 Research
 Open Access
 Published:
Some convergence results using K iteration process in \(\mathit{CAT}(0)\) spaces
Fixed Point Theory and Applications volume 2018, Article number: 11 (2018)
Abstract
In this paper, some strong and Δconvergence 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 wellknown results in the literature of fixed point theory in \(\mathit{CAT}(0)\) spaces.
Introduction
The wellknown 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 wellknown iterative processes are those of Mann [4], Ishikawa [5], Agarwal [6], Noor [7], Abbas [8], SP [9], S^{∗} [10], CR [11], NormalS [12], Picard Mann [13], PicardS [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 threestep iteration process known as the K iteration process and proved that it is converging fast as compared to all abovementioned 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 Siteration process and leading three steps PicardSiteration process. Finally, we prove some strong and Δconvergence theorems for Suzuki generalized nonexpansive mappings in the setting of \(\mathit{CAT}(0)\) spaces.
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])
For \(x,y\in X\) and let \(\xi\in[0,1]\), there exists a unique point \(s\in[ x, y]\) where \([ x, y]\) is the line segment joining x and y, such that
The notation \(((1\xi)x\oplus\xi y)\) is used for the unique point s satisfying (1).
Lemma 2.2
([13, Lemma 2.4])
For \(x,y,z\in X\) and \(\xi\in[0,1]\), we have
Let C be a nonempty closed convex subset of a \(\mathit{CAT}(0)\) space X, and let \(\{x_{n}\}\) be a bounded sequence in X. For \(x\in X\), we set
The asymptotic radius of \(\{x_{n}\}\) relative to C is given by
and the asymptotic center of \(\{x_{n}\}\) relative to C is the set
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 Δconvergent 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.
A mapping \(T:C\rightarrow C\) is called a contraction if there exists \(\xi \in(0,1)\) such that
for all \(x,y\in C\).
A mapping \(T:C\rightarrow C\) is called nonexpansive if
for all \(x,y\in C\).
In 2008, Suzuki [22] introduced a new condition on a mapping, called condition \((C)\), which is weaker than nonexpansiveness. A mapping \(T:C\rightarrow C\) is said to satisfy condition \((C)\) if for all \(x,y\in C\), we have
The mapping satisfying condition \((C)\) is called a Suzuki generalized nonexpansive mapping. The following is an example of a Suzuki generalized nonexpansive mapping which is not nonexpansive.
Example 2.4
Define a mapping \(T:[0,1]\rightarrow[0,1]\) by
We need to prove that T is a Suzuki generalized nonexpansive mapping but not nonexpansive.
If \(x=\frac{1}{11}\), \(y=\frac{1}{10}\) we see that
Hence T is not a nonexpansive mapping.
To verify that T is a Suzuki generalized nonexpansive mapping, consider the following cases:
Case I: Let \(x\in [ 0,\frac{1}{10} ) \), then \(\frac{1}{2}d(x,Tx)=\frac{12x}{2} \in ( \frac{2}{5},\frac {1}{2} ] \). For \(\frac{1}{2}d(x,Tx)\leq d(x,y)\) we must have \(\frac{12x}{2}\leq yx\), i.e., \(\frac{1}{2}\leq y\), hence \(y\in [ \frac{1}{2},1 ] \). We have
and
Hence \(\frac{1}{2}d(x,Tx)\leq d(x,y)\Longrightarrow d(Tx,Ty)\leq d(x,y)\).
Case II: Let \(x\in [ \frac{1}{10},1 ] \), then \(\frac{1}{2}d(x,Tx)=\frac{1}{2} \vert \frac{x+1}{2}x \vert =\frac {1x}{4}\in [ 0,\frac{9}{40} ] \). For \(\frac{1}{2}d(x,Tx)\leq d(x,y)\) we must have \(\frac{1x}{4}\leq \vert yx \vert \), which gives two possibilities:
(a). Let \(x< y\), then \(\frac{1x}{4}\leq yx\Longrightarrow y\geq \frac{1+3x}{4}\Longrightarrow y\in [ \frac{13}{40},1 ] \subset [ \frac{1}{10},1 ] \). So
Hence \(\frac{1}{2}d(x,Tx)\leq d(x,y)\Longrightarrow d(Tx,Ty)\leq d(x,y)\).
(b). Let \(x>y\), then \(\frac{1x}{4}\leq xy\Longrightarrow y\leq x\frac{1x}{4}=\frac{5x1}{4}\Longrightarrow y\in [ \frac {1}{8},1 ] \). Since \(y\in [ 0,1 ] \), so \(y\leq\frac {5x1}{4}\Longrightarrow x\in [ \frac{1}{5},1 ] \). So the case is \(x\in [ \frac{1}{5},1 ] \) and \(y\in [ 0,1 ] \).
Now \(x\in [ \frac{1}{5},1 ] \) and \(y\in [ \frac {1}{10},1 ] \) is already included in (a). So let \(x\in [ \frac {1}{5},1 ] \) and \(y\in [ 0,\frac{1}{10} ) \), then
For convenience, first we consider \(x\in [ \frac{1}{5},\frac {7}{8} ] \) and \(y\in [ 0,\frac{1}{10} ) \), then \(d(Tx,Ty)\leq\frac{3}{80}\) and \(d(x,y)>\frac{1}{10}\). Hence \(d(Tx,Ty)\leq d(x,y)\).
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
Let C be a nonempty subset of a \(\mathit{CAT}(0)\) space X and \(T:C\rightarrow C\) be any mapping. Then:

(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 quasinonexpansive mapping.

(iii)
[22, Lemma 7] If T is a Suzuki generalized nonexpansive mapping, then
$$ d(x,Ty)\leq3d(Tx,x)+d(x,y) $$for all \(x,y\in C\).
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])
Suppose that X is a complete \(\mathit{CAT}(0)\) space and \(x \in X\). \(\{t_{n}\}\) is a sequence in \([b, c]\) for some \(b, c \in(0, 1)\) and \(\{x_{n}\}\), \(\{y_{n}\}\) are sequences in X such that, for some \(r\geq0\), we have
then
Let \(n\geq0\) and \(\{ \xi_{n}\}\) and \(\{ \zeta_{n}\}\) be real sequences in \([0,1]\). Hussain et al. [20] introduced a new iteration process namely the K iteration process, thus:
They also proved that the K iteration process is faster than the PicardS and Siteration processes with the help of a numerical example. In order to show the efficiency of the K iteration process we use Example 2.4 with \(x_{0}=0.9\) and get Table 1. A graphic representation is given in Fig. 1. We can easily see the efficiency of the K iteration process.
Convergence results for Suzuki generalized nonexpansive mappings
In this section, we prove some strong and Δconvergence theorems of a sequence generated by a K iteration process for Suzuki generalized nonexpansive mappings in the setting of \(\mathit{CAT}(0)\) space. The K iteration process in the language of \(\mathit{CAT}(0)\) space is given by
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
Let \(p\in F(T)\) and \(z\in C\). Since T is a Suzuki generalized nonexpansive mapping,
So by Proposition 2.5(ii), we have
Using (6) we get
Similarly by using (7) we have
This implies that \(\{d(x_{n},p)\}\) is bounded and nonincreasing 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
Suppose \(F(T)\neq\emptyset\) and let \(p\in F(T)\). Then, by Theorem 3.1, \(\lim_{n\rightarrow\infty}d(x_{n},p)\) exists and \(\{x_{n}\}\) is bounded. Put
By Proposition 2.5(ii) we have
On the other hand by using (6), we have
This implies that
So
implies that
Therefore
From (9), (11), (13) and Lemma 2.7, we have \(\lim_{n\rightarrow\infty}d(Tx_{n},x_{n})=0\).
Conversely, suppose that \(\{x_{n}\}\) is bounded and \(\lim_{n\rightarrow \infty}d(Tx_{n},x_{n})=0\). Let \(p\in A(C,\{x_{n}\})\). By Proposition 2.5(iii), we have
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 Δconvergence 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 \(\{x_{n}\}\) Δconverges 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
By Lemma 2.6, we have \(F(T)\neq\emptyset\) and so by Theorem 3.1 we have \(\lim_{n\rightarrow\infty}d(Tx_{n},x_{n})=0\). Since C is compact, there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(\{x_{n_{k}}\}\) converges strongly to p for some \(p\in C\). By Proposition 2.5(iii), we have
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
By Theorem 3.1, we see that \(\lim_{n\rightarrow\infty }d(x_{n},p)\) exists for all \(p\in F(T)\) and so \(\lim_{n\rightarrow\infty}d(x_{n},F(T))\) exists. Assume that \(\lim_{n\rightarrow\infty}d(x_{n},p)=r\) for some \(r\geq0\). If \(r=0\) then the result follows. Suppose \(r>0\), from the hypothesis and condition \((I)\),
Since \(F(T)\neq\emptyset\), by Theorem 3.2, we have \(\lim_{n\rightarrow\infty}d(Tx_{n},x_{n})=0\). So (14) implies that
Since f is a nondecreasing function, from (15) we have \(\lim_{n\rightarrow\infty}d(x_{n},F(T))=0\). Thus, we have a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) and a sequence \(\{y_{k}\} \subset F(T)\) such that
So using (9), we get
Hence
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 \(x_{n}\rightarrow p\in F(T)\). □
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.
References
 1.
Saluja, G.S., Postolache, M.: Threestep iterations for total asymptotically nonexpansive mappings in \(\mathit{CAT}(0)\) spaces. Filomat 31(5), 1317–1330 (2017)
 2.
Saluja, G.S., Postolache, M., Kurdi, A.: Convergence of threestep iterations for nearly asymptotically nonexpansive mappings in \(\mathit{CAT}(k)\) spaces. J. Inequal. Appl. 2015, Article ID 156 (2015)
 3.
Abbas, M., Khan, S.H., Postolache, M.: Existence and approximation results for SKC mappings in \(\mathit{CAT}(0)\) spaces. J. Inequal. Appl. 2014, Article ID 212 (2014)
 4.
Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)
 5.
Ishikawa, S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147–150 (1974)
 6.
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)
 7.
Dhompongsa, S., Panyanak, B.: On Δconvergence theorem in \(\mathit{CAT}(0)\) spaces. Comput. Math. Appl. 56, 2572–2579 (2008)
 8.
Abbas, M., Nazir, T.: A new faster iteration process applied to constrained minimization and feasibility problems. Mat. Vesn. 66, 223–234 (2014)
 9.
Phuengrattana, W., Suantai, S.: On the rate of convergence of Mann, Ishikawa, Noor and SPiterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math. 235, 3006–3014 (2011)
 10.
Karahan, I., Ozdemir, M.: A general iterative method for approximation of fixed points and their applications. Adv. Fixed Point Theory 3(3), 510–526 (2013)
 11.
Chugh, R., Kumar, V., Kumar, S.: Strong convergence of a new three step iterative scheme in Banach spaces. Am. J. Comput. Math. 2, 345–357 (2012)
 12.
Sahu, D.R., Petrusel, A.: Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces. Nonlinear Anal., Theory Methods Appl. 74, 6012–6023 (2011)
 13.
Khan, S.H.: A Picard–Mann hybrid iterative process. Fixed Point Theory Appl. 2013, Article ID 69 (2013)
 14.
Gursoy, F., Karakaya, V.: A PicardS hybrid type iteration method for solving a differential equation with retarded argument (2014). arXiv:1403.2546v2
 15.
Thakur, B.S., Thakur, D., Postolache, M.: A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings. Appl. Math. Comput. 275, 147–155 (2016)
 16.
Thakur, B.S., Thakur, D., Postolache, M.: New iteration scheme for numerical reckoning fixed points of nonexpansive mappings. J. Inequal. Appl. 2014, Article ID 328 (2014)
 17.
Yao, Y., Postolache, M., Liou, Y.C., Yao, Z.: Construction algorithms for a class of monotone variational inequalities. Optim. Lett. 10(7), 1519–1528 (2016)
 18.
Yao, Y., Liou, Y.C., Postolache, M.: Selfadaptive algorithms for the split problem of the demicontractive operators. Optimization (2018). https://doi.org/10.1080/02331934.2017.1390747
 19.
Yao, Y., Leng, L., Postolache, M., Zheng, X.: Manntype iteration method for solving the split common fixed point problem. J. Nonlinear Convex Anal. 18(5), 875–882 (2017)
 20.
Hussain, N., Ullah, K., Arshad, M.: Fixed point approximation for Suzuki generalized nonexpansive mappings via new iteration process (2018). arXiv:1802.09888v1
 21.
Bridson, M., Heaflinger, A.: Metric Space of Nonpositive Curvature. Springer, Berlin (1999)
 22.
Suzuki, T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 340, 1088–1095 (2008)
 23.
Lawaong, W., Panyanak, B.: Approximating fixed points of nonexpansive nonself mappings in \(\mathit{CAT}(0)\) spaces. Fixed Point Theory Appl. 2010, Article ID 367274 (2010)
 24.
Basarir, M., Sahin, A.: On the strong and Δconvergence of Siteration process for generalized nonexpansive mappings on \(\mathit{CAT}(0)\) space. Thai J. Math. 12, 549–559 (2014)
 25.
Senter, H.F., Dotson, W.G.: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 44, 375–380 (1974)
 26.
Takahashi, T.: A convexity in metric spaces and nonexpansive mappings. Kodai Math. Semin. Rep. 22, 142–149 (1970)
Acknowledgements
The authors are thankful to the reviewers for their valuable comments and suggestions.
Funding
We have no funding for this project of research.
Author information
Affiliations
Contributions
All authors contributed equally. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Ullah, K., Iqbal, K. & Arshad, M. Some convergence results using K iteration process in \(\mathit{CAT}(0)\) spaces. Fixed Point Theory Appl 2018, 11 (2018). https://doi.org/10.1186/s1366301806370
Received:
Accepted:
Published:
MSC
 47H09
 47H10
Keywords
 Suzuki generalized nonexpansive mapping
 \(\mathit{CAT}(0)\) space
 K iteration process
 Δconvergence
 Strong convergence