Global stability for nonlinear dynamic equations with variable coefficients
The author considers nonlinear dynamic equations on a time scale \(\mathbb{T}\) which can be any non-empty closed subset of the real numbers that is unbounded above. This unified approach encompasses dynamical systems governed by differential equations, difference equations and combinations of both. Additionally, time delays are allowed. The considered equations are of the form \[ x^\Delta(t) = -\sum_{j=0}^n a_j(t) f_j(x(\tau_j(t)), \] where \(x^\Delta\) is the so-called delta derivative, \(\tau_1,\tau_2,\ldots,\tau_n\) are time varying time delays, \(a_1,a_2,\ldots,a_n\) are non-negative time varying coefficients, and \(f_1,f_2,\ldots,f_n\) are continuous functions such that \(f_j(0)=0\), \(f_j(x)\geq 0\) and \(f_j(-x)\geq 0\) for all \(j=0,1,\ldots,n\) and all \(x\geq 0\). Sufficient conditions are given for which the zero solution is globally asymptotically stable. The results are illustrated with two examples.
scientific article; zbMATH DE number 5299484
scientific article