Oscillation of delay differential equations via the hyper4 convergence

TL;DR

This paper establishes a precise condition under which solutions to certain delay differential equations either oscillate or tend monotonically to zero, using hyper4-iterations and Lambert's function.

Contribution

It introduces a novel method based on hyper4-iterations and Lambert's function to analyze the oscillation and convergence of solutions to delay differential equations.

Findings

Provides a sharp oscillation/convergence criterion for DDEs.

Uses hyper4-iteration convergence to Lambert's function.

Offers a new analytical approach for delay differential equations.

Abstract

A sharp condition is provided to guarantee that the (nontrivial) solutions of a DDE of the form $\dot{x}(t)+F(t,x)=0$ $t\geq 0,$ (where $F(t,\cdot)$ is an odd-like causal operator) either oscillate, or converge monotonically to zero. The method used is based on the convergence of the sequence of hyper4-iterations to the Lambert's function.