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△convergence for mixedtype total asymptotically nonexpansive mappings in hyperbolic spaces
Journal of Inequalities and Applications volume 2013, Article number: 553 (2013)
Abstract
In this paper, we prove some △convergence theorems in a hyperbolic space. A mixed AgarwalO’ReganSahu type iterative scheme for approximating a common fixed point of total asymptotically nonexpansive mappings is constructed. Our results extend some results in the literature.
MSC:47H09, 49M05.
1 Introduction and preliminaries
In this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [1]. Concretely, (X,d,W) is called a hyperbolic space if (X,d) is a metric space and W:X\times X\times [0,1]\to X is a function satisfying

(I)
\mathrm{\forall}x,y,z\in X, \mathrm{\forall}\lambda \in [0,1], d(z,W(x,y,\lambda ))\le (1\lambda )d(z,x)+\lambda d(z,y);

(II)
\mathrm{\forall}x,y\in X, \mathrm{\forall}{\lambda}_{1},{\lambda}_{2}\in [0,1], d(W(x,y,{\lambda}_{1}),W(x,y,{\lambda}_{2}))={\lambda}_{1}{\lambda}_{2}\cdot d(x,y);

(III)
\mathrm{\forall}x,y\in X, \mathrm{\forall}\lambda \in [0,1], W(x,y,\lambda )=W(y,x,(1\lambda ));

(IV)
\mathrm{\forall}x,y,z,w\in X, \mathrm{\forall}\lambda \in [0,1], d(W(x,z,\lambda ),W(y,w,\lambda ))\le (1\lambda )d(x,y)+\lambda d(z,w).
If a space satisfies only (I), 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 [3] since (I)(III) together are equivalent to (X,d,W) being a space of hyperbolic type in [3]. But it is slightly more general than the hyperbolic space defined in Reich [4] (see [1]). This class of metric spaces in [1] covers all normed linear spaces, ℝtrees in the sense of Tits, the Hilbert ball with the hyperbolic metric (see [5]), Cartesian products of Hilbert balls, Hadamard manifolds (see [4, 6]) and CAT(0) spaces in the sense of Gromov (see [7]). A thorough discussion of hyperbolic spaces and a detailed treatment of examples can be found in [1] (see also [3–5]).
A hyperbolic space is uniformly convex [8] if for u,x,y\in X, r>0 and \u03f5\in (0,2], there exists \delta \in (0,1] such that
provided that 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] is called modulus of uniform convexity if \delta =\eta (r,\u03f5) for given r>0. Besides, η is monotone if it decreases with r (for a fixed ϵ), that is,
A subset C of a hyperbolic space X is convex if W(x,y,\lambda )\in C for all x,y\in C and \lambda \in [0,1].
Let (X,d) be a metric space, and let C be a nonempty subset of X. Recall that T:C\to C is said to be a (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total asymptotically nonexpansive mapping if there exist nonnegative sequences \{{\nu}_{n}\}, \{{\mu}_{n}\} with {\nu}_{n}\to 0, {\mu}_{n}\to 0 and a strictly increasing continuous function \zeta :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with \zeta (0)=0 such that
It is well known that each nonexpansive mapping is an asymptotically nonexpansive mapping and each asymptotically nonexpansive mapping is a (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total asymptotically nonexpansive mapping.
T:C\to C is said to be uniformly LLipschitzian if there exists a constant L>0 such that
The following iteration process is a translation of the mixed AgarwalO’ReganSahu type iterative scheme introduced in [9] from Banach spaces to hyperbolic spaces. The iteration rate of convergence is similar to the Picard iteration process and faster than other fixed point iteration processes. Besides, it is independent of Mann and Ishikawa iteration processes.
where C is a nonempty closed and convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. {T}_{i}:C\to C, i=1,2, is a uniformly {L}_{i}Lipschitzian and (\{{\nu}_{n}^{(i)}\},\{{\mu}_{n}^{(i)}\},{\zeta}^{(i)})total asymptotically nonexpansive mapping, and {S}_{i}:C\to C, i=1,2, is a uniformly {\tilde{L}}_{i}Lipschitzian and (\{{\tilde{\nu}}_{n}^{(i)}\},\{{\tilde{\mu}}_{n}^{(i)}\},{\tilde{\zeta}}^{(i)})total asymptotically nonexpansive mapping such that the following conditions are satisfied:

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

(2)
There exists a constant M>0 such that {\zeta}^{(i)}(r)\le Mr, {\tilde{\zeta}}^{(i)}(r)\le Mr, \mathrm{\forall}r\ge 0, i=1,2.
Remark 1 Without loss of generality, we can assume that {T}_{i}:C\to C and {S}_{i}:C\to C, i=1,2, both are uniformly LLipschitzian and (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total asymptotically nonexpansive mappings satisfying conditions (1) and (2). In fact, letting {\nu}_{n}=max\{{\nu}_{n}^{(i)},{\tilde{\nu}}_{n}^{(i)},i=1,2\}, {\mu}_{n}=max\{{\mu}_{n}^{(i)},{\tilde{\mu}}_{n}^{(i)},i=1,2\}, L=max\{{L}_{i},{\tilde{L}}_{i},i=1,2\} and \zeta =max\{{\zeta}^{(i)},{\tilde{\zeta}}^{(i)},i=1,2\}, then {S}_{i} and {T}_{i}, i=1,2, are the required mappings.
Chang [10] proved some strong convergence theorems and △convergence theorems for approximating a common fixed point of total asymptotically nonexpansive mappings in a CAT(0) space using the mixed AgarwalO’ReganSahu type iterative scheme. More precisely, one of the results is as follows.
Theorem 1 [10]
Let C be a bounded closed and convex subset of a complete CAT(0) space X. Let {T}_{i}:C\to C and {S}_{i}:C\to C, i=1,2, be uniformly LLipschitzian and (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total asymptotically nonexpansive mappings. If \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i})\ne \mathrm{\varnothing} and the following conditions are satisfied:

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

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

(iii)
there exists a constant M>0 such that \zeta (r)\le Mr, r\ge 0;

(iv)
d(x,{T}_{i}y)\le d({S}_{i}x,{T}_{i}y) for all x,y\in C and i=1,2,
then the sequence \{{x}_{n}\} defined by (2) △converges to a common fixed point of {T}_{i} and {S}_{i}, i=1,2.
Theorem 1 can be viewed as a improvement and extension of several wellknown results in Banach spaces and CAT(0) spaces, such as [9] and [11]. Our purpose of this paper is to extend Theorem 1 from the CAT(0) spaces setting to the general setup of uniformly convex hyperbolic spaces.
Let \{{x}_{n}\} be a bounded sequence in a hyperbolic space X. For x\in X, we define
The asymptotic radius r(\{{x}_{n}\}) of \{{x}_{n}\} is given by
The asymptotic radius {r}_{C}(\{{x}_{n}\}) of \{{x}_{n}\} with respect to C\subset X is given by
The asymptotic center A(\{{x}_{n}\}) of \{{x}_{n}\} is the set
The asymptotic center {A}_{C}(\{{x}_{n}\}) of \{{x}_{n}\} with respect to C\subset X is the set
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 call x the △limit of \{{x}_{n}\}. The following lemmas are important in our paper.
Let (X,d,W) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity, and let C be a nonempty closed convex subset of X. Then every bounded sequence \{{x}_{n}\} in X has a unique asymptotic center with respect to C.
Lemma 2 [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 {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}d({x}_{n},x)\le c, {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}d({y}_{n},x)\le c and {lim}_{n\to \mathrm{\infty}}d(W({x}_{n},{y}_{n},{\alpha}_{n}),x)=c for some c\ge 0, then
Lemma 3 [12]
Let C be a nonempty closed convex subset of a uniformly convex hyperbolic space, and let \{{x}_{n}\} be a bounded sequence in C such that A(\{{x}_{n}\})=\{y\} and r(\{{x}_{n}\})=\rho. If \{{y}_{m}\} is another sequence in C such that {lim}_{m\to \mathrm{\infty}}r({y}_{m},\{{x}_{n}\})=\rho, then {lim}_{m\to \mathrm{\infty}}{y}_{m}=y.
Lemma 4 [10]
Let \{{a}_{n}\}, \{{b}_{n}\} and \{{c}_{n}\} be sequences of nonnegative numbers such that
If {\sum}_{n=1}^{\mathrm{\infty}}{b}_{n}<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{c}_{n}<\mathrm{\infty}, then {lim}_{n\to \mathrm{\infty}}{a}_{n} exists.
2 Main results
In this section, we prove our main theorems.
Theorem 2 Let C be a nonempty closed and convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let {T}_{i}:C\to C and {S}_{i}:C\to C, i=1,2, be uniformly LLipschitzian and (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total asymptotically nonexpansive mappings. If \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i})\ne \mathrm{\varnothing} and the following conditions are satisfied:

(i)
{\sum}_{n=1}^{\mathrm{\infty}}{\nu}_{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}\}\subset [a,b];

(iii)
there exists a constant M>0 such that \zeta (r)\le Mr, r\ge 0;

(iv)
d(x,{T}_{i}y)\le d({S}_{i}x,{T}_{i}y) for all x,y\in C and i=1,2,
then the sequence \{{x}_{n}\} defined by (2) △converges to a common fixed point of {T}_{i} and {S}_{i}, i=1,2.
Proof We divide our proof into three steps.
Step 1. In the sequel, we shall show that for each p\in \mathcal{F},
In fact, by conditions (1), (2), (I) and (iii), one gets
and
Combining (4) and (5), one has
and
where {\sigma}_{n}={\nu}_{n}M(1+{\alpha}_{n}(1+{\nu}_{n}M)), {\xi}_{n}=(1+{\alpha}_{n}(1+{\nu}_{n}M)){\mu}_{n}. Furthermore, using condition (i), one has
Consequently, a combination of (6), (7), (8) and Lemma 4 shows that (3) is proved.
Step 2. We claim that
In fact, it follows from (3) that {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exists for each given p\in \mathcal{F}. Without loss of generality, we assume that
By (4) and (10), one has
Noting
and
by (10) and (11), one has
Besides, by (6) one gets
which yields that
Now, by (12), (13) and Lemma 2, we have
Using the same method, we can also have that
It follows from (14), (15) and condition (iv) that
and
By virtue of (15), one has
Because we have
it follows from (15), (17) and (18) that
Combining (16) and (19), one obtains
Moreover, it follows from (16) and (19) that
This jointly with (16) and (20) yields that
and
Now by (17), (20) and (22), for each i=1,2, one gets
For each i=1,2, the combination of (15), (17), (20), (21) and (iv) yields that
Therefore, (9) is proved.
Step 3. Now we are in a position to prove the △convergence of \{{x}_{n}\}. Since \{{x}_{n}\} is bounded, by Lemma 1, it has a unique asymptotic center {A}_{C}(\{{x}_{n}\})=\{{x}^{\ast}\}. Let \{{u}_{n}\} be any subsequence of \{{x}_{n}\} with {A}_{C}(\{{u}_{n}\})=\{u\}, then by (23) and (24), for each i=1,2, we have
We claim that u\in \mathcal{F}. In fact, we define a sequence \{{z}_{m}\} in C by {z}_{m}={T}_{1}^{m}u. Then one has
By (25), one gets
which yields that
Lemma 3 shows that {lim}_{m\to \mathrm{\infty}}{T}_{1}^{m}u=u. Because {T}_{1} is uniformly continuous, we have
Hence, u\in F({T}_{1}). Using the same method, we can prove that u\in \mathcal{F}. By the uniqueness of asymptotic centers, we get that {x}^{\ast}=u. It implies that {x}^{\ast} is the unique asymptotic center of \{{u}_{n}\} for each subsequence \{{u}_{n}\} of \{{x}_{n}\}, that is, \{{x}_{n}\} △converges to {x}^{\ast}\in \mathcal{F}. The proof is completed. □
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Acknowledgements
Supported by General Project of Educational Department in Sichuan (No. 13ZB0182) and Doctor Research Foundation of Southwest University of Science and Technology (No. 11zx7130).
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Wan, LL. △convergence for mixedtype total asymptotically nonexpansive mappings in hyperbolic spaces. J Inequal Appl 2013, 553 (2013). https://doi.org/10.1186/1029242X2013553
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DOI: https://doi.org/10.1186/1029242X2013553