An oscillation theorem for the nonlinear degenerate elliptic equation in the Heisenberg group

The aim of this paper is to study the oscillation of solutions of the nonlinear degenerate elliptic equation in the Heisenberg group $H^{n}$ H n . We first derive a critical inequality in $H^{n}$ H n . Based on it, we establish a Picone-type differential inequality and a Sturm-type comparison principle. Then we obtain an oscillation theorem. Our result generalizes the related conclusions for the nonlinear elliptic equations in $R^{n}$ R n .

Oscillation Properties of Singular Quantum Trees

We discuss the possibility of generalizing the Sturm comparison and oscillation theorems to the case of singular quantum trees, that is, to Sturm-Liouville differential expressions with singular coefficients acting on metric trees and subject to some boundary and interface conditions. As there may exist non-trivial solutions of differential equations on metric trees that vanish identically on some edges, the classical Sturm theory cannot hold globally for quantum trees. However, we show that the comparison theorem holds under minimal assumptions and that the oscillation theorem holds generically, that is, for operators with simple spectra. We also introduce a special Prüfer angle, establish some properties of solutions in the non-generic case, and then extend the oscillation results to simple eigenvalues.

New Oscillation Results for Third-Order Half-Linear Neutral Differential Equations

The main purpose of this paper is to obtain criteria for the oscillation of all solutions of a third-order half-linear neutral differential equation. The main result in this paper is an oscillation theorem obtained by comparing the equation under investigation to two first order linear delay differential equations. An additional result is obtained by using a Riccati transformation technique. Examples are provided to show the importance of the main results.

Bounded Oscillation Theorem for Unstable-type Neutral Impulsive Differential Equations of the Second Order

The oscillations theory of neutral impulsive differential equations is gradually occupying a central place among the theories of oscillations of impulsive differential equations. This could be due to the fact that neutral impulsive differential equations plays fundamental and significant roles in the present drive to further develop information technology. Indeed, neutral differential equations appear in networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits). In this paper, we study the behaviour of solutions of a certain class of second-order linear neutral differential equations with impulsive constant jumps. This type of equation in practice is always known to have an unbounded non-oscillatory solution. We, therefore, seek sufficient conditions for which all bounded solutions are oscillatory and provide an example to demonstrate the applicability of the abstract result.

On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs

We consider stationary waves on nonlinear quantum star graphs, i.e., solutions to the stationary (cubic) nonlinear Schrödinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the star and classify them according to the nodal structure on each edge (i.e., the number of nodal domains or nodal points that the solution has on each edge). We discuss the relevance of these solutions in more applied settings as starting points for numerical calculations of spectral curves and put our results into the wider context of nodal counting, such as the classic Sturm oscillation theorem.