On a class of linear integro-differential equations in partial derivatives of the first order

Abstract

In this note, we study solutions of the integro-differential equation (see article). This solution depends on q arbitrary parameters. If for λ = λ' equation (51) has no solutions, then the Cauchy problem under consideration has no solutions. Finally, we note that if the determinant (40) on the manifold (39) vanishes, then system (43) is not solvable, or is solvable uniquely with respect to S and tk. Therefore, equation (46) will include arbitrary parameters. Consequently, if the initial manifold (39) is characteristic, then equation (1) has none, or has an infinite number of solutions.