Remarks on metric transforms and fixed-point theorems
© Kirk and Shahzad; licensee Springer 2013
Received: 19 October 2012
Accepted: 2 April 2013
Published: 22 April 2013
It is shown that if a mapping is a local radial contraction defined on a metric space which takes values in a metric transform of , then for many metric transforms it is also a local radial contraction (with possibly different contraction constant) relative to the original metric. Several specific examples are given. This in turn implies that the mapping has a fixed point if the space is rectifiably pathwise connected. Some results about set-valued contractions are also discussed.
In this paper, we study fixed points of mappings satisfying local contractive conditions, with a special emphasis on the following concept due to L. M. Blumenthal.
Definition 1.1 A strictly increasing concave function for which is called a metric transform.
Blumenthal has observed (see Exercise 5 on p.26 of ) that if is a metric space and if for each , where ϕ is a metric transform, then is also a metric space. He had introduced this concept earlier in  to show that the metric transform of any metric space M by , , has the Euclidean four point property, i.e., each four points of are isometric to a quadruple of points in 3-dimensional Euclidean space.
A mapping g defined on a metric space X is said to be a local radial contraction  if there exists such that for each there exists such that for all . We begin by showing that if a mapping is a local radial contraction defined on a metric space and taking values in a metric transform of , then in many instances it is also a local radial contraction (with possibly different contraction constant) relative to the original metric. Several specific examples are given. This in turn implies that the mapping has a fixed point if the space is rectifiably pathwise connected. In Section 3, we turn our attention to set-valued contractions and prove, among other things, a set-valued analog to the main result of Section 2. Finally, because it is also based on the idea of a metric transform, we revisit a counter-example given in  in somewhat more detail.
2 Local radial contractions
As noted above, a mapping g defined on a metric space X is a local radial contraction if there exists such that for each there exists such that for all . If this condition is satisfied for some independent of x then g is said to be a uniform local contraction.
Theorem 2.1 ()
Let be a complete metric space, and suppose each two points of X can be joined by a rectifiable path. Then if is a local radial contraction, g has a unique fixed point , and moreover for each .
Rakotch proved the above theorem in  under the stronger assumption that g is a local contraction in the sense that there exists such that each point of has a neighborhood such that for all . The proof of Theorem 2.1 entails showing that the metric space is complete, where ℓ is the path metric on X induced by d, and then observing that g is actually a global contraction on . (An assertion in the proof of Theorem 2.1 given in  was based on a Proposition of Holmes , which was later shown by Jungck  to be false. However, as Jungck also proved in , the assertion itself is true. Hence, the proof given in , with minor changes, is true.)
We now give a simple condition in terms of metric transforms which implies that a mapping is a local radial contraction. Notice that if ϕ is taken to be the identity mapping, the following result reduces to the definition of a local radial contraction.
- (a)For each there exists such that
- (b)There exists such that for all sufficiently small
Then g is a local radial contraction on .
In view of Theorem 2.1, we now have the following.
Theorem 2.3 Suppose, in addition to the assumptions in Theorem 2.2, X is complete and rectifiably pathwise connected. Then g has a unique fixed point , and for each .
Therefore, g is a local radial contraction on . □
- (i). Let and select . Then
Since , condition (b) follows.
- (ii), for . Then for any
and condition (b) holds for .
- (iii). Let , and set . We know that if and if then
for fixed . Let . Then . Let . Then and . If is chosen so that , then for sufficiently small. This implies that for sufficiently small, and this in turn implies that condition (b) holds.
. Let and select . Then . Let . Then and for , . This is clearly true for sufficiently small because .
Not every metric transform satisfies condition (b); provides an example. On the other hand, Proposition 2.6 below shows that the collection of metric transforms which do satisfy condition (b) are indeed numerous and complex.
Then both and are closed under functional composition.
Therefore, for sufficiently small, so it follows that . □
Finally, we observe that in Theorem 2.1 it needs only be assumed that some iterate of the mapping g is a local radial contraction. Specifically, we have the following.
Theorem 2.7 Let X be a complete metric space for which each two points can be joined by a rectifiable path, and suppose is a mapping for which is a local radial contraction for some . Then g has a unique fixed point , and for each .
Notice that g is not even assumed to be continuous. Similarly, we also have the following extension of Theorem 2.3.
Theorem 2.8 Let X be a complete metric space for which each two points can be joined by a rectifiable path, and suppose is a mapping for which satisfies the assumptions in Theorem 2.2 for some . Then g has a unique fixed point , and for each .
These results are immediate consequences of Theorems 2.1 and 2.3, and the following proposition due to Tan .
Theorem 2.9 Let X be a topological space, let , and let be a mapping for which satisfies for each . Then for each . (Also if is the unique fixed point of f, it is also the unique fixed point of g.)
Holmes makes the following claim in . If is a connected and locally connected metric space and is a uniformly continuous local radial contraction, then there exists a metric δ on X, topologically equivalent to d, such that g is a global contraction on . If true, this claim would yield Theorem 2.7 for uniformly continuous local radial contractions in a more general class of spaces. However, we append an example, taken from , which shows that this assertion is false. We do not know if it is true when is rectifiably pathwise connected.
3 Set valued contractions
A point is said to be a fixed point of T if . Our point of departure in this section is the following celebrated theorem of Nadler .
Theorem 3.1 Let be a complete metric space, and suppose be a multivalued contraction mapping. Then T has a fixed point.
Our purpose in this section is to extend Nadler’s theorem by replacing the Hausdorff metric with other metrics on which are either metrically or sequentially equivalent to H.
- (1)there exists such that
- (2)for every and ,
Thus, condition (2) is always true if is replaced with the usual Hausdorff metric H. This is precisely the fact about the Hausdorff metric that Nadler used in his proof. In fact, Nadler’s proof yields the following result. This theorem implies that every -contraction of a complete metric space X into has a fixed point .
- (1)there exists such that
- (2)if and ,
Then T has a fixed point, i.e., there exists such that .
Hence, is a Cauchy sequence, so there exists such that . It follows from (1) that . Since D and H are equivalent, . Since , it follows from the definition of Hausdorff metric that , and since Tx is closed, . □
- (1)there exists such that
- (2)if ,
Then T has a fixed point.
This implies that is a Cauchy sequence in , so exists. Since (1) implies T is continuous, . □
As noted above, (1) alone is sufficient if because (2) is redundant in this case. However, the following example shows that (1) alone is not sufficient if .
We now turn to an analog of Theorem 2.2 for set-valued mappings.
- (a)For each ,
Then for sufficiently small, T is an -uniform local multivalued contraction on .
Thus, for sufficiently small, T is an -uniform local multivalued contraction on . □
A metric space is said to be ε-chainable (where is fixed) if given there is an ε-chain joining a and b. This means there exists a finite set of points in X such that , and for all . The following result is also due to Nadler.
Theorem 3.5 ([, Theorem 6])
Let be a complete ε-chainable metric space. If is an - uniform local multivalued contraction, then T has a fixed point.
By combining the above result with Theorem 3.4 we obtain the following.
Theorem 3.6 If, in addition to the assumptions of Theorem 3.4, X is complete and connected, then T has a fixed point.
Proof A connected metric space is ε-chainable for any . □
The following example was given in . It shows that Theorem 2.3 is false if the space is merely assumed to be pathwise connected rather than rectifiably pathwise connected. This illustrates another application of the idea of metric transforms.
We first observe that is a metric space (see Proposition 4.1 below).
To deal with the case , merely take a neighborhood of x with radius less than .
Notice that the mapping of the above example is even locally contractive in the sense of Rakotch , but it is fixed-point-free. We note also that the space is topologically equivalent to with its usual metric. In particular is complete, connected, and locally connected. (A space X is said to be locally connected if given any , each neighborhood U of x contains a connected neighborhood V of x.)
Then is a metric space.
We now observe that for each and , the metric transform induces a metric on the interval . The metric space is obtained by simply ‘gluing’ the consecutive intervals at their common endpoints and applying Proposition 4.1 inductively. This results in the metric defined by (1).
The authors are grateful to the anonymous referees for their comments that considerably improved the paper. The research of N. Shahzad was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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