Regular variation and differential equations: Theory and ... · Regular variation and differential equations: Theory and methods Pavel Rehákˇ Institute of Mathematics Czech Academy

Regular variation and differential equations:Theory and methods

Pavel Rehák

Institute of MathematicsCzech Academy of Sciences

External meeting IM CAS, Brno, November 2016

Pavel Rehák (IM CAS) RV and DEs External meeting, Brno 1 / 40

Structure of the talk

Regular variationI Karamata theoryI De Haan theory

Regular variation and differential equations

Selected resultsI Emden-Fowler type systemsI Half-linear differential equations


Theory of regularly varying functionsinitiated by J. Karamata (1930). But there are also earlier works ...study of relations such that

limt→∞

f (λt)f (t)

= g(λ) ∈ (0,∞), ∀λ > 0,

together with their applications (integral transforms – Tauberiantheorems, probability theory, number theory, complex analysis,differential equations, etc.)

N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, Encyclopedia ofMathematics and its Applications, Vol. 27, Cambridge University Press, 1987.

W. Feller, An Introduction to Probability Theory and Its Applications, vol. II, Wiley,NY, 1971.

J. L. Geluk, L. de Haan, Regular Variation, Extensions and Tauberian Theorems,CWI Tract 40, Amsterdam, 1987.

E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics 508,Springer-Verlag, Berlin-Heidelberg-New York, 1976.

...Pavel Rehák (IM CAS) RV and DEs External meeting, Brno 3 / 40

DefinitionA measurable function f : [a,∞)→ (0,∞) is called regularly varying(at∞) of index ϑ if for all λ > 0,

limt→∞

f (λt)f (t)

= λϑ.

[Notation: f ∈ RV(ϑ)]If ϑ = 0, then f is called slowly varying.[Notation: f ∈ SV][RV0 means regular variation at zero.]

The Uniform Convergence TheoremIf f ∈ RV(ϑ), then the relation in the definition holds uniformly on eachcompact λ-set in (0,∞).


Representation Theorem

f is regularly varying of index ϑ if and only if

f (t) = ϕ(t) exp

{∫ t

a

δ(s)

sds

}

where ϕ(t)→ const> 0 and δ(t)→ ϑ as t →∞.

f is regularly varying of index ϑ if and only if

f (t) = tϑϕ(t) exp

{∫ t

a

ψ(s)

sds

}

where ϕ(t)→ const> 0 and ψ(t)→ 0 as t →∞.

If ϕ(t) ≡ const, then f is said to be normalized regularly varying (f ∈ NRV).


Examples of (non-)SV functionsf is regularly varying of index ϑ if and only if

f (t) = tϑL(t),

where L ∈ SV.

∏ni=1(lni t)µi , where lni t = ln lni−1 t and µi ∈ R is SV function.

2 + sin(ln2 t) and (ln Γ(t))/t are SV functions.

1t

∫ ta

1ln s ds is SV function.

SV functions may exhibit “infinite oscillation” (i.e., lim inft→∞ L(t) = 0,lim supt→∞ L(t) =∞), for example, exp

{(ln t)

13 cos(ln t)

13

}.

2 + sin t , 2 + sin(ln t) are NOT SV functions.

exp t is NOT RV function.


Extension in a logical and useful manner of the class of functions whoseasymptotic behavior is that of a power function, to functions whereasymptotic behavior is that of a power function multiplied by a factorwhich varies “more slowly” than a power function.

RV ≡ generalized power laws;(asymptotic scale invariant property)f (λt) = g(λ)f (t) f (λt) ∼ g(λ)f (t) as t →∞

Regularly varying functions have a “good behavior” with respect tointegration and many other pleasant properties.

Regularly varying functions and related objects naturally occur indifferential equations, and are very useful in the study of their qualitativeproperties.

V. Maric, Regular Variation and Differential Equations, Lecture Notes inMathematics 1726, Springer-Verlag, 2000.

...


Other selected properties

If L1,L2 ∈ SV, then

Lα1 ∈ SV (with α ∈ R), L1L2 ∈ SV, L1 + L2 ∈ SV

and

L1 ◦ L2 ∈ SV (provided L2(t)→∞).

If L ∈ SV and ϑ > 0, then

tϑL(t)→∞, t−ϑL(t)→ 0 as t →∞.


Other selected properties

(Almost monotonicity) For a positive measurable function L it holds: L ∈ SVif and only if, for every ϑ > 0, there exist a (regularly varying)nondecreasing function F and a (regularly varying) nonincreasing functionG with

tϑL(t) ∼ F (t), t−ϑL(t) ∼ G(t) as t →∞.

(Asymptotic inversion) If g ∈ RV(ϑ) with ϑ > 0, then there existsg ∈ RV(1/ϑ) with

f (g(t)) ∼ g(f (t)) ∼ t as t →∞.

Here g (an “asymptotic inverse” of f ) is determined uniquely to withinasymptotic equivalence. One version of g is the generalized inversef←(t) := inf{s ∈ [a,∞) : f (s) > t}.


Other selected properties (Karamata’s theorem)

(Karamata’s theorem; direct half)(i) If L ∈ SV and γ < −1, then∫ ∞

tsγL(s) ds ∼ 1

−γ − 1tγ+1L(t) as t →∞.

(ii) If L ∈ SV and γ > −1, then∫ t

asγL(s) ds ∼ 1

γ + 1tγ+1L(t) as t →∞.

(iii) The integral∫∞

a L(s)/s ds may or may not converge. The functionL(t) =

∫ ta L(s)/s ds or L(t) =

∫∞t L(s)/s ds is a new SV function and such

that L(t)/L(t)→ 0 as t →∞.

(Karamata’s theorem; converse half)Opposite directions in (i) and (ii).


Rapid variation

DefinitionA measurable function f : [a,∞)→ (0,∞) is called rapidly varying of index∞, we write f ∈ RPV(∞), if

limt→∞

f (λt)f (t)

=

{0 for 0 < λ < 1,∞ for λ > 1,

and is called rapidly varying of index −∞, we write f ∈ RPV(−∞), if

limt→∞

f (λt)f (t)

=

{∞ for 0 < λ < 1,0 for λ > 1.

The class of all rapidly varying solutions is denoted as RPV.

While RV functions behaved like power functions (up to a factor which varies“more slowly”), RPV functions have a behavior close to that of exponentialfunctions.Regularly bounded functions, Zygmund class, ..., RV in other settings, ...


De Haan theory, class Π

DefinitionA measurable function f ∈ [a,∞)→ R is said to belong to the class Πif there exists a function w : (0,∞)→ (0,∞) such that for all λ > 0,

limt→∞

f (λt)− f (t)w(t)

= lnλ;

we write f ∈ Π or f ∈ Π(w). The function w is called an auxiliaryfunction for f .

The class Π of functions f is, after taking absolute values, a propersubclass of SV.

The de Haan theory is both a direct generalization of the Karamata theoryand what is needed to fill certain gaps, or boundary cases, in Karamata’smain theorem.


Class Π – selected properties

If f ∈ Π, then for 0 < c < d <∞ the relation in the definition holdsuniformly for λ ∈ [c,d ].

If f ∈ Π(L), then

L(t) ∼ f (t)− 1t

∫ t

af (s) ds

as t →∞.

...


De Haan theory, class Γ

DefinitionA nondecreasing function f : R→ (0,∞) is said to belong to theclass Γ if there exists a function v : R→ (0,∞) such that for all λ ∈ R

limt→∞

f (t + λv(t))

f (t)= eλ;

we write f ∈ Γ or f ∈ Γ(v). The function v is called an auxiliary functionfor f .A function is said to belong to the class Γ−(v) if 1/f ∈ Γ(v).

It holds: Γ± ⊂ RPV(±∞)

“Inversion” of Π, but there are also other way of understanding.


Class Γ – selected properties

If f ∈ Γ, then the relation in the definition holds uniformly on compactλ-sets.

f ∈ Γ if and only if

limt→∞

f (t)∫ t

0

∫ s0 f (τ) dτ ds(∫ t

0 f (s) ds)2 = 1.

If f ∈ Γ(v), thenv(t + λv(t)) ∼ v(t)

as t →∞ locally uniformly in λ ∈ R (i.e., v ∈ SN ; self-neglecting).

...


De Haan theory, Beurling slow variation

DefinitionA measurable function f : R→ (0,∞) is Beurling slowly varying if

limt→∞

f (t + λf (t))

f (t)= 1 for all λ ∈ R;

we write f ∈ BSV.

DefinitionIf the relation holds locally uniformly in λ, then f is calledself-neglecting; we write f ∈ SN .


Class BSV – selected properties

If f ∈ BSV is continuous, then f ∈ SN .

f ∈ SN if and only if it has the representation

f (t) = ϕ(t)∫ t

0ψ(s) ds,

where limt→∞ ϕ(t) = 1 and ψ is continuous with limt→∞ ψ(t) = 0.

If f ∈ BSV is continuous, then there exists g ∈ C1 such that f (t) ∼ g(t)and g′(t)→ 0 as t →∞.

...


Regular variation and DEsTheory of RV is “non-trivially” used; assumptions, statements, proofs.

Remark: In dozens of works on DEs, their authors work with “regularvariation” without realizing it.

Linear and half-linear DEsGeluk, Grimm, Hall, Jaroš, Kusano, Maric, Omey, R., Taddei,Tanigawa, TomicQuasilinear DEs and systems (equations of Emden-Fowler,Lane-Emden, Thomas-Fermi types)Avakumovic, Evtukhov, Jaroš, Kharkov, Kusano, Manojlovic,Maric, Matucci, Radasin, R., Taliaferro, TanigawaPDEsCîrstea, Radulescu, Zhang (uniqueness and asympt. behavior forsolutions of nonlinear elliptic problems with boundary blow up);Niethammer, Pego (Ostwald ripening); Taliaferro (almost radialsymmetry)


Regular variation and DEs

Stochastic (P)DEs and Volterra equationsAppleby, Debussche, Högele, Imkeller, PattersonDifference equationsGanelius, Geronimo, Manojlovic, Kharkov, Kooman, Matucci, R.,Smith, Van Assche(incl. orthogonal polynomials)q-difference equations and dynamic equations on time scales(dependence on the graininess)R., Vítovec... Saari (n-body problem), Mijajlovic, Segan (Friedmannequations), ...


Emden-Fowler type system

Consider the quasilinear (or the generalized Emden-Fowler) equation

(Φα(y ′))′ = p(t)Φβ(y), Φλ(u) = |u|λ sgn u, (E)

where α, β > 0, p(t) > 0.

Basic classification of nonoscillatory solutions (with respect to the limitbehavior as t →∞ of a solution and its quasiderivative).

Problems of (non)existence in the classes.

A more precise description of asymptotic behavior of solutions (alongwith asymptotic formulae). In particular, we are interested in somehowdifficult classes, e.g., the strongly increasing solutions (SIS), i.e.,limt→∞ y(t) =∞ = limt→∞ y ′(t).


(Φα(y ′))′ = p(t)Φβ(y), Φλ(u) = |u|λ sgn u, (E)

Kamo and Usami (2000, 2001) assumed p(t) ∼ tσ and α > β > 0. Theyestablished conditions (in terms of α, β, σ) guaranteeing that SISsolutions y of (E) have the form

y(t) ∼ Ktγ , where K = K (α, β, σ), γ = γ(α, β, σ).

The key role was played by the asymptotic equivalence theorem whichsays, roughly speaking:If the coefficients of two equations of the form (E) are asymptoticallyequivalent, then their solutions which are “of the same class” are alsoasymptotically equivalent.

An extension of the classical results by Bellman, who assumed p(t) = tσ.


(Φα(y ′))′ = p(t)Φβ(y), Φλ(u) = |u|λ sgn u, p(t) ∼ tσ

We are interested in a generalization

A regularly varying coefficientA general coefficient in the differential termRegularly varying nonlinearitiesCoupled systemsSecond order systems of k equations (even-order scalarequations)First order systems of n equations (n may be even or odd)

– An asymptotic equivalence theorem cannot be used.

– The theory of RV in combination with some other tools finds theapplication.

– Much wider class of equations can be included.


Nonlinear system (of Emden-Fowler type)

x ′1 = a1(t)F1(x2),

x ′2 = a2(t)F2(x3),...

x ′n−1 = an−1(t)Fn−1(xn),

x ′n = an(t)Fn(x1),

(S)

n ∈ N, n ≥ 2.ai are continuous, eventually of one sign, and

|ai | ∈ RV(σi), σi ∈ R, i = 1, . . . ,n,

Fi are continuous with uFi(u) > 0 for u 6= 0, and

Fi(|·|) ∈ RV(αi), resp. Fi(|·|) ∈ RV0(αi), αi ∈ (0,∞), i = 1, . . . ,n.

A modification of our approach works also for some (moregeneral) perturbed systems.


Special cases – n-th order two term nonlinear DE’s

x (n) = p(t)Φβ(x)

or, more general,

Dqn (γn)Dqn−1(γn−1) · · ·Dq1(γ1)x(t) = p(t)Φβ(x),

where Dqi (γi)x(t) = ddt (qi(t)Φγi (x))

The order can be even as well as odd.


Special cases – equations with a generalizedLaplacian and/or an RV nonlinearity on the RHSFor example, the second order equation

(r(t)G(x ′))′ = p(t)F (x),

with

- a generalized Laplacian (may include the classical p-Laplacian operatoror the curvature operator or the relativity operator or ...)

- a regularly varying nonlinearity on the right-hand side

Typical examples of nonlinearities (we do not require monotonicity):

Fi (u) = Φα(u) = |u|α sgn u (for second order equations or systems itmay lead to the classical Laplacian operator).

Fi (u) = Φα(u)L(u), with α > 0, where L(u)→ c ∈ (0,∞), orL(u) = | ln u|γ1 | ln | ln u||γ2 .

Fi (u) = uα(A + Buβ)γ ; a special choice yieldsu√

1 + u2or

u√1− u2

.


Partial differential systemsSystem (S) includes also second order nonlinear systems of the form

(A1(t)Φλ1 (y ′1))′ = B1(t)G1(y2),

(A2(t)Φλ2 (y ′2))′ = B2(t)G2(y3),...

(Ak (t)Φλk (y ′k ))′ = Bk (t)Gk (y1),

which play important role in the study of positive radial solutions to the partialdifferential system

div(‖∇u1‖λ1−1∇u1) = ϕ1(‖z‖)G1(u2),

div(‖∇u2‖λ2−1∇u2) = ϕ2(‖z‖)G2(u3),...

div(‖∇uk‖λk−1∇uk ) = ϕk (‖z‖)Gk (u1).

Systems of Lane-Emden type.


Extremal solutions

x ′i = ai (t)Fi (xi+1), i = 1, . . . ,n, (S)

xn+1 means x1.

Conditions on coefficients: ai ∈ RV(σi ), i = 1, . . . ,n.

Conditions on nonlinearities: Fi ∈ RV(αi );nonlinearities do not need to be monotone.

“Subhomogeneity”: α1 · · ·αn < 1

Let us examine, for example, the class

SIS ={

(x1, . . . , xn) ∈ IS : limt→∞

xi (t) =∞, i = 1, . . . ,n}

;

the so-called strongly increasing solutions (or fast growing solutions).


The constants νi and hi and SV functions Li depend on the coefficients andthe nonlinearities; they can be explicitly computed.

TheoremIf νi > 0, i = 1, . . . ,n, then there exists

(x1, . . . , xn) ∈ SIS ∩ (RV(ν1)× · · · × RV(νn))

and (for every such a solution)

xi (t) ∼ hi tνi Li (t) as t →∞, i=1,. . . ,n. (AF)

If νi > 0 and , in addition, Fi = Φαi , i = 1, . . . ,n, then SIS 6= ∅ and forEVERY (x1, . . . , xn) ∈ SIS, one has

(x1, . . . , xn) ∈ RV(ν1)× · · · × RV(νn)

and (AF) holds with LF1 ≡ · · · ≡ LFn ≡ 1, LFi (t) = t−αi Fi (t), i = 1, . . . ,n.

νi = νi (σ1, . . . , σn, α1, . . . , αn), hi = hi (σ1, . . . , σn, α1, . . . , αn),

Li = Li (σ1, . . . , σn, α1, . . . , αn,La1 , . . . ,Lan ,LF1 , . . . ,LFn ).Pavel Rehák (IM CAS) RV and DEs External meeting, Brno 28 / 40

Sketch of the proof (the first part)

Properties of RV functions are frequently used ...

The Schauder-Tychonoff fixed point theorem: We obtain a solution(x1, . . . , xn) ∈ SIS such that ci tνi Li (t) ≤ xi (t) ≤ di tνi Li (t) for someconstants ci ,di , i = 1, . . . ,n.

lim inft→∞ xi (λt)/xi (t), lim supt→∞ xi (λt)/xi (t) ∈ (0,∞).

The uniform convergence theorem and some other tools yield thatlimt→∞ xi (λt)/xi (t) exists; in fact,

lim supt→∞

xi (λt)/xi (t) ≤ λνi ≤ lim inft→∞

xi (λt)/xi (t).

Thus, RV follows.

Playing with asymptotic relations and the Karamata theorem yield theasymptotic formula.


Sketch of the proof (the second part)

SIS 6= ∅ follows from the previous part. We take an arbitrary(x1, . . . , xn) ∈ SIS .

Several auxiliary (quite) technical results, the generalized AM-GMinequality and some other estimates, and the properties of RV functionsyield ci tνi Li (t) ≤ xi (t) ≤ di tνi Li (t), i = 1, . . . ,n.

xi ∈ RV(νi ) and the asymptotic formula follow by similar arguments as inthe first part.


We get back the Kamo-Usami result as a very special case.

In other (“less special”) cases the results are new; even when ai (t) ∼ tσi

(for higher order) or for second order equations (with RV coefficients).

For example, let p(t) = t%Lp(t) with Lp ∈ SV and 0 < β < 1. If%+ 1 + β(n − 1) > 0, then the equation

x (n) = p(t)Φβ(x)

possesses a solution x such that limt→∞ x (i)(t) =∞, i = 0, . . . ,n − 1,and for any such a solution there hold

x ∈ RV(%+ n1− β

)and x1−β(t) ∼ t%+nLp(t)

n∏j=1

1− β%+ n − (1− β)(j − 1)

.

Regular variation of all these solutions is normalized.

We can treat also some equations with non-RV coefficients via asuitable transformation.

Other types of solutions (decreasing, ...).

Other types of equations (perturbed equations, singular equations,superlinearity, ...)


Half-linear differential equations

(r(t)Φ(y ′))′ = p(t)Φ(y), (HL)

where r ,p ∈ C([a,∞)), r(t) > 0, Φ(u) = |u|α−1 sgn u with α > 1.

The solution space is homogeneous but not additive.

Bihari (1957, 1964), Elbert (1979), Mirzov (1976), ... dozens of authors... O. Došlý, P. R., Half-Linear Differential Equations, Elsevier, NorthHolland, 2005.

If α = 2, then (HL) reduces to a linear Sturm-Liouville equation.

Radially symmetric solutions of a certain PDE with p-Laplacian. If thedimension N = 1, then (PDE) ≡ (HL).

Special (bordeline) case of quasilinear (generalized Emden–Fowler)DEs .

α-degree functionals (variational principle, Hardy type inequalities, ...)


Solutions in the class Γ

(Φ(y ′))′ = p(t)Φ(y), (HL)

where p ∈ C([a,∞)), p(t) > 0, Φ(u) = |u|α−1 sgn u with α > 1.

TheoremIf

p−1α ∈ BSV,

then all solutions of (HL) belongs to Γ(v) or Γ−(v), where

v =

(α− 1

p

) 1α

.


Sketch of the proof(here only for increasing solutions; decreasing solutions are moredifficult)

First assume p ∈ C1. Then (p−1α )′(t)→ 0 as t →∞.

Take y such that y > 0, y ′ > 0. Set w = p−1β Φ(y ′/y).

w satisfies the generalized Riccati equation

w ′

p1α (t)

= 1− (α− 1)w

(p′(t)

αpα+1α (t)

+ wβ−1

)(β is the conjugate number to α).

w satisfies limt→∞w(t) = (α− 1)−1β .

y satisfiesy ′′(t)y(t)

y ′2(t)∼ 1

y ∈ Γ

((α−1

p

) 1α

).


Sketch of the proof (continuation)

We drop the assumption on differentiability of p.Since p ∈ BSV there exists p ∈ C1 with p(t) ∼ p(t) and(p−

1α )′(t)→ 0 as t →∞.

For ε ∈ (0,1) we consider the auxiliary equations(Φ(u′))′ = (1 + ε)p(t)Φ(u) and (Φ(v ′))′ = (1− ε)p(t)Φ(v).For increasing solutions u, v we show(

u′(t)u(t)

)α−1

p−1β (t) ∼

(1 + ε

α− 1

) 1β

,

(v ′(t)v(t)

)α−1

p−1β (t) ∼

(1− εα− 1

) 1β

.

From the theorem on differential inequalities,

limt→∞

(y ′(t)y(t)

)α−1p−

1β (t) = (α− 1)−

1β .

The rest of the proof is the same as in the previous part.


Remarks

A part of the statement is a half-linear extension of known result.The other is new also in the linear case. (The Wronskian identity,reduction of order are not at disposal, ...)

By-product: Condition for rapid variation of all solutions (openproblem solved).

By-product: There are solutions y1, y2 such that

y ′i (t) ∼ ±(

p(t)α− 1

) 1α

yi(t).

Half-linear extension of Hartman’s-Wintner’s result.

Another generalization can be made.


Complete classification of solutions

(r(t)Φ(y ′))′ = p(t)Φ(y), (HL)Karamata theory, de Haan theory, generalized Riccati technique, reciprocityprinciple, asymptotic linearization, and other tools are used to obtain

complete classification of solutionsin the framework of regular variation.

— Relations among the classes; description of the structure of the solutionspace.— Asymptotic formulae for ALL solutions.— Interesting phenomena (that are not known from the linear case) arerevealed— Quite general setting that enables us to cover many important cases.— SV components of the coefficients play an important role; more complexstructure than with “asymptotically-power” coefficients.— Sufficiency and necessity of the conditions.— New also in the linear case.

An example of the result:Pavel Rehák (IM CAS) RV and DEs External meeting, Brno 37 / 40

Consider the equation (Φ(y ′))′ = p(t)Φ(y).DS: eventually positive decreasing solutions;DSA,B: DS solutions with y → A, y ′ → B; S(N )SV : (N )SV-solutions.

TheoremLet p ∈ RV(−α). If Lp(t)→ 0 as t →∞, where Lp is SV component of p,then DS ⊂ NSV. For y ∈ DS, one has −y ∈ Π(−ty ′(t)). Moreover,(i) If

∫∞a (sp(s))

1α−1 ds =∞, then

y(t) = exp

{−∫ t

a

(sp(s)

α− 1

) 1α−1

(1 + o(1)) ds

}

as t →∞, and SSV = SNSV = DS = DS0,0.(ii) If

∫∞a (sp(s))

1α−1 ds <∞, then

y(t) = y(∞) exp

{∫ ∞t

(sp(s)

α− 1

) 1α−1

(1 + o(1)) ds

}

as t →∞, and SSV = SNSV = DS = DSB,0, B > 0.In addition, |y(∞)− y(t)| ∈ SV, Lβ−1

p (t)/(y(∞)− y(t)) = o(1).Pavel Rehák (IM CAS) RV and DEs External meeting, Brno 38 / 40

Sketch of the proof

Take y ∈ DS.

We show that ty ′(t)/y(t)→ 0 as t →∞, which implies y ∈ NSV.

From −Φ(y ′) ∈ RV(1− α), we get −y(t) ∈ Π(−ty ′(t))

Set h(t) = tα−1Φ(y ′(t))− (α− 1)∫ t

a sα−2Φ(y ′(s)) ds.

Then h ∈ Π(−(α− 1)tα−1Φ(y ′(t))) and h ∈ Π(th′(t)).

From the uniqueness of the auxiliary function up to asymptoticequivalence and h′(t) = tα−1p(t)Φ(y(t)), we gety ′(t)y(t) = −(1 + o(1))p(t), where p(t) =

(tp(t)α−1

) 1α−1 as t →∞.

We distinguish the cases∫∞ p(s) ds =∞ and

∫∞ p(s) ds <∞.Then we integrate the asymptotic relation to obtain formulas.Moreover, we find y(t)→ 0 resp. y(t)→ y(∞) ∈ (0,∞).

Properties of RV and Π functions yield the rest.


S. Matucci, P. Rehák, Extremal solutions to a system of n nonlineardifferential equations and regularly varying functions, MathematischeNachrichten 288 (2015), 1413-1430.

P. Rehák, On certain asymptotic class of solutions to second order linearq-difference equations, J. Phys. A: Math. Theor. 45 (2012) 055202.

P. Rehák, Nonlinear Differential Equations in the Framework of RegularVariation, AMathNet 2014, 207 pp.

P. Rehák, Asymptotic formulae for solutions of half-linear differentialequations, Appl. Math. Comp. 292 (2017), 165–177.

P. Rehák, V. Taddei, Solutions of half-linear differential equations in theclasses Gamma and Pi, Differ. Integral Equ. 29 (2016), 683–714.

P. Rehák, J. Vítovec, Regular variation on measure chains, NonlinearAnalysis 72 (2010), 439–448.


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Regular variation and differential equations: Theory and ... · Regular variation and differential equations: Theory and methods Pavel Rehákˇ Institute of Mathematics Czech Academy