Function bounds for solutions of Volterra equations and exponential asymptotic stability

An extension of Gronwall’s inequality is derived for the purpose of finding functions that bound the solutions of the scalar linear Volterra integro-differential equation x′(t)=−a(t)x(t)+∫0tb(t−s)x(s)ds, where a,b:[0,∞)→R are continuous functions and bb is nonnegative. These functions are used to search for conditions that drive solutions to zero and conditions under which the zero solution is exponentially asymptotically stable.