The continuation of solutions for certain nonlinear differential equations and its applications

J. Math. Anal. Appl. 329 (2007) 1127–1138 www.elsevier.com/locate/jmaa

The continuation of solutions for certain nonlinear differential equations and its applications ✩ Liqin Zhao Department of Mathematics, Beijing Normal University, Beijing 100875, PR China Received 14 April 2006 Available online 14 August 2006 Submitted by P.G.L. Leach

Abstract We give some sufficient conditions for the continuation of solutions for some nonlinear differential equations. As an application, we obtain a new criterion for the oscillation of solutions of the Liénard equation. © 2006 Elsevier Inc. All rights reserved. Keywords: Continuation; Oscillation; Nonlinear differential equation

1. Introduction The purpose of this paper is to give some sufficient conditions for the continuation of solutions of some nonlinear differential equations. This problem is motivated by the study of oscillation of Liénard equation x¨ + f1 (x)x˙ + f2 (x)x˙ 2 + g(x) = 0.

(1.1)

A solution of Eq. (1.1) is said to be oscillatory if there exists a sequence {tn } tending monotonically to ∞ such that x(tn ) = 0 (n  1). The oscillation of Liénard equation has been considered by a number of authors (see [4,6,7,10,11,17,18]). A. Constantin [4] was aware first that it was assumed tacitly in [17] that all solutions of Eq. (1.1) are continuable in the future. In fact, it was also supposed tacitly in [6,7,10,11,18] that all solutions of Eq. (1.1) are continuable in the future. So it is necessary to investigate the conditions for the continuation of solutions. ✩

Project supported by the National Natural Science Foundation of China (10301006). E-mail address: [email protected].

0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.07.019

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One can find some general results about the continuation of solutions in the books [2,15,19]. There are a few of papers on the continuation of solutions [5]. In this paper, we first investigate the continuation of some nonautonomous systems and then consider the continuation and oscillation of Liénard equation. 2. The continuation of some nonlinear differential equations In this section, we will deal with the continuation of solutions of the system  x˙ = h(y) − F (x), (2.1) y˙ = β(x)l(y) − g(x) + e(t). The system (2.1) connects Liénard equation  x˙ = h(y) − F (x), (2.2) y˙ = −g(x) + e(t), to system  x˙ = h(y) − F (x), (2.3) y˙ = l(y) − g(x) + e(t), since (2.1) reduces to (2.2) (see [1,3,20]) if β(x) ≡ 0 or l(y) ≡ 0 and it reduces to (2.3) if β(x) ≡ 1. Following R. Conti, Jiang [13] calls (2.3), with e(t) ≡ 0, a pseudolinear system under some restrictions. Let (x(t), y(t)) be the solution of (2.1) passing through the point (x0 , y0 ) at t = t0 ( 0) and (t1 , t2 ) be the maximal interval of existence for this solution. The solution (x(t), y(t)) is called to be continuable in the future if t2 = +∞. In the sequel we shall always, as standing hypotheses, assume that the following hypotheses: (H1 ) h(y), l(y), F (x), g(x) and β(x) are Lipschitz continuous in R with β(x)  0 and e(t) is continuous in R + = {t: t  0}. y (H2 ) There is M  1 such that |h(y)|  0 h(s) ds + M for all y ∈ R and y h(s) ds = +∞.

lim

|y|→+∞ 0

Theorem 2.1. Assume that system (2.1) satisfies (H1 ) and (H2 ) and the following conditions: (1) There is p > 0 such that x lim

|x|→+∞

x 0

(g(s) + β(s)F (s)) ds + p > 0 for x ∈ R and

  g(s) + β(s)F (s) ds = +∞.

0

(2) h(y)(h(y) + l(y))  0 for y ∈ R. (3) There are 0 < α  1 and k > 0 such that, for all x ∈ R, x

α     max 0, −F (x)g(x)  k g(s) + β(s)F (s) ds + p . 0

Then all solutions of system (2.1) are continuable in the future.

L. Zhao / J. Math. Anal. Appl. 329 (2007) 1127–1138

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Proof. For (x, y) ∈ R 2 and t  0, set  x y t   − 0 |e(s)| ds V (t, x, y) = e g(s) + β(s)F (s) ds + p + h(s) ds + M . 0

0

Then for all t  0, (x, y) ∈ R 2 , we have that V (t, x, y)  0 and the derivative of V (t, x, y) relative to system (2.1) is

dV

dt (2.1)

y x t

    − 0 |e(s)| ds g(s) + β(s)F (s) ds + p − e(t) =e h(s) ds + M + +e e

−

−

0

t

0 |e(s)| ds



t

0 |e(s)| ds

0

      h(y) l(y)β(x) − g(x) + e(t) + g(x) + F (x)β(x) h(y) − F (x)

  β(x) h(y)l(y) + h(y)F (x) − F 2 (x)



y



+ e(t) h(y) − h(s) ds − M − F (x)g(x) . 0

It follows from (2) that 2  1 3 h(y)l(y) + h(y)F (x) − F (x)  − h(y) − F (x) − F 2 (x)  0. 2 4 2

So, for all t  0, (x, y) ∈ R 2 ,

dV

 kV1 (x, y)α , dt (2.1) where x V1 (x, y) =

  g(s) + β(s)F (s) ds + p +

0

y h(s) ds + M. 0

Hence,

t

dV

 keα 0 |e(s)| ds dt.

α V (t, x, y) (2.1)

(2.4)

Let (x(t), y(t)) be the solution starting at the point (t0 , x0 , y0 ) and (t1 , t2 ) be the maximal interval of existence of (x(t), y(t)). Integrating (2.4) from t0 to t (< t2 ), we get ⎧ 1 [V (t, x(t), y(t))−α+1 − V (t0 , x0 , y0 )−α+1 ] ⎪ ⎪ ⎨ −α+1 u t  t0 keα 0 |e(s)| ds du, 0 < α < 1, (2.5) ⎪ u ⎪  ⎩ t ln V (t, x(t), y(t)) − ln V (t0 , x0 , y0 )  t0 keα 0 |e(s)| ds du, α = 1.

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We assert that t2 = +∞. Otherwise, suppose that t2 < +∞. It follows from Theorem 2.2 in [19] that limt→t2 − (x 2 (t) + y 2 (t)) = +∞. By (1) and (H2 ), we get  t2     lim V t, x(t), y(t) = e− 0 |e(s)| ds lim V1 t, x(t), y(t) = +∞. t→t2 −

t→t2

Let t → t2 in formula (2.5), then we can obtain a contradiction. This contradiction shows that t2 = +∞ and (x(t), y(t)) is continuable in the future. This completes the proof. 2 Theorem 2.2. Suppose that system (2.1) satisfies (H1 ) and (H2 ) and the following conditions: x x (1) There is p > 0 such that 0 g(s) ds + p > 0 for x ∈ R and lim|x|→+∞ 0 g(s) ds = +∞. (2) h(y)l(y)  0 for y ∈ R. (3) There are 0 < α  1 and k > 0 such that, for all x ∈ R,

α x   g(s) ds + p . max 0, −F (x)g(x)  k 0

Then all solutions of system (2.1) are continuable in the future. Proof. For (x, y) ∈ R 2 and t  0, set  x V (t, x, y) = e

−

t

0 |e(s)| ds

g(s) ds + p + 0

Then for all t  0, (x, y) ∈ relative to system (2.1) is

R2 ,



y

h(s) ds + M . 0

we have that V (t, x, y)  0 and the derivative of V (t, x, y)

y

x t

  dV

− 0 |e(s)| ds =e h(s) ds + M + g(s) ds + p − e(t)

dt (2.1) 0

0

    +e h(y) l(y)β(x) − g(x) + e(t) + g(x) h(y) − F (x) t    e− 0 |e(s)| ds β(x)h(y)l(y) − F (x)g(x) . −

t

0 |e(s)| ds



Hence, for all t  t0 , (x, y) ∈ R 2 x

α

y dV

k g(s) ds + p + h(s) ds + M . dt (2.2) 0

x

0

y Noting that lim(x,y)→+∞ { 0 g(s) ds + 0 h(s) ds} = +∞, similarly to the process of Theorem 2.1, we can obtain the results. This completes the proof. 2 The following corollary is an immediate consequence of Theorem 2.2 with l(y) ≡ 0. Corollary 2.1. Assume that system (2.2) satisfies (H1 ) and (H2 ) and the following conditions: (1) There is p > 0 such that

x 0

g(s) ds + p > 0 for x ∈ R and lim|x|→+∞

x 0

g(s) ds = +∞.

L. Zhao / J. Math. Anal. Appl. 329 (2007) 1127–1138

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(2) There are 0 < α  1 and k > 0 such that, for all x ∈ R, x

α   max 0, −F (x)g(x)  k g(s) ds + p . 0

Then all solutions of system (2.2) are continuable in the future. Remark 2.1. (i) For the case of e(t) ≡ 0 in (2.1), Theorem 2.1 remains valid if the hypothesis “there are 0 < α  1 and k > 0 such that, for all x ∈ R, x

α     max 0, −F (x)g(x)  k g(s) + β(s)F (s) ds + p , ” 0

is replaced by “there are ω(u) ∈ A and k > 0 such that, for all x ∈ R,  x     g(s) + β(s)F (s) ds + p , ” max 0, −F (x)g(x)  kω 0

where



A = ω ∈ C(R, R): ω(u) is nondecreasing and ω(u) > 0 for u  0, +∞ satisfying

ds = +∞ . w(s)

1

(ii) Similarly, for the case of e(t) ≡ 0 in (2.1), Theorem 2.2 remains valid if the hypothesis (3) in Theorem 2.2 is replaced by “there are ω(u) ∈ A and k > 0 such that, for all x ∈ R,  x   g(s) ds + p .” max 0, −F (x)g(x)  kω 0

(iii) Theorems 2.1 and 2.2 have no restrictions on e(t). In the study of the positive boundedness  +∞ [20] of system (2.2), it is usually assumed that “g(x)F (x)  0 for x ∈ R” and “ 0 |e(s)| ds < +∞.” Noticing the results on the boundedness of system (2.2) in [20, Theorem 1], we know that although the assumptions of Theorem 2.2 ensure that all solutions of system (2.2) are continuable in the future, some of the solutions of system (2.2) may be positively unbounded. From this point of view, we say our conditions are “exact.” To end this section, we will give a criterion on the continuation of n-dimensional systems. Consider an ordinary differential equation as a system of equations having the following form: d x  x), = X(t, dt

(t, x) ∈ R 1 × R n ,

(2.6)

where x(t) = (x1 (t), x2 (t), . . . , xn (t))T is a vector valued function of independent variable t ¯ and X(t) = (X1 , X2 , . . . , Xn )T : R 1 × R n → R n , is continuous and satisfies the local Lipschitz condition in the x variable uniformly in t.

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Theorem 2.3. Assume that there are continuous functions V (t, x) and ϕ( x ) such that, for (t, x) ∈ R 1 × R n ,     x  = +∞, (1) V (t, x)  ϕ  x , lim ϕ   x →+∞

n   ∂V  ∂V dV

+ = (t, x)Xi (t, x)  u(t)q V (t, x) , (2.7) (2) dt (2.6) ∂t ∂xi i=1

where u(t) is continuous in R with u(t)  0 and q(s) is continuous in [0, +∞) with q(s) > 0  +∞ ds and 1 q(s) = +∞. Then all solutions of system (2.6) are continuable in the future. Proof. Let φ(t; t0 , x0 ) be the solution of system (2.6) passing through the point x = x0 at t = t0 and (t1 , t2 ) be the maximal interval of existence for φ. Then for t ∈ (t1 , t2 ), the derivative of V (t, x) relative to system (2.6) is

   dV (t, φ(t; t0 , x0 ))

 u(t)q V t, φ(t; t0 , x0 ) ,

dt (2.6) dV (t, φ(t; t0 , x0 ))  u(t) dt. q(V (t, φ(t; t0 , x0 )))

(2.8)

Integrating (2.8) from t0 to t (< t2 ), we have V (t,φ(t)) 

V (t0 ,x0 )

ds  q(s)

t u(s) ds. t0

We assert that t2 = +∞. Otherwise, suppose that t2 < +∞. It follows from the theorem of continuation of solutions that limt→t2 − φ(t; t0 , x0 ) = +∞, where  ·  is any norm on Euclidean space. According to (1),     lim V t, φ(t; t0 , x0 )  lim ϕ φ(t; t0 , x0 ) = +∞. t→t2 −

t→t2 −

Hence t2

V (t,φ(t;t  0 ,x0 ))

u(s) ds  lim

t→t2 −

t0

ds = +∞. q(s)

V (t0 ,x0 )

This desired contradiction shows that t2 = +∞ and this completes the proof.

2

Corollary 2.2. Suppose that there are continuous functions V (t, x) and ϕ( x ) such that, for (t, x) ∈ R 1 × R n , V (t, x)  ϕ( x ), limx →+∞ ϕ( x ) = +∞ and the derivative of V (t, x) relative to system (2.6) is nonpositive, that is, dV dt |(2.6)  0. Then all solutions of system (2.6) are continuable in the future. Example 2.1. Let us consider system (2.2) with h(y) = c|y|p sgn y(c > 0, p  1),

e(t) = 1 + t 3 ,

L. Zhao / J. Math. Anal. Appl. 329 (2007) 1127–1138

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⎧ x  0, ⎨ x, 2 F (x) = −x + x, x ∈ [0, 1], ⎩ −x + 1, x  1,  x, x  0, g(x) = |sin x|, x  0. It is easy to see that this system satisfies the assumptions (H1 ) and (H2 ). Also we have x x x 2 π + [ x ]π |sin s| ds, x  0, π g(s) ds = 1 2 x , x  0, 2 0

and

⎧ 2 ⎪ x  0, ⎨ −x , 2 −F (x)g(x) = −(x − x )|sin x|, x ∈ [0, 1], ⎪ ⎩ 0 < (x − 1)|sin x|  [x], x > 1. Thus all conditions of Corollary 2.1 are satisfied with k = π , α = 1 and any p > 0. Hence all solutions are continuable in the future. 3. The continuation and oscillation of Liénard equation

In this section, we will study the continuation of solutions of the following system:  1 x˙ = γ (x) (h(y) − F (x)), (3.1) y˙ = −γ (x)g(x). In fact, using the transformation y = γ (x)x˙ + F (x) Liénard  xas in [12,14], one can change x equation (1.1) into (3.1) with h(y) ≡ y, γ (x) = exp( 0 f2 (s) ds) and F (x) = 0 γ (s)f1 (s) ds. Motivated by theoretical interest and plausible applications, system (3.1) has been studied intensively [12,14]. We assume that (C1 ) The functions F (x), g(x) and γ (x) are locally Lipschitz continuous on R with F (0) = 0, xg(x) > 0 (x = 0) and γ (x) > 0. (C2 ) h(y) is locally Lipschitz continuous and strictly increasing on R with h(0) = 0 and h(±∞) = ±∞. Definition 3.1. System (3.1) is said to have the properties (H+ ) if for any P0 (x0 , y0 ) with x0 > 0 and h(y0 ) > F (x0 ), the positive semi-orbit of system (3.1) starting from point P0 must cross the curve {(x, y): h(y) = F (x), x > 0}. System (3.1) is said to have the properties (H− ) if for any P0 (x0 , y0 ) with x0 < 0 and h(y0 ) < F (x0 ), the positive semi-orbit of system (3.1) starting from point P0 must cross the curve {(x, y): h(y) = F (x), x < 0}. Theorem 3.1. Assume that system (3.1) satisfies (C1 ), (C2 ), (1) or (1 ): x ˜ ˜ (1) G(±∞) = ±∞, where G(x) = 0 γ 2 (s)|g(s)| ds. There are nondecreasing function ω(u) ∈ C(R+ , R+ ) and k > 0 such that, for all x ∈ R,  x   2 max 0, −F (x)g(x)  kω γ (s)g(s) ds , 0

where ω(u) > 0 for u > 0 and

 +∞ 1

ds w(s)

= +∞.

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˜ −1 (−z))  F (G ˜ −1 (z)) for z ∈ (1 ) System (3.1) has the properties (H+ ) and (H− ), and F (G ˜ (0, min(G(±∞))). Then all solutions of system (3.1) are continuable in the future. Proof. Let (x(t), y(t)) be a solution of system (3.1) and (t1 , t2 ) be the maximal interval of existence for (x(t), y(t)). The assumptions (C1 ) and (C2 ) guarantee that (x(t), y(t)), t ∈ (t1 , t2 ), tends to the boundary of R 1 × R 2 . If the hypothesis (1) holds, it follows from Remark 2.1(ii) and system (3.1) has unique singular point (0, 0) that (x(t), y(t)) is continuable in the future. If the hypothesis (1 ) holds, then (x(t), y(t)) is positively bounded for t ∈ (t1 , t2 ). Hence t2 = +∞. This completes the proof. 2 Combing Theorem 3.1 and the qualitative behavior of the singular point (0, 0), we can achieve the following conclusion. Theorem 3.2. Suppose that all solutions of system (3.1) are continuable in the future and system (3.1) has the properties (H+ ) and (H− ). If the origin is a center, a focus, or a centro-focus, then all solutions, except the trivial one, are oscillatory. Some conditions for system (3.1) having the properties (H+ ) and (H− ) can be found in [4, 6–12,14,16]. Lemma 3.1. (See [6].) Assume that system (3.1) satisfies one of the following conditions: (1) lim supx→+∞ F (x) > −∞. (1 ) There exist N > 0 and α  1 such that F (x) < 0 for x  N and for any η > N , there exists xη > η satisfying 1 −1 h (F (xη ))

xη

γ 2 (s)g(s) ds  α. F (s)

η

Then system (3.1) has the property (H+ ) if  x 2 γ (s)g(s) ds = +∞, lim sup F (x) + 1 + F− (s) x→+∞

  F− (x) = max 0, −F (x) .

0

Lemma 3.2. (See [6].) Suppose that system (3.1) satisfies one of the following conditions: (1) lim infx→−∞ F (x) < +∞. (1 ) There exist N > 0 and α  1 such that F (x) > 0 for x  −N and for any η < −N , there exists xη < η satisfying 1 h−1 (F (xη ))

xη η

γ 2 (s)g(s) ds  α. F (s)

L. Zhao / J. Math. Anal. Appl. 329 (2007) 1127–1138

Then system (3.1) has the property (H− ) if  x 2 γ (s)g(s) ds = +∞, lim sup −F (x) + 1 + F+ (s) x→−∞

1135

  F+ (x) = max 0, F (x) .

0

Example 3.1. Consider the equation x˙ = c|y|p sgn y − F (x), where

y˙ = −dx 2n−1 ,

(3.2)



−ax 2n−1 , x  0, c, d > 0, p  1, n ∈ N, a + b  0. −bx 2n−1 , x < 0, Assume that one of the following conditions holds: F (x) =

(1) a, b < 0; (1 ) p > 2n − 1; (1

) p = 2n − 1, and   1 a 2n−1 nd  a(n + 1) c

(a > 0),

  1 b 2n−1 nd  b(n + 1) c

(b > 0).

Then all solutions are continuable in the future. Solution. Indeed, ±∞ ˜ G(x) dx = ±∞,

+∞

0

0

γ 2 (x)g(x) dx = +∞, 1 + F− (x)

−∞ 0

γ 2 (x)g(x) dx = +∞. 1 + F+ (x)

˜ −1 (−z))  F (G ˜ −1 (z)) for z ∈ (0, +∞). We assert that sysBy a + b  0, we have F (G + − tem (3.2) has the properties (H ) and (H ). Since limx→+∞ F (x) = +∞ for a < 0 and limx→−∞ F (x) = −∞ for b < 0, we just consider the cases of a > 0 and b > 0. Suppose that a > 0. Then for N = 1 and η > 1, 1 lim x→+∞ h−1 (F (x))

x η

d (x − η) γ 2 (s)g(s) ds = lim −a  1 2n−1  x→+∞ F (s) − ac p x p   1 d a −p , p = 2n − 1, = a· c +∞, p > 2n − 1.

Hence there exists xη > η such that, for x > xη , we have   1 x 2 d a −p n 1 γ (s)g(s) a · c n+1  1, ds > F (s) h−1 (F (x)) 1, η

p = 2n − 1, p > 2n − 1.

(3.3)

(3.4)

So, by (1

) and Lemma 3.1, system (3.2) has the properties (H+ ). Similarly, we can prove that system (3.2) has the properties (H− ) for b > 0. To sum up, we conclude by Theorem 3.1 that all solutions are continuable in the future. 2

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L. Zhao / J. Math. Anal. Appl. 329 (2007) 1127–1138

Example 3.2. Consider the equation x˙ = cye|y| − F (x), where



F (x) =

y˙ = −dx 4n−3 ,

−ax 2n−1 , x  0, −bx 2n−1 , x < 0,

(3.5)

c, d > 0, n ∈ N and n > 1, a + b  0.

(3.6)

Then all solutions are continuable in the future. In addition, if we assume that (2n − 1)a 2 < 4cd

(for a < 0),

(2n − 1)b2 < 4cd

(for b < 0),

(3.7)

then any nontrivial solutions are oscillatory. Solution. (i) To this end, we will check that system (3.5) satisfies conditions (C1 ), (C2 ) and (1 ) in Theorem 3.1. Indeed, ±∞ ˜ G(x) dx = ±∞,

+∞

0

0

−∞

γ 2 (x)g(x) dx = +∞, 1 + F− (x)

0

γ 2 (x)g(x) dx = +∞. 1 + F+ (x)

˜ −1 (z)) ˜ −1 (−z))  F (G F (G

for z ∈ (0, +∞). In what follows, we prove By a + b  0, we have that system (3.5) has the properties (H+ ) and (H− ). In fact, since limx→+∞ F (x) = +∞ for a < 0 and limx→−∞ F (x) = −∞ for b < 0, we just consider the cases of a > 0 and b > 0. We consider the case of a > 0 first. Let h(y) = cye|y| , h1 (y) = cy 2n−1 . Since h(y) cye−y = lim = +∞, y→−∞ h1 (y) y→−∞ cy 2n−1 lim

there is M > 0 such that h(y) < h1 (y) for y < −M. Suppose that system (3.5) fails to have the properties (H+ ). Then there is a point P0 (x0 , y0 ) ∈ R 2 with h(y0 ) > F (x0 ) and x0 > 0 such that the positive semi-orbit departing from the point P0 does not intersect the curve h(y) = F (x) at the right semi-plane. Assume that (x(t), y(t)) is the solution starting from P0 and (t1 , t2 ) be the maximal interval of existence for (x(t), y(t)). It follows from the continuation theorem [19, Theorem 2.2] and h(−∞) = −∞ that x(t) increases monotonically to +∞ as t → t2 −. Since x(t) y(t) = y0 + x0

−g(s)γ 2 (s) ds < y0 − h(y(s)) − F (s)

x(t) x0

ds 4n−3 ds → −∞ as t → t2 −, h(y0 ) + as 2n−1

there is η > 0 such that y(x) < −M for x  η, where y(x) expresses the orbit of system (3.5). Let D(x) = y(x) − h−1 1 (F (x)) for x  x0 . Then D(x) > 0

(3.8)

for all x > x0 .

On the other hand, for x > η, x D(x) = y(η) + η

  g(s)γ 2 (s) F (x) < ds − h−1 1 F (s) − h(y(s)) 

=−

 2n−1  d −a x − η2n−1 − a(2n − 1) c



1 2n−1

x

  g(s) F (x) ds − h−1 1 F (s)

η

x → −∞ as x → +∞.

L. Zhao / J. Math. Anal. Appl. 329 (2007) 1127–1138

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Hence, there is xη > η such that D(xη ) < 0. This contradicts (3.8). This desired contradiction shows that system (3.5) has the properties (H+ ). Similarly we can prove that system (3.5) has the properties (H− ) for the case of b > 0. In one word, system (3.5) satisfies the conditions (1 ) in Theorem 3.1 and all solutions are continuable in the future. (ii) We assert that under the condition (3.7), the origin is a center, a focus, or a centro-focus. It is sufficient to prove that any positive semi-orbit starting from the curve h(y) = F (x) (x = 0) must cross the positive or negative y-axis. We just need to consider the cases of a < 0 and b < 0. Since h (0) F (x) h (0)

x 0+ x

F (x) 0−

cd 1 g(s) ds = 2 > , F (s) a (2n − 1) 4 cd 1 g(s) ds = 2 > , F (s) b (2n − 1) 4

it follows from Theorem 1 in [6] that our statement is true. An application of Theorem 3.2 enables us to deduce that all nontrivial solutions are oscillatory. 2 Remark 3.1. Consider the equation x˙ = y − F (x),

y˙ = −g(x),

(3.9)

where F (x) and g(x) are the same as those in Example 3.2 and in condition (3.7) with c = 1. Similarly to Example 3.2, we can prove that all nontrivial solutions of system (3.9) are oscillatory. However, Theorem 1 in [4] is invalid to system (3.9) because the following assumption, which is an important hypothesis to ensure that all solutions are continuable in the future in Theorem 1 in [4], is not satisfied: “there is nondecreasing function ω(u) ∈ C(R+ , R+ ) such that,  x   g(s) ds , x ∈ R+ , ” (3.10) g(x) max 0, −F (x)  ω 0

where +∞ ω(u) > 0 for u > 0 and

ds = +∞. w(s)

(3.11)

1

In fact, if system (3.9) satisfies that (3.10), then there exist a nondecreasing function ω(u) ∈ C(R+ , R+ ) such that, for x > 0, a > 0,   d 6n−4 4n−2 . x ω −F (x)g(x) = adx 4n − 2 Let u =

d 4n−2 , 4n−2 x

then

3n−2

ω(u)  k1 u 2n−1 ,

 +∞

k1 > 0, u > 0,

 +∞ du du and 1 ω(u) < +∞, which contradicts to 1 ω(u) = +∞. Hence, Theorem 1 in [4] is invalid to system (3.9). So Theorem 3.2 of this paper generalizes and improves Theorem 1 in [4].

1138

L. Zhao / J. Math. Anal. Appl. 329 (2007) 1127–1138

Acknowledgments The author thanks the referees for their comments.

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