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Existence and asymptotic behavior of solutions of a boundary value problem on an infinite interval
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MATHEMATICAL
AND
COMPUTER
MODELLING
ELgEVIER
Mathematical and Computer Modelling 41 (2005) 135-157
www.elsevier.com/locate/mcm
Existence and Asymptotic Behavior of Solutions of a Boundary Value Problem on an Infinite Interval R. P. AGARWAL Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901, U.S.A. agarwal~f it. edu O. G. MUSTAFA Department of Mathematics University of Craiova A1. I. Cuza 13, Craiova, Romania oct aviangenghiz©yahoo, com YU. V. ROGOVCHENKO* Department of Mathematics Eastern Mediterranean University Famagusta, TRNC, Mersin 10, Turkey yuri. rogovchenko©emu, edu. tr
(Received and accepted August 2004) A b s t r a c t - - I n this paper, we shall study a boundary value problem on an infinite interval involving a semilinear second-order differential equation. Existence result extending recent researches is obtained by using a fixed-point theorem due to Furi and Pera. Asymptotic behavior of solutions and their first-order derivatives at infinity is discussed. Comparison with relevant known results in literature is also made. (~) 2005 Elsevier Ltd. All rights reserved. K e y w o r d s - - B o u n d a r y value problem, Infinite interval, Existence, Bounded solution, Asymptotic behavior, Furi-Pera fixed-point theorem.
1. I N T R O D U C T I O N B o u n d a r y value p r o b l e m s on infinite intervals f r e q u e n t l y o c c u r in m a t h e m a t i c a l m o d e l l i n g of v a r i o u s a p p l i e d p r o b l e m s . A s e x a m p l e s , in t h e s t u d y of u n s t e a d y flow of a g a s t h r o u g h k semiinfinite p o r o u s m e d i u m [1,2], discussion of e l e c t r o s t a t i c p r o b e m e a s u r e m e n t s in s o l i d - p r o p e l l a n t r o c k e t e x h a u s t s [3], a n a l y s i s of t h e m a s s t r a n s f e r on a r o t a t i n g d i s k in a n o n - N e w t o n i a n fluid [4], This research was initiated when O.M. and Yu.R. were visiting the Mathematisches Forschungsinstitut Oberwolfach, Germany whose warm hospitality, financial support and excellent research facilities are gratefully acknowledged. Yu.R. thanks the Abdus Salam International Centre for Theoretical Physics in Trieste, Italy, for cordial hospitality and financial support which helped to pursue the work on this paper. O.M. is indebted to Prof. Dumitru Bu~neag of the University of Craiova for constant encouragement. 0895-7177/05/$ - see front matter ~) 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2004.08.002
Typeset by JIh4S-~_aX
136
R . P , AGARWALet al.
heat transfer in the radial flow between parallel circular disks [4], investigation of the temperature distribution in the problem of phase change of solids with temperature dependent thermal conductivity [4], as well as numerous problems arising in the study of draining flows [5], circular membranes [6-8], plasma physics [9,10], radially symmetric solutions of semilinear elliptic equations [10-12], nonlinear mechanics, and non-Newtonian fluid flows [13]. In this paper, we are concerned with the boundary value problem,
~" -
m2~
=
F
(t, u),
t _ to _ O,
u (to) = u0,
lim u (t) = 0.
t-~-t-c¢
(1)
In (1), the constant m > 0 and the real-valued function F (t, u) is continuous. Problems of this type, in particular, include the well-known Holt's problem [14], u " - - ( 1 + 2 m 4 - t 2) u : 0 ' (0) =
t>O,
~,
(+~)
= 0,
which is frequently used as a test problem for checking the effectiveness of a numerical method, and the Fermi-Thomas problem [15,6], u tl - t - 1 / 2 u 3/2 : O,
t > O,
u (0) = 1,
u (+c~) -= O,
describing electrical potential in an isolated atom, for recent results see [10]. Asymptotic behavior of solutions of the linear equation, u"-(l+f(t))u=O,
t>to>O,
(2)
u" + (1 + f (t)) u = 0,
t _> to >_ 0,
(3)
and its "oscillatory" version,
has been the subject matter of numerous investigations. Although several results established for (3) (see, for instance, [17-20]), are analogous to those obtained for (2), the analysis and the techniques needed are substantially different [21, p. 55]. In fact, Bellman [22, p. 83] has emphasized that equation (2), "despite its special appearance, plays a very important role in the general theory of the second-order linear differential equations, since under very broad conditions the equation d-~
~
)
- Z(t) u = 0,
may be transformed into (2)". A similar reasoning applies to equation (3) also. The asymptotic behavior of solutions of (2) has been examined in the pioneering work of Poincar@ [23] under the sole assumption that f ( t ) --* O, as t --* +c~. Further exciting discoveries have been made by Wintner [21], Bellman [22], Ascoli [24,25], Hartman [26], and Owens [27]. The reader is also advised to note some recent contributions of Guzm£n and Pinto [28], and Pinto [29], where asymptotic behavior of nonlinear equation x' (t) = A (t) x + / (t, ~ ) ,
t > to,
and linear equation, (py')' ± f y = O,
has been investigated.
t > to,
Existence and Asymptotic Behavior
137
It is well known that all bounded solutions of the second-order linear equation, u" - m2u = O,
t > to >_ O,
(4)
tend to zero as t ~ +c~ along with their derivatives. Now, considering equation (1) as a perturbation of (4), we pose the following question in the spirit of Tricomi-Fubini theory of asymptotic integration [30, Ch. IV]: do bounded solutions of (1) carry over the decay properties of bounded solutions of (4)? Recently, an interesting existence result for the boundary value problem (1) has been obtained by O'Regan [31, Ch. 1] under rather genera/hypotheses. Although the original result has been formulated in the framework of Carath~odory theory of ordinary differential equations, here, we adapt it from [31, Theorem 1.12.3, p. 83; Example 1.12.2, p. 84] for the convenience of the reader. THEOREM 1. Let u0 be a real number and to = O. Assume that the following conditions are satisfied. (i) For each ~ > O, there exists a continuous function ~ : [0, + o o ) -~ [0, + o o ) , ~uch that [u] <_ r implies [F(t, u)[ <_ Tr(t), fOr all t > O, together with
lira e -mr fo t e'~Tr (s) ds = 0 t---r+oo
(5)
and lim
t---*+~
e m~
e-m%
(s) ds
-- 0.
(6)
(ii) There exists an M0 > 1401, such that I~1 > Mo impIies uF(t, u) >_ 0, for 311 t 2 0. Then, the boundary vMue problem (1) has a solution. There are two main approaches to the problem of existence of solutions for boundary value problems on infinite intervals. The first approach is based on a diagonalization argument discussed in detail in the monograph of Agarwal and O'Regan [al, Ch. 1], whereas the second is based on a fixed-point theorem established in 1987 by Puri and Pera [33]. In fact, Theorem 1 has been proved by employing the Furi-Pera fixed-point theorem. In the proof, the typical Bernstein-like Condition (ii), see Bernstein [34] or Dugundji and Granas [35, Lemma 7.1, Condition (i), p. 8], has been used to obtain a priori bounds for the solution of the problem (1). Since the same fixed-point theorem is used to prove the main existence result (Theorem 4) in this paper, to make the exposition self-contained, we state it here. THEOREM 2. Let E be a Fr4chet space, Q be a dosed convex subset of E, O E Q, and T : Q --* E be a continuous compact map. Assume also that, for any sequence (xj, Aj)j>I from OQ x [0,1] that converges to (x, A), where z = AT ( x ) ,
0 _< ~ < 1,
one has A~T(x3) E Q, for all j large enough. Then, T has a fixed-point in O.
Very recently, Avramescu et al. [36] have addressed the problem of asymptotic behavior of solutions for a general semilinear differential equation,
~," - a (t) ~, = F (t, ~ ) ,
t _> to _> 0,
(7)
/
assuming that the linear homogeneous differential equation, jJ /
u " - a(t) u = 0,
t > to _> 0,
(8)
138
R.P. AGARWALe t al.
has a solution f : [to, +c~) ~ (0, +c~), such that
lira f (t) = 0
and
,-.+~
lira f' (t)
,-.+oo y (t--7 = - ~
(9)
For the convenience of the reader, we only state one of the main resuks from [36]. For this, we need the function 9(t) defined by g(t)=f(t)
[ /,i 1+
t > to,
where f(t) is the solution of equation (8) satisfying (9). THEOREM 3. Let the function F(t, u) satisfy
IF(t,u)l<__h(t)lul,
t>to,
uelR,
(10)
where h : [to, +oo) ~ [0, +oo) is a continuous function. Suppose further that
if f(t)
~ f(s)h(s)ds _< C
< +c~,
(11)
and the function G(t), defined for t > to by
1//o
c (t) - g2 (t)
g2 (s) h (s) ds,
(12)
satisfies 0 _< G (t) < K < +c~
and
F
G (t) dt < +c~.
(13)
Further, assume that, for every T >_to, there exists a unique solution of the initial value problem, ~" - a (t) u = F (t, u ) , u (T) = u0,
t > to > 0, u' (T) = u:.
(14)
Then, ali bounded solutions of (7) tend to zero as t --* +co, together with their first-order derivatives. It is interesting to note that condition (11) has the same nature as the hypothesis (6) of Theorem 1, however, (13) does not necessarily imply that lim G(t) = 0.
t---*+c~
(15)
However, conditions (13) axe natural for problems of this type, cf. [21, Equations (51),(52)]. The main purpose of this paper is to obtain a new existence result for equation (7), which is similar in nature to Theorem 1, for the case where condition (15) holds. This goal is achieved by exploiting the test functions f(t) and g(t) introduced recently by Avramescu et al. in [36], and a technique which is similar to that employed by O'Regan in [31,32] and the authors in [37]. It relies essentially on the Furi-Pera fixed-point theorem. Then, by using the variation of constants formula and a pair of projection matrices, we present a result which complements Theorem 3. We conclude the paper by constructing an example of a boundary value problem where our theory can be easily applied, but Theorem 1 due to O'Regan fails.
Existence and Asymptotic Behavior
139
2. E X I S T E N C E OF B O U N D E D S O L U T I O N S AND THEIR ASYMPTOTIC BEHAVIOR We begin with the following existence result. THEOREM 4. Suppose that equation (8) has the following hypotheses are satisfied.
a
solution f(t) satisfying (9). Assume further, that
(i) For every r > O, there exists a continuous function ~-~ : [to, 4-c<~) ~ [0, 4-oo), such that [u[ <_ r implies [F(t,u)[ _< ";~(t), for a11 t >_ to, together with
and
lim ~ 1 ~ i g (s) r~ (s) ds = 0 t--.+oo
(16)
12% + f 1-L (t) f~ I (s) ~ (s) ds = o.
(17)
(ii) There exists an Mo > O, such that lul > Mo implies uF(t, u) > O, for ali t > to. Then, for ali real uo, such that luo[ < Mo, there exists ~ solution of the boundary value problem,
~" - a (t) ~ = F (t, ~ ) , u (to) = uo,
t > to _> o,
lim u (t) = O.
(18)
t---~q-oo
PROOF. The proof is similar to that of Theorem 1, due to O'Regan (see [31, Section 1.12]). Therefore, here, we confine ourselves only to crucial distinctive features, inviting the reader to consult the comprehensive exposition in the monograph by Agarwal and O'Regan [31]. In order to introduce the integral operator associated with the boundary value problem (18), we consider the linear space E = C([t0, 4-00), JR) of all continuous, real-valued functions u(t) defined on [to, 4-00). This space is endowed with the usual function operations and topology of uniform convergence of sequences of functions on compact subintervals of [to, 4-00). In fact, let (t~)n>l be a strictly increasing sequence of real numbers, such that tn > to and limn-.+oo tn = 4-00. Then, [ul,=
sup
lu(t)[,
ueE,
n>l,
tE[to,t~]
is a countable family of seminorms, see [31, p. 80]. This allows to define the metric on E by
= ~~ 1
2 - i -T-I~-~L'
and, consequently, introduce the topology, cf. [38, p. 27]. If u(t) is a solution of (18), using the variation of constants formula, we deduce that the column vector z(t) = (u(t), u'(t)) T, satisfies the integral equation, cf. [39, ca. III, Section 3], [40, formula (8.9), p. 149], or [41, Section III.6], z (t) = X (t)
1 4-
X (t) P1X -1 (s) R (s, z (s)) ds -
X (t) P2X -1 (s) R (s, z (s)) ds,
t ~ to,
where
~0
g (to)/?
c1 -- f (to----)+ ~ R(t,z(t))--
( 0 ) F(t,u(t))
'
f (s) F (s, u (s)) ds,
( I(t) X(t)= \f,(t)
g(t) ~ g'(t)/'
t > to,
(19)
R . P . AGAP~WAL et al.
140
0) 0
° 0) '
In particular, for t _> to, we obtain
u (t) = C l f (t) - f (t)
f
g (s) F (s, u (s)) ds - g (t)
I
f (s) F (s, u (s)) ds.
(20)
For r0 = 1 + M0, let
Q = { x E E : sup ,x(t), < r o } . t>to
Now, we introduce an integral operator T : Q ~ E, defined by the right-hand side of (20). To show that T has a fixed-point in Q we need to establish its continuity and compactness. However, these follow rather easily by using the techniques developed in [37, Section 4]. Here, we skip the details referring the reader to the cited paper. | Clearly, Theorem 4 yields the existence of bounded solutions of (7). Our next result addresses asymptotic behavior of bounded solutions and their first-order derivatives in more detail. THEOREM 5. Assume that F(t, u) satisfies the hypotheses of Theorem 4. Then, all bounded solutions of equation (7) tend to zero as t --* +oo, together with their first-order derivatives. PROOF. Let u(t) be a bounded solution of (7), which is defined for t > to. Then, u(t) is the unique solution of the i n i t i i value problem
y" - a (t) y = F (t, ~, (t)),
t _ to,
(21)
y (to) = ~ (to),
y' (to) = u' (to).
(22)
Now, let y(t) be a bounded solution of the linear nonhomogeneous differentia/ equation (21). Clearly, the general solution of this equation is given by
y (t) = e l l (t) + c29 (t) + yp (t),
(23)
t > to,
where yp(t) is a particular solution of (21), and the constants cl, c2 E R are arbitrary. If y(t) is the solution of the initial value problem (21), (22), i.e., y(t) = u(t), for all t > to, the constants cl and c2 can be determined by the initial conditions. Let rl --- supt>to lu(t)l. Since F(s,u(s)) satisfies Assumption (i) of Theorem 4 for r = rl, we deduce that lim - ~1 f t l g (s) F (s, u (s)) ds = 0 t-~+oo
(24)
lim ~ 1 / t °° f (s) F (s, u (s)) ds = O. t-.+o~
(25)
and
The improper integral
f
oo / (s) F ( s ,
U
(s)) ds,
converges for all t _ to, and hence, from the relation act X (t) = f (t) g' (t) - f ' (t) g (t) = 1 and (20), we can set the function yp(t) defined by
y~(t) = - / ( t )
fl
9(s)F(s,u(s)) ds-g(t)
as a particular solution of equation (21).
f(s)F(s,u(s))
ds,
t >_ to,
Existence and Asymptotic Behavior
141
Now, since the test function f(t) satisfies (9), the above conditions imply that the general solution of equation (21) has the asymptotic representation
y (t) =
(t) + o (1),
as t --* +co, where c2 is a constant. In particular, the solution u(t) of the differential equation has the same asymptotic representation. Taking into account that this solution remains bounded, whereas the test function g(t) grows without bound as t ~ +co, we conclude that if
u (t) = c2g (t) + o(1), then, it is necessary to have c2 = 0, and therefore, u(t) --* 0 as t --* +co. Furthermore, it follows from (20), that the derivative u'(t) of the bounded solution u(t) satisfies the integral equation,
u'(t) = c l f ' ( t ) - f f ( t )
g(s)F(s,u(s)) d s - g ' ( t )
f ( s ) F ( s , u ( s ) ) ds.
Finally, using the relations established in [36],
lim f'(t)git)= t-lim +oo [if(t) [ / (t) / ( t ) g (t) ] ---- 1 and lira g'( t) f ( t ) = ~ lim g' i t ) t--,+oo 2m t ~ + ~ g (t)
1 2'
(24) and (25), we conclude that u' (t) -- o (1) as t --~ +co.
|
We shall now discuss the role of conditions (16) and (17) in Theorem 5. Avramescu Theorem 20] have shown that for any bounded solution u(t) of equation (7),
(t)--]-~u(t)
_<~>tosuplu(T)I " ~
f is) h(s) ds,
et al. [36,
t>_to,
provided conditions (10) and (11) hold. Obviously, condition (17), which in this case is the same as (11), implies that for all bounded solutions of equation (7),
u'(t) - ~ u i t
) = O(1),
as t---~+co.
(26)
Thus, the characterization of asymptotic behavior of the bounded solution u(t) and its derivative u'(t) through (26) is weaker than the conclusion of Theorem 5. This indicates necessity of additional requirements for obtaining more precise information on the decay rate of bounded solutions of (7) at infinity. In fact, condition (16) provides an example of such additional requirement, while assumptions in Theorem 3 are of different type. An interesting question, which arises from this discussion, can be posed as follows. Does condition (16) imply an estimate similar to (26) if the test function f(t) and its derivative if(t) are replaced, respectively, by the second test function g(t) and its derivative g'(t)? While this problem remains unsettled, we can make some interesting observations. For this, we need to recall some useful facts from the recent paper of Avramescu et al. [36, Section 5]. Let condition (10) be replaced by a more restrictive one, h l ( t ) ] u I _ < F ( t , u ) < h 2 ( t) lul,
t>t0,
u•R,
(27)
142
R.P. A(2ARWALet al.
where hi : [to, +co) ~ [0, +oo), i --- 1, 2, are continuous functions that satisfy assumptions (11) and (13). Suppose further that the initial value problem (14) has a unique solution. Let z (t) def v' (t) - g' (t)
=
t>_to.
-Zi;v(t),
Then, looking for a solution v(t) of equation (7) that satisfies lira z (t) = l > 0,
(28)
t---*+oo
one has to assume that t -lim ~+~ ~ 1
~ i g2 (s) hi (s) ds = o.
(29)
Note that condition (29) is a weaker version of (16). The key question is whether such a solution v(t) exists. In the case when conditions (27) and (29) are satisfied, the answer is yet unknown. However, we shall present a result in this direction assuming a more flexible condition, 0 < l i m i n f z (t) _< limsup z (t) < + c o . t----~+o~ t---*+oo
(30)
THEOREM 6.
(i) Let condition (10) hold, and assume that the continuous function h(t) is such that (13) and (15) are satisfied. Then, for any solution v(t) of equation (7), lim v(t) -- - 1- I" l i m -v'(t)] .~ [~-~+o~ g(t)J
~-~+oog(t) lim z ( t ) _ _ 0 , t - ~ + ~ g (t)
and
/
=iv
~,
E
~ I~ (t)l dt g(t)
<
(31) +oc.
(ii) Suppose that in addition to (27), the following conditions are satisfied, for all t > to, Hi
ml < g, (t) < m2, - g (t) where
-< hi (t) ~ (t) '
ml 0 < Hi < 3-Q
and
and
Q = sup
h2 (t) < //2 - g (t)' ~- - to
~-_>to g ( r )
-
(32)
.
Then, for any real L # 0, there exists a solution v(t) of equation (7) such that (31) holds with lv = L, and H1 iLl < liminfz (t) _< l i m s u p z (t) _< //2 ILl m2
t~+oo
t-~+oo
(33)
rnl
PROOF. Part (i). Let v(t) be a solution of (7). Then, the variation of constants formula gives v (t) = clg (t) + g (t)
fl z (~) . g-~
d~,
t > to,
where cl -- v(to)(g(to)) -1. Combination of (34) and (7) yields 9' (t) ~' (t) + g - - ~ z
(t) = F (t, ~ (t)),
t >_ to,
(34)
Existence and Asymptotic Behavior
143
and another application of the variation of constants formula provides c2
1 Jl
z (t) = ~
+ -~
g (s) F (s, v (s)) ds,
t > to,
(35)
where c2 = z(to)g(to). Using (35), we obtain, for all t > to, the estimate
(t) l _< -17E I~1 + ~~ 1k ~ 1 fJto g(s) h(s) lv(s)l ds ~ k~]
g2(s)h(s)d~+~ 1
_< gIc~l - ~ + g -I~1 ~
(36)
~g2 (.)h(.)d..
~I=(s)l d~.
Let
(s)l ds , Z(t) = f~ilzg (s)
t>_to.
Integration of (36) over [to, t] yields
z (t) ~ Ic=l
g2 (s--'---~+ Ic~l
c (s) d~ +
C (s) Z (s) ds.
(37)
Applying Gronwall's inequality to (37), we obtain, for all t > to, Z(t)-
Iz(s)lds<-
ICl[
V(s) ds+[c2]
Therefore, the integral
G(s) ds
exp
g5
<+oo.
r °° z (s) - ds J~o g(~)
converges and it follows from (34) that limt--.+oo v(t)/g(t) E IR. Now, estimate (36) implies that z(t) = o(g(t)), as t -~ +oo, which yields lira t-*+oo g (t)
t --T-- g (t) J "
This completes the proof of Part (i). Part (ii). First, we note that integration of (32) leads to the following estimates, which are valid for t > to,
g (t)
g (s) ds < -
ml
,
g (t)
g (s)
<
-
ml
and 1 f t l g(s) d s > - - 1 [ 1 g(t) - m2
g(t0)]
g (t) J
Denote by Cb([to, +c~),R) the linear space of all continuous, bounded, real-valued functions z(t) defined on [to, +oc), endowed with the usual function operations, and equipped with the norm Izl = supt>_t o Iz(t)l. Clearly, the space E = (Cb([t0, + ~ ) , ~), I" I) is a Banach space. Let
B = { z E E : ,zI <_ 3H2 jL[ } ml We define an operator T : B ~ E as follows,
(Tz) (t) = m--; H2 IL I ~g(to) + ~ 61 J~o f g (s) F (s, v (s)) as,
144
R.P. AGARWALet al.
where
f o¢ -z~(s) ds,
v ( t ) -= Lg (t) - g (t)
and t _> to, z • B. Since //2 ]L[+}L[ 1 I(Tz)(t)lZ < .--; ~
tg 2 is) h~ I~) ds+3H2 ~1 L. ~
g2 I~) h~ is)
~
< rnl H2,LI+H2,L[ ~ 1 / t : g (s) ds + 3//21L, 1 /t{ H2 [g (s) ~ ml ~(~ < 2//2 IL]+ 3
ILl
< --]L I 2
ds
dg@T)]ds
< --ILl,
z • B,
it is clear that T(B) C B. Continuity and compactness of the operator T can be established by using the technique developed in [37, Section 4]. Now, an application of the Schauder-Tikhonov theorem implies that the operator T has a fixed-point z(t) in B. Using (27), we can refine the bounds for z(t). In fact, for t >_ to, we have
z(t) < m-~ H2 , L , ~
+ m~ H2 [L[+ 3 (\Hm--: 2 ~/ 2 fLI t - to
-
g (t)
and
z(t)>_ H21LIg(t°) m--1
~
H1
+-
m2
[
ILJ 1
g(to)J g(t)
-3
(H2~ 2 --
\ml]
ILl
t-to g@)
,
which in turn yield (33). We conclude the proof by noting that
v (t) = Lg (t) + q (t), where, for all t _> to, 3H2
Iq (t)[ _< m--~ ILl, and thus, l~ = L. | Now, we shall show that (30) yields (29). For this, we remark that the arguments employed by Avramescu et al. in [36, Section 5] cannot be used here. Let v(t) be a solution of equation (7) satisfying (30). Then, for all t > to and some c E R, we have
v (t) = cg (t) - g (t)
f
~ z (~)
-~
ds,
and there exists a T >_ to, such that, for all t >_ T, 0 < m~-1 ~'>T inf z (7) -< g(t) f t ¢¢ ~z(s) ds <
(38)
r>T
By Theorem 3, any bounded solution of (7) tends to zero as t --* +c~. Therefore, it follows from (38) that c 7~ 0. Furthermore, there exists a T1 >_ to, such that, for all t >_ T1, 1
Iv (t)l < 3
and limsup
1 /T~g 2
2 l i m s u p - =1 / ~ f(~,v(s))gis) 2 < - - lira sup z (t) < +co.
Clearly, from (39) the condition (29) is obvious.
as (39)
Existence and A s y m p t o t i c Behavior
145
The following result provides necessary and sufficient conditions on the function a(t) so that equation (8) has a solution f ( t ) which satisfies (9). Before we state and prove our theorem, we notice that so far we have assumed that two positive solutions of (8), which span the space of solutions of this differential equation, exist on the entire interval [to, +c~). However, positivity of the test functions f ( t ) and g(t) is not exploited in the proof of Theorem 5. Since any solution of the linear differential equation (8) can be continued backward for as long as the coefficient a(t) is continuous, one can replace the hypothesis of positivity of the test functions with the requirement of eventual positivity, focusing attention solely on condition (9). THEOREM 7. T h e equation (8) has a solution f : [to, +c~) ---, N satisfying (9) if and only if there exists a continuous function b : [to, +oo) --~ • such that, for all t > to and all q > O,
b (t) = a (t) and
-
m
2
ft
lim e -qt Jto eqSb (s) ds = 0. t-~+oo
(40)
PROOF. NECESSITY. Suppose that the function f ( t ) satisfies (9). Then, it cannot vanish eventually, i.e., there exists a T > to, such that If(t)l > 0, for all t _> T. For t > T, we define the function w E C I ( [ T , + c ~ ) , R ) as follows,
w (t) = g' (t) g (t) - m, where g(t) is given by g(t)=f(t)
1+
~
.
Since the test function g possesses the property lira
g' (t)
= m,
it is obvious that lira w ( t ) = 0 .
t--~+oe
Now, for t > T, let
(t) = m~ + [~, (t) + 2 m ~ (t) + ~ 2 (t)] = m 2 + b (t). a (t) = g" g (t) Using l'Hospital's rule, we obtain lira e -q*
t--*-I-oo
e q~ [w (s)] i ds =
lim [w (t)] i = 0,
t ---*-boo
i = 1, 2.
(as)
Integration by parts yields e -qt
eq~w ' (s) ds = w (t) - w (T) e q(T-t) - qe -qt
K
w (s) e qs ds,
for all t > T. Therefore, lira e -qt
t--*+~
eqSw ' (s) ds = O.
Thus, (40) follows from (41) and (42), which completes the proof of necessity.
(42)
146
R.P. AGARWAL e t al.
SUFFICIENCY.
STEP 1. First we shall show that condition (40) is equivalent to the following. (A) For any two positive numbers q and e, there exists an L0 = L0(q,E) _> to, such that, for all t > W >_ Lo,
~-q~
/;
I
eqsb(s) & < e.
(43)
Assume t h a t (40) holds. Then, for any two positive numbers q and e, there exists a T = T(q, e) > to, such that, for all t >_ T,
e -qt
eqSb (s) ds < -~.
Correspondingly, for any t > W > T, we have
< e -qW
eqSb (s) ds + e -qt
eqSb (s) ds < e.
Conversely, if (43) holds, then, for any two positive numbers q and e and, for all t > W >_ Lo, we find
e -qt
eq~b (S)
< 2'
where L0 = Lo(q,e/2). Furthermore, there exists a T = T(q,e) >__Lo such that, for all t > T,
¢ e-q* f~[~o eq'b (s) ds < 3" Consequently, for all t > T and W = Lo, we find
e -qt
eqSb (s)
< e -qt
eq% (s)
+ e -qt
eqSb (s) ds < e. o
This completes the first step. STEP 2. Fix an e E (0, 1) such that me < 1, and let
We claim that there exists a solution w(t) of the Riccati equation
~o' (t) + 2 . ~
(t) + ~o2 (t) = b (t),
t _> to,
(44)
which is defined on [L1, +c~), such that, for all t > LI, w (L1) = 0,
Iw (t)l < ms,
(45)
and lim w (t) = 0.
t--~+oo
(46)
First, we shall establish (45). For this, standard existence and uniqueness results imply that (44) with the initial condition w(L1) = 0 has a unique solution w(t), which is defined
Existence and Asymptotic Behavior
147
on [L1,L~), where L~ _< ÷oo. For the sake of contradiction, suppose that there exists an L* E (nl, n~) such that, for t C ILl, L*), [w(t)] < me and
Jw (L*)l = me.
(47)
Then, following Ascoli [25, p. 17], we can deduce that, for all t E [L1, L*],
+ e - 2 m t [ t e2m fw( )l
I (t)l _< JL1
JL1
me + (me)2 ~ i e2m(8-t) ds <
(48)
m6,
Letting t --- L* in (48), we find that Iw(L*)l < me, which contradicts assumption (47). Therefore, L~ = +co, and [w(t)l < me, for all t > LI. Thus, the validity of (45) is established. The proof of (46) is based completely on the estimate (43). The technique employed by Ascoli [25, pp. 16-17] cannot be applied here, since it requires additional assumptions on [b(t)l; however, we can use an iterative process. Let L2 def Ll -k Lo (2ra, m---~) .
Integrating equation (44) over [L2, t], we obtain e 2m8 [b (s) - w 2 (s)] ds,
w (t) = w (L2) e -2re(t-L2) + e -2mr 2
which yields, for t > L2, the estimate [w(t)l < raze -2m(t-L2) + -~ 2 -b (me) 2 -
c m(s-t) ds < me 2 +
.
JL2
-
Further, we let L3 def 1 In (4E-1) + L2 and L4 ~--~fL3-b L o ( 2 m , m---~-~). Then, for all t :> L3, we have ]w(t)l < me 2. Integration of (44) over [L4,t] yields, for t _> L4, [ : e 2re(s-t) ds < ~3mE4 + mE2e-2m(t-L4) [w (t)] < mE2e -2"~(t-L') + ~me 4 + (me2)2 JL 4
For n > 2, we introduce L2,~+1 d~f 1 In (4E-2) ÷ L2~ and L2n+2 = ~2,~+1 ÷ L0 2m,
Now, from mathematical induction, it follows that Iw(t)[ < me 2~, for all t > L2n+l. This completes the proof of our claim (46). Finally, introducing for t > L1 the test functions g(t) and f(t) as g(t)=exp
mt-k
w(s) ds ,
f(t)-~g(t)
g2(s),
1
we complete the proof. | REMARK 8. A condition similar to (40) has been obtained by Hartman [26, p. 570]. His result states that equation (8) has a solution f(t) satisfying (9), if and only if ~-~+oo [0<~<+oo 11 q- v ~
b (t) dt
= O.
(50)
However, under the hypotheses of the present paper, condition (40) has obvious preference over (50).
148
R . P . A¢AP,WAL e t
3. A N
a l.
EXAMPLE
First, we shall show that our Theorem 4 extends Theorem 1 due to O'IZegan in somewhat unexpected way. Suppose that there exists a function b(t) which satisfies (40) together with e ~ Ib(s)l ds > O.
limsup e - m r Then, equation (7) reads as
u" - m2u = H (t, u ) ,
(51)
t >_ to > o,
where H ( t , u) = b(t)u + F ( t , u). We claim that the function H ( t , u) in (51) does not satisfy Hypothesis (i) of Theorem 1. In fact, for a fixed r > 0 and for all t _> to, we have
]H (t,r)l >_ r]b(t)l - IF (t,r)] >_ rlb(t)l -'r~ (t) and limsup e -'~t
e m 8 l I t ( s , r ) l ds
t--++oo
]
It'
>_rlimsup
L
]
e'~lb(s)[ ds >0.
e-~t
t-~+oo
J to
Therefore, there exists no continuous, nonnegative function vr(t) defined on [to, +oo) such that, for all t >_ to and [u[ <_ r,
IH(t, )l < v (t) and l i m e -mr
emSvr (s) ds = O.
t--*+oo
Consequently, Theorem 1 cannot be applied in this case. In what follows, we shall show how to construct a function b(t) that possesses above-mentioned properties. Although the idea behind this construction is simple, the computational part is rather intricate. We begin with some useful facts which can be established by elementary methods. These are collected in the following two propositions which we state without proofs. P R O P O S I T I O N 9.
(1.i) Given a A > O, there exist two positive constants A1 and A2 such that, for all x > O, 1 -- e -rex
A1 <- -
1 --
e -mAx
< A2. -
-
(1.ii) For any w > 0 and any E E [0, 1), lira
e Wza-*/2 - 1
x-...oO+
X2
= O.
(1.iii) For a n y two given n u m b e r s a and 13,0 < a < t3 and for all x E [O,a -1 ln(a-l•)], 0 < e~ -
1 <_ fix.
(1.iv) For any a 6 R and for all x > O, )e~
-- 1) <
e I'~l:r -
1.
Existence and Asymptotic Behavior
149
PROPOSITION 10. Let K1, /(2, a and b berealnumbers, a < b, and let the function f : [a, b] --* be defined by
f(a,b;K1,K2)(t)= ~
[exp(Kl(b-t)2(t-a)-K2(t-a)2(b-t))-l]
•
(b - a)
Then, f has the following properties.
(2.i) f ( a ) = f(b) = O, f~(a) = K1, and f~_(b) = K2. (2.ii) For ali t e [a, b] 1 [exp ( 8 [f (t)[ < -(b- - - - a) -~
(IKI,+IK2[) (b
a)a) - lJ
-
"
(2.iii) If K1K2 < O, then sgnf(t) = sgnK1, for all t E (a, b). We are now in the position to prove the following result. THEOREM 11. Given an L > to, there exists a function w 6 CI([L, +co), R) n L I ( ( L , + c o ) , R ) which has the following properties. (i) There exists a strictly increasing sequence (ts)s> 1, satisfying lim ts = +co
n-~+oo
and
lira ( t n + l
n~+oo
-- tn)
=
O,
such that W(tn) = O. (ii) w ( t ) - ~ 0 a s t --+ + 0 %
lira sup w' (t) = +co
and
t--,-t-c~
(iii)
fit
limsup e -mr
t--*+~
lim inf w' (t) = - c o . t-*Woo
]
(52)
e ms Iw' (s)l ds = +co. L
PROOF. Fix two numbers, A > 5 and a E ( 0, 1). Consider a sequence of positive numbers (us)s>1, us --~ 0, as n --* +co, such that, for all n _>>1, Un >
2
A(n+2)
and
ltn+l <
Un ~
oo
2Un+l.
oo
2--~12
Assume also that the series Y~n=l Us diverges, and the series Y~n=l us converges. The set of sequences with these properties is nonempty and contains, for instance, (1/n)n>l. Define now sequences (Tn)n>0, (ts)s>l, (Pn)n>_l, and (Kn)n>>_l by Ts+l = T~ + Aug+l, ts=T~-u~, and
-El2 K n - = u n + 1,
To = L,
n > 0,
pn=T~+2us,
n>l,
(53)
n > 1.
Note that tt > L + 4Ul > L and lims...+oo Ts = +co. It follows from un+l + 2us _ 5un+l < Aug+l,
n > 1,
and (53) that Ps < tn+l, for n _> 1. Similarly, for n > 1, we have (A - 5) u n + l _< t s + l - p s < (A - 3) Un+l.
(54)
R . P . AGARWAL et al.
150
For t e [t~, tn+l] and n > 1, introduce the function w = w(t) by
( f(tn,pn;K~,-K~)(t), f(p~,t~+l;-K,~,K~+l)(t),
w(t)=
f o r t e [t~,p~], f o r t e ~vn,t,~+l].
This function w(t) can be defined on the entire interval [L, tl] so that it is continuously differentiable on [L, +c~). In view of (2i), for n _> 1, we have w E Cl([L,+oo), R), w(t.) = O, w'(t.) -~- K~, and w'(p, 0 = -K,~. Using (2ii), (2iii), we conclude that, for t E [t,~,pn], 0 _ ~ (t) _<
(55)
whereas for t C [pn,t~+l],
1 (t.+l
[l_exp(8(Kn+Kn+a)(t~+x_p,~)3)] p~)2
-
(56)
Now, since lim~-~+oo Kn = +c~, it follows that lim sup w' (t) _> lim w' (tn) = +c~ t---++c~
n--..*+ , ~
and liminfw' (t) _< lim w' (p~) = - c o . t---*+oo
n---*+oo
Furthermore, for n >_ 1, K . (pn
_
_
0 1 ~ .
3--e/2
tn) 3 < . . . . n+l
and 3-~/2 . (K~ + K , , + ~ ) ( t ~ + ~ _p,<)3 < ~5 (~, _ ~~J3 o -,~+~
Using (54), we deduce that, for all t e [t~, t~+l] and n >_ 1,
'w(t)I__<(9-I + (A--5)-')[exp[(130+ (N --3)3)
Un+13-'/2]J --I]Un~_I.
Since ~ < 1, from (lii), we conclude that limt_~+oo wit ) = O. Thus, Conclusions (i) and (ii) of the theorem are established. Now, we shall prove the absolute integrability of wit ). It follows from the properties of the sequence (u~)~>l, that there exists an integer N > 1, such that, for all n >_ N, 3-s/2
0<%+1
In 2
(57)
64 + (A - 3) 3.
Clearly, (57) implies that (p~ _ t~)3 _< 271n_____228K~
and
2 (t~+l - p~)3 -< 4 (K~27+InK~+I)"
Furthermore, it follows from i55) and i56) that
L
t~+l Iw (s)[ ds = In + J~,
tn
where
In =
[w (s)] ds <
-
exp
iPn
- 1
(58)
Existence and Asymptotic Behavior
151
and
&=
r
Iw (s)l ds < exp - tn+l -- Pn
Jpn
(K: + K:+I) (t~+l - p:)3
)} _ 1
Choosing in (liii), 2a =/7 = 32K~/27 and using (58), we obtain 0 <
j~
32 K
2-e/2
I~ (s)l ds <_ ~-~ n (pn - tn) 2 <_ 4 41~°+
.
(59)
On the other hand, letting in (liii), 2a = ~ = 16(Kn + K n + l ) 1 2 7 and applying (58), we get ft~+i o < .po
16 Iw(s)lds<_ ~-~ ( K n + K . + I ) ( t . + I - P . ) 2 < 2 ( ~ _
- ,~\2 ~) u .2--¢/2 +l .
Combining now (59) and (60), we find, for n _> N, the inequality 0 <
f
t~+~
Iw (s)l ds <
In 2
Consequently, we have
J~lw(s)l
d s < ln2 ~ --
e
2 - e/ 2 Un+l
e
J tn
2-~/2
?-tn .{. 1
(
~-O0,
n.~ N
i.e., the function w ( t ) is absolutely integrable. To prove Part (iii), we let yn = e -roT=
e'~
Iw' (s)l ds,
n>l.
We note that, for all t E [tj, Tj] and j >_ 1, w (t) -
1 (PJ _ tj)2 [exp ( K j (pj - tj) (pj - t) (t - tj) ) - 1]
and
w' (t) -- PJKJ- tj (PJ + tj - 2t) [l + (pj - t j ) 2 w (t)] >- PJKJ- tj (pj + t6 - 2Tj) = -3-'KJ
Furthermore, for n _> 1, 2n
-Tj
j n~
-
3m
j=n Cq
~s(em')ds=--e- 3m
' : ~~, ( e m T j -- emq ) " j=n
Thus, it follows from (li) that, for j _> 1, e mTj - e mq 1 - e-m(T~-t~) emT~ -- e mTj-1 -~- 1 -- e - m ( T j - T j - 1 )
1 -- e - m u j
1 - e-m~u~ > A1.
Hence, for n > 1, V2n >_ K,~
e-mT2~ E j=n
( emT~ -- emT~-l) = K n
[1 - exp ( - m (T2n - Tn-1))].
(60)
l:t.P. AGARWAL et al.
152
Finally, in view of the inequality 2n
T2n-Tn-1 = A Euj
~ A ( n + l)u2n > 1,
j=n
which is valid for all n > 1, we conclude that limsup~_~+~ V~ = + ~ . The proof is now complete. Now, for the Riccati equation (44), we note that for s > L,
2m]w(s)l + w
tb(s)l +
2 (s) > Iw'(s)l.
Therefore, from (52), we can conclude t h a t
f;'
limsup e - ' u t--++oo L
e r"'
Ib(s)l
ds
JL
]
If
>__limsup e -mr t--*+c~
]
e m~ Iw' (s)l ds = +c~.
In the following two propositions, we point out interesting properties of our example. PROPOSITION 12. The bound Lo in Condition (A) (suffJciency part) of Theorem 7is independent ofq. PROOF. The proposition follows from the fact that the function w(t), whose existence has been established in Theorem 11, is an absolutely integrable function t h a t tends to zero as t ~ +oo. In fact, for all t > W > L, we have
e-qt / ; eqSw(s) ds] ~ / j e -qt
I/;
I
lw(s)l ds,
eqSw 2 (s) ds < sup [Iw (T)I ] r>L
/:
Iw (s)l ds,
and
~-q~
~'
(s)
< I~ (t)l + I~ (w)] + q~-qt
~
I~ (s)l ds.
Since w(t) --* 0 as t --* +c~, for any 5 > 0, there exists a T = T(6) > L such that ]w(s)l < 6, for all s >_ T. Multiplying both sides of this inequality by exp(qs) and integrating over [W, t], for all t >_ W > T, we obtain
qe-~t
/;
e~s I~ (s)l ds < ~.
The bound in the latter inequality is clearly independent of q, which proves our claim.
I
PROPOSITION 13. We can choose the test function g(t) so that g(t) ..~ exp(mt), as t --* +co. PROOF. Writing the first equation in (49) as
g (t) = exp mt we immediately arrive at the conclusion.
~ (s) d~ , I
We conclude this section by describing several simple classes of functions that satisfy conditions (16) and (17). For this, we need the following auxiliary result.
Existence and Asymptotic Behavior
153
LEMMA 14. For given two numbers A and v, such that A E (0, 1) and 1 < v < A-1, it follows that lim [gg(t) (At)]v ~-+~
=
o.
(61)
PROOF. Let .s
(t)
or
it )
=
[g (~t)]v g (t)
Then, for t >_ to, j , it )
:
(xt~ k
g(
)
~)1
g(t) J "
Since the term in the brackets tends to (Av - 1)m < 0, as t --+ +oo, we deduce that J(t) is eventually decreasing on [t0, +oo). This yields
O < L . = lim J(t)<+cc. t---+-t-~
For the sake of contradiction, suppose that L. > 0. Applying l'Hospital's rule, we obtain t-~+~
[ g (At)
= AvL~ < L,,
g'
which is a contradiction. Thus, the conclusion of the lemma follows.
II
In what follows, p' > 1 is a number defined by (l/p) + (1/p') = 1. PROPOSITION 15. A continuous function "G(t) satisfies conditions (16) and (17) provided it belongs to one of the following classes, (i) functions that tend to zero as t ~ +oo; (ii) functions which are t~rst order derivatives of functions satisfying (i); (iii) functions which belong to LP( (to, +oo), ]~) for some p > 1. PROOF. Proof of Claims (i) and (ii) is similar to that given in Theorem 7. In the ease p = 1, Claim (iii) has been established in Avramescu et al. [36]. Here, we shall show (iii) when p > 1. To prove (16), fix a A E (0, 1). Now, an application of the HSlder inequality yields
I~ (s)l ~s =
g - ~ I~ (s)l d~ +
-< Ir~lL'((t°,+°°),a)
+
~ g - ~ I~r (s)l d~ [g(t)J
[T~ (s)l p ds
ds
~ Lg (t) J
(62)
ds
'
where the function g(t) is increasing, for all t _> T. Using l'Hospital's rule and Lemma 14, we deduce that
f < [gi41" d~ =0 ' and, respectively, "' lim t-+oo
~ Lg(t) J
ds=
~
l~m
[g(t) J
ds=
1 <+oo. rap,
The conclusion of Claim (iii) follows, by letting t --* +oo in (62). Finally, we note that (17) can be established in a similar manner. |
R . P . AGARWAL et al.
154
4. D I S C U S S I O N Here, we shall discuss how our results are related to those known in the literature. First, we observe that if g(t) is a positive, continuous function that approaches +co, not necessarily as fast as the exponential function, conditions (16) and (17) play special role in the theory of asymptotic integration, see, for instance, the papers by Faedo [42, p. 211], Ghizzetti [43, p. 30], or Hallam and Heidel [44, p. 126], where the assumption
--1 L x sp f (s) ds = 0
lim
x---*q-oo x P
0
has been fully used. We also remark that [25, Sections 1-2] gave a detailed account of the properties of the Volterra operator of the first kind defined by
(vj)
f
(t) =
e-~(~-')/(T) d-~,
t >_ O,
where f E LP((0, +co), R) for some fixed p _> 1. In the case p = 1, our proof of Part (iii) in Proposition 15 is based on the ideas of Faedo [42, p. 211], while a simple, ingenious majorization technique, which can be successfully applied to deal with (62) in some particular cases, has been suggested by Ascoli in [25, p. 15]. A more complicated problem, which remains unsettled, is concerned with finding solutions to Riccati equation (44). Assume that b(t) belongs to LP((to, +c~), •), for some p _> 1. Then, one can apply the techniques of Ascoli [25, Section 4] to compute a series expansion for the solution of equation (44), satisfying the initial condition w(0) = 0, in the form v--1
w (t) = E
wi (t) + remainder term,
i=1
where v - 1 < p < ~, wl = V2b, and n--1 =
-
v2
/=1
for n > 1. In an elegant, however, unfortunately almost forgotten paper, Owens [27] analyzed the boundary value problem for a perturbed quasi-linear differential equation
y" - A f (x, y) y = q(x, y), y (0) = 0,
x >_ O, lim y (x) = 0,
(63)
~---+-~-OO
where A > 0, f ( x , y) and q(x, y) are continuous functions satisfying (i) 0 < m < f(x, y) < M, for all x > 0, lYl - Y, and lim~-~+oo q(x, 0) = 0; (ii) f ( x , y) and q(x, y) are Lipschitz functions with respect to y, with Lipschitz constants K1 and K2; (iii) for some constant a > 0, o ~ If ( x , y ) - a ~ l
dx < ÷ c o
and
L
o~
]q(x,0)l dx < +co;
(iv) q(x, y) satisfies 2 suplq(x,y)l < Y rnA ~>_o
and
1 [4Klsup[q(x,Y)l+2K2rn] m2A x>o
<1.
Existence and Asymptotic Behavior
155
Using the method of successive approximations and two test functions ¢ and ¢ which are slightly more general than our test functions f and g, Owens established existence of a unique LLsolution of the boundary value problem (63) and obtained a priori bound for the Ll-norm of this solution. The only weakness of his method is that the test function ¢ is not computed explicitly, rather it is defined as a solution of a boundary value problem on [0, +c~); whereas, ¢ is found by using successive approximations, see [27, equation (1.12)]. For the case where a(t) is a piecewise continuous function satisfying [a(t)l < m 2, Loud [45, Section 4] obtained several interesting estimates for fundamental solutions of equation (8) by exploiting the method of successive approximations, and this technique can be also adapted for equation (7). Further extensions of Loud's results have been established by Alekseev, Levin and Levinson, see comprehensive discussion in the monograph of Coppel [39, Lemma 4, p. 116]. Application of l'Hospital's rule yields
,12 oo: (t)
Jl 7-{-4 =
g(s)
Therefore, our hypotheses can be also compared to conditions imposed by Corduneanu [46, Lemma 1, p. 104]. However, the nonlinearity in his paper should satisfy a Lipschitz condition, and thus results obtained by Corduneanu [46, Theorems 1 and 2] cannot be applied to the case discussed in this paper. Extending several results due to Coppel [39], Brauer and Wong [47] studied general problems related to those discussed in the present paper. Using the definition of g(t) and (19), one can establish existence of a constant K > 0, such that [X(t)PlX-l(s)[
to<_s<_t,
]x (t) p ~ x - l (s)l < K,
to < t <_s,
see also [47, p. 529]. Therefore, results proved by Brauer and Wong can be employed for the study of existence of solutions of equation (7) for which the limit lim -1 ln[u(t)[ t--~+c~ t exists and is finite. A straightforward application of l'Hospitai's rule allows to establish that the value of this limit is - m for the test function f(t), and m for the test function g(t) (we refer to the paper by Avramescu et al. [361 where relevant details can be found). Under assumption (10), the weakest, according to Coppel [39, Equation (4), p. 97], integrability hypothesis imposed by Brauer and Wong [47, p. 541] reads as ft+l
es=O.
(64)
This condition is considerably less restrictive compared to standard integrability conditions. However, assume that, for a nonnegative continuous function A(t), one has 0 < f t t+l ~ (s) ds < c, for t >_ T _> to. Then, according to Massera-Sch~ffer lemma (see [39, Lemma 2, p. 102], [40, Exercise 4.8, p. 76], or [48]), for all q > 0 and t _> T, max
{// e -qt
eq'A (s) ds, e qt
e-q'A (s) ds
} <- 1 - .e----"-'-~< +c~.
Thus, condition (64) is a particular case of (16) and (17), as well as of (11) and (13). We conclude the paper with an unsettled problem. Apply the fixed-point theory to study asymptotic behavior of solutions of a semilinear differential equation u" = (m 2 + b (t)) (u + n (u)),
t > to _> O,
where b(t) and R(u) are continuous functions such that IR(u)l is dominated by a suitable comparison function, whereas the only assumption on b(t) is equation (50).
156
R.P. AGARWALet al.
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36. C. Avramescu, O.C. Mustafa, S.P. Rogovchenko and Yu.V. Rogovchenko, Asymptotic integration of secondorder differential equations: Levinson-Weyl theory, Poincarg~-Perron property, Lyapunov type numbers~ and dichotomy, (submitted). 37. O.G. Mustafa and Yu.V. Rogovchenko, Clobal existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations, Nonlinear Anal. 51, 339-368, (2002). 38. K. Yosida, Functional Analysis, Second Corrected Edition, Springer-Verlag, Berlin, (1966). 39. W.A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D.C. Heath &: Comp., Boston, MA, (1965). 40. A.G. Kartsatos, Advanced Ordinary Differential Equations, Mariner Publ. Comp. Inc., Tampa, FL, (1980). 41. J.K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, (1969). 42. S. Fa~do, Propriet~ asintotiche delle soluzioni dei sistemi differenziali lineari omogenei, Ann. Mat. Pura Appl. 26, 207-215, (1947). 43. A. Ghizzetti, Un teorema sul comportamento asintotico degli integrali delle equazioni differenziali lineari omogenee, Rend. Mat. Appl. 8 (6), 28-42, (1949). 44. T.G. Hallam and J.W. Heidel, Asymptotic expansions of the subdominant solutions of a class of nonhomogenous differential equations~ J. Differential Equations 6, 125-141, (1969). 45. W.S. Loud, Boundedness and convergence of solutions of x ~ ~ cx ~ ~- g(x) = e(t)~ Duke Math. J. 24, 63-72, (1957). 46. C. Corduneanu, Sur l~existence des solutions born~es de syst~mes d~quations diff~rentielles non lin~aires, Ann. Polon. Math. 5, 103-106, (1958). 47. F. Brauer and J.S.W. Wong, On the asymptotic relationship between solutions of two systems of ordinary differential equations, J. Differential Equations 6, 527-543, (1969). 48. J.L. Massera and J.J. Sch~ffer~ Linear differential equations and functional analysis, Ann. Math. 67, 517-573, (1958).
so,z.oz~D,.zcv.
MATHEMATICAL
AND
COMPUTER
MODELLING
ELgEVIER
Mathematical and Computer Modelling 41 (2005) 135-157
www.elsevier.com/locate/mcm
Existence and Asymptotic Behavior of Solutions of a Boundary Value Problem on an Infinite Interval R. P. AGARWAL Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901, U.S.A. agarwal~f it. edu O. G. MUSTAFA Department of Mathematics University of Craiova A1. I. Cuza 13, Craiova, Romania oct aviangenghiz©yahoo, com YU. V. ROGOVCHENKO* Department of Mathematics Eastern Mediterranean University Famagusta, TRNC, Mersin 10, Turkey yuri. rogovchenko©emu, edu. tr
(Received and accepted August 2004) A b s t r a c t - - I n this paper, we shall study a boundary value problem on an infinite interval involving a semilinear second-order differential equation. Existence result extending recent researches is obtained by using a fixed-point theorem due to Furi and Pera. Asymptotic behavior of solutions and their first-order derivatives at infinity is discussed. Comparison with relevant known results in literature is also made. (~) 2005 Elsevier Ltd. All rights reserved. K e y w o r d s - - B o u n d a r y value problem, Infinite interval, Existence, Bounded solution, Asymptotic behavior, Furi-Pera fixed-point theorem.
1. I N T R O D U C T I O N B o u n d a r y value p r o b l e m s on infinite intervals f r e q u e n t l y o c c u r in m a t h e m a t i c a l m o d e l l i n g of v a r i o u s a p p l i e d p r o b l e m s . A s e x a m p l e s , in t h e s t u d y of u n s t e a d y flow of a g a s t h r o u g h k semiinfinite p o r o u s m e d i u m [1,2], discussion of e l e c t r o s t a t i c p r o b e m e a s u r e m e n t s in s o l i d - p r o p e l l a n t r o c k e t e x h a u s t s [3], a n a l y s i s of t h e m a s s t r a n s f e r on a r o t a t i n g d i s k in a n o n - N e w t o n i a n fluid [4], This research was initiated when O.M. and Yu.R. were visiting the Mathematisches Forschungsinstitut Oberwolfach, Germany whose warm hospitality, financial support and excellent research facilities are gratefully acknowledged. Yu.R. thanks the Abdus Salam International Centre for Theoretical Physics in Trieste, Italy, for cordial hospitality and financial support which helped to pursue the work on this paper. O.M. is indebted to Prof. Dumitru Bu~neag of the University of Craiova for constant encouragement. 0895-7177/05/$ - see front matter ~) 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2004.08.002
Typeset by JIh4S-~_aX
136
R . P , AGARWALet al.
heat transfer in the radial flow between parallel circular disks [4], investigation of the temperature distribution in the problem of phase change of solids with temperature dependent thermal conductivity [4], as well as numerous problems arising in the study of draining flows [5], circular membranes [6-8], plasma physics [9,10], radially symmetric solutions of semilinear elliptic equations [10-12], nonlinear mechanics, and non-Newtonian fluid flows [13]. In this paper, we are concerned with the boundary value problem,
~" -
m2~
=
F
(t, u),
t _ to _ O,
u (to) = u0,
lim u (t) = 0.
t-~-t-c¢
(1)
In (1), the constant m > 0 and the real-valued function F (t, u) is continuous. Problems of this type, in particular, include the well-known Holt's problem [14], u " - - ( 1 + 2 m 4 - t 2) u : 0 ' (0) =
t>O,
~,
(+~)
= 0,
which is frequently used as a test problem for checking the effectiveness of a numerical method, and the Fermi-Thomas problem [15,6], u tl - t - 1 / 2 u 3/2 : O,
t > O,
u (0) = 1,
u (+c~) -= O,
describing electrical potential in an isolated atom, for recent results see [10]. Asymptotic behavior of solutions of the linear equation, u"-(l+f(t))u=O,
t>to>O,
(2)
u" + (1 + f (t)) u = 0,
t _> to >_ 0,
(3)
and its "oscillatory" version,
has been the subject matter of numerous investigations. Although several results established for (3) (see, for instance, [17-20]), are analogous to those obtained for (2), the analysis and the techniques needed are substantially different [21, p. 55]. In fact, Bellman [22, p. 83] has emphasized that equation (2), "despite its special appearance, plays a very important role in the general theory of the second-order linear differential equations, since under very broad conditions the equation d-~
~
)
- Z(t) u = 0,
may be transformed into (2)". A similar reasoning applies to equation (3) also. The asymptotic behavior of solutions of (2) has been examined in the pioneering work of Poincar@ [23] under the sole assumption that f ( t ) --* O, as t --* +c~. Further exciting discoveries have been made by Wintner [21], Bellman [22], Ascoli [24,25], Hartman [26], and Owens [27]. The reader is also advised to note some recent contributions of Guzm£n and Pinto [28], and Pinto [29], where asymptotic behavior of nonlinear equation x' (t) = A (t) x + / (t, ~ ) ,
t > to,
and linear equation, (py')' ± f y = O,
has been investigated.
t > to,
Existence and Asymptotic Behavior
137
It is well known that all bounded solutions of the second-order linear equation, u" - m2u = O,
t > to >_ O,
(4)
tend to zero as t ~ +c~ along with their derivatives. Now, considering equation (1) as a perturbation of (4), we pose the following question in the spirit of Tricomi-Fubini theory of asymptotic integration [30, Ch. IV]: do bounded solutions of (1) carry over the decay properties of bounded solutions of (4)? Recently, an interesting existence result for the boundary value problem (1) has been obtained by O'Regan [31, Ch. 1] under rather genera/hypotheses. Although the original result has been formulated in the framework of Carath~odory theory of ordinary differential equations, here, we adapt it from [31, Theorem 1.12.3, p. 83; Example 1.12.2, p. 84] for the convenience of the reader. THEOREM 1. Let u0 be a real number and to = O. Assume that the following conditions are satisfied. (i) For each ~ > O, there exists a continuous function ~ : [0, + o o ) -~ [0, + o o ) , ~uch that [u] <_ r implies [F(t, u)[ <_ Tr(t), fOr all t > O, together with
lira e -mr fo t e'~Tr (s) ds = 0 t---r+oo
(5)
and lim
t---*+~
e m~
e-m%
(s) ds
-- 0.
(6)
(ii) There exists an M0 > 1401, such that I~1 > Mo impIies uF(t, u) >_ 0, for 311 t 2 0. Then, the boundary vMue problem (1) has a solution. There are two main approaches to the problem of existence of solutions for boundary value problems on infinite intervals. The first approach is based on a diagonalization argument discussed in detail in the monograph of Agarwal and O'Regan [al, Ch. 1], whereas the second is based on a fixed-point theorem established in 1987 by Puri and Pera [33]. In fact, Theorem 1 has been proved by employing the Furi-Pera fixed-point theorem. In the proof, the typical Bernstein-like Condition (ii), see Bernstein [34] or Dugundji and Granas [35, Lemma 7.1, Condition (i), p. 8], has been used to obtain a priori bounds for the solution of the problem (1). Since the same fixed-point theorem is used to prove the main existence result (Theorem 4) in this paper, to make the exposition self-contained, we state it here. THEOREM 2. Let E be a Fr4chet space, Q be a dosed convex subset of E, O E Q, and T : Q --* E be a continuous compact map. Assume also that, for any sequence (xj, Aj)j>I from OQ x [0,1] that converges to (x, A), where z = AT ( x ) ,
0 _< ~ < 1,
one has A~T(x3) E Q, for all j large enough. Then, T has a fixed-point in O.
Very recently, Avramescu et al. [36] have addressed the problem of asymptotic behavior of solutions for a general semilinear differential equation,
~," - a (t) ~, = F (t, ~ ) ,
t _> to _> 0,
(7)
/
assuming that the linear homogeneous differential equation, jJ /
u " - a(t) u = 0,
t > to _> 0,
(8)
138
R.P. AGARWALe t al.
has a solution f : [to, +c~) ~ (0, +c~), such that
lira f (t) = 0
and
,-.+~
lira f' (t)
,-.+oo y (t--7 = - ~
(9)
For the convenience of the reader, we only state one of the main resuks from [36]. For this, we need the function 9(t) defined by g(t)=f(t)
[ /,i 1+
t > to,
where f(t) is the solution of equation (8) satisfying (9). THEOREM 3. Let the function F(t, u) satisfy
IF(t,u)l<__h(t)lul,
t>to,
uelR,
(10)
where h : [to, +oo) ~ [0, +oo) is a continuous function. Suppose further that
if f(t)
~ f(s)h(s)ds _< C
< +c~,
(11)
and the function G(t), defined for t > to by
1//o
c (t) - g2 (t)
g2 (s) h (s) ds,
(12)
satisfies 0 _< G (t) < K < +c~
and
F
G (t) dt < +c~.
(13)
Further, assume that, for every T >_to, there exists a unique solution of the initial value problem, ~" - a (t) u = F (t, u ) , u (T) = u0,
t > to > 0, u' (T) = u:.
(14)
Then, ali bounded solutions of (7) tend to zero as t --* +co, together with their first-order derivatives. It is interesting to note that condition (11) has the same nature as the hypothesis (6) of Theorem 1, however, (13) does not necessarily imply that lim G(t) = 0.
t---*+c~
(15)
However, conditions (13) axe natural for problems of this type, cf. [21, Equations (51),(52)]. The main purpose of this paper is to obtain a new existence result for equation (7), which is similar in nature to Theorem 1, for the case where condition (15) holds. This goal is achieved by exploiting the test functions f(t) and g(t) introduced recently by Avramescu et al. in [36], and a technique which is similar to that employed by O'Regan in [31,32] and the authors in [37]. It relies essentially on the Furi-Pera fixed-point theorem. Then, by using the variation of constants formula and a pair of projection matrices, we present a result which complements Theorem 3. We conclude the paper by constructing an example of a boundary value problem where our theory can be easily applied, but Theorem 1 due to O'Regan fails.
Existence and Asymptotic Behavior
139
2. E X I S T E N C E OF B O U N D E D S O L U T I O N S AND THEIR ASYMPTOTIC BEHAVIOR We begin with the following existence result. THEOREM 4. Suppose that equation (8) has the following hypotheses are satisfied.
a
solution f(t) satisfying (9). Assume further, that
(i) For every r > O, there exists a continuous function ~-~ : [to, 4-c<~) ~ [0, 4-oo), such that [u[ <_ r implies [F(t,u)[ _< ";~(t), for a11 t >_ to, together with
and
lim ~ 1 ~ i g (s) r~ (s) ds = 0 t--.+oo
(16)
12% + f 1-L (t) f~ I (s) ~ (s) ds = o.
(17)
(ii) There exists an Mo > O, such that lul > Mo implies uF(t, u) > O, for ali t > to. Then, for ali real uo, such that luo[ < Mo, there exists ~ solution of the boundary value problem,
~" - a (t) ~ = F (t, ~ ) , u (to) = uo,
t > to _> o,
lim u (t) = O.
(18)
t---~q-oo
PROOF. The proof is similar to that of Theorem 1, due to O'Regan (see [31, Section 1.12]). Therefore, here, we confine ourselves only to crucial distinctive features, inviting the reader to consult the comprehensive exposition in the monograph by Agarwal and O'Regan [31]. In order to introduce the integral operator associated with the boundary value problem (18), we consider the linear space E = C([t0, 4-00), JR) of all continuous, real-valued functions u(t) defined on [to, 4-00). This space is endowed with the usual function operations and topology of uniform convergence of sequences of functions on compact subintervals of [to, 4-00). In fact, let (t~)n>l be a strictly increasing sequence of real numbers, such that tn > to and limn-.+oo tn = 4-00. Then, [ul,=
sup
lu(t)[,
ueE,
n>l,
tE[to,t~]
is a countable family of seminorms, see [31, p. 80]. This allows to define the metric on E by
= ~~ 1
2 - i -T-I~-~L'
and, consequently, introduce the topology, cf. [38, p. 27]. If u(t) is a solution of (18), using the variation of constants formula, we deduce that the column vector z(t) = (u(t), u'(t)) T, satisfies the integral equation, cf. [39, ca. III, Section 3], [40, formula (8.9), p. 149], or [41, Section III.6], z (t) = X (t)
1 4-
X (t) P1X -1 (s) R (s, z (s)) ds -
X (t) P2X -1 (s) R (s, z (s)) ds,
t ~ to,
where
~0
g (to)/?
c1 -- f (to----)+ ~ R(t,z(t))--
( 0 ) F(t,u(t))
'
f (s) F (s, u (s)) ds,
( I(t) X(t)= \f,(t)
g(t) ~ g'(t)/'
t > to,
(19)
R . P . AGAP~WAL et al.
140
0) 0
° 0) '
In particular, for t _> to, we obtain
u (t) = C l f (t) - f (t)
f
g (s) F (s, u (s)) ds - g (t)
I
f (s) F (s, u (s)) ds.
(20)
For r0 = 1 + M0, let
Q = { x E E : sup ,x(t), < r o } . t>to
Now, we introduce an integral operator T : Q ~ E, defined by the right-hand side of (20). To show that T has a fixed-point in Q we need to establish its continuity and compactness. However, these follow rather easily by using the techniques developed in [37, Section 4]. Here, we skip the details referring the reader to the cited paper. | Clearly, Theorem 4 yields the existence of bounded solutions of (7). Our next result addresses asymptotic behavior of bounded solutions and their first-order derivatives in more detail. THEOREM 5. Assume that F(t, u) satisfies the hypotheses of Theorem 4. Then, all bounded solutions of equation (7) tend to zero as t --* +oo, together with their first-order derivatives. PROOF. Let u(t) be a bounded solution of (7), which is defined for t > to. Then, u(t) is the unique solution of the i n i t i i value problem
y" - a (t) y = F (t, ~, (t)),
t _ to,
(21)
y (to) = ~ (to),
y' (to) = u' (to).
(22)
Now, let y(t) be a bounded solution of the linear nonhomogeneous differentia/ equation (21). Clearly, the general solution of this equation is given by
y (t) = e l l (t) + c29 (t) + yp (t),
(23)
t > to,
where yp(t) is a particular solution of (21), and the constants cl, c2 E R are arbitrary. If y(t) is the solution of the initial value problem (21), (22), i.e., y(t) = u(t), for all t > to, the constants cl and c2 can be determined by the initial conditions. Let rl --- supt>to lu(t)l. Since F(s,u(s)) satisfies Assumption (i) of Theorem 4 for r = rl, we deduce that lim - ~1 f t l g (s) F (s, u (s)) ds = 0 t-~+oo
(24)
lim ~ 1 / t °° f (s) F (s, u (s)) ds = O. t-.+o~
(25)
and
The improper integral
f
oo / (s) F ( s ,
U
(s)) ds,
converges for all t _ to, and hence, from the relation act X (t) = f (t) g' (t) - f ' (t) g (t) = 1 and (20), we can set the function yp(t) defined by
y~(t) = - / ( t )
fl
9(s)F(s,u(s)) ds-g(t)
as a particular solution of equation (21).
f(s)F(s,u(s))
ds,
t >_ to,
Existence and Asymptotic Behavior
141
Now, since the test function f(t) satisfies (9), the above conditions imply that the general solution of equation (21) has the asymptotic representation
y (t) =
(t) + o (1),
as t --* +co, where c2 is a constant. In particular, the solution u(t) of the differential equation has the same asymptotic representation. Taking into account that this solution remains bounded, whereas the test function g(t) grows without bound as t ~ +co, we conclude that if
u (t) = c2g (t) + o(1), then, it is necessary to have c2 = 0, and therefore, u(t) --* 0 as t --* +co. Furthermore, it follows from (20), that the derivative u'(t) of the bounded solution u(t) satisfies the integral equation,
u'(t) = c l f ' ( t ) - f f ( t )
g(s)F(s,u(s)) d s - g ' ( t )
f ( s ) F ( s , u ( s ) ) ds.
Finally, using the relations established in [36],
lim f'(t)git)= t-lim +oo [if(t) [ / (t) / ( t ) g (t) ] ---- 1 and lira g'( t) f ( t ) = ~ lim g' i t ) t--,+oo 2m t ~ + ~ g (t)
1 2'
(24) and (25), we conclude that u' (t) -- o (1) as t --~ +co.
|
We shall now discuss the role of conditions (16) and (17) in Theorem 5. Avramescu Theorem 20] have shown that for any bounded solution u(t) of equation (7),
(t)--]-~u(t)
_<~>tosuplu(T)I " ~
f is) h(s) ds,
et al. [36,
t>_to,
provided conditions (10) and (11) hold. Obviously, condition (17), which in this case is the same as (11), implies that for all bounded solutions of equation (7),
u'(t) - ~ u i t
) = O(1),
as t---~+co.
(26)
Thus, the characterization of asymptotic behavior of the bounded solution u(t) and its derivative u'(t) through (26) is weaker than the conclusion of Theorem 5. This indicates necessity of additional requirements for obtaining more precise information on the decay rate of bounded solutions of (7) at infinity. In fact, condition (16) provides an example of such additional requirement, while assumptions in Theorem 3 are of different type. An interesting question, which arises from this discussion, can be posed as follows. Does condition (16) imply an estimate similar to (26) if the test function f(t) and its derivative if(t) are replaced, respectively, by the second test function g(t) and its derivative g'(t)? While this problem remains unsettled, we can make some interesting observations. For this, we need to recall some useful facts from the recent paper of Avramescu et al. [36, Section 5]. Let condition (10) be replaced by a more restrictive one, h l ( t ) ] u I _ < F ( t , u ) < h 2 ( t) lul,
t>t0,
u•R,
(27)
142
R.P. A(2ARWALet al.
where hi : [to, +co) ~ [0, +oo), i --- 1, 2, are continuous functions that satisfy assumptions (11) and (13). Suppose further that the initial value problem (14) has a unique solution. Let z (t) def v' (t) - g' (t)
=
t>_to.
-Zi;v(t),
Then, looking for a solution v(t) of equation (7) that satisfies lira z (t) = l > 0,
(28)
t---*+oo
one has to assume that t -lim ~+~ ~ 1
~ i g2 (s) hi (s) ds = o.
(29)
Note that condition (29) is a weaker version of (16). The key question is whether such a solution v(t) exists. In the case when conditions (27) and (29) are satisfied, the answer is yet unknown. However, we shall present a result in this direction assuming a more flexible condition, 0 < l i m i n f z (t) _< limsup z (t) < + c o . t----~+o~ t---*+oo
(30)
THEOREM 6.
(i) Let condition (10) hold, and assume that the continuous function h(t) is such that (13) and (15) are satisfied. Then, for any solution v(t) of equation (7), lim v(t) -- - 1- I" l i m -v'(t)] .~ [~-~+o~ g(t)J
~-~+oog(t) lim z ( t ) _ _ 0 , t - ~ + ~ g (t)
and
/
=iv
~,
E
~ I~ (t)l dt g(t)
<
(31) +oc.
(ii) Suppose that in addition to (27), the following conditions are satisfied, for all t > to, Hi
ml < g, (t) < m2, - g (t) where
-< hi (t) ~ (t) '
ml 0 < Hi < 3-Q
and
and
Q = sup
h2 (t) < //2 - g (t)' ~- - to
~-_>to g ( r )
-
(32)
.
Then, for any real L # 0, there exists a solution v(t) of equation (7) such that (31) holds with lv = L, and H1 iLl < liminfz (t) _< l i m s u p z (t) _< //2 ILl m2
t~+oo
t-~+oo
(33)
rnl
PROOF. Part (i). Let v(t) be a solution of (7). Then, the variation of constants formula gives v (t) = clg (t) + g (t)
fl z (~) . g-~
d~,
t > to,
where cl -- v(to)(g(to)) -1. Combination of (34) and (7) yields 9' (t) ~' (t) + g - - ~ z
(t) = F (t, ~ (t)),
t >_ to,
(34)
Existence and Asymptotic Behavior
143
and another application of the variation of constants formula provides c2
1 Jl
z (t) = ~
+ -~
g (s) F (s, v (s)) ds,
t > to,
(35)
where c2 = z(to)g(to). Using (35), we obtain, for all t > to, the estimate
(t) l _< -17E I~1 + ~~ 1k ~ 1 fJto g(s) h(s) lv(s)l ds ~ k~]
g2(s)h(s)d~+~ 1
_< gIc~l - ~ + g -I~1 ~
(36)
~g2 (.)h(.)d..
~I=(s)l d~.
Let
(s)l ds , Z(t) = f~ilzg (s)
t>_to.
Integration of (36) over [to, t] yields
z (t) ~ Ic=l
g2 (s--'---~+ Ic~l
c (s) d~ +
C (s) Z (s) ds.
(37)
Applying Gronwall's inequality to (37), we obtain, for all t > to, Z(t)-
Iz(s)lds<-
ICl[
V(s) ds+[c2]
Therefore, the integral
G(s) ds
exp
g5
<+oo.
r °° z (s) - ds J~o g(~)
converges and it follows from (34) that limt--.+oo v(t)/g(t) E IR. Now, estimate (36) implies that z(t) = o(g(t)), as t -~ +oo, which yields lira t-*+oo g (t)
t --T-- g (t) J "
This completes the proof of Part (i). Part (ii). First, we note that integration of (32) leads to the following estimates, which are valid for t > to,
g (t)
g (s) ds < -
ml
,
g (t)
g (s)
<
-
ml
and 1 f t l g(s) d s > - - 1 [ 1 g(t) - m2
g(t0)]
g (t) J
Denote by Cb([to, +c~),R) the linear space of all continuous, bounded, real-valued functions z(t) defined on [to, +oc), endowed with the usual function operations, and equipped with the norm Izl = supt>_t o Iz(t)l. Clearly, the space E = (Cb([t0, + ~ ) , ~), I" I) is a Banach space. Let
B = { z E E : ,zI <_ 3H2 jL[ } ml We define an operator T : B ~ E as follows,
(Tz) (t) = m--; H2 IL I ~g(to) + ~ 61 J~o f g (s) F (s, v (s)) as,
144
R.P. AGARWALet al.
where
f o¢ -z~(s) ds,
v ( t ) -= Lg (t) - g (t)
and t _> to, z • B. Since //2 ]L[+}L[ 1 I(Tz)(t)lZ < .--; ~
tg 2 is) h~ I~) ds+3H2 ~1 L. ~
g2 I~) h~ is)
~
< rnl H2,LI+H2,L[ ~ 1 / t : g (s) ds + 3//21L, 1 /t{ H2 [g (s) ~ ml ~(~ < 2//2 IL]+ 3
ILl
< --]L I 2
ds
dg@T)]ds
< --ILl,
z • B,
it is clear that T(B) C B. Continuity and compactness of the operator T can be established by using the technique developed in [37, Section 4]. Now, an application of the Schauder-Tikhonov theorem implies that the operator T has a fixed-point z(t) in B. Using (27), we can refine the bounds for z(t). In fact, for t >_ to, we have
z(t) < m-~ H2 , L , ~
+ m~ H2 [L[+ 3 (\Hm--: 2 ~/ 2 fLI t - to
-
g (t)
and
z(t)>_ H21LIg(t°) m--1
~
H1
+-
m2
[
ILJ 1
g(to)J g(t)
-3
(H2~ 2 --
\ml]
ILl
t-to g@)
,
which in turn yield (33). We conclude the proof by noting that
v (t) = Lg (t) + q (t), where, for all t _> to, 3H2
Iq (t)[ _< m--~ ILl, and thus, l~ = L. | Now, we shall show that (30) yields (29). For this, we remark that the arguments employed by Avramescu et al. in [36, Section 5] cannot be used here. Let v(t) be a solution of equation (7) satisfying (30). Then, for all t > to and some c E R, we have
v (t) = cg (t) - g (t)
f
~ z (~)
-~
ds,
and there exists a T >_ to, such that, for all t >_ T, 0 < m~-1 ~'>T inf z (7) -< g(t) f t ¢¢ ~z(s) ds <
(38)
r>T
By Theorem 3, any bounded solution of (7) tends to zero as t --* +c~. Therefore, it follows from (38) that c 7~ 0. Furthermore, there exists a T1 >_ to, such that, for all t >_ T1, 1
Iv (t)l < 3
and limsup
1 /T~g 2
2 l i m s u p - =1 / ~ f(~,v(s))gis) 2 < - - lira sup z (t) < +co.
Clearly, from (39) the condition (29) is obvious.
as (39)
Existence and A s y m p t o t i c Behavior
145
The following result provides necessary and sufficient conditions on the function a(t) so that equation (8) has a solution f ( t ) which satisfies (9). Before we state and prove our theorem, we notice that so far we have assumed that two positive solutions of (8), which span the space of solutions of this differential equation, exist on the entire interval [to, +c~). However, positivity of the test functions f ( t ) and g(t) is not exploited in the proof of Theorem 5. Since any solution of the linear differential equation (8) can be continued backward for as long as the coefficient a(t) is continuous, one can replace the hypothesis of positivity of the test functions with the requirement of eventual positivity, focusing attention solely on condition (9). THEOREM 7. T h e equation (8) has a solution f : [to, +c~) ---, N satisfying (9) if and only if there exists a continuous function b : [to, +oo) --~ • such that, for all t > to and all q > O,
b (t) = a (t) and
-
m
2
ft
lim e -qt Jto eqSb (s) ds = 0. t-~+oo
(40)
PROOF. NECESSITY. Suppose that the function f ( t ) satisfies (9). Then, it cannot vanish eventually, i.e., there exists a T > to, such that If(t)l > 0, for all t _> T. For t > T, we define the function w E C I ( [ T , + c ~ ) , R ) as follows,
w (t) = g' (t) g (t) - m, where g(t) is given by g(t)=f(t)
1+
~
.
Since the test function g possesses the property lira
g' (t)
= m,
it is obvious that lira w ( t ) = 0 .
t--~+oe
Now, for t > T, let
(t) = m~ + [~, (t) + 2 m ~ (t) + ~ 2 (t)] = m 2 + b (t). a (t) = g" g (t) Using l'Hospital's rule, we obtain lira e -q*
t--*-I-oo
e q~ [w (s)] i ds =
lim [w (t)] i = 0,
t ---*-boo
i = 1, 2.
(as)
Integration by parts yields e -qt
eq~w ' (s) ds = w (t) - w (T) e q(T-t) - qe -qt
K
w (s) e qs ds,
for all t > T. Therefore, lira e -qt
t--*+~
eqSw ' (s) ds = O.
Thus, (40) follows from (41) and (42), which completes the proof of necessity.
(42)
146
R.P. AGARWAL e t al.
SUFFICIENCY.
STEP 1. First we shall show that condition (40) is equivalent to the following. (A) For any two positive numbers q and e, there exists an L0 = L0(q,E) _> to, such that, for all t > W >_ Lo,
~-q~
/;
I
eqsb(s) & < e.
(43)
Assume t h a t (40) holds. Then, for any two positive numbers q and e, there exists a T = T(q, e) > to, such that, for all t >_ T,
e -qt
eqSb (s) ds < -~.
Correspondingly, for any t > W > T, we have
< e -qW
eqSb (s) ds + e -qt
eqSb (s) ds < e.
Conversely, if (43) holds, then, for any two positive numbers q and e and, for all t > W >_ Lo, we find
e -qt
eq~b (S)
< 2'
where L0 = Lo(q,e/2). Furthermore, there exists a T = T(q,e) >__Lo such that, for all t > T,
¢ e-q* f~[~o eq'b (s) ds < 3" Consequently, for all t > T and W = Lo, we find
e -qt
eqSb (s)
< e -qt
eq% (s)
+ e -qt
eqSb (s) ds < e. o
This completes the first step. STEP 2. Fix an e E (0, 1) such that me < 1, and let
We claim that there exists a solution w(t) of the Riccati equation
~o' (t) + 2 . ~
(t) + ~o2 (t) = b (t),
t _> to,
(44)
which is defined on [L1, +c~), such that, for all t > LI, w (L1) = 0,
Iw (t)l < ms,
(45)
and lim w (t) = 0.
t--~+oo
(46)
First, we shall establish (45). For this, standard existence and uniqueness results imply that (44) with the initial condition w(L1) = 0 has a unique solution w(t), which is defined
Existence and Asymptotic Behavior
147
on [L1,L~), where L~ _< ÷oo. For the sake of contradiction, suppose that there exists an L* E (nl, n~) such that, for t C ILl, L*), [w(t)] < me and
Jw (L*)l = me.
(47)
Then, following Ascoli [25, p. 17], we can deduce that, for all t E [L1, L*],
+ e - 2 m t [ t e2m fw( )l
I (t)l _< JL1
JL1
me + (me)2 ~ i e2m(8-t) ds <
(48)
m6,
Letting t --- L* in (48), we find that Iw(L*)l < me, which contradicts assumption (47). Therefore, L~ = +co, and [w(t)l < me, for all t > LI. Thus, the validity of (45) is established. The proof of (46) is based completely on the estimate (43). The technique employed by Ascoli [25, pp. 16-17] cannot be applied here, since it requires additional assumptions on [b(t)l; however, we can use an iterative process. Let L2 def Ll -k Lo (2ra, m---~) .
Integrating equation (44) over [L2, t], we obtain e 2m8 [b (s) - w 2 (s)] ds,
w (t) = w (L2) e -2re(t-L2) + e -2mr 2
which yields, for t > L2, the estimate [w(t)l < raze -2m(t-L2) + -~ 2 -b (me) 2 -
c m(s-t) ds < me 2 +
.
JL2
-
Further, we let L3 def 1 In (4E-1) + L2 and L4 ~--~fL3-b L o ( 2 m , m---~-~). Then, for all t :> L3, we have ]w(t)l < me 2. Integration of (44) over [L4,t] yields, for t _> L4, [ : e 2re(s-t) ds < ~3mE4 + mE2e-2m(t-L4) [w (t)] < mE2e -2"~(t-L') + ~me 4 + (me2)2 JL 4
For n > 2, we introduce L2,~+1 d~f 1 In (4E-2) ÷ L2~ and L2n+2 = ~2,~+1 ÷ L0 2m,
Now, from mathematical induction, it follows that Iw(t)[ < me 2~, for all t > L2n+l. This completes the proof of our claim (46). Finally, introducing for t > L1 the test functions g(t) and f(t) as g(t)=exp
mt-k
w(s) ds ,
f(t)-~g(t)
g2(s),
1
we complete the proof. | REMARK 8. A condition similar to (40) has been obtained by Hartman [26, p. 570]. His result states that equation (8) has a solution f(t) satisfying (9), if and only if ~-~+oo [0<~<+oo 11 q- v ~
b (t) dt
= O.
(50)
However, under the hypotheses of the present paper, condition (40) has obvious preference over (50).
148
R . P . A¢AP,WAL e t
3. A N
a l.
EXAMPLE
First, we shall show that our Theorem 4 extends Theorem 1 due to O'IZegan in somewhat unexpected way. Suppose that there exists a function b(t) which satisfies (40) together with e ~ Ib(s)l ds > O.
limsup e - m r Then, equation (7) reads as
u" - m2u = H (t, u ) ,
(51)
t >_ to > o,
where H ( t , u) = b(t)u + F ( t , u). We claim that the function H ( t , u) in (51) does not satisfy Hypothesis (i) of Theorem 1. In fact, for a fixed r > 0 and for all t _> to, we have
]H (t,r)l >_ r]b(t)l - IF (t,r)] >_ rlb(t)l -'r~ (t) and limsup e -'~t
e m 8 l I t ( s , r ) l ds
t--++oo
]
It'
>_rlimsup
L
]
e'~lb(s)[ ds >0.
e-~t
t-~+oo
J to
Therefore, there exists no continuous, nonnegative function vr(t) defined on [to, +oo) such that, for all t >_ to and [u[ <_ r,
IH(t, )l < v (t) and l i m e -mr
emSvr (s) ds = O.
t--*+oo
Consequently, Theorem 1 cannot be applied in this case. In what follows, we shall show how to construct a function b(t) that possesses above-mentioned properties. Although the idea behind this construction is simple, the computational part is rather intricate. We begin with some useful facts which can be established by elementary methods. These are collected in the following two propositions which we state without proofs. P R O P O S I T I O N 9.
(1.i) Given a A > O, there exist two positive constants A1 and A2 such that, for all x > O, 1 -- e -rex
A1 <- -
1 --
e -mAx
< A2. -
-
(1.ii) For any w > 0 and any E E [0, 1), lira
e Wza-*/2 - 1
x-...oO+
X2
= O.
(1.iii) For a n y two given n u m b e r s a and 13,0 < a < t3 and for all x E [O,a -1 ln(a-l•)], 0 < e~ -
1 <_ fix.
(1.iv) For any a 6 R and for all x > O, )e~
-- 1) <
e I'~l:r -
1.
Existence and Asymptotic Behavior
149
PROPOSITION 10. Let K1, /(2, a and b berealnumbers, a < b, and let the function f : [a, b] --* be defined by
f(a,b;K1,K2)(t)= ~
[exp(Kl(b-t)2(t-a)-K2(t-a)2(b-t))-l]
•
(b - a)
Then, f has the following properties.
(2.i) f ( a ) = f(b) = O, f~(a) = K1, and f~_(b) = K2. (2.ii) For ali t e [a, b] 1 [exp ( 8 [f (t)[ < -(b- - - - a) -~
(IKI,+IK2[) (b
a)a) - lJ
-
"
(2.iii) If K1K2 < O, then sgnf(t) = sgnK1, for all t E (a, b). We are now in the position to prove the following result. THEOREM 11. Given an L > to, there exists a function w 6 CI([L, +co), R) n L I ( ( L , + c o ) , R ) which has the following properties. (i) There exists a strictly increasing sequence (ts)s> 1, satisfying lim ts = +co
n-~+oo
and
lira ( t n + l
n~+oo
-- tn)
=
O,
such that W(tn) = O. (ii) w ( t ) - ~ 0 a s t --+ + 0 %
lira sup w' (t) = +co
and
t--,-t-c~
(iii)
fit
limsup e -mr
t--*+~
lim inf w' (t) = - c o . t-*Woo
]
(52)
e ms Iw' (s)l ds = +co. L
PROOF. Fix two numbers, A > 5 and a E ( 0, 1). Consider a sequence of positive numbers (us)s>1, us --~ 0, as n --* +co, such that, for all n _>>1, Un >
2
A(n+2)
and
ltn+l <
Un ~
oo
2Un+l.
oo
2--~12
Assume also that the series Y~n=l Us diverges, and the series Y~n=l us converges. The set of sequences with these properties is nonempty and contains, for instance, (1/n)n>l. Define now sequences (Tn)n>0, (ts)s>l, (Pn)n>_l, and (Kn)n>>_l by Ts+l = T~ + Aug+l, ts=T~-u~, and
-El2 K n - = u n + 1,
To = L,
n > 0,
pn=T~+2us,
n>l,
(53)
n > 1.
Note that tt > L + 4Ul > L and lims...+oo Ts = +co. It follows from un+l + 2us _ 5un+l < Aug+l,
n > 1,
and (53) that Ps < tn+l, for n _> 1. Similarly, for n > 1, we have (A - 5) u n + l _< t s + l - p s < (A - 3) Un+l.
(54)
R . P . AGARWAL et al.
150
For t e [t~, tn+l] and n > 1, introduce the function w = w(t) by
( f(tn,pn;K~,-K~)(t), f(p~,t~+l;-K,~,K~+l)(t),
w(t)=
f o r t e [t~,p~], f o r t e ~vn,t,~+l].
This function w(t) can be defined on the entire interval [L, tl] so that it is continuously differentiable on [L, +c~). In view of (2i), for n _> 1, we have w E Cl([L,+oo), R), w(t.) = O, w'(t.) -~- K~, and w'(p, 0 = -K,~. Using (2ii), (2iii), we conclude that, for t E [t,~,pn], 0 _ ~ (t) _<
(55)
whereas for t C [pn,t~+l],
1 (t.+l
[l_exp(8(Kn+Kn+a)(t~+x_p,~)3)] p~)2
-
(56)
Now, since lim~-~+oo Kn = +c~, it follows that lim sup w' (t) _> lim w' (tn) = +c~ t---++c~
n--..*+ , ~
and liminfw' (t) _< lim w' (p~) = - c o . t---*+oo
n---*+oo
Furthermore, for n >_ 1, K . (pn
_
_
0 1 ~ .
3--e/2
tn) 3 < . . . . n+l
and 3-~/2 . (K~ + K , , + ~ ) ( t ~ + ~ _p,<)3 < ~5 (~, _ ~~J3 o -,~+~
Using (54), we deduce that, for all t e [t~, t~+l] and n >_ 1,
'w(t)I__<(9-I + (A--5)-')[exp[(130+ (N --3)3)
Un+13-'/2]J --I]Un~_I.
Since ~ < 1, from (lii), we conclude that limt_~+oo wit ) = O. Thus, Conclusions (i) and (ii) of the theorem are established. Now, we shall prove the absolute integrability of wit ). It follows from the properties of the sequence (u~)~>l, that there exists an integer N > 1, such that, for all n >_ N, 3-s/2
0<%+1
In 2
(57)
64 + (A - 3) 3.
Clearly, (57) implies that (p~ _ t~)3 _< 271n_____228K~
and
2 (t~+l - p~)3 -< 4 (K~27+InK~+I)"
Furthermore, it follows from i55) and i56) that
L
t~+l Iw (s)[ ds = In + J~,
tn
where
In =
[w (s)] ds <
-
exp
iPn
- 1
(58)
Existence and Asymptotic Behavior
151
and
&=
r
Iw (s)l ds < exp - tn+l -- Pn
Jpn
(K: + K:+I) (t~+l - p:)3
)} _ 1
Choosing in (liii), 2a =/7 = 32K~/27 and using (58), we obtain 0 <
j~
32 K
2-e/2
I~ (s)l ds <_ ~-~ n (pn - tn) 2 <_ 4 41~°+
.
(59)
On the other hand, letting in (liii), 2a = ~ = 16(Kn + K n + l ) 1 2 7 and applying (58), we get ft~+i o < .po
16 Iw(s)lds<_ ~-~ ( K n + K . + I ) ( t . + I - P . ) 2 < 2 ( ~ _
- ,~\2 ~) u .2--¢/2 +l .
Combining now (59) and (60), we find, for n _> N, the inequality 0 <
f
t~+~
Iw (s)l ds <
In 2
Consequently, we have
J~lw(s)l
d s < ln2 ~ --
e
2 - e/ 2 Un+l
e
J tn
2-~/2
?-tn .{. 1
(
~-O0,
n.~ N
i.e., the function w ( t ) is absolutely integrable. To prove Part (iii), we let yn = e -roT=
e'~
Iw' (s)l ds,
n>l.
We note that, for all t E [tj, Tj] and j >_ 1, w (t) -
1 (PJ _ tj)2 [exp ( K j (pj - tj) (pj - t) (t - tj) ) - 1]
and
w' (t) -- PJKJ- tj (PJ + tj - 2t) [l + (pj - t j ) 2 w (t)] >- PJKJ- tj (pj + t6 - 2Tj) = -3-'KJ
Furthermore, for n _> 1, 2n
-Tj
j n~
-
3m
j=n Cq
~s(em')ds=--e- 3m
' : ~~, ( e m T j -- emq ) " j=n
Thus, it follows from (li) that, for j _> 1, e mTj - e mq 1 - e-m(T~-t~) emT~ -- e mTj-1 -~- 1 -- e - m ( T j - T j - 1 )
1 -- e - m u j
1 - e-m~u~ > A1.
Hence, for n > 1, V2n >_ K,~
e-mT2~ E j=n
( emT~ -- emT~-l) = K n
[1 - exp ( - m (T2n - Tn-1))].
(60)
l:t.P. AGARWAL et al.
152
Finally, in view of the inequality 2n
T2n-Tn-1 = A Euj
~ A ( n + l)u2n > 1,
j=n
which is valid for all n > 1, we conclude that limsup~_~+~ V~ = + ~ . The proof is now complete. Now, for the Riccati equation (44), we note that for s > L,
2m]w(s)l + w
tb(s)l +
2 (s) > Iw'(s)l.
Therefore, from (52), we can conclude t h a t
f;'
limsup e - ' u t--++oo L
e r"'
Ib(s)l
ds
JL
]
If
>__limsup e -mr t--*+c~
]
e m~ Iw' (s)l ds = +c~.
In the following two propositions, we point out interesting properties of our example. PROPOSITION 12. The bound Lo in Condition (A) (suffJciency part) of Theorem 7is independent ofq. PROOF. The proposition follows from the fact that the function w(t), whose existence has been established in Theorem 11, is an absolutely integrable function t h a t tends to zero as t ~ +oo. In fact, for all t > W > L, we have
e-qt / ; eqSw(s) ds] ~ / j e -qt
I/;
I
lw(s)l ds,
eqSw 2 (s) ds < sup [Iw (T)I ] r>L
/:
Iw (s)l ds,
and
~-q~
~'
(s)
< I~ (t)l + I~ (w)] + q~-qt
~
I~ (s)l ds.
Since w(t) --* 0 as t --* +c~, for any 5 > 0, there exists a T = T(6) > L such that ]w(s)l < 6, for all s >_ T. Multiplying both sides of this inequality by exp(qs) and integrating over [W, t], for all t >_ W > T, we obtain
qe-~t
/;
e~s I~ (s)l ds < ~.
The bound in the latter inequality is clearly independent of q, which proves our claim.
I
PROPOSITION 13. We can choose the test function g(t) so that g(t) ..~ exp(mt), as t --* +co. PROOF. Writing the first equation in (49) as
g (t) = exp mt we immediately arrive at the conclusion.
~ (s) d~ , I
We conclude this section by describing several simple classes of functions that satisfy conditions (16) and (17). For this, we need the following auxiliary result.
Existence and Asymptotic Behavior
153
LEMMA 14. For given two numbers A and v, such that A E (0, 1) and 1 < v < A-1, it follows that lim [gg(t) (At)]v ~-+~
=
o.
(61)
PROOF. Let .s
(t)
or
it )
=
[g (~t)]v g (t)
Then, for t >_ to, j , it )
:
(xt~ k
g(
)
~)1
g(t) J "
Since the term in the brackets tends to (Av - 1)m < 0, as t --+ +oo, we deduce that J(t) is eventually decreasing on [t0, +oo). This yields
O < L . = lim J(t)<+cc. t---+-t-~
For the sake of contradiction, suppose that L. > 0. Applying l'Hospital's rule, we obtain t-~+~
[ g (At)
= AvL~ < L,,
g'
which is a contradiction. Thus, the conclusion of the lemma follows.
II
In what follows, p' > 1 is a number defined by (l/p) + (1/p') = 1. PROPOSITION 15. A continuous function "G(t) satisfies conditions (16) and (17) provided it belongs to one of the following classes, (i) functions that tend to zero as t ~ +oo; (ii) functions which are t~rst order derivatives of functions satisfying (i); (iii) functions which belong to LP( (to, +oo), ]~) for some p > 1. PROOF. Proof of Claims (i) and (ii) is similar to that given in Theorem 7. In the ease p = 1, Claim (iii) has been established in Avramescu et al. [36]. Here, we shall show (iii) when p > 1. To prove (16), fix a A E (0, 1). Now, an application of the HSlder inequality yields
I~ (s)l ~s =
g - ~ I~ (s)l d~ +
-< Ir~lL'((t°,+°°),a)
+
~ g - ~ I~r (s)l d~ [g(t)J
[T~ (s)l p ds
ds
~ Lg (t) J
(62)
ds
'
where the function g(t) is increasing, for all t _> T. Using l'Hospital's rule and Lemma 14, we deduce that
f < [gi41" d~ =0 ' and, respectively, "' lim t-+oo
~ Lg(t) J
ds=
~
l~m
[g(t) J
ds=
1 <+oo. rap,
The conclusion of Claim (iii) follows, by letting t --* +oo in (62). Finally, we note that (17) can be established in a similar manner. |
R . P . AGARWAL et al.
154
4. D I S C U S S I O N Here, we shall discuss how our results are related to those known in the literature. First, we observe that if g(t) is a positive, continuous function that approaches +co, not necessarily as fast as the exponential function, conditions (16) and (17) play special role in the theory of asymptotic integration, see, for instance, the papers by Faedo [42, p. 211], Ghizzetti [43, p. 30], or Hallam and Heidel [44, p. 126], where the assumption
--1 L x sp f (s) ds = 0
lim
x---*q-oo x P
0
has been fully used. We also remark that [25, Sections 1-2] gave a detailed account of the properties of the Volterra operator of the first kind defined by
(vj)
f
(t) =
e-~(~-')/(T) d-~,
t >_ O,
where f E LP((0, +co), R) for some fixed p _> 1. In the case p = 1, our proof of Part (iii) in Proposition 15 is based on the ideas of Faedo [42, p. 211], while a simple, ingenious majorization technique, which can be successfully applied to deal with (62) in some particular cases, has been suggested by Ascoli in [25, p. 15]. A more complicated problem, which remains unsettled, is concerned with finding solutions to Riccati equation (44). Assume that b(t) belongs to LP((to, +c~), •), for some p _> 1. Then, one can apply the techniques of Ascoli [25, Section 4] to compute a series expansion for the solution of equation (44), satisfying the initial condition w(0) = 0, in the form v--1
w (t) = E
wi (t) + remainder term,
i=1
where v - 1 < p < ~, wl = V2b, and n--1 =
-
v2
/=1
for n > 1. In an elegant, however, unfortunately almost forgotten paper, Owens [27] analyzed the boundary value problem for a perturbed quasi-linear differential equation
y" - A f (x, y) y = q(x, y), y (0) = 0,
x >_ O, lim y (x) = 0,
(63)
~---+-~-OO
where A > 0, f ( x , y) and q(x, y) are continuous functions satisfying (i) 0 < m < f(x, y) < M, for all x > 0, lYl - Y, and lim~-~+oo q(x, 0) = 0; (ii) f ( x , y) and q(x, y) are Lipschitz functions with respect to y, with Lipschitz constants K1 and K2; (iii) for some constant a > 0, o ~ If ( x , y ) - a ~ l
dx < ÷ c o
and
L
o~
]q(x,0)l dx < +co;
(iv) q(x, y) satisfies 2 suplq(x,y)l < Y rnA ~>_o
and
1 [4Klsup[q(x,Y)l+2K2rn] m2A x>o
<1.
Existence and Asymptotic Behavior
155
Using the method of successive approximations and two test functions ¢ and ¢ which are slightly more general than our test functions f and g, Owens established existence of a unique LLsolution of the boundary value problem (63) and obtained a priori bound for the Ll-norm of this solution. The only weakness of his method is that the test function ¢ is not computed explicitly, rather it is defined as a solution of a boundary value problem on [0, +c~); whereas, ¢ is found by using successive approximations, see [27, equation (1.12)]. For the case where a(t) is a piecewise continuous function satisfying [a(t)l < m 2, Loud [45, Section 4] obtained several interesting estimates for fundamental solutions of equation (8) by exploiting the method of successive approximations, and this technique can be also adapted for equation (7). Further extensions of Loud's results have been established by Alekseev, Levin and Levinson, see comprehensive discussion in the monograph of Coppel [39, Lemma 4, p. 116]. Application of l'Hospital's rule yields
,12 oo: (t)
Jl 7-{-4 =
g(s)
Therefore, our hypotheses can be also compared to conditions imposed by Corduneanu [46, Lemma 1, p. 104]. However, the nonlinearity in his paper should satisfy a Lipschitz condition, and thus results obtained by Corduneanu [46, Theorems 1 and 2] cannot be applied to the case discussed in this paper. Extending several results due to Coppel [39], Brauer and Wong [47] studied general problems related to those discussed in the present paper. Using the definition of g(t) and (19), one can establish existence of a constant K > 0, such that [X(t)PlX-l(s)[
to<_s<_t,
]x (t) p ~ x - l (s)l < K,
to < t <_s,
see also [47, p. 529]. Therefore, results proved by Brauer and Wong can be employed for the study of existence of solutions of equation (7) for which the limit lim -1 ln[u(t)[ t--~+c~ t exists and is finite. A straightforward application of l'Hospitai's rule allows to establish that the value of this limit is - m for the test function f(t), and m for the test function g(t) (we refer to the paper by Avramescu et al. [361 where relevant details can be found). Under assumption (10), the weakest, according to Coppel [39, Equation (4), p. 97], integrability hypothesis imposed by Brauer and Wong [47, p. 541] reads as ft+l
es=O.
(64)
This condition is considerably less restrictive compared to standard integrability conditions. However, assume that, for a nonnegative continuous function A(t), one has 0 < f t t+l ~ (s) ds < c, for t >_ T _> to. Then, according to Massera-Sch~ffer lemma (see [39, Lemma 2, p. 102], [40, Exercise 4.8, p. 76], or [48]), for all q > 0 and t _> T, max
{// e -qt
eq'A (s) ds, e qt
e-q'A (s) ds
} <- 1 - .e----"-'-~< +c~.
Thus, condition (64) is a particular case of (16) and (17), as well as of (11) and (13). We conclude the paper with an unsettled problem. Apply the fixed-point theory to study asymptotic behavior of solutions of a semilinear differential equation u" = (m 2 + b (t)) (u + n (u)),
t > to _> O,
where b(t) and R(u) are continuous functions such that IR(u)l is dominated by a suitable comparison function, whereas the only assumption on b(t) is equation (50).
156
R.P. AGARWALet al.
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