Numerical and Analytical Methods for Variational Inequalities and Related Problems 2013View this Special Issue
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Convergence Theorems for Total Asymptotically Nonexpansive Mappings in Hyperbolic Spaces
The purpose of this paper is to introduce the concept of total asymptotically nonexpansive mappings and to prove some -convergence theorems of the iteration process for this kind of mappings in the setting of hyperbolic spaces. The results presented in the paper extend and improve some recent results announced in the current literature.
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 , defined below, which is more restrictive than the hyperbolic type introduced in  and more general than the concept of hyperbolic space in .
A hyperbolic space is a metric space together with a mapping satisfying (i); (ii); (iii); (iv),for all and . A nonempty subset of a hyperbolic space is convex if for all and . The class of hyperbolic spaces contains normed spaces and convex subsets thereof, the Hilbert ball equipped with the hyperbolic metric , Hadamard manifolds, and CAT(0) spaces in the sense of Gromov (see ).
A hyperbolic space is uniformly convex  if for any and there exists a , such that, for all , we have provided , and .
A map , which provides such a for given and , is known as a modulus of uniform convexity of . We call monotone if it decreases with (for a fixed ), that is, for all , for all .
In the sequel, let be a metric space and let be a nonempty subset of . We will denote the fixed point set of a mapping by .
A mapping is said to be nonexpansive, if
A mapping is said to be asymptotically nonexpansive if there exists a sequence with such that
A mapping is said to be uniformly L-Lipschitzian if there exists a constant such that
Definition 1. A mapping is said to be -total asymptotically nonexpansive, if there exist nonnegative sequences with , and a strictly increasing continuous function with such that
Remark 2. From the definitions, it is clear that each nonexpansive mapping is an asymptotically nonexpansive mapping with a sequence and each asymptotically nonexpansive mapping is a -total asymptotically nonexpansive mapping with , and .
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 mentioning 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 from 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 concept of total asymptotically nonexpansive mappings and to prove some -convergence theorems of the iteration process for the approximation of total asymptotically nonexpansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results given in [6–18].
In order to define the concept -convergence in the general setup of hyperbolic spaces, we first collect some basic concepts.
Let be a bounded sequence in a hyperbolic space . For , we define a continuous functional by
The asymptotic radius of is given by
The asymptotic center of a bounded sequence with respect to is the set
This is the set of minimizers of the functional . If the asymptotic center is taken with respect to , then it is simply denoted by . 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 Leuştean  and ensures that this property also holds in a complete uniformly convex hyperbolic space.
Lemma 3 (see ). Let be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity. Then, every bounded sequence in has a unique asymptotic center with respect to any nonempty closed convex subset of .
Recall that a sequence in is said to -converge to if is the unique asymptotic center of for every subsequence of . In this case, we write - and call the - of .
A mapping is semicompact if every bounded sequence , satisfying , has a convergent subsequence.
Lemma 4 (see ). Let , , and be sequences of nonnegative real numbers satisfying If and , then the limit exists. If there exists a subsequence such that , then .
Lemma 5 (see ). Let be a uniformly convex hyperbolic space with monotone modulus of uniform convexity . Let and be a sequence in for some . If and are sequences in such that for some , then .
Lemma 6 (see ). Let be a nonempty closed convex subset of uniformly convex hyperbolic space and a bounded sequence in such that and . If is another sequence in such that , then .
2. Main Results
Theorem 7. Let be a nonempty closed convex subset of a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity . Let , be uniformly -Lipschitzian and -total asymptotically nonexpansive mappings with sequences and satisfying , , and strictly increasing function with , . Assume that , and for arbitrarily chosen , is defined as follows: where , , , , , and satisfy the following conditions: (1); (2)there exist constants with such that and ; (3)there exist a constant such that . Then, the sequence defined by (11) -converges to a common fixed point of .
Proof. The proof of Theorem 7 is divided into four steps.
Step 1. First, we prove that exists for each .
Set , , and . Since , , , . For any , by (11) we have where Substituting (13) into (12), we have Applying Lemma 4 to the inequality, we get that exist for .
Step 2. We show that .
For each , from the proof of Step 1, we know that exists. We may assume that . The case is trivial. Next, we deal with the case . From (13), we have Taking limsup on both sides in (15), we have In addition, since we have Since , it is easy to prove that It follows from Lemma 5 that On the other hand, since we have . Combined with (16), it yields that This implies that Since we have So, it follows from (25) and Lemma 5 that Observe that where It follows from (26) that Thus, from (20), (27), and (29), we have In addition, since from (20), we have Finally, since it follows from (30) and (32) that Similarly, we also can show that
Step 3. Now we prove that the sequence -converges to a common fixed point of .
In fact, since, for each , exists, this implies that the sequence is bounded, so is the sequence . Hence, by virtue of Lemma 3, has a unique asymptotic center .
Let be any subsequence of with . It follows from (34) that Now, we show that . For this, we define a sequence in by . So, we calculate Since is uniformly -Lipschitzian, from (37) we have Taking limsup on both sides of the previous estimate and using (36), we have Since , by the definition of asymptotic center of a bounded sequence with respect to and (8), this implies that , for all . Therefore, as . It follows from Lemma 6 that . As is uniformly continuous, . That is, . Similarly, we also can show that . Hence, is the common fixed point of and . Reasoning as previously mentioned by utilizing the uniqueness of asymptotic centers, we get that . Since is an arbitrary subsequence of , for all subsequence of . This proves that -converges to a common fixed point of and . This completes the proof.
The following theorem can be obtained from Theorem 7 immediately.
Theorem 8. Let be a nonempty closed convex subset of a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity . Let , , be asymptotically nonexpansive mappings with sequence satisfying . Assume that ; for arbitrarily chosen , is defined as follows: where , , and satisfy the following conditions: (1) ; (2)there exist constants with such that and . Then, the sequence defined in (40) -converges to a common fixed point of .
The authors would like to express their thanks to the editors and the referees for their helpful 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).
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