Boll. Unione Mat. Ital. (2014) 7:193–210 DOI 10.1007/s40574-014-0010-0
On the speed at which solutions of the Sturm–Liouville equation tend to zero N. A. Chernyavskaya · L. A. Shuster
Received: 17 November 2013 / Accepted: 15 August 2014 / Published online: 13 September 2014 © Unione Matematica Italiana 2014
Abstract We consider the equation − y (x) + q(x)y(x) = f (x), x ∈ R.
(1)
For a fixed p ∈ [1, ∞) and for a correctly solvable Eq. (1) in L p (R), we find a positive and continuous function α p (x) for x ∈ R such that we have a sharp by order equality y(x) = o(α p (x)), |x| → ∞, ∀y ∈ D p . Here
D p = y ∈ L p (R) : y, y ∈ AC loc (R), −y + qy ∈ L p (R) .
Keywords
Sturm–Liouville equation · Estimates of solutions
Mathematics Subject Classification
34B24 · 34C11
1 Introduction In the present paper, we consider the equation − y (x) + q(x)y(x) = f (x), x ∈ R
(1.1)
where f ∈ L p (L p (R) := L p ), p ∈ [1, ∞] (L ∞ (R) := C(R)) and 0 ≤ q ∈ L loc 1 (R).
(1.2)
N. A. Chernyavskaya Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, 84105 Beer-sheva, Israel e-mail:
[email protected] L. A. Shuster (B) Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel e-mail:
[email protected]
123
194
N. A. Chernyavskaya, L. A. Shuster
By a solution of Eq. (1.1), in the sequel we mean a function y, which is absolutely continuous together with its derivative and satisfies equality (1.1) almost everywhere in R. In addition to (1.2), we always assume that x+a def ∃a ∈ (0, ∞) : q0 (a) = inf q(t)dt > 0. (1.3) x∈R x−a
Condition (1.3) is a minimal additional requirement to (1.2) to the function q under which Eq. (1.1) is correctly solvable in the space L p , p ∈ [1, ∞] (see [8]). We recall that correct solvability of (1.1) means that the following assertions hold: (I) for every function f ∈ L p , there exists a unique solution y ∈ L p of (1.1); (II) there is an absolute constant c( p) ∈ (0, ∞) such that the solution y ∈ L p of (1.1) satisfies the inequality y p ≤ c( p) f p , ∀ f ∈ L p f p := f L p (R) . (1.4) Throughout the sequel, the symbol y stands only for solutions of (1.1) belonging to the class L p . Let us go back to (1.3). From (I)–(II), it follows (see [5]) that for p ∈ [1, ∞), the solution y ∈ L p of (1.1), regardless of f ∈ L p , satisfies the equality lim y(x) = 0
(1.5)
|x|→∞
and for p = ∞ equality (1.5) holds regardless of f ∈ C(R) if and only if x+a q(t)dt = ∞ ∀a ∈ (0, ∞) lim |x|→∞ x−a
(1.6)
(see [5]; requirement (1.6) was introduced in [13] in a different situation). In the present paper, we determine to what extent one can strengthen (1.5) for a fixed p ∈ [1, ∞]. To formulate the problem more precisely, let us introduce the set D p , p ∈ [1, ∞]: D p = {y ∈ L p : y, y ∈ AC loc (R); −y + qy ∈ L p }
(1.7)
and denote by θ (x) an arbitrary positive continuous function for x ∈ R. Clearly, under condition (1.3) the set D p consists of all solutions y ∈ L p of (1.1) which correspond to the function f from (1.1) running over the whole space L p (see (I)–(II)). Our goal consists of finding minimal additional requirements to the function θ under which, for a given p ∈ [1, ∞], we have the equality lim θ (x)y(x) = 0,
|x|→∞
∀y ∈ D p .
(1.8)
For brevity, in the sequel this problem will be called “question (1.8)” or “problem (1.8)”. Note that for Eq. (1.1), question (1.8) seems to be new (see [10] where a similar problem was studied for an equation of the first order). As in [10], an answer to this question allows one to determine the minimal speed at which solutions y(x) of Eq. (1.1) tend to zero as |x| → ∞ (see below). Let us now formulate our statements. Here we restrict ourselves to the case p = 1; see Sect. 3 of this paper for results concerning the cases p ∈ (1, ∞]. We need an auxiliary function d(x), x ∈ R which was introduced in [3]. This function is constructed from the coefficient q of Eq. (1.1). To define it, fix x ∈ R and consider the equation in d ≥ 0: √2d x+t q(ξ )dξ dt = 2. (1.9) 0
123
x−t
Solutions of the Sturm–Liouville equation tend to zero
195
It is known that under conditions (1.2)–(1.3), Eq. (1.9) has a unique finite positive solution (see Sect. 2); denote it by d(x). The function d(x) is continuous for x ∈ R (see Sect. 2). Note that the function Q(x) = d −2 (x) can be interpreted as a composed (in the sense of the theory of functions) average of the coefficient Q(ξ ), ξ ∈ R at the point ξ = x with step d(x). Indeed, denote 1 x+t Sx (q)(t) = q(ξ )dξ, x ∈ R, t > 0, 2t x−t √2η 1 t f (t)dt, η > 0. M( f )(η) = 2 η 0 Clearly, the value Sx (q)(t) is the Steklov average with step t > 0 of the function q at the point ξ = x, and the value M( f )(η) is the average of the function f (t), t ∈ (0, ∞) with step η > 0 at the point t = 0. Then from (1.9), it follows that √2d(x) x+t 1 1 Q(x) = 2 q(ξ )dξ dt = 2 d (x) 2d (x) 0 x−t √ 2d(x) x+t 1 1 = 2 t q(ξ )dξ dt = M(Sx (q))(d(x)). d (x) 0 2t x−t Definition 1.1 Let α(x) and β(x) be positive continuous functions for x ∈ R, and let p ∈ [1, ∞]. We say that the equality y(x) = o(α(x)),
|x| → ∞,
∀y ∈ D p
is sharp by order if for any function β(x), x ∈ R, such that β(x) = o(α(x)), |x| → ∞, the equality y(x) = o(β(x)),
|x| → ∞,
∀y ∈ D p
does not hold. Let us now, finally, go to our statements. Theorem 1.2 For p = 1, equality (1.8) holds if and only if σ1 < ∞. Here σ1 = sup(θ (x)d(x)).
(1.10)
x∈R
Corollary 1.3 We have a sharp by order equality y(x) = o(d(x)),
|x| → ∞,
∀y ∈ D1 .
(1.11)
Definition 1.4 [10] Suppose we are given p ∈ [1, ∞] and some positive continuous function α p (x) for x ∈ R. We say that solutions of (1.1) tend to zero as |x| → ∞ at the speed not less than o(α p (x)) if the following conditions hold: (III) y(x) = o(α p (x)),
|x| → ∞,
(IV) equality (1.12) is sharp by order.
∀y ∈ D p . (1.12)
We want to emphasize that according to Definition 1.4 and Corollary 1.3, formula (1.11) determines the speed at which solutions y ∈ D1 of (1.1) tend to zero at infinity (below for brevity we say “the speed of decrease of solutions y ∈ D1 ”) as a whole, i.e., regardless of individual properties of a concrete solution y ∈ D1 . In addition, for particular Eq. (1.1), one
123
196
N. A. Chernyavskaya, L. A. Shuster
can go to an explicit form in formula (1.11) if one can find two-sided sharp by order estimates of the function d. Here, in contrast to the study of question (1.8), a study of the function d needs only standard calculations. Indeed, conditions (1.2)–(1.3) imply d0 < ∞ where d0 = sup d(x)
(1.13)
x∈R
(see Sect. 4). Thus the study of Eq. (1.9) is reduced to applying well-known tools of local analysis that justifies our assertion. See Sect. 5 for examples of proof of explicit formulas of type (1.11) (including Eq. (1.1) with an oscillating function q). Note that in (1.12) the function α p (x), x ∈ R, does not necessarily tend to zero as |x| → ∞. For example, for the equation − z (x) + z(x) = f (x),
|x|
x ∈R
− y (x) + e y(x) = f (x),
(1.14)
x ∈ R,
(1.15)
we get (see Sect. 5) x ∈ R for (1.14), α p (x) ≡ 1,
1 α p (x) = exp − 1 − |x| x ∈ R for (1.15). 2p These relations mean that for the solution of (1.14) and (1.15) one has the relations z(x) = o(1), |x| → ∞, ∀z ∈ D p ,
1 y(x) = o exp − 1 − |x| , |x| → ∞, 2p
(1.16)
∀y ∈ D p .
(1.17)
In addition, one cannot replace (1.16) and (1.17) with the following relations: z(x) = o(ψ(x)), |x| → ∞,
1 y(x) = o ϕ(x) exp − 1 − |x| , 2p
∀z ∈ D p , |x| → ∞,
∀y ∈ D p ,
where ϕ(x) → 0, ψ(x) → 0 as |x| → ∞. To conclude this introduction, note that the obtained speed of decrease of solutions of (1.1) characterizes the whole class D1 (and not individual solutions) and therefore is minimal with respect to this class. This implies that such a value can be useful, as a priori information, for numerical inversion of concrete Eq. (1.1).
2 Preliminaries Below we present several assertions needed for the proofs (see Sect. 4). Facts mentioned in Sect. 1 are not presented again here; conditions (1.2)–(1.3) are assumed to be satisfied and are not included in the statements. Lemma 2.1 [6,7] The equation z (x) = q(x)z(x),
x ∈R
(2.1)
has a fundamental systems of solutions (FSS) {u, v} possessing the following properties: u(x) > 0,
123
v(x) > 0,
u (x) ≤ 0,
v (x) ≥ 0,
x ∈ R;
(2.2)
Solutions of the Sturm–Liouville equation tend to zero
197
v (x)u(x) − u (x)v(x) = 1 lim
x→−∞
0
−∞
dt = ρ(t)
∞
0
dt = ∞, ρ(t)
v(t)
=
1 + ρ (t) 2ρ(t)
(2.3)
v(x) u(x) = lim = 0; u(x) x→∞ v(x)
,
u (t) u(t)
(2.4)
def
ρ(t) = u(t)v(t),
|ρ (t)| < 1, v (t)
x ∈ R;
t ∈ R;
t ∈ R; =−
(2.6)
1 − ρ (t) 2ρ(t)
,
t ∈ R.
Lemma 2.2 [9] For a given x ∈ R, consider the equations in d ≥ 0: √2d x+t √2d x q(ξ )dξ dt = 1, q(ξ )dξ dt = 1; 0
x−t √ 2d
0
0 x+t
(2.7)
(2.8)
x
q(ξ )dξ dt = 2,
(2.5)
x+d
d
x−t
q(ξ )dξ = 2.
(2.9)
x−d
Each of the Eqs. (2.8)–(2.9) has a unique finite positive solution. Denote the solutions of Eq. (2.8) by d1 (x) and d2 (x), and the solutions of Eq. (2.9) by ˆ d(x) and d(x), respectively. The functions d1 (x), d2 (x) and d(x) were introduced in [3], the ˆ function d(x) was introduced by Otelbaev (see [14]). Theorem 2.3 [9] For x ∈ R we have the inequalities: √ |u (x)| v (x) 1 d1 (x); d2 (x) ≤ 2; √ ≤ v(x) u(x) 2 √ d(x) √ ≤ ρ(x) ≤ 2d(x); 2 2 √ d(x) ˆ ≤ 2d(x). √ ≤ d(x) 2
(2.10) (2.11) (2.12)
Inequalities of type (2.10) first appeared in [17] and were developed in [2,3,6,9,15]. Two-sided sharp by order estimates of the function ρ (see (2.5), (2.11)) with another, more complicated, auxiliary functions instead of d (and under some additional requirements to q), were first obtained by Otelbaev [16]. Therefore, we call all such inequalities Otelbaev inequalities. See [2,4,6,9,17,18] for further developments. Inequality (2.12) is technical. Lemma 2.4 For a given x ∈ R, consider the equation in d ≥ 0: x+d dξ = 2. x−d d(ξ )
(2.13)
ˆ ˆ Equation (2.13) has a unique finite positive solution. Denote it by d(x). The function d(x) is continuous for x ∈ R. We have the inequalities: e−4
√ 2
≤
e−8
u(t) , u(x)
√ 2
≤
√ v(t) ≤ e4 2 v(x)
√ ρ(t) ≤ e8 2 ρ(x)
for
for
ˆ |t − x| ≤ d(x),
ˆ |t − x| ≤ d(x),
(2.14) (2.15)
123
198
N. A. Chernyavskaya, L. A. Shuster
1
√ 4e8 2
≤
√ d(t) ≤ 4e8 2 d(x)
d(x)
√ 4e8 2
ˆ ≤ d(x) ≤ 4e8
for √ 2
ˆ |t − x| ≤ d(x), x ∈ R.
d(x)
(2.16) (2.17)
Remark 2.5 Although Lemma 2.4 needs a proof (see Sect. 4), it is presented here because it only contains facts which are a priori with respect to the main problem (1.8). A similar assertion first appeared in [6]. Denote by G(x, t) the Green function corresponding to Eq. (1.1): u(x)v(t), x ≥ t, G(x, t) = u(t)v(x), x ≤ t.
(2.18)
Theorem 2.6 [8] If f ∈ L p , p ∈ [1, ∞], then the solution y ∈ L p of (1.1) is of the form ∞ G(x, t) f (t)dt, x ∈ R. (2.19) y(x) = −∞
Definition 2.7 [12] Suppose that for a given function q there exist numbers a ≥ 1, b > 0, and x0 ≥ 1 such that for |x| ≥ x0 , we have the inequalities a −1 d(x) ≤ d(t) ≤ ad(x)
for √
Then the value
|t − x| ≤ bd(x).
2b γ (a, b) = a exp − 3a
(2.20)
2
(2.21)
is called the exponent of the function q. Theorem 2.8 [12] Let a function q and a number γ0 > 1 be given. Then there exists at least an exponent γ (a, b) of this function such that γ (a, b) = γ0 . Theorem 2.9 [12] Suppose that for a given function q, at least one of its exponents γ (a, b) is less than 1. Then for all x ∈ R we have sharp by order inequalities ∞ x v(t)dt ≤ H v(x)d(x), u(t)dt ≤ H u(x)d(x), (2.22) −∞
x
where a constant H ∈ [1, ∞). Remark 2.10 See Sect. 5 for methods of proofs for sharp by order estimates of the function d (see (2.9)) and checking the inequality γ (a, b) < 1.
3 Main results Recall that the solution of problem (1.8) and conclusions on the speed of decrease of solutions of (1.1) for p = 1 are presented in Sect. 1. Theorem 3.1 For p ∈ [1, ∞), let (1.8) hold. Then σ p < ∞, where σ p = sup(θ (x)d(x)2−1/ p ). x∈R
123
(3.1)
Solutions of the Sturm–Liouville equation tend to zero
199
For p = ∞, equality (1.8) holds only if lim θ (x)d 2 (x) = 0.
(3.2)
|x|→∞
Definition 3.2 Suppose that there exists a constant M ≥ 1 such that the following inequality holds: |y(x)| ≤ Md 2 (x), ∀x ∈ R, ∀y ∈ D∞ . (3.3) We say that this inequality is sharp by order if for any continuous positive function α(x), x ∈ R such that α(x)d −2 (x) → 0 as |x| → ∞, the inequality |y(x)| ≤ N α(x),
∀x ∈ R,
∀y ∈ D∞
does not hold for any constant N ≥ 1. Finally, we say that the equality y(x) = O(d 2 (x)),
x ∈ R,
y ∈ D∞
is sharp by order if inequality (3.3) is. Theorem 3.3 Suppose that for a given function q at least one of its exponents γ (a, b) is less than 1. Then for p ∈ (1, ∞), equality (1.8) holds if σ p < ∞ and in the case p = ∞, this equality holds under condition (3.2). Theorem 3.4 Suppose that the function q is such that at least one of its exponents γ (a, b) is less than 1. Then we have sharp by order equalities y(x) = o(d 2−1/ p (x)),
|x| → ∞,
y(x) = O(d 2 (x)),
∀y ∈ D p ,
x ∈ R,
p ∈ [1, ∞);
∀y ∈ D∞ .
(3.4) (3.5)
4 Proofs Proof of Lemma 2.4 For a given x ∈ R, consider the function F(d, x) x+d dξ F(d, x) = , d ≥ 0. d(ξ ) x−d Clearly, F(0, x) = 0, Fd (d, x) > 0 for d > 0 (see Lemma 2.2), and, in addition (see (2.11)): x+d x+d x+d dξ ρ(ξ ) dξ dξ 1 F(d, x) = = ≥ √ . 2 2 x−d ρ(ξ ) x−d d(ξ ) x−d d(ξ ) ρ(ξ ) Hence F(d, x) → ∞ as d → ∞ (see (2.5)), and therefore Eq. (2.13) has a unique finite ˆ ˆ positive solution. Denote this solution by d(x). Let s ∈ [0, d(x)]. Then 2=
ˆ x+d(x) ˆ x−d(x)
2=
ˆ x+d(x) ˆ x−d(x)
dξ ≤ d(ξ ) dξ ≥ d(ξ )
ˆ (x+s)+(s+d(x)) ˆ (x+s)−(s+d(x))
ˆ (x+s)+(d(x)−s)
ˆ (x+s)−(d(x)−s)
dξ , d(ξ ) dξ . d(ξ )
123
200
N. A. Chernyavskaya, L. A. Shuster
Hence ˆ ˆ + s) ≤ s + d(x) ˆ d(x) − s ≤ d(x ˆ + s) − d(x)| ˆ |d(x ≤ |s|
for
for
ˆ s ∈ [0, d(x)]
⇒
ˆ |s| ≤ d(x).
(4.1)
ˆ (For s ∈ [−d(x), 0) inequality (4.1) can be checked in a similar way). By (4.1), the ˆ function d(x) is continuous for x ∈ R. The proofs of inequalities (2.14) are all similar to each other. For example, for the solution v we use Lemma 2.1, (2.11), (2.12) and (2.13) to obtain √ v (ξ ) 1 + ρ (ξ ) 1 2 2 = ≤ ≤ , ξ ∈R ⇒ v(ξ ) 2ρ(ξ ) ρ(ξ ) d(ξ ) ˆ ˆ ˆ √ x+d(x) √ x+d(x) √ dξ dξ v(x + d(x)) ln ≤2 2 ≤2 2 = 4 2, v(x) d(ξ ) d(ξ ) ˆ x x−d(x) ˆ x x+ d(x) √ √ √ dξ dξ v(x) ≤2 2 ln ≤2 2 = 4 2. ˆ d(ξ ) d(ξ ) ˆ ˆ v(x − d(x)) x−d(x) x−d(x) ˆ ˆ These estimates and Lemma 2.1 imply, for t ∈ [x − d(x), x + d(x)] inequalities (2.14): 1
√ e4 2
≤
√ ˆ ˆ v(x − d(x)) v(t) v(x + d(x)) ≤ ≤ ≤ e4 2 . v(x) v(x) v(x)
The inequalities (2.15) now follow from the definition of the function ρ (see (2.5)). The inequalities (2.16) follow from (2.11) and (2.14). To prove (2.17), we apply (2.13), (2.14) and (2.11): 2= ≤
ˆ x+d(x) ˆ x−d(x)
1 d(x)
dξ 1 = d(ξ ) d(x)
ˆ x+d(x) ˆ x−d(x)
≥
ˆ x+d(x) ˆ x−d(x)
1 d(x)
√ 2
ˆ x−d(x)
ˆ x−d(x)
d(x) u(x)v(x) ρ(ξ ) dξ ρ(x) u(ξ )v(ξ ) d(ξ )
ˆ x+d(x) ˆ x−d(x)
d(x) u(x)v(x) ρ(ξ ) dξ ρ(x) u(ξ )v(ξ ) d(ξ )
ˆ 1 1 1 1 d(x) √ √ √ dξ = √ 2 e8 2 2 2 2e8 2 d(x)
ˆ d(x) ≤ 4e8
√ 2
⇒
ˆ d(x), x ∈ R,
dξ 1 = d(ξ ) d(x)
ˆ x+d(x)
ˆ x+d(x)
√ d(x) ˆ √ √ √ 2 2e8 2 2dξ ≤ 8e8 2 d(x)
d(x) ≤ 4e8
2=
⇒
d(x), x ∈ R.
Proof of Theorem 3.1 We shall prove that the following inequalities hold:
∞
−∞
123
G(x, t)dt ≥
ˆ x+d(x) ˆ x−d(x)
1 2 √ d (x), x ∈ R. G(x, t)dt ≥ √ 4 2e12 2
(4.2)
Solutions of the Sturm–Liouville equation tend to zero
201
Below we use (2.18), Lemma 2.4, definition (2.5) of the function ρ, and (2.11) and (2.16): x+d(x) x x+d(x) ∞ ˆ ˆ G(x, t) ≥ G(x, t)dt = u(x) v(t)dt + v(x) u(t)dt −∞
ˆ x−d(x)
ˆ x−d(x) ˆ x+d(x)
x
v(t) u(t) dt + dt = ρ(x) v(x) u(x) ˆ x−d(x) x x+d(x) ˆ x dt d 2 ˆ √ + √ ≥ ρ(x) = √ ρ(x)d(x) ˆ x−d(x) x e4 2 e4 2 e4 2 2 d(x) d(x) 1 2 √ = √ √ d (x) ⇒ (4.2). ≥ √ √ 4 2 8 2 2 2 e 4e 4 2e12 2
x
We first consider the case p = ∞. Let y0 (y0 ∈ D∞ ) be a solution of (1.1) with f (x) ≡ f 0 (x) ≡ 1, x ∈ R. Then by (1.8), (2.19) and (4.2), we have ∞ 0 = lim θ (x)y0 (x) = lim θ (x) G(x, t) f 0 (t)dt |x|→∞ |x|→∞ −∞ ∞ 1 √ G(x, t)dt ≥ √ lim θ (x)d 2 (x) ≥ 0 ⇒ (3.2). = lim θ (x) |x|→∞ −∞ 4 2e12 2 |x|→∞ Let us now turn to the case p ∈ [1, ∞). Assume the contrary: (1.8) holds for some such p but σ p = ∞. Then also σˆ p = ∞ where ˆ 2−1/ p σˆ p = sup θ (x)d(x) x∈R
(see (2.12)). Since the functions θ and dˆ are continuous (see Lemma 2.4), there exists a sequence {xn }∞ n=1 such that ˆ n )2−1/ p = n 2 n = 1, 2, . . . ; θ (xn )d(x
lim |xn | = ∞.
(4.3)
n→∞
In [8], it is shown that condition (1.3) is equivalent to the inequality dˆ0 < ∞ where ˆ dˆ0 = sup d(x) x∈R
(see Lemma 2.2). Combining this equality with (2.12), we obtain dˆ0 < ∞, where ˆ dˆ0 = sup d(x). x∈R
∞ Therefore, according to (4.3), the sequence {xn }∞ n=1 contains a subsequence {x n k }k=1 such that the segments (+)
ωk = [ωk− , ωk ],
(±)
ωk
ˆ n k ), = xn k ± d(x
k = 1, 2, . . .
do not overlap. Let us introduce the functions { f n k (t)}∞ k=1 and f 0 (t), t ∈ R: ⎧ −1/ p ⎨2 · ˆ 1 1/ p , if t ∈ ωk n 2k d(xn k ) f n k (t) = ⎩0, if t ∈ / ωk ∞ f 0 (t) = f n k (t), t ∈ R.
(4.4) (4.5)
k=1
123
202
N. A. Chernyavskaya, L. A. Shuster
Since ω ∩ ωs = ∅ for = s, we obviously have ⎧ ⎨ f n k (t), if t ∈ ωk , k = 1, 2, . . . ∞ f 0 (t) = ⎩0, ωk if t ∈ /
(4.6)
k=1
Hence p f0 p
= =
∞
−∞
| f 0 (t)| dt = p
∞
1
k=1
2n k
2p
dt ωk
∞
| f 0 (t)| dt = p
k=1 ωk ∞
ˆ nk ) d(x
=
1 2p
k=1
nk
≤
∞ k=1 ωk
| f n k (t)| p dt
∞ 1 <∞ n2
⇒
f0 ∈ L p .
n=1
Let y0 be a solution of (1.1) with right-hand side f ≡ f 0 . To estimate the values {θ (xn k )y0 (xn k )}∞ k=1 from below, we use relations (2.19), (4.2)–(4.6) and (2.15): θ (xn k )y0 (xn k ) = θ (xn k ) = θ (xn k ) 21/ p √ e4 2
∞
G(xn k , t) f 0 (t)dt θ (xn k ) G(xn k , t) f n k (t)dt = G(xn k , t)dt ˆ n k )1/ p ωk 21/ p n 2k d(x
−∞ ωk
G(xn k , t) f 0 (t)dt ≥ θ (xn k )
ωk
1/ p 1 1 n ) ˆ ˆ n k ) ≥ 2 √ θ (xn k ) d(x ρ(x ) d(x √ k d(x n nk ) 2 k 2 1/ p 1/ p 4 2 ˆ ˆ nk nk 2 2 d(xn k ) d(xn k ) e 2 2−1/ p ˆ ˆ θ (xn k ) 1 1 d(xn k ) 1 1 θ (xn k )d(xn k ) √ √ ≥ 1 1 √ · = 5 1 2 1/ p + p e4 2 d(x 8 2 + ˆ n ) 2 4e e12 2 n 2k nk k 2 22 p 1 = 5 1 √ > 0. + 2 2 p e12 2
≥
θ (xn k )
Hence θ (xn k )y0 (xn k ) → 0 as k → ∞, whereas |xn k | → ∞ as k → ∞, which is a contradiction. Hence σ p < ∞. Proof of Theorem 1.2 Necessity. This statement coincides with Theorem 3.1 for p = 1. Proof of Theorem 1.2 Sufficiency. We need some auxiliary assertions.
Lemma 4.1 For the functions d1 , d2 , defined in (2.8), the inequality (0)
dk
= sup dk (x) ≤ 4d0 , x∈R
k = 1, 2
(4.7)
holds, where d0 = sup d(x) (see (1.13)). x∈R
Proof Both inequalities in (4.7) are proved similarly. Let, say, k = 1. It is known (see [8]) ˆ ˆ that for x ∈ R the inequality η ≥ d(x) (0 ≤ η ≤ d(x)) holds if and only if x+η
x+η η q(ξ )dξ ≥ 2 η· q(ξ )dξ ≤ 2 . (4.8) x−η
123
x−η
Solutions of the Sturm–Liouville equation tend to zero
203
Therefore, by the definition of d1 (x), x ∈ R (see Lemma 2.2), we get 2=2 0
√ 2d1 (x) x
q(ξ )dξ dt ≥ 2
x−t
√
2d1 (x) x
√1 d1 (x) 2
q(ξ )dξ dt
x−t
x x− d1√(x) + d1√(x) √ 1 d1 (x) 2 2 2 2 ≥2 2 − √ d1 (x) q(ξ )dξ > √ q(ξ )dξ d1 (x) d1 (x) 1 √ √ √ 2 2 2 x− 2 2 − 2 2 x− d1 (x) 2
√ √ d1 (x) d1 (x) d1 (x) ≤ 2d x − √ ≤ 2d0 , x ∈ R, √ ≤ dˆ x − √ 2 2 2 2 2 2
⇒
where we used (2.21). This estimate implies (4.7).
Lemma 4.2 For x, t ∈ R we have the inequality
|t − x| G(x, t) ≤ cd(x) exp − . 6d0
(4.9)
Proof We shall prove inequality (4.9) for x ≥ t (the case x ≤ t is similar). From (2.10) and (4.7), it follows that 1 v (ξ ) 1 ≥ , ξ ∈R ⇒ ≥ √ v(ξ ) 6d0 2d1 (ξ ) x v(x) v (ξ ) x −t ln ⇒ = dξ ≥ v(t) v(ξ ) 6d0 t
v(t) x −t , x ≥ t. ≤ exp − v(x) 6d0
(4.10)
Below we use (2.18), (2.5), (2.11), and (4.10): G(x, t) = u(x)v(t) = u(x)v(x) ·
v(t) |t − x| v(t) . = ρ(x) · ≤ cd(x) exp − v(x) v(x) 6d0
Lemma 4.3 Let g ∈ L 1 , ν ∈ (0, ∞) and ∞ g(t) exp(−ν|t − x|)dt, (Sν g)(x) = −∞
x ∈ R.
(4.11)
Then (Sν g)(x) → 0 as |x| → ∞. Proof Let n be a natural number. Then |g(t)| exp(−ν|t − x|)dt + |g(t)| exp(−ν|t − x|)dt 0 ≤ |(Sν g)(x)| ≤ |t−x|≤n |t−x|≥n |g(t)|dt + e−νn g 1 ⇒ ≤ |t−x|≤n
0 ≤ lim |Sν g)(x)| ≤ lim |(Sν g)(x)| ≤ lim |x|→∞
|x|→∞
|x|→∞ |t−x|≤n
|g(t)|dt +e−νn g 1 = e−νn g 1 .
123
204
N. A. Chernyavskaya, L. A. Shuster
In the last inequality, let us take the limit as n → ∞. We obtain lim |Sν g)(x)| = lim |(Sν g)(x)| = 0 |x|→∞
|x|→∞
⇒
lim (Sν g)(x) = 0.
|x|→∞
Now we go to the proof of the sufficient part of Theorem 1.2. Let y ∈ D1 . Then by Theorem 2.6, there exists a function f ∈ L 1 such that equality (2.19) holds. For x ∈ R, taking into account Lemmas 4.2 and 4.3, this implies
∞ ∞ |t − x| 0 ≤ θ (x)|y(x)| ≤ θ (x) dt G(x, t)| f (t)|dt ≤ cθ (x)d(x) | f (t)| exp − 6d0 −∞ −∞ 1 ν= ∈ (0, ∞), ≤ cσ1 (Sν (| f |))(x), 6d0
and from Lemma 4.3 we have the assertion.
Proof of Corollary 1.3 Taking into account Theorem 1.2, we only have to prove that the equality (1.11) is sharp by order (see Definition 1.1). Assume the contrary: there exists a positive continuous function α(x), x ∈ R such that the following relations hold:
Then for θ (x) =
1 α(x) ,
y(x) = o(α(x)),
|x| → ∞, ∀y ∈ D1
α(x) = o(d(x)),
|x| → ∞.
x ∈ R we have (1.8):
θ (x)y(x) =
1 y(x) = o(1), α(x)
|x| → ∞,
∀y ∈ D1
and therefore, by Theorem 1.2, we have d(x) ≤ σ1 < ∞, x ∈ R α(x) i.e.,
α(x) d(x)
⇒
0<
1 α(x) ≤ , σ1 d(x)
→ 0 as |x| → ∞, which is a contradiction.
x ∈ R,
Proof of Theorem 3.4 We need the following fact.
Lemma 4.4 Let a function q be such that at least one of its exponents γ (a, b) is less than 1. Then ∞ τ −1 d 2 (x) ≤
−∞
G(x, t)dt ≤ τ d 2 (x),
x ∈ R,
(4.12)
√ √ √ where τ = max 2 2H, 4 2e12 2 .
Proof The lower estimate of (4.12) was obtained above (see (4.2)), regardless of the value of γ (a, b). In the following relations we use (2.18), (2.22), (2.5) and (2.11): ∞ x ∞ G(x, t)dt = u(x) v(t)dt + v(x) u(t)dt ≤ 2H u(x)v(x)d(x)) −∞ −∞ x √ x ∈ R. = 2Hρ(x)d(x ≤ 2 2H d 2 (x) ≤ τ d 2 (x),
123
Solutions of the Sturm–Liouville equation tend to zero
205
Let us now go to the main statement of Theorem 3.4. Let p ∈ (1, ∞) and y ∈ D p . Then there exists a function f ∈ L p such that (2.19) holds. In the following estimates, we set x ∈ R, p = p( p − 1)−1 and use (2.19), Hölder’s inequality, (4.12), (3.1), (4.9) and (4.11): ∞ 0 ≤ θ (x)|y(x)| ≤ θ (x) G(x, t)| f (t)|dt ≤ θ (x)
−∞ 1/ p
∞ −∞
·
G(x, t)dt
∞
−∞
1/ p G(x, t)| f (t)| dt p
1/ p |t − x| | f (t)| p exp − dt 6d0 −∞ 1 1 1 1 1/ p 1/ p 1 2− 1 = 2 p τ p θ (x)d(x) p Sν (| f | p )(x) ≤ 2 p τ p σ p Sν (| f | p )(x) , ν= 6d0 (4.13) 1
≤ 2pτ
1 p
2
θ (x)d(x) p
+ 1p
∞
(see (4.11)). Equality (1.8) follows from Lemma 4.3. Now let y ∈ D∞ . Then, as above, there exists a function f ∈ C(R) such that (2.19) holds. Together with (4.12), this implies ∞ G(x, t)| f (t)|dt 0 ≤ θ (x)|y(x)| ≤ θ (x) −∞ ∞ ≤ θ (x) G(x, t)dt · f C(R) ≤ cθ (x)d 2 (x) f C(R) . (4.14) −∞
Equality (1.8) follows from Theorem 3.1. Proof of Theorem 3.4 Theorem 3.4 follows from Theorem 3.3 in the same way as Corollary 1.3 follows from Theorem 1.2.
5 Examples Below we consider the equations − y (x) + e|x| y(x) = f (x),
x ∈ R,
− y (x) + (e|x| + h(x))y(x) = f (x),
(5.1)
x ∈ R,
(5.2)
α ∈ (0, ∞).
(5.3)
where, as in (1.1), f ∈ L p , p ∈ [1, ∞] and h(x) = e|x| cos(eα|x| ),
x ∈ R,
Equation (5.2) will be viewed as Eq. (5.1) with perturbation (5.3). With the help of the theorems proved above and some results from [1,11], we establish the following facts: (1) Equation (5.1) is correctly solvable in L p for all p ∈ [1, ∞]; (2) For p ∈ [1, ∞), for solutions y ∈ D p of (5.1), we have a sharp by order equality
1 y(x) = o (1−1/2 p)|x| , |x| → ∞; (5.4) e (3) For p = ∞, for solutions y ∈ D∞ of (5.1), we have a sharp by order equality
1 y(x) = O , x ∈ R; e|x|
(5.5)
123
206
N. A. Chernyavskaya, L. A. Shuster
(4) Equation (5.2) is correctly solvable in L p for all p ∈ [1, ∞] regardless of the values α ∈ (0, ∞); (5) Relations (5.4) and (5.5) hold for solutions y ∈ D p of (5.2) (respectively to p ∈ [1, ∞]) if and only if α ≥ 21 . Note that assertion (5) is an example of the influence of the oscillation of the function q on the validity of equalities (5.4)–(5.5). Let us also note that Eqs. (5.1)–(5.2) have already been studied in [1] (in a different context), and all needed technical assertions can be found in [1,11,12]. Therefore, in order not to lengthen this paper unnecessarily, we only present here a scheme of investigation of Eqs. (5.1)–(5.2) along with facts from [1] and those theorems from [1,11,12] which were used in the proofs of these facts. We want to emphasize that all the not so complicated computational part of this investigation can be found in [1]. Proof of assertion (1) Since e|x| ≥ 1, x ∈ R, for a = 1 we get x+1 inf 1dt = 2 > 0. x∈R x−1
Our assertion now follows from the main result of [8] (see (1.3) and below).
Proof of assertions (2)–(3) We shall deduce (5.4) and (5.5) from (1.11), (3.4) and (3.5). Since the validity of formula (1.11) does not depend on the values of exponents of the function e|x| , x ∈ R (see (2.21)), we have to find estimates of the function d for the function q(x) = e|x| , x ∈ R (see Lemma 2.2). Throughout the sequel the problem of estimating the function d corresponding to a given function q will be solved as follows. Since the problems of estimating the functions d and d˜ are equivalent, we use some known facts on d˜ and then use (2.12) to get similar inequalities for d automatically. Theorem 5.1 [11] Suppose that q can be represented in the form q(x) = q1 (x) + q2 (x),
x ∈R
(5.6)
where the function q1 (x) is positive for x ∈ R and twice differentiable for |x| 1, and q2 ∈ L loc 1 (R). Denote A(x) = [0, 2q1 (x)−1/2 ], x ∈ R, (5.7) x+t 1 |x| 1, (5.8) sup q (ξ )dξ , 1 (x) = q1 (x)3/2 t∈A(x) x−t 1 x+t 1 2 (x) = √ x ∈ R. q2 (ξ )dξ , (5.9) sup q1 (x) t∈A(x) x−t Then, if the following conditions hold: 1 (x) → 0,
2 (x) → 0
as |x| → ∞,
(5.10)
ˆ Eq. (2.9) has a unique finite positive solution d(x), x ∈ R. In addition, 1 + δ(x) ˆ , d(x) = √ q1 (x)
|δ(x)| ≤ 2(1 (x) + 2 (x)),
ˆ q1 (x) ≤ c, c−1 ≤ d(x)
x ∈ R.
|x| 1,
(5.11) (5.12)
Remark 5.2 Here and in the sequel the functions q(x), q1 (x), q2 (x), x ∈ R, in our examples are even. Therefore all estimates are given for x ∈ [0, ∞).
123
Solutions of the Sturm–Liouville equation tend to zero
207
To apply Theorem 5.1 to the function q(x) = e|x| , x ∈ R, set q1 = q, q2 = 0. Then 2 = 0. Let us estimate the function 1 (x) for x 1 x+t 1 1 e c eξ dξ = x/2 sup |et − e−t | ≤ x/2 sup t ≤ x/2 . (5.13) 1 (x) = 3 sup x t∈A(x) x−t e e e 2 t∈A(x) t∈A(x) e Thus 2 = 0, 1 (x) → 0 as |x| → ∞, and by (5.13) and (5.12) we have ˆ ≤ ce−|x|/2 , x ∈ R. c−1 e−|x|/2 ≤ d(x)
(5.14)
From (5.14), (1.11) and (2.12), we obtain (5.4) for p = 1. The proof of (5.4) for p ∈ (1, ∞) is based on (3.4). Therefore we first establish the inequality γ (a, b) < 1 (see Theorem 3.4) for the function q(x) = e|x| ,
x ∈ R.
Theorem 5.3 [12] Suppose that for the function q there is a function q1 such that c−1 ≤ d(x) q1 (x) ≤ c, x ∈ R.
(5.15)
Here d(x), x ∈ R, is the solution of the first equation in (2.9). Suppose the function q1 (x) is continuous for x ∈ R, continuously differentiable for |x| 1 and satisfies the relations: q1 (x) = 0, |x|→∞ q1 (x)3/2 lim
(0)
q1 > 0,
(0)
q1 = inf q1 (x). x∈R
(5.16) (5.17)
Then for any given β ≥ 1, there is x1 = x1 (β) such that for all |x| ≥ x1 (β) and t ∈ ω(x) : β β def ω(x) = x − d(x), x + d(x) , x ∈ R. (5.18) c c We have the inequalities: 1 c2
1 1 −1 d(t) . 1− ≤ ≤ c2 1 − 2β d(x) 2β
(5.19)
Here the constant c coincides with the constant from (5.15). Theorem 5.3 has a trivial consequence which, for the sake of convenience, is stated as a separate assertion. Corollary 5.4 Suppose that the function q satisfies the hypotheses of Theorem 5.3. Then there exists an exponent γ (a, b) such that γ (a, b) < 1. Proof According to Definition 2.7, we identify inequalities (5.19) and the segment (5.18) with inequalities and the segment in (2.20), obtaining
β 1 −1 a = c2 1 − , b= , β ≥ 1. 2β c Here c is an absolute constant, and the value β can be chosen arbitrarily large (change of β will only lead to another choice of the point x1 (β) in Theorem 5.3). Therefore the inequality √
2β 2 γ (a, b) = a exp − <1 3 c becomes obvious for all β 1.
123
208
N. A. Chernyavskaya, L. A. Shuster
From (5.14) and Corollary 5.4, it easily follows that the inequality γ (a, b) < 1 holds for at least one of the exponents of the function q(x) = e|x| , x ∈ R. Therefore, Theorem 3.4 implies (5.4) for p ∈ (1, ∞) and (5.5) for p = ∞. Proof of assertion (4) This fact was established in [11] with the help of the following simple result. Theorem 5.5 [1] Suppose that the function q is represented in the form (5.6) where 0 < loc q1 ∈ L loc 1 (R) and q2 ∈ L 1 (R). If the following conditions hold: (1) there is a0 ∈ (0, ∞) such that
(0)
q1 (a0 ) = inf
x+a0
x∈R x−a0
(2) τ (a) → 0 as a → ∞ where x+a τ (a) = sup q2 (t)dt x∈R
x−a
q1 (t)dt > 0
x+a
−1 q1 (t)dt
, a ≥ a0 ,
x−a
then there is a1 a0 such that q0 (a1 ) > 0 (see (1.3)). Note that to apply Theorem 5.5 to the case (see (5.2)) q(x) = e|x| + h(x),
x ∈R
in [11], the functions q1 and q2 are chosen in the following natural way: q1 (x) = e|x| ,
q2 (x) = h(x),
x ∈ R.
(5.20)
Proof of assertion (5) In [11], using Theorem 5.1 and (5.20) it is shown that for α ≥ 1/2 estimates (5.14) hold. Therefore, repeating the argument completely given in the proof of the inequality γ (a, b) < 1 in the case q(x) = e|x| , x ∈ R (see above), we conclude that this inequality remains true for α ≥ 1/2. Then, for all p ∈ [1, ∞] and α ≥ 1/2, formulas (5.4) and (5.5) for solutions y ∈ D p of (5.2) also remains true (see Theorem 3.4). It remains to show that for α ∈ (0, 1/2) and
1 θ (x) = exp 1− (5.21) |x| , x ∈ R, 2p we have σ p = ∞, p ∈ [1, ∞]. Once we establish that, we can conclude that for p ∈ [1, ∞] and α ∈ (0, 1/2) equalities (5.4) and (5.5) (respectively to p) do not hold, in view of Theorem 3.1. Towards this end, we need one more assertion. Theorem 5.6 [1] Suppose that the following conditions (1)–(2) hold: (1) the function q vanishes at the points xk , k = 1, 2, . . . and |xk | → ∞ as k → ∞. (2) the function q is absolutely continuous in R together with its derivatives q (i) , i = 1, 2, 3, and q (xk ) = 0 for all k 1. Denote
1 Ak = 0, 4 , k1 q (xk ) x+t (4) k 1, q (ξ )dξ , σk = sup 4
t∈Ak
123
x−t
Solutions of the Sturm–Liouville equation tend to zero
δk =
σk q (x
5/4 k)
209
,
k 1.
Then, if δk → 0 as k → ∞, we have 6 ˆ k) = 4 (1 + εk ), |εk | ≤ cδk , k 1. d(x q (xk ) Clearly, the function q q(x) = exp(|x|) + h(x),
x ∈R
for α ∈ (0, ∞) has zeros on (0, ∞) at the points {xk }∞ k=1 where xk =
ln[(2k + 1)π] , α
k ≥ 1.
(5.22)
In [1], using Theorem 5.6, it is shown that for all k 1 we have c−1 (2k + 1)−
1+2α 4α
ˆ k ) ≤ c(2k + 1)− ≤ d(x
1+2α 4α
,
and therefore (see (2.12)) for k 1 we obtain c−1 (2k + 1)−
1+2α 4α
≤ d(xk ) ≤ c(2k + 1)−
1+2α 4α
.
(5.23)
Hence in case (5.21) for α ∈ (0, 1/2) and p ∈ [1, ∞], we get 2− 1p
σ p = sup θ (x)d(x) x∈R
= sup e
(1− 21p )|x|
x∈R
1− 21p
≥ sup [e xk d 2 (xk )] k1
= c−1 sup(2k + 1)
1−2α 2α
2− 1p
d(x)
1− 21p
= sup[e x d 2 (x)] x∈R
1 1 1+2α 1− 2 p ≥ c−1 sup (2k + 1) α · (2k + 1)− 2α
1− 21p
k1
= ∞.
k≥1
Our assertion now follows from Theorem 3.1. Acknowledgments The authors wish to thank the anonymous referee for his very careful reading of the paper and for his helpful comments which greatly improved the exposition of the paper.
References 1. Chernyavskaya, N., El-Natanov, N., Shuster, L.: Weighted estimates for solutions of a Sturm–Louiville equation in the space L 1 (R). Proc. R. Soc. Edinb. Sect. A 141(6), 1175–1206 (2011) 2. Chernyavskaya, N., Schiff, J., Shuster, L.: A priori analysis of initial data for the Riccati equation and asymptotic properties of its solutions. Bull. Lond. Math. Soc. 41(4), 723–732 (2009) 3. Chernyavskaya, N., Shuster, L.: On the WKB-method. Diff. Uravnenija 25, 1826–1829 (1989). (in Russian) 4. Chernyavskaya, N., Shuster, L.: Estimates for Green’s function of the Sturm–Liouville operator. J. Differ. Equ. 111, 410–421 (1994) 5. Chernyavskaya, N., Shuster, L.: Solvability in L p of the Dirichlet problem for a singular nonhomogeneous Sturm–Liouville equation. Methods Appl. Anal. 5(3), 259–272 (1998) 6. Chernyavskaya, N., Shuster, L.: Estimates for the Green function of a general Sturm–Liouville operator and their applications. Proc. Am. Math. Soc. 127, 1413–1426 (1999) 7. Chernyavskaya, N., Shuster, L.: Regularity of the inversion problem for a Sturm–Liouville equation in L p . Methods Appl. Anal. 7(1), 65–84 (2000)
123
210
N. A. Chernyavskaya, L. A. Shuster
8. Chernyavskaya, N., Shuster, L.: A criterion for correct solvability of the Sturm–Liouville equation in the space L p (R). Proc. Am. Math. Soc. 130, 1043–1054 (2001) 9. Chernyavskaya, N., Shuster, L.: Classification of initial data for the Ricacati equation. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 5(8), 511–525 (2002) 10. Chernyavskaya, N., Shuster, L.: Sharp by order estimates of solutions of a simplest singular boundary value problem. Math. Nachr. 281(2), 160–170 (2008) 11. Chernyavskaya, N.A., Shuster, L.: Davies–Harrel representations, Otelbaev’s inequalities and properties of solutions of Riccati equations. J. Math. Anal. Appl. 334(2), 998–1021 (2007). (preprint, arxiv:math.CA/06.05.608, 23.05.2006) 12. Chernyavskaya, N.A., Shuster, L.: Integral inequalities for the principal fundamental system of solutions of a homogeneous Sturm–Liouville equation and their applications. Bolletino U.M.I. 5(9), no. 2, 423–448 (2012) 13. Molchanov, A.M.: Discrete spectrum conditions for self-adjoint differential equations of the second order. Trydy Mosk. Mat. Obschestva 2(2), 169–199 (1953) (Russian) 14. Mynbaev, K., Otelbaev, M.: Weighted Function Spaces and the Spectrum of Differential Operators. Nauka, Moscow (1988) 15. Oinarov, R.: Properties of the Sturm–Liouville operator in L p . Izvestiya Akad Nauk Kazakh. SSR 1, 43–47 (1990) 16. Otelbaev, M.: A criterion for the resolvent of a Sturm–Liouville operator to be a kernel. Math Notes 25, 296–297 (1979) (translation of Mat. Zametki) 17. Sapenov, M., Shuster, L.: Estimates for Green’s function and a theorem on separability of a Sturm– Liouville operator in L p , Manuscript deposited at VINITI, No. 8257–B85 18. Shuster, L.: A priori estimates for the solutions of the Sturm–Liouville equation and A. M. Molchanov’s criterion. Math. Notes 50, 746–751 (1991)
123
