PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 79(93) (2006), 51–64 DOI: 10.2298/PIM0693051J

EXISTENCE OF REGULARLY AND RAPIDLY VARYING SOLUTIONS FOR A CLASS OF THIRD ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

Jaroslav Jaroš, Kusano Takaŝi, and Vojislav Marić

To the memory of Professor Michal Greguš

Abstract. The existence of solutions belonging to the Karamata class of functions of a class of third order nonlinear differential equations is proved via some more general results on asymptotic equivalence.

1. Introduction

The theory of Karamata regularly varying functions has proved to be a powerful tool for the study of asymptotic behavior of nonoscillatory solutions of linear and nonlinear differential equations. For a variety of results produced in the framework of regular variation the reader is referred to the monograph [9] and the papers [4–8].

Most of the results found in the literature are concerned exclusively with second order differential equations, and so it is natural to raise the question as to whether higher order differential equations could be studied in the framework of regularly and rapidly varying functions.

The purpose of this paper is to provide a partial affirmative answer to this question by demonstrating the existence of regularly varying and rapidly varying solutions for third order differential equations of the type

(A)  x′′′ + 2P (t)x′ + P′(t)x = F (t, x)

2000 Mathematics Subject Classification. Primary 34C11; Secondary 34D05..

The first author is supported by grant 1/1003/04 of the Scientific Grant Agency VEGA of Slovak Republic

Communicated by Žarko Mijajlović

where P is continuously differentiable on [a, ∞), for some a > 0, F is continuous on [a, ∞) × R and

(1.1)  |F (t, x)| ⩽ G(t, x)

where G : [a, ∞) × R → R_{+}, R_{+} = [0, ∞) is a continuous function which is nondecreasing in the second variable for all t ⩾ a.

More specifically, in Section 2 we indicate a situation in which the self-adjoint linear equation

(A_{0})  y′′′ + 2P (t)y′ + P′(t)y = 0

has a fundamental set of solutions consisting of regularly and rapidly varying functions.

In Section 3 we establish conditions under which equation (A) possesses a fundamental set of solutions which are asymptotic as t → ∞ to the indicated regularly and rapidly varying solutions of equation (A_{0}).

For that purpose a more general result on asymptotic equivalence between solutions of (A) and (A_{0}) is proved (Theorems 3.1–3.3).

To reader’s benefit we summarize below the definitions and basic properties of Karamata regularly varying and rapidly varying functions which are needed in developing the main existence theorems (cf. [1]).

Definition. A measurable positive function f(t) defined in some neighborhood of infinity is said to be regularly varying (at infinity) of index ρ if

lim_{t→∞} \frac{f(λt)}{f(t)} = λ^{ρ} for any λ > 0.

The set of all regularly varying functions of index ρ is denoted by RV(ρ). If in particular ρ = 0, the symbol SV is often used for RV(0) and a member of SV is called a slowly varying function.

From the definition it follows that f(t) ∈ RV(ρ) is expressed as f(t) = t^{ρ}L(t) for some L(t) ∈ SV. So the class SV plays a central role in the theory of regular variation. Of fundamental importance are the following facts holding for any L(t) ∈ SV:

The function f(t) ∈ RV(0) if and only if it can be represented in the form

(1.2)  f(t) = c(t) exp \{ ∫_{t_{0}}^{t} \frac{δ(s)}{s}ds\}, t ⩾ t_{0},

for some t_{0} > 0 and some measurable functions c(t) and δ(t) such that

lim_{t→∞} c(t) = c_{0} ∈ (0, ∞) and lim_{t→∞} δ(t) = 0.

EXISTENCE OF REGULARLY AND RAPIDLY VARYING SOLUTIONS … 53

There holds

(1.3)  lim_{t→∞} t^{ε}L(t) = ∞ and lim_{t→∞} t^{−ε}L(t) = 0 for any ε > 0.

It is obvious that L_{i}(t) ∈ SV, f_{i}(t) ∈ RV(ρ_{i}), i = 1, 2, implies

(1.4)  L_{1}(t)L_{2}(t) ∈ SV, f_{1}(t)f_{2}(t) ∈ RV(ρ_{1} + ρ_{2}).

All functions tending to positive constants or the function Π_{1}^{n}(log_{ν} t)^{ξ_{ν}}, ξ_{ν} real and log_{ν} the ν-th iteration of the logarithm, are simple examples of slowly varying functions.

Definition. A measurable function f : [a, ∞) → (0, ∞) is called rapidly varying (at infinity) of index ∞ if

lim_{t→∞} \frac{f(λt)}{f(t)} = ∞ for λ > 1 and lim_{t→∞} \frac{f(λt)}{f(t)} = 0 for 0 < λ < 1

and is called rapidly varying (at infinity) of index −∞ if

lim_{t→∞} \frac{f(λt)}{f(t)} = 0 for λ > 1 and lim_{t→∞} \frac{f(λt)}{f(t)} = ∞ for 0 < λ < 1.

The set of all rapidly varying functions of index ∞ (or −∞) is denoted by RV(∞) (or RV(−∞)).

The function exp(at^{α}), a, α real and a ≠ 0, α > 0, is a simple example of rapidly varying function.

2. The self-adjoint equation (A_{0})

2.1. Preliminaries. Consider the second order linear differential equation

(2.1)  z′′ + \frac{1}{2}P (t)z = 0

and suppose it to be nonoscillatory. The following is known:

Lemma 2.1. [3] There exist two linearly independent solutions u(t) and v(t) of (2.1) such that

u(t)/v(t) → 0 as t → ∞,

∫_{a}^{∞} \frac{dt}{u(t)^{2}} = ∞, ∫_{a}^{∞} \frac{dt}{v(t)^{2}} < ∞, (2.2)

v(t) = u(t) ∫_{t_{0}}^{t} \frac{ds}{u(s)^{2}} , u(t) = v(t) ∫_{t}^{∞} \frac{ds}{v(s)^{2}} .

The solution u(t) satisfying (2.2) is called principal solution and v(t) is called nonprincipal one.

There is no loss of generality in assuming that both u(t) and v(t) are eventually positive.

Lemma 2.2. [2], [10] Let {u(t), v(t)} be a fundamental set of solutions on a certain interval I of equation (2.1). Then the functions y_{1}(t) = u(t)^{2}, y_{2}(t) = u(t)v(t) and y_{3}(t) = v(t)^{2} form a fundamental set of solutions on I of equation (A_{0}).

Lemma 2.3. [12] Let u(t) be a principal solution of equation (2.10). Then equation (A) can be written in the canonical factorized form:

(A′)  L_{3}x ≡ \frac{2}{u(t)^{2}} (\frac{u(t)^{2}}{2} (u(t)^{2}(\frac{1}{u(t)^{2}} x)′)′)′ = F (t, x).

To construct a fundamental set of solutions of (A) choose u(t) and v(t) as positive principal and nonprincipal solutions of (2.1). Then by Lemma 2.2 and (2.2), the set of functions

y_{1}(t) = u(t)^{2},
(2.3)  y_{2}(t) = u(t)v(t) = u(t)^{2} ∫_{a}^{t} \frac{ds}{u(s)^{2}} ,
y_{3}(t) = v(t)^{2} = u(t)^{2} ∫_{a}^{t} \frac{1}{u(s)^{2}} ∫_{a}^{s} \frac{2}{u(r)^{2}} dr ds

forms a fundamental set of solutions for the equation L_{3}x = 0 (and thus for (A_{0})). Notice also that

(2.4)  ∫_{a}^{t} \frac{1}{u(s)^{2}} ∫_{a}^{s} \frac{1}{u(r)^{2}} dr ds = \frac{1}{2} \frac{v(t)^{2}}{u(t)^{2}} .

These solutions are asymptotically ordered in the sense that

lim_{t→∞} \frac{y_{i}(t)}{y_{j}(t)} = 0 and 1 ⩽ i < j ⩽ 3.

2.2. Results. We present here some sufficient conditions which guarantee the existence of regularly and rapidly varying solutions of equation (A_{0}).

Theorem 2.1. Let c be a constant such that c < 1/4 and let ρ, σ (ρ < σ) denote the real roots of the equation

(2.5)  λ^{2} − λ + c = 0.

If P(t) is conditionally integrable on [a, ∞) and satisfies

(2.6)  lim_{t→∞} t ∫_{t}^{∞} P (s) ds = 2c,

then equation (A_{0}) has a fundamental set of regularly varying (hence nonoscillatory) solutions y_{i}(t), i = 1, 2, 3, of the form

(2.7)  y_{1}(t) = t^{2ρ}L_{1}(t), y_{2}(t) = tL_{2}(t), y_{3}(t) = t^{2σ}L_{3}(t),

where L_{1}(t) is some slowly varying function, L_{2}(t) → (1 − 2ρ)^{−1} and L_{3}(t) ∼ {(1 − 2ρ)^{2}L_{1}(t)}^{−1}(t) as t → ∞.

Proof. By [9, Theorem 1.11], condition (2.6) is necessary and sufficient for equation (2.1) to possess two linearly independent regularly varying (hence nonoscillatory) solutions of the form u(t) = t^{ρ}L_{1}(t) and v(t) = t^{σ}L_{2}(t) where L_{1}(t) is some slowly varying function and L_{2}(t) ∼ {(1 − 2ρ)L_{1}(t)}^{−1}, as t → ∞. An application of Lemma 2.2 and the use of property (1.4) of regularly varying functions give the result. □

If c = 1/4 which is the borderline case between nonoscillation and oscillation of equation (2.1), there holds

Theorem 2.2. Suppose that

lim_{t→∞} t ∫_{t}^{∞} P (s) ds = \frac{1}{2}.

Suppose furthermore that the function

ϕ(t) = t ∫_{t}^{∞} P (s) ds − \frac{1}{2}

satisfies

∫ ^{∞} \frac{|ϕ(t)|}{t} dt < ∞

and

∫ ^{∞} \frac{|ψ(t)|}{t} dt < ∞, where ψ(t) = ∫_{t}^{∞} \frac{|ϕ(s)|}{t} ds.

Then, equation (A_{0}) has a fundamental set of regularly varying (hence nonoscillatory) solutions y_{i}(t), i = 1, 2, 3, of index one and of the form

y_{1}(t) = tL_{1}(t), y_{2}(t) = t log t L_{2}(t), y_{3}(t) = t log^{2} t L_{3}(t)

where L_{1}(t) → α, α ∈ (0, ∞), L_{2}(t) → 1, L_{3}(t) → 1/α^{2} as t → ∞.

Proof. One proceeds exactly as in the proof of Theorem 2.1, using this time [9, Theorem 1.12]. □

Theorem 2.3. Let P (t) < 0 for t ⩾ a. If for each λ > 1

(2.8)  lim_{t→∞} (− t ∫_{t}^{λt} P (s) ds) = ∞,

then equation (A_{0}) has at least two rapidly varying solutions such that the first of these solutions decreases and is of the class RV(−∞) whereas the second one increases and is of the class RV(∞).

Proof. It is known (e.g., [9, Lemma 1]) that (2.1) has a (positive) decreasing solution u(t). Let v(t) be the corresponding linearly independent solution defined by (2.2). Then, by [9, Theorem 1.3], one has u(t) ∈ RV(−∞) and v(t) ∈ RV(∞) if and only if condition (2.8) holds. It is clear from the definition of rapidly varying functions that y_{1}(t) = u(t)^{2} ∈ RV(−∞) and y_{3}(t) = v(t)^{2} ∈ RV(∞) which completes the proof. □

However the third linearly independent solution y_{2}(t) = u(t)v(t) need not to be rapidly varying at all. This is shown by the example:

u(t) = e^{−t}, v(t) = e^{t}, so that y_{2}(t) = 1.

3. Regularly and rapidly varying solutions of perturbed equation (A)

The objective of this section is twofold. First, we establish sufficient conditions for equation (A) to have solutions x_{1}(t), x_{2}(t) and x_{3}(t) with the same asymptotic behavior as solutions y_{1}(t) = u(t)^{2}, y_{2}(t) = u(t)v(t) and y_{3}(t) = v(t)^{2} of equation (A_{0}), respectively. Secondly, we apply these results to construct regularly and rapidly varying solutions of (A).

3.1. Asymptotic equivalence between x_{i}(t) and y_{i}(t), i = 1, 2, 3. We prove

Theorem 3.1. If for some α > 0

(3.1)  ∫_{a}^{∞} v(t)^{2}G(t, αu(t)^{2}) dt < ∞,

then there exists an eventually positive solution x_{1}(t) of equation (A) such that for t → ∞, x_{1}(t) ∼ \frac{α}{2}u(t)^{2}.

Proof. Choose T ⩾ a such that

(3.2)  ∫_{T}^{∞} v(t)^{2}G(t, αu(t)^{2})dt ⩽ \frac{α}{2},

which is possible by (3.1). Further, define the set X_{1} by

X_{1} = {x ∈ C[ T, ∞) : 0 ⩽ x(t) ⩽ αu(t)^{2}, t ⩾ T }

and the integral operator F_{1} by

F_{1}x(t) = \frac{α}{2} u(t)^{2} + u(t)^{2} ∫_{t}^{∞} [ ∫_{t}^{s} \frac{1}{u(s_{2})^{2}} ∫_{t}^{s_{2}} \frac{1}{u(s_{1})^{2}} ds_{1} ds_{2}]u(s)^{2}F (s, x(s)) ds,

t ⩾ T.

The set X_{1} is a convex closed subset of a locally convex space C[ T, ∞) equipped with the usual metric topology of uniform convergence on compact subintervals of [ T, ∞). We will apply the Schauder–Tychonoff fixed point theorem to prove that F_{1} has a fixed element in X_{1}. To this end it suffices to verify that F_{1} maps X_{1} into itself, that F_{1} is a continuous mapping, and that F_{1}(X_{1}) is relatively compact in C[ T, ∞).

(i) Using (3.2), (2.2) and (2.4) one easily checks that x ∈ X_{1} implies F_{1}x ∈ X_{1}, that is, F_{1} is a self-map on X_{1}.

(ii) Let {x_{ν}} be a sequence of elements of X_{1} converging to x_{0} ∈ X_{1} in the topology of C[ T, ∞). Because of (3.1), the Lebesgue dominated convergence theorem can be applied to prove without difficulty that the sequence of functions {F_{1}x_{ν}(t)} converges to F_{1}x_{0}(t) uniformly on any compact subinterval of [ T, ∞). This means that the F_{1} is a continuous mapping on X_{1}.

(iii) Since the set {F_{1}x(t)} is locally uniformly bounded and locally equicontinuous on [ T, ∞), it follows from the Arzela–Ascoli lemma that F_{1}(X_{1}) is relatively compact in C[ T, ∞).

Therefore the Schauder–Tychonoff fixed point theorem ensures the existence of an element x_{1} ∈ X_{1} such that x_{1} = F_{1}x_{1}, which is equivalent to the integral equation

x_{1}(t) = \frac{α}{2} u(t)^{2}+ u(t)^{2} ∫_{t}^{∞} [ ∫_{t}^{s} \frac{1}{u(s_{2})^{2}} ∫_{t}^{s_{2}} \frac{1}{u(s_{1})^{2}} ds_{1} ds_{2}]

(3.3)  × u(s)^{2}F (s, x_{1}(s)) ds, t ⩾ T.

It is a matter of simple calculation to verify via differentiation of (3.3) that the function x_{1}(t) is a solution of the equation (A’) and hence of (A) on [ T, ∞). Also the asymptotics of x_{1}(t) is an immediate consequence of (3.3). This completes the proof. □

Theorem 3.2. If for some β > 0

(3.4)  ∫_{a}^{∞} u(t) v(t) G(t, β u(t) v(t)) dt < ∞,

then there exists an eventually positive solution x_{2}(t) of equation (A) such that for t → ∞, x_{2}(t) ∼ \frac{β}{2} u(t)v(t).

Proof. Such solution is obtained as before via the Schauder–Tychonoff theorem, as a solution of the integral equation

(3.5) x_{2}(t) = \frac{β}{2} u(t) v(t) + u(t)^{2} ∫_{T}^{t} \frac{1}{u(s)^{2}} ∫_{s}^{∞} [ ∫_{s}^{r} \frac{dσ}{u(σ)^{2}} ]u(r)^{2}F (r, x_{2}(r)) dr ds

for some T ⩾ a.

Choose T ⩾ a such that

(3.6)  ∫_{T}^{∞} u(t) v(t) G(t, β u(t) v(t)) dt ⩽ \frac{β}{2} .

Further, consider the set X_{2} ⊂ C[ T, ∞) and the integral operator F_{2} : X_{2} → C[ T, ∞) defined by

X_{2} = {x ∈ C[ T, ∞) : 0 ⩽ x(t) ⩽ β u(t) v(t), t ⩾ T }

and

F_{2}x(t) = \frac{β}{2} u(t) v(t) + u(t)^{2} ∫_{T}^{t} \frac{1}{u(s)^{2}} ∫_{s}^{∞} [ ∫_{s}^{r} \frac{dσ}{u(σ)^{2}} ]u(r)^{2}F (r, x(r)) dr ds.

Using (3.4) and proceeding exactly as in the proof of Theorem 3.1, F_{2} is shown to be a continuous mapping which sends X_{2} into a relatively compact subset of C[ T, ∞), and consequently there exists an element x_{2} ∈ X_{2} such that x_{2} = F_{2}x_{2}, which is nothing else but (3.5). That x_{2}(t) is a solution of (A) on [ T, ∞) with the desired asymptotic property, one obtains as before. This completes the proof. □

Theorem 3.3. If for some γ > 0

(3.7)  ∫_{a}^{∞} u(t)^{2}G(t, γv(t)^{2}) dt < ∞,

then there exists an eventually positive solution x_{3}(t) of (A) such that x_{3}(t) ∼ \frac{γ}{2} v(t)^{2} as t → ∞.

Proof. Again take T ⩾ a such that

(3.8)  ∫_{T}^{∞} u(t)^{2}G(t, γv(t)^{2})dt ⩽ \frac{γ}{2} .

The integral equation which generates a sought solution of (A) is this time

(3.9) x_{3}(t) = \frac{γ}{2} v(t)^{2} + u(t)^{2} ∫_{T}^{t} \frac{1}{u(s)^{2}} ∫_{s}^{t} \frac{dσ}{u(σ)^{2}} ∫_{s}^{∞} u(r)^{2}F (r, x_{3}(r)) dr ds.

Let X_{3} denote the set

X_{3} = {x ∈ C[ T, ∞) : 0 ⩽ x(t) ⩽ γv(t)^{2}, t ⩾ T }.

It is easy to see that the operator F_{3} defined by

F_{3}x(t) = \frac{γ}{2} v(t)^{2} + u(t)^{2} ∫_{T}^{t} \frac{1}{u(s)^{2}} ∫_{s}^{t} \frac{dσ}{u(σ)^{2}} ∫_{s}^{∞} u(r)^{2}F (r, x(r)) dr ds

maps X_{3} into X_{3}. Since the continuity of F_{3} and the relative compactness of F_{3}(X_{3}) are verified routinely, there exists a fixed element x_{3} of F_{3} in X_{3} which satisfies (3.9) and hence solves the equation (A). The asymptotic property of x_{3}(t) follows from (3.9). This completes the proof. □

In the case of the specialization

(3.10)  x′′′ + 2P (t)x′ + P′(t)x = Q(t)|x|^{m} sgn x,

where m > 0 is a constant and Q : [a, ∞) → R is continuous, the function G in (1.1) can be taken to be

G(t, x) = |Q(t)||x|^{m}, t ⩾ a, x ∈ R,

and conditions (3.1), (3.4) and (3.7) reduce to

(3.11)  ∫_{a}^{∞} v(t)^{2}u(t)^{2m}|Q(t)| dt < ∞,
(3.12)  ∫_{a}^{∞}[u(t)v(t)]^{m+1}|Q(t)| dt < ∞,
(3.13)  ∫_{a}^{∞} u(t)^{2}v(t)^{2m}|Q(t)| dt < ∞,

respectively.

Corollary 3.1. Condition (3.11) is sufficient for equation (3.10) to have for any α > 0 an eventually positive solution x_{1}(t) such that x_{1}(t) ∼ αu(t)^{2} as t → ∞.

Corollary 3.2. If (3.12) holds, then for any β > 0 the equation (3.10) has an eventually positive solution x_{2}(t) such that x_{2}(t) ∼ βu(t)v(t) as t → ∞.

Corollary 3.3. Condition (3.13) is sufficient for equation (3.10) to have for any γ > 0 an eventually positive solution x_{3}(t) such that x_{3}(t) ∼ γv(t)^{2} as t → ∞.

3.2. Regularly and rapidly varying solutions. If P (t) satisfies condition (2.6), then Theorem 2.1 guarantees the existence of a fundamental set of regularly varying solutions of equation (A_{0}) as given by (2.7). Then, if conditions (3.1), (3.4) and (3.7) hold, Theorems 3.1, 3.2 and 3.3, respectively, guarantee the existence of a fundamental set of solutions x_{i}(t), i = 1, 2, 3, of equation (A) such that for t → ∞, x_{i}(t) ∼ α_{i}y_{i}(t), α_{i} > 0. But this also means that x_{i}(t) are regularly varying since ϕ(t) ∼ L(t) as t → ∞ implies ϕ(t) = L^{∗}(t) where L^{∗}(t) is some slowly varying function such that L^{∗}(t) ∼ L(t) as t → ∞ [9, Proposition 7].

As an illustration, take in (A)

(3.14)  F (t, x) = t^{q}M(t)|x|^{m} sgn x, q real, m > 0, M(t) ∈ SV,

which case might occur in some applications. Here all three conditions (3.11), (3.12) and (3.13) reduce to a single one: for m > 1 (superlinear case), to q < −2σ(m − 1) − 3, for m < 1 (sublinear one), to q < −2ρ(m − 1) − 3 and for the linear case m = 1 all three conditions would reduce to q < −3. This is obtained by using (1.3), (1.4) and since ρ + σ = 1. Thus one obtains in the special case (3.14) of (A) that the fundamental set x_{i}(t), i = 1, 2, 3, exists and that for t → ∞

x_{1}(t) ∼ αt^{2ρ}L(t), x_{2}(t) ∼ βt, x_{3}(t) ∼ γt^{2σ}/L(t)^{2}

for any positive constants α, β, γ and for some slowly varying L(t).

To obtain the existence and the asymptotic behaviour of rapidly varying solutions of the perturbed equation (A) with P (t) < 0, one proceeds in the same way as above in the regularly varying case. That is: assume condition (2.7) and apply first Theorem 2.3 to obtain two rapidly varying solutions y_{1}(t) and y_{2}(t). Then assuming conditions (3.11) and (3.13), and applying Theorem 3.1 and Theorem 3.3 one concludes the existence of two rapidly varying solutions x_{1}(t) and x_{2}(t) such that for t → ∞

x_{i}(t) ∼ α_{i}y_{i}(t), α_{i} > 0, i = 1, 2.

4. Examples

Example 1. Consider the equation

(4.1) x′′′ + 2 sin(e^{t})x′ + e^{t} cos(e^{t})x = t^{q}M(t)|x|^{m} sgn x, q real, M(t) ∈ SV,

which is a special case of (A) with P (t) = sin(e^{t}). Since this P (t) satisfies (2.6) with c = 0, the second order equation

z′′ + sin(e^{t})z = 0

has linearly independent solutions u(t) ∈ RV(0) = SV and v(t) ∈ RV(1). A further analysis shows that

u(t) ∼ A_{0} and v(t) ∼ \frac{t}{A_{0}} for some constant A_{0}

[9, Example 2.1]. From the above observation, it follows that (4.1) by Section 3.2 has solutions x_{i}(t), i = 1, 2, 3, such that

x_{1}(t) ∼ α, x_{2}(t) ∼ βt, x_{3}(t) ∼ γt^{2},

for any positive α, β and γ, provided that

q < −3, q < −m − 2, q < −2m − 1,

respectively. Clearly, if m ⩾ 1 (resp. m ⩽ 1), then q < −2m − 1 (resp. q < −3) implies the satisfaction of other two conditions.

We next consider the case where the function P (t) satisfies the conditions of Theorem 2.2. The equation (2.1) with this P (t) has two linearly independent solutions u(t) and v(t) such that u(t) ∼ At^{\frac{1}{2}} and v(t) ∼ t^{\frac{1}{2}} (log t)/A for some A > 0, and so the equation (A_{0}) has a fundamental set of solutions {u(t)^{2}, u(t)v(t), v(t)^{2}} satisfying

u(t)^{2} ∼ A^{2}t, u(t)v(t) ∼ t log t, v(t)^{2} ∼ \frac{t(log t)^{2}}{A^{2}} .

It can be shown easily that the conditions (3.1), (3.4) and (3.7) formulated for this case are fulfilled if, for example, q < −m−2. If this inequality holds, our Theorems 3.1–3.3 guarantee the existence of three types of solutions x_{i}(t), i = 1, 2, 3, having the properties

x_{1}(t) ∼ αt, x_{2}(t) ∼ βt log t, x_{3}(t) ∼ γt(log t)^{2}

for any given positive constants α, β and γ.

Notice that this is in particular true if in (A) P (t) = 1/2t^{2}, in which case (2.1) is an Euler equation with solutions u(t) = t^{1/2} and v(t) = t^{1/2} log t.

Example 2. Consider the equation

(4.2)  x′′′ − 4k^{2}t^{2r}x′ − 4k^{2}rt^{2r−1}x = Q(t)|x|^{m} sgn x, t ⩾ 1,

where k > 0 is a constant and 2r is a positive integer. The associated second order linear equation is the generalized Airy’s differential equation

(4.3)  z′′ − k^{2}t^{2r}z = 0.

Since the condition (2.8) is clearly satisfied, the equation (4.3) has a fundamental set {u(t), v(t)} of eventually positive solutions, of which u(t) is a (principal) rapidly varying solution of index −∞ and v(t) is a (non-principal) rapidly varying solution of index ∞ with the asymptotic behavior

u(t) ∼ t^{−r/2} exp ( − \frac{kt^{r+1}}{r + 1} ), v(t) ∼ t^{−r/2} exp (\frac{kt^{r+1}}{r + 1} )

as t → ∞ (see [11, p. 285]). It follows that the third order linear differential equation

(4.4)  y′′′ − 4k^{2}t^{2r}y′ − 4k^{2}rt^{2r−1}y = 0, t ⩾ 1,

has a fundamental set of eventually positive, asymptotically ordered solutions y_{1}(t), y_{2}(t) and y_{3}(t) such that

y_{1}(t) = u(t)^{2} ∼ t^{−r} exp ( − \frac{2kt^{r+1}}{r + 1} ),
y_{2}(t) = u(t)v(t) ∼ t^{−r},
y_{3}(t) = v(t)^{2} ∼ t^{−r} exp (\frac{2kt^{r+1}}{r + 1} ).

The conditions (3.11), (3.12) and (3.13) reduce to

(4.5)  ∫_{1}^{∞} t^{−r(m+1)} exp \{\frac{2k(1 − m)t^{r+1}}{r + 1} \}|Q(t)| dt < ∞,
(4.6)  ∫_{1}^{∞} t^{−r(m+1)}|Q(t)| dt < ∞
(4.7)  ∫_{1}^{∞} t^{−r(m+1)} exp \{\frac{2k(m − 1)t^{r+1}}{r + 1} \}|Q(t)| dt < ∞,

respectively.

It is a matter of elementary calculation to check that, in case the function Q(t) satisfies

(4.8)  |Q(t)| ⩽ Mt^{ρ} exp(At^{r+1}), t ⩾ 1,

for some constants A, M and ρ,

(i) the condition (4.5) guaranteeing the existence of a solution x_{1}(t) ∼ αu(t)^{2} is fulfilled if either

A < \frac{2k(m − 1)}{r + 1} or A = \frac{2k(m − 1)}{r + 1} and ρ < r(m + 1) − 1;

(ii) the condition (4.6) guaranteeing the existence of a solution x_{2}(t) ∼ βu(t)v(t) is satisfied if either

A < 0 or A = 0 and ρ < r(m + 1) − 1;

(iii) the condition (4.7) guaranteeing the existence of a solution x_{3}(t) ∼ v(t)^{2} is fulfilled if either

A < \frac{2k(1 − m)}{r + 1} or A = \frac{2k(1 − m)}{r + 1} and ρ < r(m + 1) − 1.

If m = 1, that is, if the equation (4.2) is linear, then clearly A = 0 and the above conditions (i), (ii) and (iii) reduce to the single condition ρ < 2r − 1 which implies the existence of positive solutions x_{1}(t), x_{2}(t) and x_{3}(t) of all three asymptotic types.

In the inequality (4.8) instead of M = const one may take some slowly varying function M(t). Then the requested convergence may hold even for ρ = r(m+1)−1 when the exp factor disappears, but that will depend on M(t). Namely, all three conditions (4.5), (4.6) and (4.7) will reduce to

∫_{1}^{∞} \frac{M(t)}{t}dt < ∞.

Observe that in this example one can get a more precise asymptotic formula for x_{i}(t) in the case when r = 1/2 and k = 1 (i.e., when (4.3) is the Airy equation). Then

(4.9)  u(t) = t^{−1/4} exp ( − \frac{2}{3}t^{3/2})(1 + O(t^{−3/2}))
v(t) = t^{−1/4} exp (\frac{2}{3}t^{3/2})(1 + O(t^{−3/2}))

using (4.8) one can estimate the integral in (3.3) by

O(t^{− \frac{1}{2} (m+1)+ρ+1} exp ( − Bt^{3/2})), where B = \frac{2}{3}(m − 1) − A.

Hence, from (3.3), due to (4.9) one obtains for t → ∞:

a) For B > 0

y_{1}(t) = t^{−1/2} exp ( − \frac{4}{3}t^{3/2})(1 + O(t^{−3/2})).

b) For B = 0 the same formula holds when \frac{1}{2}m − ρ ⩽ 2 and

y_{1}(t) = t^{−1/2} exp ( − \frac{4}{3}t^{3/2})(1 + O(t^{− \frac{1}{2} (m+1)+ρ+1}))

when \frac{1}{2}m − ρ > 2.

By using the same reasoning one obtains analogous formulae for y_{3}(t) and y_{2}(t). The cases to be distinguished in the later case are A < 0 and A = 0, because of (4.6).

Remark. The importance of the equation of the form (A) is apparent in the study of a ladder-like electrical network with both nonlinear inductor and capacitor (see [13]). In particular, if the line possesses nonhomogeneous features and possible losses, one arrives to the equation

u_{t} + au^{p}u_{x} + bu_{xxx} + g(x)u_{x} + du = 0.

Here x and t are space and time variables, respectively, u is dimesionless voltage, g(x) is a function reflecting nonhomogeneity, p, a, b and d are real constants.

If the nonlinearity of the line is absent (i.e., p = 0), g(x) is linear (as often happens in applications), e.g., g(x) = 2dx, and the voltage is of travelling wave form u = u(x − mt), the above equation is reduced to the form (A_{0}) with P (x) = \frac{1}{b}(dx + a − m). If in addition an external force F (u, x) is present satisfying the conditions of this paper, then our results are applicable.

References

[1] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications 27, Cambridge Univ. Press, 1987.

[2] M. Greguš, Third Order Linear Differential Equations, D. Reidel, Dordrecht, 1987.

[3] P. Hartman, Ordinary Differential Equations, Wiley, New York–London–Sydney, 1964.

[4] H. Howard and V. Marić, Regularity and nonoscillation of solutions of second order linear differential equations, Bull. Acad. Serbe Sci. Arts, Classe Sci. Mat. Mat., Sci. Math. 114(22) (1997), 85–98.

[5] J. Jaroš and T. Kusano, Remarks on the existence of regularly varying solutions for second order linear differential equations, Publ. Inst. Math. (Beograd)(N.S.) 72(86) (2002), 113– 118.

[6] J. Jaroš and T. Kusano, Self-adjoint differential equations and generalized Karamata functions, Bull. Acad. Serbe Sci. Arts, Classe Sci. Math. Nat., Sci. Math. 29 (2004), 25–60.

[7] J. Jaroš, T. Kusano and T. Tanigawa, Nonoscillation theory for second order half-linear differential equations in the framework of regular variation, Results Math. 43 (2003), 129– 149.

[8] T. Kusano and V. Marić, Nonoscillatory second order linear and half-linear differential equations having regularly varying solutions, Adv. Math. Sci. Appl. 14 (2004), 351–357.

[9] V. Marić, Regular Variation and Differential Equations, Lectures Notes in Mathematics 1726, Springer-Verlag, Berlin–Heidelberg, 2000.

[10] G. Sansone, Equazioni Differenziali nel Campo Reale, Parte Prima, Nicola Zanichelli, Bologna, 1956.

[11] L. Sirovich, Techniques of Asymptotic Analysis, Springer-Verlag, New York–Heidelberg– Berlin, 1971.

[12] W. F. Trench, Canonical forms and principal systems for general disconjugate equations, Trans. Amer. Math. Soc. 189 (1974), 319–327.

[13] T. Yagi, Solitons in a nonlinear inhomogeneous transmission line, J. Phys. Soc. Japan 50 (1981), 2737–2744.

Department of Mathematical Analysis (Received 06 02 2006) Faculty of Mathematics, Physics and Informatics Comenius University 84248 Bratislava Slovakia

jaros@fmph.uniba.sk

Department of Applied Mathematics Faculty of Science Fukuoka University Fukuoka, 814-0180 Japan

tkusano@cis.fukuoka-u.ac.jp

Faculty of Technical Sciences University of Novi Sad 21000 Novi Sad Serbia

vojam@uns.ns.ac.yu