Fourth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 03, 1999, pp 1–11. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

ON OSCILLATORY SOLUTIONS OF THIRD ORDER

DIFFERENTIAL EQUATION WITH QUASIDERIVATIVES

Miroslav Bartušek

Abstract. This paper gives sufficient conditions under which all oscillatory solutions of a third order nonlinear differential equation with quasiderivatives vanish at infinity. Applications to third order differentials equation with a middle term are also given.

I. Introduction

Consider the differential equation

y^{[3]} = (\frac{1}{a_{2}} (\frac{1}{a_{1}} y′)′)′ = r(t) f(y)  (1)

where J = [0, T), T ⩽ ∞, r ∈ C^{∘}(J), f ∈ C^{∘}(R), R = (−∞, ∞), a_{i} ∈ C^{1}(J), i = 1, 2, a_{i} are positive on J,

r(t) > 0 on J,  f(x)x > 0 for x ≠ 0 ,  (H1)

and y^{[i]}, i = 0, 1, 2, 3, is the i-th quasiderivative of y defined by

y^{[0]} = y , y^{[i]} = \frac{1}{a_{i}(t)} (y^{[i−1]})′ , i = 1, 2, y^{[3]} = (y^{[2]})′ .  (2)

Let a function y : I → R have the continuous quasiderivatives up to the order 3 on I and let (1) hold on I. Then y is called a solution of (1). A solution y is called oscillatory if it is defined on J, sup_{τ⩽t<T} |y(t)| > 0 for an arbitrary τ ∈ J and if there

exists a sequence of its zeros tending to T. Denote by O the set of all oscillatory solutions of (1).

Due to the methods used, we will study two cases:

(\frac{a_{2}(t)}{a_{1}(t)})′ ⩽ 0 , t ∈ J ,  (H2)

Mathematics Subject Classifications: 34C10.

Key words: Oscillatory solutions, third order differential equations.

\begin{matrix}c \\ ◯ \end{matrix}2000 Southwest Texas State University and University of North Texas.

Published July 10, 2000.

and

(\frac{a_{2}(t)}{a_{1}(t)})′ ⩾ 0 , t ∈ J .  (H3)

A great effort has been exerted to the study of the asymptotic behaviour of oscillatory solutions of (1) and its special cases, see e.g. [1–6, 8, 10, 12].

If a_{2} ≡ a_{1} ≡ 1 and T = ∞, sufficient conditions are given in [1,10] for every oscillatory solution y of (1) to vanish at infinity, i.e.

lim_{t→∞} y(t) = 0 .  (3)

Theorem A ([10]). Let T = ∞, (H1) hold, a_{1} ≡ a_{2} ≡ 1, and 0 < M ⩽ r(t) on ℝ_{+}. If y ∈ O, then (3) holds.

The same problem is solved for (1) in [6].

Theorem B. Let (H1) and (H3) hold and let

0 < M ⩽ r(t) , a_{1}(t) a_{2}(t) ⩽ M_{1} < ∞ on J .  (4)

If y be an oscillatory solution of (1), then lim_{t→T_{−}} y(t) = 0.

Proof. The assertion is proved in [6] if T = ∞ and (4) holds on ℝ_{+}. But in the proof, the fact that J is infinite is not used; thus the statement holds for T < ∞ as well. □

The following example shows that (3) can not be valid.

Example 1. The differential equation

((e^{−t} y′)′)′ = 2 e^{−t} y , t ∈ ℝ_{+}

has an oscillatory solution y = sin t and (3) does not hold. Note that (H1) and (H2) are valid.

Besides (3), other asymptotic behaviour of oscillatory solutions of (1) with T = ∞ are often investigated. In [3, 4] we give sufficient conditions under which the sequences of the absolute values of all local extrema of y^{[i]}, i ∈ {0, 1, 2}, in a neighbourhood of ∞ are monotone for an oscillatory solution y of (1) in case r(t) ⩽ 0.

In this paper, the above mentioned results are extended to (1) under the hypothesis (H1). In the last paragraph, applications to the third order differential equation with a middle term are given.

We do not discuss the problem of the existence of oscillatory solutions of (1). It is solved in [8, 12], and for the case of usual derivatives (i.e., for a_{1} ≡ a_{2} ≡ 1), in the monographes [1] and [10] (for T = ∞).

The following lemma is a simple consequence of the definition of quasiderivatives and of (H1).

Lemma 1. Let (H1) hold and let y be a solution of (1) defined on I = [t_{1}, t_{2}] ⊂ J, t_{1} < t_{2}. Let y^{[−1]} ≡ y^{[2]}. If i ∈ {0, 1, 2} and y^{[i]}(t) > 0 (< 0) on I, then y^{[i−1]} is increasing (decreasing) on I.

Remark 1. Note that < and increasing (> and decreasing) can be replaced by ⩽ and nondecreasing (⩾ and non-increasing).

The following lemma describes the structure of oscillatory solutions of (1).

Lemma 2 ([2]). Let y ∈ O. Then sequences {t_{k}^{i}}, i = 0, 1, 2, k = 1, 2, … exist such that lim_{k→∞} t_{k}^{0} = T,

t_{k}^{0} < t_{k}^{1} < t_{k}^{2} < t_{k+1}^{0} , y^{[i]}(t_{k}^{i}) = 0 , i = 0, 1, 2 ,

\begin{aligned}(−1)^{j+1}y^{[j]}(t) y(t) & > 0 & on & (t_{k}^{0}, t_{k}^{j}) , \\ & < 0 & on & (t_{k}^{j}, t_{k+1}^{0}) , j = 1, 2; k = 1, 2, … \end{aligned}

Remark 2. Note that according to Lemmas 1 and 2, the sequences {|y(t_{k}^{1})|}_{1}^{∞}, {|y^{[1]}(t_{k}^{2})|}_{1}^{∞} and {|y^{[2]}(t_{k}^{0})|}_{1}^{∞} are the sequences of the absolute values of all local extrema of y, y^{[1]} and y^{[3]} on [t_{0}^{0}, T), respectively.

Sometimes it is useful to express (1) in an equivalent form.

Lemma 3. Let a_{0} ∈ C^{∘}(J) be positive. Then the transformation

x(t) = ∫_{0}^{t} a_{0}(s) ds , Y (x) = y(t) , t ∈ J, x ∈ [0, x^{∗}) , x^{∗} = x(T)

transforms (1) into

(\frac{1}{A_{2}} (\frac{1}{A_{1}} Y^{•} )^{•})^{•} = R(x) f(Y)  (5)

where A_{i}(x) = \frac{a_{i}(t(x))}{a_{0}(t(x))} , i = 1, 2 , R(x) = \frac{r(t(x))}{a_{0}(t(x))} , \frac{d}{dx} = ^{•} and t(x) is the inverse function to x(t). At the same time,

Y^{{i}}(x) = y^{[i]}(t) , i = 0, 1, 2, 3 ,  (6)

where

Y^{\{0\}} = Y , Y^{\{j\}} = \frac{1}{A_{j}(x)} (Y^{\{j−1\}})^{•} , j = 1, 2 , Y^{\{3\}} = (Y^{\{2\}})^{•} .

Proof. Use a direct computation or see [4]. □

2. Case (H2)

Some results will be used that are obtained for (1) under a different assumptions than (H1). Consider

(\frac{1}{b(σ)}Z′′)′ + r̄(σ) f(Z) = 0  (7)

where I ⊂ ℝ_{+}, b ∈ C^{1}(I), r̄ ∈ C^{1}(I), f ∈ C^{∘}(I), f(x)x > 0 for x ≠ 0,

b(σ) > 0 , r̄(σ) ⩾ 0 on I , f′(x) ⩾ 0 on R .

The quasiderivatives are given by

Z^{[0]} = Z , Z^{[1]} = Z′ , Z^{[2]} = \frac{Z′′}{b(σ)} .

Note that the sign of r̄ is opposite to the one of r.

Lemma 4. Let b′ ⩾ 0 and r̄′ ⩾ 0 on I. Let Z be a solution of (7) the second quasiderivatives Z^{[2]} of which has three consecutive zeros σ_{0}, σ_{1} and σ_{2} ∈ I, σ_{0} < σ_{1} < σ_{2}. Then

\sqrt{2}|Z′(σ_{2})| < |Z′(σ_{1})| .

Proof. The assertion is proved for T = ∞ and for an oscillatory solution in [4] (see Lemma 2.4 and Definition 2.1). But it follows from the proof that only information on [σ_{1}, σ_{2}] and the existence of the zero σ_{0} were used. Thus, the statement is valid under our assumptions as well. □

The following theorem investigates the asymptotic behaviour of the first and the second quasiderivatives of an oscillatory solution of (1).

Theorem 1. Let (H1) and (H2) hold. Let y ∈ O and {t_{k}^{i}}, i = 0, 1, 2, k = 1, 2, …, be given by Lemma 2.

(i) Then the sequence {|y^{[1]}(t_{k}^{2})|}_{1}^{∞} of the absolute values of all local extrema of y^{[1]} on [t_{1}^{0}, T) is decreasing.

(ii) Let r ∈ C^{1}(J), f ∈ C^{1}(R), f′ ⩾ 0 on R and (\frac{r(t)}{a_{1}(t)})′ ⩽ 0 on J.

Then lim_{t→T} y^{[1]}(t) = 0 and

|y^{[1]}(t_{k}^{2})| ⩽ 2^{\frac{1−k}{2}} |y^{[1]}(t_{1}^{2})| , k = 1, 2, …  (8)

(iii) Let r ∈ C^{1}(J), f ∈ C^{1}(R), f′ ⩾ 0 on R and (\frac{r(t)}{a_{2}(t)})′ ⩽ 0 on J.

Then the sequence {|y^{[2]}(t_{k}^{0})|}_{1}^{∞} of the absolute values of all local extrema of y^{[2]} on [t_{1}^{0}, T) is decreasing.

Proof. Note that according to Remark 2, the sequences {|y^{[1]}(t_{k}^{2})|}_{1}^{∞} and {|y^{[2]}(t_{k}^{0})|}_{1}^{∞} are the sequences of the absolute values of all local extrema of y^{[1]} and y^{[2]}, respectively.

(i) Let k ∈ {2, 3, … } and suppose, without loss of generality, that

y(t) > 0 on (t_{k}^{0}, t_{k+1}^{0}) .

Thus, according to Lemmas 1 and 2 there exists t_{k}^{∗} such that

t_{k}^{∗} ∈ (t_{k}^{0}, t_{k}^{1}) , y(t_{k}^{∗}) = y(t_{k}^{2}) ,
y is increasing (decreasing) on [t_{k}^{∗}, t_{k}^{1}] (on [t_{k}^{1}, t_{k}^{2}]) ,
y^{[1]}(t) > 0 (< 0) on [t_{k}^{∗}, t_{k}^{1}) (on t_{k}^{1}, t_{k}^{2}) ,
y^{[1]}(t_{k}^{1}) = 0 , y^{[1]}(t_{k−1}^{2}) > y^{[1]}(t_{k}^{∗}) > 0 ,
y^{[2]}(t) < 0 and |y^{[2]}| is decreasing on [t_{k}^{∗}, t_{k}^{2}) , y^{[2]}(t_{k}^{2}) = 0 .

(9)

Let φ and ψ be the inverse functions to y:

t_{k}^{∗} ⩽ φ(v) ⩽ t_{k}^{1} , y(φ(v)) = v ,
t_{k}^{1} ⩽ ψ(v) ⩽ t_{k}^{2} , y(ψ(v)) = v ,
v ∈ I = [y(t_{k}^{∗}), y(t_{k}^{1})] .

We prove by an indirect proof that

y^{[1]}(φ(v)) ⩾ |y^{[1]}(ψ(v))| , v ∈ I .  (10)

Observe that (9) yields y^{[1]}(φ(v)) > 0 and y^{[1]}(ψ(v)) < 0 for v ∈ I_{1} = [y(t_{k}^{∗}), y(t_{k}^{1})). Define

S(v) = y^{[1]}(φ(v)) − |y^{[1]}(ψ(v))| , v ∈ I .

Suppose, contrarily, that there exists v̄ ∈ I_{1} such that

S(v̄) < 0 .  (11)

Then using (2), (9) and (H2), we have

\frac{d}{dv} S(v) = \frac{y^{[2]}(φ(v))a_{2}(φ(v))}{y′(φ(v))} + \frac{y^{[2]}(ψ(v))a_{2}(ψ(v))}{y′(ψ(v))}

= \frac{y^{[2]}(φ(v))}{y^{[1]}(φ(v))} \frac{a_{2}(φ(v))}{a_{1}(φ(v))} + \frac{y^{[2]}(ψ(v))}{y^{[1]}(ψ(v))} \frac{a_{2}(ψ(v))}{a_{1}(ψ(v))}

⩽ y^{[2]}(ψ(v))\frac{a_{2}(ψ(v))}{a_{1}(ψ(v))} [\frac{1}{y^{[1]}(φ(v))} + \frac{1}{y^{[1]}(ψ(v))}] , v ∈ I_{1} .

Thus

v ∈ I_{1} , S(v) < 0 ℝ_{+}ightarrow \frac{d}{dv} S(v) < 0 .

¿From this and from (11), it is clear that

S(v) < 0 on [v̄, y(t_{k}^{1})] ,

and this contradicts S(y(t_{k}^{1})) = 0. Thus, (10) holds and using v = y(t_{k}^{∗}) in (10) and (9), y^{[1]}(t_{k−1}^{2}) > |y^{[1]}(t_{k}^{2})|.

(ii) Let t_{0} < t_{1} < t_{2}, t_{0}^{1} ⩽ t_{0} be consecutive zeros of y^{[2]}. Let us transform (1) into (5) according to Lemma 3 with a_{0} ≡ a_{1}. Then x_{i}, x_{i} = x(t_{i}), i = 0, 1, 2, are the consecutive zeros of Y^{{2}}, x_{0} < x_{1} < x_{3}.

The next transformation

σ = x_{2} − x , Y (x) = Z(σ) , x ∈ [x_{0}, x_{2}] , σ ∈ [0, x_{2} − x_{0}] ,  (12)

transforms (5) into (7) where

b(σ) = \frac{a_{2}(t(x_{2} − σ))}{a_{1}(t(x_{2} − σ))} , r̄(σ) = \frac{r(t(x_{2} − σ))}{a_{1}(t(x_{2} − σ))}

and according to (H2) and \frac{d}{dt} (\frac{r(t)}{a_{1}(t)}) ⩽ 0, we have

b′(σ) ⩾ 0 and r̄′(σ) ⩾ 0 on [0, x_{2} − x_{0}] , \frac{d}{dσ} = ′ .

As σ_{0} = 0, σ_{1} = x_{2} − x_{1}, and σ_{2} = x_{2} − x_{0} are consecutive zeros of Z^{[2]}, Lemma 4 yields

\sqrt{2}|Z′(x_{2} − x_{0})| < |Z′(x_{2} − x_{1})| .  (13)

Using (12) and (6) we have

|y^{[1]}(t_{0})| = |Y^{{1}}(x_{0})| = |Y^{•} (x_{0})| = |Z′(x_{2} − x_{0})| ,

|y^{[1]}(t_{1})| = |Y^{{1}}(x_{1})| = |Y^{•} (x_{1})| = |Z′(x_{2} − x_{1})|

and thus (13) yields \sqrt{2}|y^{[1]}(t_{1})| < |y^{[1]}(t_{0})|.

¿From this the inequality (8) holds and lim_{t→T_{−}} y^{[1]}(t) = 0.

(iii) We prove the third statement for (5) with a_{0} ≡ a_{2}

((\frac{1}{A_{1}} Y^{•} )^{•})^{•} = R(x) f(Y) ,

A_{1}(x) = \frac{a_{1}(t(x))}{a_{2}(t(x))} , R(x) = \frac{r(t(x))}{a_{2}(t(x))} , Y^{\{1\}} = \frac{1}{A_{1}(x)}Y^{•} , Y^{\{2\}} = (Y^{\{1\}})^{•};

then according to (6), it will hold for (1) too.

Applying Lemma 2 to (5), sequences {x_{k}^{i}}, k = 1, 2, …, i = 0, 1, 2 exist such that

\begin{aligned}x_{k}^{0} < x_{k}^{1} < x_{k}^{2} < x_{k+1}^{0} , k = 1, 2, …, & lim_{k→∞} x_{k}^{0} = x(T) , \\ Y^{\{i\}}(x_{k}^{i}) = 0 , (−1)^{j+1}Y^{\{j\}}(x)Y (x) & > 0 on (x_{k}^{0}, x_{k}^{j}) , \\ & < 0 on (x_{k}^{j}, x_{k+1}^{0}) , \\ k = 1, 2, …; j = 1, 2 . \end{aligned}  (14)

Let k ∈ {1, 2, … }. Put τ_{0} = x_{k}^{1}, τ_{1} = x_{k}^{2}, τ_{2} = x_{k+1}^{0}, Δ_{1} = [τ_{0}, τ_{1}], Δ_{2} = [τ_{1}, τ_{2}], δ_{1} = τ_{1} − τ_{0}, δ_{2} = τ_{2} − τ_{1} and suppose, for simplicity, that Y^{{1}}(x) ⩽ 0 on Δ_{1}. Then (14) and Lemma 1 yield

Y (x) > 0 , Y^{{1}}(x) < 0 , Y^{{2}}(x) < 0 , Y and |Y^{{2}}| are decreasing
and |Y^{{1}}| is increasing on (τ_{0}, τ_{1}) ;
Y (x) > 0 , Y^{{1}}(x) < 0 , Y^{{2}}(x) > 0 , Y and |Y^{{1}}| are decreasing
and Y^{{2}} is increasing on (τ_{1}, τ_{2}) .

(15)

The statement will be valid if we prove that

|Y^{{2}}(x_{k}^{0})| > |Y^{{2}}(τ_{0})| > Y^{{2}}(τ_{2}) .

As the first inequality follows from (14) and Lemma 1, the second one only must be proved. Thus, suppose that

|Y^{{2}}(τ_{0})| ⩽ Y^{{2}}(τ_{2}) .  (16)

According to (15) and the assumptions of the theorem, the function Y^{{2}} is concave on [τ_{0}, τ_{2}]:

(Y^{\{2\}}(x))^{••} = (Y^{\{3\}}(x))^{•} = [\frac{r(t(x))}{a_{2}(t(x))} f(Y (x))]^{•} =

= (\frac{r(t(x))}{a_{2}(t(x))})^{•} f(Y (x)) + \frac{r(t(x))}{a_{2}(t(x))} f′(Y (x))Y^{\{1\}}(x)\frac{a_{1}(t(x))}{a_{2}(t(x))} ⩽ 0 , (17)

x ∈ Δ_{1} ∪ Δ_{2} .

Thus, Y^{{2}} is above the secant line on Δ_{1} ∪ Δ_{2}, and using (14) and (15), we have

|Y^{\{1\}}(τ_{1})| = ∫_{Δ_{1}} |(Y^{\{1\}}(x))^{•}| dx = ∫_{Δ_{1}} |Y^{\{2\}}(x)| dx ⩽ |Y^{\{2\}}(τ_{0})| \frac{δ_{1}}{2} ,
|Y^{\{1\}}(τ_{1})| ⩾ Y^{\{1\}}(τ_{2}) − Y^{\{1\}}(τ_{1}) = ∫_{Δ_{2}} Y^{\{2\}}(x) dx ⩾ Y^{\{2\}}(τ_{2}) \frac{δ_{2}}{2} .

¿From this and (16),

δ_{1} ⩾ δ_{2} .  (18)

Furthermore, according to (1), (15) and (17), Y^{{3}} ⩾ 0 is decreasing on Δ_{1} ∪ Δ_{2}. From this it follows that

|Y^{{2}}(τ_{0})| = ∫_{Δ_{1}} Y^{{3}}(x) dx > Y^{{3}}(τ_{1}) δ_{1} ,
Y^{{2}}(τ_{2}) = ∫_{Δ_{2}} Y^{{3}}(x) dx < Y^{{3}}(τ_{1}) δ_{2} .

Thus, with respect to (16), δ_{1} < δ_{2} and this contradicts (18). □

The following theorem states a sufficient condition under which oscillatory solutions tend to zero as t → T.

Theorem 2. Let (H1) and (H2) hold, r ∈ C^{1}(J), f ∈ C^{1}(R), f′ ⩾ 0 on R,

(\frac{r(t)}{a_{1}(t)})′ ⩽ 0 ,  (19)

and let one of the following assumptions hold:

(i) (\frac{r(t)}{a_{2}(t)})′ ⩽ 0, 0 < M ⩽ \frac{r(t)}{a_{1}(t)} for t ∈ J ;

(ii) \frac{a_{2}(t)}{a_{1}^{2}(t)} r(t) ⩾ M > 0 for t ∈ J;

(iii) ∫_{0}^{T} a_{1}(s) ds < ∞.

If y ∈ O, then lim_{t→∞} y^{(j)}(t) = 0 for j = 0, 1.

Proof. Let y ∈ O. According to Lemma 3 with a_{0} ≡ a_{1}, it is sufficient to prove the results for(5) only:

(\frac{1}{A_{2}(x)}Y^{••})^{•} = R(x) f(Y) , \frac{d}{dx} = ^{•} ,  (20)

A_{1} ≡ 1 , A_{2}(x) = \frac{a_{2}(t(x))}{a_{1}(t(x))} , R(x) = \frac{r(t(x))}{a_{1}(t(x))} , x ∈ I = [0, x^{∗}) , x^{∗} = x(T) ,

Y^{\{1\}} = Y^{•} , Y^{\{2\}} = \frac{1}{A_{2}(x)}Y^{••} .  (21)

Denote by {x_{k}^{i}}, i = 0, 1, 2, k = 1, 2, …, the sequences given by Lemma 2 for (20) (i.e. x_{k}^{i} = t_{k}^{i}) and put

Δ_{k} = [x_{k}^{0}, x_{k}^{1}] .

Then, according to Lemmas 1 and 2,

Y^{{1}}(x) Y (x) ⩾ 0 , Y^{{2}}(x) Y (x) ⩽ 0 for x ∈ Δ_{k} ,  (22)
|Y^{{1}}| and |Y^{{2}}| are decreasing on Δ_{k} .

Furthermore, using (19),

(\frac{R(x)}{A_{1}(x)})^{•} = R^{•}(x) = (\frac{r(t)}{a_{1}(t)})′ t^{•}(x) ⩽ 0 on I ,

the assumptions of Th. 1 (ii), applied to (20), are fulfilled. Thus, lim_{x→x^{∗}} Y^{{1}}(x) = 0 and

|Y^{\{1\}}(x_{k}^{0})| ⩽ |Y^{\{1\}}(x_{k−1}^{2})| ⩽ 2^{\frac{2−k}{2}} |Y^{\{1\}}(x_{1}^{2})| , k ⩾ 2 ;  (23)

note that the first inequality follows from Lemmas 1 and 2.

We prove indirectly that

lim_{t→T} Y (t) = 0 .  (24)

Thus suppose, without loss of generality, that

|Y (x_{k}^{1})| ⩾ M_{1} > 0 , k = 1, 2, …

Then, according to Lemmas 1 and 2, there exists a sequence x̄_{k} ∈ (x_{k}^{0}, x_{k}^{1}) such that

|Y (x̄_{k})| = \frac{M_{1}}{2} , \frac{M_{1}}{2} ⩽ |Y (x)| ⩽ M_{1} on Δ̄_{k} = [x̄_{k}, x_{k}^{1}] .  (25)

Let δ_{k} = x_{k}^{1} − x̄_{k}. Using (22) and (23), we have

\frac{M_{1}}{2} ⩽ |Y (x_{k}^{1}) − Y (x̄_{k})| = ∫Δ̄_{k} |Y^{\{1\}}(x)| dx
⩽ |Y^{\{1\}}(x_{k}^{0})| δ_{k} ⩽⩽ 2^{\frac{2−k}{2}} δ_{k}|Y^{\{1\}}(x_{1}^{2})|

and thus

lim_{k→∞} δ_{k} = ∞ .  (26)

(i) According to (19) and (22),

|Y^{{2}}(x_{k}^{0})| ⩾ [Y^{{2}}(x_{k}^{1}) − Y^{{2}}(x̄_{k})] sgn Y (x_{k}^{1}) = ∫Δ̄_{k} Y^{{3}}(x) sgn Y (x) dx

= ∫Δ̄_{k} R(x) f(Y (x)) sgn Y (x) dx ⩾ Mδ_{k} min_{\frac{M_{1}}{2} ⩽s⩽M_{1}} |f(s)| > 0

and thus (26) yields lim_{k→∞} Y^{{2}}(x_{k}^{0}) = ∞ which contradicts Theorem 1 (iii). (ii) Using (22), (H2) and the assumptions, we have for x ∈ Δ̄_{k}:

A_{2}(x)|Y^{{2}}(x)| ⩾ A_{2}(x) [Y^{{2}}(x_{k}^{1}) − Y^{{2}}(x)] sgn Y (x_{k}^{1}) =

= A_{2}(x) ∫_{x}^{x_{k}^{1}} |Y^{{3}}(s)| ds ⩾ ∫_{x}^{x_{k}^{1}} R(s) A_{2}(s)|f(Y (s))| ds ⩾

⩾ MM_{2}(x_{k}^{1} − x) , M_{2} = min_{\frac{M_{1}}{2} ⩽s⩽M_{1}} |f(s)| > 0 .

¿From this and from (21),

Y^{\{1\}}(x̄_{k}) = ∫Δ̄_{k} A_{2}(x)|Y^{\{2\}}(x)| dx ⩾ MM_{2} ∫Δ̄_{k}(x_{k}^{1} − x) dx = \frac{MM_{2}}{2} δ_{k}^{2} .

Since lim_{x→x^{∗}} Y^{{1}}(x) = 0, Y^{{1}}(x̄_{k}) is bounded, say

|Y^{{1}}(x̄_{k})| ⩽ M_{3} , k = 1, 2, …,

and we can conclude that δ_{k} is bounded as well. This contradiction to (26) proves the statement.

(iii) In this case, x^{∗} < ∞ and I is bounded which contradicts (26). □

Remark 3. (i) Note that (\frac{r(t)}{a_{1}(t)})′ ⩽ 0 follows from (H2) and the fact that

(\frac{r(t)}{a_{2}(t)})′ ⩽ 0 :

(\frac{r}{a_{1}} )′ = (\frac{r}{a_{2}} \frac{a_{2}}{a_{1}} )′ = (\frac{r}{a_{2}} )′ \frac{a_{2}}{a_{1}} + \frac{r}{a_{2}} (\frac{a_{2}}{a_{1}} )′ ⩽ 0 .

(ii) The differential equation in Ex. 1 fulfills all assumptions of Th. 2 (i) with the exception of 0 < M ⩽ \frac{r(t)}{a_{1}(t)}.

3. Case (H3)

In this section (1) will be studied under the assumption (H3).

Theorem B gives us a sufficient condition for every oscillatory solution to vanish at T. We generalize this result as follows.

Theorem 3. Let (H1) and (H3) hold and let

M ∈ (0, ∞), a_{1}(t) a_{2}(t) ⩽ Mr^{2}(t), t ∈ J .

Then for every oscillatory solution y of (1), lim_{t→T^{−}} y(t) = 0.

Proof. Using Lemma 3 with a_{0} = r, the statement follows from Theorem B applied to (5). □

Remark 4. (i) It is proved in [6] that if (H3) holds, then \{\sqrt{\frac{a_{1}}{a_{2}}} |y^{[1]}||_{t=t_{k}^{2}}\}_{k=1}^{∞} is a decreasing sequence.

(ii) Note that Theorem B is the special case of Theorem 3. Furthermore, if

a_{1}(t) = a_{2}(t) = r(t) = e^{−t}, J = ℝ_{+} ,

then the assumptions of Theorem 3 are fulfilled and the ones of Theorem 3 not. Thus Theorem 3 is a generalization of Theorem B.

4. Applications

We apply the previous results to the equation

y′′′ + q(t)y′ = s(t) f(y)  (27)

where q ∈ C^{∘}(ℝ_{+}), s ∈ C^{∘}(ℝ_{+}), f ∈ C(R),

s(t) > 0 on ℝ_{+} , f(x)x > 0 for x ≠ 0 .  (H4)

A solution y of (27) is called oscillatory if it is defined on ℝ_{+}, sup_{τ⩽t<∞} |y(t)| > 0 for

every τ ∈ ℝ_{+} and there exists a sequence of zeros of y tending to ∞.

Let h be a positive solution on [τ, ∞), τ ∈ ℝ_{+}, of the equation

h′′ + q(t)h = 0 .  (28)

Then (27) is equivalent to (1) (see [5] or make a direct computation) on J = [τ, ∞), where T = ∞,

\begin{matrix}a_{1}(t) = h(t) , a_{2}(t) = \frac{1}{h^{2}(t)} , r(t) = s(t) h(t) , \\ y^{[1]} = \frac{y′}{h} , y^{[2]} = h^{2}(y^{[1]})′ . \end{matrix}  (29)

Thus (H1) is satisfied, (H2) holds if h is increasing, and (H3) holds if h is decreasing.

Theorem 4. Let (H4) hold,

q(t) ⩽ 0 , s(t) ⩾ M > 0 for t ∈ [M_{1}, ∞)

and ∫_{0}^{∞}t|q(t)| dt < ∞ where M and M_{1} are positive constants. Then every oscillatory

solution of (27) tends to zero as t → ∞.

Proof. If follows from [11] and from ∫_{0}^{∞}t|q(t)| dt < ∞ that (28) is non-oscillatory

and there exists a positive solution h of (27) that is decreasing for large t and lim_{t→∞} h(t) = h_{0} ∈ (0, ∞). Thus, the conclusion follows from Theorem 3. □

On oscillatory solutions 11

Theorem 5. Let (H4) hold, s ∈ C^{1}(ℝ_{+}), f ∈ C^{1}(R), f′ ⩾ 0 on R,

q(t) ⩾ 0 , 0 < M ⩽ s(t) , s′(t) ⩽ 0 for t ∈ [M_{1}, ∞) ,

and ∫_{0}^{∞}tq(t) dt < ∞ where M and M_{1} are positive constants. Then every oscillatory

solution of (27) tends to zero as t → ∞ along with its first derivative.

Proof. It follows from [7] and from ∫_{0}^{t} tq(t) dt < ∞ that (28) is nonoscillatory and there exists a positive solution h of (28) that is increasing for large t and lim_{t→∞} h(t) = h_{0} ∈ (0, ∞). Then (27) is equivalent (1) and (29). Thus, the statement

follows from Theorem 2 (ii) and the fact that lim_{t→∞} y′(t) = lim_{t→∞} y^{[1]}(t) h(t) = 0 (see Theorem 1 (ii)). □

Remark 5. Theorems 4 and 5 expand the results obtained in [9].

References

[1] Bartušek M., Asymptotic Properties of Oscillatory Solutions of Differential Equations of the n-th Order, Folia FSN. Univ. Masar. Brunensis, Masaryk University, Brno, 1992.

[2] Bartušek M., On the structure of solutions of a system of three differential inequalities, Arch. Math. (Brno), 30, 1994, 117–130.

[3] Bartušek M., Asymptotic behaviour of oscillatory solutions of n-th order differential equations with quasiderivatives, Czech. Math. J., 47(122), 1997, 245–259.

[4] Bartušek M., On unbounded oscillatory solutions of n-th order differential equations with quasiderivatives, Nonlinear Analysis, 30, 1997, 1595–1605.

[5] Bartušek M., Došlá Z., Oscillatory criteria for nonlinear third order differential equations with quasiderivatives, Diff. Eqs. Dyn. Syst., 3, 1995, 251–268.

[6] Bartušek M., Osička, J., Asymptotic behaviour of solutions of a third-order nonlinear differential equation, Nonlinear Analysis, 34, 1998, 653–664.

[7] Cecchi M., Marini M., Villari G., Integral criteria for a classification of solutions of linear differential equations, Diff. Eqs., 99, 1992, 381–397.

[8] Chanturia T., On monotone and oscillatory solutions of ordinary differential equations of higher order, Ann. Pol Math. XXXVII, 1980, 93–111 (In Russian).

[9] Greguš M., Greguš M. jr., Asymptotic properties of solutions of a certain nonanatomous nonlinear differential equation of the third order, Boll. U.M.I (7) 7-A (1993), 341–350.

[10] Kiguradze I. T., Chanturia T., Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Acad. Press, Dordrecht, 1993.

[11] Marini M., Zezza P., On the asymptotic behaviour of the solutions of a class of second-order linear differential equations, J. Diff. Eqs., 28, 1978, 1–17.

[12] Trench W. F., Oscillatory properties of perturbed disconjugate equations, Proc. Amer. Math. Soc. 52, 1975, 147–155.

Miroslav Bartušek

Department of Mathematics, Masaryk University Janáčkovo nám. 2a, 662 95 Brno, Czech Republic

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