Let *X* be a set. A binary relation \u2aaf\subset X\times X on *X* is called *transitive* if for any x,y,z\in X,

x\u2aafy\u2aafz\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}x\u2aafz,

(2.1)

*reflexive* if

\mathrm{\forall}x\in X,\phantom{\rule{1em}{0ex}}x\u2aafx,

(2.2)

and *antisymmetric* if for any x,y\in X,

x\u2aafy\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}y\u2aafx\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}x=y.

(2.3)

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 \mathrm{\Phi}(A) be the set of functions \varphi :A\to {\mathbb{R}}_{+}. Let \varphi ,\psi \in \mathrm{\Phi}(A). We write \varphi =0 if \varphi (a)=0 for each a\in A, and \varphi \le \psi if \varphi (a)\le \psi (a) for each a\in A. The expressions \varphi +\psi and max\{\varphi ,\psi \} are defined respectively by

\mathrm{\forall}a\in A,\phantom{\rule{1em}{0ex}}(\varphi +\psi )(a)=\varphi (a)+\psi (a),

(2.4)

\mathrm{\forall}a\in A,\phantom{\rule{1em}{0ex}}(max\{\varphi ,\psi \})(a)=max\{\varphi (a),\psi (a)\}.

(2.5)

For {\{{\varphi}_{i}\}}_{i\in \mathbb{N}}\subset \mathrm{\Phi}(A), we write {\varphi}_{i}\to 0 if {\varphi}_{i}(a)\to 0 as i\uparrow \mathrm{\infty} for each a\in A (we omit ‘as i\uparrow \mathrm{\infty}’ from here on).

Let d:X\times X\times A\to {\mathbb{R}}_{+}; the dependence of *d* on (x,y,a)\in X\times X\times A is expressed by d(x,y)(a). We treat the expression d(x,y) as a function from *A* to {\mathbb{R}}_{+}; more precisely, d(x,y) is the function \varphi \in \mathrm{\Phi}(A) given by \varphi (a)=d(x,y)(a) for all a\in A. Under the conventions described in the previous paragraph, for any x,y,{x}^{\prime},{y}^{\prime}\in X and {\{{x}_{i}\}}_{i\in \mathbb{N}},{\{{y}_{i}\}}_{i\in \mathbb{N}}\subset X, we have the following relations:

d(x,y)=0\phantom{\rule{1em}{0ex}}\u27fa\phantom{\rule{1em}{0ex}}\mathrm{\forall}a\in A,\phantom{\rule{1em}{0ex}}d(x,y)(a)=0,

(2.6)

d(x,y)\le d({x}^{\prime},{y}^{\prime})\phantom{\rule{1em}{0ex}}\u27fa\phantom{\rule{1em}{0ex}}\mathrm{\forall}a\in A,\phantom{\rule{1em}{0ex}}d(x,y)(a)\le d({x}^{\prime},{y}^{\prime})(a),

(2.7)

d({x}_{i},{y}_{i})\to 0\phantom{\rule{1em}{0ex}}\u27fa\phantom{\rule{1em}{0ex}}\mathrm{\forall}a\in A,\phantom{\rule{1em}{0ex}}d({x}_{i},{y}_{i})(a)\to 0.

(2.8)

The expressions d(x,y)+d({x}^{\prime},{y}^{\prime}) and max\{d(x,y),d({x}^{\prime},{y}^{\prime})\} are defined as in (2.4) and (2.5).

We say that *d* is *identifying* if for any x,y\in X,

d(x,y)=0\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}x=y,

(2.9)

*reflexive* if

\mathrm{\forall}x\in X,\phantom{\rule{1em}{0ex}}d(x,x)=0,

(2.10)

and *symmetric* if

\mathrm{\forall}x,y\in X,\phantom{\rule{1em}{0ex}}d(x,y)=d(y,x).

(2.11)

We say that *d* satisfies the *triangle inequality* if

\mathrm{\forall}x,y,z\in X,\phantom{\rule{1em}{0ex}}d(x,z)\le d(x,y)+d(y,z).

(2.12)

We say that *d* is *one-dimensional* if d(x,y)(a) does not depend on *a* for any x,y\in X. If *d* is one-dimensional, then we treat *d* as a function from X\times X to {\mathbb{R}}_{+}. 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 d:X\times X\times A\to {\mathbb{R}}_{+}. Even though ⪯ is merely a binary relation, we regard it as a type of order.

We say that a sequence {\{{x}_{i}\}}_{i\in \mathbb{N}} is *increasing* if {x}_{i}\u2aaf{x}_{i+1} for all i\in \mathbb{N}. We say that a function f:D\to \mathbb{R} with D\subset \mathbb{R} is *increasing* if f(x)\le f(y) for any x,y\in D with x\le y.

We say that *d* is *regular* if for any x,y,z\in X with x\u2aafy\u2aafz, we have

max\{d(x,y),d(y,z)\}\le d(x,z).

(2.13)

This means that if x\u2aafy, then d(x,y) increases as *x* ‘decreases’ or *y* ‘increases’.

**Example 2.1** Let X=\mathbb{R}. Let ⪯ be the usual partial order on ℝ. For x,y\in X, define d(x,y)=|x-y|. Then *d* is one-dimensional, a metric, and regular.

**Example 2.2** Let *X* be the set of functions on ℝ. Let A=\mathbb{R}. For f,g\in X, write f\u2aafg if f\le g. Then ⪯ is a partial order. For f,g\in X and a\in A, define d(f,g)(a)=|f(a)-g(a)|. Then *d* is not one-dimensional, but *d* is identifying, reflexive, symmetric, regular, and satisfies the triangle inequality.

**Example 2.3** Let (S,\mathcal{S}) be a measurable space. Let *X* be the set of finite measures on *S*. For \mu ,\nu \in X, write \mu \u2aaf\nu if \mu (B)\le \nu (B) for each B\in \mathcal{S}. Then ⪯ is a partial order. Let *A* be the set of bounded measurable functions from *S* to ℝ. For \mu ,\nu \in X and f\in A, define d(\mu ,\nu )(f)=|\int f\phantom{\rule{0.2em}{0ex}}d\mu -\int f\phantom{\rule{0.2em}{0ex}}d\nu |. Then *d* is not one-dimensional, but *d* is identifying, reflexive, symmetric, regular, and satisfies the triangle inequality.

**Example 2.4** Let X={\mathbb{R}}^{2}. For x,y\in X, write x\u2aafy if \parallel x\parallel \le \parallel y\parallel, where \parallel \cdot \parallel is the Euclidian norm. Then ⪯ is a preorder, but it is not a partial order since it fails to be antisymmetric. For x,y\in X, let d(x,y)=\parallel x-y\parallel. Then *d* is a metric, but not regular. For example, (1/2,0)\u2aaf(0,1)\u2aaf(1,0), but d((0,1),(1,0))=\sqrt{2}>d((1/2,0),(1,0))=1/2.

**Example 2.5** Let X={\mathbb{R}}^{2}. For x,y\in X, write x\u2aafy if x\le y componentwise. Define *d* as in Example 2.4. Then *d* is a metric and regular.

**Example 2.6** Let X={\mathbb{R}}^{2}. For x,y\in X, write x\u2aafy if {x}_{1}<{y}_{1} or if {x}_{1}={y}_{1} and {x}_{2}\le {y}_{2}, where x=({x}_{1},{x}_{2}), *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, (0,0)\u2aaf(1,100)\u2aaf(2,0), but d((0,0),(1,100))>100>d((0,0),(2,0))=2.

A self-map T:X\to X is called *order-preserving* if for any x,y\in X,

x\u2aafy\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}Tx\u2aafTy.

(2.14)

A *fixed point* of *T* is an element x\in X such that Tx=x. We say that a fixed point {x}^{\ast} of *T* is *globally stable* if

\mathrm{\forall}x\in X,\phantom{\rule{1em}{0ex}}d({T}^{i}x,{x}^{\ast})\to 0.

(2.15)

Note that if {x}^{\ast} 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 i\in \mathbb{N}, we have d(x,{x}^{\ast})=d({T}^{i}x,{x}^{\ast})\to 0; thus x={x}^{\ast}.

We say that T:X\to X is *asymptotically contractive* if

\mathrm{\forall}x,y\in X,\phantom{\rule{1em}{0ex}}d({T}^{i}x,{T}^{i}y)\to 0.

(2.16)

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).