Bulletin T.CXXVII de l’Académie Serbe des Sciences et des Arts - 2003 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 28

ASYMPTOTICS OF SOME CLASSES OF NONOSCILLATORY SOLUTIONS OF SECOND-ORDER HALF-LINEAR DIFFERENTIAL EQUATIONS

K. TAKAŜI, V. MARIĆ, T. TANIGAWA

(Presented at the 2nd Meeting, held on March 28, 2003)

A b s t r a c t. The precise asymptotic behaviour at infinity of some classes of nonoscillatory solutions of the half-linear differential equations is determined.

AMS Mathematics Subject Classification (2000): 34D05

Key Words: half-linear equations,regular solutions,asymptotics of solutions

0. Introduction

Let α > 0 be a constant and let q : [0, ∞) → ℝ be a continuous function which is conditionally integrable in the sense that

∫_{0}^{∞} q(t)dt = lim_{T →∞} ∫_{0}^{T} q(s)ds exists and is finite.

We consider the half-linear differential equation

(|y′|^{α−1}y′)′ + q(t)|y|^{α−1}y = 0, t ≥ 0,  (A)

and derive the precise asymptotic behaviour of some classes of its nonoscillatory solutions y(t) meaning, as usual, that we construct a positive, continuous function φ(t) defined on a positive half-axis such that y(t)/φ(t) → 1 as t → ∞, denoted as y(t) ∼ φ(t).

In particular, we treat in that respect the nonoscillatory solutuions of (A) which belong to the class of slowly varying functions in the sense of Karamata [1], which is of frequent occurrence in various branches of mathematical analysis.

For brevity, we use the canonical representation of these as the definition.

Definition 0.1. A positive measurable function L(t) defined on (0, ∞) is slowly varying if and only if it can be written in the form

L(t) = c(t) exp \{∫_{t_{0}}^{t} \frac{ε(s)}{s}ds\} , t ≥ t_{0},

for some t_{0} > 0, where c(t) and ε(t) are such that for t → ∞

c(t) → c ∈ (0, ∞) and ε(t) → 0.

If c(t) is identically a positive constant, then L(t) is called normalized.

The present work is the first attempt at scrutinizing the asymptotic behaviour of slowly varying solutions of the half-linear differential eqautions. Note that the asymptotic analysis of slowly varying solutions for the linear equation y′′ + q(t)y = 0, which is a special case of (A) with α = 1, has been made by several authors; see e.g. [2, 3, 5, 6]

1. Results

The existence of nonoscillatory solutions of (A) is essentially proved (for c(t) = c) in [4, Lemma 2.2], but we present the proof here for the reader’s benefit. We put

E(α) = \frac{α^{α}}{(α + 1)^{α+1}} ,

which is referred to as the generalized Euler constant with respect to (A), and make use of the asterisk notation:

ξ^{γ∗} = |ξ|^{γ−1}ξ = |ξ|^{γ}sgn ξ for ξ ∈ ℝ and γ > 0.

Theorem 1.1. Put

Q(t) = ∫_{t}^{∞} q(s)ds  (1.1)

and suppose that there exists a continuous function P : [t_{0}, ∞) → (0, ∞), t_{0} ≥ 0, such that lim_{t→∞} P (t) = 0 and

|Q(t)| ≤ P (t), t ≥ t_{0},  (1.2)
∫_{t}^{∞} P (s)^{1+ \frac{1}{α}} ds ≤ \frac{1}{α}c(t)^{\frac{1}{α}} P (t), t ≥ t_{0},  (1.3)

where c(t) is a continuous nonincreasing function satisfying

0 < c(t) ≤ c < E(α), t ≥ t_{0},  (1.4)

for some constant c. Then the equation (A) has a nonoscillatory solution of the form

y(t) = exp \{∫_{t_{0}}^{t}[v(s) + Q(s)]^{\frac{1}{α} ∗}ds\} , t ≥ t_{0},  (1.5)

where v(t) is a solution of the integral equation

v(t) = α ∫_{t}^{∞} |v(s) + Q(s)|^{1+ \frac{1}{α}} ds, t ≥ t_{0},  (1.6)

satisfying

v(t) = O(P (t)) as t → ∞.  (1.7)

P r o o f. Consider the function y(t) defined by (1.5). It is easy to see that y(t) is a solution of (A) if v(t) is chosen in such a way that u(t) = v(t)+Q(t) satisfies the generalized Riccati equation

u′ + α|u|^{1+ \frac{1}{α}} + q(t) = 0, t ≥ t_{0}.  (1.8)

This requirement yields the differential equation for v(t):

v′ + α|v + Q(t)|^{1+ \frac{1}{α}} = 0, t ≥ t_{0},  (1.9)

from which the equation (1.6) follows via integration over [t, ∞) under the additional condition lim_{t→∞} v(t) = 0.

We shall show that a unique solution of (1.6) of the desired kind indeed exists by using the Banach contraction theorem. Let C_{P} [t_{0}, ∞) denote the set of all continuous functions v(t) on [t_{0}, ∞) such that

∥v∥_{P} = sup_{t≥t_{0}} \frac{|v(t)|}{P (t)} < ∞.  (1.10)

It is clear that C_{P} [t_{0}, ∞) is a Banach space with the norm ∥ · ∥_{P}. Define the set V ⊂ C_{P} [t_{0}, ∞) and the mapping F : V → C_{P} [t_{0}, ∞) by

V = {v ∈ C_{P} [t_{0}, ∞) : ∥v(t)∥_{P} ≤ α, t ≥ t_{0}}  (1.11)

and

Fv(t) = α ∫_{t}^{∞} |v(s) + Q(s)|^{1+ \frac{1}{α}} ds, t ≥ t_{0},  (1.12)

respectively. If v ∈ V, then

|Fv(t)| ≤ α(1 + α)^{1+ \frac{1}{α}} ∫_{t}^{∞} P (s)^{1+ \frac{1}{α}} ds ≤ (1 + α)^{1+ \frac{1}{α}} c(t)^{\frac{1}{α}} P (t), t ≥ t_{0},

from which it follows, in view of (1.4), that

∥Fv∥_{P} ≤ (1 + α)^{1+ \frac{1}{α}} c^{\frac{1}{α}} < (1 + α)^{1+ \frac{1}{α}} E(α)^{\frac{1}{α}} = α.  (1.13)

This shows that F maps V into itself. If v_{1}, v_{2} ∈ V, then, using the mean value theorem, we have

|Fv_{1}(t) − Fv_{2}(t)| ≤ α ∫_{t}^{∞} ||v_{1}(s) + Q(s)|^{1+ \frac{1}{α}} − |v_{2}(s) + Q(s)|^{1+ \frac{1}{α}} | ds

≤ α (1 + \frac{1}{α}) ∫_{t}^{∞}[(1 + α)P (s)]^{\frac{1}{α}} |v_{1}(s) − v_{2}(s)|ds

= (1 + α)^{1+ \frac{1}{α}} ∫_{t}^{∞} P (s)^{1+ \frac{1}{α}} \frac{|v_{1}(s) − v_{2}(s)|}{P (s)}ds

≤ (1 + α)^{1+ \frac{1}{α}} \frac{1}{α}c(t)^{\frac{1}{α}} P (t)∥v_{1} − v_{2}∥_{P} , t ≥ t_{0},

which implies that

∥Fv_{1} − Fv_{2}∥_{P} ≤ \frac{1}{α}(1 + α)^{1+ \frac{1}{α}} c^{\frac{1}{α}} ∥v_{1} − v_{2}∥_{P} .  (1.14)

Since \frac{1}{α}(1 + α)^{1+ \frac{1}{α}} c^{α} < 1 (cf. (1.13)), we conclude that F is a contraction mapping on V.

The contraction mapping principle then guarantees the existence of a unique element v ∈ V such that v = Fv, which clearly is a solution of the integral equation (1.6). Then, the function y(t) given by (1.5) with this v(t) gives a solution of (A) on [t_{0}, ∞). That v(t) satisfies (1.7) is a consequence of the fact that v ∈ V. This completes the proof.

Corollary 1.1.The equation (A) has a normalized slowly varying solution if

lim_{t→∞} t^{α} ∫_{t}^{∞} q(s)ds = 0.  (1.15)

P r o o f. Here, one can take the function c(t) from Theorem 1.1 to be

c(t) = sup_{s≥t} |s^{α} ∫_{s}^{∞} q(r)dr| .  (1.16)

Then c(t) is nonincreasing and tends to zero as t → ∞. Choose t_{0} > 0 so that

c(t) < E(α) and |Q(t)| ≤ \frac{c(t)}{t^{α}} for t ≥ t_{0}.

The second inequality holds due to (1.15). Take in Theorem 1.1 P (t) = c(t)/t^{α}. Then (1.2) holds and

∫_{t}^{∞} P (s)^{1+ \frac{1}{α}} ds = ∫_{t}^{∞} [\frac{c(s)}{s^{α}}]^{1+ \frac{1}{α}} ds ≤ \frac{c(t)^{1+ \frac{1}{α}}}{αt} = \frac{1}{α}c(t)^{\frac{1}{α}} P (t), t ≥ t_{0}.

Consequently, by Theorem 1.1, (A) has a nonoscillatory solution y(t) of the form (1.5) on [t_{0}, ∞) with v(t) satisfying (1.7). Since

t^{α}v(t) = O(t^{α}P (t)) = o(1) and t^{α}Q(t) = O(t^{α}P (t)) = o(1)

as t → ∞, y(t) can be rewritten as

y(t) = exp \{∫_{t_{0}}^{t} \frac{ε(s)}{s}ds\} , t ≥ t_{0},

with ε(t) = [t^{α}(v(t) + Q(t))]^{\frac{1}{α} ∗} = o(1) as t → ∞ due to Definition 0.1. This completes the proof.

Theorem 1.2. Suppose that the hypotheses of Theorem 1.1 are satisfied. Suppose furthermore that there exists a positive integer n such that

∫ ^{∞} c(t)^{\frac{n}{α}} P (t)^{\frac{1}{α}} dt < ∞ if 0 < α ≤ 1,  (1.17)

∫ ^{∞} c(t)^{\frac{n}{α^{2}}} P (t)^{\frac{1}{α}} dt < ∞ if α > 1.  (1.18)

Then, for the solution (1.5) of the equation (A), the following asymptotic formula holds for t → ∞

y(t) ∼ A exp \{∫_{t_{0}}^{t}[v_{n−1}(s) + Q(s)]^{\frac{1}{α} ∗}ds\} ,  (1.19)

where A is a positive constant. Here the sequence {v_{n}(t)} of successive approximations is defined by

v_{0}(t) = 0, v_{n}(t) = α ∫_{t}^{∞} |v_{n−1}(s) + Q(s)|^{1+ \frac{1}{α}} ds, n = 1, 2, · · · . (1.20)

P r o o f. Let y(t) be the solution (1.5) of (A) obtained in Theorem 1.1. Recall that the function v(t) used in (1.5) has been constructed as the fixed element in C_{P} [t_{0}, ∞) of the contractive mapping F defined by (1.12). Th e standard proof of the contraction mapping principle shows that the sequence {v_{n}(t)} defined by (1.20) converges to v(t) uniformly on [t_{0}, ∞). To see how fast v_{n}(t) approaches v(t) we proceed as follows. First, note that |v_{n}(t)| ≤ αP (t), t ≥ t_{0}, n = 1, 2, · · ·. By definition, we have

|v_{1}(t)| = α ∫_{t}^{∞} |Q(s)|^{1+ \frac{1}{α}} ds ≤ α ∫_{t}^{∞} P (s)^{1+ \frac{1}{α}} ds ≤ c(t)^{\frac{1}{α}} P (t),

and

|v_{2}(t) − v_{1}(t)| ≤ α ∫_{t}^{∞} ||v_{1}(s) + Q(s)|^{1+ \frac{1}{α}} − |Q(s)|^{1+ \frac{1}{α}} | ds

≤ α (1 + \frac{1}{α}) ∫_{t}^{∞}[(1 + α)P (s)]^{1+ \frac{1}{α}} |v_{1}(s)|ds

≤ (α + 1)^{1+ \frac{1}{α}} ∫_{t}^{∞} c(s)^{\frac{1}{α}} P (s)^{1+ \frac{1}{α}} ds ≤ (α + 1)^{1+ \frac{1}{α}} c(t)^{\frac{1}{α}} ∫_{t}^{∞} P (s)^{1+ \frac{1}{α}} ds

≤ \frac{1}{α}(α + 1)^{1+ \frac{1}{α}} c(t)^{\frac{2}{α}} P (t) ≤ E(α)^{\frac{1}{α}} [\frac{c(t)}{E(α)}]^{\frac{2}{α}} P (t)

for t ≥ t_{0}. Assuming that

|v_{n}(t) − v_{n−1}(t)| ≤ E(α)^{\frac{1}{α}} [\frac{c(t)}{E(α)}]^{\frac{n}{α}} P (t), t ≥ t_{0}  (1.21)

for some n ∈ ℕ, we compute |v_{n+1}(t) − v_{n}(t)| ≤ α ∫_{t}^{∞} ||Q(s) + v_{n}(s)|^{1+ \frac{1}{α}} − |Q(s) + v_{n−1}(s)|^{1+ \frac{1}{α}} | ds

≤ α (1 + \frac{1}{α}) ∫_{t}^{∞}[(1 + α)P (s)]^{\frac{1}{α}} |v_{n}(s) − v_{n−1}(s)|ds

= (α + 1)^{1+ \frac{1}{α}} ∫_{t}^{∞} E(α)^{\frac{1}{α}} [\frac{c(s)}{E(α)}]^{\frac{n}{α}} P (s)^{1+ \frac{1}{α}} ds

= (α + 1)^{1+ \frac{1}{α}} E(α)^{\frac{1}{α}} [\frac{c(t)}{E(α)}]^{\frac{n}{α}} ∫_{t}^{∞} P (s)^{1+ \frac{1}{α}} ds

≤ (α + 1)^{1+ \frac{1}{α}} E(α)^{\frac{1}{α}} [\frac{c(t)}{E(α)}]^{\frac{n}{α}} \frac{1}{α}c(t)^{\frac{1}{α}} P (t)

= E(α)^{\frac{1}{α}} [\frac{c(t)}{E(α)}]^{\frac{n+1}{α}} P (t), t ≥ t_{0},

which establishes the truth of (1.21) for all integers n ∈ ℕ.

Now we have

v(t) = v_{n−1}(t) + r_{n}(t)

with

r_{n}(t) = ∑_{k=n}^{∞}[v_{k}(t) − v_{k−1}(t)],

from which, due to (1.21), it follows that

|v(t) − v_{n−1}(t)| ≤ ∑_{k=n}^{∞} E(α)^{\frac{1}{α}} [\frac{c(t)}{E(α)}]^{\frac{k}{α}} P (t)

≤ E(α)^{\frac{1}{α}} [\frac{c(t)}{E(α)}]^{\frac{n}{α}} ∑_{k=0}^{∞} (\frac{c}{E(α)})^{k} P (t) (1.22)

= E(α) [\frac{c(t)}{E(α)}]^{\frac{n}{α}} \frac{E(α)}{E(α) − c}P (t) = Kc(t)^{\frac{n}{α}} P (t)

for t ≥ t_{0}, where K is a constant depending only on α and n.

Using (1.5) and (1.22), we obtain

\frac{y(t)}{exp \{∫_{t_{0}}^{t}[Q(s) + v_{n−1}(s)]^{\frac{1}{α} ∗}ds\}}

(1.23)

= exp \{∫_{t_{0}}^{t} ([Q(s) + v(s)]^{\frac{1}{α} ∗} − [Q(s) + v_{n−1}(s)]^{\frac{1}{α} ∗}) ds\} .

Let 0 < α ≤ 1. Then, by the mean value theorem and (1.22),

|[Q(t) + v(t)]^{\frac{1}{α} ∗} − [Q(t) + v_{n−1}(t)]^{\frac{1}{α} ∗}| ≤ \frac{1}{α}[(1 + α)P (t)]^{\frac{1}{α} −1}|v(t) − v_{n−1}(t)|

(1.24) ≤ Lc(t)^{\frac{n}{α}} P (t)^{\frac{1}{α}} , t ≥ t_{0},

where L is a constant depending on α and n.

Let α > 1. Then, using (1.22) and the inequality |a^{θ} − b^{θ}| ≤ 2|a − b|^{θ} holding for θ ∈ (0, 1) and a, b ∈ ℝ, we see that

|[Q(t) + v(t)]^{\frac{1}{α} ∗} − [Q(t) + v_{n−1}(t)]^{\frac{1}{α} ∗}| ≤ 2|v(t) − v_{n−1}(t)|^{\frac{1}{α}}

(1.25) ≤ Mc(t)^{\frac{n}{α^{2}}} P (t)^{\frac{1}{α}} , t ≥ t_{0},

where M is a constant depending on α and n.

Combining (1.23) with (1.24) or (1.25) according as 0 < α ≤ 1 or α > 1, and using (1.17) or (1.18), we conculde that the right-hand side of (1.23) tends to a constant A > 0 as t → ∞, which implies that y(t) has the des ired asymptotic behaviour (1.19). This completes the proof.

Corollary 1.2. Suppose that (1.15) holds and that the function c(t) defined by (1.16) satisfies

∫ ^{∞} \frac{c(t)^{\frac{n+1}{α}}}{t} dt < ∞ if 0 < α ≤ 1,  (1.26)

∫ ^{∞} \frac{c(t)^{\frac{n+α}{α^{2}}}}{t} dt < ∞ if α > 1.  (1.27)

Then the formula (1.19) holds for the slowly varying solution y(t) of (A).

P r o o f. The conclusion follows from Theorem 1.2 combined with the observation that in this case c(t)^{\frac{n}{α}} P (t)^{\frac{1}{α}} = c(t)^{\frac{n+1}{α}} /t and c(t)^{\frac{n}{α^{2}}} P (t) = c(t)^{\frac{n+α}{α^{2}}} /t according to whether 0 < α ≤ 1 and α > 1.

2. Examples

Two examples illustrating our main results will be given below.

Example 2.1. Consider the equation

(|y′|^{α−1}y′)′ + kt^{β} sin(t^{γ})|y|^{α−1}y = 0, t ≥ 1,  (2.1)

where k, α, β and γ are positive constants satisfying

γ > 1 + α + β.  (2.2)

Since

∫_{t}^{∞} s^{β} sin(s^{γ})ds = \frac{1}{γ} t^{1+β−γ} cos(t^{γ}) + \frac{1 + β − γ}{γ} ∫_{t}^{∞} s^{β−γ} cos(s^{γ})ds,

there exists a positive constant K such that

|∫_{t}^{∞} ks^{β} sin(s^{γ})ds| ≤ Kt^{1+β−γ}, t ≥ 1,  (2.3)

which, in view of (2.2), implies that

lim_{t→∞} t^{α} ∫_{t}^{∞} ks^{β} sin(s^{α})ds = 0.

Therefore, the equation (2.1) has a slowly varying solution y(t) by Corollary 1.1.

In this case the function c(t) defined by (1.16) can be taken to be c(t) = Kt^{1+α+β−γ}. Since c(t) satisfies both (1.26) and (1.27) for any n ∈ ℕ because of (2.2), from Corollary 1.2 for n = 1 we conclude that th e slowly varying solution y(t) of (2.1) has the asymptotic behaviour

y(t) ∼ A exp \{∫_{t_{0}}^{t} (∫_{s}^{∞} kr^{β} sin(r^{γ})dr)^{\frac{1}{α} ∗} ds\} as t → ∞,  (2.4)

which is equivalent to y(t) ∼ A_{0} (constant), since the integral in the braces in (2.4) converges as t → ∞ because of (2.3).

Example 2.2. Consider the equation

(|y′|^{α−1}y′)′ + \frac{a + b sin t}{t^{β}(log t)^{γ}}|y|^{α−1}y = 0, t ≥ e,  (2.5)

where the constants appearing in (2.5) are positive except for a, and satisfy β ≥ α + 1 and |a| < b.

I) We first suppose that a ≠ 0. Note that, for β > 1,

Q(t) = ∫_{t}^{∞} \frac{a + b sin s}{s^{β}(log s)^{γ}}ds = \frac{a}{β − 1}t^{1−β}(log t)^{−γ} [1 + O (\frac{1}{t})] .  (2.6)

Let β > α + 1. Then, (Q(t))^{\frac{1}{α} ∗} is absolutely integrable on [e, ∞) and t^{α}Q(t) → 0 as t → ∞. Corollary 1.1 then implies that (2.5) possesses a slowly varying solution y(t).

The function c(t) = (2|a|/(β − 1))t^{1+α−β}(log t)^{−γ} defined by (1.16) satisfies the conditions (1.26) and (1.27) for any n ∈ ℕ, so that, by Corollary 1.2 with n = 1, y(t) enjoys the asymptotic property

y(t) ∼ A exp \{∫_{t_{0}}^{t}(Q(s))^{\frac{1}{α} ∗}ds\} ∼ A_{0} as t → ∞.

Let β = α + 1. We see that t^{α}Q(t) → 0 as t → ∞ also in this case, so that (2.5) has a slowly varying solution y(t). As easily verified, the function c(t) = (2|a|/α)(log t)^{−γ} satisfies the conditi ons (1.26) and (1.27) become, respectively,

∫ ^{∞} t^{−1}(log t)^{− \frac{(n+1)γ}{α}} dt < ∞ (0 < α ≤ 1)  (2.7)

and

∫ ^{∞} t^{−1}(log t)^{− \frac{(n+α)γ}{α^{2}}} dt < ∞ (α > 1),  (2.8)

which are fulfilled if one determines n to satify

n > \frac{α − γ}{γ} (0 < α ≤ 1) or n > \frac{α(α − γ)}{γ} (α > 1).  (2.9)

For practical use write (2.9) as

γ > \frac{α}{n + 1} (0 < α ≤ 1) or γ > \frac{α^{2}}{n + α} (α > 1),

which is equivalent to

γ > α max \{\frac{1}{n + 1}, \frac{α}{n + α}\} .  (2.10)

Obviously, the range γ > α max \{\frac{1}{2}, \frac{α}{1+α}\} is such that (2.10) i.e., (2.9) holds for n = 1, so that Corollary 1.2 can be applied with n = 1, leading to

y(t) ∼ A exp \{∫_{t_{0}}^{t}(Q(s))^{\frac{1}{α} ∗}ds\}

(2.11)

∼ A′ exp \{(\frac{a}{α})^{\frac{1}{α} ∗} ∫_{t_{0}}^{t} s^{−1}(log s)^{− \frac{γ}{α}} ds\} as t → ∞,

from which it readily follows that

y(t) ∼ A_{0} if γ > α

and

y(t) ∼ A_{0}(log t)^{δ}, δ = (\frac{a}{α})^{\frac{1}{α} ∗} if γ = α.

Arguing in the same way, we conclude that (2.9) holds for n = 2 in the range

α max \{\frac{1}{3}, \frac{α}{2 + α}\} < γ ≤ α max \{\frac{1}{2}, \frac{α}{1 + α}\} .  (2.12)

Then, the conclusion of Corollary 1.2 holds with n = 2, that is,

y(t) ∼ A exp \{∫_{t_{0}}^{t}[v_{1}(s) + Q(s)]^{\frac{1}{α} ∗}ds\} as t → ∞,  (2.13)

where v_{1}(t) = α ∫_{t}^{∞} |Q(s)|^{1+ \frac{1}{α}} ds. Using (2.6), we have

v_{1}(t) = α ∫_{t}^{∞} |\frac{a}{α}s^{−α}(log s)^{−γ} [1 + O (\frac{1}{s})]|^{1+ \frac{1}{α}} ds

(2.14)

= |\frac{a}{α}|^{1+ \frac{1}{α}} t^{−α}(log t)^{−γ(1+ \frac{1}{α})} [1 + O (\frac{1}{log t})].

Putting

w_{1}(t) = |\frac{a}{α}|^{1+ \frac{1}{α}} t^{−α}(log t)^{−γ(1+ \frac{1}{α})},  (2.15)

we claim that

y(t) ∼ A′ exp \{∫_{t_{0}}^{t}[w_{1}(s) + Q(s)]^{\frac{1}{α} ∗}ds\} as t → ∞.  (2.16)

In fact, if α > 1, then

∫_{t_{0}}^{t} |[v_{1}(s) + Q(s)]^{\frac{1}{α} ∗} − [w_{1}(s) + Q(s)]^{\frac{1}{α} ∗}| ds

(2.17)

≤ 2 ∫_{t_{0}}^{t} |v_{1}(s) − w_{1}(s)|^{\frac{1}{α} ∗}ds ≤ K ∫_{t_{0}}^{t} s^{−1}(log s)^{− \frac{γ}{α}(1+ \frac{1}{α})− \frac{1}{α}} ds,

where K is a constant depending on α and a. Since γ > α/(α+2) by (2.12),

\frac{γ}{α} (1 + \frac{1}{α}) + \frac{1}{α} > 1 + \frac{2}{α(α + 2)} > 1,

which implies that the last integral in (2.17) converges as t → ∞. If 0 < α ≤ 1, then, using the inequality |v_{1}(t)|, |w_{1}(t)| ≤ αP (t) = 2|a|t^{−α}(log t)^{−γ} already known, we obtain

∫_{t_{0}}^{t} |[v_{1}(s) + Q(s)]^{\frac{1}{α} ∗} − [w_{1}(s) + Q(s)]^{\frac{1}{α} ∗}| ds

≤ M_{1} ∫_{t_{0}}^{t}[s^{−α}(log s)^{−γ}]^{\frac{1}{α} −1}|v_{1}(s) − w_{1}(s)|ds

(2.18)

≤ M_{2} ∫_{t_{0}}^{t}[s^{−α}(log s)^{−γ}]^{\frac{1}{α} −1}s^{−α}(log s)^{−γ(1+ \frac{1}{α})−1}ds

= M_{3} ∫_{t_{0}}^{t} s^{−1}(log s)^{− \frac{2γ}{α} −1}ds,

the last integral of which clearly converges as t → ∞. Here M_{i}, i = 1, 2, 3, are constants depending only on α and a. Combining (2.16) with (2.15) establishes the asymptotic formula for t → ∞

y(t) ∼ A′′ exp \{∫_{t_{0}}^{t} [|\frac{a}{α}|^{1+ \frac{1}{α}} s^{−α}(log s)^{−γ(1+ \frac{1}{α})} + \frac{a}{α}s^{−α}(log s)^{−γ}]^{\frac{1}{α} ∗} ds\} .

(2.19)

Observe that when specialized to the case α = 1, (2.19) reduces to the following formulas obtained in [3], cf. [5, p.67],

y(t) ∼ A_{1}(log t)^{a^{2}} exp \{2a(log t)^{\frac{1}{2}} \}  if γ = \frac{1}{2},

y(t) ∼ A_{1} exp \{\frac{a}{1 − γ} (log t)^{1−γ}\} exp \{\frac{a^{2}}{1 − 2γ} (log t)^{1−2γ}\} if \frac{1}{3} < γ < \frac{1}{2}.

Let α = \frac{1}{2}, for example. Then, (2.19) implies

y(t) ∼ A_{2}(log t)^{32|a|^{3}a} exp \{8a^{2}(log t)^{\frac{1}{2}} \}  if γ = \frac{1}{4},

y(t) ∼ A_{2} exp \{\frac{32|a|^{3}a}{1 − 4γ}(log t)^{1−4γ}\} exp \{\frac{4a^{2}}{1 − 2γ} (log t)^{1−2γ}\} if \frac{1}{6} < γ < \frac{1}{4}.

II) Next we consider the equation (2.5) with a = 0, that is,

(|y′|^{α−1}y′)′ + \frac{b sin t}{t^{β}(log t)^{γ}} |y|^{α−1}y = 0, t ≥ e,  (2.20)

where b > 0 is a constant. We suppose that β ≥ α. In this case we have

Q(t) = bt^{−β}(log t)^{−γ} cos t + O(t^{−β−1}(log t)^{−γ}) as t → ∞,

and t^{α}Q(t) → 0 as t → ∞, which implies that (2.20) possesses a slowly varying solution y(t).

If β > α, then, by taking c(t) = 2bt^{α−β}(log t)^{−γ}, we see that (1.26) and (1.27) are satisfied for all n ∈ ℕ, and so from Corollary 1.2 with n = 1 it follows that y(t) ∼ A_{0} as t → ∞ since [Q(t)]^{\frac{1}{α} ∗} is integrable on [e, ∞).

If β = α, then c(t) = 2b(log t)^{−γ} satisfies (1.26) and (1.27) if and only if (2.10) holds. Consequently, if γ > α max \{\frac{1}{2}, \frac{α}{1+α}\}, then Corollary 1.2 is applicable to the c ase n = 1 and, using the conditional integrability of [Q(t)]^{\frac{1}{α} ∗} which is implied by that of t^{−1}(log t)^{− \frac{α}{γ}} cos t, we conclude that y(t) ∼ A_{0} as t → ∞. Furthermore, if γ satisfies (2.12), then from Corollary 1.2 with n = 2 we obtain (2.13), which, with the use of the fact

v_{1}(t)=α ∫_{t}^{∞} |Q(s)|^{1+ \frac{1}{α}} ds = |b|^{1+ \frac{1}{α}} t^{−α}(log t)^{−γ(1+ \frac{1}{α})}| cos t|^{1+ \frac{1}{α}} (1+O (\frac{1}{log t}))

as t → ∞, yields the following asymptotic formula for y(t):

y(t) ∼ A′ exp \{ ∫_{t_{0}}^{t} [|b|^{1+ \frac{1}{α}} s^{−α}(log s)^{−γ(1+ \frac{1}{α})}| cos s|^{1+ \frac{1}{α}} (2.21)

+ bs^{−α}(log s)^{−γ} cos s]^{\frac{1}{α} ∗}ds\}.

When specialized to the case α = 1, (2.21) reduces to

y(t) ∼ A_{1}(log t)^{\frac{b^{2}}{2}} if γ = \frac{1}{2}

y(t) ∼ A_{1} exp \{\frac{b^{2}}{2(1 − 2γ)}(log t)^{1−2γ}\} if \frac{1}{3} < γ < \frac{1}{2},

which have been obtained in [5, p. 68]. Letting α = \frac{1}{3} in (2.21), an elementary calculation shows that

y(t) ∼ A_{2}(log t)^{\frac{3}{8} b^{6}} if γ = \frac{1}{6}

y(t) ∼ A_{2} exp \{\frac{3b^{2}}{8(1 − 6γ)}(log t)^{1−6γ}\} if \frac{1}{9} < γ < \frac{1}{6}.

REFERENCES

[1] N. H. B i n g h a m, C. M. G o l d i e and J. L. T e u g e l s, Regular Variation, Encyclopedia of Mathematics and its Applications 27, Cambridge Univ. Press, 1987.

[2] J. L. G e l u k, On slowly varying solutions of the linear second order differential equations, Publ. Inst. Math. (Beograd) 48 (62) (1990), 52 – 60.

[3] H. H o w a r d, V. M a r i ć and Z. R a d a š i n, Asymptotics of nonoscillatory solutions of second order linear differential equations, Zbornik Rad. Prir. Mat. Fak. Univ. Novi Sad, Ser. Mat. 20, 1 (1990), 107 – 116.

[4] J. J a r o š, T. K u s a n o and T. T a n i g a w a, Nonoscillation theory for second order half-linear differential equations in the framework of regular variation, Results Math. (to appear).

[5] V. M a r i ć, Regular Variation and Differential Equations, Lecture Notes in Mathematics 1726, Springer-Verlag, Berlin-Heidelberg-New York, 2000.

[6] V. M a r i ć and M. T o m i ć, Slowly varying solutions of second order linear differential equations, Publ. Inst. Math. (Beograd) 58 (72) (1995), 129 – 136.

Kusano Takasi Department of Applied Mathematics Faculty of Science Fukuoka University,8-l9-l Nanakuma Jonan-Ku,Fukuoka 8l4-0l80 Japan Tomoyuki Tanigawa Department of Mathematics Toyama National College of Technology l3Hongo-cho Toyama 939-8630 Japan

Vojislav Marić Serbian Academy of Sciences and Arts Knez Mihajlova 35 ll000 Beograd Serbia and Montenegro