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Approximating fixed points of α-nonexpansive mappings in uniformly convex Banach spaces and spaces
Fixed Point Theory and Applications volume 2013, Article number: 57 (2013)
Abstract
An existence theorem for a fixed point of an α-nonexpansive mapping of a nonempty bounded, closed and convex subset of a uniformly convex Banach space has been recently established by Aoyama and Kohsaka with a non-constructive argument. In this paper, we show that appropriate Ishikawa iterate algorithms ensure weak and strong convergence to a fixed point of such a mapping. Our theorems are also extended to spaces.
AMS Subject Classification:54E40, 54H25, 47H10, 37C25.
1 Introduction
The purpose of this paper is to study fixed point theorems of α-nonexpansive mappings of spaces. A metric space X is a space if it is geodesically connected, and if every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane (see Section 4 for the precise definition). Our approach is to prove firstly weak and strong convergence theorems for Ishikawa iterations of α-nonexpansive mappings in uniformly convex Banach spaces. Then, we extend the results to spaces.
Here are the details. Let E be a (real) Banach space and let C be a nonempty subset of E. Let be a mapping. Denote by the set of fixed points of T, i.e., . We say that T is nonexpansive if for all x, y in C, and that T is quasi-nonexpansive if and for all x in C and y in .
The concept of nonexpansivity of a map T from a convex set C into C plays an important role in the study of the Mann-type iteration given by
Here, is a real sequence in satisfying some appropriate conditions, which is usually called a control sequence. A more general iteration scheme is the Ishikawa iteration given by
where the sequences and satisfy some appropriate conditions. In particular, when all , the Ishikawa iteration (1.2) becomes the standard Mann iteration (1.1). Let T be nonexpansive and let C be a nonempty closed and convex subset of a uniformly convex Banach space E satisfying the Opial property. Takahashi and Kim [1] proved that, for any initial data in C, the sequence of iterations defined by the Ishikawa iteration (1.2) converges weakly to a fixed point of T, with appropriate choices of control sequences and .
Following Aoyama and Kohsaka [2], a mapping is said to be α-nonexpansive for some real number if
Clearly, 0-nonexpansive maps are exactly nonexpansive maps. Moreover, T is Lipschitz continuous whenever . An example of a discontinuous α-nonexpansive mapping (with ) has been given in [2]. See also Example 3.6(b).
An existence theorem for a fixed point of an α-nonexpansive mapping T of a nonempty bounded, closed and convex subset C of a uniformly convex Banach space E has been recently established by Aoyama and Kohsaka [2] with a non-constructive argument. In Section 3, we show that, under mild conditions on the control sequences and , the fixed point set is nonempty if and only if the sequence obtained by the Ishikawa iteration (1.2) is bounded and . In this case, converges weakly or strongly to a fixed point of T.
In Section 5, we establish the existence result of an α-nonexpansive mapping in a -space in parallel to [2]. We then extend the convergence theorems obtained in Section 3 to the case of spaces, as we planned.
2 Preliminaries
Let E be a (real) Banach space with the norm and the dual space . Denote by the strong convergence of a sequence to x in E and by the weak convergence. The modulus δ of the convexity of E is defined by
for every ϵ with . A Banach space E is said to be uniformly convex if for every . Let . The norm of E is said to be Gâteaux differentiable if for each x, y in S, the limit
exists. In this case, E is called smooth. If the limit (2.1) is attained uniformly in x, y in S, then E is called uniformly smooth. A Banach space E is said to be strictly convex if whenever and . It is well-known that E is uniformly convex if and only if is uniformly smooth. It is also known that if E is reflexive, then E is strictly convex if and only if is smooth; for more details, see [3].
A Banach space E is said to satisfy the Opial property [4] if, for every weakly convergent sequence in E, we have
for all y in E with . It is well known that all Hilbert spaces, all finite dimensional Banach spaces and the Banach spaces () satisfy the Opial property, while the uniformly convex spaces () do not; see, for example, [4–6].
Let be a bounded sequence in a Banach space E. For any x in E, we set
The asymptotic radius of relative to a nonempty closed and convex subset C of E is defined by
The asymptotic center of relative to C is the set
It is well known that if E is uniformly convex, then consists of exactly one point; see [7, 8].
Lemma 2.1 Let C be a nonempty subset of a Banach space E. Let be an α-nonexpansive mapping for some such that . Then T is quasi-nonexpansive. Moreover, is norm closed.
Proof Let and . Then we have
Therefore,
This inequality ensures the closedness of . □
Lemma 2.2 Let C be a nonempty subset of a Banach space E. Let be an α-nonexpansive mapping for some . Then the following assertions hold.
-
(i)
If , then
-
(ii)
If , then
Proof
-
(i)
Observe
This implies that
(ii) Observe
This implies that
□
Proposition 2.3 (Demiclosedness principle)
Let C be a subset of a Banach space E with the Opial property. Let be an α-nonexpansive mapping for some . If converges weakly to z and , then . That is, is demiclosed at zero, where I is the identity mapping on E.
Proof Since converges weakly to z and , both and are bounded. Let . If , then in view of Lemma 2.2(i),
If , then in view of Lemma 2.2(ii),
These relations imply
From the Opial property, we obtain . □
The following result has been proved in [9].
Lemma 2.4 Let be a fixed real number. If E is a uniformly convex Banach space, then there exists a continuous strictly increasing convex function with such that
for all x, y in and .
Recently, Aoyama and Kohsaka [2] proved the following fixed point theorem for α-nonexpansive mappings of Banach spaces.
Lemma 2.5 Let C be a nonempty closed and convex subset of a uniformly convex Banach space E. Let be an α-nonexpansive mapping for some . Then the following conditions are equivalent.
-
(i)
There exists x in C such that is bounded.
-
(ii)
.
3 Fixed point and convergence theorems in Banach spaces
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
In view of (3.1), we conclude by applying Lemma 3.1 that
It follows that
From the property of g, we deduce that
In the same manner, we also obtain that
On the other hand, from (1.2) we get
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
Case (i): If , then (3.5) becomes
since all are in . We then derive from (3.3) that
Case (ii): If , then (3.5) becomes
We then derive from (3.3) again that
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
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 .
4 An existence result in spaces
Let be a metric space. A geodesic path joining x to y in X (or briefly, a geodesic from x to y) is a map c from a closed interval into X such that , , and for all t, in . In particular, c is an isometry and . The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denoted by . 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 a uniquely geodesic if there exists exactly one geodesic joining x and y for each x, y in X. A subset Y of X is said to be convex if Y includes every geodesic segment joining any two of its points.
A geodesic triangle in a geodesic space consists of three points , , in X (the vertices of Δ), together with a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for a geodesic triangle in a geodesic space is a triangle in the Euclidean plane together with a one-to-one correspondence from Δ onto such that it is an isometry on each of the three segments. A geodesic space X is said to be a space if all geodesic triangles Δ satisfy the inequality:
It is easy to see that a space is uniquely geodesic.
It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a space. Other examples include inner product spaces, ℝ-trees (see, for example, [11]), Euclidean building (see, for example, [12]), and the complex Hilbert ball with a hyperbolic metric (see, for example, [8]). For a thorough discussion on other spaces and on the fundamental role they play in geometry, see, for example, [12–14].
We collect some properties of spaces. For more details, we refer the readers to [15–17].
Lemma 4.1 [16]
Let be a space. Then the following assertions hold.
-
(i)
For x, y in X and t in , there exists a unique point z in such that
(4.1)We use the notation for the unique point z satisfying (4.1).
-
(ii)
For x, y in X and t in , we have
The notion of asymptotic centers in a Banach space can be extended to a space as well by simply replacing the distance defined by with the one defined by the metric . In particular, in a space, consists of exactly one point whenever C is a closed and convex set and is a bounded sequence; see [[18], Proposition 7].
A sequence in a space X is said to Δ-converge to x in X if x is the unique asymptotic center of for every subsequence of . In this case, we write , and we call x the Δ-limit of .
Lemma 4.3 [19]
Every bounded sequence in a complete space X has a Δ-convergent subsequence.
Lemma 4.4 [21]
Let C be a closed and convex subset of a complete space X. If is a bounded sequence in C, then the asymptotic center of is in C.
Lemma 4.5 [22]
Let X be a complete space and let . Suppose that and for . If for some we have
then .
Recall that the Ishikawa iteration in spaces is described as follows: For any initial point in C, we define the iterates by
where the sequences and satisfy some appropriate conditions.
We introduce the notion of α-nonexpansive mappings of spaces.
Definition 4.6 Let C be a nonempty subset of a space X and let . A mapping is said to be α-nonexpansive if
The following is the counterpart to Lemma 2.5. However, we do not know if the compactness assumption can be removed from the negative α case.
Lemma 4.7 Let C be a nonempty closed and convex subset of a complete space X. Let be an α-nonexpansive mapping for some . In the case , we have if and only if is bounded for some x in C. If C is compact, we always have .
Proof Assume first that . The necessity is obvious. We verify the sufficiency. Suppose that is bounded for some x in C. Set for . By the boundedness of , there exists z in X such that . It follows from Lemma 4.4 that . Furthermore, we have
This implies
Thus,
Consequently, , ensuring that .
Next, we assume and C is compact. In particular, T is continuous and the sequence of for any x in C is bounded. In what follows, we adapt the arguments in [2] with slight modifications.
Let μ be a Banach limit, i.e., μ is a bounded unital positive linear functional of such that . Here, s is the left shift operator on . We write for the value of with in as usual. In particular, . As showed in [[2], Lemmas 3.1 and 3.2], we have
and
defines a continuous function from C into ℝ.
By compactness, there exists y in C such that . Suppose that there is another z in C such that . Let m be the midpoint in the geodesic segment joining y to z. In view of Lemma 4.1, we see that g is convex. Thus, too. Observing the comparison triangles in , we have
Consequently,
This amounts to say
Since , we have . Finally, it follows from (4.3) that . By uniqueness, we have . □
The proofs of the following results are similar to those in Sections 2 and 3.
Lemma 4.8 Let C be a nonempty subset of a space X. Let be an α-nonexpansive mapping for some such that . Then T is quasi-nonexpansive.
Lemma 4.9 Let C be a nonempty closed and convex subset of a space X. Let be an α-nonexpansive mapping for some . Then the following assertions hold.
-
(i)
If , then
-
(ii)
If , then
Lemma 4.10 Let C be a nonempty closed and convex subset of a space X. Let be an α-nonexpansive mapping for some . Let a sequence with in C be defined by (4.2) such that and are arbitrary sequences in . Let . Then the following assertions hold:
-
(1)
for .
-
(2)
exists.
-
(3)
exists.
Lemma 4.11 [15]
Let C be a nonempty convex subset of a space X and let be a quasi-nonexpansive map whose fixed point set is nonempty. Then is closed, convex and hence contractible.
The following result is deduced from Lemmas 4.8 and 4.11.
Lemma 4.12 Let C be a nonempty convex subset of a space X and let be an α-nonexpansive mapping with a nonempty fixed point set for some . Then is closed, convex, and hence contractible.
Lemma 4.13 Let C be a nonempty closed and convex subset of a complete space X and let be an α-nonexpansive mapping for some . If is a sequence in C such that and for some z in X, then and .
Proof It follows from Lemma 4.4 that .
Let . By Lemma 4.9(i), we deduce that
for all n in ℕ. Thus we have
Let . Then, by Lemma 4.9(ii), we have
for all n in ℕ. This implies again that
By the uniqueness of asymptotic centers, . □
5 Fixed point and convergence theorems in spaces
In this section, we extend our results in Section 3 to spaces.
Theorem 5.1 Let C be a nonempty closed and convex subset of a complete space X and let be an α-nonexpansive mapping for some . Let and be sequences in such that for a subsequence of . In the case , we assume also that . Let be a sequence with in C defined by (4.2). Then the fixed point set if and only if is bounded and .
Proof Suppose that and z in is arbitrarily chosen. By Lemma 4.10, exists and is bounded. Let
It follows from Lemmas 4.8 and 4.1(ii) that
Thus, we have
On the other hand, it follows from (4.2) and (5.1) that
In view of (5.1)-(5.3) and Lemma 4.5, we conclude that
By simply replacing with in the proof of Theorem 3.2, we have the desired result . The proof in the converse direction follows similarly. □
Theorem 5.2 Let C be a nonempty closed and convex subset of a complete space X and let be an α-nonexpansive mapping for some . Let and be sequences in such that for a subsequence of . In the case , we assume also that . Let be a sequence with in C defined by (4.2). If , then Δ-converges to a fixed point of T.
Proof It follows from Theorem 5.1 that is bounded and . Denote by , where the union is taken over all subsequences of . We prove that . Let . Then there exists a subsequence of such that . In view of Lemmas 4.3 and 4.4, there exists a subsequence of such that for some v in C. Since , Lemma 4.13 implies that . By Lemma 4.10, exists. We claim that . For else, the uniqueness of asymptotic centers implies that
which is a contradiction. Thus, we have and hence .
Now, we prove that Δ-converges to a fixed point of T. It suffices to show that consists of exactly one point. Let be a subsequence of . In view of Lemmas 4.3 and 4.4, there exists a subsequence of such that for some v in C. Let and . By the argument mentioned above, we have and . We show that . If it is not the case, then the uniqueness of asymptotic centers implies that
which is a contradiction. Thus we have the desired result. □
Theorem 5.3 Let C be a nonempty compact convex subset of a complete space X and let be an α-nonexpansive mapping for some . Let and be sequences in such that for a subsequence of . In the case , we assume also that . Let be a sequence with in C defined by (4.2). Then converges in metric to a fixed point of T.
Proof Using Lemmas 4.7 and 4.9 and replacing with in the proof of Theorem 3.4, we conclude the desired result. □
As in the proof of Theorem 3.5, we can verify the following result.
Theorem 5.4 Let C be a nonempty compact convex subset of a complete space X and let be an α-nonexpansive mapping for some . Let and be sequences in such that for a subsequence of . In the case , we assume also that . Let be a sequence with in C defined by (4.2). If T satisfies condition (I), then converges in metric to a fixed point of T.
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
The authors would like to thank the referees for their careful reading and valuable suggestions. This research was partially supported by the Grants NSC 99-2115-M-110-007-MY3 (for N.-C. Wong) and NSC 99-2221-E-037-007-MY3 (for J.-C. Yao).
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Naraghirad, E., Wong, NC. & Yao, JC. Approximating fixed points of α-nonexpansive mappings in uniformly convex Banach spaces and spaces. Fixed Point Theory Appl 2013, 57 (2013). https://doi.org/10.1186/1687-1812-2013-57
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DOI: https://doi.org/10.1186/1687-1812-2013-57
Keywords
- α-nonexpansive mapping
- fixed point
- Ishihawa iteration algorithm
- uniformly convex Banach space
- spaces