New Common Fixed Points for Total Asymptotically Nonexpansive Mapping in CAT(0) Space

Strong and ∆-convergence for a two-step iteration process utilizing asymptotically nonexpansive and total asymptotically nonexpansive noneslf mappings in the CAT(0) spaces have been studied. As well, several strong convergence theorems under semi-compact and condition (M) have been proved. Our results improve and extend numerous familiar results from the existing literature.


Introduction:
A metric space G is a CAT(0) space, if it is geodesically connected and if each geodesic triangle in G is at least as thin as its comparison triangle in the Euclidean plane. Some typical examples of CAT(0) spaces are R-trees, Pre-Hilbert space and Euchlidean buldings (1).
Fixed point theory in CAT(0) spaces was foremost initialized through Kirk (1). He proved that each nonexpansive (single valued) mapping defined on a bounded closed convex subset of a complete CAT(0) spaces permanently has fixed point. Therefore, the fixed point theory for single valued as well multivalued mappings in CAT(0) spaces has intensively been evolved by numerous authors. The convergence for nonexpansive mappings in CAT(0) spaces was studied by . Thereafter, Khan and Abbas (3) studied the strong and ∆-convergance in CAT(0) space for an iteration process that is indepent of the Ishikawa iteration process. As well, several of these results obtained for two nonexpansive mappins. It is important to remember that fixed point theorems in CAT(0) space can be stratified to graph theory, computer science and biology (1). Let ( ) be a metric space and ( ) . A geodesic path from , this means an isometry , -(, -) such as ( ) ( ) The image of every geodesic path between is called geodesic segment. Each point y in the segment is appeared by ( ) , -, -* ( ) , -+ The space ( ) is called a geodesic if each two points of are jouned through a geodesic segment, and is uniquely geodesic if there exists properly one geodesic jouning for every . A subset of is called convex if has each geodesic segment joining any two points in (4)(5)(6).
A geodesic triangle ( ) is a geodesic metric space ( ) that consists of three points (the vertices ) and a geodesic segment between every pair of vertices (the edges of ) A comparison triangle ̅ ( ̅̅̅ ̅̅̅ ̅̅̅) in for ( ) a triangle in 2-dimensional Euclidean plane with ̅̅̅ ̅̅̅ ̅̅̅ such as where | | is the Eulidean norm on ( ) CAT(0): A geodesic space is called CAT(0) space if whole geodesic triangles achieve the following comparison axiom.
Let be a geodesic triangle in G and ̅ be a comparison triangle for Therefore, is called to If are the points in CAT(0) and if ( ) therefore the CAT(0) inequality leads Which is the (CN) inequality of Bruhat and Tits. In verity, a geodesic space is a CAT(0) space if it accomplishes (CN) (3).
The asymptotic radius ( * + ) * + is given through ( * + ) * ( * +) +, and the asymptotic center (* +) * + is defined as (* +) * ( * +) ( * + )+ It is familiar that in CAT(0) space, (* +) has punctually one point. Numerous iteration processes have been structured and suggested in order to approximate fixed points. The Picard iteration for a mapping is defined by (1) The modified Mann iteration is considered by Schu (5), as below ( ) (2) Where * + ( ) The modified Ishikawa iteration is studied by Tan and Xu (5), as below Recently, Sahin and Basarir (5) modified the above iteration in a CAT(0) space, as follows The following iteration has been studied by M. R. Yadava (9) for common fixed points of two self mappins , , -This iteration as well decreases to Mann iteration when or Inspired and motivated by the work of M. R. Yadava (6), the iteration (6) for common fixed points of two mapping asymptotically naonexpansive and total asymptotically nonexpansive nonself mappings in a CAT(0) space is modified, as follows.
Deem E to be a nonempty closed convex subset of a complete CAT(0) space G, to be an asymptotically nonexpansive and to be a total asymptotically nonexpansive mappings. Presume that * + is a sequence produced by and is a nonexpansive retraction of onto .
In this paper, a new iteration for approximating a common fixed point of asymptotically nonexpansive and total asymptotically nonexpensive nonself mappings is constructed. Some strong convergence theorems and ∆convergence theorem under appropriate conditions like semi-compact and condition (M) in CAT(0) spaces are proved. As well, numerical example to elucidate our work is provided.

Preliminaries
A mapping is called total asymptotically nonexpansive (11) if * + * + and a strictly nondecreasing A mapping is called total asymptotically nonexpansive nonself (11) if * + * + and a strictly nondecreasing ) The notion of asymptotcally nonexpansive mapping was foremost introduced by Gloebel and Kirk. Therefore Alber et al. introduced the class of total asymptotically nonexpensive, which generalizes some classes of mappings that are spans of asymptotically nonexpensive. Several authours have been extensively studied these classes of mappings (6). Definition (2)(13): A sequence * + a CAT(0) space is called ∆-convergence to is the unique asymptotic center of * + subsequence * + * + Here, note down is the ∆-limit of * + Note that given * + * + to and through the uniqueness of the asymptotic center that gives ( ) ( ) Therefore, each CAT(0) space achieves the Opial property.

Lemma (3)(8):
Let be a CAT(0) space and Presume * + is a sequence in ,for several ( ) * + * + are sequences in such as . Lemma (4)(6): Let * + * + * + be the sequences of positive numbers such as If there is a subsequence * + * + such as Lemma (5)(14): Each bounded sequence in a complete CAT(0) space holds a ∆-convergence subsequence. Lemma (6)(15): If is closed convex subset of a complete CAT(0) space and if * + is bounded sequence in , thus the asympitotic center of * + is in Theorem (7)(11): Let be a closed convex subset of a complete CAT(0) space . Let T be a mapping accomplishing one of the following conditions:  is an asymptotically nonexpansive mapping with a sequence * + , )  is an asymptotically nonexpansive nonself mapping  is a total asymptotically nonexpansive mapping Let * + be a bounded sequence in such as ( )

The convergence results
In this part, ∆-convergance and some strong convergnce theorems by using iteration (7) for asymptotically nonexpansive and total asymptotically nonexpansive nonself mappings in CAT(0) spaces are proved. Theorem (8): Let be a nonempty closed convex subset of a complete CAT(0) space . Let be a uniformly L-lipschitzain and asymptotically nonexpansive and be a uniformly Llipschitzain total asymptotically nonexpansive nonself mappings with ( ) . Presume that{ + is defined by (7). If Thus, the sequence { + is ∆-convergence to a several points ( ( ) ( )) Proof: Step 1: Step 2: Next, proving that ( ) ( ) In verity, it follows up from step (1) that for each given ( ) exists. Presume that

That gives ( ( ) )
In turn, (8) and (9), that deduces Notice that punctually one point. Hence,{ + is ∆-convergence to a common fixed point of Theorem (9): Under the presumption of Theorem (8). Therefore the sequence { + is defined by : is ∆-convergence to a common fixed point of .

Numerical example
Our results through the following example is elucidated (Table 1) Example (14): Deem with its usual metric, so is as well complete CAT(0) space. Let , -, which is a closed bounded convex subset of . Define two mappings ( ) ( ) So is asymptotically nonexpansive mapping with * + and is a total asymptotically nonexpansive nonself mapping with , . Obviously, ( ) * + ( ) of the mappings Put ( ). By using Matlab,the iteration which is defined by (1) for initial points Lastly, the convergence demeanors of the iteration (7) is appeared in Fig. 1. The consequence is that converges to zero.