Simple fixed point results for order-preserving self-maps and applications to nonlinear Markov operators
© Kamihigashi and Stachurski; licensee Springer. 2013
Received: 3 September 2013
Accepted: 1 December 2013
Published: 27 December 2013
Consider a preordered metric space . Suppose that if . We say that a self-map T on X is asymptotically contractive if as for all . We show that an order-preserving self-map T on X has a globally stable fixed point if and only if T is asymptotically contractive and there exist such that for all and . We establish this and other fixed point results for more general spaces where d consists of a collection of distance measures. We apply our results to order-preserving nonlinear Markov operators on the space of probability distribution functions on ℝ.
The majority of fixed point theorems require a space that is complete in some sense. Fixed point theorems based on the metric approach such as the celebrated Banach contraction principle and its numerous extensions commonly assume a complete metric space (see, e.g., ). Results based on the order-theoretic approach such as Tarski’s fixed point theorem and the Knaster-Tarski fixed point theorem typically require a complete lattice or a chain-complete partially ordered space (see, e.g., ). These two approaches are combined in the growing literature on fixed point theory for partially ordered complete metric spaces (e.g., [3–11]), where completeness still plays an indispensable role.
However, there are various situations in which it is fairly easy to construct a good candidate for a fixed point even if the underlying space may not be complete. For example, consider a self-map on a space of real-valued functions on some set. Then an increasing sequence of functions majorized by a common function converges pointwise to some function in the same space. If this pointwise limit turns out to be a good candidate for a fixed point, then there is no need to verify that the entire space is complete or chain-complete.
In this paper we develop simple fixed point results for order-preserving self-maps on a space equipped with a transitive binary relation and a collection of distance measures. Most of our results assume the existence of a good candidate for a fixed point instead of completeness. Some of our results use the condition that the self-map in question is asymptotically contractive, which means in our terminology that two distinct points are mapped arbitrarily close to each other after sufficiently many iterations. In the case of Markov operators induced by Markov chains, this property is an implication of the order-theoretic mixing condition introduced in , which is a natural property of various stochastic processes (see [12, 13]). We show that asymptotic contractiveness is not only a useful condition for showing the existence of a fixed point, but also a necessary condition for the existence of a globally stable fixed point.
In practice, a candidate for a fixed point must be constructed or must be shown to exist. If the underlying space is a complete metric space, then the limit of a certain Cauchy sequence serves as a good candidate. This classical approach is still common in the recent literature on fixed points of order-preserving self-maps on partially ordered complete metric spaces (e.g., [3–6, 8, 10, 11]). For comparison purposes, we establish a fixed point result for such spaces as a consequence of our general results.
To illustrate how a candidate fixed point can be constructed in practice, we consider nonlinear Markov operators on the space of probability distribution functions on ℝ. We provide a simple sufficient condition for the existence of a globally stable fixed point.
A binary relation is called a preorder if it is transitive and reflexive. A preorder ⪯ is called a partial order if it is antisymmetric.
For , we write if as for each (we omit ‘as ’ from here on).
The expressions and are defined as in (2.4) and (2.5).
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 .
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 .
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 .
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).
3 Fixed point results
Let X and A be sets. Let ⪯ be a binary relation on X. Let . Let . In this section we maintain the following assumptions.
Assumption 3.1 T is order-preserving.
Assumption 3.2 ⪯ is transitive.
Assumption 3.3 d is identifying.
Assumption 3.4 d is regular.
The following theorem is the most fundamental of our fixed point results.
Then is a fixed point of T.
where the convergence holds by (3.1). It follows that ; thus is a fixed point of T since d is identifying. □
The above proof generalizes the fixed point argument used in . Under additional assumptions, conditions (3.1)-(3.3) are also necessary for the existence of a fixed point.
Theorem 3.2 Suppose that ⪯ is reflexive. Suppose further that d is reflexive. Then T has a fixed point if and only if there exist satisfying (3.1)-(3.3).
Proof The ‘if’ part follows from Theorem 3.1. For the ‘only if’ part, let be a fixed point of T. Then since ⪯ and d are reflexive, (3.1)-(3.3) trivially hold with . □
Let us now consider global stability of a fixed point. We start with a simple consequence of asymptotic contractiveness.
Lemma 3.1 Suppose that T is asymptotically contractive and has a fixed point . Then is globally stable.
For global stability, let be arbitrary. Again by (2.16) with , we obtain (2.15). Hence is globally stable. □
Theorem 3.3 Suppose that T is asymptotically contractive. Suppose further that there exist satisfying (3.2) and (3.3). Then is a globally stable fixed point of T.
Proof Since T is asymptotically contractive, x and satisfy (3.1). Thus, by Theorem 3.1, is a fixed point of T. Global stability follows from Lemma 3.1. □
Theorem 3.4 Suppose that ⪯ is reflexive. Suppose further that d is symmetric and satisfies the triangle inequality. Then T has a globally stable fixed point if and only if T is asymptotically contractive and there exist satisfying (3.2) and (3.3).
Thus (2.16) holds; i.e., T is asymptotically contractive. By reflexivity of ⪯, (3.2) and (3.3) hold with . □
4 The case of a complete metric space
In this section, in addition to Assumptions 3.1-3.4, we maintain the following assumptions.
Assumption 4.1 is a complete metric space.
Assumption 4.2 For any increasing sequence converging to some , we have for all .
Assumption 4.3 For any increasing sequence converging to some , if there exists such that for all , then .
Assumptions 4.2 and 4.3 hold if ⪯ is closed (i.e., a closed subset of ). To see this, let be an increasing sequence converging to some . Then given any , we have for all ; thus letting , we obtain . Furthermore, if there exists such that for all , then letting yields .
Then T has a fixed point.
where the first inequality in (4.6) holds by (4.3) (with ) and regularity of d. Since are arbitrary, it follows that is Cauchy.
Thus (3.2) holds. Condition (3.1) follows from (4.7) and (4.1) with and . From (4.7) we have for all . Thus by Assumption 4.3, . Hence (3.3) holds. It follows by Theorem 3.1 that is a fixed point of T. □
Suppose further that there exists satisfying (4.2). Then T has a fixed point.
Thus (4.1) holds. Let be as in the proof of Theorem 4.1. It is shown in  that is Cauchy, so that it converges to some . By Assumption 4.2, we have for all . Thus (4.3) holds with . Now the conclusion follows by Theorem 4.1. □
The core part of the proof of [, Theorem 2.1] is to show that is Cauchy. Since this can in fact be done without Assumptions 3.4 and 4.3, the corresponding part of [, Theorem 2.1] is not directly comparable to Theorem 4.1. The same remark applies to [, Theorem 2.1]. In [3, 8], instead of Assumptions 3.4 and 4.3, the recursive structure of (4.8) or (4.9) is utilized to show that is Cauchy and that its limit is a fixed point. See, e.g., [3–6, 8, 10] for extensions.
5 Nonlinear Markov operators
In this section we consider the case in which T is a self-map on the space of probability distribution functions on ℝ. Such a map is often called a nonlinear Markov operator; linear Markov operators are often associated with Markov chains. Since our approach does not require linearity, we allow T to be nonlinear. The analysis of this section can be extended to Markov chains on considerably more general spaces than ℝ along the lines of [12, 13, 16].
Note that ⪯ is a partial order. This partial order is known as ‘stochastic dominance’. We also write if for all . Hence if and only if .
It is easy to see that Assumptions 3.2-3.4 hold under (5.3) and (5.4), and that d is symmetric and satisfies the triangle inequality.
It is shown in [, Theorem 3.1] that T is asymptotically contractive if it is the linear Markov operator on F associated with an ‘order mixing’ Markov chain. Informally, a Markov chain is order mixing if given any two independent versions and of the same chain with different initial conditions, we have at least once with probability one. This is a natural property of various stochastic processes; see [12, 13].
The following result is a restatement of Theorem 3.4.
The next result provides a sufficient condition for the existence of satisfying (5.5) and (5.6).
Then T has a globally stable fixed point .
where the infimum is taken pointwise. By construction, satisfies (5.5). We verify that , and that (5.6) holds.
To see that , note that since each is increasing, so is . From (5.7)-(5.9) it follows that . Thus for all . Since and , we have and . We see that is right continuous or, equivalently, upper semicontinuous (given that is increasing) because the pointwise infimum of a family of upper semicontinuous functions is upper semicontinuous (see [, p.43]).
It remains to verify (5.6). Since for all , we have for all . Taking the pointwise infimum of the right-hand side over and noticing that is decreasing with respect to ≤, we obtain ; i.e., . □
One way to ensure the existence of satisfying (5.8) is by assuming that is ‘tight’ (with viewed as a sequence of probability measures). In this case, has a weak limit, which can be used as an upper bound on with respect to ⪯. This is the approach taken in .
Although (5.7) and (5.8) imply that is tight, Theorem 5.2 does not follow from [, Theorem 3.1, Lemma 6.5]. First of all, T can be nonlinear here. Second, asymptotic contractiveness is weaker than the ‘order mixing’ condition. Third, T is not assumed to be ‘bounded in probability’ here.
If one assumes that in addition to (5.7) and (5.8), then T maps into itself, where is the set of functions such that . In this case, the existence of a fixed point can be shown by applying the Knaster-Tarski fixed point theorem [, p.16] to the restriction of T to . However, since we do not assume that here, T need not be a self-map on . Thus Theorem 5.2 does not follow from the Knaster-Tarski fixed point theorem.
Financial support from ARC Discovery Outstanding Researcher Award DP120100321 and the Japan Society for the Promotion of Science is gratefully acknowledged.
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