^{*}\) such that \(\zeta(r) \leq M^{*}r\) for all \(r\geq0\).
Proof
Since T is uniformly LLipschitzian, we have T is continuous. By using Theorem 1.9, we get \(F(T)\neq\emptyset\). Next, we will divide the proof into three steps.
This completes the proof. □
By using the conclusion in Step 1 of Theorem 2.1 and the same technique as in the proof of Theorem 3.2 of Thakur et al. [9], we get the strong convergence result (Theorem 2.2). Then, in order to avoid repetition, the details are omitted.
Theorem 2.2
 (S_{1}):

\(\sum_{n=1}^{\infty} v_{n} < \infty\) and \(\sum_{n=1}^{\infty} u_{n} < \infty\);
 (S_{2}):

there exist constants \(a, b\) with \(0 < a\leq\alpha_{n} \leq b < 1\) for all \(n\in\mathbb{N}\) and \(0< a(1b) \leq\frac{1}{2}\);
 (S_{3}):

there exists a constant \(M^{*}\) such that \(\zeta(r) \leq M^{*}r\) for all \(r\geq0\).
In [15], Senter and Dotson introduced the concept of special self mapping as follows.
Definition 2.3
([15])
Using the result in Step 2 of Theorem 2.1 with Condition (I) and the same technique as in the proof of Theorem 3.3 of Thakur et al. [9], we now state the following strong convergence result for total asymptotically nonexpansive mappings without the proof.
Theorem 2.4
 (S_{1}):

\(\sum_{n=1}^{\infty} v_{n} < \infty\) and \(\sum_{n=1}^{\infty} u_{n} < \infty\);
 (S_{2}):

there exist constants a, b with \(0 < a\leq\alpha_{n} \leq b < 1\) for all \(n\in\mathbb{N}\) and \(0< a(1b) \leq\frac{1}{2}\);
 (S_{3}):

there exists a constant \(M^{*}\) such that \(\zeta(r) \leq M^{*}r\) for all \(r\geq0\);
 (S_{4}):

T satisfies Condition (I).
3 Numerical example
In this section, using Example 3.1, we will compare the convergence of the modified PicardIshikawa hybrid iteration process (2.1) with the modified Mann iteration process (1.10) and the modified PicardMann hybrid iteration process (1.9).
Example 3.1
Iterates of modified Mann, modified PicardMann hybrid, and modified PicardIshikawa hybrid iterations for \(\pmb{x_{1}=1.5}\)
Iterate  The modified Mann iteration process  The modified PicardMann hybrid iteration  The modified PicardIshikawa hybrid iteration 

\(x_{1}\)  1.50000000000000  1.50000000000000  1.50000000000000 
\(x_{2}\)  1.13188130791299  0.95198821855406  1.00000000000000 
\(x_{3}\)  1.04396043597100  1.00000000000000  1.00000000000000 
\(x_{4}\)  1.01099010899275  1.00000000000000  1.00000000000000 
\(x_{5}\)  1.00219802179855  1.00000000000000  1.00000000000000 
\(x_{6}\)  1.00036633696643  1.00000000000000  1.00000000000000 
\(x_{7}\)  1.00005233385235  1.00000000000000  1.00000000000000 
\(x_{8}\)  1.00000654173154  1.00000000000000  1.00000000000000 
\(x_{9}\)  1.00000072685906  1.00000000000000  1.00000000000000 
\(x_{10}\)  1.00000007268591  1.00000000000000  1.00000000000000 
\(x_{11}\)  1.00000000660781  1.00000000000000  1.00000000000000 
\(x_{12}\)  1.00000000055065  1.00000000000000  1.00000000000000 
\(x_{13}\)  1.00000000004236  1.00000000000000  1.00000000000000 
\(x_{14}\)  1.00000000000303  1.00000000000000  1.00000000000000 
\(x_{15}\)  1.00000000000020  1.00000000000000  1.00000000000000 
Iterates of modified Mann, modified PicardMann hybrid, and modified PicardIshikawa hybrid iterations for \(\pmb{x_{1}=1.9}\)
Iterate  The modified Mann iteration process  The modified PicardMann hybrid iteration  The modified PicardIshikawa hybrid iteration 

\(x_{1}\)  1.90000000000000  1.90000000000000  1.90000000000000 
\(x_{2}\)  1.13027756377320  0.95262315543817  1.00000000000000 
\(x_{3}\)  1.04342585459107  1.00000000000000  1.00000000000000 
\(x_{4}\)  1.01085646364777  1.00000000000000  1.00000000000000 
\(x_{5}\)  1.00217129272955  1.00000000000000  1.00000000000000 
\(x_{6}\)  1.00036188212159  1.00000000000000  1.00000000000000 
\(x_{7}\)  1.00005169744594  1.00000000000000  1.00000000000000 
\(x_{8}\)  1.00000646218074  1.00000000000000  1.00000000000000 
\(x_{9}\)  1.00000071802008  1.00000000000000  1.00000000000000 
\(x_{10}\)  1.00000007180201  1.00000000000000  1.00000000000000 
\(x_{11}\)  1.00000000652746  1.00000000000000  1.00000000000000 
\(x_{12}\)  1.00000000054395  1.00000000000000  1.00000000000000 
\(x_{13}\)  1.00000000004184  1.00000000000000  1.00000000000000 
\(x_{14}\)  1.00000000000299  1.00000000000000  1.00000000000000 
\(x_{15}\)  1.00000000000020  1.00000000000000  1.00000000000000 
Next, we will give an example to show the nontrivial difference between the rate of convergence of the modified PicardIshikawa hybrid iteration process (2.1) with the modified PicardMann hybrid iteration process (1.9).
Example 3.2
Declarations
Acknowledgements
The authors gratefully acknowledge the financial support provided by Thammasat University Research Fund under the TU Research Scholar, Contract No. 2/10/2559.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Bridson, M, Haefliger, A: Metric Spaces of NonPositive Curvature. Springer, Berlin (1999) View ArticleMATHGoogle Scholar
 Kirk, WA: A fixed point theorem in \(\operatorname{CAT}(0)\) spaces and Rtrees. Fixed Point Theory Appl. 2004(4), 309316 (2004) MathSciNetView ArticleMATHGoogle Scholar
 Bruhat, F, Tits, J: Groupes réductifs sur un corps local. I. Données radicielles valuées. Publ. Math. Inst. Hautes Études Sci. 41, 5251 (1972) MathSciNetView ArticleGoogle Scholar
 Dhompongsa, S, Kirk, WA, Sims, B: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal., Theory Methods Appl. 65(4), 762772 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 35, 171174 (1972) MathSciNetView ArticleMATHGoogle Scholar
 Alber, YI, Chidume, CE, Zegeye, H: Approximating fixed points of total asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2006, Article ID 10673 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Chang, SS, Wang, L, Lee, HWJ, Chan, CK, Yang, L: Demiclosed principle and Δconvergence theorems for total asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces. Appl. Math. Comput. 219(5), 26112617 (2012) MathSciNetMATHGoogle Scholar
 Panyanak, B: On total asymptotically nonexpansive mappings in \(\operatorname{CAT}(\kappa)\) spaces. J. Inequal. Appl. 2014, Article ID 336 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Thakur, BS, Thakur, D, Postolache, M: Modified PicardMann hybrid iteration process for total asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2015, Article ID 140 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Qihou, L: Iterative sequences for asymptotically quasinonexpansive mappings with error member. J. Math. Anal. Appl. 259(1), 1824 (2001) MathSciNetView ArticleMATHGoogle Scholar
 Kirk, WA, Panyanak, B: A concept of convergence in geodesic spaces. Nonlinear Anal., Theory Methods Appl. 68(12), 36893696 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Dhompongsa, S, Kirk, WA, Panyanak, B: Nonexpansive setvalued mappings in metric and Banach spaces. J. Nonlinear Convex Anal. 8(1), 3545 (2007) MathSciNetMATHGoogle Scholar
 Dhompongsa, S, Panyanak, B: On Δconvergence theorems in \(\operatorname{CAT}(0)\) spaces. Comput. Math. Appl. 56(10), 25722579 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Nanjaras, B, Panyanak, B: Demiclosed principle for asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces. Fixed Point Theory Appl. 2010, Article ID 268780 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Senter, HF, Dotson, WG: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 44, 375380 (1974) MathSciNetView ArticleMATHGoogle Scholar
 Kim, GE: Strong convergence to fixed point of a total asymptotically nonexpansive mapping. Fixed Point Theory Appl. 2013, Article ID 302 (2013) MathSciNetView ArticleMATHGoogle Scholar