具有脉冲的无限时滞系统的持久性与全局吸引性

kj(s)xj(t − s)ds, 1 ≤ i, j ≤ 2.
0
K
t
xi(t) =
hikxi(0) exp( Pi(s)ds) > 0, i = 1, 2.
0<tk <t
0
Ún1.2µ [5]b m ∈ P C[R+, R] §…§ ØëY:•t = tk¿…3t = tk(k = 1, 2, ...)´†ë Y " …k
…k."
-P C(J, R) = {φ : J → R, t = tk ž§φ ´ëY " …φ(t−k )Úφ(t+k )•3§¿…φ(t−k ) = φ(tk), k = 1, 2, ...}”dóÀ‡©•§ Ä ½nŒ•§XÚ(1.1) k˜‡•˜)x(t) = x(t, x0) ∈
P C([0, +∞), R+ × R+).
x˙ i(t) ≥ xi(t)[riL − aijM (Mj + ε) − (aiiM + biM (Mj + ε))xi(t)], i = j. •Ä±eXÚ
u˙ i(t) = ui(t)[riL − aijM (Mj + ε) − (aiiM + biM (Mj + ε))ui(t)], i = j, ui(t+k ) = hikui(tk), k = 1, 2, ...
1. Úó
3)ÔêÆ¥Logistic.®² @•´˜«›©-‡ .§NõUõ .®² Ñ éÐ (Ø" '•`§du‚¸ G!5K•Ú¢Dσ§žmò´Ò •Ä? Logistic . [1]- [3]"•C§3©z [4]¥§ŠöïÄ ˜aóÀž¢Logistic . ±È5Ú ÛáÚ5" ©·‚ïĘaäkóÀÚÕž¢ XÚ ±È5Ú ÛáÚ5§¤ (JU?Úí2 ®k ©z (Ø"
(1.2)
Ù¥x1(t)Úx2(t)´«+x1Úx2 3žmtž• «+—ݧri(t), aij(t),bi(t)(i, j = 1, 2)´ëY¼ê…
DOI: 10.12677/pm.2019.93050
378
nØêÆ
ÜX , oï|
éu¤kt > 0k.¶éu?¿‰½ ëY¼êf (t)§-fLÚfM ©OL«inf 0≤t<+∞ f (t) Úsup
Received: Apr. 26th, 2019; accepted: May 6th, 2019; published: May 21st, 2019
Abstract
In this paper, we study a system with impulsive and infinite delay. By using the comparison theorem of impulsive differential equations and constructing some suitable Lyapunov functionals, we discuss the permanence and global attractivity of the model.
0≤t<+∞
f (t)
¶ki
:
[0, +∞)
→
(0, +∞)(i
= 1, 2)´ëYؼê§=
+∞ 0
ki(s)ds
=
1; 0
< t1
<
t2
<
... < tk < tk+1 < ...´óÀž•…limk→+∞ tk = +∞ óÀ6Ä{hik : k = 1, 2, ...}(i = 1, 2) ´ S
(2.1)
¦
e−λi2(t−s) ≤
hik ≤ e−λi1(t−s).
s<tk <t
éuXÚ(1.1) (1.2) ?¿ )X(t) = (x1(t), x2(t))T ·‚Œ±
Ù¥§
mi ≤ lim inf xi(t) ≤ lim sup xi(t) ≤ Mi, i = 1, 2.
t→+∞
t→+∞
u˙ = g(t, u), t = tk, k = 1, 2, ..., u(t+k ) = φk(u(tk)), tk > t0, k = 1, 2, ..., u(t+0 ) = u0,
(1.4)
3[t0, +∞)þ•3"Km(t+0 ) ≤ u0 ¿›Xm(t) ≤ r(t), t ≥ t0. e Ø ª(1.3)‡ Ò§-p(t)• X Ú(1.4)3[t0, +∞) •
ÜX , oï|
Á‡
T©ïÄ äkóÀ Õž¢XÚ ±È5Ú ÛáÚ5" |^óÀ‡©•§Ø ª±9˜ E| ¤ E XÚ´±Y)• " EÜ· Lyapunov¼êÚ˜ ©ÛE|y²Ù ÛáÚ 5§·‚ (Jí2ÚU? ƒ'©z (J"
'…c
óÀ§ž¢§±È5§ ÛáÚ5
Copyright c 2019 by author(s) and Hans Publishers Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). /licenses/by/4.0/
=k§
a
lim sup y(t) ≤
,
t→+∞
r − λ2
ÓnŒ §
lim inf x(t) ≥ r − λ2 .
t→+∞
a
¤±§
y(t) ≥ y(T )e−(r−λ1)(t−T ) + a [1 − e−(r−λ1)(t−T )]. r − λ1
=k§ Úny."
a
lim inf y(t) ≥
,
t→+∞
ds
≤ y(T )eλ2(t−T )e−r(t−T ) + a
t T
e−(r−λ2)(t−s)ds
≤
y(T )e−(r−λ2)(t−T )
+
a r−λ2
[e−(r−λ2
)(t−t)
−
e(−(r−λ2)(t−T )]
=
y(T )e−(r−λ2)(t−T )
+
a r−λ2
[1
−
e−(r−λ2)(t−T )]
Permanence and Global Attractivity of an Impulsive Infinite Delay System
Ruyue Zhang, Jianli Li
Department of Mathematics, Hunan Normal University, Changsha Hunan
a
t→+∞
t→+∞
a
y²µ
x(t)
=
1 §KXÚz•
y(t)
é?¿T > 0§k
dy(t) dt
= −ry(t) + a, t = tk,
y(t+k )
=
1 hk
y
(tk
),
k
=
1, 2, ...
¤±§
y(t)
= y(T )
T <tk<t
1 hk
e−r(t−T
)
+
a
t T
s<tk <t
1 hk
e−r(t−s)
r − λ1
lim sup x(t) ≤ r − λ1 .
t→+∞
a
DOI: 10.12677/pm.2019.93050
380
nØêÆ
ÜX , oï|
5µdÚn ^‡(1.5)'©z [4] Lemma 2.1 ^‡(2.2)•Ð§Ïd·‚ (JU? ®k© z (Ø"
Ún1.4µ [6]éu?¿y ∈ P C([0, +∞), R+)-k : [0, +∞) → (0, +∞)•˜‡ëYؼê§=
Dm(t) ≤ g(t, m(t)), t = tk, k = 1, 2, ..., m(t+k ) ≤ φk(m(tk)), k = 1, 2, ...,
(1.3)
Ù¥g ∈ C[R+ × R+, R], φk ∈ C[R, R]¿…φk(u)éuz˜‡k = 1, 2, ... 'uu´Ø~ "-r(t)•ó À‡©•§ •Œ)
ω˙ i(t) = ωi(t)[riM − aiiLωi(t)], ωi(t+k ) = hikωi(tk), k = 1, 2, ...
d^‡(2.1)´
λi1 < riM §¤±dÚn1.3·‚Œ±
lim sup xi(t)
t→+∞
≤
lim sup ωi(t)
t→+∞
≤
Mi
=
riM − λi1 . aiiL
Ún1.1µ -x(t) = (x1(t), x2(t))T •XÚ(1.1) (1.2) ?¿˜‡)§Kxi(t) > 0, i = 1, 2, t ≥ 0. y²µ (1.1) 1i‡•§Œ± ¤Xe/ª
x˙ i(t) = Pi(t)xi(t), t = tk, i = 1, 2,
Ù¥
+∞
Pi(t) = ri(t) − aii(t)xi(t) − aij(t)xj(t) − bi(t)xi(t)