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Massera-Type theorem and asymptotically periodic Logistic equations
Nonlinear Analysis: Real World Applications 7 (2006) 1268 – 1283 www.elsevier.com/locate/na
Massera-Type theorem and asymptotically periodic Logistic equations夡 Gao Haiyina, b , Wang Kec , Wei Fengyinga,∗ , Ding Xiaohuac a School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, PR China b College of Applied Science, Changchun University, Changchun 130022, Jilin, PR China c Department of Mathematics, Harbin Institute of Technology, Weihai, 264209, Shandong, PR China
Received 17 September 2005; accepted 15 November 2005
Abstract Some important properties of asymptotically periodic functions were studied in this paper. Sufficient conditions of existence of globally stable asymptotically periodic solution were obtained. Then, Massera-Type theorems were discussed for one-dimensional, two-dimensional, higher-dimensional asymptotically periodic systems. Finally, global stability of periodic Logistic equations and asymptotically periodic Logistic equations were considered, respectively. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 34K25; 43A60 Keywords: Massera-Type theorem; Asymptotically periodic Logistic equations; Global stability
1. Introduction Periodic systems are a kind of very important systems in many scientific fields. A lot of work about periodic systems have been extensively discussed by many experts in recent years [1–6]. However, so far, the research work on asymptotically periodic systems is much fewer than periodic ones. In fact, asymptotically periodic systems describe our world more realistic and more accurate than periodic ones to some extent. Therefore, for asymptotically periodic systems, establishing the criteria of existence of asymptotically periodic solution is important and necessary. It is well known that Logistic equations play an important role in biological mathematics. But up to now, fewer results have been obtained for asymptotically periodic Logistic equations than for periodic ones. First of all, the definition of asymptotically periodic function is recalled, and after that, the authors investigate some important properties of asymptotically periodic functions, and have obtained the conclusions that asymptotically periodic Logistic equations have an globally stable asymptotically periodic solution. In 1950, Massera [1] considered scalar periodic systems x˙ = f (t, x),
(1.1)
夡 Supported by NNSF of PR China (Nos. 10171010 and 10201005), Key Project on Science and Technology of Education Ministry of PR China (No. 01061). ∗ Corresponding author. E-mail address: [email protected] (W. Fengying).
1468-1218/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2005.11.008
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where f : R × R → R is continuous, and f (t, x) = f (t + , x), > 0. He obtained a famous result for system (1.1), that is, if initial value solution of system (1.1) was unique, and system (1.1) had a positively bounded solution, then system (1.1) had an periodic solution. So far, within the knowledge of the authors, classical Massera-Type theorem had never been generalized to asymptotically periodic case. Obviously, such research work to extend Massera theorem to asymptotically periodic equations is important and necessary. Next, some Massera-Type theorems were discussed on scalar, two-dimensional and higher-dimensional asymptotically periodic systems, respectively. Finally, global stability of periodic Logistic equations and asymptotically periodic Logistic equations were considered, respectively. 2. Asymptotically periodic functions In this section, we introduced some concepts and notations as preliminary. Let R + = [0, +∞), R = (−∞, +∞). Definition 2.1. If f ∈ C(R + , R) and f (t) = g(t) + (t), and g(t) is continuous T-periodic function and limt→+∞ (t) = 0, then f (t) is called to be an asymptotically T-periodic function. Lemma 2.1. f is an asymptotically periodic function if and only if f is a bounded function and there exists T > 0 such that lim (f (t + T ) − f (t)) = 0.
(2.1)
t→+∞
Proof. (⇒) If f is an asymptotically periodic function, then we can suppose that f (t) = g(t) + (t) as in the Definition 2.1. Obviously f (t) is bounded on R + and lim (f (t + T ) − f (t)) = lim (g(t + T ) + (t + T ) − g(t) − (t))
t→+∞
t→+∞
= lim ((t + T ) − (t)) = 0. t→+∞
R+
(⇐) Suppose that f : → R is a bounded continuous function and expression (2.1) holds. Let fn (t) = f (t + nT ) and Fn (t) = supk n fk (t). Obviously, fn is a bounded continuous function on R + , and Fn (t) is also a bounded continuous function on R + with Fn+1 (t) Fn (t), t ∈ R + . From Dini theorem, the sequence {Fn (t)} uniformly converges to a bounded continuous function defined on R + on → (t) as n → +∞. any given bounded close subinterval of R + . This is denoted by Fn (t)− →F Consequently, for any given t ∈ R + , {Fn (t)} uniformly converges to F (t) on interval [t, t + T ]. By (2.1) and Heine theorem, we know that lim (f (t + T + nT ) − f (t + nT )) = 0.
n→+∞
Thus, we have F (t + T ) − F (t) =
lim (Fn (t + T ) − Fn (t))
n→+∞
= lim
n→+∞
sup f (t + T + kT ) − sup f (t + kT )
k n
k n
= lim sup (f (t + T + kT ) − f (t + kT )) n→+∞ k n
= lim sup(f (t + T + kT ) − f (t + kT )). k→+∞
It follows that F (t + T ) − F (t) = lim sup(f (t + (n + 1)T ) − f (t + nT )) n→+∞
= lim (f (t + (n + 1)T ) − f (t + nT )) = 0. n→+∞
Thus F (t) is a T-periodic function on R + .
(2.2)
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To show {Fn (t)} uniformly converges to F (t) on R + , let In = [nT , (n + 1)T ] and A(m, n) = sup (Fn (t) − F (t)). t∈Im
Since F (t) is a T-periodic function, A(m, n) = sup
t∈Im
sup f (t + kT ) − F (t) = sup
k n
t∈I0
sup f (t + kT ) − F (t)
k m+n
= A(0, m + n). It is easy to see that lims→+∞ A(0, s) = 0. For any given > 0, there exists an N ∈ N such that for any k N , A(0, s) = |A(0, s)| < .
(2.3)
By (2.3), for any t ∈ Im , when n N and then n + m > N, we obtain that A(m, n) = A(0, m + n) = sup |Fn (t) − F (t)| < . t∈Im
Since t ∈ R + is arbitrary, we have supt∈R + |Fn (t) − F (t)| < , for n N , that is, {Fn (t)} uniformly converges to F (t) on R + . By the similar method, letting Hn (t) = inf fk (t) = inf f (t + kT ), k n
k n
we can prove that {Hn (t)} uniformly converges to a continuous T-periodic function H (t) on R + . Clearly, Hn (t)Fn (t), and then H (t)F (t). By (2.1), for any given > 0, there exists an N ∈ N, such that for any n N , |fn+1 (t) − fn (t)| < , 2
t ∈ R+,
and fN (t) −
< Hn+1 (t)Fn+1 (t) < fN (t) + , 2 2
and then |Fn+1 (t) − Hn+1 (t)| < ,
t ∈ R + , nN .
It follows that |F (t) − H (t)|,
t ∈ R+,
Since > 0 is arbitrary, F (t) ≡ H (t), t ∈ R + . Next we will prove that f (t) − F (t) → 0, as t → +∞. Because of Fn (t) = sup fk (t) fn (t) inf fk (t)Hn (t), k n
k n
t ∈ R+,
and Fn (t), Hn (t) uniformly converge to F (t) on R + , by Sandwich theorem, we have → (t), fn (t)− →F
t ∈ R+.
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Thus, for any given > 0, there exists an N ∈ N, such that for any n N , |fn (t) − F (t)| < , 2
t ∈ R+,
and |f (t) − F (t)||f (t) − fN (t)| + |fN (t) − F (t)| < |f (t) − f (t + N T )| + . 2
(2.4)
From expression (2.1), for the above > 0, there exists T > 0 such that when t > T , |f (t + T ) − f (t)| <
. 2N
It follows that |f (t) − f (t + N T )| |f (t) − f (t + T )| + |f (t + T ) − f (t + 2T )| + · · · + |f (t + N T ) − f (t + (N − 1)T )| < . 2 Then, expression (2.4) implies that |f (t) − F (t)| <
+ = , 2 2
t >T,
and then limt→+∞ (f (t) − F (t)) = 0. From the fact that f (t) = F (t) + (f (t) − F (t)) and Definition 2.1, f (t) is an asymptotically periodic function. The proof is complete. Lemma 2.2. If f (t) and g(t) are asymptotically periodic functions, then f (t)g(t) is also an asymptotically periodic function. It is easy to check, so we omit the proof here.
3. Massera-Type theorems Definition 3.1. If f (t, x) ∈ C(R + × R n , R) and f (t, x) = g(t, x) + (t, x), and g(t, x) is continuous T-periodic function in t and limt→+∞ (t, x) = 0 uniformly on x ∈ H ⊂ , where is an open set on R n and H is a compact set, then f (t, x) is said to be an asymptotically T-periodic function in t. Suppose that f (t, x) is an asymptotically T-periodic function in t with f (t, x) = g(t, x) + (t, x) as in Definition 3.1. Consider the following two equations: u = f (t, u),
u ∈ ,
(3.1)
u = g(t, u),
u ∈ .
(3.2)
Theorem 3.1. If Eq. (3.1) has a positively bounded solution x(t), t ∈ R + , and Eq. (3.2) is globally stable, then Eq. (3.2) has an mT-periodic solution. If Eq. (3.1) has a positively bounded solution x(t), t ∈ R + , and Eq. (3.2) satisfies the uniqueness of solution with initial value, and any bounded solution of Eq. (3.2) is globally stable, then x(t) is an mT-asymptotically periodic solution of Eq. (3.1), where m ∈ N.
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Proof. Let |x(t)| M, t ∈ R + , and xn (t) = x(t + nT ), fn (t, x) = f (t + nT , x). Letting = t + nT , it is easy to show that dxn (t) dx(t + nT ) dx(t + nT ) dx() = = = dt dt d(t + nT ) d = f (, x()) = f (t + nT , x(t + nT )) = fn (t, xn (t)). Then x = xn (t) is a solution of equation x = fn (t, x). Noting that f (t, x) = g(t, x) + (t, x),
(3.3)
and g(t, x) is a continuous T-periodic function in t, it follows that |g(t, x)|
max
(t,x)∈[0,T ]×[−M,M]
|g(t, x)| < ∞.
By the known condition limt→+∞ (t, x) = 0, for x ∈ [−M, M], one derives that there exists an A > 0, such that for any t A, and x ∈ [−M, M], it follows that |(t, x)| 1. And again (t, x) is continuous function, |(t, x)|
max
(t,x)∈[0,A]×[−M,M]
|(t, x)| + 1 < ∞.
Thus f (t, x) is bounded on R + × [−M, M]. Then fn (t, x) is uniformly bounded on R + × [−M, M]. Therefore, the sequence {xn (t)} is an uniformly bounded and equicontinuous function sequence on R + . Let I be a finite closed subinterval on R + . There exists a subsequence {xni (t)} of {xn (t)} such that {xni (t)} uniformly converges to y(t) on I, that is, → xni (t)− →y(t),
i → +∞, t ∈ I .
It follows that
xni (t) = xni (0) +
t 0
fni (s, xni (s)) ds,
(3.4)
i = 1, 2, . . . , t ∈ I .
(3.5)
By expression (3.3) f (s, xni (s)) = g(s, xni (s)) + (s, xni (s)), then we obtain that fni (s, xni (s)) = f (s + ni T , xni (s)) = g(s + ni T , xni (s)) + (s + ni T , xni (s)) = g(s, xni (s)) + (s + ni T , xni (s)).
(3.6)
By expression (3.4) it is implied that → g(s, xni (s))− →g(s, y(s)),
i → +∞, s ∈ I ,
→ i → +∞, s ∈ I , by expression (3.6), one sees that again from (s + ni T , xni (s))− →0, → fni (s, xni (s))− →g(s, y(s)),
i → +∞, s ∈ I .
Letting i → +∞ on both sides of (3.5), one then derives t y(t) = y(0) + g(s, y(s)) ds, t ∈ I . 0
Thus, y(t) is a solution that satisfies the initial condition u(0) = y(0) of Eq. (3.2). Let yk (t) = y(t + kT ),
k = 1, 2, . . . .
(3.7)
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Obviously, yk (t) is the solution of Eq. (3.2) with the initial condition u(0) = y(0). Sequence {yk (t)} is uniformly bounded and equicontinuous on R + , then there exists an uniformly convergent subsequence {yki (t)}, i = 1, 2, . . . on any closed interval of R + , that is, → →z(t), yki (t)−
i → +∞.
Let m = k2 − k1 . Because Eq. (3.2) is globally stable, |y(t + mT ) − y(t)| → 0,
t → +∞.
(3.8)
→ →z(t), i → +∞, we have Noting that yki (t)− y(mT + ki T ) = yki (mT ) → z(mT ),
i → +∞,
that is, yki +m (0) → z(mT ),
i → +∞.
(3.9)
On the other hand, from expression (3.8), we get |y(ki T + mT ) − y(ki T )| → 0,
i → +∞,
that is, |yki +m (0) − yki (0)| → 0,
i → +∞.
(3.10)
From the expression yki (0) → z(0), i → +∞, it implies that |yki +m (0) − z(0)| |yki +m (0) − yki (0)| + |yki (0) − z(0)| → 0,
i → +∞,
therefore yki +m (0) → z(0), i → +∞. Applying (3.9) one then derives that z(0) = z(mT ). In other words, z(t) is an mT-periodic solution of Eq. (3.2). As in the proof above, xn (t) is the solution of equation
x = fn (t, x) = g(t, x) + (t + nT , xn (t)), (3.11) x(0) = x(nT ). Obviously, → →g(t, x), fn (t, x)−
n → +∞, x ∈ H .
Since {xn (t)} is uniformly bounded and equicontinuous on R + , there exists a uniformly convergent subsequence {xni (t)} on any finite closed interval of R + . Assume that → →y(t), xni (t)−
i → +∞, t ∈ I ,
where I is any finite subinterval of R + . By (3.11), one can get that t fni (s, xni (s)) ds. xni (t) = xni (0) + 0
Letting i → +∞ yields t g(s, y(s)) ds, y(t) = y(0) + 0
and then y(t) is the bounded solution of equation
u = g(t, u), u(0) = y(0).
(3.12)
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By the proof above, one can obtain that y(t) is a globally stable mT-periodic solution of Eq. (3.12). Then for any > 0, there exists T > 0 such that when t > T , |xn1 (t) − y(t)| < , that is, |x(t + n1 T ) − y(t)| = |x(t + n1 T ) − y(t + n1 T )| < . Now letting A = T + n1 T , we then obtain that |x(t) − y(t)| <
for t A.
It follows that y(t) is an asymptotically mT-periodic solution of Eq. (3.1). The proof is complete.
Theorem 3.2. Suppose n = 1 in Theorem 3.1 and f is an asymptotically T-periodic function in t with f (t, x) = g(t, x) + (t, x), and g(t, x), (t, x) satisfy g(t + T , x) = g(t, x), T > 0, limt→+∞ (t, x) = 0, x ∈ I ⊂ ⊂ R, where I is any finite closed interval on . If equation u = f (t, u),
u∈
(3.13)
has a positively bounded solution x(t), t ∈ R + , and equation u = g(t, u),
u∈
(3.14)
satisfies the uniqueness of solution with initial value, and any bounded solution of Eq. (3.14) is globally stable, then x(t) is a an asymptotically T-periodic solution of Eq. (3.13). Proof. Suppose |x(t)| M, t ∈ R + . Let xn (t) = x(t + nT ), fn (t, x) = f (t + nT , x), n = 1, 2, . . ., then dxn (t) = fn (t, xn (t)), dt that is to say, x = xn (t) is the solution of equation x = fn (t, x). Noting that f (t, x) = g(t, x) + (t, x), and g(t, x) is a continuous T-periodic function in t, and (t, x) is a continuous function with limt→+∞ (t, x) = 0, x ∈ [−M, M], then |g(t, x)|
max
(t,x)∈[0,T ]×[−M,M]
|g(t, x)| < ∞,
and there exists an A > 0, such that for any t A, x ∈ [−M, M], |(t, x)| 1, it follows that |(t, x)|
max
(t,x)∈[0,A]×[−M,M]
|(t, x)| + 1 < ∞,
therefore |f (t, x)| |g(t, x)| + |(t, x)| < ∞. Hence f (t, x) is bounded on R + × [−M, M]. Thus fn (t, x) is uniformly bounded on R + × [−M, M]. From Arzela–Ascoli lemma, there exists a continuous function y(t) defined on R + and a subsequence {xni (t)} of {xn (t)} such that {xni (t)} uniformly converges to a continuous function y(t) on any finite closed subinterval J of R + , which means that → →y(t), xni (t)−
i → +∞, t ∈ J .
One sees that xni (t) = xni (0) +
t 0
fni (s, xni (s)) ds,
(3.15)
i = 1, 2, . . . .
(3.16)
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From the fact that f (s, xni (s)) = g(s, xni (s)) + (s, xni (s)), we get fni (s, xni (s)) = f (s + ni T , xni (s)) = g(s, xni (s)) + (s + ni T , xni (s)).
(3.17)
By expression (3.15), it is easy to obtain that → →g(s, y(s)), g(s, xni (s))−
i → +∞, s ∈ J ,
applying the condition limi→+∞ (s + ni T , xni (s)) = 0, s ∈ J , and from expression (3.17), it follows that → →g(s, y(s)), fni (s, xni (s))−
i → +∞, s ∈ J .
Now letting i → +∞, (3.16) implies that t g(s, y(s)) ds, t ∈ J . y(t) = y(0) +
(3.18)
0
Since J is an arbitrary given interval of R + , for any t ∈ R + , expression (3.18) holds. Thus, y(t) is a bounded solution of equation
u (t) = g(t, u), (3.19) u(0) = y(0). From (3.15), it then yields |y(t)|M, t ∈ R + , where M > 0 is a constant. By Massera theorem, Eq. (3.19) has a T-periodic solution y(t). In view of the proof of Theorem 3.2, one sees that y(t) is globally stable, then for any > 0, there exists T > 0 such that when t > T , |xni (t) − y(t)| < , this deduces that |x(t + ni T ) − y(t)| = |x(t + ni T ) − y(t + ni T )| < . Taking B = T + ni T such that when t B, |x(t) − y(t)| < . Then we obtain that y(t) is an asymptotically T-periodic solution of (3.13). The proof is complete.
Theorem 3.3. Suppose that n = 2 in Theorem 3.1 and f (t, x) is an asymptotically T-periodic function with f (t, x) = g(t, x) + (t, x), and g(t, x) is a continuous T-periodic function in t, and (t, x) is continuous function with limt→+∞ (t, x) = 0, x ∈ I ⊂ , where I is any compact region of ⊂ R 2 . If equation u = f (t, u),
u∈
(3.20)
has a positively bounded solution x(t), t ∈ R + , and equation u = g(t, u),
u∈
(3.21)
satisfies the uniqueness of solution with initial value, and any bounded solution of Eq. (3.21) is globally stable, then x(t) is an asymptotically T-periodic solution of Eq. (3.20). Proof. Suppose |x(t)|M, for t ∈ R + . Let xn (t) = x(t + nT ), fn (t, x) = f (t + nT , x).
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It is easy to check that x = xn (t) is a solution of equation x = fn (t, x). From the known conditions of this theorem, one can then obtain that |f (t, x)|
max
(t,x)∈[0,T ]×[−M,M]
|g(t, x)| +
max
(t,x)∈[0,A]×[−M,M]
|(t, x)| + 1 < ∞.
Thus f (t, x) is bounded on R + × [−M, M], and so fn (t, x) is uniformly bounded on R + × [−M, M]. Therefore {xn (t)} is uniformly bounded and equicontinuous on R + . By Arzela–Ascoli theorem, there exist a continuous function y(t) defined on R + and a subsequence {xni (t)} of {xn (t)}, such that {xni (t)} uniformly converges to y(t) on any finite closed interval J of R + . This means → →y(t), xni (t)−
i → +∞, t ∈ J .
(3.22)
It is easy to show that xni (t) = xni (0) +
t 0
fni (s, xni (s)) ds,
i = 1, 2, . . . .
(3.23)
From the expression f (s, xni (s)) = g(s, xni (s)) + (s, xni (s)), one can then derive that fni (s, xni (s)) = f (s + ni T , xni (s)) = g(s, xni (s)) + (s + ni T , xni (s)).
(3.24)
Applying (3.22), one sees that → →g(s, y(s)), g(s, xni (s))−
i → +∞, s ∈ J ,
again, noting that limi→+∞ (s + ni T , xni (s)) = 0, s ∈ J , and by (3.24), it follows that → →g(s, y(s)), fni (s, xni (s))−
i → +∞, s ∈ J .
Now letting i → +∞ on both sides of (3.23), we obtain that t g(s, y(s)) ds, t ∈ J . y(t) = y(0) +
(3.25)
0
Since J is arbitrary interval of R + , for any t ∈ R + , (3.25) holds. In other words, y(t) is a bounded solution of equation
u (t) = g(t, u), (3.26) u(0) = y(0). From (3.22), immediately, we know that |y(t)|M, t ∈ R + , where M > 0 is a constant. If all positively solutions of two-dimensional T-periodic equation u = g(t, u) are bounded, then by Massera theorem, it has a T-periodic solution. Since any bounded solution y(t) of Eq. (3.21) is globally stable, in other words, for any > 0, there exists T > 0 such that when t > T , |xni (t) − y(t)| < , this means that |x(t + ni T ) − y(t)| = |x(t + ni T ) − y(t + ni T )| < . Take B = T + ni T such that when t B, |x(t) − y(t)| < . Thus y(t) is an asymptotically T-periodic solution of (3.20). The proof is complete.
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4. Asymptotically periodic Logistic equations Theorem 4.1. If r(t) and k(t) are all asymptotically T-periodic functions with r(t) > 0, k(t) > 0, then x(t) x (t) = r(t)x(t) 1 − k(t)
(4.1)
has an asymptotically T-periodic solution which is globally asymptotically stable. Proof. The solution of Eq. (4.1) is t −1 s e− 0 r(t−u) du h(t − s) ds , 0
where h(t) = r(t)/k(t) and denote it by f (t)−1 . By Lemma 2.2, we can only show that t s f (t) = e− 0 r(t−u) du h(t − s) ds 0
is an asymptotically T-periodic function. Since r(t), k(t) are all asymptotically T-periodic functions, h(t) is an asymptotically T-periodic function. It is easy to check that t M M |f (t)| Me−s ds = (1 − e−t ) , 0 where |h(t)|M, t ∈ R + . Assuming that r(t)N, k(t) N, t ∈ R + such that as t → +∞ t t s r(t − s) f (t) = e− 0 r(t−u) du e−Ns ds = 2 (1 − e−Nt ) → 2 > 0, ds k(t − s) N N N 0 0 it follows that lim inf f (t) t→+∞
> 0. N2
(4.2)
Obviously, f (t) is a positive continuous function on R + . By (4.2), then there exists > 0 such that f (t) > 0, t ∈ Therefore f −1 (t) is bounded continuous function. To show f (t + T ) − f (t) → 0, as t → +∞. The expression f (t + T ) − f (t) can also be expressed as f (t + T ) − f (t) = I1 + I2 + I3 , where t+T s e− 0 r(t+T −u) du h(t + T − s) ds, I1 = t t s I2 = e− 0 r(t+T −u) du [h(t + T − s) − h(t − s)] ds, 0 t s s I3 = [e− 0 r(t+T −u) du − e− 0 r(t−u) du ]h(t − s) ds.
R+.
0 s
We note that |e− 0 r(t+T −u) du h(t + T − s)|Me−s , and then t+T M Me−s ds = [e−t − e−(t+T ) ], |I1 | t therefore limt→+∞ I1 = 0. Since h(t) is an asymptotically T-periodic function, let (t) = h(t + T ) − h(t) → 0
as t → +∞,
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then
t
I2 =
e−
s
r(t+T −u) du
0
t
(t − s) ds =
0
e−
t−s 0
r(t+T −u) du
(s) ds.
0
Taking absolute value from two sides then yields
t
|I2 |
e
−
t−s 0
r(t+T −u) du
t
|(s)| ds
e
0
−(t−s)
0
1 |(s)| ds = t e
t
es |(s)| ds,
0
again taking limits, it implies that 1 t→+∞ et
t
lim |I2 | lim
t→+∞
If
+∞ s e |(s)| ds < ∞, then 0
1 t→+∞ et
t
es |(s)| ds = 0.
lim
If
es |(s)| ds.
0
0
+∞ s e |(s)| ds = ∞, then by L’ Hospital’s rule 0
1 lim t→+∞ et
et |(t)| 1 = lim |(t)| = 0. t→+∞ et t→+∞
t
es |(s)| ds = lim
0
Therefore, limt→+∞ I2 = 0. Since r(t) is an asymptotically T-periodic function, let (t) = r(t + T ) − r(t) → 0, as t → +∞.
t
I3 =
0
0
= =
t t
[e− [e− e−
s 0
r(t+T −u) du
− e−
s
0 (r(t−u)+(t−u)) du
s 0
r(t−u) du
[e−
s 0
s 0
r(t−u) du
− e−
s
(t−u) du
0
]h(t − s) ds
r(t−u) du
]h(t − s) ds
− 1]h(t − s) ds,
0
one then obtains that I3 M 0
=M
0
t t t
e− e−
s 0
s 0
r(t−u) du
|e−
r(t−u) du
|e−
t−s
s 0
(t−u) du
t
t−s
(u) du
− 1| ds
− 1| ds
t
e− 0 r(t−u) du |e− s (u) du − 1| ds 0 t
t
Me−t es e− s (u) du − 1 ds.
=M
0
(4.3)
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The derivative of the right part of (4.3) is d t s − t (u) du e |e s − 1| ds dt 0 t t j = es |e− s (u) du − 1| ds jt 0 t t j = es [(e− s (u) du − 1)2 ]1/2 ds jt 0 t t t − s (u) du − 1)e− s (u) du (−(t)) s 2(e e ds = t 0 2 (e− s (u) du − 1)2 t t t = es sign(e− s (u) du − 1)e− s (u) du (−(t)) ds. 0
By (4.3) and L’ Hospital’s rule, we can then derive that t M d s − st (u) du lim I3 lim · e |e − 1| ds t→+∞ t→+∞ et dt 0 t M s − st (u) du lim e e ds · |(t)| t→+∞ et 0 t t M = lim es e− s (u) du ds · lim |(t)|. t t→+∞ e t→+∞ 0 Let − (t)=max(0, −(t)). Obviously, − (t)0. Noting that (t) → 0, as t → +∞, one then sees that − (t) → 0, as t → +∞. Further, from e−
t s
(u) du
it follows that M 0 t e
t − e s (u) du ,
t
s −
e e 0
t s
(u) du
M ds t e
t
t − es e s (u) du ds.
0
Taking limits for the above expression, t t M M s − st (u) du s st − (u) du 0 lim e e ds lim e e ds. t→+∞ et 0 t→+∞ et 0 Let
t
F (t) = 0
t − es e s (u) du ds,
A = lim
t→+∞
M et
t
e s e −
t s
(u) du
(4.4)
ds.
0
Obviously, F (t) is a monotonous increasing function. If F (+∞) < ∞, then the right part of (4.4) is 0. If F (+∞)=∞, then by L’ Hospital’s rule, t t − M lim es e s (u) du ds t t→+∞ e 0 t M t s st − (u) du − = lim e + e e (t) ds t→+∞ 2 et 0 t M M s st − (u) du = lim + lim e e ds · − (t) t→+∞ 2 t→+∞ 2 et 0 t M − (t) M s st − (u) du e e ds · lim . = 2 + lim t→+∞ et 0 t→+∞
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G. Haiyin et al. / Nonlinear Analysis: Real World Applications 7 (2006) 1268 – 1283 −
Applying the conclusion that limt→+∞ (t) = 0, we get that t M M s st − (u) du lim e e ds = 2 . t→+∞ et 0 In other words, 0 A M/2 and then yields limt→+∞ I3 = A limt→+∞ |(t)| = 0, therefore limt→+∞ (f (t + T ) − f (t)) = 0. The proof is complete. 5. Global stability and ultimate boundedness The global stability and ultimate boundedness of nonautonomous Logistic equation are discussed in this section. Next we will, respectively, consider periodic, asymptotically periodic Logistic equation x(t)
x (t) = r(t)x(t) 1 − , (5.1) k(t) x(t)
x (t) = [r(t) + (t)]x(t) 1 − , (5.2) k(t) + (t) where r(t), k(t) are continuous periodic functions with (t) → 0, (t) → 0 as t → +∞. Assume that 0 < r∗ r(t) r ∗ < ∞, 0 < k∗ k(t) k ∗ < ∞, 0 < ∗ r(t) + (t) ∗ < ∞, 0 < ∗ k(t) + (t) ∗ < ∞. The solutions of Eqs. (5.1), (5.2) with initial value x0 at t0 , respectively, are −1 t t t r(s) − r(s) ds x1 (t, t0 , x0 ) = x0 e t0 + e− s r(u) du , ds k(s) t0 −1 t t t r(s) + (s) − [r(s)+(s)] ds x2 (t, t0 , x0 ) = x0 e t0 + e− s [r(u)+(u)] du . ds k(s) + (s) t0 Theorem 5.1. Eqs. (5.1) and (5.2) are global stability. Proof. Let x(t) be any solution of Eq. (5.1), X(t) = ln x(t), X ∗ (t) = ln x ∗ (t). Liapunov function for Eq. (5.1) is V1 (t, X(t), X∗ (t)) = |X(t) − X ∗ (t)|. It is easy to check that V1 (t, X(t), X∗ (t)) is differentiable and large enough. Now computing right-hand upper derivative V1 (t, X(t), X∗ (t)) along Eq. (5.1), X(t) − X ∗ (t) x(t) ˙ x˙ ∗ (t)
∗ V1 (t, X(t), X (t))| = − |X(t) − X ∗ (t)| x(t) x ∗ (t) r(t) X(t) − X ∗ (t) = − (x(t) − x ∗ (t)) k(t) |X(t) − X ∗ (t)| r(t) X X(t) − X ∗ (t) ∗ = − (e (t) − eX (t) ) . k(t) |X(t) − X ∗ (t)| Then there exists a function (t), which lies between X(t) and X ∗ (t), such that eX (t) − eX
∗ (t)
= e(t) (X(t) − X ∗ (t)),
therefore r(t) (t) |X(t) − X ∗ (t)|2 e k(t) |X(t) − X ∗ (t)| r(t) (t) e |X(t) − X ∗ (t)| 0. = − k(t)
V1 (t, X(t), X∗ (t))| = −
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The above expression is an equality if and only if X = X ∗ , that is, x(t) = x ∗ (t), thus V1 (t, X(t), X∗ (t)) is negative. Therefore Eq. (5.1) is bounded. If x(t) = x ∗ (t), then V1 = 0, from LaSalle invariant theorem, the maximum positive invariant set is a single point x(t) = x ∗ (t), therefore x(t) = x ∗ (t) is global asymptotically stable. Let x(t) be any solution of Eq. (5.2), X(t) = ln x(t), X ∗∗ (t) = ln x ∗∗ (t). Next Liapunov function of Eq. (5.2) is V2 (t, X(t), X∗∗ (t)) = |X(t) − X ∗∗ (t)|. It is easy to see that V2 (t, X(t), X∗∗ (t)) is differentiable and large enough. We compute right-hand upper derivative V2 (t, X(t), X∗∗ (t)) along Eq. (5.2) X(t) − X ∗∗ (t) x(t) ˙ x˙ ∗∗ (t) V2 (t, X(t), X∗∗ (t))| = − |X(t) − X ∗∗ (t)| x(t) x ∗∗ (t) r(t) X(t) − X ∗∗ (t) = − (x(t) − x ∗∗ (t)) k(t) |X(t) − X ∗∗ (t)| r(t) X X(t) − X ∗∗ (t) ∗∗ = − (e (t) − eX (t) ) . k(t) |X(t) − X ∗∗ (t)| Then there exists a function (t), which lies between X(t) and X ∗∗ (t), such that eX (t) − eX
∗∗ (t)
= e (t) (X(t) − X ∗∗ (t)),
therefore V2 (t, X(t), X∗∗ (t))| = −
r(t) (t) |X(t) − X ∗∗ (t)|2 r(t) (t) e =− e |X(t) − X ∗∗ (t)| 0. k(t) |X(t) − X ∗∗ (t)| k(t)
The above expression is an equality if and only if X = X ∗∗ , that is, x(t) = x ∗∗ (t), thus V2 (t, X(t), X∗∗ (t)) is negative. Therefore Eq. (5.2) is bounded. If x(t) = x ∗∗ (t), then V2 = 0, from LaSalle invariant theorem, the maximum positive invariant set is a single point x(t) = x ∗∗ (t), therefore x(t) = x ∗∗ (t) is global asymptotically stable. The proof is complete. Definition 5.1. Consider the system x˙ = f (t, x). For any > 0, (t0 , x0 ) ∈ J ×B , there exists T =T (t0 , x0 ) > 0 such that when t t0 +T , it follows that x(t, t0 , x0 ) ∈ BM , then the solution x(t, t0 , x0 ) is said to be ultimately bounded, where J = [t0 , +∞), B = {x : |x| }, M > 0 is a constant. Theorem 5.2. Eqs. (5.1) and (5.2) are ultimate boundedness. Proof. To check the solution of (4.1) is ultimately bounded, by Definition 5.1, just to determine the related T1 = T1 (t0 , x0 ) > 0. Let |x1 (t, t0 , x0 )|M1 , where M1 is a positive number. That is,
−1
t
− t r(s) ds t − s r(u) du r(s)
M1 .
x0 e t0 ds + e
k(s)
t0 In view of biological meaning of models, we omit the invalid case, and only investigate the following case: t r∗ 1 x0 e−(t−t0 )r∗ + e−(t−s)r∗ ds , k M ∗ 1 t0 that is, x0 −
r∗ k∗ r∗
e−(t−t0 )r∗
1 r∗ − . M1 k∗ r∗
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When x0 −
r∗ < 0, k∗ r∗
1 r∗ − < 0, M1 k∗ r∗
1 r∗ r∗ − < x0 − M1 k ∗ r∗ k∗ r∗
hold, then e−(t−t0 )r∗ D1 , where 1 r∗ r ∗ −1 D1 = x0 − − , M1 k ∗ r∗ k∗ r∗ taking logarithm from two sides and then −(t − t0 )r∗ ln D1 , it is easy to see that t − t0 −
ln D1 . r∗
Taking T1 = T1 (t0 , x0 ) = −
ln D1 > 0. r∗
Again by Definition 5.1, it yields that any solution of Eq. (5.1) through (t0 , x0 ) is ultimately bounded. From [6], if T k(t) > 0, 0 r(s) ds > 0, then Eq. (5.1) has a unique positive solution T
x ∗ (t) = e 0 r(s) ds − 1
t+T t
r(s) − t r() d ds e s k(s)
−1 ,
moreover, x ∗ (t) is globally stable for any solution x1 (t, t0 , x0 ) with initial value x(0) = x0 > 0 and lim |x1 (t, t0 , x0 ) − x ∗ (t)| = 0.
t→+∞
In other words, for any > 0, there exists T1 > 0, such that for any t > T1 , it follows that |x1 (t, t0 , x0 ) − x ∗ (t)| < . Taking T1∗ = max{T1 , T1 } > 0, by Definition 5.1, the solution x ∗ (t) of Eq. (5.1) is ultimately bounded. Similarly, for Eq. (5.2), we just have to find T2 = T2 (t0 , x0 ) > 0. Let |x2 (t, t0 , x0 )| M2 , by the analogous discussion, it follows that t ∗ 1 x0 e−(t−t0 ) ∗ + e−(t−s) ∗ ds . M ∗ 2 t0 We denote D2 = D2 (x0 ) :=
1 ∗ − M2 ∗ ∗
∗ −1 x0 − . ∗ ∗
Taking T2 = T2 (t0 , x0 ) = −
ln D2 > 0, ∗
in terms of Definition 5.1, any solution of Eq. (5.2) through (t0 , x0 ) is ultimately bounded. From [1], if inf t∈R r(t) > 0, supt∈R r(t) < ∞, inf t∈R k(t) > 0, supt∈R k(t) < ∞ are satisfied, then Eq. (5.2) has a global stable solution −1 t s ∗∗ − 0 [r(t−u)+(t−u)] du r(t − s) + (t − s) ds e , x (t) = k(t − s) + (t − s) 0 and the solution x2 (t, t0 , x0 ) of Eq. (5.2) with positive initial value x0 satisfies lim |x2 (t, t0 , x0 ) − x ∗∗ (t)| = 0.
t→+∞
G. Haiyin et al. / Nonlinear Analysis: Real World Applications 7 (2006) 1268 – 1283
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In other words, for any > 0, there exists T2 > 0 such that for any t > T2 , it follows that |x ∗∗ (t) − x2 (t, t0 , x0 )| < . Now taking T2∗ = max{T2 , T2 } > 0, from Definition 5.1, the solution x ∗∗ (t) of Eq. (5.2) is ultimately bounded. The proof is complete. References [1] B.D. Coleman, Nonautonomous Logistic equation as models of the adjustment of populations to environmental change, J. Math. Biosci. 45 (3–4) (1979) 159–173. [2] M. Fan, K. Wang, Optimal harvesting policy for single population with periodic coefficients, J. Math. Biosci. 152 (2) (1998) 165–177. [3] X. Liao, The Theory, Method and Application of Stability, Southeast University Press, WuHan, 1999. [4] Z. Ma, Population Ecology on Mathematics Mold and Research, Anhui Education Press, 1996. [5] J.L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J. 17 (1950) 457–475. [6] K. Wang, Extension of Massera theorem, J. Acta Math. Sci. 10 (1990) 197–199.
Massera-Type theorem and asymptotically periodic Logistic equations夡 Gao Haiyina, b , Wang Kec , Wei Fengyinga,∗ , Ding Xiaohuac a School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, PR China b College of Applied Science, Changchun University, Changchun 130022, Jilin, PR China c Department of Mathematics, Harbin Institute of Technology, Weihai, 264209, Shandong, PR China
Received 17 September 2005; accepted 15 November 2005
Abstract Some important properties of asymptotically periodic functions were studied in this paper. Sufficient conditions of existence of globally stable asymptotically periodic solution were obtained. Then, Massera-Type theorems were discussed for one-dimensional, two-dimensional, higher-dimensional asymptotically periodic systems. Finally, global stability of periodic Logistic equations and asymptotically periodic Logistic equations were considered, respectively. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 34K25; 43A60 Keywords: Massera-Type theorem; Asymptotically periodic Logistic equations; Global stability
1. Introduction Periodic systems are a kind of very important systems in many scientific fields. A lot of work about periodic systems have been extensively discussed by many experts in recent years [1–6]. However, so far, the research work on asymptotically periodic systems is much fewer than periodic ones. In fact, asymptotically periodic systems describe our world more realistic and more accurate than periodic ones to some extent. Therefore, for asymptotically periodic systems, establishing the criteria of existence of asymptotically periodic solution is important and necessary. It is well known that Logistic equations play an important role in biological mathematics. But up to now, fewer results have been obtained for asymptotically periodic Logistic equations than for periodic ones. First of all, the definition of asymptotically periodic function is recalled, and after that, the authors investigate some important properties of asymptotically periodic functions, and have obtained the conclusions that asymptotically periodic Logistic equations have an globally stable asymptotically periodic solution. In 1950, Massera [1] considered scalar periodic systems x˙ = f (t, x),
(1.1)
夡 Supported by NNSF of PR China (Nos. 10171010 and 10201005), Key Project on Science and Technology of Education Ministry of PR China (No. 01061). ∗ Corresponding author. E-mail address: [email protected] (W. Fengying).
1468-1218/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2005.11.008
G. Haiyin et al. / Nonlinear Analysis: Real World Applications 7 (2006) 1268 – 1283
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where f : R × R → R is continuous, and f (t, x) = f (t + , x), > 0. He obtained a famous result for system (1.1), that is, if initial value solution of system (1.1) was unique, and system (1.1) had a positively bounded solution, then system (1.1) had an periodic solution. So far, within the knowledge of the authors, classical Massera-Type theorem had never been generalized to asymptotically periodic case. Obviously, such research work to extend Massera theorem to asymptotically periodic equations is important and necessary. Next, some Massera-Type theorems were discussed on scalar, two-dimensional and higher-dimensional asymptotically periodic systems, respectively. Finally, global stability of periodic Logistic equations and asymptotically periodic Logistic equations were considered, respectively. 2. Asymptotically periodic functions In this section, we introduced some concepts and notations as preliminary. Let R + = [0, +∞), R = (−∞, +∞). Definition 2.1. If f ∈ C(R + , R) and f (t) = g(t) + (t), and g(t) is continuous T-periodic function and limt→+∞ (t) = 0, then f (t) is called to be an asymptotically T-periodic function. Lemma 2.1. f is an asymptotically periodic function if and only if f is a bounded function and there exists T > 0 such that lim (f (t + T ) − f (t)) = 0.
(2.1)
t→+∞
Proof. (⇒) If f is an asymptotically periodic function, then we can suppose that f (t) = g(t) + (t) as in the Definition 2.1. Obviously f (t) is bounded on R + and lim (f (t + T ) − f (t)) = lim (g(t + T ) + (t + T ) − g(t) − (t))
t→+∞
t→+∞
= lim ((t + T ) − (t)) = 0. t→+∞
R+
(⇐) Suppose that f : → R is a bounded continuous function and expression (2.1) holds. Let fn (t) = f (t + nT ) and Fn (t) = supk n fk (t). Obviously, fn is a bounded continuous function on R + , and Fn (t) is also a bounded continuous function on R + with Fn+1 (t) Fn (t), t ∈ R + . From Dini theorem, the sequence {Fn (t)} uniformly converges to a bounded continuous function defined on R + on → (t) as n → +∞. any given bounded close subinterval of R + . This is denoted by Fn (t)− →F Consequently, for any given t ∈ R + , {Fn (t)} uniformly converges to F (t) on interval [t, t + T ]. By (2.1) and Heine theorem, we know that lim (f (t + T + nT ) − f (t + nT )) = 0.
n→+∞
Thus, we have F (t + T ) − F (t) =
lim (Fn (t + T ) − Fn (t))
n→+∞
= lim
n→+∞
sup f (t + T + kT ) − sup f (t + kT )
k n
k n
= lim sup (f (t + T + kT ) − f (t + kT )) n→+∞ k n
= lim sup(f (t + T + kT ) − f (t + kT )). k→+∞
It follows that F (t + T ) − F (t) = lim sup(f (t + (n + 1)T ) − f (t + nT )) n→+∞
= lim (f (t + (n + 1)T ) − f (t + nT )) = 0. n→+∞
Thus F (t) is a T-periodic function on R + .
(2.2)
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To show {Fn (t)} uniformly converges to F (t) on R + , let In = [nT , (n + 1)T ] and A(m, n) = sup (Fn (t) − F (t)). t∈Im
Since F (t) is a T-periodic function, A(m, n) = sup
t∈Im
sup f (t + kT ) − F (t) = sup
k n
t∈I0
sup f (t + kT ) − F (t)
k m+n
= A(0, m + n). It is easy to see that lims→+∞ A(0, s) = 0. For any given > 0, there exists an N ∈ N such that for any k N , A(0, s) = |A(0, s)| < .
(2.3)
By (2.3), for any t ∈ Im , when n N and then n + m > N, we obtain that A(m, n) = A(0, m + n) = sup |Fn (t) − F (t)| < . t∈Im
Since t ∈ R + is arbitrary, we have supt∈R + |Fn (t) − F (t)| < , for n N , that is, {Fn (t)} uniformly converges to F (t) on R + . By the similar method, letting Hn (t) = inf fk (t) = inf f (t + kT ), k n
k n
we can prove that {Hn (t)} uniformly converges to a continuous T-periodic function H (t) on R + . Clearly, Hn (t)Fn (t), and then H (t)F (t). By (2.1), for any given > 0, there exists an N ∈ N, such that for any n N , |fn+1 (t) − fn (t)| < , 2
t ∈ R+,
and fN (t) −
< Hn+1 (t)Fn+1 (t) < fN (t) + , 2 2
and then |Fn+1 (t) − Hn+1 (t)| < ,
t ∈ R + , nN .
It follows that |F (t) − H (t)|,
t ∈ R+,
Since > 0 is arbitrary, F (t) ≡ H (t), t ∈ R + . Next we will prove that f (t) − F (t) → 0, as t → +∞. Because of Fn (t) = sup fk (t) fn (t) inf fk (t)Hn (t), k n
k n
t ∈ R+,
and Fn (t), Hn (t) uniformly converge to F (t) on R + , by Sandwich theorem, we have → (t), fn (t)− →F
t ∈ R+.
G. Haiyin et al. / Nonlinear Analysis: Real World Applications 7 (2006) 1268 – 1283
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Thus, for any given > 0, there exists an N ∈ N, such that for any n N , |fn (t) − F (t)| < , 2
t ∈ R+,
and |f (t) − F (t)||f (t) − fN (t)| + |fN (t) − F (t)| < |f (t) − f (t + N T )| + . 2
(2.4)
From expression (2.1), for the above > 0, there exists T > 0 such that when t > T , |f (t + T ) − f (t)| <
. 2N
It follows that |f (t) − f (t + N T )| |f (t) − f (t + T )| + |f (t + T ) − f (t + 2T )| + · · · + |f (t + N T ) − f (t + (N − 1)T )| < . 2 Then, expression (2.4) implies that |f (t) − F (t)| <
+ = , 2 2
t >T,
and then limt→+∞ (f (t) − F (t)) = 0. From the fact that f (t) = F (t) + (f (t) − F (t)) and Definition 2.1, f (t) is an asymptotically periodic function. The proof is complete. Lemma 2.2. If f (t) and g(t) are asymptotically periodic functions, then f (t)g(t) is also an asymptotically periodic function. It is easy to check, so we omit the proof here.
3. Massera-Type theorems Definition 3.1. If f (t, x) ∈ C(R + × R n , R) and f (t, x) = g(t, x) + (t, x), and g(t, x) is continuous T-periodic function in t and limt→+∞ (t, x) = 0 uniformly on x ∈ H ⊂ , where is an open set on R n and H is a compact set, then f (t, x) is said to be an asymptotically T-periodic function in t. Suppose that f (t, x) is an asymptotically T-periodic function in t with f (t, x) = g(t, x) + (t, x) as in Definition 3.1. Consider the following two equations: u = f (t, u),
u ∈ ,
(3.1)
u = g(t, u),
u ∈ .
(3.2)
Theorem 3.1. If Eq. (3.1) has a positively bounded solution x(t), t ∈ R + , and Eq. (3.2) is globally stable, then Eq. (3.2) has an mT-periodic solution. If Eq. (3.1) has a positively bounded solution x(t), t ∈ R + , and Eq. (3.2) satisfies the uniqueness of solution with initial value, and any bounded solution of Eq. (3.2) is globally stable, then x(t) is an mT-asymptotically periodic solution of Eq. (3.1), where m ∈ N.
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Proof. Let |x(t)| M, t ∈ R + , and xn (t) = x(t + nT ), fn (t, x) = f (t + nT , x). Letting = t + nT , it is easy to show that dxn (t) dx(t + nT ) dx(t + nT ) dx() = = = dt dt d(t + nT ) d = f (, x()) = f (t + nT , x(t + nT )) = fn (t, xn (t)). Then x = xn (t) is a solution of equation x = fn (t, x). Noting that f (t, x) = g(t, x) + (t, x),
(3.3)
and g(t, x) is a continuous T-periodic function in t, it follows that |g(t, x)|
max
(t,x)∈[0,T ]×[−M,M]
|g(t, x)| < ∞.
By the known condition limt→+∞ (t, x) = 0, for x ∈ [−M, M], one derives that there exists an A > 0, such that for any t A, and x ∈ [−M, M], it follows that |(t, x)| 1. And again (t, x) is continuous function, |(t, x)|
max
(t,x)∈[0,A]×[−M,M]
|(t, x)| + 1 < ∞.
Thus f (t, x) is bounded on R + × [−M, M]. Then fn (t, x) is uniformly bounded on R + × [−M, M]. Therefore, the sequence {xn (t)} is an uniformly bounded and equicontinuous function sequence on R + . Let I be a finite closed subinterval on R + . There exists a subsequence {xni (t)} of {xn (t)} such that {xni (t)} uniformly converges to y(t) on I, that is, → xni (t)− →y(t),
i → +∞, t ∈ I .
It follows that
xni (t) = xni (0) +
t 0
fni (s, xni (s)) ds,
(3.4)
i = 1, 2, . . . , t ∈ I .
(3.5)
By expression (3.3) f (s, xni (s)) = g(s, xni (s)) + (s, xni (s)), then we obtain that fni (s, xni (s)) = f (s + ni T , xni (s)) = g(s + ni T , xni (s)) + (s + ni T , xni (s)) = g(s, xni (s)) + (s + ni T , xni (s)).
(3.6)
By expression (3.4) it is implied that → g(s, xni (s))− →g(s, y(s)),
i → +∞, s ∈ I ,
→ i → +∞, s ∈ I , by expression (3.6), one sees that again from (s + ni T , xni (s))− →0, → fni (s, xni (s))− →g(s, y(s)),
i → +∞, s ∈ I .
Letting i → +∞ on both sides of (3.5), one then derives t y(t) = y(0) + g(s, y(s)) ds, t ∈ I . 0
Thus, y(t) is a solution that satisfies the initial condition u(0) = y(0) of Eq. (3.2). Let yk (t) = y(t + kT ),
k = 1, 2, . . . .
(3.7)
G. Haiyin et al. / Nonlinear Analysis: Real World Applications 7 (2006) 1268 – 1283
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Obviously, yk (t) is the solution of Eq. (3.2) with the initial condition u(0) = y(0). Sequence {yk (t)} is uniformly bounded and equicontinuous on R + , then there exists an uniformly convergent subsequence {yki (t)}, i = 1, 2, . . . on any closed interval of R + , that is, → →z(t), yki (t)−
i → +∞.
Let m = k2 − k1 . Because Eq. (3.2) is globally stable, |y(t + mT ) − y(t)| → 0,
t → +∞.
(3.8)
→ →z(t), i → +∞, we have Noting that yki (t)− y(mT + ki T ) = yki (mT ) → z(mT ),
i → +∞,
that is, yki +m (0) → z(mT ),
i → +∞.
(3.9)
On the other hand, from expression (3.8), we get |y(ki T + mT ) − y(ki T )| → 0,
i → +∞,
that is, |yki +m (0) − yki (0)| → 0,
i → +∞.
(3.10)
From the expression yki (0) → z(0), i → +∞, it implies that |yki +m (0) − z(0)| |yki +m (0) − yki (0)| + |yki (0) − z(0)| → 0,
i → +∞,
therefore yki +m (0) → z(0), i → +∞. Applying (3.9) one then derives that z(0) = z(mT ). In other words, z(t) is an mT-periodic solution of Eq. (3.2). As in the proof above, xn (t) is the solution of equation
x = fn (t, x) = g(t, x) + (t + nT , xn (t)), (3.11) x(0) = x(nT ). Obviously, → →g(t, x), fn (t, x)−
n → +∞, x ∈ H .
Since {xn (t)} is uniformly bounded and equicontinuous on R + , there exists a uniformly convergent subsequence {xni (t)} on any finite closed interval of R + . Assume that → →y(t), xni (t)−
i → +∞, t ∈ I ,
where I is any finite subinterval of R + . By (3.11), one can get that t fni (s, xni (s)) ds. xni (t) = xni (0) + 0
Letting i → +∞ yields t g(s, y(s)) ds, y(t) = y(0) + 0
and then y(t) is the bounded solution of equation
u = g(t, u), u(0) = y(0).
(3.12)
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By the proof above, one can obtain that y(t) is a globally stable mT-periodic solution of Eq. (3.12). Then for any > 0, there exists T > 0 such that when t > T , |xn1 (t) − y(t)| < , that is, |x(t + n1 T ) − y(t)| = |x(t + n1 T ) − y(t + n1 T )| < . Now letting A = T + n1 T , we then obtain that |x(t) − y(t)| <
for t A.
It follows that y(t) is an asymptotically mT-periodic solution of Eq. (3.1). The proof is complete.
Theorem 3.2. Suppose n = 1 in Theorem 3.1 and f is an asymptotically T-periodic function in t with f (t, x) = g(t, x) + (t, x), and g(t, x), (t, x) satisfy g(t + T , x) = g(t, x), T > 0, limt→+∞ (t, x) = 0, x ∈ I ⊂ ⊂ R, where I is any finite closed interval on . If equation u = f (t, u),
u∈
(3.13)
has a positively bounded solution x(t), t ∈ R + , and equation u = g(t, u),
u∈
(3.14)
satisfies the uniqueness of solution with initial value, and any bounded solution of Eq. (3.14) is globally stable, then x(t) is a an asymptotically T-periodic solution of Eq. (3.13). Proof. Suppose |x(t)| M, t ∈ R + . Let xn (t) = x(t + nT ), fn (t, x) = f (t + nT , x), n = 1, 2, . . ., then dxn (t) = fn (t, xn (t)), dt that is to say, x = xn (t) is the solution of equation x = fn (t, x). Noting that f (t, x) = g(t, x) + (t, x), and g(t, x) is a continuous T-periodic function in t, and (t, x) is a continuous function with limt→+∞ (t, x) = 0, x ∈ [−M, M], then |g(t, x)|
max
(t,x)∈[0,T ]×[−M,M]
|g(t, x)| < ∞,
and there exists an A > 0, such that for any t A, x ∈ [−M, M], |(t, x)| 1, it follows that |(t, x)|
max
(t,x)∈[0,A]×[−M,M]
|(t, x)| + 1 < ∞,
therefore |f (t, x)| |g(t, x)| + |(t, x)| < ∞. Hence f (t, x) is bounded on R + × [−M, M]. Thus fn (t, x) is uniformly bounded on R + × [−M, M]. From Arzela–Ascoli lemma, there exists a continuous function y(t) defined on R + and a subsequence {xni (t)} of {xn (t)} such that {xni (t)} uniformly converges to a continuous function y(t) on any finite closed subinterval J of R + , which means that → →y(t), xni (t)−
i → +∞, t ∈ J .
One sees that xni (t) = xni (0) +
t 0
fni (s, xni (s)) ds,
(3.15)
i = 1, 2, . . . .
(3.16)
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From the fact that f (s, xni (s)) = g(s, xni (s)) + (s, xni (s)), we get fni (s, xni (s)) = f (s + ni T , xni (s)) = g(s, xni (s)) + (s + ni T , xni (s)).
(3.17)
By expression (3.15), it is easy to obtain that → →g(s, y(s)), g(s, xni (s))−
i → +∞, s ∈ J ,
applying the condition limi→+∞ (s + ni T , xni (s)) = 0, s ∈ J , and from expression (3.17), it follows that → →g(s, y(s)), fni (s, xni (s))−
i → +∞, s ∈ J .
Now letting i → +∞, (3.16) implies that t g(s, y(s)) ds, t ∈ J . y(t) = y(0) +
(3.18)
0
Since J is an arbitrary given interval of R + , for any t ∈ R + , expression (3.18) holds. Thus, y(t) is a bounded solution of equation
u (t) = g(t, u), (3.19) u(0) = y(0). From (3.15), it then yields |y(t)|M, t ∈ R + , where M > 0 is a constant. By Massera theorem, Eq. (3.19) has a T-periodic solution y(t). In view of the proof of Theorem 3.2, one sees that y(t) is globally stable, then for any > 0, there exists T > 0 such that when t > T , |xni (t) − y(t)| < , this deduces that |x(t + ni T ) − y(t)| = |x(t + ni T ) − y(t + ni T )| < . Taking B = T + ni T such that when t B, |x(t) − y(t)| < . Then we obtain that y(t) is an asymptotically T-periodic solution of (3.13). The proof is complete.
Theorem 3.3. Suppose that n = 2 in Theorem 3.1 and f (t, x) is an asymptotically T-periodic function with f (t, x) = g(t, x) + (t, x), and g(t, x) is a continuous T-periodic function in t, and (t, x) is continuous function with limt→+∞ (t, x) = 0, x ∈ I ⊂ , where I is any compact region of ⊂ R 2 . If equation u = f (t, u),
u∈
(3.20)
has a positively bounded solution x(t), t ∈ R + , and equation u = g(t, u),
u∈
(3.21)
satisfies the uniqueness of solution with initial value, and any bounded solution of Eq. (3.21) is globally stable, then x(t) is an asymptotically T-periodic solution of Eq. (3.20). Proof. Suppose |x(t)|M, for t ∈ R + . Let xn (t) = x(t + nT ), fn (t, x) = f (t + nT , x).
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It is easy to check that x = xn (t) is a solution of equation x = fn (t, x). From the known conditions of this theorem, one can then obtain that |f (t, x)|
max
(t,x)∈[0,T ]×[−M,M]
|g(t, x)| +
max
(t,x)∈[0,A]×[−M,M]
|(t, x)| + 1 < ∞.
Thus f (t, x) is bounded on R + × [−M, M], and so fn (t, x) is uniformly bounded on R + × [−M, M]. Therefore {xn (t)} is uniformly bounded and equicontinuous on R + . By Arzela–Ascoli theorem, there exist a continuous function y(t) defined on R + and a subsequence {xni (t)} of {xn (t)}, such that {xni (t)} uniformly converges to y(t) on any finite closed interval J of R + . This means → →y(t), xni (t)−
i → +∞, t ∈ J .
(3.22)
It is easy to show that xni (t) = xni (0) +
t 0
fni (s, xni (s)) ds,
i = 1, 2, . . . .
(3.23)
From the expression f (s, xni (s)) = g(s, xni (s)) + (s, xni (s)), one can then derive that fni (s, xni (s)) = f (s + ni T , xni (s)) = g(s, xni (s)) + (s + ni T , xni (s)).
(3.24)
Applying (3.22), one sees that → →g(s, y(s)), g(s, xni (s))−
i → +∞, s ∈ J ,
again, noting that limi→+∞ (s + ni T , xni (s)) = 0, s ∈ J , and by (3.24), it follows that → →g(s, y(s)), fni (s, xni (s))−
i → +∞, s ∈ J .
Now letting i → +∞ on both sides of (3.23), we obtain that t g(s, y(s)) ds, t ∈ J . y(t) = y(0) +
(3.25)
0
Since J is arbitrary interval of R + , for any t ∈ R + , (3.25) holds. In other words, y(t) is a bounded solution of equation
u (t) = g(t, u), (3.26) u(0) = y(0). From (3.22), immediately, we know that |y(t)|M, t ∈ R + , where M > 0 is a constant. If all positively solutions of two-dimensional T-periodic equation u = g(t, u) are bounded, then by Massera theorem, it has a T-periodic solution. Since any bounded solution y(t) of Eq. (3.21) is globally stable, in other words, for any > 0, there exists T > 0 such that when t > T , |xni (t) − y(t)| < , this means that |x(t + ni T ) − y(t)| = |x(t + ni T ) − y(t + ni T )| < . Take B = T + ni T such that when t B, |x(t) − y(t)| < . Thus y(t) is an asymptotically T-periodic solution of (3.20). The proof is complete.
G. Haiyin et al. / Nonlinear Analysis: Real World Applications 7 (2006) 1268 – 1283
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4. Asymptotically periodic Logistic equations Theorem 4.1. If r(t) and k(t) are all asymptotically T-periodic functions with r(t) > 0, k(t) > 0, then x(t) x (t) = r(t)x(t) 1 − k(t)
(4.1)
has an asymptotically T-periodic solution which is globally asymptotically stable. Proof. The solution of Eq. (4.1) is t −1 s e− 0 r(t−u) du h(t − s) ds , 0
where h(t) = r(t)/k(t) and denote it by f (t)−1 . By Lemma 2.2, we can only show that t s f (t) = e− 0 r(t−u) du h(t − s) ds 0
is an asymptotically T-periodic function. Since r(t), k(t) are all asymptotically T-periodic functions, h(t) is an asymptotically T-periodic function. It is easy to check that t M M |f (t)| Me−s ds = (1 − e−t ) , 0 where |h(t)|M, t ∈ R + . Assuming that r(t)N, k(t) N, t ∈ R + such that as t → +∞ t t s r(t − s) f (t) = e− 0 r(t−u) du e−Ns ds = 2 (1 − e−Nt ) → 2 > 0, ds k(t − s) N N N 0 0 it follows that lim inf f (t) t→+∞
> 0. N2
(4.2)
Obviously, f (t) is a positive continuous function on R + . By (4.2), then there exists > 0 such that f (t) > 0, t ∈ Therefore f −1 (t) is bounded continuous function. To show f (t + T ) − f (t) → 0, as t → +∞. The expression f (t + T ) − f (t) can also be expressed as f (t + T ) − f (t) = I1 + I2 + I3 , where t+T s e− 0 r(t+T −u) du h(t + T − s) ds, I1 = t t s I2 = e− 0 r(t+T −u) du [h(t + T − s) − h(t − s)] ds, 0 t s s I3 = [e− 0 r(t+T −u) du − e− 0 r(t−u) du ]h(t − s) ds.
R+.
0 s
We note that |e− 0 r(t+T −u) du h(t + T − s)|Me−s , and then t+T M Me−s ds = [e−t − e−(t+T ) ], |I1 | t therefore limt→+∞ I1 = 0. Since h(t) is an asymptotically T-periodic function, let (t) = h(t + T ) − h(t) → 0
as t → +∞,
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then
t
I2 =
e−
s
r(t+T −u) du
0
t
(t − s) ds =
0
e−
t−s 0
r(t+T −u) du
(s) ds.
0
Taking absolute value from two sides then yields
t
|I2 |
e
−
t−s 0
r(t+T −u) du
t
|(s)| ds
e
0
−(t−s)
0
1 |(s)| ds = t e
t
es |(s)| ds,
0
again taking limits, it implies that 1 t→+∞ et
t
lim |I2 | lim
t→+∞
If
+∞ s e |(s)| ds < ∞, then 0
1 t→+∞ et
t
es |(s)| ds = 0.
lim
If
es |(s)| ds.
0
0
+∞ s e |(s)| ds = ∞, then by L’ Hospital’s rule 0
1 lim t→+∞ et
et |(t)| 1 = lim |(t)| = 0. t→+∞ et t→+∞
t
es |(s)| ds = lim
0
Therefore, limt→+∞ I2 = 0. Since r(t) is an asymptotically T-periodic function, let (t) = r(t + T ) − r(t) → 0, as t → +∞.
t
I3 =
0
0
= =
t t
[e− [e− e−
s 0
r(t+T −u) du
− e−
s
0 (r(t−u)+(t−u)) du
s 0
r(t−u) du
[e−
s 0
s 0
r(t−u) du
− e−
s
(t−u) du
0
]h(t − s) ds
r(t−u) du
]h(t − s) ds
− 1]h(t − s) ds,
0
one then obtains that I3 M 0
=M
0
t t t
e− e−
s 0
s 0
r(t−u) du
|e−
r(t−u) du
|e−
t−s
s 0
(t−u) du
t
t−s
(u) du
− 1| ds
− 1| ds
t
e− 0 r(t−u) du |e− s (u) du − 1| ds 0 t
t
Me−t es e− s (u) du − 1 ds.
=M
0
(4.3)
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The derivative of the right part of (4.3) is d t s − t (u) du e |e s − 1| ds dt 0 t t j = es |e− s (u) du − 1| ds jt 0 t t j = es [(e− s (u) du − 1)2 ]1/2 ds jt 0 t t t − s (u) du − 1)e− s (u) du (−(t)) s 2(e e ds = t 0 2 (e− s (u) du − 1)2 t t t = es sign(e− s (u) du − 1)e− s (u) du (−(t)) ds. 0
By (4.3) and L’ Hospital’s rule, we can then derive that t M d s − st (u) du lim I3 lim · e |e − 1| ds t→+∞ t→+∞ et dt 0 t M s − st (u) du lim e e ds · |(t)| t→+∞ et 0 t t M = lim es e− s (u) du ds · lim |(t)|. t t→+∞ e t→+∞ 0 Let − (t)=max(0, −(t)). Obviously, − (t)0. Noting that (t) → 0, as t → +∞, one then sees that − (t) → 0, as t → +∞. Further, from e−
t s
(u) du
it follows that M 0 t e
t − e s (u) du ,
t
s −
e e 0
t s
(u) du
M ds t e
t
t − es e s (u) du ds.
0
Taking limits for the above expression, t t M M s − st (u) du s st − (u) du 0 lim e e ds lim e e ds. t→+∞ et 0 t→+∞ et 0 Let
t
F (t) = 0
t − es e s (u) du ds,
A = lim
t→+∞
M et
t
e s e −
t s
(u) du
(4.4)
ds.
0
Obviously, F (t) is a monotonous increasing function. If F (+∞) < ∞, then the right part of (4.4) is 0. If F (+∞)=∞, then by L’ Hospital’s rule, t t − M lim es e s (u) du ds t t→+∞ e 0 t M t s st − (u) du − = lim e + e e (t) ds t→+∞ 2 et 0 t M M s st − (u) du = lim + lim e e ds · − (t) t→+∞ 2 t→+∞ 2 et 0 t M − (t) M s st − (u) du e e ds · lim . = 2 + lim t→+∞ et 0 t→+∞
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Applying the conclusion that limt→+∞ (t) = 0, we get that t M M s st − (u) du lim e e ds = 2 . t→+∞ et 0 In other words, 0 A M/2 and then yields limt→+∞ I3 = A limt→+∞ |(t)| = 0, therefore limt→+∞ (f (t + T ) − f (t)) = 0. The proof is complete. 5. Global stability and ultimate boundedness The global stability and ultimate boundedness of nonautonomous Logistic equation are discussed in this section. Next we will, respectively, consider periodic, asymptotically periodic Logistic equation x(t)
x (t) = r(t)x(t) 1 − , (5.1) k(t) x(t)
x (t) = [r(t) + (t)]x(t) 1 − , (5.2) k(t) + (t) where r(t), k(t) are continuous periodic functions with (t) → 0, (t) → 0 as t → +∞. Assume that 0 < r∗ r(t) r ∗ < ∞, 0 < k∗ k(t) k ∗ < ∞, 0 < ∗ r(t) + (t) ∗ < ∞, 0 < ∗ k(t) + (t) ∗ < ∞. The solutions of Eqs. (5.1), (5.2) with initial value x0 at t0 , respectively, are −1 t t t r(s) − r(s) ds x1 (t, t0 , x0 ) = x0 e t0 + e− s r(u) du , ds k(s) t0 −1 t t t r(s) + (s) − [r(s)+(s)] ds x2 (t, t0 , x0 ) = x0 e t0 + e− s [r(u)+(u)] du . ds k(s) + (s) t0 Theorem 5.1. Eqs. (5.1) and (5.2) are global stability. Proof. Let x(t) be any solution of Eq. (5.1), X(t) = ln x(t), X ∗ (t) = ln x ∗ (t). Liapunov function for Eq. (5.1) is V1 (t, X(t), X∗ (t)) = |X(t) − X ∗ (t)|. It is easy to check that V1 (t, X(t), X∗ (t)) is differentiable and large enough. Now computing right-hand upper derivative V1 (t, X(t), X∗ (t)) along Eq. (5.1), X(t) − X ∗ (t) x(t) ˙ x˙ ∗ (t)
∗ V1 (t, X(t), X (t))| = − |X(t) − X ∗ (t)| x(t) x ∗ (t) r(t) X(t) − X ∗ (t) = − (x(t) − x ∗ (t)) k(t) |X(t) − X ∗ (t)| r(t) X X(t) − X ∗ (t) ∗ = − (e (t) − eX (t) ) . k(t) |X(t) − X ∗ (t)| Then there exists a function (t), which lies between X(t) and X ∗ (t), such that eX (t) − eX
∗ (t)
= e(t) (X(t) − X ∗ (t)),
therefore r(t) (t) |X(t) − X ∗ (t)|2 e k(t) |X(t) − X ∗ (t)| r(t) (t) e |X(t) − X ∗ (t)| 0. = − k(t)
V1 (t, X(t), X∗ (t))| = −
G. Haiyin et al. / Nonlinear Analysis: Real World Applications 7 (2006) 1268 – 1283
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The above expression is an equality if and only if X = X ∗ , that is, x(t) = x ∗ (t), thus V1 (t, X(t), X∗ (t)) is negative. Therefore Eq. (5.1) is bounded. If x(t) = x ∗ (t), then V1 = 0, from LaSalle invariant theorem, the maximum positive invariant set is a single point x(t) = x ∗ (t), therefore x(t) = x ∗ (t) is global asymptotically stable. Let x(t) be any solution of Eq. (5.2), X(t) = ln x(t), X ∗∗ (t) = ln x ∗∗ (t). Next Liapunov function of Eq. (5.2) is V2 (t, X(t), X∗∗ (t)) = |X(t) − X ∗∗ (t)|. It is easy to see that V2 (t, X(t), X∗∗ (t)) is differentiable and large enough. We compute right-hand upper derivative V2 (t, X(t), X∗∗ (t)) along Eq. (5.2) X(t) − X ∗∗ (t) x(t) ˙ x˙ ∗∗ (t) V2 (t, X(t), X∗∗ (t))| = − |X(t) − X ∗∗ (t)| x(t) x ∗∗ (t) r(t) X(t) − X ∗∗ (t) = − (x(t) − x ∗∗ (t)) k(t) |X(t) − X ∗∗ (t)| r(t) X X(t) − X ∗∗ (t) ∗∗ = − (e (t) − eX (t) ) . k(t) |X(t) − X ∗∗ (t)| Then there exists a function (t), which lies between X(t) and X ∗∗ (t), such that eX (t) − eX
∗∗ (t)
= e (t) (X(t) − X ∗∗ (t)),
therefore V2 (t, X(t), X∗∗ (t))| = −
r(t) (t) |X(t) − X ∗∗ (t)|2 r(t) (t) e =− e |X(t) − X ∗∗ (t)| 0. k(t) |X(t) − X ∗∗ (t)| k(t)
The above expression is an equality if and only if X = X ∗∗ , that is, x(t) = x ∗∗ (t), thus V2 (t, X(t), X∗∗ (t)) is negative. Therefore Eq. (5.2) is bounded. If x(t) = x ∗∗ (t), then V2 = 0, from LaSalle invariant theorem, the maximum positive invariant set is a single point x(t) = x ∗∗ (t), therefore x(t) = x ∗∗ (t) is global asymptotically stable. The proof is complete. Definition 5.1. Consider the system x˙ = f (t, x). For any > 0, (t0 , x0 ) ∈ J ×B , there exists T =T (t0 , x0 ) > 0 such that when t t0 +T , it follows that x(t, t0 , x0 ) ∈ BM , then the solution x(t, t0 , x0 ) is said to be ultimately bounded, where J = [t0 , +∞), B = {x : |x| }, M > 0 is a constant. Theorem 5.2. Eqs. (5.1) and (5.2) are ultimate boundedness. Proof. To check the solution of (4.1) is ultimately bounded, by Definition 5.1, just to determine the related T1 = T1 (t0 , x0 ) > 0. Let |x1 (t, t0 , x0 )|M1 , where M1 is a positive number. That is,
−1
t
− t r(s) ds t − s r(u) du r(s)
M1 .
x0 e t0 ds + e
k(s)
t0 In view of biological meaning of models, we omit the invalid case, and only investigate the following case: t r∗ 1 x0 e−(t−t0 )r∗ + e−(t−s)r∗ ds , k M ∗ 1 t0 that is, x0 −
r∗ k∗ r∗
e−(t−t0 )r∗
1 r∗ − . M1 k∗ r∗
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When x0 −
r∗ < 0, k∗ r∗
1 r∗ − < 0, M1 k∗ r∗
1 r∗ r∗ − < x0 − M1 k ∗ r∗ k∗ r∗
hold, then e−(t−t0 )r∗ D1 , where 1 r∗ r ∗ −1 D1 = x0 − − , M1 k ∗ r∗ k∗ r∗ taking logarithm from two sides and then −(t − t0 )r∗ ln D1 , it is easy to see that t − t0 −
ln D1 . r∗
Taking T1 = T1 (t0 , x0 ) = −
ln D1 > 0. r∗
Again by Definition 5.1, it yields that any solution of Eq. (5.1) through (t0 , x0 ) is ultimately bounded. From [6], if T k(t) > 0, 0 r(s) ds > 0, then Eq. (5.1) has a unique positive solution T
x ∗ (t) = e 0 r(s) ds − 1
t+T t
r(s) − t r() d ds e s k(s)
−1 ,
moreover, x ∗ (t) is globally stable for any solution x1 (t, t0 , x0 ) with initial value x(0) = x0 > 0 and lim |x1 (t, t0 , x0 ) − x ∗ (t)| = 0.
t→+∞
In other words, for any > 0, there exists T1 > 0, such that for any t > T1 , it follows that |x1 (t, t0 , x0 ) − x ∗ (t)| < . Taking T1∗ = max{T1 , T1 } > 0, by Definition 5.1, the solution x ∗ (t) of Eq. (5.1) is ultimately bounded. Similarly, for Eq. (5.2), we just have to find T2 = T2 (t0 , x0 ) > 0. Let |x2 (t, t0 , x0 )| M2 , by the analogous discussion, it follows that t ∗ 1 x0 e−(t−t0 ) ∗ + e−(t−s) ∗ ds . M ∗ 2 t0 We denote D2 = D2 (x0 ) :=
1 ∗ − M2 ∗ ∗
∗ −1 x0 − . ∗ ∗
Taking T2 = T2 (t0 , x0 ) = −
ln D2 > 0, ∗
in terms of Definition 5.1, any solution of Eq. (5.2) through (t0 , x0 ) is ultimately bounded. From [1], if inf t∈R r(t) > 0, supt∈R r(t) < ∞, inf t∈R k(t) > 0, supt∈R k(t) < ∞ are satisfied, then Eq. (5.2) has a global stable solution −1 t s ∗∗ − 0 [r(t−u)+(t−u)] du r(t − s) + (t − s) ds e , x (t) = k(t − s) + (t − s) 0 and the solution x2 (t, t0 , x0 ) of Eq. (5.2) with positive initial value x0 satisfies lim |x2 (t, t0 , x0 ) − x ∗∗ (t)| = 0.
t→+∞
G. Haiyin et al. / Nonlinear Analysis: Real World Applications 7 (2006) 1268 – 1283
1283
In other words, for any > 0, there exists T2 > 0 such that for any t > T2 , it follows that |x ∗∗ (t) − x2 (t, t0 , x0 )| < . Now taking T2∗ = max{T2 , T2 } > 0, from Definition 5.1, the solution x ∗∗ (t) of Eq. (5.2) is ultimately bounded. The proof is complete. References [1] B.D. Coleman, Nonautonomous Logistic equation as models of the adjustment of populations to environmental change, J. Math. Biosci. 45 (3–4) (1979) 159–173. [2] M. Fan, K. Wang, Optimal harvesting policy for single population with periodic coefficients, J. Math. Biosci. 152 (2) (1998) 165–177. [3] X. Liao, The Theory, Method and Application of Stability, Southeast University Press, WuHan, 1999. [4] Z. Ma, Population Ecology on Mathematics Mold and Research, Anhui Education Press, 1996. [5] J.L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J. 17 (1950) 457–475. [6] K. Wang, Extension of Massera theorem, J. Acta Math. Sci. 10 (1990) 197–199.