Compact almost automorphic dynamics of non-autonomous differential equations with exponential dichotomy and applications to biological models with delay
Alan Chávez Nelson Aragonés Manuel Pinto Ulices Zavaleta


Compact almost automorphic dynamics of non-autonomous differential equations with exponential dichotomy and applications to biological models with delay]Compact almost automorphic dynamics of non-autonomous differential equations with exponential dichotomy and applications to biological models with delay










[1]Alan Chávezajchavez@unitru.edu.pe

2]Nelson Aragonésnaragones@unitru.edu.pe
These authors contributed equally to this work.

3]Manuel Pintopintoj.uchile@gmail.com
These authors contributed equally to this work.

4]Ulices Zavaletaauzavaleta@unitru.edu.pe
These authors contributed equally to this work.

*[1]OASIS and GRACOCC research groups, Instituto de investigación en Matemáticas, Facultad de Ciencias Físicas y Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II S/N, Trujillo, 13011, La Libertad, Perú

[2,4]OASIS research group, Instituto de investigación en Matemáticas, Facultad de Ciencias Físicas y Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II S/N, Trujillo, 13011, La Libertad, Perú

[3]Departamento de Matem'aticas, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Santiago, 7800003, Región Metropolitana, Chile






In the present work, we prove that, if A(·) is a compact almost automorphic matrix and the system

    x'(t) = A(t)x(t)  ,

possesses an exponential dichotomy with Green function G(·, ·), then its associated system

    y'(t) = B(t)y(t)  ,

where B(·) ∈ H(A) (the hull of A(·)) also possesses an exponential dichotomy. Moreover, the
Green function G(·, ·) is compact Bi-almost automorphic in ℝ^2, this implies that G(·, ·) is  Δ_2 - like uniformly continuous, where Δ_2 is the principal diagonal of ℝ^2, an important ingredient in the proof of invariance of the compact almost automorphic function space under convolution product with kernel G(·, ·). Finally, we study the existence of a positive compact almost automorphic solution of non-autonomous differential equations of biological interest having non-linear harvesting terms and mixed delays.


















[MSC Classification]34A30, 34A34, 34C27.

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§ INTRODUCTION


Let us consider the non-autonomous differential equation

    x'(t) =A(t)x(t) + f(t)  ,

and suppose that the associate system

    x'(t)= A(t)x(t)

possesses an exponential dichotomy, then the unique bounded solution to equation (<ref>) is

    x(t)=∫_ℝG (t,s)f(s) ds  ,

where G(·, ·) is the Green function associated to (<ref>). If  A(·) is an almost
 automorphic matrix and f is an almost automorphic function, then the solution x is almost automorphic if G(·,·) is
 Bi-almost automorphic. To prove that G(·,·) is Bi-almost automorphic is not trivial. For example, in infinite dimension, in order to do that, it is important to add as hypothesis that the resolvent operator also be almost automorphic, moreover it is necessary a delicate analysis of the behaviour of the Yosida approximations of the evolution family A(·), see for instance  <cit.>. In contrasts, in the finite dimensional case, A. Coronel and co-authors have reported in <cit.> the notion of integrable Bi-almost automorphy and proved that G(·,·) is integrable Bi-almost automorphy if a projection matrix P commutes with the fundamental matrix solution of  (<ref>).





The Bi-almost periodicity/automorphy notion of a continuous function of two variables, goes back at least to the work of T. J. Xiao et. al. <cit.>. This notion is very useful in the study of almost periodic/automorphic solutions of non-autonomous differential equations (see the works  <cit.> and references cited therein) and also of integral equations with kernel Γ(t,s) depending on two variables, see for instance <cit.>.


The main purpose of the present work is to prove, in finite dimension, that if A(·) is a compact almost  automorphic matrix, and






the system (<ref>) possesses  an (α, K, P)-exponential dichotomy (Definition <ref>), then the system

    y'(t)= B(t)y(t)  ,

possesses an (α, K, P_0)-exponential dichotomy, where B(·)∈ H(A) and H(A) stands for the hull of A(·) (see comments after Definition <ref>), this is the analogous result to the case of almost periodic systems, see <cit.>. We also prove, without the techniques and the machinary of the infinite dimensional case, that the associated Green function G(·,·) is compact Bi-almost automorphic  (Theorem <ref> in section 3).

Notice that, since G(·, ·) is discontinuous in the principal diagonal of ℝ^2, thus continuous almost everywhere, then it is measurable and therefore G(·, ·) does not fall into the notion of Bi-almost periodicity/automorphy introduced in <cit.> which is given for continuous functions. This motivates the introduction, in this work, of the notion of (compact) Bi-almost automorphy for measurable functions. Also, we are able to prove that, although G(·, ·) is not uniformly continuous (Proposition <ref>), it is Δ_2-like uniformly continuous (Lemma <ref>), where Δ_2 :={ (t,t)∈ℝ^2   :   t ∈ℝ}⊂ℝ^2 denotes the principal diagonal of ℝ^2.

Also we study the existence of a positive compact almost automorphic solution of the following  biological model with non-linear harvesting terms and mixed delays 




    u̇(t)=
    -α(t)u(t)+∑_i=1^nβ_i(t)f_i(λ_i(t)u(t-τ_i(t))+b(t)H(u(t-σ(t)))  .



The present work is organized as follows: in section <ref>, we introduce the notion of compact Bi-almost automorphy for measurable functions and revisit some preliminary facts on compact almost automorphic functions. In section <ref> we prove the main result of the present work, thus Theorem <ref> and Lemma <ref>. In section <ref>, we use the Δ_2-like uniform continuity of G(·, ·) to show invariance of the almost automorphic function space under convolutions whose kernel is the Green function G(·, ·). Finally, in section <ref> we prove the existence of a unique positive compact almost automorphic solution of the delay differential equation (<ref>). 






§ PRELIMINARIES

In the present work, 𝕏 is a real or complex Banach spaces with norm ||·||_𝕏
, ||·|| denotes the norm of matrices,  ||·||_∞ denotes the supremum norm and |·| is the absolute value.

A continuous function f:→, is compact almost
automorphic if, for every sequence s'_n⊂ℝ, there
exists a subsequence s_n⊆s'_n and a function
f̃: → such that the following limits hold


    lim_n→ + ∞sup_t∈𝒞||f(t+s_n)-f̃(t)||_𝕏=0,  lim_n→ +
    ∞sup_t∈𝒞||f̃(t-s_n)-f(t)||_𝕏=0  .

where 𝒞⊂ℝ, is compact.

If in (<ref>) the limits are pointwise in ℝ, f is called almost automorphic.
We denote by 𝒦AA(, ) the apace of compact almost automorphic functions and by AA(,) the space of almost automorphic functions. 

AA(,) and 𝒦AA(,) are Banach spaces under the supremum norm and 𝒦AA(,)⊂ AA(,) holds, see <cit.>.

For an almost automorphic function f, its Hull, denoted by H(f), is the set of all functions f̃ that satisfy Definition <ref>.














The following is a useful characterization of compact almost automorphic functions on the real line. For the multidimensional euclidean space, see <cit.>.

 A continuous function f:ℝ→𝕏 is compact almost automorphic if and only if it is almost automorphic and uniformly continuous.


In the following proposition, we summarize some properties of compact almost automorphic function, for further details see for instance <cit.> and the books <cit.>.


Let f,g ∈𝒦 AA (ℝ; 𝕏), then


  * If 𝕏 be a Banach algebra with norm ||· ||_𝕏,  addition +_𝕏 and multiplication ×_𝕏, then 𝒦 AA(ℝ,𝕏) is also a Banach algebra with norm ||· ||_∞ and  the operations: if f,g ∈𝒦 AA(ℝ,𝕏), then

    (f+ g)(t):=f(t) +_𝕏 g(t)  ,     t ∈ℝ  ,


    (f· g)(t):=f(t) ×_𝕏 g(t)  ,     t ∈ℝ  .



  * f is bounded  and, if f̃ is the limit function in definition <ref>, then

    ||f||_∞=||f̃||_∞  .



  * If F: 𝕏→𝕐 be a continuous function, then F∘ f:ℝ→𝕐 is compact almost automorphic.


  * The range of f is relatively compact.


  * If a∈𝒦 AA (ℝ; ℝ), then F(t)=f(t-a(t)) is compact almost automorphic from ℝ to 𝕏.








Next, we provide the definition of Bi-almost automorphy for measurable functions.


Let f:×→  be a measurable function, f is compact Bi-almost automorphic if, for every sequence s'_n_n∈⊂ there exist a subsequence s_n⊆s'_n  and a function f̃: ×→  such that, if 𝕂⊂× is compact, the following limits hold

    lim_n→ + ∞sup_(t,s)∈𝕂||f(t+s_n, s+s_n)- f̃(t,s)||_𝕏=0  , 
       lim_n→ + ∞sup_(t,s)∈𝕂||f̃(t-s_n, s-s_n)-f(t,s)||_𝕏=0  .

When the limits are pointwise in ×, f is called Bi-almost automorphic.



Let Φ(· ) be a fundamental matrix of system  (<ref>), then (<ref>) has an exponential dichotomy with parameters (α, K, P), if there exist positive constants α, K and
a projection P (P^2=P) such that ||G(t,s)||≤
Ke^-αt-s,  t,s∈. Where G(· , ·) is the Green function defined by:

    G(t,s):=
    Φ(t)PΦ^-1(s),     t≥ s , 
    
    -Φ(t)(I-P)Φ^-1(s),     t < s  .

In this case, we say that system (<ref>) has an (α, K, P)-exponential dichotomy.



<cit.>
We say that a measurable function C:×→ is λ-bounded if,  there exists a positive function λ:×→ such that, for every τ∈ we have

    ||C(t+τ,s+τ )||_𝕏≤λ(t,s)  .






If G(·,·) is the Green function defined in Definition <ref>; then, G(·,·) is measurable and λ-bounded, for λ(t,s)=Ke^-α|t-s| .




Let us suppose that the Bi-almost automorphic function C:×→𝕏  is λ-bounded.

Then, its limit function C̃:×→𝕏 (see definition <ref>) satisfies:

    ||C̃(t+τ,s+τ )||_𝕏≤λ(t,s)   ,   ∀τ∈ℝ  .




Let {s_n'}⊂ℝ be arbitrary. Since C is Bi-almost automorphic, there exist a subsequence {s_n}⊂{s_n'} and a function C̃ such that the following pointwise limits hold







    C̃(t,s ):=lim_n→ +∞C(t+s_n,s+s_n )   ,     C(t,s)=lim_n→ +∞C̃(t-s_n,s-s_n )  .
 



On the other hand, we have

    ||C̃(t+τ,s+τ )||_𝕏≤ ||C̃(t+τ,s+τ )-C(t+τ+s_n,s+τ+s_n )||_𝕏+ λ(t,s) .








Now, taking the limit as n→ +∞ in the last inequality 
we obtain the result.


This Lemma improves <cit.>, where it was proved that C̃(·, ·) satisfies the inequality ||C̃(t ,s  )||_𝕏≤λ(t,s)   ,     (t,s)∈ℝ×ℝ. This Lemma will be invoked in section <ref>.  Further details on almost automorphic functions can be found in <cit.> and on exponential dichotomy in <cit.>.

























§ BI-COMPACT ALMOST AUTOMORPHY OF THE GREEN FUNCTION G(· , ·)






Our main result in this section is the following Theorem:


 Suppose that system (<ref>) has an (α, K, P)-exponential  dichotomy with fundamental matrix Φ(t),
Φ(0)=I and A(·) is compact almost automorphic, that is:
Given and arbitrary sequence s'_n⊂ there exists a
subsequence s_n⊂s'_n such that the following
limits hold uniformly on compacts sets of the real line

    lim_n→ +∞ A(t+s_n) = :B(t),  lim_n→
    +∞B(t-s_n)= A(t)  .

Then, there exists a projection matrix P_0, such that the system

    y'(t)=B(t)y(t)  ,

has an  (α, K, P_0)-exponential  dichotomy. Furthermore, the associated Green function of system (<ref>) is compact Bi-almost automorphic.



Before to prove Theorem <ref>, we stablish the following Lemma:

 Under hypothesis of Theorem <ref>, there exists a subsequence {ζ_n}⊂{s_n} such that, the following limits hold

    lim_n→ + ∞sup_t∈𝒞||Φ_n(t)-Ψ(t)||=0,  lim_n→ +
    ∞sup_t∈𝒞||Φ_n^-1(t)-Ψ^-1(t)||=0  ,






where 𝒞⊂ℝ is compact, Φ_n(t)=Φ(t+ζ_n)Φ^-1(ζ_n) and Ψ is a fundamental matrix of (<ref>).



(Proof of Lemma <ref>)

It is sufficient to consider the compact interval 𝒞=[-a, a], with a>0.

From hypothesis, there exists a subsequence {s_n}⊂{s_n'} such that the limits in (<ref>) hold. For this subsequence, let us consider the sequence Φ_n^*(t)=Φ(t+s_n)Φ^-1(s_n), n∈ℕ; and notice that for each n ∈, Φ_n^*(0)=I and Φ_n^*(·) is a fundamental matrix of the system:

    x'(t)=A(t+s_n)x(t)  .

Then,  by direct integration it follows that

    Φ_n^*(t)= I+∫_0^tA(u+s_n)Φ_n^*(u)du.










Let M>0 be the supremum of A(·); then, by the Gronwall-Bellman Lemma we have

    Φ_n^*(t)≤Ie^Mt≤Ie^Ma  .

That is, Φ_n^* is uniformly bounded on 𝒞. Now, since

    ( Φ_n^*(t))' = A(t+s_n)Φ_n^*(t),  for n∈  and t∈𝒞  ,

then the sequence of derivatives ( Φ_n^*(·))'  is uniformly bounded over 𝒞.
It follows from Arzela-Ascoli Theorem that
Φ_n^* has a subsequence Φ_n (i.e. there
exists an associated subsequence  ζ_n⊆s_n) such that Φ_n(t)=Φ(t+ζ_n)Φ^-1(ζ_n) and is uniformly convergent on 𝒞.
That is, there exists Ψ such that

    lim_n→ +∞Φ_n=Ψ, uniformly on 𝒞.

On the other hand, since the subsequence {Φ_n} satisfies (<ref>), then

    Φ'_n(t)=A(t+s_n) Φ_n(t),   n∈,  t∈𝒞 .

Therefore, Φ'_n converges uniformly over 𝒞 to B(t)Ψ(t). Consequently, Ψ(·) is differentiable and moreover

    lim_n→ +∞Φ'_n(t)=Ψ'(t) over 𝒞 .

Since Ψ'(t)=B(t) Ψ (t) and Ψ(0)=I, then Ψ  is a fundamental matrix of (<ref>). That is, Ψ (·) is non singular, and hence  Ψ^-1 exists  on  𝒞.

Now we prove that the following limit holds uniformly on 𝒞

    lim_n→ +∞Φ^-1_n(t)=Ψ^-1(t)  .

Firstly, notice that Φ^-1_n
satisfies the equation

    x'(t)= -x(t)A(t+ζ_n)  ;

and, by direct integration we have

    Φ^-1_n(t)= I - ∫_0^tΦ^-1_n(u)A(u+ζ_n)du .

Similarly,

    Ψ^-1(t)= I - ∫_0^tΨ^-1(u)B(u)du  .

Then,

    Φ_n^-1(t)-Ψ^-1(t)   ≤   ∫_0^t Ψ^-1(u)B(u)
    -Φ^-1_n(u)A(u+ζ_n)du 
       ≤   ∫_0^t Ψ^-1(u) -Φ^-1_n(u)duA_∞ + 
        +    ∫_0^tA(u+ζ_n)-B(u)duΨ^-1_∞ .

Now, since the limits in (<ref>) are uniform in 𝒞; given ϵ >0, there exists N_0∈ such that for all n≥ N_0 :

    sup_t∈𝒞A(t+ζ_n)-B(t)<ϵ .

Therefore, from inequality (<ref>) we have, for n≥ N_0

    Φ_n^-1(t)-Ψ^-1(t)≤ϵ
    aΨ^-1_∞+ ∫_0^t Ψ^-1(u)
    -Φ^-1_n(u)du A_∞  ;

and, from the Gronwall-Bellman Lemma again, we conclude

    Φ_n^-1(t)-Ψ^-1(t)≤ϵ
    aΨ^-1_∞ e^A_∞ a  .


Next, we prove the main result of this section.

(Proof of Theorem <ref>)

Let  Φ be a fundamental matrix of
(<ref>) and

    G(t,s):=
    Φ(t)PΦ^-1(s),     t≥ s ,
    
    -Φ(t)(I-P)Φ^-1(s),     t < s .

its associated Green function such that ||G(t,s)||≤ Ke^-αt-s,  t,s∈.

The proof will be divided into three steps.

Step 1. The system (<ref>) has an (α, K, P_0)-exponential dichotomy.

Because of Lemma <ref>, there exists a subsequence {ζ_n}⊂{s_n} such that the following limits are uniform on compact subsets of ℝ:

    lim_n→ +∞Φ_n(t)=Ψ(t)     and    lim_n→ +∞Φ_n^-1(t)=Ψ^-1(t)  ,

where Φ_n(t)=Φ(t+ζ_n)Φ^-1(ζ_n) and Ψ is a fundamental matrix of (<ref>).









Note that Φ(ζ_n)PΦ^-1(ζ_n)≤
K and Φ(ζ_n)QΦ^-1(ζ_n)≤ K, where Q=I-P. Thus, the sequences of matrices {Φ(ζ_n)PΦ^-1(ζ_n) }  and {Φ(ζ_n)QΦ^-1(ζ_n) } are bounded; therefore,





there exists a subsequence
η_n⊂ζ_n such that:

    lim_n→ +∞Φ(η_n)PΦ^-1(η_n)=: P_0  ,
       lim_n→ +∞Φ(η_n)QΦ^-1(η_n)=: Q_0  .

Notice that  P_0^2=P_0 and P_0+Q_0=I. Now, taking n→ +∞  in the following inequalities:

    ( Φ(t+η_n)Φ^-1(η_n) ) Φ(η_n)PΦ^-1(η_n) ( Φ(s+η_n)Φ^-1(η_n) )^-1≤
    Ke^-α(t-s),   t≥ s
       ( Φ(t+η_n)Φ^-1(η_n) ) Φ(η_n)QΦ^-1(η_n) ( Φ(s+η_n)Φ^-1(η_n) )^-1≤
    Ke^-α(s-t),   t < s  ,

we conclude that

    Ψ(t)P_0Ψ^-1(s)≤ Ke^-α(t-s),  t≥ s  ,
       Ψ(t)Q_0Ψ^-1(s)≤ Ke^-α(s-t),  t < s  .

That is, there exists a projection matrix P_0, such that the homogeneous system (<ref>) has an (α, K, P_0)-exponential dichotomy with Green function:

    G̃(t,s):=
    Ψ(t)P_0Ψ^-1(s),     t≥ s  ,
    
    -Ψ(t)Q_0Ψ^-1(s),     t < s  .


Step 2. The Green function G(·, ·)   defined in (<ref>) is Bi-almost automorphic.

Let { s_n'} be an arbitrary sequence. Then, from hypothesis and Step 1, there exists a subsequence {η_n}⊂{s_n'} such that the following pointwise limit holds

    lim_n→ +∞G(t+η_n, s+η_n)=G̃(t,s)  ;

where,

    G(t+η_n,s+η_n):=
    Φ_n(t)P_nΦ_n^-1(s),     t≥ s
    
    -Φ_n(t)Q_nΦ_n^-1(s),     t < s  ,

Φ_n(t)=Φ(t+η_n)Φ^-1(η_n), P_n:=Φ(η_n)PΦ^-1(η_n),   Q_n:=Φ(η_n)QΦ^-1(η_n)
 and G̃(·, ·) was defined in (<ref>).












We claim that, there exists a subsequence ξ_n⊆η_n such
that the following limit is attained

    lim_n→ +∞G̃(t-ξ_n, s-ξ_n)=G(t,s)  .

In fact, since Ψ( · ), with Ψ(0)=I,  is a fundamental matrix of (<ref>), then for each n∈ℕ, Ψ_n(t):=Ψ(t-η_n)Ψ^-1(-η_n)  is a fundamental matrix of the system

    z'(t)=B(t-η_n)z  .

Arguing as in the Step 1, we can find a subsequence {ξ_n}⊂{η_n } such that











    lim_n→ +∞Ψ(-ξ_n)P_0Ψ^-1(-ξ_n)=:P̃_1,
       lim_n→ +∞Ψ(-ξ_n)Q_0Ψ^-1(-ξ_n)=: Q̃_1  .

Notice that P̃_1^2= P̃_1,  P̃_1+
Q̃_1=I; and, as in the proof of Lemma <ref>, we have that the limits lim_n→ +∞B(t-ξ_n)= A(t) and  lim_n→
+∞Ψ_n(t)= Υ(t) are uniform on compact subsets of , where  Υ(t)  is a
fundamental matrix of system (<ref>), note that Υ(0)=I. Then,
there exists a non singular matrix C such that   Υ(t)=Φ(t)C, 
and because Υ(0)=Φ(0)=I, we have   C=I,  therefore  
Υ(t)=Φ(t). On the other hand, we know that the projection for an exponential dichotomy is unique, then  P̃_1=P and
Q̃_1=Q. All this reasoning implies that

    lim_n→ + ∞G̃(t-ξ_n,s-ξ_n)=G(t,s)  ,

for each point (t,s)∈ℝ^2, as we claimed.












Step 3: The Green function G(·, ·)   defined in (<ref>) is compact Bi-almost automorphic.

From  Lemma <ref>, there exists a subsequence {η_n}⊂{s_n} such that if 𝒞=[a,b] is a compact subset  of ℝ, then the following limits hold:

    lim_n→ +∞sup_t∈𝒞Φ_n(t) -Ψ(t)=0     and    lim_n→ +∞sup_t∈𝒞Φ_n^-1(t)-Ψ^-1(t)=0  ,

where, Φ_n(t)=Φ(t+η_n)Φ^-1(η_n) and Ψ is a fundamental matrix of the system in (<ref>). Also, there exist positive constants C_1, C_2, C_3, such that for every n∈ℕ and  t∈𝒞, we have:

    Φ_n(t)≤ C_1, Φ_n^-1(t)≤ C_2,  Ψ(t)≤ C_3  .

Furthermore,

    P_n=Φ(η_n)PΦ^-1(η_n)→ P_0  ,  as  n→ +∞  ,
       Q_n=Φ(η_n)QΦ^-1(η_n)→ Q_0  ,  as  n→ +∞  .

We claim that, if 𝕂=[a,b]× [a,b]⊂ℝ^2, then

    lim_n→ +∞sup_(t,s)∈𝕂||G(t+η_n,s+η_n)-G̃(t,s)||=0  ,

where, G(t+η_n,s+η_n) was defined in (<ref>). In fact,

1) If t≥ s, then












    G(t+s_n,s+s_n)-G̃(t,s)=Φ_n(t)P_nΦ_n^-1(s)-Ψ(t)P_0Ψ^-1(s)=

    
    =Φ_n(t)(P_n-P_0+P_0)Φ_n^-1(s)-Ψ(t)P_0Ψ^-1(s)=

    
    =Φ_n(t)(P_n-P_0)Φ_n^-1(s)+Φ_n(t)P_0Φ_n^-1(s)-Ψ(t)P_0Ψ^-1(s)=

    
    =Φ_n(t)(P_n-P_0)Φ_n^-1(s)+(Φ_n(t)-Ψ(t)+Ψ(t))P_0Φ_n^-1(s)-Ψ(t)P_0Ψ^-1(s)=

    
    =Φ_n(t)(P_n-P_0)Φ_n^-1(s)+(Φ_n(t)-Ψ(t))P_0Φ_n^-1(s)+Ψ(t)P_0Φ_n^-1(s)-Ψ(t)P_0Ψ^-1(s)=

    
    =Φ_n(t)(P_n-P_0)Φ_n^-1(s)+(Φ_n(t)-Ψ(t))P_0Φ_n^-1(s)+Ψ(t)P_0(Φ_n^-1(s)-Ψ^-1(s))≤

    
    ≤Φ_n(t) P_n-P_0 Φ_n^-1(s)+Φ_n(t)-Ψ(t) P_0 Φ_n^-1(s)+

    
    +Ψ(t) P_0 Φ_n^-1(s)-Ψ^-1(s)<

    
    <A'P_n-P_0+B'Φ_n(t)-Ψ(t)+C'Φ_n^-1(s)-Ψ^-1(s)  ,

for some positive constants A^',B^', C^'. Thus,

    G(t+s_n,s+s_n)-G̃(t,s)    <     A'P_n-P_0+B'sup_t∈ [a,b]Φ_n(t)-Ψ(t)+
       +     C'sup_s∈ [a,b]Φ_n^-1(s)-Ψ^-1(s)  .


2) If t<s, then (as in the previous computation) we have

    G(t+s_n,s+s_n)-G̃(t,s)    =    Φ_n(t)Q_nΦ_n^-1(s)-Ψ(t)Q_0Ψ^-1(s)<
       <   A^'Q_n-Q_0 +  B^'sup_t∈ [a,b]Φ_n(t)-Ψ(t)+ 
       +    C^'sup_s∈ [a,b]Φ_n^-1(s)-Ψ^-1(s)  ,



Therefore, from 1), 2), (<ref>), (<ref>) and (<ref>) we conclude (<ref>).

Analogously, a subsequence {ξ_n}⊂{η_n} can be obtained  such that, on compact sets  𝕂⊂ℝ^2, the following limit holds

    n→ +∞limsup_(t,s)∈𝕂||G̃(t-ξ_n,s-ξ_n)-G(t,s)||=0  .






































































 In the work <cit.>, the authors assume that the fundamental matrix solution Φ of the system 

    x'(t)=A(t)x(t)  ,

commutes with the projection P in order to prove the integral Bi-almost automorphy of the Green function G(·, ·); in our result, namely Theorem <ref>, this hypothesis is not necessary.


 From Step 1 and Step 2 of the previous proof, we conclude that Theorem <ref> can be stated in the almost automorphic category; thus, for systems

    x'(t)=A(t)x(t)  ,

in which A(·) is an almost automorphic matrix.



Although G(·, ·) is measurable,  discontinuous  and compact Bi-almost automorphic, it does not contradicts the result of Veech <cit.>, where he proved that on locally compact groups, Haar measurable almost automorphic functions are continuous. To the contrary, this put in light a crucial difference between fully almost automorhic functions, where we take sequences in the full euclidean group ℝ^2 (in this case), and the measurable (compact) Bi-almost automorphic ones, in which the sequences belongs to the proper subgroup Δ_2 ⊂ℝ^2, which is the principal diagonal of ℝ^2.


The following Lemma is crucial in the proof of invariance of the compact almost automorphic function space under convolution products whose kernel is the Green function G(·, ·) (Theorem <ref>).

 (Δ_2-like uniform continuity of the Green function G(· , ·)) Let {t_n} and {s_n} be real sequences such that |t_n-s_n|→ 0, as n→ +∞; then, for each (t,s) ∈ℝ^2 we have  G(t+t_n,s+t_n)-G(t+s_n,s+s_n)→ 0, as n→ +∞.


Let us denote by 𝒰:={(t,s) ∈ℝ^2  :  t < s} and by 𝒱:={(t,s) ∈ℝ^2  :  t ≥ s}, also define G^2(t,s):=-Φ(t)QΦ^-1(s) and G^1(t,s):=Φ(t)PΦ^-1(s). Since 𝒰, 𝒱 constitute a partition of ℝ^2, then any point (t,s) is either in 𝒰 or in 𝒱. Let us suppose that (t,s)∈𝒰 and take the sequence

    α_n(t,s)=G^2(t+t_n,s+t_n)-G^2(t+s_n,s+s_n)  .

We claim that α_n(t,s) → 0, when n→ +∞. In fact, let {α_n'(t,s)}⊂{α_n(t,s)} be a convergent subsequence which converges to ψ_0(t,s), where

    α_n'(t,s)=G^2(t+t_n',s+t_n')-G^2(t+s_n',s+s_n')  ,

where {t_n'}⊂{t_n} and {s_n'}⊂{s_n}. We must prove that ψ_0(t,s)=0.

Since G^2(· , · ) is compact Bi-almost automorphic on 𝒰, there exists a subsequence {s_n”}⊂{s_n'} and a function G̃^2 such that the following limits holds:

    lim_n→ +∞sup_(t,s) ∈𝕂 G^2(t+s_n”,s+s_n”)- G̃^2(t,s)    =   0  , 
    lim_n→ +∞sup_(t,s) ∈𝕂G̃^2(t-s_n”,s-s_n”)- G^2(t,s)    =   0  ;

where, 𝕂 is a compact subset of 𝒰. As a byproduct, the limit function G̃^2 is continuous on 𝒰.

Since {|t_n-s_n|} is bounded, there exists R>0 such that t_n-s_n ∈ [-R,R] (a compact interval in ℝ). Therefore the points (t_n-s_n,t_n-s_n)∈ L_R, where L_R:={(z,z)∈ℝ^2  :  |z|≤√(2)R} a compact subset of (the diagonal of) ℝ^2, which implies that L_R(t,s):=(t,s)+L_R is a compact subset of 𝒰. Now, from the inequalities:

    α_n”(t,s)     =      G^2(t+t_n”,s+t_n”)-G^2(t+s_n”,s+s_n”)
       ≤    G^2(t+t_n”-s_n”+s_n”,s+t_n”-s_n”+s_n”)-G̃^2(t+t_n”- s_n”,s+t_n”- s_n”)  + 
       +   G̃^2(t+t_n”- s_n”,s+t_n”- s_n”)-G̃^2(t,s)+G̃^2(t,s)-G^2(t+s_n”,s+s_n”) 
       ≤   sup_(z_1,z_2)∈ L_R(t,s) G^2(z_1+s_n”,z_2+s_n”)-G̃^2(z_1,z_2) +
       +   G̃^2(t+t_n”- s_n”,s+t_n”- s_n”)-G̃^2(t,s)+G̃^2(t,s)-G^2(t+s_n”,s+s_n”)   ,

from (<ref>) and the continuity of G̃^2 we conclude that α_n”(t,s) → 0, as n→ +∞. This implies that ψ_0(t,s)=0; therefore, α_n(t,s) → 0 when n→ +∞, as claimed.

Analogously, it can be shown that if (t,s)∈𝒱, then

    lim_n→ +∞ G^1(t+t_n,s+t_n)-G^1(t+s_n,s+s_n) =0  .


In the proof of this Lemma, only the fact that each branch of the Green function, namely the functions  G^1 and G^2, being compact Bi-almost automorphic in their respective domains, was utilized, without requiring the full compact Bi-almost automorphy of G(· , ·).

That is, the previous Lemma is true only assuming that G(· , ·) is compact Bi-almost automorphic relative to the following collection

    Λ:={𝕂⊂ℝ^2  :  𝕂 is   compact  and 𝕂∩Δ_2 = ∅ } .

 Compact Bi-almost automorphic relative to Λ, means that in Definition <ref>  compact sets are taken in Λ.










§ COMPACT ALMOST AUTOMORPHIC SOLUTIONS OF NON-AUTONOMOUS DIFFERENTIAL EQUATIONS




 §.§ invariance of 𝒦AA under convolution product with kernel G(· , ·)



Let A(·) be a compact almost automorphic matrix and suppose that system (<ref>) has an (α, K, P)-exponential dichotomy with Green function G(·,·). Then the operator 𝒢_1 defined by 

    𝒢_1 u (t):=∫_G(t,s)u(s)ds  ,





leaves invariant the space 𝒦AA(ℝ; ℂ^p).












Let  u∈𝒦AA(ℝ; ℂ^p), by Proposition <ref> it is sufficient to prove that 𝒢 u is almost automorphic and uniformly continuous. We will accomplish this in two steps.

Step 1: 𝒢 u is almost automorphic. Let {s'_n} be any real sequence. Since G(·,·) is  compact  Bi-almost automorphic and u is compact almost automorphic, there exists a subsequence  {s_n}⊆{s'_n} such that on compact subsets of
ℝ×ℝ the following uniform limits holds:

    lim_n→ +∞ G(t+s_n,s+s_n)=:G̃(t,s),  lim_n→ +∞G̃(t-s_n,s-s_n)=G(t,s);

and on compact subsets of ℝ the following uniform limits holds

    lim_n→ +∞ u(t+s_n)=:ũ(t),  lim_n→ +∞ũ(t-s_n)=u(t).


In particular the previous limits are pointwise. Let v(t)=𝒢 u(t); then, since G(·,·) is λ-bounded, for λ(t,s)=Ke^-α|t-s|  , t,s ∈ℝ, by Lemma <ref> and using the Lebesgue's Dominate Convergence Theorem, we have

    lim_n→ +∞ v(t+s_n)=ṽ(t),  lim_n→ +∞ṽ(t-s_n)=v(t)  ,

where

    ṽ(t):=∫_-∞^+∞G̃(t,s)ũ(s)ds.

What we have proved is that v=𝒢 u is almost automorphic.

Step 2: 𝒢 u is uniformly continuous. Let {t_n},{s_n } be two sequences in ℝ such that |t_n-s_n| → 0 when n→ +∞, then

    |𝒢 u(t_n)-𝒢 u(s_n)|   ≤   ∫_ℝG(t_n, s+t_n)-G(s_n,s+s_n)|u(s+t_n)|ds 
       +   ∫_ℝG(s_n,s+s_n) |u(s+t_n)-u(s+s_n)|ds
       :=   I_1(n)+I_2(n)  .

Now, by Lemma <ref>, the Δ_2-like uniform continuity of G(·,·), the uniform continuity of u and the Lebesgue's Dominated Convergence Theorem, we conclude

    lim_n→ +∞I_1(n)=0   = lim_n→ +∞I_2(n)   .



The proof of the following Theorem, is analogous to the one of Theorem <ref>.


Let A(·) be a compact almost automorphic matrix and suppose that system (<ref>) has an (α, K, P)-exponential dichotomy with Green function G(·,·). Then the operator 𝒢_2 defined by 

    𝒢_2 u (t):=∫_-∞^t G(t,s)u(s)ds  ,

leaves invariant the space 𝒦AA(ℝ; ℂ^p).





 In contrast with the almost automorphic scenario, it is not enough that G(·,·) be Bi-almost automorphy and λ-bounded, for λ(t,s)=Ke^-α|t-s| (for instance), to show the invariance of the space 𝒦AA(ℝ; ℂ^p) under the convolution products in Theorems <ref> and <ref>; it seems very important also the property of being Δ_2-like uniformly continuous, which is a property that meets the Green function G(·,·). The Δ_2-like uniform  continuity of G(·,·) is a hypothesis that has not been considered in <cit.>. Note that, the aforementioned hypothesis cannot be concluded from the Bi-almost automorphy and λ-boundedness of G(·,·). 




The proof of the following Proposition is immediate

If a(·) is compact almost automorphic, then 

    G_1(t,s):=∫_s^t a(ξ)dξ  ,    and    G_2(t,s):=exp( ∫_s^t a(ξ)dξ)  ,

are Bi-almost automorphics and Δ_2-like uniformly continuous.








 §.§ compact almost automorphic solutions

Finally, the following theorem arrives


If the linear system (<ref>) has an (α, K, P)-exponential
dichotomy with Green function G(· , ·), then the system 

    x'(t)=A(t)x(t)+f(t)

where A(·) is a compact almost automorphic matrix and f ∈𝒦AA(ℝ; ℂ^p), has a unique compact almost automorphic solution x, given by

    x(t)=∫_G(t,s)f(s)ds  .

Moreover, if Φ(·) is the fundamental solution of system (<ref>) and P the projection, then, the solution x is given by

    x(t)=∫_-∞^t Φ(t)PΦ^-1(s)f(s)ds+∫_t^∞Φ(t)(I-P)Φ^-1(s)f(s)ds  .


Obviously, the previous Proposition can be stated for almost automorphic systems, instead of purely compact almost automorphic ones.


















§ THE EXISTENCE OF A POSITIVE COMPACT ALMOST AUTOMORPHIC SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS OF BIOLOGICAL INTEREST.


In this section, in the spirit of the papers <cit.>, we study the existence of a  positive compact almost automorphic solution of the following biological model with non-linear harvesting terms and mixed delays 




    u̇(t)=
    -α(t)u(t)+∑_i=1^nβ_i(t)f_i(λ_i(t)u(t-τ_i(t))+b(t)H(u(t-σ(t)))  .

This model codify, at least, the cases of:


  * Nicholson: f_i(z)=ze^-z  .

  * Lasota-Wazewska: f_i(z)=e^-z  .

  * Mackey-Glass: f_i,m(z)=z1+z^m  .

Note that, in equation (<ref>), we are considering a harvesting term with variable delay, this case was not considered in the work <cit.>, but it has been considered in the work <cit.>.

The following are the necessary hypothesis on the coefficients of
(<ref>) to obtain our results:


  A1 The functions α(t), b(t),σ(t), β_i(t),
λ_i(t), τ_i(t) for i =1, 2, ⋯, n are positive compact almost automorphic functions on ℝ.

  A2 The functions α(·), β_i(·), λ_i(·) are bounded away from zero.

  A3 The function H:^+_0 ↦^+_0 is bounded and  Lipschitz continuous, i.e. there exists a positive constant L_H
such that:

    |H(u)-H(v)| < L_H|u-v|  for u,v ∈^+_0



If f is a real bounded continuous function, defined in its domain D_f⊂ℝ, then f and f are defined by

    f:=sup_t∈ D_f{ f(t)}  ,   f:=inf_t∈ D_f{ f(t)}  .

Also define the positive constants τ and λ as follows 

    τ= max{max_1≤ i ≤ nτ_i, σ}  ,  λ= max_1≤ 1≤
    nλ_i  .
 
Denote by C([-τ,0], ℝ) the Banach space of continuous functions from [-τ,0] to ℝ under the norm ||ϕ||_τ=sup_θ∈ [-τ,0]|ϕ(θ)|. Let 𝒞^+ be the cone of non-negative functions in C([-τ,0], ℝ), that is

    𝒞^+={ϕ∈  C([-τ,0], ℝ)  :  ϕ(t)≥ 0}  ,

also define the set

    𝒞_0^+={ϕ∈𝒞^+   :  ϕ(0)> 0 }  .

Due to biological interpretations, the set of admissible
initial conditions for equation (<ref>) is considered in 𝒞_0^+.

If x(·) is defined in the interval [t_0-τ,σ] with t_0, σ∈ℝ, then the function x_t ∈ C([-τ,0], ℝ) is defined by x_t(θ):=x(t+θ), for all θ∈ [-τ, 0]. Therefore, the initial condition for equation (<ref>) is 

    x_t_0=ϕ  ,   ϕ∈𝒞_0^+  .




A solution of the IVP (<ref>) - (<ref>) is denoted by x(t;t_0,ϕ), but for convenience, we will denote it by x(t). We assume the existence of a unique solution for equation (<ref>). 
The proof of the following Proposition follows by the comparison principle

 If conditions A1-A3 holds, then any solution x of equation (<ref>) with initial condition ϕ∈𝒞_0^+, satisfies

    0≤∑_i=1^nβ_i f_i +b Hα≤lim inf_t→ +∞x(t) ≤lim sup_t→ +∞x(t) ≤∑_i=1^nβ_i f_i +b Hα  .


In particular, Proposition <ref> means that all solutions of equation (<ref>) are biological meaningful.

We state the following additional assumptions:


  A4 For all 1≤ i ≤ n, the functions f_i:^+_0 ^+_0 are Lipschitz continuous, non-negative and they yield its maximum
value over ^+_0, i.e. f_i=f_i(m_i^*) with
m^*_i∈^+_0 and f_i is non-increasing function for x > m^*_i.

  A5 There exist two positive constants γ_1 and γ_2 such that:

    0≤m^*/λ<γ_1<1/α∑_i=1^nβ_i  
    f_i(λγ_2)+b H  ,
       1/α∑_i=1^n
    β_i f_i+ b H <
    γ_2  ;

where, m^*= max_1≤ 1≤
nm^*_i.

  A6 For all 1≤ i ≤ n, there exist positive numbers ℓ_f_i which are the Lipschitz constants of f_i on [m^*, +∞).



Let  Ω_0:={ϕ ∈𝒞^+  :  γ_1 <ϕ(t)< γ_2  ,   t ∈ [-τ,0] }. If conditions A1-A5 are fulfilled; then, for every ϕ∈Ω_0, the solution x(t) of equation (<ref>) satisfies

    γ_1 < x(t) < γ_2  ,   t ∈ [t_0, ζ(ϕ))

and its existence interval can be extended to [t_0, +∞).


Firstly, let us proof that x(t) < γ_2  ,   t ∈ [t_0, ζ(ϕ)). If it were not true, then there must be t_1 ∈ [t_0, ζ(ϕ)) such that

    x(t_1)=γ_2,  and   x(t)<γ_2  ,   ∀ t∈ [t_0-τ,t_1)  ;

then, using A5, we have the following contradiction

    0    ≤    x'(t_1)=-α(t_1)u(t_1)+∑_i=1^nβ_i(t_1)f_i(λ_i(t_1)u(t_1-τ_i(t_1)))+b(t_1)H(u(t_1-σ(t_1))
       <   -αγ_2 +∑_i=1^n
    β_i f_i+ b H
       <    0  .

Similarly, suppose that the inequality γ_1 < x(t) ,   t ∈ [t_0, ζ(ϕ)) does not holds; then,  there exists t_2 ∈ [t_0, ζ(ϕ)) such that

    x(t_2)=γ_1,  and   x(t)>γ_1  ,   ∀ t∈ [t_0-τ,t_2)  ;

then, using A5, we have the following contradiction

    0    ≥    x'(t_2)=-α(t_2)u(t_2)+∑_i=1^nβ_i(t_2)f_i(λ_i(t_2)u(t_2-τ_i(t_2)))+b(t_2)H(u(t_2-σ(t_2))
       >     -αγ_1 +∑_i=1^nβ_i  
    f_i(λγ_2)+b H
       >    0  .

The fact that ζ(ϕ)=+∞, is a consequence of Theorem 3.2 in <cit.>.





 Assume A1-A6 hold. If



    bL_H+∑_i=1^nβ_i λ_i ℓ_f_i<α

holds, then equation (<ref>) has a unique compact almost automorphic solution in the set

    Ω =u∈ KAA(, ^+)  :  γ_1≤ u(t)≤γ_2,    ∀  
    t∈  .



Let us define the operator

    (𝔐u)(t):=∫_-∞^te^-∫_s^tα(ξ)dξ(Nu)(s)ds  ,

where, 
N:C() C() is given by

    N(u)(s)= ∑_i=1^nβ_i(s)f_i(λ_i(s)u(s-τ_i(s)))+
    b(s)H(u(s-σ(s)))  .


Now we prove that Ω  is closed, 𝔐( Ω) ⊂Ω and 𝔐 is a contraction; after that, the conclusion will follow from the Banach fixed point Theorem.



  * Ω is closed. In fact, let the sequence {u_n}⊂Ω such that u_n → u uniformly on ℝ. Then, given ϵ >0, there exist N_0 ∈ℕ such that

    ||u_n-u||_∞<ϵ  ,   ∀ n≥ N_0  .

Therefore, for all t ∈ℝ and ∀ n≥ N_0, we have

    |u(t)| ≤ ||u_n-u||_∞+|u_n(t)| < ϵ+γ_2  ,


    γ_1 -ϵ <u_n(t) -ϵ < u(t)  ;

which means that u ∈Ω.


  * 𝔐( Ω) ⊂Ω. If u is compact almost automorphic, then from the properties in Proposition <ref>, we have that Nu is compact almost automorphic and from Proposition <ref> and Theorem <ref>, we have that 𝔐(u) is compact almost automorphic too.  Moreover, if u ∈Ω, we have:

    ∑_i=1^nβ_i  
    f_i(λγ_2)+b H≤(  Nu ) (s) ≤∑_i=1^n
    β_if_i+ b H  ;

therefore,

    γ_1< 1α∑_i=1^nβ_i  
    f_i(λγ_2)+b H≤∫_-∞^te^-∫_s^tα(ξ)dξ(  Nu )(s)ds ≤1α∑_i=1^n
    β_if_i+ b H <γ_2  .



  * 𝔐  is a contraction. This follows from the inequality

    ||𝔐(u)-𝔐(v)||_∞≤κ ||u-v||_∞  ,  ∀ u,v ∈Ω  ;

where,

    κ:=1αbL_H+∑_i=1^nβ_i λ_i ℓ_f_i <1  .







































































































































































Acknowledgements

The authors wish to thank the anonymous reviewers for their helpful comments and suggestions, which have considerably improved the version of this work.







§ DECLARATIONS






  * Funding: A. Chávez, N. Aragonés, U. Zavaleta and M. Pinto have been partially supported by research grand 038-2021-FONDECYT-Perú. A. Chávez and M. Pinto have also been partially supported by CONCYTEC through the PROCIENCIA program under the E041-2023-01




  * Conflict of interest: The authors declare no conflict of interest.



  * Data availability: Not applicable.



  * Author contribution: The results were obtained by A.Ch., N.A., M.P. and U.Z. during many discussions, the authors have participated equally. Typing was done mostly by A.Ch. All authors corrected the manuscript.


























































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