necessary and isocline applications vertical for...

Funkcialaj Ekvacioj, 33 (1990) 19-38

On a Dynamical System in the Lienard Plane.Necessary and Sufficient Conditions for the Intersection

with the Vertical Isocline and Applications*

By

Gabriele VILLARI and Fabio ZANOLIN(University of Firenze and University of Udine, Italy)

1. Introduction

This paper is devoted to the study of the qualitative behaviour of thesolutions of the autonomous planar system of Lienard type

$¥dot{¥mathrm{X}}=y-F(x)$

(1.1)$¥dot{y}=-g(x)$ .

Such system has been widely investigated in the literature concerningnonlinear oscillations for differential equations (see [19], [18], [3], [9], [26],

[27] $)$ . In particular, system (1.1) may be used in the study of the scalarLienard equation

(1.2) $¥ddot{x}+f(x)¥dot{¥mathrm{x}}+g(x)=0$ ,

setting

$F(¥mathrm{x}):=¥int_{0}^{¥mathrm{x}}f(u)du$ .

As discussed in some recent papers (see [16], [8], [10], [22]), a significant pointis to find conditions ensuring that the trajectories of system (1.1) intersect thevertical isocline, that is the curve $y=F(x)$. Indeed, this property is a crucialstep for the proof of the existence of oscillatory and periodic solutions of system(1.1). Moreover, dealing with the periodically forced Lienard equation

(1.3) $¥ddot{x}+(f(¥mathrm{x})+p(x))¥dot{x}+g(x)=e(t)=e(t+T)$ ,

an usual technique for producing $¥mathrm{T}$-periodic solutions is to find a Jordan curve

* Work performed under the auspices of GNAFA-CNR and partially supported by the grant$¥mathrm{M}¥mathrm{P}¥mathrm{I}$ 60%, $¥mathrm{Z}¥mathrm{A}¥mathrm{N}6¥mathrm{Q}4$ .The results of this paper have been presented at the “I. $¥mathrm{C}.¥mathrm{N}$.O. ’87” (Budapest, August 17-23,1987)

20 Gabriele VILLARI and Fabio ZANOLIN

in the phase-plane which is crossed from the exterior to the interior by thetrajectories of the considered system. Such Jordan curve may be built up bytrajectory paths of (1.1) or other comparison systems. Even in this case, theabove mentioned conditions give useful applications.

In the study of (1.1) or (1.2), the classical assumption on the restoring term$g(x)$ is

$xg(x)>0$ for $x$ $¥neq 0$ .

Under this hypothesis, conditions for the intersection with the vertical isoclinewere obtained by many authors ([6], [15], [25], [16], [8], [10], [22]). Inparticular, J. R. Graef [8] gave necessary and sufficient conditions under thehypothesis

$xF(x)>0$ for $|x|$ large.

Such result was improved by T. Hara and T. Yoneyama [10] and Gab. Villari[22]. In [22], no restrictions on the sign of $F(x)$ are required, but $F(x)$ mustbe bounded below for $x$ positive and bounded above for $x$ negative.

Our main theorem for system (1.1) provides a necessary and sufficientcondition under the more general assumption

$¥lim_{x¥rightarrow}¥sup_{+¥infty}F(x)>-¥infty$ and $¥lim_{¥mathrm{x}¥rightarrow}¥inf_{-¥infty}F(x)<+¥infty$ .

Such result completes the analogous one by T. Hara and T. Yoneyama [10]who gave only the sufficient condition.

For the general case, we discuss other assumptions which, even if are notnecessary and sufficient, nevertheless seem to be very sharp.

The plan of the paper is the following.In section 2 we study the problem of the intersection with the isocline

$y=F(x)$. The corresponding results, as well known, may be used for obtainingoscillatory theorems and necessary and sufficient conditions for the existence ofa global center at the origin. For the first case, assumptions preventing thepossibility that the origin is a stable node will be necessary, while, furtherconditions of symmetry may be used for the second problem (see [6], [16],[21], [10] $)$ . In section 3 the previous results are employed for the studyof (non-trivial) periodic solutions of the Lienard equation (1.2). The forcedequation (1.3) is studied in section 4. The theorems in sections 3 and 4 areachieved introducing two properties named (H) and (K). The former concernsthe existence of a comparison system with the property that all the trajectoriessufficiently far from the origin are closed. The latter one assumes the existenceof a trajectory for system (1.1) which comes from the infinity and it is oscilla-tory for increasing time. It is worthy to observe that almost all the resultsknown in the literature may be deduced from one of these conditions.

A Dynamical System in the Lienard Plane 21

Preliminary results from this research were presented at Equadiff ’85 (Brno,August 1985) and I.C.M. ’86 (Berkeley, August 1986) (see also [14]).

2. Intersection with the curve $y =F(x)$

Consider the Lienard system

$¥dot{X}=y-F(x)$

(2. 1)$¥dot{y}=-g(¥mathrm{x})$

with: $F$, $g:R¥rightarrow R$ continuous functions and satisfying the property of unique-ness for the solutions to the Cauchy problems associated to equation (2.1),and

(2.2) $xg(x)>0$ , for every $x$ $¥neq 0$ .

Without loss of generality, we also assume

$F(0)=0$ .

For any point $P=(x_{0}, y_{0})¥in R^{2}$ , we denote by $¥gamma^{+}(P)$ and $¥gamma^{-}(P)$ the positive,respectively negative, semitrajectory (semiorbit) of (2.1) passing through $P$ . Wealso denote by $a$ the curve $y=F(¥mathrm{x})$ .

In this section, we are concerned with conditions ensuring the intersectionof $¥gamma^{+}(P)$ (respectively $¥gamma^{-}(P)$ ) with $a$ , for any fixed $P¥in R^{2}$ .

With respect to the intersection of $¥gamma^{+}(P)$ with $¥alpha$ , we observe that, by astandard inspection of the direction of the field, it sufficies to consider the cases

(2.3) $x_{0}¥geqq 0$ , $¥mathcal{Y}0>F(x_{0})$

and

(2.4) $x_{0}¥leqq 0$ , $¥mathcal{Y}0<F(x_{0})$ .

For any $F$, define$F_{+}(x):=¥max¥{0, F(x)¥}$ ,

$F_{¥_}(x):=¥max¥{0, -F(x)¥}$ ,

and observe that $F(¥mathrm{x})$ may be written as

$F(¥mathrm{x})=F_{+}(x)-F_{¥_}(x)$ .

Finally, it is convenient to introduce the functions

$¥Gamma_{+}(x):=¥int_{0}^{x}(1+F_{+}(¥xi))^{-1}g(¥xi)d¥xi$


and

$¥Gamma_{-}(x):=¥int_{¥mathrm{o}}^{¥mathrm{x}}(1+F_{-}(¥xi))^{-1}g(¥xi)d¥xi$ ,

which play a crucial role in all this section.The following result extends a previous theorem of the first author (see [22,

Th. 1]), to the case in which $F(x)$ is not bounded below for $x$ positive orbounded above for $x$ negative.

Theorem 2.1. Assume

(2.5) $¥lim_{x¥rightarrow}¥sup_{¥dagger¥infty}F(x)>-¥infty$ .

Then, for every $P=(x_{0}, y_{0})$ verifying (2.3), $¥gamma^{+}(P)$ intersects the curve $y=F(x)$, at$(x, F(x))$ with $x$ $>x_{0}$ , if and only if(2.6) $¥lim_{¥mathrm{x}¥rightarrow}¥sup_{+¥infty}$

$[¥Gamma_{¥_}(x)+F(x)]=+¥infty$ .

Assume

(2.7) $¥lim_{¥mathrm{x}¥rightarrow}¥inf_{-¥infty}F(x)<+¥infty$ .

Then, for every $P=(x_{0}, y_{0})$ verifying (2.4), $¥gamma^{+}(P)$ intersects the curve $y=F(x)$, at$(x, F(x))$ with $x$ $<x_{0}$ , if and only if(2.8) $¥lim_{¥mathrm{x}¥rightarrow}¥sup_{-¥infty}$

$[¥Gamma_{+}(x)-F(x)]=+¥infty$ .

Proof. We examine only the first situation, as the second one may betreated exactly in the same way.I. Sufficiency.

If condition (2.6) holds, we distinguish the following cases:

I.a) $¥lim_{¥mathrm{x}¥rightarrow}¥sup_{+¥infty}F(x)=+¥infty$ .

Here it is sufficient to repeat the proof in [22, p. 271].

I.b) $¥lim_{x¥rightarrow}¥sup_{+¥infty}F(x)<+¥infty$ .

In this case, condition (2.6) implies that

$¥int_{0}^{+¥infty}(1+F_{-}(x))^{-1}g(x)dx=+¥infty$ .

Suppose, by contradiction, that $¥gamma^{+}(P)$ does not intersect $a$ . Then, arguing


like in [10, p. 179], we have that

$ x(t)¥rightarrow+¥infty$ as $t¥rightarrow T^{-}$

being [0, $T[(T¥leqq¥infty)$ the right maximal interval of existence of the solution$(x(t), y(t))$ of (2. 1) with $(x(0), y(0))=P$. We have

$F(x(t))<y(t)¥leqq ¥mathcal{Y}¥mathrm{o}$ , for all $t$ $¥geqq 0$ .

Hence

$¥dot{x}(t)=y(t)-F(x(t))¥leqq y_{0}-F_{+}(x(t))+F_{¥_}(x(t))$

$¥leqq|y_{0}|+F_{¥_}(¥mathrm{x}(t))¥leqq K(1+F_{¥_}(x(t)))$ ,

with $K=¥max¥{1, |y_{0}|¥}$ . Then the slope of $y$ with respect to $x$ , which is givenby $y^{¥prime}=dy/dx$ , can be evaluated as follows

$y^{¥prime}(x)¥leqq-g(x)/K(1+F_{¥_}(x))$ .

Integrating from 0 to $x$ $=x(t)$, we obtain

$ y(x)-¥mathcal{Y}0¥leqq-(1/K)¥int_{0}^{x}(1+F_{-}(¥xi))^{-1}g(¥xi)d¥xi$ ,

and hence,

$-¥infty<¥lim_{x¥rightarrow}¥sup_{+¥infty}F(x)¥leqq¥lim_{¥mathrm{x}¥rightarrow}¥sup_{+¥infty}y(¥mathrm{x})=¥lim_{x¥rightarrow+¥infty}y(x)=-¥infty$ ,

which is a contradiction.$¥mathrm{I}¥mathrm{I}$ . Necessity.

Assume now that

$¥lim_{x¥rightarrow}¥sup_{+¥infty}[¥Gamma_{-}(¥mathrm{x})+F(x)]<+¥infty$ .

This implies that there exist $H$, $K>0$ such that

$F(x)<K$ , for $x$ $¥geqq 0$

and

$¥sup_{¥mathrm{x}¥geqq 0}¥Gamma_{-}(x)=H<+¥infty$ .

Arguing like in [22, p. 271], we consider the curves defined by

$V(x, y)=¥frac{1}{2}(y-K)^{2}+¥Gamma_{¥_}(x)=c=$ constant.

The curves which intersect the $¥mathrm{y}$ -axis with $y¥geqq¥sqrt{2H}+K$, do not intersect theline $y=K$ . We have


$¥dot{V}=g(x)(1+F_{¥_}(x))^{-1}(y-F(x))-(y-K)g(x)$

$=g(x)(1+F_{¥_}(x))^{-1}(y-F(x)-(y-K)(1+F_{¥_}(x)))$

$=g(x)(1+F_{¥_}(x))^{-1}(K-F(x)-(y-K)F_{¥_}(x))$

$=g(x)(1+F_{¥_}(x))^{-1}(K-F_{+}(x)+F_{¥_}(x)(1-y+K))$ .

Consider the curve $¥beta:V(x, y)=H$, which intersects the $¥mathrm{y}$-axis at the point$(0, ¥sqrt{(2H)}+K)$, and evaluate $¥dot{V}$ along $¥beta$ . We obtain

$¥dot{V}=g(x)(1+F_{¥_}(x))^{-1}(K-F_{+}(x)+F_{¥_}(x)(1-(H-¥Gamma_{¥_}(x))^{1/2}¥sqrt{2}))$ .

Then, by the choice of $H$ and $K$, we get $¥dot{V}>0$, for each $X¥geqq¥overline{X}$ (with $¥overline{x}$

sufficiently large, such that $¥Gamma_{¥_}(¥overline{x})>H-(1/2))$ . This implies that, for $x$ $¥geqq¥overline{X}$, thetrajectories of system (2.1) are bounded away from $¥beta$ . In particular, considerthe point $P_{1}:=(¥overline{x}, K+(H-¥Gamma_{¥_}(¥overline{x}))^{1/2}¥sqrt{2})$ belonging to $¥beta$ . Clearly, $¥gamma^{-}(P_{1})$ in-tersects the $¥mathrm{y}$-axis at a point $P_{2}=(0,¥overline{y})$ , with $¥overline{y}>K$ . Therefore, for any $P=$

$(0, y_{0})$, with $¥mathcal{Y}0¥geqq¥overline{y}$, $¥gamma^{+}(P)$ is bounded away from $y=K$ and hence it does notintersect $a$ . The proof is complete. $¥square $

Remark 2.1. When $F(x)$ is bounded below for $x¥geqq 0$ (respectively, boundedabove for $x$ $¥leqq 0$), it can be easily seen that

$¥int_{0}^{+¥infty}(1+F_{¥_}(x))^{-1}g(x)dx=+¥infty$ if and only if $¥int_{0}^{+¥infty}g(x)dx$ $=+¥infty$

(respectively,

$¥int_{0}^{-¥infty}(1+F_{+}(x))^{-1}g(x)dx=+¥infty$ if and only if $¥int_{0}^{-¥infty}g(x)dx$ $=+¥infty)$ .

Hence, conditions (2.6) and (2.8) may be written, respectively, like

$¥lim_{¥mathrm{x}¥rightarrow}¥sup_{¥dagger¥infty}[G(x)+F(¥mathrm{x})]=+¥infty$ ,

and

$¥lim_{x¥rightarrow}¥sup_{-¥infty}[G(¥mathrm{x})-F(x)]=+¥infty$ ,

with

$ G(¥mathrm{x}):=¥int_{0}^{x}g(¥xi)d¥xi$ .

Therefore, this result contains Theorem 1 in [22]. With respect to [10],Theorem 2.1 provides a necessary condition for the existence of a global centerat the origin, improving Theorem 4.1 in [10].


The only case which remains to be investigated is the situation in which

(2.9) $¥lim_{¥mathrm{x}¥rightarrow¥dagger¥infty}F(x)=-¥infty$ ,

respectively,

(2. 10) $¥lim_{x¥rightarrow-¥infty}F(x)=+¥infty$ .

Consider a point $P=(x_{0}, y_{0})$ satisfying (2.3). As a first step we findconditions guaranteeing that $¥gamma^{+}(P)$ will not intersect the curve $a$ . Such as-sumptions may be used to produce a necessary condition. Again, we producea curve $¥beta$ such that $¥gamma^{+}(P)$ will be bounded away from $¥alpha$ .

Consider a continuously differentiable function $A:[d,$ $+¥infty[¥rightarrow R$, for some$d¥geqq 0$, and let $¥beta$ be the graph of $y=K-A(x)$, with $K>0$ a suitable constant.We require

(2.11) $A^{¥prime}(x)¥geqq 0$ and $K-A(x)>F(x)$ , for every $x¥geqq d$ .

Then we have the following

Lemma 2.1. Assume (2.11) and suppose that there exists $¥overline{x}¥geqq d$ such that

(2.12) $A(x)A^{¥prime}(x)+F(x)A^{¥prime}(x)+g(x)¥leqq 0$ , for every $x¥geqq¥overline{x}$ ,

then there is $¥overline{y}>0$ such that, for all $P=(0, y_{0})$ with $¥mathcal{Y}0¥geqq¥overline{y}$, $¥gamma^{+}(P)$ does notintersect $a$ .

$Proo/$. Consider the function

$V(x, y)=A(¥mathrm{x})+y$ ,

defining the curve $¥beta$ , via $V(x, y)=K$. The time rate of change of $V$ along asolution trajectory of equation (2. 1) is

$¥dot{V}=A^{¥prime}(x)¥dot{x}+¥dot{y}=A^{¥prime}(x)(y-F(x))-g(x)$ .

Evaluating $¥dot{V}$ on $¥beta$ , we have

$¥dot{V}=A^{¥prime}(¥mathrm{x})(K-A(x)-F(x))-g(x)$

$=A^{¥prime}(x)K-A(x)A^{¥prime}(x)-F(x)A^{¥mathit{1}}(x)-g(x)$

$¥geqq-A(x)A^{¥prime}(x)-F(x)A^{¥prime}(x)-g(x)¥geqq 0$

for $X¥geqq¥overline{X}$. This means that for any $P=(¥mathrm{x}_{0}, y_{0})$ with $x_{0}¥geqq¥overline{x}$, $¥mathcal{Y}0¥geqq K-A(x_{0})$ ,$¥gamma^{¥dotplus}(P)$ does not reach points below $¥beta$ , that is the set $¥{(x, y):x ¥geqq¥overline{x}, y¥geqq K-A(¥mathrm{x})¥}$

is positively invariant. Now, consider the point $P_{1}=(¥overline{x}, K-A(¥overline{x}))$ belongingto $¥beta$ and follows $¥gamma^{-}(P_{1})$ . Observe that $¥gamma^{-}(P_{1})$ intersects the $¥mathrm{y}$-axis at a point


$P_{2}=(0,¥overline{y})$ and then we have the desired result (by (2.11)) arguing like inTheorem 2.1. $¥square $

It is clear from the proof, that the hypothesis (2.12) may be replaced by

(2.13) $A(x)A^{¥prime}(x)+F(x)A^{¥prime}(x)+g(x)¥leqq KA^{¥prime}(x)$ , for every $X¥geqq¥overline{X}$ .

Remark 2.2. The previous lemma may have many applications dependingby the choice of the function $A(x)$ .

For instance, taking

$A(¥mathrm{x})=¥sqrt{2G(x)}$ , $ G(x):=¥int_{0}^{¥mathrm{x}}g(¥xi)d¥xi$ ,

by easy computations, we have that (2.11) and (2.13) are satisfied provided that

$¥lim_{¥mathrm{x}¥rightarrow}¥sup_{+¥infty}[2¥sqrt{2G(x)}+F(x)]<+¥infty$ .

This results provides a necessary condition for $¥gamma^{+}(P)$ crossing $a$ , in the light ofFilippov’s theorem [6], [19, p. 321]. In fact, we get that

if $¥gamma^{+}(P)$ intersects $a$ , for any $P=(x_{0}, y_{0})$ satisfying (2.3), then

(2. 14) $¥lim_{¥mathrm{x}¥rightarrow}¥sup_{¥dagger¥infty}[2¥sqrt{2G(x)}+F(x)]=+¥infty$ .

We recall that A. F. Filippov [6] obtained the sufficient condition:if

(2.15) $¥lim_{x¥rightarrow+¥infty}[a¥sqrt{2G(x)}+F(x)]=+¥infty$ for $0<a<2$ ,

then $¥gamma^{+}(P)$ intersects $a$ , for any $P=(x_{0}, y_{0})$ satisfying (2.3).On the other hand, taking

$ A(x)=2¥int_{b}^{¥mathrm{x}}|F(¥xi)|^{-1}g(¥xi)d¥xi$ , with $F(x)<0$ for $x$ $¥geqq b$ ,

we have that (2.11) and (2.12) are satisfied provided that

$|F(x)|^{-1}¥int_{b}^{x}|F(¥xi)|^{-1}g(¥xi)d¥xi¥leqq 1/4$ , for $x¥geqq c¥geqq b$ .

This is precisely the hypothesis in [10, Th. 4.3]. With the same choice of $A(x)$,

it can be easily checked that (2.11) and (2.13) are satisfied provided that

$¥mathrm{f}1_{b}^{¥mathrm{x}}|F(¥xi)|^{-1}g(¥xi)d¥xi+F(x)]$ is bounded from above on [$b$ , $+¥infty$ [. Thus we

can get a necessary condition, in the case in which (2.9) is assumed, as follows:if $¥gamma^{+}(P)$ intersects $a$ , for any $P=(x_{0}, y_{0})$ satisfying (2.3), then


(2. 16) $¥lim_{x¥rightarrow}¥sup_{+¥infty}¥mathrm{f}1_{b}^{x}|F(¥xi)|^{-1}g(¥xi)d¥xi+F(x)]=+¥infty$

(compare with (2. 18) in Theorem 2.2, below).We recall that T. Hara and T. Yoneyama in [10, Th. 4.2] obtained a

sufficient condition which may be written (in the case of validity of (2.9)) as:if

(2.16) $¥lim_{¥chi¥rightarrow+¥infty}¥mathrm{H}_{c}^{x}|F(¥xi)|^{-1}g(¥xi)d¥xi+F(x)]=+¥infty$ , for $¥mathrm{o}<a<4$ ,

then $¥gamma^{+}(P)$ intersects $a$ , for any $P=(x_{0}, y_{0})$ satisfying (2.3).We finally observe that all the results proved and stated in Lemma 2.1

and in the present Remark 2.2, may be easily adapted to the case in which $P=$

$(x_{0}, y_{0})$ satisfies condition (2.4). In particular, assumptions (2.11) and (2.12) willbe replaced by

$A^{¥prime}(x)¥leqq 0$ and $A(x)-K<F(x)$ , for every $x$ $¥leqq-d$

and by$A(¥mathrm{x})A^{¥prime}(x)-F(x)A^{¥prime}(x)+g(x)¥geqq 0$ , for every $x¥leqq-¥overline{x}$ ,

respectively.

In the light of Remark 2.2, we give the following result.

Theorem 2.2. If, for every $P=(x_{0}, y_{0})$ verifying (2.3), $¥gamma^{+}(P)$ intersects thecurve $y=F(x)$, then

(2. 18) $¥lim_{x¥rightarrow}¥sup_{+¥infty}[4¥Gamma_{¥_}(x)+F(x)]=+¥infty$ .

If, for every $P=(x_{0}, y_{0})$ verifying (2.4), $¥gamma^{+}(P)$ intersects the curve $y=F(x)$, then

(2. 19) $¥lim_{¥mathrm{x}¥rightarrow}¥sup_{-¥infty}[4¥Gamma_{+}(x)-F(x)]=+¥infty$ .

Proof. Again, we prove only the first part of the theorem, as the secondone is treated in the same way.

Suppose, by contradiction that

$¥lim_{¥chi¥rightarrow}¥sup_{¥dagger¥infty}[4¥Gamma_{-}(x)+F(x)]<+¥infty$ .

This implies that there is a constant $H>0$ such that

(2.20) $F(x)<H-4¥Gamma_{¥_}(¥mathrm{x})$ , for all $x$ $¥geqq 0$ .

Following Lemma 2.1, we define

$A(x)=2¥Gamma_{-}(x)$ .


We have $A^{¥prime}(x)¥geqq 0$ and $K-A(x)>F(x)$ for all $x$ $¥geqq 0$ and $K¥geqq H$, so that (2.11)is verified. Now,

$A(¥mathrm{x})A^{¥prime}(x)+F(x)A^{¥prime}(x)+g(x)$

$=(1+F_{¥_}(x))^{-1}g(x)(4¥Gamma_{¥_}(x)+2F(x))+g(x)$

$=g(x)(1+F_{¥_}(x))^{-1}(4¥Gamma_{¥_}(x)+2F(x)+1+F_{¥_}(x))$

$=g(x)(1+F_{¥_}(x))^{-1}(4¥Gamma_{¥_}(x)+F(x)+F_{+}(x)-F_{¥_}(x)+1+F_{¥_}(x))$

$=g(x)(1+F_{¥_}(x))^{-1}(4¥Gamma_{¥_}(x)+F(x)+F_{+}(x)+1)$

$<g(x)(1+F_{¥_}(x))^{-1}(H+F_{+}(x)+1)$

$¥leqq g(x)(1+F_{¥_}(x))^{-1}K=KA^{¥prime}(x)$ ,

for any $K¥geqq H+1+¥sup¥{F_{+}(x):x¥geqq 0¥}$ (observe that $ K<+¥infty$ by (2.20)). Hencewe have that (2.13) holds for every $x¥geqq 0$ . So that, by Lemma 2.1, we find acontradiction. $¥square $

Remark 2.3. Under the assumption (2.5) (respectively (2.7)), we note thatcondition (2.18) and condition (2.6) are equivalent (respectively (2.19) and (2.8)are equivalent).

Now we consider a sufficient condition for the intersection of $¥gamma^{+}(P)$ withthe curve $a$ .

Theorem 2.3. If there is a constant $0<a<4$ such that

(2.21) $¥lim_{¥chi¥rightarrow+¥infty}[a¥Gamma_{¥_}(x)+F(x)]=+¥infty$ ,

then, for every $P=(x_{0}, y_{0})$ verifying (2.3), $¥gamma^{+}(P)$ intersects the curve $y=F(x)$ at$(¥mathrm{x}, F(x))$ with $x>x_{0}$ . If there is a constant $0<a<4$ such that

(2.22) $¥lim_{x¥rightarrow-¥infty}[a¥Gamma_{+}(x)-F(x)]=+¥infty$ ,

then, for every $P=(x_{0}, y_{0})$ verifying (2.4), $¥gamma^{+}(P)$ intersects the curve $y=F(x)$ at$(x, F(x))$ with $x<x_{0}$ .

$Proo/$. As usual, we examine only the first situation. Assume that condi-tion (2.21) holds and let $P$ verify (2.3). We distinguish the following cases:

I) $¥lim_{x¥rightarrow}¥sup_{+¥infty}F(¥mathrm{x})>-¥infty$ .

In this case, we have only to observe that assumption (2.21) impliescondition (2.6) of Theorem 2.1 and therefore the result is proved.

$¥mathrm{I}¥mathrm{I})$

$¥lim_{x¥rightarrow+¥infty}F(x)=-¥infty$ .


Let $b>0$ be such that $F(x)<0$ for $x$ $¥geqq b$ . Then

$¥Gamma_{-}(x)=K(b)+¥int_{b}^{x}(1+|F(¥xi)|)^{-1}g(¥xi)d¥xi$ ,

there $K(b)=¥int_{0}^{b}(1+F_{¥_}(x))^{-1}g(x)dx$ is a constant. Hence, for any fixed $b$ ,

(2.21) may be written as

(2.23) $¥lim_{x¥rightarrow+¥infty}[¥int_{b}^{x}(1+|F(¥xi)|)^{-1}g(¥xi)d¥xi-(1/a)|F(x)|]=+¥infty$ .

Observe that $|F(x)|/(1+|F(x)|)¥leqq 1$ , for all $x$ .

Therefore, for $b$ sufficiently large, from (2.23) we get

$+¥infty=¥lim_{¥mathrm{x}¥rightarrow+¥infty}[¥int_{b}^{x}¥frac{g(¥xi)}{|F(¥xi)|}¥cdot¥frac{|F(¥xi)|}{1+|F(¥xi)|}d¥xi-(1/a)|F(x)|]$

$¥leqq¥lim_{x¥rightarrow+¥infty}[¥int_{b}^{x}|F(¥xi)|^{-1}g(¥xi)d¥xi-(1/a)|F(x)|]$ .

Then, we have

$¥lim_{x¥rightarrow+¥infty}[¥int_{b}^{x}|F(¥xi)|^{-1}g(¥xi)d¥xi-(1/a)|F(x)|]=+¥infty$ .

Thus there exists a number $b_{1}>b$, such that

$[¥int_{b}^{x}|F(¥xi)|^{-1}g(¥xi)d¥xi-(1/a)|F(x)|]¥geqq 0$ , for $x¥geqq b_{1}$ .

This implies condition $(A_{6})$ in [10, Lemma 4.2] which gives the desired result.Note that, with a similar argument, it may be shown that assumption $(A_{6})$

in [10] is indeed equivalent to our condition (2.21) under (2.9). $¥square $

Remark 2.4. As mentioned above, when (2.9) or (2.10) hold, our result isequivalent to that of T. Hara and T. Yoneyama [10]. However, conditions(2.21) and (2.22) seem to be interesting because they are in line with those ofTheorem 2.1 (which are necessary and sufficient). This suggests the problem toknow whether hypotheses (2.5) and (2.7) of Theorem 2.1 may be dropped, inorder to have the most general possible conditions.

We finally observe that analogous theorems may be obtained with respectto the problem of intersection of $¥gamma^{-}(P)$ with the curve $a$ . Such results will beused in the next section. More precisely, we have, for a point $P=(x_{0}, y_{0})$ ,verifying


(2.24) $x_{0}¥geqq 0$ , $¥mathcal{Y}0<F(x_{0})$

or

(2.25) $x_{0}¥leqq 0$ , $¥mathcal{Y}0>F(x_{0})$ ,

the following results.

Theorem 2.4. Assume

1 $ x¥mathrm{i}¥mathrm{m}¥inf F(x)¥rightarrow+¥infty<+¥infty$ .

Then, for every $P=(x_{0}, y_{0})$ verifying (2.24), $¥gamma^{-}(P)$ intersects the curve $y=F(x)$,at $(x, F(x))$ with $x>x_{0}$ , if and only if(2.26) $¥lim_{¥chi¥rightarrow}¥sup_{¥dagger¥infty}$

$[¥Gamma_{+}(x)-F(x)]=+¥infty$ .

Assume$¥lim_{x¥rightarrow}¥sup_{-¥infty}F(x)>-¥infty$ .

Then, for every $P=(x_{0}, y_{0})$ verifying (2.25), $¥gamma^{-}(P)$ intersects the curve $y=F(x)$,at $(x, F(x))$ with $x<x_{0}$ , if and only if(2.27) $¥lim_{x¥rightarrow}¥sup_{-¥infty}$

$[¥Gamma_{¥_}(x)+F(x)]=+¥infty$ .

Theorem 2.5. If, for every $P=(x_{0}, y_{0})$ verifying (2.24), $¥gamma^{-}(P)$ intersects thecurve $y=F(x)$, at $(x, F(x))$ with $x>x_{0}$ , then

(2.28) $¥lim_{x¥rightarrow}¥sup_{+¥infty}[4¥Gamma_{+}(¥mathrm{x})-F(x)]=+¥infty$ .

If, for every $P=(x_{0}, y_{0})$ verifying (2.25), $¥gamma^{-}(P)$ intersects the curve $y=F(x)$, at$(x, F(x))$ with $x$ $<x_{0}$ , then

(2.29) $¥lim_{¥chi¥rightarrow}¥sup_{-¥infty}[4¥Gamma_{¥_}(x)+F(x)]=+¥infty$ .

Theorem 2.6. If there is a constant $0<a<4$ such that

(2.30) $¥lim_{x¥rightarrow+¥infty}[a¥Gamma_{+}(x)-F(x)]=+¥infty$ ,

then, for every $P=(x_{0}, y_{0})$ verifying (2.24), $¥gamma^{-}(P)$ intersects the curve $y=F(x)$, at$(x, F(x))$ with $x$ $>x_{0}$ . If there is a constant $0<a<4$ such that

(2.31) $¥lim_{x¥rightarrow-¥infty}[a¥Gamma_{¥_}(x)+F(x)]=+¥infty$ ,

then, for every $P=(x_{0}, y_{0})$ verifying (2.25), $¥gamma^{-}(P)$ intersects the curve $y=F(x)$, at$(x, F(x))$ with $x$ $<x_{0}$ .


The proof of these results is straightforward, following the same argumentspreviously employed, and so it is omitted.

3. Periodic solutions of Lienard equation

We consider in this section the Lienard autonomous equation

(3. 1) $¥ddot{X}+f(x)¥dot{x}+g(x)=0$ .

We assume through the section that /, $g:R¥rightarrow R$ are continuous functionswith $g$ locally lipschitzian and $xg(x)>0$ for every $x¥neq 0$ .

It is well known that equation (3.1) may be written as a system in theLienard plane, that is

$¥dot{X}=y-F(x)$

(3.2)$¥dot{y}=-g(x)$ ,

with $F(x):=¥int_{0}^{x}f(u)du$ , or as a system in the phase plane, that is

$¥dot{¥mathrm{x}}=y$

(3. 3)$¥dot{y}=-f(¥mathrm{x})y-g(x)$ .

Clearly, systems (3.2) and (3.3) are equivalent. For instance, the nonlineartransformation $T:(x, y)¥rightarrow(x, y+F(x))$ carries system (3.2) into (3.3). We notethat the assumptions on $f$ and $g$ guarantee the uniqueness and (local) existencefor the solutions of Cauchy problems associated to (3.2).

We say that system (3.2) has the property (H) if it possesses a closedtrajectory $¥gamma$ surrounding the origin and containing in its interior all the criticalpoints of the system and such that each trajectory outside $¥gamma$ is closed. Condi-tions for $f$ and $g$ ensuring that property (H) holds can be found in [6], [15],[21], [10], [23]. In Lemma 3.1 and Theorem 3.1 we restrict ourselves to thecase in which

(3.4) $f$ and $g$ are odd functions,

which guarantee that the trajectories of system (3.2) are symmetric with respectto the $¥mathrm{y}$-axis. We also note that, from our assumptions on $g$ , the only criticalpoint of (3.2) is the origin.

The results of the preceding section may be used in connection with thefollowing lemma.

Lemma 3.1. Assume (3.4) and suppose that there is $¥overline{y}>0$ , such that, foreach $P=(0, y_{0})$,


$¥gamma^{+}(P)$ intersects $a$ for $¥mathcal{Y}0¥geqq¥overline{y}$

(3.5) and

$¥gamma^{-}(P)$ intersects $a$ for $¥mathcal{Y}0¥leqq-¥overline{y}$ .

Then system (3.2) has $t/ie$ property (H).

The proof is achieved with a standard argument and the results of theprevious section can be applied in order to guarantee the validity of theproperty (3.5).

More precisely, (3.5) holds in each of the following situations:

1. $¥lim_{¥chi¥rightarrow}¥sup_{+¥infty}F(x)=+¥infty$ , $¥lim_{¥chi¥rightarrow}¥inf_{+¥infty}F(x)=-¥infty$ ;

2. $|F(x)|¥leqq K$ for $x¥geqq 0$ , $¥lim_{x¥rightarrow+¥infty}G(x)=+¥infty$ (see [22]);

3. $¥lim_{¥mathrm{x}¥rightarrow+¥infty}F(x)=+¥infty$ , $¥lim_{x¥rightarrow+¥infty}[a¥Gamma_{+}(x)-F(x)]=+¥infty$ $(0<a<4)$ ;

4. $¥lim_{¥chi¥rightarrow+¥infty}F(x)=-¥infty$ , $¥lim_{x¥rightarrow+¥infty}[a¥Gamma_{-}(x)+F(¥mathrm{x})]=+¥infty$ $*(0<a<4)$;

5. .$¥lim_{x¥rightarrow}¥sup_{+¥infty}F(x)=+¥infty$ , $¥lim_{x¥rightarrow}¥inf_{¥dagger¥infty}F(x)=k¥in R$ ,

$¥lim_{¥chi¥rightarrow}¥sup_{+¥infty}[¥Gamma_{+}(¥mathrm{x})-F(x)]=+¥infty$ ;

6. $¥lim_{x¥rightarrow}¥sup_{+¥infty}F(x)=k¥in R$ , $¥lim_{x¥rightarrow}¥inf_{+¥infty}F(x)=-¥infty$ ,

$¥lim_{x¥rightarrow}¥sup_{-¥infty}[¥Gamma_{-}(x)+F(x)]=+¥infty$ .

The cases listed above contain all the possible situations which can be con-sidered using the theorems in section 2 and give a fairly complete description ofthe assumption (3.5).

On the line of [21], we can produce the following result of existence ofperiodic solutions for the Lienard equation.

Theorem 3.1. Consider the equation

(3.6) $¥ddot{x}+(f(x)+p(x))¥dot{x}+g(x)=0$ ,

with $f$, $g$ , $p:R¥rightarrow R$ continuous functions and $g$ locally lipschitzian with $xg(x)>0$

for $x$ $¥neq 0$ . If system (3.2) verifies property (H) and$(a_{1})$ the origin is a source for (3.6),$(a_{2})$ there exists $¥delta>0$ such that $p(x)>0$ for $|x|¥geqq¥delta$ and, for some $¥delta_{1}>¥delta$ ,

$¥int_{-¥delta_{1}}^{¥delta_{1}}p(x)dx>0$ .

Then equation (3.6) has at least a nontrivial periodic solution.


Proof. Arguing like in [21, Th. 1], we may produce a bounded semi-trajectory for the system

$¥dot{x}=y$

(3.7)$¥dot{y}=-(f(¥mathrm{x})+p(x))y-g(¥mathrm{x})$ .

As the origin is a source, the Poincare-Bendixson theorem implies that system(3.7) has at least one limit cycle, and this completes the proof. $¥square $

Clearly, Lemma 3.1 can be used in order to have the property (H). Inparticular, under the assumptions of Lemma 3.1, condition $(a_{1})$ is satisfied if

$x¥int_{¥mathrm{o}}^{X}p(¥xi)d¥xi<0$ for $¥mathrm{o}<|x|<¥epsilon$ ($¥epsilon$ small enough).

We finally observe that property (H) is verified for the system

$¥dot{¥mathrm{x}}=y$

$¥dot{y}=-g(x)$ ,

when $¥lim_{|x|¥rightarrow+¥infty}G(x)=+¥infty$ (such case was considered in [21, Th. 1]). There-fore Theorem 3.1 is an extension of the classical results of N. Levinson andO. K. Smith [11] and A. Dragilev [5].

Now we use the necessary conditions for the intersection of $¥gamma^{+}(P)$ (respect-

ively $¥gamma^{-}(P))$ with the curve $a$ .

We say that system (3.2) has the property (K) if there is $¥overline{y}¥neq 0$ such that$¥gamma^{-}(0,¥overline{y})$ does not intersect $a$ and $¥gamma^{+}(0,¥overline{y})$ is oscillatory.

Conditions for $f$ and $g$ which guarantee that property (K) holds can befound in [7], [20] and using the results in section 2. Then we have thefollowing theorem.

Theorem 3.2. Consider system (3.2) with $f$, $g:R¥rightarrow R$ continuous, $g$ locallylipschitzian with $xg(x)>0$ for $x$ $¥neq 0$. If

$(b_{1})$ the origin is a source,$(b_{2})$ property (K) holds,

then there is at least one limit cycle of (3.2), that is, a nontrivial periodic solutionof equation (3.1).

Proof. Arguing like in [20], we can produce a bounded semitrajectory ofsystem (3.2). In fact, in the light of property (K), we claim that for $¥overline{P}=(0,¥overline{y})$ ,$¥gamma^{+}(¥overline{P})$ intersects the $¥mathrm{y}$-axis in $P_{1}=(0, y_{1})$, with $y_{1}¥overline{y}<0$ . Then we observethat $¥gamma^{+}(P_{1})=¥gamma^{+}(¥overline{P})$ intersects $a$ and, successively, again the $¥mathrm{y}$-axis at a point$P_{2}=(0, y_{2})$. Note that $|y_{2}|<|¥overline{y}|$ since $¥gamma^{-}(¥overline{P})$ does not intersects $a$ . There-fore we have proved that $¥gamma^{+}(¥overline{P})$ is bounded. As the origin is a source, thePoincare-Bendixson theorem implies that system (3.2) has at least a limitcycle. $¥square $


If the origin is a source, one may easily verify that any semitrajectory $¥gamma^{+}(P)$

which intersects $a$ , intersects the $¥mathrm{y}$-axis, too. In this situation we can proceedas follows.

Consider a $¥mathrm{p}¥mathrm{o}¥mathrm{i}¥dot{¥mathrm{n}}¥mathrm{t}¥overline{P}=(0,¥overline{y})$ with, for instance, $¥overline{y}<0$ . If condition (2.28) isnot fulfilled, $¥gamma^{-}(¥overline{P})$ does not intersect $a$ in $x>0$, provided that $|¥overline{y}|$ is largeenough. If, on the other hand, (2.22), or (2.7)?(2.8), holds, then $¥gamma^{+}(¥overline{P})$ intersects$a$ in $x<0$ and so, also the $¥mathrm{y}$-axis in $P_{1}=(0, y_{1})$ with $y_{1}>0$. In order to verify(K) it remains to prove that $¥gamma^{+}(P_{1})$ intersects $a$ . As (2.28) is not satisfied, $F(x)$

must be bounded below for $x$ $¥geqq 0$ and then the intersection of $¥gamma^{+}(P_{1})$ with $a$ ispossible provided that $¥lim¥sup_{¥chi¥rightarrow+¥infty}[G(x)+F(x)]=+¥infty$ (see [22]). In thisway, we have obtained the following

Corollary 3.1. Consider system (3.2) with the usual regularity assumptionson $f$ and $g$ , and $xg(x)>0$ for $x¥neq 0$. Suppose

$(c_{1})$ $¥mathrm{x}F(x)<0$ for $ 0<|x|<¥epsilon$,$(c_{2})$ $4¥Gamma_{+}(x)-F(x)¥leqq K$, for every $x$ $¥geqq 0$,$(c_{3})$ $¥lim_{x¥rightarrow¥_¥infty}[a¥Gamma_{+}(x)-F(x)]=+¥infty$ , with $0<a<4$,$(c_{4})$ $¥lim_{x¥rightarrow+¥infty}G(x)=+¥infty$ .

Then system (3.2) $/ias$ at least a limit cycle.

We observe that the case in which

$G(x)¥leqq H$ , for $x$ $¥geqq 0$ ,

is already considered in [22, Th. 3].

Combining the results of section 2, one may produce other corollarieswhich are omitted for simplicity.

4. The forced case

We now consider the periodically forced Lienard equation

(4. 1) $¥ddot{x}+f(x)¥dot{x}+g(x)=e(t)=e(t+T)$ .

It is well known that the existence of a $¥mathrm{T}$-periodic solution for equation (4.1)may be achieved with topological methods in function spaces (see [12], [17],[1], [4] $)$ . For an extensive bibliography we refer to [13].

Here we employ the classical technique of finding a flow-invariant region inthe plane for the system

$¥dot{x}=y$

(4.2)$¥dot{y}=-f(x)y-g(x)+e(t)$ ,

or the system


$¥dot{X}=y-F(x)+E(t)$(4. 3)

$¥dot{y}=-¥tilde{g}(x)$ ,

with $¥tilde{g}(x):=g(x)-¥overline{e}$ and $E(t):=¥int_{¥mathrm{o}}^{t}(e(s)-¥overline{e})ds$ , where $¥overline{e}:=(1/T)¥int_{¥mathrm{o}}^{T}e(t)dt$ is the

mean value in a period of the forcing term $e(¥cdot)$ . In this way, the existence of a$¥mathrm{T}$-periodic solution for (4.2) or (4.3) is obtained by producing a fixed point forthe associated Poincare map.

We assume, through this section that /, $g$, $e:R¥rightarrow R$ are continuous func-tions, with $e(¥cdot)$ $¥mathrm{T}$-periodic, $g$ locally lipschitzian and

(4.4) $¥lim_{|x|¥rightarrow+¥infty}g(x)$ sign $x$ $=+¥infty$ .

Assumption (4.4) ensures that

$(g(x)- ¥text{{¥it ¥^{e}}})x>0$ , for $|x|$ large enough ,

with $ e¥mathrm{A}:=¥max$ $¥{|e(t)|:t¥in[0, T]¥}$ .As is proved in [24], we observe that Theorem 3.1 can be extended to this

case.

Theorem 4.1. Assume the regularity conditions and (4.4). If system (3.2)verifies condition (//), then the equation

$¥ddot{x}+(f(x)+p(x))¥dot{x}+g(x)=e(t)$

(with $p:R¥rightarrow R$ continuous), has a $T$-periodic solution, provided that there exist$¥epsilon$ , $¥delta>0$ such that

$ p(x)¥geqq¥epsilon$ for $|x|¥geqq¥delta$ .

The result is proved studying system (4.2). However, it is clear that (3.3)verifies the property (H) if and only if system (3.2) fulfils the same prop-erty. As one can see in [24], the assumption (H) plays a fundamental role inthe proof of this theorem. The results of section 2 give us new possibilities toproduce (H).

Using system (4.3), the property (K) may be employed as well. First of allwe observe that $E(¥cdot)$ is a $¥mathrm{T}$-periodic function and therefore it is bounded. Wedefine

$E_{1}=¥min¥{E(t):t ¥in[0, T]¥}$ , $E_{2}=¥max¥{E(t):t¥in[0, T]¥}$ ,

and consider the auxiliary autonomous systems

$¥dot{X}=y-F(x)+E_{1}$(4. 5)

$¥dot{y}=-¥tilde{g}(x)$ ,


$¥dot{X}=y-F(x)+E_{2}$

(4.6)$¥dot{y}=-¥tilde{g}(x)$ .

A comparison with their respective slopes shows that when $x¥tilde{g}(x)>0$, that is(by (4.4)) for $|x|¥geqq d>0$ ($d$ being a suitable constant), then the trajectories ofsystem (4.3) enter the trajectories of system (4.6) for $x$ $¥geqq d$, while the trajectoriesof (4.3) enter the trajectories of (4.5) for $x¥leqq-d$. For this reason, we pro-duce the property (K) working with system (4.5) for $x¥leqq-d$ and system (4.6)for $x$ $¥geqq d$ . A possible result which can be obtained in this manner, is thefollowing.

Theorem 4.2. Assume the regularity conditions and (4.4). Suppose:$(k_{1})$ $4¥Gamma_{+}(x)-F(x)¥leqq K$ , for every $x¥geqq 0$,$(k_{2})$ $¥lim[a¥Gamma_{+}(x)-F(x)]=+¥infty$ , with $0<a<4$.

$¥mathrm{x}¥rightarrow-¥infty$

Then equation (4.1) $/zas$ at feast a $T$-periodic solution.

Proof. First of all we notice that the properties $(k_{1})$ and $(k_{2})$ do notchange if we replace $F(x)$ with $F(x)-E_{1}$ or $F(x)-E_{2}$ and replace $g(x)$ with$¥tilde{g}(x)$. Following Theorem 3.2 we may produce a flow-invariant region homeo-morphic to a closed disc.

Consider a suitable strip $|x|¥leqq d$ in the plane, such that $¥tilde{g}(x)¥mathrm{x}>0$ for$|x|¥geqq d$ .

Property $(k_{1})$ implies that there is $¥overline{y}<0$ such that for every $y¥leqq¥overline{y}$,if we consider the system (4.6), then $¥gamma^{-}(d, y)$ does not intersect the curve$y=F(x)-E_{2}$ . Choose a point $P_{1}=(d, y_{1})$ with this property. In the strip$|x|¥leqq d$ , $y^{¥prime}=dy/dx$ $=-¥tilde{g}(¥mathrm{x})/(y-F(x)+E(t))$ is bounded (for $|y|$ large enough),so that we can find a suitable point $P_{2}=(-d, y_{2})$, with $y_{2}<y_{1}$ , such that thesolutions of (4.3) which pass through $P_{1}$ at any time are bounded below by thesegment $¥overline{P_{1}P_{2}}$ (see [19]). We consider now system (4.5) and observe that $¥gamma^{+}(P_{2})$

intersects the $(-¥mathrm{d})$-axis at a point $P_{3}=(-d, y_{3})$ with $y_{3}>0$ (without lossof generality, due to hypothesis $(k_{2}))$. Again, the solutions of system (4.3)which pass through $P_{3}$ at any time, are bounded above by the segment $¥overline{P_{3}P_{4}}$ ,where $P_{4}=(d, y_{4})$ is a suitable point with $y_{4}>y_{3}$ . Observe, from (4.4), that

$¥int_{0}^{X}¥tilde{g}(¥xi)d¥xi¥rightarrow+¥infty$ for $ x¥rightarrow+¥infty$ . Therefore, in the light of Theorem 3.2, if we

consider system (4.6), we have that $¥gamma^{+}(P_{4})$ intersects the curve $y=F(x)-E_{2}$ ata point $P_{5}=(¥mathrm{x}_{5}, y_{5})$ , $y_{5}=F(x_{5})-E_{2}$ . Finally, let $P_{6}=(¥mathrm{x}_{¥sigma}, y_{¥sigma})$ , with $x_{6}=x_{5}$

and $P_{6}¥in¥gamma^{-}(P_{1})$.

The Jordan $¥mathrm{c}¥mathrm{u}¥mathrm{r}¥mathrm{v}¥mathrm{e}¥nearrow¥overline{P_{1}P_{2}P_{3}P_{4}P_{5}P_{6}P_{1}}$ is the boundary of the required flow-invariant region. The Brouwer fixed point theorem may be invoked to get theexistence of a $¥mathrm{T}$-periodic solution. $¥square $


Remark 4.1. We observe that for any point $P¥in R^{2}$ , we can find a suitableregion $V$, as constructed in the proof of Theorem 4.1, such that $P¥in V$ There-fore, we get also that all the solutions of system (4.2) are bounded for $t$ $¥geqq 0$ .

As mentioned at the end of section 3, we can use the results of section 2 togive other similar theorems, suitably changing assumptions $(k_{1})$ and $(k_{2})$ . Alsothe condition (4.4) may be relaxed, maintaining the property that $x¥tilde{g}(x)>0$ for$|x|$ large.

Remark 4.2. The Brouwer fixed point theorem may also be applied whena negatively invariant region is produced. Thus one can easily obtain resultscombining the theorems in section 2 in order to have that $¥gamma^{-}(P_{1})$ intersects thecharacteristic curves, while $¥gamma^{+}(P_{2})$ does not.

Addendum. In the recent paper “A necessary and sufficient conditionfor oscillation of the generalized Lienard equation” (Ann. Mat. Pura Appl.,154(1989), 223-230), T. Hara, T. Yoneyama and J. Sugie have obtained aresult which (although stated in a different manner) is essentially Theorem2.1 of the present article. However, we stress the fact that the two resultsare obtained independently and, moreover, all the other theorems andapplications are completely different.

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Springer-Verlag, Berlin, 1963.[4] Cesari, L. and Kannan, R., Periodic solutions in the large of Lienard systems with forcing

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Equations, 12 (1972), 34-62.[9] Hale, J. K., Ordinary Differential Equations, Wiley Intersci., New York, 1969.[10] Hara, T. and Yoneyama, T., On the global center of generalized Lienard equation and its

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[12] Mawhin, J., An extension of a theorem of A. C. Lazer on forced nonlinear oscillations,J. Math. Anal. AppL, 40 (1972), 20-29.

[13] Omari, P., Villari, Gab. and Zanolin, F., Periodic solutions of the Lienard equation withone-sided growth restrictions, J. Differential Equations, 67 (1987), 278-293.

[14] Omari, P., Villari, Gab. and Zanolin, F., A survey of recent applications of fixed pointtheory to periodic solutions of the Lienard equation, in Fixed point theory and its appica-tions (R. F. Brown, ed.), 171-178. Contemporary Math., vol. 72, Amer. Math. Soc., Provi-dence R. I., 1988.

[15] Opial, Z., Sur un theoreme de A. Filippov, Ann. Polon. Math., 5 (1958), 67-75.[16] Ponzo, P. J. and Wax, N., On periodic solutions of the system $¥dot{x}=y$ ? $F(x),¥dot{y}=-g(x)$,

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chungen, Cremonese, Roma, 1963.[19] Sansone, G. and Conti, R., Non-linear differential equations, Pergamon Press, 1964.[20] Villari, Gab., Periodic solutions of Lienard’s equation, J. Math. Anal. AppL, 86 (1982),

379-386.[21] Villari, Gab., On the existence of periodic solutions for Lienard’s equation, Nonlinear

Anal., 7 (1983), 71-78.[22] Villari, Gab., On the qualitative behaviour of solutions of Lienard equation, J. Differential

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periodic solutions, Int. J. Nonlinear Mech., 23 (1988), 1-7.[24] Villari, Gab and Zanolin, F., On forced nonlinear oscillations of a second order equation

with strong restoring term, Funkcial. Ekvac., 31 (1988), 383-395.[25] Wu Zhuo-qun, Existence of limit cycles of nonlinear oscillating differential equations, J.

Northeast China Peoples Univ., 2 (1956), 33-46 (Chinese).[26] Ye Yan-qian, et al., Theory of Limit Cycles, Amer. Math. Soc., Providence R.I., 1986.[27] Zhang Zhi-fen, Ding Tong-ren, et al., Qualitative Theory of Ordinary Differential Equations,

Press of Science, Beijing, 1985 (Chinese).

nuna adreso:Gabriele VillariIstituto Matematico “U. Dini”Universita Viale Morgagni 67/A50134 FirenzeItaly

Fabio ZanolinDipartimento di Matematica eInformaticaUniversita, via Zanon 6-833100 UdineItaly

(Ricevita la 14-an de marto, 1988)(Reviziita la 21-an de februaro, 1989)

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