Stability Results for Neutral Integro-Differential Equations with Multiple Functional Delays

Abstract

Necessary and sufficient conditions for the zero solution of the nonlinear neutral integro-differential equation<br />begin{eqnarray*}<br />&&frac{d}{dt}Big(r(t)Big[x(t)+Q(t, x(t-g_1(t)),...,x(t-g_N(t)))Big]Big)<br /> &&= -a(t)x(t)+ sum^{N}_{i=1}int^{t}_{t-g_i(t)}k_i(t,s)f_i(x(s))ds<br /> end{eqnarray*}<br /> to be asymptotically stable are obtained. In the process we invert the integro-differential equation and obtain an equivalent integral equation. The contraction mapping principle is used as the main mathematical tool for establishing the necessary and sufficient conditions.