# Two-Parametric Conditionally Oscillatory Half-Linear Differential Equations

Authors: Ondřej Došlý; Simona Fišnarová
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study perturbations of the nonoscillatory half-linear differential equation(r(t)Φ(x′))′+c(t)Φ(x)=0, Φ(x):=|x|p-2x, p>1. We find explicit formulas for the functions r̂, ĉ such that the equation [(r(t)+λr̂(t))Φ(x′)]′+[c(t)+μĉ(t)]Φ(x)=0 is conditionally oscillatory, that is, there exists a constant γ such that  the previous equation is oscillatory if μ-λ>γ and nonoscillatory if μ-λ<γ. The obtained results extend the previous results concerning two-parametric perturbations of the half-linear Euler differential equation.


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## Body

## 1. Introduction

Conditionally  oscillatory equations play an important role in the oscillation theory of the Sturm-Liouville second-order differential equation(1.1)(r(t)x′)′+c(t)x=0,
with positive continuous functions r, c. Equation (1.1) with λc instead of c is said to be conditionally oscillatory if there exists λ0>0, the so-called oscillation constant of (1.1), such that this equation is oscillatory for λ>λ0 and nonoscillatory for λ<λ0. A typical example of a conditionally oscillatory equation is the Euler differential equation(1.2)x′′+λt2x=0,
which has the oscillation constant λ0=1/4 as can be verified by a direct computation when looking for solutions of (1.2) in the form x(t)=tα. This leads to the classical Kneser (non)oscillation criterion which states that   (1.1) with r(t)≡1 is oscillatory provided(1.3)liminf⁡t→∞t2c(t)>14,
and nonoscillatory if(1.4)limsup⁡t→∞t2c(t)<14.
This shows that the potential c(t)=t-2 is the border line between oscillation and nonoscillation. Note that the concept of conditional oscillation of (1.1) was introduced in [1].The linear oscillation theory extends almost verbatim to the half-linear differential equation(1.5)(r(t)Φ(x′))′+c(t)Φ(x)=0,Φ(x):=|x|p-2x,p>1,
including the definition of conditional oscillation. The half-linear version of Euler equation (1.2) is the equation (1.6)(Φ(x′))′+λtpΦ(x)=0,
which has the oscillation constant λ0=γp:=((p-1)/p)p, and (non)oscillation criteria (1.3), (1.4) extend in a natural way to (1.5) with r(t)≡1. A complementary concept to the conditional oscillation is the concept of strong (non)oscillation. Equation (1.5) with λc instead of c is said to be strongly (non)oscillatory if it is (non)oscillatory for every λ>0. Sometimes, strongly oscillatory equations are regarded as conditionally oscillatory with the oscillation constant λ0=0 and strongly nonoscillatory as conditionally oscillatory with the oscillation constant λ0=∞. We refer to [2] for results along this line.In our paper, we are motivated by a statement presented in [3, 4], where the two-parametric perturbation of the Euler differential equation with the critical coefficient(1.7)(Φ(x′))′+γptpΦ(x)=0
is investigated. It is shown there that the equation(1.8)[(1+λlog⁡2t)Φ(x′)]′+[γptp+μtplog⁡2t]Φ(x)=0
is oscillatory if μ-γpλ>μp:=(1/2)((p-1)/p)p-1 and nonoscillatory in the opposite case. Note that an important role in proving the results of [4] is played by the fact that we know explicitly the solution h(t)=t(p-1)/p of (1.7).Here, we treat the problem of conditional oscillation in the following general setting. We suppose that (1.5) is nonoscillatory and that h is its eventually positive solution. We find explicit formulas for the functions r̂, ĉ such that the equation(1.9)[(r(t)+λr̂(t))Φ(x′)]′+[c(t)+μĉ(t)]Φ(x)=0
is conditionally oscillatory, that is, there exists a constant γ such that (1.9) is oscillatory if μ-λ>γ and nonoscillatory if μ-λ<γ.The setup of the paper is as follows. In the next section, we present some statements of the half-linear oscillation theory. Section3 is devoted to the so-called modified Riccati equation associated with (1.5) and (1.9). The main result of the paper, the construction of the functions r̂, ĉ such that (1.9) is two-parametric conditionally oscillatory, is presented in Section 4.


## 2. Auxiliary Results

As we have already mentioned in the previous section, the linear oscillation theory extends almost verbatim to half-linear equation (1.5). The word “almost” reflects the fact that not all linear methods can be extended to (1.5), some results for (1.5) are the same as those for (1.1), but to prove them, one has to use different methods than in the linear case. A typical method of this kind is the following transformation formula. If f(t)≠0 is a sufficiently smooth function and functions x, y are related by the formula x=f(t)y, then we have the identity(2.1)f(t)[(r(t)x′)′+c(t)x]=(R(t)y′)′+C(t)y,
where (2.2)R(t)=r(t)f2(t),C(t)=f(t)[(r(t)f′(t))′+c(t)f(t)].
In particular, x is a solution of (1.1) if and only if y is a solution of the equation (Ry′)′+Cy=0. The transformation identity (2.1) does not extend to (1.5).To illustrate the meaning of this fact in the conditional oscillation of (1.1) and (1.5), suppose that   (1.1) is nonoscillatory and let h be its so-called principal solution (see [5, Chapter  XI]), that is, a solution such that ∫∞r-1(t)h-2(t)dt=∞. We would like to find a function ĉ such that the equation (2.3)(r(t)x′)′+(c(t)+μĉ(t))x=0
is conditionally oscillatory and to find its oscillation constant. The transformation x=h(t)y transforms (1.1) into the one term equation (r(t)h2(t)y′)′=0 and the transformation of independent variable s=∫tr-1(τ)h-2(τ)dτ further to the equation   d2y/ds2=0. Now, from (1.2), we know that the “right” perturbation term in the last equation is  1/s2 with the oscillation constant 1/4. Substituting back for s, we get the conditionally oscillatory equation (2.4)(R(t)y′)′+μR(t)(∫tR-1(s)ds)2y=0,R(t)=r(t)h2(t),
and the back transformation y=h-1(t)x results in the conditionally oscillatory equation (2.5)(r(t)x′)′+[c(t)+μh2(t)R(t)(∫tR-1(s)ds)2]x=0,
with the oscillation constant  μ0=1/4. The previous result is one of the main statements of [6], but it was proved there by a different method.In the next section, we will show how to modify this method to be applicable to half-linear equations. At this moment, we present the result of [7] with the classical (i.e., one  parametric) conditional oscillation of (1.5). Let h be a positive solution of (1.5) such that h′(t)≠0 for large t. We denote(2.6)R(t):=r(t)h2(t)|h′(t)|p-2,G(t):=r(t)h(t)Φ(h′(t)),(2.7)ĉ(t)=1|h(t)|pR(t)(∫tR-1(s)ds)2.Theorem 2.1.
Suppose that (1.5) possesses a nonoscillatory solution h such that h′(t)≠0 for large t, and R, G are given by (2.6). If
(2.8)∫∞dtR(t)=∞,liminf⁡t→∞|G(t)|>0,
then the equation
(2.9)(r(t)Φ(x′))′+[c(t)+μĉ(t)]Φ(x)=0
is conditionally oscillatory,  and its oscillation constant is μ0=1/2q, where q is the conjugate exponent to p, that is,   1/p+1/q=1.Note that in the linear casep=2, the function f(t)=h(t)∫tr-1(τ)h-2(τ)dτ is a solution of (2.9) with μ=μ0=1/4. In the general half-linear case, an explicit solution of (2.9) is no longer known, but we are able to “estimate” this solution. The next statement, which is also taken from [7], presents a result along this line.Theorem 2.2.
Suppose that (2.8) holds and let f(t)=h(t)(∫tR-1(s)ds)1/p,  then a solution of (2.9) with μ=1/2q is of the form
(2.10)x(t)=f(t)(1+O((∫tR-1(s)ds)-1)),
and (suppressing the argument t)
(2.11)f[(rΦ(f′))′+(c+12qhpR(∫tR-1)2)Φ(f)]=-(p-1)(p-2)G′G2(∫tR-1)-(p-1)(p-2)3p3G3(∫tR-1)2[(p-3)G′+2pr|h′|p]+O(G-3(∫tR-1)-3)[G′pG2-(p3-4p2+11p-6)h′2p3h-1qR(∫tR-1)],
as t→∞.The last statement presented in this section is the so-calledreciprocity principle. Let x be a solution of (1.5) and let u:=rΦ(x′) be its quasiderivative, then u is a solution of the reciprocal equation(2.12)(c1-q(t)Φ-1(u′))′+r1-q(t)Φ-1(u)=0,
where Φ-1(u)=|u|q-2u is the inverse function of Φ.


## 3. Modified Riccati Equation

Suppose thatλ and r̂ in (1.9) are such that r(t)+λr̂(t)>0. Let x(t)≠0 in an interval I be a solution of (1.9), and let w=(r+λr̂)Φ(x′/x). Then, w solves in I the “standard” Riccati equation(3.1)w′+c(t)+μĉ(t)+(p-1)[r(t)+λr̂(t)]1-q|w|q=0.
More precisely, the following statement holds.Lemma 3.1 ([8, Theorem  2.2.1]).
The following statements are equivalent:(i)
equation (1.9) is nonoscillatory;(ii)
equation (3.1) has a solution on an interval [T,∞);(iii)
there exists a continuously differentiable functionw such that
(3.2)w′+c(t)+μĉ(t)+(p-1)[r(t)+λr̂(t)]1-q(t)|w|q≤0
on an interval [T,∞).In the linear case, ifx is a solution of (1.1), x=f(t)y, and v=rf2y′/y is the Riccati variable corresponding to the equation on the right-hand side in (2.1), then v=f2(w-wf) where w=rx′/x,  wf=rf′/f. This suggests to investigate the function v=fp(w-wf) in the half-linear case, and this leads to the modified Riccati equation introduced in the next statement which is taken from [4] with a modification from [3].Lemma 3.2.
Suppose thatf is a positive differentiable function, wf=(r+λr̂)Φ(f′/f), and  w is a continuously differentiable function, and put v=fp(w-wf), then the following identity holds:
(3.3)fp(t)[w′+c(t)+μĉ(t)+(p-1)(r(t)+λr̂(t))1-q|w|q]=v′+f(t)[l(f(t))+l̂(f(t))]+(p-1)(r(t)+λr̂(t))1-qf-q(t)G(t,v),
where
(3.4)l(f)=(r(t)Φ(f′))′+c(t)Φ(f),l̂(f)=λ(r̂(t)Φ(f′))′+μĉ(t)Φ(f),(3.5)G(t,v)=|v+Ω(t)|q-qΦ-1(Ω(t))v-|Ω(t)|q,Ω:=(r+λr̂)fΦ(f′).
In particular, if w is a solution of  (3.1), then v is a solution of the modified Riccati equation
(3.6)v′+f(t)[l(f(t))+l̂(f(t))]+(p-1)(r(t)+λr̂(t))1-qf-q(t)G(t,v)=0.
Conversely, if v is a solution of (3.6), then w=wf+f-pv is a solution of (3.1).Observe that in casef≡1, the modified Riccati equation (3.6) reduces to the standard Riccati equation (3.1).Next, we will investigate the function𝒢 in (3.5). First, we present a result from [4, Lemmas  5 and 6].Lemma 3.3.
The function𝒢 defined in (3.5) has the following properties.  (i)
𝒢(t,v)≥0 with the equality if and only if v=0.(ii)
Ifq≥2, one has the inequality
(3.7)G(t,v)≥q2|Ω(t)|q-2v2.Now, we concentrate on an estimate of the function𝒢 in case q<2.Lemma 3.4.
Suppose thatq<2 and limt→∞|Ω(t)|=∞, then there is a constant β>0 such that for v∈(-∞,-v0], v0>0, and large t(3.8)G(t,v)≥β|Ω(t)|q-2|v|q.Proof.
Consider the function(3.9)H(t,v)={G(t,v)|v|q,forv≠0,0,forv=0.
First of all,
(3.10)lim⁡v→±∞H(t,v)=1,lim⁡v→0H(t,v)=0.
Now, we compute local extrema of ℋ with respect to v. We have (suppressing the argument t)
(3.11)Hv=1|v|2q{[qΦ-1(v+Ω)-qΦ-1(Ω)]|v|q-qΦ-1(v)[|v+Ω|q-qΦ-1(Ω)v-|Ω|q]}=qv2Φ-1(v){vΦ-1(v+Ω)-vΦ-1(Ω)-|v+Ω|q+qΦ-1(Ω)v+|Ω|q}=qv2Φ-1(v){-ΩΦ-1(v+Ω)+(q-1)Φ-1(Ω)v+|Ω|q}.
Denote 𝒩(v) the function in braces on the last line of the previous computation. We have 𝒩(0)=0,
(3.12)N′(v)=-(q-1)Ω|v+Ω|q-2+(q-1)Φ-1(Ω)=(q-1)Ω[-|v+Ω|q-2+|Ω|q-2]=0
if and only if v=0 and v=-2Ω, and
(3.13)N′′(v)=-(q-1)(q-2)Ω|v+Ω|q-3sgn⁡(v+Ω).
This means that v=0 is the local minimum and v=-2Ω is the local maximum of the function 𝒩. Using this result, an examination of the graph of the function ℋ shows that this function has the local minimum at v=0 and a local maximum in the interval (-∞,-2Ω) if Ω>0, and this maximum is in (-2Ω,∞) if Ω<0. Next, denote v* the value for which ℋ(t,v*)=1. Consequently, for any v0>0, it follows from (3.10) that
(3.14)inf⁡v∈(-∞,-v0]H(t,v)=H(t,-v¯)=1|v¯|q[|Ω-v¯|q+qv¯Φ-1(Ω)-|Ω|q],
where
(3.15)v¯={-v*if-v0<v*<0,v0otherwise.
Next, we want to investigate the dependence of this infimum on Ω when |Ω|→∞. To this end, we investigate the function F(x)=|x-a|q+qaΦ-1(x)-|x|q for x→±∞, a∈ℝ being a parameter. We have (using the expansion formula for (1+x)α)
(3.16)F(x)=Φ-1(x){|x-a|q-|x|qΦ-1(x)+qa}=Φ-1(x){x[(1-ax)q-1]+qa}=Φ-1(x)((q2)a2x+o(x-1))=a2(q2)|x|q-2(1+o(1)),
as |x|→∞. Consequently, if limt→∞|Ω(t)|=∞, there exists a constant β>0 such that (3.8) holds.Now, we are ready to formulate a complement of [9, Theorem  2] which is presented in that paper under the assumption that the function Ω is bounded.Theorem 3.5.
Letf be a positive continuously differentiable function such that f′(t)≠0 for large t. Suppose that ∫∞ℛ-1(t)dt=∞, where  ℛ=(r+λr̂)f2|f′|p-2, C(t)≥0 for large t, and limt→∞|Ω(t)|=∞, then all possible proper solutions (i.e., solutions which exist on some interval of the form [T,∞)) of the equation
(3.17)v′+C(t)+(p-1)(r(t)+λr̂(t))1-qf-q(t)G(t,v)=0
are nonnegative.Proof.
First consider the caseq<2. Let v0>0 be arbitrary. By Lemma 3.4, there exists T0∈ℝ and β>0 such that for t≥T0 and v∈(-∞,-v0],
(3.18)(p-1)(r+λr̂)1-qf-qG(t,v)≥β(p-1)(r+λr̂)1-qf-q|Ω|q-2|v|q=(p-1)β|v|qR.
Suppose that v is the solution of (3.17) such that v(t0)=-v0 for some t0≥T0, then
(3.19)v′+C(t)+(p-1)β|v|qR(t)≤0,
for t≥t0 for which the solution v exists. Now, we use the same argument as in the proof of Theorem  2 in [9]. Consider the equation
(3.20)z′+C(t)+(p-1)β|z|qR(t)=0.
This is the standard Riccati equation corresponding to the half-linear equation
(3.21)(Rp-1(t)Φ(x′))′+βp-1C(t)Φ(x)=0.
Assumptions of theorem imply, by [8, Corollary  4.2.1], that all proper solutions of (3.20) are nonnegative. It means that any solution of (3.20) which starts with a negative initial condition blows down to -∞ in a finite time. Inequality (3.19) implies that if z is the solution of (3.20) satisfying z(t0)=v(t0)=-v0, that is, z starts with the same initial value as the solution v of (3.17), then v decreases faster than z. In particular, if z blows down to -∞ at a finite time, then v does as well. This means that all proper solutions of (3.17), if any, are nonnegative.
In caseq≥2, we proceed in a similar way. We use (3.7) and we compare (3.17) with the equation
(3.22)z′+C(t)+p2z2R(t)=0,
which is the standard Riccati equation corresponding to the linear equation
(3.23)(R(t)x′)′+p2C(t)x=0.
Then, reasoning in the same way as in case q<2, we obtain the conclusion that all proper solutions of (3.17) are nonnegative also in this case.


## 4. Two-Parametric Conditional Oscillation

Recall thath is a positive solution of (1.5) such that h′(t)≠0 for large t, g=rΦ(h′) is its quasiderivative, R, G are given by (2.6), and ĉ is given by (2.7). Recall also that the quasiderivative g is a solution of the reciprocal equation (2.12), denote by(4.1)G̃:=c1-qgΦ-1(g′)=-rhΦ(h′),R̃:=c1-qg2|g′|q-2=r2|h′|2p-2chp-2
the “reciprocal” analogues of G and R, and define (4.2)r̂(t)=1|h′(t)|pR̃(t)(∫tR̃-1(s)ds)2.Our main result reads as follows.Theorem 4.1.
Suppose that conditions (2.8) hold. Further, suppose that
(4.3)lim⁡t→∞r̂(t)r(t)=0,
and that there exist limits
(4.4)lim⁡t→∞r(t)|h′(t)|pc(t)hp(t),lim⁡t→∞(r̂(t)Φ(f′(t)))′ĉ(t)Φ(f(t)),
the second one being finite, where f(t)=h(t)(∫tR-1(s)ds)1/p. If  μ-λ<1/2q, then   (1.9) is nonoscillatory; if  μ-λ>1/2q, then it is oscillatory.Proof.
First consider the caseμ=0 in (1.9), that is, we consider the equation
(4.5)[(r(t)+λr̂(t))Φ(x′)]′+c(t)Φ(x)=0.
The quantities G̃ and R̃ defined in (4.1) satisfy
(4.6)G̃=-rhΦ(h′)=-G,R̃=r2|h′|2p-2chp-2=-h(rΦ(h′))2(rΦ(h′))′,
hence, integrating by parts,
(4.7)∫tR̃-1(s)ds=-∫t1h(s)[r(s)Φ(h′(s))]′[r(s)Φ(h′(s))]2ds=1h(t)r(t)Φ(h′(t))+∫th′(s)h2(s)1r(s)Φ(h′(s))ds=1G(t)+∫tR-1(s)ds.
Consequently, conditions (2.8) imply that corresponding conditions for G̃ and R̃ also hold. This means, in view of Theorem 2.1 (applied to the reciprocal equation (2.12)), that the equation
(4.8)(c1-q(t)Φ-1(u′))′+[r1-q(t)+λ|g(t)|qR̃(t)(∫tR̃-1(s)ds)2]Φ-1(u)=0
is oscillatory for λ>1/2p and nonoscillatory in the opposite case.
The reciprocal equation to (4.5) is the equation
(4.9)(c1-q(t)Φ-1(u′))′+(r(t)+λr̂(t))1-qΦ-1(u)=0.
Since (4.3) holds, we have
(4.10)(r+λr̂)1-q=r1-q(1+λr̂r)1-q=r1-q(1+(1-q)λr̂r+o(r̂r)),
as t→∞. Hence, we can rewrite (4.9) in the following form:
(4.11)(c1-q(t)Φ-1(u′))′+r1-q(t)(1+(1-q)λr̂(t)r(t)+o(r̂(t)r(t)))Φ-1(u)=0.
Let λ>-1/2q what is equivalent to λ(1-q)<1/2p, then, in view of (4.3), there exists λ̃ such that λ(1-q)<λ̃<1/2p, hence, for large t,
(4.12)r1-q(1+λ(1-q)r̂r+o(r̂r))<r1-q(1+λ̃r̂r)=r1-q+λ̃|g|qR̃(∫tR̃-1(s)ds)2.
This means that the equation
(4.13)(c1-q(t)Φ-1(u′))′+[r1-q(t)+λ̃|g(t)|qR̃(t)(∫tR̃-1(s)ds)2]Φ-1(u)=0
is a majorant of (4.9) and this majorant is nonoscillatory by Theorem 2.1 applied to (4.8). So (4.9) is also nonoscillatory, and hence (4.5) is nonoscillatory as well. The same argument implies oscillation of (4.5) if λ<-1/2q.
Now, we turn our attention to the general caseμ≠0. Let f:=h(∫tR-1(s)ds)1/p, and consider the term
(4.14)f[l(f)+l̂(f)]
appearing in the modified Riccati equation (3.6), where the operators ℓ, ℓ̂ are defined by (3.4). In order to use the asymptotic formula from Theorem 2.2, we write f[ℓ(f)+ℓ̂(f)]=A+B, where
(4.15)A=f[(rΦ(f′))′+(c+12qĉ)Φ(f)],B=f[λ(r̂Φ(f′))′+(μ-12q)ĉΦ(f)].
LetL∈ℝ be the second limit in (4.4), that is,
(4.16)(r̂Φ(f′))′=LĉΦ(f)(1+o(1))ast⟶∞.
The leading term in the expression A is constG′G-2(∫tR-1(s)ds)-1 by Theorem 2.2, while, concerning the asymptotics of B,
(4.17)B=fĉΦ(f)[Lλ+μ-12q+o(1)]=1R(∫tR-1(s)ds)[Lλ+μ-12q+o(1)],
as t→∞. The existence of the first limit in (4.4) implies that there exists the limit
(4.18)lim⁡t→∞G′(t)G-2(t)R-1(t)=lim⁡t→∞r(t)h2(t)|h′(t)|p-2(r(t)|h′(t)|p-c(t)hp(t))(r(t)h(t)Φ(h′(t)))2=1-lim⁡t→∞c(t)hp(t)r(t)|h′(t)|p.
The limit in (4.18) must be 0, which follows from the l'Hospital rule and the fact that the integral of R-1 is divergent, while the integral of G′G-2 is convergent by the second assumption in (2.8). This means that the term B dominates A; hence,  A(t)+B(t)>0 for large t if Lλ+μ-1/2q>0 and A(t)+B(t)<0 for large t if Lλ+μ-1/2q<0.
Now, it remains to prove that these inequalities imply (non)oscillation of (1.9) and that L=-1.
To prove the nonoscillation, letLλ+μ-1/2q<0, that is, A(t)+B(t)<0 for large t, and let 𝒢 be defined by (3.5). By Lemma 3.3(i) v=0 is a solution of the inequality
(4.19)v′+A(t)+B(t)+(p-1)(r(t)+λr̂(t))1-qf-q(t)G(t,v)≤0,
for large t,  and by identity (3.3) in Lemma 3.2 we obtain that w=(r+λr̂)Φ(f′/f) satisfies the Riccati inequality (3.2), that is, (1.9) is nonoscillatory by Lemma 3.1(iii).
To prove the oscillation, letLλ+μ-1/2q>0, that is,  A(t)+B(t)>0 for large t. Observe that for t→∞(4.20)∫tfp(s)ĉ(s)ds=∫t1R(s)(∫tR-1(τ)dτ)ds=log⁡(∫tR-1(s)ds)⟶∞,
and hence ∫∞B(t)dt=∞, which consequently means that ∫∞(A(t)+B(t))dt=∞. Here, we have used the fact that the integral of the leading term in A and also integrals of other terms in the asymptotic formula of Theorem 2.2 are convergent, see [7, page 161]. Suppose, on the contrary, that (1.9) is nonoscillatory. Then by Lemma 3.1, there exists a solution w of the associated Riccati equation (3.1) for large t and, by Lemma 3.2, the function v=fp(w-wf), where wf=(r+λr̂)Φ(f′/f), is a solution of the modified Riccati equation (3.6) for large t. Integrating (3.6), we get
(4.21)v(T)-v(t)=∫Tt(A(s)+B(s))ds+(p-1)∫Tt(r(s)+λr̂(s))1-qf-q(s)G(s,v(s))ds.
Now,  we use Theorem 3.5. In view of (2.8) and (4.3),  we have for t→∞,
(4.22)|Ω(t)|=(r(t)+λr̂(t))f(t)|Φ(f′(t))|=r(t)(1+o(1))h(t)(∫tR-1(s)ds)1/p|Φ(h′(t))|(∫tR-1(s)ds)(p-1)/p×(1+1pG(t)(∫tR-1(s)ds))p-1=|G(t)|(∫tR-1(s)ds)(1+o(1))⟶∞,R(t)=(r(t)+λr̂(t))f2(t)|f′(t)|p-2=r(t)(1+o(1))h2(t)(∫tR-1(s)ds)2/p|h′(t)|p-2(∫tR-1(s)ds)(p-2)/p×(1+1pG(t)(∫tR-1(s)ds))p-2=R(t)(∫tR-1(s)ds)(1+o(1)),
and hence
(4.23)∫tdsR(s)⟶∞ast⟶∞.
Consequently, v(t)≥0 by Theorem 3.5. This means that the left-hand side in (4.21) is bounded above as t→∞, while the right-hand side tends to ∞ which yields the required contradiction proving that (1.9) is oscillatory if Lλ+μ>1/2q.
Finally, consider again the caseμ=0. In that case, we proved in the first part of the proof that (1.9) is oscillatory or nonoscillatory depending on whether λ<-1/2q or λ>-1/2q. This shows that the second limit in (4.4) must be −1.Remark 4.2.
(i) From the proof of Theorem4.1, it follows that if the first limit in (4.4) exists, then conditions (2.8) imply that this limit is 1, and the assumptions of the theorem imply that if the second limit in (4.4) exists and is finite, then it is −1.
(ii) Theorem4.1 can be applied to the Euler equation (1.7), and one can obtain the same result for (1.8) as in [4, Corollary  3]. Indeed, in this case, we have h(t)=t(p-1)/p, r=1, c(t)=γpt-p, where γp=((p-1)/p)p and by a direct computation
(4.24)G(t)=(p-1p)p-1,R(t)=R̃(t)=(p-1p)p-2t,
hence,
(4.25)ĉ(t)=[(t(p-1)/p)p(p-1p)p-2t[(pp-1)p-2log⁡t]2]-1=(pp-1)2-pt-plog⁡-2t,r̂(t)=[(p-1pt-1/p)p(p-1p)p-2t[(pp-1)p-2log⁡t]2]-1=(pp-1)2log⁡-2t,
which mean that conditions (2.8) and (4.3) are satisfied. Concerning the limits in (4.4), we have
(4.26)r|h′(t)|p=(p-1p)pt-1=c(t)hp(t),
that is, the first limit in (4.4) is 1. Next,
(4.27)f(t)=(pp-1)(p-2)/pt(p-1)/plog⁡1/pt,
and consequently,
(4.28)ĉ(t)Φ(f(t))=(pp-1)2-pt-plog⁡-2t[(pp-1)(p-2)/pt(p-1)/plog⁡1/pt]p-1=(pp-1)-1+2/pt-2+1/plog⁡-1-1/pt,f′(t)=(pp-1)(p-2)/p[p-1pt-1/plog⁡1/pt+1pt-1/plog⁡1/p-1t]=(p-1p)2/pt-1/plog⁡1/pt[1+1p-1log⁡-1t].
Using this formula,
(4.29)r̂(t)Φ(f′(t))=(pp-1)2/pt-1+1/plog⁡-1-1/pt[1+log⁡-1t+p-22log⁡-2t+O(log⁡-2t)],
and hence,
(4.30)(r̂(t)Φ(f′(t)))′=(pp-1)2/p[-p-1pt-2+1/plog⁡-1-1/pt(1+O(log⁡-1t))-p+1pt-2+1/plog⁡-2-1/pt(1+O(log⁡-1t))+t-2+1/plog⁡-1-1/ptO(log⁡-2t)p-1p]=-(pp-1)2/p-1t-2+1/plog⁡-1-1/pt(1+O(log⁡-1t)),
as t→∞. This means that the second limit in (4.4) is −1. According to Theorem 4.1, we obtain that the equation
(4.31)[(1+λ(pp-1)21log⁡2t)Φ(x′)]′+[γptp+μ(pp-1)2-p1tplog⁡2t]Φ(x)=0
is nonoscillatory if μ-λ<1/2q and oscillatory if μ-λ>1/2q. If we denote λ̃=λ(p/(p-1))2 and μ̃=μ(p/(p-1))2-p, we see that   (1.8) (with λ̃, μ̃ instead of λ, μ, resp.) is nonoscillatory if μ̃-γpλ̃<(1/2)((p-1)/p)p-1, and it is oscillatory if μ̃-γpλ̃>(1/2)((p-1)/p)p-1, that is, we have the statement from [4].
(iii) In [3], it is proved that (1.8) is nonoscillatory also in the limiting case μ-γpλ=μp. We conjecture that we have also the same situation in the general case, that is,   (1.9) is nonoscillatory also in the case μ-λ=1/2q.



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*Source: 182827-2011-02-20.xml*