SINGULAR BOUNDARY-VALUE PROBLEMS FOR ORDINARY SECOND-ORDER DIFFERENTIAL EQUATIONS I. T. Kiguradze and B. L. Shekhter
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SINGULAR BOUNDARY-VALUE PROBLEMS FOR ORDINARY SECOND-ORDER DIFFERENTIAL EQUATIONS I. T. Kiguradze and B. L. Shekhter
UDC 517.927
This article gives an exposition of the fundamental results of the theory of boundary-value problems for ordinary second-order differential equations having singularities with respect to the independent variable or one of the phase variables. In particular criteria are given for solvability and unique solvability of two-point boundary-value problems and problems concerning bounded and monotonic solutions. Several specific problems are considered which arise in applications (atomic physics, field theory, boundary-layer theory of a viscous incompressible fluid, etc.) Introduction The systematic study of initial and boundary-value problems for an ordinary second-order differential equation
,,"= f(t,,,,,,')
(0.1)
having singularities with respect to the independent variable or one of the phase variables has only a thirtyyear history, although such problems began to arise quite long ago in applications. For example, as early as the beginning of the century, in a paper of Emden [74] devoted to the equilibrium of a sphere formed from a polytropic gas, there arose a singular Cauchy problem ,," =
t
- ,,',
u(o) = c 0 > 0 ,
u'(o) = o.
Nevertheless for a long time mathematicians limited themselves to the study of singular problems of a specific type and did not make any attempts to work out more or less general methods of investigation. In the well-known monograph of Sansone ([51], p. 349) in particular it is stated that because of the absence of any general theory of solvability of the singular Cauchy problem, the existence of a solution of the abovementioned problem from the paper of Emden can be established "only through direct study" of the corresponding differential equation. At present the singular Canchy problem has been studied with considerable completeness not only for Eq. (0.1) but also for differential equations and systems of higher orders [11, 12, 49, 59]. The theory of singular boundary-value problems for Eq. (0.1) is also quite far advanced. The present work is devoted to an exposition of the fundamentals of this theory using the example of two-point problems and the problems of bounded and monotone solutions. In the first chapter (w167 we consider two-point problems of the form
uCa+) =
c1,
u c'-1) (b-) = c~,
(0.2,)
where i E {1,2} and - o o < a < b < +or The basis of w is formed by the results of [7, 16, 61, 82, 83], which are concerned with the existence and uniqueness of solutions of the linear differential equation u" = pl ( t ) u + p2(t)
' + p0 (t),
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 30, pp. 105-201, 1987. 2340
0090-4104/88/4302-2340512.50 9 1988 Plenum Publishing Corporation
satisfying the boundary conditions (0.2,) and (0.22). Here we do not exclude the case when all the functions pj : (a, b) -+ It, (j -- 0,1, 2), fail to be summable on the closed interval [a, b], having singularities at its endpoints. w167and 3 are devoted to the study of two-point boundary-value problems for Eq. (0.1) when f : (a, b) x I{ 2 --* R has nonintegrable singularities with respect to the first argument at the points a and b. In w we consider the case when Eq. (0.1) is comparable to a linear equation in a certain sense. The study of boundary-value problems for regular equations of such type dates from the papers of Picard [92], Tonelli [102], and Epheser [75]. Of subsequent investigations we note [26, 47, 68]. The existence and uniqueness theorems we give for solutions of the problems (0.1), (0.2,), (i = 1, 2), are a certain modification of the results of the papers [15, 82]. In addition, in w we discuss the question of nonuniqueness conditions for solutions of the problems mentioned. Following [28, 48, 60], we establish criteria for existence of at least a given number of solutions of these problems and study their oscillation properties. When uniqueness does not hold, such properties are frequently of interest from the point of view of applications (cf., for example, [86]). In w we study equation (0.1) with, right-hand side rapidly growing on the phase variables. The fundamental :results on the solvability of boundary-value problems for such equations in the regular case are due to S. N. Bernshtein [3], Nagumo [88], and Epheser [75]. These results have been generalized in various directions by many authors (cf. [4, 8, 30, 56] and the literature indicated in these works). In [13, 14] an approach to the study of singular two-point boundary-value problems is suggested based on a priori estimates of the solutions of one-sided differential inequalities. We adhere to this approach in our exposition of the results of w In w we consider the question of the existence and uniqueness of a solution of the equation (0.1) satisfying the conditions
u(a+)=O,
u(i-')(b-)=O,
u(t)>0
fora 0
forO 0, a > 0, and c < 0 [80]. Problems of this type have been studied by A. D. Myshkis and G. V. Shcherbina [46, 64-66]. w is devoted to the particular boundary-value problem ~" = - ~ - ~ ' + ~ - I ~ l ' s g n ~ , t r = o, ~ ( + c o ) = o,
(0.7) (0.8)
where "I and A are real numbers with A > 0. This problem arises in nonlinear field theory in the study of elementary particle interactions. In addition, it is encountered in a variety of other areas of physics, in 2342
particular in the statistical theory of the nucleus and in nonlinear optics (cf., for example, [1, 2, 9, 87, 89] and the literature mentioned in these works). Equation (0.7) has been especially intensively studied in the case "7 = 2. In this case the transformation ~J
=
gA
A~t
,o.yl
-
t
brings (0.7) into the form
r
1
= , -
sgnv.
(0.10)
If v : (0, + c o ) - + t t is a solution of (0.10), then ,(0+)
= 0,
,(+co)
= 0
(0.11)
and, in addition, lira I~(t)l < +co, t--*O
(0.12)
t
then the function u : (0, +co) -+ It defined by relation (0.9) is a solution of the problem (0.7), (0.8) with "7=2. As Nehari has shown [89], for 1 < ), < 5 the problem (0.10), (0.11) has a positive solution on (0, +co), which for A ~< 4 satisfies (0.12), while for A = 5 this problem has no nonzero solutions. However, the method applied ill [89] attests that for all ), >/5 there are no such solutions (cf. also [63, 96]). Nehari's result was strengthened by Ryder [95], who showed that for 1 < A < 5 the problem (0.10), (0.11) has a solution with any preassigned number of zeros in (0, +co) (for), E (1, 4) the analogous assertion was established by V. P. Shirikov [63]). However, the question whether the solutions constructed in [89] and [95] satisfy the condition (0.12) for A > 4 remained open. A positive answer to this question was given by Sansone [96]. In this way it was established that if '7 = 2, 1 < ), < 5, and l E {0, 1,... }, then the problem (0.7), (0.8) has a solution with exactly l zeros in (0, +co). In the general case, i.e., when "7 may differ from 2, the problem (0.7), (0.8) was studied by Kurtz [87], who established (although the proof was not completely correct for i ~ 0) that for any nonnegative integer l there exists a solution having exactly l zeros on (0, +co), provided 1 ~< "7 ~< 2 and 1 < A ~< 3. In w we study solvability and properties of the solutions of the problem (0.7), (0.8) for all real '7 and positive A, following the paper [99]. In the present paper we use the following notation:
R+ = [0, +co),
R"=.R•
(,=2,3,...);
u(t+) and u(t-) denote respectively the right- and left-hand one-sided limits of the function u at the point t; C(I), where I C R is the set of continuous functions u : I --, R; C ~([t~, t2]) is the set of functions u : [t~, t2] --* R that are absolutely continuous together with their first derivative; /:,([tl,t2]) is the set of summable functions u : [t,,t2] --* R; C~oc (I) and L~oo(I), where I C R is an open or half-open interval, is the set of functions u : I --* R whose restrictions to any closed interval [t~, t~] c I belong to the class ~1 ([tl, t2 ]) and L([tl, t2 ]) respectively; K([t~,t2] • D), where D C R " , n E { 1 , 2 , . . . } , (R t = R) is the Carathdodory class, i.e., the set of functions f : [tl,t2] x D -* R such that f(.,x~,...,x,): [tt,t2] -* R is measurable for all ( x z , . . . , x , ) E D, f(t,.,...,.) : D ~ R is continuous for almost all t E [tl,t2], and
sup {IfC-,xl,...,x.)l : ( x l , . . . , x . )
e
D0} E L([tl,t2])
for any compact set Do C D;
2343
K~or (I • D), where I C R is an open or half-open interval and D c R " , n G {1, 2 , . . . }, is the set of functions f : I • D --* R whose restrictions to the set [t~,t~] • D belong to K([tl,t2] • D) for any closed interval [tx,t2] C I; g~ • 2) is the set of functions f : ( t l , t ~ ) x R 2 ~ R for which the mapping t ~ f(t,x(t),y(t)) is measurable for any x,y G C((tl,t2)). We shall call a function u : (a, b) --. R a solution of Eq. (0.1) in the interval (a, b) if it belongs to C11or((a, b)) and satisfies (0.1) almost everywhere on this interval. PROBLEMS w
Chapter 1 ON A FINITE INTERVAL
Linear Equations The present section is devoted to the linear singular differential equation
(1.1)
u" = p1(t)u + P2Ct)u' + po(t) under boundary conditions of the following two types:
u(a+) =
cx,
u ( b - ) = c2
(1.21)
u'Cb-) = c2,
(1.22)
and u(a+) = c,,
where c r R, (j = 1, 2). Before stating the restrictions imposed on the functions pl, p2, and p0, we introduce the transformation a : L,oc ((a, b)) --, C((a, b)) defined by the equality
a(p)(t) = exp
[/i
]
per) dr .
2
If a(p) e L([a, b]), we set
,,,(p)(O - ~(P)(O
oCp)(Odr ,,(p)Cr) dr
and
a2(p)(t)
-
1 f[ a(p)(r) dr
a(~(t)
on(a,b). In studying the problem (1.1), (1.21) we shall assume that
Pj,P2 e L,or
a(p~) e L([a,b]),
pjcrl(p2 ) e L([a,b])
(j : 0 , 1 ) ,
(1.31)
(j = 0,1).
(1.3~)
and in studying the problem (1.1), (1.22) we shall assume that Pj,P2 e L,or ((a,b]),
a(p2) e L([a,b]),
pja2(p2) e L([a,b])
Relations (1.31) hold if, for example,
Ip~(Ol< [(t_,=)(b_O],§
(i=o,1),
6
Ip2(t)l -< ~ + ~
6
+ b--~'
and relations (1.32) hold if A
Ip~(t)l -< (t-a),+~
2344
(i =0,1),
Ip2(0J -< ~ + - $- - a
for a < t < b,
where ), > 0 and 0 ~ 6 < 1. 1.1. T h e H o m o g e n e o u s E q u a t i o n . In this subsection we study the behavior of solutions of the singular homogeneous equation u" : p, (t)u + P, Ct)u' (1.4) near points where its coefficients have singularities. In what follows we shall pay special attention to certain classes of such equations having no nontrivial solutions under the boundary conditions u(a+) = 0,
u(b-)=O
(1.,.51)
u(a+) = 0,
u'(b-) = 0,
(1.52)
or for these classes also play an important role in the study of nonlinear boundary-value problems. In studying Eq. (1.4), we shall assume that either
Pl,P2 E L,or
a(p2) e L([a,b]),
pliYl(p2) E L([a,b]),
(1.61)
Pl,P2 e LIoc (Ca, b]),
6rCp2) e L(la, b]),
plO'2 (p2) e L([a,b]).
(1.62)
or
LEMMA1.1. Suppose conditions (1.61) hold. Then 1. Equation (1.4) has a solution u satisfying the initial conditions . ( a + ) : o,
with ~(t) = o
(/;
lim .'(t) t-.. a(p2)Ct) = 1,
o(p,)C~)d~
)
as
t-,
(1.7)
a;
(1.8)
2. any solution ~ of this equation linearly independent of u has a finite nonzero limit fi(a+); 3. for any bounded continuously differential function v : (a, b) --+ R lira inf u(t)lv'(t)l ,--o ~(p2)(t)
_
(1.9)
o.
PROOF: Let tk e (a,a--{--b-), (k = 1,2,...), and tk ---~ a as k --+ q-oo. F o r e a c h n a t u r a l n u m b e r k w e define the function uk on the interval [tk, b) as the solution of Eq. (1.4) under the initial conditions u(tk) = 0,
u'Ctk)=a(p2)(tk).
Then
u~(t) =o(P2)(t)[l + / i Pl(r) aCP2)(r)uk(r) dr]
fortkO
fora o, and so
_~'(t)
I~'(t)l > ~---;:~-
u~r)
oo
I~Ct)l/>
for a < t ~ o~,
~(~o)
Iv'Cr)ldr-I~(~o)1/> 61n
uCt)
Iv(~o)l
for ~ < t < ~o,
which is impossible if the function v is bounded. The lemma is now proved. Using an elementary cha~ge of the independent variable in Eq. (1.4) we can obtain the following assertion from Lemma 1.1. LEMMA 1.1'. Suppose conditions (1.61) hold. Then 1. Equation (1.4) has a solution u satisfying the initial conditions
u(b-) : 0,
2346
lira o(p2)(t) u'(t) ----1, t-.b
(1.131)
with u(t) = 0
o(p2)(r) dr
)
as
t -+ b;
2. any solution ~ of this equation linearly independent of u has a finite nonzero limit ~(b-); 3. for any bounded continuously differentiable function v : (a, b) ~ t t liminf u ( t ) l v ' ( t ) l
,-.b
- o.
oCp2)Ct)
We remark that if conditions (1.62) hold, then the coefficients of Eq. 1.4 have no nonintegrable singularities at the point b. Consequently we can arbitrarily prescribe the values of the solution and its first derivative at this point. In particular, we shall need the solution of this equation under the initial conditions
u(b-) = 1,
u'(b-) = 0
(1.132)
below. We note also that, as shown in [83], if the second of conditions (1.22) is replaced by lim
u'(t) Cp2)Ct) - c2,
then the problem (1.1), (1.22) can be considered even in the case when the functions Pl, P2, and P0 are not integrable at the point b. 1.2. T h e G r e e n ' s F u n c t i o n . As in the regular case, there exists a close connection between the unique solvability of ~he boundary-value problem (1.1), (1.2d) and the absence of nontrivial solutions of the homogeneous problem (1.4), (1.5,), (i = 1,2). We now turn to the study of this connection. DEFINITION 1.1. Let i E {1,2}. A function ,~ : (a,b) x (a,b) -* R is called a Green's function of the problem (1.4), (1.5d) if for any r e Ca, b): 1. The function u(t) = ~(t, r) is continuous on (a,b) and satisfies the boundary conditions (1.5i); 2. The restriction of u to the intervals (a,r) and (r,b) is a solution of mq. (1.4); 3.
1.
LEMMA 1.2. Let i E {1,2); let the conditions (1.6i) be satisfied, and let the problem (1.4), (1.5i) have no nontriviaI solutions. Then there exists a unique Green's function ~ of this problem and
_ ~(t,r) =
u2(t)Ul(r) u2(a+)a(p2)(r) utCt)u2(r)
for a < r < t < b, (1.14) for a < t < r < b,
-2(a+)o(p2)(r)
where ul and u2 are solutions of the initial-value problems (1.4), (1.7) and (1.4), (1,13i) respectively. PROOF: As we verified above, under the hypotheses of the lemma there exist solutions ul and u2 of the initial-value problems (1.4), (1.7) and (1.4), (1.13i). Since the boundary-value problem (1.4), (1.5i) has no nontrivial sohltions, u2 (a+) ~ 0. It is clear that the function .~ given by equality (1.14) satisfies the conditions I and 2 in the definition of a Green's function. We shall show that it also satisfies condition 3 of the definition, i.e., that w(t) = u2 (a+), where wCt) = C x ( t ) u 2 ( t ) - u ~ ( t ) u l ( t ) for a < t < b.
o(p2)Ct) Indeed by L e m m a 1.1 we have w(a+) = u2(a+), and by Liouville's formula w(t) = const. Thus ff is a Green's function for the problem (1.4), (1.5i). Uniqueness follows from the unique solvability of this problem. The l e m m a is now proved. The basic proposition that we shall prove relating to the problems (1.1), (1.21) and (1.1), (1.22) amounts to the following.
2347
THEOREM 1.1. I f i E {1, 2} and conditions (1.3,) hold, then the problem (1.1), (1.2,) is uniquely solvable ff and only ff the homogeneous problem (1.4), (1.5,) has no nontriviM solutions. If the latter holds, then the solution u of the problem (1.1), (1.2,) can be represented by Green's formula
~ab u(t)=uo(t)+
~(t,r)po(r)dr
(1.15)
fora o
ql(t) />pl(t),
fora10 for a < t < b.
(1.18) (1.19)
Then the solution v of Eq. (1.16) under the initial conditions v(a+) = 0 ,
lim - v'(t) - - aCq2)Ct)
1
(1.20)
t-~'~
satisfies the inequalities
v(t)>0
fora O f o r a < t < b ,
2348
(1.22)
then also v'(t) > 0
fora10 for a < t < t* and w(t*) < 0, and this contradicts (1.24). Thus, v(t) > 0 f o r a < t < b .
(1.25)
Consequently
w'(t) >>.0 for a < t < b
(1.26)
and the (possibly infinite) limit w(b-) exists. If v(b-) = 0, then, according to (1.25) and Lemma 1.1', lim v'Ct) < 0, t--.b a(q~)(t)
lim inf v(t)lu'(t)l - 0 .
t--,~
a(q2)(t)
But since, by (1.18), u(b-) > 0, we find from these relations that w(b-) < 0, which, by (1.26), contradicts (1.24). Therefore v(b-) > 0. Using this inequality, in the case when i = 2, we easily obtain v'(b-) > O. Finally, if (1.22) holds, then, since w(t) >~ 0 for a < t < b, we find from (1.18) and (1.25) that (1.23) holds. The lemma is now proved. We remark that when the coefficients of the equations (1.4) and (1.16) are continuous on the closed interval [a, b], pz (t) - q2 (t), and i = 1, Lemma 1.3 is essentially the known comparison theorem of Sturm ([56], p. 394). LEMMA 1.3'. Let i E {1,2}, and let relations (1.61), (1.17i), and (1.18) hold, where u is a solution of Eq. (1.4). Further suppose v is a solution of the problem (1.16), (1.20), and
ql(t) >>.pl(t),
[q2(t)-p2(t)]v'(t) >>.0 for a < t < b.
Then inequalities (1.21) hold. Moreover if condition (1.23) does not hold, then condition (1.22) also cannot hold. To verify this proposition it suffices to repeat the proof of Lemma 1.3 only replacing the factor 1/a(q2 ) (t) by 1/o(p2)(t) in the equality defining the function w.
2349
1.4. T h e Sets V1 ((a, b)) a n d 172((a, b)) a n d t h e i r S t r u c t u r e . We shall consider the classes of equations (1.4) whose coefficients pl and p~ satisfy the inequalities
pj,(t)
fora O. Moreover if the relations (1.30i) and (1.31) hold and for any measurable functions q, q~ : (a, a) --. It satisfying (1.41) the problem (1.42), (1.5~) has no nontrivial solutions, then (1.40) holds. PROOF: The first part of the theorem follows directly from Lemma 1.3, as we have already noted above. Let us prove the second part. Suppose (1.30~) and (1.31) hold and the problem (1.42), (1.51) has no nontrivial solutions when the coefficients of Eq. (1.42) are measurable and satisfy (1.41). Then, in particular,
(pX,pl + Ipll,pn,p
2)
v CCa, b)).
But since the solution of the problem
=Ca+)=0,
lira
u'Ct)
tr(p2,)C t) - 1
has no zeros in the interval (a, b), this relation, according to Theorem 1.2, implies (1.40). The theorem is now proved. In order to bring o u t c e r t a i n other properties of the sets Vlo((a,b)) and V2o((a,b)) we introduce the following definitions. DEFINITION 1.5. Let c E (a, b]. We shall say that the vector-valued function (pl ,P2) E (a, b) --* R 2 belongs to the set V~'0((a, b), c) if conditions (1.61) hold and the solution tt of the problem (1.4), (1.7) satisfies the inequalities u'(t)(c - t) >l O f o r a < t < b , u(b-) > O. (1.43) DEFINITION 1.6. We shall say that the vector-valued function (Pl,P2) : (a, b) --+ R 2 belongs to the set V~o((a,b)) if conditions (1.62) hold and the solution u of the problem (1.4), (1.7) satisfies the inequalities u'(t)>0 2354
fora0
fora~ p2 (t)
for a < t < b,
then (Pl,P~,P) E V2o((a,b)).
(1.51)
2355
Moreover, if (1.48) and (1.51) hold, then (1.50) also holds. To conclude this subsection we note that when C1.47) holds, Eqs. (1.42), whose coefficients satisfy (1.41), are nonoscillating on the closed interval [a, b], i.e., have no nontrivial solutions under the boundary conditions u ( t l + ) = 0,
u ( t 2 - ) = 0,
for any tl,t2 E [a, b], tl < t2. There is an extensive literature on regular nonoscillating equations (eL, for example, [73, 94]). But if (1.50) holds, then Eqs. C1.42) with qj ELloe{(a,b]), qj(t) >lpj(t) f o r a < t < b have no nontrivial solutions satisfying the boundary conditions
(3"=1,2),
~(tl+) = 0, ~'(t2-) = 0 for any tl, t2 E [a, b), tl < t2. These facts follow from Lemma 1.3. 1.5. S o m e V e c t o r - V a l u e d F u n c t i o n s B e l o n g i n g to t h e Classes V~((a, b)). In this subsection we shall present a variety of effective criteria guaranteeing the inclusion (1.33). LEMMA 1.51. Suppose relations (1.301) and (1.31) hold and suppose there exists a point d E (a,b) such
that /dp-~ (t) / t exp [ / " P21(s) ds] dr dt O for a < t < to, where to is some point in the interval (a, b0). Assume that to E (a, d]. By Lemma 1.1, /~=sup Obviously
{
u'(t)exp
[/o
]
u'(to) = O,
p2(r) dr :a 0 for a < t < b, i.e., (1.54) holds. But if (1.53) does not hold, t h e n there exists a point c E Ca, b0) for which z(t) < 0 for c < t < b0 and so u'(t) < 0 for c < t < b, so t h a t (1.54) cannot hold. T h e l e m m a is now proved. In exactly the same way we can prove
2357
LEMMA1.62. Suppose lz, l~ 6 It. and the function g : (a, b) ~ (0, +oo) is absolutely continuous on each closed subinterval of the interval (a, b] and summable on the interval [a, b]. Then the condition b
I(ll,l,)
fa
>
g(t) dt
(1.55)
is necessary and sufficient for the relation r
(Ixg',-12g + g ) 6 V~o((a,b)). As Opial [91] has shown, in the case of nonnegative lx and 12 relation (1.55) follows from the inequality l l h 2 + 212h < 7r2,
where
b
h= 2
fa
g(t) dt.
Setting g(t) = (t - a) -~
for a < t < b,
we obtain the following corollary of Lemmas 1.61 and 1.62. COROLLARY. Let A 6 [0, 1) and 11,12 6 R. Then the condition
I(lx 12) >1 ( b - a ) Z - x '
(1.56)
1-),
is necessary and sufficient for the vector-wMued function (PI,P2) : (a,b) ---} R 2 to belong to the set V~o((a,b),b), where Pl (t) = -tl
(t - a ) - 2 ~ ,
p~Ct) = -t2(t
- a) -~
-
A t-a
- -
for a < t < b,
and if the inequality (1.56) is strict, the condition is necessary and sufficient for the function to belong to % CCa,b))LEMMA 1.7. / r e 6 (a,b), l j , m j 6 R, (j = 1,2), and the function g is the same as in Lemma 1.6,, then the inclusion (Pz,P2) 6 V:o((a,b),c ), (1.57) where
p,(t) = - l x g ' (t),
v,(t) = -12g(t) + 9'(t) g(t---ff for a < t ~< e,
Pl (t) = --17~1g 2 (t),
p2(t) = m2g(t) + 9'(t) g(t---~ for c < t < b,
holds if and only if mz >1O, I(l,,12) = and
(1.58)
j[cb /(lr'/'t 1 , I'1"12) >
2358
g(t)dt,
gCt) dr.
(1.59)
PROOF OF NECESSITY: Let u be a solution of the problem (1.4), (1.7). Then, according to (1.43), we have (pl,p2) e V~0((a,c),c ) \ V~0 ((a,c)). Hence, taking account of Lemmas 1.61 and 1.62, we obtain equality (1.58). In addition, it follows from (1.36) and (1.43) that ml is nonnegative. If (1.59) ,does not hold, then, denoting by b0 a point of the interval (c, b] for which
I(ml,m2) :
fe b~g(t)dt,
carrying out the change of variable t = b0 + c - t' in (1.4), and again applying Lemmas 1.61 and 1.6~, we verify that u(bo-) = 0. This, however, contradicts (1.57). Consequently, inequality (1.59) holds. The necessity is now proved. Sufficiency is proved analogously. COROLLARY. Ire E (a,b), $i E [0, 1), ll,m1 E R, (3" = 1,2), then the inclusion
(Px,P2) e V~o((a,b),c ), where pl(t) = -ll(t-
a)
plCt) = - m l ( b - t )
,
t -$1 a
p (t) =
l~ (t - a)-X'
for a < t ~< c,
$2 -2x~ , p2(t)= b------t + m 2 ( b - t ) -x2
forc.O, i(11,/2) -- (C-- a) 1-)q 1 - A1
I ( m l , m 2 ) > ( b - c) 1-~2 '
1 -
$2
To prove this it suffices to set g(t)
J" ( b - c ) - ~ 2 ( t - a ) -~' (c a) -x' ( b - t) -x=
fora b -
a.
We remark that for g(t) ---- 1 this assertion is the theorem of Vallde Poussin [103] already mentioned. LEMMA 1.8. Suppose i C {1,2}, k is a natural number, the function g is the same as in Lemma 1.6i, and l u , 11~, and 12 are constants with 112 ~O. Then the condition
(2k + 1-i)I(112,-12)
>.O
forat
(1.77)
u'(tl) = O.
b
jft~(p2)(r) dr
w(t) = r
f o r a < t < b.
Since w is a solution of the equation
b
w' = p2 (t)w + [pl (t)u(t)
+ po (t)]
ft
a(p2)(r)
d r - u'(t)a(p2)(t),
upon applying (1.68), (1.69), and (1.77), we obtain
,,,(to) >.0,
u(t,)~,(u'(t2)) 0 for to < t < t*,
lu'(t*)l < r.
Then (3.10) implies the inequality for to < t < t * .
u"(t)rl,(u'(t)) >>.-w(u'(t))(h(t) + lu'Ct)l)
From this, in view of the choice of the constant r*, we obtain lu'(t0)l ~< r*. In a completely analogous manner one can show that this inequality holds also in the case when (3.13) holds. The lemma is now proved. LEMMA 3.3. Let a < a < ao < bo < ~ < b, and let ro a n d r benonnegativenumbers, h : [ a , b ] - + R + a sumrnable function, and w a Nagumo function. Then there is a positive constant r* such that for any points tl e [a,a], t2 E [/~,b] and any function u 9 Cl([tl,t2] ) the inequalities (3.5), u'(t)Tlr(ul(t))sgn(t--ao)<xw(u'(t))h(t)+lu~(t)l)
fortE(tl,ao)U(bo,t2)
(3.14)
and
u"(t),Tr(lu'(t)l)sgnu(t ) >>.-w(u'(t))(h(t) + lu'(t)l) for ot < t ' - h ~
lu'Ct)l 2 f o r t l < t -ho(s) - gCs)lr
Ir and
]u'(t)] 1 0 be constants, and let ho E L~or and hi E L([a, b]) be nonnegative functions with ho summable with respect to the weight t - a . Then there exists a nonnegative function r* e ((Ca, b])~L([a,b]) such that estimate (3.17) holds whenever [tl,t2] c [a,b] and the function u E Ol([tl,t2]) satisties, together with (3.5), the conditions
u" (t)rl,(u'(t)) >~ -ho(t) -
t - a + hi (t)
lu'(t)[- h2lu'(t)l'
for tt < t < t2
a~,2d
lu'Ct,)l < r.
Lemmas 3.3 and 3.5 imply
2377
LEMMA 3.6. Let a < a < ao < b0 < / ~ < b, A E [0,b - a), let r0 >1 O, r >>.O, h2 >1 0 be constants and h o e Lloc((a,b)) and h I e L([a,b]) nonnegative functions with ho summable with respect to the weight ( t - a ) • ( b - t ) . Then there exists a nonnegative function r* E C ( (a,b) ) n L([a,b]) such that estimate (3.17) holds whenever tl E (a,a], t2 E [/~,b) and the function u E Cl([tl,t~]), satisaes, together with (3.5), the conditions
u"(t)rl, Cu'(t))sgn(t-ao) Cl,
3 (i-1)
(b-) >/c2).
Then s is called a lower (resp.upper) function of the problem (3.1), (3.2,). The following lemma holds; it is a simple modification of the Scorza Dragoni Theorem (cf. [51], p. 110). LEMMA 3.7. Let i E {1,2} and suppose there exist a lower function Sl and an upper function s~ of the problem (3.1), (3.2i) such that Sl(t ) • s2(t ) for a < t < b (3.22) and
If(t,x,y)l 0, z;.(tl) > 0, (3.31) with a 1 continuous at the point tl. Since aj is nondecreasing and fj is continuous, conditions (3.29)-(3.31) guarantee that there exists a point t2 E (tl, b] such that zj(t)>0, z;(t)>0 f o r t , < t < t 2
2379
and
,; (t,-) =o.
(3.32)
Without loss of generality we may assume that
~i(t)
o - t r C t W ( t )
where r = max
for tl < t < t2,
}
ujCt)l + I,,)(t)l): tl < t < t2 . "j----1
Then,
u'(t2)>/u'(tl)exp(-ft:21r(r)dr)
> O,
which contradicts condition (3.49). The theorem is now proved. The following proposition is proved in an analogous manner.
2383
THEOREM 3.52. Let the function f be nondecreasing on the second argument and satisfy condition (3.48) for any r e a + , where t, e L,oo ((a,b)) and lim s u p /Jtt o l ' ( r ) d r < + c ~ t-.b Then the problem
(3.t), (3.22)
fora 0 and h E Lloc ((a, b)) is a nonpositive function. Special cases of (4.4) are the equations mentioned in the introduction t2 v," (4.5) 32u 2 and
ut'--
1--t
(4.6)
u
4.1. O s c i l l a t i o n L e m m a s . In this subsection we present some propositions on the oscillation properties of the equation v" = g(t)v, (4.7) where the function g : (a, b) --* R_ is summable either with weight (t - a) (b - t) or with weight t - a. These properties will be needed for the exposition that follows. We denote by v~ the solution of Eq. (4.7) satisfying the initial conditions
vCa+) =0,
r
1.
The values of ve and v P 8 at the point b will be taken as their left-hand limits. An immediate corollary of Lemmas 1.51 and 1.52 is 2384
LEMMA 4 . 1 .
Let i E {1,2} and
~
b(t - ~ ) ( b - t)~-i Ig(t)l dt ~< ( b - ~)2-i,
Then vg(i-1) has no zeros in (a, b]. Integral criteria for vg and v~ to have at least one zero in (a, b] were established by Korshinkova [24, 25] and Lomtatidze [36]. We present two lemmas from [36]. LEMMA 4.21. Let g : (a, b) --+ R _ be summable with respect to the weight (t - a)(b - t) and
1 t-a
t(~'-a)'(b-r)lg(r)ldr+-gi-~
(r-a)(b-r)'lgC~)ldr>~b-a
fora
yER
(4.12)
hold, where g and h : (a,b) -~ R_ are summable with weight (t - a)(b - t), v~ having at least one zero on (a,b] and vh having no zeros on (a,b]. Then the problem (4.1), (4.21) is solvable. PROOF: By Theorem 1.1 the equation
v" = h(t)v has a solution v such that
1 vCa+) = vCb-) = -~'
v(t) >
1
-~o
for a < t < b.
In view of (4.12) it is obvious that v is an upper function for Eq. (4.1). According to L e m m a 4.1, for any sufficiently small r > 0 we have
fora1 0 Let
for a < t 1 0 and r6 >~0 such that
if(t,x,y)l18lo(t)
for tl ~< t < bo,
where /o(t) = 1 + ftl [exp ( f t [ / 2 ( f ) d r ) + f / e x p
12(f) d~)ll (~) d~] dr.
(jft
However, this last inequality contradicts condition (5.22), for 1+
lim ~
,-.b
l(t) to(t)
< +co.
The contradiction so obtained proves inequality (5.23). The inequality
u(t)u'(t) >10
for a < t l O, u'(t) 1g(t)u'(t)
for to ~< t < b.
Therefore u"(t)>g(t)u'(t)
forto ~
--oo
(5.28)
fora0, u'(t)>O fort>O. THEOREM 5.7.
The problem (5.8), (5.9) is solvable.
PROOF: We first consider the case when p > 2. We set sl (t) = O, s2 (t) = 2, and define the function f by the equalities 3x
f(t'z'Y)=b--t
p'
--
-~-2~] 1/2
b2
z' Y' + ( 1 - pzZ~' 2x ( 1 - ~ - ) + ~J + (b _--~)z for
mad (5.31). It is cleat that f E Kloc ([0,b) • I t ' ) ,
l(t)
9
and 82 are lower and upper functions of Eq. (5.1) and 4 6 2 inequality (5.161) holds, where a = 0, r = 0, 0J(y) = 1 + lYl, hi(t) - (b_t)2 + -~ and h2(t) - p2 - 4 " 81
Therefore a~cording to Theorem 5.31 Eq. (5.8) has a solution u satisfying the conditions
u(O+) = 2,
O ~ u(t) ~< 2 f o r 0 < t < b .
(5.32)
It is clear that
u(t) > 0
for0 0for some to > 0, then by (5.35) we find
,,'(t)/> ,,'(to)
and ,'o/> ,,(t)/> ,,'(to)Ct-~o)
for t > to.
The contradiction so obtained shows that u satisfies conditions (5.41). The theorem is now proved.
2397
COROLLARY 5 . 1 .
Let
f(t,x,y) . --W(y)(hl(t) § h=Ct)lYl)
for t > O,
0 < = < to,
y < -r,
where a, p, to, and r are positive numbers, A 9 C1, +oo), hi 9 L,or ((0, +eo)), h, 9 C((0, § Nagumo function. Then for any c 9 [0,r0] the problem (5.1), (5.41) has at least one solution.
(5.37) and w is a
PROOF: Choose c > 0 and ~ 9 (0,a) so t h a t
" z(t) dt > ro, where
(5.38)
1
.(t) = [~ + , t ~-I ] , - , . Without loss of generality we may assume that ~C") > r.
We set rl = z(O),
a(y) =
l
1
for y t> - r i ,
2- y rl 0
for - 2rl < y < - r l ,
(5.39)
for y ~< - 2 r l ,
](t,x,y) = { a(y)f(t,x,y)
f(t, =,y)
for 0 < t 1~,
and consider the differential equation
.,t= ](t,.,.').
(5.41)
Since ri > r, the inequalities
](t,=,y) < ~(y)(L (t) § L(t)lyl)
for 0 < t < ~,
0 ~< z ~< ro,
y ~< - 2 r i
and
](t,=,y) >1 - ~ ( u ) ( L ( O
+ ~,,(Olyl)
for t > o,
o < = < ro,
y < -2ri
foUow from (5.3~) and (5.4O), where/,i(t) = 0,/,,(t) = Ih~(#)l for 0 < t < # and/,,(t) = Ih~(t)l for t > #, (k = 1,2). According to Theorem 5.8i if c E [0, r0] the problem (5.41), (5.4i) has a solution u. We shall show that
ut(t) >>.- z ( t )
for 0 < t / - p t ;~-= I~,'(t)l:'
for to < t < t,.
From this, taking account of (5.43), we find 1
I,,'(ti)l/>
[pt; -1 + I,,'(to)l '-~'
-
pt~o-1 ],-.x > z(tl).
Therefore it is clear that tl = a and inequality (5.44) holds on [to, a]. On the other hand, according to (5.38) arid (5,.44) ro > uCto) - u C a ) = -
fo
u'Cr)dr >
/;
zCr) dr > ro.
The contradiction so obtained proves inequality (5.42). But it follows from (5.40) and (5.42) that u is a solution of Eq. (5.1). The corollary is now proved. COROLLARY 5.2. Let f
6
Klor (R+
x
R+
f(t,x,y)>~O
x
R _ ) and let fort>0,
x~>0,
y~/0. Then there exists a positive number r0 such that for any c e [0,r0] the problem (5.1), (5.41) has at least one solution. PROOF: Let r I be an arbitrary positive number and h(t) = sup { I f ( t , z , y ) l :
0 ~< :r ~< n ,
- 2 r , ~< y ~< 0}.
(5.45)
Choose numbers fl > 0 and r0 6 (0, r,) such that ro + f f hi (t) dt < r,, and define the function ] by means of equalities (5.39) and (5.40). It is clear that ](t,x,y)=O f o r O < t < ; ~ , o ~ = < r 0 ,
(5.46)
y>.O f o r t > 0 ,
0~<x~ 0, A > 1. W e note that this case is the most important from the point of view of applications.
THEOREM 6 . 1 . Let 7 > 0, A > 1 and let u : (0, +oo) ~ R be a nontrivial solution of the problem (6.1), (6.2). Then u is nonoscillating, uCO(t)~CL1)~et-~/2e -t
ast~+oo,
(in0,1)
(6.6)
and 11 : 12:13 = [7 + 3 - A ( - / - 1)]: [2(A + 1)]: [(7 + 1)(A - 1)],
(6.7)
where e is a nonzero constant and I1 =
/o
t~u2(t) dt,
I2 =
/o
t
lu(t)l
TM
dt,
& =
/o
t~u"(t) dt.
A direct consequence of (6.7) is the
2403
COROLLARY. For "7 > 1 and A/> (7 + 3)/("7 - 1) the problem (6.1), (6.2) has no nontriviaI solutions. PROOF OF THEOREM 6.1 : If the solution u of the problem (6.1), (6.2) is oscillating, there exists a point to E (0,+co) such that 0 < lu(to)l < 1, u'(to) : 0 and so V,(to) < 0 also, which contradicts Lemma 6.4. Thus u is nonoscillating. Further, by (6.2) and Lemma 6.4,
xu'(t) for all sufficiently large t, which, as is easy to verify, implies the inequality
lu(t)[ < Ke -'12
for t > 0,
where K is some constant. Consequently by Mattell's Theorem (cf., for example [18], p. 107), the linear differential equation
r = _2r t
+, _ lu(t)l
-I
has a fundamental system of solutions (~, v0) such that ~(')(t) ~ t -~/n
e', ~(i)(t)N
( _ l ) i t - ~ / 2 e-t
as t --* +co,
(i=0,1).
(6.8)
Since u is also a solution of this equation and satisfies (6.2), there exists a nonzero constant c for which u(t) =- C6o(t). Relation (6.6) follows from this. Finally, substituting the solution u in Eq. (6.1), multiplying both sides of the equality thus obtained by t 7+1 u'(t), and then integrating them over the interval (0, +oo), taking account of (6.2) and (6.6), we arrive at the equality (7 -{- 1)It
2('7 A ++ II) /2 + ( 7 - I)13 = 0.
(6.9)
The same procedure, only multiplying by tTu(t) gives the equality It
--/2+
13 = 0.
(6.10)
Relation (6.7) follows from (6.9) and (6.10). The theorem is now proved. THEOREM 6.2. / f e i t h e r 7 > l a n d 1 < A < (7 + 3)/(7 - 1 ) o r 0 < 7 ~< 1 a n d A > 1, then for any nonnegative integer I the problem (6.1), (6.2) has a solution u : (0, +co) --~ R with exactly I zeros. +co +co PROOF: Let I E {0, 1,... }. We construct sequences (t~),=l and (T,),= t such that the relations
1 O0,
(n:l,2,...).
Hence for any natural number n by the differential inequality theorem we have 1
1
1
0 < V,. (t) + ~ ~< t-~-(tt + {) and so
1
l-'(t)l< ~-v/2/z+l
for 0 < t 1. Using the reasoning applied in deducing (6.9) and (6.10), we verify the equalities (7 + 1)11,
2(7 + 1) 12. + (71)I,. = 2(t~n+1Vu,, (in) - V.,. (1)) ,~+I
and
/i. - 1 2 . + ls. = u~.(1)u.(1),
(6.15)
where n E { 1 , 2 , . . . } and I.,
=
x2. =
t
lu.Ct)l T M
tit,
Is,, =
t~u'~2Ct)dt.
2405
Consequently 3 + ff - A('7 - 1) I~. = 2(I1. - t~+1V.n (t.) -t- V(u. (1)) -t- ('7 - 1)u~ (1)un (1). ),+1 On the other hand, according to HSlder's inequality,
T21(~+1) I1. ~--
2~
We remark that according to (6.13), (6.16), (6.23), and the Cauchy-Schwarz inequality
lu.(t)l
~< lu.(1)l +
lu'.(r)ldr To,
(6.30)
where
L = --M r o e + K m
t ~12 e( ' - = ~ ) t dt).
(6.31)
J1
Obviously
u.CTo) :.CTo) < O, I .(To)l < (1
(6.32)
and I-.(t)l < Ke -t 2408
for 1 ~< t < To,
(6.33)
whenever n ~> no and no is a sufficiently large natural number. Fix n E {no, no + 1,... } and denote by So the smallest zero of u , in the interval (To, Tn]. As follows from (6.11) and (6.32),
"`,,(t)u'(t) ~-~--~--iu.(t)
for so ~l,,.(so)l(s-
so).
Thus, according to (6.29), So 1 s 2 As follows from (6.38), (6.39), (6.41), and Le~ma 6.3,
,~-1 u~ ' (t) > u .2 (s) -~ + 1 = lu~Cso) = 2 V u
(6.42)
.(so)> ~V,,,,(t)
for s~ 1. Then a nontrivial solution of the problem (6.1), (6.2) does not vanish on (0,+oo), and for a n y t o 6 It+ and u0 6 [-1,1] \ {0} there exists a solution u of this problem such that u(to+) = uo, u(t)u'(t) < 0 for t > to. (6.44) PROOF: The first part of the conclusion of this theorem follows from L e m m a 6.4. If to e 1%+ and Uo 6 [-1, 1] \ {0}, then, by Theorem 5.81 and L e m m a 6.1, Eq. (6.1) has a solution u : (0, +oo) --+ 1~ satisfying the conditions (6.44). Obviously u(+oo) = u'(-boo) = 0. By L e m m a 6.3 V~,(t) < V~(+oo) = 0. Hence
(~ ~-1) 1/(A-l) luCt)[
o,
and, solving Eq. (6.1) as a linear equation in u', we verify that u'(0-b) = 0. The theorem is now proved. REMARK. Setting to = 0 in (6.44), we obtain a solution that is monotonic on all of (0, -boo]. But for 7 < 0 and A > 1 the problem (6.1), (6.2) can also have nonmonotonic solutions. For example, it is easy to verify that if ~/< - 1 , to > 0, and uo = 1, then the solutions of (6.1), (6.44) are not monotonic on (0, -boo). We now consider the case when :~ ~< 1. From L e m m a 3 of [98] and L e m m a 6.4 above one can deduce LEMMA 6 . 5 .
IrA < 1 then all nontrivial solutions of the problem (6.1), (6.2) are oscillating.
THEOREM 6 . 5 . Let ",/> 0 and )~ < 1. Then for any Uo 6 ( - 1 , 1) there exists a solution u of the problem (6.1), (6.2) such that u(0+) = Uo. Moreover if u is a nontrivial solution of the problem (6.1), (6.2), then u is oscillating and u(0-b) 6 ( - 1 , 1 ) \ {0}. PROOF: Fix u0 e (-1,1) \ {0}. According to the corollary of Theorem 5.1 of [18], Eq. (6.1) has a solution u : (0, -boo) -+ R satisfying the initial conditions = uo,
u ' ( o + ) = o.
If to > 0 and lu01 < lu(t0)[ < 1, then Vu(to) >~ Vu(0-b), which contradicts L e m m a 6.3. Therefore lu(t)l < luo[ < 1 for t > 0, the function V~ defined by equality (6.3) is positive exists and is finite. Consequently
(6.45)
on (0,-boo), and by L e m m a 6.3 the limit ~ = Vu (-boo)
0 < Z < vu (t) < v , ( 0 + )
for t > o.
(6.46)
Suppose u is not oscillating. Then by (6.1) and (6.45) u is monotonic in some neighborhood of -boo and u(-boo) = 0. This, according to Lemma 6.5, contradicts our assumption, so that u is oscillating.
2411
Set
T=
2T M
(6.47)
~x (1 - u~-;~)"
From (6.46) we have
u'2(tn) > 2~ (n=1,2,...), where T ~
tl < t2,
#
for , 1 . < t < , 2 .
(Y : 1, 2; n = 1, 2 , . . . ) .
(6.49)
According to (6.1) and (6.45),
I-'(t.)l < lu'(sln)[ + tn - 81n (n = 1 , 2 , . . . ) . Therefore, taking account of (6.48) and (6.49), we obtain s,. - ,1. > t. -,1.
~> v ~ ( v ~ -
(6.50)
(n = 1 , 2 , . . . ) .
1)
Further, from (6.45), (6.46), and (6.49), it follows that
1 -I'('i") [~,+1 f12 ~---$--i _
0,
(6.53)
Sin
and re(t) : m a x { n : S2,~ ~< t}. By (6.50) and (6.52),
s2n - sin /> Sln
v~Cv~-1)
(n=1,2,...).
311 -[- ( n - - 1 ) ~
-}-oo
If 3 > 0, then the series ~].,=1 (s2, - s l , ) / s l , diverges and, according to (6.53), V~(t) < 0 for sufficiently large t, which contradicts (6.46). Thus 3 = 0, i.e., u satisfies the boundary conditions (6.2). Now consider an arbitrary nontrivial soluton u of the problem (6.1), (6.2). According to Lemmas 6.3 and 6.5, u is oscillating and u(0+) r 0. If we assume that lu(0+)l > 1, then from (6.1) we obtain the relation u(t)u' (t) > 0 for t > 0, contradicting the definition of u. The equality lu(0+)i = 1 is also impossible, since it leads to the identity ]u(t)l - 1 (cf., for example, Theorem 5.2 of [18]). Consequently u(0+) E (-1,1) \ {0}. The theorem is now proved. To complete our study of all the values of q and A we are interested in, it remains only to prove the following proposition. THEOREM 6.6. If either A < 1 and ~t