GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 3, 1995, 241-258

KNESER-TYPE OSCILLATION CRITERIA FOR SELF-ADJOINT TWO-TERM DIFFERENTIAL EQUATIONS

ONDŘEJ DOŠLÝ AND JAN OSIČKA

Abstract. It is proved that the even-order equation y^{(2n)}+p(t)y = 0 is (n, n) oscillatory at ∞ if

lim_{t→∞}(−1)^{n} log t ∫_{t}^{∞} s^{2n−1} (p(s) + \frac{μ_{2n}}{s^{2n}} ) ds < −K_{n},

where K_{n} = (−1)^{n−1} \frac{1}{2} \frac{d^{2}}{dλ^{2}} P_{2n}(λ)|_{λ= \frac{2n−1}{2}} , P (λ) = λ(λ − 1) … (λ −
2n + 1), μ_{2n} = P (\frac{2n−1}{2}).

1. INTRODUCTION

In this paper we deal with the oscillation properties of two-term differential equation of even order

y^{(2n)} + p(t)y = 0,  (1.1)

where t ∈ I = [1, ∞), p(t) ∈ C(I). The literature dealing with this problem is very voluminous; recall the monographs [1–5] and the references given therein.

If we study the oscillation properties of (1.1) from the point of view of the calculus of variations, the following definition plays an important role.

Definition 1.1. Two points t_{1}, t_{2} are said to be (n, n) conjugate relative to (1.1) if there exists a nontrivial solution of (1.1) such that y^{(i)}(t_{1}) = 0 = y^{(i)}(t_{2}), i = 0, …, n − 1.

The oscillation properties of linear equations related to this definition are studied in [3, 5], and recent references concerning this topic may be found in the survey paper [6].

241

1991 Mathematics Subject Classification. 34C10.

Key words and phrases. Kneser constant, conjugate points, Markov system, principal system.

1072-947X/95/0500-0241$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

If one is interested in factorization of the differential operator on the lefthand-side of (1.1) and similar problems, another definition of disconjugacy of (1.1) introduced by Levin and Nehari has to be considered.

Definition 1.2. Equation (1.1) is said to be disconjugate on an interval I_{0} ⊆ I whenever any nontrivial solution of (1.1) has at most (2n − 1) zeros on I_{0}. Equation (1.1) is said to be eventually disconjugate if there exists c ∈ I such that (1.1) is disconjugate on (c, ∞).

To distinquish between the oscillation properties defined by Definition 1.1 and the disconjugacy, oscillation, etc. defined by Definition 1.2, we shall refer to the latter as LN-disconjugacy, LN-oscillation and to the former as (n, n) disconjugacy, (n, n) oscillation, etc. Clearly, if (1.1) is LNdisconjugate on an interval I_{0} ⊆ I it is also (n, n)-disconjugate on this interval. In this paper the principal concern is the oscillation behavior of (1.1) in the sense of Definition 1.1, but if the function p(t) does not change sign for large t, the oscillation properties of (1.1) according to Definition 1.1 are very close to that given by Definition 1.2; for more details see [1].

Recall that Kneser-type oscillation criteria for (1.1) are criteria which compare equation (1.1) with the Euler equation

y^{(2n)} − \frac{μ_{2n}}{t^{2n}} y = 0,  (1.2)

where μ_{2n} = P_{2n}(\frac{2n−1}{2}) and

P_{2n}(x) = x(x − 1) … (x − 2n + 1) .  (1.3)

Criteria of this kind for (1.1) and a partial differential equation

(−Δ)^{n}u + p(x)u = 0,

where x = (x_{1}, …, x_{m}) ∈ ℝ^{m} and Δ denotes the Laplace operator, have been studied in [7–9], among others.

The paper is organized as follows. In the next section we summarize the properties of solutions of self-adjoint, even-order, differential equations and their relation to the linear Hamiltonian systems (LHS). The main result of this paper – the Kneser-type oscillation criterion for (1.1) – is given in Section 3. Section 4 is devoted to remarks and comments concerning the results, and the last section contains some technical lemma needed in the proofs of all the statements given in this paper.

2. PRELIMINARY RESULTS

First of all, recall the basic properties of the Euler equation (1.2). The algebraic equation P_{2n}(x) = 0 has 2n real roots x_{i} = i − 1, i = 1, …, 2n. The function y = P_{2n}(x) has exactly n local minima and (n − 1) local maxima and its graph is symmetric with respect to the line x = \frac{2n−1}{2}. The equation

P_{2n}(λ) = P_{2n} (\frac{2n − 1}{2})  (2.1)

has exactly 2n − 2 simple roots; denote them by λ_{1} < λ_{2} < ⋯ < λ_{n−1} < 2n − 1 − λ_{n−1} < 2n − 1 − λ_{n−2} < ⋯ < 2n − 1 − λ_{1} and one double root λ_{n} = \frac{2n−1}{2}. The solutions of (1.2) are of the form y_{i} = t^{λ_{i}}, i = 1, …, n − 1, y_{n} = t^{\frac{2n−1}{2}} , y_{n+1} = t^{\frac{2n−1}{2}} log t, y_{n+i+1} = t^{2n−1−λ_{i}}, i = 1, …, n − 1. Observe that these solutions form the so-called Markov system of solutions on I_{0} = (1, ∞), which means that the Wronskians

W (y_{1}, …, y_{k}) = |\begin{matrix}y_{1} & … & y_{k} \\ ⋮ & ⋮ \\ y_{1}^{(k−1)} & … & y_{k}^{(k−1)} \end{matrix}| ,

k = 1, …, 2n, are positive throughout I_{0}. Moreover, these solutions form the so-called hierarchical system of functions, i.e., y_{i} = o(y_{i+1}) as t → ∞, i = 1, …, 2n − 1.

Equation (1.1) is the special form of the self-adjoint even-order differential equation

∑_{k=0}^{n}(−1)^{k} (p_{k}(t)y^{(k)})^{(k)} = 0 ,  (2.2)

which is closely related to the linear Hamiltonian system

u′ = Au + B(t)v, v′ = C(t)u − A^{T} v,  (2.3)

where u, v : I → ℝ^{n}, A, B, C : I → ℝ^{n×n}, the superscript T stands for the transpose of the matrix indicated and the matrices B, C are symmetric, i.e., B = B^{T} , C = C^{T}. More precisely, let y be a solution of (2.2). Set u_{i} = y^{(i−1)}, i = 1, …, n, v_{n} = p_{n}y^{(n)}, v_{n−i} = −v′_{n−i+1} + p_{n−i}y^{(n−i)}, i = 1, …, n − 1, u = (u_{1}, …, u_{n})^{T}, v = (v_{1}, …, v_{n})^{T}. The n-dimensional vectors u, v are solutions of the LHS of (2.3), where

B(t) = diag{0, …, 0, p_{n}^{−1}(t)} , C(t) = diag{p_{0}(t), …, p_{n−1}(t)} ,

A = \begin{cases}1 & for j = i + 1, i = 1, …, n − 1 \\ 0 & otherwise . \end{cases}

We say that the solution y of (2.2) generates the solution (u, v) of (2.3).

Now consider the matrix analogy of (2.3)

U′ = AU + B(t)V, V′ = C(t)U − A^{T} V.  (2.4)

A self-conjugate solution (U, V) of (2.4) (i.e. U^{T} (x)V (x) ≡ V^{T} (x)U(x); alternative terminology is self-conjoined [11] or isotropic [2]; our terminology is due to [13]) is said to be principal (nonprincipal) at a point b if the matrix U is nonsingular near b and

lim_{x→b} (∫_{d}^{x} U^{−1}(s)B(s)U^{T −1}(s)ds)^{−1} = 0

(lim_{x→b} (∫_{d}^{x} U^{−1}(s)B(s)U^{T −1}(s)ds)^{−1} = M) ,

M being a nonsingular n×n matrix, for some d ∈ I which is sufficiently close to b. A principal solution of (2.4) at b is determined uniquely up to a right multiple by a nonsingular n × n matrix. Let y_{1}, …, y_{n} be solutions of (2.2) and let (u_{1}, v_{1}), …, (u_{n}, v_{n}) be the solutions of (2.3) generated by y_{1}, …, y_{n}. If the vectors u_{1}, …, u_{n}, v_{1}, …, v_{n} form the columns of the solution (U, V) of (2.4) we say that this solution is generated by the solutions y_{1}, …, y_{n} of (2.2). Solutions y_{1}, …, y_{n} of (2.2) are said to form the principal system of solutions if the solution (U, V) of the associated LHS generated by y_{1}, …, y_{n} is principal.

Using the concept of principal system of solutions of a self-adjoint even order differential equation, the following oscillation criterion was proved in [10].

Theorem A. Let y_{1}, …, y_{n} be a principal system of solutions of the equation

(r(t)y^{(n)})^{(n)} = 0  (2.5)

at b and let (U, V) be the solution of the matrix linear Hamiltonian system corresponding to (2.5) generated by y_{1}, …, y_{n}. If there exists c = (c_{1}, …, c_{n})^{T} ∈ ℝ^{n} such that for h = c_{1}y_{1} + ⋯ + c_{n}y_{n}

lim sup_{t→b} \frac{∫_{t}^{b} qh^{2}}{c^{T} (∫^{t} U^{−1}BU^{T −1})^{−1}c} < −1  (2.6)

where B = diag{0, …, 0, r^{−1}(t)}, then the equation

(−1)^{n} (r(t)y^{(n)})^{(n)} + q(t)y = 0  (2.7)

is (n, n)-oscillatory at the right end point b of the interval (a, b).

(Recall that equation (2.7) is said to be (n, n)-oscillatory at b if in any neighborhood of b there exists at least one pair of conjugate points.) In this paper we prove an oscillation criterion for (1.1) which is principally similar to the criterion for equation (2.7) given by Theorem A. In Theorem A equation (2.7) is viewed as a perturbation one-term equation (2.5), and it is proved that if the function q is sufficiently negative (i.e., (2.6) holds), then (2.7) is oscillatory. Here we apply this idea in a modified form to equation (1.1); this equation is considered as a perturbation of the two-term Euler equation (1.2). Comparing Theorem A with our criterion, here we are able to compute explicitly the term whose analog in Theorem A is the term

c^{T} (∫^{t} U^{−1}BU^{T −1})^{−1} c, hence our criterion is more practical.

3. OSCILLATION CRITERION

The key idea of the proof of the following oscillation criterion for (1.1) consists in application of the variational principle given in Lemma 5.1. In particular, for arbitrarily large t_{0} ∈ ℝ, we construct a nontrivial function y ∈ W^{2,n}(t_{0}, ∞), supp y ⊂ (t_{0}, ∞) such that

I(y; t_{0}, ∞) = ∫_{t_{0}}^{∞} [(y^{(n)})^{2} + (−1)^{n}p(t)y^{2}(t)] dt ≤ 0 .  (3.1)

Theorem 3.1. Suppose that

lim_{t→∞}(−1)^{n} log t ∫_{t}^{∞} s^{2n−1} (p(s) + \frac{μ_{2n}}{s^{2n}} ) ds < −K_{n},  (3.2)

where

K_{n} = (−1)^{n−1} \frac{1}{2} \frac{d^{2}}{dλ^{2}} P_{2n}(λ)|_{λ= \frac{2n−1}{2}} .  (3.3)

Then equation (1.1) is (n, n)-oscillatory at ∞.

Proof. Let t_{0} ∈ (1, ∞) be arbitrary and define the test function as follows

y(t) = \begin{cases}0, & t ∈ [1, t_{0}], \\ f(t), & t ∈ [t_{0}, t_{1}], \\ t^{\frac{2n−1}{2}} , & t ∈ [t_{1}, t_{2}], \\ g(t), & t ∈ [t_{2}, t_{3}], \\ 0, & t ∈ [t_{3}, ∞), \end{cases}

where f, g are the solutions of (1.2) satisfying the boundary conditions

f^{(i)}(t_{0}) = 0, f^{(i)}(t_{1}) = (t^{\frac{2n−1}{2}} )^{(i)} |_{t=t_{1}}, g^{(i)}(t_{2}) = (t^{\frac{2n−1}{2}} )^{(i)} |_{t=t_{2}},

g^{(i)}(t_{3}) = 0, i = 0, …, n − 1 .

These solutions exist uniquely in view of Lemma 5.3. The points 1 < t_{0} < t_{1} < t_{2} < t_{3} will be specified later. We shall show that I(y; t_{0}, ∞) < 0 if t_{1}, t_{2}, t_{3} are sufficiently large.

Let λ_{i}, i = 1, …, n − 1, λ_{n} = \frac{2n−1}{2} be the first n roots (ordered by size) of the equation (2.1). Denote y_{i} = t^{λ_{i}}, i = 1, …, n − 1, y_{n} = t^{\frac{2n−1}{2}} , and

U = (\begin{matrix}y_{1} & … & y_{n} \\ ⋮ & & ⋮ \\ y_{1}^{(n−1)} & … & y_{n}^{(n−1)} \end{matrix}) ,

V = (\begin{matrix}(−1)^{n−1}y_{1}^{(2n−1)} & … & (−1)^{n−1}y_{n}^{(2n−1)} \\ ⋮ & & ⋮ \\ −y_{1}^{(n+1)} & … & −y_{n}^{(n+1)} \\ y_{1}^{(n)} & … & y_{n}^{(n)} \end{matrix}) .

Then by Lemma 5.6 y_{1}, …, y_{n} form the principal system of solutions of (1.2) at ∞, and (U, V) is the principal solution of the LHS of the associated matrix LHS. By Lemma 5.2

\begin{aligned}u_{1}(t) & = U(t) ∫_{t_{0}}^{t} U^{−1}BU^{T −1}ds (∫_{t_{0}}^{t_{1}} U^{−1}BU^{T −1}ds)^{−1} e_{n}, \\ v_{1}(t) & = (V (t) ∫_{t_{0}}^{t} U^{−1}BU^{T −1}ds + U^{T −1}(t)) (∫_{t_{0}}^{t_{1}} U^{−1}BU^{T −1}ds)^{−1} e_{n}, \\ u_{2}(t) & = U(t) ∫_{t}^{t_{3}} U^{−1}BU^{T −1}ds (∫_{t_{2}}^{t_{3}} U^{−1}BU^{T −1}ds)^{−1} e_{n}, \\ v_{2}(t) & = (V (t) ∫_{t}^{t_{3}} U^{−1}BU^{T −1}ds − U^{T −1}(t)) (∫_{t_{2}}^{t_{3}} U^{−1}BU^{T −1}ds)^{−1} e_{n}, \end{aligned} (3.4)

where e_{n} = (0, …, 0, 1)^{T} ∈ ℝ^{n}, are solutions of the vector LHS corresponding to (1.2) and according to Lemma 5.2 f(t) = e_{1}^{T} u_{1}(t), g(t) = e_{1}^{T} u_{2}(t), e_{1} = (1, 0, …, 0)^{T} ∈ ℝ^{n}.

Using Lemma 5.4 we have

∫_{t_{0}}^{t_{1}} [(f^{(n)}(t))^{2} − (−1)^{n} \frac{μ_{2n}}{t^{2n}} f^{2}(t)] dt = v_{1}^{T} (t_{1})u_{1}(t_{1}) =

= e_{n}^{T} V (t_{1})U(t_{1})e_{n} + e_{n}^{T} (∫_{t_{0}}^{t_{1}} U^{−1}BU^{T −1}ds)^{−1} e_{n} and by Lemma 5.9

e_{n}^{T} (∫_{t_{0}}^{t_{1}} U^{−1}BU^{T −1}ds)^{−1} e_{n} = \frac{K_{n}}{log t_{1} + M} ,

where M is a positive real constant (its value may be computed explicitly, but it is not important and K_{n} is given by (3.3)). Similarly,

∫_{t_{1}}^{t_{2}} [(y_{n}^{(n)}(t))^{2} − (−1)^{n} \frac{μ_{2n}}{t^{2n}} y_{n}^{2}(t)] dt = e_{n}^{T} V (t_{2})U(t_{2})e_{n} − e_{n}^{T} V (t_{1})U(t_{1})e_{n} ,

∫_{t_{2}}^{t_{3}} [(g^{(n)}(t))^{2} − (−1)^{n} \frac{μ_{2n}}{t^{2n}} g^{2}(t)] dt =

= e_{n}^{T} (∫_{t_{2}}^{t_{3}} U^{−1}BU^{T −1}ds)^{−1} e_{n} − e_{n}^{T} V (t_{2})U(t_{2})e_{n} .

Computing the integrals

∫_{t_{0}}^{t_{1}} [p(t) + \frac{μ_{2n}}{t^{2n}} ] f^{2}(t) dt, ∫_{t_{2}}^{t_{3}} [p(t) + \frac{μ_{2n}}{t^{2n}} ] g^{2}(t) dt,

we proceed as follows. The function f is a solution of (1.2), hence it can be expressed in the form f = c_{1}y_{1} + ⋯ + c_{n}y_{n} + c_{n+1}y_{n+1} + ⋯ + c_{2n}y_{2n}, c_{i} ∈ ℝ, i = 1, …, 2n. It follows that

(\frac{f}{y_{n}} )′ = c_{1} (\frac{y_{1}}{y_{n}} )′ + ⋯ + c_{n−1} (\frac{y_{n−1}}{y_{n}} )′ +

+ c_{n+1} (\frac{y_{n+1}}{y_{n}} )′ + ⋯ + c_{2n} (\frac{y_{2n}}{y_{n}} )′ =

= c_{1}(λ_{1} − \frac{2n − 1}{2})t^{λ_{1}− \frac{2n+1}{2}} + ⋯ +

+ c_{n−1}(λ_{n−1} − \frac{2n − 1}{2})t^{λ_{n−1}− \frac{2n+1}{2}} +

+ c_{n+1}(log t)′ + ⋯ + c_{2n}(\frac{2n − 1}{2} − λ_{1})t^{\frac{2n−3}{2} −λ_{1}}.

Since the functions (y_{1}/y_{n})′, …, (y_{n−1}/y_{n})′, (y_{n+1}/y_{n})′, …, (y_{2n}/y_{n})′ form the Markov system of solutions of certain (2n − 1)-order linear differential equation, by Lemma 5.5 this equation is LN-disconjugate on (1, ∞). As (f/y_{n})′ has zeros of multiplicity (n − 1) both at t = t_{0} and t = t_{1}, this function does not vanish in the interval (t_{0}, t_{1}), i.e., the function f/y_{n} is increasing in this interval. By the second mean value theorem of integral calculus there exists ξ_{1} ∈ (t_{0}, t_{1}) such that

∫_{t_{0}}^{t_{1}} (p(t) + \frac{μ_{2n}}{t^{2n}} ) f^{2}(t) dt = ∫_{t_{0}}^{t_{1}} (p(t) + \frac{μ_{2n}}{t^{2n}} ) y_{n}^{2}(t) (\frac{f}{y_{n}(t)})^{2} dt =

248 ONDŘEJ DOŠLÝ AND JAN OSIČKA

= ∫_{ξ_{1}}^{t_{1}} (p(t) + \frac{μ_{2n}}{t^{2n}} ) y_{n}^{2}(t) dt.

Similarly, the function g/y_{n} is decreasing on (t_{2}, t_{3}) and there exists ξ_{2} ∈ (t_{2}, t_{3}) such that

∫_{t_{2}}^{t_{3}} (p(t) + \frac{μ_{2n}}{t^{2n}} ) g^{2}(t) dt = ∫_{t_{2}}^{t_{3}} (p(t) + \frac{μ_{2n}}{t^{2n}} ) y_{n}^{2}(t) (\frac{g(t)}{y_{n}(t)})^{2} dt =

= ∫_{t_{2}}^{ξ_{2}} (p(t) + \frac{μ_{2n}}{t^{2n}} ) y_{n}^{2}(t) dt.

Using these computations and Lemma 5.9 we get

I(y; 1, ∞) = I(y; t_{0}, t_{3}) = ∫_{t_{0}}^{t_{1}} [(f^{(n)}(t))^{2} − (−1)^{n} \frac{μ_{2n}}{t^{2n}} f^{2}(t)] dt +

+∫_{t_{1}}^{t_{2}}[(y_{n}^{(n)}(t))^{2}−(−1)^{n} \frac{μ_{2n}}{t^{2n}} y_{n}^{2}(t)] dt+∫_{t_{2}}^{t_{3}}[(g^{(n)}(t))^{2}−(−1)^{n} \frac{μ_{2n}}{t^{2n}} g^{2}(t)] dt+

+ (−1)^{n} ∫_{t_{0}}^{t_{1}} [p(t) + \frac{μ_{2n}}{t^{2n}} ] f^{2}(t) dt + (−1)^{n} ∫_{t_{1}}^{t_{2}} [p(t) + \frac{μ_{2n}}{t^{2n}} ] y_{n}^{2}(t) dt +

+ (−1)^{n} ∫_{t_{2}}^{t_{3}} [p(t) + \frac{μ_{2n}}{t^{2n}} ] g^{2}(t) dt = e_{n}^{T} (∫_{t_{0}}^{t_{1}} U^{−1}BU^{T −1}dt)^{−1} e_{n} +

+ e_{n}^{T} (∫_{t_{2}}^{t_{3}} U^{−1}BU^{T −1}dt)^{−1} e_{n} + (−1)^{n} ∫_{ξ_{1}}^{ξ_{2}} [p(t) + \frac{μ_{2n}}{t^{2n}} ] y_{n}^{2}(t) dt =

= e_{n}^{T} (∫_{t_{0}}^{t_{1}}U^{−1}BU^{T −1}dt)^{−1} e_{n} [1+(−1)^{n} \frac{∫_{ξ1}^{ξ2} (p(t) + \frac{μ_{2n}}{t^{2n}} ) y_{n}^{2}(t) dt}{e_{n}^{T} (∫_{t_{0}}^{t_{1}} U^{−1}BU^{T −1} dt)^{−1} e_{n}}+

+\frac{e_{n}^{T} (∫_{t_{2}}^{t_{3}} U^{−1}BU^{T −1} dt)^{−1} e_{n}}{e_{n}^{T} (∫_{t_{0}}^{t_{1}} U^{−1}BU^{T −1} dt)^{−1} e_{n}} ] ≤ e_{n}^{T} (∫_{t_{0}}^{t_{1}} U^{−1}BU^{T −1}dt)^{−1} e_{n} ×

×[1+(−1)^{n} \frac{∫_{ξ1}^{ξ2}(p(t)+ \frac{μ_{2n}}{t^{2n}} ) y_{n}^{2}(t) dt}{e_{n}^{T} (∫_{t_{0}}^{ξ_{1}}U^{−1}BU^{T−1} dt)^{−1} e_{n}} + \frac{e_{n}^{T} (∫_{t_{2}}^{t_{3}}U^{−1}BU^{T −1}dt)^{−1} e_{n}}{e_{n}^{T} (∫_{t_{0}}^{t_{1}}U^{−1}BU^{T −1}dt)^{−1} e_{n}} ]=

= \frac{K_{n}}{log t_{1} + M} [1 + (−1)^{n} \frac{log ξ_{1} + M}{log ξ_{1}} \frac{log ξ_{1}}{K_{n}} ∫_{ξ_{1}}^{ξ_{2}} (p(t) + \frac{μ_{2n}}{t^{2n}} ) y_{n}^{2}(t) dt+

+\frac{log t_{1} + M}{K_{n}} e_{n}^{T} (∫_{t_{2}}^{t_{3}} U^{−1}BU^{T −1} dt)^{−1} e_{n}] = \frac{K_{n}}{log t_{1} + M} \frac{log ξ_{1} + M}{log ξ_{1}} ×

× [\frac{log ξ_{1}}{log ξ_{1} + M} + (−1)^{n} \frac{log ξ_{1}}{K_{n}} ∫_{ξ_{1}}^{ξ_{2}} (p(t) + \frac{μ_{2n}}{t^{2n}} ) y_{n}^{2}(t) dt+

+\frac{log t_{1} + M}{log ξ_{1} + M} · \frac{log ξ_{1}}{K_{n}} · e_{n}^{T} (∫_{t_{2}}^{t_{3}} U^{−1}BU^{T −1}dt)^{−1} e_{n}] .

The inequality in this computation is justified by the fact that according to (3.2)

(−1)^{n} ∫_{ξ_{1}}^{ξ_{2}} [p(t) + \frac{μ_{2n}}{t^{2n}} ] y_{n}^{2}(t) dt < 0

if ξ_{1} and ξ_{2} are sufficiently large and hence

\frac{(−1)^{n} ∫_{ξ_{1}}^{ξ_{2}} [p(t) + \frac{μ_{2n}}{t^{2n}} ] y_{n}^{2}(t) dt}{e_{n}^{T} (∫_{t_{0}}^{t_{1}} U^{−1}BU^{T −1}dt)^{−1} e_{n}} ≤ \frac{(−1)^{n} ∫_{ξ_{1}}^{ξ_{2}} [p(t) + \frac{μ_{2n}}{t^{2n}} ] dt}{e_{n}^{T} (∫_{t_{0}}^{ξ_{1}} U^{−1}BU^{T −1}dt)^{−1} e_{n}}

for ξ_{1} ≤ t_{1}. Now let ϵ > 0 be such that the limit in (3.2) is less than −K_{n} − 4ϵ. Since lim_{t→∞}(K_{n} log t + M)/K_{n} log t = 1, we have

\frac{log ξ_{1}}{log ξ_{1} + M} < 1 + ϵ

if t_{0} is sufficiently large. According to (3.2) t_{2} > t_{1} can be chosen such that

\frac{log ξ_{1}}{K_{n}} ∫_{ξ_{1}}^{ξ_{2}} [p(t) + \frac{μ_{2n}}{t^{2n}} ] y_{n}^{2}(t) dt < −1 − 2ϵ,

whenever ξ_{2} > t_{2}. Finally, since lim_{t_{3}→∞} (∫_{t_{2}}^{t_{3}} U^{−1}BU^{T −1}dt)^{−1} = 0, t_{3} can be chosen such that

\frac{log t_{1} + M}{log ξ_{1} + M} · \frac{log ξ_{1}}{K_{n}} · e_{n}^{T} (∫_{t_{2}}^{t_{3}} U^{−1}BU^{T −1}ds)^{−1} e_{n} < ϵ .

Summarizing all estimates, we have

I(y; t_{0}, t_{3}) < \frac{K_{n}}{log t_{1} + M} · \frac{log ξ_{1} + M}{log ξ_{1}} (1 + ϵ − 1 − 2ϵ + ϵ) ≤ 0

and according to Lemma 5.1 equation (1.1) is (n, n)-oscillatory at ∞.

4. REMARKS

i) In [11] we studied the problem of existence of (n, n)-conjugate points in an interval (a, b) and proved the following theorem.

Theorem B. Let y_{1}, …, y_{m}, 1 ≤ m ≤ n, be solutions of (2.5) which are contained in principal systems of solutions both at a and b. If there exist c_{1}, …, c_{m} ∈ ℝ such that

lim sup_{t_{1}↓a,t_{2}↑b} ∫_{t_{1}}^{t_{2}} q(t) (c_{1}y_{1}(t) + ⋯ + c_{m}y_{m}(t))^{2} dt < 0

then equation (2.7) is (n, n)-conjugate on (a, b).

A slight modification of the proof of this theorem applies also to equation (1.1) considered as a perturbation of (1.2) for t ∈ (0, ∞). Observe that y_{n} = t^{\frac{2n−1}{2}} is the only solution (up to a multiple by a nonzero real constant) of (1.2) which is contained in the principal systems of solutions both at t = 0 and t = ∞; hence we have the following statement.

Theorem 4.1. Suppose that

lim sup_{t_{1}↓0,t_{2}↑∞}(−1)^{n} ∫_{t_{1}}^{t_{2}} t^{\frac{2n−1}{2}} [p(t) + \frac{μ_{2n}}{t^{2n}} ] dt < 0.

Then (1.1) is (n, n) conjugate on the interval (0, ∞).

(ii) Taking into consideration more general test functions which are linear combinations of the principal solutions, we have the following more general statement whose proof is analogous to that of Theorem 3.1.

Theorem 4.2. Let λ_{1}, …, λ_{n−1} be the first (ordered by size) (n−1) roots of (2.1),

h(t) = c_{1}t^{λ_{1}} + ⋯ + c_{n−1}t^{λ_{n−1}} + c_{n}t^{\frac{2n−1}{2}} ,  (4.1)

where c_{1}, …, c_{n} ∈ ℝ and l = max {j ∈ {1, …, n}, c_{j} ≠ 0}. If

lim_{t→∞}(−1)^{n}t^{2n−1−2λ_{l}} ∫_{t}^{∞} [p(s) + \frac{μ_{2n}}{s^{2n}} ] h^{2}(s) ds < −K̃_{n},

where

K̃_{n} = \frac{1}{2} (\frac{2n − 1}{2} − λ_{l}) ∏_{k=1}^{n−1}(2n − 1 − λ_{k} − λ_{l})^{2}

in the case l < n, and

lim_{t→∞}(−1)^{n} log t ∫_{t}^{∞} [p(s) + \frac{μ_{2n}}{s^{2n}} ] h^{2}(s) ds < −K_{n},

where K_{n} is the same as in Theorem 3.1 for l = n, then (1.1) is (n, n)- oscillatory at ∞.

(iii) In [1] the following LN-oscillation criterion for (1.1) has been proved.

Theorem C [1, Theorem 2.3, Theorem 2.4]. Let one of the following two conditions be satisfied:

(i)

lim_{t→∞} log t ∫_{t}^{∞} s^{k−1} [−p(s) + \frac{m_{k}^{∗}}{s^{k}} ]_{−} ds = ∞,  (4.2_{1})
∫_{1}^{∞} s^{k−1} log^{2} s [−p(s) + \frac{m_{k}^{∗}}{s^{k}} ]_{+} ds < ∞,  (4.2_{2})

(ii)

lim_{t→∞} log t ∫_{t}^{∞} s^{k−1} [−p(s) − \frac{m_{∗k}}{s^{k}} ]_{+} ds = ∞,  (4.3_{1})
∫_{1}^{∞} s^{k−1} log^{2} s [−p(s) − \frac{m_{∗k}}{s^{k}} ]_{−} ds < ∞,  (4.3_{2})

where m_{k}^{∗}, m_{∗k} are the least local maxima of the polynomials

P^{∗}(λ) = −λ(λ − 1) … (λ − k), P_{∗}(λ) = λ(λ − 1) … (λ − k + 1)

respectively, and [f(t)]_{±} = max{±f(t), 0}. Then the equation y^{(k)} + p(t)y = 0 is LN-oscillatory.

If k = 2n, Theorem 3.1 gives a sufficient condition even for the (n, n)- oscillation of (1.1) which is weaker than given by Theorem C. Indeed, for n even μ_{2n} = P_{∗} (\frac{2n−1}{2}) is the least local minimum of P_{∗}(λ) and for n odd μ_{2n} is the greatest local minimum of P_{2n}(λ) = −P^{∗}(λ), i.e., the least local maximum of P^{∗}(λ) = −P_{2n}(λ). Hence, for n even (4.2_{1}) reads

lim_{t→∞} log t ∫_{t}^{∞} s^{2n−1} [−p(s) − \frac{μ_{2n}}{s^{2n}} ]_{−} ds =

= lim_{t→∞} log t ∫_{t}^{∞} s^{2n−1} [p(s) + \frac{μ_{2n}}{s^{2n}} ]_{+} ds = ∞ .  (4.4)

On the other hand, (4.2_{2}) gives

∞>∫_{1}^{∞} s^{2n−1} log^{2} s [−p(s)− \frac{μ_{2n}}{s^{2n}} ]_{+} ds>log t ∫_{t}^{∞} s^{2n−1} [p(s)+ \frac{μ_{2n}}{s^{2n}} ]_{−} ds.

Consequently, this inequality and (4.4) give

lim_{t→∞} log t ∫_{t}^{∞} s^{2n−1} [p(s) + \frac{μ_{2n}}{s^{2n}} ] ds = −∞

which is a stronger condition than (3.2). (Under the assumption p(t)+ \frac{μ_{2n}}{t^{2n}} ≤ 0 for large t, this condition is proved to be sufficient for oscillation of (1.1) in [3].) If n is odd, we get a similar conclusion from (4.3_{1}), (4._{2}). Note also that if the function p(t) does not change sign for large t, the LN-oscillation properties and (n, n)-oscillation properties of (1.1) are essentially the same (see [1]).

(iv) Ideas similar to those used here for ordinary differential equations may be used in a modified form in order to investigate oscillation and spectral properties of the singular differential operators associated with the partial differential equation

(−Δ)^{n}u + a(x)u = 0,

where x = (x_{1}, …, x_{m}) ∈ ℝ^{m} and Δ = ∑_{i}^{m} \frac{∂^{2}}{∂x_{i}^{2}} ; cf. [7–9, 12]. We hope to explore this idea elsewhere.

5. Technical Lemmas

In this section we give some technical lemmas needed in the previous sections. We start with the fundamental variational lemma.

Lemma 5.1 ([7]). Equation (2.2) is conjugate on I_{0} = (c, d) ⊆ I if and

only if there exists a nontrivial function y ∈ W^{∘} _{n}^{2}(I_{0}) (W^{∘} _{n}^{2} is the Sobolev space of functions for which y, …, y^{(n−1)} are absolutely continuous on I_{0}, y_{n} ∈ L^{2}(I_{0}) and supp y ⊂ I_{0}) such that

I(y; c, d) = ∫_{c}^{d} [∑_{k=0}^{n} p_{k}(t)(y^{(k)}(t))^{2}] dt ≤ 0 .

Lemma 5.2 ([2]). Let (U, V) be a self-conjugate solution of (2.4) such that U is nonsingular on some subinterval I_{0} ⊆ I. Then

(U_{1}, V_{1}) = (U ∫_{d}^{t} U^{−1}BU^{T −1} dt, V ∫_{d}^{t} U^{−1}BU^{T −1} dt + U^{T −1}) , d ∈ I,

is also a self-conjugate solution of (2.3) and W = V U^{−1} is a solution of the Riccati matrix differential equation

W′ + A^{T} W + W A + W B(t)W − C(t) = 0 .

Lemma 5.3 ([2]). Let (1.1) be disconjugate on I_{0} = (c, d) ⊂ I and let t_{1}, t_{2} ∈ I_{0}, α = (α_{1}, …, α_{n}), β = (β_{1}, …, β_{n}) ∈ ℝ^{n} be arbitrary. There exists a unique solution y of (2.2) such that

y^{(i−1)}(t_{1}) = α_{i} , y^{(i−1)}(t_{2}) = β_{i} , i = 1, …, n .

Lemma 5.4 ([5]). Let y be a solution of (2.2) and (u, v) be the solution of the associated LHS of (2.3) generated by y. Then

∫_{a}^{b} [∑_{k=0}^{n} p_{k}(t) (y^{(k)}(t)^{2}] dt = u^{T} (b)v(b) − u^{T} (a)v(a) .

Now recall briefly the oscillation properties of the linear differential equation

y^{(n)} + q_{n−1}(t)y^{(n−1)} + ⋯ + q_{0}(t)y = 0 .  (5.1)

The proofs of these statements may be found in [2, Chap. III].

Lemma 5.5. Equation (5.1) is LN-disconjugate on I = (b, ∞) if and only if there exists a Markov system of solutions of (5.1) on I. This system can be found in such a way that it satisfies the additional conditions

y_{i} > 0 for large t, i = 1, …, n ,

y_{k−1} = o(y_{k}) for t → ∞ , k = 2, …, n, (5.2)

i.e., it forms a hierarchical system as t → ∞.

Lemma 5.6. Let equation (2.2) be eventually LN-disconjugate at ∞ and let y_{1}, …, y_{2n} be a Markov system of solutions of this equation satisfying (5.2) (with n replaced by 2n). Then y_{1}, …, y_{n} form a principal system of solutions of (2.2) at ∞.

Lemma 5.7. Let y_{1}, …, y_{n} ∈ C^{n}, r ∈ C^{n}, and r ≠ 0. Then

W (ry_{1}, …, ry_{n}) = r^{n}W (y_{1}, …, y_{n}) .

In particular, if y_{1} ≠ 0, we have

W (y_{1}, …, y_{n}) = y_{1}^{n}W ((y_{2}/y_{1})′, …, (y_{n}/y_{1})′) .

Lemma 5.8. Let u_{1} = t^{α_{1}}, …, u_{n} = t^{α_{n}}, α_{i} ∈ ℝ, i = 1, …, n. Then

W (y_{1}, …, y_{n}) = ∏_{1≤i<j}^{n}(α_{j} − α_{i})t^{∑_{k=1}^{n} α_{k}− \frac{n(n−1)}{2}} .  (5.3)

Proof. Using Lemma 5.7 we have

W (t^{α_{1}}, …, t^{α_{n}}) = t^{nα_{1}}W ((t^{α_{2}−α_{1}})′ , …, (t^{α_{n}−α_{1}})^{,})′ =
= t^{nα_{1}−(n−1)}(α_{2} − α_{1}) … (α_{n} − α_{1})W (t^{α_{2}−α_{1}}, …, t^{α_{n}−α_{1}})

and repeating the same argument (n − 1)-times we get (5.3).

We finish this paper with an evaluation of the constants K_{n}, K̃_{n} in Theorems 3.1 and 4.2. Let μ_{1}, …, μ_{n}, ν_{1}, …, ν_{n} ∈ ℝ, μ_{i} + ν_{j} ≠ 0, i, j = 1, …, n. Denote

D(μ_{1}, …, μ_{n}; ν_{1}, …, ν_{n}) = |\begin{matrix}(μ_{1} + ν_{1})^{−1} & … & (μ_{1} + ν_{n})^{−1} \\ ⋮ & ⋮ \\ (μ_{n} + ν_{1})^{−1} & … & (μ_{n} + ν_{n})^{−1} \end{matrix}| .

Then by a direct computation we have

D(μ_{1}, …, μ_{n}; ν_{1}, …, ν_{n}) = \frac{∏_{1≤k<l≤n}(μ_{k} − μ_{l})(ν_{k} − ν_{l})}{∏_{k,l=1}^{n}(μ_{k} + ν_{l})} .  (5.4)

Lemma 5.9. Let y_{1} =t^{λ_{1}}, …, y_{n−1} =t^{λ_{n−1}}, y_{n} =t^{\frac{2n−1}{2}} , i.e., y_{1}, …, y_{n} is the principal system of solutions of (1.2) at ∞ and let B =diag{0, …, 0, 1} ∈ ℝ^{n×n}. If U denotes the Wronski matrix of y_{1}, …, y_{n}, then

(∫ ^{t} U^{−1}BU^{T −1}ds)_{n,n}^{−1} = \frac{(−1)^{n−1} \frac{1}{2} \frac{d^{2}}{dλ^{2}} P_{2n}(λ)|_{λ= \frac{2n−1}{2}}}{log t + M}

(∫ ^{t} U^{−1}BU^{T −1}ds)_{i,i}^{−1} ∼ t^{2n−1−2λ_{i}} \frac{2n − 1 − 2λ_{i}}{4} ∏_{k=1}^{n−1}(2n − 1 − λ_{k} − λ_{i})^{2}

as t → ∞. Here the relation f(t) ∼ g(t) as t → ∞ means lim_{t→∞} \frac{f(t)}{g(t)} = 1.

Proof. We have (U^{−1}BU^{T −1})_{i,j} = (U^{−1})_{i,n}(U^{−1})_{j,n}, and using the rule for computation of the entries of the inverse matrix, we obtain

(U^{−1})_{i,n} = (−1)^{n+i} \frac{W (y_{1}, …, ŷ_{i}, …, y_{n})}{W (y_{1}, …, y_{n})},  (5.5)

where the circumflex ˆ means that the denoted component is missing, i.e., W (y_{1}, …, ŷ_{i}, …, y_{n}) = W (y_{1}, …, y_{i−1}, y_{i+1}, …, y_{n}). Substituting y_{1} = t^{λ_{1}}, …, y_{n−1} = t^{λ_{n−1}}, y_{n} = t^{\frac{2n−1}{2}} into (5.5) and using Lemma 5.8, we have

(U^{−1}BU^{T −1})_{i,j} = (−1)^{i+j} \frac{W (y_{1}, …, ŷ_{i}, …, y_{n})W (y_{1}, …, ŷ_{j}, …, y_{n})}{[W (y_{1}, …, y_{n})]^{2}} =

= (−1)^{i+j} \frac{t^{2n−2−λ_{i}−λ_{j}}}{∏_{\begin{matrix}1≤k≤n \\ k≠i \end{matrix}} |λ_{i} − λ_{k}| ∏_{\begin{matrix}1≤k≤n \\ k≠j \end{matrix}} |λ_{j} − λ_{k}|}.

Denote

A_{i} = ∏_{\begin{matrix}1≤k≤n \\ k≠i \end{matrix}} |λ_{i} − λ_{k}| .

Then

(∫ ^{t} U^{−1}BU^{T −1}ds)_{i,j} = \begin{cases}A_{n}^{−2} log t, & if i + j = 2n \\ \frac{(−1)^{i+j}t^{2n−1−λ_{i}−λ_{j}}}{A_{i}A_{j}(2n−1−λ_{i}−λ_{j})} , & if i + j < 2n \end{cases}

(since ∫^{t} U^{−1}BU^{T −1}ds → ∞ as t → ∞, for computation of its inverse the lower limit in the integral is immaterial). Further denote

a_{i} = \frac{2n − 1}{2} − λ_{i}, γ = ∑_{k=1}^{n−1} a_{k}, A = ∏_{k=1}^{n} A_{k} .

We have

D = |∫^{t} U^{−1}BU^{T −1}ds| = \frac{t^{γ}}{A^{2}} |\begin{matrix}\frac{1}{2a_{1}} & \frac{−1}{a_{1}+a_{2}} & … & \frac{(−1)^{n+1}}{a_{1}+a_{n}} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ \frac{(−1)^{n+1}}{a_{n}+a_{1}} & \frac{(−1)^{n+2}}{a_{n}+a_{2}} & … & log t \end{matrix} | .

Let

D̃(μ_{1}, …, μ_{n}; ν_{1}, …, ν_{n}) = |\begin{matrix}\frac{1}{μ_{+}ν_{1}} & \frac{−1}{μ_{1}+ν_{2}} & … & \frac{(−1)^{n+1}}{μ_{1}+ν_{n}} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ \frac{(−1)^{n+1}}{μ_{n}+ν_{1}} & \frac{(−1)^{n+2}}{μ_{n}+ν_{2}} & … & \frac{1}{μ_{n}+ν_{n}} \end{matrix} |

and let D̃(μ_{1}, …, μ̂_{i}, …, μ_{n}; ν_{1}, …, ν̂_{j}, …, ν_{n}) denote the subdeterminant of D̃(μ_{1}, …, μ_{n}; ν_{1}, …, ν_{n}), where the ith row and the jth column are supressed. Then

D̃(μ_{1}, …, μ_{n}; ν_{1}, …, ν_{n}) = D(μ_{1}, …, μ_{n}; ν_{1}, …, ν_{n}),

D̃(μ_{1}, .., μ̂_{i}, .., μ_{n}; ν_{1}, .., ν̂_{j}, .., ν_{n}) = (−1)^{i+j}D(μ_{1}, .., μ̂_{i}, .., μ_{n}; ν_{1}, .., ν̂_{j}, .., ν_{n})

and

D = \frac{t^{γ}}{A} [log t D(a_{1}, …, a_{n−1}; a_{1}, …, a_{n−1}) + M̃]

(the precise value of the constant M̃ is not important).

Now compute the entries of the matrix (∫^{t} U^{−1}BU^{T −1}ds)^{−1}.

(∫ ^{t} U^{−1}BU^{T −1}ds)_{n,n}^{−1} = \frac{1}{D} det (∫ ^{t} U^{−1}BU^{T −1}ds)_{i,j=1}^{n−1} =

256 ONDŘEJ DOŠLÝ AND JAN OSIČKA

= \frac{t^{γ} ∏_{k=1}^{n−1} A_{k}^{−2}D(a_{1}, …, a_{n−1}; a_{1}, …, a_{n−1})}{t^{γ}A^{−2}(D(a_{1}, …, a_{n−1}; a_{1}, …, a_{n−1}) log t + M̃)} = \frac{A_{n}^{2}}{log t + M} .

Further, (∫ ^{t} U^{−1}BU^{T −1}ds)_{i,n}^{−1} = (∫ ^{t} U^{−1}BU^{T −1}ds)_{n,i} =

= \frac{1}{D} t^{γ−a_{i}} \frac{A_{i}A_{n}}{A} D̃(a_{1}, …, a_{n−1}; a_{1}, …, â_{i}, …, a_{n}) =

= A_{i}A_{n}t^{λ_{i}− \frac{2n−1}{2}} (−1)^{n+i} \frac{D(a_{1}, …, a_{n−1}; a_{1}, …, â_{i}, …, a_{n})}{D(a_{1}, …, a_{n−1}; a_{1}, …, a_{n−1}) log t + M}

and for i, j < n

(∫ ^{t} U^{−1}BU^{T −1}ds)_{i,j}^{−1} = \frac{1}{D} \frac{t^{γ−a_{i}−a_{j}}A_{i}A_{j}}{A} ×

× |\begin{matrix}\frac{1}{2a_{1}} & … & \frac{(−1)^{j}}{a_{1}+a_{j−1}} & \frac{(−1)^{j+2}}{a_{1}+a_{j+1}} & … & \frac{(−1)^{n+1}}{a_{1}+a_{n}} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ \frac{(−1)^{i}}{a_{i−1}+a_{1}} & … & & & … & \frac{(−1)^{n+i−1}}{a_{i−1}+a_{n}} \\ \frac{(−1)^{i+2}}{a_{i+1}+a_{1}} & … & & & … & \frac{(−1)^{n+i+1}}{a_{i+1}+a_{n}} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ \frac{(−1)^{n+1}}{a_{n}+a_{1}} & … & \frac{(−1)^{n+j−1}}{a_{n}+a_{j−1}} & \frac{(−1)^{n+j+1}}{a_{n}+a_{j+1}} & … & log t \end{matrix} | =

= (−1)^{i+j}A_{i}A_{j}t^{λ_{i}+λ_{j}−2n+1} ×

×\frac{log t D(a_{1}, …, â_{i}, …, a_{n−1}; a_{1}, …, â_{j}, …, a_{n−1}) + L}{log t D(a_{1}, …, a_{n−1}; a_{1}, …, a_{n−1}) + M} .

For computation of the diagonal entries, using (5.4) we have

A_{i}^{2} \frac{D(a_{1}, …, â_{i}, …, a_{n−1}; a_{1}, …, â_{i}, …, a_{n})}{D(a_{1}, …, a_{n−1}; a_{1}, …, a_{n−1})} = \frac{∏_{k=1}^{n−1}(a_{k} + a_{i})^{2}A_{i}^{2}}{2a_{i} ∏_{\begin{matrix}1≤k≤n−1 \\ k≠i \end{matrix}}(a_{k} − a_{i})^{2}} =

= \frac{1}{2} \frac{a_{i}^{2}}{2a_{i}} ∏_{k=1}^{n−1}(a_{k} + a_{i})^{2} = \frac{1}{2} (\frac{2n − 1}{2} − λ_{i}) ∏_{k=1}^{n−1}(2n − 1 − λ_{k} − λ_{i})^{2} .

Finally,

A_{n}^{2} = ∏_{k=1}^{n−1} |a_{n} − a_{k}|^{2} = ∏_{k=1}^{n−1} (\frac{2n − 1}{2} − λ_{k})^{2} =

= (−1)^{n−1} ∏_{k=1}^{n−1}(λ − λ_{k})(λ − 2n + 1 + λ_{k})|_{λ= \frac{2n−1}{2}} =

= (−1)^{n−1} lim_{λ→ \frac{2n−1}{2}} \frac{P_{2n}(λ) − μ_{2n}}{(λ − \frac{2n−1}{2})^{2}} = \frac{1}{2} \frac{d^{2}}{dλ^{2}} P_{2n}(λ)|_{λ= \frac{2n−1}{2}} .

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(Received 10.02.1994)

Authors’ address: Masaryk University Faculty of Science Janačkovo nám, 2a, 662 95 BRNO Czech Republic