PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 88(102) (2010), 1–20 DOI: 10.2298/PIM1002001T

REGULARLY VARYING SOLUTIONS OF

PERTURBED EULER DIFFERENTIAL EQUATIONS AND RELATED FUNCTIONAL DIFFERENTIAL EQUATIONS

Kusano Takaŝi and Vojislav Marić

Communicated by Stevan Pilipović

Abstract. An asymptotic analysis in the framework of Karamata regularly varying functions is performed for the solutions of second order linear differential and functional differential equations in the critical case i.e., when condition (1.5) as given below, holds.

1. Introduction

As is witnessed by the recent book [9] theory of regular variation (in the sense of Karamata) has made it possible to develop a new significant aspect of asymptotic analysis of linear and nonlinear ordinary differential equations.

For the reader’s convenience we recall here the definition and some basic properties of regularly varying functions. A measurable function f : [0, ∞) → (0, ∞) is said to be regularly varying of index ρ ∈ R if

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

The totality of regularly varying functions of index ρ is denoted by RV(ρ). We often use the symbol SV to denote RV(0) and call members of SV slowly varying functions. By definition any function f(t) ∈ RV(ρ) can be expressed as f(t) = t^{ρ}L(t) with L(t) ∈ SV. One of the most important properties of RV(ρ)-functions is the following representation theorem.

Theorem 1.1. f(t) ∈ RV(ρ) if and only if f(t) is expressed in the form

(1.1)  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) = ρ.

2010 Mathematics Subject Classification: Primary 34C11, 34K06; Secondary 26A12.

If c(t) ≡ c_{0} in (1.1), then the function f(t) is called a normalized regularly varying function of index ρ, and the totality of such functions form an important subclass, denoted by n-RV(ρ), of RV(ρ). The symbol n-RV stands for n-RV(0) and its members are referred to as normalized slowly varying functions.

Let L(t) ∈ SV. What is the asymptotic behavior of L(t) as t → ∞? It may occur that lim_{t→∞} L(t) = const > 0, that is, L(t) is asymptotic to a positive constant as t → ∞. In this case L(t) is said to be a trivial SV-function. Naturally L(t) may exhibit a different behavior: lim_{t→∞} L(t) = 0, or lim_{t→∞} L(t) = ∞, or otherwise, in which case L(t) is said to be a nontrivial SV-function. Thus it is possible that a nontrivial SV-function may grow to infinity or decay to zero as t tends to ∞, but its behavior at infinity is severely restricted as the following theorem asserts.

Theorem 1.2. Let L(t) be a slowly varying function. Then, for any ε > 0,

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

Useful is the following theorem due to Karamata on the asymptotic behavior of integrals involving slowly varying functions.

Theorem 1.3. Let L(t) be a slowly varying function.

(i) If γ > −1, then ∫_{t_{0}}^{t} s^{γ}L(s) ds ∼ \frac{t^{γ+1}}{γ + 1}L(t) as t → ∞.

(ii) If γ < −1, then ∫_{t}^{∞} s^{γ}L(s) ds ∼ −\frac{t^{γ+1}}{γ + 1}L(t) as t → ∞.

Here the symbol ∼ is used to denote the asymptotic equivalence:

F (t) ∼ G(t) ⟺ lim_{t→∞} \frac{F (t)}{G(t)} = 1.

For a comprehensive exposition of theory of regular variation and its applications to various branches of mathematical analysis the reader is referred to the book [1].

We are now in a position to state an example of typical results on the study of asymptotic behavior of solutions of the linear ordinary differential equation

(A)  x′′ + q(t)x = 0

in the framework of regular variation.

Theorem 1.4. (Howard and Marić [4]) Suppose that q(t) is a continuous function which is integrable on [a, ∞). Let a constant c ∈ (−∞, \frac{1}{4}) be given and let λ_{0} and λ_{1}, λ_{0} < λ_{1}, be the real roots of the quadratic equation

(1.2)  λ^{2} − λ + c = 0.

Then, equation (A) has a fundamental set of solutions {x_{0}(t), x_{1}(t)} such that x_{0}(t) ∈ n-RV(λ_{0}), x_{1}(t) ∈ n-RV(λ_{1}) if and only if

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

After the publication of [9] attempts have been made by the present authors [6, 7, 8] to establish the existence of regularly varying solutions to functional differential equations with deviating arguments of the form

(B)  x′′(t) + q(t)x(g(t)) = 0,

where q(t) and g(t) are continuous on [a, ∞), and g(t) is increasing. There holds

Theorem 1.5. [8] Let c, λ_{0} and λ_{1} be as in Theorem 1.4 with c replaced by −c. Suppose that q(t) is eventually negative and that g(t) satisfies g(t) < t, g(t) → ∞, as t → ∞ and g(t)/t → 1, as t → ∞.

Then, equation (B) has two solutions x_{0}(t) ∈ n-RV(λ_{0}) and x_{1}(t) ∈ n-RV(λ_{1}) if and only if (1.3) is satisfied.

We note that if in particular c = 0, in which case λ_{0} = 0 and λ_{1} = 1, the conclusion of Theorem 1.5 holds if the last condition on q(t) is replaced by lim sup_{t→∞} t/g(t) < ∞.

It should be noticed that the constant c in Theorem 1.4 is not allowed to exceed 1/4. In fact, let q(t) be eventually positive, and suppose that c > 1/4. Then, the roots of (1.2) are imaginary and so equation (A) is oscillatory by the well-known oscillation criterion of Hille [2]. A question then arises: What happens if c = 1/4? This is a critical case in the sense that equation (A) may or may not be oscillatory as is shown by the equation

(1.4)  x′′ + q(t)x = 0, q(t) = \frac{1}{4t^{2}} + \frac{d}{(t log t)^{2}} ,

where d is a positive constant. It is clear that q(t) satisfies (1.3) with c = 1/4 and it is known that (1.4) is oscillatory or nonoscillatory according as d > 1/4 or d ⩽ 1/4. Therefore, additional conditions are needed to ensure oscillation or non-oscillation of equation (A) in case c = 1/4; see e.g., the books [3] and [10]. A criterion for non-oscillation of (A) in the language of regular variation, applicable to this case, has been given by Howard and Marić in [4].

Theorem 1.6. [4] Let q(t) be continuous and integrable on [a, ∞). Suppose that

(1.5)  lim_{t→∞} t ∫_{t}^{∞} q(s) ds = \frac{1}{4}.

Put ϕ(t) = t ∫_{t}^{∞} q(s) ds − \frac{1}{4}, and let the integral ∫^{∞} |ϕ(t)|/tdt converge. Suppose moreover that

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

Then equation (A) possesses a fundamental set of solutions {x_{0}(t), x_{1}(t)} such that

x_{0}(t) = t^{1/2}L_{0}(t), L_{0}(t) ∈ n-RV, lim_{t→∞} L_{0}(t) ∈ (0, ∞),

x_{1}(t) = t^{1/2} log tL_{1}(t), L_{1}(t) ∈ n-RV, lim_{t→∞} L_{1}(t) ∈ (0, ∞).

The objective of this paper is first (in Section 2) to make a more detailed analysis of regularly varying solutions of equation (A) under the condition (1.5), obtaining a generalized version of Theorem 1.6, and then (in Section 3) to make use of the results for (A) to establish the existence of similar solutions for the functional differential equation (B) with q(t) satisfying (1.5). In what follows the function q(t) is assumed to be integrable over [a, ∞) unless stated otherwise.

In analyzing the non-oscillation and asymptotic behavior of pertinent equations in the critical case a crucial role is played by the concept of generalized regularly varying functions which was introduced by Jaroš and Kusano [5].

A measurable function f : [0, ∞) → (0, ∞) is said to be a (generalized) regularly varying function of index ρ with respect to log t if f(t) is expressed as f(t) = g(log t) for some regularly varying function g(τ) of index ρ in the sense of Karamata. We use the symbol RV_{log t}(ρ) to denote the set of all regularly varying functions of index ρ with respect to log t. As before, SV_{log t} stands for RV_{log t}(0), and members of SV_{log t} are called (generalized) slowly varying functions with respect to log t. If, in the representation f(t) = g(log t), g(τ) is in n-RV(ρ) (or in n-RV), then f(t) is termed a normalized regularly varying function of index ρ (or a normalized slowly varying function) with respect to log t. The set of all normalized RV(ρ)- or SV-functions with respect to log t will be denoted by n-RV_{log t}(ρ) or n-RV_{log t}.

Since the composition of a slowly varying function and log t is clearly slowly varying, we see that SV_{log t} ⊂ SV and RV_{log t}(ρ) ⊂ RV(ρ). It should be noted, however, that both inclusions are proper. For example, log t ∈ SV but log t ∉ SV_{log t} because log t ∈ RV_{log t}(1).

The representation theorem for RV(ρ)-functions combined with the above definition provides a characterization for the class RV_{log t}(ρ).

Theorem 1.7. f(t) ∈ RV_{log t}(ρ) if and only if f(t) is expressed in the form

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

for some t_{0} > 1 and some measurable functions c(t) and δ(t) such that

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

We note that f(t) ∈ n-RV_{log t}(ρ) if and only if c(t) ≡ c_{0} in the representation formula (1.6).

2. Regularly varying solutions of (A) in the critical case

We begin by considering equation (A) with q(t) satisfying condition (1.5). In this case equation (A) may well be called a perturbed Euler differential equation.

First observe that a regularly varying solution of (A), if any, must have the regularity index 1/2. This is an immediate consequence of the following lemma.

Lemma 2.1. If (A) has an n-RV(ρ)-solution, then it holds that

(2.1)  lim_{t→∞} t ∫_{t}^{∞} q(s) ds = ρ(1 − ρ).

Proof. Let x(t) ∈ n-RV(ρ) be a solution of (A) on [t_{0}, ∞). Define u(t) = x′(t)/x(t) and v(t) = tu(t). It is known that u(t) satisfies the Riccati equation

(2.2)  u′ + u^{2} + q(t) = 0, t ⩾ t_{0}.

On the other hand, using the representation theorem (Theorem 1.1), we see that x(t) ∈ n-RV(ρ) satisfies

lim_{t→∞} v(t) = lim_{t→∞} \frac{tx′(t)}{x(t)} = ρ,

which implies in particular that u(t) → 0 as t → ∞ and that u(t)^{2} = v(t)^{2}/t^{2} is integrable on [t_{0}, ∞). From (2.2) it then follows that q(t) is integrable on [t_{0}, ∞), and integration of (2.2) from t to ∞ yields

u(t) = ∫_{t}^{∞} u(s)^{2}ds + ∫_{t}^{∞} q(s) ds, t ⩾ t_{0},

which, written for v(t), takes the form

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

Letting t → ∞ in (2.3), we conclude that (2.1) holds as desired. □

Our task therefore is to seek RV(1/2)-solutions of equation (A) under condition (1.5). We are based on the simple fact that the change of variables x = t^{1/2}y transforms (A) into the differential equation

(2.4)  (ty′)′ + t(q(t) − \frac{1}{4t^{2}} )y = 0.

Thus, to obtain a solution of (A) belonging to RV(1/2) of (A) it suffices to verify the existence of an SV-solution y(t) of (2.4) and form the function x(t) = t^{1/2}y(t). As is shown in [5], the class of generalized regularly varying functions with respect to log t, which is smaller than the classical regularly varying functions, is a wellsuited framework for the asymptotic analysis of (2.4). So, our attention will be directed to the existence of RV_{log t}(ρ)-solutions of differential equations of the form

(2.5)  (ty′)′ + p(t)y = 0,

where p(t) is continuous and integrable on [a, ∞).

Basic to the subsequent development is the following existence theorem.

Theorem 2.1. Let d ∈ (−∞, \frac{1}{4}) be a given constant and let μ_{0} and μ_{1}, μ_{0} < μ_{1}, be the real roots of the quadratic equation

(2.6)  μ^{2} − μ + d = 0.

Then, equation (2.5) has a fundamental set of solutions {y_{0}(t), y_{1}(t)} such that

(2.7)  y_{0}(t) ∈ n-RV_{log t}(μ_{0}), y_{1}(t) ∈ n-RV_{log t}(μ_{1})

if and only if

(2.8)  lim_{t→∞} log t ∫_{t}^{∞} p(s) ds = d.

Proof. (The “only if” part) Suppose that (2.5) possesses a pair of solutions on [t_{0}, ∞) satisfying (2.7). Let y(t) stands for y_{i}(t), i = 0 or 1. Put

(2.9)  u(t) = \frac{ty′(t)}{y(t)}, v(t) = u(t) log t.

Using the representation for y(t)

y(t) = y(t_{0}) exp\{∫_{t_{0}}^{t} \frac{δ(s)}{s log s} ds\}, lim_{t→∞} δ(t) = μ,

where μ = μ_{i}, i = 0 or 1, we see that

lim_{t→∞} v(t) = lim_{t→∞} u(t) log t = lim_{t→∞} δ(t) = μ,

which implies that u(t) → 0 as t → ∞ and u(t)^{2}/t = v(t)^{2}/t(log t)^{2} is integrable on [t_{0}, ∞). Consequently, integrating from t to ∞ the Riccati equation

(2.10)  u′ + \frac{u^{2}}{t} + p(t) = 0, t ⩾ t_{0},

satisfied by u(t), we obtain

u(t) = ∫_{t}^{∞} \frac{u(s)^{2}}{s} ds + ∫_{t}^{∞} p(s) ds,

which, in view of (2.9), can be transformed into

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

Passing to the limit as t → ∞, we conclude that

lim_{t→∞} log t ∫_{t}^{∞} p(s) ds = μ − μ^{2} = d,

ensuring the truth of (2.8).

(The “if” part) Suppose that (2.8) is satisfied. First we will be concerned with the existence of a solution of (2.5) belonging to n-RV_{log t}(μ_{0}). Note that μ_{0} < 1/2. Put

P (t) = log t ∫_{t}^{∞} p(s) ds − d.

Let l be a positive constant such that

\frac{2(μ_{0} + 2)}{1 − 2μ_{0}}l < 1 if μ_{0} > 0, \frac{2|μ_{0}| + 2}{1 + 2|μ_{0}|}l < 1 if μ_{0} < 0

and choose t_{0} ⩾ a so that |P (t)| ⩽ l^{2}, t ⩾ t_{0}. Define

V = {v ∈ C_{0}[t_{0}, ∞) : |v(t)| ⩽ l, t ⩾ t_{0}},

where C_{0}[t_{0}, ∞) denotes the set of all continuous functions on [t_{0}, ∞) that tend to 0 as t → ∞. Clearly, C_{0}[t_{0}, ∞) is a Banach space with the norm ∥v∥_{0} = sup_{t⩾t_{0}} |v(t)|.

Consider the integral operator F defined by

Fv(t) = (log t)^{1−2μ_{0}} ∫_{t}^{∞} \frac{2μ_{0}P (s) + (v(s) + P (s))^{2}}{s(log s)^{2−2μ_{0}}} ds, t ⩾ t_{0}.

It can be shown easily that v ∈ V implies |Fv(t)| ⩽ l, t ⩾ t_{0}, and that v_{1}, v_{2} ∈ V implies ∥Fv_{1} −Fv_{2}∥_{0} ⩽ 4l/(1−2μ_{0})·∥v_{1} −v_{2}∥_{0}. This shows that F is a contraction mapping on V, and so there exists a unique v ∈ V such that v = Fv, that is, v = v(t) is a solution of the integral equation

(2.11)  v(t) = (log t)^{1−2μ_{0}} ∫_{t}^{∞} \frac{2μ_{0}P (s) + (v(s) + P (s))^{2}}{s(log s)^{2−2μ_{0}}} ds, t ⩾ t_{0}.

With this v(t) we construct the function

y_{0}(t) = exp\{∫_{t_{0}}^{t} \frac{μ_{0} + v(s) + P (s)}{s log s} ds\}, t ⩾ t_{0},

and claim that y_{0}(t) is a solution of equation (2.5) in RV_{log t}(μ_{0}). That y_{0}(t) ∈ RV_{log t}(μ_{0}) is a consequence of Theorem 1.7 since μ_{0} + v(t) + P (t) → μ_{0} as t → ∞. To show that y_{0}(t) is a solution of (2.5) it suffices to verify that the function u(t) = (μ_{0} + v(t) + P (t))/ log t satisfies the Riccati equation (2.10) on [t_{0}, ∞). It is a matter of elementary calculation to see that (2.10) can then be transformed into

(\frac{v(t)}{(log t)^{1−2μ_{0}}} )′ + \frac{2μ_{0}P (t) + (v(t) + P (t))^{2}}{t(log t)^{2−2μ_{0}}} = 0,

But this is the differential equation that follows from differentiation of the integral equation (2.11). This establishes the existence of an RV_{log t}(μ_{0})-solution of equation (2.5). □

Up to this point the smaller root μ_{0} of (2.6) has been tacitly assumed to be nonzero. We remark here that if μ_{0} = 0, which occurs in the case d = 0, the construction of the solution y_{0}(t) of equation (2.5) becomes slightly simpler, and proceeds as follows. We let a constant 0 < l < 1/4 be fixed, choose T > a so that |P (t)| ⩽ l for t ⩾ T, solve, with the help of the contraction mapping principle, the integral equation

v(t) = log t ∫_{t}^{∞} \frac{(v(s) + P (s))^{2}}{s(log s)^{2}} ds, t ⩾ T,

in the set V = {v ∈ C_{0}[T, ∞) : 0 ⩽ v(t) ⩽ l, t ⩾ T }, and finally form the function

y_{0}(t) = exp\{∫_{T}^{t} \frac{v(s) + P (s)}{s log s} ds\}, t ⩾ T.

Then y_{0}(t) is shown to be an SV_{log t}-solution of (2.5).

Next, we turn to the construction of an RV_{log t}(μ_{1})-solution of (2.5). We note that μ_{1} > 1/2. Let m be a positive constant such that

m < \frac{1}{4} and \frac{2(μ_{1} + 2)}{2μ_{1} − 1}m ⩽ 1.

Choose t_{1} ⩾ a so that |P (t)| ⩽ m^{2}, t ⩾ t_{1}, and consider the set W and the integral operator G defined by W = {w ∈ C_{0}[t_{1}, ∞) : |w(t)| ⩽ m, t ⩾ t_{1}}, and

Gw(t) = (log t)^{1−2μ_{1}} ∫_{t_{1}}^{t} s^{−1}(log s)^{2μ_{1}−2}{2μ_{1}P (s) + (w(s) − P (s))^{2}} ds, t ⩾ t_{1}.

It is straightforward to check that w ∈ W implies |Gw(t)| ⩽ m, t ⩾ t_{1}, and that w_{1}, w_{2} ∈ W implies ∥Gw_{1} − Gw_{2}∥_{0} ⩽ 4m/(2μ_{1} − 1)·∥w_{1} − w_{2}∥_{0}. This shows that G is a contraction on W, so that there exists a w ∈ W such that w = Gw, which is equivalent to the integral equation (2.12)

w(t) = (log t)^{1−2μ_{1}} ∫_{t_{1}}^{t} s^{−1}(log s)^{2μ_{1}−2}{2μ_{1}P (s) + (w(s) − P (s))^{2}} ds, t ⩾ t_{1}.

Let y_{1}(t) be defined with this w(t) by

y_{1}(t) = exp\{∫_{t_{1}}^{t} \frac{μ_{1} − w(s) + P (s)}{s log s} ds\}, t ⩾ t_{1}.

On the one hand, y_{1}(t) ∈ RV_{log t}(μ_{1}) since −w(t) + P (t) → 0 as t → ∞, and on the other, y_{1}(t) is a solution of equation (2.5) since u(t) = (μ_{1} − w(t) + P (t))/ log t satisfies the Riccati equation (2.10) on [t_{1}, ∞). In fact, substituting u(t) for (2.10) yields

((log t)^{2μ_{1}−1}w(t))′ = (log t)^{2μ_{1}−2}{2μ_{1}P (t) + (w(t) − P (t))^{2}},

which follows from direct differentiation of (2.12). Thus condition (2.8) is also sufficient for equation (2.5) to possess an RV_{log t}(μ_{1})-solution. This completes the proof of Theorem 2.1.

The main result of this section is obtained as a corollary to Theorem 2.1.

Theorem 2.2. Let d, μ_{0} and μ_{1} be as in Theorem 2.1and suppose the function t(q(t)−1/4t^{2}) is integrable over (t, ∞). Then equation (A) possesses a fundamental set of solutions {x_{0}(t), x_{1}(t)} such that

(2.13)  x_{i}(t) = t^{1/2}y_{i}(t), y_{i}(t) ∈ n-RV_{log t}(μ_{i}), i = 0, 1,

if and only if

(2.14)  lim_{t→∞} log t ∫_{t}^{∞} s(q(s) − \frac{1}{4s^{2}} ) ds = d.

Proof. Observe that equation (A) has a fundamental set of solutions x_{i}(t), i = 0, 1, as described in (2.13) if and only if the functions y_{i}(t) = t^{−1/2}x_{i}(t), i = 0, 1, are solutions of (2.4) belonging to RV_{log t}(μ_{i}), i = 0, 1, and apply Theorem 2.1 to equation (2.4) which is a special case of (2.5) with p(t) = t(q(t) − 1/4t^{2}). □

Remark 2.1. It should be noticed that condition (2.14) automatically implies (1.5). In fact, (2.14) guarantees the existence of regularly varying solutions of index \frac{1}{2} for equation (A), which means by Lemma 2.1 that (1.5) must be satisfied.

Example 2.1. Consider the perturbed Euler equation

(2.15)  x′′ + q(t)x = 0, q(t) = \frac{1}{4t^{2}} + \frac{d(t)}{(t log t)^{2}} ,

where d(t) is a positive continuous function on [e, ∞) such that lim_{t→∞} d(t) = d ∈ (0, \frac{1}{4}). Since

lim_{t→∞} log t ∫_{t}^{∞} s(q(s) − \frac{1}{4s^{2}} ) ds = lim_{t→∞} log t ∫_{t}^{∞} \frac{d(s)}{s(log s)^{2}} ds = d,

it follows from Theorem 2.2 that equation (2.15) has two linearly independent solutions x_{i}(t) = t^{1/2}y_{i}(t), i = 0, 1, as described in (2.13).

Example 2.2. Our next example is the following.

(2.16)  x′′ + q(t)x = 0, q(t) = \frac{1}{4t^{2}} + \frac{1}{t^{2}(log t)^{2} log log t}.

An easy calculation shows that (2.14) holds for d = 0 and so Theorem 2.2 ensures the existence of two linearly independent solutions x_{(}t) = t^{1/2}y_{i}(t), i = 0, 1, such that y_{0}(t) ∈ SV_{log t} and y_{1}(t) ∈ RV_{log t}(1). Since t^{1/2} log log t is a solution of (2.16), one can take x_{0}(t) = t^{1/2} log log t and x_{1}(t) = x_{0}(t) ∫_{T}^{t} ds/x_{0}(s)^{2}, T > e being sufficiently large. As is easily seen,

t^{−1/2}x_{1}(t) = log log t ∫_{T}^{t} \frac{ds}{s(log log s)^{2}} ∼ \frac{log t}{log log t} ∈ RV_{log t}(1).

Observe that Theorem 1.6 is not applicable to either of these examples.

3. Regularly varying solutions of (B) in the critical case

We now turn our attention to functional differential equations with retarded argument of the form (B)

x′′(t) + q(t)x(g(t)) = 0,

where q(t) and g(t) are positive and continuous on [a, ∞), and g(t) is increasing and satisfies g(t) < t and lim_{t→∞} g(t) = ∞. We look for regularly varying solutions of (B) under the assumption that

(3.1)  lim_{t→∞} t ∫_{t}^{∞} q(s) ds = \frac{1}{4}.

In view of Theorem 1.5 it is natural to expect that the nonoscillatory nature of (B) would be similar to that of the ordinary differential equation (A) provided the retarded argument g(t) is a small perturbation of t, or more specifically that condition (3.1) would ensure the existence of RV(\frac{1}{2})-solutions as described in Theorem 2.2 if g(t) is “sufficiently close” to t in some sense. The objective of this section is to demonstrate the truth of the above expectation by proving the following theorem.

Theorem 3.1. Suppose that q(t) − 1/4t^{2} is eventually positive and that g(t) has the property that

(3.2)  \frac{g(t)}{t} = 1 + O(\frac{1}{t^{α}} ) as t → ∞,

for some α > 0. Let a constant d ∈ [0, \frac{1}{4}) be fixed and let μ_{0} and μ_{1}, μ_{0} < μ_{1}, be the roots of the quadratic equation

(3.3)  μ^{2} − μ + d = 0.

Then, equation (B) possesses two n-RV(\frac{1}{2})-solutions x_{i}(t), i = 0, 1, such that

(3.4)  x_{i}(t) = t^{1/2}y_{i}(t), y_{i}(t) ∈ n-RV_{log t}(μ_{i}), i = 0, 1,

10 KUSANO AND MARIĆ

if and only if

(3.5) lim_{t→∞} log t ∫_{t}^{∞} s(q(s) − \frac{1}{4s^{2}} ) ds = d.

Proof. (The “only if” part) Suppose that (B) has two regularly varying solutions x(t) = t^{1/2}y(t) as described in (3.4), where y(t) stands for y_{i}(t), i = 0 or 1. Then, y(t) is an RV_{log t}(μ_{i})-solution, i = 0 or 1, of the differential equation

(3.6)  (ty′(t))′ + t((\frac{g(t)}{t})^{1/2}q(t)y(g(t)) − \frac{1}{4t^{2}} y(t)) = 0,

which can be regarded as a linear ordinary differential equation

(3.7)  (ty′(t))′ + q_{y}(t)y(t) = 0, q_{y}(t) = t((\frac{g(t)}{t})^{1/2} \frac{y(g(t))}{y(t)}q(t) − \frac{1}{4t^{2}} ).

From the “only if” part of Theorem 2.2 we see that

(3.8) lim_{t→∞} log t ∫_{t}^{∞} s((\frac{g(s)}{s})^{1/2} \frac{y(g(s))}{y(s)}q(s) − \frac{1}{4s^{2}} ) ds = d.

We now rewrite the integrand of (3.8) as follows:

(3.9) t((\frac{g(t)}{t})^{1/2} \frac{y(g(t))}{y(t)}q(t) − \frac{1}{4t^{2}} )

= t(\frac{g(t)}{t})^{1/2} \frac{y(g(t))}{y(t)}(q(t) − \frac{1}{4t^{2}} ) + ((\frac{g(t)}{t})^{1/2} \frac{y(g(t))}{y(t)} − 1)\frac{1}{4t}.

Since y(t) ∈ n-RV_{log t}(μ_{i}), y(t) is expressed in the form

y(t) = exp\{∫_{t_{0}}^{t} \frac{δ(s)}{s log s} ds\}, lim_{t→∞} δ(t) = μ_{i},

for some t_{0} > 1, so that

\frac{y(g(t))}{y(t)} = exp\{− ∫_{g(t)}^{t} \frac{δ(s)}{s log s} ds\},

from which, noting that 0 ⩽ δ(t) ⩽ 1 for sufficiently large t, we have

(3.10)  \frac{log g(t)}{log t} ⩽ \frac{y(g(t))}{y(t)} ⩽ 1 for all large t

Since (3.2) implies

(3.11)  \frac{log g(t)}{log t} = 1 + O(\frac{1}{t^{α} log t}) and (\frac{g(t)}{t})^{1/2} = 1 + O(\frac{1}{t^{α}} )

as t → ∞, we find from (3.10) that

(\frac{g(t)}{t})^{1/2} \frac{y(g(t))}{y(t)} = 1 + O(\frac{1}{t^{α}} ) as t → ∞,

which implies that

(3.12)  lim_{t→∞} log t ∫_{t}^{∞}((\frac{g(s)}{s})^{1/2} \frac{y(g(s))}{y(s)} − 1)\frac{1}{4s} ds = 0.

Then, combining (3.9) with (3.8) and (3.12), we conclude that

lim_{t→∞} log t ∫_{t}^{∞} s(\frac{g(s)}{s})^{1/2} \frac{y(g(s))}{y(s)}(q(s) − \frac{1}{4s^{2}} ) ds = d,

from which (3.5) can be derived easily.

(The “if” part) Suppose that (3.5) holds. We begin with the case where d > 0, so that the roots of (3.3) satisfies 0 < μ_{0} < \frac{1}{2} < μ_{1} < 1. By means of the identity

t((\frac{g(t)}{t})^{1/2} \frac{log g(t)}{log t}q(t) − \frac{1}{4t^{2}} )

= (\frac{g(t)}{t})^{1/2} \frac{log g(t)}{log t}t(q(t) − \frac{1}{4t^{2}} ) + ((\frac{g(t)}{t})^{1/2} \frac{log g(t)}{log t} − 1)\frac{1}{4t},

it can be shown that

(3.13)  lim_{t→∞} log t ∫_{t}^{∞}((\frac{g(s)}{s})^{1/2} \frac{log g(s)}{log s}q(s) − \frac{1}{4s^{2}} ) ds = d.

In fact, using the relation

(\frac{g(t)}{t})^{1/2} \frac{log g(t)}{log t} = 1 + O(\frac{1}{t^{α}} ) as t → ∞,

(cf. (3.11)), we see that

lim_{t→∞} log t ∫_{t}^{∞}((\frac{g(s)}{s})^{1/2} \frac{log g(s)}{log s} − 1)\frac{1}{4s} ds = 0

and

lim_{t→∞} log t ∫_{t}^{∞}(\frac{g(s)}{s})^{1/2} \frac{log g(s)}{log s}s(q(s) − \frac{1}{4s^{2}} ) ds = d,

from which (3.13) follows immediately.

Our first task is to prove the existence of an RV(\frac{1}{2})-solution x_{0}(t) of equation (A) such that

x_{0}(t) = t^{1/2}y_{0}(t), y_{0}(t) ∈ n-RV_{log t}(μ_{0}).

Let l be a positive constant such that

\frac{2(μ_{0} + 2)}{1 − 2μ_{0}}l < 1,

and choose T > a so large that g(T) ⩾ a and

(3.14)  |log t ∫_{t}^{∞} s(q(s) − \frac{1}{4s^{2}} ) ds − d| ⩽ l^{2},

and

(3.15)  |log t ∫_{t}^{∞} s((\frac{g(s)}{s})^{1/2} \frac{log g(s)}{log s}q(s) − \frac{1}{4s^{2}} ) ds − d| ⩽ l^{2},

for t ⩾ T. Let Ξ denote the set consisting of all functions ξ(t)∈C[g(T), ∞)∩C^{1}[T, ∞) that are expressed in the form

(3.16)  ξ(t) = 1, g(T) ⩽ t ⩽ T,  ξ(t) = exp\{∫_{T}^{t} \frac{δ_{ξ}(s)}{s log s} ds\}, t ⩾ T,

where δ_{ξ}(t) satisfies 0 ⩽ δ_{ξ}(t) ⩽ 1 and lim_{t→∞} δ_{ξ}(t) = μ_{0}. The set Ξ can be regarded as a closed and convex subset of the locally convex space C^{1}[T, ∞) of all continuously differentiable functions on [T, ∞) with the metric topology of uniform convergence of functions and their derivatives on compact subintervals of [T, ∞). Let {ξ_{n}(t)} be a sequence in Ξ consisting of

(3.17)  ξ_{n}(t) = exp\{∫_{T}^{t} \frac{δ_{n}(s)}{s log s} ds\}, t ⩾ T, n = 1, 2, …,

where δ_{n}(t) satisfies 0 ⩽ δ_{n}(t) ⩽ 1, n = 1, 2, .., and lim_{t→∞} δ_{n}(t) = μ_{0}. Suppose that {ξ_{n}(t)} and {ξ′_{n}(t)} converge, respectively, to ξ(t) and ξ′(t) on compact subintervals of [T, ∞). Since by (3.17)

(3.18)  t log t\frac{ξ′_{n}(t)}{ξ_{n}(t)} = δ_{n}(t), t ⩾ T, n = 1, 2, …,

we find that

δ(t) := lim_{n→∞} δ_{n}(t) = t log t\frac{ξ′(t)}{ξ(t)}, t ∈ [T, ∞).

It is clear that δ(t) is continuous on [T, ∞), and satisfies 0 ⩽ δ(t) ⩽ 1 and lim_{t→∞} δ_{n}(t) = μ_{0}. It follows that

ξ(t) = exp\{∫_{T}^{t} \frac{δ(s)}{s log s} ds\}, t ⩾ T,

which implies that ξ(t) ∈ Ξ, showing that Ξ is a closed subset of C^{1}[T, ∞). To make sure the convexity of Ξ let ξ_{1}(t), …, ξ_{N}(t) be N functions in Ξ and let c_{1}, …, c_{N} be N positive constants such that ∑_{k=1}^{N} c_{k} = 1. Then, using (3.18), we see that the function ∑_{k=1}^{N} c_{k}ξ_{k}(t) is expressed in the form

(3.19)  ∑_{k=1}^{N} c_{k}ξ_{k}(t) = exp\{∫_{T}^{t} \frac{δ_{N}(s)}{s log s} ds\}, t ⩾ T,

in terms of the continuous function δ_{N}(t) defined by

δ_{N}(t) = \frac{∑_{k=1}^{N} c_{k}δ_{k}(t)ξ_{k}(t)}{∑_{k=1}^{N} c_{k}ξ_{k}(t)}.

As easily checked, 0 ⩽ δ_{N}(t) ⩽ 1 and lim_{t→∞} δ_{N}(t) = μ_{0}, and hence (3.19) ensures that ∑_{k=1}^{N} c_{k}ξ_{k}(t) ∈ Ξ, proving that Ξ is a convex set.

Another important property of Ξ is that

(3.20)  \frac{log g(t)}{log t} ⩽ \frac{ξ(g(t))}{ξ(t)} ⩽ 1, t ⩾ T,

for all ξ(t) ∈ Ξ. This is an immediate consequence of the relation

\frac{ξ(g(t))}{ξ(t)} = exp\{− ∫_{g(t)}^{t} \frac{δ_{ξ}(s)}{s log s} ds\}, 0 ⩽ δ_{ξ}(t) ⩽ 1,

following from (3.16). In fact, if t ⩾ T_{1}, where T_{1} > T is such that T = g(T_{1}), then we have

exp\{− ∫_{g(t)}^{t} \frac{δ_{ξ}(s)}{s log s} ds\} ⩾ exp\{− ∫_{g(t)}^{t} \frac{1}{s log s} ds\} = \frac{log g(t)}{log t},

and if T ⩽ t ⩽ T_{1}, then we have

\frac{ξ(g(t))}{ξ(t)} ⩾ exp\{− ∫_{T}^{t} \frac{1}{s log s} ds\} = \frac{log T}{log t} ⩾ \frac{log g(t)}{log t}.

Using (3.20), we find that for all ξ(t) ∈ Ξ (3.21)

t((\frac{g(t)}{t})^{1/2} \frac{log g(t)}{log t}q(t) − \frac{1}{4t^{2}} ) ⩽ t((\frac{g(t)}{t})^{1/2} \frac{ξ(g(t)}{ξ(t)}q(t) − \frac{1}{4t^{2}} ) ⩽ t(q(t) − \frac{1}{4t^{2}} ).

For simplicity we put for any ξ(t) ∈ Ξ

(3.22)  q_{ξ}(t) = t((\frac{g(t)}{t})^{1/2} \frac{ξ(g(t))}{ξ(t)}q(t) − \frac{1}{4t^{2}} ),

and

(3.23)  Q_{ξ}(t) = log t ∫_{t}^{∞} q_{ξ}(s) ds − d.

Because of (3.14) and (3.15) we have |Q_{ξ}(t)| ⩽ l^{2}, t ⩾ T, ξ(t) ∈ Ξ. This inequality makes it possible to apply Theorem 2.1 to each member of the family of ordinary differential equations

(3.24)  (ty′)′ + q_{ξ}(t)y = 0, ξ(t) ∈ Ξ,

obtaining a solution y_{ξ}(t) in n-RV_{log t}(μ_{0}) of the form

(3.25)  y_{ξ}(t) = exp\{∫_{T}^{t} \frac{μ_{0} + v_{ξ}(s) + Q_{ξ}(s)}{s log s} ds\}, t ⩾ T,

where v_{ξ}(t) solves the integral equation

(3.26)  v_{ξ}(t) = (log t)^{1−2μ_{0}} ∫_{t}^{∞} \frac{2μ_{0}Q_{ξ}(s) + (v_{ξ}(s) + Q_{ξ}(s))^{2}}{s(log s)^{2−2μ_{0}}} ds, t ⩾ T,

and |v_{ξ}(t)| ⩽ l for t ⩾ T and lim_{t→∞} v_{ξ}(t) = 0.

Our final procedure is to show that the set {y_{ξ}(t) : ξ(t) ∈ Ξ} contains at least one member which provides an RV_{log t}(μ_{0})-solution of the retarded differential equation (3.7) with the help of the Schauder–Tychonoff fixed point theorem. To this end we define Φ to be the mapping which assigns to each ξ(t) ∈ Ξ the function Φξ(t) given by

(3.27)  Φξ(t) = 1 for g(T) ⩽ t ⩽ T,  Φξ(t) = y_{ξ}(t) for t ⩾ T.

(i) Φ maps Ξ into itself. This follows immediately from (3.25). Put δ_{ξ}(t) = μ_{0} + v_{ξ}(t) + Q_{ξ}(t). Since |v_{ξ}(t) + Q_{ξ}(t)| ⩽ 2l, t ⩾ T, v_{ξ}(t) + Q_{ξ}(t) → 0 as t → ∞ and μ_{0} ∈ (0, \frac{1}{2}), we have 0 ⩽ δ_{ξ}(t) ⩽ 1 for t ⩾ T and lim_{t→∞} δ_{ξ}(t) = μ_{0}, which ensures that Φξ(t) ∈ Ξ.

(ii) Φ(Ξ) is relatively compact in C^{1}[T, ∞). In virtue of (3.25) we obtain for any ξ(t) ∈ Ξ and for t ⩾ T

y_{ξ}(t) ⩽ exp\{∫_{T}^{t} \frac{1}{s log s} ds\} = \frac{log t}{log T} ,

|y′_{ξ}(t)| = y_{ξ}(t)\frac{|μ_{0} + v_{ξ}(t) + Q_{ξ}(t)|}{t log t} ⩽ \frac{1}{T log T} ,

and

|(ty′_{ξ}(t))′| = |q_{ξ}(t)|y_{ξ}(t) ⩽ t(q(t) + \frac{1}{4t^{2}} )\frac{log t}{log T} .

This clearly establishes, via the Arzela–Ascoli lemma, the relative compactness of the set Φ(Ξ).

(iii) Φ is a continuous mapping. Let {ξ_{n}(t)} be a sequence in Ξ converging to ξ(t) in C^{1}[T, ∞). This means that ξ_{n}(t) → ξ(t) and ξ′_{n}(t) → ξ′_{n}(t) as n → ∞ uniformly on compact subintervals of [T, ∞). We have to prove that

(3.28)  Φξ_{n}(t) → Φξ(t) and (Φξ_{n})′(t) → (Φξ)′(t)

uniformly on any compact subinterval of [T, ∞). Using (3.25), we obtain for t ⩾ T

|Φξ_{n}(t) − Φξ(t)| = |y_{ξ_{n}}(t) − y_{ξ}(t)|

⩽ |exp\{∫_{T}^{t} \frac{μ_{0} + v_{ξ_{n}}(s) + Q_{ξ_{n}}(s)}{s log s} ds\} − exp\{∫_{T}^{t} \frac{μ_{0} + v_{ξ}(s) + Q_{ξ}(s)}{s log s} ds\}|

⩽ \frac{log t}{log T} ∫_{T}^{t} \frac{|v_{ξ_{n}}(s) − v_{ξ}(s)| + |Q_{ξ_{n}}(s) − Q_{ξ}(s)|}{s log s} ds,

and

|(Φξ_{n})′(t) − (Φξ)′(t)| = |y′_{ξ_{n}}(t) − y′_{ξ}(t)|

= |y_{ξ_{n}}(t)\frac{μ_{0} + v_{ξ_{n}}(t) + Q_{ξ_{n}}(t)}{t log t} − y_{ξ}(t)\frac{μ_{0} + v_{ξ}(t) + Q_{ξ}(t)}{t log t}|

⩽ |(y_{ξ_{n}}(t) − y_{ξ}(t))\frac{μ_{0} + v_{ξ_{n}}(t) + Q_{ξ_{n}}(t)}{t log t} + y_{ξ}(t)\frac{(v_{ξ_{n}}(t) − v_{ξ}(t)) + (Q_{ξ_{n}}(t) − Q_{ξ}(t))}{s log s}|

⩽ \frac{|y_{ξ_{n}}(t) − y_{ξ}(t)|}{t log t} + \frac{log t}{log T} \frac{|v_{ξ_{n}}(t) − v_{ξ}(t)| + |Q_{ξ_{n}}(t) − Q_{ξ}(t)|}{t log t}.

Consequently, in order to verify (3.28) it suffices to prove that the two sequences

(3.29)  \frac{|v_{ξ_{n}}(t) − v_{ξ}(t)|}{t log t},  \frac{|Q_{ξ_{n}}(t) − Q_{ξ}(t)|}{t log t}

converge to 0 uniformly on compact subintervals of [T, ∞). As a matter of fact, one can prove the uniform convergence on [T, ∞) of the sequences

(3.30)  \frac{|v_{ξ_{n}}(t) − v_{ξ}(t)|}{log t} and \frac{|Q_{ξ_{n}}(t) − Q_{ξ}(t)|}{log t}.

As for the second sequence in (3.30) we have from (3.23)

\frac{|Q_{ξ_{n}}(t) − Q_{ξ}(t)|}{log t} ⩽ ∫_{t}^{∞} |q_{ξ_{n}}(s) − q_{ξ}(s)|ds,

and estimate the integrand as follows:

(3.31) |q_{ξ_{n}}(s) − q_{ξ}(s)| ⩽ (\frac{g(s)}{s})^{1/2}|\frac{ξ_{n}(g(s))}{ξ_{n}(s)} − \frac{ξ(g(s))}{ξ(s)}|s(q(s) − \frac{1}{4s^{2}} )

+ |(\frac{g(s)}{s})^{1/2} \frac{ξ_{n}(g(s))}{ξ_{n}(s)} − 1|\frac{1}{4s} + |(\frac{g(s)}{s})^{1/2} \frac{ξ(g(s))}{ξ(s)} − 1|\frac{1}{4s}

⩽ 2s(q(s) − \frac{1}{4s^{2}} ) + 2(1 − (\frac{g(s)}{s} \frac{log g(t)}{log t})^{1/2})\frac{1}{4s},

which shows that |q_{ξ_{n}}(t) − q_{ξ}(t)| is bounded by an integrable function on [T, ∞) independent of n. Since q_{ξ_{n}}(t) → q_{ξ}(t) as n → ∞ for every t ⩾ T, from the Lebesgue convergence theorem it follows that |Q_{ξ_{n}}(t) − Q_{ξ}(t)|/ log t converges to 0 uniformly on [T, ∞), and hence so does the second sequence in (3.29). To examine the first sequence in (3.31) we proceed as follows. Using (3.26), we have (3.32)

\frac{|v_{ξ_{n}}(t) − v_{ξ}(t)|}{(log t)^{1−2μ_{0}}} ⩽ 4l ∫_{t}^{∞} \frac{|v_{ξ_{n}}(s) − v_{ξ}(s)|}{s(log s)^{2−2μ_{0}}} ds + (4l + 2μ_{0}) ∫_{t}^{∞} \frac{|Q_{ξ_{n}}(s) − Q_{ξ}(s)|}{s(log s)^{2−2μ_{0}}} ds

for t ⩾ T. Putting

z(t) = ∫_{t}^{∞} \frac{|v_{ξ_{n}}(s) − v_{ξ}(s)|}{s(log s)^{2−2μ_{0}}} ds,

(3.32) is transformed into the differential inequality

(3.33)  ((log t)^{4l}z(t))′ ⩾ −\frac{4l + 2μ_{0}}{t(log t)^{1−4l}} ∫_{t}^{∞} \frac{|Q_{ξ_{n}}(s) − Q_{x}i(s)|}{s(log s)^{2−2μ_{0}}} ds, t ⩾ T.

We now integrate (3.33) from t to ∞. Noting that (log t)^{4l}z(t) → 0 as t → ∞ and that the right-hand side of (3.33) is integrable over [T, ∞), we obtain

(3.34)  z(t) ⩽ \frac{4l + 2μ_{0}}{4l(log t)^{4l}} ∫_{t}^{∞} \frac{|Q_{ξ_{n}}(s) − Q_{ξ}(s)|}{s(log s)^{2−2μ_{0}−4l}} ds, t ⩾ T.

Note that 2 − 2μ_{0} − 4l > 1. Using (3.34) in (3.32), we have

\frac{|v_{ξ_{n}}(t) − v_{ξ}(t)|}{(log t)^{1−2μ_{0}}}

⩽ \frac{4l + 2μ_{0}}{(log t)^{4l}} ∫_{t}^{∞} \frac{|Q_{ξ_{n}}(s) − Q_{ξ}(s)|}{s(log s)^{2−2μ_{0}−4l}} ds + (4l + 2μ_{0}) ∫_{t}^{∞} \frac{|Q_{ξ_{n}}(s) − Q_{ξ}(s)|}{s(log s)^{2−2μ_{0}}} ds

⩽ \frac{2(4l + 2μ_{0})}{(log t)^{4l}} ∫_{t}^{∞} \frac{|Q_{ξ_{n}}(s) − Q_{ξ}(s)|}{s(log s)^{2−2μ_{0}−4l}} ds

for t ⩾ T. This shows that |v_{ξ_{n}}(t) − v_{ξ}(t)|/(log t)^{1−2μ_{0}} converges to 0 uniformly on [T, ∞), and hence so does the sequence |v_{ξ_{n}}(t)−v_{ξ}(t)|/ log t. We therefore conclude that the mapping Φ defined by (3.27) is continuous in the topology of C^{1}[T, ∞). Thus all the hypotheses of the Schauder–Tychonoff fixed point theorem are fulfilled, and so there exists ξ_{0}(t) ∈ Ξ such that ξ_{0}(t) = Φξ_{0}(t), that is, ξ_{0}(t) = y_{ξ_{0}}(t) for t ⩾ T. This means that ξ_{0}(t) satisfies the linear differential equation

(tξ_{0}(t))′ + t((\frac{g(t)}{t})^{1/2} \frac{ξ_{0}(g(t))}{ξ_{0}(t)}q(t) − \frac{1}{4t^{2}} )ξ_{0}(t) = 0,

which is rewritten as

(tξ_{0}(t))′ + t((\frac{g(t)}{t})^{1/2}q(t)ξ_{0}(g(t)) − \frac{1}{4t^{2}} ξ_{0}(t)), t ⩾ T.

This implies that equation (3.6) has an RV(μ_{0})-solution ξ_{0}(t) existing on [T, ∞). The function x_{0}(t) = t^{1/2}ξ_{0}(t) then provides a solution of equation (B) enjoying the first property in (3.4) with i = 0.

Next, we will be concerned with the construction of a solution x_{1}(t) of equation (B) such that x_{1}(t) = t^{1/2}y_{1}(t) with y_{(}t) ∈ RV(μ_{1}). Note that μ_{1} ∈ (\frac{1}{2}, 1). Let m be any positive constant such that

\frac{2(2μ_{1} + 2)}{2μ_{1} − 1}m ⩽ 1,

and choose T > a large enough so that g(T) ⩾ a,

(3.35)  |log t ∫_{t}^{∞} s(q(s) − \frac{1}{4s^{2}} ) ds − d| ⩽ m^{2},

and

(3.36)  |log t ∫_{t}^{∞}((\frac{g(s)}{s})^{1/2} \frac{log g(s)}{log s}q(s) − \frac{1}{4s^{2}} ) ds − d| ⩽ m^{2},

for t ⩾ T. Define H to be the set of all functions C[g(T), ∞)∩C^{1}[T, ∞) such that

(3.37)  η(t) = 1, g(T) ⩽ t ⩽ T;  η(t) = exp\{∫_{T}^{t} \frac{δ_{η}(s)}{s log s} ds\}, t ⩾ T,

where δ_{η}(t) is continuous and satisfies 0 ⩽ δ_{η}(t) ⩽ 1 and lim_{t→∞} δ_{η}(t) = μ_{1}. As in the preceding case the set H can be regarded as a closed convex subset of the locally convex space C^{1}[T, ∞). Also note that there hold the inequalities (3.20) and (3.21) with ξ(t) replaced by η(t). Put

q_{η}(t) = t((\frac{g(t)}{t})^{1/2} \frac{η(g(t))}{η(t)}q(t) − \frac{1}{4t^{2}} ),

and

Q_{η}(t) = log t ∫_{t}^{∞} q_{η}(s) ds − d.

Since Q_{η}(t) satisfies |Q_{η}(t)| ⩽ m^{2}, t ⩾ T, for all η(t) ∈ H, by (3.35) and (3.36), applying Theorem 2.1 to the family of linear ordinary differential equations

(ty′)′ + q_{η}(t)y = 0, η(t) ∈ H,

we obtain for each η ∈ H a solution Y_{η}(t) ∈ RV_{log t}(μ_{1}) having the expression

(3.38)  Y_{η}(t) = exp\{∫_{T}^{t} \frac{μ_{1} − w_{η}(s) + Q_{η}(s)}{s log s} ds\}, t ⩾ T,

where w_{η}(t) is a solution of the integral equation

(3.39)  w_{η}(t) = (log t)^{1−2μ_{1}} ∫_{T}^{t} \frac{2μ_{1}Q_{η}(s) + (w_{η}(s) − Q_{η}(s))^{2}}{s(log s)^{2−2μ_{1}}} ds,

satisfying |w_{η}(t)| ⩽ m for t ⩾ T and lim_{t→∞} w_{η}(t) = 0.

We define by Ψ the mapping which assigns to each η(t) ∈ H the function

Ψη(t) = 1, g(T) ⩽ t ⩽ T, Ψη(t) = Y_{η}(t), t ⩾ T.

Our final step is to verify that Ψ is continuous and sends H into a relatively compact subset of H so that the Schauder–Tychonoff fixed point theorem is applicable to Ψ. The same arguments as employed in the case of Φ defined by (3.27) are sufficient for this purpose. To avoid tedious duplications only a brief mention will be made of the continuity of Ψ. Let {η_{n}(t)} be a sequence in H converging to η(t) in C^{1}[T, ∞). Our goal is attained if it is shown that

(3.40)  Y_{η_{n}}(t) → Y_{η}(t) and Y′_{η_{n}}(t) → Y′_{η}(t) as n → ∞.

uniformly on any compact subinterval of [T, ∞). In view of (3.38) we see that (3.40) is assured if the two sequences

\frac{|w_{η_{n}}(t) − w_{η}(t)|}{log t} and \frac{|Q_{η_{n}}(t) − Q_{η}(t)|}{log t}

converge to 0 uniformly on compact subintervals of [T, ∞). We need only to consider the first sequence in (3.41). From (3.39) we obtain the inequality

(3.41) \frac{|w_{η_{n}}(t) − w_{η}(t)|}{(log t)^{1−2μ_{1}}}

⩽ 4m ∫_{T}^{t} \frac{|w_{η_{n}}(s) − w_{η}(s)|}{s(log s)^{2−2μ_{1}}} ds + (4m + 2μ_{1}) ∫_{T}^{t} \frac{|Q_{η_{n}}(s) − Q_{η}(s)|}{s(log s)^{2−2μ_{1}}} ds, t ⩾ T.

Using the function

z(t) = ∫_{T}^{t} \frac{|w_{η_{n}}(s) − w_{η}(s)|}{s(log s)^{2−2μ_{1}}} ds,

we transform (3.41) into

(\frac{z(t)}{(log t)^{4m}} )′ ⩽ \frac{4m + 2μ_{1}}{t(log t)^{4m+1}} ∫_{T}^{t} \frac{|Q_{η_{n}}(s) − Q_{η}(s)|}{s(log s)^{2−2μ_{1}}} ds, t ⩾ T,

which, after integration over [T, t], yields

(3.42)  z(t) ⩽ \frac{4m + 2μ_{1}}{4m} (log t)^{4m} ∫_{T}^{t} \frac{|Q_{η_{n}}(s) − Q_{η}(s)|}{s(log s)^{2−2μ_{1}+4m}} ds,

for t ⩾ T. Combining (3.41) with (3.42), we have

\frac{|w_{η_{n}}(t) − w_{η}(t)|}{log t} ⩽ \frac{4m + 2μ_{1}}{(log t)^{2μ_{1}−4m}} ∫_{T}^{t} \frac{|Q_{η_{n}}(s) − Q_{η}(s)|}{s(log s)^{2−2μ_{1}+4m}} ds

+ (4m + 2μ_{1})(log t)^{2μ_{1}} ∫_{T}^{t} \frac{|Q_{η_{n}}(s) − Q_{η}(s)|}{s(log s)^{2−2μ_{1}}} ds, t ⩾ T,

whence it follows that

\frac{|w_{η_{n}}(t) − w_{η}(t)|}{log t} ⩽ 2(4m + 2μ_{1}) ∫_{T}^{t} \frac{|Q_{η_{n}}(s) − Q_{η}(s)|}{s(log s)^{2}} ds, t ⩾ T.

This ensures the desired convergence of the sequence |w_{η_{n}}(t) − w_{η}(t)|/ log t.

Thus the continuity of the mapping Ψ has been assured. Let η_{1}(t) ∈ H be a fixed point of Ψ : η_{1}(t) = Ψη_{1}(t). Since η_{1}(t) = Y_{η_{1}}(t) for t ⩾ T, η_{1}(t) satisfies the differential equation

(tη′_{1}(t))′ + t((\frac{g(t)}{t})^{1/2} \frac{η_{1}(g(t))}{η_{1}(t)}q(t) − \frac{1}{4t^{2}} )η_{1}(t) = 0,

or

(tη′_{1}(t))′ + t((\frac{g(t)}{t})^{1/2}q(t)η_{1}(g(t)) − \frac{1}{4t^{2}} η_{1}(t)) = 0, t ⩾ T.

Since η_{1}(t) ∈ RV_{log t}(μ_{1}), the function x_{1}(t) defined by x_{1}(t) = t^{1/2}η_{1}(t) gives a second solution of equation (B).

It remains to deal with the case where d = 0. In this case the roots of the quadratic equation (3.2) are μ_{0} = 0 and μ_{1} = 1, and we are required to look for the solutions y_{i}(t) ∈ n-RV_{log t}(i), i = 0, 1, of equation (3.6). Such solutions can be found in essentially the same manner as in the previous case d > 0. Only a brief explanation will be given about how to construct a solution y_{0}(t) ∈ n-RV(0) = n-RV. This time, instead of (3.16), we choose the set Ξ consisting of all functions ξ(t) ∈ C[g(T), ∞)∩C^{1}[T, ∞) such that

ξ(t) = 1, g(T) ⩽ t ⩽ T; ξ(t) = exp\{∫_{T}^{t} \frac{δ_{ξ}(s)}{s log s} ds\}, t ⩾ T,

where δ_{ξ}(t) satisfies −1 ⩽ δ_{ξ}(t) ⩽ 1 and lim_{t→∞} δ_{ξ}(t) = 0, and T > a will be determined later. It is clear that

\frac{log g(t)}{log t} ⩽ \frac{ξ(g(t))}{ξ(t)} ⩽ \frac{log t}{log g(t)}, t ⩾ T,

for all ξ(t) ∈ Ξ. Consequently,

(3.43) t((\frac{g(t)}{t})^{1/2} \frac{log g(t)}{log t}q(t) − \frac{1}{4t^{2}} ) ⩽ t((\frac{g(t)}{t})^{1/2} \frac{ξ(g(t))}{ξ(t)}q(t) − \frac{1}{4t^{2}} )

⩽ t((\frac{g(t)}{t})^{1/2} \frac{log t}{log g(t)}q(t) − \frac{1}{4t^{2}} ), t ⩾ T.

Let a constant 0 < l < \frac{1}{4} be fixed, and choose T > a so that g(T) ⩾ a and both (3.15) (with l^{2} replaced by l) and the following inequality are satisfied:

(3.44)  |log t ∫_{t}^{∞} s((\frac{g(s)}{s})^{1/2} \frac{log s}{log g(s)}q(s) − \frac{1}{4s^{2}} ) ds − d| ⩽ l, t ⩾ T.

Therefore, by (3.15), (3.43) and (3.44), the function Q_{ξ}(t) defined by (3.23) and (3.22) satisfies |Q_{ξ}(t)| ⩽ l, t ⩾ T, for all ξ(t) ∈ Ξ, because of which Theorem 2.2 guarantees, for every ξ(t) ∈ Ξ, the existence of a solution y_{ξ}(t) ∈ SV_{log t} of equation (3.24) having the representation

y_{ξ}(t) = exp\{∫_{T}^{t} \frac{v_{ξ}(s) + Q_{ξ}(s)}{s log s} ds\}, t ⩾ T,

where v_{ξ}(t) solves the integral equation

v_{ξ}(t) = log t ∫_{t}^{∞} \frac{(v_{ξ}(s) + Q_{ξ}(s))^{2}}{s(log s)^{2}} ds, t ⩾ T.

Let us define the mapping Φ by (3.27) and repeat the same (somewhat simplified) argument as that applied to the case of positive d. Then, via the Schauder-Tychonoff fixed point theorem, we are led to the conclusion that Φ has a fixed point ξ_{0}(t) ∈ Ξ, which generates a solution x_{0}(t) = t^{1/2}ξ_{0}(t) of (A) with the desired first property.

In order to obtain a solution x_{1}(t) of (A) with the second property of (3.4) (with μ_{1} = 1), we need only to define H replacing (3.37) to be the set of all functions η(t) ∈ C[g(T), ∞)∩C^{1}[T, ∞) such that

η(t) = 1, g(T) ⩽ t ⩽ T; η(t) = exp\{∫_{T}^{t} \frac{δ_{η}(s)}{s log s} ds\}, t ⩾ T,

where δ_{η}(t) satisfies 0 ⩽ δ_{η}(t) ⩽ 2 and lim_{t→∞} δ_{η}(t) = 1, and apply almost the same argument as in the case of d > 0 on the basis of the inequality

(\frac{log g(t)}{log t})^{2} ⩽ \frac{η(g(t))}{η(t)} ⩽ \frac{log t}{log g(t)}, t ⩾ T,

holding for all η(t) ∈ H. No further explanation will be necessary. This completes the proof of Theorem 3.1. □

Example 3.1. Consider the retarded differential equation

(3.45)  x′′(t) + q(t)x(g(t)) = 0, q(t) = \frac{1}{4t^{2}} + \frac{d(t)}{(t log t)^{2}} , t ⩾ e,

where g(t) is a retarded argument satisfying the conditions given in Theorem 3.1 and d(t) is a positive continuous function such that lim_{t→∞} d(t) = d ∈ [0, \frac{1}{4}). Since condition (3.5) holds for q(t) (cf. Example 2.1), by Theorem 3.1 equation (3.45) possesses two solutions x_{i}(t), i = 0, 1, such that t^{−1/2}x_{i}(t) ∈ RV_{log t}(μ_{i}), i = 0, 1, where μ_{i}, i = 0, 1, are the roots of (3.3). Admissible retarded arguments g(t) include t − τ, t − t^{θ}, t − log t, where τ and θ < 1 are positive constants.

Example 3.2. Consider the retarded differential equation

(3.46)  x′′(t) + q(t)x(t − τ) = 0,

where

q(t) = \frac{1}{4t^{3/2}(t − τ)^{1/2}} (\frac{log t}{log(t − τ)})^{1/8} + \frac{7}{64t^{3/2}(t − τ)^{1/2}(log t)^{15/8}(log(t − τ))^{1/8}} .

It ia a matter of elementary computation to see that

lim_{t→∞} log t ∫_{t}^{∞} s(q(s) − \frac{1}{4s^{2}} ) ds = \frac{7}{64}.

This implies via Theorem 3.1 that equation (3.46) possesses two solutions x_{i}(t) = t^{1/2}y_{i}(t), i = 0, 1, such that y_{0}(t) ∈ RV_{log t}(\frac{1}{8}) and y_{1}(t) ∈ RV_{log t}(\frac{7}{8}). One such solution is x_{0}(t) = t^{1/2}(log t)^{1/8}.

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Department of Applied Mathematics (Received 10 09 2009) Faculty of Science Fukuoka University Fukuoka, 841-0180 Japan

 kusanot@zj8.so-net.ne.jp

Serbian Academy of Sciences and Arts Kneza Mihaila 35 11000 Beograd, Serbia

 vojam@uns.ac.rs