On the structure of the solutions of Volterra integral equations with degenerate kernel (Q1912439)

Expositions of the theory of linear integral equations frequently begin with Fredholm equations with degenerate kernel, while Volterra equations with degenerate kernel are encountered only rarely in the literature. In the present paper we study linear Volterra integral equations of first and second kind with a degenerate kernel of the form \[ a(x) \nu(x)+ \sum^n_{j=1} b_j (x) \int^x_\alpha c_j (t) \nu(t) dt= \mu(x), \qquad \alpha< x< \beta \tag{1} \] and equations with a degenerate kernel of the more general form \[ a(x) \nu(x)+ \sum^{n_0}_{j=1} b_j (x) \int^x_\alpha c_j (t) \nu(t) dt- \sum^n_{j= n_0+ 1} b_j(x) \int^\beta_x c_j(t) \nu (t) dt= \mu (x). \] The last equation is a Fredholm equation with degenerate kernel having a discontinuity on the diagonal, but it is easily reduced to a Volterra equation of the form (1). Formulas are obtained that determine the structure of the solutions of the equations considered.