On the Ranges of Certain Damped Nonlinear Differential Equations (*).

:E. :bT. ])A~CEI¢ (Armidale, Australia)

Summary. - I n this paper, we prove the existence of periodic solutions of some second order damped nonlinear di]#rential equations with ]orcing term. I n m a n y cases, our results are, in a sense, best l~ossible.

~¢Iany authors have considered the existence of periodic solutions with period

of the equations

(1) - ~"(t) ÷ ~(x ' ( t ) ) ÷ g(x(t)) = h(t)

and

(2) - x"(t) + / ( x ( t ) ) x'(t) + g(x(t)) = h(t) .

See for example [6], [7], [8], [9], [10], [11], [13], and [14]. In this paper , we obtain some addit ional results of this type b y combining a l ternat ive methods with an e lementa ry existence resul t for a simple differential equat ion and est imates for so- lutions of certain differential equations. Our results i l lustrate once again how these methods can be used to improve classical results and, in some cases, to obtain best possible results. Unlike most o ther results obta ined b y a l ternat ive methods, it turns out t ha t the range is enclosed by two

282 ]]. ~ . DA~CE~: On the ranges o/ certain dampe nonlinear, etc.

We assume throughout the paper tha t ], g and ~ are continuous functions from R to R and, for i = 1, 2, we let

:K~ = {/~ L~[O, ~] : eq. (i) has a solution on [0, a]

with x(O)----x(a) and x ' (O)= x'(a)}.

Define WL to be {u eL~[0, x]: u" e / ~ [ 0 , ~], u(0) = u(a), u'(0) = u'(z)}, where the derivatives are distributional derivatives. This is a Banach space for the norm Ilu]]~ = IIu]]~ + ][u~II~, where I] [], is the usual norm on ZP[0, ~]. t0inally, define

H~: W~ -->L~[0, ~] by H~(x)(t) = --x"( t) + F(x'(t)) + g(x(t)) .

H~ is defined analogously. Thus £~ is the range of H~.

1. - A related first order problem.

In this section, we discuss the existence of periodic solutions with period ~ of eq. (1) when g = 0 (~nd h ~L¢~[O, ~l). Eqa iwlen t ly , we discuss the existence of

solutions of the equation

(3) -- v ' + F(v) = h

on [0,~] with v(O)=v(~) and f v = O . Yuet 0

0

Define H3: W-->L~[O, ~] by H a ( v ) = - v'-}-F(v) and let £3 be the range of H3 (i.e. :K3 is the set of h a L°'[O, ~] for which eq. (3) has a solution in W). We first obta.in

a useful estimate.

LE~w~ 1. - I] v ~ C[O, ~], v'a L~[O, ~], ~(0) = v(a) and -- v'-t- F(v) = ~ -t- e where # ~ R and e ~ L~[0, 7@ then IIv'][~< I[eIl~ and Iv(s) - v(t)]< I s - t[~llell~ ]or s,

t e [0, ~]. I~ v e w , then l lvlI~

E. 1~. D~z~CE~: On the ranges o] certain damped nonlinear, etc. 283

If ] eL°°[0, s], let t)/----i 7~-J] and define Eo ~ (/~/~=[0, s] : 1'] = 0}. We now

obtain the main result of this section.

THEOREM 1. -- (i) There is a continuous function r: Eo--> R such tha t ~--- - {r(u) + u: u

(ii) I f u e E o , ]r(u)I~/3(~) is continuous. Define v(~) by v(~)( t )= v(t, o~, fl(a)). By the second inequal i ty in Lemma 1 (with v----v(~), #----/3(~) and e = u), Iv(cx)(s)- v(oc)(t)[ ~½]lu]]~, v(~)(t) > 0 on [0, ~] and thus fv(~) > O. Similarly, 0

if ~ < - - :~]luU2, fv(o~) < 0. Hence, by continuity, there exists ~ o e R such tha t 0

fV(~o) = 0. Thus V(~o) e W and H3(v(~o)) = fl(~o) + u. Hence /3(~o) -~ u e ~ , as 0

required. We finally obtain some estimates for v(ao) which will be needed in Step 2. By

the thi rd inequali ty in Lemma 1, Uv(~o)ll~

284 E.N. DANCE~: On the ranges of eertai damped nonlinear, etc.

for all y in R. By Step 1, there exist v~ e W and ti~ e R such t h a t - v~' Jr xt

F.(v.) = ti. + ~ an~ II~.tl~ 0 for x near x0: Thus ( v - w)(x) > 0 if x is near Xo bu t greater t han xo" Hence we have a contradict ion and thus Step 3 follows.

:By Steps 2 and 3, we see tha t , for each u e Eo, there exists a unique t i e R for

which fl ÷ u E~3. We denote this unique ti b y r(u).

STEP 4. - W e prove Theorem l(ii). B y the ~rgument used to establish eq. (4),

we see tha t

(5) ] r (u) t w i n W a s n--~ ~ w h e r e - - w ' ~ F ( w ) - - - - l ÷ u . Thus l : r ( u ) . Thus we have a contradict ion and hence r is continuous at u. This completes the proof of Theorem 1.

REmArKS. -- 1) B y a more careful proof, it is possible to show tha t a similar result holds if Z~[0~ ~] is replaced by Ll[0, ~]. The k e y est imate to prove this is obtained by mult iplying eq. (3) by sgn F(v) . (A similar idea occurs in the proof of Theorem 3(ii).)

Similar bounds still hold.

2) I f F is monotone, an a l ternat ive proof of Steps 1 and 2 can be given b y using a result of Brezis [1, p. 65]. I f iv is cont inuously differentiable, Theorem 1 can be proved by using the implicit funct ion theorem (and some of our estimates).

E . N . DANCER: On the ranges of certain damped nonlinear, etc. 285

This m e t hod can be used to show t h a t r is smooth if F is and tha t , in the case where /~ is real analyt ic , r'(u) = 0 if and only if e i ther u = 0 or F is affine.

We need the following l e m m a in §3. Note tha t , if v, w ~L~[0, ~], v>~w means

t h a t v(t)>~w(t) a.e. on [0, ~] and v > w means t h a t v>~w and {t e [0, z ] : v ( t )> w(t)} has posi t ive measure .

I~E~'m~A 2. -- I] w ~ W, u E Eo, ~ ~ I t and H 3 ( w ) < a ~ u, then o:>~r(u).

PROOF. - Choose v ~ W such t h a t H3(v) : r(u) ~ u. I f ~ < r(u), we can obta in a cont radic t ion b y the same a r g u m e n t as in S tep 3 of the proof of Theorem 1. Thus a>r(u) as required.

I~E~ARKS. -- 1) A similar resul t holds wi th the inequalit ies reversed.

2) L e m m a 2 can be used to p rove t h a t

I r (u l ) - r(u2)l< [[ul-- u~]I~o for ul, u 2 e E o .

I t can also be used in examples to obta in explicit es t imates for r(u) (by choosing sui table funct ions w).

3) I f F is cont inuously differentiable (or more general ly satisfies a local one- sided IApschitz condition), a more careful a r g u m e n t shows t h a t ~ > r(u) if Ha(w)

< ~ ~ u and t h a t H3 is 1-1. Final ly, i t is of in teres t to s t u d y the behav iour of r. We men t ion a few results

of this type. They are der ived f rom the formula zr(u) = f ( F ( v ) -- kv), where k ~ R , 0

v e W and Ha(v) = r(u) ~ u. Firs t ly , r(0) = / ~ ( 0 ) . B y choosing sui table funct ions

v, one finds t h a t r is bounded on Eo if and only if there is a k in R such t h a t 2~(y) - - ky is bounded on R and, in this case,

(6) inf {r(u) : e = inf { F ( y ) - y e R}.

Moreover, the in f imum is achieved if and only if F ( y ) - ky>~F(O) on /1. (Similar results bold for the supremtlm.) Similarly, r is bounded below if and only ff there is a k in R such t h a t F(y) -- ky is bounded below on R. There is a (more complicated) m e t h o d to calculate the inf imum in this case. (Eq. (6) still holds if there is a unique k for which F(y) -- ky is bounded below, while, if F is convex, the in f imum is /~(0).

Moreover, the inf imum is achieved if and only if there is a k such t h a t ~V(y) - - ky>~F(0) on IR (in which case it is F(0)) . Final ly , if Y is real analyt ic , r has a local m a x i m u m (minimum) a t zero if and only if F ( y ) - ~'(O)y has a local m a x i m u m (minimum)

a t zero. I t would be of in teres t to find addi t ional proper t ies of r.

286 E . N . DA~CE~: On the ranges of certain damped nonlinear, etc.

2 . - T h e m a i n r e s u l t s .

We obtain our main theorems. We assume that the limits I* = l~m g(y) exist (possibly ± co).

Tm~ORE~ 2. - Suppose that I - and I + are finite and I + ..:> I - . Then

(~ + u: t i e R , U e Eo, ~+ q- r(u) > /~ > / - ÷ r(u)}_cat~.

To prove this theorem, we use degree theory. We need a lemma to s tar t our h o m o t o p y a rgument ~n4 we need some a priori bounds. We use ordinary Leray- Sehauder degree theory though we could just as easily use M~whin's coincidence

degree ([8]). Le t 0~ = (v e W~: I]v]]= < r} and, if a > 0 and h e L~[0, ~], define T: W~--~ W2¢~

by Tx -~ x - 2 - 1 ( a x - F(x ' ) -- g(x) -~ h), where S: W~ -+Z~[0, ~] is defined by Sx = -- x" ~ ax.

LE~C~WA 3. - Assume that IF(y)I < K on R, I + an~ I - are finite, I + - K - llhll~ > 0 and I - ~ K ~- ]If]]~ < 0. Then, if r is sufficiently large, dog (T, 0, 0~) is defined and n o n - z e r o .

P R O O F . - This follows easily f rom 1%emarks 2 and 3 af ter Theorem 1 in [2]. All the conditions there are obviously satisfied except for one. We must show tha t if K1 > 0, then there is a y > 0 such tha t

(7) ( - - F((~ + v)') -- g(~ + v) + h, ~> < 0

if ~ e R , [~I>7, v ~ W L a n d " . Uv][~ is the usual scalar p roduc t on L~[0, z]. l~emember tha t , in the no ta t ion of [2], N(A) is the set of constant func- tions.) By our assumptions on g, we can choose F > 0 such t h a t g ( y ) > K + IIh]I~ if Y > 7 - - K 1 and g ( y ) < - - K - - [IhU~ if y < - - 7 + K1. Since lIvI]~

E . N . D~_~C:GR: On the ranges o/ certain damped nonlinear, etc. 287

t ha t l ( y ) - - > # > K + ] I + ] + I I - I + IIhll~ as y - ~ (where ~ = ~ + u ) . Define V: W2oo × [0, 1] --> W~ by

V(x, s) ----- x - S - l [ a x - ~ ( x ' ) - (1 -- s ) g ( x ) - sl(x) -{- h i .

I t is easy to show tha t the mapping (x, s) -+ x - V(x, s) is completely continuous. Lemma 3 implies that , if r is large, deg (V( , 1), 0, 0,) is defined and non-zero. Hence,

/ t ! if we prove t ha t there is ~n r ' > 0 such t h a t V(x, s) ~ 0 for IIx]loo>r and 0

288 E . N . DA~CE~: On the ranges o/ certain damped nonlinear, etc.

the funct ion corresponding to r when F is replaced b y F~. Suppose t h a t u e Eo. Then there is a v in W such t h a t -- v ' ÷ F(v) = r(u) ~- u. Hence, if n > I]v[]~, -- v ' ÷

F~(v) = r ( u ) ÷ u. Hence, b y Theorem 1, r ~ ( u ) = r(u) if n is large. I f 1 + > f i - r(u) > I - , then, since r~(u) = r(u) for n large, I+ > ti -- r~(u) > I - if n is large.

Thus, b y L e m m a 4, there is an x~ in W~ such t h a t

(11) - x'.' + F.(x' . ) + g(x. ) = ti + u

if n is large. B y L e m m a 1 (with # = ti, e = u - g(x.) and F replaced b y F . ) , we

see t h a t

llx. I I~< ~ " I l u - g(x.)II~ < g~

(since g is bounded on R). Thus, ff n > Ka, F.(x'~) = E(x;) and eq. (11) becomes

- x". + F(x:) + g(x.) = ti + u .

Hence ti ÷ u e : ~ , as required. We now consider the case where g is ei ther bounded above or below.

TE~OREZ~ 3. -- (i) Suppose that I + is / ini te and I - = - ~ . Then

(ti + u: t i eR , Ue Eo, ti < r(u) + I+} c~1 .

(ii) Suppose that I + is / inite, I - = ÷ c~ and either tZlto,~ )

/~l(-~,o] is bounded below. Then

{ti ÷ u: t i eR , ueE0 , t i > r(u) + 1+}_c:~i.

is bounded below or

We need the following t r ivial l emma.

L~M~A 5. - Z /A c [0, ~], w e LI[0, ~] and w(t)> -- Mon A, thenf lwl ~ - n and .g~(y)-= g ( - - n ) if y < - - n . Bo th pa r t s of Theorem 3 are p roved b y app ly ing

Theorem 2 to

(12) - x" ÷ F(x') ÷ g°(x) = ti ÷ u ,

and then establishing a bound for x.

(i) Assume t h a t u e Eo and /3 e :~ such t h a t ti -- r(u) < I +. I f n is large, we see b y app ly ing Theorem 2 t ha t eq. (12) has a solution x . in W~. B y our assumpt ions

on g and the definition of g~, there is an m < 0 such t h a t

(13) g~(y) < t - - ][uH~-- F(0)

E. Ig. D±~gE~: On the ranges o~ certain damped nonlinear, etc. 289

if 7,. - m. Choose t ~ [0,~r] such tha t x,~(t)>~x~(t.) on [0,~r]. Assume by way of contradict ion t ha t x , ( t n ) < m for some n > - m. Since x ~ ( t : ) = 0 and x: = g~(x~) -F F(x'~) -- f l - u, eq. (13) and a simple calculation imply t h a t x:(t) < 0 for ~lmost all t near t.~: This is impossible because x~ achieves its min imum at t~: Thus x~( t~)>m if n ~ - - m . Hence, if n ~ - - m , x ~ ( t ) > m ~ - - n on [0, ~] and thus g~(x~(t)) --= g(x,(t)) on [0, ~r]. Hence, since x= is a solution of eq. (13), $~ is also a solution of eq. (1). Thus fl -F u e :K1, as required.

(ii) The proof of this is similar bu t a li t t le more complicated. Assume t h a t u ~ Eo and f l e R such tha t fl -- r(u) > I+. I f n is large, we see b y applying Theorem 2 (or more s tr ict ly its analogue when I - > I+) t h a t eq. (12) has a solution x~. We consider the case where F][0,~) is bounded below. The other case is similar. Define s: R - + R by s(y) = 1 if y~>0 and s(y) = 0 otherwise. Then s is measurable. JJet

y

q ( y ) - ~ f s . By a result well-klmwn in par t ia l differential equations (cp. [15, 0

f ! f f I / • L e m m a 1.1]), q(x~) is differentiable ~.e. and q(x.(t)) = s(x.( t))x.( t) a.e. Since q(x.) is absolutely continuous (in fact , Lipschitz) on [0, ~]~ it follows t h a t

(since x~'(z) = x'.(0)). to ~r, we see tha t

~r

' " ( 4 s ( x . ) x . = q ( x : ( = ) ) - q ( o ) ) = o 0

Hence if we mul t ip ly eq. (12) b y s(x:) and integrate f rom 0

(14) f(F(J) + Jr u)< II/ + An An

where A~ = {t ~ [0, ;r]: x'~(t) >I 0}. By our assumptions on F and g, there is a K1 e R such t ha t J~(x'~(t)) -F g~(x~(t))>~gx on A~. This inequal i ty , Le mma 5 and eq. (14) imply tha t there is a K~ > 0 such t h a t

(15) flF(x:) -~- g.(x.) l < K 2 . An

Suppose t ha t w e [0, ~r] such t ha t x'~(w) > O. Le t v d e n o t e t h e largest point in [-- Jr, w) such t ha t x.'~(v) = 0 (where we have ex tended x~ to [-- ~r, 0) b y periodicity). Then x'~(t) >7 0 on Iv, w] and thus Iv, w] c_ A~. Hence

~dJ

-- I x . < 1:1 ~) A n

< f[[F(x:) -k gdx.)[ ÷ [fl -F A n

K a .

1 9 - .Annali di Matemat~ca

290 E. :N. DA~CE~: On the ranges of certain damped nonlinear, etc.

Here we have used tha t x~ is a solution of eq. (12) to obtain the second last inequal i ty and eq. (15) to obtain the last. Hence

(16) x~(t)~Kd. Since x ~ ( 0 ) = x~(z), this inequal i ty and (16) imply t ha t x ~ ( t ) ~ K d - - K ~ on [0, 7~]. Hence if n > [K~-- Karl , g,(x,(t)) = g(x~(t)) on [0, ~]. Hence, as in pa r t (i), fl -~ u ~ : ~ , as required.

t~Eh~:ARKS ON WHEellESS 2 AND 3. -- 1) Wi th a more careful proof, it is possible to assume tha t u ¢ Eli0, z] in bo th Theorems 2 and 3 a n d t o weaken the addit ional assumption on F in Theorem 3(ii) to ei ther

lim inf y-l(log y)-~F(y) > -- co or lim sup y-~(log [Y[)-~F(Y) < c~. y - > c o y - - > - c o

The key idea in the proof of this last resul t is to first reduce to the case where _F(y) < 0 or R (by obtaining an est imate as in the proof of Theorem3(i i ) except t ha t a slightly different funct ion s is used) and then consider the possible growth of

½(x'(t)) 2 - f g (as in [3]). The extra assumption of F can be removed in the case where o

lira sup ]y-~ g(y)[ < 1 and in m a n y cases where F grows more rapidly than g. How-

ever, we 4o not know whether i t can be ent i re ly removed.

2) As usual, similar results hold for the other cases where at least one of I - and I + are finite and I - V: I +. Theorem 2 can be improved. I f g is bounded on R, g(y)>~J+ for y large positive, g ( y ) K J - for y large negat ive and J-~fl--r(u)>~J-. This is p roved by applying Theorem 2 to the equat ion where g is replaced b y g -~ ( l /n) 1 (where 1 is defined in essentially the same way as in the proof of Le mma 4) and then passing to the limit.

3) Wi th some care, our methods can be used to prove similar results if g and F depend on t provided t ha t there is a funct ion q: R -~ {-- 1, 1} such t h a t q(y)F(t, y) is bounded below on [0, z] × R . (Est imates are obta ined by ~ similar me thod to tha t in the proof of Theorem 3(ii).) For example, if g(t, y) is bounded on [0, x ] × R , g(t, y) -+ I±(t) as y ~ ~ c~ uniformly in t and u e Eo, then fl ~- u ~ ~1 provided tha t

[PI+- fi + r ( u - OI+)] [ H - - fl + r ( ~ - QI-)] < 0

(where, for v e L~[0, z], Qv • v - Pv and P is defined in § 1). I n this case, it can also be p roved tha t , for each u ~ Eo, {fl E R: fl -~ u ~ ~i} is non-empty .

4) Analogous results hold for eq. (2) if r(u) is replaced b y the zero function. Unlike the case of eq. (1), no addit ional assumption on f is needed for any of the results.

:E. ~ . DANCEI~: On the ranges o/ certain damped nonlinear, etc. 291

The proofs are similar (but easier) except t h a t we obtain bounds differently. I f we in tegra te eq. (2) on [0, ~] and use Lemma 5, we obtain a bound for lid(X)]I~. By

Y

in tegrat ing eq. (2), we then see t h a t x ' ÷ ~(x) ~ c ÷ ~, where /~(y) : f f , e e R and 0

we have a bound for [I41[ ~. Lemma 1 then implies a bound for Ux'I]~ and it is then easy to obtain addit ional bounds as before. The resul t in the case where I - and I +

are finite was previously obta ined in [7].

3. - Best possible results and uniqueness.

We first show that~ under an addi t ional assumption, equal i ty holds in Theorem 2.

THEOREM 4. -- Suppose that the assumpt ions of Theorem 2 hold and that I - ~

< g(y) < I + on R . Then

~ = (fl + u: f l e R , u e ~0, ~+ + r(u) > fl > I - + r(u)}.

PROOF. -- B y Theorem 2, i t suffices to show tha t , if fl e R, u ~ ~o ~nd fl ÷ u e :K~, then I + > fl - - r(u) > I - . I f fl ÷ ue:K~, then there is an x in W~ such tha t H~(x) = f l ÷ u . B y our ~ssumption on g, there is an s > 0 such t h a t g ( x ( t ) ) > ~ I - ÷ e on [0, z]. Hence we see t ha t

- x" + ~(x') < (fl + u) - ( I - + ~) -= ( f l - I - - ~) + u .

Thus, by Lemma 2 (with c ~ - - - - f l - - I - - - e ) , we see t h a t f l - - I - - - s > r ( u ) . Hence, since e > O, fl - - r(u) > I - . Similarly I + > f l - - r(u). This completes the proof.

I~E~ARKS. -- 1) A similar result holds concerning the best possible na ture of Theorem 3 and of the resul t in l~emark 3 af te r Theorem 3. Thus, if in addit ion to the assumptions there , Z-(t) < g(t, y) < I+(t) on [0, ~] × R , then

R1 : {fl + u: f ie R, u e Eo, e I + + r ( u - Q1 +) > fl > ~ - + r(u - Q I - ) ) .

(Note tha t , for each u e Eo, there exists a t least one fl satisfying the two inequalities.)

2) I f F is cont inuously differentiable, our methods can be used to discuss the case where I - < g(y) • > I - + r(u)).

3) I f g is bounded on R , g(y) > I + for some y in R (or g(y) ~ I - for some y in t5) and g and /~ are cont inuously differentiable on R, the me thod of sub- and

292 E . N . DANCER: On the ranges o/ certain damped nonlinear, etc.

super-solut ions can be used as in § 4 of [4] to find addi t ional proper t ies of : ~ . Fo r example , if g(y) > I - on R and g(y) > I + for all y large posit ive, i t tu rns out t h a t

there is a cont inuous funct ion 5: Eo -* R such t h a t l(u) > I + -~ r(u) on Eo,

aq = + u: u e z- + r (u )<

and the equat ion H~(x) = fi -~ u has a t least two solutions if I + + r(u) < fl < l(u).

(Weaker resul ts hold if we only assume t h a t g ( y ) ~ I + for some y.) There is one po in t to be noted. To apply the m e t h o d of sub- and super-solutions, we first app ly it wi th a t runca ted f f (us in the proof of Theorem 2) and then use L e m m a 1 to pass

to the limit. The m e t h o d of sub- and super-solutions can also be used (as in [4])

to s tudy proper t ies of :K~ when g(y) -~ c~ (or - - c~) as [Yl--> c~ (provided t h a t we impose a condit ion on F similar to t h a t in Theorem 3(ii)).

4) As usuM, similar resul ts hold for eq. (2).

We now discuss uniqueness. For this, we assume that ~ and g are cont inuously di]ferentiable ( though F and g locally Lipschitz would suffice for m a n y of our results).

L~.~tTc~A 6. - Suppose that a, d ~ L~[O, zr], x ~ W ~ , d < 0 and x" q- ax 'q - dx -= O.

Then x = O.

P~ooF. - We can ex tend a, d and x to [0, 2~] b y extending periodically. Thus, since ~(0) = x(er), e i ther x or - - x has a non-negat ive m a x i m u m in (0, 2er). Hence b y the m a x i m u m principle (cp [12], Theorem 1.3), x is cons tan t on [0, 2~r]. (They

assume a l i t t le more regular i ty t han we have bu t their proof is val id under our as- sumptions.) Hence, since x"q - ax 'q - dx = O, dx = O. Since d :~ 0 and x is constant ,

i t follows t h a t x = 0.

TKEOnE~ 5. -- I / g'(y) > 0 on R , then /or each h ~ 5~ there is a unique x in W ~

such that H~(x)--~ h.

P R O O F . - - I f X~, X~ e W~ and H~(xl) = H~(x~), we see b y a simple calculat ion t h a t

w ~- x ~ - x~ satisfies - - w"q- aw 'q - bw = O, where

/ [x ' l ( t ) - x'2(t)J-~[F(X'l(t)) - - F(x'~(t))] a(t) /

x'l(t) ¢: x'2(t)

otherwise

and b is defined analogously. Since g'(y) ~ 0 on R, the mean-vMue theo rem implies t h a t b(t) ~ 0 on [0, ~]. Hence, b y L e m m a 6, w - = O, i.e., xl = x~. This completes

the proof.

I~E~ARKS. -- 1) B y a more careful proof, one sees t h a t uniqueness holds if g ' ( y ) ~ 0 on R un4 g' does not van ish on a n y interval . Conversely, i t is easy to show thu t

uniqueness fails (for some h ~ :g.1) if g' vanishes on an interval .

E. ~ . DANCER: On the ranges of certain damped nonlinear, etc. 293

2) A degree a rgument similar to t h a t ment ioned ~t the end of § 1 of [4] implies tha t uniqueness fails if there exist yz, y ~ R such t h a t g'(y~)~ 0 ~ g'(y~).

3) The corresponding problem when g ' ( y ) ~ 0 on R seems much more com- pl icated and we only ment ion a few par t ia l results. As before, we are in teres ted in the case where a t least one of I + and I - is finite. I f g'(y) ~ ~ 4 for some y in R, t hen the methods ment ioned in [4] and a pe r tu rba t ion a rgument can be used to show tha t there is an ~ > 0 (depending on g) such t h a t uniqueness fails whenever IF'(y) l~0 (~ g'(y)>~ -- 1 on R or (ii) IF'(y)]~>c > 0 and 0 > g'(y) > -- 2c on R. We conjecture t ha t uniqueness fails if y-lg(y) __> ~ (or y - > - c~). ( I t can be shown tha t this conjecture is t rue if F(y) ~ cy, g is twice cont inuously differentiab]e and g " ( y ) ¢ 0 on R.)

4) Similar results hold for eq. (2) if we replace F ' by f. (In deriving these, it is usually convenient to consider the adjoint of any linearization.)

4. - S o m e other results .

In this section, we wish to briefly discuss some other results. Assume that g(y) s g n y - ~ - - ~ as [Yl--> ~ , that the limits F ~ = l~m F(y) exist (possibly ~ ~ )

y

and that F÷ F-. (For eq (2), is de ned by F(y)=f i . ) 0

A perusal of the proofs of §§ 5.3 and 5.5 in [13] shows tha t , if F + -- F - -~ :[: c~, then ~1 and ~ are bo th L~[0, z] (and, for eq. (2), we m a y replace L~[0, ~] b y L~[0, ~]). I f F + and F - are bo th finite, a similar perusal shows tha t h ~ if sup h

inf h ~ ½ IF + -- F - t and t ha t h ~ :~2 if sup E -- inf E ~ IF + -- F - I, where E(t)

=f(Qh). (Q was defined in § 2.) In each case, one also finds tha t an appropr ia te o

degree is non-zero.

Henceforth, we assume that F + and F - are finite. Then eqs. (1) and (2) behave ra ther like eq. (1) with no damping t e rm present . Thus we can obtain results by the methods of [2] and [3]. (For some results, we need the ext ra condit ion tha t Ill is bounded and integrable on R.) I t is sometimes more convenient to consider H2 as a mapping of

(x e C~[0, z] : x is periodic} into (h': h e C[0, ~]: h is periodic},

where the der ivat ive is a dis tr ibut ional derivative. This is convenient because, for these spaces, f (x)x ' is a bounded per tu rba t ion and the existence theorem for the init ial-value problem still holds. We do no t a t t e m p t to obtain results in this way in detail bu t only briefly discuss two occasions where interest ing differences f rom the undamped problem emerge.

294 E. IV. D,~CEn: On the ranges o] certain damped nonlinear, ete.

Firstly, assume that n is a positive integer, g(y) ~ -- 4n~y ~- p(y), p is bounded on R and the limits p~== l im p(y) exist. I f (i) (undamped c~se) /~----0 and p-~<

y-->=t= ~a

E . ~-. D A ~ C ~ : On the ranges o] vertain gamped na~,~tinear, etc. 295

[3] E. •. DANCE~, Boundary-value problems for weaklynonlinear ordinary di]]erential equa- tions, Bull. Austral . Math. Soc., ]5 (1976), 10p. 321-328,

[4] E. N. DANC~, On the ranges of certain weakly nonlinear elliptic partial diNercntial equa- tions, to appear in J. Math. Pures et Appliqu4es.

[5] E. N. DANC~, Asymptotics applied to nonlinear boundary-value p~vblems, Bull. Austral . Math. Sot., 18 (1978), pp . 29-35.

[6] P. F~D~ICKSO_~ - A. L A z ~ , Necessary and su]]ieient damping in a second order oscil- lator, J. Differential Eqs., 5 (1969), Pl). 262-270.

[7] S. Fu~lK - J. MAW~IN, Periodic solutions o] some nonlinear di]]erential equations o] higher order, Cas. P~st. Mat., 10O (1975), 10p. 276-283.

[8] R. GAINES - J . MAWmN, Coincidence degree, and nonlinear di]]erential equations, Springer- Verlag, Berlin (1977).

[9] A. LAZER, On Svhauder's ]ixed point theorem and ]creed second order nonlinear oscilla- tions, J. Math. Anal. Appl. , 21 (1968), pp. 421-425.

[10] A. LAZER - D. LEACh, Bounded perturbations o] ]orced harmonic oscillations at resonance, Ann. Mat. Pu ra Appl . , 82 (1969), pp. 49-68.

[11] J. MAWHIN, An extension o] a theorem o] A. C. ~azer on ]creed nonlinear oscillations, J. Math. Anal . Appl . , 40 (1972), pp. 20-29.

[12] M. Pt~oT~I~ - H. W]~I~CBE~G~R, Maxiraum principles in di]]erential equations, Prentice Hall , Englewood Cliffs (1967).

[13] R. REIssIo - G. SA~SONE - R. COATI, Qualitative Theorie nichtlinearer Di]]erentialglei. chungen, Edizioni Cremonese, Roma (1963).

[14] R. REIsSxO, Extension o] some results concerning the generalized Lidnard equation, Ann. Mat. Pu ra Appl . , 104 (1975), pp. 269-281.

[I5] G. STA~n'ACCmA, ~quations elliptiques du second ordre ~ eoe]]ieients discontinus, Univer- s i ty o~ Montreal Press, Montreal (1966).