GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 4, 1996, 301-314

OSCILLATORY CRITERIA FOR NONLINEAR nTH-ORDER DIFFERENTIAL EQUATIONS WITH QUASIDERIVATIVES

MIROSLAV BARTUŠEK

Abstract. Sufficient conditions are given for the existence of oscillatory proper solutions of a differential equation with quasiderivatives L_{n}y = f(t, L_{0}y, …, L_{n−1}y) under the validity of the sign condition f(t, x_{1}, …, x_{n})x_{1} ≤ 0, f(t, 0, x_{2}, …, x_{n}) = 0 on ℝ_{+} × ℝ^{n}.

1. Introduction

Consider the nth-order differential equation

L_{n}y(t) = f(t, L_{0}y, L_{1}y, …, L_{n−1}y) in 𝔻 = ℝ_{+} × ℝ^{n},  (1)

where n ≥ 2, ℝ_{+} = [0, ∞], ℝ = (−∞, ∞), L_{i}y is the ith quasiderivative of y defined as

\begin{matrix}L_{0}y(t) = \frac{y(t)}{a_{0}(t)}, L_{i}y(t) = \frac{(L_{i−1}y(t))′}{a_{i}(t)} , i = 1, 2, …, n − 1, \\ L_{n}y(t) = (L_{n−1}y(t))′ , \end{matrix}  (2)

functions a_{i} ∈ C^{∘}(ℝ_{+}) are positive, and f : 𝔻 → ℝ fulfills the local Carathéodory conditions.

Throughout the paper we assume that

f(t, x_{1}, …, x_{n}) x_{1} ≤ 0, f(t, 0, x_{2}, …, x_{n}) = 0 in 𝔻 .  (3)

Definition. A function y : [0, T) → ℝ, T ∈ (0, ∞], is called a solution of (1) if (1) is valid for almost all t ∈ [0, T). It is called noncontinuable if either T = ∞ or T < ∞, and

lim sup_{t→T} ∑_{i=0}^{n−1} |L_{i}y(t)| = ∞ .

301 Let y : [0, T) → ℝ, T ≤ ∞, be a noncontinuable solution of (1). It is said to be proper if T = ∞ and sup_{τ≤t<∞} |y(t)| > 0 for all τ ∈ ℝ_{+}. It is said to be singular of the first (second) kind if t^{∗} ∈ (0, ∞) exists such that

1991 Mathematics Subject Classification. 34C10.

Key words and phrases. Oscillatory solutions, existence criteria, differential equation of nth order.

1072-947X/96/0700-0301$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

y ≡ 0 in [t^{∗}, ∞), sup_{0≤t≤t^{∗}} ∑_{i=0}^{n−1} |L_{i}y(t)| > 0

(if T < ∞). A proper solution y is said to be oscillatory if a sequence {t_{k}}_{0}^{∞} exists such that t_{k} ∈ ℝ_{+}, lim_{k→∞} t_{k} = ∞ and y(t_{k}) = 0 holds. Otherwise, it is called nonoscillatory.

Many authors studied the problem of structure and properties of proper nonoscillatory solutions of (1) (see, e.g., [1]–[3]). But as regards proper oscillatory solutions, their existence is proved only in the cases where n ≥ 3 and a_{i} ≡ 1 (see [4]–[6]), or n = 3 (see [1]).

Definition. Equation (1) has property A if every proper solution y is oscillatory for even n and it is either oscillatory or

lim_{t→∞} L_{i}y(t) = 0 monotonically, i = 0, 1, …, n − 1,  (4)

for odd n.

Similarly to a differential equation without quasiderivatives (a_{i} ≡ 1), it is possible to use the following way to prove the existence of proper oscillatory solutions: If

1^{∘} there exists no singular solution of the 1st kind; 2^{∘} there exists no singular solution of the 2st kind; 3^{∘} (1) has Property A; 4^{∘} the initial conditions of y at zero are choosen such that (4) is not

valid,

then y is oscillatory proper.

Sufficient conditions for the validity of relations 1^{∘}, 2^{∘}, 4^{∘} can be easily obtained similarly to the case a_{i} ≡ 1 (see later). Very profound results concerning 3^{∘} are given in [7].

In our paper we generalize the results which could be obtained by this approach. Especially, we shall weaken conditions 1^{∘} and 3^{∘}.

Sometimes, we will suppose that

a_{n}(t)|x_{1}|^{λ} ≦ |f(t, x_{1}, …, x_{n})| in 𝔻,  (5)

NONLINEAR DIFFERENTIAL EQUATIONS. OSCILLATORY CRITERIA 303

where 0 < λ ≤ 1, a_{n} ∈ L_{loc}(ℝ_{+}), a_{n} ≥ 0;

∫_{0}^{∞}a_{i}(t) dt = ∞ , i = 1, 2, …, n − 1 ,  (6)
|f(t, x_{1}, …, x_{n})| ≤ h(t)ω( ∑_{i=1}^{n} |x_{i}|) in 𝔻,  (7)

where h ∈ L_{loc}(ℝ_{+}), ω ∈ C^{∘}(ℝ_{+}), ω(x) > 0 for x > 0, ∫_{0}^{∞} \frac{dt}{ω(t)} = ∞;

|f(t, x_{1}, …, x_{n})| ≤ A(t)g(|x_{1}|) in ℝ_{+} × [−ε, ε]^{n},  (8)

where ε > 0, A ∈ L_{loc}(ℝ_{+}), g ∈ C^{∘}[0, ε], g(0) = 0, g(x) > 0 for x > 0,

∫_{0}^{ε} \frac{dt}{g(t)} = ∞;

\{ \begin{matrix}let \frac{a_{1}}{a_{2}} ∈ C^{1}(ℝ_{+}) for n = 3 , \\ a_{1} ∈ C^{1}(ℝ_{+}), a_{2} ∈ C^{1}(ℝ_{+}), \frac{a_{3}}{a_{1}} ∈ C^{2}(ℝ_{+}) for n = 4 \\ and let for n > 4 an index l ∈ \{1, 2, …, n − 4\} exist \\ such that a_{l+j}, j = 1, 2, are absolutely continuous and \\ a′_{l+j}, j = 1, 2, are locally bounded from below. \end{matrix}  (9)

Notation. If b_{i} ∈ C^{∘}(I), then

I^{∘}(t) ≡ 1, I^{k}(t, b_{1}, …, b_{k}) = ∫_{0}^{t} b_{1}(s)I^{k−1}(s, b_{2}, …, b_{k}) ds, t ∈ I .

Put a_{nj+i}(t) = a_{i}(t), j ∈ {…, −1, 0, 1, … }, i ∈ {0, 1, …, n}, N = {1, 2, … }.

2. Main results

Further, we shall investigate a solution y of (1) that satisfies the initial conditions

l ∈ {0, 1, …, n − 1} , τ ∈ {−1, 1} , τL_{i}y(0) > 0, i = 0, 1, …, l , (10)
τL_{j}y(0) < 0, j = l + 1, …, n − 1,

and we shall prove that this solution is oscillatory proper under the validity of certain assumptions.

Theorem 1. Let λ ∈ (0, 1) and let (5), (7), and (9) be valid. Let

∫_{0}^{∞}a_{i+1}(τ_{i+1}) ∫_{0}^{τ_{i+1}}a_{i+2}(τ_{i+2}) ∫_{0}^{τ_{i+2}}⋯ ∫_{0}^{τ_{n−1}}a_{n}(τ_{n})[ ∫_{0}^{τ_{n}}a_{n+1}(τ_{n+1}) …

∫_{0}^{τ_{i+n−1}}a_{i+n}(τ_{i+n}) dτ_{i+n} … dτ_{n+1}]^{λ} × dτ_{n} … dτ_{i+1} = ∞ ,  (11)
i = 0, 1, …, n − 1 .

Then any solution y of (1) that fulfills the Cauchy initial conditions (10) is oscillatory proper.

Theorem 2. Let λ = 1, (5), (6), and (7) hold. Let

lim sup_{t→∞} I^{1}(a_{n−1}) ∫_{t}^{∞} \frac{I^{n−1}(s, a_{1}, … a_{n−1})}{I^{1}(s, a_{n−1})}a_{n}(s) ds > 1 .  (12)

Further, let either (9) or (8) hold.

Then any solution y of (1), that fulfills the Cauchy initial conditions (10) is oscillatory proper.

Theorem 3. Let (6), (7) be valid and let functions a_{n} ∈ L_{loc}(ℝ_{+}), b ∈ C^{∘}(ℝ_{+}) exist such that ∫_{0}^{∞} a_{n}(t) dt = ∞, b(0) = 0, b(x) > 0 for x > 0, b is nondecreasing, and

a_{n}(t)b(|x_{1}|) ≤ |f(t, x_{1}, …, x_{n})| in 𝔻 .

Further, let either (9) or (8) be valid. Then any solution y of (1) that fulfills (10) is oscillatory proper.

3. Proof of main results

Let us define two special types of solutions of (1) that will be encountered later.

Type I (τ): y : [0, τ) → ℝ, 0 < τ ≤ ∞ and sequences {t_{k}^{i}}, {t̄_{k}^{n−1}}, k ∈ ℕ, i ∈ {0, 1, …, n − 1} exist such that lim_{k→∞} t_{k}^{∘} = τ,

0 ≤ t_{k}^{0} < t_{k}^{n−1} ≤ t̄_{k}^{n−1} < t_{k}^{n−2} ⋯ < t_{k}^{1} < t_{k+1}^{0} ,

NONLINEAR DIFFERENTIAL EQUATIONS. OSCILLATORY CRITERIA 305

L_{i}y(t_{k}^{i}) = 0, i = 0, 1, …, n − 2, L_{n−1}y(t) = 0 for t ∈ [t_{k}^{n−1}, t̄_{k}^{n−1}], k ∈ ℕ,

L_{i}y(t)L_{0}y(t) > 0 for t ∈ (t_{k}^{0}, t_{k}^{i}) , i = 0, 1, …, n − 1,
< 0 for t ∈ (t_{k}^{i}, t_{k+1}^{0}) , i = 0, 1, …, n − 2,
< 0 for t ∈ (t̄_{k}^{n−1}, t_{k+1}^{0}) , i = n − 1, k ∈ ℕ .

If τ < ∞, then lim_{t→τ} L_{i}y(t) = 0, i = 0, 1, …, n − 1.

Type II (s ): y : ℝ_{+} → ℝ, s ∈ {0, 1, …, n − 1}, τ ∈ ℝ_{+},

L_{j}y(t)L_{s}y(t) ≥ 0 for j ∈ {0, 1, …, s}
≤ 0 for j ∈ {s + 1, …, n − 1} ,
L_{m}y(t) ≠ 0, m ∈ {0, 1, …, n − 2}, t ∈ [τ, ∞) .

Remark. Any solution y of Type I (∞) (of Type II (s)) is oscillatory proper (nonoscillatory proper). If we define y ≡ 0 on [τ, ∞), then any solution y of Type I(τ), τ < ∞ is singular of the first kind.

Lemma 1. Let J = [t_{1}, t_{2}] ⊂ ℝ_{+}, t_{1} < t_{2} and y : J → ℝ be a solution of (1).

(a) If j ∈ {1, 2, …, n}, L_{j}y(t) ≥ 0 (≤ 0) in J, then L_{j−1}y is nondecreasing (nonincreasing) in J;

(b) if j ∈ {1, 2, …, n}, L_{j}y(t) > 0 (< 0) in J, then L_{j−1} y is increasing (decreasing) in J;

(c) if L_{0}y(t) ≥ 0 (≤ 0) in J, then L_{n−1} y is nonincreasing (nondecreasing) in J.

Proof.

(a) Let L_{j}y(t) ≥ 0 in J. Then according to (2) either (L_{j−1}y(t))′ = a_{j}(t)L_{j}y(t) ≥ 0, j < n or (L_{n−1}y(t))′ = L_{n}y(t) ≥ 0 holds.

(b), (c) The proof is similar, only (3) must be used instead of (2) in (c).

Lemma 2. Let y : ℝ_{+} → ℝ be a solution of (1) which satisfies (10). Then one of the following possibilities holds:

(a) y is of Type I (∞)

(b) there exists τ ∈ (0, ∞) such that y is of Type I(τ) in [0, τ).

(c) there exists i ∈ {0, …, n − 1} such that y is of Type II (i).

Proof. First suppose that y satisfies the Cauchy initial conditions

σL_{i}y(0) > 0, i = 0, 1, …, n − 1 .  (13)

According to Lemma 1 σL_{i}y > 0, i = 0, 1, …, n − 1, in some right neighborhood of t = 0, and σL_{j}y, j = 0, 1, …, n − 2, are nondecreasing (σL_{n−1}y is nondecreasing) until σL_{j+1}y ≥ 0 (σL_{0}y ≥ 0). Thus either y is of Type II (n − 1) or numbers t^{n}, t̄^{n} exist such that

0 < t^{n} ≤ t̄^{n}, σL_{j}y(t) > 0 in [0, t̄^{n}], j ∈ {0, 1, …, n − 2},
σL_{n−1}y(t) > 0 in [0, t^{n}), σL_{n−1}y(t) ≡ 0 in [t^{n}, t̄^{n}] ,

σL_{j}y(t) > 0, σL_{n−1}y < 0 in some right neighborhood of t = t̄^{n} .

By the same procedure it can be proved that either y is of Type II (s), s ∈ {0, …, n − 2}, or numbers t^{j}, j ∈ {0, 1, …, n − 2}, exist such that

t̄^{n−1} < t^{n−2} < ⋯ < t^{0}, σL_{i}y(t^{i}) = 0, σL_{i}y > 0 in (t^{i+1}, t^{i}) ,
σL_{m}y > 0, σL_{k}y < 0 in (t^{i+1}, t^{i}],
m ∈ {0, 1, … i − 1}, k ∈ {i + 1, … n − 1},

and

σL_{i}y < 0, i ∈ {0, 1, …, n−1} in some right neighborhood of t^{0}. (14)

Thus (13) is valid in this neighborhood and the statement follows by repeating the considerations in the case (13). Note that in the case Type I(τ), τ < ∞, the relations lim_{t→τ} L_{i}y(t) = 0, i = 0, 1, …, n − 1, must be valid because y is defined in ℝ_{+}.

Further, let (10) be valid. By the use of (13), (14) we see that the same initial conditions are valid in some t^{∗}, t^{∗} ∈ [0, t^{0}], in the previous part of the proof. Thus the statement of the lemma can be proved similarly.

Remark. Let y : [0, τ) → ℝ, τ < ∞, be a noncontinuable solution. Then the statement of Lemma 2 is valid, too, if (a) is changed into

(a′) y is of Type I(τ) with the exception of lim_{t→τ} L_{i}y(t) = 0, i = 0, 1, …, n − 1, and if Type II (s) is defined only on [0, τ).

Lemma 3 ([6, Lemma 9.2]). Let c_{0} ≥ 0, t_{0} ∈ I ⊂ ℝ_{+}, h ∈ L_{loc}(I), h ≥ 0, ω ∈ C^{0}(ℝ_{+}), ω(x) > 0 for x > c_{0}, ∫_{c_{0}}^{∞} \frac{ds}{ω(s)} < ∞. Then for every continuous function x(t) : I → ℝ_{+} which satisfies

x(t) ≤ c_{0} + [ ∫_{t_{0}}^{t} h(τ)ω(x(τ)) dτ] sign(t − t_{0}), t ∈ I ,

we have

x(t) ≤ Ω^{−1}(| ∫_{t_{0}}^{t} h(τ) dτ|), t ∈ I,

where Ω^{−1} is the inverse function of Ω(s) = ∫_{c_{0}}^{s} \frac{dτ}{ω(τ)}.

Lemma 4. Let (7) hold. Then there exists no singular solution of (1) of the second kind.

The lemma can be proved analogously to Lemma 4 in [7].

Lemma 5 (see [7], Lemma 1.5 and Consequence 1.2). Let ω : (0, ∞) → ℝ_{+} be continuous, nondecreasing and h ∈ L_{loc}(ℝ_{+}), h ≥ 0, such that

∫_{0}^{∞}h(t) dt = ∞, ∫_{0}^{1} \frac{dx}{ω(x)} < ∞ .

Then the differential inequality u′ + a(t)ω(u) ≤ 0 has no proper positive solution in ℝ_{+}.

Lemma 6. Let (5) be valid and one of the following conditions hold:

(a) λ = 1, (6) and (12) hold

(b) λ ∈ (0, 1), (11) holds.

Then there exists no solution of (1) of Type II(i), i = 0, 1, …, n − 1.

Proof. (a) With respect to (6) no solution of (1) of Type II(i), i = 0, 1, …, n − 2, exists (see [3]). The fact that there exists no solution of Type II (n − 1) is proved by Chanturia [7] in the proof of Theorem 3.5.

(b) We prove indirectly that a solution of Type II(s), s ∈ {0, 1, …, n−1}, does not exist. Thus suppose, without loss of generality, that a solution of (1) y : ℝ_{+} → ℝ exists such that T ∈ ℝ_{+},

\begin{matrix}L_{i}y(t) ≥ 0 , i = 0, 1, …, s; L_{j}y(t) ≤ 0 , j = s + 1, … n − 1 , \\ L_{m}y(t) ≠ 0 , m = 0, 1, …, n − 2 , t ≥ T . \end{matrix} (15)

Then according to Lemma 1 and (3)

|L_{i}y| is nondecreasing for i ∈ {0, 1, …, n − 1} , i ≠ s ,
L_{s}y is nonincreasing in [T, ∞) .

(16)

Further, by the use of (2), (5), (15), (16)

\begin{matrix}|L_{i}y(t)| ≥ ∫_{T}^{∞}a_{i+1}(s)|L_{i+1}y(s)| ds , i = 0, 1, …, n − 2 , \\ |L_{n−1}y(t)| ≧ ∫_{T}^{∞}|L_{n}y(s)| ds ≧ ∫_{T}^{∞}a_{n}(s)|L_{0}y(s)|^{λ} ds , \\ − (L_{s}y(t))′ = a_{s+1}(t)|L_{s+1}y(t)| for s ∈ \{0, 1, …, n − 2\} , \\ − (L_{s}y(t))′ = −L_{n}y(t) ≥ a_{n}(t)(L_{0}y(t))^{λ} for s = n − 1 . \end{matrix} (17)

308 MIROSLAV BARTUŠEK

From this and (17) we have for t ∈ [T, ∞)

|L_{s+1}y(t)| ≧

≧ ∫_{T}^{t} a_{s+2}(τ_{s+2}) ∫_{T}^{τ_{s+2}}a_{s+3}(τ_{s+3})⋯ ∫_{T}^{τ_{n−2}}a_{n−1}(τ_{n−1})|L_{n−1}y(τ_{n−1})| ≧

≧ ∫_{T}^{t} a_{s+2}(τ_{i+2})⋯ ∫_{T}^{τ_{n−2}}a_{n−1}(τ_{n−1}) ∫_{T}^{τ_{n−1}}a_{n}(τ_{n}) ×

×[ ∫_{T}^{τ_{n}}a_{1}(s_{1}) ∫_{T}^{τ_{1}}⋯ ∫_{T}^{τ_{s−1}}a_{s}(τ_{s})L_{s}y(τ_{s})]^{λ} dτ_{s} … dτ_{1}dτ_{n} … dτ_{s+2} ≤

≤ Z_{s}(t, T)(L_{s}y(t))^{λ}, s = 0, 1, …, n − 2,

|L_{0}y(t)| ≥ Z_{n−1}(t, T)L_{n−1}y(t) (for s = n − 1),

where

Z_{s}(t, T) = ∫_{T}^{t} a_{s+2}(τ_{i+2})⋯ ∫_{T}^{τ_{n−1}}a_{n}(τ_{n})[ ∫_{T}^{τ_{n}}a_{1}(τ_{1}) …

⋯ ∫_{T}^{τ_{s−1}}a_{s}(τ_{s})dτ_{s} … dτ_{1}]^{λ}dτ_{n} dτ_{s+2}, s = 0, 1, …, n − 2 ,

Z_{n−1}(t, T) = ∫_{T}^{t} a_{1}(τ_{1}) ∫_{T}^{τ_{1}}a_{2}(τ_{2})⋯ ∫_{T}^{τ_{n−2}}a_{n−1}(τ_{n−1}) dτ_{n−1} … dτ_{1}

(for s = n − 1) .

It follows from (17) that

(L_{s}y(t))′ + a_{s+1}(t)Z_{s}^{β}(t, T)(L_{s}y(t))^{λ} ≤ 0, t ∈ [T, ∞),

where β = 1 for s ∈ {0, 1, …, n − 2}, β = λ for s = n − 1. As according to (11)

Z_{s}(∞, T) = Z_{s}(∞, 0) = ∞,

we get the contradiction to Lemma 5 if L_{s}y(t) > 0 in [T, ∞). Thus with respect to (17)

s = n − 1, L_{n−1}y(t) ≡ 0 on [τ, ∞), τ ∈ [T, ∞), is the last case which has to be considered. In that case, according to (17), (16),

0 = −(L_{n−1}y(t))′ ≧ a_{n}(t)(L_{0}y(t))^{λ} ,
a_{n}(t) = 0 for almost all t ∈ [τ, ∞) .

The contradiction to (11), i = n−1, proves the statement of the lemma.

Remark.

(a) The idea of the proof (b) is due to Kiguradze [5] (for the nth-order differential equation); see [7], too.

(b) In [7] sufficient conditions for equation (1) to have Property A are given. For example, (1) has Property A if (5), (6), λ = 1,

lim sup_{t→∞} \frac{I^{n−i}(t, a_{n−1}, …, a_{i})}{I^{n−i−1}(t, a_{n−1}, …, a_{i+1})} ×

× ∫_{t}^{∞} \frac{I^{n−i−1}(s, a_{n−1}, …, a_{i+1})I^{i}(s, a_{1}, …, a_{i}))}{I^{1}(s, a_{i})}a_{n}(s) ds > 1 (18)

for i = 1, 2, …, n − 1, 2|(i + n) and ∫_{0}^{∞}I^{n−1}(t, a_{n−1}, …, a_{1})a_{n}(t) dt = ∞

holds.

It is evident that if (1) has Property A then solutions of Type II (i), i = 0, 1, …, n − 1, do not exist. Condition (12) is the same as (19) for i = n−1. Assumptions of Lemma 6 are weaker see the following example. A similar situation exists for 0 < λ < 1. Moreover, in [7] an extra assumption is made in this case.

Example. Consider equation (1) with (5) where n = 6, a_{0} = a_{1} = a_{2} = a_{3} = a_{4} = 1, a_{5} = \frac{1}{t+1}, a_{6} = \frac{1}{(t+1)^{5}} . Then condition (11) is true, but (19) is not true for i = 3. Thus solutions of Type II(i), i = 0, 1, …, 5, do not exist; at the same time the above results of (5) do not guarantee Property A for (1).

Lemma 7. Let (6) hold and functions a_{n} ∈ L_{loc}(ℝ_{+}), g ∈ C^{0}(ℝ_{+}) exist such that g(0) = 0, g(x) > 0 for x > 0, g is nondecreasing, ∫_{0}^{∞} a_{n}(t) dt = ∞, and

a_{n}(t)g(|x_{1}|) ≤ |f(t, x_{1}, …, x_{n})| in 𝔻 .

Then there exists no solution of (1) of Type II (i), i = 0, 1, …, n − 1.

Proof. According to [3] and (6) no solution of Type II (i), i = 0, 1, …, n−2, exists. Let y be a solution of (1) of Type II (n − 1). Then according to Lemma 1 |L_{n−1}y| is nonincreasing and

∞ > |L_{n−1}y(∞) − L_{n−1}y(T)| = ∫_{T}^{∞}|L_{n}y(s)| ds ≧

≧ ∫_{T}^{∞}a_{n}(t)g(|L_{0}y(s)|) ds ≥ g(|L_{0}y(T)|) ∫_{T}^{∞}a_{n}(s) ds = ∞ .

The contradiction proves the lemma.

Lemma 8. Let (8) be valid. Then there exists no singular solution of (1) of the first kind.

Proof. Let on the contrary a solution y of (1) of the first kind exist. Then numbers τ, τ_{1} ∈ ℝ_{+}, τ_{1} < τ, exist such that

\begin{matrix}ϱ(τ_{1}) > 0, L_{i}y ≡ 0 on [τ, ∞), i = 0, 1, …, n − 1, \\ where ϱ(t) = ∑_{i=0}^{n−1} |L_{i}y(t)| . \end{matrix}  (19)

Then by the use of (2) and (8)

|L_{i}y(t)| ≤ ∫_{t}^{τ} a_{i+1}(s)|L_{i+1}y(s)| ds, i = 0, 1, …, n − 2 ,

|L_{n−1}y(t)| ≤ ∫_{t}^{τ} |L_{n}y(s)| ds ,

|L_{i}y(t)| ≦

≦ ∫_{t}^{τ} a_{i+1}(s_{i+1}) ∫_{s_{i+1}}^{τ} a_{i+2}⋯ ∫_{s_{n−2}}^{τ} a_{n−1}(s_{n−1}) ∫_{s_{n−1}}^{τ} |L_{n}y(s_{n})| ds_{n} … ds_{i+1} ≤

≤ [ ∏_{j=i+1}^{n−1} ∫_{τ_{1}}^{τ} a_{j}(s) ds] ∫_{t}^{τ} |L_{n}y(s)| ds, i = 0, 1, …, n − 2 ,

ϱ(t) ≤ C ∫_{t}^{τ} |L_{n}y(s)| ds ≤ C ∫_{t}^{τ} A(s)g(ϱ(s)) ds , t ∈ [τ_{1}, τ],

where

C = ∑_{i=0}^{n−2} ∏_{j=i+1}^{n−1} ∫_{τ_{1}}^{τ} a_{j}(s) ds + 1 .

Then it follows from Lemma 3 that

∫_{0}^{ϱ(τ_{1})}\frac{ds}{g(s)} ≤ C ∫_{τ_{1}}^{τ} A(s) ds < ∞,

which contradicts (8) and (19).

Lemma 9. Let y be a solution of (1) defined in ℝ_{+} that satisfies the initial conditions (10). Let (9) be valid. Then y is not of Type I(τ) for τ < ∞.

Proof. For n = 3, 4 the statement follows from [8] and [9]. Let n > 4. Let on the contrary a solution y of Type I(τ), τ < ∞ exist. It follows from the assumptions of the lemma that an interval Λ = [τ_{1}, τ], τ_{1} < τ, exists, for which we have

\frac{max_{t∈Λ} a_{e} · max_{t∈Λ} a_{e+1}}{min_{t∈Λ} a_{e} · min_{t∈Λ} a_{e+1}} ≤ \frac{5}{4} , a_{e+1}(t)a_{e+2}(t) + [a′_{e+1}(t)]_{−} ∫_{Λ} a_{e+2}(s) ds > 0 ,

a_{e+2}(t)a_{e+3}(t) + [a′_{e+2}(t)]_{−} ∫_{Λ} a_{e+3}(s) ds > 0,

(20)

where [g(t)]_{−} = min(0, g(t)).

Use the same notation as in the definition of Type I(τ). According to lim_{t→τ} L_{e}y(t) = 0 there exists k_{0} ∈ ℕ such that

|L_{e}y(t_{k_{0}}^{e+1})| > |L_{e}y(t_{k_{0}+1}^{e+1})| > 0 , t_{k_{0}}^{e+1} > τ_{1} .  (21)

Denote t_{k_{0}}^{e+1} = t_{1}, t_{k_{0}}^{e} = t_{2}, t_{k_{0}}^{e−1} = t_{3}, Λ_{1} = t_{2} − t_{1}, Λ_{2} = t_{3} − t_{2}. Then it follows from (21) and from the definition of Type I(τ) that (we choose

312 MIROSLAV BARTUŠEK

L_{e−1}(t_{2}) > 0 for simplicity)

L_{e−1}y > 0 in [t_{1}, t_{3}), L_{e−1}y(t_{3}) = 0 , L_{e−1} is increasing (decreasing) in [t_{1}, t_{2}]( in [t_{2}, t_{3}]) , L_{e}y > 0 in [t_{1}, t_{2}) , L_{e}y(t_{2}) = 0 , L_{e}y < 0 in (t_{2}, t_{3}] , L_{e} is decreasing in [t_{0}, t_{3}] , L_{e+1}y(t_{1}) = 0 , L_{e+1}y < 0 in (t_{1}, t_{3}], L_{e+1}y is decreasing in [t_{0}, t_{3}] L_{e+j}y < 0 and L_{e+j}y is decreasing in [t_{0}, t_{3}], j = 2, 3 .

(22)

From this and (21), (22)

L_{e}y(t_{1}) > |L_{e}y(t_{3})|,  (23)

L_{e+1}y(t) = ∫_{t_{1}}^{t} a_{e+2}(s)L_{e+2}y(s) ds ≥ L_{e+2}y(t) ∫_{Λ} a_{e+2}(s) ds , t ∈ [t_{1}, t_{3}],

[L_{e}y(t)]′′ = [a_{e+1}L_{e+1}y(t)]′ = a_{e+1}(t)a_{e+2}(t)L_{e+2}y(t) +

+a′_{e+1}(t)L_{e+1}y(t) ≤ a_{e+1}(t)a_{e+2}(t)L_{e+2}y(t) + +[a′_{e+2}(t)]_{−}L_{e+1}y(t) ≤ L_{e+2}y(t)[a_{e+1}(t)a_{e+2}(t) + [a′_{e+2}(t)]_{−}

− ∫_{Λ} a_{e+2}(s) ds] < 0 , t ∈ [t_{1}, t_{3}] .  (24)

Thus

L_{e}y is concave in [t_{1}, t_{3}] .  (25)

We can prove similarly that

L_{e+1}y is concave in [t_{1}, t_{3}] .  (26)

Further, by the use of (23), (25)

L_{e−1}y(t_{2}) = ∫_{t_{2}}^{t_{3}} a_{e}(s)|L_{e}y(s)| ds ≤ max_{s∈Λ} a_{e}(s)|L_{e}y(t_{3})|\frac{Λ_{2}}{2} ,

L_{e−1}y(t_{2}) ≧ L_{e−1}y(t_{2}) − L_{e−1}y(t_{1}) = ∫_{t_{1}}^{t_{2}} a_{e}(s)L_{e}y(s) ds ≥

≥ min_{s∈Λ} a_{e}(s)L_{e}y(t_{1})\frac{Λ_{1}}{2} .

Thus, according to (24)

1 ≤ \frac{|L_{e}y(t_{3})|}{L_{e}y(t_{1})} \frac{max_{s∈Λ} a_{e}(s)}{min_{s∈Λ} a_{e}(s)} \frac{Λ_{2}}{Λ_{1}} < \frac{max_{s∈Λ} a_{e}(s)}{min_{s∈Λ} a_{e}(s)} \frac{Λ_{2}}{Λ_{1}} .  (27)

According to (23), (26)

L_{e}(t_{1}) = ∫_{t_{1}}^{t_{2}} a_{e+1}(s)|L_{e+1}y(s)| ds ≦ |L_{e+1}y(t_{2})|\frac{Λ_{1}}{2} max_{s∈Λ} a_{e+1}(s) ,

|L_{e}(t_{3})| = ∫_{t_{2}}^{t_{3}} a_{e+1}(s)|l_{e+1}y(s)| ds ≧ |L_{e+1}y(t_{2})|Λ_{2} min_{s∈Λ} a_{e+1}(s) .

Thus, according to (24), (27) and (23)

1 < \frac{Λ_{1}}{2Λ_{2}} \frac{max_{s∈Λ} a_{e+1}(s)}{min_{s∈Λ} a_{e+1}(s)} ≤ \frac{1}{2} \frac{max_{s∈Λ} a_{e+1}(s) max_{s∈Λ} a_{e}(s)}{min_{s∈Λ} a_{e+1}(s) min_{s∈Λ} a_{e}(s)} ≤ \frac{5}{8}.

The contradiction proves the statement of the lemma.

Proof of Theorem 1. According to Lemmas 2, 6, and 9 y is of Type I(∞) and by the use of Lemma 4 it is proper.

Proof of Theorem 2. The statement is a consequence of Lemmas 2, 4, 6, 8, and 9.

Proof of Theorem 3. It follows from Lemmas 4, 8, and 9 that y is proper and according to Lemma 7 it is of Type I(∞).

Acknowledgements

M. Bartušek was partially supported by grant 201/93/0452 of the Czech Grant Agency.

References

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(Received 20.11.1994)

Author’s address: Faculty of Sciences Masaryk University Janačkovo nám. 2a, 662 95 Brno Czech Republic