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On the necessary and sufficient conditions for the boundedness of the solutions of the nonlinear oscillating equation
Nonlinear
Analysis,
Theory,
Methods
&Applications,
Vol. 23, No. 11, pp. 1467-1475, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/94 $7.00+ .@I
Pergamon 0362-546X(94)30021-8
ON THE NECESSARY AND SUFFICIENT CONDITIONS FOR THE BOUNDEDNESS OF THE SOLUTIONS OF THE NONLINEAR OSCILLATING EQUATION? LIHONG
HUANG
Department of Applied Mathematics, Hunan University, Changsha, Hunan 410082, People’s Republic of China (Received
19 April 1993; received for publication
27 January
1994)
Nonlinear oscillating equation, necessary and sufficient condition, boundedness.
Key words andphrases:
1. INTRODUCTION
In this paper we consider the nonlinear oscillating equation R + f(x)h(i)i
+ g(x)&)
= 0
(1.1)
or the equivalent system dx/dt = y,
dy/dt = -f(x)Mr)u
- g(x)W),
(1.2)
where f and g are continuous on R and h, and k are continuous positive on R, and they satisfy the conditions which ensure the existence of a unique solution to the initial problem of (1.2). For ease of exposition, throughout this paper we will adopt the following notation X y 1 du, G(x) = g(u) du, H(Y) = s0 s o h(u) y K(Y) =
u __
.Io k(u)
du,
W, Y) = G(x) + K(Y).
If h(y) = k(y) = 1 for ally E R, then (1.2) becomes the well-known Lienard system dx/dt = y,
dy/dt = -f(x)y
- g(x).
(1.3)
The problems on the boundedness of solutions of (1.2) and (1.3) have been extensively studied by many authors (see, for example, [l-12]). In particular, in 1965 and 1970, Burton [l, 91 gave the following results. 1.1 [9]. (i) Suppose that the conditions (A) f(x) 1 0 for all x E R, (B) xg(x) > 0 for all x # 0 hold. Then all solutions of (1.3) are bounded if and only if
THEOREM
lim (G(x) + IF(x = +a~. (1.4) I-+*= (ii) [l]. Suppose that k(y) = 1 for all y E R, and assumptions (A) and (B) hold. Then all solutions of (1.2) are bounded if and only if (1.4) holds. t This work was partially supported by National Natural Science Foundation of China. 1467
LIHONGHUANG
1468
In 1972, Heidel [2] established a set of sufficient conditions solutions of (1.2).
for the boundedness
of all
THEOREM1.2 [2]. Suppose that (A), (B) and (1.4) hold and (C) lim K(y) = +a. y-*CO Then all solutions of (1.2) are bounded. In 1987, Sugie [3] replaced the Signum condition (B) with the condition (B’) there exist constants G, > 0 and X,, > 0 such that (i) G(x) 2 -Go for all x E R, (ii) G(x) < lim sup G(u) for all x 2 X0, IL++m
(iii) G(x) < lim sup G(u) for all x I -X,, . u---m
He obtained the following result. THEOREM1.3 [3]. Suppose that assumptions (A), (B’) and (C) hold. Then all solutions of (1.2) are bounded if and only if limsup(lF(x)j x++m
+ G(x)] = +a.
(1.5)
As is well known, assumption (A) plays an important role in the proof of the above theorems 1. l-l .3. Since the authors of [l-3, 91 use the function V(x, y) and the fact that I&(x, y) = -f(x)h(y)y2/k(y) I 0 which follows from (A) to deduce the boundedness of y(t) first, then they use this result to discuss the boundedness of x(t). It is clear that the boundedness of y(t) cannot be proved by this method if (A) is relaxed to (A*) for some X0 > 0, f(x) 2 0 if 1x12 X0, since ti~1.2j(x,y) is a Signum now. Recently, Zhou [5] considered the boundedness of solutions of (1.2) under the assumption (A*). He introduced the following results (i.e. theorems l-4 in [5]). THEOREM1.4 [5]. (i) Suppose that assumptions (A*), (B’) (i) and (C) hold, and the following conditions: (D) lim N(y) = *co, (E) iG*l%ge positive Y,, there is a constant r > 0 such that k(y)/h(y) I r for all IyI L y0, (F) there exist constants CI> X0, b > X0 such that F(b) - F(-a) > 0, are also satisfied. Then all solutions of (1.2) are bounded if and only if (1.5) holds. (ii) Suppose that assumptions (A), (B’) (i) and (C) hold. Then all solutions of (1.2) are bounded if and only if (1.5) is satisfied. (iii) Suppose that (A) and (B’) (i) hold. Then all solutions of (1.3) are bounded if and only if (1.5) is satisfied. (iv) Suppose that (A*), (B’) (i) and (F) are satisfied. Then all solutions of (1.3) are bounded if and only if (1.5) holds. However, by carefully examining the proofs of the theorems in [5], we find that there exist certain errors in Zhou’s proofs. Now let us consider the following system dx/dt = y,
dy/dt = -3x2y - (-2xePX2).
(1.6)
1469
Nonlinear oscillating equation
It is easy to see that all conditions in theorem 1.4 are satisfied and (1.5) also holds. However, for system (1.6), the origin is the only critical point and is a saddle point. It is obvious that (1.6) possesses unbounded solutions. This example illustrates that the results (i)-(iv) in theorem 1.4 are not true. Thus, it is worth while to re-establish necessary and sufficient conditions for the boundedness of solutions of (1.2) when (A) is relaxed to (A*) and the Signum condition (B) does not hold. This is the main purpose of our paper. 2. LEMMAS
In this section we give some lemmas which simplify the proofs of our main results in Section 3. LEMMA 2.1.
Let (D) hold and . yh(y) $Iflmko = foe.
(E*)
Then, for any 4 < p and W > 0, there exists r > 0 such that for all s B r the positive semitrajectory: (x(t), y(t)) of (1.2) passing through the point (q, s) at time t,, must intersect the line x = p at some time t, > t,, and y(t) > W for all t E [to, tl]. Proof. Let A4 = maxi 1f(x), Ig(x)l: q 5 x 5 p). In view of (E*), it is easy to see that there exists fi > 0 such that yh(y)/k(y) 2 p for all y 2 W. Again according to (D) and the fact that H(y) is strictly increasing, which follows from dH(y)/dy = l/h(y) > 0 for ally E R, it is certain that there exists r > 0 such that for all s L r H(s) > H(W)
+ M(p - q)(l + l//3).
(2.1)
Suppose that the conclusions of lemma 2.1 were not true. Then there exists s, 1 r such that the positive semitrajectory (x(t), y(t)) of (1.2) starting from (q, sO)at time t,, must belong to one of the following two possible cases. (i) There exists t* > to such that x(t*) E (q,p], y(t*) = Wand y(t) > W for all t E (to, t*). (ii) x(t) E (q,p) and y(t) > W for all t > t,. If case (i) occurred, then we would have H(W)
- k&s,) = -
x(t*)g(x(u))k(y(u)) dx(u) 2 -M(p y(u)h(y(u)) 4
f(u) du -
4
- q)(l + l/P).
This inequality gives Ms,) 5 H(W) + M(P - 4)(1 + l/P), which contradicts (2.1). Therefore, case (i) is impossible. We next assume that case (ii) occurs, and let (x(t), y(t)) be defined on a right-maximal interval [to, T) (T may be +m). Since there are no critical points in the region ((x, y): q I x 5 p, y 2 W), it is certain that li:, s;p y(t) = + cc). On the other hand, we have MY(O)
-
ff(s,) = -
4
f(u)du
-
xw g(x(~)w(Y@)) (wu) 5 M(P - 4)(1 + l/P) y(4h(y(u))
I4
1470
LIHONG
HUANG
which with (D) imply lim sup y(t) < +oo. However, this contradicts the fact lim sup y(t) = +a~. t+T
Hence, case (ii) is also fmiossible.
This completes the proof of lemma 2.1.
By using an exactly similar fashion, we can prove the following lemma 2.2. Let (D) and (E*) hold. Then, for any q < p and W > 0, there exists r > 0 such that for all s 2 r the positive semitrajectory (x(t), y(t)) of (1.2) passing through the point (p, -s) at time to must intersect the line x = q at some time t, > t, , and y(t) < - W for all t E [to, tl]. LEMMA 2.2.
2.3. Suppose that (A*), (C), (D), (E*) and (1.5) hold and (B*) There exist constants Go > 0 and X0 > 0 such that
LEMMA
(i) G(x) 2 -G,, for all x E R, (ii) G(x) I lim sup G(u) for all x 1 X0, Zl++CC
(iii) G(x) I lim sup G(u) for all x 5 -X,, . u---m
Then, for any (xO,yO) E ((x,y): x 1 0, y > 01, the positive semitrajectory of (1.2) passing through (x0, yo) must intersect the positive x-axis or tend to some point (x*, 0) satisfying g(x*) = 0 and x* > 0. Proof. Suppose that there exists some point (x0, y,J E ((x, y): x 1 0, y > 0) such that the positive semitrajectory of (1.2) starting from the point (x0, y,,) at time to does not intersect the positive x-axis and does not tend to some point (x *, 0) satisfying g(x*) = 0 and x* > 0. We denote by L+ and (x(t),y(t)) this positive semitrajectory and its coordinate at time t, respectively. Let L+ be defined on a right-maximal interval [to, T) (T may be +oo). Then we have dx(t)/dt > 0 and y(t) > 0 for all t E [to, T). Therefore, it is certain that (i) lim x(t) < +co t-+T
and lim supy(t)
= +oo or (ii) limx(t)
= +oo.
By Ikmma 2.1, it is easy to Hee’that case (i) is impossible. Now we let (ii) hold. Then there exists t, ?I to such that x(tl) 2 X0 (where X0 is the same as in (A*)) and x(t) > X,, for all t E (tl, T). It follows from (A*) that I&,(x(t),
y(t)) = -f(x)Y2~(Y)MY)
5 0
for all t E [tl, T).
(2.2)
for all t E [tl, T).
(2.3)
Consequently, G@(t)) + K(y(t)) 5 G@(M) + K)‘(t,))
Since K(y) > 0 for all y # 0, it follows from (2.3) that lim sup G(x(t)) = lim sup G(x) < +oo. F(x) = +m. Again from (1.5) it follows that lim suplF(x)( = + 00, whiih ind (A*) irnpl;-Gk X’+m x++oO Let N = lim sup G(x) and V(t) = V(x(t), y(t)). We next show that V(t) > N for all t E [tl , T). x+ +m
Suppose not. Then there exists s1 E [tl , T) such that V(s,) I N. If V(s,) < N, then from (2.2) it follows that V(t) I V(s,) < N for all t E [sl, T).This implies lim sup G(x) I V(s,) < N, X-+m which contradicts N = lim sup G(x). If V(s,) = N, then the above argument and (2.2) also x-++m imply that V(t) = Nfor all t E [sl, T). We notice that there exists a sequence (r,J tending to +oo such that f(r,J > 0. In fact, if there exists r 2 X0 such that f(x) = 0 for all x L r, then
Nonlinear oscillating equation
1471
F(x) I F(r) < +oo, which contradicts
lim F(x) = + 00. Therefore, it follows from (2.2) and x* +m the above argument that there exists s3 > s, > si such that P(Q) < 0 and V(s3) < V(s,) = N, which is a contradiction to V(t) = N for all t E [sl, 7’). Therefore, V(t) > N for all t E [tl , T). Again from (2.3), (B*)(i) and (C), it follows that lim s;py(t) < +a, and so there exist constants I > 0 and 6 > 0 such that for all t E [to, T) _
h(Y(o)/k(Y(t)) 1 2
and
l/k(y(t))
5 6.
(2.4)
t E 14, n.
(2.5)
Thus, we have YW m(t))
6u
5
du = dy2(t)/2,
s0 In view of (B*) (ii), we have that K(y(t))
Therefore,
= V(t) - G(x(t)) 2 V(t) - N > 0,
for all t E [tl, T).
by (2.5) we obtain
IYWI2 Jww Since @(x(t)) = f(x(t))y(t),
- N)/6
for all t E [t, , T).
(2.6)
it follows from (2.4) and (2.6) that
ri =-f(x(t))Y2(t)h(r(t))/k(y(t)) 5 -Ad2(
V(t) - N)/G@(x(t)),
t E ItI, 0
We integrate both sides of the above inequality from t, to t E [tl , T) and obtain 2+(t)
- N - 2dV(t,)
- N I -Am(F(x(t))
- F(x(t,)).
This implies F(x(t)) I F(x(tI)) + F
(d’(t,)
- N - dJ’(t) - N),
t E 14, n.
However, this contradicts the fact that $x(t)
= +oo and lim F(x) = +oo. Hence case (ii) is x+ +co also impossible. This completes the proof of lemma 2.3. By an analogous argument, we can prove the following lemma.
LEMMA2.4. Suppose that (A*), (B*), (C), (D), (E*) and (1.5) are satisfied. Then, for any (x0, yo) E ((x, y): x I 0, y < 01, the positive semitrajectory of (1.2) passing through (x0, yo) must intersect the negative x-axis or tend to some point (x*, 0) satisfying g(x*) = 0 and x* < 0.
Remark 2.1. In the proof of lemma 2.3, assumption (B*) (i) is only used to derive (2.4). However, if we consider system (1.3), then (2.4) holds naturally when we choose I = 6 = 1. Therefore, even if the first member (B*) (i) of (B*) is not assumed, lemmas 2.3 and 2.4 are also true for system (1.3).
1472
LIHONG HUANG 3. RESULTS
In this section, we first give a sufficient condition for the boundedness of solutions of (1.2) (theorem 3.1) and a sufficient condition to guarantee the existence of an unbounded solution of (1.2) (theorem 3.2), and then from these a necessary and sufficient condition for the boundedness of solutions of (1.2) follows (theorem 3.3). THEOREM3.1. Suppose that the conditions (A*), (B*), (C), (D), (E*), (F) and (1.5) are satisfied. Then all solutions of (1.2) are bounded. Proof. It is sufficient to show that all positive Suppose not. Then there exists some point semitrajectory of (1.2) starting from A, at time (x(t), y(t)) this unbounded positive semitrajectory lemmas 2.1-2.4, it is easy to see that ,!,+(A,,) is a Let K = max&l, M
= maxtI.fWl,
IYJ,
semitrajectories of (1.2) are bounded. A, = (x0, yO) E R2 such that the positive t,, is unbounded. We denote by L+(A,) and and its coordinate at time t, respectively. By clockwise spiral surrounding the origin. a, b, &I
k(x):
+
1,
1x15 W
In view of (F), we may choose a constant W, 2 WI such that -(F(b)
- F(-a))
+ 2W,M/W,
< 0.
(3.1)
Again by (E*), it is certain that there exists W, L W2 such that lYh(Y)/k(Y)l 2 w,
for all IyI 5 W, .
(3.2)
By using a similar argument as in the proof of lemma 2.1, we can prove that there exists a constant r > W, such that H(s) > H(W,) + 2W,M(l
+ l/W,)
for all s 1 r.
We next show that there exists t, > to such that x(ti) = -WI and y(ti) > r. For ease of exposition, in the following discussion we denote by L+(P) the positive semitrajectory of (1.2) passing through an arbitrary point P. Let P, = (-WI, r + 1). Then from lemma 2.1 it follows that L+(P,) intersects the line x = W, 0 at some point P2 and the segmental arc P, P2 of L+(P,) is situated above the line y = W,. By lemma 2.3, L+(P,) also intersects the positive x-axis at some point P3 = (xp, , 0) or tends to P3. It is certain that the segmental arc w3 of L+(P,) is on the right-hand side of the line x = WI and xp3 > W, . According to lemma 2.2, we may choose a point P4 = (xp,, yp,) on the line x = X& such that L+(P,) intersects the line x = -WI at some point Ps and the segmental arc m5 of L+(P,) is situated below the line y = -W, . By lemma 2.4, L+(P,) also intersects the negative x-axis at some point P6 = (xp,, 0) or tends to Ph. It is certain that the segmental arc Ps P6 of L+(P,) is on the left-hand side of the line x = -WI and xps < -WI . Again in view of lemma 2.1, we may choose a point P7 = (xp,, yp,) on the line x = xp6 such that L+(P7) n intersects the line x = -WI at some point Ps and the segmental arc P,P, of L+(P,) is situated above the line y = r + 1. In view of the choice of W (i = 1,2,3), it is easy to see that A, lies w--in the interior of the closed curve Pi P2 Ps U P3 P4 U P4 Ps P6 U P6P, n P, Ps U Ps PI. Since L+(A,) is unbounded and on the line segments P3P4 and PsP, we have dx/dt < 0 and
Nonlinear oscillating equation
1473
dx/dt > 0, respectively, it is obvious that L+(A,) must intersect the line segment PsP, . Therefore, there exists t, > to such that x(tr) = -W, and y(tJ > r. By an exactly similar fashion, we can prove that there exists t: > to such that x(t,*> = W, and y(tl*) < --t* Since L+(&) is unbounded and is a clockwise spiral, it is easy to see, from the above argument and lemmas 2.1-2.4, that on L+(A,) there exist the points AI = (-WI, yAl), AZ = (W, ,yA2),
A, = (W, ,Y,+),
4
= C-W, ,Y,+)
and
A, = (- Wl,yA5)
such
that
the
segmental arc A,A2 of L+(A,) is situated above the line y = W,, the segmental arc A,A3 of L+(A,) is on the right-hand side of the line x = WI, the segmental arc A,& of L+(A,) is e situated below the line y = - W, , the segmental arc &A5 of L’(A,) is on the left-hand side of the line x = -WI and y,, > yAl. On the other hand, by (A*), (3.1) and (3.2), along the segmental arc AIA2 of L+(A,) we have
s s s Wl
w1&aMYW)
WY&) - MYA,) = - _w .fWdu -
-w,
1
b
I-
_=
5 -(F(b)
f(u) du -
Y(WYW)
w1gw)w(Y(u))
-w,
&.(@
&@)
Y@P(Y@))
- F(--a)) + 2MW,/W, < 0.
This implies that y,, > yA2 (> W,) and so K(Y&) > K(YAJ.
(3.3)
By an analogous argument, we can prove y,+ < yAI (< -W,) and so NY&) > K(y,4,).
(3.4)
Again, since li(l,zj(x, y) 5 0 for all 1x12 WI, it follows that G(K)
+ K(y,,)
2 G(W) + K(YA,)
and
G(-K)
+ K(Y,J 2 G(-W,) + K(Y,+).
Thus, we have K(YAJ 2 K(YAJ
and
NY,,)
2 NY,,).
(3.5)
From (3.3)-(3.5), it follows that K(yA,) > K(y& and so yA1 > y,, . However, this contradicts the fact yA5 > y,, . Thus the proof of theorem 3.1 is now complete. 3.1. By remark 2.1 and the proof of theorem 3.1, it is clear that theorem 3.1 is also true for system (1.3) even if condition (B*) (i) is not assumed.
Remark
THEOREM 3.2.
Suppose that conditions (A*), (B*) (i) and (C) hold, furthermore, SupF(x) < +=J
and
X20
Sup G(x) < +=J X2.0
(3.6)
or inf F(x) > -w
and
XSO
Then there exists an unbounded
Sup G(x) < +oo. X50
solution of (1.2).
(3.7)
LIHONG HUANG
1474
Proof. We only consider the case that (3.6) holds, since the case that (3.7) holds can be treated similarly. Let N = SUP,,~ G(x). Then, in view of (C), we may choose a constant w > 1 such that
s
++’ Y dy > 2(N + G,).
Now let M = max 15ys ,{h(y)/k(y)]. X0 is given in (A*)) such that
(3.8)
k(y)
1
Then, by (3.6) it is certain that there exists x* > X0 (where
+a
1
f(x) dx <
N*.
Lx*
We now consider the positive semitrajectory t,, and let (x(t), y(t)) be its coordinate at time Otherwise, it is certain that there exist 1 < y(t) I w and x(t) L x* for all t E [tl, t2). %)
l -dy=Y
.iXV,)
wk(y)
j-(x(t))
of (1.2) starting from the point (x*, w) at time t. We will prove that y(t) > 1 for all t 2 to. t, > f, 2 t, such that y(f2) = 1, y(tJ = w, On the other hand, we have
y(tMy(t)) dx(t) k(y(t))
-
x@2) g(x(t)) dx(t)
s x(t1)
+m L
-wM
>
x*
f(x) dx - GMfd) + GW,))
-wM
-N-G,
= -2(N + G,). This contradicts (3.8). Therefore, y(t) > 1 for all t 2 to. Again, since dx/dt = y > 0 if y > 0 and (1.2) has no critical points in the region 1(x, y): x > 0, y 2 l), it is easy to see that the positive semitrajectory of (1.2) passing through (x*, w) is unbounded. Theorem 3.2 is proved. By theorems 3.1 and (3.2), we obtain the following theorem. THEOREM3.3. Suppose that conditions (A*), (B*), (C), (D), (E*) and (F) are satisfied. Then all solutions of (1.2) are bounded if and only if (1 S) holds. Remark 3.2. For the system (1.3), the conditions Remark 3.3. If (A) holds andf(x) Remark 3.4. Obviously,
(C), (D) and (E*) hold naturally.
+ 0, then (F) is automatically
satisfied.
(E) implies (E*), (B) implies (B’), and (B’) implies (B*). On the other hand, though Sugie [3] suggests that condition (B’) in theorem 1.3 can be relaxed to (B”) there exist constants G,, > 0, X0 > 0 and sequences (r,) tending to +oo and (w,) tending to --oo such that G(x) 1 -G, for all x E R, G(x) I lim sup G(u) for all x 1 X0, ll++CC G(x) 5 lim sup G(u) for all x I -X,, , f(r,,) > 0, G(r,J < lim sup G(x), f(w,) > 0 and u---co x* +m G( w,) < lim sup G(x), x+--m
Nonlinear oscillating equation
1475
he also points out “however, this is not so easy to see and seems to require several additional pages for the proof. Hence, we omit it.” (See remark 3.5 in [3].) However, it should be noted that assumption (B*) in our theorems is weaker than (B”). REFERENCES 1. BURTON T. A., On the equation x + f(x)h(x)x + g(x) = e(t), Annali mat. puru uppl. 85, 37-48 (1970). 2. HEIDEL J. W., A Lyapunov function for a generalized Lienard equation, J. math. Analysis Applic. 39, 192-197 (1972). 3. SUGIE J., On the generalized Lienard equation without the signurn condition, J. math. Anulysis Applic. 128, 80-91 (1987). 4. SUGIE J., On the boundedness of solutions of the generalized Lienard equation without the Signum condition, Nonlinear Analysis 11,1391-1397 (1987). 5. ZHOU Y. R., On the boundedness of the solutions of the nonlinear oscillating equation, J. math. Analysis Applic. 164, 9-20 (1992). 6. GRAEF J. R., On the generalized Lienard equation with negative damping, J. dif$ Eqns 12, 34-62 (1972). 7. VILLARI G., On the qualitative behaviour of solutions of Lienard equation, J. diff. Eqns 67, 269-277 (1987). 8. KATO J., On a boundedness condition for solutions of a generalized Litnard equation, J. d$f. Eqns 65, 269-286 (1986). 9. BURTON T. A., The generalized Lienard equation, SIAM J. Control Optim. 3, 223-230 (1965). 10. ZHANG B., On the retarded Litnard equation, Proc. Am. math. Sot. 115, 779-785 (1992). 11. ZHANG B., Boundedness and stability of solutions of the retarded Litnard equation with negative damping, Nonlinear Analysis 20, 303-3 13 (1993). 12. SANSONE A. & CONTI R., Nonlinear Differentiul Equations. Macmillan, New York (1964).
Analysis,
Theory,
Methods
&Applications,
Vol. 23, No. 11, pp. 1467-1475, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/94 $7.00+ .@I
Pergamon 0362-546X(94)30021-8
ON THE NECESSARY AND SUFFICIENT CONDITIONS FOR THE BOUNDEDNESS OF THE SOLUTIONS OF THE NONLINEAR OSCILLATING EQUATION? LIHONG
HUANG
Department of Applied Mathematics, Hunan University, Changsha, Hunan 410082, People’s Republic of China (Received
19 April 1993; received for publication
27 January
1994)
Nonlinear oscillating equation, necessary and sufficient condition, boundedness.
Key words andphrases:
1. INTRODUCTION
In this paper we consider the nonlinear oscillating equation R + f(x)h(i)i
+ g(x)&)
= 0
(1.1)
or the equivalent system dx/dt = y,
dy/dt = -f(x)Mr)u
- g(x)W),
(1.2)
where f and g are continuous on R and h, and k are continuous positive on R, and they satisfy the conditions which ensure the existence of a unique solution to the initial problem of (1.2). For ease of exposition, throughout this paper we will adopt the following notation X y 1 du, G(x) = g(u) du, H(Y) = s0 s o h(u) y K(Y) =
u __
.Io k(u)
du,
W, Y) = G(x) + K(Y).
If h(y) = k(y) = 1 for ally E R, then (1.2) becomes the well-known Lienard system dx/dt = y,
dy/dt = -f(x)y
- g(x).
(1.3)
The problems on the boundedness of solutions of (1.2) and (1.3) have been extensively studied by many authors (see, for example, [l-12]). In particular, in 1965 and 1970, Burton [l, 91 gave the following results. 1.1 [9]. (i) Suppose that the conditions (A) f(x) 1 0 for all x E R, (B) xg(x) > 0 for all x # 0 hold. Then all solutions of (1.3) are bounded if and only if
THEOREM
lim (G(x) + IF(x = +a~. (1.4) I-+*= (ii) [l]. Suppose that k(y) = 1 for all y E R, and assumptions (A) and (B) hold. Then all solutions of (1.2) are bounded if and only if (1.4) holds. t This work was partially supported by National Natural Science Foundation of China. 1467
LIHONGHUANG
1468
In 1972, Heidel [2] established a set of sufficient conditions solutions of (1.2).
for the boundedness
of all
THEOREM1.2 [2]. Suppose that (A), (B) and (1.4) hold and (C) lim K(y) = +a. y-*CO Then all solutions of (1.2) are bounded. In 1987, Sugie [3] replaced the Signum condition (B) with the condition (B’) there exist constants G, > 0 and X,, > 0 such that (i) G(x) 2 -Go for all x E R, (ii) G(x) < lim sup G(u) for all x 2 X0, IL++m
(iii) G(x) < lim sup G(u) for all x I -X,, . u---m
He obtained the following result. THEOREM1.3 [3]. Suppose that assumptions (A), (B’) and (C) hold. Then all solutions of (1.2) are bounded if and only if limsup(lF(x)j x++m
+ G(x)] = +a.
(1.5)
As is well known, assumption (A) plays an important role in the proof of the above theorems 1. l-l .3. Since the authors of [l-3, 91 use the function V(x, y) and the fact that I&(x, y) = -f(x)h(y)y2/k(y) I 0 which follows from (A) to deduce the boundedness of y(t) first, then they use this result to discuss the boundedness of x(t). It is clear that the boundedness of y(t) cannot be proved by this method if (A) is relaxed to (A*) for some X0 > 0, f(x) 2 0 if 1x12 X0, since ti~1.2j(x,y) is a Signum now. Recently, Zhou [5] considered the boundedness of solutions of (1.2) under the assumption (A*). He introduced the following results (i.e. theorems l-4 in [5]). THEOREM1.4 [5]. (i) Suppose that assumptions (A*), (B’) (i) and (C) hold, and the following conditions: (D) lim N(y) = *co, (E) iG*l%ge positive Y,, there is a constant r > 0 such that k(y)/h(y) I r for all IyI L y0, (F) there exist constants CI> X0, b > X0 such that F(b) - F(-a) > 0, are also satisfied. Then all solutions of (1.2) are bounded if and only if (1.5) holds. (ii) Suppose that assumptions (A), (B’) (i) and (C) hold. Then all solutions of (1.2) are bounded if and only if (1.5) is satisfied. (iii) Suppose that (A) and (B’) (i) hold. Then all solutions of (1.3) are bounded if and only if (1.5) is satisfied. (iv) Suppose that (A*), (B’) (i) and (F) are satisfied. Then all solutions of (1.3) are bounded if and only if (1.5) holds. However, by carefully examining the proofs of the theorems in [5], we find that there exist certain errors in Zhou’s proofs. Now let us consider the following system dx/dt = y,
dy/dt = -3x2y - (-2xePX2).
(1.6)
1469
Nonlinear oscillating equation
It is easy to see that all conditions in theorem 1.4 are satisfied and (1.5) also holds. However, for system (1.6), the origin is the only critical point and is a saddle point. It is obvious that (1.6) possesses unbounded solutions. This example illustrates that the results (i)-(iv) in theorem 1.4 are not true. Thus, it is worth while to re-establish necessary and sufficient conditions for the boundedness of solutions of (1.2) when (A) is relaxed to (A*) and the Signum condition (B) does not hold. This is the main purpose of our paper. 2. LEMMAS
In this section we give some lemmas which simplify the proofs of our main results in Section 3. LEMMA 2.1.
Let (D) hold and . yh(y) $Iflmko = foe.
(E*)
Then, for any 4 < p and W > 0, there exists r > 0 such that for all s B r the positive semitrajectory: (x(t), y(t)) of (1.2) passing through the point (q, s) at time t,, must intersect the line x = p at some time t, > t,, and y(t) > W for all t E [to, tl]. Proof. Let A4 = maxi 1f(x), Ig(x)l: q 5 x 5 p). In view of (E*), it is easy to see that there exists fi > 0 such that yh(y)/k(y) 2 p for all y 2 W. Again according to (D) and the fact that H(y) is strictly increasing, which follows from dH(y)/dy = l/h(y) > 0 for ally E R, it is certain that there exists r > 0 such that for all s L r H(s) > H(W)
+ M(p - q)(l + l//3).
(2.1)
Suppose that the conclusions of lemma 2.1 were not true. Then there exists s, 1 r such that the positive semitrajectory (x(t), y(t)) of (1.2) starting from (q, sO)at time t,, must belong to one of the following two possible cases. (i) There exists t* > to such that x(t*) E (q,p], y(t*) = Wand y(t) > W for all t E (to, t*). (ii) x(t) E (q,p) and y(t) > W for all t > t,. If case (i) occurred, then we would have H(W)
- k&s,) = -
x(t*)g(x(u))k(y(u)) dx(u) 2 -M(p y(u)h(y(u)) 4
f(u) du -
4
- q)(l + l/P).
This inequality gives Ms,) 5 H(W) + M(P - 4)(1 + l/P), which contradicts (2.1). Therefore, case (i) is impossible. We next assume that case (ii) occurs, and let (x(t), y(t)) be defined on a right-maximal interval [to, T) (T may be +m). Since there are no critical points in the region ((x, y): q I x 5 p, y 2 W), it is certain that li:, s;p y(t) = + cc). On the other hand, we have MY(O)
-
ff(s,) = -
4
f(u)du
-
xw g(x(~)w(Y@)) (wu) 5 M(P - 4)(1 + l/P) y(4h(y(u))
I4
1470
LIHONG
HUANG
which with (D) imply lim sup y(t) < +oo. However, this contradicts the fact lim sup y(t) = +a~. t+T
Hence, case (ii) is also fmiossible.
This completes the proof of lemma 2.1.
By using an exactly similar fashion, we can prove the following lemma 2.2. Let (D) and (E*) hold. Then, for any q < p and W > 0, there exists r > 0 such that for all s 2 r the positive semitrajectory (x(t), y(t)) of (1.2) passing through the point (p, -s) at time to must intersect the line x = q at some time t, > t, , and y(t) < - W for all t E [to, tl]. LEMMA 2.2.
2.3. Suppose that (A*), (C), (D), (E*) and (1.5) hold and (B*) There exist constants Go > 0 and X0 > 0 such that
LEMMA
(i) G(x) 2 -G,, for all x E R, (ii) G(x) I lim sup G(u) for all x 1 X0, Zl++CC
(iii) G(x) I lim sup G(u) for all x 5 -X,, . u---m
Then, for any (xO,yO) E ((x,y): x 1 0, y > 01, the positive semitrajectory of (1.2) passing through (x0, yo) must intersect the positive x-axis or tend to some point (x*, 0) satisfying g(x*) = 0 and x* > 0. Proof. Suppose that there exists some point (x0, y,J E ((x, y): x 1 0, y > 0) such that the positive semitrajectory of (1.2) starting from the point (x0, y,,) at time to does not intersect the positive x-axis and does not tend to some point (x *, 0) satisfying g(x*) = 0 and x* > 0. We denote by L+ and (x(t),y(t)) this positive semitrajectory and its coordinate at time t, respectively. Let L+ be defined on a right-maximal interval [to, T) (T may be +oo). Then we have dx(t)/dt > 0 and y(t) > 0 for all t E [to, T). Therefore, it is certain that (i) lim x(t) < +co t-+T
and lim supy(t)
= +oo or (ii) limx(t)
= +oo.
By Ikmma 2.1, it is easy to Hee’that case (i) is impossible. Now we let (ii) hold. Then there exists t, ?I to such that x(tl) 2 X0 (where X0 is the same as in (A*)) and x(t) > X,, for all t E (tl, T). It follows from (A*) that I&,(x(t),
y(t)) = -f(x)Y2~(Y)MY)
5 0
for all t E [tl, T).
(2.2)
for all t E [tl, T).
(2.3)
Consequently, G@(t)) + K(y(t)) 5 G@(M) + K)‘(t,))
Since K(y) > 0 for all y # 0, it follows from (2.3) that lim sup G(x(t)) = lim sup G(x) < +oo. F(x) = +m. Again from (1.5) it follows that lim suplF(x)( = + 00, whiih ind (A*) irnpl;-Gk X’+m x++oO Let N = lim sup G(x) and V(t) = V(x(t), y(t)). We next show that V(t) > N for all t E [tl , T). x+ +m
Suppose not. Then there exists s1 E [tl , T) such that V(s,) I N. If V(s,) < N, then from (2.2) it follows that V(t) I V(s,) < N for all t E [sl, T).This implies lim sup G(x) I V(s,) < N, X-+m which contradicts N = lim sup G(x). If V(s,) = N, then the above argument and (2.2) also x-++m imply that V(t) = Nfor all t E [sl, T). We notice that there exists a sequence (r,J tending to +oo such that f(r,J > 0. In fact, if there exists r 2 X0 such that f(x) = 0 for all x L r, then
Nonlinear oscillating equation
1471
F(x) I F(r) < +oo, which contradicts
lim F(x) = + 00. Therefore, it follows from (2.2) and x* +m the above argument that there exists s3 > s, > si such that P(Q) < 0 and V(s3) < V(s,) = N, which is a contradiction to V(t) = N for all t E [sl, 7’). Therefore, V(t) > N for all t E [tl , T). Again from (2.3), (B*)(i) and (C), it follows that lim s;py(t) < +a, and so there exist constants I > 0 and 6 > 0 such that for all t E [to, T) _
h(Y(o)/k(Y(t)) 1 2
and
l/k(y(t))
5 6.
(2.4)
t E 14, n.
(2.5)
Thus, we have YW m(t))
6u
5
du = dy2(t)/2,
s0 In view of (B*) (ii), we have that K(y(t))
Therefore,
= V(t) - G(x(t)) 2 V(t) - N > 0,
for all t E [tl, T).
by (2.5) we obtain
IYWI2 Jww Since @(x(t)) = f(x(t))y(t),
- N)/6
for all t E [t, , T).
(2.6)
it follows from (2.4) and (2.6) that
ri
V(t) - N)/G@(x(t)),
t E ItI, 0
We integrate both sides of the above inequality from t, to t E [tl , T) and obtain 2+(t)
- N - 2dV(t,)
- N I -Am(F(x(t))
- F(x(t,)).
This implies F(x(t)) I F(x(tI)) + F
(d’(t,)
- N - dJ’(t) - N),
t E 14, n.
However, this contradicts the fact that $x(t)
= +oo and lim F(x) = +oo. Hence case (ii) is x+ +co also impossible. This completes the proof of lemma 2.3. By an analogous argument, we can prove the following lemma.
LEMMA2.4. Suppose that (A*), (B*), (C), (D), (E*) and (1.5) are satisfied. Then, for any (x0, yo) E ((x, y): x I 0, y < 01, the positive semitrajectory of (1.2) passing through (x0, yo) must intersect the negative x-axis or tend to some point (x*, 0) satisfying g(x*) = 0 and x* < 0.
Remark 2.1. In the proof of lemma 2.3, assumption (B*) (i) is only used to derive (2.4). However, if we consider system (1.3), then (2.4) holds naturally when we choose I = 6 = 1. Therefore, even if the first member (B*) (i) of (B*) is not assumed, lemmas 2.3 and 2.4 are also true for system (1.3).
1472
LIHONG HUANG 3. RESULTS
In this section, we first give a sufficient condition for the boundedness of solutions of (1.2) (theorem 3.1) and a sufficient condition to guarantee the existence of an unbounded solution of (1.2) (theorem 3.2), and then from these a necessary and sufficient condition for the boundedness of solutions of (1.2) follows (theorem 3.3). THEOREM3.1. Suppose that the conditions (A*), (B*), (C), (D), (E*), (F) and (1.5) are satisfied. Then all solutions of (1.2) are bounded. Proof. It is sufficient to show that all positive Suppose not. Then there exists some point semitrajectory of (1.2) starting from A, at time (x(t), y(t)) this unbounded positive semitrajectory lemmas 2.1-2.4, it is easy to see that ,!,+(A,,) is a Let K = max&l, M
= maxtI.fWl,
IYJ,
semitrajectories of (1.2) are bounded. A, = (x0, yO) E R2 such that the positive t,, is unbounded. We denote by L+(A,) and and its coordinate at time t, respectively. By clockwise spiral surrounding the origin. a, b, &I
k(x):
+
1,
1x15 W
In view of (F), we may choose a constant W, 2 WI such that -(F(b)
- F(-a))
+ 2W,M/W,
< 0.
(3.1)
Again by (E*), it is certain that there exists W, L W2 such that lYh(Y)/k(Y)l 2 w,
for all IyI 5 W, .
(3.2)
By using a similar argument as in the proof of lemma 2.1, we can prove that there exists a constant r > W, such that H(s) > H(W,) + 2W,M(l
+ l/W,)
for all s 1 r.
We next show that there exists t, > to such that x(ti) = -WI and y(ti) > r. For ease of exposition, in the following discussion we denote by L+(P) the positive semitrajectory of (1.2) passing through an arbitrary point P. Let P, = (-WI, r + 1). Then from lemma 2.1 it follows that L+(P,) intersects the line x = W, 0 at some point P2 and the segmental arc P, P2 of L+(P,) is situated above the line y = W,. By lemma 2.3, L+(P,) also intersects the positive x-axis at some point P3 = (xp, , 0) or tends to P3. It is certain that the segmental arc w3 of L+(P,) is on the right-hand side of the line x = WI and xp3 > W, . According to lemma 2.2, we may choose a point P4 = (xp,, yp,) on the line x = X& such that L+(P,) intersects the line x = -WI at some point Ps and the segmental arc m5 of L+(P,) is situated below the line y = -W, . By lemma 2.4, L+(P,) also intersects the negative x-axis at some point P6 = (xp,, 0) or tends to Ph. It is certain that the segmental arc Ps P6 of L+(P,) is on the left-hand side of the line x = -WI and xps < -WI . Again in view of lemma 2.1, we may choose a point P7 = (xp,, yp,) on the line x = xp6 such that L+(P7) n intersects the line x = -WI at some point Ps and the segmental arc P,P, of L+(P,) is situated above the line y = r + 1. In view of the choice of W (i = 1,2,3), it is easy to see that A, lies w--in the interior of the closed curve Pi P2 Ps U P3 P4 U P4 Ps P6 U P6P, n P, Ps U Ps PI. Since L+(A,) is unbounded and on the line segments P3P4 and PsP, we have dx/dt < 0 and
Nonlinear oscillating equation
1473
dx/dt > 0, respectively, it is obvious that L+(A,) must intersect the line segment PsP, . Therefore, there exists t, > to such that x(tr) = -W, and y(tJ > r. By an exactly similar fashion, we can prove that there exists t: > to such that x(t,*> = W, and y(tl*) < --t* Since L+(&) is unbounded and is a clockwise spiral, it is easy to see, from the above argument and lemmas 2.1-2.4, that on L+(A,) there exist the points AI = (-WI, yAl), AZ = (W, ,yA2),
A, = (W, ,Y,+),
4
= C-W, ,Y,+)
and
A, = (- Wl,yA5)
such
that
the
segmental arc A,A2 of L+(A,) is situated above the line y = W,, the segmental arc A,A3 of L+(A,) is on the right-hand side of the line x = WI, the segmental arc A,& of L+(A,) is e situated below the line y = - W, , the segmental arc &A5 of L’(A,) is on the left-hand side of the line x = -WI and y,, > yAl. On the other hand, by (A*), (3.1) and (3.2), along the segmental arc AIA2 of L+(A,) we have
s s s Wl
w1&aMYW)
WY&) - MYA,) = - _w .fWdu -
-w,
1
b
I-
_=
5 -(F(b)
f(u) du -
Y(WYW)
w1gw)w(Y(u))
-w,
&.(@
&@)
Y@P(Y@))
- F(--a)) + 2MW,/W, < 0.
This implies that y,, > yA2 (> W,) and so K(Y&) > K(YAJ.
(3.3)
By an analogous argument, we can prove y,+ < yAI (< -W,) and so NY&) > K(y,4,).
(3.4)
Again, since li(l,zj(x, y) 5 0 for all 1x12 WI, it follows that G(K)
+ K(y,,)
2 G(W) + K(YA,)
and
G(-K)
+ K(Y,J 2 G(-W,) + K(Y,+).
Thus, we have K(YAJ 2 K(YAJ
and
NY,,)
2 NY,,).
(3.5)
From (3.3)-(3.5), it follows that K(yA,) > K(y& and so yA1 > y,, . However, this contradicts the fact yA5 > y,, . Thus the proof of theorem 3.1 is now complete. 3.1. By remark 2.1 and the proof of theorem 3.1, it is clear that theorem 3.1 is also true for system (1.3) even if condition (B*) (i) is not assumed.
Remark
THEOREM 3.2.
Suppose that conditions (A*), (B*) (i) and (C) hold, furthermore, SupF(x) < +=J
and
X20
Sup G(x) < +=J X2.0
(3.6)
or inf F(x) > -w
and
XSO
Then there exists an unbounded
Sup G(x) < +oo. X50
solution of (1.2).
(3.7)
LIHONG HUANG
1474
Proof. We only consider the case that (3.6) holds, since the case that (3.7) holds can be treated similarly. Let N = SUP,,~ G(x). Then, in view of (C), we may choose a constant w > 1 such that
s
++’ Y dy > 2(N + G,).
Now let M = max 15ys ,{h(y)/k(y)]. X0 is given in (A*)) such that
(3.8)
k(y)
1
Then, by (3.6) it is certain that there exists x* > X0 (where
+a
1
f(x) dx <
N*.
Lx*
We now consider the positive semitrajectory t,, and let (x(t), y(t)) be its coordinate at time Otherwise, it is certain that there exist 1 < y(t) I w and x(t) L x* for all t E [tl, t2). %)
l -dy=Y
.iXV,)
wk(y)
j-(x(t))
of (1.2) starting from the point (x*, w) at time t. We will prove that y(t) > 1 for all t 2 to. t, > f, 2 t, such that y(f2) = 1, y(tJ = w, On the other hand, we have
y(tMy(t)) dx(t) k(y(t))
-
x@2) g(x(t)) dx(t)
s x(t1)
+m L
-wM
>
x*
f(x) dx - GMfd) + GW,))
-wM
-N-G,
= -2(N + G,). This contradicts (3.8). Therefore, y(t) > 1 for all t 2 to. Again, since dx/dt = y > 0 if y > 0 and (1.2) has no critical points in the region 1(x, y): x > 0, y 2 l), it is easy to see that the positive semitrajectory of (1.2) passing through (x*, w) is unbounded. Theorem 3.2 is proved. By theorems 3.1 and (3.2), we obtain the following theorem. THEOREM3.3. Suppose that conditions (A*), (B*), (C), (D), (E*) and (F) are satisfied. Then all solutions of (1.2) are bounded if and only if (1 S) holds. Remark 3.2. For the system (1.3), the conditions Remark 3.3. If (A) holds andf(x) Remark 3.4. Obviously,
(C), (D) and (E*) hold naturally.
+ 0, then (F) is automatically
satisfied.
(E) implies (E*), (B) implies (B’), and (B’) implies (B*). On the other hand, though Sugie [3] suggests that condition (B’) in theorem 1.3 can be relaxed to (B”) there exist constants G,, > 0, X0 > 0 and sequences (r,) tending to +oo and (w,) tending to --oo such that G(x) 1 -G, for all x E R, G(x) I lim sup G(u) for all x 1 X0, ll++CC G(x) 5 lim sup G(u) for all x I -X,, , f(r,,) > 0, G(r,J < lim sup G(x), f(w,) > 0 and u---co x* +m G( w,) < lim sup G(x), x+--m
Nonlinear oscillating equation
1475
he also points out “however, this is not so easy to see and seems to require several additional pages for the proof. Hence, we omit it.” (See remark 3.5 in [3].) However, it should be noted that assumption (B*) in our theorems is weaker than (B”). REFERENCES 1. BURTON T. A., On the equation x + f(x)h(x)x + g(x) = e(t), Annali mat. puru uppl. 85, 37-48 (1970). 2. HEIDEL J. W., A Lyapunov function for a generalized Lienard equation, J. math. Analysis Applic. 39, 192-197 (1972). 3. SUGIE J., On the generalized Lienard equation without the signurn condition, J. math. Anulysis Applic. 128, 80-91 (1987). 4. SUGIE J., On the boundedness of solutions of the generalized Lienard equation without the Signum condition, Nonlinear Analysis 11,1391-1397 (1987). 5. ZHOU Y. R., On the boundedness of the solutions of the nonlinear oscillating equation, J. math. Analysis Applic. 164, 9-20 (1992). 6. GRAEF J. R., On the generalized Lienard equation with negative damping, J. dif$ Eqns 12, 34-62 (1972). 7. VILLARI G., On the qualitative behaviour of solutions of Lienard equation, J. diff. Eqns 67, 269-277 (1987). 8. KATO J., On a boundedness condition for solutions of a generalized Litnard equation, J. d$f. Eqns 65, 269-286 (1986). 9. BURTON T. A., The generalized Lienard equation, SIAM J. Control Optim. 3, 223-230 (1965). 10. ZHANG B., On the retarded Litnard equation, Proc. Am. math. Sot. 115, 779-785 (1992). 11. ZHANG B., Boundedness and stability of solutions of the retarded Litnard equation with negative damping, Nonlinear Analysis 20, 303-3 13 (1993). 12. SANSONE A. & CONTI R., Nonlinear Differentiul Equations. Macmillan, New York (1964).