Global stability for a class of functional differential equations with distributed delay and non-monotone bistable nonlinearity
In this paper, the following functional differential equation is considered: \[ x'(t) = -f(x(t))+\int^{\tau}_0h(a)g(x(t-\tau))da, t>0; \; x(t)=\phi(t), -\tau\leq t\leq 0, \] where \(f\) and \(g\) are functions from \(\mathbb{R}\) to \(\mathbb{R}\) satisfying \(f(0) = g(0) = 0\) and the equation \(f(x) = g(x)\) has two positive roots \(x_2 > x_1 >0\). Under the assumption \(\int^{\tau}_0h(a)da =1\), the above equation has equilibria \(0, x_1\) and \(x_2\). Then, various criteria are established for \(0\) and \(x_2\) to be globally asymptotically stable by using Lyapunov functional methods. Note that the global asymptotic stability of \(0\) and that of \(x_2\) have restrictions on the magnitude of \(x(t)\). Comparing this work with those in the references, this deals with more general cases and it is a valuable extension to the existing theory.
scientific article; zbMATH DE number 7390457
scientific article