Solutions of Riemann–Weber type half-linear differential equation

summary:We establish an asymptotic formula for a pair of linearly independent solutions of the subcritical Riemann–Weber type half-linear differential equation. We also complement the results of the author and M. Ünal, Acta Math. Hungar. 120 (2008), 147–163, where the equation was considered in the critical case

Hille-Wintner type comparison kriteria for half-linear second order differential equations

summary:We establish Hille-Wintner type comparison criteria for the half-linear second order differential equation $$\left(r(t)\Phi (x^{\prime })\right)^{\prime }+c(t)\Phi (x)=0,\quad \Phi (x)=|x|^{p-2}x\,,\ p>1\,,$$ where this equation is viewed as a perturbation of another equation of the same form

Disconjugacy and Transformations for Symplectic Systems

We examine transformations and diconjugacy for general symplectic systems which include as special cases linear Hamiltonian difference systems and Sturm-Liouville difference equations of higher order. We give a Reid roundabout theorem for these systems and also for reciprocal symplectic systems. Particularly, we investigate a connection between eventual disconjugacy of linear Hamiltonian difference systems and their reciprocals. Finally, we present a dinsconjugacy-preserving transformation of a Sturm-Liouville equation of higher order which transforms this equation into another one of the same order

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

summary:The aim of this contribution is to study the role of the coefficient $r$ in the qualitative theory of the equation $(r(t)\Phi (y^{\Delta}))^{\Delta} +p(t)\Phi (y^{\sigma})=0$, where $\Phi (u)=|u|^{\alpha -1}\mathop{\rm sgn}u$ with $\alpha >1$. 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 type technique, which are supplemented by some new observations

The Ratio of Eigenvalues of the Dirichlet Eigenvalue Problem for Equations with One-Dimensional p

We establish an estimate for the ratio of eigenvalues of the Dirichlet eigenvalue problem for the equation with one-dimensional p-Laplacian involving a nonnegative unimodal (single-well) potential