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The purpose of this paper is to introduce the concept of

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 [

A hyperbolic space is a metric space

A hyperbolic space is uniformly convex [

A map

In the sequel, let

A mapping

A mapping

A mapping

A mapping

From the definitions, it is clear that each nonexpansive mapping is an asymptotically nonexpansive mapping with a sequence

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

In order to define the concept

Let

The

The

This is the set of minimizers of the functional

Let

Recall that a sequence

A mapping

Let

Let

Let

Let

there exist constants

there exist a constant

The proof of Theorem

Set

For each

In fact, since, for each

Let

The following theorem can be obtained from Theorem

Let

Take

This completes the proof of Theorem

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).