Some convergence results using K iteration process in \(\mathit{CAT}(0)\) spaces
- Kifayat Ullah^{1}Email author,
- Kashif Iqbal^{1} and
- Muhammad Arshad^{2}
https://doi.org/10.1186/s13663-018-0637-0
© The Author(s) 2018
Received: 11 January 2018
Accepted: 20 March 2018
Published: 16 April 2018
Abstract
In this paper, some strong and Δ-convergence results are proved for Suzuki generalized nonexpansive mappings in the setting of \(\mathit{CAT}(0)\) spaces using the K iteration process. We also give an example to show the efficiency of the K iteration process. Our results are the extension, improvement and generalization of many well-known results in the literature of fixed point theory in \(\mathit{CAT}(0)\) spaces.
Keywords
MSC
1 Introduction
The well-known Banach contraction theorem uses the Picard iteration process for approximation of fixed point. Numerical computation of fixed points for suitable classes of contractive mappings, on appropriate geometric framework, is an active research area nowadays [1–3]. Many iterative processes have been developed to approximate fixed points of different type of mappings. Some of the well-known iterative processes are those of Mann [4], Ishikawa [5], Agarwal [6], Noor [7], Abbas [8], SP [9], S^{∗} [10], CR [11], Normal-S [12], Picard Mann [13], Picard-S [14], Thakur New [15] and so on. These processes have a wide rang of applications to general variational inequalities or equilibrium problems as well as to split feasibility problems [16–19]. Recently, Hussain et al. [20] introduced a new three-step iteration process known as the K iteration process and proved that it is converging fast as compared to all above-mentioned iteration processes. They use a uniformly convex Banach space as a ground space and prove strong and weak convergence theorems. On the other hand, we know that every Banach space is a \(\mathit{CAT}(0)\) space.
Motivated by the above, in this paper, first we develop an example of Suzuki generalized nonexpansive mappings which is not nonexpansive. We compare the speed of convergence of the K iteration process with the leading two steps S-iteration process and leading three steps Picard-S-iteration process. Finally, we prove some strong and Δ-convergence theorems for Suzuki generalized nonexpansive mappings in the setting of \(\mathit{CAT}(0)\) spaces.
2 Preliminaries
For details as regards \(\mathit{CAT}(0)\) spaces please see [21]. Some results are recalled here for \(\mathit{CAT}(0)\) space X.
Lemma 2.1
([7])
The notation \(((1-\xi)x\oplus\xi y)\) is used for the unique point s satisfying (1).
Lemma 2.2
([13, Lemma 2.4])
Just like in uniformly convex Banach spaces, it is well known that \(A(C,\{x_{n}\})\) consists of exactly one point in a complete \(\mathit{CAT}(0)\) space.
Definition 2.3
In \(\mathit{CAT}(0)\) space X, a sequence \(\{x_{n}\}\) is said to be Δ-convergent to \(s\in X\) if s is the unique asymptotic center of \(\{u_{x}\}\) for every subsequence \(\{u_{x}\}\) of \(\{x_{n}\}\). In this case we write \(\Delta \text{-}\lim_{n}x_{n}=s\) and call s the \(\Delta\text{-}\lim\) of \(\{x_{n}\}\).
A point p is called a fixed point of a mapping T if \(T(p)=p\), and \(F(T)\) represents the set of all fixed points of the mapping T. Let C be a nonempty subset of a \(\mathit{CAT}(0)\) space X.
Example 2.4
To verify that T is a Suzuki generalized nonexpansive mapping, consider the following cases:
Case II: Let \(x\in [ \frac{1}{10},1 ] \), then \(\frac{1}{2}d(x,Tx)=\frac{1}{2} \vert \frac{x+1}{2}-x \vert =\frac {1-x}{4}\in [ 0,\frac{9}{40} ] \). For \(\frac{1}{2}d(x,Tx)\leq d(x,y)\) we must have \(\frac{1-x}{4}\leq \vert y-x \vert \), which gives two possibilities:
(b). Let \(x>y\), then \(\frac{1-x}{4}\leq x-y\Longrightarrow y\leq x-\frac{1-x}{4}=\frac{5x-1}{4}\Longrightarrow y\in [ -\frac {1}{8},1 ] \). Since \(y\in [ 0,1 ] \), so \(y\leq\frac {5x-1}{4}\Longrightarrow x\in [ \frac{1}{5},1 ] \). So the case is \(x\in [ \frac{1}{5},1 ] \) and \(y\in [ 0,1 ] \).
Next consider \(x\in [ \frac{7}{8},1 ] \) and \(y\in [ 0,\frac{1}{10} ) \), then \(d(Tx,Ty)\leq\frac{1}{10}\) and \(d(x,y)>\frac{72}{80}\). Hence \(d(Tx,Ty)\leq d(x,y)\). So \(\frac{1}{2}d(x,Tx)\leq d(x,y)\Longrightarrow d(Tx,Ty)\leq d(x,y)\).
Hence T is a Suzuki generalized nonexpansive mapping.
We now list some basic results.
Proposition 2.5
- (i)
[22, Proposition 1] If T is nonexpansive then T is a Suzuki generalized nonexpansive mapping.
- (ii)
[22, Proposition 2] If T is a Suzuki generalized nonexpansive mapping and has a fixed point, then T is a quasi-nonexpansive mapping.
- (iii)[22, Lemma 7] If T is a Suzuki generalized nonexpansive mapping, thenfor all \(x,y\in C\).$$ d(x,Ty)\leq3d(Tx,x)+d(x,y) $$
Lemma 2.6
([22, Theorem 5])
Let C be a weakly compact convex subset of a \(\mathit{CAT}(0)\) space X. Let T be a mapping on C. Assume that T is a Suzuki generalized nonexpansive mapping. Then T has a fixed point.
Lemma 2.7
([23, Lemma 2.9])
Sequences generated by K, Picard-S- and S-iteration processes
K | Picard-S | S | |
---|---|---|---|
\(x_{0}\) | 0.9 | 0.9 | 0.9 |
\(x_{1}\) | 0.9875 | 0.975 | 0.95 |
\(x_{2}\) | 0.998561026471100 | 0.994244105884402 | 0.976976423537605 |
\(x_{3}\) | 0.999840932805849 | 0.998727462446794 | 0.989819699574350 |
\(x_{4}\) | 0.999982839247306 | 0.999725427956896 | 0.995606847310338 |
\(x_{5}\) | 0.999998178520930 | 0.999941712669962 | 0.998134805438786 |
\(x_{6}\) | 0.999999808905464 | 0.999987769949668 | 0.999217276778764 |
\(x_{7}\) | 0.999999980126971 | 0.999997456252294 | 0.999674400293643 |
\(x_{8}\) | 0.999999997947369 | 0.999999474526643 | 0.999865478820511 |
\(x_{9}\) | 0.999999999789148 | 0.999999892043912 | 0.999944726482773 |
\(x_{10}\) | 0.999999999978438 | 0.999999977920327 | 0.999977390415280 |
\(x_{11}\) | 0.999999999997803 | 0.999999995501064 | 0.999990786179471 |
\(x_{12}\) | 0.999999999999777 | 0.999999999086208 | 0.999996257108584 |
\(x_{13}\) | 0.999999999999977 | 0.999999999814902 | 0.999998483680543 |
\(x_{14}\) | 0.999999999999998 | 0.999999999962595 | 0.999999387160191 |
\(x_{15}\) | 1 | 0.999999999992457 | 0.999999752825556 |
\(x_{16}\) | 1 | 0.999999999998482 | 0.999999900490241 |
\(x_{17}\) | 1 | 0.999999999999695 | 0.999999960003588 |
\(x_{18}\) | 1 | 0.999999999999939 | 0.999999983947466 |
\(x_{19}\) | 1 | 0.999999999999988 | 0.999999993565774 |
\(x_{20}\) | 1 | 0.999999999999997 | 0.999999997424076 |
3 Convergence results for Suzuki generalized nonexpansive mappings
Theorem 3.1
Let C be a nonempty closed convex subset of a complete \(\mathit{CAT}(0)\) space X and \(T:C\rightarrow C\) be a Suzuki generalized nonexpansive mapping with \(F(T)\neq\emptyset\). For arbitrarily chosen \(x_{0}\in C\), let the sequence \(\{x_{n}\}\) be generated by (5) then \(\lim_{n\rightarrow \infty}d(x_{n},p)\) exists for any \(p\in F(T)\).
Proof
This implies that \(\{d(x_{n},p)\}\) is bounded and non-increasing for all \(p\in F(T)\). Hence \(\lim_{n\rightarrow\infty}d(x_{n},p)\) exists, as required. □
Theorem 3.2
Let C, X, T and \(\{x_{n}\}\) be as in Theorem 3.1, where \(\{ \xi_{n}\}\) and \(\{ \zeta_{n}\}\) are sequences of real numbers in \([a,b]\) for some a, b with \(0< a\leq b<1\). Then \(F(T)\neq\emptyset\) if and only if \(\{x_{n}\}\) is bounded and \(\lim_{n\rightarrow\infty}d(Tx_{n},x_{n})=0\).
Proof
From (9), (11), (13) and Lemma 2.7, we have \(\lim_{n\rightarrow\infty}d(Tx_{n},x_{n})=0\).
This implies that \(Tp\in A(C,\{x_{n}\})\). Since X is uniformly convex, \(A(C,\{x_{n}\})\) is a singleton and hence we have \(Tp=p\). Hence \(F(T)\neq \emptyset\). □
The proof of the following Δ-convergence theorem is similar to the proof of [24, Theorem 3.3].
Theorem 3.3
Let C, X, T and \(\{x_{n}\}\) be as in Theorem 3.2 with \(F(T)\neq\emptyset\). Then \(\{x_{n}\}\) Δ-converges to a fixed point of T.
Next we prove the strong convergence theorem.
Theorem 3.4
Let C, X, T and \(\{x_{n}\}\) be as in Theorem 3.2 such that C is compact subset of X. Then \(\{x_{n}\}\) converges strongly to a fixed point of T.
Proof
Letting \(k\rightarrow\infty\), we get \(Tp=p\), i.e., \(p\in F(T)\). By Theorem 3.1, \(\lim_{n\rightarrow\infty}d(x_{n},p)\) exists for every \(p\in F(T)\) and so the \(x_{n}\) converge strongly to p. □
A strong convergence theorem using condition I introduced by Senter and Dotson [25] is as follows.
Theorem 3.5
Let C, X, T and \(\{x_{n}\}\) be as in Theorem 3.2 with \(F(T)\neq\emptyset\). If T satisfies condition \((I)\), then \(\{x_{n}\}\) converges strongly to a fixed point of T.
Proof
This shows that \(\{y_{k}\}\) is a Cauchy sequence in \(F(T)\) and so it converges to a point p. Since \(F(T)\) is closed, \(p\in F(T)\) and then \(\{x_{n_{k}}\}\) converges strongly to p. Since \(\lim_{n\rightarrow \infty}d(x_{n},p)\) exists, we have \(x_{n}\rightarrow p\in F(T)\). □
4 Conclusions
The extension of the linear version of fixed point results to nonlinear domains has its own significance. To achieve the objective of replacing a linear domain with a nonlinear one, Takahashi [26] introduced the notion of a convex metric space and studied fixed point results of nonexpansive mappings in this framework. This initiated the study of different convexity structures on a metric space. Here we extend a linear version of convergence results to the fixed point of a mapping satisfying condition C for the newly introduced K iteration process [20] to nonlinear \(\mathit{CAT}(0)\) spaces.
Declarations
Acknowledgements
The authors are thankful to the reviewers for their valuable comments and suggestions.
Funding
We have no funding for this project of research.
Authors’ contributions
All authors contributed equally. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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