A condition for the unique solvability of nonlocal boundary value problems for systems of functional-differential equations

Abstract

When considering non-local boundary value problems for functional-differential equations, when the derivative of the desired function is contained in the right side, one could use the resolvent of the integral equation. But, as is known, the resolvent of an integral equation of the second kind of the Fredholm type cannot always be uniquely determined. In some cases, you can use the properties of the kernel of the integro-differential equation. In this paper, we consider a nonlocal boundary value problem for systems of integro-differential equations with involution, when the kernel of the integral term containing the derivative has a partial derivative. Using the properties of an involutive transformation, the problem is reduced to the study of a multipoint boundary value problem for systems of integro-differential equations. The parameterization method proposed by Professor D. Dzhumabaev was applied to this problem. New parameters are introduced, and based on these parameters, we pass to new variables. When passing to new variables, we obtain the initial conditions for the initial equation. With the help of this condition, it is possible to determine the solution of the resulting Cauchy problem, as well as the system of linear equations. Applying the Fredholm theory to solve the obtained systems of integral equations, i.e. the unique solvability of the problem under study, we reduce to the reversibility of the matrix, which depends on the initial data. An example was shown as an illustration of the proposed method.