Let be a metric space and let denote the family of nonempty, closed and bounded subsets of X. For let and . The usual Hausdorff distance between A and B is defined as
A mapping is called a multivalued contraction mapping if there exists a constant such that
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.
One example of a metric on which is metrically equivalent to the Hausdorff metric H is the metric , which was introduced in . is defined by setting
Clearly, is metrically equivalent to the Hausdorff metric:
A multivalued mapping is called an -contraction if
there exists such that
for every and ,
It follows immediately from the definition of the Hausdorff metric H that if and if . Then for each there exists such that
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 .
Theorem 3.2 Let be a complete metric space, and let D be any metric on which is sequentially equivalent to the Hausdorff metric H. Suppose satisfies
if and ,
Then T has a fixed point, i.e., there exists such that .
Proof (cf. ) By saying D is sequentially equivalent to H, we mean that for and ,
Suppose . Select and . By (2) and (1), there exists such that
Similarly, there exists such that
In general, for each 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, . □
Remark 3.3 The point valued analog of Theorem 3.2 is rather trivial. Let be a complete metric space, and let ρ be any metric on X which is sequentially equivalent to the d. Suppose satisfies
there exists such that
Then T has a fixed point.
Combining (1) and (2), we have
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 .
Example Take with the metric: . Define by setting . Clearly, T has no fixed point. However, if then , and
We now turn to an analog of Theorem 2.2 for set-valued mappings.
A mapping is said to be an -uniform local multi-valued contraction (where and ) if for , . This definition, given in , is modelled after a concept introduced by Edelstein in .
Theorem 3.4 Let be a metric space and . Suppose there exists a metric transform ϕ and such that the following conditions hold:
For each ,
There exists such that for sufficiently small,
Then for sufficiently small, T is an -uniform local multivalued contraction on .
Proof Let , and observe that
Now suppose there exists such that for t sufficiently small,
Then for sufficiently small,
and since ϕ is strictly increasing this in turn implies
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 . □