 Research
 Open Access
 Published:
Mixed type iteration for total asymptotically nonexpansive mappings in hyperbolic spaces
Fixed Point Theory and Applications volume 2013, Article number: 353 (2013)
Abstract
The purpose of this paper is to introduce the concept of total asymptotically nonexpansive mappings and to prove some Δconvergence theorems of the mixed type iteration process to approximating a common fixed point for two asymptotically nonexpansive mappings and two total asymptotically nonexpansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results announced in the current literature.
MSC:47H09, 47H10.
1 Introduction and preliminaries
Most of the problems in various disciplines of science are nonlinear in nature, whereas fixed point theory proposed in the setting of normed linear spaces or Banach spaces majorly depends on the linear structure of the underlying spaces. A nonlinear framework for fixed point theory is a metric space embedded with a ‘convex structure’. The class of hyperbolic spaces, nonlinear in nature, is a general abstract theoretic setting with rich geometrical structure for metric fixed point theory. The study of hyperbolic spaces has been largely motivated and dominated by questions about hyperbolic groups, one of the main objects of study in geometric group theory.
Throughout this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [1], defined below, which is more restrictive than the hyperbolic type introduced in [2] and more general than the concept of hyperbolic space in [3].
A hyperbolic space is a metric space (X,d) together with a mapping W:{X}^{2}\times [0,1]\to X satisfying

(i)
d(u,W(x,y,\alpha ))\le \alpha d(u,x)+(1\alpha )d(u,y);

(ii)
d(W(x,y,\alpha ),W(x,y,\beta ))=\alpha \beta d(x,y);

(iii)
W(x,y,\alpha )=W(y,x,(1\alpha ));

(iv)
d(W(x,z,\alpha ),W(y,w,\alpha ))\le (1\alpha )d(x,y)+\alpha d(z,w)
for all x,y,z,w\in X and \alpha ,\beta \in [0,1]. A nonempty subset K of a hyperbolic space X is convex if W(x,y,\alpha )\in K for all x,y\in K and \alpha \in [0,1]. The class of hyperbolic spaces contains normed spaces and convex subsets thereof, the Hilbert ball equipped with the hyperbolic metric [4], Hadamard manifolds as well as CAT(0) spaces in the sense of Gromov (see [5]).
A hyperbolic space is uniformly convex [6] if for any r>0 and \u03f5\in (0,2] there exists \delta \in (0,1] such that for all u,x,y\in X, we have
provided d(x,u)\le r, d(y,u)\le r and d(x,y)\ge \u03f5r.
A map \eta :(0,\mathrm{\infty})\times (0,2]\to (0,1], which provides such \delta =\eta (r,\u03f5) for given r>0 and \u03f5\in (0,2], is known as a modulus of uniform convexity of X. We call η monotone if it decreases with r (for fixed ϵ), i.e., \mathrm{\forall}\u03f5>0, \mathrm{\forall}{r}_{2}\ge {r}_{1}>0 (\eta ({r}_{2},\u03f5)\le \eta ({r}_{1},\u03f5)).
In the sequel, let (X,d) be a metric space, and let K be a nonempty subset of X. We shall denote the fixed point set of a mapping T by F(T)=\{x\in K:Tx=x\}.
A mapping T:K\to K is said to be nonexpansive if
A mapping T:K\to K is said to be asymptotically nonexpansive if there exists a sequence \{{k}_{n}\}\subset [0,\mathrm{\infty}) with {k}_{n}\to 0 such that
A mapping T:K\to K is said to be uniformly LLipschitzian if there exists a constant L>0 such that
Definition 1.1 A mapping T:K\to K is said to be (\{{\mu}_{n}\},\{{\xi}_{n}\},\rho )total asymptotically nonexpansive if there exist nonnegative sequences \{{\mu}_{n}\}, \{{\xi}_{n}\} with {\mu}_{n}\to 0, {\xi}_{n}\to 0 and a strictly increasing continuous function \rho :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with \rho (0)=0 such that
Remark 1.1 From the definitions, it is to know that each nonexpansive mapping is an asymptotically nonexpansive mapping with a sequence \{{k}_{n}=0\}, and each asymptotically nonexpansive mapping is a (\{{\mu}_{n}\},\{{\xi}_{n}\},\rho )total asymptotically nonexpansive mapping with {\xi}_{n}=0, and \rho (t)=t, t\ge 0.
The existence of fixed points of various nonlinear mappings has relevant applications in many branches of nonlinear analysis and topology. On the other hand, there are certain situations where it is difficult to derive conditions for the existence of fixed points for certain types of nonlinear mappings. It is worth to mention that fixed point theory for nonexpansive mappings, a limit case of a contraction mapping when the Lipschitz constant is allowed to be 1, requires tools far beyond metric fixed point theory. Iteration schemas are the only main tool for analysis of generalized nonexpansive mappings. Fixed point theory has a computational flavor as one can define effective iteration schemas for the computation of fixed points of various nonlinear mappings. The problem of finding a common fixed point of some nonlinear mappings acting on a nonempty convex domain often arises in applied mathematics.
The purpose of this paper is to introduce the concepts of total asymptotically nonexpansive mappings and to prove some Δconvergence theorems of the mixed type iteration process for approximating a common fixed point of two asymptotically nonexpansive mappings and two total asymptotically nonexpansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results given in [6–19].
In order to define the concept of Δconvergence 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 r(\{{x}_{n}\}) of \{{x}_{n}\} is given by
The asymptotic center {A}_{k}(\{{x}_{n}\}) of a bounded sequence \{{x}_{n}\} with respect to K\subset X is the set
This is the set of minimizers 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}\}). It is known that uniformly convex Banach spaces and 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 [20] and ensures that this property also holds in a complete uniformly convex hyperbolic space.
Lemma 1.1 [20]
Let (X,d,W) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity. Then every bounded sequence \{{x}_{n}\} in X has a unique asymptotic center with respect to any nonempty closed convex subset K of X.
Recall that a sequence \{{x}_{n}\} in X is said to Δconverge 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 \mathrm{\Delta}\text{}{lim}_{n\to \mathrm{\infty}}{x}_{n}=x and call x the \mathrm{\Delta}\text{}\text{limit} of \{{x}_{n}\}.
A mapping T:K\to K is semicompact if every bounded sequence \{{x}_{n}\}\subset K satisfying d({x}_{n},T{x}_{n})\to 0 has a convergent subsequence.
Lemma 1.2 [9]
Let \{{a}_{n}\}, \{{b}_{n}\} and \{{\delta}_{n}\} be sequences of nonnegative real numbers satisfying
If {\sum}_{n=1}^{\mathrm{\infty}}{\delta}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{b}_{n}<\mathrm{\infty}, then the limit {lim}_{n\to \mathrm{\infty}}{a}_{n} exists. If there exists a subsequence \{{a}_{{n}_{i}}\}\subset \{{a}_{n}\} such that {a}_{{n}_{i}}\to 0, then {lim}_{n\to \mathrm{\infty}}{a}_{n}=0.
Lemma 1.3 [12]
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 c\ge 0, then {lim}_{n\to \mathrm{\infty}}d({x}_{n},{y}_{n})=0.
Lemma 1.4 [12]
Let K be a nonempty closed convex subset of uniformly convex hyperbolic space and \{{x}_{n}\} be a bounded sequence in K such that A(\{{x}_{n}\})=\{y\} and r(\{{x}_{n}\})=\zeta. If \{{y}_{m}\} is another sequence in K such that {lim}_{m\to \mathrm{\infty}}r({y}_{m},\{{x}_{n}\})=\zeta, then {lim}_{m\to \mathrm{\infty}}{y}_{m}=y.
2 Main results
Theorem 2.1 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let {T}_{i}:K\to K, i=1,2, be a uniformly {L}_{i}Lipschitzian and (\{{\mu}_{n}^{i}\},\{{\xi}_{n}^{i}\},{\rho}^{i})total asymptotically nonexpansive mapping with sequence \{{\mu}_{n}^{i}\} and \{{\xi}_{n}^{i}\} satisfying {lim}_{n\to \mathrm{\infty}}{\mu}_{n}^{i}=0, {lim}_{n\to \mathrm{\infty}}{\xi}_{n}^{i}=0 and a strictly increasing function {\rho}^{i}:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with {\rho}^{i}(0)=0, i=1,2, let {S}_{i}:K\to K, i=1,2, be a uniformly \tilde{{L}_{i}}Lipschitzian and asymptotically nonexpansive mapping with sequence \{{k}_{n}^{i}\} satisfying {lim}_{n\to \mathrm{\infty}}{k}_{n}^{i}=0. Assume that \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i})\ne \mathrm{\varnothing}, and for arbitrarily chosen {x}_{1}\in K, \{{x}_{n}\} is defined as follows:
where \{{\mu}_{n}^{i}\}, \{{\xi}_{n}^{i}\}, {\rho}^{i}, {k}_{n}^{i}, i=1,2, \{{\alpha}_{n}\} and \{{\beta}_{n}\} satisfy the following conditions:

(1)
{\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}^{i}<\mathrm{\infty}, {\sum}_{n=1}^{\mathrm{\infty}}{\xi}_{n}^{i}<\mathrm{\infty}, {\sum}_{n=1}^{\mathrm{\infty}}{k}_{n}^{i}<\mathrm{\infty}, i=1,2;

(2)
There exist constants a,b\in (0,1) with 0<b(1a)\le \frac{1}{2} such that \{{\alpha}_{n}\}\subset [a,b] and \{{\beta}_{n}\}\subset [a,b];

(3)
There exists a constant {M}^{\ast}>0 such that {\rho}^{i}(r)\le {M}^{\ast}r, r>0, i=1,2;

(4)
d(x,{T}_{i}y)\le d({S}_{i}x,{T}_{i}y) for all x,y\in K and i=1,2.
Then the sequence \{{x}_{n}\} defined by (2.1) Δconverges to a common fixed point of \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i}).
Proof Set L=max\{{L}_{i},\tilde{{L}_{i}},i=1,2\}, {\mu}_{n}=max\{{\mu}_{n}^{i},{k}_{n}^{i},i=1,2\} and {\xi}_{n}=max\{{\xi}_{n}^{i},i=1,2\}, \rho =max\{{\rho}^{i},i=1,2\}. By condition (1), we know that {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}, {\sum}_{n=1}^{\mathrm{\infty}}{\xi}_{n}<\mathrm{\infty}. The proof of Theorem 2.1 is divided into four steps.
Step 1. First we prove that {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exists for each p\in \mathcal{F}.
For any given p\in \mathcal{F}, since {T}_{i}, i=1,2, is a total asymptotically nonexpansive mapping and {S}_{i}, i=1,2, is an asymptotically nonexpansive mapping, by condition (3) and (2.1), we have
where
Substituting (2.3) into (2.2) and simplifying it, we have
where {\delta}_{n}={\mu}_{n}(1+{\alpha}_{n}{M}^{\ast}(1+{\beta}_{n}+{\mu}_{n}+{\beta}_{n}{\mu}_{n}{M}^{\ast})), {b}_{n}=(1+{\beta}_{n}+{\beta}_{n}{\mu}_{n}{M}^{\ast}){\alpha}_{n}{\xi}_{n}. Since {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}, {\sum}_{n=1}^{\mathrm{\infty}}{\xi}_{n}<\mathrm{\infty} and condition (2), it follows from Lemma 1.2 that {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exists for p\in \mathcal{F}.
Step 2. We show that
For each p\in \mathcal{F}, from the proof of Step 1, we know that {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exists. We may assume that {lim}_{n\to \mathrm{\infty}}d({x}_{n},p)=c\ge 0. If c=0, then the conclusion is trivial. Next, we deal with the case c>0. From (2.3), we have
Taking lim sup on both sides in (2.6), we have
In addition, since
and
then we have
and
Since {lim}_{n\to \mathrm{\infty}}d({x}_{n+1},p)=c, it is easy to prove that
It follows from (2.8)(2.10) and Lemma 1.3 that
By the same method, we can also prove that
By virtue of condition (4), it follows from (2.11) and (2.12) that
and
From (2.1) and (2.12) we have
Observe that
It follows from (2.14) and (2.15) that
This together with (2.13) implies that
On the other hand, from (2.11) and (2.16), we have
Hence from (2.17) and (2.18), we have that
In addition, since
from (2.13) and (2.19), we have
Finally, for all i=1,2, we have
It follows from (2.14), (2.17) and (2.20) that
By virtue of condition (4), d({S}_{i}{x}_{n},{T}_{i}^{n}{x}_{n})\le d({S}_{i}^{n}{x}_{n},{T}_{i}^{n}{x}_{n}), we have
it follows from (2.12), (2.14), (2.17) and (2.18) that
Step 3. Now we prove that the sequence \{{x}_{n}\} Δconverges to a common fixed point of \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i}).
In fact, for each p\in F, {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exists. This implies that the sequence \{d({x}_{n},p)\} is bounded, so is the sequence \{{x}_{n}\}. Hence, by virtue of Lemma 1.1, \{{x}_{n}\} has a unique asymptotic center {A}_{K}(\{{x}_{n}\})=\{x\}.
Let \{{u}_{n}\} be any subsequence of \{{x}_{n}\} with {A}_{K}(\{{u}_{n}\})=\{u\}. It follows from (2.5) that
Now, we show that u\in F({T}_{i}). For this, we define a sequence \{{z}_{n}\} in K by {z}_{j}={T}_{i}^{j}u. So we calculate
Since {T}_{i} is uniformly LLipschitzian, from (2.22) we have
Taking lim sup on both sides of the above estimate and using (2.21), we have
And so
Since {A}_{K}(\{{u}_{n}\})=\{u\}, by the definition of asymptotic center {A}_{K}(\{{u}_{n}\}) of a bounded sequence \{{u}_{n}\} with respect to K\subset X, we have
This implies that
Therefore we have
It follows from Lemma 1.4 that {lim}_{j\to \mathrm{\infty}}{T}_{i}^{j}u=u. As {T}_{i} is uniformly continuous, so that {T}_{i}u={T}_{i}({lim}_{j\to \mathrm{\infty}}{T}_{i}^{j}u)={lim}_{j\to \mathrm{\infty}}{T}_{i}^{j+1}u=u. That is, u\in F({T}_{i}). Similarly, we also can show that u\in F({S}_{i}). Hence, u is the common fixed point of {T}_{i} and {S}_{i}. Reasoning as above by utilizing the uniqueness of asymptotic centers, we get that x=u. Since \{{u}_{n}\} is an arbitrary subsequence of \{{x}_{n}\}, therefore A(\{{u}_{n}\})=\{u\} for all subsequence \{{u}_{n}\} of \{{x}_{n}\}. This proves that \{{x}_{n}\} Δconverges to a common fixed point of \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i}). This completes the proof. □
The following theorem can be obtained from Theorem 2.1 immediately.
Theorem 2.2 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let {T}_{i}:K\to K, i=1,2, be a uniformly {L}_{i}Lipschitzian and asymptotically nonexpansive mapping with sequence \{{t}_{n}^{i}\}\subset [1,\mathrm{\infty}) satisfying {lim}_{n\to \mathrm{\infty}}{t}_{n}^{i}=1, and {S}_{i}:K\to K, i=1,2, be a nonexpansive mapping. Assume that \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i}), for arbitrarily chosen {x}_{1}\in K, \{{x}_{n}\} is defined as follows:
where \{{t}_{n}^{i}\}, i=1,2, \{{\alpha}_{n}\} and {\beta}_{n} satisfy the following conditions:

(1)
{\sum}_{n=1}^{\mathrm{\infty}}({t}_{n}^{i}1)<\mathrm{\infty}, i=1,2;

(2)
There exist constants a,b\in (0,1) with 0<b(1a)\le \frac{1}{2} such that \{{\alpha}_{n}\}\subset [a,b] and \{{\beta}_{n}\}\subset [a,b];

(3)
d(x,{T}_{i}y)\le d({S}_{i}x,{T}_{i}y) for all x,y\in K and i=1,2.
Then the sequence \{{x}_{n}\} defined in (2.23) Δconverges to a common fixed point of \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i}).
Proof Take {\rho}^{i}(t)=t, t\ge 0, {\xi}_{n}^{i}=0, {\mu}_{n}^{i}={t}_{n}^{i}1, {k}_{n}^{i}=0, i=1,2, in Theorem 2.1. Since all the conditions in Theorem 2.1 are satisfied, it follows from Theorem 2.1 that the sequence \{{x}_{n}\} Δconverges to a common fixed point of \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i}).
This completes the proof of Theorem 2.2. □
Theorem 2.3 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let {T}_{i}:K\to K, i=1,2, be a uniformly {L}_{i}Lipschitzian and asymptotically nonexpansive mapping with sequence \{{t}_{n}^{i}\}\subset [1,\mathrm{\infty}) satisfying {lim}_{n\to \mathrm{\infty}}{t}_{n}^{i}=1. Assume that \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i}), for arbitrarily chosen {x}_{1}\in K, \{{x}_{n}\} is defined as follows:
where \{{t}_{n}^{i}\}, i=1,2, \{{\alpha}_{n}\} and {\beta}_{n} satisfy the following conditions:

(1)
{\sum}_{n=1}^{\mathrm{\infty}}({t}_{n}^{i}1)<\mathrm{\infty}, i=1,2;

(2)
There exist constants a,b\in (0,1) with 0<b(1a)\le \frac{1}{2} such that \{{\alpha}_{n}\}\subset [a,b] and \{{\beta}_{n}\}\subset [a,b].
Then the sequence \{{x}_{n}\} defined in (2.24) Δconverges to a common fixed point of \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i}).
Proof Take {\rho}^{i}(t)=t, t\ge 0, {\xi}_{n}^{i}=0, {\mu}_{n}^{i}={t}_{n}^{i}1, {S}_{i}=I, i=1,2, in Theorem 2.1. Since all the conditions in Theorem 2.1 are satisfied, it follows from Theorem 2.1 that the sequence \{{x}_{n}\} Δconverges to a common fixed point of \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i}).
This completes the proof of Theorem 2.3. □
References
Kohlenbach U: Some logical metatheorems with applications in functional analysis. Trans. Amer. Math. Soc. 2004, 357(1):89–128.
Kuhfittig PKF: Common fixed points of nonexpansive mappings by iteration. Pacific J. Math. 1981, 97(1):137–139. 10.2140/pjm.1981.97.137
Reich S, Shafrir I: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 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 N, Haefliger A: Metric Spaces of NonPositive Curvature. Springer, Berlin; 1999.
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
Agarwal RP, O’Regan D, Sahu DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear convex Anal. 2007, 8(1):61–79.
Chang SS, Cho YJ, Zhou H: Demiclosed principal and weak convergence problms for asymptotically nonexpansive mappings. J. Korean. Math. Soc. 2001, 38(6):1245–1260.
Chang SS, Wang L, Lee HWJ, et al.:Total asymptotically nonexpansive mappings in a CAT(0) space demiclosed principle and Δconvergence theorems for total asymptotically nonexpansive mappings in a CAT(0) space. Appl. Math. Comput 2012, 219: 2611–2617. 10.1016/j.amc.2012.08.095
Fukharuddin H, Khan AR: Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces. Comput. Math. Appl. 2007, 53: 1349–1360. 10.1016/j.camwa.2007.01.008
Gu F, Fu Q: Strong convergence theorems for common fixed points of multistep iterations with errors in Banach spaces. J. Inequal. Appl. 2009., 2009: Article ID 819036 10.1155/2009/819036
Khan AR, Fukharuddin H, Kuan 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 AR, Khamsi MA, Fukharuddin H:Strong convergence of a general iteration scheme in CAT(0) spaces. Nonlinear Anal. 2011, 74: 783–791. 10.1016/j.na.2010.09.029
Osilike MO, Aniagbosor SC: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. Math. Compt. Model. 2000, 32: 1181–1191. 10.1016/S08957177(00)001990
Sahin A, Basarir M: On the strong convergence of a modified S iteration process for asymptotically quasinonexpansive mappings in a CAT(0) space. Fixed Point Theory Appl. 2013., 2013: Article ID 12 10.1186/16871812201312
Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884
Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 1991, 158: 407–413. 10.1016/0022247X(91)90245U
Tan KK, Xu HK: Fixed point iteration process for asymptotically nonexpansive mappings. Proc. Amer. Math. Soc. 1994, 122(3):733–739. 10.1090/S00029939199412039935
Yao Y, Liou YC: New iterative schemes for asymptotically quasinonexpansive mappings. J. Inequal. Appl. 2010., 2010: Article ID 934692 10.1155/2010/934692
Leustean L: Nonexpansive iteration in uniformly convex W hyperbolic spaces. Contemporary Mathematics 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.
Acknowledgements
The authors would like to express their thanks to the editors and referees for their useful comments and suggestions. This work is supported by Scientific Research Fund of SiChuan Provincial Education Department (No. 11ZA222) and the Natural Science Foundation of Yibin University (No. 2012S07) and the National Natural Science foundation of China (Grant No. 11361070).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly to this research work. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Zhao, Lc., Chang, Ss. & Kim, J.K. Mixed type iteration for total asymptotically nonexpansive mappings in hyperbolic spaces. Fixed Point Theory Appl 2013, 353 (2013). https://doi.org/10.1186/168718122013353
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718122013353
Keywords
 asymptotically nonexpansive mapping
 total asymptotically nonexpansive mapping
 common fixed point
 hyperbolic space