Electronic Journal of Qualitative Theory of Differential Equations Proc. 8th Coll. QTDE, 2008, No. 7 1-14; http://www.math.u-szeged.hu/ejqtde/

PRINCIPAL SOLUTION OF HALF-LINEAR DIFFERENTIAL EQUATION: LIMIT AND INTEGRAL CHARACTERIZATION

ZUZANA DOŠLÁ AND ONDŘEJ DOŠLÝ

Abstract. We investigate integral and limit characterizations of the principal solution of the nonoscillatory half-linear differential equation

(r(t)Φ(x′))′ + c(t)Φ(x) = 0, Φ(x) = |x|^{p−2}, p > 1. In particular, we supplement and extend the results of the previous papers [2, 3, 5, 8, 11].

1. Introduction

In this paper we deal with the second order half-linear differential equation

(r(t)Φ(x′))′ + c(t)Φ(x) = 0, Φ(x) = |x|^{p−2}, p > 1, (1)

where r, c are continuous functions for t ≥ 0 and r(t) > 0.

The qualitative theory of (1) attracted considerable attention in recent years. It was shown that many properties of solutions of (1) are very similar to those of the linear Sturm-Liouville differential equation (which is the special case p = 2 in (1))

(r(t)x′)′ + c(t)x = 0. (2)

On the other hand, several phenomena have been indicated, where the behavior of solutions of (1) and (2) is completely different. We refer to the recent book [12] for the comprehensive treatment of the theory of half-linear differential equations. For the reader who is not familiar with the elements of the theory of half-linear equations let us recall at least that the terminology half-linear equation comes from the Hungarian mathematicians I. Bihari [1] and Á. Elbert [13] and it is motivated by the fact that the solution space of (1) has just one half of the properties which characterize linearity, namely homogeneity, but generally not additivity.

Our principal concern is to investigate properties of the so-called principal solution of (1). The concept of the principal solution of the linear equation (2) was introduced

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 7, p. 1 by Leighton and Morse [18] and basic properties of this solution were investigated by Hartman (see [16] for a basic survey). It was shown, that nonoscillatory equation (2) (i.e., there exists T ∈ ℝ, such that any nontrivial solution of (2) has at most one zero point on [T, ∞)) possesses a unique (up to a nonzero multiplicative factor) solution x̃, called the principal solution, with the property that

Date: June 10, 2008.

1991 Mathematics Subject Classification. 34C10.

Key words and phrases. Half-linear differential equation; principal solution, integral characterization, limit characterization, Riccati equation.

Research supported by the Research Project MSM0022162409 of the Ministry of Education of the Czech Republic, and Grant 201/07/0145 of the Czech Grant Agency.

The paper is in final form and no version of it is submitted for publication elsewhere.

lim_{t→∞} \frac{x̃(t)}{x(t)} = 0 (3)

for any solution x linearly independent of x̃. An equivalent characterization of the principal solution x̃ is

∫ ^{∞} \frac{dt}{r(t)x̃^{2}(t)} = ∞ (4)

since this integral is convergent for any solution linearly independent of x̃.

A closer examination of the treatment given in [16, Chap. XI] reveals that characterizations (3) and (4) are based on the additivity of the solution space and the Wronskian identity for solutions of (2), respectively, but none of these two properties extends to (1). However, as observed Mirzov [19] and later independently Elbert and Kusano [15], there is another equivalent characterization of the principal solution of (2) (less known than (3) and (4)) that does extend to (1). It is based on the fact that the solution w̃ := rx̃′/x̃ of the Riccati equation

w′ + c(t) + \frac{w^{2}}{r(t)} = 0 (5)

corresponding to the principal solution x̃ of (1) is smaller than any other solution of (5) for large t .

Mirzov [19] observed that a similar situation holds for the half-linear equation (1) and the associated Riccati equation

w′ + c(t) + (p − 1)\frac{|w|^{q}}{r^{q−1}(t)} = 0, q = \frac{p}{p − 1}, (6)

which is related to (1) by the Riccati substitution

w̃(t) = r(t)Φ(x̃′(t)/x̃(t)).

More precisely, Mirzov showed that among all solutions w of (6) there exists the minimal one w̃, minimal in the sense that for any other solution w of (6) we have

(7)  w̃(t) < w(t) for large t.

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 7, p. 2 The principal solution of (1) is that solution x̃ which “generates” the minimal solution w̃ via the Riccati substitution, i.e., it is given by the formula

x̃(t) = C exp {∫ ^{t} r^{1−q}(s)Φ^{−1}(w̃(s)) ds} ,

where Φ^{−1}(x) = |x|^{q−2}x is the inverse function of Φ. Obviously, the principal solution is unique up to a constant factor.

Comparing the above three properties of principal solutions of (2) – the “Riccati property” (7), the limit property (3) and the integral property (4), the last one is specific: it involves just one solution x̃, without comparing it with other solutions. The extension of the integral property to half-linear equations seems to be a difficult problem. As we will show, the universal integral characterization which would contain only one solution likely does not exist in general.

In this paper we continue in the investigation initiated in [2, 3, 4, 5, 6, 8, 11], where various integral characterizations of the principal solution of (1) have been offered and the limit characterization (3) has been investigated (see [2]–[5]). Our main result shows how the limit characterization (3) of the principal solution x̃ is related with two types of integral characterizations. We also present an example showing that one of the results of the paper [8] is not correct.

We will use the following notation:

J_{r} = ∫ ^{∞} \frac{dt}{r^{q−1}(t)}, J_{c} = ∫ ^{∞} |c(t)| dt.

Note that if both integrals J_{r} and J_{c} are divergent, then (1) is oscillatory, see [12].

2. Survey of the known results

We recall known results concerning integral and limit characterizations of the principal solution of (1). The first attempt to find an integral characterization of the principal solution of (1) was made in the paper of Mirzov [19]. To introduce it, denote

F (t) := \begin{cases}\frac{1−|t|^{q}}{1−t} + Φ^{−1}(1 − t) & t ≠ 1, \\ q & t = 1, \end{cases}

and

m_{∗} = min{F (t) : t ∈ [0, 1]}, m^{∗} = max{F (t) : t ∈ [0, 1]}.

Mirzov’s integral characterization of the pricipal solution of (1) is given in the next proposition.

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 7, p. 3 Proposition 1. Suppose that J_{r} = ∞. If x̃ is the principal solution, then

∫ ^{∞} \frac{dt}{r^{q−1}(t)|x̃(t)|^{m_{∗}}} = ∞,

and, conversely, if

∫ ^{∞} \frac{dt}{r^{q−1}(t)|x̃(t)|^{m^{∗}}} = ∞,

then x̃ is the principal solution of (1).

However, the exponents m_{∗}, m^{∗} coincide (and m_{∗} = m^{∗} = 2) only in the linear case p = 2 and generally m^{∗} > m_{∗}, so Mirzov’s integrals characterize principal solution of (1) equivalently only in the linear case.

The second attempt to find an integral characterization of the principal solution of (1) comes from the paper [8]. The main result of that paper reads as follows.

Proposition 2. Let x̃ be a solution of (1) such that x̃′(t) ≠ 0 for large t. Then we have the following statements:

(i) Let p ∈ (1, 2]. If

I(x̃) := ∫ ^{∞} \frac{dt}{r(t)x̃^{2}(t)|x̃′(t)|^{p−2}} = ∞, (8)

then x̃ is the principal solution of (1). (ii) Let p ≥ 2. If x̃ is the principal solution of (1) then (8) holds. (iii) Suppose that p ≥ 2, J_{r} = ∞, the function γ(t) := ∫_{t}^{∞} c(s) ds exists, and

γ(t) ≥ 0, but γ(t) ≢ 0 for large t. Then x̃ is the principal solution of (1) if and only if (8) holds.

Note that the statement (iii) of the previous proposition is stated in [8] without the assumption p ≥ 2. As the next example shows, the implication: “x̃ is the principal solution ⟹ I(x̃) = ∞” may fail to hold for p ∈ (1, 2).

Example 1. Consider the equation

((x′)^{1/2})′ + \frac{15t^{−3/2}}{(t^{9} − 1)^{1/2}} x^{1/2} = 0, t ≥ 1. (9)

This equation has a solution x̃(t) = 1 − 1/t^{9} which satisfies x̃(t) → 1 and its quasiderivative

x̃^{[1]} := (x̃′)^{1/2} = 3/t^{5} → 0

as t → ∞. According to the uniqueness result of [17] (see also [3, Theorem B]), all bounded solutions of (1) are uniquely determined up to a multiplicative constant.

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 7, p. 4 Hence,

lim_{t→∞} \frac{x̃(t)}{x(t)} = 0 (10)

for any solution x ≠ λx̃. Suppose that x̃ is not a principal solution of (9) and let x_{0} be a positive principal solution. By (7) we have

x′_{0}(t)/x_{0}(t) < x̃′(t)/x̃(t) for large t,

that is the function x̃/x_{0} is positive and increasing for large t. On the other hand, we have by (10) that x̃/x_{0} → 0, which is a contradiction. Thus x̃ is the principal solution of (9). However

I(x̃) = ∫ ^{∞} \frac{x̃′(t)}{x̃^{2}(t)x̃^{[1]}(t)} dt ∼ ∫ ^{∞}(x̃′(t))^{1/2} dt ∼ 3 ∫_{1}^{∞} \frac{dt}{t^{5}} < ∞.

Hence, Proposition 2, part (iii), is not generally true for p ∈ (1, 2).

Another integral characterization of the principal solution has been suggested in the papers [2, 3, 4], and the main results of these papers along this line are summarized in the next statement.

Proposition 3. Suppose that either (i) c(t) < 0 for large t, or (ii) both integrals J_{r}, J_{c} are convergent.

Then a solution x̃ of (1) is principal if and only if

∫ ^{∞} \frac{dt}{r^{q−1}x̃^{2}(t)} = ∞. (11)

Note that if J_{r} = ∞, c(t) > 0, and p > 2, then the principal solution of (1) need not satisfy (11), as the next example shows.

Example 2. Consider the half-linear Euler equation

(Φ(x′))′ + (\frac{γ}{t} )^{p} Φ(x) = 0, γ = (p − 1)/p

The solution x̃(t) = t^{γ} of this equation is principal (see e.g. [12, p. 40]) but does not satisfy (11) for p > 2. Note that any linearly independent nonprincipal solution is given by the asymptotic formula

x(t) = t^{γ}(log t)^{\frac{2}{p}} (1 + O(log^{−1} t)),

i.e., the limit characterization of the principal solution (10) holds in this case. EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 7, p. 5

When c(t) > 0 for large t and one of the integrals J_{r} or J_{c} is divergent, the integral characterization of the principal solution of (1) has been studied in [3, 5] and is summarized in the following two statements (see [5, Theorem 2 and Theorem 3]):

Proposition 4. Suppose that c(t) > 0 for large t, (1) is nonoscillatory and J_{r} = ∞. In addition, if both integrals

∫ ^{∞} \frac{1}{r^{q−1}(t)}(∫_{t}^{∞} c(s) ds)^{q−1} dt, ∫ ^{∞} c(t) (∫ ^{t} \frac{ds}{r^{q−1}(s)})^{p−1} dt (12)

are divergent, assume p > 2. Then a solution x̃ of (1) is principal if and only if

∫ ^{∞} \frac{c(t)(x̃(t))^{p−2}}{(r(t)Φ(x̃′(t))^{2}} dt = ∞.

Proposition 5. Suppose that c(t) > 0 for large t, (1) is nonoscillatory and J_{c} = ∞. In addition, if both integrals

∫ ^{∞} \frac{1}{r^{q−1}(t)}(∫ ^{t} c(s) ds)^{q−1} dt, ∫ ^{∞} c(t) (∫_{t}^{∞} \frac{ds}{r^{q−1}(s)})^{p−1} dt (13)

are divergent, assume p < 2. Then a solution x̃ of (1) is principal if and only if (8) holds.

Concerning the limit characterization of the half-linear principal solution, we have following statement proved in [2] (the case c(t) < 0), [3, 5] (the case c(t) > 0), and [4] (the case when c is allowed to change its sign).

Proposition 6. Suppose that one of the following conditions holds:

(i) c(t) < 0 for large t;

(ii) Both integrals J_{r}, J_{c} are convergent;

(iii) Assumptions of Proposition 4 or 5 are satisfied. Then a solution x̃ of (1) is principal if and only if

lim_{t→∞} \frac{x̃(t)}{x(t)} = 0 (14)

for every solution x of (1) linearly independent of x̃.

We complete this section by the following statement which is a new result so we present it including the proof.

Proposition 7. Suppose that c(t) > 0 for large t and one of the following conditions holds:

(i) J_{r} = ∞ and both integrals in (12) are convergent; (ii) J_{c} = ∞ and both integrals in (13) are convergent.

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 7, p. 6 If x̃ is the principal solution of (1), then

∫ ^{∞} \frac{dt}{r^{q−1}|x̃(t)|^{m}} = ∞ (15)

for any m > 1. Conversely, if x̃ is a solution of (1) satisfying (15) for some m > 1, then x̃ is the principal solution.

Proof. First assume (i). By [5, Corolary 1], the principal solution x̃ tends to a nonzero constant and so (15) holds for any m > 1. By [6, Theorem 7] and [5, Corolary 1], any linearly independent solution x satisfies

lim_{t→∞} \frac{x(t)}{∫^{t} r^{1−q}(s)ds} = c, 0 < |c| < ∞,

thus

∫ ^{∞} \frac{dt}{r^{q−1}(t)|x(t)|^{m}} < ∞ ⟺ ∫ ^{∞} \frac{r^{1−q}(t)}{(∫^{t} r^{1−q}(s)ds)^{m}} dt < ∞

for any m > 1. Concerning the case (ii), we proceed by a similar way using the fact that the principal solution x̃ and any other solution x satisfy

lim_{t→∞} \frac{x̃(t)}{∫_{t}^{∞} r^{1−q}(s)ds} = c, 0 < |c| < ∞, lim_{t→∞} x(t) = d, 0 < |d| < ∞.

□

Finally note that integral characterization (8) and limit characterization (14) have been studied for the first time in [14, Theorem 4] for perturbed half-linear Euler equations.

3. Auxiliary statements

Now we present some auxiliary results concerning certain functions of two and three variables which we will need later. An important role in our investigation is played by the functions

Q(u, v) := \frac{1}{p}|u|^{q} − Φ^{−1}(u)v + \frac{1}{q}|v|^{q},
W (u, v) := \frac{[Φ^{−1}(v) − Φ^{−1}(u)](v − u)}{Q(u, v)},

and

H(x, u, v) := \frac{1}{r^{q−1}(t)x^{p}} \frac{Q(u, v)}{(u − v)^{2}} .

Observe that for p = 2 we have

Q(u, v) = \frac{1}{2}(v − u)^{2}, W (u, v) ≡ 2, and H(x, u, v) = \frac{1}{2r(t)x^{2}} .

Lemma 1. There exist positive constants K_{1} < K_{2} such that

0 < K_{1} ≤ W (u, v) ≤ K_{2} (16)

for every u, v ∈ ℝ.

Proof. We have W (0, v) = q for v ∈ ℝ, hence (16) holds for u = 0. If u ≠ 0, let z = v/u. Then

W (u, v) = F (z) := \frac{|z|^{q} − Φ^{−1}(z) − z + 1}{\frac{1}{q}|z|^{q} − z + \frac{1}{p}}

and it suffices to show that the function F satisfies the inequalities stated in the Lemma. This function is nonnegative since

|z|^{q} − Φ^{−1}(z) − z + 1 = (Φ^{−1}(z) − 1)(z − 1) ≥ 0

and \frac{1}{q}|z|^{q} − z + \frac{1}{p} ≥ 0 (Young’s inequality), and the required inequalities follow from the fact that at the only zero point z = 1 of the denominator, we have

lim_{z→1} \frac{|z|^{q} − Φ^{−1}(z) − z + 1}{\frac{1}{q}|z|^{q} − z + \frac{1}{p}} = \frac{q(q − 1) + (q − 1)(q − 2)}{q(q − 1)} = \frac{2(q − 1)}{q},

i.e., this is a removable discontinuity, and from finiteness of the limit

lim_{z→±∞} \frac{|z|^{q} − Φ^{−1}(z) − z + 1}{\frac{1}{q}|z|^{q} − z + \frac{1}{p}} = q.

The proof is complete. □

The next statement is a technical result concerning a certain identity for solutions of (1) and of the associated Riccati equation (6).

Lemma 2. Let x, y be nonoscillatory solutions of (1) and let w_{x}, w_{y} be the associated solutions of (6). Then

(\frac{1}{|x(t)|^{p}(w_{y}(t) − w_{x}(t))})′ = pH(x(t), w_{x}(t), w_{y}(t)). (17)

Proof. Without loss of generality we may suppose that x(t) > 0 for large t. Denote by f(t) = x^{p}(t)(w_{y}(t) − w_{x}(t)). Then, we have

f′(t) = pΦ(x)x′(w_{y} − w_{x}) + x^{p}(w′_{y} − w′_{x})
= pΦ(x)r^{1−q}Φ^{−1}(w_{x})x(w_{y} − w_{x}) − (p − 1)x^{p}r^{1−q}(|w_{y}|^{q} − |w_{x}|^{q})
= x^{p}r^{1−q} [pΦ^{−1}(w_{x})(w_{y} − w_{x}) − (p − 1)(|w_{y}|^{q} − |w_{x}|^{q})] ,

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 7, p. 8 and hence

−(\frac{1}{f(t)})′ = \frac{f′(t)}{f^{2}(t)} = \frac{x^{p}r^{1−q} [pΦ^{−1}(w_{x})(w_{y} − w_{x}) − (p − 1)(|w_{y}|^{q} − |w_{x}|^{q})]}{x^{2p}(w_{y} − w_{x})^{2}}
= −\frac{(p − 1)|w_{y}|^{q} + pw_{y}Φ^{−1}(w_{x}) − |w_{x}|^{q}}{r^{q−1}x^{p}(w_{y} − w_{x})^{2}}
= −\frac{p}{r^{q−1}x^{p}} \frac{Q(w_{x}, w_{y})}{(w_{y} − w_{x})^{2}}
= −pH(x, w_{x}, w_{y})

and this completes the proof. □

4. Main result

Now we are in a position to formulate the main result of our paper. This result relates various quantities involving nonoscillatory solutions of (1) and associated solutions of the Riccati equation (6). As corollaries, it provides a unified view on the various characterizations of the principal solutions of (1) presented in Section 2.

Theorem 1. Let x, y be a pair of independent solutions of (1) and let w_{x}, w_{y} be the associated solutions of (6). Then the following integral and limit relations are equivalent:

(i)

∫ ^{∞} H(x(t), w_{x}(t), w_{y}(t)) dt = ∞;

(ii)

∫ ^{∞} \frac{1}{r^{q−1}(t)|x(t)|^{p}} \frac{Φ^{−1}(w_{y}(t)) − Φ^{−1}(w_{x}(t))}{w_{y}(t) − w_{x}(t)} dt = ∞;

(iii)

lim_{t→∞} |x(t)|^{p}(w_{y}(t) − w_{x}(t)) = 0 and w_{x}(t) < w_{y}(t) for large t;

(iv)

lim_{t→∞} \frac{x(t)}{y(t)} = 0.

Proof. According to the homogeneity of the solution space of (1), we can suppose that the relationship between x, y and w_{x}, w_{y} is

(18)  x(t) = exp {∫_{T}^{t} r^{1−q}(s)Φ^{−1}(w_{x}(s)) ds} ,
y(t) = exp {∫_{T}^{t} r^{1−q}(s)Φ^{−1}(w_{y}(s)) ds} ,

for some T ∈ ℝ.

(i) ⟺ (ii): Using Lemma 1 we get

\frac{1}{K_{2}} \frac{Φ^{−1}(v) − Φ^{−1}(u)}{v − u} ≤ \frac{Q(u, v)}{(u − v)^{2}} ≤ \frac{1}{K_{1}} \frac{Φ^{−1}(v) − Φ^{−1}(u)}{v − u}

for every u, v ∈ ℝ. From this we get

\frac{1}{K_{2}} F (x(t), w_{x}(t), w_{y}(t)) ≤ H(x(t), w_{x}(t), w_{y}(t)) ≤ \frac{1}{K_{1}} F (x(t), w_{x}(t), w_{y}(t)),

where

F (x(t), w_{x}(t), w_{y}(t)) = \frac{Φ^{−1}(w_{y}(t)) − Φ^{−1}(w_{x}(t))}{r^{q−1}(t)|x(t)|^{p}(w_{y}(t) − w_{x}(t))}.

Integrating these inequalities, the equivalency between (i) and (ii) follows.

(i) ⟺ (iii): Using Lemma 2, we get by integration of (17) from T to t, T < t,

\frac{1}{|x(s)|^{p}(w_{y}(s) − w_{x}(s))}|_{T}^{t} = p ∫_{T}^{t} H(x(s), w_{x}(s), w_{y}(s)) ds. (19)

Letting t → ∞, we get the conclusion.

(i) ⟺ (iv): According to (18), we have for some T ∈ ℝ

\frac{y(t)}{x(t)} = exp \{∫_{T}^{t} r^{1−q}(s) [Φ^{−1}(w_{y}(s)) − Φ^{−1}(w_{x}(s))] ds\} .

Hence, it suffices to show the equivalency between (i) and

∫_{T}^{∞} r^{1−q}(t) [Φ^{−1}(w_{y}(t)) − Φ^{−1}(w_{x}(t))] dt = ∞. (20)

First suppose that (i) holds. Using the L’Hospital rule in the form lim inf(f/g) ≥ lim inf(f′/g′), we get

L := lim inf_{t→∞} \frac{∫_{T}^{t} r^{1−q}(s) [Φ^{−1}(w_{y}(s)) − Φ^{−1}(w_{x}(s))] ds}{log (∫_{T}^{t} H(x(s), w_{x}(s), w_{y}(s)) ds)}

≥ lim inf_{t→∞} (\frac{[Φ^{−1}(w_{y}(t)) − Φ^{−1}(w_{x}(t))] (w_{y}(t) − w_{x}(t))^{2}}{px^{−p}(t) Q(w_{x}(t), w_{y}(t))} ∫_{T}^{t} H(x(s), w_{x}(s), w_{y}(s)) ds)

= \frac{1}{p} lim inf_{t→∞} (x^{p}(t)[w_{y}(t) − w_{x}(t)]W (w_{x}(t), w_{y}(t)) ∫_{T}^{t} H(x(s), w_{x}(s), w_{y}(s)) ds) .

Lemma 2 yields (19) and thus

x^{p}(t)(w_{y}(t) − w_{x}(t)) = \frac{1}{K + ∫_{T}^{t} H(x(s), w_{x}(s), w_{y}(s)) ds},

where K is a real constant. Taking into account (16), we have

L ≥ \frac{1}{p} lim inf_{t→∞} \frac{∫_{T}^{t} H(x(s), w_{x}(s), w_{y}(s)) ds}{K + ∫_{T}^{t} H(x(s), w_{x}(s), w_{y}(s)) ds}W (w_{x}(t), w_{y}(t)) ≥ \frac{K_{1}}{p} > 0.

Consequently, (i) implies (20).

Conversely, suppose that (20) holds. Proceeding similarly as above,

L̃ := lim inf_{t→∞} \frac{log (∫_{T}^{t} H(x(s), w_{x}(s), w_{y}(s)) ds)}{∫_{T}^{t} r^{1−q}(s) [Φ^{−1}(w_{y}(s)) − Φ^{−1}(w_{x}(s))] ds}

≥ p lim inf_{t→∞} \frac{K + ∫_{T}^{t} H(x(s), w_{x}(s), w_{y}(s)) ds}{(∫_{T}^{t} H(x(s), w_{x}(s), w_{y}(s)) ds) W (w_{x}(t), w_{y}(t))}

≥ \frac{p}{K_{2}} lim_{t→∞} \frac{K + ∫_{T}^{t} H(x(s), w_{x}(s), w_{y}(s)) ds}{∫_{T}^{t} H(x(s), w_{x}(s), w_{y}(s)) ds} > 0.

Thus, (20) implies (i). □

As an immediate consequence of Theorem 1 we have the following statement.

Corollary 1. Let x be a solution of (1) and w_{x} be the associated solution of (6). If any of conditions (i) – (iv) of Theorem 1 holds for every solution y of (1) and the associated solution w_{y} of (6), then x is the principal solution of (1).

Remark 1. In the linear case, i.e. p = 2, the conditions (i) and (ii) of Theorem 1 are the same and reduces to (4).

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 7, p. 11 Corollary 2. Let any of assumptions (i)-(iii) of Proposition 6 hold, let x̃ be a principal solution of (1), and w̃ be the associated minimal solution of (6). Then

lim_{t→∞} |x̃(t)|^{p}(w(t) − w̃(t)) = 0

for any other solution w of (6).

Based on the linear case, Corollary 2, and explicitly computable examples, we conjecture that the converse statement to Corollary 1 holds.

Conjecture 1. Suppose that x̃ is the principal solution of (1) and w̃ is the associated solution of (6). Then all conditions (i) – (iv) of Theorem 1 hold for any other solution w of (6).

In Section 2 we have presented various characterizations of the principal solutions of (1). Now, using condition (ii) from Theorem 1, we can relate these characterizations if we have at our disposal some additional information about asymptotic behavior of solutions of (6).

Corollary 3. Let x̃ be a solution of (1) such that x̃′(t) ≠ 0 for large t and w̃ be the associated solution of (6). Suppose that there exists a constant K > 0 such that

lim sup_{t→∞} |\frac{w(t)}{w̃(t)}| < K (21)

for every solution w of (6). Then the following statements are valid:

(i) (8) holds if and only if (10) holds for any solution x of (1) linearly independent of x̃.

(ii) If (8) holds, then x̃ is the principal solution of (1).

(iii) If p ≥ 2, then x̃ is the principal solution of (1) if and only if (8) holds.

Proof. By Theorem 1, it is sufficient to prove that condition (8) is equivalent to condition (ii) of this theorem for any solution w. We have

\frac{1}{r(t)x̃^{2}(t)|x̃′(t)|^{p−2}} = \frac{1}{r^{q−1}(t)|x̃(t)|^{p}} \frac{Φ^{−1}(w̃(t))}{w̃(t)}

and

\frac{Φ^{−1}(w(t)) − Φ^{−1}(w̃(t))}{w(t) − w̃(t)} = \frac{Φ^{−1}(w̃(t))}{w̃(t)}G(\frac{w(t)}{w̃(t)})

where G(t) = \frac{Φ^{−1}(t)−1}{t−1}. Since the function G is is positive, bounded and bounded away from zero on the interval [−K, K], there exist positive constants m_{1}, m_{2} such that

m_{1} ≤ G(\frac{w(t)}{w̃(t)}) ≤ m_{2}.

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 7, p. 12 Thus,

\frac{m_{1}}{r(t)x̃^{2}(t)|x̃′(t)|^{p−2}} ≤ \frac{1}{r^{q−1}(t)|x̃(t)|^{p}} \frac{Φ^{−1}(w̃(t))}{w̃(t)}G(\frac{w(t)}{w̃(t)}) ≤ \frac{m_{2}}{r(t)x̃^{2}(t)|x̃′(t)|^{p−2}} .

Hence (ii) is equivalent to (8). Now the conclusion follows from Corollary 1 and the parts (ii)–(iv) of Theorem 1. □

Remark 2. The statement (i) of Corollary 3 does not hold without assumption (21), as Example 1 illustrates. More precisely, one can check that the minimal solution of the corresponding Riccati equation to (9) satisfies w̃(t) ∼ 3t^{−4}. By [6, Theorems 4,7] any other solution w satisfies w(t) ∼ t^{−1/2}, thus (21) is not satisfied.

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(Received August 23, 2007)

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

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

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

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