BEHAVIOR OF SOLUTIONS OF NONLINEAR FUNCTIONAL VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS

The authors consider the nonlinear functional Volterra integro-differential equation with multiple delays

$$ x^{\prime}(t) = - a(t)x(t) + \sum_{i=1}^n \int_{t-\tau_i}^{t} b_i(t,s)f_i(x(s))ds. $$

They give sufficient conditions so that solutions are bounded, belong to $L^1$, or belong to $L^2$. They also prove the stability and global asymptotic stability of the zero solution. Their technique of proof involves defining appropriate Lyapunov functionals.