Study of systems of Hammerstein integral equations of the first kind

The article addresses systems of Hammerstein integral equations of the first kind, in which the determinant of the non-diagonal matrix-kernel is identically equal to zero.The class of problems under consideration has fundamental differences from standard cases: the solution may not exist, be non-unique, or depend on high derivatives of the input data. In terms of matrix pencils, sufficient conditions for the local existence of a unique solution in the class of continuous functions are formulated. Illustrative examples are given. Difficulties arising in the construction of numerical methods for solving these problems are discussed.