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 Δ 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.


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].
(2)  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.
In order to define the concept Δ-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 (⋅, { }) : → [0, ∞) 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 [19] and ensures that this property also holds in a complete uniformly convex hyperbolic space.
Proof. The proof of Theorem 7 is divided into four steps.
For each ∈ ( 1 ) ∩ ( 2 ), from the proof of Step 1, we know that lim → ∞ ( , ) exists. We may assume that lim → ∞ ( , ) = ≥ 0. The case = 0 is trivial. Next, we deal with the case > 0. From (13), we have Taking limsup on both sides in (15), we have In addition, since we have Since lim → ∞ ( +1 , ) = , it is easy to prove that It follows from Lemma 5 that On the other hand, since we have So, it follows from (25) and Lemma 5 that Observe that In addition, since it follows from (30) and (32) that Similarly, we also can show that Step 3. Now we prove that the sequence { } Δ-converges to a common fixed point of ( 1 ) ∩ ( 2 ). In fact, since, for each ∈ , lim → ∞ ( , ) exists, this implies that the sequence { ( , )} is bounded, so is the sequence { }. Hence, by virtue of Lemma 3, { } has a unique asymptotic center Now, we show that ∈ ( 1 ). For this, we define a sequence { } in by = 1 . So, we calculate  = . That is, ∈ ( 1 ). Similarly, we also can show that ∈ ( 2 ). Hence, is the common fixed point of 1 and 2 . Reasoning as previously mentioned by utilizing the uniqueness of asymptotic centers, we get that = . Since Proof. Take ( ) = , ≥ 0, = 0, = − 1 in Theorem 7. Since all conditions in Theorem 7 are satisfied, it follows from Theorem 7 that the sequence { } Δ-converges to a common fixed point of := ⋂ 2 =1 ( ). This completes the proof of Theorem 8.