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

Document Type : Original Article

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Department of Mathematics and Statistics, University of Cape Coast, Cape Coast, Ghana.

Abstract

Necessary and sufficient conditions for the zero solution of the nonlinear neutral integro-differential equation
\begin{eqnarray*}
&&\frac{d}{dt}\Big(r(t)\Big[x(t)+Q(t, x(t-g_1(t)),...,x(t-g_N(t)))\Big]\Big)\\
&&= -a(t)x(t)+ \sum^{N}_{i=1}\int^{t}_{t-g_i(t)}k_i(t,s)f_i(x(s))ds
\end{eqnarray*}
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.

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