This paper is concerned with the nonlinear boundary value problems for first order integro-differential equations with impulsive integral conditions. By using of the method of lower and upper solutions coupled with the monotone iterative technique, we give conditions for the existence of extremal solutions.

For quasilinear integro-diierential equations of the form u t ?aA(u) = f, where a is a scalar singular integral kernel that behaves like t ? , 1 2 < 1 and A is a second order quasilinear elliptic operator in divergence form, solutions are found for which A(u) is integrable over space and time.

This paper presents models of ultimate ruin theory in the form of 1st order integro-differential equations and systems of such equations (cf. Albrecher and Boxma (2004), Dickson and Gray (1984), Lin, Willmot and Drekic (2003)), and gives some numerical results obtained from the direct solution of one of them (Albrecher and Boxma (2004)).