On the Systems of Romanovskij Type Linear Integral Equations with Partial Integrals

Systems of linear integral equations are studied in spaces of continuous and continuously differentiable on the square of vector functions.
The systems considered in the paper contain matrix partially integral operators and Romanovsky matrix operators. Systems of equations with such operators are not Fredholm in any of the spaces mentioned, even in the general case of given smooth kernels. The paper considers systems of equations with kernels from the space of vector functions continuous on a square with values in the space of functions summable on an interval.
Theorem 2 contains conditions under which the Fredholm property of a system of linear integral equations of Romanovskii type with partial integrals in the space of continuous vector functions is equivalent to the invertibility of a simpler system of linear integral equations with partial integrals. In obtaining these conditions, the S.M. Nikol’skii theorem on the representation of the Fredholm operator as a sum of reversible and compact operators was used. Specific classes of kernels for which the statement of Theorem 2 is true are given, a special case of a system of linear integral Romanovski type equations with partial integrals is considered, for which the Fredholm system is equivalent to the invertibility of linear integral equations with a parameter for each parameter value. Theorem 5 contains the Fredholm conditions for a system of integral equations of the Romanovsky type with partial integrals and continuously differentiable kernels in the space of continuously differentiable vector functions.
Keywords: systems of Romanovskij type linear integral equations, partial integrals, fredholmness of systems, invertibility of systems, matrix operators and equations, kernels of potential type.