# Sections of polar actions

Juan Manuel Lorenzo-Naveiro, Ivan Solonenko  Department of Mathematics, King’s
College London, United Kingdom<EMAIL_ADDRESS>Department of
Mathematics, University of Santiago de Compostela, Spain<EMAIL_ADDRESS>
# Sections of polar actions

Juan Manuel Lorenzo-Naveiro, Ivan Solonenko  Department of Mathematics, King’s
College London, United Kingdom<EMAIL_ADDRESS>Department of
Mathematics, University of Santiago de Compostela, Spain<EMAIL_ADDRESS>
###### Abstract.

In this short note we provide an elementary proof of the folklore result in
the theory of isometric Lie group actions on Riemannian manifolds asserting
that sections of polar actions are totally geodesic.

Let $G$ be a connected Lie group acting properly and isometrically on a
complete Riemannian manifold $M$. A complete immersed submanifold
$\Sigma\subseteq M$ is called a section of the action $G\curvearrowright M$ if
it meets all the orbits perpendicularly. More formally:

  1. $\bullet$

$\Sigma\cap(G\cdot p)\neq\varnothing$ for all $p\in M$.

  2. $\bullet$

$T_{p}\Sigma$ and $T_{p}(G\cdot p)$ are orthogonal subspaces of $T_{p}M$ for
all $p\in\Sigma$.

An action is called polar if it admits a section. Quite often, the hypotheses
on $\Sigma$ are strengthened by assuming that it is embedded and/or closed.
For the sake of generality, we will not make those assumptions. The aim of
this note is to provide an elementary proof of the following well-known
result:

######  Main Theorem.

Let $\Sigma\subseteq M$ be a section of a polar action $G\curvearrowright M$.
Then $\Sigma$ is a totally geodesic submanifold of $M$.

Although this is a very basic fact in the theory of isometric actions, the
literature on the topic does not appear to be teeming with its various proofs.
Some of the classical papers that are usually cited as containing a proof of
this result are [5] by Palais and Terng (as well as their subsequent book [6])
and [7] by Szenthe. The authors of these works only prove that a section
$\Sigma$ is totally geodesic at those points which are regular in $M$ (i.e.
whose $G$-orbits are principal). One would then expect $\Sigma$ to be globally
totally geodesic because the regular points form an open and dense subset
$M_{\mathrm{reg}}$ of $M$ and the second fundamental form of $\Sigma$ is a
smooth tensor field. The problem here is that $\Sigma\cap M_{\mathrm{reg}}$
may not, in theory, be dense in $\Sigma$. While some authors (c.f. [1,
Exercise 4.9 (iii)], [3, Proposition 1.3 (a)], [4, Theorem 30.9 (4)]) comment
briefly on why this statement should hold, it appears that no detailed proofs
are currently available. In this note, we follow a route proposed in [2] and
give a complete proof that sections are always totally geodesic. In
particular, we show that $\Sigma\cap M_{\mathrm{reg}}$ is indeed dense in
$\Sigma$.

We will use a number of standard facts about proper isometric actions:

  1. (1)

[4, Definition 29.2] For any $p\in M$, there exists $\varepsilon>0$ such that
the restriction of the normal exponential map $\exp^{\perp}\colon\nu(G\cdot
p)\to M$ to $\nu^{\varepsilon}(G\cdot p)=\left\\{\hskip
0.29999pt{\xi\in\nu(G\cdot p)\mid||\xi||<\varepsilon}\hskip
0.29999pt\right\\}$ is a diffeomorphism onto an open neighborhood of $G\cdot
p$, which we denote $U^{\varepsilon}(G\cdot p)$ (here $\nu(G\cdot p)$ stands
for the normal bundle of the orbit $G\cdot p$). This is a version of the
tubular neighborhood theorem adapted to an extrinsically homogeneous
submanifold. For any such $\varepsilon$, the submanifold
$S_{p}^{\varepsilon}=\exp^{\perp}(\nu_{p}^{\varepsilon}(G\cdot p))$ is a slice
of the action at $p$. Note that $S_{p}^{\varepsilon}\cap
S_{q}^{\varepsilon}=\varnothing$ for any point $q\in G\cdot p$ other than $p$.

  2. (2)

[2, Corollary 2.2.2] An orbit $G\cdot p$ is principal if and only if the slice
representation $G_{p}\to\mathrm{O}(\nu_{p}(G\cdot p)),g\mapsto g_{*p}$, is
trivial.

  3. (3)

[4, Theorem 29.14] Regular points of $G\curvearrowright M$ form an open and
dense subset of $M$.

  4. (4)

[2, Section 2.3.1] If the action of $G$ is polar and $\Sigma$ is a section,
then $\dim\Sigma$ coincides with the cohomogeneity of the action, which we
denote by $\text{cohom}(G\curvearrowright M)$. In particular, if $G\cdot p$ is
a principal or exceptional orbit, we have $T_{p}\Sigma=\nu_{p}(G\cdot p)$.

Whenever we write $S_{p}^{\varepsilon}$ or $U^{\varepsilon}(G\cdot p)$, we
assume that $\varepsilon$ is small enough as in (1) above. We start by
recalling the proof of the fact that sections are totally geodesic at their
regular points.

######  Proposition 1 ([2, Theorem 2.3.2]).

If $\Sigma$ is a section of a polar action $G\curvearrowright M$, then its
second fundamental form vanishes at all the regular points of $M$ lying in
$\Sigma$.

###### Proof.

Let $p\in\Sigma$ be any point such that $G\cdot p$ has maximal dimension. For
any $\xi\in\nu_{p}\Sigma=T_{p}(G\cdot p)$, we can find an element $X$ in the
Lie algebra $\mathfrak{g}$ of $G$ such that

$\xi=X^{*}_{p},\;\text{where}\hskip
3.99994ptX^{*}_{q}=\dfrac{d}{dt}\bigg{|}_{t=0}\exp(tX)\cdot q.$

Here, $\exp$ denotes the Lie exponential map of $G$. Now, $X^{*}$ is a Killing
vector field, so its covariant derivative is skew-symmetric. Moreover, $X^{*}$
is everywhere orthogonal to $\Sigma$ by polarity. Let $\mathbb{II}$ be the
second fundamental form of $\Sigma$. For any $v\in T_{p}\Sigma$, we have
$\langle\mathbb{II}(v,v),\xi\rangle=-\langle v,\nabla_{v}X^{*}\rangle=0$.
Since $\mathbb{II}(v,v)$ is tangent to $G\cdot p$, we obtain
$\mathbb{II}(v,v)=0$. Polarizing, we get $\mathbb{II}=0$. Note that we have
actually proved that $\mathbb{II}$ vanishes not only at the regular points of
$\Sigma$, but also at those whose orbits are exceptional. ∎

The key to finishing the proof is to see that, in fact, regular points are
dense in any section. Since the second fundamental form is smooth, the
preceding proposition implies that it will then vanish globally. We will need
a few lemmas.

######  Lemma 2.

Let $p\in M$ be arbitrary. Then $G_{q}\subseteq G_{p}$ for all $q\in
S^{\varepsilon}_{p}$.

###### Proof.

For any $g\in G_{q}$, we have $q=g\cdot q\in S^{\varepsilon}_{p}\cap
S^{\varepsilon}_{g\cdot p}\neq\varnothing$, which implies $g\cdot p=p$ and
therefore $g\in G_{p}$. ∎

######  Corollary 3.

Every $p\in M$ has an open neighborhood $U$ such that the orbit type of any
$q\in U$ is greater than or equal to that of $p$, i.e. $G_{q}$ is conjugated
to a subgroup of $G_{p}$.

###### Proof.

Just take $U=U^{\varepsilon}(G\cdot p)=G\cdot S_{p}^{\varepsilon}$. Since
points on the same orbit have the same orbit type, everything follows from
Lemma 2. ∎

######  Lemma 4 ([2, Exercise 2.11.4]).

For each $p\in M$, the cohomogeneity of the slice representation at $p$
coincides with the cohomogeneity of the action $G\curvearrowright M$.

###### Proof.

Let $\varepsilon$ be small enough as before and let
$\xi\in\nu_{p}^{\varepsilon}(G\cdot p)$. We begin by proving that
$(G_{p})_{\xi}=G_{\exp(\xi)}$. The diffeomorphism
$\exp^{\perp}\colon\nu_{p}^{\varepsilon}(G\cdot p)\to
U_{p}^{\varepsilon}(G\cdot p)=S_{p}^{\varepsilon}$ is $G_{p}$-equivariant,
which implies that $(G_{p})_{\xi}=(G_{p})_{\exp(\xi)}\subseteq G_{\exp(\xi)}$.
But now it follows from Lemma 2 that $G_{\exp(\xi)}\subseteq G_{p}$, so
$(G_{p})_{\exp(\xi)}$ simply coincides with $G_{\exp(\xi)}$. We thus have:

$\displaystyle\mathrm{codim}_{M}(G\cdot\exp(\xi))$ $\displaystyle=\dim M-(\dim
G-\dim G_{\exp(\xi)})$ $\displaystyle=\dim M-(\dim G-\dim G_{p}+\dim
G_{p}-\dim(G_{p})_{\xi})$ $\displaystyle=(\dim M-\dim(G\cdot
p))-\dim(G_{p}\cdot\xi)$ $\displaystyle=\mathrm{codim}_{\nu_{p}(G\cdot
p)}(G_{p}\cdot\xi).$

Consequently, $\mathrm{cohom}(G_{p}\curvearrowright\nu_{p}(G\cdot
p))=\mathrm{cohom}(G\curvearrowright U^{\varepsilon}(G\cdot p))$. But since
the regular points of $G\curvearrowright M$ are dense in $M$, there is one in
$U^{\varepsilon}(G\cdot p)$, hence $\mathrm{cohom}(G\curvearrowright
U^{\varepsilon}(G\cdot p))=\mathrm{cohom}(G\curvearrowright M)$. ∎

######  Lemma 5 ([2, Exercise 2.11.5]).

Let $p\in M$ be such that $G\cdot p$ is a nonprincipal orbit, and let
$V\subseteq\nu_{p}(G\cdot p)$ be the subspace of fixed points for the slice
representation at $p$. Then $\dim V$ is strictly smaller than the
cohomogeneity of the action $G\curvearrowright M$.

###### Proof.

Let $x\in\nu_{p}(G\cdot p)$ be a regular point of the slice representation.
Write it as $x=y+z$, where $y\in V$ and $z\in V^{\perp}$. Since the slice
representation is orthogonal, it leaves $V^{\perp}$ invariant. Moreover,
$V^{\perp}$ is a subspace of positive dimension because the slice
representation at $p$ in nontrivial by assumption. The codimension of
$G_{p}\cdot x$ in $\nu_{p}(G\cdot p)$ is equal to the cohomogeneity of the
slice representation, which is also the cohomogeneity of the action. Notice
that $G_{p}\cdot x=y+G_{p}\cdot z$, which means that $G_{p}\cdot x$ and
$G_{p}\cdot z$ have the same dimension. Since $G_{p}\cdot z$ is compact and
contained in $V^{\perp}$, it must be of positive codimension in $V^{\perp}$.
Altogether, we conclude:

$\displaystyle\mathrm{cohom}(G\curvearrowright M)$
$\displaystyle=\mathrm{codim}_{\nu_{p}(G\cdot p)}(G_{p}\cdot x)$
$\displaystyle=\mathrm{codim}_{\nu_{p}(G\cdot p)}(G_{p}\cdot z)$
$\displaystyle=\dim V+\mathrm{codim}_{V^{\perp}}(G_{p}\cdot z)>\dim V,$

finishing the proof. ∎

######  Lemma 6.

Assume that $\Sigma$ is a section whose set of regular points is not dense.
Then there is a nonempty open subset $\Omega\subseteq\Sigma$ such that all of
its points have the same orbit type but none of them is regular.

###### Proof.

By our assumption, there must be a nonempty open subset $\Xi\subseteq\Sigma$
containing no regular points. Let $\Xi^{\prime}\subseteq\Xi$ stand for the
subset of points whose isotropy subgroups have the smallest possible dimension
among all points of $\Xi$. Pick $p\in\Xi^{\prime}$ such that $G_{p}$ has the
smallest possible number of connected components among all points of
$\Xi^{\prime}$ (this is possible because all the isotropy subgroups of $G$ are
compact). By Corollary 3, there is a neighborhood $U$ of $p$ in $M$ such that
the orbit type of every $q\in U$ is greater than or equal to that of $p$. By
construction, $\Omega=U\cap\Xi\subseteq\Sigma$ consists of nonregular points
of the same orbit type. ∎

Now we are ready to prove the main result.

###### Proof of the main theorem.

By Proposition 1, it suffices to prove that $M_{\mathrm{reg}}\cap\Sigma$ is
dense in $\Sigma$. Suppose this is not the case. Then by Lemma 6, there exists
a nonempty open subset $\Omega\subseteq\Sigma$ all of whose points have the
same nonprincipal orbit type. Pick any $p\in\Omega$, take $\varepsilon>0$
small enough as before, and denote $S=S_{p}^{\varepsilon}$. Shrinking $\Omega$
if necessary, we may assume that it is an embedded submanifold of $M$ and lies
in $U^{\varepsilon}(G\cdot p)$.

Let $F\colon G\times S\twoheadrightarrow U^{\varepsilon}(G\cdot p)$ be the
smooth map defined by $F(g,x)=g\cdot x$. Since the orbits in
$U^{\varepsilon}(G\cdot p)$ intersect the slices $S_{q}^{\varepsilon}\;(q\in
G\cdot p)$ transversely, $F$ is a smooth submersion. Given
$(g,s),(g^{\prime},s^{\prime})\in G\times S$, one has
$F(g,s)=F(g^{\prime},s^{\prime})$ if and only if $s=g^{-1}g^{\prime}\cdot
s^{\prime}$. Since the latter lies both in $S$ and $g^{-1}g^{\prime}\cdot S$,
we deduce that $g^{-1}g^{\prime}\in G_{p}$. It follows that the fibers of $F$
coincide with the orbits of the action

$G_{p}\curvearrowright G\times S,\quad h\cdot(g,s)=(gh^{-1},h\cdot s),$

so $F$ induces a diffeomorphism111Note that $G\times_{G_{p}}S$ is the fiber
bundle over $G/G_{p}\cong G\cdot p$ with a model fiber $S$ associated to the
principal $G_{p}$-bundle $G\twoheadrightarrow G/G_{p}\cong G\cdot p$. between
$G\times_{G_{p}}S=(G\times S)/G_{p}$ and $U^{\varepsilon}(G\cdot p)$. We now
consider the restriction $\widetilde{F}\colon
F^{-1}(\Omega)\twoheadrightarrow\Omega$, which is still a submersion. We claim
that the smooth map $\beta\colon F^{-1}(\Omega)\to S$ given by $\beta(g,s)=s$
is constant along the fibers of $\widetilde{F}$. Indeed, consider elements
$(g,s),(g^{\prime},s^{\prime})\in F^{-1}(\Omega)$ such that $g\cdot
s=g^{\prime}\cdot s^{\prime}$. Since $g\cdot s\in\Omega$, the points $s$ and
$p$ have the same orbit type, and since $G_{s}\subseteq G_{p}$, we conclude
that $G_{s}=G_{p}$. Furthermore, we know that there exists some $h\in
G_{p}=G_{s}$ such that $(g^{\prime},s^{\prime})=(gh^{-1},h\cdot
s)=(gh^{-1},s)$, so $s=s^{\prime}$ as claimed. For this reason, $\beta$
factors through $\widetilde{F}$ to a smooth map $f\colon\Omega\to S$ such that
$f(\widetilde{F}(g,s))=s$.

We assert that $f$ is an injective immersion. It suffices to prove that it is
an immersion at $p$, since shrinking $\Omega$ once again would let us assume
it is an injective immersion globally. In order to see this, we will show that
$f_{*p}$ is just the inclusion
$T_{p}\Omega=T_{p}\Sigma\hookrightarrow\nu_{p}(G\cdot p)=T_{p}S$. Indeed, let
$\pi\colon G\to G\cdot p$ be the orbit map at $p$, and let $\mathfrak{g}$
denote the Lie algebra of $G$. Consider the following commutative diagram:

$\textstyle{T_{(e,p)}(F^{-1}(\Omega))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{F}_{*(e,p)}}$$\scriptstyle{\beta_{*(e,p)}}$$\textstyle{T_{p}\Omega\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*p}}$$\textstyle{T_{p}S}$

Observe that $F_{*(e,p)}\colon\mathfrak{g}\oplus T_{p}S\to T_{p}M=T_{p}(G\cdot
p)\oplus T_{p}S$ decomposes as $\pi_{*e}\colon\mathfrak{g}\to T_{p}(G\cdot p)$
and $\mathrm{Id}\colon T_{p}S\to T_{p}S$. Given $X\in T_{p}\Omega$, there is
some $Y\in T_{(e,p)}(F^{-1}(\Omega))$ such that $X=\widetilde{F}_{*(e,p)}(Y)$.
This $Y$ must be of the form $X^{\prime}+X$, where
$X^{\prime}\in\mathfrak{g}$. But since $\beta$ is just the projection onto the
second factor of $G\times S$, we have
$f_{*p}(X)=\beta_{*(e,p)}(X^{\prime}+X)=X$.

To conclude, consider the submanifold $\widetilde{\Omega}=f(\Omega)\subseteq
S$. A point of $\widetilde{\Omega}$ is of the form $f(g\cdot s)=s$, where
$(g,s)\in G\times S$ is such that $g\cdot s\in\Omega$. As we saw earlier,
$G_{s}=G_{p}$, so $G_{p}\cdot s=\\{s\\}$. This means that $\widetilde{\Omega}$
is pointwise fixed by $G_{p}$, hence so is
$T_{p}\widetilde{\Omega}\subseteq\nu_{p}(G\cdot p)$. Since

$\dim
T_{p}\widetilde{\Omega}=\dim\widetilde{\Omega}=\dim\Omega=\dim\Sigma=\mathrm{cohom}(G\curvearrowright
M),$

we arrive at a contradiction by Lemma 5. ∎

## References

  * [1] Alexandrino, M. M., Bettiol, R.G.: _Lie groups and geometric aspects of group actions_. Springer, Cham (2015)
  * [2] Berndt, J., Console, S., Olmos, C. E.: _Submanifolds and holonomy_ , 2nd edn. CRC Press, Boca Raton (2016)
  * [3] Grove, K., Ziller, W.: Polar manifolds and actions. J. Fixed Point Theory Appl. (2012). https://doi.org/10.1007/s11784-012-0087-y
  * [4] Michor, P. W.: _Topics in Differential Geometry_. Amer. Math. Soc., Providence (2008)
  * [5] Palais, R.S., Terng, C.-L.: A general theory of canonical forms. Trans. Amer. Math. Soc. (1987). https://doi.org/10.1090/S0002-9947-1987-0876478-4
  * [6] Palais, R.S., Terng, C.-L.: Critical point theory and submanifold geometry. Lecture Notes in Mathematics, 1353, Springer-Verlag, Berlin (1988)
  * [7] Szenthe, J.: Orthogonally transversal submanifolds and the generalizations of the Weyl group. Period. Math. Hungar. (1984). https://doi.org/10.1007/BF02454161


††thanks: The research of the second author has been supported by the grant
GA20-11846S of the Czech Science Foundation. The first author has been
partially supported by Gnampa, National Institute for Advanced Mathematics
(INdAM)

26A12

# On increasing solutions of half-linear delay differential equations

Serena Matucci  Department of Mathematics and Informatics “U. Dini”,
University of Florence, I-50139 Florence, Italy<EMAIL_ADDRESS>Pavel
Řehák  Institute of Mathematics, FME, Brno University of Technology, Technická
2, Brno, CZ-61669, Czech Republic<EMAIL_ADDRESS>
###### Abstract.

We establish conditions guaranteeing that all eventually positive increasing
solutions of a half-linear delay differential equation are regularly varying
and derive precise asymptotic formulae for them. The results here presented
are new also in the linear case and some of the observations are original also
for non-functional equations. A substantial difference between the delayed and
non-delayed case for eventually positive decreasing solutions is pointed out.

###### keywords:

Half-linear differential equation, delayed differential equation, increasing
solution, asymptotic behavior, regular variation

###### 1991 Mathematics Subject Classification:

34K25

paper submitted to _Mathematics for applications_

##  1\. Introduction

Let us consider the equation

$(r(t)\Phi(y^{\prime}))^{\prime}=p(t)\Phi(y(\tau(t)))$ (1.1)

where $r,p$ are positive continuous functions on $[a,\infty)$ and
$\Phi(u)=|u|^{\alpha-1}\operatorname{sgn}u$ with $\alpha>1$. Throughout the
paper we assume (unless not stated otherwise) that $\tau$ satisfies the
following conditions:

$\tau\in C^{1},\ \tau^{\prime}>0,\ \tau(t)\leq t,$ (1.2)

and

$\limsup_{t\to\infty}\frac{t}{\tau(t)}<\infty.$ (1.3)

The above conditions imply $\lim_{t\to\infty}\tau(t)=\infty$, and are
fulfilled by standard examples of delay, e.g., $\tau(t)=t-\sigma$ with
$\sigma>0$, or $\tau(t)=\lambda t$ with $\lambda\in(0,1)$. Note that, in
contrast to the linear case where an equation with a general delay can be
transformed into an equation with a constant delay (see [15]), in the half-
linear case it makes a good sense to consider a general $\tau$. Solutions of
(1.1) are understood in the classical sense, i.e., a solution $y$ of (1.1) is
a $C^{1}$ function defined in an interval $I\subseteq[a,\infty)$, such that
$r\Phi(y^{\prime})\in C^{1}(I)$ and $y$ satisfies (1.1) in $I$.

Our aim is to describe asymptotic behavior of all eventually positive
increasing solutions of (1.1) via the theory of regular variation. Regularly
varying functions have been shown to be a very useful tool in studying
asymptotic properties of various type of differential equations, see, in
particular, the monograph [14]. Linear and half-linear ordinary differential
equations have been considered in the framework of regular variation e.g. in
[2, 3, 5, 7, 8, 17, 18, 19]. Linear and half-linear functional differential
equations have been studied in this framework in [5, 9, 10, 11, 13, 16, 20,
21]. The typical result in the works [9, 10, 11, 13, 20, 21] is the existence
of a regularly varying solution, obtained by means of a topological approach,
as, for example, the Schauder-Tychonoff fixed point theorem. Our approach is
different. We deal with the entire class of eventually positive increasing
solutions (which we assume to be non empty), and we show that it is a subset
of the class of regularly varying solutions. In addition, we derive precise
asymptotic formula for each solution. All our results are new also in the
(functional) linear case, and some of them are new in the non-functional
(half-linear as well as linear) case. The comparison with existing results is
discussed in remarks after the main results.

The paper is organized as follows. In the next section we give some basic
information on equation (1.1) and recall selected facts from the theory of
regular variation, which will include also De Haan class $\Pi$. The main
results are presented in Section 3. In addition of showing regular variation
of increasing solutions and deriving asymptotic formulae, we briefly discuss
also decreasing solutions; in particular, we show that under reasonable
assumptions they may exhibit a quite different behavior when compared with the
solutions of non-delayed equations. Directions for a possible future research
are discusses as well. The last section contains all the proofs.

##  2\. Preliminaries

As usual, the relation $f(t)\sim g(t)$ as $t\to\infty$ means
$\lim_{t\to\infty}f(t)/g(t)=1$, the relation $f(t)\asymp g(t)$ as $t\to\infty$
means that $\exists c_{1},c_{2}\in(0,\infty)$ such that $c_{1}g(t)\leq
f(t)\leq c_{2}g(t)$ for large $t$, and $f(t)=o(g(t))$ as $t\to\infty$ means
that $\lim_{t\to\infty}f(t)/g(t)=0$.

As for nonoscillatory solutions of (1.1) (i.e., the solutions which are
eventually of one sign), without loss of generality, we work just with
positive solutions, i.e., with the class

$\mathcal{S}=\\{y:y(t)\text{ is a positive solution of }\eqref{e}\text{ for
large }t\\}.$

We wish to include our results into the framework of a standard classification
of nonoscillatory solutions, which is given in what follows. Because of the
sign conditions on the coefficients, all positive solutions of (1.1) are
eventually monotone, therefore any such a solution belongs to one of the
following disjoint classes:

$\displaystyle{\mathcal{IS}}=\left\\{y\in\mathcal{S}:y^{\prime}(t)>0\text{ for
large }t\right\\},\ \
{\mathcal{DS}}=\left\\{y\in\mathcal{S}:y^{\prime}(t)<0\text{ for large
}t\right\\}.$

As for the nonemptiness of these classes as well as of the subclasses defined
below, general conditions are not known, as far as we know. Partial results
related to the problem of the existence of nonoscillatory solutions were
obtained in [12]. Further, as a by-product of the investigation of half-linear
delay differential equations in the framework of regular variation via a fixed
point approach, some existence theorems under a setting which is close to our
ones can be found e.g. in [11, 20].

We shall focus on the class ${\mathcal{IS}}$. This class can be divided into
the mutually disjoint subclasses:

$\displaystyle{\mathcal{IS}}_{\infty}=\left\\{y\in{\mathcal{IS}}:\lim_{t\to\infty}y(t)=\infty\right\\},\
\
{\mathcal{IS}}_{B}=\left\\{y\in{\mathcal{IS}}:\lim_{t\to\infty}y(t)=b\in\mathbb{R}\right\\},$

Define the so-called quasiderivative $y^{[1]}$ of $y\in\mathcal{S}$ by
$y^{[1]}=r\Phi(y^{\prime})$. We introduce the following convention

$\displaystyle{\mathcal{IS}}_{u,v}=\left\\{y\in{\mathcal{IS}}:\lim_{t\to\infty}y(t)=u,\
\ \lim_{t\to\infty}y^{[1]}(t)=v\right\\}.$

where $u,v\in\\{B,\infty\\}$. If $u=B$ [$v=B$] we mean that the value of the
corresponding limit is a real nonzero number. Using this convention we further
distinguish the following types of solutions which form subclasses in
${\mathcal{IS}}_{B}$ and ${\mathcal{IS}}_{\infty}$ (we list only those ones
that are not a-priori excluded):

${\mathcal{IS}}_{B,B},{\mathcal{IS}}_{B,\infty},{\mathcal{IS}}_{\infty,B},{\mathcal{IS}}_{\infty,\infty}.$
(2.1)

In some places we need to emphasize that the classes and subclasses of
eventually positive increasing solutions are associated with a particular
equation, say equation ($\ast$); then we write

${\mathcal{IS}}^{(\ast)},\ {\mathcal{IS}}_{B}^{(\ast)}$, etc. (2.2)

In the second part of this section we recall basic information on the Karamata
theory of regularly varying functions and the de Haan class $\Pi$; for more
information see the monographs [1, 4, 6].

A measurable function $f:[a,\infty)\to(0,\infty)$ is called regularly varying
(at infinity) of index $\vartheta$, $\vartheta\in\mathbb{R}$, if

$\lim_{t\to\infty}{f(\lambda t)}/{f(t)}=\lambda^{\vartheta}\ \ \text{ for
every $\lambda>0$;}$ (2.3)

we write $f\in{\mathcal{RV}}(\vartheta)$. If $\vartheta=0$, then we speak
about slowly varying functions; we write $f\in{\mathcal{SV}}$, thus
${\mathcal{SV}}={\mathcal{RV}}(0)$. By ${\mathcal{RV}}$ we mean either
abbreviation of “regularly varying” or
${\mathcal{RV}}=\bigcup_{\vartheta\in\mathbb{R}}{\mathcal{RV}}(\vartheta)$.

A function $f\in{\mathcal{RV}}(\vartheta)$ if and only if there exists a
function $L\in{\mathcal{SV}}$ such that $f(t)=t^{\vartheta}L(t)$ for every
$t.$ The slowly varying component of $f\in{\mathcal{RV}}(\vartheta)$ will be
denoted by $L_{f}$, i.e., $L_{f}(t):={f(t)}/{t^{\vartheta}}.$

The following result (the so-called Uniform Convergence Theorem, see e.g. [1])
is one of the most fundamental theorems in the theory. Many important
properties of ${\mathcal{RV}}$ functions follow from it.

######  Proposition 2.1.

If $f\in{\mathcal{RV}}(\vartheta)$, then the relation (2.3) holds uniformly on
each compact $\lambda$-set in $(0,\infty)$.

Another important result in the theory of ${\mathcal{RV}}$ functions is the
Representation Theorem (see e.g. [1]).

######  Proposition 2.2.

$f\in{\mathcal{RV}}(\vartheta)$ if and only if

$f(t)=\varphi(t)t^{\vartheta}\exp\left\\{\int_{t_{0}}^{t}\frac{\psi(s)}{s}\,{\mathrm{d}}s\right\\},$
(2.4)

$t\geq t_{0}$, for some $t_{0}>0$, where $\varphi,\psi$ are measurable with
$\lim_{t\to\infty}\varphi(t)=C\in(0,\infty)$ and $\lim_{t\to\infty}\psi(t)=0$.
A function $f\in{\mathcal{RV}}(\vartheta)$ can alternatively be represented as

$f(t)=\varphi(t)\exp\left\\{\int_{t_{0}}^{t}\frac{\omega(s)}{s}\,{\mathrm{d}}s\right\\},$

$t\geq t_{0}$, for some $t_{0}>0$, where $\varphi,\omega$ are measurable with
$\lim_{t\to\infty}\varphi(t)=C\in(0,\infty)$ and
$\lim_{t\to\infty}\omega(t)=\vartheta$.

A regularly varying function $f$ is said to be normalized regularly varying,
and we write $f\in{\mathcal{NRV}}(\vartheta),$ if $\varphi(t)\equiv C$ in
(2.4) or in the alternative representation. If (2.4) holds with $\vartheta=0$
and $\varphi(t)\equiv C$, we say that $f$ is normalized slowly varying, and we
write $f\in{\mathcal{NSV}}.$ Clearly, if $f$ is a $C^{1}$ function and
$\lim_{t\to\infty}{tf^{\prime}(t)}/{f(t)}=\vartheta,$ then
$f\in{\mathcal{NRV}}(\vartheta).$ Conversely, if
$f\in{\mathcal{NRV}}(\vartheta)\cap C^{1}$, then
$\lim_{t\to\infty}tf^{\prime}(t)/f(t)=\vartheta$.

The classes of regularly varying solutions of (1.1) are defined as follows:

$\displaystyle{\mathcal{S}}_{{\mathcal{SV}}}={\mathcal{S}}\cap{\mathcal{SV}},\quad{\mathcal{S}}_{{\mathcal{RV}}}(\vartheta)={\mathcal{S}}\cap{\mathcal{RV}}(\vartheta),$
$\displaystyle{\mathcal{S}}_{{\mathcal{NSV}}}={\mathcal{S}}\cap{\mathcal{NSV}},\quad{\mathcal{S}}_{{\mathcal{NRV}}}(\vartheta)={\mathcal{S}}\cap{\mathcal{NRV}}(\vartheta).$

The Karamata Integration Theorem (see e.g. [1, 4]) plays a very important role
in our theory. Its statement can be summarized as follows.

######  Proposition 2.3.

Let $L\in{\mathcal{SV}}.$

  * (i)

If $\vartheta<-1$, then $\int_{t}^{\infty}s^{\vartheta}L(s)\,{\mathrm{d}}s\sim
t^{\vartheta+1}L(t)/(-\vartheta-1)$ as $t\to\infty$.

  * (ii)

If $\vartheta>-1$, then $\int_{a}^{t}s^{\vartheta}L(s)\,{\mathrm{d}}s\sim
t^{\vartheta+1}L(t)/(\vartheta+1)$ as $t\to\infty$.

  * (iii)

If $\int_{a}^{\infty}L(s)/s\,{\mathrm{d}}s$ converges, then
$\widetilde{L}(t)=\int_{t}^{\infty}L(s)/s\,{\mathrm{d}}s$ is a
${\mathcal{SV}}$ function; if $\int_{a}^{\infty}L(s)/s\,{\mathrm{d}}s$
diverges, then $\widetilde{L}(t)=\int_{a}^{t}L(s)/s\,{\mathrm{d}}s$ is a
${\mathcal{SV}}$ function; in both cases, ${L(t)}/{\widetilde{L}(t)}\to 0$ as
$t\to\infty$.

Here are further useful properties of ${\mathcal{RV}}$ functions. The proofs
of (i)–(viii) are either easy or can be found in [1, 4]. The proof of (ix) can
be found in the last section.

######  Proposition 2.4.

  * (i)

If $f\in{\mathcal{RV}}(\vartheta)$, then $\ln f(t)/\ln t\to\vartheta$ as
$t\to\infty$. It then clearly implies that $\lim_{t\to\infty}f(t)=0$ provided
$\vartheta<0$, or $\lim_{t\to\infty}f(t)=\infty$ provided $\vartheta>0$.

  * (ii)

If $f\in{\mathcal{RV}}(\vartheta)$, then
$f^{\alpha}\in{\mathcal{RV}}(\alpha\vartheta)$ for every
$\alpha\in\mathbb{R}$.

  * (iii)

If $f_{i}\in{\mathcal{RV}}(\vartheta_{i})$, $i=1,2$, $f_{2}(t)\to\infty$ as
$t\to\infty$, then $f_{1}\circ
f_{2}\in{\mathcal{RV}}(\vartheta_{1}\vartheta_{2})$.

  * (iv)

If $f_{i}\in{\mathcal{RV}}(\vartheta_{i})$, $i=1,2$, then
$f_{1}+f_{2}\in{\mathcal{RV}}(\max\\{\vartheta_{1},\vartheta_{2}\\})$.

  * (v)

If $f_{i}\in{\mathcal{RV}}(\vartheta_{i})$, $i=1,2$, then
$f_{1}f_{2}\in{\mathcal{RV}}(\vartheta_{1}+\vartheta_{2})$.

  * (vi)

If $f_{1},\dots,f_{n}\in{\mathcal{RV}}$, $n\in\mathbb{N}$, and
$R(x_{1},\dots,x_{n})$ is a rational function with nonnegative coefficients,
then $R(f_{1},\dots,f_{n})\in{\mathcal{RV}}$.

  * (vii)

If $L\in{\mathcal{SV}}$ and $\vartheta>0$, then $t^{\vartheta}L(t)\to\infty,$
$t^{-\vartheta}L(t)\to 0$ as $t\to\infty$.

  * (viii)

If $f\in{\mathcal{RV}}(\vartheta)$, $\vartheta\neq 0$, then there exists $g\in
C^{1}$ with $g(t)\sim f(t)$ as $t\to\infty$ and such that
$tg^{\prime}(t)/g(t)\to\vartheta$, whence $g\in{\mathcal{NRV}}(\vartheta)$.
Moreover, $g$ can be taken such that
$|g^{\prime}|\in{\mathcal{NRV}}(\vartheta-1)$.

  * (ix)

If $|f^{\prime}|\in{\mathcal{RV}}(\vartheta)$, $\vartheta\in\mathbb{R}$, with
$f^{\prime}$ being eventually of one sign and $f(t)\to 0$ or $f(t)\to\infty$
as $t\to\infty$, then $f\in{\mathcal{NRV}}(\vartheta+1)$.

Some other properties of ${\mathcal{RV}}$ functions, needed in the proofs of
the main results, are presented in some auxiliary lemmas in the last section.

Finally, we recall the definition of the De Haan class $\Pi$, together with
some useful properties.

A measurable function $f:[a,\infty)\to\mathbb{R}$ is said to belong to the
class $\Pi$ if there exists a function $w:(0,\infty)\to(0,\infty)$ such that
for $\lambda>0$

$\lim_{t\to\infty}\frac{(f(\lambda t)-f(t))}{w(t)}=\ln\lambda;$ (2.5)

we write $f\in\Pi$ or $f\in\Pi(w)$. The function $w$ is called an auxiliary
function for $f$. The class $\Pi$, after taking absolute values, forms a
proper subclass of ${\mathcal{SV}}$.

Next we give some properties of the class $\Pi$. The proofs of (i)–(ii) can be
found in the monographs [4, 6]. For (iii) see e.g. [17].

######  Proposition 2.5.

  * (i)

If $f\in\Pi(v),$ then $v(t)\sim
f(t)-\frac{1}{t}\int_{a}^{t}f(s)\,{\mathrm{d}}s$ as $t\to\infty$.

  * (ii)

If $f\in\Pi$, then $\lim_{t\to\infty}f(t)=:f(\infty)\leq\infty$ exists. If the
limit is infinite, then $f\in{\mathcal{SV}}$. If the limit is finite, then
$f(\infty)-f(t)\in{\mathcal{SV}}$.

  * (iii)

If $f^{\prime}\in{\mathcal{RV}}(-1)$, then $f\in\Pi(tf^{\prime}(t))$.

##  3\. Main results

Denote

$G(t)=\Phi^{-1}\left(\frac{tp(t)}{r(t)}\right)=\left(\frac{tp(t)}{r(t)}\right)^{\beta-1},$

where $\beta$ is the conjugate number of $\alpha$, i.e., $1/\alpha+1/\beta=1$.
We start by establishing conditions guaranteeing slow variation of increasing
solutions, for which we also derive asymptotic formulae.

######  Theorem 3.1.

(I) Assume that $\int_{a}^{\infty}p(s)\,{\mathrm{d}}s=\infty$ and

$\lim_{t\to\infty}\frac{t^{\alpha-1}}{r(t)}\int_{a}^{t}p(s)\,{\mathrm{d}}s=0.$
(3.1)

Then ${\mathcal{IS}}\subset{\mathcal{NSV}}$. If, in addition,
$p\in{\mathcal{RV}}(\delta)$ with $\delta>-1$, then for any
$y\in{\mathcal{IS}}$ the following hold.

(i) If $\int_{a}^{\infty}G(s)\,{\mathrm{d}}s=\infty$, then $y$ satisfies the
formula

$y(t)=\exp\left\\{\int_{a}^{t}(1+o(1))\frac{G(s)}{\Phi^{-1}(\delta+1)}\,{\mathrm{d}}s\right\\}$
(3.2)

as $t\to\infty$. Moreover,
${\mathcal{S}}_{{\mathcal{NSV}}}={\mathcal{S}}_{{\mathcal{SV}}}={\mathcal{IS}}={\mathcal{IS}}_{\infty,\infty}$.

(ii) If $\int_{a}^{\infty}G(s)\,{\mathrm{d}}s<\infty$, then $y$ satisfies the
formula

$y(t)=N\exp\left\\{-\int_{t}^{\infty}(1+o(1))\frac{G(s)}{\Phi^{-1}(\delta+1)}\,{\mathrm{d}}s\right\\}$
(3.3)

as $t\to\infty$, where $N:=\lim_{t\to\infty}y(t)\in(0,\infty)$. Moreover,
${\mathcal{S}}_{{\mathcal{NSV}}}={\mathcal{S}}_{{\mathcal{SV}}}={\mathcal{IS}}={\mathcal{IS}}_{B,\infty}$.

(II) Let conditions at point (I) be satisfied and let
$r\in{\mathcal{RV}}(\gamma)$ hold. Then $\gamma\geq\alpha+\delta$. For any
$y\in{\mathcal{IS}}$ it holds $y(t)\in\Pi(ty^{\prime}(t))$ provided
$\gamma=\alpha+\delta$. Moreover, if $\gamma=\alpha+\delta$ and
$\int_{a}^{\infty}G(s)\,{\mathrm{d}}s<\infty$, or $\gamma>\alpha+\delta$, then

$N-y(t)\sim\frac{N}{\Phi^{-1}(\delta+1)}\int_{t}^{\infty}G(s)\,{\mathrm{d}}s\in{\mathcal{RV}}((\delta+1-\gamma)(\beta-1)+1)$
(3.4)

as $t\to\infty$. In particular, if $\gamma=\alpha+\delta$, then
$|N-y|\in{\mathcal{SV}}$ and

$\frac{L_{p}^{\beta-1}(t)}{L_{r}^{\beta-1}(t)(N-y(t))}=o(1)$ (3.5)

as $t\to\infty$.

Condition (3.1) is necessary in a certain sense. More precisely, the following
lemma holds.

######  Lemma 3.2.

Let $r\in{\mathcal{RV}}(\gamma)$ with $\gamma>\alpha-1$, and $\tau(t)\asymp t$
as $t\to\infty$. If there exists $y\in{\mathcal{IS}}\cap{\mathcal{NSV}}$, then
(3.1) holds. If, in addition, $p\in{\mathcal{RV}}(\delta)$, then

$\lim_{t\to\infty}t^{\alpha+\delta-\gamma}\frac{L_{p}(t)}{L_{r}(t)}=0.$ (3.6)

Note that if

$p\in{\mathcal{RV}}(\delta),\ \ r\in{\mathcal{RV}}(\delta+\alpha)$ (3.7)

hold, then

$G(t)=\frac{1}{t}\left(\frac{L_{p}(t)}{L_{r}(t)}\right)^{\beta-1}\in{\mathcal{RV}}(-1).$
(3.8)

Observe that (3.7) along with $\delta>-1$ yield $\gamma>\alpha-1$ (i.e., the
assumption from Lemma 3.2). Further, condition (3.1) (as well as condition
(3.6)) reduces to

$\lim_{t\to\infty}\frac{L_{p}(t)}{L_{r}(t)}=0,$ (3.9)

which is (3.6) with $\gamma=\delta+\alpha$.

From the non-functional case (see [19]) we know that
$\int_{a}^{\infty}r^{1-\beta}(s)\,{\mathrm{d}}s=\infty$ and

$\lim_{t\to\infty}\frac{t^{\alpha-1}}{r(t)}\int_{t}^{\infty}p(s)\,{\mathrm{d}}s=0$
(3.10)

imply ${\mathcal{DS}}\subset{\mathcal{NSV}}$. Under the additional condition
$p\in{\mathcal{RV}}(\delta)$ with $\delta<-1$, asymptotic formulae similar to
(3.2) and (3.3) can be established. In particular, ${\mathcal{SV}}$ solutions
must be sought among decreasing solutions. This result can be understood as a
certain complement to Theorem 3.1. However, as we will see, here we encounter
a quite big difference between the functional and non-functional case. For
instance, let $r(t)=1$, $\alpha=2$, and $\tau(t)=t-1$ in (1.1). Then
$y(t)=e^{-t^{2}}$ is a solution of (1.1) which takes here the form
$y^{\prime\prime}=p(t)y(t-1)$, where $p(t)=(4t^{2}-2)e^{1-2t}$. Clearly,
$\lim_{t\to\infty}t^{2}p(t)=0$, and this condition implies also
$\lim_{t\to\infty}t\int_{t}^{\infty}p(s)\,{\mathrm{d}}s=0$, i.e., condition
(3.10). However, $y\in{\mathcal{DS}}$, but $y\not\in{\mathcal{RV}}$. Thus we
see that (in contrast to the situation for ${\mathcal{IS}}$ discussed in
Theorem 3.1), for decreasing solutions in our framework, qualitative behavior
may substantially change when passing from ordinary to functional equations.
An open problem is whether or not
$\int_{a}^{\infty}r^{1-\beta}\,{\mathrm{d}}s=\infty$ and (3.10) imply
${\mathcal{DS}}\subset{\mathcal{NSV}}$ when assuming, in addition, that
$p\in{\mathcal{RV}}(\delta)$ with $\delta<-1$.

Note that if $p\in{\mathcal{RV}}(\delta)$ with $\delta\neq-1$, then both (3.1)
when $\delta>-1$, and (3.10) when $\delta<-1$, reduce to the condition

$\lim_{t\to\infty}\frac{t^{\alpha}p(t)}{r(t)}=0,$

which can easily be seen from the Karamata theorem.

We proceed with a complementary case with respect to Theorem 3.1 in the sense
that we study increasing solutions when $\delta<-1$ under the conditions (3.7)
and (3.1) (which yield (3.9)). We shall prove regular variation of these
solutions where the index is equal to

$\varrho:=\frac{-1-\delta}{\alpha-1},$ (3.11)

and derive asymptotic formulae. Denote

$H_{\tau}(t)=(t\tau^{\prime}(t))^{\alpha-1}\frac{p(t)}{r(\tau(t))}.$

If (3.7) and the first condition in (3.12) hold, then
$H_{\tau}\in{\mathcal{RV}}(-1)$ and $H_{\tau}$ can be written as

$H_{\tau}(t)=\frac{1}{t}\left(\frac{t}{\tau(t)}\right)^{\delta+\alpha}(\tau^{\prime}(t))^{\alpha-1}\frac{L_{p}(t)}{L_{r}(\tau(t))}.$

Since the convergence/divergence of the integrals $\int_{a}^{\infty}G$ and
$\int_{a}^{\infty}H_{\tau}$ plays an important role, the following example is
of interest.

######  Example 3.3.

Assume for simplicity that $\tau^{\prime}(t)\sim\lambda$ with
$\lambda\in(0,1]$ as $t\to\infty$; note that $\tau(t)=\lambda t$ or
$\tau(t)=t-\sigma$ both satisfy this condition. Then

$H_{\tau}(t)\sim\frac{\lambda^{-\delta-1}t^{\alpha-1}p(t)}{r(\tau(t))}\sim\frac{\lambda^{-\delta-1}t^{\alpha-1}p(t)}{r(t)}=\frac{\lambda^{-\delta-1}}{t}\cdot\frac{L_{p}(t)}{L_{r}(t)}$

as $t\to\infty$ by Lemma 4.1 provided (3.7) holds. Recall that $G$ takes here
the form (3.8). Taking $r,p$ such that $L_{p}(t)/L_{r}(t)\sim\ln^{\gamma}t$ as
$t\to\infty$, $\gamma\in(-\infty,0)$, we see that (3.9) is fulfilled.
Moreover, if $\alpha<2$ and $-1<\gamma<1-\alpha$ then
$\int_{a}^{\infty}G<\infty$ while $\int_{a}^{\infty}H_{\tau}=\infty$, and if
$\alpha>2$ and $1-\alpha<\gamma<-1$ then $\int_{a}^{\infty}G=\infty$ while
$\int_{a}^{\infty}H_{\tau}<\infty$. Note that in the linear case (i.e.,
$\alpha=2$), $\lambda^{-\delta-1}G(t)\sim H_{\tau}(t)$ as $t\to\infty$, and
thus half-linear equations exhibit more complex behavior.

######  Theorem 3.4.

Assume that (3.7) holds and $\delta<-1$. Let

$(r^{1-\beta}\circ\tau)\tau^{\prime}\in{\mathcal{RV}}(\delta(1-\beta)-\beta)\
\ \text{and}\ \
\left(\frac{L_{p}(t)}{L_{r}(t)}\right)^{\beta-1}\tau^{\prime}(t)\to 0$ (3.12)

as $t\to\infty$. Then ${\mathcal{IS}}\subset{\mathcal{NRV}}(\varrho)$, where
$\varrho$ is defined in (3.11), and $y^{[1]}(t)\in\Pi(tp(t)\Phi(y(\tau(t))))$
for every $y\in{\mathcal{IS}}$. In addition, for any $y\in{\mathcal{IS}}$ the
following hold.

(i) If $\int_{a}^{\infty}H_{\tau}(s)\,{\mathrm{d}}s=\infty$, then

$y(t)=tr^{1-\beta}(t)\exp\left\\{\int_{a}^{t}(1+o(1))\frac{\beta-1}{\Phi(\varrho)}H_{\tau}(s)\,{\mathrm{d}}s\right\\}$
(3.13)

as $t\to\infty$. Moreover,
${\mathcal{S}}_{{\mathcal{NRV}}}(\varrho)={\mathcal{S}}_{{\mathcal{RV}}}(\varrho)={\mathcal{IS}}={\mathcal{IS}}_{\infty,\infty}$.

(ii) If $\int_{a}^{\infty}H_{\tau}(s)\,{\mathrm{d}}s<\infty$, then

$y(t)=A+\int_{a}^{t}M^{\beta-1}r^{1-\beta}(s)\exp\left\\{-\int_{s}^{\infty}(1+o(1))\frac{\beta-1}{\varrho^{\alpha-1}}H_{\tau}(\xi)\,{\mathrm{d}}\xi\right\\}\,{\mathrm{d}}s$
(3.14)

as $t\to\infty$, where $M=\lim_{t\to\infty}y^{[1]}(t)\in(0,\infty)$, for some
$A\in\mathbb{R}$, and
${\mathcal{S}}_{{\mathcal{NRV}}}(\varrho)={\mathcal{S}}_{{\mathcal{RV}}}(\varrho)={\mathcal{IS}}={\mathcal{IS}}_{\infty,B}$.
Moreover, $M-y^{[1]}\in{\mathcal{SV}}$ and

$\frac{t^{\delta+\alpha}(\tau^{\prime}(t))^{\alpha-1}L_{p}(t)}{\tau^{\delta+\alpha}(t)L_{r}(t)(M-y^{[1]}(t))}=o(1)$
(3.15)

as $t\to\infty$.

######  Remark 3.5.

(i) In fact – as a closer examination of the proof shows – because of the
first additional condition in (3.12), we could assume only
$p\in{\mathcal{RV}}(\delta)$ instead of (3.7). Yet, for the proof of regular
variation of solutions and asymptotic formulae, we could drop the requirement
of $p$ being regularly varying (similarly as $r$ does not need to be regularly
varying in Theorem 3.1). But, for simplicity and since we want to express some
formulae in terms of $L_{p},L_{r}$, and thus being in a better correspondence
with Theorem 3.1, we prefer to take the stronger assumption.

(ii) In view of Proposition 2.4, the former condition in (3.12) is implied by
$\tau^{\prime}\in{\mathcal{SV}}$ (provided we assume
$r\in{\mathcal{RV}}(\delta+\alpha)$). The latter condition in (3.12) is
implied by (3.9), and $\limsup_{t\to\infty}\tau^{\prime}(t)<\infty$ (provided
we assume (3.7)).

(iii) Similarly as in Lemma 3.2, we can obtain a necessary condition in the
setting of Theorem 3.4. In particular, if $\delta<-1$, $\tau(t)\asymp t$ as
$t\to\infty$, (3.7) is satisfied, and
$\mathcal{S}_{\mathcal{NRV}}(\varrho)\neq\emptyset$, then (3.9) holds.

In [13, 21] (see also [20]), half-linear differential equations with both
retarded and advanced arguments are considered. Necessary and sufficient
conditions for the existence of a (generalized) ${\mathcal{SV}}$ solution and
a (generalized) ${\mathcal{RV}}(1)$ solution are established. Note that the
conditions on the coefficients are more general than ours and takes an
integral form similar to (3.1) and (3.10). The conditions on the delayed
argument are slightly more special than our ones. The methods in [13, 21] are
based on the results for the associated equation without deviating arguments
and the Schauder-Tychonoff fixed point theorem. We emphasize that while in
[13, 20, 21] the existence of ${\mathcal{RV}}$ solutions is studied, here we
attempt to deal with all positive increasing solutions.

Our results are essentially new also in the linear case, i.e., when
$\alpha=2$. The linear version of the existence results mentioned in the
previous item can be found in [9, 10, 11].

If $\tau(t)=t$, Theorems 3.1 and 3.4 reduce to an improvement of the results
in [17, 19]; note that in those papers, conditions (3.7) and (3.9) are assumed
throughout.

Next we establish a generalization of the previous theorems. As we will see,
as a by-product, we solve the natural problem arising from Theorem 3.1 and
3.4, namely the missing (“critical”) case $\delta=-1$. Also some equations
that do not have ${\mathcal{RV}}$ coefficients can be treated by generalized
theorems.

We distinguish the two cases, namely

$\int_{a}^{\infty}r^{1-\beta}(s)\,{\mathrm{d}}s=\infty$ (3.16)

and

$\int_{a}^{\infty}r^{1-\beta}(s)\,{\mathrm{d}}s<\infty.$ (3.17)

In the former case we denote
$R_{D}(t)=\int_{a}^{t}r^{1-\beta}(s)\,{\mathrm{d}}s$ and in the latter case we
denote $R_{C}(t)=\int_{t}^{\infty}r^{1-\beta}(s)\,{\mathrm{d}}s$. Further,
$R_{D}^{-1}$ stands for the inverse of $R_{D}$ and $Q^{-1}$ is the inverse of
$Q$, where $Q=1/R_{C}$. It is easy to see that $(R_{C}\circ Q^{-1})(s)=1/s$.

We first give a generalization of Theorem 3.4. Denote

$\tau_{D}=R_{D}\circ\tau\circ R_{D}^{-1},\ \ p_{D}=(pr^{\beta-1})\circ
R_{D}^{-1},$ (3.18)

and

$q_{D}=((\tau^{\prime}_{D}\circ R_{D})R_{D})^{\alpha-1}p.$

The following set of conditions will play a role:

$\limsup_{t\to\infty}\frac{t}{\tau_{i}(t)}<\infty,\
\tau_{i}^{\prime}\in{\mathcal{SV}},\
\limsup_{t\to\infty}\tau^{\prime}_{i}(t)<\infty,$ (3.19)

where $i=D$ or $i=C$ according to whether $\delta<-1$ or $\delta>-1$,
respectively.

######  Theorem 3.6.

Assume that (3.16) holds, $p_{D}\in{\mathcal{RV}}(-\alpha)$, and
$\lim_{t\to\infty}L_{p_{D}}(t)=0$. Let (1.2) and (3.19) with $i=D$ be
fulfilled. Then for any $y\in{\mathcal{IS}}$ one has $y\circ
R^{-1}_{D}\in{\mathcal{NRV}}(1)$ and the following hold.

(i) If $\int_{a}^{\infty}q_{D}(s)\,{\mathrm{d}}s=\infty$, then

$y(t)=R_{D}(t)\exp\left\\{\int_{a}^{t}(1+o(1))(\beta-1)q_{D2}(s)\,{\mathrm{d}}s\right\\}$
(3.20)

as $t\to\infty$ and $y\in{\mathcal{IS}}_{\infty,\infty}$.

(ii) If $\int_{a}^{\infty}q_{D}(s)\,{\mathrm{d}}s<\infty$, then

$y(t)=A+\int_{a}^{t}M^{\beta-1}r^{1-\beta}(s)\exp\left\\{-\int_{s}^{\infty}(1+o(1))(\beta-1)q_{D}(u)\,{\mathrm{d}}u\right\\}\,{\mathrm{d}}s$
(3.21)

as $t\to\infty$, and $y\in{\mathcal{IS}}_{\infty,B}$, with
$\lim_{t\to\infty}y^{[1]}(t)=M\in(0,\infty)$, for some $A\in\mathbb{R}$. In
addition, $|y^{[1]}\circ R_{D}^{-1}-N|\in{\mathcal{SV}}$ and

$\frac{(\tau^{\prime}_{D}(R_{D}(t)))^{\alpha-1}R_{D}^{\alpha}(t)p(t)r^{\beta-1}(t)}{M-y^{[1]}(t)}=o(1)$
(3.22)

as $t\to\infty$.

The next result is a complement of Theorem 3.6, and generalizes Theorem 3.1.
Denote

$\tau_{C}=Q\circ\tau\circ Q^{-1},\ \ p_{C}=(R_{C}^{2}pr^{\beta-1})\circ
Q^{-1}$ (3.23)

and

$q_{C}=(R_{C}pr^{\beta-1})^{\beta-1}.$

######  Theorem 3.7.

Assume that (3.17) holds, $p_{C}\\!\in\\!{\mathcal{RV}}(\alpha-2)$, and
$\lim_{t\to\infty}L_{p_{C}}(t)=0$. Let (1.2) and (3.19) with $i=C$ be
fulfilled. Then for any $y\in{\mathcal{IS}}$ one has $y\circ
Q^{-1}\in{\mathcal{NSV}}$ and the following hold.

(i) If $\int_{a}^{\infty}q_{C}(s)\,{\mathrm{d}}s=\infty$, then

$y(t)=\exp\left\\{\int_{a}^{t}(1+o(1))(\beta-1)^{\beta-1}q_{C}(s)\,{\mathrm{d}}s\right\\}$
(3.24)

as $t\to\infty$ and $y\in{\mathcal{IS}}_{\infty,\infty}$.

(ii) If $\int_{a}^{\infty}q_{C}(s)\,{\mathrm{d}}s<\infty$, then

$y(t)=N\exp\left\\{-\int_{t}^{\infty}(1+o(1))(\beta-1)^{\beta-1}q_{C}(s)\,{\mathrm{d}}s\right\\}$
(3.25)

as $t\to\infty$ and $y\in{\mathcal{IS}}_{B,\infty}$, with
$\lim_{t\to\infty}y(t)=N\in(0,\infty)$. In addition, $|y\circ
Q^{-1}-N|\in{\mathcal{SV}}$ and

$\frac{R^{\alpha}_{C}(t)p(t)r^{\beta-1}(t)}{\Phi(y(t)-N)}=o(1)$ (3.26)

as $t\to\infty$.

######  Remark 3.8.

In view of Lemma 4.2-(ii), if $\tau$ satisfies (1.2), (1.3),
$\tau^{\prime}\in{\mathcal{SV}}$,
$\limsup_{t\to\infty}\tau^{\prime}(t)<\infty$, and
$r\in{\mathcal{RV}}(\delta+\alpha)$, then (3.19) (which is assumed in Theorems
3.6 and 3.7) is fulfilled, where $i=D$ or $i=C$ according to whether
$\delta<-1$ or $\delta>-1$, respectively.

Theorems 3.6 and 3.7 are indeed generalizations of the previous theorems. As
easily seen from Lemma 4.2 (see also Remark 3.8), the assumptions of Theorem
3.4 and Theorem 3.1 (supposing here (3.7)) imply the ones of Theorem 3.6 and
Theorem 3.7, respectively. Asymptotic formulae in Theorems 3.1 and 3.4 can be
obtained from the general ones by applying the Karamata integration theorem
(Proposition 2.3) to $R_{D}$ and $R_{C}$.

Theorem 3.6 and Theorem 3.7 allow us to obtain asymptotic formulae also when
the coefficients of the equation are not regularly varying. For example, let
$p(t)=e^{\gamma t}t^{\omega}$, $r(t)=e^{\gamma t}$ with $\gamma<0$ and
$\omega<0$. Then (3.16) holds and

$p_{D}(s)=\frac{1}{((s+K)\gamma(1-\beta))^{\alpha}}\ln^{\omega}\left((s+K)\gamma(1-\beta)\right)^{\frac{1}{\gamma(1-\beta)}}$

for some $K\in(0,\infty)$. Therefore, $p_{D}\in{\mathcal{RV}}(-\alpha)$ and
$L_{p_{D}}(s)\to 0$ as $s\to\infty$.

Theorem 3.6 and Theorem 3.7 enable us to cover the missing cases in Theorems
3.1 and 3.4 in the sense of the critical case $\delta=-1$. Let us illustrate
this fact if, for simplicity, $r(t)=t^{\alpha-1}$. Assume, in addition, that
$p\in{\mathcal{RV}}(-1)$. Then (3.7) with $\delta=-1$ holds. It is easy to see
that (3.16) is fulfilled. We have $R_{D}(t)=\ln t$ (taking $a=1$) and
$R_{D}^{-1}(s)=\exp s$. Since
$p_{D}(s)=p(e^{s})(e^{s})^{(\alpha-1)(\beta-1)}=e^{s}p(e^{s})=L_{p}(e^{s})$,
we get $p_{D}\in{\mathcal{RV}}(-\alpha)$ provided

$L_{p}\circ\exp\in{\mathcal{RV}}(-\alpha).$ (3.27)

Under this condition, $L_{p_{D}}(\ln
t)=s^{\alpha}p_{D}(s)=s^{\alpha}L_{p}(e^{s})=L_{p}(t)\ln^{\alpha}t$ (with
$t=e^{s}$). Hence, $\lim_{t\to\infty}L_{p_{D}}(t)=0$ is implied by

$\lim_{t\to\infty}L_{p}(t)\ln^{\alpha}t=0.$ (3.28)

Thus the assumptions of Theorem 3.6 are fulfilled provided (3.27) and (3.28)
hold, and we get $y\circ\exp\in{\mathcal{NRV}}(1)$ for $y\in{\mathcal{IS}}$;
in fact, all eventually positive increasing solutions of (1.1) are slowly
varying. As for the asymptotic formulae, note that

$q_{D}(t)=\left(\frac{t\tau^{\prime}(t)}{\tau(t)}\right)^{\alpha-1}p(t)\ln^{\alpha-1}s.$

Take, for example,

$p(t)=\frac{1}{t}\cdot\frac{1}{\ln^{\alpha}t}\cdot\frac{1}{\ln^{\omega}\ln
t},$

$\omega>0$. Then

$(L_{p}\circ\exp)(t)=\frac{1}{t^{\alpha}\ln^{\omega}t}\in{\mathcal{RV}}(-\alpha).$

Assume $s\tau^{\prime}(s)\asymp\tau(s)$ as $s\to\infty$. Then

$q_{D}(t)=\left(\frac{t\tau^{\prime}(t)}{\tau(t)}\right)^{\alpha-1}\frac{\ln^{-\omega}\ln
t}{t\ln t}\asymp\frac{\ln^{-\omega}\ln t}{t\ln t}$

as $t\to\infty$. Hence, $\int_{a}^{\infty}q_{D}$ diverges [converges] when
$\omega\leq 1$ [$\omega>1$].

Many formulae in the previous two theorems (and their applications) can be
expressed in terms of generalized regularly varying functions; this concept
was introduced in [7] and has been used in several papers, see, for instance,
[21]. Generalized regular variation is defined as follows: a function $f$ is
regularly varying of index $\vartheta\in\mathbb{R}$ with respect to $\omega\in
C^{1}$, with $\omega^{\prime}>0$ and $\lim_{t\to\infty}\omega(t)=\infty$, if
$f\circ\omega^{-1}\in{\mathcal{RV}}(\vartheta)$. Denote the set of all
regularly varying function of index $\vartheta$ with respect to $\omega$ by
${\mathcal{RV}}_{\omega}(\vartheta)$. Hence, for example, instead of $y\circ
R_{D}^{-1}\in{\mathcal{NRV}}(1)$ we could write
$y\in{\mathcal{NRV}}_{R_{D}}(1)$, instead of $p_{D}\in{\mathcal{RV}}(-\alpha)$
we could write $pr^{1-\beta}\in{\mathcal{RV}}_{R_{D}}(-\alpha)$, and so on.

If $\tau(t)=t$, then Theorem 3.6 (Theorem 3.7) reduces to [18, Theorem 5]
([18, Theorem 6]).

In the last paragraph of this section we indicate some directions for a
possible future research.

(i) One of the open problems is to obtain a similar theory for equation (1.1)
with the opposite sign condition on the coefficient $p$. In this framework, it
would be new even in the ordinary (i.e., non-functional) case.

(ii) A natural problem is to complete our theory also for decreasing
solutions. As we could see in the observation after formula (3.10), the
difference between functional and non-functional case is “bigger” than for
increasing solutions. We conjecture that, for example,
$\int_{a}^{\infty}r^{1-\beta}(s)\,{\mathrm{d}}s=\infty$,
$p\in{\mathcal{RV}}(\delta)$ with $\delta<-1$, and (3.10) imply
${\mathcal{DS}}\subset{\mathcal{NSV}}$; the assumption on regular variation of
$p$ (or some other restriction on $p$) cannot be omitted. Once we have
guaranteed slow variation of decreasing solutions, asymptotic formulae similar
to the above ones can be obtained.

(iii) Another natural problem is to examine advanced equations, i.e., the case
in which the condition $\tau(t)\leq t$ is replaced by $\tau(t)\geq t$. Here
the situation is “reversed” in the sense that decreasing solutions are easier
to be handled than increasing ones. For instance, it is not difficult to show
that $\int_{a}^{\infty}r^{1-\beta}(s)\,{\mathrm{d}}s=\infty$,
$p\in{\mathcal{RV}}(\delta)$ with $\delta<-1$, and (3.10) (i.e., the example
of the setting is the same as in the previous item) imply
${\mathcal{DS}}\subset{\mathcal{NSV}}$. Such a statement would be a half-
linear extension of [5, Theorem 5.1] which deals with linear advanced
equations. To the best of authors’ knowledge, this paper is the first one
where functional differential equations are analyzed in the framework of
regular variation. As for the corresponding result for an advanced equation
when (3.1) with $\delta>-1$ holds, nothing is known about slow variation of
all increasing solutions even in the linear case. On the other hand,
practically all arguments which are used to obtain asymptotic formulae, work
with appropriate modifications also for advanced equations.

(iv) Since we assume that ${L_{p}(t)}/{L_{r}(t)}={t^{\alpha}p(t)}/{r(t)}$
tends to zero as $t\to\infty$, (i.e., condition (3.9)), it is natural to
examine also the condition $\lim_{t\to\infty}{t^{\alpha}p(t)}/{r(t)}$ $=C>0$
or its generalization in the sense of (3.1) or (3.10). In the case of
equations without deviating argument, see [18], we can use suitable
transformations which lead to a linear second order equation. Then the results
on ${\mathcal{SV}}$ solutions can be applied to the transformed equation. A
similar method however is not known for the associated functional differential
equations (even in the linear case).

##  4\. Proofs of the main results

In order to prove the main theorems, first we derive several auxiliary
statements.

###### Proof of Proposition 2.4-(ix).

(ix) If $f^{\prime}\in{\mathcal{RV}}(-1)$ and $f(t)\to\infty$ as $t\to\infty$,
then
$f(t)=f(a)+\int_{a}^{t}f^{\prime}(s)\,{\mathrm{d}}s\sim\int_{a}^{t}f^{\prime}(s)\,{\mathrm{d}}s$.
Hence, since $tf^{\prime}(t)\in{\mathcal{SV}}$,

$tf^{\prime}(t)/f(t)\sim
tf^{\prime}(t)/\int_{a}^{t}f^{\prime}(s)\,{\mathrm{d}}s\to 0$

as $t\to\infty$ by Proposition 2.3-(iii). Similarly we proceed when
$|f^{\prime}|\in{\mathcal{RV}}(-1)$ and $f(t)\to 0$ as $t\to\infty$. As for
the case $\vartheta\neq-1$, we use again similar arguments and apply
Proposition 2.3-(i) when $\vartheta<-1$, or Proposition 2.3-(ii) when
$\vartheta>-1$. ∎

######  Lemma 4.1.

Let $f,g$ be defined on $[a,\infty)$.

(i) If $f\in{\mathcal{SV}}$ and $g(t)\asymp t$ as $t\to\infty$, then
$f(g(t))\sim f(t)$ and $f\circ g\in{\mathcal{SV}}$.

(ii) If $f\in{\mathcal{RV}}(\vartheta)$, $\vartheta\in\mathbb{R}$, and
$g(t)\asymp t$ as $t\to\infty$, then $f(g(t))\asymp f(t)$.

(iii) If $f\in{\mathcal{RV}}(\vartheta)$, $\vartheta\in\mathbb{R}$, and
$g(t)\sim t$ as $t\to\infty$, then $f(g(t))\sim f(t)$.

###### Proof.

(i) Since $c\leq g(t)/t\leq d$, $t\in[b,\infty)$, for some $0<c<d<\infty$ and
$b\geq a$, in view of the Uniform Convergence Theorem, we have

$\frac{f(g(t))}{f(t)}=\frac{f((g(t)/t)t)}{f(t)}\to 1$

as $t\to\infty$, i.e., $f(g(t))\sim f(t)$. Therefore, for every $\lambda>0$,

$\frac{f(g(\lambda t))}{f(g(t))}\sim\frac{f(\lambda t)}{f(t)}\sim 1$

as $t\to\infty$, i.e., $f\circ g\in{\mathcal{SV}}$.

(ii) Let $L(t)=f(t)/t^{\vartheta}$. Then $L\in{\mathcal{SV}}$ and

$\frac{f(g(t))}{f(t)}=\left(\frac{g(t)}{t}\right)^{\vartheta}\frac{L(g(t))}{L(t)},$
(4.1)

In view of (i), we have

$c_{1}\leq c_{2}\frac{L(g(t))}{L(t)}\leq\frac{f(g(t))}{f(t)}\leq
d_{2}\frac{L(g(t))}{L(t)}\leq d_{1},$

$t\in[b,\infty)$, for some $c_{1},c_{2},d_{1},d_{2}\in(0,\infty)$ and $b\geq
a$.

(iii) As above, let $L(t)=f(t)/t^{\vartheta}$. Then $L\in{\mathcal{SV}}$ and,
in view of (4.1) and (i), we have

$\frac{f(g(t))}{f(t)}\sim\frac{L(g(t))}{L(t)}\sim 1$

as $t\to\infty$. ∎

######  Lemma 4.2.

Let $\tau_{i},p_{i}$, $i=C,D$, are defined as in (3.18) and (3.23).

(i) If (3.7) and (3.9) hold, then $p_{D}\in{\mathcal{RV}}(-\alpha)$ and
$L_{p_{D}}(s)\to 0$ as $s\to\infty$ provided $\delta<-1$, while
$p_{C}\in{\mathcal{RV}}(\alpha-2)$ and $L_{p_{C}}(s)\to 0$ as $s\to\infty$
provided $\delta>-1$.

(ii) Let $\tau$ satisfy (1.2). Then $\tau_{i}$ fulfills

$\tau_{i}\in C^{1},\ \tau^{\prime}_{i}>0,\ \tau_{i}(t)\leq t,$ (4.2)

where $i=D$ or $i=C$ according to whether (3.16) or (3.17) holds,
respectively. If, in addition, (1.3) holds, $\tau^{\prime}\in{\mathcal{SV}}$,
$\limsup_{t\to\infty}\tau^{\prime}(t)<\infty$, and
$r\in{\mathcal{RV}}(\delta+\alpha)$, then (3.19) holds, where $i=D$ or $i=C$
according to whether $\delta<-1$ or $\delta>-1$, respectively.

###### Proof.

(i) The statement follows from Proposition 2.4, the Karamata integration
theorem (Proposition 2.3), and the fact that
$L_{p}(t)/L_{r}(t)\sim\varrho^{\alpha}(L_{p_{D}}\circ R_{D})(t)$ as
$t\to\infty$ provided $\delta<-1$ (the case $\delta>-1$ is similar). The
details are left to the reader.

(ii) We will consider only the case $i=D$, since the case $i=C$ is similar.
Since $\tau,R_{D},R_{D}^{-1}$ are $C^{1}$ functions with positive derivatives,
$\tau_{D}\in C^{1}$ and $\tau_{D}^{\prime}>0$ follow. Further, from
$\tau(t)\leq t$, we have $\tau(R_{D}^{-1}(s))\leq R^{-1}_{D}(s)$, and since
$R_{D}$ is increasing, we obtain $\tau_{D}(s)\leq s$. From now on assume that
$r\in{\mathcal{RV}}(\delta+\alpha)$. Then $R_{D}\in{\mathcal{RV}}(\varrho)$ by
Proposition 2.4 and the Karamata theorem (Proposition 2.3). If $M>0$ exists,
such that $t\leq M\tau(t)$, then $R^{-1}_{D}(s)\leq M\tau(R_{D}^{-1}(s))$.
Consequently, $s\leq R_{D}(M\tau(R^{-1}_{D}(s)))\leq N\tau_{D}(s)$, for some
$N>0$, where the existence of $N$ is guaranteed by regular variation of
$R_{D}$. Hence, $\limsup_{t\to\infty}t/\tau_{D}(t)<\infty$ follows. Since
$\tau^{\prime}\in{\mathcal{SV}}$ and $R_{D}\in{\mathcal{RV}}(\varrho)$, we
have $\tau\in{\mathcal{RV}}(1)$ and $R_{D}^{-1}\in{\mathcal{RV}}(1/\varrho)$.
Thus, in view of Proposition 2.4,

$\displaystyle\tau^{\prime}_{D}$ $\displaystyle=(R^{\prime}_{D}\circ\tau\circ
R^{-1}_{D})(\tau^{\prime}\circ R^{-1})(R^{-1})^{\prime}$
$\displaystyle\in{\mathcal{RV}}\left((\varrho-1)\cdot
1\cdot\frac{1}{\varrho}+0\cdot\frac{1}{\varrho}+\left(\frac{1}{\varrho}-1\right)\right)={\mathcal{SV}}.$

Finally, assume that $\tau^{\prime}(t)\leq M$ for some $M>0$. In view of Lemma
4.1, we have $r^{1-\beta}(\tau(t))/r^{1-\beta}(t)\asymp 1$ as $t\to\infty$.
Hence,

$\tau^{\prime}_{D}(s)=r^{1-\beta}(\tau(R^{-1}_{D}(s)))\tau^{\prime}(R^{-1}_{D}(s))\frac{1}{r^{1-\beta}(R^{-1}_{D}(s))}\leq
N$

for some $N>0$.

∎

######  Lemma 4.3.

Let $y$ be a solution of (1.1) and let $\tau$ be differentiable. Then
$u=Cr\Phi(y^{\prime})$, $C\in\mathbb{R}$, satisfies the reciprocal equation

$(\widetilde{r}(t)\Phi^{-1}(u^{\prime}))^{\prime}=\widetilde{p}(t)\Phi^{-1}(u(\tau(t))),$
(4.3)

where

$\widetilde{r}(t)=p^{1-\beta}(t),\quad\widetilde{p}(t)=\tau^{\prime}(t)r^{1-\beta}(\tau(t)).$
(4.4)

If $p\in{\mathcal{RV}}(\delta)$ and
$(r^{1-\beta}\tau^{\prime}\circ\tau)\in{\mathcal{RV}}(\widetilde{\delta})$
(where the latter condition is implied e.g. by
$r\in{\mathcal{RV}}(\delta+\alpha)$ and $\tau^{\prime}\in{\mathcal{SV}}$),
then $\widetilde{p}\in{\mathcal{RV}}(\widetilde{\delta})$,
$\widetilde{r}\in{\mathcal{RV}}(\widetilde{\delta}+\beta)$, where
$\widetilde{\delta}=\delta(1-\beta)-\beta$. If, moreover, (3.7) holds, then

$\frac{L_{\widetilde{p}}(t)}{L_{\widetilde{r}}(t)}\asymp\left(\frac{L_{p}(t)}{L_{r}(t)}\right)^{\beta-1}\tau^{\prime}(t)$
(4.5)

as $t\to\infty$.

###### Proof.

Since $y$ is a solution of (1.1), from the definition of $u$ we have
$u^{\prime}(t)=p(t)\Phi(y(\tau))$. Hence
$y(\tau(t))=\Phi^{-1}(1/p(t))\Phi^{-1}(u(t))$. Further,
$y^{\prime}(t)=\Phi^{-1}(1/t(t))$ $\Phi^{-1}(u(t))$ and
$(y\circ\tau)^{\prime}=(y^{\prime}\circ\tau)\tau^{\prime}$. Now it is easy to
see that $u$ satisfies (4.3). The fact that
$\widetilde{p}\in{\mathcal{RV}}(\widetilde{\delta})$ and
$\widetilde{r}\in{\mathcal{RV}}(\widetilde{\delta}+\beta)$ follows from
Proposition 2.4. In view of Lemma 4.1, we have

$\frac{L_{\widetilde{p}}(t)}{L_{\widetilde{r}}(t)}=\frac{t^{\beta}\tau^{\prime}(t)r^{1-\beta}(\tau(t))}{p^{1-\beta}(t)}\asymp\left(\frac{t^{\alpha}p(t)}{r(t)}\right)^{\beta-1}\tau^{\prime}(t)$

as $t\to\infty$, which implies (4.5). ∎

######  Lemma 4.4.

Put $s=\varphi(t)$ and $x(s)=y(\varphi^{-1}(s))$, where $\varphi$ is a
differentiable function such that $\varphi^{\prime}(t)\neq 0$, equation (1.1)
is transformed into the equation

$\frac{{\mathrm{d}}}{{\mathrm{d}}s}\left(\widehat{r}(s)\Phi\left(\frac{{\mathrm{d}}x}{{\mathrm{d}}s}\right)\right)=\widehat{p}(s)\Phi(x(\widehat{\tau}(s))),$
(4.6)

where

$\widehat{p}=\frac{p\circ\varphi^{-1}}{\varphi^{\prime}\circ\varphi^{-1}},\ \
\widehat{r}=(r\circ\varphi^{-1})\Phi(\varphi^{\prime}\circ\varphi^{-1}),\ \
\widehat{\tau}=\varphi\circ\tau\circ\varphi^{-1}.$

Further,

$x^{[1]}(s):=\widehat{r}(s)\Phi\left(\frac{{\mathrm{d}}x}{{\mathrm{d}}s}(s)\right)=y^{[1]}(t).$
(4.7)

If (3.7), (3.16), and $\varphi=R_{D}$ hold, then $\widehat{r}=1$ and
$\widehat{p}=(pr^{\beta-1})\circ R_{D}^{-1}\in{\mathcal{RV}}(-\alpha)$. If
(3.7), (3.17), and $\varphi=Q$ hold, then $\widehat{r}(s)=s^{2\alpha-2}$ and
$\widehat{p}=(R_{C}^{2}pr^{\beta-1})\circ Q^{-1}\in{\mathcal{RV}}(\alpha-2)$.

###### Proof.

The form of the transformed equation follows from the fact that
$x(\varphi(t))$$=y(t)$ and
$\frac{{\mathrm{d}}}{{\mathrm{d}}t}=\varphi^{\prime}(t)\frac{{\mathrm{d}}}{{\mathrm{d}}s}$.
The indices of regular variation of $\widetilde{p}$ in both cases can easily
be computed via Proposition 2.4 and the Karamata theorem (Proposition 2.3). ∎

###### Proof of Theorem 3.1.

(I) Take $y\in{\mathcal{IS}}$. Integrating (1.1) from $b$ to $t$, where $b$ is
so large that $y(\tau(s))>0$ for $s\geq b$, we get

$\displaystyle r(t)\Phi(y^{\prime}(t))$
$\displaystyle=r(b)\Phi(y^{\prime}(b))+\int_{b}^{t}p(s)\Phi(y(\tau(s)))\,{\mathrm{d}}s$
(4.8) $\displaystyle\geq
r(b)\Phi(y^{\prime}(b))+\Phi(y(\tau(b)))\int_{b}^{t}p(s)\,{\mathrm{d}}s\to\infty$

as $t\to\infty$, since $\int_{a}^{\infty}p(s)\,{\mathrm{d}}s=\infty$. Thus
$\lim_{t\to\infty}y^{[1]}(t)=\infty$. The equality in (4.8) and the divergence
of the integral lead to the existence of $B>0$ such that

$\displaystyle r(t)\Phi(y^{\prime}(t))$ $\displaystyle\leq
B\int_{b}^{t}p(s)\Phi(y(\tau(s)))\,{\mathrm{d}}s\leq
B\Phi(y(\tau(t)))\int_{b}^{t}p(s)\,{\mathrm{d}}s$ $\displaystyle\leq
B\Phi(y(t))\int_{b}^{t}p(s)\,{\mathrm{d}}s$

for large $t$. Hence,

$0<\left(\frac{ty^{\prime}(t)}{y(t)}\right)^{\alpha-1}\leq\frac{Bt^{\alpha-1}}{r(t)}\int_{b}^{t}p(s)\,{\mathrm{d}}s,$

where the right-hand side tends to zero as $t\to\infty$ by our assumptions.
Therefore, $ty^{\prime}(t)/y(t)\to 0$ as $t\to\infty$, and so
$y\in{\mathcal{NSV}}$. Note that conditions (1.2), (1.3) imply $\tau\asymp t$.
Hence, in view of Lemma 4.1, we have $y(t)\sim y(\tau(t))$ as $t\to\infty$.
Moreover, if $p\in{\mathcal{RV}}(\delta)$ with $\delta>-1$, then
$p\Phi(y)\in{\mathcal{RV}}(\delta)$ by Proposition 2.4. Thus, recalling the
equality in (4.8), Proposition 2.3 yields

$\displaystyle r(t)\Phi(y^{\prime}(t))$
$\displaystyle\sim\int_{b}^{t}p(s)\Phi(y(\tau(s)))\,{\mathrm{d}}s\sim\int_{b}^{t}p(s)\Phi(y(s))\,{\mathrm{d}}s$
$\displaystyle=\int_{b}^{t}s^{\delta}L_{p}(s)y^{\alpha-1}(s)\,{\mathrm{d}}s$
$\displaystyle\sim\frac{1}{\delta+1}t^{\delta+1}L_{p}(t)y^{\alpha-1}(t)=\frac{1}{\delta+1}tp(t)y^{\alpha-1}(t)$

as $t\to\infty$. This implies that $y^{[1]}\in{\mathcal{RV}}(\delta+1)$,
$\delta+1>0$, therefore $\lim_{t\to\infty}y^{[1]}(t)=\infty$. Further, from
the above asymptotic relation we have

$\frac{y^{\prime}(t)}{y(t)}=(1+o(1))\Phi^{-1}\left(\frac{tp(t)}{(\delta+1)r(t)}\right)$
(4.9)

as $t\to\infty$.

If $\int_{a}^{\infty}G(s)\,{\mathrm{d}}s=\infty$, then we integrate (4.9) from
$b$ to $t$, to get

$\ln y(t)=\ln
y(b)+\int_{b}^{t}(1+o(1))\frac{G(s)}{\Phi^{-1}(\delta+1)}\,{\mathrm{d}}s=\int_{a}^{t}(1+o(1))\frac{G(s)}{\Phi^{-1}(\delta+1)}\,{\mathrm{d}}s,$
(4.10)

where the latter equality is true thanks to the divergence of
$\int_{a}^{\infty}G(s)\,{\mathrm{d}}s$. Indeed, it easily follows that for
$\varepsilon(t)\to 0$ there are $\varepsilon_{1}(t),\varepsilon_{2}(t)\to 0$
(as $t\to\infty$) such that

$c+\int_{b}^{t}(1+\varepsilon(s))G(s)\,{\mathrm{d}}s=(1+\varepsilon_{1}(t))\int_{a}^{t}(1+\varepsilon(s))G(s)\,{\mathrm{d}}s=\int_{a}^{t}(1+\varepsilon_{2}(s))G(s)\,{\mathrm{d}}s,$

where $c$ is constant. Relation (4.10) readily implies formula (3.2).
Moreover, we have $\lim_{t\to\infty}y(t)=\infty$, and so
${\mathcal{IS}}\subseteq{\mathcal{IS}}_{\infty,\infty}$.

If $\int_{a}^{\infty}G(s)\,{\mathrm{d}}s<\infty$, then we integrate (4.9) from
$t$ to $\infty$, to get

$\ln N-\ln
y(t)=\int_{t}^{\infty}(1+o(1))\frac{G(s)}{\Phi^{-1}(\delta+1)}\,{\mathrm{d}}s,$

where $N=\lim_{t\to\infty}y(t)$, and formula (3.3) follows. Clearly, $N$ has
to be finite because of convergence of the integral, and so
${\mathcal{IS}}\subseteq{\mathcal{IS}}_{B,\infty}$.

Now we show that ${\mathcal{SV}}$ solutions cannot decrease (when
$\delta>-1$), i.e., ${\mathcal{S}}_{{\mathcal{SV}}}\subseteq{\mathcal{IS}}$.
Take $y\in{\mathcal{DS}}$. Since $y^{[1]}$ is negative increasing, there
exists $\lim_{t\to\infty}y^{[1]}(t)=K\in(-\infty,0]$. Suppose now that
$y\in{\mathcal{SV}}$. Then, in view of Lemma 4.1 and Proposition 2.4,
$py^{\alpha-1}\circ\tau\in{\mathcal{RV}}(\delta)$. Hence,
$\int_{b}^{t}p(s)\Phi(y(\tau(s)))\,{\mathrm{d}}s\to\infty$ as $t\to\infty$
since $\delta>-1$. This is however a contradiction, which can be seen from the
equality in (4.8). Since $\mathcal{S}={\mathcal{IS}}\cup{\mathcal{DS}}$, we
have proved that ${\mathcal{S}}_{{\mathcal{SV}}}\subseteq{\mathcal{IS}}$. The
other relations between the classes
${\mathcal{IS}},{\mathcal{S}}_{{\mathcal{SV}}},{\mathcal{S}}_{{\mathcal{NSV}}}$,
and ${\mathcal{IS}}_{x,\infty}$ with $x=\infty$ or $x=B$ are clear.

(II) If $p\in{\mathcal{RV}}(\delta)$ and $r\in{\mathcal{RV}}(\gamma)$, then,
because of Proposition 2.3, condition (3.1) reads as

$\lim_{t\to\infty}\frac{t^{\alpha+\delta-\gamma}L_{p}(t)}{L_{r}(t)}=0,$

from which we necessarily obtain $\gamma\geq\alpha+\delta$ by Proposition
2.4-(vii). Let $\gamma=\alpha+\delta$ and $y\in{\mathcal{IS}}$. Then, in view
of Lemma 4.1 and Proposition 2.4, $y\circ\tau\in{\mathcal{SV}}$, thus
$(r\Phi(y^{\prime}))^{\prime}=p\Phi(y\circ\tau)\in{\mathcal{RV}}(\delta)$.
Since
$r(t)\Phi(y^{\prime}(t))\sim\int_{t}^{\infty}p(s)y\Phi(\tau(s))\,{\mathrm{d}}s$
as $t\to\infty$, we have $r\Phi(y^{\prime})\in{\mathcal{RV}}(\delta+1)$ by
Proposition 2.3, and so
$\Phi(y^{\prime})\in{\mathcal{RV}}(\delta+1-\delta-\alpha)={\mathcal{RV}}(1-\alpha)$.
Thus $y^{\prime}\in{\mathcal{RV}}(-1)$ by Proposition 2.4 and
$y\in\Pi(ty^{\prime}(t))$ by Proposition 2.5. Notice that condition
$\gamma>\alpha+\delta$ implies the convergence of
$\int_{a}^{\infty}G(s)\,{\mathrm{d}}s$. Thus, if $\gamma>\alpha+\delta$ or
$\gamma=\alpha+\delta$ and $\int_{a}^{\infty}G(s)\,{\mathrm{d}}s<\infty$, as
we already know, $\lim_{t\to\infty}y(t)=N\in(0,\infty)$, and from (4.9),

$y^{\prime}(t)\sim\frac{1}{\Phi^{-1}(\delta+1)}G(t)y(t)\sim\frac{N}{\Phi^{-1}(\delta+1)}G(t)\in{\mathcal{RV}}((\delta+1-\gamma)(\beta-1))$

as $t\to\infty$. Integrating this relation from $t$ to $\infty$ and using
Proposition 2.3, we obtain (3.4). If, in addition, $\gamma=\alpha+\delta$,
then, in view of $y\in\Pi$, we get $N-y\in{\mathcal{SV}}$ by Proposition 2.5;
in our case this can easily be seen also from (3.4). Formula (3.5) follows
from Proposition 2.3-(iii) since
$G(t)=\frac{1}{t}\left(\frac{L_{p}(t)}{L_{r}(t)}\right)^{\beta-1}$. ∎

###### Proof of Lemma 3.2.

Take $y\in{\mathcal{IS}}\cap{\mathcal{NSV}}$. Set $w=r\Phi(y^{\prime}/y)$.
Then $w$ is eventually positive and satisfies

$w^{\prime}(t)-\frac{\Phi(y(\tau(t)))}{\Phi(y(t))}p(t)+(\alpha-1)r^{1-\beta}(t)w^{\beta}(t)=0$
(4.11)

for large $t$. We have

$0<\frac{t^{\alpha-1}}{r(t)}w(t)=\left(\frac{ty^{\prime}(y)}{y(t)}\right)^{\alpha-1}\to
0$

as $t\to\infty$ since $y\in{\mathcal{NSV}}$. Further,
$t^{\alpha-1}/r(t)=t^{\alpha-1-\gamma}/L_{r}(t)\in{\mathcal{RV}}(\alpha-1-\gamma)$.
By our assumptions, $t^{\alpha-1}/r(t)\to 0$ as $t\to\infty$. Denote

$\Psi(t)=\frac{t^{\alpha-1}}{r(t)}\int_{b}^{t}r^{1-\beta}(s)w^{\beta}(s)\,{\mathrm{d}}s,$

where $b\geq a$ is such that $y(t)>0$ and $y^{\prime}(t)>0$ for $t\geq b$. If
$\int_{b}^{\infty}r^{1-\beta}(s)w^{\beta}(s)\,{\mathrm{d}}s<\infty$, then
clearly $\Psi(t)\to 0$ as $t\to\infty$. Let
$\int_{b}^{\infty}r^{1-\beta}(s)w^{\beta}(s)\,{\mathrm{d}}s=\infty$. Without
loss of generality we may assume $r\in{\mathcal{NRV}}(\gamma)\cap C^{1}$.
Indeed, if $r$ is not normalized or is not in $C^{1}$, then we can take
$\tilde{r}\in{\mathcal{NRV}}(\gamma)\cap C^{1}$ with $\tilde{r}(t)\sim r(t)$
as $t\to\infty$, and we have

$\Psi(t)\sim\frac{t^{\alpha-1}}{\tilde{r}(t)}\int_{b}^{t}\tilde{r}^{1-\beta}(s)(\tilde{r}(s)\Phi(y^{\prime}(s)/y(s)))^{\beta}{\mathrm{d}}s.$

The L’Hospital rule yields

$\displaystyle\lim_{t\to\infty}\Psi(t)$
$\displaystyle=\lim_{t\to\infty}\frac{r^{1-\beta}(t)w^{\beta}(t)}{r^{\prime}(t)t^{1-\alpha}+(1-\alpha)r(t)t^{-\alpha}}$
$\displaystyle=\lim_{t\to\infty}\frac{(ty^{\prime}(t)/y(t))^{\alpha}}{tr^{\prime}(t)/r(t)+(1-\alpha)}=\frac{0}{\gamma-\alpha+1}=0.$

Integrating (4.11) from $b$ to $t$ and multiplying by $t^{\alpha-1}/r(t)$, we
obtain

$\frac{t^{\alpha-1}}{r(t)}w(t)-\frac{t^{\alpha-1}}{r(t)}w(b)=\frac{t^{\alpha-1}}{r(t)}\int_{b}^{t}p(s)\Phi\left(\frac{y(\tau(s))}{y(s)}\right)\,{\mathrm{d}}s+(\alpha-1)\Psi(t).$
(4.12)

In view of the previous observations, from (4.12) we obtain

$\frac{t^{\alpha-1}}{r(t)}\int_{b}^{t}p(s)\Phi\left(\frac{y(\tau(s))}{y(s)}\right)\,{\mathrm{d}}s\to
0$ (4.13)

as $t\to\infty$. Since $y\in{\mathcal{SV}}$ and $\tau(t)\asymp t$ as
$t\to\infty$, by Lemma 4.1, there exists $K>0$ such that $y(\tau(t))/y(t)\geq
K$ for $t\geq b$, which, in view of (4.13), implies (3.1). Assuming in
addition $p\in{\mathcal{RV}}(\delta)$, then (3.6) follows by Proposition 2.3.
∎

###### Proof of Theorem 3.4.

For the coefficients $\widetilde{p},\widetilde{r}$ defined by (4.4) we have
that $\widetilde{p}\in{\mathcal{RV}}(\widetilde{\delta})$ and
$\widetilde{r}\in{\mathcal{RV}}(\widetilde{\delta}+\beta)$, where
$\widetilde{\delta}:=\delta(1-\beta)-\beta$, and
$\lim_{t\to\infty}L_{\widetilde{p}}(t)/L_{\widetilde{r}}(t)=0$ thanks to Lemma
4.3 and (3.12).

Take $y\in{\mathcal{IS}}$ and let $u=r\Phi(y^{\prime})$. Then
$u\in{\mathcal{IS}}^{\eqref{re}}$, recalling the notation (2.2). We have
$\widetilde{\delta}>-1$, and thus we can apply Theorem 3.1 to equation (4.3)
to obtain $u\in{\mathcal{NSV}}$ and $u\in\Pi(tu^{\prime})$. Hence,
$y^{\prime}\in{\mathcal{RV}}((-\delta-\alpha)(\beta-1))$ by Proposition 2.4.
Since $\delta<-1$, we have $(-\delta-\alpha)(\beta-1)>-1$, and thus
$\int_{b}^{\infty}y^{\prime}(s)\,{\mathrm{d}}s=\infty$. Consequently,
$y(t)\sim
y(t)-y(b)=\int_{b}^{t}y^{\prime}(s)\,{\mathrm{d}}s\in{\mathcal{NRV}}(\varrho)$
as $t\to\infty$, where
$\varrho=(-\delta-1)/(\alpha-1)=(-\delta-\alpha)(\beta-1)+1$. Moreover,
$y^{[1]}\in\Pi(tu^{\prime}(t))=\Pi(tp(t)\Phi(y(\tau(t))))$ and
$y\in{\mathcal{IS}}_{\infty}$ since the index $\varrho$ of regular variation
is positive. We have shown that
${\mathcal{IS}}\subset{\mathcal{NRV}}(\varrho)$. Next we derive asymptotic
formulae by applying again Theorem 3.1.

If
$\int^{\infty}(s\widetilde{p}(s)\widetilde{r}(s))^{\alpha-1}{\mathrm{d}}s=\infty$,
then

$u(t)=\exp\left\\{\int_{a}^{t}(1+o(1))\Phi\left(\frac{s\widetilde{p}(s)}{(\widetilde{\delta}+1)\widetilde{r}(s)}\right)\,{\mathrm{d}}s\right\\}$
(4.14)

as $t\to\infty$ and $u\in{\mathcal{IS}}_{\infty}^{\eqref{re}}$.

If
$\int^{\infty}(s\widetilde{p}(s)\widetilde{r}(s))^{\alpha-1}{\mathrm{d}}s<\infty$,
then

$u(t)=M\exp\left\\{-\int_{t}^{\infty}(1+o(1))\Phi\left(\frac{s\widetilde{p}(s)}{(\widetilde{\delta}+1)\widetilde{r}(s)}\right)\,{\mathrm{d}}s\right\\}$
(4.15)

as $t\to\infty$ and $u\in{\mathcal{IS}}_{B}^{\eqref{re}}$, where
$M=\lim_{t\to\infty}u(t)\in(0,\infty)$.

Since

$\left(\frac{t\widetilde{p}(t)}{\widetilde{r}(t)}\right)^{\alpha-1}=H_{\tau}(t)\
\ \text{and}\ \ \widetilde{\delta}+1=\varrho,$

from (4.14) and $y^{\prime}(t)\sim\varrho y(t)/t$, under the assumption
$\int_{a}^{\infty}H_{\tau}=\infty$, we get

$y(t)=(1+o(1))\frac{1}{\varrho}tr^{1-\beta}(t)\exp\left\\{\int_{a}^{t}(1+o(1))\frac{\beta-1}{\varrho^{\alpha-1}}H_{\tau}(s)\,{\mathrm{d}}s\right\\}$
(4.16)

as $t\to\infty$. Because of divergence of $\int_{a}^{\infty}H_{\tau}$, the
expression $(1+o(1))/{\varrho}$ can be included into the $(1+o(1))$ term
inside the integral in (4.16) (similarly as in (4.10)), and thus we obtain
formula (3.13). Since $y\in{\mathcal{IS}}_{\infty}$ and
$u\in{\mathcal{IS}}_{\infty}^{\eqref{re}}$, we have
$y\in{\mathcal{IS}}_{\infty,\infty}$.

Now assume that $\int_{a}^{\infty}H_{\tau}<\infty$. Then (4.15) implies

$y^{\prime}(t)=r^{1-\beta}(t)M^{\beta-1}\exp\left\\{-\int_{t}^{\infty}(1+o(1))\frac{\beta-1}{\varrho^{\alpha-1}}H_{\tau}(s)\,{\mathrm{d}}s\right\\}$

as $t\to\infty$. Integrating from $b$ to $t$, replacing the lower limit in the
integral by $a$ and $y(b)$ by a suitable $A$, we obtain formula (3.14).
Moreover $y\in{\mathcal{IS}}_{\infty,B}$, in view of
$y\in{\mathcal{IS}}_{\infty}$ and $u\in{\mathcal{IS}}_{B}^{\eqref{re}}$. From
Theorem 3.1 we know that

$|M-u|\in{\mathcal{SV}}\ \ \text{and}\ \
\frac{L_{\widetilde{p}}^{\alpha-1}(t)}{L_{\widetilde{r}}^{\alpha-1}(M-u(t))}=o(1)$

as $t\to\infty$. Noting that, by Lemma 4.1,

$\frac{L_{\widetilde{p}}^{\alpha-1}(t)}{L_{\widetilde{r}}^{\alpha-1}}\sim\left(\frac{t}{\tau(t)}\right)^{\delta+\alpha}(\tau^{\prime}(t))^{\alpha-1}\frac{L_{p}(t)}{L_{r}(t)}$

as $t\to\infty$, we get $|M-y^{[1]}|\in{\mathcal{SV}}$ and (3.15) easily
follow. The equalities among the subclasses follow from the relations
${\mathcal{IS}}\subseteq{\mathcal{S}}_{{\mathcal{NRV}}}(\varrho)\subseteq{\mathcal{S}}_{{\mathcal{RV}}}(\varrho)\subseteq{\mathcal{IS}}\subseteq{\mathcal{IS}}_{\infty,x}\subseteq{\mathcal{IS}}$,
$x=B$ or $x=\infty$ according to whether $\int_{a}^{\infty}H_{\tau}$ converges
or diverges, respectively; note a regularly varying solution of (1.1) with a
positive index is necessarily increasing. ∎

###### Proof of Theorem 3.6.

First note that, in view of Lemma 4.2-(ii), (1.2) implies (4.2) with $i=D$.
Take $y\in{\mathcal{IS}}$. Then $x(s)=y(t)$, with $s=R_{D}(t)$, satisfies
equation (4.6), where $\widehat{r}=1$, $\widehat{p}=p_{D}$,
$\widehat{\tau}=\tau_{D}$, and $x\in{\mathcal{IS}}^{\eqref{te}}$, see Lemma
4.4. Note that the interval $[a,\infty)$ is transformed into $[b,\infty)$ for
some $b$. Since $\widehat{r}\in{\mathcal{SV}}$ and
$\widehat{p}\in{\mathcal{RV}}(-\alpha)$, we can apply Theorem 3.4 on equation
(4.6); in fact, here $\delta=-\alpha<-1$. We get $x\in{\mathcal{NRV}}(1)$
since $\delta=-\alpha$ implies $\varrho=1$. Moreover,

$x(s)=s\exp\left\\{\int_{b}^{s}(1+o(1))(\beta-1)(u\widehat{\tau}^{\prime}(u))^{\alpha-1}\widehat{p}(u)\,{\mathrm{d}}u\right\\}$
(4.17)

as $s\to\infty$, with $x\in{\mathcal{IS}}_{\infty,\infty}^{\eqref{te}}$
provided
$\int_{b}^{\infty}(u\widehat{\tau}^{\prime}(u))^{\alpha-1}\widehat{p}(u)\,{\mathrm{d}}u=\infty$,
while

$x(s)=B+\int_{b}^{s}M^{\beta-1}\exp\left\\{-\int_{\xi}^{\infty}(1+o(1))(\beta-1)(u\widehat{\tau}^{\prime}(u))^{\alpha-1}\widehat{p}(u)\,{\mathrm{d}}u\right\\}\,{\mathrm{d}}\xi$
(4.18)

as $s\to\infty$, for some $B\in\mathbb{R}$, with
$x\in{\mathcal{IS}}_{\infty,B}^{\eqref{te}}$ provided
$\int_{b}^{\infty}(u\widehat{\tau}^{\prime}(u))^{\alpha-1}\widehat{p}(u)\,{\mathrm{d}}u=\infty$.
Since $\widehat{p}=p_{D}$ and $\widehat{\tau}=\tau_{D}$, substitutions
$v=R^{-1}_{D}(u)$ and $\eta=R^{-1}_{D}(\xi)$ in the integrals (4.17) and
(4.18) yield formulae (3.20) and (3.21). The fact that
$y\in{\mathcal{IS}}_{\infty,i}$, $i\in\\{B,\infty\\}$, follows from
$x\in{\mathcal{IS}}_{\infty,i}^{\eqref{te}}$ and (4.7), recalling that
$x(R^{-1}_{D}(s))=y(t)$, $\widehat{p}=p_{D}$, and $\widehat{\tau}=\tau_{D}$.
From $|M-x^{[1]}|\in{\mathcal{SV}}$ we obtain $|M-y^{[1]}\circ
R_{D}^{-1}|\in{\mathcal{SV}}$. In view of Lemma 4.2 and condition (3.19), we
have $\tau_{D}(s)\asymp s$ as $s\to\infty$. Hence, expressing relation (3.15)
in terms of our setting, we obtain

$\frac{(\widehat{\tau}^{\prime}(s))^{\alpha-1}L_{\widehat{p}}(s)}{M-x^{[1]}(s)}=o(1)$

as $s\to\infty$. Recalling that
$L_{\widehat{p}}(s)=s^{\alpha}\widehat{p}(s)=R_{D}^{\alpha}(t)p(t)r^{\beta-1}(t)$,
we get (3.22). ∎

###### Proof of Theorem 3.7.

The proof is similar to that of Theorem 3.6. Here, for $y\in{\mathcal{S}}$, we
make the substitution $x(s)=y(t)$ with $s=Q(t)$. In view of Lemma 4.4, $x$
satisfies equation (4.6), where
$\widehat{r}=s^{2\alpha-2}\in{\mathcal{RV}}(2\alpha-2)$,
$\widehat{p}=p_{C}\in{\mathcal{RV}}(\alpha-2)$, and $\widehat{\tau}=\tau_{C}$.
From Lemma 4.2-(ii), conditions (4.2) with $i=C$ holds. Since, in fact,
$\delta=\alpha-2>-1$, we can apply Theorem 3.1 to equation (4.6) where we
consider $x\in{\mathcal{IS}}^{\eqref{te}}$ (i.e., $y\in{\mathcal{IS}}$). The
details are left to the reader. ∎

## References

  * [1] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, Vol. 27, Cambridge University Press, 1987.
  * [2] O. Došlý, P. Řehák, Half-linear Differential Equations, Elsevier, North Holland, 2005.
  * [3] J. L. Geluk, On slowly varying solutions of the linear second order differential equation, in: Proceedings of the Third Annual Meeting of the International Workshop in Analysis and its Applications. Publ. Inst. Math. (Beograd) (N.S.) 48 (1990), 52–60.
  * [4] J. L. Geluk, L. de Haan, Regular Variation, Extensions and Tauberian Theorems, CWI Tract 40, Amsterdam, 1987.
  * [5] L. J. Grimm, L. M. Hall, Self adjoint differential equations and Karamata functions, in: Proceedings of the Third International Colloquium on Differential Equations (Plovdiv, 1992), 77–87, VSP, Utrecht, 1993.
  * [6] L. de Haan, On Regular Variation and its Applications to the Weak Convergence of Sample Extremes, Mathematisch Centrum Amsterdam, 1970\.
  * [7] J. Jaroš, T. Kusano, Self-adjoint differential equations and generalized Karamata functions, Bull. Cl. Sci. Math. Nat. Sci. Math. 29 (2004), 25–60.
  * [8] J. Jaroš, T. Kusano, T. Tanigawa, Nonoscillation theory for second order half-linear differential equations in the framework of regular variation, Results Math. 43 (2003), 129–149.
  * [9] T. Kusano, V. Marić, On a class of functional differential equations having slowly varying solutions, Publ. Inst. Math. (Beograd) (N.S.) 80 (2006), 207–217.
  * [10] T. Kusano, V. Marić, Slowly varying solutions of functional differential equations with retarded and advanced arguments, Georgian Math. J. 14 (2007), 301–314.
  * [11] T. Kusano, V. Marić, Regularly varying solutions to functional differential equations with deviating argument, Bull. Cl. Sci. Math. Nat. Sci. Math. 32 (2007), 105–128.
  * [12] T. Kusano, B. S. Lalli, On oscillation of half-linear functional-differential equations with deviating arguments, Hiroshima Math. J. 24 (1994), 549–563.
  * [13] J. Manojlović, T. Tanigawa, Regularly varying solutions of half-linear differential equations with retarded and advanced arguments, Math. Slovaca 65 (2015), 1361–1402.
  * [14] V. Marić, Regular Variation and Differential Equations, Lecture Notes in Mathematics 1726, Springer-Verlag, Berlin-Heidelberg-New York, 2000.
  * [15] F. Neuman, Transformation and canonical forms of functional-differential equations, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 349–357.
  * [16] M. Pituk, Oscillation of a linear delay differential equation with slowly varying coefficient, Appl. Math. Letters 73 (2017), 29–36.
  * [17] P. Řehák, On some methods in half-linear asymptotic theory, Electron. J. Differential Equations, 2016 (2016), No. 267, 1–27.
  * [18] P. Řehák, Asymptotic formulae for solutions of half-linear differential equations, Appl. Math. Comp 292 (2017), 165–177.
  * [19] P. Řehák, V. Taddei, Solutions of half-linear differential equations in the classes Gamma and Pi, Differen. Integral Equat. 29 (2016), 683–714.
  * [20] T. Tanigawa, Regularly varying solutions of half-linear functional differential equations with retarded arguments, Acta Math. Hungar. 120 (2008), 53–78.
  * [21] T. Tanigawa, Generalized regularly varying solutions of second order nonlinear differential equations with deviating arguments, Mem. Differ. Equ. Math. Phys. 57 (2012), 123–162.