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Modified Picard-Mann hybrid iteration process for total asymptotically nonexpansive mappings
- Balwant Singh Thakur^{1},
- Dipti Thakur^{1} and
- Mihai Postolache^{2}Email author
https://doi.org/10.1186/s13663-015-0391-5
© Thakur et al. 2015
- Received: 8 January 2015
- Accepted: 30 July 2015
- Published: 14 August 2015
Abstract
In this paper, using the modified hybrid Picard-Mann iteration process, we establish Δ-convergence and strong convergence theorems for total asymptotically nonexpansive mappings on a \(CAT(0)\) space. Results established in the paper extend and improve a number of results in the literature. A numerical example is also given to examine the fastness of the proposed iteration process under different control conditions and initial points.
Keywords
- total asymptotically nonexpansive mappings
- Δ-convergence and strong convergence theorem
- \(CAT(0)\) space
MSC
- 47H09
- 47H10
1 Introduction
Let K be a nonempty, closed and convex subset of a normed linear space E. A mapping \(T\colon K\to K\) is said to be nonexpansive if \(\Vert Tx-Ty\Vert \leq \Vert x-y\Vert \) for every x, y in K. In the last four decades, many papers have appeared in the literature on the iteration methods to approximate fixed points of a nonexpansive mapping, cf. [1–6] and the references therein. In the meantime, some generalizations of nonexpansive mappings have appeared, namely asymptotically nonexpansive mapping [7], asymptotically nonexpansive type mapping [8], asymptotically nonexpansive mappings in the intermediate sense [9].
Iterative approximation of fixed points of total nonexpansive mappings has also been studied by [12–15].
On the other hand, in 2003, Kirk [17, 18] initiated the study of fixed point theory in metric spaces with nonpositive curvature. He showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete \(CAT(0)\) space always has a fixed point. After his work, fixed point theory in \(CAT(0)\) spaces has been rapidly developed and many papers have appeared [19–24]. Nanjaras and Panyanak [25] proved the demiclosed principle for asymptotically nonexpansive mappings in \(CAT(0)\) space and obtained a Δ-convergence theorem for the Mann iteration. Abbas et al. [26] proved the demiclosed principle for asymptotically nonexpansive mappings in the intermediate sense and established convergence theorems, Tang et al. [27] proved the demiclosed principle for total asymptotically nonexpansive mappings in a \(CAT(0)\) space and obtained Δ-convergence theorem.
We establish some strong and Δ-convergence results of the iterative process (1.3) for total asymptotically nonexpansive mappings on a \(CAT(0)\) space. Our results extend and improve the corresponding results of Chang et al. [28], Nanjaras and Panyanak [25] and others.
2 Preliminaries
Throughout the paper, we denote by \(\mathbb{N}\) the set of positive integers and by \(\mathbb{R}\) the set of real numbers.
The following lemma plays an important role in our paper.
Lemma 2.1
([23], Lemma 2.4)
Let C be a nonempty subset of a \(CAT(0)\) space X.
Definition 2.1
Definition 2.2
Definition 2.3
Definition 2.4
From the definitions, we see that each nonexpansive mapping is an asymptotically nonexpansive mapping with a sequence \(\{k_{n} =1\}\), and each asymptotically nonexpansive mapping is a \((\{\nu_{n}\}, \{\mu_{n}\}, \zeta)\)- total asymptotically nonexpansive mapping with \(\mu_{n}=0\), \(\nu_{n}=k_{n}-1\), \(n \geq1\) and \(\zeta(t)=t\), \(t\geq0\), and each asymptotically nonexpansive mapping is a uniformly L-Lipschitzian mapping with \(L=\sup\{k_{n}\}\), \(n\geq1\).
A sequence \(\{x_{n}\}\) in X is said to Δ-converge to \(p \in X\) if p is the unique asymptotic center of \(\{u_{n}\}\) for every subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\). In this case, we write \(\Delta\mbox{-}\!\lim x_{n}=p\) and call p the Δ-limit of \(\{x_{n}\}\).
Lemma 2.2
- (i)
Every bounded sequence in a complete \(CAT(0)\) space always has a Δ-convergent subsequence [24], p.3690.
- (ii)
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 [29], Proposition 2.1.
- (iii)
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 \(p = u\) [23], Lemma 2.8.
The following results are useful to prove our main result.
Lemma 2.3
([25], Lemma 4.5)
Lemma 2.4
([30], Lemma 2)
Lemma 2.5
([28], Theorem 2.8)
Let C be a closed convex subset of a complete \(CAT(0)\) space X and let \(T \colon C\to C\) be a total asymptotically nonexpansive and uniformly L-Lipschitzian mapping. Let \(\{x_{n}\}\) be a bounded sequence in C such that \(\lim_{n\to\infty}d(x_{n}, Tx_{n}) = 0\) and \(\Delta\mbox{-}\!\lim_{n\to\infty}x_{n} = p\). Then \(Tp = p\).
The following existence result is also needed.
Lemma 2.6
([31], Corollary 3.2)
Let C be a nonempty bounded closed convex subset of a complete \(CAT(0)\) space X. If \(T \colon C\to C\) is a continuous total asymptotically nonexpansive mapping, then T has a fixed point.
3 Main results
We now establish a △-convergence result for the modified Picard-Mann hybrid iterative process.
Theorem 3.1
- (1)
\(\sum_{n=1}^{\infty}\nu_{n} <\infty\); \(\sum_{n=1}^{\infty }\mu_{n} <\infty\);
- (2)
there exist constants \(a, b \in(0,1)\) with \(0 < b(1- a) \leq \frac{1}{2}\) such that \(\{\alpha_{n}\} \subset[a, b]\);
- (3)
there exists a constant \(M^{*} > r\) such that \(\zeta(r) \leq M^{*}r\), \(r \geq0\);
Proof
Since T is Lipschitz continuous, \(F(T)\neq\emptyset\) by Lemma 2.6. We divide the proof of Theorem 3.1 into three steps.
Step-I. First we prove that \(\lim_{n\to\infty}d(x_{n}, p)\) exists for each \(p \in F(T)\).
It follows from condition (i) and Lemma 2.4 that \(\lim_{n\to \infty} d(x_{n}, p)\) exists.
Proof of Step-III. Let \(u \in W_{\Delta}(x_{n})\). Then there exists a subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\) such that \(A(\{u_{n}\}) = \{u\}\). By Lemma 2.2(i), there exists a subsequence \(\{v_{n}\}\) of \(\{u_{n}\}\) such that \(\Delta\mbox{-}\!\lim_{n\to\infty} v_{n} = v \in C\). By Lemma 2.5, \(v \in F(T)\). Since \(\{d (u_{n}, v)\}\) converges, by Lemma 2.2(iii), \(u = v\). This shows that \(W_{\Delta}(x_{n}) \subseteq F(T)\).
Now we prove that \(W_{\Delta}(x_{n})\) consists of exactly one point. Let \(\{u_{n}\}\) be a subsequence of \(\{x_{n}\}\) with \(A(\{u_{n}\}) = \{u\}\) and let \(A(\{x_{n}\}) = \{x\}\). We have already seen that \(u = v\) and \(v\in F(T)\). Finally, since \(\{d(x_{n}, v)\}\) converges, by Lemma 2.2(iii), we have \(x = v \in F(T)\). This shows that \(W_{\Delta}(x_{n}) = \{x\}\). □
We now establish some strong convergence results.
Theorem 3.2
Proof
Necessity is obvious. Conversely, suppose that \(\liminf_{n\to \infty}d(x_{n},F(T))= 0\). As proved in Step-I of Theorem 3.1, \(\lim_{n\to\infty} d(x_{n}, F(T))\) exists for all \(p \in F(T)\). Thus, by hypothesis, \(\lim_{n\to\infty} d(x_{n}, F(T)) = 0\).
Senter and Dotson [32], p.375, introduced the concept of Condition (I) as follows.
Definition 3.1
It is weaker than demicompactness for a nonexpansive mapping T defined on a bounded set. Since every completely continuous mapping \(T\colon K\to K\) is continuous and demicompact, so it satisfies Condition (I). Recently, Kim [33] gave an interesting example of total asymptotically nonexpansive self-mapping satisfying Condition (I).
Example 1
([33], Example 3.7)
Here T is a uniformly continuous and total asymptotically nonexpansive mapping with \(F(T)= \{1 \}\). Also T satisfies Condition (I), but T is not Lipschitzian and hence it is not an asymptotically nonexpansive mapping.
Using Condition (I), we now establish the following strong convergence result for total asymptotically nonexpansive mapping.
Theorem 3.3
Let X, C, T, \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{x_{n}\}\) satisfy the hypothesis of Theorem 3.1 and let T be a mapping satisfying Condition (I). Then the sequence \(\{x_{n}\}\) generated by (1.3) converges strongly to a fixed point of T.
Proof
4 Numerical example
In this section, using Example 1 of a total asymptotically nonexpansive mapping, we compare the convergence of modified Picard-Mann hybrid iteration process (1.3) with the modified Mann iteration process (1.1).
Iterates of modified Mann and modified Picard-Mann hybrid iterations
Iterate | \(\boldsymbol {x_{1}=1.1}\) | \(\boldsymbol {x_{1}=1.5}\) | \(\boldsymbol {x_{1}=1.9}\) | |||
---|---|---|---|---|---|---|
Iteration ( 1.1 ) | Iteration ( 1.3 ) | Iteration ( 1.1 ) | Iteration ( 1.3 ) | Iteration ( 1.1 ) | Iteration ( 1.3 ) | |
\(x_{2}\) | 1.03218253804965 | 0.98903980508798 | 1.13188130791299 | 0.95198821855406 | 1.13027756377320 | 0.95262315543817 |
\(x_{3}\) | 1.01072751268322 | 1.00000000000000 | 1.04396043597100 | 1.00000000000000 | 1.04342585459107 | 1.00000000000000 |
\(x_{4}\) | 1.00268187817080 | 1.00000000000000 | 1.01099010899275 | 1.00000000000000 | 1.01085646364777 | 1.00000000000000 |
\(x_{5}\) | 1.00053637563416 | 1.00000000000000 | 1.00219802179855 | 1.00000000000000 | 1.00217129272955 | 1.00000000000000 |
\(x_{6}\) | 1.00008939593903 | 1.00000000000000 | 1.00036633696643 | 1.00000000000000 | 1.00036188212159 | 1.00000000000000 |
\(x_{7}\) | 1.00001277084843 | 1.00000000000000 | 1.00005233385235 | 1.00000000000000 | 1.00005169744594 | 1.00000000000000 |
\(x_{8}\) | 1.00000159635605 | 1.00000000000000 | 1.00000654173154 | 1.00000000000000 | 1.00000646218074 | 1.00000000000000 |
\(x_{9}\) | 1.00000017737289 | 1.00000000000000 | 1.00000072685906 | 1.00000000000000 | 1.00000071802008 | 1.00000000000000 |
\(x_{10}\) | 1.00000001773729 | 1.00000000000000 | 1.00000007268591 | 1.00000000000000 | 1.00000007180201 | 1.00000000000000 |
\(x_{11}\) | 1.00000000161248 | 1.00000000000000 | 1.00000000660781 | 1.00000000000000 | 1.00000000652746 | 1.00000000000000 |
\(x_{12}\) | 1.00000000013437 | 1.00000000000000 | 1.00000000055065 | 1.00000000000000 | 1.00000000054395 | 1.00000000000000 |
\(x_{13}\) | 1.00000000001034 | 1.00000000000000 | 1.00000000004236 | 1.00000000000000 | 1.00000000004184 | 1.00000000000000 |
\(x_{14}\) | 1.00000000000074 | 1.00000000000000 | 1.00000000000303 | 1.00000000000000 | 1.00000000000299 | 1.00000000000000 |
\(x_{15}\) | 1.00000000000005 | 1.00000000000000 | 1.00000000000020 | 1.00000000000000 | 1.00000000000020 | 1.00000000000000 |
\(x_{16}\) | 1.00000000000000 | 1.00000000000000 | 1.00000000000001 | 1.00000000000000 | 1.00000000000001 | 1.00000000000000 |
\(x_{17}\) | 1.00000000000000 | 1.00000000000000 | 1.00000000000000 | 1.00000000000000 | 1.00000000000000 | 1.00000000000000 |
Comparison of fastness for different control conditions
\(\boldsymbol {\alpha_{n}}\) | Least number of iterate to reach the fixed point 1 | |||||
---|---|---|---|---|---|---|
\(\boldsymbol {x_{1}=1.1}\) | \(\boldsymbol {x_{1}=1.5}\) | \(\boldsymbol {x_{1}=1.9}\) | ||||
Iteration ( 1.1 ) | Iteration ( 1.3 ) | Iteration ( 1.1 ) | Iteration ( 1.3 ) | Iteration ( 1.1 ) | Iteration ( 1.3 ) | |
0.3 | \(x_{87}\) | \(x_{3}\) | \(x_{91}\) | \(x_{3}\) | \(x_{92}\) | \(x_{3}\) |
0.5 | \(x_{45}\) | \(x_{3}\) | \(x_{47}\) | \(x_{3}\) | \(x_{47}\) | \(x_{3}\) |
0.84 | \(x_{19}\) | \(x_{2}\) | \(x_{21}\) | \(x_{2}\) | \(x_{21}\) | \(x_{2}\) |
0.95 | \(x_{13}\) | \(x_{2}\) | \(x_{14}\) | \(x_{2}\) | \(x_{14}\) | \(x_{2}\) |
\(\frac{n}{n+1}\) | \(x_{16}\) | \(x_{3}\) | \(x_{17}\) | \(x_{3}\) | \(x_{17}\) | \(x_{3}\) |
\(1-\frac{1}{\sqrt{n+1}}\) | \(x_{26}\) | \(x_{3}\) | \(x_{27}\) | \(x_{3}\) | \(x_{28}\) | \(x_{3}\) |
\(\frac{1}{\sqrt{n+1}}-\frac{1}{(n+1)^{2}}\) | \(x_{235}\) | \(x_{3}\) | \(x_{258}\) | \(x_{3}\) | \(x_{261}\) | \(x_{3}\) |
\(\frac{1}{\sqrt{n+5}}\) | \(x_{265}\) | \(x_{3}\) | \(x_{288}\) | \(x_{3}\) | \(x_{295}\) | \(x_{3}\) |
\(\frac{1}{\sqrt{2n+5}}\) | \(x_{492}\) | \(x_{3}\) | \(x_{537}\) | \(x_{3}\) | \(x_{555}\) | \(x_{3}\) |
From Table 1 and Table 2, it is clear that for a total asymptotically nonexpansive mapping, modified Mann iteration (1.1) is very much sensitive about the choice of initial point and control condition \(\alpha_{n}\), whereas the behavior of proposed iteration (1.3) is consistent.
Declarations
Acknowledgements
The authors wished to express their thanks to the anonymous referee for helpful comments which improved the paper. The first author is supported by the Chhattisgarh Council of Science and Technology, India (MRP-2015). The second author would like to thank the Rajiv Gandhi National Fellowship, University Grants Commission, Government of India under the grant (F1-17.1/2011-12/RGNF-ST-CHH-6632).
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
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