Lemma 3.1 Let C be a nonempty closed and convex subset of a Banach space E. Let be an α-nonexpansive mapping for some . Let a sequence with in C be defined by the Ishikawa iteration (1.2) such that and are arbitrary sequences in . Suppose that the fixed point set contains an element z. Then the following assertions hold.
-
(1)
for all .
-
(2)
exists.
-
(3)
exists, where denotes the distance from x to .
Proof
In view of Lemma 2.1, we conclude that
Consequently,
This implies that is a bounded and nonincreasing sequence. Thus, exists.
In the same manner, we see that is also a bounded nonincreasing real sequence, and thus converges. □
Theorem 3.2 Let C be a nonempty closed and convex subset of a uniformly convex Banach space E. Let be an α-nonexpansive mapping for some . Let and be sequences in and let be a sequence with in C defined by the Ishikawa iteration (1.2).
-
1.
If is bounded and , then the fixed point set .
-
2.
Assume . Then is bounded, and the following hold.
Case 1: .
-
(a)
when .
-
(b)
when .
Case 2: .
-
(a)
when
-
(b)
when and .
Proof Assume that is bounded and . There is a bounded subsequence of such that . Suppose . Let . If , then, by Lemma 2.2(i), we have
This implies that
If , then, by Lemma 2.2(ii), we have
This implies again that
Thus, we have in all cases
This means that . By the uniform convexity of E, we conclude that .
Conversely, let and let . It follows from Lemma 3.1 that exists and hence is bounded. In view of Lemmas 2.1 and 2.4, we obtain a continuous strictly increasing convex function with such that
(3.1)
In view of (3.1), we conclude by applying Lemma 3.1 that
It follows that
From the property of g, we deduce that
(3.2)
In the same manner, we also obtain that
(3.3)
On the other hand, from (1.2) we get
(3.4)
Observing (3.4), we see that the assertions about the case follow from (3.2) and (3.3).
In what follows, we discuss the case . Assume first . By Lemma 2.1 and (3.3), we see that . Since T is α-nonexpansive, in view of (3.4), we obtain
(3.5)
Case (i): If , then (3.5) becomes
since all are in . We then derive from (3.3) that
(3.6)
Case (ii): If , then (3.5) becomes
We then derive from (3.3) again that
(3.7)
Finally, we assume instead. By (3.2) we have subsequences and of and , respectively, such that
Replacing by the number and dealing with the subsequences and in (3.6) and (3.7), we will arrive at the desired conclusion that . This gives . □
Theorem 3.3 Let C be a nonempty closed and convex subset of a uniformly convex Banach space E with the Opial property. Let be an α-nonexpansive mapping with a nonempty fixed point set for some . Let and be sequences in and let be a sequence with in C defined by the Ishikawa iteration (1.2).
Assume that , and assume, in addition, if . Then converges weakly to a fixed point of T.
Proof It follows from Theorem 3.2 that is bounded and . The uniform convexity of E implies that E is reflexive; see, for example, [3]. Then there exists a subsequence of such that as . In view of Proposition 2.3, we conclude that . We claim that as . Suppose on the contrary that there exists a subsequence of converging weakly to some q in C with . By Proposition 2.3, we see that . Lemma 3.1 says that exists for all z in . The Opial property then implies
This is a contradiction. Thus , and the desired assertion follows. □
Theorem 3.4 Let C be a nonempty compact and convex subset of a uniformly convex Banach space E. Let be an α-nonexpansive mapping for some . Let and be sequences in .
When , we assume . When , we assume either
Let be a sequence with in C defined by the Ishikawa iteration (1.2). Then converges strongly to a fixed point z of T.
Proof Since C is bounded, it follows from Lemma 2.5 that the fixed point set of T is nonempty. In view of Theorem 3.2, the sequence is bounded and . By the compactness of C, there exists a subsequence of converging strongly to some z in C, and . In particular, is bounded. Let . If , then, in view of Lemma 2.2(i), we obtain
Therefore,
If , then, in view of Lemma 2.2(ii), we obtain
Therefore,
It follows that . Thus we have . By Lemma 3.1, exists. Therefore, z is the strong limit of the sequence . □
Let C be a nonempty closed and convex subset of a Banach space E. A mapping is said to satisfy condition (I) [10] if
there exists a nondecreasing function with and for all such that
Using Theorem 3.2, we can prove the following result.
Theorem 3.5 Let C be a nonempty closed and convex subset of a uniformly convex Banach space E. Let be an α-nonexpansive mapping with a nonempty fixed point set for some . Let and be sequences in . When , we assume . When , we assume either
Let be a sequence with in C defined by the Ishikawa iteration (1.2). If T satisfies condition (I), then converges strongly to a fixed point z of T.
Proof
It follows from Theorem 3.2 that
Therefore, there is a subsequence of such that
Since T satisfies condition (I), with respect to the sequence , we obtain
This implies that, there exist a subsequence of , denoted also by , and a sequence in such that
(3.8)
In view of Lemma 3.1, we have
This implies
Consequently, is a Cauchy sequence in . Due to the closedness of in E (see Lemma 2.1), we deduce that for some z in . It follows from (3.8) that . By Lemma 3.1, we see that exists. This forces . □
The following examples explain why we need to impose some conditions on the control sequences in previous theorems.
Examples 3.6 (a) Let be defined by . Then T is a 0-nonexpansive (i.e., nonexpansive) mapping. Setting all , the Ishikawa iteration (1.2) provides a sequence
no matter how we choose . Unless , we can never reach the unique fixed point 0 of T via .
(b) Let be defined by
Then T is a -nonexpansive mapping. Indeed, for any x in and , we have
The other cases can be verified similarly. It is worth mentioning that T is neither nonexpansive nor continuous. Setting all , the Ishikawa iteration (1.2) provides a sequence
For any arbitrary starting point in , we have and
Consider two possible choices of the values of :
Case 1. If we set , , then and , the unique fixed point of T.
Case 2. If we set , , then and . Unless , we can never reach the unique fixed point 0 of T via .