Relation with Degree Theory

Chapter III

Relation with Degree Theory 1

The

Periodic

Problem

It has been seen that the periodic problem

u"= f(t,u), u(a) = u(b), u'(a) = u'(b)

(1.1)

is equivalent to the fixed point problem

u = Tu,

(1.2)

where T : g([a, b]) ~ C([a, b]) is defined from

j~ab (Tu)(t) "-

G(t, s)[f(s, u(s)) - u(s)] ds

(1.3)

and G(t, s) is the Green's function corresponding to (I-1.4). In this chapter, we associate the set

gt = {u e C([a, b]) I Vt e [a, b], a(t) < u(t) < fl(t)}

(1.4)

to the pair of lower and upper solutions (a, fl) and wish to consider the degree deg(I - T, gt). To this end we have to reinforce the concepts of lower and upper solutions so that the boundary of ~ does not contain solutions of (1.1). This modification amounts to requiring that the solutions cannot be tangent to the curves u=a(t) and u = Z ( t ) 135

136

III. R E L A T I O N W I T H D E G R E E T H E O R Y

respectively from above or from below and motivates the definitions we introduce in the following section. For problems with a nonlinearity f which depends on the derivative

ul'- f(t,u,u'), u(a) = u(b), u'(a) - u'(b),

(1.5)

the corresponding fixed point problem reads

u(t) - ( T u ) ( t ) ' -

~ b G(t, s)[f (s, u(s), u'(s)) - u(s)] ds,

where T'Cl([a,b]) -~ Cl([a,b]). Given lower and upper solutions a and/3, and some R > 0, we shall compute the degree deg(I - T, ~), where

~t - {u e cl([a,b]) l Vt e [a,b], ~(t) < u(t) < D(t), [u'(t)l < R}. As above, this requirement will force us to reinforce the concepts of lower and upper solutions and furthermore to choose R to be an a priori bound on the derivative u I. 1.1

The Strict Lower and Upper

Solutions

D e f i n i t i o n s 1.1. A C 2 or W2,1-1oweF solution a of (1.5) (resp. (1.1)) is said to be strict if every solution u of (1.5) (resp. (1.1)) with u > a is such that u(t) > a(t) on [a, b].

Similarly, a C 2 or W2,X-upper solution ~ of (1.5) (resp. (1.1)) is said to be strict if every solution u of (1.5) (resp. (1.1)) with u < ~ is such that u(t) < Z(t) on [a, b]. The classical way to obtain such a notion in the case of a continuous f and a or/3 C C2([a, b]) is described in the following proposition. P r o p o s i t i o n 1.1. Let f " [a, b] x I~2 -~ l~ be continuous and a e C2([a, b]) be such that (a) for a n t e [a, b], c~"(t) > f(t,c~(t),a'(t)); (b) a(a) = a(b), a'(a) >_ a'(b). Then a is a strict C2-lower solution of (1.5).

1. T H E P E R I O D I C

137

PROBLEM

Proof. Let u be a solution of (1.5) such t h a t u _> a and assume by contradiction t h a t min (u(t) - c~(t)) - u(to) - (~(to) = O.

tE[a,b]

We have u ' ( t o ) - a'(to) = 0 which follows from assumption (b) in case to = a or b. Hence, we obtain the contradiction

0 <_ u " ( t o ) - a"(to) = f(to, c~(to), a ' ( t o ) ) - a"(to) < O.

K]

Using the same a r g u m e n t we obtain the corresponding result for upper solutions. P r o p o s i t i o n 1.2. Let f " [a, b] x ]~2 ___+R be continuous and 13 E C2([a, b]) be such that (a) for all t c [a,b], 13"(t) < f ( t , t3(t),~'(t)); (b) ~(a) = ~(b), ~'(a) <_ ~'(b). Then ~ is a strict C2-upper solution of (1.5). If f is not continuous but LP-Carathe~odory, these last results no longer hold. In fact, even the stronger condition for a.e. t e [a,b],

a " ( t ) >_ f ( t , c ~ ( t ) , a ' ( t ) ) + 1

(1.6)

does not prevent solutions u of (1.5) to be t a n g e n t to the curve u - a(t) from above. Such a situation occurs, for example, for the bounded function

f ( t , u ) "= - 1 , u 2 + sin t :=1 + sin t ' := - sin t,

if u < - 1 , if - 1


if sin t < u,

if we consider a(t) - - 1 , u(t) - sint, a motivates the following proposition.

0 and b = 27r. This r e m a r k

P r o p o s i t i o n 1.3. Let f : [a, b] x ]R2 --~ ]R be an L1-Carathdodory function. Let ~ E C([a, b]) be such that ~(a) - ~(b) and consider its periodic extension on IR defined by (~(t) = c~(t + b - a). A s s u m e that c~ is not a solution of (1.5) and for any to E IR, either (a) D _ a ( t o ) < D+a(to) or (b) there exist an open interval Io and ~o > 0 such that to E Io, (~ C W2'l(Io) and for a.e. t C Io, all u E [c~(t), c~(t)+c0] and all v c [c~'(t)-e0, a ' ( t ) + e 0 ] ,

___ S(t, Then a is a strict W 2 ' l - l o w e r solution of (1.5).

138

III. R E L A T I O N

WITH DEGREE

THEORY

Proof. The function a is a W2'l-lower solution since clearly it satisfies Definition I-6.1. Let u be a solution of (1.5) such that u > a. As a is not a solution, there exists t* such that u(t*) > a(t*). Extend u and a by periodicity and assume by contradiction that to = inf{t > t * [ u ( t ) = a(t)} exists. As c~ - u is m a x i m a at to, we have D_a(to) - u'(to) >_ D+a(to) ul(to). Therefore, assumption (b) applies. We have that u'(to)-C~'(to) = 0 and there exist Io and eo > 0 according to (b). It follows that we can choose tl C I0 with tl < to such that u ' ( t l ) - o/(tl) ( 0 and for every t e It1, to[ <

<

Hence, for almost every t C It1, to[, we can write

a"(t) > f(t, u(t), u'(t)), which leads to the contradiction

0 < ( u ' - c ~ ' ) ( t o ) - (u'-c~')(t~) =

[ f ( t , u ( t ) , u ' ( t ) ) - c ~ " ( t ) ] d t < O.

D

1

In the same way we can prove the following result on strict upper solutions. 1.4. Let f ' [ a , b] x 11(2 -~ R be an L1-Carathdodory function. Let ~ C C([a, b]) be such that ~(a) = ~(b) and consider its periodic extension on R defined by ~(t) = ~(t + b - a). Assume that ~ is not a solution of (1.5) and for any to c R, either (a) D - ~ ( t o ) > D+~(to) or (b) there exist an open interval Io and co > 0 such that to c Io, ~ c W2'1(Io) and for a.e. t c Io, all u c [~(t)-co, ~(t)] and all v c [~'(t)-eo, ~'(t)+eo], ~"(t) < f ( t , u, v). Then ~ is a strict W2'l-upper solution of (1.5).

Proposition

Observe t h a t in the continuous case, if a satisfies the conditions of Proposition 1.1, then it satisfies the conditions of Proposition 1.3. The converse, however, does not hold since Proposition 1.3 does not require strict inequalities. To be able to say that a function a which satisfies (1.6) is a strict lower solution, we need some more regularity on f. In particular we have the following result.

1. T H E P E R I O D I C

139

PROBLEM

1.5. Let f : [a, b] • R ~ ~ R be an L1-Carathdodory function that satisfies the assumption ( A ) f o r all to E [a, b], (uo, vo) c R 2 and e > O, there exists 5 > 0 such that

Proposition

It-tol < 5, lu-uol < ~, Iv-vol < ~ ~ Let A > 0 and a c W 2 ' l ( a , b) be such that

If(t,u,v)-f(t,

uo,vo)l < ~.

c~"(t) >_ f ( t , a ( t ) , a ' ( t ) ) + A, c~(a) - a(b), c~'(a) >__a'(b). Then a is a strict W2'l-lower solution of (1.5). Remark Notice t h a t in this proposition, Condition (A) does not imply t h a t f is continuous. For example, we can take f(t, u, v) = g(u, v ) + h(t) w i t h g continuous and h C Ll(a,b) or f ( t , u , v) = g(u,v)h(t) with g continuous and h C L ~ ( a , b).

Proof. Let us deduce we will prove t h a t for such t h a t to c Io and cd(t) - co _< v < cg(t)

this proposition from Proposition 1.3. To this end, any to c R, there exist an open interval I0 and e0 > 0 for a.e. t E I0, for all u, v with a ( t ) _< u < a ( t ) + e0, + ~o, we have

~" (t) >_ f (t, u, v). Let to C IR be fixed. Define uo = a(to), vo - c~'(to) and e - A/2. We deduce then from A s s u m p t i o n (A) a 5 > 0 such t h a t

I t - tol < ~, I ~ - ~(to)l < ~, I ~ - ~'(to)l < =~ If(t, u, v) - f(t, a(to), a'(to))] < A/2. Let now 7/< 5 be such t h a t for any t E [to - r/, to + rl],

I~(t)- ~(to)l < ~/2,

I~'(t)- ~'(to)i < ~/2.

Choose Io = ]to - rl, to + 7/[ and eo = a/2. T h e result follows now since c~"(t) - f(t, u, v) = cd'(t) - f(t, a(t), c~'(t))

+ f ( t , c ~ ( t ) , c ~ ' ( t ) ) - f(t,a(to),C~'(to)) +f(t,c~(to),a'(to)) - f(t, u, v) >__ A -

A / 2 - A / 2 = O.

D

In this last proposition, we can weaken the assumption a E W2'l(a, b) and allow angles as previously. Also, we can make Condition (A) one-sided and impose (A') for all to E [a, b], (uo, vo) E II~2, and e > 0, there exists 5 > 0 such t h a t

It- tol < 5, I ~ -

~ol < ~, ~ e [~,,,~ + ~[,

I V l - VoI < 5, 1"/22- VOI < 5

~

:(t,'/.t2,V2)- f(t,?.tl,Vl)

< s

140

III. R E L A T I O N

WITH DEGREE

THEORY

For strict upper solutions we have a similar result. 1.6. Let f 9 [a, b] • R 2 -~ R be an L1-Carathdodory function that satisfies Assumption (A) in Proposition 1.5. Let B > 0 and ~ E W2'l(a, b) be such that

Proposition

~"(t) <_ f ( t , ~ ( t ) , ~ ' ( t ) ) - B, ~(a) = ~(b), ~'(a) < ~'(b). Then ~ is a strict W2'l-upper solution of (1.5). E x e r c i s e 1.1. Generalize Propositions 1.5 and 1.6 to allow corners in c~ and ~ as in Propositions 1.3 and 1.4. Another simple and interesting situation concerns the case where f satisfies a Lipschitz condition in v and a one-sided Lipschitz condition in u. Such assumptions are classical in studying monotone iterative methods. 1.7. Let f " [a, b] • R 2 ~ R be an L1-Carathdodory function such that for some k, 1 c L l(a, b; R +),

Proposition

(a) for a.e. t E [a, b], all U l, U2 C ~ with U l ~ U2 and v C R, /(t,

- f(t,

v) _< k(t)(

2 -

(b) for a.e. t c [a, b] and all u, Vl, v 2 C ~, I f ( t , U, V2) -- f ( t , U,

Vl)[ ~

l ( t ) [ v 2 --

Vl[.

Then every W2,1-1ower solution ~ (resp. every W2'l-upper solution ~) of (1.5) which is not a solution is a strict W2,1-1ower solution (resp. a strict W2,1-upper solution). Proof. Let u be a solution of (1.5) such t h a t u > a. As in the proof of Proposition 1.3, we find an open interval I0, to c I0 and tl c I0 with tl < to such t h a t u(to) = ~(to), u'(to) - c~'(to), u ' ( t l ) - c~'(tl) < 0 and for a.e. tCIo a"(t) > f ( t , a ( t ) , a ' ( t ) ) . Let w = u -

a and observe that, on It1, to[,

- w " +l(t)sgn(w'(t)) w' + k(t)w >_ - f ( t , u ( t ) , u ' ( t ) ) + f ( t , a ( t ) , a ' ( t ) ) +l(t)[u'(t) - a'(t)l + k(t)(u(t) - a(t)) ~0.

1. T H E P E R I O D I C

PROBLEM

141

Defining

P(t) =

-

~tt l(s)sgn(w'(s))

ds,

1

Q(t) = - ft t eP(~)k(s) ds,

R(t) = -

It t Q(s)e -P(~) ds, 1

1

we compute d

d--~[(w'(t )

-~ Q(t)w(t))e R(t)] = eR(t)[eP(t)(w"(t) -- l(t)]w'(t)[- k(t)w(t)) - e-P(t)Q2(t)w(t)]

e P(t)

<0. Hence, we have the contradiction

0 >_ (w'(t)e p(t) + Q(t)w(t))eR(t)]tt: = - w ' ( t , ) > O.

[:]

E x e r c i s e 1.2. Assume f satisfies the assumptions of Proposition 1.7 and a is a w2'l-lower solution (resp. fl is a w 2 ' l - u p p e r solution) of (1.5) which is not a solution. Prove that any W2'l-upper solution/3 >_ a (resp. W 2'llower solution a _ a. R e m a r k 1.1. In Proposition 1.7, we can assume (a) and (b) to hold only in a neighbourhood of { ( t , a ( t ) , a ' ( t ) ) l t 9 [a,b] such that a'(t) exists} (resp. {(t,13(t),~'(t))lt 9 [a,b] such that/~'(t) exists}). However this proposition does not hold without the Lipschitz conditions. Even a one-sided H61der condition is not enough as shown by the following example u " - 121ul 1/2 ' u ( - 1 ) = u(1), u ' ( - 1 ) = u'(1).

(1.7)

Here ~(t) = t 4 is an upper solution which is not a solution of (1.7). On the other hand, u(t) = 0 is a solution such that u(0) - 13(0). Hence, ~ is not strict. 1.2

Existence

and Multiplicity

Results

Now we can prove the key result of this section. 1.8. Let solutions of problem (resp. B C [a, b]) to Let E be defined

Theorem

a and 13 e C([a,b]) be strict w2'l-lower and upper (1.5) such that on [a, b], a(t) < ~(t). Define A c [a,b]

be the set of points where a (resp. 13) is derivable. by

E := {(t,u,v) 9 [a,b] x R 2 1 a ( t ) < u
(1.8)

142

III. R E L A T I O N W I T H D E G R E E T H E O R Y

and p, q E [1, co] be such that-~1 + ~1 = 1. Assume f " E ~ R satisfies LP-Carathdodory conditions and there exists N C L 1(a, b), N > 0 such that for a.e. t E A (resp. for a.e. t c B ) f(t,a(t),a'(t)) > -N(t)

(resp. f ( t , ~ ( t ) , ~ ' ( t ) ) < N(t)).

Assume moreover there exist ~ E C(R+,R+), r E LP(a, b) and R > 0 such that j~oR sl/q ~ - ~ d s > IIr ) -mina(t))l/qt (1.9) and that the function f satisfies one of the one-sided Nagumo conditions (a), (b), (c), or (d) in Theorem I-6.1. Then deg(I - T, gt) = 1, (1.10) where T : CX([a, b]) --, Ca([a, b]) is defined by

(Tu)(t) .=

G(t, s)[f(s, u(s), u'(s)) - u(s)] ds,

(1.11)

G(t, s) is the Green's function corresponding to (I-1.4) and fl is given by

a - {u e cl([a,b])lVt e [a,b], a(t) < u(t) < ~(t), lu'(t)l < R}.

(1.12)

In particular, the problem (1.5) has at least one solution u c W2'p(a,b) such that for all t E [a, b] a(t) < u(t) < fl(t). Proof. Increasing N if necessary, we can assume N(t) >_ If(t, u, v)l if t E [a, b], a(t) < u _< ~(t) and Ivl <_ R. Define then f(t, u, v) = max{min{f(t, 9/(t, u), v),N(t)}, - N ( t ) } , wl(t, 5) = X A ( t ) m a x l f ( t , a ( t ) ' a'(t) + v ) - f ( t , a ( t ) , a ' ( t ) ) I 1.1<~ w2(t 5) = X B ( t ) m a x If(t,~(t) ~'(t) + v) -- f ( t ~(t),~'(t)) I, '

Ivl<_a

'

'

where ~ is defined from (I-1.3), XA and XB are the characteristic functions of the sets A and B. It is clear that wi are L1-Carath~odory functions, non-decreasing in 5 such that wi(t, 0) = 0 and Iwi(t, 5)1 < 2N(t). We consider now the modified problem

u" - u = f(t, u, u') - w(t, u), u(a) = u(b), u'(a) = u'(b), where

w(t, u) - / ~ ( t ) - w2(t, u -/3(t)), ~U~

= ~(t) + ~l(t, ~ ( t ) - ~),

if ~ > ~ ( t ) ,

if a(t) < u < ~(t), if u < a(t).

(1.13)

1. T H E P E R I O D I C

PROBLEM

143

This problem is equivalent to the fixed point problem U -- Tu,

where T " (J1 ([a, b]) ~ C 1([a, b]) is defined by -

a(t,

s)[i(s,

-

Observe that 7~ is completely continuous and there exists/~ large enough so that a C B(O,R) and T(Cl([a,b])) C B(O,/~). Hence we have, by the properties of the degree, d e g ( I - T, B(0,/~)) = 1. We know that every fixed point u of T is a solution of (1.13). Arguing as in the proof of Theorem I-6.6, we see that c~ < u < ~ and Ilu'll~ < R. As c~ and/3 are strict, a < u < ~. Hence, every fixed point of T is in ft and by the excision property we obtain d e g ( I - T, f~) - d e g ( I - 2~, f~) - deg(I - 7~, B(0,/~)) - 1. Existence of a solution u such that for all t E [a, b], <

<

follows now from the properties of the degree.

[2]

A generalization concerns the use of several lower and upper solutions. 1.9. Let c~ e C([a,b]) (i = 1 , . . . , n ) a n d ~j e C([a,b]) ( j 1 , . . . , m) be respectively W2,1-lower and upper solutions of (1.5). Assume a ' = max ai and ~ ' min ~j

Theorem

l <:i
l <_j<_m

satisfy a(t) < Z(t) on [a, b] and are strict, i.e. any solution u of (1.5) with a <_ u < 0 such that for all i and a.e. t E Ai (resp. for all j and a.e. t E B j ) f(t,c~i(t),a~(t)) >_ - N ( t )

(resp. f ( t , ~ 3 ( t ) , ~ ( t ) ) <_ N(t)).

Assume moreover there exist ~ c C(R+,R+o ), ~ E L~(a, b) and R > 0 that satisfy (1.9) and that the function f satisfies one of the one-sided Nagumo conditions (a), (b), (c), or (d) in Theorem I-6.1.

144

III. R E L A T I O N W I T H D E G R E E T H E O R Y Then

deg(I - T, ~) = 1,

where T : C 1 ([a, b]) ~ C 1 ([a, b]) is defined by (1.11) and 12 is given by (1.12). In particular, the problem (1.5) has at least one solution u C W2'P(a, b) such that for all t E [a, b]

~(t) < ~(t) < Z(t). E x e r c i s e 1.3. Prove Theorem 1.9 paraphrasing the argument of the previous proof and using the idea of Theorem I-6.10. In case f is independent of u', this result reduces to T h e o r e m 1.10. Let ai e C([a, b]) (i - 1 , . . . , n ) and 13j e C([a, b]) (j = 1 , . . . , m ) be respectively W2,1-1ower and upper solutions of,(1.1). Assume a : = max ai and /3"= min 13j l "(i
l ~_j<_m

satisfy a(t) < j3(t) on [a, b] and are strict (see Theorem 1.9). Let E "= {(t, u) E [a,b] x R ] mini hi(t) < u < maxj/3j(t)} and assume f " E ~ R satisfies L1-Carathdodory conditions. Then deg(I - T, ~) = 1, where T ' C ( [ a , b]) --~ C([a,b]) is defined by (1.3) and ~ is given by (1.4). In particular, the problem (1.1) has at least one solution u C w2'l(a,b) such that for all t E [a, b]

~(t) < ~(t) < 9(t). Observe that we can replace the Nagumo conditions by any condition so that an a priori bound in the space C([a, b]) on solutions u of the corresponding modified problem implies an a priori bound on ]lu']l~ which is the case for the Rayleigh and the Li~nard equation. Consider first the Rayleigh equation u" + g(~') + h(t, ~, ~') = o, u(a) = u(b), u'(a) = u'(b).

T h e o r e m 1.11. Let a solutions of (1.14) such (1.8), g E C(R) and h for some H c L2(a,b), (t,u,v) C E ,

(1.14)

and ~ E C([a,b]) be strict W2'l-lower and upper that on [a,b], a(t) < j3(t). Let E be defined by : E ~ R be a Carathdodory function such that for a.e. t e [a,b], and for all (u,v) e 11(2 with ]h(t, u, v)] < H(t).

1. T H E P E R I O D I C

PROBLEM

145

Thgn deg(I - T, ~2) -- 1,

where f~ is defined by (1.12) (with R > 0 large enough), T " CX([a,b])

C1 ([a, b]) is defined by b (Tu)(t) := - J = G(t,s)[g(u'(s)) + h(s,u(s),u'(s)) + C(s)u(s)] ds,

G(t, s) is the Green's function of ~" - c(t)u

- f(t),

(115)

u(a) - u(b), u'(a) = u'(b), and C E Ll(a, b) is chosen such that C(t) > Ig(0)] + 1 + 3H(t) on [a, b]. In particular, the problem (1.14) has at least one solution u c W2'2(a, b) such that for all t E [a, b]

~(t) < u(t) < ~(t). Proof. The proof repeats the arguments of Theorem I-6.8. Using the notations therein, observing that a and/3 are strict, and denoting ~~1 -- {u E c l ( [ a , b ] ) l l l u l l ~ < p, Ilu'll~ < R}, we have d e g ( I - T, ~2) = d e g ( I - T~, f~) = d e g ( I - T1, a l ) = d e g ( I - To, ~1) = deg(I - To, B(O, Ro)) - 1. This concludes the proof.

V1

The next result concerns the Li4nard equation

u" + g(u)u' + h(t, u) - O, u(a) = u(b), u'(a) = u'(b).

(1.16)

T h e o r e m 1.12. Let a and ~ E C([a, b]) be strict W2'l-lower and upper solutions of (1.16) such that on [a, b], a(t) < ~(t). Let E = {(t, u) E [a, b] x IR I a(t ) _< u _< /3(t)}, g C C(IR) and h" E ~ IR be an L1-Carathdodory function. Then deg(I - T, ft) = 1,

where ft is defined by (1.12) (with R > 0 large enough), T " Cl([a,b]) ---+

C l([a, b]) is defined by (ru)(t) .= -

Z

a(t,

s)[g(~(~))~'(~) + / ( s , ~(~)) + C(s)~(~)] as,

146

III. R E L A T I O N W I T H D E G R E E T H E O R Y

G(t, s) is the Green's function of (1.15) and C E L l(a, b) is chosen such that for a.e. t e [a,b], ]g(c~(t))] < C(t), ]g(fi(t))] < C(t) and for every (t, u) e E, If(t, u)l < C(t). In particular, the problem (1.16) has at least one solution u c w2'l(a, b) such that for all t c [a, b]

~(t) < ~(t) < ~(t). Proof. The proof repeats the arguments of Theorem 1.11 together with Theorem I-6.9. I-1 A first multiplicity result that we can deduce from Theorem 1.8 is obtained when we have two pairs of lower and upper solutions. In this case, we can prove existence of a third solution.

T h e o r e m 1.13. (The Three Solutions Theorem) Let Oll, ~1 and ~2, fi2 C C([a,b]) be two pairs of W2'i-lower and upper solutions of (1.5) such that on [a, b] C~l(t) _~ ~l(t), C~l(t) ~ ~2(t), ~2(t) ~ ~2(t) and there exists to C [a, b] with

a2(to) > ~1 (to). Assume further fil and a2 are strict W2'l-upper and lower solutions. Define

A~ c [a, b] ( ~ p . B, c [a, hi) to b~ th~ ~ t of point~ ~ h ~ ~ (r~p. ~ ) i~ derivable. Let E be defined by E "= { ( t , u , v ) c [a,b] x R 2 ] min a~(t) < u < maxfii(t)} i--1,2 -- -- i=1,2

(1.17)

and p, q c [1, c~] be such that-~1 + ~1 = 1. Assume f " E ~ R satisfies LP-Carathdodory conditions and there exists N C Ll(a,b), N > 0 such that for i = 1,2 and a.e. t C Ai (resp. a.e. t c Bi) f(t,c~(t),t~(t)) > -g(t)

(resp. f(t, fli(t),fi~(t)) < Y ( t ) ) .

Assume moreover there exist ~ c C(R+,Ro+), ~ c LP(a,b) and R > 0 such that (1.9) holds and f satisfies one of the one-sided Nagumo conditions (a), (b), (c), or (d) in Theorem 1-6.1, with ~ = ot1 and ~--~2. Then the problem (1.5) has at least three solutions Ul,U2,U3 C W2'p(a, b) such that for all t c [a, b]

al (t) < ul (t) < ill(t), ol2(t) < u2(t) _~ fl2(t), ul(t) _ u3(t) _~ u2(t) and there exist t l, t2 C [a, b] with

u3(tl) > fll(tl), Zt3(t2)< Ol2(t2).

1. T H E P E R I O D I C

147

PROBLEM

Notice that the conditions u3(tl) > /31(tl) and u3(t2) < a2(t2) is a localization condition that implies t h a t u3 ~ u] and u3 ~ u2.

Proof. Define gi(t, u, v) = / ( t , cez(t), v) -- ai(t) -- Wli(t, at(t) -- u), = y(t, u, v) - u,

if u < ai(t), if u >__ai(t),

hi(t, u, v) = / ( t , Z~(t), v) - Z~(t) + w2~(t, u -/3~(t)), = f ( t , u, v) - u,

if u >/3,(t), if u
where !

w~,(t, 5) - XA, (t) m a x I f ( t a~(t) a~(t) + v) -- f ( t , a~(t), a~(t))l, Ivl<~ ' ' w2i(t, 5) = XB, (t) m a x If(t, 13~(t), ~ ( t ) + v) - f(t,/3~(t), fl~(t))l, Ivl<~

XA~ and XB~ are the characteristic functions of the sets Ai and Bi. We consider now the modified problem u" - u -- F(t, u, u'), u(a) = u(b), u'(a) = u'(b),

(1.18)

where

F(t, u, v) - gl (t, u, v), = f ( t , u, v ) - u, = h2(t, u, v),

if U ~ OL1 (t), if ax(t) < u < 132(t), if fl2(t) < u.

Let us choose k so that fll ~ 132 "J-k and a l b]) - ~ C l ([a, b]) by

k < a2, and define T 9

C 1 ([a,

(Tu)(t) =

G(t, s ) F ( s , u(s), u'(s)) ds,

where G(t, s) is the Green's function corresponding to (I-1.4).

Step 1 - Computation of d e g ( I - T, f~1,1), where Ftl,1 = {u c cl([a,b]) l Vt c [ a , b ] , a l ( t ) -

k < u(t) < ~l(t), lu'(t)l < R}.

Define the alternative modified problem u " - u = F ( t , u, u ' ) ,

u(a) = u(b), u'(a) - u'(b),

(~.19)

where

F(t, u, u') = gl (t, u, v), if u <__a l (t), = f(t,u,v)u, if Oil ( t ) < u < min{fll(t), fl2(t)}, = m a x { h i (t, u, v), h2(t, u, v)}, if min{fll(t), fl2(t)} < u.

III. R E L A T I O N W I T H D E G R E E T H E O R Y

148

Define next T" C' ([a, b]) --, C l([a, b]) by (~)(t)

-

a(t, s)P(~, ~(~), ~'(~)) a~.

For any A C [0, 1], we consider then the homotopy Tx - AT + (1 - A)T. Claim 1 9 If A c [0, 1] and u is a fixed point of T~, we have a l <_ u <_ ~2. This result follows from the usual maximum principle argument as in Step 3 of the proof of Theorem I-6.6. Claim 2 9 If A c [0, 1] and u C ~1,1 is a fixed point of 7"~, we have u
d e g ( I - T, fl1,1) = d e g ( I - T~, fl1,1) = d e g ( I - T, t2~,1) = 1. Step 2 - deg(I - T, g/2,2) - 1, where

fl2,2 - {u e cl([a,b]) l Vt e [a,b],a2(t) < u(t) < 132(t) + k, lu'(t)[ < R}. The proof of this result parallels the proof of Step 1. There exist three solutions ~

Step 3 -

(i = 1,2, 3) of (1.18) such that

OzI -- k < Ul < ~1, 01.2 < u2
and there exist t l, t2 C [a, b] with

U3(tl) > fll(tl), ?~3(t2) < ~2(t2).

The two first solutions are obtained from the fact that d e g ( I - T,~1,1) = 1

and

d e g ( I - T,~2,2) - 1.

Define ~1,2 -- {U C cl([a, b]) ] Vt ~ [a, b], C~1( t ) - k < u(t) < fl2(t)+k, [u'(t)[ < R}. We have 1 - deg(I - T, f~ 1,2) = deg(I - T, ~1,1) + deg(I - T, g/2,2) + deg(I - T, ~1,2\(~1,1 tJ ~2,2)) which implies d e g ( I - T, 9,1,2 \ (t~1,1 t2 f l 2 , 2 ) ) - - 1 and the existence of u3 E g/1,2 \ (t~1,1 tJ t~2,2) follows.

1. T H E P E R I O D I C

PROBLEM

149

Step 4 - There exist solutions u~ (i = 1, 2, 3) of (1.5) such that 0/1 ~ Ul < ~1, 0/2 < U2 ~ ~2, Ul ~ U3 ~ U2

and there exist t l, t2 c [a, b], with

u3(tl) > ~1 (tl), u3(t2) < 0/2(t2). We know that solutions u of (1.18) are such that 0/1__u_<~2

and

[[u'll~ <__R,

i.e. they are solutions of (1.5). Next, from Theorem I-6.11, we know there exist extremal solutions Umin and Umax of (1.5) in [0/1,~2]. The claim follows then with ul = Umin, u2 -- Umax and u3 = ~3. [--I Observe that in this theorem ul _ min{/~l,/~2} and u2 >_ max{a1,0/2}. E x a m p l e 1.1. In Example I-1.4, we have proved that if h C C([0,27~]) is such that []hilL1 _ 3, I/t[ < cos(6[[h[[L, ), the problem u" + sin u - h(t), ~(0) = ~ ( 2 ~ ) , ~'(0) - ~ ' ( 2 ~ ) ,

(1.20)

has at least one solution. With Theorem 1.13, it is now easy to extend this example to functions h c L 1(0, 27r) and to complement it as follows. Let w to be the solution of w" = h(t),

w(0) - w(2r

w'(0) = w'(2~), ~ - 0,

and

~x(t) = - ~ - + ~(t), ~ ( t ) = ~ + w(t),

~ (t) = - ~ + w(t), ~2(t) = ~- + w(t).

Using Propositions 1.5, 1.6 and Theorem 1.13, we find three solutions of (1.20), i.e. 0/1 < Ul < ~1, 0/2 < U2 < ~2 and 0/1 < U3 < ~2 with u3(tl) _~ /~1(tl) and u3(t2) _< 0/2(t2) for some tl and t2 C [0,2~]. Notice that Ul might be u2 - 2~ but u3 ~= ul mod 2~. Hence, this problem has at least two geometrically different solutions. E x a m p l e 1.2. Consider the equation (1 + e c o s t ) u ' -

(2esint)u' + asinu = 4esint

(1.21)

which describes the motions of a satellite in the plane of its orbit around its center of mass. Here u is twice the angle between the radius vector of the satellite and one of its principal axes of inertia lying in the plane of the orbit, t is the true anomaly, and a and e are parameters such that la[ _ 3 and0
150

III. R E L A T I O N

WITH

DEGREE

THEORY

Let us prove t h a t for all a and e e I - 1, 1[ with [e I < Jail4, the equation (1.21) has at least two 27r-periodic solutions which do not differ by a multiple of 27r. To this end we apply, in case a > 0, T h e o r e m 1.13 with

a n d o b t a i n t h r e e solutions ~1 < u l < D1, a2 < u2
and

C~1 < U3 < /~2.

Notice t h a t u3 ~ ~1 a n d c~2 ~ u3 which implies u3 ~- U l m o d 27r. O n the o t h e r h a n d , U l m i g h t be u2 - 27r so t h a t we o b t a i n only two geometrically different solutions of (1.21). A similar a r g u m e n t holds if a < 0. T h e s a m e conclusion holds for all a and e C R, with lel < e0, where e0 = 0 . 2 9 8 2 . . . is t h e positive root of 9e 4 + ( 1 7 + ~ ) 41 62 - ( 1 + ~ ) =41 0. T h e s e conditions improve t h e preceding ones if a is small. To prove this result in case a >_ 0, let

z(t) = 2

jfot ( (1--e2)3/2 - 1) ds, (l+ecoss)2

be a 27r-periodic solution of (1 + e c o s t ) z " - ( 2 e s i n t ) z ' = 4 e s i n t . If we can find a c o n s t a n t c such t h a t on [0, 27r] (t) = z ( t ) + c e

]o,

this function is a strict lower solution as follows from P r o p o s i t i o n 1.1. T h e result follows t h e n from T h e o r e m 1.13 with a l < D1 = a l + 7r < a2 = a l + 27r
[1 111

27r2 [ 2~3e2

&i2(t) dt = -5--

T h e function a l = 51 + 7r/2 works.

(i-~)~/~ - 2

71.2


1. T H E P E R I O D I C Example

PROBLEM

151

1.3. Consider the problem Ut' + CU' -- U 3 -~- 3u = p(t), =

=

(1.22)

where p 6 L ~ (0, 2u). Claim 1 - For all p 6 L~176 27r), (1.22) has at least one solution.

This result follows from Theorem I-6.9, where a l < - 1 and ~2 > 1 are constants such that a 3 - 3al < -[[p[[o~ and ~3 _ 3/32 > [[p[[oo. Claim 2 - For all p 6 L ~ ( O , 2~) such that [[PI[~ < 2, (1.22) has at least three solutions. This result follows from Theorem 1.13, where a l and ~2

are chosen as in Claim 1, 31 = - 1 and a2 = 1. The functions 31 and a2 are strict upper and lower solutions as follows from Propositions 1.6 and 1.5. Claim 3 - For all p 6 L ~ ( 0 , 2 u ) such that [[p][~ = 2, (1.22) has at least two solutions. Let ai and ~i (i = 1, 2) be chosen as in Claim 2. The

existence of solutions ui 6 [ai, ~ ] follows then from Theorem I-6.9. This result is best possible as follows from the case c ~: 0 and p constant. Here, we can give an exact count of the number of solutions. Multiplying the equation (1.22) by u' and integrating we obtain lit'IlL2 = 0. Hence, in this case, the problem admits only constant solutions. It is then easy to see that it has exactly (a) three solutions for p 6] - 2, 2[, (b) two solutions for p = - 2 or 2, (c) one solution for p 6] - c ~ , - 2 [ or p 6 ]2, +c~[. This situation is represented at figure 1.

-2

2

Fig. 1 : Diagram of the solutions of (1.22)

III. R E L A T I O N

152

WITH DEGREE

THEORY

We can generalize this example and complement Exercice I-6.4. E x e r c i s e 1.4. Consider the problem

u" + g(u)sgn(u')lu'l ~ + f ( t , u) - 8(t), u(0) = u(27r), u'(0) = u'(27r),

(1.23)

where f is an LP-CarathSodory function that satisfies Condition (A) in Proposition 1.5, g is continuous, and s c LC~(0,27r). Assume 0 < r <

2 - 1/p, lim

u----* ~ r

f (t, u) - - o c ,

lim

U---~-- r

f (t, u) - +oc,

uniformly in t and there exist real numbers v < u such that f(t, v) < f(t, u). Prove t h a t for any s e LCr 27r) with f(t, v) < s(t) < f(t, u), problem (1.23) has at least three solutions.

Hint 9 Use constant upper and lower solutions 9 ~1 < 131 = v < c~2 - u < ~2. E x e r c i s e 1.5. Adapt Theorem 1.13 to the Rayleigh and the Lidnard equations ( 1 . 1 4 ) a n d (1.16). Another way to obtain multiplicity results is to exhibit domains f~ 1 D f~ such that deg(I - T, f~ 1 ) = O and deg(I - T, Ft) = 1, which is the basic idea of the following theorem. 1.14. Let k > 0 and ~, ~ C C([a,b]) be respectively a W 2'~lower solution and a strict W2'l-upper solution of the problem (1.5) with c~ < ~ < k. Define A c [a, b] (resp. B c [a, b]) to be the set of points where c~ (resp. fl) is derivable. 1 + ~1 = 1 , and E = {(t u , v ) e [a,b] x R 2 l Let p, q C [1, oc] with -~ a(t) <_ u}. Assume f " E ~ R satisfies LP-Carathdodory conditions and there exists N c Ll(a,b), N > 0 such that for a.e. t C A (resp. for a.e. teB)

Theorem

f(t,c~(t),c~'(t)) > - Y ( t )

(resp. y(t, fl(t),Z'(t)) < Y ( t ) ) .

Let R > 0, ~ ~ C(R +, R+) and r ~ LP(a, b) satisfy

~0R ~81/q ( ~ ds

> II~bllLp(k - min

and suppose that for a.e. t C [a, b] and all (u, v) E R 2 with c~(t) < u < k, f (t, u, v) < r

1. T H E P E R I O D I C

PROBLEM

153

A s s u m e at last that f o r every solution u c W2'l(a, b) of

u" <_ y(t, u, u'), (1.24)

u(a) = u(b), u'(a) - u'(b),

with u >_ a we have u < k on [a,b]. Then the problem (1.5) has at least two solutions Ul, u2 c W2'p(a, b) such that f o r all t C [a, b]

~(t) <_ ~ ( t ) < ~(t),

~ ( t ) <_ u2(t) < k

and for some t l C [a, b],

u2(tl) > ~(tl). Proof. Define y(t,

~, ~)

-

y(t,-y~

(t,

~), ~),

and w l ( t , ~ ) = X A ( t ) m a x I f ( t c~(t) a ' ( t ) + v) -- f ( t a ( t ) a'(t)) I Ivl<~ ' ' ' ' '

where 71(t, u) - max{c~(t), u} and ~ a is the characteristic function of the set A. We consider now the modified problem

~"

-

~ = ](t,

~, ~')

- ~(t,

~) -

~,

u(a) - u(b), u'(a) - u'(b),

(1.25)

where s > 0 and w(t, u) - c~(t) + wl (t, c~(t) - u),

= u,

if u < a(t), if u __ c~(t).

Solutions of (1.25) solve the fixed point problem u(t) = ( T u ) ( t ) + s,

where

(ru)(t) and

a(t, s) is

=

a ( t , s ) [ f (t, u, u') - ~o(t, u)] ds

the Green's function corresponding to (I-1.4).

Claim 1 - Solutions u of (1.25), with s >_ O, are such that c~ <_ u < k. The claim ~ _< u follows from the argument used in the proof of Theorem I-6.6. Hence u is a solution of (1.24) and the claim u < k is an assumption. Claim 2 - Solutions u of(1.25), with s >_ O, are such that Ilu'll~ < R. From Claim 1, a __ u < k. Arguing then as in Theorem I-6.6, Claim 2 follows from Proposition I-4.8.

154

III. R E L A T I O N

WITH DEGREE

THEORY

Claim 3 - deg(I - T, 9t) = 1, where f~ = {u e Cl([a, b]) ] Yt e [a, b], a ( t ) 1 < u(t) < /3(t), ]u'(t)[ < R}. Notice that a - 1 is a lower solution of (1.25) and from Claim 1, this lower solution is strict. The claim follows then from Theorem 1.8. Claim 4 - d e g ( I - T , f~l) = 0, where ~-~1 --" {U e Cl([a,b]) [ Vt e [ a , b ] , a ( t ) 1 < u(t) < k, ]u'(t)[ < R}. Solutions u of (1.25) are such that u-Tu+s.

As further they are in gtl, it is clear that I[u-Tul[oo - s is a priori bounded. Hence for s = so large enough there is no solution of (1.25). It follows that deg(I - T, 9tl) = d e g ( I - T -

so, 9tl) = 0.

Conclusion - From Claim 3, there exists a solution Ul c Ft. By Theorem I-6.11, we can choose ux to be the minimal solution in 9t. Next, by excision, we deduce from the previous claims that

deg(I - T, 9tl \ s

- -1,

and there exists a second solution u2 C ~1 \ f~. If u2 ~ Ul, Ul and u2 are upper solution and we deduce from I-6.10 existence of a solution u3 with C~ ~ U 3 __~ min{ul, u2} which contradicts ul to be minimal. V-1 Notice t h a t we cannot use a one-sided Nagumo condition such as

/(t, u, v) >_ to obtain an a priori bound on the derivative of solutions of (1.25) which is uniform in s > 0 (see Claim 2). Observe also that solutions of (1.24) are regular upper solutions of (1.5). E x e r c i s e 1.6. Write and prove a result similar to the preceding theorem so t h a t there exist solutions us with a

and

-k
E x e r c i s e 1.7. Paraphrase Theorem 1.14 for the Rayleigh and the Li~nard equations (1.14) and (1.16). If f does not depend on the derivative u I, Theorem 1.14 reduces to the following.

1. T H E P E R I O D I C

PROBLEM

155

T h e o r e m 1.15. Let k > O, ~, ~ E C([a, b]) be respectively a W2'l-lower solution and a strict W2,1-upper solution of (1.1) such that ~ < ~ < k. Let E = {(t,u) E [a,b] • R I a(t) <_ u} and assume f 9 E ~ R is an L1-Carathdodory function. Suppose moreover that for all solutions u E W 2'1(a, b) of

u" <_ f ( t , u), u(a) - u(b), u'(a) - u'(b), with u >_ a we have u < k on [a,b]. Then the problem (1.1) has at least two solutions Ul, ~2 E W2'l(a,b) such that for all t E [a, b]

c~(t) ~ ul(t) ~ ~(t),

Ul(t) ~ lt2(t ) ~ k

and for some t l E [a, b],

u2(tl) > ~(tl). E x a m p l e 1.4. In E x a m p l e I-1.2, we have proved that if h E C([a, b]) is such t h a t for all t E [ a , b ] , - 1 / 4 _< h(t) <_ O, then a l ( t ) = - 1 and a 2 ( t ) = 0 are lower solutions and ~l(t) = - 1 / v ~ and ~2(t) = 1 / v ~ are upper solutions of u" + u 4 - u 2 - h(t), u(a) = u(b), u' (a) = u' (b). (1.26) If we reinforce the condition on h assuming - 1 / 4 < h(t) <_ 0 for all t c [a, b], we can use T h e o r e m 1.15 and prove the existence of a third solution u3 such t h a t for some tl E [a, b], u3(tl) > 1/v/2. To this end, we apply this theorem with a = 0 and ~ = 1/v/-2. The upper solution ~ is strict from Proposition 1.2. It remains to prove an a priori b o u n d on the positive upper solutions of (1.26). Observe t h a t U 4 - - U 2 ~ _1~ on R+ . Hence, if u is an upper solutions of (1.26), < h(t)

-

< h(t) + 88<

1

Moreover mint u(t) = u(to) < 1 as otherwise u"(t) < 0 on [a,b] which contradicts the periodicity of u. E x t e n d i n g u and h by periodicity and as u'(to) = 0 we have, for all t E [to, to + b - a[,

u(t) -- u(to) +

u " ( s ) ( t - s)ds<_ 1 + -~(b - a) 2,

which proves the required a priori bound. I f w e assume further - 1 / 4 < h(t) < 0 for all t E [a,b], the lower solution c~ = 0 is also strict as follows from Proposition 1.1 and we conclude, using

III. R E L A T I O N W I T H D E G R E E T H E O R Y

156

Theorem 1.13, that this problem has a fourth solution u4 such that for all

t e [~, hi, Ul ~ U n ~ U 2 .

Notice at last that if h is constant, the four solutions are easy to exhibit. 2 2.1

The The

Dirichlet Strict

Problem Lower and Upper

Solutions

Consider the Dirichlet problem

~" = / ( t , ~, ~'), u(a) = o, u(b) = O,

(2.1)

where f is an LP-Carath@odory function. It has been seen that this problem is equivalent to the fixed point problem

u(t) = (Tu)(t) := fab G(t,

s)f(s,u(s), u'(s)) ds,

(2.2)

where G(t, s) is the Green's function corresponding to

u " = f(t), u(a) = O, u(b) = O.

(2.3)

In this chapter, we consider the degree of I - T for an open set ~t of functions u that lie between the lower and upper solutions c~ and ~. However, we want to allow a and ~ to satisfy the boundary conditions. In that case, the set

ft = {u E C~([a, bl) l llu'll~ < R, Mt e ]a, b[, a(t) < u(t) < fl(t)} is not open in C~([a, b]). Similarly, if / does not depend on u', we cannot use the set = {u C C0([a, b]) I Vt c ]a, b[, a(t) < u(t) < fl(t)}, which might not be open in C0([a, b]). A way out is to impose some additional condition on the function at the boundary points t = a and t = b. For example, the set

(u e C]([a,b]) l ll~'ll~ < R, Vt e ]a, b[, c~(t) < u(t) < ~(t), D+a(a) < u'(a) < D+~(a), D_a(b) > u'(b) > D-fl(b)}

2. T H E D I R I C H L E T P R O B L E M

157

is open in Cl([a, b]). To work in a space of continuous function, we need to generalize the conditions a(t) < u(t) on ]a,b[, D+c~(a) < u'(a) and D - a ( b ) > u'(b) for such functions. Let us first introduce the following definition. D e f i n i t i o n 2.1. Let u, v C C([a, b]). We write u >.- v or v -.< u if

u(t) > v(t),

on ]a, b[,

D+u(a) > D+v(a), D-u(b) < D_v(b),

if u(a) = v(a), if u(b) = v(b),

or, which is equivalent, if there exists c > 0 such that for any t C [a, b] -

>

where e(t) := sin(Tr ~-a) t - a is the eigenfunction corresponding to the first eigenvalue of the problem u" + )~u = O,

u(a) - O, u(b) = O.

The next step is to define the concept of strict lower and upper solutions. D e f i n i t i o n s 2.2. A lower solution a of (2.1) is said strict if every solution u of (2.1) with a <_ u is such that a -< u. A n upper solution/3 of (2.1) is said strict if every solution u of (2.1) with u <_ ~ is such that u -< ~. Observe that the notion of strict lower and upper solutions we have defined with the strict order -< is based on the same idea that the corresponding notion we have introduced for the periodic case. In this last case, a lower solution a is said strict if every solution u of the periodic problem with a <_ u is such t h a t a < u, i.e. if there exists e > 0 such that for any t e [a, b] u ( t ) - a(t) >_ c 1. t-a The connection is obvious if we notice t h a t sin(Trb_--~) and 1 are the first eigenfunctions of the corresponding eigenvalue problems

u" + )~u = O,

u(a) = O, u(b) - O,

and

u" + )~u - O,

u(a) = u(b), u'(a) = u'(b).

More generally, a lower solution a is strict if for every solution u with c~ _< u, there exists c > 0 such t h a t c~(t)+ c9~1(t) <_ u(t), where 9~1 is the first eigenfunction of the corresponding eigenvalue problem. Most of the results we present for the Dirichlet problem extend to the separated boundary conditions. Such generalizations are left to the reader as exercises.

158

III. R E L A T I O N

WITH DEGREE

THEORY

A first result concerns lower and upper solutions which are C 2. P r o p o s i t i o n 2.1. Let f : [a, b] • IR2 ~ R be continuous and a E C([a, b])M C2(]a, b D be such that (a) for all t E ]a, hi, a"(t) > f ( t , c~(t), ~'(t)); (b) for to E {a, b}, either a(to) < 0 or c~(to) = 0, a e C2(]a, b[ U {to}), and a"(to) > f(to, a(to), c~'(to)). Then a is a strict C2-1ower solution of (2.1).

Proof. From the assumptions, a is a C2-1ower solution of (2.1). Let then u be a solution of (2.1) such t h a t a < u and assume, by contradiction, that for any c = l / n , there exists tn E [a, b] such that u(t

) -

<

1 sin(r~).

(2.4)

It follows there exists a subsequence of (tn)n that converges to a point to such that u(to) = a(to). If to E ]a, b[, we have u'(to) = a'(to). On the other hand, if to = a, we know t h a t a E C2([a, b D. Further, we deduce from (2.4) that -~) _ a(a) U(tn) -- u(a) < a(tn) + n1 s i n ( r t ~b-a tn -- a -tn -- a This result implies t h a t u'(a) _< a ' (a). As further u - a is m i n i m u m at t = a, we also have u'(a) >__ a'(a). Hence, u'(a) = a ' ( a ) . A similar reasoning applies if to = b so t h a t in all cases u'(to) - a'(to) = 0. At last, we obtain the contradiction 0 <_ u " ( t o ) - a " ( t o ) = f(to, a ( t o ) , a ' ( t o ) ) - a " ( t o ) < O. [3 In a similar way, we can write the following. P r o p o s i t i o n 2.2. Let f " [a, b] x R 2 --+ R be continuous and ~ E C([a, b])M C2(]a, b D be such that (a) for all t E]a,b[, ~"(t) < f(t,~(t),13'(t)); (b) for to e {a, b}, either Z(to) > 0 or Z(to) -- O, Z E C2(]a,b[U {to}) and Z"(to) < f ( t o , ~ ( t o ) , Z ' ( t o ) ) . Then ~ is a strict C2-upper solution of (2.1). Notice t h a t an upper solution/3 such that

~"(t) < f ( t , ~ ( t ) , ~ ' ( t ) )

on ]a,b[,

~(a) >_ O,

~(b) > 0

is not necessarily strict. Consider for example the problem (2.1) defined on [a, b] - [0, 1] with

f ( t , u, u') - 0, = 7u/t 2 = 7t,

if u __ 0, if 0 < u < t 3 if u > t 3.

159

2. T H E D I R I C H L E T P R O B L E M

The function u(t) = 0 is a solution and ~(t) = t 3 >__ u(t) is an upper solution which is not strict (as ~(0) = u(0) and ~'(0) = u'(0)) but satisfies ~"(t) < f ( t , ~ ( t ) , ~ ' ( t ) ) on ]0, 1], ~(0) - 0 and ~(1) > 0. In the Carath~odory case, we can use the following propositions. P r o p o s i t i o n 2.3. Let f 9 [a, b] x I~2 --~ R be an L l-Carathgodory function. A s s u m e that ~ E C([a, b]) is not a solution of (2.1) and that (a) for all to E ]a,b[, either D _ ~ ( t o ) < D+~(to) or there exist an open interval Io C [a, b] and c > 0 such that to c Io, c~ E W2'l(I0) and for a.e. t c Io, all u C [c~(t), a(t) + esin(Tr t-ab_--~)] and all v E [ a ' ( t ) - e, c~'(t)+ c], c~"(t) >_ f ( t , u, v): (b) either a(a) < 0 or a(a) = 0 and there exist ~ > 0 such that a C w 2 ' l ( a , a + c) and for a.e. t E [a, a + ~], all u c [c~(t), c~(t) + e sin(~~t-a)] and all v E -

+

>_ f ( t , u, (c)

< o

or a(b) = 0 and there exist c > 0 such that a E w 2 ' l ( b - e,b) and for a.e. t E [b - c,b], all u E [a(t), c~(t)+ e sin(~b_--~)]t-a and all v E -

+

~" (t) > f (t, u, v). Then c~ is a strict W 2 ' l - l o w e r solution of (2.1). Proof. Notice first t h a t c~ satisfies Definition 1I-2.1 and therefore is a W 2'llower solution. Let u be a solution of (2.1) such t h a t u > a. Arguing by contradiction as in Proposition 2.1 there exists a sequence (tn)n t h a t satisfies (2.4) and t h a t converges to a point to such t h a t u(to) = c~(t0) and u'(to) = ~'(to). As a is not a solution, we can find t* such that u(t*) > a(t*). Assume to < t* and define tl = ma x { t < t* I t ( t ) - a(t)}. We prove then that u(t l ) = c~(tl), u ' ( t l ) = a ' ( t l ) and fix e > 0 from the assumptions. Next, for t >_ tl near enough tl, u(t) e [a(t) , a ( t ) + e sin ( T rt-a y)]

and

u' (t) E [ a ' ( t ) - c ,

a' ( t ) + c] .

Hence, we compute u'(t) - a'(t) --

S

[f(s, u(s), u'(s)) - a ' ( s ) ] ds < 0

1

for t > t l, near enough t l which contradicts the definition of t l. A similar argument holds if to > t*. [:3

160

III. R E L A T I O N

WITH DEGREE

THEORY

Strict W2,1-upper solutions can be obtained from a similar proposition. P r o p o s i t i o n 2.4. Let f " [a, b] x R 2 -o R be an L1-Carathdodory function. Assume that 13 E C([a, b]) is not a solution of (2.1) and that (a) for all to e ]a, b[, either D - ~ ( t o ) > D+~(to) or there exist an open interval Io C [a, b] and c > 0 such that to E Io, E W2'1(I0) and for a.e. t c Io, all u E [fl(t) - esin(Tr t-ab_--~),fl(t)] and all v E [ f l ' ( t ) - c, f l ' ( t ) + e],

~"(t) <_ f ( t , u, v); (b) either ~ ( a ) > 0 or ~(a) = 0 and there exist e > 0 such that ~ C W2'~(a,a + c) and for a.e. t E [a,a + e], all u E [ ~ ( t ) - e s i n ( n ~ ) , ~ ( t ) ] and all v E [~'(t) - ~, ~'(t) + ~], ~"(t) _< S(t, ~, v); (c) e~the~ Z(b) > 0 or ~(b) = 0 and there exist c > 0 such that ~ C W 2 ' l ( b - c,b) and for a.e. t E [ b - c , b ] , a l l u E [ ~ ( t ) - e s i n ( ~ b--'~), t-~ ~(t)] and all v C [9'(t) - ~, ~ ' ( t ) + ~],

~"(t) < I(t, ~, v). Then ~ is a strict W2,1-upper solution of (2.1). In these propositions the curves u = a(t) and u - fl(t) can have angles, provided their opening is from above for a and from below for/3. Also, a a n d / 3 can be zero at the end points a and b. In this case, the second alternative imposes some second order condition near these end points but restricted to some angular region above the curve u = a(t) and below the curve u - ~(t). Notice at last that the first alternative in (b) and (c), i.e. the strict inequalities on a and fl at the end points, can be interpreted as a zero order condition and the first alternative in (a), the angular condition, as a first order one in order to prevent solutions to be tangent to a or/3. Observe also that, in the continuous case, if a satisfies the conditions of Proposition 2.1, it satisfies the conditions of Proposition 2.3. The converse, however, does not hold since Proposition 2.3 does not require strict inequalities. As in the periodic case, we can give conditions on f to ensure t h a t lower or upper solutions t h a t satisfies conditions like

a"(t) >_ f ( t , a(t), a'(t)) + A or

t3"(t) <_ f(t,13(t),fl'(t)) - A, with A > 0, are strict lower or upper solution.

2. T H E D I R I C H L E T P R O B L E M

161

2.5. Let f : [a, b] x R 2 ~ ll~ be an L1-Carathdodory function that satisfies the assumption (B) for all to E [a, b], (u0, v0) E R 2 and e > O, there exists 5 > 0 such that

Proposition

I t - t o l < 5, ]U1 --UO] < (~, U2 E [Ul,Ul -4- 5sin(Tr t-aV:---d)[,

I~)1- V01 < 5, I v 2 - Y01 < 5 :=> f ( t , U2, v2)- f ( t , u l , V l ) <

e.

Let A > 0 and a E W 2'1(a, b) be such that a"(t) >__f ( t , a ( t ) , a ' ( t ) ) + A, a(a) <_ O, a(b) <_ O. Then a is a strict lower solution of (2.1). Proof. The proof follows as for Proposition 1.5.

El

Notice t h a t Condition (B) holds if f ( t , u , v ) = g ( u , v ) + h(t) with g continuous and h E Ll(a,b) or if f(t, u, v) = g(u, v)h(t) with g continuous and h C Lee(a, b). 2.6. Let f : [ a , b ] x R 2 ~ it( be an L1-Carathdodory function that satisfies Condition (B) of Proposition 2.5. Let B > 0 and ~ C W2'l(a, b) be such that

Proposition

/~"(t) <_ f(t,~(t),/3'(t)) - B, ~(a) >_ O, ~(b) >_ O. Then fl is a strict upper solutions of (2.1). Notice t h a t it is easy to generalize these results using lower and upper solutions with corners. As for the periodic problem, we can s t u d y cases where f satisfies a Lipschitz condition in v and a one-sided Lipschitz condition in u. 2.7. Let f : [a, b] x R 2 ---, R be an L1-Carathdodory function such that for some k, l C L l(a, b; R +),

Proposition

(a) for a.e. t E [a, b], all Ul, U2 E ]~ with Ul <_ U2 and v E R, f(t, u2, v) - f(t, Ul, v) ~ k(t) (u2 - Ul); (b) for a.e. t E [a, b] and all u, Vl, v2 E R, If(t, u, v2) - f(t, u, vl)I _< l(t)]v2 - vii. Let ~ (resp. ~) be a lower (resp. upper) solution of (2.1) which is not a solution and assume

162

III. R E L A T I O N W I T H D E G R E E T H E O R Y

(c) either c~(a) < 0 (rasp. ~(a) > O) or c~(a) = 0 (rasp. ~(a) = O) and there exists an interval I0 = [a, c[C [a,b] such that a E W2'1(I0) (rasp. ~ E W2'1(Io)) and for a.e. t C Io, a"(t) > f(t,c~(t),a'(t))

(resp. ~"(t) < f ( t , ~ ( t ) , ~ ' ( t ) ) ) ;

(d) either a(b) < 0 (rasp. ~(b) > O) or a(b) = 0 (rasp. ~(b) = O) and there exists an interval Io = ]c, b] C [a,b] such that c~ E W2'1(I0) (rasp. ~ E W2'1(Io)) and for a.e. t c Io, a"(t) > f ( t , a ( t ) , a ' ( t ) )

(resp. ~"(t) < f(t, fl(t),~'(t))).

Then c~ (rasp. ~) is a strict lower solution (rasp. a strict upper solution) of (2.1). Proof. Let u be a solution of (2.1) such that u > a. We argue as in Proposition 2.1 to exhibit a point to E [a,b] such that u(to) = c~(t0) and u'(to) = c~'(t0). Next, we conclude as in Proposition 1.7. KI R e m a r k 2.1. As for periodic problem, we can see from the problem (1.7) that this proposition does not hold without the Lipschitz conditions on f. We can however assume (a) and (b) to hold only in a neighbourhood of {(t,c~(t),c~'(t)) i t e [a,b] such that c~'(t) exists} (rasp. {(t,~(t),/~'(t)) i t e [a,b] such that ~'(t)exists}). E x e r c i s e 2.1. Assume f satisfies the assumptions of Proposition 2.7 and ~,/3 >_ ~ are W2,1-1ower and upper solutions of (2.1). Assume further c~ or is not a solution and satisfies conditions (c) and (d) of Proposition 2.7. Prove then that a -
2.2

E x i s t e n c e and M u l t i p l i c i t y R e s u l t s

The main result of this section relates strict lower and upper solutions with degree theory. T h e o r e m 2.8. Let ~ and ~ E C([a, b]) be strict lower and upper solutions of the problem (2.1) such that ~ -~ ~. Define A c [a, b] (rasp. B c [a, b]) to be the set of points where c~ (rasp. ~) is derivable. Let E be defined by (1.8), Z(a) - a(b) Z(b) - a(a)} r m&x{ b-a ' b-a

and let p, q E [1, c~] be such that s~

163

2. T H E D I R I C H L E T P R O B L E M

Assume f " E ~ R satisfies LP-Carathdodory conditions and there exists N E Ll(a, b), N > 0 such that for a.e. t C A (resp. for a.e. t E B ) f ( t , a ( t ) , a ' ( t ) ) >__-N(t)

(resp. f ( t , ~ ( t ) , ~ ' ( t ) ) <_ N(t)).

Assume moreover there exist ~ E C(R +, R+), r C LP(a, b) and R > r such that

~rR 81/q ds

> II~b[Iip(mtaxfl(t ) - mina(t))l/qt

and that the .function f satisfies for a.e. t c [a, b] and all (u, v) with (t,u,v) ~ E, lY(t, ~, ~)l < r Then deg(I - T, fl) = 1, where T" C] ([a, b]) ~ C~ ([a, b]) is defined by

(ru)(t) . with

a(t, s) the

G(t, s)S(s, u(s), ~'(s)) as,

Green's function corresponding to (2.3) and

a = {u E C~([a, bl) i a ~ u -~/3, Iiu'll~ < R}.

(2.5)

In particular, the problem (2.1) has at least one solution u E W2'p(a, b) such that Proof. Increasing Y if necessary, we can assume Y(t) > If(t, u,v)] if t E [a, b], a(t) _ u <_ ~(t) and I v l _ R. Define then f(t, u, v) = max{min{f(t, ~,(t, u), v), Y ( t ) } , - Y ( t ) } , wl(t,5) = XA(t) max ]f(t,a(t),a'(t) + v ) - f(t,a(t),a'(t))[, Ivl_<~ w2(t, 5) = X B ( t ) m a x l f ( t , ~ ( t ) fl'(t) + v ) - f ( t ~(t) fl'(t))[ Ivl<_~

'

'

'

'

where 7 is defined from (I-1.3), Xa and XB are the characteristic functions of the sets A and B. We consider now the modified problem

~" = / ( t , ~, ~') - ~(t, ~), u(a) = O, u(b) = O, where ~ ( t , ~) = - ~ ( t ,

~ - ~(t)),

= ~1 (t, ~ ( t ) - u),

if u > ~(t), if a(t) <_ u <_ ~(t), if u < a ( t ) .

(2.6)

164

III. R E L A T I O N W I T H D E G R E E T H E O R Y

This problem is equivalent to the fixed point problem u = Tu,

where T" C(~([a, b]) ~ C~ ([a, b]) is defined by (Tu)(t) =

/ab

G(t, s)[f (s, u(s), u'(s)) - w(s, u(s))] ds.

Observe that T is completely continuous and we can find/~ large enough so that f~ C B(0,/~) and T(C~([a,b])) C B(O,[~). Hence we have, by the properties of the degree, deg(I - T, B(0, R)) - 1. We know that every fixed point of T is a solution of (2.6) and arguing as in Step 3 and 4 of the proof of Theorem II-2.1 we prove a < u
Yl

follows now from the properties of the degree.

E x e r c i s e 2.2. Consider Theorem 2.8 with a one-sided Nagumo condition. E x e r c i s e 2.3. Extend Theorem lem u" alu(a) blu(b)

2.8 to the separated boundary value prob= f ( t , u, u'), - a2u'(a) = Ao, + b2u'(b) = Bo,

in caseAo, B o C R , al, bs c R , a2, b 2 c R + , a 2 + a ~ > 0 a n d b ~ + b 2

2>0.

Hint 9 See [31].

In the case where f does not depend on the derivative u t, this result reduces to the following theorem.

2. T H E D I R I C H L E T P R O B L E M

165

T h e o r e m 2.9. Let a and ~ E C([a,b]) be strict W2'l-lower and upper solutions of the problem u" = f(t, u), u(a) - O, u(b) = O,

(2.7)

such that a -~ ~. Define E := {(t, u) E [a, b] x R ] a(t) _ u _< ~(t)} and assume f : E ~ R is an L1-Carathdodory function. Then, for R > 0 large enough, deg(I - T, f~) = 1, where fl is defined by (2.5), T " C~ ([a, bl) -~ C~ ([a, b]) by t~ b

(Tu)(t) " - / a

G(t, s)f(s, u(s)) ds

and G(t, s) is the Green's function corresponding to the problem (2.3). In particular, the problem (2.7) has at least one solution u E W2'l(a, b) such that In this theorem, it is essential to define T on C~ ([a, b]), since ft* := {u E Co([a, bl) l a -~ u ~ fl} is not open in the C-topology. As in Section II-4 we can also consider the case of an A-Carath6odory function f. As already observed, in that case, we look for solutions in W2'~4(a, b) and it is not meaningful to define the corresponding fixed point problem on C~ ([a, b]). This remark forces us to make stronger assumptions on strict lower and upper solutions. T h e o r e m 2.10. Let a and ~ E C([a,b]) be strict w2'S-lower and upper solutions of the problem (2.7) such that a(a) < O < ~(a),

a(b) < O < ~(b)

and

Vt E]a,b[, a(t) < ~(t).

Define E := {(t,u) E [a,b] x R l a ( t ) _< u _ B(t)} and assume f : E - , R is an A-Carath~odory function. Then deg(I - T, fl) = 1, where fl = {u e C0([a, b]) I Vt E [a, b], a(t) < u(t) < ~(t)},

166

III. R E L A T I O N W I T H D E G R E E T H E O R Y

T : Co([a, b ] ) ~ Co([a,b]) is defined by (Tu)(t) "=

G(t, s)f(s, u(s)) ds

and G(t, s) is the Green's function corresponding to the problem (2.3). In particular, the problem (2.7) has at least one solution u E W2'A(a, b) such that for all t E [a, b],

~(t) < ~(t) < ~(t). Exercise 2.4. Prove Theorem 2.10. The simplest multiplicity result that we can deduce from Theorem 2.8 is obtained when we have two pairs of strict lower and upper solutions. T h e o r e m 2.11. (The Three Solutions Theorem) Let al, ~1 and a2, [32 C C([a,b]) be two pairs of W2,1-1ower and upper solutions of (2.1) such that on [a, b] ~ (t) < ~l(t), ~l(t) < ~2(t), ~2(t) < ~2(t) and there exists to E [a, b] with

c~2(to) > Z~(to). Assume further ~1 and a2 are strict W2'l-upper and lower solutions. Define Ai C [a, b] (resp. Bi C [a, b]) to be the set of points where ai (resp. ~i) is derivable. 1 + 1 = 1 and r = Let E be defined by (1.17), p, q e [1, c~] be such that -~ max{ ~2(a)-al(b) b-a ' /32(b)-c~l(a) -6"Z_a }. Suppose f " E ~ ]R is an LP-Carathdodory function and there exists N E Ll(a,b), N > 0 such that for i = 1,2 and a.e. t E Ai (resp. a.e. t E Bi) f(t, ai(t), a~(t)) >_ - N ( t )

(resp. f(t, ~ ( t ) , ~ ( t ) ) <_ Y(t)).

Assume moreover there exist ~ C C(R+,R+), r c LP(a, b) and R > r such that R sl/q - ~ ds > [[r ) -- minal(t))l/qt holds and for a.e. t e [a, b] and all (u, v) with (t, u, v) e E, If(t, u, v)l <_ r Then the problem (2.1) has at least three solutions Ul,U2,U3 E W2'p(a, b) such that a1<__ ul -< ~l, a2 -< u2 <_ ~2, ul <_ ua <_ u2

2. T H E D I R I C H L E T P R O B L E M

167

and there exist t l, t2 c [a, b] with

u3(t ) >

<

E x e r c i s e 2.5. Prove the preceding result adapting the argument of Theorem 1.13. As in the periodic case, another way to obtain multiplicity result is to exhibit domains ~1 ~ f~ such t h a t deg(I - T, ft~) = 0

and

deg(I - T, ft) = 1.

Such a result can be obtained if we assume that we have an a priori bound on all the upper solutions which satisfies the boundary conditions. Here we assume f does not depend on the derivative u ~. T h e o r e m 2.12. Let k > 0 and ~, ~ c C([a, b]) be respectively a W2'l-lower solution and a strict W 2 ' l - u p p e r solution of (2.7) with c~ < ~ < k. Let E = {(t,u) e [a,b] • R I c~(t) <__ u} and f 9 E ~ R be an L 1Carathdodory function. A s s u m e moreover that for all solutions u C W2'l(a, b) of

u" < :(t, u), u(a) = O, u(b) - O, with u >_ ~ we have u < k on [a,b]. Then the problem (2.7) has at least two solutions ul and u2 C W2'l(a, b) such that ~ <_ ul -~ ~, ul <_ u2 and there exists to C [a, b] with

> Z(to). Proof. Consider the modified problem u" = f (t, ~1 (t, u) )

s,

-

(2.s)

u(a) = O, u(b) = O, where ~/l(t,u) = m a x { a ( t ) , u } . problem

Solutions of (2.8) solve the fixed point

u(t) = ( T u ) ( t ) where

s

(t-a)(b-t) 2

'

~ab (Tu)(t) =

G(t, s ) f ( t , ~/1 (t, u)) ds

168

III. R E L A T I O N

WITH DEGREE

THEORY

and G(t, s) is the Green's function corresponding to (II-4.2). As in the proof of Theorem 1.14, we prove that solutions of (2.8) with s _> 0 are such t h a t a _< u < k . Hence IITu - u[[~ = s (b8a)2 is a priori bounded, which implies t h a t for s - so large enough, (2.8) has no solution. Now we deduce from the equation that there exists some R > 0 so that solutions u of (2.8) with 0 < s _ so satisfy [lu'[l~ _ R. The rest of the proof follows as in Theorem 1.14. [:3 Again we can consider the case where f satisfies A-Carath~odory conditions. T h e o r e m 2.13. Let k > 0 and ~, /3 E C([a, b]) be respectively a W2'l-lower solution and a strict W2,~-upper solution of (2.7) with a < 13 < 0 and/3(b) > O. Let E - {(t,u) E [a,b] x R [ a(t) < u) and f 9 E ~ R be an ACarathdodory function. Assume moreover that for all solutions u C W 2'1(a, b) of

u" < f (t, u(a) = O, u(b) = O, with u > c~ we have u < k on [a, b]. Then the problem (2.7) has at least two solutions ua and u2 E W2'A(a, b) such that for all t E [a, b] ol(t) < ul(t) < ~(t), ~tl(t) ~ u2(t )

and there exists to E [a, b] with

2(to) > E x e r c i s e 2.6. Prove the above theorem.

3 3.1

Non

Well-ordered

Lower

and

Upper

Solutions

Introduction

We noticed already in Chapter I that the method of lower and upper solutions depends strongly on the ordering c~ <_/3. Consider for example the periodic problem u" + u = sin t, (3.1) =

=

Clearly this problem has no solution although a(t) - 1 is a lower solution and/3(t) - - 1 < a(t) is an upper solution. On the other hand, lower and

3. T H E N O N W E L L - O R D E R E D

CASE

169

upper solutions a , / ~ satisfying the reversed ordering condition/~ <__a arise naturally in situations where the corresponding problem has a solution. As a very simple example, we can consider the linear problem 2 = sin t, u" + ~u -

-

The functions a = 3 and/~ - _ 3 are lower and upper solutions such t h a t a >_/~. Notice, however, t h a t the unique solution u(t) - - 3 s i n t does not lie between the lower and the upper solution. More generally, we can consider the problem u" + Au - sin t, -

(3.2)

-

where A lies between the two first eigenvalues of the corresponding periodic problem, i.e. 0 < A < 1. It is easy to see that, in such a case, there exist constant lower and upper solutions of (3.2) which are in the reversed order. Further, the solution u of (3.2) lies between all constant upper and lower solution, 13 < u < a, if and only if 0 < A < 89 We shall see in Section 3.5 t h a t if we reinforce this condition, i.e. A < ~,r such a result holds true for any problem

u" + Au - f (t), =

=

where f is continuous. More precisely, the solution lies between every pair of lower and upper solutions with a >_ ~. This limitation is optimal since we can find, for A > 88 forcings f and lower and upper solutions a > such that the corresponding solution does not lie between 13 and a. The use of the method of lower and upper solutions is strongly related to the interaction of the nonlinearity with the eigenvalues of the corresponding linear problem. Consider the Dirichlet problem

u" + f ( t , u) = 0, u(a) - O, u(b) - O,

(3.3)

together with lower and upper solutions a and 13 such that

a"(t) + f(t, a(t)) >_ O, /3"(t) + f ( t , ~(t)) <_ O,

a(a) <_ O, a(b) <_ O, /3(a) >_ O, ~(b) >_ O.

Let us assume a _< 13 and

f (t, u) - f (t, v) > A. m

(3.4)

III. R E L A T I O N W I T H D E G R E E T H E O R Y

170

It follows that

-(~"(t)-

a"(t)) >_ f ( t , ~ ( t ) ) - f(t,a(t)) >_ A ( ~ ( t ) - a ( t ) )

and integrating twice by part, we obtain

A

<_ ~ (~(t) - a(t)) sin (~ ~_~)dt t--a

(~(t)-a(t))sin(Tr t-a~_~)dt.

If furthermore A > AI, where A1 = (b_---~)2 is the first eigenvalue of the Dirichlet problem, then this inequality implies a = ~ which is then a solution of (3.3). This observation means that it is not possible to work the method of lower and upper solutions, with a _< fl, if the nonlinearity satisfies (3.4) with A > AI, i.e. if it is "above the first eigenvalue". This is a fundamental limitation of the method. A dual conclusion holds if we assume a and/~ satisfy the reversed order a _ / 3 and

(3.5)

f (t, u) - f (t, v) < )~. 'U - - U

In this case, we deduce a(a) =/3(a) - O , a(b) =/3(b) = O,

-(c~"(t) - Z"(t)) <_/(t, ~(t)) - / ( t , Z(t)) <_ ,x(c~(t) - Z(t)) and (a'-

fl,)2 ds - A

~ab

(a - fl)2 ds

ba

:-f~

b

(a - ~3)"(a - ~) ds - A fa (c~-/3)2ds <-- O"

If A < A1, the above inequality implies a = / 3 and this function is a solution of (3.3). It follows now that it is not possible to work a method of lower and upper solutions, with /3 _< a, if the nonlinearity satisfies (3.5) with A < A1, i.e. if it is "under the first eigenvalue". It is easy to see that the same conclusions hold for other boundary value problems such as the periodic one

u" + f ( t , u) = O, ~(0) = u(2~), u'(0) = ~'(2~). Here the first eigenvalue is A1 - 0 and the method does not work with a 0 or with/3 _ a, if the nonlinearity satisfies (3.5) with A < 0. For the periodic problem we can complete this remark. If we assume the lower and upper solutions, a and ~, to be not "well-ordered", i. e. a(to) > ~(t0) for some to E [0, 2~], and

f (t, u) U--V

f (t, v) <_ 14,

3. T H E N O N W E L L - O R D E R E D

CASE

171

we can prove a >__13. Assume by contradiction a ( t l ) < 3(tl) for some tl. Extending u "- a - j3 by periodicity, we define then Sl " - i n f { t _ to [u > 0 on It, t0]} and s2 := sup{t >_ to l u > 0 on [t0, t]} < Sl + 27r. The function u satisfies the Dirichlet problem u" + q(t)u = a(t),

U(S 1) = 0, U(S2) -- 0,

where q(t) < ~1 and a >_ 0. It is easy to see that the first eigenvalue A of the problem u" + q(t)u + Au - O,

u(sy) = O, u ( s 2 ) = 0 ,

is positive. It follows then from the maximum principle (Theorem A-5.2) that u < 0 on ]Sl, s2[ which is a contradiction. The existence of lower an upper solutions a, /3 such that a >_ 3 is not sufficient to guarantee the solvability of this problem which is clear from Problem (3.1). The difficulty there comes from the interference of the nonlinearity with the second eigenvalue of the problem A2 = 1. In the theorems we present in this section, we will put assumptions so that the nonlinearity remains "below this second eigenvalue'. 3.2

A Periodic

Non-resonance

Problem

Consider the problem u" +

= f ( t , u, u'), = 0,

where f is bounded and B represents the boundary conditions. This problem can be thought of as a perturbation of the linear eigenvalue problem u" + Au = O, B ( u ( a ) , u(b), u'(a), u'(b)) = O. In this section, we shall apply the method of lower and upper solutions to such problems when A is the first eigenvalue. Our first result concerns the periodic problem u" = f ( t , u, u'), u(a) - u(b), u'(a) = u'(b)

(3.6)

with one-sided boundedness of the nonlinearity. Here the first eigenvalue is A1 = 0 .

172

III. R E L A T I O N

WITH DEGREE

THEORY

T h e o r e m 3.1. Let a and ~ C C([a, b]) be W2,1-lower and upper solutions of (3.6) such that a ~ 13. Define A C [a, b] (rasp. B C [a, b]) to be the set of points where a (rasp. 13) is derivable. Assume f 9 [a, b] x R 2 ~ R is an L1-Carathdodory function and there exists N C Ll(a, b), N > 0 such that for a.e. t C A (rasp. a.e. t c B )

f ( t , a ( t ) , a ' ( t ) ) >_ - N ( t )

(rasp. f ( t , ~ ( t ) , ~ ' ( t ) ) <_ N(t)).

Assume further that for some h E Ll(a,b) either f (t, u, v) < h(t)

on [a, b] x R 2,

f (t, u, v) >_ h(t)

on [a, b] x IR2.

or

Then there exists a solution u of (3.6) in 8 := {u E Cl([a,b])13tl,t2 c [a,b], u(tl) _ 13(tl), u(t2) __ a(t2)}.

(3.7)

Proof. For each r _> 2 (b - a) 2, we define

f~(t, ~, ~) - f(t, ~, v),

if lul < r,

= (1 + r - l u l ) f ( t , u , v )

+ ( l u l - r) u,

_ u

if r < iul < r + 1,

ifr+l
r'

and consider the problem ~" = f~(t, ~, u'), u(a) = u(b), u'(a) = u'(b).

(3.8)

Claim - There exists k > 0 such that for any r >_ 2 ( b - a) 2, solutions u of (3.8), which are in S, are such that Ilulloo _< k. Consider the case f ( t , u, v) <_ h(t) on [a, b] x R 2. Let u E S be a solution of (3.8) and let to, t l and t2 c [a, b] be such that

~'(t0) = 0,

u(t~) __ ~(t,) _> -I1~11~

~nd

u(t2) __ ~(t2) < I1~11~.

Extending u by periodicity, we can write for t C [to, to + b - a]

u'(t) - -

f

to+b-a

fr(s, u(s), u'(s)) ds >_ - I l h l l L ,

-

2I1,,11~ (b-~)-

Jt

It follows that for t C It1, t l + b - a]

u(t) = u(tl) +

u ' ( s ) d s > -I1~]1 o o -IlhllLl(b - a ) - Ilull~ 2

3. T H E N O N W E L L - O R D E R E D

CASE

173

and for t E It2 - b + a, t2]

=

(t2)

u'(s) ds < I1~11~ + IlhllL 1 (b- a) + I1~11~ 2

-

Hence, we have

Iiull

2(11 11 + IIz ll + IlhllL,(b- a))=: k.

A similar a r g u m e n t holds if f(t, u, v) >_ h(t).

Conclusion - Consider the problem (3.8), with r > max{k, 2 ( b - a ) 2 } . It is easy to see t h a t a l : - r - 2 and/32 - r + 2 are lower and upper solutions. Assume ~ is not a strict upper solution. There exists then a solution u of (3.8) such t h a t u <_ /3 and for some tl C [a,b], u ( t l ) = ~(tl). As further c~ ~ / 3 , there exists t2 E [a, b] such t h a t a(t2) > fl(t2). It follows t h a t c~(t2) > u(t2), u C S, and we deduce from the claim t h a t Ilul]~ < k. Hence, u is a solution of (3.6) in S. We come to the same conclusion if c~ is not a strict lower solution. Suppose now that/31 - fl and a2 - c~ are strict upper and lower solutions. We deduce then from T h e o r e m 1.13 the existence of three solutions of (3.8) one of them, u, being such t h a t for some tl, t2 e [a, b], u(tl) > fl(tl) and u(t2) < a(t2). Hence, u E S and from the claim Ilull~ < k. This implies t h a t u solves (3.6) and proves the theorem. !-] Example

3.1. The problem

u" = f(u),

u ( - 1 ) -- u(1), u ' ( - 1 ) -- u'(1),

where

f(u)-

121ul 1/2, if u _ 4, =28-u, if4
is such t h a t ~(t) : t 4 and c~(t) - 28 > /3(t) are respectively upper and lower solutions. The solutions in the corresponding set S are the constant functions 0 and 28 but are not in the interior of S. A possible generalization concerns the use of several lower and upper solutions. E x e r c i s e 3.1. Let c~i e C([a,b]) (i : 1 , . . . , n ) and /3j E C([a,b]) (j 1 , . . . , m) be respectively W2,1-1ower and upper solutions of (3.6) such t h a t c ~ ' - m a x c~i ~ 1 3 : = min j3j. l
l<_j<__m

174

III. R E L A T I O N

WITH DEGREE

THEORY

Assume f " [a, b] x ]R2 --, IR is an Carath6odory function such that for some h C L l(a, b) either

f(t, u, v) <_ h(t) on [a, b] x R 2, or

f(t, u, v) >_ h(t) on [a,b] x R 2. Prove there exists a solution u C S of (3.6), where S is defined in (3.7). In a similar way, we can consider the Li6nard equation

~" + g(~)u' + y(t, ~) = o,

(3.9)

u(a) = u(b), u'(a) = u'(b).

T h e o r e m 3.2. Let c~ and fl C C([a, b]) be W2'l-lower and upper solutions of (3.9) such that c~ y~ ft. Assume g e C(R) and f " [a, b] x R ---, IR is an L1-Carathdodory function such that for some h E L l(a, b) either

f (t, u) < h(t)

on [a, b] x R,

f (t, u) > h(t)

on [a, b] x R.

or

Then there exists a solution u E S of (3.9), where S is defined in (3.7). Proof. For each r > 8 ( b - a) 2, we define f~(t, u) -- f (t, u), = (1 + r = -~

if ]u] < r,

lul)f(t, u) - ( l u l - r)~, if r < lul < r + 1, if r + 1 <

lul,

and consider the problem

~" + g(u)u' +/~(t, u) = o, u(a) = u(b), u'(a) - u'(b).

(3.10)

Claim - There exists k > 0 such that for any r >_ 8 ( b - a) 2, solutions u of (3.10), which are in S, are such that Ilull~ < k. Consider the case f (t, u) >_ h(t) on [a, b] • R. Let u E $ be a solution of (3.10). Integrating (3.10), or multiplying this equation by u and integrating, we obtain respectively

f b f~(t, ~(t)) dt = o

and

f bu'2(t) dt = f bfr(t, u(t))u(t)

dt.

3. T H E N O N W E L L - O R D E R E D

175

CASE

It follows t h a t

b [{u'][~2 = fa fr(t, u(t))(u(t) - [[u]{oo)dt <_ 2([[h[lL~ + Ilu~~176 (b - a))[lul[~,

i.e.

Ilull~ Ilu'll L~ <- (211hllLll]ull~) '/2 + 2(b-a)'/~" Define now t l and t2 E [a, b] to be such t h a t

u(tl) ~ ~(tl) ~_ --[{~[[oo

u(t2) ~_ a(t2) _~ [[a[[oo.

and

E x t e n d i n g u by periodicity, we can write for t c It l, t l - [ - b - a]

u(t) ----u ( t l ) +

u'(s) ds >_ -{{13[[oo - [lu'llL2( b -- a) 1/2 1

> -IIZ[{ o o - (2(b - a)llhll L, I]ullc r )1/2 __ [lulloo 2

_

and for t C It2 - b + a, t2]

u(t) = u(t2) -

u'(s) ds < ]Jail _

c~

+ (2(b - a)l]h[]L IlU]loo) 1/2 -~- Ilull~176 1

2

"

Hence, we obtain for some k > 0

I1 11 < k. A similar a r g u m e n t holds if f ( t , u) <_ h(t).

Conclusion - The rest of the proof repeats the argument used in T h e o r e m 3.1. 3.3

Kl Interaction

with

the

First

FuSfk Curve

T h e reason of imposing in T h e o r e m 3.1 a boundedness assumption, such as f ( t , u, v) < h(t), is to make sure the nonlinearity does not interfere with the second eigenvalue of the corresponding linear problem. An alternative way to prevent such an interference is to assume some a s y m p t o t i c control on the quotient f(t, u, v ) / u as [u{ goes to infinity. We can generalize further assuming different behaviours of this quotient as u goes to plus or minus infinity. The following t h e o r e m concerns such a generalization. To simplify, we consider the problem

u" + f (t, u) = O, u(a) - u(b), u'(a) - u' (b).

(3.11)

III. R E L A T I O N W I T H D E G R E E T H E O R Y

176

T h e o r e m 3.3. Let a and ~ E C([a, b]) be W2'l-lower and upper solutions of (3.11) such that a ~ ~. Let f 9 [a, b] x R --~ R be an L1-Carathdodory function such that for some functions a+ <_ O, b+ >_ 0 in Ll(a,b), a+ (t) < lim inf f (t, u) <_ lim sup f (t, u) <_ b+ (t), u-*=kc~

U

u~-t-co

(3.12)

U

uniformly in t C [a, b]. A s s u m e further that for any p, q E Ll(a,b), with a+ < p << b+ and a_ < q < b_, the nontrivial solutions of u" + p(t)u + - q ( t ) u - - O, u(a) = u(b), u'(a) = u'(b),

(3.13)

where u + (t) = max{u(t), 0} and u - ( t ) = m a x { - u ( t ) , 0}, do not have zeros. Then the problem (3.11) has at least one solution u C $, where S is defined in (3.7). Proof. Step 1 - Claim 9 There exists c > 0 so that for any p, q c L1 (a, b), with a+ - ~ <_ p <_ b+ + c and a_ - e <_ q <_ b_ + c, the nontrivial solutions of (3.13) do not have zeros. If the claim were wrong, there would exist sequences (Pn)n, (qn)n C i l ( a , b ) , (tn)n C [a,b] and (Un)n C w 2 ' i ( a , b ) so that a+ - 1 / n <_ Pn <_ b+ + 1/n, a_ - 1 / n <_ qn ~_ b_ + 1 / n and Un is a solution of (3.13) (with p - Pn and q - qn) such that IlUnllCl - 1 and un(tn) = 0. Going to subsequences we can assume, using the DunfordPettis Theorem (see [43]), Pn ~

P, an ~

q in

Ll(a,b),

un ~ u in C'([a,b]),

t n --'+ to.

It follows that a+ <_ p < b+, a_ _ q < b_, u is a solution of (3.13) and u(t0) - 0 which contradicts the assumption. Step 2 - The modified problem.

Let us choose R > 0 large enough so that

a+ - c <_ g+(t, u) - f (t, u) <_ b+ + ~, u

for u >_ R,

a_ - e < g _ ( t , u ) = f ( t , u ) <_ b_ + e, for u < - R u and extend these functions on [a, b] x R so that these inequalities remain valid. As f is L1-Carath~odory, there exists k c L l(a, b) such that f ( t , u) = g+(t, u)u + - g_(t, u ) u - + h(t, u) and Ih(t, u)l _< k(t).

3. T H E N O N W E L L - O R D E R E D

CASE

177

Next, for each r >_ 1, we define g~ (t, u) - g+ (t, u), =(l+r-lul)g+(t,u), = 0,

if lu] < r, ifr_
hr(t, u) = h(t, u), = (1 + r -

if lu[ < r, if r _ [ul < r + 1, if r + 1 _< lul

lul)h(t, u),

0,

and consider the modified problem u" + g+ (t, u)u + - gr (t, u ) u - + hr(t, u) = O, u(a) - u(b), u' (a) - u'(b).

(3.14)

Step 3 - Claim 9 There exists k > 0 such that, for any r > k, solutions u of (3.14), which are in 8, are such that [[ul]~ < k. Assume by contradiction, there exist sequences (rn)n and (Un)n C S, where rn >_ n and Un is a solution of (3.14) with r = r n such that IlUnllcx~ ~ n. As Un C ~, there exist sequences (tln)n and (t2n)n C [a,b] such that Un(tln) ~_ ~(tln) and Un(t2n) <_ oe(t2n). Consider now the functions vn = un/llunlloo which solve the problems II

+

(t,

_

(t,

hr u ( t , u n )

+ I

__ O,

-

I

vn(a) = vn(b), Vn(a ) = vn(b). Going to subsequence, we can assume as above grT(',Un)

--~ p ,

grn(',Un)

--~

q in Ll(a,b),

Vn ~ v in Cl([a,b]),

tin ---+tl,

hr.(t,Un)llunll~-~ 0 in L l(a, b), t2n --+ t2.

It follows that v satisfies (3.13) and by assumption has no zeros. Hence, we come to a contradiction since v(tl) >_ 0 and v(t2) _< 0 which implies v has a zero. Conclusion - We deduce from Theorem 3.1 that problem (3.14) with r > max{llall~, [[~lloo, k} has a solution u C S and conclude from Step 3. K] In this theorem we control asymptotically the nonlinearity using the functions a+, b+. Next we impose some admissibility condition on the box [a+, b+] x [a_, b_] which is to assume that for any (p, q) e [a+, b+] x [a_, b_], the nontrivial solutions of problem (3.13) do not have zeros. Such a condition implies that the nonlinearity does not interfere with the second eigenvalue A2 of the periodic problem, i.e. (A2, A2) r [a+, b+] x [a_, b_].

III. R E L A T I O N W I T H D E G R E E T H E O R Y

178

This remark can be completed considering the second curve of the Fu~ik spectrum. The Fu(:fk spectrum is the set ~- of points (#, ~) c R 2 such that the problem u"+#u +-vu=0, u(a) = u(b), u'(a) = u'(b), has nontrivial solutions. From explicit computations of the solution it is easy to see that (x)

n=l

where

J=x = {(~,o) 1~ c n~} u {(o, ~ ) 1 ~ c R} and

~. = ((~,~) I ~

bDa + ~1 D- ~(,_,)),

~-

2,3,...

The following proposition relates the admissibility of the box [a+, b+] x [a_, b_] with the Fu~ik spectrum. Proposition

3.4. Let (#, u) c Y=2 and p, q c LX(a, b). A s s u m e that

p(t) < # ,

q(t) < u,

for a.e. t E [a,b]

and for some set I C [a, b] of positive measure p(t) < # ,

q(t) < ~,

for a.e. t e l .

Then the nontrivial solutions of problem (3.13) have no zeros. Proof. Assume there exists a nontrivial solution u which has a zero. Extend u by periodicity and let to and t I be consecutive zeros such that u is positive on ]to,t:[. Define v(t) = sin(vrfi(t - to)) and compute (uv' - vu')lttl - ~tl 1(p(t) - #)u(t)v(t) dt. Iftl-to<

~",

we come to a contradiction

0 < -v(t:)u'(tl)

< O.

Hence tl - to > ~ " and we only have equality in case p(t) = # on [to, tl]. Similarly, we prove the distance between two consecutive zeros tl and t2 with u negative on ]tl,t2[ is such that t 2 - tl > ~ with equality if and only if q(t) = v on [tl, t2]. It follows that

b-a>t2-to>

7~

~+-~

~"

=b-a.

3. T H E N O N W E L L - O R D E R E D

CASE

179

These inequalities imply that t 2 - t o - b - a , p(t) = # on [to, t~] and q(t) = v on [tl,t2] which contradicts the assumptions. E] R e m a r k Assumption (3.12) can be replaced by the following: For every ~ > 0, there is % E L ~(a, b) such that, for a.e. t E [a, b] and every uER,

f (t, u) = g+(t, u)u + - g_(t, u ) u - + h(t, u), with g+ and h, L l-Carath~odory functions satisfying a+ (t) - e <_ g+ (t, u) _< b+ (t) + e and

Lh(t, u)[ <__%(t). There is a vast literature concerning system which are asymptotic to positively homogeneous problem as in the above theorems. Most of these results can be adapted to a situation where there exist non well-ordered lower and upper solutions. We present here some extensions as exercises. E x e r c i s e 3.2. Let of (3.11) such that Let f 9 [a, b] • some functions a+

c~ and ~ E C([a, b]) be W2'l-lower and upper solutions ~ ~ ft. R ~ R be an L1-Carathdodory function such that for <_ O, b_ >_ 0 in Ll(a, b),

a_ (t) <_ liminf f (t, u) <_ limsup f (t, u) <_ b_ (t), U----*--O0

U

u----*--CX~

U

a+(t) <_ liminf f(t, u) , u---* + c ~

U

uniformly in t E [a, b]. Assume further that for any q c L l(a, b), with q <_ b_ and any { E [a, b[, the problem u" + q(t)u = O, u(t--) = O, u({ + b - a) = O, has only the trivial solution (where q(t) = q ( t - ( b - a ) ) f o r t E ]b, { + b - a ] ) . Prove then that the problem (3.11) has at least one solution u E S, where $ is defined by (3.7). Hint" See [95].

180

III. R E L A T I O N W I T H D E G R E E T H E O R Y

Consider the problem u" + g(u) = h(t), u(a) - u(b), u'(a) - u'(b).

(3.15)

For such a problem, we can obtain similar result using conditions on the potential G(u) - f o g(s)ds. In this case lower bounds on g(~) are not U necessary anymore but we cannot prove the solution is in S. The following problem concerns such a result. E x e r c i s e 3.3. Let a and ~ E C([a, b]) be W2'l-lower and upper solutions of (3.15). Let h C L~176 b), g E C(R) be such that for some (#, v) e ~2 lim sup g(u) < # and lim sup g(u) <_ v, u---~+oo

liminf u---~ + c ~

U

2G( ) U2

u--*-oo

<#

or liminf

U

2G( )

u---* - o o

u2


Prove then that the problem (3.15) has at least one solution. Hint 9 In case a > ~ see [147]. The proof without this condition can be built using the ideas in [79].

3.4

Multiplicity

Results

Existence of several solutions are studied in Sections 1.2 and 2.2. Some additional results can be obtained using non well-ordered lower and upper solutions. The following result complements Theorem 3.3. T h e o r e m 3.5. Let al and a2 C C([a, b]) be w2'l-lower solutions of (3.11) and/3 e C([a,b]) be a strict W2,1-upper solution such that al < ~ and Assume f " [a, b] • IR ---, R is an L1-Carathdodory function such that for some function a+ C L1 (a, b) ,

liminf f ( t , u) >_ a+(t), u--*+oo

(3.16)

U

uniformly in t E [a, b]. Then the problem (3.11) has at least two solutions Ul and u2 such that a l < Ul < ~,

u2 c S

and

ul < u2,

where S is defined in (3.7) with a - max{at, a2}.

3. T H E N O N W E L L - O R D E R E D

CASE

181

Proof. For any r > max{ll~lll~, 11~211~, I1~11~}, we consider the modified problem u" + f~(t, u) = O, u(a) = u(b), u' (a) = u' (b),

(3.17)

where f r ( t , U) :

f ( t , OL1 ( t ) ) -- U -~- Oz1 ( t ) ,

=/(t, = (1 + r -~ 0,

u ) f ( t , u),

if u < a l (t), ifal(t)
Claim 1 - Every solution of (3.17) is such that u >_ al. This result follows from the usual maximum principle argument as it is used, for example, in the proof of Theorem I-3.1. Claim 2 - There exists k > Oso that for a n y r > ma~{ll~lli~, 11~211~, IIZ~ll~} and any solution u e S of (3.17), we have Ilulloo < k. As u e S, there exist to and t l such that u(to)--

min u(t)<_ u(tl)__ a ( t l ) < t~[a,b]

I1~11~.

Further, we deduce from the asymptotic character of f that there exists 5+ and h C Ll(a, b) such that for a.e. t E [a, b] and all u >_ al(t), fr(t, u) >_ - [ a + ( t ) u + h(t)].

Hence, we have for t C [to, to + b - a], u(t) = u(to) -

fr(s, u(s))(t - s ) d s

<_ Ilall~ + IlhlLL~ (b - a) + (b - a)

5+ (s)u(s) ds

and the claim follows from Gronwall's Lemma. C o n c l u s i o n - Consider the problem (3.17) for

r > max{llOllllc~, Ilce211c~, IIr k), where k is given in Claim 2. It follows from Theorem I-3.4 that there exists a solution Ul of (3.17) which is minimal in [al,~]. Hence, a l ( t ) _< ul(t) _< r which implies this solution solves also (3.11). Next, we can apply Exercise 3.1 to obtain a solution u2 E S of (3.17). Prom Claim 1, u2 >_ a l , and from Claim 2, u2 _ k < r. Therefore u2 is a solution of (3.11).

182

III. R E L A T I O N W I T H D E G R E E T H E O R Y

At last, notice that if u2 ~ Ul, Ul and u2 are upper solutions of (3.17) and we deduce from Theorem I-3.2 the existence of a solution, u3 with O/1 ~ U3 ~ min{ul, u2} which contradicts ul to be minimal. !"1 Assumptions such as (a) f ( t , u) is nonnegative or (b) f ( t , 0) _ 0 and f ( t , Ul) - f ( t , U2) _~ - M ( u l - u2), if Ul _> U2 ~ 0, imply the asymptotic behaviour (3.16) and we can choose a l - 0. Hence, existence of nonnegative upper and lower solutions implies existence of two nonnegative solutions one of them being nontrivial. Observe that the existence of the third solution of u" + u 4 - u 2 = h(t), u(a) = u(b), u'(a) - u'(b), i f - 1 / 4 < h(t) N 0 obtained in Example 1.4 can be deduced from Theorem 3.5 in a quite obvious way. We can extend Theorem 3.5 to the Li~nard equation (3.9). T h e o r e m 3.6. Let C~l and ~2 e C([a, b]) be w2'l-lower solutions of (3.9) and ~ C C([a,b]) be a strict W2,1-upper solution such that ~1 ~_ ~ and

~2~. Assume g e C(R) and f " In, b] • ]R ---. IR is an L1-Carathdodory function such that for some function h C L l(a, b), lim inf f (t, u) >_ h(t), ~t ---* -~- OO

uniformly in t C [a, b]. Then the problem (3.9) has at least two solutions Ul and u2 such that (~1 ~_Ul < ~,

u2 c S

and

Ul ~_u2,

where S is defined in (3.7) with a = max{a1, a2}. Proof. The proof is exactly the same as for Theorem 3.5 except Claim 2 which repeats the argument used to prove the claim in Theorem 3.2. W1 E x e r c i s e 3.4. Consider the Li~nard equation

u" + f (u)u' + g(u) + h(t, u, u') = O, u(a) - u(b), u'(a) = u'(b),

(3.18)

where f, g : R --~ R are continuous functions and h : [a, b] • R 2 ~ R is continuous and bounded. Prove that if inf~ [f(s)l > 0, the existence of a lower and an upper solution implies the solvability of (3.18).

Hint : See [235].

3. T H E N O N W E L L - O R D E R E D 3.5

Lower

and

Upper

Solutions

in the

183

CASE Reversed

Order

In case a > / ~ on [a,b], we can ask whether the solution of (3.11) satisfies the "strong localization" /3 <_ u <_ c~ on [a, b] as in the well ordered case. This is not always the case under the assumptions of T h e o r e m 3.3 as shown by problem (3.2). B u t it is true if f (t, u) - f (t, v) < u-v -

()2 7r

b

a

"

This analysis is the content of this section. T h e o r e m 3.7. Let a a n d / 3 C W 2 ' l ( a , b ) be lower of (3.11) such that a > /~, E = {(t,u) e [a,b] x R and f 9 E ~ IR be an L1-Carathdodory function. b-a) ~] such that for a.e . t e [a,b] and all u e k e 10 , (~--f(t,a(t))

and upper solutions [ /~(t) < u < a ( t ) } A s s u m e there exists [/~(t) , c~(t)] ,

- k s ( t ) < f ( t , u) - ku < f ( t , / ~ ( t ) ) - k/3(t).

Then the problem (3.11) has at least one solution u c w Z ' l ( a , b ) such that f o r all t E [a, b]

~(t) <__u(t) <_ ,~(t). Proof. We can assume t h a t a and fl are not solutions; otherwise, the result is trivial. Let us consider the modified problem u" + f ( t , u) = 0, u(a) - u(b), u'(a) = u'(b),

(3.19)

where

?(t, ~) - f(t, ~(t)) + k ( u - ~(t)), = f ( t , u),

= f(t, ~(t)) + k ( u - / 3 ( t ) ) ,

if ~ >_ ~ ( t ) ,

if/~(t) ~ u ~ ~(t), if u _
By T h e o r e m 3.3 and the R e m a r k after Proposition 3.4, problem (3.19) has a solution u c S. It remains to prove the announced localization. Let us prove u _< c~. Observe t h a t v = ~ - u satisfies v" + kv = (~" + f ( t , ~)) + ( f ( t , u) - ku) - ( f ( t , ~) - k~) =: h(t), v(a) - v(b) = O, v'(a) - v'(b) >__0 where h E L t ( a , b ) is such t h a t h _> 0, h ~ 0. By Corollary A-6.3, v >_ 0, i.e. u < a. In the same way, we prove u >_/~. [::]

184

III. R E L A T I O N

WITH DEGREE

THEORY

R e m a r k Observe that the limitation on k in Theorem 3.7 is optimal if we want the strong localization. Consider the problem u" + k u = f ( t ) , u ( a ) - u(b), u'(a) = u'(b),

with k > (b---~ 9 Let c, d be such t h a t ( lr )2 < C2 < k < d 2

and

c7r + ~ = b - a ,

and define f(t)

71"

:= k -c J sin c ( t - a)

iftE[a,a+c],

"9- - k-d2d s i n d ( t - - d ~+ - d ~ - a ) ,

i f t E ] a + ~ , " b] .

This problem has/3 = 0 and a = { max{ kc-~2' d2-k d ) as upper and lower solutions satisfying the reversed ordering/3 _ a. But its solution is given by u ( t ) : - ~1 sin c(t - a), 9-- d ! s i n d ( t - c + - d

~

if t E [a, a + ~-] " -

a ),

iftE]a+-g,

" b]

which is not one-signed, i.e. it is not between the upper solution/3 = 0 and the lower solution. E x e r c i s e 3.5. Prove t h a t under the assumptions of Theorem 3.7, if for some k E ]0, ( ~ ~ )2], for a.e. t E [a, b], and for all u, v E [~(t), a(t)], u # v f (t, u) - f (t, v) < k, U--V

then the solutions of (3.11) in [~,a] are ordered. solutions Umin, Umax E W 2 ' l ( a , b) such that

Prove also there are

~ Umin ~_~Umax ~_~6~,

and any other solution u of (3.11) such t h a t / 3 _ u _ a satisfies Umin ~ U ~ Umax.

3. T H E N O N W E L L - O R D E R E D 3.6

T h e Dirichlet

185

CASE

Problem

In this Section 3, we have worked out the periodic boundary value problem. These results can be extended to deal with other boundary conditions. We present here such an extension to the Dirichlet problem. The theorems we state are the counterpart of previously obtained results and their proof repeats previous arguments. Our first concern is a non-resonance result for the problem 7r )2 it/ u" + (~-a u = f ( t , u, ), ~(a) = 0, ~(b) = 0,

where (

(3.20)

)2 is the first eigenvalue of the corresponding eigenvalue problem u" + Au = 0, u(a) = 0, u(b) = 0.

(3.21)

Theorem 3.1 can then be modified as follows. T h e o r e m 3.8. A s s u m e c~ and [3 E C([a,b]) are W2'l-lower and upper solutions of (3.20) such that a ~ /3. Suppose f " [a,b] x R 2 - , R is a Carathdodory function such that for some h E Ll(a, b) If(t, u, v)l < h(t),

on [a, b] • R 2.

Then there exist a solution u of (3.20) in $ := adhcol {u e Clo([a, b])l ?tl,t2 e [a,b], u(ti) > 13(tl), u ( t 2 ) < ol(t2)}. (3.22) Proof. For each r >_ 1, we define

/~(t, u, v) = / ( t , u, v),

if lul < r,

= (1 + r - l u l ) f ( t ,

u, v) + ( l u l - r ) ~ ,

if r _< lul < r + 1,

if r + 1 < lul,

_ u_

and consider the problem u" + (b-~) 2u -- f~(t, u, u'), ~(a) = 0, ~(b) = 0.

(3.23)

t-~ Claim 1 - For some C > O, (~(t) -~ C sin (Tr~_~) t-a and Z(t) ~- - C sin(~ ~-a)" Let r > max{ll~l[~ , [I/~1]~}. Define w to be the solution of w"+(')2w=

~ V,

w(a)=r+2,

w(b)=r+2,

e(t) = sin(it t-a ), choose k > 0 such that ke + w > r + 2 on [a, b] and define ~2 = - a l = ke + w. Observe that/~2 and a l are upper and lower solutions of (3.23). Hence the claim follows from Remark II-2.1.

III. R E L A T I O N W I T H D E G R E E T H E O R Y

186

Claim 2 - There exists k > 0 such that for any r > k, solutions u of (3.23), which are in S, are such that I[u[]oo < k. Assume by contradiction there exist sequences (rn)n and (un)n C S, where rn >_ n and un is a solution of (3.23) with r - rn, such that Ilu,~lloo > n. Let us write

~ ( t ) = e~(t) + ~ ( t ) . ~a b

(tn(t)e(t) dt = 0. We deduce from Proposition A-4.5 that for

where

some K > 0 b

-

~-a) un(t)le(t)dt

< K

Lf~,~(t un(t) u'~(t))[e(t)dt < K

/a

(h(t) + ~ ) e ( t ) d t .

Hence, for some C1 > 0 and n large enough, we can write

II~nil

(3.24)

< c1(1 + ~ )

Let to be such that u-!n (to) = 0. It follows that

7r )2 Ifin(s)[ + [frn (s, un(s), Un(S))])ds[ / [fi'(t)[ < ] ~i ((~-:-~ < C2(1 + I"n__A) u 7"n for some (?2 > 0. Assume now that for some subsequence lim rink - +co. This assumpk---*c<~ tion implies

~(t)

> (~n~ -C311e~llc')~(t) > ( ~

-C4(1 +

~))~(t)

for some C3 and C4 > 0. Hence, for nk large enough, unk ~- a which contradicts Unk E $. In a similar way, we prove that no subsequence of (rn)n goes to - c o . It follows that r~ and also, using (3.24), Ilunll~ are bounded which contradicts the assumption.

C o n c l u s i o n - Let r > m~x{ll~ll~, II~lloo,k}. As in Claim 1, we define f12 - - a l -- ke + w and conclude as in the proof of Theorem 3.1. [2] Next we consider the problem of interaction with Fu6fk spectrum. A result similar to Theorem 3.3 holds for the Dirichlet problem

u" u(a)

+ =

f ( t , u) = 0, O, u(b) = O.

(3.25)

3. T H E N O N W E L L - O R D E R E D

CASE

187

T h e o r e m 3.9. Let ~, /3 e C([a, b]) be W 2 ' l - l o w e r and upper solutions of (3.25) such that a ~ Z. Let f " [a, b] • R ~ R be an L1-Carathdodory function such that for some functions a+ _< ( ~---~---b_a), b:l: > 7r )2 in Ll(a, b),

a+(t) <_ l i m i n f / ( t , u) _< limsup/(t, u---* =l=c~

U

u---* =l=c~

u) < b+(t), U

uniformly in t E [a, b]. Assume further that for any p, q E Ll(a,b), with a+ < p <_ b+ and a_ < q < b_, the nontrivial solutions of u" + p(t)u + - q ( t ) u - = O, u(a) = O, u(b) = O,

(3.26)

where u + ( t ) = max{u(t), 0} and u - ( t ) = m a x { - u ( t ) , 0}, do not have interior zeros. Then the problem (3.25) has at least one solution u c S, where S is defined by (3.22).

E x e r c i s e 3.6. Prove the above theorem. The condition on p, q, can be related with the Fu5l"k spectrum as in Proposition 3.4. Roughly speaking, the nontrivial solutions of the problem (3.26) do not have zeros if the box [a+, b+] • [a_, b_] lies under the second Fu~fk curve J=2 of the Dirichlet problem. At last, we can write multiplicity theorems for (3.25) which paraphrase results worked out in Section 3.4. We state here a situation with two lower solutions and a strict upper one. T h e o r e m 3.10. Let c~t, ~2 E C([a,b]) be W2,1-lower solutions of (3.25) and ~ G C([a,b]) be a strict W2,t-upper solution such that as <_ ~ and (~2 ~ ~. Assume f " [a, b] • R ---, R is an L t-Carathdodory function such that for some function a+ c Ll(a,b), (3.16) holds uniformly in t G [a,b]. Then the problem (3.25) has at least two solutions u t and u2 such that (~1 <_ Ul -< ~,

u2 c S

and

ut <_ u2,

where $ is defined in (3.22) with ~ - max{c~l, ~2}.

E x e r c i s e 3.7. Prove Theorem 3.10.

188

III. R E L A T I O N W I T H D E G R E E T H E O R Y

Hint 9 Adapt the proof of Theorem 3.5. In Claim 2, notice that u implies the existence of sequences (Un)n and (tn)n such that Un ~ Cl([a,b]) and un(t~) < a(t~) _< Ce(t~). Here e(t) = sin(Trt-a~--~)and such that a <_ Ce as in Remark II-2.1. We deduce then the existence such that u(to) <_ Ca(to) and u ' ( t o ) = Ce'(to) and compute u(t) <_ Ca(to) + Ce'(to)(t - to) -

E u C of

,S in is to

fr(s, u(s))(t - s)ds.

The proof follows then as in Theorem 3.5. In very much the same way, we can state a dual result with two upper solutions and a strict lower one. T h e o r e m 3.11. Let a e C([a, b]) be a strict W2'l-lower solution of (3.25) and ~1,132 E C([a, b]) be W2'l-upper solutions such that a < ~2 and a ~ ~1. A s s u m e f 9 [a, b] x R ~ R is an L1-Carathdodory function such that for some function a_ c L1 (a, b) , liminf u--+--~

f(t, u) >_ a_(t), U

uniformly in t C [a, b]. Then the problem (3.25) has at least two solutions U l and u2 such that ul E $,

a <_ u2 -~ ~2

and

Ul <_ u2,

where S is defined in (3.22) with ~ - min{~l, ~2 }.

E x e r c i s e 3.8. Extend the results of this section to the separated boundary value problem u" = f (t, u),

alu(a)- a2?zt(a) --0, blu(b) + b2u'(b) = O,

in c a s e a l , bl e R , a2, b2e]R + , a 2 + a 2 > 0 a n d b

2+b22>0.