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
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. □
NOTES
1Such a sequence exists if X is locally compact and second countable.
2Let 
be a metric space. A mapping 
is said to be contractive (strictly contractive) if 
for each 
with
. In our result, we use this mapping, but the following well-known result (for a detailed argument, see Goebel and Kirk [8], Kirk [9]) is not applicable since it assume a compact metric space.
Theorem: Let 
be a compact metric space and let 
be a contractive mapping. Then 
has a unique fixed point
, and moreover, for each
,
.
3Note that 
for any
.