Let X be a set. A binary relation on X is called transitive if for any ,
and antisymmetric if for any ,
A binary relation is called a preorder if it is transitive and reflexive. A preorder ⪯ is called a partial order if it is antisymmetric.
Let A be a set. Let be the set of functions . Let . We write if for each , and if for each . The expressions and are defined respectively by
For , we write if as for each (we omit ‘as ’ from here on).
Let ; the dependence of d on is expressed by . We treat the expression as a function from A to ; more precisely, is the function given by for all . Under the conventions described in the previous paragraph, for any and , we have the following relations:
The expressions and are defined as in (2.4) and (2.5).
We say that d is identifying if for any ,
and symmetric if
We say that d satisfies the triangle inequality if
We say that d is one-dimensional if does not depend on a for any . If d is one-dimensional, then we treat d as a function from to . If d is one-dimensional, identifying, reflexive, symmetric, and satisfies the triangle inequality, then d is called a metric.
In what follows, the set X is assumed to be equipped with a binary relation ⪯ and a function . Even though ⪯ is merely a binary relation, we regard it as a type of order.
We say that a sequence is increasing if for all . We say that a function with is increasing if for any with .
We say that d is regular if for any with , we have
This means that if , then increases as x ‘decreases’ or y ‘increases’.
Example 2.1 Let . Let ⪯ be the usual partial order on ℝ. For , define . Then d is one-dimensional, a metric, and regular.
Example 2.2 Let X be the set of functions on ℝ. Let . For , write if . Then ⪯ is a partial order. For and , define . Then d is not one-dimensional, but d is identifying, reflexive, symmetric, regular, and satisfies the triangle inequality.
Example 2.3 Let be a measurable space. Let X be the set of finite measures on S. For , write if for each . Then ⪯ is a partial order. Let A be the set of bounded measurable functions from S to ℝ. For and , define . Then d is not one-dimensional, but d is identifying, reflexive, symmetric, regular, and satisfies the triangle inequality.
Example 2.4 Let . For , write if , where is the Euclidian norm. Then ⪯ is a preorder, but it is not a partial order since it fails to be antisymmetric. For , let . Then d is a metric, but not regular. For example, , but .
Example 2.5 Let . For , write if componentwise. Define d as in Example 2.4. Then d is a metric and regular.
Example 2.6 Let . For , write if or if and , where , etc. This binary relation ⪯ is a lexicographic order, which is a partial order. Define d as in Example 2.4. Then d is a metric, but not regular. For example, , but .
A self-map is called order-preserving if for any ,
A fixed point of T is an element such that . We say that a fixed point of T is globally stable if
Note that if is a globally stable fixed point of T, then T has no other fixed point as long as d is identifying. To see this, note that if T has another fixed point x, then for any , we have ; thus .
We say that is asymptotically contractive if
The term ‘asymptotically contractive’ has been used in different senses in the literature (e.g., [14, 15]). Our usage of the term can be justified by noting that (2.16) is an asymptotic property as well as an implication of well-known contraction properties; see (4.8) and (4.9).