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 Open Access
Fixed point approximation for \(SKC\)mappings in hyperbolic spaces
 Jong Kyu Kim^{1}Email author and
 Samir Dashputre^{2}
https://doi.org/10.1186/s1366001508680
© Kim and Dashputre 2015
 Received: 8 July 2015
 Accepted: 14 October 2015
 Published: 26 October 2015
Abstract
In this paper, we introduce the class of \(SKC\)mappings, which is a generalization of the class of Suzukigeneralized nonexpansive mappings, and we prove the strong and Δconvergence theorems of the Siteration process which is generated by \(SKC\)mappings (Karapinar and Tas in Comput. Math. Appl. 61:33703380, 2011) in uniformly convex hyperbolic spaces. As uniformly convex hyperbolic spaces contain Banach spaces as well as \(\operatorname {CAT}(0)\) spaces, our results can be viewed as an extension and generalization of several wellknown results in Banach spaces as well as \(\operatorname {CAT}(0)\) spaces.
Keywords
 \(SKC\)mappings
 Δconvergence
 uniformly convex hyperbolic spaces
 Siteration process
MSC
 47H09
 47H10
 49M05
1 Introduction
First, we give some definitions for the main results.
Definition 1.1
 (i)nonexpansive iffor all \(x, y \in C\);$$\Vert Tx  Ty \Vert \leq \Vert x  y \Vert , $$
 (ii)quasinonexpansive iffor each \(p \in F(T)\) and for all \(x\in C\), where \(F(T)= \{x \in C; Tx = x \}\) denotes the set of fixed points of T.$$\Vert Tx  p \Vert \leq \Vert x  p \Vert , $$
In 2008, Suzuki [1] introduced a class of singlevalued mappings, called Suzukigeneralized nonexpansive mappings, as follows.
Definition 1.2
From the following examples, we know that condition \((C)\) is weaker than nonexpansiveness and stronger than quasinonexpansiveness, that is, every nonexpansive mapping T satisfies condition \((C)\) and if the mapping T satisfies condition \((C)\) with \(F(T) \neq\phi\), then it is a quasinonexpansive.
Example 1.3
[1]
Example 1.4
[1]
In [1], Suzuki proved the existence of the fixed point and convergence theorems for mappings satisfying condition \((C)\) in Banach spaces. In the same space setting under certain conditions Dhompongsa et al. [2] improved the results of Suzuki [1] and obtained a fixed point result for mappings with condition \((C)\).
In 2011, Karapınar et al. [3], proposed some new classes of mappings which significantly generalized the notion of Suzukitype nonexpansive mappings as follows.
Definition 1.5
 (i)a SuzukiCiric mapping SCC [3] ifwhere \(M(x, y) = \max\{d(x, y), d( x, Tx), d( y, Ty), d( x, Ty), d( y, Tx) \} \) for all \(x, y \in C\);$$\frac{1}{2} d(Tx, Ty) \leq d(x, y)\quad \text{implies}\quad d(Tx, Ty) \leq M(x, y), $$
 (ii)a SuzukiKC mapping SKC ifwhere \(N(x, y) = \max \{d(x, y),\frac{d(x, Tx) + d(y, Ty)}{2}, \frac {d(x, Ty)+ d(y, Tx)}{2} \} \) for all \(x, y \in C\);$$\frac{1}{2} d(Tx, Ty) \leq d( x, y)\quad \text{implies}\quad d(Tx, Ty) \leq N(x, y), $$
 (iii)a KannanSuzuki mapping KSC iffor all \(x, y \in C\);$$\frac{1}{2} d(Tx, Ty) \leq d(x, y)\quad \text{implies}\quad d(Tx, Ty) \leq \frac{d(x, Tx) + d(y, Ty)}{2} $$
 (iv)a ChatterjeaSuzuki mapping CSC iffor all \(x, y \in C\).$$\frac{1}{2} d(T x, T y) \leq d(x, y)\quad \text{implies}\quad d(Tx, Ty) \leq \frac{d(y, Tx) + d(x, Ty)}{2} $$
From the above definition, it is clear that every nonexpansive mapping satisfies condition \(SKC\), but the converse is not true, as becomes clear from the following examples.
Example 1.6
[3]
Example 1.7
[3]
Example 1.8
In the framework of \(\operatorname {CAT}(0)\) spaces one gave some characterization of existing fixed point results for mappings with condition \((C)\). In [4], Abbas et al. extended the result of Nanjaras et al. [5] for the class of \(SKC\)mappings and proved some strong and Δconvergence results for a finite family of \(SKC\)mappings using an Ishikawatype iteration process in the framework of \(\operatorname {CAT}(0)\) spaces (see [4]).
On the other hand, the following fixed point iteration processes have been extensively studied by many authors for approximating either fixed points of nonlinear mappings (when these mappings are already known to have fixed points) or solutions of nonlinear operator equations.
(M) The Mann iteration process (see [6, 7]) is defined as follows:
 (M_{1}):

\(0 \leq\alpha_{n} <1 \),
 (M_{2}):

\(\lim_{n\to \infty} \alpha_{n} = 0\),
 (M_{3}):

\(\sum_{n =1}^{\infty} \alpha_{n} = \infty\).
(I) The Ishikawa iteration process (see [6, 8]) is defined as follows:
 (I_{1}):

\(0\leq\alpha_{n} \leq \beta_{n} < 1\),
 (I_{2}):

\(\lim_{n \to\infty} \beta_{n} =0\),
 (I_{3}):

\(\sum_{n =1}^{\infty} \alpha_{n} \beta_{n} = \infty\).
It is clear that the process (M) is not a special case of the process (I) because of condition (I_{1}). In some papers (see [9–13]) condition (I_{1}) \(0\leq\alpha_{n} \leq\beta_{n} < 1\) has been replaced by the general condition (\(\mathrm{I}_{1}^{'}\)) \(0< \alpha _{n}, \beta_{n} <1\). With this general setting, the process (I) is a natural generalization of the process (M). It is observed that, if the process (M) is convergent, then the process (I) with condition (\(\mathrm{I}_{1}^{'}\)) is also convergent under suitable conditions on \(\alpha_{n}\) and \(\beta_{n}\).
It is easy to see that neither the process (M) nor the process (I) reduces to an Siteration process and vice versa. Thus, the Siteration process is independent of the Mann [7] and Ishikawa [8] iteration processes (see [6, 14, 15]).
It is observed that the rate of convergence of the Siteration process is similar to the Picard iteration process, but faster than the Mann iteration process for a contraction mapping (see [6, 14, 15]).
On the other hand, in [16], Leuştean proved that \(\operatorname {CAT}(0)\) spaces are uniformly convex hyperbolic spaces with a modulus of uniform convexity \(\eta(r, \varepsilon) = \frac{\varepsilon^{2}}{8}\) quadratic in ε. Therefore, we know that the class of uniformly convex hyperbolic spaces is a generalization of both uniformly convex Banach spaces and \(\operatorname {CAT}(0)\) spaces.
We consider the following definition of a hyperbolic space introduced by Kohlenbach [17], and, also, Zhao et al. [18] and Kim et al. [19] got some convergence results in a hyperbolic space setting.
Definition 1.9
 (W1)
\(d( u, W(x, y, \alpha)) \leq\alpha d( u, x) + ( 1\alpha ) d( u, y)\);
 (W2)
\(d(W(x, y, \alpha), W(x, y,\beta)) = \vert\alpha \beta \vert d( x, y)\);
 (W3)
\(W(x, y, \alpha) = W(y, x, 1\alpha)\);
 (W4)
\(d( W(x, z, \alpha), W(y, w, \alpha))\leq(1\alpha) d(x, y)+\alpha d(z, w)\),
A metric space is said to be a convex metric space in the sense of Takahashi [20], where a triple \((X, d, W) \) satisfies only (W1) (see [21]). We get the notion of the space of hyperbolic type in the sense of Goebel and Kirk [22], where a triple \((X, d, W)\) satisfies (W1)(W3). Condition (W4) was already considered by Itoh [23] under the name of ‘condition III’ and it is used by Reich and Shafrir [24] and Kirk [25] to define their notions of hyperbolic spaces.
The class of hyperbolic spaces include normed spaces and convex subsets thereof, the Hilbert space ball equipped with the hyperbolic metric [26], the Hadamard manifold, and the \(\operatorname {CAT}(0)\) spaces in the sense of Gromov (see [27]).
A mapping \(\eta: (0, \infty) \times(0, 2] \to(0,1]\) providing such a \(\delta= \eta(r, \varepsilon)\) for given \(r>0\) and \(\varepsilon \in(0, 2]\), is called a modulus of uniform convexity. We say that η is monotone if it decreases with r for fixed ε.
The purpose of this paper is to prove some strong and Δconvergence theorems of the Siteration process which is generated by \(SKC\)mappings in uniformly convex hyperbolic spaces. Our results can be viewed as an extension and a generalization of several wellknown results in Banach spaces as well as \(\operatorname {CAT}(0)\) spaces (see [1–6, 15, 21, 28–30]).
2 Preliminaries
First, we give the concept of Δconvergence and some of its basic properties.
If the asymptotic radius and the asymptotic center are taken with respect to X, then these are simply denoted by \(r_{a}( X, \{x_{n}\}) = r_{a}( \{x_{n}\})\) and \(\operatorname {AC}(X, \{x_{n}\})= \operatorname {AC}(\{x_{n}\})\), respectively. We know that, for \(x \in X\), \(r_{a}( x, \{x_{n}\}) = 0 \) if and only if \(\lim_{n \to\infty} x_{n} = x\).
It is well known that every bounded sequence has a unique asymptotic center with respect to each closed convex subset in uniformly convex Banach spaces and even \(\operatorname {CAT}(0)\) spaces.
The following lemma is due to Leuştean [31] and we know that this property also holds in a complete uniformly convex hyperbolic space.
Lemma 2.1
[31]
Let \((X, d, W)\) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Then every bounded sequence \(\{x_{n}\}\) in X has a unique asymptotic center with respect to any nonempty closed convex subset C of X.
Definition 2.2
A sequence \(\{x_{n}\}\) in a hyperbolic space X is said to be Δconvergent to \(x \in X\), if x is the unique asymptotic center of \(\{u_{n}\}\) for every subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\). In this case, we write Δ\(\lim_{n} x_{n} = x\) and we call x the Δlimit of \(\{x_{n}\}\).
Recall that a bounded sequence \(\{x_{n}\}\) in a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity η is said to be regular if \(r_{a}(X, \{x_{n}\}) = r_{a}(X, \{ u_{n}\})\) for every subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\).
Lemma 2.3
[33]
3 Main results
Now we will give the definition of \(Fej\acute{e}r\) monotone sequences.
Definition 3.1
Example 3.2
Let C be a nonempty subset of X and let \(T : C \to C\) be a quasinonexpansive (in particular, nonexpansive) mapping such that \(F(T) \neq\phi\). Then the Picard iterative sequence \(\{x_{n}\}\) is \(Fej\acute{e}r\) monotone with respect to \(F(T)\).
Proposition 3.3
[19]
 (1)
\(\{x_{n}\}\) is bounded;
 (2)
the sequence \(\{d(x_{n}, p)\}\) is decreasing and convergent for all \(p \in F(T)\).
We now define the Siteration process in hyperbolic spaces (see [19]):
We can easily prove the following lemma from the definition of \(SKC\)mapping.
Lemma 3.4
Let C be a nonempty closed convex subset of a hyperbolic space X and let \(T :C \to C\) be an \(SKC\)mapping. If \(\{x_{n}\}\) is a sequence defined by (3.1), then \(\{x_{n}\}\) is \(Fej\acute{e}r\) monotone with respect to \(F(T)\).
Proof
Lemma 3.5
Let C be a nonempty closed convex subset of a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity η and let \(T :C \to C\) be an \(SKC\)mapping. If \(\{x_{n}\}\) is the sequence defined by (3.1), then \(F(T)\) is nonempty if and only if \(\{x_{n}\}\) is bounded and \(\lim_{n \to\infty}d(x_{n}, Tx_{n}) = 0\).
Proof
Conversely, suppose that \(\{x_{n}\}\) is bounded and \(\lim_{n \to\infty} d(x_{n}, Tx_{n}) = 0\).
Now, we are in a position to prove the Δconvergence theorem.
Theorem 3.6
Let C be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η and let \(T : C \to C\) be an \(SKC\)mapping with \(F(T) \neq\phi\). If \(\{x_{n}\}\) is the sequence defined by (3.1), then the sequence \(\{x_{n}\}\) is Δconvergent to a fixed point of T.
Proof
Remark 3.7
Theorem 3.6 is an extension of Theorem 3.3 of Abbas et al. [4] from \(\operatorname {CAT}(0)\) space to a uniformly convex hyperbolic space. Theorem 3.6 also holds for the \(KSC\), \(SCC\), and \(CSC\)mappings.
Now, we will introduce the strong convergence theorems in hyperbolic spaces.
Theorem 3.8
Proof
Next, we will give one more strong convergence theorem by using Theorem 3.8. We recall the definition of condition (I) introduced by Senter and Doston [34].
Theorem 3.9
Let C be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η and let \(T : C \to C\) be an \(SKC\)mapping with condition (I) and \(F(T) \neq\phi\). Then the sequence \(\{x_{n}\}\) which is defined by (3.1) converges strongly to some fixed point of T.
Proof
Remark 3.10
Theorems 3.6, 3.8, and 3.9 improve and extend the previous known results from Banach spaces and \(\operatorname {CAT}(0)\) spaces to uniformly convex hyperbolic spaces (see [1–6, 15, 21, 28–30], in particular, Theorems 3.1, 3.2, 3.3, and 3.4 in [4]). In our results, we considered the Siteration which is faster than the other iteration process to approximate the fixed point of underlying mapping in the framework of uniformly convex hyperbolic spaces.
4 Numerical example
Example 4.1
It is easy to see that T satisfies the \(SKC\) condition and \(0 \in C\) is a fixed point of T. It is observed that it satisfies all conditions in Theorem 3.6. Let \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) be constant sequences such that \(\alpha_{n} = \beta_{n} = \frac{1}{2}\) for all \(n \geq0\). From the definition of T the following cases arise.
Declarations
Acknowledgements
This work was supported by the Basic Science Research Program through the National Research Foundation (NRF) Grant funded by Ministry of Education of the republic of Korea (2014046293).
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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