Highly oscillatory solutions of a Neumann problem for a pp-laplacian equation

We deal with a boundary value problem of the form $-\epsilon(\phi_p(\epsilon u'))'+a(x)W'(u)=0,\quad u'(0)=0=u'(1),$ where $\phi_p(s) = \vert s \vert^{p-2} s$ for $s \in \mathbb{R}$ and $p>1$, and $W:[-1,1] \to {\mathbb R}$ is a double-well potential. We study the limit profile of solutions when $\epsilon \to 0^+$ and, conversely, we prove the existence of nodal solutions associated with any admissible limit profile when $\epsilon$ is small enough

On oscillatory solutions of certain difference equations

Some difference equations with deviating arguments are discussed in the context of the oscillation problem. The aim of this paper is to present the sufficient conditions for oscillation of solutions of the equations discussed

Towards a resolution of the Buchanan-Lillo conjecture

Buchanan and Lillo both conjectured that oscillatory solutions of the first-order delay differential equation with positive feedback $x^{\prime }(t)=p(t)x(\tau (t))$, $t\geq 0$, where $0\leq p(t)\leq 1$, $0\leq t-\tau (t)\leq 2.75+\ln2,t\in \mathbb{R},$ are asymptotic to a shifted multiple of a unique periodic solution. This special solution was known to be uniform for all nonautonomous equations, and intriguingly, can also be described from the more general perspective of the mixed feedback case (sign-changing $p$). The analog of this conjecture for negative feedback, $p(t)\leq0$, was resolved by Lillo, and the mixed feedback analog was recently set as an open problem. In this paper, we investigate the convergence properties of the special periodic solutions in the mixed feedback case, characterizing the threshold between bounded and unbounded oscillatory solutions, with standing assumptions that $p$ and $\tau$ are measurable, $\tau (t)\leq t$ and $\lim_{t\rightarrow \infty }\tau (t)=\infty$. We prove that nontrivial oscillatory solutions on this threshold are asymptotic (differing by $o(1)$) to the special periodic solutions for mixed feedback, which include the periodic solution of the positive feedback case. The conclusions drawn from these results elucidate and refine the conjecture of Buchanan and Lillo that oscillatory solutions in the positive feedback case $p(t)\geq0$, would differ from a multiple, translation, of the special periodic solution, by $o(1)$.Comment: 27 page