Bulletin T.CXXIX de l’Académie serbe des sciences et des arts - 2004 Classe des aciences mathématiques et naturelles sciences mathématiques, N\underline{o} 29

SELF-ADJOINT DIFFERENTIAL EQUATIONS AND GENERALIZED KARAMATA FUNCTIONS

J. JAROŠ, T. KUSANO

(Presented at the 1st Meeting, held on February 27, 2004)

A b s t r a c t. Howard and Marić have recently developed nice nonoscillation theorems for the differential equation

y′′ + q(t)y = 0  (∗)

by means of regularly varying functions in the sense of Karamata. The purpose of this paper is to show that their results can be fully generalized to differential equations of the form

(p(t)y′)′ + q(t)y = 0  (∗∗)

by using the notion of generalized Karamata functions, which is needed to comprehend how delicately the asymptotic behavior of solutions of (∗∗) is affected by the function p(t).

AMS Mathematics Subject Classification (2000): Primary 34C11; Secondary 26A12

Key Words: Self-adjoint differential equation, Karamata functions, generalized Karamata functions, asymptotic behavior

0. Introduction

One of the remarkable modern achievements in the qualitative study of ordinary differential equations is the nonoscillation theory of the second order linear differential equation

y′′ + q(t)y = 0, t ≥ a,  (A)

that was built by Marić, Tomić and others by means of regularly varying functions introduced by Karamata; see e.g. the papers [4, 5, 8 – 10]. For an excellent survey of this subject and related topics the reader is referred to the monograph of Marić [7].

It is natural that there have already been attempts at extending the results for (A) to the more general self-adjoint equation

(p(t)y′)′ + q(t)y = 0, t ≥ a;  (B)

see e.g. the papers [2, 7]. However, the generalization so far seems to be far from being complete, and it would be worthwhile to examine the possibility of generalizing the known results for (A) to (B) as far as possible within the framework of Karamata functions.

The objective of this paper is to show that this kind of generalization can actually be carried out provided the classes of functions in which the solutions of (B) are sought are replaced by those of generalized Karamata functions reflecting the essential role played by the differential operator (p(t)y′)′. To be more specific, we will take up the paper by Howard and Marić [4] dealing with (A) and demonstrate that all the results therein can be transplanted to the new field of generalized Karamata functions to yield nontrivial nonoscillation theorems for the equation (B).

We assume that in (B) the functions p : [a, ∞) → (0, ∞) and q : [a, ∞) → R are continuous; q(t) is allowed to be oscillatory in the sense that it takes both positive and negative values in any neighborhood of infinity.

It is essential to distinguish the following two cases for the function p(t):

∫_{a}^{∞} \frac{dt}{p(t)} = ∞,  (0.1)

∫_{a}^{∞} \frac{dt}{p(t)} < ∞.  (0.2)

We will use extensively the functions

P (t) = ∫_{a}^{t} \frac{ds}{p(s)}  in case (0.1) hods,  (0.3)

and

π(t) = ∫_{t}^{∞} \frac{ds}{p(s)}  in case (0.2) holds.  (0.4)

Our main results are formulated and proved in Sections 2 and 3 devoted, respectively, to the cases where p(t) satisfies (0.1) and (0.2). We assume that q(t) and π(t)^{2}q(t) are integrable on [a, ∞) in Sections 2 and 3 and establish nonoscillation criteria which altogether show the delicate dependence of the asymptotic growth or decay of solutions of (B) considered as generalized Karamata functions upon the values of the limit of the functions P (t) ∫_{t}^{∞} q(s)ds and \frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q(s)ds as t → ∞. Our results exhibit a distinctive duality existing between the cases (0.1) and (0.2). The definitions and basic properties of generalized Karamata functions are summarized in Section 1 as natural generalizations of those of the classical Karamata functions which are listed in the Appendix. Some examples illustrating the main results are presented at the end of Sections 2 and 3.

1. Generalized Karamata functions

The purpose of this preparatory section is to set up the framework of positive functions which is suitable for the asymptotic analysis of the selfadjoint differential equation (B). This is done by properly generalizing the class of regularly varying functions in the sense of Karamata, based on the observation that the behavior of solutions of (B) depends heavily on the function P (t) or π(t) given by (0.3) or (0.4), respectively. In the generalization use is made of a positive function R(t) which is continuously differentiable on [t_{0}, ∞) and satisfies

R′(t) > 0 for t ≥ t_{0} and lim_{t→∞} R(t) = ∞.

The inverse function of R(t) is denoted by R^{−1}(t). In later sections it will turn out that R(t) = P (t) or R(t) = 1/π(t) is the best choice of R(t) for the analysis of the equation (B) with p(t) subject to (0.1) or (0.2), respectively.

The definitions and some basic properties of generalized Karamata functions now follow. They appear as exact parallels of those developed in the original theory of Karamata functions, which will be listed in the Appendix.

Definition 1.1. A measurable function f : [t_{0}, ∞) → (0, ∞) is said to be slowly varying with respect to R(t) if the function f ∘R^{−1}(t) is slowly varying in the sense of Definition A.1. The totality of slowly varying functions with respect to R(t) is denoted by SV_{R}.

Definition 1.2. A measurable function g : [t_{0}, ∞) → (0, ∞) is said to be regularly varying with index α with respect to R(t) if the function g ∘ R^{−1}(t) is regularly varying with index α in the sense of Definition A.2. The set of all regularly varying functions with index α with respect to R(t) is denoted by RV_{R}(α).

Of fundamental importance is the following representation theorem for the generalized slowly and regularly varying functions, which is an immediate consequence of Theorem A.1.

Proposition 1.1.(i) A function f(t) is slowly varying with respect to R(t) if and only if it can be expressed in the form

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

(1.1)

where c(t) and ε(t) are measurable functions such that

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

(ii) A function g(t) is regularly varying with index α with respect to R(t) if and only if it has the representation

g(t) = c(t) exp \{∫_{t_{0}}^{t} \frac{R′(s)}{R(s)}δ(s)ds\}
= c(t) exp {∫_{t_{0}}^{t}(log R(s))′δ(s)ds} , t ≥ t_{0},

(1.2)

where c(t) and δ(t) are measurable functions such that

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

If the function c(t) in (1.2) (or (1.3)) is identically a constant on [t_{0}, ∞), then the function f(t) (or g(t)) is called normalized slowly varying (or normalized regularly varying with index α) with respect to R(t). The totality of such functions is denoted by n-SV_{R} (or n-RV_{R}(α)).

It is easy to see that if g(t) ∈ RV_{R}(α) (or n-RV_{R}(α)), then g(t) = [R(t)]^{α}f(t) for some f(t) ∈ SV_{R} (or n-SV_{R}).

Proposition 1.2. (i) f(t) ∈ SV_{R} implies f(t)^{β} ∈ SV_{R} for any β ∈ R. f_{1}(t), f_{2}(t) ∈ SV_{R} implies f_{1}(t)f_{2}(t) ∈ SV_{R}.

(ii) Let f(t) ∈ SV_{R}. Then, for any γ > 0,

lim_{t→∞} R(t)^{γ}f(t) = ∞, lim_{t→∞} R(t)^{−γ}f(t) = 0.  (1.3)

(iii) g(t) ∈ RV_{R}(α) and β ∈ R implies g(t)^{β} ∈ RV_{R}(αβ). g_{i}(t) ∈ RV_{R}(α_{i}), i = 1, 2, implies g_{1}(t)g_{2}(t) ∈ RV_{R}(max(α_{1}, α_{2})).

Proposition 1.3. A positive measurable function f(t) belongs to n-SV_{R} if and only if, for any γ > 0, R(t)^{γ}f(t) is ultimately increasing and R(t)^{−γ}f(t) is ultimately decreasing.

Propositions 1.2 and 1.3 follow readily from Theorems A.2 and A.3, respectively. Karamata’s theorem (Theorem A.4) is generalized in the following manner.

Proposition 1.4. (i) If γ > −1, then for any f(t) ∈ SV_{R},

∫_{t_{0}}^{t} R′(s)R(s)^{γ}f(s)ds ∼ \frac{R(t)^{γ+1}f(t)}{γ + 1} as t → ∞.  (1.4)

(ii) If γ < −1, then for any f(t) ∈ SV_{R}, ∫_{t_{0}}^{∞} R′(t)R(t)^{γ}f(t)dt < ∞, and

∫_{t}^{∞} R′(s)R(s)^{γ}f(s)ds ∼ − \frac{R(t)^{γ+1}f(t)}{γ + 1} as t → ∞.  (1.5)

Here and hereafter the notation φ(t) ∼ ψ(t) as t → ∞ is used to mean the asymptotic equivalence of φ(t) and ψ(t) : lim_{t→∞} ψ(t)/φ(t) = 1.

Definition 1.3. A measurable function h : [t_{0}, ∞) → (0, ∞) is called regularly bounded with respect to R(t) if the function h∘R^{−1}(t) is regularly bounded in the sense of Definition A.3.

An immediate consequence of Theorem A.5 is the following

Proposition 1.5. A function h(t) is regularly bounded with respect to R(t) if and only if it has the representation

h(t) = exp \{η(t) + ∫_{t_{0}}^{t} \frac{R′(s)}{R(s)}ξ(s)ds\} ,  (1.6)

where ξ(t) and η(t) are bounded measurable functions on [t_{0}, ∞).

The totality of regularly bounded functions with respect to R(t) is denoted by RO_{R}.

It can be verified that the integral and the product of functions in RO_{R} remain in this class.

Remark. It would be of interest to observe that there exists a function which is slowly varying in the generalized sense but is not slowly varying in the sense of Karamata, so that, roughly speaking, the class of generalized Karamata functions is larger than that of classical Karamata functions. In fact, using the notation

exp_{0} t = t, exp_{n} t = exp(exp_{n−1} t),
log_{0} t = t, log_{n} t = log(log_{n−1} t), n = 1, 2, ⋯,

we define the functions ϕ_{n}(t) and f_{n}(t) for n ∈ ℤ by

ϕ_{n}(t) = exp_{n} t, ϕ_{−n}(t) = log_{n} t, n = 0, 1, 2, ⋯,

and

f_{n}(t) = 2 + sin ϕ_{n}(t), n = 0, ±1, ±2, · · · .

Since ϕ_{n}^{−1}(t) = ϕ_{−n}(t) and ϕ_{m} ∘ ϕ_{n}(t) = ϕ_{m+n}(t) for any m, n ∈ ℤ, we have

f_{n} ∘ ϕ_{m}^{−1}(t) = f_{n−m}(t)

for any n, m ∈ ℤ, from which, by taking into account the fact that

f_{n}(t) ∈ SV for n ≦ −2 and f_{n}(t) ∉ SV for n ≧ −1,

we conclude that

f_{n}(t) ∉ SV and f_{n}(t) ∈ SV_{ϕ_{m}} if n ≧ −1 and m ≧ n + 2.

2. Nonoscillation criteria (The first case)

A) We begin the study of nonoscillation of the equation (B) with the case where the function p(t) satisfies the condition (0.1). We assume that the function q(t) is integrable on [a, ∞) and satisfies one of the four conditions listed below:

−\frac{1}{4} < lim inf_{t→∞} P (t) ∫_{t}^{∞} q(s)ds ≤ lim sup_{t→∞} P (t) ∫_{t}^{∞} q(s)ds < \frac{1}{4},  (2.1)

lim_{t→∞} P (t) ∫_{t}^{∞} q(s)ds = 0,  (2.2)

−∞ < lim_{t→∞} P (t) ∫_{t}^{∞} q(s)ds < \frac{1}{4},  (2.3)

lim_{t→∞} P (t) ∫_{t}^{∞} q(s)ds = \frac{1}{4},  (2.4)

where P (t) is defined by (0.3). Note that lim_{t→∞} P (t) = ∞ by (0.1).

Our objective here is to prove four nonoscillation theorems which show how subtly the asymptotic behaviour of all solutions of (B) is affected by the values of the limits appearing in (2.1)–(2.4).

Theorem 2.1. If (2.1) holds, then the equation (B) is nonoscillatory and all of its solutions are regularly bounded with respect to P (t).

Theorem 2.2. If (2.2) holds, then the equation (B) is nonoscillatory and has a fundamental set of solutions {y_{1}(t), y_{2}(t)} such that

y_{1}(t) ∈ n-SV_{P} , y_{2}(t) ∈ n-RV_{P} (1).  (2.5)

It will be shown that these theorems follow immediately from the next existence principle.

Lemma 2.1. Suppose there exists a continuous function Q̃ : [t_{0}, ∞) → (0, ∞) with the properties that lim_{t→∞} Q̃(t) = 0,

|∫_{t}^{∞} q(s)ds| ≤ Q̃(t), t ≥ t_{0},  (2.6)

∫_{t}^{∞} \frac{Q̃(s)^{2}}{p(s)} ds ≤ cQ̃(t), t ≥ t_{0},  (2.7)

where c ∈ (0, \frac{1}{4}) is a constant. Then, the equation (B) is nonoscillatory and has a solution of the form

y(t) = exp \{∫_{t_{0}}^{t} \frac{Q(s) + v(s)}{p(s)}ds\} , t ≥ t_{0},  (2.8)

where Q(t) = ∫_{t}^{∞} q(s)ds and v(t) = O(Q̃(t)) as t → ∞.

P r o o f o f L e m m a 2.1. We look for a solution y(t) of (B) expressed in the form (2.8), which amounts to requiring that the function u(t) = Q(t) + v(t) satisfies the Riccati equation

u′ + \frac{u^{2}}{p(t)} + q(t) = 0, t ≥ t_{0}.  (2.9)

The differential equation for v(t) then reads:

v′ + \frac{(Q(t) + v(t))^{2}}{p(t)} = 0, t ≥ t_{0},  (2.10)

which, combined with the requirement lim_{t→∞} v(t) = 0, yields the integral equation

v(t) = ∫_{t}^{∞} \frac{(Q(s) + v(s))^{2}}{p(s)} ds, t ≥ t_{0}.  (2.11)

We will solve (2.11) by means of the contraction mapping principle. Let C_{Q}[t_{0}, ∞) denote the set of all continuous functions v(t) on [t_{0}, ∞)

such that

∥v∥_{Q} = sup_{t≥t_{0}} \frac{|v(t)|}{Q̃(t)} < ∞.  (2.12)

Clearly, C_{Q}[t_{0}, ∞) is a Banach space equipped with the norm ∥v∥_{Q}. Define the set V ⊂ C_{Q}[t_{0}, ∞) and the integral operator F acting on V as follows:

V = {v ∈ C_{Q}[t_{0}, ∞) : |v(t)| ≤ Q̃(t), t ≥ t_{0}} ,  (2.13)

Fv(t) = ∫_{t}^{∞} \frac{(Q(s) + v(s))^{2}}{p(s)} ds, t ≥ t_{0}.  (2.14)

If v ∈ V, then, using (2.6) and (2.7), we have

|Fv(t)| ≤ 4 ∫_{t}^{∞} \frac{Q̃(s)^{2}}{p(s)} ds ≤ 4cQ̃(t) ≤ Q̃(t), t ≥ t_{0},

so that Fv ∈ V. Furthermore, if v, w ∈ V, then

|Fv(t) − Fw(t)| ≤ ∫_{t}^{∞} \frac{2|Q(s)| + |v(s)| + |w(s)|}{p(s)}|v(s) − w(s)|ds

≤ 4 ∫_{t}^{∞} \frac{Q̃(s)^{2}}{p(s)} \frac{|v(s) − w(s)|}{Q̃(s)}ds ≤ 4∥v − w∥_{Q} ∫_{t}^{∞} \frac{Q̃(s)^{2}}{p(s)} ds

≤ 4c∥v − w∥_{Q} Q̃(t), t ≥ t_{0},

from which it follows that

∥Fv − Fw∥_{Q} ≤ 4c∥v − w∥_{Q}.

By the contraction mapping principle there exists a unique function v ∈ V such that v = Fv, which clearly is a solution of the integral equation (2.11), and hence, of the differential equation (2.10). Therefore, the function y(t) defined by (2.8) with this v(t) provides the desired solution of (B) on [t_{0}, ∞). That v(t) = O(Q̃(t)) as t → ∞ is a consequence of the fact that v ∈ V. This completes the proof of Lemma 2.1.

P r o o f o f T h e o r e m 2.1. Assume that (2.1) holds. Then, there exist positive constants c < \frac{1}{4} and t_{0} such that

|P (t) ∫_{t}^{∞} q(s)ds| ≤ c, that is, |∫_{t}^{∞} q(s)ds| ≤ \frac{c}{P (t)}, t ≥ t_{0}.

Put Q̃(t) = c/P (t). Then,

∫_{t}^{∞} \frac{Q̃(s)^{2}}{p(s)} ds = ∫_{t}^{∞} \frac{1}{p(s)} (\frac{c}{P (s)})^{2} ds = \frac{c^{2}}{P (t)} = cQ̃(t), t ≥ t_{0},

and so by Lemma 2.1 (B) has a solution y_{1}(t) of the form

y_{1}(t) = exp \{∫_{t_{0}}^{t} \frac{Q(s) + v(s)}{p(s)}ds\} , t ≥ t_{0},

with v(t) satisfying v(t) = O(1/P (t)) as t → ∞. Rewriting y_{1}(t) as

y_{1}(t) = exp \{∫_{t_{0}}^{t} \frac{P (s)(Q(s) + v(s))}{p(s)P (s)}ds\}

and noting that η(t) = P (t)(Q(t) + v(t)) is bounded on [t_{0}, ∞), we see that y_{1}(t) is regularly bounded with respect to P (t).

The second (linearly independent) solution of (B) is given by

y_{2}(t) = y_{1}(t) ∫_{t_{0}}^{t} \frac{ds}{p(s)y_{1}(s)^{2}} or y_{2}(t) = y_{1}(t) ∫_{t}^{∞} \frac{ds}{p(s)y_{1}(s)^{2}} (2.15)

according to whether y_{1}(t) is a principal solution or a non-principal solution of (B) in the sense that

∫_{t_{0}}^{∞} \frac{dt}{p(t)y_{1}(t)^{2}} = ∞, or ∫_{t_{0}}^{∞} \frac{dt}{p(t)y_{1}(t)^{2}} < ∞,  (2.16)

respectively. In either case it can be verified without difficulty that y_{2}(t) is also regularly bounded with respect to P (t) by using the fact that the integral and the product of regularly bounded functions with respect to P (t) is also a function of the same type. Finally, all solutions of (B) are regularly bounded with respect to P (t) since they are linear combinations of y_{1}(t) and y_{2}(t). This completes the proof of Theorem 2.1.

P r o o f o f T h e o r e m 2.2. Assume that (2.2) holds. Put

c(t) = sup_{s≥t} |P (s) ∫_{s}^{∞} q(r)dr| , t ≥ t_{0}.

It is clear that c(t) is decreasing to zero as t → ∞ and satisfies

|∫_{t}^{∞} q(s)ds| ≤ \frac{c(t)}{P (t)}, t ≥ t_{0}.

The function Q̃(t) = c(t)/P (t) satisfies

∫_{t}^{∞} \frac{Q̃(s)^{2}}{p(s)} ds = ∫_{t}^{∞} \frac{1}{p(s)} (\frac{c(s)}{P (s)})^{2} ds ≤ \frac{c(t)^{2}}{P (t)} = c(t)Q̃(t), t ≥ t_{0}.

Consequently, Lemma 2.1 ensures that (B) has a solution y_{1}(t) of the form

y_{1}(t) = exp \{∫_{t_{0}}^{t} \frac{P (s)(Q(s) + v(s))}{p(s)P (s)}ds\} , t ≥ t_{0},

which is slowly varying with respect to P (t) since the function ε(t) = P (t)(Q(t) + v(t)) → 0 as t → ∞ by construction. We claim that y_{1}(t) is a principal solution of (B) in the sense of (2.16). In fact, since lim_{t→∞} P (t)^{\frac{1}{2}} /y_{1}(t)^{2} = ∞ by Proposition 1.2-(ii) (note that 1/y_{1}(t)^{2} ∈ SV_{P}),

there is a constant m > 0 such that P (t)^{\frac{1}{2}} /y_{1}(t)^{2} ≥ m for t ≥ t_{0}, and we have

∫_{t_{0}}^{t} \frac{ds}{p(s)y_{1}(s)^{2}} = ∫_{t_{0}}^{t} \frac{P (s)^{− \frac{1}{2}}}{p(s)} \frac{P (s)^{\frac{1}{2}}}{y_{1}(s)^{2}} ds ≥ m ∫_{t_{0}}^{t} \frac{P (s)^{− \frac{1}{2}}}{p(s)} ds
= 2m [P (t)^{\frac{1}{2}} − P (t_{0})^{\frac{1}{2}} ] → ∞ as t → ∞.

The second (linearly independent) solution y_{2}(t) of (B) is determined by the first formula in (2.15). Applying first Proposition 1.4-(i), we see that

∫_{t_{0}}^{t} \frac{ds}{p(s)y_{1}(s)^{2}} ∼ \frac{P (t)}{y_{1}(t)^{2}} as t → ∞,

and then using Proposition 1.2-(i), we conclude that

y_{2}(t) ∼ \frac{P (t)}{y_{1}(t)} ∈ n-RV_{P} (1) as t → ∞.

This completes the proof of Theorem 2.2.

B) A different approach makes it possible to generalize Theorem 2.2 so as to cover the case where the condition (2.3) is satisfied.

Let c ∈ (−∞, \frac{1}{4}) and denote by λ_{1} and λ_{2}, λ_{1} < λ_{2}, the real roots of the quadratic equation

λ^{2} − λ + c = 0.  (2.17)

Theorem 2.3. The equation (B) is nonoscillatory and has a fundamental set of solutions {y_{1}(t), y_{2}(t)} such that

y_{1}(t) ∈ n-RV_{P} (λ_{1}), y_{2}(t) ∈ n-RV_{P} (λ_{2})

if and only if

lim_{t→∞} P (s) ∫_{t}^{∞} q(s)ds = c.  (2.18)

P r o o f. (The “only if”part) Suppose that (B) has solutions y_{i}(t) belonging to n-RV_{P} (λ_{i}), i = 1, 2, which can be expressed as

y_{i}(t) = exp \{∫_{t_{0}}^{t} \frac{δ_{i}(s)}{p(s)P (s)}ds\} , lim_{t→∞} δ_{i}(t) = λ_{i}, i = 1, 2. (2.19)

Put u_{i}(t) = p(t)y′_{i}(t)/y_{i}(t), i = 1, 2. We integrate the Riccati equation (2.9) satisfied by u_{i}(t) and noting in view of (2.19) that u_{i}(t) = δ_{i}(t)/P (t) → 0 as t → ∞, we obtain

u_{i}(t) = ∫_{t}^{∞} \frac{u_{i}(s)^{2}}{p(s)} ds + ∫_{t}^{∞} q(s)ds, t ≥ t_{0}, i = 1, 2,

whence it follows that the functions v_{i}(t) = P (t)u_{i}(t) satisfy

v_{i}(t) = P (t) ∫_{t}^{∞} \frac{v_{i}(s)^{2}}{p(s)P (s)^{2}} ds + P (t) ∫_{t}^{∞} q(s)ds, t ≥ t_{0}, i = 1, 2.

Passing to the limit as t → ∞ in the above and noting that lim_{t→∞} v_{i}(t) = λ_{i}, we conclude that

lim_{t→∞} P (t) ∫_{t}^{∞} q(s)ds = λ_{i} − λ_{i}^{2} = c, i = 1, 2.

(The “if”part) Assume that (2.18) holds. Put Φ(t) = P (t) ∫_{t}^{∞} q(s)ds−c. The desired solutions will be sought in the form

y_{i}(t) = exp \{∫_{t_{i}}^{t} \frac{Φ(s) + λ_{i} + v_{i}(s)}{p(s)P (s)}ds\}  (2.20)

for some t_{i} > a and v_{i}(t) such that lim_{t→∞} v_{i}(t) = 0. The requirement that u_{i}(t) = (Φ(t) + λ_{i} + v_{i}(t))/P (t) satisfy (2.9) yields the differential equations for v_{i}(t):

v′_{i} + \frac{2Φ(t) + 2λ_{i} − 1}{p(t)P (t)}v_{i} + \frac{Φ(t)^{2} + 2λ_{i}Φ(t) + v_{i}^{2}}{p(t)P (t)} = 0.  (2.21)

It is convenient to rewrite (2.21) as

(ρ_{i}(t)v_{i})′ + \frac{ρ_{i}(t)}{p(t)P (t)} [Φ(t)^{2} + 2λ_{i}Φ(t) + v_{i}^{2}] = 0,  (2.22)

where

ρ_{i}(t) = exp \{∫_{1}^{t} \frac{2Φ(s) + 2λ_{i} − 1}{p(s)P (s)}ds\} , i = 1, 2.  (2.23)

Let us first consider the case i = 1. Note that ρ_{1}(t) ∈ n-RV_{P} (2λ_{1} − 1). Since 2λ_{1} − 1 < 0, lim_{t→∞} ρ_{1}(t) = 0 by Proposition 1.2-(ii), and integrating (2.22) from t to ∞, we obtain

v_{1}(t) = \frac{1}{ρ_{1}(t)} ∫_{t}^{∞} \frac{ρ_{1}(s)}{p(s)P (s)} [Φ(s)^{2} + 2λ_{1}Φ(s) + v_{1}(s)^{2}] ds,  (2.24)

which will be solved below by means of the contraction mapping theorem. For this purpose we need the following two properties of ρ_{1}(t), both of which are immediate consequences of L’Hospital’s rule:

lim_{t→∞} \frac{1}{ρ_{1}(t)} ∫_{t}^{∞} \frac{ρ_{1}(s)}{p(s)P (s)}ds = \frac{1}{1 − 2λ_{1}} > 0,  (2.25)

lim_{t→∞} \frac{1}{ρ_{1}(t)} ∫_{t}^{∞} \frac{ρ_{1}(s)}{p(s)P (s)}h(s)ds = 0 if h(t) ∈ C[a, ∞) and lim_{t→∞} h(t) = 0.

(2.26)

Choose a positive constant ε_{1} such that 8ε_{1}/(1 − 2λ_{1}) < 1 and let t_{1} > a large enough so that

|Φ(t)^{2} + 2λ_{1}Φ(t)| ≤ ε_{1}^{2}, t ≥ t_{1},  (2.27)

and

\frac{1}{ρ_{1}(t)} ∫_{t}^{∞} \frac{ρ_{1}(s)}{p(s)P (s)}ds ≤ \frac{2}{1 − 2λ_{1}} , t ≥ t_{1}.  (2.28)

Let C_{0}[t_{1}, ∞) denote the Banach space of continuous functions on [t_{1}, ∞) tending to zero as t → ∞ with the norm ∥v∥_{0} = sup_{t≥t_{1}} |v(t)|. Define

V_{1} = {v ∈ C_{0}[t_{1}, ∞) : |v(t)| ≤ ε_{1}, t ≥ t_{1}} ,  (2.29)

F_{1}v(t) = \frac{1}{ρ_{1}(t)} ∫_{t}^{∞} \frac{ρ_{1}(s)}{p(s)P (s)} [Φ(s)^{2} + 2λ_{1}Φ(s) + v(s)^{2}] ds, t ≥ t_{1}.

(2.30)

It can be shown that F_{1} is a contraction mapping on V_{1}. In fact, if v ∈ V_{1}, then

|F_{1}v(t)| ≤ 2ε_{1}^{2} \frac{1}{ρ_{1}(t)} ∫_{t}^{∞} \frac{ρ_{1}(s)}{p(s)P (s)}ds ≤ \frac{4ε_{1}^{2}}{1 − 2λ_{1}} ≤ \frac{ε_{1}}{2} , t ≥ t_{1},

and if v, w ∈ V, then

\begin{matrix}|F_{1}v(t) − F_{1}w(t)| & ≤ \frac{1}{ρ_{1}(t)} ∫_{t}^{∞} \frac{ρ_{1}(s)}{p(s)P (s)} (|v(s)| + |w(s)|) |v(s) − w(s)|ds \\ < 2ε_{1}∥v − w∥_{0} \frac{1}{ρ_{1}(t)} ∫_{t}^{∞} \frac{ρ_{1}(s)}{p(s)P (s)}ds ≤ \frac{4ε_{1}}{1 − 2λ_{1}} ∥v − w∥_{0}, t ≥ t_{1}, \end{matrix}

which implies that ∥Fv − Fw∥_{0} ≤ \frac{1}{2}∥v − w∥_{0}.

Therefore, there exists a unique element v_{1} ∈ V_{1} such that v_{1} = F_{1}v_{1}. Since v_{1}(t) is a solution of (2.24), and hence of (2.21), the function y_{1}(t) defined by (2.20) with this v_{1}(t) gives a solution of the differential equation (B) on [t_{1}, ∞). That y_{1}(t) ∈ n-RV_{P} (λ_{1}) follows from the fact that Φ(t) + λ_{1} + v_{1}(t) → λ_{1} as t → ∞ (cf. Proposition 1.1-(ii)).

The solution y_{1}(t) can be represented as y_{1}(t) = P (t)^{λ_{1}}f_{1}(t), f_{1}(t) ∈ n-SV_{P}, so that we see that there is a constant M > 0 such that y_{1}(t)^{2} = P (t) · P (t)^{2λ_{1}−1}f_{1}(t)^{2} ≤ MP (t), t ≥ t_{1}, because P (t)^{2λ_{1}−1}f_{1}(t)^{2} → 0 as t → ∞ by Proposition 1.2-(ii). It follows that

∫_{t_{1}}^{t} \frac{ds}{p(s)y_{1}(s)^{2}} ≥ \frac{1}{M} ∫_{t_{1}}^{t} \frac{ds}{p(s)P (s)} = \frac{1}{M} log \frac{P (t)}{P (t_{1})} → ∞ as t → ∞,

which ensures that y_{1}(t) is a principal solution of (B).

Our next task is to examine the case i = 2. In this case, the desired solution y_{2}(t) ∈ n-RV_{P} (λ_{2}) of (B) is given by (2.20) after having obtained v_{2}(t) as a solution to the first order differential equation (2.22) (i = 2) formed by means of the function ρ_{2}(t) defined by (2.23). It is clear that lim_{t→∞} ρ_{2}(t) = ∞ since 2λ_{2} − 1 > 0 (cf. Proposition 1.2-(ii)). Besides, using L’Hospital’s rule we see that ρ_{2}(t) satisfies

lim_{t→∞} \frac{1}{ρ_{2}(t)} ∫_{t_{2}}^{t} \frac{ρ_{2}(s)}{p(s)P (s)}ds = \frac{1}{2λ_{2} − 1} > 0,  (2.31)

lim_{t→∞} \frac{1}{ρ_{2}(t)} ∫_{t_{2}}^{t} \frac{ρ_{2}(s)}{p(s)P (s)}h(s)ds = 0 if h(t) ∈ C[a, ∞) and lim_{t→∞} h(t) = 0,

(2.32)

for any fixed t_{2} > a.

Let ε_{2} be a positive constant such that 8ε_{2}/(2λ_{2} − 1) < 1 and choose t_{2} > a so that

|Φ(t)^{2} + 2λ_{2}Φ(t)| ≤ ε_{2}^{2}, t ≥ t_{2},  (2.33)
\frac{1}{ρ_{2}(t)} ∫_{t_{2}}^{t} \frac{ρ_{2}(s)}{p(s)P (s)}ds ≤ \frac{2}{2λ_{2} − 1}, t ≥ t_{2}.  (2.34)

It is a matter of easy calculation to show that the integral operator F_{2} given by

F_{2}v(t) = −\frac{1}{ρ_{2}(t)} ∫_{t_{2}}^{t} \frac{ρ_{2}(s)}{p(s)P (s)} [Φ(s)^{2} + 2λ_{2}Φ(s) + v(s)^{2}] ds, t ≥ t_{2},

(2.35)

is a contraction mapping acting on the set

V_{2} = {v ∈ C_{0}[t_{2}, ∞) : |v(t)| ≤ ε_{2}, t ≥ t_{2}} .  (2.36)

Let v ∈ V_{2} be the fixed element of F_{2}. Then it solves the integral equation

v_{2}(t) = −\frac{1}{ρ_{2}(t)} ∫_{t_{2}}^{t} \frac{ρ_{2}(s)}{p(s)P (s)} [Φ(s)^{2} + 2λ_{2}Φ(s) + v_{2}(s)^{2}] ds, t ≥ t_{2}, (2.37)

and a fortiori the differential equation (2.22). This function v_{2}(t) is used to determine the solution y_{2}(t) of (B) by (2.20). That y_{2}(t) ∈ n-RV_{P} (λ_{2}) is a consequence of the fact that Φ(t) + λ_{2} + v_{2}(t) → λ_{2} as t → ∞.

Let y_{2}(t) = P (t)^{λ_{2}}f_{2}(t), f_{2}(t) ∈ n-SV_{P}. Since P (t)^{\frac{2λ_{2}−1}{2}} f_{2}(t) → ∞ as t → ∞ by Proposition 1.2-(ii), we have

y_{2}(t)^{2} = P (t)^{\frac{2λ_{2}+1}{2}} · P (t)^{\frac{2λ_{2}−1}{2}} f_{2}(t) ≥ mP (t)^{\frac{2λ_{2}+1}{2}} , t ≥ t_{2},

for some constant m > 0, and so we find

∫_{t_{2}}^{t} \frac{ds}{p(s)y_{2}(s)^{2}} ≤ \frac{1}{m} ∫_{t_{2}}^{t} \frac{ds}{p(s)P (s)^{\frac{2λ_{2}+1}{2}}} < \frac{2}{2λ_{2} − 1}P (t_{2})^{\frac{1−2λ_{2}}{2}} , t ≥ t_{2},

which implies that y_{2}(t) is a non-principal solution of (B). Thus the proof of Theorem 2.3 is complete.

Remark 2.1. In the proof of the “if”part of Theorem 2.3 both of the two linearly independent solutions y_{1}(t) and y_{2}(t) of (B) have been constructed under the condition (2.3) by means of the contraction mapping principle. Actually, it suffices to establish the existence of either of them, because if y_{1}(t) (or y_{2}(t)) has been found first, then the other one y_{2}(t) (or y_{1}(t)) can be obtained by the formula

y_{2}(t) = y_{1}(t) ∫_{t_{2}}^{t} \frac{ds}{p(s)y_{1}(s)^{2}} (y_{1}(t) = y_{2}(t) ∫_{t}^{∞} \frac{ds}{p(s)y_{2}(s)^{2}} )

and it can be concluded that y_{1}(t) ∈ n-RV_{P} (λ_{1}) implies y_{2}(t) ∈ n-RV_{P} (λ_{2}) (or y_{2}(t) ∈ n-RV_{P} (λ_{2}) implies y_{1}(t) ∈ n-RV_{P} (λ_{1})) from (i) (or (ii)) of the generalized Karamata theorem (Proposition 1.4).

C) Our consideration in this subsection is devoted to the case where the condition (2.4) is satisfied for the equation (B). This case is critical in the sense that nothing definite can be said about the oscillation or nonoscillation of (B) unless additional conditions are placed on p(t) and q(t). Simple but nontrivial conditions guaranteeing the nonoscillation of (B) in this critical case are given in the following

Theorem 2.4. Assume that (2.4) holds. Put

Φ(t) = P (t) ∫_{t}^{∞} q(s)ds − \frac{1}{4}  (2.38)

40 J. Jaroš, T. Kusano

and suppose that

∫ ^{∞} \frac{|Φ(t)|}{p(t)P (t)}dt < ∞,  (2.39)
∫ ^{∞} \frac{Ψ(t)}{p(t)P (t)}dt < ∞, where Ψ(t) = ∫_{t}^{∞} \frac{|Φ(s)|}{p(s)P (s)}ds.  (2.40)

Then the equation (B) possesses a fundamental set of solutions {y_{1}(t), y_{2}(t)} such that y_{i}(t) ∈ n-RV_{P} (\frac{1}{2}) , i = 1, 2, and

y_{1}(t) = P (t)^{\frac{1}{2}} f_{1}(t), y_{2}(t) = P (t)^{\frac{1}{2}} log P (t)f_{2}(t),  (2.41)

where f_{i}(t) ∈ n-SV_{P} and lim_{t→∞} f_{i}(t) = f_{i}(∞) ∈ (0, ∞), i = 1, 2, with f_{1}(∞)f_{2}(∞) = 1.

P r o o f. We seek a solution of (B) expressed in the form

y_{1}(t) = exp \{∫_{t_{0}}^{t} \frac{Φ(s) + \frac{1}{2} + v(s)}{p(s)P (s)}ds\} .  (2.42)

The same argument as in the proof of the “if”part of Theorem 2.3 leads to the differential equation for v(t):

(ρ(t)v)′ + \frac{ρ(t)}{p(t)P (t)} [Φ(t)^{2} + Φ(t) + v^{2}] = 0,  (2.43)

where ρ(t) is given by

ρ(t) = exp \{− ∫_{1}^{t} \frac{2Φ(s)}{p(s)P (s)}ds\} .  (2.44)

Let m > 0 be a constant such that |Φ(t)^{2} + Φ(t)| ≤ mΦ(t), t ≥ a, which is possible by (2.4). Noting that lim_{t→∞} ρ(t) = const > 0 by (2.39) and using (2.40), we see that there exists t_{0} > a such that

\frac{ρ(s)}{ρ(t)} ≤ 2, s ≥ t ≥ t_{0},  (2.45)
32m ∫_{t_{0}}^{∞} \frac{Ψ(s)}{p(s)P (s)}ds ≤ 1.  (2.46)

Let C_{Ψ}[t_{0}, ∞) denote the Banach space of all continuous functions on [t_{0}, ∞) such that sup_{t≥t_{0}}{|v(t)|/Ψ(t)} < ∞, and let V ⊂ C_{Ψ}[t_{0}, ∞) and G :

V → C_{Ψ}[t_{0}, ∞) be defined by

V = {v ∈ C_{Ψ}[t_{0}, ∞) : |v(t)| ≤ 4mΨ(t), t ≥ t_{0}} ,  (2.47)
Gv(t) = \frac{1}{ρ(t)} ∫_{t}^{∞} \frac{ρ(s)}{p(s)P (s)} [Φ(s)^{2} + Φ(s) + v(s)^{2}] ds, t ≥ t_{0}. (2.48)

We show that G is a contraction mapping on V. In fact, if v ∈ V, then

|Gv(t)| ≤ \frac{1}{ρ(t)} ∫_{t}^{∞} \frac{ρ(s)}{p(s)P (s)} [m|Φ(t)| + 16m^{2}Ψ(s)^{2}] ds

≤ 2m ∫_{t}^{∞} \frac{|Φ(s)|}{p(s)P (s)}ds + 32m^{2}Ψ(t) ∫_{t}^{∞} \frac{Ψ(s)}{p(s)P (s)}ds

≤ 2mΨ(t) + mΨ(t) ≤ 4mΨ(t), t ≥ t_{0},

and if v, w ∈ V, then

|Gv(t) − Gw(t)| ≤ \frac{1}{ρ(t)} ∫_{t}^{∞} \frac{ρ(s)}{p(s)P (s)}(|v(s)| + |w(s)|)|v(s) − w(s)|ds

≤ 16m ∫_{t}^{∞} \frac{Ψ(s)}{p(s)P (s)}|v(s) − w(s)|ds

= 16m ∫_{t}^{∞} \frac{Ψ(s)^{2}}{p(s)P (s)} \frac{|v(s) − w(s)|}{Ψ(s)}ds

≤ 16m ∫_{t_{0}}^{∞} \frac{Ψ(s)}{p(s)P (s)}ds∥v − w∥_{Ψ}Ψ(t), t ≥ t_{0},

which implies that ∥Gv − Gw∥_{Ψ} ≤ \frac{1}{2}∥v − w∥_{Ψ}.

Let v ∈ V be the fixed element of G. Then it satisfies the integral equation

v(t) = \frac{1}{ρ(t)} ∫_{t}^{∞} \frac{ρ(s)}{p(s)P (s)} [Φ(s)^{2} + Φ(s) + v(s)^{2}] ds, t ≥ t_{0}, (2.49)

and hence the differential equation (2.43) for t ≥ t_{0}. It follows that the function y_{1}(t) defined by (2.42) with this v(t) provides a solution of the equation (B) on [t_{0}, ∞).

To see that y_{1}(t) ∈ n-RV_{P} (\frac{1}{2}) it suffices to rewrite y_{1}(t) as

y_{1}(t) = P (t)^{\frac{1}{2}} f_{1}(t), f_{1}(t) = exp \{∫_{t_{0}}^{t} \frac{Φ(s) + v(s)}{p(s)P (s)}ds\} ,

and note that f_{1}(t) ∈ n-SV_{P} since Φ(t) + v(t) → 0 as t → ∞. Furthermore, f_{1}(t) has a positive limit as t → ∞ since (2.39) and (2.40) ensure that (Φ(t) + v(t))/p(t)P (t) is absolutely integrable on [t_{0}, ∞). Using this fact, it is shown that the integral of 1/p(t)y_{1}(t)^{2} on [t_{0}, ∞) diverges, which implies that y_{1}(t) is a principal solution of (B).

The second (linearly independent) solution y_{2}(t) given by

y_{2}(t) = y_{1}(t) ∫_{t_{0}}^{t} \frac{ds}{p(s)y_{1}(s)^{2}} = P (t)^{\frac{1}{2}} f_{1}(t) ∫_{t_{0}}^{t} \frac{ds}{p(s)P (s)f_{1}(s)^{2}} , t ≥ t_{0},

satisfies, via L’Hospital’s rule,

\frac{y_{2}(t)}{P (t)^{\frac{1}{2}} log P (t)} ∼ \frac{1}{f_{1}(∞)} \frac{∫_{t_{0}}^{t} \frac{ds}{p(s)P (s)}}{log P (t)} ∼ \frac{1}{f_{1}(∞)} as t → ∞.

Therefore, y_{2}(t) can be expressed as y_{2}(t) = P (t)^{\frac{1}{2}} log P (t)f_{2}(t) with f_{2}(t) ∈ n-SV_{P} such that lim_{t→∞} f_{2}(t) = 1/f_{1}(∞). Thus the proof is complete.

D) Example 2.1. The Hermite equation

(e^{−t^{2}}y′)′ + λe^{−t^{2}}y = 0, t ≥ 0,  (2.50)

λ ∈ R being a parameter, is a special case of (B) with p(t) = e^{−t^{2}} and q(t) = λe^{−t^{2}}. The function defined by (0.3) is P (t) = ∫_{0}^{t} e^{s^{2}}ds. Since

lim_{t→∞} P (t) ∫_{t}^{∞} q(s)ds = lim_{t→∞} (∫_{0}^{t} e^{s^{2}}ds) (∫_{t}^{∞} λe^{−s^{2}}ds) = 0,

by Theorem 2.2 the equation (2.50) is nonoscillatory for any λ and has a fundamental set of solutions {y_{1}(t), y_{2}(t)} such that y_{1}(t) ∈ n-SV_{P} , y_{2}(t) ∈ n-RV_{P} (1).

Let H_{n}(t) be the Hermite polynomial of degree n ∈ N, which is a solution of (2.50) for the case λ = 2n. That y_{1}(t) = H_{n}(t) ∈ n-SV_{P} follows from Proposition 1.3, since, for any γ > 0, P (t)^{γ}H_{n}(t) is increasing to ∞ and P (t)^{−γ}H_{n}(t) is decreasing to 0 as t → ∞.

Example 2.2. Consider the equation

(ty′)′ + [kt^{α} sin(t^{β}) + \frac{c}{t(log t)^{2}} ] y = 0, t ≥ e,  (2.51)

where k, α and β are positive constants. In this case the function P (t) defined by (0.3) can be taken to be P (t) = log t.

Suppose that β > 1 + α. Then, we have

∫_{t}^{∞} s^{α} sin(s^{β})ds = \frac{1}{β} t^{1+α−β} cos(t^{β})+ \frac{1 + α − β}{β} ∫_{t}^{∞} s^{α−β} cos(s^{β})ds, (2.52)

from which it follows that

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

and

lim_{t→∞} log t ∫_{t}^{∞} [ks^{α} sin(s^{β}) + \frac{c}{s(log s)^{2}} ] ds = c.  (2.53)

Let c ∈ (−∞, \frac{1}{4}). Then, by Theorem 2.3, the equation (2.51) is nonoscillatory and has a pair of solutions {y_{1}(t), y_{2}(t)} such that y_{i}(t) ∈ n-RV_{log t}(λ_{i}), i = 1, 2, where λ_{1}, λ_{2} (λ_{1} < λ_{2}) are the real roots of λ^{2} − λ + c = 0.

Now let c = \frac{1}{4}. Put

Φ(t) = log t ∫_{t}^{∞} [ks^{α} sin(s^{β}) + \frac{1}{4s(log s)^{2}} ] ds − \frac{1}{4} = log t ∫_{t}^{∞} ks^{α} sin(s^{β})ds.

We then see from (2.52) that

\frac{|Φ(t)|}{tP (t)} = \frac{1}{t} |∫_{t}^{∞} ks^{α} sin(s^{β})ds| = O (t^{α−β}) as t → ∞,

which implies that |Φ(t)|/tP (t) is integrable on [e, ∞) and

Ψ(t) = ∫_{t}^{∞} \frac{|Φ(s)|}{sP (s)}ds = O (t^{1+α−β}) as t → ∞.

Since α − β < −1, Ψ(t)/tP (t) = O (t^{α−β}) / log t is integrable on [e, ∞), and Theorem 2.4 guarantees the nonoscillation of (2.51) as well as the existence of two linearly independent solutions of the form

y_{1}(t) = (log t)^{\frac{1}{2}} f_{1}(t), y_{2}(t) = (log t)^{\frac{1}{2}} log(log t)f_{2}(t),

with f_{i}(t) ∈ n-SV_{log t}, i = 1, 2, satisfying f_{2}(t) ∼ 1/f_{1}(t) as t → ∞ and

lim_{t→∞} f_{1}(t) = f_{1}(∞) ∈ (0, ∞).

We notice that the coefficient q(t) = kt^{α} sin(t^{β}) + \frac{c}{t(log t)^{2}} is strongly oscillating in the sense that

lim inf_{t→∞} q(t) = −∞ and lim sup_{t→∞} q(t) = ∞.

3. Nonoscillation criteria (The second case)

A) We now turn to the study of equation (B) in the case where p(t) satisfies the condition (0.2). We assume here that the function π(t)^{2}q(t) is integrable on [a, ∞), where π(t) is defined by (0.4), and establish four nonoscillation criteria corresponding to the following additional conditions on the integral of π(t)^{2}q(t):

−\frac{1}{4} < lim inf_{t→∞} \frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q(s)ds ≤ lim sup_{t→∞} \frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q(s)ds < \frac{1}{4}; (3.1)

lim_{t→∞} \frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q(s)ds = 0;  (3.2)
−∞ < lim_{t→∞} \frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q(s)ds < \frac{1}{4};  (3.3)
lim_{t→∞} \frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q(s)ds = \frac{1}{4}.  (3.4)

Theorem 3.1. If (3.1) holds, then the equation (B) is nonoscillatory and all of its solutions are regularly bounded with respect to 1/π(t).

Theorem 3.2. If (3.2) holds, then the equation (B) is nonoscillatory and has a fundamental set of solutions {y_{1}(t), y_{2}(t)} such that

y_{1}(t) ∈ n-RV_{\frac{1}{π}} (−1), y_{2}(t) ∈ n-SV_{\frac{1}{π}} .  (3.5)

These two theorems are counterparts of Theorems 2.1 and 2.2 and follow readily from the lemma below which parallels Lemma 2.1.

Lemma 3.1. Suppose that there exists a continuous function Q̃ : [t_{0}, ∞) → (0, ∞) with the properties that lim_{t→∞} Q̃(t) = 0,

|∫_{t}^{∞} π(s)^{2}q(s)ds| ≤ Q̃(t), t ≥ t_{0},  (3.6)

∫_{t}^{∞} \frac{Q̃(s)^{2}}{p(s)π(s)^{2}} ds ≤ cQ̃(t), t ≥ t_{0},  (3.7)

where c ∈ (0, \frac{1}{4}) is a constant. Then, the equation (B) is nonoscillatory and has a solution of the form

y(t) = exp \{∫_{t_{0}}^{t} \frac{Q(s) − π(s) + v(s)}{p(s)π(s)^{2}}ds\} ,  (3.8)

where Q(t) = ∫_{t}^{∞} π(s)^{2}q(s)ds and v(t) = O(Q̃(t)) as t → ∞.

P r o o f o f L e m m a 3.1. A nonoscillatory solution y(t) of (B) will be sought in the form (3.8). It suffices to determine v(t) so that u(t) = (Q(t) − π(t) + v(t))/π(t)^{2} satisfies the Riccati equation (2.9). The differential equation for v(t) then reads:

v′ + \frac{(Q(t) + v)^{2}}{p(t)π(t)^{2}} = 0, t ≥ t_{0},  (3.9)

which, upon integrating with the additional requirement lim_{t→∞} v(t) = 0, yields

v(t) = ∫_{t}^{∞} \frac{(Q(s) + v(s))^{2}}{p(s)π(s)^{2}} ds, t ≥ t_{0}.  (3.10)

As in the proof of Lemma 2.1 let C_{Q}[t_{0}, ∞) denote the set of all continuous functions u(t) on [t_{0}, ∞) such that (2.12) is satisfied, and let the set V of continuous functions and the integral operator F acting on V be defined by

V = {v ∈ C_{Q}[t_{0}, ∞) : |v(t)| ≤ Q̃(t), t ≥ t_{0}}  (3.11)

and

Fv(t) = ∫_{t}^{∞} \frac{(Q(s) + v(s))^{2}}{p(s)π(s)^{2}} ds, t ≥ t_{0}.  (3.12)

It is easy to verify that F maps V into itself and satisfies

∥Fv − Fw∥_{Q} ≤ 4c∥v − w∥_{Q},

so that F has a unique fixed element v ∈ V. This function v = v(t) satisfies (3.10), and hence (3.9), on [t_{0}, ∞), and so the function y(t) defined by (3.8) with this v(t) gives rise to a solution of (B) on [t_{0}, ∞) with the required property. This completes the proof.

P r o o f o f T h e o r e m 3.1. The condition (3.1) is equivalent to the existence of positive constants c < 1/4 and t_{0} > a such that

|\frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q(s)ds| ≤ c or |∫_{t}^{∞} π(s)^{2}q(s)ds| ≤ cπ(t), t ≥ t_{0}.

Put Q̃(t) = cπ(t). Then, it satisfies

∫_{t}^{∞} \frac{Q̃(s)^{2}}{p(s)π(s)^{2}} ds = c^{2} ∫_{t}^{∞} \frac{ds}{p(s)} = c^{2}π(t) = cQ̃(t), t ≥ t_{0},

and so, by Lemma 3.1, (B) has a nonoscillatory solution y_{1}(t) of the form

y_{1}(t) = exp \{∫_{t_{0}}^{t} \frac{Q(s) − π(s) + v(s)}{p(s)π(s)^{2}}ds\} , t ≥ t_{0}.

Rewriting y_{1}(t) as

y_{1}(t) = exp \{∫_{t_{0}}^{t} \frac{\frac{1}{π(s)}(Q(s) − π(s) + v(s))}{p(s)π(s)}ds\}

and noting that (Q(t) − π(t) + v(t))/π(t) is bounded on [t_{0}, ∞), we see that y_{1}(t) is regularly bounded with respect to 1/π(t). The second (linearly independent) solution y_{2}(t) given by (2.15) is also regularly bounded with respect to 1/π(t). The proof of Theorem 3.1 is complete.

P r o o f o f T h e o r e m 3.2. Define c(t) by

c(t) = sup_{s≥t} \frac{1}{π(s)} |∫_{s}^{∞} π(r)^{2}q(r)dr| .

Then c(t) decreases to 0 as t → ∞ and there exists t_{0} > a such that c(t) < 1/4 and

|∫_{t}^{∞} π(s)^{2}q(s)ds| ≤ c(t)π(t), t ≥ t_{0}.

If we put Q̃(t) = c(t)π(t), then

∫_{t}^{∞} \frac{Q̃(s)^{2}}{p(s)π(s)^{2}} ds ≤ c(t)^{2} ∫_{t}^{∞} \frac{ds}{p(s)} = c(t)^{2}π(t) = c(t)Q̃(t), t ≥ t_{0}.

Therefore, Lemma 3.1 is applicable to this case and it follows that (B) possesses a solution y_{1}(t) of the form

y_{1}(t) = exp \{∫_{t_{0}}^{t} \frac{Q(s) − π(s) + v(s)}{p(s)π(s)^{2}}ds\} , t ≥ t_{0},

which can be written as

y_{1}(t) = c_{0}π(t) exp \{∫_{t_{0}}^{t} \frac{Q(s) + v(s)}{p(s)π(s)^{2}}ds\} , t ≥ t_{0}

for some constant c_{0} > 0. This implies that y_{1}(t) ∈ n-RV_{\frac{1}{π}} (−1), since (Q(t) + v(t))/π(t) = O(c(t)) as t → ∞. This solution is a principal solution of (B). To see this, let y_{1}(t) = π(t)f_{1}(t), f_{1}(t) ∈ n-SV_{\frac{1}{π}}. Choose a positive constant ε < 1. Then, by Proposition 1.2-(ii), lim_{t→∞}(1/π(t))^{ε}f_{1}(t)^{−2} = ∞,

so that there exists a constant m > 0 such that (1/π(t))^{ε}f_{1}(t)^{−2} ≥ m for t ≥ t_{0}. We then have

∫_{t_{0}}^{t} \frac{ds}{p(s)y_{1}(s)^{2}} ≥ m ∫_{t_{0}}^{t} \frac{ds}{p(s)π(s)^{2−ε}} = \frac{m}{1 − ε} [π(t)^{ε−1} − π(t_{0})^{ε−1}] → ∞, t → ∞,

which implies that y_{1}(t) is a principal solution of (B) as claimed.

Let y_{2}(t) be the second solution of (B) defined by the first formula in (2.15). Using the relation

∫_{t_{0}}^{t} \frac{ds}{p(s)y_{1}(s)^{2}} = ∫_{t_{0}}^{t} (\frac{1}{π(s)})′ \frac{ds}{f_{1}(s)^{2}} ∼ \frac{1}{π(t)f_{1}(t)^{2}} as t → ∞,

following from Proposition 1.4-(i), we arrive at the desired conclusion that

y_{2}(t) ∼ \frac{1}{f_{1}(t)} ∈ n-SV_{\frac{1}{π}} as t → ∞.

This completes the proof.

B) We now take up the condition (3.3) which can be written as

lim_{t→∞} \frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q(s)ds = c ∈ (−∞, \frac{1}{4}) .  (3.13)

Theorem 3.3. Let c ∈ (−∞, \frac{1}{4}) and let μ_{1} and μ_{2}, μ_{1} < μ_{2}, denote the two real roots of the quadratic equation

μ^{2} + μ + c = 0.  (3.14)

The equation (B) is nonoscillatory and has a fundamental set of solutions {y_{1}(t), y_{2}(t)} such that

y_{1}(t) ∈ n-RV_{\frac{1}{π}} (μ_{1}), y_{2}(t) ∈ n-RV_{\frac{1}{π}} (μ_{2})

if and only if the condition (3.13) is satisfied.

P r o o f. (The “only if”part) Suppose that there exist linearly independent solutions y_{i}(t) ∈ n-RV_{\frac{1}{π}} (μ_{i}), i = 1, 2, which have representations

y_{i}(t) = exp \{∫_{t_{i}}^{t} \frac{δ_{i}(s)}{p(s)π(s)}ds\} , lim_{t→∞} δ_{i}(t) = μ_{i}, i = 1, 2.  (3.15)

Put u_{i}(t) = p(t)y′_{i}(t)/y_{i}(t) and v_{i}(t) = π(t)u_{i}(t), i = 1, 2. In view of (3.15) we see that v_{i}(t) → μ_{i} as t → ∞ and from the Riccati equations (2.9) satisfied by u_{i}(t) we have the differential equations for v_{i}(t):

(π(t)v_{i})′ + \frac{2v_{i} + v_{i}^{2}}{p(t)} + π(t)^{2}q(t) = 0, t ≥ t_{i}, i = 1, 2.

Integration of the above equation from t to ∞ yields

v_{i}(t) = \frac{1}{π(t)} ∫_{t}^{∞} \frac{2v_{i}(s) + v_{i}(s)^{2}}{p(s)} ds+\frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q(s)ds, t ≥ t_{i}, i = 1, 2.

(3.16)

Letting t → ∞ in (3.16), we conclude that

μ_{i} = 2μ_{i} + μ_{i}^{2} + lim_{t→∞} \frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q(s)ds,

which clearly implies (3.13).

(The “if”part) Suppose that (3.3) holds. We put

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

and try to find a solution of (B) of the form

y_{i}(t) = exp \{∫_{t_{i}}^{t} \frac{φ(s) + μ_{i} + v_{i}(s)}{p(s)π(s)}ds\} .  (3.17)

The functions v_{i}(t) should be determined from the requirement that u_{i}(t) = (φ(t) + μ_{i} + v_{i}(t))/π(t) satisfy the Riccati equation (2.9). A straightforward calculation leads to the differential equations for v_{i}(t):

v′_{i} + \frac{2φ(t) + 2μ_{i} + 1}{p(t)π(t)}v_{i} + \frac{φ(t)^{2} + (2μ_{i} + 1)φ(t) + v_{i}^{2}}{p(t)π(t)} = 0,  (3.18)

which are conveniently rewritten as follows:

(ρ_{i}(t)v_{i})′ + \frac{ρ_{i}(t)}{p(t)π(t)} [φ(t)^{2} + (2μ_{i} + 1)φ(t) + v_{i}^{2}] = 0,  (3.19)

where

ρ_{i}(t) = exp \{∫_{1}^{t} \frac{2φ(s) + 2μ_{i} + 1}{p(s)π(s)}ds\} , i = 1, 2.  (3.20)

We first consider the case i = 1. Then, ρ_{1}(t) ∈ n-RV_{\frac{1}{π}} (2μ_{1} + 1) from (3.20), which implies that lim_{t→∞} ρ_{1}(t) = 0 since 2μ_{1} + 1 < 0. To obtain a solution v_{1}(t) of (3.19) such that lim_{t→∞} v_{1}(t) = 0 it suffices to solve the integral equation

v_{1}(t) = \frac{1}{ρ_{1}(t)} ∫_{t}^{∞} \frac{ρ_{1}(s)}{p(s)π(s)} [φ(s)^{2} + (2μ_{1} + 1)φ(s) + v_{1}(s)^{2}] ds. (3.21)

It is easy to show, via L’Hospital’s rule, that:

lim_{t→∞} \frac{1}{ρ_{1}(t)} ∫_{t}^{∞} \frac{ρ_{1}(s)}{p(s)π(s)}ds = −\frac{1}{2μ_{1} + 1} > 0,  (3.22)

lim_{t→∞} \frac{1}{ρ_{1}(t)} ∫_{t}^{∞} \frac{ρ_{1}(s)}{p(s)π(s)}h(s)ds = 0 if h(t) ∈ C[a, ∞) and lim_{t→∞} h(t) = 0.

(3.23)

Let ε_{1} be a positive constant such that −8ε_{1}/(2μ_{1} + 1) < 1, and choose t_{1} > a large enough so that

|φ(t)^{2} + (2μ_{1} + 1)φ(t)| ≤ ε_{1}^{2}, t ≥ t_{1},  (3.24)

and

\frac{1}{ρ_{1}(t)} ∫_{t}^{∞} \frac{ρ_{1}(s)}{p(s)π(s)}ds ≤ −\frac{2}{2μ_{1} + 1}, t ≥ t_{1}.  (3.25)

Consider the set V_{1} of continuous functions and the integral operator F_{1} defined by

V_{1} = {v ∈ C_{0}[t_{1}, ∞) : |v(t)| ≤ ε_{1}, t ≥ t_{1}} ,  (3.26)

F_{1}v(t) = \frac{1}{ρ_{1}(t)} ∫_{t}^{∞} \frac{ρ_{1}(s)}{p(s)π(s)} [φ(s)^{2} + (2μ_{1} + 1)φ(s) + v(s)^{2}] ds, t ≥ t_{1}.

(3.27)

Then it can be shown that F_{1}(V_{1}) ⊂ V_{1} and

∥F_{1}v − F_{1}w∥_{0} ≤ −\frac{4}{2μ_{1} + 1}ε_{1}∥v − w∥_{0} ≤ \frac{1}{2}∥v − w∥_{0}, v, w ∈ V_{1}.

The contraction mapping principle then ensures the existence of a fixed element v_{1} ∈ V_{1} of F_{1}, which gives rise to a solution of (3.21) on [t_{1}, ∞). Thus a solution y_{1}(t) of (B) is obtained by using this v_{1}(t) in (3.15) and it belongs to n-RV_{\frac{1}{π}} (μ_{1}) because lim_{t→∞} v_{1}(t) = 0.

Let y_{1}(t) = (1/π(t))^{μ_{1}}f_{1}(t), f_{1}(t) ∈ n-SV_{\frac{1}{π}}. Since 2μ_{1}+1 < 0 and f_{1}(t) ∈

n-SV_{\frac{1}{π}}, there exists a constant k > 0 such that (1/π(t))^{− \frac{2μ_{1}+1}{2}} f_{1}(t)^{−2} ≥ k for t ≥ t_{1} (cf. Proposition 1.2-(ii)). Using this inequality, we have

∫_{t_{1}}^{t} \frac{ds}{p(s)y_{1}(s)^{2}} = ∫_{t_{1}}^{t} \frac{1}{p(s)} (\frac{1}{π(s)})^{−2μ_{1}} f_{1}(s)^{−2}ds

= ∫_{t_{1}}^{t} (\frac{1}{π(s)})′ (\frac{1}{π(s)})^{−2μ_{1}−2} f_{1}(s)^{−2}ds

= ∫_{t_{1}}^{t} (\frac{1}{π(s)})′ (\frac{1}{π(s)})^{−μ_{1}− \frac{3}{2}} (\frac{1}{π(s)})^{− \frac{2μ_{1}+1}{2}} f_{1}(s)^{−2}ds

≥ k ∫_{t_{1}}^{t} (\frac{1}{π(s)})′ (\frac{1}{π(s)})^{−μ_{1}− \frac{3}{2}} ds

= −\frac{k}{μ_{1} + \frac{1}{2}} [(\frac{1}{π(t)})^{−μ_{1}− \frac{1}{2}}−(\frac{1}{π(t_{1})})^{−μ_{1}− \frac{1}{2}} ] → ∞ as t → ∞,

which implies that y_{1}(t) is a principal solution of (B)

We next consider the case i = 2. The function ρ_{2}(t) given by (3.20) (i = 2) satisfies lim_{t→∞} ρ_{2}(t) = ∞ and has the properties that

lim_{t→∞} \frac{1}{ρ_{2}(t)} ∫_{t_{2}}^{t} \frac{ρ_{2}(s)}{p(s)π(s)}ds = \frac{1}{2μ_{2} + 1} > 0,  (3.28)

lim_{t→∞} \frac{1}{ρ_{2}(t)} ∫_{t_{2}}^{t} \frac{ρ_{2}(s)}{p(s)π(s)}h(s)ds = 0 if h(t) ∈ C[a, ∞) and lim_{t→∞} h(t) = 0,

(3.29)

for any fixed t_{2} > a, which are verifiable by means of L’Hospital’s rule.

Let ε_{2} be a positive constant such that 8ε_{2}/(2μ_{2} +1) ≤ 1 and take t_{2} > a large enough so that

|φ(t)^{2} + (2μ_{2} + 1)φ(t)| ≤ ε_{2}^{2}, t ≥ t_{2},  (3.30)
\frac{1}{ρ_{2}(t)} ∫_{t_{2}}^{t} \frac{ρ_{2}(s)}{p(s)π(s)}ds ≤ \frac{2}{2μ_{2} + 1}, t ≥ t_{2}.  (3.31)

Consider the integral operator F_{2} defined by

F_{2}v(t) = −\frac{1}{ρ_{2}(t)} ∫_{t_{2}}^{t} \frac{ρ_{2}(s)}{p(s)π(s)} [φ(s)^{2} + (2μ_{2} + 1)φ(s) + v(s)^{2}] ds, t ≥ t_{2},

(3.32)

on the set

V_{2} = {v ∈ C_{0}[t_{2}, ∞) : |v(t)| ≤ ε_{2}, t ≥ t_{2}} .  (3.33)

It is shown that F_{2} sends V_{2} into itself and satisfies

∥F_{2}v − F_{2}w∥_{0} < \frac{4ε_{2}}{2μ_{2} + 1}∥v − w∥_{0} ≤ \frac{1}{2}∥v − w∥_{0}, v, w ∈ V_{2}.

Therefore F_{2} has a unique fixed element v ∈ V_{2}, which solves the integral equation

v_{2}(t) = −\frac{1}{ρ_{2}(t)} ∫_{t_{2}}^{t} \frac{ρ_{2}(s)}{p(s)π(s)} [φ(s)^{2} + (2μ_{2} + 1)φ(s) + v_{2}(s)^{2}] ds, t ≥ t_{2}.

(3.34)

Since v_{2}(t) satisfies the differential equation (3.21) (i = 2), the function y_{2}(t) defined by (3.15) with this v_{2}(t) gives a solution of (B) on [t_{2}, ∞) belonging to n-RV_{\frac{1}{π}} (μ_{2}). Let y_{2}(t) = (1/π(t))^{μ_{2}}f_{2}(t), f_{2}(t) ∈ n-SV_{\frac{1}{π}}. Noting that there

is a constant l > 0 such that (1/π(t))^{\frac{2μ_{2}+1}{2}} f_{2}(t)^{−2} ≤ l, t ≥ t_{2}, by Proposition 1.2-(ii), we obtain

∫_{t_{2}}^{t} \frac{ds}{p(s)y_{2}(s)^{2}} = ∫_{t_{2}}^{t} \frac{1}{p(s)} (\frac{1}{π(s)})^{−2μ_{2}} f_{2}(s)^{−2}ds

= ∫_{t_{2}}^{t} (\frac{1}{π(s)})′ (\frac{1}{π(s)})^{−3μ_{2}− \frac{5}{2}} (\frac{1}{π(s)})^{\frac{2μ_{2}+1}{2}} f_{2}(s)^{−2}ds

≤ l ∫_{t_{2}}^{t} (\frac{1}{π(s)})′ (\frac{1}{π(s)})^{−3μ_{2}− \frac{5}{2}} ds

≤ \frac{l}{3μ_{2} + \frac{3}{2}} (\frac{1}{π(t_{2})})^{−3μ_{2}− \frac{3}{2}} , t ≥ t_{2},

which shows that y_{2}(t) is a non-principal solution of (B). This completes the proof of Theorem 3.3.

Remark 3.1. We observe that to prove Theorem 3.3 it suffices to construct either of the solutions y_{1}(t) and y_{2}(t) of (B). In fact, if a principal solution y_{1}(t) ∈ n-RV_{\frac{1}{π}} (μ_{1}) has been found, then the second solution y_{2}(t) is given by y_{2}(t) = y_{1}(t) ∫_{t_{1}}^{t} ds/p(s)y_{1}(s)^{2}. Noting that y_{1}(t) = (1/π(t))^{μ_{1}}f_{1}(t), f_{1}(t) ∈ n-SV_{\frac{1}{π}}, we find

∫_{t_{1}}^{t} \frac{ds}{p(s)y_{1}(s)^{2}} = ∫_{t_{1}}^{t} (\frac{1}{π(s)})′ (\frac{1}{π(s)})^{−2μ_{1}−2} f_{1}(s)^{−2}ds

∼ \frac{−1}{2μ_{1} + 1} (\frac{1}{π(t)})^{−2μ_{1}−1} f_{1}(t)^{−2} as t → ∞

by the generalized Karamata theorem (Proposition 1-3), whence we conclude that

y_{2}(t) ∼ \frac{−1}{2μ_{1} + 1} (\frac{1}{π(t)})^{−μ_{1}−1} f_{1}(t)^{−1} = \frac{−1}{2μ + 1} (\frac{1}{π(t)})^{μ_{2}} f(t) as t → ∞.

This shows that y_{2}(t) ∈ n-RV_{\frac{1}{π}} (μ_{2}).

On the other hand, if a non-principal solution y_{2}(t) ∈ n-RV_{\frac{1}{π}} (μ_{2}) has been found, then the second solution y_{1}(t) determined by y_{1}(t) = y_{2}(t) ∫_{t}^{∞} ds/ p(s)y_{2}(s)^{2} belongs to n-RV_{\frac{1}{π}} (μ_{1}). Since y_{2}(t) = (1/π(t))^{μ_{2}}f_{2}(t), f_{2}(t) ∈ n-SV_{\frac{1}{π}}, the generalized Karamata theorem implies that

∫_{t}^{∞} \frac{ds}{p(s)y_{2}(s)^{2}} = ∫_{t}^{∞} (\frac{1}{π(s)})′ (\frac{1}{π(s)})^{−2μ_{2}−2} f_{2}(s)^{−2}ds

∼ \frac{1}{2μ_{2} + 1} (\frac{1}{π(t)})^{−2μ_{2}−1} f_{2}(t)^{−2} as t → ∞,

from which it follows that y_{1}(t) ∈ n-RV_{\frac{1}{π}} (μ_{1}) as desired.

C) The following theorem is a counterpart of Theorem 2.4 in Section 2.

Theorem 3.4. Assume that (3.4) holds. Put

Φ(t) = \frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q(s)ds − \frac{1}{4}  (3.35)

and suppose that

∫ ^{∞} \frac{|Φ(t)|}{p(t)π(t)}dt < ∞,  (3.36)

∫ ^{∞} \frac{Ψ(t)}{p(t)π(t)}dt < ∞, where Ψ(t) = ∫_{t}^{∞} \frac{|Φ(s)|}{p(s)π(s)}ds.  (3.37)

Then, the equation (B) possesses a fundamental set of solutions {y_{1}(t), y_{2}(t)} such that y_{i}(t) ∈ n-RV_{\frac{1}{π}} (−\frac{1}{2}) , i = 1, 2, and

y_{1}(t) = π(t)^{\frac{1}{2}} f_{1}(t), y_{2}(t) = π(t)^{\frac{1}{2}} log \frac{1}{π(t)}f_{2}(t),  (3.38)

where f_{i}(t) ∈ n-SV_{\frac{1}{π}} and lim_{t→∞} f_{i}(t) = f_{i}(∞) ∈ (0, ∞), i = 1, 2, f_{1}(∞)f_{2}(∞) = 1.

P r o o f. The function defined by

y(t) = exp \{∫_{t_{0}}^{t} \frac{Φ(s) − \frac{1}{2} + v(s)}{p(s)π(s)}ds\}  (3.39)

will be a solution of (B) if the function u(t) = (Φ(t) − \frac{1}{2} + v(t))/π(t) satisfies the Riccati equation (2.9) on some interval [t_{0}, ∞). We then have the differential equation for v(t) :

v′ + \frac{2Φ(t)}{p(t)π(t)}v + \frac{Φ(t)^{2} + Φ(t) + v^{2}}{p(t)π(t)} = 0,  (3.40)

which can be rewritten as

(ρ(t)v)′ + \frac{ρ(t)}{p(t)π(t)} [Φ(t)^{2} + Φ(t) + v^{2}] = 0,  (3.41)

where

ρ(t) = exp \{∫_{1}^{t} \frac{2Φ(s)}{p(s)π(s)}ds\} .  (3.42)

Let m > 0 be a constant such that

|Φ(t)^{2} + Φ(t)| ≤ mΦ(t), t ≥ 0,  (3.43)

and let t_{0} > 0 be large enough so that

\frac{ρ(s)}{ρ(t)} ≤ 2 for s ≥ t ≥ t_{0},  (3.44)

and

32m ∫_{t_{0}}^{∞} \frac{Ψ(s)}{p(s)π(s)}ds ≤ 1,  (3.45)

which is possible because of (3.36) and (3.37), respectively.

Define

V = {v ∈ C_{Ψ}[t_{0}, ∞) : |v(t)| ≤ 4mΨ(t), t ≥ t_{0}} ,  (3.46)
Gv(t) = \frac{1}{ρ(t)} ∫_{t}^{∞} \frac{ρ(s)}{p(s)π(s)} [Φ(s)^{2} + Φ(s) + v(s)^{2}] ds, t ≥ t_{0}. (3.47)

It is easily checked that v ∈ V implies Gv ∈ V and v, w ∈ V implies

∥Gv − Gw∥_{Ψ} ≤ \frac{1}{2}∥v − w∥_{Ψ}.

Consequently, there exists a unique element v_{1} ∈ V such that v_{1} = Gv_{1}, which is a solution of the integral equation

v_{1}(t) = \frac{1}{ρ(t)} ∫_{t}^{∞} \frac{ρ(s)}{p(s)π(s)} [Φ(s)^{2} + Φ(s) + v_{1}(s)^{2}] ds t ≥ t_{0}. (3.48)

Since (3.48) is an integrated version of (3.41), the function y_{1}(t) given by (3.39) with this v_{1}(t) is a solution of (B) on [t_{0}, ∞) belonging to n-RV_{\frac{1}{π}} (−\frac{1}{2}). Representing y_{1}(t) as

y_{1}(t) = (\frac{1}{π(t)})^{− \frac{1}{2}} f_{1}(t) = π(t)^{\frac{1}{2}} f_{1}(t), f_{1}(t) = exp \{∫_{t_{0}}^{t} \frac{Φ(s) + v(s)}{p(s)π(s)}ds\} ,

(3.49)

we see from (3.36) and (3.37) that lim_{t→∞} f_{1}(t) = f_{1}(∞) ∈ (0, ∞). Thus y_{1}(t) is a principal solution of (B). The second (non-principal) solution y_{2}(t) of (B) is determined by y_{2}(t) = y_{1}(t) ∫_{t_{0}}^{t} ds /p(s)y_{1}(s)^{2} . Since ∫_{t_{0}}^{t} ds /p(s)y_{1}(s)^{2} ∼ f_{1}(∞)^{−2} log(1/π(t)) as t → ∞, we conclude that y_{2}(t) has the form y_{2}(t) = π(t)^{\frac{1}{2}} log(1/π(t)) · f_{2}(t) with f_{2}(t) ∈ n-SV_{\frac{1}{π}} such that f_{2}(∞) = 1/f_{1}(∞). This completes the proof.

Example 3.1. It is known [3] that the Weber equation

(e^{\frac{t^{2}}{2}} y′)′ − 2λe^{\frac{t^{2}}{2}} y = 0, λ ∈ R,  (3.50)

has a pair of solutions {y_{1}(t), y_{2}(t)} satisfying

y_{1}(t) ∼ t^{−1−2λ}e^{− \frac{t^{2}}{2}} , y_{2}(t) ∼ t^{2λ} as t → ∞.  (3.51)

This equation is a special case of (B) with p(t) = e^{t^{2}/2} and q(t) = −2λe^{t^{2}/2}. The function p(t) satisfies (0.2) and generates π(t) = ∫_{t}^{∞} e^{−s^{2}/2}ds by (0.4). Since

π(t) = ∫_{t}^{∞} e^{− \frac{s^{2}}{2}} ds ∼ t^{−1}e^{− \frac{t^{2}}{2}} as t → ∞  (3.52)

and

lim_{t→∞} \frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q(s)ds = −2λ lim_{t→∞} \frac{∫_{t}^{∞} (∫_{s}^{∞} e^{− \frac{r^{2}}{2}} dr)^{2} e^{\frac{s^{2}}{2}} ds}{∫_{t}^{∞} e^{− \frac{s^{2}}{2}} ds} = 0,

we can apply Theorem 3.2 to conclude that there exists a fundamental set of solutions \{η_{1}(t), η_{2}(t)\} of (3.50) such that η_{1}(t) ∈ n-RV_{\frac{1}{π}} (−1), η_{2}(t) ∈ n-SV_{\frac{1}{π}}, that is,

η_{1}(t) = (∫_{t}^{∞} e^{− \frac{s^{2}}{2}} ds) f_{1}(t), η_{2}(t) = f_{2}(t)  (3.53)

for some f_{i}(t) ∈ n-SV_{\frac{1}{π}}, i = 1, 2, such that f_{2}(t) ∼ 1/f_{1}(t) as t → ∞. No further specific information on f_{i}(t) is included in Theorem 3.2.

Comparison of (3.51) with the asymptotic relation

η_{1}(t) ∼ t^{−1} (∫_{t}^{∞} e^{− \frac{s^{2}}{2}} ds) f_{1}(t), η_{2}(t) ∼ \frac{1}{f_{1}(t)} as t → ∞,

which, in view of (3.52), is equivalent to (3.53), shows that the function f_{1}(t) can be taken to be f_{1}(t) = t^{−2λ}. That actually t^{−2λ} ∈ n-SV_{\frac{1}{π}} follows from Proposition 1.3, based on the observation that, for every γ > 0,

(\frac{1}{π(t)})^{−γ} t^{−2λ} = (te^{\frac{t^{2}}{2}} )^{−γ} t^{−2λ} is eventually decreasing

and

(\frac{1}{π(t)})^{γ} t^{−2λ} = (te^{\frac{t^{2}}{2}} )^{γ} t^{−2λ} is eventually increasing

Example 3.2. We present here an interpretation of the solutions of the Legendre equation

((t^{2} − 1)y′)′ − λ(λ + 1)y = 0, λ > 0,  (3.54)

from the viewpoint of generalized Karamata functions.

The function p(t) = t^{2} − 1 clearly satisfies (0.2) for t > 1 and generates by (0.4) the function

π(t) = \frac{1}{2} log \frac{t + 1}{t − 1} ∼ \frac{1}{t} as t → ∞.  (3.55)

Since q(t) = λ(λ + 1) satisfies

lim_{t→∞} \frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q(s)ds = −λ(λ + 1),

we assert from Theorem 3.3 that (3.54) has a pair of of solutions {y_{1}(t), y_{2}(t)} such that

y_{1}(t) ∈ n-RV_{\frac{1}{π}} (−(λ + 1)), y_{2}(t) ∈ n-RV_{\frac{1}{π}} (λ).

We have used the fact that μ_{1} = −(λ + 1), μ_{2} = λ are the real roots of the quadratic equation μ^{2} + μ − λ(λ + 1) = 0. It should be observed because of (3.55) that the class of regularly varying functions with index α with respect to 1/π(t) coincides with that of regularly varying functions with index α in the sense of Karamata. Thus, the equation (3.54) has a fundamental set of solutions {y_{1}(t), y_{2}(t)} such that

y_{1}(t) ∈ n-RV(−(λ + 1)), y_{2}(t) ∈ n-RV(λ).

The above observation endorses the classical result [6] from the theory of special functions that the Legendre functions of the first and second kinds, denoted, respectively, by P_{λ}(t) and Q_{λ}(t), which are linearly independent solutions of (3.54), have the asymptotic properties

P_{λ}(t) ∼ \frac{Γ(λ + \frac{1}{2})}{\sqrt{π} Γ(λ + 1)}(2t)^{λ} ∈ n-RV(λ) as t → ∞,

Q_{λ}(t) ∼ \frac{\sqrt{π} Γ(λ + 1)}{Γ(λ + \frac{3}{2})}(2t)^{−λ−1} ∈ n-RV(−(λ + 1)) as t → ∞.

Example 3.3. Consider the equation

(e^{αt}y′)′ + [ke^{βt} sin(e^{γt}) + α^{2}ce^{αt}] y = 0, t ≥ 0.  (3.56)

where α, β, γ, c and k are positive constants. We put

p(t) = e^{αt}, q_{0}(t) = ke^{βt} sin(e^{γt}), q(t) = q_{0}(t) + α^{2}ce^{αt}.

Suppose that α − β + γ > 0. Using the function π(t) = e^{−αt}/α defined by (0.4), we obtain

∫_{t}^{∞} π(s)^{2}q_{0}(s)ds = \frac{k}{α^{2}} ∫_{t}^{∞} e^{(−2α+β)s} sin(e^{γs})ds

= \frac{k}{α^{2}γ} e^{−(2α−β+γ)t} cos(e^{γt}) − \frac{2α − β + γ}{α^{2}γ} ∫_{t}^{∞} e^{−(2α−β+γ)s} cos(e^{γs})ds,

(3.57) which implies that

lim_{t→∞} \frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q_{0}(s)ds = 0

and

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

Let c ∈ (−∞, \frac{1}{4}). Then, it follows from Theorem 3.3 that the equation (3.56) is nonoscillatory and possesses a pair of solutions {y_{1}(t), y_{2}(t)} such that y_{i}(t) ∈ n-RV_{αe^{αt}}(μ_{i}), i = 1, 2, where μ_{1}, μ_{2} (μ_{1} < μ_{2}) are the real roots of μ^{2} + μ + c = 0.

Now let c = \frac{1}{4} and define

Φ(t) = \frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q(s)ds − \frac{1}{4} = \frac{1}{π(t)} ∫_{t}^{∞} π(s)^{2}q_{0}(s)ds.

In view of (3.57) we see that Φ(t) satisfies

\frac{|Φ(t)|}{p(t)π(t)} = O(e^{−(α−β+γ)t}) as t → ∞,

which implies that |Φ(t)|/p(t)π(t) is integrable on [0, ∞) and

Ψ(t) = ∫_{t}^{∞} \frac{|Φ(s)|}{p(s)π(s)}ds = O(e^{−(α−β+γ)t}) as t → ∞.

Since the function Ψ(t) is also integrable on [0, ∞), from Theorem 3.4 it follows that (3.56) is nonoscillatory and has a fundamental set of solutions {y_{1}(t), y_{2}(t)} satisfying

y_{1}(t) ∼ αe^{\frac{α}{2} t}f_{1}(t), y_{2}(t) ∼ \frac{α^{2}te^{\frac{α}{2} t}}{f_{1}(t)} as t → ∞,

where f_{1}(t) is a slowly varying function with respect to αe^{αt} such that

lim_{t→∞} f_{1}(t) = f_{1}(∞) ∈ (0, ∞).

We note that if β > α, then the function q(t) in (3.56) is strongly oscillating in the sense that

lim inf_{t→∞} q(t) = −∞ and lim sup_{t→∞} q(t) = ∞.

Appendix

We summarize here definitions and properties of some of the basic classes of Karamata functions which have been generalized in Section 1 so as to be an appropriate means for the analysis of self-adjoint differential equations of the form (B). For the proofs of the theorems stated below the reader is referred to the books [1, 11].

Definition A.1. A measurable function f : [t_{0}, ∞) → (0, ∞) is said to be slowly varying if it satisfies

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

Definition A.2. A measurable function g : [t_{0}, ∞) → (0, ∞) is said to be regularly varying with index α ∈ R if it satisfies

lim_{t→∞} \frac{g(λt)}{g(t)} = λ^{α} for every λ > 0.

One of the fundamental properties of slowly and regularly varying functions is the following representation theorem.

Theorem A.1. (i) A positive measurable function f(t) is slowly varying if and only if it can be expressed in the form

f(t) = c(t) exp \{∫_{t_{0}}^{t} \frac{ε(s)}{s}ds\} , t ≥ t_{0},  (A.1)

where c(t) and ε(t) are measurable functions such that

lim_{t→∞} c(t) = c ∈ (0, ∞), lim_{t→∞} ε(t) = 0.

(ii) A positive measurable function g(t) is regularly varying with index α if and only if it has the representation

g(t) = c(t) exp \{∫_{t_{0}}^{t} \frac{δ(s)}{s}ds\} , t ≥ t_{0},  (A.2)

where c(t) and δ(t) are measurable functions such that

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

If the function c(t) in (A.1) (or (A.2)) is identically a constant on [t_{0}, ∞), then f(t) (or g(t)) is called a normalized slowly varying function (or a normalized regularly varying function with index α). The totality of slowly varying (or normalized slowly varying) functions is denoted by SV (or n-SV) and the totality of regularly varying (or normalized regularly varying) functions with index α is denoted by RV(α) (or n-RV(α)). It is easy to see that g(t) ∈ RV(α)(or g(t) ∈ n-RV(α)) if and only if g(t) = t^{α}f(t) for some f(t) ∈ SV (or f(t) ∈ n-SV).

Theorem A.2. (i) f(t) ∈ SV implies f(t)^{β} ∈ SV for any β ∈ R. f_{1}(t), f_{2}(t) ∈ SV implies f_{1}(t)f_{2}(t) ∈ SV.

(ii) If f(t) ∈ SV, then for any γ > 0,

lim_{t→∞} t^{γ}f(t) = ∞, lim_{t→∞} t^{−γ}f(t) = 0.  (A.3)

(iii) g(t) ∈ RV(α) and β ∈ R imply g(t)^{β} ∈ RV(αβ). g_{i}(t) ∈ RV(α_{i}), i = 1, 2, implies g_{1}(t)g_{2}(t) ∈ RV(max(α_{1}, α_{2})).

Theorem A.3. (Bojanic and Karamata) A positive measurable function f(t) belongs to n-SV if and only if, for every γ > 0, t^{γ}f(t) is ultimately increasing and t^{−γ}f(t) is ultimately decreasing.

Theorem A.4. (Karamata) (i) If ν > −1, then for any f(t) ∈ SV,

∫_{t_{0}}^{t} s^{ν}f(s)ds ∼ \frac{t^{ν+1}f(t)}{ν + 1} as t → ∞.  (A.4)

(ii) If ν < −1, then for any f(t) ∈ SV, ∫_{t_{0}}^{∞} s^{ν}f(s)ds < ∞, and

∫_{t}^{∞} s^{ν}f(s)ds ∼ − \frac{t^{ν+1}f(t)}{ν + 1} as t → ∞.  (A.5)

Here the notation φ(t) ∼ ψ(t) as t → ∞ is used to mean lim_{t→∞} ψ(t)/φ(t) = 1.

Definition A.3. A measurable function h : [t_{0}, ∞) → (0, ∞) is called regularly bounded if it satisfies

0 < lim inf_{t→∞} \frac{h(λt)}{h(t)} ≤ lim sup_{t→∞} \frac{h(λt)}{h(t)} < ∞ for every λ ≥ 1.

Theorem A.5. A function h(t) is regularly bounded if and only if it has the representation

h(t) = exp \{η(t) + ∫_{t_{0}}^{t} \frac{ξ(s)}{s}ds\} , t ≥ t_{0},  (A.6)

where ξ(t) and η(t) are bounded and measurable on [t_{0}, ∞). The set of all regularly bounded functions is denoted by RO. It can be shown that the integral and the product of functions in RO remain in this class.

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Jaroslav Jaroš Department of Mathematical Analysis Faculty of Mathematics, Physics and Informatics Comenius University 842 15 Bratislava, Slovakia Kusano Takaŝi Department of Applied Mathematics Faculty of Science Fukuoka University 8 – 19 – 1 Nanakuma, Jonan-ku, Fukuoka, 814 – 0180 Japan