Metric fixed point theory for nonexpansive mappings defined on unbounded sets
© Alghamdi et al.; licensee Springer. 2014
Received: 26 March 2014
Accepted: 1 July 2014
Published: 22 July 2014
It is standard practice in metric fixed point theory to reduce fixed point questions for mappings defined on unbounded sets to the bounded case. Many of these results are couched in a Banach space framework and involve bounded orbits. We examine these results in a somewhat broader metric context here.
Keywordsnonexpansive mappings fixed points spaces
If a closed convex subset has the fixed point property for all nonexpansive self-mappings, then is it necessarily bounded? This has long been an open question in metric fixed point theory. The answer is ‘yes’ if X is a Hilbert space (see ). It has been shown recently (see ) that the failure of the fixed point property for every unbounded convex closed set is not a characteristic of Hilbert spaces; more precisely, for every unbounded closed convex set in , there exists a fixed point free nonexpansive self-mapping of the set. On the other hand, it is obvious that nontrivial nonexpansive mappings defined on unbounded sets may have fixed points. Consider, for example, simple rotations in the plane. However, in this case the mapping has bounded orbits. Indeed, the following result is found in the original 1965 paper of Kirk .
Theorem 1.1 Suppose K is a nonempty closed and convex subset of a reflexive Banach space, and suppose K has a normal structure. Suppose is a nonexpansive mapping, and suppose is bounded for some (hence all) . Then f has a fixed point.
The proof rests on the following fact (also proved in ).
Lemma 1.1 Suppose K is a convex subset of a normed linear space and suppose is nonexpansive. If is bounded for some , then some bounded convex subset of K is mapped into itself by f.
The above observations served as motivation for the following result.
Theorem 1.2 (Theorem 3.1 of )
Let C be a closed convex subset of a Banach space X, let be a finite commuting family of nonexpansive self-mappings of C, and suppose is bounded for some and all . Then there is a nonempty bounded closed and convex subset of C which is mapped into itself by each member of .
This theorem in conjunction with Theorem 4 of  assures that in the setting of Theorem 1.1 finite commuting families of nonexpansive mappings with bounded orbits always have a common fixed point.
It is our objective in this paper to examine when analogs of the above results hold in broader contexts, and whether they hold for more general classes of mappings.
2 The setting
The results in this paper will depend strongly on the notions of metric convexity. The following definition is discussed in detail by Kohlenbach in .
- (i)and ,
- (ii)and ,
and , ;
- (iv)and ,
is called the metric segment joining x and y (condition (iii) ensures that is an isometric image of the real line interval ). Hyperbolic spaces include all normed linear spaces and convex subsets thereof, as well as all spaces in the sense of Gromov (see ). Another important class of hyperbolic spaces are the so-called Busemann spaces (see ). These are precisely the hyperbolic spaces that are uniquely geodesic . (We will not invoke condition (iv) in this paper.) For fixed point theory in these spaces, we refer the reader to [12–18].
We say that a subset K of a Takahashi convex metric space is convex if for all and . For some of our results discussed below this is all that is needed. With this convention all closed and open metric balls are convex and the intersection of any family of convex sets is also convex. We use to denote the closed ball centered at with radius . We adopt the customary notation and write .
We begin with an abstract version of Lemma 1.1.
Lemma 3.1 Suppose K is a convex subset of a Takahashi convex metric space and suppose is nonexpansive. If is bounded for some , then some bounded convex subset of K is mapped into itself by f.
Then . Also for each k, so is nonempty. Clearly is convex and bounded (). Therefore is an increasing sequence of uniformly bounded convex sets in K. It follows that is a bounded convex set which is invariant under f. □
A Takahashi convex metric space X is said to have the FPP if every bounded closed convex subset of X has the fixed point property for nonexpansive mappings. In view of Lemma 3.1 the following is immediate.
Theorem 3.1 Let X be a Takahashi convex metric space which has the FPP, and let K be a nonempty closed and convex subset of X. Suppose is a nonexpansive mapping, and suppose is bounded for some . Then f has a fixed point.
Remark 1 In connection with Theorem 3.1 the following example is noteworthy.
so T has bounded orbits.
4 Eventually nonexpansive maps
In this section we point out that Theorem 3.1 extends to a wider class of mappings in more restricted settings.
for all . If , T is said to be asymptotically nonexpansive. If for n sufficiently large, T is said to be eventually nonexpansive (see ). The following lemma is obtained by slightly adjusting the argument in the proof of Lemma 3.1.
Lemma 4.1 Suppose K is a convex subset of a Takahashi convex metric space and suppose is a mapping which is eventually nonexpansive. If is bounded for some , then there exist and a bounded convex subset of K which is mapped into itself by each of the mappings , .
Also for each k, so is nonempty. Clearly is convex and bounded (). Therefore is an increasing sequence of uniformly bounded convex sets in K. It follows that is bounded and convex.
Since f is eventually nonexpansive, there exists such that for . Thus . So, for m sufficiently large, . In particular, is a bounded closed convex subset of K which is invariant under . □
Theorem 4.1 Let X be a reflexive or separable Banach space which has the FPP, let K be a closed convex subset of X, and suppose is eventually nonexpansive. If is bounded for some , then f has a fixed point.
Proof Let W be as in Lemma 4.1. Then in particular is a bounded closed convex set which in invariant under the commuting nonexpansive mappings and . One can now apply a classical result of Bruck  to conclude that and have a common fixed point which is necessarily a fixed point of f. □
ℝ-trees (or metric trees) are a class of hyperbolic spaces which have interesting geometric properties.
there is a unique geodesic (metric) segment denoted by joining each pair of points x and y in X; and
Theorem 4.1 extends to complete ℝ-trees without any additional assumptions.
Theorem 4.2 Let be a complete ℝ-tree, let K be a closed convex subset of X, and suppose is a mapping which is eventually nonexpansive and for which is bounded for some . Then f has a fixed point.
Theorem 4.2 is an immediate consequence of the following two facts. Proposition 4.1 was first proved in . For convenience of the reader, we repeat the proof here.
Theorem 4.3 ()
for all , where . Then T has a fixed point.
Proposition 4.1 Let be a metric space and suppose is eventually uniformly Lipschitzian for a sequence , and suppose T has a bounded orbit. If , then all orbits of T are bounded.
Then , where . Since y is arbitrary, all orbits of T are bounded. □
Remark Some form of asymptotic control over the behavior of the mapping is needed for the validity of Proposition 4.1, even if the mapping is continuous and X is the real line. It is easy to construct continuous mappings of the real line that have exactly one fixed point and all other orbits are unbounded. However, it is shown in  that if T is assumed to be continuous in Theorem 4.3, then the assumption that may be replaced with the much weaker assumption that .
5 Bounded orbits of families of mappings
Our next theorem is an analog of Theorem 3.1 in  which is formulated there in a Banach space setting. Note that commutativity of appears (at least in some sense) to be essential to the proof. As noted in , this result shows that the assumption of strict convexity is not needed for Theorem 4 of . (As Bula remarks in , this theorem is not true for infinite families.)
Theorem 5.1 Let be a Takahashi convex metric space, and let K be a convex subset of X, and let be a finite commutative family of nonexpansive self-mappings of K. Suppose is bounded for some (and hence all) and each . Then there is a bounded convex subset of K which is left invariant by each member of .
and . (Notice that here we use the fact that the mappings f and g commute.) It follows that S is a bounded convex set which is invariant under both f and g.
Theorem 5.1 has a different proof if the space X is of hyperbolic type. For this we need the following fact. Recall that a mapping f of a metric space into itself is said to be asymptotically regular if for each , . The following is a consequence of results of ; also see .
Proposition 5.1 Let K be a bounded convex subset of a space of hyperbolic type, and suppose is nonexpansive. Fix , and define by setting . Then is asymptotically regular. In particular, .
Therefore it is also the case that , and thus for all .
The proof is now completed as in the first proof. □
Remark 2 An interesting feature of the second proof is that it is only necessary to assume that f and g commute on the set for some rather than on the entire domain.
6 A condition of Djebali-Hammache
In  the authors present some new versions of fixed point theorems for nonexpansive mappings defined on closed, convex subsets of Banach spaces which are not necessarily bounded. In this section we discuss a result which they compare with Theorem 2.4 of  (see below). The following definition and notation are taken from .
Definition 6.1 Let Q be a nonempty closed convex subset of a Banach space X. A mapping is said to have the property if there exists a nonempty bounded closed convex subset such that .
(Implicit in the above is the assumption also that .)
By the Banach contraction mapping theorem, this set is always nonempty if f is nonexpansive and Q is a nonempty convex subset of X which contains the origin.
This set is denoted by when A depends explicitly on some function f.
Then f has a fixed point.
This theorem is an immediate consequence of the following lemma (Lemma 3.1 in ).
Lemma 6.1 Under the assumptions of Theorem 6.1, .
A Banach space X is said to have the FPP if each of its bounded closed convex subsets has the fixed point property for nonexpansive self-mappings. The following is Theorem 2.4 of .
Theorem 6.2 Let X be a Banach space which has the FPP, let C be a closed convex subset of X, and suppose is a nonexpansive mapping for which is nonempty and bounded for some . Then f has a fixed point.
The authors of  compare Theorem 6.1 with Theorem 6.2 and remark that in Theorem 6.1 the boundedness of is relaxed and the assumption that the space has the FPP is dropped. However, the added condition in Theorem 6.1 that the mapping satisfies property in conjunction with the fact that implies immediately reduces Theorem 6.1 to the bounded case. Also, the nonexpansiveness of f is used in the proof of Theorem 6.1 only to guarantee the existence of an approximate fixed point sequence for f and to guarantee that f is continuous. Finally, condition (6.2) in Theorem 6.1 is deceptively strong and, as the following result shows, Theorem 6.1 is essentially trivial.
Then S is finite. (Thus, f can have no nontrivial approximate fixed point sequence.)
We omit the details since a more general theorem is proved below.
There is a weaker version of condition (6.3) that is somewhat more realistic. In fact this appears to be the version the authors of  actually use in their applications.
In the following denotes a metric space, and . For , and for and , denote their Banach space analogs defined above with replacing .
Then . In particular, if f is continuous, then f has a fixed point.
which contradicts (6.4). □
Now suppose u and v are fixed points of f with . Choose so that . Since and , it must be the case that . Thus condition (6.4) implies that the fixed point set of f is always discrete.
for all (see ).
If f satisfies Suzuki’s condition (C), then f has a fixed point.
for all n. So converges to and hence . □
Remark If Q is a bounded convex subset of a Banach space and f satisfies Suzuki’s condition (C), then f always has a bounded approximate fixed point sequence by Lemma 6 of .
7 Historical comment about bounded orbits
and moreover, if and , then .
Theorem 7.1 ()
If R is a straight G-space (has unique metric segments) which has convex spheres, and if ϕ is a motion of R (an isometry of R onto itself) for which is bounded for some , then ϕ has a fixed point.
It was subsequently shown in Kirk  that it suffices to assume only that some subsequence of is bounded in Theorem 7.1, an assumption later shown by Całka  to be (nontrivially) equivalent to the original.
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for financial support. The authors are grateful to anonymous reviewers for useful comments.
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