Uniformly Continuous 𝐿¹ Solutions of Volterra Equations and Global Asymptotic Stability
The scalar linear Volterra integro-differential equation
is investigated, where a and b are continuous functions. Liapunov functionals are constructed in order to obtain sufficient conditions so that solutions of (E) are absolutely Riemann integrable on [0,∞) and have bounded derivatives. Then some of these conditions are replaced with less stringent ones while others are eliminated altogether. Under the new conditions, it is shown that one of the Liapunov functionals is uniformly continuous which in turn implies that solutions of (E) are uniformly continuous. We then employ Barbălat’s lemma to prove the zero solution of (E) is stable and that all solutions of (E) approach zero as 𝓉 → ∞. Examples illustrated with numerical solutions are provided.