A time scales version of a Wirtinger-type inequality and applications

Hilscher, Roman Šimon

doi:10.1016/s0377-0427(01)00447-2

Cited by 39 publications

(19 citation statements)

References 27 publications

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“…Very early after the concept of time scales was introduced, equations of type (1.1) have started to be studied, see Erbe and Hilger [9]. Among others, some effort has been devoted to extensions of Hille-Nehari criteria and other related topics to time scales, like Kneser's criteria and oscillatory properties of Euler's equation, see Bohner and Saker [4], Bohner andÜnal [5], Erbe et al [10], Hilscher [13], andŘehák [22,23]. The results in quoted papers which are related to our subject are interesting and valuable (the claims come as consequences of various techniques and they may serve as a good inspiration) but the problem is that they contain restrictions that disable examination of many remaining important cases.…”

Section: ) With Continuous Coefficients R(t) > 0 and P(t)mentioning

confidence: 99%

How the constants in Hille-Nehari theorems depend on time scales

Řehák

2006

Adv. Differ Equ.

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We present criteria of Hille-Nehari-type for the linear dynamic equation (r(t)y Δ ) Δ + p(t)y σ = 0, that is, the criteria in terms of the limit behavior of (As a particular important case, we get that there is a (sharp) critical constant in those criteria which belongs to the interval [0,1/4], and its value depends on the graininess μ and the coefficient r. Also we offer some applications, for example, criteria for strong (non-) oscillation and Kneser-type criteria, comparison with existing results (our theorems turn out to be new even in the discrete case as well as in many other situations), and comments with examples.

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Section: ) With Continuous Coefficients R(t) > 0 and P(t)mentioning

confidence: 99%

How the constants in Hille-Nehari theorems depend on time scales

Řehák

2006

Adv. Differ Equ.

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“…In recent years, there have been established quite many results that are related to an extension of Hille-Nehari theorems (or to an examination of Euler type equations) to other or more general settings, in particular, discrete, half-linear or time scales ones, see [5,6,7,8,11,13,14,15,16,21,22,26]. However, the results presented there usually contain certain restrictions that disable examination of many important cases.…”

Section: Q (R(t)d Q Y(t)) + P(t)y(qt)mentioning

confidence: 99%

“…A crucial point which leads to a "variable critical constant", is that we do not force the "problematic term" occuring in a generalized Riccati dynamic equation (see (13)) to behave like almost in the continuous case which was a usual procedure in previous approaches.…”

Section: Q (R(t)d Q Y(t)) + P(t)y(qt)mentioning

confidence: 99%

A critical oscillation constant as a variable of time scales for half-linear dynamic equations

Řehák

2010

Mathematica Slovaca

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ABSTRACT. We present criteria of Hille-Nehari type for the half-linear dynamic equation (r(t)Φ(y ∆ )) ∆ + p(t)Φ(y σ ) = 0 on time scales. As a particular important case we get that there is a a (sharp) critical constant which may be different from what is known from the continuous case, and its value depends on the graininess of a time scale and on the coefficient r. As applications we state criteria for strong (non)oscillation, examine generalized Euler type equations, and establish criteria of Kneser type. Examples from q-calculus, a Hardy type inequality with weights, and further possibilities for study are presented as well. Our results unify and extend many existing results from special cases, and are new even in the well-studied discrete case. IntroductionConsider the half-linear dynamic equationwhere Φ(u) = |u| α−1 sgn u with α > 1, 1/r(t) > 0 and p(t) are rd-continuous functions defined on a time scale interval [a, ∞), a ∈ T, and a time scale T is assumed to be unbounded from above. This equation covers a large variety of 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 34C10, 34K11; Secondary 39A11, 39A12, 39A13. K e y w o r d s:

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“…This is possible only when f is a constant. But if f were a nonzero constant function, this would be inconsistent with the convergence of 10) and (2.1) follows from (2.10) in the same way as (2.6) does from (2.5). Now we prove that the constant factor is the best possible provided µ(t)/t → 0 as t → ∞.…”

Section: Denote ϕ(T) = F(t)/(t−a)mentioning

confidence: 99%

“…To discuss this question, first recall some known results on linear equations. Note that, for example, in [4,10] (3.17) More precisely, the inequality…”

Section: Remark 310 (I)mentioning

confidence: 99%

Hardy inequality on time scales and its application to half-linear dynamic equations

Řehák¹

2005

J Inequal Appl

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496 Hardy inequality on time scales a generalized Euler dynamic equation. Those results turn out to be new even in the special linear case. The questions how the graininess of the time scale affects the (non)oscillation of the equation, as well as some other related topics, are also discussed there.Before we present our main result, let us recall some essentials about time scales. In 1988, Hilger [9] introduced the calculus on time scales which unifies continuous and discrete analysis. A time scale T is an arbitrary nonempty closed subset of the real numbers R. We define the forward jump operator σ by σ(t) := inf{s ∈ T : s > t}, and the graininess µ of the time scale T by µ(t) := σ(t) − t. A point t ∈ T is said to be right-dense, rightscattered, if σ(t) = t, σ(t) > t, respectively. We denote f σ := f • σ. For a function f : T → R the delta derivative is defined by f ∆ (t) := lim s→t,σ(s) =t

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A time scales version of a Wirtinger-type inequality and applications

Cited by 39 publications

References 27 publications

How the constants in Hille-Nehari theorems depend on time scales

How the constants in Hille-Nehari theorems depend on time scales

A critical oscillation constant as a variable of time scales for half-linear dynamic equations

Hardy inequality on time scales and its application to half-linear dynamic equations

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