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Dynamics of a Periodic Differential Equation with a Singular Nonlinearity of Attractive Type
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARITCLE NO.
202, 1027]1039 Ž1996.
0358
Dynamics of a Periodic Differential Equation with a Singular Nonlinearity of Attractive Type U Pedro Martınez-Amores and Pedro J. TorresU ´
Departamento de Matematica ´ Aplicada, Uni¨ ersidad de Granada, 18071 Granada, Spain Communicated by Jack K. Hale Received January 24, 1995; revised November 29, 1995
INTRODUCTION In this paper we shall consider a general class of equations with a restoring force of attractive type that includes xY q cxX q
1 xa
s pŽ t . ,
Ž 1.
where c G 0, a ) 0, and p is a continuous T-periodic function for some T ) 0. We are interested in the existence and stability of positive Tperiodic solutions of Ž1.. The existence of T-periodic solutions of this class of equations where the restoring force is a singular nonlinearity that becomes infinite in zero has been proved by Lazer and Solimini w2x for the case without friction Ž c s 0. and by Habets and Sanchez w1x for the damped case. However, the ´ stability properties of these solutions have been less studied. It is well known that for the autonomous case Ž pŽ t . ' p 0 ) 0., this equation has a unique saddle point. In this paper we prove that the dynamics of the periodic equation Ž1. is similar to the autonomous case. For our study is fundamental a Massera’s convergence theorem in R n given by Smith in w6x. The paper is divided in three sections. In Section 1, the main results are stated. It is seen that Ž1. has a unique unstable periodic solution w . U
Supported by D. G. C. Y. T. PB92-0953, M.E.C., Spain, and E.E.C. project ERBCHRXCT94-0555. 1027 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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MART´ ıNEZ-AMORES AND TORRES
Moreover, for the damped case it is proved that there exists a global stable manifold of this periodic solution which is determined by a strictly decreasŽbeing Rq w .. such that this graph divides the ing map h: R ª Rq 0 0 s 0, q` plane of initial conditions in two open sets, one of them Žset A in Fig. 1. corresponds to the solutions that tend to q` as t ª q`, and the other Žset B. to the solutions that goes to zero in finite time, that is, the ones that are ‘‘absorbed’’ by the singularity. When the potential V Ž x . is infinite as x ª 0q Žcase a G 1., we have that h: R ª Rq Žbeing Rqs Ž0, q`.. is a homeomorphism, whereas if V Ž x . is finite as x ª 0q Žcase 0 - a - 1. there exists ¨ s g R such that hŽ ¨ s . s 0. Similar results are showed for the unstable manifold. As a consequence, there are no homoclinic points. We prove these results in Section 2. For this, the study of the behavior of the solutions and its derivatives according to its maximal interval of existence Ž wy, wq ., y` F wy- wqF q`, is fundamental. Some properties of comparison of solutions are given which are needed in the proofs. Finally, in Section 3, we remark that the previous results are true for the undamped case.
1. MAIN RESULTS Let us consider the forced Lienard equation, ´ xY Ž t . q f Ž x Ž t . . xX Ž t . q g Ž x Ž t . . s p Ž t . ,
Ž 2.
where f, g: Rqª Rq are continuous and p: R ª R is continuous and T-periodic. Moreover, we assume that f and g have continuous derivative.
FIG. 1. The semiplane of initial conditions.
1029
DYNAMICS OF A PERIODIC EQUATION
In w2x for f s 0 and in w1x for arbitrary f and under the condition lim g Ž x . s q`,
lim g Ž x . s 0
xª0 q
Ž 3.
xªq`
the method of upper and lower solutions has been used to prove that a necessary and sufficient condition for the existence of positive T-periodic solutions of Ž2. is that the mean value p s Ž1rT .H0T pŽ t . dt ) 0. The study of the stability can be done by using the topological degree as in w5x. However, we shall use the following lemma of linearization of Ž2. at a periodic solution. LEMMA 1.
Let r g C 1 ŽR., q g C ŽR. T-periodic functions such that r Ž t . G 0, q Ž t . - 0
; t g R.
If m 1 , m 2 are the characteristic multipliers of the equation X
yY q Ž r Ž t . y . q q Ž t . y s 0 then 0 - m 2 - 1 - m 1. Proof. Multipliers verify
m2 y D m q W s 0, where D is the discriminant and W s eyH 0 r Ž t . dt g Ž0, 1x. We must prove that D ) 1 q W. Let r, q the mean values of r and q, and T
rl Ž t . s l r q Ž 1 y l . r Ž t . ,
ql Ž t . s l q q Ž 1 y l . q Ž t .
;l g w 0, 1 x .
Consider the equation X
yY q Ž rl Ž t . y . q ql Ž t . y s 0. D s DŽ l. is continuous by continuous dependence theorem, and clearly DŽ1. ) 1 q W, so we will prove that DŽ l. / 1 q W, ;l g w0, 1x. If it is false, then there exists a nontrivial T-periodic solution c . If c ) 0, integrating Eq. Ž1. obtains a contradiction, so c must change sign. In consequence, we can fix t 0 - t 1 such that c Ž t 0 . s c Ž t 1 . s 0, c Ž t . ) 0, ; t g Ž t 0 , t 1 .. Integrating in Ž t 0 , t 1 . we have 0 ) c X Ž t1 . y c X Ž t 0 . q
t1
Ht q Ž s . c Ž s . ds s 0, l
0
another contradiction.
MART´ ıNEZ-AMORES AND TORRES
1030
This lemma shows that if Ž2. has a unique T-periodic solution then this solution must be unstable. A result of this type was given in w3x for equations of the form xY s g Ž t, x . under some conditions on g. In this context we have the following result. THEOREM 1. Assume that f G 0 and Ž3. hold. If p ) 0 and g is strictly decreasing then Ž2. has a unique positi¨ e T-periodic solution which is unstable. Proof. By Lemma 1, it is enough to prove that Ž2. has a unique T-periodic solution. Suppose, by contradiction, that x 1 and x 2 are two different T-periodic solutions of Ž2.. If z Ž t . s x 1Ž t . y x 2 Ž t . does not change of sign in w0, T x, we have a contradiction only subtracting the respective equations and integrating in Ž0, T .. If z Ž t . vanishes at some t 0 g w0, T x then there must be at least other zero of z Ž t . on w0, T x. Indeed, if z Ž t 0 . s 0 and z Ž t . ) 0 for t / t 0 , z has a minimum at t 0 and zX Ž t 0 . s 0. Hence x 1Ž t . s x 2 Ž t ., t g w0, T x, by uniqueness. So, let t 0 and t 1 be two successive zeros of z on w0, T x and assume that Ž z t . - 0 over Ž t 0 , t 1 .. Using the monotonicity of g we obtain zY Ž t . q f Ž x 1 Ž t . . xX1 Ž t . y f Ž x 2 Ž t . . xX2 Ž t . - 0,
t g Ž t 0 , t1 . .
Since zX Ž t 0 . - 0 and zX Ž t 1 . ) 0, an integration over Ž t 0 , t 1 . gives a contradiction. The same conclusion is obtained if z Ž t . - 0, t g w t 0 , t 1 x. From now we assume that all the conditions in Theorem 1 hold and that w Ž t . is the T-periodic solution given in this theorem. The following theorem describes the asymptotic behavior of trajectories of Ž2.. THEOREM 2.
Assume that 0 - m s inf f F sup f s M - q`
Ž 4.
and MF
2 q '2 2 y '2
m.
Ž 5.
If x Ž t . is a solution of Ž2. defined on w t 0 , q`. then lim tªq`
x Ž t . y w Ž t . s 0,
lim tªq`
or lim x Ž t . s q`.
tªq`
xX Ž t . y w X Ž t . s 0,
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DYNAMICS OF A PERIODIC EQUATION
Proof. We show first that the assumptions of Theorem 2 in w6x are satisfied. Making the change of variable y s xX q F Ž x ., where F X Ž x . s f Ž x ., and taking Ps
1
y1 , 0
c 2 y1
ž
c ) 0,
/
one can see that condition ŽH3. in w6x Žsee w4x. is equivalent to the matrix
As
g X Ž x . y cf Ž x . q l c f Ž x. q c 2
f Ž x. q c 2
yl
yl
y1
0
l ) 0,
,
be negative definite for some l and c, that is, g X Ž x . y cf Ž x . q l c - 0 and X
det Ž A . s yg Ž x . q cf Ž x . y l c y
ž
f Ž x. q c 2
2
yl
/
) 0.
Since g X Ž x . - 0, these inequalities hold if and only if c q 2 l q 2'l c G M, c q 2 l y 2'l c F m, and
l F m. Now, taking c s K l and maximizing K, these inequalities hold if Ž4. and Ž5. are satisfied. From w6x it follows that lim t ªq`w x Ž t . y w Ž t .x s 0, lim t ªq`w xX Ž t . y X w Ž t .x s 0 or that there exists a sequence t n4 ª q` such that some of the following cases hold:
Ž a.
x Ž t n . ª 0;
Ž b.
xX Ž t n . ª q`,
Ž c.
x Ž t n . ª q`.
If Ža. holds, from the equation we obtain that xY Ž t n . q f Ž x Ž t n .. xX Ž t n . ª y` as t n ª q`. Suppose that there exists a sequence tn ª q` such that x Žtn . ª K ) 0. One can interlace t n4 and tn4 and take t n such that x Ž t . has a local minimum at t n . Then xX Ž t n . s 0 and xY Ž t n . G 0, so xY Ž t n . cannot tend to y` as t n ª q`. Hence x Ž t . ª 0 as t ª q`. Now, by the
MART´ ıNEZ-AMORES AND TORRES
1032
mean value theorem, there exists other sequence that we call again t n4 such that xX Ž t n . ª 0. As above, we get lim t ªq` xX Ž t . s 0. A new application of the mean value theorem gives a sequence, called t n4 , such that xY Ž t n . ª 0. Evaluating the equation in t n and taking limits we have a contradiction. If Ža. does not hold and xX Ž t n . ª q` Žresp. y`. as t n ª q` then Y x Ž t n . q g Ž x Ž t n .. ª y` Žresp. q`. as t n ª q`. Supposing that there exists a sequence tn ª q` such that xX Žtn . ª K - q` Žresp. K ) y`. we can take xY Ž t n . s 0 and then g Ž x Ž t n .. ª q` Žresp. y`., which is not possible. Hence lim t ªq` < xX Ž t .< s q`. Since lim t ªq` x Ž t . / 0, we have that g Ž x Ž t .. is bounded for t G t 0 and if lim t ªq` xX Ž t . s "` then lim t ªq` xY Ž t . s .`, a contradiction. If Žc. holds, we are going to prove that lim t ªq` x Ž t . s q`. There exists some t 0 such that z Ž t . s x Ž t . y w Ž t . ) 0 for all t ) t 0 Žsee Proposition 1 of Section 2.. As w Ž t . is bounded, we have that lim t ªq` x Ž t . s q` is equivalent to lim t ªq` z Ž t . s q`. If this is not true, then there exist sequences z Ž t n .4 of maxima of z and z Žtn .4 of minima of z such that lim z Ž t n . s q`,
t nªq`
lim z Ž tn . s K - q`,
tnªq`
with t n - tn - t nq1 for all n. Since g is strictly decreasing, we have that zY Ž t . q f Ž x Ž t . . xX Ž t . y f Ž w Ž t . . w X Ž t . s g Ž w Ž t . . y g Ž x Ž t . . ) 0 ; t ) t0 . Hence, integrating over x t n , tnw,
HxxŽ t t. f Ž s . ds y HwwŽ t t. f Ž s . ds s HwxŽtt . f Ž s . ds y HwxŽ tt . f Ž s . ds ) 0, Ž n. n
Ž n. n
Ž n.
Ž n.
n
n
so Mz Žtn . ) mz Ž t n ., and we have a contradiction when n ª q`. When the solution x Ž t . is defined on w t 0 , wq. , wq- q`, there exists a sequence t n ª wq, t n g w t 0 , wq. such that some of the following cases hold:
Ž aX .
x Ž t n . ª q`,
Ž bX .
xX Ž t n . ª q`,
Ž cX .
x Ž t n . ª 0.
As above, one can show that only ŽcX . holds and that then lim t ª w q x Ž t . s 0. From Corollary 2.1 in w6x we can use the same reasonings for the solutions defined on Žy`, t 0 x or on Ž wy, t 0 x, wy) y`. So, we have the following description of the behavior of the solutions of Ž2. according to its maximal interval of existence: Ži. On Žy`, q`., lim t ªq`w x Ž t . y w Ž t .x s 0 or lim t ªq` x Ž t . s q` and, on the other hand, lim t ªy`w x Ž t . y w Ž t .x s 0 or lim t ªy` x Ž t . s q`.
1033
DYNAMICS OF A PERIODIC EQUATION
Žii. On Ž wy, q`., lim t ª w y x Ž t . s 0 and either lim t ªq`w x Ž t . y w Ž t .x s 0 or lim t ªq` x Ž t . s q`. Žiii. On Žy`, wq. , lim t ª w q x Ž t . s 0 and either lim t ªy`w x Ž t . y w Ž t .x s 0 or lim t ªy` x Ž t . s q`. Živ. On Ž wy, wq ., lim t ª w y x Ž t . s 0 s lim t ª w q x Ž t .. The behavior of xX Ž t . is given by the following lemma. LEMMA 2. According to the maximal inter¨ al of existence of the solution x Ž t ., we ha¨ e that lim t ª w q xX Ž t . - 0, lim t ª w y xX Ž t . ) 0, and these limits are finite if and only if the potential V Ž x . s Hx1 g Ž s . ds is finite as x ª 0q. Moreo¨ er, lim sup t ª "` xX Ž t . and lim inf t ª "` xX Ž t . are finite for any potential. Proof. We only show the first assertion. Integrating the equation over w t 0 , wqy e x with e ) 0, we obtain xX Ž wqy e . s xX Ž t 0 . q
w qy e
Ht
p Ž s . ds y
0
y
q
HxxŽ t w. ye f Ž s . ds Ž
.
0
q
w ye
Ht
g Ž x Ž s . . ds.
0
It is then clear that lim e ª 0 xX Ž wqy e . exists and must be negative. Further, this limit is finite if and only if V Ž0q. is finite. Denote by x Ž t, x 0 , ¨ . the solution of Ž2. such that x Ž0, x 0 , ¨ . s x 0 , xX Ž0, x 0 , ¨ . s ¨ . The main results of this paper describe the global dynamics of Eq. Ž2. and its proofs are given in the next section. THEOREM 3. Assume that the conditions of Theorem 1 and Ž4., Ž5. are satisfied and that V Ž0q. s q`. Then there exists a strictly decreasing continuous function h: R ª Rq such that x Ž t, hŽ ¨ ., ¨ . ª w Ž t . as t ª q`. If x 0 - hŽ ¨ . then lim t ª w q x Ž t . s 0 and if x 0 ) hŽ ¨ . then lim t ªq` x Ž t . s q`. Moreo¨ er, lim ¨ ªy ` hŽ ¨ . s q` and lim ¨ ªq ` hŽ ¨ . s 0. THEOREM 4. Assume that the conditions of Theorem 1 and Ž4., Ž5. are satisfied and that V Ž0q. - q`. Then there exists a function h: Žy`, ¨ s x ª Rq 0 in the same conditions of the pre¨ ious theorem, but now there exists a ‘‘critical ¨ elocity’’ ¨ s such that hŽ ¨ s . s 0. Remark 1. Theorems 3 and 4 give a function such that its graph has the property that the solutions of Ž2. with initial conditions in the graph approach the periodic solution w Ž t . as t ª q`. This graph is the section
1034
MART´ ıNEZ-AMORES AND TORRES
on t s 0 of the global stable manifold of w Ž t ., that can be defined as W sŽ w. s
D Ft Ž graph Ž h . . ,
tgR
where Ft denotes the flow generated by Ž2. and graph Ž h . s Ž x 0 , ¨ . g Rq 0 = R: h Ž ¨ . s x 0 4 . The existence of the stable manifold can be obtained from general results of hyperbolic manifolds via the Poincare ´ map since w Ž t . is a hyperbolic solution Žsee w5x. in the sense that the Floquet multipliers have modulus different from 1. However, our proofs are done working directly on the Eq. Ž2. and, moreover, we describe the geometry of these hyperbolic manifolds. Remark 2. Note that the choice of the initial time Ž0 in our case. is not essential in the proofs. So, for any other initial time, that is, any other section of the global stable manifold W s Ž w Ž t .., the plane of initial conditions has the same structure. Analogous results can be proved looking towards the past. In this case, we have the following theorem. THEOREM 5. Assume that the conditions of Theorem 1 and Ž4., Ž5. are satisfied and that V Ž0q. s q`. Then the set
Ž x 0 , ¨ . g Rq= R: x Ž t , x 0 , ¨ . y w Ž t . ª 0; as t ª y`4 is the set of initial conditions Ž the section on t s 0. of the global unstable manifold of w Ž t . and is the graph of a strictly increasing function j : R ª Rq such that lim ¨ ªy ` j Ž ¨ . s 0, lim ¨ ªq ` j Ž ¨ . s q`. Moreo¨ er, if x 0 - j Ž ¨ . then x Ž t, x 0 , ¨ . ª 0 as t ª wy and if x 0 ) j Ž ¨ . then x Ž t, x 0 , ¨ . ª q` as t ª y`. When V Ž0q. is finite, the function j Ž ¨ . has the property that there exists ¨ u g R such that j Ž ¨ u . s 0. Note that the intersection of h and j determine the initial conditions of the periodic solution.
2. PROOFS To prove the theorems of Section 1 we need to point out some facts. The first is a comparison result of two solutions of Ž2.. In this section we shall assume that all the conditions on f, g in Theorems 1 and 2 hold.
1035
DYNAMICS OF A PERIODIC EQUATION
PROPOSITION 1. point in common.
Any couple of different solutions of Ž2. has at most one
Proof. Suppose x 1Ž t ., x 2 Ž t . are two different solutions of Ž2. such that x1Ž t0 . s x 2 Ž t0 . ,
x 1Ž t1 . s x 2 Ž t1 .
and x 1Ž t . ) x 2 Ž t . for all t gx t 0 , t 1w. Then, z Ž t . s x 1Ž t . y x 2 Ž t . satisfies zY Ž t . q f Ž x 1 Ž t . . xX1 Ž t . y f Ž x 2 Ž t . . xX2 Ž t . s g Ž x 2 Ž t . . y g Ž x 1 Ž t . . ) 0 ; t gx t 0 , t1 w , so integrating over x t 0 , t 1w we have that zX Ž t 1 . ) zX Ž t 0 ., a contradiction because z Ž t . ) 0 for any t gx t 0 , t 1w. PROPOSITION 2. Let x Ž t . s x Ž t, x 0 , ¨ 0 ., x 1Ž t . s x Ž t, x 1 , ¨ 1 . be solutions of Ž2. such that x 0 - x 1. If x 1 y x 0 G Ž ¨ 0 y ¨ 1 .rm then x Ž t . - x1Ž t .
; t G 0.
Proof. If the conclusion fails to hold, then there must be some t 1 ) 0 such that x Ž t 1 . s x 1Ž t 1 .. Set z Ž t . s x Ž t . y x 1Ž t .. Then z Ž t . - 0, t g w0, t 1 . and z Ž t 1 . s 0. As in the proof of Theorem 1 we have zY Ž t . q f Ž x Ž t . . xX Ž t . y f Ž x 1 Ž t . . xX1 Ž t . - 0
; t g 0, t 1 .
and an integration over w0, t 1 . gives zX Ž t 1 . y ¨ 0 q ¨ 1 q
Hxx t
Ž 1.
0
f Ž s . ds y
Hxx
1Ž t 1 .
f Ž s . ds - 0.
1
So zX Ž t 1 . - ¨ 0 y ¨ 1 y Hxx01 f Ž s . ds F ¨ 0 y ¨ 1 y mŽ x 1 y x 0 . F 0, but it is not possible. Remark 1. By Proposition 1, Proposition 2 holds true if x 0 s x 1 and
¨ 0 - ¨ 1.
Remark 2. The same conclusion is obtained for t F 0 if x 1 y x 0 F Ž ¨ 0 y ¨ 1 .rM. The following lemma asserts that two solutions of Ž2. with the same initial velocity cannot be very close. LEMMA 3. Let x Ž t . s x Ž t, x 0 , ¨ ., x 1 s x Ž t, x 1 , ¨ . be solutions of Ž2. defined on w0, q`.. Gi¨ en d ) 0, there exists e ) 0 such that if x 1 y x 0 ) d , then there exists a sequence tn ª q` such that x 1Žtn . y x Žtn . ) e for large n.
MART´ ıNEZ-AMORES AND TORRES
1036
The same conclusion is true for x Ž t . s x Ž t, x 0 , ¨ 0 . and x 1Ž t . s x Ž t, x 0 , ¨ 1 . with ¨ 1 y ¨ 0 ) d . Proof. Set z Ž t . s x 1Ž t . y x Ž t .. Then z Ž t . ) 0 ; t g w0, q`., by Proposition 2. If the conclusion is not true, z Ž t . ª 0 when t ª q`. By the mean value theorem there must be a sequence tn ª q` such that zX Žtn . ª 0 and zX Žtn . - 0. As in the previous proof, an integration on Ž0, tn . gives z X Ž tn . y z X Ž 0 . q
Hxx
1Žt n .
f Ž s . ds y
1
Hxx t
Ž n.
f Ž s . ds ) 0.
0
Since zX Ž0. s 0, we get for large n
HxxŽt t.
1Ž n .
f Ž s . ds G
n
Hx
x1
f Ž s . ds
0
and, hence, M Ž x 1Žtn . y x Žtn .. G mŽ x 1 y x 0 .. Taking e - d mrM we are done. If x Ž t . s w Ž t . and x Ž t . y w Ž t . ª 0 as t ª q`, we get 0 G mŽ x 1 y x 0 ., but this is not possible. When x 0 s x 1 and ¨ 1 y ¨ 0 ) d we obtain M Ž x 1Žtn . y x Žtn .. G ¨ 1 y ¨ 0 and it is enough to take e - drM. LEMMA 4.
Fixed ¨ g R, the set A s x 0 g Rq : lim x Ž t , x 0 , ¨ . s q`
½
tªq`
5
is a nonempty open inter¨ al. Proof. If A is empty then, we can choose x 1 g Rq such that x 1 y w Ž0. G Ž w X Ž0. y ¨ .rm, and applying Proposition 2 and Theorem 2 we obtain a contradiction. If x 0 g A and x 1 ) x 0 then x Ž t, x 1 , ¨ . ª q` as t ª q`, by Proposition 2. Hence A is an interval. To prove that A is open, let x 0 g A and C s lim sup xX Ž t , x 0 , ¨ . ,
c s lim inf xX Ž t , x 0 , ¨ . . tªq`
tªq`
Given R ) Ž ¨ y c q 1.rm q x 0 , we have x Ž nT, x 0 , ¨ . ) R and c y 1 xX Ž nT, x 0 , ¨ . - C q 1 for some positive integer n. By continuous dependence there exists d ) 0 such that if < x 0 y x 1 < - d then x Ž nT, x 1 , ¨ . ) R and c y 1 - xX Ž nT, x 1 , ¨ . - C q 1. If we define x 1Ž t . s x Ž t q nT, x 1 , ¨ . then x1Ž 0. y x 0 ) R y x 0 )
¨ ycq1
m
X
)
¨ y x1Ž 0.
m
.
Hence, by Proposition 2, x 1Ž t . ª q` as t ª q` and x 1 g A.
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DYNAMICS OF A PERIODIC EQUATION
LEMMA 5.
Fixed ¨ g R, the set B s x 0 g Rq : limq x Ž t , x 0 , ¨ . s 0
½
tªw
5
is an open inter¨ al. Moreo¨ er, if V Ž0q. is infinite then B is nonempty. Proof. Let pM s max t g w0, T x pŽ t . and x M ) 0 such that g Ž x M . s pM . We claim that a local minimum of any solution x Ž t . of Ž2. is larger than x M . If not, x Ž t . would have a minimum at t s t 0 with x Ž t 0 . - x M , then xX Ž t 0 . s 0, xY Ž t 0 . G 0 and g Ž x Ž t 0 .. ) pM . But this is impossible, because xY Ž t 0 . q f Ž x Ž t 0 . . xX Ž t 0 . q g Ž x Ž t 0 . . s p Ž t 0 . F pM . Now, if x 0 g B, for R ) 0 there exists t 0 such that x Ž t 0 , x 0 , ¨ . - R and xX Ž t 0 , x 0 , ¨ . - 0. By continuous dependence, there exists d ) 0 such that if < x 0 y x 1 < - d then x Ž t 0 , x 1 , ¨ . - R and xX Ž t 0 , x 1 , ¨ . - 0. Taking R s x M we obtain that x Ž t, x 1 , ¨ . has not a minimum for t ) t 0 and, hence, x Ž t, x 1 , ¨ . ª 0 as t ª wq. So B is open. To prove that B is nonempty we see that if we have Hxx0M g Ž s . ds ) pM x M q ¨ 2r2 then x Ž t . ª 0 as t ª wq. In fact, if this is not the case, let t 0 ) 0 be such that x Ž t . - x M , xX Ž t . ) 0 ; t g Ž0, t 0 .. Since xX Ž t . s
p Ž t . y g Ž x Ž t . . y xY Ž t . f Ž xŽ t. .
)0
; t g Ž 0, t 0 . ,
we have xY Ž t . q g Ž x Ž t .. - pM , t g Ž0, t 0 .. An integration over w0, t 0 x after multiplying by xX Ž t . gives
Hx
xM
g Ž s . ds - pM Ž x M y x 0 . y
0
xX Ž t 0 . 2
2
q
¨2
2
- pM x M q
¨2
2
and we get a contradiction. In a similar way, one can obtain that for a fixed x 0 g Rq, the set A1 s ¨ g R: lim x Ž t , x 0 , ¨ . s q`
½
tªq`
5
is a nonempty open interval. COROLLARY 1. If the potential is infinite, for any fixed ¨ g R Ž resp. x 0 g Rq. there exists a unique x 0 g Rq Ž resp. ¨ g R. such that x Ž t, x 0 , ¨ . y w Ž t . ª 0 as t ª q`. Proof. By Lemmas 4 and 5, A and B are nonempty open intervals, and by Proposition 2 we have that sup B F inf A. If we take x 0 g wsup B, inf A x, then x Ž t, x 0 , ¨ . is bounded and defined until q`, so using Theorem 2 we conclude that x Ž t, x 0 , ¨ . y w Ž t . ª 0 as t ª q`. Moreover, by Lemma 3, sup B s inf A, so x 0 exists and it is unique.
MART´ ıNEZ-AMORES AND TORRES
1038
Proof of Theorem 3. Using the previous lemmas, the function h: R ª Rq defined by h Ž ¨ . s sup B s inf A is well defined and lim t ªq`w x Ž t, hŽ ¨ ., ¨ . y w Ž t .x s 0. Let see that h is strictly decreasing. If it is not true, there exists ¨ 0 - ¨ 2 such that hŽ ¨ 0 . F hŽ ¨ 2 .. Let x 0 s hŽ ¨ 0 .. Then lim t ªq`w x Ž t, x 0 , ¨ 0 . y w Ž t .x s 0 and since x 0 - hŽ ¨ 2 . Žthe equality is not possible because of Corollary 1., lim t ª w q x Ž t, x 0 , ¨ 2 . s 0, a contradiction with Proposition 1. If h is not continuous at ¨ , let x 0 such that hŽ ¨ q. - x 0 - hŽ ¨ y. . By Corollary 1, there exists ¨ 0 g R such that hŽ ¨ 0 . s x 0 . If ¨ 0 - ¨ Žresp. ¨ 0 ) ¨ . then hŽ ¨ 0 . ) hŽ ¨ y. ) x 0 Žresp. hŽ ¨ 0 . - hŽ ¨ q. - x 0 ., a contradiction. h is decreasing and bounded below, so 'lim ¨ ªq ` hŽ ¨ ., and it is zero because in the other case A1 would be empty for any x 0 minor that this limit. In the same way the limit lim ¨ ªy ` hŽ ¨ . exists, and modifying slightly Lemma 3, we see that it is infinite. Finally, for the proof of Theorem 4 we prove that if the potential is finite in zero then there exists ¨ s g R such that for all ¨ G ¨ s we have that B is empty. In fact, let Ž x 1 , ¨ 1 . g Rq= R be the initial conditions of w Ž t . at t s 0. Take x 0 - x 1 such that x 1 y x 0 F Ž ¨ 0 y ¨ 1 .rM, where ¨ 0 s hy1 Ž x 0 ., ¨ 1 s hy1 Ž x 1 .. By Proposition 2, x Ž t, x 0 , ¨ 0 . - w Ž t ., t F 0. Hence x Ž t, x 0 , ¨ 0 . ª 0 as t ª wyŽ x 0 , ¨ 0 .. By continuation of solutions theorem, we have that lim sup wy Ž x, hy1 Ž x . . F wy Ž x 1 , ¨ 1 . s y`. xªxy 1
Hence, there exists ¨ and x 0 s hŽ ¨ . such that wy Ž x 0 , ¨ . s ykT, a multiple of T. Now, if we take a sequence ¨ n ª ¨ as n ª q` with ¨ n - ¨ , for the solutions x Ž t, hŽ ¨ n ., ¨ n . we have lim sup wy Ž h Ž ¨ n . , ¨ n . F ykT . nªq`
So, the points x ŽykT, hŽ ¨ n ., ¨ n ., xX ŽykT, hŽ ¨ n ., ¨ n . are the initial conditions at t s 0 of x Ž t y kT, hŽ ¨ n ., ¨ n . and x Ž t , x Ž ykT , h Ž ¨ n . , ¨ n . , xX Ž ykT , h Ž ¨ n . , ¨ n . . y w Ž t . s x Ž t y kT , h Ž ¨ n . , ¨ n . y w Ž t . ª 0
DYNAMICS OF A PERIODIC EQUATION
1039
as t ª q`. Hence, if we take limits in n, by continuous dependence of initial conditions, the point Ž x ŽykT, x 0 , ¨ ., xX ŽykT, x 0 , ¨ .. is on graphŽ h.. But x ŽykT, x 0 , ¨ . s 0 and xX ŽykT, x 0 , ¨ . is finite by Lemma 2, so if we let xX ŽykT, x 0 , ¨ . s ¨ s , then hŽ ¨ s . s 0 and, consequently, for all ¨ G ¨ s the set B is empty.
3. CASE WITHOUT FRICTION All the previous results can be obtained for the case without friction Ž f ' 0. by similar reasonings. For the sake of brevity, we will not repeat all the proofs because they are very similar; for example, in Theorem 1 we only have to take
l s 0,
Ps
ž
0 y1
y1 0
/
and we have the assumptions of Theorem 1 in w6x.
ACKNOWLEDGMENTS We thank the referee for his suggestions that have improved the presentation of the paper. We also are grateful to R. Ortega for his suggestions and comments on the contents of this paper.
REFERENCES 1. P. Habets and L. Sanchez, Periodic solutions of some Lienard equations with singularities, ´ Proc. Amer. Math. Soc. 109 Ž1990., 1035]1044. 2. A. C. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc. 99 Ž1987., 109]114. 3. J. Mawhin and C. Munoz, ˜ Application du degre´ topologique `a l’estimation du nombre des solutions periodiques d’equations differentielles, Ann. Mat. Pura Appl. 46, No. 4 Ž1973., ´ ´ 1]19. 4. R. Ortega, Topological degree and stability of periodic solutions for certain differential equations, J. London Math. Soc. 42, Ž1990., 505]516. 5. R. Ortega, Some applications of the topological degree to stability theory, in ‘‘Topological Methods in Differential Equations and Inclusions,’’ NATO ASI Ser. 6, Kluwer Academic, Dordrecht, 1994. 6. R. A. Smith, Massera’s convergence theorem for periodic nonlinear differential equations, J. Math. Anal. Appl. 120 Ž1986., 679]708.
202, 1027]1039 Ž1996.
0358
Dynamics of a Periodic Differential Equation with a Singular Nonlinearity of Attractive Type U Pedro Martınez-Amores and Pedro J. TorresU ´
Departamento de Matematica ´ Aplicada, Uni¨ ersidad de Granada, 18071 Granada, Spain Communicated by Jack K. Hale Received January 24, 1995; revised November 29, 1995
INTRODUCTION In this paper we shall consider a general class of equations with a restoring force of attractive type that includes xY q cxX q
1 xa
s pŽ t . ,
Ž 1.
where c G 0, a ) 0, and p is a continuous T-periodic function for some T ) 0. We are interested in the existence and stability of positive Tperiodic solutions of Ž1.. The existence of T-periodic solutions of this class of equations where the restoring force is a singular nonlinearity that becomes infinite in zero has been proved by Lazer and Solimini w2x for the case without friction Ž c s 0. and by Habets and Sanchez w1x for the damped case. However, the ´ stability properties of these solutions have been less studied. It is well known that for the autonomous case Ž pŽ t . ' p 0 ) 0., this equation has a unique saddle point. In this paper we prove that the dynamics of the periodic equation Ž1. is similar to the autonomous case. For our study is fundamental a Massera’s convergence theorem in R n given by Smith in w6x. The paper is divided in three sections. In Section 1, the main results are stated. It is seen that Ž1. has a unique unstable periodic solution w . U
Supported by D. G. C. Y. T. PB92-0953, M.E.C., Spain, and E.E.C. project ERBCHRXCT94-0555. 1027 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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MART´ ıNEZ-AMORES AND TORRES
Moreover, for the damped case it is proved that there exists a global stable manifold of this periodic solution which is determined by a strictly decreasŽbeing Rq w .. such that this graph divides the ing map h: R ª Rq 0 0 s 0, q` plane of initial conditions in two open sets, one of them Žset A in Fig. 1. corresponds to the solutions that tend to q` as t ª q`, and the other Žset B. to the solutions that goes to zero in finite time, that is, the ones that are ‘‘absorbed’’ by the singularity. When the potential V Ž x . is infinite as x ª 0q Žcase a G 1., we have that h: R ª Rq Žbeing Rqs Ž0, q`.. is a homeomorphism, whereas if V Ž x . is finite as x ª 0q Žcase 0 - a - 1. there exists ¨ s g R such that hŽ ¨ s . s 0. Similar results are showed for the unstable manifold. As a consequence, there are no homoclinic points. We prove these results in Section 2. For this, the study of the behavior of the solutions and its derivatives according to its maximal interval of existence Ž wy, wq ., y` F wy- wqF q`, is fundamental. Some properties of comparison of solutions are given which are needed in the proofs. Finally, in Section 3, we remark that the previous results are true for the undamped case.
1. MAIN RESULTS Let us consider the forced Lienard equation, ´ xY Ž t . q f Ž x Ž t . . xX Ž t . q g Ž x Ž t . . s p Ž t . ,
Ž 2.
where f, g: Rqª Rq are continuous and p: R ª R is continuous and T-periodic. Moreover, we assume that f and g have continuous derivative.
FIG. 1. The semiplane of initial conditions.
1029
DYNAMICS OF A PERIODIC EQUATION
In w2x for f s 0 and in w1x for arbitrary f and under the condition lim g Ž x . s q`,
lim g Ž x . s 0
xª0 q
Ž 3.
xªq`
the method of upper and lower solutions has been used to prove that a necessary and sufficient condition for the existence of positive T-periodic solutions of Ž2. is that the mean value p s Ž1rT .H0T pŽ t . dt ) 0. The study of the stability can be done by using the topological degree as in w5x. However, we shall use the following lemma of linearization of Ž2. at a periodic solution. LEMMA 1.
Let r g C 1 ŽR., q g C ŽR. T-periodic functions such that r Ž t . G 0, q Ž t . - 0
; t g R.
If m 1 , m 2 are the characteristic multipliers of the equation X
yY q Ž r Ž t . y . q q Ž t . y s 0 then 0 - m 2 - 1 - m 1. Proof. Multipliers verify
m2 y D m q W s 0, where D is the discriminant and W s eyH 0 r Ž t . dt g Ž0, 1x. We must prove that D ) 1 q W. Let r, q the mean values of r and q, and T
rl Ž t . s l r q Ž 1 y l . r Ž t . ,
ql Ž t . s l q q Ž 1 y l . q Ž t .
;l g w 0, 1 x .
Consider the equation X
yY q Ž rl Ž t . y . q ql Ž t . y s 0. D s DŽ l. is continuous by continuous dependence theorem, and clearly DŽ1. ) 1 q W, so we will prove that DŽ l. / 1 q W, ;l g w0, 1x. If it is false, then there exists a nontrivial T-periodic solution c . If c ) 0, integrating Eq. Ž1. obtains a contradiction, so c must change sign. In consequence, we can fix t 0 - t 1 such that c Ž t 0 . s c Ž t 1 . s 0, c Ž t . ) 0, ; t g Ž t 0 , t 1 .. Integrating in Ž t 0 , t 1 . we have 0 ) c X Ž t1 . y c X Ž t 0 . q
t1
Ht q Ž s . c Ž s . ds s 0, l
0
another contradiction.
MART´ ıNEZ-AMORES AND TORRES
1030
This lemma shows that if Ž2. has a unique T-periodic solution then this solution must be unstable. A result of this type was given in w3x for equations of the form xY s g Ž t, x . under some conditions on g. In this context we have the following result. THEOREM 1. Assume that f G 0 and Ž3. hold. If p ) 0 and g is strictly decreasing then Ž2. has a unique positi¨ e T-periodic solution which is unstable. Proof. By Lemma 1, it is enough to prove that Ž2. has a unique T-periodic solution. Suppose, by contradiction, that x 1 and x 2 are two different T-periodic solutions of Ž2.. If z Ž t . s x 1Ž t . y x 2 Ž t . does not change of sign in w0, T x, we have a contradiction only subtracting the respective equations and integrating in Ž0, T .. If z Ž t . vanishes at some t 0 g w0, T x then there must be at least other zero of z Ž t . on w0, T x. Indeed, if z Ž t 0 . s 0 and z Ž t . ) 0 for t / t 0 , z has a minimum at t 0 and zX Ž t 0 . s 0. Hence x 1Ž t . s x 2 Ž t ., t g w0, T x, by uniqueness. So, let t 0 and t 1 be two successive zeros of z on w0, T x and assume that Ž z t . - 0 over Ž t 0 , t 1 .. Using the monotonicity of g we obtain zY Ž t . q f Ž x 1 Ž t . . xX1 Ž t . y f Ž x 2 Ž t . . xX2 Ž t . - 0,
t g Ž t 0 , t1 . .
Since zX Ž t 0 . - 0 and zX Ž t 1 . ) 0, an integration over Ž t 0 , t 1 . gives a contradiction. The same conclusion is obtained if z Ž t . - 0, t g w t 0 , t 1 x. From now we assume that all the conditions in Theorem 1 hold and that w Ž t . is the T-periodic solution given in this theorem. The following theorem describes the asymptotic behavior of trajectories of Ž2.. THEOREM 2.
Assume that 0 - m s inf f F sup f s M - q`
Ž 4.
and MF
2 q '2 2 y '2
m.
Ž 5.
If x Ž t . is a solution of Ž2. defined on w t 0 , q`. then lim tªq`
x Ž t . y w Ž t . s 0,
lim tªq`
or lim x Ž t . s q`.
tªq`
xX Ž t . y w X Ž t . s 0,
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DYNAMICS OF A PERIODIC EQUATION
Proof. We show first that the assumptions of Theorem 2 in w6x are satisfied. Making the change of variable y s xX q F Ž x ., where F X Ž x . s f Ž x ., and taking Ps
1
y1 , 0
c 2 y1
ž
c ) 0,
/
one can see that condition ŽH3. in w6x Žsee w4x. is equivalent to the matrix
As
g X Ž x . y cf Ž x . q l c f Ž x. q c 2
f Ž x. q c 2
yl
yl
y1
0
l ) 0,
,
be negative definite for some l and c, that is, g X Ž x . y cf Ž x . q l c - 0 and X
det Ž A . s yg Ž x . q cf Ž x . y l c y
ž
f Ž x. q c 2
2
yl
/
) 0.
Since g X Ž x . - 0, these inequalities hold if and only if c q 2 l q 2'l c G M, c q 2 l y 2'l c F m, and
l F m. Now, taking c s K l and maximizing K, these inequalities hold if Ž4. and Ž5. are satisfied. From w6x it follows that lim t ªq`w x Ž t . y w Ž t .x s 0, lim t ªq`w xX Ž t . y X w Ž t .x s 0 or that there exists a sequence t n4 ª q` such that some of the following cases hold:
Ž a.
x Ž t n . ª 0;
Ž b.
xX Ž t n . ª q`,
Ž c.
x Ž t n . ª q`.
If Ža. holds, from the equation we obtain that xY Ž t n . q f Ž x Ž t n .. xX Ž t n . ª y` as t n ª q`. Suppose that there exists a sequence tn ª q` such that x Žtn . ª K ) 0. One can interlace t n4 and tn4 and take t n such that x Ž t . has a local minimum at t n . Then xX Ž t n . s 0 and xY Ž t n . G 0, so xY Ž t n . cannot tend to y` as t n ª q`. Hence x Ž t . ª 0 as t ª q`. Now, by the
MART´ ıNEZ-AMORES AND TORRES
1032
mean value theorem, there exists other sequence that we call again t n4 such that xX Ž t n . ª 0. As above, we get lim t ªq` xX Ž t . s 0. A new application of the mean value theorem gives a sequence, called t n4 , such that xY Ž t n . ª 0. Evaluating the equation in t n and taking limits we have a contradiction. If Ža. does not hold and xX Ž t n . ª q` Žresp. y`. as t n ª q` then Y x Ž t n . q g Ž x Ž t n .. ª y` Žresp. q`. as t n ª q`. Supposing that there exists a sequence tn ª q` such that xX Žtn . ª K - q` Žresp. K ) y`. we can take xY Ž t n . s 0 and then g Ž x Ž t n .. ª q` Žresp. y`., which is not possible. Hence lim t ªq` < xX Ž t .< s q`. Since lim t ªq` x Ž t . / 0, we have that g Ž x Ž t .. is bounded for t G t 0 and if lim t ªq` xX Ž t . s "` then lim t ªq` xY Ž t . s .`, a contradiction. If Žc. holds, we are going to prove that lim t ªq` x Ž t . s q`. There exists some t 0 such that z Ž t . s x Ž t . y w Ž t . ) 0 for all t ) t 0 Žsee Proposition 1 of Section 2.. As w Ž t . is bounded, we have that lim t ªq` x Ž t . s q` is equivalent to lim t ªq` z Ž t . s q`. If this is not true, then there exist sequences z Ž t n .4 of maxima of z and z Žtn .4 of minima of z such that lim z Ž t n . s q`,
t nªq`
lim z Ž tn . s K - q`,
tnªq`
with t n - tn - t nq1 for all n. Since g is strictly decreasing, we have that zY Ž t . q f Ž x Ž t . . xX Ž t . y f Ž w Ž t . . w X Ž t . s g Ž w Ž t . . y g Ž x Ž t . . ) 0 ; t ) t0 . Hence, integrating over x t n , tnw,
HxxŽ t t. f Ž s . ds y HwwŽ t t. f Ž s . ds s HwxŽtt . f Ž s . ds y HwxŽ tt . f Ž s . ds ) 0, Ž n. n
Ž n. n
Ž n.
Ž n.
n
n
so Mz Žtn . ) mz Ž t n ., and we have a contradiction when n ª q`. When the solution x Ž t . is defined on w t 0 , wq. , wq- q`, there exists a sequence t n ª wq, t n g w t 0 , wq. such that some of the following cases hold:
Ž aX .
x Ž t n . ª q`,
Ž bX .
xX Ž t n . ª q`,
Ž cX .
x Ž t n . ª 0.
As above, one can show that only ŽcX . holds and that then lim t ª w q x Ž t . s 0. From Corollary 2.1 in w6x we can use the same reasonings for the solutions defined on Žy`, t 0 x or on Ž wy, t 0 x, wy) y`. So, we have the following description of the behavior of the solutions of Ž2. according to its maximal interval of existence: Ži. On Žy`, q`., lim t ªq`w x Ž t . y w Ž t .x s 0 or lim t ªq` x Ž t . s q` and, on the other hand, lim t ªy`w x Ž t . y w Ž t .x s 0 or lim t ªy` x Ž t . s q`.
1033
DYNAMICS OF A PERIODIC EQUATION
Žii. On Ž wy, q`., lim t ª w y x Ž t . s 0 and either lim t ªq`w x Ž t . y w Ž t .x s 0 or lim t ªq` x Ž t . s q`. Žiii. On Žy`, wq. , lim t ª w q x Ž t . s 0 and either lim t ªy`w x Ž t . y w Ž t .x s 0 or lim t ªy` x Ž t . s q`. Živ. On Ž wy, wq ., lim t ª w y x Ž t . s 0 s lim t ª w q x Ž t .. The behavior of xX Ž t . is given by the following lemma. LEMMA 2. According to the maximal inter¨ al of existence of the solution x Ž t ., we ha¨ e that lim t ª w q xX Ž t . - 0, lim t ª w y xX Ž t . ) 0, and these limits are finite if and only if the potential V Ž x . s Hx1 g Ž s . ds is finite as x ª 0q. Moreo¨ er, lim sup t ª "` xX Ž t . and lim inf t ª "` xX Ž t . are finite for any potential. Proof. We only show the first assertion. Integrating the equation over w t 0 , wqy e x with e ) 0, we obtain xX Ž wqy e . s xX Ž t 0 . q
w qy e
Ht
p Ž s . ds y
0
y
q
HxxŽ t w. ye f Ž s . ds Ž
.
0
q
w ye
Ht
g Ž x Ž s . . ds.
0
It is then clear that lim e ª 0 xX Ž wqy e . exists and must be negative. Further, this limit is finite if and only if V Ž0q. is finite. Denote by x Ž t, x 0 , ¨ . the solution of Ž2. such that x Ž0, x 0 , ¨ . s x 0 , xX Ž0, x 0 , ¨ . s ¨ . The main results of this paper describe the global dynamics of Eq. Ž2. and its proofs are given in the next section. THEOREM 3. Assume that the conditions of Theorem 1 and Ž4., Ž5. are satisfied and that V Ž0q. s q`. Then there exists a strictly decreasing continuous function h: R ª Rq such that x Ž t, hŽ ¨ ., ¨ . ª w Ž t . as t ª q`. If x 0 - hŽ ¨ . then lim t ª w q x Ž t . s 0 and if x 0 ) hŽ ¨ . then lim t ªq` x Ž t . s q`. Moreo¨ er, lim ¨ ªy ` hŽ ¨ . s q` and lim ¨ ªq ` hŽ ¨ . s 0. THEOREM 4. Assume that the conditions of Theorem 1 and Ž4., Ž5. are satisfied and that V Ž0q. - q`. Then there exists a function h: Žy`, ¨ s x ª Rq 0 in the same conditions of the pre¨ ious theorem, but now there exists a ‘‘critical ¨ elocity’’ ¨ s such that hŽ ¨ s . s 0. Remark 1. Theorems 3 and 4 give a function such that its graph has the property that the solutions of Ž2. with initial conditions in the graph approach the periodic solution w Ž t . as t ª q`. This graph is the section
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MART´ ıNEZ-AMORES AND TORRES
on t s 0 of the global stable manifold of w Ž t ., that can be defined as W sŽ w. s
D Ft Ž graph Ž h . . ,
tgR
where Ft denotes the flow generated by Ž2. and graph Ž h . s Ž x 0 , ¨ . g Rq 0 = R: h Ž ¨ . s x 0 4 . The existence of the stable manifold can be obtained from general results of hyperbolic manifolds via the Poincare ´ map since w Ž t . is a hyperbolic solution Žsee w5x. in the sense that the Floquet multipliers have modulus different from 1. However, our proofs are done working directly on the Eq. Ž2. and, moreover, we describe the geometry of these hyperbolic manifolds. Remark 2. Note that the choice of the initial time Ž0 in our case. is not essential in the proofs. So, for any other initial time, that is, any other section of the global stable manifold W s Ž w Ž t .., the plane of initial conditions has the same structure. Analogous results can be proved looking towards the past. In this case, we have the following theorem. THEOREM 5. Assume that the conditions of Theorem 1 and Ž4., Ž5. are satisfied and that V Ž0q. s q`. Then the set
Ž x 0 , ¨ . g Rq= R: x Ž t , x 0 , ¨ . y w Ž t . ª 0; as t ª y`4 is the set of initial conditions Ž the section on t s 0. of the global unstable manifold of w Ž t . and is the graph of a strictly increasing function j : R ª Rq such that lim ¨ ªy ` j Ž ¨ . s 0, lim ¨ ªq ` j Ž ¨ . s q`. Moreo¨ er, if x 0 - j Ž ¨ . then x Ž t, x 0 , ¨ . ª 0 as t ª wy and if x 0 ) j Ž ¨ . then x Ž t, x 0 , ¨ . ª q` as t ª y`. When V Ž0q. is finite, the function j Ž ¨ . has the property that there exists ¨ u g R such that j Ž ¨ u . s 0. Note that the intersection of h and j determine the initial conditions of the periodic solution.
2. PROOFS To prove the theorems of Section 1 we need to point out some facts. The first is a comparison result of two solutions of Ž2.. In this section we shall assume that all the conditions on f, g in Theorems 1 and 2 hold.
1035
DYNAMICS OF A PERIODIC EQUATION
PROPOSITION 1. point in common.
Any couple of different solutions of Ž2. has at most one
Proof. Suppose x 1Ž t ., x 2 Ž t . are two different solutions of Ž2. such that x1Ž t0 . s x 2 Ž t0 . ,
x 1Ž t1 . s x 2 Ž t1 .
and x 1Ž t . ) x 2 Ž t . for all t gx t 0 , t 1w. Then, z Ž t . s x 1Ž t . y x 2 Ž t . satisfies zY Ž t . q f Ž x 1 Ž t . . xX1 Ž t . y f Ž x 2 Ž t . . xX2 Ž t . s g Ž x 2 Ž t . . y g Ž x 1 Ž t . . ) 0 ; t gx t 0 , t1 w , so integrating over x t 0 , t 1w we have that zX Ž t 1 . ) zX Ž t 0 ., a contradiction because z Ž t . ) 0 for any t gx t 0 , t 1w. PROPOSITION 2. Let x Ž t . s x Ž t, x 0 , ¨ 0 ., x 1Ž t . s x Ž t, x 1 , ¨ 1 . be solutions of Ž2. such that x 0 - x 1. If x 1 y x 0 G Ž ¨ 0 y ¨ 1 .rm then x Ž t . - x1Ž t .
; t G 0.
Proof. If the conclusion fails to hold, then there must be some t 1 ) 0 such that x Ž t 1 . s x 1Ž t 1 .. Set z Ž t . s x Ž t . y x 1Ž t .. Then z Ž t . - 0, t g w0, t 1 . and z Ž t 1 . s 0. As in the proof of Theorem 1 we have zY Ž t . q f Ž x Ž t . . xX Ž t . y f Ž x 1 Ž t . . xX1 Ž t . - 0
; t g 0, t 1 .
and an integration over w0, t 1 . gives zX Ž t 1 . y ¨ 0 q ¨ 1 q
Hxx t
Ž 1.
0
f Ž s . ds y
Hxx
1Ž t 1 .
f Ž s . ds - 0.
1
So zX Ž t 1 . - ¨ 0 y ¨ 1 y Hxx01 f Ž s . ds F ¨ 0 y ¨ 1 y mŽ x 1 y x 0 . F 0, but it is not possible. Remark 1. By Proposition 1, Proposition 2 holds true if x 0 s x 1 and
¨ 0 - ¨ 1.
Remark 2. The same conclusion is obtained for t F 0 if x 1 y x 0 F Ž ¨ 0 y ¨ 1 .rM. The following lemma asserts that two solutions of Ž2. with the same initial velocity cannot be very close. LEMMA 3. Let x Ž t . s x Ž t, x 0 , ¨ ., x 1 s x Ž t, x 1 , ¨ . be solutions of Ž2. defined on w0, q`.. Gi¨ en d ) 0, there exists e ) 0 such that if x 1 y x 0 ) d , then there exists a sequence tn ª q` such that x 1Žtn . y x Žtn . ) e for large n.
MART´ ıNEZ-AMORES AND TORRES
1036
The same conclusion is true for x Ž t . s x Ž t, x 0 , ¨ 0 . and x 1Ž t . s x Ž t, x 0 , ¨ 1 . with ¨ 1 y ¨ 0 ) d . Proof. Set z Ž t . s x 1Ž t . y x Ž t .. Then z Ž t . ) 0 ; t g w0, q`., by Proposition 2. If the conclusion is not true, z Ž t . ª 0 when t ª q`. By the mean value theorem there must be a sequence tn ª q` such that zX Žtn . ª 0 and zX Žtn . - 0. As in the previous proof, an integration on Ž0, tn . gives z X Ž tn . y z X Ž 0 . q
Hxx
1Žt n .
f Ž s . ds y
1
Hxx t
Ž n.
f Ž s . ds ) 0.
0
Since zX Ž0. s 0, we get for large n
HxxŽt t.
1Ž n .
f Ž s . ds G
n
Hx
x1
f Ž s . ds
0
and, hence, M Ž x 1Žtn . y x Žtn .. G mŽ x 1 y x 0 .. Taking e - d mrM we are done. If x Ž t . s w Ž t . and x Ž t . y w Ž t . ª 0 as t ª q`, we get 0 G mŽ x 1 y x 0 ., but this is not possible. When x 0 s x 1 and ¨ 1 y ¨ 0 ) d we obtain M Ž x 1Žtn . y x Žtn .. G ¨ 1 y ¨ 0 and it is enough to take e - drM. LEMMA 4.
Fixed ¨ g R, the set A s x 0 g Rq : lim x Ž t , x 0 , ¨ . s q`
½
tªq`
5
is a nonempty open inter¨ al. Proof. If A is empty then, we can choose x 1 g Rq such that x 1 y w Ž0. G Ž w X Ž0. y ¨ .rm, and applying Proposition 2 and Theorem 2 we obtain a contradiction. If x 0 g A and x 1 ) x 0 then x Ž t, x 1 , ¨ . ª q` as t ª q`, by Proposition 2. Hence A is an interval. To prove that A is open, let x 0 g A and C s lim sup xX Ž t , x 0 , ¨ . ,
c s lim inf xX Ž t , x 0 , ¨ . . tªq`
tªq`
Given R ) Ž ¨ y c q 1.rm q x 0 , we have x Ž nT, x 0 , ¨ . ) R and c y 1 xX Ž nT, x 0 , ¨ . - C q 1 for some positive integer n. By continuous dependence there exists d ) 0 such that if < x 0 y x 1 < - d then x Ž nT, x 1 , ¨ . ) R and c y 1 - xX Ž nT, x 1 , ¨ . - C q 1. If we define x 1Ž t . s x Ž t q nT, x 1 , ¨ . then x1Ž 0. y x 0 ) R y x 0 )
¨ ycq1
m
X
)
¨ y x1Ž 0.
m
.
Hence, by Proposition 2, x 1Ž t . ª q` as t ª q` and x 1 g A.
1037
DYNAMICS OF A PERIODIC EQUATION
LEMMA 5.
Fixed ¨ g R, the set B s x 0 g Rq : limq x Ž t , x 0 , ¨ . s 0
½
tªw
5
is an open inter¨ al. Moreo¨ er, if V Ž0q. is infinite then B is nonempty. Proof. Let pM s max t g w0, T x pŽ t . and x M ) 0 such that g Ž x M . s pM . We claim that a local minimum of any solution x Ž t . of Ž2. is larger than x M . If not, x Ž t . would have a minimum at t s t 0 with x Ž t 0 . - x M , then xX Ž t 0 . s 0, xY Ž t 0 . G 0 and g Ž x Ž t 0 .. ) pM . But this is impossible, because xY Ž t 0 . q f Ž x Ž t 0 . . xX Ž t 0 . q g Ž x Ž t 0 . . s p Ž t 0 . F pM . Now, if x 0 g B, for R ) 0 there exists t 0 such that x Ž t 0 , x 0 , ¨ . - R and xX Ž t 0 , x 0 , ¨ . - 0. By continuous dependence, there exists d ) 0 such that if < x 0 y x 1 < - d then x Ž t 0 , x 1 , ¨ . - R and xX Ž t 0 , x 1 , ¨ . - 0. Taking R s x M we obtain that x Ž t, x 1 , ¨ . has not a minimum for t ) t 0 and, hence, x Ž t, x 1 , ¨ . ª 0 as t ª wq. So B is open. To prove that B is nonempty we see that if we have Hxx0M g Ž s . ds ) pM x M q ¨ 2r2 then x Ž t . ª 0 as t ª wq. In fact, if this is not the case, let t 0 ) 0 be such that x Ž t . - x M , xX Ž t . ) 0 ; t g Ž0, t 0 .. Since xX Ž t . s
p Ž t . y g Ž x Ž t . . y xY Ž t . f Ž xŽ t. .
)0
; t g Ž 0, t 0 . ,
we have xY Ž t . q g Ž x Ž t .. - pM , t g Ž0, t 0 .. An integration over w0, t 0 x after multiplying by xX Ž t . gives
Hx
xM
g Ž s . ds - pM Ž x M y x 0 . y
0
xX Ž t 0 . 2
2
q
¨2
2
- pM x M q
¨2
2
and we get a contradiction. In a similar way, one can obtain that for a fixed x 0 g Rq, the set A1 s ¨ g R: lim x Ž t , x 0 , ¨ . s q`
½
tªq`
5
is a nonempty open interval. COROLLARY 1. If the potential is infinite, for any fixed ¨ g R Ž resp. x 0 g Rq. there exists a unique x 0 g Rq Ž resp. ¨ g R. such that x Ž t, x 0 , ¨ . y w Ž t . ª 0 as t ª q`. Proof. By Lemmas 4 and 5, A and B are nonempty open intervals, and by Proposition 2 we have that sup B F inf A. If we take x 0 g wsup B, inf A x, then x Ž t, x 0 , ¨ . is bounded and defined until q`, so using Theorem 2 we conclude that x Ž t, x 0 , ¨ . y w Ž t . ª 0 as t ª q`. Moreover, by Lemma 3, sup B s inf A, so x 0 exists and it is unique.
MART´ ıNEZ-AMORES AND TORRES
1038
Proof of Theorem 3. Using the previous lemmas, the function h: R ª Rq defined by h Ž ¨ . s sup B s inf A is well defined and lim t ªq`w x Ž t, hŽ ¨ ., ¨ . y w Ž t .x s 0. Let see that h is strictly decreasing. If it is not true, there exists ¨ 0 - ¨ 2 such that hŽ ¨ 0 . F hŽ ¨ 2 .. Let x 0 s hŽ ¨ 0 .. Then lim t ªq`w x Ž t, x 0 , ¨ 0 . y w Ž t .x s 0 and since x 0 - hŽ ¨ 2 . Žthe equality is not possible because of Corollary 1., lim t ª w q x Ž t, x 0 , ¨ 2 . s 0, a contradiction with Proposition 1. If h is not continuous at ¨ , let x 0 such that hŽ ¨ q. - x 0 - hŽ ¨ y. . By Corollary 1, there exists ¨ 0 g R such that hŽ ¨ 0 . s x 0 . If ¨ 0 - ¨ Žresp. ¨ 0 ) ¨ . then hŽ ¨ 0 . ) hŽ ¨ y. ) x 0 Žresp. hŽ ¨ 0 . - hŽ ¨ q. - x 0 ., a contradiction. h is decreasing and bounded below, so 'lim ¨ ªq ` hŽ ¨ ., and it is zero because in the other case A1 would be empty for any x 0 minor that this limit. In the same way the limit lim ¨ ªy ` hŽ ¨ . exists, and modifying slightly Lemma 3, we see that it is infinite. Finally, for the proof of Theorem 4 we prove that if the potential is finite in zero then there exists ¨ s g R such that for all ¨ G ¨ s we have that B is empty. In fact, let Ž x 1 , ¨ 1 . g Rq= R be the initial conditions of w Ž t . at t s 0. Take x 0 - x 1 such that x 1 y x 0 F Ž ¨ 0 y ¨ 1 .rM, where ¨ 0 s hy1 Ž x 0 ., ¨ 1 s hy1 Ž x 1 .. By Proposition 2, x Ž t, x 0 , ¨ 0 . - w Ž t ., t F 0. Hence x Ž t, x 0 , ¨ 0 . ª 0 as t ª wyŽ x 0 , ¨ 0 .. By continuation of solutions theorem, we have that lim sup wy Ž x, hy1 Ž x . . F wy Ž x 1 , ¨ 1 . s y`. xªxy 1
Hence, there exists ¨ and x 0 s hŽ ¨ . such that wy Ž x 0 , ¨ . s ykT, a multiple of T. Now, if we take a sequence ¨ n ª ¨ as n ª q` with ¨ n - ¨ , for the solutions x Ž t, hŽ ¨ n ., ¨ n . we have lim sup wy Ž h Ž ¨ n . , ¨ n . F ykT . nªq`
So, the points x ŽykT, hŽ ¨ n ., ¨ n ., xX ŽykT, hŽ ¨ n ., ¨ n . are the initial conditions at t s 0 of x Ž t y kT, hŽ ¨ n ., ¨ n . and x Ž t , x Ž ykT , h Ž ¨ n . , ¨ n . , xX Ž ykT , h Ž ¨ n . , ¨ n . . y w Ž t . s x Ž t y kT , h Ž ¨ n . , ¨ n . y w Ž t . ª 0
DYNAMICS OF A PERIODIC EQUATION
1039
as t ª q`. Hence, if we take limits in n, by continuous dependence of initial conditions, the point Ž x ŽykT, x 0 , ¨ ., xX ŽykT, x 0 , ¨ .. is on graphŽ h.. But x ŽykT, x 0 , ¨ . s 0 and xX ŽykT, x 0 , ¨ . is finite by Lemma 2, so if we let xX ŽykT, x 0 , ¨ . s ¨ s , then hŽ ¨ s . s 0 and, consequently, for all ¨ G ¨ s the set B is empty.
3. CASE WITHOUT FRICTION All the previous results can be obtained for the case without friction Ž f ' 0. by similar reasonings. For the sake of brevity, we will not repeat all the proofs because they are very similar; for example, in Theorem 1 we only have to take
l s 0,
Ps
ž
0 y1
y1 0
/
and we have the assumptions of Theorem 1 in w6x.
ACKNOWLEDGMENTS We thank the referee for his suggestions that have improved the presentation of the paper. We also are grateful to R. Ortega for his suggestions and comments on the contents of this paper.
REFERENCES 1. P. Habets and L. Sanchez, Periodic solutions of some Lienard equations with singularities, ´ Proc. Amer. Math. Soc. 109 Ž1990., 1035]1044. 2. A. C. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc. 99 Ž1987., 109]114. 3. J. Mawhin and C. Munoz, ˜ Application du degre´ topologique `a l’estimation du nombre des solutions periodiques d’equations differentielles, Ann. Mat. Pura Appl. 46, No. 4 Ž1973., ´ ´ 1]19. 4. R. Ortega, Topological degree and stability of periodic solutions for certain differential equations, J. London Math. Soc. 42, Ž1990., 505]516. 5. R. Ortega, Some applications of the topological degree to stability theory, in ‘‘Topological Methods in Differential Equations and Inclusions,’’ NATO ASI Ser. 6, Kluwer Academic, Dordrecht, 1994. 6. R. A. Smith, Massera’s convergence theorem for periodic nonlinear differential equations, J. Math. Anal. Appl. 120 Ž1986., 679]708.