Journal of Inequalities in Pure and
Applied Mathematics
SEPARATION AND DISCONJUGACY
R.C. BROWN
Department of Mathematics,
University of Alabama-Tuscaloosa,
AL 35487-0350, USA.
EMail: dbrown@gp.as.ua.edu
volume 4, issue 3, article 56,
2003.
Received 21 November, 2002;
accepted 25 March, 2003.
Communicated by: A.M. Fink
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
130-02
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Abstract
We show that certain properties of positive solutions of disconjugate second
order differential expressions M[y] = −(py0 )0 + qy imply the separation of the
minimal and maximal operators determined by M in L2 (Ia ) where Ia = [a, ∞),
a > −∞, i.e., the property that M[y] ∈ L2 (Ia ) ⇒ qy ∈ L2 (Ia ). This result
will allow the development of several new sufficient conditions for separation
and various inequalities associated with separation. Some of these allow for
rapidly oscillating q. It is shown in particular that expressions M with W KB
solutions are separated, a property leading to a new proof and generalization
of a 1971 separation criterion due to Everitt and Giertz. A final result shows
that the disconjugacy of M − λq2 for some λ > 0 implies the separation of M.
2000 Mathematics Subject Classification: Primary: 26D10, 34C10; Secondary
34L99, 47E05
Key words: Separation, Symmetric second order differential operator, Disconjugacy,
Limit-point.
The author wishes to thank Don B. Hinton for his encouragement and support in the
preparation of this paper
Separation and Disconjugacy
R.C. Brown
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1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
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1.
Introduction
Consider the symmetric second order differential expression
M [y] := −(py 0 )0 + qy
(1.1)
where p > 0, p0 and q are continuous on the interval Ia = [a, ∞), a > −∞. M
is said to be disconjugate if every nontrivial real solution has at most one zero
in Ia . A sufficient condition (from Sturm’s comparison theorem) for disconjugacy is that q ≥ 0, and in this case one can show existence of two positive
solutions u1 and u2 of M [y] = 0 on Ia , called the principal and nonprincipal
solution respectively, such that u01 ≤ 0 and u02 > 0 on Ia . More generally, M
is disconjugate on Ia if and only if there exists a positive solution u on the interior of Ia . For proofs of these facts and additional discussion see Hartman [15,
Corollaries 6.1 and 6.4].
Recall also that M determines several differential operators in the Hilbert
space L2 (Ia ). In particular the “preminimal” and “maximal” operators L00 and
L are given by M [y] for y in the domains D00 ≡ C0∞ (Ia ), the space of infinitely
differentiable functions with compact support in the interior of Ia and
0
2
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D = {y ∈ L (Ia ) ∩ ACloc (Ia ) : py ∈ ACloc (Ia ); M [y] ∈ L (Ia )},
where ACloc stands for the real locally absolutely continuous functions on Ia
and L2 (Ia ) denotes the usual Hilbert space associated with equivalence classes
of Lebesgue square integrable functions f, g having norm and inner product
Z
21
Z
2
, [f, g] :=
f ḡ.
kf k =
|f |
Ia
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The “minimal operator” L0 with domain D0 is then defined as the closure of L00 .
With the above definitions one can show that
(i) C0∞ (Ia ) ⊂ D00 ⊂ D0 ⊂ D,
(ii) L00∗ = L∗0 = L,
(iii) L∗ = L0 ,
(iv) D00 , D0 , and D are dense in L2 (Ia ).
The regularity assumptions made in this paper on p and q are stronger than
necessary to properly define L0 , L. In general one needs only to assume the
so-called “minimal conditions" that p−1 and q are locally integrable on (a, ∞).
In this case C0∞ (Ia ) may not be contained in D00 but the properties (ii)–(iv) will
still hold. The maximal and minimal operators L and L0 can also be defined
relative to an arbitrary interval (a, b) where −∞ ≤ a < b ≤ ∞. If p−1 , q
are Lebesgue integrable on some interval (a, c) or (c, b) for a < c < b then a
or b are said to be “regular"; otherwise they are “singular". (Infinite endpoints
however are considered singular even if p−1 , q are integrable on (a, b).) Thus in
our setting a is regular and ∞ is singular–we signal this by writing Ia = [a, ∞)
rather than (a, b).
M is limit-point or LP at ∞ if there is at most one solution of M [y] = 0
which is in L2 (Ia ), and limit-circle or LC at the point if both solutions are so
integrable. This can be shown equivalent to each of the following properties
(i) {y, z}(∞) := limx→∞ (ypz̄ 0 − py 0 z̄)(x) = 0 for all y, z ∈ D.
Separation and Disconjugacy
R.C. Brown
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(ii) D = D0 ⊕ span(φ1 , φ2 ), where φ1 , φ2 ∈ D and have compact support in
Ia . Thus D is a two dimensional extension of D0 .
It is clear that if M is disconjugate it is LP at ∞ since the nonprincipal
solution u2 ∈
/ L2 (Ia ). A stronger condition at ∞ than LP is strong limit-point
or SLP which means
lim (pyz̄)(x) = 0
x→∞
∀ y, z ∈ D. For a thorough development of these operator theoretic ideas see
Naimark, [17, §17]. Discussion of the SLP concept may be found in Everitt,
[7].
We turn now to the central concern of this paper.
Definition 1.1. M is said to be separated on D0 or on D—equivalently L0 or
L is separated—if qy ∈ L2 (Ia ). (Obviously also by application of the triangle
inequality (py 0 )0 ∈ L2 (Ia ).)
The following is an exercise in the Closed Graph Theorem (see e.g. [16]).
Proposition 1.1. Separation on D0 or D is equivalent to the inequality
(1.2)
Ak(py 0 )0 k + Ckqyk ≤ KkM [y]k + Lkyk.
for nonnegative constants A, C, K and L.
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R.C. Brown
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The next result shows some connections between LP or SLP at ∞ and
separation. Its proof may be found in [2].
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Proposition 1.2. If M is separated on D0 then it is separated on D if M is LP
at ∞. On the other hand, if M is separated on D then it is SLP at ∞.
J. Ineq. Pure and Appl. Math. 4(3) Art. 56, 2003
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Remark 1.1. Two immediate consequences of Proposition 1.2 are (i) if M is
LC at ∞ then it is not separated, (ii) if M is LP but not SLP at ∞ then M is
not separated on D0 .
Several criteria for separation of M given by Everitt and Giertz in a series of
pioneering papers [8] – [12], also see Everitt, Giertz, and Weidmann [13], and
Atkinson [1]. More recent results (that include weighted cases) may be found
in Brown and Hinton [2],[3]. We quote three typical results.
Theorem A (Brown and Hinton [2]). If p−1 is locally integrable on Ia , pq ≥ 0,
q(x) is locally absolutely continuous, and
1/2 0 p q (x) (1.3)
q 3/2 (x) ≤ θ < 2,
Separation and Disconjugacy
R.C. Brown
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on Ia then M is separated on D.
Remark 1.2. The original version of Theorem A with p = 1 and q > 0 is due
to Everitt and Giertz [11]. The case of nontrivial p but θ < 1 is given in [9].
Theorem B (Brown and Hinton [3]). Suppose that p−1 is locally integrable
on Ia , pq ≥ 0, and q, p are twice differentiable on Ia . Then M is separated on
D0 if
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0 0
(1.4)
lim sup
x→∞
(pq )
≤ θ < 2.
q2
Remark 1.3. Note that in the case p = 1 both Theorems A and B work for a
wide class of increasing q such as q(x) = exp(x), q(x) = exp(xn ) for n > 0,
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q(x) = exp(exp(· · · exp(x)) · · · ), etc. On the other hand, both theorems fail if
q is rapidly oscillatory, e.g., q(x) = exp(x)(1 + sin(exp(x)). Note also that a
consequence of Theorem B is that if p = 1 and q 00 ≤ 0 (i.e., q is concave down)
then M is separated.
Theorem C (Brown and Hinton [2]). Suppose p−1 ∈ L1loc (Ia ), pq ≥ 0, q is
differentiable. Then M is separated on D0 if either
Z ∞ 0
1
q
(1.5)
sup (x − a)
= K1 <
2
q
4
x∈Ia
x
R.C. Brown
or
Z
(1.6)
sup (x − a)
x∈Ia
∞
(q 0 )2 = K2 < ∞.
x
Remark 1.4. In this theorem we see that separation holds for any p satisfying
weak conditions provided that q is of slow enough growth. For example q(x) =
xβ , β < 12 , satisfies (1.5) and q(x) = K log(x) satisfies (1.6). These facts
should not be particularly surprising since if q = 1 then M would be separated
for any p; consequently one can conjecture that the same ought to be true if q
has slow enough growth.
Recently Chernyavskaya and Schuster,[4] have given necessary and sufficient conditions using averaging techniques due to Otelbaev for the inequalities
(1.7)
(1.8)
Separation and Disconjugacy
KkM [y]kp,R ≥ ky 00 kp,R + kqykp,R
≥ krykp,R ,
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where the norms are Lp norms on R, q ≥ 1 and is locally integrable, r > 0 is
locally p integrable, M [y] = −y 00 + qy ∈ Lp (R), and 1 ≤ p ≤ ∞. Note that
(1.7) or (1.8) can hold on the Lp analog of D only if M has no Lp or r-weighted
Lp solutions. Although the conditions in [4] seem challenging to implement
they can be applied to rapidly oscillating potentials such as
(1.9)
q(x) = exp(|x|) + exp(|x|)(1 + sin(exp(|x|))
for which both Theorems A and B fail.
In this paper we show that certain pointwise properties of a positive solution
of a disconjugate expression M imply that M is separated on D. This means
in particular that separation occurs if M has a fundamental set of solutions,
sometimes called W KB solutions, with a particular asymptotic behavior at ∞.
Since the existence of W KB solutions follows from certain integral conditions
satisfied by p and q, we are led to a test for separation that includes a wellknown 1971 result of Everitt and Giertz as a special case. We also show that
our approach leads to several other sufficient conditions for separation which
do not require verification of properties of positive solutions of M . Some of
these will work for rapidly oscillating potentials similar to (1.9). We look also
at conditions that ensure that the mapping associated with the inequality
√ (1.10)
hy ≤ KkM [y]k + Lkyk
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is compact where h is a weight, i.e., a positive locally integrable function, which
in turn will lead to a more general inequality (see (2.17) below) than (1.10). We
also investigate “perturbation” results: if M1 [y] = −(py 0 )0 + q1 y is separated,
J. Ineq. Pure and Appl. Math. 4(3) Art. 56, 2003
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when is the same true of M2 [y] = −(py 0 )0 + q2 y when in some sense q2 is
“close” to q1 ?
Although our tests for separation hold only in L2 (Ia ) and are sufficient but
not necessary, they are easy to apply. Moreover we consider nontrivial p and on
occasion allow q to be negative or even unbounded below which is a more general setting than in [4]. Finally, as already mentioned, the inequalities (such as
(2.17) below) associated with separation may be more complicated than (1.7)–
(1.8).
We use the following notational conventions in the paper. Positive constants
will be denoted by capital letters with or without subscripts such as C, K, K1 ,
etc. The value of a constant may change from line to line without a change in
the symbol denoting it. If f and g are functions f ∼ g denotes the asymptotic
equivalence of f and g, i.e., limx→∞ f /g = 1. L2 (w; Ia ) is the standard wweighted Hilbert space with norm and inner product
Z
kf kw =
2
w|f |
Ia
12
Z
,
[f, g]w =
wf ḡ,
Ia
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R.C. Brown
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where w is a weight. The class of Lebesgue integrable or locally Lebesgue
integrable functions on Ia will be denoted by L(Ia ) or Lloc (Ia ).
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Remark 1.5. The Hilbert space theory (see e.g. [17] of the operators L0 and
L is usually developed on complex domains. Thus D is the space of locally
absolutely continuous complex valued functions f on Ia such that f and M [f ]
belong to L2 (Ia ) with similar changes in the definitions of D00 and D0 . All the
standard closure and adjoint properties of L0 and L remain true in both cases.
Since the chief tool in our development is the concept of disconjugacy which is
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defined only for real-valued solutions of M , we will derive conditions for the
separation of M only for real D0 and D. However all our results go over to the
complex case. This is seen from observation that if f = f1 + if2 ∈ D then
kM [f ]k2 = kM [f1 ]k2 + kM [f2 ]k2 ,
kqf k2 = kqf1 k2 + kqf2 k2 .
Therefore M (f ), qf ∈ L2 (Ia ) ↔ M [f1 ], M [f2 ], qf1 , qf2 ∈ L2 (Ia ).
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2.
Main Results
Theorem 2.1. Let p > 0, q be C 1 functions. Suppose M [y] = −(py 0 )0 + qy has
a positive solution on the interior of Ia such that
(2.1)
(2.2)
(2.3)
(pu0 )0 u ≡ qu2 ≤ 2p(u0 )2 ,
(1 − δ)(u0 )2 ≤ u00 u, δ ∈ [0, 1/3),
p0 u0 ≥ 0.
Then q ≥ 0 and M is separated on D.
Separation and Disconjugacy
R.C. Brown
Proof. We need only show that M is separated on D0 . Because M is disconjugate and as will be seen below (see (2.9)) q ≥ 0, M is LP at ∞ and separation
on D will follow by Proposition 1.2; in this case by Proposition 1.1 y will satisfy
an inequality of the form
kqyk2 ≤ Ckyk2 + DkM [y]k2
for certain positive constants C, D.
Let z(t) = −u0 /u. Then z satisfies the Riccati-type equation
(2.4)
(pz)0 = pz 2 − q.
Since
(2.5)
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(pz)0 =
−u(pu0 )0 + p(u0 )2
u2
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(2.1) – (2.3) is equivalent to the properties
−pz 2 ≤ (pz)0 ,
z 0 ≤ δz 2 ,
p0 z ≤ 0.
(2.6)
(2.7)
(2.8)
To see this, note that from the definition of z and (2.5)
−u(pu0 )0
p(u0 )2
≤
u2
u2
0 2
−u(pu0 )0 + p(u0 )2
p(u )
⇔− 2 ≤
u
u2
0
2
⇔ (pz) ≥ −pz .
(2.1) ⇔ −2
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R.C. Brown
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Also
−uu00
(u0 )2
≥
u2
u2
(u0 )2
−uu00 + (u0 )2
⇔δ 2 ≥
u
u2
2
0
⇔ δz ≥ z .
(2.2) ⇔ −(1 − δ)
Finally, the definition of z and (2.3) clearly implies that p0 z ≤ 0.
Next define the operators
L(y) = y 0 + zy,
L∗ (y) = −y 0 + zy
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where y ∈ C0∞ (Ia ).
We now derive sufficient conditions for the “separation” of L∗ . We have
kL∗ (y)k2 = [L∗ (y), L∗ (y)]
= [LL∗ (y), y]
= [−y 00 + (z 2 + z 0 )y, y]
Z
=
(y 0 )2 + (z 2 + z 0 )y 2 .
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Since p0 z ≤ 0 we see that
0
R.C. Brown
0
0
0
0
(pz) = p z + pz ⇒ pz ≥ (pz) ≥ −pz
⇒ z 0 ≥ −z 2 .
2
Because z 0 + z 2 is nonnegative the inequality
kL∗ (y)k2 ≥ ky 0 k2
holds. By the triangle inequality it also follows that
kzyk2 ≤ 4kL∗ (y)k2 .
The remaining step is use the separation of L∗ to show that M restricted to
∞
C0 (Ia ) is also separated. We first observe that
L∗ (pL(y)) = −(py 0 + pzy)0 + z(py 0 + pzy)
= −(py 0 )0 + [−(pz)0 + pz 2 ]y
= −(py 0 )0 + qy.
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A consequence of (2.7) – (2.8) is that
−(pz)0 + pz 2 = −pz 0 − p0 z + pz 2
≥ −pz 0 + pz 2
≥ pz 2 (1 − δ)
≥ 0.
Therefore both
(2.9)
q ≥ 0 and
(pz)0 ≤ δpz 2 .
R.C. Brown
Now also
(2.10)
kM [y]k2 = [L∗ (pL)(y), L∗ (pL)(y)]
= kL∗ (pL(y))k2
1
≥ kz(pL(y))k2
4
1 ∗
= [L ((zp)2 L(y)), y]
4
1
=
−((zp)2 y 0 )0 + (z 4 p2 − (z 3 p2 )0 y, y
4Z
1
[(zp)2 (y 0 )2 + (z 4 p2 − (z 3 p2 )0 )y 2 ].
=
4 Ia
Hence since p0 z ≤ 0 and z 0 ≤ δz 2 ,
(2.11)
Separation and Disconjugacy
z 4 p2 − (z 3 p2 )0 = z 4 p2 − 3z 2 z 0 p2 − 2z 2 p(p0 z)
≥ (1 − 3δ)z 4 p2 .
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But (pz)0 ≥ −pz 2 , so
pz 2 = q + (pz)0 ≥ q − pz 2 .
Hence also pz 2 ≥ q/2. Combining this with (2.10) and (2.11) gives the inequality
(2.12)
kM [y]k2 ≥
1 √ 0 2 1 − 3δ
k pqy k +
kqyk2 ,
8
4
Separation and Disconjugacy
which immediately yields the separation inequality
(2.13)
16
kM [y]k2 ≥ kqyk2 .
1 − 3δ
A closure argument (cf. [2, Lemma 1]) shows that the same inequalities are
true on the minimal domain D0 .
Remark 2.1.
(i) It is well-known that the existence of a positive solution u, the existence of
a continuously differentiable solution z of the inequality z 0 + z 2 /p + q ≤
0, or the identity M [y] = L∗ (pL(y)) for y having a continuous second
derivative are each equivalent to the disconjugacy of M on I; see e.g.
[15, Corollary 6.1, Theorem 7.2] or Coppel [5, p.6].
(ii) We may require that both the conditions q ≥ 0 and (2.1) – (2.3) hold
“eventually”, i.e. on Ia0 for sufficiently large a0 > a. In this case the
restriction of M to Ia0 will be separated on its maximal domain. Since q is
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bounded on (a, a0 ] it is immediate that separation holds also for Ia (cf. [2,
Remark 1 and Proposition 2]) although the corresponding inequality may
be of the form (1.1) rather than (2.13).
(iii) If we retrace the proof of Theorem 2.1 with p = 1 (2.1) – (2.3) becomes
(2.14) (1 − δ)(u0 )2 ≤ u00 u ≤ 2(u0 )2
⇔ (1 − δ)(u0 )2 ≤ qu2 ≤ 2(u0 )2 ,
δ ∈ [0, 1/3) ,
with a corresponding change in (2.6) – (2.8).
Separation and Disconjugacy
(iv) If q is positive and u satisfies (2.1) or (2.2) then u0 is strictly positive or
negative, for if u0 (x0 ) = 0 either u(x0 ) = 0 or one of (pu0 )0 or u vanishes
at x0 . In either case q > 0 ⇒ u(x0 ) = 0, implying that u ≡ 0.
In the remainder of the paper “separated” means separated on D unless the
restriction to D0 is stated. Also, in proving separation inequalities on D0 such
as (1.2) we will generally start with y ∈ C0∞ (Ia ) and omit the routine closure
argument which extends the inequality to D0 .
We now show that information about the asymptotic behavior of positive
solutions of M [y] = 0 can yield criteria for separation based on the stable
conditions of p and q.
Theorem 2.2. Suppose that p, q are positive and twice differentiable with p
nonnegative or nonpositive. Set
Z xr
q
t(x) :=
,
p
a
−1/4
µ(x) := (pq)
,
0
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√
and assume that limx→∞ t(x) = ∞, µ(pµ0 )0 ∈ L1 (Ia ), and lim supx→∞ |p0 |/ pq =
δ < 31 . Then M is separated.
Proof. By Coppel [6, Theorem 13], M has fundamental solutions u such that
for x → ∞
u0 ∼= ±(pµ)−1 exp(±t(x)).
u ∼= µ exp(±t(x)),
It follows that (pu0 )0 ∼ qy1 and so
0 0
2
r
u(pu ) ∼ qu ∼
p
exp(±2t(x)) ∼ p(u0 )2 .
q
Clearly (2.1) is satisfied on Ia0 for sufficiently large a0 > a. To derive (2.2)
observe that the asymptotic equivalence of p(u0 )2 and (pu0 )0 implies that
(u0 )2 ∼ u00 u +
p0 0
u u.
p
But
(p0 /p)(u0 u/(u0 )2 ) =
p0 u
u
∼ p0 p
0
pu
p q/pu2
1
|p0 |
∼ p 0 µ2 ≤ √ ≤ δ + <
pq
3
as x → ∞. Thus for > 0 and on some Ia0 with a0 sufficiently large we have
that
(u0 )2 ≤ (u00 u + (δ + )(1 + )(u0 )2 ) ⇒ (1 − (δ + )(1 + ))(u0 )2 ≤ u00 u
Separation and Disconjugacy
R.C. Brown
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which obviously implies (2.2) if is small enough. Finally, if p0 ≥ 0 we choose
y1 = µ(x) exp(t(x)) and if p0 ≤ 0 we choose y1 = µ exp(−t(x)). In either case
(2.3) holds. By Remark 2.1 (ii), the fact that M is LP at ∞, and Theorem 2.1,
separation follows.
In 1970 [8] Everitt and Giertz showed:
Corollary 2.3. If p = 1, q ≥ d > 0, and
Z
q −1/4 (q −1/4 )00 < ∞,
Ia
Separation and Disconjugacy
R.C. Brown
then M is separated.
Proof. Evidently this condition is a special case of Theorem 2.1 with p = 1, cf.
[6, Theorem 14].
Remark 2.2. The hypothesis of Corollary 2.3 can be shown to be equivalent to
(see [6, p. 122]
Z
−3/2 00 q
q < ∞,
Ia
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unless q(x) ∼ cx−4 and q 0 (x) ∼ −4cx−5 for c a positive constant. But in this
case M is trivially separated on Ia if a > 0.
Close
A similar result using the asymptotic properties of solutions but requiring
less smoothness on q is given by:
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Theorem 2.4. Suppose that p = 1, q ≥ d > 0 is differentiable, and
Z
|q 0 |
<∞
3r/2−1/2
Ia q
for some r, 1 ≤ r ≤ 2. Then M is separated.
Proof. By a result of Hartman and Winter [15, p. 320] M has solutions u such
that
Separation and Disconjugacy
u ∼ q −1/4 exp(±t(x)),
R.C. Brown
u0 ∼ ±q 1/2 u,
Rx√
Rx
where t(x) = a q. Since u00 ∼ q 3/4 exp(± a q 1/2 ), it is clear that (2.14) is
satisfied on some Ia0 , a0 > a.
In most cases however it is difficult to verify (2.1) – (2.3) or (2.14) directly,
which motivates us to seek an equivalent formulation of Theorem 2.1 for which
knowledge of properties of positive solutions of M [y] = 0 is not required.
Theorem 2.5. Let p > 0 and z be C 1 (I) functions. Then if (2.6) – (2.8) hold
and q = pz 2 − (pz)0 . M [y] is separated and the inequality (2.13) holds.
Proof. The fact that Theorem
2.1 implies Theorem 2.5 is clear. On the other
R
− z
hand, if we set u = e
, then u is a positive solution of M [y] = 0. z = −u0 /u,
and the conditions (2.1) – (2.3) hold as they are equivalent to (2.6) – (2.8). Thus
all the assumptions of Theorem 2.1 are satisfied.
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Theorem 2.6. Suppose that M is separated, q ≥ d > 0, and that h is a weight.
Assume further that either
q2
=∞
x→∞ h
lim
(2.15)
or limx→∞ h = ∞, and
(2.16)
√
K k py 0 k ≥ hθ/2 y for some θ > 1 and all y ∈ C0∞ . Let GM (y) := {(y, M [y]), y ∈ D},
equipped with the graph norm. Then the mapping λ : GM → L2 (h; I) given by
λ(GM (y) = y is compact, and M satisfies an inequality of the form
√ (2.17)
kM [y]k + K()kyk ≥ hy on D for > 0.
Proof. If (2.15) holds and M is separated, then by (1.2) of Proposition 1.1 there
is an inequality of the form
(2.18)
L
K
kyk + kM [y]k ≥ kqyk
C
C
Z 2 12
q
2
=
hy
h
Ia
Z ∞
12
2
≥n
hy
.
xn
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for any positive integer and where the sequence {xn } → ∞. Let λn : GM →
L2 (Ian ) be given by the characteristic function on Ian composed with λ, where
Ian = [a, xn ]. Since the solutions of M [y] = 0 and q are continuous on Ian ,
a Green’s function argument shows that the maps λn : GM → L2 (h; Ian ) are
compact. By (2.18) the λn converge in operator norm to a compact limit λ.
Also since q ≥ d > 0, q is closed, considered as a multiplication operator
q̃ : L2 (Ia ) → L2 (Ia ), and since M is separated D ⊂ D(q̃). In this situation
Corollary V.3.8 of Goldberg [14, p. 123] applies and gives (2.17).
Under the second set of conditions we have from the Cauchy-Schwartz inequality, integration by parts, and since limx→∞ h = ∞ that on some Ia0 , a0 > a,
and for y ∈ C0∞ (Ia0 ) that
k(py 0 )0 k khθ/2 yk ≥ k(py 0 )0 k kyk
≥ [(py 0 )0 , y]
√
= k py 0 k2
≥ K −2 khθ/2 yk2 .
Hence
k(py 0 )0 k ≥ K −2 khθ/2 yk
√
≥ K −2 kh(θ−1)/2 hyk
√
≥ K −2 nk hyk(xn ,∞) .
Since M is separated we obtain from (1.2) the inequality
√ L
K
0 0
−1 kyk + kM [y]k ≥ k(py ) k ≥ K n hy C
C
(xn ,∞)
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on the C0∞ (Ia ) functions and therefore also on D0 ; the proof that λ restricted to
D0 is compact continues as in the first part. But since D is a finite dimensional
extension of D0 , λ is also compact.
Remark 2.3. Following Everitt and Giertz [8] we say that q is in the class P (γ)
or q ∈ P (γ) if whenever y ∈ D then |q|γ ∈ L2 (Ia ). Thus the separation of M
on D is equivalent to q ∈ P (1). It is also easy to verify by thinking of q = q1 +q2
where q1 (x) ≤ 1 and q2 (x) > 1 that q ∈ P (γ) ⇒ q ∈ P (β) for any β ∈ (0, γ].
Suppose now q ∈ P (1) and limx→∞ q = ∞. Then from the first part of Theorem
2.6 not only will q ∈ P (θ), θ < 1, but the “compactness” inequality (2.17) will
hold if h = q θ . If M is separated, q → ∞, and (2.16) holds for h = q 2
and θ > 1 then q ∈ P (θ), and we have the interesting consequence that the
mapping λ : GM → L2 (q; Ia ) is compact. In general, if q ∈ P (γ) and q → ∞
then λ : GM → L2 (q β ; Ia ) is compact.
A disadvantage of Theorem 2.5 is that although q has the form pz 2 − (pz)0 ,
since M is disconjugate, it may be difficult to determine z and to verify (2.6)
– (2.8). We attempt to remedy this problem in the next three corollaries and
obtain additional usable tests.
0 0
(pz12 − (pz1 )0 )
Corollary 2.7. If M1 [y] = −(py ) + q1 y where q1 (z1 ) =
satisfies
the hypotheses of Theorem 2.5 and M1,c [y] = −(py 0 )0 + q1,c y where q1,c (z1 ) =
(pc2 z12 − (pcz1 )0 )y, where c > 1 then M1,c [y] is separated. More generally, if g
is a differentiable function such that g, g 0 ≥ 0 and
(2.19)
g 0 (x)x2
≤1
g(x)2
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then if q2 = z22 − z20 where z2 = g(z1 ) M2 is separated. Conversely, if M2 [y]
satisfies the hypotheses of Theorem 2.5 and
g 0 (x)x2
≥ 1,
g(x)2
(2.20)
then M1 is separated.
Proof. Let z2 = cz1 . Then since c > 1, z2 satisfies (2.6) – (2.8) and q2 =
pz22 − (pz2 )0 . Also p0 z2 ≤ 0. Separation follows by Theorem 2.5.
For the second part, since z2 satisfies (2.6) – (2.8) and by (2.19) we have that
z20 = g 0 (z1 )z10 ≥ −g 0 (z1 )z12 ≥ −g(z1 )2 = −z22
≤ δg 0 (z1 )z12 ≤ δg(z1 )2 = δz22 .
Separation and Disconjugacy
R.C. Brown
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Thus z2 satisfies (2.6) – (2.8) and we can again apply Theorem 2.5. On the other
hand, using (2.20)
g(z1 )2
≥ −z12 ,
g 0 (z1 )
δg(z1 )2
≤ δz12 .
z20 ≤ δz22 ⇔ z10 ≤ 0
g (z1 )
z20 ≥ −z22 ⇔ z10 ≥ −
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√
1
2
3 3
Example 2.1. Let p = 1, z1 (x) = x, and q1 (x) = x − 2 x− 2 . If a > 2 ,
then (2.6) – (2.8) is satisfied for some δ < 13 . If g(x) = exp(x2 ), (2.19) is
1
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satisfied for say a > 2. Taking z2 (x) = g(z1 ) = exp(x) we get that q2 (x) =
exp(2x) − exp(x) and there is an inequality of the form
KkM2 [y]k ≥ kq2 yk
on D0 defined on Ia . That M2 is separated on D0 also follows from Theorem A,
but the inequality seems new.
The next two lemmas are useful.
Lemma 2.8. Suppose that M1 [y] = −(py 0 )0 + q1 y is separated on D0 . If
q2
lim sup < 1 + γ,
x→∞ q1
q2
lim inf
> 1 − γ,
x→∞ q1
where γ is sufficiently small, then M2 [y] is also separated on D0 .
Proof. Choose a0 large enough so that on Ia0
q2
− 1 < γ.
q1
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R.C. Brown
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Since M2 [y] = M1 (y) + (q2 − q1 )y by the triangle inequality and inequality
(1.2) we have that
q
2
Lkyk + KkM2 [y]k + K q1 q1 − 1 y ≥ KkM1 [y]k + Lkyk
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for y ∈ C0∞ . Hence on Ia0
Lkyk + KkM2 [y]k + Kγkq1 yk ≥ Ckq1 yk
≥ C(1 + γ)−1 kq2 yk.
Thus
Lkyk + KkM2 [y]k ≥ dkq2 yk,
where d = (1 + γ)−1 (C − Kγ), which is positive for small enough γ.
Lemma 2.9. Suppose that M1 [y] = −(py 0 )0 + q1 y satisfies the separation inequality (2.17) with h = q12 for any > 0 on D0 . If also there are constants
K1 , K2 > 0 such that K1 ≤ |q1 /q2 | ≤ K2 then M2 [y] = −(py 0 )0 + q2 y satisfies
the same separation inequality on D0 with h = q22 for sufficiently small > 0.
Proof. Since
M2 [y] = M1 [y] + q2
q1
1−
q2
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R.C. Brown
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y
for y ∈ C0∞ (Ia ), we arrive at the inequality
q
1
+ K()kyk ≥ kM1 [y]k + K()kyk
kM2 [y]k + q
1
−
y
2
q2
≥ kq1 yk ≥ K1 kq2 yk
for any > 0. Hence also
kM2 [y]k + K()kyk ≥ d1 kq2 yk,
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where d1 = (K1 − (1 + K2 ) > 0 for small enough .
Remark 2.4. Taking q2 = −q1 and K1 = K2 = 1 in Lemma 2.9, we see that if
M1 satisfies (2.17) then so does M2 which means that we can have separation
for a potential q which is negative and unbounded below provided the expression constructed with potential |q| satisfies (2.17).
Example 2.2. Suppose p(x) = 1 and q1 (x) = exp(x). Then by Theorem A or B
M1 is separated. Let tn (x) = exp(exp(· · · exp(x)) · · · ) be a n-fold iteration of
exp(x) and set q2 (x) = exp(x)(1 + sin(tn (x)), > 0. Then Theorems A and
B do not apply because (1.3) and (1.4) are unbounded. However, by Lemma 2.8
M2 is separated if is sufficiently small. Clearly tn (x) can be replaced by any
other rapidly increasing function.
Example 2.3. Let p1 (x) = exp(x) and q1 (x) = x1/3 on Ia . By Theorem C M1 is
separated. It is easy to verify that p1 and q1 satisfy the Muckenhoupt condition
Z ∞
Z x
−1
sup
p1
q1θ < ∞, θ > 1,
x∈Ia
x
a
and therefore (cf. Opić and Kufner [18, Theorem 6.2]) the Hardy inequalθ/2
ity Kk exp(x/2)y 0 k ≥ kq1 yk holds on C0∞ . Therefore from the second part
of Theorem 2.6 we obtain an inequality of the form (2.17). If now q2 (x) =
−q1 (x)(2 + sin(exp(xn )) we will have from Lemma 2.9 the same kind of inequality but with
M2 [y] = −(exp(x)y 0 )0 − x1/3 (2 + sin(exp(xn ))y.
Theorem 2.10. If p > 0, z is a C 1 function, p0 z ≤ 0 and
−K1 z 2 ≤ z 0 ≤ K2 z 2
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R.C. Brown
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for positive constants K1 , K2 then the operators
M1,c [y] = −(py 0 )0 + q1,c y,
M2,c [y] = −(py 0 )0 + q2,c y,
where q1,c = c2 pz 2 − c(pz)0 and q2,c = c2 pz 2 are separated for sufficiently large
c ≥ 1.
Proof. To prove that M1,c is separated we retrace the proof of Theorem 2.1. Let
Lc (y) = y 0 + czy and L∗c (y) = −y 0 + czy, where y ∈ C0∞ (I). Then
Z ∞
∗
2
kLc (y)k =
(y 0 )2 + (cz 0 + c2 z 2 )y 2 .
Separation and Disconjugacy
R.C. Brown
1
If c ≥ K1 , then cz 0 + c2 z 2 ≥ 0 and as before,
kczk2 ≤ 4kL∗c [y]k2 .
Likewise L∗c (pLc (y)) = M1,c [y] and
q1,c = −pcz 0 − p0 cz + pc2 z 2
≥ −pcz 0 + pc2 z 2
≥ pcz 2 (c − K2 ) ≥ 0
if c > K2 . From the definition of q1,c we also have that (pz)0 ≤ K2 pz 2 . And so
Z
1
2
[(czp)2 (y 0 )2 + (c4 z 4 p2 − ((cz)3 p2 )0 y 2 ]
kM1,c [y]k ≥
4
Z Ia
≥
[c4 z 4 p2 − 3c3 z 2 z 0 p2 ]y 2
Ia
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Z
3K2
≥ 1−
c4 p2 z 4 y 2
c
Ia
for c > 3K2 . Now also
k(pcz)0 yk + kM2,c [y]k ≥ kM1,c [y]k ≥ K3 c2 pz 2 y where K3 =
p
1 − 3K2 /c, so that
kM2,c [y]k ≥ K3 c2 pz 2 y − k(pcz)0 yk
r
r !
3K2
K2 c2 pz 2 y .
≥
1−
−
c
c
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R.C. Brown
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Since the constant is positive for large enough c the inequality (2.13) for M2,c [y]
is established. Since
q1,c
= (1 − (pz)0 (c2 pz 2 )
q2,c
K2
≤ 1+ 2
c
K2
≥ 1− 2 ,
c
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Lemma 2.8 may be applied to conclude that M1,c is separated and satisfies an
inequality like (2.13).
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p
Example 2.4. If p0 is of constant sign, let z = −sgn(p0 ) q/p then p0 z ≤ 0 as
required and q2,c = c2 q. A calculation shows that the hypothesis of Theorem
2.10 becomes
0
p
p1/2 q 0
−2K1 ≥ √ − 3/2 sgn(p0 ) ≤ 2K2 .
pq
q
Equivalently we can require that
0
p
p1/2 q 0 η = sup √ − 3/2 < ∞
pq
q
x∈Ia
Separation and Disconjugacy
R.C. Brown
to conclude that Md [y] = −(py 0 )0 + dqy is separated for sufficiently large d.
For example, if p(x) = q(x) = exp(x2 ) both Theorem A and B fail for any Md
yet η = 0 and so we have an inequality of the form
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Kk − (exp(x2 )y 0 )0 + d exp(x2 )yk ≥ kd exp(x2 )yk
Contents
for large enough d.
Corollary 2.11. Let p, z, h, and g be functions such that p > 0 and p, z are C 1 ,
p0 z ≤ 0, z 0 ≤ δz 2 for δ ∈ [0, 1/3), h ≥ d > 0, g is bounded, and
h(pz)0 = 0,
(2.21)
lim x→∞ pz 2 0 0
2
(2.22)
lim pz = ∞,
x→∞
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0
then M1 [y] = −(py ) + q1 y, where q1 = pz − (pz) is separated on D and
M2 [y] = −(py 0 )0 + q2 , where q2 = pz 2 + hg(pz)0 is separated on at least on D0 .
If we assume additionally that
2
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then the inequalities
kMi [y]k + K()kyk ≥ kqjθ yk
hold for 1 ≤ i, j ≤ 2 and θ < 1 on D if i = 1 and on D0 if i = 2.
Proof. Since
h(pz)0 (pz)0 pz 2 ≥ d pz 2 ,
(pz)0 /pz 2 → 0 as x → ∞ which implies that for Ia0 = [a0 , ∞) and a0 sufficiently large, −pz 2 ≤ (pz)0 . Since the assumptions of Theorem 2.5 are satisfied,
M1 [y] is separated on D relative Ia0 and by Remark 2.1(ii) also on Ia . Since
Separation and Disconjugacy
q1
1 − (pz)0 /pz 2
= lim
=1
x→∞ q2
x→∞ 1 − hg(pz)0 /pz 2
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lim
the separation of M2 and M3 on D0 follows from Lemma 2.8.
To prove the second claim, a calculation will show that
qi2
= lim (pz 2 )2(1−θ) T (z, p, θ) = ∞,
x→∞ q 2θ
x→∞
j
lim
1 ≤ i, j ≤ 2,
where T (z, p, θ) is a term going to 1 as x → ∞. For example,
2
q32
1 + g(pz)0 /pz 2
2 2(1−θ)
lim
= lim (pz )
= ∞.
x→∞ q22
x→∞
(1 + hg(pz)0 /pz 2 )θ
The inequalities follow from the second part of Theorem 2.6.
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Example 2.5. Set p(x) = exp(x/3), z(x) = − exp(x/3), h(x) = exp((1 −
3)x/3), and g(x) = − sin(tn (x)), where tn (x) is as in Example 2.3. Then
x
1
1
2x
2x
0
0
p z = − exp
≤ δ exp
≤ 0, z = − exp
= z2,
3
3
3
3
3
and (2.21) holds. Then
x 0
2
0
M [y] := − exp
y + exp(x) 1 + exp(−x) sin(tn (x)) y
3
3
is separated on D0 . Since M is LP at ∞ the separation actually holds on D.
The final result of this paper is quite different from Theorem 2.1 but it reinforces the connection between disconjugacy and separation. In addition, the
proof is quite elementary.
Theorem 2.12. Let p > 0 and q ≥ d > 0 be continuous. Suppose that M λ [y] =
−(py 0 )0 + (q − λq 2 )y is disconjugate on Ia for some λ > 0. Then M [y] =
−(py 0 )0 + qy is separated.
Proof. It is well known (see e.g. [15, Theorem 6.2]) that the disconjugacy of
M λ is equivalent to the positive definiteness of the functional
Z
λ
Q [y] =
p|y 0 |2 + (q − λq 2 )|y|2 for y ∈ C0∞ (Ia ).
Ia
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In other words, we must have the inequality
Z
Z
0
0 2
2
(2.23)
Q [y] ≡
p|y | + qy ≥ λ
q 2 |y|2
Ia
Ia
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with equality holding if and only if y = 0. Now consider the expression Mq2 =
q −2 [−py 0 )0 +qy], where y is an appropriate function in L2 (q 2 ; Ia ). If y ∈ C0∞ (Ia )
then the Cauchy-Schwartz inequality and (2.23) yields that
kMq2 [y]kq2 kykq2 ≥ Q0 [y] ≥ λkyk2q2 ≡ λkqyk2 .
It follows that the inequality
d−2 kM [y]k ≥ kMq2 [y]kq2 ≥ λkqyk
Separation and Disconjugacy
C0∞
holds on the
functions and also therefore on D0 . Because M is LP at ∞
we again conclude that it is separated on D.
Remark 2.5.
(2.24)
(i) If p−1 ∈ L(Ia ) and
Z ∞
Z
−1
sup
p
x∈Ia
x
a
R.C. Brown
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x
λq 2 − q
<
1
4
2 then (cf. Example 2.4) the Hardy inequality p1/2 y 0 ≥ (λq 2 − q)1/2 y holds on C0∞ (Ia ) with equality if and only if y = 0. This inequality implies
the positive definiteness of Qλ .
(ii) If w ≥ d > is a weight and we require that (−py 0 )0 + (q − λw) be disconjugate, then the proof of Theorem 2.12 will yield the inequality
√ (2.25)
d−1 kM [y]k ≥ λ wy .
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Substituting w for q 2 in (2.24) will give a sufficient condition for (2.25).
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References
[1] F.V. ATKINSON, On some results of Everitt and Giertz, Proc. Royal Soc.
Edinburgh Sect. A, 71A (1972/3), 151–158.
[2] R.C. BROWN, D.B. HINTON AND M.F. SHAW, Some separation criteria
and inequalities associated with linear second order differential operators;
in Function Spaces and Applications (D.E. Edmunds, et al., Eds.), Narosa
Publishing House, New Delhi, 2000, 7–35.
Separation and Disconjugacy
[3] R.C. BROWN AND D.B. HINTON, Two separation criteria for second
order ordinary or partial differential operators, Math. Bohem., 124 (1999),
273–292.
[4] N. CHERNYAVSKAYA AND L. SCHUSTER, Weight summability of solutions of the Sturm-Liouville equation, J. Differential Equations, 151
(1999), 456–473.
[5] W.A. COPPEL, Disconjugacy, Lecture Notes in Mathematics, Vol. 220
(A. Dold and B. Eckmann, Eds.), Springer-Verlag, Berlin, Heidelberg, and
New York, 1971.
[6] W.A. COPPEL, Stability and Asymptotic Behavior of Differential Equations, Heath Mathematical Monographs (D.V. Widder, Ed.), D. C. Heath
and Company Boston, 1965.
[7] W.N. EVERITT, A note on the Dirichlet conditions for second-order differential expressions, Canad. J. Math., 28(2) (1976), 312–320.
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[8] W.N. EVERITT AND M. GIERTZ, Some properties of the domains of
certain differential operators, Proc. London Math. Soc., 23(3) (1971), 301–
324.
[9] W.N. EVERITT AND M. GIERTZ, Some inequalities associated with the
domains of ordinary differential operators, Math. Z., 126 (1972), 308–
328.
[10] W.N. EVERITT AND M. GIERTZ, On limit-point and separation criteria
for linear differential expressions, Proceedings of the 1972 Equadiff Conference, Brno, 1972, 31–41.
Separation and Disconjugacy
[11] W.N. EVERITT AND M. GIERTZ, Inequalities and separation for certain
ordinary differential operators, Proc. London Math. Soc. (3), 28(3) (1974),
352–372.
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[12] W.N. EVERITT AND M. GIERTZ, Inequalities and separation for
Schrodinger type operators in L2 Rn , Proc. Royal Soc. Edinburgh Sect.
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[14] S. GOLDBERG, Unbounded Linear Operators Theory and Applications,
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[15] P. HARTMAN, Ordinary Differential Equations, Second Edition,
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[16] R. KAUFFMAN, On the limit-n classification of ordinary differential operators with positive coefficients, Proc. Lond. Math. Soc., 35(3) (1977),
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[17] M.A. NAIMARK, Linear Differential Operators, Part II, Frederick Ungar,
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[18] B. OPIĆ AND A. KUFNER, Hardy-type Inequalities, Longman Scientific
and Technical, Harlow, Essex, UK, 1990.
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