In this paper we point out some corrections needed in [1].
Recently, a geometric notion of seminormal structure has been introduced as follows.
Definition 1 (see [1]). A convex pair in a Banach space is said to have seminormal structure if, for any bounded, closed, and convex pair with , there exits such that It has been remarked in [1] that the pair has seminormal structure if and only if has normal structure in the sense of Brodskiĭ and Mil’man [2]. We revise this remark as follows. If the pair has seminormal structure, then has normal structure in the sense of Brodskiĭ and Mil’man. Indeed, if the set has normal structure, then may not have seminormal structure. We illustrate this with the following example.
Example 2. Let with the usual metric and let . Then has normal structure because is a nonempty, bounded, closed, and convex subset of the uniformly convex Banach space . Suppose and . Then ; that is, does not have seminormal structure.
The following notion has also been given in [1].
Definition 3 (see [1]). A nonempty, bounded, closed, and convex pair of a normed linear space is said to have property (D) provided that for each nonempty, closed, and convex pair one has In [1], the following proposition has been obtained to derive Corollary 5 (see Corollary 12 in [1]).
Proposition 4 (see Proposition 11 in [1]). Let be a nonempty, bounded, closed, and convex pair in a uniformly convex Banach space such that has the property (D). Then has seminormal structure.
Corollary 5 (see Corollary 12 in [1]). Let be a nonempty, bounded, closed, and convex pair in a uniformly convex Banach space such that has the property (D). Assume that is a cyclic relatively nonexpansive mapping. Then has a fixed point.
In the following, we give a counterexample to Proposition 4 which suggests that the result of Corollary 5 should be revised.
Example 6. Let with the usual metric and let and . It is clear that has the property (D). Now, consider and and suppose . Then
that is, does not have seminormal structure.
Using an argument similar to that in the proof of Proposition 11 in [1], we are able to correct Corollary 5 as follows.
Corollary 7. Let be a nonempty, bounded, closed, and convex pair in a uniformly convex Banach space such that has the property (D). If is a cyclic relatively nonexpansive mapping, then either is nonempty and has a fixed point in or has a best proximity point.
Proof. Suppose denotes the collection of all nonempty, closed, and convex pairs such that is cyclic on and there exists a pair for which . Note that . By using Zorn’s lemma we can see that has a minimal element, say . If , then is a nonempty, bounded, closed, and convex subset of a uniformly convex Banach space and is a nonexpansive mapping. Thus has a fixed point and we are finished. So, we assume that . We now consider the following cases.
Case 1. If , we may assume that . Consequently, there exists such that . Since is a cyclic relatively nonexpansive mapping, we have
This implies that has a best proximity point.
Case 2. If , by an argument similar to that in Proposition 11 of [1], we conclude that there exists a pair such that . By analogous proof of Theorem 8 in [1], we obtain that , which is a contradiction.