1. Introduction
In [
1,
2], A. M. Rubinov introduced a discrete disperse dynamical system which is generated by a set-valued self-mapping of a compact metric space. This dynamical system was investigated in [
1,
2,
3,
4,
5,
6,
7]. It has a prototype in the economic growth theory [
1,
2,
8,
9]. In particular, it is an abstract extension of the classical von Neumann–Gale model [
1,
2,
8,
9]. This dynamical system is described by a compact metric space of states and a transition operator which is set-valued. Such dynamical systems correspond to certain models of economic dynamics [
1,
2,
8,
9]. More precisely, in [
1,
2,
3], the description of global attractors for certain dynamical systems was obtained; the uniform convergence of trajectories to global attractors was studied in [
4] and the behavior of trajectories under the presence of computational errors was analyzed in [
5], while, in [
6], analogous results were obtained for systems with a Lyapunov function. These results are collected in our recent book [
7].
In the present paper, we study the convergence and structure of trajectories of the continuous-time analog of this dynamical system generated by a differential inclusion. In particular, we show that, if the differential inclusion has a certain symmetric property, its turnpike possesses the corresponding symmetric property.
We introduce a global attractor (turnpike) for our dynamical system which is the closure of the set of all limit points of all trajectories; we show that all trajectories on an infinite interval converge to this set and that all trajectories on finite and sufficiently large intervals spend most of the time in a small neighborhood of the turnpike. If we know a finite number of approximate trajectories of our system, then we know the turnpike and this information can be useful if we need to find new trajectories of our system or their approximations. We believe that our results can be extended to the case of perturbed trajectories of our system.
It should be mentioned that the turnpike phenomenon holds for many problems in various areas of research [
7,
9,
10,
11,
12,
13,
14,
15,
16].
Let
be the
n-dimensional Euclidean space equipped with the inner product
which induces the Euclidean norm
and let
X be a nonempty closed set in
equipped with the relative topology.
Let us denote, by
, the set of all natural numbers. For every point
and every positive real number
r, let us put
Let us suppose that
. We define
The mapping F is upper semicontinuous at a point if, for every open set N which contains , there is an open neighborhood of the point in the space X for which
The mapping F is upper semicontinuous if it is upper semicontinuous at each point .
For the proof of the following result see Proposition 2 of [
17].
Proposition 1. Let us assume that F is upper semicontinuous and that is closed for every . Then, is closed in .
It is easy to see that the next result is true.
Proposition 2. Let us assume that the set is closed in , , V is a neighborhood of the point in the space X and that the set is bounded. Then, the mapping F is upper semicontinuous at the point .
Let
. A function
is called a trajectory if it is absolutely continuous (a. c.) and
We denote, by , the collection of all trajectories .
Let . A function is a trajectory if, for every , its restriction to the interval belongs to . we denote, by , the set of all trajectories .
A function is a trajectory if, for each pair of numbers , its restriction to the interval belongs to . We denote, by , the collection of all trajectories .
In the sequel, we suppose that the following assumption holds.
(A1) The mapping F is upper semicontinuous and is a compact, convex set for every point .
In view of Proposition 1, the set is closed in .
Proposition 3. The mapping F is bounded on bounded sets. In other words, for every positive number , there is a positive number such that Proof. Let
. We show that there exists
such that (2) holds. Let us assume the contrary. Then, for every
, there is
such that
In view of (3), we may assume, without loss of generality, that there is a limit
Since the mapping
F is upper semicontinuous, there is an open neighborhood
U of the point
x in the space
X such that, for each point
, we have
This contradicts (4) and completes the proof of Proposition 3. □
In our study, we apply the following two theorems (see Theorem 4 on page 13 of [
17] and Theorem 1 on page 60 of [
17], respectively).
Theorem 1. Let be an interval, for every , be an a. c. function such that, for each real number , the sequence is bounded and let a positive function satisfyfor a. e. real numbers and every . Then, there are a subsequence and an a. c. function such that converges to the function x uniformly over compact subsets of the interval I and that the functions converges weakly to the function in as . Theorem 2. Let be an interval, for every , and be Lebesgue measurable functions such that, for a. e. real numbers and every open neighborhood N of zero in the space , there is such that, for every natural number , Let us suppose that converges a. e. to the function and converges to y weakly in as . Then, for a. e. real number , Proposition 3 implies the next proposition.
Proposition 4. Let us assume that and is an a. c. function which satisfiesThen, the function x is Lipschitz on In addition to (A1), the following assumption (A2) is assumed to be satisfied everywhere below.
(A2) For every positive number M, there is a positive number such that, for every positive number T and each function which satisfies , the equation is valid for every number .
Note that (A2) holds for models of economic growth which are prototypes of our dynamical system [
7,
9].
The next result, which is deduced from Theorems 1 and 2, plays an important role in our study.
Theorem 3. Let us assume that , for each , and that the set is bounded. Then, there exist a subsequence and a trajectory such that converges to x as uniformly on and converges to as weakly in .
Proof. There exists
such that
By (6) and (A2), there is a positive number
for which
It follows, from Proposition 3 and Equations (1) and (7), that there is a positive number
for which
Theorem 1 and Equation (
8) imply that there exist a subsequence
and an a. c. function
such that
converges to
x uniformly over
as
and
converges weakly to
in
as
. Combined with Theorem 2, this convergence implies that
Theorem 3 is proved. □
2. The Results
We begin with the next theorem. It will be proved in
Section 3.
Theorem 4. Let be a nonempty closed bounded set. Then, the following properties are equivalent:
(1) There exists a function such that .
(2) For every , there exists a function such thatand for every . Corollary 1. The following properties are equivalent:
(1) There exists a function .
(2) For every , there is a function such thatand Corollary 2. Let . Then, the following properties are equivalent:
(1) There exists a function such that .
(2) For every , there is a function such thatand for all . In the sequel, we assume that there exists a function .
In view of (A2),
. Evidently,
is a closed subset of
X. In the literature, the set
is called a global attractor of
a. Note that, in [
1,
2],
is called a turnpike set of
a.
For every point
and every nonempty set
, we define
The following proposition is proved in
Section 3.
Proposition 5. For every function , The following theorem is proved in
Section 4.
Theorem 5. Let be positive real numbers. Then, there is a positive number such that for every function which satisfiesthere is a number for which The following proposition is proved in
Section 5.
Proposition 6. Let . Then, there exists such that and The following theorem, which is proved in
Section 6, is our main result.
Theorem 6. The following properties are equivalent:
(1) If satisfiesthen for all . (2) For each , there exists such that, for each which satisfies and , the inequalityholds for all Properties (1) and (2) usually hold for models of economic dynamics, which are prototypes of our dynamical system [
1,
2,
8,
9]. In particular, it holds for the von Neumann–Gale model generated by a monotone process of convex type which was studied in [
18].
Property (2) is the turnpike property, which is well known in mathematical economics. It was discovered by Samuelson in 1948 (see [
19]) and further analyzed for optimal trajectories of models of economic dynamics. See, for example, [
2,
8,
9] and the references mentioned there. Recently, it was shown that the turnpike phenomenon holds for many important classes of problems arising in various areas of research [
7,
9,
10,
11,
12,
13,
14,
15,
16]. For related infinite horizon problems, see [
9,
20,
21,
22,
23,
24,
25,
26,
27].
Proposition 7. Let us assume that property (1) of Theorem 6 holds and that satisfies . Then, for all .
The following theorem is our second main result, which is also proved in
Section 7. It shows that, if the starting point of the trajectory is closed to the turnpike, then the trajectory leaves the turnpike only when
t is closed to the right endpoint
T of the domain.
Theorem 7. Let us suppose that property (1) of Theorem 6 holds and that are positive real numbers. Then, there exist positive real numbers such that, for every function satisfying andthe inequalityis valid for every Now, we show that, if the set-valued mapping
F has a certain symmetric property, then the turnpike
possesses the corresponding symmetric property. Let us assume that
is a linear invertible mapping such that
is the identity mapping in
. Clearly,
. Let us assume that
and
Let
. Then, for a. e.
,
and
. This implies that
Since
is the identity mapping in
, the inclusion above implies that
Thus, .