The Karamata integration theorem on time scales and its applications in dynamic and difference equations

Řehák, P.

doi:10.1016/j.amc.2018.06.023

Cited by 7 publications

(17 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As in the proof of (a), we have

\int_{t}^{\infty} u false(s false) normalΔ s = \sum_{j = k}^{\infty} y false(j false) normalΔ τ false(j false)

, and so

\int^{\infty} u false(s false) normalΔ s < \infty

. Applying Řehák,, theorem 1 we get

\int_{t}^{\infty} u false(s false) normalΔ s \in {scriptN scriptS scriptV}_{T}

and

t u false(t false) = o false(\int_{t}^{\infty} u false(s false) normalΔ s false)

. The statement now clearly follows, in view of the relations t = τ ( k ), u = y ∘ τ −1 , and

\int_{t}^{\infty} u false(s false) normalΔ s = \sum_{j = k}^{\infty} y false(j false) normalΔ τ false(j false)

.…”

Section: Regularly Varying Sequences With Respect To τmentioning

confidence: 65%

“…Since (with t = τ ( k ))

{140%true \int}^{​}_{t}^{\infty} u (s) Δ s = {140%true \int}^{​}_{t}^{\infty} y (τ^{- 1} (s)) Δ s = \underset{s \in {[t, \infty)}_{double-struckT}}{140%true \sum} y (τ^{- 1} (s)) μ (s) = {140%true \sum}_{j = k}^{\infty} y (j) μ (τ (j)) = {140%true \sum}_{j = k}^{\infty} y (j) Δ τ (j),

we get

\sum_{j = k}^{\infty} y false(j false) normalΔ τ false(j false) \sim τ false(k false) u false(τ false(k false) false)

as k → ∞ , which yields the desired formula. (b) The proof is similar to that of (a) utilizing Řehák ,. theorem 1 (c) Let

\sum_{j = m}^{\infty} y false(j false) normalΔ τ false(j false) < \infty

.…”

Section: Regularly Varying Sequences With Respect To τmentioning

confidence: 76%

“…In view of Proposition , we obtain

y \notin {scriptR scriptV}_{double-struckZ}^{τ}

. On the other hand, we have

k normalΔ y false(k false) false/ y false(k false) = k false(y false(k false) false/ y false(k false) false) \exp ((), \sum_{j = m}^{k} 1 false/ false(j \ln^{γ} j false) - \sum_{j = m}^{k - 1} 1 false/ false(j \ln^{γ} j false)) = k false(\exp false(1 false/ k \ln^{γ} k false) - 1 false) \sim k false(1 false/ false(k \ln^{γ} k false) false) \to 0

as k → ∞ , and so

y \in {scriptS scriptV}_{Z}

. (iii)We have

y \in false(scriptN false) {scriptR scriptV}_{double-struckZ}^{τ} false(ϑ false)

if and only if

y \circ τ^{- 1} \in false(scriptN false) {scriptR scriptV}_{T} false(ϑ false)

if and only if

y false(τ^{- 1} false(t false) false) = t^{ϑ} \overset{x}{true} false(t false)

\overset{x}{true} \in false(scriptN false) {scriptS scriptV}_{T}

, where the former equivalence follows from Proposition , while the latter one follows from Řehák ,. proposition 1 Thus,

y \in false(scriptN false) {scriptR scriptV}_{double-struckZ}^{τ} false(ϑ false)

can be characterized as y ( τ −1 ( t )) = t ϑ x ( τ −1 ( t )), where

x = \overset{x}{true} \circ τ \in false(scriptN false) {scriptS scriptV}_{double-struckZ}^{τ}

.…”

Section: Regularly Varying Sequences With Respect To τmentioning

confidence: 91%

“…Similarly, we get the claim for

{scriptN scriptS scriptV}_{double-struckZ}^{τ}

sequences using the characterization of

{scriptN scriptS scriptV}_{T}

functions that take the form with η ( t ) = C and ϑ = 0. (v)By Řehák, proposition 1 and Proposition , with t = τ ( k ), we get y ( k + 1) = ( y ∘ τ −1 )( σ ( t ))∼( y ∘ τ −1 )( t ) = y ( k ) as t → ∞ (i.e. k → ∞ ). (vi)By Řehák, proposition 1 and Proposition , with t = τ ( k ), we get y ( k )/ τ ϑ − ε ( k ) = ( y ∘ τ −1 )( t )/ t ϑ − ε → ∞ as t → ∞ . The other part of the statement follows similarly. (vii)By Proposition , we have

u : = y \circ τ^{- 1} \in {scriptR scriptV}_{T} false(ϑ false)

, where

double-struckT = τ false(Z_{m} false)

.…”

Section: Regularly Varying Sequences With Respect To τmentioning

confidence: 93%

“…The concept of regular variation on time scales is motivated by the above mentioned definition of discrete regular variation, which involves the difference operator. Recall (see Řehák) that an rd‐continuous function

f : double-struckT \to false(0, \infty false)

is (normalized) regularly varying of index ϑ on time scale

double-struckT

, and we write

f \in false(scriptN false) {scriptR scriptV}_{T} false(ϑ false)

, if there exists a positive rd‐continuously delta differentiable function g satisfying f ( t )∼ g ( t ) ( f ( t ) = g ( t )) and tg Δ ( t )/ g ( t )→ ϑ as t → ∞ ; (normalized) slow variation on

double-struckT

means

false(scriptN false) {scriptR scriptV}_{T} false(0 false) = : false(scriptN false) {scriptS scriptV}_{T}

. Here, g Δ ( t ) stands for the delta derivative of g at t ; ie, for any ε > 0, there is a neighborhood U of t such that |( g ( σ ( t )) − g ( s ) − g Δ ( t )( σ ( t ) − s )| ≤ ε | σ ( t ) − s | for all s ∈ U .…”

Section: Regularly Varying Sequences With Respect To τmentioning

confidence: 99%

See 4 more Smart Citations

Refined discrete regular variation and its applications

Řehák

2019

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

We introduce a new class of the so-called regularly varying sequences with respect to and state its properties. This class, on one hand, generalizes regularly varying sequences. On the other hand, it refines them and makes it possible to do a more sophisticated analysis in applications. We show a close connection with regular variation on time scales; thanks to this relation, we can use the existing theory on time scales to develop discrete regular variation with respect to . We reveal also a connection with generalized regularly varying functions.As an application, we study asymptotic behavior of solutions to linear difference equations; we obtain generalization and extension of known results. The theory also yields, in some way, a new view on the tests for convergence and divergence of series; we establish the statement that generalizes Raabe test and Bertrand test. KEYWORDS asymptotic behavior, difference equation, Karamata theorem, regularly varying sequence, time scale, Raabe testwill be presented in Section 3. We also indicate directions for further research. Note that (1) can be written as the three-term recurrence relation a(k)y(k + 2) + b(k)y(k + 1) + c(k)y(k) = 0 (and vice versa). In Section 4, we show how Math Meth Appl Sci. 2019;42:6009-6020.wileyonlinelibrary.com/journal/mma

show abstract

“…As in the proof of (a), we have