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Pseudo compact almost automorphic solutions to a family of delay differential equations

  • Feng-Xia Zheng and Hong-Xu Li EMAIL logo
Published/Copyright: December 11, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

In this article, a family of delay differential equations with pseudo compact almost automorphic coefficients is considered. By introducing a concept of Bi-pseudo compact almost automorphic functions and establishing the properties of these functions, and using Halanay’s inequality and Banach fixed point theorem, some results on the existence, uniqueness and global exponential stability of pseudo compact automorphic solutions of the equations are obtained. Our results extend some recent works. Moreover, an example is given to illustrate the validity of our results.

Keywords: delay differential equations; pseudo compact almost automorphic solutions; Bi-pseudo compact almost automorphic; global exponential stability
MSC 2010: 34C27; 34K14

1 Introduction

The study of almost periodic type solutions to differential equations have attracted many researchers (see, e.g., [1–10] and the references therein). The space of pseudo compact almost automorphic functions (p.k.a.a.) includes pseudo almost periodic functions (p.a.p.), almost periodic functions (a.p.) and compact almost automorphic functions (k.a.a.). However, as far as we know, there are few results on compact almost automorphic type solutions to differential equations.

In the study by Abbas et al. [11], by applying the properties of p.k.a.a. and bi-almost automorphic (Bi-a.a.), and the Banach fixed point theorem, Abbas et al. obtain the results of p.k.a.a. solutions to the following delay differential equation:

(1.1) u ˙ ( t ) = − α ( t ) u ( t ) + ∑ i = 1 n β i ( t ) F i ( λ i ( t ) u ( t − τ i ( t ) ) ) + b ( t ) H ( u ( t ) ) ,

where the coefficient function α : R → R does not consider ergodic perturbation, i.e., α is positive a.a.

Lots of results on equation (1.1) and its analogue equations do not consider the ergodic perturbation of coefficients, such as [2], by applying the properties of p.a.p. and the Banach fixed point theorem, Chérif obtain the results of p.a.p. solutions to Nicholson’s blowflies model with mixed delays of the form:

(1.2) u ˙ ( t ) = − α ( t ) u ( t ) + ∑ i = 1 n β i ( t ) u ( t − τ i ) e − ω i ( t ) u ( t − τ i ) − b ( t ) u ( t − σ ) + β 0 ( t ) ∫ − τ 0 K ( t , s ) u ( t + s ) e − u ( t + s ) d s ,

where α : R → R is positive a.p.

It is realistic to consider that the coefficients have an ergodic perturbation [12]. Coronel et al. [13] remark that in the context of differential equations with almost automorphic coefficients, the Bi-almost automorphic (Bi-a.a.) property of the Green function is fundamental to prove the existence of solutions of the same class. Naturally, it is necessary to consider the Green function with Bi-compact almost automorphic (Bi-k.a.a.) property and Bi-ergodic components to prove the existence of solutions to differential equations with pseudo compact almost automorphic coefficients. On the other hand, it is meaningful to consider mixed delays with ergodic components.

Motivated by the aforementioned results, we introduce the concept of Bi-pseudo compact almost automorphic functions (Bi-p.k.a.a.) and establish some basic properties for these functions. Then by applying the properties of p.k.a.a. and Bi-p.k.a.a., and the Banach fixed point theorem, we study the p.k.a.a. solutions to the following delay differential equation:

(1.3) u ˙ ( t ) = − α ( t ) u ( t ) + ∑ i = 1 n β i ( t ) f i ( t , u ( t − τ i ( t ) ) ) + b ( t ) H ( u ( t − σ ) ) + β 0 ( t ) ∫ − τ 0 K ( t , s ) f 0 ( t , u ( t + s ) ) d s ,

where α : R → R is positive p.k.a.a.

It is clear that (1.3) includes both (1.1) and (1.2). In our results, by using the properties of p.k.a.a and Bi-p.k.a.a., the conditions for some useful results and the existence of p.k.a.a. solutions to equation (1.3) are weaker than those in the existing literature (see Remarks 2.6, 2.15, and 3.7). Moreover, by using Halanay’s inequality, the globally exponential stability of the positive p.k.a.a. solution to equation (1.3) does not need to add a condition to the existence conditions as in [9] (Remark 4.6). Therefore, our results extend some known results.

The outline of this article is as follows. In Section 2, we recall the concept and properties of pseudo compact almost automorphic functions, define Bi-pseudo compact almost automorphic functions, and establish some properties of these functions. Sections 3 and 4 contain some results on the existence, uniqueness, and global exponential stability of pseudo compact almost automorphic solution to equation (1.3). In Section 5, we provide an example to illustrate the validity of our results. Finally, a conclusion is drawn in Section 6.

2 Preliminaries

Let ( V , ‖ ⋅ ‖ ) , ( U , ‖ ⋅ ‖ ) be two Banach spaces, and BC ( R , V ) ( resp. BC ( R × U , V ) ) denotes the space of bounded continuous functions f : R → V ( resp. f : R × U → V ) . Denote ‖ f ‖ ∞ ≔ sup t ∈ R ‖ f ( t ) ‖ , ‖ f ‖ μ ≔ sup − μ ≤ t ≤ 0 ‖ f ( t ) ‖ and R + ≔ [ 0 , ∞ ) .

Definition 2.1

[11,14–17] (i) A continuous function f : R → V is said to be almost automorphic (a.a.) if for any sequence { ξ n ′ } n = 1 ∞ ⊂ R , there exists a subsequence { ξ n } n = 1 ∞ of { ξ n ′ } n = 1 ∞ such that

lim n → ∞ f ( t + ξ n ) = f ˜ ( t )

is well defined for all t ∈ R , and

lim n → ∞ f ˜ ( t − ξ n ) = f ( t )

for all t ∈ R . Denote by A A ( R , V ) the space of all such functions. A continuous function f : R × U → V is said to be almost automorphic if f ( t , x ) is almost automorphic in t ∈ R uniformly for all x in any bounded subset of U . Denote by A A ( R × U , V ) the space of all such functions.

If the aforementioned limits hold uniformly in compact subsets of R , then f is said to be compact almost automorphic (k.a.a.). Denote by KAA ( R , V ) , the space of all such functions. A continuous function f : R × U → V is said to be compact almost automorphic if f ( t , x ) is compact almost automorphic in t ∈ R uniformly for all x in any bounded subset of U . Denote by KAA ( R × U , V ) the space of all such functions.

(ii) A continuous function f : R → V (resp. f : R × U → V ) is said to be pseudo almost automorphic (p.a.a.) if f is decomposed as follows:

f = f 1 + f 2 ,

where f 1 ∈ A A ( R , V ) (resp. A A ( R × U , V ) ) and f 2 ∈ PAP 0 ( R , V ) (resp. f 2 ∈ PAP 0 ( R × U , V ) ), which is defined by

PAP 0 ( R , V ) = f 2 ∈ BC ( R , V ) : lim r → ∞ 1 2 r ∫ − r r ‖ f 2 ( s ) ‖ d s = 0 resp. PAP 0 ( R × U , V ) = f 2 ∈ BC ( R × U , V ) : lim r → ∞ 1 2 r ∫ − r r ‖ f 2 ( s , x ) ‖ d s = 0 uniformly for all x in any bounded subset of U .

Denote by PAA ( R , V ) (resp. PAA ( R × U , V ) ) the space of all such functions. The functions f 1 and f 2 are, respectively, called the almost automorphic and the ergodic components of f .

(iii) A continuous function f : R → V (resp. f : R × U → V ) is said to be pseudo compact almost automorphic (p.k.a.a.) if f is decomposed as follows:

f = f 1 + f 2 ,

where f 1 ∈ KAA ( R , V ) (resp. KAA ( R × U , V ) ) and f 2 ∈ PAP 0 ( R , V ) (resp. f 2 ∈ PAP 0 ( R × U , V ) ). Denote by PKAA ( R , V ) (resp. PKAA ( R × U , V ) ) the space of all such functions. The functions f 1 and f 2 are, respectively, called the compact almost automorphic and the ergodic components of f .

Remark 2.2

Let f = f 1 + f 2 ∈ PKAA ( R , V ) with f 1 ∈ KAA ( R , V ) and f 2 ∈ PAP 0 ( R , V ) . In view of the properties of p.a.a., it follows immediately that f is bounded and { f 1 ( t ) : t ∈ R } ⊂ { f ( t ) : t ∈ R } ¯ since PKAA ( R , V ) is a subspace of PAA ( R , V ) . Moreover, from the definition of k.a.a., we see that KAA ( R , V ) is translation-invariant. So the space PKAA ( R , V ) is translation-invariant since PAP 0 ( R , V ) is translation-invariant. See [15,16] for more details on p.a.a.

Lemma 2.3

[11,18] The following assertions hold:

  1. PKAA ( R , V ) is a Banach space with the supremum norm.

  2. The decomposition of a pseudo compact almost automorphic function is unique.

  3. Let f ∈ PKAA ( R , R ) and g ∈ PKAA ( R , R ) , then f g ∈ PKAA ( R , R ) .

  4. A function f : R → V is k.a.a. if and only if it is a.a. and uniformly continuous.

By [15, Lemma 4.36, Theorem 6.8], we can obtain the following result.

Lemma 2.4

Let f = f 1 + f 2 ∈ PKAA ( R × V , V ) with f 1 ∈ KAA ( R × V , V ) , f 2 ∈ PAP 0 ( R × V , V ) , and x ↦ f ( t , x ) be uniformly continuous in any bounded subset of V uniformly for t ∈ R . If x ∈ PKAA ( R , V ) , then f ( ⋅ , x ( ⋅ ) ) ∈ PKAA ( R , V ) .

Lemma 2.5

If f ∈ PKAA ( R , R ) , τ ∈ PKAA ( R , R ) ∩ C 1 ( R , R + ) and τ ′ ( t ) ≤ τ * < 1 , then f ( ⋅ − τ ( ⋅ ) ) ∈ PKAA ( R , R ) .

Proof

Since f , τ ∈ PKAA ( R , R ) , f and τ can be expressed as f = f 1 + f 2 and τ = τ 1 + τ 2 , respectively, where f 1 , τ 1 ∈ KAA ( R , R ) and f 2 , τ 2 ∈ PAP 0 ( R , R ) . Denote

F 1 ( t ) = f 1 ( t − τ 1 ( t ) ) , F 2 ( t ) = f 2 ( t − τ ( t ) ) , and F 3 ( t ) = f 1 ( t − τ ( t ) ) − f 1 ( t − τ 1 ( t ) ) .

Then

f ( t − τ ( t ) ) = F 1 ( t ) + F 2 ( t ) + F 3 ( t ) .

It follows from [19, Lemma 7] that F 1 ∈ KAA ( R , R ) . By the same arguments as in the proof of [11, Lemma 4], it is easy to obtain that F 2 ∈ PAP 0 ( R , R ) . Now it remains to prove that F 3 ∈ PAP 0 ( R , R ) . By Lemma 2.3 (iv), we have f 1 is uniform continuity, then for any ε > 0 , there exists a constant δ > 0 such that for all t ′ , t ″ ∈ R , ∣ t ′ − t ″ ∣ < δ ,

(2.1) ∣ f 1 ( t ′ ) − f 1 ( t ″ ) ∣ < ε 2 .

Since τ 2 ∈ PAP 0 ( R , R ) , it follows from [20, Lemma 1.1] that

lim r → ∞ 1 2 r meas ( M r , ε ) = 0 ,

where meas ( ⋅ ) denotes the Lebesgue measure and M r , ε = { t ∈ [ − r , r ] : ‖ τ 2 ( t ) ‖ = ‖ τ ( t ) − τ 1 ( t ) ‖ ≥ ε } . Particularly, one can find δ > 0 such that r > δ ,

(2.2) 1 2 r meas ( [ − r , r ] ∩ M r , δ ) < ε 4 ‖ f 1 ‖ ∞ .

Thus, we can deduce from (2.1) and (2.2) that

1 2 r ∫ − r r ∣ F 3 ( t ) ∣ d t = 1 2 r ∫ − r r ∣ f 1 ( t − τ ( t ) ) − f 1 ( t − τ 1 ( t ) ) ∣ d t = 1 2 r ∫ [ − r , r ] ∩ M r , δ ∣ f 1 ( t − τ ( t ) ) − f 1 ( t − τ 1 ( t ) ) ∣ d t + 1 2 r ∫ [ − r , r ] \ M r , δ ∣ f 1 ( t − τ ( t ) ) − f 1 ( t − τ 1 ( t ) ) ∣ d t < ε 2 + ε 2 = ε ,

which means F 3 ∈ PAP 0 ( R , R ) .□

Remark 2.6

Lemma 2.5 shows that the condition “ τ ∈ KAA ( R , R ) ∩ C 1 ( R , R + ) ” in [11] can be improved by the condition “ τ ∈ PKAA ( R , R ) ∩ C 1 ( R , R + ) .”

Next we recall and introduce some notations, which will be used to obtain our main results.

Definition 2.7

(i) A continuous function G : R × R → V is said to be Bi-almost automorphic (Bi-a.a.) if for any sequence { ξ n ′ } n = 1 ∞ ⊂ R , there exists a subsequence { ξ n } n = 1 ∞ of { ξ n ′ } n = 1 ∞ such that

lim n → ∞ G ( t + ξ n , s + ξ n ) = G ˜ ( t , s ) , lim n → ∞ G ˜ ( t − ξ n , s − ξ n ) = G ( t , s )

for all ( t , s ) ∈ R 2 . Denote by B A A ( R × R , V ) the space of all such functions.

If the aforementioned limits hold uniformly in compact regions of R 2 , then G is said to be Bi-compact almost automorphic (Bi-k.a.a.). Denote by BKAA ( R × R , V ) the space of all such functions.

(ii) A continuous function G : R × R → V is said to be Bi-pseudo almost automorphic (Bi-p.a.a.) if G is decomposed as follows:

G = G 1 + G 2 ,

where G 1 ∈ B A A ( R × R , V ) and G 2 ∈ BPAP 0 ( R × R , V ) , which is defined by

BPAP 0 ( R × R , V ) = G 2 ∈ BC ( R × R , V ) : lim r → ∞ 1 2 r ∫ − r r ‖ G 2 ( w + t , w + s ) ‖ d w = 0 for all ( t , s ) ∈ R 2 .

Denote by BPAA ( R × R , V ) the space of all such functions. The functions G 1 and G 2 are, respectively, called the Bi-a.a. and the Bi-ergodic components of G .

(iii) A continuous function G : R × R → V is said to be Bi-pseudo compact almost automorphic (Bi-p.k.a.a.) if G is decomposed as follows:

G = G 1 + G 2 ,

where G 1 ∈ BKAA ( R × R , V ) and G 2 ∈ BPAP 0 ( R × R , V ) . Denote by BPKAA ( R × R , V ) the space of all such functions. The functions G 1 and G 2 are, respectively, called the Bi-k.a.a. and the Bi-ergodic components of G .

Remark 2.8

  1. (i) is given in [21], (ii) is given in [22].

  2. Let G ∈ BKAA ( R × R , V ) . It follows from the definition of Bi-k.a.a. that G ˜ is continuous.

  3. Let G ( t , s ) = g ( t − s ) for some continuous function g : R × R → V . Then it is easy to verify that G ∈ BKAA ( R × R , V ) .

  4. Let G = G 1 + G 2 ∈ BPKAA ( R × R , V ) with G 1 ∈ BKAA ( R × R , V ) and G 2 ∈ PAP 0 ( R × R , V ) . By the same arguments as in the proof of [22, Proposition 2.8], it is easy to obtain that for every ( t , s ) ∈ R 2 , { G 1 ( t + r , s + r ) : r ∈ R } ⊂ { G ( t + r , s + r ) : r ∈ R } ¯ and the decomposition of G ( t + ⋅ , s + ⋅ ) is unique.

Definition 2.9

A continuous function G : R × R → V is said to be Bi-uniformly continuous if for any sequences { t n ′ } , { t n ″ } , { s n ′ } and { s n ″ } such that t n ′ − s n ′ = t n ″ − s n ″ and ∣ t n ′ − s n ′ ∣ → 0 as n → ∞ , implies

‖ G ( t n ′ , t n ″ ) − G ( s n ′ , s n ″ ) ‖ → 0

as n → ∞ . That is, for any ε > 0 , there exists a δ = δ ( ε ) > 0 such that x 1 − x 2 = y 1 − y 2 , and ∣ x 1 − x 2 ∣ < δ implies

‖ G ( x 1 , y 1 ) − G ( x 2 , y 2 ) ‖ < ε .

Remark 2.10

  1. It is clear that G is Bi-uniformly continuous if G : R × R → V is uniformly continuous. But the converse is not true. For instance, it is easy to see that G ( x , y ) = x sin ( x − y ) is Bi-uniformly continuous but not uniformly continuous.

  2. If G : R × R → V is Bi-uniformly continuous, and if lim m → ∞ G ( t + ξ m , s + ξ m ) = G ˜ ( t , s ) for all ( t , s ) ∈ R 2 and some { ξ m } m = 1 ∞ ⊂ R , then G ˜ is Bi-uniformly continuous. Indeed, by the Bi-uniformly continuity of G , for any sequences { t n ′ } , { t n ″ } , { s n ′ } , and { s n ″ } such that t n ′ − s n ′ = t n ″ − s n ″ and ∣ t n ′ − s n ′ ∣ → 0 as n → ∞ , implies

    ‖ G ( t n ′ , t n ″ ) − G ( s n ′ , s n ″ ) ‖ → 0

    as n → ∞ . Then

    ‖ G ( t n ′ + ξ m , t n ″ + ξ m ) − G ( s n ′ + ξ m , s n ″ + ξ m ) ‖ → 0

    as n → ∞ uniformly for { ξ m } . Thus,

    lim n → ∞ ‖ G ˜ ( t n ′ , t n ″ ) − G ˜ ( s n ′ , s n ″ ) ‖ = lim n → ∞ lim m → ∞ ‖ G ( t n ′ + ξ m , t n ″ + ξ m ) − G ( s n ′ + ξ m , s n ″ + ξ m ) ‖ = lim m → ∞ lim n → ∞ ‖ G ( t n ′ + ξ m , t n ″ + ξ m ) − G ( s n ′ + ξ m , s n ″ + ξ m ) ‖ = 0 .

That is, G ˜ is Bi-uniformly continuous.

Lemma 2.11

A function G : R × R → V is Bi-k.a.a. if and only if it is Bi-a.a. and Bi-uniformly continuous.

Proof

Let G is Bi-a.a. and Bi-uniformly continuous. Then for any sequence { ξ n ′ } n = 1 ∞ ⊂ R , there exists a subsequence { ξ n } n = 1 ∞ of { ξ n ′ } n = 1 ∞ such that

lim n → ∞ G ( t + ξ n , s + ξ n ) = G ˜ ( t , s ) , lim n → ∞ G ˜ ( t − ξ n , s − ξ n ) = G ( t , s )

for all ( t , s ) ∈ R 2 . Noticing that G ˜ is Bi-uniformly continuous since G is Bi-uniformly continuous (Remark 2.10), it is easy to verify that the aforementioned limits hold uniformly for ( t , s ) in compact regions of R 2 . That is, G is Bi-k.a.a.

On the other hand, if G is Bi-k.a.a., then G is Bi-a.a. It remains to show that G is Bi-uniformly continuous. Take sequences { t n ′ } , { t n ″ } , { s n ′ } and { s n ″ } such that t n ′ − s n ′ = t n ″ − s n ″ and ∣ t n ′ − s n ′ ∣ → 0 as n → ∞ . Let

α n = ‖ G ( t n ′ , t n ″ ) − G ( s n ′ , s n ″ ) ‖ ,

we need to prove α n → 0 as n → ∞ . Now it is sufficient to prove that every subsequence α ˆ n = ‖ G ( t ˆ n ′ , t ˆ n ″ ) − G ( s ˆ n ′ , s ˆ n ″ ) ‖ of α n has a subsequence α ˜ n = ‖ G ( t ˜ n ′ , t ˜ n ″ ) − G ( s ˜ n ′ , s ˜ n ″ ) ‖ such that α ˜ n → 0 as n → ∞ . Let

{ ξ n } = { t n ′ − s n ′ } = { t n ″ − s n ″ } , { ξ ˆ n } = { t ˆ n ′ − s ˆ n ′ } = { t ˆ n ″ − s ˆ n ″ } ,

where { ξ ˆ n } ⊂ { ξ n } . Since G is Bi-k.a.a., there exists a subsequence { ξ ˜ n } = { t ˜ n ′ − s ˜ n ′ } = { t ˜ n ″ − s ˜ n ″ } of { ξ ˆ n } such that

(2.3) lim n → ∞ G ( t + ξ ˜ n , s + ξ ˜ n ) = G ˜ ( t , s ) , lim n → ∞ G ˜ ( t − ξ ˜ n , s − ξ ˜ n ) = G ( t , s )

uniformly for ( t , s ) in compact regions of R 2 . Meanwhile, since the function G ˜ is continuous (Remark 2.8 (II)), and ∣ t ˜ n ′ − s ˜ n ′ ∣ = ∣ t ˜ n ″ − s ˜ n ″ ∣ → 0 as n → ∞ , we have

(2.4) ‖ G ˜ ( s ˜ n ′ , s ˜ n ″ ) − G ˜ ( s ˜ n ′ − ( t ˜ n ′ − s ˜ n ′ ) , s ˜ n ″ − ( t ˜ n ″ − s ˜ n ″ ) ) ‖ → 0

as n → ∞ . Then from (2.3) and (2.4), we have

α ˜ n = ‖ G ( t ˜ n ′ , t ˜ n ″ ) − G ( s ˜ n ′ , s ˜ n ″ ) ‖ ≤ ‖ G ( t ˜ n ′ − s ˜ n ′ + s ˜ n ′ , t ˜ n ″ − s ˜ n ″ + s ˜ n ″ ) − G ˜ ( s ˜ n ′ , s ˜ n ″ ) ‖ + ‖ G ˜ ( s ˜ n ′ , s ˜ n ″ ) − G ˜ ( s ˜ n ′ − ( t ˜ n ′ − s ˜ n ′ ) , s ˜ n ″ − ( t ˜ n ″ − s ˜ n ″ ) ) ‖ + ‖ G ˜ ( s ˜ n ′ − ( t ˜ n ′ − s ˜ n ′ ) , s ˜ n ″ − ( t ˜ n ″ − s ˜ n ″ ) ) − G ( s ˜ n ′ , s ˜ n ″ ) ‖ ≤ ‖ G ( ξ ˜ n + s ˜ n ′ , ξ ˜ n + s ˜ n ″ ) − G ˜ ( s ˜ n ′ , s ˜ n ″ ) ‖ + ‖ G ˜ ( s ˜ n ′ , s ˜ n ″ ) − G ˜ ( s ˜ n ′ − ( t ˜ n ′ − s ˜ n ′ ) , s ˜ n ″ − ( t ˜ n ″ − s ˜ n ″ ) ) ‖ + ‖ G ˜ ( s ˜ n ′ − ξ ˜ n , s ˜ n ″ − ξ ˜ n ) − G ( s ˜ n ′ , s ˜ n ″ ) ‖ → 0

as n → ∞ . Thus, we showed that α n = ‖ G ( t n ′ , t n ″ ) − G ( s n ′ , s n ″ ) ‖ → 0 as n → ∞ . That is, G is Bi-uniformly continuous.□

We note that the property of Ψ α ( t , s ) = e − ∫ s t α ( r ) d r is essential to prove the existence of solution to equation (1.3). Similar to [22, Proposition 3.10], we can obtain the following result.

Proposition 2.12

Let α = α 1 + α 2 ∈ PKAA ( R , R ) with α 1 ∈ KAA ( R , R ) and α 2 ∈ PAP 0 ( R , R ) . Then Ψ α is Bi-p.k.a.a., where Ψ α 1 is Bi-k.a.a. with Ψ α 1 ( t , s ) = e − ∫ s t α 1 ( r ) d r , and Ψ α − Ψ α 1 is Bi-ergodic.

Proof

Since α 1 ∈ KAA ( R , R ) , then for any sequence { ξ n ′ } n = 1 ∞ ⊂ R , there exists a subsequence { ξ n } n = 1 ∞ of { ξ n ′ } n = 1 ∞ such that lim n → ∞ α 1 ( r + ξ n ) = α ˜ 1 ( r ) uniformly for r in compact sets of R . Thus, for all ( t , s ) ∈ R 2 ,

lim n → ∞ Ψ α 1 ( t + ξ n , s + ξ n ) = lim n → ∞ e − ∫ s + ξ n t + ξ n α 1 ( r ) d r = lim n → ∞ e − ∫ s t α 1 ( r + ξ n ) d r = e − ∫ s t α ˜ 1 ( r ) d r = Ψ α ˜ 1 ( t , s )

uniformly in compact regions of R 2 . Similarly, we can easily obtain that

lim n → ∞ Ψ α ˜ 1 ( t − ξ n , s − ξ n ) = Ψ α 1 ( t , s )

uniformly in compact regions of R 2 . That is, Ψ α ∈ BKAA ( R × R , R ) . The remains proof is similar to [22, Proposition 3.10 (ii)], and we omit the details here.□

Remark 2.13

Let Φ α ( t , s ) = e ∫ s t α ( r ) d r and α = α 1 + α 2 ∈ PKAA ( R , R ) with α 1 ∈ KAA ( R , R ) and α 2 ∈ PAP 0 ( R , R ) , by the same arguments as in the proof of Proposition 2.12, it is easy to obtain that Φ α is Bi-p.k.a.a., where Φ α 1 is Bi-k.a.a. with Φ α 1 ( t , s ) = e ∫ s t α 1 ( r ) d r , and Φ α − Φ α 1 is Bi-ergodic.

Proposition 2.14

If the function u ∈ PKAA ( R , R ) and there is a Bi-p.k.a.a. function g : R × R → R such that

∣ g ( t , s ) ∣ ≤ c e − α ( t − s ) , t ≥ s ,

where c and α are two positive constants, then the function

ϱ ( t ) = ∫ − ∞ t g ( t , s ) u ( s ) d s

belongs to PKAA ( R , R ) .

Proof

Since u ∈ PKAA ( R , R ) and g ∈ BPKAA ( R × R , R ) , u and g can be expressed as u = u 1 + u 2 and g = g 1 + g 2 , respectively, where u 1 ∈ KAA ( R , R ) , u 2 ∈ PAP 0 ( R , R ) , g 1 ∈ BKAA ( R × R , R ) , g 2 ∈ BPAP 0 ( R × R , R ) . Denote ϱ ( t ) = ϱ 1 ( t ) + ϱ 2 ( t ) + ϱ 3 ( t ) , where

ϱ 1 ( t ) = ∫ − ∞ t g 1 ( t , s ) u 1 ( s ) d s , t ∈ R , ϱ 2 ( t ) = ∫ − ∞ t g 1 ( t , s ) u 2 ( s ) d s , t ∈ R , ϱ 3 ( t ) = ∫ − ∞ t g 2 ( t , s ) u ( s ) d s , t ∈ R .

Now we can complete the proof by the following two steps.

Step 1. We prove that ϱ 1 ∈ KAA ( R , R ) . Since u 1 ∈ KAA ( R , R ) and g 1 ( t , s ) is Bi-k.a.a., for any sequence { t n ′ } n = 1 ∞ ⊂ R , there exists a subsequence { t n } n = 1 ∞ of { t n ′ } n = 1 ∞ such that

(2.5) ∣ u 1 ( t n + s ) − u ˜ 1 ( s ) ∣ → 0 , ∣ u ˜ 1 ( s − t n ) − u 1 ( s ) ∣ → 0

as n → ∞ for each t ∈ R . And

(2.6) ∣ g 1 ( t + t n , s + t n ) − g ˜ 1 ( t , s ) ∣ → 0 , ∣ g ˜ 1 ( t − t n , s − t n ) − g 1 ( t , s ) ∣ → 0

as n → ∞ for each t , s ∈ R . By Remark 2.8 (IV), we have

sup r ∈ R ∣ g 1 ( t + r , s + r ) ∣ ≤ sup r ∈ R ∣ g ( t + r , s + r ) ∣ ≤ c e − α ( t − s ) , t ≥ s .

Then we have

(2.7) ∣ g 1 ( t + ⋅ , s + ⋅ ) ∣ ≤ c e − α ( t − s ) , t ≥ s .

Pose ϱ ˜ 1 ( t ) = ∫ − ∞ t g ˜ 1 ( t , s ) u ˜ 1 ( s ) d s . Notice that ‖ u ˜ 1 ‖ ∞ ≤ ‖ u 1 ‖ ∞ ≤ ‖ u ‖ ∞ . Then by Lebesgue’s dominated convergence theorem and (2.5)–(2.7), we obtain

∣ ϱ 1 ( t + t n ) − ϱ 1 ˜ ( t ) ∣ = ∫ − ∞ t g 1 ( t + t n , s + t n ) u 1 ( t n + s ) d s − ∫ − ∞ t g ˜ 1 ( t , s ) u ˜ 1 ( s ) d s ≤ ∫ − ∞ t ( g 1 ( t + t n , s + t n ) − g ˜ 1 ( t , s ) ) u ˜ 1 ( s ) d s + ∫ − ∞ t g 1 ( t + t n , s + t n ) ( u 1 ( t n + s ) − u ˜ 1 ( s ) ) d s ≤ ‖ u ‖ ∞ ∫ − ∞ t ( g 1 ( t + t n , s + t n ) − g ˜ 1 ( t , s ) ) d s + ∫ − ∞ t c e − α ( t − s ) ( u 1 ( t n + s ) − u ˜ 1 ( s ) ) d s → 0

as n → ∞ for all t ∈ R . Similarly, we can easily obtain that ∣ ϱ ˜ 1 ( t − t n ) − ϱ 1 ( t ) ∣ → 0 as n → ∞ for all t ∈ R . That is, ϱ 1 ∈ A A ( R , R ) .

Moreover, we can show that ϱ 1 is uniformly continuous. It follows from Lemma 2.3 (iv) and Lemma 2.11 that ∣ t n − s n ∣ → 0 as n → ∞ ,

∣ g 1 ( t n , t n + r ) − g 1 ( s n , s n + r ) ∣ → 0 , ∣ u 1 ( t n + r ) − u 1 ( s n + r ) ∣ → 0

as n → ∞ for all r ∈ R . Thus,

∣ ϱ 1 ( t n ) − ϱ 1 ( s n ) ∣ = ∫ − ∞ t n g 1 ( t n , s ) u 1 ( s ) d s − ∫ − ∞ s n g 1 ( s n , s ) u 1 ( s ) d s = ∫ − ∞ 0 g 1 ( t n , t n + r ) u 1 ( t n + r ) d r − ∫ − ∞ 0 g 1 ( s n , s n + r ) u 1 ( s n + r ) d r ≤ ∫ − ∞ 0 ( g 1 ( t n , t n + r ) − g 1 ( s n , s n + r ) ) u 1 ( t n + r ) d r + ∫ − ∞ 0 g 1 ( s n , s n + r ) ( u 1 ( t n + r ) − u 1 ( s n + r ) ) d r ≤ ‖ u ‖ ∞ ∫ − ∞ 0 ( g 1 ( t n , t n + r ) − g 1 ( s n , s n + r ) ) d r + ∫ − ∞ 0 c e − α r ( u 1 ( t n + r ) − u 1 ( s n + r ) ) d r → 0

as n → ∞ by Lebesgue’s dominated convergence theorem. That is, ϱ 1 ∈ KAA ( R , R ) by Lemma 2.3 (iv).

Step 2. We prove that ϱ 2 ∈ PAP 0 ( R , R ) and ϱ 3 ∈ PAP 0 ( R , R ) . By applying Fubini theorem, Lebesgue’s dominated convergence theorem and the translation invariance property of ergodic function u 2 , we have

lim r → ∞ 1 2 r ∫ − r r ∣ ϱ 2 ( t ) ∣ d t = lim r → ∞ 1 2 r ∫ − r r ∫ − ∞ t g 1 ( t , s ) u 2 ( s ) d s d t = lim r → ∞ 1 2 r ∫ − r r ∫ 0 ∞ g 1 ( t , t − s ) u 2 ( t − s ) d s d t ≤ ∫ 0 ∞ c e − α s lim r → ∞ 1 2 r ∫ − r r ∣ u 2 ( t − s ) ∣ d t d s = 0 .

Then by applying Fubini theorem, Lebesgue’s dominated convergence theorem and g 2 ∈ PAP 0 ( R × R , R ) , we obtain

lim r → ∞ 1 2 r ∫ − r r ∣ ϱ 3 ( t ) ∣ d t ≤ lim r → ∞ 1 2 r ∫ − r r ∫ − ∞ r g 2 ( t , s ) u ( s ) d s d t = lim r → ∞ 1 2 r ∫ − r r ∫ − ∞ 0 g 2 ( t , t + s ) u ( t + s ) d s d t ≤ ∫ − ∞ 0 ‖ u ‖ ∞ lim r → ∞ 1 2 r ∫ − r r ∣ g 2 ( t , t + s ) ∣ d t d s = 0 .

This means ϱ 2 , ϱ 3 ∈ PAP 0 ( R , R ) .□

Remark 2.15

Proposition 2.14 shows that the condition “there is a Bi-a.a. function g : R × R → R ” in [11, Corollary 1] can be improved by the condition “there is a Bi-p.k.a.a. function g : R × R → R .”

3 Existence and uniqueness of p.k.a.a solution

We consider the delay differential equation (1.3). It is used to model the population growth of species. u ( t ) denotes the density of the population, α ( t ) denotes the death rate of the population, M ( t , u ) = ∑ i = 1 n β i ( t ) f i ( t , u ( t − τ i ( t ) ) ) + β 0 ( t ) ∫ − τ 0 K ( t , s ) f 0 ( t , u ( t + s ) ) d s denotes the birth function, K ( t , s ) denotes the delay kernel, and b ( t ) H ( u ( t − σ ) ) means the immigration function or harvesting function, respectively, if b ( t ) is nonnegative or nonpositive.

Throughout this article, given a bounded continuous function f defined on R , denote

f ¯ = sup t ∈ R { f ( t ) } , f ̲ = inf t ∈ R { f ( t ) } ,

and also we denote

τ ¯ ≔ max 1 ≤ i ≤ n { τ i ¯ } , μ = max { τ ¯ , τ , σ } .

In the population model, only nonnegative solutions are biologically meaningful. We consider the following initial condition

(3.1) u t 0 = φ , φ ∈ C 0 ,

where C 0 defined by

C 0 = { φ ∈ BC ( [ − μ , 0 ] , R + ) , φ ( 0 ) > 0 } .

Now we make the following assumptions:

  1. α , β 0 , β i , τ i : R → R are positive p.k.a.a. with α ̲ > 0 , β i ̲ > 0 for 1 ≤ i ≤ n and b : R → R is p.k.a.a.

  2. K : R × [ − τ , 0 ] → R , ( t , s ) ↦ K ( t , s ) is uniformly continuous and positive p.k.a.a. in t ∈ R for each s ∈ [ − τ , 0 ] .

  3. H : R + → R + is Lipschitz continuous, i.e., there exists a positive constant L H such that

    ∣ H ( u ) − H ( v ) ∣ ≤ L H ∣ u − v ∣ , for u , v ∈ R + .

    In addition, we suppose that H ( 0 ) = 0 .

  4. For all 0 ≤ i ≤ n , f i : R × R → R are nonnegative p.k.a.a., x ↦ f i ( t , x ) is Lipschitzian uniformly for t ∈ R , and f i reaches its maximum value in R × R + , i.e., f i ¯ = sup t ∈ R , x ∈ R + f i ( t , x ) = f i ( t i , n i + ) for some t i ∈ R , n i + ∈ R + .

  5. There exist two positive constants γ 1 and γ 2 such that

    n + < γ 1 < 1 α ¯ ∑ i = 1 n β i ̲ f i − + b ̲ L H γ 2 , 1 α ̲ ∑ i = 1 n β i ¯ f i ¯ + β 0 ¯ f 0 ¯ ‖ K ‖ ∞ τ + b ¯ L H γ 2 < γ 2 ,

    where n + = max 1 ≤ i ≤ n { n i + } , f i − = inf t ∈ R , x ∈ [ n + , γ 2 ] f i ( t , x ) .

  6. For all 0 ≤ i ≤ n , there exist positive constants L f i such that for all x , y ∈ [ n + , ∞ ) ,

    ∣ f i ( t , x ) − f i ( t , y ) ∣ ≤ L f i ∣ x − y ∣ , t ∈ R .

Remark 3.1

  1. Assume that (A2) holds. It is easy to obtain that K is bounded on R × [ − τ , 0 ] .

  2. In [11], the nonlinear term F i ( λ i ( t ) u ) = u e − λ i ( t ) u in the Nicholson model and F i ( λ i ( t ) u ) = e − λ i ( t ) u in the Lasota-Wazewska model satisfy the following assumption:

(A4′) For all 1 ≤ i ≤ n , F i : R + → R + are Lipschitz continuous and reach its maximum value in R + , i.e., F i ¯ = sup x ∈ R + F i ( x ) = F i ( m i * ) for some m i * ∈ R + , and F i is nonincreasing in x > m i * .

We note that (A4) is satisfied by lots of functions. For example, sin u e − λ ( t ) u in the general Nicholson model satisfies (A4), but this function does not satisfy (A4′). Besides, u e − λ ( t ) u and e − λ ( t ) u also satisfy (A4). Consequently, the model we considered includes not only the mixed Nicholson model and the Lasota-Wazewska model but also other useful mixed models, such as the mixed general Nicholson model and the Lasota-Wazewska model.

Lemma 3.2

Assume that (A2) holds. Let f ∈ PKAA ( R × R , R ) such that u ↦ f ( t , u ) is Lipschitzian uniformly for t ∈ R . If u ∈ PKAA ( R , R ) , then the function ϕ ( t ) = ∫ − τ 0 K ( t , s ) f ( t , u ( t + s ) ) d s belongs to PKAA ( R , R ) .

Proof

By using the composition theorem of p.k.a.a. (Lemma 2.4) and translation invariant (Remark 2.2), one can deduce easily that for all s ∈ R the function ψ : t ↦ f ( t , u ( t + s ) ) belongs to PKAA ( R , R ) . Then, ψ can be expressed as follows:

ψ = ψ 1 + ψ 2 ,

where ψ 1 ∈ KAA ( R , R ) and ψ 2 ∈ PAP 0 ( R , R ) . Since K : R × [ − τ , 0 ] → R is positive p.k.a.a. in t ∈ R for each s ∈ [ − τ , 0 ] , i.e., K ( ⋅ , s ) ∈ PKAA ( R , R ) for each s ∈ [ − τ , 0 ] , K can be expressed as K = K 1 + K 2 , where K 1 ( ⋅ , s ) ∈ KAA ( R , R ) and K 2 ( ⋅ , s ) ∈ PAP 0 ( R , R ) for each s ∈ [ − τ , 0 ] . Denote ϕ ( t ) = ϕ 1 ( t ) + ϕ 2 ( t ) + ϕ 3 ( t ) , where

ϕ 1 ( t ) = ∫ − τ 0 K 1 ( t , s ) ψ 1 ( t ) d s , t ∈ R , ϕ 2 ( t ) = ∫ − τ 0 K 1 ( t , s ) ψ 2 ( t ) d s , t ∈ R , ϕ 3 ( t ) = ∫ − τ 0 K 2 ( t , s ) ψ ( t ) d s , t ∈ R .

Now we can complete the proof by the following two steps.

Step 1. We prove that ϕ 1 ∈ KAA ( R , R ) . Since ψ 1 ∈ KAA ( R , R ) and K 1 ( ⋅ , s ) ∈ KAA ( R , R ) for each s ∈ [ − τ , 0 ] , for any sequence { t n ′ } n = 1 ∞ ⊂ R , there exists a common subsequence { t n } n = 1 ∞ of { t n ′ } n = 1 ∞ such that

(3.2) ∣ ψ 1 ( t + t n ) − ψ ˜ 1 ( t ) ∣ → 0 , ∣ ψ ˜ 1 ( t − t n ) − ψ 1 ( t ) ∣ → 0

as n → ∞ for all t ∈ R . And

(3.3) ∣ K 1 ( t + t n , s ) − K ˜ 1 ( t , s ) ∣ → 0 , ∣ K ˜ 1 ( t − t n , s ) − K 1 ( t , s ) ∣ → 0

as n → ∞ for all t ∈ R and for all s ∈ [ − τ , 0 ] . Pose

ϕ ˜ 1 ( t ) = ∫ − τ 0 K ˜ 1 ( t , s ) ψ ˜ 1 ( t ) d s , t ∈ R .

Notice that ‖ ψ ˜ 1 ‖ ∞ ≤ ‖ ψ 1 ‖ ∞ ≤ ‖ ψ ‖ ∞ and ‖ K ˜ 1 ‖ ∞ ≤ ‖ K 1 ‖ ∞ ≤ ‖ K ‖ ∞ by Remarks 3.1 and 2.2. Then

∣ ϕ 1 ( t + t n ) − ϕ 1 ˜ ( t ) ∣ = ∫ − τ 0 K 1 ( t + t n , s ) ψ 1 ( t + t n ) d s − ∫ − τ 0 K ˜ 1 ( t , s ) ψ ˜ 1 ( t ) d s ≤ ∫ − τ 0 ( K 1 ( t + t n , s ) − K ˜ 1 ( t , s ) ) ψ ˜ 1 ( t ) d s + ∫ − τ 0 K 1 ( t + t n , s ) ( ψ 1 ( t + t n ) − ψ ˜ 1 ( t ) ) d s ≤ ‖ ψ ‖ ∞ ∫ − τ 0 ( K 1 ( t + t n , s ) − K ˜ 1 ( t , s ) ) d s + ‖ K ‖ ∞ ∫ − τ 0 ( ψ 1 ( t + t n ) − ψ ˜ 1 ( t ) ) d s → 0

as n → ∞ for all t ∈ R by Lebesgue’s dominated convergence theorem and (3.2)–(3.3). Similarly, we can easily obtain that ∣ ϕ ˜ 1 ( t − t n ) − ϕ 1 ( t ) ∣ → 0 as n → ∞ for all t ∈ R . That is, ϕ 1 ∈ A A ( R , R ) .

Moreover, we can show that ϕ 1 is uniformly continuous. By Lemma 2.3 (iv), we have for any sequences { t n } and { s n } such that ∣ t n − s n ∣ → 0 as n → ∞ ,

∣ K 1 ( t n , s ) − K 1 ( s n , s ) ∣ → 0 , ∣ ψ 1 ( t n ) − ψ 1 ( s n ) ∣ → 0

as n → ∞ for all s ∈ [ − τ , 0 ] since ψ 1 ∈ KAA ( R , R ) and K 1 ( ⋅ , s ) ∈ KAA ( R , R ) for each s ∈ [ − τ , 0 ] . Thus,

∣ ϕ 1 ( t n ) − ϕ 1 ( s n ) ‖ = ∫ − τ 0 K 1 ( t n , s ) ψ 1 ( t n ) d s − ∫ − τ 0 K 1 ( s n , s ) ψ 1 ( s n ) d s ≤ ∫ − τ 0 ( K 1 ( t n , s ) − K 1 ( s n , s ) ) ψ 1 ( t n ) d s + ∫ − τ 0 K 1 ( s n , s ) ( ψ 1 ( t n ) − ψ 1 ( s n ) ) d s ≤ ‖ ψ ‖ ∞ ∫ − τ 0 ( K 1 ( t n , s ) − K 1 ( s n , s ) ) d s + ‖ K ‖ ∞ ∫ − τ 0 ( ψ 1 ( t n ) − ψ 1 ( s n ) ) d s → 0

as n → ∞ by Lebesgue’s dominated convergence theorem. Then ϕ 1 ∈ K A A   ( R , R ) by Lemma 2.3 (iv).

Step 2. We prove that ϕ 2 ∈ PAP 0 ( R , R ) and ϕ 3 ∈ PAP 0 ( R , R ) . Since ψ 2 ∈ PAP 0 ( R , R ) , by applying Fubini theorem and Lebesgue’s dominated convergence theorem, we have

lim r → ∞ 1 2 r ∫ − r r ∣ ϕ 2 ( t ) ∣ d t = lim r → ∞ 1 2 r ∫ − r r ∫ − τ 0 K 1 ( t , s ) ψ 2 ( t ) d s d t ≤ ∫ − τ 0 ‖ K ‖ ∞ lim r → ∞ 1 2 r ∫ − r r ∣ ψ 2 ( t ) ∣ d t d s = 0 .

Moreover, since K 2 ( ⋅ , s ) ∈ PAP 0 ( R , R ) for each s ∈ [ − τ , 0 ] , by applying Fubini theorem and Lebesgue’s dominated convergence theorem, we obtain

lim r → ∞ 1 2 r ∫ − r r ∣ ϕ 3 ( t ) ∣ d t ≤ lim r → ∞ 1 2 r ∫ − r r ∫ − τ 0 K 2 ( t , s ) ψ ( t ) d s d t ≤ ∫ − τ 0 ‖ ψ ‖ ∞ lim r → ∞ 1 2 r ∫ − r r ∣ K 2 ( t , s ) ∣ d t d s = 0 .

This means ϕ 2 , ϕ 3 ∈ PAP 0 ( R , R ) .□

Lemma 3.3

Let Ω 0 = { φ : φ ∈ BC ( [ − μ , 0 ] , R + ) , γ 1 < φ ( t ) < γ 2 , t ∈ [ − μ , 0 ] } , where γ 1 and γ 2 are given in (A5). Assume that (A1–A5) hold. Then, for any φ ∈ Ω 0 , the solution u ( t , t 0 , φ ) of equation (1.3) satisfies

γ 1 < u ( t , t 0 , φ ) < γ 2 , t ∈ [ t 0 , ζ ( φ ) )

and the existence interval of each solution to equation (1.3) can be extended to [ t 0 , ∞ ) .

Proof

The proof is inspired by [9, Lemma 3.1]. Denote u ( t ) = u ( t , t 0 , φ ) . Let [ t 0 , t * ) ⊆ [ t 0 , ζ ( φ ) ) . We claim that

(3.4) 0 < u ( t ) < γ 2 , ∀ t ∈ [ t 0 , t * ) .

Suppose (3.4) does not hold, then there exists t 1 ∈ ( t 0 , t * ) such that

u ( t 1 ) = γ 2 and 0 < u ( t ) < γ 2 , ∀ t ∈ [ t 0 − μ , t 1 ) .

Notice that

u ′ ( t 1 ) = lim t → t 1 u ( t ) − u ( t 1 ) t − t 1 ≥ 0 .

On the other hand, in view of (A1)–(A5), we obtain

u ′ ( t 1 ) = − α ( t 1 ) u ( t 1 ) + ∑ i = 1 n β i ( t 1 ) f i ( t 1 , u ( t − τ i ( t 1 ) ) ) + b ( t 1 ) H ( u ( t 1 − σ ) ) + β 0 ( t 1 ) ∫ − τ 0 K ( t 1 , s ) f 0 ( t 1 , u ( t 1 + s ) ) d s ≤ − α ̲ γ 2 + ∑ i = 1 n β i ¯ f i ¯ + b ¯ L H γ 2 + β 0 ¯ f 0 ¯ ‖ K ‖ ∞ τ < 0 ,

which is a contraction and then (3.4) holds.

Next we prove that

(3.5) u ( t ) > γ 1 , ∀ t ∈ [ t 0 , ζ ( φ ) ) .

Suppose (3.5) does not hold, then there exists t 2 ∈ ( t 0 , ζ ( φ ) ) such that

u ( t 2 ) = γ 1 and u ( t ) > γ 1 , ∀ t ∈ [ t 0 − μ , t 2 ) .

Notice that

u ′ ( t 2 ) = lim t → t 2 u ( t ) − u ( t 2 ) t − t 2 ≤ 0 .

On the other hand, in view of (A1)–(A5), we obtain

u ′ ( t 2 ) = − α ( t 2 ) u ( t 2 ) + ∑ i = 1 n β i ( t 2 ) f i ( t 2 , u ( t 2 − τ i ( t 2 ) ) ) + b ( t 2 ) H ( u ( t 2 − σ ) ) + β 0 ( t 2 ) ∫ − τ 0 K ( t 2 , s ) f 0 ( t 2 , u ( t 2 + s ) ) d s ≥ − α ¯ γ 1 + ∑ i = 1 n β i ̲ f i − + b ̲ L H γ 2 > 0 ,

which is a contraction and then (3.5) holds. Thus, it follows from continuation theorem [23, Theorem 2.3.1] that the existence interval of each solution to equation (1.3) can be extended to [ t 0 , ∞ ) .□

To achieve our main result, we define the operator Γ on PKAA ( R , R ) by:

( Γ u ) ( t ) = ∫ − ∞ t e − ∫ s t α ( ξ ) d ξ ( Nu ) ( s ) d s ,

where Nu is defined as follows:

(3.6) ( Nu ) ( s ) = ∑ i = 1 n β i ( s ) f i ( s , u ( s − τ i ( s ) ) ) + b ( s ) H ( u ( s − σ ) ) + β 0 ( s ) ∫ − τ 0 K ( s , ξ ) f 0 ( s , u ( s + ξ ) ) d ξ .

Theorem 3.4

Assume that (A1)–(A6) hold. If

(3.7) τ i ∈ C 1 ( R , R + ) and τ i ′ ( t ) ≤ τ * < 1 , i = 1 , 2 , … , n ,

and

(3.8) r = ∑ i = 1 n β i ¯ L f i + β 0 ¯ ‖ K ‖ ∞ τ L f 0 + b ¯ L H α ̲ < 1 ,

equation (1.3) has a unique solution in Ω = { u : u ∈ PKAA ( R , R ) , γ 1 ≤ u ( t ) ≤ γ 2 } .

Proof

It follows from Lemma 3.3 that the solution of equation (1.3) with initial condition (3.1) is in Ω , i.e., for all t ∈ [ t 0 , ∞ ) , γ 1 < u ( t ) < γ 2 whenever u satisfies (1.3) and (3.1). Obviously, the solution of equation (1.3) is a fixed point of the mapping Γ in Ω . Let us now prove that

Γ : PKAA ( R , R ) → PKAA ( R , R ) .

In fact, for any u ∈ PKAA ( R , R ) , Lemma 2.5 shows that

u ( ⋅ − τ i ( ⋅ ) ) ∈ PKAA ( R , R ) , i = 1 , 2 , … , n ,

since τ i ∈ PKAA ( R , R ) ∩ C 1 ( R , R + ) and τ ′ ( t ) ≤ τ * < 1 . Then it follows from composition Theorem (Lemma 2.4) that

f ( ⋅ , u ( ⋅ − τ i ( ⋅ ) ) ) ∈ PKAA ( R , R ) ,

and we obtain from [11, Theorem1] and Remark 2.2 that

H ( u ( ⋅ − σ ) ) ∈ PKAA ( R , R ) .

We now apply Lemma 2.3 (iii) and Lemma 3.2, and conclude that

β i ( ⋅ ) f i ( ⋅ , u ( ⋅ − τ i ( ⋅ ) ) ) ∈ PKAA ( R , R ) , i = 1 , 2 , … , n , β 0 ( ⋅ ) ∫ − τ 0 K ( ⋅ , ξ ) f 0 ( ⋅ , u ( ⋅ + ξ ) ) d ξ ∈ PKAA ( R , R ) ,

and

b ( ⋅ ) H ( u ( ⋅ − σ ) ) ∈ PKAA ( R , R ) ,

which yield that Nu belongs to PKAA ( R , R ) . Notice that e − ∫ s t α ( ξ ) d ξ is Bi-p.k.a.a. since α ∈ PKAA ( R , R ) by Proposition 2.12. Thus, by Proposition 2.14, it follows that

t ↦ ∫ − ∞ t e − ∫ s t α ( ξ ) d ξ ( Nu ) ( s ) d s belongs to PKAA ( R , R )

since e − ∫ s t α ( ξ ) d ξ ≤ e − α ̲ ( t − s ) , t ≥ s . Thus, Γ : PKAA ( R , R ) → PKAA ( R , R ) .

For any u ∈ Ω , in view of (A1)–(A5), we obtain

( Nu ) ( s ) = ∑ i = 1 n β i ( s ) f i ( s , u ( s − τ i ( s ) ) ) + b ( s ) H ( u ( s − σ ) ) + β 0 ( s ) ∫ − τ 0 K ( s , ξ ) f 0 ( s , u ( s + ξ ) ) d ξ ≤ ∑ i = 1 n β i ¯ f i ¯ + β 0 ¯ f 0 ¯ ‖ K ‖ ∞ τ + b ¯ L H γ 2 .

Then,

( Γ u ) ( t ) = ∫ − ∞ t e − ∫ s t α ( ξ ) d ξ ( Nu ) ( s ) d s ≤ ∫ − ∞ t e − α ̲ ( t − s ) ( Nu ) ( s ) d s ≤ ∑ i = 1 n β i ¯ f i ¯ + β 0 ¯ f 0 ¯ ‖ K ‖ ∞ τ + b ¯ L H γ 2 α ̲ < γ 2 .

On the other hand,

( Nu ) ( s ) = ∑ i = 1 n β i ( s ) f i ( s , u ( s − τ i ( s ) ) ) + b ( s ) H ( u ( s − σ ) ) + β 0 ( s ) ∫ − τ 0 K ( s , ξ ) f 0 ( s , u ( s + ξ ) ) d ξ ≥ ∑ i = 1 n β i ̲ f i − + b ̲ L H γ 2 .

Then,

( Γ u ) ( t ) = ∫ − ∞ t e − ∫ s t α ( ξ ) d ξ ( Nu ) ( s ) d s ≥ ∫ − ∞ t e − α ¯ ( t − s ) ( Nu ) ( s ) d s ≥ ∑ i = 1 n β i ̲ f i − + b ̲ L H γ 2 α ¯ > γ 1 .

That is, Γ maps Ω into Ω .

Now for any u , v ∈ Ω , by (A6), we obtain

∣ ( Nu ) ( s ) − ( N v ) ( s ) ∣ ≤ ∑ i = 1 n β i ( s ) ( f i ( s , u ( s − τ i ( s ) ) ) − f i ( s , v ( s − τ i ( s ) ) ) ) + β 0 ( s ) ∫ − τ 0 K ( s , ξ ) ( f 0 ( s , u ( s + ξ ) ) − f 0 ( s , v ( s + ξ ) ) ) d ξ + ∣ b ( s ) ( H ( u ( s − σ ) ) − H ( v ( s − σ ) ) ) ∣ ≤ ∑ i = 1 n β i ¯ L f i + β 0 ¯ ‖ K ‖ ∞ τ L f 0 + b ¯ L H ‖ u − v ‖ ∞ .

Then,

∣ ( Γ u ) ( t ) − ( Γ v ) ( t ) ∣ = ∫ − ∞ t e − ∫ s t α ( ξ ) d ξ [ ( Nu ) ( s ) − ( N v ) ( s ) ] d s ≤ ∫ − ∞ t e − α ̲ ( t − s ) ∣ ( Nu ) ( s ) − ( N v ) ( s ) ∣ d s ≤ ∑ i = 1 n β i ¯ L f i + β 0 ¯ ‖ K ‖ ∞ τ L f 0 + b ¯ L H α ̲ ‖ u − v ‖ ∞ .

This together with (3.8) implies that

‖ ( Γ u ) − ( Γ v ) ‖ ∞ ≤ ∑ i = 1 n β i ¯ L f i + β 0 ¯ ‖ K ‖ ∞ τ L f 0 + b ¯ L H α ̲ ‖ u − v ‖ ∞ = r ‖ u − v ‖ ∞ .

By Banach contraction mapping principle, Γ has a unique fixed point in Ω . Hence, equation (1.3) has a unique p.k.a.a. solution in Ω .□

Now we consider equation (1.1), where f i ( t , u ) = F i ( λ i ( t ) u ) for i = 1 , 2 , … , n . We assume that F i satisfies (A4′) and the following condition:

(A6′) For all 1 ≤ i ≤ n , there exist positive constants L F i such that for all x , y ∈ [ m * , ∞ ) ,

∣ F i ( x ) − F i ( y ) ∣ ≤ L F i ∣ x − y ∣ ,

where m * = max 1 ≤ i ≤ n m i * .

From Theorem 3.4, we obtain the following corollary.

Corollary 3.5

Assume that (A1), (A3), (A4′), and (A6′) hold. If

λ i ∈ PKAA ( R , R ) and λ i ̲ > 0 , τ i ∈ C 1 ( R , R + ) and τ i ′ ( t ) ≤ τ * < 1

for i = 1 , 2 , … , n and

r = ∑ i = 1 n β i ¯ λ i ¯ L F i + b ¯ L H α ̲ < 1 ,

equation (1.1)  has a unique solution in

Ω = { u : u ∈ PKAA ( R , R ) , γ 1 ≤ u ( t ) ≤ γ 2 } ,

where

max 1 ≤ i ≤ n m i * λ i ̲ < γ 1 < 1 α ¯ ∑ i = 1 n β i ̲ F i ( λ i ¯ γ 2 ) + b ̲ L H γ 2 , 1 α ̲ ∑ i = 1 n β i ¯ F i ¯ + b ¯ L H γ 2 < γ 2 .

Proof

In view of (A4′), we obtain

f i ¯ = F i ¯ = F i ( m i * ) with m i * = λ i ( t i ) n i + for some t i ∈ R .

So

n + = max 1 ≤ i ≤ n { n i + } ≤ max 1 ≤ i ≤ n m i * λ i ̲ .

Moreover, if max 1 ≤ i ≤ n m i * λ i ̲ < γ 1 , (A4′) implies that

f i − = F i ( λ i ¯ γ 2 )

since m i * < λ i ̲ u ≤ λ i ( t ) u ≤ λ i ¯ γ 2 . From ( A 6 ′ ) , we deduce that for all u , v ∈ [ n + , ∞ ) ,

∣ f i ( t , u ) − f i ( t , v ) ∣ = ∣ F i ( λ i ( t ) u ) − F i ( λ i ( t ) v ) ∣ ≤ L F i ∣ λ i ( t ) ( u − v ) ∣ ≤ L F i λ i ¯ ∣ u − v ∣ .

That is, L f i = L F i λ i ¯ . Thus, it follows from Theorem 3.4 that equation (1.1) has a unique positive solution in

Ω = { u : u ∈ PKAA ( R , R ) , γ 1 ≤ u ( t ) ≤ γ 2 } ,

where

□ max 1 ≤ i ≤ n m i * λ i ̲ < γ 1 < 1 α ¯ ∑ i = 1 n β i ̲ F i ( λ i ¯ γ 2 ) + b ̲ L H γ 2 , 1 α ̲ ∑ i = 1 n β i ¯ F i ¯ + b ¯ L H γ 2 < γ 2 .

Next we consider equation (1.2), where f 0 ( t , u ) = u e − u , f i ( t , u ) = u e − λ i ( t ) u , i = 1 , 2 , … , n , H ( u ( t ) ) = u ( t ) . From Theorem 3.4, we obtain the following corollary.

Corollary 3.6

Assume that (A1) and (A2) hold. If

λ i ∈ PKAA ( R , R ) and λ i ̲ > 0 , i = 1 , 2 , … , n ,

and

r = ∑ i = 1 n β i ¯ + β 0 ¯ ‖ K ‖ ∞ τ + b ¯ e 2 e 2 α ̲ < 1 ,

equation (1.2) has a unique solution in

Ω = { u : u ∈ PKAA ( R , R ) , γ 1 ≤ u ( t ) ≤ γ 2 } ,

where

1 min λ i ̲ < γ 1 < 1 α ¯ ∑ i = 1 n β i ̲ γ 2 e − λ i ¯ γ 2 − b ¯ γ 2 , 1 e α ̲ ∑ i = 1 n β i ¯ 1 λ i ̲ + β 0 ¯ ‖ K ‖ ∞ τ < γ 2 .

Proof

Obviously, (A3) and (A4) hold. By a simple computation, we have

f i ¯ = inf t ∈ R , s ∈ R f i ( t , u ) = 1 λ i ̲ e − 1 with n i + = 1 λ i ̲ , n + = max 1 ≤ i ≤ n { n i + } = 1 min λ i ̲ , f i − = inf t ∈ R , u ∈ [ n + , γ 2 ] f i ( t , u ) = γ 2 e − λ i ¯ γ 2 .

Notice that for all u , v ∈ [ 1 , ∞ ) ,

∣ u e − u − v e − v ∣ ≤ 1 e 2 ∣ u − v ∣ .

Then

∣ f i ( t , u ) − f i ( t , v ) ∣ = ∣ u e − λ i ( t ) u − v e − λ i ( t ) v ∣ ≤ 1 λ i ( t ) 1 e 2 ∣ λ i ( t ) ( u − v ) ∣ ≤ 1 e 2 ∣ u − v ∣ .

That is, L f i = 1 e 2 . Moreover, “ − b ( t ) ≤ 0 ,” which present the rate of extraction of the population, it follows from Theorem 3.4 that equation (1.2) has a unique positive solution in

Ω = { u : u ∈ PKAA ( R , R ) , γ 1 ≤ u ( t ) ≤ γ 2 } ,

where

□ 1 min λ i ̲ < γ 1 < 1 α ¯ ∑ i = 1 n β i ̲ γ 2 e − λ i ¯ γ 2 − b ¯ γ 2 , 1 e α ̲ ∑ i = 1 n β i ¯ 1 λ i ̲ + β 0 ¯ ‖ K ‖ ∞ τ < γ 2 .

Remark 3.7

  1. Theorem 3.4 shows that we extend the result of [11]. In [11], the model (without mixed delays) that is a mixture of the Nicholson model and the Lasota-Wazewska model was researched. Other useful population models can be considered in our results, such as mixed general Nicholson model and Lasota-Wazewska model with mixed delays (Example 5.1).

  2. Theorem 3.4 shows that we extend the result of [2,4,6,8,9,24]. In [2,4,9], the result of p.a.p. solutions to Nicholson’s blowflies model was obtained. In [6,8, 24], the result of periodic, a.p. and p.a.p. solutions to the Lasota-Wazewska model was obtained. In our result, we consider the p.k.a.a. solutions to a general model.

  3. Comparing with condition [11, A1], Corollary 3.5 shows that the condition “ α , τ i : R → R are positive k.a.a.” can be improved by the condition “ α , τ i : R → R are positive p.k.a.a.”

  4. Notice that n i + ≤ m i * λ i ̲ ≤ m * λ ̲ , where m * = max 1 ≤ i ≤ n { m i * } and λ ̲ = min 1 ≤ i ≤ n { λ i ̲ } . Therefore, n + ≤ m * λ ̲ . Comparing with condition [11, A5], Corollary 3.5 shows that we assume a lower bound for γ 1 . That is, the condition “ m * λ ̲ < γ 1 ” can be improved by the condition “ n + < γ 1 .”

  5. Comparing with [2], Corollary 3.6 shows that the ergodic components of α and delay kernel K can be considered in our result.

4 Global exponential stability and global exponential attractivity of p.k.a.a solution

In this section, we discuss the global exponential stability and global exponential attractivity conditions of the p.k.a.a solution to equation (1.3). Our result is based on Halanay’s inequality (Lemma 4.3).

Definition 4.1

[1] Let u * ( t ) , u ( t ) be solutions of equation (1.3) with initial condition u * ( s ) = φ * ( s ) and u ( s ) = ϕ ( s ) , s ∈ [ − μ , 0 ] .

  1. Suppose that there exists constant λ > 0 and M φ > 1 such that

    ∣ u ( t ) − u * ( t ) ∣ ≤ M φ ‖ φ − φ * ‖ μ e − λ t , t ≥ 0 ,

    where ‖ φ − φ * ‖ μ = sup − μ ≤ s ≤ 0 ∣ φ ( s ) − φ * ( s ) ∣ . Then u * is said to be globally exponential stable.

  2. Suppose that there exists ε > 0 such that

    e ε t ∣ u ( t ) − u * ( t ) ∣ → 0 ( t → + ∞ ) .

    Then u * is said to be globally exponential attractive.

Remark 4.2

Global exponential stability implies global exponential attractivity [1].

Lemma 4.3

[25] (Halanay’s inequality) Let t 0 be a real number and μ be a nonnegative number. If v : [ t 0 − μ , ∞ ) → R + satisfies

d d t v ( t ) ≤ − α v ( t ) + β sup s ∈ [ t − μ , t ] v ( s ) , t ≥ t 0 ,

where α and β are constants with α > β > 0 , then

v ( t ) ≤ ∣ v t 0 ∣ μ e − η ( t − t 0 ) , for t ≥ t 0 ,

where ∣ v t 0 ∣ μ = sup − μ ≤ θ ≤ 0 v t 0 ( θ ) and η is the unique positive solution of

η = α − β e η μ .

Theorem 4.4

Assume that (A1)–(A6) hold, also (3.7) and (3.8) hold. Then the unique p.k.a.a. solution of equation (1.3) in Ω is globally exponential stable.

Proof

Let u * ( t ) be the p.k.a.a. solution of (1.3) given in Theorem 3.4 and u ( t ) be an arbitrary solution of equation (1.3). Pose

z ( t ) = u ( t ) − u * ( t ) ,

with the initial condition

z ( s ) = θ ( s ) , s ∈ [ − μ , 0 ] .

Then

(4.1) z ˙ ( t ) = − α ( t ) z ( t ) + ( Nu ) ( t ) − ( Nu * ) ( t ) ,

where Nu is defined as (3.6).

∣ ( Nu ) ( s ) − ( Nu * ) ( s ) ∣ = ∑ i = 1 n β i ( t ) ( f i ( s , u ( s − τ i ( s ) ) ) − f i ( s , u * ( s − τ i ( s ) ) ) ) + β 0 ( t ) ∫ − τ 0 K ( s , ξ ) ( f 0 ( s , u ( s + ξ ) ) − f 0 ( s , u * ( s + ξ ) ) ) d ξ + b ( s ) H ( u ( s − σ ) − H ( u * ( s − σ ) ) ) ≤ ∑ i = 1 n β i ¯ L f i + β 0 ¯ ‖ K ‖ ∞ τ L f 0 + b ¯ L H sup r ∈ [ s − μ , s ] ∣ z ( r ) ∣ .

Then, for t ≥ 0 , we have

∣ z ( t ) ∣ = z ( 0 ) e − ∫ 0 t α ( ξ ) d ξ + ∫ 0 t e − ∫ s t α ( ξ ) d ξ ( ( Nu ) ( s ) − ( Nu * ) ( s ) ) d s ≤ ∣ z ( 0 ) ∣ e − ∫ 0 t α ( ξ ) d ξ + ∫ 0 t e − ∫ s t α ( ξ ) d ξ ∣ ( Nu ) ( s ) − ( Nu * ) ( s ) ∣ d s ≤ ∫ 0 t e − α ̲ ( t − s ) ∑ i = 1 n β i ¯ L f i + β 0 ¯ ‖ K ‖ ∞ τ L f 0 + b ¯ L H sup r ∈ [ s − μ , s ] ∣ z ( r ) ∣ d s + ‖ θ ‖ μ e − α ̲ t .

Thus,

∣ z ˙ ( t ) ∣ ≤ − α ̲ ∣ z ( t ) ∣ + ∑ i = 1 n β i ¯ L f i + β 0 ¯ ‖ K ‖ ∞ τ L f 0 + b ¯ L H sup r ∈ [ t − μ , t ] ∣ z ( r ) ∣ .

Notice that α ̲ > ∑ i = 1 n β i ¯ L f i + β 0 ¯ ‖ K ‖ ∞ τ L f 0 + b ¯ L H > 0 by (3.8), then it follows from Halanay’s inequality that there exists positive constants η such that

∣ z ( t ) ∣ ≤ ‖ θ ‖ μ e − η t , for t ≥ 0 ,

where η is the unique positive solution of

η = α − ∑ i = 1 n β i ¯ L f i + β 0 ¯ ‖ K ‖ ∞ τ L f 0 + b ¯ L H e η μ .

That is,

∣ u ( t ) − u * ( t ) ∣ ≤ ‖ θ ‖ μ e − η t , for t ≥ 0 .

Thus, there exists M > 1 such that

∣ u ( t ) − u * ( t ) ∣ ≤ ‖ θ ‖ μ e − η t ≤ M ‖ θ ‖ μ e − η t , for t ≥ 0 .

Hence, the p.k.a.a. solution u * ( t ) of equation (1.3) is globally exponentially stable.□

By Remark 4.2, we can obtain the following corollary.

Corollary 4.5

Assume that (A1)–(A6) hold, also (3.7) and (3.8) hold. Then the unique p.k.a.a. solution of equation (1.3) in Ω is globally attractive.

Remark 4.6

In [9], an additional condition is added to the existence conditions to obtain the global exponential stability by constructing a Lyapunov function. In [2], the globally attractivity of the positive solution is obtained without adding an additional condition to the existence conditions by inequality technique, but the exponential stability is not obtained. In our result, we obtain the globally exponential stability of the positive solution without adding an additional condition to the existence conditions in view of Halanay’s inequality.

5 Example

To illustrate our results, we consider a population model of mixed type with mixed delays in this section.

Example 5.1

We consider the following model with harvesting:

(5.1) u ˙ ( t ) = − α ( t ) u ( t ) + β i ( t ) ∑ i = 1 2 f i ( t , u ( t − τ i ( t ) ) ) + b ( t ) H ( u ( t − σ ) ) + β 0 ( t ) ∫ − τ 0 K ( t , s ) f 0 ( t , u ( t + s ) ) d s ,

where

β 1 ( t ) = 12 e 1.1 , β 2 ( t ) = e , β 0 ( t ) = 1 , b ( t ) = − 1 100 e , H ( u ( t − σ ) ) = u ( t − 1 ) , K ( t , s ) = e − ∣ cos ( t ) ∣ − 1 1 + t 2 + s , τ = 1 .

Let α 1 ( n ) = sign ( cos ( 2 2 π n ) ) , α 1 ( t ) is the linear extension of α 1 ( n ) over R . Then

α ( t ) = 14 + 1 10 α 1 ( t ) + 1 1 + t 2

is p.k.a.a. but not pseudo almost periodic since α 1 ( t ) is k.a.a but not almost periodic [26]. Notice that cos ( t ) + cos ( 2 t ) is almost periodic but not periodic. Then

τ 1 ( t ) = τ 2 ( t ) = 0.8 + 0.1 cos ( t ) + 0.1 cos ( 2 t ) + 1 1 + t 2

is p.k.a.a. but not pseudo periodic since the space of k.a.a. functions include almost periodic functions.

Case 1. We consider a model that is a mixture of Nicholson type with f 1 ( t , u ) = u e − 1.8 u and Lasota-Wazewska type with f 2 ( t , u ) = e − 0.7 u . Obviously, ( A 1 ) − ( A 4 ) hold. By a simple computation, we obtain

τ 1 ′ ( t ) = τ 2 ′ ( t ) = − 0.1 sin ( t ) − 0.1 * 2 sin ( t ) − 2 t ( 1 + t 2 ) 2 < 0.25 + 9 8 3 < 1 . α ¯ = 15.1 , α ̲ = 13.9 , β 1 ¯ = β 1 ̲ = 12 e 1.1 , β 2 ¯ = β 2 ̲ = e , β 0 ¯ = 1 , τ ¯ = 2 , b ̲ = − 1 100 e , L H = 1 , ‖ K ‖ ∞ = 1 . f 0 ¯ = e − 1 , f 1 ¯ = 1 1.8 e − 1 , f 2 ¯ = 1 , n 1 + = 1 1.8 , n 2 + = 0 , m 1 * = 1 , m 2 * = 0 , n + = max 1 1.8 , 0 = 1 1.8 , m * = max { 1 , 0 } = 1 , λ ̲ = min { 1.8 , 0.7 } = 0.7 .

Notice that for all u , v ∈ [ 1 1.8 , ∞ ) ,

∣ u e − u − v e − v ∣ ≤ 1 e 2 ∣ u − v ∣ , ∣ e − u − e − v ∣ ≤ 1 e 1 1.8 ∣ u − v ∣ .

It follows that for all u , v ∈ [ 1 1.8 , ∞ ) ,

L f 0 = e − 2 , L f 1 = e − 2 , L f 2 = 0.7 e 1 1.8 .

Since we consider a model with harvesting ( b ( t ) ≤ 0 ), we obtain

1 α ̲ ∑ i = 1 2 β i ¯ f i ¯ + β 0 ¯ f 0 ¯ ‖ K ‖ ∞ τ = 12 e 1.1 × 1 1.8 e − 1 + e + e − 1 13.9 ≈ 0.7521 < γ 2 ,

Now we fix γ 2 = 0.755 , it follows that

f 1 − = 0.755 e − 1.8 * 0.755 ≈ 0.1940 , f 2 − = e − 0.7 * 0.755 ≈ 0.5895 .

Thus,

0.5556 ≈ 1 1.8 = n + < γ 1 < 1 α ¯ ∑ i = 1 n β i ̲ f i − + b ̲ L H γ 2 ≈ 0.5690 .

We fix γ 1 = 0.5557 . That is, condition (A5) holds. Finally, we obtain

r = ∑ i = 1 n β i ¯ L f i + β 0 ¯ ‖ K ‖ ∞ τ L f 0 + b ¯ L H α ̲ ≈ 0.4395 < 1

Hence, all the conditions of Theorems 3.4 and 4.4 hold. Therefore, the model (5.1) has a unique p.k.a.a. solution that is globally exponentially stable (Figure 1 (a)) in the region

Ω 1 = { u : u ∈ PKAA ( R , R ) , 0.5557 ≤ u ( t ) ≤ 0.755 } .

Case 2. We consider a new model that is a mixture of general Nicholson type with f 1 ( t , u ) = sin u e − 1.8 u and Lasota-Wazewska type with f 2 ( t , u ) = e − 0.7 u . By a simple computation, we obtain

f 1 ¯ ≈ 0.1949 , f 2 ¯ = 1 , with n 1 + = arctan 1 1.8 , n 2 + = 0 , n + = max { arctan 1 1.8 , 0 } = arctan 1 1.8 .

Notice that for all u , v ∈ [ arctan 1 1.8 , ∞ ) ,

∣ u e − u − v e − v ∣ ≤ 1 e 2 ∣ u − v ∣ , ∣ e − u − e − v ∣ ≤ 1 e arctan 1 1.8 ∣ u − v ∣

and

∣ sin ( u ) e − 1.8 u − sin ( v ) e − 1.8 v ∣ ≤ cos ( θ ) − 1.8 sin ( θ ) e 1.8 θ θ = arctan ( 3.5 ) ∣ u − v ∣ .

It follows that for all u , v ∈ [ arctan 1 1.8 , ∞ ) ,

L f 0 = e − 2 , L f 1 = cos ( θ ) − 1.8 sin ( θ ) e 1.8 θ θ = arctan ( 3.5 ) ≈ 0.1611 , L f 2 = 0.7 e arctan 1 1.8 .

Since we consider a model with harvesting ( b ( t ) ≤ 0 ), we obtain

1 α ̲ ∑ i = 1 2 β i ¯ f i ¯ + β 0 ¯ f 0 ¯ ‖ K ‖ ∞ τ = 12 e 1.1 × 0.1949 + e + e − 1 13.9 ≈ 0.7275 < γ 2 ,

Now we fix γ 2 = 0.728 , it follows that

f 1 − = sin ( 0.728 ) e − 1.8 * 0.728 ≈ 0.1801 , f 2 − = e − 0.7 * 0.728 ≈ 0.6031 .

Thus,

0.5071 ≈ arctan 1 1.8 = n + < γ 1 < 1 α ¯ ∑ i = 1 n β i ̲ f i − + b ̲ L H γ 2 ≈ 0.5364 .

We fix γ 1 = 0.51 . That is, condition (A5) holds. Finally, we obtain

r = ∑ i = 1 n β i ¯ L f i + β 0 ¯ ‖ K ‖ ∞ τ L f 0 + b ¯ L H α ̲ ≈ 0.5102 < 1 .

Hence, all the conditions of Theorems 3.4 and 4.4 hold. Therefore, model (5.1) has a unique p.k.a.a. solution that is globally exponentially stable (Figure 1 (b)) in the region

Ω 2 = { u : u ∈ PKAA ( R , R ) , 0.51 ≤ u ( t ) ≤ 0.728 } .

Figure 1 Graphs of model (5.1).
Figure 1

Graphs of model (5.1).

Remark 5.1

  1. This example shows that our results are applicable not only to mixed Nicholson model and Lasota-Wazewska model but also to mixed general Nicholson model and Lasota-Wazewska model with mixed delays.

  2. Comparing with [11], we consider the ergodic components of α and the mixed delays with ergodic components. Moreover, we assume a lower bound for γ 1 in case 1 since n + = 1 1.8 < m * λ ̲ = 1 0.7 .

6 Conclusion

We study the existence, uniqueness, and global exponential stability for the pseudo compact automorphic solutions of a family of delay differential equations. By using the properties of p.k.a.a and Bi-p.k.a.a., some novel existence conditions for p.k.a.a. solutions of equations are obtained. Especially, the assumption for coefficients of equations “k.a.a.” can be weaken as “p.k.a.a.” In addition, by using Halanay’s inequality, the conditions for the global exponential stability of pseudo compact automorphic solutions of the equations without adding an additional condition to the existence conditions. The techniques and methods that we use here to study the existence of solutions can be extended to other models. Finally, an example is given to illustrate the validity of our results.

Acknowledgement

We would like to thank the the editors and referees greatly for their valuable comments and suggestions, which improve the quality of this article.

  1. Funding information: This work was supported by a Grant of NNSF of China (No. 11971329).

  2. Author contributions: All authors contributed to this research.

  3. Conflict of interest: The authors declare that they have no conflict of interest.

  4. Data availability statement: No data are associated with this research.

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Received: 2023-09-23
Revised: 2024-05-11
Accepted: 2024-06-17
Published Online: 2024-12-11

© 2024 the author(s), published by De Gruyter

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  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
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  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
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  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
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  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
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  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
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  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
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  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
https://doi.org/10.1515/dema-2024-0074
Keywords for this article
delay differential equations; pseudo compact almost automorphic solutions; Bi-pseudo compact almost automorphic; global exponential stability
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Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
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  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
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  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
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  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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