Inequalities and exponential stability and instability in finite delay Volterra integro-differential equations (Q1928215)

The authors investigate stability properties of the integral-delay type functional equation \[ x'(t)= p(t) x(t)- \int^t_{t-\tau} q(t,s) x(s)\,ds,\quad t\geq 0, \] where \(\tau>0\) is a constant, and \(q: [0,\infty)\times [-\tau,\infty)\to \mathbb{R}\), and \(p:[0,\infty)\to \mathbb{R}\), by means of Lyapunov functionals like \[ V(t)= \Biggl[x(t)- \int^t_{t-\tau} A(t,s)x(s)\,ds\Biggr]^2+ \int^0_{-\tau} \int^t_{t+s} A^2(t,z)x^2(z)\,dz\,ds, \] obtaining an interesting estimate for the solutions. Under suitable conditions on data, the following estimate is obtained: \[ |x(t)|^2\leq 2\Biggl({2\alpha-1\over \alpha-1}\Biggr) VJ(t_0)\exp\Biggl(\int^{t-(\alpha- 1)\tau/\alpha}_{t_0} [p(s)- A(s,s)]\,ds\Biggr), \] where \[ A(t,s):= \int^\tau_{t-s} q(u+ s,s)\,du\quad\text{for }t\in [0,\infty)\text{ and }s\in[-\tau,\infty). \] A comparison equation is obtained for \(V(t)\), namely \(V'(t)= Q(t)V(t)\), which is used to derive properties for the solutions \(Q(t)= p(t)- A(t,t)\). A condition for instability has the form \[ \int^\infty_{t_0} A^2(s,s)\,ds= \infty. \]