Rend. Sem. Mat. Univ. Pol. Torino Vol. 48°, 2 (1990)

Dynamical Systems and O.D.E.

M.Cecchi - M.Marini

ON THE CONTINUABILITY AND BOUNDEDNESS OF SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS

A b s t r a c t . In this paper the functional differential equation

*"+/(«,s(0(oy(M*))=o

is considered without coercitivity assumptions at infinity on the function / . Necessary

and sufficient conditions for the existence of solutions with certain asymptotic

properties are established.

Introduction

Consider the following functional..differential equation

x" + f(l,x(g(t)),x'(k(t))) = 0 (1)

and the special case

x" + r(0v**.(9(0), V W))) = 0 (1')

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where:

' g : [a,-foo) —• HI, g continuous, g(+oo) = -f oo ;

h : [a,+00) —* lit, h continuous, /i(-f-oo) = +00 ;

r : [a,-f-00) —* 1R, r continuous, r(t) > 0 ;

/ : [o,4-oo) x IR —• 1R, / continuous, f(t,u,v) > 0

for M > 0, v > 0, / E [a, + 0 0 ) ;

, <p; JR —• 1R, y? continuous ,c^(w,,v) > 0 for u > 0, v > 0 .

We are interested in the existence of positive increasing bounded solutions of (1) which are defined in some neighborhood of infinity (+00).

The equation (1) and the corresponding ordinary equation

x" + f(t,x,x') = 0 (1")

have been considered by many authors ([1], [2], [3], [5]-[22]).

Indeed equations (1) and ( l ' ) have wide applicability and arise in a variety of fields such as nonlinear mechanics, nuclear physics, boundary layer phenomena for viscous fluids, etc. An ample bibliography on the subject may be found in [5], [13], [17], [22].

Many results on the oscillatory and nonoscillatory behavior of the solutions of equations of the type (1) and ( l " ) are obtained by integral averaging techniques: the problem of finding conditions ensuring the continuability of such solutions has not often been considered. Indeed, it is assumed that every solution under consideration is continuable to the right, and is nontrivial, i.e. every solution x — x(t) must be defined on some ray [tx, -f 00) for some tx > /,Q, and sup {|a:(<)| : t > T} > 0 for every T > tx > t.Q (see, e.g., [8], [9], [10], [11], [19], [20]).

The aim of this paper is to give necessary and sufficient conditions for the existence of solutions x == x(t) with certain asymptotic properties on [tx,-\-oo). We shall use a topological approach which also gives us the continuability of such solutions. By means integral averaging techniques we will obtain a necessary condition in order that the equation (1) has eventually bounded

197

positive increasing solutions. Such a condition only involves the integral behavior of a suitable function q(t). Furthermore, by using a topological approach, also the sufficiency of the above condition is proved.

Finally our results are extended to the functional differential equation:

\p(tW(t))' + f(t,x(g(t)),x'(h(t))) = Q (2)

where

p : [a,-foo) —• 111, p continuous, p(t) > 0 . 7/2)

When the function p is continuous but does not have a continuous derivative, (2) is interpreted as the first order functional system:

f *'w = W)y{t)

\y\l) = -f(t,X(g(t))^y(k(t))).

In order to obtain results on the asymptotic nature of nonoscillatory solutions, the majority of the authors have often assumed the function / to be monotone and globally either superlinear or sublinear.

In this paper we shall not make this assumption, and, hence, the results here obtained extend some of previous results given in [15, Chap. 4 §5] and, when 7i = 2, given in [13].

1. Main result

We now prove the existence of eventually positive bounded increasing solutions of the functional equation (1) by means the existence of certain solutions of a suitable ordinary differential equation, which is obtained by a linearization device. A similar approach, but with a different linearization, has been used in [5], to prove the asymptotic decay of a class of solutions of a nonlinear equation with delay.

We will also use a general result of continuity and compactness of operators defined by a linearization device which reduces the existence of

198

solutions of a boundary value problem for differential equations in noncompact intervals to the existence of suitable a-priori bounds [4]. For sake of completeness we will first briefly illustrate this approach.

Consider the following boundary value problem

x E S

where J is a real interval, possibly unbounded, ifi : Jx 111 —• lit is a continuous, function and S is a nonempty subset of the Frechet space C (J,lit) of all real functions defined in J with continuous derivative. Assume that there exists a subset Q of C 1 (.7, lit) such that for every u G Q the boundary value problem

|>" = v>(*,u(0V(0) ,4)

xes [ }

admits a unique solution. Let T : Q —> S the operator which assigns to any u G Q the unique solution x = Tu'of (4).

Obviously the fixed points of T are solutions for the boundary value problem (3). Now, in-order, to use the Schauder-TychonofT fixed point Theorem, one has to show that:

i) there exists a closed convex subset 0 such that T maps Q into itself;

ii) T is continuous with relatively compact image.

Clearly condition i) is mainly related to the existence of a suitable a-priori estimates; on the other hand condition ii) is strictly connected with the topological structure of the space ''C}(J, lit). Now although in many case T may be discontinuous or noncompact, we have proved in [4] that the existence of suitable a priori bounds for the linear boundary value problem (4), guarantees the continuity and compactness of T in 17.

The following holds [4]:

THEOREM A. Consider the boundary value problems (3), (4) with the above

conditions. Assume that:

a) there exists a nonempty, bounded, closed, convex subset it of C (J,Hl) such that for any u G 0 there exists a unique solution x = T(u) of the problem (4), and T(u) 6 12;

199

b) for every {un} C ft and for every {x„} C T(Q) such that xn = Tun, Cl Cl

n E N, if un —• u and xn —• x, then x G 5. Then the problem (3) admits at least one solution defined on the interval J.

We can now prove the following

THEOREM 1. Assume conditions H\). Suppose that there exists c > 0 such that

y-foo / sq(s)ds < -foo where

If ) q(t) = max. / ( / , - t t ,u) , t > a . }

§<«<c 0<v<c

Then equation (1) has eventually bounded positive increasing solutions.

Proof. From 7/3) it follows that

/ q(s)ds < +00 and / J(s,UjV)ds-< -foo J a J a

c for - < u < c, 0 < v < c.

Choose IQ > a, t\ > t$ > a such that

r+00 c r+00

/ (s ~ to)q(s)ds < - , / g(s)(is < c •

and #(/) > /0 , h(t) > t0 for t > t \ . '

Let # and /i be the functions given by

' 9(t) for t > ti

<g{h) for t 0 < t < h 5(0 = {

( hit) for * > t\ h{i) -

[/?,(/,!) for t0 < t < t{

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Consider the subset U of C ' Q ^ O J + O O J J I R ) defined by

Q= {ue ( ^ ( [ /n ,+00) , l i t ) : £ < ti(l) < c , 0 < u'(t) < c, / > /,0}

Clearly 12 is a nonempty, closed, bounded, convex subset of C*([IQ, +00),1R).

For every u ^ f i w e consider now the boundary value problem:

' 5 i )

, 52)

53)

Clearly (5) has only one solution for any u £ Q. Therefore we may define an operator

T r a - C ^ m + . o o J i l R )

that associates to every u 6 il the unique solution y = T(u) of (5). We will apply the Theorem A to prove the existence of a fixed point of the operator T, which will be, clearly, a solution on [%, -foo) of the functional differential equation:

x" = -f(i,x(g(t)), AH*))h W

and so a solution on [/1,-foo) of (1).

From the variation of constants formula we have:

c /"+°° y(t) = 2 + (* - <o) y f(s,Hg(s)), u'(h(s)))ds

+ f (s-to)f{s,u(g(s))yu'(h(s)))ds.

*" = -/(«, # ) ) , «'(J(I)))

2/(<o) = 5

y+OO

!/'(<o)= / / (a ,u( j (*) , t i ' ( f t («)))d«. •/*o

(5)

c Obviously y(t) > - . Moreover:

•4-00

2/(0 < 5 + / (* " <o)/(«, «(5(*) j, ti'(£(*)))cte

+ I (s-to)f(syu{S(s))yy!(h(s)))ds

< 5 + / (*-*o)Mtt(SW),t*'(/iW))d * Jtn

Hence

Moreover

and so

+ /.(< -«o)/(», «(*(*)), *'(&(«)))<*«

< - + 2 / (« - «o)/(«,«(j(«)), u'(h(s)))ds

c f+°° c c < 2 + 2 / (3-t0)q(s)ds<- + 2- = c.

§ <•»(«) < «

/

+OO

/(*,«(g(«)),«'(A(«)))rf«,

/

+ OO

q(s)ds<c.

Then T(ft) C ft.

It remains to prove that condition b) of Theorem A holds, i.e. Cl Cl

{«»»} C ft, un -—• u, yn = Tun, yn —* y, then y satisfies 52) and 53). Clearly y(t0) = - .

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Since: r + OO

0n(*o) = / f(s,un(g(s)), u'n(h(s)))ds , Jto

taking into account that u (E fi, from the Lebesgue dominated convergence Theorem we get that 53) holds. Therefore there exists a fixed point x of the operator T, so that a solution on [<i,+oo) of the functional differential equation (1).

Finally, taking into account that for every u & tt the function y = Tu is an eventually bounded positive increasing function, we obtain the result. •

When n = 2 Theorem 1 extends a recent result in [13].

2. T h e e q u a t i o n b ( * > ' ( 0 1 ' + / (M(0(O)»* ' (MO)) = °-

Consider now the functional differential equation

[P(*)*'(0]'+.'/(*, *(»(0).*'(MO)) = o (2)

under the above assumptions for the functions p,f,g and h.

The following results on the existence of eventually bounded positive increasing solutions hold:

THEOREM 2. Assume conditions H\) and H2) and:

lim.infp(0 > 0 i/4) t—*-foo

Suppose that there exists c > 0 sucii that

•-f 00

I Q(S) / ~T~\dr ds < -f-00 where Ja Ja P(r)

q(t) = max f(t,u, v) . |<«<c 0<v<c

Ih)

Then equation (2) has eventually bounded positive increasing solutions.

203

Proof. The proof is quite similar to this one of Theorem 1. We hint only the principal steps.

From 7/5) it follows that

/ q(s)ds < -f-00 and / f(s,u,v)ds < + Ja Ja

00

for such - < w < c , 0 < v < c.

Let K = inf p(t). From H2) and i/4) we have K > 0. <£[a,+oo)

Choose <Q > ct» ^l > 0̂ >'fl such that

y-foo /"s 1 c 1 / + 0 0

k q{s)Lw)drds<*Kk q{s)ds<c

and #(<) > to,h(t) > to for t > t\.

Let #,/i be the functions as given in the proof of Theorem 1.

Let us proceed in the same manner as before: for every « E f i consider the boundary value problem

(W)y'(t)}' = - / ( « , % ( < ) ) , u'(A(0)).

:v(«o) = 2

1 /"fo° J/'(*0) = ^ - T / f{8,u(U»)),Ak»W» •

(7)

Let 1/ = Tu the unique solution of (7).

From the variation of constants formula we have:

y = Tu = l + J [/(*,«(*(«)). AH»)))J' ^ j d

+ [Jt ^jdr} [J*™ f(S,u(g(s)), u'(h(s)))dsj .

ds

204

Using the same argument as before we obtain

\<y{t)<c,

and •-f oo 1 f^°°

0 < y\t) <— q{s)ds < c . K Jt0

Finally in order to obtain the existence of a fixed point of the operator T it suffices to proceed in the same manner as in the proof of Theorem 1. •

If 7/4) fails, then Theorem 2 cannot be applied. Moreover in this case the existence of eventually bounded positive increasing solutions of (2) may be obtained assuming in addition that the function / : (t,u,v) —• IR. is bounded with respect to v when u is near to zero. In fact the following holds:

THEOREM 3. Assume conditions II\), H2)' Suppose that there exists c > 0 and a continuous function q\ : ja,-foo) —» Ul such that

c f(t,u,v) < <7i(t) for t > a, - < u < c, v > 0 and

/•-fOO rS 1 ^ 6 J

Then the equation (2)..has eventua,Ily bounded positive increasing solutions.

Proof. We hint only the principal steps of the proof.

Let us proceed as in the proof of Theorem 1: choose to,t\ such that

• + OO rS

/ 91W / -7-rdrds < -

r-foo / qi(s)ds < c,

Jtn

and g(t)> to, h(t) > t0 for t> t{.

205

Let g,h be as before, and let Q be the subset of C*([tQ,+oo),IR) defined by

il= {u e C 1 ([*<,,+oo),]R) : ~ < u(t) < c, 0 < ti'(i) < -^-,t > t0} .

By the same argument as that given in the proof of Theorem 1, we obtain the result. •

•+oo ^

it) Theorem 3 extends recent results in [15, Chap. 4 §5] concerning the existence of eventually bounded positive increasing solutions of an equation of the type (2).

Notice that such a result does not require that / ~r\dt — +oo; so

3. Necessary conditions

In this section we examine the boundedness of positive solutions of (1). If conditions 7/3) fails, then there may exist equations of the type (1) with unbounded positive increasing solutions, as the following example shows:

EXAMPLE. Let us consider the functional equation:

X" + 41* -14<3/2 V j ( x ( g ( < ) ) ) = ° ' > 2 ' ( 8 )

where ip is a real continuous function such that V'M = eu — 1 for u > 0,

and g(t) — -ln2t. For this equation assumption 773) does not hold, while the

monotone increasing function x(t) = y/t is a solution of (8) for t > 2.

We shall give now some conditions ensuring that every positive increasing solution of (1) is unbounded. For the special case l ') such a result implies that the condition

/ tr(t)dt < +00 777) J a

is, in some sense, also necessary in order to (]/) has eventually bounded positive increasing solutions.

The following holds:

206

THEOREM 4. Assume condition 7/j). Suppose that there exists d > 0 such that for every compact A', K C (0,-foo):

r+oo / sqj({s)ds = +oo , where

Ja

qK(t) = mm f(t,u,v) .

ve[o,d]

Then every eventually positive increasing solution of (1) is unbounded.

Proof. Let x be an eventually positive increasing solution of (1). Assume that x is bounded. Hence we have a;(+oo) = lim x(t) = L, L < -f oo. Let

t—v-f-oo

t0 £ [a,+oo) such that: z( / ) > 0, £(<?(/)) > 0, x'(t) > 0, x'(h(t)) > 0 for every Z > /o- Then for every / > IQ xn(t) < 0 and, from the bounded ness of #, we obtain lim x'(t) = 0. Hence, without loss of generality we can assume that

t—•-f-oo L

0 <V-( / i (0) '< d, - < *(#(0) < X" for every t > t0. Let

qL(i) = min / ( l , t i , v ) ; -ue[±,L] v€[0,rf]

integrating equation (1) in (/,< -f T), < > JQ> w e have:

ri-f-T x'(t) = I f(s,x(g(s));x'(h(s)))ds + *'(* + T ) >

> / qL(s)ds + x'(t + T).-

Then as T —*• + oo from (9) we get:

(9)

/

+oo ?£(*)rf«, (10)

+oo which gives a contradiction if ft qi(s)ds — -foo

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If the integral is finite, say /^ qi(s)ds < -foo, then from (10), integrating in [<o^]> we obtain

't f-j-OO ri r-too x(t) > x(t0) + / / qL(r)drds

J to Js ft y-foo

= x(t0) + (s - t0)qL(s)ds -h (i- t0) qi(r)di J to Jt

rt /*+°° •>x(to)+ sqi(s)ds-to • qL(s)ds>

Jtn Jtn to Jto

Taking into account assumption 11%) we have again a contradiction. Hence x is unbounded. •

In order to make condition 7/g) more transparent it suffices to apply the above result to simpler equation ( l ' ) .

COROLLARY 1. Assume condition II\). Suppose that

/•-foo

/ tr(t)dt = +oo (7/9) J a

Then every eventually positive increasing solution of.(l') is unbounded.

The proof easily follows from Theorem 4.

Such a result completes also a recent criterion in [6].

Finally from Theorem 1 and Corollary 1 we have the following:

THEOREM 5. Assume condition 7/j). Equation (l') has an eventually bounded positive increasing solution if and only if condition IIj) holds.

The proof follows immediately from Theorem 1 and Corollary 1.

Such a result extends recent criteria in [15, Chap. 4 §5] and, when n = 2, in [13].

Finally we point out that the results in this section may be conveniently translated to the more general equation (2) with minor changes.

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Mariella CECCIII

Universita di Siena,

Via del Capitano 15, 53100 Siena, Italy.

Mauro MARINI

Universita di Firenze,

Via di S.Marta 3, 50139 Firenze, Italy.

Lavoro pervenuto in redazione il 18.1.91