arXiv subjects

John Ioannis Stavroulakis

Publications and source records attributed to John Ioannis Stavroulakis.

9 recordsLinked to original sources

arXiv2026

Oscillation of Second-Order Delay Equations with Positive Feedback

We study oscillation of the second-order delay equation \[ x''(t)=x(τ(t)), \] with a nondecreasing delayed argument $τ(t)\le t$ satisfying $τ(t)\overset{t\to\infty}{\to}\infty$. Set \begin{equation*} a=\liminf_{t\to\infty}(t-τ(t)),\qquad b=\limsup_{t\to\infty}(t-τ(t)). \end{equation*} We describe a continuous strictly decreasing function \begin{equation} β:[0,2/e]\longrightarrow[2/e,\sqrt2],\qquad β(0)=\sqrt2,\qquadβ(2/e)=2/e, \notag \end{equation} such that, if the equation has an eventually positive decreasing solution, then \begin{equation} a\leq2/e,\qquad b\leβ(a). \notag \end{equation} The bound $b\leβ(a)$ is sharp and interpolates between the classical $\limsup$ threshold and the $\liminf$ autonomous threshold. A detailed spectral analysis gives a simple interpretation of the result and links it directly to the oscillation theory of first-order delay equations.

math.DS↗

arXiv2026

A quantitative Schur comparison theorem for curves in CAT(k) spaces

We obtain a quantitative form of Schur's comparison theorem for curves with finite total curvature in CAT(k) spaces. This sharpens and extends the classical arm and bow lemmas in Euclidean space, as well as their Riemannian analogues. The proof is based on a comparison formula for curves in model planes, expressed in terms of curvature measures and a notion of moment arm borrowed from mechanics. Another ingredient is a refinement of Reshetnyak's theorem that controls the curvature of the majorizing curve.

math.DG↗

arXiv2026

Total absolute curvature and rigidity of surfaces in Cartan-Hadamard manifolds

We show that closed surfaces with minimal total absolute curvature in Cartan-Hadamard 3-manifolds bound flat convex bodies. This generalizes Chern-Lashof's theorem for surfaces in Euclidean space and solves a problem posed by Gromov in 1985. Our proof is based on an isometric embedding construction via holonomy, and uses Pogorelov's theory of surfaces with bounded extrinsic curvature. Along the way, we obtain a regularity result for convex hulls and a Schur-type comparison theorem for curves in Cartan-Hadamard manifolds.

math.DG↗

arXiv2026

A local isoperimetric inequality for balls with nonpositive curvature

We show that small perturbations of the metric of a ball in Euclidean n-space to metrics with nonpositive curvature do not reduce the isoperimetric ratio. Furthermore, the isoperimetric ratio is preserved only if the perturbation corresponds to a homothety of the ball. These results establish a sharp local version of the Cartan-Hadamard conjecture.

math.DG↗

arXiv2026

Total curvature of convex hypersurfaces in Cartan-Hadamard manifolds

We show that if the curvature of a Cartan-Hadamard $n$-manifold is constant near a convex hypersurface $Γ$, then the total Gauss-Kronecker curvature $\mathcal{G}(Γ)$ is not less than that of any convex hypersurface nested inside $Γ$. This extends Borbély's monotonicity theorem in hyperbolic space. It follows that $\mathcal{G}(Γ)$ is bounded below by the volume of the unit sphere in Euclidean space $\mathbf{R}^n$.

math.DG↗

arXiv2025

Dimension estimates for nonlinear nonautonomous systems in arbitrary normed spaces

We calculate explicit estimates for the dimension of trajectories satisfying a certain growth bound. We generalize the classic results of Kurzweil by considering nonlinear nonautonomous and uniformly compact dynamical systems on normed spaces over arbitrary fields. We furthermore refine the results in the case of delay equations, greatly simplifying the relevant growth bounds.

math.DS↗

arXiv2023

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 $x^{\prime }(t)=p(t)x(τ(t))$, $t\geq 0$, where $0\leq p(t)\leq 1$, $0\leq t-τ(t)\leq 2.75+\ln2,t\in \mathbb{R},$ 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 $τ$ are measurable, $τ(t)\leq t$ and $\lim_{t\rightarrow \infty }τ(t)=\infty$. 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 $o(1)$.

math.DS↗

arXiv2023

Semicycles and correlated asymptotics of oscillatory solutions to second-order delay differential equations

We obtain several new comparison results on the distance between zeros and local extrema of solutions for the second order delay differential equation \begin{equation*} x^{\prime \prime }(t)+p(t)x(t-τ(t))=0,~~t\geq s\text{ }\ \end{equation*} where $τ:\mathbb{R}\rightarrow \lbrack 0,+\infty )$, $p:\mathbb{R}% \rightarrow \mathbb{R}$ are Lebesgue measurable and uniformly essentially bounded, including the case of a sign-changing coefficient. We are thus able to calculate upper bounds on the semicycle length, which guarantee that an oscillatory solution is bounded or even tends to zero. Using the estimates of the distance between zeros and extrema, we investigate the classification of solutions in the case $p(t)\leq 0,t\in \mathbb{R}.$

math.DS↗

arXiv2020

Stability and oscillation of linear delay differential equations

There is a close connection between stability and oscillation of delay differential equations. For the first-order equation $$ x^{\prime}(t)+c(t)x(τ(t))=0,~~t\geq 0, $$ where $c$ is locally integrable of any sign, $τ(t)\leq t$ is Lebesgue measurable, $\lim_{t\rightarrow\infty}τ(t)=\infty$, we obtain sharp results, relating the speed of oscillation and stability. We thus unify the classical results of Myshkis and Lillo. We also generalise the $3/2-$stability criterion to the case of measurable parameters, improving $1+1/e$ to the sharp $3/2$ constant.

math.DS↗