A Multiparameter Family of Solutions to the Volterra Linear Integral Equation of the First Kind

Abstract

We study the Volterra integral equation of the first kind with an integral operator of order \(n\), a singularity, and a sufficiently smooth kernel in a certain Banach space with weight. It reduces to an integro-differential equation with two terms in the left-hand side. The first term corresponds to an equation for which an explicitly multiparameter family of solutions is constructed. For the second term we obtain an equation with an operator whose norm in an arbitrary Banach space is arbitrarily small near zero. Such splitting of the integral operator allows constructing a particular and general solutions to the integro-differential equations in the corresponding Banach space in the form of convergent series. Thus, under certain restrictions on the operator pencil corresponding to a given integral operator, a multiparameter family of solutions is constructed for the original integral equation.