On the Ishikawa iteration processes for multivalued mappings in some spaces
© Panyanak; licensee Springer. 2014
Received: 4 July 2013
Accepted: 9 October 2013
Published: 2 January 2014
The purpose of this paper is to prove the strong convergence of the Ishikawa iteration processes for some generalized multivalued nonexpansive mappings in the framework of CAT(1) spaces. Our results extend the corresponding results given by Shahzad and Zegeye (Nonlinear Anal. 71:838-844, 2009), Puttasontiphot (Appl. Math. Sci. 4:3005-3018, 2010), Song and Cho (Bull. Korean Math. Soc. 48:575-584, 2011) and many others.
Roughly speaking, a space is a geodesic space of bounded curvature. The precise definition is given below. Here CAT means the initials of three mathematician’s names (E Cartan, AD Alexandrov and A Toponogov) who have made important contributions to the understanding of curvature via inequalities for the distance function, and κ is a real number that we impose as the curvature bound of the space.
Fixed point theory in spaces was first studied by Kirk [1, 2]. His works were followed by a series of new works by many authors (see, e.g., [3–12]) mainly focusing on CAT(0) spaces. Since any space is a space for (see [, p.165]), all results for CAT(0) spaces immediately apply to any space with . Notice also that all spaces (with appropriate sizes) are uniformly convex metric spaces in the sense of . Thus, the results in  concerning uniformly convex metric spaces also hold in spaces as well.
The first result concerning the convergence of an Ishikawa iteration process for multivalued mappings was proved by Sastry and Babu  in a Hilbert space. Panyanak  extended the result of Sastry and Babu to a uniformly convex Banach space. Since then the strong convergence of the Ishikawa iteration processes for multivalued mappings has been rapidly developed and many of papers have appeared (see, e.g., [31–36]). Among other things, Shahzad and Zegeye  defined two types of Ishikawa iteration processes as follows.
They proved, under some suitable assumptions, that the sequence defined by (A) and (B) converges strongly to a fixed point of T. In 2010, Puttasontiphot  gave analogous results to those of Shahzad and Zegeye in complete CAT(0) spaces.
In this paper, we extend Puttasontiphot’s results to the setting of spaces with .
be nonexpansive if for all ;
- (ii)be quasi-nonexpansive if and
- (iii)satisfy condition (I) if there is a nondecreasing function with , for such that
- (iv)be hemicompact if for any sequence in E such that
there exists a subsequence of and such that .
A point is called a fixed point of T if . We denote by the set of all fixed points of T.
The following lemma can be found in . We observe that the boundedness of the images of T is superfluous.
for all ;
In this case, we write . The space 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 E of X is said to be convex if E includes every geodesic segment joining any two of its points.
Let , then is called a D-geodesic space if any two points of X with their distance smaller than D are joined by a geodesic segment. Notice that is a geodesic space if and only if it is a D-geodesic space.
with metric , (see [, Proposition 2.1]).
if then is the Euclidean space ;
if then is obtained from by multiplying the distance function by the constant .
(see [, Lemma 2.14]). We call the triangle having vertices , , in a comparison triangle of . Notice that it is unique up to an isometry of , and we denote it by . A point is called a comparison point for if .
Definition 2.3 A metric space is called a space if it is -geodesic and any geodesic triangle in X with satisfies the inequality.
It follows from [, Proposition 1.4] that spaces are uniquely geodesic spaces. In this paper, we consider spaces with . Since most of the results for such spaces are easily deduced from those for CAT(1) spaces, in what follows, we mainly focus on CAT(1) spaces. The following lemma is a consequence of Proposition 3.1 in .
for any and any points .
The following lemma is also needed.
Lemma 2.5 
Let be a nonnegative real sequence such that is bounded. Then has a subsequence which converges to zero.
3 Main results
We begin this section by proving a crucial lemma.
Lemma 3.1 Let be a CAT(1) space with convex metric, E be a nonempty closed convex subset of X, and be a quasi-nonexpansive mapping with and for each . Let be the sequence of Ishikawa iterates defined by (A) (replacing + with ⊕). Then exists for each .
This shows that the sequence is decreasing and bounded below. Thus exists for any . □
Now, we prove the strong convergence of the Ishikawa iteration process defined by (A).
Theorem 3.2 Let be a complete CAT(1) space with convex metric and , E be a nonempty closed convex subset of X, and be a quasi-nonexpansive mapping with and for each . Let and be the sequence of Ishikawa iterates defined by (A) (replacing + with ⊕). If T satisfies condition (I), then converges strongly to a fixed point of T.
Thus, . Also, as . Since T satisfies condition (I), we have . The proof of the remaining part follows the proof of Theorem 3.2 in , therefore we omit it. □
Theorem 3.3 Let be a complete CAT(1) space with convex metric and , E be a nonempty closed convex subset of X, and be a quasi-nonexpansive mapping with and for each . Assume that (i) ; (ii) ; (iii) , and let be the sequence of Ishikawa iterates defined by (A) (replacing + with ⊕). If T is hemicompact and continuous, then converges strongly to a fixed point of T.
This implies that since is closed. Thus exists by Lemma 3.1 and hence q is the limit of itself. □
To avoid the restriction of T, that is, for , we use the iteration process defined by (B).
Theorem 3.4 Let be a complete CAT(1) space with convex metric and , E be a nonempty closed convex subset of X, and be a multivalued mapping with and is quasi-nonexpansive. Let and be the sequence of Ishikawa iterates defined by (B) (replacing + with ⊕). If T satisfies condition (I), then converges strongly to a fixed point of T.
It follows that and this implies . Applying Theorem 3.2 to the map , we can conclude that the sequence defined by (B) converges to a point . This completes the proof. □
The following theorem is an analogue of Theorem 3 in .
Theorem 3.5 Let be a complete CAT(1) space with convex metric and , E be a nonempty closed convex subset of X, and be a hemicompact mapping with and is quasi-nonexpansive and continuous. Assume that (i) ; (ii) ; (iii) , and let be the sequence of Ishikawa iterates defined by (B) (replacing + with ⊕). Then converges strongly to a fixed point of T.
The hemicompactness of follows from that of T. The conclusion follows from Theorem 3.3. □
The author thanks Chiang Mai University for financial support.
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