Some recent results concerning properties of solutions of the half-linear second order differential equation $${\left(r\left(t\right)\Phi \left(x^{\text{'}}\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(x\right)=0,\Phi \left(x\right):={\left|x\right|}^{p-2}x,p>1,(*)$$ are presented. A particular attention is paid to the oscillation theory of $(*)$. Related problems are also discussed.

In this paper we consider linear Hamiltonian differential systems without the controllability (or normality) assumption. We prove the Rayleigh principle for these systems with Dirichlet boundary conditions, which provides a variational characterization of the finite eigenvalues of the associated self-adjoint eigenvalue problem. This result generalizes the traditional Rayleigh principle to possibly abnormal linear Hamiltonian systems. The main tools...

In this paper we consider linear Hamiltonian differential systems without the controllability (or normality) assumption. We prove the Rayleigh principle for these systems with Dirichlet boundary conditions, which provides a variational characterization of the finite eigenvalues of the associated self-adjoint eigenvalue problem. This result generalizes the traditional Rayleigh principle to possibly abnormal linear Hamiltonian systems. The main tools...

We introduce the concept of the regular (nonoscillatory) half-linear second order differential equation $${\left(r\left(t\right)\Phi \left(x^{\text{'}}\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(x\right)=0,\Phi \left(x\right):={\left|x\right|}^{p-2}x,p>1(*)$$ and we show that if (*) is regular, a solution $x$ of this equation such that $x^{\text{'}}\left(t\right)\ne 0$ for large $t$ is principal if and only if $${\int }^{\infty }\frac{dt}{r\left(t\right)x^2\left(t\right){\left|x^{\text{'}}\left(t\right)\right|}^{p-2}}=\infty .$$ Conditions on the functions $r,c$ are given which guarantee that (*) is regular.

Sufficient conditions for oscillation of a certain nonlinear trinomial third order differential equation are proved.

We consider the half-linear differential equation of the form $$\left(p\left(t\right)\right|x^{\text{'}}{|}^{\alpha }\mathrm{sgn}{x}^{\text{'}}{)}^{\text{'}}+q\left(t\right){\left|x\right|}^{\alpha }\mathrm{sgn}x=0,t\ge t_0,$$ under the assumption that $p{\left(t\right)}^{-1/\alpha }$ is integrable on $[t_0,\infty )$. It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as $t\to \infty$.

We are concerned with the uniqueness problem for solutions to the second order ODE of the form $x^{\text{'}\text{'}}+f(x,t)=0$, subject to appropriate initial conditions, under the sole assumption that $f$ is non-decreasing with respect to $x$, for each $t$ fixed. We show that there is non-uniqueness in general; on the other hand, several types of reasonable additional assumptions make the problem uniquely solvable. The interest in this problem comes, among other, from the study of oscillations of lumped parameter systems with implicit...

This paper deals with the asymptotic behavior as $t\to \infty$ of solutions $u$ to the forced Preisach oscillator equation $\ddot{w}\left(t\right)+u\left(t\right)=\psi \left(t\right)$, $w=u+𝒫\left[u\right]$, where $𝒫$ is a Preisach hysteresis operator, $\psi \in L^{\infty }(0,\infty )$ is a given function and $t\ge 0$ is the time variable. We establish an explicit asymptotic relation between the Preisach measure and the function $\psi$ (or, in a more physical terminology, a balance condition between the hysteresis dissipation and the external forcing) which guarantees that every solution remains bounded for all times. Examples show...