Abstract: In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter λ = 0 and generalized periodic boundary conditions are introduced in a Lebesgue space which consists of functions with vector valued in a general Banach space, and then describe the solution of these abstract boundary value problem by the abstract linear integral operator of Volterra type. We call this process the integral operator solving process.

Key words: abstract kinetic equation with the first kind of critical parameter, boundary
value problem of abstract kinetic equation, generalized periodic boundary conditions, abstract linear integral operator of Volterra type, integral operator solving process