Existence theorems for L^-solutions of the Hammerstein integral equation in Banach spaces

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXX (1990)

Andrzej Soltysiak and Stanislaw Szufla (Poznan)

Existence theorems for L^-solutions

of the Hammerstein integral equation in Banach spaces

Dedicated to Professor Wlady slaw Orlicz on the occasion o f his 85th birthday

1. Introduction. Let X, Y be Banach spaces and let D be a bounded domain in the Euclidean space Rv. For an arbitrary iV-function (p we shall denote by Ьуф , X ) the Orlicz space of all strongly measurable functions u: D-+X for which the number

INI* = inf{r > 0: { (p(\\u{s)\\/r)ds ^ 1}

D

is finite. It is well known that ( L ^ D , X), Ц-Ц^) is a Banach space (cf. [7, 5]).

The symbol E^D , X) will stand for the closure in L ^D , X) of the set of all bounded functions. For properties of the spaces L ^D , X) see [5], pp. 76-106.

In this paper we give sufficient conditions for the existence of a solution

■xeL^D, of the non-linear integral equation (1) x(t) = p(t) + A J K(t, s)g(s, x(s))ds,

D

where К is a strongly measurable function from D x D into the Banach space f£ (Y, X) of all bounded endomorphisms from Y into X and g: D x X -+ Y is a function satisfying the Carathéodory conditions. The functions К and g will be assumed to satisfy the classical conditions (see [5]) which guarantee that the integral operator F defined by

F{x)(t) = J K(t, s)g(s, x(s))ds (teD, x e E ^ D , X))

D

maps the space E ^D , X) into itself.

In a finite-dimensional space X these conditions automatically ensure the existence of an L^-solution of (1). If the dimension of the space X is infinite, then this is not so and some other conditions must be added to get this solution.

12 — Commentationes Math. 30.1

In this paper we shall assume that the function g satisfies a certain Lipschitz condition with respect to the Kuratowski measure of noncompact­

ness a. Recall that the Kuratowski measure of a bounded subset A of X, denoted by a {A), is the infimum of positive numbers e for which there exists a finite covering of A by sets of diameter less than e. Fundamental properties of a are given in [1] and [9].

As usual we write Ü(D, X ) for the space of all strongly measurable functions и : D-+X with \\u\\t = §D\\u(f)\\dt < oo and C(D, X) for the Banach space of all continuous functions и: D->X with the ordinary supremum norm.

The Lebesgue measure on Rv will be denoted by g. Without loss of generality we shall always assume that all functions from I}(D, X) are extended to Rv by putting u(t) = 0 for t outside D.

The present paper extends the result of [10]. It should also be noted that similar results for the Volterra integral equations were proved in [8].

2. Prerequisites. Assume that M, N are complementary А-functions. We introduce the following conditions concerning g and K.

Cl. (s, x) I—>g(s, x) is a function from D x X into Y which is continuous in x for a.e. seD and strongly measurable in s for every xeX .

C2. ||gf(s, x)|| ^ ft(s) + H(||x||) for seD and x e X , where b e L ^ D , R) and H is a nonnegative, nondecreasing, continuous function defined on R+.

C3. 1° N satisfies the condition A', i.e. there exist /?, u0 ^ 0 such that N(uv) ^ f}N(u)N(v) for

, v ^ u0.

2° \\K\\

E

(D2,R).

3° (p is an А-function and there exist со, у, u0 ^ 0 such that N(coH(u)) ^ y(p(u) ^ yM(u) for

^ u0.

C4. 1° A satisfies the condition A3, i.e. there exist 0 such that N(fhi) ^ uN(u) for

^ u0.

2° \\K\\eEM(D2,R).

3° There exist ft, u0 > 0 such that

H(u) ^ f}M(u)/u for и ^ u0.

4° (p is an А-function satisfying the condition A' and such that Я <p(M(WK(l, s»))dtds < 00

D2

(existence of cp follows from 1° and 2°).

C5. 1° ^(p is an А -function and the function ^N satisfies the condition A2, i.e. there exist u0 ^ 0 such that

A (2

) ^ f}N(u) for u0.

2° There exists у > 0 such that

H(u) ^ уА_1(<р(м)) for и ^ 0.

3° || K(t, *)|| e E M(D, R)îor a.e. te D a n d the function t\-^\\K(t, -)||M belongs to E^D , R).

It is well known (cf. [5], Th. 19.1, Th. 19.2, L. 16.3, and Th. 17.6) that conditions C1-C3 imply that the integral operator F maps the unit ball in L^D , X) into E^D , X), and conditions Cl, C2 plus C4 or C5 imply that F is a mapping of E^D , X) into itself. #

Before passing to further considerations we shall prove a certain lemma concerning general Banach function spaces. Suppose that В is a Banach fundamental space of functions on D endowed with the norm ||-||B (in the terminology of [3], p. 139). For an arbitrary Banach space Z denote by B(Z) the space of all strongly measurable functions и: D->Z such that \\

\[

B. It is a Banach space with respect to the norm \\u\\B = || || u\\ ||B.

Le m m a 1!

Assume that В satisfies the condition

(A) x „ \ 0 implies II

J

^O

and moreover L00 <= B. Then C(D, Z) is dense in B(Z).

P ro o f. Suppose xeB(Z). Then there exists a sequence (xn) of simple functions such that ||x„(t)|| ^ 2 ||x (t)|| and x„-+x a.e. on D. Condition (A) implies that the function x has absolutely continuous norm in B. Hence by the inequality ||x„(t) — x(f)|| ^ 3 ||x(t)|| we find that the sequence (x„ — x) has equi-absolutely continuous norms in B. Consequently ||x„ — x|| B — > 0, which means that L°° is dense in B(Z).

Now, suppose that x e B ( Z ) is bounded. Let ||x(t)|| ^ a. By Lusin’s theorem there exists a sequence (xn) of continuous functions such that l|x„(t)ll ^ a an(i the difference x(t) — xn{t) is zero outside a set Dn <= D with p{Dn) < 1/n. Thus ||x x n||g = ||(

-

„)

„II

^ 2a||xDJ |B^ 0 as n-> oo and we are done.

We introduce a function к defined by k{t) = 2\\K(t, -)||

f°r teD. Observe that condition C3, 2° implies kel}(D , R). Reasoning as in [5], p. 183, one can show that from C4, 2° and 4° it follows that k e E ^ D , R) while obviously the same holds true when C5 is assumed. Since E^D , R) cz L1{D, R), we conclude that \\K\\ belongs to the mixed norm space Емл = L1 [£ M] (see [3], p. 401). It is clear that Емл satisfies condition (A) and therefore, in view of Lemma 1, C(D, j£?(Y, X)j is dense in £ M>1(j£?(Y, A)). This implies

(2) Hm j \\K(t + h, -) — K(t, ’)\\Mdt = 0.

h-0 D

Now we present some inequalities concerning the operator F. We

distinguish three cases.

I. If C3 holds, then by Lemma 19.1 of [5] there exists a constant C such that for every measurable subset T of D and x e L ^ D , X), \\х\\9 ^ 1, we have

(3) И а д Ы 1 < ^ С № т х л 1 и . .

Moreover, there exist со, у, u0 > 0 such that (4) ||Я (М )||„ < - ( 1 + J JV(»H(||x(S)||))ds)

CO D

< - ( l +Я(соЯ(и0)) + у J <p(||x(s)||)ds)

CO D

for every x e L ^ D , X) with W*h< 1-

II. If C4 is satisfied, then by Theorem 19.2 of [5] there exists a constant C such that for every measurable subset T of D and an arbitrary x e L ^ D , X) we have

(5)

+ ||Я ( М ) у .

Moreover, as 3° of C4 implies that there exist со, q, u0 > 0 such that N(œH(u)) ^ rju for и ^ u0, we see that for every x e L ^ D , X)

(6) ||я(||х||)||л « - ( l + J

CO D

< — (1 + ЛГ(соЯ(м0)) +ц J ||x(s)|| ds).

со D

III. If C5 is assumed, then for an arbitrary x e E ^ D , X) we have

(7) ||H(||x||)||* < у||ЛГ-1 ЫЫ1))1и ^ У + У J ç>(l|x(s)||)ds.

D

Moreover, by C2 and Holder’s inequality we obtain

(8) ||F(x)(t)|| ^ /с(0(||Ь|^ + ||Я(||х||)||^) for a.e. teD , and consequently

(9) II^WZrlU< 11^г11Л11*^ + И11х11)У

for every measurable subset T of D and each x e E ^ D , X). Furthermore, by Holder’s inequality, from C2 it follows that for every t&D such that K (t,-)eE M(D, R) we have

(10) j ||K(t, s)g(s, x(s))|| ds ^ 2 1|^:(t, -)z.p Нлг(It^II

+

p

where P is an arbitrary measurable subset of D and x e Еф(0, X).

For simplicity we introduce the following notation:

L1 = LHD, X), Lv = L,(D, X), Ev = E9(D, X),

B'v

= {xe£„:

By if C3 holds, Ey if C4 or C5 holds.

Now we show that F is a continuous mapping of Qv into Ev. Let x„, x0 eQy and lim ^ ^ ||хи- х 0||„ = 0. Suppose that ||F(xII) - F ( x 0)||fl, does not con­

verge to zero as n -» oo. Then there are e > 0 and a subsequence (xnj) such that

and limj-^oo xnj(t) = x 0(t) for a.e. teD. As x„->x0 in Qv, the sequences (JDç)(||x„(s)||)ds) and (Jz> II x„(s) || ds) are bounded. This implies, in view of (10), (4), (6), and (7), that for a.e. teD the sequence (||iC(t, s)g(s, x„(s))||) is equi-inte- grable on D. Since for a.e. teD, lint,--.,*, K(t, s)g(s, xnj(s)) = K(t, s)g(s, x0(s)) for a.e. seD, applying the Vitali convergence theorem we get lim ^ ^ F(x„.)(0 = F(x0)(t) for a.e. teD. On the other hand, from the inequalities: (3) if C3 holds, (5) and (6) if C4 is satisfied, and respectively (9) and (7) if C5 is assumed, it follows that the sequence (F(xnj)) has equi-absolutely continuous norms in Ly. This implies lim ,-^ \\F(xnj} — -F(x0)||<p = 0, in contradiction with (11).

3. Main results.

Th e o r e m

1. Assume that conditions Cl, C2, and C3 are satisfied. Let

a measurable function w: D->R+ be such that the map t ^ W ( t ) — ||^ ( t,‘)wllo0 îS well defined for a.e. te D and belongs to L 1(D, R). Moreover, assume that for every в > 0 and for each bounded subset Z of X there exists a closed subset De of D such that p(D\D£) < г and

for an arbitrary closed subset T of De. Then for every p e E with ||p|| < 1 there exists a positive number

such that for every X

R with \X\ <

equation (1) has a solution x e £ f

Th e o r e m

2. Assume that conditions Cl, C2, and one of C4, C5 are fulfilled.

Let a measurable function w: D-+R+ be such that the map t\-+W(t) = \\K(t, -)w||^

is well defined for a.e. te D and belongs to L ^D , R), where ф is the complemen­

tary N -function to (p. I f (12) holds true, then for every

Е9 there exists a positive number g such that for each

with

< g equation (1) has a solution xeE^.

P ro o f. Let an arbitrary peE ^ be fixed from now on. The function p has the norm ||p|| < 1 when C3 is assumed. Choose a positive g in the following way:

(11) ||F(xB/)-F(xo)||„ > e for j = 1, 2 ,...

( 12 ) a(p(Tx Z)) < sup w(s)a(Z)

s e T

I. If C3 holds, then

II. If C4 is fulfilled, then

e = m ln(

||JC||„

A)>)9’ 211^11,)’

where $ denotes the norm of the identity operator from Lç into L 1.

III. Assume C5 holds. Denote by Q the set of all q > 0 for which there exists r > 0 such that

I

Ilp(0ll

N + yr))dt ^ Г.

D

Then

e=min(supe’^ x )

Fix XeR with |Я| <

. Define a mapping G by the formula G(x) = p + XF(x) for xeQ y. Choose a positive number r such that: r = 1 if C3 holds,

W P h + W c m ^

\\b\\N + — {l + N(œH(u0)) + r]Sr)

if C4 is satisfied, and

I

<p(\\p{t)\\ + №\k(t)(y+\\b\\N + yr))dt

r

D

if C5 is assumed.

Set

\x\\v

r} if C3 or C4 holds,

\v if C5 holds,

where U

{xeE^: JD(p(||x(s)||)^s

r}.

If C3 is assumed, then by (3) we get

IIGMM 1Ни+1ВДК||м^ Ilpiu+^||X||M^1

for

II. If C4 is satisfied, then by (5) and (6) we obtain

IIG(x)||„« Hpt ⁺ |/l|C||K||soM \\b\\N + -(l+ N (a)H (u0j) + ri f ||л

_ (U D >(s)\\ds)

^ i \ p K + w c m ^ oM II ^Ь К ^N + + N (œH(uo)) + rlS IIх II J

a l|p||»+WC!!K||voM н ы и + ^ + ^ ш Ж и 0^ ^ ) ^Г

for x eB .

III. Under assumption C5 inequalities (8) and (7) imply l|G(x)(OII ^ \\p{t)\\ + \l\k(t)(\\b\\N + y + y J <p(||x(s)||)ds)

D

< \\p(t)\\ + \MHt){\\b\\N + y + yr) and so

J (p(\\G(x)(t)\\)dt < J q>(\\p{t)\\ + \l\k{t){\\b\\N + y + yr))dt < r

D D

for x e U . Hence G(U) c. U. Consequently G(t7) c G(U) c Ü.

Notice that Ü is a bounded, closed, convex subset of Ev and U a Brv+1.

Moreover, there exists a number A such that

If C3 or C4 is fulfilled (13) follows immediately from (4) and (6). Under assumption C5 inequality (7) implies that

On the other hand, in view of Theorem 17.6 in [5], we see that the mapping Ж defined by (Jfu)(t) = H(\u(t)\) for u e E ^ D , R) and teD maps continuously E^D, R) into E

(D, R). This obviously implies that (13) holds true also in this case.

Inequalities (8) and (13) yield

(14) ||G(x)(t)|| < m(t) for x e B and teD , where m{t) = \\p(t)\\+A\X\k{t).

Moreover, by our assumptions and Holder’s inequality it follows that Г2||Х((,-)11„(11Ыи + ||Я (||х||)У

(13) for all x e B .

||b|U + ||H(||x||)||N sj A for all x e V .

\\F(x){t + h)-F(x)(t)\\ < 1 if teD and t + 2||X(t+A, -)-K (t, •)||м(«Ыя + ||я (М 1 )У

if t, t + heD.

From this, in view of (13), we get

(15) \\G(x){t + h) — G(x)(t)|| ^ d(t, h) for x e B , teD , and \h\ sufficiently small, where

Relation (2) implies evidently that

lim J d(t, h)dt = 0

and

J J d(t, s)dtds < oo,

D x Q r

where Qr denotes the closed ball in Rv with centre 0 and radius r.

We shall use the following notation. For an arbitrary set F of functions from D into X denote by v the function defined by v(t) = a(F(r)) for te D (under the convention that a(Z) = oo if Z is unbounded), where V{t) = {u(t): ueV}.

Furthermore, for an arbitrary subset T of D put V(T) = {u(t): we F, te T }. In [10] the following lemma was proved.

Le m m a 2.

Let V be a subset of Ü such that

m(t

and \\x(t + h)

— x(t)|| ^ d(t, h) for all x e V ,t e D , and small \h\, where m and d are the functions from (14) and (15). Then

(a )

L1(D,R);

(b) aj(F) ^ $Dv(t)dt;

(c) for every e > 0 there exists a closed subset De of D such that H(D\DË) < s and for each closed subset T of De there exists

T such that

* ( V ( T ) ) = v(t) .

Assume that F is a subset of В such that

(16) F = convG(F) or F = G (F )u { 0 } .

We want to show that F is relatively compact in L^. Relations (14), (15), and (16) imply that F satisfies all assumptions of Lemma 2. Thus the function t\-^v(t) = a(F(t)) is integrable on D and the Kuratowski measure of noncom­

pactness in L1 of the set V, cifV), has the estimate

(17) M F K J v(t)dt.

D

Fix

> 0 and te D such that \\K(t, *)||M < oo. By the Lusin theorem, the Egorov theorem, and Lemma 2 we choose a closed subset Z of D such that H(D\Z) < e, 2Л ||K(t, ')%

\

W

< £> the functions K(t, •), w, v are continuous on Z, and for each closed subset T of Z there exists те T such that a(F(T)) = v(z).

Putting

{ K(t, s)g(s, V(s))ds = {f K(t, s)g(s, x(s))ds: x e V } ,

T T

we get

G(F)(t) c p(t) + /l|X (£, s)g(s, V(s))ds + À J K(t, s)g(s, F(s))rfs,

Z D \ Z

and consequently

(18) a(G(F)(£)) < |2|a(jX(£, s)g(s, F(s))rfs) + |2|a( J K{t, s)g(s, F(s))Js).

Z D \Z

Since Z is compact and the functions K(t, •), v, w are continuous on Z, there

and and

J J d{t, s)dtds < oo, DxQr

l|K(t,

<7)11 ^ e/3/ï2, jr(s) —u(<r)| < e/3/ï2,

|w(s) — w(cr)| < e/3/?2 for s, <7GZ with \s — a\ < Ô.

Further, we choose nonempty closed sets Z l5 ..., Z„ such that

n

Z = (J Z f, diam Z f ^ Ô, fi(Zt n Z j) = 0 for i, j = 1, ..., n, i ф j.

i= 1

Then for every i, 1 ^ i < n, there exist тг, sf, (Tie Z i with the properties (21) a(K(Zi)) = и(т{), sup ||X(t, s)|| = \\K(t, sJH, sup w(s) = w(<7{).

seZ i seZi

On the other hand, from (12) it follows that there exists a closed subset De of D such that g(D\D£) < e, 2A \\K(t, -)

\

.W

< e> and

(22) a(j(T x ViZ,))) s; supw(s)x(V(Z,))

seT

for i = 1, ..., n and an arbitrary closed subset T of De. Put S = Z n D e, = Z i r\De, and Pf = {K(t, s)g(s, x): s e S i? x e F (Z ()}. By the continuity of K(t, •) on Z we find that {K(t, s): s e S j is compact. Therefore

« ( Л К sup ||K(r,s)||«(9(Si xK (Zi))),

seSi

and further, by (21) and (22) we get

(23) a ( P ^ \ \ K ( t 9s M w ( a M ^ - In view of the mean value theorem we have

jK (t, s)g(s, x(s))ds = £ J K{t, s)g(s, x(s))dss £ //(5,.)

Р,.

S i = l Si i ⁼ 1

for each xeV, Thus by (23) and the corresponding properties of a a ($K{t, s)g(s, V(s))ds) < Z //(Sf)a(convP.) = £ М ^)а(Рг)

S i =1 i = 1

< Z ^(5f)||^(t, s J llw ^ X ii).

i = 1 But from (19) and (20) it follows that

p(Si)||K(t, < J ||X(r, s)||w(s)t;(s)ds + e/i(Si).

Si

Hence

a(jK (t, s)g(s, F(s))ds) < Z j ||K(t, s)|| w(s)t>(s)ds + £ eM(S‘)

S i = l Si i = 1

= J ||K(t, s)|| w(s)v{s)ds + eg(S),

and finally

(24) a ($K{t, s)g(s, V(s))ds) ^ J || K(t, s)|| w(s)v(s)ds + eg{D).

s D

Further, for an arbitrary x e K by Holder’s inequality and (13), we get

|| J

K { t

s)g(s, x

A \\K{t, -)X

\

W

D \Z

which implies

a(

K(t, s)g(s, F(s))ds) < 2г.

D \ Z

Similarly

a(

K(t , s)g(s, V(s))ds)^2e z\s

since Z \ S c= D\De.

From the above inequalities, (24), (18), and the inclusion

$K( t , s)g(s, V(sj)ds a $K(t , s)g(s, V{s))ds+ f K{t, s)g(s, V(s))ds

z s z\s

it follows that

a(G(F)(t)) ^ |A| J || K(t, s)|| w(s)i;(s)ds + 4|A|e + |A|6ju(D).

D

As г is arbitrary, we obtain

a(G(F)(t)) ^ |A[ J || K(t, s)||w(s)y(s)ds.

D

Moreover, in view of (16) and (14), we have

(25) a(F(0) ^ a(G(F)(t)) ^ 2m{t).

Thus

(26) v{t) ^ |A| J ||K{t, s)|| w{s)v{s))ds

D

Now we consider two cases.

I. Assume C3 is satisfied. Then m eÜ (D , R) and, by (25), veI}(D, R).

Notice that (26) holds for a.e. teD. By our assumptions on w and by Holder’s inequality we get

v(t) ^ |A| W(t)||i;||1 for a.e. teD . This implies

Ni^lAIII^IUNi.

Since IAIIIWH

< 1, we obtain ||t;|| А = 0 .

II. If C4 or C5 is fulfilled, then m e £ (p(D, R) and, by (25),

E

(D, R). A

above we obtain v(t) < 2|A| W(t) for a.e. te D and further

2|A||| WII^Hull,,. Since 2\Â\ || WW^ < 1, we have ||u||^ = 0 and so as before.

In both these cases, in view of (17), a 1(F) = 0.

On the other hand, from (3), (5), and (9) it follows that V has equi- -absolutely continuous norms in Ev. Hence V is relatively compact in E .

The above considerations imply that the mapping G satisfies the assump­

tions of the following theorem ([12], Th. 1).:

Let В be a bounded, closed, and convex subset of a Banach space such that 0

B, and let G be a continuous mapping of В into itself I f the implication

V = convG(K) or V = G(F)u{0} => V is relatively compact holds for every subset V of B, then G has a fixed point.

This result is a modification of the famous Sadovskii fixed point theorem.

Therefore we conclude that there exists

B such that G(x) = x. Clearly x e E y and x is a solution of equation (1).

4. Concluding remarks. It is easy to see that our results remain true if we replace the Kuratowski measure of noncompactness a by the Hausdorff measure f.

One can easily check that if the function g is of the form g = gx + g2 where (i) for every bounded subset Z of X and an arbitrary e > 0 there exists a closed subset De of D such that p(D\De) < e and the set g1(Dex Z) is relatively compact in У;

(ii) \\g2(s, x) — g2(s,

< w(s)||x — y|| for all x , y e X and a.e. seD, then g satisfies condition (12) with the measure a replaced by the Hausdorff measure /?.

It is an open problem whether instead of (12) we can take (27) ot(g(s, Z )) ^ w(s)a(Z) for a.e. seD.

However, if we carry out the reasoning similar to that in [11] (which relies on Heinz’s well-known result [2] on measurability of the function £i—►a(I/ (f)), and Monch’s fixed point theorem ([6], Th. 2.1)) with (27) instead of (12), we can prove that the assertions of Theorems 1 and 2 remain true if

is replaced by

q

/2.

5. Appendix. We shall show that Theorem 2, with the assumption similar to C5, also remains true for an Orlicz space L ^D , X ) generated by a general­

ized IV-function q>: R + x D^>R+. Recall that L^(D, X) is the space of all strongly measurable functions u: D ^ X for which the number

N 1 , = inf{r > 0: J <p(||u(OII/r, t)dt ^ 1}

D

is finite. It is well known that (L ^ D , X), || • ||^> is a Banach space (see [4]). As

before we also consider the space E^D , X).

Now assume that

1° M, N: R +x D ^ R + are complementary iV-functions, M satisfies the following condition:

(*) J M(u, t)dt < oo for all и > 0,

D

and N satisfies the condition Л2.

2° (p : R + x D ->R+ is an iV-function satisfying condition (*) and such that и ^ cq>(u, t) + a(t) for all и ^ 0 and a.e. teD ,

where c is a positive number and ael}{D, R). Let \]/ be the complementary function to (p.

3° \\K(t, -)||

R) for a.e. teD and the function ft—»\\K(t, -)||M belongs to £ ф(/), Л).

4° ||gf(s,

)|| < y(s, ||x||) where (s, u)h->y(s, u) is a function from D x R + into R + , measurable in s and continuous in u, and there exist f , rj > 0 and bel}(D, R), b ^ 0, such that

N(fiy(s, и), s) < rj<p(u, s) + b(s) for all и ^ 0 and a.e. seD.

Th e o r e m 3.

Let a measurable function

D->R+ be such that the function

t\—► W (t) = ||K(t, -)w||^ is well defined for a.e. teD and belongs to L ^D , R). I f the above assumptions are fulfilled and the function g satisfies (12), then for every p e E v there exists a positive number

such that for all

R with \

\ <

equation (1) has a solution

P ro o f. Fix a function p e E Denote by Q the set of all q > 0 for which there exists r > 0 such that

j <

(llp(OII +(g/0)fc(t)(l + WbWx+rir), t)dt ^ r.

D

Let

= min (sup Q, 1/21| И^П^). Now the proof runs similarly to the proof of Theorem 2 under assumption C5 if only we replace ||b|U +II^CWDH

by

||y(-, IMDHtf and use the inequality

\\y(-, II^IDtU < 4(1 + J b(s)ds + rj j <jp(||x(s)||, s)ds).

P

To make our argument self-contained we must prove that the super­

position operator у defined by y(u)(s) = y(s, |u(s)|) for u e E ^ D , R), seD, maps continuously E^D , R) into EN(D, R).

First we shall show that

(28) If a set Z c E ^D , R) has equi-absolutely continuous norms in Еф, then

y(Z) has equi-absolutely continuous norms in EN.

As N satisfies Д2, there exists со > 0 such that N(2u, s) ^ coN(u, s) for seD and и ^ 0. For a given e > 0 we choose a positive integer к in such a way that l/2k~1 p < e. Since Z has equi-absolutely continuous norms in Еф, we take Ô > 0 such that

f b(s)xT(s)ds + cokri WuXrWcp ^ 1 and ilw^xlU < 1

D

for all u e Z and each measurable subset T of D with ц(Т) < Ô. Hence, by 4°,

\\2kpy{-,

< 1+ J N(2kPy(s, \u(s)\)xT(s), s)ds

D

^ 1 + cofc J N(Py(s, |u(s)|)^r (s), s)ds

D

^ 1 +œk J b(s)xT(s)ds + (Dkrj J (p(\u(s)\xT(s), s)ds

D D

< 1 +ык J b(s)xT(s)ds + wkri \\uxT\\v ^ 2,

D

so that ||y(M)Xrll;v ^ l/2k~1 P < e for u e Z and each measurable subset T of D such that ц(Т) < Ô.

Let un, U

GE^D, R) and lim ,,^ \\un — м0||ф = 0. Suppose that

\\y(un) - y ( u 0)WN does not converge to 0 as oo. Then there are £ > 0 and a subsequence (unj) such that

(29) lly(^)-y(w 0)lliv > £ for j = 1 , 2 , . . .

and limj

un.(s) = u0(s) for a.e. seD. Clearly we have lirn ,-^ y(u^Xs)

= y(u0)(s) for a.e. seD. Moreover, since lim,,-^ \\un — м0||^ = 0 implies that the sequence (w„) has equi-absolutely continuous norms in E(p, from (28) it follows that the sequence (y(un)) has equi-absolutely continuous norms in EN. From this we conclude that lint/-*, ||y(w„.) — y(w0)||N = 0, in contradiction with (29).

References

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Univ. Padova 75 (1986), 1-14.

INSTYTUT MATEMATYKI, UNIWERSYTET IM. A. MICKIEWICZA INSTITUTE O F MATHEMATICS, A. MICKIEWICZ UNIVERSITY MATEJKI 48/49, 60-679 POZNAN, POLAND