On st,ability and L v solutions of o r d i n a r y differential equations. (1) THOMAS G. HALLAM: (U. S. A.) (*)

Summary. - The usual definition of the stability of a solution of a system of ordinary diffe. rential equations is extended by introd~¢cin~ two positive control functions. These functi. ons are used to control the +'ate of growth of the initial position of the solution a~d the rate of growth of the solution. Definitions and results are also given for the cot. responding analog~ees of boundedness, weak bo~endedness, and uniform properties of the so. tions of differential equations. The problem of determining when solutions of certain linear and weokIy q~onlinear differential equations lie in a modified Lp-space is also considered.

1. -

Introduction.

The purpose of this article is twofold: first, we will introduce a seemin. gly natural and apparently new modification of the definition of stability and indicate some consequences of this extension. Secondly, the problem of dete. rrnining when the solution~ of certain linear and weakly nonlinear differential equations are in the space L[[o, ~ ) (actually, a modified L P - - s p a c e ) will be considered. This joint development is somewhat inherent in this general question since, as the results below signify, (~uniform stability T> may be regarded as a limiting case in the investigation of LP - - solutions of differential equations. W e will now give a brief historical sketch of the problem and indicate its development as determined in the references. The relationships between the solutions of the differential equations tl.1)

d : c / d t = A(I)x, t ~ to ;

(1.2)

d y / d t = A(t)y + b(t), t ~ t o ;

(1.3)

d z / d t --~ A(t)z + f(t, z), t ~ t o ;

have been widely investigated since the stability perturbation problem was formulated by PERRON [13] in 1930. COPPEL [5], using a fundamental result of MASSENA and SC~A)~FER fill, developed an effective criteria to determine the stability or instability of the zero solution of Equation (1.3}. (1) This research was suploorted by the l~ational Science Foundation under grant GP8921. (*) Entrata in Redazione iI 13 maggio 1969,

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T. G. HALLAM: On stability and L p solutions o/ ordinary, etc.

As a motivation for his hypothesis, he also investigates the connection between a fundamental matrix XIt) associated with (1.1) and the admissibility of the pair of BANACrr spaces (L ~, L~t for the Equation [1.2). In particular, COPPEL proved: The pair (L ~, L ~) is admissible for (1.2); that is, the inhomogeneous linear equation {1.2) has at least one bounded solution for every bounded, continuous function b(t), if and only if there exists supplementary projections [1, P2 and a positive constant K such that for t~>to

tO

t

This result was also obtained independently by BRIDGLAND in [43. In his re. cent monograph [6], p. 131, COPPEL discusses the admissibility of the pair (L1,L~). Along this same line, CoaTI [7] considers a projected Lq norm on the expression in (1.41 and demonstrates the admissibility of the pair (LP, L -) where p-1 + q - l ~ 1. In this article, in a more general setting but by similar methods as used by COPPEL, BRIDC~LA.ND, and CoN~I, we obtain a condition which yields the admissibility of the pair (q)p, Wrt where q)P, Wr are snitably modified L P, L r spaces (see definitions below). An investigation of weakly nonlinear systems which possess a similar <
T. O. HALLAM: On stability and L p solutions o] ordinary, etc.

309

matrix at t = t ~ , will be written X(t; h). The modified LP-space which we will frequently use is defined as follows: Let +---~b(t) be a positive continuous function from I to R 1. The BA~AC~[ space IFe(X) consists of all measurable functions x(.) from I to the m-dimensional Euclidean space X with the property that I + - l ( . ) x ( . ) l e L p ( I ~. The norm of

,J?-~(s) x(s)I e ds ~, 1 ~ p < ~ and sup I q~-~(t) x(t) I when p = ~ .

x e We is

to ~

A

t

to

corresponding space (bp(X) with the associated function ~0--~~(t) will also be used. In our utilization of We(X), the underlying base space X will be evident from the context; for this reason we will usually depress the X and write We instead of WP(X). A subspace Wo which consists of all elements of W ~ such that [~b-l(t) x(t)]--->0 as t-->c<~ is also useful in our considerations. The pair of BaNACK spaces (B, D) is called admissible for the equation (1.2) if for every b e B , Equation (1.2) has at least one D-solution (i.e., a solution in D). W e define a family F of functions (from which the function f of Equation (1.3) will be chosen) to consist of all continuous //~-valued functions defined on I X R " which satisfy the following properties: (1.5) There exists a scalar valued function ). such that If(t, zl) --

f (t, z~) [ ~ X(t) Izl -- z21

for all z~, z2 ~ / / " . (1.6) The function If(., 0)1 is in (I)p. In (1.6), if p is fixed in an argument it will be specified; otherwise, it is to be regarded as arbitrary, l ~ p ~ c ~ . 2.-

Stability.

In the development of the theory of systems of ordinary differential equations various scalar valued functions have played an important role. Two more of these scalar functions will now be introduced into a concept which occupies a distinguished position in the field of differential equations, namely the concept or stability. For the succeeding definitions, we consider a system of differential equations

(2.~)

dw/dt---=g(t, wt, t ~ t o ,

where w, g are n - v e c t o r s and g(t, 0)=--0 for t~__to. The zero solution of (2.1)

310

T. G. HALLAM: On stability and L~ solutions of ordinary, etc.

is called stable provided for each given ~ > 0 there exists a ~=~ltl, ~) such that whenever t w l ] < ~ then the trajectory w(t; tl,wl) through (ll,w~} exists and ] w ( t ; t ~ , w ~ ) [ < ~ for t ~ t l . This definition will now be extended by introducing positive continuous control functions ~ = ~ ( t } , ~ = ~tt) in the following manner. DEFINITION 2.l. - The solution w ~ 0 of t2.1) will be called ('¢, ¢)-stable if given any ~ > 0 there exists a positive n u m b e r 8~ =8~(lz, ,) such that if w~ satisfies the inequality [w~l < ~9~(t0 then w(t; t~, w~) exists and satisfies the inequality [w(t; t~, w~)l < ~¢(t~ for all t~__t~. DEF[N~IO~ 2.2. - The solution w ~ 0 will be called uniformly (?, ¢)-sta. ble if the ~ero solution of (2.1) is (% ¢)-stable and ~-----~(~); that is, ~.~ is independent of initial time t~. REMARk: 2.1. - The above definitions reduce to the usual <> and (~uniform stability >> if we take ¢f~ ~ = 1 as the control functions in Definitions 2.l and 2.2 respectively. From the Definitions 2.1 and 2.2 the notions of <) (provided g is linear in w) may also be obtained by taking respectively, z(t} ~ 1 and ¢ ( t ) = o(1} as t--->c~, in Definiton 2.1 and ~(t)=~(t)=e-% for some a > 0 , in Definiti'on 2.2. The next definitions are obvious (% ¢)-analogues of equiboundedness (see YosI~IZAWA, [18], p. 36) and weak boundedness (see HALLAM, [9]).

D E F I ~ I O ~ 2.3. - The system (2.1) is (% ¢)-bounded if given any a > 0 there exists a ~+-----~¢(tl, ~) > 0 such that whenever l w~ I < a ~(t~) then w(t; t~, w~) exists and [w(t; t~, w~) I <~¢~(t) for all t ~ t ~ . DEH~O~ 2.4. - The system (2.1) is uniformly (% +)-bounded if (2.1) is (% +)-bounded and ~+ is independent of t~. DE]~X~Z~ZON 2.5. - The system (2.1) is (?, ~)-weakly bounded if there exist functions 7 ~ 7~tt~) and p¢= p+(t0 such that whenever t w~ t < 7~(6) ~(t0 then w~t; 6, wl) exists and satisfies Iw(t; t~, w~)[ < p+(t~)¢(t) for t ~ 6 . D E ~ ] ~ O ~ 2.6. - The system (2.1) is uniformly (% ~)-rceakly bounded if (2.1) is (?, ~)-weakly bounded and the functions 7~, p+ are independent of 6. It is clear from the definitions that if system (2.1) satisfies either Definition 2.1 or Definition 2.3 then Definition 2.5 is also satisfied by (2.1). An analogous statement applies for the <~uniform)> properties. As the next results will be specialized to Equation (l.l), we note by the Principle of Superposition we may discuss the (% ¢)-stability of (1.1) rather than the (% ~)-stabitity of the ~ero solution of (1.1). For the linear homogene-

T. G. HALLAM: On stability and L p solutions o] ordinary, etc.

311

ous differential equation (1.1) we obtain the following equivalences which are well k n o w n for ordinary stability; see [9]. TltEOI:tEM 2.1. - The following conditions are equivalent:

{2.2) The system (1.1) is (~, ,~)-slable. (2.3) The system (1.1) is (% ~}-bounded. (2.4)

The system (1.1) is (% q~)-weakly bounded.

PRooF. - To show that the theorem is valid we will establish the impli. cations (2.2}-+ {2.4)-+ (2.3)---> (2.2}. As r e m a r k e d above, we have the implication ~2.2).-->(2.4) being trivially satisfied. To verify that (2.4)-+[2.3) we suppose that the functions 7~ and ~+ play their roles as in the Definition 2.5. Then, since x(t; t~, x~)~-X~t; tl)x~, we have

Ix(t; t~, x~)l= IX(l; t~)x~ I< e+ +(0 whenever t x~t ~ 7¢ ~(6). This implies that

IX(l; t~) ~(l,)!~r( ~~++its; hence, if {x~t ~ ac~(t~) then

I~c(t; t~, ~,i)1 <- Ix(t; t~) ~(t~) f t ~-~ (t~)~1 f --1

which establishes (2.3) with ~+ ='1,7 ~ ~+ a. Next, we consider the implication: {2.3)-+(2.2). Let ~, ~4 be as in Defini. tion 2.3; then, writing

Ix(t; 6, x~)l = IX(t; 6) ~16~ ~-%) xl f we obtain

Ix(t; t~) ~(t~) l <_ ~-~ ~+ +(t~, t ~ t~. Therefore, by taking ~ ¢ : ¢ ~ 1 , (2.2) follows; this completes the proof of T h e o r e m 2.1. The next result is an analogue of Theorem 2.1 in the uniform (% +)-stable case; however, in this instance, we are able to obtain additional information on the relationships between Equations (1.1), (1.2), and (1.3}. THEOREM 2.2. - The following conditions are equivalent: (2.5)

System (1.1) is uniformly (~; ~)-stable.

(2.6)

System (1.1) is uniformly (% +)-bounded.

(2.7)

System (1.1) is uniformly (~, q;)-weakly bounded.

312

T. G. HALLAM: On stability and L p solutions o/ ordinary, etc. (2.8)

The function r(tj=]X(t; s ) ~ ( s ) l i s in W% t ~ s ~ t o ; exists an M > 0 such that

that is, there

Ix(t; s) ~(s) I ~ ~ +ttl, t ~ s >_ to .

(2.9)

All solutions of Equation (1.2) are in class ~ class 091 .

for every b in

(2.10) All solutions of (1.3) are in class W" for every function f in F where it is required that ) , 1 - = ~ and t f{', 0)1 are in 09~. PRoo:F. - The proo[ that the conditions (2.5), (2.6), (2.7), and (2.8) are all equivalent follows by the arguments used in Tneorem 2.1; in those proofs, note that in this instance all of the functions ~.+, ~+, %, ~+ are independent of the initial time t~. To complete the proof, we establish the circular implication: (2.10)-+(2.9) (2.8) -+ (2.10). Since, for any b in ff91, b is also in class F as defined by condition (2.10) we obtain (2.9} immediately. Suppose now that (2.9)holds; then, the choice of b(t)~O implies that X(t; h) is in class ~"=. H e n c e , from the equation -+

t

(2.1 1)

+.-~(tty{t) = ~.-l(t~ X(t; tl) yl + f ~-l(tl X(t; s) b',s) ds t1

we conclude from (2.9) that t

f +-1(t) x(t; s) b(s)ds

(2.12)

tl

is bounded for all b(t} in 091. An application of the BANAC:K-STEINHAUS Theorem (see BELL)L~N [2] P. 517 for a particularly appropriate formulationl to (2.12) implies that (2.8) is satisfied. It remains to establish the implication (2.8)-+(2.10). By the variation of parameters equation, for any f e F , any solution z(t) of (1.3) may be written as

(2. ~3)

+-1(0 z(t) = ¢,-1¢t) x(t; t~l z~ t

+ f c,-ltt) xf~; s) f(s, ~(s))ds, t~h. t1

Applying (2.8) and the defining properties of class F in (2.10) we have t

1+-l(t)

+ M f <(s>'
T. G. HALLAM: On stability and L ~ solutions o/ordinary', etc.

313

where

An application of the BELLMA~-GRo~w.~LL inequality to the above inequality establishes that z~ q;~ and completes the proof of Theorem 2.2. The next r e m a r k is concerned with the existence of qY*~-solutions of Equation (1.3). Although the hypothesis involving weak boundedness is apparently new, the comparison technique used in the proof is well- known. RBMAIaK 2.2 - Suppose that condition (2 8) is satisfied; let the function f of Equation (1.3) satisfy the inequality

(2.14)

~-~(t)[f(t, z)[
whore g(t, .) is nondecreasing. If the scalar differential equation

(2.15)

v' = g(t, v)

is (1,1)-weakly bounded then any solution z(t) of (1.3) with ]z(t0t sufficiently small is in W ~. PaOO:F. - From the Equation (2.13) it follows that

1+-'(t) ~(t) t ~ z~f r-'(t,)z, 1+ ~1yg (s, t '~-'(s) z(s) t) as. to

Therefore~ applying an integral inequality, (see for example, COPI~EL [6] p. 35), we obtain

1+-qt) ~(t) l ~ v(t) where v(t)is the maximal solution of (2.15) satisfying the initial condition v(tl)=Ml~-l(t~)zl!. The hypothesis that (2.15) is (1,1)-weakly bounded implies that there exists a T > 0 and a p > 0 such that if [v(t~)I
then any

solution z(t) of (1.3) having sufficiently small initial condition is in W ~'. In particular, if r = 1 then all solutions are in W 0~.

Annali di Matematica

40

314

T. G. HALLAM:

On stability and L p solutions of ordinary, etc.

PROOF. - The differential equation ded provided

la(t)dt~e~

v'= a(t)V, r ~ l ,

is (1,1)-weakly boun.

as may be verified directly by solving the differen-

tim equation. Since

~-'(t) l f(t, zit)) 1<_ ~-~(t) ~(t) +~(ttl +-~(t) z(t) i~ this remark is a consequence of R e m a r k 2.2. REMARK 2.4. - If, under the conditions of R e m a r k 2.3, we have also that

Xfl ) is in W0~ then any solution z(t) of (1.3) which is in Wzc is also in W% O P R O O F . - For given ~ 0

(2.16)

T~tl

select

sufficiently large so that

-5I~1~-1{t~) z(h) lf T-a(t) ~(t) +~(t)dt < el2; T

here p is the bound on the solutions of the comparison equation whose exi. stence is guaranteed by the weak boundedncss hypothesis. If l ~ T, we have

I+-l(t) ~(t) t ~ l +-t(t) x(t; l~) ~(tl) l

(2.17)

T

+ f L+-1(0 xtt; s) f~s, ~(s))Lds tl

+ f l ~-~(tt x(t; s) f(s, z(s)t Lds. T

The first two terms on the right of (2.17) can be made small by choosing t large; in particular, let 2'1 be such that if t ~ T I

]+-1(0 x(t; h) z(tl) l < ~/4 and T

f l+-'(t) x(t; s) f(s, z(s))i ds < ~/4. tl

The last term on the right in {2.17) is majorized by (2.16). The combination of these inequalities shows that z(l) is in W0~. As a concluding remark on {% ~)-stability we again note that only (1,1}stability (usual stability) and uniform (e-% e-~')-stabili|y (exponential asympto-

T. G. HALLAM: On stability and L p sohttions o] ordinary, etc.

315

tic stability) have been widely investigated. Physically, it appears that in certain circumstances, other types of (~, +)-stability might be equally as useful. The introduction of the functions ~, ,~ also permits a discussion of the stability of a linear system of differential equations with constant coefficients when the characteristic equation has multiple roots. For example, if the linear system (1.1} with A(t) constant has a characteristic root with real part a, a > 0 , of multiplicity n then the function ~ may be taken as P-~ e -~.

3. - W~ S o l u t i o n s .

Section 2 developed some results related to finding solutions of a differential equation which lie in a given space ~'". In this section our c o n t r i b u t i o n is in finding solutions which lie in the modified L~-spaces t p , l ~ r ~ . Criteria will be developed to determine when there exist tIYr-solutions and, in the extreme case, when all solutions are in ~ . To this end we shall use the following notation and terminology. Let X~ denote the subspace of R ~ consisting of all vectors which are initial values at t ~ t o of tIY~-solutions of (1. U. Let X2 designate a subspace of R = complementary to X~ and suppose P~, P: are corresponding projections of R ~ onto .7(1. X2 respectively. Associated with these projections we have the matrices X(t)P,X-~(s), i--~ 1, 2, which will be denoted by X~(t; 8), i~-.~--1, 2. For the r e m a i n d e r of this article p is a given number, l ~ p ~ , and q its conjugate where p-~ A - q - ~ 1. Define

eft) -~

I;

11

t Xfl; s} ~,(s) Iqds q-, t ~ to;

to

~2(t) =

[f

1

x2(t; s) v(s)f~ ds ;, t ~ to;

where for utility and convention we assume that when p----t and~ hence, q~-cx~ the above expressions are ~l(t) =

sup

Xl(l; s) ~(s) l;

o2(t) =

sup Ij x2(t, s) ~(s)1.

In the next result the family of functions F as defined by the previously specified properties (1.5)and (1.6)wilt be considered. A further restriction

T. G. HALLAM: On stability and L ~ solutions o] ordinary, etc.

316

which will be placed upon this family is that there exist constants a > l , ~ ' ~ 1 with ~-:~-~ (£)-~ ~ l such that k~ ~ ~ , where ). is the LIPSOttlTZ function of condition (1.5}, is in ~aq/~-~ AffP~'r/~-~'. The existence of a pair a, a' which have this property is guaranteed if r ~ p . For in tiffs instance, q - ~ - ~ r - ~ l ; anti, hence, any choice of a, £ such that ~z~q, : ¢ ' ~ r will suffice. Also, when r .~p, we understand that a = q, £ = r and ~P~q/~-a------~'~/~-~" = (I)~ . With respect to members of this family F, we obtain the following theorem on the existence of W~-sotutions of (1.3).

THEORE~ 3.1.

-

If the functions p~(t) exist and are in class q;~ then there

exists a 1P-solution of (1.3) for every choice of f in F. PI~OOF. The proof is similar to the ones given by 0OPPEL [6], Co~rt [7], and STAIKOS [14] who considered the limiting case L ~. We will restrict our development to the eases 1 ~ r < ~ because the techniques used for the space ~ requires only a slight modification of the above mentioned arguments for L% We define for z 6 W~, -

t Ts(t) = f Xl(t;

8) f(s,

z(8))d8

--

fx2(t;

s) f(s,

8(8))ds.

The following inequality shows that T is well defined and maps W~ into itself. t

to

to

V

1

to

i 1+-~(v) X2(v; s) ~(s) l I ~-~(s) f(s, o) 1ds I~dv ;

+ to

+

t "

~t

v

(;/

I ]+-~(v) XI(v; s) ~(s) I ~-~(s) Z(s) +(s) f+-l(s) z(s) lds l~dv 7

~0

+

tO

i I+-1(v) X2(v; s) ?(s)[cFI(s) [(s) +(s) I+-~{s) z(s) i ds {rdv ; to

v

if(', o)!¢1 ~ I ¢ + If(', o)I,~t ~ I ~ t

t~ ~q

~

v

to

T. G. HALLAM: On stability and L p solutions o] ordinary, etc. v

v

to

+

1

tO

[f(f (f[

)' (f{

t

tj

•

'

317

1+-1(v) X2(v; s) ~(s))~ ds ~

~

]~°" )' ) 11

~-'(s) ),:(s) T ds ~'.

v

,l(f

~-:(s) X:(s) T ds);~

v

t +-:(s) z(s)Z~'~"ds ~-~ tt dt 7

v

where 0~-:+(£}-~=1, :3-:+ {~')-:= l, y-: + (Y')-: = 1, and ~----q~-:, y = r ( r - - £ ) . Continuing the above inequality, we have t

1

~o

t

to

tO

t

~0

e,~

v

wherein

B:----if(', 0) 1,~ liP: iy + IP~Iv~];

Next, we show that T is a contraction on W~. The defining properties of F and a procedure as used above establishes the inequality

+-:(t) t Tz~(t) -- Tz2tt) l ~ B2~-:(t) ~ : ( t ) I f [ ~-:(s) [ z:(s} - z2(s) I]r ds -; to

318

T. G. HALLAM: On stability and L p solutions of ordinary, etc.

&n application of the MINKOWSKI inequality leads to

to

to

If to is sufficiently large (and we assume that this is the situation), 0 < 1, and T is a contraction mapping. H e n c e , there is a solution z e t I ;~ of the equation z = Tz. It is readily verified that such a function is also a solution of Equation (l.3). This completes the proof of Theorem 3.1. REMARK 3.1. - The specialization of Theorem 3.1 to Equation (l.2) leads to the following result. If ~.(t) exist and are in t p , i~---1, 2, then the pair ((I)p, W~) is admissible for Equation (1.2}. In fact, for any b in (I)P, the solution y{t) of (1.2) given by

to

t

satisfies the inequality

l Y(t} ] ~ ]b I,~ [~(t) q- ~2(t) ] . RE~A.RK 3.2. - In general, the converse of the result in Remark 3.1 is not true; although for the space ~ r _ ~ L ~ it has been demonstrated to be va. lid by COATI [7] (see also, COPPEL for the spaces ~ p = L 1 of L~}. As a counterexample to the general statement, consider the sFaces ~ p = ~ ' ~ = L ~. For every b in L ~, the differential equation

d x / dt --~ x -t- b(t)

(3.1) has an Ll-solution given by

x(tl = -- f e

b(s) ds.

t

The fact that ~v is in L 1 may be seen from the following inequality.

tD

~0

t

T. G. HALLAM: On stability and L ~ solutions of ordinary, etc.

319

fo-s,b(s,t

dt ds

~0 tO

~ ;b(s) l ds . Hence, the pair (L ~, L ~) is admissible for Equation (3.[). However, the functions p1, p2 are given by ~1 ~ 0, p2 ~ l; but~ ~2 is not in L 1. REMARK 3.3. - The proof given for the above theorem can be extended to demonstrate that there is a homeomorphism between the W'-solutions of Equation (1.2) and those of the equation

(3.2)

~'= A(O~ + b(t) + f(t, z);

for if y(t) is any function in Wr then the above proof shows there exists a unique solution of the equation (3.3)

z(l) = y(t) .4- Tz(t).

Farthermore, z'(t) - - y'(t) .--- A(t)[ z(t) - - y(t)] -4- f(t, z(t) ), so that y(t) is a solution o[ ~1.2) if and only if z(t) is a solution of (3.2). The continuity follows from subtracting the equation zo=yo-[-Tzo from (3.3) and applying the MI~KOWSKI inequality. In the instance that the space (I;'~--L ~ this result appears in COppEr, [6], p. 76. EXAMPLE. AS a simple illustration of a system which exhibits the behavior demonstrated in the previous theorem, we consider the homogeneous system -

t L dtj 8x2

wherein :¢> 1, ~ ( a - - 1 ) q

t~>to>O, 0

~t ~-~ - - ~t -~

X2

-1, and ( ~ - - 1 ) r > q. T h i s system has i e- ~

f-~ 0

0 ) e,~ t-~ J

320

T. G. HALLAM: On stability and L p solutions o] ordinary, etc.

as a f u n d a m e n t a l m a t r i x ; a n d as projections associated w i t h the subspaces X l , X2 we have

pt=(o o), p =(o 10

A r o u t i n e c a l c u l a t i o n shows that

to

and 0o

p2(t) =

If

It

X2(t; s)lqds q ~ K 2 t - ~

t

which implies that both pl, p: are in L ~. N e x t , the e x t r e m e case when all the solutions of E q u a t i o n (1.1) are in q;~ will be c o n s i d e r e d . F o r the space L ~, this p r o b l e m h a s been considered by PERRO~ [13], B E ~ L ~ L ~ [2], CONTI [7], STAIKOS [14], and others. S i m i l a r results for the space W ~ were developed by HALLAM in [8]. In o u r c o n s i d e r a t i o n s we will use a ~P sqace a n a l o g u e of a l e m m a due to COPPEL [5] whtch was valid for the space L ~. LEMMA 3.1. - I f

~(0 =

Ixlt; s) ~(s)Iq ds to

is in W r lhen IX(t)[ is also in the space W r. PROOF. - Defi~:e hit)-~-~q(t)IX(t)i- q ; t h e n we write t

f h(s)dsX(t)=f I~-l,s) X(s)i-q X(t)X-1(8)~st) ~-l(s) X(s)as to

to

Using the ~7~0LDER i n e q u a l i t y , we obtain t

13.41

t x(~tl<--

h(s) ds-q-~(O. to

T. O. HALLAM: On stability and L p solutions o{ ordinary, etc.

Therefore, it suffices to show that

I;

t

321

11

h(s)ds-~ is bounded.

to t

Denoting by ~(t)-~fh(s)ds, from (3.4) we obtain to

1

(3.5}

[ ~(tt ]~-~ tx(t) l ~-~(t)~ ~-~(tl ~(t}.

Using the fact that 1

[ ~'(t)]-7 = IX(t) ~-~(t) t (3.5) becomes 't where to(t)= ~-~(t) ~(t). An integration leads to the inequality t

t

to

to

for some constant K; this completes the proof of the lemma. In the next result, the function ) q = ~ . ~ is required to be in the class ¢~ where z - - ~ if p----r and ~ = p r / r - p when r > p . t

TI=IEOREM 3.2. - I f the function ~(t)----- [Xlt; s)¢~(s)Iq ds

is in the class

,o W r, the~ all of the solutions of (1.3) are in the class Wr, r ~ p ,

for every f in F.

P R O O F . - W e shall only give the proof for q and r finite since the boundedness cases have been amply discussed by the previously mentioned authors. Even though our results appear to be new for the ~'~-case the modification required by the introduction of the scalar functions are routine. In the course of the proof, we will use a useful result due to WILLETT and Wol,TG [17] which we state as LEMI~A 3.2. - Let the functions v(t)uP(t), v(t)wP(t), and v(t)UPo(t) be locally

integrable functions on I = fro, c~). Then, the following inequatily for 1 ~ p < c~

(;

t

u(t} ~__uo(t) -~ w(t)

)1

v(s) u~(s) ds ~ (t e I)

to

AnnaIi di Matematica

41

322

T. O. HALLAM: On stability and L ~ solutions o] ordinary, etc.

implies that t

v(s) u~(s) e(s) ds -~ (f v(s) up(s) as 7 ~ to

(teI),

]

1 - - (1 - - e(t)}~

to

where t

to

b~rom the defining property of F and the variation of parametels equation w e ]flare

13.61

Iz(Ol~ {xit; to) zol t

+ flxtt; s) i[lf(s, O) I + X(s)[z(s)l] ds. to

By applying I~()LDERS inequality, we have

f

Ixtt; s).~(s)l{ ~-x(s) f(s, O~{ds

(3.7)

to

B~ ~(0 , B~ = ! f( ., o)t+p;

and t

f

(3.8)

iX(l; s) ~(s) [¢~-~(s) X~(s) t ~-'(s) z(s) l ds

t,,

t

t0

Inserting (3.7} and (3.8) into (3.6) yields

t+-1(0 z(OI ~ T+-~(t) x¢; to) zo i

(3.9t + B1 +-1(0

{ ft

] 1

[

1

~(,) + +-'(t)~(t, { |[ ~-1'8)xl(s)]p i +-t(8)z(8){p ds{;. to

T. G. HALLAM: On stability and L p solutions o/ ordinary, etc.

323

First, we will consider the case r = p ; here, i l ( t ) E ~ ~ (let B2 be an upper bound) so an application of the W I L L E ~ - W o ~ Lemma 3.2 leads to

(a.lo)

[ +-'(t) z(t) l ~Uo(t) + Bz+-~(t) 9(t) u~(t),

where

uo(t) = [ +-,(t) x(t; to) zo t + B1 ~(t) , and t

u,(t)

:

If':(', '-, ( "~k'('),',',~v)"l ~ l_f~_°x,(_,,..#(~>,~)t~ 'o

'o

t

tO

By virtue of Lemma 3.1 and the hypothesis, Uo is in q~r; from this it is easy to see that udt) is bounded. It now follows from (3.10) that z ( t ) 6 ~ in the case r - - p . If r : ~ p , we apply the MI~KOWSKI inequality in (3.9) to obtain t

If

(3.11)

I +--1(s)

~(8)I r d~ ~ ~ I X ( t ) I ~ , r ~ X(go)~0 I + B1 [~ [vr

I~

to t

v

to

1

to

By the H(iLDER inequality, for ~ - 1 + { ~ , ) - 1 = 1 with ~ = r / r - - p , v

v

[>1(8, ~ (8,I. + l(s, °~s, .~:.~

~ ..(f+.(8,

to

to

Putting this result into the inequality (3.11) leads to t

1

~o

t

~o v

•

(;,, I,I,l > ]~ -1 8 z s

tO

rds d v 7 ,

ols, t dsl

324

T. G. HALLAM: On stability and L p solutions of ordinary, etc.

where B3 is the sum of the first two terms on the right in (3.11) and B4~--An application of L e m m a 3.2 yields the following continuation of the above inequality. t

If ¢0

ao

1

I ds ~

t1-- l1 - - c x p i f

B4

t0

tlll-I

This shows that z ~ W ~ and completes tbe proof of the theorem REMARK 3 . 4 . - The restriction r ~ p in Theorem 3.2 is unnecessary if Equation (1.2) is considered rather than Equation {1.3). The next result presents another simple application to ordinary differential equations of L e m m a 3,2. The perturbed equation (1.3) will be considered where we shall require that

(3.12)

If(t, z)l ~,(t)lzl,

rather than the Lipschitz condition (1.5) of the family F. RE~AR~: 3.5. - Let p(t) be in ~r and {3.12) be satisfied, if ),t(t)~-)qt),~-l(t) is in class (I)~, with ~-~-~ when p-~-~r and z - ~ - p r / r - - p when p < r , then all of the solutions of (1.3) are in W~. PROOF. - The details of the proof are quite similar to the last part of Theorem 3.2. W e shall only sketch the argument here. F r o m the variation of parameters formula we obtain for t ~ t ~ t o ~

f

11

l ~{t) l <_ IxIt) t x-~It~) ~ i + ~(t) ( [ ~-~(s) x~(s) ? [ +-~(s) z(s) l~ as -; to t

tO

The use of Lemmas 3.1 and 3.2 yields the desired conclusion. It should also be pointed out that the very general BANACg space results of t t A ~ I A S r and ONUO~IO [10] could be applied here to obtain similar results.

T. G. HALLAM: On stability and L ~ solutions o] ordinary, etc.

325

REFERENCES

[1] ]~. A. ~NTOSIEWICZ and P. DxvIs, Some implications of Liap,~nov's conditions for sta['2]

[3] [4] [5] [6] [7] [8] [9] [tO] [11] [12] [t3] [1~] [15] [16] [17]

[18]

bility, J. Rational Meeh Anal. 3([95~)~ i47457. g . BULL~{AX, On an application of a Banach-Steinhaus theorem to the study of the bo~dedness of solutions of non,linear differential and difference equations, Ann. Math. 49(1948), 515-522. T. I~t. BglDaL.~ND JR., Asy~nptotic behavior of the solutions of nonlinear differential equations, Proc. Amer. Math. Soc. 13(1962), 373-377. - - - - , 0~ the bo~ndedness and uniform boundedness of solutions of nonhomogeneous systems, J. Math. Anal. Appl. 12(t965). 471-487. W . A. CoPP~L, 0 a the stability of ordinary differential equations~ J. London Math. Soc. 38{1963), 255-260. -- .--, Stability and asymptotic behavior of dif/erential equations, D. C. Heath, Boston, 1965. R. CONT,, 0n the bo~tndedness of solutions of ordinary differential equations, Funkcialaj Ekvacioj 9(1966), 23-.26. T. @ K~LLAH~ 0 a the asymptotic growth of the solutions of a system of nonhomogeneous line:~r differential equations, J. Math. Anal. Appl., 25(1969), 254-2~5. - - - - , YVeakly bounded systems of differential equations~ Proc. Amer. Math. Soc., 19 (1968), 1242-1246. P. tthaT~hN and ~ . ONucmc~ On the asymptotic integration of ordinary differential equations~ Pac. J. Math. 13(1963), 517-573. J. L. MhSSEaA and J. J. SCHAFFnR, Linear differential equations and functional ann. lysis I, Ann. Math. 67(1958), 517-573. - - - - and - - - - , Linear differential equations and functional analysis IV, Math. Ann139(1960), 287-342. O. PEaRON, Die stabilitatsfrage bet differe*~tialgleichu~gen, Math. Z. 32(1930), 703-728, V. A. STA~KOS, A note on the boundedness of solutions of ordinary differential equations. Boll. U . M . I . 2(1068), 256-261. A. S'raAuss, Liapunov functions and L p solutions of differential equations, Trans. Amer. Math. Soo. ]19{1965), 37-50. D. WILLETT~ Nonlinear vector integral equ.~ltions as contraction mappings, Arch. Rational. ]~[ech. Anal. 15(196t), 7~!-86. D. ~ILLETT and J. S. W . W o s ~ , On the discrete analog~tes of some generalizations of Gronwall's inequality, Monatsh. Math. 69(~65), 362-367. T. ~ffOSI-IIZAWA,Stability Theory by Liap~nov~s Second Method~ Math. Soe. of Japan, 1966.