Electronic Journal of Qualitative Theory of Differential Equations Proc. 8th Coll. QTDE, 2008, No. 15 1-11; http://www.math.u-szeged.hu/ejqtde/

ASYMPTOTIC FORMULAS FOR SOLUTIONS OF HALF-LINEAR EULER-WEBER EQUATION

ZUZANA PÁTÍKOVÁ

Abstract. We establish improved asymptotic formulas for nonoscillatory solutions of the half-linear Euler-Weber type differential equation

(Φ(x′))′ + [\frac{γ_{p}}{t^{p}} + \frac{μ_{p}}{t^{p} log^{2} t}] Φ(x) = 0, Φ(x) := |x|^{p−2}x, p > 1

with critical coefficients

γ_{p} = (\frac{p − 1}{p})^{p} , μ_{p} = \frac{1}{2} (\frac{p − 1}{p})^{p−1} ,

where this equation is viewed as a perturbation of the half-linear Euler equation.

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

1. Introduction

The aim of this paper is to present asymptotic formulas for solutions of the halflinear Euler-Weber type differential equation

(Φ(x′))′ + [\frac{γ_{p}}{t^{p}} + \frac{μ_{p}}{t^{p} log^{2} t}] Φ(x) = 0, (1)

where γ_{p} = (\frac{p−1}{p})^{p} , μ_{p} = \frac{1}{2} (\frac{p−1}{p})^{p−1} . This equation is a special case of a general half-linear second order differential equation

(r(t)Φ(x′))′ + c(t)Φ(x) = 0, (2)

where Φ(x) := |x|^{p−1} sgn x, p > 1, and r, c are continuous functions, r(t) > 0 (in the studied equation (1) we have r(t) ≡ 1). Let us recall that similarly as in the linear case, which is a special case of (2) for p = 2 and equation (2) then reduces to the linear Sturm-Liouville differential equation

(r(t)x′)′ + c(t)x = 0,

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 15, p. 1 even in the half-linear (non)oscillation theory equation (2) can be classified as oscillatory if every its nontrivial solution has infinitely many zeros tending to infinity and as nonoscillatory otherwise. The classical approach to half-linear equation (2) is to regard it as a perturbation of the one-term equation

1991 Mathematics Subject Classification. 34C10.

Key words and phrases. Half-linear differential equation, half-linear Euler equation, half-linear Euler-Weber equation, modified Riccati equation.

Research supported by the grant 201/07/0145 of the Grant Agency of the Czech Republic.

(r(t)Φ(x′))′ = 0.

Our approach is slightly modified, we use the perturbation principle introduced in [9], [3] and applied in [2], [5], [7], [8], [14], [15]. According to this concept equation (2) can be seen as a perturbation of a general (nonoscillatory) half-linear equation

(r(t)Φ(x′))′ + c̃(t)Φ(x) = 0, (3)

i.e., (2) can be rewritten in the form

(r(t)Φ(x′)′ + c̃(t)Φ(x) + (c(t) − c̃(t))Φ(x) = 0.

From this point of view, the studied equation (1) can be considered as a perturbation of the half-linear Euler equation

(Φ(x′))′ + \frac{γ_{p}}{t^{p}} Φ(x) = 0. (4)

This equation is nonoscillatory and γ_{p} is the so-called critical coefficient, critical in that sence that if it is replaced by any bigger constant, such equation becomes oscillatory, and for less constants nonoscillation is preserved.

Half-linear Euler-Weber equation (1) was studied by Elbert and Schneider in [9]. They derived the asymptotic formulas for its two linearly independent solutions in the forms

x_{1}(t) = t^{\frac{p−1}{p}} log^{\frac{1}{p}} t (1 + o(1)) as t → ∞
x_{2}(t) = t^{\frac{p−1}{p}} log t^{\frac{1}{p}} (log(log t))^{\frac{2}{p}} (1 + o(1)) as t → ∞.

Our results show that the terms (1 + o(1)) are special slowly varying functions.

2. Preliminaries

Let q be the conjugate number of p, i.e., \frac{1}{p} + \frac{1}{q} = 1. Let x be a solution of nonoscillatory equation (2), then the following Riccati type first order differential equation

w′ + c(t) + (p − 1)r^{1−q}(t)|w|^{q} = 0 (5)

holds, where w(t) = r(t)Φ(x′/x). It is well known from the (non)oscillation theory for half-linear equations (see e.g. [1, p. 171]), that equation (2) is nonoscillatory if and only if there exists a solution of the Riccati equation (5) on some interval of the form [T, ∞).

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 15, p. 2

Using the approach of perturbations, it is convenient to deal with the so-called modified Riccati equation (which was derived e.g. in [4]), whose solvability is again equivalent to nonoscillation of equation (2).

Let Φ^{−1} be the inverse function of Φ(x), h(t) be a (positive) solution of (3), and w_{h}(t) = r(t)Φ(h′/h) be the solution of the Riccati equation associated with (3). The modified Riccati equation then reads as

((w − w_{h})h^{p})′ + (C(t) − c̃(t))h^{p} + pr^{1−q}h^{p}P (Φ^{−1}(w_{h}), w) = 0, (6)

where

P (u, v) := \frac{|u|^{p}}{p} − uv + \frac{|v|^{q}}{q} ≥ 0,

with the equality P (u, v) = 0 if and only if v = Φ(u). Observe that if c̃(t) ≡ 0 and h(t) ≡ 1, then (6) reduces to (5) and this is also the reason why we call this equation modified Riccati equation.

Let us recall that a positive measurable function L(t) defined on (0, ∞) is said to be a slowly varying function in the sence of Karamata (see e.g. [11], [12]) if it satisfies

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

From the representation theorem for slowly varying functions (see [10]) we know that they are in the form

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

for some t_{0} > 0, where l(t) and ε(t) are measurable functions such that

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

If l(t) is identically a positive constant, we say that L(t) is a normalized slowly varying function.

Asymptotic formulas for some nonoscillatory solutions of the general perturbed Euler equation

(Φ(x′))′ + \frac{γ_{p}}{t^{p}} Φ(x) + g(t)Φ(x) = 0, (7)

were studied in [13] and the following two statements were proved there.

Theorem 1. Suppose that

c(t) := \frac{γ_{p}}{t^{p}} + g(t) ≥ 0 for large t,

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 15, p. 3 the integral ∫^{∞} g(t)t^{p−1} dt converges, and let

c := lim_{t→∞} log t ∫_{t}^{∞} g(s)s^{p−1} ds < μ_{p} (8)

holds. Then (7) possesses a pair of solutions

x_{i}(t) = t^{\frac{p−1}{p}} (log t)^{ν_{i}} L_{i}(t),

where λ_{i} := (\frac{p−1}{p})^{p−1} ν_{i} are roots of the equation

\frac{λ^{2}}{4μ_{p}} − λ + c = 0 (9)

and L_{i}(t) are normalized slowly varying functions of the form L_{i}(t) = exp \{∫^{t} \frac{ε_{i}(s)}{s log s} ds\} with ε_{i}(t) → 0 for t → ∞, i = 1, 2.

Taking g(t) := \frac{μ_{p}}{t^{p} log^{2} t} we have a special perturbation of half-linear Euler equation (4) with lim_{t→∞} log t ∫_{t}^{∞} g(s)s^{p−1} ds = μ_{p}. Then we have in some sence a limit case of (8) in Theorem 1 and the quadratic equation (9) has just one real zero. The derivation of the asymptotic formula for the principal solution of Euler-Weber equation (1) can be made in a similar manner as in the proof of Theorem 1 (see [13]).

Theorem 2. Equation (1) has a solution satisfying the asymptotic formula

x_{1}(t) = t^{\frac{p−1}{p}} (log t)^{\frac{1}{p}} L_{1}(t), (10)

where L_{1}(t) is a normalized slowly varying function in the form L_{1}(t) = exp \{∫^{t} \frac{ε_{1}(s)}{s log s} ds\} and ε_{1}(t) → 0 as t → ∞.

3. Main result

As the main result we introduce the asymptotic formula for the second solution of the Euler-Weber equation (1), which is linearly independent to the principal one stated in Theorem 2. This is also the answer to the open problem conjecturing such result presented in [13].

Theorem 3. Equation (1) has a solution satisfying the asymptotic formula

x_{2}(t) = t^{\frac{p−1}{p}} log t^{\frac{1}{p}} (log(log t))^{\frac{2}{p}} L_{2}(t),

where L_{2}(t) is a normalized slowly varying function in the form

L_{2}(t) = exp \{∫ ^{t} \frac{ε_{2}(s)}{s log s log(log s)} ds\}

and ε_{2}(t) → 0 as t → ∞.

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 15, p. 4 Proof. First we formulate the modified Riccati equation associated with (1). Let w be a solution of the Riccati equation

w′ + \frac{γ_{p}}{t^{p}} + \frac{μ_{p}}{t^{p} log^{2} t} + (p − 1)|w|^{q} = 0. (11)

Since \frac{γ_{p}}{t^{p}} + \frac{μ_{p}}{t^{p} log^{2} t} ≥ 0 for large t, from [6, Cor. 4.2.1] we have w(t) ≥ 0 for large t. Let

w_{h}(t) = Φ (\frac{h′}{h} ) = (\frac{p − 1}{p})^{p−1} t^{1−p}

be the solution of Riccati equation associated with (4) generated by the solution h(t) = t^{\frac{p−1}{p}}, and denote

v(t) = (w(t) − w_{h}(t))h^{p}(t) = t^{p−1} (w − (\frac{p − 1}{p})^{p−1} t^{1−p}) . (12)

Modified Riccati equation (6), where c̃(t) = \frac{γ_{p}}{t^{p}} , C(t) = \frac{γ_{p}}{t^{p}} + \frac{μ_{p}}{t^{p} log^{2} t}, has then the form

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

which, by an easy calculation, arrives at

v′ + \frac{μ_{p}}{t log^{2} t} + \frac{p − 1}{t}G(v) = 0, (13)

where

G(v) = |v + (\frac{p − 1}{p})^{p−1}|^{q} − v − (\frac{p − 1}{p})^{p} ,

with the equality G(v) = 0 if and only if v = 0.

Now we show that v(t) → 0 for t → ∞. Integrating (13) from T to t, T ≤ t, we have

[(\frac{p − 1}{p})^{p−1} − t^{p−1}w]_{T}^{t} = ∫_{T}^{t} \frac{μ_{p}}{s log^{2} s} ds + (p − 1) ∫_{T}^{t} \frac{G(v)}{s} ds

Letting t → ∞ and taking into account that w(t) ≥ 0 for large t,

[(\frac{p − 1}{p})^{p−1} − t^{p−1}w]_{T}^{∞} ≤ T^{p−1}w(T).

Hence

∫_{T}^{∞} \frac{μ_{p}}{s log^{2} s} ds + (p − 1) ∫_{T}^{∞} \frac{G(v)}{s} ds ≤ T^{p−1}w(T)

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 15, p. 5 and since the integral ∫_{T}^{∞} \frac{μ_{p}}{s log^{2} s} ds converges, the integral ∫_{T}^{∞} \frac{G(v)}{s} ds converges too. This means that lim_{t→∞} v(t) exists and v(t) → 0, since if v(t) → v_{0} ≠ 0, then G(v(t)) → G(v_{0}) > 0 which contradicts the convergence of ∫_{T}^{∞} \frac{G(v)}{s} ds.

Let us investigate the behavior of the function G(v). By L’Hospital’s rule (used twice) we have

lim_{v→0} \frac{G(v)}{v^{2}} = \frac{q − 1}{2} (\frac{p}{p − 1})^{p−1} = \frac{q − 1}{4μ_{p}}.

Hence, for every ε > 0 there exists δ > 0 such that

(\frac{q − 1}{4μ_{p}} − ε) v^{2} ≤ G(v) ≤ (\frac{q − 1}{4μ_{p}} + ε) v^{2}. (14)

for v satisfying |v| < δ. Similarly for \frac{∂G}{∂v}, as

lim_{v→0} \frac{\frac{∂G}{∂v}}{v} = (q − 1) (\frac{p}{p − 1})^{p−1} = \frac{q − 1}{2μ_{p}},

to every ε > 0 one can find δ > 0 such that

(\frac{q − 1}{2μ_{p}} − ε) v ≤ \frac{∂G}{∂v} ≤ (\frac{q − 1}{2μ_{p}} + ε) v (15)

as |v| < δ.

We assume that a solution of modified Riccati equation (13) is in the form

v(t) = \frac{2μ_{p} log(log t) + 4μ_{p} + z(t)}{log t log(log t)}.

Then for its derivative we have

v′(t) = \frac{(2μ_{p} \frac{1}{t log t} + z′(t)) log t log(log t) − (2μ_{p} log(log t) + 4μ_{p} + z(t))(\frac{1}{t} log(log t) + \frac{1}{t})}{log^{2} t log^{2}(log t)}

and substituing into the modified Riccati equation (13) we get the equation

z′(t) − \frac{z(t)}{t log t log(log t)}

+ \frac{−4μ_{p} − 4μ_{p} log(logt)−μ_{p} log^{2}(logt)−z(t) log(logt) + (p − 1)G(v) log^{2}t log^{2}(logt)}{t log t log(log t)} = 0,

which can be rewritten as

z′(t) + \frac{z(t)}{t log t log(log t)}

+ \frac{−4μ_{p}−4μ_{p} log(logt)−μ_{p} log^{2}(logt)−2z(t)−z(t) log(logt)+(p−1)G(v)log^{2}t log^{2}(logt)}{t logt log(logt)} =0.

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 15, p. 6 If we denote

r(t) = exp \{∫ ^{t} \frac{1}{s log s log(log s)} ds\}

then the previous equation is equivalent to

(16) (r(t)z(t))′ + r(t)\frac{1}{t log t log(log t)}H(z, t) = 0,

where

H(z,t)=−4μ_{p}−4μ_{p} log(logt)−μ_{p} log^{2}(logt)−2z(t)−z(t) log(logt)+(p−1)G(v) log^{2}t log^{2}(logt).

Let C_{0}[T, ∞) denote the set of all continuous functions on [T, ∞) tending to zero as t → ∞; concrete T will be specified later. C_{0}[T, ∞) is a Banach space with the norm ∥z∥ = sup{|z(t)| : t ≥ T }. We consider the integral operator

F z(t) = −\frac{1}{r(t)} ∫ ^{t} \frac{r(s)}{s log s log(log s)}H(z, s) ds

on the set

V = {z ∈ C_{0}[T, ∞) : |z(t)| < ε_{1}, t ≥ T },

where ε_{1}, T are suitably chosen (will be specified later). Now our aim is to show that the operator F is a contraction on the set V and maps V to itself.

First we show that ∫^{t} \frac{r(s)}{s log s log(log s)} ds diverges. We have r(t) → ∞ for t → ∞ and

lim_{t→∞} ∫ ^{t} \frac{r(s)}{s log s log(log s)} ds = lim_{t→∞}[log(log s)]^{t} = ∞

Furthermore,

r′(t) = r(t)\frac{1}{t log t log(log t)}

and by L’Hospital’s rule we have

lim_{t→∞} \frac{1}{r(t)} ∫ ^{t} \frac{r(s)}{s log s log(log s)} ds = \frac{\frac{r(t)}{t log t log(log t)}}{r′(t)} = 1 > 0.

Let T_{1} be large enough such that

(17) \frac{1}{r(t)} ∫ ^{t} \frac{r(s)}{s log s log(log s)} ds < 2

for t ≥ T_{1}.

Let ε_{1} > 0, such that

2 (ε_{1} + \frac{ε_{1}}{4μ_{p}} ) ≤ 1 (18)

and

2 (\frac{ε_{1}}{2μ_{p}} + ε_{1}) < \frac{1}{2}. (19)

Let T_{2} be such that |z(t)| < ε_{1}^{2} for t ≥ T_{2}.

In order to show that H(z, t) → 0 for t → ∞, the estimates for H(z, t) are:

|H(z, t)| = |−4μ_{p} − 4μ_{p} log(log t) − μ_{p} log^{2}(log t) − 2z(t) − z(t) log(log t)

+(p − 1)G(v) log^{2} t log^{2}(log t)|

≤ |−4μ_{p} − 4μ_{p} log(log t) − μ_{p} log^{2}(log t) − 2z(t) − z(t) log(log t) + \frac{v^{2}}{4μ_{p}} log^{2} t log^{2}(log t)|

+ |(p − 1) log^{2} t G(v) − \frac{v^{2}}{4μ_{p}} log^{2} t log^{2}(log t)|

= |\frac{z^{2}}{4μ_{p}} | + |(p − 1) log^{2} t G(v) − \frac{v^{2}}{4μ_{p}} log^{2} t log^{2}(log t)| ,

where the definition of v have been used in the step between the second and the third line of the above computation. Now, according to (14), the second term in the last expression is arbitrarily small for small v, i.e., as v(t) → 0 for t → ∞, there exists T_{3} large enough such that for t ≥ T_{3}

|\frac{z^{2}}{4μ_{p}} | + |(p − 1) log^{2} t G(v) − \frac{v^{2}}{4μ_{p}} log^{2} t log^{2}(log t)| ≤ \frac{ε_{1}^{2}}{4μ_{p}} + ε_{1}^{2}

for t ≥ max{T_{2}, T_{3}}.

Similarly, we will need an estimate for the difference |H(z_{1}, t) − H(z_{2}, t)|. Using the mean value theorem (with z ∈ V such that min{z_{1}(t), z_{2}(t)} ≤ z(t) ≤ max{z_{1}(t), z_{2}(t)}) we have (suppressing the argument t in the functions z, z_{1}, z_{2})

|H(z_{1}, s)−H(z_{2}, s)| = |(z_{2}− z_{1})(2+ log(logt))+(p−1) log^{2}t log^{2}(log t)\frac{∂G(v, z)}{∂z}(z_{1}− z_{2})|

≤ ∥z_{1} − z_{2}∥ (|−(2 + log(log t)) + \frac{v}{2μ_{p}} log t log(log t)|
+ |(p − 1) log^{2} t log^{2}(log t) \frac{∂G(v, z)}{∂z} − \frac{v}{2μ_{p}} log t log(log t)|)

= ∥z_{1} − z_{2}∥ (|\frac{z}{2μ_{p}} | + |(p − 1) log t log(log t) \frac{∂G(v, z)}{∂v} − \frac{v}{2μ_{p}} log t log(log t)|)

≤ ∥z_{1} − z_{2}∥ (|\frac{z}{2μ_{p}} | + ε_{1}) ≤ ∥z_{1} − z_{2}∥ (\frac{ε_{1}}{2μ_{p}} + ε_{1})

EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 15, p. 8 for t ≥ max{T_{2}, T_{4}}, where T_{4} is such that |v(t)| < δ for t ≥ T_{4} (such T_{4} exists because of (15)).

We take T = max{T_{1}, T_{2}, T_{3}, T_{4}}. Then

|F z(t)| ≤ \frac{1}{r(t)} ∫_{t}^{∞} \frac{r(s)}{s log s log(log s)}|H(z, s)| ds ≤ (ε_{1}^{2} + \frac{ε_{1}^{2}}{4μ_{p}} ) \frac{1}{r(t)} ∫_{t}^{∞} \frac{r(s)}{s log(log s)} ds

< 2 (ε_{1}^{2} + \frac{ε_{1}^{2}}{4μ_{p}} ) ≤ ε_{1}

using (18) and hence F maps V to itself.

Next we show that F is a contraction. We have (using the definition of F)

|F z_{1}(t) − F z_{2}(t)| = \frac{1}{r(t)} ∫_{t}^{∞} \frac{r(s)}{s log(log s)}|H(z_{1}, s) − H(z_{2}, s)| ds

≤ ∥z_{1} − z_{2}∥ (\frac{ε_{1}}{2μ_{p}} + ε_{1}) \frac{1}{r(t)} ∫_{t}^{∞} \frac{r(s)}{s log s log(log s)} ds,

which is, according to (17) and (19), less than \frac{1}{2}∥z_{1}−z_{2}∥ and hence F is a contraction. By the Banach fixed point theorem, F has a fixed point z that satisfies z = F z, i.e.

z(t) = −\frac{1}{r(t)} ∫ ^{t} \frac{r(s)}{s log s log(log s)}H(z, s) ds.

Differentiating the last equality we see that z(t) is a solution of (16) and hence v(t) = \frac{2μ_{p} log(log t)+4μ_{p}+z(t)}{log t log(log t)} is a solution of modified Riccati equation (13).

Now, for the solution of Riccati equation (11) w(t), we have

w(t) = t^{1−p} (v + (\frac{p − 1}{p})^{p−1}) = t^{1−p} (\frac{2μ_{p} log(log t) + 4μ_{p} + z(t)}{log t log(log t)} + (\frac{p − 1}{p})^{p−1})

and the solution x of equation (1) is

x(t) = exp ∫ ^{t} Φ^{−1}(w(s)) ds.

Furthermore,

Φ^{−1}(w) = Φ^{−1} (t^{1−p} (\frac{p − 1}{p})^{p−1} [(\frac{p − 1}{p})^{1−p} v + 1])

= \frac{1}{t} \frac{p − 1}{p} [(\frac{p − 1}{p})^{1−p} v + 1]^{q−1} = \frac{1}{t} \frac{p − 1}{p} [1 + (q − 1) (\frac{p − 1}{p})^{1−p} v + o(v)]

= \frac{1}{t} \frac{p − 1}{p} + \frac{1}{p} (\frac{p − 1}{p})^{1−p} \frac{2μ_{p} log(log t) + 4μ_{p} + z(t)}{log t log(log t)} + o(\frac{v}{t})

= \frac{p − 1}{p} \frac{1}{t} + \frac{\frac{1}{p}}{t log t} + \frac{\frac{2}{p}}{t log t log(log t)} + \frac{\frac{1}{p} (\frac{p−1}{p})^{1−p} z(t) + o(2μ_{p} log(log t) + 4μ_{p} + z(t))}{t log t log(log t)}.

Denote

\frac{1}{p} (\frac{p − 1}{p})^{1−p} z(t) + o(2μ_{p} log(log t) + 4μ_{p} + z(t)) = ε_{2}(t),

then the solution of (1) is in the form

x(t) = exp ∫ ^{t} Φ^{−1}(w(s)) ds = t^{\frac{p−1}{p}} (log t)^{\frac{1}{p}} (log(log t))^{\frac{2}{p}} exp \{∫ ^{t} \frac{ε_{2}(s)}{s log s log(log s)} ds\}

and the statement is proved. □

Remark 1. Let us denote that functions L_{1}(t), L_{2}(t) from Theorems 2, 3 are in some sence “more slowly varying” than standard slowly varying functions in the sence of Karamata. The function L_{1}(t) remains slowly varying even after the substitution u = log s in the integrated term, and such subtitution can be used even twice in the function L_{2}(t) without a change of the property of slowly variability.

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(Received August 7, 2007)

Department of Mathematics, Tomas Bata University in Zlín, Nad Stráněmi 4511, 760 05 Zlín, Czech Republic

  E-mail address: patikova@fai.utb.cz