Another type of Mann iterative scheme for two mappings in a complete geodesic space
© Kimura and Nakagawa; licensee Springer. 2014
Received: 15 October 2013
Accepted: 23 January 2014
Published: 13 February 2014
In this paper, we show a Δ-convergence theorem for a Mann iteration procedure in a complete geodesic space with two quasinonexpansive and Δ-demiclosed mappings. The proposed method is different from known procedures with respect to the order of taking the convex combination.
The fixed point approximation has been studied in a variety of ways and its results are useful for the other studies. In 1953, Mann  introduced an iteration procedure for approximating fixed points of a nonexpansive mapping T in a Hilbert space. Later, Reich  discussed this iteration procedure in a uniformly convex Banach space whose norm is Fréchet differentiable. In 1998, Takahashi and Tamura  considered an iteration procedure with two nonexpansive mappings and obtained weak convergence theorems for this procedure in a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable. On the other hand, in 2008, Dhompongsa and Panyanak  proved the following theorem.
where is a sequence in , with the restrictions that diverges and . Then Δ-converges to a fixed point of T.
In this paper, we show an analogous result to Theorem 1.1 using the iterative scheme (1) in a complete space with two quasinonexpansive and Δ-demiclosed mappings. We also deal with the image recovery problem for two closed convex sets.
Let X be a metric space. For , a mapping is said to be a geodesic if c satisfies , and for all . An image of c is called a geodesic segment joining x and y. For , X is said to be an r-geodesic metric space if, for any with , there exists a geodesic segment . In particular, if a segment is unique for any with , then X is said to be a uniquely r-geodesic metric space. In what follows, we always assume for any . Thus, we say X is a geodesic metric space instead of a -geodesic metric space. For the more general case, see .
Let X be a uniquely geodesic metric space. A geodesic triangle is defined by . Let M be the two-dimensional unit sphere in . For , a triangle is called a comparison triangle of if , , . Further, for any and , if satisfies and , then z is denoted by . A point is called a comparison point of if . X is said to be a space if, for any and its comparison points , the inequality holds.
Let X be a geodesic metric space and a bounded sequence of X. For , we put . The asymptotic radius of is defined by . Further, the asymptotic center of is defined by . If, for any subsequences of , , i.e., their asymptotic center consists of the unique element , then we say Δ-converges to and we denote it by .
Let X be a metric space. A mapping is said to be a nonexpansive if T satisfies for any . The set of fixed points of T is denoted by . Further, a mapping with is said to be a quasinonexpansive if T satisfies for any and . Moreover, T is said to be Δ-demiclosed if, for any bounded sequence and satisfying and , we have .
3 Tools for the main results
In this section, we introduce some tools for using the main theorem.
Theorem 3.1 (Kimura and Satô )
Corollary 3.2 (Kimura and Satô )
Theorem 3.3 (Espínola and Fernández-León )
consists of exactly one point;
has a Δ-convergent subsequence.
Theorem 3.4 (Kimura and Satô )
Let X be a metric space and T a mapping from X into itself. If T is a nonexpansive with , then T is quasinonexpansive and Δ-demiclosed.
The following lemmas are important properties of real numbers and they are easy to show. Thus, we omit the proofs.
Lemma 3.5 Let δ be a real number such that and , real sequences satisfying , and . Then .
Lemma 3.6 Let and , bounded real sequences satisfying , and . Then .
Lemma 3.7 Let and be bounded real sequences satisfying . Then .
4 The main result
In this section, we show the main result.
for . Then Δ-converges to a common fixed point of S and T.
So, we get for all and there exists .
and the proof is finished. So, we may assume or for all .
Since for , we have and thus, . Therefore, we get and we also get . It implies and .
This is a contradiction. Hence we get . Therefore, we have Δ-converges to a common fixed point of S and T. □
By Theorem 3.4, we know that a nonexpansive mapping having a fixed point satisfies the assumptions in Theorem 4.1. Thus, we get the following result.
for . Then Δ-converges to a common fixed point of S and T.
5 An application to the image recovery
The image recovery problem is formulated as to find the nearest point in the intersection of family of closed convex subsets from a given point by using corresponding metric projection of each subset. In this section, we consider this problem for two subsets of a complete space.
for . Then Δ-converges to a fixed point of the intersection of and .
The authors thank the anonymous referees for their valuable comments and suggestions. The first author is supported by Grant-in-Aid for Scientific Research No. 22540175 from the Japan Society for the Promotion of Science.
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