Orlicz type category results for differential equations in Banach spaces

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIII (1983) ROCZNIKI POLSKIEGO TOWARZYSTWA MATE MATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXIII (1983)

F. S.

(Firenze) and J.

(Krakôw)

Orlicz type category results for differential equations in Banach spaces

Dedicated to Prof. W Orlicz Let £ be a separable real Banach space with norm ||. Let / = [0, 1].

We denote by B(x0, r) the closed ball in a metric space SC, with center at x 0 and radius r > 0. When SC = E and x 0 = 0 wé put for brevity В — B(0, r).

If 'zlf z2 are continuous functions defined on [a, b] with values in E, we set Iki — = sup {\z1( t ) - z 2(t)\\ t e [ a , h]}.

We say that a map /: I x B -►£ is Carathéodory, if /(•, x) is measurable on / and / (t, •) is continuous on B. Observe that for any / which is Carathéodory, the function mf (-) = sup |/(•, x)| is measurable. We consider

xeB

M = {/: I x В -* E\ f is Carathéodory and J mf (s) ds < + oo}, i

<£ = {/ e M\ f is locally lipschitzian on I x B }.

For/, g e M we put

o ( f,g ) = j sup |/(t, x )-g {t, x)| dt.

/ x eB

We say that / is equivalent to g, in symbols / ~ g, if o (f, g) = 0. Clearly ~ is I an equivalence relation. Then we set Ш — Ji\^ and for any f & J i we denote

by F the equivalence class which contains /. For F, Ge$R, we set

q

{F, G) — o (f, g), f e F , g eG .

It is easy to see that

(F, G) is well defined since the right-hand side is independent of the particular choice of the representatives of the equivalence classes and, moreover, g is a metric on УЯ.

Lemma

1. Under g,

is a complete metric space.

Proof. The proof can be obtained using the same technique as in the finite dimensional case ([4], Lemma 3).

Denote by 2 the set of all such that G has a representative g e l f .

2 is dense in

Proof. Let F eïïfl and let e > 0. Let f e F . l t suffices to show that there is g e I f such that a (f, g) < e.

Let mf correspond to /. There is r\ > 0 such that for any measurable set J e l with g{J) < rj (g denotes the Lebesgue measure on I) we have

(1)

J mf {s) ds < 5 e.

By a generalization of Scorza-Dragoni’s Theorem due to Castaing ([1], Remark 2), there is a compact set A a 1 such that g(I\A) < rj and / is continuous on A x B. In addition we can assume that mf is continuous on A.

Observe that for each (r, x ) e A x B we have |/(r, x)|^M, where M

= max mf {t).

teA

Clearly, there is a compact set К c 1\A such that

(

2

Define

g((I\A)\K) <

4M 4-1

(t ,x ) e A x B , 0 , (r, x ) e K x B .

By Dugundji’s Theorem [3] / can be extended to a continuous function / defined on / x В and such that |/ (t, x)| ^ M for each (t, x ) e l x B . Obviously

J e Л . f

By Lasota and Yorke’s Lemma ([ 6 ], .Lemma 1) there exists g e l f such that

(3) sup {|/(r, x) — g(t, x)|| (t, x ) e I x B } < i e.

Now, using (3) we have

ff(/,

^ a (/,/) + <7 (/, g) ^ J sup I f ( t , x) J { t , x)| dt +

A x eB

+ J sup |/(f, x )-/ (r, x)| dt+ J sup if ( t , x )-/ (f, x)| d t + i e.

К X€B (I\A)\K xeB

Since on A x B , f = f while on K x B ,/ = 0 we have

о {f,

^ S mf (s) d s+ J SUP I/(L x)ldt + J sup I/(r, x)| d f+ i e,

К (I\A)\K x eB (I\A)\K xeB

Orlicz type category results 195

and using ( 1 ) and ( 2 ) we get

<*(/, 9) < i e + i E + M 1 + i e < £, which completes the proof.

Let К be a closed non-empty subset contained in 5(0, rf 2). We consider the space x V endowed with metric

max {

{F, G), \u — u|}, (F , u), {G, x V.

We observe that 2JÎ x V is a complete metric space.

For any (F, и )еШ х V we consider the Cauchy problem

(4) x' = F (f, x), x(0) = «.

In the sequel this problem will be denoted by (F, u}.

An absolutely continuous function x: [0, a) -+ В (a > 0) is called a solution of (4) if and only if it satisfies the integral equation

t

x(r) ~ u + J/(s, x{s))ds for each 0 ^ t < a, о

where / is any representative of F.

A continuous function x£ : [0, a] -» В (a > 0) is called an e-solution of (4) if it satisfies the inequality

t

|xe( f ) - u - J / ( s , xe(s))ds| ^ £ for each 0 < t ^ a, о

where / is any representative of F.

We denote by xF,u (resp. xf,u) a solution (resp, an £-solution) of problem

! F, u}.

Let

1 Ik

ЭДк = {Fe9W| J mf (s)ds ^ i r for f e F ] , Qk = 40lk n £ . о

Notice that sJRk as a closed subset of ЭД1 is complete, and £k is dense in ЭД1к.

Moreover, for any (G, u )e 2 k x V, xG,u certainly exists and is unique on Ik = [0, l/к]. In addition, for any (F, u)esJ>JtkxK there are e-solutions of ргоЫещ (F, и} defined on lk.

Lemma 3.

Let (G, v )e 2 k x V and let xG,v be the unique solution o f problem {G, f}, which is defined on l k. Then, for each £ > 0 there are <5G>t,(£) > 0 and

e g,v( £ ) > 0 s u c h t h a t

(F, u)eB((G, v), <5 0 ,Д£)), 0 < e < eGtV(Ç) imply ||xf— xG’lJ||/jt ^ ç.

Proof. We adapt an argument from ([2], Lemma 2.2). Let (G, r )e £ k x V and let xG,v be the solution of problem {G, defined on Ik. Denote by g the (unique) locally lipschitzian function belonging to G. Let L > 0 and rj > 0 correspond to g and xG,v according to Lemma 2.1 of [2]. Let £ > 0. Define (5) S0 = min {§ ge~Llk, £ £e~L/k}, e0 = min ge~L/k, \ ^e~L,k}.

Let (F , u)eB({G, v), Ô0) and let 0 < г < 8 0. Let J = {fe/k| ||хр,“ —

^ 4/2}. Let a = su p J. Clearly J is non-empty and closed. We claim that a = 1/k. To see this, suppose the contrary, i.e., a < 1 /к.

Let 0 < a < a — \/k be such that

a + a

j mg(s)ds< ^ rj.

O L For re [a, a + c] we have

(6) |xf ■ “ (г) - Xe’’ (r)| s: |xf•“ (r) - xf •“ («)[+|xf • " (a) - xc- » | + |x°--(a)-x0-» (r)|.

Furthermore, if f e F and g e G is locally lipschitzian, we have

\ .

(7) |xf’“ ( f ) - x f ’u (a)| < |xf*“ (t ) - u - $ f ( s , xf’u (s))ds| + o

+ |xf •“ (a) - и - ] f (s, x f (s)) flfs| + J I/ (s, xf(s))| ds

0 a

^ 2s + j |/(s, xf,u (s)) g (s, xf’"(s))|i/s + J jg(s, x f’u(s))jds

a a

a + a

^ 2 £ + <50+ j mg(s)ds < 2a + 00+^rj ^ f rj.

On the other hand

f X + f f

( 8 ) |xG’c( a ) - x G’,'(f)| ^ J \g(s, xG’”(s))|ds < J mg(s)ds <

a a

Then from ( 6 ), using (7), ( 8 ) and the definition of a, we obtain

|xf,u (t) xG,v (f)| < I = r\, t G [a, a + (j].

Using the last inequality and the definition of a, by an argument similar to that of Lemma 2.2 of [2], we obtain for any te [0 , a + cr]

(9) | x f (t) — xG,v (f)| ^ (e 0 + 2<50) eu,

which by virtue of (5) furnishes a contradiction. Thus a = 1 /к.

Put àG v(à) = ô 0 and £G>„(£) = eo- Then from (9) (which clearly holds for

every t e l k) by virtue of (5) the statement of the lemma follows.

Orlicz type category results 197

Lemma 4.

Let T)k be the set o f all {F, и)еШ к x V such that the problem {F, u} has exactly one solution xF,u defined on Ik which in addition depends continuously on the initial data, i.e. if (F„, u„) ->(F, u), (F„, ип)еШ к x V, and if the problem {F n, u„} has a solution defined on Ik, then ||.xFn,u" —xF,“||/k -> 0 as n -*■

Then 5)k is a residual set in ЭД1к x V.

Proof. Using Lemma 3 and the argument of [2] one can prove that the set

n и

((

, v), &а, л т )

i = 1 (G,t;)elik x V

(here <5G(„(l/i) corresponds to (G, v) and 1/t according to Lemma 3), which is residual in 9Jlk xV, is contained in s 2 )k.

From Lemma 4 and the Kuratowski-Ulam’s Theorem [5] we obtain

1. Let fflk x V be defined as above. Then there exists a set sJJÎk residual in such that for each F еШ к there is a set VF residual in V, which has the property: for each ueVF the problem {F ,u } has a unique solution defined on Ik and, in addition, un-*u (u„eVF) implies j|;tF»’“» — xF,“||/(, -> 0 as n -> + 00 .

00

Using Theorem 1 and the fact that x V — (J 9Jîk x V, one can prove

the following *=1

Theorem 2.

Let

V be defined as above. Then, there exists a set

residual in 9Л, such that for every F e 9J1° there is a residual subset VF o f V which has the property : for every ueVF, the problem {F ,u } has a unique solution which is defined on [0, /1] (/? > 0, depending on F) and this solution depends continuously on ueVF.

References

[1 ] C. C a s ta in g , Sur le graphe d ’une multi-application Souslinienne, Séminaire de Mathématique 1969, Faculté des Sciences de Montpellier.

[2 ] F. S. De B la s i and J. M y jak , G eneric flow s gen erated by contiuous vector field s in B anach spaces, Advances in Math, (to appear).

[3 ] J. D u g u n d ji, An extension o f Tietze’s theorem, Pacific J. Math. 1 (1951), p. 353-367.

[4 ] M. K is ie le w ic z , T he Orlicz type theorem fo r differential-integral equations with a lagging argument, Ann. Polon. Math. 31 (1975), p. 65-71.

[5 ] K. K u r a to w s k i and M. U la m , Quelques propriétés topologiques du produit com binatoire, Fund. Math. 19 (1932), 248-251.

[6 ] A. L a s o t a and J. A. Y o r k e , T he generic property o f existence o f solutions o f differential equations in B an ach spaces, J. Difer. Eq. 13 (1973), p. 1-12.

[7 ] W. O rlic z , Zur T heorie der Differentialgleichung y’ = / ( x , y), Bull. Acad. Polon. Sci. Sér.

A (1932), p. 221-228.