Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy

Definition 2.1

(Function class C ) Let us call C ∈ C , if C meets the assumptions in the following:

1. C ≥ 0 is a monotone nondecreasing function and admits at most finite discontinuous jump points on every compact interval of R .

2. C ( 0 ) = C ( 0 + ) = 0 , and there is a nonnegative constant N such that for any x ≠ y , for all γ ∈ K [ C ( x ) ] , η ∈ K [ C ( y ) ] ,

   (2.3) γ − η x − y ≤ N ,

   in which

   K [ C ( x ) ] = [ min { C ( x − ) , C ( x + ) } , max { C ( x − ) , C ( x + ) } ] .

Remark 2.1

Since the controlling function considered in model (1.2) is discontinuous, the usual Lipschitz condition is no longer satisfied, so the unilateral Lipschitz condition (2.3) is imposed to investigate the exponential stability of model (1.2), which has been successfully used to study the neural network and biomathematical models. Moreover, it is readily seen from Assumptions (1)–(2) that

Definition 2.2

(Filippov solution) Function x ( t ) is said to be a solution of model (1.2) on [ − τ + , η ( φ ) ) ( η ( φ ) ∈ ( t 0 , ∞ ] ) if

1. x ( t ) is continuous on [ − τ + , η ( φ ) ) and absolutely continuous on each compact interval of [ t 0 , η ( φ ) ) ;

2. there exists a function γ ( t ) : [ − τ + , η ( φ ) ) → R that is measurable satisfying γ ( t ) ∈ K [ C ( x ( t ) ) ] for a.a. t ∈ [ − τ + , η ( φ ) ) . Moreover, it satisfies

   (2.5) x ′ ( t ) = − α ( t ) x ( t ) + β ( t ) x n ( t − τ ( t ) ) e − δ ( t ) x ( t − τ ( t ) ) − b ( t ) γ ( t ) ,

   for a.a. t ∈ [ t 0 , η ( φ ) ) .

Remark 2.2

As is shown in [19], for all C ∈ C , the following set-valued map

has nonempty compact convex values. Besides, it is upper semicontinuous and hence is measurable. It follows from the measurable selection theorem that there is a bounded measurable function γ ( t ) ∈ K [ C ( x ( t ) ) ] such that (2.5) holds.

Definition 2.3

Assume that f ( t ) : R → R is continuous. f ( t ) is said to be almost periodic on R , if, for arbitrary ε > 0 , the set T ( f , ε ) = { w : ∣ f ( t + w ) − f ( t ) ∣ < ε , t ∈ R } is relatively dense, that is, for arbitrary ε > 0 , one can find a real number l = l ( ε ) > 0 such that, for each interval with length l ( ε ) , there is a number w = w ( ε ) belonging to this interval such that ∣ f ( t + w ) − f ( t ) ∣ < ε for all t ∈ R . The set of all almost periodic functions is represented by AP ( R , R ) .

Definition 2.4

Suppose x ∈ R n and Q ( t ) is a n -th order continuous matrix defined on R . The linear system

is said to admit an exponential dichotomy on R , if there are positive constants k , α , and projection P such that the fundamental solution matrix X ( t ) of (2.7) meets

in which I is the identity matrix.

Lemma 2.1

If system (2.7) has an exponential dichotomy, then the following almost periodic system

admits a unique almost periodic solution x ( t ) , and

Lemma 2.2

Suppose c i is almost periodic, which is defined on R and

Then, the linear system

satisfies exponential dichotomy on R .

Lemma 2.3

[21,22] If

and β > ζ > 0 , then as t ≥ t 0 , the following inequality holds:

where μ is the unique positive root to the following algebra equation:

Lemma 2.4

Let C ∈ C , then the maximal right existence interval of each solution x ( t ; t 0 , φ ) for problem (1.2)–(2.1) is [ t 0 − τ + , ∞ ) and x ( t ; t 0 , φ ) for t ∈ [ t 0 , ∞ ) . Moreover, limsup t → ∞ x ( t ; t 0 , φ ) ≤ n n e − n α − β + ( δ − ) n .

Proof

For convenience, let us, respectively, denote x ( t ; t 0 , φ ) and η ( t 0 , φ ) by x ( t ) and η ( φ ) .

We first prove that

Otherwise, there is t 1 ∈ ( t 0 , η ( φ ) ) satisfying x ( t 1 ) = 0 and x ( t ) > 0 for t ∈ [ t 0 , t 1 ) . It follows from C ∈ C that

This is a contradiction, which means that (2.8) is true. We then prove that η ( φ ) = ∞ . Considering (2.8) and sup x ∈ [ 0 , + ∞ ) x n e − x = n n e − n , we have

which produces

This, combined with (2.8) and the continuation theorem [18, p. 78, Theorem 2], means that η ( φ ) = ∞ .

At last, one finds from η ( φ ) = ∞ and (2.9) that limsup t → ∞ x ( t ) ≤ n n e − n α − β + ( δ − ) n . The proof is completed.□

Lemma 3.1

(Positive invariance) Let 0 < n < 1 . If there is a positive constant M > κ 1 satisfies

and

Then, the set C 0 = { φ ∣ φ ∈ C , κ 1 < φ ( t ) < M , for any t ∈ [ − τ + , 0 ] } is positively invariant.

Proof

Suppose φ ∈ C 0 . According to Lemma 2.3, model (1.2) with initial condition (2.1) admits a unique solution x ( t ; t 0 , φ ) on [ t 0 − τ + , ∞ ) and x ( t ; t 0 , φ ) > 0 for t ∈ [ t 0 − τ + , ∞ ) . For simplicity, we denote x ( t ; t 0 , φ ) by x ( t ) . It is required to testify that

First, we claim that

Argued by contradiction, there is t 1 ∈ ( t 0 , η ( φ ) ) such that

It follows from (2.4), (2.5), (3.1), (3.4), and the fact sup x ∈ R + x n e − x = n n e − n that

which is impossible. This proves (3.3).

Second, we show that

If (3.5) does not hold, there is t 2 ∈ ( t 0 , ∞ ) satisfying

Then,

accordingly,

Combining (2.4), (2.5), (3.1) with (3.7) yields

which is a contradiction. Thus, (3.5) holds. (3.3) together with (3.5) implies that the main results of Lemma 3.1 hold.□

Theorem 3.2

If the assumptions in Lemma 3.1  hold, assume further that

in which L ( u ) = e − u u n − 1 ( n − u ) . Then, model (1.2) has a unique almost periodic solution in

Proof

For each ϕ ∈ AP ( R , R ) , consider the following equation:

where γ ϕ ( t ) ∈ K [ C ( ϕ ( t ) ) ] is measurable. It is easy to see that M [ α ] > 0 , one finds from Lemma 2.2 that the following linear equation

has exponential dichotomy on R . By Lemma 2.1, equation (3.9) just possesses an almost periodic solution, which is represented as

Set

evidently, Ω ⊂ AP ( R , R ) is a closed set.

Define

Apparently, to show that model (1.2) has a unique almost periodic solution, one only needs to prove that Γ admits a fixed point in Ω .

Let us first testify that the operator Γ is a self-mapping from Ω to itself. Actually, for each ϕ ∈ Ω , by (3.1), (3.11), and sup x ∈ R + x n e − x = n n e − n , one has

Meanwhile, in regard to (2.4) and (3.1), one has

which, and (3.12), implies that the mapping Γ is a self-mapping from Ω to Ω .

Next we demonstrate that the mapping Γ is contractive on Ω . For every ϕ 1 , ϕ 2 ∈ Ω , we obtain

that is,

In (3.13), we have used the following fact: Denote L ( u ) = e − u u n − 1 ( n − u ) , it is easy to see that

one has

Since ρ < 1 in (3.14), it means that Γ is contractive. We then conclude from the Banach fixed point theorem that Γ admits a unique fixed point which is exactly the solution to model (1.2) in Ω ∈ AP ( R , R ) . This completes the proof of Theorem 3.2.□

Theorem 3.3

Under the assumptions of Theorem 3.2, then the unique almost periodic solution of model (1.2) is globally exponentially stable.

Proof

Let x ( t ) be any solution of model (1.2) and x * ( t ) be an almost periodic solution of model (1.2), consider the following Lyapunov function:

in which e ( t ) = x ( t ) − x * ( t ) . In view of the monotonicity of controlling function, one obtains