|jxJh -> ^ : x) On the asymptotic relationship between solutions of linear and quasilinear differential equations in Banach spaces

ROCZNIKI POLSKIEGO TOWARZYSTWA MA ТЕМATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVI (1986)

St a n i s l a w Sz u f l a

(Poznan)

On the asymptotic relationship between solutions o f linear and quasilinear differential equations in Banach spaces

Let J = [0,

and let £ be a Banach space with norm || ||. We consider the problem

(1) x ' = A { t ) x + f { t , x) thought of as a perturbation of

(2) x' = A(t)x.

More precisely, we give some conditions which guarantee that for each bounded solution w: J -> E of (2) there exist a ^ 0 and a solution и:

x )

£ of (1) such that

lim \\u(t) — w(f)|| = 0.

t --*■ GO

For the case E — R" this problem was investigated by many authors (see e.g.

[2 ], [3], [5], [6], [13]).

In our paper we use some of the notation, definitions and results from [7]. We introduce the following denotations:

Ё — the space of continuous linear mappings E E;

H — the Lebesgue measure in R ;

Ll (J,E) — the space of Bochner integrable functions x : J -> E with the norm |jxJh = J||x(s)|| ds;

j

L(J, E) — the space of strongly measurable functions x: J -> E, Bochner integrable on every finite subinterval J' of J, with the topology of the convergence in the mean on every such J', i.e., convergence in L1 (J', E) of the restrictions to J';

В = B(J, E) — the Banach function space of all strongly measurable functions x: J -+ E such that j|x||eB(J, R), provided with the norm ||x||B

= ||lWl||e> where B{J, R) is a Banach function space such that:

(i) B(J, R)

L(J, R) and B(J

R) is stronger than L(J, R);

(ii) B(J, R) is not stronger than L l (J, R);

156 S. Szufl a

(iii) B [J, R) contains all essentially bounded functions with compact support;

(iv) if ueB(J, R) and v is a real-valued measurable function on J such that |r| ^ |

|, then veB(J, R) and ||у||в ^ ||u||B;

(v) if (u„) is a sequence in B(J, R) such* thatlim un(t) = 0 for almost

n -+ 00

every teJ, and (un) has equi-absolutely continuous norms in B{J, R), then lim |к/[0,а]11в = 0 for any a > 0.

n-> 00

Let B' denote the associate space to В (cf. [7], p. 50).

Further, let AeL{J, Ё), and let E0 be the set of all points of E which are values for r = 0 of bounded solutions of (2). We assume that E0 is closed and has a closed complement Ex. Let P be the projection of E onto E0.

Moreover, let U: J -> Ê be the solution of the differential equation U'

= A( t) U with the initial condition U (0) = I. By Theorem 62.D of [7] we have the inequality

(3) ||1/(г)Р||<ЛГ||Р|к~* for re J,

where N and

are some positive numbers.

Throughout this paper we shall assume that:

1° For each be В there exists at least one bounded solution of

(4)

x ' = A(t)x + b(t);

let к > 0 be a constant such that for each be В equation

has a unique bounded solution x with x(0 )e E 1, and this solution satisfies sup j|x(r)|| < Ai|/?||B (the existence of к follows from Theorem 51.E of [7]);

teJ

2° p: J ^ J is a nondecreasing continuous function such that p(r) < r for r > 0;

3° h is a nonnegative function belonging to B{J, R) and k\\h\\B ^ 1;

4°

(r, r)-»co(r, r) is a function from J2 into J such that:

(a) со is measurable in t and nondecreasing in r;

(b) for each r > 0 the function œ{-, r) belongs to B(J, R), and lim ||ш( -, r)xz\\e = 0 and lim ||o>(-, r)x[d. J|B = 0;

f l ( Z ) - * 0 d ~>ao

5° (r, x)-> f (r, x) is a function from J x E into E such that / is strongly measurable in r and continuous in x, and

||/(r, x)|K co{t, ||x||) for all (t , x ) e J x E .

Th e o r e m

1. I f conditions l°-5° hold and

IIf ( t , x ) - f ( t , y)|| ^ h{t)p(\\x-y\\)

for all x, ye E and almost all teJ, then for each bounded solution w: J -> E of (2) there exists a unique bounded solution u: J -> E of ( 1) such that Pu(0)

= w(0), and lim ||w(f) — w (f)|| = 0.

t->00

Proof. This result follows immediately from (3) and Theorems 1 and 3 of [10].

In what follows we shall need a function G defined by

\ U ( t ) P U - l (s) i f 0 < s < r , [ , S) \ - U { t ) ( I - P ) U ~ l (s) if s > r > 0.

We shall assume that

6° G(t, -)eB' and ||G(f, -)I|B < к for all teJ.

It is known (cf. [7 ], p. 163) that 6° implies Г, but it is an open question whether 1° implies 6°. If E is finite-dimensional, then 1° and 6° are equivalent.

Let a be the Kuratowski measure of noncompactness in E (cf. [4 ], [1]).

Th e o r e m

2. I f conditions 2°-6° hold and for any a > 0, e > 0 and any bounded subset X of E there exists a closed subset D of [0,a] such that

>u([0, a]\D) <

and

(6)

x( f( TxX) ) Hs\iph( r) pix( X) )

te T

for each closed subset T of D, then for each bounded solution w: J -> E of (2) there exist a > 0 and a solution u: [a, oo)->£ of (1) such that

limIMO —w(r)|| = 0.

t -*ao

Proof. Let w be a bounded solution of (2), and let r > 0 be such that

||w(r)|| ^ r for teJ. By 4° there exists a > 0 such that

(7)

2r)x[a(00)||B < r/k.

Let Ja = [a, oo). Denote by C the space of continuous functions x: Ja-+E with the topology of uniform convergence on compact subintervals of Ja, and by V — the set of all x e C which satisfy the inequalities

1И0Н ^ 2r for t e J a and

t t

(8) ||x(f) —

(

)|| ^ 2r J||v4(s)|| ds+ |tu(s, 2r)ds for a ^ т ^ t.

X X

It is clear that V is a closed convex subset of C.

Consider the mapping F defined by 00

F(x)(t) = w(t)+ j G(t, s)f(s, x (s))ds

V, teJJ.

158 S. Szufl a

By 4°-6°, (7) and the Holder inequality we have 00

(9) l|F(x)(f)|| < 1И011+ j \\G(t, s)||w(s, 2r)ds

a

^ _r + ||G(r, -)IUIM\ 2r) _{Х[а.оо\\\в} < _2r

and

(10) \\F(x)(t)-F(y)(t)\\ < k\\f{-, * )- / (• , y)\\B

for all x, ye V and t e J a. Moreover, the function z — F{x) is a solution of the differential equation z' = A ( t ) z + f ( t , x(t)). Hence, by (9), 5° and the mean value theorem, we obtain the inequality

t t

||F(x)(r)-F(x)(T)|| < 2r J’||/l (s)|| ds + J'co(s, 2r)ds

Г г

for x e V and a ^ г ^ t. This shows that F maps V into itself.

We shall prove that F is continuous. Let (x„) be a sequence in V such that lim xn = i in C From 4° it follows that for a given s > 0 we can

n-*<x>

choose d ^ a such that ||co ( *, 2r)/Idi00(||B ^ e/2k. On the other hand, lim Ц/(s, xa{s))-f(s, x(s))|| = 0 and \\f(s, xn( s))-f(s, x(.v))|| ^ 2co(s, 2r) rt -* 00

for seJa. Owing to 4° and (v), from this we deduce that lim ||(/ (•, x „)-/ (-, x ))Z[MI||B = 0,

/1 -+ oo

and therefore there is n0 such that

!!(/(*• *„)“ / ( S

<

for

Thus for n ^ n0 we have

tl/(•, x „)-/ (-, x)||B < ||(/(-, x „)-/ (-, x))xIe.d]||B + |M'> 2г)Хы.Лв < Ф * and consequently, by (10),

l|F(x„)(f)-F(x)(f)|| ^ e for n ^ n0 and t e J u.

For any subset X of V put ft(X) = sup \cc(X(r)): t e J a), where A(r)

= Jx(t): xe A }. Using (8), the Ascoli’s theorem and corresponding properties of a, it can be easily shown that

/(A ) = 0 <=>X is relatively compact in C, X c Y = > P { X ) ^ P ( Y ) ,

P( \ u ) u X) = P( X) (u e V ),

P( X) = P( X) , P (conv A) = 0(A).

Now we shall prove that

(11) P( F( X) ) ^ рф( Х) ) for each subset X of V.

Let X be a subset of V. Fix te Ja and e > 0. By 4° we can choose d ^ t such that A: ||co( -, 2r)/[daJ|B ^ e. Analogously as in the proof of inequality (8) in [11], it can be shown that the assumptions of Theorem 2 imply the inequality

a ((jG (f, s) f ( st x(s))ds: x e X } ) ^ j'||G(r, s)|| h(s) p ( x ( X (s)j)ds.

a a

Moreover,

|||G(f, s)f(s, x(s))Js|| ^ ||G(f, *)||B' ||

( *, 2r)x[d,oo)\\B ^ £ for each xe V, d

and

F( X) ( t) c= x(t) + JJ G(r, s) f( s, x(s))ds: xeX| + ( f G(f, s) f( s, x(s))ds: x e X ]

From the above we deduce that

a(F(20(f)) ^ f||G(f, s)\\h(s)p(ct(X{s)))ds + 2c

a

^ j ||G(f, s)\\ h(s)ds ■ p(P(X)) + 2£.

a As e is arbitrary, this implies

* ( F ( X m ) « I ||G(f, s)|| h(s)ds p(IHX)).

a

Therefore, by 3°, 6° and the Holder inequality, we get a (F (X )(f))«* | | % p (jS (X ))«p (0 (X )) for teJ„, and finally

P ( F ( X ) ) ^ p ( P { X ) ) .

We define a sequence (>’„) by y0 = 0, yH+l = F(y„) (n = 0, 1, ...). Let Y

— [yn: n — 0, 1 , }. As Y = F {Y) u {0}, we have p( F( Y) ) = p(Y). By 2° and

(11) this implies that P( Y) = 0, i.e., У is relatively compact in C. Let Q be the

set of all limit points of (>’„). Then Q = F{Q). Denote by Q the family of all

subsets X of V such that conv F( X) о X and Q с X. It can be easily

verified that the intersection H of all sets of Q is nonempty and H

160 S. S z u f l a

= conv F (H). Hence fi (H) = fi (F (H)), and consequently, by 2° and (11), H is compact. Applying now the Schauder-Tichonov theorem to the mapping F\H, we conclude that there exists ue H such that и — F (и). It is clear that и is a bounded solution of (1) on Ja.

It remains to show that

(12) lim ||w(f)-w(f)|| = 0.

Г —*00

By 4°, for a given г > 0 there exists d ^ a such that ||co( *, 2r)x[d oo)||B ^ e/2k.

On the other hand, from (3) it follows that there exists t0 ^ d such that

||t/(f)P||-||J'L_1 (*)/(*, U(s))ds\\ ^ e/2 for t ^ t0.

a

Since for t ^ d we have

d oo

u( t) - w{t) = U ( t ) P $ U - l (s)f(s, u(s))ds + j G(t, s)f(s, u{s))ds,

a d

by 5°, 6° and the Hôlder inequality from this we infer that

ll«W -w(r)l! «||t/(0P||||j'C/“ 1(s)/(s, «(s))</s|| + ||G(f, ■)||B.||w(', 2r)xM. J I B< c

a

for г ^ t0- This proves (12).

T

3. Suppose that the assumptions of Theorem 2 are fulfilled and there exist nonnegative functions m, qeL( J, R) such that

(13) ||/(f, x)|| m(t) + q(t)\\x\\ for all {t, x) eJ xE.

Then for each bounded solution w: J -» E of (2) there exists a solution v:

J -> E of (1) such that lim ||t? (f) — w(r)|| =0.

t -+0C

Proof. Let w: J -> E be a bounded solution of (2), and let u: [a, oo) -> E be a solution of (1) given by Theorem 2. Consider the Cauchy problem (14) x' = g( t,x) , x(0) = U ~1 (a)u(a),

where g{t, x) = — U ~ 1{a — t ) f ( a — t, U(a — t) x) for te[0 , a] and xeE. Put c = max (||l/~1(f)||: 0 ^ t ^ a} and d — max {||l/(f)||: 0 < t ^ a}, and fix a bounded subset X of E. Let Y = { U ( t ) x: 0 ^ t ^ a, x e X} . As the set U ([0, a]) is compact, we have

(15) 0L(Y)^da{X).

It follows from (6) that for a given e > 0 there exists a closed subset D of [0, a] such that p([0, a] \D) <£ and

ct(f ( Tx T)) ^ sup h(t)<x(Y)

teT

(

16

for each closed subset T of D. Hence, owing to (15), (16) and the compactness of (У_1([0, a]), for any closed subset T of D we get

a № _1(0/(r, U{t)x): teT, xe X }) ^ ca(/(Tx Y)) ^ c sup h(t)x(Y)

teT

^ cd sup h(t)a(X).

teT

Therefore >u ([0, a]\(a — D)) < e and I

ot(g(TxX)) ^ cd sup h(a — t)cc(X) for each closed subset T of a — D.

teT

By Theorem 2 of [11], from this it follows that for each те[0, a) and yeE there exists a solution x of the Cauchy problem

x' = g( t, x) , х(т) = у,

defined on some interval [т, т + <5]. On the other hand, by (13), we have

||g(t, x)|| ^ cm(a — t) + cdq(a — t)\\x\\ (0 ^ t ^ a, xeE),

so that any solution x of (14) defined on [0, rj), rj < a, satisfies the inequality t

||x(r)|| ^ ||x(0)|| + §(cm(a-s) + cdq{a-s)\\x(s)\\)ds.

о Thus, by Gronwall’s lemma,

11 t

||x(r)|| ^(||x(0)|| + §cm(a — s)ds)exp(\cdq(a — s)ds) for îe [0 , rj).

о

From the above we deduce that there exists a solution x of (14) defined on [0, a]. Consequently, the function t - + y(t) — U { a - t ) x ( a - t ) satisfies the integral equation

i

y(t) - U{t) U ~1 (a)u{a)+ §U(t) U ~ 1 (s)/(s, y{s))ds for te[0 , a],

a

and finally the function v defined by

v(t) = y(t) u(t)

if 0 ^ t ^ a, if t ^ a, is a solution of (1).

Remarks: 1. If и: [a,

-» E is a bounded solution of (1), then routine computations show that the function w defined by

O O

w(f) = u{t)— J G(t, s)f(s, u(s))ds (t ^ a)

a

П — Prace matematyczne 26.1

162 S. S z u f l a

is a solution of (2). Arguing similarly as in the proof of (12), we conclude that lim ||w(t) — u (Oil = 0.

t -► ao

2. Condition (6) is completely natural, because it holds if / = f x + f 2, where f x satisfies (5) and f 2 is completely continuous.

3. For the case В

1) and p{r

cr (c < 1), Theorem 1 reduces to a result from [6].

4. In Theorems 1, 2 or 3, В may be any Orlicz space Lv generated by an N-function cp (cf. [8]).

5. Putting, in Theorem 2, E = Rn and В = M p (p > 1), where M p is the i + i

space of measurable functions x: J —► Rn with sup( j ||x(s)||pds)1/p < x , we t^O t

obtain Theorem 1 of [5] (because in this case

oo i + 1

l|G(f, •)||e- ^ E ( f HG(C s)\\qds)ll4t where q = p/(p- 1)).

/=0 /

\

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