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Some convergence results for modified SPiteration scheme in hyperbolic spaces
Fixed Point Theory and Applications volume 2014, Article number: 133 (2014)
Abstract
In this paper, we prove some strong and Δconvergence theorems for a modified SPiteration scheme for total asymptotically nonexpansive mappings in hyperbolic spaces by employing recent technical results of Khan et al. (Fixed Point Theory Appl. 2012:54, 2012). The results presented here extend and improve some wellknown results in the current literature.
MSC:47H09, 47H10.
1 Introduction and preliminaries
Iterative schemes play a key role in approximating fixed points of nonlinear mappings. Structural properties of the underlying space, such as strict convexity and uniform convexity, are very much needed for the development of iterative fixed point theory in it. Hyperbolic spaces are general in nature and inherit rich geometrical structure suitable to obtain new results in topology, graph theory, multivalued analysis and metric fixed point theory.
Kohlenbach [1] introduced the hyperbolic spaces, defined below, which play a significant role in many branches of mathematics.
A hyperbolic space (X,d,W) is a metric space (X,d) together with a mapping W:X\times X\times [0,1]\to X satisfying
(W1) d(z,W(x,y,\lambda ))\le (1\lambda )d(z,x)+\lambda d(z,y),
(W2) d(W(x,y,{\lambda}_{1}),W(x,y,{\lambda}_{2}))={\lambda}_{1}{\lambda}_{2}d(x,y),
(W3) W(x,y,\lambda )=W(y,x,(1\lambda )),
(W4) d(W(x,z,\lambda ),W(y,w,\lambda ))\le (1\lambda )d(x,y)+\lambda d(z,w)
for all x,y,z,w\in X and \lambda ,{\lambda}_{1},{\lambda}_{2}\in [0,1].
A subset K of a hyperbolic space X is convex if W(x,y,\lambda )\in K for all x,y\in K and \lambda \in [0,1]. If a space satisfies only (W1), it coincides with the convex metric space introduced by Takahashi [2]. The concept of hyperbolic spaces in [1] is more restrictive than the hyperbolic type introduced by Goebel et al. [3] since (W1)(W3) together are equivalent to (X,d,W) being a space of hyperbolic type in [3]. Also it is slightly more general than the hyperbolic space defined by Reich et al. [4]. The class of hyperbolic spaces in [1] contains all normed linear spaces and convex subsets thereof, ℝtrees, the Hilbert ball with the hyperbolic metric (see [5]), Cartesian products of Hilbert balls, Hadamard manifolds and CAT(0) spaces (see [6]) as special cases. Recently, the concept of puniformly convexity has been defined by Naor et al. [7] and its nonlinear version for p=2 in hyperbolic spaces was studied by Khan [8]. Any CAT(0) space is 2uniformly convex (see [9]).
The following example accentuates the importance of hyperbolic spaces.
Let {B}_{H} be an open unit ball in a complex Hilbert space (H,\u3008\cdot \u3009) w.r.t. the metric (also known as the Kobayashi distance)
where
Then ({B}_{H},{k}_{{B}_{H}},W) is a hyperbolic space where W(x,y,\lambda ) defines a unique point z in a unique geodesic segment [x,y] for all x,y\in {B}_{H}. For more on hyperbolic spaces and a detailed treatment of examples, we refer the readers to [1].
A hyperbolic space (X,d,W) is said to be

(i)
strictly convex [2] if for any x,y\in X and \lambda \in [0,1], there exists a unique element z\in X such that d(z,x)=\lambda d(x,y) and d(z,y)=(1\lambda )d(x,y);

(ii)
uniformly convex [10] if for all u,x,y\in X, r>0 and \epsilon \in (0,2], there exists \delta \in (0,1] such that d(W(x,y,\frac{1}{2}),u)\le (1\delta )r whenever d(x,u)\le r, d(y,u)\le r and d(x,y)\ge \epsilon r.
A mapping \eta :(0,\mathrm{\infty})\times (0,2]\to (0,1] providing such \delta =\eta (r,\epsilon ) for given r>0 and \epsilon \in (0,2] is called modulus of uniform convexity. We call η monotone if it decreases with r (for a fixed ε). A uniformly convex hyperbolic space is strictly convex (see [11]).
Let K be a nonempty subset of a metric space (X,d) and T be a selfmapping on K. Denote by F(T)=\{x\in K:T(x)=x\} the set of fixed points of T and d(x,F(T))=inf\{d(x,p):p\in F(T)\}. A selfmapping T is said to be

(1)
nonexpansive if d(Tx,Ty)\le d(x,y) for all x,y\in K;

(2)
asymptotically nonexpansive if there exists a sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}) with {k}_{n}\to 1 such that d({T}^{n}x,{T}^{n}y)\le {k}_{n}d(x,y) for all x,y\in K and n\ge 1;

(3)
uniformly LLipschitzian if there exists a constant L>0 such that d({T}^{n}x,{T}^{n}y)\le Ld(x,y) for all x,y\in K and n\ge 1;

(4)
total asymptotically nonexpansive if there exist nonnegative real sequences \{{\mu}_{n}\}, \{{v}_{n}\} with {\mu}_{n}\to 0, {v}_{n}\to 0 and a strictly increasing continuous function \zeta :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with \zeta (0)=0 such that
d({T}^{n}x,{T}^{n}y)\le d(x,y)+{v}_{n}\zeta (d(x,y))+{\mu}_{n}
for all x,y\in K and n\ge 1 (see [[12], Definition 2.1]).
It follows from the above definitions that each nonexpansive mapping is an asymptotically nonexpansive mapping with {k}_{n}=1, \mathrm{\forall}n\ge 1 and that each asymptotically nonexpansive mapping is a total asymptotically nonexpansive mapping with {v}_{n}={k}_{n}1, {\mu}_{n}=0, \mathrm{\forall}n\ge 1, \zeta (t)=t, \mathrm{\forall}t\ge 0. Moreover, each asymptotically nonexpansive mapping is a uniformly LLipschitzian mapping with L={sup}_{n\in \mathbb{N}}\{{k}_{n}\}. However, the converse of these statements is not true, in general.
It has been shown that every total asymptotically nonexpansive mapping defined on a nonempty bounded closed convex subset of a complete uniformly convex hyperbolic space always has a fixed point (see [[13], Theorem 3.1]).
The following iteration process is a translation of the SPiteration scheme introduced in [14] from Banach spaces to hyperbolic spaces. The SPiteration is equivalent to Mann, Ishikawa, Noor iterations and converges faster than the others for the class of continuous and nondecreasing functions (see [14]).
where K is a nonempty, closed and convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity and T:K\to K is a uniformly LLipschitzian and total asymptotically nonexpansive mapping.
Inspired and motivated by Khan et al. [15], Khan [16], Fukharuddin and Khan [17], Wan [18], Zhao et al. [19] and Zhao et al. [20], we prove some strong and △convergence theorems of the modified SPiteration process for approximating a fixed point of total asymptotically nonexpansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results given in [14, 21].
The concept of △convergence in a metric space was introduced by Lim [22] and its analogue in CAT(0) spaces was investigated by Dhompongsa and Panyanak [23]. In order to define the concept of △convergence in the general setup of hyperbolic spaces, we first collect some basic concepts.
Let \{{x}_{n}\} be a bounded sequence in a hyperbolic space X. For x\in X, we define a continuous functional r(\cdot ,\{{x}_{n}\}):X\to [0,\mathrm{\infty}) by
The asymptotic radius \rho =r(\{{x}_{n}\}) of \{{x}_{n}\} is given by
The asymptotic center of \{{x}_{n}\} with respect to a subset K of X is defined as follows:
This is the set of minimizer of the functional r(\cdot ,\{{x}_{n}\}). If the asymptotic center is taken with respect to X, then it is simply denoted by A(\{{x}_{n}\}).
Recall that a sequence \{{x}_{n}\} in X is said to △converge 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 \u25b3\text{}{lim}_{n\to \mathrm{\infty}}{x}_{n}=x and call x as △limit of \{{x}_{n}\}.
It is known that uniformly convex Banach spaces and even CAT(0) spaces enjoy the property that ‘bounded sequences have unique asymptotic centers with respect to closed convex subsets.’ The following lemma is due to Leustean [24] and ensures that this property also holds in a complete uniformly convex hyperbolic space.
Lemma 1 [[24], Proposition 3.3]
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 K of X.
In the sequel, we shall need the following results.
Lemma 2 [[15], Lemma 2.5]
Let (X,d,W) be a uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Let x\in X and \{{\alpha}_{n}\} be a sequence in [a,b] for some a,b\in (0,1). If \{{x}_{n}\} and \{{y}_{n}\} are sequences in X such that
for some r\ge 0, then
Lemma 3 [[15], Lemma 2.6]
Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space and let \{{x}_{n}\} be a bounded sequence in K such that A(\{{x}_{n}\})=\{y\} and r(\{{x}_{n}\})=\rho. If \{{y}_{m}\} is another sequence in K such that {lim}_{m\to \mathrm{\infty}}r({y}_{m},\{{x}_{n}\})=\rho, then {lim}_{m\to \mathrm{\infty}}{y}_{m}=y.
Lemma 4 [[25], Lemma 2]
Let \{{a}_{n}\}, \{{b}_{n}\} and \{{\delta}_{n}\} be sequences of nonnegative real numbers such that
If {\sum}_{n=1}^{\mathrm{\infty}}{\delta}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{b}_{n}<\mathrm{\infty}, then {lim}_{n\to \mathrm{\infty}}{a}_{n} exists.
2 Main results
We begin with Δconvergence of the modified SPiterative sequence \{{x}_{n}\} defined by (1.1) for total asymptotically nonexpansive mappings in hyperbolic spaces.
Theorem 1 Let K be a nonempty, closed and convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let T:K\to K be a uniformly LLipschitzian and total asymptotically nonexpansive mapping with F(T)\ne \mathrm{\varnothing}. If the following conditions are satisfied:

(i)
{\sum}_{n=1}^{\mathrm{\infty}}{v}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty};

(ii)
there exist constants a,b\in (0,1) such that \{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subset [a,b];

(iii)
there exists a constant M>0 such that \zeta (r)\le Mr, \mathrm{\forall}r\ge 0,
then the sequence \{{x}_{n}\} defined by (1.1), Δconverges to a fixed point of T.
Proof We divide our proof into three steps.
Step 1. First we prove that the following limits exist:
Since T is a total asymptotically nonexpansive mapping, by condition (iii), we get
and
Substituting (2.2) into (2.3) and simplifying it, we have
Similarly, we obtain
Combining (2.4) and (2.5), we have
and so
where {\sigma}_{n}={v}_{n}M({\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}+{v}_{n}M({\alpha}_{n}{\beta}_{n}+{\beta}_{n}{\gamma}_{n}+{\alpha}_{n}{\gamma}_{n}+{\alpha}_{n}{\beta}_{n}{\gamma}_{n}{v}_{n}M)) and {\xi}_{n}={\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}+{v}_{n}M({\alpha}_{n}{\beta}_{n}+{\beta}_{n}{\gamma}_{n}+{\alpha}_{n}{\gamma}_{n}+{\alpha}_{n}{\beta}_{n}{\gamma}_{n}{v}_{n}M). By virtue of condition (i),
By Lemma 4, {lim}_{n\to \mathrm{\infty}}d({x}_{n},F(T)) and {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exist for each p\in F(T).
Step 2. Next we prove that {lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0.
In fact, it follows from (2.1) that {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exists for each given p\in F(T). We may assume that {lim}_{n\to \mathrm{\infty}}d({x}_{n},p)=r. The case r=0 is trivial. Next, we deal with the case r>0. Taking lim sup on both sides in inequality (2.4), we have
Since
we have
In addition,
With the help of (2.7)(2.9) and Lemma 2, we have
On the other hand, since
we have {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}} d({y}_{n},p)\ge r. Combined with (2.7), it yields that {lim}_{n\to \mathrm{\infty}}d({y}_{n},p)=r. This implies that
Taking lim sup on both sides in inequality (2.2), we have
Since
we have
With the help of (2.11)(2.13) and Lemma 2, we have
By the same method, we can also prove that
By (2.10), we get
In a similar way, we have
and
It follows that
Since T is uniformly LLipschitzian, therefore we obtain
Hence, (2.15) and (2.16) imply that
Step 3. Now we prove that the sequence \{{x}_{n}\} △converges to a fixed point of T.
In fact, for each p\in F(T), {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exists. This implies that the sequence \{{x}_{n}\} is bounded. Hence by virtue of Lemma 1, \{{x}_{n}\} has a unique asymptotic center {A}_{K}(\{{x}_{n}\})=\{x\}. Let \{{u}_{n}\} be any subsequence of \{{x}_{n}\} such that {A}_{K}(\{{u}_{n}\})=\{u\}. Then, by (2.17), we have
We claim that u\in F(T). In fact, we define a sequence \{{z}_{m}\} in K by {z}_{m}={T}^{m}u. So, we calculate
Since T is uniformly LLipschitzian, from (2.19), we have
Taking lim sup on both sides of the above estimate and using (2.18), we have
This implies that r({z}_{m},\{{u}_{n}\})r(u,\{{u}_{n}\})\to 0 as m\to \mathrm{\infty}. It follows from Lemma 3 that {lim}_{m\to \mathrm{\infty}}{T}^{m}u=u. Utilizing the uniform continuity of T, we have that
Hence u\in F(T). Moreover, {lim}_{n\to \mathrm{\infty}}d({x}_{n},u) exists by (2.1). Suppose that x\ne u. By the uniqueness of asymptotic centers, we have
a contradiction. Hence x=u. Since \{{u}_{n}\} is an arbitrary subsequence of \{{x}_{n}\}, therefore A(\{{u}_{n}\})=\{u\} for all subsequences \{{u}_{n}\} of \{{x}_{n}\}, that is, \{{x}_{n}\} △converges to x\in F(T). The proof is completed. □
We now discuss the strong convergence of the modified SPiteration for total asymptotically nonexpansive mappings in hyperbolic spaces.
Theorem 2 Let K, X, T and \{{x}_{n}\} be the same as in Theorem 1. Suppose that conditions (i)(iii) in Theorem 1 are satisfied. Then \{{x}_{n}\} converges strongly to some p\in F(T) if and only if {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0.
Proof If \{{x}_{n}\} converges to p\in F(T), then {lim}_{n\to \mathrm{\infty}}d({x}_{n},p)=0. Since 0\le d({x}_{n},F(T))\le d({x}_{n},p), we have {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0.
Conversely, suppose that {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0. It follows from (2.1) that {lim}_{n\to \mathrm{\infty}}d({x}_{n},F(T)) exists. Thus by hypothesis {lim}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0.
Next, we show that \{{x}_{n}\} is a Cauchy sequence. In fact, it follows from (2.6) that for any p\in F(T),
where {\sum}_{n=1}^{\mathrm{\infty}}{\sigma}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{\xi}_{n}<\mathrm{\infty}. Hence for any positive integers n, m, we have
Since for each x\ge 0, 1+x\le {e}^{x}, we have
where N={e}^{{\sum}_{i=1}^{\mathrm{\infty}}{\sigma}_{i}}<\mathrm{\infty}. Therefore we have
This shows that \{{x}_{n}\} is a Cauchy sequence in K. Since K is a closed subset in a complete hyperbolic space X, it is complete. We can assume that \{{x}_{n}\} converges strongly to some point {p}^{\star}\in K. It is easy to prove that F(T) is a closed subset in K, so is F(T). Since {lim}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0, we obtain {p}^{\star}\in F(T). This completes the proof. □
Remark 1 In Theorem 2, the condition {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}} d({x}_{n},F(T))=0 may be replaced with {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}} d({x}_{n},F(T))=0.
Example 1 Let ℝ be the real line with the usual norm \cdot  and let K=[1,1]. Define two mappings {T}_{1},{T}_{2}:K\to K by
and
It is proved in [[26], Example 3.1] that both {T}_{1} and {T}_{2} are asymptotically nonexpansive mappings with {k}_{n}=1, \mathrm{\forall}n\ge 1. Therefore they are total asymptotically nonexpansive mappings with {v}_{n}={\mu}_{n}=0, \mathrm{\forall}n\ge 1, \zeta (t)=t, \mathrm{\forall}t\ge 0. Additionally, they are uniformly LLipschitzian mappings with L=1. Clearly, F({T}_{1})=\{0\} and F({T}_{2})=\{x\in K;0\le x\le 1\}. Set
Thus, the conditions of Theorem 1 are fulfilled. Therefore the results of Theorem 1 and Theorem 2 can be easily seen.
Example 2 Let ℝ be the real line with the usual norm \cdot  and let K=[0,\mathrm{\infty}). Define two mappings {S}_{1},{S}_{2}:K\to K by {S}_{1}(x)=sinx and {S}_{2}(x)=x. It is proved in [[27], Example 1] that both {S}_{1} and {S}_{2} are total asymptotically nonexpansive mappings with {v}_{n}=\frac{1}{{n}^{2}}, {\mu}_{n}=\frac{1}{{n}^{3}}, \mathrm{\forall}n\ge 1. Additionally, they are uniformly LLipschitzian mappings with L=1. Clearly, F({S}_{1})=\{0\} and F({S}_{2})=\{x\in K;0\le x<\mathrm{\infty}\}. Let \{{\alpha}_{n}\}, \{{\beta}_{n}\} and \{{\gamma}_{n}\} be the same as in (2.20). Similarly, the conditions of Theorem 1 are satisfied. So, the results of Theorem 1 and Theorem 2 also can be received.
Recall that a mapping T from a subset K of a metric space (X,d) into itself is semicompact if every bounded sequence \{{x}_{n}\}\subset K satisfying d({x}_{n},T{x}_{n})\to 0 as n\to \mathrm{\infty} has a strongly convergent subsequence.
Senter and Dotson [[28], p.375] introduced the concept of condition (I) as follows.
A mapping T:K\to K with F(T)\ne \mathrm{\varnothing} is said to satisfy condition (I) if there exists a nondecreasing function f:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with f(0)=0 and f(r)>0 for all r\in (0,\mathrm{\infty}) such that
By using the above definitions, we obtain the following strong convergence theorems.
Theorem 3 Under the assumptions of Theorem 1, if T is semicompact, then \{{x}_{n}\} converges strongly to a fixed point of T.
Proof It follows from (2.1) that \{{x}_{n}\} is a bounded sequence. Also, by (2.17), we have {lim}_{n\to \mathrm{\infty}} d({x}_{n},T{x}_{n})=0. Then, by the semicompactness of T, there exists a subsequence \{{x}_{{n}_{k}}\}\subset \{{x}_{n}\} such that \{{x}_{{n}_{k}}\} converges strongly to some point p\in K. Moreover, by the uniform continuity of T, we have
This implies that p\in F(T). Again, by (2.1), {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exists. Hence p is the strong limit of the sequence \{{x}_{n}\}. As a result, \{{x}_{n}\} converges strongly to a fixed point p of T. □
Theorem 4 Under the assumptions of Theorem 1, if T satisfies condition (I), then \{{x}_{n}\} converges strongly to a fixed point of T.
Proof By virtue of (2.1), {lim}_{n\to \mathrm{\infty}}d({x}_{n},F(T)) exists. Further, by condition (I) and (2.17), we have
That is, {lim}_{n\to \mathrm{\infty}} f(d({x}_{n},F(T)))=0. Since f is a nondecreasing function satisfying f(0)=0 and f(r)>0 for all r\in (0,\mathrm{\infty}), it follows that {lim}_{n\to \mathrm{\infty}} d({x}_{n},F(T))=0. Now Theorem 2 implies that \{{x}_{n}\} converges strongly to a point p in F(T). □
Remark 2 Theorems 14 contain the corresponding theorems proved for asymptotically nonexpansive mappings when {v}_{n}={k}_{n}1, {\mu}_{n}=0, \mathrm{\forall}n\ge 1, \zeta (t)=t, \mathrm{\forall}t\ge 0.
References
Kohlenbach U: Some logical metatheorems with applications in functional analysis. Trans. Am. Math. Soc. 2004, 357(1):89–128.
Takahashi W: A convexity in metric spaces and nonexpansive mappings. Kodai Math. Semin. Rep. 1970, 22: 142–149. 10.2996/kmj/1138846111
Goebel K, Kirk WA: Iteration processes for nonexpansive mappings. Contemp. Math. 21. In Topological Methods in Nonlinear Functional Analysis. Edited by: Singh SP, Thomeier S, Watson B. Am. Math. Soc., Providence; 1983:115–123.
Reich S, Shafrir I: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal., Theory Methods Appl. 1990, 15: 537–558. 10.1016/0362546X(90)90058O
Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings. Dekker, New York; 1984.
Bridson M, Haefliger A: Metric Spaces of NonPositive Curvature. Springer, Berlin; 1999.
Naor, A, Silberman, L: Poincaré inequalities, embeddings, and wild groups (2010) arXiv: 1005.4084v1 [math.GR]
Khan AR: Common fixed point and solution of nonlinear functional equations. Fixed Point Theory Appl. 2013., 2013: Article ID 290 10.1186/168718122013290
Khamsi MA, Khan A: Inequalities in metric spaces with applications. Nonlinear Anal. 2011, 74: 4036–4045. 10.1016/j.na.2011.03.034
Shimizu T, Takahashi W: Fixed points of multivalued mappings in certain convex metric spaces. Topol. Methods Nonlinear Anal. 1996, 8: 197–203.
Leustean L: A quadratic rate of asymptotic regularity for CAT(0)spaces. J. Math. Anal. Appl. 2007, 325: 386–399. 10.1016/j.jmaa.2006.01.081
Chang SS, Wang L, Joesph Lee HW, Chan CK, Yang L: Demiclosed principle and Δconvergence theorems for total asymptotically nonexpansive mappings in CAT(0) spaces. Appl. Math. Comput. 2012, 219: 2611–2617. 10.1016/j.amc.2012.08.095
Fukharuddin, H, Kalsoom, A, Khan, MAA: Existence and higher arity iteration for total asymptotically nonexpansivemappings in uniformly convex hyperbolic spaces (2013) arXiv: 1312.2418v2 [math.FA]
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. 2011, 235: 3006–3014. 10.1016/j.cam.2010.12.022
Khan AR, Fukharuddin H, Khan MAA: An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 54 10.1186/16871812201254
Khan MAA: Convergence analysis of a multistep iteration for a finite family of asymptotically quasinonexpansive mappings. J. Inequal. Appl. 2013., 2013: Article ID 423 10.1186/1029242X2013423
Fukharuddin H, Khan MAA: Convergence analysis of a general iteration schema of nonlinear mappings in hyperbolic spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 238 10.1186/168718122013238
Wan LL: △Convergence for mixedtype total asymptotically nonexpansive mappings in hyperbolic spaces. J. Inequal. Appl. 2013., 2013: Article ID 553 10.1186/1029242X2013553
Zhao LC, Chang SS, Kim JK: Mixed type iteration for total asymptotically nonexpansive mappings in hyperbolic spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 353 10.1186/168718122013353
Zhao LC, Chang SS, Wang XR: Convergence theorems for total asymptotically nonexpansive mappings in hyperbolic spaces. J. Appl. Math. 2013., 2013: Article ID 689765
Şahin A, Başarır M: On the strong and △convergence of SPiteration on a CAT(0) space. J. Inequal. Appl. 2013., 2013: Article ID 311 10.1186/1029242X2013311
Lim TC: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 1976, 60: 179–182. 10.1090/S0002993919760423139X
Dhompongsa S, Panyanak B: On △convergence theorems in CAT(0) spaces. Comput. Math. Appl. 2008, 56(10):2572–2579. 10.1016/j.camwa.2008.05.036
Leustean L: Nonexpansive iterations in uniformly convex W hyperbolic spaces. Contemp. Math. 513. In Nonlinear Analysis and Optimization I: Nonlinear Analysis. Edited by: Leizarowitz A, Mordukhovich BS, Shafrir I, Zaslavski A. Am. Math. Soc., Providence; 2010:193–209.
Qihou L: Iterative sequences for asymptotically quasinonexpansive mappings with error member. J. Math. Anal. Appl. 2001, 259: 18–24. 10.1006/jmaa.2000.7353
Guo W, Cho YJ, Guo W: Convergence theorems for mixed type asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2012., 2012: Article ID 224 10.1186/168718122012224
Kzıltunç H, Yolaçan E: Strong convergence theorems for two total asymptotically nonexpansive nonself mappings in Banach spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 90 10.1186/16871812201390
Senter HF, Dotson WG: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 1974, 44: 375–380. 10.1090/S00029939197403466088
Acknowledgements
The authors would like to thank the editor and referees for their careful reading and valuable comments and suggestions which led to the present form of the paper. This paper was supported by Sakarya University Scientific Research Foundation (Project number: 20130200003).
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Şahin, A., Başarır, M. Some convergence results for modified SPiteration scheme in hyperbolic spaces. Fixed Point Theory Appl 2014, 133 (2014). https://doi.org/10.1186/168718122014133
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DOI: https://doi.org/10.1186/168718122014133
Keywords
 hyperbolic space
 fixed point
 total asymptotically nonexpansive mapping
 modified SPiteration scheme
 strong convergence
 Δconvergence