
In this paper, by using the Mawhin's continuation theorem, some easily verifiable sufficient conditions are obtained to guarantee the existence of almost periodic solutions of impulsive non-autonomous Lotka-Volterra predator-prey system with harvesting terms. Our result corrects the result obtained in [
Citation: Li Wang, Hui Zhang, Suying Liu. On the existence of almost periodic solutions of impulsive non-autonomous Lotka-Volterra predator-prey system with harvesting terms[J]. AIMS Mathematics, 2022, 7(1): 925-938. doi: 10.3934/math.2022055
[1] | Qi Xiao, Jin Zhong . Characterizations and properties of hyper-dual Moore-Penrose generalized inverse. AIMS Mathematics, 2024, 9(12): 35125-35150. doi: 10.3934/math.20241670 |
[2] | Waleed Mohamed Abd-Elhameed, Amr Kamel Amin, Nasr Anwer Zeyada . Some new identities of a type of generalized numbers involving four parameters. AIMS Mathematics, 2022, 7(7): 12962-12980. doi: 10.3934/math.2022718 |
[3] | Faik Babadağ . A new approach to Jacobsthal, Jacobsthal-Lucas numbers and dual vectors. AIMS Mathematics, 2023, 8(8): 18596-18606. doi: 10.3934/math.2023946 |
[4] | Changsheng Luo, Jiagui Luo . Complete solutions of the simultaneous Pell equations (a2+1)y2−x2=y2−bz2=1. AIMS Mathematics, 2021, 6(9): 9919-9938. doi: 10.3934/math.2021577 |
[5] | Cencen Dou, Jiagui Luo . Complete solutions of the simultaneous Pell's equations (a2+2)x2−y2=2 and x2−bz2=1. AIMS Mathematics, 2023, 8(8): 19353-19373. doi: 10.3934/math.2023987 |
[6] | Faik Babadağ, Ali Atasoy . A new approach to Leonardo number sequences with the dual vector and dual angle representation. AIMS Mathematics, 2024, 9(6): 14062-14074. doi: 10.3934/math.2024684 |
[7] | Elahe Mehraban, T. Aaron Gulliver, Salah Mahmoud Boulaaras, Kamyar Hosseini, Evren Hincal . New sequences from the generalized Pell p−numbers and mersenne numbers and their application in cryptography. AIMS Mathematics, 2024, 9(5): 13537-13552. doi: 10.3934/math.2024660 |
[8] | Bin Zhou, Xiujuan Ma, Fuxiang Ma, Shujie Gao . Robustness analysis of random hyper-networks based on the internal structure of hyper-edges. AIMS Mathematics, 2023, 8(2): 4814-4829. doi: 10.3934/math.2023239 |
[9] | Waleed Mohamed Abd-Elhameed, Anna Napoli . New formulas of convolved Pell polynomials. AIMS Mathematics, 2024, 9(1): 565-593. doi: 10.3934/math.2024030 |
[10] | Ümit Tokeşer, Tuğba Mert, Yakup Dündar . Some properties and Vajda theorems of split dual Fibonacci and split dual Lucas octonions. AIMS Mathematics, 2022, 7(5): 8645-8653. doi: 10.3934/math.2022483 |
In this paper, by using the Mawhin's continuation theorem, some easily verifiable sufficient conditions are obtained to guarantee the existence of almost periodic solutions of impulsive non-autonomous Lotka-Volterra predator-prey system with harvesting terms. Our result corrects the result obtained in [
Dual numbers were first given by Clifford (1845–1879), and some properties of those were studied in the geometrical investigation, and Kotelnikov [1] introduced their first applications. Study applied to line geometry and kinematics dual numbers and dual vectors [2]. He demonstrated that the directed lines of Euclidean 3-space and the points of the dual unit sphere in D3 have a one-to-one relationship. Field theory also relies heavily on these numbers [3]. The most intriguing applications of dual numbers in field theory are found in a number of Wald publications [4]. Dual numbers have contemporary applications in kinematics, dynamics, computer modeling of rigid bodies, mechanism design, and kinematics [5,6,7].
Complex numbers have significant advantages in derivative computations. However, the second derivative computations lost these advantages [8]. J. A. Fike developed the hyper-dual numbers to solve this issue [9]. These numbers may be used to calculate both the first and second derivatives while maintaining the benefits of the first derivative using complex numbers. Furthermore, it is demonstrated that this numerical approach is appropriate for open kinematic chain robot manipulators, sophisticated software, and airspace system analysis and design [10].
In the literature, sequences of integers have an important place. The most famous of these sequences have been demonstrated in several areas of mathematics. These sequences have been researched extensively because of their complex characteristics and deep connections to several fields of mathematics. The Fibonacci and Lucas sequences and their related numbers are of essential importance due to their various applications in biology, physics, statistics, and computer science [11,12,13]. Many authors were interested in introducing and investigating several generalizations and modifications of Fibonacci and Lucas sequences. The authors investigated two classes that generalize Fibonacci and Lucas sequences, and they utilized them to compute some radicals in reduced forms. Panwar [14] defined the generalized k-Fibonacci sequence as
Fk,n=pkFk,n−1+qFk,n−2, |
with initial conditions Fk,0=a and Fk,1=b. If a=0,k=2,p=q=b=1, the classic Pell sequence and for a=b=2,k=2,p=q=1, Pell-Lucas sequences appear.
The Pell numbers are the numbers of the following integer sequence:
0,1,2,5,12,29,70,169,408,985,2378,... |
The sequence of Pell numbers, which is denoted by Pn is defined as the linear reccurence relation
Pn=2Pn−1+Pn−2,P0=0,P1=1, n≥2. |
The integer sequence of Pell-Lucas numbers denoted by Qn is given by
2,2,6,14,34,82,198,478,1154,2786,6726,..., |
with the same reccurence relation
Qn=2Qn−1+Qn−2,Q0=Q1=2, n≥2. |
The characteristic equation of these numbers is x2−2x−1=0, with roots α=1+√2 and β=1−√2 and the Binet's forms of these sequences are given as[15,16,17,18],
Pn=αn−βnα−β | (1.1) |
and
Qn=αn+βn. | (1.2) |
The set of dual numbers is defined as
D={d=a+εa∗∣a,a∗∈R,ε2=0,ε≠0}. |
The set of hyper-dual numbers is
˜D={γ=γ0+γ1ε+γ2ε∗+γ3εε∗∣γ0,γ1,γ2,γ3∈R}, |
or can be rewritten as
˜D={γ=d+ε∗d∗∣d,d∗∈D}, |
where ε, ε∗ and εε∗ are hyper-dual units that satisfy
(ε)2=(ε∗)2=0,ε≠ε∗≠0,εε∗=ε∗ε. |
This set forms commutative and associative algebra over both the dual and real numbers [8,9,10].
The square root of a hyper-dual number γ can be defined by
√γ=√γ0+γ12√γ0ε+γ22√γ0ε∗+(γ32√γ0−γ1γ24γ0√γ0)εε∗. | (1.3) |
A hyper-dual vector is any vector of the form
→γ=→γ0+→γ1ε+→γ2ε∗+→γ3εε∗, |
where →γ0,→γ1,→γ2,→γ3 are real vectors, this vector can be rewritten as →γ=→d+ε∗→d∗, where →d and →d∗ are dual vectors. Let →γ and →δ be hyper-dual vectors, then their scalar product is defined as
⟨→γ,→δ⟩HD=⟨→γ0,→δ0⟩+(⟨→γ0,→δ1⟩+⟨→γ1,→δ0⟩)ε+(⟨→γ0,→δ2⟩+⟨→γ2,→δ0⟩)ε∗+(⟨→γ0,→δ3⟩+⟨→γ1,→δ2⟩+⟨→γ2,→δ1⟩+⟨→γ3,→δ0⟩)εε∗, | (1.4) |
which continents inner products of real vectors.
Let f(x0+x1ε+x2ε∗+x3εε∗) be a hyper-dual function, then
f(x0+x1ε+x2ε∗+x3εε∗)=f(x0)+x1f(x0)ε+x2f′(x0)ε∗+(x3f′(x0)+x1x2f″(x0))εε∗. | (1.5) |
Suppose →γ, →δ and Φ be unit hyper-dual vectors and hyper-dual angle respectively then by using (1.5) the scalar product can be written as
⟨→γ,→δ⟩HD=cosΦ=cosϕ−ε∗ϕ∗sinϕ=(cosψ−εψ∗sinψ)−ε∗ϕ∗(sinψ+εψ∗cosψ), | (1.6) |
where ϕ and ψ are, respectively, dual and real angles.
The norm of a hyper-dual vector →γ is given by
‖→γ‖HD=‖→γ0‖+⟨→γ0,→γ1⟩‖→γ0‖ε+⟨→γ0,→γ2⟩‖→γ0‖ε∗+(⟨→γ0,→γ3⟩‖→γ0‖+⟨→γ1,→γ2⟩‖→γ0‖−⟨→γ0,→γ1⟩⟨→γ0,→γ2⟩‖→γ0‖3)εε∗, |
for ‖→γ0‖≠0. If ‖→γ‖HD=1 that is ‖→γ0‖=1 and ⟨→γ0,→γ1⟩=⟨→γ0,→γ2⟩=⟨→γ0,→γ3⟩=⟨→γ1,→γ2⟩=0, then →γ is a unit hyper-dual vector.
In this paper, we introduce the hyper-dual Pell and the hyper-dual Pell-Lucas numbers, which provide a natural generalization of the classical Pell and Pell-Lucas numbers by using the concept of hyper-dual numbers. We investigate some basic properties of these numbers. We also define a new vector and angle, which are called hyper-dual Pell vector and angle. We give properties of these vectors and angles to exert in the geometry of hyper-dual space.
In this section, we define the hyper-dual Pell and hyper-dual Pell-Lucas numbers and then demonstrate their fundamental identities and properties.
Definition 2.1. The nth hyper-dual Pell HPn and hyper-dual Pell-Lucas HQn numbers are defined respectively as
HPn=Pn+Pn+1ε+Pn+2ε∗+Pn+3εε∗ | (2.1) |
and
HQn=Qn+εQn+1+ε∗Qn+2+εε∗Qn+3, | (2.2) |
where Pn and Qn are nth Pell and Pell-Lucas numbers.
The few hyper-dual Pell and hyper-dual Pell-Lucas numbers are given as
HP1=1+2ε+5ε∗+12εε∗,HP2=2+5ε+12ε∗+29εε∗,... |
and
HQ1=2+6ε+14ε∗+34εε∗,HQ2=6+14ε+34ε∗+82εε∗,... |
Theorem 2.1. The Binet-like formulas of the hyper-dual Pell and hyper-dual Pell-Lucas numbers are given, respectively, by
HPn=φnφ_−ψnψ_φ−ψ | (2.3) |
and
HQn=φnφ_+ψnψ_, | (2.4) |
where
φ_=1+φε+φ2ε∗+φ3εε∗,ψ_=1+ψε+ψ2ε∗+ψ3εε∗. | (2.5) |
Proof. From (2.1) and the Binet formula of Pell numbers, we obtain
HPn=Pn+Pn+1ε+Pn+2ε∗+Pn+3εε∗=φn−ψnφ−ψ+φn+1−ψn+1φ−ψε+φn+2−ψn+2φ−ψε∗+φn+3−ψn+3φ−ψεε∗=φn(1+φε+φ2ε∗+φ3εε∗)φ−ψ−ψn(1+ψε+ψ2ε∗+ψ3εε∗)φ−ψ=φnφ_−ψnψ_φ−ψ. |
On the other hand, using (2.2) and the Binet formula of Pell-Lucas numbers we obtain
HQn=Qn+Qn+1ε+Qn+2ε∗+Qn+3εε∗=(φn+ψn)+(φn+1+ψn+1)ε+(φn+2+ψn+2)ε∗+(φn+3+ψn+3)εε∗=φn(1+φε+φ2ε∗+φ3εε∗)+ψn(1+ψε+ψ2ε∗+ψ3εε∗)=φnφ_+ψnψ_. |
Theorem 2.2. (Vajda-like identities) For non-negative integers m, n, and r, we have
HPmHPn−HPm−rHPn+r=(−1)n+1Pm−n−rPr(1+2ε+6ε∗+12εε∗),HQmHQn−HQm−rHQn+r=(−1)nQm−n−(−1)n+rQm−n−2r(1+2ε+6ε∗+12εε∗). |
Proof. By using the Binet-like formula of hyper-dual Pell numbers, we obtain
HPmHPn−HPm−rHPn+r=(φmφ_−ψmψ_φ−ψ)(φnφ_−ψnψ_φ−ψ)−(φm−rφ_−ψm−rψ_φ−ψ)(φn+rφ_−ψn+rψ_φ−ψ)=(φr−ψr)(φnψm−r−ψnφm−r)(φ−ψ)2φ_ψ_=−(φm−n−r−ψm−n−r)(φr−ψr)(φ−ψ)2φ_ψ_, |
and by using (1.1), we obtain
HPmHPn−HPm−rHPn+r=(−1)n+1Pm−n−rPr(1+2ε+6ε∗+12εε∗). |
Similarly for hyper-dual Pell-Lucas numbers, we can obtain
HQmHQn−HQm−rHQn+r=(φmφ_+ψmψ_)(φnφ_+ψnψ_)−(φm−rφ_+ψm−rψ_)(φn+rφ_+ψn+rψ_)=φ_ψ_(φm−n+ψm−n−φm−n−2r−ψm−n−2r). |
Using (1.2) and (2.5),
HQmHQn−HQm−rHQn+r=(−1)nQm−n−(−1)n+rQm−n−2r(1+2ε+6ε∗+12εε∗). |
Thus, we obtain the desired results.
Theorem 2.3. (Catalan-like identities) For non negative integers n and r, with n≥r, we have
HPn−rHPn+r−HP2n=(−1)n−rP2r(1+2ε+6ε∗+12εε∗),HQn−rHQn+r−HQ2n=8(−1)n−rP2r(1+2ε+6ε∗+12εε∗). |
Proof. From (2.3), we obtain
HPn−rHPn+r−HP2n=(φn−rφ_−ψn−rψ_φ−ψ)(φn+rφ_−ψn+rψ_φ−ψ)−(φnφ_−ψnψ_φ−ψ)2=φnψn8φ_ψ_(2−ψrφ−r−ψ−rφr)=(−1)n−rφ_ψ_(φr−ψrφ−ψ)2, |
and by using (1.1) and (2.5), we will have
HPn−rHPn+r−HP2n=(−1)n−rP2r(1+2ε+6ε∗+12εε∗). |
On the other hand, from (2.4) and (2.5) we obtain
HQn−rHQn+r−HQ2n=(φn−rφ_+ψn−rψ_)(φn+rφ_+ψn+rψ_)−(φnφ_+ψnψ_)2=φ_ψ_(φn−rψn+r+φn+rψn−r−2ψnφn)=8(−1)n−rφ_ψ_(φr−ψrφ−ψ)2=8(−1)n−rP2r(1+2ε+6ε∗+12εε∗). |
Corollary 2.1. (Cassini-like identities) For non-negative integer n, we have
HPn−1HPn+1−HP2n=(−1)n−1(1+2ε+6ε∗+12εε∗),HQn−1HQn+1−HQ2n=8(−1)n−1(1+2ε+6ε∗+12εε∗). |
Proof. We can get the result by taking r=1 in Theorem 2.3.
Theorem 2.4. (d'Ocagne-like identities) For non-negative integers n and m,
HPm+1HPn−HPmHPn+1=(−1)mPn−m(1+2ε+6ε∗+12εε∗),HQm+1HQn−HQmHQn+1=8(−1)nPm−n(1+2ε+6ε∗+12εε∗). |
Proof. Using (1.1), (2.3), and (2.5), we have
HPm+1HPn−HPmHPn+1=(φm+1φ_−ψm+1ψ_φ−ψ)(φnφ_−ψnψ_φ−ψ)−(φmφ_−ψmψ_φ−ψ)(φn+1φ_−ψn+1ψ_φ−ψ)=(φ−ψ)(φnψm−φmψn)φ_ψ_=(−1)mPn−m(1+2ε+6ε∗+12εε∗). |
Using (1.2), (2.4) and (2.5), we have
HQm+1HQn−HQmHQn+1=8(−1)nPm−n(1+2ε+6ε∗+12εε∗). |
In this section, we introduce hyper-dual Pell vectors and hyper-dual Pell angle. We will give geometric properties of them.
Definition 3.1. The nth hyper-dual Pell vector is defined as
→HPn=→Pn+→Pn+1ε+→Pn+2ε∗+→Pn+3εε∗, |
where →Pn=(Pn,Pn+1,Pn+2) is a real Pell vector. The hyper-dual Pell vector →HPn can be rewritten in terms of dual Pell vectors →Pn and →P∗n as
→HPn=(→Pn+→Pn+1ε)+(→Pn+2+→Pn+3ε)ε∗=→Pn+ε∗→P∗n. |
Theorem 3.1. The scalar product of hyper-dual Pell vectors →HPn and →HPm is
⟨→HPn,→HPm⟩=7Qn+m+28−(−1)mQn−m8+(7Qn+m+34−(−1)mQn−m4)ε+(7Qn+m+44−3(−1)mQn−m4)ε∗+(7Qn+m+52−3(−1)mQn−m2)εε∗. | (3.1) |
Proof. By using (1.4), we can write
⟨→HPn,→HPm⟩=⟨→Pn,→Pm⟩+(⟨→Pn,→Pm+1⟩+⟨→Pn+1,→Pm⟩)ε+(⟨→Pn,→Pm+2⟩+⟨→Pn+2,→Pm⟩)ε∗+(⟨→Pn,→Pm+3⟩+⟨→Pn+1,→Pm+2⟩+⟨→Pn+2,→Pm+1⟩+⟨→Pn+3,→Pm⟩)εε∗. | (3.2) |
Now we calculate the above inner products for real Pell vectors →Pn and →Pm by using Binet's formula of Pell numbers as
⟨→Pn,→Pm⟩=PnPm+Pn+1Pm+1+Pn+2Pm+2=(φn−ψnφ−ψ)(φm−ψmφ−ψ)+(φn+1−ψn+1φ−ψ)(φm+1−ψm+1φ−ψ)+(φn+2−ψn+2φ−ψ)(φm+2−ψm+2φ−ψ)=φn+m+ψn+m(φ−ψ)2+φn+m+2+ψn+m+2(φ−ψ)2+φn+m+4+ψn+m+4(φ−ψ)2−(φnψm+φmψn)φ−mψ−m(φ−ψ)2φ−mψ−m=18(Qn+m+Qn+m+2+Qn+m+4+(−1)mQn−m)=7Qn+m+28−(−1)mQn−m8. |
⟨→Pn,→Pm+1⟩=7Qn+m+38+(−1)mQn−m−18,⟨→Pn+1,→Pm⟩=7Qn+m+38−(−1)mQn−m+18,⟨→Pn,→Pm+2⟩=7Qn+m+48−(−1)mQn−m−28,⟨→Pn+2,→Pm⟩=7Qn+m+48−(−1)mQn−m+28,⟨→Pn,→Pm+3⟩=7Qn+m+58+(−1)mQn−m−38,⟨→Pn+1,→Pm+2⟩=7Qn+m+58−(−1)mQn−m−18,⟨→Pn+2,→Pm+1⟩=7Qn+m+58+(−1)mQn−m+18,⟨→Pn+3,→Pm⟩=7Qn+m+58−(−1)mQn−m+38. |
By substituting these equalities in (3.2), we obtain the result.
Example 3.1. Let →HP1=(1,2,5)+(2,5,12)ε+(5,12,29)ε∗+(12,29,70)εε∗ and →HP0=(0,1,2)+(1,2,5)ε+(2,5,12)ε∗+(5,12,29)εε∗ be the hyper-dual Pell vectors. The scalar product of →HP1 and →HP0 are
⟨→HP1,→HP0⟩=7Q3−Q18+7Q4−Q14ε+7Q5−3Q14ε∗+7Q6−3Q12εε∗=12+59ε+142ε∗+690εε∗. |
By the other hand
⟨→HP1,→HP0⟩=⟨→P1,→P0⟩+(⟨→P1,→P1⟩+⟨→P2,→P0⟩)ε+(⟨→P1,→P2⟩+⟨→P3,→P0⟩)ε∗+(⟨→P1,→P3⟩+⟨→P2,→P2⟩+⟨→P3,→P1⟩+⟨→P4,→P0⟩)εε∗=12+(30+29)ε+(72+70)ε∗+(174+173+174+169)εε∗=12+59ε+142ε∗+690εε∗. |
The results are the same as we expected.
Corollary 3.1. The norm of →HPn is
‖→HPn‖2=⟨→HPn,→HPn⟩=7Q2n+28−(−1)n4+(7Q2n+34−(−1)n2)ε+(7Q2n+44−3(−1)n2)ε∗+(7Q2n+52−3(−1)n)εε∗. | (3.3) |
Proof. The proof is clear from taking m=n in (3.1).
Example 3.2. Find the norm of →HP1=(1,2,5)+(2,5,12)ε+(5,12,29)ε∗+(12,29,70)εε∗.
If we take n=1 in (3.3) and use (1.3), then we will get
‖→HP1‖=√7Q48+14+(7Q54+12)ε+(7Q64+32)ε∗+(7Q72+3)εε∗=√30+144ε+348ε∗+1676εε∗=√30+72√30ε+174√30ε∗+7345√30εε∗. |
From (1.6) and (3.1), the following cases can be given for the scalar product of hyper-dual Pell vectors →HPn and →HPm.
Case 3.1. Assume that cosϕ=0 and ϕ∗≠0, then ψ=π2, ψ∗=0, therefore
⟨→HPn,→HPm⟩=−ε∗ϕ∗=(7Qm+n+44−3(−1)mQn−m4)ε∗+(7Qm+n+52−3(−1)mQn−m2)εε∗, |
then, we get
ϕ∗=(−1)m(32+ε)−74(Qm+n+4+2εQm+n+5) |
and corresponding dual lines d1 and d2 are perpendicular such that they do not intersect each other; see Figure 1.
Case 3.2. Assume that ϕ∗=0 and ϕ≠0, then we obtain
⟨→HPn,→HPm⟩=cosϕ=(7Qm+n+28−(−1)mQn−m8)+(7Qm+n+34−(−1)mQn−m4)ε, |
therefore
ϕ=arccos((7Qm+n+28−(−1)mQn−m8)+(7Qm+n+34−(−1)mQn−m4)ε), |
and corresponding dual lines d1 and d2 intersect each other; see Figure 2.
Case 3.3. Assume that cosϕ=0 and ϕ∗=0, then ψ=π2 and ψ∗=0, therefore
⟨→HPn,→HPm⟩=0, |
and dual lines d1 and d2 intersect each other at a right angle; see Figure 3.
Case 3.4. Assume that ϕ=0 and ϕ∗=0, then
⟨→HPn,→HPm⟩=1, |
in this case corresponding dual lines d1 and d2 are parallel; see Figure 4.
In the present study, we introduce two families of hyper-dual numbers with components containing Pell and the Pell-Lucas numbers. First, we define hyper-dual Pell and Pell-Lucas numbers. Afterwards, by means of the Binet's formulas of Pell and Pell-Lucas numbers, we investigate identities such as the Binet-like formulas, Vajda-like, Catalan-like, Cassini-like, and d'Ocagne-like identities. After that, we define hyper-dual Pell vector and angle with some properties and geometric applications related to them. In the future it would be valuable to replicate a similar exploration and development of our findings on hyper-dual numbers with Pell and Pell-Lucas numbers. These results can trigger further research on the subjects of the hyper-dual numbers, vector, and angle to carry out in the geometry of dual and hyper-dual space.
Faik Babadağ and Ali Atasoy: Conceptualization, writing-original draft, writing-review, editing. All authors have read and approved the final version of the manuscript for publication.
The authors declare that they have no conflict of interest.
[1] | D. D. Bainov, P. S. Simeonov, Impulsive differential equations: Periodic solution and applications, London: Longman, 1993. doi: 10.1201/9780203751206. |
[2] | V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, Singpore: World Scientific, 1989. doi: 10.1142/0906. |
[3] | A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, Singpore: World Scientific, 1995. doi: 10.1142/2892. |
[4] |
X. N. Liu, L. S. Chen, Complex dynamics of Holling type Ⅱ Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos Solitons Fractals, 16 (2003), 311–320. doi: 10.1016/S0960-0779(02)00408-3. doi: 10.1016/S0960-0779(02)00408-3
![]() |
[5] |
Y. H. Xia, Global analysis of an impulsive delayed Lotka-Volterra competition system, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1597–1616. doi: 10.1016/j.cnsns.2010.07.014. doi: 10.1016/j.cnsns.2010.07.014
![]() |
[6] |
M. Liu, K. Wang, Asymptotic behavior of a stochastic nonautonomous Lotka-Volterra competitive system with impulsive perturbations, Math. Comput. Model., 57 (2013), 909–925. doi: 10.1016/j.mcm.2012.09.019. doi: 10.1016/j.mcm.2012.09.019
![]() |
[7] |
Q. Wang, B. X. Dai, Y. M. Chen, Multiple periodic solutions of an impulsive predator-prey model with Holling-type Ⅳ functional response, Math. Comput. Model., 49 (2009), 1829–1836. doi: 10.1016/j.mcm.2008.09.008. doi: 10.1016/j.mcm.2008.09.008
![]() |
[8] |
X. L. Hu, G. R. Liu, J. R. Yan, Existence of multiple positive periodic solutions of delayed predator-prey models with functional responses, Comput. Math. Appl., 52 (2006), 1453–1462. doi: 10.1016/j.camwa.2006.08.030. doi: 10.1016/j.camwa.2006.08.030
![]() |
[9] |
K. H. Zhao, Y. Ye, Four positive periodic solutions to a periodic Lotka-Volterra predatory-prey system with harvesting terms, Nonlinear Anal. Real World Appl., 11 (2010), 2448–2455. doi: 10.1016/j.nonrwa.2009.08.001. doi: 10.1016/j.nonrwa.2009.08.001
![]() |
[10] | K. H. Zhao, Y. K. Li, Multiple positive periodic solutions to a non-autonomous Lotka-Volterra predator-prey system with harvesting terms, Electron. J. Differ. Equ., 49 (2011), 1–11. |
[11] |
C. J. Xu, P. l. Li, Y. Guo, Global asymptotical stability of almost periodic solutions for a nonautonomous competing model with time-varying delays and feedback controls, J. Biol. Dyn., 13 (2019), 407–421. doi: 10.1080/17513758.2019.1610514. doi: 10.1080/17513758.2019.1610514
![]() |
[12] |
C. J. Xu, P. L. Li, Y. C. Pang, Existence and exponential stability of almost periodic solutions for neutral type BAM neural networks with distributed leakage delays, Math. Methods Appl. Sci., 40 (2017), 2177–2196. doi: 10.1002/mma.4132. doi: 10.1002/mma.4132
![]() |
[13] |
C. J. Xu, M. X. Liao, P. L. Li, Z. X. Liu, S. Yuan, New results on pseudo almost periodic solutions of quaternion-valued fuzzy cellular neural networks with delays, Fuzzy Sets Syst., 411 (2021), 25–47. doi: 10.1016/j.fss.2020.03.016. doi: 10.1016/j.fss.2020.03.016
![]() |
[14] |
G. Stamov, I. Stamov, A. Martynyuk, T. Stamov, Almost periodic dynamics in a new class of impulsive reaction-diffusion neural networks with fractional-like derivatives, Chaos Solitons Fractals, 143 (2021), 110647. doi: 10.1016/j.chaos.2020.110647. doi: 10.1016/j.chaos.2020.110647
![]() |
[15] |
C. Xu, M. Liao, P. Li, Q. Xiao, S. Yuan, A new method to investigate almost periodic solutions for an Nicholson's blowflies model with time-varying delays and a linear harvesting term, Math. Biosci. Eng., 16 (2019), 3830–3840. doi: 10.3934/mbe.2019189. doi: 10.3934/mbe.2019189
![]() |
[16] |
E. Kaslik, S. Sivasundaram, Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions, Nonlinear Anal. Real World Appl., 13 (2012), 1489–1497. doi: 10.1016/j.nonrwa.2011.11.013. doi: 10.1016/j.nonrwa.2011.11.013
![]() |
[17] |
J. O. Alzabut, G. T. Stamov, E. Sermutlu, Positive almost periodic solutions for a delay logarithmic population model, Math. Comput. Model., 53 (2011), 161–167. doi: 10.1016/j.mcm.2010.07.029. doi: 10.1016/j.mcm.2010.07.029
![]() |
[18] |
Y. Xie, X. Li, Almost periodic solutions of single population model with hereditary effects, Appl. Math. Comput., 203 (2008), 690–697. doi: 10.1016/j.amc.2008.05.085. doi: 10.1016/j.amc.2008.05.085
![]() |
[19] |
Y. K. Li, Y. Ye, Multiple positive almost periodic solutions to an impulsive non-autonomous Lotka-Volterra predator-prey system with harvesting terms, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3190–3201. doi: 10.1016/j.cnsns.2013.03.014. doi: 10.1016/j.cnsns.2013.03.014
![]() |
[20] |
L. Wang, M. Yu, Favard's theorem of piecewise continuous almost periodic functions and its application, J. Math. Anal. Appl., 413 (2014), 35–46. doi: 10.1016/j.jmaa.2013.11.029. doi: 10.1016/j.jmaa.2013.11.029
![]() |
[21] | C. Y. Zhang, Almost periodic type function and ergodicity, Springer, 2003. |
[22] | C. Y. He, Almost periodic differential equations, Beijing: Higher Education Press, 1992. |
[23] | C. Corduneanu, Almost periodic functions, 2 Eds, Chelsea, New York, 1989. |
[24] | T. Diagana, Almost automorphic type and almost periodic type functions in abstract spaces, New York: Springer-Verlag, 2013. doi: 10.1007/978-3-319-00849-3. |
[25] | A. M. Fink, Almost periodic differential equations, Berlin: Springer-Verlag, 1974. doi: 10.1007/BFb0070324. |
[26] | M. Kostić, Almost periodic and almost automorphic type solutions to integro-differential equations, Berlin: De Gruyter, 2019. doi: 10.1515/9783110641851. |
[27] | B. M. Levitan, Almost periodic functions, Gostekhi zdat, in Russian, Moscow, 1953. |
[28] | G. M. N'Guerekata, Almost automorphic and almost periodic functions in abstract spaces, Boston: Springer, 2001. doi: 10.1007/978-1-4757-4482-8. |
[29] | R. E. Gaines, J. L. Mawhin, Coincidence degree and nonlinear differential equation, Berlin: Springer-Verlay, 1977. doi: 10.1007/BFb0089537. |