Solvability of systems of Volterra integral equations of the first kind with piecewise continuous kernels (Q1946914)

The author considers the following system of integral equations: \[ \int_0^tK(t,s)x(s)ds=f(t),\quad 0<t\leq T, \] where the matrix kernel \(K(t,s)\) of dimension \(m\times m\) has discontinuity points of the first kind on curves \( s=\alpha_i(t)\), \(i=1,2,\dots,n-1,\) lying in the compact region \(0\leq s\leq t\leq T\). The kernel and the right part are continuous and have continuous derivatives with respect to \(t\) in the corresponding domains. The homogeneous system may have nontrivial solutions. In this paper, the author proves the existence of a continuous solution depending on free parameters, establishes sufficient conditions for the existence of a unique continuous solution and constructs an asymptotic approximation for solutions of the system.