# Perturbations of Half-Linear Euler Differential Equation and Transformations of Modified Riccati Equation

Authors: Ondřej Došlý; Hana Funková
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We investigate transformations of the modified Riccati differential equation and the obtained results we apply in the investigation of oscillatory properties of perturbed half-linear Euler differential equation. A perturbation is also allowed in the differential term.


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## Body

## 1. Introduction

The half-linear Euler differential equation(1.1)(Φ(x′))′+γptpΦ(x)=0,Φ(x):=|x|p-2x,p>1,
with the so-called oscillation constantγp:=((p-1)/p)p-1 plays an important role in the oscillation theory of the half-linear differential equation
(1.2)(r(t)Φ(x′))′+c(t)Φ(x)=0,
with the continuous functions r, c, and r(t)>0. The reason is that (1.1) represents a kind of borderline between oscillation and nonoscillation in the half-linear oscillation theory. More precisely, if r(t)=1 in (1.2), then this equation is oscillatory provided
(1.3)liminft→∞tpc(t)>γp,
and nonoscillatory if
(1.4)limsupt→∞tpc(t)<γp,
see, for example, [1]. Formulas (1.3), (1.4) show what “borderline” means. The potential c(t)=γp/tp “separates” potentials c in (1.2) with r(t)≡1 for which this equation is oscillatory or nonoscillatory. Criteria (1.3), (1.4) can be extended to the general case r(t)≢1. In this general setting, the Kneser type criterion is formulated in terms of the lower and upper limit of the expression
(1.5)rq-1(t)(∫tr1-q(s)ds)pc(t),
if ∫∞r1-q(t)dt=∞, q=p/(p-1) being the conjugate exponent of p, and of the expression
(1.6)rq-1(t)(∫t∞‍r1-q(s)ds)pc(t),
if ∫∞r1-q(t)dt<∞. The constant γp in this criterion remains the same. In the linear case p=2, (1.3) and (1.4) are the classical Kneser (non)oscillation criteria, see [2].Our investigation is mainly motivated by the papers [3–5]. In [4], perturbations of (1.1) of the form,
(1.7)(Φ(x′))′+[γptp+∑j=1n‍βjtpLogj2t]Φ(x)=0,
were investigated. Here, the notation
(1.8)Logkt=∏j=1klogjt,logkt=logk-1(logt),log1t=logt
is used. It was shown that the crucial role in (1.7) plays the constant μp:=(1/2)((p-1)/p)p-1. In particular, if n=1 in (1.7), that is, this equation reduces to the so-called Riemann-Weber half-linear differential equation, then this equation is oscillatory if β1>μp and nonoscillatory in the opposite case. In general, if βj=μp for j=1,…,n-1, then (1.7) is oscillatory if and only if βn>μp.In [5], the perturbations of the linear Euler differential equation were investigated and a perturbation was also allowed in the term-involving derivative. More precisely, the differential equation
(1.9)[(1+∑j=1n‍αjLogj2t)x′]′+[14t2+∑j=1n‍βjt2Logj2t]x=0
was considered. It was shown that if there exists k∈{1,…,n} such that βj-αj/4=1/4 for j=1,…,k-1, and βk-αk≠1/4, then (1.9) is oscillatory if and only if βk-αk/4>1/4. If βj-αj=1/4 for all j=1,…,n, then (1.11) is nonoscillatory. This result was partially extended to half-linear equations in [3]. There,
(1.10)[(1+αlog2t)Φ(x′)]′+[γptp+βtplog2t]Φ(x)=0
was investigated and it was shown that (1.10) is oscillatory if and only β-αγp>μp. For some related results see also [6].In this paper we deal with perturbations of the Euler half-liner differential equation in full generality. We consider(1.11)[(1+∑j=1n‍αjLogj2t)Φ(x′)]′+[γptp+∑j=1n‍βjtpLogj2t]Φ(x)=0,
and we find an explicit formula for the relationship between constants αj, βj in (1.11) which implies (non)oscillation of this equation. In the last section of the paper we explain why perturbations are just in the above considered form. Our result is based on a new method which consists in transformations of the modified Riccati equations associated with (1.2). The main result along this line is established in Section 3, while its application to the perturbed Euler equation is presented in Section 4. In the last section we present some remarks and comments concerning the results of our paper. In the next section we recall some essentials of the half-linear oscillation theory.


## 2. Preliminaries

It is a well-known fact that many of the results of the linear oscillation theory can be directly extended to half-linear equation (1.2), even if, in contrast to the case p=2 (then (1.2) is a linear equation), the additivity of the solution space is lost and only homogeneity remains. In particular, the so-called Riccati technique, consisting in the relationship between (1.2) and its associated Riccati type equation (related to (1.2) by the substitution w=rΦ(x'/x))
(2.1)R[w](t):=w'+c(t)+(p-1)r1-q(t)|w|q=0,q:=pp-1
extends almost literally to (1.2). More precisely, the following statement holds (see [1, Theorem 2.2.1]).Proposition 2.1.
Equation (1.2) is nonoscillatory if and only if there exists a differentiable function w such that R[w](t)=0 for large t.Themodified Riccati equation associated with (1.2) is introduced explicitly in [7], but it can be found implicitly already in some earlier papers, for example, [8–10]. Suppose that (1.2) is nonoscillatory (i.e., every its nontrivial solution is eventually positive or negative) and let h be a positive differentiable function. Consider the substitution
(2.2)v=hp(t)w-G(t),G(t):=r(t)h(t)Φ(h′(t)),
where w is a solution of (2.1). Then v is a solution of the modified Riccati equation(2.3)v'+c~(t)+(p-1)r1-q(t)h-q(t)H(v,G(t))=0,
with
(2.4)H(v,G):=|v+G|q-qΦ-1(G)v-|G|q,Φ-1(s)=|s|q-2s being the inverse function of Φ, and
(2.5)c~(t)=h(t)[(r(t)Φ(h′(t)))′+c(t)Φ(h(t))].
Note that the function H(v,G) satisfies H(v,G)≥0 for every v,G∈ℝ and H(v,G)=0=Hv(v,G) if and only if v=0. Observe also that Riccati equation (2.1) is a special case of (2.3) with h(t)≡1, that is, G(t)≡0.In the investigation of perturbations of the half-linear Euler equation we will need the following criteria for (non)existence of aproper solution of (2.3). Recall that a solution v of (2.3) is called proper if it exists on some interval [T,∞). Nonexistence of a proper solution of (2.3) is equivalent to oscillation of (1.2) since it eliminates (via the transformation w=h-p(v+G)) proper solutions of (2.1). For more details concerning this method, as well as the proof of the next two propositions, we refer to [11].For the sake of the later application, we will write (2.3) in the form
(2.6)v'+C(t)+(p-1)R-1(t)H(v,G(t))=0,
with continuous functions C, R, and R(t)>0.Proposition 2.2.
(i) IfC(t)≤0 for large t, then (2.6) possesses a (nonnegative) proper solution.
In the remaining part of the proposition suppose that(2.7)liminft→∞|G(t)|>0,C(t)≥0,forlarget.
Denote
(2.8)ℛ(t)=R-1(t)|G(t)|q-2,
and suppose that
(2.9)∫∞ℛ(t)dt=∞,∫∞C(t)dt<∞.
(ii) If(2.10)limsupt→∞(∫tℛ(s)ds)(∫t∞‍C(s)ds)<12q,
then (2.6) has a proper solution.
(iii) If(2.11)liminft→∞(∫tℛ(s)ds)(∫t∞‍C(s)ds)>12q,
then (2.6) possesses no proper solution.Proposition 2.3.
Together with (2.6) consider the equation of the same form
(2.12)v'+D(t)+(p-1)R-1(t)H(v,G(t))=0,
with the function D satisfying D(t)≥C(t) for large t. If the (majorant) equation (2.12) has a proper solution, then (2.6) has a proper solution as well.Next, we recall basic properties of solutions of the “critical” half-linear Euler and Riemann-Weber differential equations as presented, for example, in [4]. Consider the half-linear Euler differential equation
(2.13)(Φ(x′))′+γtpΦ(x)=0.
This equation is nonoscillatory if and only if γ≤γp=((p-1)/p)p. In the critical case γ=γp, (2.13) has the solution h(t)=t(p-1)/p, and every linearly independent solution is asymptotically equivalent (up to a multiplicative factor) to the function x(t)=t(p-1)/plog2/pt. The Riemann-Weber half-linear differential equation
(2.14)(Φ(x′))′+[γptp+μtplog2t]Φ(x)=0
is nonoscillatory if and only if μ≤μp=(1/2)((p-1)/p)p-1. In the critical case μ=μp, (2.14) has the (so-called principal) solution which is asymptotically equivalent (up to a multiplicative factor) to the function h(t)=t(p-1)/plog1/pt, and every linearly independent solution is asymptotically equivalent to the function x(t)=t(p-1)/plog1/ptlog2/p(logt), see [4].Finally, we recall the transformation method of the investigation of (1.9) which we extend in a modified form to half-linear equations. The Sturm-Liouville differential equation
(2.15)(r(t)x′)′+c(t)x=0
is the special case p=2 in (1.2). The transformation x=f(t)y gives the identity (suppressing the argument t)
(2.16)f[(rx′)′+cx]=(rf2y′)′+f[(rf′)′+cf]y.
In particular, if f(t)≠0, then x is a solution of (2.15) if and only if y is a solution of the equation
(2.17)(r(t)f2(t)y′)′+f(t)[(r(t)f′(t))′+c(t)f(t)]y=0.
Let us emphasize at this moment that we have in disposal no half-linear version of transformation identity (2.16).Let us denote(2.18)r(t)=1+∑j=1n‍αjLogj2t,c(t)=14t2+∑j=1n‍βjt2Logj2t.
First we apply the transformation x=ty to (1.9). Using (2.16) and the fact that f(t)=t is a solution of the critical Euler linear equation x′′+(1/4t2)x=0, we find that y is a solution of the equation
(2.19)(tr(t)y′)′+[∑j=1n‍βj-αj/4tLogj2t]y=0.
Now, we change the independent variable t↦et, the resulting equation is
(2.20)(r(et)y′)′+[∑j=1n‍βj-αj/4t2Logj-12t]y=0.
Here we take Log0t=1. Equation (2.20) is oscillatory by Kneser oscillation criterion if β1-α1/4>1/4 and nonoscillatory if β1-α1/4<1/4. Indeed, since (r(et)~1 as t→∞, we have in (1.5) with p=2(2.21)r(et)(∫tr-1(es)ds)2~t2,
as t→∞, and hence
(2.22)limt→∞r(et)(∫tr-1(es)ds)2[∑j=1∞‍βj-αj/4t2Logj-12t]=β1-α14.
If β1-α1/4=1/4, we can repeat the previous transformations and we obtain
(2.23)(r(e2(t))y′)′+[β2-α2/4t2+⋯+βn-αn/4t2Logn-22t]y=0,
here e2(t):=eet. Now it should be clear how one can obtain the result of [5] concerning oscillation of (1.9). We repeat the transformation of dependent variable y↦ty followed by the change of independent variable t↦et as long as the condition βj-αj/4=1/4 is satisfied.As we have emphasized above, we have no half-linear version of the linear transformation identity (2.16). Consequently, the above procedure cannot be applied directly to (1.2). However, as observed, for example, in [6, 11], the modified Riccati equation in the linear case p=2 is
(2.24)v'+h[(rh′)′+ch]+v2rh2=0,
which is just the Riccati equation associated with differential equation (2.17). Hence, modified Riccati equation can be regarded, in a certain sense, as a half-linear substitution for the linear transformation identity (2.16). This is just the idea which we develop in the next section and apply it in the investigation of the perturbed Euler equation.


## 3. Transformation of Modified Riccati Equation

As a starting point of this section we consider the modified Riccati equation in the form(3.1)v'+C(t)+(p-1)R-1(t)H(v,G(t))=0,
where the function H is given by (2.4), the functions R, C are supposed to be continuous and R(t)>0. In this equation, we call the function C the absolute term (since this term does not contain the unknown function v).We consider the transformation(3.2)z=fp(t)v-U(t),
with a positive differentiable function f and with a function U which we determine as follows. We have (again suppressing the argument t, this argument we will suppress also now and then in the next parts of the paper) the following:
(3.3)z′=pf′f(z+U)+fp{-C-(p-1)R-1[-qΦ-1(G)f-p(z+U)-|G|qf-pq|z+U+fpG|q-qΦ-1(G)f-p(z+U)-|G|q]}-U'=-(p-1)R-1f-q|z+U+fpG|q+p[f'f+R-1Φ-1(G)]z+p[f'f+R-1Φ-1(G)]U-(p-1)fpR-1|G|q-U′.
Next we determine the function U in such a way that the differential equation for z is again an equation of the form (3.1) (in which H(0,G)=0=Hv(0,G)). Denote Ω:=U+fpG. The terms on the fourth line of the previous computation
(3.4)-(p-1)R-1f-q|z+Ω|q+p[f′f+R-1Φ-1(G)]z
we will take as the first two terms in the function of the same form as H in (3.1). Differentiating (3.4) with respect to z, substituting z=0, and setting the obtained expression equal to zero, we obtain
(3.5)R-1f-qΦ-1(Ω)=f′f+R-1Φ-1(G),
hence
(3.6)Ω=fΦ(Rf′+fΦ-1(G)).
Consequently, we obtain the transformed modified Riccati equation
(3.7)z′+C~+(p-1)R-1f-q[|z+Ω|q-qΦ-1(Ω)z-|Ω|q]=0,
where
(3.8)C~=-p(f′f+R-1Φ-1(G))U+fpC-(p-1)R-1fp|G|q+(p-1)R-1f-q|Ω|q+U′,(3.9)U=-fpG+fΦ(Rf′+fΦ-1(G)).


## 4. Perturbations of Euler Differential Equation

Now we apply the results of the previous section to the perturbed Euler half-linear differential equation(4.1)[(∑j=0n‍αjLogj2t)Φ(x′)]′+(∑j=0n‍βjtpLogj2t)Φ(x)=0,
where α0=1, β0=γp:=((p-1)/p)p and
(4.2)Logjt=∏i=1jlogit,logjt=log(logj-1t),
with Log0t=1.To simplify the next computations, we denote(4.3)r(t)=∑j=0n‍αjLogj2t,c(t)=∑j=0n‍βjtpLogj2t.
The Riccati equation associated with (4.1) is
(4.4)w′+c(t)+(p-1)r1-q(t)|w|q=0.In order to better understand the next transformation procedure, we recommend the reader to compare it with the linear transformation idea described at the end of the previous section. The transformation(4.5)v1=tp-1w-U1,
with U1 specified later, transforms (4.4) into
(4.6)v'+c~1(t)t+p-1tr1-q(t)H(v,Ω1(t))=0,
with c~1/t given by (3.8), that is, c~1(t)/t=C~(t) with f(t)=t(p-1)/p, R=rq-1, and G=0. This means that c~1/t=X1+Y1+Z1+U1′+fpc, where
(4.7)U1=-fpG+fΦ(Rf′+fΦ-1(G))=rΓp,Γp:=(p-1p)p-1,X1=-p(f′f+R-1Φ-1(G))U1=-(p-1)t-1rΓp,Y1=-(p-1)R-1fp|G|q=0,Ω1=fΦ(Rf′+fΦ-1(G))=rΓp,Z1=(p-1)R-1f-q|Ω1|q=(p-1)r1-qt-1rqγp=(p-1)rt-1γp.
Hence, by a direct computation we obtain
(4.8)c~1(t)t=∑j=1n‍BjtLogj2t+O(t-1log-3t),
where
(4.9)Bj=βj-αjγp.In (4.6), with the above given c~1(t)/t, we change the independent variable t↦et and the resulting equation is
(4.10)v1′+c1(t)+(p-1)r1-q(et)[|v1+Ω1(et)|q-qΦ-1(Ω1(et))v-Ω1q(et)]=0,
with
(4.11)c1(t):=c~1(et)=B1t2+B2t2log2t+⋯+Bnt2Logn-12t+O(t-3).As the next step, we consider the modified Riccati equation(4.12)v1′+B1t2+B2t2log2t+⋯+Bnt2Logn-12t+O(t-3)+(p-1)r~11-q(t)H(v,Ω~1(t))=0,
where now
(4.13)r~1(t):=r(et)=1+α1t2+α2t2log2t+⋯+αnt2Logn-12t,Ω~1(t):=Ω1(et)=r(et)Γp=r~1(t)Γp.We apply the transformationv2=tv1-U2, the quantity U2 is again determined in such a way that we obtain a modified Riccati equation containing H type function for v2. Hence, using the results from formula (3.8), with f(t)=t1/p, G(t)=Ω~1(t)=r~1(t)Γp, and R-1(t)=r~11-q(t), we have
(4.14)Ω2=fΦ(Rf′+fΦ-1(Ω~1))=t1/pΦ(1pr~1q-1t(1/p)-1+t1/pΦ-1(r~1Γp))=r~1tΓp(1+1(p-1)t)p-1,
and using the binomial expansion
(4.15)U2=-fpΩ~1+Ω2=-tr~1Γp+r~1tΓp(1+1(p-1)t)p-1=r~1Γp[1+p-22(p-1)t+O(t-2)].
Futher,
(4.16)X2=-p(f'f+R-1Φ-1(Ω~1))U2=-p(1pt1/p-1t1/p+r~11-qΦ-1(r~1Γ))r~1Γ(1+p-22(p-1)t+O(t-2))=r~1Γ(-1t-p-22(p-1)t2-(p-1)-p-22t+O(t-2)),Y2=-(p-1)R-1fp|Ω~1|q=-(p-1)r~11-qtr~1qγ=-t(p-1)γr~1,Z2=(p-1)R-1f-q|Ω2|q=(p-1)r~11-qt-q/pr~1qγtq(1+1(p-1)t)p=(p-1)γr~1t+pγr~1+p2tγr~1+O(t-2).
Hence, the absolute term in the resulting modified Riccati equation is
(4.17)c~2(t)t∶=X2+Y2+Z2+U2′+t(∑j=1n‍Bjt2Logj-12)=r~1{p-22t[-(p-1)γ+(p-1)γ]+[-(p-1)Γ+pγ]+1t[-Γ-p-22Γ+p2γ]+O(t-2)}+∑j=1n‍BjtLogj-12t=r~1[-μpt+O(t-2)]+B1t+⋯+BntLogn-12t=1t(-μp+B1)+B2tlog2t+⋯+BntLogn-12t+O(t-2).
Observe that the O term in U2 and later in other Uj can be differentiated because of its special form. Hence, if B1=μp, we obtain
(4.18)v2′+B2tlog2t+⋯+BntLogn-12t+O(t-2)+(p-1)r~1q-1(t)t1-qH(v2,Ω2)=0.
In this equation we apply again the change of independent variable t↦et and the resulting equation is
(4.19)v2′+c2(t)+(p-1)r~11-q(et)e(2-q)tH(v2,Ω~2)=0
with c2(t)=c~1(et), Ω~2(t)=Ω2(et), and
(4.20)c2(t)=B2t2+⋯+Bnt2Logn-22t+O(e-t).
We use the notation
(4.21)r~2(t)=r~1(et)et,…,r~k(t)=r~k-1(et)et,
in the next computations. With this notation, we have
(4.22)v2′+B2t2+⋯Bnt2Logn-22t+O(e-t)+(p-1)r~21-qetH(v2,Ω~2)=0.
We apply the transformation v3=tv2-U3 to (4.22). We obtain
(4.23)v3′+c~3(t)t+(p-1)r~21-qett1-qH(v3,Ω3)=0,
where, with f(t)=t1/p and R-1(t)=r~21-qet,
(4.24)Ω3=fΦ(Rf′+Φ-1(Ω~2))=r~2tΓp(1+1(p-1)et+1(p-1)tet)p-1,(4.25)c~3(t)t:=X3+Y3+Z3+U3′+tc2(t),
with
(4.26)U3=-tΩ~2+Ω3=tr~2Γp{-(1+1(p-1)et)p-1+(1+1(p-1)et+1(p-1)tet)p-1}=tr~2Γp{-(1+1et+p-22(p-1)e2t+O(e-3t))+1+1et+1tet+p-22(p-1)e2t(1+1t)2+O(e-3t)}=r~2Γp{1et+p-2(p-1)e2t+p-22(p-1)te2t+O(te-3t)},X3=-p(f'f+R-1Φ-1(Ω~2))U3=-[1t+(p-1)et(1+1(p-1)et)]r~2Γp×[1et+p-2(p-1)e2t+p-22(p-1)te2t+O(te-3t)]=-r~2Γp[1tet+p-2(p-1)te2t+p-22(p-1)t2e2t+(p-1)+p-2et+p-22tet+1et+p-2(p-1)e2t+p-22(p-1)te2t+O(te-2t)]=-r~2Γp[(p-1)+p-1et+p2tet+O(te-2t)],Y3=-(p-1)r~21-qett1-qΩ~2q=-(p-1)r~21-qett1-qr~2qΓpqtq[1+1(p-1)et]p=-(p-1)γpr~2ett[1+p(p-1)et+p2(p-1)e2t+O(e-2t)]=r~2[-(p-1)γptet-pγpt-p2γptet+O(te-2t)],Z3=(p-1)r~21-qett1-qΩ3q=(p-1)r~21-qett1-qtqr~2qγp[1+1(p-1)et+1(p-1)tet]p=(p-1)tr~2γpet[1+p(p-1)et+p(p-1)tet+p2(p-1)e2t(1+1t)2+O(e-3t)]=r~2[(p-1)γpett+pγpt+pγp+pt2etγp+pγp1et+p2γp1tet+O(te-2t)].
Substituting into (4.25) the above computed quantities, we have
(4.27)c~3(t)t=B2-μpt+B3tlog2t+⋯+BntLogn-22t+O(te-t).
Consequently, if B2=μp=(1/2)((p-1)/p)p-1, we obtain
(4.28)v3′+∑j=3n‍BjtLogj-22t+O(te-t)+(p-1)r~21-q(t)ett1-qH(v3,Ω3(t))=0.
In this equation, the change of independent variable t↦et results
(4.29)v3′+c3(t)+(p-1)r~31-q(t)E2(t)H(v3,Ω~3(t))=0,c3(t):=c~3(et).
Here, and also in the sequel, we use the notation
(4.30)e1(t):=et,…,en(t):=en-1(et),En(t):=en(t)⋯e1(t),
where n is the integer in (4.1).Now we are already in a position to make the induction step in transformations of modified Riccati equations. We suppose thatBj=μp for j=1,…,k-2 for some k∈{3,…,n}, so we have
(4.31)vk-1′+ck-1(t)+(p-1)r~k-11-q(t)Ek-2(t)H(vk-1,Ω~k-1(t))=0,
with
(4.32)Ω~k-1(t)=Γpr~k-1(t)(1+1+E1(t)+⋯+Ek-3(t)Ek-2(t))p-1,(4.33)ck-1(t)=Bk-1t2+⋯+Bnt2Logn-k+12t+O(tEk-33(t)Ek-2(t)),
where r~k is given by (4.21). We will also use the notation
(4.34)rk(t):=rk-1(et),r1(t):=r~1(t)=r(et).
Then r~k(t)=rk(t)Ek-1(t) and rk(t)=r(ek(t)) with r given by (4.3).We putvk=tvk-1-Uk. We have
(4.35)Uk=-tΩ~k-1+Ωk=tr~k-1Γp{-[1+1+⋯+Ek-3(p-1)Ek-2]p-1+[1+1+⋯+Ek-3(p-1)Ek-2+1(p-1)tEk-2]p-1}=rk-1Γp[1+(p-2)(1+⋯+Ek-3)(p-1)Ek-2+p-22(p-1)tEk-2+O(tEk-33Ek-22)],
and with f(t)=t1/p, R-1=r~k-11-qEk-2 and Ω~k-1 given by (4.32)
(4.36)Xk=-p(f'f+R-1Φ-1(Ω~k-1))Uk=-r~k-1Γp[1t+(p-1)Ek-2+(1+⋯+Ek-3)]×[1Ek-2+(p-2)(1+⋯+Ek-3)(p-1)Ek-22+p-22(p-1)tEk-22+O(tEk-33Ek-22)],=-rk-1Γp[(p-1)Ek-2+(p-1)(1+⋯+Ek-3)+p2t+(p-2)(1+⋯+Ek-3)2(p-1)Ek-2+O(tEk-33Ek-2)],Yk=-(p-1)R-1fpΩ~k-1q=-(p-1)r~k-11-qEk-2tr~k-1qγp(1+1+⋯+Ek-3(p-1)Ek-2)p=-rk-1[(p-1)γptEk-22-pγptEk-2(1+⋯+Ek-3)+pγp2t(1+⋯+Ek-3)2+O(tEk-33Ek-2)],Zk=(p-1)R-1f-qΩ~k-1q=(p-1)r~k-1Ek-2tγp[1+(1+⋯+Ek-3)(p-1)Ek-2+1(p-1)tEk-2]p=rk-1[(tEk-33Ek-2)(p-1)γptEk-22+pγptEk-2(1+⋯+Ek-3)+pγpEk-2+pγp2t(1+⋯+Ek-3)2+pγp(1+⋯+Ek-3)+pγp2t+O(tEk-33Ek-2)].
Then, using that (p-1)Γp=pγp and
(4.37)p2(γp-Γp)=-12(p-1p)p-1=-μp,
we have
(4.38)Xk+Yk+Zk=-μpt+O(tEk-33Ek-2).
The last formula is the result of a direct computation where one needs to show that all terms with the faster growth than t-1 vanish. Further, again by a direct computation
(4.39)Uk′={rk-1Γp[1+(p-2)(1+⋯+Ek-3)(p-1)Ek-2+p-22(p-1)tEk-2+O(tEk-33Ek-22)]}′=O(tEk-32Ek-2).Consequently, in the resulting modified Riccati equation forvk(4.40)vk′+c~k(t)t+(p-1)R-1(t)H(vk,Ωk(t))=0,
with R-1(t)=r~k-1-1(t)Ek-2(t)t1-q we have
(4.41)c~k(t)t:=Xk+Yk+Zk+Uk′+tck-1=Bk-1-μpt+∑j=kn‍BjtLogj+1-k2+O(tEk-33Ek-2),
as t→∞.Now we can summarize the previous computations as follows.Theorem 4.1.
Suppose that there existsk∈{2,…,n} such that
(4.42)βj-γpαj=μp,μp=12(p-1p)p-1,j=1,…,k-1,
and βk-γpαk≠0. Then (1.11) is oscillatory if βk-γpαk>μp and nonoscillatory if βk-γpαk<μp. If (4.42) holds for all j=1,…,n, (1.11) is nonoscillatory.Proof.
We apply Proposition2.2 to the modified Riccati equation (4.40) for vk. In this equation, with the notation from Proposition 2.2,
(4.43)ℛ=R-1|Ωk|q-2~tq-2r~k-1q-2Γpq-2r~k-11-qEk-2t1-q=rk-1t-1qp-2~qp-2t-1,
hence ∫tℛ(s)ds~qp-2logt and
(4.44)∫t∞‍ck(s)ds~Bklogt,Bk=βk-αkγp.
Here (and also earlier), f~g for a pair of functions f, g means limt→∞(f(t)/g(t))=1. Consequently, if Bkqp-2>1/(2q), what happens if and only if Bk>μp, modified Riccati equation (4.40) for vk has no proper solution in view of Proposition 2.2 (iii). Now, via the “back” transformations
(4.45)vj-1=-Uj+vjt,j=2,…,k,w=-U1+t(1-p)/pv1,
the same holds for the Riccati equation associated with (1.11) and hence this equation is oscillatory by Proposition 2.1.
IfBk<μp, nonoscillation of (1.11) follows from parts (i) (when Bk<0) and (ii) (when 0≤Bk<μp) of Proposition 2.2 since the existence of a proper solution of the modified Riccati equation for vk+1 implies the existence of a proper solution for the Riccati equation (4.4) associated with (1.11), hence this equation is nonoscillatory by Proposition 2.1.
Finally, if (4.42) holds for all j=1,…,n, then the absolute term in the modified Riccati equation for vn+1 is d(t):=c~n+1(t)/t=O(tEn-23(t)/En-1(t)) and replacing d by its nonnegative part d+=max{0,d}, we get a majorant of the modified Riccati equation for vn+1 (in the sense of Proposition 2.3). The function d+ satisfies the same asymptotic estimate as d. To estimate the integral ∫t∞‍d+(s)ds we proceed as follows. We have, via the substitution en-2(s)=u, En-2(s)ds=du, using the inequality logju≤u, and followed by integration by parts,
(4.46)∫t∞‍sEn-23(s)En-1(s)ds=∫t∞‍s(e1(s)⋯en-2(s))3en-1(s)⋯e1(t)ds=∫en-2(t)∞‍logn-2ulogn-3u⋯logu·ueudu≤∫en-2(u)∞‍un-1eudu~-un-1e-u|en-2(u)∞=en-2n-1(u)en-1(u).
Consequently,
(4.47)limt→∞logt∫t∞‍d+(s)ds=0,
hence the modified Riccati equation with for vn+1 with d+ instead of c~n+1(t)/t possesses a proper solution by Proposition 2.2 and, by Proposition 2.3, the Riccati equation for vn+1 has the same property. This implies that (1.11) is nonoscillatory using the same argument as in the previous part of the proof.


## 5. Remarks and Comments

(i) In the previous section, we applied successively the transformationvk=tvk-1-Uk to the modified Riccati equation, followed by the change of independent variable t↦et. This change of the independent variable was motivated by the linear case and also by the fact that upon this transformation the modified Riccati equation simplifies. Without this change of independent variable, the transformation procedure can be “reformulated” as follows. As shown at the beginning of the previous section, the transformation (2.2), that is, v=hp(t)w-G(t), G(t)=r(t)h(t)Φ(h'(t)), transforms the Riccati equation (2.1) associated with (1.2) into the modified Riccati equation (2.3). The transformation (3.2), that is, z=fp(t)v-U(t), transforms (2.3) into an equation of the same form, with the function C~ given by a relatively complicated formula (3.8). The composition of these transformations gives
(5.1)z=(f(t)h(t))pw-(fp(t)G(t)+U(t)),
and by a direct computation, using (3.9), we have fpG+U=rfhΦ((fh)′). So, the resulting modified Riccati equation for z is just the modified Riccati equation resulting from (2.1) via (2.2) with h replaced by fh. In this equation, the function c~ is given by (2.5) with h replaced by fh, that is, c~=fh[(rΦ((fh)′)′+cΦ(fh)].Now, consider the function(5.2)h(t)=t(p-1)/p(Lognt)1/p=t(p-1)/plog1/pt⋯logn1/pt.
In view of the previous consideration, the application of transformation (2.2) with this h can be decomposed into the successive transformations v1=tp-1w-G, vj=logj-1tvj-Uj, j=2,…,k. Hence, the successive transformations treated in the previous section can be replaced by just one transformation, with the transformation function (5.2).This idea has been used in [3] in the case that n=1 in (1.11) and in (5.2). However, as shown in the computations of that paper (where also substantially the results of [12] have been used), this method is technically complicated even in this relatively simple case. This is also the reason why we developed the method of successive transformations of modified Riccati equation presented in the previous section.(ii) The reason why the perturbation terms in (1.11) are just αj/Logj2t in the differential term and βj/tpLogj2t by Φ(x) is motivated by the fact that in this form they “match together”. More precisely if we replace some of them by a term with a faster asymptotic growth, then this term “overrules” the remaining terms and the equation becomes (non)oscillatory for any positive value of the corresponding parameter αj or βj. On the other hand, functions with slower asymptotic growth have no influence on the oscillatory behavior. These considerations are closely related to concepts of strong (non)oscillation of half-linear equations as treated for example in [13].(iii) In [6], and partially also in [11], we have considered
(5.3)[(r(t)+λr~(t))Φ(x′)]′+[c(t)+μc~(t)]Φ(x)=0
as a perturbation of (1.2). We found assumptions of the functions r, r~, c, c~ (which are satisfied in case of the perturbed Euler equation) which guarantee that there exists a constant γ such that (5.3) is oscillatory if μ-λ>γ and nonoscillatory if μ-λ<γ. The limiting case μ-λ=γ remained undecided, mainly because of technical computational problems. In view of perturbations of Euler equation with n=1r(t)=1, r~(t)=γplog-2t, c(t)=t-p, and c~(t)=t-plog-2t (then γ=μp) we hope to prove that (5.3) is nonoscillatory also in the limiting case μ-λ=γ. We also hope that the method of transformations of modified Riccati equation elaborated in this paper can be applied to treat the “multiparametric” general case, not only for perturbations of Euler equation.



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*Source: 738472-2012-10-24.xml*