Elbert-type comparison theorems for a class of nonlinear Hamiltonian systems

Abstract

Picone-type identities are established for a pair of solutions $(x,y)$ and $(X,Y)$ of the respective systems of the form \begin{equation} x' = r(t)x + p(t)\varphi_{1/\alpha} (y), \qquad y' = - q(t)\varphi_\alpha (x) - r(t)y, \tag{1.1} \end{equation} and \begin{equation} X' = R(t)X + P(t)\varphi_{1/\alpha}(Y), \qquad Y' = - Q(t)\varphi_\alpha (X) - R(t)Y, \tag{1.2} \end{equation} where $\alpha$ is a positive constant, $p, q, r, P, Q$ and $R$ are continuous functions on an interval $J$ and $\varphi_\gamma (u)$ denotes the odd function in $u \in \mathbb{R}$ defined by $$ \varphi_\gamma (u) = |u|^\gamma \operatorname{sgn} u = |u|^{\gamma-1}u, \qquad \gamma > 0. $$ The identities are used to prove Sturmian comparison and separation results for components of solutions of systems (1.1) and (1.2).

Keywords

• hamiltonian system
• picone's identity
• sturmian comparison