Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 13, 1-21, http://www.math.u-szeged.hu/ejqtde/

OSCILLATION AND NONOSCILLATION OF PERTURBED HIGHER ORDER EULER-TYPE DIFFERENTIAL EQUATIONS

SIMONA FIŠNAROVÁ

Abstract. Oscillatory properties of even order self-adjoint linear differential equations in the form

∑_{k=0}^{n}(−1)^{k}ν_{k} (\frac{y^{(k)}}{t^{2n−2k−α}} )^{(k)} = (−1)^{m} (q_{m}(t)y^{(m)})^{(m)} , ν_{n} := 1,

where m ∈ {0, 1}, α ∉ {1, 3, …, 2n − 1} and ν_{0}, …, ν_{n−1}, are real constants satisfying certain conditions, are investigated. In particular, the case when q_{m}(t) = \frac{β}{t^{2n−2m−α} ln^{2} t} is studied.

1. Introduction

The aim of this paper is to investigate oscillatory behavior of the even order self adjoint differential equation

(1)  L_{ν}(y) = (−1)^{m} (q_{m}(t)y^{(m)})^{(m)} , m ∈ {0, 1, …, n − 1},

where q_{m} is a continuous function and L_{ν} is the Euler differential operator

L_{ν}(y) := ∑_{k=0}^{n}(−1)^{k}ν_{k} (\frac{y^{(k)}}{t^{2n−2k−α}} )^{(k)} , α ∉ \{1, 3, …, 2n − 1\}, ν_{n} := 1.

Moreover, we suppose that ν_{0}, …, ν_{n−1} are real constants such that the characteristic polynomial of the Euler equation

(2)  L_{ν}(y) = 0,

i.e., the polynomial

(3)  P (λ) := ∑_{k=0}^{n}(−1)^{k}ν_{k} ∏_{j=1}^{k}(λ − j + 1)(λ − 2n + j + α)

1991 Mathematics Subject Classification. 34C10. Key words and phrases. Self-adjoint differential equation, oscillation and nonoscillation criteria,

variational method, conditional oscillation.

Research supported by the grant 201/04/0580 of the Czech Grant Agency and by the grant A1163401/04 of the Grant Agency of the Academy of Sciences of the Czech Republic.

EJQTDE, 2005, No. 13, p. 1 has a double root at \frac{2n−1−α}{2} and 2n − 2 distinct real roots. This is equivalent to the following two conditions.

(4)  \{ \begin{matrix}The polynomial \\ ∑_{k=0}^{n}(−1)^{k}ν_{k} ∏_{j=1}^{k} (z − \frac{(2n+1−2j−α)^{2}}{4} ) \\ has n − 1 distinct positive roots, \end{matrix}

and

(5)  ∑_{k=0}^{n} \frac{ν_{k}}{4^{k}} ∏_{j=1}^{k}(2n + 1 − 2j − α)^{2} = 0.

We use the usual convention that the product ∏_{j=1}^{k} equals 1 when k < j.

Note that the assumptions imposed on ν_{0}, …, ν_{n−1} mean that (2) is nonoscillatory, since it has the so-called ordered system of solutions (a fundamental system of positive solutions y_{1}, …, y_{2n} satisfying y_{i} = o(y_{i+1}) as t → ∞, i = 1, …, 2n − 1)

y_{1} = t^{α_{1}}, …, y_{n−1} = t^{α_{n−1}}, y_{n} = t^{α_{0}} = t^{\frac{2n−1−α}{2}} , (6)

y_{n+1} = t^{α_{0}} ln t = t^{\frac{2n−1−α}{2}} ln t, y_{n+2} = t^{2n−1−α−α_{n−1}}, …, y_{2n} = t^{2n−1−α−α_{1}},

where α_{1}, …, α_{n−1}, α_{0}, α_{n+1}, …, α_{2n} are the roots of (3), ordered by size.

Note also that the problem of (non)oscillation of Euler differential equation (2) is treated in [15, §30, §40]. It is known that (2) is nonoscillatory if and only if its coefficients ν_{0}, …, ν_{n−1} belong to a certain closed convex subset R_{ν} of ℝ^{n} which can be described using a transformation which converts (2) into an equation with constant coefficients. For example, if n = 2 and α = 0, then R_{ν} is the set of coefficients ν_{0}, ν_{1} satisfying ν_{0} ≥ −\frac{9}{4}ν_{1} − \frac{9}{16} for ν_{1} ≥ −\frac{5}{2} and ν_{0} ≥ \frac{1}{4}(2 − ν_{1})^{2} for ν_{1} ≤ −\frac{5}{2} and (4)–(5) mean that we consider the linear part of the boundary ν_{0} + \frac{9}{4}ν_{1} + \frac{9}{16} = 0, ν_{1} > −\frac{5}{2}. Our restriction (4)–(5) on the coefficients ν_{0}, …, ν_{n−1} is forced by the method used in this paper, which is based on Polya’s factorization of the operator L_{ν}. It is an open problem as how to investigate oscillatory properties of perturbed Euler operators when assumptions (4)–(5) are not satisfied; this problem is a subject of the present investigation. We refer to [15] for a more detailed treatment of the (non)oscillation of (2).

As a consequence of the presented oscillation and nonoscillation criteria, we deal with the oscillation of (1) in case q_{m}(t) = \frac{β}{t^{2n−2m−α} ln^{2} t}, m ∈ \{0, 1\}, β ∈ ℝ. In particular, we show that, under the conditions (4) and (5), equation(1) is nonoscillatory if and only if β ≤ ν̃_{n,α} in case m = 0; it is oscillatory if β > \frac{16ν̃_{n,α}}{(2n−1−α)^{2}} and nonoscillatory

EJQTDE, 2005, No. 13, p. 2 if β < \frac{4ν̃_{n,α}}{(2n−1−α)^{2}} in case m = 1, where

(7)  ν̃_{n,α} := ∑_{l=1}^{n} \{\frac{ν_{n+1−l}}{4^{n+1−l}} ∏_{k=l}^{n}(2k − 1 − α)^{2} ∑_{k=l}^{n} \frac{1}{(2k − 1 − α)^{2}} \} .

This paper can be regarded as a continuation of some recent papers where the two-term differential equation in the form

(−1)^{n} (t^{α}y^{(n)})^{(n)} = p(t)y, α ∉ {1, 3, …, 2n − 1},

has been investigated, see [4, 5, 6, 8, 10, 11, 12, 13, 14, 16]. Namely, we extend the results of [8] and also of [7] dealing with (1) in the case where n = 2 and α = 0.

Similarly, as in the above mentioned papers, we use the methods based on the factorization of disconjugate operators, variation techniques, and the relationship between self-adjoint equations and linear Hamiltonian systems.

The paper is organized as follows. In the next sections we recall necessary definitions and some preliminary results. Our main results, the oscillation and nonoscillation criteria for (1), are contained in Section 3 and Section 4. In the last section we formulate some technical results needed in the proofs.

2. Preliminaries

Here, we present some basic results which we will apply in the next section. We will need a statement concerning factorization of formally self-adjoint differential operators. Consider the equation

(8)  L(y) := ∑_{k=0}^{n}(−1)^{k} (r_{k}(t)y^{(k)})^{(k)} = 0, r_{n}(t) > 0.

Lemma 1. ([1]) Suppose that equation (8) possesses a system of positive solutions y_{1}, …, y_{2n} such that Wronskians W (y_{1}, …, y_{k}) ≠ 0, k = 1, …, 2n, for large t. Then the operator L (given by the left-hand side of (8)) admits the factorization for large t

L(y) = \frac{(−1)^{n}}{a_{0}(t)} (\frac{1}{a_{1}(t)} (… \frac{r_{n}(t)}{a_{n}^{2}(t)} (\frac{1}{a_{n−1}(t)} … \frac{1}{a_{1}(t)} (\frac{y}{a_{0}(t)})′ …)′ …)′)′ ,

where

a_{0} = y_{1}, a_{1} = (\frac{y_{2}}{y_{1}} )′ , a_{i} = \frac{W (y_{1}, …, y_{i+1})W (y_{1}, …, y_{i−1})}{W^{2}(y_{1}, …, y_{i})}, i = 1, …, n − 1,

and a_{n} = \frac{1}{a_{0} ··· a_{n−1}} .

Using the previous result we can factor the differential operator L_{ν}. The proof of the following statement is almost the same as that of [8, Lemma 2.2].

EJQTDE, 2005, No. 13, p. 3 Lemma 2. Let α ≠ {1, 3, …, 2n − 1} and suppose that (4) and (5) hold. Then we have, for any sufficiently smooth function y,

(9) L_{ν}(y) = \frac{(−1)^{n}}{a_{0}(t)} (\frac{1}{a_{1}(t)} (… \frac{t^{α}}{a_{n}^{2}(t)} (\frac{1}{a_{n−1}(t)} … \frac{1}{a_{1}(t)} (\frac{y}{a_{0}(t)})′ …)′ …)′)′ ,

where

a_{0}(t) = t^{α_{0}}, a_{k}(t) = t^{α_{k}−α_{k−1}−1}, k = 1, …, n − 1, a_{n}(t) = t^{(n−1)−α_{n−1}},

with α_{0} = \frac{2n−1−α}{2} and α_{1} < ⋯ < α_{n−1} the first roots (ordered by their size) of the polynomial P (λ) given by (3).

Now we recall basic oscillatory properties of self-adjoint differential equations (8). These properties can be investigated within the scope of the oscillation theory of linear Hamiltonian systems (LHS)

(10)  x′ = A(t)x + B(t)u, u′ = C(t)x − A^{T} (t)u,

where A, B, C are n × n matrices with B, C symmetric. Indeed, if y is a solution of (8) and we set

x = ( y ; y′ ; ⋮ ; y^{(n−1)} ) , u = ( (−1)^{n−1}(r_{n}y^{(n)})^{(n−1)} + ⋯ + r_{1}y′ ; ⋮ ; −(r_{n}y^{(n)})′ + r_{n−1}y^{(n−1)} ; r_{n}y^{(n)} );

then (x, u) solves (10) with A, B, C given by

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

A = A_{i,j} = \{ \begin{matrix}1, if j = i + 1, i = 1, …, n − 1, \\ 0, elsewhere. \end{matrix}

In this case we say that the solution (x, u) of (10) is generated by the solution y of (8). Moreover, if y_{1}, …, y_{n} are solutions of (8) and the columns of the matrix solution (X, U) of (10) are generated by the solutions y_{1}, …, y_{n}, we say that the solution (X, U) is generated by the solutions y_{1}, …, y_{n}.

Recall that two different points t_{1}, t_{2} are said to be conjugate relative to system (10) if there exists a nontrivial solution (x, u) of this system such that x(t_{1}) = 0 = x(t_{2}). Consequently, by the above mentioned relationship between (8) and (10), these points are conjugate relative to (8) if there exists a nontrivial solution y of this equation such that y^{(i)}(t_{1}) = 0 = y^{(i)}(t_{2}), i = 0, 1, …, n − 1. System (10) (and hence also equation (8)) is said to be oscillatory if for every T ∈ ℝ there exists a pair of points

EJQTDE, 2005, No. 13, p. 4 t_{1}, t_{2} ∈ [T, ∞) which are conjugate relative to (10) (relative to (8)), in the opposite case (10) (or (8)) is said to be nonoscillatory.

We say that a conjoined basis (X, U) of (10) (i.e., a matrix solution of this system with n×n matrices X, U satisfying X^{T} (t)U(t) = U^{T} (t)X(t) and rank (X^{T} , U^{T})^{T} = n) is the principal solution of (10) if X(t) is nonsingular for large t and for any other conjoined basis (X̄, Ū) such that the (constant) matrix X^{T} Ū − U^{T} X̄ is nonsingular, lim_{t→∞} X̄^{−1}(t)X(t) = 0 holds. The last limit equals zero if and only if

(11)  lim_{t→∞} (∫ ^{t} X^{−1}(s)B(s)X^{T −1}(s) ds)^{−1} = 0

([17]). A principal solution of (10) is determined uniquely up to a right multiple by a constant nonsingular n × n matrix. If (X, U) is the principal solution, any conjoined basis (X̄, Ū) such that the matrix X^{T} Ū − U^{T} X̄ is nonsingular is said to be a nonprincipal solution of (10). Solutions y_{1}, …, y_{n} of (8) are said to form the principal (nonprincipal) system of solutions if the solution (X, U) of the associated linear Hamiltonian system generated by y_{1}, …, y_{n} is a principal (nonprincipal) solution. Note that if (8) possesses a fundamental system of positive solutions y_{1}, …, y_{2n} satisfying y_{i} = o(y_{i+1}) as t → ∞, i = 1, …, 2n − 1, (the so-called ordered system of solutions), then the “small” solutions y_{1}, …, y_{n} form the principal system of solutions of (8).

Using the relation between (8), (10) and the so-called Roundabout Theorem for linear Hamiltonian systems (see e.g. [17]), one can easily prove the following variational lemma.

Lemma 3. ([15]) Equation (8) is nonoscillatory if and only if there exists T ∈ ℝ such that

F(y; T, ∞) := ∫_{T}^{∞} [∑_{k=0}^{n} r_{k}(t)(y^{(k)}(t))^{2}] dt > 0

for any nontrivial y ∈ W^{n,2}(T, ∞) with a compact support in (T, ∞).

We will also need the following Wirtinger-type inequality.

Lemma 4. ([15]) Let y ∈ W^{1,2}(T, ∞) have a compact support in (T, ∞) and let M be a positive differentiable function such that M′(t) ≠ 0 for t ∈ [T, ∞). Then

∫_{T}^{∞} |M′(t)|y^{2} dt ≤ 4 ∫_{T}^{∞} \frac{M^{2}(t)}{|M′(t)|}y′^{2} dt.

The following statement can be proved using repeated integration by parts, similarly as in [6, Lemma 4].

EJQTDE, 2005, No. 13, p. 5 Lemma 5. Let y ∈ W_{0}^{n,2}(T, ∞) have a compact support in (T, ∞) and (4)–(5) hold. Then

∫_{T}^{∞}[t^{α} (y^{(n)})^{2}+ \frac{ν_{n−1}}{t^{2−α}} (y^{(n−1)})^{2}+ ⋯ + \frac{ν_{1}}{t^{2n−2−α}} (y′)^{2}+ \frac{ν_{0}}{t^{2n−α}} y^{2}] dt

= ∫_{T}^{∞} \frac{t^{α}}{a_{n}} \{[\frac{1}{a_{n−1}} (\frac{1}{a_{n−2}} (… \frac{1}{a_{1}} (\frac{y}{a_{0}} )′)′ …)′]′\}^{2} dt,

where a_{0}, …, a_{n} are given in Lemma 2.

We finish this section with one general oscillation criterion based on the concept of principal solutions. The proof of this statement can be found in [2]. Let us consider the equation

(12)  L(y) = M(y),

where

M(y) = ∑_{k=0}^{m}(−1)^{k}(r̃_{k}(t)y^{(k)})^{(k)}, m ∈ {1, …, n − 1}

and r̃_{j}(t) ≥ 0 for large t.

Proposition 1. Suppose (8) is nonoscillatory and y_{1}, …, y_{n} is the principal system of solutions of this equation. Equation (12) is oscillatory if there exists c = (c_{1}, …, c_{n})^{T} ∈ ℝ^{n} such that

lim sup_{t→∞} \frac{∫_{t}^{∞}[∑_{k=0}^{m} r̃_{k}(t)(h^{(k)}(s))^{2}]ds}{c^{T} (∫^{t} X^{−1}(s)B(s)X^{T}^{−1}(s)ds)^{−1} c} > 1, h := c_{1}y_{1} + ⋯ + c_{n}y_{n},

where (X, U) is the solution of the linear Hamiltonian system associated with (8) generated by y_{1}, …, y_{n}.

3. Oscillation and nonoscillation criteria for (1) in case m = 0

In this section, we deal with (1) in the case m = 0, i.e., with the equation

(13)  L_{ν}(y) = q_{0}(t)y.

We start with a nonoscillation criterion for (13).

Theorem 1. Suppose that (4)–(5) hold and ν̃_{n,α} is given by (7). If the second order equation

(14)  (tu′)′ + \frac{1}{4ν̃_{n,α}} t^{2n−1−α}q_{0}(t)u = 0,

EJQTDE, 2005, No. 13, p. 6 is nonoscillatory, then (13) is also nonoscillatory.

Proof. Let T ∈ ℝ be such that the statement of Lemma 3 holds for (14) and let y ∈ W^{n,2}(T, ∞) be any function with compact support in (T, ∞). Using Lemma 5, Wirtinger’s inequality (Lemma 4), which we apply (n − 1)-times, and Lemma 6 from the last section, we obtain

∫_{T}^{∞}[t^{α} (y^{(n)})^{2}+ \frac{ν_{n−1}}{t^{2−α}} (y^{(n−1)})^{2}+ ⋯ + \frac{ν_{1}}{t^{2n−2−α}} (y′)^{2}+ \frac{ν_{0}}{t^{2n−α}} y^{2}] dt

≥ ∏_{k=1}^{n−1} (\frac{2n − 1 − α}{2} − α_{k})^{2} ∫_{T}^{∞} t [(\frac{y}{a_{0}} )′]^{2} dt

= 4ν̃_{n,α} ∫_{T}^{∞} t [(\frac{y}{t^{\frac{2n−1−α}{2}}} )′]^{2} dt.

Hence, we have

∫_{T}^{∞}[t^{α} (y^{(n)})^{2}+ \frac{ν_{n−1}}{t^{2−α}} (y^{(n−1)})^{2}+ ⋯ + \frac{ν_{1}}{t^{2n−2−α}} (y′)^{2}+ \frac{ν_{0}}{t^{2n−α}} y^{2} − q_{0}(t)y^{2}] dt

≥ 4ν̃_{n,α} ∫_{T}^{∞} \{t [(\frac{y}{t^{\frac{2n−1−α}{2}}} )′]^{2} − \frac{1}{4ν̃_{n,α}} q_{0}(t)y^{2}\} dt

= 4ν̃_{n,α} ∫_{T}^{∞} \{t [(\frac{y}{t^{\frac{2n−1−α}{2}}} )′]^{2} − \frac{1}{4ν̃_{n,α}} t^{2n−1−α}q_{0}(t) (\frac{y}{t^{\frac{2n−1−α}{2}}} )^{2}\} dt > 0,

according to Lemma 3, since (14) is nonoscillatory (take u = y/t^{\frac{2n−1−α}{2}} ) and consequently, nonoscillation of (13) follows from this Lemma as well. □

Theorem 2. Let q_{0}(t) ≥ 0 for large t, ν̃_{n,α} is the constant given by (7), conditions (4)–(5) hold, and

(15)  ∫ ^{∞} (q_{0}(t) − \frac{ν̃_{n,α}}{t^{2n−α} ln^{2} t}) t^{2n−1−α} ln tdt = ∞.

Then (13) is oscillatory.

Proof. Let T ∈ ℝ be arbitrary, T < t_{0} < t_{1} < t_{2} < t_{3} (these values will be specified later). We show that for t_{2}, t_{3} sufficiently large, there exists a function 0 ≢ y ∈ W^{n,2}(T, ∞) with compact support in (T, ∞) and such that

F(y; T, ∞)

:= ∫_{T}^{∞}[t^{α} (y^{(n)})^{2}+ \frac{ν_{n−1}}{t^{2−α}} (y^{(n−1)})^{2}+ ⋯ + \frac{ν_{1}}{t^{2n−2−α}} (y′)^{2}+ \frac{ν_{0}}{t^{2n−α}} y^{2} − q_{0}(t)y^{2}] dt ≤ 0

EJQTDE, 2005, No. 13, p. 7 and then, nonoscillation of (1) will be a consequence of Lemma 3. We construct the function y as follows:

y(t) = \{ \begin{matrix}0, & t ≤ t_{0}, \\ f(t), & t_{0} ≤ t ≤ t_{1}, \\ h(t), & t_{1} ≤ t ≤ t_{2}, \\ g(t), & t_{2} ≤ t ≤ t_{3}, \\ 0, & t ≥ t_{3}, \end{matrix}

where

h(t) = t^{\frac{2n−1−α}{2}} \sqrt{ln t},

f ∈ C^{n}[t_{0}, t_{1}] is any function such that

f^{(j)}(t_{0}) = 0, f^{(j)}(t_{1}) = h^{(j)}(t_{1}), j = 0, …, n − 1

and g is the solution of (2) satisfying the boundary conditions

(16)  g^{(j)}(t_{2}) = h^{(j)}(t_{2}), g^{(j)}(t_{3}) = 0, j = 0, …, n − 1.

Denote

K := ∫_{t_{0}}^{t_{1}}[t^{α} (f^{(n)})^{2}+ \frac{ν_{n−1}}{t^{2−α}} (f^{(n−1)})^{2}+ ⋯ + \frac{ν_{1}}{t^{2n−2−α}} (f′)^{2}+ \frac{ν_{0}}{t^{2n−α}} f^{2} − q_{0}(t)f^{2}] dt.

By a direct computation and using Lemma 7, we have for k = 1, …, n,

h^{(k)}(t) = t^{\frac{2n−1−α}{2} −k} [\frac{1}{2^{k}} ∏_{l=n−k+1}^{n}(2l − 1 − α)\sqrt{ln t} + \frac{A_{k}}{\sqrt{ln t}} + \frac{B_{k}}{\sqrt{ln^{3} t}} + o(ln^{− \frac{3}{2}} t)]

as t → ∞, where

(17)  A_{k} = ∑_{j=1}^{k}(−1)^{j−1}\binom{k}{j}\frac{a_{j}}{2^{k−j}} ∏_{l=n−k+j+1}^{n}(2l − 1 − α),

(18)  B_{k} = ∑_{j=1}^{k}(−1)^{j−1}\binom{k}{j}\frac{b_{j}}{2^{k−j}} ∏_{l=n−k+j+1}^{n}(2l − 1 − α).

Consequently,

(h^{(k)})^{2} = t^{2n−2k−1−α} [\frac{ln t}{4^{k}} ∏_{l=n−k+1}^{n}(2l − 1 − α)^{2} + \frac{A_{k}}{2^{k−1}} ∏_{l=n−k+1}^{n}(2l − 1 − α)

+ \frac{B_{k}}{2^{k−1} ln t} ∏_{l=n−k+1}^{n}(2l − 1 − α) + \frac{A_{k}^{2}}{ln t} + 2\frac{A_{k}B_{k}}{ln^{2} t} + \frac{B_{k}^{2}}{ln^{3} t} + O (ln^{−3} t)]

as t → ∞.

EJQTDE, 2005, No. 13, p. 8 Since

∫ \frac{(h^{(k)})^{2}}{t^{2n−2k−α}} dt = \frac{1}{4^{k}} ∏_{l=n−k+1}^{n}(2l − 1 − α)^{2} ∫ \frac{ln t}{t}dt + \frac{A_{k}}{2^{k−1}} ∏_{l=n−k+1}^{n}(2l − 1 − α) ln t

+ (A_{k}^{2} + \frac{B_{k}}{2^{k−1}} ∏_{l=n−k+1}^{n}(2l − 1 − α)) ∫ \frac{dt}{t ln t} + o(1),

as t → ∞, k = 0, …, n (take A_{0} = B_{0} = 0), we obtain

∫_{t_{1}}^{t_{2}}[t^{α} (h^{(n)})^{2}+ \frac{ν_{n−1}}{t^{2−α}} (h^{(n−1)})^{2}+ ⋯ + \frac{ν_{1}}{t^{2n−2−α}} (h′)^{2}+ \frac{ν_{0}}{t^{2n−α}} h^{2}] dt

= (∑_{k=0}^{n} \frac{ν_{k}}{4^{k}} ∏_{j=1}^{k}(2n + 1 − 2j − α)^{2}) ∫_{t_{1}}^{t_{2}} \frac{ln t}{t}dt + K̃_{n,α} ln t_{2} + ν̂_{n,α} ∫_{t_{1}}^{t_{2}} \frac{dt}{t ln t}

+L_{1} + o(1)

= K̃_{n,α} ln t_{2} + ν̂_{n,α} ∫_{t_{1}}^{t_{2}} \frac{dt}{t ln t} + L_{1} + o(1),

as t_{2} → ∞, L_{1} ∈ ℝ, where we have used (5) and denoted

(19)  K̃_{n,α} := ∑_{k=1}^{n} \frac{A_{k}ν_{k}}{2^{k−1}} ∏_{l=n−k+1}^{n}(2l − 1 − α),

(20)  ν̂_{n,α} := ∑_{k=1}^{n} ν_{k} [A_{k}^{2} + \frac{B_{k}}{2^{k−1}} ∏_{l=n−k+1}^{n}(2l − 1 − α)] .

Concerning the interval [t_{2}, t_{3}], since q_{0}(t) ≥ 0 for large t, we have

∫_{t_{2}}^{t_{3}}[t^{α} (g^{(n)})^{2}+ \frac{ν_{n−1}}{t^{2−α}} (g^{(n−1)})^{2}+ ⋯ + \frac{ν_{1}}{t^{2n−2−α}} (g′)^{2}+ \frac{ν_{0}}{t^{2n−α}} g^{2} − q_{0}(t)g^{2}] dt

≤ ∫_{t_{2}}^{t_{3}}[t^{α} (g^{(n)})^{2}+ \frac{ν_{n−1}}{t^{2−α}} (g^{(n−1)})^{2}+ ⋯ + \frac{ν_{1}}{t^{2n−2−α}} (g′)^{2}+ \frac{ν_{0}}{t^{2n−α}} g^{2}] dt.

Next we use the relationship between equation (2) and corresponding LHS (10). Since g is a solution of (2), we have

x = ( g ; g′ ; ⋮ ; g^{(n−1)} ) , u = ( (−1)^{n−1}(t^{α}g^{(n)})^{(n−1)} + ⋯ + ν_{1}t^{α−2n+2}g′ ; ⋮ ; −(t^{α}g^{(n)})′ + ν_{n−1}t^{α−2}g^{(n−1)} ; t^{α}g^{(n)} )

and

B(t) = diag{0, …, 0, t^{−α}}, C(t) = diag{ν_{0}t^{α−2n}, …, ν_{n−1}t^{α−2}},

A = A_{i,j} = \{ \begin{matrix}1, if j = i + 1, i = 1, …, n − 1, \\ 0, elsewhere. \end{matrix}

Then, using conditions (16),

∫_{t_{2}}^{t_{3}}[t^{α} (g^{(n)})^{2}+ \frac{ν_{n−1}}{t^{2−α}} (g^{(n−1)})^{2}+ ⋯ + \frac{ν_{1}}{t^{2n−2−α}} (g′)^{2}+ \frac{ν_{0}}{t^{2n−α}} g^{2}] dt

= ∫_{t_{2}}^{t_{3}}[u^{T} (t)B(t)u(t) + x^{T} (t)C(t)x(t)]dt

= ∫_{t_{2}}^{t_{3}}[u^{T} (t)(x′(t) − Ax(t)) + x^{T} (t)C(t)x(t)]dt

= u^{T} (t)x(t)|_{t_{2}}^{t_{3}} + ∫_{t_{2}}^{t_{3}} x^{T} (t)[−u′(t) − A^{T} u(t) + C(t)x(t)]dt

= −u^{T} (t_{2})x(t_{2}).

Further, let (X, U) be the principal solution of the LHS associated with (2). Then (X̃, Ũ) defined by

X̄(t) = X(t) ∫_{t}^{t_{3}} X^{−1}(s)B(s)X^{T −1}(s)ds,
Ū(t) = U(t) ∫_{t}^{t_{3}} X^{−1}(s)B(s)X^{T −1}(s)ds − X^{T −1}(t)

is also a conjoined basis of this LHS, and according to (16), if we let

h̃ = (h, h′, …, h^{(n−1)})^{T} ,

we obtain

x(t) = X(t) ∫_{t}^{t_{3}} X^{−1}(s)B(s)X^{T −1}(s)ds
× (∫_{t_{2}}^{t_{3}} X^{−1}(s)B(s)X^{T −1}(s)ds)^{−1} X^{−1}(t_{2})h̃(t_{2}),
u(t) = (U(t) ∫_{t}^{t_{3}} X^{−1}(s)B(s)X^{T −1}(s)ds − X^{T −1}(t))
× (∫_{t_{2}}^{t_{3}} X^{−1}(s)B(s)X^{T −1}(s)ds)^{−1} X^{−1}(t_{2})h̃(t_{2}),

EJQTDE, 2005, No. 13, p. 10 (see e.g. [1]) and hence

−u^{T} (t_{2})x(t_{2}) = h̃^{T} (t_{2})X^{T −1}(t_{2}) (∫_{t_{2}}^{t_{3}} X^{−1}(s)B(s)X^{T −1}(s)ds)^{−1} X^{−1}(t_{2})h̃(t_{2})
−h̃^{T} (t_{2})U(t_{2})X^{−1}(t_{2})h̃(t_{2}).

Using the fact that the principal solution (X, U) is generated by

y_{1} = t^{α_{1}}, …, y_{n−1} = t^{α_{n−1}}, y_{n} = t^{\frac{2n−1−α}{2}} ,

where α_{1}, …, α_{n−1}, α_{0} = \frac{2n−1−α}{2} are the first roots (ordered by size) of (3), by a direct computation (similarly to that in [8, Theorem 3.2]), we get

h̃^{T} (t_{2})U(t_{2})X^{−1}(t_{2})h̃(t_{2}) = K̂_{n,α} ln t_{2} + L_{2} + o(1), as t_{2} → ∞,

where L_{2} is a real constant and

(21)  K̂_{n,α} := ∑_{k=1}^{n} \{\frac{ν_{k}}{2^{2k−1}} ∏_{l=n−k+1}^{n}(2l − 1 − α)^{2} ∑_{l=n−k+1}^{n} \frac{1}{2l − 1 − α}\} .

If we summarize all the above computations, we obtain

F(y; T, ∞) ≤ K + K̃_{n,α} ln t_{2} + ν̂_{n,α} ∫_{t_{1}}^{t_{2}} \frac{dt}{t ln t} + L_{1} + o(1) − ∫_{t_{1}}^{t_{2}} q_{0}(t)h^{2}(t)dt

+h̃^{T} (t_{2})X^{T −1}(t_{2}) (∫_{t_{2}}^{t_{3}} X^{−1}(s)B(s)X^{T −1}(s)ds)^{−1} X^{−1}(t_{2})h̃(t_{2})

−K̂_{n,α} ln t_{2} − L_{2} − o(1), as t_{2} → ∞.

It follows from Lemma 9 that K̃_{n,α} = K̂_{n,α} and ν̃_{n,α} = ν̂_{n,α}, and according to (15), it is possible to choose t_{2} > t_{1} so large that

ν̂_{n,α} ∫_{t_{1}}^{t_{2}} \frac{dt}{t ln t} − ∫_{t_{1}}^{t_{2}} q_{0}(t)h^{2}(t)dt ≤ −(K + L_{1} − L_{2} + 2)

and that the sum of all the terms o(1) is less than 1. Moreover, since (X, U) is the principal solution, we can choose t_{3} > t_{2} such that

h̃^{T} (t_{2})X^{T −1}(t_{2}) (∫_{t_{2}}^{t_{3}} X^{−1}(s)B(s)X^{T −1}(s)ds)^{−1} X^{−1}(t_{2})h̃(t_{2}) ≤ 1.

All together means that

F(y; T, ∞) ≤ K − (K + L_{1} − L_{2} + 2) + L_{1} + 1 + 1 − L_{2} = 0,

and hence (1) is oscillatory. □ EJQTDE, 2005, No. 13, p. 11 Corollary 1. Suppose that (4) and (5) hold. The equation

(22)  L_{ν}(y) = \frac{β}{t^{2n−α} ln^{2} t}y

is nonoscillatory if and only if β ≤ ν̃_{n,α}.

Proof. If β > ν̃_{n,α}, then

∫ ^{∞} (q_{0}(t) − \frac{ν̃_{n,α}}{t^{2n−α} ln^{2} t}) t^{2n−1−α} ln tdt = ∫ ^{∞} \frac{β − ν̃_{n,α}}{t ln t} dt = ∞

and (22) is oscillatory according to Theorem 2.

On the other hand, since the second order equation

(tu′)′ + \frac{μ}{t ln^{2} t}u = 0

is nonoscillatory for μ ≤ \frac{1}{4}, we have nonoscillation of

(tu′)′ + \frac{1}{4ν̃_{n,α}} t^{2n−1−α} \frac{β}{t^{2n−α} ln^{2} t}u = 0

for \frac{β}{4ν̃_{n,α}} ≤ \frac{1}{4}, i.e., for β ≤ ν̃_{n,α}. The nonoscillation of (22) for these ν follows from Theorem 1. □

4. Oscillation and nonoscillation criteria for (1) in case m = 1

In this section we turn our attention to the equation

(23)  L_{ν}(y) = −(q_{1}(t)y′)′.

Theorem 3. Suppose that (4) and (5) hold and 4ν̃_{n,α} > t^{2n−2−α}q_{1}(t) for large t. If the second order equation

(24) [t (1 − \frac{1}{4ν̃_{n,α}} t^{2n−2−α}q_{1}(t)) u′]′ − \frac{2n − 1 − α}{8ν̃_{n,α}}t^{\frac{2n−1−α}{2}} (q_{1}(t)t^{\frac{2n−3−α}{2}} )′ u = 0

is nonoscillatory, then equation (23) is also nonoscillatory.

Proof. Let T ∈ ℝ be such that the statement of Lemma 3 holds for (24) and let y ∈ W^{n,2}(T, ∞) with compact support in (T, ∞) be arbitrary. We show that the quadratic functional associated with (23) is positive for any nontrivial y ∈ W^{n,2}(T, ∞) with compact support in (T, ∞) by using the same argument as in the proof of

EJQTDE, 2005, No. 13, p. 12 Theorem 1 and the transformation of this functional by the substitution y = t^{\frac{2n−1−α}{2}} u (applied to the term ∫_{T}^{∞} q_{1}(t)(y′)^{2}dt) as follows

∫_{T}^{∞}[t^{α} (y^{(n)})^{2}+ \frac{ν_{n−1}}{t^{2−α}} (y^{(n−1)})^{2}+ ⋯ + \frac{ν_{1}}{t^{2n−2−α}} (y′)^{2}+ \frac{ν_{0}}{t^{2n−α}} y^{2} − q_{1}(t)(y′)^{2}] dt

≥ 4ν̃_{n,α} ∫_{T}^{∞} t [(\frac{y}{t^{\frac{2n−1−α}{2}}} )′]^{2} dt − ∫_{T}^{∞} q_{1}(t)(y′)^{2}dt

= 4ν̃_{n,α} ∫_{T}^{∞} t(u′)^{2}dt

− ∫_{T}^{∞} [q_{1}(t)t^{2n−1−α}(u′)^{2} − \frac{2n − 1 − α}{2}t^{\frac{2n−1−α}{2}} (q_{1}(t)t^{\frac{2n−3−α}{2}} )′ u^{2}] dt

= 4ν̃_{n,α} ∫_{T}^{∞} \{(t − \frac{1}{4ν̃_{n,α}} q_{1}(t)t^{2n−1−α}) (u′)^{2}

+ \frac{1}{4ν̃_{n,α}} \frac{2n − 1 − α}{2}t^{\frac{2n−1−α}{2}} (q_{1}(t)t^{\frac{2n−3−α}{2}} )′ u^{2}\} dt > 0.

□

The following oscillation criterion is based on Proposition 1 and we prove it similarly to [10, Theorem 4.1].

Theorem 4. Let q_{1}(t) ≥ 0 for large t, (4), (5) hold and

lim sup_{t→∞} ln t ∫_{t}^{∞} q_{1}(s)s^{2n−3−α}ds > \frac{16ν̃_{n,α}}{(2n − 1 − α)^{2}} .

Then (23) is oscillatory.

Proof. First, recall that equation (2) is nonoscillatory and its ordered system of solutions is

y_{1} = t^{α_{1}}, …, y_{n−1} = t^{α_{n−1}}, y_{n} = t^{α_{0}} = t^{\frac{2n−1−α}{2}} , (25)

ỹ_{1} = t^{α_{0}} ln t = t^{\frac{2n−1−α}{2}} ln t, ỹ_{2} = t^{2n−1−α−α_{n−1}}, …, ỹ_{n} = t^{2n−1−α−α_{1}}.

Let (X, U) denote the principal solution of LHS associated with (2) generated by y_{1}, …, y_{n} and let (X̃, Ũ) be the solution of this LHS generated by ỹ_{1}, …, ỹ_{n}. According to Proposition 1, which we apply to equation (23) by taking c = (0, …, 0, 1)^{T} , we have that (23) is oscillatory if

lim sup_{t→∞} \frac{\frac{(2n−1−α)^{2}}{4} ∫_{t}^{∞} q_{1}(s)s^{2n−3−α}ds}{(∫^{t} X^{−1}(s)B(s)X^{T}^{−1}(s)ds)_{n,n}^{−1}} > 1.

It remains to show that (∫^{t} X^{−1}(s)B(s)X^{T}^{−1}(s)ds)_{n,n}^{−1} ∼ \frac{4ν̃_{n,α}}{ln t} as t → ∞. (Here by

the symbol f(t) ∼ g(t) we mean lim_{t→∞} \frac{f(t)}{g(t)} = 1.) Since

(X^{−1}(t)X̃(t))′ = −X^{−1}(t) [A(t)X(t) + B(t)U(t)] X^{−1}(t)X̃(t)
+X^{−1}(t) [A(t)X̃(t) + B(t)Ũ(t)]
= X^{−1}(t)B(t)X^{T}^{−1}(t)X^{T} (t)Ũ(t) − X^{−1}(t)B(t)X^{T}^{−1}(t)U^{T} (t)X̃(t)
= X^{−1}(t)B(t)X^{T}^{−1}(t)L,

where L := X^{T} (t)Ũ(t) − U^{T} (t)X̃(t), we have

(∫ ^{t} X^{−1}(s)B(s)X^{T}^{−1}(s)ds)^{−1} = LX̃^{−1}(t)X(t).

It follows from Lemma 10 below that

(∫ ^{t} X^{−1}(s)B(s)X^{T}^{−1}(s)ds)_{n,n}^{−1} = (LX̃^{−1}(t)X(t))_{n,n}

= ∑_{j=1}^{n} L_{n,j} (X̃^{−1}(t)X(t))_{j,n} = ∑_{j=1}^{n} L_{n,j} \frac{W̃_{j,n}(t)}{W̃ (t)},

where W̃_{j,n}(t) := W (ỹ_{1}, …, ỹ_{j−1}, y_{n}, ỹ_{j+1}, …, ỹ_{n}) and W̃ (t) := W (ỹ_{1}, …, ỹ_{n}). (Here L_{i,j}, i, j = 1, …, n denote the entries of L.) Consequently,

∑_{j=1}^{n} L_{n,j} \frac{W̃_{j,n}(t)}{W̃ (t)} ∼ L_{n,l} \frac{W̃_{l,n}(t)}{W̃ (t)}, l := min \{j ∈ \{1, …, n\}, L_{n,j} ≠ 0\} ,

as t → ∞, since lim_{t→∞} \frac{W̃_{k,n}(t)}{W̃_{l,n}(t)} = 0 if l < k, by Lemma 11 below. By a direct computation (see Lemma 13), we obtain

L_{n,1} = x_{[n]}^{T}ũ_{[1]} − u_{[n]}^{T}x̃_{[1]} = 4ν̃_{n,α},

where x_{[n]}, u_{[n]} denote the n-th column of X, U respectively and x̃_{[1]}, ũ_{[1]} denote the first column of X̃, Ũ respectively. It means that we take l = 1, since ν̃_{n,α} is positive (see Lemma 6). Next, we compute the above mentioned wronskians. Using Lemma

EJQTDE, 2005, No. 13, p. 14 12, we have

W̃_{1,n}(t) = W (t^{\frac{2n−1−α}{2}} , t^{2n−1−α−α_{n−1}}, …, t^{2n−1−α−α_{1}})
= t^{\frac{n(2n−1−α)}{2}} W (1, t^{\frac{2n−1−α}{2} −α_{n−1}}, …, t^{\frac{2n−1−α}{2} −α_{1}})
= ∏_{k=1}^{n−1} (α_{0} − α_{k}) t^{\frac{n(2n−1−α)}{2}} W (t^{\frac{2n−3−α}{2} −α_{n−1}}, …, t^{\frac{2n−3−α}{2} −α_{1}}) ,

and similarly

W̃ (t) = W (t^{\frac{2n−1−α}{2}} ln t, t^{2n−1−α−α_{n−1}}, …, t^{2n−1−α−α_{1}})
= t^{\frac{n(2n−1−α)}{2}} W (ln t, t^{\frac{2n−1−α}{2} −α_{n−1}}, …, t^{\frac{2n−1−α}{2} −α_{1}})
= t^{\frac{n(2n−1−α)}{2}} \{ln t ∏_{k=1}^{n−1} (α_{0} − α_{k}) W (t^{\frac{2n−3−α}{2} −α_{n−1}}, …, t^{\frac{2n−3−α}{2} −α_{1}})

−t^{\frac{2n−1−α}{2} −α_{n−1}} ∏_{k=1}^{n−2} (α_{0} − α_{k}) W (t^{−1}, t^{\frac{2n−3−α}{2} −α_{n−2}}, …, t^{\frac{2n−3−α}{2} −α_{1}})

+ ⋮

+(−1)^{n+1}t^{\frac{2n−1−α}{2} −α_{1}} ∏_{k=2}^{n−1} (α_{0} − α_{k}) W (t^{−1}, t^{\frac{2n−3−α}{2} −α_{n−1}}, …, t^{\frac{2n−3−α}{2} −α_{2}})\}

∼ ∏_{k=1}^{n−1} (α_{0} − α_{k}) t^{\frac{n(2n−1−α)}{2}} W (t^{\frac{2n−3−α}{2} −α_{n−1}}, …, t^{\frac{2n−3−α}{2} −α_{1}}) ln t,

as t → ∞, by Lemma 11. Finally,

\frac{W̃_{1,n}(t)}{W̃ (t)} ∼ \frac{1}{ln t},

as t → ∞ and the proof is completed. □ Remark 1. By Hille’s nonoscillation criterion, the equation

(r(t)x′)′ + c(t)x = 0,

where c(t) ≥ 0 for large t and ∫^{∞} r^{−1}(t)dt = ∞, is nonoscillatory if

lim_{t→∞} (∫ ^{t} r^{−1}(s)ds) (∫_{t}^{∞} c(s)ds) < \frac{1}{4}.

EJQTDE, 2005, No. 13, p. 15 If we apply this criterion to (24), we obtain that (24) is nonoscillatory if

∫ ^{∞} t^{−1} (4ν̃_{n,α} − t^{2n−2−α}q_{1}(t))^{−1} ds = ∞, (2n−1−α) (q_{1}(t)t^{\frac{2n−3−α}{2}} )′ ≤ 0 for large t

and

lim_{t→∞} (∫ ^{t} s^{−1} (4ν̃_{n,α} − s^{2n−2−α}q_{1}(s))^{−1} ds)

× (∫_{t}^{∞} s^{\frac{2n−1−α}{2}} |(q_{1}(s)s^{\frac{2n−3−α}{2}} )′| ds) < \frac{1}{2|2n − 1 − α|}.

Corollary 2. Let (4) and (5) hold and set q_{1}(t) = \frac{β}{t^{2n−2−α} ln^{2} t}. Then (23) is oscillatory if β > \frac{16ν̃_{n,α}}{(2n−1−α)^{2}} and nonoscillatory if β < \frac{4ν̃_{n,α}}{(2n−1−α)^{2}} .

Proof. If β > \frac{16ν̃_{n,α}}{(2n−1−α)^{2}} , then lim sup_{t→∞} ln t ∫_{t}^{∞} q_{1}(s)s^{2n−3−α}ds = β > \frac{16ν̃_{n,α}}{(2n−1−α)^{2}} and (23)

is oscillatory according to Theorem 4. If 0 ≤ β < \frac{4ν̃_{n,α}}{(2n−1−α)^{2}} , the nonoscillation of (23) is a consequence of Theorem 3 in view of Remark 1. Indeed, one can verify, by a direct computation, that all the assumptions are satisfied and that

∫_{t}^{∞} s^{\frac{2n−1−α}{2}} |(q_{1}(s)s^{\frac{2n−3−α}{2}} )′| ds = \{ \begin{matrix}\frac{|2n−1−α|}{2} \frac{β}{ln t} + \frac{β}{ln^{2} t}, & if 2n − 1 − α > 0, \\ \frac{|2n−1−α|}{2} \frac{β}{ln t} − \frac{β}{ln^{2} t}, & if 2n − 1 − α < 0 \end{matrix}

and

∫ ^{t} s^{−1} (4ν̃_{n,α} − s^{2n−2−α}q_{1}(s))^{−1} ds = \frac{1}{4ν̃_{n,α}} ln t + o(ln t).

Hence,

lim_{t→∞} (∫ ^{t} s^{−1} (4ν̃_{n,α} − s^{2n−2−α}q_{1}(s))^{−1} ds) (∫_{t}^{∞} s^{\frac{2n−1−α}{2}} |(q_{1}(s)s^{\frac{2n−3−α}{2}} )′| ds)

= \frac{|2n − 1 − α|}{8ν̃_{n,α}}β < \frac{1}{2|2n − 1 − α|}.

If β < 0, then nonoscillation of (23) follows from comparing this equation with the nonoscillatory equation (2). □

We can apply Proposition 1 to equation (1) for arbitrary m ∈ {1, …, n − 1} as well. If we choose c = (0, …, 0, 1)^{T} and noting that

(y_{n}^{(m)})^{2} = \frac{1}{4^{m}} (2n − 1 − α)^{2}(2n − 3 − α)^{2} ⋯ (2n + 1 − 2m − α)^{2}t^{2n−1−2m−α},

where y_{n} = t^{\frac{2n−1−α}{2}} is the solution of (2), we get the following statement. EJQTDE, 2005, No. 13, p. 16 Theorem 5. Suppose (4) and (5) hold, q_{m}(t) ≥ 0 for large t, and

lim sup_{t→∞} ln t ∫_{t}^{∞} q(s)s^{2n−1−2m−α}ds > 4^{m+1}ν̃_{n,α} ∏_{j=1}^{m} \frac{1}{(2n + 1 − 2j − α)^{2}} .

Then (1) is oscillatory.

Corollary 3. Let (4) and (5) hold and set q_{m}(t) = \frac{β}{t^{2n−2m−α} ln^{2} t}. Then (1) is oscillatory if β > 4^{m+1}ν̃_{n,α} ∏_{j=1}^{m} \frac{1}{(2n+1−2j−α)^{2}} .

5. Technical results

Lemma 6. Let α_{k}, k = 1, …, n − 1, be the first n − 1 roots (ordered by size) of the polynomial (3) and α_{0} = \frac{2n−1−α}{2}. Then

4ν̃_{n,α} = ∏_{k=1}^{n−1} (\frac{2n − 1 − α}{2} − α_{k})^{2} ,

where ν̃_{n,α} is given by (7).

Proof. The substitution μ = λ − \frac{2n−1−α}{2} converts the polynomial P (λ) (given by (3)) into the polynomial

Q(μ) = ∑_{k=0}^{n}(−1)^{k}ν_{k} ∏_{j=1}^{k} (μ^{2} − \frac{(2n + 1 − 2j − α)^{2}}{4} ) ,

whose roots are μ = 0 (double), μ = ±β_{k}, where β_{k} = \frac{2n−1−α}{2} − α_{k}, k = 1, …, n − 1. This means, that

Q(μ) = (−1)^{n}μ^{2}(μ^{2} − β_{1}^{2}) ⋯ (μ^{2} − β_{n−1}^{2}).

Comparing the coefficients of μ^{2} in both expressions for Q(μ), we obtain the assertion of this Lemma. □

Lemma 7. ([8]) For arbitrary k ∈ ℕ,

(\sqrt{ln t})^{(k)} = \frac{(−1)^{k−1}}{t^{k}} (\frac{a_{k}}{\sqrt{ln t}} + \frac{b_{k}}{\sqrt{ln^{3} t}} + o (ln^{− \frac{3}{2}} t)) ,

where a_{k}, b_{k} are given by the recursion

a_{1} = \frac{1}{2}, a_{k+1} = ka_{k}; b_{1} = 0, b_{k+1} = kb_{k} + \frac{a_{k}}{2} ,

or explicitly by

(26)  a_{k} = a_{1} ∏_{j=1}^{k−1} j = \frac{1}{2}(k − 1)!, b_{k} = \frac{(k − 1)!}{4} ∑_{j=1}^{k−1} \frac{1}{j}, k ≥ 2.

Lemma 8. ([8]) Let a_{j}, b_{j} be given by (26) and set

A_{k,ᾱ} := ∑_{j=1}^{k}(−1)^{j−1}\binom{k}{j}\frac{a_{j}}{2^{k−j}} ∏_{l=j+1}^{k}(2l − 1 − ᾱ)

and

B_{k,ᾱ} := ∑_{j=1}^{k}(−1)^{j−1}\binom{k}{j}\frac{b_{j}}{2^{k−j}} ∏_{l=j+1}^{k}(2l − 1 − ᾱ).

Then, for arbitrary k ∈ ℕ,

(27)  A_{k,ᾱ} = \frac{1}{2^{k}} ∏_{l=1}^{k}(2l − 1 − ᾱ) ∑_{l=1}^{k} \frac{1}{2l − 1 − ᾱ}

and

(28)  \frac{1}{4^{k}} ∏_{l=1}^{k}(2l − 1 − ᾱ)^{2} ∑_{l=1}^{k} \frac{1}{(2l − 1 − ᾱ)^{2}} = A_{k,ᾱ}^{2} + \frac{B_{k,ᾱ}}{2^{k−1}} ∏_{l=1}^{k}(2l − 1 − ᾱ).

Lemma 9. Let K̃_{n,α}, K̂_{n,α} ν̃_{n,α} and ν̂_{n,α} be given by (19), (21), (7) and (20). Then

K̃_{n,α} = K̂_{n,α}

and

ν̃_{n,α} = ν̂_{n,α}.

Proof. To prove the first equality, it suffices to show that

A_{k} = \frac{1}{2^{k}} ∏_{l=n−k+1}^{n}(2l − 1 − α) ∑_{l=n−k+1}^{n} \frac{1}{2l − 1 − α}, k = 1, …, n.

This follows from the definition of A_{k} by (17) and from formula (27) of Lemma 8, where we take ᾱ = α + 2k − 2n.

Concerning the second identity, we need to show that for k = 1, …, n,

\frac{1}{4^{k}} ∏_{l=n−k+1}^{n}(2l − 1 − α)^{2} ∑_{l=n−k+1}^{n} \frac{1}{(2l − 1 − α)^{2}} = A_{k}^{2} + \frac{B_{k}}{2^{k−1}} ∏_{l=n−k+1}^{n}(2l − 1 − α),

which holds according to (28) if we let ᾱ = α + 2k − 2n. □

Lemma 10. ([1]) Let y_{1}, …, y_{n}, ỹ_{1}, …, ỹ_{n} ∈ C^{n−1} be a system of linearly independent functions and let X, X̃ be the Wronski matrices of y_{1}, …, y_{n}, and ỹ_{1}, …, ỹ_{n}, respectively. Then

[X̃^{−1}X]_{i,j} = \frac{W (ỹ_{1}, …, ỹ_{i−1}, y_{j}, ỹ_{i+1}, …, ỹ_{n})}{W (ỹ_{1}, …, ỹ_{n})}.

Lemma 11. ([1]) Let y_{1}, …, y_{m} ∈ C^{m−1} be an ordered system of functions (at ∞) and let i_{1}, …, i_{k} ∈ {1, …, m} be such that i_{1} < i_{2} < ⋯ < i_{k}, j_{1}, …, j_{k} ∈ {1, …, m}, and j_{1} < j_{2} < ⋯ < j_{k}. Then

lim_{t→∞} \frac{W (y_{i_{1}}, …, y_{i_{k}})}{W (y_{j_{1}}, …, y_{j_{k}})} = 0, k = 1, …, m,

whenever i_{1} ≤ j_{1}, …, i_{k} ≤ j_{k} and at least one of the inequalities is strict.

Lemma 12. ([1]) Let y_{1}, …, y_{m} ∈ C^{m−1}, r ∈ C^{m−1} and r ≠ 0. Then

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

Lemma 13. Let y_{1}, …, y_{n}, ỹ_{1}, …, ỹ_{n} be an ordered system of solutions of (2) given by (25) and let L := X^{T} (t)Ũ(t) − U^{T} (t)X̃(t), where (X, U) is the principal solution of LHS associated with (2) generated by y_{1}, …, y_{n} and (X̃, Ũ) is the solution of this LHS generated by ỹ_{1}, …, ỹ_{n}. Let L_{i,j}, i, j = 1, …, n be the entries of L. Then

L_{n,1} = 4ν̃_{n,α},

where 4ν̃_{n,α} is given by (7).

Proof. We have L_{n,1} = x_{[n]}^{T}ũ_{[1]} − u_{[n]}^{T}x̃_{[1]}, where x_{[n]}, u_{[n]} denote the n-th column of X, U respectively and x̃_{[1]}, ũ_{[1]} denote the first column of X̃, Ũ respectively, i.e.,

x_{[n]} = ( y_{n} ; y′_{n} ; ⋮ ; y_{n}^{(n−1)} ) , u_{[n]} = ( (−1)^{n−1}(t^{α}y_{n}^{(n)})^{(n−1)} + ⋯ + ν_{1}t^{α−2n+2}y′_{n} ; ⋮ ; −(t^{α}y_{n}^{(n)})′ + ν_{n−1}t^{α−2}y_{n}^{(n−1)} ; t^{α}y_{n}^{(n)} )

and

x̃_{[1]} = ( ỹ_{1} ; ỹ′_{1} ; ⋮ ; ỹ_{1}^{(n−1)} ) , ũ_{[1]} = ( (−1)^{n−1}(t^{α}ỹ_{1}^{(n)})^{(n−1)} + ⋯ + ν_{1}t^{α−2n+2}ỹ′_{1} ; ⋮ ; −(t^{α}ỹ_{1}^{(n)})′ + ν_{n−1}t^{α−2}ỹ_{1}^{(n−1)} ; t^{α}ỹ_{1}^{(n)} ) ,

where y_{n} = t^{\frac{2n−1−α}{2}} and ỹ_{1} = t^{\frac{2n−1−α}{2}} ln t. By a direct computation and using the fact that L is a constant matrix (so that we don’t need to take into account the terms

EJQTDE, 2005, No. 13, p. 19 with ln t), we obtain L_{n,1} = C_{1} − C_{2}, where

C_{1} = \frac{∏_{k=1}^{n−1}(2k − 1 − α)^{2}}{2^{2n−2}} + \frac{ν_{n−1} ∏_{k=2}^{n−1}(2k − 1 − α)^{2}}{2^{2n−4}} + ⋯ + \frac{ν_{2}}{2^{2}} (2n − 3 − α)^{2} + ν_{1}

+ ⋮

+ ∑_{k=2}^{n} \frac{1}{2k − 1 − α} [\frac{∏_{k=1}^{n}(2k − 1 − α)^{2}}{2^{2n−2}(3 − α)} + \frac{ν_{n−1} ∏_{k=2}^{n}(2k − 1 − α)^{2}}{2^{2n−4}(3 − α)} ]

+\frac{∏_{k=1}^{n}(2k − 1 − α)^{2}}{2^{2n−2}(1 − α)} ∑_{k=1}^{n} \frac{1}{2k − 1 − α}

and

C_{2} = \frac{1}{2n − 1 − α} [\frac{∏_{k=1}^{n}(2k − 1 − α)^{2}}{2^{2n−2}(2n − 3 − α)} + ⋯ + \frac{ν_{2}}{2^{2}} (2n − 1 − α)^{2}(2n − 3 − α)]

+ ⋮

+ ∑_{k=3}^{n} \frac{1}{2k − 1 − α} [\frac{∏_{k=1}^{n}(2k − 1 − α)^{2}}{2^{2n−2}(3 − α)} + \frac{ν_{n−1} ∏_{k=2}^{n}(2k − 1 − α)^{2}}{2^{2n−4}(3 − α)} ]

+ ∑_{k=2}^{n} \frac{1}{2k − 1 − α} \frac{∏_{k=1}^{n}(2k − 1 − α)^{2}}{2^{2n−2}(1 − α)} .

Hence,

C_{1} − C_{2} = \frac{1}{4^{n−1}} ∏_{k=1}^{n}(2k − 1 − α)^{2} ∑_{k=1}^{n} \frac{1}{(2k − 1 − α)^{2}}

+\frac{ν_{n−1}}{4^{n−2}} ∏_{k=2}^{n}(2k − 1 − α)^{2} ∑_{k=2}^{n} \frac{1}{(2k − 1 − α)^{2}}

+ ⋮

+\frac{ν_{2}}{4} (2n − 1 − α)^{2}(2n − 3 − α)^{2} (\frac{1}{(2n − 1 − α)^{2}} + \frac{1}{(2n − 3 − α)^{2}} )

+ν_{1}

= 4ν̃_{n,α}.

□

EJQTDE, 2005, No. 13, p. 20

References

[1] W. A. Coppel, Disconjugacy, Lectures Notes in Math., No. 220, Springer Verlag, Berlin-Hei-delberg 1971.

[2] O. Došlý, Oscillation criteria and the discreteness of the spectrum of self-adjoint, even order, differential operators, Proc. Roy. Soc. Edinburgh 119A (1991), 219–232.

[3] O. Došlý, Oscillation criteria for self-adjoint linear differential equations, Math. Nachr. 166 (1994), 141-153.

[4] O. Došlý, Oscillation and spectral properties of a class of singular self-adjoint differential operators, Math. Nachr. 188 (1997), 49-68.

[5] O. Došlý, Constants in oscillation theory of higher order Sturm-Liouville differential equations, Electron. J. Differ. Equ. 2002 (2002), No. 34, pp.1-12.

[6] O. Došlý, Oscillatory properties of fourth order Sturm-Liouville differential equations, Acta Univ. Palacki. Olomouc., Fac. Rer. Nat. Mathematica 41 (2002), 49-59.

[7] O. Došlý, Oscillation and spectral properties of self-adjoint even order differential operators with middle terms, E. J. Qualitative Theory Diff. Equ., Proc. 7’th Coll. Qualitative Theory of Diff. Equ., No. 7. (2004), pp.1-21.

[8] O. Došlý, S. Fišnarová, Oscillation and nonoscillation of solutions to even order self-adjoint differential equations, Electron. J. Differ. Equ. 2003 (2003), No. 115, pp.1-21.

[9] O. Došlý, J. Osička, Kneser-type oscillation criteria for self-adjoint, two terms, differential equations, Georgian J. Math. 2 (1995), 241-258.

[10] O. Došlý, J. Osička, Oscillation and nonoscillation of higher order self-adjoint differential equations, Czech. Math. J. 52 (127) (2002), 833-849.

[11] O. Došlý, J. Osička, Oscillatory properties of higher order Sturm-Liouville differential equations, Studies Univ. Žilina, Math,. Ser. 15 (2002), 25-40.

[12] F. Fiedler, Oscillation criteria for a class of 2n-order ordinary differential operators, J. Differential Equations, 42 (1982), 155-188.

[13] F. Fiedler, Oscillation criteria for a special class of 2n-order ordinary differential equations, Math. Nachr. 131 (1987), 205-218.

[14] S. Fišnarová, Oscillatory properties of fourth order self-adjoint differential equations, Arch. Math. 40 (2004), 457-469.

[15] I. M. Glazman, Direct Methods of Qualitative Analysis of Singular Differential Operators, Davey, Jerusalem 1965.

[16] E. Müller-Pfeiffer, An oscillation theorem for self-adjoint differential equations, Math. Nachr. 108 (1982), 79-92.

[17] W. T. Reid, Sturmian Theory for Ordinary Differential Equations, Springer Verlag, New York-Heidelberg-Berlin 1980.

(Received September 10, 2004)

Department of Mathematics, Masaryk University, Janáčkovo nám. 2a, CZ-662 95 Brno.

E-mail address: simona@math.muni.cz

EJQTDE, 2005, No. 13, p. 21