Regular variation on measure chains

Nonlinear Analysis 72 (2010) 439–448

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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

Regular variation on measure chains Pavel Řehák a,∗ , Jiří Vítovec b a

Institute of Mathematics, Academy of Sciences of the Czech Republic, Žižkova 22, 61662 Brno, Czech Republic

b

Department of Mathematical Analysis, Faculty of Science, Masaryk University Brno, Kotlářská 2, 61137 Brno, Czech Republic

article

info

Article history: Received 11 November 2008 Accepted 18 June 2009 MSC: 26A12 26A99 34C11 39A11 39A12

abstract In this paper we show how the recently introduced concept of regular variation on time scales (or measure chains) is related to a Karamata type definition. We also present characterization theorems and an embedding theorem for regularly varying functions defined on suitable subsets of reals. We demonstrate that for a ‘‘reasonable’’ theory of regular variation on time scales, certain additional condition on a graininess is needed, which cannot be omitted. We establish a number of elementary properties of regularly varying functions. As an application, we study the asymptotic properties of solution to second order dynamic equations. © 2009 Elsevier Ltd. All rights reserved.

Keywords: Regularly varying function Regularly varying sequence Measure chain Time scale Embedding theorem Representation theorem Second order dynamic equation Asymptotic properties

1. Introduction Recall that a measurable function f : [a, ∞) → (0, ∞) is said to be regularly varying of index ϑ , ϑ ∈ R, if it satisfies lim

x→∞

f (λx) f (x)

= λϑ for all λ > 0;

(1)

we write f ∈ RVR (ϑ). If ϑ = 0, then f is said to be slowly varying. Fundamental properties of regularly varying functions are that relation (1) holds uniformly on each R x compact λ-set in (0, ∞) and f ∈ RVR (ϑ) if and only if it may be written in the form f (x) = ϕ(x)xϑ exp a η(s)/s ds , where ϕ and η are measurable with ϕ(x) → C ∈ (0, ∞) and η(x) → 0 as x → ∞; see [1–4]. In the basic theory of regularly varying sequences two main approaches are known. First, the approach by Karamata [5], based on a counterpart of the continuous definition: A positive sequence {fk }, k ∈ {a, a + 1, . . .} ⊂ Z, is said to be regularly varying of index ϑ , ϑ ∈ R, if lim

k→∞

∗

f[λk] fk

= λϑ for all λ > 0,

Corresponding author. E-mail addresses: [email protected] (P. Řehák), [email protected] (J. Vítovec).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.06.078

(2)

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P. Řehák, J. Vítovec / Nonlinear Analysis 72 (2010) 439–448

where [u] denotes the integer part of u. Second, the approach by Galambos and Seneta [6], based on a purely sequential definition: A positive sequence {fk } is said to be regularly varying of index ϑ if there exists a positive sequence {αk } satisfying fk ∼ C αk and limk→∞ k (1 − αk−1 /αk ) = ϑ, C being a positive constant. In [7] it was shown that these two definitions are equivalent. In [8] we suggest to replace (equivalently) the second condition in the latter definition by limk→∞ kn∆αk /αk = oϑ . A regularly varying sequence can be represented as fk = ϕk kϑ

Qk−1 j=a

1 + ψj /j , see [8], or as fk = ϕk kϑ exp




Pk−1 j =a

ψj /j ,

where ϕk → C ∈ (0, ∞) and ψk → 0 as k → ∞, see [6,7]. For further reading on the discrete case we refer, e.g., to [9]. Recall that the theory of regular variation can be viewed as the study of relations similar to (1) or (2), together with their wide applications, see, e.g., [1,2,4,8,10–12]. There is a very practical way how regularly varying functions can be understood: Extension in a logical and useful manner of the class of functions whose asymptotic behavior is that of a power function, to functions where asymptotic behavior is that of a power function multiplied by a factor which varies ‘‘more slowly’’ than a power function. In [6,7], see also [13], the so-called embedding theorem was established (and the converse result holds as well): If {yk } is a regularly varying sequence, then the function R (of a real variable), defined by R(x) = y[x] , is regularly varying. Such a result makes it then possible to apply the continuous theory to the theory of regularly varying sequences. However, the development of a discrete theory, analogous to the continuous one, is not generally close, and sometimes far from a simple imitation of arguments for regularly varying functions, as noticed and demonstrated in [7]. Simply, the embedding theorem is just one of powerful tools, but sometimes it is not immediate that from a continuous results its discrete counterpart is easily obtained thanks to the embedding; sometimes it is even not possible to use this tool and the discrete theory requires a specific approach, different from the continuous one. Recall that the calculus on time scales (or, more generally, on measure chains) deals essentially with functions defined on nonempty closed subsets of R, see [14,15]. Hence, it unifies and extends usual calculus and quantum (q- or h-) calculi. Also it helps to describe and understand discrepancies between individual cases. A theory of regular variation on time scales offers something more than the embedding result, and has the following advantages: Once there is proved a result on a general time scale, it automatically holds for the continuous and the discrete case, without any other effort. Moreover, at the same time, the theory works also on other time scales which may be different from the ‘‘classical’’ ones. In [16] we introduced the concept of regular variation on time scales, the form of which is motivated by a modification of the purely sequential criterion mentioned above. There we also derived a simple characterization of regularly varying functions, and investigated regularly varying behavior of decreasing solutions to a second order linear dynamic equation. In the present paper we show how the definition from [16] is connected to a Karamata type definition, where the latter one is motivated by (1) and (2). Further, under certain conditions, we establish an embedding result which relates a general time scales theory with the continuous theory. We also derive two important representation formulas and state a number of useful elementary properties of regularly varying functions on time scales. We show that to obtain a reasonable theory, from a certain point of view, we need an additional requirement on the graininess of a time scale, which cannot be improved. Roughly speaking, if the graininess is sufficiently small (µ(t ) = o(t )), then we get a continuous like (or a discrete like) theory. If the graininess is as in the q-calculus, then, as shown in [17], not only that basic results are differently looking, but in some important aspects we got a surprising simplifications comparing with the ‘‘classical’’ theories. Finally, for graininesses which are somehow large, the theory fails. As an application, we study the asymptotic properties of solutions of second order dynamic equations. 2. Preliminaries We assume that the reader is familiar with the notion of time scales. Thus note just that T, σ , f σ , µ, f ∆ , a f ∆ (s) ∆s, and ef (t , a) stand for time scale, forward jump operator, f ◦ σ , graininess, delta derivative of f , delta integral of f from a to b, and generalized exponential function, respectively. Recall that the concept of integration on time scales can be developed in various usual manners, e.g., Newton, or Riemann, or Lebesgue. See [14], which is the initiating paper of the time scale theory, and the monograph [15] containing a lot of information on time scale calculus. Before we give the first definition, note that in some parts below the conditions on smoothness can be somehow relaxed. But we do not do it since our theory focuses on a generalization in the sense of a ‘‘domain of definition’’ rather than considering ‘‘badly behaving’’ functions. In this paper, T is assumed to be unbounded above. In [16] we introduced the concept of regular variation on T in the following way.

Rb

Definition 1. A measurable function f : T → (0, ∞) is said to be regularly varying of index ϑ , ϑ ∈ R, if there exists a positive rd-continuously delta differentiable function α satisfying f (t ) ∼ C α(t )

and

lim

t →∞

t α ∆ (t )

α(t )

= ϑ,

(3)

C being a positive constant; we write f ∈ RV T (ϑ). If ϑ = 0, then f is said to be slowly varying, we write f ∈ SV T . Using elementary properties of linear first order dynamic equations and generalized exponential functions, the following representation was established in [16]: f ∈ RV T (ϑ) if and only if it has the representation f (t ) = ϕ(t )eδ (t , a),

(4)

P. Řehák, J. Vítovec / Nonlinear Analysis 72 (2010) 439–448

441

where ϕ is a positive function tending to a positive constant and δ is an rd-continuous positively regressive function satisfying limt →∞ t δ(t ) = ϑ . 3. Main results We start with a representation theorem. Theorem 1. (i) Let µ(t ) = o(t ). It holds f ∈ RVT (ϑ) if and only if it has a representation f (t ) = ϕ(t )t ϑ eη (t , a)

(5)

where ϕ is a positive measurable function tending to a positive constant and an rd-continuous η satisfies limt →∞ t η(t ) = 0. If ϑ = 0, then the condition µ(t ) = o(t ) can be omitted and (5) coincides with representation (4). (ii) Let µ(t ) = o(t ). It holds f ∈ RVT (ϑ) if and only if it has a representation ϑ

f (t ) = ϕ(t )t exp

Z

t

ψ(s) s

a



∆s ,

(6)

where ϕ is a positive measurable function tending to a positive constant and an rd-continuous ψ satisfies limt →∞ ψ(t ) = 0. If ϑ = 0, then the condition µ(t ) = o(t ) can be replaced by µ(t ) = O(t ). Proof. We show the implications (4) ⇒ (5) ⇒ (6) ⇒ (3). From (4), f (t ) ∼ C1 t ϑ L(t ), where C1 > 0 and L(t ) = eδ (t , a)t −ϑ . Consequently, tL∆ (t )

= t δ(t )

L(t )



ϑ

t

σ (t )

−ϑ



t

σ (t )

ξ (t ) σ (t )

ϑ−1

= o(1)

as t → ∞, where t ≤ ξ (t ) ≤ σ (t ), since 1 ≤ ξ (t )/t ≤ 1 + µ(t )/t = 1 + o(1). Hence L ∈ SV T , and so L(t ) = C2 eη (t , a) with C2 > 0 and limt →∞ t η(t ) = 0. This o n n implies (5). o From (5) we have f (t ) ∼ C3 t ϑ exp

Rt ψ(s)/s ∆s H (t ), where C3 > 0, H (t ) = e ψ (t , a) exp − a ψ(s)/s ∆s , and t nR o t limt →∞ ψ(t ) = 0. We show that limt →∞ H (t ) = 1. We have e ψ (t , a) = exp a ξµ(s) (ψ(s)/s) ∆s , where Rt a

t

ξµ(t )



ψ(t )



t

 =

ln (µ(t )ψ(t )/t + 1) /µ(t ) ψ(t )/t

for µ(t ) > 0 for µ(t ) = 0.

In view of the equalities limx→0 ln(x + 1)/x = 1 and limt →∞ µ(t )ψ(t )/t = 0, we get ξµ(t ) (ψ(t )/t ) ∼ ψ(t )/t. Consequently, limt →∞ H (t ) = 1, and so f has representation (6). nR o t From (6) we have f (t ) = ϕ(t )α(t ), where α(t ) = t ϑ exp a ψ(s)/s ∆s with limt →∞ ψ(t ) = 0. Then, at a rightscattered t, t α ∆ (t )

α(t )

=

(t ϑ )∆ t ϑ−1

 +

σ (t ) t

ϑ

exp(µ(t )ψ(t )/t ) − 1

µ(t )ψ(t )/t

ψ(t ),

while, at a right-dense t, t α ∆ (t )/α(t ) = ϑ + ψ(t ). Thanks to µ(t ) = o(t ) (resp. µ(t ) = O(t ) for ϑ = 0) we get limt →∞ t α ∆ /α(t ) = ϑ , and so f satisfies (3).  From the last theorem, f ∈ RVT (ϑ) iff f (t ) = t ϑ L(t ), where L ∈ SVT . Next we prove that Definition 1 implies the following Karamata type definition. Definition 2 (Karamata Type Definition). A measurable function f : T → (0, ∞) satisfying lim

t →∞

f (τ (λt )) f (t )

= λϑ

(7)

uniformly on each compact λ-set in (0, ∞), where τ : R → T is defined as τ (t ) = max{s ∈ T : s ≤ t }, is said to be regularly varying of index ϑ (ϑ ∈ R) in the sense of Karamata. We write f ∈ KRV T (ϑ). Theorem 2. Let µ(t ) = o(t ). If f ∈ RVT (ϑ), then f ∈ KRVT (ϑ). If ϑ = 0, then the condition µ(t ) = o(t ) can be replaced by µ(t ) = O(t ).

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P. Řehák, J. Vítovec / Nonlinear Analysis 72 (2010) 439–448

Proof. We want to show that (7) holds uniformly for λ from a compact subinterval [c , d] of (0, ∞). We confine our attention to λ ≥ 1 (i.e., 1 ≤ c < d); the case λ ∈ (0, 1) being handled similarly. From Theorem 1 we have (6). Hence, f (τ (λt )) f (t )

ϕ(τ (λt )) = ϕ(t )



τ (λt )

ϑ

τ (λt )

Z exp

t

ψ(s) s

t



∆s .

(8)

Clearly, ϕ(τ (λt ))/ϕ(t ) → 1 as t → ∞ uniformly for λ ∈ [c , d]. To prove that (τ (λt )/t )ϑ → λϑ as t → ∞ uniformly on the λ-set [c , d], it is sufficient to show that supλ∈[c ,d] |λt /τ (λt ) − 1| → 0 as t → ∞. First note that for x ∈ R, x ≥ a, τ (x) ≤ x ≤ σ (τ (x)), and thus 1 ≤ x/τ (x) ≤ 1 + µ(τ (x))/τ (x) = o(1) as x → ∞. We have

  λt µ(τ (λt )) µ(τ (Λ(t )t )) σ (τ (λt )) − 1 ≤ sup − 1 ≤ sup ≤ τ (λt ) τ (ct ) λ∈[c ,d] τ (λt ) λ∈[c ,d] τ (λt ) λ∈[c ,d] µ(τ (Λ(t )t )) τ (Λ(t )t ) µ(τ (Λ(t )t )) τ (dt ) = · ≤ · = o(1) τ (Λ(t )t ) τ (ct ) τ (Λ(t )t ) τ (ct ) sup

as t → ∞, where Λ : T → [c , d] is a suitable function. The uniform convergence to 1 of the last term in (8) follows from τ (λt )

Z

sup

λ∈[c ,d]

ψ(s) s

t

Z ∆s ≤ sup

τ (λt )

|ψ(s)| s

λ∈[c ,d] t

≤ (τ (dt ) − t ) sup

∆s ≤

t

|ψ(s)|

s ≥t

τ (dt )

Z

s

|ψ(s)| s

≤ t (d − 1) sup s≥t

∆s

|ψ(s)| s

≤ (d − 1) sup |ψ(s)| = o(1) s≥t

as t → ∞.



Before showing that the Karamata type definition makes an embedding possible, and, consequently, implies Definition 1, we prove a useful lemma. Lemma 1. Let µ(t ) = O(t ). If f ∈ KRVT (ϑ), then f σ (t )/f (t ) → 1 as t → ∞. Proof. If µ(t ) = O(t ), then M ∈ N exists such that 0 ≤ µ(t )/t ≤ M − 1 for all t ∈ T. Hence 1 ≤ (t + µ(t ))/t ≤ M and thus 1 ≤ σ (t )/t ≤ M

for all t ∈ T

(9)

and 1/M ≤

p

t /σ (t ) ≤ 1

for all t ∈ T.

(10)

We distinguish two cases. (i) ϑ = 0. By Definition 2 lim f (τ (Λ(t )t ))/f (t ) = 1

t →∞

where Λ : T → R is a bounded function. Thanks to (9) we may take the function Λ(t ) = σ (t )/t. Then we get 1 = lim

t →∞

f (τ (t σ (t )/t )) f (t )

= lim

f σ (t )

t →∞

f (t )

.

(ii) ϑ 6= 0. From Definition 2 we have limt →∞ f (τ (t /λ))/f (t ) = (1/λ)ϑ for all λ > 0 and thus we get lim

t →∞

f (τ (τ (λt )/λ)) f (τ (λt ))

 ϑ =

1

λ

for all λ > 0.

(11)

Multiplying (7) by (11) and shifting t to σ (t ) we obtain lim

t →∞

f (τ (τ (λσ (t ))/λ)) f σ (t )

= 1 for all λ > 0.

Hence, lim

t →∞

f (τ (τ (Λ(t )σ (t ))/Λ(t ))) f σ (t )

=1

(12)

P. Řehák, J. Vítovec / Nonlinear Analysis 72 (2010) 439–448

where Λ : T → R is a bounded function. Thanks to (10) we may take the function Λ(t ) =

τ



τ (Λ(t )σ (t )) Λ(t )

443

√

t /σ (t ). Since


!
! √ τ t /σ (t )σ (t ) t σ (t ) τ =τ =τ √ √ t /σ (t ) t /σ (t ) ! r  p σ (t ) =τ t t σ (t ) = t =τ √



t

we can rewrite (12) to limt →∞ f (t )/f σ (t ) = 1.



Remark 1. If f ∈ RV T (ϑ), then the property limt →∞ f σ (t )/f (t ) = 1 is almost immediate. Indeed, f (t ) = ϕ(t )α(t ), where limt →∞ ϕ(t ) = C > 0 and α σ (t )/α(t ) = (α(t ) + µ(t )α ∆ (t ))/α(t ) = 1 + (µ(t )/t )t α ∆ (t )/α(t ) = 1 + o(t ). Theorem 3 (Embedding Theorem). Assume that T satisfies



every large t ∈ T is isolated and µ is either bounded or eventually nondecreasing with µ(t ) = O(t ) as t → ∞.

(13)

If f ∈ KRVT (ϑ), then the function R : R → R defined by R(x) := f (τ (x)) satisfies R ∈ RVR (ϑ). Proof. We can write lim

x→∞

R(λx) R(x)

= lim

f (τ (λx))

·

f (τ (λτ (x)))

f (τ (x)) f (τ (λτ (x))) f (τ (λx)) . = λϑ lim x→∞ f (τ (λτ (x))) x→∞

= lim

x→∞

f (τ (λτ (x))) f (τ (x))

·

f (τ (λx)) f (τ (λτ (x)))

The theorem will be proved, if we show that lim

x→∞

f (τ (λx)) f (τ (λτ (x)))

= 1 for all λ > 0.

(14)

Due to [1, Theorem 1.4.3] it is enough to show that (14) holds for all λ in a set of positive measure. Next we show that for every λ from a suitably chosen set of positive measure there exists A = A(λ) ∈ R such that card (τ (λτ (x)), τ (λx)) ≤ A for large x ∈ R. If we show it, we can apply k-times (k ≤ A) Lemma 1 (with the use of an obvious transitivity property) and hereby relation (14) will be verified. (i) Let µ(t ) < H (H ∈ R) for large t. Then x − τ (x) < H and λx − λτ (x) < λH for all λ > 0, hence τ (λx) − τ (λτ (x)) < (λ + 1)H < ∞ for large x. Therefore, there exists A ∈ R such that card (τ (λτ (x)), τ (λx)) ≤ A for large x and λ > 0. (ii) Suppose that limt →∞ µ(t ) = ∞ and the function µ(t ) is nondecreasing for large t. Let λ > N, where N ∈ N satisfies σ (τ (x))/τ (x) ≤ N for all x (this N exists, see (9)). Hence σ (τ (x))/τ (x) ≤ λ and therefore, x < λτ (x) for all x ∈ R. Using this inequality we can write

τ (λx) ≤ ≤ ≤ =

τ (λσ (τ (x))) = τ (λτ (x) + λµ(τ (x))) ≤ λτ (x) + λµ(τ (x)) τ (λτ (x)) + µ(τ (λτ (x))) + λµ(τ (x)) τ (λτ (x)) + µ(τ (λτ (x))) + λµ(τ (λτ (x))) τ (λτ (x)) + (λ + 1)µ(τ (λτ (x))).

Hence



   τ (λτ (x)), τ (λx) ⊆ τ (λτ (x)), τ (λτ (x)) + (λ + 1)µ(τ (λτ (x)))

and thus for x˜ ∈ R, x˜ := τ (λτ (x)), we get

(τ (λτ (x)), τ (λx)) ⊆ (˜x, x˜ + (λ + 1)µ(˜x)). It is easy to see that for large x˜ there is card (˜x, x˜ + (λ + 1)µ(˜x)) < [λ + 1] (where [λ + 1] denotes the integer part of number λ + 1), because the graininess at every point σ (t ) is greater (or the same) than the graininess at point t (for large t).  Later we give comments to additional conditions on T, like µ(t ) = o(t ), or µ(t ) = O(t ), or (13). The next result can be understood as a converse of the previous one, in view of Theorem 2. Condition (13) does not need to be assumed. Theorem 4. Let µ(t ) = o(t ). If f : T → R and f (t ) = R(t ) for t ∈ T, where R ∈ RVR (ϑ), then f ∈ RVT (ϑ). If ϑ = 0, then the condition µ(t ) = o(t ) can be replaced by µ(t ) = O(t ).

444

P. Řehák, J. Vítovec / Nonlinear Analysis 72 (2010) 439–448

Proof. If R ∈ RV R (ϑ), then we have R(x) = ϕ(x)xϑ exp a ψ(s)/s ds , where limx→∞ ϕ(x) = C > 0, limx→∞ ψ(x) = 0, ψ may be taken as continuous, and a ∈ T, see e.g. [2]. Then we get, for t ∈ T, t ≥ a, f (t ) = ϕ(t )t ϑ α(t ), where n o

R x

Rt

α(t ) = exp

a



ψ(s)/s ds . Further α ∆ (t ) = exp(η(t ))G(t ), where R σ (u)

G(t ) = lim

t

u→t

ψ(s)/s ds (σ (u) − t )

and t

Z

ψ(s) s

a

σ (t )

Z ds −

|ψ(s)|

ds ≤ η(t ) ≤

s

t

ψ(s)

t

Z

s

a

σ (t )

Z ds + t

|ψ(s)| s

ds.

If t is right-scattered, then using the Mean Value Theorem, G(t ) =

σ (t )

Z

1

µ(t )

ψ(s) s

t

ds =

ψ(ω(t )) , ω(t )

where t ≤ ω(t ) ≤ σ (t ). If t is right-dense, then the L’Hospital rule yields G(t ) = ψ(t )/t. Hence, t α ∆ (t )

α(t )

=

t ψ(ω(t ))

ω(t )

exp(η(t ))

· exp

nR

t a

o. ψ(s)/s ds

Since 1 ≤ ω(t )/t ≤ 1 + µ(t )/t, we have that t /ω(t ) is bounded. Moreover, σ (t )

Z t

|ψ(s)| s

ds = |ψ(ζ (t ))| ln(1 + µ(t )/t ) = o(1),

where t ≤ ζ (t ) ≤ σ (t ). Consequently, limt →∞ t α ∆ (t )/α(t ) = 0, and so α ∈ SV . Hence we have f ∈ RV (ϑ), in view of Theorem 1.  Remark 2. In the last proof we may proceed in an alternative way, where we come to the regularly varying of index ϑ function f which is represented by ϑ

f (t ) = ϕ(t )t exp

(Z

˜ s) ψ(

t

s

a

) ∆s

with

˜ t ) = lim ψ( u→t

t

σ ( u) − t

σ (u)

Z t

ψ(s) s

ds = o(1).

Theorems 2–4 imply the following equivalence between Definitions 1 and 2. Theorem 5. Let T satisfy (13) with µ(t ) = o(t ). Then f ∈ RVT (ϑ) if and only if f ∈ KRVT (ϑ). If ϑ = 0, then the condition µ(t ) = o(t ) can be replaced by µ(t ) = O(t ). Remark 3. (i) A reader might wonder whether the condition µ(t ) = o(t ) (or µ(t ) = O(t )), which repeatedly appears in our assumptions, can be omitted to obtain a general theory of regular variation which is applicable on any time scales; in particular, on time scales with a ‘‘large’’ graininess. The following observations show that without some modifications, this is not possible. First recall that we want f (t ) = t ϑ , ϑ 6= 0, being a typical example of a regularly varying function of index ϑ . However, for instance, with T = qN0 = {qk : k ∈ N0 }, where q > 1, we have f σ (t )/f (t ) → qϑ 6= 1, and so the property from Lemma 1 fails to hold. Also tf ∆ (t )/f (t ) → (qϑ − 1)/(q − 1), which is different from the expected value ϑ . Moreover, for f (t ) = t ϑ , (7) holds only when λ = qj , j ∈ Z. Among others this means that even if Lemma 1 and Theorem 3 hold for µ(t ) = O(t ), it is senseful to assume µ(t ) = o(t ) when ϑ 6= 0. Note that with suitable modifications, a reasonable theory of regular variation on qN0 (or on qZ ) can be established, see [17]; it is worthy to mention that the theory shows some interesting and surprising simplifications comparing with that on T = R or T = Z, since T = qN0 is somehow natural setting for a characterization of regularly varying behavior. If T is such that, e.g., µ(tk ) ∼ Ctk and µ(sk ) = o(sk ), where C > 0 and {tk }, {sk } ⊂ T with limk→∞ tk = limk→∞ sk = ∞, then the situation is not transparent, and there is no reasonable theory on such T. Also, if the graininess is eventually ‘‘very big’’ (or a combination of ‘‘very big’’ and ‘‘small’’), then the theory gives

= {2p : k ∈ N0 } with p > 1. Take f (t ) = t ϑ with ϑ > 0 and set  ϑ  k−1 k ϑ k k pk λ = 1/2. Then, with t = 2 , k ∈ N0 , f (τ (λt ))/f (t ) = τ (2p /2)/2p = 2p /2p → 0 as t → ∞ resp. k → ∞. N0

no proper results. Indeed, for instance, let T = 2p

k

Here not only that the value of the limit is different from the expected value 2−ϑ , but it is equal to zero and an information

P. Řehák, J. Vítovec / Nonlinear Analysis 72 (2010) 439–448

445

on the index is lost. Note that the value of that limit equaling to zero or infinity is the phenomenon, which is related to the so-called rapid variation in the continuous case (a theory of rapid variation on time scales has not been developed yet). k

k−1

Observe also that f (τ (λt ))/f (t ) → 0 as t → ∞ for any λ ∈ (0, 1). Indeed, if λ ∈ (0, 1), then τ (λ2p ) = 2p for k sufficiently large. Finally observe that tf ∆ (t )/f (t ) = t ((t p )ϑ − t ϑ )/(t ϑ (t p − t )) = (t ϑ(p−1) − 1)/(t p−1 − 1). Hence, this fraction tends to 1 or ∞ or 0 according to whether ϑ is equal to 1 or is strictly greater than 1 or is strictly less than 1, respectively. In two eventualities this value is different from ϑ . It still remains to discuss whether the condition µ(t ) = O(t ) N0 can be omitted (when ϑ = 0). Again, let T = 2p with p > 1. Take f (t ) = ln t. We expect that f ∈ SV . Indeed, we have ∆ p p tf (t )/f (t ) = t (ln t − ln t )/((t − 1) ln t ) = (p − 1)t /(t p − 1) → 0 as t → ∞. On the other hand, with λ ∈ (0, 1), t = 2k , k

k−1

k

k

for sufficiently large t, f (τ (λt ))/f (t ) = ln τ (λ2p )/ ln 2p = ln 2p / ln 2p = pk−1 /pk = 1/p 6= 1, where 1 is the expected value, in view of slow variation. Note that since µ(t ) = t p − t, the condition µ(t ) = O(t ) is not fulfilled. From the above observations we conclude that it is advisable to distinguish and consider the two cases µ(t ) = o(t ) and µ(t ) ∼ Ct (or, better for a practice, µ(t ) = Ct) with C > 0 in the theory of regular variation on time scales. Concerning just a slow variation, it is sufficient to consider only one general case, namely µ(t ) = O(t ). (ii) Note that Theorem 3 (and hereby the if part of Theorem 5) requires an additional condition on the graininess, namely (13) (generally with µ(t ) = o(t )). This condition is not too restrictive Pnregarding to practical purposes. Indeed, e.g., T = hN = {hn : n ∈ N} with h > 0, or T = {Hn : n ∈ N} with H0 = 0, Hn = k=1 1/k for n ∈ N, or T = Nα0 = {nα : n ∈ N0 } β

with 0 < α < 1 all have a bounded graininess, while T = N0 with β > 1 has an unbounded increasing graininess satisfying µ(t ) = o(t ). On the other hand, we conjecture that (13) is not needed in Theorems 3 and 5, and can be simply relaxed to a natural condition µ(t ) = o(t ) (resp. µ(t ) = O(t )). Another improvement which we believe could work is an omission of the uniformity in Definition 2; this property should follow immediately from the existence of (7). Before we list elementary properties of regularly varying functions we introduce the concept of a normalized regular variation. Definition 3. An rd-continuously delta differentiable function function f : T → (0, ∞) is said to be normalized regularly varying of index ϑ , ϑ ∈ R, if it satisfies lim

tf ∆ (t )

t →∞

f (t )

= ϑ;

we write f ∈ N RV T (ϑ). If ϑ = 0, then f is said to be normalized slowly varying; we write f ∈ N SV T . For the concept of normalized regular variation in the continuous case see, e.g., [18]. In [8] we dealt with normalized regularly varying sequences. Note that the remark on p. 53 of [1] saying that the distinction between ordinary and normalized regular variation disappears on passing from functions to sequences is not true (take, e.g., fk = 1 + (−1)k /k). See also [7, p. 96]. Proposition 1. Let µ(t ) = o(t ), resp. µ(t ) = O(t ) if the index of regular variation is nonzero resp. zero. Then regularly varying functions on T have the following properties. (i) For f ∈ N RV T (ϑ) in representation formulas (4)–(6), it holds ϕ(t ) ≡ const > 0. Moreover, f ∈ N RVT (ϑ) iff f (t ) = t ϑ L(t ), where L ∈ N SVT . (ii) Let f ∈ RV T (ϑ). Then limt →∞ ln f (t )/ ln t = ϑ . This implies limt →∞ f (t ) = 0 if ϑ < 0 and limt →∞ f (t ) = ∞ if ϑ > 0. (iii) Let f ∈ RV T (ϑ). Then limt →∞ f (t )/t ϑ−ε = ∞ and limt →∞ f (t )/t ϑ+ε = 0 for every ε > 0. (iv) Let f ∈ RV T (ϑ1 ) and g ∈ RV T (ϑ2 ). Then fg ∈ RV T (ϑ1 + ϑ2 ), 1/f ∈ RV T (−ϑ1 ), and f γ ∈ RV T (γ ϑ). The same holds if RV T is replaced by N RV T . (v) Let f ∈ RV T (ϑ). If f is convex, then it is decreasing provided ϑ ≤ 0, and it is increasing provided ϑ > 0. A concave f is increasing. If f ∈ N RV T (ϑ), then it is decreasing provided ϑ < 0 and it is increasing provided ϑ > 0. (vi) Let f ∈ RV T (ϑ), ϑ ∈ R, be rd-continuously delta differentiable. If f is convex or concave, then f ∈ N RV T (ϑ). (vii) (Zygmund type characterization) Let µ(t ) = o(t ). For a positive rd-continuously delta differentiable function f , f ∈ N RV T (ϑ) if and only if f (t )/t γ is eventually increasing for each γ < ϑ and f (t )/t ζ is eventually decreasing for each ζ > ϑ. Proof. (i) We know that f ∈ RV T (ϑ) can be written as f (t ) = t ϑ L(t ), where L ∈ SV T . The statement follows from tf ∆ (t ) f (t )

=

t (t ϑ )∆ tϑ

 +

σ (t )

ϑ

tL∆ (t ) L(t )

t

since for ϑ 6= 0 we have t (t ϑ )∆ /t ϑ → ϑ and (σ (t )/t )ϑ → 1 as t → ∞. (ii) From representation (6) we have ln f (t ) ln t

=

ln ϕ(t ) ln t

Rt +ϑ +

a

ψ(s)/s ∆s ln t

.

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We claim that the last term tends to zero as t → ∞. This follows from the fact that ψ(t ) → 0 and ln t can be written as Rt (1 + |O(1)|)/s ∆s. Indeed, at a right-scattered t we have (ln t )∆ = (1/t )[ln(1 + µ(t )/t )]/(µ(t )/t ). (iii) This follows from representation (5) and part (ii) of this proposition. (iv) This follows from representation (6). (v) First note that the convexity of f implies clearly its eventual monotonicity. Similarly, the concavity implies that f is increasing; if f were decreasing then it cannot be eventually positive. Next we show that a convex f ∈ SV T is decreasing. By a contradiction assume that f is increasing. Thanks to convexity we then have f (t ) ≥ Mt for large t and for some M > 0. But now f cannot be slowly varying by (iii) of this proposition. Similarly we proceed when ϑ < 0 and f is convex. If f is convex with ϑ > 0, then it tends to ∞ and hence must be increasing. The claim for a normalized function follows from (i) of this proposition. (vi) By (v) of this proposition, one of the conditions must eventually hold (a) f is convex, decreasing, or (b) f is concave, increasing, or (c) f is convex, increasing. Case (a) was proved in [19]. Case (b) is very similar to (a). To prove (c) we use

R τ (λt )

arguments also similar to (a): From Theorem 2, f satisfies (7). Using the equality f (τ (λt )) − f (t ) = t f ∆ (s) ∆s, monotonicity properties of f ∆ , and (7) it is not difficult to show that lim supt →∞ tf ∆ (t )/f (t ) ≤ limλ→1+ (λϑ −1)/(λ−1) = ϑ and lim inft →∞ tf ∆ (t )/f (t ) ≥ limλ→1− (1 − λϑ )/(1 − λ) = ϑ . (vii) By (iv) and (v) of this proposition f (t )/t γ ∈ N RV T (ϑ −γ ) is increasing and f (t )/t ζ ∈ N RV T (ϑ −ζ ) is decreasing. Conversely, from (f (t )/t γ )∆ > 0 we get tf ∆ (t )/f (t ) > (t γ )∆ /t γ −1 = γ (1 + o(1)). Similarly, tf ∆ (t )/f (t ) < ζ (1 + o(1)). The statement follows by choosing γ and ζ arbitrarily close to ϑ .  4. Applications Consider the linear second order dynamic equation y∆∆ = p(t )yσ ,

(15)

where p is a positive rd-continuous function on a time scale interval [a, ∞). Some basic qualitative properties of (15) can be found in [15]. In this section we provide information about asymptotic behavior of all solutions of (15) as t → ∞. First note that all nontrivial solutions of (15) are nonoscillatory (i.e., eventually of one sign) and eventually monotone. Because of linearity, without loss of generality, we may consider just positive solutions of (15); we denote this set as M. Thanks to the monotonicity, the set M can be further split in the two classes M+ and M− , where

M+ = {y ∈ M : ∃ty ∈ T such that y(t ) > 0, y∆ (t ) > 0 for t ≥ ty }, M− = {y ∈ M : y(t ) > 0, y∆ (t ) < 0}. These classes are always nonempty. In [16,19] we established necessary and sufficient conditions for all positive decreasing solutions of (15) to be regularly varying. Here we want to apply the above developed theory of regular variation, and complete the results from [16,19] in the sense of increasing solutions. For related results in special settings see [8,11,12,17]. Theorem 6. Let µ(t ) = o(t ). (i) Eq. (15) has a fundamental set of solutions u(t ) = L(t ) ∈ SV T ,

v(t ) = t L˜ (t ) ∈ RV T (1)

if and only if ∞

Z

p(s) ∆s = 0.

lim t

t →∞

(16)

t

Moreover, L, L˜ ∈ N SV T with L˜ (t ) ∼ 1/L(t ). All positive decreasing solutions of (15) belong to N SV T and all positive increasing solutions of (15) belong to N RV T (1). (ii) Eq. (15) has a fundamental set of solutions u(t ) = t ϑ1 L(t ) ∈ RV T (ϑ1 ),

v(t ) = t ϑ2 L˜ (t ) ∈ RV (ϑ2 )

if and only if ∞

Z

p(s) ∆s = A > 0,

lim t

t →∞

(17)

t

where ϑ1 < 0 and ϑ2 = 1 − ϑ1 are the roots of the equation ϑ 2 − ϑ − A = 0. Moreover L, L˜ ∈ N SV T with L˜ (t ) ∼ 1/((1 − 2ϑ1 )L(t )). All positive decreasing solutions of (15) belong to N RV T (ϑ1 ) and all positive increasing solutions of (15) belong to N RV T (ϑ2 ).

P. Řehák, J. Vítovec / Nonlinear Analysis 72 (2010) 439–448

447

Proof. Parts (i) and (ii) of the theorem will be proved simultaneously assuming A ≥ 0 in (17) and, consequently, ϑ1 ≤ 0.

R∞

Sufficiency. Let limt →∞ t t p(s) ∆s = A. From [16], if u is a positive decreasing solution (which always exists), then u ∈ N RV (ϑ1 ). Hence, u(t ) = t ϑ1 L(t ), where L ∈ N SV by (i) of Proposition 1. Put z = 1/u2 . Then z ∈ N RV T (−2ϑ1 ) by (iv) of Proposition 1. Moreover, z (t ) ∼ 1/(u(t )uσ (t )) as t → ∞ by Lemma 1 or Remark 1. A second linearly independent Rt solution v of (15) is given by v(t ) = u(t ) a 1/(u(s)uσ (s)) ∆s. Taking into account that u is decreasing (recessive), it holds

1/(u(s)uσ (s)) ∆s = ∞. Further, tz (t ) → ∞ as t → ∞ by (iv) and (ii) of Proposition 1. The time scale L’Hospital rule now yields

R∞ a

lim

t /u(t )

v(t )

t →∞

= lim R t t →∞

a

tz (t ) 1/(u(s)uσ (s)) ∆s

= lim

t →∞

z (t ) + σ (t )z ∆ (t ) 1/(u(t )uσ (t ))

∆

= 1 + lim

t →∞

σ (t )z (t ) = 1 − 2ϑ1 z (t )

since µ(t ) = o(t ). Hence (1 − 2ϑ1 )v(t ) ∼ t /u(t ) = t 1−ϑ1 /L(t ). Consequently, v(t ) = t 1−ϑ1 L˜ (t ), where L˜ (t ) ∼ 1/[(1 − ϑ1 )L(t )] and L˜ ∈ SV T by (iv) of Proposition 1. This implies v ∈ RV T (ϑ2 ) by (i) of Proposition 1. Further, since (the solution) v is convex, it is increasing by (v) of Proposition 1, thus it is normalized by (vi) of Proposition 1, and therefore L˜ is normalized too by (i) of Proposition 1. Necessity. See the proof in [16] and note that in view of the convexity, a solution u ∈ RV T (ϑ1 ) necessarily decreases and a solution v ∈ RV T (ϑ2 ) necessarily increases by (v) of Proposition 1. Remark 4. (i) It is sufficient to assume just µ(t ) = O(t ) for the statement: A positive decreasing solution of (15) belongs to SV T if and only if (16) holds. In general the condition µ(t ) = o(t ) cannot be omitted in Theorem 6, see also Remark 3. (ii) Using arguments similar R ∞ to those in [16] we can show that if a positive (increasing) solution v of (15) belongs to RV T (ϑ2 ), then limt →∞ t t p(s) ∆s = A. It means that this condition is necessary for any of the conditions u ∈ RV T (ϑ1 ) and v ∈ RV T (ϑ2 ). (iii) In the if parts of Theorem 6, conditions (16) and (17) can be replaced by the simpler ones limt →∞ t 2 p(t ) = 0 and limt →∞ t 2 p(t ) = A, respectively. (iv) In view of Theorem 6 and (ii) of Proposition 1, with the notation − M− SV = M ∩ N SV T , − M− RV (ϑ1 ) = M ∩ N RV T (ϑ1 ),

ϑ1 < 0,

+ M+ RV (ϑ2 ) = M ∩ N RV T (ϑ2 ),

ϑ2 > 0,

M0 = {y ∈ M : lim y(t ) = 0}, −

−

t →∞

M+ ∞

= {y ∈ M : lim y(t ) = ∞}, +

t →∞

we can claim + + + M− = M− SV ⇐⇒ (16) ⇐⇒ M = MRV (1) = M∞ , − + + + M− = M− RV (ϑ1 ) = M0 ⇐⇒ (17) ⇐⇒ M = MRV (ϑ2 ) = M∞ .

(v) The case where the limit in (17) (or the limit of a similar expression) equals ∞ is also of interest. Such a condition is conjectured to guarantee the existence of rapidly varying like solutions. A theory of rapid variation on time scales has been mentioned in Remark 3 as an open problem. Acknowledgements The first author was supported by the Grants KJB100190701 of the Grant Agency of ASCR and 201/07/0145 of the Czech Grant Agency, and by the Institutional Research Plan AV0Z010190503. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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