A Fixed Point Theorem and an Application to Bellman Operators

This study introduces a new definition of a metric that corresponds with the topology of uniform convergence on any compact set, and shows both the existence of a unique fixed point of some operator by using this metric and that the iteration of such an operator results in convergence to this fixed point. We demonstrate that this result can be applied to Bellman operators in many situations involving economic dynamics.

• Bellman Operator; Uniform Convergence

1. Introduction

Dynamic programming (DP) is an important tool in economic dynamics because many models in which a representative agent maximizes a discounted sum of utilities can be treated as a DP problem. In this context, a fixed point of a Bellman operator plays a significant role and fixed point theorems for contraction mappings (Banach [1]) are usually used for this problem (see Le Van [2], Stokey and Lucas [3]). Recently, fixed point theorems of order-type, such as Knaster-Tarski (e.g., Aliprantis and Border [4], Granas and Dugundji [5]), have also been used for this issue (see Kamihigashi [6], Le Van and Vailakis [7]).

This study treats a fixed point theorem of the former. However, the metric we use is different from those in past research. Although most related research uses the uniform norm as the metric, we treat a new metric that corresponds with the topology of uniform convergence on any compact set. Our main results focus on two points. First, we show that there exists a unique fixed point of some operator. Second, we show that the iteration of such an operator results in convergence to this fixed point. This fixed point theorem can be applied Bellman operators in many dynamic economic systems.

The rest of the paper is organized as follows. In the next section, we introduce our framework and state our basic result. In Section 3, we present an application of our theorem to Bellman operators. In the appendix, we give an additional result on our metric.

2. Framework and Basic Results

Let X be a Hausdorff space and suppose that there exists an increasing sequence of compact sets in X such that¹

For any real-valued functions on, let

and let be the set of all functions such that. Then is a pseudo-metric on. Define

for any and d is a metric of. In the Appendix, we will verify that, for any sequences and, if and only if converges to f uniformly on any compact subset of X.

The following theorem holds.

Theorem 1: Suppose that satisfies the following two conditions:

1) For any and any, if for any;

2) There exists such that, for any

and.

Choose any and define and for any. Then converges to a unique fixed point of with respect to d.

Proof: Choose any. By definition of, we have

for any. By (1) and (2),

Hence, we have. By symmetry, we can verify that. Thus,

for all, and hence

for any $n$. Therefore, if,

If are two distinct fixed points of T, then

which is a contradiction. Thus, T has at most one fixed point².

Next, for any and,

Therefore, if, then

and thus is a Cauchy sequence. Hence, converges to some real number denoted by. Then

and thus

Hence, the function is in.

Now, for any,

as, and thus for any. Choose any, and choose any such that. We have already shown that, for any sufficiently large,

for all. Then³

and thus.

Now, T is a Lipschitz function on d and is thus continuous. Hence,. Meanwhile, since ,. Thus, and so f^* is a fixed point of T. This completes the proof.

3. Application to Bellman Operators

Let be a Hausdorff space, let be a correspondence from into and let be a real-valued function on, and define

.

We call the operator B a Bellman operator. Consider the following problem:

Let denote the maximum value for the above problem. It is well-known that under several conditions, is a fixed point of the Bellman operator.

Then we can show the following theorem.

Theorem 2: Suppose that 1) is real-valued and continuous on;

2) for any.

Then is a mapping from into. Further, for any, if and for any, then converges to a unique fixed point of with respect to.

Note that the conditions of Theorem 2 are not so strict. In many economic models, the following conditions are satisfied:

1);

2) There exists such that, if, then for any;

3) For any,;

4) is non-increasing in;

5) is continuous in.

Under these conditions, we can show that , and thus condition 1) of Theorem 2 is satisfied. Also, by setting, condition 2) of Theorem 2 is satisfied. Hence Theorem 2 is applicable.

Proof: By 2), B satisfies 1) and 2) of Theorem 1. Hence, it suffices to show that B is a mapping from into. By 1), we have. Choose any. As in the proof of Theorem 1, we can show that

Then

which implies that. This completes the proof. □

4. Conclusion

In this paper, we introduced a new fixed point theorem and showed that it can be applied to the Bellman operator of several economic models. The claim of our theorem includes not only the existence of fixed point but also the convergence result on iteration. By using our result, one can get value function from iterative application of the Bellman operator in a wide class of dynamic economic models.

5. Acknowledgements

The authors are grateful to Hiroyuki Ozaki for his helpful comments and suggestion. This research is partially supported by Keio/Kyoto Joint Global Center of Excellence Program Raising Market Quality-Integrated Design of “Market Infrastructure”.

Appendix. Additional Notes on Our Metric

In this section, we prove the following theorem.

Theorem A: Suppose satisfies the assumption in Section 2 and we define as in Section 2. Suppose also that is a sequence in and that. Then if and only if

for any compact set.

Proof of Theorem A: If the latter holds, then we have for any. Therefore,

.

Conversely, suppose that. For any compact set, is an open covering of, and thus there exist such that

Since, we have. Therefore,

which completes the proof. □

References

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