John Ioannis Stavroulakis
Department of Mathematics, Ariel University, Ariel, Israel
Department of Mathematics, University of Pannonia, Veszprém, Hungary
The Buchanan-Lillo Conjecture: problems and challenges
Absztrakt:
Consider the first-order delay differential equation
\[
x^{\prime}(t)=p(t)x(\tau(t)),t\geq t_{0}\in
\mathbb{R}
\]
where $p\colon \mathbb{R} \rightarrow \mathbb{R} $, $\tau\colon \mathbb{R} \rightarrow \mathbb{R} $, $\tau(t)\leq t$, $\lim_{t\rightarrow\infty}\tau(t)=\infty$, and $p,\tau$ are measurable. For fixed feedback (sign of $p$), there exists a uniform bound ($3/2$ for negative feedback, and $2.75+\ln2$ for positive feedback) on the maximum delay $\tau_{m}:=\sup_{t\geq0}(t-\tau(t))$ which guarantees that all oscillatory functions which solve such an equation with smaller $\tau_{m}$, tend to zero. Buchanan conjectured that when $\tau_{m}=3/2$ ($2.75+\ln2$ ), all nontrivial oscillatory solutions tend to unique periodic functions $\varpi^{-},\left(\varpi^{+}\right) $, which are uniform for all nonautonomous equations and were first described by Myshkis and Soboleva. While this was proven by Lillo in the case of negative feedback, the case of positive feedback remains open. Intriguingly, \ $\varpi^{+}$ can also be described from the perspective of the threshold periodic solutions of the mixed feedback equation (sign-changing
$p$) for a fixed oscillation speed. We discuss the mixed feedback analog, the open question of convergence in the positive feedback case, and possible approaches to the problem.
