The ”freezing” method for ordinary differential equations is extended to the Volterra integral equations in a Banach space of the type x(t)− ∫ t 0 K(t, t− s)x(s)ds = f(t) (t ≥ 0), where K(t, s) is an operator valued function ”slowly” varying in the first argument. Besides, sharp explicit stability conditions are derived. Subject Classification: 45M10, 45N05

In this article, the homotopy perturbation method is proposed to solve systems of Volterra integral equations of the first kind. Theoretical proposes are presented briefly, so that this paper can be read independently. To illustrate the method some examples are presented. The results reveal that the homotopy perturbation method is very effective and simple and gives the exact solutions.

The qualitative and quantitative analyses of numerical methods delay differential equations (DDEs) are now quite well understood, as reflected in the recent monograph by Bellen and Zennaro (2003). This is in remarkable contrast to the situation in the numerical analysis of more general Volterra functional equations in which delays occur in connection with memory terms described by Volterra inte...