On Systems of Integro-Differential and Integral Equations with Identically Singular Matrix Multiplying the Principal Part

We consider linear homogeneous systems of integro-differential and integral equations with Volterra and Fredholm matrix kernels with zero initial conditions. The case is studied where the unknown vector function depends on one (integro-differential systems) or two (systems of integral equations) arguments and the matrix multiplying the principal part is square and identically singular. We point out the fundamental difference between the systems in question and systems solved for the principal part: there exists not only a trivial solution. In terms of matrix pencils and polynomials, we state sufficient conditions under which problems for the systems in question have only the trivial solution. Illustrative examples are given.