The study of fixed point theory in CAT(0) spaces of various classes of mappings using different iterative schemes have been the focus of vigorous research paper for many authors. In the last three years many research paper have been published on the iterative approximation of fixed points in CAT(0) space, using several iteration process such as Mann and Ishikawa iteration process. The fixed point theory for single-value and multivalued mappings in CAT(0) spaces has been rapidly developed and much research paper have published (see [3], [4], [5], [6], [7], [8], [11], [12], [13], [14]). A CAT (0) space is geodesic space for which each geodesic triangle is at least as ’thin’ as its comparison triangle in the Euclidean plane. A notion of convergence introduced independently several years ago by Lim and Kuezumow is shown in CAT(0) space to be very similar to the usual weak convergence in Banach space.

In 1976 Lim [15] introduced a concept of convergence in a general metric space setting which he called -convergence. Kirk and Panyanak [13] specialized this concept to CAT(0) space and showed that many Banach space results involving weak convergence have precise analogs in the setting. Dompongs and Panyanak [6] continued to work in this direction. Their results involved Mann and Ishikawa iteration scheme involving one mapping.

In 2010, Puttasontiphot, T. [17] has prove strong convergence theorem to approximate fixed points of quasi-nonexpansive multivalued mappings by the Mann and Ishikawa iteration process in the frame work of CAT(0) spaces. Fixed point theory in CAT(0) spaces was first studied by Kirk (see [9], [10]). He showed that every nonexpansive mapping defined on a bounded, closed, convex subset of a complete CAT(0) space always has a fixed point.

In this paper, inspired by the above fact, we prove and strong convergence theorems of the two step iteration scheme defined by Thianwan [19] for nonexpansive mappings in the CAT(0) spaces.

(1.1)

Where , are appropriate sequences in [0, 1]. If for all , then (1.1) reduces to Mann iteration scheme:

(1.2)

where is in (0, 1).

2. PRELIMINARIES:

Let (X, d) be a metric space. A geodesic path joining to (or, more briefly, a geodesic from x to y) is a map c from the closed interval to X such that c(0) = x, , and for all . In particular, c is an isometry and . The image of c is called a geodesic (or metric) segment joining x and y. When it is unique this geodesic segment is denoted by [x, y]. A geodesic triangle in a geodesic metric space (X, d) consists of three points in X (the vertices of ) and geodesic segment between each pair of vertices (the edges of ). A comparison triangle for the geodesic triangle in (X, d) is a triangle in the Euclidean plane such that for , .

Now, we give the following definitions which will be used in the our main result:

Definition 2.1. Let be a geodesic triangle in X and let be a comparison triangle for . then is said to satisfy the CAT(0) inequality if for all and all comparison points , then

If are points in a CAT(0) space and if is the midpoint of the segment then the CAT(0) inequality implies

. (2.1)

This if the (2.1) inequality of Bruhat and Tits [2]. In fact (see [1]), a geodesic space is a CAT(0) space if and only if satisfies the (2.1) inequality.

Let be a bounded sequence in CAT(0) space X. For , we set

The asymptotic radius of is defined by

and the asymptotic center of of is the set

Definition 2.2. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each . A subset is said to be convex if Y includes every geodesic segment joining any two of its points.

Definition 2.3 ([13], [15]). A sequence in a metric space X is said to - converge to if x is the unique asymptotic center of {un} for every subsequence of . In the case we write − and call x the −limit of .

Definition 2.4. Suppose K be a nonempty subset of a CAT(0) space X and a mapping is said to be nonexpansive if for all . We know that a point is a fixed point of T if Tx = x, i.e., .

Definition 2.5. A mapping is said to satisfy condition (I) [18] if there exists a nondecreasing function with f(0) = 0 at f(r) > 0 such that for all , where .

Now, some elementary lemmas about CAT(0) spaces which will be used in the proofs of our main results:

Lemma 2.1 ([13]). Every bounded sequence in a complete CAT(0) space always has a -convergence subsequence.

Lemma 2.2 ([5]). If K is a closed convex subset of a complete CAT(0) space and if is a bounded sequence in K, then the asymptotic center of is in K.

Lemma 2.3 ([6]). Let (X, d) be a CAT(0) space.

(i) For and , there exists a unique point such that

and . (LL)

We use the notation for the unique point z satisfying (LL).

(ii) For and , we have

(iii) For x, y 2 X and t 2 [0, 1], we have

3. MAIN RESULTS:

In this section, we have proved and strong convergence theorem and find approximate

common fixed points of self mappings T. In the consequence, F denotes the set of common fixed point of the mappings F(T).

Lemma 3.1. Let K be a nonempty bounded, closed, convex subset of a complete CAT(0) space X and suppose be a nonexpansive mapping with , where F(T) is the set of fixed points of F(T). Let and be the sequences defined by (1.1), where and be real sequences in [0, 1]. For a given , then exists for all .

Proof. Suppose and and using the iteration scheme (1.1) and Lemma 2.3, we have

(3.1)

and again using iteration scheme (1.1), we get

(3.2)

Substituting (3.2) into (3.1), we obtain

. (3.3)

This prove that is a non-increasing and bounded sequence and hence

exists. This completes the proof.

Theorem 3.1. Let K be a nonempty closed convex subset of a complete CAT(0) space X and let be a nonexpansive mapping with and and be the sequences in for some with restriction . Suppose the sequence defined by the iteration (1.1). Then

Proof. Suppose and . By the lemma 3.1, exists. Then by the lemma 2.3 (iii), we obtain

(3.4)

and

(3.5)

Substring (3.5) into (3.4), we obtain

(3.6)

This implies

, (3.7)

and so

This implies that . This completes the proof.

Theorem 3.2. Let K be a nonempty closed convex subset of complete CAT(0) space X and let be satisfies condition (C). Let be the sequence defined by (1.2) with restriction and , where for some . Then −converges to a fixed point of T.

Proof. From Theorem 3.1, . Suppose where the union is taken over all subseqeunces of such that . We claim that . Let , then there exists a subsequence of such that . By Lemma 2.1 and 2.2 there exists a subsequence of such that , since , then by Lemma 3.1 and exists, we claim that u = v. Suppose not, by the uniqueness of asymptotic centers.

a contradiction, and hence . To show that converges to a fixed point of T, if suffices to show that consists of exactly on point. Let be a subsequence of by lemma 2.1 and 2.2 there exists subsequence and such that . Let and . We get seen that u = v and . we can complete the proof by showing that x = v. Suppose not, since is convergent, then by uniqueness of asymptotic centers,

a contradiction, and hence the conclusion.

Theorem 3.3. Assume that X, K, T, are as in Theorem 3.2. If T satisfies condition I, then converges strongly to a fixed point of T.

Proof. Since T satisfies condition (I), we get . So there is a subsequence of such that for some for all . By Lemma 3.1, we obtain

Hence

This show that is a Cauchy sequences in F(T). Since F(T) is closed in X, there exists a point such that . It follows that . Since exists then it must be the . This competes the proof.

The following result is immediate sequel of our strong convergence theorem.

Corollary 3.1. Let K be a nonempty closed convex subset of a complete CAT(0) space X and let be a nonexpansive mapping with and be the sequence in for some with restriction . Suppose the sequence defined by the iteration (1.2). Then .

4. REFERENCES:

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[2] Bruhat, F. and Tits, J., Gropes r´eductifs sur un corps local. I. Donn´ees radicielles valu´ees, Inst. Houtes ´E tudes Sci., Publ. Math. 41(1972), 5-251.

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[14] Laokul, T. and Panyanak, B., Approximating fixed points of nonexpansive mappings in CAT(0)spaces, Int. Journal of Math. Anal. 3(2009), 1305-1315.

[15] Lim, T.C., Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60(1976), 179-182.

[16] Mann, W.R., 1953, ”Mean value methods in iteration,” Proc. Amer. Math. Soc. 4, pp.506-510.

[17] Puttasontiphot, T., Mann and Ashikawa iteration schemes for multivalued mappings in CAT(0) spaces, Appl. Math. Sci., Vol.4, No.61(2010), 3005-3018.

[18] Senter H.F. and Dotson W.G., Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44(1974), 375-38.

[19] Thianwan, S., Common fixed points of the new iterations for two asymptotically nonexpansive nonself-mappings in a Banch space, J. Omput Appl. Math. 224(2009), 688-695.

Received on 10.03.2013 Accepted on 12.04.2013

Research J. Science and Tech 5(3): July- Sept., 2013 page 368-374