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Oscillatory properties of second order half-linear difference equations

Pavel Řehák — 2001

Czechoslovak Mathematical Journal

We study oscillatory properties of the second order half-linear difference equation Δ ( r k | Δ y k | α - 2 Δ y k ) - p k | y k + 1 | α - 2 y k + 1 = 0 , α > 1 . ( HL ) It will be shown that the basic facts of oscillation theory for this equation are essentially the same as those for the linear equation Δ ( r k Δ y k ) - p k y k + 1 = 0 . We present here the Picone type identity, Reid Roundabout Theorem and Sturmian theory for equation (HL). Some oscillation criteria are also given.

Second order linear q -difference equations: nonoscillation and asymptotics

Pavel Řehák — 2011

Czechoslovak Mathematical Journal

The paper can be understood as a completion of the q -Karamata theory along with a related discussion on the asymptotic behavior of solutions to the linear q -difference equations. The q -Karamata theory was recently introduced as the theory of regularly varying like functions on the lattice q ℕ 0 : = { q k : k ∈ ℕ 0 } with q > 1 . In addition to recalling the existing concepts of q -regular variation and q -rapid variation we introduce q -regularly bounded functions and prove many related properties. The q -Karamata theory is then...

A role of the coefficient of the differential term in qualitative theory of half-linear equations

Pavel Řehák — 2010

Mathematica Bohemica

The aim of this contribution is to study the role of the coefficient r in the qualitative theory of the equation ( r ( t ) Φ ( y Δ ) ) Δ + p ( t ) Φ ( y σ ) = 0 , where Φ ( u ) = | u | α - 1 sgn u with α > 1 . We discuss sign and smoothness conditions posed on r , (non)availability of some transformations, and mainly we show how the behavior of r , along with the behavior of the graininess of the time scale, affect some comparison results and (non)oscillation criteria. At the same time we provide a survey of recent results acquired by sophisticated modifications of the Riccati...

Regulární variace: od škálové invariance ke konvergenčním testům

Pavel Řehák — 2023

Pokroky matematiky, fyziky a astronomie

Článek se snaží přiblížit některé aspekty teorie regulární variace. Jde o pojem z klasické analýzy, který má bohatou historii a četné aplikace v teorii pravděpodobnosti, teorii čísel, integrálních transformacích, komplexní analýze, diferenciálních rovnicích, teorii her či teorii grafů. Regulárně měnící se funkce mají souvislost s mnoha matematickými pojmy, včetně škálové invariance, kterou náš výklad začíná, či konvergenčními testy pro nekonečné řady, kterými náš výklad končí. V průběhu výkladu...

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