Limit periodic homogeneous linear difference systems

Self Cite

The research contained in this paper belongs to the qualitative theory of dynamic equations on time scales. Via the detailed analysis of solutions of the associated Riccati equation and an advanced averaging technique, we provide the description of domain of nonoscillation of very general equations. The results are formulated and proved for half‐linear equations (i.e., equations connected to PDEs with one dimensional p‐Laplacian) on time scales. Nevertheless, we obtain new results also for linear difference equations. Moreover, the combination of the presented results with previous ones shows that many useful equations are conditionally oscillatory. Such equations are ideal as testing and comparison equations in real‐world models which are beyond capabilities of known oscillation and nonoscillation criteria often.

𝕋 = ℤ

𝕋

.…”

Section: Nonoscillation Criteriamentioning

confidence: 77%

Nonoscillation of half‐linear dynamic equations on time scales

Hasil

Kiseľák

Pospíšil

et al. 2021

Self Cite

“…Note that, in Hasil and Veselý, 45 there is mentioned that Equation (1.2) is oscillatory for some functions R,S also satisfying (5.4). It is based on constructions published in Veselý 55 (see also studies [56][57][58][59] in the discrete case). However, the existence of such coefficients R,S follows from results about equations with perturbations as well (see Elbert and Schneider 60 and also other studies [61][62][63][64] ).…”

Section: Corollaries and Commentsmentioning

confidence: 99%

Non‐oscillation of linear and half‐linear differential equations with unbounded coefficients

Šišoláková

2020

We deal with Euler-type half-linear second-order differential equations, and our intention is to derive conditions in order their non-trivial solutions are non-oscillatory. This paper connects to the article P. Hasil, J. Šišoláková, M. Veselý: Averaging technique and oscillation criterion for linear and half-linear equations, Appl. Math. Lett. 92 (2019), 62-69, where the corresponding oscillatory counterpart is studied and an oscillation criterion is established. The used effective technique for this investigation is the combination of the generalized adapted Prüfer angle and the modified Riccati transformation. This paper is completed by two corollaries and an example concerning the linear case when we obtain new results as well.

“…We remark that the above described problem about the threshold case is documented for difference equations as well. The relevant constructions of almost periodic sequences can be found in Veselý (and they are used, eg, in Hasil and Veselý. (d) General timescale. Many timescales can be redefined and supplemented to periodic ones and many models work with periodic timescales naturally (seasonal rhythm, signals, etc.).…”

Section: Open Problemsmentioning

confidence: 99%

Section: Open Problemsmentioning

confidence: 99%

Oscillation result for half‐linear dynamic equations on timescales and its consequences

Hasil

Veselý

2019

Self Cite

We study oscillatory properties of half‐linear dynamic equations on timescales. Via the combination of the Riccati technique and an averaging method, we find the domain of oscillation for many equations. The presented main result is not the conversion of a known result from the theory of differential or difference equations, ie, we obtain new results for the timescales double-struckT=double-struckR (for differential equations) and double-struckT=double-struckZ (for difference equations). Half‐linear equations generalize linear equations (in fact, they coincide with certain one‐dimensional PDEs with p‐Laplacian), but the main result is new also for linear differential and difference equations. The corresponding corollaries and examples are given as well.