Boundedness and global exponential stability for delayed differential equations with applications (Q833139)

The authors study a system of functional differential equations of the form \[ \dot x_i(t)= -\rho_i(t, x_t)[b_i(x_i(t))+ f_i(t, x_t)], \] \(t\geq 0\), \(i=1,\dots, n\), where \(\rho_i:[0,\infty)\times S\to (0,\infty)\), \(b_i: \mathbb{R}\to \mathbb{R}\), \(f_i:[0,\infty)\times S\to\mathbb{R}\) are continuous on some set \(S\subset C_n\) of admissible initial conditions, and where \(C_n\) stands for the space of continuous functions from \([-\tau, 0]\) to \(\mathbb{R}^n\), \(\tau> 0\), equipped with the supremum norm. They show that if \(S\) contains an equilibrium point \(x^*\), \(\text{inf}\{\rho_i(t, \varphi): t\geq t_0,\varphi\in S\}> 0\), \(i= 1,\dots, n\), and certain other conditions are satisfied, then the equilibrium point \(x^*\) is globally exponentially stable.